0\\end{array}\\right.","}$ Considering that we are concerned with electrical potential and thus $y$ negative and rearranging and replacing the half-angle formula we have $\\frac{d y}{dx} = - 2 \\kappa \\sinh \\frac{y}{2}$ This formula to will be useful to derive an equation for net charge on the diffuse layer, $\\sigma $ .","The net charge is the integral of the charge density on the $x$ interval of zero to infinity.","$\\sigma = \\int _0^\\infty \\rho dx$ Replacing the Poisson-Boltzmann equation into the surface charge equation we have $\\sigma = \\int _0^\\infty (- \\frac{\\epsilon }{4 \\pi } \\frac{d^2 \\psi }{dx^2}) dx = - \\frac{\\epsilon }{4 \\pi } \\int _0^\\infty \\frac{d^2 \\psi }{dx^2} dx$ The integration is over the second derivative, resulting $\\sigma = -\\frac{\\epsilon }{4 \\pi } \\bigg (\\frac{d \\psi }{dx}\\bigg )\\bigg |_{x=0}^{x=\\infty }$ But the electrical potential is constant at $x=\\infty $ such that the term $d\\psi /dx$ vanishes when evaluated at infinity, resulting $\\sigma = \\frac{\\epsilon }{4 \\pi } \\bigg (\\frac{d \\psi }{dx}\\bigg )_{x=0}$ Expressing sigma in terms of $y$ $\\sigma = \\frac{\\epsilon k T }{4 \\pi |z_i| e } \\bigg (\\frac{d y}{dx}\\bigg )_{x=0}$ Now we can replace the equation for $dy/dx$ derived previously and the condition that $y = y_0$ at $x = 0$ resulting in $\\sigma = -\\frac{\\epsilon k T \\kappa }{2 \\pi |z_i| e } \\sinh \\frac{y_0}{2}$ Which in terms of electrical potential and replacing $\\kappa $ is $\\sigma = -\\frac{\\epsilon k T }{2 \\pi |z_i| e } \\sqrt{\\frac{8 \\pi z_i^2 e^2 n_{0i}^2}{\\epsilon k T}} \\sinh \\frac{\\frac{|z_i| e \\psi _0}{kT}}{2} = - \\sqrt{\\frac{2 \\epsilon k T n_{0i}^2}{\\pi }} \\sinh \\frac{|z_i| e \\psi _0}{2kT}$ The surface charge, $\\sigma _0$ has the same magnitude of the sum over the diffuse layer with opposite sign $\\sigma _0 = \\sqrt{\\frac{2 \\epsilon k T n_{0i}^2}{\\pi }} \\sinh \\frac{|z_i| e \\psi _0}{2kT}$ This is the Gouy-Chapman relationship for the surface charge density [39].","The charge density if a function of the surface potential and ion concentration among other factors.","In variable charged oxides the surface potential is not constant and varies with pH.","Considering small surface potential, the last equation can be simplifiedNote that the hyperbolic sin expands to $ \\sinh y = y + y^3/3!","+ y^5/5!","+ ...$ thus, for small values of $y$ higher order terms may be neglected to [39] $\\sigma _0 \\approx \\frac{\\epsilon \\kappa }{4 \\pi } \\psi _0$ and the surface potential can be determined by the activity of the potential determining ion $a_{pdi}$ in relation to its activity at the point of zero charge (pzc) $\\psi _0 \\pm \\frac{RT}{zF} \\ln {\\frac{a_{pdi}}{(a_{pdi})_{pzc}}}$ If $H^+$ is the potential determining ion as is the case in oxide surfaces, the relationship between pH and hydrogen activity is $ pH = - log[H^+]$ and the equation can be written as $\\psi _0 = \\frac{-2.303 RT}{F} (pH - pH_{pzc})$ Combining the electrical potential and surface charge equations, the following equation is valid $\\sigma _0 \\approx \\frac{-2.303 RT \\epsilon \\kappa }{4\\pi F} (pH - pH_{pzc})$ In which $F$ is the Faraday constant.","On a side note, the rigorous derivation above was provided by Kenneth L. Babcock one of the fathers of soil chemistry [6].","Much of soil chemistry interfaces with soil physics on the subjects of thermodynamics and transport and retention of solutes.","One of the most important soil chemists in recent history Garrison Sposito was a student of Kenneth Babcock.","Sposito gave fundamental contributions to soil physics no only on thermodynamics and solute chemistry but also regarding water transport and scaling of soil properties and processes.","We have illustrated that soils and rock differ in terms of their mineralogical composition and by the fact that rocks are consolidated materials.","Rocks differ from sedimentary deposits in terms of genesis and mineralogy and by the fact that soils have organic matter in their composition.","The amount of organic matter varies with soil type, but according to soil science criteria, mineral soils have around 5% organic matter on mass basis for average soils.","The concept of organic soils is a somewhat loose definition based on a more or less arbitrary limit of organic matter or carbon content [87].","The classification of organic soils varies according to engineering classification systems [70] and can vary depending on several specific criteria within a soil classification system used in soil science [87].","Because soils occupy only an thin layer of the Earth surface, usually a meter or two to a few meters in depth at the most in most cases, they are under direct influence of the weather and organisms.","Soils are usually rich in organisms, both macro and micro, and plants and are continually affected by their actions.","Soils are dynamic and porosity and structure are continually being altered by the actions of roots, ants, termites and other burrowing animals of larger size.","Rocks and sedimentary deposits can reach kilometers in depth on the Earth's crust and are very little affected by organisms and surface conditions in most cases.","The transition zone between soils and subsurface materials is called the saprolite and can reach several meters in depth under extreme weathering condition on the tropics over certain rocks.","Soils constitute for the most part the unsaturated zone, where the pores are saturated by both gas and water, although it is possible that the water table can be located within the soil profile.","Below the water table is the saturated zone where the pores are saturated with water.","Most aquifers are located in sedimentary deposits and to some extent porous or fractured rocks.","Transport and retention of water, contaminants, oil and gas in rocks and sedimentary deposits is usually the object of study of hydrogeology and petroleum engineering while soil physics is often concerned with soils and other unsaturated media.","However, the mathematical analytical and numerical framework between these disciplines is similar since the equations that govern transport processes in these materials have the same physical basis if they are not the same.","Much of the governing equations of processes of transport originated from the realm of mechanics and later chemical engineering.","In the next chapter we will try to provide a mathematical framework for the equations that govern transport processes in soils and other natural porous media.","ll $ \\psi $ Electric potential $ \\psi _0 $ Electric potential at the surface of the charged plane $ x $ Distance from the charged plane $ z $ Ion valence $ e $ Elementary charge $ k $ Boltzmann constant $ T $ Absolute temperature $ \\rho $ Charge density $ \\epsilon $ Absolute permittivity of the medium $ \\kappa ^{-1} $ Debye length $ \\text{e} $ Euler's number $ ln $ Logarithm base e $ log $ Logarithm base 10 $ c_1, c_2, C_1, C_2 $ General constants for differential equations solutions $ y $ General variable for mathematical manipulations $ y_0 $ Value of $y$ at $y = 0$ $ u, v $ Variables for integration by substitution $ sgn $ Sign function $ \\sigma $ Net charge at the diffuse layer $ \\sigma _0 $ Surface charge $ R $ Molar gas constant $ F $ Faraday constant $ a_{pdi} $ Activity of a potential determining ion (pdi) $ pH_{pzc} $ pH at the point of zero charge (pzc) Up until now we have been discussing properties of the solid matrix of porous media.","Some of these properties are directly affected by water content and can be altered by engineering, hydrological, agricultural as well as a wide range of natural processes, both on short and long term.","We have not discussed so far any type of flux of liquid of gaseous phases through porous materials.","Detailed discussions of mass flow through porous media are reserved to the chapters dealing with saturated and unsaturated hydrodynamics of porous media, and other transport phenomena.","Technically speaking, we will deal with another property of the solid matrix on the next chapter, the grain size distribution, so there should not be an immediate need for equations governing fluid flow.","However, the majority of methods for determination of grain size distribution of porous media rely on settling in liquid and, occasionally, gaseous materials.","To understand the dynamics of settling we need to understand the fundamental equations of fluid dynamics which control how settling occurs in a viscous media.","The knowledge discussed in this chapter will also be the basis for modeling flow problems in open conducts and to a large extent flow in porous media.","This chapter will introduce you to the beauty and brutality that is the mathematics involved in fluid mechanics and in theoretical soil physics.","If you are already familiar with tensor analysis in areas such as continuum mechanics or general relativity you should be able to follow this chapter without major issues.","Consulting the bibliography might help with issues specific to fluid mechanics that might arise.","If you are seeing the application of tensor analysis for the first time it might be worth consulting an introductory text on tensors.","When dealing with fluid flow, two formulations are usually discussed in the literature, the Lagrangian and the Eulerian specifications, after the great mathematicians Joseph-Louis Lagrange and Leonard Euler Although Euler was responsible for both formulations, see for example Landau and Lifshitz (1987, p. 5) [57]..","The Lagrangian formulation follows from the idea of tracking individual particles or the center of mass of a group of particles from classical mechanics.","In the context of fluid flow the movement of a single water molecule could be tracked in relation to an initial time $t_0$ and position $\\mathbf {s}$ , where, as seen in Chapter REF .","$\\mathbf {s} = s_x \\mathbf {i} + s_y \\mathbf {j} + s_z \\mathbf {k}$ Thus, in the Lagrangian formulation, the velocity of a tracked particle or center of mass of a group of particles is $\\mathbf {v}(\\mathbf {s}, t)$ or, considering the vector components $\\mathbf {v}(s_x, s_y, s_z, t)$ .","It is a consensus that the Lagrangian formulation is not mathematically convenient for the majority of fluid flow problems and thus, it will not be discussed further.","The formulation normally adopted is more akin to that used in electromagnetism in the sense that the flow quantities are defined as a vector field with independent variables position in space and time, $\\mathbf {x}$ , and t. The vector velocity field in the Eulerian formulation is traditionally represented as $\\mathbf {u}(\\mathbf {x}, t) $ , where $\\mathbf {u} = u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}$ often also represented as $\\mathbf {u} = u \\mathbf {x} + v \\mathbf {y} + w \\mathbf {z}$ such that $\\mathbf {u}(\\mathbf {x}, t) $ can be written as $\\mathbf {u}(u_x, u_y, u_z,t) $ in the first case and $\\mathbf {u}(u, v, w,t) $ in the second.","The difference between $\\mathbf {v}$ and $\\mathbf {u}$ is that in the latter we are now no longer interested in an individual particle or the center of mass of a group of particles, what we have is a velocity field in space and time.","Now at any position in space and at a given time there is a velocity value assigned.","Imagine a 2D river or stream seen from above, in the Lagrangian river we would track the position of individual particles as a function of time, while on the Eulerian river we are not worried about individual particles but we have a velocity field that covers the entire space of the river, such that if we chose an arbitrary point $\\mathbf {x_1}$ and $t_1$ on the surface of the river, the velocity at that point is given by $\\mathbf {u}(\\mathbf {x_1}, t_1)$ .","One advantage of the Eulerian formulation is now obvious as there is no need to track an exorbitant number of particles to model the flow of a river or stream.","Another major advantage of the Eulerian formulation is that it can be used to model many other important fluid properties such as density and pressure over space and time, however it is important to point out that these are scalar fields given by $\\rho (\\mathbf {x}, t)$ and $P(\\mathbf {x}, t)$ ,C.L.","for density and pressure, respectively, noting that $\\mathbf {u}$ at any point in space and time is itself a vector, while $\\rho $ and $P$ are scalars.","From the Eulerian formulation we now find the acceleration of a fluid element.","The acceleration of the fluid element is not given directly by $ \\partial \\mathbf {u}/ \\partial t $ as each fluid element travels through the vector field as times varies, being subjected to velocity variation in space along the vector field in addition to time.","The correct formulation of the acceleration of a material element is given by $\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u}$ Using the notation introduced in Chapter REF and because $\\mathbf {u}$ is a vector the equation above can be written as $\\frac{\\partial }{\\partial t} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) \\cdot [\\frac{\\partial }{\\partial x}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) \\mathbf {i} + \\\\ + \\frac{\\partial }{\\partial y}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) \\mathbf {j} + \\frac{\\partial }{\\partial z} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k})\\mathbf {k}]$ which, considering the dot product $\\mathbf {u} \\cdot \\nabla \\mathbf {u}$ , can be written as $\\frac{\\partial }{\\partial t} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + [u_x \\frac{\\partial }{\\partial x}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + u_y \\frac{\\partial }{\\partial y}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + \\\\ + u_z \\frac{\\partial }{\\partial z} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k})]$ With some effortAs opposed to \"it can easily be shown\" it can be shown that the vector equation above can be written as a set of equations.","You will see that most of the fluid dynamics equations in two and three-dimensional spaces are actually sets of equations and it is important that you familiarize yourself with the steps in converting from vector form to components and vice-versa when doableIn several fluid equations, multiple tensor components are involved, and it is not practical to do these operations without using vector and tensor identities.","Thus in Cartesian coordinates, the components in $x$ , $y$ and $z$ are $\\frac{\\partial u_x}{\\partial t} + [u_x \\frac{\\partial u_x}{\\partial x} + u_y \\frac{\\partial u_x}{\\partial y} + u_z \\frac{\\partial u_x}{\\partial z} ] \\\\\\frac{\\partial u_y}{\\partial t} + [u_x \\frac{\\partial u_y}{\\partial x} + u_y \\frac{\\partial u_y}{\\partial y} + u_z \\frac{\\partial u_y}{\\partial z} ]\\\\\\frac{\\partial u_z}{\\partial t} + [u_x \\frac{\\partial u_z}{\\partial x} + u_y \\frac{\\partial u_z}{\\partial y} + u_z \\frac{\\partial u_z}{\\partial z} ]$ The quantity $\\mathbf {u} \\cdot \\nabla \\mathbf {u} $ is the dot product of a vector (first order tensor) and a second order tensor and is a vector, such that $ \\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u} $ is the sum of two vectors and is itself a vector.","The derivative form found in Equation 4.4 has such an importance in fluid dynamics that it constitutes a mathematical operator $\\boxed{\\frac{D}{D t} = \\frac{\\partial }{\\partial t} + \\mathbf {u} \\cdot \\nabla }$ such that for any scalar or vector quantity $f$ we have $\\frac{D f}{D t} = \\frac{\\partial f}{\\partial t} + \\mathbf {u} \\cdot \\nabla f$ In which $ \\frac{\\partial f}{\\partial t} $ is the temporal rate of change at a specific location and $\\mathbf {u} \\cdot \\nabla f $ is the convective rate of change due to transport of the fluid element from one point to another in space.","If $f$ is constant throughout the space under consideration $\\frac{D f}{D t} = \\frac{\\partial f}{\\partial t} + \\mathbf {u} \\cdot \\nabla f = 0$ and there is no acceleration of the material elements of fluid.","The conservation equation, or mass conservation equation, is often referred to the continuity equation on the literature.","In agreement with G. K. Batchelor, we do not think that continuity is an appropriate term and will refrain from using it in this text.","The idea of conservation of mass, be it a gas, a fluid or a particulate solid, is a very intuitive one.","Imagine a garden hose, if you consider a given length of the hose, say one meter, once you turn on the faucet and the flux out of the faucet becomes constant, and if there are no leaks along the hose, the amount of water that enters on one end is equal to the amount that exits on the other end, per unit time.","This condition in which the amounts in and out of the system are constant and equal is called steady state and is very important in groundwater hydrology.","This simple example is a basic application of the conservation equation, if there are no losses or gains within the system, matter is conserved.","Suppose now you have a steady state condition and there are leaks in your garden hose within the length considered.","The mass conservation equation still applies, the amount in is equal to the sum of the amounts on the other end of the hose plus any leaks, per unit time.","In real scenarios we often have to deal with sources or sinks within the volume considered, this is especially true in the case of reactive and/or volatile transport in which there can be loss of gain within the consider volume due to adsorption, condensation, degradation and many other physico-chemical processes.","A simple example of a sink (loss) would be evaporation of water in a reservoir.","Let us now treat this problem from a mathematical point of view.","There are at least two approaches to treat this problem, one is deceitfully simple and direct and is based on vector calculus.","The other is based on algebraAs in algebraic manipulations in a broader sense and appears to be very complex although being very simple.","The first one can be found in advanced fluid dynamics and mechanics, physics and applied mathematics books, while the other is preferred in introductory engineering and in soil physics books.","The conceptual model is the same in both cases, an element of volume in space, in which flow is considered through an element of surface area.","It is easy for us to consider water, but you will see later that the conservation equation applies to any substance in transport processes in porous media.","In the vector treatment we are not initially concerned with the geometry of the volume, other than it should be a closed space.","Initially, however, it should be much easier to imagine the volume as a cube or as a sphere or other similar closed surfacesIn mathematics, the topology of the space is of interest, and can have implications in terms of the mathematical description.","This is, however, beyond the scope of this text.","Starting with a volume $V$ enclosed by a surface $A$ completely filled with a liquid of density $\\rho $ , the mass of water within the volume is given by $\\int \\rho dV$ This should come of no surprise, as by integrating the volume elements $dV$ over the entire volume we are left with $V$ , and obviously the water density in [M L$^{-3}$ ] multiplied by the total volume in [L$^{3}$ ] is the mass M of water within the enclosed volume.","The amount of water flowing through an element of surface area $dA$ $\\int \\rho \\mathbf {u} \\cdot \\mathbf {n} dA$ Where $\\mathbf {n}$ is the unit normal in dAC.L..","Note that the velocity, $\\mathbf {u}$ , in [L T$^{-1}$ ] times water density, $\\rho $ , in [M L$^{-3}$ ], over the entire surface, $A$ , in [L$^{2}$ ] is the rate of mass transfer across the surface in [M T$^{-1}$ ].","Now the rate of variation of the mass within the volume with time can be written in differential form as $\\frac{\\partial }{\\partial t} \\int \\rho dV$ in units of [M T$^{-1}$ ].","If this doesn't make much sense to you don't worry, you can find more information about this construction in vector calculus or calculus 3 books.","Note that as we defined Equation 4.8 as the total mass within the volume, Equation 4.10 is nothing more than the variation of mass with time within the volume.","Now, from our discussion of conservation laws we can agree that the variation of mass within a volume is the net rate of flow across its surface, and thus the conservation of mass implies that $\\frac{\\partial }{\\partial t} \\int \\rho dV = - \\int \\rho \\mathbf {u} \\cdot \\mathbf {n} dA$ So that $\\frac{\\partial }{\\partial t} \\int \\rho dV + \\int \\rho \\mathbf {u} \\cdot \\mathbf {n} dA = 0$ Now all the vector calculus tools discussed in Chapter REF will come in handy.","We can use the divergence theorem to transform a surface integral into a volume integral.","Applying it to the second term in Equation 4.12 results in $\\frac{\\partial }{\\partial t} \\int \\rho dV + \\int \\nabla \\cdot (\\rho \\mathbf {u}) dV = 0$ Which can be written as $\\int [ \\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mathbf {u})] dV = 0$ Because the integration must hold for any volume, the integrand must vanish and we are left with one of the fundamental equations of fluid dynamics, the mass conservation (or continuity) equation $\\boxed{ \\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mathbf {u}) = 0 }$ Now for the derivation that does not need vector calculus.","Imagine a cube with sides $\\Delta x$ , $\\Delta y$ and $\\Delta z$ .","Consider a velocity component $u_x$ in the direction $x$ .","The area which is perpendicular to the velocity component $u_x$ is $\\Delta y\\Delta z$ .","In other words, the velocity component $u_x$ enters into the cube in the side $\\Delta y\\Delta z$ .","Now, the rate of mass accumulation in the direction $x$ is the amount of mass that enters on one end on the direction $x$ and leaves on the other, in that same direction.","The amount of mass that enters in one end is equal to the velocity component $u_x$ times the density of the fluid $\\rho $ times the area $ \\Delta y \\Delta z$ , note that $\\frac{m_{in-x}}{t} = \\rho u_x \\Delta y \\Delta z = \\frac{\\mathrm {M}}{L^3} \\frac{\\mathrm {L}}{T} L^2 = \\frac{\\mathrm {M}}{T}$ The amount that exits on the other end will depend on the variation of the velocity of the fluid through the direction $x$ within the volume, if the fluid slows down, the velocity decreases and vice-versa.","Therefore, in the outflow we have to consider the rate of variation of velocity within the volume such that, the outflow mass in the $x$ direction is $\\frac{m_{out-x}}{t} = ( \\rho u_x + \\frac{\\partial \\rho u_x}{\\partial x}\\Delta x) \\Delta y \\Delta z$ Therefore, the amount of mass that accumulates or is lost per unit time due to flux in the $x$ direction is given by $\\frac{\\partial }{\\partial t} M_{x} = \\rho u_x \\Delta y \\Delta z - (\\rho u_x + \\frac{\\partial \\rho u_x}{\\partial x}\\Delta x) \\Delta y \\Delta z$ Which results in $\\frac{\\partial }{\\partial t} M_{x} = - \\frac{\\partial \\rho u_x}{\\partial x}\\Delta x \\Delta y \\Delta z$ By symmetry it is not difficult to show that the mass lost or accumulated due to flux on the directions $y$ and $z$ is $\\frac{\\partial }{\\partial t} M_{y} = - \\rho \\frac{\\partial \\rho u_y}{\\partial y}\\Delta x \\Delta y \\Delta z$ $\\frac{\\partial }{\\partial t} M_{z} = - \\rho \\frac{\\partial \\rho u_z}{\\partial z}\\Delta x \\Delta y \\Delta z$ and that the total mass accumulated or loss within the cube per unit time due to flow in the three directions is $\\frac{\\partial }{\\partial t} M_{xyz} = - (\\frac{\\partial \\rho u_x}{\\partial x}+ \\frac{\\partial \\rho u_y}{\\partial y}+ \\frac{\\partial \\rho u_z}{\\partial z})\\Delta x \\Delta y \\Delta z$ Since the density of the cube is $\\rho $ and its volume is $\\Delta x \\Delta y \\Delta z$ and rearranging $ \\rho = M/V$ to $ M = \\rho V$ we have $\\frac{\\partial }{\\partial t} \\rho \\Delta x \\Delta y \\Delta z = - (\\frac{\\partial \\rho u_x}{\\partial x}+\\frac{\\partial \\rho u_y}{\\partial y}+\\frac{\\partial \\rho u_z}{\\partial z})\\Delta x \\Delta y \\Delta z$ or $\\frac{\\partial \\rho }{\\partial t} = - (\\frac{\\partial \\rho u_x}{\\partial x}+\\frac{\\partial \\rho u_y}{\\partial y}+\\frac{\\partial \\rho u_z}{\\partial z})$ Using the vector operators, this equation can be written as $\\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mathbf {u}) = 0$ which is the same as Equation 4.15.","The derivation using geometric considerations is favored in soil physics and hydrology books while the vector calculus derivation is found in fluid mechanics and dynamics books for physicists and mathematicians.","If the density of the fluid is constant, $\\rho $ does not vary with time and $\\partial \\rho /\\partial t = 0 $ and we arrive at an important case of the conservation equation $\\boxed{\\nabla \\cdot \\mathbf {u} = 0}$ Stating in words, the divergence of the velocity field is zero for an incompressible fluid.","For most practical purposes, water can be assumed as incompressible and Equation 4.23 will be valid for many flow problems.","Much like for mass, the conservation laws are valid for other extensive properties in fluid mechanics including energy and momentum and this will allow us to derive the equations at the core of fluid mechanics and transport phenomena.","Recall from introductory physics that Newton's second law can be written either as $\\mathbf {F} = m\\mathbf {a}$ or as $\\mathbf {F} = \\frac{d \\mathbf {p}}{dt}$ where the vector quantity $\\mathbf {p}$ is the linear momentum and is equal to $ m \\mathbf {v} $ , for an object of mass $m$ moving at velocity $\\mathbf {v}$ .","Considering an element of fluid of volume $V$ , the momentum corresponding to the entire volume is given by $\\int \\mathbf {u} \\rho dV$ Notice that velocity is in [L T$^{-1}$ ], density is in [M L$^{-3}$ ] and volume is in [L$^{3}$ ] such that the units are consistent with those of momentum [M L T$^{-1}$ ].","The rate of change of momentum for the elementary volume can be represented by the derivative of the relationship above with respect to time, the material derivative in this case.","If the density can be assumed as constant $\\int \\frac{D\\mathbf {u}}{Dt} \\rho dV$ Now, as specified by Newton's second law of motion, the rate of change of momentum in an object is equal to the sum of forces acting on the object.","There are two important points here to be considered because we are dealing with fluid mechanics and not with a point particle.","The first is that the total momentum is the sum of the individual momentum of each infinitesimal element of fluid.","If the system was modeled as a system of individual particles the integral on the previous equations could be replaced by a summation of the momenta of the $n$ particles that compose the system.","The second point is that the forces acting in a volume of fluid can be divided into body (or volume) forces such as those generated by gravity and electromagnetic fields and surface (of contact) forces, which act directly over the surface enclosing the volume element.","Surface forces are ubiquitous in continuum mechanics and are an important concept in soil physics both in treating the solid and liquid phases and their interactions.","To understand surface forces it is convenient to explore the concept of the stress tensor.","Imagine an element of volume of fluid immersed in a fluid or other continuous matter, the matter around the element of fluid exerts a contact force across the entire surface of the fluid element.","For the sake of simplicity, consider a volume element of cubic shape in Cartesian coordinates system.","Each face can be subjected to forces that are perpendicular (i.e.","normal) to the surface, in one direction, or parallel (tangential) to the surface in two directions.","For example, the face that lies on the $x-y$ plane can be subjected to a perpendicular force on the $z$ direction and to two parallel forces on the $x$ and $y$ directions.","Thus, the cube can be subjected to three forces in each face for each of its six faces.","If each surface of the cube is characterized by a surface area, we can define each component of the stress acting on the surface as force over area.","Thus, on the $z$ direction lies the $x-y$ plane with a normal stress component $\\sigma _{zz}$ and two perpendicular stress components, one in the $x$ direction, represented by $\\sigma _{zx}$ and another in the $y$ direction represented by $\\sigma _{zy}$ .","This leaves us with three unique stress components in each direction, $x$ , $y$ and $z$ , with a total of nine stress components completely defining the stresses acting on the element.","The nine elements define the stress tensor $ \\sigma _{ij} =\\begin{bmatrix}\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\\end{bmatrix}$ The components where $i = j$ (i.e.","$\\sigma _{xx}$ , $\\sigma _{yy}$ , $\\sigma _{zz}$ ) are the normal stress components, while the components where $i \\ne j$ are the tangential stress components.","The tangential stress components correspond to shearing stresses seen in solids which can cause isovolumetric deformations.","It is convenient now to introduce the Kronecker delta operator $\\delta _{ij} ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if} ~ i \\ne j, \\\\1 & \\text{if} ~ i = j \\\\\\end{array}\\right.","}$ Such that the normal stresses on the fluid element can be represented as $\\sigma _{ij} = \\delta _{ij} \\sigma _{ij}$ It is important to point out that if the fluid element is at rest, the static fluid pressure is defined as $\\sigma _{ij} = -p \\delta _{ij}$ Applying this equation to the stress tensor results in $ \\sigma _{ij} =\\begin{bmatrix}-p & 0 & 0 \\\\0 & -p & 0 \\\\0 & 0 & -p \\\\\\end{bmatrix}$ Which is nothing more than the well known principle from hydrostatics which states that for a small volume immersed in a fluid (such as in water for example), the pressure is constant across its surface.","Keep in mind that for a moving fluid element the story is quite different.","Now the total surface force exerted on the volume element is the sum of the stresses acting in each surface element with area $dS$ and normal $n_j$ and can be written as $\\int \\sigma _{ij} n_{j} dS$ which can be transformed to a volume integral using the divergence theorem $\\int \\frac{\\partial \\sigma _{ij}}{\\partial x_j} dV$ From Newton's law expressed as conservation of momentum, for the element of volume of the fluid, we can write the total momentum as the sum of forces acting on the fluid, including the summation of all body forces and the surface forces $\\int \\frac{D u_i}{Dt} \\rho dV = \\int F_i \\rho dV + \\int \\frac{\\partial \\sigma _{ij}}{\\partial x_j} dV$ As usual, the integral vanishes identically and we are left with $\\frac{D u_i}{Dt} \\rho = F_i \\rho + \\frac{\\partial \\sigma _{ij}}{\\partial x_j}$ In which $F_i$ is the sum of all body forces acting on the fluid element, i.e.","gravitational, electromagnetic, etc.","Expanding this equation in three dimensional Cartesian coordinates using Einstein's summation convention we have $\\frac{D u_x}{Dt} \\rho = F_x \\rho + \\frac{\\partial \\sigma _{xx}}{\\partial x} + \\frac{\\partial \\sigma _{xy}}{\\partial y} + \\frac{\\partial \\sigma _{xz}}{\\partial z}\\\\\\frac{D u_y}{Dt} \\rho = F_y \\rho + \\frac{\\partial \\sigma _{yx}}{\\partial x} + \\frac{\\partial \\sigma _{yy}}{\\partial y} + \\frac{\\partial \\sigma _{yz}}{\\partial z}\\\\\\frac{D u_z}{Dt} \\rho = F_z \\rho + \\frac{\\partial \\sigma _{zx}}{\\partial x} + \\frac{\\partial \\sigma _{zy}}{\\partial y} + \\frac{\\partial \\sigma _{zz}}{\\partial z}$ These equations show that the acceleration in each direction is caused by the forces acting on that direction and the components of the stress tensor acting on the face perpendicular to the direction of the acceleration.","With some inspection it can also become clear that the components of the stress tensor can cause deformation during motion depending of the nature of the shearing components (i.e.","$i \\ne j$ ).","With that in mind, the stress tensor can be divided into two components, one equivalent to the static pressure experienced by a stationary volume of fluid and a non-isotropic component which originates due to motion of the fluid element $\\sigma _{ij} = -p \\delta _{ij} + d_{ij}$ Which can be represented as $\\begin{bmatrix}\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\\end{bmatrix}=\\begin{bmatrix}-p & 0 & 0 \\\\0 & -p & 0 \\\\0 & 0 & -p \\\\\\end{bmatrix}+\\begin{bmatrix}d_{xx} & d_{xy} & d_{xz} \\\\d_{yx} & d_{yy} & d_{yz} \\\\d_{zx} & d_{zy} & d_{zz} \\\\\\end{bmatrix}$ Now $d_{ij}$ is called the deviatoric stress tensor and is the non-isotropic, tangential, component of stresses.","Considering the components of the stress tensor and expanding the material derivative operator, the unabridged equation of motion for an element of fluid in three-dimensional Cartesian coordinates is $\\rho (\\frac{\\partial u_x}{\\partial t}+u_x \\frac{\\partial u_x}{\\partial x}+u_y\\frac{\\partial u_x}{\\partial y}+u_z\\frac{\\partial u_x}{\\partial z}) = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\frac{\\partial d_{xx}}{\\partial x} + \\frac{\\partial d_{xy}}{\\partial y} + \\frac{\\partial d_{xz}}{\\partial z}\\\\\\rho (\\frac{\\partial u_y}{\\partial t}+u_x \\frac{\\partial u_y}{\\partial x}+u_y\\frac{\\partial u_y}{\\partial y}+u_z\\frac{\\partial u_y}{\\partial z}) = F_y \\rho -\\frac{\\partial p}{\\partial y}+ \\frac{\\partial d_{yx}}{\\partial x} + \\frac{\\partial d_{yy}}{\\partial y} + \\frac{\\partial d_{yz}}{\\partial z}\\\\\\rho (\\frac{\\partial u_z}{\\partial t}+u_x \\frac{\\partial u_z}{\\partial x}+u_y\\frac{\\partial u_z}{\\partial y}+u_z\\frac{\\partial u_z}{\\partial z}) = F_z \\rho -\\frac{\\partial p}{\\partial z} + \\frac{\\partial d_{zx}}{\\partial x} + \\frac{\\partial d_{zy}}{\\partial y} + \\frac{\\partial d_{zz}}{\\partial z}$ The vector form, being simpler and more elegant, is referred to as the Cauchy momentum equation after Augustin-Louis Cauchy $\\boxed{\\rho \\frac{D \\mathbf {u}}{D t} = \\mathbf {F} \\rho - \\nabla p + \\nabla \\cdot \\mathbf {d}}$ The full momentum equation is a highly nonlinear partial differential equation and knowledge of the stress tensor and body forces to which the fluid volume are subjected would be necessary to find a solution for this equation, if at all possible.","Two of the most fundamental equations of fluid mechanics work with simplifications and assumptions over the momentum equation.","The first developed from Newton's laws was derived by Euler and a more fundamental one which is thought to account for most fluid flow phenomena was developed by Claude-Louis Navier and Georges Gabriel Stokes less than a hundred years later.","The Euler equations were derived by Leonhard Euler himself in 1755 for the motion of an inviscid fluid, or a fluid without viscosity.","The complete set of equations of motion, including the viscosity term, were derived in the XIX century, culminating from the works of Claude-Louis Navier and Sir George Gabriel Stokes, after which they are named.","The Euler equations can be understood as a special case of the Navier-Stokes equations and both sets of equations are applications of the Newton laws of motion to fluid flow, and both can be understood as special cases of the Cauchy equation of motion within the Newtonian framework.","Up until now we have not discussed viscosity on any broader sense, most of us know the colloquial definition of viscosity used in daily language.","A so called viscous fluid has more resistance to flow as we might have experienced with grease and honey as compared to water, for example.","A higher viscosity reflects, in essence, higher internal friction on the fluid, or a larger magnitude of the forces between its elemental particles.","Mathematically, viscosity can be represented as the coefficient of proportionality between a tangential stress and the rate of variation of velocity of a fluid in a direction perpendicular to that of the applied stress.","The most simple explanation is that found on most of the literature and we can make use of the stress tensor already introduced.","Imagine you have a two-dimensional element of fluid within a system of horizontal and vertical axes $x$ and $y$ .","Now imagine that in the upper surface of the fluid element a stress $\\sigma _{xy}$ parallel to the surface and perpendicular to the $y$ axis is applied.","If the fluid has any internal friction, any imaginary fluid layer parallel to the surface layer where the stress is being applied will want to move with it.","For a rigid material with infinite viscosity over a frictionless surface, every imaginary layer will move at the same velocity and the material will move as a block.","Now if the material is not rigid the gradient of velocity in the $y$ direction will be proportional to the stress applied and as mentioned above, the viscosity is the coefficient of proportionality $\\sigma _{xy} = \\mu \\frac{d }{dy} u_x$ The more viscous the material, the more a tangential stress applied will propagate to the lower layersC.L.. One of the most important assumptions is that this relationship is linear, or alternatively, that the viscosity is constant in this equation.","This is the case for the so called Newtonian fluids, if the relationship is nonlinear the fluid is called non-Newtonian and a different framework needs to be adopted for finding a general equation of motion.","Now suppose an ideal fluid has no viscosity, in this case there is no sense in speaking of tangential components because there is no internal friction, the deviatoric stress term in the Cauchy equation is zero and we are left with $\\boxed{\\rho \\frac{D \\mathbf {u}}{D t} = \\mathbf {F} \\rho - \\nabla p }$ This is Euler's equation of motion and is valid for ideal fluids.","We can further simplify Euler's equation disregarding body forces, when possible, so that acceleration is caused only by surface forces $\\rho \\frac{D \\mathbf {u}}{D t} = - \\nabla p$ Expanding the material derivative term $\\rho (\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u}) = - \\nabla p$ Euler's equation can be also used to derive the equation for hydrostatic pressure in a fluid in equilibrium.","In this cases there is no motion and the acceleration term represented by the material derivative is zero.","For the hydrostatic pressure to exist it is necessary the action of a body force, represented here by gravitation force $ \\mathbf {F} = \\mathbf {g} $ $\\nabla p = \\rho \\mathbf {g}$ Expanding $\\frac{\\partial p}{\\partial x} \\mathbf {i} + \\frac{\\partial p}{\\partial y} \\mathbf {j} + \\frac{\\partial p}{\\partial z} \\mathbf {k} = \\rho (g_x \\mathbf {i} + g_y \\mathbf {j} + g_z \\mathbf {k})$ And considering that Earth's gravitational force acts in $z$ only, and considering $z$ positive upwards $\\frac{\\partial p}{\\partial z} = -\\rho g$ Which can be readily integrated as for $\\rho $ a $g$ constant as $\\int \\partial p = - \\rho g \\int \\partial z$ Resulting in $p = - \\rho g z + \\text{constant}$ Applying the boundary condition that $p = p_0$ when $z = 0$ (fluid surface) results $p = p_0 - \\rho g z$ Which will increase considering that the absolute value of $z$ increases with depth and that $z$ is negative in the coordinate system adopted.","The problem with Euler's equation, except for special applications, is that most liquids will have some degree of viscosity and the non-isotropic components of the stress tensor need to be considered on the analysis.","The Navier-Stokes equations are the most general form of the governing equation of fluid motion and can be interpreted as a special case of Cauchy momentum equation.","The transition from the Cauchy momentum equation to the Navier-Stokes equations requires that the non-isotropic stress tensor is expressed in a form amenable to the solution of the differential equation using analytical or numerical methods.","Following the language of Batchelor [8], it is necessary to find a relationship between the deviatoric stress tensor and the local properties of the fluid.","As with a lot of transport relations in physics, one possible simplification is to assume a linear relationship between the deviatoric stress and the velocity gradient.","Keep in mind that this assumption carries profound implications in all areas of physics that deal with transport phenomena, either of matter, energy, momentum, electrical current, heat, ... and should not be taken lightly, I urge you to consult the references for details.","The hypothesis of a linear relation between the deviatoric stress tensor and the velocity gradients can be expressed analytically as $d_{ij} = A_{ijkl} \\frac{\\partial u_k}{\\partial x_l}$ Now $A_{ijkl}$ is a fourth order tensor with 81 individual components, from $A_{xxxx}$ to $A_{zzzz}$ in Cartesian $x$ , $y$ and $z$ coordinates.","The full expansion of this equation is not practical in this text due to space and formatting considerations, but a reduced form is provided, considering Einstein's summation convention $& d_{xx} = A_{xxxx} \\frac{\\partial u_x}{\\partial x} + A_{xxxy} \\frac{\\partial u_x}{\\partial y} + A_{xxxz} \\frac{\\partial u_x}{\\partial z} + \\cdots + A_{xxzz} \\frac{\\partial u_z}{\\partial z}\\\\& d_{xy} = A_{xyxx} \\frac{\\partial u_x}{\\partial x} + A_{xyxy} \\frac{\\partial u_x}{\\partial y} + A_{xyxz} \\frac{\\partial u_x}{\\partial z} + \\cdots + A_{xyzz} \\frac{\\partial u_z}{\\partial z}\\\\& \\vdots \\\\& d_{zz} = A_{zzxx} \\frac{\\partial u_x}{\\partial x} + A_{zzxy} \\frac{\\partial u_x}{\\partial y} + A_{zzxz} \\frac{\\partial u_x}{\\partial z} + \\cdots + A_{zzzz} \\frac{\\partial u_z}{\\partial z}$ There are nine coefficients in each line and nine lines, one for each component of the deviatoric stress tensor, totaling the 81 coefficients of the $A_{ijkl}$ fourth-order tensor.","As we have done before, it is convenient to write the deviatoric stress tensor as the sum of its symmetric and antisymmetric counterparts $d_{ij} = A_{ijkl} d_{ij}^{s} + A_{ijkl} d_{ij}^{a}$ or $d_{ij} = A_{ijkl} e_{kl} + A_{ijkl} \\xi _{ij}$ where $e_{kl} = \\frac{1}{2}(\\frac{\\partial u_k}{\\partial u_l} + \\frac{\\partial u_l}{\\partial u_k})$ is a symmetric second-order tensor, which in $x$ , $y$ and $z$ Cartesian coordinates can be represented as $\\frac{1}{2}\\begin{bmatrix}(\\frac{\\partial u_x}{\\partial x} + \\frac{\\partial u_x}{\\partial x}) & (\\frac{\\partial u_x}{\\partial y} + \\frac{\\partial u_y}{\\partial x}) & (\\frac{\\partial u_x}{\\partial z} + \\frac{\\partial u_z}{\\partial x}) \\\\(\\frac{\\partial u_y}{\\partial x} + \\frac{\\partial u_x}{\\partial y}) & (\\frac{\\partial u_y}{\\partial y} + \\frac{\\partial u_y}{\\partial y}) & (\\frac{\\partial u_y}{\\partial z} + \\frac{\\partial u_z}{\\partial y}) \\\\(\\frac{\\partial u_z}{\\partial x} + \\frac{\\partial u_x}{\\partial z}) & (\\frac{\\partial u_z}{\\partial y} + \\frac{\\partial u_y}{\\partial z}) & (\\frac{\\partial u_z}{\\partial z} + \\frac{\\partial u_z}{\\partial z}) \\\\\\end{bmatrix}$ Verify that the quantities that are opposed to each other across the diagonal are the same, and this is what is mean be symmetric in this case.","The antisymmetric component is $\\xi _{kl} = \\frac{1}{2}(\\frac{\\partial u_k}{\\partial u_l} - \\frac{\\partial u_l}{\\partial u_k})$ and can be represented as $\\frac{1}{2}\\begin{bmatrix}(\\frac{\\partial u_x}{\\partial x} - \\frac{\\partial u_x}{\\partial x}) & (\\frac{\\partial u_x}{\\partial y} - \\frac{\\partial u_y}{\\partial x}) & (\\frac{\\partial u_x}{\\partial z} - \\frac{\\partial u_z}{\\partial x}) \\\\(\\frac{\\partial u_y}{\\partial x} - \\frac{\\partial u_x}{\\partial y}) & (\\frac{\\partial u_y}{\\partial y} - \\frac{\\partial u_y}{\\partial y}) & (\\frac{\\partial u_y}{\\partial z} - \\frac{\\partial u_z}{\\partial y}) \\\\(\\frac{\\partial u_z}{\\partial x} - \\frac{\\partial u_x}{\\partial z}) & (\\frac{\\partial u_z}{\\partial y} - \\frac{\\partial u_y}{\\partial z}) & (\\frac{\\partial u_z}{\\partial z} - \\frac{\\partial u_z}{\\partial z}) \\\\\\end{bmatrix}$ Verify that the quantities across the diagonals are different (reverse sign) and this is what is meant by antisymmetric in this context.","Verify also that the sum of the symmetric and antisymmetric counterparts reduces to the second-order tensor $\\frac{\\partial u_k}{\\partial x_l}$ .","The antisymmetric component can further be written as $\\xi _{kl} = -\\frac{1}{2}A_{ijkl} \\epsilon _{klm} \\omega _{m}$ The operator $\\epsilon _{klm}$ is a third-order tensor, with 27 components, where only six are non-zero, called the Levi-Civita symbol with the following property: $\\epsilon _{klm} ={\\left\\lbrace \\begin{array}{ll}+1 & \\text{if} ~(k, l, m)~\\text{is an even permutation} \\\\-1 & \\text{if} ~(k, l, m)~\\text{is an odd permutation} \\\\0 & \\text{if any index is repeated}\\end{array}\\right.","}$ Notice that the choice of $k, l, m$ is incidental to the problem, most often on the literature $i, j, k$ or numbers are used.","Even permutations in this case are $(k,l,m)$ , $(l,m,k)$ and $(m, k, l)$ and odd permutations are $(m, l, k)$ , $(l, k, m)$ and $(k, m, l)$ .","The vector $\\omega _{m}$ has a fundamental role in fluid mechanics, and it is called the vorticity of the fluid.","The components of the vorticity vector in $x$ , $y$ and $z$ Cartesian coordinates are $\\omega _x = \\frac{\\partial u_z}{\\partial y} - \\frac{\\partial u_y}{\\partial z} \\\\\\omega _y = \\frac{\\partial u_x}{\\partial z} - \\frac{\\partial u_z}{\\partial x} \\\\\\omega _z = \\frac{\\partial u_y}{\\partial x} - \\frac{\\partial u_x}{\\partial y}$ Which can be obtained from the curl of the $\\textbf {u}$ vector field (recall Chapter REF ) $\\omega = \\nabla \\times \\mathbf {u}$ A fluid with zero curl is called irrotational.","The mathematical concept of rotation is directly linked to the physical idea of rotation in fluid flow.","The idea of rotational and irrotational flows is rarely discussed in soil physics, largely because of the assumptions adopted related to the nature of the flow as we will discuss later.","Now the deviatoric stress tensor can be rewritten as $d_{ij} = A_{ijkl} e_{kl} - \\frac{1}{2} A_{ijkl} \\epsilon _{klm} \\omega _m$ In simple liquids, $A_{ijkl}$ , is an isotropic tensor, that is, the molecular structure of the fluid is isotropic.","In this case, $A_{ijkl}$ , can be written as a summation of products of delta tensorsObviously, there is much more to this, the reader is encouraged to consult a book on tensor analysis and Aris [4] $A_{ijkl} = \\mu \\delta _{ik}\\delta _{jl} + \\mu ^{\\prime } \\delta _{il}\\delta _{jk} + \\mu ^{\\prime \\prime } \\delta _{ij}\\delta _{kl}$ Because $A_{ijkl}$ is symmtrical in $i$ and $j$ , it is required that $\\mu = \\mu ^{\\prime } $ such that $A_{ijkl} = 2\\mu \\delta _{ik}\\delta _{jl} + \\mu ^{\\prime \\prime } \\delta _{ij}\\delta _{kl}$ The requirement that $A_{ijkl}$ is also symmetrical in $k$ and $l$ requires that the term $\\omega $ is dropped in the deviatoric stress tensor, so that by replacing into it we have $d_{ij} = 2\\mu \\delta _{ik}\\delta _{jl} e_{kl} + \\mu ^{\\prime \\prime } \\delta _{ij}\\delta _{kl} e_{kl}$ Because the rules of manipulation of Kronecker delta tensors, the relationship can be simplified to $d_{ij} = 2\\mu e_{ij} + \\mu ^{\\prime \\prime } \\nabla \\cdot e_{kk}~\\delta _{ij}$ Where $e_{kk} = \\nabla \\cdot \\mathbf {u} $ is the rate of expansion.","Since the deviatoric stress tensor does not contribute to the mean normal stresses $d_{ii} = (2\\mu + 3\\mu ^{\\prime \\prime }) e_{ii} = 0$ Such that $(2\\mu + 3\\mu ^{\\prime \\prime }) = 0$ Solving for $\\mu ^{\\prime \\prime }$ and replacing into the deviatoric stress tensor C.L.","results $d_{ij} = 2\\mu (e_{ij} - \\frac{1}{3} e_{ii}~\\delta _{ij})$ We now have an expression for the deviatoric stress tensor which can be used to define the stress tensor.","Replacing into the stress tensorC.L.","results $\\sigma _{ij} = -p \\delta _{ij} + 2\\mu (e_{ij} - \\frac{1}{3} e_{ii}~\\delta _{ij})$ Which can then be replaced into the equation of motion resulting in the set of equations $\\rho \\frac{D u_x}{D t} = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\frac{\\partial }{\\partial x}[ 2 \\mu (e_{xx} - \\frac{1}{3} e_{xx} \\delta _{xx})]+ \\\\ \\frac{\\partial }{\\partial y}[ 2 \\mu (e_{xy} - \\frac{1}{3} e_{xx} \\delta _{xy})]+ \\frac{\\partial }{\\partial z}[ 2 \\mu (e_{xz} - \\frac{1}{3} e_{xx} \\delta _{xz})]$ $\\rho \\frac{D u_y}{D t} = F_y \\rho -\\frac{\\partial p}{\\partial y}+ \\frac{\\partial }{\\partial x}[ 2 \\mu (e_{yx} - \\frac{1}{3} e_{yy} \\delta _{yx})]+ \\\\ \\frac{\\partial }{\\partial y}[ 2 \\mu (e_{yy} - \\frac{1}{3} e_{yy} \\delta _{yy})]+ \\frac{\\partial }{\\partial z}[ 2 \\mu (e_{yz} - \\frac{1}{3} e_{yy} \\delta _{yz})]$ $\\rho \\frac{D u_z}{D t} = F_z \\rho -\\frac{\\partial p}{\\partial z}+ \\frac{\\partial }{\\partial x}[ 2 \\mu (e_{zx} - \\frac{1}{3} e_{zz} \\delta _{zx})]+ \\\\ \\frac{\\partial }{\\partial y}[ 2 \\mu (e_{zy} - \\frac{1}{3} e_{zz} \\delta _{zy})]+ \\frac{\\partial }{\\partial z}[ 2 \\mu (e_{zz} - \\frac{1}{3} e_{zz} \\delta _{zz})]$ These are the Navier-Stokes equations and they account for the description of viscous fluid flow.","The viscosity term in the full form presented above is not independent of position and the equation is valid for compressible fluids as well as incompressible fluids.","If we consider that the viscosity is constant throughout the flow field, the term $\\mu $ can be placed outside the derivative terms.","Another simplification that can be used for modeling flow in porous media is incompressibility.","For an incompressible fluid $\\nabla \\cdot \\mathbf {u} = e_{ii} = 0 $ and the term $ e_{ii} \\delta _{ij} $ is dropped, resulting in $\\rho \\frac{D u_x}{D t} = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\mu \\frac{\\partial }{\\partial x} (e_{xx})+\\mu \\frac{\\partial }{\\partial y}(e_{xy})+ \\mu \\frac{\\partial }{\\partial z} (e_{xz})\\\\\\rho \\frac{D u_y}{D t} = F_y \\rho -\\frac{\\partial p}{\\partial y}+ \\mu \\frac{\\partial }{\\partial x}(e_{yx})+\\mu \\frac{\\partial }{\\partial y}(e_{yy})+ \\mu \\frac{\\partial }{\\partial z}(e_{yz})\\\\\\rho \\frac{D u_z}{D t} = F_z \\rho -\\frac{\\partial p}{\\partial z}+ \\mu \\frac{\\partial }{\\partial x}(e_{zx})+\\mu \\frac{\\partial }{\\partial y}(e_{zy})+ \\mu \\frac{\\partial }{\\partial z}(e_{zz})$ Note that, in this case, each partial derivative of $e_{ij}$ can be written as $\\frac{\\partial }{\\partial x_j} e_{ij} = \\frac{\\partial }{\\partial x_j} \\frac{1}{2}(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}) = \\frac{1}{2}\\frac{\\partial ^2 u_i}{\\partial x_j\\partial x_j} + \\frac{1}{2}\\frac{\\partial }{\\partial x_i}\\frac{\\partial u_j}{\\partial x_j} = \\frac{1}{2} \\nabla ^2 u_i + \\frac{1}{2} \\frac{\\partial }{\\partial x_i}(\\nabla \\cdot \\mathbf {u})$ the incompressibility condition reduces the Navier-Stokes equations to $\\rho \\frac{D u_x}{D t} = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\mu ( \\frac{\\partial ^2 u_x}{\\partial x^2} + \\frac{\\partial ^2 u_x}{\\partial y^2} + \\frac{\\partial ^2 u_x}{\\partial z^2})\\\\\\rho \\frac{D u_y}{D t} = F_y \\rho -\\frac{\\partial p}{\\partial y} + \\mu ( \\frac{\\partial ^2 u_y}{\\partial x^2} + \\frac{\\partial ^2 u_y}{\\partial y^2} + \\frac{\\partial ^2 u_y}{\\partial z^2})\\\\\\rho \\frac{D u_z}{D t} = F_z \\rho -\\frac{\\partial p}{\\partial z} + \\mu ( \\frac{\\partial ^2 u_z}{\\partial x^2} + \\frac{\\partial ^2 u_z}{\\partial y^2} + \\frac{\\partial ^2 u_z}{\\partial z^2})$ which can be written in vector form and by expanding the material derivative as $\\boxed{\\rho (\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u} ) = \\mathbf {F} \\rho - \\nabla p + \\mu \\nabla ^2 \\mathbf {u}}$ This is the fundamental equation that governs the flow of incompressible fluids and many important results can be derived by solving it for a specified set of boundary conditions.","We will used it over subsequent chapters to investigate settling and water flow phenomena.","What we will call Stokes equation is a special case of the Navier-Stokes equation for a set of specified conditions.","Because we will use Stokes equation as a starting point for particle settling and flow in porous media derivations, it will be specified here.","Under steady-state flow or flows in which the velocity variation is small enough that it can be considered constant and in conditions where the inertia is small as compared to the pressure and viscous terms the acceleration term on the left side of the Navier-Stokes equation can be neglected.","If the body forces can also be neglected due to being small or the direction of flow in relation to a gravitational field for example, the governing equations become $\\boxed{\\nabla p = \\mu \\nabla ^2 \\mathbf {u}}$ $\\nabla \\cdot \\mathbf {u} = 0$ The second of those being the conservation law described previously.","Bernoulli equation can be derived from Euler equations by using the simplifying assumptions of steady and isentropic flow.","Steady flow means that the velocity is constant in any particular point in the flow field, i.e.","does not vary with time, such that $ \\frac{\\partial \\mathbf {u}}{\\partial t} = 0 $ .","Isentropic means that the entropy $s$ is constant throughout the flow field.","To derive Bernoulli equation we need to first express Euler equation in terms of enthalpy.","In thermodynamics, specific enthalpy can be expressed as $dh = Tds + vdP$ But, because entropy is constant, $ds = 0 $ and $dh = vdp$ In this equation $h$ is the specific enthalpy or enthalpy per unit mass $h = H/M$ , $s$ is the specific entropy, $s = S/M$ and $v$ is the specific volume $v = V/M$ .","Since we know that density of a fluid is mass per unit volume, the specific volume can be written as $v = 1/\\rho $ and $dh = \\frac{1}{\\rho }dp$ which can be generalized to $\\nabla h = \\frac{1}{\\rho } \\nabla p$ and replaced into Euler equation resulting $\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u} = - \\nabla h + \\mathbf {F} \\rho $ We now use a vector calculus identity referred to as the vector dot del operator $\\mathbf {u} \\cdot \\nabla \\mathbf {u} = \\frac{1}{2} \\nabla u^2 - \\mathbf {u} \\times (\\nabla \\times \\mathbf {u} )$ by replacing into the “enthalpy” Euler equation, and with the condition that the velocity does not very in time specified previously, we have $\\mathbf {u} \\times (\\nabla \\times \\mathbf {u} ) = - \\nabla (h + \\frac{1}{2} u^2) + \\mathbf {F}$ Flow conditions require that the curl terms on the left side is zero.","We can also rewrite the force term as the negative of the gradient of a potential $\\Psi $ $0 = - \\nabla (h + \\frac{1}{2} u^2) - \\nabla \\Psi $ and lump the terms into a single gradient operator $\\nabla (h + \\frac{1}{2} u^2 + \\Psi ) = 0$ Replacing the expression for the enthalpy previously defined and considering $\\Psi $ as the force potential to Earth's gravitational field $\\Psi = \\rho g$ , we are left with $\\boxed{\\frac{p}{\\rho } + \\frac{1}{2} u^2 + \\rho g = \\text{constant}}$ This equation states that the left hand sum is constant along any streamline.","A streamline is a line whose tangent at any point gives the direction of the velocity vector at that point.","Streamlines are defined by the system of equations $\\frac{dx}{u_x} = \\frac{dy}{u_y} = \\frac{dz}{u_z}$ Reynolds number is known in soil physics and applied hydraulics and hydrology as a means of establishing if a given flow regime is laminar or turbulent.","It is much more than that, it derives from writing the Navier-Stokes equation in a dimensionless form and conveys important information about the flow regime and scaling of flow processes and can be used in laboratory to design experiments to measure flow parameters that could not be measured otherwise, on account of the law of similarity.","We start with the Navier-Stokes equations in an incompressible and homogeneous fluid with regard to density.","The body force is ignored, but the same principles applied to the other terms can be used if one wishes to proceed including the body force $\\rho (\\frac{\\partial u_i}{\\partial t} + u_j \\frac{\\partial u_i}{\\partial x_j} ) = -\\frac{\\partial p}{\\partial x_i} + \\mu \\frac{\\partial ^2 u_i}{\\partial x_j \\partial x_i}$ Each quantity on the Navier-Stokes equation is now written in a dimensionless form by defining some representative length L and velocity U, with the new dimensionless variables $\\mathbf {u}^{\\prime }$ , $t^{\\prime }$ and $\\mathbf {x}^{\\prime }$ $\\mathbf {u}^{\\prime } = \\frac{\\mathbf {u}}{U}$ $\\mathbf {x}^{\\prime } = \\frac{\\mathbf {x}}{L}$ $t^{\\prime } = \\frac{tU}{L}$ Note that U/L has units of 1/time since [L T$^{-1}$ ] / [L] = [T$^{-1}$ ].","Pressure can also be normalized using $p^{\\prime } = \\frac{p}{\\rho U^2}$ Since $\\rho U^2$ has units of [M L$^{-3}$ ] [L T$^{-1}$ ]$^2$ = [M L$^{-1}$ T$^{-2}$ ] which is equivalent to pressure.","Substituting the dimensionless quantities on the Navier-Stokes $\\rho (\\frac{\\partial u^{\\prime }_i U}{\\partial t^{\\prime }L/U} + u^{\\prime }_j U \\frac{\\partial u^{\\prime }_i U}{\\partial x^{\\prime }_j L} ) = -\\frac{\\partial p^{\\prime }\\rho U^2}{\\partial x^{\\prime }_i L} + \\mu \\frac{\\partial ^2 u^{\\prime }_i U}{\\partial x^{\\prime }_j L \\partial x^{\\prime }_i L}$ Remembering that constant terms can be brought out of the derivatives and reorganizing $\\frac{U^2}{L}\\rho (\\frac{\\partial u^{\\prime }_i}{\\partial t^{\\prime }} + u^{\\prime }_j \\frac{\\partial u^{\\prime }_i}{\\partial x^{\\prime }_j} ) = -\\frac{U^2}{L}\\frac{\\partial p^{\\prime }\\rho }{\\partial x^{\\prime }_i} + \\mu \\frac{U}{L^2}\\frac{\\partial ^2 u^{\\prime }_i}{\\partial x^{\\prime }_j \\partial x^{\\prime }_i} \\\\\\frac{\\partial u^{\\prime }_i}{\\partial t^{\\prime }} + u^{\\prime }_j \\frac{\\partial u^{\\prime }_i}{\\partial x^{\\prime }_j} = -\\frac{\\partial p^{\\prime }}{\\partial x^{\\prime }_i} + \\frac{\\mu }{\\rho U L}\\frac{\\partial ^2 u^{\\prime }_i}{\\partial x^{\\prime }_j \\partial x^{\\prime }_i}$ The dimensionless quantity $\\boxed{Re = \\frac{\\rho U L}{\\mu }}$ is called the Reynolds number, being introduced by Stokes and later named after Osborne Reynolds who popularized its use.","The dimensionless form of the Navier-Stokes equation can be written in vector notation as $\\frac{\\partial \\mathbf {u}^{\\prime }}{\\partial t^{\\prime }} + \\mathbf {u}^{\\prime }\\cdot \\nabla \\mathbf {u}^{\\prime } = -\\nabla p^{\\prime } + \\frac{1}{Re} \\nabla ^2 \\mathbf {u}^{\\prime }$ The Reynolds number is a ratio of inertial forces, that tend to keep the fluid in motion, and viscous forces, which resist movement.","In fluids in which the Reynolds number is low, there is large contribution of inertial forces.","If inertial forces are much larger than viscous forces the flow can become unstable and vortices and other flow instabilities can develop.","Under such conditions the assumption that the flow is smooth, constant and “continuous” is no longer valid and the flow can no longer be considered laminar.","If inertial forces are very high in respect to viscous forces, fully turbulent flow might develop.","ll $ \\mathbf {s} $ Vector position function $ \\mathbf {u}, \\mathbf {v} $ Vector velocity functions $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k}, $ Unit vectors in Cartesian coordinates $ \\mathbf {x}, \\mathbf {y}, \\mathbf {z}, $ Unit vectors in Cartesian coordinates $ s_x, s_y, s_z $ Components of the position vector in Cartesian coordinates $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ u, v, w $ Components of the velocity vector in Cartesian coordinates $ t $ Time $ \\frac{D}{D t} $ Substantial (or material) derivative $ f $ Generalized function $ \\rho $ Density $ V $ Volume $ A $ Area $ \\mathbf {n} $ Normal unit vector $ m, M $ Mass $ x, y, z, $ Generalized Cartesian coordinates $ \\mathbf {F} $ Force vector $ \\mathbf {a} $ Acceleration vector $ \\mathbf {p} $ Momentum vector $ \\sigma $ Stress $ \\delta _{ij} $ Kroenecker delta $ p $ Pressure $ S $ surface $ d_{ij} $ Deviatoric stress tensor $ \\mathbf {d} $ Deviatoric stress tensor, vector form $ \\mathbf {g} $ Gravitational body force vector $ g_x, g_y, g_z $ Components of the gravitational body force vector in Cartesian coordinates $ A_{ijkl} $ Fourth-order tensor $ e_{kl} $ Symmetric component of the deviatoric stress tensor $ \\xi _{kl} $ Antisymmetric component of the deviatoric stress tensor $ \\epsilon _{klm} $ Levi-Civita symbol $ \\omega $ Vorticity vector $ \\omega _x, \\omega _y, \\omega _w $ Components of the vorticity vector in Cartesian coordinates $ e_{kk} $ Rate of expansion $ \\mu $ Viscosity $ h $ Specific enthalpy $ s $ Specific entropy $ v $ Specific volume $ T $ Temperature $ P $ Pressure $ S $ Entropy $ H $ Enthalpy $ \\Psi $ Potential $ \\mathbf {u^{\\prime }} $ Dimensionless velocity vector $ \\mathbf {x^{\\prime }} $ Dimensionless position vector $ t^{\\prime } $ Dimensionless time $ p^{\\prime } $ Dimensionless pressure $ Re $ Reynolds number $ L $ Length $ U $ Velocity There are two basic methods for the determination of particle size distribution of soils and sediments.","For coarser particles such as gravel and cobbles, and sands to some extent, grain size distributions or grain size fractions can be determined using sieves.","In essence what is done is to disaggregate the particles, if they are aggregated at all, using physical and or chemical methods, pass the material through a set of sieves of decreasing apertures and to weigh the dried fraction retained in each sieve and to express this as a percent of the total dried mass of the material used.","The sieve aperture used for each class interval is a convention, and the textural classes vary between disciplines in soil science, Earth science and engineering.","For particles finer than sand, sieving does not work adequately, first because it might not be feasible to create sieves with apertures small enough for the particles, which might have diameters on the micrometer range, second because it might be virtually impossible to disaggregate these particles in the dried state, and third because even if these particles could be disaggregated and precision micrometer aperture sized sieves created, they might cause the clogging of the apertures of the sieve which at that size range would act analogous as pores.","For smaller particles, most methods of particle size determination rely on settling of the dispersed particles in a fluid, usually water.","Particles are assumed as perfectly spherical, non-interacting, slowly settling in an infinitely dispersed solution.","Under these conditions the Navier-Stokes equation solution for a sphere settling in a viscous fluid is the solution to the problem.","The fact that the settling is slow under low Reynolds number guarantees that the flow around the sphere is laminar (Figure REF a).","Under turbulent flows, analytical solutions to the Navier-Stokes equations, such as the one presented below, might not be feasible due to the complexity of the chaotic fluctuations of the flow regime.","In any case even before turbulence developed, the settling velocity might be large enough such that the inertial component of the Navier-Stokes equation is not negligible, and analytic solutions might not be achievable.","Figure: Illustration of sphere settling under laminar regime.The solution of the sphere settling problem was provided by G.G.","Stokes [92], many authors present interpretations of Stokes solution which are more or less similar to the original and to each-other [66], [57], [8], [1].","The solution given by Acheson [1] is presented here for reasons of clarity and conciseness.","For slow, steady-state settling conditions the mathematical problems is to find an analytical solution to the Stokes and conservation equations presented in Chapter REF .","$\\nabla p = \\mu \\nabla ^2 \\mathbf {u}$ $\\nabla \\cdot \\mathbf {u} = 0$ Not surprisingly, the problem of flow past a sphere is solved by using spherical coordinates.","Thus the vector $\\mathbf {u} = \\langle u_x, u_y, u_z \\rangle = u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}$ is written in spherical coordinates as $\\mathbf {u} = \\langle u_r(r, \\theta ), u_{\\theta }(r, \\theta ), 0 \\rangle $ The conservation equation $ \\nabla \\cdot \\mathbf {u}$ is satisfied by a Stokes stream function $\\Psi (r , \\theta )$ which satisfies the conditions $u_r = \\frac{1}{r^2 \\sin {\\theta }} \\frac{\\partial \\Psi }{\\partial \\theta }, ~ u_{\\theta } = \\frac{1}{r \\sin {\\theta }} \\frac{\\partial \\Psi }{\\partial r}$ Now, the curl of $\\mathbf {u}$ in spherical coordinates (consult Chapter REF ) is $\\nabla \\times \\mathbf {u} = \\langle 0, 0, \\frac{1}{r \\sin {\\theta }} E^2 \\Psi \\rangle $ With $E^2$ representing the differential operator $E^2 = \\frac{\\partial ^2}{\\partial r^2} + \\frac{\\sin {\\theta }}{r^2} \\frac{\\partial }{\\partial \\theta } (\\frac{1}{\\sin {\\theta }} \\frac{\\partial }{\\partial \\theta } )$ By virtue of vector identities, Stokes equation can be written as $\\nabla p = -\\mu \\nabla \\times (\\nabla \\times \\mathbf {u})$ in which the gradient operators in $p$ and after $\\mu $ can be written in spherical coordinates as $\\nabla = \\langle \\frac{\\partial }{\\partial r}, \\frac{1}{r} \\frac{\\partial }{\\partial \\theta }, \\frac{1}{r \\sin {\\theta }} \\frac{\\partial }{\\partial \\phi } \\rangle $ from Equations 5.3, 5.5. and 5.6 the components in $\\mathbf {r}$ and $\\theta $ can be obtained $\\frac{\\partial p}{\\partial r} = \\frac{\\mu }{r^2 \\sin {\\theta }} \\frac{\\partial }{\\partial \\theta } E^2 \\Psi ,~\\frac{1}{r} \\frac{\\partial p}{\\partial \\theta } = \\frac{-\\mu }{r \\sin {\\theta }} \\frac{\\partial }{\\partial r} E^2 \\Psi $ The detailed procedure for cross differentiation considering a stream function $\\Psi $ is given by Lamb [66].","Acheson [1] explains it in terms of the pressure being eliminated by cross-differentiation which implies that $E(E^2\\Psi ) = 0$ such thatObviously there is much more to this, the user is encouraged to consult Stokes (1901)[92] and Lamb (1932)[66], for an in depth discussion.","Stokes (1901) is a reprint of the original and Lamb's discussion draws directly from the source and the level of detail provided is unmatched in modern literature (Horace Lamb was a student of Sir George Stokes at Trinity College, University of Cambridge).","There is a very elegant derivation in Landau and Lifshitz [57] based largely on vector calculus operators.","Although precise, Landau and Lifshitz and Batchelor [8] derivations rely heavily in implicit arguments.","We tried to find a compromise between conciseness and clarity and most likely have not been entirely successful to say the least.","$[\\frac{\\partial ^2}{\\partial r^2} + \\frac{\\sin {\\theta }}{r^2} \\frac{\\partial }{\\partial \\theta } (\\frac{1}{\\sin {\\theta }} \\frac{\\partial }{\\partial \\theta } ) ]^2 \\Psi = 0$ The boundary conditions on the settling sphere are $u_r = u_{\\theta } = 0 ~\\text{at} ~r = a$ $u_r = U\\cos {\\theta },~u_{\\theta } = -U\\sin {\\theta } ~\\text{as} ~r \\rightarrow a$ Because the decomposition of the velocity components in $\\mathbf {r}$ and $\\theta $ .","The latter conditions along with Equation 5.2, results, by integration in the condition $\\Psi = \\frac{U}{2} r^2 \\sin ^2{\\theta } ~\\text{as} ~r \\rightarrow a$ Now we search for a solution to the partial differential equation 5.8.","The theory on solution to partial differential equations plays a fundamental role in analytical methods in fluid mechanics and soil physics, reader unfamiliar with it are encouraged to consult Boas [14] for a concise introduction and Asmar [5] for a more detailed discussion.","Equation 5.8 is satisfied by a solution of the form $\\Psi = \\sin ^2{\\theta } f(r)$ provided that $[\\frac{d^2}{dr^2} - \\frac{2}{r^2}] f(r) = 0$ with solution of the form $f(r) = \\frac{A}{r} + Br + Cr^2 + Dr^4$ with A, B, C and D constants.","Substituting Equation 5.14 into 5.12 results $\\Psi = \\sin ^2{\\theta } [ \\frac{A}{r} + Br + Cr^2 + Dr^4]$ The values of the constant can be obtained by applying the boundary conditions previously defined to Equation 5.15, resulting $\\Psi = \\frac{U}{2}[ r^2 + \\frac{a^3}{2r} - \\frac{3ar}{2}] \\sin ^2{\\theta }$ By combining $\\mathbf {r}$ component of Equation 5.7 , Equation 5.4 and Equation 5.16 and integrating from $r = 0$ from $r = \\infty $ a pressure function is obtained $p = p_{\\infty } - \\frac{3}{2} \\frac{\\mu U a}{r^2} \\cos {\\theta }$ The stress components on a sphere settling in a viscous fluid are $\\tau _r = -p + 2\\mu \\frac{\\partial u_r}{\\partial r}, ~ \\tau _{\\theta } = \\mu r \\frac{\\partial }{\\partial r}(\\frac{u_{\\theta }}{r}) + \\frac{\\mu }{r} \\frac{\\partial u_r}{\\partial \\theta }, ~\\tau _{\\phi } = 0$ From Equations 5.2, 5.16 and 5.18 we have $\\tau _r = -p_{\\infty } + \\frac{3}{2} \\frac{\\mu U}{a}\\cos {\\theta }$ $\\tau _{\\theta } = - \\frac{3}{2} \\frac{\\mu U}{a}\\sin {\\theta }$ The resulting stress vector on the sphere is $\\tau _{net} = \\tau _r \\cos {\\theta } - \\tau _{\\theta }\\sin {\\theta } = -p_{\\infty }\\cos {\\theta } + \\frac{3}{2} \\frac{\\mu U}{a}$ and the total drag force on the sphere can be calculated by integrating the net force in spherical coordinates $F_{D} = \\int _0^{2\\pi } \\int _0^{\\pi } a^2~\\tau _{net}~\\sin {\\theta }~d \\theta d \\phi $ $F_{D} = \\int _0^{2\\pi } \\int _0^{\\pi } a^2~(-p_{\\infty }\\cos {\\theta } + \\frac{3}{2} \\frac{\\mu U}{a})~\\sin {\\theta }~d \\theta d \\phi \\\\F_{D} = \\int _0^{2\\pi } \\int _0^{\\pi } a^2~(-p_{\\infty } ~\\sin {\\theta }~\\cos {\\theta })~d \\theta d \\phi + \\int _0^{2\\pi } \\int _0^{\\pi } (a~\\frac{3}{2} \\mu U~\\sin {\\theta })~d \\theta d \\phi \\\\F_{D} = \\int _0^{2\\pi } a^2~(-p_{\\infty } ~\\frac{\\sin ^2{\\theta }}{2})\\bigg |_0^{\\pi }~d \\phi + \\int _0^{2\\pi } -(a~\\frac{3}{2} \\mu U~\\cos {\\theta })\\bigg |_0^{\\pi } ~ d \\phi \\\\$ Remembering that $\\sin {0}$ = 0 and $\\sin {\\pi } = 0$ and that $\\cos {0}$ = 1 and $\\cos {\\pi } = -1$ , the first term on the right side is equal to zero $F_{D} = \\int _0^{2\\pi } (2 a~\\frac{3}{2} \\mu U) ~ d \\phi \\\\F_{D} = \\int _0^{2\\pi } (a 3 \\mu U) ~ d \\phi \\\\F_{D} = (a 3 \\mu U) \\phi \\bigg |_0^{2\\pi } \\\\$ $\\boxed{F_{D} = 6 \\pi \\mu U a}$ This is Stokes formula for the drag on a sphere settling at low velocities in a viscous fluid.","It is valid for creep flow, often called Stokes flow, in which the inertial forces are much smaller than the viscous forces.","Flow is laminar, meaning that fluid particles follow smooth trajectories, without turbulent mixing.","It is often conceptualized as continuous flow lines around the object which is moving through the fluid or around which the fluid is moving.","The terminal settling velocity can be found by the force balance in the $z$ direction since the sphere is settling in direction of the gravitational force, against viscosity and buoyancy in a stationary fluid.","$\\sum F_z = F_D + F_b + F_g = ma$ Buoyancy force is proportional to the mass of fluid displaced by the settling object $F_b = m_{fluid} g$ While the gravitational force is $F_g = m_{sphere} g$ But the volume of displaced fluid and the volume of the sphere are the same and equal to $V_{sphere} = \\frac{4}{3} \\pi R^3$ If the density of the fluid is $\\rho _{fluid} = \\frac{m_{fluid}}{V_{fluid}}$ and that of the sphere is $\\rho _{sphere} = \\frac{m_{sphere}}{V_{sphere}}$ the buoyancy force can be written as $F_b = \\rho _{fluid} \\frac{4}{3} \\pi R^3 g$ while the gravitational force is $F_g = \\rho _{sphere} \\frac{4}{3} \\pi R^3 g$ Now in the drag force equation $a$ is R, the radius of the sphere and the terminal translational velocity can be written as $v_s$ such that $\\boxed{F_{D} = 6 \\pi \\mu v_s R}$ Replacing all the forces in Equation 5.24 and remembering that the terminal velocity is constant, i.e., there is no acceleration, $ma = 0$ , and $\\sum F_z = 6 \\pi \\mu v_s R + \\rho _{fluid} \\frac{4}{3} \\pi R^3 g - \\rho _{sphere} \\frac{4}{3} \\pi R^3 g = 0$ Where the coordinates systems establishes that downward forces have negative sign.","Solving for $v_s$ $6 \\pi \\mu v_s R = \\rho _{sphere} \\frac{4}{3} \\pi R^3 g - \\rho _{fluid} \\frac{4}{3} \\pi R^3 g \\\\6 \\pi \\mu v_s R = \\frac{4}{3}(\\rho _{sphere} - \\rho _{fluid}) \\pi R^3 g \\\\v_s = \\frac{2}{9} \\frac{ (\\rho _{sphere} - \\rho _{fluid}) g}{\\mu } R^2$ Since the sphere in this case is the solid particles settling, or solid minerals, we can rewrite using notation from previous chapters as $\\boxed{v_s = \\frac{2}{9} \\frac{ (\\rho _{p} - \\rho _{f}) g}{\\mu } R^2}$ This equation is know as Stokes law of settling and it is used to model the settling behavior of infinitely dispersed spherical solid particles settling in laminar regime.","Terminal velocity is a function of the squared radius of the particle, therefore, everything else constant in a viscous fluid, larger particles settle faster.","Settling velocity is also controlled by the densities of the fluid and the solid, viscosity and gravity.","Fluid density and viscosity are controlled by temperature and empirical laws should need to be used to estimate these quantities.","Stokes law of settling is valid for laminar flow, where $Re \\ll 1$ The limits are determined empirically so the critical value of the Reynolds number might depend on the nature of the problem being studied.","More specifically, there might be transitional flow regimes between laminar and fully turbulent.","The Reynolds number a settling spherical particle is $Re = \\frac{2\\rho _f R v_s}{\\mu }$ Considering a maximum Reynolds number for laminar flow in settling of $Re \\approx 1$ (this might be a conservative value), the maximum radius of a smooth spherical particle falling under laminar flow regime is $\\frac{\\mu }{2\\rho _{f} R} = \\frac{2}{9} \\frac{ (\\rho _{p} - \\rho _{f}) g}{\\mu } R^2 \\\\$ $R = \\@root 3 \\of { \\frac{9 \\mu ^2}{4 \\rho _f(\\rho _s-\\rho _f)g }}$ For water density of $\\rho \\approx 1000 $ kg m$^{-3}$ , and viscosity at 25 $^{o}$ C of $\\mu \\approx 0.00089 $ Pa s$^{-1}$ , the maximum radius of a smooth spherical quartz particle with $\\rho _s \\approx 2650 $ kg m$^{-3}$ and $g = 9.80665 m~s^{-2}$ falling under laminar flow regime is approximately $R = 4.793 \\times 10^{-5}~m $ or $R = 0.048~mm$ , which translates into a diameter of approximately $0.1~mm$ .","On the other hand of the spectrum, for colloidal particles, Stokes law is no longer valid because under real circumstances, the solution is not infinitely dispersed and the particles are subjected to thermal motion (also known as Brownian motion).","As smaller particles settle, a concentration gradient is established along the length of the settling column and a concentration gradient will keep particles in suspension as long as there is no coagulation or aggregation in suspension [60], [76].","In other words, the thermal motion is able to overcome gravitational forcesAlbert Einstein's Ph.D. dissertation was on Brownian motion, before he was made famous by his studies in relativity and the photoelectric effect.","The Péclet number is another dimensionless number in hydrodynamics which expresses the ratio between advective and diffusive transport rates [76].","If $Pe \\ll 1$ , diffusive transport predominates and settling is perturbed by thermal motion.","Considering $Pe = 1$ the radius of a sphere settling is given by $R = [\\frac{3 k T}{4 \\pi (\\rho _s - \\rho _f)g}]^{\\frac{1}{4}}$ In which $k = 1.380649 \\times 10^{-23}~J~K^{-1}$ is the Boltzmann constant and all other variables were previously defined.","Using the values for a spherical silicate mineral settling in water presented for the previous calculations, the radius above which diffusive transports can be consider to start affecting settling is $ r = 4.96 \\times 10^{-7} ~m$ or $r = 0.4965~\\mu m$ , or approximately $1~\\mu m$ .","The conclusion is that Stokes law for an average spherical mineral particle settling in water at room temperature is valid for diameters above that of colloidal particles and below that of fine sand particles.","Below the colloidal limit settling is hindered by thermal motion and above the fine sand limit settling is under turbulent regime.","Stokes law only applies to creeping (laminar) flow.","In reality, the dynamics of particles on the clay size range is much more complicated because of surface forces among the interacting particles.","These forces can cause aggregation of particles in suspension to which Stokes law applies to some degree and settling via coagulation and other mechanisms.","Stokes law in particle size characterization of soils and fine sediments assumes that aggregation and coagulation do not apply and the analysis should be disregarded if these processes are observed.","We saw in Chapter REF that when charged particles are put in contact with an electrolyte solution an Electric Double Layer (EDL) develops with charged ions adsorbed onto the surface by electrostatic forces.","The concentration of ions is larger at the surface and at certain concentrations the EDL extends far enough that repulsive forces predominated between charged particles and the colloidal suspension remains dispersed.","However, this is not the entire story.","Between particles, atoms and molecules, dispersion forcesThe term dispersion forces here refers to intermolecular and interatomic forces and not to particles in dispersion originating from quantum fluctuations arise.","These dispersion forces are grouped as what is called van der Waals forces (vdW) and for the systems we are concerned are attractive in nature.","If the electrolyte concentration is increased, the EDL is compressed and the vdW forces which are stronger than the repulsive forces in the short range predominate and the colloidal particles can aggregate and coagulate.","This can happen with dispersed soil solutions depending on the valence and concentration of the electrolyte as clays are charged particles of small diameter.","The theory of balance of these attractive and repulsive forces in colloidal suspensions is called DLVO after Boris Derjaguin, Lev Landau, Evert Verwey and Theodoor Overbeek who developed it, the first two working together and the last two independently.","Considering the interaction of two charged spheres of radius R in a 1:1 electrolyte solution, the energy of interaction due to electrostatic double layer forces is given by [52] $W_{EDL} = \\frac{64 \\pi k T R \\rho _{\\infty }}{\\kappa } \\tanh ^2 (\\frac{z e \\psi _0 }{4kT}) \\exp {(-\\kappa D^{\\prime })}$ in which the eletrochemical parameters are the same as in Chapter 3, i.e.","$\\rho _{\\infty }$ is the electrolyte concentration in bulk solution, $\\psi _0$ is the surface potential, $z$ is the valence of the ions, $k$ is the Boltzmann constant, $T$ is the absolute temperature, $\\kappa $ is a parameter related to the thickness of the double layer, and $e$ is the electronic charge, and $R$ is the spheres radius.","While the attractive vdW force is given for two spheres of the same radius as [52] $W_{vdW} = \\frac{A_HR}{12 D^{\\prime }}$ in which $D^{\\prime }$ is the separation distance usually in $nm$ or angstrons (Å) and $A_H$ is the Hamaker constant $A_H = \\pi ^2 C \\rho _1 \\rho _2$ In which $\\rho _1$ and $\\rho _1$ are the number densities of the interacting spheres and $C$ is the London constantFor the time being we will refrain from exploring the derivations of these relationships as they might require exploring quantum mechanical perturbation theory.","In the case of equal particles $\\rho _1 = \\rho _2$ and $\\rho _1\\rho _2 = \\rho ^2$ .","The Hamaker constant has an important role in the adsorption of water films in surfaces and is fundamental for modeling dry water adsorption in soils.","The pair interaction energy considering the two forces is $W(D^{\\prime }) = W_{EDL} - W_{vdW}$ Negative values of the pair interaction energy (usually given in J) indicate potential for van der Waals attraction and coagulation.","However the relationship is complex and depending on several of the factors seen in the preceding equations including temperature, which affects the kinetic energy of the colloidal particles, surface potential, which might change with factors such as pH as we discussed in Chapter REF and others.","The proper analysis must be done in terms of the energy barrier that needs to be overcome for coagulation to occur.","The electrolyte concentration at which coagulation occurs is the critical coagulation concentration (ccc) which plays a critical role in grain size analysis of soils, especially in the grain size analysis of oxidic soils which are variable charged and in which the analysis is almost always performed under dispersion using sodium hydroxide or other sodium salts [60].","If the constraints of settling velocity, lower limit of settling and dispersing agent concentration are observed, then grain size analysis by settling methods can be performed normally.","For the determination of particle size distribution and the proportion of the particle fractions we need some more or less arbitrary scale for classification of grain sizes (Table REF ).","For soils, the soil must be first dispersed chemically and or physically using salt solutions such as sodium hydroxide and/or sodium hexametaphoshate at concentrations high enough to create repulsive EDL forces but not above the ccc.","Mechanical energy is applied to the samples by stirring, ultrasonic waves, gas bubbles or shaking.","Several combinations of chemical and physical methods are possible, some more efficient than others [40].","Table: Classification of soil particles based on diameter .After dispersion, the sample is again vigorously stirred and the amount of particles in suspension is measured at prescribed times directly, by collecting a volumetric sample of the dispersion, or indirectly, by using a hydrometer or by transmission/diffraction to electromagnetic radiation methods as in laser and x-ray methods.","The most common direct method is the pipette method in which a sample is taken and oven dried and the mass in suspension measured using a precision scale.","The mass in suspension is then extrapolated for the entire volume.","As for the indirect methods the most commonly used is the hydrometer method.","Hydrometers measure the amount or particles in suspension by buoyancy, the greater the mass in suspension the greater the buoyancy force, as particles settle, the hydrometer will tend gradually sink in response to settling of particles until settling stops, when the hydrometer will be reading the amount of particles in suspension under equilibrium, or the density of the dispersing solution if the material is composed only of coarser particles.","Most hydrometers are nothing more than a weighed glass tube with a bulbous shape and a scale.","The scale is calibrated at the solution level and will read zero for pure water at a given temperature and as the mass of particles in suspension increases, the reading will increase in the downward direction indicating the mass of particles in suspension, usually in grams, per unit volume, usually liters.","For any settling based method, the settling velocity is given by Stokes law and the mass of particles greater than a given radius (or diameter) in suspension will depend on the sampling depth and on the calculated settling time.","From basic physics the velocity of a particle is given by $v_s = \\frac{\\Delta S}{\\Delta t}$ Because we are concerned with a spherical particle settling under Stokes regime, and therefore at a constant terminal velocity, $v_s$ is the settling velocity, $S$ is the settling depth and $t$ is the time for the particle to reach $S$ .","Replacing the velocity in Stokes law $\\frac{\\Delta S}{\\Delta t} = \\frac{2}{9} \\frac{ (\\rho _{p} - \\rho _{f}) g}{\\mu } R^2$ $\\Delta t = \\frac{9 \\mu \\Delta S}{2 (\\rho _{p} - \\rho _{f}) g R^2}$ and because it is convenient to set $t_0$ and $S_0$ to zero we have $\\boxed{t = \\frac{9 \\mu S}{2 (\\rho _{p} - \\rho _{f}) g R^2}}$ This equation implies that at time $t$ , all particles with radius greater than $R$ will have settled to a depth greater than $S$ .","In addition to the settling depth and the particle radius, the time for a reading for determining particle size distribution using a given method is dependent on the fluid viscosity, the settling mineral particle density, the density of the fluid and gravitation acceleration.","Particle density, discussed in Chapter REF , can be measured with relatively simple methods, while fluid viscosity and density will require more specialized high precision techniques.","It is much more convenient to perform the measurements at a standard temperature to which water viscosity and density have tabulated values or to use published empirical relationships for water viscosity and density, for example $\\mu = 1 \\times 10^{-6} [280.68 (\\frac{T}{300})^{-1.9} + 511.45 (\\frac{T}{300})^{-7.7} + 61.131 (\\frac{T}{300})^{-19.6} + 0.45903 (\\frac{T}{300})^{-40.0} ]$ in which $\\mu $ is water dynamic viscosity in Pa s and T is the temperature in Kelvin in the range of 253.15 K to 383.15 K [51], and $\\rho _f = 999.974950 [1 - \\frac{(Tc - 3.983035)^2(Tc + 301.797)}{522528.9(Tc + 69.34881)}]$ in which $\\rho _f $ is the density of water in kg m$^{-3}$ and Tc is the temperature in $^{o}$ C in the range of 0 $^{o}$ C to 40 $^{o}$ C [94].","For example, using the equations above at 25°C = 298.15 K, $\\mu = 0.000889997~ Pa~ s$ , $\\rho _f = 997.047~ kg~ m^{-3}$ , and using $\\rho _s $ and $g$ as in previous calculations, for the silt $D = 0.05~mm$ and clay $D = 0.002~mm$ limits, the time for all the particles with diameter greater than $D$ to settle 0.1 m (10 cm) will be $t_{clay} = 6.86~h$ and $ t_{silt} = 39.53~s $ .","Thus, a reading at $t_{silt}$ will indicate the amount of particles finer than 0.05 mm in suspension and the reading at $t_{clay}$ will indicate the amount of particles finer than 0.002 mm in diameter in suspension.","The particle size distribution curve is the measured cumulative mass fraction of particles in suspension (usually in %) at a given time plotted as a function of the logarithm of particle diameter corresponding to each time, given by $D = \\sqrt{\\frac{18 \\mu S}{ (\\rho _{p} - \\rho _{f}) g t}}$ Examples of hypothetical granulometric curves for clay and sandy soils are illustrated in Figure REF Figure: Grain size distribution curves for hypothetical clay and sandy soils.In soil science, and soil physics to some degree, once the mass based fraction of particles in the general classes clay ($D< 0.002~mm$ ), silt ($0.002~mm \\le D < 0.05~mm$ ) and sand ($0.05~ mm \\le D < 2.00~ mm$ ) is determined, the data can be entered into a ternary plot generally referred to as textural triangle and the nominal soil texture class determined (Figure REF ).","The nominal texture classes are a relic from soil pedology and are an attempt to classify soils using the “feeling” method in the field.","The feeling method can be useful to discern the general texture classes after extensive training of the field technician.","In broader terms, the soil sample is disagregatted, moistened to provide some degree of dispersion by manipulating it using the fingers and rubbed between the index and thumb so that the soil texture is literally felt by the person.","It can be useful as an expedite field method for estimating soil texture and for soil classification where laboratory methods might not be available.","Figure: Textural triangle for determination of soil texture class.","The direction of data entry for each class is indicated by the arrows.Grain size distribution and soil texture are important properties of soils because they will directly or indirectly affect almost all soil physical and chemical phenomena.","Aggregation and pore size distribution will affect water transport and storage as well as aeration and susceptibility to compaction.","The amount of clay will affect surface area the reactivity of soils as we discussed previously.","For a given mineralogy, increasing clay content, will tend to increase plasticity and susceptibility to compaction and compression under certain conditions.","ll $ p $ Pressure $ \\mu $ Viscosity $ \\mathbf {u} $ Vector velocity function $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k}$ Unit vectors in Cartesian coordinates $ \\mathbf {r}, \\theta , \\phi $ Unit vectors in spherical coordinates $ u_r, u_{\\theta }, u_{\\phi } $ Components of the velocity vector in spherical coordinates $ r, \\theta , phi $ Length and angle components in spherical coordinates $ \\Psi $ Stokes stream function $ E^2 $ Differential operator $ a $ Radius $ A, B, C, D $ Constants $ \\tau $ Stress components $ F_D $ Drag force $ U $ Velocity $ F_b $ Buoyancy force $ F_g $ Gravitational force $ m $ Mass $ \\rho $ Density $ R $ Radius $ v_s $ Settling velocity $ Re $ Reynolds number $ Pe $ Péclet number $ T $ Temperature $ k $ Boltzmann constant $ W_{EDL} $ Energy of interaction due to double layer forces $ \\rho _{\\infty } $ Electrolyte concentration in bulk solution $ \\kappa $ Parameter related to the thickness of the double layer $ z $ Valence of the ion $ e $ Electronic charge $ W_{vdW} $ Attractive van der Waals force $ A_H $ Hamaker constant $ D^{\\prime } $ Separation distance $ \\rho _1, \\rho _2 $ Number densities of interacting spheres $ C $ London constant $ W(D^{\\prime }) $ Pair interaction energy $ S $ Settling distance $ T_c $ Temperature in Celsius $ D $ Particle diameter Because soils and sedimentary deposits are composed of solid, unconsolidated particles, mechanical stresses caused by foundations, plant growth, traffic of machines, animals and people, overburden of confining layers, water in pores, clay expansion and contraction and many other natural and man-made processes can cause reorganization of particles and depending on the circumstances, compaction, consolidation or compression or increase in pore volume by processes such as loosening.","Before we consider aggregation of soil primary particles and organic matter, it is convenient to treat porous media as an idealized stacking of spheres.","This is directly applied to round, homogeneous, coarse grained particles in a few soils and sedimentary deposits and can be an important tool in modeling water retention and transport phenomena, but it is less representative to heterogeneous particle size distributions and as the particle shapes deviate from an ideal sphere.","Treating porous media as a set of equally sized spherical particles, the porosity or pore fraction of the material will depend on how the spheres are organized in space.","There are more and less efficient ways of packing spheres, a pile of oranges in a supermarket tray is an example of a packing system.","The packing models are organized in lattices which can be defined as a repetitive arrangement in space.","The lattices seem in this chapter are directly applied to the organization of atoms in space seen in Chapter REF and the structure of minerals is organized in lattices, often following the systems below, but usually with more than one type of chemical element, each element represented by different spheres [44].","The systems described below can be applied to soil aggregates to some extent.","Soil aggregates are usually less resistant to soil stresses than most primary particles and will be discussed in this chapter.","The most common sphere packing arrangements are described below.","The packing density is derived from geometric arguments.","I. Cubic Lattice In the case of cubic lattice, or simple cubic lattice, the spheres touch and the distance between the spheres centers is two times the radius ($R$ ).","The volume of the unit cell considered is the cube that unites the points at the centers of the eight spheres (Figure REF ).","Considering the side of the cube as $A = 2R$ , the volume of the unit cell, the volume occupied by spheres and the packing density, given by the volume occupied by the spheres over the unit cell volume are given by Figure: Cubic lattice.$V_{cell} = A^3 = (2R)^3 = 8R^3 \\\\V_{spheres} = 8 \\cdot \\frac{1}{8} \\cdot \\frac{4}{3} \\pi R^3 = \\frac{4}{3} \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{\\frac{4}{3} \\pi R^3}{8R^3}$ $\\eta = \\frac{\\pi }{6} = 0.52359 ...$ In which $V_{cell}$ is the volume of the unit cell, $V_{sphere}$ is the corresponding volume of spheres located within the unit cell and $\\eta $ is dimensionless packing density.","Notice that within the unit cell there are eight $1/8$ of spheres such that the total volume of spheres within the unit cell is $8 \\cdot 1/8 = 1$ sphere.","Similar logic will be used in the following packing schemes.","II.","Face-Centered Cubic In this arrangement, an additional sphere is located on the center of each face (Figure REF ).","Now the side $A$ is no longer $2R$ as additional space is needed between the spheres in the corners to accommodate the spheres on the faces.","Notice that the simplified ball-and-stick representation is used, but each corner and face is occupied by a large sphere of radius $R$ .","The color red in indicates the corner spheres and the light green indicates the face spheres (Figure REF ).","Figure: Face-centered cubic packing.Because of the additional space, the side of the unit cell is given by $A = 2 \\sqrt{2} R$ and the volumes and packing density can be calculated as $V_{cell} = A \\cdot A \\cdot A = (2 \\sqrt{2} R)^3 = 16 \\sqrt{2} R^3\\\\V_{spheres} = (8 \\cdot \\frac{1}{8} + 6 \\cdot \\frac{1}{2}) \\cdot \\frac{4}{3} \\pi R^3 = \\frac{16}{3} \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{\\frac{16}{3} \\pi R^3}{16 \\sqrt{2} R^3}$ $\\eta = \\frac{\\pi }{3 \\sqrt{2}} = 0.74048 ...$ Notice that the packing density is considerably higher and the porosity, given by $ 1 - \\eta $ is considerably lower, around 25%, when compared to the simple cubic lattice.","III.","Body-Centered Cubic In this case there is one sphere in each corner and one sphere (represented in blue) in the center of the structure (Figure REF ).","Notice that the sides $A$ are still equal on the cube, but the diagonals are formed by the radius of the spheres in the transverse corners plus the diameter of the sphere in the center ($2R$ ) totaling $4R$ .","From the triangle formed by the height $A$ , the diagonal $4R$ , the diagonals in each face can be calculated from simple geometric arguments resulting in $d = \\sqrt{16 R^2 - A^2}$ .","Again the relationship between $A$ and $R$ can be calculated from the diagonal of each face and the sides $A$ using simple geometric arguments resulting in $ A = 4R/\\sqrt{3}$ .","The volumes and packing density can then be calculated as before Figure: Body-centered cubic packing.$V_{cell} = (\\frac{4R}{\\sqrt{3}})^3 = \\frac{64}{3\\sqrt{3}} R^3 \\\\V_{spheres} = (8 \\cdot \\frac{1}{8} + 1) \\cdot \\frac{4}{3} \\pi R^3 = \\frac{8}{3} \\pi R^3\\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{\\frac{8}{3} \\pi R^3}{\\frac{64}{3\\sqrt{3}} R^3}$ $\\eta = \\frac{\\pi \\sqrt{3}}{8} = 0.68017 ...$ IV.","Hexagonal Lattice Now on the two hexagonal packing schemes presented here, the geometric polyhedron considered is no longer a cube but an hexagonal prism.","Looking at one layer of packed spheres from above it is clear that the center of the spheres can be linked to form an imaginary hexagon (Figure REF ).","The height of the hexagonal prism will depend of how the additional layers of spheres are stacked.","If the additional layer of spheres is staked directly on top of the first layer, an hexagonal packing lattice is formed (Figure REF ).","Notice that because the spheres on the base touch, the base edge, $s$ , is $2R$ and because the next layer is stacked directly above the first layer, the height of the hexagonal prism linking the center of the group of spheres is also $2R$ .","Recalling that the volume of an hexagonal prism is given by $ (3 \\sqrt{3}/2) s^2 h $ , the volume of the unit cell is the volume of the hexagonal prism and the packing density can be calculated as $V_{cell} = \\frac{3}{2} \\cdot \\sqrt{3} \\cdot s^2 \\cdot h = \\frac{3}{2} \\cdot \\sqrt{3} \\cdot (2R)^2 \\cdot (2R) = 12 \\sqrt{3} R^3\\\\V_{spheres} = 3 \\cdot \\frac{4}{3} \\pi R^3 = 4 \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{4 \\pi R^3}{12 \\sqrt{3} R^3}$ $\\eta = \\frac{\\pi }{3 \\sqrt{3}} = 0.64599 ...$ Notice that there are 12 spheres in the corners of the hexagonal prism and each one contributes to 1/6th of the volume within the unit cell, plus two half spheres in the upper and lower faces, thus we have $ 12 \\cdot 1/6 + 2 \\cdot 1/2 = 3$ spheres.","Figure: Spheres plane in an hexagonal arrangement.Figure: Hexagonal lattice.V.","Hexagonal Close Packing The hexagonal close packing is similar to hexagonal packing but there are three spheres located in between the two staked planes (Figure REF a).","It is important to note that these three spheres will also form a plane of spheres when multiple layers are considered in three dimensional space, in addition to the two planes of spheres represented in red (Figure REF a).","The calculation of the volume of the unit cell is more complicated than in the hexagonal packing lattice because the distance between the centers of the spheres in the two planes is no longer $2R$ .","Consider a tetrahedron formed by the central sphere in the base plus two spheres on the edges forming the base of the tetrahedron and one of the spheres in the middle plane as the upper vertex (Figure REF b).","Considering the sides of the base of the tetrahedron as $a$ , the distance to the point of projection of the vertex sphere $b$ can be calculated from the internal angle and the position at the middle point of the side of the base $a/2$ (Figure REF c) Figure: Hexagonal close packing.$\\cos {30^{o}} &= \\frac{a/2}{b}\\\\b &= \\frac{a}{\\sqrt{3}}$ The height $d$ to the plane of spheres in the middle of the unit cell is then given by (Figure REF d) $a^2 &= d^2 + b^2 = d^2 + (\\frac{a}{\\sqrt{3}})^2 = d^2 + (\\frac{a^2}{3})\\\\d &= a \\frac{\\sqrt{2}}{\\sqrt{3}}$ Because the sides of edge of the base of the hexagonal prism is $a = 2R$ and the total distance between the two planes in red in Figure REF a is $2d = 2 \\cdot 2R \\frac{\\sqrt{2}}{\\sqrt{3}}$ , the total height of the hexagonal prism are $h^{\\prime }_T = 4R \\frac{\\sqrt{2}}{\\sqrt{3}}$ and the volumes and packing density of the hexagonal close packing lattice are given by $V_{cell} = \\frac{3}{2} \\cdot \\sqrt{3} \\cdot (2R)^2 \\cdot (4R\\frac{\\sqrt{2}}{\\sqrt{3}}) = 24R^3\\sqrt{2} \\\\V_{spheres} = 6 \\cdot \\frac{4}{3} \\pi R^3 = 8 \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{8 \\pi R^3}{24R^3\\sqrt{2}}$ $\\eta = \\frac{\\pi }{3 \\sqrt{2}} = 0.74048 ...$ Notice that the packing density is the same as the Face Centered Cubic.","In essence the difference between these two systems is geometrical, as in the way the layers of spheres are conceptualized and organized.","One system can be transformed into the other by rotation of coordinates.","In soil science, aggregation refers to the union of primaryThe meaning here is not as in primary minerals but in the basic building blocks of soil aggregates mineral particles clay, silt and sand with organic compounds and various amorphous materials forming clusters with shapes and sizes that are more or less constant in certain naturally occurring soil types and depths, in response to soils forming processes.","Soil structure generally refers to the shape, size and structural stability of these aggregates and larger domains formed by the set of aggregates in a given soil.","In natural soils, structure is a direct product of soil forming processes, namely climate, organisms, relief, parent material and time.","In fine textured soils, close to surface, the structure tends to be composed of larger aggregates and with higher porosity, especially in humid tropical climates where there is higher biological activity of plants, animals and other organisms.","Aggregates in natural soils also tend to be more stable than in areas disturbed by human activity.","Human activity, especially agriculture, tends to disturb soil structure, destroying large size aggregates due to tillage and chemical modification on soil upper layers.","The destruction of aggregates and soil structure is usually a physical and a chemical process.","Traffic of animals, humans and machinery tends to compress and break soil aggregates, causing increases in bulk density and decrease in porosity, creating an environment that is less adequate to biological activity which further causes reduction in creation of new aggregates and pores, and loss of organic matter.","Agricultural practices also employ a variety of tillage implements which in some cases are directly designed to destroy soil structure for sowing operations, while others tend to destroy soil structure indirectly.","Chemically, the input of ionic chemicals such as fertilizers and other contaminants can cause dispersion by the surface chemistry mechanisms discussed in Chapter REF which decreases soil structural stability on one end while on the other some compounds can also kill soil organisms, contributing to loss of organic matter and biological activity, further increasing soil structural degradation.","We will not discuss tillage systems and soil quality in this book.","There are a lot of sociological, ecological, philosophical and political issues related to soil quality, tillage and agriculture that rely on personal opinion of different groups none of which we have any interest in supporting.","Water retention due to capillary phenomena tends to be controlled by the pore size distribution and pore shape in soils and these are to a larger extent controlled by the size and shape of soil aggregates.","In most soils aggregation creates at least two pore domains, the intra-aggregate pores and inter-aggregate pores, intra-aggregate pores tends to be larger and usually hold water by capillary phenomena at lower energies of retention when compared to inter-aggregates smaller pores.","In soils with high amounts of clay, these inter-aggregate pores tend to be very small and as such, water retention in them can have contribution of both capillary and adsorption phenomena.","Water retention can be modeled in terms of aggregation and pore size distribution in self-similar and fractal models for example [77].","Formation of aggregates is a slow and complex process dependent on several factors.","There are several theories of soil aggregate formation (e.g.","[95]), but overall it is thought that aggregation in most soils is dependent on biological activity and organic matter which creates physical and chemical cementing agents contributing to uniting primary particles in larger structural domains.","Biological activity can directly contribute to formation of aggregates as some organisms such as termites and earthworms produce aggregate like structures by ingesting or manipulating soil organic matter and mineral particles.","From the chemical point of view, charged surfaces in soil particles, colloidal organic matter, and ions can create aggregating and dispersing forces during aggregate formation.","Higher valence cations tend to favor aggregation while monovalent high hydrated radius such as sodium tend to favor dispersionAs we seen, this depends also on the concentration in solution.","Physically, stresses caused by roots and organisms and mechanical stresses caused by natural processes such as soil drying, expansion and shrinking of soil minerals due to loss of interlayer water as well as capillary forces related to drying can also contribute to aggregation of primary particles.","Soil structure, shapes, sizes and stability of aggregates, total porosity and pore size distribution are all affected by stresses applied to soils by human activities.","The modifications caused by these activities will in turn affect soil functioning.","In many engineering applications such as in the construction of earth dams and foundations, the increase in bulk density and reduction of porosity to a minimum are usually desired while in agriculture, where the soil is viewed as a medium to plant growth, porosity, pore size and distribution must be at an ideal state for root growth, water storage, water and air circulation and biological activity.","Much of the theory behind soil porosity reduction and increase in bulk density in response to applied stresses comes form geotechnical engineering, specific from the discipline of Soil Mechanics.","In soil mechanics, compaction is the increase in density of a soil by reduction of the pore volume occupied by air, consolidation is a reduction in volume of a fully saturated, low permeability soil due to drainage of soil water on the pores, and compression is a reduction of volume of a soil under compressive stress [29], [33].","Although we have not discussed soil permeability to water, it can be understood as a coefficient which measured how easy it is for water to flow out of the soil under a pressure gradient.","On the geotechnical engineering literature, consolidation applies to low permeability soils, such as soils with high phylossilicate clays content, such that there is a time lag between the application of the load and the extrusion of water from pores and soil settlement [33].","We have studied the concept of stress tensor applied to fluids in porous media in Chapter REF and now we need to apply the same concept to stresses in a solid (Figure REF ).","Imagining a volume of soil shaped like a cube, forces can be applied to each surface can be parallel or perpendicular to each face.","The force component divided by the area over which it is acting acting defines the stress $\\sigma $ (Figure REF ).","As with the stress tensor in a fluid, the stress components in each direction, $x$ , $y$ and $z$ can be further decomposed into three components, one for each original direction.","Therefore, the stress components acting on the face perpendicular to the direction $x$ are $\\sigma _{xx}$ , $\\sigma _{xy}$ and $\\sigma _{xz}$ and so forth for the other three directions.","The full stress tensor is, as before $\\sigma _{ij} =\\begin{bmatrix}\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\\end{bmatrix}$ The perpendicular (or normal) stresses $ \\sigma _{ii} = \\sigma _{xx} = \\sigma _{yy} = \\sigma _{zz} $ are equivalent to pressures, or force over area ($F/A$ ) acting over each surfaces while the stresses parallel to each face $ \\sigma _{ij}, ~ i \\ne j$ are called the shearing stresses, also $F/A$ , in units analogous to pressure units.","Stresses applied by foundations, traffic of animals and machinery, tillage and other agricultural operations are usually multi-directional and have normal and shearing components, especially in non-rigid, granular porous materials such as soils and sedimentary deposits.","In some cases the stresses can be modeled as unidimensional pressures as in modeling compaction by agricultural machinery and animal traffic.","Figure: Components of stress in a tridimensional solid.If we imagine a foundation, a tire or an animal foot applying a vertical force to a soil, the response of the soil will depend on the magnitude of the applied force and on the mechanical strength of the soil.","The mechanical strength of the soil will depend directly and indirectly on a series of factors such as soil structure, grain and aggregate strength, grain size distribution, porosity, water content, mineralogy, story of previous applied loads and many other internal mechanisms.","If the applied stress is greater than the mechanical strength of the soil, deformation will occur.","Considering only unidimensional compressive stresses on a soil, deformation will correspond to a decrease in porosity.","There are different mechanisms and regimes of deformation in soils, a soil can suffer elastic deformation when the soil returns to its original state after the force is removed or plastic deformation when the soils does not returns to its original state after the force is removed.","On a macro scale, in soils subjected to unequal stresses in multiple directions, rupture or compressive failure can also occur, although these concepts are more applied to rigid solids.","However, internally, rupture of aggregates due to compressive stresses is an important mechanism of soil compaction, even under uniaxial loads.","As larger aggregates break the overall strength of the soil decreases and the volume of pores also decreases, as the structural units are destroyed and the pores filled with smaller aggregates or loose particles.","Dry soils are more subjected to elastic deformation and rupture while wet soils tend to become plastic above a given water content depending on specific characteristics of the soil, like grain size distribution and mineralogy.","Compaction in dry soils can be caused by reorganization of aggregates and individual particles, as seen in the packing models at the beginning of this chapter, rupture of aggregates and in some cases of individual mineral particles under load, and elastic deformation of aggregates and individual particles (usually sand and coarser fractions) under elevated loads.","Usually aggregates in dry soil tend to be more brittle, i.e.","they can fracture with little deformation, while as the water content increases, water molecules act as lubricants within particles and aggregates, contributing to plastic deformation.","The set of critical water contents at which a fine soil transition to different consistency states is called Atterberg limits.","The Atterberg limits are used in engineering and in agricultural sciences to predict soil behavior in the field for construction, traffic and tillage.","In simple terms the principal soil conceptual consistency limits considered in soil physics and soil mechanics are [33] Shrinkage limit: Most fine texture soils, especially soils rich in smectites, shrink as the soil dries.","This shrinking can be thought to be related to two main mechanisms, first in phylossilicate clays, water is lost from the interlayer planes causing shrinkage at the mineral structure level which reflects on the macrostructure, second, drying tends to cause internal stresses between particles and aggregates due to forces related to meniscus in pore water water.","These forces can pull the particles together as the soil dries and the radius of the meniscus becomes smaller increasing the capillary forces.","The shrinkage limit is the water content below which decrease in water content will not cause further volume reduction of the soil (i.e.","shrinkage).","A soil in a water content below the shrinkage limit is said to be in a solid state and it is said to be in a semisolid state at water contents above the shrinkage limit and below the plastic limit.","Plastic limit: The water content below which the soil ceases to display plastic behavior.","In practical terms the soil will start to crumble or fracture if manipulated when below the plastic limit.","Above the plastic limit the soil is subjected to plastic deformation and is in a plastic state.","Liquid limit: It is the water content above which the soil changes from a plastic state to a liquid sate.","In practical terms, above the liquid limit the water content is such that the soil will begin to flow similar to a liquid under stress.","Shrinkage limit: Most fine texture soils, especially soils rich in smectites, shrink as the soil dries.","This shrinking can be thought to be related to two main mechanisms, first in phylossilicate clays, water is lost from the interlayer planes causing shrinkage at the mineral structure level which reflects on the macrostructure, second, drying tends to cause internal stresses between particles and aggregates due to forces related to meniscus in pore water water.","These forces can pull the particles together as the soil dries and the radius of the meniscus becomes smaller increasing the capillary forces.","The shrinkage limit is the water content below which decrease in water content will not cause further volume reduction of the soil (i.e.","shrinkage).","A soil in a water content below the shrinkage limit is said to be in a solid state and it is said to be in a semisolid state at water contents above the shrinkage limit and below the plastic limit.","Plastic limit: The water content below which the soil ceases to display plastic behavior.","In practical terms the soil will start to crumble or fracture if manipulated when below the plastic limit.","Above the plastic limit the soil is subjected to plastic deformation and is in a plastic state.","Liquid limit: It is the water content above which the soil changes from a plastic state to a liquid sate.","In practical terms, above the liquid limit the water content is such that the soil will begin to flow similar to a liquid under stress.","Atterberg limits are strongly dependent on mineralogy and soil texture.","The main goal in agricultural soils is that traffic and other agricultural operations are to be performed below the plastic limit.","Plastic deformations are related to structural degradation and compaction that is usually expensive and labor intensive to recover.","Above the liquid limit machinery can sink and become stuck in some conditions.","In practice however, as the soil dries the mechanical resistance can become such that it is impossible or not economically viable to perform tillage or sowing operations.","Thus, the limits for trafficability and workability in soils can be narrow, either due to excess water content, turning the soil plastic and subjected to compaction or consolidation, or due to water content that is too low for efficient tillage operations.","Soils that are too plastic, such as those rich in smectites, are often too sticky, requiring high amounts of energy for operations or even being impossible to till under certain conditions In many practical applications, either agricultural, military or geotechnical engineering, a simple and easy approach to estimate soil mechanical resistance is the use of cone penetrometers.","The principle is very simple, a rod with a cone at its end is inserted into the soil and the mechanical resistance in terms of stress or force required is measured.","Devices based on this principle are called penetrometers.","There are many different designs of penetrometers, either analogical or with digital sensors, for measuring force and data collecting.","Soil mechanical resistance measured by a penetrometer can give an idea of the mechanical resistance of the soil to withstand loads and can be used to estimate trafficability.","In agricultural sciences, the resistance to penetration or cone index is used as a proxy to the mechanical resistance encountered by plant roots.","The mechanisms of root growth are complex and multiple and it is wrong to attribute critical limits of soil resistance to root growth based on penetrometer readings.","In practical terms, a penetrometer can be used as one parameters, among many, for assessing soil compaction and decision making in terms of compaction remediation measures.","Penetrometer resistance measurements should never be used without checking soil water content and bulk density as it is an exponential function of these two soil variables (Figure REF ).","An empirical relationship between soil penetration resistance, bulk density and water content usually encountered on the literature is [62] $SR = a_0 \\rho ^{a_1} \\theta ^{a_2}$ In which $SR$ is the soil mechanical resistance, usually in stress (analogous to pressure) units, $\\rho $ is soil bulk density, $\\theta $ is the volumetric water content, and $a0$ , $a1$ and $a2$ are empirical parameters commonly fitted using regression techniques such as least squares regression.","This equation has useful practical applications for adjusting $SR$ to a given water content and bulk density so that comparisons between different areas and conditions can be made, and also has pedagogical applications, illustrating the exponential relationship between $SR$ and $\\theta $ and $\\rho $ .","However, you must be aware that it is an empirical relationship and has no physical basis and was not derived based on first principles and thus does not explain the physical basis of the relationship between resistance, density and water content.","As such, it cannot be extrapolated to different soils, or to conditions not observed on the fitting data.","First principles derivations of the $SR = f(\\rho , \\theta )$ relationship should be possible accounting for internal stresses of the soil provided by cohesive forces between particles and within aggregates, and adhesive forces generated by water in contact with solid surfaces.","These forces will also depend on soil texture, mineralogy, organic matter content among other soil properties.","Figure: Soil penetration resistance as a function of bulk density and volumetric water content based on empirical data.","The parameters of Equation 6.8 used are within the ranges for real soils, a 0 =0.01a_{0} = 0.01, a 1 =5.0a_{1} = 5.0 and a 2 =-1.0a_{2} = -1.0.ll $ V_{cell} $ Volume of the unit cell $ V_{spheres} $ Volume of spheres inside the unit cell $ A $ Side length in cubic unit cells $ R $ Radius of spheres $ \\eta $ Packing density $ s $ Side length in an hexagonal prism $ h $ Height of the hexagonal prism $ a, b, c $ Parameters for calculation in an hexagonal close packing system $ h^{\\prime }_T $ Height of the unit cell in an hexagonal close packing system $ \\sigma _{ij} $ Stress tensor $ SR $ Soil penetration resistance $ \\rho $ Soil bulk density $ \\theta $ Volumetric water content $ a_0, a_1, a_2 $ Empirical parameters on the soil penetration resistance function On the bulk of a pure, static liquid in thermal equilibrium, all molecules are subjected to the same forces, they interact with their neighbors through intermolecular surface forces which, when averaged over time, are constant in all directions and thus, the resultant of these forces is zero.","This is not true when there is a discontinuity between the fluid and a different fluid, liquid, solid or gas.","On an interface with a gas, the balance of forces is altered such that a resulting force on the direction of the interior of the fluid appears.","Because the molecules at the interface are surrounded by other liquid molecules at the liquid end, but not at the gas end, resulting cohesion forces pulling the molecules in direction of the liquid appear.","Intermolecular forces also generate the adhesion and adsorption forces between the liquid and solid surfaces.","The surface tension at the interface can be conceptualized using the thermodynamics or the mechanics frameworks.","The mechanics framework states that, because of these interfacial forces, the external work at constant volume and temperature, $\\delta W$ , required to increase the surface area by an infinitesimal amount $dA$ is directly proportional to the increase in area with the proportionality coefficient being called surface tensionThe symbol $\\delta $ with respect to W instead of d is because of thermodynamic considerations, $d\\hspace*{-0.80002pt}\\bar{}\\hspace*{1.00006pt}$ is also often used.","Heat and work in thermodynamics are imperfect differentials.","See Callen (2005).","$\\delta W = \\gamma dA$ In which if the energy $W$ is in $J$ and the area in $m^2$ , and the surface tension coefficient needs to have units of $J~m^{-2}$ , which is consistent with the definition of energy per unit surface area.","The surface tension is very often conceptualized using the wire and soap bar analogy and experiment.","Imagine that you have a square wire with one of the sides movable, now imagine a soap bubble is created within the square, if the square has a fixed dimension $l$ and a movable dimension $x$ , the area can be increased by increasing or decreasing the length of $x$ , such that $dA = ldx$ this implies that $\\delta W = \\gamma l dx$ In this case, the surface tension coefficient is usually conceptualized as a force per unit length in $N~m^{-1}$ .","Note that $1~J = 1~N~m$ such that $1~N~m^{-1}$ = $1~J~m^{-2}$ Now lets consider the liquid, the gas and the interface as a thermodynamical system.","Because the variables of interest in this system are temperature and volume, the thermodynamic potential of interest here is the Helmholtz free energy, $\\mathcal {F}$ [24], [75] $d\\mathcal {F} = -SdT - pdV + \\mu _i dN_i$ Since the temperature and number of moles are kept constant, $dT$ and $dN_i$ are zero, and the equation reduces to $d\\mathcal {F} = - pdV$ Since we know from thermodynamics that $\\delta W = - pdV$ we have, for an interface $d\\mathcal {F} = d\\mathcal {F}_1 + d\\mathcal {F}_2 + d\\mathcal {F}_3 = - \\delta W_1 - \\delta W_2 - \\delta W_3$ in which 1 and 2 refer to the solid and liquid phases, respectively, and 3 to the interface phase.","Considering that Equation 7.2 is known experimentally, and that the work in that case is negative because it is carried by the interface instead of by an external agent we have, after some reorganizing $- \\delta W_1 - \\delta W_2 - \\delta W_3 = - p_1dV_1 - p_2dV_2 + \\gamma dA \\\\- p_1dV_1 - p_2dV_2 + d F_3 = - p_1dV_1 - p_2dV_2 + \\gamma dA \\\\d \\mathcal {F}_3 = \\gamma dA \\\\\\gamma = \\frac{d \\mathcal {F}_3}{dA}$ Rewriting in a more elegant symbology $\\boxed{\\gamma = \\bigg (\\frac{\\partial \\mathcal {F}_{int}}{\\partial A}\\bigg )_T}$ The surface tension coefficient is the change in Helmholtz free energy of the interface ($\\mathcal {F}_{int}$ ) per unit area at constant temperature [75].","Notice that we have not yet discussed the concept of thermodynamic potential because it would take us to a long detour which is not strictly necessary at this point.","This long detour will be necessary when we discuss the concept of soil water potentials later in this chapter.","Capillary theory is based on developments by Thomas YoungYoung, T. 1805.","An essay on the cohesion of fluids.","Transactions of the Royal Society of London.","95:65-87. who laid the descriptive foundations of the process, Pierre LaplaceLaplace, M. 1805.","Traité de mécanique céleste.","Tome Quatrième.","Courcier, Paris.","who provided a rigorous initial mathematization and Carl Fredrich GaussGavss, C.F.","1830.","Principia generalia theoriae figvrae flvidorvm in statv aeqvilibrii.","Gottingae.","who unified both descriptions in terms of the virtual work principle, based on the principle of least action.","The works are based on a rigorous mathematical description and occupy several pages of works in French for Laplace and Latin for Gauss.","Like most of what is now known as the core of soil physics, simplifications and ad-hoc derivations came later in order to provide more palatable descriptions.","The Young-Laplace equation is a differential equation of the form $\\Delta p = \\gamma \\nabla \\cdot \\mathbf {n}$ in which $\\nabla \\cdot \\mathbf {n}$ represents the mean curvature and can be calculated using differential geometry for different shapes of the interfacial surface.","For reasons of simplicity and elegance we will present here the derivations of Landau and Lifshitz [57] followed by that of Defay and Prigogine [34].","Suppose you have a curved surface with two principal radius of curvature $R_1$ and $R_2$ on an interface between two media.","A limiting case would be a sphere in which $R_1 \\equiv R_2$ .","An element of surface area $dA$ is defined in the interface between the two mediums (Figure REF a).","The area element can be calculated as the product of two elements of circumference of a circle, defined in terms of the two radii $R_1$ and $R_2$ as $dl_1$ and $dl_2$ .","Suppose now that the surface area is increased by displacing the surface by an amount $\\delta \\zeta $ in the direction of the normal of the surface $dA$ (Figure REF ).","The two elements of circumference are now $dl_1^{\\prime }$ and $dl_2^{\\prime }$ and the surface area is increased from $dA = dl_1dl_2$ to $dA^{\\prime } = dl_1^{\\prime }dl_2^{\\prime }$ .","An element of volume between the surfaces is $dV = \\delta \\zeta dA$ .","If the pressures on the two media are $p_1$ and $p_2$ , the work necessary for the volume increase is related to the pressure difference between the two surfaces by $\\delta W_v = \\int p dV = \\int (-p_1 + p_2) dV = \\int (-p_1 + p_2) \\delta \\zeta dA$ and the total work is the work necessary for the volume increase added to the work necessary for surface expansion, as described on the previous section, thus $\\delta W = \\delta W_v + \\delta W_s = \\int p dV + \\gamma \\delta A = -\\int (p_1 - p_2) \\delta \\zeta dA + \\gamma \\delta A$ in which $\\delta A$ is a change in the area of surface.","As we have two areas $dA$ and $dA^{\\prime }$ , the change in surface area after displacement, $\\delta A$ , is needed.","Suppose that the length element $dl_1$ is increased to $dl_1^{\\prime }$ after the $\\delta \\zeta $ displacement in the direction of the normal.","Considering an element of angle $\\theta $ , and using the circumference of a circle formula (Figure REF b) we have $dl_1 = \\theta R_1$ and $dl_1^{\\prime } = \\theta (R_1 + \\delta \\zeta )$ Isolating $\\theta $ in the first equation and replacing into the second we have $dl_1^{\\prime } = \\frac{dl_1}{R_1} (R_1 + \\delta \\zeta ) = (1 + \\frac{\\delta \\zeta }{R_1}) dl_1$ Using the same argument in $dl_2$ $dl_2^{\\prime } = \\frac{dl_2}{R_2} (R_2 + \\delta \\zeta ) = (1 + \\frac{\\delta \\zeta }{R_2}) dl_2$ Now $dA^{\\prime }$ can be calculated in terms of $dl_1$ and $dl_2$ $dA^{\\prime } & = dl_1^{\\prime }dl_2^{\\prime } \\\\& = (1 + \\frac{\\delta \\zeta }{R_1}) dl_1 (1 + \\frac{\\delta \\zeta }{R_2}) dl_2 \\\\& = dl_1dl_2 + dl_1dl_2 \\frac{\\delta \\zeta }{R_2} + dl_1dl_2 \\frac{\\delta \\zeta }{R_1} +dl_1dl_2 \\frac{\\delta \\zeta }{R_1} \\frac{\\delta \\zeta }{R_2}$ Neglecting the much smaller second order term $\\delta \\zeta \\delta \\zeta = \\delta \\zeta ^2$ $dA^{\\prime } = (1 + \\frac{\\delta \\zeta }{R_1} + \\frac{\\delta \\zeta }{R_2}) dl_1dl_2 \\\\$ Now the element change in area as the surface is displaced is $da & = dA^{\\prime } - dA \\\\& = (1 + \\frac{\\delta \\zeta }{R_1} + \\frac{\\delta \\zeta }{R_2}) dl_1dl_2 - dl_1dl_2 \\\\& = (\\frac{\\delta \\zeta }{R_1} + \\frac{\\delta \\zeta }{R_2}) dl_1dl_2$ The total change in area of the surface of separation is $\\delta A = \\int da = \\int (\\frac{1}{R_1} + \\frac{1}{R_2}) \\delta \\zeta dA$ and replacing into Equation 7.10 $\\delta W = -\\int (p_1 - p_2) \\delta \\zeta dA + \\gamma \\int (\\frac{1}{R_1} + \\frac{1}{R_2}) \\delta \\zeta dA$ Because the displacement is required to occur in thermodynamic equilibrium, $\\delta W = 0$ and we can write $\\int [ (p_1 - p_2) - \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2}) ] \\delta \\zeta dA = 0$ In which, as in other cases seen before, the integral vanishes identically, thus $\\boxed{\\Delta p = p_1 - p_2 = \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2})}$ This is the Young-Laplace equation for a curved interface with constant radii $R_1$ and $R_2$ between two media.","For a sphere in which $R \\equiv R_1 \\equiv R_2$ it reduces to $\\boxed{\\Delta p = p_1 - p_2 = \\frac{2 \\gamma }{R}}$ Figure: Schematics for the energy derivation of the Young-Laplace equation.The derivation from Defay and Prigogine [34] is simpler and relies on mechanical and geometrical arguments.","Consider a non-spherical cap with principal radii of curvature $R_1$ and $R_2$ (Figure REF ).","Because of surface tension, a line element $\\delta l$ at A is subjected to a force $F_A = \\gamma \\delta l$ .","The projection of $F_A$ on the normal P-N is then $\\sin {\\phi } = \\frac{proj_{F_A}}{\\gamma \\delta l} \\\\proj_{F_A} = \\gamma \\delta l \\sin {\\phi }$ Using the small angle approximation on the triangle A-P-N it is possible to define $\\sin {\\phi } \\approx \\tan {\\phi } \\approx \\phi \\approx \\frac{r}{R_1}$ and $proj_{F_A} = \\gamma \\frac{r}{R_1} \\delta l$ The same argument for points B, C and D results in $proj_{F_B} = \\gamma \\frac{r}{R_1} \\delta l$ $proj_{F_C} = \\gamma \\frac{r}{R_2} \\delta l$ $proj_{F_D} = \\gamma \\frac{r}{R_2} \\delta l$ The resulting force on the N direction on the cap is the sum of the force components of each point in the N direction $F = proj_{F_A} + proj_{F_B} + proj_{F_C} + proj_{F_D} = \\gamma \\frac{r}{R_1} \\delta l + \\gamma \\frac{r}{R_1} \\delta l + \\gamma \\frac{r}{R_2} \\delta l + \\gamma \\frac{r}{R_2} \\delta l$ $F = 2 r \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2}) \\delta l$ Figure: Schematics for the mechanics derivation of the Young-Laplace equation (Adapted from ).The total force around the border of the cap can be calculated by integrating in $\\delta l$ .","Since there are four quadrants and the perimeter is treated as that of a circle, the integration is made in the 0 to $\\pi /2$ interval.","The element of length needs to be transformed into an element of angle using the arc length formula $\\delta l = r \\delta \\theta $ .","$F_{Do} = \\int _0^{\\frac{\\pi }{2}} 2r\\gamma (\\frac{1}{R_1} + \\frac{1}{R_2}) r \\delta \\theta = \\gamma \\pi r^2 (\\frac{1}{R_1} + \\frac{1}{R_2})$ For the cap to be in equilibrium, the force downward caused by surface tension has to be in equilibrium with the upward force which in this case is caused by the pressure gradient between the two sides of the interface $F_U = (p_1 - p_2) \\pi r^2$ Note that the pressure difference is force over area and $\\pi r^2$ is an area such that we have the resulting force.","On equilibrium $F_U = F_{Do}$ $F_U = F_{Do} = (p_1 - p_2) \\pi r^2 = \\gamma \\pi r^2 (\\frac{1}{R_1} + \\frac{1}{R_2})$ And we again arrive at the Young-Laplace equation $p_1 - p_2 = \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2})$ The Young-Dupré equation defines the equilibrium condition, more specifically the contact angle, when three homogeneous phases are placed in contact and are in thermodynamic equilibrium.","The condition applies to an interface solid-liquid-gas where the shape of a liquid drop is defined by the balance of the surface tensions on the three interfaces, liquid-solid, liquid-gas and gas solid.","In general liquids in contact with solids can behave as wetting or nonwetting phases.","If the fluid behaves as a wetting phase it will tend to maximize the surface area of contact with the solid (Figure REF a) while nonwetting phases will tend to minimize the contact area with the solid (Figure REF b).","Water over a hydrophilic surface is an example of a wetting phase while it is a nonwetting phase when in contact with hydrophobic surfaces such as surfaces covered with wax.","The simplest derivation of the force balance is based on the force equilibrium on the $x$ plane (Figure REF ).","For a wetting phase, for example, the surface tension on the direction of the solid-gas interface $\\gamma _{13}$ is balanced by summation of the forces on the opposite direction, the surface tension on the solid-liquid interface $\\gamma _{12}$ and the $x$ component of the surface tension on the liquid-gas interface, $\\gamma _{23} \\cos {\\theta }$ $\\boxed{\\gamma _{13} = \\gamma _{12} + \\gamma _{23} \\cos {\\theta }}$ $\\theta $ in this case is the contact angle between the liquid-gas and solid-liquid interfaces.","Figure: Angle of contact on an interface solid (1) - liquid (2) - gas (3).","(a) Is a wetting liquid θ<90 o \\theta < 90 ^{o} and (b) is a nonwetting liquid θ>90 o \\theta > 90 ^{o}.The physical interpretation of the capillary rise phenomenon is a controversial issue.","The force balance interpretation given in many textbooks is criticized in terms of accuracy, alternative mechanical and thermodynamic interpretations being preferred.","The force balance interpretations due to pressure difference will be presented here due to its simplicity.","Consider water rising to the level h in a fine glass tube (Figure REF ), the pressures at points A and C are equal to atmospheric pressure being exerted at the surface of the liquid, $P_A = P_C = P_{atm}$ , and following Pascal's principle, $P_C = P_D = P_{atm}$ .","Capillary rise and the Young-Laplace equation indicate that there is a pressure difference between the two sides of the curved surface represented by the meniscus between A and B.","Because there is capillary rise, the pressure is lower below the surface of the meniscus, in B, than above the meniscus, in A, such that $P_B < P_A$ .","The capillary rise also indicates that the pressure in D is greater than in B, such that a pressure gradient exists producing an upwards force.","The condition of equilibrium in terms of force is $(P_D - P_B - \\rho g h) \\pi r^2 = 0$ In other words, the upwards force $P_D \\pi r^2 $ = force/area $\\cdot $ area = force, is balanced by the weight of the water column $\\rho g h \\pi r^2$ = P $\\cdot $ area = force/area $\\cdot $ area = force, and the force exerted at B.","Because of the equilibrium condition, $P_D - P_B = P_A - P_B $ and $P_A - P_B = \\rho g h$ substituting the Young-Laplace equation for the pressure gradient on the interface $\\frac{2 \\gamma }{R} = \\rho g h$ This equation can be written in terms of the radius of the capillary tube instead of the radius of the spherical meniscus by considering the geometrical relation in Figure REF $R = \\frac{r}{\\cos {\\theta }}$ The capillary rise equation results from replacing $R$ into the Young-Laplace equation $\\boxed{h = \\frac{2 \\gamma \\cos {\\theta }}{\\rho g r}}$ The rise considered is up to the lowest point of the meniscus, the volume of water above that point being neglected from the weight of the column because it is usually much less than the weight of the rest of the column.","Figure: Capillary rise.We will leave capillary rise and curved interfaces aside for a little while in order to discuss the theory of soil water potentials.","Potential theory of soil water is nothing more than an application of one of the most fundamental principles in physics - and in nature for that matter -, the principle of conservation of energy, to soil systems.","We already discussed several phenomena in terms of energy throughout the book, now we need to formalize how energy plays a role in transport and storage of water in porous media.","Energy is a fundamental concept on the ontology of physics and it is often easier to be understood intuitively than to be given an actual definition.","The concept of energy in classical mechanics is a good starting point for an intuitive discussion.","We all know that a body will fall under the force of gravity if it is released close to Earth's surface.","The nature of gravitational force which accelerates the body in direction of the center of Earth is complex and the most accepted explanation is given in terms of Einstein's general relativity and curvature of space-time.","The theory is beyond the scope of this book and for the sake of simplicity we can accept that such force exists as has been experimentally verified.","One of the most important ideas in the concept of energy is that it can only be defined in relation some reference state or condition.","When we lift a body 10 m into the air close to Earth's surface we are defining Earth's surface itself as the reference level.","We could lift a body 10 m above the surface in Mars, on the Moon or above some point in an inertial space station trillions of kilometers away from the nearest star or planet and use any of these points as a reference level and calculate stored energy.","The gravitational force near Earth is defined experimentally as $\\mathbf {F} = -m \\mathbf {g}$ The negative sign indicates that the force is downwards in a coordinate system where upwards is defined as positive.","If we defined Earth's surface at a particular point and we wish to increase the elevation of an object with mass, work needs to be done against this force.","If the object is then released, work is done by the gravitational field as the object falls to a new equilibrium condition.","The work done against gravitational force by elevating an object with mass m from the surface reference point where h is set to zero to a height h is $W = \\int _{0}^{h} \\mathbf {F} dh = \\int _{0}^{h} -m \\mathbf {g} dh = -m \\mathbf {g} \\bigg \\vert _{0}^{h} = -m \\mathbf {g} h - (-m \\mathbf {g}0) = -m \\mathbf {g} h$ If the force is conservative, or path independent, the work done is directly related to an energy gradient as $W = -\\Delta U$ Thus the gravitational potential energy of a body lifted to a height h above a reference level where height is arbitrarily set as zero is $U = mgh$ A body falls from height because it has mechanical energy stored.","Water in rivers, streams, pipes, and pores in soils and rocks also moves because of an energy gradient.","In pipes, rivers and streams, the flow can be fast enough that there is a contribution from kinetic energy to the overall energy conservation calculations, as evidenced by the use of the Bernoulli equation, but this is usually not the case in small pores where the flow is slow enough that the kinetic energy contribution can be ignored.","In water in soils and rocks, many different forms of energy can be involved.","In Chapter REF , where we discussed the fundamental equations of fluid mechanics, we saw that pressure gradients are involved in the flow of fluids, and we also saw that gravitational forces can be involved along with pressure gradients.","In porous media, especially in unsaturated porous media, things can be a little more complicated, in both statics and dynamics, a variety of other forces are involved, including complex intermolecular and surface forces.","Because of the irregularity and often random distribution of pores and solids, other governing laws in addition to the Navier-Stokes equations might be necessary for modeling such systems.","Although in this section we discuss the role of potentials in statics, a truly static condition is virtually impossible in natural porous media.","Even a block of wet soil in laboratory conditions might be constantly subjected to potential gradients with the surrounding air and thermal fluctuations within the soil mass, which will cause energy gradients and water movement in response to these gradients.","However, the assumption of a transitory static condition is very important for the use of methods of determination of potential and water content in porous materials.","To discuss the state of energy of water in contact with solids in rocks and soils we need to address a different form of energy, other than the mechanical forms gravitational potential energy and kinetic energy, the internal energy.","The internal energy, contrary to forms of energy which are related to external fields and motion of the system as whole, is the energy within the system as defined by a particular state in comparison to a reference state in which the energy is arbitrarily taken as zero.","In these equilibrium states, the system can be completely characterized macroscopically by the internal energy, U, the volume, V, and the number of moles of its chemical components, $N_{1}$ , $N_{2}$ , ..., $N_{i}$ .","Another extensive function that can be posited for equilibrium states of a system is the entropy, SInterpretations of the concept of entropy in popular media and introductory courses are most often inadequate, if not incorrect.","Entropy must be defined on a physical and mathematical basis, and different definitions can be given based on different subfields of physics, we will address the issue in later editions.","For now refer to [42], [24].","The entropy is a function of the other extensive properties $S = S(U, V, N_i)$ and because the entropy function is continuous, differentiable and is a monotonically increasing function of energy, it can be solved for energy resulting in $U = U(S, V, N_i)$ Now the internal energy and entropy can be stated for a multiphased porous media as long as it is in a state of equilibriumNon-equilibrium thermodynamics plays an important role in transport phenomena, but for the sake of simplicity it is necessary to assume that states of equilibrium are attained for a reasonable simple treatment of the problem, and in such condition they contain all information necessary to characterize the system.","By computing the differential of the energy form we have [24] $dU = \\left(\\frac{\\partial U}{\\partial S}\\right)_{V, N_i} dS + \\left(\\frac{\\partial U}{\\partial V}\\right)_{S, N_i} dV + \\sum _i \\left(\\frac{\\partial U}{\\partial N_i}\\right)_{V, N_j} dN_i$ From this equation the well known intensive properties temperature, pressure and chemical potential of the $i^{th}$ chemical species can be defined by the following partial derivatives [24] $T = \\left(\\frac{\\partial U}{\\partial S}\\right)_{V, N_i}$ $-P = \\left(\\frac{\\partial U}{\\partial V}\\right)_{S, N_i}$ $\\mu _i = \\left(\\frac{\\partial U}{\\partial N_i}\\right)_{S, V, N_j}$ The internal energy relationship can then be written as $dU = T dS -P dV + \\mu _i dN_i$ For a partially saturated porous media composed of a solid, liquid and a gas, the chemical potential can be defined for each phase, assuming that there are no different species within each phase, such that $dU = T dS -P dV + \\mu _{solid} dN_{solid} + \\mu _{liquid} dN_{liquid} + \\mu _{gas} dN_{gas}$ T, P and $\\mu _i$ are partial derivatives of functions of S, V and $N_i$ and because of that they are themselves functions of these quantities.","These functional relationships are called equations of state.","For a natural porous media system we are interested in the equation of state which describes the chemical potential of pore water $\\mu _w = \\mu _w(S, V, N_1, N_2, ...)$ However, $S$ , $V$ and $N_i$ are inconvenient variables to control in experimental studies in partially saturated soils and rocks.","Since the chemical potential of water in this case was derived from the energy function, these variables also appear on the equation state.","One alternative is to change the independent variables on the water chemical potential equation of state by performing a Legendre transformation in order to obtain new functions called thermodynamic potentials which can be a function of one or more more easily measured and controlled intensive parameters under experimental conditions.","Examples of thermodynamic potentials are the internal energy itself, Helmholtz free energy, enthalpy and Gibbs free energy, although others can be derived.","A convenient potential for porous media studies is the Gibbs (free) energy $G$ .","In the case of the Gibbs energy we want to replace both S and V as independent variables and in this case G is defined by differential of $U$ minus the two conjugate pairs one wishes to replace, $TS$ and $-PV$ [42] $dG = d(U - TS - (-PV)) = d(U -TS + PV) = dU - TdS - SdT + PdV + VdP$ Recalling dU = TdS - pdV $dG = TdS - PdV + \\sum _i \\mu _i dN_i - TdS - SdT + PdV + VdP$ arriving at $dG = - SdT + VdP + \\sum _i \\mu _i dN_i$ In which the Gibbs free energy is a function of $T$ , $P$ and $N_i$ $G = G(T, P, N_i)$ As with the internal energy function, we can also define intensive parameters and equations of state for the Gibbs free energy $-S = \\left(\\frac{\\partial G}{\\partial T}\\right)_{P, N_i}$ $V = \\left(\\frac{\\partial G}{\\partial P}\\right)_{T, N_i}$ $\\mu _i = \\left(\\frac{\\partial G}{\\partial N_i}\\right)_{T, P, N_j}$ This last term is the partial molar Gibbs energy of component $i$ and is defined at constant T, P and number of moles of any other constituents, $N_j$ [15].","Thus, for the component $i$ , the equation of state is a function of the form $\\mu _i = \\mu _i(T, P, N_j)$ with differential $d\\mu _i = \\left(\\frac{\\partial \\mu _i}{\\partial T}\\right)_{P, N_j} dT + \\left(\\frac{\\partial \\mu _i}{\\partial P}\\right)_{T, N_j} dP + \\sum _j \\left(\\frac{\\partial \\mu _i}{\\partial N_j}\\right)_{T, P} dN_j$ There are many ways of using these equations in thermodynamic systems, the user having freedom within the constraints of the theory to choose the independent variables.","In soils and other porous systems we are usually concerned with chemical potential of the water in the pores and surfaces $\\mu _w$ .","It is convenient to modify the chemical potential to include the mas based or gravimetric water content $\\theta _g$ , which is essentially the mass based water fraction.","Under these conditions, the chemical potential of water can be defined as a function of $T$ , $P$ , $\\theta _g$ and the number of moles of each dissolved solute fraction $N_j$ [7], [15], [95]An alternative derivation can be found in [90] $\\mu _w = \\mu _w(T, P, \\theta _g, N_j)$ thus, $d\\mu _w = \\left(\\frac{\\partial \\mu _w}{\\partial T}\\right)_{P, \\theta _g, N_j} dT + \\left(\\frac{\\partial \\mu _w}{\\partial P}\\right)_{T, \\theta _g, N_j} dP + \\left(\\frac{\\partial \\mu _w}{\\partial \\theta _g}\\right)_{T, P, N_j} d\\theta _g + \\sum _j \\left(\\frac{\\partial \\mu _w}{\\partial N_j}\\right)_{T, P, \\theta _g} dN_j$ In the soil physics literature what is defined as water potential ($\\psi _w$ ) is the chemical potential of water in soil in reference to a Standard State, usually pure water at 293.15 K and 101325 Pa ($\\mu _w^0$ ) $\\boxed{\\psi _w \\equiv d\\mu _w \\equiv \\mu _w - \\mu _w^0}$ Make sure to understand this concept, it is one of the most fundamental principles in applied soils physics and, along with other forms energy and potential gradients, defines how water is stored and moves in soils, and the principles of operation of various field and laboratory devices such as tensiometers, pressure plate apparatus, psychrometers and what the water retention curve is.","Most soil physicists and vadose zone hydrologists take this basic knowledge for granted and it is impossible to have a firm grasp on anything related to soil water phenomena without understanding these concepts.","Since $\\mu _w$ is an intensive property in units of energy/amount (usually $J~kg^{-1}$ or $J~mol^{-1}$ ) the water potential has the same units.","Having defined what the water potential $\\psi _w$ is we now have to define its components as usually encountered on the soil physics literatureWe mention soil physics because these concepts evolved within and along with early soil physics.","It is interesting to follow the evolution of the concepts and discussions on journal articles published by N.E.","Edlefsen and A.B.C.","Anderson, the Gardner brothers, G.H.","Bolt, K.L.","Babcock, R.D.","Miller, S. Takagi and many of their collaborators and students during the 1940s, 50s and 60s.","Because$j$ here has the same function as $i$ from when this function was first presented $\\mu _w = \\left(\\frac{\\partial G}{\\partial N_j}\\right)$ the coefficients in $dP$ and in $dT$ in Equation 7.54 can be transformed by maintaining the appropriate quantities constant and changing the order of differentiation as [15], [24] $\\frac{\\partial \\mu _w}{\\partial T} = \\frac{\\partial }{\\partial T}(\\frac{\\partial G}{\\partial N_j}) = \\frac{\\partial }{\\partial N_j} (\\frac{\\partial G}{\\partial T}) = \\frac{\\partial }{\\partial N_j}(-S) = -\\frac{\\partial S}{\\partial N_j} = -S_w$ $\\frac{\\partial \\mu _w}{\\partial P} = \\frac{\\partial }{\\partial P} (\\frac{\\partial G}{\\partial N_j}) = \\frac{\\partial }{\\partial N_j} (\\frac{\\partial G}{\\partial P}) = \\frac{\\partial }{\\partial N_j}(V) = \\frac{\\partial V}{\\partial N_j} = V_w$ in which $V_w$ is the partial molar volume of water and $S_w$ is the partial molar entropy of pore water.","Replacing into the original formula we arrive at $\\boxed{d\\mu _w = -S_w dT + V_w dP + \\left(\\frac{\\partial \\mu _w}{\\partial \\theta _g}\\right)_{T, P, N_j} d\\theta _g + \\sum _j \\left(\\frac{\\partial \\mu _w}{\\partial N_j}\\right)_{T, P, \\theta _g} dN_j}$ This is the thermodynamic equation that defines the potentials of soil water, it is possible to arrive at this equation using different thermodynamic potentials although Gibbs free energy and the Groenevelt-Parlange [41] potentials are most often employed in the soil physics literature [90].","Using the soil physics $\\psi $ notation, this equation is usually written as [95] $\\boxed{\\psi _w = -S_w dT + \\psi _p + \\psi _m + \\psi _o}$ Considering isothermal conditions, $dT = 0$ , and the equation reduces to $\\boxed{\\psi _w =\\psi _p + \\psi _m + \\psi _o}$ In reality, many field and laboratory applications of these equations must consider isothermal conditions, not doing so would require some entropy function for the temperature potential, $\\psi _T$ , to be considered, which would cause great mathematical complications.","The $\\psi $ terms in this equation are the most commonly defined components of soil water potential $\\psi _w$ , the pressure potential $\\psi _p = V_w dP$ the matric potential $\\psi _m = \\left(\\frac{\\partial \\mu _w}{\\partial \\theta _g}\\right)_{T, P, N_j} d\\theta _g$ and the osmotic potential $\\psi _o = \\sum _j \\left(\\frac{\\partial \\mu _w}{\\partial N_j}\\right)_{T, P, \\theta _g} dN_j$ The pressure potential is usually associated with hydraulic pressure in saturated soils below the water table, however it can be positive or negativeA body of water can be under tension, and not only under compressive stresses.","Under the water table the pressure potential corresponds to the hydraulic pressure plus the atmospheric pressure, at the water table the hydraulic pressure is zero and the second component of pressure is the atmospheric pressure.","Because we define the standard state at atmospheric pressure, the pressure potential is zero at the water table.","In unsaturated soils the liquid pressure is zero and the soil water is at atmospheric pressure.","The soil matric potential is negative and is related to the capillary and adsorptive forces exerted by the soil solid phase on soil water.","Everything else constant, the water potential in an unsaturated soil is lower than that of free water at the same height and elevation, so that the state of energy of soil water is lower than that of free water.","In simpler terms, water molecules in contact with soil solid particles are “less free to move” because of the forces exerted by these surfaces.","At higher water contents, the adsortion forces of water molecules with surfaces play a small role when compared to capillary forces.","As the soil dries and the water is restricted to thin films around the particles and very small pores, the adsorption component cannot be neglected and the matric potential will be some function of the capillary potential and the adsorption potential.","Detailed discussion of the components of capillary and adsorptive forces can be found in [72], [96].","The osmotic (or solute) potential is the reduction of water potential in contact with dissolved solids, or more specifically, solutes.","Water molecules in contact with dissolved solids have lower energy, thus, as with cells and in other semi-permeable membranes, a larger concentration within the porous membrane or the cell will tend to draw water from the exterior until a pressure equilibrium is achieved in response to the osmotic forces.","Different authors report different soil water potentials.","This is a complex issue and as we mentioned before, the potentials will depend on thermodynamic theoretical considerations such as the thermodynamic potential used to derive the water potentials, see for example the discussions on pneumatic, submergence and overburden potentials, among others in [95], [90].","Because we initially expressed the water potential as a partial molar Gibbs energy $\\partial G/\\partial N_j$ , the units of the water potential are in $J~mol^{-1}$ , and this water potential can be defined as the molar water potential.","However, this unit is not often used in soil physics and groundwater hydrology, the most used units are derived from this basic unit.","Considering the molar mass of water $M_w$ in $kg~mol^{-1}$ , we can divide the soil molar water potential by it, defining the water potential in units of $J~kg^{-1}$ $\\frac{\\psi _w}{M_w} \\rightarrow \\frac{J~mol^{-1}}{kg~mol^{-1}} = \\frac{J}{kg}$ Perhaps the most common form encountered in soil physics is in terms of pressure units, or the soil water pressure.","The water potential in joules per mass is converted to pressure units by multiplying by water density $\\psi _w \\rho _w \\rightarrow \\frac{J }{kg} \\frac{kg}{m^{-3}} = \\frac{J}{m^{3}} = \\frac{N~m}{m^{3}} = \\frac{N}{m^{2}} = Pa$ Pascal ($Pa$ ) and its equivalents $kPa$ , $hPa$ and $MPa$ are frequently used in soil physics, especially when dealing with matric potentials measured using tensiometers and pressure plates.","The third most commonly used conversion of water potential is in terms of head units or length, or the soil water head.","To convert to length units, the water potential in joules per mass is divided by the gravitational acceleration, $g$ $\\frac{\\psi _w}{g} \\rightarrow \\frac{J~kg^{-1} }{m~s^{-2}} = \\frac{N~m~kg^{-1} }{m~s^{-2}} = \\frac{N~kg^{-1} }{ s^{-2}} = \\frac{kg~m~s^{-2}~kg^{-1} }{ s^{-2}} = m$ Recall for the above derivations that $1~J = 1~N~m$ and that $1~N = 1~kg~m~s^{-2}$ .","The head units are common in hydrogeology and to some extent in soil physics, especially for the pressure potential.","Sposito [90] warned that the different units should not obscure the physical interpretation of the problem.","No matter in what unit it is expressed, the soil water potential represent an gradient of energy per unit amount of water which drives movement of soil water.","Because water close to Earth's surface is subjected to a gravitational field and gravitational forces, the description of the potentials is incomplete without considering the effect of gravity.","For a system under the effect of gravity the gravitational potential $ \\phi = gz$ must be added to the chemical potential of water to completely define the state of said water.","The complete treatment of the inclusion of the gravitational potential into the thermodynamic treatment of soil water is complex and a more detailed analysis of the methods can be found in Babcock [6].","The general theoretical account of including the gravitational potential with the chemical potential can be found in Guggenheim [42].","For nowA complete treatment is planned to be included in later editions and for simplicity we will follow the approach of [95] and simply add the soil water gravitational potential to the other potentials asNotice that this relationship can be easily demonstrated to hold by considering the gravitational potential energy $ U = mgh$ and solving the density of water $\\rho _w = m_w/V$ for $m_w$ and replacing into the gravitational potential energy.","The soil water gravitational potential then is simply $U/V \\rightarrow J~m^{-3} = N~m^{-2} = Pa $ $\\psi _g = \\rho _w g z$ In which $z$ is the vertical elevation.","Considering now the potentials in pressure units, the total potential of soil water is the sum of the components of water potential and the gravitational potential, plus any other potentials that might be relevant under specific conditions $\\boxed{\\psi _t = \\psi _p + \\psi _m + \\psi _o + \\psi _g + ....}$ Remember that a potential gradient can be understood as a potential to perform work, movement of water will occur as a response to a gradient of total potential.","In a water body under static conditions the total potential must be the same everywhere.","Notice that the individual components might not be equal, but the sum of the components must be the same everywhere, otherwise there is a potential gradient and the water will flow in response to such gradient.","Consider the points $A$ , $B$ and $C$ at heights $a$ , $b$ and $c$ on the vessel in Figure REF .","If the water level is at $h$ and we consider the gravitational reference at the dotted line at the bottom, the gravitational potential, in head units, at each point is $\\psi _{gA} = a \\\\\\psi _{gB} = b \\\\\\psi _{gC} = c$ and the pressure potential is $\\psi _{pA} = h - a \\\\\\psi _{pB} = h - b \\\\\\psi _{pC} = h - c$ Since only these two potentials are being considered, the total potential $\\psi _t = \\psi _p + \\psi _g$ is $\\psi _{tA} = (h - a) + a = h \\\\\\psi _{tB} = (h - b) + b = h\\\\\\psi _{tC} = (h - c) + c = h$ Thus the total potential is constant and equal to $h$ everywhere.","The same principle is valid for saturated soils, porous rocks and sedimentary deposits, if the total potential is constant everywhere the fluid is static.","Figure: Total potential in a vessel.Consider now a porous medium which is half saturated and half unsaturated (Figure REF ).","Above the water table the water column is under tension following the mechanisms described for a capillary tube.","We showed that the height of capillary rise was given by $h = \\frac{2 \\gamma \\cos {\\theta }}{\\rho g r}$ Figure: Total potential in a partially saturated porous medium.Because water at each point above the water table is under tension as demonstrated for the capillary rise equation derivation, the capillary potential in head units at any point above the water table can be shown to be equivalent to the height above the water tableThis offers no mathematical or physical proof of any kind and is presented as a statement.","A mathematical proof is being pursued $\\psi _c = -h$ Considering that the matric potential can be approximated by the capillary potential $\\psi _m \\approx \\psi _c$ , the matric potential, in head units, will correspond to the negative of the height of water column above the water table.","Thus, for a porous medium that is half saturated and half unsaturated (Figure REF ) we can calculate the gravitational, pressure, matric and total potentials at the points $A$ , $B$ and $C$ .","The gravitational potential is $\\psi _{gA} = a \\\\\\psi _{gB} = b \\\\\\psi _{gC} = c$ the pressure potential is $\\psi _{pA} = b - a \\\\\\psi _{pB} = 0 \\\\\\psi _{pC} = 0$ the matric potential is $\\psi _{mA} = 0 \\\\\\psi _{mB} = 0 \\\\\\psi _{mC} = -(c - b)$ such that the total potential, $\\psi _t = \\psi _g + \\psi _g + \\psi _m $ , is $\\psi _{tA} = a + (b - a) + 0 = b\\\\\\psi _{tB} = b + 0 + 0 = b\\\\\\psi _{tC} = c + 0 -(c - b) = b$ Thus the total potential is constant and equal to $b$ on the whole system, as expected.","Most devices for measuring soil water potential fall into four categories: 1) suction devices, such as Haines apparatus or tension tables, 2) tensiometers, 3) pressure plates or pressure chamber apparatus and 4) psychrometers [31].","Methods 1, 2 and 3 rely on similar physical principles, the soil sample is put into contact with a porous membrane which is contact with a water reservoir.","In 1 the pressure in the water reservoir is then decreased by decreasing the elevation of the opening of a water column in contact with the porous membrane which in turn is in contact with the soil sample.","As the pressure decreases, water in an initially saturated soil sample will flow to the reservoir, and from there to the outflow tube, until equilibrium is reached.","In head units, the capillary potential and thus the matric potential are effectively the negative of the decrease in elevation of the opening of the water column with respect to the porous membrane.","$\\psi _m \\approx \\psi _c = -h$ If the soil sample is relatively small in height, less than one to two cm [31], at equilibrium the matric potential of soil water is the same as the porous membrane.","Different porous membranes can be used, depending on the application of the method and the desired matric potential range.","Fine sand, kaolin clay, paper, porous stone and others can be found on the literature.","Often in the literature the term tension is used instead or matric potential, tension ($\\tau $ ) being simply the negative value of the matric potential $\\tau = -\\psi _m$ Tensiometers (category 2) function in a similar way, except that the pressure on the water reservoir is not controlled by the user as in the case of hanging water column devices.","The device is usually composed of a tube full of water with a porous cup on one end.","The porous cup is put into contact with soil and if the soil is unsaturated, water will flow from the device and into the soil until equilibrium is reached.","When equilibrium is reached, the water within the tube is in a state of tension and the tension or pressure can be measured using a variety of methods.","Historically, the tension was measured by the decrease in height of the water level in a tube connected to the porous medium, however, because this would require a very long column, a liquid a much greater density, mercury (Hg) was used in contact with a water reservoir, resulting in the need of much smaller columns and in devices that could be conveniently deployed to the field.","Later, due to the extreme environmental toxicity or Hg, and for convenience, analog pressure gauges were used.","Although analog gauges are still used, with the decrease in cost of electronics in the last decades, digital pressure and tension sensors connected to data logging devices are now common in research and in agricultural and environmental monitoring applications.","At equilibrim, and considering the water reservoir and soil at the same elevation, the sum of the pressure and matric potentials are going to be the same on the soil and on the water reservoir [95] $\\psi _{ps} + \\psi _{ms} = \\psi _{pr} + \\psi _{mr}$ Where the subscripts $s$ and $r$ stand for soil and water reservoir, respectively.","However, the pressure potential is zero if the soil is unsaturated and the matric potential is zero in the water reservoir as there is no soil matrix such that $\\psi _{ms} = \\psi _{pr}$ Therefore, the (negative) pressure reading corresponding to the water on the reservoir is the soil matric potential.","Notice that atmospheric pressure still acts on the unsaturated soil, but since the pressure reading on the device is with respect to the atmospheric pressure (by definition the gauge pressure), the two terms cancel out and there is no need to consider atmospheric pressure.","Because the boiling point of water at pressures below atmospheric pressures decreases as the pressure difference increases at constant temperature, the maximum practical tension possible at the sea level and at average surface temperature is around $8~m$ , although in laboratory conditions, using de-areated water, higher tensions are possible [102].","In order to increase the range of pressures (or tensions), the soil sample and porous membrane can be put inside a pressurized chamber (category 3).","As the pressure inside the chamber increases, the capillary pressure of the water in the porous membrane decreases as the pores will gradually drain according to their radius.","At higher pressures, water exists in smaller pores of small radius, such that the radius of curvature of the water meniscus is also very small, corresponding to very small capillary pressures (higher numbers in absolute value, or in terms of tension).","The soil samples in hydraulic contact with the porous membrane will lose water until equilibrium is reached True equilibrium is never reached in reality, but fluxes are microscopic only or gradients small enough so that they can be disregarded.","If the height of the soil sample is small, the gravitational potential can be neglected and the tension on the soil sample is the gauge pressure within the chamber.","The porous membrane, usually a porous ceramic plate, is in contact with a water reservoir which is open to the atmosphere such that, as before $\\psi _{ps} + \\psi _{ms} = \\psi _{pr} + \\psi _{mr}$ and the matric and pressure potential are now zero on the water reservoir such that $\\psi _{ms} = -\\psi _{ps}$ In which $\\psi _{ps}$ is effectively the gauge pressure within the chamber.","Pressure chamber devices can reach tensions well beyond those of tension devices.","The limit usually considered is more or less arbitrarily set at $15~m$In part because of the idea that there is some universal permanent wilting point.","At pressures above this value, adsorption forces start to play a more prominent role and the pressure gradient alone is not enough to further remove water from the soil.","The water that cannot be removed by pressure plate apparatus is often referred to residual water content and can often be removed using other methods.","Psychromters, hygrometes and potentiameters (category 4) are, generally speaking, devices that measure the water potential based on the Kelvin equation $\\psi _{w} = \\frac{R_g T}{V_w} \\ln {\\frac{p}{p_0}}$ in which the water potential measured is the sum of the matric and osmotic potential, $R_g$ is the universal gas constant, T is the absolute temperature, $V_w$ is the molar volume of water, $p$ is the air actual vapor pressure and $p_0$ is the air saturation vapor pressure, and $p/p_0$ is the relative humidity of the air in contact with the sample.","The water activity is also used in some devices and is, by definition, $a_w \\equiv p/p_0$ .","The water potential measured by these devices will not necessarily correspond to the water potential measured by pressure and suction devices, since the nature of the components measured are different.","One of the main advantages of psychrometric and other similar devices is that they can measure potentials at extremely low water contents, well beyond what is possible with the pressure chamber, well below -1.5 MPa.","Details on these devices can be found in [3], [82] The gravimetric method, based on weighing a soil or porous material sample, oven drying at 105 °C for at least 24 h and reweighing the oven dried sample and calculating the mass of water divided the mass of oven dried material is the standard method for measuring water content and it is used to calibrate indirect water content estimation methods.","The gravimetric method is a direct method because the mass of water is directly measured by weighing.","Because it is a direct method, if conducted properly using scales with high enough accuracy and precision, results in errors, both systematic and random, much smaller than indirect methods.","In indirect methods, the water content is estimated indirectly and they are usually calibrated for a particular soil or condition using the gravimetric method.","Errors related to calibration, or lack of thereof, and instrumental errors, both on the sensors and electronics usually result in lower precision and/or accuracy when compared to direct methods.","There are several different types of indirect methods for measuring water content, the two most common types in use today being methods using radioactive sources, such as neutrons and gamma radiation, and general electromagnetic methods including time domain reflectometry (TDR), capacitance and other devices [31].","The neutron thermalization method historically has been an important method for water content measurement and monitoring in the field.","Once calibrated it provides fast and accurate measurements of soil water content.","Physically, the method relies on a radioctive source that emits fast neutrons, with kinetic energy on the MeVeV = electron-volt; 1 $eV = 1.60217646 \\cdot 10^{-19}~J$ range and speed on the $10^4~km~s^{-1}$ order of magnitude, and a detector of slow neutrons, with energies less than about 1 eV [45].","The fast neutrons emitted by the source collide with hydrogen atoms (H) in water molecules.","Because H has a high scattering cross section, it can absorb or scatter fast neutrons, causing attenuation of the energies and correspondingly the velocity of the neutrons.","The higher the water content, more hydrogen nuclei are available for collisions, attenuating fast neutrons and producing slow neutrons which are then counted by the detector.","Of course other atoms can attenuate neutrons, but because H present in water is particularly effective in doing so in comparison to other atoms commonly found in soils, it can be used to water content estimation in most soils.","The calibration is performed by direct measurement of volumetric water content around the sensing volume and corresponding slow neutron count ratios [45] $C_R = \\frac{x}{x_s}$ in which $x$ is the count in the measured material and $x_s$ is a standard count.","The count ratio can then be related to the volumetric water content by a calibration equation of the form $\\theta _v = b_0 + b_1 C_R$ in which $b_0$ and $b_1$ are empirical parameters obtained from statistical or mathematical fitting procedures and $\\theta _v$ is the volumetric water content [45].","Other forms of calibration equations can be found on the literature for different materials and water content ranges.","The principle of operation of most electromagnetic methods including TDR, frequency domain reflectometry (FDR), ground penetrating radar (GPR), capacitance, impedance and others are based on the principle that the relative permittivityReferred to as the dielectric constant in older literature of water is much greater than that of other soil components, including air, minerals and organic matter.","The relative permittivity of a material is a ratio of the absolute permittivity of the material to that of vacuum $\\epsilon _r = \\frac{\\epsilon }{\\epsilon _0}$ Because the absolute permittivity of the material, $\\epsilon $ , and that of vacuum have both the same units, usually $F~m^{-1}$ , and the permittivity of vacuum is a physical constant $\\epsilon _0 = 8.8541878128(13)\\cdot 10^{-12} ~ F~m^{-1}$ , $\\epsilon _r$ is a dimensionless normalized quantity, with values around 80 for water, 1 for air and around 5 for quartz rich soils [80].","The relative permittivity of the porous media as a whole will be some form of average of the individual permittivities of each phase, the term apparent permittivity ($\\epsilon _{r(app)}$ ) being used on the literature [63].","Because the solid phase is usually constant in fractional volume for most porous media applications and because the relativity permittivity of one of the dynamic phases, water, is much greater than that of the other, air, under unsaturated conditions the apparent permittivity will be a function of the volumetric water fraction, i.e.","the volumetric water content.","Thus, for most practical applications, water content measured using direct methods, usually gravimetric, can be related to apparent relativity permittivity measurements obtained using electromagnetic measuring devices, generating calibration equations.","The most common empirical calibration equations found in the literature are third degree polynomials, square root and logarithmic [64] $\\theta _v = a_0 + a_1 \\epsilon _{r(app)} + a_2 \\epsilon _{r(app)}^2 + a_3 \\epsilon _{r(app)}^3$ $\\theta _v = a_0 + a_1 \\sqrt{\\epsilon _{r(app)}}$ $\\theta _v = a_0 + a_1 \\log {(\\epsilon _{r(app)})}$ in which $a_0$ to $a_3$ are empirical fitting parameters, fitted for each individual equation and $\\log $ is the natural logarithm.","Most commercial devices come with “universal” calibration equations from the manufacturers, but it is almost always recommended that the device should be calibrated for the individual soils or porous materials in which is going to be used.","Water content can also be estimated using more physically based models based on effective medium approximations theory [58], [63].","There are several different designs and electronics for electromagnetic water content measuring sensors.","The relative permittivity is in reality an imaginary number and is highly dependent on the frequency of operation of the device in terms of the frequency of the electromagnetic radiation emitted.","For most soils, but not all, the relative apparent permittivity can be approximated by the real component of the complex permittivity.","A detailed discussion of measuring water content with electromagnetic devices is beyond the scope of this book.","The user is highly encouraged to study electromagnetic theory before using any electromagnetic device for soil and porous media measurements, there are many misconceptions related to these devices.","For more information the reader is referred to [99], [31], [63].","There are many other methods used for estimating water content, including methods based on remote sensors, thermal methods and other ionizing radiation methods.","These methods will not be discussed in this book, as they are not as broadly employed as the methods discussed above.","The user interested in these specific methods should consult [31] and the references listed therein.","For a set of tubes of different radius $r_i$ , there is an exact relationship between the percent saturation and the capillary pressure.","At the capillary pressure $Pc_i$ , all tubes of radius greater than $r_i$ will have drained, following the Young-Laplace equation $Pc_{i} =\\frac{ 2\\gamma }{r_i}$ thus the volume of water retained will correspond to the sum of the volumes of water retained in all tubes of radius smaller or equal than $r_i$ .","But the volume will also depend on the capillary rise for each individual tube which is related to the radius by the capillary rise equation For the angle $\\theta = 0$ and $\\cos theta = 1$.","$h_{i} =\\frac{ 2\\gamma }{\\rho g r_i}$ Considering the volume of each tube as that of a cylinder $V_{i} =\\pi r_i^2 h_i$ and replacing the capillary rise equation $V_{i} = \\pi r_i^2 \\frac{ 2\\gamma }{\\rho g r_i} = \\frac{ 2\\gamma \\pi r_i}{\\rho g }$ Now the volume of water is the sum of the volumes of water in all $i$ pores with radius equal or smaller than $r_i$ $V_{water} =\\sum _i \\pi r_i^2 \\frac{ 2\\gamma }{\\rho g r_i} = \\sum _i \\frac{ 2\\gamma \\pi r_i}{\\rho g }$ and the fractional water content is simply the volume of water divided by the total volume of all tubes in the system $\\theta _{f} =\\frac{ V_{water}}{V_{t} }$ Thus, for each capillary pressure $Pc_{j}$ corresponding to a radius $r_j$ , the fractional water content can be directly calculated from the sum of the volumes of water in all pores with radius less or equal than $r_j$ An explicit mathematical relationship can be derived by manipulating summation terms, this will be explored in following editions.","This is what is called the capillary bundle model for water retention.","It is also an important model for hydraulic conductivity, as we will discuss in later chapters.","The capillary bundle model is a useful tool to model and understand transport and retention of water in porous media.","In this approach, the pore space is simplified as a set of continuous tubes of constant radius in each pore.","Because pores are irregular, do not have constant radius, are usually not cylindrical, and because forces related to other than simple capillary phenomena act in complex natural porous systems, there is no direct way of calculating water saturation from matric potential and empirical and semi-empirical models are necessary.","Empirical models rely on fitting to observed data of water content and matric potential, in other words, there is no analytical expression to calculate the volumetric water content by knowing only the matric potential, water content data is necessary.","The water retention curve is nothing more than the empirical relationship between water content and matric potential obtained from laboratory and field data.","The relationship has been observed to follow a characteristic sigmoid function (S-shaped) when the matric potential is considered in log scale.","To avoid mathematical and graphing difficulties it is convenient to consider the absolute values of matric potential ($|\\psi _m|$ ), which as we have seen is a negative quantity, when dealing with water retention equations.","There has been several efforts to derive analytical expressions for the water retention curve.","These have important practical applications in irrigation and environmental modeling and because, in most cases, an unsaturated hydraulic conductivity function was derived from the water retention curve (e.g.","[19], [25], [97], [56]).","Some of the models most often found on the literature are presented below.","We will leave the related unsaturated hydraulic conductivity functions for later chapters.","One of the earliest models that saw wide application is that of Brooks and Corey (BC) [19] For simplicity and to be consistent we will refer to soil water potential as $\\psi $ on the water retention equations $\\theta _v ={\\left\\lbrace \\begin{array}{ll}\\theta _r + (\\theta _s - \\theta _r) (\\frac{\\psi _b}{|\\psi |})^\\lambda , & \\text{for } |\\psi | > \\psi _b \\\\\\theta _s, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.","}$ The BC equation, like many water retention models, is a piecewise equation, meaning that it is segmented into two parts.","Its parameters are the saturation water content, $\\theta _s$ , the residual water content, $\\theta _r$ the air entry value, often called bubbling pressure, $\\psi _b$ , and the pore size distribution parameter, $\\lambda $ .","Most water retention models can be defined in terms of saturation and residual water contents.","The saturation water content correspond to water content at a matric potential of zero, whenever it is defined, where in theory the pore space is completely water saturated.","The residual water content is the remaining water content at very high values of matric tensionOr very low values if you are thinking in potential, remember that tension is defined as the negative of the potential, where further increase in tension does not cause decrease in water content.","In most models, residual water content is an asymptotic parameter reached by taking tension to infinity.","The bubbling pressure is the matric tension in which air starts to enter the pore system, usually at relatively low matric tensions for most soils.","Below the bubbling pressure, increase in tension does not cause a corresponding decrease in water content.","Most water retention models will have parameters related to pore size distribution since, as we saw, water retention is controlled by the distribution of pores.","In the BC model the piecewise condition is defined by the $\\psi _b$ parameter.","At tensions below or equal to $\\psi _b$ the water content is constant and equal to saturation water content, while at tension above $\\psi _b$ water content decreases following a power law of $\\lambda $ .","Most water retention equations can be expressed in terms of the effective saturation, given by $S_e = \\frac{\\theta _v - \\theta _r}{\\theta _s - \\theta _r}$ The BC equation, for example, can be expressed as $S_e ={\\left\\lbrace \\begin{array}{ll}(\\frac{\\psi _b}{|\\psi |})^\\lambda , & \\text{for } |\\psi | > \\psi _b \\\\1, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.","}$ Using the effective saturation normalizes the water content between zero and one, making it possible to compare the shapes of the curves for different materials.","Using the effective saturation can, in some cases, make fitting easier using computer methods by reducing the number of fitting parameters.","The Campbell equation is in essence the BC equation, but defining the effective saturation as $\\theta /\\theta _s$ [25] $\\theta ={\\left\\lbrace \\begin{array}{ll}\\theta _s (\\frac{\\psi _b}{|\\psi |})^b, & \\text{for } |\\psi | > \\psi _b \\\\\\theta _s, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.","}$ In which the physical meaning is the same as the BC equation, and $b$ is essentially $\\lambda $ .","Brutsaert [21] proposed an equation of the form $S_e = \\frac{a}{a + \\psi ^b}$ where a and b are empirical parameters.","Brutsaert's equation was later improved upon by van Genuchten, resulting in one of the most frequently used water retention equations in soil physics and hydrology [97] $\\theta _v = \\theta _r + \\frac{(\\theta _s - \\theta _r)}{[1 + (\\alpha \\psi )^n]^m}$ In which $m$ and $n$ are shape parameters related to the pore size distribution.","It is often convenient to define the Mualem [67] restriction as $m = 1 - \\frac{1}{n}$ Reducing the number of empirical parameters to four.","Kosugui [56] derived an equation based on a lognormal pore size distribution $\\theta _v ={\\left\\lbrace \\begin{array}{ll}\\theta _r + \\frac{(\\theta _s - \\theta _r)}{2} \\text{erfc}[ \\frac{\\log (\\psi _b-\\psi )/(\\psi _b - \\psi _0) - \\sigma ^2}{\\sqrt{2} \\sigma }] , & \\text{for } |\\psi | > \\psi _b \\\\\\theta _s, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.","}$ with two new parameters, $\\sigma $ , the standard deviation of the transformed pore capillary pressure distribution, and $\\psi _0$ , the capillary pressure at the inflection point.","Kosugui's model uses the complementary error function (erfc), defined as [14] $\\text{erf}(x) = \\frac{2}{\\sqrt{\\pi }}\\int _0^x e^{-t^2} dt$ $\\text{erfc}(x) = 1 - \\text{erf}(x) = \\frac{2}{\\sqrt{\\pi }}\\int _x^\\infty e^{-t^2} dt$ In which erf is the error function.","However, the erfc is calculated numerically in almost all practical applications and is available as a function in most programming languages.","Thus, the user does not need to use the integral form when fitting water retention data.","The BC, Brutsaert-van Genuchten and Kosugui equation are fitted to simulated oxisol data and shown in Figure REF Figure: Commonly used water retention models fitted to simulated oxisol data.ll $ W $ Thermodynamical work $ \\gamma $ Surface tension $ A, A^{\\prime }, a $ Area $ l $ length $ dx $ Infinitesimal increase in $x$ $ \\mathcal {F} $ Helmholtz (free) energy $ S $ Entropy $ p, P $ Pressure, except when specified otherwise $ \\mu $ Chemical potential of a substance $ N_{i} $ Number of moles of a substance $ T $ Temperature $ \\mathbf {n} $ Unit normal vector $ R, r $ Radius $ l_1, l_2, l^{\\prime }_1, l^{\\prime }_2 $ Elements of arc $ \\zeta $ Unspecified displacement $ \\theta , \\phi $ Angles $ proj_F $ Component of a force $ F $ Force $ \\rho $ Generalized density $ \\rho , \\rho _w $ Water density $ g $ Earth's gravitational acceleration $ U $ Potential of a force $ m $ Mass $ h $ Height $ G $ Gibbs (free) energy $ \\theta _g $ Gravimetric water content $ \\theta _v $ Volumetric water content $ V_w $ Partial molar volume of water $ S_w $ Partial molar entropy of water $ \\psi $ Soil potential $ \\psi _w $ Soil water potential $ \\psi _p $ Soil pressure potential $ \\psi _m $ Soil matric potential $ \\psi _o $ Soil osmotic potential $ \\psi _c, P_c $ Soil capillary potential $ \\psi _g $ Soil water gravitational potential $ \\psi _t $ Total soil potential $ z $ Elevation $ \\tau $ Soil water tension $ p, p0 $ Actual air vapor pressure and air saturation vapor pressure on Kelvin's equation $ R_g $ Universal gas constant on Kelvin's equation $ a w $ Water activity on Kelvin's equation $ C_R $ Count rate $ a, b, a_0, a_1, _2, A_3, b_0, b_1, m, n $ Empirical parameters $ \\epsilon _r $ Relative permittivity $ \\epsilon $ Absolute permittivity of a material $ \\epsilon _0 $ Permittivity of free space $ \\theta _f $ Fractional water content $ V_{water} $ Volume of water $ V_t $ Total volume $ \\theta _r $ Residual volumetric water content $ \\theta _s $ Saturation volumetric water content $ \\psi _b $ Air entry value $ \\lambda $ pore size distribution parameter $ S_e $ Effective saturation $ \\alpha $ Inverse air entry value $ \\psi _0 $ Capillary pressure at the inflection point of Kosugi's equation $ \\sigma $ Standard deviation of transformed capillary pressure distribution in Kosugi's equation $ \\text{erf} $ Error function $ t $ Generalized variable in the definition of erf $ \\text{erfc} $ Complementary error function As you will see in the following sections, although there are analytical solutions for the Navier-Stokes equations to simple cases of laminar flow in open conduits, for flow in porous media, the statement given by Morris Muskat ([69] Muskat, 1937, p.55) summarizes the problem “Unfortunately, however, in spite of the justifiable simplification of neglecting the inertial forces - due to the small velocities generally characteristic of flow through porous media - the mathematical difficulties of applying these equations to porous media are for practical purposes entirely unsurmountable.\"","Although theoretical derivations of Darcy's law from the Navier-Stokes equations have been proposed over the last decades, a solution for random porous media will invariable depend on some stochastic mathematical method, such as averaging or homogenization theory, a direct analytical derivation for a random and irregular porous medium remains a problem of difficulty on par with that of modelling turbulence.","In simpler words, the random nature of the behaviour of fluid flows cannot be directly solved analytically in all but the simplest cases.","We will be mentioning Muskat's work on several occasion throughout this chapter.","He was a trained theoretical physicist who earned his Ph.D. at California Institute of Technology in 1929 with a thesis on “The Continuous Spectra of Hydrogen Like Atoms” [68].","After graduating, he joined the petroleum industry were he applied his training in problems of multi-phase flow in porous media, making important contributions to the mathematics of flow in porous media in the process, especially by modifying and generalizing the original form of Darcy's equation.","His original contributions remain an important topic not only in engineering and applied mathematics, but also in pure mathematics (e.g.","[2]).","His point is that there are no analytical solutions of the Navier-Stokes equation for true random porous media.","On top of that, there is no guarantee that a solution exists for many fluid mechanics problems.","This remains an open problem in mathematics in the form of the Navier-Stokes existence and smoothness problem one of the so called millennium problems for which you can win a large sum of money you manage to solve it.","Getting back to Darcy's law, it is an empirical law, initially based on experimental data.","There has been many a posteriori attempts to derive Darcy's law from first principles, including averaging the Navier-Stokes equations, but the fact remain that most of these make several assumptions along the way and are for the most part largely heuristic, there are no first principles derivation from a direct analytical solution of the Navier-Stokes equations and the hydraulic conductivity remains an empirical constant.","Before we get too much ahead of ourselves, as you haven't been formally introduced to Darcy's law, let's look at a few cases of flow through free conduits where there is nothing but the fluid going through the conduit.","In porous media, as we have discussed, there is a solid porous matrix inside a volume through which the fluid is moving.","Under these conditions, the fluid does not occupy the entire inner volume and is forced to flow through an often very complex geometry of pores.","This is one of the main factors which makes flow thorough porous media so mathematically complex.","In this section, two analytical solutions of the Navier-Stokes will be discussed.","The first is that for flow through two infinite and flat surfaces parallel to each-other and the second is for flow through a cylindrical conduit.","The second is often seen in soil physics and transport of fluid through porous media books as an introduction to Darcy's law, given that the pore network of a porous material can be generalized to a network of tubes, from which an approximation to Darcy's law can be obtained Which as an approximation, does not adequatelly describes real porous media.","Case I.","Flow between two infinite plates Let us start be recalling the Navier-Stokes equations in vector form $\\frac{\\partial \\mathbf {u}}{ \\partial t} + (\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} = \\mathbf {F} -\\frac{1}{\\rho } \\nabla p + \\frac{\\mu }{\\rho } \\nabla ^{2} \\mathbf {u}$ For steady state flow, the velocity field does not vary with time, so $\\mathbf {u}$ is constant with time everywhere such that $\\frac{\\partial \\mathbf {u}}{ \\partial t} = 0$ and $(\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} = \\mathbf {F} -\\frac{1}{\\rho } \\nabla p + \\frac{\\mu }{\\rho } \\nabla ^{2} \\mathbf {u}$ There are many conditions in which body forces can be neglected, such as in purely horizontal flow of a fluid in which the horizontal dimension is much larger than the vertical direction in which a gravitational field acts or if the contact forces are much larger than the body forces.","Ignoring the body forces results in $(\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} = -\\frac{1}{\\rho } \\nabla p + \\frac{\\mu }{\\rho } \\nabla ^{2} \\mathbf {u}$ Now we have to take a look at each term, the term $ (\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} $ was discussed earlier in the context of the material derivative, based on that discussion, it is possible to show that it represents the set of equations $u_x \\frac{\\partial u_x}{\\partial x} + u_y \\frac{\\partial u_x}{\\partial y} + u_z \\frac{\\partial u_x}{\\partial z}$ $u_x \\frac{\\partial u_y}{\\partial x} + u_y \\frac{\\partial u_y}{\\partial y} + u_z \\frac{\\partial u_y}{\\partial z}$ $u_x \\frac{\\partial u_z}{\\partial x} + u_y \\frac{\\partial u_z}{\\partial y} + u_z \\frac{\\partial u_z}{\\partial z}$ and it is called the inertial or convective term, due to change of velocity as the fluid element is transported to different positions in the velocity field.","The first term on the right side is the pressure gradient and represent the contact forces which cause movement of the fluid if the gradient is not equal to zero.","Recalling the gradient operator, $ \\nabla p$ can be represented as $\\nabla p = \\frac{\\partial p}{\\partial x} \\mathbf {i} + \\frac{\\partial p}{\\partial y} \\mathbf {j} + \\frac{\\partial p}{\\partial x} \\mathbf {k}$ Do not worry about the density $\\rho $ , it is assumed as constant and thus can be manipulated to either side of the equation.","To have an intuitive understanding of the pressure gradient, think about a garden hose connected to a reservoir a few meters above ground, it is the pressure gradient between both ends of the hose that causes the water to flow out of the end where the overall pressure is lower (assuming the hose is perpendicular in relation to the gravitational field), but in this case the gradient is unidimensional, say $\\partial p/\\partial x$ .","The second term on the right is the viscosity or diffusion term, recalling the Laplacian operator, it can be written as $\\mu \\nabla ^2 \\mathbf {u} = \\mu [\\nabla \\cdot \\nabla (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k})]$ which in Cartesian coordinates resolves in three components on the directions $\\mathbf {i}$ , $\\mathbf {j}$ and $\\mathbf {k}$ $\\mu (\\frac{\\partial ^2 u_x}{\\partial x^2} + \\frac{\\partial ^2 u_x}{\\partial y^2} + \\frac{\\partial ^2 u_x}{\\partial z^2} )$ $\\mu (\\frac{\\partial ^2 u_y}{\\partial x^2} + \\frac{\\partial ^2 u_y}{\\partial y^2} + \\frac{\\partial ^2 u_y}{\\partial z^2} )$ $\\mu (\\frac{\\partial ^2 u_z}{\\partial x^2} + \\frac{\\partial ^2 u_z}{\\partial y^2} + \\frac{\\partial ^2 u_z}{\\partial z^2} )$ or, simply $\\mu \\nabla ^2 \\mathbf {u} = \\mu (\\nabla ^2 u_x \\mathbf {i} + \\nabla ^2 u_y \\mathbf {j} + \\nabla ^2 u_x \\mathbf {k})$ For a fluid moving slowly, the inertial forces are much smaller than the viscous forces, such that $ \\mu \\nabla ^2 \\mathbf {u} \\gg \\mathbf {u} \\cdot \\nabla \\mathbf {u} $ and the latter can be ignored resulting in a much simpler form of the Navier-Stokes equations $\\nabla p = \\mu \\nabla ^{2} \\mathbf {u}$ Or, in Cartesian coordinates $\\frac{\\partial p}{\\partial x} \\mathbf {i} + \\frac{\\partial p}{\\partial y} \\mathbf {j} + \\frac{\\partial p}{\\partial z} \\mathbf {k} = \\mu [ \\nabla \\cdot \\nabla (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) ]$ Now many analytical solutions of the Navier-Stokes equations can be derived from this simplification, for a broad range of experimental conditions.","The important point here is that at low flow velocities, when the viscous forces are much greater than the inertial forces the flow is often laminar (for many fluids found in daily life, the most important being water, this is true, it might not be if viscosity is a lot larger or smaller than expected values).","If the flow velocity is too high and/or viscosity too low a phenomenon known as turbulence might occur and it might become impossible to solve the differential equations analytically.","We can finally state our problem as the “laminar flow of water between two smooth parallel plates in which the pressure gradient is in the $x$ direction, the plates are infinitely long on the $y$ direction and the distance $d$ between the plates, through which the water flows, is oriented in the $z$ direction”.","If you are not familiar with solutions of physics problems in textbooks and theoretical models, you might wonder why would anyone assume anything as infinite and how on earth would this translate into a practical application.","The infinite plates simplification is a common occurrence in electromagnetism and other areas of physics and engineering as it becomes much easier to solve the differential equations that arise from the geometry of the problem.","The simple fact of assuming flow through a square tube results in a partial differential equation with series solution Check any partial differential equations or mathematical methods for physicists book and see how convoluted the whole thing is.","First if the pressure gradient is in the $x$ direction, there are no pressure gradients on the $y$ or $z$ directions and the equation further simplifies to $\\frac{\\partial p}{\\partial x} \\mathbf {i} = \\mu [ \\nabla ^2 (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) ]$ Second, if the flow is in the $x$ direction, and is steady-state, the velocity does not vary on that direction, and if the plates are infinite in $y$ , as stated it also does not vary in $y$ such that $\\frac{\\partial p}{\\partial x} \\mathbf {i} = \\mu \\frac{\\partial ^2 u_x}{\\partial z^2} \\mathbf {i}$ Showing that $ \\mu \\nabla ^2 \\mathbf {u} = \\mu \\frac{\\partial ^2 u_z}{\\partial z^2} \\mathbf {i} $ when the flow velocity does not vary in $x$ or $y$ is left as an exercise to the reader.","A simple solution can be found if the pressure gradient in the $x$ direction is constant.","Omitting the unit vectors the equation to be solved is $\\frac{\\partial p}{\\partial x} = \\mu \\frac{\\partial ^2 u_x}{\\partial z^2}$ Whose solution can be found by separating the variables and integrating $\\int \\partial ^2 u_z = \\int \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\partial z^2$ $\\int \\partial u_z = \\int (\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} z + C_1) \\partial z$ $u_z = \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{z^2}{2} + C_1 z + C_2$ As always, we need the boundary conditions to solve the equation, in this case, because of the viscosity condition, the velocity is zero at the surface of the plates $u_z = 0 ~\\text{at} ~z = 0, ~u_z = 0 ~\\text{at} ~z = h{^{\\prime }}$ The first condition gives $C_2 = 0$ , while applying the second $0 = \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}^2}{2} + C_1 h{^{\\prime }} + 0 \\\\C_1 h = -\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}^2}{2}$ $C_1 = -\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}}{2}$ Replacing into Equation 8.5 results $u_z = \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{z^2}{2} -\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}}{2} z$ $\\boxed{u_z = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} z(z - h{^{\\prime }})}$ The discharge (or volumetric flux) passing through a section can be calculated by integrating the flow velocity over the area (or length in this case) $Q = \\int _0^{h{^{\\prime }}} u dz \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} \\int _0^{h{^{\\prime }}} z(z - h{^{\\prime }}) dz \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} (\\int _0^{h{^{\\prime }}} z^2 dz - \\int _0^{h{^{\\prime }}} zh{^{\\prime }} dz) \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} ( \\frac{z^3}{3}\\bigg |_0^{h{^{\\prime }}} - h^{\\prime } \\frac{z^2}{2}\\bigg |_0^{h{^{\\prime }}}) \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} ( \\frac{h{^{\\prime }}^3}{3} - \\frac{h{^{\\prime }}^3}{2}) \\\\$ $\\boxed{Q = - \\frac{1}{12\\mu } \\frac{\\partial p}{\\partial x} h{^{\\prime }}^3 \\\\}$ Dimensional analysis shows that in this case the formula results in an “areal discharge” in [L$^2$ T$^{-1}$ ].","Note that this has to be the case because of the infinite plates assumption.","Case II.","Flow in a cylindrical conduit - Poiseuille flow We can start with Equation 8.5, the process before hand is the same as discussed in the previous section.","The pressure gradient driving the flow is still in the $x$ direction but now the circular geometry in the cylider cross section requires both $y$ and $z$ components $\\frac{\\partial p}{\\partial x} \\mathbf {i} = \\mu (\\frac{\\partial ^2 u_y}{\\partial y^2}\\mathbf {j} + \\frac{\\partial ^2 u_z}{\\partial z^2} \\mathbf {k})$ This equation can be much more easily solved in cylindrical coordinates in which the Laplacian operator can be written as $\\nabla ^2 = \\frac{1}{r} \\frac{\\partial }{\\partial r}( r \\frac{\\partial }{\\partial r}) + \\frac{1}{r^2} \\frac{\\partial ^2 }{\\partial \\theta ^2} + \\frac{\\partial ^2 }{\\partial z^2}$ However, the flow velocity components do not vary in $z$ nor with the polar angle $\\theta $ along the cylindrical conduit, varying only with the radius of the tube $r$ , such that $\\nabla ^2 = \\frac{1}{r} \\frac{\\partial }{\\partial r}( r \\frac{\\partial }{\\partial r})$ In this case, Equation 8.10 can be written in cylindrical coordinates as $\\frac{\\partial p}{\\partial x} = \\mu \\frac{1}{r} \\frac{\\partial }{\\partial r}( r \\frac{\\partial u_r}{\\partial r})$ Now integration can be performed in a single variable $r$ instead of on $y$ and $z$ which simplifies the process.","$\\int \\partial ( r \\frac{\\partial u_r}{\\partial r}) = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\int r \\partial r\\\\r \\frac{\\partial u_r}{\\partial r} = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{r^2}{2} + C_1\\\\\\int \\partial u_r = \\int (\\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{r}{2} + \\frac{C_1}{r}) \\partial r \\\\$ $u_r = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{r^2}{4} + C_1 \\log {r} + C_2 \\\\$ The boundary conditions in this case are that the velocity is zero at the walls of the tube, as before, because of the viscosity condition, the velocity is maximum at center of the tube, at $r = 0$ .","$u_r = 0 ~\\text{at} ~r = a, ~u_r = u_{max} ~\\text{at} ~ r = 0$ Another important condition is that the velocity is finite at the center of the tube.","This implies that $C_1$ has to be zero, otherwise $C_1 \\log {0}$ results in a singularity.","Applying the first boundary condition and with $C_1 = 0 $ $0 = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{a^2}{4} + 0 + C_2 \\\\$ $C_2 = - \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{a^2}{4}$ Replacing into Equation 8.14 $u_r = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} (r^2 - a^2)$ The discharge (volumetric flux) can now be calculated by integrating the velocity function in $y$ and $z$ $Q = \\int _0^y \\int _0^z u dydz$ But again, because of the circular geometry of the tube's cross section, this is much more easily done by converting to polar coordinates, taking into account that $dydz$ is an area element $dA$ , which in polar coordinates is $ r dr d\\phi $ .","There should be no mystery here, if you take a circle of radius r and rotate this radius around 360° you will draw an area.","Rewriting the volumetric flux equation and solving $Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } \\int _0^a (r^2 - a^2) r dr d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } \\int _0^a (r^3 - r a^2) dr d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } (\\frac{r^4}{4} - \\frac{r^2}{2} a^2)\\bigg |_0^a d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } (-\\frac{a^4}{4}) d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } (-\\frac{a^4}{4}) d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} (-\\frac{a^4}{4}) \\phi \\bigg |_0^{2\\pi } \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} (-\\frac{a^4}{4}) 2\\pi \\\\$ $\\boxed{Q = -\\frac{\\pi }{8\\mu }\\frac{\\partial p}{\\partial x} a^4 \\\\}$ in which $Q$ is the discharge in [L$^3$ T$^{-1}$ ] as expected.","Henry Darcy was a french engineer who, while working on water distribution system for the city of Dijon in the 1800s, wrote a monumental report concerning various aspects of water supply and distribution [32].","In one of the appendixes of the document he presents an equation that describes the flow of water through a column filled with sand under a hydraulic gradient (Figure REF ) $q = k \\frac{s}{e} [h + e \\mp h_0]$ In which $q$ was the water flux, $e$ is the height of the sand column, $s$ is the cross sectional area of the column, $h$ is the water height in the upper part of the column (on the inlet), $h_0$ is the water height on the inferior part of the column (on the outlet).","Figure: Schematics of sand column used in the original Darcy experiment (adapted from ).","(a) is the water inlet, (b) and (d) are manometers, (c) is the sand column, (e) is the water outlet, (f) is the support, and (g) is a reservoir for water collecting and measuring outflow volume.If the water height is zero at the outlet, $h_0 = 0$ $q = k \\frac{s}{e} [h + e]$ He realized that the water flux through the sand column was proportional to the pressure difference between the inlet and the outlet and he defined the constant of proportionality as a coefficient k, dependent on the permeability of the sand (“dependant de la permeabilité de la couche”).","So $k$ in this context was defined as a permeability, dependent on the physical characteristics of the solid phase.","In contemporary terms, the height of water at the inlet and outlet is effectively the pressure potential in terms of height of water.","Considering the potentials at the inlet and outlet, and if we establish the gravitational reference at the bottom of the column, we have at the inlet (top of the column): $\\psi _{gi} = e \\\\\\psi _{pi} = h + P_0\\\\\\psi _{ti} = e + h + P_0$ At the outlet (bottom of the column): $\\psi _{go} = 0 \\\\\\psi _{po} = h_0 + P_0\\\\\\psi _{to} = h_0 + P_0$ in which $P_0$ is the atmospheric pressure.","Therefore, Darcy's law can be expressed as $q = k A\\frac{\\psi _{ti} - \\psi _{to}}{L} = k s \\frac{(e + h + P_0) - (h_0 + P_0)}{e} \\\\= k s\\frac{(e + h + P_0) - (h_0 + P_0)}{e} = k s\\frac{(e + h)}{e}$ for $L = e$ , the height of the column, $A = s$ the cross sectional area of the column and $h_0 = 0$ .","The unidimensional form of Darcy's law, as applied to a column filled with saturated porous media is represented in modern notation in introductory soil physics and groundwater hydrology books as $\\boxed{Q = K_s A\\frac{\\psi _{ti} - \\psi _{to}}{L} = Q = K_s A\\frac{\\Delta \\psi _{t}}{L}}$ for total potential expressed in terms of height of water.","A more precise general representation is in terms of differentials, thus for a one dimensional column oriented in some arbitrary $x$ direction, Darcy's law is $\\boxed{Q = -K_s A\\frac{d \\psi _{t}}{dx}}$ in which $Q$ is the water discharge in L$^3$ T$^{-1}$ , $K_s$ is now defined as the saturated hydraulic conductivity in L T$^{-1}$ and the term $\\frac{d \\psi _{t}}{dx}$ is termed the hydraulic gradient or the potential difference over the length between the two points in which potential is being measured, in L L$^{-1}$ .","The negative sign, source of much confusion in soil physics, means that the flow is opposite to the direction of the gradient.","In other words, the flow is always from the high total potential to the low total potential.","As with the original Darcy formulation, these equations can be expressed in terms of height of water on the inlet and outlet of a saturated column filled with some porous material.","If the column is oriented on the $z$ direction, here defined as in the direction of the gravitational field, we have $Q = -K_s A \\frac{d \\psi _{t}}{dz} = -K_s A \\frac{ (\\psi _{pi} + \\psi _{gi}) - (\\psi _{po} + \\psi _{go}) }{dz}$ If the length of the column is $z$ and establishing the gravitational potential at the bottom of the column at $z = 0$ , we have $Q = -K_s A \\frac{ (\\psi _{pi} + z) - (\\psi _{po} + 0) }{z}$ Representing the pressure potential in terms of height $h_i$ $Q = -K_s A \\frac{ (h_i-h_o) + z}{z} = -K_s A (\\frac{ h_i-h_o}{z} + 1)$ If the pressure potential is zero on the outlet $Q = -K_s A (\\frac{ h_i}{z} + 1)$ It is easy to show that if the column is oriented perpendicular to the gravitational field in some arbitrary $x$ (or $y$ ) direction, Darcy's law in terms of the height of water on the inlet and outlet will be $Q = -K_s A (\\frac{ h_i - h_o}{x})$ For $h_o$ = 0 this reduces to $Q = -K_s A \\frac{ h_i}{x}$ where $x$ (or $y$ ) represent the horizontal length of the column and $z$ the vertical length.","It is clear that Darcy's law is a linear relationship between discharge and gradient with slope equal to the saturated hydraulic conductivity.","Darcy's law, along with Fourier's law of heat conduction, Ohm's law in electromagnetism, Fick's law of molecular diffusion and Poiseuille's law is one of the linear gradient transport laws discovered in the 1800s [84].","Although Darcy only mentions Poiseuille's law in his report, it is very likely that he was aware and was influenced by the physics of the time in creating his empirical law [84].","Darcy's law can be written in terms of a flux density $q$ , similar to the other laws of physics mentioned, by dividing the discharge $Q$ by the area $A$ $q = \\frac{Q}{A} = -K_s A \\frac{d \\psi _t}{dx}$ Darcy obviously knew about and investigated Poiseuille's, as he was an engineer concerned with water distribution, which would obviously require studying water flow in pipes.","Analyzing Poiseuille's solution for flow in a cylindrical conduit, $Q = -\\frac{\\pi }{8\\mu }\\frac{\\partial p}{\\partial x} a^4$ we can rewrite this equation by reorganizing it $Q = -\\frac{\\pi a^2 a^2}{8\\mu }\\frac{\\partial p}{\\partial x}$ and introducing $A = \\pi a^2$ as the area of the tube $Q = - A\\frac{\\pi a^2}{8\\mu }\\frac{\\partial p}{\\partial x}$ Rewriting the term $\\frac{\\pi a^2}{8\\mu }$ as some proportionality constant $k^*$ $Q = - A k^*\\frac{\\partial p}{\\partial x}$ Poiseulle's law can be interpreted as some limiting case of Darcy's law for when there is no porous media within the pipe.","The presence of a porous medium causes additional stresses in the fluid that decrease the discharge under the same potential gradient.","Because natural porous media usually have complex grain size distribution, grain shape and rugosity, resulting in complex pore size distribution and geometry, the empirical constant in Darcy's law, $K_s$ , has to be determined experimentally, there is no analytical solution of Darcy's law from the Navier-Stokes equation.","There are a few studies that use stochastic considerations to derive Darcy's law from the Navier-Stokes equations, but as discussed previously, they must rely in some averaging mechanism, unless the porous media model is extremelly simplistic.","In a stratified media, such as a soil profile, sedimentary rock or deposit, or laboratory column, where the direction of flux is perpendicular to the layering, the flow will be determined by some form of averaging of the individual hydraulic conductivity of the layers Here as a direct average, in a much more simplistic way than what was discussed previously for averaging of the Navier-Stokes equation at the pore scale.","Consider a composite column, composed of two independently homogeneous but different materials $A$ and $B$ with heights $x_A$ and $x_B$ (REF ).","The flux is in the $z$ direction and the total potential is measured on the points $a$ , $b$ and $c$ as $\\psi _{ta}$ , $\\psi _{tb}$ and $\\psi _{tc}$ .","Darcy's law is defined for material $A$ in terms of the flux between points $a$ and $b$ $q_{ab} = -K_{sA} \\frac{\\psi _{tA} - \\psi _{tB}}{x_A}$ and for material $B$ in terms of the flux between points $b$ and $c$ $q_{bc} = -K_{sB} \\frac{\\psi _{tB} - \\psi _{tC}}{x_B}$ Figure: Darcy's law in a composite column with individually homogeneous but different materials AA and BB.As point $b$ is located in the interface between the two materials, the total potential is equal and we can solve both equations for $\\psi _{tB}$ .","$\\psi _{tB} = \\psi _{tA} + \\frac{q_{ab}x_A}{ K_{sA}}$ $\\psi _{tB} = \\psi _{tC} - \\frac{q_{bc}x_B}{ K_{sB}}$ Replacing $\\psi _{tB}$ in any of the equations and solving for the total potential difference between the column's inlet in $a$ and the outlet in $c$ $\\psi _{tA} - \\psi _{tC} = - (\\frac{q_{bc}x_B}{ K_{sB}} + \\frac{q_{ab}x_A}{ K_{sA}})$ For constant (steady-state) saturated flow, for which Darcy's law is being considered, the flux in any point within the length of the composite column is constant, and thus $q_{ab} = q_{bc} = q_{ac} = q$ , and $\\psi _{tA} - \\psi _{tC} = - q(\\frac{x_B}{ K_{sB}} + \\frac{x_A}{ K_{sA}})$ Finally solving for $q$ we arrive at an expression for a composite material $q = -\\frac{ (\\psi _{tA} - \\psi _{tC})}{(\\frac{x_B}{ K_{sB}} + \\frac{x_A}{ K_{sA}})}$ The term $\\frac{1}{(\\frac{x_B}{ K_{sB}} + \\frac{x_A}{ K_{sA}})} = \\frac{K_{sA}K_{sB}}{x_BK_{sA} + x_AK_{sB}}$ is a form of weighed mean of the hydraulic conductivity of the layers with the height of each layer acting as a weighing term.","In practical terms, especially in laboratory, saturated hydraulic conductivity can be determined using constant head and falling head permeametersAs we will see, hydraulic conductivity and permeability are different things, however, these devices historically have been called permeameters.","As they can also be used to measure the permeability of the medium, we will maintain the use of the terminology.","In constant head permeameters, the total potential or hydraulic head is kept constant in both ends of the column.","The hydraulic head must be equal or greater than zero on both ends.","If the total potential is negative the water will be under tension, the water content might be below saturation and hydraulic conductivity is no longer constant.","Saturated hydraulic conductivity is calculated by measuring discharge or flux density as $K_s = -\\frac{QL}{A(h_i - h_o)}$ for a horizontal column, and $K_s = -\\frac{Q}{A[(h_i - h_o)/L +1]}$ for a vertical column with the gravitational reference chosen at the outlet.","Note that the flow can be in any direction including upwards, the direction being indicated by the sign of the hydraulic conductivity.","The second common type of laboratory method for measuring saturated hydraulic conductivity are falling head permeameters.","In the falling head method the hydraulic head is not constant but falls from a height $h_0$ to a height $h$ in time.","Falling head permeameters are convenient when high hydraulic heads are required [10].","To derive the expression for measuring saturated hydraulic conductivity, consider a pemeameter connected to a tube filled with water on its base (Figure REF ).","The flow is upwards and the discharge is measured at the outlet on the water level on the upper part of the column.","The tube has diameter $d_t$ , radius $r_t$ and area $A_t$ and the column filled with porous media has diameter $d_c$ , radius $r_c$ and area $A_c$ .","The inflow on the smaller tube is given by [37] $Q_{in} = -A_t \\frac{dh}{dt}$ while on the column the flow obeys Darcy's law $Q_{out} = \\frac{K A_c h}{L}$ Under saturated flow conditions the discharge into the column has to be equal to the discharge out of the column $Q_{in} = Q_{out}$ $-A_t \\frac{dh}{dt} = \\frac{K A_c h}{L}$ Reorganizing the terms and integrating $\\frac{dh}{h} = - \\frac{A_c K}{A_t L} dt$ $\\int \\frac{dh}{h} = - \\int \\frac{A_c K}{A_t L} dt$ $\\log {|h|} \\bigg |_{h}^{h_{0}} = - \\frac{A_c K}{A_t L} t \\bigg |_{t_{0}}^t$ By applying the boundary conditions $t = 0$ at $h = h_0$ , and considering $h$ as always positive we arrive at $\\log {h} - \\log {h_{0}} = - \\frac{A_c K}{A_t L} t$ Solving for $K$ and considering the area of the small tube as $A_t = \\pi r_t^2 $ and the area of the soil column as $A_t = \\pi r_t^2$ Remember that $\\log x - \\log y = \\log {x/y}$ and notice that we multiplied both sides by $-1$ to eliminate the negative sign in $K$ .","$K = \\frac{r_t^2 L}{r_c^2 t} \\log \\frac{h_0}{h}$ or, in terms of diameter $K = \\frac{d_t^2 L}{d_c^2 t} \\log \\frac{h_0}{h}$ For alternative methods and equipment for measuring saturated hydraulic conductivity on the field and on the laboratory, the user is referred to [31].","Figure: Diagram illustrating a generic the falling head permeameter.Up until now we have avoided dealing with vectors by working with flow in one dimension.","A column filled with a porous medium saturated with water, where the diameter of the column is small and where the walls are impermeable can be treated as an unidimensional system and the flux density vector is oriented in the general direction of water flow.","In soils and sedimentary deposits, aquifers, porous rocks and many other natural materials, flow is often not constrained to one dimension.","This is especially true on the field, where water and contaminant transport occur in large scales and in response to total potential gradients, but also in laboratory, for large enough samples and when heterogeneity cannot be disregarded.","Under such conditions, not only the flux density is a vector but the gradient is also a vector, and often hydraulic conductivity is a second order tensor.","Recalling Chapter REF , Darcy's law can be generalized as $\\mathbf {q} = \\mathbf {K} \\cdot \\nabla \\psi _t$ In three-dimensional Cartesian coordinates, the flux density is decomposed as $\\mathbf {q} = q_x \\mathbf {i} + q_y \\mathbf {j} + q_x \\mathbf {k}$ while the three-dimensional gradient is $\\nabla = \\frac{\\partial \\psi _t}{\\partial x} \\mathbf {i} + \\frac{\\partial \\psi _t}{\\partial y} \\mathbf {j} + \\frac{\\partial \\psi _t}{\\partial z} \\mathbf {k}$ Under conditions of anisotropy, where hydraulic conductivity varies with direction in the material, saturated hydraulic conductivity is a second rank tensor with $3^2 = 9$ elements $\\mathbf {K} = K_{ij} =\\begin{pmatrix}K_{xx} & K_{xy} & K_{xz}\\\\K_{yx} & K_{yy} & K_{yz}\\\\K_{zx} & K_{zy} & K_{zz}\\end{pmatrix}$ With the conductivity tensor, Darcy's law can be written as $\\begin{pmatrix}q_{x}\\\\q_{y} \\\\q_{z}\\end{pmatrix}=\\begin{pmatrix}K_{xx} & K_{xy} & K_{xz}\\\\K_{yx} & K_{yy} & K_{yz}\\\\K_{zx} & K_{zy} & K_{zz}\\end{pmatrix}\\cdot \\begin{pmatrix}\\partial \\psi _t/\\partial x\\\\\\partial \\psi _t/\\partial y \\\\\\partial \\psi _t/\\partial z\\end{pmatrix}$ Therefore, the components of flux density in $x$ , $y$ and $z$ are $q_x = K_{xx} \\frac{\\partial \\psi _t}{\\partial x} + K_{xy} \\frac{\\partial \\psi _t}{\\partial y} + K_{xz}\\frac{\\partial \\psi _t}{\\partial z}$ $q_y = K_{yx} \\frac{\\partial \\psi _t}{\\partial x} + K_{yy} \\frac{\\partial \\psi _t}{\\partial y} + K_{yz}\\frac{\\partial \\psi _t}{\\partial z}$ $q_z = K_{zx} \\frac{\\partial \\psi _t}{\\partial x} + K_{zy} \\frac{\\partial \\psi _t}{\\partial y} + K_{zz}\\frac{\\partial \\psi _t}{\\partial z}$ The hydraulic conductivity tensor is a symmetrical tensor, meaning that $K_{ij} = K_{ji}$ such that the tensor reduces to six different components.","There are profound implications related to these equations.","The fact that the conductivity tensor is symmetrical is by no means a trivial issue and is a property of most of the transport equations and related to the theory know as the Onsager reciprocal relations, in the realm of non-equilibrium thermodynamics.","Another important property of the conductivity tensor is that it is assumed as orthotropic, meaning that there are three principal directions for the conductivity vectors which are orthogonal to eachother.","If this condition is valid, it is possible to orientate the axis in such a way that the conductivity tensor is reduced to three principal directions, such that $K_ij = 0$ for $i \\ne j$ and $K_{ii}$ for $i = j$ $\\mathbf {K} = K_{ij} =\\begin{pmatrix}K_{xx} & 0 & 0\\\\0 & K_{yy} & 0\\\\0 & 0 & K_{zz}\\end{pmatrix}$ In practice, this means that it is possible to orientate a sample of an anisotropic media in laboratory conditions such that only the three principal permeability directions need to be measured.","If the medium is isotropic, there is no difference between $K_ii$ measured in any direction and $K$ is reduced to the constant hydraulic conductivity scalar $\\mathbf {K} = K_{ij} = K\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{pmatrix}$ Suppose now that the saturated hydraulic conductivity does not vary with direction at each point in space, but the value, even being equal in all directions, is different in different points in space.","In this case, the saturated hydraulic conductivity is a scalar function in space $K(x,y,z)$ and Darcy's law is $\\mathbf {q} = -K(x,y,z) \\nabla \\psi _t$ A medium where the hydraulic conductivity does not vary with direction at a given point in space is said to be isotropic and if the scalar hydraulic conductivity does vary in space is called heterogeneous.","If the saturated hydraulic conductivity does not vary with direction and is the same in all points in space the medium is said to be homogenous and isotropic and Darcy's law simplifies to $\\mathbf {q} = -K \\nabla \\psi _t$ A consistent representation of Darcy's law requires that the potentials are expressed in terms of hydraulic height [L].","This guarantees that the gradient $\\frac{\\partial \\psi _t}{\\psi x}$ or $y$ or $z$ is nondimentional [L L$^{-1}]$ and that the flux density has the same units of the saturated hydraulic conductivity [L T$^{-1}$ ].","It is convenient now to fully introduce the notation used before for the total potential in height units as $h_t = h_p + h_g$ in which $h_t$ is the total potential, $h_p$ is the pressure potential and $h_g$ is the gravitational potential, all in [L] units.","Using this notation and considering the $z$ direction as the direction of the gravitational force, the components of Darcy's law in $x$ , $y$ and $z$ are $q_x = -K \\frac{\\partial h_t}{\\partial x} = -K \\frac{h_{ti} - h_{to}}{x - x_0} = -K \\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{x - x_0}$ $q_y = -K \\frac{\\partial h_t}{\\partial y} = -K \\frac{h_{ti} - h_{to}}{y - y_0} = -K \\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{y - y_0}$ $q_z = -K \\frac{\\partial h_t}{\\partial y} = -K \\frac{h_{ti} - h_{to}}{z - z_0} = -K \\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{z - z_0}$ where the subscripts $i$ and $o$ stand for inlet and outlet.","Setting the gravitational reference at $x_0$ , $y_0$ and $z_0$ , on the outlet, the gravitational potential will be $z_0$ in both the inlet and outlet in $x$ and $y$ and in the outlet only in $z$ .","If the column height is $z$ , and the inlet is located in $z$ this will be the gravitational potential in the inlet for the vertical column.","Using these conditions, the components of Darcy's law in three dimensions are $q_x = -K \\frac{(h_{pi} + z_0) - (h_{po} + z_0)}{x - x_0} = \\frac{\\partial h_{p}}{\\partial x}$ $q_y = -K \\frac{(h_{pi} + z_0) - (h_{po} + z_0)}{y - y_0} = \\frac{\\partial h_{p}}{\\partial y}$ $q_z = -K \\frac{(h_{pi} + z) - (h_{po} + z_0)}{z - z_0} = -K \\frac{(h_{pi} - h_{po}) + (z - z_0)}{z - z_0} = -K (\\frac{\\partial h_{p} + \\partial z}{\\partial z})$ $q_z = -K (\\frac{\\partial h_{p}}{\\partial z} + 1)$ Suppose now the total potential is expressed in pressure units.","Using the $z$ coordinate for convenience $q_z = -K \\frac{\\partial \\psi _t}{\\partial z}$ it becomes clear that the the gradient is no longer dimensionless and the equation is not consistent in terms of units.","Expressing the total potential in terms of the sum of the potentials $q_z = -K \\frac{\\partial (\\psi _p + \\psi _g)}{\\partial z} = -K \\frac{\\partial (\\rho _w g h + \\rho _w g z)}{\\partial z}$ and using the definitions from Chapter REF , it becomes clear that for consistence of units to be achieved, the saturated hydraulic conductivity would have to be divided by $\\rho _w g$ $q_z = -\\frac{K}{\\rho _w g} \\frac{\\partial (\\rho _w g h + \\rho _w g z)}{\\partial z}$ Therefore, for a liquid of constant density, Darcy's law in its original form is retrieved by placing the constant terms outside of the derivative $q_z = -\\frac{K \\rho _w g}{\\rho _w g} \\frac{\\partial (h + z)}{\\partial z} = -K \\frac{\\partial (h + z)}{\\partial z}$ Although stated otherwise in his report, the saturated hydraulic conductivity as defined by Darcy in his original experiment is a composite property of the solid and liquid phases [69].","Although this is convenient in soil physics where the phases concerned are usually soil and water, it might not be convenient in groundwater hydrology and petroleum engineering, where different and often multiple phases are concerned and where the characteristics of the porous material might be of concern.","Suppose you define a saturated conductivity term for water using a column experiment.","You could repeat the same experiment with other fluids, for example oil, salt solution, gasoline, and for each fluid you would obtain one saturated conductivity.","To circumvent this issue, and using dimensional analysis, the saturated conductivity term was rewritten in terms of the properties of the fluid being used, namely viscosity and density and a new coefficient, called intrinsic permeability or simply permeability $\\kappa $ , which depends on the solid phase alone, thus [73], [101], [69] $K = \\frac{\\kappa \\rho _f g}{\\mu }$ In which $\\rho _f$ and $\\mu $ are the density and viscosity of a fluid moving through the porous materialThese laws should be valid for viscous, incompressible newtonian fluids in general.","Thus, Darcy's law can be expressed in terms of pressure using the permeability as $q_z = -\\frac{\\kappa \\rho _f g}{\\mu } \\frac{1}{\\rho _f g} \\frac{\\partial (\\rho _f g h + \\rho _f g z)}{\\partial z} = -\\frac{\\kappa }{\\mu } \\frac{\\partial (\\rho _f g h + \\rho _f g z)}{\\partial z}$ and in terms of length for the potential as $q_z = -\\frac{\\kappa \\rho _f g}{\\mu } \\frac{\\partial ( h + z)}{\\partial z}$ For an arbitrary $x$ direction perpendicular to Earth's gravitational field these reduce to $q_x = - \\frac{\\kappa }{\\mu } \\frac{\\partial (\\rho _f g h)}{\\partial z} = - \\frac{\\kappa }{\\mu } \\frac{\\partial P}{\\partial z}$ and $q_x = -\\frac{\\kappa \\rho _f g}{\\mu } \\frac{\\partial h}{\\partial x}$ Note that the similarity between Darcy's law and Poiseuille's equation is much more evident when the pressure form is considered.","In general, the permeability is a tensor with the same properties of the saturated hydraulic conductivity, i.e.","for an anisotropic medium $\\mathbf {k} = k_{ij} =\\begin{pmatrix}k_{xx} & k_{xy} & k_{xz}\\\\k_{yx} & k_{yy} & k_{yz}\\\\k_{zx} & k_{zy} & k_{zz}\\end{pmatrix}$ which can be orientated in the direction of the principal axes $\\mathbf {k} = k_{ij} =\\begin{pmatrix}k_{xx} & 0 & 0\\\\0 & k_{yy} & 0\\\\0 & 0 & k_{zz}\\end{pmatrix}$ and for an isotropic medium $\\mathbf {k} = k_{ij} = k\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{pmatrix}$ We have seen that the flux density vector in Darcy's law is the discharge over area.","Because the pore geometry is complex, a variety of shapes and sizes of pores are available for flow in porous materials.","The flux density in Darcy's law is not the velocity of flow of fluid in the pores.","Because the flow only takes place within the pores, the average flow velocity is the flux density divided by the porosity.","$v = \\frac{Q}{An} = \\frac{q}{n}$ in which $n$ is the porosity.","This relationship is valid because for an homogeneous porous material, area porosity is equivalent to the volumetric porosity [10].","In other words, considering a cross section of the column, only part of the area is available for flow and for a principle equivalent to Bernoulli's law, the flux velocity at each point will be higher for smaller pores, within the limitations of viscosity and wall effects.","The difference in velocities in a porous material will be an important mechanism in contaminant transport.","Like many other equations and laws in fluid mechanics, Darcy's law is valid for laminar flow.","Laminar flow here meaning the same as we discussed when analyzing the problem of a sphere settling under gravity.","The flow is slow enough that there is no turbulence or vortexes.","Poiseuille's law also has a restriction of laminar flow.","As we seen for the sphere problem, one way of identifying an approximate limit between laminar and turbulent flow is by using the Reynolds number.","For flow in a cylindrical conduit, the Reynolds number is $Re = \\frac{\\rho _f u d_h}{\\mu }$ In which $d_h$ is the hydraulic diameter of the pipe, simple the inner diameter if the tube is circular.","For Darcy's law, this can be approximated by $Re = \\frac{\\rho _f u d^{\\prime }}{\\mu }$ In which $d^{\\prime }$ is some representative length dimension of the porous matrix representing the elementary channels of the porous medium, usually a parameter related to average grain diameter or other representative grain diameters [10] such as the diameter with 10$\\%$ , 30$\\%$ or or 50$\\%$ of the material passing through a sieve of mesh equal to the diameter of interest.","Darcy's law implies a linear relationship between flux density and hydraulic gradient.","The experimentally determined $R_e$ values for the transition zone from laminar to turbulent flow for Darcy's law ranges from 1 to 10, with laminar flow for values less than 1, turbulent flow for values above 10 [10].","For values above 10 the relationship between flux density and gradient is no longer linear and Darcy's law cannot be used.","As the flow velocity increases, inertia terms are no longer negligible.","At the transition zone, the Darcy-Forchheimer equation can be used, presented here in one dimension as [86].","$-\\frac{\\partial p}{\\partial x} = \\frac{\\mu }{\\kappa }q + \\beta \\rho _f q^2$ In which $\\beta $ is the Forchheimer coefficient or the inertial permeability.","Note the quadratic term on the flux density.","If the flow is too slow, for very small gradients and for some very fine soils and sediments, surface forces (Chapter REF ) can become important as well as viscous effects.","It has been speculated that under such conditions water can display non-newtonian behavior and non-darcyan flow is observed [53], [10].","Saturated hydraulic conductivity is a function of total porosity and pore size distribution.","As such it is a function of bulk density and grain size distribution.","Compaction and consolidations will decrease the total porosity and will tend to reduce saturated hydraulic conductivity.","In soil science saturated hydraulic conductivity can be used as an indicator of soil degradation.","Grain size distribution controls not only pore size distribution but also influences bulk density.","Coarse grained materials can have lower total porosity but the average pore diameter will be greater, resulting in larger values of saturated hydraulic conductivity.","For soils rich in 2:1 layer silicates, high clay content will result in lower saturated hydraulic conductivity, because of the smaller average pore radius and in some cases because of expansion of the silicate clays.","For soils rich in iron and aluminum oxides, the saturated hydraulic conductivity is often very high, due to the microaggregate structure with two pore domains, micropores and mesopores, and macropores.","Under saturated conditions the bulk of the water transport will take part in macropores.","In sediments and rocks, grain size distribution and degree of consolidation will control saturated hydraulic conductivity.","In highly consolidated igneous and metamorphic rocks, water transport can occur in fractures and hydraulic conductivity can be highly directional.","Saturated hydraulic conductivity in the field is not a normally distributed variable and this should be accounted for when analyzing data.","Bear [10] reports saturated hydraulic conductivity values ranging from $10^{2}~cm~s^{-1}$ for clean gravel to $10^{-11}~cm~s^{-1}$ for impervious rocks.","The same source reports values ranging from $10^{-3}~cm~s^{-1}$ for sandy soils to $10^{-11}~cm~s^{-1}$ for dispersive clay soils.","Because of the relationship of saturated hydraulic conductivity with porous media parameters, various attempts have been made to derive equations to calculate the saturated hydraulic conductivity from physical properties of the material.","These models vary from physically based but approximated models based on Poiuseuille's equation to purely empirical models based on statistical regression.","One model commonly found in engineering and hydrogeology is the Kozeny-Carman equationDerivations based on the original sources should be provided in later editions $k = \\frac{n^3}{(1-n)^2} \\frac{c_0}{M_s^2}$ in which $\\kappa $ is the permeability, $M_s$ the the specific surface area with respect to unit volume of porous medium, $n$ is the porosity and $c_0$ is the Kozeny's constant, dependent on the geometrical form of the channels on the model [10].","The term $f(n) = \\frac{n^3}{(1-n)^2}$ is the porosity factor.","It has been found that $c_0 = 1/5$ resulted good agreement with measured data, resulting in $k = \\frac{n^3}{(1-n)^2} \\frac{1}{5M_s^2}$ Defining the mean particle size as $d_m = M_s/6$ , the Kozeny-Carman equation can be written in terms mean of particle diameter as $k = \\frac{n^3}{(1-n)^2} \\frac{d_m^2}{180}$ In Chapter REF we derived a mass conservation equation (continuity equation) of the form $\\frac{\\partial \\rho }{\\partial t} = - \\nabla \\cdot (\\rho \\mathbf {u})$ Suppose that water is being accumulated in the pore space of the representative volume of interest.","If the media is completely saturated and the density of water is considered constant, $\\frac{\\partial \\rho }{\\partial t} = 0$ and $\\nabla \\cdot (\\rho \\mathbf {u}) = \\rho \\nabla \\cdot \\mathbf {u}$ and, as before, we arrive at $\\nabla \\cdot \\mathbf {u} = 0$ For flow in porous media, the velocity term is replaced by the flux density $\\nabla \\cdot \\mathbf {q} = 0$ We already know that the flux density is represented by Darcy's law, thus, expanding the components of flux and replacing Darcy's law $(\\frac{\\partial }{\\partial x} \\mathbf {i} + \\frac{\\partial }{\\partial y} \\mathbf {j} + \\frac{\\partial }{\\partial y} \\mathbf {k}) \\cdot (q_x \\mathbf {i} + q_y \\mathbf {j} + q_z \\mathbf {k}) = 0\\\\(\\frac{\\partial q_x}{\\partial x} + \\frac{\\partial q_y}{\\partial y} + \\frac{\\partial q_z}{\\partial z}) = 0 \\\\(\\frac{\\partial }{\\partial x}(-K_x \\frac{\\partial h}{\\partial x}) + \\frac{\\partial }{\\partial y}(-K_y \\frac{\\partial h}{\\partial y}) + \\frac{\\partial }{\\partial z} (-K_z \\frac{\\partial h}{\\partial z}) ) = 0\\\\$ dropping the parentheses and multiplying both sides by $-1$ results in $\\frac{\\partial }{\\partial x}(K_x \\frac{\\partial h}{\\partial x}) + \\frac{\\partial }{\\partial y}(K_y \\frac{\\partial h}{\\partial y}) + \\frac{\\partial }{\\partial z} (K_z \\frac{\\partial h}{\\partial z}) ) = 0$ This equation is valid for a porous media where the unsaturated hydraulic conductivity cannot be assumed as constant.","Because for a truly anisotropic media, the saturated hydraulic conductivity is a second rank tensor, a more general representation would be $\\nabla \\cdot (\\mathbf {K} \\cdot \\nabla {h}) = 0$ Back to more general cases, if the saturated hydraulic conductivities in each direction does not vary with the position in space, but the conductivities in each direction are different, we have $ K_x \\ne K_y \\ne K_z $ all constant, and $K_x \\frac{\\partial h^2}{\\partial ^2 x} + K_y \\frac{\\partial ^2 h}{\\partial y^2} + K_z \\frac{\\partial ^2 h}{\\partial z^2} = 0$ Finally, if the saturated hydraulic conductivity is constant everywhere, independent of direction, such that $ K_x \\ne K_y \\ne K_z = K $ we have $K \\frac{\\partial h^2}{\\partial ^2 x} + K \\frac{\\partial ^2 h}{\\partial y^2} + K \\frac{\\partial ^2 h}{\\partial z^2} = 0$ and dividing both sides by $K$ we arrive at $\\boxed{\\frac{\\partial h^2}{\\partial ^2 x} + \\frac{\\partial ^2 h}{\\partial y^2} + \\frac{\\partial ^2 h}{\\partial z^2} = 0}$ This is a partial differential equation know as Laplace equation.","It is ubiquitous in physics and mathematics and can be represented using the aptly named Laplace operator ($\\nabla ^2$ ), of which we have made extensive usage in this text $\\boxed{\\nabla ^2 h = 0}$ Laplace's equation is a second-order partial differential equation that appears in electrostatics, gravitation, head conduction, transport phenomena and many other problems in applied physics, mathematics and engineering.","As discussed previously, there is vast amount of literature covering methods of solution of partial differential equations, the user being encourage to consult the references.","Analytical solutions in three and two dimensions can be found in the literature.","As an example we will show how Laplace's equation can be solved in one-dimension to find the distribution of $h$ along a length of saturated porous media.","Let us consider flow in the $x$ directionTo avoid any complication with implicit gravitational field, such that h is constant in $y$ and $z$ and the partial derivatives are zero $\\frac{\\partial h^2}{\\partial ^2 x} = 0 \\\\\\int \\partial h^2 = \\int 0 \\partial ^2 x \\\\\\int \\partial h = \\int C_1 \\partial x \\\\$ $h = C_1 x + C_2 \\\\$ To find the values of the constants one just needs to repeat the same procedure of applying the boundary conditions.","Supposing, for example that $h = h_1$ at $x = 0$ and $h = h_2$ at $x = h$ , we have $h = \\frac{(h_2 - h_1)}{h} x + h_1 \\\\$ This shows that $h$ varies linearly along the length of the $x$ gradient.","Laplace's equation solutions in 2D and 3D are highly dependent on the geometry of the problem, and can be challenging.","For example, A general solution of Laplace's equation in a square geometry with boundary conditions with $h = 0$ in all sides except the right vertical wall will have the form $h(x, y) = \\sum _{n = 1}^\\infty c_n \\sinh (\\frac{n \\pi x}{b}) sin(\\frac{n \\pi y}{b})$ with $c_n = \\frac{2}{b \\sinh (\\frac{n \\pi a}{b})} \\int _0^b g_2(y) sin(\\frac{n \\pi y}{b}) dy$ in which $g_2(y)$ is a function that defines the nonzero boundary conditionhttp://www.math.ubc.ca/~peirce/M257_316_2012_Lecture_24.pdf.","ll $ p $ Pressure $ \\mu $ Viscosity $ \\mathbf {u} $ Vector velocity function $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k} $ Unit vectors in Cartesian coordinates $ x, y, z$ Generalized Cartesian coordinates $ C_1, C_2 $ Constants $ h $ Distance between plates $ Q $ Discharge $ r $ Radial dimension $ \\phi $ Angle in polar coordinates $ q $ Water flux $ h^{\\prime } $ Height of water above a given point (hydraulic head) $ A, s $ Cross section area $ L, e $ Length $ \\psi $ Water potential $ P $ Pressure $ k , K_s $ Saturated hydraulic conductivity $ d_t $ Tube diameter $ r_t $ Tube radius $ A_t $ Tube cross sectional area $ d_c $ Column diameter $ r_c $ Column radius $ A_c $ Column cross sectional area $ t $ Time $ \\mathbf {q} $ Flux density vector $ \\mathbf {K} $ Saturated hydraulic conductivity tensor, vector form $ q_x, q_y, q_z $ Components of the flux density vector in Cartesian coordinates $ K_{ij} $ Saturated hydraulic conductivity tensor $ \\rho _w $ Water density $ \\rho _f $ Fluid density $ g $ Earth's gravitational acceleration $ k $ Intrinsic permeability $ \\mathbf {k} $ Permeability tensor, vector form $ k_{ij} $ Permeability tensor $ n $ Porosity $ Re $ Reynolds number $ \\beta $ Forchheimer coefficient $ M_s $ Specific surface area $ c_0 $ Kozeny's constant $ d_m $ Mean particle size $ \\rho $ Generalized density Edgar Buckingham, one of the forefathers of soil physics, wrote a 61 page report to the U.S. Department of agriculture which set the foundations of what today is the theory of unsaturated flow in soils [22].","Using an analogy to other physical transport laws, he postulated that the capillary current density (equivalent to $q$ in modern notation) is proportional to the capillary potential gradient.","Adapting to the notation used in this book and in one dimension His $\\lambda $ was replaced with $\\beta $ as we will need the former later $q = \\beta \\frac{\\partial \\psi _c}{\\partial x}$ Based on his observations and theoretical analysis he concluded that the capillary conductivity, $\\beta $ , was not constant and was a function of water content $\\beta = f(\\theta )$ .","His conductivity law can be understood as a generalization of Darcy's law to unsaturated porous media, thus, writing the total potential in terms of hydraulic head, $h$ , and $\\beta $ as an unsaturated hydraulic conductivity, which is no longer constant but dependent on water content, Darcy's law can be written, for the one dimensional case as $\\boxed{q = K(\\theta ) \\frac{\\partial h}{\\partial x}}$ This is the one dimensional Darcy-Buckingham equation.","Contrary to the saturated case, the unsaturated hydraulic conductivity is an empirical function of water content, and not a constant which can be determined experimentally.","There has been a lot of effort in developing these empirical functions, with a lot of the fundamental equations being derived in the XX century.","Many of the water retention functions presented in Chapter REF were essentially derived to arrive at an unsaturated hydraulic conductivity function.","One of the fundamental differences between Darcy's law and the Darcy-Buckingham equation is that in all, or at least in some part of the porous media, the water content is below saturation.","This implies that the pressure potential is zero and the matric potential is the component of interest along with the gravitational potential in the total potential.","Instead of applying pressure, represented by a height of water column on the inlet and outlet of the column filled with soil or other porous media of interest, the outlet and inlet can be under tension or suction, by placing the water level below the height at the point of interest.","Generally speaking, the pressure potential can be positive on the inlet and negative on the outlet which causes the column to be partially saturated in part of its extension and saturated on the rest.","The most common example, given in many soil physics textbooks is a column which is under tension on both ends (Figure REF ).","Usually, a porous plate is used on both ends of the column, similar to the system on a tensiometer.","Figure: Unsaturated horizontal column.Flow is from point A from point B (right to left).","By fixing the gravitational reference ($z = 0$ ) at the center of the column, where the inlet and outlet tubes are located, the matric potential is equal to the distance between the center of the column and the water level on the recipients connected to the inlet and outlet.","The recipient connected to the outlet is fed from a water source to guarantee that the level of water is constant.","Another way of achieving this, which is commonly used in laboratory, is to use a Mariotte's bottle.","The scheme illustrated in Figure REF is valid as long as the column's height is small enough not to create significant gravitational potentials on the $z$ direction, this principle also being valid for saturated columns.","The problem is critical in unsaturated soils because gravitational potential gradients can induce total potential gradients on the $z$ direction which induce hydraulic conductivity and velocity gradients.","The gradient on the $x$ direction follows the decrease in water content along the column.","For simple illustrative examples, if the gradient small and the column can be assumed as small enough on the $x$ length, the unsaturated hydraulic conductivity can be approximated as the average between the inlet and outlet.","In real world applications an unsaturated hydraulic conductivity function is necessary.","Because unsaturated hydraulic conductivity was theorized to be a function of water content and because water content has been experimentally shown to be a function of matric potential, hydraulic conductivity can be expressed as a function of matric potential $\\boxed{q = -K(h_m) \\frac{\\partial h}{\\partial x}}$ in which $h_m = \\psi _m$ is, as before, the matric potential expressed in height units.","Expanding the total potential, we arrive at similar equations as for the saturated hydraulic conductivity in a general horizontal $x$ or $y$ direction $\\boxed{q = -K(h_m) \\frac{\\partial h_m}{\\partial x}}$ $\\boxed{q = -K(h_m) \\frac{\\partial h_m}{\\partial y}}$ and on the $z$ direction, parallel to the gravitational field $\\boxed{q = -K(h_m) (\\frac{\\partial h_m}{\\partial z} + 1)}$ Applying the Darcy-Buckingham equation to Figure REF , and expressing the total potential as $h_t$ for clarity $q = -K(h_m) (\\frac{h_{t,A} - h_{t,B}}{x_A - x_B}) = -K(h_m) (\\frac{(h_{m,A}-h_{g,A}) - (h_{m,B}-h_{g,B})}{x_A - x_B})$ Because $h_{g,A} = h_{g,B} = 0$ for a horizontal column and making $x_A - x_B = L$ the length of the column $q = -K(h_m) (\\frac{(h_{A}-0) - (h_{B}-0)}{L}) = -K(h_m) (\\frac{h_{A} - h_{B}}{L})$ where $h_A$ and $h_B$ are the distances between the center of the column, where the inlet and outlet are located, and the respective water levels at the reservoirs.","We now know the transport equation for unsaturated porous media.","The relationship indicated by the Darcy-Buckingham equation is physically and mathematically more complex than Darcy's law because introduces a hitherto unknown relationship between unsaturated hydraulic conductivity and water content or matric potential.","Early experimental investigations indicated a nonlinear relationship between unsaturated hydraulic conductivity and water content or matric potential.","As we have already indicated, unsaturated hydraulic conductivity functions have been derived from the water retention curve function.","The unsaturated hydraulic conductivity functions can be derived from the water retention function using the frameworks proposed by Burdine [23], [19], [97] Remember from the previous chapter that $\\Theta $ is the relative saturation equal to $(\\theta - \\theta _r)/(\\theta _s - \\theta _r)$ $K_r(\\Theta ) = \\Theta ^2 \\int _0^{\\Theta } \\frac{d \\Theta }{\\psi _m^2(\\Theta )}\\bigg / \\int _0^1 \\frac{d \\Theta }{\\psi _m^2(\\Theta )}$ and Mualem [67], [97] $K_r(\\Theta ) = \\Theta ^{1/2} [\\int _0^{\\Theta } \\frac{d \\Theta }{\\psi _m(\\Theta )}\\bigg / \\int _0^1 \\frac{d \\Theta }{\\psi _m(\\Theta )}]^2$ Details about the derivation of these formulas can be found in [23], [19], [67].","They are essentially based on pore size distribution considerations.","The symbology used in these equations is the same as the one used in the previous chapter.","Note that we are using $\\psi _m$ here so not to avoid being constrained by units, since these equations do not necessarily require matric potential in height units.","The new term $K_r$ is the relative unsaturated hydraulic conductivity $K_r(\\Theta ) = \\frac{K(\\Theta )}{K_s}$ varying from 0 to 1, and in which $K_s $ is the saturated hydraulic conductivity.","To obtain the unsaturated hydraulic conductivity function it is necessary to solve the water retention function for $\\psi _m$ , insert it on the Burdine or Mualem equations and solve the integrals.","For the Brooks and Corey equation [19], the process is straightforward.","Expressing the Brooks and Corey in terms of the relative saturation $\\Theta $ $\\Theta = (\\frac{\\psi _b}{\\psi _m})^{\\lambda }$ we first solve for the matric potential $\\psi _m = (\\frac{\\psi _b}{\\Theta ^{1/\\lambda }})$ then plug into the Burdine equation $K_r(\\Theta ) = \\Theta ^2 \\int _0^{\\Theta } \\frac{d \\Theta }{(\\frac{\\psi _b}{\\Theta ^{1/\\lambda }})^2}\\bigg / \\int _0^1 \\frac{d \\Theta }{(\\frac{\\psi _b}{\\Theta ^{1/\\lambda }})^2}\\\\= \\Theta ^2 \\int _0^{\\Theta } \\frac{\\Theta ^{2/\\lambda } d \\Theta }{\\psi _b^2} \\bigg / \\int _0^1 \\frac{\\Theta ^{2/\\lambda } d \\Theta }{\\psi _b^2}$ because $\\psi _b$ is constant $K_r(\\Theta ) = \\Theta ^2 \\int _0^{\\Theta } \\Theta ^{2/\\lambda } d \\Theta \\bigg / \\int _0^1 \\Theta ^{2/\\lambda } d \\Theta $ The solution of the integral form in the numerator and denominator is trivial, i.e.","$\\int _a^b x^{2/c} dx = \\frac{x^{2/c+1}}{2/c+1} \\bigg |_a^b$ thus $K_r(\\Theta ) = \\Theta ^2 \\frac{[\\Theta ^{2/\\lambda +1}]_0^{\\Theta }}{[\\Theta ^{2/\\lambda +1}]_0^1}$ Applying the integration limits $K_r(\\Theta ) = \\Theta ^2 \\frac{\\Theta ^{2/\\lambda +1}}{1} = \\Theta ^2 \\Theta ^{2/\\lambda +1}$ or $K_r(\\Theta ) = \\Theta ^{2/\\lambda +3}$ Recalling that $\\Theta $ is the effective saturation and $K_r$ is the relative unsaturated conductivity, the equation can be written in terms of volumetric water content and unsaturated hydraulic conductivity $K(\\theta ) = K_s (\\frac{\\theta - \\theta _r}{\\theta _s - \\theta _r})^{2/\\lambda +3}$ To retrieve the matric potential form of unsaturated hydraulic conductivity just replace the original Brooks and Corey function for $\\Theta $ $K(\\psi _m) = K_s (\\frac{\\psi _b}{\\psi _m})^{2 +3 \\lambda }$ These last two equations are the Brooks and Corey unsaturated hydraulic conductivity equations derived using Burdine's approach.","The formulas can also be derived using Mualem's formulas by the same process.","For Brutsaert-van Genuchten type equations the process is not as straightforward.","Following [97], we can solve the effective saturation form of the water retention function resulting in $\\psi _m = [\\frac{1}{\\alpha } (\\frac{1}{\\Theta ^{1/m}} - 1)^{1/n}]$ Replacing into the Mualem function $K_r(\\Theta ) = \\Theta ^{1/2} [\\int _0^{\\Theta } \\frac{d \\Theta }{[\\frac{1}{\\alpha } (\\frac{1}{\\Theta ^{1/m}} - 1)^{1/n}]}\\bigg / \\int _0^1 \\frac{d \\Theta }{[\\frac{1}{\\alpha } (\\frac{1}{\\Theta ^{1/m}} - 1)^{1/n}]}]^2$ Doing some simple algebraic manipulations and placing the constant $1/\\alpha $ term outside of the integral $K_r(\\Theta ) = \\Theta ^{1/2} [\\int _0^{\\Theta } (\\frac{\\Theta ^{1/m} }{1 - \\Theta ^{1/m}})^{1/n} d \\Theta \\bigg / \\int _0^1 (\\frac{\\Theta ^{1/m} }{1 - \\Theta ^{1/m}})^{1/n} d \\Theta ]^2$ The issue here is the solution of the integral $\\int (\\frac{\\Theta ^{1/m} }{1 - \\Theta ^{1/m}})^{1/n} d \\Theta $ The integral on the denominator is the same equation with different bounds.","This equations does not have a closed form analytical solution.","This means that the solution is in form of special functions, there is no simple expression in which you can plug the constants and obtain results without using numerical methods or approximations to the special functions.","In the last chapter we encountered the complementary error function when dealing with water retention functions.","Other important special functions are the gamma and beta functions and their incomplete counterparts [14].","To find an analytical solution to this equation van Genuchten [97] used manipulations of the incomplete beta function [14].","The first step is apply the transformation $\\Theta = y^m \\\\d\\Theta = m y^{m-1} dy$ which after replaced into the Equation REF results $\\int _0^{y^m} (\\frac{(y^m)^{1/m} }{1 - (y^m)^{1/m}})^{1/n} m y^{m-1} dy$ simplifying and rearranging $m\\int _0^{y^m} y^{1/n + m -1} (1-y)^{-1/n} dy$ This equation is a form of the incomplete beta function [14].","These functions have a very important role in physics and mathematics and have been studied for hundreds of years, dating at least to the time of Euler and Legendre.","There is no closed form analytical solution for the general case, but it is possible to introduce simplifications such that a solution can be found.","The next simplification was to propose that for all integers $k = m-1+1/n$ the integration could be performed directly.","For the particular case $k=0$ $m = 1-1/n$ We already seen this term in the last chapter when studying the Brutsaert-van Genuchten equation.","It was then called the Mualem restriction, the reason why being apparent now.","Applying this restriction to the incomplete beta function representation allows a straightforward solution of the integral.","For simplicity we will replace $n$ rewriting the equation in terms of $m$ , thus $m\\int _0^{y^m} y^{m -m} (1-y)^{m-1} dy = m\\int _0^{y^m} (1-y)^{m-1} dy$ The solution to this equation is very simple and can be found using the $u$ substitution technique studied in introductory calculus books.","Let $u = 1-y \\\\du = -dy$ we have $-m\\int _0^{y^m} u^{m-1} du$ Integrating in $u$ $-m \\frac{u^{m-1+1}}{m-1+1} |_0^{y^m} = u^{m} |_0^{y^m}$ Substituting $y$ back $-(1-y)^{m} |_0^{y^m}$ At the beginning of the derivation we established that $\\Theta = y^m$ , replacing the original variable $-(1-\\Theta ^{1/m})^{m} |_0^\\Theta = 1-(1-\\Theta ^{1/m})^{m}$ The exact same steps apply to the denominator, however the limits of integration result in $-(1-\\Theta ^{1/m})^{m} |_0^1 = 1$ Replacing the solutions of the integrals into the Mualem expression (Equation REF ) $K_r(\\Theta ) = \\Theta ^{1/2} [\\frac{1-(1-\\Theta ^{1/m})^{m} }{1}]^2$ and finally $K_r(\\Theta ) = \\Theta ^{1/2} [1-(1-\\Theta ^{1/m})^{m} ]^2$ with $m = 1-1/n \\\\0 < m < 1$ The van Genuchten-Mualem hydraulic conductivity function can be written in terms of $\\theta $ and $K(\\theta )$ as $K(\\theta ) = K_s (\\frac{\\theta -\\theta _r}{\\theta _s-\\theta _r})^{1/2} [1-(1-(\\frac{\\theta -\\theta _r}{\\theta _s-\\theta _r})^{1/m})^{m} ]^2$ The matric potential form can be retrieved by replacing the Brutsaert-van Genuchten in terms of $\\Theta $ Be careful not to mix the the subscript “m” on the matric potential symbol $\\psi _m$ with the empirical parameter “m” $K_r(\\psi _m) = \\frac{[1-(\\alpha \\psi _m)^{nm}((1+(\\alpha \\psi _m)^n)^{-m}]^2}{[1+(\\alpha \\psi _m)^n]^{m/2}}$ the derivation being left to the reader as an exercise.","The van Genuchten unsaturated hydraulic conductivity functions can also be derived from the Burdine expression and the Brooks and Corey equations can be derived from the Mualem expression.","In fact many new equations have been derived using these expressions.","There are other approaches to derive unsaturated hydraulic conductivity functions.","A comparison of the Brooks and Corey [19] and van Genuchten [97] relative unsaturated conductivity functions is shown in Figure REF for $\\lambda = 0.47$ and $m = 0.77$ .","Figure: Relative unsaturated hydraulic conductivity as a function of effective saturation for the van Genuchten equation with m=0.77m = 0.77 and the Brooks and Corey equation with λ=0.47\\lambda = 0.47.In the previous chapter we derived the Laplace equation for a saturated porous media from the mass conservation equation (continuity equation).","To derive the conservation equation for unsaturated porous media we again start from the generalized mass conservation equation presented in Chapter REF .","$\\frac{\\partial \\rho }{\\partial t} = - \\nabla \\cdot (\\rho \\mathbf {u})$ The original derivation of this equation considered a control volume full of water or other Newtonian viscous fluid.","We retrieved Laplace's equation on the last chapter by replacing the density of the fluid as the density of water $\\rho _w$ and by considering water as an incompressible fluid such that its density is constant resulting in it vanishing from the conservation equation.","Note that for a saturated porous medium, the volume available for water storage is less than the total volume of interest and is equal to the porosity of the medium.","For saturated porous media, assuming the porosity and water density as constant has the same implications as if the media was simply a volume filled with water, the rate of variation of water content with time is always zero.","For unsaturated porous media these assumptions might not be valid as the volume (or mass) of water within the pore space might vary with time.","In this case we are not immediately concerned with the variation of the density of fluid within the volume of interest, but we are concerned with the variation of the concentration of the fluid within the porous media, thus we can rewrite the conservation equation as $\\frac{\\partial c_w}{\\partial t} = - \\nabla \\cdot (\\rho _w \\mathbf {u})$ in which $c_w = m_w/V_t$ is now the water concentration in mass $m_w$ per unit of total volume of the porous medium $V_t$ $\\frac{\\partial \\frac{m_w}{V_t}}{\\partial t} = - \\nabla \\cdot (\\rho _w \\mathbf {u})$ We know that the density of water is $\\rho _w = m_w/V_w$ , thus solving for $m_w$ and replacing into the conservation equation $\\frac{\\partial \\frac{\\rho _w V_w}{V_t}}{\\partial t} = - \\nabla \\cdot (\\rho _w \\mathbf {u})$ Assuming that the density of water is constant, it can be placed outside the derivative and divergence terms $\\rho _w \\frac{\\partial \\frac{V_w}{V_t}}{\\partial t} = - \\rho _w \\nabla \\cdot \\mathbf {u}$ Recalling that $V_w/V_t$ is the volumetric water content $\\theta $ , and assuming that for a porous medium the velocity term $\\mathbf {u}$ can be represented by the flux density $\\mathbf {q}$ $\\frac{\\partial \\theta }{\\partial t} = - \\nabla \\cdot \\mathbf {q}$ With volumetric water content varying with time, this is a partial differential equation that is difficult to solve analytically without certain assumptions and simplifications.","If the water content does not vary with time $\\partial \\theta / \\partial t = 0$ and $\\nabla \\cdot \\mathbf {q} = 0$ Note, however, that many of the simplifying assumptions used for saturated flow are not valid because $K$ and $\\theta $ are no longer constant in space, even under steady state conditions for a partially saturated porous medium.","Before deriving the general equation that governs unsaturated flow in porous media we need to find a general form of the Darcy-Buckingham's equation.","Theoretically, the most general form of Darcy-Buckingham's equation for an anisotropic, heterogeneous porous media would include a vector valued function in space and matric potential for $K$ The tensor form for anisotropic and heterogeneous porous media is speculative, the mathematical implications and the calculations have not being investigated in depth by the author $\\mathbf {q} = -\\mathbf {K}(\\mathbf {r}, \\psi _m) \\cdot \\nabla \\psi _t$ in which $\\mathbf {r} = x~ \\mathbf {i} + y~ \\mathbf {j}+ z~\\mathbf {k}$ For an homogeneous and anisotropic medium, unsaturated hydraulic conductivity does not vary in space, only with direction at a given point.","In this case the components of the conductivity tensor are scalar functions of matric potential alone $\\mathbf {K}(\\psi _m) = -K(\\psi _m)_{ij} =\\begin{pmatrix}K(\\psi )_{xx} & K(\\psi _m)_{xy} & K(\\psi _m)_{xz}\\\\K(\\psi )_{yx} & K(\\psi _m)_{yy} & K(\\psi _m)_{yz}\\\\K(\\psi )_{zx} & K(\\psi _m)_{zy} & K(\\psi _m)_{zz}\\end{pmatrix}$ For an heterogeneous and isotropic system, the unsaturated hydraulic conductivity is no longer a tensor but a scalar function of space and matric potential, such that $K(x,y,z,\\psi _m)$ and $\\mathbf {q} = -K(x, y, z, \\psi _m) \\nabla \\psi _t$ The simplest case is when hydraulic conductivity is a function of matric potential only everywhere $\\mathbf {q} = -K(\\psi _m) \\nabla \\psi _t$ Richards [78] derived an equation governing the fluids in porous media by replacing the Darcy-Buckingham equation into the mass conservation equation for water $\\boxed{\\frac{\\partial \\theta }{\\partial t} = \\nabla \\cdot ( K(\\psi _m) \\nabla \\psi _t )}$ This is known as the Richards or Richardson-Richards equation.","Richardson derived an analogous equations and published it in a small section of a book dealing with numerical methods for weather prediction [79].","The equation derived by Richardson was not derived from Darcy's law but included a “conductance of the channels connecting opposite faces” in a volume of soil.","Whether Richards was aware of Richardson's work is unknown, although some authors believe he wasn't [55] Richards did not cite Richardson in his original work, but it is entirely possible that his ideas could have reached Richards, ideas seem to float through several means within a scientific epoch.","Expansion of the divergence and gradient terms results in $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(\\psi _m) \\frac{\\partial \\psi _t}{\\partial x}] + \\frac{\\partial }{\\partial y} [K(\\psi _m) \\frac{\\partial \\psi _t}{\\partial y}] + \\frac{\\partial }{\\partial z} [K(\\psi _m) \\frac{\\partial \\psi _t}{\\partial z}]$ The unsaturated hydraulic conductivity is not constant and cannot be placed outside the derivative.","Writing in terms of the total potential in height of water $h = h_m + h_g$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) \\frac{\\partial h}{\\partial x}] + \\frac{\\partial }{\\partial y} [K(h_m) \\frac{\\partial h}{\\partial y}] + \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial h}{\\partial z}]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) \\frac{\\partial (h_m + h_g)}{\\partial x}] + \\frac{\\partial }{\\partial y} [K(h_m) \\frac{\\partial (h_m + h_g)}{\\partial y}] + \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial (h_m + h_g)}{\\partial z}]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) (\\frac{\\partial h_m}{\\partial x} + \\frac{\\partial h_g}{\\partial x})] + \\frac{\\partial }{\\partial y} [K(h_m) (\\frac{\\partial h_m}{\\partial y} + \\frac{\\partial h_g}{\\partial y})] + \\frac{\\partial }{\\partial z} [K(h_m) (\\frac{\\partial h_m}{\\partial z} + \\frac{\\partial h_g}{\\partial z})]$ Considering that the gradient of the gravitational potential is zero in $x$ and $y$ , i.e.","the gravitational potential is constant such that $\\partial h_g/\\partial x = 0 $ and $\\partial h_g/\\partial y = 0$ and that in length units the gravitational potential difference in $z$ is equal to $\\partial z$ , Richardson-Richards equations in $x$ , $y$ and $z$ are $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) \\frac{\\partial h_m}{\\partial x} ]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial y} [K(h_m) \\frac{\\partial h_m}{\\partial y}]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [K(h_m) (\\frac{\\partial h_m}{\\partial z} + \\frac{\\partial z}{\\partial z})] = \\frac{\\partial }{\\partial z} [K(h_m) (\\frac{\\partial h_m}{\\partial z} + 1)]$ Richardson-Richards equation is a nonlinear differential equation which has analytical solutions only in very specific cases which might be physically realistic.","One idea to put the equation in a form which could be solved more easily is to include a diffusivity term.","Diffusion type equations have been studied for centuries and before high processing power computers were available, any simplification would help immensely in terms of simplifying an equation to find analytical solutions or to solve it numerically.","Klute [54] noted an artifice used by Buckingham [22] and later by [27], [28] in order to write the Darcy-Buckingham and Richardson-Richards equations in terms of a hydraulic diffusivity term.","First let us rewrite Richardson-Richards equation in the $z$ direction by simple using the distributive property of multiplication and recalling that the unsaturated hydraulic conductivity can be written as a function of volumetric water content $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [K(\\theta ) \\frac{\\partial h_m}{\\partial z}] + \\frac{\\partial K(\\theta ) }{\\partial z}$ Note that in the gradient term $\\frac{\\partial h_m}{\\partial z}$ if we can assume that the matric potential is a unique function of water content [71], then, by the chain rule for an implicit function [14] $\\frac{\\partial h_m}{\\partial z} =\\frac{\\partial h_m}{\\partial \\theta } \\frac{\\partial \\theta }{\\partial z}$ Replacing the expanded differential into the Richardson-Richards equation $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [K(\\theta ) \\frac{\\partial h_m}{\\partial \\theta } \\frac{\\partial \\theta }{\\partial z}]+ \\frac{\\partial K(\\theta ) }{\\partial z}$ The term $D(\\theta ) = K(\\theta ) \\frac{\\partial h_m}{\\partial \\theta }$ is now called the hydraulic diffusivity or soil water diffusivity [28], [54] and the Richardson-Richards equation can be written in water content form as $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [D(\\theta ) \\frac{\\partial \\theta }{\\partial x}] + \\frac{\\partial K(\\theta ) }{\\partial z}$ Conversely, Richardson-Richards equation can also be written in the matric potential form by defining the implicit differential for $\\partial \\theta /\\partial t$ and replacing into the left side of the equation in terms of matric potential [50] $\\frac{\\partial \\theta }{\\partial h_m}\\frac{\\partial h_m}{\\partial t} = \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial h_m}{\\partial z}] + \\frac{\\partial K(h_m) }{\\partial z}$ By defining a new function $C(\\theta ) = \\frac{\\partial \\theta }{\\partial h_m}$ called the hydraulic capacity, we can write the Richardson-Richards equation in terms of matric potential only as $C(h_m)\\frac{\\partial h_m}{\\partial t} = \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial h_m}{\\partial z}] + \\frac{\\partial K(h_m) }{\\partial z}$ The hydraulic capacity is the slope of the water retention function.","The reasoning for reducing the number of variables in these equations is that it should simplify the process of solution either analytically, when an analytical solution is possible, or numerically.","Because of the functional relationship between water content and matric potential, and unsaturated hydraulic conductivity, the functions discussed in Chapter REF and in this chapter can be used when solving these equations.","Because these functions exist, there is also a functional relationship between hydraulic diffusivity and water content.","Notice that based on the discussion on the previous paragraphs, the hydraulic diffusivity can be written as [50] $D(\\theta ) = \\frac{K(\\theta )}{C(\\theta )}$ It is also worth noticing that the Darcy-Buckingham equation can be written in terms of diffusivity.","In a generalized horizontal $x$ direction $q = D(\\theta ) \\frac{\\partial \\theta }{\\partial x}$ Beware, however, of the interpretation of the hydraulic diffusivity [53], [71].","The introduction of the diffusivity form made it easier to solve the equation in certain cases, but the hydraulic diffusivity does not have the same meaning as a diffusion term in heat and solute dispersion processes.","Keep in mind that many of these equations require a unique relationship between the variables unsaturated hydraulic conductivity, diffusivity and water content and/or matric potential, and between water content and matric potential.","This assumption is not physically realistic because of hysteresis, the relationship between water content and matric potential is not unique in real natural porous media and dependents on the history of the processes, either if a drying or wetting processe is being considering.","In addition, for wetting-drying cycles within a given water content range, there might by multiple possible paths (hysteresis loops).","In other words, there is a non-unique relationship between water content and matric potential for real porous media, for a given matric potential there might be at least two corresponding water content values depending on the wetting history.","Bruce and Klute [20] described a procedure do find the diffusivity function following a procedure that Ludwig Boltzmann [16] used to solve diffusion type equations, now known as the Boltzmann transformation.","Recall the diffusivity form of the Richardson-Richards equation in a generalized horizontal $x$ direction $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [D(\\theta ) \\frac{\\partial \\theta }{\\partial x} ]$ Boltzmann [16] showed that diffusion type equations can be solved by introducing a new variable $\\Lambda = x t^{-1/2}$ For the diffusivity equation of concern in soil and porous media transport, the volumetric water content is now a function of the new variable [53] $\\theta = f(\\Lambda )$ The idea now is to replace $\\Lambda $ into the Richardson-Richards equation in order to reduce it from a partial differential equation in $\\theta $ , $t$ and $x$ to a ordinary differential equation in $\\Lambda $ and $\\theta $ .","The first step is to calculate the partial derivatives of $\\Lambda $ in $x$ and $t$ $\\frac{\\partial \\Lambda }{\\partial x} = \\frac{1}{\\sqrt{t}}$ $\\frac{\\partial \\Lambda }{\\partial t} = -\\frac{x}{2t^{3/2}} = -\\frac{\\Lambda }{2t}$ Next, the water content partial derivatives can be expanded by the chain rule.","Recall that, by the definition of $\\Lambda $ , the unsaturated hydraulic conductivity is an implicit function of it such that $\\frac{\\partial \\theta }{\\partial x} = \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\partial \\Lambda }{\\partial x}$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\partial \\Lambda }{\\partial t}$ Since we have already calculated the partial derivatives of $\\Lambda $ it is just a matter of replacing them into the two equations above $\\frac{\\partial \\theta }{\\partial x} = \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{1}{\\sqrt{t}}$ $\\frac{\\partial \\theta }{\\partial t} = -\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t}$ Replacing these into the Richardson-Richards equation results in $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{\\partial }{\\partial x} [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{1}{\\sqrt{t}}]$ Because the partial derivatives on the right side no longer depend on $t$ they can be placed outside of the differentials $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{1}{\\sqrt{t}} \\frac{\\partial }{\\partial x} [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ We can now expand the partial derivative in $x$ by the chain rule $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{1}{\\sqrt{t}} \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ] \\frac{\\Lambda }{\\partial x}$ and, since we already know $\\partial \\Lambda / \\partial x$ $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{1}{\\sqrt{t}} \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]\\frac{1}{\\sqrt{t}}$ and finally [20] $-\\frac{\\Lambda }{2} \\frac{\\partial \\theta }{\\partial \\Lambda } = \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ An experimentally based solution to this ordinary differential equation was proposed by [20] for the boundary conditions $\\theta = \\theta _i~ \\text{for}~ x > 0, t = 0 \\\\\\theta = \\theta _s~ \\text{for}~ x = 0, t \\ge 0$ in which $\\theta _i$ is the initial water content of the system and $\\theta _s$ is the saturated water content.","These conditions are valid for an infinitely long column, such that the wetting front will not reach the other end.","In practical terms, a long column filled with porous material at constant initial water content is placed in contact with a water source, such that the water content will increase with time.","After an elapsed time $t$ , the $\\Lambda $ function can be evaluated along the length of the column.","The $\\Lambda $ function would later be called the sorptivity of the soil or porous media.","Under these conditions, the $\\Lambda $ form of Richardson-Richards equation can be integrated in the limits of initial water content to water content at a position $x$ and from $\\Lambda $ at the initial condition to $\\Lambda $ evaluated at $x$ $-\\int _{\\theta _i}^{\\theta _x}\\frac{\\Lambda }{2} \\frac{\\partial \\theta }{\\partial \\Lambda } = \\int _{\\Lambda _i}^{\\Lambda _x} \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ which immediately simplifies to $-\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = \\int _{\\Lambda _i}^{\\Lambda _x} \\partial [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ Noticing that $\\Lambda _i = \\Lambda _\\infty $ corresponds to $\\theta _i$ and that $\\Lambda _x$ corresponds to $\\theta _x$ , a change of variables can be performed on the right integral $-\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = \\int _{\\theta _i}^{\\theta _x} \\partial [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ or $-\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _i}^{\\theta _x} = [D(\\theta _x) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _x} - [D(\\theta _i) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _i}$ Because $\\theta _i$ is constant, the partial derivative $\\partial \\theta / \\partial \\Lambda $ evaluated at $\\theta _i$ is zero $\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = - [D(\\theta _x) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _x}$ Thus, the diffusivity can be calculated numerically and experimentally by [20] $D(\\theta _x) = -\\frac{1}{2} (\\frac{\\partial \\Lambda }{\\partial \\theta } )_{\\theta _x} \\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta $ The procedure is described in [53] and on the original publication, among others [31].","You have seen that the mathematical complexity of the models for describing unsaturated flow is substantially greater than that of saturated flow equations.","The partial differential equations used includes the variables $K$ , $D$ , $\\theta $ and $h_m$ in addition to spatial coordinates.","These variables are usually related to each other by nonlinear functions, which must be considered when searching for the solution.","In many real world applications, in transient flow conditions and in dimensions greater that one, these equations might not have analytical solutions or can only be solved analytically using unrealistic boundary conditions.","One example of simplification was described on the previous section.","When analytical solutions are not possible, numerical solutions are used.","In numerical methods the equations are solved numerically through approximation methods.","In fact, much of fluid mechanics and many areas of applied physics and groundwater hydrology now rely heavily on numerical methods.","A whole subarea of fluid mechanics called computational fluid mechanics is now on the frontier of knowledge in the field.","Simple numerical solutions can be computed by hand but today and for real world scenarios, most applications will use a computer.","Knowledge of informatics and programming languages is essential to applied numerical methods in soil physics and groundwater hydrology.","ll $\\beta $ Capillary conductivity defined by Buckingham [22] $\\psi _c$ Capillary potential $x$ Generalized horizontal direction or coordinate $q$ Flux density $K(\\theta )$ Unsaturated hydraulic conductivity as a function of volumetric water content $h$ Total potential in length units $h_m$ Matric potential in length units $K(h_m)$ Unsaturated hydraulic conductivity as a function of matric potential $z$ Generalized vertical direction or coordinate $h_t$ Total potential in length units $h_g$ Gravitational potential in length units $K_r(\\Theta ) $ Unsaturated hydraulic conductivity as a function of effective saturation $\\Theta $ Effective saturation $= (\\theta - \\theta _r)/(\\theta _s - \\theta _r)$ $\\psi _m$ Matric potential in general units $\\lambda $ Pore distribution parameter of Brooks and Corey [19] $a, b, c$ Arbitrary constants $\\theta $ Volumetric water content $\\theta _r$ Residual water content $K_s$ Saturated hydraulic conductivity $y$ Parameter for transformation on the van Genuchten [97] solution or arbitrary horizontal direction or coordinate elsewhere $k$ Generic parameter for solving the Mualem equation in the van Genuchten framework $m$ Shape parameter for the van Genuchten equation $n$ Shape parameter for the van Genuchten equation $\\alpha $ Inverse of air entry pressure for the van Genuchten equation $\\rho $ Arbitrary density of a material for the generalized conservation equation $\\mathbf {u}$ Velocity vector for fluid flow in general $c_w$ Water concentration within a porous medium $\\rho _w$ Water density $t$ Time $m_w$ Water mass $V_w$ Water volume within a porous medium $\\mathbf {q}$ Flux density in a porous medium $\\mathbf {K}(\\mathbf {r}, \\psi _m) $ Unsaturated hydraulic conductivity tensor $\\mathbf {r}$ Position vector $K(x,y,z, \\psi _m) $ Unsaturated hydraulic conductivity scalar function $D(\\theta )$ Diffusivity function $C(\\theta )$ Water (hydraulic) capacity function $ = \\partial \\theta /\\partial \\psi _m$ $\\Lambda $ Variable introduced for the Boltzmann transformation In the last two chapters we studied the equation of conservation of water for porous media under saturated and unsaturated conditions.","To study conservation of other types of mass in porous media, similar principles can be applied.","Soluble solids are of immense interest in groundwater hydrology and soil physics because a large part of the groundwater and soil contaminants are soluble solids.","Examples include pesticides, organic molecules, viruses, bacteria, radionuclides, ammonium, nitrates, phosphates and many others.","In agriculture, fertilizers and other plant nutrients can be also transported through soil with water fluxes and by diffusion.","Many compounds can act as either plant nutrients or contaminants, depending on where they are and on the amount and on intrinsic factors of soils and aquifers.","The process can be also complicated as many solids are volatile and/or can be adsorbed onto the solid phase, suffer transformations, decay and many other processes.","Suppose you have a volume of a porous media either soil, an aquifer or other geologic material.","Suppose now that we are concerned with the dynamics of a single soluble solid.","The concentration of the solid within the porous media is $C$ .","One of the mechanisms by which the contaminant can be transported in and out of the volume of interest is by advection.","Advective transport is the mass transport with the flow of the fluid in which it is dissolved.","When the conservation equation was derived, we saw that the flux density was given by $\\mathbf {q}$ .","The mass flux of the dissolved solid being transported in an unsaturated medium is given by $\\mathbf {J}_{adv} = \\theta \\mathbf {v} C$ in which $\\theta $ is the volumetric water content [L$^3$ L$^{-3}$ ], $\\mathbf {v}$ is the water velocity within the pores [L T$^{-1}$ ], as discussed in the previous chapter and $C$ is the concentration of the dissolved solid in the liquid phase [M L$^{-3}$ ].","Dimensional analysis indicates that the vector $\\mathbf {J}_{adv}$ is the mass of the dissolved solid entering (or exiting) per unit area per unit time [M L$^{-2}$ T$^{-1}$ ].","If there was no diffusion or hydrodynamic dispersion, an unrealistic scenario for dissolved solids in porous media, the conservation equation would be of the form $\\frac{\\partial (\\theta C)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C) $ However, dissolved solids are subject to diffusion due to concentration gradients.","As we discussed, regions of high concentration of dissolved solids will result in lower water potential, in this scenario diffusion tends to create equilibrium of concentration throughout the fluid, subjected that there are no other constraints regarding other potentials e.g.","thermal and electrostatic.","Considering an element of static, incompressible fluid, not within a porous medium, the diffusion of a soluble solid in the fluid can be modeled using Fick's law $\\mathbf {J}_{diff} = - D_d \\nabla C$ $\\mathbf {J}_{diff}$ is the diffusive flux [M L$^{-2}$ T$^{-1}$ ], $D_d$ is the diffusion coefficient in [L$^{2}$ T$^{-1}$ ] and $\\nabla C$ is the concentration gradient.","Within a porous media, not all the space is available for diffusion.","In a saturated porous media, the space occupied by solid particles is not available for diffusion, only the water saturated pore space.","Even within a saturated porous medium, it might be that not all of the pore space is available to transport due to the fact that not all pores might be connected, and dead end pores [38].","In unsaturated porous media, only the fraction of space saturated with water is available for transport [12].","To account for the geometry of the pore space available for fluid flow and its modification with changing water content, a second rank tortuosity tensor is included into Fick's law of diffusion [11], [12] $\\mathbf {J}_{diff} = - D_d \\mathbf {T}(\\theta ) \\cdot \\nabla C$ The product of the diffusion coefficient by the tortuosity tensor can be written as a tensor, defined as the coefficient of diffusion for a a given phase in a partially saturated porous medium [12] $\\mathbf {J}_{diff} = - \\mathbf {D}^*(\\theta ) \\cdot \\nabla C$ where $ \\mathbf {D}^*(\\theta ) = D_d\\mathbf {T}(\\theta )$ An analysis of the diffusion and tortuosity tensors will be left for later editions, the interested reader should consult [11], [12].","In essence this equation shows that the concentration of soluble solids will still tend to equilibrum due to concentration gradients in partially saturated porous media, but the volume available for diffusive movement will be limited by the volume occupied by water, and the path for movement will be constrained by the geometry of the porous space.","Because of the geometry of the solid and pore space, the velocity and paths of water flux in porous media tends to be highly heterogeneous.","In general, the more heterogeneous the porous media is in terms of grain size distribution and solid particles shapes and surface rugosity, the more heterogeneous the flux behavior will be, specially if the shapes, and rugosity distributions are more or less random.","Grain and pore size distributions are still random but tend to follow certain statistical distributions and this has been taken advantage for modeling flow and transport using stochastic models.","Because of the irregular distribution of velocities within the pores, the irregularity of the pore paths, a third transport mechanism called mechanical dispersion is also important.","Even within a single pore, because of viscosity and surface forces, the velocity will be maximum at the center of the pore and zero at the surface if the pore is modelled as two infinite, parallel plates or as a cylinder, as we have seen when deriving Hagen-Poiseuille's equation in Chapter REF .","All these sub-mechanisms cause mixing of the dissolved solid when the water in which it is dissolved is flowing through a porous medium.","Dispersion has been shown experimentally to follow a Fick's diffusion type law [12] $\\mathbf {J}_{disp} = - \\mathbf {D} \\cdot \\nabla C$ in which $\\mathbf {D}$ is the dispersion coefficient, also a second rank tensor.","Considering the three forms of transport, the total flux of a dissolved solid in a partially saturated porous medium is $\\mathbf {J}_{tot} =\\mathbf {J}_{adv} + \\mathbf {J}_{diff} + \\mathbf {J}_{disp} = \\theta \\mathbf {v} C - (\\mathbf {D}^*(\\theta ) \\cdot \\nabla C + \\mathbf {D} \\cdot \\nabla C)$ The diffusion and mechanical dispersion components can be lumped into a single tensor called the coefficient of hydrodynamic dispersion $\\mathbf {D}_h$ $\\mathbf {D}_h = \\mathbf {D} + \\mathbf {D}^*(\\theta )$ such that the total flux can be written as $\\mathbf {J}_{tot} =\\mathbf {J}_{adv} + \\mathbf {J}_{diff} + \\mathbf {J}_{disp} = \\theta \\mathbf {v} C - \\mathbf {D}_h \\cdot \\nabla C$ In theory our work is almost be done, considering the conservation of mass and the total flux of the dissolved solid, the conservation equation should be something like $\\frac{\\partial (\\theta C)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C - \\mathbf {D}^*(\\theta ) \\cdot \\nabla C - \\mathbf {D} \\cdot \\nabla C)$ In reality, the behavior of chemicals in porous media is much more complicated than that.","Many chemicals can transition from one phase to another, for example, several compounds exist as a solid or as a liquid or as both depending on chemical equilibrium conditions, other chemicals can be adsorbed onto solid surfaces by several different mechanisms, including those discussed in Chapter REF , while some are subjected to both adsorption and volatilization.","In addition to that, some chemicals can be created or destroyed while being transported, for example, radionuclides can decay generating different isotopes or elements, organic matter can be transformed or destroyed by organisms and other bio-degradation processes, and both mechanisms can be effectively creating new chemicals in the environment.","Because of that, a general equation of transport of chemicals in porous media must include terms for exchanges among phases, and sources and sinks of compounds, a source meaning a term for creation of a substance and a sink a term for the elimination of a given substance Obviously conservation laws still apply, a substance is eliminated by being transformed into other forms of matter or energy and created from other forms of matter or energy.","Following the language of [12] the mass balance equation for a single chemical species in a porous media including sources and sinks is $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C - \\mathbf {D}^*(\\theta ) \\cdot \\nabla C - \\mathbf {D} \\cdot \\nabla C) + \\theta \\rho _{f} \\Gamma _{f} + \\rho \\Gamma _s$ in which the total mass balance term in time $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t}$ now accounts for the rate of variation in time of the amount adsorbed into the solid phase in the form of the component $\\frac{\\partial \\rho F}{\\partial t}$ where $F$ is the mass adsorbed into the solid phase per unit mass of solid phase and $\\rho $ is the bulk density of the porous medium.","The source (or sink) term of the chemical in the liquid phase is given by $\\theta \\rho _f \\Gamma _f$ in which $\\theta $ is, as before, the volumetric water content, $\\rho _f$ is the density of water, or of the liquid phase in general, and $\\Gamma _f$ is the rate of creation (or destruction if the signal is negative) of the chemical per unit mass of water [M T$^{-1}$ M$^{-1}$ ].","The source (or sink) term for the chemical in the solid phase is given by $\\rho \\Gamma _s$ in which $\\Gamma _s$ is the rate of creation (or destruction if the signal is negative) per unit mass of solid phase [M T$^{-1}$ M$^{-1}$ ].","In theory, the transport equation can be further expanded to include transport in the gas phase, by including a source term in the gas phase and the vapor diffusive flux [47].","The form without the gas transport is commonly used in soil physics and hydrology and has been called the advection-dispersion or advective-dispersive equation when the lumped hydrodynamic dispersion term is considered $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C - \\mathbf {D}_h \\cdot \\nabla C) + \\theta \\rho _{f} \\Gamma _f + \\rho \\Gamma _s$ The full equation with the tensor components in three dimensions is obviously too complex for any analytical treatment.","Solutions for several cases are available on the literature, in general [12], [38] and for particular cases [98] Another fundamental reference for the mathematics of diffusion and numerical methods is the book by John Crank [30], Crank along with Phyllis Nicolson also created an important finite differences scheme for solving differential equations.","One obvious simplification, especially useful for laboratory column experiments and field studies of soil profiles or geological strata is to consider a single dimension.","Thus, in a generalized horizontal $x$ direction $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t} = - \\frac{\\partial }{\\partial x} (\\theta \\mathbf {v} C) + \\frac{\\partial }{\\partial x} (\\mathbf {D}_h \\frac{\\partial C}{\\partial x}) + \\theta \\rho _{f} \\Gamma _f + \\rho \\Gamma _s$ For one dimensional flow of a non-adsorbing, chemical which suffers no degradation and is not created within the porous media the sources the terms can be discarded.","An example would be dissolved sodium chloride (NaCl) being transported in water through a column composed of silica sand.","A further simplification can be applied if the hydrodynamic dispersion coefficient is constant for the fluid and can be placed outside the partial derivative.","With all these simplifications, the conservation equation is $\\frac{\\partial (\\theta C)}{\\partial t} = - \\frac{\\partial }{\\partial x} (\\theta v_x C) + D_h \\frac{\\partial ^2 C}{\\partial x^2}$ If the water content and velocity are constant they can be placed outside the derivative terms and $\\frac{\\partial C}{\\partial t} = - v_x \\frac{\\partial C}{\\partial x} + D \\frac{\\partial ^2 C}{\\partial x^2}$ where $D = D_h/\\theta $ .","This equation can be solved for different boundary conditions.","For an infinite uni-dimensional domain saturated with water with initial chemical concentration $C = 0$ in which a solution containing a chemical (or a tracer) is introduced in the inlet with concentration $C = C_0$ subjected to the following the initial condition $C(x,0) = 0 ~ \\text{at}~ x \\ge 0$ and boundary conditions $C(0,t) = C_0~ \\text{for}~ t \\ge 0 \\\\C(\\infty ,t) = 0~ \\text{for}~ t \\ge 0$ the solution is provided by Ogata and Banks [74] The solution spans at least three pages in the original and might be included in further editions $\\frac{C(x,t)}{C_0} = \\frac{1}{2} [\\text{erfc}(\\frac{x-v_xt}{2\\sqrt{Dt}}) + \\exp {(\\frac{v_x x}{D})}\\text{erfc}(\\frac{x+v_xt}{2\\sqrt{Dt}})]$ where erfc is the complementary error function presented in Chapter REF .","For a cylinder filled with a static fluid in which a pulse of solution with concentration $C_0$ is injected at $x=0$ , the concentration will spread symmetrically from the point of application with time following [30] $C(t) = \\frac{M}{2\\sqrt{\\pi D t}} \\exp {(-\\frac{x^2}{4Dt})}$ in which $M$ is the mass of substance deposited at time $t= 0$ at $x = 0$ .","This solution has been adapted to a cylinder filled with porous media in which the fluid is moving with velocity $v_x$ by considering a relative displacement term $x - v_x t$ [10], [81] $\\frac{C(x,t)}{C_0} = \\frac{1}{2\\sqrt{\\pi D t}} \\exp {[-\\frac{(x-v_x t)^2}{4Dt}]}$ Jury and Sposito [48] working within a different framework arrived at a solution of similar form with the exception of a cubic time term and an additional $x$ length term $\\frac{C(x,t)}{C_0} = \\frac{x}{2\\sqrt{\\pi D t^3}} \\exp {[-\\frac{(x-v_x t)^2}{4Dt}]}$ Equations REF , REF and REF can be applied to finite column of length $L$ to predict the relative concentration $C/C_0$ at $x = L$ with time $t$ .","Simulated values of $C/C_0$ as a function of time for a $100~cm$ saturated horizontal column with solution moving with velocity $v_x = 2.0~cm~h^{-1}$ for the hydrodynamic dispersion coefficients $D = D_h/\\theta $ of $0.5~cm^2~h^{-1}$ , $5.0~cm^2~h^{-1}$ , $10.0~cm^2~h^{-1}$ and $50.0~cm^2~h^{-1}$ are presented for each of the solutions.","Equation REF predicts an increase in the concentration outlet from zero to one with the time for solute to first appear on the outlet depending on the diffusivity (Figure REF ).","These types curves are called breakthrough curves because the model predicts the time and behavior for the solute to break through the column.","With increasing hydrodynamic dispersion coefficient, the contribution of dispersion and diffusion to solute transport increases, ultimately resulting in some of the solute breaking through earlier than the average velocity of the fluid alone would predict.","Dispersion and diffusion also cause a dilution of the front with respect to the concentration of the applied pulse.","Small values of $D$ indicate little contribution of diffusion and dispersion as can be observed from the increase in the slope of the curve at lower values of $D$ .","At a theoretical $D=0$ the solute pulse would move exactly at the velocity of the fluid flow and would appear at once at the end of the column without change in concentration.","This type of theoretical flow is called piston flow, because it would mimic the behavior of a fluid being pushed by a piston, as in a syringe for example.","Figure: Breakthrough curves for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~ h^{-1} and different hydrodynamic dispersion parameter values.For the same simulation parameters used for Equation REF Equations REF and REF predict the behavior of a pulse of a solution of concentration $C_0$ with time (Figures REF , REF ).","The two equations predict similar results as $D$ decreases.","Equation REF tends to predict higher concentrations as $D$ is increased when compared to Equation REF .","For a theoretical $D = 0$ both equations would predict a single instantaneous pulse, sometimes modeled as a dirac delta function.","As $D$ increases, an increase in the spreading of the pulse is observed, the outlet concentration reaching smaller maximum output concentrations.","Figure: Behaviour of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm ~h^{-1} and different hydrodynamic dispersion parameter values .Figure: Behavior of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~h^{-1} and different hydrodynamic dispersion parameter values .Suppose now you have a chemical that does not have any sources of sinks but can be adsorbed into the solid phase, as it is the case with ions in solution and charged compounds in soils.","In one dimension, the mass balance equation for a single species can be written as [47], [12] $\\frac{\\partial (\\theta C)}{\\partial t} + \\frac{\\partial \\rho F}{\\partial t} = \\frac{\\partial }{\\partial x} D_h \\frac{\\partial C}{\\partial x} - \\frac{\\partial }{\\partial x} (\\theta v_x C)$ For a saturated medium, the volumetric water content $\\theta $ is constant and equal to the total porosity $\\phi $ and can be placed outside of the partial derivatives.","Assuming also the hydrodynamic dispersion coefficient and flow velocity as constant, the equation simplifies to $\\phi \\frac{\\partial C}{\\partial t} + \\frac{\\partial \\rho F}{\\partial t} = D_h \\frac{\\partial ^2 C}{\\partial x^2} - \\phi v_x \\frac{\\partial C}{\\partial x}$ If the mass adsorbed onto the solid phase is a linear function of the concentration in the liquid phase, we can define the following relationship $F = K_d C$ This equation describes a linear adsorption isotherm with $K_d$ being the partition coefficient between the solid and liquid phases, and it is the slope of the relationship between $F$ and $C$ .","Replacing it into the mass balance equation $\\phi \\frac{\\partial C}{\\partial t} + \\frac{\\partial \\rho K_d C }{\\partial t} = D_h \\frac{\\partial ^2 C}{\\partial x^2} - \\phi v_x \\frac{\\partial C}{\\partial x}$ Because $K_d$ does not depend on the concentration and the bulk density of the porous medium it is assumed as constant $\\phi \\frac{\\partial C}{\\partial t} + \\rho K_d\\frac{\\partial C }{\\partial t} = D_h \\frac{\\partial ^2 C}{\\partial x^2} - \\phi v_x \\frac{\\partial C}{\\partial x}$ Reorganizing and rearranging $(1 + \\frac{\\rho K_d}{\\phi })\\frac{\\partial C }{\\partial t} = D \\frac{\\partial ^2 C}{\\partial x^2} - v_x \\frac{\\partial C}{\\partial x}$ In which $D_h = D/\\phi $ .","Defining a coefficient of retardation $R_d$ [12] $R_d \\equiv 1 + \\frac{\\rho K_d}{\\phi }$ the mass balance equation is $R_d\\frac{\\partial C }{\\partial t} = D \\frac{\\partial ^2 C}{\\partial x^2} - v_x \\frac{\\partial C}{\\partial x}$ The retardation coefficient is a number equal to or greater than zero.","There are different adsorption isotherms, including nonlinear relationships between $F$ and $C$ and in the mass balance equation needs to be derived for each specific case.","For linear adsorption isotherms, the retardation coefficient appears in the breakthrough and pulse solutions as a linear term [85] $\\frac{C(x,t)}{C_0} = \\frac{1}{2} [\\text{erfc}(\\frac{R_d x-v_xt}{2\\sqrt{D R_d t}}) + \\exp {(\\frac{v_x x}{D})}\\text{erfc}(\\frac{R_d x+v_xt}{2\\sqrt{DR_d t}})]$ $\\frac{C(x,t)}{C_0} = \\frac{x}{2\\sqrt{\\pi D t^3 / R_d }} \\exp {(-\\frac{(R_d x-v_x t)^2}{4DR_d t})}$ The retardation coefficient, as the name indicates, controls the rate of retardation or a delay in the transport of the chemical throughout the column and will tend to increase the width of the pulse.","The effect of the retardation coefficient in Equation REF is illustrated for $R_d = 0.25$ , $0.50$ and $1.0$ (Figure REF ).","Figure: Behavior of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~h^{-1} and different retardation factors.Note that for unsaturated porous media the retardation factor is a function of volumetric water content.","There are numerous other cases and solutions that could be covered including unsaturated cases, presence of mobile and immobile fluid phases, partition among gas, solid and liquid phases, radioactive decay and others.","For the time being the interested reader is encouraged to consult [98], [12], [38], [85].","The framework for discussing gas transport in porous media is the same as that of general chemical transport discussed in the previous section.","Technically, gas transport could have been included with chemical transport, especially for gases that partition between liquid and gas phases and can be adsorbed into solids.","In this section we will provide a brief discussion of the specifics of gas transport.","As with other types of mass transport, gases can be transported by advection and diffusion.","Gas diffusion is controlled by Fick's law, one of the general transport laws in physics discussed in previous sections.","For a binary mixture of gases, the mole flux of the component of interest is given by [100] $\\mathbf {J}_g^{mole} = -C_g \\mathbf {D}_{g(ij)} \\nabla \\chi _g$ in which $C_g$ is the concentration of the gas, $\\mathbf {D}_g$ is the diffusion coefficient tensor and $\\chi _g$ is the mole fraction of the component gas of interest.","This relationship can be written in terms of a mass flux as $\\mathbf {J}_g^{mass} = -\\rho _g \\mathbf {D}_{g(ij)} \\nabla \\omega _g$ in which $\\omega _g$ is the mass fraction of the component of interest.","Now $\\mathbf {J}_g$ and $\\mathbf {J}_g^{mole}$ indicate the flux relative to the average mole or mass velocity of the mixture and not to stationary coordinates [13], [100].","Supposing that the diffusion coefficient is constant through the medium, it does not need to be treated as a tensor and $\\mathbf {J}_g^{mass} = -\\rho _g D_g \\nabla \\omega _g$ As we have discussed for water and chemicals, the general law for a volume devoid of solid phase is not valid when the diffusion or advection is taking place within a porous medium.","If the porous medium was completely devoid of liquid, i.e.","completely dry, the solid particles would still occupy part of the volume and generate a complex and tortuous pore space through which the gas would need to navigate.","For a completely saturated porous medium, the gas phase would need to diffuse through the liquid, the rates being often orders of magnitude lower than that in air.","If the pore space is partially saturated with water, only a fraction of the pore space inversely proportional to water content is available for diffusion and advection.","Thus, Fick's law needs to be adapted to porous media.","Under such conditions the diffusion coefficient is replaced by an effective diffusion coefficient $D_g^* = \\beta D_g$ in which $\\beta $ is a porous media factor $\\beta = \\phi S_g \\tau $ where $\\phi $ is the porosity, $S_g$ is the effective gas saturation, from zero to one, and $\\tau $ is the tortuosity factor.","The tortuosity factor can be written as $\\tau = \\phi ^{1/3} S_g^{7/3}$ which for a media completely saturated with gas simplifies to $\\tau = \\phi ^{1/3}$ In soil physics it is more common that the gradient is expressed in terms of concentration, which is effectively the density multiplied by the mass fraction, thus $\\mathbf {J}_g = - D_g^* \\nabla C_g$ For gas diffusion in soil, the soil gas diffusion coefficient is $ D_g^*=\\tau D_g$ and the tortuosity if often expressed as $ \\tau = 0.66 \\theta _a $ in which $\\theta _a$ is the soil volumetric air content [53], [47].","For advective gas transport, Darcy's law is valid [47], [100] $\\mathbf {q}_g = - \\frac{\\kappa _g}{\\mu _g} (\\nabla P_g - \\rho _g \\mathbf {g})$ here written in terms of the permeability of the media to the gas ($\\kappa _g$ ), the viscosity of the gas ($\\mu _g$ ), the density of the gas ($\\rho _g$ ) and the pressure gradient ($\\nabla P_g$ ) and in which, as defined before, $q_g$ is the flux density, but for the gas in this case.","The formula can be also written in terms of a gas conductivity $K_g = k_g/\\mu _g$ parameter $\\mathbf {q}_g = - K_g (\\nabla P_g - \\rho _g \\mathbf {g})$ In general, the gravity term can be neglected because the density of most gases is low, thus $\\mathbf {q}_g = - K_g \\nabla P_g$ for pressure in length units.","Because advective flow of gases is usually not an important transport mechanism in soil physics and groundwater hydrology, except in very specific conditions, transport of gases in porous media is modeled as a diffusion processBeware that this might not be the case when modeling natural gas reservoirs.","We can now rewrite the general transport equation for a non-adsorbing gas for a generalized direction $z$ , neglecting advective transport, as $\\frac{\\partial (\\theta _g C_g)}{\\partial t} = \\frac{\\partial }{\\partial z} \\mathbf {J}_g + \\theta _g \\rho _{g} \\Gamma _g$ $\\frac{\\partial (\\theta _g C)}{\\partial t} = \\frac{\\partial }{\\partial z} (\\mathbf {D}^*_g \\frac{\\partial C^g}{\\partial x}) + \\theta _g \\rho _{g} \\Gamma _g$ and if the volumetric air content and the soil gas diffusion coefficient are constant $\\theta _g \\frac{\\partial C_g}{\\partial t} = D^*_g \\frac{\\partial ^2 C_g}{\\partial z^2} + \\theta _g \\rho _{g} \\Gamma $ In many cases this is a very simple differential equation whose techniques of solution were generally shown in Chapter REF .","Thus, for transport of a inert gas (no sources of sinks) through a column of length $L$ saturated with inert gas with concentration $C_0$ at the inlet and zero at the outlet, the solution is [47] $C_g(z) = C_0 (1 - \\frac{z}{L})$ while for steady-state transport, in which the gas concentration does not vary with time (i.e.","$ \\partial C_g/\\partial t = 0 $ ) with a consumption term (e.g.","oxygen in soil environment) the solution is [47] $C_g(z) = C_0 + \\frac{\\theta _g \\rho _g \\Gamma }{2 D^*_g} (z^2 + 2Lz)$ Because the calculations are simple the previous derivations are left as an exercise to reader.","The same conservation principles for mass applied in the previous sections generally apply for heat transport.","However, heat transport is a form of energy transport and some particularities need to be consider.","Like before, heat can be transported by diffusion and advection, but in the case of advection, heat is transported with transport of a solid, gas or liquid.","The amount of heat stored in a material is given by [53] $Q = \\rho c V T$ in which $Q$ is the amount of thermal energy stored in the material, in joules or calories, $\\rho $ is the density of the material in [M L$^{-3}$ ], $c$ is the mass based specific heat capacity or the heat capacity $C_h$ divided by the amount of mass of the material, $V$ is the volume of material and $T$ is the temperature.","Because the heat capacity is given in energy per temperature, the mass based specific heat capacity is in energy times temperature$^{-1}$ mass$^{-1}$ , representing thus the amount of energy necessary to increase the temperature of one unit mass of the material in one unit of temperature, usually in (calories) cal $g^{-1}$ $^{o}C^{-1}$ or in $J$ $kg^{-1}$ $K^{-1}$ .","As with the other transport laws discussed throughout this material, thermal conduction law is given by $\\mathbf {J}_T = -\\frac{\\kappa _T}{\\rho c} \\nabla T$ in which $\\kappa _T$ is the thermal conductivity.","Defining now a thermal diffusivity term as $D_T = -\\frac{\\kappa _T}{\\rho c}$ $\\mathbf {J}_T = - D_T \\nabla T$ This is Fourier's law of heat conduction and it is analogous to Fick's law for gases, Darcy's law for fluids and Ohm's law for electricity.","The driver gradient is now not a gradient in terms of concentration or potential but in terms of temperature.","As before, we can define a general conservation equation for heat [47] $\\frac{\\partial T}{\\partial t} = - \\frac{\\partial J_T}{\\partial z} + \\gamma _T $ in which $\\gamma _T$ is a general heat source or sink term.","Following the procedures for the other conservation laws, and considering the thermal diffusivity term as constant, the heat flux $J_T$ can be replaced into the conservation equation, and ignoring heat sources and sinks $\\frac{\\partial T}{\\partial t} = D_T \\frac{\\partial ^2 T}{\\partial z^2} $ Because the temperature at surface ($z = 0$ ) varies sinusoidally due to daily and annual variations of temperature and solar heat influx, a solution for temperature variation in depth is of the form [53] $T(z, t) = T_a + A \\exp {(-\\sqrt{\\frac{\\omega }{2D_T}}z)} \\sin {[\\omega t - \\sqrt{\\frac{\\omega }{2D_T}}z + \\xi ]} $ in which $z$ is the depth within the material or porous media, $t$ is time, $T_a$ is the average temperature, $A$ is the amplitude of the sinusoidal wave components, $\\omega = 2\\pi /P$ , where $P$ is the period, and $\\xi $ is the phase constant.","The temperature variation with depth up to $100.0~cm$ , for a soil with thermal diffusivity $D_T = 0.015~cm^2~s^{-1}$ , phase constant $\\xi = -7\\pi /12$ , $T_a = 25.0~^{o} C$ and $A = 10.0~^{o} C$ is illustrated in Figure REF , where $\\omega $ was calculated considering a period of $24~h$ or $86400~ s$ .","Note that the amplitude of temperature is damped with depth, being close to zero at $100~ cm$ and maximum at surface.","Figure: Variation of temperature with time over a period of two days up to a depth of 100cm100~ cm for a soil with D T =0.015cm 2 s -1 D_T = 0.015~cm^2~s^{-1} .ll $\\mathbf {J}_{adv} $ Advective flux density for a dissolved solid being transported $\\theta $ Volumetric water content $\\mathbf {v} $ Fluid velocity vector $C $ Dissolved chemical concentration $D_d $ Generalized diffusion coefficient $\\mathbf {J}_{diff} $ Diffusive flux density for a dissolved solid being transported $\\mathbf {T}(\\theta ) $ Tortuosity tensor $\\mathbf {D}^*(\\theta ) $ Coefficient of diffusion for a dissolved solid being transported in porous media $\\mathbf {D} $ Dispersion tensor in porous media $\\mathbf {D}_{h} $ Hydrodynamic dispersion tensor $\\mathbf {J}_{tot} $ Total flux density for a dissolved solid being transported in porous media $\\mathbf {J}_{disp} $ Dispersive flux density for a dissolved solid being transported in porous media $F $ Mass of chemical absorbed per unit mass of solid phase $\\rho _f $ Fluid density $\\rho $ Bulk density of the porous media $\\Gamma $ Rate of creation or destruction of the chemical in the liquid phase $\\Gamma _s $ Rate of creation of destruction of the chemical adsorbed into the solid phase $v_x $ Velocity in a generalized $x$ direction $D_h $ Hydrodynamic dispersion coefficient $D $ Dispersion coefficient corrected to volumetric water content $= D_h/\\theta $ $\\phi $ Total porosity $K_d $ Coefficient of partition $R_d $ Retardation term $t $ Time $\\mathbf {J}_{g}^{mole} $ Mole flux for a gaseous component $\\mathbf {D}_{g(ij)} $ Gas diffusion coefficient tensor $ \\chi _g$ Mole fraction of the gaseous component of interest $ \\omega $ Mass fraction of the component of interest $\\rho _g $ Density of the gas $ D_g$ Gas diffusion coefficient $ D^*_g $ Effective gas diffusion coefficient in porous media $ \\beta $ Porous media factor $ S_g $ Effective gas saturation $ \\tau $ Tortuosity factor $ \\kappa _g $ Permeability of the media to a gas $ \\mu _g $ Viscosity of the gas $ P_g $ Gas pressure $ \\mathbf {g} $ Acceleration of gravity $ \\mathbf {q}_g $ Advective flux density of a gas in porous media $ K_g $ Gas conductivity $\\Gamma _g $ Rate of creation of destruction of the gas $C_g $ Concentration of the gas $ L $ Length of a column filled with porous media $\\theta _g$ Gas volumetric content $ Q $ Energy stored in the form of heat $ c $ Specific heat capacity $ V$ Volume of a material $ T$ Temperature $\\mathbf {J}_{T} $ Heat flux $ \\kappa _T $ Thermal conductivity $ D_T $ Thermal diffusivity $ \\omega $ $ = 2 \\pi / P $ , where $P$ is the period $ T_a$ Average temperature $ A $ Sinusoidal wave amplitude $ \\xi $ Phase constant As we have seen throughout this book, many of the models that describe physical phenomena are in the form of partial differential equations which might not have have simple analytical solutions for modeling real world phenomena, and very often might not have any analytical solutions at all.","When analytical solutions are not available, numerical approximations are an extremely useful tool in modeling real world phenomena.","With today's computers availability and processing power, numerical methods can be easily implemented using computational tools.","Large computational codes, both open source and proprietary, are used for modeling subsurface flow and transport in soils, geological deposits and watersheds.","Even larger models for matter and energy balance are available for Earth's scale modeling, but these are somewhat beyond the scope of this book.","One example of numerical model that all of us encounter in daily life is weather prediction.","Weather prediction is based on a complicated set of differential equations, based on a large set of parameters which cannot be solved analytically for larger domains such as a city, state, country or the whole world.","For weather prediction the variables available are the past observations and current observations.","These values will be then the initial conditions of the model, as we have seen for the solution of many differential equations.","The domains might also be subjected to a set of prescribed boundary conditions.","Given past predictions, and boundary and initial conditions, a numerical model can predict the weather in the future.","This type of prediction based on initial and boundary conditions using numerical solutions to differential equations is called forward modeling.","The stability, accuracy and range of the predictions is dependent on a series of factors including, accuracy of past and current predictions, nature of the differential equations being solved, numerical method being used and many others.","Another type of modeling frequently used in soil physics and groundwater hydrology is called inverse modeling.","In inverse modeling we have a set of observations and we wish to fit a particular empirical model to data to make predictions in the same system from which the data was obtained or in similar systems.","Inverse modeling can be used to make predictions in time and/or space but it is fundamentally different from numerical forward modeling because the empirical data is necessary a priori.","One example is a dataset of pairs water content and matric potential values which is fit to a water retention curve equation.","Fitting is usually achieved using statistical techniques such as least squares, maximum likelihood or Bayesian regression, among others.","The parameterized model can be then used to monitor water content from matric potential data in soils, vadose zone or other media for agricultural applications and environmental modeling.","Both types of models can be improved, calibrated and validated using independent data.","In the weather prediction model, once data has been collected, it can be compared to the numerical predictions and error indicators can be computed.","The model used and computation technique can be adapted accordingly.","In the case of the inverse model, once the water retention equation is fit to data, it can be compared to data collected in nearby sites using the same or similar laboratory techniques and the results be compared.","Once a large enough data set is compiled, stochastic estimation of the empirical parameters that are representative of the area based on more or less arbitrary statistical and mathematical indicators can be determined.","Over the next sections we will provide an extremely condensed account of curve fitting techniques and numerical methods used in soil physics and groundwater hydrology.","For the time being only finite differences schemes will discussed for numerical methods for solving differential equations, because they are somewhat simpler from the mathematical point of view and can be easily implemented in most programming languages.","The equations presented in this material are used to model real world data.","Consider for example the water retention curve and soil mechanical resistance equations.","In the first case data from the water retention curve can be obtained using the gravimetric method for water content, and pressure plates or tensiometers for soil matric potential.","For the soil mechanical resistance to penetration equation the soil resistance can be obtained by cone penetrometers, bulk density by the core method and water content by the gravimetric method.","After data collection, the equations can be fit to data using statistical or mathematical techniques.","Most of the equations encountered in soil physics and groundwater hydrology are nonlinear.","The meaning of nonlinear is not directly related to the shape of the curve when data is plotted, nonlinear is related to the parameters of the equation.","Consider a general linear equation, essentially a first degree polynomial $y = a x + b$ where $y$ is the dependent variable, $x$ is the independent variable and $a$ and $b$ are fitting parameters.","For pairs of $y$ and $x$ the procedure to fit a linear regression model is exact.","In other words, there are exact analytical formulas that can be used to obtain the values of $a$ and $b$ A theoretical analysis is beyond the scope of this material and can be found in introductory statistics and generalized linear models textbooks $a = \\frac{\\sum (x_i - \\bar{x})(y_i - \\bar{y})}{\\sum (x_i - \\bar{x})^2}$ $b = \\frac{\\sum y_i - a \\sum x_i }{n}$ in which $x_i$ and $y_i$ are the $i^{th}$ observation and $\\bar{x}$ and $\\bar{y}$ the averages for the variables $x$ and $y$ , and $n$ is the number of observations.","For polynomials of degree greater than one, although the shape of the curve of a plot is not linear, the fitting procedure is still exact and the equation is not considered a nonlinear equation.","The definition of a nonlinear regression model is related to nonlinearity in relation to the parameters.","In other words, a true nonlinear function is nonlinear with respect to its parameters.","Consider a second degree polynomial.","$y = a x^2 + b x + c$ The partial derivatives with respect to the parameters are $\\frac{\\partial y}{\\partial a} = x^2$ $\\frac{\\partial y}{\\partial b} = x$ None of the derivatives are functions of the parameters.","Consider now a water retention function of the form $\\Theta = \\frac{A}{A + |\\psi _m|^B}$ in which $\\Theta = S_e$ .","The partial derivatives with respect to the parameters are $\\frac{\\partial \\Theta }{\\partial A} = \\frac{|\\psi _m|^B}{(A + |\\psi _m|^B)^2}$ $\\frac{\\partial \\Theta }{\\partial B} = -\\frac{A|\\psi _m|^B \\log {|\\psi _m|}}{(A + |\\psi _m|^B)^2}$ It is easy to see that the partial derivatives with respect to the each parameter are functions of the parameters.","These types of functions cannot be fit to data by straightforward analytical and computational methods as in the case of linear and polynomial regression.","Nonlinear regression methods are required to fit nonlinear models.","Nonlinear regression methods rely on numerical methods based on successive approximations using iterative methods such as the Newton-Raphson, Gauss-Newton and Levenberg-Marquardt methods.","In each case an initial guess of the values of the parameters is provided by the user and the procedure tries to converge to an optimal parameter vector where the sum of the squares of the differences between the observed and predicted $y$ values is minimized.","This is a general description of one of the most common methods for fitting nonlinear regression models, nonlinear least squares [9].","Although there are other methods for fitting nonlinear models such as maximum likelihood, Bayesian estimation and other robust methods [9], nonlinear least squares is still the most common methods in commercial and noncommercial software packages.","A generalized nonlinear model can be written as a function of not only the independent vector $\\mathbf {x} $ but also of a parameter vector $\\mathbf {\\Omega } $ [9] $y_i = f(\\mathbf {x}; \\mathbf {\\Omega }) + \\epsilon _i$ in which $\\epsilon _i$ is random error term with mathematical expectation zero In practical terms the average is zero.","The procedure used must find a least squares estimate of $\\mathbf {\\Omega } $ , denoted by $\\hat{\\mathbf {\\Omega }} $ which minimizes the sum of squares [9], [83] $SSQ(\\mathbf {\\Omega }) = || \\mathbf {y} - \\mathbf {x} \\mathbf {\\Omega } ||^2 = \\sum _{i=1}^ {n} [y_i - f(x_i; \\mathbf {\\Omega })]^2$ In simple terms, $y_i$ are the measured values and $f(x_i; \\mathbf {\\Omega })$ are the values predicted by the function.","Thus, $y_i - f(x_i; \\mathbf {\\Omega })$ are the differences between measured and predicted values, often called residuals or errors which are sought to be minimized.","Because of their numerical nature, nonlinear regression models can be unstable and convergence to an optimal solution might not always happen.","As the complexity of the model and the number of parameters increases, the likelihood that the search procedure will diverge and not find the minimum on the parameters surface increases.","Because of that, a topic of research in statistics, numerical methods and computation is to find algorithms with adequate properties for nonlinear regression.","The search procedures discussed here try to find an optimal parameter vector by an iterative procedure of the type [9], [83] $\\mathbf {\\Omega } ^{(\\phi +1)} = \\mathbf {\\Omega } ^{(\\phi )} + \\delta ^{(\\phi )}$ In which $\\phi $ can be understood as the iteration number, thus the estimate of the $\\mathbf {\\Omega }$ vector at $\\phi +1$ is conditionally based on the estimate at $\\phi $ plus a step term $\\delta $ at the iteration $\\phi $ .","For the Gauss-Newton method [9] $\\delta ^{(\\phi )} = - (\\mathbf {J}^{(\\phi )T} \\mathbf {J}^{(\\phi )})^{-1} \\mathbf {J}^{(\\phi )T} \\mathbf {r}^{(\\phi )T}$ $\\mathbf {J}$ is the Jacobian matrix estimates, the Jacobian is the matrix of the derivatives with respect to the parameters, $T$ indicates a transposed vector or matrix, $-1$ indicates the inverse of the matrix and $\\mathbf {r}$ is a vector of residuals, i.e.","observed minus predicted values at a given iteration level.","Obviously there is much more to this, although the actual application to a simple function might not be very complicated from the programming standpoint.","A simple implementation of the Gauss-Newton method will be provided next.","Consider the full implementation of the iterative procedure for the parameter vector using the Gauss-Newton step $\\mathbf {\\Omega } ^{(\\phi +1)} = \\mathbf {\\Omega } ^{(\\phi )} - (\\mathbf {J}^{(\\phi )T} \\mathbf {J}^{(\\phi )})^{-1} \\mathbf {J}^{(\\phi )T} \\mathbf {r}^{(\\phi )T}$ Suppose now we want to fit a simple water retention function of the form $\\theta = \\frac{a}{|\\psi _m|^b}$ in which $\\theta $ is the volumetric water content and $\\psi _m$ is the matric potential, in generalized units not relevant for this example, and $a$ and $b$ are fitting parameters.","We wish to fit the model to the fictitious data set $\\theta = \\lbrace 0.4, 0.38, 0.35, 0.3, 0.28, 0.23, 0.22, 0.21 \\rbrace $ $\\psi _m = \\lbrace -1, -5, -10, -50, -100, -1000, -5000, -10000 \\rbrace $ $\\theta $ and $\\psi _m$ are vectors containing the observed water content and matric potential estimates.","We start the Gauss-Newton method with an initial guess for the parameter vector, say $a = 0.5$ and $b = 0.2$ .","Ideally the initial parameter vector should be defined taking into consideration the physical meaning of the parameters and/or the user should plot predicted and estimated values in the same graph and try to estimate the approximate best fit visually, so the parameter vector is close to the optimum values.","The parameter vector at the zeroth iteration contains the initial guess, thus $\\Omega ^{(0)} =\\begin{bmatrix}0.50\\\\0.20\\end{bmatrix}$ The Jacobian matrix is the matrix containg the partial derivatives of the residuals vector $\\mathbf {r} = \\theta - \\hat{\\theta }$ with respect to the parameters, where $\\hat{\\theta }$ is the vector containing the model estimates based on the parameter vector.","In simpler term, for each observation $i$ , the residuals function is $r_i = \\theta _i- \\frac{a}{|\\psi _{m,i}|^b}$ Thus the partial derivatives with respect to the parameters are $\\frac{\\partial r_i}{\\partial a} = -|\\psi _{m,i}|^{-b}$ $\\frac{\\partial r_i}{\\partial b} = a|\\psi _{m,i}|^{-b}\\log {|\\psi _{m,i}|}$ The Jacobian matrix is then populated as $J_{i,2} = \\lbrace \\frac{\\partial r_i}{\\partial a}, \\frac{\\partial r_i}{\\partial b} \\rbrace $ ThusWe are approximating everything to four decimals whenever possible to save space, however, it is almost never a good idea to work with low precision approximations in numerical methods $\\mathbf {J}^{(0)} =\\begin{bmatrix}-1.00000 & 0.00000 \\\\-0.72478 & 0.58324 \\\\-0.63096 & 0.72642 \\\\-0.45731 & 0.89449 \\\\-0.39811, & 0.91668 \\\\-0.25119, & 0.86757 \\\\-0.18206 & 0.77530 \\\\-0.15849, & 0.72987\\end{bmatrix}$ with transpose $\\begin{split}& \\mathbf {J}^{(0)T} = \\\\&\\begin{bmatrix}-1.0000& -0.7248& -0.6310& -0.4573& -0.3981& -0.2512& -0.1821& -0.1585 \\\\0.0000& 0.5832& 0.7264& 0.8945& 0.9167& 0.8676& 0.7753& 0.7299\\end{bmatrix}\\end{split}$ The product $\\mathbf {J}^{(0)T}\\mathbf {J}^{(0)}$ is then a $ 2 \\times 2$ matrix At this point it might convenient for the reader to review an introductory linear algebra book $\\mathbf {J}^{(0)T}\\mathbf {J}^{(0)} =\\begin{bmatrix}2.4124 & -2.1298 \\\\-2.1298 & 4.3948\\end{bmatrix}$ With inverse likewise calculated using linear algebra procedures $(\\mathbf {J}^{(0)T}\\mathbf {J}^{(0)})^{-1} =\\begin{bmatrix}0.7245 & 0.3511 \\\\0.3511 & 0.3977\\end{bmatrix}$ The residuals vector is $\\mathbf {r}^{(0)} =\\begin{bmatrix}-0.1000 \\\\0.0176 \\\\0.0345 \\\\0.0713 \\\\0.0809 \\\\0.1044 \\\\0.1290 \\\\0.1308 \\\\\\end{bmatrix}$ resulting in a Gauss-Newton step vector $\\delta ^{0} = (\\mathbf {J}^{(0)T} \\mathbf {J}^{(0)})^{-1} \\mathbf {J}^{(0)T} \\mathbf {r}^{(0)T}$ of $\\delta ^{(0)} =\\begin{bmatrix}0.1107 \\\\0.1582\\end{bmatrix}$ Noting that $\\mathbf {\\Omega } ^{(1)} = \\Omega ^{(0)} - \\delta ^{(0)}$ results in $\\mathbf {\\Omega } ^{(1)} =\\begin{bmatrix}0.3893 \\\\0.0418\\end{bmatrix}$ with $SSQ = 0.067$ .","Repeating the procedure for five iterations results in $a = 0.4099$ and $b = 0.0761$ and $SSQ = 0.00075715$ .","For comparison, a numerical procedure from a popular free software spreadsheet resulted in the same values after 40 iterations.","The Levenberg-Marquardt method improves upon the Gauss-Newton method allowing for search in two direction and for singular or ill conditioned matrices [9], [83] $\\delta ^{(\\phi )} = - (\\mathbf {J}^{(\\phi )T} \\mathbf {J}^{(\\phi )} + \\eta ^{(\\phi )} \\mathbf {D}^{(\\phi )} )^{-1} \\mathbf {J}^{(\\phi )T} \\mathbf {r}^{(\\phi )T}$ in which $\\mathbf {D}^{(\\phi )}$ is a diagonal matrix with positive elements and $\\eta $ is a damping factor.","The Levenberg-Marquardt procedure is usually efficient for Brutsaert-van Genuchten type equations, although the Gauss-Newton procedure is usually adequate if the number of parameters is reduced and if the initial guess is reasonably close to optimal values.","A program for fitting the Brutsaert-van Genuchten to soil water retention data is presented in Appendix REF (Figure REF ).","The Levenberg-Marquardt is invoked from a program module and thus was not written from scratch.","Figure: Brutsaert-van Genuchten equation (BvG) fit to simulated water retention data from the program in Appendix .The procedures described above can be also used in soil mechanical resistance equations of the form $SR = a \\rho ^b \\theta ^c$ Note that the partial derivatives indicate that this function is nonlinear $\\frac{\\partial SR}{\\partial a} = \\rho ^b \\theta ^c$ $\\frac{\\partial SR}{\\partial b} = ba\\theta ^c\\rho ^{(b-1)}$ $\\frac{\\partial SR}{\\partial c} = ca\\rho ^b \\theta ^{(c-1)}$ However, this equation is intrinsically linear since it can be transformed into a linear function.","In this case the transformation is achieved by applying logarithms at both sides $\\log {SR} = \\log {a} + b \\log {\\rho } + c \\log { \\theta }$ which can be fitted by multiple linear regression with fitting parameters $a$ , $b$ and $c$ .","Note, however, that linearization of intrinsically linear functions can result in undesirable statistical properties [9], [59] In order to approximate differential equations using finite difference methods, it is necessary to use a mathematical tool called Taylor series.","Taylor series allow us to expand a infinitely differentiable function around a point using its derivatives.","It is a fundamental tool in physics and mathematics, having applications in several different areas, especially in the approximation of functions which are too difficult or impossible to solve analytically.","Consider a real of imaginary valued function $f(x)$ which we want to expand around the point $a$ , in which $a$ is a real or complex number.","The Taylor series expansion is $f(x) = f(a) + \\frac{(x-a)}{1!}","\\frac{d }{dx} f(a) + \\frac{(x-a)^2}{2!}","\\frac{d^2 }{dx^2} f(a) + \\frac{(x-a)^3}{3!","}\\frac{d^3 }{dx ^3} f(a)+ ...$ which can be written in summation form as $f(x) = \\sum _{0}^{\\infty } \\frac{(x-a)^n}{n!}","\\frac{d^n }{dx ^n} f(a)$ An important case of the Taylor series is when the expansion is done around zero, being called Maclaurin series $f(x) = f(0) + \\frac{x}{1!}","\\frac{d }{dx} f(0)+ \\frac{x^2}{2!}","\\frac{d^2 }{dx^2}f(0) + \\frac{x^3}{3!","}\\frac{d^3 }{dx ^3} f(0)+ ...$ For most applications in physics, the higher order terms (higher powers) are small and can be neglected, thus resulting in a simple and often precise approximation of mathematical functions.","This was particularly important before the advent of electronic calculators and digital computers, when the estimations of functions which today are viewed as trivial such as the value of $\\pi $ where difficult tasks.","Examples of Taylor series approximations for a few simple functions are given below [93] $e^x = 1 + x + \\frac{x^2}{2} + \\frac{x^3}{6} + \\frac{x^4}{24} + ...$ $\\log (1 + x) = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + ...$ $(1 + x)^n = 1 + nx + \\frac{n(n-1)}{2} x^2 + \\frac{n(n-1)(n-2)}{6} x^3 + ...$ We rewrite Laplace equation for total water potential for a two dimensional domain $\\frac{\\partial ^2 h}{\\partial x^2} + \\frac{\\partial ^2 h}{\\partial y^2} = 0$ As we have discussed, this equation does have an analytical solution in terms of series.","However, the numerical solution of this equation provides a convenient introduction to finite difference methods.","Our goal here is to approximate the second order partial derivatives within a domain.","Let's define the domain as a region in space, here in the horizontal plane, with dimensions $L_x$ and $L_y$ .","The domain can be discretized in the form of a regular grid with nodes spaced $\\Delta x$ in the $x$ direction and $\\Delta y$ in the $y$ direction (Figure REF ).","Figure: Discretization for the finite differences solution of the 2D Laplace equation.The second order partial derivatives in the $x$ direction can be approximated between two adjacent points, where the next point is located in the positive $x$ direction of $h^j_i$ , by Taylor series expansion [46] $\\begin{split}h^j_{i+1} = & h^j_{i} + (x^j_{i+1} - x^j_{i}) \\frac{\\partial h}{\\partial x} + \\frac{(x^j_{i+1} - x^j_{i})^2}{2!}","\\frac{\\partial ^2 h}{\\partial x^2} + \\\\ & \\frac{(x^j_{i+1} - x^j_{i})^3}{3!}","\\frac{\\partial ^3 h}{\\partial x^3} + ...\\end{split}$ and for points is located in the negative $x$ direction of $h(i,j)$ by $\\begin{split}h^j_{i-1} = & h^j_{i} - (x^j_{i-1} - x^j_{i}) \\frac{\\partial h}{\\partial x} + \\frac{(x^j_{i-1} - x^j_{i})^2}{2!}","\\frac{\\partial ^2 h}{\\partial x^2} - \\\\ & \\frac{(x^j_{i-1} - x^j_{i})^3}{3!}","\\frac{\\partial ^3 h}{\\partial x^3} + ...\\end{split}$ Adding the two equations and noting that for our regular grid $(x^j_{i+1} - x^j_{i}) = \\Delta x$ $(x^j_{i-1} - x^j_{i}) = -\\Delta x$ results in $h^j_{i+1} + h^j_{i-1} = 2h^j_{i} + (\\Delta x)^2 \\frac{\\partial ^2 h}{\\partial x^2} + \\mathcal {O}((\\Delta x)^3)$ In which $\\mathcal {O}((\\Delta x)^n)$ are higher order terms represented using the big “o” notation $\\mathcal {O}$ .","If we ignore error terms with order greater than two, which tends to be increasingly smaller as the order of the term increases, the approximation to the second order derivative in $x$ is $\\frac{\\partial ^2 h}{\\partial x^2} = \\frac{h^j_{i+1} + h^j_{i-1} - 2h^j_{i}}{(\\Delta x)^2}$ The same procedure results in the approximation in $y$ $\\frac{\\partial ^2 h}{\\partial y^2} = \\frac{h^{j+1}_{i} + h^{j-1}_{i} - 2h^{j}_{i}}{(\\Delta y)^2}$ Replacing the approximations into the Laplace equation and solving for $h^j_i $ allows us to approximate the values of the total potential inside the domain, given the boundary conditions of fixed total potentials at the upper, lower, right and left boundaries using the formula $h^j_{i} = \\frac{1}{4} [h^j_{i+1} + h^j_{i-1} + h^{j+1}_{i} + h^{j-1}_{i}]$ The solution is obtained by numerical methods.","A program for solving the 2D Laplace equation using the Jacobi iterative method is presented in Appendix REF .","A surface plot The figure was produced with Gnuplot: http://www.gnuplot.info/ of the solution for $h_{left} = 20~cm$ , $h_{right} = 1~cm$ , $h_{top} = 5~cm$ and $h_{bottom} = 0~cm$ is presented in Figure REF .","Figure: Numerical solution of the 2D Laplace equation from the program in Appendix .The discretization of an 1D partial differential equation in time and space follows a similar scheme as 2D discretization in space.","In diffusion type equations (e.g.","advective-dispersive, Richards-Richardson, heat equation) besides the second order partial derivative in space, there is a first order derivative in time.","Recall the 1D advective-dispersive equation for constant flux velocity and diffusion coefficient $\\frac{\\partial C}{\\partial t} = D \\frac{\\partial C}{\\partial x} - v \\frac{\\partial ^2 C}{\\partial x^2}$ We already know from the previous section that the second order partial derivatives in space can be approximated by $\\frac{\\partial ^2 C}{\\partial x^2} = \\frac{C^j_{i+1} + C^j_{i-1} - 2C^j_{i}}{(\\Delta x)^2}$ Obviously for concentration $C$ in this case.","To find the approximation of first order partial derivatives, the Taylor series expansion equations as in Equations REF REF are subtracted.","Ignoring higher order terms and solving for the partial derivatives of interest, the discretization of the first order partial derivatives in space and time for the advective-dispersive equation are $\\frac{\\partial C}{\\partial x} = \\frac{C^j_{i+1} + C^j_{i-1}}{(2\\Delta x)}$ $\\frac{\\partial C}{\\partial t} = \\frac{C^j_{i+1} + C^j_{i-1}}{(2\\Delta t)}$ The discretization is now in space and time, where $i$ is the increment in space and $j$ the increment in time.","Care must be taking in writing programs regarding the order of $i$ and $j$ in loops.","For the discretization in time, the condition at $t=0$ is now the initial contition and the conditions in the inlet and outlet of the 1D domain are the boundary conditions (Figure REF ).","Figure: Discretization for the finite differences solution of the 1D advective-dispersive equation.By replacing the partial derivatives in time and space, the finite differences equation to be solved for the 1D advective-dispersive equation is $C^j_{i} = \\Delta t C^{j-1}_{i} + D\\frac{\\Delta t}{\\Delta x^2} (C^{j-1}_{i-1} - 2C^{j-1}_{i} + C^{j-1}_{i+1} ) - v\\frac{\\Delta t }{2 \\Delta x} (C^{j-1}_{i+1} - C^{j-1}_{i-1} )$ One of the issues of the numerical solution of the advective-dispersive equation is that it is valid for infinitely long columns.","To circumvent this issue a program was written to solve the equation for a long column, but the outlet concentration measurements were made a fixed point along the length $x$ .","The comparison of the numerical and analytical solutions of the advective-dispersive equation for concentration measured at $50~cm$ from the inlet is presented in Figure REF .","The column length for the numerical simulations was $L = 100~cm$ , the flow velocity was $v = 2.0~cm~h^{-1} $ , the dispersion coefficient $D = D_h/\\theta = 5.0~cm^2~h^{-1} $ and the simulation time was $100~h$ .","The program is listed in the Appendix REF .","Figure: Comparison of numerical and analytical solutions of the 1D advective-dispersive equation from the program in Appendix .By the same principles illustrated for the Laplace and advective-dispersive equations, the Richardson-Richards can be discretized to be solved by explicit methods.","Recalling the 1D matric potential (here represented as $\\psi $ for simplicity) form of the Richardson-Richards equation in the general coordinate $z$ [50] $C^*(\\psi )\\frac{\\partial \\psi }{\\partial t} = \\frac{\\partial }{\\partial z} [K(\\psi ) (\\frac{\\partial \\psi }{\\partial z} + 1 )]$ The main issue here is that the unsaturated hydraulic conductivity is not constant and cannot be placed outside of the derivative terms.","The hydraulic capacity function also needs to be defined for this equation to be solved numerically.","Following the discussion in Chapter REF we will define the following water retention and unsaturated hydraulic conductivity functions [50] $\\theta (\\psi ) = \\theta _r + \\frac{\\alpha (\\theta _s - \\theta _r)}{\\alpha + |\\psi |^{\\beta }}$ $K(\\psi ) = \\frac{A}{A + |\\psi |^B}$ Recalling that the hydraulic capacity function is $C^*(\\psi ) = [\\alpha h_m (\\theta _s - \\theta _r) \\psi ^{\\beta -1}] \\frac{\\beta (\\alpha + \\psi ^ {\\beta })}{\\alpha + \\psi ^{\\beta }}$ The discretization for the explicit solution is [50] $\\begin{split}{\\psi }^{j+1}_{i} = & {\\psi }^{j}_{i} + \\frac{\\Delta t}{{C^*}^{j}_{i} \\Delta z} [ K^{j}_{i+1/2} (\\frac{{\\psi }^{j}_{i+1} - {\\psi }^{j}_{i}}{\\Delta z} -1 ) \\\\& - K^{j}_{i-1/2} (\\frac{{\\psi }^{j}_{i} - {\\psi }^{j}_{i-1}}{\\Delta z} -1 ) ]\\end{split}$ where the unsaturated hydraulic conductivity at intermediary point in nodes is $K^{j}_{i+1/2} = \\frac{K^{j}_{i+1}+ K^{j}_{i}}{2}$ $K^{j}_{i-1/2} = \\frac{K^{j}_{i}+ K^{j}_{i-1}}{2}$ Thus, the numerical solution can be achieved by applying the discretization schemes above while simultaneously evaluating the functions $K(\\psi )$ and $^*C(\\psi )$ at the $\\psi ^j_i$ points.","The grid is analogous to the one described for the advective-dispersive equation.","A program for solving the 1D Richardson-Richards equation using an explicit scheme is presented in Appendix REF .","The results for the parameters $\\alpha = 1.611 \\times 10^{6}~cm^{-1}$ , $\\theta _s = 0.287~cm^3~cm^{-3} $ , $\\theta _r = 0.075~cm^3~cm^{-3}= $ and $\\beta = 3.96$ are compared to the Philip's quasi-analytical solution presented by [50] for an $80~cm$ 1D soil column initially at $\\psi = -61.5~cm$ ($t=0$ ) and with $\\psi = -20.73~cm$ at soil surface and $\\psi = -61.5~cm$ at the bottom of the column (or profile) for $t > 0$ (Figure REF ).","The unsaturated hydraulic conductivity parameters were $ A = 0.00944$ and $B = 4.74$ .","Figure: Comparison of numerical and quasi-analytical solutions of the 1D Richardson-Richards equation from the program in Appendix .","The quasi-analytical solution follows Philip's method and is presented in .Although explicit methods are mathematically simpler and easier to implement from the computational point of view, they are notoriously unstable for most real world problems.","The discretization scheme is then limited by the stability of the method.","For most application implicit methods are preferred, these methods result in a set of equations which needs to be solved using matrix manipulation algorithms.","In the implicit discretization of the 1D equation, there is no simpler equation which can be inserted into a code to solve the problem, a simultaneous solution for the different nodes is sought.","Two popular choices of discretization are the Backward Time Centered Space (BTCS) $(-\\frac{\\gamma }{\\Delta x^2}) u^{j+1}_{i-1} + (\\frac{1}{\\Delta t} + \\frac{2\\gamma }{\\Delta x^2}) u^{j+1}_{i} + (-\\frac{\\gamma }{\\Delta x^2}) u^{j+1}_{i+1} = \\frac{1}{\\Delta t} u^{j}_{i}$ and the Crank-Nicolson scheme [30] A very good introduction to these methods can be found in https://web.cecs.pdx.edu/~gerry/class/ME448/ from which these discretization schemes were transcribed.","$\\begin{split}(-\\frac{\\gamma }{2\\Delta x^2}) u^{j+1}_{i-1} + (\\frac{1}{\\Delta t} + & \\frac{\\gamma }{\\Delta x^2}) u^{j+1}_{i} + (-\\frac{\\gamma }{2\\Delta x^2}) u^{j+1}_{i+1} = \\\\& \\frac{\\gamma }{2\\Delta x^2}) u^{j}_{i-1} + (\\frac{1}{\\Delta t} - \\frac{\\gamma }{\\Delta x^2}) u^{j}_{i} + \\frac{\\gamma }{2\\Delta x^2} u^{j}_{i+1}\\end{split}$ In which $u^j_i$ is the temperature at each grid point, $\\gamma = \\kappa _T/ (\\rho c_p)$ is the thermal diffusivity and the other parameters are the same as previously defined.","These numerical schemes can be applied to the Richardson-Richards and other partial differential equations.","One common strategy of solution is the diagonal matrix solver, often called the Thomas algorithm [26].","Technically, the explicit scheme can be thought as an special case of the discretization of the heat equation called the Backward Time Centered Space (BTCS).","The Crank-Nicolson and other more complex schemes are recommended over BTCS and (Forward Time Ceneterd Space) FTCS for real world applications.","The main advantages of simpler methods, specially the FTCS is that it can be used as a teaching tool for a first introduction to finite differences methods, as it is done in most numerical methods courses and books.","In many problems other even more complicated schemes, both mathematically and computationally are employed, namely Finite Elements, Finite Volume and Lattice Boltzmann methods A simple introduction to these should be included in further editions.","Most modern computational fluid mechanics, soil physics and groundwater hydrology numerical codes are based on one or more of these methods.","ll $ y $ Generic dependent variable $ x $ Generic independent variable $ a, b, c, d, e $ Generic fitting parameters $ y_0 $ Generic $y$ coordinate $ x_0 $ Generic $x$ coordinate $ \\Theta $ Effective saturation $ = (\\theta - \\theta _s)/(\\theta _s - \\theta _r)$ $ \\theta $ Volumetric water content $ \\theta _s $ Saturation volumetric water content $ \\theta _r $ Residual volumetric water content $ A, B $ Unsaturated hydraulic conductivity function empirical parameters $ \\psi _m $ Matric potential in unspecified units $ SSQ $ Sum of squares of the residuals $ \\Omega $ Parameter vector $ \\mathbf { y} $ Observed dependent variable values vector $ \\mathbf { x} $ Observed independent variable values vector $ \\delta $ Step in nonlinear regression procedures $ \\mathbf { J} $ Jacobian matrix $ \\mathbf { r} $ Predicted values vector $ \\mathbf { D} $ Diagonal matrix $ \\phi $ Iteration level $ \\eta $ Damping factor $ SR $ Soil mechanical resistance to penetration $ \\rho $ Bulk density $ h $ Total potential in length units $ i, j$ Integer variables for step in space and/or time $ \\Delta x, \\Delta y, \\Delta t $ Discretization interval in time and space $ C $ Concentration $ v $ Flow velocity $ D $ Dispersion coefficient corrected to volumetric water content $= D_h/\\theta $ $ t $ Time $ h_m $ Matric potential in length units $ \\psi $ Matric potential in length units $ C^*(h_m) $ Hydraulic capacity function $ = \\partial \\theta /\\partial h_m$ $ \\theta (h_m) $ Water retention function $ K(h_m) $ Unsaturated hydraulic conductivity function $ \\alpha , \\beta $ Water retention function fitting parameters $ u$ Temperature at specified grid points $ y^j_i$ Specifies the value of the property at $i$ and $j$ for any function $y$ $ \\gamma $ Thermal diffusivity $ \\kappa _T $ Thermal conductivity $ c_p$ Specific heat capacity tocchapterBibliography $\\alpha $ , $A $ : alpha $\\beta $ , $B $ : beta $\\gamma $ , $\\Gamma $ : gamma $\\delta $ , $\\Delta $ : delta $\\epsilon $ ,$\\varepsilon $ , $E $ : epsilon $\\zeta $ , $ Z $ : zeta $\\eta $ , $H $ : eta $\\theta $ , $\\Theta $ : theta $\\iota $ , $I $ : iota $\\kappa $ , $ K $ : kappa $\\lambda $ , $\\Lambda $ : lambda $\\mu $ , $ M $ : mu $\\nu $ , $ N $ : nu $\\xi $ , $ \\Xi $ : xi $o$ , $ O $ : omicron $\\pi $ , $ \\Pi $ : pi $\\rho $ , $ R $ : rho $\\sigma $ , $ \\varsigma $ , $ \\Sigma $ : sigma $\\tau $ , $ T $ : tau $\\upsilon $ , $ Y $ : upsilon $\\phi $ , $\\varphi $ , $ \\Phi $ : phi $\\chi $ , $X$ : chi $\\psi $ , $ \\Psi $ : psi $\\omega $ , $\\Omega $ , $ \\Phi $ : omega T.P.","Leão in Oct 5 2021, light revision in Aug 29 2022 Fits the Brutsaert-van Genuchten equation to data using the Leverberg-Marquardt algorithm Language: Python 3.8.10 This code does not check for regression diagnostics Import the necessary packages import numpy as np from scipy.optimize import curvefit, leastsquares import matplotlib.pyplot as plt Data psi = np.array([20, 40, 60, 80, 100, 200, 500, 800, 1000, 3000, 5000, 15000]) theta = np.array([0.41, 0.409, 0.406, 0.41268, 0.41045, 0.39708, 0.3899, 0.3525, 0.3203, 0.18478, 0.14497, 0.13]) Define initial parameters (Initial guess for the nonlinear regression procedure) thetas = 0.4 thetar = 0.1 alpha = 0.001 n = 2.0 m = 0.5 Defines the nonlinear regression model to be fit def vg(x, qs, qr, alpha, n, m): return qr + (qs - qr)/(1+ (alpha*psi)**n)**m Puts the initial parametes into the \"InitialParams\" vector InitPar = [thetas, thetar, alpha, n, m] Curve fitting procedure - \"lm\" is the Leverberg-Marquardt algorithm popt, pcov = curvefit(vg, psi, theta, p0=InitPar, method='lm') Creates values of psi for plotting the predicted data xd2 = np.linspace(10, 30000, 10000) Uses the parameters from the fit to create the data for the predicted plot thetap2 = popt[1] + (popt[0] - popt[1])/(1+ (popt[2]*xd2)**popt[3])**popt[4] computes the standard deviation of the parameters perr = np.sqrt(np.diag(pcov)) print(\"Best fit paramaters, thetas, thetar, alpha, n and m \", popt) print(\"Covariance of the parameters \", popt) print(\"Standard deviation of the parameters \", perr) Plot of the data and best fit line plt.plot(psi, theta, \"o\", label = \"Simulated data\") plt.plot(xd2, thetap2, \":\", label = \"van Genuchten (1980)\") plt.ylim(0, 0.45) plt.ylabel('Volumetric water content [vol/vol]') plt.xlabel('Matric potential [|cm|]') legend = plt.legend(loc='lower left', shadow=True, fontsize='large') plt.xscale(\"log\") plt.show() // Solution to the laplace equation 2D using an explicit finite differences scheme // T.P.","Leao on 5 out 2021 light revision on Aug 29 2022 // Language: C++ include include include include int main() // define the simulation parameters double dx = 5.0; double dy = 5.0; double X = 100.00; // x length double Y = 100.00; // y length int nx = X/dx+1; int ny = Y/dy+1; int i, j; // define the grid double h[nx][ny]; // initialize the array with zeros for(i=0; i < nx; i++) for(j=0; j < ny; j++) h[i][j] = 0.0; std::cout << \"————————————-\" << std::endl; // define the boundary conditions for(i=0; i < nx; i++) h[i][0] = 5.0; h[i][ny-1] = 1.0; for(j=1; j < ny; j++) h[0][j] = 20.0; h[nx-1][j] = 0; // Solves the system double d[1000], diff; double tol = 1e-5; std::cout << \"Iterations and diff \" << std::endl; for (int iter = 0; iter <100; iter++) // Maximum number of iterations for(i=1; i < nx-1; i++) for(j=1; j < ny-1; j++) h[i][j] = 0.25 * (h[i+1][j] + h[i-1][j] + h[i][j+1] + h[i][j-1]); d[iter] = h[1][1]; diff = d[iter] - d[iter-1]; if (diff < tol) break; std::cout << iter << \"\" << diff << std::endl; // Print iteration and difference between sucessive iterations at h[1][1] std::cout << \"Solution in gnuplot format \" << std::endl; for(i=0; i < nx; i++) for(j=0; j < ny; j++) std::cout << std::setprecision(6) << h[i][j] << \"\"; // Prints solution in terminal std::cout << std::endl; return 0; // Solution to the advective dispersive 1D equation using an explicit method // T.P.","Leao 01 out 2021 light revision 29 aug 2022 // Language: C++ include include include include int main() // define the simulation parameters double dx = 1.0; double dtau = 0.1; double TMAX = 500.0; double L = 100.0; double D = 5; double V = 2.0; int nx = L/dx+1; int ntau = TMAX/dtau+1; int i, j; // define the grid double C[ntau][nx]; // initialize the array with zeros for(i=0; i < nx; i++) for(j=0; j < ntau; j++) C[j][i] = 0.0; std::cout << \"————————————-\" << std::endl; // define the initial conditions for(i=0; i < nx; i++) C[0][i] = 0.0; // define the boundary conditions // for(j=1; j < ntau; j++) C[j][0] = 1.0; C[j][nx-1] = 0.0; // this is the explicit solution for(j=0; j < ntau-1; j++) for(i=1; i < nx-1; i++) C[j+1][i] = C[j][i] + dtau/pow(dx,2)*D*(C[j][i-1] -2*C[j][i] +C[j][i+1]) - V *dtau/(2*dx)*(C[j][i+1] - C[j][i-1]); std::cout << \" Explicit solution to the 1D advective-dispersion equation in time \"; std::cout << \"\"; std::cout << \" \"; std::cout << \"\"; for(j=0; j < ntau; j++) std::cout << \"| \"; for(i=0; i < nx; i++) std::cout << C[j][i] << \" \" ; return 0; // Solves the 1D richards equatio using and explicit finite differences scheme // This code has been validaded using Haverkamp 1977 analytical solutions using Philip's model // The discretization also follows Haverkamp 77 // In 2 out 2021 - TPL, light revision 29 aug 2022 // Language: C++ include include include include // static parameters for Warwick hydraulic functions, see Zarba 1988 MIT Masters define thetas 0.287 define thetar 0.075 define alpha 1.611e06 define beta1 3.96 define Ks 9.44e-3 define A 1.175e06 define B 4.74 using namespace std; // define the functions for water retention and hydraulic conductivity double theta(double h) double theta; theta = alpha * (thetas-thetar)/(alpha+ pow(abs(h), beta1)) + thetar; return theta; double K(double h) double K; K = Ks * A/(A + (pow(abs(h), B))); return K; double C(double h) double C; C = alpha*(thetas-thetar) * pow(abs(h), beta1-1) * beta1/((alpha+ pow(abs(h), beta1))* (alpha+ pow(abs(h), beta1))); return C; int main() double dz = 1.0; double dt = 0.4; int j, i; double L = 80.0; double tmax = 360; int nx = L/dz + 1; int ntau = tmax/dt + 1; double h[ntau][nx];//, K[ntau][nx], C[ntau][nx], q[ntau][nx]; for (j = 1; j < ntau; j++) h[j][0] = -20.73; h[j][nx-1] = -61.5; for (i = 0; i < nx; i++) h[0][i] = -61.5; for (j=0; j < ntau-1; j++) for (i=1; i < nx-1; i++) h[j+1][i] = h[j][i] + dt*1.0/(C(h[j][i])*dz) * ( (K(h[j][i+1])+K(h[j][i]) )/2.0 * ((h[j][i+1]-h[j][i])/dz -1.0 ) - (K(h[j][i])+K(h[j][i-1]) )/2.0 * ((h[j][i]-h[j][i-1])/dz - 1.0 )); for (j = 0; j < ntau; j++) for (i = 0; i < nx; i++) cout << i << \"\" << h[j][i]<<\"\"; cout << endl; // saves output to *.dat file std::ofstream outdata; std::ofstream output; outdata.open(\"rich1d.dat\"); for (int j=0; j < ntau; j++)outdata << std::endl; //std::cout << j << std::endl; for (int i=0; i < nx; i++) outdata << h[j][i] <<\"\"; outdata.close(); cout << endl; cout << \"—————————\" << endl; cout << \"Results saved to rich1d.dat\" << endl; return 0;"]],"string":"[\n [\n \"Physics of Soil and Other Natural Porous Media\"\n ],\n [\n \"Abstract The purpose of this monograph is to review the early theoretical basis of what is known today as soil physics and to serve as a textbook for intermediate porous media physics or transport in porous media graduate courses.\"\n ],\n [\n \"colorlinks=true, linkcolor=blue, filecolor=blue, citecolor = black, urlcolor=red, tocchapterPreface The purpose of this monograph is to review the early theoretical basis of what is known today as soil physics and to serve as a textbook for intermediate level porous media physics or transport in porous media graduate courses.\",\n \"In writing this material we tried to avoid as much as possible the presentation and discussion of empirical methods lacking a rigorous physical basis.\",\n \"Because much of agricultural soil physics is still based on empirical methods, it was not possible to avoid discussing water retention and unsaturated hydraulic conductivity equations, which still lack a rigorous theoretical basis.\",\n \"The fully empirical soil resistance equation is also discussed in a few sections.\",\n \"We also avoided relying on soil physics books opting to consult original sources in fluid mechanics, mathematics, engineering and porous media physics, from which much of soil physics is largely derived.\",\n \"Because of that we understand that “Physics of Soil and Other Natural Porous Media” is a more appropriately title than “Soil Physics”, which imply restriction in methods and applications.\",\n \"We are deeply indebted in writing this manuscript to the early predecessors that laid the foundations for porous media physics and fluid dynamics, namely the mathematicians L. Euler, G.G.\",\n \"Stokes, H. Lamb, G.I.\",\n \"Taylor and G.I.\",\n \"Batchelor, the physicists I. Newton, L.F. Richardson, E. Buckingham, M. Muskat, L.A. Richards, L.D.\",\n \"Landau, E.M. Lifshitz, A.E.\",\n \"Scheidegger, the engineers H. Darcy, J.\",\n \"Bear and his collaborators, G. de Josselin de Jong, J. Philip.\",\n \"We also became aware of a profound contributions to transport phenomena in porous media of a class unknown to me and largely unsung in the soil science community, the chemical engineers, listing here R. Aris, H. Brenner, F.A.L.\",\n \"Dullien, J.C. Slattery, R.B.\",\n \"Bird, W.E.\",\n \"Stewart and E.N.\",\n \"Lightfoot.\",\n \"Not forgotten is the dedication of many mathematicians, computer scientists and physicist who worked on numerical and computational methods.\",\n \"Of course there are many people with significant contributions to theoretical soil physics and transport phenomena in porous media, scattered in many publications for at least two centuries, too many to mention.\",\n \"We plan in gradually introducing newer literature and expanding the mathematical foundations in later editions, always aiming at maintaining the physical basis of the processes and phenomena and avoiding purely empirical methods.\",\n \"Because of the nature of this material and the amount of mathematics required, many mistakes and misspellings are expected in earlier drafts.\",\n \"Potential users are encouraged to reports mistakes and to provide suggestions and corrections to improve the material.\",\n \"Drafts will be indicated as fractional numbers on the cover, while main releases are indicated as integers.\",\n \"A quantity is said to be a scalar if it is not associated with both magnitude and direction.\",\n \"One example of a scalar is the temperature at a given point in space.\",\n \"If you are interested in the temperature at a given point at a given time, the measurement is associated with a single value.\",\n \"In this case, if we use the symbol T to represent temperature and the measured value is in Kelvin units, it can be represented as $T = 376.7 K$ Now if you record several instances of the temperature in space and/or time you have a set of all the temperatures recorded and temperature is now a variable.\",\n \"$T = \\\\left\\\\lbrace 376.7 , 375.6 , 370.7 , 369.9 , ..., N \\\\right\\\\rbrace $ where $N$ is the $n^{th}$ temperature measurement.\",\n \"In general you would want to associate the variable of interest with a system of spacial coordinates and/or time measurements.\",\n \"Imagine that your thermometer is placed in a fixed position on a weather station and records temperature at regular intervals throughout the year.\",\n \"Each temperature measurement is now associated with a time measurement, usually the date and the time in hours, minutes and seconds.\",\n \"A table of such measurements would look like Table REF .\",\n \"Table: Temperature measurements recorded at different time intervals.Now we can assign the temperature as a function of time at that point in space.\",\n \"In reality, temperature is not a function of time, nothing really is a function of time, temperature varies because the position of the sun in the sky and its influence on incident radiation, season of the year, cloud cover, convective air flow and many other factors, but sometimes it is easier for us to model it simply as a function of time.\",\n \"Temperature as a function of time can be represented as $T = f(t)$ or simply as $T(t)$ .\",\n \"One class of empirical functions that can be used to model temperature is $T(t) = T_{0} + \\\\alpha \\\\cos {(\\\\omega t + \\\\phi )}$ In this equation temperature and time are variables since each represents a set of values, while $T_{0}$ , $\\\\alpha $ , $\\\\omega $ and $\\\\phi $ are parameters which can be obtained by fitting the equation to data.\",\n \"Fitting can be achieved using statistical techniques such as least-squares regression or could also be done visually for such a simplistic equation, as long as one is aware of the potential limitations of doing so.\",\n \"The parameters values obtained by fitting the equation to observed data are usually refer to fitting parameters or empirical parameters.\",\n \"In this case each has a specific mathematical meaning.\",\n \"$\\\\alpha $ is the amplitude of oscillation of the sinusoidal wave produced, $T_{0}$ is the value in which the predictions are centered, $\\\\omega $ is the angular frequency and $ \\\\phi $ is the phase lag.\",\n \"Equation 1.3 is said to be univariate as temperature $T$ depends on time alone.\",\n \"Notice that the parameters determine the shape of the function but $T$ is not a function of the parameters since they do not vary.\",\n \"$T$ is a functions of time, $t$ , alone in this case (Figure REF ).\",\n \"Figure: Example of a sinusoidal function of time with parameters T 0 T_{0} = 15.5, α\\\\alpha = 10, ω\\\\omega = 2 and φ\\\\phi = 10.Given any parameterization of Equation 1.3, one might be interested in the rate of variation of temperature as a function of time.\",\n \"The rate of variation of a given variable as a function of the same independent variable can be calculated using the first derivative of the function.\",\n \"There are many symbolic representations for the derivative of a function.\",\n \"For example, given a generic function $y = f(x) $ its first derivative can be represented using Lagrange's notation as $y^{\\\\prime }(x)$ and by Leibniz's notation as $dy/dx$ .\",\n \"Newton's dot notation, common in classical mechanics is usually reserved for position as a function of time, $x(t)$ for example, such that its velocity is $v = \\\\dot{x}(t) $ and its acceleration is $ a = \\\\dot{v}(t)= \\\\ddot{x}(t) $ .\",\n \"In classical mechanics $x$ , $v$ and $a$ are usually vector quantities which were not denoted as such here because we haven't talked about vectors yet.\",\n \"We shall use Leibniz's notation in this book and as such the first derivative of Equation 1.3 is $\\\\frac{ dT(t)}{dt} =-A \\\\omega \\\\sin {(\\\\omega t + \\\\phi )}$ If one is interested on the rate of variarion of the rate of variation of the temperature as a function of time, what is needed is the second derivative of the function, represented using the same notation as $\\\\frac{ dT^{2}(t)}{dt^{2}} =-A \\\\omega ^{2} \\\\cos {(\\\\omega t + \\\\phi )}$ and so forth.\",\n \"For now you don't have to worry about the rules of differentiation, we will get to that in a bit, what you should understand first is the geometrical meaning of the derivative.\",\n \"Observe the graph of the first derivative of Figure REF .\",\n \"The dashed line in Figure REF is the rate of change of temperature as a function of time.\",\n \"Notice that at the maximum and minimum values of the original function (Point A), the value of the derivative is zero (Point B) while at the inflection points, where the curvature of the first graphs changes from upwards to downwards and vice-versa, the rate of change, or the derivative function, is at its absolute maximum values.\",\n \"Let's now analyze a simple example from classical mechanics.\",\n \"Suppose an object is relased from rest from a height of 10 meters.\",\n \"Following Newton's laws the governing equation is $Y(t) = Y_{0} + v_{0} t - \\\\frac{1}{2}g t^{2}$ The only variables in this equation are $Y$ and $t$ , everything else is either a constant or can assumed to be constant if the height of fall is small compared to the radius of Earth, as is the case with gravitational acceleration.\",\n \"As before, we are treating gravitation acceleration as a scalar, since we are assuming it as a constant and are only interested in its effect in the $Y$ direction.\",\n \"$Y_{0}$ is the initial height from which the object is released, $v_{0}$ is the initial velocity, Y is the predicted height and t is time.\",\n \"If the initial height is $Y_{0} = 10 \\\\ m $ and if the object is released from rest, such that $v_{0} = 0~m~s^{-1} $ , we have $Y(t) = 10 - \\\\frac{1}{2}g t^{2}$ Notice that the height $Y$ decreases as a function of time, as it is expected, since the object is falling on Earth's gravitational field.\",\n \"At one point $Y$ will reach zero, which is when the object reaches the surface.\",\n \"It is also important to notice that the equation predicts negative heights.\",\n \"This means only that the object has passed a surface that we assumed as a reference point, such that all gravitational potential energy considerations are made with reference to that point, if we were interested in that.\",\n \"It turns out we aren't right now but that will be fundamental when we study the energy of water in porous media.\",\n \"If we plot $Y(t)$ as a function of time you will see that position $Y$ decreases quadratically with time, since time is raised to the power of two in this equation.\",\n \"$Y$ is a polynomial function of time in this case.\",\n \"Extracting the roots of such polynomials can provide important information in physics problems, as we shall discuss later.\",\n \"For now suppose you want the velocity as a function of time.\",\n \"We know that the velocity is the rate of variation of distance over time and this can be expressed in derivative form as $\\\\frac{dY(t)}{dt} = v(t) = - g t$ The velocity is a linear function of time, meaning that it increases at a constant rate as the object falls.\",\n \"As you might expect the rate with which the velocity increases is the gravitational acceleration of Earth, $g$ .\",\n \"To further make that clear let us calculate the rate of change of velocity with time, or the acceleration $\\\\frac{dY^{2}(t)}{dt^{2}} = \\\\frac{dv(t)}{dt} = a = - g$ With that, one can be convinced that an object falling under Earth's gravitational field falls with an acceleration equal to Earth's gravitational acceleration.\",\n \"Suppose now that all you known is that an object is falling under a gravitational field equal to $g$ and you have no equation of motion.\",\n \"From Equation 1.9, we know that the rate of variation of velocity of the object is $\\\\frac{dv(t)}{dt} = - g$ As before we also know that the object was released from a height of 10 m with initial velocity equal to zero.\",\n \"We can rearrange Equation 1.10 by multiplying both sides of the equality by $dt$ $dv(t) = - g dt$ This is one of the simplest possible cases of what is called a differential equation and to remove the differential from both sides all we need to do is integrate it $\\\\int dv(t) = \\\\int - g dt = - g \\\\int dt$ Since g is a constant.\",\n \"As this is an indefinite integral, the solution includes a constant of integration $C_{1}$ .\",\n \"$v(t) = - g t + C_{1}$ Constants will routinely appear in the solutions of differential equations in physical problems.\",\n \"To find the value of the constant one needs specific conditions of the problem under evaluation.\",\n \"These can be called initial conditions or, in many applications, boundary conditions.\",\n \"We know here that when the velocity is zero, the time is also zero, in other words when the experiment was set, the very moment when the object was released from rest it was at a height of 10 m. Therefore, our initial conditions are $v = 0,\\\\ t = 0$ Substitution in Equation 1.13 results $0 = - g 0 + C_{1} \\\\Rightarrow C_{1} = 0$ Now from Equations 1.8, 1.13 and 1.15 $dY(t) = v(t)dt = - g t dt$ By repeating the integration procedure we arrive at $Y(t) = - \\\\frac{1}{2}g t^{2} + C_{2}$ Since we know that the height is 10 m when time is zero we arrive at $Y(t) = - \\\\frac{1}{2}g t^{2} + 10 = 10 - \\\\frac{1}{2}g t^{2}$ Which is the exact same equation we started with (Equation 1.7).\",\n \"The objective of the steps from Equations 1.6 to 1.18 were two-fold, first to show the relationship between derivation and integration and how such operations can appear on the solutions of physical problems, and second, mainly on the steps covered from Equations 1.10 to 1.18, to show the logic behind what most students outside the fields of physics and mathematics understand by mathematical modeling.\",\n \"In a broad sense, mathematical modeling is viewed as the solution of a differential equation applied to a set of initial or boundary conditions.\",\n \"As physicists and mathematicians know well simple solutions to these problems are rare if they exist at all.\",\n \"Nevertheless, the strategy used here can be applied to simplifications of mechanics and dynamics problems, such as uniform flux through an isotropic medium with constant boundary conditions.\",\n \"Derivation and integration are at the core of physics and applied mathematics.\",\n \"These techniques were developed along with the disciplines of physics and mathematics during the “Scientific Revolution\\\" on around the years 1600-1700s to solve real problems in mechanics by the likes of Newton and Lebniz, followed by improvements and innovations by mathematicians such as Laplace, Lagrange and Gauss.\",\n \"The discipline of fluid dynamics was developed shortly after, based on the theoretical insights provided by Isaac Newton.\",\n \"Leonard Euler, contemporary of Laplace and Lagrange laid the foundation of much of what is now known as fluid dynamics, followed by fundamental developments by George G. Stokes and Claude-Louis Navier which fathered the equation at the heart of fluid mechanics, the Navier-Stokes equations.\",\n \"Now that we have an idea how differentiation and integration can be used to solve physical problems, lets look at some of the basic rules and symbols associated with these techniques.\",\n \"For a function $f(x)$ the following basic rules of differentiation apply.\",\n \"The derivative of a constant is zero $\\\\frac{d}{dx}c = 0$ In other words, there can be no rate of variation of a constant function $f(x) = c$ , the value remains constant with time and is equal to c. The most basic rule of differentiation can be written as $\\\\frac{d}{dx} x^{n} = nx^{n-1}$ For any $ n \\\\in \\\\mathbb {R} $ .\",\n \"A constant multiplied by any function can be written out of the differentiation $\\\\frac{d}{dx} cf(x) = c \\\\frac{d}{dx} f(x)$ For $ c $ constant.\",\n \"If we have two functions of $x$ , say $u(x)$ and $v(x)$ the following sum, product and quotient rules apply.\",\n \"$\\\\frac{d}{dx} (u + v) = \\\\frac{du}{dx} + \\\\frac{dv}{dx}$ $\\\\frac{d}{dx} (uv) = u\\\\frac{dv}{dx} + v\\\\frac{du}{dx}$ $\\\\frac{d}{dx} (\\\\frac{u}{v}) = \\\\frac{v\\\\frac{du}{dx} - u\\\\frac{dv}{dx}}{v^{2}}$ A few examples are $\\\\frac{d}{dx}2 = 0$ $\\\\frac{d}{dx} x^{3} = 3x^{3-1} = 3x^{2}$ $\\\\frac{d}{dx} 2f = 2 \\\\frac{d}{dx}x^{3} = 6x^{2}$ $\\\\frac{d}{dx} (2x^{3} + 3x^{2}) = 6x^{2} + 6x$ $\\\\frac{d}{dx} (x^{4}x^{3}) = x^{4}3x^{2} + 4x^{3}x^{3} = 3x^{6} + 4x^{6} = x^{4}x^{3}$ It should be obvious that $x^{4}x^{3}$ could have been written as $x^{7}$ and derived as such, but we wish to limit our examples to the derivatives of univariate functions.\",\n \"We will use a similar approach below for simplicity.\",\n \"$\\\\frac{d}{dx} (\\\\frac{x^{4}}{x^{3}}) = \\\\frac{x^{3}4x^{3}-x^{4}3x^{2}}{(x^{3})^{2}} = \\\\frac{x^{6}}{x^{6}} = 1$ Other simple derivatives that might not directly follow these rules are $\\\\frac{d}{dx} \\\\sin {x} = \\\\cos {x}$ $\\\\frac{d}{dx} \\\\cos {x} = -\\\\sin {x}$ $\\\\frac{d}{dx} \\\\tan {x} = sec^{2} x$ $\\\\frac{d}{dx} e^{x} = e^{x}$ $\\\\frac{d}{dx} \\\\log {x} = \\\\frac{1}{x}$ $\\\\frac{d}{dx} \\\\frac{1}{x} = -\\\\frac{1}{x^{2}}$ Our objective here is not to teach elementary calculus, as you should have been exposed to that in intro math courses, but to review the symbols that should be necessary in later chapters.\",\n \"The relationship between integration and differentiation is summarized by the fundamental theorem of calculus $F(x) = \\\\int _{a}^{x} f(t)dt$ $\\\\frac{d}{dx}F(x) = \\\\frac{d}{dx} \\\\int _{a}^{x} f(t)dt = f(x)$ and $\\\\int _{a}^{b} f(x)dx = F(a) - F(b)$ For $f(x)$ continuous on the interval $[a,b]$ and $F(x)$ continuous on the interval $[a,b]$ and differentiable in $(a,b)$ .\",\n \"$F(x)$ in this case is called the primitive of $f(x)$ .\",\n \"In other words $F(x)$ is any function that when differentiated results in $f(x)$ .\",\n \"For our purposes, what you have to take from the theorem is that, for a function $f(x)$ which follows the conditions stated above, its integral is $F(x)$ , which can be reverted back to $f(x)$ by differentiation.\",\n \"We have demonstrated an example of this process in Equations 1.6 to 1.18, but bear in mind that this discussion is as far as possible from a mathematical proof, which we will not provide here.\",\n \"The interested reader can find more about proofs in mathematical books on the field of analysis.\",\n \"Now, intuitively, integration can be understood as the “inverse path\\\" of differentiation, so if the derivative of any function $x^{n}$ is given by $\\\\frac{d}{dx} x^{n} = nx^{n-1}$ for $n \\\\ne -1$ , we can retrieve the original function by adding 1 on the exponent and dividing the derivative by $n$ such that $\\\\frac{nx^{n-1+1}}{n-1+1} = x^{n}$ and thus, at least operationally, the integral of a function $x^{n}$ can be computed as $\\\\int x^{n} dx = \\\\frac{x^{n+1}}{n+1} + C$ Where $C$ is an integration constant and $dx$ indicates that the integration is performed on the variable $x$ .\",\n \"The integral in equation (1.42) is said to be unbounded because there are no limits of integration.\",\n \"By rule (1.19) any constant when differentiated will be equal to zero and in this case if we move from right to left in equation (1.42) by differentiation is not difficult to see that the original function will result if differentiation is carried out.\",\n \"Rarely in physics and applied mathematics we will encounter such simple functions.\",\n \"In many cases the functions to be integrated are much more complicated and often, especially when dealing with differential equations, might not have an analytical or closed form solution.\",\n \"In water and other transport phenomena such equations are the rule, and they are solved using a different approach that uses approximations and computational methods.\",\n \"These numerical solutions are at the core of not only fluid dynamics in the form of computational fluid dynamics (CFD), but are also an essential component of modern soil physics, hydrology and hydrogeology.\",\n \"For simple univariate integrals the rules and techniques below apply $\\\\int x^{n}dx = \\\\frac{x^{n+1}}{n+1} + C$ $\\\\int x^{2}dx = \\\\frac{x^{3}}{3} + C$ $\\\\int dx = x + C$ $\\\\int x^{3}dx = \\\\frac{x^{4}}{4} + C$ $\\\\int \\\\sin {x} \\\\ dx = -\\\\cos {x} + C$ $\\\\int \\\\cos {x} \\\\ dx = sin{x} + C$ $\\\\int \\\\tan {x} \\\\ dx = -\\\\log {\\\\cos {x}} + C$ $\\\\int e^{x} \\\\ dx = e^{x} + C$ $\\\\int \\\\log {x} \\\\ dx = x \\\\log {x} - x + C$ $\\\\int \\\\frac{1}{x} \\\\ dx = \\\\log {x} + C$ In introductory calculus books you learn about integration as a Riemann sum and as the area under a curve $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\sum _{k=1}^{n} f(c_{k}) \\\\Delta {x} = J = \\\\int _{a}^{b} f(x) dx$ In this equation, when the length of the intervals $\\\\Delta {x}$ goes to zero and the number of $k$ intervals goes to infinite, the sum can be interpreted as an integral over the entire interval from $a$ to $b$ and the result is the area under a curve, $J$ .\",\n \"We will not discuss this right now but it is important that you get familiarized with sums and their role in calculus, because they make frequent appearance in properties of granular materials and on the solutions of differential equations in advanced soil physics.\",\n \"Consider a very simple example of a line described by (REF ) $f(x) = y = \\\\frac{x}{2} + 2$ Figure: Graph of the function f(x)=y=x 2+2f(x) = y = \\\\frac{x}{2} + 2 .If you wish to calculate the area of the curve on the interval from $x = 2$ to $x = 8$ you could simply calculate the area of the trapeze bounded on its upper end by the line and in its lower end by the x axis and at its left and right limits by the lines $x = 2$ and $x = 8$ resulting in 120.\",\n \"Notice that you would get the same result if you divide the interval from 2 to 8 in 6 equal intervals of length 1 in $x$ , calculate the area of each and sum the areas.\",\n \"This is exactly what it is done by the left hand side of equation (1.53).\",\n \"Now by integration, the area under the line is given by (REF ) $J = \\\\int _{2}^{8} 2x + 10 dx = [x^{2} + 10x]\\\\Biggr |_{2}^{8} = [8^{2} + 10\\\\times 8] - [2^{2} + 10\\\\times 2] = 120$ It is obvious from the previous discussion that you don't need integration to calculate the area under a straight line, but it is also clear that you cannot use the formulas created for regular geometric figures for calculating the area under curves and this is the appeal of integration.\",\n \"You should be familiarized with integration techniques from your intro math courses such as substitution, integration by parts and partial fractions.\",\n \"If you haven't had any college level math you are urged to take at least a semester to study it before any serious endeavor in soil physics.\",\n \"True soil physics requires vector calculus and mathematical methods of physics, and whoever tells you - “you don't need these subjects to understand soil physics\\\" - does not know what soil physics is.\",\n \"When integration is performed in a bounded interval such as in equation (1.55), the integral is said to be a definite integral and there is no need for a constant of integration as is the case with indefinite integrals.\",\n \"Some properties of the indefinite integral are $\\\\int _{b}^{a} f(x) dx = -\\\\int _{a}^{b} f(x) dx$ $\\\\int _{a}^{a} f(x) dx = 0$ $\\\\int _{a}^{b} kf(x) dx = k\\\\int _{a}^{b} f(x) dx$ $\\\\int _{a}^{b} (f(x) \\\\pm g(x))dx = \\\\int _{a}^{b} f(x) dx \\\\pm \\\\int _{a}^{b} g(x) dx$ $\\\\int _{a}^{b} f(x) dx + \\\\int _{b}^{c} f(x) dx = \\\\int _{a}^{c} f(x) dx$ Where $f(x)$ and $g(x)$ are both univariate functions of $x$ , and $k$ is a constant.\",\n \"You must have learned in high school physics that a vector differs from a scalar in a sense that it has both direction and magnitude.\",\n \"Now if we think of soil physical properties we might think of a range of properties that are scalars such as soil bulk density, soil temperature and soil water content.\",\n \"Now when we think of flow of water and gases inside porous media, most often than not it is as important to know the magnitude of the water or gas being transported, or in other words the bulk mass being transported by unit of time, as it is important to know the direction in which they are being transported.\",\n \"Suppose you have an irrigation project and you calculate exactly the volume of water that need to be provided for each plant for adequate growth and productivity, now suppose that you provide this water below the root zone so that it is mostly lost by downwards flow, you would be then providing the right magnitude of water but the direction of the flow would be inadequate.\",\n \"Suppose you have a rainfall and you are concerned with aquifer recharge and sediment loss by erosion.\",\n \"If most of the water flows vertically it is likely that groundwater recharge would be maximized and erosion minimized, and thus you wanna take measures for this to happen such as conserving natural vegetation or using sustainable land management practices.\",\n \"Let's say $x$ is a scalar, if we say that $x = 10$ that is all we need to describe it, meaning that its magnitude is 10.\",\n \"Now before talking about direction is important to define a system of coordinates.\",\n \"The familiar system of coordinates that you might be familiar with is the two dimensional Cartesian space or the plane.\",\n \"Suppose you and your friend are pushing a large box on an empty room.\",\n \"Imagine that you are pushing the box by applying a force parallel to two of the walls and your friend is applying a force perpendicular to the force you are applying.\",\n \"Let's call the direction which you are applying the force x so that the wall parallel to that directions has the coordinates of x, while the direction in which your friend is applying the force is y and the wall parallel to that direction defines the y axis.\",\n \"If none of the forces applied is zero, the object will move at an angle to each of the applied forces because the vector sum of the forces has a resulting force $\\\\mathbf {F}$ defined as $\\\\mathbf {F} = F_{x} \\\\mathbf {i} + F_{y} \\\\mathbf {j}$ In this equation $F_{x}$ and $F_{y}$ are the components of the force F in the directions x and y or the forces applied by you and your friend.\",\n \"The symbols $\\\\mathbf {i}$ and $\\\\mathbf {j}$ are used to indicate unit vectors, and their role is to indicate in which directions the components of the vector are acting.\",\n \"The unit vectors have magnitude 1 and can be represented by different notations.\",\n \"If we have three forces being applied in a three dimensional space, with coordinates x, y and z equation (1.61) can be written as $\\\\mathbf {F} = F_{x} \\\\mathbf {i} + F_{y} \\\\mathbf {j} + F_{z} \\\\mathbf {k}$ In fluid dynamics is not unusual that the unit vectors are represented using a different notation.\",\n \"A velocity vector in a three dimensional space is usually written as $\\\\mathbf {v} = v_{x} \\\\mathbf {x} + v_{y} \\\\mathbf {y} + v_{z} \\\\mathbf {z}$ Where $\\\\mathbf {i} = \\\\mathbf {x}$ , $\\\\mathbf {j} = \\\\mathbf {y}$ and $\\\\mathbf {k} = \\\\mathbf {z}$ .\",\n \"Now in your handwritten notebook or in a board it is not convenient to write letters as boldface as notation for representing vectors.\",\n \"In handwritten form but also in a few books, vectors are represented with arrows draw above the symbol while unit vectors are represented by the circumflex accent over the symbol, known in mathematics and colloquial language as hat “”.\",\n \"Using this notation, equations (1.62) and (1.63) can be written as $\\\\vec{F} = F_{x} \\\\hat{i} + F_{y} \\\\hat{j} + F_{z} \\\\hat{k}$ Another notation commonly seen in fluid dynamics is $\\\\vec{v} = v_{x} \\\\hat{x} + v_{y} \\\\hat{y} + v_{z} \\\\hat{z}$ In this material I will use the boldface representation because I feel it looks more elegant in print and electronic format.\",\n \"In more advanced physics and mathematics books these vectors could also be represented as $\\\\mathbf {F} = \\\\langle F_{x}, F_{y}, F_{z} \\\\rangle $ $\\\\mathbf {v} = \\\\langle v_{x}, v_{y}, v_{z} \\\\rangle $ and we might make use of this notation for higher dimension vectors (i.e.\",\n \"higher than three dimensions).\",\n \"Suppose now that in Cartesian space there is a vector a defined as $\\\\mathbf {a} = a_{x} \\\\mathbf {i} + a_{y} \\\\mathbf {j}$ as represented in Figure REF .\",\n \"Figure: Vector 𝐚\\\\mathbf {a} with components 𝐚 𝐱 \\\\mathbf {a_{x}} and 𝐚 𝐲 \\\\mathbf {a _{y}}.You will notice that the components of the vector are the projections of $\\\\mathbf {a}$ over the axis in the directions $\\\\mathbf {i}$ and $\\\\mathbf {j}$ .\",\n \"The length of the vector or its magnitude is given by $|\\\\mathbf {a}| = \\\\sqrt{a_{x}^2 + a_{y}^2}$ The proof is immediate from elementary mathematics and the user is encouraged to consult calculus and linear algebra books for further information.\",\n \"A few properties of vectors are listed below $\\\\mathbf {a} + \\\\mathbf {b} = \\\\mathbf {b} + \\\\mathbf {a}$ $\\\\mathbf {a} + \\\\mathbf {0} = \\\\mathbf {a}$ $k(\\\\mathbf {a} + \\\\mathbf {b}) = k\\\\mathbf {a} + k\\\\mathbf {b}$ $cd\\\\mathbf {a} = c(d\\\\mathbf {a})$ $\\\\mathbf {a} + (\\\\mathbf {b} + \\\\mathbf {c}) = (\\\\mathbf {a} + \\\\mathbf {b}) + \\\\mathbf {c}$ $\\\\mathbf {a} + (-\\\\mathbf {a}) = \\\\mathbf {0}$ $(c + d)\\\\mathbf {a} = c\\\\mathbf {a} + d\\\\mathbf {a}$ $1\\\\mathbf {a} = \\\\mathbf {a}$ Where $\\\\mathbf {a}$ , $\\\\mathbf {b}$ and $\\\\mathbf {c}$ are vectors and and $c$ and $d$ are scalars.\",\n \"In all vectors shown until now the components are treated as scalars as the direction is specified by the unit vectors.\",\n \"In many cases the components themselves are functions in x, y and z such that $\\\\mathbf {V}(x,y,z) = u(x,y,z) \\\\ \\\\mathbf {i} + v(x,y,z) \\\\ \\\\mathbf {j} + w(x,y,z) \\\\ \\\\mathbf {k}$ or $\\\\mathbf {V}(x,y,z) = \\\\langle u(x,y,z), v(x,y,z), w(x,y,z) \\\\rangle $ In rectangular coordinates each of these components can represent an equation which is itself a function of $x$ , $y$ and $z$ .\",\n \"Equation (1.78, 1.79) represent vector fields where in each point in three dimensional space a vector can be drawn specifying the field.\",\n \"A simple example would be a vector field where $u(x,y,z) = 2xyz$ $v(x,y,z) = x^{2} + yz$ $w(x,y,z) = x - y + 2z$ which can be written as $\\\\mathbf {V}(x,y,z) = 2xyz \\\\ \\\\mathbf {i} + (x^{2} + yz) \\\\ \\\\mathbf {j} + (x - y + 2z) \\\\ \\\\mathbf {k}$ If we define two vectors $\\\\mathbf {a}$ and $\\\\mathbf {b}$ $\\\\mathbf {a} = a_{x} \\\\mathbf {i} + a_{y} \\\\ \\\\mathbf {j} + a_{z} \\\\ \\\\mathbf {k}$ $\\\\mathbf {b} = b_{x} \\\\mathbf {i} + b_{y} \\\\ \\\\mathbf {j} + b_{z} \\\\ \\\\mathbf {k}$ regular mathematical operations of sum and subtractions of these two vectors and multiplication by scalars are defined by properties (1.70) to (1.77), see for example property (1.76) $(c + d)\\\\mathbf {a} = (c + d)(a_{x} \\\\mathbf {i} + a_{y} \\\\ \\\\mathbf {j} + a_{z} \\\\ \\\\mathbf {k})\\\\\\\\= (c + d)a_{x} \\\\mathbf {i} + (c + d) a_{y} \\\\ \\\\mathbf {j} + (c + d)a_{z} \\\\ \\\\mathbf {k}\\\\\\\\= ca_{x} \\\\mathbf {i} + c a_{y} \\\\ \\\\mathbf {j} + ca_{z} \\\\ \\\\mathbf {k} + da_{x} \\\\mathbf {i} + d a_{y} \\\\ \\\\mathbf {j} + da_{z} \\\\ \\\\mathbf {k}\\\\\\\\= c \\\\mathbf {a} + d \\\\mathbf {a}$ You might have noticed that nothing is said about multiplication of vectors by vectors in those properties.\",\n \"This is because regular multiplication and division operations used for scalars are not valid for vectors, they require their own rules.\",\n \"The first of such rules is the scalar or dot product, represented by the symbol “$\\\\cdot {}$ ” $\\\\mathbf {a} \\\\cdot \\\\mathbf {b} = a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z}$ The demonstration requires some further explanation as you might not be familiar or might not recall the properties of unit vectors $\\\\mathbf {a} \\\\cdot \\\\mathbf {b} =(a_{x}\\\\mathbf {i} + a_{y}\\\\mathbf {j} + a_{z}\\\\mathbf {k}) \\\\cdot (b_{x}\\\\mathbf {i} + b_{y}\\\\mathbf {j} + b_{z}\\\\mathbf {k}) \\\\\\\\= a_{x}\\\\mathbf {i} \\\\cdot b_{x}\\\\mathbf {i} +a_{x}\\\\mathbf {i} \\\\cdot b_{y}\\\\mathbf {j} +a_{x}\\\\mathbf {i} \\\\cdot b_{z}\\\\mathbf {k} +a_{y}\\\\mathbf {i} \\\\cdot b_{x}\\\\mathbf {i} +a_{y}\\\\mathbf {i} \\\\cdot b_{y}\\\\mathbf {j} + \\\\\\\\+ a_{y}\\\\mathbf {i} \\\\cdot b_{z}\\\\mathbf {k} +a_{z}\\\\mathbf {i} \\\\cdot b_{x}\\\\mathbf {i} +a_{z}\\\\mathbf {i} \\\\cdot b_{y}\\\\mathbf {j} +a_{z}\\\\mathbf {i} \\\\cdot b_{z}\\\\mathbf {k} \\\\\\\\= a_{x}b_{x}\\\\mathbf {i} \\\\cdot \\\\mathbf {i} +a_{x}b_{y}\\\\mathbf {i} \\\\cdot \\\\mathbf {j} +a_{x}b_{z}\\\\mathbf {i} \\\\cdot \\\\mathbf {k} +a_{y}b_{x}\\\\mathbf {i} \\\\cdot \\\\mathbf {i} +a_{y}b_{y}\\\\mathbf {i} \\\\cdot \\\\mathbf {j} + \\\\\\\\+ a_{y}b_{z}\\\\mathbf {i} \\\\cdot \\\\mathbf {k} +a_{z}b_{x}\\\\mathbf {i} \\\\cdot \\\\mathbf {i} +a_{z}b_{y}\\\\mathbf {i} \\\\cdot \\\\mathbf {j} +a_{z}b_{z}\\\\mathbf {i} \\\\cdot \\\\mathbf {k} \\\\\\\\= a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z}$ This is because the dot product operates following a distributive property, and for the unit vectors defined as $i = \\\\langle 1,0,0 \\\\rangle $ , $j = \\\\langle 0,1,0 \\\\rangle $ and $k = \\\\langle 0,0,1 \\\\rangle $ $\\\\mathbf {i} \\\\cdot \\\\mathbf {i} = \\\\mathbf {i} \\\\cdot \\\\mathbf {i} = \\\\mathbf {i} \\\\cdot \\\\mathbf {i} = 1$ $\\\\mathbf {i} \\\\cdot \\\\mathbf {j} = \\\\mathbf {i} \\\\cdot \\\\mathbf {k} = \\\\mathbf {j} \\\\cdot \\\\mathbf {k} = 0$ A special case of the dot product in two dimensions is the Pythagoras theorem, which as we already seen it provides the length of a vector $\\\\mathbf {a} \\\\cdot \\\\mathbf {a} = |\\\\mathbf {a}|^{2} = a_{x}^2 + a_{y}^2$ Back to three dimensions, the dot product can be written as $\\\\mathbf {a} \\\\cdot \\\\mathbf {b} = |\\\\mathbf {a}| |\\\\mathbf {b}| \\\\cos {\\\\theta }$ Where $\\\\cos {\\\\theta }$ is the angle between the two vectors.\",\n \"There is much more information about the geometric interpretation of the dot product in calculus and linear algebra books.\",\n \"This information is important for the interpretation of vector quantities in soil physics and groundwater hydrology but it would be beyond the scope of this book to present it.\",\n \"What is important now is to understand the definitions of these vector operators because they are required for understanding vector calculus operators and the fundamental transport equations.\",\n \"The second operator is the cross or vector product, represented by the symbol “$\\\\times $ ” $\\\\mathbf {a} \\\\times \\\\mathbf {b} =(a_{y} b_{z} - a_{z} b_{y}) \\\\mathbf {i} +(a_{z} b_{x} - a_{x} b_{z}) \\\\mathbf {j} +(a_{x} b_{y} - a_{y} b_{x}) \\\\mathbf {k}$ or $\\\\mathbf {a} \\\\times \\\\mathbf {b} = \\\\ \\\\langle a_{y}b_{z} - a_{z}b_{y}, a_{z}b_{x} - a_{x}b_{z}, a_{x}b_{y} - a_{y}b_{x} \\\\rangle $ The demonstration can be done in a similar way as we did for the dot product $\\\\mathbf {a} \\\\times \\\\mathbf {b} =(a_{x}\\\\mathbf {i} + a_{y}\\\\mathbf {j} + a_{z}\\\\mathbf {k}) \\\\times (b_{x}\\\\mathbf {i} + b_{y}\\\\mathbf {j} + b_{z}\\\\mathbf {k}) \\\\\\\\= a_{x}\\\\mathbf {i} \\\\times b_{x}\\\\mathbf {i} +a_{x}\\\\mathbf {i} \\\\times b_{y}\\\\mathbf {j} +a_{x}\\\\mathbf {i} \\\\times b_{z}\\\\mathbf {k} +a_{y}\\\\mathbf {i} \\\\times b_{x}\\\\mathbf {i} +a_{y}\\\\mathbf {i} \\\\times b_{y}\\\\mathbf {j} + \\\\\\\\+ a_{y}\\\\mathbf {i} \\\\times b_{z}\\\\mathbf {k} +a_{z}\\\\mathbf {i} \\\\times b_{x}\\\\mathbf {i} +a_{z}\\\\mathbf {i} \\\\times b_{y}\\\\mathbf {j} +a_{z}\\\\mathbf {i} \\\\times b_{z}\\\\mathbf {k} \\\\\\\\= a_{x}b_{x}\\\\mathbf {i} \\\\times \\\\mathbf {i} +a_{x}b_{y}\\\\mathbf {i} \\\\times \\\\mathbf {j} +a_{x}b_{z}\\\\mathbf {i} \\\\times \\\\mathbf {k} +a_{y}b_{x}\\\\mathbf {j} \\\\times \\\\mathbf {i} +a_{y}b_{y}\\\\mathbf {j} \\\\times \\\\mathbf {j} + \\\\\\\\+ a_{y}b_{z}\\\\mathbf {j} \\\\times \\\\mathbf {k} +a_{z}b_{x}\\\\mathbf {k} \\\\times \\\\mathbf {i} +a_{z}b_{y}\\\\mathbf {k} \\\\times \\\\mathbf {j} +a_{z}b_{z}\\\\mathbf {k} \\\\times \\\\mathbf {k} \\\\\\\\$ and because the geometrical properties of the cross product are different than those of the dot product such that $\\\\mathbf {i} \\\\times \\\\mathbf {i} = \\\\mathbf {j} \\\\times \\\\mathbf {j} = \\\\mathbf {k} \\\\times \\\\mathbf {k} = 0$ $\\\\mathbf {i} \\\\times \\\\mathbf {j} = \\\\mathbf {k}$ $\\\\mathbf {j} \\\\times \\\\mathbf {k} = \\\\mathbf {i}$ $\\\\mathbf {k} \\\\times \\\\mathbf {i} = \\\\mathbf {j}$ $\\\\mathbf {j} \\\\times \\\\mathbf {i} = -\\\\mathbf {k}$ $\\\\mathbf {k} \\\\times \\\\mathbf {j} = -\\\\mathbf {i}$ $\\\\mathbf {i} \\\\times \\\\mathbf {k} = -\\\\mathbf {j}$ we have $\\\\mathbf {a} \\\\times \\\\mathbf {b} =a_{x}b_{x} 0 +a_{x}b_{y}\\\\mathbf {k} +a_{x}b_{z}(-\\\\mathbf {j}) +a_{y}b_{x}(-\\\\mathbf {k}) +a_{y}b_{y}0 + \\\\\\\\+ a_{y}b_{z}\\\\mathbf {i} +a_{z}b_{x}\\\\mathbf {j} +a_{z}b_{y}(-\\\\mathbf {i}) +a_{z}b_{z}0 \\\\\\\\= (a_{y} b_{z} - a_{z} b_{y}) \\\\mathbf {i} +(a_{z} b_{x} - a_{x} b_{z}) \\\\mathbf {j} +(a_{x} b_{y} - a_{y} b_{x}) \\\\mathbf {k}$ Notice that the result of scalar product is, as the name implies, a scalar while the vector product is, as you might have guessed, a vector.\",\n \"In practice the vector product might be calculated from $\\\\mathbf {a} \\\\times \\\\mathbf {b} = \\\\begin{vmatrix}\\\\mathbf {i} & \\\\mathbf {j} & \\\\mathbf {k} \\\\\\\\a_{x} & a_{y} & a_{z} \\\\\\\\b_{x} & b_{y} & b_{z} \\\\\\\\\\\\end{vmatrix}$ such that $\\\\mathbf {a} \\\\times \\\\mathbf {b} =a_{y}b_{z} \\\\mathbf {i} +a_{z}b_{x} \\\\mathbf {j} +a_{x}b_{y} \\\\mathbf {k} -a_{z}b_{y} \\\\mathbf {i} -a_{x}b_{z} \\\\mathbf {j} -a_{y}b_{x} \\\\mathbf {k} \\\\\\\\= (a_{y} b_{z} - a_{z} b_{y}) \\\\mathbf {i} +(a_{z} b_{x} - a_{x} b_{z}) \\\\mathbf {j} +(a_{x} b_{y} - a_{y} b_{x}) \\\\mathbf {k}$ Note that there are different approaches to calculating the determinant of a matrix so we are omitting a few steps.\",\n \"You can consult any calculus or linear algebra book you if you are unfamiliar with the procedure.\",\n \"Notice that in many countries such procedures are taught in secondary school so you might want to consult such books as well for a more didactic introduction.\",\n \"The vector product can also be written as $\\\\mathbf {a} \\\\times \\\\mathbf {b} = |\\\\mathbf {a}| |\\\\mathbf {b}| \\\\sin {\\\\theta }$ Where $\\\\sin {\\\\theta }$ is the sin of the angle between the two vectors.\",\n \"As with the dot product the reader is encouraged to consult its geometrical properties and interpretation.\",\n \"The properties of the scalar and vector products are summarized below $\\\\mathbf {a} \\\\cdot \\\\mathbf {a} = |\\\\mathbf {a}|^{2}$ $\\\\mathbf {a} \\\\cdot (\\\\mathbf {b} + \\\\mathbf {c}) = \\\\mathbf {a} \\\\cdot \\\\mathbf {b} + \\\\mathbf {a} \\\\cdot \\\\mathbf {c}$ $\\\\mathbf {a} \\\\cdot \\\\mathbf {b} = \\\\mathbf {b} \\\\cdot \\\\mathbf {a}$ $(c\\\\mathbf {a}) \\\\cdot \\\\mathbf {b} = c(\\\\mathbf {a} \\\\cdot \\\\mathbf {b}) = \\\\mathbf {a} \\\\cdot (c\\\\mathbf {b})$ $\\\\mathbf {0} \\\\cdot \\\\mathbf {a} = 0$ $\\\\mathbf {a} \\\\times \\\\mathbf {b} = -\\\\mathbf {b} \\\\times \\\\mathbf {a}$ $(c\\\\mathbf {a}) \\\\times \\\\mathbf {b} = c(\\\\mathbf {a} \\\\times \\\\mathbf {b}) = \\\\mathbf {a} \\\\times (c\\\\mathbf {b})$ $\\\\mathbf {a} \\\\times (\\\\mathbf {b} + \\\\mathbf {c}) =\\\\mathbf {a} \\\\times \\\\mathbf {b} + \\\\mathbf {a} \\\\times \\\\mathbf {c}$ $(\\\\mathbf {a} \\\\times \\\\mathbf {b}) + \\\\mathbf {c} =\\\\mathbf {a} \\\\times \\\\mathbf {c} + \\\\mathbf {b} \\\\times \\\\mathbf {c}$ $\\\\mathbf {a} \\\\cdot (\\\\mathbf {b} + \\\\mathbf {c}) =(\\\\mathbf {a} \\\\times \\\\mathbf {b}) \\\\cdot \\\\mathbf {c}$ $\\\\mathbf {a} \\\\times (\\\\mathbf {b} + \\\\mathbf {c}) =(\\\\mathbf {a} \\\\cdot \\\\mathbf {c})\\\\mathbf {b} - (\\\\mathbf {a} \\\\cdot \\\\mathbf {b})\\\\mathbf {c}$ Before delving into vector calculus operators we need to take a step back and look what how derivation and integration work when you have a multivariate function.\",\n \"As we've seen, an univariate function has a dependent variable that it is a function of a single independent variable as is the case with the equation $y = x$ .\",\n \"A multivariate function is one which is a function of two or more independent variables, for example, each component of the vector field in (1.83) is a multivariate function of the variables $x$ , $y$ and $z$ as shown in (1.80), (1.81) and (1.82).\",\n \"You already know a few basic rules to differentiate univariate functions, but how to proceed with multivariate functions?\",\n \"The tools for solving such problems are in the realm of multivariate calculus and are the partial derivatives and multiple integrals.\",\n \"Most engineering, mathematics and physics undergraduate programs devote an entire course for these subjects, usually calculus III or multivariate calculus.\",\n \"The idea behind partial differentiation is apparently simple if we ignore rigorous mathematical proofs and theorems behind it as we will do in this material.\",\n \"If you have a multivariate function you can take the derivative with respect to a single variable and treat the other variables as constants as in (1.21).\",\n \"When taking partial derivatives the symbol “$\\\\partial $ ” is used instead of “$d$ ”.\",\n \"For example, if we wish to differentiate (1.80) with respect to $x$ we have $\\\\frac{\\\\partial }{\\\\partial x} u(x,y,z) = \\\\frac{\\\\partial }{\\\\partial x}2xyz = yz \\\\frac{\\\\partial x}{\\\\partial x} 2x = 2yz$ Now let's say you want to derivate (1.81) with respect to $y$ $\\\\frac{\\\\partial }{\\\\partial y} v(x,y,z) = \\\\frac{\\\\partial }{\\\\partial y} (x^{2} + yz) = \\\\frac{\\\\partial }{\\\\partial y} x^{2} + \\\\frac{\\\\partial }{\\\\partial y} yz = 0 + z = z$ Because here $x$ is a constant and any constant squared is still a constant, and $z$ is another constant.\",\n \"Other representations of a partial derivative are for example $(\\\\frac{\\\\partial }{\\\\partial x} u)_{y,z}$ and $\\\\frac{\\\\partial }{\\\\partial x} u \\\\Bigm | _{y,z}$ Where in both cases the differentiation is performed with respect to $x$ with $y$ and $z$ held constant.\",\n \"Such notation is most commonly used Thermodynamics, a field which makes extensive use of multivariate calculus and interlaces with both fluid dynamics and soil physics.\",\n \"Earlier we saw that the first derivative is the rate of variation of a function, intuitively, for a straight line $y = ax + b$ The first derivative is the slope of the line $\\\\frac{dy}{dx} = a$ For a second degree polynomial, the rate of variation is not a constant, but it is a straight line so that the slope of the quadratic curve varies with $x$ $y = ax^{2} + bx + c$ The first derivative is the slope of the line $\\\\frac{dy}{dx} = 2ax + b$ Notice that the derivatives can be used to identify critical points of functions as you might learn in any intro calculus books.\",\n \"A function in two or more dimensions will have slopes in each particular direction as represented by the partial derivatives.\",\n \"Lets consider a function $w(x,y,z)$ $w = x^{2} + 2xy + z^{3}$ with partial derivatives in $x$ , $y$ and $z$ $\\\\frac{\\\\partial w}{\\\\partial x} = 2x + 2y$ $\\\\frac{\\\\partial w}{\\\\partial y} = 2x$ $\\\\frac{\\\\partial w}{\\\\partial z} = 3z^{2}$ Which are the slopes of the function in the directions $x$ and $y$ .\",\n \"From our previous discussions it should be apparent that these partial derivatives represent the slopes of the function on the directions $ \\\\mathbf {i}$ , $ \\\\mathbf {j}$ and $ \\\\mathbf {k}$ and as such we can represent these partial derivatives as a vector function at any point in space (if it exists) as $\\\\nabla {w(x,y,z)} =\\\\frac{\\\\partial w}{\\\\partial x} = (2x + 2y) \\\\mathbf {i} +\\\\frac{\\\\partial w}{\\\\partial y} = (2x) \\\\mathbf {j} +\\\\frac{\\\\partial w}{\\\\partial z} = (3z^{2}) \\\\mathbf {k}$ Where the symbol “$\\\\nabla $ ” is called “nabla” or “del” and is a representation for a mathematical operator called gradient.\",\n \"The gradient of a scalar or vector function is a vector function and can be generalized as $\\\\boxed{\\\\nabla {} =\\\\frac{\\\\partial w}{\\\\partial x} \\\\mathbf {i} +\\\\frac{\\\\partial w}{\\\\partial y} \\\\mathbf {j} +\\\\frac{\\\\partial w}{\\\\partial z} \\\\mathbf {k}}$ applied to any scalar function $f$ results $\\\\boxed{\\\\nabla {f} =\\\\frac{\\\\partial w}{\\\\partial x}f \\\\ \\\\mathbf {i} +\\\\frac{\\\\partial w}{\\\\partial y}f \\\\ \\\\mathbf {j} +\\\\frac{\\\\partial w}{\\\\partial z}f \\\\ \\\\mathbf {k}}$ Now for a vector $\\\\mathbf {u} = u \\\\ \\\\mathbf {i} + v \\\\ \\\\mathbf {j} + w \\\\ \\\\mathbf {k}$ The gradient in rectangular coordinates is a second rank tensor $\\\\nabla {\\\\mathbf {u}} =\\\\begin{pmatrix}\\\\frac{\\\\partial u}{\\\\partial x} & \\\\frac{\\\\partial v}{\\\\partial x} & \\\\frac{\\\\partial w}{\\\\partial x}\\\\\\\\\\\\frac{\\\\partial u}{\\\\partial y} & \\\\frac{\\\\partial v}{\\\\partial y} & \\\\frac{\\\\partial w}{\\\\partial y} \\\\\\\\\\\\frac{\\\\partial u}{\\\\partial z} & \\\\frac{\\\\partial v}{\\\\partial z} & \\\\frac{\\\\partial w}{\\\\partial z}\\\\end{pmatrix}$ Although this vector gradient makes an appearance in Navier-Stokes equations, it is often not necessary to expand it.\",\n \"The reader of this material should focus on the gradient of scalar functions whose importance will be apparent when we deal of water flux in porous media.\",\n \"Now the dot product of the gradient operator over a vector, say $u$ , defines the divergence of the vector $\\\\boxed{\\\\nabla {\\\\cdot \\\\mathbf {u}} =\\\\frac{\\\\partial u}{\\\\partial x} +\\\\frac{\\\\partial v}{\\\\partial y} +\\\\frac{\\\\partial w}{\\\\partial z} }$ Which is a scalar field.\",\n \"The demonstration is straightforward from the definition of $\\\\nabla $ and $\\\\mathbf {u}$ and from the dot product properties $\\\\nabla {\\\\cdot \\\\mathbf {u}} =(\\\\frac{\\\\partial }{\\\\partial x} \\\\mathbf {i} +\\\\frac{\\\\partial }{\\\\partial y} \\\\mathbf {j} +\\\\frac{\\\\partial }{\\\\partial z} \\\\mathbf {k}) \\\\cdot (u \\\\mathbf {i} + v \\\\mathbf {j} + w \\\\mathbf {k}) \\\\\\\\= \\\\frac{\\\\partial u}{\\\\partial x} \\\\mathbf {i} \\\\cdot \\\\mathbf {i} +\\\\frac{\\\\partial v}{\\\\partial y} \\\\mathbf {j} \\\\cdot \\\\mathbf {j}+\\\\frac{\\\\partial w}{\\\\partial z} \\\\mathbf {k} \\\\cdot \\\\mathbf {k} =\\\\frac{\\\\partial u}{\\\\partial x} +\\\\frac{\\\\partial v}{\\\\partial y} +\\\\frac{\\\\partial w}{\\\\partial z}$ Following the same logic we can apply the vector product of the gradient to any vector function yielding the curl of a vector, $\\\\mathbf {u}$ for example $\\\\boxed{\\\\nabla {\\\\times \\\\mathbf {u}} =(\\\\frac{\\\\partial w}{\\\\partial y}-\\\\frac{\\\\partial v}{\\\\partial z}) \\\\mathbf {i} +(\\\\frac{\\\\partial u}{\\\\partial z}-\\\\frac{\\\\partial w}{\\\\partial x}) \\\\mathbf {j} +(\\\\frac{\\\\partial v}{\\\\partial x}-\\\\frac{\\\\partial u}{\\\\partial y}) \\\\mathbf {k}}$ Which is a also vector field.\",\n \"As we learned with the dot product the determinant form is the easiest way to remember how to calculate the curl of a vector function $\\\\nabla {\\\\times \\\\mathbf {u}} =\\\\begin{vmatrix}\\\\mathbf {i} & \\\\mathbf {j} & \\\\mathbf {k}\\\\\\\\\\\\frac{\\\\partial }{\\\\partial x} & \\\\frac{\\\\partial }{\\\\partial y} & \\\\frac{\\\\partial }{\\\\partial z} \\\\\\\\u & v & w\\\\end{vmatrix}$ As with the scalar and vector products there are several properties that define operations for the gradient, divergence and curl, including operations among them.\",\n \"One such operation which appears very frequently in applied mathematics and physics is the divergence of a gradient.\",\n \"$\\\\boxed{\\\\nabla {\\\\cdot (\\\\nabla {f)} =\\\\frac{\\\\partial ^{2} f}{\\\\partial x^{2}} +\\\\frac{\\\\partial ^{2} f}{\\\\partial y^{2}} +\\\\frac{\\\\partial ^{2} f}{\\\\partial z^{2}} }}$ The demonstration is also somewhat trivial $\\\\nabla {\\\\cdot (\\\\nabla {f})} =(\\\\frac{\\\\partial }{\\\\partial x} \\\\mathbf {i}+\\\\frac{\\\\partial }{\\\\partial y} \\\\mathbf {j}+\\\\frac{\\\\partial }{\\\\partial z} \\\\mathbf {k}) \\\\cdot (\\\\frac{\\\\partial f}{\\\\partial x} \\\\mathbf {i}+\\\\frac{\\\\partial f}{\\\\partial y} \\\\mathbf {j}+\\\\frac{\\\\partial f}{\\\\partial z} \\\\mathbf {k}) \\\\\\\\= \\\\frac{\\\\partial ^{2} f}{\\\\partial x^{2}} \\\\mathbf {i} \\\\cdot \\\\mathbf {i} +\\\\frac{\\\\partial f ^{2}}{\\\\partial y^{2}} \\\\mathbf {j} \\\\cdot \\\\mathbf {j}+\\\\frac{\\\\partial w^{2}}{\\\\partial z^{2}} \\\\mathbf {k} \\\\cdot \\\\mathbf {k} \\\\\\\\= \\\\frac{\\\\partial ^{2} f}{\\\\partial x^{2}} +\\\\frac{\\\\partial ^{2} f}{\\\\partial y^{2}} +\\\\frac{\\\\partial ^{2} f}{\\\\partial z^{2}}$ The operator $\\\\nabla {}^{2} = \\\\nabla {} \\\\cdot \\\\nabla {}$ is called the Laplace operator because of the equation of the form $\\\\nabla {}^{2}f = \\\\frac{\\\\partial ^{2} f}{\\\\partial x^{2}} +\\\\frac{\\\\partial ^{2} f}{\\\\partial y^{2}} +\\\\frac{\\\\partial ^{2} f}{\\\\partial z^{2}} = 0$ which is of the utmost importance not only for soil physics and fluid mechanics but to most areas of physics.\",\n \"The Laplace operator also operates in vectors such that $\\\\nabla {}^{2} \\\\mathbf {u}= \\\\nabla {}^{2} u \\\\ \\\\mathbf {i} + \\\\nabla {}^{2} v \\\\ \\\\mathbf {i} + \\\\nabla {}^{2} w \\\\ \\\\mathbf {i}$ Understanding the gradient, divergence, curl and Laplacian is fundamental for the understanding of fluid mechanics and therefore soil physics and underground hydrology.\",\n \"Understanding these operators along with their properties and interrelations and the fundamental theorems of vectors calculus is almost all that is needed before presenting the fundamental equations of fluid mechanics.\",\n \"If you decide to consult the literature be aware that in many older books the operators are presented as del($f$ ), grad($f$ ) or grad $f$ for the gradient of a scalar function $f$ , div($\\\\mathbf {u}$ ) or div $\\\\mathbf {u}$ for the divergence of a vector function $\\\\mathbf {u}$ , curl($\\\\mathbf {u}$ ), curl $\\\\mathbf {u}$ , rot($\\\\mathbf {u}$ ) or rot $\\\\mathbf {u}$ for the curl of a vector $\\\\mathbf {u}$ and div(grad $f$ ) for the Laplacian of a scalar function $f$ .\",\n \"The “rot” in curl is because in many countries the curl is called rotational or one its translated similar words.\",\n \"The word rotational as you might imagine comes from rotation and the divergence is obviously related to something that diverges.\",\n \"The meaning is exactly as implied, these operators were developed in the realm of the physics of fluids.\",\n \"Most of the mathematics used in fluid dynamics was used to describe the pervasive ether (or æther) before Einstein and his contemporaries proved that it was not a necessary entity for the description of the the cosmos.\",\n \"A few vector operators identities are offered without proof $\\\\nabla {(fg)} = f (\\\\nabla {g}) + g (\\\\nabla {f})$ $\\\\nabla {(\\\\mathbf {A} \\\\cdot \\\\mathbf {B} )} = \\\\mathbf {A} \\\\times (\\\\nabla \\\\times \\\\mathbf {B}) + \\\\mathbf {B} \\\\times (\\\\nabla \\\\times \\\\mathbf {A}) + (\\\\mathbf {A} \\\\cdot \\\\nabla ) \\\\mathbf {B} + (\\\\mathbf {B} \\\\cdot \\\\nabla )\\\\mathbf {A}$ $\\\\nabla \\\\cdot (f \\\\mathbf {A}) = f(\\\\nabla \\\\cdot \\\\mathbf {A}) + \\\\mathbf {A} \\\\cdot (\\\\nabla {f})$ $\\\\nabla \\\\cdot (\\\\mathbf {A} \\\\times \\\\mathbf {B}) =\\\\mathbf {B} \\\\cdot (\\\\nabla \\\\times \\\\mathbf {A}) - \\\\mathbf {A} \\\\cdot (\\\\nabla \\\\times \\\\mathbf {B})$ $\\\\nabla \\\\times (f\\\\mathbf {A}) =f(\\\\nabla \\\\times \\\\mathbf {A}) - \\\\mathbf {A} \\\\times (\\\\nabla f)$ $\\\\nabla \\\\times (\\\\mathbf {A} \\\\times \\\\mathbf {B}) = (\\\\mathbf {B} \\\\cdot \\\\nabla )\\\\mathbf {A} - (\\\\mathbf {A} \\\\cdot \\\\nabla ) \\\\mathbf {B} + \\\\mathbf {A}(\\\\nabla \\\\cdot \\\\mathbf {B}) - \\\\mathbf {B}(\\\\nabla \\\\cdot \\\\mathbf {A})$ $\\\\nabla \\\\cdot (\\\\nabla \\\\times \\\\mathbf {A}) = 0$ $\\\\nabla \\\\times (\\\\nabla f) = 0$ $\\\\nabla \\\\times (\\\\nabla \\\\times \\\\mathbf {A}) = \\\\nabla (\\\\nabla \\\\times \\\\mathbf {A}) - \\\\nabla ^{2}\\\\mathbf {A}$ For now it is also important to say that integration can also be performed in multidimensional space, we have seen that integration in one dimension can be represented as $\\\\int x dx = \\\\frac{x^{2}}{2} + C$ now integration in two dimensions can be understood in terms of a surface integral, for example $\\\\int \\\\int xy dx dy$ Similar for what was discussed for partial derivatives, the integration is performed on each variable per step, considering the others as a constant $\\\\int \\\\int xy dx dy = \\\\frac{x^{2}}{2}\\\\int y dy + C_{1} \\\\\\\\= \\\\frac{x^{2}}{2}\\\\frac{y^{2}}{2} + C_{1}y + C_{2} \\\\\\\\= \\\\frac{x^{2}y^{2}}{4} + C_{1}y + C_{2}$ The same principles being applied to simple integrals in three or more dimensions.\",\n \"An integral in three dimensions is an integral in volume, for example $\\\\int \\\\int \\\\int xyz dx dy dz = C_{1} x y + C_{2} y + C_{3} + 1/8 x^{2} y^{2} z^{2}$ Of course integrals of such simple functions are rarely seen in physics and applied mathematics.\",\n \"More often than not more advanced integration techniques and system of coordinates transformation are needed for solving the equations.\",\n \"The most common systems of coordinates used being polar, cylindrical and spherical coordinates.\",\n \"The user is encouraged to consult the literature for transformations in system of coordinates and for the gradient, divergence, curl and Laplacian in those coordinate systems.\",\n \"Another integral that is common in physics and applied math is the integral over a curve or line integral, in which the integration is performed over a curve $\\\\int _{a}^{b} \\\\mathbf {v} \\\\cdot d \\\\mathbf {l}$ which for a closed loop is usually represented as $\\\\oint \\\\mathbf {v} \\\\cdot d \\\\mathbf {l}$ Integrals over lines, regions (surface) and volumes (domains) can also be represented by the symbols $\\\\oint _C, \\\\ \\\\ \\\\int _R, \\\\ \\\\ \\\\int _S, \\\\ \\\\ \\\\int _V, \\\\ \\\\ \\\\int _D,$ Such representations appear in three fundamental theorems of vector calculus which are used very often in fluid mechanics, electromagnetism and other areas of applied mathematics and physics [49], the divergence (or Gauss) theorem $\\\\boxed{\\\\iint _S \\\\mathbf {F} \\\\cdot \\\\mathbf {n} \\\\ d\\\\sigma = \\\\iiint _D \\\\nabla \\\\cdot \\\\mathbf {F} \\\\ dV}$ Green's theorem in normal form $\\\\boxed{\\\\oint _C \\\\mathbf {F} \\\\cdot \\\\mathbf {n} \\\\ ds = \\\\iint _R \\\\nabla \\\\cdot \\\\mathbf {F} \\\\ dA}$ and in tangential form $\\\\boxed{\\\\oint _C \\\\mathbf {F} \\\\cdot d \\\\mathbf {r} = \\\\iint _R \\\\nabla \\\\times \\\\mathbf {F} \\\\ \\\\cdot \\\\mathbf {k} \\\\ dA}$ and Stokes' theorem $\\\\boxed{\\\\oint _C \\\\mathbf {F} \\\\cdot d \\\\mathbf {r} = \\\\iint _S \\\\nabla \\\\times \\\\mathbf {F} \\\\ \\\\cdot \\\\mathbf {n} \\\\ d \\\\sigma }$ When working with the algebra of general relativity and other theoretical physics topics, Albert Einstein introduced a notation aimed at simplifying the representation of summation terms.\",\n \"As you might know, a sum is represented by the Greek letter $\\\\Sigma $ , so a sum of three $x$ is terms can be represented as $y = \\\\sum _{i=1}^n x_i = x_1 + x_2 + x_3$ Now imagine working through hundreds or thousands of equation and having to repeat this symbology over and over.\",\n \"To simplify the notation Einstein omitted the $\\\\Sigma $ symbol from the sums.\",\n \"Thus, in Einstein's notation, the sum above can be simplified to $y = x_i = x_1 + x_2 + x_3 + ...$ as the number of terms in the sum can be omitted for most practical purposes.\",\n \"If there is a single index in the equation, say $i$ , the sum is over $i$ $y =a_i x_i = a_1 x_1 + a_2x_2 +a_2 x_3 + ...$ If there are two or more indexes, say $i$ and $j$ i, the summation is on the repeated indexes, for example [14] $y_i = a_{ij} x_j = a_{i1} x_1 + a_{i2} x_2 + a_{i3} x_3 + ...$ $\\\\frac{\\\\partial u}{\\\\partial x_i}\\\\frac{\\\\partial x_i}{\\\\partial y} = \\\\frac{\\\\partial u}{\\\\partial x_1}\\\\frac{\\\\partial x_1}{\\\\partial y}+\\\\frac{\\\\partial u}{\\\\partial x_2}\\\\frac{\\\\partial x_2}{\\\\partial y}+ \\\\frac{\\\\partial u}{\\\\partial x_3}\\\\frac{\\\\partial x_3}{\\\\partial y}+ ...$ Although Einstein's notation might take some time to get used to, it is very convenient to deal with a large number of equations, especially when dealing with tensors, and it will be used extensively in this material.\",\n \"As a final warning to the reader, if most or all of the content in this chapter is new to you, this material is far from enough for you to learn the subject as it condenses material that is usually seen in three one semester undergraduate calculus courses.\",\n \"You should study and review the subject on your own if you wish to be proficient in porous media physics.\",\n \"If you are learning soil physics for the first time as a soil scientist, agronomist or other life sciences field you should use this chapter to become familiar with the notation and language used in this book and you can consult it whenever necessity arises throughout this book.\",\n \"ll $ T $ Temperature $ f $ Generalized function $ T(t) $ Time dependent temperature $ T_0 $ Temperature where the function is centered $ \\\\alpha $ Amplitude $ \\\\omega $ Angular frequency $ \\\\phi $ Phase lag $ t $ Time $ Y $ Height $ Y_0 $ Initial height $ v_0 $ Initial velocity $ g $ Earth's gravitational acceleration $ k, c, C, C_1, C_2 $ Constants $ n $ Integer number $ x, y, v $ Generalized position and velocity scalar variables $ \\\\dot{x} $ First derivative of position, or velocity $ \\\\ddot{x} $ Second derivative of position, or acceleration $a$ $ F $ Generalized function $ F(x) $ Generalized function of x $ f(t) $ Generalized function of t $ f(x), g(x) $ Generalized functions of x $ J $ Area under a curve $ \\\\vec{F} $ Generalized vector function $ F_x, F_y, F_z $ Components of $\\\\vec{F}$ in Cartesian coordinates $ \\\\mathbf {i}, \\\\mathbf {j}, \\\\mathbf {k}, \\\\hat{i}, \\\\hat{j}, \\\\hat{k}, \\\\hat{x}, \\\\hat{y}, \\\\hat{z}, $ Different representations of unit vectors in Cartesian coordinates $ \\\\vec{v} $ Velocity vector $ v_x, v_y, v_z $ Components of $\\\\vec{v}$ in Cartesian coordinates $ u, v, w $ Alternative representation of components of $\\\\vec{v}$ in Cartesian coordinates $ \\\\mathbf {a}, \\\\mathbf {b}, \\\\mathbf {A}, \\\\mathbf {B}, \\\\mathbf {V} $ Generalized vectors $ c, d $ Generalized scalars $ a_x, a_y $ Cartesian components of $\\\\mathbf {a}$ $ |\\\\mathbf {a}| $ Norm of $\\\\mathbf {a}$ $ |\\\\mathbf {b}| $ Norm of $\\\\mathbf {b}$ $ d $ Univariate derivative operator $ \\\\partial $ Partial derivative operator $ \\\\int $ Integral operator $ \\\\sum $ Summation operator $ \\\\nabla $ Gradient $ \\\\nabla \\\\cdot $ Divergence $ \\\\nabla \\\\times $ Curl $ \\\\nabla ^2 $ Laplacian $ \\\\mathbf {n} $ Unit normal vector $ \\\\mathbf {r} $ Generalized position vector $ A $ Area element $ V $ Volume element $ \\\\sigma $ Surface area element The first requirement for a material to be considered a porous media is obviously that it is composed of a solid but not necessarily rigid matrix that is not continuous in space such that the spaces between the solid fractions are pores.\",\n \"Right now we are considering those pores as empty spaces, disregarding the possibility that they can be filled by a material of a different nature than that of the solid matrix.\",\n \"There is a large amount of natural and artificial porous materials on Earth and by a large extent on the universe.\",\n \"A common bath or kitchen sponge is a perfect example of a porous material whose matrix is not rigid and not granular.\",\n \"These set of distinctions regarding the mechanical nature of the matrix, i.e.\",\n \"rigid, elastic, plastic, porous, etc., and the nature of the pores, i.e.\",\n \"connected or non-connected, is fundamental for the understanding of the physical properties of the media and we will talk some of them throughout this book.\",\n \"Natural porous media include tissues from animals, vegetables, fungi and a range of other organisms, while artificial porous media of significance for large scale transport phenomena in nature include concrete, asphalt and other building materials.\",\n \"A lot of the theory discussed in this book might apply to these artificial materials, especially regarding water and contaminant transport and storage.\",\n \"Asphalt and concrete used for pavement can under certain circumstances be considered low porosity and low permeability porous media and thus control the infiltration of water in urban environments and roads.\",\n \"The understanding of water storage and dynamics in foods, animal feed, grains, paper, and a variety of material is fundamental for their conservation and theories regarding water storage and determination in soils and rocks can be readily applied to food products.\",\n \"These materials are however beyond the scope of this book.\",\n \"We are interested here in three types of natural porous materials that cover immense extensions in both area and volume on Earth and are the basis of water and chemical elements cycles, water storage, agriculture, and environmental impacting activities, the materials we are concerned are soils, sediments and rocks.\",\n \"As with the last chapter a true introduction to these topics would require several one semester courses covering each of the subjects, including intro soil science and geology courses, soil genesis, mineralogy (both soil and rock), igneous and metamorphic petrology, and sedimentology and stratigraphy courses.\",\n \"Thus, we can only give a brief introduction and provide the reader with a terminology to be used in the rest of the book.\",\n \"While the last chapter might be superficial to the student of mathematics and physics and might be too advanced to the life sciences student, in this chapter the opposite is true, the challenge here is to present a palatable introduction on the subjects of soil science and geology for those who might not have had any introduction to the subject, as with the last chapter this is not without the risk of displeasing both groups.\",\n \"There are different types of porous media, as was mentioned before a common kitchen sponge is an example of a nonrigid and nongranular porous media.\",\n \"This means that the geometry and volume of the pores might not be constant as the material is wetted and dried, even excluding here the possibility that the sponge is compressed by the hand.\",\n \"Because water is a polar, almost non-compressible media, it can apply forces to the sponge surfaces that might cause it to deform.\",\n \"In the saturated case, water can transmit forces and stresses on the material that holds it, and as it dries the water molecules exert capillary forces which might cause surfaces to come close together.\",\n \"Soils and sediments are a class of porous materials which are both granular and nonrigid, meaning that the grains can be reorganized in response to applied stresses, altering pore volume and pore sizes distribution.\",\n \"Whether you subscribe to a philosophical view of an objective reality or not, this entire book deals with an idealized representation of what a porous medium is.\",\n \"Throughout it we will make a series of simplifications and assumptions in order to be create a conceptual and physical representation of porous materials that is suitable for mathematical treatment.\",\n \"A representation is a simplification of a material which is often infinitely complex and impossible to be represented mathematically if every shape, every composition, every topological aspect is uniquely considered.\",\n \"One of the fundamental assumptions that lie at the core of solid and fluid mechanics and that is inherited into soil physics is the continuum hypothesis.\",\n \"Water, air, soils and rocks are composed of discrete particles.\",\n \"In the case of water, these particles are water molecules that could be further discretized into hydrogen and oxygen atoms which could be further discretized and so onElementary particles physics and quantum effects and uncertainty will not be dealt in this book unless in brief explanatory notes such as this one.\",\n \"For the most part, and maybe for the time being, soil physics can be entirely formulated within a classical mechanics framework.. Soils, rocks and sedimentary deposits are for the most part composed of minerals, minerals on the other hand are composed of atoms occupying positions in a regular lattice.\",\n \"On a nanoscopic scales, if we measured a cross sectional area of a still frame of water or gas flowing or of a mineral and calculate the areal occupation of space we could fall into a molecule or atom, empty space or a boundary between the two.\",\n \"For most materials the areal density at this scale would be highly irregular depending on where the area is evaluated.\",\n \"As we increase the measured area, moving into macroscopic measured areas, the areal density will tend to a representation of the average areal occupation of the material (Figure REF ).\",\n \"The principle holds if we consider a given property being measured over a volume of the material.\",\n \"The minimum volume in which the measured property is a representation of the statistical properties of the macroscopic area is the representative elementary volume (REV) and corresponds to, by analogy, the approximate point on the plot in Figure REF where the line oscillations are negligible.\",\n \"Figure: Illustration of the Representative Elementary Volume (REV) concept.\",\n \"AA is the measured area and the circles represent solid particles.\",\n \"As the measured area increases, the areal density will tend to the average of the system.Compared to a pure liquid or gas, a porous medium is a much more complicated system as we often have multiple phases interacting.\",\n \"Each phase can be treated as a continuous medium on its own which interacts with the surrounding continuous phases.\",\n \"For some properties the entire system can be treated as a single continuous medium, including all phases.\",\n \"Often, when dealing with flow problems, fluids and gases are treated as dynamical phases while the solid matrix is treated as static, and the overall flow properties are defined by the interactions between the phases, while for measuring properties such as bulk mass, density, or temperature at a given time, for example, the individual composition of each phase might be neglected and the system might be treated as a whole.\",\n \"A system can be closed, isolated or open, closed systems do not interact with their surroundings, and there is no exchange of matter or energy, in closed systems there might be exchange of heat or other forms of energy through the walls of the system and its surroundings and in open systems there might be exchange of matter and energy of the system and its surroundings.\",\n \"Depending on the scenario, any representation can apply to natural porous media.\",\n \"When modeling transport phenomena, the system is usually considered open, such open system are sometimes understood to be analogous to the concept of control volume in engineering.\",\n \"We will use the traditional definitions of homogeneity and isotropy to classify porous media as a system.\",\n \"First a homogeneous system can be defined as one with only one component and one phase, and where its physical properties do not vary with space.\",\n \"Pure water in liquid state at 25 $^{o}$ C is a homogeneous substance.\",\n \"Of course, achieving a complete state of purity in water might not be feasible, and defining a completely homogeneous density and temperature system for water might also not be realistic, for one because of the constraints of the continuum hypothesis discussed before.\",\n \"But this would lead us back to the philosophical argument which, if explored to the full extent, might making writing this book impossible.\",\n \"So we will established that a substance that can be thought as composed of only one constituent and a single phase is homogeneous.\",\n \"Substances composed of two or more phases and/or two or more separate constituents are heterogeneous.\",\n \"A mixture of water and ice is heterogeneous while a mixture of water and salt is homogeneous if temperature, density etc.\",\n \"do not vary within the system.\",\n \"Soils, rocks and sediments are invariable heterogeneous systems, although it is often useful to assume that soils where the physical in which the properties are at near equilibrium everywhere are homogeneous systems.\",\n \"Even dry sand composed only of quartz grains might be thought as heterogeneous because it is a mixture of quartz and air.\",\n \"In nature, most soils are heterogeneous mixtures composed of various minerals, amorphous inorganic compounds, organic matter, air and a solution composed of water and various dissolved substances.\",\n \"In later chapters we will discuss homogeneity in respect to dynamical physical properties, but that will be in a different context.\",\n \"A porous media can therefore be thought as a system which can be subdivided into subsystems.\",\n \"For many purposes these systems can be modeled using thermodynamics.\",\n \"Classical thermodynamics is concerned with the macroscopic properties of matter.\",\n \"In this sense it usually deals with properties that represent an average of microscopic systems.\",\n \"The bridge between classical thermodynamics and the microscopic world is given by the discipline of statistical mechanics.\",\n \"Many derivations in fluid mechanics and soil physics are grounded in classical thermodynamics.\",\n \"The entire theory of water retention in soils is derived from classical thermodynamics, although we are led to believe that it could be derived from first principles from statistical mechanics.\",\n \"Classical thermodynamics does not take into account quantum effects which can be studied in statistical mechanics.\",\n \"On a side note, general relativity can also be accounted in fluid mechanics, especially when dealing with gas ejects in astrophysics, in the discipline of relativistic fluid mechanics.\",\n \"However, water and other fluids in porous media in Earth's gravitational fields, move slowly enough and are studied in scales large enough that classical mechanics and thermodynamics usually suffice.\",\n \"You could however use the theories presented in this book to study fluid flow in Mars or other planets, as long as the conditions of scale and velocity hold.\",\n \"In thermodynamicsClassical thermodynamics if not specified otherwise.\",\n \"simple systems can be macroscopically characterized by extensive properties such as internal energy, volume and number of moles.\",\n \"What is meant by extensive variable is that if the system is subdivided into subsystems, the magnitude of the extensive variable into each subsystem is less than the total.\",\n \"Extensive variables depend on the extension of the system.\",\n \"Intensive variables can be defined as ratios of extensive variables and do not vary with extension or location where the measurements are taken in an homogeneous system in equilibrium.\",\n \"Thus, temperature can be defined as the partial derivative of internal energy, $U$ , with respect to entropy, $S$ , while volume, $V$ , and number of moles, N$_r$ , of dissolved substances are held constant, represented as $T = \\\\left( \\\\frac{\\\\partial U}{\\\\partial S} \\\\right)_{V, N_1, ..., N_r}$ The others being pressure, $P$ , defined as the partial derivative of internal energy with respect to volume, V, with S and N$_r$ constant, and the electrochemical potential of a given component, defined as the internal energy divided by the number of moles of the component, N$_j$ , while keeping the number of moles of other components, N$_r$ , constant, along with $S$ and $V$ .\",\n \"$P = -\\\\left( \\\\frac{\\\\partial U}{\\\\partial V} \\\\right)_{S, N_1, ..., N_r}$ $\\\\mu _j = -\\\\left( \\\\frac{\\\\partial U}{\\\\partial N_j} \\\\right)_{S, N_1, ..., N_r}$ We can generalize the concept of intensive and extensive variable to commonly measured properties of materials, such as mass and volume, generating a number of well known intensive properties, the most common of which probably being density, represented in this book by the Greek letter $\\\\rho $ (Appendix A), defined by the mass of a substance, $M$ , divided by its volume, $V$ $\\\\rho = \\\\frac{M}{V}$ It is often convenient to define an area density as some extensive property divided by the area (as we did when explaining the REV concept), and a linear density as some extensive property divided by length.\",\n \"For example, a thin wire can have its linear density calculated by dividing its weight by its length.\",\n \"Measurements and calculations of physical systems are often linked to some system of units.\",\n \"The most common systems are the International System of Units (SI), the CGS (centimeter-gram-seconds) and to some extent the British Imperial Units.\",\n \"In this book, whenever possible we will adhere to the international system of units.\",\n \"The SI uses seven base units, the second (s) for time, the meter (m) for length, the kilogram (kg) for mass, the ampere (A) for electric current, the kelvin (K) for temperature, the mole (mol) for amount of substance and the candela (cd) for luminous intensity.\",\n \"A number of derived units can be defined from the SI, for example the newton (N = kg m s$^{-2}$ ) for force, the pascal (Pa = kg m$^{-1}$ s$^{-2}$ ) for pressure and stress and the joule (J = kg m$^{2}$ s$^{-2}$ ) for energy, work and heat.\",\n \"It is convenient, however, for many derivations and for checking consistency or units, to express the dimensions of physical quantities independent of a system of units.\",\n \"For elegance, in this book, whenever dimensions are considered when expressing a physical quantity, square brackets [] will be used.\",\n \"The elementary dimensions and symbols most often used in soil physics and groundwater hydrology are length [L], mass [M], time [T] and absolute temperature [$\\\\Theta $ ].\",\n \"Therefore, using the base length dimension L, length is represented as [L], area [L$^2$ ] and volume [L$^3$ ].\",\n \"The consistence of dimensions and units can be calculated for a physical quantity.\",\n \"For pressure for example we have $P = \\\\frac{\\\\text{force}}{\\\\text{area}} = \\\\frac{N}{m^2} = \\\\frac{kg~m~s^{-2}}{m^2} = \\\\frac{kg~m}{m^2~s^{2}} = \\\\frac{kg}{m~s^{2}} = \\\\text{Pa}$ Alternatively, in terms of dimensions $P = \\\\frac{[M~L~T^{-2}]}{[L^2]} = \\\\frac{[M~L]}{[L^2 T^{2}]} = \\\\frac{[M]}{[L~T^{2}]} = [M L^{-1} T^{-2}]$ The last two terms on the right being representations of the same thing, any of which can be chosen for convenience.\",\n \"A range of intensive properties are defined in soil physics, soil mechanics and geology which are useful to characterize porous materials.\",\n \"We will, for the most part, use the formulas and definitions encountered in soil physics but several are analogous or are simple adaptations from soil mechanics or chemistry.\",\n \"Dry bulk densitybecause wet bulk density is rarely considered in soil physics, the term “bulk density\\\" often refers to “dry bulk density\\\".\",\n \"is the mass of dried material over volume.\",\n \"Dry bulk density can be measured by collecting an undisturbed sample of the material, by using a coring device and measuring the overall volume of the core.\",\n \"The sample is usually oven dried at 105 $^{o}$ C for 24h to 48h to determine its dried mass be weighing.\",\n \"The measured volume includes pores and therefore, the more porous the material is the lower its bulk density is.\",\n \"$\\\\rho _b = \\\\frac{m_s}{V_t}$ In which $\\\\rho _b$ is the dry bulk density in [M L$^{-3}$ ], $m_s$ is the dry soil mass [M] and V$_t$ is the total volume [L$^3$ ].\",\n \"Bulk density can be modified by artificial compaction or consolidation due to applied loads or by natural processes such as pore filling with finer particles or diagenetic precipitation or consolidation in sedimentary deposits.\",\n \"Due to low natural bulk densities soils can be easily compacted by agricultural machinery, grazing, a process which is directly linked to soil degradation.\",\n \"Bulk density is determined by porosity, grain size and mineralogy of the solid particles.\",\n \"In most soils, bulk densities ranges from less than 1000 $kg~m^3$ in oxidic and organic soils under natural conditions to more than 1300 $kg~m^3$ in sandy or severely compacted clay soils.\",\n \"In sedimentary deposits the density will depend on factors such as mineralogical composition, age of deposition and depth below overburden layers.\",\n \"Particle density is analogous to the “true\\\" density of the material, considering only the mass and volume occupied by the solid particles.\",\n \"For accurate measurement of particle density, accurate measurement of solid particles volume is needed.\",\n \"This is usually achieved by liquid or gas displacement methods, where the volume occupied by solid particles is equivalent to the displaced volume of liquid or gas at a given temperature.\",\n \"$\\\\rho _s = \\\\frac{m_s}{V_s}$ In which $\\\\rho _s$ is the (solid) particle density in [M L$^{-3}$ ], $m_s$ is the dry soil mass [M] and V$_p$ is the volume occupied by solid particles [L$^3$ ].\",\n \"Particle density represents a physical averaging of the individual densities of the components of the solid phase.\",\n \"Soil and sediment particle density is often assumed as 2650 kg m$^{-3}$ , the density of the mineral quartz, which is abundant in many soils and rocks.\",\n \"However the density can be lower in materials rich in organic particles and higher in materials rich in certain heavy iron and titanium oxides.\",\n \"The volumetric porosity is the volume of pores per unit total volume of soil including solids and pores.\",\n \"It is the fraction of the material that is occupied by voids.\",\n \"In this calculation we are not concerned if the voids are occupied by gas or water.\",\n \"$\\\\phi = \\\\frac{V_v}{V_t}$ In which $\\\\phi $ is the total porosity [L$^{3}$ L$^{-3}$ ] (meaning volume of pores per total volume and thus the units are often not canceled), $V_v$ volume of voids, or pores [L$^3$ ] and V$_t$ is the total volume of the porous media [L$^3$ ].\",\n \"Because it might be difficult to accurately measure the volume of voids in soil and as particle and bulk densities are routinely measured, it is convenient to calculate the total porosity as $\\\\phi = 1 - \\\\frac{\\\\rho _b}{\\\\rho _b}$ Equation 2.10 can be obtained by isolating the volumes in Equations 2.7 and 2.8, replacing into Equation 2.9 and assuming that the dry soil masses are the same.\",\n \"Bulk density and total porosity are sensitive to compaction, pore clogging and alterations in pore structure, either anthropogenic or caused by natural processes.\",\n \"Traffic by machinery, humans and animals and loads by buildings and roads can directly affect bulk density and porosity, which will in turn affect fluxes of matter and energy in porous materials.\",\n \"Particle distribution, shape and sorting can also affect bulk density and porosity.\",\n \"Density and porosity are important parts of the characterization of the solid matrix and pore space.\",\n \"Another important property used for the characterization of porous materials is the specific surface area (SSA).\",\n \"$SSA = \\\\frac{A_s}{m_s}$ In which $A_s$ is the surface area of the particles and $m_s$ is the mass of particles.\",\n \"The specific surface area is in [L$^{2}$ M$^{-1}$ ] and represents the surface area of solid particles per unit mass of particles.\",\n \"It is usually expressed in $m^{2}~g^{-1}$ or $m^{2}~kg^{-1}$ , although the former is usually preferred because it results in smaller numbers.\",\n \"Surface area varies with grain size, roundness and surface rugosity, and with granulometric distribution of a material, and materials composed of finer particles will usually have larger specific surfaces areas.\",\n \"Surface area influences the reactivity of the material to water and contaminant retention via adsorption and other mechanisms and in the natural accumulation of organic matter.\",\n \"Like many other properties discussed in this book, SSA is an operational concept that depends on the method used for its determination.\",\n \"It can be determined by adsorption of nitrogen, water, ethylene glycol monoethyl ether (EGME) and other substances in vapor phase, and values in the range of 1 to 200 $m^{2}~g^{-1}$ can be found in soils depending on grain size distribution [61].\",\n \"The most common method found in the literature is nitrogen ($N_2$ ) adsorption using BETFrom the authors of the study that presented the adsorption isotherms used Brunauer, Emmett and Teller method.\",\n \"However, BET could underestimate SSA in certain materials because the $N_2$ molecule might be too large to enter interlayer spaces and some very small pores [61].\",\n \"Equally important in soil physics in the quantification of the the amount and volume of the fraction occupied by fluid.\",\n \"In theory the fraction occupied by different fluids can be calculated in a similar fashion.\",\n \"Fluids of interest in porous media physics are, in addition to water, air, natural gas, petroleum, and a wide range of contaminants such as DNAPLDense non-aqueous phase liquids, LNAPLLight non-aqueous phase liquids, industrial oils and a range of compounds dissolved in water at various concentrations.\",\n \"We will use the symbol $\\\\theta $ for representing water mass and volume content in porous media because it has been traditionally used in soil physics, but the calculation can be extended to other dense fluids, and to some extent to gases in the case of volumetric saturation.\",\n \"The gravimetric water content is the mass of water per mass of dry soil or sediment material.\",\n \"It is usually measured by weighing (gravimetrically) a portion of material before and after drying in an oven at 105 $^{o}$ C for 24h to 48h.\",\n \"$\\\\theta _g = \\\\frac{m_w}{m_s}$ In which $\\\\theta _g$ is the gravimetric water content [M M$^{-1}$ ], meaning mass of water per mass of dry soil or sediment and thus the units are often not canceled, m$_a$ is the mass of water [M] and m$_s$ is mass of dry soil or sediment [M].\",\n \"Gravimetric water content in excess of 1 kg kg$^{-1}$ is possible for highly porous materials such as volcanic ash and materials rich in organic matter.\",\n \"Because gravimetric water content does not take into consideration the volume available for water storage and total pore space, it is much more useful to define a volumetric water content, or the volume of water per unit volume of porous media $\\\\theta = \\\\frac{V_w}{V_t}$ In which $\\\\theta $ is the volumetric water content [L$^{3}$ L$^{-3}$ ], meaning volume of water divided by total volume of soil or sediment and thus the units are often not canceled, V$_a$ volume of water [L$^{3}$ ] and V$_t$ is the total volume of dry soil or sediment [L$^{3}$ ], including pores.\",\n \"Defining the density of water as $\\\\rho _w = \\\\frac{m_w}{V_w}$ and isolating the volumes in Equations 2.8 and 2.14 and replacing into Equation 2.13 we have $\\\\theta = \\\\frac{m_w/\\\\rho _w}{m_s/\\\\rho _b} = \\\\frac{\\\\rho _b}{\\\\rho _w} \\\\frac{m_w}{m_s}$ which with Equation 2.9 can be written as $\\\\theta = \\\\frac{\\\\rho _b}{\\\\rho _w} \\\\theta _g$ For simplicity, water content is usually calculated from this equation because bulk density and gravimetric water content are routinely measured in laboratory.\",\n \"The density of water can be calculated from temperature, as we will see in later chaptersAlthough it is often assumed as 1 g cm$^{-3}$ or 1000 kg m$^{-3}$ , depending on the unit system adopted.. Another quantity seen in soil physics, soil mechanics and petroleum engineering is the degree of saturation.\",\n \"The degree of saturation is the volume of water divided by the volume of voids $S_w = \\\\frac{V_w}{V_v} = \\\\frac{\\\\theta }{\\\\phi }$ The degree of saturation can theoretically vary from 0 in a dry media to 1 in a completely water saturated media and it is usually represented as a dimensionless number.\",\n \"Another quantity used in soil mechanics is the void ratio $e = \\\\frac{V_v}{V_s}$ In which V$_s$ is the volume of solids [L$^{3}$ ].\",\n \"It is left to the reader to prove the following relationships $e = \\\\frac{\\\\phi }{1 - \\\\phi }$ $\\\\phi = \\\\frac{e}{1 + e}$ The void ratio is used in geotechnical engineering to evaluate volume changing in foundations under loading.\",\n \"ll $ T $ Temperature $ U $ Internal energy $ S $ Entropy $ N_r $ Number of moles of the rth substance $ P $ Pressure $ V$ Volume $ \\\\mu _j$ Chemical potential of the jth substance $ \\\\rho $ Generalized density $ M$ Generalized mass $ [L]$ Generalized length unit $ [M]$ Generalized mass unit $ [T]$ Generalized time unit $ [\\\\Theta ]$ Generalized temperature unit $ \\\\rho _b$ Dry bulk density $ m_s$ Dry soil (or sediment) mass $ V_t$ Total volume $ \\\\rho _s$ Solid particles density $ V_s$ Solid particles volume $ \\\\phi $ Volumetric porosity $ V_v$ Volume of voids (pores) $ SSA$ Specific surface area $ A_s$ Surface area $ \\\\theta _g$ Gravimetric water content $ m_w$ Mass of water $ \\\\theta $ Volumetric water content $ \\\\rho _w$ Water density $ V_w$ Volume of water $ S_w$ Degree of saturation $ e$ Void ratio Conceptually, a soil is a material that has been subjected to the process, or set of processes, known as pedogenesis.\",\n \"Large fractions of Earth's crust is composed of rocks, these rocks are usually formed under intense pressure and/or temperature conditions.\",\n \"Rocks are divided into three main types igneous, metamorphic and sedimentary.\",\n \"Igneous rocks are formed when magma is cooled, either within Earth's crust or when it flows to the surface of Earth in the form of lava.\",\n \"Metamorphic rocks are formed when any type of rock is subjected to intense pressure and/or temperature conditions within the crust such that minerals are subjected to mineralogical and petrological transformations without complete fusion of all minerals of the rock at high temperatures.\",\n \"Sedimentary rocks are those formed by sediments that are usually transported to an accumulation area and then subjected to pressure and temperature within ranges of values usually much lower than those required for the formation of metamorphic rocks, finally transforming into a sedimentary rock by a set of processes called diagenesis.\",\n \"Because those rocks are formed at high temperatures and pressures when compared to those at Earth's surface they tend to be thermodynamically unstable when exposed to lower pressures, temperatures and chemical environment close to the surface.\",\n \"In addition rocks near surface are often exposed to rainwater, yearly and daily temperature variation and a variety of compounds and forces produced by organisms in the interface with the atmosphere.\",\n \"The transformations related to the conditions at Earth's surface caused by all these processes is called weathering.\",\n \"At the right conditions, weathering can evolve into pedogenesis and the minerals present in the rocks can be altered or reprecipitated into forms stable at atmospheric conditions.\",\n \"While rocks can be composed of a range of minerals, usually dependent on rock genesis, soils usually concentrate phylossilicate clays, iron and aluminum oxides and tectossilicates such as quartz and feldspars, depending on weathering conditions.\",\n \"These forms tend to be much more stable in soil environment than other minerals found in rocks.\",\n \"Of course soils are a complex material with a large variety of minerals of different species, but most soils will have a predominance of oxides, phylossilicate clays and quartz in different proportions in addition to the organic phase.\",\n \"Because soils and sedimentary deposits and to some extent partially weathered rocks of all types tend to be more porous, compressible and with an interconnected pore network than other rock types, the material presented in this book tends to apply more directly to these materials.\",\n \"However, fractures and pores of different geometries can be found in other types of rocks and with the necessary theoretical constraints, the models can be applied to these materials as well.\",\n \"Soils and sedimentary deposits, in particular, are unconsolidated media and therefore, the individuals particles and aggregates can be easily disturbed by a series of processes such as compaction, penetration of roots and organisms, traffic, and many others, while rocks are consolidated materials being able to withstand mechanical stresses of larger magnitude.\",\n \"The degree of consolidation of rocks varies with genesis and mineralogical composition, but it is safe to say that most rocks under normal conditions will act as a rigid matrix when considering flow and transport phenomena.\",\n \"We mentioned that soils are formed from weathering and pedogenesis of rocks due to the minerals present in rocks being thermodynamically unstable in Earth's superficial environment.\",\n \"Technically the process is a lot more complex, soils can not only form from rocks but also from sediments and from previously existing soils, and under influence of anthropogenic activities such as deposition of matter and superficial disturbance.\",\n \"A given soil is in equilibrium with its environment, including climate, natural vegetation and geomorphological setting.\",\n \"If any of these factors is significantly altered a soil can evolve, devolve, be complete buried or completely destroyed by erosional processes over time.\",\n \"Likewise, sediment and rock formation is very complex and dependent on several geological and geomorphological processes and on factors such as Earth's internal dynamic and climate.\",\n \"Because soil, rock and sediment forming processes are very complex and covered in different fields of soil science, geology and physical geography, discussing each one of them is beyond the scope of this book.\",\n \"An important issue in natural porous media physics is the composition of solid phase where mechanical and transport phenomena are being considered.\",\n \"Rocks, sedimentary deposits and soils of interest are largely composed of minerals.\",\n \"A mineral is a compound that has a more or less constant chemical composition between samples of the same mineral, has an internal structure that is organized and is formed naturally in Earth or other astronomical objects.\",\n \"A rock is composed of one or more minerals and is consolidated, i.e.\",\n \"cannot be easily disassembled by hand or using manual tools such as spades or a geological hammer.\",\n \"From the soil physics point of view, soils are much more complicated materials than rocks because they are very heterogeneous in composition and are not consolidated, being easily deformed and disaggregated with much less effort than that needed for most rocks.\",\n \"Soils possess varying grain size distributions, proportions of inorganic mineral and amorphous materials, organic phases, and gas and solution.\",\n \"Large variability in the composition of the mineral phase can be found depending on climate, parent material and other soil forming processes.\",\n \"The other factors that makes soils so complex is that the colloidal solid phase is most often electrically charged, mainly due to inorganic and organic superficial groups and isomorphic substitution in crystalline lattices.\",\n \"Because of that, the small particles have the ability to interact with charged ions in solution.\",\n \"Modeling contaminant retention and transport in soils can be much more challenging in soils than in sedimentary rocks composed of a single coarse grained mineral phase, such as in sandstones and sand deposits, where the density of charges is negligible and the specific surface area is very small compared to that of most soils.\",\n \"Because soils are so complex, they cannot be fully characterized and studied in detail in laboratory or in lattice, physicochemical or thermodynamic simulations as geologists do with individual minerals and simple rocks and physicists do with synthetic crystals.\",\n \"Although soil physics is traditionally constrained to the domains of fluid phenomena and soil mechanics, much can be done by using the same approach and techniques used by condensed matter physicists.\",\n \"The fact that the study and characterization of soils has been almost entirely relegated to soil science and engineering during most of its history reflects the fact that the complexity and heterogeneity of the material prevents, for the most part, the physicist's approach which is often concerned with materials that are simple enough that they can be investigated and modeled using available laboratory and mathematical methods.\",\n \"With appropriate simplifications and with modern physical methods, soils can be studied in the framework of condensed matter physics.\",\n \"Although we mentioned that soils are more complex from the physical point of view than most rocks and sedimentary deposits, there are geological settings where the line between soils, rocks and sediments is blurred, either because of a physical mixing of different materials due to pedogenetic and geological conditions and/or because a system were mechanical stresses and transport phenomena is considered has interfaces between different materials.\",\n \"One example of a material intermediary between soils and rocks is the saprolite which is a partially weathered rock theoretically in its way to become a soil.\",\n \"Another example is sedimentary rocks and deposits composed of phylossilicate clays.\",\n \"These materials can result in chemical behavior when interacting with contaminants more closely related to soils than to inert materials.\",\n \"It is common in natural environments that soils occupy the most superficial layers in Earth's crust followed by saprolite, slightly weathered or fragmented or fractured rock followed by consolidated rock.\",\n \"A dissolved contaminant, for example, will have to interact with soil and saprolite before it reaches deeper saturated layers in porous rock, which can compose aquifers where water is being extracted.\",\n \"Ultimately the mineralogical composition of soils, rocks and saprolite will influence, if not control, the interaction with contaminants, water transport and storage properties as well as mechanical behavior.\",\n \"Other than inorganic materials, organic deposits, which constitute some soils and geological deposits, present an unique challenge from the physics and engineering point of view, as they tend to be very loose, compressible, can be easily lost due to decomposition, and can be very reactive when it comes to interacting with chemicals, depending on the nature of the organic fractions involved.\",\n \"Water retention and transport is particularly challenging to be modeled in these materials as the tend to have very low densities, can be very porous and can absorb water on a content greater than the solid phase on a mass base.\",\n \"We will focus on materials that are mostly inorganic throughout this book, as these are materials comprise the Earth's crust upper layer in most of the planet, while organic deposits are restricted to certain geological and pedogenetic settings.\",\n \"One of the disciplines in which condensed matter physics and mineralogy intersect is crystallography.\",\n \"Most of us have an intuitive knowledge of what a crystal is, it is usually associated with “rock crystals” such as quartz.\",\n \"Early physicists or natural philosophers observed that many minerals had an external shape that was more or less constant between different samples of the same mineral [44].\",\n \"They speculated that the external shape is the result of the internal organization or the packing of minuscule particles which compose matter.\",\n \"Later, these minuscule particles would be recognized as atoms and the internal organization of these atoms would be experimentally confirmed by x-ray diffraction [18].\",\n \"You might have noticed that most varieties of quartz that people collect are formed by elongated crystals.\",\n \"If you count the number of faces in these crystals and measure the internal angle between faces you will find six and approximately 120$^{o}$ .\",\n \"This is reflect of the internal atomic structure of quartz, which in ideal conditions is composed entirely of silicon and oxygen atoms.\",\n \"Crystals are then materials that have a highly organized internal structure which repeats in space, and crystallography is the branch of science dedicated to the study of crystals.\",\n \"Materials which have a random or non-organized arrangement of atoms are called amorphous.\",\n \"Crystals can be synthesized in laboratory or can be formed naturally such as in soil and rock forming processes where atoms reorganized in response to a change in thermodynamic conditions.\",\n \"These inorganic crystals that are formed under natural conditions and which can be identified as having a more or less constant chemical composition between different species and whose internal arrangement and physical properties make it possible to recognized as a distinct species from other crystals in the environment are called minerals.\",\n \"Minerals in rocks originate from rock forming processes or they can be inherited from a different rock when they pass through the rock cycle.\",\n \"Rocks are primarily classified based on origin as igneous, sedimentary and metamorphic.\",\n \"Igneous rocks are formed from cooling of magma within Earth or lava in Earth's surface.\",\n \"As the materials cools, bonds are formed between atoms and the structure organizes itself based on the affinity between atoms, remaining concentration and atomic radius relationships.\",\n \"The issue of radius ratio is usually modeled using the sphere models presented Chapter REF , but for atoms instead of mineral particles or aggregates as in that case.\",\n \"Because igneous rocks form from magma or lava which are molten materials, the minerals are newly formed when the rock finishes cooling.\",\n \"This is not always the case in other types of rocks.\",\n \"Sedimentary rocks are formed from erosion, transport, deposition and consolidation of sediments which came from another source, usually either a rock that was weathered or a soil or sedimentary deposit which was eroded.\",\n \"In reality, sedimentary rocks can be also formed from precipitation of ions in solution with or without the influence of organisms.\",\n \"Sedimentary rocks can be composed of minerals that were formed in other rocks or other conditions and were weathered and transported forming a new rock via consolidation without altering the chemical composition and crystalline structure or by minerals that were formed during diagenesis.\",\n \"Metamorphic rocks are formed by solid state transformations in which part or all of the minerals in a rock of any type are transformed into new minerals by the action of pressure and/or temperature at greater intensities than those that occur during the genesis of sedimentary rocks.\",\n \"During formation of metamorphic rocks there can be re-composition and/or reorganization of the internal structure of minerals, thus forming new species.\",\n \"Partial fusion of some minerals is allowed, but if the rock is completely molten and later recrystallized it would be classified as an igneous rock.\",\n \"Rocks are composed of minerals, there are rocks composed of only one mineral species, such as some marbles and sandstones and rocks composed of more than one type of mineral such as basalts and granites.\",\n \"There are minerals that are more commonly found in each type of rock and this is used in the classification and identification of rocks.\",\n \"The chemical composition of the most common minerals reflects the composition of Earth's continental crust in which oxygen, silicon, aluminum, iron, calcium, sodium, potassium and magnesium account for approximately 98% of Earth's crust on a mass basis.\",\n \"Oxygen and silicon alone account for more than 70% of the mass.\",\n \"Minerals are classified based on the chemical composition usually represented as a chemical formula.\",\n \"Although some minerals are composed by a single element bonded via ionic or covalent bond, diamond (C), graphite (C), sulfur (S), gold (Au) being a few examples, most minerals of interest in rocks, sediments and soils are composed of one or more cations bonded to an anion or anionic group.\",\n \"These cations and anions will bond to form stable configurations based on valence, affinity, abundance and radius ratio and according to thermodynamic laws.\",\n \"The two most common elements on Earth's crust are an anion and a cation, oxygen ($O^{2-}$ ) and silicon ($Si^{4+}$ ).\",\n \"Based on this configuration one would expect that a mineral with formula $SiO_2$ with one silicon bonding with two oxygens would be a stable configuration.\",\n \"Although $SiO_2$ is the formula of a common silicate mineral, the radius ratio precludes the coordination of only two oxygen atoms around the silicon.\",\n \"In reality, each silicon atom coordinates four oxygen atoms and shares one electron with each oxygen via a predominately covalent bond.\",\n \"This structure with one silicon coordinating four oxygen atoms is represented as a polyhedron called the silicon tetrahedron (Figure REF (a)) and forms the basis of the silicate minerals which are extremely common in soils, sediments and rocks.\",\n \"Because each oxygen has only one electron shared with the silicon atom, one more electron needs to be shared in each corner of the tetrahedron for physicochemical stability to occur.\",\n \"In silicates the stability can be achieved by other cations forming “bridges” between tetrahedrons defining the class of nesossilicates, by sharing the apical oxygen between two tetrahedra forming a double tetrahedra in the sorossilicates, by sharing two oxygens between adjacent tetrahedra forming ring of three or six tetrahedra in the cyclossilicates, by sharing two or three oxygens forming chains in the inossilicates, by sharing the three basal oxygens forming sheets of tetrahedrons in the phyillossilicates (Figure REF (b)) or by sharing all four oxygens forming complex tridimensional networks in the tectossilicates.\",\n \"The chemical formula in each of these classes is defined by the occurrence of $Si_nO_m$ in the formula in which $n$ and $m$ are whole numbers or fractions.\",\n \"The class of phylossilicates is particularly important in soils and a few sedimentary rocks as these clay minerals can define or heavily influence the physical and chemical behavior or the material in which they occur, they can also have surface charges and can interact with ions in solution.\",\n \"A few examples of silicates common in rocks are quartz ($SiO_2$ ), orthoclase feldspar ($KAlSi_3O_8$ ), plagioclase feldspars ranging from calcium the plagioclase anorthite ($NaAlSi_3O_8$ ) to the sodium plagioclase albite ($CaAl_2Si_2O_8$ ), olivine ($(Mg, Fe)_2SiO_4$ ) and various pyroxenes.\",\n \"These minerals are commonly found in igneous rocks and can appear in sedimentary and metamorphic rocks as an inheritance from the original material which either suffered diagenesis or metamorphism.\",\n \"Other mineral classes are found in many types of rocks usually in smaller quantities, a few examples being oxides such as zircon ($ZrSiO_4$ ), ilmenite ($FeTiO_3$ ) and rutile ($TiO_2$ ) and sulfides such as spharelite ($ZnS$ ) and galene ($PbS$ ).\",\n \"In sedimentary rocks, besides quartz which is common in sandstones and also in sedimentary deposits, it is also relatively common to find minerals that formed in solution via precipitaion such as calcite and aragonite (both $CaCO_3$ , but with different internal structures), dolomite ($CaMg(CO_3)_2$ ), gypsum ($CaSO_4\\\\cdot 2H_2O$ ), anhydrite ($CaSO_4$ ) and salts such as halite ($NaCl$ ) and sylvite ($NaCl$ ).\",\n \"These minerals are usually composed primarily by ionic bonds and tend to be less resistant to weathering.\",\n \"Figure: Single silicon tetrahedron (a) and tetrahedral sheet (b) and sigle octahedra (c) and octahedral sheet (d) (Adapted from ).Soils are formed by weathering and pedogenesis over rocks, sedimentary deposits or either previously formed soils.\",\n \"They can inherit minerals from the soil forming material that have gone through no discernible mineralogical transformation except size reduction and fragmentation.\",\n \"These minerals are called primary minerals and the amount in soil will depend on parent material and climate.\",\n \"Common primary minerals found in soils are quartz, feldspars, primary micas (a group of phylossilicates).\",\n \"Heavy minerals such as rutile, anatase (also $TiO_2$ ) and magnetite ($Fe_3O_4$ ), tend to occur in very minor quantities in some soils.\",\n \"More often than not new minerals are created during pedogenesis and these minerals control how the soil functions, its porosity, pore geometry, mechanical properties and how it interacts with water, gas and with solutes.\",\n \"The minerals that are formed during pedogenesis are often referred to as secondary minerals while the minerals inherited from parent material are the primary minerals.\",\n \"On the tropics where average annual temperatures and precipitation are elevated, at least in part of the year, soils tend to accumulate iron and aluminum oxides, oxi-hydroxides and hydroxides as secondary minerals, commonly hematite ($Fe_2O_3$ ), goethite ($FeOOH$ ) and gibbsite ($Al(OH)_3$ ), and usually referred to as a group as iron and aluminum oxides.\",\n \"Iron oxides usually give strong red, orange and yellow hues to soils and are usually not restrictive to agricultural use of soils from a physical standpoint but can be severely restrictive from a soil fertility standpoint under natural conditions.\",\n \"Soils rich in oxides are challenging for modeling water and contaminant transport and storage because they usually are associated with unique pore geometry and distribution and because surface hydroxi groups ($OH$ ) these minerals are subjected to protonation and deprotonation according to solution pH following the reactions [89] $M - OH + H^+ \\\\rightleftharpoons M - OH_2^+$ $M - OH \\\\rightleftharpoons M - O^- + H^+$ Thus, under acidic conditions the OH groups protonate, generating a positive charges surfaces with ability to attract anions while at higher pHs the OH groups deprotonate generating negatively charged surfaces with the ability to attract cations.\",\n \"At an equilibrium point where there is an equilibrium between positive and negative sites the points of zero charge (PCZ) of the compound can be found.\",\n \"In soils this behavior occurs in iron and aluminum oxides such as hematite, goethite and gibbsite where M represents iron (Fe) or aluminum (Al) metals usually on octahedral sites on the mineral structure.\",\n \"However these variable charged sites can be found in organic compounds and in silicates to some degree.\",\n \"Roughly speaking, the PZC of a soil represents a weighed average of PZCs of individual components.\",\n \"Cation exchange capacity (CEC) and anion exchange capacity (AEC) in soils are defined by the predominance of negative or positive surface sites.\",\n \"AEC is mainly observed in highly weathered oxidic soils in the tropics and can revert to CEC as soil pH increases.\",\n \"The other major group of secondary minerals in soils are the phylossilicates or clay minerals.\",\n \"Although these minerals can occur as primary minerals in sedimentary deposits and rocks, they occur in almost all soils in various quantities depending on parent material and pedogenetic conditions.\",\n \"Clay minerals along with iron and aluminum oxides can be found on the clay fraction of soils while primary minerals usually occur in the silt and sand fractions.\",\n \"Phylossilicate minerals in soils are layered or sheet minerals where there is an alternation of sheets formed by silica tetrahedrons and octahedral layers where metallic cations such as aluminum or magnesium coordinate six hydroxides forming an octahedral polyhedron of coordination (Figure REF (c)) in which some of its hydroxides are shared among adjacent octahedrons (Figure REF (d)).\",\n \"Phylossilicate clays are classified according to the proportion of octahedral (Figure REF (a)) to tetrahedral (Figure REF (b)) layers.\",\n \"In 2:2 layer silicates, often simplified to 1:1, two octahedral layers bound to two tetrahedral layers (Figure REF (b)), while in 4:2 layer silicates, often simplified to 2:1, two tetrahedral layers are each bound to one octahedral layer forming two pairs of 2:1 structures (Figure REF (c)).\",\n \"The space between the pairs of sheets in the 1:1 configuration and pair of groups of sheets in the 2:1 configuration is called in the interlayer space and the distance between the groups of 1:1 or 2:1 sheets can vary according to environmental conditions in some types of minerals.\",\n \"The most common 1:1 layer mineral in tropical soils is kaolinite ($Al_2Si_2O_5(OH)_4$ ) and because the space between layers does not vary and because it has little isomorphic substitution it has relatively low specific surface area, ion retention and exchange capacity.\",\n \"The structure of kaolinite is presented in Figure REF illustrating the tetrahedral and octahedral sheets .\",\n \"The distance between the basal planes of the two 1:1 layers does not vary with environmental conditions and is around 0.7 nm.\",\n \"On the octahedral sheets some of the hydroxides are replaced with oxygen and these are shared with silicon tetrahedrons forming the mineral structure (Figure REF ).\",\n \"Isomorphic substitution is a process that occurs during the mineral formation in which the central cation is replaced by a cation of similar atomic radios but with different valence.\",\n \"Thus if aluminum ($Al^{3+}$ ) is replaced by iron in its reduced state ($Fe^{2+}$ ) in specific point within the octahedral sheet, a charge deficit will appear due to the resulting unbalanced negative charges in hydroxides.\",\n \"These charges are the basis of the chemical reactivity and will control the dynamics of nutrients and contaminants in soils and other materials rich in phylossilicate clays.\",\n \"The most common 2:1 phylossilicate minerals in soils are from the smectite group, particularly montmorillonite $((Na,Ca)_{0.33}(Al,Mg)_2(Si_4O_{10})(OH)_2 \\\\cdot nH_2O)$ , and vermiculite $((Mg,Fe^{2+},Fe^{3+})_3[(Al,Si)_4O_{10}](OH)_2 \\\\cdot 4H_2O$ ).\",\n \"Montmorillonite is highly expansible, meaning that the interlayer space can be increase when water enters the space, has high specific surface area, is very reactive towards adsorption of water and solutes and can have shrinking swelling behavior according to soil water content, being extremely hard when dry and extremely plastic and sticky when wet.\",\n \"Smectite's structure is illustrated in Figure REF showing the range of variation of the distance between the two 2:1 structure.\",\n \"Isomorphic substitution on cation on the octahedral layer is common in smectites (Figure REF ) and other 2:1 minerals and because of that the surface charge and CEC of these minerals is much larger than oxides and 1:1 minerals.\",\n \"The interlayer can be filled with water and cations under wet conditions, expanding the mineral on a microscopic scale and consequently the material on a macroscopic scale.\",\n \"Another less common 2:1 mineral is vermiculite which is less expansive than montmorillonite.\",\n \"Illite $ (K,H_3O)(Al,Mg,Fe)_2(Si,Al)_4O_{10}[(OH)_2,(H_2O)]$ ) can also occur in some soils in minor amounts depending on genesis and parent material, but the interlayer space in this mineral usually does not vary and it tends to be less reactive than either montmorillonite or vermiculite.\",\n \"There are also hydroxi-interlayered minerals in which the interlayer space is occupied by octahedral sheets or natural polymers, but these are of restricted occurrence in terms of amount and representative and are of interest only in specific cases.\",\n \"Anyone interested in soil physics or in unsaturated zone hydrogeology must be proficient in soil chemistry and mineralogy.\",\n \"It is impossible to study these soils and sediments without understanding how the solid phase interacts with fluids and gases, these disciplines cannot be dissociated.\",\n \"The reader is strongly recommended to consult books on the subject of soil chemistry and mineralogy [91], [36].\",\n \"Figure: Representation of the tetrahedral (a) and octahedral (b) sheets and 1:1 (c) and 2:1 (d) layer minerals.Figure: Kaolinite structure (Adapted from ).Figure: Smectite structure (Adapted from ).Because soil secondary minerals, colloidal organic matter and some amorphous compounds are charged, an electric field around those compounds results which can attract ions in solution.\",\n \"The liquid phase solution in this case is in the pores around solid particles and the volume occupied by it can range from complete saturation of the pores to thin films over solid surfaces.\",\n \"Ions in solution will form a distribution based on the equilibrium of the electrostatic attractive and repulsive forces in particle surfaces and the concentration gradient in solution.\",\n \"The interaction of the charged surfaces and ions results in a structure called Electric Double Layer (EDL).\",\n \"The presence of an EDL and its relationship with the different types of surface sites in soil minerals and organic compounds will influence the behavior of clay particles in aqueous solution as in the case of grain size analysis and in the stability of the soils in the field.\",\n \"The mechanisms of dispersion and coagulation play a fundamental role in soil dispersion both in laboratory and under field conditions.\",\n \"The surface chemistry will also determine parameters for solute transport and retention in soils.\",\n \"The solutes of interest include inorganic and organic compounds in fertilizers, polar and non-polar pesticides, viruses, bacteria and other pathogenic microorganisms, heavy metals, radionuclides and many other.\",\n \"Solutes can also be transported attached to dispersed clays and other colloids in what is called colloid-facilitated transport.\",\n \"The EDL can be modeled using the diffuse double layer theory, which assumes the mineral surfaces as a plane with an even distribution of charges.\",\n \"Under these conditions, the distribution of charges at a distance $x$ from the surface plane under equilibrium conditions follows the Boltzmann distribution [6], [39] $n_i = n_{0i} \\\\exp {(\\\\frac{-z e \\\\psi (x)}{ kT})}$ In which $n_i$ is the ion concentration of the ith species at a distance $x$ from the surface, $\\\\psi (x)$ is the electrical potential at the same distance, $n_{0i}$ is the ion concentration of the ith species in bulk solution, $z$ is the ion valence, $e$ is the elementary charge, $k$ is the Boltzmann constant and $T$ is the absolute temperature.\",\n \"The charge density at any point in the diffuse layer is given by $\\\\rho = \\\\sum z_i e n_i$ In which $i$ indicates a summation term over every ion species.\",\n \"The variation of the electrical field strength with distance is given by a differential equation know as the Poisson equation $\\\\frac{d^2 \\\\psi }{dx^2} = - \\\\frac{4 \\\\pi }{\\\\epsilon } \\\\rho $ in which $\\\\epsilon $ is the absolute permittivity of the medium.\",\n \"For symmetrical electrolytes, the Poisson equation can be expressed in terms of ion concentration and charge density by combining it with the concentration distribution and charge density equations, resulting in the Poisson-Boltzmann equation $\\\\frac{d^2 \\\\psi }{dx^2} = - \\\\frac{4 \\\\pi }{\\\\epsilon } \\\\sum z_i e n_{0i} \\\\exp {(-\\\\frac{z_i e \\\\psi }{kT})}$ For suspensions consisting of a single symmetrical electrolyte the summation expands to This can be achieved by considering the valence of the positively charged ions as $z_{+} = +|z|$ and the valence of the negatively charged ions as $z_{-} = -|z|$ .\",\n \"Note that a symmetrical electrolyte has the same proportion of cations and anions, the most simple example being NaCl which dissociated to Na$^{+}$ and Cl$^{-}$ , therefore, their concentrations in solution are also equal, i.e.\",\n \"$n_{0+} = n_{0-}$ $\\\\frac{d^2 \\\\psi }{dx^2} = - \\\\frac{4 \\\\pi |z_i| n_{0i}}{\\\\epsilon } [ \\\\exp {(-\\\\frac{|z_i| e \\\\psi }{kT})} - \\\\exp {(\\\\frac{|z_i| e \\\\psi }{kT})} ]$ To solve this differential equation several simplifications and manipulations are necessary.\",\n \"The first is to define $y = \\\\frac{|z_i| e \\\\psi }{kT}$ resulting in $\\\\frac{d^2 y}{dy^2} = - \\\\frac{4 \\\\pi z_i^2 e^2 n_{0i}}{\\\\epsilon k T} [ \\\\exp {(-y)} - \\\\exp {(y)} ]$ It is also useful to expand the exponential term using Taylor series Note that $\\\\exp {(-y)} - \\\\exp {(y)}$ can also be written as $-2 \\\\sinh (y) $ and $\\\\sinh (y)$ can be written as a Taylor series expansion, as it will be used again later $\\\\exp {(-y)} - \\\\exp {(y)} = -2y - \\\\frac{y^3}{3} - \\\\frac{y^5}{60} + ...$ If the solution is dilute the Debye-Hückel $-\\\\frac{|z_i| e \\\\psi }{kT} \\\\ll 1$ approximation is used and higher order terms on the series can be neglected.\",\n \"Under such conditions the expansion can be simplified to $\\\\exp {(-y)} - \\\\exp {(y)} \\\\approx -2y$ Replacing the expanded exponential into the original equation $\\\\frac{d^2 y}{dy^2} = - \\\\frac{4 \\\\pi z_i^2 e^2 n_{0i}}{\\\\epsilon k T} (-2y) = \\\\frac{8 \\\\pi z_i^2 e^2 n_{0i}}{\\\\epsilon k T} y$ We can now define $\\\\kappa ^2 = \\\\frac{8 \\\\pi z_i^2 e^2 n_{0i}^2}{\\\\epsilon k T}$ The parameter $\\\\kappa $ is an important parameter in surface chemistry and its inverse $\\\\kappa ^{-1}$ is the Debye length or distance over which the electric potential decays to 1/e e here is Euler’s number e = 2.71828....\",\n \"The differential equation to be solved then simplifies to $\\\\frac{d^2 y}{dy^2} = \\\\kappa ^2 y$ This is a second order linear ordinary differential equation with general solution The solution to this differential equation would require a lengthy detour into differential equations theory and will be provided in future editions $y = c_1 \\\\exp (\\\\kappa x) + c_2 \\\\exp (-\\\\kappa x)$ The constants can be determined by applying the boundary conditions to the problem.\",\n \"The first boundary condition states that the electrical potential $\\\\psi = 0 $ at $x = \\\\infty $ .\",\n \"This implies that $y = 0$ at $x = \\\\infty $ $0 = c_1 \\\\exp (\\\\kappa \\\\infty ) + c_2 \\\\exp (-\\\\kappa \\\\infty ) = c_1 \\\\exp (\\\\kappa \\\\infty ) + 0$ $c_1 = 0$ The second is that the $\\\\psi = \\\\psi _0$ and thus $y = y_0$ at x = 0 $y_0 = 0 \\\\exp (\\\\kappa 0) + c_2 \\\\exp (-\\\\kappa 0) = c_2$ Replacing the constant on the general solution $y = y_0 \\\\exp (-\\\\kappa x)$ Therefore $\\\\psi = \\\\psi _0 \\\\exp (-\\\\kappa x)$ This is the electrical potential at the electric double layer following the Gouy-Chapman model with the Debye-Hückel approximation.\",\n \"Notice that this solution is valid for the potential in relation to an infinite plane.\",\n \"For different geometries and where the solute concentration is not low enough for the Debye-Hückel the solution for the Poisson-Boltzmann equation is not trivial (see for example [91] Chapter 5).\",\n \"The potential is $\\\\psi _0$ at the surface and deceases exponentially as the distance $x$ from the surface increases.\",\n \"It is a simplistic model but it is usually to explain double layer compression under high ionic strengths and the theory of surface forces balance that controls dispersion and coagulation of clays.\",\n \"This theory plays a fundamental role in the physicochemistry of salt affected soils, erosion and sedimentation and grain size analysis.\",\n \"Let us now consider an alternative treatments to Equation REF .\",\n \"From hyperbolic geometry identities, the hyperbolic sin function can be written in terms of exponential functions as $\\\\ \\\\sinh y = - \\\\frac{\\\\exp {(-y)} - \\\\exp {(y)}}{2}$ Thus, replacing it into Equation REF results in $\\\\frac{d^2 y}{dx^2} = \\\\frac{8 \\\\pi z_i^2 e^2 n_{0i}}{\\\\epsilon k T} \\\\sinh y$ And from the definition of $\\\\kappa $ $\\\\frac{d^2 y}{dx^2} = \\\\kappa ^2 \\\\sinh y$ This is a nonlinear second-order differential equation and fortunately we only need to find how it reduces to a first-order equation on integration.\",\n \"The reason will become apparent in further steps.\",\n \"To reduce the equation to first order we multiply both sides by $dy/dx$ and integrate [14] $\\\\int \\\\frac{d^2 y}{dx^2}\\\\frac{d y}{dx} = \\\\int \\\\kappa ^2 \\\\sinh y \\\\frac{d y}{dx}$ Substituting $u = dy/dx$ we have $du = d^2y/dx^2$ on the left side and making $v = y$ we have $dv = d^2y/dx^2$ on the left side $\\\\int u du = \\\\int \\\\kappa ^2 \\\\sinh v dv$ which results in $\\\\frac{u^2}{2} = \\\\kappa ^2 \\\\cosh v + C_1$ Where $C_1$ is an integration constant.\",\n \"Replacing u and v we have $(\\\\frac{d y}{dx})^2 = 2 \\\\kappa ^2 \\\\cosh y + C_2$ in which $C_2 = 2C_1$ and it does not alter the solution to the problem.\",\n \"Applying the square root to both sides $\\\\frac{d y}{dx} = \\\\sqrt{ 2 \\\\kappa ^2 \\\\cosh y + C_2}$ Applying the boundary condition that the electrical potential and thus $y$ is zero when $x = \\\\infty $ i.e.\",\n \"bulk solution and that $dy/dx = 0$ at that point, we find that $C_2 = -2\\\\kappa ^2$ and $\\\\frac{d y}{dx} = \\\\sqrt{ 2 \\\\kappa ^2 \\\\cosh y - 2 \\\\kappa ^2}$ Finally we consider another geometric identity of the hyperbolic functions, the half-angle formula $\\\\sinh \\\\frac{y}{2} = sgn(y) \\\\sqrt{ \\\\frac{\\\\cosh y - 1}{2}}$ In which $sgn$ is the sign function with the following properties $sgn (x)={\\\\left\\\\lbrace \\\\begin{array}{ll}-1 & \\\\text{if} ~ x < 0\\\\\\\\0 & \\\\text{if} ~ x = 0\\\\\\\\1 & \\\\text{if} ~ x > 0\\\\end{array}\\\\right.\",\n \"}$ Considering that we are concerned with electrical potential and thus $y$ negative and rearranging and replacing the half-angle formula we have $\\\\frac{d y}{dx} = - 2 \\\\kappa \\\\sinh \\\\frac{y}{2}$ This formula to will be useful to derive an equation for net charge on the diffuse layer, $\\\\sigma $ .\",\n \"The net charge is the integral of the charge density on the $x$ interval of zero to infinity.\",\n \"$\\\\sigma = \\\\int _0^\\\\infty \\\\rho dx$ Replacing the Poisson-Boltzmann equation into the surface charge equation we have $\\\\sigma = \\\\int _0^\\\\infty (- \\\\frac{\\\\epsilon }{4 \\\\pi } \\\\frac{d^2 \\\\psi }{dx^2}) dx = - \\\\frac{\\\\epsilon }{4 \\\\pi } \\\\int _0^\\\\infty \\\\frac{d^2 \\\\psi }{dx^2} dx$ The integration is over the second derivative, resulting $\\\\sigma = -\\\\frac{\\\\epsilon }{4 \\\\pi } \\\\bigg (\\\\frac{d \\\\psi }{dx}\\\\bigg )\\\\bigg |_{x=0}^{x=\\\\infty }$ But the electrical potential is constant at $x=\\\\infty $ such that the term $d\\\\psi /dx$ vanishes when evaluated at infinity, resulting $\\\\sigma = \\\\frac{\\\\epsilon }{4 \\\\pi } \\\\bigg (\\\\frac{d \\\\psi }{dx}\\\\bigg )_{x=0}$ Expressing sigma in terms of $y$ $\\\\sigma = \\\\frac{\\\\epsilon k T }{4 \\\\pi |z_i| e } \\\\bigg (\\\\frac{d y}{dx}\\\\bigg )_{x=0}$ Now we can replace the equation for $dy/dx$ derived previously and the condition that $y = y_0$ at $x = 0$ resulting in $\\\\sigma = -\\\\frac{\\\\epsilon k T \\\\kappa }{2 \\\\pi |z_i| e } \\\\sinh \\\\frac{y_0}{2}$ Which in terms of electrical potential and replacing $\\\\kappa $ is $\\\\sigma = -\\\\frac{\\\\epsilon k T }{2 \\\\pi |z_i| e } \\\\sqrt{\\\\frac{8 \\\\pi z_i^2 e^2 n_{0i}^2}{\\\\epsilon k T}} \\\\sinh \\\\frac{\\\\frac{|z_i| e \\\\psi _0}{kT}}{2} = - \\\\sqrt{\\\\frac{2 \\\\epsilon k T n_{0i}^2}{\\\\pi }} \\\\sinh \\\\frac{|z_i| e \\\\psi _0}{2kT}$ The surface charge, $\\\\sigma _0$ has the same magnitude of the sum over the diffuse layer with opposite sign $\\\\sigma _0 = \\\\sqrt{\\\\frac{2 \\\\epsilon k T n_{0i}^2}{\\\\pi }} \\\\sinh \\\\frac{|z_i| e \\\\psi _0}{2kT}$ This is the Gouy-Chapman relationship for the surface charge density [39].\",\n \"The charge density if a function of the surface potential and ion concentration among other factors.\",\n \"In variable charged oxides the surface potential is not constant and varies with pH.\",\n \"Considering small surface potential, the last equation can be simplifiedNote that the hyperbolic sin expands to $ \\\\sinh y = y + y^3/3!\",\n \"+ y^5/5!\",\n \"+ ...$ thus, for small values of $y$ higher order terms may be neglected to [39] $\\\\sigma _0 \\\\approx \\\\frac{\\\\epsilon \\\\kappa }{4 \\\\pi } \\\\psi _0$ and the surface potential can be determined by the activity of the potential determining ion $a_{pdi}$ in relation to its activity at the point of zero charge (pzc) $\\\\psi _0 \\\\pm \\\\frac{RT}{zF} \\\\ln {\\\\frac{a_{pdi}}{(a_{pdi})_{pzc}}}$ If $H^+$ is the potential determining ion as is the case in oxide surfaces, the relationship between pH and hydrogen activity is $ pH = - log[H^+]$ and the equation can be written as $\\\\psi _0 = \\\\frac{-2.303 RT}{F} (pH - pH_{pzc})$ Combining the electrical potential and surface charge equations, the following equation is valid $\\\\sigma _0 \\\\approx \\\\frac{-2.303 RT \\\\epsilon \\\\kappa }{4\\\\pi F} (pH - pH_{pzc})$ In which $F$ is the Faraday constant.\",\n \"On a side note, the rigorous derivation above was provided by Kenneth L. Babcock one of the fathers of soil chemistry [6].\",\n \"Much of soil chemistry interfaces with soil physics on the subjects of thermodynamics and transport and retention of solutes.\",\n \"One of the most important soil chemists in recent history Garrison Sposito was a student of Kenneth Babcock.\",\n \"Sposito gave fundamental contributions to soil physics no only on thermodynamics and solute chemistry but also regarding water transport and scaling of soil properties and processes.\",\n \"We have illustrated that soils and rock differ in terms of their mineralogical composition and by the fact that rocks are consolidated materials.\",\n \"Rocks differ from sedimentary deposits in terms of genesis and mineralogy and by the fact that soils have organic matter in their composition.\",\n \"The amount of organic matter varies with soil type, but according to soil science criteria, mineral soils have around 5% organic matter on mass basis for average soils.\",\n \"The concept of organic soils is a somewhat loose definition based on a more or less arbitrary limit of organic matter or carbon content [87].\",\n \"The classification of organic soils varies according to engineering classification systems [70] and can vary depending on several specific criteria within a soil classification system used in soil science [87].\",\n \"Because soils occupy only an thin layer of the Earth surface, usually a meter or two to a few meters in depth at the most in most cases, they are under direct influence of the weather and organisms.\",\n \"Soils are usually rich in organisms, both macro and micro, and plants and are continually affected by their actions.\",\n \"Soils are dynamic and porosity and structure are continually being altered by the actions of roots, ants, termites and other burrowing animals of larger size.\",\n \"Rocks and sedimentary deposits can reach kilometers in depth on the Earth's crust and are very little affected by organisms and surface conditions in most cases.\",\n \"The transition zone between soils and subsurface materials is called the saprolite and can reach several meters in depth under extreme weathering condition on the tropics over certain rocks.\",\n \"Soils constitute for the most part the unsaturated zone, where the pores are saturated by both gas and water, although it is possible that the water table can be located within the soil profile.\",\n \"Below the water table is the saturated zone where the pores are saturated with water.\",\n \"Most aquifers are located in sedimentary deposits and to some extent porous or fractured rocks.\",\n \"Transport and retention of water, contaminants, oil and gas in rocks and sedimentary deposits is usually the object of study of hydrogeology and petroleum engineering while soil physics is often concerned with soils and other unsaturated media.\",\n \"However, the mathematical analytical and numerical framework between these disciplines is similar since the equations that govern transport processes in these materials have the same physical basis if they are not the same.\",\n \"Much of the governing equations of processes of transport originated from the realm of mechanics and later chemical engineering.\",\n \"In the next chapter we will try to provide a mathematical framework for the equations that govern transport processes in soils and other natural porous media.\",\n \"ll $ \\\\psi $ Electric potential $ \\\\psi _0 $ Electric potential at the surface of the charged plane $ x $ Distance from the charged plane $ z $ Ion valence $ e $ Elementary charge $ k $ Boltzmann constant $ T $ Absolute temperature $ \\\\rho $ Charge density $ \\\\epsilon $ Absolute permittivity of the medium $ \\\\kappa ^{-1} $ Debye length $ \\\\text{e} $ Euler's number $ ln $ Logarithm base e $ log $ Logarithm base 10 $ c_1, c_2, C_1, C_2 $ General constants for differential equations solutions $ y $ General variable for mathematical manipulations $ y_0 $ Value of $y$ at $y = 0$ $ u, v $ Variables for integration by substitution $ sgn $ Sign function $ \\\\sigma $ Net charge at the diffuse layer $ \\\\sigma _0 $ Surface charge $ R $ Molar gas constant $ F $ Faraday constant $ a_{pdi} $ Activity of a potential determining ion (pdi) $ pH_{pzc} $ pH at the point of zero charge (pzc) Up until now we have been discussing properties of the solid matrix of porous media.\",\n \"Some of these properties are directly affected by water content and can be altered by engineering, hydrological, agricultural as well as a wide range of natural processes, both on short and long term.\",\n \"We have not discussed so far any type of flux of liquid of gaseous phases through porous materials.\",\n \"Detailed discussions of mass flow through porous media are reserved to the chapters dealing with saturated and unsaturated hydrodynamics of porous media, and other transport phenomena.\",\n \"Technically speaking, we will deal with another property of the solid matrix on the next chapter, the grain size distribution, so there should not be an immediate need for equations governing fluid flow.\",\n \"However, the majority of methods for determination of grain size distribution of porous media rely on settling in liquid and, occasionally, gaseous materials.\",\n \"To understand the dynamics of settling we need to understand the fundamental equations of fluid dynamics which control how settling occurs in a viscous media.\",\n \"The knowledge discussed in this chapter will also be the basis for modeling flow problems in open conducts and to a large extent flow in porous media.\",\n \"This chapter will introduce you to the beauty and brutality that is the mathematics involved in fluid mechanics and in theoretical soil physics.\",\n \"If you are already familiar with tensor analysis in areas such as continuum mechanics or general relativity you should be able to follow this chapter without major issues.\",\n \"Consulting the bibliography might help with issues specific to fluid mechanics that might arise.\",\n \"If you are seeing the application of tensor analysis for the first time it might be worth consulting an introductory text on tensors.\",\n \"When dealing with fluid flow, two formulations are usually discussed in the literature, the Lagrangian and the Eulerian specifications, after the great mathematicians Joseph-Louis Lagrange and Leonard Euler Although Euler was responsible for both formulations, see for example Landau and Lifshitz (1987, p. 5) [57]..\",\n \"The Lagrangian formulation follows from the idea of tracking individual particles or the center of mass of a group of particles from classical mechanics.\",\n \"In the context of fluid flow the movement of a single water molecule could be tracked in relation to an initial time $t_0$ and position $\\\\mathbf {s}$ , where, as seen in Chapter REF .\",\n \"$\\\\mathbf {s} = s_x \\\\mathbf {i} + s_y \\\\mathbf {j} + s_z \\\\mathbf {k}$ Thus, in the Lagrangian formulation, the velocity of a tracked particle or center of mass of a group of particles is $\\\\mathbf {v}(\\\\mathbf {s}, t)$ or, considering the vector components $\\\\mathbf {v}(s_x, s_y, s_z, t)$ .\",\n \"It is a consensus that the Lagrangian formulation is not mathematically convenient for the majority of fluid flow problems and thus, it will not be discussed further.\",\n \"The formulation normally adopted is more akin to that used in electromagnetism in the sense that the flow quantities are defined as a vector field with independent variables position in space and time, $\\\\mathbf {x}$ , and t. The vector velocity field in the Eulerian formulation is traditionally represented as $\\\\mathbf {u}(\\\\mathbf {x}, t) $ , where $\\\\mathbf {u} = u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}$ often also represented as $\\\\mathbf {u} = u \\\\mathbf {x} + v \\\\mathbf {y} + w \\\\mathbf {z}$ such that $\\\\mathbf {u}(\\\\mathbf {x}, t) $ can be written as $\\\\mathbf {u}(u_x, u_y, u_z,t) $ in the first case and $\\\\mathbf {u}(u, v, w,t) $ in the second.\",\n \"The difference between $\\\\mathbf {v}$ and $\\\\mathbf {u}$ is that in the latter we are now no longer interested in an individual particle or the center of mass of a group of particles, what we have is a velocity field in space and time.\",\n \"Now at any position in space and at a given time there is a velocity value assigned.\",\n \"Imagine a 2D river or stream seen from above, in the Lagrangian river we would track the position of individual particles as a function of time, while on the Eulerian river we are not worried about individual particles but we have a velocity field that covers the entire space of the river, such that if we chose an arbitrary point $\\\\mathbf {x_1}$ and $t_1$ on the surface of the river, the velocity at that point is given by $\\\\mathbf {u}(\\\\mathbf {x_1}, t_1)$ .\",\n \"One advantage of the Eulerian formulation is now obvious as there is no need to track an exorbitant number of particles to model the flow of a river or stream.\",\n \"Another major advantage of the Eulerian formulation is that it can be used to model many other important fluid properties such as density and pressure over space and time, however it is important to point out that these are scalar fields given by $\\\\rho (\\\\mathbf {x}, t)$ and $P(\\\\mathbf {x}, t)$ ,C.L.\",\n \"for density and pressure, respectively, noting that $\\\\mathbf {u}$ at any point in space and time is itself a vector, while $\\\\rho $ and $P$ are scalars.\",\n \"From the Eulerian formulation we now find the acceleration of a fluid element.\",\n \"The acceleration of the fluid element is not given directly by $ \\\\partial \\\\mathbf {u}/ \\\\partial t $ as each fluid element travels through the vector field as times varies, being subjected to velocity variation in space along the vector field in addition to time.\",\n \"The correct formulation of the acceleration of a material element is given by $\\\\frac{\\\\partial \\\\mathbf {u}}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u}$ Using the notation introduced in Chapter REF and because $\\\\mathbf {u}$ is a vector the equation above can be written as $\\\\frac{\\\\partial }{\\\\partial t} (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) + (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) \\\\cdot [\\\\frac{\\\\partial }{\\\\partial x}(u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) \\\\mathbf {i} + \\\\\\\\ + \\\\frac{\\\\partial }{\\\\partial y}(u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) \\\\mathbf {j} + \\\\frac{\\\\partial }{\\\\partial z} (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k})\\\\mathbf {k}]$ which, considering the dot product $\\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u}$ , can be written as $\\\\frac{\\\\partial }{\\\\partial t} (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) + [u_x \\\\frac{\\\\partial }{\\\\partial x}(u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) + u_y \\\\frac{\\\\partial }{\\\\partial y}(u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) + \\\\\\\\ + u_z \\\\frac{\\\\partial }{\\\\partial z} (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k})]$ With some effortAs opposed to \\\"it can easily be shown\\\" it can be shown that the vector equation above can be written as a set of equations.\",\n \"You will see that most of the fluid dynamics equations in two and three-dimensional spaces are actually sets of equations and it is important that you familiarize yourself with the steps in converting from vector form to components and vice-versa when doableIn several fluid equations, multiple tensor components are involved, and it is not practical to do these operations without using vector and tensor identities.\",\n \"Thus in Cartesian coordinates, the components in $x$ , $y$ and $z$ are $\\\\frac{\\\\partial u_x}{\\\\partial t} + [u_x \\\\frac{\\\\partial u_x}{\\\\partial x} + u_y \\\\frac{\\\\partial u_x}{\\\\partial y} + u_z \\\\frac{\\\\partial u_x}{\\\\partial z} ] \\\\\\\\\\\\frac{\\\\partial u_y}{\\\\partial t} + [u_x \\\\frac{\\\\partial u_y}{\\\\partial x} + u_y \\\\frac{\\\\partial u_y}{\\\\partial y} + u_z \\\\frac{\\\\partial u_y}{\\\\partial z} ]\\\\\\\\\\\\frac{\\\\partial u_z}{\\\\partial t} + [u_x \\\\frac{\\\\partial u_z}{\\\\partial x} + u_y \\\\frac{\\\\partial u_z}{\\\\partial y} + u_z \\\\frac{\\\\partial u_z}{\\\\partial z} ]$ The quantity $\\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u} $ is the dot product of a vector (first order tensor) and a second order tensor and is a vector, such that $ \\\\frac{\\\\partial \\\\mathbf {u}}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u} $ is the sum of two vectors and is itself a vector.\",\n \"The derivative form found in Equation 4.4 has such an importance in fluid dynamics that it constitutes a mathematical operator $\\\\boxed{\\\\frac{D}{D t} = \\\\frac{\\\\partial }{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla }$ such that for any scalar or vector quantity $f$ we have $\\\\frac{D f}{D t} = \\\\frac{\\\\partial f}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla f$ In which $ \\\\frac{\\\\partial f}{\\\\partial t} $ is the temporal rate of change at a specific location and $\\\\mathbf {u} \\\\cdot \\\\nabla f $ is the convective rate of change due to transport of the fluid element from one point to another in space.\",\n \"If $f$ is constant throughout the space under consideration $\\\\frac{D f}{D t} = \\\\frac{\\\\partial f}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla f = 0$ and there is no acceleration of the material elements of fluid.\",\n \"The conservation equation, or mass conservation equation, is often referred to the continuity equation on the literature.\",\n \"In agreement with G. K. Batchelor, we do not think that continuity is an appropriate term and will refrain from using it in this text.\",\n \"The idea of conservation of mass, be it a gas, a fluid or a particulate solid, is a very intuitive one.\",\n \"Imagine a garden hose, if you consider a given length of the hose, say one meter, once you turn on the faucet and the flux out of the faucet becomes constant, and if there are no leaks along the hose, the amount of water that enters on one end is equal to the amount that exits on the other end, per unit time.\",\n \"This condition in which the amounts in and out of the system are constant and equal is called steady state and is very important in groundwater hydrology.\",\n \"This simple example is a basic application of the conservation equation, if there are no losses or gains within the system, matter is conserved.\",\n \"Suppose now you have a steady state condition and there are leaks in your garden hose within the length considered.\",\n \"The mass conservation equation still applies, the amount in is equal to the sum of the amounts on the other end of the hose plus any leaks, per unit time.\",\n \"In real scenarios we often have to deal with sources or sinks within the volume considered, this is especially true in the case of reactive and/or volatile transport in which there can be loss of gain within the consider volume due to adsorption, condensation, degradation and many other physico-chemical processes.\",\n \"A simple example of a sink (loss) would be evaporation of water in a reservoir.\",\n \"Let us now treat this problem from a mathematical point of view.\",\n \"There are at least two approaches to treat this problem, one is deceitfully simple and direct and is based on vector calculus.\",\n \"The other is based on algebraAs in algebraic manipulations in a broader sense and appears to be very complex although being very simple.\",\n \"The first one can be found in advanced fluid dynamics and mechanics, physics and applied mathematics books, while the other is preferred in introductory engineering and in soil physics books.\",\n \"The conceptual model is the same in both cases, an element of volume in space, in which flow is considered through an element of surface area.\",\n \"It is easy for us to consider water, but you will see later that the conservation equation applies to any substance in transport processes in porous media.\",\n \"In the vector treatment we are not initially concerned with the geometry of the volume, other than it should be a closed space.\",\n \"Initially, however, it should be much easier to imagine the volume as a cube or as a sphere or other similar closed surfacesIn mathematics, the topology of the space is of interest, and can have implications in terms of the mathematical description.\",\n \"This is, however, beyond the scope of this text.\",\n \"Starting with a volume $V$ enclosed by a surface $A$ completely filled with a liquid of density $\\\\rho $ , the mass of water within the volume is given by $\\\\int \\\\rho dV$ This should come of no surprise, as by integrating the volume elements $dV$ over the entire volume we are left with $V$ , and obviously the water density in [M L$^{-3}$ ] multiplied by the total volume in [L$^{3}$ ] is the mass M of water within the enclosed volume.\",\n \"The amount of water flowing through an element of surface area $dA$ $\\\\int \\\\rho \\\\mathbf {u} \\\\cdot \\\\mathbf {n} dA$ Where $\\\\mathbf {n}$ is the unit normal in dAC.L..\",\n \"Note that the velocity, $\\\\mathbf {u}$ , in [L T$^{-1}$ ] times water density, $\\\\rho $ , in [M L$^{-3}$ ], over the entire surface, $A$ , in [L$^{2}$ ] is the rate of mass transfer across the surface in [M T$^{-1}$ ].\",\n \"Now the rate of variation of the mass within the volume with time can be written in differential form as $\\\\frac{\\\\partial }{\\\\partial t} \\\\int \\\\rho dV$ in units of [M T$^{-1}$ ].\",\n \"If this doesn't make much sense to you don't worry, you can find more information about this construction in vector calculus or calculus 3 books.\",\n \"Note that as we defined Equation 4.8 as the total mass within the volume, Equation 4.10 is nothing more than the variation of mass with time within the volume.\",\n \"Now, from our discussion of conservation laws we can agree that the variation of mass within a volume is the net rate of flow across its surface, and thus the conservation of mass implies that $\\\\frac{\\\\partial }{\\\\partial t} \\\\int \\\\rho dV = - \\\\int \\\\rho \\\\mathbf {u} \\\\cdot \\\\mathbf {n} dA$ So that $\\\\frac{\\\\partial }{\\\\partial t} \\\\int \\\\rho dV + \\\\int \\\\rho \\\\mathbf {u} \\\\cdot \\\\mathbf {n} dA = 0$ Now all the vector calculus tools discussed in Chapter REF will come in handy.\",\n \"We can use the divergence theorem to transform a surface integral into a volume integral.\",\n \"Applying it to the second term in Equation 4.12 results in $\\\\frac{\\\\partial }{\\\\partial t} \\\\int \\\\rho dV + \\\\int \\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u}) dV = 0$ Which can be written as $\\\\int [ \\\\frac{\\\\partial \\\\rho }{\\\\partial t} + \\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u})] dV = 0$ Because the integration must hold for any volume, the integrand must vanish and we are left with one of the fundamental equations of fluid dynamics, the mass conservation (or continuity) equation $\\\\boxed{ \\\\frac{\\\\partial \\\\rho }{\\\\partial t} + \\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u}) = 0 }$ Now for the derivation that does not need vector calculus.\",\n \"Imagine a cube with sides $\\\\Delta x$ , $\\\\Delta y$ and $\\\\Delta z$ .\",\n \"Consider a velocity component $u_x$ in the direction $x$ .\",\n \"The area which is perpendicular to the velocity component $u_x$ is $\\\\Delta y\\\\Delta z$ .\",\n \"In other words, the velocity component $u_x$ enters into the cube in the side $\\\\Delta y\\\\Delta z$ .\",\n \"Now, the rate of mass accumulation in the direction $x$ is the amount of mass that enters on one end on the direction $x$ and leaves on the other, in that same direction.\",\n \"The amount of mass that enters in one end is equal to the velocity component $u_x$ times the density of the fluid $\\\\rho $ times the area $ \\\\Delta y \\\\Delta z$ , note that $\\\\frac{m_{in-x}}{t} = \\\\rho u_x \\\\Delta y \\\\Delta z = \\\\frac{\\\\mathrm {M}}{L^3} \\\\frac{\\\\mathrm {L}}{T} L^2 = \\\\frac{\\\\mathrm {M}}{T}$ The amount that exits on the other end will depend on the variation of the velocity of the fluid through the direction $x$ within the volume, if the fluid slows down, the velocity decreases and vice-versa.\",\n \"Therefore, in the outflow we have to consider the rate of variation of velocity within the volume such that, the outflow mass in the $x$ direction is $\\\\frac{m_{out-x}}{t} = ( \\\\rho u_x + \\\\frac{\\\\partial \\\\rho u_x}{\\\\partial x}\\\\Delta x) \\\\Delta y \\\\Delta z$ Therefore, the amount of mass that accumulates or is lost per unit time due to flux in the $x$ direction is given by $\\\\frac{\\\\partial }{\\\\partial t} M_{x} = \\\\rho u_x \\\\Delta y \\\\Delta z - (\\\\rho u_x + \\\\frac{\\\\partial \\\\rho u_x}{\\\\partial x}\\\\Delta x) \\\\Delta y \\\\Delta z$ Which results in $\\\\frac{\\\\partial }{\\\\partial t} M_{x} = - \\\\frac{\\\\partial \\\\rho u_x}{\\\\partial x}\\\\Delta x \\\\Delta y \\\\Delta z$ By symmetry it is not difficult to show that the mass lost or accumulated due to flux on the directions $y$ and $z$ is $\\\\frac{\\\\partial }{\\\\partial t} M_{y} = - \\\\rho \\\\frac{\\\\partial \\\\rho u_y}{\\\\partial y}\\\\Delta x \\\\Delta y \\\\Delta z$ $\\\\frac{\\\\partial }{\\\\partial t} M_{z} = - \\\\rho \\\\frac{\\\\partial \\\\rho u_z}{\\\\partial z}\\\\Delta x \\\\Delta y \\\\Delta z$ and that the total mass accumulated or loss within the cube per unit time due to flow in the three directions is $\\\\frac{\\\\partial }{\\\\partial t} M_{xyz} = - (\\\\frac{\\\\partial \\\\rho u_x}{\\\\partial x}+ \\\\frac{\\\\partial \\\\rho u_y}{\\\\partial y}+ \\\\frac{\\\\partial \\\\rho u_z}{\\\\partial z})\\\\Delta x \\\\Delta y \\\\Delta z$ Since the density of the cube is $\\\\rho $ and its volume is $\\\\Delta x \\\\Delta y \\\\Delta z$ and rearranging $ \\\\rho = M/V$ to $ M = \\\\rho V$ we have $\\\\frac{\\\\partial }{\\\\partial t} \\\\rho \\\\Delta x \\\\Delta y \\\\Delta z = - (\\\\frac{\\\\partial \\\\rho u_x}{\\\\partial x}+\\\\frac{\\\\partial \\\\rho u_y}{\\\\partial y}+\\\\frac{\\\\partial \\\\rho u_z}{\\\\partial z})\\\\Delta x \\\\Delta y \\\\Delta z$ or $\\\\frac{\\\\partial \\\\rho }{\\\\partial t} = - (\\\\frac{\\\\partial \\\\rho u_x}{\\\\partial x}+\\\\frac{\\\\partial \\\\rho u_y}{\\\\partial y}+\\\\frac{\\\\partial \\\\rho u_z}{\\\\partial z})$ Using the vector operators, this equation can be written as $\\\\frac{\\\\partial \\\\rho }{\\\\partial t} + \\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u}) = 0$ which is the same as Equation 4.15.\",\n \"The derivation using geometric considerations is favored in soil physics and hydrology books while the vector calculus derivation is found in fluid mechanics and dynamics books for physicists and mathematicians.\",\n \"If the density of the fluid is constant, $\\\\rho $ does not vary with time and $\\\\partial \\\\rho /\\\\partial t = 0 $ and we arrive at an important case of the conservation equation $\\\\boxed{\\\\nabla \\\\cdot \\\\mathbf {u} = 0}$ Stating in words, the divergence of the velocity field is zero for an incompressible fluid.\",\n \"For most practical purposes, water can be assumed as incompressible and Equation 4.23 will be valid for many flow problems.\",\n \"Much like for mass, the conservation laws are valid for other extensive properties in fluid mechanics including energy and momentum and this will allow us to derive the equations at the core of fluid mechanics and transport phenomena.\",\n \"Recall from introductory physics that Newton's second law can be written either as $\\\\mathbf {F} = m\\\\mathbf {a}$ or as $\\\\mathbf {F} = \\\\frac{d \\\\mathbf {p}}{dt}$ where the vector quantity $\\\\mathbf {p}$ is the linear momentum and is equal to $ m \\\\mathbf {v} $ , for an object of mass $m$ moving at velocity $\\\\mathbf {v}$ .\",\n \"Considering an element of fluid of volume $V$ , the momentum corresponding to the entire volume is given by $\\\\int \\\\mathbf {u} \\\\rho dV$ Notice that velocity is in [L T$^{-1}$ ], density is in [M L$^{-3}$ ] and volume is in [L$^{3}$ ] such that the units are consistent with those of momentum [M L T$^{-1}$ ].\",\n \"The rate of change of momentum for the elementary volume can be represented by the derivative of the relationship above with respect to time, the material derivative in this case.\",\n \"If the density can be assumed as constant $\\\\int \\\\frac{D\\\\mathbf {u}}{Dt} \\\\rho dV$ Now, as specified by Newton's second law of motion, the rate of change of momentum in an object is equal to the sum of forces acting on the object.\",\n \"There are two important points here to be considered because we are dealing with fluid mechanics and not with a point particle.\",\n \"The first is that the total momentum is the sum of the individual momentum of each infinitesimal element of fluid.\",\n \"If the system was modeled as a system of individual particles the integral on the previous equations could be replaced by a summation of the momenta of the $n$ particles that compose the system.\",\n \"The second point is that the forces acting in a volume of fluid can be divided into body (or volume) forces such as those generated by gravity and electromagnetic fields and surface (of contact) forces, which act directly over the surface enclosing the volume element.\",\n \"Surface forces are ubiquitous in continuum mechanics and are an important concept in soil physics both in treating the solid and liquid phases and their interactions.\",\n \"To understand surface forces it is convenient to explore the concept of the stress tensor.\",\n \"Imagine an element of volume of fluid immersed in a fluid or other continuous matter, the matter around the element of fluid exerts a contact force across the entire surface of the fluid element.\",\n \"For the sake of simplicity, consider a volume element of cubic shape in Cartesian coordinates system.\",\n \"Each face can be subjected to forces that are perpendicular (i.e.\",\n \"normal) to the surface, in one direction, or parallel (tangential) to the surface in two directions.\",\n \"For example, the face that lies on the $x-y$ plane can be subjected to a perpendicular force on the $z$ direction and to two parallel forces on the $x$ and $y$ directions.\",\n \"Thus, the cube can be subjected to three forces in each face for each of its six faces.\",\n \"If each surface of the cube is characterized by a surface area, we can define each component of the stress acting on the surface as force over area.\",\n \"Thus, on the $z$ direction lies the $x-y$ plane with a normal stress component $\\\\sigma _{zz}$ and two perpendicular stress components, one in the $x$ direction, represented by $\\\\sigma _{zx}$ and another in the $y$ direction represented by $\\\\sigma _{zy}$ .\",\n \"This leaves us with three unique stress components in each direction, $x$ , $y$ and $z$ , with a total of nine stress components completely defining the stresses acting on the element.\",\n \"The nine elements define the stress tensor $ \\\\sigma _{ij} =\\\\begin{bmatrix}\\\\sigma _{xx} & \\\\sigma _{xy} & \\\\sigma _{xz} \\\\\\\\\\\\sigma _{yx} & \\\\sigma _{yy} & \\\\sigma _{yz} \\\\\\\\\\\\sigma _{zx} & \\\\sigma _{zy} & \\\\sigma _{zz} \\\\\\\\\\\\end{bmatrix}$ The components where $i = j$ (i.e.\",\n \"$\\\\sigma _{xx}$ , $\\\\sigma _{yy}$ , $\\\\sigma _{zz}$ ) are the normal stress components, while the components where $i \\\\ne j$ are the tangential stress components.\",\n \"The tangential stress components correspond to shearing stresses seen in solids which can cause isovolumetric deformations.\",\n \"It is convenient now to introduce the Kronecker delta operator $\\\\delta _{ij} ={\\\\left\\\\lbrace \\\\begin{array}{ll}0 & \\\\text{if} ~ i \\\\ne j, \\\\\\\\1 & \\\\text{if} ~ i = j \\\\\\\\\\\\end{array}\\\\right.\",\n \"}$ Such that the normal stresses on the fluid element can be represented as $\\\\sigma _{ij} = \\\\delta _{ij} \\\\sigma _{ij}$ It is important to point out that if the fluid element is at rest, the static fluid pressure is defined as $\\\\sigma _{ij} = -p \\\\delta _{ij}$ Applying this equation to the stress tensor results in $ \\\\sigma _{ij} =\\\\begin{bmatrix}-p & 0 & 0 \\\\\\\\0 & -p & 0 \\\\\\\\0 & 0 & -p \\\\\\\\\\\\end{bmatrix}$ Which is nothing more than the well known principle from hydrostatics which states that for a small volume immersed in a fluid (such as in water for example), the pressure is constant across its surface.\",\n \"Keep in mind that for a moving fluid element the story is quite different.\",\n \"Now the total surface force exerted on the volume element is the sum of the stresses acting in each surface element with area $dS$ and normal $n_j$ and can be written as $\\\\int \\\\sigma _{ij} n_{j} dS$ which can be transformed to a volume integral using the divergence theorem $\\\\int \\\\frac{\\\\partial \\\\sigma _{ij}}{\\\\partial x_j} dV$ From Newton's law expressed as conservation of momentum, for the element of volume of the fluid, we can write the total momentum as the sum of forces acting on the fluid, including the summation of all body forces and the surface forces $\\\\int \\\\frac{D u_i}{Dt} \\\\rho dV = \\\\int F_i \\\\rho dV + \\\\int \\\\frac{\\\\partial \\\\sigma _{ij}}{\\\\partial x_j} dV$ As usual, the integral vanishes identically and we are left with $\\\\frac{D u_i}{Dt} \\\\rho = F_i \\\\rho + \\\\frac{\\\\partial \\\\sigma _{ij}}{\\\\partial x_j}$ In which $F_i$ is the sum of all body forces acting on the fluid element, i.e.\",\n \"gravitational, electromagnetic, etc.\",\n \"Expanding this equation in three dimensional Cartesian coordinates using Einstein's summation convention we have $\\\\frac{D u_x}{Dt} \\\\rho = F_x \\\\rho + \\\\frac{\\\\partial \\\\sigma _{xx}}{\\\\partial x} + \\\\frac{\\\\partial \\\\sigma _{xy}}{\\\\partial y} + \\\\frac{\\\\partial \\\\sigma _{xz}}{\\\\partial z}\\\\\\\\\\\\frac{D u_y}{Dt} \\\\rho = F_y \\\\rho + \\\\frac{\\\\partial \\\\sigma _{yx}}{\\\\partial x} + \\\\frac{\\\\partial \\\\sigma _{yy}}{\\\\partial y} + \\\\frac{\\\\partial \\\\sigma _{yz}}{\\\\partial z}\\\\\\\\\\\\frac{D u_z}{Dt} \\\\rho = F_z \\\\rho + \\\\frac{\\\\partial \\\\sigma _{zx}}{\\\\partial x} + \\\\frac{\\\\partial \\\\sigma _{zy}}{\\\\partial y} + \\\\frac{\\\\partial \\\\sigma _{zz}}{\\\\partial z}$ These equations show that the acceleration in each direction is caused by the forces acting on that direction and the components of the stress tensor acting on the face perpendicular to the direction of the acceleration.\",\n \"With some inspection it can also become clear that the components of the stress tensor can cause deformation during motion depending of the nature of the shearing components (i.e.\",\n \"$i \\\\ne j$ ).\",\n \"With that in mind, the stress tensor can be divided into two components, one equivalent to the static pressure experienced by a stationary volume of fluid and a non-isotropic component which originates due to motion of the fluid element $\\\\sigma _{ij} = -p \\\\delta _{ij} + d_{ij}$ Which can be represented as $\\\\begin{bmatrix}\\\\sigma _{xx} & \\\\sigma _{xy} & \\\\sigma _{xz} \\\\\\\\\\\\sigma _{yx} & \\\\sigma _{yy} & \\\\sigma _{yz} \\\\\\\\\\\\sigma _{zx} & \\\\sigma _{zy} & \\\\sigma _{zz} \\\\\\\\\\\\end{bmatrix}=\\\\begin{bmatrix}-p & 0 & 0 \\\\\\\\0 & -p & 0 \\\\\\\\0 & 0 & -p \\\\\\\\\\\\end{bmatrix}+\\\\begin{bmatrix}d_{xx} & d_{xy} & d_{xz} \\\\\\\\d_{yx} & d_{yy} & d_{yz} \\\\\\\\d_{zx} & d_{zy} & d_{zz} \\\\\\\\\\\\end{bmatrix}$ Now $d_{ij}$ is called the deviatoric stress tensor and is the non-isotropic, tangential, component of stresses.\",\n \"Considering the components of the stress tensor and expanding the material derivative operator, the unabridged equation of motion for an element of fluid in three-dimensional Cartesian coordinates is $\\\\rho (\\\\frac{\\\\partial u_x}{\\\\partial t}+u_x \\\\frac{\\\\partial u_x}{\\\\partial x}+u_y\\\\frac{\\\\partial u_x}{\\\\partial y}+u_z\\\\frac{\\\\partial u_x}{\\\\partial z}) = F_x \\\\rho -\\\\frac{\\\\partial p}{\\\\partial x} + \\\\frac{\\\\partial d_{xx}}{\\\\partial x} + \\\\frac{\\\\partial d_{xy}}{\\\\partial y} + \\\\frac{\\\\partial d_{xz}}{\\\\partial z}\\\\\\\\\\\\rho (\\\\frac{\\\\partial u_y}{\\\\partial t}+u_x \\\\frac{\\\\partial u_y}{\\\\partial x}+u_y\\\\frac{\\\\partial u_y}{\\\\partial y}+u_z\\\\frac{\\\\partial u_y}{\\\\partial z}) = F_y \\\\rho -\\\\frac{\\\\partial p}{\\\\partial y}+ \\\\frac{\\\\partial d_{yx}}{\\\\partial x} + \\\\frac{\\\\partial d_{yy}}{\\\\partial y} + \\\\frac{\\\\partial d_{yz}}{\\\\partial z}\\\\\\\\\\\\rho (\\\\frac{\\\\partial u_z}{\\\\partial t}+u_x \\\\frac{\\\\partial u_z}{\\\\partial x}+u_y\\\\frac{\\\\partial u_z}{\\\\partial y}+u_z\\\\frac{\\\\partial u_z}{\\\\partial z}) = F_z \\\\rho -\\\\frac{\\\\partial p}{\\\\partial z} + \\\\frac{\\\\partial d_{zx}}{\\\\partial x} + \\\\frac{\\\\partial d_{zy}}{\\\\partial y} + \\\\frac{\\\\partial d_{zz}}{\\\\partial z}$ The vector form, being simpler and more elegant, is referred to as the Cauchy momentum equation after Augustin-Louis Cauchy $\\\\boxed{\\\\rho \\\\frac{D \\\\mathbf {u}}{D t} = \\\\mathbf {F} \\\\rho - \\\\nabla p + \\\\nabla \\\\cdot \\\\mathbf {d}}$ The full momentum equation is a highly nonlinear partial differential equation and knowledge of the stress tensor and body forces to which the fluid volume are subjected would be necessary to find a solution for this equation, if at all possible.\",\n \"Two of the most fundamental equations of fluid mechanics work with simplifications and assumptions over the momentum equation.\",\n \"The first developed from Newton's laws was derived by Euler and a more fundamental one which is thought to account for most fluid flow phenomena was developed by Claude-Louis Navier and Georges Gabriel Stokes less than a hundred years later.\",\n \"The Euler equations were derived by Leonhard Euler himself in 1755 for the motion of an inviscid fluid, or a fluid without viscosity.\",\n \"The complete set of equations of motion, including the viscosity term, were derived in the XIX century, culminating from the works of Claude-Louis Navier and Sir George Gabriel Stokes, after which they are named.\",\n \"The Euler equations can be understood as a special case of the Navier-Stokes equations and both sets of equations are applications of the Newton laws of motion to fluid flow, and both can be understood as special cases of the Cauchy equation of motion within the Newtonian framework.\",\n \"Up until now we have not discussed viscosity on any broader sense, most of us know the colloquial definition of viscosity used in daily language.\",\n \"A so called viscous fluid has more resistance to flow as we might have experienced with grease and honey as compared to water, for example.\",\n \"A higher viscosity reflects, in essence, higher internal friction on the fluid, or a larger magnitude of the forces between its elemental particles.\",\n \"Mathematically, viscosity can be represented as the coefficient of proportionality between a tangential stress and the rate of variation of velocity of a fluid in a direction perpendicular to that of the applied stress.\",\n \"The most simple explanation is that found on most of the literature and we can make use of the stress tensor already introduced.\",\n \"Imagine you have a two-dimensional element of fluid within a system of horizontal and vertical axes $x$ and $y$ .\",\n \"Now imagine that in the upper surface of the fluid element a stress $\\\\sigma _{xy}$ parallel to the surface and perpendicular to the $y$ axis is applied.\",\n \"If the fluid has any internal friction, any imaginary fluid layer parallel to the surface layer where the stress is being applied will want to move with it.\",\n \"For a rigid material with infinite viscosity over a frictionless surface, every imaginary layer will move at the same velocity and the material will move as a block.\",\n \"Now if the material is not rigid the gradient of velocity in the $y$ direction will be proportional to the stress applied and as mentioned above, the viscosity is the coefficient of proportionality $\\\\sigma _{xy} = \\\\mu \\\\frac{d }{dy} u_x$ The more viscous the material, the more a tangential stress applied will propagate to the lower layersC.L.. One of the most important assumptions is that this relationship is linear, or alternatively, that the viscosity is constant in this equation.\",\n \"This is the case for the so called Newtonian fluids, if the relationship is nonlinear the fluid is called non-Newtonian and a different framework needs to be adopted for finding a general equation of motion.\",\n \"Now suppose an ideal fluid has no viscosity, in this case there is no sense in speaking of tangential components because there is no internal friction, the deviatoric stress term in the Cauchy equation is zero and we are left with $\\\\boxed{\\\\rho \\\\frac{D \\\\mathbf {u}}{D t} = \\\\mathbf {F} \\\\rho - \\\\nabla p }$ This is Euler's equation of motion and is valid for ideal fluids.\",\n \"We can further simplify Euler's equation disregarding body forces, when possible, so that acceleration is caused only by surface forces $\\\\rho \\\\frac{D \\\\mathbf {u}}{D t} = - \\\\nabla p$ Expanding the material derivative term $\\\\rho (\\\\frac{\\\\partial \\\\mathbf {u}}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u}) = - \\\\nabla p$ Euler's equation can be also used to derive the equation for hydrostatic pressure in a fluid in equilibrium.\",\n \"In this cases there is no motion and the acceleration term represented by the material derivative is zero.\",\n \"For the hydrostatic pressure to exist it is necessary the action of a body force, represented here by gravitation force $ \\\\mathbf {F} = \\\\mathbf {g} $ $\\\\nabla p = \\\\rho \\\\mathbf {g}$ Expanding $\\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {i} + \\\\frac{\\\\partial p}{\\\\partial y} \\\\mathbf {j} + \\\\frac{\\\\partial p}{\\\\partial z} \\\\mathbf {k} = \\\\rho (g_x \\\\mathbf {i} + g_y \\\\mathbf {j} + g_z \\\\mathbf {k})$ And considering that Earth's gravitational force acts in $z$ only, and considering $z$ positive upwards $\\\\frac{\\\\partial p}{\\\\partial z} = -\\\\rho g$ Which can be readily integrated as for $\\\\rho $ a $g$ constant as $\\\\int \\\\partial p = - \\\\rho g \\\\int \\\\partial z$ Resulting in $p = - \\\\rho g z + \\\\text{constant}$ Applying the boundary condition that $p = p_0$ when $z = 0$ (fluid surface) results $p = p_0 - \\\\rho g z$ Which will increase considering that the absolute value of $z$ increases with depth and that $z$ is negative in the coordinate system adopted.\",\n \"The problem with Euler's equation, except for special applications, is that most liquids will have some degree of viscosity and the non-isotropic components of the stress tensor need to be considered on the analysis.\",\n \"The Navier-Stokes equations are the most general form of the governing equation of fluid motion and can be interpreted as a special case of Cauchy momentum equation.\",\n \"The transition from the Cauchy momentum equation to the Navier-Stokes equations requires that the non-isotropic stress tensor is expressed in a form amenable to the solution of the differential equation using analytical or numerical methods.\",\n \"Following the language of Batchelor [8], it is necessary to find a relationship between the deviatoric stress tensor and the local properties of the fluid.\",\n \"As with a lot of transport relations in physics, one possible simplification is to assume a linear relationship between the deviatoric stress and the velocity gradient.\",\n \"Keep in mind that this assumption carries profound implications in all areas of physics that deal with transport phenomena, either of matter, energy, momentum, electrical current, heat, ... and should not be taken lightly, I urge you to consult the references for details.\",\n \"The hypothesis of a linear relation between the deviatoric stress tensor and the velocity gradients can be expressed analytically as $d_{ij} = A_{ijkl} \\\\frac{\\\\partial u_k}{\\\\partial x_l}$ Now $A_{ijkl}$ is a fourth order tensor with 81 individual components, from $A_{xxxx}$ to $A_{zzzz}$ in Cartesian $x$ , $y$ and $z$ coordinates.\",\n \"The full expansion of this equation is not practical in this text due to space and formatting considerations, but a reduced form is provided, considering Einstein's summation convention $& d_{xx} = A_{xxxx} \\\\frac{\\\\partial u_x}{\\\\partial x} + A_{xxxy} \\\\frac{\\\\partial u_x}{\\\\partial y} + A_{xxxz} \\\\frac{\\\\partial u_x}{\\\\partial z} + \\\\cdots + A_{xxzz} \\\\frac{\\\\partial u_z}{\\\\partial z}\\\\\\\\& d_{xy} = A_{xyxx} \\\\frac{\\\\partial u_x}{\\\\partial x} + A_{xyxy} \\\\frac{\\\\partial u_x}{\\\\partial y} + A_{xyxz} \\\\frac{\\\\partial u_x}{\\\\partial z} + \\\\cdots + A_{xyzz} \\\\frac{\\\\partial u_z}{\\\\partial z}\\\\\\\\& \\\\vdots \\\\\\\\& d_{zz} = A_{zzxx} \\\\frac{\\\\partial u_x}{\\\\partial x} + A_{zzxy} \\\\frac{\\\\partial u_x}{\\\\partial y} + A_{zzxz} \\\\frac{\\\\partial u_x}{\\\\partial z} + \\\\cdots + A_{zzzz} \\\\frac{\\\\partial u_z}{\\\\partial z}$ There are nine coefficients in each line and nine lines, one for each component of the deviatoric stress tensor, totaling the 81 coefficients of the $A_{ijkl}$ fourth-order tensor.\",\n \"As we have done before, it is convenient to write the deviatoric stress tensor as the sum of its symmetric and antisymmetric counterparts $d_{ij} = A_{ijkl} d_{ij}^{s} + A_{ijkl} d_{ij}^{a}$ or $d_{ij} = A_{ijkl} e_{kl} + A_{ijkl} \\\\xi _{ij}$ where $e_{kl} = \\\\frac{1}{2}(\\\\frac{\\\\partial u_k}{\\\\partial u_l} + \\\\frac{\\\\partial u_l}{\\\\partial u_k})$ is a symmetric second-order tensor, which in $x$ , $y$ and $z$ Cartesian coordinates can be represented as $\\\\frac{1}{2}\\\\begin{bmatrix}(\\\\frac{\\\\partial u_x}{\\\\partial x} + \\\\frac{\\\\partial u_x}{\\\\partial x}) & (\\\\frac{\\\\partial u_x}{\\\\partial y} + \\\\frac{\\\\partial u_y}{\\\\partial x}) & (\\\\frac{\\\\partial u_x}{\\\\partial z} + \\\\frac{\\\\partial u_z}{\\\\partial x}) \\\\\\\\(\\\\frac{\\\\partial u_y}{\\\\partial x} + \\\\frac{\\\\partial u_x}{\\\\partial y}) & (\\\\frac{\\\\partial u_y}{\\\\partial y} + \\\\frac{\\\\partial u_y}{\\\\partial y}) & (\\\\frac{\\\\partial u_y}{\\\\partial z} + \\\\frac{\\\\partial u_z}{\\\\partial y}) \\\\\\\\(\\\\frac{\\\\partial u_z}{\\\\partial x} + \\\\frac{\\\\partial u_x}{\\\\partial z}) & (\\\\frac{\\\\partial u_z}{\\\\partial y} + \\\\frac{\\\\partial u_y}{\\\\partial z}) & (\\\\frac{\\\\partial u_z}{\\\\partial z} + \\\\frac{\\\\partial u_z}{\\\\partial z}) \\\\\\\\\\\\end{bmatrix}$ Verify that the quantities that are opposed to each other across the diagonal are the same, and this is what is mean be symmetric in this case.\",\n \"The antisymmetric component is $\\\\xi _{kl} = \\\\frac{1}{2}(\\\\frac{\\\\partial u_k}{\\\\partial u_l} - \\\\frac{\\\\partial u_l}{\\\\partial u_k})$ and can be represented as $\\\\frac{1}{2}\\\\begin{bmatrix}(\\\\frac{\\\\partial u_x}{\\\\partial x} - \\\\frac{\\\\partial u_x}{\\\\partial x}) & (\\\\frac{\\\\partial u_x}{\\\\partial y} - \\\\frac{\\\\partial u_y}{\\\\partial x}) & (\\\\frac{\\\\partial u_x}{\\\\partial z} - \\\\frac{\\\\partial u_z}{\\\\partial x}) \\\\\\\\(\\\\frac{\\\\partial u_y}{\\\\partial x} - \\\\frac{\\\\partial u_x}{\\\\partial y}) & (\\\\frac{\\\\partial u_y}{\\\\partial y} - \\\\frac{\\\\partial u_y}{\\\\partial y}) & (\\\\frac{\\\\partial u_y}{\\\\partial z} - \\\\frac{\\\\partial u_z}{\\\\partial y}) \\\\\\\\(\\\\frac{\\\\partial u_z}{\\\\partial x} - \\\\frac{\\\\partial u_x}{\\\\partial z}) & (\\\\frac{\\\\partial u_z}{\\\\partial y} - \\\\frac{\\\\partial u_y}{\\\\partial z}) & (\\\\frac{\\\\partial u_z}{\\\\partial z} - \\\\frac{\\\\partial u_z}{\\\\partial z}) \\\\\\\\\\\\end{bmatrix}$ Verify that the quantities across the diagonals are different (reverse sign) and this is what is meant by antisymmetric in this context.\",\n \"Verify also that the sum of the symmetric and antisymmetric counterparts reduces to the second-order tensor $\\\\frac{\\\\partial u_k}{\\\\partial x_l}$ .\",\n \"The antisymmetric component can further be written as $\\\\xi _{kl} = -\\\\frac{1}{2}A_{ijkl} \\\\epsilon _{klm} \\\\omega _{m}$ The operator $\\\\epsilon _{klm}$ is a third-order tensor, with 27 components, where only six are non-zero, called the Levi-Civita symbol with the following property: $\\\\epsilon _{klm} ={\\\\left\\\\lbrace \\\\begin{array}{ll}+1 & \\\\text{if} ~(k, l, m)~\\\\text{is an even permutation} \\\\\\\\-1 & \\\\text{if} ~(k, l, m)~\\\\text{is an odd permutation} \\\\\\\\0 & \\\\text{if any index is repeated}\\\\end{array}\\\\right.\",\n \"}$ Notice that the choice of $k, l, m$ is incidental to the problem, most often on the literature $i, j, k$ or numbers are used.\",\n \"Even permutations in this case are $(k,l,m)$ , $(l,m,k)$ and $(m, k, l)$ and odd permutations are $(m, l, k)$ , $(l, k, m)$ and $(k, m, l)$ .\",\n \"The vector $\\\\omega _{m}$ has a fundamental role in fluid mechanics, and it is called the vorticity of the fluid.\",\n \"The components of the vorticity vector in $x$ , $y$ and $z$ Cartesian coordinates are $\\\\omega _x = \\\\frac{\\\\partial u_z}{\\\\partial y} - \\\\frac{\\\\partial u_y}{\\\\partial z} \\\\\\\\\\\\omega _y = \\\\frac{\\\\partial u_x}{\\\\partial z} - \\\\frac{\\\\partial u_z}{\\\\partial x} \\\\\\\\\\\\omega _z = \\\\frac{\\\\partial u_y}{\\\\partial x} - \\\\frac{\\\\partial u_x}{\\\\partial y}$ Which can be obtained from the curl of the $\\\\textbf {u}$ vector field (recall Chapter REF ) $\\\\omega = \\\\nabla \\\\times \\\\mathbf {u}$ A fluid with zero curl is called irrotational.\",\n \"The mathematical concept of rotation is directly linked to the physical idea of rotation in fluid flow.\",\n \"The idea of rotational and irrotational flows is rarely discussed in soil physics, largely because of the assumptions adopted related to the nature of the flow as we will discuss later.\",\n \"Now the deviatoric stress tensor can be rewritten as $d_{ij} = A_{ijkl} e_{kl} - \\\\frac{1}{2} A_{ijkl} \\\\epsilon _{klm} \\\\omega _m$ In simple liquids, $A_{ijkl}$ , is an isotropic tensor, that is, the molecular structure of the fluid is isotropic.\",\n \"In this case, $A_{ijkl}$ , can be written as a summation of products of delta tensorsObviously, there is much more to this, the reader is encouraged to consult a book on tensor analysis and Aris [4] $A_{ijkl} = \\\\mu \\\\delta _{ik}\\\\delta _{jl} + \\\\mu ^{\\\\prime } \\\\delta _{il}\\\\delta _{jk} + \\\\mu ^{\\\\prime \\\\prime } \\\\delta _{ij}\\\\delta _{kl}$ Because $A_{ijkl}$ is symmtrical in $i$ and $j$ , it is required that $\\\\mu = \\\\mu ^{\\\\prime } $ such that $A_{ijkl} = 2\\\\mu \\\\delta _{ik}\\\\delta _{jl} + \\\\mu ^{\\\\prime \\\\prime } \\\\delta _{ij}\\\\delta _{kl}$ The requirement that $A_{ijkl}$ is also symmetrical in $k$ and $l$ requires that the term $\\\\omega $ is dropped in the deviatoric stress tensor, so that by replacing into it we have $d_{ij} = 2\\\\mu \\\\delta _{ik}\\\\delta _{jl} e_{kl} + \\\\mu ^{\\\\prime \\\\prime } \\\\delta _{ij}\\\\delta _{kl} e_{kl}$ Because the rules of manipulation of Kronecker delta tensors, the relationship can be simplified to $d_{ij} = 2\\\\mu e_{ij} + \\\\mu ^{\\\\prime \\\\prime } \\\\nabla \\\\cdot e_{kk}~\\\\delta _{ij}$ Where $e_{kk} = \\\\nabla \\\\cdot \\\\mathbf {u} $ is the rate of expansion.\",\n \"Since the deviatoric stress tensor does not contribute to the mean normal stresses $d_{ii} = (2\\\\mu + 3\\\\mu ^{\\\\prime \\\\prime }) e_{ii} = 0$ Such that $(2\\\\mu + 3\\\\mu ^{\\\\prime \\\\prime }) = 0$ Solving for $\\\\mu ^{\\\\prime \\\\prime }$ and replacing into the deviatoric stress tensor C.L.\",\n \"results $d_{ij} = 2\\\\mu (e_{ij} - \\\\frac{1}{3} e_{ii}~\\\\delta _{ij})$ We now have an expression for the deviatoric stress tensor which can be used to define the stress tensor.\",\n \"Replacing into the stress tensorC.L.\",\n \"results $\\\\sigma _{ij} = -p \\\\delta _{ij} + 2\\\\mu (e_{ij} - \\\\frac{1}{3} e_{ii}~\\\\delta _{ij})$ Which can then be replaced into the equation of motion resulting in the set of equations $\\\\rho \\\\frac{D u_x}{D t} = F_x \\\\rho -\\\\frac{\\\\partial p}{\\\\partial x} + \\\\frac{\\\\partial }{\\\\partial x}[ 2 \\\\mu (e_{xx} - \\\\frac{1}{3} e_{xx} \\\\delta _{xx})]+ \\\\\\\\ \\\\frac{\\\\partial }{\\\\partial y}[ 2 \\\\mu (e_{xy} - \\\\frac{1}{3} e_{xx} \\\\delta _{xy})]+ \\\\frac{\\\\partial }{\\\\partial z}[ 2 \\\\mu (e_{xz} - \\\\frac{1}{3} e_{xx} \\\\delta _{xz})]$ $\\\\rho \\\\frac{D u_y}{D t} = F_y \\\\rho -\\\\frac{\\\\partial p}{\\\\partial y}+ \\\\frac{\\\\partial }{\\\\partial x}[ 2 \\\\mu (e_{yx} - \\\\frac{1}{3} e_{yy} \\\\delta _{yx})]+ \\\\\\\\ \\\\frac{\\\\partial }{\\\\partial y}[ 2 \\\\mu (e_{yy} - \\\\frac{1}{3} e_{yy} \\\\delta _{yy})]+ \\\\frac{\\\\partial }{\\\\partial z}[ 2 \\\\mu (e_{yz} - \\\\frac{1}{3} e_{yy} \\\\delta _{yz})]$ $\\\\rho \\\\frac{D u_z}{D t} = F_z \\\\rho -\\\\frac{\\\\partial p}{\\\\partial z}+ \\\\frac{\\\\partial }{\\\\partial x}[ 2 \\\\mu (e_{zx} - \\\\frac{1}{3} e_{zz} \\\\delta _{zx})]+ \\\\\\\\ \\\\frac{\\\\partial }{\\\\partial y}[ 2 \\\\mu (e_{zy} - \\\\frac{1}{3} e_{zz} \\\\delta _{zy})]+ \\\\frac{\\\\partial }{\\\\partial z}[ 2 \\\\mu (e_{zz} - \\\\frac{1}{3} e_{zz} \\\\delta _{zz})]$ These are the Navier-Stokes equations and they account for the description of viscous fluid flow.\",\n \"The viscosity term in the full form presented above is not independent of position and the equation is valid for compressible fluids as well as incompressible fluids.\",\n \"If we consider that the viscosity is constant throughout the flow field, the term $\\\\mu $ can be placed outside the derivative terms.\",\n \"Another simplification that can be used for modeling flow in porous media is incompressibility.\",\n \"For an incompressible fluid $\\\\nabla \\\\cdot \\\\mathbf {u} = e_{ii} = 0 $ and the term $ e_{ii} \\\\delta _{ij} $ is dropped, resulting in $\\\\rho \\\\frac{D u_x}{D t} = F_x \\\\rho -\\\\frac{\\\\partial p}{\\\\partial x} + \\\\mu \\\\frac{\\\\partial }{\\\\partial x} (e_{xx})+\\\\mu \\\\frac{\\\\partial }{\\\\partial y}(e_{xy})+ \\\\mu \\\\frac{\\\\partial }{\\\\partial z} (e_{xz})\\\\\\\\\\\\rho \\\\frac{D u_y}{D t} = F_y \\\\rho -\\\\frac{\\\\partial p}{\\\\partial y}+ \\\\mu \\\\frac{\\\\partial }{\\\\partial x}(e_{yx})+\\\\mu \\\\frac{\\\\partial }{\\\\partial y}(e_{yy})+ \\\\mu \\\\frac{\\\\partial }{\\\\partial z}(e_{yz})\\\\\\\\\\\\rho \\\\frac{D u_z}{D t} = F_z \\\\rho -\\\\frac{\\\\partial p}{\\\\partial z}+ \\\\mu \\\\frac{\\\\partial }{\\\\partial x}(e_{zx})+\\\\mu \\\\frac{\\\\partial }{\\\\partial y}(e_{zy})+ \\\\mu \\\\frac{\\\\partial }{\\\\partial z}(e_{zz})$ Note that, in this case, each partial derivative of $e_{ij}$ can be written as $\\\\frac{\\\\partial }{\\\\partial x_j} e_{ij} = \\\\frac{\\\\partial }{\\\\partial x_j} \\\\frac{1}{2}(\\\\frac{\\\\partial u_i}{\\\\partial x_j} + \\\\frac{\\\\partial u_j}{\\\\partial x_i}) = \\\\frac{1}{2}\\\\frac{\\\\partial ^2 u_i}{\\\\partial x_j\\\\partial x_j} + \\\\frac{1}{2}\\\\frac{\\\\partial }{\\\\partial x_i}\\\\frac{\\\\partial u_j}{\\\\partial x_j} = \\\\frac{1}{2} \\\\nabla ^2 u_i + \\\\frac{1}{2} \\\\frac{\\\\partial }{\\\\partial x_i}(\\\\nabla \\\\cdot \\\\mathbf {u})$ the incompressibility condition reduces the Navier-Stokes equations to $\\\\rho \\\\frac{D u_x}{D t} = F_x \\\\rho -\\\\frac{\\\\partial p}{\\\\partial x} + \\\\mu ( \\\\frac{\\\\partial ^2 u_x}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 u_x}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 u_x}{\\\\partial z^2})\\\\\\\\\\\\rho \\\\frac{D u_y}{D t} = F_y \\\\rho -\\\\frac{\\\\partial p}{\\\\partial y} + \\\\mu ( \\\\frac{\\\\partial ^2 u_y}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 u_y}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 u_y}{\\\\partial z^2})\\\\\\\\\\\\rho \\\\frac{D u_z}{D t} = F_z \\\\rho -\\\\frac{\\\\partial p}{\\\\partial z} + \\\\mu ( \\\\frac{\\\\partial ^2 u_z}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 u_z}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 u_z}{\\\\partial z^2})$ which can be written in vector form and by expanding the material derivative as $\\\\boxed{\\\\rho (\\\\frac{\\\\partial \\\\mathbf {u}}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u} ) = \\\\mathbf {F} \\\\rho - \\\\nabla p + \\\\mu \\\\nabla ^2 \\\\mathbf {u}}$ This is the fundamental equation that governs the flow of incompressible fluids and many important results can be derived by solving it for a specified set of boundary conditions.\",\n \"We will used it over subsequent chapters to investigate settling and water flow phenomena.\",\n \"What we will call Stokes equation is a special case of the Navier-Stokes equation for a set of specified conditions.\",\n \"Because we will use Stokes equation as a starting point for particle settling and flow in porous media derivations, it will be specified here.\",\n \"Under steady-state flow or flows in which the velocity variation is small enough that it can be considered constant and in conditions where the inertia is small as compared to the pressure and viscous terms the acceleration term on the left side of the Navier-Stokes equation can be neglected.\",\n \"If the body forces can also be neglected due to being small or the direction of flow in relation to a gravitational field for example, the governing equations become $\\\\boxed{\\\\nabla p = \\\\mu \\\\nabla ^2 \\\\mathbf {u}}$ $\\\\nabla \\\\cdot \\\\mathbf {u} = 0$ The second of those being the conservation law described previously.\",\n \"Bernoulli equation can be derived from Euler equations by using the simplifying assumptions of steady and isentropic flow.\",\n \"Steady flow means that the velocity is constant in any particular point in the flow field, i.e.\",\n \"does not vary with time, such that $ \\\\frac{\\\\partial \\\\mathbf {u}}{\\\\partial t} = 0 $ .\",\n \"Isentropic means that the entropy $s$ is constant throughout the flow field.\",\n \"To derive Bernoulli equation we need to first express Euler equation in terms of enthalpy.\",\n \"In thermodynamics, specific enthalpy can be expressed as $dh = Tds + vdP$ But, because entropy is constant, $ds = 0 $ and $dh = vdp$ In this equation $h$ is the specific enthalpy or enthalpy per unit mass $h = H/M$ , $s$ is the specific entropy, $s = S/M$ and $v$ is the specific volume $v = V/M$ .\",\n \"Since we know that density of a fluid is mass per unit volume, the specific volume can be written as $v = 1/\\\\rho $ and $dh = \\\\frac{1}{\\\\rho }dp$ which can be generalized to $\\\\nabla h = \\\\frac{1}{\\\\rho } \\\\nabla p$ and replaced into Euler equation resulting $\\\\frac{\\\\partial \\\\mathbf {u}}{\\\\partial t} + \\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u} = - \\\\nabla h + \\\\mathbf {F} \\\\rho $ We now use a vector calculus identity referred to as the vector dot del operator $\\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u} = \\\\frac{1}{2} \\\\nabla u^2 - \\\\mathbf {u} \\\\times (\\\\nabla \\\\times \\\\mathbf {u} )$ by replacing into the “enthalpy” Euler equation, and with the condition that the velocity does not very in time specified previously, we have $\\\\mathbf {u} \\\\times (\\\\nabla \\\\times \\\\mathbf {u} ) = - \\\\nabla (h + \\\\frac{1}{2} u^2) + \\\\mathbf {F}$ Flow conditions require that the curl terms on the left side is zero.\",\n \"We can also rewrite the force term as the negative of the gradient of a potential $\\\\Psi $ $0 = - \\\\nabla (h + \\\\frac{1}{2} u^2) - \\\\nabla \\\\Psi $ and lump the terms into a single gradient operator $\\\\nabla (h + \\\\frac{1}{2} u^2 + \\\\Psi ) = 0$ Replacing the expression for the enthalpy previously defined and considering $\\\\Psi $ as the force potential to Earth's gravitational field $\\\\Psi = \\\\rho g$ , we are left with $\\\\boxed{\\\\frac{p}{\\\\rho } + \\\\frac{1}{2} u^2 + \\\\rho g = \\\\text{constant}}$ This equation states that the left hand sum is constant along any streamline.\",\n \"A streamline is a line whose tangent at any point gives the direction of the velocity vector at that point.\",\n \"Streamlines are defined by the system of equations $\\\\frac{dx}{u_x} = \\\\frac{dy}{u_y} = \\\\frac{dz}{u_z}$ Reynolds number is known in soil physics and applied hydraulics and hydrology as a means of establishing if a given flow regime is laminar or turbulent.\",\n \"It is much more than that, it derives from writing the Navier-Stokes equation in a dimensionless form and conveys important information about the flow regime and scaling of flow processes and can be used in laboratory to design experiments to measure flow parameters that could not be measured otherwise, on account of the law of similarity.\",\n \"We start with the Navier-Stokes equations in an incompressible and homogeneous fluid with regard to density.\",\n \"The body force is ignored, but the same principles applied to the other terms can be used if one wishes to proceed including the body force $\\\\rho (\\\\frac{\\\\partial u_i}{\\\\partial t} + u_j \\\\frac{\\\\partial u_i}{\\\\partial x_j} ) = -\\\\frac{\\\\partial p}{\\\\partial x_i} + \\\\mu \\\\frac{\\\\partial ^2 u_i}{\\\\partial x_j \\\\partial x_i}$ Each quantity on the Navier-Stokes equation is now written in a dimensionless form by defining some representative length L and velocity U, with the new dimensionless variables $\\\\mathbf {u}^{\\\\prime }$ , $t^{\\\\prime }$ and $\\\\mathbf {x}^{\\\\prime }$ $\\\\mathbf {u}^{\\\\prime } = \\\\frac{\\\\mathbf {u}}{U}$ $\\\\mathbf {x}^{\\\\prime } = \\\\frac{\\\\mathbf {x}}{L}$ $t^{\\\\prime } = \\\\frac{tU}{L}$ Note that U/L has units of 1/time since [L T$^{-1}$ ] / [L] = [T$^{-1}$ ].\",\n \"Pressure can also be normalized using $p^{\\\\prime } = \\\\frac{p}{\\\\rho U^2}$ Since $\\\\rho U^2$ has units of [M L$^{-3}$ ] [L T$^{-1}$ ]$^2$ = [M L$^{-1}$ T$^{-2}$ ] which is equivalent to pressure.\",\n \"Substituting the dimensionless quantities on the Navier-Stokes $\\\\rho (\\\\frac{\\\\partial u^{\\\\prime }_i U}{\\\\partial t^{\\\\prime }L/U} + u^{\\\\prime }_j U \\\\frac{\\\\partial u^{\\\\prime }_i U}{\\\\partial x^{\\\\prime }_j L} ) = -\\\\frac{\\\\partial p^{\\\\prime }\\\\rho U^2}{\\\\partial x^{\\\\prime }_i L} + \\\\mu \\\\frac{\\\\partial ^2 u^{\\\\prime }_i U}{\\\\partial x^{\\\\prime }_j L \\\\partial x^{\\\\prime }_i L}$ Remembering that constant terms can be brought out of the derivatives and reorganizing $\\\\frac{U^2}{L}\\\\rho (\\\\frac{\\\\partial u^{\\\\prime }_i}{\\\\partial t^{\\\\prime }} + u^{\\\\prime }_j \\\\frac{\\\\partial u^{\\\\prime }_i}{\\\\partial x^{\\\\prime }_j} ) = -\\\\frac{U^2}{L}\\\\frac{\\\\partial p^{\\\\prime }\\\\rho }{\\\\partial x^{\\\\prime }_i} + \\\\mu \\\\frac{U}{L^2}\\\\frac{\\\\partial ^2 u^{\\\\prime }_i}{\\\\partial x^{\\\\prime }_j \\\\partial x^{\\\\prime }_i} \\\\\\\\\\\\frac{\\\\partial u^{\\\\prime }_i}{\\\\partial t^{\\\\prime }} + u^{\\\\prime }_j \\\\frac{\\\\partial u^{\\\\prime }_i}{\\\\partial x^{\\\\prime }_j} = -\\\\frac{\\\\partial p^{\\\\prime }}{\\\\partial x^{\\\\prime }_i} + \\\\frac{\\\\mu }{\\\\rho U L}\\\\frac{\\\\partial ^2 u^{\\\\prime }_i}{\\\\partial x^{\\\\prime }_j \\\\partial x^{\\\\prime }_i}$ The dimensionless quantity $\\\\boxed{Re = \\\\frac{\\\\rho U L}{\\\\mu }}$ is called the Reynolds number, being introduced by Stokes and later named after Osborne Reynolds who popularized its use.\",\n \"The dimensionless form of the Navier-Stokes equation can be written in vector notation as $\\\\frac{\\\\partial \\\\mathbf {u}^{\\\\prime }}{\\\\partial t^{\\\\prime }} + \\\\mathbf {u}^{\\\\prime }\\\\cdot \\\\nabla \\\\mathbf {u}^{\\\\prime } = -\\\\nabla p^{\\\\prime } + \\\\frac{1}{Re} \\\\nabla ^2 \\\\mathbf {u}^{\\\\prime }$ The Reynolds number is a ratio of inertial forces, that tend to keep the fluid in motion, and viscous forces, which resist movement.\",\n \"In fluids in which the Reynolds number is low, there is large contribution of inertial forces.\",\n \"If inertial forces are much larger than viscous forces the flow can become unstable and vortices and other flow instabilities can develop.\",\n \"Under such conditions the assumption that the flow is smooth, constant and “continuous” is no longer valid and the flow can no longer be considered laminar.\",\n \"If inertial forces are very high in respect to viscous forces, fully turbulent flow might develop.\",\n \"ll $ \\\\mathbf {s} $ Vector position function $ \\\\mathbf {u}, \\\\mathbf {v} $ Vector velocity functions $ \\\\mathbf {i}, \\\\mathbf {j}, \\\\mathbf {k}, $ Unit vectors in Cartesian coordinates $ \\\\mathbf {x}, \\\\mathbf {y}, \\\\mathbf {z}, $ Unit vectors in Cartesian coordinates $ s_x, s_y, s_z $ Components of the position vector in Cartesian coordinates $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ u, v, w $ Components of the velocity vector in Cartesian coordinates $ t $ Time $ \\\\frac{D}{D t} $ Substantial (or material) derivative $ f $ Generalized function $ \\\\rho $ Density $ V $ Volume $ A $ Area $ \\\\mathbf {n} $ Normal unit vector $ m, M $ Mass $ x, y, z, $ Generalized Cartesian coordinates $ \\\\mathbf {F} $ Force vector $ \\\\mathbf {a} $ Acceleration vector $ \\\\mathbf {p} $ Momentum vector $ \\\\sigma $ Stress $ \\\\delta _{ij} $ Kroenecker delta $ p $ Pressure $ S $ surface $ d_{ij} $ Deviatoric stress tensor $ \\\\mathbf {d} $ Deviatoric stress tensor, vector form $ \\\\mathbf {g} $ Gravitational body force vector $ g_x, g_y, g_z $ Components of the gravitational body force vector in Cartesian coordinates $ A_{ijkl} $ Fourth-order tensor $ e_{kl} $ Symmetric component of the deviatoric stress tensor $ \\\\xi _{kl} $ Antisymmetric component of the deviatoric stress tensor $ \\\\epsilon _{klm} $ Levi-Civita symbol $ \\\\omega $ Vorticity vector $ \\\\omega _x, \\\\omega _y, \\\\omega _w $ Components of the vorticity vector in Cartesian coordinates $ e_{kk} $ Rate of expansion $ \\\\mu $ Viscosity $ h $ Specific enthalpy $ s $ Specific entropy $ v $ Specific volume $ T $ Temperature $ P $ Pressure $ S $ Entropy $ H $ Enthalpy $ \\\\Psi $ Potential $ \\\\mathbf {u^{\\\\prime }} $ Dimensionless velocity vector $ \\\\mathbf {x^{\\\\prime }} $ Dimensionless position vector $ t^{\\\\prime } $ Dimensionless time $ p^{\\\\prime } $ Dimensionless pressure $ Re $ Reynolds number $ L $ Length $ U $ Velocity There are two basic methods for the determination of particle size distribution of soils and sediments.\",\n \"For coarser particles such as gravel and cobbles, and sands to some extent, grain size distributions or grain size fractions can be determined using sieves.\",\n \"In essence what is done is to disaggregate the particles, if they are aggregated at all, using physical and or chemical methods, pass the material through a set of sieves of decreasing apertures and to weigh the dried fraction retained in each sieve and to express this as a percent of the total dried mass of the material used.\",\n \"The sieve aperture used for each class interval is a convention, and the textural classes vary between disciplines in soil science, Earth science and engineering.\",\n \"For particles finer than sand, sieving does not work adequately, first because it might not be feasible to create sieves with apertures small enough for the particles, which might have diameters on the micrometer range, second because it might be virtually impossible to disaggregate these particles in the dried state, and third because even if these particles could be disaggregated and precision micrometer aperture sized sieves created, they might cause the clogging of the apertures of the sieve which at that size range would act analogous as pores.\",\n \"For smaller particles, most methods of particle size determination rely on settling of the dispersed particles in a fluid, usually water.\",\n \"Particles are assumed as perfectly spherical, non-interacting, slowly settling in an infinitely dispersed solution.\",\n \"Under these conditions the Navier-Stokes equation solution for a sphere settling in a viscous fluid is the solution to the problem.\",\n \"The fact that the settling is slow under low Reynolds number guarantees that the flow around the sphere is laminar (Figure REF a).\",\n \"Under turbulent flows, analytical solutions to the Navier-Stokes equations, such as the one presented below, might not be feasible due to the complexity of the chaotic fluctuations of the flow regime.\",\n \"In any case even before turbulence developed, the settling velocity might be large enough such that the inertial component of the Navier-Stokes equation is not negligible, and analytic solutions might not be achievable.\",\n \"Figure: Illustration of sphere settling under laminar regime.The solution of the sphere settling problem was provided by G.G.\",\n \"Stokes [92], many authors present interpretations of Stokes solution which are more or less similar to the original and to each-other [66], [57], [8], [1].\",\n \"The solution given by Acheson [1] is presented here for reasons of clarity and conciseness.\",\n \"For slow, steady-state settling conditions the mathematical problems is to find an analytical solution to the Stokes and conservation equations presented in Chapter REF .\",\n \"$\\\\nabla p = \\\\mu \\\\nabla ^2 \\\\mathbf {u}$ $\\\\nabla \\\\cdot \\\\mathbf {u} = 0$ Not surprisingly, the problem of flow past a sphere is solved by using spherical coordinates.\",\n \"Thus the vector $\\\\mathbf {u} = \\\\langle u_x, u_y, u_z \\\\rangle = u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}$ is written in spherical coordinates as $\\\\mathbf {u} = \\\\langle u_r(r, \\\\theta ), u_{\\\\theta }(r, \\\\theta ), 0 \\\\rangle $ The conservation equation $ \\\\nabla \\\\cdot \\\\mathbf {u}$ is satisfied by a Stokes stream function $\\\\Psi (r , \\\\theta )$ which satisfies the conditions $u_r = \\\\frac{1}{r^2 \\\\sin {\\\\theta }} \\\\frac{\\\\partial \\\\Psi }{\\\\partial \\\\theta }, ~ u_{\\\\theta } = \\\\frac{1}{r \\\\sin {\\\\theta }} \\\\frac{\\\\partial \\\\Psi }{\\\\partial r}$ Now, the curl of $\\\\mathbf {u}$ in spherical coordinates (consult Chapter REF ) is $\\\\nabla \\\\times \\\\mathbf {u} = \\\\langle 0, 0, \\\\frac{1}{r \\\\sin {\\\\theta }} E^2 \\\\Psi \\\\rangle $ With $E^2$ representing the differential operator $E^2 = \\\\frac{\\\\partial ^2}{\\\\partial r^2} + \\\\frac{\\\\sin {\\\\theta }}{r^2} \\\\frac{\\\\partial }{\\\\partial \\\\theta } (\\\\frac{1}{\\\\sin {\\\\theta }} \\\\frac{\\\\partial }{\\\\partial \\\\theta } )$ By virtue of vector identities, Stokes equation can be written as $\\\\nabla p = -\\\\mu \\\\nabla \\\\times (\\\\nabla \\\\times \\\\mathbf {u})$ in which the gradient operators in $p$ and after $\\\\mu $ can be written in spherical coordinates as $\\\\nabla = \\\\langle \\\\frac{\\\\partial }{\\\\partial r}, \\\\frac{1}{r} \\\\frac{\\\\partial }{\\\\partial \\\\theta }, \\\\frac{1}{r \\\\sin {\\\\theta }} \\\\frac{\\\\partial }{\\\\partial \\\\phi } \\\\rangle $ from Equations 5.3, 5.5. and 5.6 the components in $\\\\mathbf {r}$ and $\\\\theta $ can be obtained $\\\\frac{\\\\partial p}{\\\\partial r} = \\\\frac{\\\\mu }{r^2 \\\\sin {\\\\theta }} \\\\frac{\\\\partial }{\\\\partial \\\\theta } E^2 \\\\Psi ,~\\\\frac{1}{r} \\\\frac{\\\\partial p}{\\\\partial \\\\theta } = \\\\frac{-\\\\mu }{r \\\\sin {\\\\theta }} \\\\frac{\\\\partial }{\\\\partial r} E^2 \\\\Psi $ The detailed procedure for cross differentiation considering a stream function $\\\\Psi $ is given by Lamb [66].\",\n \"Acheson [1] explains it in terms of the pressure being eliminated by cross-differentiation which implies that $E(E^2\\\\Psi ) = 0$ such thatObviously there is much more to this, the user is encouraged to consult Stokes (1901)[92] and Lamb (1932)[66], for an in depth discussion.\",\n \"Stokes (1901) is a reprint of the original and Lamb's discussion draws directly from the source and the level of detail provided is unmatched in modern literature (Horace Lamb was a student of Sir George Stokes at Trinity College, University of Cambridge).\",\n \"There is a very elegant derivation in Landau and Lifshitz [57] based largely on vector calculus operators.\",\n \"Although precise, Landau and Lifshitz and Batchelor [8] derivations rely heavily in implicit arguments.\",\n \"We tried to find a compromise between conciseness and clarity and most likely have not been entirely successful to say the least.\",\n \"$[\\\\frac{\\\\partial ^2}{\\\\partial r^2} + \\\\frac{\\\\sin {\\\\theta }}{r^2} \\\\frac{\\\\partial }{\\\\partial \\\\theta } (\\\\frac{1}{\\\\sin {\\\\theta }} \\\\frac{\\\\partial }{\\\\partial \\\\theta } ) ]^2 \\\\Psi = 0$ The boundary conditions on the settling sphere are $u_r = u_{\\\\theta } = 0 ~\\\\text{at} ~r = a$ $u_r = U\\\\cos {\\\\theta },~u_{\\\\theta } = -U\\\\sin {\\\\theta } ~\\\\text{as} ~r \\\\rightarrow a$ Because the decomposition of the velocity components in $\\\\mathbf {r}$ and $\\\\theta $ .\",\n \"The latter conditions along with Equation 5.2, results, by integration in the condition $\\\\Psi = \\\\frac{U}{2} r^2 \\\\sin ^2{\\\\theta } ~\\\\text{as} ~r \\\\rightarrow a$ Now we search for a solution to the partial differential equation 5.8.\",\n \"The theory on solution to partial differential equations plays a fundamental role in analytical methods in fluid mechanics and soil physics, reader unfamiliar with it are encouraged to consult Boas [14] for a concise introduction and Asmar [5] for a more detailed discussion.\",\n \"Equation 5.8 is satisfied by a solution of the form $\\\\Psi = \\\\sin ^2{\\\\theta } f(r)$ provided that $[\\\\frac{d^2}{dr^2} - \\\\frac{2}{r^2}] f(r) = 0$ with solution of the form $f(r) = \\\\frac{A}{r} + Br + Cr^2 + Dr^4$ with A, B, C and D constants.\",\n \"Substituting Equation 5.14 into 5.12 results $\\\\Psi = \\\\sin ^2{\\\\theta } [ \\\\frac{A}{r} + Br + Cr^2 + Dr^4]$ The values of the constant can be obtained by applying the boundary conditions previously defined to Equation 5.15, resulting $\\\\Psi = \\\\frac{U}{2}[ r^2 + \\\\frac{a^3}{2r} - \\\\frac{3ar}{2}] \\\\sin ^2{\\\\theta }$ By combining $\\\\mathbf {r}$ component of Equation 5.7 , Equation 5.4 and Equation 5.16 and integrating from $r = 0$ from $r = \\\\infty $ a pressure function is obtained $p = p_{\\\\infty } - \\\\frac{3}{2} \\\\frac{\\\\mu U a}{r^2} \\\\cos {\\\\theta }$ The stress components on a sphere settling in a viscous fluid are $\\\\tau _r = -p + 2\\\\mu \\\\frac{\\\\partial u_r}{\\\\partial r}, ~ \\\\tau _{\\\\theta } = \\\\mu r \\\\frac{\\\\partial }{\\\\partial r}(\\\\frac{u_{\\\\theta }}{r}) + \\\\frac{\\\\mu }{r} \\\\frac{\\\\partial u_r}{\\\\partial \\\\theta }, ~\\\\tau _{\\\\phi } = 0$ From Equations 5.2, 5.16 and 5.18 we have $\\\\tau _r = -p_{\\\\infty } + \\\\frac{3}{2} \\\\frac{\\\\mu U}{a}\\\\cos {\\\\theta }$ $\\\\tau _{\\\\theta } = - \\\\frac{3}{2} \\\\frac{\\\\mu U}{a}\\\\sin {\\\\theta }$ The resulting stress vector on the sphere is $\\\\tau _{net} = \\\\tau _r \\\\cos {\\\\theta } - \\\\tau _{\\\\theta }\\\\sin {\\\\theta } = -p_{\\\\infty }\\\\cos {\\\\theta } + \\\\frac{3}{2} \\\\frac{\\\\mu U}{a}$ and the total drag force on the sphere can be calculated by integrating the net force in spherical coordinates $F_{D} = \\\\int _0^{2\\\\pi } \\\\int _0^{\\\\pi } a^2~\\\\tau _{net}~\\\\sin {\\\\theta }~d \\\\theta d \\\\phi $ $F_{D} = \\\\int _0^{2\\\\pi } \\\\int _0^{\\\\pi } a^2~(-p_{\\\\infty }\\\\cos {\\\\theta } + \\\\frac{3}{2} \\\\frac{\\\\mu U}{a})~\\\\sin {\\\\theta }~d \\\\theta d \\\\phi \\\\\\\\F_{D} = \\\\int _0^{2\\\\pi } \\\\int _0^{\\\\pi } a^2~(-p_{\\\\infty } ~\\\\sin {\\\\theta }~\\\\cos {\\\\theta })~d \\\\theta d \\\\phi + \\\\int _0^{2\\\\pi } \\\\int _0^{\\\\pi } (a~\\\\frac{3}{2} \\\\mu U~\\\\sin {\\\\theta })~d \\\\theta d \\\\phi \\\\\\\\F_{D} = \\\\int _0^{2\\\\pi } a^2~(-p_{\\\\infty } ~\\\\frac{\\\\sin ^2{\\\\theta }}{2})\\\\bigg |_0^{\\\\pi }~d \\\\phi + \\\\int _0^{2\\\\pi } -(a~\\\\frac{3}{2} \\\\mu U~\\\\cos {\\\\theta })\\\\bigg |_0^{\\\\pi } ~ d \\\\phi \\\\\\\\$ Remembering that $\\\\sin {0}$ = 0 and $\\\\sin {\\\\pi } = 0$ and that $\\\\cos {0}$ = 1 and $\\\\cos {\\\\pi } = -1$ , the first term on the right side is equal to zero $F_{D} = \\\\int _0^{2\\\\pi } (2 a~\\\\frac{3}{2} \\\\mu U) ~ d \\\\phi \\\\\\\\F_{D} = \\\\int _0^{2\\\\pi } (a 3 \\\\mu U) ~ d \\\\phi \\\\\\\\F_{D} = (a 3 \\\\mu U) \\\\phi \\\\bigg |_0^{2\\\\pi } \\\\\\\\$ $\\\\boxed{F_{D} = 6 \\\\pi \\\\mu U a}$ This is Stokes formula for the drag on a sphere settling at low velocities in a viscous fluid.\",\n \"It is valid for creep flow, often called Stokes flow, in which the inertial forces are much smaller than the viscous forces.\",\n \"Flow is laminar, meaning that fluid particles follow smooth trajectories, without turbulent mixing.\",\n \"It is often conceptualized as continuous flow lines around the object which is moving through the fluid or around which the fluid is moving.\",\n \"The terminal settling velocity can be found by the force balance in the $z$ direction since the sphere is settling in direction of the gravitational force, against viscosity and buoyancy in a stationary fluid.\",\n \"$\\\\sum F_z = F_D + F_b + F_g = ma$ Buoyancy force is proportional to the mass of fluid displaced by the settling object $F_b = m_{fluid} g$ While the gravitational force is $F_g = m_{sphere} g$ But the volume of displaced fluid and the volume of the sphere are the same and equal to $V_{sphere} = \\\\frac{4}{3} \\\\pi R^3$ If the density of the fluid is $\\\\rho _{fluid} = \\\\frac{m_{fluid}}{V_{fluid}}$ and that of the sphere is $\\\\rho _{sphere} = \\\\frac{m_{sphere}}{V_{sphere}}$ the buoyancy force can be written as $F_b = \\\\rho _{fluid} \\\\frac{4}{3} \\\\pi R^3 g$ while the gravitational force is $F_g = \\\\rho _{sphere} \\\\frac{4}{3} \\\\pi R^3 g$ Now in the drag force equation $a$ is R, the radius of the sphere and the terminal translational velocity can be written as $v_s$ such that $\\\\boxed{F_{D} = 6 \\\\pi \\\\mu v_s R}$ Replacing all the forces in Equation 5.24 and remembering that the terminal velocity is constant, i.e., there is no acceleration, $ma = 0$ , and $\\\\sum F_z = 6 \\\\pi \\\\mu v_s R + \\\\rho _{fluid} \\\\frac{4}{3} \\\\pi R^3 g - \\\\rho _{sphere} \\\\frac{4}{3} \\\\pi R^3 g = 0$ Where the coordinates systems establishes that downward forces have negative sign.\",\n \"Solving for $v_s$ $6 \\\\pi \\\\mu v_s R = \\\\rho _{sphere} \\\\frac{4}{3} \\\\pi R^3 g - \\\\rho _{fluid} \\\\frac{4}{3} \\\\pi R^3 g \\\\\\\\6 \\\\pi \\\\mu v_s R = \\\\frac{4}{3}(\\\\rho _{sphere} - \\\\rho _{fluid}) \\\\pi R^3 g \\\\\\\\v_s = \\\\frac{2}{9} \\\\frac{ (\\\\rho _{sphere} - \\\\rho _{fluid}) g}{\\\\mu } R^2$ Since the sphere in this case is the solid particles settling, or solid minerals, we can rewrite using notation from previous chapters as $\\\\boxed{v_s = \\\\frac{2}{9} \\\\frac{ (\\\\rho _{p} - \\\\rho _{f}) g}{\\\\mu } R^2}$ This equation is know as Stokes law of settling and it is used to model the settling behavior of infinitely dispersed spherical solid particles settling in laminar regime.\",\n \"Terminal velocity is a function of the squared radius of the particle, therefore, everything else constant in a viscous fluid, larger particles settle faster.\",\n \"Settling velocity is also controlled by the densities of the fluid and the solid, viscosity and gravity.\",\n \"Fluid density and viscosity are controlled by temperature and empirical laws should need to be used to estimate these quantities.\",\n \"Stokes law of settling is valid for laminar flow, where $Re \\\\ll 1$ The limits are determined empirically so the critical value of the Reynolds number might depend on the nature of the problem being studied.\",\n \"More specifically, there might be transitional flow regimes between laminar and fully turbulent.\",\n \"The Reynolds number a settling spherical particle is $Re = \\\\frac{2\\\\rho _f R v_s}{\\\\mu }$ Considering a maximum Reynolds number for laminar flow in settling of $Re \\\\approx 1$ (this might be a conservative value), the maximum radius of a smooth spherical particle falling under laminar flow regime is $\\\\frac{\\\\mu }{2\\\\rho _{f} R} = \\\\frac{2}{9} \\\\frac{ (\\\\rho _{p} - \\\\rho _{f}) g}{\\\\mu } R^2 \\\\\\\\$ $R = \\\\@root 3 \\\\of { \\\\frac{9 \\\\mu ^2}{4 \\\\rho _f(\\\\rho _s-\\\\rho _f)g }}$ For water density of $\\\\rho \\\\approx 1000 $ kg m$^{-3}$ , and viscosity at 25 $^{o}$ C of $\\\\mu \\\\approx 0.00089 $ Pa s$^{-1}$ , the maximum radius of a smooth spherical quartz particle with $\\\\rho _s \\\\approx 2650 $ kg m$^{-3}$ and $g = 9.80665 m~s^{-2}$ falling under laminar flow regime is approximately $R = 4.793 \\\\times 10^{-5}~m $ or $R = 0.048~mm$ , which translates into a diameter of approximately $0.1~mm$ .\",\n \"On the other hand of the spectrum, for colloidal particles, Stokes law is no longer valid because under real circumstances, the solution is not infinitely dispersed and the particles are subjected to thermal motion (also known as Brownian motion).\",\n \"As smaller particles settle, a concentration gradient is established along the length of the settling column and a concentration gradient will keep particles in suspension as long as there is no coagulation or aggregation in suspension [60], [76].\",\n \"In other words, the thermal motion is able to overcome gravitational forcesAlbert Einstein's Ph.D. dissertation was on Brownian motion, before he was made famous by his studies in relativity and the photoelectric effect.\",\n \"The Péclet number is another dimensionless number in hydrodynamics which expresses the ratio between advective and diffusive transport rates [76].\",\n \"If $Pe \\\\ll 1$ , diffusive transport predominates and settling is perturbed by thermal motion.\",\n \"Considering $Pe = 1$ the radius of a sphere settling is given by $R = [\\\\frac{3 k T}{4 \\\\pi (\\\\rho _s - \\\\rho _f)g}]^{\\\\frac{1}{4}}$ In which $k = 1.380649 \\\\times 10^{-23}~J~K^{-1}$ is the Boltzmann constant and all other variables were previously defined.\",\n \"Using the values for a spherical silicate mineral settling in water presented for the previous calculations, the radius above which diffusive transports can be consider to start affecting settling is $ r = 4.96 \\\\times 10^{-7} ~m$ or $r = 0.4965~\\\\mu m$ , or approximately $1~\\\\mu m$ .\",\n \"The conclusion is that Stokes law for an average spherical mineral particle settling in water at room temperature is valid for diameters above that of colloidal particles and below that of fine sand particles.\",\n \"Below the colloidal limit settling is hindered by thermal motion and above the fine sand limit settling is under turbulent regime.\",\n \"Stokes law only applies to creeping (laminar) flow.\",\n \"In reality, the dynamics of particles on the clay size range is much more complicated because of surface forces among the interacting particles.\",\n \"These forces can cause aggregation of particles in suspension to which Stokes law applies to some degree and settling via coagulation and other mechanisms.\",\n \"Stokes law in particle size characterization of soils and fine sediments assumes that aggregation and coagulation do not apply and the analysis should be disregarded if these processes are observed.\",\n \"We saw in Chapter REF that when charged particles are put in contact with an electrolyte solution an Electric Double Layer (EDL) develops with charged ions adsorbed onto the surface by electrostatic forces.\",\n \"The concentration of ions is larger at the surface and at certain concentrations the EDL extends far enough that repulsive forces predominated between charged particles and the colloidal suspension remains dispersed.\",\n \"However, this is not the entire story.\",\n \"Between particles, atoms and molecules, dispersion forcesThe term dispersion forces here refers to intermolecular and interatomic forces and not to particles in dispersion originating from quantum fluctuations arise.\",\n \"These dispersion forces are grouped as what is called van der Waals forces (vdW) and for the systems we are concerned are attractive in nature.\",\n \"If the electrolyte concentration is increased, the EDL is compressed and the vdW forces which are stronger than the repulsive forces in the short range predominate and the colloidal particles can aggregate and coagulate.\",\n \"This can happen with dispersed soil solutions depending on the valence and concentration of the electrolyte as clays are charged particles of small diameter.\",\n \"The theory of balance of these attractive and repulsive forces in colloidal suspensions is called DLVO after Boris Derjaguin, Lev Landau, Evert Verwey and Theodoor Overbeek who developed it, the first two working together and the last two independently.\",\n \"Considering the interaction of two charged spheres of radius R in a 1:1 electrolyte solution, the energy of interaction due to electrostatic double layer forces is given by [52] $W_{EDL} = \\\\frac{64 \\\\pi k T R \\\\rho _{\\\\infty }}{\\\\kappa } \\\\tanh ^2 (\\\\frac{z e \\\\psi _0 }{4kT}) \\\\exp {(-\\\\kappa D^{\\\\prime })}$ in which the eletrochemical parameters are the same as in Chapter 3, i.e.\",\n \"$\\\\rho _{\\\\infty }$ is the electrolyte concentration in bulk solution, $\\\\psi _0$ is the surface potential, $z$ is the valence of the ions, $k$ is the Boltzmann constant, $T$ is the absolute temperature, $\\\\kappa $ is a parameter related to the thickness of the double layer, and $e$ is the electronic charge, and $R$ is the spheres radius.\",\n \"While the attractive vdW force is given for two spheres of the same radius as [52] $W_{vdW} = \\\\frac{A_HR}{12 D^{\\\\prime }}$ in which $D^{\\\\prime }$ is the separation distance usually in $nm$ or angstrons (Å) and $A_H$ is the Hamaker constant $A_H = \\\\pi ^2 C \\\\rho _1 \\\\rho _2$ In which $\\\\rho _1$ and $\\\\rho _1$ are the number densities of the interacting spheres and $C$ is the London constantFor the time being we will refrain from exploring the derivations of these relationships as they might require exploring quantum mechanical perturbation theory.\",\n \"In the case of equal particles $\\\\rho _1 = \\\\rho _2$ and $\\\\rho _1\\\\rho _2 = \\\\rho ^2$ .\",\n \"The Hamaker constant has an important role in the adsorption of water films in surfaces and is fundamental for modeling dry water adsorption in soils.\",\n \"The pair interaction energy considering the two forces is $W(D^{\\\\prime }) = W_{EDL} - W_{vdW}$ Negative values of the pair interaction energy (usually given in J) indicate potential for van der Waals attraction and coagulation.\",\n \"However the relationship is complex and depending on several of the factors seen in the preceding equations including temperature, which affects the kinetic energy of the colloidal particles, surface potential, which might change with factors such as pH as we discussed in Chapter REF and others.\",\n \"The proper analysis must be done in terms of the energy barrier that needs to be overcome for coagulation to occur.\",\n \"The electrolyte concentration at which coagulation occurs is the critical coagulation concentration (ccc) which plays a critical role in grain size analysis of soils, especially in the grain size analysis of oxidic soils which are variable charged and in which the analysis is almost always performed under dispersion using sodium hydroxide or other sodium salts [60].\",\n \"If the constraints of settling velocity, lower limit of settling and dispersing agent concentration are observed, then grain size analysis by settling methods can be performed normally.\",\n \"For the determination of particle size distribution and the proportion of the particle fractions we need some more or less arbitrary scale for classification of grain sizes (Table REF ).\",\n \"For soils, the soil must be first dispersed chemically and or physically using salt solutions such as sodium hydroxide and/or sodium hexametaphoshate at concentrations high enough to create repulsive EDL forces but not above the ccc.\",\n \"Mechanical energy is applied to the samples by stirring, ultrasonic waves, gas bubbles or shaking.\",\n \"Several combinations of chemical and physical methods are possible, some more efficient than others [40].\",\n \"Table: Classification of soil particles based on diameter .After dispersion, the sample is again vigorously stirred and the amount of particles in suspension is measured at prescribed times directly, by collecting a volumetric sample of the dispersion, or indirectly, by using a hydrometer or by transmission/diffraction to electromagnetic radiation methods as in laser and x-ray methods.\",\n \"The most common direct method is the pipette method in which a sample is taken and oven dried and the mass in suspension measured using a precision scale.\",\n \"The mass in suspension is then extrapolated for the entire volume.\",\n \"As for the indirect methods the most commonly used is the hydrometer method.\",\n \"Hydrometers measure the amount or particles in suspension by buoyancy, the greater the mass in suspension the greater the buoyancy force, as particles settle, the hydrometer will tend gradually sink in response to settling of particles until settling stops, when the hydrometer will be reading the amount of particles in suspension under equilibrium, or the density of the dispersing solution if the material is composed only of coarser particles.\",\n \"Most hydrometers are nothing more than a weighed glass tube with a bulbous shape and a scale.\",\n \"The scale is calibrated at the solution level and will read zero for pure water at a given temperature and as the mass of particles in suspension increases, the reading will increase in the downward direction indicating the mass of particles in suspension, usually in grams, per unit volume, usually liters.\",\n \"For any settling based method, the settling velocity is given by Stokes law and the mass of particles greater than a given radius (or diameter) in suspension will depend on the sampling depth and on the calculated settling time.\",\n \"From basic physics the velocity of a particle is given by $v_s = \\\\frac{\\\\Delta S}{\\\\Delta t}$ Because we are concerned with a spherical particle settling under Stokes regime, and therefore at a constant terminal velocity, $v_s$ is the settling velocity, $S$ is the settling depth and $t$ is the time for the particle to reach $S$ .\",\n \"Replacing the velocity in Stokes law $\\\\frac{\\\\Delta S}{\\\\Delta t} = \\\\frac{2}{9} \\\\frac{ (\\\\rho _{p} - \\\\rho _{f}) g}{\\\\mu } R^2$ $\\\\Delta t = \\\\frac{9 \\\\mu \\\\Delta S}{2 (\\\\rho _{p} - \\\\rho _{f}) g R^2}$ and because it is convenient to set $t_0$ and $S_0$ to zero we have $\\\\boxed{t = \\\\frac{9 \\\\mu S}{2 (\\\\rho _{p} - \\\\rho _{f}) g R^2}}$ This equation implies that at time $t$ , all particles with radius greater than $R$ will have settled to a depth greater than $S$ .\",\n \"In addition to the settling depth and the particle radius, the time for a reading for determining particle size distribution using a given method is dependent on the fluid viscosity, the settling mineral particle density, the density of the fluid and gravitation acceleration.\",\n \"Particle density, discussed in Chapter REF , can be measured with relatively simple methods, while fluid viscosity and density will require more specialized high precision techniques.\",\n \"It is much more convenient to perform the measurements at a standard temperature to which water viscosity and density have tabulated values or to use published empirical relationships for water viscosity and density, for example $\\\\mu = 1 \\\\times 10^{-6} [280.68 (\\\\frac{T}{300})^{-1.9} + 511.45 (\\\\frac{T}{300})^{-7.7} + 61.131 (\\\\frac{T}{300})^{-19.6} + 0.45903 (\\\\frac{T}{300})^{-40.0} ]$ in which $\\\\mu $ is water dynamic viscosity in Pa s and T is the temperature in Kelvin in the range of 253.15 K to 383.15 K [51], and $\\\\rho _f = 999.974950 [1 - \\\\frac{(Tc - 3.983035)^2(Tc + 301.797)}{522528.9(Tc + 69.34881)}]$ in which $\\\\rho _f $ is the density of water in kg m$^{-3}$ and Tc is the temperature in $^{o}$ C in the range of 0 $^{o}$ C to 40 $^{o}$ C [94].\",\n \"For example, using the equations above at 25°C = 298.15 K, $\\\\mu = 0.000889997~ Pa~ s$ , $\\\\rho _f = 997.047~ kg~ m^{-3}$ , and using $\\\\rho _s $ and $g$ as in previous calculations, for the silt $D = 0.05~mm$ and clay $D = 0.002~mm$ limits, the time for all the particles with diameter greater than $D$ to settle 0.1 m (10 cm) will be $t_{clay} = 6.86~h$ and $ t_{silt} = 39.53~s $ .\",\n \"Thus, a reading at $t_{silt}$ will indicate the amount of particles finer than 0.05 mm in suspension and the reading at $t_{clay}$ will indicate the amount of particles finer than 0.002 mm in diameter in suspension.\",\n \"The particle size distribution curve is the measured cumulative mass fraction of particles in suspension (usually in %) at a given time plotted as a function of the logarithm of particle diameter corresponding to each time, given by $D = \\\\sqrt{\\\\frac{18 \\\\mu S}{ (\\\\rho _{p} - \\\\rho _{f}) g t}}$ Examples of hypothetical granulometric curves for clay and sandy soils are illustrated in Figure REF Figure: Grain size distribution curves for hypothetical clay and sandy soils.In soil science, and soil physics to some degree, once the mass based fraction of particles in the general classes clay ($D< 0.002~mm$ ), silt ($0.002~mm \\\\le D < 0.05~mm$ ) and sand ($0.05~ mm \\\\le D < 2.00~ mm$ ) is determined, the data can be entered into a ternary plot generally referred to as textural triangle and the nominal soil texture class determined (Figure REF ).\",\n \"The nominal texture classes are a relic from soil pedology and are an attempt to classify soils using the “feeling” method in the field.\",\n \"The feeling method can be useful to discern the general texture classes after extensive training of the field technician.\",\n \"In broader terms, the soil sample is disagregatted, moistened to provide some degree of dispersion by manipulating it using the fingers and rubbed between the index and thumb so that the soil texture is literally felt by the person.\",\n \"It can be useful as an expedite field method for estimating soil texture and for soil classification where laboratory methods might not be available.\",\n \"Figure: Textural triangle for determination of soil texture class.\",\n \"The direction of data entry for each class is indicated by the arrows.Grain size distribution and soil texture are important properties of soils because they will directly or indirectly affect almost all soil physical and chemical phenomena.\",\n \"Aggregation and pore size distribution will affect water transport and storage as well as aeration and susceptibility to compaction.\",\n \"The amount of clay will affect surface area the reactivity of soils as we discussed previously.\",\n \"For a given mineralogy, increasing clay content, will tend to increase plasticity and susceptibility to compaction and compression under certain conditions.\",\n \"ll $ p $ Pressure $ \\\\mu $ Viscosity $ \\\\mathbf {u} $ Vector velocity function $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ \\\\mathbf {i}, \\\\mathbf {j}, \\\\mathbf {k}$ Unit vectors in Cartesian coordinates $ \\\\mathbf {r}, \\\\theta , \\\\phi $ Unit vectors in spherical coordinates $ u_r, u_{\\\\theta }, u_{\\\\phi } $ Components of the velocity vector in spherical coordinates $ r, \\\\theta , phi $ Length and angle components in spherical coordinates $ \\\\Psi $ Stokes stream function $ E^2 $ Differential operator $ a $ Radius $ A, B, C, D $ Constants $ \\\\tau $ Stress components $ F_D $ Drag force $ U $ Velocity $ F_b $ Buoyancy force $ F_g $ Gravitational force $ m $ Mass $ \\\\rho $ Density $ R $ Radius $ v_s $ Settling velocity $ Re $ Reynolds number $ Pe $ Péclet number $ T $ Temperature $ k $ Boltzmann constant $ W_{EDL} $ Energy of interaction due to double layer forces $ \\\\rho _{\\\\infty } $ Electrolyte concentration in bulk solution $ \\\\kappa $ Parameter related to the thickness of the double layer $ z $ Valence of the ion $ e $ Electronic charge $ W_{vdW} $ Attractive van der Waals force $ A_H $ Hamaker constant $ D^{\\\\prime } $ Separation distance $ \\\\rho _1, \\\\rho _2 $ Number densities of interacting spheres $ C $ London constant $ W(D^{\\\\prime }) $ Pair interaction energy $ S $ Settling distance $ T_c $ Temperature in Celsius $ D $ Particle diameter Because soils and sedimentary deposits are composed of solid, unconsolidated particles, mechanical stresses caused by foundations, plant growth, traffic of machines, animals and people, overburden of confining layers, water in pores, clay expansion and contraction and many other natural and man-made processes can cause reorganization of particles and depending on the circumstances, compaction, consolidation or compression or increase in pore volume by processes such as loosening.\",\n \"Before we consider aggregation of soil primary particles and organic matter, it is convenient to treat porous media as an idealized stacking of spheres.\",\n \"This is directly applied to round, homogeneous, coarse grained particles in a few soils and sedimentary deposits and can be an important tool in modeling water retention and transport phenomena, but it is less representative to heterogeneous particle size distributions and as the particle shapes deviate from an ideal sphere.\",\n \"Treating porous media as a set of equally sized spherical particles, the porosity or pore fraction of the material will depend on how the spheres are organized in space.\",\n \"There are more and less efficient ways of packing spheres, a pile of oranges in a supermarket tray is an example of a packing system.\",\n \"The packing models are organized in lattices which can be defined as a repetitive arrangement in space.\",\n \"The lattices seem in this chapter are directly applied to the organization of atoms in space seen in Chapter REF and the structure of minerals is organized in lattices, often following the systems below, but usually with more than one type of chemical element, each element represented by different spheres [44].\",\n \"The systems described below can be applied to soil aggregates to some extent.\",\n \"Soil aggregates are usually less resistant to soil stresses than most primary particles and will be discussed in this chapter.\",\n \"The most common sphere packing arrangements are described below.\",\n \"The packing density is derived from geometric arguments.\",\n \"I. Cubic Lattice In the case of cubic lattice, or simple cubic lattice, the spheres touch and the distance between the spheres centers is two times the radius ($R$ ).\",\n \"The volume of the unit cell considered is the cube that unites the points at the centers of the eight spheres (Figure REF ).\",\n \"Considering the side of the cube as $A = 2R$ , the volume of the unit cell, the volume occupied by spheres and the packing density, given by the volume occupied by the spheres over the unit cell volume are given by Figure: Cubic lattice.$V_{cell} = A^3 = (2R)^3 = 8R^3 \\\\\\\\V_{spheres} = 8 \\\\cdot \\\\frac{1}{8} \\\\cdot \\\\frac{4}{3} \\\\pi R^3 = \\\\frac{4}{3} \\\\pi R^3 \\\\\\\\\\\\eta = \\\\frac{V_{sphere}}{V_{cell}} = \\\\frac{\\\\frac{4}{3} \\\\pi R^3}{8R^3}$ $\\\\eta = \\\\frac{\\\\pi }{6} = 0.52359 ...$ In which $V_{cell}$ is the volume of the unit cell, $V_{sphere}$ is the corresponding volume of spheres located within the unit cell and $\\\\eta $ is dimensionless packing density.\",\n \"Notice that within the unit cell there are eight $1/8$ of spheres such that the total volume of spheres within the unit cell is $8 \\\\cdot 1/8 = 1$ sphere.\",\n \"Similar logic will be used in the following packing schemes.\",\n \"II.\",\n \"Face-Centered Cubic In this arrangement, an additional sphere is located on the center of each face (Figure REF ).\",\n \"Now the side $A$ is no longer $2R$ as additional space is needed between the spheres in the corners to accommodate the spheres on the faces.\",\n \"Notice that the simplified ball-and-stick representation is used, but each corner and face is occupied by a large sphere of radius $R$ .\",\n \"The color red in indicates the corner spheres and the light green indicates the face spheres (Figure REF ).\",\n \"Figure: Face-centered cubic packing.Because of the additional space, the side of the unit cell is given by $A = 2 \\\\sqrt{2} R$ and the volumes and packing density can be calculated as $V_{cell} = A \\\\cdot A \\\\cdot A = (2 \\\\sqrt{2} R)^3 = 16 \\\\sqrt{2} R^3\\\\\\\\V_{spheres} = (8 \\\\cdot \\\\frac{1}{8} + 6 \\\\cdot \\\\frac{1}{2}) \\\\cdot \\\\frac{4}{3} \\\\pi R^3 = \\\\frac{16}{3} \\\\pi R^3 \\\\\\\\\\\\eta = \\\\frac{V_{sphere}}{V_{cell}} = \\\\frac{\\\\frac{16}{3} \\\\pi R^3}{16 \\\\sqrt{2} R^3}$ $\\\\eta = \\\\frac{\\\\pi }{3 \\\\sqrt{2}} = 0.74048 ...$ Notice that the packing density is considerably higher and the porosity, given by $ 1 - \\\\eta $ is considerably lower, around 25%, when compared to the simple cubic lattice.\",\n \"III.\",\n \"Body-Centered Cubic In this case there is one sphere in each corner and one sphere (represented in blue) in the center of the structure (Figure REF ).\",\n \"Notice that the sides $A$ are still equal on the cube, but the diagonals are formed by the radius of the spheres in the transverse corners plus the diameter of the sphere in the center ($2R$ ) totaling $4R$ .\",\n \"From the triangle formed by the height $A$ , the diagonal $4R$ , the diagonals in each face can be calculated from simple geometric arguments resulting in $d = \\\\sqrt{16 R^2 - A^2}$ .\",\n \"Again the relationship between $A$ and $R$ can be calculated from the diagonal of each face and the sides $A$ using simple geometric arguments resulting in $ A = 4R/\\\\sqrt{3}$ .\",\n \"The volumes and packing density can then be calculated as before Figure: Body-centered cubic packing.$V_{cell} = (\\\\frac{4R}{\\\\sqrt{3}})^3 = \\\\frac{64}{3\\\\sqrt{3}} R^3 \\\\\\\\V_{spheres} = (8 \\\\cdot \\\\frac{1}{8} + 1) \\\\cdot \\\\frac{4}{3} \\\\pi R^3 = \\\\frac{8}{3} \\\\pi R^3\\\\\\\\\\\\eta = \\\\frac{V_{sphere}}{V_{cell}} = \\\\frac{\\\\frac{8}{3} \\\\pi R^3}{\\\\frac{64}{3\\\\sqrt{3}} R^3}$ $\\\\eta = \\\\frac{\\\\pi \\\\sqrt{3}}{8} = 0.68017 ...$ IV.\",\n \"Hexagonal Lattice Now on the two hexagonal packing schemes presented here, the geometric polyhedron considered is no longer a cube but an hexagonal prism.\",\n \"Looking at one layer of packed spheres from above it is clear that the center of the spheres can be linked to form an imaginary hexagon (Figure REF ).\",\n \"The height of the hexagonal prism will depend of how the additional layers of spheres are stacked.\",\n \"If the additional layer of spheres is staked directly on top of the first layer, an hexagonal packing lattice is formed (Figure REF ).\",\n \"Notice that because the spheres on the base touch, the base edge, $s$ , is $2R$ and because the next layer is stacked directly above the first layer, the height of the hexagonal prism linking the center of the group of spheres is also $2R$ .\",\n \"Recalling that the volume of an hexagonal prism is given by $ (3 \\\\sqrt{3}/2) s^2 h $ , the volume of the unit cell is the volume of the hexagonal prism and the packing density can be calculated as $V_{cell} = \\\\frac{3}{2} \\\\cdot \\\\sqrt{3} \\\\cdot s^2 \\\\cdot h = \\\\frac{3}{2} \\\\cdot \\\\sqrt{3} \\\\cdot (2R)^2 \\\\cdot (2R) = 12 \\\\sqrt{3} R^3\\\\\\\\V_{spheres} = 3 \\\\cdot \\\\frac{4}{3} \\\\pi R^3 = 4 \\\\pi R^3 \\\\\\\\\\\\eta = \\\\frac{V_{sphere}}{V_{cell}} = \\\\frac{4 \\\\pi R^3}{12 \\\\sqrt{3} R^3}$ $\\\\eta = \\\\frac{\\\\pi }{3 \\\\sqrt{3}} = 0.64599 ...$ Notice that there are 12 spheres in the corners of the hexagonal prism and each one contributes to 1/6th of the volume within the unit cell, plus two half spheres in the upper and lower faces, thus we have $ 12 \\\\cdot 1/6 + 2 \\\\cdot 1/2 = 3$ spheres.\",\n \"Figure: Spheres plane in an hexagonal arrangement.Figure: Hexagonal lattice.V.\",\n \"Hexagonal Close Packing The hexagonal close packing is similar to hexagonal packing but there are three spheres located in between the two staked planes (Figure REF a).\",\n \"It is important to note that these three spheres will also form a plane of spheres when multiple layers are considered in three dimensional space, in addition to the two planes of spheres represented in red (Figure REF a).\",\n \"The calculation of the volume of the unit cell is more complicated than in the hexagonal packing lattice because the distance between the centers of the spheres in the two planes is no longer $2R$ .\",\n \"Consider a tetrahedron formed by the central sphere in the base plus two spheres on the edges forming the base of the tetrahedron and one of the spheres in the middle plane as the upper vertex (Figure REF b).\",\n \"Considering the sides of the base of the tetrahedron as $a$ , the distance to the point of projection of the vertex sphere $b$ can be calculated from the internal angle and the position at the middle point of the side of the base $a/2$ (Figure REF c) Figure: Hexagonal close packing.$\\\\cos {30^{o}} &= \\\\frac{a/2}{b}\\\\\\\\b &= \\\\frac{a}{\\\\sqrt{3}}$ The height $d$ to the plane of spheres in the middle of the unit cell is then given by (Figure REF d) $a^2 &= d^2 + b^2 = d^2 + (\\\\frac{a}{\\\\sqrt{3}})^2 = d^2 + (\\\\frac{a^2}{3})\\\\\\\\d &= a \\\\frac{\\\\sqrt{2}}{\\\\sqrt{3}}$ Because the sides of edge of the base of the hexagonal prism is $a = 2R$ and the total distance between the two planes in red in Figure REF a is $2d = 2 \\\\cdot 2R \\\\frac{\\\\sqrt{2}}{\\\\sqrt{3}}$ , the total height of the hexagonal prism are $h^{\\\\prime }_T = 4R \\\\frac{\\\\sqrt{2}}{\\\\sqrt{3}}$ and the volumes and packing density of the hexagonal close packing lattice are given by $V_{cell} = \\\\frac{3}{2} \\\\cdot \\\\sqrt{3} \\\\cdot (2R)^2 \\\\cdot (4R\\\\frac{\\\\sqrt{2}}{\\\\sqrt{3}}) = 24R^3\\\\sqrt{2} \\\\\\\\V_{spheres} = 6 \\\\cdot \\\\frac{4}{3} \\\\pi R^3 = 8 \\\\pi R^3 \\\\\\\\\\\\eta = \\\\frac{V_{sphere}}{V_{cell}} = \\\\frac{8 \\\\pi R^3}{24R^3\\\\sqrt{2}}$ $\\\\eta = \\\\frac{\\\\pi }{3 \\\\sqrt{2}} = 0.74048 ...$ Notice that the packing density is the same as the Face Centered Cubic.\",\n \"In essence the difference between these two systems is geometrical, as in the way the layers of spheres are conceptualized and organized.\",\n \"One system can be transformed into the other by rotation of coordinates.\",\n \"In soil science, aggregation refers to the union of primaryThe meaning here is not as in primary minerals but in the basic building blocks of soil aggregates mineral particles clay, silt and sand with organic compounds and various amorphous materials forming clusters with shapes and sizes that are more or less constant in certain naturally occurring soil types and depths, in response to soils forming processes.\",\n \"Soil structure generally refers to the shape, size and structural stability of these aggregates and larger domains formed by the set of aggregates in a given soil.\",\n \"In natural soils, structure is a direct product of soil forming processes, namely climate, organisms, relief, parent material and time.\",\n \"In fine textured soils, close to surface, the structure tends to be composed of larger aggregates and with higher porosity, especially in humid tropical climates where there is higher biological activity of plants, animals and other organisms.\",\n \"Aggregates in natural soils also tend to be more stable than in areas disturbed by human activity.\",\n \"Human activity, especially agriculture, tends to disturb soil structure, destroying large size aggregates due to tillage and chemical modification on soil upper layers.\",\n \"The destruction of aggregates and soil structure is usually a physical and a chemical process.\",\n \"Traffic of animals, humans and machinery tends to compress and break soil aggregates, causing increases in bulk density and decrease in porosity, creating an environment that is less adequate to biological activity which further causes reduction in creation of new aggregates and pores, and loss of organic matter.\",\n \"Agricultural practices also employ a variety of tillage implements which in some cases are directly designed to destroy soil structure for sowing operations, while others tend to destroy soil structure indirectly.\",\n \"Chemically, the input of ionic chemicals such as fertilizers and other contaminants can cause dispersion by the surface chemistry mechanisms discussed in Chapter REF which decreases soil structural stability on one end while on the other some compounds can also kill soil organisms, contributing to loss of organic matter and biological activity, further increasing soil structural degradation.\",\n \"We will not discuss tillage systems and soil quality in this book.\",\n \"There are a lot of sociological, ecological, philosophical and political issues related to soil quality, tillage and agriculture that rely on personal opinion of different groups none of which we have any interest in supporting.\",\n \"Water retention due to capillary phenomena tends to be controlled by the pore size distribution and pore shape in soils and these are to a larger extent controlled by the size and shape of soil aggregates.\",\n \"In most soils aggregation creates at least two pore domains, the intra-aggregate pores and inter-aggregate pores, intra-aggregate pores tends to be larger and usually hold water by capillary phenomena at lower energies of retention when compared to inter-aggregates smaller pores.\",\n \"In soils with high amounts of clay, these inter-aggregate pores tend to be very small and as such, water retention in them can have contribution of both capillary and adsorption phenomena.\",\n \"Water retention can be modeled in terms of aggregation and pore size distribution in self-similar and fractal models for example [77].\",\n \"Formation of aggregates is a slow and complex process dependent on several factors.\",\n \"There are several theories of soil aggregate formation (e.g.\",\n \"[95]), but overall it is thought that aggregation in most soils is dependent on biological activity and organic matter which creates physical and chemical cementing agents contributing to uniting primary particles in larger structural domains.\",\n \"Biological activity can directly contribute to formation of aggregates as some organisms such as termites and earthworms produce aggregate like structures by ingesting or manipulating soil organic matter and mineral particles.\",\n \"From the chemical point of view, charged surfaces in soil particles, colloidal organic matter, and ions can create aggregating and dispersing forces during aggregate formation.\",\n \"Higher valence cations tend to favor aggregation while monovalent high hydrated radius such as sodium tend to favor dispersionAs we seen, this depends also on the concentration in solution.\",\n \"Physically, stresses caused by roots and organisms and mechanical stresses caused by natural processes such as soil drying, expansion and shrinking of soil minerals due to loss of interlayer water as well as capillary forces related to drying can also contribute to aggregation of primary particles.\",\n \"Soil structure, shapes, sizes and stability of aggregates, total porosity and pore size distribution are all affected by stresses applied to soils by human activities.\",\n \"The modifications caused by these activities will in turn affect soil functioning.\",\n \"In many engineering applications such as in the construction of earth dams and foundations, the increase in bulk density and reduction of porosity to a minimum are usually desired while in agriculture, where the soil is viewed as a medium to plant growth, porosity, pore size and distribution must be at an ideal state for root growth, water storage, water and air circulation and biological activity.\",\n \"Much of the theory behind soil porosity reduction and increase in bulk density in response to applied stresses comes form geotechnical engineering, specific from the discipline of Soil Mechanics.\",\n \"In soil mechanics, compaction is the increase in density of a soil by reduction of the pore volume occupied by air, consolidation is a reduction in volume of a fully saturated, low permeability soil due to drainage of soil water on the pores, and compression is a reduction of volume of a soil under compressive stress [29], [33].\",\n \"Although we have not discussed soil permeability to water, it can be understood as a coefficient which measured how easy it is for water to flow out of the soil under a pressure gradient.\",\n \"On the geotechnical engineering literature, consolidation applies to low permeability soils, such as soils with high phylossilicate clays content, such that there is a time lag between the application of the load and the extrusion of water from pores and soil settlement [33].\",\n \"We have studied the concept of stress tensor applied to fluids in porous media in Chapter REF and now we need to apply the same concept to stresses in a solid (Figure REF ).\",\n \"Imagining a volume of soil shaped like a cube, forces can be applied to each surface can be parallel or perpendicular to each face.\",\n \"The force component divided by the area over which it is acting acting defines the stress $\\\\sigma $ (Figure REF ).\",\n \"As with the stress tensor in a fluid, the stress components in each direction, $x$ , $y$ and $z$ can be further decomposed into three components, one for each original direction.\",\n \"Therefore, the stress components acting on the face perpendicular to the direction $x$ are $\\\\sigma _{xx}$ , $\\\\sigma _{xy}$ and $\\\\sigma _{xz}$ and so forth for the other three directions.\",\n \"The full stress tensor is, as before $\\\\sigma _{ij} =\\\\begin{bmatrix}\\\\sigma _{xx} & \\\\sigma _{xy} & \\\\sigma _{xz} \\\\\\\\\\\\sigma _{yx} & \\\\sigma _{yy} & \\\\sigma _{yz} \\\\\\\\\\\\sigma _{zx} & \\\\sigma _{zy} & \\\\sigma _{zz} \\\\\\\\\\\\end{bmatrix}$ The perpendicular (or normal) stresses $ \\\\sigma _{ii} = \\\\sigma _{xx} = \\\\sigma _{yy} = \\\\sigma _{zz} $ are equivalent to pressures, or force over area ($F/A$ ) acting over each surfaces while the stresses parallel to each face $ \\\\sigma _{ij}, ~ i \\\\ne j$ are called the shearing stresses, also $F/A$ , in units analogous to pressure units.\",\n \"Stresses applied by foundations, traffic of animals and machinery, tillage and other agricultural operations are usually multi-directional and have normal and shearing components, especially in non-rigid, granular porous materials such as soils and sedimentary deposits.\",\n \"In some cases the stresses can be modeled as unidimensional pressures as in modeling compaction by agricultural machinery and animal traffic.\",\n \"Figure: Components of stress in a tridimensional solid.If we imagine a foundation, a tire or an animal foot applying a vertical force to a soil, the response of the soil will depend on the magnitude of the applied force and on the mechanical strength of the soil.\",\n \"The mechanical strength of the soil will depend directly and indirectly on a series of factors such as soil structure, grain and aggregate strength, grain size distribution, porosity, water content, mineralogy, story of previous applied loads and many other internal mechanisms.\",\n \"If the applied stress is greater than the mechanical strength of the soil, deformation will occur.\",\n \"Considering only unidimensional compressive stresses on a soil, deformation will correspond to a decrease in porosity.\",\n \"There are different mechanisms and regimes of deformation in soils, a soil can suffer elastic deformation when the soil returns to its original state after the force is removed or plastic deformation when the soils does not returns to its original state after the force is removed.\",\n \"On a macro scale, in soils subjected to unequal stresses in multiple directions, rupture or compressive failure can also occur, although these concepts are more applied to rigid solids.\",\n \"However, internally, rupture of aggregates due to compressive stresses is an important mechanism of soil compaction, even under uniaxial loads.\",\n \"As larger aggregates break the overall strength of the soil decreases and the volume of pores also decreases, as the structural units are destroyed and the pores filled with smaller aggregates or loose particles.\",\n \"Dry soils are more subjected to elastic deformation and rupture while wet soils tend to become plastic above a given water content depending on specific characteristics of the soil, like grain size distribution and mineralogy.\",\n \"Compaction in dry soils can be caused by reorganization of aggregates and individual particles, as seen in the packing models at the beginning of this chapter, rupture of aggregates and in some cases of individual mineral particles under load, and elastic deformation of aggregates and individual particles (usually sand and coarser fractions) under elevated loads.\",\n \"Usually aggregates in dry soil tend to be more brittle, i.e.\",\n \"they can fracture with little deformation, while as the water content increases, water molecules act as lubricants within particles and aggregates, contributing to plastic deformation.\",\n \"The set of critical water contents at which a fine soil transition to different consistency states is called Atterberg limits.\",\n \"The Atterberg limits are used in engineering and in agricultural sciences to predict soil behavior in the field for construction, traffic and tillage.\",\n \"In simple terms the principal soil conceptual consistency limits considered in soil physics and soil mechanics are [33] Shrinkage limit: Most fine texture soils, especially soils rich in smectites, shrink as the soil dries.\",\n \"This shrinking can be thought to be related to two main mechanisms, first in phylossilicate clays, water is lost from the interlayer planes causing shrinkage at the mineral structure level which reflects on the macrostructure, second, drying tends to cause internal stresses between particles and aggregates due to forces related to meniscus in pore water water.\",\n \"These forces can pull the particles together as the soil dries and the radius of the meniscus becomes smaller increasing the capillary forces.\",\n \"The shrinkage limit is the water content below which decrease in water content will not cause further volume reduction of the soil (i.e.\",\n \"shrinkage).\",\n \"A soil in a water content below the shrinkage limit is said to be in a solid state and it is said to be in a semisolid state at water contents above the shrinkage limit and below the plastic limit.\",\n \"Plastic limit: The water content below which the soil ceases to display plastic behavior.\",\n \"In practical terms the soil will start to crumble or fracture if manipulated when below the plastic limit.\",\n \"Above the plastic limit the soil is subjected to plastic deformation and is in a plastic state.\",\n \"Liquid limit: It is the water content above which the soil changes from a plastic state to a liquid sate.\",\n \"In practical terms, above the liquid limit the water content is such that the soil will begin to flow similar to a liquid under stress.\",\n \"Shrinkage limit: Most fine texture soils, especially soils rich in smectites, shrink as the soil dries.\",\n \"This shrinking can be thought to be related to two main mechanisms, first in phylossilicate clays, water is lost from the interlayer planes causing shrinkage at the mineral structure level which reflects on the macrostructure, second, drying tends to cause internal stresses between particles and aggregates due to forces related to meniscus in pore water water.\",\n \"These forces can pull the particles together as the soil dries and the radius of the meniscus becomes smaller increasing the capillary forces.\",\n \"The shrinkage limit is the water content below which decrease in water content will not cause further volume reduction of the soil (i.e.\",\n \"shrinkage).\",\n \"A soil in a water content below the shrinkage limit is said to be in a solid state and it is said to be in a semisolid state at water contents above the shrinkage limit and below the plastic limit.\",\n \"Plastic limit: The water content below which the soil ceases to display plastic behavior.\",\n \"In practical terms the soil will start to crumble or fracture if manipulated when below the plastic limit.\",\n \"Above the plastic limit the soil is subjected to plastic deformation and is in a plastic state.\",\n \"Liquid limit: It is the water content above which the soil changes from a plastic state to a liquid sate.\",\n \"In practical terms, above the liquid limit the water content is such that the soil will begin to flow similar to a liquid under stress.\",\n \"Atterberg limits are strongly dependent on mineralogy and soil texture.\",\n \"The main goal in agricultural soils is that traffic and other agricultural operations are to be performed below the plastic limit.\",\n \"Plastic deformations are related to structural degradation and compaction that is usually expensive and labor intensive to recover.\",\n \"Above the liquid limit machinery can sink and become stuck in some conditions.\",\n \"In practice however, as the soil dries the mechanical resistance can become such that it is impossible or not economically viable to perform tillage or sowing operations.\",\n \"Thus, the limits for trafficability and workability in soils can be narrow, either due to excess water content, turning the soil plastic and subjected to compaction or consolidation, or due to water content that is too low for efficient tillage operations.\",\n \"Soils that are too plastic, such as those rich in smectites, are often too sticky, requiring high amounts of energy for operations or even being impossible to till under certain conditions In many practical applications, either agricultural, military or geotechnical engineering, a simple and easy approach to estimate soil mechanical resistance is the use of cone penetrometers.\",\n \"The principle is very simple, a rod with a cone at its end is inserted into the soil and the mechanical resistance in terms of stress or force required is measured.\",\n \"Devices based on this principle are called penetrometers.\",\n \"There are many different designs of penetrometers, either analogical or with digital sensors, for measuring force and data collecting.\",\n \"Soil mechanical resistance measured by a penetrometer can give an idea of the mechanical resistance of the soil to withstand loads and can be used to estimate trafficability.\",\n \"In agricultural sciences, the resistance to penetration or cone index is used as a proxy to the mechanical resistance encountered by plant roots.\",\n \"The mechanisms of root growth are complex and multiple and it is wrong to attribute critical limits of soil resistance to root growth based on penetrometer readings.\",\n \"In practical terms, a penetrometer can be used as one parameters, among many, for assessing soil compaction and decision making in terms of compaction remediation measures.\",\n \"Penetrometer resistance measurements should never be used without checking soil water content and bulk density as it is an exponential function of these two soil variables (Figure REF ).\",\n \"An empirical relationship between soil penetration resistance, bulk density and water content usually encountered on the literature is [62] $SR = a_0 \\\\rho ^{a_1} \\\\theta ^{a_2}$ In which $SR$ is the soil mechanical resistance, usually in stress (analogous to pressure) units, $\\\\rho $ is soil bulk density, $\\\\theta $ is the volumetric water content, and $a0$ , $a1$ and $a2$ are empirical parameters commonly fitted using regression techniques such as least squares regression.\",\n \"This equation has useful practical applications for adjusting $SR$ to a given water content and bulk density so that comparisons between different areas and conditions can be made, and also has pedagogical applications, illustrating the exponential relationship between $SR$ and $\\\\theta $ and $\\\\rho $ .\",\n \"However, you must be aware that it is an empirical relationship and has no physical basis and was not derived based on first principles and thus does not explain the physical basis of the relationship between resistance, density and water content.\",\n \"As such, it cannot be extrapolated to different soils, or to conditions not observed on the fitting data.\",\n \"First principles derivations of the $SR = f(\\\\rho , \\\\theta )$ relationship should be possible accounting for internal stresses of the soil provided by cohesive forces between particles and within aggregates, and adhesive forces generated by water in contact with solid surfaces.\",\n \"These forces will also depend on soil texture, mineralogy, organic matter content among other soil properties.\",\n \"Figure: Soil penetration resistance as a function of bulk density and volumetric water content based on empirical data.\",\n \"The parameters of Equation 6.8 used are within the ranges for real soils, a 0 =0.01a_{0} = 0.01, a 1 =5.0a_{1} = 5.0 and a 2 =-1.0a_{2} = -1.0.ll $ V_{cell} $ Volume of the unit cell $ V_{spheres} $ Volume of spheres inside the unit cell $ A $ Side length in cubic unit cells $ R $ Radius of spheres $ \\\\eta $ Packing density $ s $ Side length in an hexagonal prism $ h $ Height of the hexagonal prism $ a, b, c $ Parameters for calculation in an hexagonal close packing system $ h^{\\\\prime }_T $ Height of the unit cell in an hexagonal close packing system $ \\\\sigma _{ij} $ Stress tensor $ SR $ Soil penetration resistance $ \\\\rho $ Soil bulk density $ \\\\theta $ Volumetric water content $ a_0, a_1, a_2 $ Empirical parameters on the soil penetration resistance function On the bulk of a pure, static liquid in thermal equilibrium, all molecules are subjected to the same forces, they interact with their neighbors through intermolecular surface forces which, when averaged over time, are constant in all directions and thus, the resultant of these forces is zero.\",\n \"This is not true when there is a discontinuity between the fluid and a different fluid, liquid, solid or gas.\",\n \"On an interface with a gas, the balance of forces is altered such that a resulting force on the direction of the interior of the fluid appears.\",\n \"Because the molecules at the interface are surrounded by other liquid molecules at the liquid end, but not at the gas end, resulting cohesion forces pulling the molecules in direction of the liquid appear.\",\n \"Intermolecular forces also generate the adhesion and adsorption forces between the liquid and solid surfaces.\",\n \"The surface tension at the interface can be conceptualized using the thermodynamics or the mechanics frameworks.\",\n \"The mechanics framework states that, because of these interfacial forces, the external work at constant volume and temperature, $\\\\delta W$ , required to increase the surface area by an infinitesimal amount $dA$ is directly proportional to the increase in area with the proportionality coefficient being called surface tensionThe symbol $\\\\delta $ with respect to W instead of d is because of thermodynamic considerations, $d\\\\hspace*{-0.80002pt}\\\\bar{}\\\\hspace*{1.00006pt}$ is also often used.\",\n \"Heat and work in thermodynamics are imperfect differentials.\",\n \"See Callen (2005).\",\n \"$\\\\delta W = \\\\gamma dA$ In which if the energy $W$ is in $J$ and the area in $m^2$ , and the surface tension coefficient needs to have units of $J~m^{-2}$ , which is consistent with the definition of energy per unit surface area.\",\n \"The surface tension is very often conceptualized using the wire and soap bar analogy and experiment.\",\n \"Imagine that you have a square wire with one of the sides movable, now imagine a soap bubble is created within the square, if the square has a fixed dimension $l$ and a movable dimension $x$ , the area can be increased by increasing or decreasing the length of $x$ , such that $dA = ldx$ this implies that $\\\\delta W = \\\\gamma l dx$ In this case, the surface tension coefficient is usually conceptualized as a force per unit length in $N~m^{-1}$ .\",\n \"Note that $1~J = 1~N~m$ such that $1~N~m^{-1}$ = $1~J~m^{-2}$ Now lets consider the liquid, the gas and the interface as a thermodynamical system.\",\n \"Because the variables of interest in this system are temperature and volume, the thermodynamic potential of interest here is the Helmholtz free energy, $\\\\mathcal {F}$ [24], [75] $d\\\\mathcal {F} = -SdT - pdV + \\\\mu _i dN_i$ Since the temperature and number of moles are kept constant, $dT$ and $dN_i$ are zero, and the equation reduces to $d\\\\mathcal {F} = - pdV$ Since we know from thermodynamics that $\\\\delta W = - pdV$ we have, for an interface $d\\\\mathcal {F} = d\\\\mathcal {F}_1 + d\\\\mathcal {F}_2 + d\\\\mathcal {F}_3 = - \\\\delta W_1 - \\\\delta W_2 - \\\\delta W_3$ in which 1 and 2 refer to the solid and liquid phases, respectively, and 3 to the interface phase.\",\n \"Considering that Equation 7.2 is known experimentally, and that the work in that case is negative because it is carried by the interface instead of by an external agent we have, after some reorganizing $- \\\\delta W_1 - \\\\delta W_2 - \\\\delta W_3 = - p_1dV_1 - p_2dV_2 + \\\\gamma dA \\\\\\\\- p_1dV_1 - p_2dV_2 + d F_3 = - p_1dV_1 - p_2dV_2 + \\\\gamma dA \\\\\\\\d \\\\mathcal {F}_3 = \\\\gamma dA \\\\\\\\\\\\gamma = \\\\frac{d \\\\mathcal {F}_3}{dA}$ Rewriting in a more elegant symbology $\\\\boxed{\\\\gamma = \\\\bigg (\\\\frac{\\\\partial \\\\mathcal {F}_{int}}{\\\\partial A}\\\\bigg )_T}$ The surface tension coefficient is the change in Helmholtz free energy of the interface ($\\\\mathcal {F}_{int}$ ) per unit area at constant temperature [75].\",\n \"Notice that we have not yet discussed the concept of thermodynamic potential because it would take us to a long detour which is not strictly necessary at this point.\",\n \"This long detour will be necessary when we discuss the concept of soil water potentials later in this chapter.\",\n \"Capillary theory is based on developments by Thomas YoungYoung, T. 1805.\",\n \"An essay on the cohesion of fluids.\",\n \"Transactions of the Royal Society of London.\",\n \"95:65-87. who laid the descriptive foundations of the process, Pierre LaplaceLaplace, M. 1805.\",\n \"Traité de mécanique céleste.\",\n \"Tome Quatrième.\",\n \"Courcier, Paris.\",\n \"who provided a rigorous initial mathematization and Carl Fredrich GaussGavss, C.F.\",\n \"1830.\",\n \"Principia generalia theoriae figvrae flvidorvm in statv aeqvilibrii.\",\n \"Gottingae.\",\n \"who unified both descriptions in terms of the virtual work principle, based on the principle of least action.\",\n \"The works are based on a rigorous mathematical description and occupy several pages of works in French for Laplace and Latin for Gauss.\",\n \"Like most of what is now known as the core of soil physics, simplifications and ad-hoc derivations came later in order to provide more palatable descriptions.\",\n \"The Young-Laplace equation is a differential equation of the form $\\\\Delta p = \\\\gamma \\\\nabla \\\\cdot \\\\mathbf {n}$ in which $\\\\nabla \\\\cdot \\\\mathbf {n}$ represents the mean curvature and can be calculated using differential geometry for different shapes of the interfacial surface.\",\n \"For reasons of simplicity and elegance we will present here the derivations of Landau and Lifshitz [57] followed by that of Defay and Prigogine [34].\",\n \"Suppose you have a curved surface with two principal radius of curvature $R_1$ and $R_2$ on an interface between two media.\",\n \"A limiting case would be a sphere in which $R_1 \\\\equiv R_2$ .\",\n \"An element of surface area $dA$ is defined in the interface between the two mediums (Figure REF a).\",\n \"The area element can be calculated as the product of two elements of circumference of a circle, defined in terms of the two radii $R_1$ and $R_2$ as $dl_1$ and $dl_2$ .\",\n \"Suppose now that the surface area is increased by displacing the surface by an amount $\\\\delta \\\\zeta $ in the direction of the normal of the surface $dA$ (Figure REF ).\",\n \"The two elements of circumference are now $dl_1^{\\\\prime }$ and $dl_2^{\\\\prime }$ and the surface area is increased from $dA = dl_1dl_2$ to $dA^{\\\\prime } = dl_1^{\\\\prime }dl_2^{\\\\prime }$ .\",\n \"An element of volume between the surfaces is $dV = \\\\delta \\\\zeta dA$ .\",\n \"If the pressures on the two media are $p_1$ and $p_2$ , the work necessary for the volume increase is related to the pressure difference between the two surfaces by $\\\\delta W_v = \\\\int p dV = \\\\int (-p_1 + p_2) dV = \\\\int (-p_1 + p_2) \\\\delta \\\\zeta dA$ and the total work is the work necessary for the volume increase added to the work necessary for surface expansion, as described on the previous section, thus $\\\\delta W = \\\\delta W_v + \\\\delta W_s = \\\\int p dV + \\\\gamma \\\\delta A = -\\\\int (p_1 - p_2) \\\\delta \\\\zeta dA + \\\\gamma \\\\delta A$ in which $\\\\delta A$ is a change in the area of surface.\",\n \"As we have two areas $dA$ and $dA^{\\\\prime }$ , the change in surface area after displacement, $\\\\delta A$ , is needed.\",\n \"Suppose that the length element $dl_1$ is increased to $dl_1^{\\\\prime }$ after the $\\\\delta \\\\zeta $ displacement in the direction of the normal.\",\n \"Considering an element of angle $\\\\theta $ , and using the circumference of a circle formula (Figure REF b) we have $dl_1 = \\\\theta R_1$ and $dl_1^{\\\\prime } = \\\\theta (R_1 + \\\\delta \\\\zeta )$ Isolating $\\\\theta $ in the first equation and replacing into the second we have $dl_1^{\\\\prime } = \\\\frac{dl_1}{R_1} (R_1 + \\\\delta \\\\zeta ) = (1 + \\\\frac{\\\\delta \\\\zeta }{R_1}) dl_1$ Using the same argument in $dl_2$ $dl_2^{\\\\prime } = \\\\frac{dl_2}{R_2} (R_2 + \\\\delta \\\\zeta ) = (1 + \\\\frac{\\\\delta \\\\zeta }{R_2}) dl_2$ Now $dA^{\\\\prime }$ can be calculated in terms of $dl_1$ and $dl_2$ $dA^{\\\\prime } & = dl_1^{\\\\prime }dl_2^{\\\\prime } \\\\\\\\& = (1 + \\\\frac{\\\\delta \\\\zeta }{R_1}) dl_1 (1 + \\\\frac{\\\\delta \\\\zeta }{R_2}) dl_2 \\\\\\\\& = dl_1dl_2 + dl_1dl_2 \\\\frac{\\\\delta \\\\zeta }{R_2} + dl_1dl_2 \\\\frac{\\\\delta \\\\zeta }{R_1} +dl_1dl_2 \\\\frac{\\\\delta \\\\zeta }{R_1} \\\\frac{\\\\delta \\\\zeta }{R_2}$ Neglecting the much smaller second order term $\\\\delta \\\\zeta \\\\delta \\\\zeta = \\\\delta \\\\zeta ^2$ $dA^{\\\\prime } = (1 + \\\\frac{\\\\delta \\\\zeta }{R_1} + \\\\frac{\\\\delta \\\\zeta }{R_2}) dl_1dl_2 \\\\\\\\$ Now the element change in area as the surface is displaced is $da & = dA^{\\\\prime } - dA \\\\\\\\& = (1 + \\\\frac{\\\\delta \\\\zeta }{R_1} + \\\\frac{\\\\delta \\\\zeta }{R_2}) dl_1dl_2 - dl_1dl_2 \\\\\\\\& = (\\\\frac{\\\\delta \\\\zeta }{R_1} + \\\\frac{\\\\delta \\\\zeta }{R_2}) dl_1dl_2$ The total change in area of the surface of separation is $\\\\delta A = \\\\int da = \\\\int (\\\\frac{1}{R_1} + \\\\frac{1}{R_2}) \\\\delta \\\\zeta dA$ and replacing into Equation 7.10 $\\\\delta W = -\\\\int (p_1 - p_2) \\\\delta \\\\zeta dA + \\\\gamma \\\\int (\\\\frac{1}{R_1} + \\\\frac{1}{R_2}) \\\\delta \\\\zeta dA$ Because the displacement is required to occur in thermodynamic equilibrium, $\\\\delta W = 0$ and we can write $\\\\int [ (p_1 - p_2) - \\\\gamma (\\\\frac{1}{R_1} + \\\\frac{1}{R_2}) ] \\\\delta \\\\zeta dA = 0$ In which, as in other cases seen before, the integral vanishes identically, thus $\\\\boxed{\\\\Delta p = p_1 - p_2 = \\\\gamma (\\\\frac{1}{R_1} + \\\\frac{1}{R_2})}$ This is the Young-Laplace equation for a curved interface with constant radii $R_1$ and $R_2$ between two media.\",\n \"For a sphere in which $R \\\\equiv R_1 \\\\equiv R_2$ it reduces to $\\\\boxed{\\\\Delta p = p_1 - p_2 = \\\\frac{2 \\\\gamma }{R}}$ Figure: Schematics for the energy derivation of the Young-Laplace equation.The derivation from Defay and Prigogine [34] is simpler and relies on mechanical and geometrical arguments.\",\n \"Consider a non-spherical cap with principal radii of curvature $R_1$ and $R_2$ (Figure REF ).\",\n \"Because of surface tension, a line element $\\\\delta l$ at A is subjected to a force $F_A = \\\\gamma \\\\delta l$ .\",\n \"The projection of $F_A$ on the normal P-N is then $\\\\sin {\\\\phi } = \\\\frac{proj_{F_A}}{\\\\gamma \\\\delta l} \\\\\\\\proj_{F_A} = \\\\gamma \\\\delta l \\\\sin {\\\\phi }$ Using the small angle approximation on the triangle A-P-N it is possible to define $\\\\sin {\\\\phi } \\\\approx \\\\tan {\\\\phi } \\\\approx \\\\phi \\\\approx \\\\frac{r}{R_1}$ and $proj_{F_A} = \\\\gamma \\\\frac{r}{R_1} \\\\delta l$ The same argument for points B, C and D results in $proj_{F_B} = \\\\gamma \\\\frac{r}{R_1} \\\\delta l$ $proj_{F_C} = \\\\gamma \\\\frac{r}{R_2} \\\\delta l$ $proj_{F_D} = \\\\gamma \\\\frac{r}{R_2} \\\\delta l$ The resulting force on the N direction on the cap is the sum of the force components of each point in the N direction $F = proj_{F_A} + proj_{F_B} + proj_{F_C} + proj_{F_D} = \\\\gamma \\\\frac{r}{R_1} \\\\delta l + \\\\gamma \\\\frac{r}{R_1} \\\\delta l + \\\\gamma \\\\frac{r}{R_2} \\\\delta l + \\\\gamma \\\\frac{r}{R_2} \\\\delta l$ $F = 2 r \\\\gamma (\\\\frac{1}{R_1} + \\\\frac{1}{R_2}) \\\\delta l$ Figure: Schematics for the mechanics derivation of the Young-Laplace equation (Adapted from ).The total force around the border of the cap can be calculated by integrating in $\\\\delta l$ .\",\n \"Since there are four quadrants and the perimeter is treated as that of a circle, the integration is made in the 0 to $\\\\pi /2$ interval.\",\n \"The element of length needs to be transformed into an element of angle using the arc length formula $\\\\delta l = r \\\\delta \\\\theta $ .\",\n \"$F_{Do} = \\\\int _0^{\\\\frac{\\\\pi }{2}} 2r\\\\gamma (\\\\frac{1}{R_1} + \\\\frac{1}{R_2}) r \\\\delta \\\\theta = \\\\gamma \\\\pi r^2 (\\\\frac{1}{R_1} + \\\\frac{1}{R_2})$ For the cap to be in equilibrium, the force downward caused by surface tension has to be in equilibrium with the upward force which in this case is caused by the pressure gradient between the two sides of the interface $F_U = (p_1 - p_2) \\\\pi r^2$ Note that the pressure difference is force over area and $\\\\pi r^2$ is an area such that we have the resulting force.\",\n \"On equilibrium $F_U = F_{Do}$ $F_U = F_{Do} = (p_1 - p_2) \\\\pi r^2 = \\\\gamma \\\\pi r^2 (\\\\frac{1}{R_1} + \\\\frac{1}{R_2})$ And we again arrive at the Young-Laplace equation $p_1 - p_2 = \\\\gamma (\\\\frac{1}{R_1} + \\\\frac{1}{R_2})$ The Young-Dupré equation defines the equilibrium condition, more specifically the contact angle, when three homogeneous phases are placed in contact and are in thermodynamic equilibrium.\",\n \"The condition applies to an interface solid-liquid-gas where the shape of a liquid drop is defined by the balance of the surface tensions on the three interfaces, liquid-solid, liquid-gas and gas solid.\",\n \"In general liquids in contact with solids can behave as wetting or nonwetting phases.\",\n \"If the fluid behaves as a wetting phase it will tend to maximize the surface area of contact with the solid (Figure REF a) while nonwetting phases will tend to minimize the contact area with the solid (Figure REF b).\",\n \"Water over a hydrophilic surface is an example of a wetting phase while it is a nonwetting phase when in contact with hydrophobic surfaces such as surfaces covered with wax.\",\n \"The simplest derivation of the force balance is based on the force equilibrium on the $x$ plane (Figure REF ).\",\n \"For a wetting phase, for example, the surface tension on the direction of the solid-gas interface $\\\\gamma _{13}$ is balanced by summation of the forces on the opposite direction, the surface tension on the solid-liquid interface $\\\\gamma _{12}$ and the $x$ component of the surface tension on the liquid-gas interface, $\\\\gamma _{23} \\\\cos {\\\\theta }$ $\\\\boxed{\\\\gamma _{13} = \\\\gamma _{12} + \\\\gamma _{23} \\\\cos {\\\\theta }}$ $\\\\theta $ in this case is the contact angle between the liquid-gas and solid-liquid interfaces.\",\n \"Figure: Angle of contact on an interface solid (1) - liquid (2) - gas (3).\",\n \"(a) Is a wetting liquid θ<90 o \\\\theta < 90 ^{o} and (b) is a nonwetting liquid θ>90 o \\\\theta > 90 ^{o}.The physical interpretation of the capillary rise phenomenon is a controversial issue.\",\n \"The force balance interpretation given in many textbooks is criticized in terms of accuracy, alternative mechanical and thermodynamic interpretations being preferred.\",\n \"The force balance interpretations due to pressure difference will be presented here due to its simplicity.\",\n \"Consider water rising to the level h in a fine glass tube (Figure REF ), the pressures at points A and C are equal to atmospheric pressure being exerted at the surface of the liquid, $P_A = P_C = P_{atm}$ , and following Pascal's principle, $P_C = P_D = P_{atm}$ .\",\n \"Capillary rise and the Young-Laplace equation indicate that there is a pressure difference between the two sides of the curved surface represented by the meniscus between A and B.\",\n \"Because there is capillary rise, the pressure is lower below the surface of the meniscus, in B, than above the meniscus, in A, such that $P_B < P_A$ .\",\n \"The capillary rise also indicates that the pressure in D is greater than in B, such that a pressure gradient exists producing an upwards force.\",\n \"The condition of equilibrium in terms of force is $(P_D - P_B - \\\\rho g h) \\\\pi r^2 = 0$ In other words, the upwards force $P_D \\\\pi r^2 $ = force/area $\\\\cdot $ area = force, is balanced by the weight of the water column $\\\\rho g h \\\\pi r^2$ = P $\\\\cdot $ area = force/area $\\\\cdot $ area = force, and the force exerted at B.\",\n \"Because of the equilibrium condition, $P_D - P_B = P_A - P_B $ and $P_A - P_B = \\\\rho g h$ substituting the Young-Laplace equation for the pressure gradient on the interface $\\\\frac{2 \\\\gamma }{R} = \\\\rho g h$ This equation can be written in terms of the radius of the capillary tube instead of the radius of the spherical meniscus by considering the geometrical relation in Figure REF $R = \\\\frac{r}{\\\\cos {\\\\theta }}$ The capillary rise equation results from replacing $R$ into the Young-Laplace equation $\\\\boxed{h = \\\\frac{2 \\\\gamma \\\\cos {\\\\theta }}{\\\\rho g r}}$ The rise considered is up to the lowest point of the meniscus, the volume of water above that point being neglected from the weight of the column because it is usually much less than the weight of the rest of the column.\",\n \"Figure: Capillary rise.We will leave capillary rise and curved interfaces aside for a little while in order to discuss the theory of soil water potentials.\",\n \"Potential theory of soil water is nothing more than an application of one of the most fundamental principles in physics - and in nature for that matter -, the principle of conservation of energy, to soil systems.\",\n \"We already discussed several phenomena in terms of energy throughout the book, now we need to formalize how energy plays a role in transport and storage of water in porous media.\",\n \"Energy is a fundamental concept on the ontology of physics and it is often easier to be understood intuitively than to be given an actual definition.\",\n \"The concept of energy in classical mechanics is a good starting point for an intuitive discussion.\",\n \"We all know that a body will fall under the force of gravity if it is released close to Earth's surface.\",\n \"The nature of gravitational force which accelerates the body in direction of the center of Earth is complex and the most accepted explanation is given in terms of Einstein's general relativity and curvature of space-time.\",\n \"The theory is beyond the scope of this book and for the sake of simplicity we can accept that such force exists as has been experimentally verified.\",\n \"One of the most important ideas in the concept of energy is that it can only be defined in relation some reference state or condition.\",\n \"When we lift a body 10 m into the air close to Earth's surface we are defining Earth's surface itself as the reference level.\",\n \"We could lift a body 10 m above the surface in Mars, on the Moon or above some point in an inertial space station trillions of kilometers away from the nearest star or planet and use any of these points as a reference level and calculate stored energy.\",\n \"The gravitational force near Earth is defined experimentally as $\\\\mathbf {F} = -m \\\\mathbf {g}$ The negative sign indicates that the force is downwards in a coordinate system where upwards is defined as positive.\",\n \"If we defined Earth's surface at a particular point and we wish to increase the elevation of an object with mass, work needs to be done against this force.\",\n \"If the object is then released, work is done by the gravitational field as the object falls to a new equilibrium condition.\",\n \"The work done against gravitational force by elevating an object with mass m from the surface reference point where h is set to zero to a height h is $W = \\\\int _{0}^{h} \\\\mathbf {F} dh = \\\\int _{0}^{h} -m \\\\mathbf {g} dh = -m \\\\mathbf {g} \\\\bigg \\\\vert _{0}^{h} = -m \\\\mathbf {g} h - (-m \\\\mathbf {g}0) = -m \\\\mathbf {g} h$ If the force is conservative, or path independent, the work done is directly related to an energy gradient as $W = -\\\\Delta U$ Thus the gravitational potential energy of a body lifted to a height h above a reference level where height is arbitrarily set as zero is $U = mgh$ A body falls from height because it has mechanical energy stored.\",\n \"Water in rivers, streams, pipes, and pores in soils and rocks also moves because of an energy gradient.\",\n \"In pipes, rivers and streams, the flow can be fast enough that there is a contribution from kinetic energy to the overall energy conservation calculations, as evidenced by the use of the Bernoulli equation, but this is usually not the case in small pores where the flow is slow enough that the kinetic energy contribution can be ignored.\",\n \"In water in soils and rocks, many different forms of energy can be involved.\",\n \"In Chapter REF , where we discussed the fundamental equations of fluid mechanics, we saw that pressure gradients are involved in the flow of fluids, and we also saw that gravitational forces can be involved along with pressure gradients.\",\n \"In porous media, especially in unsaturated porous media, things can be a little more complicated, in both statics and dynamics, a variety of other forces are involved, including complex intermolecular and surface forces.\",\n \"Because of the irregularity and often random distribution of pores and solids, other governing laws in addition to the Navier-Stokes equations might be necessary for modeling such systems.\",\n \"Although in this section we discuss the role of potentials in statics, a truly static condition is virtually impossible in natural porous media.\",\n \"Even a block of wet soil in laboratory conditions might be constantly subjected to potential gradients with the surrounding air and thermal fluctuations within the soil mass, which will cause energy gradients and water movement in response to these gradients.\",\n \"However, the assumption of a transitory static condition is very important for the use of methods of determination of potential and water content in porous materials.\",\n \"To discuss the state of energy of water in contact with solids in rocks and soils we need to address a different form of energy, other than the mechanical forms gravitational potential energy and kinetic energy, the internal energy.\",\n \"The internal energy, contrary to forms of energy which are related to external fields and motion of the system as whole, is the energy within the system as defined by a particular state in comparison to a reference state in which the energy is arbitrarily taken as zero.\",\n \"In these equilibrium states, the system can be completely characterized macroscopically by the internal energy, U, the volume, V, and the number of moles of its chemical components, $N_{1}$ , $N_{2}$ , ..., $N_{i}$ .\",\n \"Another extensive function that can be posited for equilibrium states of a system is the entropy, SInterpretations of the concept of entropy in popular media and introductory courses are most often inadequate, if not incorrect.\",\n \"Entropy must be defined on a physical and mathematical basis, and different definitions can be given based on different subfields of physics, we will address the issue in later editions.\",\n \"For now refer to [42], [24].\",\n \"The entropy is a function of the other extensive properties $S = S(U, V, N_i)$ and because the entropy function is continuous, differentiable and is a monotonically increasing function of energy, it can be solved for energy resulting in $U = U(S, V, N_i)$ Now the internal energy and entropy can be stated for a multiphased porous media as long as it is in a state of equilibriumNon-equilibrium thermodynamics plays an important role in transport phenomena, but for the sake of simplicity it is necessary to assume that states of equilibrium are attained for a reasonable simple treatment of the problem, and in such condition they contain all information necessary to characterize the system.\",\n \"By computing the differential of the energy form we have [24] $dU = \\\\left(\\\\frac{\\\\partial U}{\\\\partial S}\\\\right)_{V, N_i} dS + \\\\left(\\\\frac{\\\\partial U}{\\\\partial V}\\\\right)_{S, N_i} dV + \\\\sum _i \\\\left(\\\\frac{\\\\partial U}{\\\\partial N_i}\\\\right)_{V, N_j} dN_i$ From this equation the well known intensive properties temperature, pressure and chemical potential of the $i^{th}$ chemical species can be defined by the following partial derivatives [24] $T = \\\\left(\\\\frac{\\\\partial U}{\\\\partial S}\\\\right)_{V, N_i}$ $-P = \\\\left(\\\\frac{\\\\partial U}{\\\\partial V}\\\\right)_{S, N_i}$ $\\\\mu _i = \\\\left(\\\\frac{\\\\partial U}{\\\\partial N_i}\\\\right)_{S, V, N_j}$ The internal energy relationship can then be written as $dU = T dS -P dV + \\\\mu _i dN_i$ For a partially saturated porous media composed of a solid, liquid and a gas, the chemical potential can be defined for each phase, assuming that there are no different species within each phase, such that $dU = T dS -P dV + \\\\mu _{solid} dN_{solid} + \\\\mu _{liquid} dN_{liquid} + \\\\mu _{gas} dN_{gas}$ T, P and $\\\\mu _i$ are partial derivatives of functions of S, V and $N_i$ and because of that they are themselves functions of these quantities.\",\n \"These functional relationships are called equations of state.\",\n \"For a natural porous media system we are interested in the equation of state which describes the chemical potential of pore water $\\\\mu _w = \\\\mu _w(S, V, N_1, N_2, ...)$ However, $S$ , $V$ and $N_i$ are inconvenient variables to control in experimental studies in partially saturated soils and rocks.\",\n \"Since the chemical potential of water in this case was derived from the energy function, these variables also appear on the equation state.\",\n \"One alternative is to change the independent variables on the water chemical potential equation of state by performing a Legendre transformation in order to obtain new functions called thermodynamic potentials which can be a function of one or more more easily measured and controlled intensive parameters under experimental conditions.\",\n \"Examples of thermodynamic potentials are the internal energy itself, Helmholtz free energy, enthalpy and Gibbs free energy, although others can be derived.\",\n \"A convenient potential for porous media studies is the Gibbs (free) energy $G$ .\",\n \"In the case of the Gibbs energy we want to replace both S and V as independent variables and in this case G is defined by differential of $U$ minus the two conjugate pairs one wishes to replace, $TS$ and $-PV$ [42] $dG = d(U - TS - (-PV)) = d(U -TS + PV) = dU - TdS - SdT + PdV + VdP$ Recalling dU = TdS - pdV $dG = TdS - PdV + \\\\sum _i \\\\mu _i dN_i - TdS - SdT + PdV + VdP$ arriving at $dG = - SdT + VdP + \\\\sum _i \\\\mu _i dN_i$ In which the Gibbs free energy is a function of $T$ , $P$ and $N_i$ $G = G(T, P, N_i)$ As with the internal energy function, we can also define intensive parameters and equations of state for the Gibbs free energy $-S = \\\\left(\\\\frac{\\\\partial G}{\\\\partial T}\\\\right)_{P, N_i}$ $V = \\\\left(\\\\frac{\\\\partial G}{\\\\partial P}\\\\right)_{T, N_i}$ $\\\\mu _i = \\\\left(\\\\frac{\\\\partial G}{\\\\partial N_i}\\\\right)_{T, P, N_j}$ This last term is the partial molar Gibbs energy of component $i$ and is defined at constant T, P and number of moles of any other constituents, $N_j$ [15].\",\n \"Thus, for the component $i$ , the equation of state is a function of the form $\\\\mu _i = \\\\mu _i(T, P, N_j)$ with differential $d\\\\mu _i = \\\\left(\\\\frac{\\\\partial \\\\mu _i}{\\\\partial T}\\\\right)_{P, N_j} dT + \\\\left(\\\\frac{\\\\partial \\\\mu _i}{\\\\partial P}\\\\right)_{T, N_j} dP + \\\\sum _j \\\\left(\\\\frac{\\\\partial \\\\mu _i}{\\\\partial N_j}\\\\right)_{T, P} dN_j$ There are many ways of using these equations in thermodynamic systems, the user having freedom within the constraints of the theory to choose the independent variables.\",\n \"In soils and other porous systems we are usually concerned with chemical potential of the water in the pores and surfaces $\\\\mu _w$ .\",\n \"It is convenient to modify the chemical potential to include the mas based or gravimetric water content $\\\\theta _g$ , which is essentially the mass based water fraction.\",\n \"Under these conditions, the chemical potential of water can be defined as a function of $T$ , $P$ , $\\\\theta _g$ and the number of moles of each dissolved solute fraction $N_j$ [7], [15], [95]An alternative derivation can be found in [90] $\\\\mu _w = \\\\mu _w(T, P, \\\\theta _g, N_j)$ thus, $d\\\\mu _w = \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial T}\\\\right)_{P, \\\\theta _g, N_j} dT + \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial P}\\\\right)_{T, \\\\theta _g, N_j} dP + \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial \\\\theta _g}\\\\right)_{T, P, N_j} d\\\\theta _g + \\\\sum _j \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial N_j}\\\\right)_{T, P, \\\\theta _g} dN_j$ In the soil physics literature what is defined as water potential ($\\\\psi _w$ ) is the chemical potential of water in soil in reference to a Standard State, usually pure water at 293.15 K and 101325 Pa ($\\\\mu _w^0$ ) $\\\\boxed{\\\\psi _w \\\\equiv d\\\\mu _w \\\\equiv \\\\mu _w - \\\\mu _w^0}$ Make sure to understand this concept, it is one of the most fundamental principles in applied soils physics and, along with other forms energy and potential gradients, defines how water is stored and moves in soils, and the principles of operation of various field and laboratory devices such as tensiometers, pressure plate apparatus, psychrometers and what the water retention curve is.\",\n \"Most soil physicists and vadose zone hydrologists take this basic knowledge for granted and it is impossible to have a firm grasp on anything related to soil water phenomena without understanding these concepts.\",\n \"Since $\\\\mu _w$ is an intensive property in units of energy/amount (usually $J~kg^{-1}$ or $J~mol^{-1}$ ) the water potential has the same units.\",\n \"Having defined what the water potential $\\\\psi _w$ is we now have to define its components as usually encountered on the soil physics literatureWe mention soil physics because these concepts evolved within and along with early soil physics.\",\n \"It is interesting to follow the evolution of the concepts and discussions on journal articles published by N.E.\",\n \"Edlefsen and A.B.C.\",\n \"Anderson, the Gardner brothers, G.H.\",\n \"Bolt, K.L.\",\n \"Babcock, R.D.\",\n \"Miller, S. Takagi and many of their collaborators and students during the 1940s, 50s and 60s.\",\n \"Because$j$ here has the same function as $i$ from when this function was first presented $\\\\mu _w = \\\\left(\\\\frac{\\\\partial G}{\\\\partial N_j}\\\\right)$ the coefficients in $dP$ and in $dT$ in Equation 7.54 can be transformed by maintaining the appropriate quantities constant and changing the order of differentiation as [15], [24] $\\\\frac{\\\\partial \\\\mu _w}{\\\\partial T} = \\\\frac{\\\\partial }{\\\\partial T}(\\\\frac{\\\\partial G}{\\\\partial N_j}) = \\\\frac{\\\\partial }{\\\\partial N_j} (\\\\frac{\\\\partial G}{\\\\partial T}) = \\\\frac{\\\\partial }{\\\\partial N_j}(-S) = -\\\\frac{\\\\partial S}{\\\\partial N_j} = -S_w$ $\\\\frac{\\\\partial \\\\mu _w}{\\\\partial P} = \\\\frac{\\\\partial }{\\\\partial P} (\\\\frac{\\\\partial G}{\\\\partial N_j}) = \\\\frac{\\\\partial }{\\\\partial N_j} (\\\\frac{\\\\partial G}{\\\\partial P}) = \\\\frac{\\\\partial }{\\\\partial N_j}(V) = \\\\frac{\\\\partial V}{\\\\partial N_j} = V_w$ in which $V_w$ is the partial molar volume of water and $S_w$ is the partial molar entropy of pore water.\",\n \"Replacing into the original formula we arrive at $\\\\boxed{d\\\\mu _w = -S_w dT + V_w dP + \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial \\\\theta _g}\\\\right)_{T, P, N_j} d\\\\theta _g + \\\\sum _j \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial N_j}\\\\right)_{T, P, \\\\theta _g} dN_j}$ This is the thermodynamic equation that defines the potentials of soil water, it is possible to arrive at this equation using different thermodynamic potentials although Gibbs free energy and the Groenevelt-Parlange [41] potentials are most often employed in the soil physics literature [90].\",\n \"Using the soil physics $\\\\psi $ notation, this equation is usually written as [95] $\\\\boxed{\\\\psi _w = -S_w dT + \\\\psi _p + \\\\psi _m + \\\\psi _o}$ Considering isothermal conditions, $dT = 0$ , and the equation reduces to $\\\\boxed{\\\\psi _w =\\\\psi _p + \\\\psi _m + \\\\psi _o}$ In reality, many field and laboratory applications of these equations must consider isothermal conditions, not doing so would require some entropy function for the temperature potential, $\\\\psi _T$ , to be considered, which would cause great mathematical complications.\",\n \"The $\\\\psi $ terms in this equation are the most commonly defined components of soil water potential $\\\\psi _w$ , the pressure potential $\\\\psi _p = V_w dP$ the matric potential $\\\\psi _m = \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial \\\\theta _g}\\\\right)_{T, P, N_j} d\\\\theta _g$ and the osmotic potential $\\\\psi _o = \\\\sum _j \\\\left(\\\\frac{\\\\partial \\\\mu _w}{\\\\partial N_j}\\\\right)_{T, P, \\\\theta _g} dN_j$ The pressure potential is usually associated with hydraulic pressure in saturated soils below the water table, however it can be positive or negativeA body of water can be under tension, and not only under compressive stresses.\",\n \"Under the water table the pressure potential corresponds to the hydraulic pressure plus the atmospheric pressure, at the water table the hydraulic pressure is zero and the second component of pressure is the atmospheric pressure.\",\n \"Because we define the standard state at atmospheric pressure, the pressure potential is zero at the water table.\",\n \"In unsaturated soils the liquid pressure is zero and the soil water is at atmospheric pressure.\",\n \"The soil matric potential is negative and is related to the capillary and adsorptive forces exerted by the soil solid phase on soil water.\",\n \"Everything else constant, the water potential in an unsaturated soil is lower than that of free water at the same height and elevation, so that the state of energy of soil water is lower than that of free water.\",\n \"In simpler terms, water molecules in contact with soil solid particles are “less free to move” because of the forces exerted by these surfaces.\",\n \"At higher water contents, the adsortion forces of water molecules with surfaces play a small role when compared to capillary forces.\",\n \"As the soil dries and the water is restricted to thin films around the particles and very small pores, the adsorption component cannot be neglected and the matric potential will be some function of the capillary potential and the adsorption potential.\",\n \"Detailed discussion of the components of capillary and adsorptive forces can be found in [72], [96].\",\n \"The osmotic (or solute) potential is the reduction of water potential in contact with dissolved solids, or more specifically, solutes.\",\n \"Water molecules in contact with dissolved solids have lower energy, thus, as with cells and in other semi-permeable membranes, a larger concentration within the porous membrane or the cell will tend to draw water from the exterior until a pressure equilibrium is achieved in response to the osmotic forces.\",\n \"Different authors report different soil water potentials.\",\n \"This is a complex issue and as we mentioned before, the potentials will depend on thermodynamic theoretical considerations such as the thermodynamic potential used to derive the water potentials, see for example the discussions on pneumatic, submergence and overburden potentials, among others in [95], [90].\",\n \"Because we initially expressed the water potential as a partial molar Gibbs energy $\\\\partial G/\\\\partial N_j$ , the units of the water potential are in $J~mol^{-1}$ , and this water potential can be defined as the molar water potential.\",\n \"However, this unit is not often used in soil physics and groundwater hydrology, the most used units are derived from this basic unit.\",\n \"Considering the molar mass of water $M_w$ in $kg~mol^{-1}$ , we can divide the soil molar water potential by it, defining the water potential in units of $J~kg^{-1}$ $\\\\frac{\\\\psi _w}{M_w} \\\\rightarrow \\\\frac{J~mol^{-1}}{kg~mol^{-1}} = \\\\frac{J}{kg}$ Perhaps the most common form encountered in soil physics is in terms of pressure units, or the soil water pressure.\",\n \"The water potential in joules per mass is converted to pressure units by multiplying by water density $\\\\psi _w \\\\rho _w \\\\rightarrow \\\\frac{J }{kg} \\\\frac{kg}{m^{-3}} = \\\\frac{J}{m^{3}} = \\\\frac{N~m}{m^{3}} = \\\\frac{N}{m^{2}} = Pa$ Pascal ($Pa$ ) and its equivalents $kPa$ , $hPa$ and $MPa$ are frequently used in soil physics, especially when dealing with matric potentials measured using tensiometers and pressure plates.\",\n \"The third most commonly used conversion of water potential is in terms of head units or length, or the soil water head.\",\n \"To convert to length units, the water potential in joules per mass is divided by the gravitational acceleration, $g$ $\\\\frac{\\\\psi _w}{g} \\\\rightarrow \\\\frac{J~kg^{-1} }{m~s^{-2}} = \\\\frac{N~m~kg^{-1} }{m~s^{-2}} = \\\\frac{N~kg^{-1} }{ s^{-2}} = \\\\frac{kg~m~s^{-2}~kg^{-1} }{ s^{-2}} = m$ Recall for the above derivations that $1~J = 1~N~m$ and that $1~N = 1~kg~m~s^{-2}$ .\",\n \"The head units are common in hydrogeology and to some extent in soil physics, especially for the pressure potential.\",\n \"Sposito [90] warned that the different units should not obscure the physical interpretation of the problem.\",\n \"No matter in what unit it is expressed, the soil water potential represent an gradient of energy per unit amount of water which drives movement of soil water.\",\n \"Because water close to Earth's surface is subjected to a gravitational field and gravitational forces, the description of the potentials is incomplete without considering the effect of gravity.\",\n \"For a system under the effect of gravity the gravitational potential $ \\\\phi = gz$ must be added to the chemical potential of water to completely define the state of said water.\",\n \"The complete treatment of the inclusion of the gravitational potential into the thermodynamic treatment of soil water is complex and a more detailed analysis of the methods can be found in Babcock [6].\",\n \"The general theoretical account of including the gravitational potential with the chemical potential can be found in Guggenheim [42].\",\n \"For nowA complete treatment is planned to be included in later editions and for simplicity we will follow the approach of [95] and simply add the soil water gravitational potential to the other potentials asNotice that this relationship can be easily demonstrated to hold by considering the gravitational potential energy $ U = mgh$ and solving the density of water $\\\\rho _w = m_w/V$ for $m_w$ and replacing into the gravitational potential energy.\",\n \"The soil water gravitational potential then is simply $U/V \\\\rightarrow J~m^{-3} = N~m^{-2} = Pa $ $\\\\psi _g = \\\\rho _w g z$ In which $z$ is the vertical elevation.\",\n \"Considering now the potentials in pressure units, the total potential of soil water is the sum of the components of water potential and the gravitational potential, plus any other potentials that might be relevant under specific conditions $\\\\boxed{\\\\psi _t = \\\\psi _p + \\\\psi _m + \\\\psi _o + \\\\psi _g + ....}$ Remember that a potential gradient can be understood as a potential to perform work, movement of water will occur as a response to a gradient of total potential.\",\n \"In a water body under static conditions the total potential must be the same everywhere.\",\n \"Notice that the individual components might not be equal, but the sum of the components must be the same everywhere, otherwise there is a potential gradient and the water will flow in response to such gradient.\",\n \"Consider the points $A$ , $B$ and $C$ at heights $a$ , $b$ and $c$ on the vessel in Figure REF .\",\n \"If the water level is at $h$ and we consider the gravitational reference at the dotted line at the bottom, the gravitational potential, in head units, at each point is $\\\\psi _{gA} = a \\\\\\\\\\\\psi _{gB} = b \\\\\\\\\\\\psi _{gC} = c$ and the pressure potential is $\\\\psi _{pA} = h - a \\\\\\\\\\\\psi _{pB} = h - b \\\\\\\\\\\\psi _{pC} = h - c$ Since only these two potentials are being considered, the total potential $\\\\psi _t = \\\\psi _p + \\\\psi _g$ is $\\\\psi _{tA} = (h - a) + a = h \\\\\\\\\\\\psi _{tB} = (h - b) + b = h\\\\\\\\\\\\psi _{tC} = (h - c) + c = h$ Thus the total potential is constant and equal to $h$ everywhere.\",\n \"The same principle is valid for saturated soils, porous rocks and sedimentary deposits, if the total potential is constant everywhere the fluid is static.\",\n \"Figure: Total potential in a vessel.Consider now a porous medium which is half saturated and half unsaturated (Figure REF ).\",\n \"Above the water table the water column is under tension following the mechanisms described for a capillary tube.\",\n \"We showed that the height of capillary rise was given by $h = \\\\frac{2 \\\\gamma \\\\cos {\\\\theta }}{\\\\rho g r}$ Figure: Total potential in a partially saturated porous medium.Because water at each point above the water table is under tension as demonstrated for the capillary rise equation derivation, the capillary potential in head units at any point above the water table can be shown to be equivalent to the height above the water tableThis offers no mathematical or physical proof of any kind and is presented as a statement.\",\n \"A mathematical proof is being pursued $\\\\psi _c = -h$ Considering that the matric potential can be approximated by the capillary potential $\\\\psi _m \\\\approx \\\\psi _c$ , the matric potential, in head units, will correspond to the negative of the height of water column above the water table.\",\n \"Thus, for a porous medium that is half saturated and half unsaturated (Figure REF ) we can calculate the gravitational, pressure, matric and total potentials at the points $A$ , $B$ and $C$ .\",\n \"The gravitational potential is $\\\\psi _{gA} = a \\\\\\\\\\\\psi _{gB} = b \\\\\\\\\\\\psi _{gC} = c$ the pressure potential is $\\\\psi _{pA} = b - a \\\\\\\\\\\\psi _{pB} = 0 \\\\\\\\\\\\psi _{pC} = 0$ the matric potential is $\\\\psi _{mA} = 0 \\\\\\\\\\\\psi _{mB} = 0 \\\\\\\\\\\\psi _{mC} = -(c - b)$ such that the total potential, $\\\\psi _t = \\\\psi _g + \\\\psi _g + \\\\psi _m $ , is $\\\\psi _{tA} = a + (b - a) + 0 = b\\\\\\\\\\\\psi _{tB} = b + 0 + 0 = b\\\\\\\\\\\\psi _{tC} = c + 0 -(c - b) = b$ Thus the total potential is constant and equal to $b$ on the whole system, as expected.\",\n \"Most devices for measuring soil water potential fall into four categories: 1) suction devices, such as Haines apparatus or tension tables, 2) tensiometers, 3) pressure plates or pressure chamber apparatus and 4) psychrometers [31].\",\n \"Methods 1, 2 and 3 rely on similar physical principles, the soil sample is put into contact with a porous membrane which is contact with a water reservoir.\",\n \"In 1 the pressure in the water reservoir is then decreased by decreasing the elevation of the opening of a water column in contact with the porous membrane which in turn is in contact with the soil sample.\",\n \"As the pressure decreases, water in an initially saturated soil sample will flow to the reservoir, and from there to the outflow tube, until equilibrium is reached.\",\n \"In head units, the capillary potential and thus the matric potential are effectively the negative of the decrease in elevation of the opening of the water column with respect to the porous membrane.\",\n \"$\\\\psi _m \\\\approx \\\\psi _c = -h$ If the soil sample is relatively small in height, less than one to two cm [31], at equilibrium the matric potential of soil water is the same as the porous membrane.\",\n \"Different porous membranes can be used, depending on the application of the method and the desired matric potential range.\",\n \"Fine sand, kaolin clay, paper, porous stone and others can be found on the literature.\",\n \"Often in the literature the term tension is used instead or matric potential, tension ($\\\\tau $ ) being simply the negative value of the matric potential $\\\\tau = -\\\\psi _m$ Tensiometers (category 2) function in a similar way, except that the pressure on the water reservoir is not controlled by the user as in the case of hanging water column devices.\",\n \"The device is usually composed of a tube full of water with a porous cup on one end.\",\n \"The porous cup is put into contact with soil and if the soil is unsaturated, water will flow from the device and into the soil until equilibrium is reached.\",\n \"When equilibrium is reached, the water within the tube is in a state of tension and the tension or pressure can be measured using a variety of methods.\",\n \"Historically, the tension was measured by the decrease in height of the water level in a tube connected to the porous medium, however, because this would require a very long column, a liquid a much greater density, mercury (Hg) was used in contact with a water reservoir, resulting in the need of much smaller columns and in devices that could be conveniently deployed to the field.\",\n \"Later, due to the extreme environmental toxicity or Hg, and for convenience, analog pressure gauges were used.\",\n \"Although analog gauges are still used, with the decrease in cost of electronics in the last decades, digital pressure and tension sensors connected to data logging devices are now common in research and in agricultural and environmental monitoring applications.\",\n \"At equilibrim, and considering the water reservoir and soil at the same elevation, the sum of the pressure and matric potentials are going to be the same on the soil and on the water reservoir [95] $\\\\psi _{ps} + \\\\psi _{ms} = \\\\psi _{pr} + \\\\psi _{mr}$ Where the subscripts $s$ and $r$ stand for soil and water reservoir, respectively.\",\n \"However, the pressure potential is zero if the soil is unsaturated and the matric potential is zero in the water reservoir as there is no soil matrix such that $\\\\psi _{ms} = \\\\psi _{pr}$ Therefore, the (negative) pressure reading corresponding to the water on the reservoir is the soil matric potential.\",\n \"Notice that atmospheric pressure still acts on the unsaturated soil, but since the pressure reading on the device is with respect to the atmospheric pressure (by definition the gauge pressure), the two terms cancel out and there is no need to consider atmospheric pressure.\",\n \"Because the boiling point of water at pressures below atmospheric pressures decreases as the pressure difference increases at constant temperature, the maximum practical tension possible at the sea level and at average surface temperature is around $8~m$ , although in laboratory conditions, using de-areated water, higher tensions are possible [102].\",\n \"In order to increase the range of pressures (or tensions), the soil sample and porous membrane can be put inside a pressurized chamber (category 3).\",\n \"As the pressure inside the chamber increases, the capillary pressure of the water in the porous membrane decreases as the pores will gradually drain according to their radius.\",\n \"At higher pressures, water exists in smaller pores of small radius, such that the radius of curvature of the water meniscus is also very small, corresponding to very small capillary pressures (higher numbers in absolute value, or in terms of tension).\",\n \"The soil samples in hydraulic contact with the porous membrane will lose water until equilibrium is reached True equilibrium is never reached in reality, but fluxes are microscopic only or gradients small enough so that they can be disregarded.\",\n \"If the height of the soil sample is small, the gravitational potential can be neglected and the tension on the soil sample is the gauge pressure within the chamber.\",\n \"The porous membrane, usually a porous ceramic plate, is in contact with a water reservoir which is open to the atmosphere such that, as before $\\\\psi _{ps} + \\\\psi _{ms} = \\\\psi _{pr} + \\\\psi _{mr}$ and the matric and pressure potential are now zero on the water reservoir such that $\\\\psi _{ms} = -\\\\psi _{ps}$ In which $\\\\psi _{ps}$ is effectively the gauge pressure within the chamber.\",\n \"Pressure chamber devices can reach tensions well beyond those of tension devices.\",\n \"The limit usually considered is more or less arbitrarily set at $15~m$In part because of the idea that there is some universal permanent wilting point.\",\n \"At pressures above this value, adsorption forces start to play a more prominent role and the pressure gradient alone is not enough to further remove water from the soil.\",\n \"The water that cannot be removed by pressure plate apparatus is often referred to residual water content and can often be removed using other methods.\",\n \"Psychromters, hygrometes and potentiameters (category 4) are, generally speaking, devices that measure the water potential based on the Kelvin equation $\\\\psi _{w} = \\\\frac{R_g T}{V_w} \\\\ln {\\\\frac{p}{p_0}}$ in which the water potential measured is the sum of the matric and osmotic potential, $R_g$ is the universal gas constant, T is the absolute temperature, $V_w$ is the molar volume of water, $p$ is the air actual vapor pressure and $p_0$ is the air saturation vapor pressure, and $p/p_0$ is the relative humidity of the air in contact with the sample.\",\n \"The water activity is also used in some devices and is, by definition, $a_w \\\\equiv p/p_0$ .\",\n \"The water potential measured by these devices will not necessarily correspond to the water potential measured by pressure and suction devices, since the nature of the components measured are different.\",\n \"One of the main advantages of psychrometric and other similar devices is that they can measure potentials at extremely low water contents, well beyond what is possible with the pressure chamber, well below -1.5 MPa.\",\n \"Details on these devices can be found in [3], [82] The gravimetric method, based on weighing a soil or porous material sample, oven drying at 105 °C for at least 24 h and reweighing the oven dried sample and calculating the mass of water divided the mass of oven dried material is the standard method for measuring water content and it is used to calibrate indirect water content estimation methods.\",\n \"The gravimetric method is a direct method because the mass of water is directly measured by weighing.\",\n \"Because it is a direct method, if conducted properly using scales with high enough accuracy and precision, results in errors, both systematic and random, much smaller than indirect methods.\",\n \"In indirect methods, the water content is estimated indirectly and they are usually calibrated for a particular soil or condition using the gravimetric method.\",\n \"Errors related to calibration, or lack of thereof, and instrumental errors, both on the sensors and electronics usually result in lower precision and/or accuracy when compared to direct methods.\",\n \"There are several different types of indirect methods for measuring water content, the two most common types in use today being methods using radioactive sources, such as neutrons and gamma radiation, and general electromagnetic methods including time domain reflectometry (TDR), capacitance and other devices [31].\",\n \"The neutron thermalization method historically has been an important method for water content measurement and monitoring in the field.\",\n \"Once calibrated it provides fast and accurate measurements of soil water content.\",\n \"Physically, the method relies on a radioctive source that emits fast neutrons, with kinetic energy on the MeVeV = electron-volt; 1 $eV = 1.60217646 \\\\cdot 10^{-19}~J$ range and speed on the $10^4~km~s^{-1}$ order of magnitude, and a detector of slow neutrons, with energies less than about 1 eV [45].\",\n \"The fast neutrons emitted by the source collide with hydrogen atoms (H) in water molecules.\",\n \"Because H has a high scattering cross section, it can absorb or scatter fast neutrons, causing attenuation of the energies and correspondingly the velocity of the neutrons.\",\n \"The higher the water content, more hydrogen nuclei are available for collisions, attenuating fast neutrons and producing slow neutrons which are then counted by the detector.\",\n \"Of course other atoms can attenuate neutrons, but because H present in water is particularly effective in doing so in comparison to other atoms commonly found in soils, it can be used to water content estimation in most soils.\",\n \"The calibration is performed by direct measurement of volumetric water content around the sensing volume and corresponding slow neutron count ratios [45] $C_R = \\\\frac{x}{x_s}$ in which $x$ is the count in the measured material and $x_s$ is a standard count.\",\n \"The count ratio can then be related to the volumetric water content by a calibration equation of the form $\\\\theta _v = b_0 + b_1 C_R$ in which $b_0$ and $b_1$ are empirical parameters obtained from statistical or mathematical fitting procedures and $\\\\theta _v$ is the volumetric water content [45].\",\n \"Other forms of calibration equations can be found on the literature for different materials and water content ranges.\",\n \"The principle of operation of most electromagnetic methods including TDR, frequency domain reflectometry (FDR), ground penetrating radar (GPR), capacitance, impedance and others are based on the principle that the relative permittivityReferred to as the dielectric constant in older literature of water is much greater than that of other soil components, including air, minerals and organic matter.\",\n \"The relative permittivity of a material is a ratio of the absolute permittivity of the material to that of vacuum $\\\\epsilon _r = \\\\frac{\\\\epsilon }{\\\\epsilon _0}$ Because the absolute permittivity of the material, $\\\\epsilon $ , and that of vacuum have both the same units, usually $F~m^{-1}$ , and the permittivity of vacuum is a physical constant $\\\\epsilon _0 = 8.8541878128(13)\\\\cdot 10^{-12} ~ F~m^{-1}$ , $\\\\epsilon _r$ is a dimensionless normalized quantity, with values around 80 for water, 1 for air and around 5 for quartz rich soils [80].\",\n \"The relative permittivity of the porous media as a whole will be some form of average of the individual permittivities of each phase, the term apparent permittivity ($\\\\epsilon _{r(app)}$ ) being used on the literature [63].\",\n \"Because the solid phase is usually constant in fractional volume for most porous media applications and because the relativity permittivity of one of the dynamic phases, water, is much greater than that of the other, air, under unsaturated conditions the apparent permittivity will be a function of the volumetric water fraction, i.e.\",\n \"the volumetric water content.\",\n \"Thus, for most practical applications, water content measured using direct methods, usually gravimetric, can be related to apparent relativity permittivity measurements obtained using electromagnetic measuring devices, generating calibration equations.\",\n \"The most common empirical calibration equations found in the literature are third degree polynomials, square root and logarithmic [64] $\\\\theta _v = a_0 + a_1 \\\\epsilon _{r(app)} + a_2 \\\\epsilon _{r(app)}^2 + a_3 \\\\epsilon _{r(app)}^3$ $\\\\theta _v = a_0 + a_1 \\\\sqrt{\\\\epsilon _{r(app)}}$ $\\\\theta _v = a_0 + a_1 \\\\log {(\\\\epsilon _{r(app)})}$ in which $a_0$ to $a_3$ are empirical fitting parameters, fitted for each individual equation and $\\\\log $ is the natural logarithm.\",\n \"Most commercial devices come with “universal” calibration equations from the manufacturers, but it is almost always recommended that the device should be calibrated for the individual soils or porous materials in which is going to be used.\",\n \"Water content can also be estimated using more physically based models based on effective medium approximations theory [58], [63].\",\n \"There are several different designs and electronics for electromagnetic water content measuring sensors.\",\n \"The relative permittivity is in reality an imaginary number and is highly dependent on the frequency of operation of the device in terms of the frequency of the electromagnetic radiation emitted.\",\n \"For most soils, but not all, the relative apparent permittivity can be approximated by the real component of the complex permittivity.\",\n \"A detailed discussion of measuring water content with electromagnetic devices is beyond the scope of this book.\",\n \"The user is highly encouraged to study electromagnetic theory before using any electromagnetic device for soil and porous media measurements, there are many misconceptions related to these devices.\",\n \"For more information the reader is referred to [99], [31], [63].\",\n \"There are many other methods used for estimating water content, including methods based on remote sensors, thermal methods and other ionizing radiation methods.\",\n \"These methods will not be discussed in this book, as they are not as broadly employed as the methods discussed above.\",\n \"The user interested in these specific methods should consult [31] and the references listed therein.\",\n \"For a set of tubes of different radius $r_i$ , there is an exact relationship between the percent saturation and the capillary pressure.\",\n \"At the capillary pressure $Pc_i$ , all tubes of radius greater than $r_i$ will have drained, following the Young-Laplace equation $Pc_{i} =\\\\frac{ 2\\\\gamma }{r_i}$ thus the volume of water retained will correspond to the sum of the volumes of water retained in all tubes of radius smaller or equal than $r_i$ .\",\n \"But the volume will also depend on the capillary rise for each individual tube which is related to the radius by the capillary rise equation For the angle $\\\\theta = 0$ and $\\\\cos theta = 1$.\",\n \"$h_{i} =\\\\frac{ 2\\\\gamma }{\\\\rho g r_i}$ Considering the volume of each tube as that of a cylinder $V_{i} =\\\\pi r_i^2 h_i$ and replacing the capillary rise equation $V_{i} = \\\\pi r_i^2 \\\\frac{ 2\\\\gamma }{\\\\rho g r_i} = \\\\frac{ 2\\\\gamma \\\\pi r_i}{\\\\rho g }$ Now the volume of water is the sum of the volumes of water in all $i$ pores with radius equal or smaller than $r_i$ $V_{water} =\\\\sum _i \\\\pi r_i^2 \\\\frac{ 2\\\\gamma }{\\\\rho g r_i} = \\\\sum _i \\\\frac{ 2\\\\gamma \\\\pi r_i}{\\\\rho g }$ and the fractional water content is simply the volume of water divided by the total volume of all tubes in the system $\\\\theta _{f} =\\\\frac{ V_{water}}{V_{t} }$ Thus, for each capillary pressure $Pc_{j}$ corresponding to a radius $r_j$ , the fractional water content can be directly calculated from the sum of the volumes of water in all pores with radius less or equal than $r_j$ An explicit mathematical relationship can be derived by manipulating summation terms, this will be explored in following editions.\",\n \"This is what is called the capillary bundle model for water retention.\",\n \"It is also an important model for hydraulic conductivity, as we will discuss in later chapters.\",\n \"The capillary bundle model is a useful tool to model and understand transport and retention of water in porous media.\",\n \"In this approach, the pore space is simplified as a set of continuous tubes of constant radius in each pore.\",\n \"Because pores are irregular, do not have constant radius, are usually not cylindrical, and because forces related to other than simple capillary phenomena act in complex natural porous systems, there is no direct way of calculating water saturation from matric potential and empirical and semi-empirical models are necessary.\",\n \"Empirical models rely on fitting to observed data of water content and matric potential, in other words, there is no analytical expression to calculate the volumetric water content by knowing only the matric potential, water content data is necessary.\",\n \"The water retention curve is nothing more than the empirical relationship between water content and matric potential obtained from laboratory and field data.\",\n \"The relationship has been observed to follow a characteristic sigmoid function (S-shaped) when the matric potential is considered in log scale.\",\n \"To avoid mathematical and graphing difficulties it is convenient to consider the absolute values of matric potential ($|\\\\psi _m|$ ), which as we have seen is a negative quantity, when dealing with water retention equations.\",\n \"There has been several efforts to derive analytical expressions for the water retention curve.\",\n \"These have important practical applications in irrigation and environmental modeling and because, in most cases, an unsaturated hydraulic conductivity function was derived from the water retention curve (e.g.\",\n \"[19], [25], [97], [56]).\",\n \"Some of the models most often found on the literature are presented below.\",\n \"We will leave the related unsaturated hydraulic conductivity functions for later chapters.\",\n \"One of the earliest models that saw wide application is that of Brooks and Corey (BC) [19] For simplicity and to be consistent we will refer to soil water potential as $\\\\psi $ on the water retention equations $\\\\theta _v ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\theta _r + (\\\\theta _s - \\\\theta _r) (\\\\frac{\\\\psi _b}{|\\\\psi |})^\\\\lambda , & \\\\text{for } |\\\\psi | > \\\\psi _b \\\\\\\\\\\\theta _s, & \\\\text{for } |\\\\psi | \\\\le \\\\psi _b\\\\end{array}\\\\right.\",\n \"}$ The BC equation, like many water retention models, is a piecewise equation, meaning that it is segmented into two parts.\",\n \"Its parameters are the saturation water content, $\\\\theta _s$ , the residual water content, $\\\\theta _r$ the air entry value, often called bubbling pressure, $\\\\psi _b$ , and the pore size distribution parameter, $\\\\lambda $ .\",\n \"Most water retention models can be defined in terms of saturation and residual water contents.\",\n \"The saturation water content correspond to water content at a matric potential of zero, whenever it is defined, where in theory the pore space is completely water saturated.\",\n \"The residual water content is the remaining water content at very high values of matric tensionOr very low values if you are thinking in potential, remember that tension is defined as the negative of the potential, where further increase in tension does not cause decrease in water content.\",\n \"In most models, residual water content is an asymptotic parameter reached by taking tension to infinity.\",\n \"The bubbling pressure is the matric tension in which air starts to enter the pore system, usually at relatively low matric tensions for most soils.\",\n \"Below the bubbling pressure, increase in tension does not cause a corresponding decrease in water content.\",\n \"Most water retention models will have parameters related to pore size distribution since, as we saw, water retention is controlled by the distribution of pores.\",\n \"In the BC model the piecewise condition is defined by the $\\\\psi _b$ parameter.\",\n \"At tensions below or equal to $\\\\psi _b$ the water content is constant and equal to saturation water content, while at tension above $\\\\psi _b$ water content decreases following a power law of $\\\\lambda $ .\",\n \"Most water retention equations can be expressed in terms of the effective saturation, given by $S_e = \\\\frac{\\\\theta _v - \\\\theta _r}{\\\\theta _s - \\\\theta _r}$ The BC equation, for example, can be expressed as $S_e ={\\\\left\\\\lbrace \\\\begin{array}{ll}(\\\\frac{\\\\psi _b}{|\\\\psi |})^\\\\lambda , & \\\\text{for } |\\\\psi | > \\\\psi _b \\\\\\\\1, & \\\\text{for } |\\\\psi | \\\\le \\\\psi _b\\\\end{array}\\\\right.\",\n \"}$ Using the effective saturation normalizes the water content between zero and one, making it possible to compare the shapes of the curves for different materials.\",\n \"Using the effective saturation can, in some cases, make fitting easier using computer methods by reducing the number of fitting parameters.\",\n \"The Campbell equation is in essence the BC equation, but defining the effective saturation as $\\\\theta /\\\\theta _s$ [25] $\\\\theta ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\theta _s (\\\\frac{\\\\psi _b}{|\\\\psi |})^b, & \\\\text{for } |\\\\psi | > \\\\psi _b \\\\\\\\\\\\theta _s, & \\\\text{for } |\\\\psi | \\\\le \\\\psi _b\\\\end{array}\\\\right.\",\n \"}$ In which the physical meaning is the same as the BC equation, and $b$ is essentially $\\\\lambda $ .\",\n \"Brutsaert [21] proposed an equation of the form $S_e = \\\\frac{a}{a + \\\\psi ^b}$ where a and b are empirical parameters.\",\n \"Brutsaert's equation was later improved upon by van Genuchten, resulting in one of the most frequently used water retention equations in soil physics and hydrology [97] $\\\\theta _v = \\\\theta _r + \\\\frac{(\\\\theta _s - \\\\theta _r)}{[1 + (\\\\alpha \\\\psi )^n]^m}$ In which $m$ and $n$ are shape parameters related to the pore size distribution.\",\n \"It is often convenient to define the Mualem [67] restriction as $m = 1 - \\\\frac{1}{n}$ Reducing the number of empirical parameters to four.\",\n \"Kosugui [56] derived an equation based on a lognormal pore size distribution $\\\\theta _v ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\theta _r + \\\\frac{(\\\\theta _s - \\\\theta _r)}{2} \\\\text{erfc}[ \\\\frac{\\\\log (\\\\psi _b-\\\\psi )/(\\\\psi _b - \\\\psi _0) - \\\\sigma ^2}{\\\\sqrt{2} \\\\sigma }] , & \\\\text{for } |\\\\psi | > \\\\psi _b \\\\\\\\\\\\theta _s, & \\\\text{for } |\\\\psi | \\\\le \\\\psi _b\\\\end{array}\\\\right.\",\n \"}$ with two new parameters, $\\\\sigma $ , the standard deviation of the transformed pore capillary pressure distribution, and $\\\\psi _0$ , the capillary pressure at the inflection point.\",\n \"Kosugui's model uses the complementary error function (erfc), defined as [14] $\\\\text{erf}(x) = \\\\frac{2}{\\\\sqrt{\\\\pi }}\\\\int _0^x e^{-t^2} dt$ $\\\\text{erfc}(x) = 1 - \\\\text{erf}(x) = \\\\frac{2}{\\\\sqrt{\\\\pi }}\\\\int _x^\\\\infty e^{-t^2} dt$ In which erf is the error function.\",\n \"However, the erfc is calculated numerically in almost all practical applications and is available as a function in most programming languages.\",\n \"Thus, the user does not need to use the integral form when fitting water retention data.\",\n \"The BC, Brutsaert-van Genuchten and Kosugui equation are fitted to simulated oxisol data and shown in Figure REF Figure: Commonly used water retention models fitted to simulated oxisol data.ll $ W $ Thermodynamical work $ \\\\gamma $ Surface tension $ A, A^{\\\\prime }, a $ Area $ l $ length $ dx $ Infinitesimal increase in $x$ $ \\\\mathcal {F} $ Helmholtz (free) energy $ S $ Entropy $ p, P $ Pressure, except when specified otherwise $ \\\\mu $ Chemical potential of a substance $ N_{i} $ Number of moles of a substance $ T $ Temperature $ \\\\mathbf {n} $ Unit normal vector $ R, r $ Radius $ l_1, l_2, l^{\\\\prime }_1, l^{\\\\prime }_2 $ Elements of arc $ \\\\zeta $ Unspecified displacement $ \\\\theta , \\\\phi $ Angles $ proj_F $ Component of a force $ F $ Force $ \\\\rho $ Generalized density $ \\\\rho , \\\\rho _w $ Water density $ g $ Earth's gravitational acceleration $ U $ Potential of a force $ m $ Mass $ h $ Height $ G $ Gibbs (free) energy $ \\\\theta _g $ Gravimetric water content $ \\\\theta _v $ Volumetric water content $ V_w $ Partial molar volume of water $ S_w $ Partial molar entropy of water $ \\\\psi $ Soil potential $ \\\\psi _w $ Soil water potential $ \\\\psi _p $ Soil pressure potential $ \\\\psi _m $ Soil matric potential $ \\\\psi _o $ Soil osmotic potential $ \\\\psi _c, P_c $ Soil capillary potential $ \\\\psi _g $ Soil water gravitational potential $ \\\\psi _t $ Total soil potential $ z $ Elevation $ \\\\tau $ Soil water tension $ p, p0 $ Actual air vapor pressure and air saturation vapor pressure on Kelvin's equation $ R_g $ Universal gas constant on Kelvin's equation $ a w $ Water activity on Kelvin's equation $ C_R $ Count rate $ a, b, a_0, a_1, _2, A_3, b_0, b_1, m, n $ Empirical parameters $ \\\\epsilon _r $ Relative permittivity $ \\\\epsilon $ Absolute permittivity of a material $ \\\\epsilon _0 $ Permittivity of free space $ \\\\theta _f $ Fractional water content $ V_{water} $ Volume of water $ V_t $ Total volume $ \\\\theta _r $ Residual volumetric water content $ \\\\theta _s $ Saturation volumetric water content $ \\\\psi _b $ Air entry value $ \\\\lambda $ pore size distribution parameter $ S_e $ Effective saturation $ \\\\alpha $ Inverse air entry value $ \\\\psi _0 $ Capillary pressure at the inflection point of Kosugi's equation $ \\\\sigma $ Standard deviation of transformed capillary pressure distribution in Kosugi's equation $ \\\\text{erf} $ Error function $ t $ Generalized variable in the definition of erf $ \\\\text{erfc} $ Complementary error function As you will see in the following sections, although there are analytical solutions for the Navier-Stokes equations to simple cases of laminar flow in open conduits, for flow in porous media, the statement given by Morris Muskat ([69] Muskat, 1937, p.55) summarizes the problem “Unfortunately, however, in spite of the justifiable simplification of neglecting the inertial forces - due to the small velocities generally characteristic of flow through porous media - the mathematical difficulties of applying these equations to porous media are for practical purposes entirely unsurmountable.\\\"\",\n \"Although theoretical derivations of Darcy's law from the Navier-Stokes equations have been proposed over the last decades, a solution for random porous media will invariable depend on some stochastic mathematical method, such as averaging or homogenization theory, a direct analytical derivation for a random and irregular porous medium remains a problem of difficulty on par with that of modelling turbulence.\",\n \"In simpler words, the random nature of the behaviour of fluid flows cannot be directly solved analytically in all but the simplest cases.\",\n \"We will be mentioning Muskat's work on several occasion throughout this chapter.\",\n \"He was a trained theoretical physicist who earned his Ph.D. at California Institute of Technology in 1929 with a thesis on “The Continuous Spectra of Hydrogen Like Atoms” [68].\",\n \"After graduating, he joined the petroleum industry were he applied his training in problems of multi-phase flow in porous media, making important contributions to the mathematics of flow in porous media in the process, especially by modifying and generalizing the original form of Darcy's equation.\",\n \"His original contributions remain an important topic not only in engineering and applied mathematics, but also in pure mathematics (e.g.\",\n \"[2]).\",\n \"His point is that there are no analytical solutions of the Navier-Stokes equation for true random porous media.\",\n \"On top of that, there is no guarantee that a solution exists for many fluid mechanics problems.\",\n \"This remains an open problem in mathematics in the form of the Navier-Stokes existence and smoothness problem one of the so called millennium problems for which you can win a large sum of money you manage to solve it.\",\n \"Getting back to Darcy's law, it is an empirical law, initially based on experimental data.\",\n \"There has been many a posteriori attempts to derive Darcy's law from first principles, including averaging the Navier-Stokes equations, but the fact remain that most of these make several assumptions along the way and are for the most part largely heuristic, there are no first principles derivation from a direct analytical solution of the Navier-Stokes equations and the hydraulic conductivity remains an empirical constant.\",\n \"Before we get too much ahead of ourselves, as you haven't been formally introduced to Darcy's law, let's look at a few cases of flow through free conduits where there is nothing but the fluid going through the conduit.\",\n \"In porous media, as we have discussed, there is a solid porous matrix inside a volume through which the fluid is moving.\",\n \"Under these conditions, the fluid does not occupy the entire inner volume and is forced to flow through an often very complex geometry of pores.\",\n \"This is one of the main factors which makes flow thorough porous media so mathematically complex.\",\n \"In this section, two analytical solutions of the Navier-Stokes will be discussed.\",\n \"The first is that for flow through two infinite and flat surfaces parallel to each-other and the second is for flow through a cylindrical conduit.\",\n \"The second is often seen in soil physics and transport of fluid through porous media books as an introduction to Darcy's law, given that the pore network of a porous material can be generalized to a network of tubes, from which an approximation to Darcy's law can be obtained Which as an approximation, does not adequatelly describes real porous media.\",\n \"Case I.\",\n \"Flow between two infinite plates Let us start be recalling the Navier-Stokes equations in vector form $\\\\frac{\\\\partial \\\\mathbf {u}}{ \\\\partial t} + (\\\\mathbf {u} \\\\cdot \\\\nabla ) \\\\mathbf {u} = \\\\mathbf {F} -\\\\frac{1}{\\\\rho } \\\\nabla p + \\\\frac{\\\\mu }{\\\\rho } \\\\nabla ^{2} \\\\mathbf {u}$ For steady state flow, the velocity field does not vary with time, so $\\\\mathbf {u}$ is constant with time everywhere such that $\\\\frac{\\\\partial \\\\mathbf {u}}{ \\\\partial t} = 0$ and $(\\\\mathbf {u} \\\\cdot \\\\nabla ) \\\\mathbf {u} = \\\\mathbf {F} -\\\\frac{1}{\\\\rho } \\\\nabla p + \\\\frac{\\\\mu }{\\\\rho } \\\\nabla ^{2} \\\\mathbf {u}$ There are many conditions in which body forces can be neglected, such as in purely horizontal flow of a fluid in which the horizontal dimension is much larger than the vertical direction in which a gravitational field acts or if the contact forces are much larger than the body forces.\",\n \"Ignoring the body forces results in $(\\\\mathbf {u} \\\\cdot \\\\nabla ) \\\\mathbf {u} = -\\\\frac{1}{\\\\rho } \\\\nabla p + \\\\frac{\\\\mu }{\\\\rho } \\\\nabla ^{2} \\\\mathbf {u}$ Now we have to take a look at each term, the term $ (\\\\mathbf {u} \\\\cdot \\\\nabla ) \\\\mathbf {u} $ was discussed earlier in the context of the material derivative, based on that discussion, it is possible to show that it represents the set of equations $u_x \\\\frac{\\\\partial u_x}{\\\\partial x} + u_y \\\\frac{\\\\partial u_x}{\\\\partial y} + u_z \\\\frac{\\\\partial u_x}{\\\\partial z}$ $u_x \\\\frac{\\\\partial u_y}{\\\\partial x} + u_y \\\\frac{\\\\partial u_y}{\\\\partial y} + u_z \\\\frac{\\\\partial u_y}{\\\\partial z}$ $u_x \\\\frac{\\\\partial u_z}{\\\\partial x} + u_y \\\\frac{\\\\partial u_z}{\\\\partial y} + u_z \\\\frac{\\\\partial u_z}{\\\\partial z}$ and it is called the inertial or convective term, due to change of velocity as the fluid element is transported to different positions in the velocity field.\",\n \"The first term on the right side is the pressure gradient and represent the contact forces which cause movement of the fluid if the gradient is not equal to zero.\",\n \"Recalling the gradient operator, $ \\\\nabla p$ can be represented as $\\\\nabla p = \\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {i} + \\\\frac{\\\\partial p}{\\\\partial y} \\\\mathbf {j} + \\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {k}$ Do not worry about the density $\\\\rho $ , it is assumed as constant and thus can be manipulated to either side of the equation.\",\n \"To have an intuitive understanding of the pressure gradient, think about a garden hose connected to a reservoir a few meters above ground, it is the pressure gradient between both ends of the hose that causes the water to flow out of the end where the overall pressure is lower (assuming the hose is perpendicular in relation to the gravitational field), but in this case the gradient is unidimensional, say $\\\\partial p/\\\\partial x$ .\",\n \"The second term on the right is the viscosity or diffusion term, recalling the Laplacian operator, it can be written as $\\\\mu \\\\nabla ^2 \\\\mathbf {u} = \\\\mu [\\\\nabla \\\\cdot \\\\nabla (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k})]$ which in Cartesian coordinates resolves in three components on the directions $\\\\mathbf {i}$ , $\\\\mathbf {j}$ and $\\\\mathbf {k}$ $\\\\mu (\\\\frac{\\\\partial ^2 u_x}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 u_x}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 u_x}{\\\\partial z^2} )$ $\\\\mu (\\\\frac{\\\\partial ^2 u_y}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 u_y}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 u_y}{\\\\partial z^2} )$ $\\\\mu (\\\\frac{\\\\partial ^2 u_z}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 u_z}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 u_z}{\\\\partial z^2} )$ or, simply $\\\\mu \\\\nabla ^2 \\\\mathbf {u} = \\\\mu (\\\\nabla ^2 u_x \\\\mathbf {i} + \\\\nabla ^2 u_y \\\\mathbf {j} + \\\\nabla ^2 u_x \\\\mathbf {k})$ For a fluid moving slowly, the inertial forces are much smaller than the viscous forces, such that $ \\\\mu \\\\nabla ^2 \\\\mathbf {u} \\\\gg \\\\mathbf {u} \\\\cdot \\\\nabla \\\\mathbf {u} $ and the latter can be ignored resulting in a much simpler form of the Navier-Stokes equations $\\\\nabla p = \\\\mu \\\\nabla ^{2} \\\\mathbf {u}$ Or, in Cartesian coordinates $\\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {i} + \\\\frac{\\\\partial p}{\\\\partial y} \\\\mathbf {j} + \\\\frac{\\\\partial p}{\\\\partial z} \\\\mathbf {k} = \\\\mu [ \\\\nabla \\\\cdot \\\\nabla (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) ]$ Now many analytical solutions of the Navier-Stokes equations can be derived from this simplification, for a broad range of experimental conditions.\",\n \"The important point here is that at low flow velocities, when the viscous forces are much greater than the inertial forces the flow is often laminar (for many fluids found in daily life, the most important being water, this is true, it might not be if viscosity is a lot larger or smaller than expected values).\",\n \"If the flow velocity is too high and/or viscosity too low a phenomenon known as turbulence might occur and it might become impossible to solve the differential equations analytically.\",\n \"We can finally state our problem as the “laminar flow of water between two smooth parallel plates in which the pressure gradient is in the $x$ direction, the plates are infinitely long on the $y$ direction and the distance $d$ between the plates, through which the water flows, is oriented in the $z$ direction”.\",\n \"If you are not familiar with solutions of physics problems in textbooks and theoretical models, you might wonder why would anyone assume anything as infinite and how on earth would this translate into a practical application.\",\n \"The infinite plates simplification is a common occurrence in electromagnetism and other areas of physics and engineering as it becomes much easier to solve the differential equations that arise from the geometry of the problem.\",\n \"The simple fact of assuming flow through a square tube results in a partial differential equation with series solution Check any partial differential equations or mathematical methods for physicists book and see how convoluted the whole thing is.\",\n \"First if the pressure gradient is in the $x$ direction, there are no pressure gradients on the $y$ or $z$ directions and the equation further simplifies to $\\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {i} = \\\\mu [ \\\\nabla ^2 (u_x \\\\mathbf {i} + u_y \\\\mathbf {j} + u_z \\\\mathbf {k}) ]$ Second, if the flow is in the $x$ direction, and is steady-state, the velocity does not vary on that direction, and if the plates are infinite in $y$ , as stated it also does not vary in $y$ such that $\\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {i} = \\\\mu \\\\frac{\\\\partial ^2 u_x}{\\\\partial z^2} \\\\mathbf {i}$ Showing that $ \\\\mu \\\\nabla ^2 \\\\mathbf {u} = \\\\mu \\\\frac{\\\\partial ^2 u_z}{\\\\partial z^2} \\\\mathbf {i} $ when the flow velocity does not vary in $x$ or $y$ is left as an exercise to the reader.\",\n \"A simple solution can be found if the pressure gradient in the $x$ direction is constant.\",\n \"Omitting the unit vectors the equation to be solved is $\\\\frac{\\\\partial p}{\\\\partial x} = \\\\mu \\\\frac{\\\\partial ^2 u_x}{\\\\partial z^2}$ Whose solution can be found by separating the variables and integrating $\\\\int \\\\partial ^2 u_z = \\\\int \\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\partial z^2$ $\\\\int \\\\partial u_z = \\\\int (\\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} z + C_1) \\\\partial z$ $u_z = \\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{z^2}{2} + C_1 z + C_2$ As always, we need the boundary conditions to solve the equation, in this case, because of the viscosity condition, the velocity is zero at the surface of the plates $u_z = 0 ~\\\\text{at} ~z = 0, ~u_z = 0 ~\\\\text{at} ~z = h{^{\\\\prime }}$ The first condition gives $C_2 = 0$ , while applying the second $0 = \\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{h{^{\\\\prime }}^2}{2} + C_1 h{^{\\\\prime }} + 0 \\\\\\\\C_1 h = -\\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{h{^{\\\\prime }}^2}{2}$ $C_1 = -\\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{h{^{\\\\prime }}}{2}$ Replacing into Equation 8.5 results $u_z = \\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{z^2}{2} -\\\\frac{1}{\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{h{^{\\\\prime }}}{2} z$ $\\\\boxed{u_z = \\\\frac{1}{2\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} z(z - h{^{\\\\prime }})}$ The discharge (or volumetric flux) passing through a section can be calculated by integrating the flow velocity over the area (or length in this case) $Q = \\\\int _0^{h{^{\\\\prime }}} u dz \\\\\\\\Q = \\\\frac{1}{2\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} \\\\int _0^{h{^{\\\\prime }}} z(z - h{^{\\\\prime }}) dz \\\\\\\\Q = \\\\frac{1}{2\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} (\\\\int _0^{h{^{\\\\prime }}} z^2 dz - \\\\int _0^{h{^{\\\\prime }}} zh{^{\\\\prime }} dz) \\\\\\\\Q = \\\\frac{1}{2\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} ( \\\\frac{z^3}{3}\\\\bigg |_0^{h{^{\\\\prime }}} - h^{\\\\prime } \\\\frac{z^2}{2}\\\\bigg |_0^{h{^{\\\\prime }}}) \\\\\\\\Q = \\\\frac{1}{2\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} ( \\\\frac{h{^{\\\\prime }}^3}{3} - \\\\frac{h{^{\\\\prime }}^3}{2}) \\\\\\\\$ $\\\\boxed{Q = - \\\\frac{1}{12\\\\mu } \\\\frac{\\\\partial p}{\\\\partial x} h{^{\\\\prime }}^3 \\\\\\\\}$ Dimensional analysis shows that in this case the formula results in an “areal discharge” in [L$^2$ T$^{-1}$ ].\",\n \"Note that this has to be the case because of the infinite plates assumption.\",\n \"Case II.\",\n \"Flow in a cylindrical conduit - Poiseuille flow We can start with Equation 8.5, the process before hand is the same as discussed in the previous section.\",\n \"The pressure gradient driving the flow is still in the $x$ direction but now the circular geometry in the cylider cross section requires both $y$ and $z$ components $\\\\frac{\\\\partial p}{\\\\partial x} \\\\mathbf {i} = \\\\mu (\\\\frac{\\\\partial ^2 u_y}{\\\\partial y^2}\\\\mathbf {j} + \\\\frac{\\\\partial ^2 u_z}{\\\\partial z^2} \\\\mathbf {k})$ This equation can be much more easily solved in cylindrical coordinates in which the Laplacian operator can be written as $\\\\nabla ^2 = \\\\frac{1}{r} \\\\frac{\\\\partial }{\\\\partial r}( r \\\\frac{\\\\partial }{\\\\partial r}) + \\\\frac{1}{r^2} \\\\frac{\\\\partial ^2 }{\\\\partial \\\\theta ^2} + \\\\frac{\\\\partial ^2 }{\\\\partial z^2}$ However, the flow velocity components do not vary in $z$ nor with the polar angle $\\\\theta $ along the cylindrical conduit, varying only with the radius of the tube $r$ , such that $\\\\nabla ^2 = \\\\frac{1}{r} \\\\frac{\\\\partial }{\\\\partial r}( r \\\\frac{\\\\partial }{\\\\partial r})$ In this case, Equation 8.10 can be written in cylindrical coordinates as $\\\\frac{\\\\partial p}{\\\\partial x} = \\\\mu \\\\frac{1}{r} \\\\frac{\\\\partial }{\\\\partial r}( r \\\\frac{\\\\partial u_r}{\\\\partial r})$ Now integration can be performed in a single variable $r$ instead of on $y$ and $z$ which simplifies the process.\",\n \"$\\\\int \\\\partial ( r \\\\frac{\\\\partial u_r}{\\\\partial r}) = \\\\frac{1}{\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\int r \\\\partial r\\\\\\\\r \\\\frac{\\\\partial u_r}{\\\\partial r} = \\\\frac{1}{\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{r^2}{2} + C_1\\\\\\\\\\\\int \\\\partial u_r = \\\\int (\\\\frac{1}{\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{r}{2} + \\\\frac{C_1}{r}) \\\\partial r \\\\\\\\$ $u_r = \\\\frac{1}{\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{r^2}{4} + C_1 \\\\log {r} + C_2 \\\\\\\\$ The boundary conditions in this case are that the velocity is zero at the walls of the tube, as before, because of the viscosity condition, the velocity is maximum at center of the tube, at $r = 0$ .\",\n \"$u_r = 0 ~\\\\text{at} ~r = a, ~u_r = u_{max} ~\\\\text{at} ~ r = 0$ Another important condition is that the velocity is finite at the center of the tube.\",\n \"This implies that $C_1$ has to be zero, otherwise $C_1 \\\\log {0}$ results in a singularity.\",\n \"Applying the first boundary condition and with $C_1 = 0 $ $0 = \\\\frac{1}{\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{a^2}{4} + 0 + C_2 \\\\\\\\$ $C_2 = - \\\\frac{1}{\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\frac{a^2}{4}$ Replacing into Equation 8.14 $u_r = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} (r^2 - a^2)$ The discharge (volumetric flux) can now be calculated by integrating the velocity function in $y$ and $z$ $Q = \\\\int _0^y \\\\int _0^z u dydz$ But again, because of the circular geometry of the tube's cross section, this is much more easily done by converting to polar coordinates, taking into account that $dydz$ is an area element $dA$ , which in polar coordinates is $ r dr d\\\\phi $ .\",\n \"There should be no mystery here, if you take a circle of radius r and rotate this radius around 360° you will draw an area.\",\n \"Rewriting the volumetric flux equation and solving $Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\int _0^{2\\\\pi } \\\\int _0^a (r^2 - a^2) r dr d\\\\phi \\\\\\\\Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\int _0^{2\\\\pi } \\\\int _0^a (r^3 - r a^2) dr d\\\\phi \\\\\\\\Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\int _0^{2\\\\pi } (\\\\frac{r^4}{4} - \\\\frac{r^2}{2} a^2)\\\\bigg |_0^a d\\\\phi \\\\\\\\Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\int _0^{2\\\\pi } (-\\\\frac{a^4}{4}) d\\\\phi \\\\\\\\Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} \\\\int _0^{2\\\\pi } (-\\\\frac{a^4}{4}) d\\\\phi \\\\\\\\Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} (-\\\\frac{a^4}{4}) \\\\phi \\\\bigg |_0^{2\\\\pi } \\\\\\\\Q = \\\\frac{1}{4\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} (-\\\\frac{a^4}{4}) 2\\\\pi \\\\\\\\$ $\\\\boxed{Q = -\\\\frac{\\\\pi }{8\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} a^4 \\\\\\\\}$ in which $Q$ is the discharge in [L$^3$ T$^{-1}$ ] as expected.\",\n \"Henry Darcy was a french engineer who, while working on water distribution system for the city of Dijon in the 1800s, wrote a monumental report concerning various aspects of water supply and distribution [32].\",\n \"In one of the appendixes of the document he presents an equation that describes the flow of water through a column filled with sand under a hydraulic gradient (Figure REF ) $q = k \\\\frac{s}{e} [h + e \\\\mp h_0]$ In which $q$ was the water flux, $e$ is the height of the sand column, $s$ is the cross sectional area of the column, $h$ is the water height in the upper part of the column (on the inlet), $h_0$ is the water height on the inferior part of the column (on the outlet).\",\n \"Figure: Schematics of sand column used in the original Darcy experiment (adapted from ).\",\n \"(a) is the water inlet, (b) and (d) are manometers, (c) is the sand column, (e) is the water outlet, (f) is the support, and (g) is a reservoir for water collecting and measuring outflow volume.If the water height is zero at the outlet, $h_0 = 0$ $q = k \\\\frac{s}{e} [h + e]$ He realized that the water flux through the sand column was proportional to the pressure difference between the inlet and the outlet and he defined the constant of proportionality as a coefficient k, dependent on the permeability of the sand (“dependant de la permeabilité de la couche”).\",\n \"So $k$ in this context was defined as a permeability, dependent on the physical characteristics of the solid phase.\",\n \"In contemporary terms, the height of water at the inlet and outlet is effectively the pressure potential in terms of height of water.\",\n \"Considering the potentials at the inlet and outlet, and if we establish the gravitational reference at the bottom of the column, we have at the inlet (top of the column): $\\\\psi _{gi} = e \\\\\\\\\\\\psi _{pi} = h + P_0\\\\\\\\\\\\psi _{ti} = e + h + P_0$ At the outlet (bottom of the column): $\\\\psi _{go} = 0 \\\\\\\\\\\\psi _{po} = h_0 + P_0\\\\\\\\\\\\psi _{to} = h_0 + P_0$ in which $P_0$ is the atmospheric pressure.\",\n \"Therefore, Darcy's law can be expressed as $q = k A\\\\frac{\\\\psi _{ti} - \\\\psi _{to}}{L} = k s \\\\frac{(e + h + P_0) - (h_0 + P_0)}{e} \\\\\\\\= k s\\\\frac{(e + h + P_0) - (h_0 + P_0)}{e} = k s\\\\frac{(e + h)}{e}$ for $L = e$ , the height of the column, $A = s$ the cross sectional area of the column and $h_0 = 0$ .\",\n \"The unidimensional form of Darcy's law, as applied to a column filled with saturated porous media is represented in modern notation in introductory soil physics and groundwater hydrology books as $\\\\boxed{Q = K_s A\\\\frac{\\\\psi _{ti} - \\\\psi _{to}}{L} = Q = K_s A\\\\frac{\\\\Delta \\\\psi _{t}}{L}}$ for total potential expressed in terms of height of water.\",\n \"A more precise general representation is in terms of differentials, thus for a one dimensional column oriented in some arbitrary $x$ direction, Darcy's law is $\\\\boxed{Q = -K_s A\\\\frac{d \\\\psi _{t}}{dx}}$ in which $Q$ is the water discharge in L$^3$ T$^{-1}$ , $K_s$ is now defined as the saturated hydraulic conductivity in L T$^{-1}$ and the term $\\\\frac{d \\\\psi _{t}}{dx}$ is termed the hydraulic gradient or the potential difference over the length between the two points in which potential is being measured, in L L$^{-1}$ .\",\n \"The negative sign, source of much confusion in soil physics, means that the flow is opposite to the direction of the gradient.\",\n \"In other words, the flow is always from the high total potential to the low total potential.\",\n \"As with the original Darcy formulation, these equations can be expressed in terms of height of water on the inlet and outlet of a saturated column filled with some porous material.\",\n \"If the column is oriented on the $z$ direction, here defined as in the direction of the gravitational field, we have $Q = -K_s A \\\\frac{d \\\\psi _{t}}{dz} = -K_s A \\\\frac{ (\\\\psi _{pi} + \\\\psi _{gi}) - (\\\\psi _{po} + \\\\psi _{go}) }{dz}$ If the length of the column is $z$ and establishing the gravitational potential at the bottom of the column at $z = 0$ , we have $Q = -K_s A \\\\frac{ (\\\\psi _{pi} + z) - (\\\\psi _{po} + 0) }{z}$ Representing the pressure potential in terms of height $h_i$ $Q = -K_s A \\\\frac{ (h_i-h_o) + z}{z} = -K_s A (\\\\frac{ h_i-h_o}{z} + 1)$ If the pressure potential is zero on the outlet $Q = -K_s A (\\\\frac{ h_i}{z} + 1)$ It is easy to show that if the column is oriented perpendicular to the gravitational field in some arbitrary $x$ (or $y$ ) direction, Darcy's law in terms of the height of water on the inlet and outlet will be $Q = -K_s A (\\\\frac{ h_i - h_o}{x})$ For $h_o$ = 0 this reduces to $Q = -K_s A \\\\frac{ h_i}{x}$ where $x$ (or $y$ ) represent the horizontal length of the column and $z$ the vertical length.\",\n \"It is clear that Darcy's law is a linear relationship between discharge and gradient with slope equal to the saturated hydraulic conductivity.\",\n \"Darcy's law, along with Fourier's law of heat conduction, Ohm's law in electromagnetism, Fick's law of molecular diffusion and Poiseuille's law is one of the linear gradient transport laws discovered in the 1800s [84].\",\n \"Although Darcy only mentions Poiseuille's law in his report, it is very likely that he was aware and was influenced by the physics of the time in creating his empirical law [84].\",\n \"Darcy's law can be written in terms of a flux density $q$ , similar to the other laws of physics mentioned, by dividing the discharge $Q$ by the area $A$ $q = \\\\frac{Q}{A} = -K_s A \\\\frac{d \\\\psi _t}{dx}$ Darcy obviously knew about and investigated Poiseuille's, as he was an engineer concerned with water distribution, which would obviously require studying water flow in pipes.\",\n \"Analyzing Poiseuille's solution for flow in a cylindrical conduit, $Q = -\\\\frac{\\\\pi }{8\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x} a^4$ we can rewrite this equation by reorganizing it $Q = -\\\\frac{\\\\pi a^2 a^2}{8\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x}$ and introducing $A = \\\\pi a^2$ as the area of the tube $Q = - A\\\\frac{\\\\pi a^2}{8\\\\mu }\\\\frac{\\\\partial p}{\\\\partial x}$ Rewriting the term $\\\\frac{\\\\pi a^2}{8\\\\mu }$ as some proportionality constant $k^*$ $Q = - A k^*\\\\frac{\\\\partial p}{\\\\partial x}$ Poiseulle's law can be interpreted as some limiting case of Darcy's law for when there is no porous media within the pipe.\",\n \"The presence of a porous medium causes additional stresses in the fluid that decrease the discharge under the same potential gradient.\",\n \"Because natural porous media usually have complex grain size distribution, grain shape and rugosity, resulting in complex pore size distribution and geometry, the empirical constant in Darcy's law, $K_s$ , has to be determined experimentally, there is no analytical solution of Darcy's law from the Navier-Stokes equation.\",\n \"There are a few studies that use stochastic considerations to derive Darcy's law from the Navier-Stokes equations, but as discussed previously, they must rely in some averaging mechanism, unless the porous media model is extremelly simplistic.\",\n \"In a stratified media, such as a soil profile, sedimentary rock or deposit, or laboratory column, where the direction of flux is perpendicular to the layering, the flow will be determined by some form of averaging of the individual hydraulic conductivity of the layers Here as a direct average, in a much more simplistic way than what was discussed previously for averaging of the Navier-Stokes equation at the pore scale.\",\n \"Consider a composite column, composed of two independently homogeneous but different materials $A$ and $B$ with heights $x_A$ and $x_B$ (REF ).\",\n \"The flux is in the $z$ direction and the total potential is measured on the points $a$ , $b$ and $c$ as $\\\\psi _{ta}$ , $\\\\psi _{tb}$ and $\\\\psi _{tc}$ .\",\n \"Darcy's law is defined for material $A$ in terms of the flux between points $a$ and $b$ $q_{ab} = -K_{sA} \\\\frac{\\\\psi _{tA} - \\\\psi _{tB}}{x_A}$ and for material $B$ in terms of the flux between points $b$ and $c$ $q_{bc} = -K_{sB} \\\\frac{\\\\psi _{tB} - \\\\psi _{tC}}{x_B}$ Figure: Darcy's law in a composite column with individually homogeneous but different materials AA and BB.As point $b$ is located in the interface between the two materials, the total potential is equal and we can solve both equations for $\\\\psi _{tB}$ .\",\n \"$\\\\psi _{tB} = \\\\psi _{tA} + \\\\frac{q_{ab}x_A}{ K_{sA}}$ $\\\\psi _{tB} = \\\\psi _{tC} - \\\\frac{q_{bc}x_B}{ K_{sB}}$ Replacing $\\\\psi _{tB}$ in any of the equations and solving for the total potential difference between the column's inlet in $a$ and the outlet in $c$ $\\\\psi _{tA} - \\\\psi _{tC} = - (\\\\frac{q_{bc}x_B}{ K_{sB}} + \\\\frac{q_{ab}x_A}{ K_{sA}})$ For constant (steady-state) saturated flow, for which Darcy's law is being considered, the flux in any point within the length of the composite column is constant, and thus $q_{ab} = q_{bc} = q_{ac} = q$ , and $\\\\psi _{tA} - \\\\psi _{tC} = - q(\\\\frac{x_B}{ K_{sB}} + \\\\frac{x_A}{ K_{sA}})$ Finally solving for $q$ we arrive at an expression for a composite material $q = -\\\\frac{ (\\\\psi _{tA} - \\\\psi _{tC})}{(\\\\frac{x_B}{ K_{sB}} + \\\\frac{x_A}{ K_{sA}})}$ The term $\\\\frac{1}{(\\\\frac{x_B}{ K_{sB}} + \\\\frac{x_A}{ K_{sA}})} = \\\\frac{K_{sA}K_{sB}}{x_BK_{sA} + x_AK_{sB}}$ is a form of weighed mean of the hydraulic conductivity of the layers with the height of each layer acting as a weighing term.\",\n \"In practical terms, especially in laboratory, saturated hydraulic conductivity can be determined using constant head and falling head permeametersAs we will see, hydraulic conductivity and permeability are different things, however, these devices historically have been called permeameters.\",\n \"As they can also be used to measure the permeability of the medium, we will maintain the use of the terminology.\",\n \"In constant head permeameters, the total potential or hydraulic head is kept constant in both ends of the column.\",\n \"The hydraulic head must be equal or greater than zero on both ends.\",\n \"If the total potential is negative the water will be under tension, the water content might be below saturation and hydraulic conductivity is no longer constant.\",\n \"Saturated hydraulic conductivity is calculated by measuring discharge or flux density as $K_s = -\\\\frac{QL}{A(h_i - h_o)}$ for a horizontal column, and $K_s = -\\\\frac{Q}{A[(h_i - h_o)/L +1]}$ for a vertical column with the gravitational reference chosen at the outlet.\",\n \"Note that the flow can be in any direction including upwards, the direction being indicated by the sign of the hydraulic conductivity.\",\n \"The second common type of laboratory method for measuring saturated hydraulic conductivity are falling head permeameters.\",\n \"In the falling head method the hydraulic head is not constant but falls from a height $h_0$ to a height $h$ in time.\",\n \"Falling head permeameters are convenient when high hydraulic heads are required [10].\",\n \"To derive the expression for measuring saturated hydraulic conductivity, consider a pemeameter connected to a tube filled with water on its base (Figure REF ).\",\n \"The flow is upwards and the discharge is measured at the outlet on the water level on the upper part of the column.\",\n \"The tube has diameter $d_t$ , radius $r_t$ and area $A_t$ and the column filled with porous media has diameter $d_c$ , radius $r_c$ and area $A_c$ .\",\n \"The inflow on the smaller tube is given by [37] $Q_{in} = -A_t \\\\frac{dh}{dt}$ while on the column the flow obeys Darcy's law $Q_{out} = \\\\frac{K A_c h}{L}$ Under saturated flow conditions the discharge into the column has to be equal to the discharge out of the column $Q_{in} = Q_{out}$ $-A_t \\\\frac{dh}{dt} = \\\\frac{K A_c h}{L}$ Reorganizing the terms and integrating $\\\\frac{dh}{h} = - \\\\frac{A_c K}{A_t L} dt$ $\\\\int \\\\frac{dh}{h} = - \\\\int \\\\frac{A_c K}{A_t L} dt$ $\\\\log {|h|} \\\\bigg |_{h}^{h_{0}} = - \\\\frac{A_c K}{A_t L} t \\\\bigg |_{t_{0}}^t$ By applying the boundary conditions $t = 0$ at $h = h_0$ , and considering $h$ as always positive we arrive at $\\\\log {h} - \\\\log {h_{0}} = - \\\\frac{A_c K}{A_t L} t$ Solving for $K$ and considering the area of the small tube as $A_t = \\\\pi r_t^2 $ and the area of the soil column as $A_t = \\\\pi r_t^2$ Remember that $\\\\log x - \\\\log y = \\\\log {x/y}$ and notice that we multiplied both sides by $-1$ to eliminate the negative sign in $K$ .\",\n \"$K = \\\\frac{r_t^2 L}{r_c^2 t} \\\\log \\\\frac{h_0}{h}$ or, in terms of diameter $K = \\\\frac{d_t^2 L}{d_c^2 t} \\\\log \\\\frac{h_0}{h}$ For alternative methods and equipment for measuring saturated hydraulic conductivity on the field and on the laboratory, the user is referred to [31].\",\n \"Figure: Diagram illustrating a generic the falling head permeameter.Up until now we have avoided dealing with vectors by working with flow in one dimension.\",\n \"A column filled with a porous medium saturated with water, where the diameter of the column is small and where the walls are impermeable can be treated as an unidimensional system and the flux density vector is oriented in the general direction of water flow.\",\n \"In soils and sedimentary deposits, aquifers, porous rocks and many other natural materials, flow is often not constrained to one dimension.\",\n \"This is especially true on the field, where water and contaminant transport occur in large scales and in response to total potential gradients, but also in laboratory, for large enough samples and when heterogeneity cannot be disregarded.\",\n \"Under such conditions, not only the flux density is a vector but the gradient is also a vector, and often hydraulic conductivity is a second order tensor.\",\n \"Recalling Chapter REF , Darcy's law can be generalized as $\\\\mathbf {q} = \\\\mathbf {K} \\\\cdot \\\\nabla \\\\psi _t$ In three-dimensional Cartesian coordinates, the flux density is decomposed as $\\\\mathbf {q} = q_x \\\\mathbf {i} + q_y \\\\mathbf {j} + q_x \\\\mathbf {k}$ while the three-dimensional gradient is $\\\\nabla = \\\\frac{\\\\partial \\\\psi _t}{\\\\partial x} \\\\mathbf {i} + \\\\frac{\\\\partial \\\\psi _t}{\\\\partial y} \\\\mathbf {j} + \\\\frac{\\\\partial \\\\psi _t}{\\\\partial z} \\\\mathbf {k}$ Under conditions of anisotropy, where hydraulic conductivity varies with direction in the material, saturated hydraulic conductivity is a second rank tensor with $3^2 = 9$ elements $\\\\mathbf {K} = K_{ij} =\\\\begin{pmatrix}K_{xx} & K_{xy} & K_{xz}\\\\\\\\K_{yx} & K_{yy} & K_{yz}\\\\\\\\K_{zx} & K_{zy} & K_{zz}\\\\end{pmatrix}$ With the conductivity tensor, Darcy's law can be written as $\\\\begin{pmatrix}q_{x}\\\\\\\\q_{y} \\\\\\\\q_{z}\\\\end{pmatrix}=\\\\begin{pmatrix}K_{xx} & K_{xy} & K_{xz}\\\\\\\\K_{yx} & K_{yy} & K_{yz}\\\\\\\\K_{zx} & K_{zy} & K_{zz}\\\\end{pmatrix}\\\\cdot \\\\begin{pmatrix}\\\\partial \\\\psi _t/\\\\partial x\\\\\\\\\\\\partial \\\\psi _t/\\\\partial y \\\\\\\\\\\\partial \\\\psi _t/\\\\partial z\\\\end{pmatrix}$ Therefore, the components of flux density in $x$ , $y$ and $z$ are $q_x = K_{xx} \\\\frac{\\\\partial \\\\psi _t}{\\\\partial x} + K_{xy} \\\\frac{\\\\partial \\\\psi _t}{\\\\partial y} + K_{xz}\\\\frac{\\\\partial \\\\psi _t}{\\\\partial z}$ $q_y = K_{yx} \\\\frac{\\\\partial \\\\psi _t}{\\\\partial x} + K_{yy} \\\\frac{\\\\partial \\\\psi _t}{\\\\partial y} + K_{yz}\\\\frac{\\\\partial \\\\psi _t}{\\\\partial z}$ $q_z = K_{zx} \\\\frac{\\\\partial \\\\psi _t}{\\\\partial x} + K_{zy} \\\\frac{\\\\partial \\\\psi _t}{\\\\partial y} + K_{zz}\\\\frac{\\\\partial \\\\psi _t}{\\\\partial z}$ The hydraulic conductivity tensor is a symmetrical tensor, meaning that $K_{ij} = K_{ji}$ such that the tensor reduces to six different components.\",\n \"There are profound implications related to these equations.\",\n \"The fact that the conductivity tensor is symmetrical is by no means a trivial issue and is a property of most of the transport equations and related to the theory know as the Onsager reciprocal relations, in the realm of non-equilibrium thermodynamics.\",\n \"Another important property of the conductivity tensor is that it is assumed as orthotropic, meaning that there are three principal directions for the conductivity vectors which are orthogonal to eachother.\",\n \"If this condition is valid, it is possible to orientate the axis in such a way that the conductivity tensor is reduced to three principal directions, such that $K_ij = 0$ for $i \\\\ne j$ and $K_{ii}$ for $i = j$ $\\\\mathbf {K} = K_{ij} =\\\\begin{pmatrix}K_{xx} & 0 & 0\\\\\\\\0 & K_{yy} & 0\\\\\\\\0 & 0 & K_{zz}\\\\end{pmatrix}$ In practice, this means that it is possible to orientate a sample of an anisotropic media in laboratory conditions such that only the three principal permeability directions need to be measured.\",\n \"If the medium is isotropic, there is no difference between $K_ii$ measured in any direction and $K$ is reduced to the constant hydraulic conductivity scalar $\\\\mathbf {K} = K_{ij} = K\\\\begin{pmatrix}1 & 0 & 0\\\\\\\\0 & 1 & 0\\\\\\\\0 & 0 & 1\\\\end{pmatrix}$ Suppose now that the saturated hydraulic conductivity does not vary with direction at each point in space, but the value, even being equal in all directions, is different in different points in space.\",\n \"In this case, the saturated hydraulic conductivity is a scalar function in space $K(x,y,z)$ and Darcy's law is $\\\\mathbf {q} = -K(x,y,z) \\\\nabla \\\\psi _t$ A medium where the hydraulic conductivity does not vary with direction at a given point in space is said to be isotropic and if the scalar hydraulic conductivity does vary in space is called heterogeneous.\",\n \"If the saturated hydraulic conductivity does not vary with direction and is the same in all points in space the medium is said to be homogenous and isotropic and Darcy's law simplifies to $\\\\mathbf {q} = -K \\\\nabla \\\\psi _t$ A consistent representation of Darcy's law requires that the potentials are expressed in terms of hydraulic height [L].\",\n \"This guarantees that the gradient $\\\\frac{\\\\partial \\\\psi _t}{\\\\psi x}$ or $y$ or $z$ is nondimentional [L L$^{-1}]$ and that the flux density has the same units of the saturated hydraulic conductivity [L T$^{-1}$ ].\",\n \"It is convenient now to fully introduce the notation used before for the total potential in height units as $h_t = h_p + h_g$ in which $h_t$ is the total potential, $h_p$ is the pressure potential and $h_g$ is the gravitational potential, all in [L] units.\",\n \"Using this notation and considering the $z$ direction as the direction of the gravitational force, the components of Darcy's law in $x$ , $y$ and $z$ are $q_x = -K \\\\frac{\\\\partial h_t}{\\\\partial x} = -K \\\\frac{h_{ti} - h_{to}}{x - x_0} = -K \\\\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{x - x_0}$ $q_y = -K \\\\frac{\\\\partial h_t}{\\\\partial y} = -K \\\\frac{h_{ti} - h_{to}}{y - y_0} = -K \\\\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{y - y_0}$ $q_z = -K \\\\frac{\\\\partial h_t}{\\\\partial y} = -K \\\\frac{h_{ti} - h_{to}}{z - z_0} = -K \\\\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{z - z_0}$ where the subscripts $i$ and $o$ stand for inlet and outlet.\",\n \"Setting the gravitational reference at $x_0$ , $y_0$ and $z_0$ , on the outlet, the gravitational potential will be $z_0$ in both the inlet and outlet in $x$ and $y$ and in the outlet only in $z$ .\",\n \"If the column height is $z$ , and the inlet is located in $z$ this will be the gravitational potential in the inlet for the vertical column.\",\n \"Using these conditions, the components of Darcy's law in three dimensions are $q_x = -K \\\\frac{(h_{pi} + z_0) - (h_{po} + z_0)}{x - x_0} = \\\\frac{\\\\partial h_{p}}{\\\\partial x}$ $q_y = -K \\\\frac{(h_{pi} + z_0) - (h_{po} + z_0)}{y - y_0} = \\\\frac{\\\\partial h_{p}}{\\\\partial y}$ $q_z = -K \\\\frac{(h_{pi} + z) - (h_{po} + z_0)}{z - z_0} = -K \\\\frac{(h_{pi} - h_{po}) + (z - z_0)}{z - z_0} = -K (\\\\frac{\\\\partial h_{p} + \\\\partial z}{\\\\partial z})$ $q_z = -K (\\\\frac{\\\\partial h_{p}}{\\\\partial z} + 1)$ Suppose now the total potential is expressed in pressure units.\",\n \"Using the $z$ coordinate for convenience $q_z = -K \\\\frac{\\\\partial \\\\psi _t}{\\\\partial z}$ it becomes clear that the the gradient is no longer dimensionless and the equation is not consistent in terms of units.\",\n \"Expressing the total potential in terms of the sum of the potentials $q_z = -K \\\\frac{\\\\partial (\\\\psi _p + \\\\psi _g)}{\\\\partial z} = -K \\\\frac{\\\\partial (\\\\rho _w g h + \\\\rho _w g z)}{\\\\partial z}$ and using the definitions from Chapter REF , it becomes clear that for consistence of units to be achieved, the saturated hydraulic conductivity would have to be divided by $\\\\rho _w g$ $q_z = -\\\\frac{K}{\\\\rho _w g} \\\\frac{\\\\partial (\\\\rho _w g h + \\\\rho _w g z)}{\\\\partial z}$ Therefore, for a liquid of constant density, Darcy's law in its original form is retrieved by placing the constant terms outside of the derivative $q_z = -\\\\frac{K \\\\rho _w g}{\\\\rho _w g} \\\\frac{\\\\partial (h + z)}{\\\\partial z} = -K \\\\frac{\\\\partial (h + z)}{\\\\partial z}$ Although stated otherwise in his report, the saturated hydraulic conductivity as defined by Darcy in his original experiment is a composite property of the solid and liquid phases [69].\",\n \"Although this is convenient in soil physics where the phases concerned are usually soil and water, it might not be convenient in groundwater hydrology and petroleum engineering, where different and often multiple phases are concerned and where the characteristics of the porous material might be of concern.\",\n \"Suppose you define a saturated conductivity term for water using a column experiment.\",\n \"You could repeat the same experiment with other fluids, for example oil, salt solution, gasoline, and for each fluid you would obtain one saturated conductivity.\",\n \"To circumvent this issue, and using dimensional analysis, the saturated conductivity term was rewritten in terms of the properties of the fluid being used, namely viscosity and density and a new coefficient, called intrinsic permeability or simply permeability $\\\\kappa $ , which depends on the solid phase alone, thus [73], [101], [69] $K = \\\\frac{\\\\kappa \\\\rho _f g}{\\\\mu }$ In which $\\\\rho _f$ and $\\\\mu $ are the density and viscosity of a fluid moving through the porous materialThese laws should be valid for viscous, incompressible newtonian fluids in general.\",\n \"Thus, Darcy's law can be expressed in terms of pressure using the permeability as $q_z = -\\\\frac{\\\\kappa \\\\rho _f g}{\\\\mu } \\\\frac{1}{\\\\rho _f g} \\\\frac{\\\\partial (\\\\rho _f g h + \\\\rho _f g z)}{\\\\partial z} = -\\\\frac{\\\\kappa }{\\\\mu } \\\\frac{\\\\partial (\\\\rho _f g h + \\\\rho _f g z)}{\\\\partial z}$ and in terms of length for the potential as $q_z = -\\\\frac{\\\\kappa \\\\rho _f g}{\\\\mu } \\\\frac{\\\\partial ( h + z)}{\\\\partial z}$ For an arbitrary $x$ direction perpendicular to Earth's gravitational field these reduce to $q_x = - \\\\frac{\\\\kappa }{\\\\mu } \\\\frac{\\\\partial (\\\\rho _f g h)}{\\\\partial z} = - \\\\frac{\\\\kappa }{\\\\mu } \\\\frac{\\\\partial P}{\\\\partial z}$ and $q_x = -\\\\frac{\\\\kappa \\\\rho _f g}{\\\\mu } \\\\frac{\\\\partial h}{\\\\partial x}$ Note that the similarity between Darcy's law and Poiseuille's equation is much more evident when the pressure form is considered.\",\n \"In general, the permeability is a tensor with the same properties of the saturated hydraulic conductivity, i.e.\",\n \"for an anisotropic medium $\\\\mathbf {k} = k_{ij} =\\\\begin{pmatrix}k_{xx} & k_{xy} & k_{xz}\\\\\\\\k_{yx} & k_{yy} & k_{yz}\\\\\\\\k_{zx} & k_{zy} & k_{zz}\\\\end{pmatrix}$ which can be orientated in the direction of the principal axes $\\\\mathbf {k} = k_{ij} =\\\\begin{pmatrix}k_{xx} & 0 & 0\\\\\\\\0 & k_{yy} & 0\\\\\\\\0 & 0 & k_{zz}\\\\end{pmatrix}$ and for an isotropic medium $\\\\mathbf {k} = k_{ij} = k\\\\begin{pmatrix}1 & 0 & 0\\\\\\\\0 & 1 & 0\\\\\\\\0 & 0 & 1\\\\end{pmatrix}$ We have seen that the flux density vector in Darcy's law is the discharge over area.\",\n \"Because the pore geometry is complex, a variety of shapes and sizes of pores are available for flow in porous materials.\",\n \"The flux density in Darcy's law is not the velocity of flow of fluid in the pores.\",\n \"Because the flow only takes place within the pores, the average flow velocity is the flux density divided by the porosity.\",\n \"$v = \\\\frac{Q}{An} = \\\\frac{q}{n}$ in which $n$ is the porosity.\",\n \"This relationship is valid because for an homogeneous porous material, area porosity is equivalent to the volumetric porosity [10].\",\n \"In other words, considering a cross section of the column, only part of the area is available for flow and for a principle equivalent to Bernoulli's law, the flux velocity at each point will be higher for smaller pores, within the limitations of viscosity and wall effects.\",\n \"The difference in velocities in a porous material will be an important mechanism in contaminant transport.\",\n \"Like many other equations and laws in fluid mechanics, Darcy's law is valid for laminar flow.\",\n \"Laminar flow here meaning the same as we discussed when analyzing the problem of a sphere settling under gravity.\",\n \"The flow is slow enough that there is no turbulence or vortexes.\",\n \"Poiseuille's law also has a restriction of laminar flow.\",\n \"As we seen for the sphere problem, one way of identifying an approximate limit between laminar and turbulent flow is by using the Reynolds number.\",\n \"For flow in a cylindrical conduit, the Reynolds number is $Re = \\\\frac{\\\\rho _f u d_h}{\\\\mu }$ In which $d_h$ is the hydraulic diameter of the pipe, simple the inner diameter if the tube is circular.\",\n \"For Darcy's law, this can be approximated by $Re = \\\\frac{\\\\rho _f u d^{\\\\prime }}{\\\\mu }$ In which $d^{\\\\prime }$ is some representative length dimension of the porous matrix representing the elementary channels of the porous medium, usually a parameter related to average grain diameter or other representative grain diameters [10] such as the diameter with 10$\\\\%$ , 30$\\\\%$ or or 50$\\\\%$ of the material passing through a sieve of mesh equal to the diameter of interest.\",\n \"Darcy's law implies a linear relationship between flux density and hydraulic gradient.\",\n \"The experimentally determined $R_e$ values for the transition zone from laminar to turbulent flow for Darcy's law ranges from 1 to 10, with laminar flow for values less than 1, turbulent flow for values above 10 [10].\",\n \"For values above 10 the relationship between flux density and gradient is no longer linear and Darcy's law cannot be used.\",\n \"As the flow velocity increases, inertia terms are no longer negligible.\",\n \"At the transition zone, the Darcy-Forchheimer equation can be used, presented here in one dimension as [86].\",\n \"$-\\\\frac{\\\\partial p}{\\\\partial x} = \\\\frac{\\\\mu }{\\\\kappa }q + \\\\beta \\\\rho _f q^2$ In which $\\\\beta $ is the Forchheimer coefficient or the inertial permeability.\",\n \"Note the quadratic term on the flux density.\",\n \"If the flow is too slow, for very small gradients and for some very fine soils and sediments, surface forces (Chapter REF ) can become important as well as viscous effects.\",\n \"It has been speculated that under such conditions water can display non-newtonian behavior and non-darcyan flow is observed [53], [10].\",\n \"Saturated hydraulic conductivity is a function of total porosity and pore size distribution.\",\n \"As such it is a function of bulk density and grain size distribution.\",\n \"Compaction and consolidations will decrease the total porosity and will tend to reduce saturated hydraulic conductivity.\",\n \"In soil science saturated hydraulic conductivity can be used as an indicator of soil degradation.\",\n \"Grain size distribution controls not only pore size distribution but also influences bulk density.\",\n \"Coarse grained materials can have lower total porosity but the average pore diameter will be greater, resulting in larger values of saturated hydraulic conductivity.\",\n \"For soils rich in 2:1 layer silicates, high clay content will result in lower saturated hydraulic conductivity, because of the smaller average pore radius and in some cases because of expansion of the silicate clays.\",\n \"For soils rich in iron and aluminum oxides, the saturated hydraulic conductivity is often very high, due to the microaggregate structure with two pore domains, micropores and mesopores, and macropores.\",\n \"Under saturated conditions the bulk of the water transport will take part in macropores.\",\n \"In sediments and rocks, grain size distribution and degree of consolidation will control saturated hydraulic conductivity.\",\n \"In highly consolidated igneous and metamorphic rocks, water transport can occur in fractures and hydraulic conductivity can be highly directional.\",\n \"Saturated hydraulic conductivity in the field is not a normally distributed variable and this should be accounted for when analyzing data.\",\n \"Bear [10] reports saturated hydraulic conductivity values ranging from $10^{2}~cm~s^{-1}$ for clean gravel to $10^{-11}~cm~s^{-1}$ for impervious rocks.\",\n \"The same source reports values ranging from $10^{-3}~cm~s^{-1}$ for sandy soils to $10^{-11}~cm~s^{-1}$ for dispersive clay soils.\",\n \"Because of the relationship of saturated hydraulic conductivity with porous media parameters, various attempts have been made to derive equations to calculate the saturated hydraulic conductivity from physical properties of the material.\",\n \"These models vary from physically based but approximated models based on Poiuseuille's equation to purely empirical models based on statistical regression.\",\n \"One model commonly found in engineering and hydrogeology is the Kozeny-Carman equationDerivations based on the original sources should be provided in later editions $k = \\\\frac{n^3}{(1-n)^2} \\\\frac{c_0}{M_s^2}$ in which $\\\\kappa $ is the permeability, $M_s$ the the specific surface area with respect to unit volume of porous medium, $n$ is the porosity and $c_0$ is the Kozeny's constant, dependent on the geometrical form of the channels on the model [10].\",\n \"The term $f(n) = \\\\frac{n^3}{(1-n)^2}$ is the porosity factor.\",\n \"It has been found that $c_0 = 1/5$ resulted good agreement with measured data, resulting in $k = \\\\frac{n^3}{(1-n)^2} \\\\frac{1}{5M_s^2}$ Defining the mean particle size as $d_m = M_s/6$ , the Kozeny-Carman equation can be written in terms mean of particle diameter as $k = \\\\frac{n^3}{(1-n)^2} \\\\frac{d_m^2}{180}$ In Chapter REF we derived a mass conservation equation (continuity equation) of the form $\\\\frac{\\\\partial \\\\rho }{\\\\partial t} = - \\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u})$ Suppose that water is being accumulated in the pore space of the representative volume of interest.\",\n \"If the media is completely saturated and the density of water is considered constant, $\\\\frac{\\\\partial \\\\rho }{\\\\partial t} = 0$ and $\\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u}) = \\\\rho \\\\nabla \\\\cdot \\\\mathbf {u}$ and, as before, we arrive at $\\\\nabla \\\\cdot \\\\mathbf {u} = 0$ For flow in porous media, the velocity term is replaced by the flux density $\\\\nabla \\\\cdot \\\\mathbf {q} = 0$ We already know that the flux density is represented by Darcy's law, thus, expanding the components of flux and replacing Darcy's law $(\\\\frac{\\\\partial }{\\\\partial x} \\\\mathbf {i} + \\\\frac{\\\\partial }{\\\\partial y} \\\\mathbf {j} + \\\\frac{\\\\partial }{\\\\partial y} \\\\mathbf {k}) \\\\cdot (q_x \\\\mathbf {i} + q_y \\\\mathbf {j} + q_z \\\\mathbf {k}) = 0\\\\\\\\(\\\\frac{\\\\partial q_x}{\\\\partial x} + \\\\frac{\\\\partial q_y}{\\\\partial y} + \\\\frac{\\\\partial q_z}{\\\\partial z}) = 0 \\\\\\\\(\\\\frac{\\\\partial }{\\\\partial x}(-K_x \\\\frac{\\\\partial h}{\\\\partial x}) + \\\\frac{\\\\partial }{\\\\partial y}(-K_y \\\\frac{\\\\partial h}{\\\\partial y}) + \\\\frac{\\\\partial }{\\\\partial z} (-K_z \\\\frac{\\\\partial h}{\\\\partial z}) ) = 0\\\\\\\\$ dropping the parentheses and multiplying both sides by $-1$ results in $\\\\frac{\\\\partial }{\\\\partial x}(K_x \\\\frac{\\\\partial h}{\\\\partial x}) + \\\\frac{\\\\partial }{\\\\partial y}(K_y \\\\frac{\\\\partial h}{\\\\partial y}) + \\\\frac{\\\\partial }{\\\\partial z} (K_z \\\\frac{\\\\partial h}{\\\\partial z}) ) = 0$ This equation is valid for a porous media where the unsaturated hydraulic conductivity cannot be assumed as constant.\",\n \"Because for a truly anisotropic media, the saturated hydraulic conductivity is a second rank tensor, a more general representation would be $\\\\nabla \\\\cdot (\\\\mathbf {K} \\\\cdot \\\\nabla {h}) = 0$ Back to more general cases, if the saturated hydraulic conductivities in each direction does not vary with the position in space, but the conductivities in each direction are different, we have $ K_x \\\\ne K_y \\\\ne K_z $ all constant, and $K_x \\\\frac{\\\\partial h^2}{\\\\partial ^2 x} + K_y \\\\frac{\\\\partial ^2 h}{\\\\partial y^2} + K_z \\\\frac{\\\\partial ^2 h}{\\\\partial z^2} = 0$ Finally, if the saturated hydraulic conductivity is constant everywhere, independent of direction, such that $ K_x \\\\ne K_y \\\\ne K_z = K $ we have $K \\\\frac{\\\\partial h^2}{\\\\partial ^2 x} + K \\\\frac{\\\\partial ^2 h}{\\\\partial y^2} + K \\\\frac{\\\\partial ^2 h}{\\\\partial z^2} = 0$ and dividing both sides by $K$ we arrive at $\\\\boxed{\\\\frac{\\\\partial h^2}{\\\\partial ^2 x} + \\\\frac{\\\\partial ^2 h}{\\\\partial y^2} + \\\\frac{\\\\partial ^2 h}{\\\\partial z^2} = 0}$ This is a partial differential equation know as Laplace equation.\",\n \"It is ubiquitous in physics and mathematics and can be represented using the aptly named Laplace operator ($\\\\nabla ^2$ ), of which we have made extensive usage in this text $\\\\boxed{\\\\nabla ^2 h = 0}$ Laplace's equation is a second-order partial differential equation that appears in electrostatics, gravitation, head conduction, transport phenomena and many other problems in applied physics, mathematics and engineering.\",\n \"As discussed previously, there is vast amount of literature covering methods of solution of partial differential equations, the user being encourage to consult the references.\",\n \"Analytical solutions in three and two dimensions can be found in the literature.\",\n \"As an example we will show how Laplace's equation can be solved in one-dimension to find the distribution of $h$ along a length of saturated porous media.\",\n \"Let us consider flow in the $x$ directionTo avoid any complication with implicit gravitational field, such that h is constant in $y$ and $z$ and the partial derivatives are zero $\\\\frac{\\\\partial h^2}{\\\\partial ^2 x} = 0 \\\\\\\\\\\\int \\\\partial h^2 = \\\\int 0 \\\\partial ^2 x \\\\\\\\\\\\int \\\\partial h = \\\\int C_1 \\\\partial x \\\\\\\\$ $h = C_1 x + C_2 \\\\\\\\$ To find the values of the constants one just needs to repeat the same procedure of applying the boundary conditions.\",\n \"Supposing, for example that $h = h_1$ at $x = 0$ and $h = h_2$ at $x = h$ , we have $h = \\\\frac{(h_2 - h_1)}{h} x + h_1 \\\\\\\\$ This shows that $h$ varies linearly along the length of the $x$ gradient.\",\n \"Laplace's equation solutions in 2D and 3D are highly dependent on the geometry of the problem, and can be challenging.\",\n \"For example, A general solution of Laplace's equation in a square geometry with boundary conditions with $h = 0$ in all sides except the right vertical wall will have the form $h(x, y) = \\\\sum _{n = 1}^\\\\infty c_n \\\\sinh (\\\\frac{n \\\\pi x}{b}) sin(\\\\frac{n \\\\pi y}{b})$ with $c_n = \\\\frac{2}{b \\\\sinh (\\\\frac{n \\\\pi a}{b})} \\\\int _0^b g_2(y) sin(\\\\frac{n \\\\pi y}{b}) dy$ in which $g_2(y)$ is a function that defines the nonzero boundary conditionhttp://www.math.ubc.ca/~peirce/M257_316_2012_Lecture_24.pdf.\",\n \"ll $ p $ Pressure $ \\\\mu $ Viscosity $ \\\\mathbf {u} $ Vector velocity function $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ \\\\mathbf {i}, \\\\mathbf {j}, \\\\mathbf {k} $ Unit vectors in Cartesian coordinates $ x, y, z$ Generalized Cartesian coordinates $ C_1, C_2 $ Constants $ h $ Distance between plates $ Q $ Discharge $ r $ Radial dimension $ \\\\phi $ Angle in polar coordinates $ q $ Water flux $ h^{\\\\prime } $ Height of water above a given point (hydraulic head) $ A, s $ Cross section area $ L, e $ Length $ \\\\psi $ Water potential $ P $ Pressure $ k , K_s $ Saturated hydraulic conductivity $ d_t $ Tube diameter $ r_t $ Tube radius $ A_t $ Tube cross sectional area $ d_c $ Column diameter $ r_c $ Column radius $ A_c $ Column cross sectional area $ t $ Time $ \\\\mathbf {q} $ Flux density vector $ \\\\mathbf {K} $ Saturated hydraulic conductivity tensor, vector form $ q_x, q_y, q_z $ Components of the flux density vector in Cartesian coordinates $ K_{ij} $ Saturated hydraulic conductivity tensor $ \\\\rho _w $ Water density $ \\\\rho _f $ Fluid density $ g $ Earth's gravitational acceleration $ k $ Intrinsic permeability $ \\\\mathbf {k} $ Permeability tensor, vector form $ k_{ij} $ Permeability tensor $ n $ Porosity $ Re $ Reynolds number $ \\\\beta $ Forchheimer coefficient $ M_s $ Specific surface area $ c_0 $ Kozeny's constant $ d_m $ Mean particle size $ \\\\rho $ Generalized density Edgar Buckingham, one of the forefathers of soil physics, wrote a 61 page report to the U.S. Department of agriculture which set the foundations of what today is the theory of unsaturated flow in soils [22].\",\n \"Using an analogy to other physical transport laws, he postulated that the capillary current density (equivalent to $q$ in modern notation) is proportional to the capillary potential gradient.\",\n \"Adapting to the notation used in this book and in one dimension His $\\\\lambda $ was replaced with $\\\\beta $ as we will need the former later $q = \\\\beta \\\\frac{\\\\partial \\\\psi _c}{\\\\partial x}$ Based on his observations and theoretical analysis he concluded that the capillary conductivity, $\\\\beta $ , was not constant and was a function of water content $\\\\beta = f(\\\\theta )$ .\",\n \"His conductivity law can be understood as a generalization of Darcy's law to unsaturated porous media, thus, writing the total potential in terms of hydraulic head, $h$ , and $\\\\beta $ as an unsaturated hydraulic conductivity, which is no longer constant but dependent on water content, Darcy's law can be written, for the one dimensional case as $\\\\boxed{q = K(\\\\theta ) \\\\frac{\\\\partial h}{\\\\partial x}}$ This is the one dimensional Darcy-Buckingham equation.\",\n \"Contrary to the saturated case, the unsaturated hydraulic conductivity is an empirical function of water content, and not a constant which can be determined experimentally.\",\n \"There has been a lot of effort in developing these empirical functions, with a lot of the fundamental equations being derived in the XX century.\",\n \"Many of the water retention functions presented in Chapter REF were essentially derived to arrive at an unsaturated hydraulic conductivity function.\",\n \"One of the fundamental differences between Darcy's law and the Darcy-Buckingham equation is that in all, or at least in some part of the porous media, the water content is below saturation.\",\n \"This implies that the pressure potential is zero and the matric potential is the component of interest along with the gravitational potential in the total potential.\",\n \"Instead of applying pressure, represented by a height of water column on the inlet and outlet of the column filled with soil or other porous media of interest, the outlet and inlet can be under tension or suction, by placing the water level below the height at the point of interest.\",\n \"Generally speaking, the pressure potential can be positive on the inlet and negative on the outlet which causes the column to be partially saturated in part of its extension and saturated on the rest.\",\n \"The most common example, given in many soil physics textbooks is a column which is under tension on both ends (Figure REF ).\",\n \"Usually, a porous plate is used on both ends of the column, similar to the system on a tensiometer.\",\n \"Figure: Unsaturated horizontal column.Flow is from point A from point B (right to left).\",\n \"By fixing the gravitational reference ($z = 0$ ) at the center of the column, where the inlet and outlet tubes are located, the matric potential is equal to the distance between the center of the column and the water level on the recipients connected to the inlet and outlet.\",\n \"The recipient connected to the outlet is fed from a water source to guarantee that the level of water is constant.\",\n \"Another way of achieving this, which is commonly used in laboratory, is to use a Mariotte's bottle.\",\n \"The scheme illustrated in Figure REF is valid as long as the column's height is small enough not to create significant gravitational potentials on the $z$ direction, this principle also being valid for saturated columns.\",\n \"The problem is critical in unsaturated soils because gravitational potential gradients can induce total potential gradients on the $z$ direction which induce hydraulic conductivity and velocity gradients.\",\n \"The gradient on the $x$ direction follows the decrease in water content along the column.\",\n \"For simple illustrative examples, if the gradient small and the column can be assumed as small enough on the $x$ length, the unsaturated hydraulic conductivity can be approximated as the average between the inlet and outlet.\",\n \"In real world applications an unsaturated hydraulic conductivity function is necessary.\",\n \"Because unsaturated hydraulic conductivity was theorized to be a function of water content and because water content has been experimentally shown to be a function of matric potential, hydraulic conductivity can be expressed as a function of matric potential $\\\\boxed{q = -K(h_m) \\\\frac{\\\\partial h}{\\\\partial x}}$ in which $h_m = \\\\psi _m$ is, as before, the matric potential expressed in height units.\",\n \"Expanding the total potential, we arrive at similar equations as for the saturated hydraulic conductivity in a general horizontal $x$ or $y$ direction $\\\\boxed{q = -K(h_m) \\\\frac{\\\\partial h_m}{\\\\partial x}}$ $\\\\boxed{q = -K(h_m) \\\\frac{\\\\partial h_m}{\\\\partial y}}$ and on the $z$ direction, parallel to the gravitational field $\\\\boxed{q = -K(h_m) (\\\\frac{\\\\partial h_m}{\\\\partial z} + 1)}$ Applying the Darcy-Buckingham equation to Figure REF , and expressing the total potential as $h_t$ for clarity $q = -K(h_m) (\\\\frac{h_{t,A} - h_{t,B}}{x_A - x_B}) = -K(h_m) (\\\\frac{(h_{m,A}-h_{g,A}) - (h_{m,B}-h_{g,B})}{x_A - x_B})$ Because $h_{g,A} = h_{g,B} = 0$ for a horizontal column and making $x_A - x_B = L$ the length of the column $q = -K(h_m) (\\\\frac{(h_{A}-0) - (h_{B}-0)}{L}) = -K(h_m) (\\\\frac{h_{A} - h_{B}}{L})$ where $h_A$ and $h_B$ are the distances between the center of the column, where the inlet and outlet are located, and the respective water levels at the reservoirs.\",\n \"We now know the transport equation for unsaturated porous media.\",\n \"The relationship indicated by the Darcy-Buckingham equation is physically and mathematically more complex than Darcy's law because introduces a hitherto unknown relationship between unsaturated hydraulic conductivity and water content or matric potential.\",\n \"Early experimental investigations indicated a nonlinear relationship between unsaturated hydraulic conductivity and water content or matric potential.\",\n \"As we have already indicated, unsaturated hydraulic conductivity functions have been derived from the water retention curve function.\",\n \"The unsaturated hydraulic conductivity functions can be derived from the water retention function using the frameworks proposed by Burdine [23], [19], [97] Remember from the previous chapter that $\\\\Theta $ is the relative saturation equal to $(\\\\theta - \\\\theta _r)/(\\\\theta _s - \\\\theta _r)$ $K_r(\\\\Theta ) = \\\\Theta ^2 \\\\int _0^{\\\\Theta } \\\\frac{d \\\\Theta }{\\\\psi _m^2(\\\\Theta )}\\\\bigg / \\\\int _0^1 \\\\frac{d \\\\Theta }{\\\\psi _m^2(\\\\Theta )}$ and Mualem [67], [97] $K_r(\\\\Theta ) = \\\\Theta ^{1/2} [\\\\int _0^{\\\\Theta } \\\\frac{d \\\\Theta }{\\\\psi _m(\\\\Theta )}\\\\bigg / \\\\int _0^1 \\\\frac{d \\\\Theta }{\\\\psi _m(\\\\Theta )}]^2$ Details about the derivation of these formulas can be found in [23], [19], [67].\",\n \"They are essentially based on pore size distribution considerations.\",\n \"The symbology used in these equations is the same as the one used in the previous chapter.\",\n \"Note that we are using $\\\\psi _m$ here so not to avoid being constrained by units, since these equations do not necessarily require matric potential in height units.\",\n \"The new term $K_r$ is the relative unsaturated hydraulic conductivity $K_r(\\\\Theta ) = \\\\frac{K(\\\\Theta )}{K_s}$ varying from 0 to 1, and in which $K_s $ is the saturated hydraulic conductivity.\",\n \"To obtain the unsaturated hydraulic conductivity function it is necessary to solve the water retention function for $\\\\psi _m$ , insert it on the Burdine or Mualem equations and solve the integrals.\",\n \"For the Brooks and Corey equation [19], the process is straightforward.\",\n \"Expressing the Brooks and Corey in terms of the relative saturation $\\\\Theta $ $\\\\Theta = (\\\\frac{\\\\psi _b}{\\\\psi _m})^{\\\\lambda }$ we first solve for the matric potential $\\\\psi _m = (\\\\frac{\\\\psi _b}{\\\\Theta ^{1/\\\\lambda }})$ then plug into the Burdine equation $K_r(\\\\Theta ) = \\\\Theta ^2 \\\\int _0^{\\\\Theta } \\\\frac{d \\\\Theta }{(\\\\frac{\\\\psi _b}{\\\\Theta ^{1/\\\\lambda }})^2}\\\\bigg / \\\\int _0^1 \\\\frac{d \\\\Theta }{(\\\\frac{\\\\psi _b}{\\\\Theta ^{1/\\\\lambda }})^2}\\\\\\\\= \\\\Theta ^2 \\\\int _0^{\\\\Theta } \\\\frac{\\\\Theta ^{2/\\\\lambda } d \\\\Theta }{\\\\psi _b^2} \\\\bigg / \\\\int _0^1 \\\\frac{\\\\Theta ^{2/\\\\lambda } d \\\\Theta }{\\\\psi _b^2}$ because $\\\\psi _b$ is constant $K_r(\\\\Theta ) = \\\\Theta ^2 \\\\int _0^{\\\\Theta } \\\\Theta ^{2/\\\\lambda } d \\\\Theta \\\\bigg / \\\\int _0^1 \\\\Theta ^{2/\\\\lambda } d \\\\Theta $ The solution of the integral form in the numerator and denominator is trivial, i.e.\",\n \"$\\\\int _a^b x^{2/c} dx = \\\\frac{x^{2/c+1}}{2/c+1} \\\\bigg |_a^b$ thus $K_r(\\\\Theta ) = \\\\Theta ^2 \\\\frac{[\\\\Theta ^{2/\\\\lambda +1}]_0^{\\\\Theta }}{[\\\\Theta ^{2/\\\\lambda +1}]_0^1}$ Applying the integration limits $K_r(\\\\Theta ) = \\\\Theta ^2 \\\\frac{\\\\Theta ^{2/\\\\lambda +1}}{1} = \\\\Theta ^2 \\\\Theta ^{2/\\\\lambda +1}$ or $K_r(\\\\Theta ) = \\\\Theta ^{2/\\\\lambda +3}$ Recalling that $\\\\Theta $ is the effective saturation and $K_r$ is the relative unsaturated conductivity, the equation can be written in terms of volumetric water content and unsaturated hydraulic conductivity $K(\\\\theta ) = K_s (\\\\frac{\\\\theta - \\\\theta _r}{\\\\theta _s - \\\\theta _r})^{2/\\\\lambda +3}$ To retrieve the matric potential form of unsaturated hydraulic conductivity just replace the original Brooks and Corey function for $\\\\Theta $ $K(\\\\psi _m) = K_s (\\\\frac{\\\\psi _b}{\\\\psi _m})^{2 +3 \\\\lambda }$ These last two equations are the Brooks and Corey unsaturated hydraulic conductivity equations derived using Burdine's approach.\",\n \"The formulas can also be derived using Mualem's formulas by the same process.\",\n \"For Brutsaert-van Genuchten type equations the process is not as straightforward.\",\n \"Following [97], we can solve the effective saturation form of the water retention function resulting in $\\\\psi _m = [\\\\frac{1}{\\\\alpha } (\\\\frac{1}{\\\\Theta ^{1/m}} - 1)^{1/n}]$ Replacing into the Mualem function $K_r(\\\\Theta ) = \\\\Theta ^{1/2} [\\\\int _0^{\\\\Theta } \\\\frac{d \\\\Theta }{[\\\\frac{1}{\\\\alpha } (\\\\frac{1}{\\\\Theta ^{1/m}} - 1)^{1/n}]}\\\\bigg / \\\\int _0^1 \\\\frac{d \\\\Theta }{[\\\\frac{1}{\\\\alpha } (\\\\frac{1}{\\\\Theta ^{1/m}} - 1)^{1/n}]}]^2$ Doing some simple algebraic manipulations and placing the constant $1/\\\\alpha $ term outside of the integral $K_r(\\\\Theta ) = \\\\Theta ^{1/2} [\\\\int _0^{\\\\Theta } (\\\\frac{\\\\Theta ^{1/m} }{1 - \\\\Theta ^{1/m}})^{1/n} d \\\\Theta \\\\bigg / \\\\int _0^1 (\\\\frac{\\\\Theta ^{1/m} }{1 - \\\\Theta ^{1/m}})^{1/n} d \\\\Theta ]^2$ The issue here is the solution of the integral $\\\\int (\\\\frac{\\\\Theta ^{1/m} }{1 - \\\\Theta ^{1/m}})^{1/n} d \\\\Theta $ The integral on the denominator is the same equation with different bounds.\",\n \"This equations does not have a closed form analytical solution.\",\n \"This means that the solution is in form of special functions, there is no simple expression in which you can plug the constants and obtain results without using numerical methods or approximations to the special functions.\",\n \"In the last chapter we encountered the complementary error function when dealing with water retention functions.\",\n \"Other important special functions are the gamma and beta functions and their incomplete counterparts [14].\",\n \"To find an analytical solution to this equation van Genuchten [97] used manipulations of the incomplete beta function [14].\",\n \"The first step is apply the transformation $\\\\Theta = y^m \\\\\\\\d\\\\Theta = m y^{m-1} dy$ which after replaced into the Equation REF results $\\\\int _0^{y^m} (\\\\frac{(y^m)^{1/m} }{1 - (y^m)^{1/m}})^{1/n} m y^{m-1} dy$ simplifying and rearranging $m\\\\int _0^{y^m} y^{1/n + m -1} (1-y)^{-1/n} dy$ This equation is a form of the incomplete beta function [14].\",\n \"These functions have a very important role in physics and mathematics and have been studied for hundreds of years, dating at least to the time of Euler and Legendre.\",\n \"There is no closed form analytical solution for the general case, but it is possible to introduce simplifications such that a solution can be found.\",\n \"The next simplification was to propose that for all integers $k = m-1+1/n$ the integration could be performed directly.\",\n \"For the particular case $k=0$ $m = 1-1/n$ We already seen this term in the last chapter when studying the Brutsaert-van Genuchten equation.\",\n \"It was then called the Mualem restriction, the reason why being apparent now.\",\n \"Applying this restriction to the incomplete beta function representation allows a straightforward solution of the integral.\",\n \"For simplicity we will replace $n$ rewriting the equation in terms of $m$ , thus $m\\\\int _0^{y^m} y^{m -m} (1-y)^{m-1} dy = m\\\\int _0^{y^m} (1-y)^{m-1} dy$ The solution to this equation is very simple and can be found using the $u$ substitution technique studied in introductory calculus books.\",\n \"Let $u = 1-y \\\\\\\\du = -dy$ we have $-m\\\\int _0^{y^m} u^{m-1} du$ Integrating in $u$ $-m \\\\frac{u^{m-1+1}}{m-1+1} |_0^{y^m} = u^{m} |_0^{y^m}$ Substituting $y$ back $-(1-y)^{m} |_0^{y^m}$ At the beginning of the derivation we established that $\\\\Theta = y^m$ , replacing the original variable $-(1-\\\\Theta ^{1/m})^{m} |_0^\\\\Theta = 1-(1-\\\\Theta ^{1/m})^{m}$ The exact same steps apply to the denominator, however the limits of integration result in $-(1-\\\\Theta ^{1/m})^{m} |_0^1 = 1$ Replacing the solutions of the integrals into the Mualem expression (Equation REF ) $K_r(\\\\Theta ) = \\\\Theta ^{1/2} [\\\\frac{1-(1-\\\\Theta ^{1/m})^{m} }{1}]^2$ and finally $K_r(\\\\Theta ) = \\\\Theta ^{1/2} [1-(1-\\\\Theta ^{1/m})^{m} ]^2$ with $m = 1-1/n \\\\\\\\0 < m < 1$ The van Genuchten-Mualem hydraulic conductivity function can be written in terms of $\\\\theta $ and $K(\\\\theta )$ as $K(\\\\theta ) = K_s (\\\\frac{\\\\theta -\\\\theta _r}{\\\\theta _s-\\\\theta _r})^{1/2} [1-(1-(\\\\frac{\\\\theta -\\\\theta _r}{\\\\theta _s-\\\\theta _r})^{1/m})^{m} ]^2$ The matric potential form can be retrieved by replacing the Brutsaert-van Genuchten in terms of $\\\\Theta $ Be careful not to mix the the subscript “m” on the matric potential symbol $\\\\psi _m$ with the empirical parameter “m” $K_r(\\\\psi _m) = \\\\frac{[1-(\\\\alpha \\\\psi _m)^{nm}((1+(\\\\alpha \\\\psi _m)^n)^{-m}]^2}{[1+(\\\\alpha \\\\psi _m)^n]^{m/2}}$ the derivation being left to the reader as an exercise.\",\n \"The van Genuchten unsaturated hydraulic conductivity functions can also be derived from the Burdine expression and the Brooks and Corey equations can be derived from the Mualem expression.\",\n \"In fact many new equations have been derived using these expressions.\",\n \"There are other approaches to derive unsaturated hydraulic conductivity functions.\",\n \"A comparison of the Brooks and Corey [19] and van Genuchten [97] relative unsaturated conductivity functions is shown in Figure REF for $\\\\lambda = 0.47$ and $m = 0.77$ .\",\n \"Figure: Relative unsaturated hydraulic conductivity as a function of effective saturation for the van Genuchten equation with m=0.77m = 0.77 and the Brooks and Corey equation with λ=0.47\\\\lambda = 0.47.In the previous chapter we derived the Laplace equation for a saturated porous media from the mass conservation equation (continuity equation).\",\n \"To derive the conservation equation for unsaturated porous media we again start from the generalized mass conservation equation presented in Chapter REF .\",\n \"$\\\\frac{\\\\partial \\\\rho }{\\\\partial t} = - \\\\nabla \\\\cdot (\\\\rho \\\\mathbf {u})$ The original derivation of this equation considered a control volume full of water or other Newtonian viscous fluid.\",\n \"We retrieved Laplace's equation on the last chapter by replacing the density of the fluid as the density of water $\\\\rho _w$ and by considering water as an incompressible fluid such that its density is constant resulting in it vanishing from the conservation equation.\",\n \"Note that for a saturated porous medium, the volume available for water storage is less than the total volume of interest and is equal to the porosity of the medium.\",\n \"For saturated porous media, assuming the porosity and water density as constant has the same implications as if the media was simply a volume filled with water, the rate of variation of water content with time is always zero.\",\n \"For unsaturated porous media these assumptions might not be valid as the volume (or mass) of water within the pore space might vary with time.\",\n \"In this case we are not immediately concerned with the variation of the density of fluid within the volume of interest, but we are concerned with the variation of the concentration of the fluid within the porous media, thus we can rewrite the conservation equation as $\\\\frac{\\\\partial c_w}{\\\\partial t} = - \\\\nabla \\\\cdot (\\\\rho _w \\\\mathbf {u})$ in which $c_w = m_w/V_t$ is now the water concentration in mass $m_w$ per unit of total volume of the porous medium $V_t$ $\\\\frac{\\\\partial \\\\frac{m_w}{V_t}}{\\\\partial t} = - \\\\nabla \\\\cdot (\\\\rho _w \\\\mathbf {u})$ We know that the density of water is $\\\\rho _w = m_w/V_w$ , thus solving for $m_w$ and replacing into the conservation equation $\\\\frac{\\\\partial \\\\frac{\\\\rho _w V_w}{V_t}}{\\\\partial t} = - \\\\nabla \\\\cdot (\\\\rho _w \\\\mathbf {u})$ Assuming that the density of water is constant, it can be placed outside the derivative and divergence terms $\\\\rho _w \\\\frac{\\\\partial \\\\frac{V_w}{V_t}}{\\\\partial t} = - \\\\rho _w \\\\nabla \\\\cdot \\\\mathbf {u}$ Recalling that $V_w/V_t$ is the volumetric water content $\\\\theta $ , and assuming that for a porous medium the velocity term $\\\\mathbf {u}$ can be represented by the flux density $\\\\mathbf {q}$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = - \\\\nabla \\\\cdot \\\\mathbf {q}$ With volumetric water content varying with time, this is a partial differential equation that is difficult to solve analytically without certain assumptions and simplifications.\",\n \"If the water content does not vary with time $\\\\partial \\\\theta / \\\\partial t = 0$ and $\\\\nabla \\\\cdot \\\\mathbf {q} = 0$ Note, however, that many of the simplifying assumptions used for saturated flow are not valid because $K$ and $\\\\theta $ are no longer constant in space, even under steady state conditions for a partially saturated porous medium.\",\n \"Before deriving the general equation that governs unsaturated flow in porous media we need to find a general form of the Darcy-Buckingham's equation.\",\n \"Theoretically, the most general form of Darcy-Buckingham's equation for an anisotropic, heterogeneous porous media would include a vector valued function in space and matric potential for $K$ The tensor form for anisotropic and heterogeneous porous media is speculative, the mathematical implications and the calculations have not being investigated in depth by the author $\\\\mathbf {q} = -\\\\mathbf {K}(\\\\mathbf {r}, \\\\psi _m) \\\\cdot \\\\nabla \\\\psi _t$ in which $\\\\mathbf {r} = x~ \\\\mathbf {i} + y~ \\\\mathbf {j}+ z~\\\\mathbf {k}$ For an homogeneous and anisotropic medium, unsaturated hydraulic conductivity does not vary in space, only with direction at a given point.\",\n \"In this case the components of the conductivity tensor are scalar functions of matric potential alone $\\\\mathbf {K}(\\\\psi _m) = -K(\\\\psi _m)_{ij} =\\\\begin{pmatrix}K(\\\\psi )_{xx} & K(\\\\psi _m)_{xy} & K(\\\\psi _m)_{xz}\\\\\\\\K(\\\\psi )_{yx} & K(\\\\psi _m)_{yy} & K(\\\\psi _m)_{yz}\\\\\\\\K(\\\\psi )_{zx} & K(\\\\psi _m)_{zy} & K(\\\\psi _m)_{zz}\\\\end{pmatrix}$ For an heterogeneous and isotropic system, the unsaturated hydraulic conductivity is no longer a tensor but a scalar function of space and matric potential, such that $K(x,y,z,\\\\psi _m)$ and $\\\\mathbf {q} = -K(x, y, z, \\\\psi _m) \\\\nabla \\\\psi _t$ The simplest case is when hydraulic conductivity is a function of matric potential only everywhere $\\\\mathbf {q} = -K(\\\\psi _m) \\\\nabla \\\\psi _t$ Richards [78] derived an equation governing the fluids in porous media by replacing the Darcy-Buckingham equation into the mass conservation equation for water $\\\\boxed{\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\nabla \\\\cdot ( K(\\\\psi _m) \\\\nabla \\\\psi _t )}$ This is known as the Richards or Richardson-Richards equation.\",\n \"Richardson derived an analogous equations and published it in a small section of a book dealing with numerical methods for weather prediction [79].\",\n \"The equation derived by Richardson was not derived from Darcy's law but included a “conductance of the channels connecting opposite faces” in a volume of soil.\",\n \"Whether Richards was aware of Richardson's work is unknown, although some authors believe he wasn't [55] Richards did not cite Richardson in his original work, but it is entirely possible that his ideas could have reached Richards, ideas seem to float through several means within a scientific epoch.\",\n \"Expansion of the divergence and gradient terms results in $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} [K(\\\\psi _m) \\\\frac{\\\\partial \\\\psi _t}{\\\\partial x}] + \\\\frac{\\\\partial }{\\\\partial y} [K(\\\\psi _m) \\\\frac{\\\\partial \\\\psi _t}{\\\\partial y}] + \\\\frac{\\\\partial }{\\\\partial z} [K(\\\\psi _m) \\\\frac{\\\\partial \\\\psi _t}{\\\\partial z}]$ The unsaturated hydraulic conductivity is not constant and cannot be placed outside the derivative.\",\n \"Writing in terms of the total potential in height of water $h = h_m + h_g$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} [K(h_m) \\\\frac{\\\\partial h}{\\\\partial x}] + \\\\frac{\\\\partial }{\\\\partial y} [K(h_m) \\\\frac{\\\\partial h}{\\\\partial y}] + \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) \\\\frac{\\\\partial h}{\\\\partial z}]$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} [K(h_m) \\\\frac{\\\\partial (h_m + h_g)}{\\\\partial x}] + \\\\frac{\\\\partial }{\\\\partial y} [K(h_m) \\\\frac{\\\\partial (h_m + h_g)}{\\\\partial y}] + \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) \\\\frac{\\\\partial (h_m + h_g)}{\\\\partial z}]$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} [K(h_m) (\\\\frac{\\\\partial h_m}{\\\\partial x} + \\\\frac{\\\\partial h_g}{\\\\partial x})] + \\\\frac{\\\\partial }{\\\\partial y} [K(h_m) (\\\\frac{\\\\partial h_m}{\\\\partial y} + \\\\frac{\\\\partial h_g}{\\\\partial y})] + \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) (\\\\frac{\\\\partial h_m}{\\\\partial z} + \\\\frac{\\\\partial h_g}{\\\\partial z})]$ Considering that the gradient of the gravitational potential is zero in $x$ and $y$ , i.e.\",\n \"the gravitational potential is constant such that $\\\\partial h_g/\\\\partial x = 0 $ and $\\\\partial h_g/\\\\partial y = 0$ and that in length units the gravitational potential difference in $z$ is equal to $\\\\partial z$ , Richardson-Richards equations in $x$ , $y$ and $z$ are $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} [K(h_m) \\\\frac{\\\\partial h_m}{\\\\partial x} ]$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial y} [K(h_m) \\\\frac{\\\\partial h_m}{\\\\partial y}]$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) (\\\\frac{\\\\partial h_m}{\\\\partial z} + \\\\frac{\\\\partial z}{\\\\partial z})] = \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) (\\\\frac{\\\\partial h_m}{\\\\partial z} + 1)]$ Richardson-Richards equation is a nonlinear differential equation which has analytical solutions only in very specific cases which might be physically realistic.\",\n \"One idea to put the equation in a form which could be solved more easily is to include a diffusivity term.\",\n \"Diffusion type equations have been studied for centuries and before high processing power computers were available, any simplification would help immensely in terms of simplifying an equation to find analytical solutions or to solve it numerically.\",\n \"Klute [54] noted an artifice used by Buckingham [22] and later by [27], [28] in order to write the Darcy-Buckingham and Richardson-Richards equations in terms of a hydraulic diffusivity term.\",\n \"First let us rewrite Richardson-Richards equation in the $z$ direction by simple using the distributive property of multiplication and recalling that the unsaturated hydraulic conductivity can be written as a function of volumetric water content $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [K(\\\\theta ) \\\\frac{\\\\partial h_m}{\\\\partial z}] + \\\\frac{\\\\partial K(\\\\theta ) }{\\\\partial z}$ Note that in the gradient term $\\\\frac{\\\\partial h_m}{\\\\partial z}$ if we can assume that the matric potential is a unique function of water content [71], then, by the chain rule for an implicit function [14] $\\\\frac{\\\\partial h_m}{\\\\partial z} =\\\\frac{\\\\partial h_m}{\\\\partial \\\\theta } \\\\frac{\\\\partial \\\\theta }{\\\\partial z}$ Replacing the expanded differential into the Richardson-Richards equation $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [K(\\\\theta ) \\\\frac{\\\\partial h_m}{\\\\partial \\\\theta } \\\\frac{\\\\partial \\\\theta }{\\\\partial z}]+ \\\\frac{\\\\partial K(\\\\theta ) }{\\\\partial z}$ The term $D(\\\\theta ) = K(\\\\theta ) \\\\frac{\\\\partial h_m}{\\\\partial \\\\theta }$ is now called the hydraulic diffusivity or soil water diffusivity [28], [54] and the Richardson-Richards equation can be written in water content form as $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial x}] + \\\\frac{\\\\partial K(\\\\theta ) }{\\\\partial z}$ Conversely, Richardson-Richards equation can also be written in the matric potential form by defining the implicit differential for $\\\\partial \\\\theta /\\\\partial t$ and replacing into the left side of the equation in terms of matric potential [50] $\\\\frac{\\\\partial \\\\theta }{\\\\partial h_m}\\\\frac{\\\\partial h_m}{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) \\\\frac{\\\\partial h_m}{\\\\partial z}] + \\\\frac{\\\\partial K(h_m) }{\\\\partial z}$ By defining a new function $C(\\\\theta ) = \\\\frac{\\\\partial \\\\theta }{\\\\partial h_m}$ called the hydraulic capacity, we can write the Richardson-Richards equation in terms of matric potential only as $C(h_m)\\\\frac{\\\\partial h_m}{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [K(h_m) \\\\frac{\\\\partial h_m}{\\\\partial z}] + \\\\frac{\\\\partial K(h_m) }{\\\\partial z}$ The hydraulic capacity is the slope of the water retention function.\",\n \"The reasoning for reducing the number of variables in these equations is that it should simplify the process of solution either analytically, when an analytical solution is possible, or numerically.\",\n \"Because of the functional relationship between water content and matric potential, and unsaturated hydraulic conductivity, the functions discussed in Chapter REF and in this chapter can be used when solving these equations.\",\n \"Because these functions exist, there is also a functional relationship between hydraulic diffusivity and water content.\",\n \"Notice that based on the discussion on the previous paragraphs, the hydraulic diffusivity can be written as [50] $D(\\\\theta ) = \\\\frac{K(\\\\theta )}{C(\\\\theta )}$ It is also worth noticing that the Darcy-Buckingham equation can be written in terms of diffusivity.\",\n \"In a generalized horizontal $x$ direction $q = D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial x}$ Beware, however, of the interpretation of the hydraulic diffusivity [53], [71].\",\n \"The introduction of the diffusivity form made it easier to solve the equation in certain cases, but the hydraulic diffusivity does not have the same meaning as a diffusion term in heat and solute dispersion processes.\",\n \"Keep in mind that many of these equations require a unique relationship between the variables unsaturated hydraulic conductivity, diffusivity and water content and/or matric potential, and between water content and matric potential.\",\n \"This assumption is not physically realistic because of hysteresis, the relationship between water content and matric potential is not unique in real natural porous media and dependents on the history of the processes, either if a drying or wetting processe is being considering.\",\n \"In addition, for wetting-drying cycles within a given water content range, there might by multiple possible paths (hysteresis loops).\",\n \"In other words, there is a non-unique relationship between water content and matric potential for real porous media, for a given matric potential there might be at least two corresponding water content values depending on the wetting history.\",\n \"Bruce and Klute [20] described a procedure do find the diffusivity function following a procedure that Ludwig Boltzmann [16] used to solve diffusion type equations, now known as the Boltzmann transformation.\",\n \"Recall the diffusivity form of the Richardson-Richards equation in a generalized horizontal $x$ direction $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial x} ]$ Boltzmann [16] showed that diffusion type equations can be solved by introducing a new variable $\\\\Lambda = x t^{-1/2}$ For the diffusivity equation of concern in soil and porous media transport, the volumetric water content is now a function of the new variable [53] $\\\\theta = f(\\\\Lambda )$ The idea now is to replace $\\\\Lambda $ into the Richardson-Richards equation in order to reduce it from a partial differential equation in $\\\\theta $ , $t$ and $x$ to a ordinary differential equation in $\\\\Lambda $ and $\\\\theta $ .\",\n \"The first step is to calculate the partial derivatives of $\\\\Lambda $ in $x$ and $t$ $\\\\frac{\\\\partial \\\\Lambda }{\\\\partial x} = \\\\frac{1}{\\\\sqrt{t}}$ $\\\\frac{\\\\partial \\\\Lambda }{\\\\partial t} = -\\\\frac{x}{2t^{3/2}} = -\\\\frac{\\\\Lambda }{2t}$ Next, the water content partial derivatives can be expanded by the chain rule.\",\n \"Recall that, by the definition of $\\\\Lambda $ , the unsaturated hydraulic conductivity is an implicit function of it such that $\\\\frac{\\\\partial \\\\theta }{\\\\partial x} = \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\partial \\\\Lambda }{\\\\partial x}$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\partial \\\\Lambda }{\\\\partial t}$ Since we have already calculated the partial derivatives of $\\\\Lambda $ it is just a matter of replacing them into the two equations above $\\\\frac{\\\\partial \\\\theta }{\\\\partial x} = \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{1}{\\\\sqrt{t}}$ $\\\\frac{\\\\partial \\\\theta }{\\\\partial t} = -\\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\Lambda }{2t}$ Replacing these into the Richardson-Richards equation results in $-\\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\Lambda }{2t} = \\\\frac{\\\\partial }{\\\\partial x} [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{1}{\\\\sqrt{t}}]$ Because the partial derivatives on the right side no longer depend on $t$ they can be placed outside of the differentials $-\\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\Lambda }{2t} = \\\\frac{1}{\\\\sqrt{t}} \\\\frac{\\\\partial }{\\\\partial x} [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]$ We can now expand the partial derivative in $x$ by the chain rule $-\\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\Lambda }{2t} = \\\\frac{1}{\\\\sqrt{t}} \\\\frac{\\\\partial }{\\\\partial \\\\Lambda } [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ] \\\\frac{\\\\Lambda }{\\\\partial x}$ and, since we already know $\\\\partial \\\\Lambda / \\\\partial x$ $-\\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } \\\\frac{\\\\Lambda }{2t} = \\\\frac{1}{\\\\sqrt{t}} \\\\frac{\\\\partial }{\\\\partial \\\\Lambda } [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]\\\\frac{1}{\\\\sqrt{t}}$ and finally [20] $-\\\\frac{\\\\Lambda }{2} \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } = \\\\frac{\\\\partial }{\\\\partial \\\\Lambda } [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]$ An experimentally based solution to this ordinary differential equation was proposed by [20] for the boundary conditions $\\\\theta = \\\\theta _i~ \\\\text{for}~ x > 0, t = 0 \\\\\\\\\\\\theta = \\\\theta _s~ \\\\text{for}~ x = 0, t \\\\ge 0$ in which $\\\\theta _i$ is the initial water content of the system and $\\\\theta _s$ is the saturated water content.\",\n \"These conditions are valid for an infinitely long column, such that the wetting front will not reach the other end.\",\n \"In practical terms, a long column filled with porous material at constant initial water content is placed in contact with a water source, such that the water content will increase with time.\",\n \"After an elapsed time $t$ , the $\\\\Lambda $ function can be evaluated along the length of the column.\",\n \"The $\\\\Lambda $ function would later be called the sorptivity of the soil or porous media.\",\n \"Under these conditions, the $\\\\Lambda $ form of Richardson-Richards equation can be integrated in the limits of initial water content to water content at a position $x$ and from $\\\\Lambda $ at the initial condition to $\\\\Lambda $ evaluated at $x$ $-\\\\int _{\\\\theta _i}^{\\\\theta _x}\\\\frac{\\\\Lambda }{2} \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } = \\\\int _{\\\\Lambda _i}^{\\\\Lambda _x} \\\\frac{\\\\partial }{\\\\partial \\\\Lambda } [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]$ which immediately simplifies to $-\\\\frac{1}{2}\\\\int _{\\\\theta _i}^{\\\\theta _x} \\\\Lambda \\\\partial \\\\theta = \\\\int _{\\\\Lambda _i}^{\\\\Lambda _x} \\\\partial [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]$ Noticing that $\\\\Lambda _i = \\\\Lambda _\\\\infty $ corresponds to $\\\\theta _i$ and that $\\\\Lambda _x$ corresponds to $\\\\theta _x$ , a change of variables can be performed on the right integral $-\\\\frac{1}{2}\\\\int _{\\\\theta _i}^{\\\\theta _x} \\\\Lambda \\\\partial \\\\theta = \\\\int _{\\\\theta _i}^{\\\\theta _x} \\\\partial [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]$ or $-\\\\frac{1}{2}\\\\int _{\\\\theta _i}^{\\\\theta _x} \\\\Lambda \\\\partial \\\\theta = [D(\\\\theta ) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]_{\\\\theta _i}^{\\\\theta _x} = [D(\\\\theta _x) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]_{\\\\theta _x} - [D(\\\\theta _i) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]_{\\\\theta _i}$ Because $\\\\theta _i$ is constant, the partial derivative $\\\\partial \\\\theta / \\\\partial \\\\Lambda $ evaluated at $\\\\theta _i$ is zero $\\\\frac{1}{2}\\\\int _{\\\\theta _i}^{\\\\theta _x} \\\\Lambda \\\\partial \\\\theta = - [D(\\\\theta _x) \\\\frac{\\\\partial \\\\theta }{\\\\partial \\\\Lambda } ]_{\\\\theta _x}$ Thus, the diffusivity can be calculated numerically and experimentally by [20] $D(\\\\theta _x) = -\\\\frac{1}{2} (\\\\frac{\\\\partial \\\\Lambda }{\\\\partial \\\\theta } )_{\\\\theta _x} \\\\int _{\\\\theta _i}^{\\\\theta _x} \\\\Lambda \\\\partial \\\\theta $ The procedure is described in [53] and on the original publication, among others [31].\",\n \"You have seen that the mathematical complexity of the models for describing unsaturated flow is substantially greater than that of saturated flow equations.\",\n \"The partial differential equations used includes the variables $K$ , $D$ , $\\\\theta $ and $h_m$ in addition to spatial coordinates.\",\n \"These variables are usually related to each other by nonlinear functions, which must be considered when searching for the solution.\",\n \"In many real world applications, in transient flow conditions and in dimensions greater that one, these equations might not have analytical solutions or can only be solved analytically using unrealistic boundary conditions.\",\n \"One example of simplification was described on the previous section.\",\n \"When analytical solutions are not possible, numerical solutions are used.\",\n \"In numerical methods the equations are solved numerically through approximation methods.\",\n \"In fact, much of fluid mechanics and many areas of applied physics and groundwater hydrology now rely heavily on numerical methods.\",\n \"A whole subarea of fluid mechanics called computational fluid mechanics is now on the frontier of knowledge in the field.\",\n \"Simple numerical solutions can be computed by hand but today and for real world scenarios, most applications will use a computer.\",\n \"Knowledge of informatics and programming languages is essential to applied numerical methods in soil physics and groundwater hydrology.\",\n \"ll $\\\\beta $ Capillary conductivity defined by Buckingham [22] $\\\\psi _c$ Capillary potential $x$ Generalized horizontal direction or coordinate $q$ Flux density $K(\\\\theta )$ Unsaturated hydraulic conductivity as a function of volumetric water content $h$ Total potential in length units $h_m$ Matric potential in length units $K(h_m)$ Unsaturated hydraulic conductivity as a function of matric potential $z$ Generalized vertical direction or coordinate $h_t$ Total potential in length units $h_g$ Gravitational potential in length units $K_r(\\\\Theta ) $ Unsaturated hydraulic conductivity as a function of effective saturation $\\\\Theta $ Effective saturation $= (\\\\theta - \\\\theta _r)/(\\\\theta _s - \\\\theta _r)$ $\\\\psi _m$ Matric potential in general units $\\\\lambda $ Pore distribution parameter of Brooks and Corey [19] $a, b, c$ Arbitrary constants $\\\\theta $ Volumetric water content $\\\\theta _r$ Residual water content $K_s$ Saturated hydraulic conductivity $y$ Parameter for transformation on the van Genuchten [97] solution or arbitrary horizontal direction or coordinate elsewhere $k$ Generic parameter for solving the Mualem equation in the van Genuchten framework $m$ Shape parameter for the van Genuchten equation $n$ Shape parameter for the van Genuchten equation $\\\\alpha $ Inverse of air entry pressure for the van Genuchten equation $\\\\rho $ Arbitrary density of a material for the generalized conservation equation $\\\\mathbf {u}$ Velocity vector for fluid flow in general $c_w$ Water concentration within a porous medium $\\\\rho _w$ Water density $t$ Time $m_w$ Water mass $V_w$ Water volume within a porous medium $\\\\mathbf {q}$ Flux density in a porous medium $\\\\mathbf {K}(\\\\mathbf {r}, \\\\psi _m) $ Unsaturated hydraulic conductivity tensor $\\\\mathbf {r}$ Position vector $K(x,y,z, \\\\psi _m) $ Unsaturated hydraulic conductivity scalar function $D(\\\\theta )$ Diffusivity function $C(\\\\theta )$ Water (hydraulic) capacity function $ = \\\\partial \\\\theta /\\\\partial \\\\psi _m$ $\\\\Lambda $ Variable introduced for the Boltzmann transformation In the last two chapters we studied the equation of conservation of water for porous media under saturated and unsaturated conditions.\",\n \"To study conservation of other types of mass in porous media, similar principles can be applied.\",\n \"Soluble solids are of immense interest in groundwater hydrology and soil physics because a large part of the groundwater and soil contaminants are soluble solids.\",\n \"Examples include pesticides, organic molecules, viruses, bacteria, radionuclides, ammonium, nitrates, phosphates and many others.\",\n \"In agriculture, fertilizers and other plant nutrients can be also transported through soil with water fluxes and by diffusion.\",\n \"Many compounds can act as either plant nutrients or contaminants, depending on where they are and on the amount and on intrinsic factors of soils and aquifers.\",\n \"The process can be also complicated as many solids are volatile and/or can be adsorbed onto the solid phase, suffer transformations, decay and many other processes.\",\n \"Suppose you have a volume of a porous media either soil, an aquifer or other geologic material.\",\n \"Suppose now that we are concerned with the dynamics of a single soluble solid.\",\n \"The concentration of the solid within the porous media is $C$ .\",\n \"One of the mechanisms by which the contaminant can be transported in and out of the volume of interest is by advection.\",\n \"Advective transport is the mass transport with the flow of the fluid in which it is dissolved.\",\n \"When the conservation equation was derived, we saw that the flux density was given by $\\\\mathbf {q}$ .\",\n \"The mass flux of the dissolved solid being transported in an unsaturated medium is given by $\\\\mathbf {J}_{adv} = \\\\theta \\\\mathbf {v} C$ in which $\\\\theta $ is the volumetric water content [L$^3$ L$^{-3}$ ], $\\\\mathbf {v}$ is the water velocity within the pores [L T$^{-1}$ ], as discussed in the previous chapter and $C$ is the concentration of the dissolved solid in the liquid phase [M L$^{-3}$ ].\",\n \"Dimensional analysis indicates that the vector $\\\\mathbf {J}_{adv}$ is the mass of the dissolved solid entering (or exiting) per unit area per unit time [M L$^{-2}$ T$^{-1}$ ].\",\n \"If there was no diffusion or hydrodynamic dispersion, an unrealistic scenario for dissolved solids in porous media, the conservation equation would be of the form $\\\\frac{\\\\partial (\\\\theta C)}{\\\\partial t} = -\\\\nabla \\\\cdot (\\\\theta \\\\mathbf {v} C) $ However, dissolved solids are subject to diffusion due to concentration gradients.\",\n \"As we discussed, regions of high concentration of dissolved solids will result in lower water potential, in this scenario diffusion tends to create equilibrium of concentration throughout the fluid, subjected that there are no other constraints regarding other potentials e.g.\",\n \"thermal and electrostatic.\",\n \"Considering an element of static, incompressible fluid, not within a porous medium, the diffusion of a soluble solid in the fluid can be modeled using Fick's law $\\\\mathbf {J}_{diff} = - D_d \\\\nabla C$ $\\\\mathbf {J}_{diff}$ is the diffusive flux [M L$^{-2}$ T$^{-1}$ ], $D_d$ is the diffusion coefficient in [L$^{2}$ T$^{-1}$ ] and $\\\\nabla C$ is the concentration gradient.\",\n \"Within a porous media, not all the space is available for diffusion.\",\n \"In a saturated porous media, the space occupied by solid particles is not available for diffusion, only the water saturated pore space.\",\n \"Even within a saturated porous medium, it might be that not all of the pore space is available to transport due to the fact that not all pores might be connected, and dead end pores [38].\",\n \"In unsaturated porous media, only the fraction of space saturated with water is available for transport [12].\",\n \"To account for the geometry of the pore space available for fluid flow and its modification with changing water content, a second rank tortuosity tensor is included into Fick's law of diffusion [11], [12] $\\\\mathbf {J}_{diff} = - D_d \\\\mathbf {T}(\\\\theta ) \\\\cdot \\\\nabla C$ The product of the diffusion coefficient by the tortuosity tensor can be written as a tensor, defined as the coefficient of diffusion for a a given phase in a partially saturated porous medium [12] $\\\\mathbf {J}_{diff} = - \\\\mathbf {D}^*(\\\\theta ) \\\\cdot \\\\nabla C$ where $ \\\\mathbf {D}^*(\\\\theta ) = D_d\\\\mathbf {T}(\\\\theta )$ An analysis of the diffusion and tortuosity tensors will be left for later editions, the interested reader should consult [11], [12].\",\n \"In essence this equation shows that the concentration of soluble solids will still tend to equilibrum due to concentration gradients in partially saturated porous media, but the volume available for diffusive movement will be limited by the volume occupied by water, and the path for movement will be constrained by the geometry of the porous space.\",\n \"Because of the geometry of the solid and pore space, the velocity and paths of water flux in porous media tends to be highly heterogeneous.\",\n \"In general, the more heterogeneous the porous media is in terms of grain size distribution and solid particles shapes and surface rugosity, the more heterogeneous the flux behavior will be, specially if the shapes, and rugosity distributions are more or less random.\",\n \"Grain and pore size distributions are still random but tend to follow certain statistical distributions and this has been taken advantage for modeling flow and transport using stochastic models.\",\n \"Because of the irregular distribution of velocities within the pores, the irregularity of the pore paths, a third transport mechanism called mechanical dispersion is also important.\",\n \"Even within a single pore, because of viscosity and surface forces, the velocity will be maximum at the center of the pore and zero at the surface if the pore is modelled as two infinite, parallel plates or as a cylinder, as we have seen when deriving Hagen-Poiseuille's equation in Chapter REF .\",\n \"All these sub-mechanisms cause mixing of the dissolved solid when the water in which it is dissolved is flowing through a porous medium.\",\n \"Dispersion has been shown experimentally to follow a Fick's diffusion type law [12] $\\\\mathbf {J}_{disp} = - \\\\mathbf {D} \\\\cdot \\\\nabla C$ in which $\\\\mathbf {D}$ is the dispersion coefficient, also a second rank tensor.\",\n \"Considering the three forms of transport, the total flux of a dissolved solid in a partially saturated porous medium is $\\\\mathbf {J}_{tot} =\\\\mathbf {J}_{adv} + \\\\mathbf {J}_{diff} + \\\\mathbf {J}_{disp} = \\\\theta \\\\mathbf {v} C - (\\\\mathbf {D}^*(\\\\theta ) \\\\cdot \\\\nabla C + \\\\mathbf {D} \\\\cdot \\\\nabla C)$ The diffusion and mechanical dispersion components can be lumped into a single tensor called the coefficient of hydrodynamic dispersion $\\\\mathbf {D}_h$ $\\\\mathbf {D}_h = \\\\mathbf {D} + \\\\mathbf {D}^*(\\\\theta )$ such that the total flux can be written as $\\\\mathbf {J}_{tot} =\\\\mathbf {J}_{adv} + \\\\mathbf {J}_{diff} + \\\\mathbf {J}_{disp} = \\\\theta \\\\mathbf {v} C - \\\\mathbf {D}_h \\\\cdot \\\\nabla C$ In theory our work is almost be done, considering the conservation of mass and the total flux of the dissolved solid, the conservation equation should be something like $\\\\frac{\\\\partial (\\\\theta C)}{\\\\partial t} = -\\\\nabla \\\\cdot (\\\\theta \\\\mathbf {v} C - \\\\mathbf {D}^*(\\\\theta ) \\\\cdot \\\\nabla C - \\\\mathbf {D} \\\\cdot \\\\nabla C)$ In reality, the behavior of chemicals in porous media is much more complicated than that.\",\n \"Many chemicals can transition from one phase to another, for example, several compounds exist as a solid or as a liquid or as both depending on chemical equilibrium conditions, other chemicals can be adsorbed onto solid surfaces by several different mechanisms, including those discussed in Chapter REF , while some are subjected to both adsorption and volatilization.\",\n \"In addition to that, some chemicals can be created or destroyed while being transported, for example, radionuclides can decay generating different isotopes or elements, organic matter can be transformed or destroyed by organisms and other bio-degradation processes, and both mechanisms can be effectively creating new chemicals in the environment.\",\n \"Because of that, a general equation of transport of chemicals in porous media must include terms for exchanges among phases, and sources and sinks of compounds, a source meaning a term for creation of a substance and a sink a term for the elimination of a given substance Obviously conservation laws still apply, a substance is eliminated by being transformed into other forms of matter or energy and created from other forms of matter or energy.\",\n \"Following the language of [12] the mass balance equation for a single chemical species in a porous media including sources and sinks is $\\\\frac{\\\\partial (\\\\theta C + \\\\rho F)}{\\\\partial t} = -\\\\nabla \\\\cdot (\\\\theta \\\\mathbf {v} C - \\\\mathbf {D}^*(\\\\theta ) \\\\cdot \\\\nabla C - \\\\mathbf {D} \\\\cdot \\\\nabla C) + \\\\theta \\\\rho _{f} \\\\Gamma _{f} + \\\\rho \\\\Gamma _s$ in which the total mass balance term in time $\\\\frac{\\\\partial (\\\\theta C + \\\\rho F)}{\\\\partial t}$ now accounts for the rate of variation in time of the amount adsorbed into the solid phase in the form of the component $\\\\frac{\\\\partial \\\\rho F}{\\\\partial t}$ where $F$ is the mass adsorbed into the solid phase per unit mass of solid phase and $\\\\rho $ is the bulk density of the porous medium.\",\n \"The source (or sink) term of the chemical in the liquid phase is given by $\\\\theta \\\\rho _f \\\\Gamma _f$ in which $\\\\theta $ is, as before, the volumetric water content, $\\\\rho _f$ is the density of water, or of the liquid phase in general, and $\\\\Gamma _f$ is the rate of creation (or destruction if the signal is negative) of the chemical per unit mass of water [M T$^{-1}$ M$^{-1}$ ].\",\n \"The source (or sink) term for the chemical in the solid phase is given by $\\\\rho \\\\Gamma _s$ in which $\\\\Gamma _s$ is the rate of creation (or destruction if the signal is negative) per unit mass of solid phase [M T$^{-1}$ M$^{-1}$ ].\",\n \"In theory, the transport equation can be further expanded to include transport in the gas phase, by including a source term in the gas phase and the vapor diffusive flux [47].\",\n \"The form without the gas transport is commonly used in soil physics and hydrology and has been called the advection-dispersion or advective-dispersive equation when the lumped hydrodynamic dispersion term is considered $\\\\frac{\\\\partial (\\\\theta C + \\\\rho F)}{\\\\partial t} = -\\\\nabla \\\\cdot (\\\\theta \\\\mathbf {v} C - \\\\mathbf {D}_h \\\\cdot \\\\nabla C) + \\\\theta \\\\rho _{f} \\\\Gamma _f + \\\\rho \\\\Gamma _s$ The full equation with the tensor components in three dimensions is obviously too complex for any analytical treatment.\",\n \"Solutions for several cases are available on the literature, in general [12], [38] and for particular cases [98] Another fundamental reference for the mathematics of diffusion and numerical methods is the book by John Crank [30], Crank along with Phyllis Nicolson also created an important finite differences scheme for solving differential equations.\",\n \"One obvious simplification, especially useful for laboratory column experiments and field studies of soil profiles or geological strata is to consider a single dimension.\",\n \"Thus, in a generalized horizontal $x$ direction $\\\\frac{\\\\partial (\\\\theta C + \\\\rho F)}{\\\\partial t} = - \\\\frac{\\\\partial }{\\\\partial x} (\\\\theta \\\\mathbf {v} C) + \\\\frac{\\\\partial }{\\\\partial x} (\\\\mathbf {D}_h \\\\frac{\\\\partial C}{\\\\partial x}) + \\\\theta \\\\rho _{f} \\\\Gamma _f + \\\\rho \\\\Gamma _s$ For one dimensional flow of a non-adsorbing, chemical which suffers no degradation and is not created within the porous media the sources the terms can be discarded.\",\n \"An example would be dissolved sodium chloride (NaCl) being transported in water through a column composed of silica sand.\",\n \"A further simplification can be applied if the hydrodynamic dispersion coefficient is constant for the fluid and can be placed outside the partial derivative.\",\n \"With all these simplifications, the conservation equation is $\\\\frac{\\\\partial (\\\\theta C)}{\\\\partial t} = - \\\\frac{\\\\partial }{\\\\partial x} (\\\\theta v_x C) + D_h \\\\frac{\\\\partial ^2 C}{\\\\partial x^2}$ If the water content and velocity are constant they can be placed outside the derivative terms and $\\\\frac{\\\\partial C}{\\\\partial t} = - v_x \\\\frac{\\\\partial C}{\\\\partial x} + D \\\\frac{\\\\partial ^2 C}{\\\\partial x^2}$ where $D = D_h/\\\\theta $ .\",\n \"This equation can be solved for different boundary conditions.\",\n \"For an infinite uni-dimensional domain saturated with water with initial chemical concentration $C = 0$ in which a solution containing a chemical (or a tracer) is introduced in the inlet with concentration $C = C_0$ subjected to the following the initial condition $C(x,0) = 0 ~ \\\\text{at}~ x \\\\ge 0$ and boundary conditions $C(0,t) = C_0~ \\\\text{for}~ t \\\\ge 0 \\\\\\\\C(\\\\infty ,t) = 0~ \\\\text{for}~ t \\\\ge 0$ the solution is provided by Ogata and Banks [74] The solution spans at least three pages in the original and might be included in further editions $\\\\frac{C(x,t)}{C_0} = \\\\frac{1}{2} [\\\\text{erfc}(\\\\frac{x-v_xt}{2\\\\sqrt{Dt}}) + \\\\exp {(\\\\frac{v_x x}{D})}\\\\text{erfc}(\\\\frac{x+v_xt}{2\\\\sqrt{Dt}})]$ where erfc is the complementary error function presented in Chapter REF .\",\n \"For a cylinder filled with a static fluid in which a pulse of solution with concentration $C_0$ is injected at $x=0$ , the concentration will spread symmetrically from the point of application with time following [30] $C(t) = \\\\frac{M}{2\\\\sqrt{\\\\pi D t}} \\\\exp {(-\\\\frac{x^2}{4Dt})}$ in which $M$ is the mass of substance deposited at time $t= 0$ at $x = 0$ .\",\n \"This solution has been adapted to a cylinder filled with porous media in which the fluid is moving with velocity $v_x$ by considering a relative displacement term $x - v_x t$ [10], [81] $\\\\frac{C(x,t)}{C_0} = \\\\frac{1}{2\\\\sqrt{\\\\pi D t}} \\\\exp {[-\\\\frac{(x-v_x t)^2}{4Dt}]}$ Jury and Sposito [48] working within a different framework arrived at a solution of similar form with the exception of a cubic time term and an additional $x$ length term $\\\\frac{C(x,t)}{C_0} = \\\\frac{x}{2\\\\sqrt{\\\\pi D t^3}} \\\\exp {[-\\\\frac{(x-v_x t)^2}{4Dt}]}$ Equations REF , REF and REF can be applied to finite column of length $L$ to predict the relative concentration $C/C_0$ at $x = L$ with time $t$ .\",\n \"Simulated values of $C/C_0$ as a function of time for a $100~cm$ saturated horizontal column with solution moving with velocity $v_x = 2.0~cm~h^{-1}$ for the hydrodynamic dispersion coefficients $D = D_h/\\\\theta $ of $0.5~cm^2~h^{-1}$ , $5.0~cm^2~h^{-1}$ , $10.0~cm^2~h^{-1}$ and $50.0~cm^2~h^{-1}$ are presented for each of the solutions.\",\n \"Equation REF predicts an increase in the concentration outlet from zero to one with the time for solute to first appear on the outlet depending on the diffusivity (Figure REF ).\",\n \"These types curves are called breakthrough curves because the model predicts the time and behavior for the solute to break through the column.\",\n \"With increasing hydrodynamic dispersion coefficient, the contribution of dispersion and diffusion to solute transport increases, ultimately resulting in some of the solute breaking through earlier than the average velocity of the fluid alone would predict.\",\n \"Dispersion and diffusion also cause a dilution of the front with respect to the concentration of the applied pulse.\",\n \"Small values of $D$ indicate little contribution of diffusion and dispersion as can be observed from the increase in the slope of the curve at lower values of $D$ .\",\n \"At a theoretical $D=0$ the solute pulse would move exactly at the velocity of the fluid flow and would appear at once at the end of the column without change in concentration.\",\n \"This type of theoretical flow is called piston flow, because it would mimic the behavior of a fluid being pushed by a piston, as in a syringe for example.\",\n \"Figure: Breakthrough curves for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~ h^{-1} and different hydrodynamic dispersion parameter values.For the same simulation parameters used for Equation REF Equations REF and REF predict the behavior of a pulse of a solution of concentration $C_0$ with time (Figures REF , REF ).\",\n \"The two equations predict similar results as $D$ decreases.\",\n \"Equation REF tends to predict higher concentrations as $D$ is increased when compared to Equation REF .\",\n \"For a theoretical $D = 0$ both equations would predict a single instantaneous pulse, sometimes modeled as a dirac delta function.\",\n \"As $D$ increases, an increase in the spreading of the pulse is observed, the outlet concentration reaching smaller maximum output concentrations.\",\n \"Figure: Behaviour of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm ~h^{-1} and different hydrodynamic dispersion parameter values .Figure: Behavior of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~h^{-1} and different hydrodynamic dispersion parameter values .Suppose now you have a chemical that does not have any sources of sinks but can be adsorbed into the solid phase, as it is the case with ions in solution and charged compounds in soils.\",\n \"In one dimension, the mass balance equation for a single species can be written as [47], [12] $\\\\frac{\\\\partial (\\\\theta C)}{\\\\partial t} + \\\\frac{\\\\partial \\\\rho F}{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial x} D_h \\\\frac{\\\\partial C}{\\\\partial x} - \\\\frac{\\\\partial }{\\\\partial x} (\\\\theta v_x C)$ For a saturated medium, the volumetric water content $\\\\theta $ is constant and equal to the total porosity $\\\\phi $ and can be placed outside of the partial derivatives.\",\n \"Assuming also the hydrodynamic dispersion coefficient and flow velocity as constant, the equation simplifies to $\\\\phi \\\\frac{\\\\partial C}{\\\\partial t} + \\\\frac{\\\\partial \\\\rho F}{\\\\partial t} = D_h \\\\frac{\\\\partial ^2 C}{\\\\partial x^2} - \\\\phi v_x \\\\frac{\\\\partial C}{\\\\partial x}$ If the mass adsorbed onto the solid phase is a linear function of the concentration in the liquid phase, we can define the following relationship $F = K_d C$ This equation describes a linear adsorption isotherm with $K_d$ being the partition coefficient between the solid and liquid phases, and it is the slope of the relationship between $F$ and $C$ .\",\n \"Replacing it into the mass balance equation $\\\\phi \\\\frac{\\\\partial C}{\\\\partial t} + \\\\frac{\\\\partial \\\\rho K_d C }{\\\\partial t} = D_h \\\\frac{\\\\partial ^2 C}{\\\\partial x^2} - \\\\phi v_x \\\\frac{\\\\partial C}{\\\\partial x}$ Because $K_d$ does not depend on the concentration and the bulk density of the porous medium it is assumed as constant $\\\\phi \\\\frac{\\\\partial C}{\\\\partial t} + \\\\rho K_d\\\\frac{\\\\partial C }{\\\\partial t} = D_h \\\\frac{\\\\partial ^2 C}{\\\\partial x^2} - \\\\phi v_x \\\\frac{\\\\partial C}{\\\\partial x}$ Reorganizing and rearranging $(1 + \\\\frac{\\\\rho K_d}{\\\\phi })\\\\frac{\\\\partial C }{\\\\partial t} = D \\\\frac{\\\\partial ^2 C}{\\\\partial x^2} - v_x \\\\frac{\\\\partial C}{\\\\partial x}$ In which $D_h = D/\\\\phi $ .\",\n \"Defining a coefficient of retardation $R_d$ [12] $R_d \\\\equiv 1 + \\\\frac{\\\\rho K_d}{\\\\phi }$ the mass balance equation is $R_d\\\\frac{\\\\partial C }{\\\\partial t} = D \\\\frac{\\\\partial ^2 C}{\\\\partial x^2} - v_x \\\\frac{\\\\partial C}{\\\\partial x}$ The retardation coefficient is a number equal to or greater than zero.\",\n \"There are different adsorption isotherms, including nonlinear relationships between $F$ and $C$ and in the mass balance equation needs to be derived for each specific case.\",\n \"For linear adsorption isotherms, the retardation coefficient appears in the breakthrough and pulse solutions as a linear term [85] $\\\\frac{C(x,t)}{C_0} = \\\\frac{1}{2} [\\\\text{erfc}(\\\\frac{R_d x-v_xt}{2\\\\sqrt{D R_d t}}) + \\\\exp {(\\\\frac{v_x x}{D})}\\\\text{erfc}(\\\\frac{R_d x+v_xt}{2\\\\sqrt{DR_d t}})]$ $\\\\frac{C(x,t)}{C_0} = \\\\frac{x}{2\\\\sqrt{\\\\pi D t^3 / R_d }} \\\\exp {(-\\\\frac{(R_d x-v_x t)^2}{4DR_d t})}$ The retardation coefficient, as the name indicates, controls the rate of retardation or a delay in the transport of the chemical throughout the column and will tend to increase the width of the pulse.\",\n \"The effect of the retardation coefficient in Equation REF is illustrated for $R_d = 0.25$ , $0.50$ and $1.0$ (Figure REF ).\",\n \"Figure: Behavior of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~h^{-1} and different retardation factors.Note that for unsaturated porous media the retardation factor is a function of volumetric water content.\",\n \"There are numerous other cases and solutions that could be covered including unsaturated cases, presence of mobile and immobile fluid phases, partition among gas, solid and liquid phases, radioactive decay and others.\",\n \"For the time being the interested reader is encouraged to consult [98], [12], [38], [85].\",\n \"The framework for discussing gas transport in porous media is the same as that of general chemical transport discussed in the previous section.\",\n \"Technically, gas transport could have been included with chemical transport, especially for gases that partition between liquid and gas phases and can be adsorbed into solids.\",\n \"In this section we will provide a brief discussion of the specifics of gas transport.\",\n \"As with other types of mass transport, gases can be transported by advection and diffusion.\",\n \"Gas diffusion is controlled by Fick's law, one of the general transport laws in physics discussed in previous sections.\",\n \"For a binary mixture of gases, the mole flux of the component of interest is given by [100] $\\\\mathbf {J}_g^{mole} = -C_g \\\\mathbf {D}_{g(ij)} \\\\nabla \\\\chi _g$ in which $C_g$ is the concentration of the gas, $\\\\mathbf {D}_g$ is the diffusion coefficient tensor and $\\\\chi _g$ is the mole fraction of the component gas of interest.\",\n \"This relationship can be written in terms of a mass flux as $\\\\mathbf {J}_g^{mass} = -\\\\rho _g \\\\mathbf {D}_{g(ij)} \\\\nabla \\\\omega _g$ in which $\\\\omega _g$ is the mass fraction of the component of interest.\",\n \"Now $\\\\mathbf {J}_g$ and $\\\\mathbf {J}_g^{mole}$ indicate the flux relative to the average mole or mass velocity of the mixture and not to stationary coordinates [13], [100].\",\n \"Supposing that the diffusion coefficient is constant through the medium, it does not need to be treated as a tensor and $\\\\mathbf {J}_g^{mass} = -\\\\rho _g D_g \\\\nabla \\\\omega _g$ As we have discussed for water and chemicals, the general law for a volume devoid of solid phase is not valid when the diffusion or advection is taking place within a porous medium.\",\n \"If the porous medium was completely devoid of liquid, i.e.\",\n \"completely dry, the solid particles would still occupy part of the volume and generate a complex and tortuous pore space through which the gas would need to navigate.\",\n \"For a completely saturated porous medium, the gas phase would need to diffuse through the liquid, the rates being often orders of magnitude lower than that in air.\",\n \"If the pore space is partially saturated with water, only a fraction of the pore space inversely proportional to water content is available for diffusion and advection.\",\n \"Thus, Fick's law needs to be adapted to porous media.\",\n \"Under such conditions the diffusion coefficient is replaced by an effective diffusion coefficient $D_g^* = \\\\beta D_g$ in which $\\\\beta $ is a porous media factor $\\\\beta = \\\\phi S_g \\\\tau $ where $\\\\phi $ is the porosity, $S_g$ is the effective gas saturation, from zero to one, and $\\\\tau $ is the tortuosity factor.\",\n \"The tortuosity factor can be written as $\\\\tau = \\\\phi ^{1/3} S_g^{7/3}$ which for a media completely saturated with gas simplifies to $\\\\tau = \\\\phi ^{1/3}$ In soil physics it is more common that the gradient is expressed in terms of concentration, which is effectively the density multiplied by the mass fraction, thus $\\\\mathbf {J}_g = - D_g^* \\\\nabla C_g$ For gas diffusion in soil, the soil gas diffusion coefficient is $ D_g^*=\\\\tau D_g$ and the tortuosity if often expressed as $ \\\\tau = 0.66 \\\\theta _a $ in which $\\\\theta _a$ is the soil volumetric air content [53], [47].\",\n \"For advective gas transport, Darcy's law is valid [47], [100] $\\\\mathbf {q}_g = - \\\\frac{\\\\kappa _g}{\\\\mu _g} (\\\\nabla P_g - \\\\rho _g \\\\mathbf {g})$ here written in terms of the permeability of the media to the gas ($\\\\kappa _g$ ), the viscosity of the gas ($\\\\mu _g$ ), the density of the gas ($\\\\rho _g$ ) and the pressure gradient ($\\\\nabla P_g$ ) and in which, as defined before, $q_g$ is the flux density, but for the gas in this case.\",\n \"The formula can be also written in terms of a gas conductivity $K_g = k_g/\\\\mu _g$ parameter $\\\\mathbf {q}_g = - K_g (\\\\nabla P_g - \\\\rho _g \\\\mathbf {g})$ In general, the gravity term can be neglected because the density of most gases is low, thus $\\\\mathbf {q}_g = - K_g \\\\nabla P_g$ for pressure in length units.\",\n \"Because advective flow of gases is usually not an important transport mechanism in soil physics and groundwater hydrology, except in very specific conditions, transport of gases in porous media is modeled as a diffusion processBeware that this might not be the case when modeling natural gas reservoirs.\",\n \"We can now rewrite the general transport equation for a non-adsorbing gas for a generalized direction $z$ , neglecting advective transport, as $\\\\frac{\\\\partial (\\\\theta _g C_g)}{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} \\\\mathbf {J}_g + \\\\theta _g \\\\rho _{g} \\\\Gamma _g$ $\\\\frac{\\\\partial (\\\\theta _g C)}{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} (\\\\mathbf {D}^*_g \\\\frac{\\\\partial C^g}{\\\\partial x}) + \\\\theta _g \\\\rho _{g} \\\\Gamma _g$ and if the volumetric air content and the soil gas diffusion coefficient are constant $\\\\theta _g \\\\frac{\\\\partial C_g}{\\\\partial t} = D^*_g \\\\frac{\\\\partial ^2 C_g}{\\\\partial z^2} + \\\\theta _g \\\\rho _{g} \\\\Gamma $ In many cases this is a very simple differential equation whose techniques of solution were generally shown in Chapter REF .\",\n \"Thus, for transport of a inert gas (no sources of sinks) through a column of length $L$ saturated with inert gas with concentration $C_0$ at the inlet and zero at the outlet, the solution is [47] $C_g(z) = C_0 (1 - \\\\frac{z}{L})$ while for steady-state transport, in which the gas concentration does not vary with time (i.e.\",\n \"$ \\\\partial C_g/\\\\partial t = 0 $ ) with a consumption term (e.g.\",\n \"oxygen in soil environment) the solution is [47] $C_g(z) = C_0 + \\\\frac{\\\\theta _g \\\\rho _g \\\\Gamma }{2 D^*_g} (z^2 + 2Lz)$ Because the calculations are simple the previous derivations are left as an exercise to reader.\",\n \"The same conservation principles for mass applied in the previous sections generally apply for heat transport.\",\n \"However, heat transport is a form of energy transport and some particularities need to be consider.\",\n \"Like before, heat can be transported by diffusion and advection, but in the case of advection, heat is transported with transport of a solid, gas or liquid.\",\n \"The amount of heat stored in a material is given by [53] $Q = \\\\rho c V T$ in which $Q$ is the amount of thermal energy stored in the material, in joules or calories, $\\\\rho $ is the density of the material in [M L$^{-3}$ ], $c$ is the mass based specific heat capacity or the heat capacity $C_h$ divided by the amount of mass of the material, $V$ is the volume of material and $T$ is the temperature.\",\n \"Because the heat capacity is given in energy per temperature, the mass based specific heat capacity is in energy times temperature$^{-1}$ mass$^{-1}$ , representing thus the amount of energy necessary to increase the temperature of one unit mass of the material in one unit of temperature, usually in (calories) cal $g^{-1}$ $^{o}C^{-1}$ or in $J$ $kg^{-1}$ $K^{-1}$ .\",\n \"As with the other transport laws discussed throughout this material, thermal conduction law is given by $\\\\mathbf {J}_T = -\\\\frac{\\\\kappa _T}{\\\\rho c} \\\\nabla T$ in which $\\\\kappa _T$ is the thermal conductivity.\",\n \"Defining now a thermal diffusivity term as $D_T = -\\\\frac{\\\\kappa _T}{\\\\rho c}$ $\\\\mathbf {J}_T = - D_T \\\\nabla T$ This is Fourier's law of heat conduction and it is analogous to Fick's law for gases, Darcy's law for fluids and Ohm's law for electricity.\",\n \"The driver gradient is now not a gradient in terms of concentration or potential but in terms of temperature.\",\n \"As before, we can define a general conservation equation for heat [47] $\\\\frac{\\\\partial T}{\\\\partial t} = - \\\\frac{\\\\partial J_T}{\\\\partial z} + \\\\gamma _T $ in which $\\\\gamma _T$ is a general heat source or sink term.\",\n \"Following the procedures for the other conservation laws, and considering the thermal diffusivity term as constant, the heat flux $J_T$ can be replaced into the conservation equation, and ignoring heat sources and sinks $\\\\frac{\\\\partial T}{\\\\partial t} = D_T \\\\frac{\\\\partial ^2 T}{\\\\partial z^2} $ Because the temperature at surface ($z = 0$ ) varies sinusoidally due to daily and annual variations of temperature and solar heat influx, a solution for temperature variation in depth is of the form [53] $T(z, t) = T_a + A \\\\exp {(-\\\\sqrt{\\\\frac{\\\\omega }{2D_T}}z)} \\\\sin {[\\\\omega t - \\\\sqrt{\\\\frac{\\\\omega }{2D_T}}z + \\\\xi ]} $ in which $z$ is the depth within the material or porous media, $t$ is time, $T_a$ is the average temperature, $A$ is the amplitude of the sinusoidal wave components, $\\\\omega = 2\\\\pi /P$ , where $P$ is the period, and $\\\\xi $ is the phase constant.\",\n \"The temperature variation with depth up to $100.0~cm$ , for a soil with thermal diffusivity $D_T = 0.015~cm^2~s^{-1}$ , phase constant $\\\\xi = -7\\\\pi /12$ , $T_a = 25.0~^{o} C$ and $A = 10.0~^{o} C$ is illustrated in Figure REF , where $\\\\omega $ was calculated considering a period of $24~h$ or $86400~ s$ .\",\n \"Note that the amplitude of temperature is damped with depth, being close to zero at $100~ cm$ and maximum at surface.\",\n \"Figure: Variation of temperature with time over a period of two days up to a depth of 100cm100~ cm for a soil with D T =0.015cm 2 s -1 D_T = 0.015~cm^2~s^{-1} .ll $\\\\mathbf {J}_{adv} $ Advective flux density for a dissolved solid being transported $\\\\theta $ Volumetric water content $\\\\mathbf {v} $ Fluid velocity vector $C $ Dissolved chemical concentration $D_d $ Generalized diffusion coefficient $\\\\mathbf {J}_{diff} $ Diffusive flux density for a dissolved solid being transported $\\\\mathbf {T}(\\\\theta ) $ Tortuosity tensor $\\\\mathbf {D}^*(\\\\theta ) $ Coefficient of diffusion for a dissolved solid being transported in porous media $\\\\mathbf {D} $ Dispersion tensor in porous media $\\\\mathbf {D}_{h} $ Hydrodynamic dispersion tensor $\\\\mathbf {J}_{tot} $ Total flux density for a dissolved solid being transported in porous media $\\\\mathbf {J}_{disp} $ Dispersive flux density for a dissolved solid being transported in porous media $F $ Mass of chemical absorbed per unit mass of solid phase $\\\\rho _f $ Fluid density $\\\\rho $ Bulk density of the porous media $\\\\Gamma $ Rate of creation or destruction of the chemical in the liquid phase $\\\\Gamma _s $ Rate of creation of destruction of the chemical adsorbed into the solid phase $v_x $ Velocity in a generalized $x$ direction $D_h $ Hydrodynamic dispersion coefficient $D $ Dispersion coefficient corrected to volumetric water content $= D_h/\\\\theta $ $\\\\phi $ Total porosity $K_d $ Coefficient of partition $R_d $ Retardation term $t $ Time $\\\\mathbf {J}_{g}^{mole} $ Mole flux for a gaseous component $\\\\mathbf {D}_{g(ij)} $ Gas diffusion coefficient tensor $ \\\\chi _g$ Mole fraction of the gaseous component of interest $ \\\\omega $ Mass fraction of the component of interest $\\\\rho _g $ Density of the gas $ D_g$ Gas diffusion coefficient $ D^*_g $ Effective gas diffusion coefficient in porous media $ \\\\beta $ Porous media factor $ S_g $ Effective gas saturation $ \\\\tau $ Tortuosity factor $ \\\\kappa _g $ Permeability of the media to a gas $ \\\\mu _g $ Viscosity of the gas $ P_g $ Gas pressure $ \\\\mathbf {g} $ Acceleration of gravity $ \\\\mathbf {q}_g $ Advective flux density of a gas in porous media $ K_g $ Gas conductivity $\\\\Gamma _g $ Rate of creation of destruction of the gas $C_g $ Concentration of the gas $ L $ Length of a column filled with porous media $\\\\theta _g$ Gas volumetric content $ Q $ Energy stored in the form of heat $ c $ Specific heat capacity $ V$ Volume of a material $ T$ Temperature $\\\\mathbf {J}_{T} $ Heat flux $ \\\\kappa _T $ Thermal conductivity $ D_T $ Thermal diffusivity $ \\\\omega $ $ = 2 \\\\pi / P $ , where $P$ is the period $ T_a$ Average temperature $ A $ Sinusoidal wave amplitude $ \\\\xi $ Phase constant As we have seen throughout this book, many of the models that describe physical phenomena are in the form of partial differential equations which might not have have simple analytical solutions for modeling real world phenomena, and very often might not have any analytical solutions at all.\",\n \"When analytical solutions are not available, numerical approximations are an extremely useful tool in modeling real world phenomena.\",\n \"With today's computers availability and processing power, numerical methods can be easily implemented using computational tools.\",\n \"Large computational codes, both open source and proprietary, are used for modeling subsurface flow and transport in soils, geological deposits and watersheds.\",\n \"Even larger models for matter and energy balance are available for Earth's scale modeling, but these are somewhat beyond the scope of this book.\",\n \"One example of numerical model that all of us encounter in daily life is weather prediction.\",\n \"Weather prediction is based on a complicated set of differential equations, based on a large set of parameters which cannot be solved analytically for larger domains such as a city, state, country or the whole world.\",\n \"For weather prediction the variables available are the past observations and current observations.\",\n \"These values will be then the initial conditions of the model, as we have seen for the solution of many differential equations.\",\n \"The domains might also be subjected to a set of prescribed boundary conditions.\",\n \"Given past predictions, and boundary and initial conditions, a numerical model can predict the weather in the future.\",\n \"This type of prediction based on initial and boundary conditions using numerical solutions to differential equations is called forward modeling.\",\n \"The stability, accuracy and range of the predictions is dependent on a series of factors including, accuracy of past and current predictions, nature of the differential equations being solved, numerical method being used and many others.\",\n \"Another type of modeling frequently used in soil physics and groundwater hydrology is called inverse modeling.\",\n \"In inverse modeling we have a set of observations and we wish to fit a particular empirical model to data to make predictions in the same system from which the data was obtained or in similar systems.\",\n \"Inverse modeling can be used to make predictions in time and/or space but it is fundamentally different from numerical forward modeling because the empirical data is necessary a priori.\",\n \"One example is a dataset of pairs water content and matric potential values which is fit to a water retention curve equation.\",\n \"Fitting is usually achieved using statistical techniques such as least squares, maximum likelihood or Bayesian regression, among others.\",\n \"The parameterized model can be then used to monitor water content from matric potential data in soils, vadose zone or other media for agricultural applications and environmental modeling.\",\n \"Both types of models can be improved, calibrated and validated using independent data.\",\n \"In the weather prediction model, once data has been collected, it can be compared to the numerical predictions and error indicators can be computed.\",\n \"The model used and computation technique can be adapted accordingly.\",\n \"In the case of the inverse model, once the water retention equation is fit to data, it can be compared to data collected in nearby sites using the same or similar laboratory techniques and the results be compared.\",\n \"Once a large enough data set is compiled, stochastic estimation of the empirical parameters that are representative of the area based on more or less arbitrary statistical and mathematical indicators can be determined.\",\n \"Over the next sections we will provide an extremely condensed account of curve fitting techniques and numerical methods used in soil physics and groundwater hydrology.\",\n \"For the time being only finite differences schemes will discussed for numerical methods for solving differential equations, because they are somewhat simpler from the mathematical point of view and can be easily implemented in most programming languages.\",\n \"The equations presented in this material are used to model real world data.\",\n \"Consider for example the water retention curve and soil mechanical resistance equations.\",\n \"In the first case data from the water retention curve can be obtained using the gravimetric method for water content, and pressure plates or tensiometers for soil matric potential.\",\n \"For the soil mechanical resistance to penetration equation the soil resistance can be obtained by cone penetrometers, bulk density by the core method and water content by the gravimetric method.\",\n \"After data collection, the equations can be fit to data using statistical or mathematical techniques.\",\n \"Most of the equations encountered in soil physics and groundwater hydrology are nonlinear.\",\n \"The meaning of nonlinear is not directly related to the shape of the curve when data is plotted, nonlinear is related to the parameters of the equation.\",\n \"Consider a general linear equation, essentially a first degree polynomial $y = a x + b$ where $y$ is the dependent variable, $x$ is the independent variable and $a$ and $b$ are fitting parameters.\",\n \"For pairs of $y$ and $x$ the procedure to fit a linear regression model is exact.\",\n \"In other words, there are exact analytical formulas that can be used to obtain the values of $a$ and $b$ A theoretical analysis is beyond the scope of this material and can be found in introductory statistics and generalized linear models textbooks $a = \\\\frac{\\\\sum (x_i - \\\\bar{x})(y_i - \\\\bar{y})}{\\\\sum (x_i - \\\\bar{x})^2}$ $b = \\\\frac{\\\\sum y_i - a \\\\sum x_i }{n}$ in which $x_i$ and $y_i$ are the $i^{th}$ observation and $\\\\bar{x}$ and $\\\\bar{y}$ the averages for the variables $x$ and $y$ , and $n$ is the number of observations.\",\n \"For polynomials of degree greater than one, although the shape of the curve of a plot is not linear, the fitting procedure is still exact and the equation is not considered a nonlinear equation.\",\n \"The definition of a nonlinear regression model is related to nonlinearity in relation to the parameters.\",\n \"In other words, a true nonlinear function is nonlinear with respect to its parameters.\",\n \"Consider a second degree polynomial.\",\n \"$y = a x^2 + b x + c$ The partial derivatives with respect to the parameters are $\\\\frac{\\\\partial y}{\\\\partial a} = x^2$ $\\\\frac{\\\\partial y}{\\\\partial b} = x$ None of the derivatives are functions of the parameters.\",\n \"Consider now a water retention function of the form $\\\\Theta = \\\\frac{A}{A + |\\\\psi _m|^B}$ in which $\\\\Theta = S_e$ .\",\n \"The partial derivatives with respect to the parameters are $\\\\frac{\\\\partial \\\\Theta }{\\\\partial A} = \\\\frac{|\\\\psi _m|^B}{(A + |\\\\psi _m|^B)^2}$ $\\\\frac{\\\\partial \\\\Theta }{\\\\partial B} = -\\\\frac{A|\\\\psi _m|^B \\\\log {|\\\\psi _m|}}{(A + |\\\\psi _m|^B)^2}$ It is easy to see that the partial derivatives with respect to the each parameter are functions of the parameters.\",\n \"These types of functions cannot be fit to data by straightforward analytical and computational methods as in the case of linear and polynomial regression.\",\n \"Nonlinear regression methods are required to fit nonlinear models.\",\n \"Nonlinear regression methods rely on numerical methods based on successive approximations using iterative methods such as the Newton-Raphson, Gauss-Newton and Levenberg-Marquardt methods.\",\n \"In each case an initial guess of the values of the parameters is provided by the user and the procedure tries to converge to an optimal parameter vector where the sum of the squares of the differences between the observed and predicted $y$ values is minimized.\",\n \"This is a general description of one of the most common methods for fitting nonlinear regression models, nonlinear least squares [9].\",\n \"Although there are other methods for fitting nonlinear models such as maximum likelihood, Bayesian estimation and other robust methods [9], nonlinear least squares is still the most common methods in commercial and noncommercial software packages.\",\n \"A generalized nonlinear model can be written as a function of not only the independent vector $\\\\mathbf {x} $ but also of a parameter vector $\\\\mathbf {\\\\Omega } $ [9] $y_i = f(\\\\mathbf {x}; \\\\mathbf {\\\\Omega }) + \\\\epsilon _i$ in which $\\\\epsilon _i$ is random error term with mathematical expectation zero In practical terms the average is zero.\",\n \"The procedure used must find a least squares estimate of $\\\\mathbf {\\\\Omega } $ , denoted by $\\\\hat{\\\\mathbf {\\\\Omega }} $ which minimizes the sum of squares [9], [83] $SSQ(\\\\mathbf {\\\\Omega }) = || \\\\mathbf {y} - \\\\mathbf {x} \\\\mathbf {\\\\Omega } ||^2 = \\\\sum _{i=1}^ {n} [y_i - f(x_i; \\\\mathbf {\\\\Omega })]^2$ In simple terms, $y_i$ are the measured values and $f(x_i; \\\\mathbf {\\\\Omega })$ are the values predicted by the function.\",\n \"Thus, $y_i - f(x_i; \\\\mathbf {\\\\Omega })$ are the differences between measured and predicted values, often called residuals or errors which are sought to be minimized.\",\n \"Because of their numerical nature, nonlinear regression models can be unstable and convergence to an optimal solution might not always happen.\",\n \"As the complexity of the model and the number of parameters increases, the likelihood that the search procedure will diverge and not find the minimum on the parameters surface increases.\",\n \"Because of that, a topic of research in statistics, numerical methods and computation is to find algorithms with adequate properties for nonlinear regression.\",\n \"The search procedures discussed here try to find an optimal parameter vector by an iterative procedure of the type [9], [83] $\\\\mathbf {\\\\Omega } ^{(\\\\phi +1)} = \\\\mathbf {\\\\Omega } ^{(\\\\phi )} + \\\\delta ^{(\\\\phi )}$ In which $\\\\phi $ can be understood as the iteration number, thus the estimate of the $\\\\mathbf {\\\\Omega }$ vector at $\\\\phi +1$ is conditionally based on the estimate at $\\\\phi $ plus a step term $\\\\delta $ at the iteration $\\\\phi $ .\",\n \"For the Gauss-Newton method [9] $\\\\delta ^{(\\\\phi )} = - (\\\\mathbf {J}^{(\\\\phi )T} \\\\mathbf {J}^{(\\\\phi )})^{-1} \\\\mathbf {J}^{(\\\\phi )T} \\\\mathbf {r}^{(\\\\phi )T}$ $\\\\mathbf {J}$ is the Jacobian matrix estimates, the Jacobian is the matrix of the derivatives with respect to the parameters, $T$ indicates a transposed vector or matrix, $-1$ indicates the inverse of the matrix and $\\\\mathbf {r}$ is a vector of residuals, i.e.\",\n \"observed minus predicted values at a given iteration level.\",\n \"Obviously there is much more to this, although the actual application to a simple function might not be very complicated from the programming standpoint.\",\n \"A simple implementation of the Gauss-Newton method will be provided next.\",\n \"Consider the full implementation of the iterative procedure for the parameter vector using the Gauss-Newton step $\\\\mathbf {\\\\Omega } ^{(\\\\phi +1)} = \\\\mathbf {\\\\Omega } ^{(\\\\phi )} - (\\\\mathbf {J}^{(\\\\phi )T} \\\\mathbf {J}^{(\\\\phi )})^{-1} \\\\mathbf {J}^{(\\\\phi )T} \\\\mathbf {r}^{(\\\\phi )T}$ Suppose now we want to fit a simple water retention function of the form $\\\\theta = \\\\frac{a}{|\\\\psi _m|^b}$ in which $\\\\theta $ is the volumetric water content and $\\\\psi _m$ is the matric potential, in generalized units not relevant for this example, and $a$ and $b$ are fitting parameters.\",\n \"We wish to fit the model to the fictitious data set $\\\\theta = \\\\lbrace 0.4, 0.38, 0.35, 0.3, 0.28, 0.23, 0.22, 0.21 \\\\rbrace $ $\\\\psi _m = \\\\lbrace -1, -5, -10, -50, -100, -1000, -5000, -10000 \\\\rbrace $ $\\\\theta $ and $\\\\psi _m$ are vectors containing the observed water content and matric potential estimates.\",\n \"We start the Gauss-Newton method with an initial guess for the parameter vector, say $a = 0.5$ and $b = 0.2$ .\",\n \"Ideally the initial parameter vector should be defined taking into consideration the physical meaning of the parameters and/or the user should plot predicted and estimated values in the same graph and try to estimate the approximate best fit visually, so the parameter vector is close to the optimum values.\",\n \"The parameter vector at the zeroth iteration contains the initial guess, thus $\\\\Omega ^{(0)} =\\\\begin{bmatrix}0.50\\\\\\\\0.20\\\\end{bmatrix}$ The Jacobian matrix is the matrix containg the partial derivatives of the residuals vector $\\\\mathbf {r} = \\\\theta - \\\\hat{\\\\theta }$ with respect to the parameters, where $\\\\hat{\\\\theta }$ is the vector containing the model estimates based on the parameter vector.\",\n \"In simpler term, for each observation $i$ , the residuals function is $r_i = \\\\theta _i- \\\\frac{a}{|\\\\psi _{m,i}|^b}$ Thus the partial derivatives with respect to the parameters are $\\\\frac{\\\\partial r_i}{\\\\partial a} = -|\\\\psi _{m,i}|^{-b}$ $\\\\frac{\\\\partial r_i}{\\\\partial b} = a|\\\\psi _{m,i}|^{-b}\\\\log {|\\\\psi _{m,i}|}$ The Jacobian matrix is then populated as $J_{i,2} = \\\\lbrace \\\\frac{\\\\partial r_i}{\\\\partial a}, \\\\frac{\\\\partial r_i}{\\\\partial b} \\\\rbrace $ ThusWe are approximating everything to four decimals whenever possible to save space, however, it is almost never a good idea to work with low precision approximations in numerical methods $\\\\mathbf {J}^{(0)} =\\\\begin{bmatrix}-1.00000 & 0.00000 \\\\\\\\-0.72478 & 0.58324 \\\\\\\\-0.63096 & 0.72642 \\\\\\\\-0.45731 & 0.89449 \\\\\\\\-0.39811, & 0.91668 \\\\\\\\-0.25119, & 0.86757 \\\\\\\\-0.18206 & 0.77530 \\\\\\\\-0.15849, & 0.72987\\\\end{bmatrix}$ with transpose $\\\\begin{split}& \\\\mathbf {J}^{(0)T} = \\\\\\\\&\\\\begin{bmatrix}-1.0000& -0.7248& -0.6310& -0.4573& -0.3981& -0.2512& -0.1821& -0.1585 \\\\\\\\0.0000& 0.5832& 0.7264& 0.8945& 0.9167& 0.8676& 0.7753& 0.7299\\\\end{bmatrix}\\\\end{split}$ The product $\\\\mathbf {J}^{(0)T}\\\\mathbf {J}^{(0)}$ is then a $ 2 \\\\times 2$ matrix At this point it might convenient for the reader to review an introductory linear algebra book $\\\\mathbf {J}^{(0)T}\\\\mathbf {J}^{(0)} =\\\\begin{bmatrix}2.4124 & -2.1298 \\\\\\\\-2.1298 & 4.3948\\\\end{bmatrix}$ With inverse likewise calculated using linear algebra procedures $(\\\\mathbf {J}^{(0)T}\\\\mathbf {J}^{(0)})^{-1} =\\\\begin{bmatrix}0.7245 & 0.3511 \\\\\\\\0.3511 & 0.3977\\\\end{bmatrix}$ The residuals vector is $\\\\mathbf {r}^{(0)} =\\\\begin{bmatrix}-0.1000 \\\\\\\\0.0176 \\\\\\\\0.0345 \\\\\\\\0.0713 \\\\\\\\0.0809 \\\\\\\\0.1044 \\\\\\\\0.1290 \\\\\\\\0.1308 \\\\\\\\\\\\end{bmatrix}$ resulting in a Gauss-Newton step vector $\\\\delta ^{0} = (\\\\mathbf {J}^{(0)T} \\\\mathbf {J}^{(0)})^{-1} \\\\mathbf {J}^{(0)T} \\\\mathbf {r}^{(0)T}$ of $\\\\delta ^{(0)} =\\\\begin{bmatrix}0.1107 \\\\\\\\0.1582\\\\end{bmatrix}$ Noting that $\\\\mathbf {\\\\Omega } ^{(1)} = \\\\Omega ^{(0)} - \\\\delta ^{(0)}$ results in $\\\\mathbf {\\\\Omega } ^{(1)} =\\\\begin{bmatrix}0.3893 \\\\\\\\0.0418\\\\end{bmatrix}$ with $SSQ = 0.067$ .\",\n \"Repeating the procedure for five iterations results in $a = 0.4099$ and $b = 0.0761$ and $SSQ = 0.00075715$ .\",\n \"For comparison, a numerical procedure from a popular free software spreadsheet resulted in the same values after 40 iterations.\",\n \"The Levenberg-Marquardt method improves upon the Gauss-Newton method allowing for search in two direction and for singular or ill conditioned matrices [9], [83] $\\\\delta ^{(\\\\phi )} = - (\\\\mathbf {J}^{(\\\\phi )T} \\\\mathbf {J}^{(\\\\phi )} + \\\\eta ^{(\\\\phi )} \\\\mathbf {D}^{(\\\\phi )} )^{-1} \\\\mathbf {J}^{(\\\\phi )T} \\\\mathbf {r}^{(\\\\phi )T}$ in which $\\\\mathbf {D}^{(\\\\phi )}$ is a diagonal matrix with positive elements and $\\\\eta $ is a damping factor.\",\n \"The Levenberg-Marquardt procedure is usually efficient for Brutsaert-van Genuchten type equations, although the Gauss-Newton procedure is usually adequate if the number of parameters is reduced and if the initial guess is reasonably close to optimal values.\",\n \"A program for fitting the Brutsaert-van Genuchten to soil water retention data is presented in Appendix REF (Figure REF ).\",\n \"The Levenberg-Marquardt is invoked from a program module and thus was not written from scratch.\",\n \"Figure: Brutsaert-van Genuchten equation (BvG) fit to simulated water retention data from the program in Appendix .The procedures described above can be also used in soil mechanical resistance equations of the form $SR = a \\\\rho ^b \\\\theta ^c$ Note that the partial derivatives indicate that this function is nonlinear $\\\\frac{\\\\partial SR}{\\\\partial a} = \\\\rho ^b \\\\theta ^c$ $\\\\frac{\\\\partial SR}{\\\\partial b} = ba\\\\theta ^c\\\\rho ^{(b-1)}$ $\\\\frac{\\\\partial SR}{\\\\partial c} = ca\\\\rho ^b \\\\theta ^{(c-1)}$ However, this equation is intrinsically linear since it can be transformed into a linear function.\",\n \"In this case the transformation is achieved by applying logarithms at both sides $\\\\log {SR} = \\\\log {a} + b \\\\log {\\\\rho } + c \\\\log { \\\\theta }$ which can be fitted by multiple linear regression with fitting parameters $a$ , $b$ and $c$ .\",\n \"Note, however, that linearization of intrinsically linear functions can result in undesirable statistical properties [9], [59] In order to approximate differential equations using finite difference methods, it is necessary to use a mathematical tool called Taylor series.\",\n \"Taylor series allow us to expand a infinitely differentiable function around a point using its derivatives.\",\n \"It is a fundamental tool in physics and mathematics, having applications in several different areas, especially in the approximation of functions which are too difficult or impossible to solve analytically.\",\n \"Consider a real of imaginary valued function $f(x)$ which we want to expand around the point $a$ , in which $a$ is a real or complex number.\",\n \"The Taylor series expansion is $f(x) = f(a) + \\\\frac{(x-a)}{1!}\",\n \"\\\\frac{d }{dx} f(a) + \\\\frac{(x-a)^2}{2!}\",\n \"\\\\frac{d^2 }{dx^2} f(a) + \\\\frac{(x-a)^3}{3!\",\n \"}\\\\frac{d^3 }{dx ^3} f(a)+ ...$ which can be written in summation form as $f(x) = \\\\sum _{0}^{\\\\infty } \\\\frac{(x-a)^n}{n!}\",\n \"\\\\frac{d^n }{dx ^n} f(a)$ An important case of the Taylor series is when the expansion is done around zero, being called Maclaurin series $f(x) = f(0) + \\\\frac{x}{1!}\",\n \"\\\\frac{d }{dx} f(0)+ \\\\frac{x^2}{2!}\",\n \"\\\\frac{d^2 }{dx^2}f(0) + \\\\frac{x^3}{3!\",\n \"}\\\\frac{d^3 }{dx ^3} f(0)+ ...$ For most applications in physics, the higher order terms (higher powers) are small and can be neglected, thus resulting in a simple and often precise approximation of mathematical functions.\",\n \"This was particularly important before the advent of electronic calculators and digital computers, when the estimations of functions which today are viewed as trivial such as the value of $\\\\pi $ where difficult tasks.\",\n \"Examples of Taylor series approximations for a few simple functions are given below [93] $e^x = 1 + x + \\\\frac{x^2}{2} + \\\\frac{x^3}{6} + \\\\frac{x^4}{24} + ...$ $\\\\log (1 + x) = x - \\\\frac{x^2}{2} + \\\\frac{x^3}{3} - \\\\frac{x^4}{4} + ...$ $(1 + x)^n = 1 + nx + \\\\frac{n(n-1)}{2} x^2 + \\\\frac{n(n-1)(n-2)}{6} x^3 + ...$ We rewrite Laplace equation for total water potential for a two dimensional domain $\\\\frac{\\\\partial ^2 h}{\\\\partial x^2} + \\\\frac{\\\\partial ^2 h}{\\\\partial y^2} = 0$ As we have discussed, this equation does have an analytical solution in terms of series.\",\n \"However, the numerical solution of this equation provides a convenient introduction to finite difference methods.\",\n \"Our goal here is to approximate the second order partial derivatives within a domain.\",\n \"Let's define the domain as a region in space, here in the horizontal plane, with dimensions $L_x$ and $L_y$ .\",\n \"The domain can be discretized in the form of a regular grid with nodes spaced $\\\\Delta x$ in the $x$ direction and $\\\\Delta y$ in the $y$ direction (Figure REF ).\",\n \"Figure: Discretization for the finite differences solution of the 2D Laplace equation.The second order partial derivatives in the $x$ direction can be approximated between two adjacent points, where the next point is located in the positive $x$ direction of $h^j_i$ , by Taylor series expansion [46] $\\\\begin{split}h^j_{i+1} = & h^j_{i} + (x^j_{i+1} - x^j_{i}) \\\\frac{\\\\partial h}{\\\\partial x} + \\\\frac{(x^j_{i+1} - x^j_{i})^2}{2!}\",\n \"\\\\frac{\\\\partial ^2 h}{\\\\partial x^2} + \\\\\\\\ & \\\\frac{(x^j_{i+1} - x^j_{i})^3}{3!}\",\n \"\\\\frac{\\\\partial ^3 h}{\\\\partial x^3} + ...\\\\end{split}$ and for points is located in the negative $x$ direction of $h(i,j)$ by $\\\\begin{split}h^j_{i-1} = & h^j_{i} - (x^j_{i-1} - x^j_{i}) \\\\frac{\\\\partial h}{\\\\partial x} + \\\\frac{(x^j_{i-1} - x^j_{i})^2}{2!}\",\n \"\\\\frac{\\\\partial ^2 h}{\\\\partial x^2} - \\\\\\\\ & \\\\frac{(x^j_{i-1} - x^j_{i})^3}{3!}\",\n \"\\\\frac{\\\\partial ^3 h}{\\\\partial x^3} + ...\\\\end{split}$ Adding the two equations and noting that for our regular grid $(x^j_{i+1} - x^j_{i}) = \\\\Delta x$ $(x^j_{i-1} - x^j_{i}) = -\\\\Delta x$ results in $h^j_{i+1} + h^j_{i-1} = 2h^j_{i} + (\\\\Delta x)^2 \\\\frac{\\\\partial ^2 h}{\\\\partial x^2} + \\\\mathcal {O}((\\\\Delta x)^3)$ In which $\\\\mathcal {O}((\\\\Delta x)^n)$ are higher order terms represented using the big “o” notation $\\\\mathcal {O}$ .\",\n \"If we ignore error terms with order greater than two, which tends to be increasingly smaller as the order of the term increases, the approximation to the second order derivative in $x$ is $\\\\frac{\\\\partial ^2 h}{\\\\partial x^2} = \\\\frac{h^j_{i+1} + h^j_{i-1} - 2h^j_{i}}{(\\\\Delta x)^2}$ The same procedure results in the approximation in $y$ $\\\\frac{\\\\partial ^2 h}{\\\\partial y^2} = \\\\frac{h^{j+1}_{i} + h^{j-1}_{i} - 2h^{j}_{i}}{(\\\\Delta y)^2}$ Replacing the approximations into the Laplace equation and solving for $h^j_i $ allows us to approximate the values of the total potential inside the domain, given the boundary conditions of fixed total potentials at the upper, lower, right and left boundaries using the formula $h^j_{i} = \\\\frac{1}{4} [h^j_{i+1} + h^j_{i-1} + h^{j+1}_{i} + h^{j-1}_{i}]$ The solution is obtained by numerical methods.\",\n \"A program for solving the 2D Laplace equation using the Jacobi iterative method is presented in Appendix REF .\",\n \"A surface plot The figure was produced with Gnuplot: http://www.gnuplot.info/ of the solution for $h_{left} = 20~cm$ , $h_{right} = 1~cm$ , $h_{top} = 5~cm$ and $h_{bottom} = 0~cm$ is presented in Figure REF .\",\n \"Figure: Numerical solution of the 2D Laplace equation from the program in Appendix .The discretization of an 1D partial differential equation in time and space follows a similar scheme as 2D discretization in space.\",\n \"In diffusion type equations (e.g.\",\n \"advective-dispersive, Richards-Richardson, heat equation) besides the second order partial derivative in space, there is a first order derivative in time.\",\n \"Recall the 1D advective-dispersive equation for constant flux velocity and diffusion coefficient $\\\\frac{\\\\partial C}{\\\\partial t} = D \\\\frac{\\\\partial C}{\\\\partial x} - v \\\\frac{\\\\partial ^2 C}{\\\\partial x^2}$ We already know from the previous section that the second order partial derivatives in space can be approximated by $\\\\frac{\\\\partial ^2 C}{\\\\partial x^2} = \\\\frac{C^j_{i+1} + C^j_{i-1} - 2C^j_{i}}{(\\\\Delta x)^2}$ Obviously for concentration $C$ in this case.\",\n \"To find the approximation of first order partial derivatives, the Taylor series expansion equations as in Equations REF REF are subtracted.\",\n \"Ignoring higher order terms and solving for the partial derivatives of interest, the discretization of the first order partial derivatives in space and time for the advective-dispersive equation are $\\\\frac{\\\\partial C}{\\\\partial x} = \\\\frac{C^j_{i+1} + C^j_{i-1}}{(2\\\\Delta x)}$ $\\\\frac{\\\\partial C}{\\\\partial t} = \\\\frac{C^j_{i+1} + C^j_{i-1}}{(2\\\\Delta t)}$ The discretization is now in space and time, where $i$ is the increment in space and $j$ the increment in time.\",\n \"Care must be taking in writing programs regarding the order of $i$ and $j$ in loops.\",\n \"For the discretization in time, the condition at $t=0$ is now the initial contition and the conditions in the inlet and outlet of the 1D domain are the boundary conditions (Figure REF ).\",\n \"Figure: Discretization for the finite differences solution of the 1D advective-dispersive equation.By replacing the partial derivatives in time and space, the finite differences equation to be solved for the 1D advective-dispersive equation is $C^j_{i} = \\\\Delta t C^{j-1}_{i} + D\\\\frac{\\\\Delta t}{\\\\Delta x^2} (C^{j-1}_{i-1} - 2C^{j-1}_{i} + C^{j-1}_{i+1} ) - v\\\\frac{\\\\Delta t }{2 \\\\Delta x} (C^{j-1}_{i+1} - C^{j-1}_{i-1} )$ One of the issues of the numerical solution of the advective-dispersive equation is that it is valid for infinitely long columns.\",\n \"To circumvent this issue a program was written to solve the equation for a long column, but the outlet concentration measurements were made a fixed point along the length $x$ .\",\n \"The comparison of the numerical and analytical solutions of the advective-dispersive equation for concentration measured at $50~cm$ from the inlet is presented in Figure REF .\",\n \"The column length for the numerical simulations was $L = 100~cm$ , the flow velocity was $v = 2.0~cm~h^{-1} $ , the dispersion coefficient $D = D_h/\\\\theta = 5.0~cm^2~h^{-1} $ and the simulation time was $100~h$ .\",\n \"The program is listed in the Appendix REF .\",\n \"Figure: Comparison of numerical and analytical solutions of the 1D advective-dispersive equation from the program in Appendix .By the same principles illustrated for the Laplace and advective-dispersive equations, the Richardson-Richards can be discretized to be solved by explicit methods.\",\n \"Recalling the 1D matric potential (here represented as $\\\\psi $ for simplicity) form of the Richardson-Richards equation in the general coordinate $z$ [50] $C^*(\\\\psi )\\\\frac{\\\\partial \\\\psi }{\\\\partial t} = \\\\frac{\\\\partial }{\\\\partial z} [K(\\\\psi ) (\\\\frac{\\\\partial \\\\psi }{\\\\partial z} + 1 )]$ The main issue here is that the unsaturated hydraulic conductivity is not constant and cannot be placed outside of the derivative terms.\",\n \"The hydraulic capacity function also needs to be defined for this equation to be solved numerically.\",\n \"Following the discussion in Chapter REF we will define the following water retention and unsaturated hydraulic conductivity functions [50] $\\\\theta (\\\\psi ) = \\\\theta _r + \\\\frac{\\\\alpha (\\\\theta _s - \\\\theta _r)}{\\\\alpha + |\\\\psi |^{\\\\beta }}$ $K(\\\\psi ) = \\\\frac{A}{A + |\\\\psi |^B}$ Recalling that the hydraulic capacity function is $C^*(\\\\psi ) = [\\\\alpha h_m (\\\\theta _s - \\\\theta _r) \\\\psi ^{\\\\beta -1}] \\\\frac{\\\\beta (\\\\alpha + \\\\psi ^ {\\\\beta })}{\\\\alpha + \\\\psi ^{\\\\beta }}$ The discretization for the explicit solution is [50] $\\\\begin{split}{\\\\psi }^{j+1}_{i} = & {\\\\psi }^{j}_{i} + \\\\frac{\\\\Delta t}{{C^*}^{j}_{i} \\\\Delta z} [ K^{j}_{i+1/2} (\\\\frac{{\\\\psi }^{j}_{i+1} - {\\\\psi }^{j}_{i}}{\\\\Delta z} -1 ) \\\\\\\\& - K^{j}_{i-1/2} (\\\\frac{{\\\\psi }^{j}_{i} - {\\\\psi }^{j}_{i-1}}{\\\\Delta z} -1 ) ]\\\\end{split}$ where the unsaturated hydraulic conductivity at intermediary point in nodes is $K^{j}_{i+1/2} = \\\\frac{K^{j}_{i+1}+ K^{j}_{i}}{2}$ $K^{j}_{i-1/2} = \\\\frac{K^{j}_{i}+ K^{j}_{i-1}}{2}$ Thus, the numerical solution can be achieved by applying the discretization schemes above while simultaneously evaluating the functions $K(\\\\psi )$ and $^*C(\\\\psi )$ at the $\\\\psi ^j_i$ points.\",\n \"The grid is analogous to the one described for the advective-dispersive equation.\",\n \"A program for solving the 1D Richardson-Richards equation using an explicit scheme is presented in Appendix REF .\",\n \"The results for the parameters $\\\\alpha = 1.611 \\\\times 10^{6}~cm^{-1}$ , $\\\\theta _s = 0.287~cm^3~cm^{-3} $ , $\\\\theta _r = 0.075~cm^3~cm^{-3}= $ and $\\\\beta = 3.96$ are compared to the Philip's quasi-analytical solution presented by [50] for an $80~cm$ 1D soil column initially at $\\\\psi = -61.5~cm$ ($t=0$ ) and with $\\\\psi = -20.73~cm$ at soil surface and $\\\\psi = -61.5~cm$ at the bottom of the column (or profile) for $t > 0$ (Figure REF ).\",\n \"The unsaturated hydraulic conductivity parameters were $ A = 0.00944$ and $B = 4.74$ .\",\n \"Figure: Comparison of numerical and quasi-analytical solutions of the 1D Richardson-Richards equation from the program in Appendix .\",\n \"The quasi-analytical solution follows Philip's method and is presented in .Although explicit methods are mathematically simpler and easier to implement from the computational point of view, they are notoriously unstable for most real world problems.\",\n \"The discretization scheme is then limited by the stability of the method.\",\n \"For most application implicit methods are preferred, these methods result in a set of equations which needs to be solved using matrix manipulation algorithms.\",\n \"In the implicit discretization of the 1D equation, there is no simpler equation which can be inserted into a code to solve the problem, a simultaneous solution for the different nodes is sought.\",\n \"Two popular choices of discretization are the Backward Time Centered Space (BTCS) $(-\\\\frac{\\\\gamma }{\\\\Delta x^2}) u^{j+1}_{i-1} + (\\\\frac{1}{\\\\Delta t} + \\\\frac{2\\\\gamma }{\\\\Delta x^2}) u^{j+1}_{i} + (-\\\\frac{\\\\gamma }{\\\\Delta x^2}) u^{j+1}_{i+1} = \\\\frac{1}{\\\\Delta t} u^{j}_{i}$ and the Crank-Nicolson scheme [30] A very good introduction to these methods can be found in https://web.cecs.pdx.edu/~gerry/class/ME448/ from which these discretization schemes were transcribed.\",\n \"$\\\\begin{split}(-\\\\frac{\\\\gamma }{2\\\\Delta x^2}) u^{j+1}_{i-1} + (\\\\frac{1}{\\\\Delta t} + & \\\\frac{\\\\gamma }{\\\\Delta x^2}) u^{j+1}_{i} + (-\\\\frac{\\\\gamma }{2\\\\Delta x^2}) u^{j+1}_{i+1} = \\\\\\\\& \\\\frac{\\\\gamma }{2\\\\Delta x^2}) u^{j}_{i-1} + (\\\\frac{1}{\\\\Delta t} - \\\\frac{\\\\gamma }{\\\\Delta x^2}) u^{j}_{i} + \\\\frac{\\\\gamma }{2\\\\Delta x^2} u^{j}_{i+1}\\\\end{split}$ In which $u^j_i$ is the temperature at each grid point, $\\\\gamma = \\\\kappa _T/ (\\\\rho c_p)$ is the thermal diffusivity and the other parameters are the same as previously defined.\",\n \"These numerical schemes can be applied to the Richardson-Richards and other partial differential equations.\",\n \"One common strategy of solution is the diagonal matrix solver, often called the Thomas algorithm [26].\",\n \"Technically, the explicit scheme can be thought as an special case of the discretization of the heat equation called the Backward Time Centered Space (BTCS).\",\n \"The Crank-Nicolson and other more complex schemes are recommended over BTCS and (Forward Time Ceneterd Space) FTCS for real world applications.\",\n \"The main advantages of simpler methods, specially the FTCS is that it can be used as a teaching tool for a first introduction to finite differences methods, as it is done in most numerical methods courses and books.\",\n \"In many problems other even more complicated schemes, both mathematically and computationally are employed, namely Finite Elements, Finite Volume and Lattice Boltzmann methods A simple introduction to these should be included in further editions.\",\n \"Most modern computational fluid mechanics, soil physics and groundwater hydrology numerical codes are based on one or more of these methods.\",\n \"ll $ y $ Generic dependent variable $ x $ Generic independent variable $ a, b, c, d, e $ Generic fitting parameters $ y_0 $ Generic $y$ coordinate $ x_0 $ Generic $x$ coordinate $ \\\\Theta $ Effective saturation $ = (\\\\theta - \\\\theta _s)/(\\\\theta _s - \\\\theta _r)$ $ \\\\theta $ Volumetric water content $ \\\\theta _s $ Saturation volumetric water content $ \\\\theta _r $ Residual volumetric water content $ A, B $ Unsaturated hydraulic conductivity function empirical parameters $ \\\\psi _m $ Matric potential in unspecified units $ SSQ $ Sum of squares of the residuals $ \\\\Omega $ Parameter vector $ \\\\mathbf { y} $ Observed dependent variable values vector $ \\\\mathbf { x} $ Observed independent variable values vector $ \\\\delta $ Step in nonlinear regression procedures $ \\\\mathbf { J} $ Jacobian matrix $ \\\\mathbf { r} $ Predicted values vector $ \\\\mathbf { D} $ Diagonal matrix $ \\\\phi $ Iteration level $ \\\\eta $ Damping factor $ SR $ Soil mechanical resistance to penetration $ \\\\rho $ Bulk density $ h $ Total potential in length units $ i, j$ Integer variables for step in space and/or time $ \\\\Delta x, \\\\Delta y, \\\\Delta t $ Discretization interval in time and space $ C $ Concentration $ v $ Flow velocity $ D $ Dispersion coefficient corrected to volumetric water content $= D_h/\\\\theta $ $ t $ Time $ h_m $ Matric potential in length units $ \\\\psi $ Matric potential in length units $ C^*(h_m) $ Hydraulic capacity function $ = \\\\partial \\\\theta /\\\\partial h_m$ $ \\\\theta (h_m) $ Water retention function $ K(h_m) $ Unsaturated hydraulic conductivity function $ \\\\alpha , \\\\beta $ Water retention function fitting parameters $ u$ Temperature at specified grid points $ y^j_i$ Specifies the value of the property at $i$ and $j$ for any function $y$ $ \\\\gamma $ Thermal diffusivity $ \\\\kappa _T $ Thermal conductivity $ c_p$ Specific heat capacity tocchapterBibliography $\\\\alpha $ , $A $ : alpha $\\\\beta $ , $B $ : beta $\\\\gamma $ , $\\\\Gamma $ : gamma $\\\\delta $ , $\\\\Delta $ : delta $\\\\epsilon $ ,$\\\\varepsilon $ , $E $ : epsilon $\\\\zeta $ , $ Z $ : zeta $\\\\eta $ , $H $ : eta $\\\\theta $ , $\\\\Theta $ : theta $\\\\iota $ , $I $ : iota $\\\\kappa $ , $ K $ : kappa $\\\\lambda $ , $\\\\Lambda $ : lambda $\\\\mu $ , $ M $ : mu $\\\\nu $ , $ N $ : nu $\\\\xi $ , $ \\\\Xi $ : xi $o$ , $ O $ : omicron $\\\\pi $ , $ \\\\Pi $ : pi $\\\\rho $ , $ R $ : rho $\\\\sigma $ , $ \\\\varsigma $ , $ \\\\Sigma $ : sigma $\\\\tau $ , $ T $ : tau $\\\\upsilon $ , $ Y $ : upsilon $\\\\phi $ , $\\\\varphi $ , $ \\\\Phi $ : phi $\\\\chi $ , $X$ : chi $\\\\psi $ , $ \\\\Psi $ : psi $\\\\omega $ , $\\\\Omega $ , $ \\\\Phi $ : omega T.P.\",\n \"Leão in Oct 5 2021, light revision in Aug 29 2022 Fits the Brutsaert-van Genuchten equation to data using the Leverberg-Marquardt algorithm Language: Python 3.8.10 This code does not check for regression diagnostics Import the necessary packages import numpy as np from scipy.optimize import curvefit, leastsquares import matplotlib.pyplot as plt Data psi = np.array([20, 40, 60, 80, 100, 200, 500, 800, 1000, 3000, 5000, 15000]) theta = np.array([0.41, 0.409, 0.406, 0.41268, 0.41045, 0.39708, 0.3899, 0.3525, 0.3203, 0.18478, 0.14497, 0.13]) Define initial parameters (Initial guess for the nonlinear regression procedure) thetas = 0.4 thetar = 0.1 alpha = 0.001 n = 2.0 m = 0.5 Defines the nonlinear regression model to be fit def vg(x, qs, qr, alpha, n, m): return qr + (qs - qr)/(1+ (alpha*psi)**n)**m Puts the initial parametes into the \\\"InitialParams\\\" vector InitPar = [thetas, thetar, alpha, n, m] Curve fitting procedure - \\\"lm\\\" is the Leverberg-Marquardt algorithm popt, pcov = curvefit(vg, psi, theta, p0=InitPar, method='lm') Creates values of psi for plotting the predicted data xd2 = np.linspace(10, 30000, 10000) Uses the parameters from the fit to create the data for the predicted plot thetap2 = popt[1] + (popt[0] - popt[1])/(1+ (popt[2]*xd2)**popt[3])**popt[4] computes the standard deviation of the parameters perr = np.sqrt(np.diag(pcov)) print(\\\"Best fit paramaters, thetas, thetar, alpha, n and m \\\", popt) print(\\\"Covariance of the parameters \\\", popt) print(\\\"Standard deviation of the parameters \\\", perr) Plot of the data and best fit line plt.plot(psi, theta, \\\"o\\\", label = \\\"Simulated data\\\") plt.plot(xd2, thetap2, \\\":\\\", label = \\\"van Genuchten (1980)\\\") plt.ylim(0, 0.45) plt.ylabel('Volumetric water content [vol/vol]') plt.xlabel('Matric potential [|cm|]') legend = plt.legend(loc='lower left', shadow=True, fontsize='large') plt.xscale(\\\"log\\\") plt.show() // Solution to the laplace equation 2D using an explicit finite differences scheme // T.P.\",\n \"Leao on 5 out 2021 light revision on Aug 29 2022 // Language: C++ include include include include int main() // define the simulation parameters double dx = 5.0; double dy = 5.0; double X = 100.00; // x length double Y = 100.00; // y length int nx = X/dx+1; int ny = Y/dy+1; int i, j; // define the grid double h[nx][ny]; // initialize the array with zeros for(i=0; i < nx; i++) for(j=0; j < ny; j++) h[i][j] = 0.0; std::cout << \\\"————————————-\\\" << std::endl; // define the boundary conditions for(i=0; i < nx; i++) h[i][0] = 5.0; h[i][ny-1] = 1.0; for(j=1; j < ny; j++) h[0][j] = 20.0; h[nx-1][j] = 0; // Solves the system double d[1000], diff; double tol = 1e-5; std::cout << \\\"Iterations and diff \\\" << std::endl; for (int iter = 0; iter <100; iter++) // Maximum number of iterations for(i=1; i < nx-1; i++) for(j=1; j < ny-1; j++) h[i][j] = 0.25 * (h[i+1][j] + h[i-1][j] + h[i][j+1] + h[i][j-1]); d[iter] = h[1][1]; diff = d[iter] - d[iter-1]; if (diff < tol) break; std::cout << iter << \\\"\\\" << diff << std::endl; // Print iteration and difference between sucessive iterations at h[1][1] std::cout << \\\"Solution in gnuplot format \\\" << std::endl; for(i=0; i < nx; i++) for(j=0; j < ny; j++) std::cout << std::setprecision(6) << h[i][j] << \\\"\\\"; // Prints solution in terminal std::cout << std::endl; return 0; // Solution to the advective dispersive 1D equation using an explicit method // T.P.\",\n \"Leao 01 out 2021 light revision 29 aug 2022 // Language: C++ include include include include int main() // define the simulation parameters double dx = 1.0; double dtau = 0.1; double TMAX = 500.0; double L = 100.0; double D = 5; double V = 2.0; int nx = L/dx+1; int ntau = TMAX/dtau+1; int i, j; // define the grid double C[ntau][nx]; // initialize the array with zeros for(i=0; i < nx; i++) for(j=0; j < ntau; j++) C[j][i] = 0.0; std::cout << \\\"————————————-\\\" << std::endl; // define the initial conditions for(i=0; i < nx; i++) C[0][i] = 0.0; // define the boundary conditions // for(j=1; j < ntau; j++) C[j][0] = 1.0; C[j][nx-1] = 0.0; // this is the explicit solution for(j=0; j < ntau-1; j++) for(i=1; i < nx-1; i++) C[j+1][i] = C[j][i] + dtau/pow(dx,2)*D*(C[j][i-1] -2*C[j][i] +C[j][i+1]) - V *dtau/(2*dx)*(C[j][i+1] - C[j][i-1]); std::cout << \\\" Explicit solution to the 1D advective-dispersion equation in time \\\"; std::cout << \\\"\\\"; std::cout << \\\" \\\"; std::cout << \\\"\\\"; for(j=0; j < ntau; j++) std::cout << \\\"| \\\"; for(i=0; i < nx; i++) std::cout << C[j][i] << \\\" \\\" ; return 0; // Solves the 1D richards equatio using and explicit finite differences scheme // This code has been validaded using Haverkamp 1977 analytical solutions using Philip's model // The discretization also follows Haverkamp 77 // In 2 out 2021 - TPL, light revision 29 aug 2022 // Language: C++ include include include include // static parameters for Warwick hydraulic functions, see Zarba 1988 MIT Masters define thetas 0.287 define thetar 0.075 define alpha 1.611e06 define beta1 3.96 define Ks 9.44e-3 define A 1.175e06 define B 4.74 using namespace std; // define the functions for water retention and hydraulic conductivity double theta(double h) double theta; theta = alpha * (thetas-thetar)/(alpha+ pow(abs(h), beta1)) + thetar; return theta; double K(double h) double K; K = Ks * A/(A + (pow(abs(h), B))); return K; double C(double h) double C; C = alpha*(thetas-thetar) * pow(abs(h), beta1-1) * beta1/((alpha+ pow(abs(h), beta1))* (alpha+ pow(abs(h), beta1))); return C; int main() double dz = 1.0; double dt = 0.4; int j, i; double L = 80.0; double tmax = 360; int nx = L/dz + 1; int ntau = tmax/dt + 1; double h[ntau][nx];//, K[ntau][nx], C[ntau][nx], q[ntau][nx]; for (j = 1; j < ntau; j++) h[j][0] = -20.73; h[j][nx-1] = -61.5; for (i = 0; i < nx; i++) h[0][i] = -61.5; for (j=0; j < ntau-1; j++) for (i=1; i < nx-1; i++) h[j+1][i] = h[j][i] + dt*1.0/(C(h[j][i])*dz) * ( (K(h[j][i+1])+K(h[j][i]) )/2.0 * ((h[j][i+1]-h[j][i])/dz -1.0 ) - (K(h[j][i])+K(h[j][i-1]) )/2.0 * ((h[j][i]-h[j][i-1])/dz - 1.0 )); for (j = 0; j < ntau; j++) for (i = 0; i < nx; i++) cout << i << \\\"\\\" << h[j][i]<<\\\"\\\"; cout << endl; // saves output to *.dat file std::ofstream outdata; std::ofstream output; outdata.open(\\\"rich1d.dat\\\"); for (int j=0; j < ntau; j++)outdata << std::endl; //std::cout << j << std::endl; for (int i=0; i < nx; i++) outdata << h[j][i] <<\\\"\\\"; outdata.close(); cout << endl; cout << \\\"—————————\\\" << endl; cout << \\\"Results saved to rich1d.dat\\\" << endl; return 0;\"\n ]\n]"},"id":{"kind":"string","value":"2210.07716"}}},{"rowIdx":2993,"cells":{"text":{"kind":"list like","value":[["HGARN: Hierarchical Graph Attention Recurrent Network for Human Mobility\n Prediction"],["Abstract Human mobility prediction is a fundamental task essential for various applications, including urban planning, transportation services, and location recommendation.","Existing approaches often ignore activity information crucial for reasoning human preferences and routines, or adopt a simplified representation of the dependencies between time, activities and locations.","To address these issues, we present Hierarchical Graph Attention Recurrent Network (HGARN) for human mobility prediction.","Specifically, we construct a hierarchical graph based on all users' history mobility records and employ a Hierarchical Graph Attention Module to capture complex time-activity-location dependencies.","This way, HGARN can learn representations with rich contextual semantics to model user preferences at the global level.","We also propose a model-agnostic history-enhanced confidence (MaHec) label to focus our model on each user's individual-level preferences.","Finally, we introduce a Recurrent Encoder-Decoder Module, which employs recurrent structures to jointly predict users' next activities (as an auxiliary task) and locations.","For model evaluation, we test the performances of our Hgarn against existing SOTAs in recurring and explorative settings.","The recurring setting focuses more on assessing models' capabilities to capture users' individual-level preferences.","In contrast, the results in the explorative setting tend to reflect the power of different models to learn users' global-level preferences.","Overall, our model outperforms other baselines significantly in the main, recurring, and explorative settings based on two real-world human mobility data benchmarks.","Source codes of HGARN are available at https://github.com/YihongT/HGARN."],["Introduction","Human mobility is critical for various downstream applications such as urban planning, transportation services, location recommendation, and epidemic management.","The ability to model and accurately predict future human mobility can inform us about important public policy decisions for society's betterment, such as promoting social integration, encouraging innovation, and maximizing productivity [29].","Central to human mobility modeling is the problem of next location prediction, i.e., the prediction of where an individual is going next, which has received great attention.","On the one hand, the increasing prevalence of mobile devices and the popularity of location-based social networks (LBSNs) provide unprecedented data sources for mining individual-level mobility traces and preferences [38].","On the other hand, the advancement of AI and machine learning offers a plethora of analytical tools for modeling human mobility.","These innovations have supercharged the development of human mobility models in the past decade, especially for the next location prediction.","Figure: An illustration of two human mobility trajectories.","Activities are essential in affecting human travel decisions and in modeling multiple dependencies.Traditional approaches studied human mobility over sequential transitions, where several approaches explored using Markov Chains (Mc) [10], [28], [25] to model transition patterns over location sequences.","Later, recurrent neural network-based [13] models demonstrated better predictive power over Mc-based methods, where pioneering works employed recurrent structures to model temporal periodicity [9] and spatial regularity [36].","Some other Rnn-based methods incorporate spatial and temporal contexts [21], [30] into the Rnn update process to boost models' performances.","In addition, due to the great success of the Transformer architecture [31], the attention mechanism has also been adopted in several works [22], [12] to better model sequences and obtain good prediction results.","In recent years, graph-based approaches leveraged graph representation learning [37], [3] and graph neural networks (Gnns) [15] to model user preferences [20] and the spatial-temporal relationships [8], [22] between locations, obtaining rich representations [33], [27] to improve the performance of the next location prediction.","While these works rely on mining individual location sequences for human mobility prediction, they rarely consider the underlying travel behavior structure and activity information.","In the travel behavior literature, an individual's decision to visit a location is usually considered to be derived from their daily activity patterns [2].","As a result, important inter-dependencies exist between activity participation and location visitation behaviors, which are often overlooked.","Furthermore, the number of possible activity categories is much smaller than the number of locations, making it easier to obtain a robust estimate for the former.","With data regarding human activity patterns becoming increasingly accessible, it is both behaviorally meaningful and computationally efficient to incorporate such activity information and its spatial-temporal dependencies in human mobility modeling.","Figure REF shows a couple of human mobility trajectories reflecting time-activity-location dependencies.","For example, when the time is approaching noon, one user may decide to dine at a nearby restaurant, and another may go to the movie theater for a specific starting time.","These examples illustrate that activities are usually scheduled according to the time of day, affecting the choice of which location to visit next, as also evidenced by Figure REF , where people usually go to restaurants at noon and mostly return home or go to the gym at night.","However, only a few works have considered adding activity information (e.g., location categories) to improve next location prediction.","Yu et al.","[39] used activities along with spatial distances to reduce search space (i.e., candidate locations).","Huang et al.","[14] proposed a Cslsl which comprises an Rnn-based [7] structure, where the time, activity, and location are predicted sequentially to model human travel decision logic.","However, the design of Cslsl oversimplifies the time-activity-location dependencies.","Given data sparsity and behavioral uncertainties [44], the time prediction tends to be more challenging [43], which can further compromise the prediction of activities and locations.","To the best of our knowledge, no efforts have been made to model the sophisticated time-activity-location dependencies and leverage them to improve the next location prediction.","In this paper, we present Hierarchical Graph Attention Recurrent Network (Hgarn) for the next location prediction.","Specifically, we first construct a hierarchical graph based on time-activity-location history mobility records of all users and employ a Hierarchical Graph Attention Module to capture complex temporal-view activity-activity, affiliated-view activity-location, spatial-view location-location dependencies.","In this way, Hgarn can learn representations with rich contextual semantics to model user preferences at the global level.","Correspondingly, a model-agnostic history-enhanced confidence (MaHec) label is proposed to guide our model to learn each user’s individual-level preferences.","We finally introduce a Recurrent Encoder-Decoder Module.","The module takes the sequence of user embeddings combined with the learned hierarchical graph representations as input to jointly predict a user's next activity (as the auxiliary task) and location (as the main task).","Specifically, the former can be residually incorporated into the latter.","In such a design, our model can leverage the learned time-activity-location dependencies to benefit both global- and individual-level human mobility modeling, and use predicted next activity distribution to facilitate model performance for personalized next location prediction.","In summary, our paper makes the following contributions: [leftmargin=*] We propose a Hierarchical Graph to represent the temporal-view activity-activity, affiliated-view activity-location, spatial-view location-location dependencies.","To the best of our knowledge, among the few approaches incorporating activity information into the next location prediction, this is the first work to model the dependencies of time, activities and locations by leveraging a Hierarchical Graph.","We design a Hierarchical Graph Attention Recurrent Network (i.e., Hgarn), which contains a hierarchical graph attention module to model dependencies between time, activities, and locations to capture users' global-level preferences, and a recurrent encoder-decoder module to incorporate the hierarchical graph representations into sequence modeling and utilize next activity prediction to boost the next location prediction.","We introduce a simple yet effective model-agnostic history-enhanced confidence (MaHec) label to guide our model to learning each user's individual-level preferences, which enables the model to focus more on relevant locations in their history trajectories when predicting their next locations.","Extensive experiments are conducted using two real-world location-based social networks (LBSN) check-in datasets.","Specifically, we evaluate the prediction performance of our Hgarn against existing SOTAs in the main, recurring, and explorative settings.","Our work is the first to separately evaluate the next location prediction models' recurring and explorative prediction performances.","The results show that the Hgarn can significantly outperform existing methods in all experimental settings.","Figure: A workflow of the proposed Hgarn."],["Related work","The next location prediction is essentially about sequence modeling since successive location visits are usually correlated [5], [40].","Traditional Mc-based methods often incorporate other techniques, such as matrix factorization [28] and activity-based modeling [25] to capture mobility patterns and make predictions.","However, Mc-based methods are limited in capturing long-term dependencies or predicting exploratvie human mobility.","Deep learning-based approaches consider the next location prediction a sequence-to-sequence task and obtain better prediction results than traditional methods.","The majority of existing deep models are based on Recurrent neural networks [13], [6] (Rnn).","Strnn [21] is a pioneering work that incorporates spatial-temporal features between consecutive human visits into Rnn models to predict human mobility.","Stgn [42] adds spatial and temporal gates to Lstm to learn users interests.","Flashback [36] leverages spatial and temporal intervals to compute an aggregated past Rnn hidden states for predictions.","Lstpm [30] introduces a non-local network and a geo-dilated Lstm to model users’ long- and short-term preferences.","Some approaches use the attention mechanism to improve their performances.","DeepMove [9] leveraged attention mechanisms combined with an Rnn module to capture users’ long- and short-term preferences.","Arnn [12] uses a knowledge graph to find related neighbor locations and model the sequential regularity of check-ins through attentional Rnn.","Stan [22] extracts relative spatial-temporal information between consecutive and non-consecutive locations through a spatio-temporal attention network for locations predictions.","In addition, some efforts incorporate contextual information [17] such as geographical information [18], dynamic-static [24], text content about locations [4] into sequence modeling.","Recently, graph-based models have been designed to enrich contextual features, which can reflect global preferences.","LBSN2Vec [37] performs random walks on a hypergraph to learn embeddings for next location and friendship predictions.","STP-UDGAT [20] uses Gat to learn location relationships from both local and global views based on constructed spatial, temporal, and preference graphs.","HMT-GRN [19] learned several user-region matrices of different granularity levels to alleviate the data sparsity issue and efficiently predict next locations.","Gcdan [8] uses dual attention to model the high-order sequential dependencies and graph convolutions to mitigate the data sparsity issue.","Graph-Flashback [27] uses knowledge graph embedding to construct transition graph, and applies Gcn to refine graph representations and combines with Flashback to make predictions.","Some other works focused on different aspects, activity-aware methods like [1] model the activities with a weighted category hierarchy (WCH), CatDM [39] uses activities along with spatial distances to reduce search space, Cslsl [14] proposed an Rnn-based causal structure to capture human travel decision logic.","However, most existing methods ignore activity information and cannot effectively model the time-activity-location dependencies, which are essential for predicting and understanding human mobility."],["Preliminaries","This section introduces definitions relevant to this study and then formulates the next location prediction problem.","We use notations $U=\\left\\lbrace u^{i} \\right\\rbrace _{i=1}^{|U|}$ , $L=\\lbrace l^{i}\\rbrace _{i=1}^{|L|}$ , $C=\\lbrace c^{i}\\rbrace _{i=1}^{|C|}$ , and $T=\\lbrace t^{i}\\rbrace _{i=1}^{|T|}$ to denote the sets of users, locations, activities and time series, respectively.","In particular, we denote a user $u\\in U$ 's sets of locations, activities and time series in a temporal order as $L_{u}=\\lbrace l^{i}_{u} \\rbrace _{i=1}^{|L_{u}|}$ , $C_{u}=\\lbrace c^{i}_{u} \\rbrace _{i=1}^{|C_{u}|}$ , and $T_{u}=\\lbrace t^{i}_{u} \\rbrace _{i=1}^{|T_{u}|}$ , where $\\delta _{u}^{i}$ may not equal to $\\delta ^{i}$ , $\\delta \\in \\lbrace U, L, C, T\\rbrace $ .","Definition 1 (Mobility Record) We use the notation $r$ to denote a single human mobility record.","Specifically, the $i$ th record of a given user $u \\in U$ is represented by a tuple $r_{u}^{i}=(u, c_{u}^{i}, l_{u}^{i}, t_{u}^{i})$ .","Each record tuple comprises a user $u \\in U$ , an activity $c_{u}^{i} \\in C_{u}$ , a location $l_{u}^{i} \\in L_{u}$ and the visit time $t_{u}^{i} \\in T_{u}$ .","Definition 2 (Trajectory) A trajectory is a sequence of mobility records that belongs to a user $u \\in U$ , denoted by $R_{u}=\\lbrace r_{u}^{i}\\rbrace _{i=1}^{|R_{u}|}$ .","Each trajectory $R_{u}$ can be divided into a history trajectory $R_{u}^{his}=\\lbrace r_{u}^{i}\\rbrace _{i=1}^{|R_{u}|-1}$ and the final record $r_{u}^{|R_{u}|}$ .","In this way, we can obtain the user $u$ 's activity trajectories $R_{u}^{C}={\\lbrace c_{u}^{i}\\rbrace }_{i=1}^{|R_{u}|}$ , location trajectories $R_{u}^{L}={\\lbrace l_{u}^{i}\\rbrace }_{i=1}^{|R^{u}|}$ , and time trajectories $R_{u}^{T}={\\lbrace t_{u}^{i}\\rbrace }_{i=1}^{|R_{u}|}$ with their history trajectories $R_{u}^{C, his}$ , $R_{u}^{L, his}$ and $R_{u}^{T, his}$ .","Given the definitions above, the next location prediction problem can be formulated as follows: Problem 1 (Next Location Prediction) Given a user $u \\in U$ 's history trajectory $R_{u}^{his}$ as input, we consider $u$ 's next record as its future state.","The human mobility prediction task $\\mathcal {T}$ maps $u$ 's history trajectory $R_{u}^{his}$ to $u$ 's next location $l_{u}^{|R_{u}|}$ in the future.","The above-described process can be summarized as follows: $R_{u}^{his}\\stackrel{\\mathcal {T}(\\cdot ;\\theta )}{\\longrightarrow }l_{u}^{|R_{u}|},$ where $\\theta $ is the parameters of mapping $\\mathcal {T}$ ."],["Methodology","The Hgarn's workflow is demonstrated in Figure REF .","The raw data is first encoded in the embedding layer and then input to the hierarchical graph attention module to model multi-dependencies.","Finally, the user's personalized embeddings are fused with the learned hierarchical graph representations and input to the recurrent encoder-decoder module to make predictions.","In the following sections, we will elaborate on the details of our Hgarn."],["Embedding Layer","We first introduce our embedding layer, which aims to assign trainable embeddings to each unique mobility record element, i.e., the user, activity, location and time.","It is worth noting that all these elements are discrete values except for time, and it is necessary to split the continuous time into discrete intervals to facilitate the encoding of time.","In this work we discretize the 24 hours of the day into $|T^{h}|$ time slots $T^{h}=\\lbrace h\\rbrace _{h=1}^{|T^{h}|}$ and another variable $T^{w}=\\lbrace w\\rbrace _{w=1}^{|T^{w}|}$ indicating the day of the week.","Note that all $t \\in T$ can be written in the form of $t=(h, w)$ .","We obtain the embeddings of each user, activity, location and time $e_{u} \\in \\mathbb {R}^{d^u}$ , $e_{c}, e_{l} \\in \\mathbb {R}^{d}$ , $e_{t} \\in \\mathbb {R}^{d^t}$ by multiplying their one-hot vectors with the corresponding trainable transformation matrices, where $d^u$ , $d$ and $d^t$ represent embedding dimensions.","$e_{U} \\in \\mathbb {R}^{|U| \\times {d^u}}$ , $e_{C} \\in \\mathbb {R}^{|C| \\times {d}}$ , $e_{L} \\in \\mathbb {R}^{|L| \\times {d}}$ and $e_{T} \\in \\mathbb {R}^{|T| \\times {d^t}}$ are used to denote the embedding matrices for the user, activity, location and time, respectively.","$e_{r_{u}^{i}}=\\left(e_{u}, e_{c_{u}^{i}}, e_{l_{u}^{i}}, e_{t_{u}^{i}}\\right)$ is used to denote a user $u$ 's embeddings of $i$ th record."],["Hierarchical Graph Attention Module","The hierarchical graph attention module consists of two parts: hierarchical graph construction and hierarchical graph attention (HGA) networks on the graph for multi-dependencies modeling.","Figure: The design of hierarchical graph attention module."],["Hierarchical Graph Construction","Given that a spatial network is naturally a graph and recent advances in graph neural networks [23], [32], we choose to model the location-location, location-activity and activity-activity dependencies with a hierarchical graph.","The constructed hierarchical graph is shown in Figure REF , which includes three layers: location layer, localized-activity layer, and activity layer, where the localized-activity layer is adopted to suppress noise aggregated from the location layer.","We formally describe the hierarchical graph with notation $G=(V,E)$ , where $V$ $=$ $V^{L} \\cup V^{C} \\cup V^{C^\\prime }$ and $E=\\lbrace A^{L}$ ,$A^{C}$ ,$A^{LC^\\prime }$ ,$A^{CC^\\prime }\\rbrace $ .","Specifically, $V^{L}$ and $V^{C}$ represents the location node-set and the activity node-set.","$V^{C^\\prime }$ indicates the localized-activity node-set.","$E$ comprises four adjacency matrices that denote connectivity between (1) two location nodes, (2) two activity nodes, (3) a location node and a localized-activity node, (4) an activity node and a localized-activity node.","For the location adjacency matrix $A^{L}$ , we utilize the geographical distance to determine whether two nodes are linked.","We employ the haversine formula to compute distances between locations.","Given $l^{i}$ and $l^{j}$ and their GPS data, $A^{L} \\in \\mathbb {R}^{|L| \\times |L|}$ is as: $A^{L}_{l^{i}, l^{j}}={\\left\\lbrace \\begin{array}{ll}1, & Haversine \\left(\\text{GPS}^{i}, \\text{GPS}^{j} \\right) < D^{h}\\\\0, & \\text{otherwise}\\end{array}\\right.","},$ where $D^{h}$ is a hyperparameter to control the distance threshold to influence the connectivity between locations.","The construction of $A^{C}$ is based on history trajectories of all users.","Intuitively, the dependency between two activities can be measured by the frequency of co-occurrence in the same time interval $h \\in T^{h}$ .","However, if we directly consider the activity co-occurrence frequency based on all trajectories (regardless of the user), it may lead to unrelated activities being interrelated (e.g., check-in at subway stations and gyms both often occur in the evening), due to the difference in user preferences.","Instead, a more reasonable way is to learn the inter-activity dependencies based on activity co-occurrence within individual-level trajectory sets.","Therefore, we can traverse the history trajectories of each user $u \\in U$ and calculate the co-occurrence matrix $M^{C} \\in \\mathbb {R}^{|C| \\times |C|}$ : $M^{C}_{c_{u}^{i}, c_{u}^{j}}=\\sum _{u \\in U}\\sum _{i=1}^{|R_{u}^{his}|}\\sum ^{|R_{u}^{his}|}_{j=i+1} \\mathbf {1}_{h}\\left(h_{u}^{i}, h_{u}^{j}\\right),$ $\\mathbf {1}_{h}\\left(h_{u}^{i}, h_{u}^{j}\\right)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } {h}_{u}^{i} = {h}_{u}^{j} \\\\0, & \\text{otherwise}\\end{array}\\right.","},$ where $\\mathbf {1}_{h}(\\cdot )$ is an indicator function.","Larger elements in the $M^{C}$ indicate stronger dependencies between the corresponding two activities.","For the sake of simplicity and suppression of correlation between some of the less relevant activities, we calculate the adjacency between each activity pair $(c^{i}, c^{j})$ to attain $A^{C} \\in \\mathbb {R}^{|C| \\times |C|}$ : $A^{C}_{c^{i}, c^{j}}={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } M^{C}_{c^{i}, c^{j}} > \\text{mean}\\left(M^{C}\\right) \\\\0, & \\text{otherwise}\\end{array}\\right.","},$ The matrix $A^{LC^{\\prime }}$ defines the adjacency between nodes of the location layer and nodes of the localized-activity layer.","Each node of $V^{L}$ is linked to only one node of $V^{C^\\prime }$ , representing the corresponding activity category at that location.","In contrast, each node of $V^{C^\\prime }$ may be linked to multiple nodes of $V^{L}$ , as several locations can share the same activity.","Formally, we define $A^{LC^{\\prime }}_{L} \\in \\mathbb {R}^{|L| \\times |C|}$ based on the affiliations of locations and activities, where each row corresponds to a location and each column an activity.","Additionally, we construct the adjacency matrix $A^{LC^{\\prime }} \\in \\mathbb {R}^{(|L|+|C|) \\times (|L|+|C|)}$ based on $A^{LC^{\\prime }}_{L}$ in the following block matrix form: $A^{LC^{\\prime }}=\\begin{bmatrix}O_{L} & A^{LC^{\\prime }}_{L} \\\\{(A^{LC^{\\prime }}_{L})}^{\\top } & O_{C}\\end{bmatrix},$ where ${(A^{LC^{\\prime }})}^{\\top }$ is the transpose of $A^{LC^{\\prime }}$ , $O_{L} \\in \\mathbb {R}^{|L| \\times |L|}$ and $O_{C} \\in \\mathbb {R}^{|C| \\times |C|}$ are two zero matrices.","The function of the localized-activity layer is to suppress noise from the location layer aggregated to the activity layer, so we simply define that each node in the localized-activity layer is connected to the node in the activity layer representing the same activity.","Mathematically, similar to Equation REF , we have $A^{CC^\\prime } \\in \\mathbb {R}^{2|C| \\times 2|C|}$ : $A^{CC^\\prime }=\\begin{bmatrix}O_{C} & I_{C} \\\\I_{C} & O_{C}\\end{bmatrix},$ where $I_{C} \\in \\mathbb {R}^{|C| \\times |C|}$ is an identity matrix."],["Hierarchical Graph Attention Networks","Inspired by recent work conducted deep learning on hierarchical graphs [11], [34], [41], we design a hierarchical graph attention network (HGA) to model the hierarchical graph we just defined.","Since the importance of locations within a certain distance is different for a specific location, we employ graph attention mechanisms to model spatial-view location-location dependencies: $H^{L}=\\textsc {\\textsc {Gat}}_{L}\\left(e_{L}, A^{L}\\right),$ where $\\textsc {\\textsc {Gat}}(\\cdot )$ is a standard implementation of [32], $H^{L} \\in \\mathbb {R}^{|L| \\times d^{g}}$ is the learned representations as the output of the $\\textsc {Gat}_{L}$ .","To integrate location information into representation learning of activities and suppress the noise aggregated to the nodes of the activity layer, we introduce the localized-activity layer to pre-aggregate location embeddings.","We first concatenate $e_L$ and $e_C$ to obtain the fused embedding matrix $e_{LC} \\in \\mathbb {R}^{(|L|+|C|) \\times d} = {\\begin{bmatrix}e_L & e_C\\end{bmatrix}}^{\\top }$ .","Then the localized-activity process is implemented as: $H^{LC^{\\prime }}=\\textsc {Gat}_{LC}\\left(e_{LC}, A^{LC^{\\prime }}\\right),$ where $H^{LC^{\\prime }} \\in \\mathbb {R}^{(|L|+|C|) \\times d^{g}}$ is the output of $\\textsc {Gat}_{LC}$ , we further slice out the first $|L|$ rows of matrix $H^{LC^{\\prime }}$ to obtain the pre-aggregated representation matrix ${H^{C^{\\prime }}} \\in \\mathbb {R}^{|C| \\times d^{g}}$ .","The learned representation $H^{C^{\\prime }}$ is again concatenated with activity embeddings $e_C$ as the $\\textsc {Gat}_C$ 's input $e_{CC}={\\begin{bmatrix}e_C & H^{C^{\\prime }}\\end{bmatrix}}^{\\top }$ .","It is worth noting that for all nodes in the activity layer, we could simultaneously aggregate information from neighbors in the localized-activity layer and neighbors in the same layer by simply modifying the matrix $A^{CC^\\prime }$ to: $A^{CC^\\prime }_{new}=\\begin{bmatrix}A^{C} & I_{C} \\\\I_{C} & O_{C}\\end{bmatrix}.$ We still employ a similar strategy to update the representation of the nodes in the activity layer: ${H^{C}}^\\prime =\\textsc {Gat}_{C}\\left(e_{CC}, A^{CC^\\prime }_{new}\\right),$ where ${H^{C}}^\\prime \\in \\mathbb {R}^{2|C| \\times d^{g}}$ is the learned representation from $\\textsc {Gat}_{C}$ , we slice out the last $|C|$ rows of matrix ${H^{C}}^\\prime $ to obtain the updated activity node representation $H^{C} \\in \\mathbb {R}^{|C| \\times d^{g}}$ ."],["Recurrent Encoder-Decoder Module","The recurrent encoder-decoder module 's structure is shown in Figure REF .","Given a user $u \\in U$ 's history trajectory, the input of the activity and the location encoder at $i$ th iteration are implemented as: $X^{C,i}_{u}= {e_{u}} \\Vert e_{t_{u}^{i}} \\Vert e_{c_{u}^{i}} \\Vert H^{C}_{c_{u}^{i}},$ $X^{L,i}_{u}= {e_{u}} \\Vert e_{t_{u}^{i}} \\Vert e_{l_{u}^{i}} \\Vert H^{C}_{c_{u}^{i}} \\Vert H^{L}_{l_{u}^{i}},$ where $\\Vert $ is the concatenation operation, $H^{C}_{c_{u}^{i}}$ and $H^{L}_{l_{u}^{i}}$ are the learned hierarchical graph representation of the activity node $c_{u}^{i}$ and the location node $l_{u}^{i}$ , the concatenated $X^{C,i}_{u}$ and $X^{L,i}_{u}$ are the inputs of the two encoders at the $i$ th iteration.","Figure: Recurrent encoder-decoder module's architecture.Both activity and location encoders are implemented with Lstm, where hidden states updating process between $h^{\\cdot ,i}_{u}$ and $h^{\\cdot ,i-1}_{u}$ can be illustrated as: $\\begin{aligned}p_{u}^{i}, f_{u}^{i}, g_{u}^{i}, o_{u}^{i} =\\text{split} ( ( & W_{x}X^{\\cdot ,i}_{u}+b_{x} ) + ( W_{h}h^{\\cdot ,i-1}_{u}+b_{h} ) ),\\end{aligned}$ $c^{\\cdot , i}_{u} = \\sigma \\left(f_{u}^{i}\\right) \\odot c^{\\cdot , i-1}_{u} + \\sigma \\left(p_{u}^{i}\\right) \\odot \\tanh \\left(g_{u}^{i}\\right),$ $h^{\\cdot , i}_{u} = \\sigma \\left(o_{u}^{i}\\right) \\odot \\tanh \\left(c^{\\cdot , i}_{u}\\right),$ where “$\\cdot $ ” can be either $C$ or $L$ .","$p_{u}^{i}$ , $f_{u}^{i}$ , $g_{u}^{i}$ , $o_{u}^{i}$ are the input, forget, cell and output gates.","$c^{\\cdot , i}_{u}$ is the cell state at iteration $i$ .","$\\sigma (\\cdot )$ and $\\tanh (\\cdot )$ are the sigmoid and tanh activation functions.","$W_{x}$ , $W_{h}$ , $b_{x}$ and $b_{h}$ are the trainable weights shared by inputs and $\\odot $ is the Hadamard product.","Initial hidden states for $h^{\\cdot , 1}_{u}$ and $c^{\\cdot , 1}_{u}$ are set to zeros.","After obtaining final hidden states of activity and location encoder as $h^{C}_{u}$ and $h^{L}_{u}$ , we implement our activity decoder as a multi-layer perceptron (Mlp) to get the next activity logits $\\widetilde{h^{C}_{u}} \\in \\mathbb {R}^{|C|}$ : $\\widetilde{h^{C}_{u}}=\\textsc {Mlp}_{C}\\left(h^{C}_{u}\\right).$ We finally residually combine the obtained activity logits with the encoded $h^{L}_{u}$ to learn the location logits $\\widetilde{h^{L}_{u}} \\in \\mathbb {R}^{|L|}$ : $\\begin{aligned}\\widetilde{h^{L}_{u}}=& \\lambda _{r}\\cdot \\textsc {Mlp}_{L}^{r}\\left(h^{L}_{u}\\right)+ (1-\\lambda _{r})\\cdot \\textsc {Mlp}_{L}\\left(\\textsc {Mlp}_{L}^{h}\\left(h^{L}_{u} \\Vert h^{C}_{u}\\right) \\Vert \\widetilde{h^{C}_{u}}\\right),\\end{aligned}$ where $\\lambda _{r}$ is a factor that controls the residual connections to trades off different features."],["In this section we introduce the model-agnostic history-enhanced confidence label and explain the optimization process of Hgarn."],["Model-Agnostic History-Enhanced Confidence Label Construction","As human mobility trajectories exhibit repetitive nature (i.e., recurring mobility), existing models [9], [8] often try to capture periodicity by applying attention mechanisms to all user trajectories, have limited performances and interpretability.","To overcome this, we modify the original label to get the model-agnostic history-enhanced confidence label (MaHec label, in short), which can guide our model to focus on relevant user trajectories.","Specifically, for each location $l^{i}\\in L$ , we differentiate its confidence for a user $u$ 's next location in two types: ${\\textsc {MaHec}}_{l^{i}}^{u}={\\left\\lbrace \\begin{array}{ll}w^{c}, & \\text{if } l^{i}=l_{u}^{|R_{u}|} \\\\\\left(1-w^{c}\\right) \\cdot f_{l^{i}}^{u}, & \\text{otherwise}\\end{array}\\right.","},$ $f_{l^{i}}^{u}=\\frac{1}{|R_{u}^{his}|}\\sum _{{l^{i}}\\in L} \\mathbf {1}_{R^{L,his}_{u}}\\left(l^{i}\\right),\\text{ }\\text{ }\\mathbf {1}_{R^{L,his}_{u}}\\left(l^{i}\\right)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } l^{i} \\in R^{L,his}_{u} \\\\0, & \\text{if } l^{i} \\notin R^{L,his}_{u}\\end{array}\\right.","},$ where $\\mathbf {1}_{R^{L,his}_{u}}(\\cdot )$ is an indicator function and $w^{c}\\in [0,1]$ is a hyperparameter that indicates the confidence of the $u$ 's ground truth label and $f_{l^{i}}^{u}$ denotes the user $u$ 's history visit frequency to $l^{i}$ .","Then the MaHec label for the user $u$ 's next location is as: ${\\textsc {MaHec}}_{L}^{u}=\\left({\\textsc {MaHec}}_{l^{i}}^{u},\\right)_{i=1}^{|L|}\\in \\mathbb {R}^{|L|},$ where each element in ${\\textsc {MaHec}}_{L}^{u}$ represents the confidence that the user $u$ decides to pick as its next location.","Similarly, we conduct the same operations for user activity trajectories to obtain ${\\textsc {MaHec}}_{C}^{u}$ ."],["Model Optimization","Since next location prediction is a classification problem, we transform $\\widetilde{h^{L}_{u}}$ to the probability distribution $\\widehat{h^{L}_{u}} \\in \\mathbb {R}^{|L|}$ of all locations by $\\widehat{h^{L}_{u}}=\\text{Softmax}\\left(\\widetilde{h^{L}_{u}}\\right)$ .","Given ${\\textsc {MaHec}}_{L}^{u}$ and $\\widehat{h^{L}_{u}}$ , we compute $\\mathcal {L}_L$ based on cross-entropy loss: $\\mathcal {L}_L=- \\frac{1}{|U|}\\sum _{u \\in U} \\sum _{i=1}^{|L|}{\\textsc {MaHec}}^{u}_{l^{i}}\\cdot \\log \\left(\\widehat{h^{L, i}_{u}}\\right),$ where $\\widehat{h^{L, i}_{u}}$ is the $i$ th element of $\\widehat{h^{L}_{u}}$ and we compute the next activity loss $\\mathcal {L}_{C}$ based on same above described operations.","Finally, we could train our Hgarn end-to-end with a total loss function: $\\mathcal {L}=\\lambda _{L}\\cdot \\mathcal {L}_{L}+\\lambda _{C}\\cdot \\mathcal {L}_{C}$ $\\lambda _{L}$ and $\\lambda _{C}$ are hyperparameters that trade off different loss terms."],["Experiments","In this section, we compare our Hgarn with existing SOTAs on two real-world LBSN check-in datasets."],["Datasets","We adopt two LBSN datasets [38] containing Foursquare check-in records in New York City (NYC) and Tokyo (TKY) from April 12, 2012 to February 16, 2013, including 227,428 check-ins for NYC and 573,703 check-ins for TKY.","The location distributions of NYC and TKY datasets are shown in Figure REF in the appendix.","Users and locations with less than 10 records are removed following previous work, and after cleaning, NYC and TKY have 308 and 286 activities, respectively.","We divide the data into a training and testing sets in a ratio of 8:2, following the setting of [42].","Key data summary statistics are listed in Table REF .","Table: Main performance comparisons of different methods in two real-world datasets."],["Baselines & Experimental Details","The following baseline methods are considered in experiments: [leftmargin=*] Mc [28]: Mc is a widely used sequential prediction approach which models transition patterns based on visited locations.","Strnn [21]: Strnn is an Rnn-based model that incorporates the spatial-temporal contexts by leveraging transition matrices.","DeepMove [9]: DeepMove uses attention mechanisms and an Rnn module to capture users' long- and shot-term preferences.","Lstpm [30]: Lstpm introduces a non-local network and a geo-dilated Lstm to model users' long- and short-term preferences.","Flashback [36]: Flashback is an Rnn-based model that leverages spatial and temporal intervals to compute an aggregated hidden state from past hidden states for prediction.","Plspl [35]: Plspl incorporates activity information to learn user preferences and utilizes two Lstms to capture long-term and short-term preferences.","PG2Net [16]: PG2Net learns users' group and personalized preferences with spatial-temporal dependencies and attention-based Bi-Lstm.","Gcdan [8]: Gcdan leverages graph convolution to learn spatial-temporal representations and use dual-attention to model the sequential dependencies.","Cslsl [14]: Cslsl employs multi-task learning to model decision logic and two Lstms to capture long- and short-term preferences.","Graph-Flashback [27]: Graph-Flashback adds Gcn to Flashback to enrich learned transition graph representations constructed based on defined similarity functions over embeddings from the existing Knowledge Graph Embedding method.","For all baselines and our method, we adopt two commonly-employed metrics in prior works: Rec@K (Recall) and NDCG@K (Normalized Discounted Cumulative Gain).","Specific formulas of these two metrics are defined as: $Recall@K=\\frac{1}{|U|}\\sum _{u \\in U}\\frac{|l_{u}^{|R_{u}|} \\cap \\widehat{l_{u, K}^{|R_{u}|}}|}{|l_{u}^{|R_{u}|}|}$ and $NDCG@K=\\frac{1}{|U|}\\sum _{u \\in U}\\sum _{i=1}^{K}\\frac{|l_{u}^{|R_{u}|} \\cap \\widehat{l_{u, i}^{|R_{u}|}}|}{\\log (i+1)}$ , where $\\widehat{l^{\\cdot }_{u, k}}$ indicates the top $k$ predicted locations.","We port all the baselines to our run time environment for fair comparisons based on their open-source codes.","We carefully tuned their hyperparameters to get the best results.","Additionally, unlike previous works that only evaluate overall model performances (under the main setting), we also conduct experiments under the recurring and explorative settings for more comprehensive performance evaluation against the existing SOTAs.","For the main and recurring settings, we choose $K=\\lbrace 1, 5, 10\\rbrace $ for evaluation.","As the performance is generally poorer under the explorative setting, we set $K=\\lbrace 10, 20\\rbrace $ .","For the choice of hyperparameters, we set both $\\lambda _{L}$ and $\\lambda _{C}$ to 1, $\\lambda _{r}$ to 0.6 for both datasets.For embedding dimensions, we set $d=200$ , $d^{u}=10$ , $d^{t}=30$ , $d^{g}=50$ and the dimension of encoders' hidden states are set to 600.","Detailed reproducibility information can be found in supplementary materials."],["Main Results","Table REF shows the main performance comparisons of different methods for the next location prediction.","Our Hgarn achieves state-of-the-art performances on all metrics for both datasets.","Specifically, Hgarn outperforms the best baseline approach by 12-19% on Recall@$K$ and NDCG@$K$ for NYC, and 11-20% for TKY.","Its advantages become more significant as $K$ increases, validating the effectiveness of the hierarchical graph modeling and MaHec label for the next location prediction task.","Table: Results under the recurring setting.In addition, we also evaluate different models separately in the recurring and explorative settings.","In the recurring setting, based on results in Table REF , our model shows an improvement of (10.8%, 23.4%, 31.6%, 10.8%, 19.1%, 20.1%) and (9.4%, 19.2%, 19.7%, 9.4%, 16.5%, 16.9%) on all metrics (R@1, R@5, R@10, N@1, N@5, N@10) compared to the best baseline on the two datasets.","Hgarn outperforms all baselines significantly in the recurring setting.","Table: Results under the explorative setting.As shown in Table REFThe Mc's results are all zeros and thus deleted from the table., the overall results in the explorative setting are much lower than those in the main and recurring settings, which is intuitive because of the inherent difficulty of predicting unseen locations.","A possible approach to improving the prediction in the explorative setting is to model the dependencies between locations.","In addition, due to the larger number of locations in the TKY dataset, the hierarchical graph modeling may introduce noise, making our model less effective in ranking the predicted locations.","The above hypotheses may also be why our model performs better in Recall but are not consistently better in NDCG."],["Ablation Study","To study each component of Hgarn, we conduct an ablation study considering the following four variants of Hgarn: (1) Hgarn w/o HGat contains only the recurrent encoder-decoder module for the next location prediction.","(2) Hgarn w/o AGat: This variant's hierarchical graph attention module contains only the location layer and corresponding graph attention networks.","(3) Hgarn w/o Res is the variant that removes the residual connection of the recurrent encoder-decoder module.","(4) Hgarn w/o MaHec is the variant that leverages original labels to optimize our model.","The ablation study results are shown in Figure REF .","It is found that all metrics get improved as more components are included.","The gradual increase in prediction results from Hgarn w/o HGat to Hgarn w/o AGat to Hgarn is a fine-grained demonstration of the effectiveness of Gat in each layer of the hierarchical graph.","In addition, the model's performance improvements by adding the MaHec label is significant when $K$ are large, verifying its effectiveness under simplicity."],["Hyperparameter Sensitivity Analysis","We further study the sensitivity of a few key parameters by varying each parameter while keeping others constant.","(1) $D^{h}$ affects the location dependencies, too high or too low will lead to decreased performances.","The best results were obtained at $D^{h}=1$ km on NYC and optimal at $D^{h}=0.1$ km on TKY, this is probably because the geographic space and the distance between locations are larger in NYC than in TKY, as in Figure REF .","(2) $w^{c}$ A affects the model's attention to history locations.","The results show an upward and then downward trend as $w^{c}$ rises on both datasets, indicating that both too large and too small dependence on locations in the history trajectory is detrimental to the model's performance.","(3) $\\lambda _{r}$ affects how much activity information is fused in predicting the next location.","Intuitively, a $\\lambda _{r}$ too high would introduce noise, and too low may result in ineffective utilization of activity information.","The results align with our conjecture that the model achieves optimal performance with $\\lambda _{r}$ at 0.6 for both NYC and TKY datasets."],["How does ","To understand the mechanism of MaHec label, we randomly selected two human trajectories from two datasets and visualized the predicted probability difference of “Hgarn” and “Hgarn w/o MaHec”.","Figure REF shows the probability change, where the purple line represents the locations in the current user's history trajectory.","With MaHec labels, the probability distribution of the next location predicted by the model increases in most of the visited locations, demonstrating that our MaHec labels can effectively guide the model to pay attention to the user's history trajectory when predicting the next location.","The mechanism of MaHec can also interpret, to some extent, why the prediction performances of our Hgarn can far exceed that of the baseline methods, especially under the recurring setting."],["What the Hierarchical Graph learned?","Unlike other methods that have difficulty interpreting learned higher-order spatial-temporal dependencies, our Hgarn can somewhat understand the dependencies between activities through the learned Hierarchical Graph.","We visualize one attention head of $\\textsc {Gat}_{C}$ 's sliced attention matrix to analyze the learned activity-activity dependencies.","In Figure REF , We select four activity pairs to show the related activities and their corresponding attention scores.","These activity pairs are intuitive and consistent with common sense human travel, such as the high dependencies between Gyms and Stadiums, Bus Stops and Travel Lounges.","These results have important implications for understanding human activity patterns and predicting the next location."],["Conclusion","Both travel behavior theories and empirical evidence suggest that human mobility patterns largely depend on the need to participate in activities at different times of the day.","Therefore, it is crucial to consider the latter when modeling the former.","In this paper, we propose a Hierarchical Graph Attention Recurrent Network (Hgarn) for activity-aware human mobility prediction.","Specifically, Hgarn introduces hierarchical graph attention mechanisms on a constructed hierarchical graph to model time-activity-location dependencies, and employ next activity prediction as an auxiliary task to further improve the main task of the next location prediction.","In addition, we propose a simple yet powerful MaHec label that can guide our model to flexibly tradeoff the importance of history locations when predicting future locations.","Finally, we perform comprehensive experiments to demonstrate the superiority of Hgarn, this is the first work to evaluate existing models and our Hgarn under recurring and explorative settings to our best knowledge.","We find that introducing activity information can effectively improve the model's prediction accuracy.","In addition, our results show that the existing models have unsatisfactory prediction performances under the explorative setting.","We hope our work could spark the attention of future work on explorative mobility prediction.","For future work, we consider adding a ranking-based component to organize high-probability candidate locations and incorporate human travel decision logic modeling to make more interpretable predictions."],["Appendix","In this section, we provide more details, visualizations and experimental results."],["Experimental Supplementary","Figure REF demonstrated the location distribution of NYC and TKY datasets.","Figure: Location distributions of NYC and TKY.Figure REF displayed two more probability distribution change cases with the effect of MaHec labels.","Figure: More visualizations of predicted locations probaility distributions on two datasets."],["Introduction's supplementary","Given the recent emphasis on modeling contextual information (e.g., spatio-temporal) in mobility prediction approaches, we find little work analyzing changes in model prediction performances other than adding or removing relevant components (i.e., ablation studies) and parameter sensitivities.","However, it is well-known that human mobility exhibits two different tendencies, i.e., returning to \"old\" places and exploring \"new\" ones [26].","Inspired by this dichotomy, we can define recurring and explorative mobility based on whether a user' next location is seen before, as illustrated in Figure REF .","Human mobility trajectories contain both recurring and explorative tendencies, although different users may have different mixtures of the two.","When data is sparse, the latter may seem more pronounced, exacerbating the challenge of the next location prediction.","Therefore, when evaluating those mobility prediction models, it is necessary to assess their predictive power under both recurring and explorative settings in addition to the main setting.","Figure: Activity distribution statistics in New York and Tokyo at two different time periods.Figure REF demonstrated a strong temporal dependency between activities, and we remove TKY's top-1 activity Train Station for better visualization."],["Reproducibility","In this section, to address the reproducibility of our experiments, we provide the specific information about computing devices and detailed hyperparameter settings used in our experiments.","All models (our Hgarn and other baselines) with learnable parameters are trained on a desktop with Intel Core i9-10990K CPU @3.7GHz $\\times $ 10, 2666MHz $\\times $ 2 $\\times $ 16GB RAM, GeForce RTX 3080 $\\times $ 2, 500GB SSD.","We implement Hgarn based on Pytorch.","Parameters of Hgarn are randomly initialized and optimized by the Adam optimizer with a learning rate of 2e-4, decaying by 0.8 with each epoch and we train Hgarn by 80 epochs.","For hyperparameter settings, we set $\\lambda _{L}$ =$\\lambda _{C}$ =1, and dimensions $d^{g}=50$ , $d=200$ , $d^{u}=20$ , $d^{t}=30$ .","Location and activity encoders have hidden states with dimensions of 600.","The above settings remain the same in all experiment settings.","For the main and recurring settings of all datasets, we employ 2 attention heads and set the dropout to 0.1, while for the explorative setting of all datasets, we let the number of attention heads be 1 and set the dropout to 0.6 to prevent overfitting.","For the NYC main and recurring settings, we set $D^{h}$ to 1, $w^{c}$ to 0.8, and $\\lambda _{r}$ to 0.6, while for the explorative setting, $D^{h}$ , $w^{c}$ , and $\\lambda _{r}$ is set to 0.1, 0.9, and 1, respectively.","for the TKY main and recurring settings, $D^{h}$ , $w^{c}$ , and $\\lambda _{r}$ are set to 0.1, 0.6 and 0.6 for TKY's main and recurring, respectively, and 0.1, 1, 0.5 for the explorative setting.","(All notations mentioned above, their corresponding explanations could be found in the main text)"],["Full Numerical Results","In this section, we show the complete numerical results for the figures presented in the main part.","Table REF contains complete numerical results for $D^{h}$ 's sensitivity experiments.","Table REF contains complete numerical results for $\\lambda _{r}$ 's sensitivity experiments.","Table REF contains complete numerical results for $w^{c}$ 's sensitivity experiments.","Table REF contains complete numerical results for our ablation study.","Detailed analysis of the relevant experimental results can be found in the experimental section of the main part.","Table: Full results of w c w^{c}'s sensitivity experiments.Table: Full results of D h D^{h}'s sensitivity experiments.Table: Full results of λ r \\lambda _{r}'s sensitivity experiments.Table: Full results of the ablation study."]],"string":"[\n [\n \"HGARN: Hierarchical Graph Attention Recurrent Network for Human Mobility\\n Prediction\"\n ],\n [\n \"Abstract Human mobility prediction is a fundamental task essential for various applications, including urban planning, transportation services, and location recommendation.\",\n \"Existing approaches often ignore activity information crucial for reasoning human preferences and routines, or adopt a simplified representation of the dependencies between time, activities and locations.\",\n \"To address these issues, we present Hierarchical Graph Attention Recurrent Network (HGARN) for human mobility prediction.\",\n \"Specifically, we construct a hierarchical graph based on all users' history mobility records and employ a Hierarchical Graph Attention Module to capture complex time-activity-location dependencies.\",\n \"This way, HGARN can learn representations with rich contextual semantics to model user preferences at the global level.\",\n \"We also propose a model-agnostic history-enhanced confidence (MaHec) label to focus our model on each user's individual-level preferences.\",\n \"Finally, we introduce a Recurrent Encoder-Decoder Module, which employs recurrent structures to jointly predict users' next activities (as an auxiliary task) and locations.\",\n \"For model evaluation, we test the performances of our Hgarn against existing SOTAs in recurring and explorative settings.\",\n \"The recurring setting focuses more on assessing models' capabilities to capture users' individual-level preferences.\",\n \"In contrast, the results in the explorative setting tend to reflect the power of different models to learn users' global-level preferences.\",\n \"Overall, our model outperforms other baselines significantly in the main, recurring, and explorative settings based on two real-world human mobility data benchmarks.\",\n \"Source codes of HGARN are available at https://github.com/YihongT/HGARN.\"\n ],\n [\n \"Introduction\",\n \"Human mobility is critical for various downstream applications such as urban planning, transportation services, location recommendation, and epidemic management.\",\n \"The ability to model and accurately predict future human mobility can inform us about important public policy decisions for society's betterment, such as promoting social integration, encouraging innovation, and maximizing productivity [29].\",\n \"Central to human mobility modeling is the problem of next location prediction, i.e., the prediction of where an individual is going next, which has received great attention.\",\n \"On the one hand, the increasing prevalence of mobile devices and the popularity of location-based social networks (LBSNs) provide unprecedented data sources for mining individual-level mobility traces and preferences [38].\",\n \"On the other hand, the advancement of AI and machine learning offers a plethora of analytical tools for modeling human mobility.\",\n \"These innovations have supercharged the development of human mobility models in the past decade, especially for the next location prediction.\",\n \"Figure: An illustration of two human mobility trajectories.\",\n \"Activities are essential in affecting human travel decisions and in modeling multiple dependencies.Traditional approaches studied human mobility over sequential transitions, where several approaches explored using Markov Chains (Mc) [10], [28], [25] to model transition patterns over location sequences.\",\n \"Later, recurrent neural network-based [13] models demonstrated better predictive power over Mc-based methods, where pioneering works employed recurrent structures to model temporal periodicity [9] and spatial regularity [36].\",\n \"Some other Rnn-based methods incorporate spatial and temporal contexts [21], [30] into the Rnn update process to boost models' performances.\",\n \"In addition, due to the great success of the Transformer architecture [31], the attention mechanism has also been adopted in several works [22], [12] to better model sequences and obtain good prediction results.\",\n \"In recent years, graph-based approaches leveraged graph representation learning [37], [3] and graph neural networks (Gnns) [15] to model user preferences [20] and the spatial-temporal relationships [8], [22] between locations, obtaining rich representations [33], [27] to improve the performance of the next location prediction.\",\n \"While these works rely on mining individual location sequences for human mobility prediction, they rarely consider the underlying travel behavior structure and activity information.\",\n \"In the travel behavior literature, an individual's decision to visit a location is usually considered to be derived from their daily activity patterns [2].\",\n \"As a result, important inter-dependencies exist between activity participation and location visitation behaviors, which are often overlooked.\",\n \"Furthermore, the number of possible activity categories is much smaller than the number of locations, making it easier to obtain a robust estimate for the former.\",\n \"With data regarding human activity patterns becoming increasingly accessible, it is both behaviorally meaningful and computationally efficient to incorporate such activity information and its spatial-temporal dependencies in human mobility modeling.\",\n \"Figure REF shows a couple of human mobility trajectories reflecting time-activity-location dependencies.\",\n \"For example, when the time is approaching noon, one user may decide to dine at a nearby restaurant, and another may go to the movie theater for a specific starting time.\",\n \"These examples illustrate that activities are usually scheduled according to the time of day, affecting the choice of which location to visit next, as also evidenced by Figure REF , where people usually go to restaurants at noon and mostly return home or go to the gym at night.\",\n \"However, only a few works have considered adding activity information (e.g., location categories) to improve next location prediction.\",\n \"Yu et al.\",\n \"[39] used activities along with spatial distances to reduce search space (i.e., candidate locations).\",\n \"Huang et al.\",\n \"[14] proposed a Cslsl which comprises an Rnn-based [7] structure, where the time, activity, and location are predicted sequentially to model human travel decision logic.\",\n \"However, the design of Cslsl oversimplifies the time-activity-location dependencies.\",\n \"Given data sparsity and behavioral uncertainties [44], the time prediction tends to be more challenging [43], which can further compromise the prediction of activities and locations.\",\n \"To the best of our knowledge, no efforts have been made to model the sophisticated time-activity-location dependencies and leverage them to improve the next location prediction.\",\n \"In this paper, we present Hierarchical Graph Attention Recurrent Network (Hgarn) for the next location prediction.\",\n \"Specifically, we first construct a hierarchical graph based on time-activity-location history mobility records of all users and employ a Hierarchical Graph Attention Module to capture complex temporal-view activity-activity, affiliated-view activity-location, spatial-view location-location dependencies.\",\n \"In this way, Hgarn can learn representations with rich contextual semantics to model user preferences at the global level.\",\n \"Correspondingly, a model-agnostic history-enhanced confidence (MaHec) label is proposed to guide our model to learn each user’s individual-level preferences.\",\n \"We finally introduce a Recurrent Encoder-Decoder Module.\",\n \"The module takes the sequence of user embeddings combined with the learned hierarchical graph representations as input to jointly predict a user's next activity (as the auxiliary task) and location (as the main task).\",\n \"Specifically, the former can be residually incorporated into the latter.\",\n \"In such a design, our model can leverage the learned time-activity-location dependencies to benefit both global- and individual-level human mobility modeling, and use predicted next activity distribution to facilitate model performance for personalized next location prediction.\",\n \"In summary, our paper makes the following contributions: [leftmargin=*] We propose a Hierarchical Graph to represent the temporal-view activity-activity, affiliated-view activity-location, spatial-view location-location dependencies.\",\n \"To the best of our knowledge, among the few approaches incorporating activity information into the next location prediction, this is the first work to model the dependencies of time, activities and locations by leveraging a Hierarchical Graph.\",\n \"We design a Hierarchical Graph Attention Recurrent Network (i.e., Hgarn), which contains a hierarchical graph attention module to model dependencies between time, activities, and locations to capture users' global-level preferences, and a recurrent encoder-decoder module to incorporate the hierarchical graph representations into sequence modeling and utilize next activity prediction to boost the next location prediction.\",\n \"We introduce a simple yet effective model-agnostic history-enhanced confidence (MaHec) label to guide our model to learning each user's individual-level preferences, which enables the model to focus more on relevant locations in their history trajectories when predicting their next locations.\",\n \"Extensive experiments are conducted using two real-world location-based social networks (LBSN) check-in datasets.\",\n \"Specifically, we evaluate the prediction performance of our Hgarn against existing SOTAs in the main, recurring, and explorative settings.\",\n \"Our work is the first to separately evaluate the next location prediction models' recurring and explorative prediction performances.\",\n \"The results show that the Hgarn can significantly outperform existing methods in all experimental settings.\",\n \"Figure: A workflow of the proposed Hgarn.\"\n ],\n [\n \"Related work\",\n \"The next location prediction is essentially about sequence modeling since successive location visits are usually correlated [5], [40].\",\n \"Traditional Mc-based methods often incorporate other techniques, such as matrix factorization [28] and activity-based modeling [25] to capture mobility patterns and make predictions.\",\n \"However, Mc-based methods are limited in capturing long-term dependencies or predicting exploratvie human mobility.\",\n \"Deep learning-based approaches consider the next location prediction a sequence-to-sequence task and obtain better prediction results than traditional methods.\",\n \"The majority of existing deep models are based on Recurrent neural networks [13], [6] (Rnn).\",\n \"Strnn [21] is a pioneering work that incorporates spatial-temporal features between consecutive human visits into Rnn models to predict human mobility.\",\n \"Stgn [42] adds spatial and temporal gates to Lstm to learn users interests.\",\n \"Flashback [36] leverages spatial and temporal intervals to compute an aggregated past Rnn hidden states for predictions.\",\n \"Lstpm [30] introduces a non-local network and a geo-dilated Lstm to model users’ long- and short-term preferences.\",\n \"Some approaches use the attention mechanism to improve their performances.\",\n \"DeepMove [9] leveraged attention mechanisms combined with an Rnn module to capture users’ long- and short-term preferences.\",\n \"Arnn [12] uses a knowledge graph to find related neighbor locations and model the sequential regularity of check-ins through attentional Rnn.\",\n \"Stan [22] extracts relative spatial-temporal information between consecutive and non-consecutive locations through a spatio-temporal attention network for locations predictions.\",\n \"In addition, some efforts incorporate contextual information [17] such as geographical information [18], dynamic-static [24], text content about locations [4] into sequence modeling.\",\n \"Recently, graph-based models have been designed to enrich contextual features, which can reflect global preferences.\",\n \"LBSN2Vec [37] performs random walks on a hypergraph to learn embeddings for next location and friendship predictions.\",\n \"STP-UDGAT [20] uses Gat to learn location relationships from both local and global views based on constructed spatial, temporal, and preference graphs.\",\n \"HMT-GRN [19] learned several user-region matrices of different granularity levels to alleviate the data sparsity issue and efficiently predict next locations.\",\n \"Gcdan [8] uses dual attention to model the high-order sequential dependencies and graph convolutions to mitigate the data sparsity issue.\",\n \"Graph-Flashback [27] uses knowledge graph embedding to construct transition graph, and applies Gcn to refine graph representations and combines with Flashback to make predictions.\",\n \"Some other works focused on different aspects, activity-aware methods like [1] model the activities with a weighted category hierarchy (WCH), CatDM [39] uses activities along with spatial distances to reduce search space, Cslsl [14] proposed an Rnn-based causal structure to capture human travel decision logic.\",\n \"However, most existing methods ignore activity information and cannot effectively model the time-activity-location dependencies, which are essential for predicting and understanding human mobility.\"\n ],\n [\n \"Preliminaries\",\n \"This section introduces definitions relevant to this study and then formulates the next location prediction problem.\",\n \"We use notations $U=\\\\left\\\\lbrace u^{i} \\\\right\\\\rbrace _{i=1}^{|U|}$ , $L=\\\\lbrace l^{i}\\\\rbrace _{i=1}^{|L|}$ , $C=\\\\lbrace c^{i}\\\\rbrace _{i=1}^{|C|}$ , and $T=\\\\lbrace t^{i}\\\\rbrace _{i=1}^{|T|}$ to denote the sets of users, locations, activities and time series, respectively.\",\n \"In particular, we denote a user $u\\\\in U$ 's sets of locations, activities and time series in a temporal order as $L_{u}=\\\\lbrace l^{i}_{u} \\\\rbrace _{i=1}^{|L_{u}|}$ , $C_{u}=\\\\lbrace c^{i}_{u} \\\\rbrace _{i=1}^{|C_{u}|}$ , and $T_{u}=\\\\lbrace t^{i}_{u} \\\\rbrace _{i=1}^{|T_{u}|}$ , where $\\\\delta _{u}^{i}$ may not equal to $\\\\delta ^{i}$ , $\\\\delta \\\\in \\\\lbrace U, L, C, T\\\\rbrace $ .\",\n \"Definition 1 (Mobility Record) We use the notation $r$ to denote a single human mobility record.\",\n \"Specifically, the $i$ th record of a given user $u \\\\in U$ is represented by a tuple $r_{u}^{i}=(u, c_{u}^{i}, l_{u}^{i}, t_{u}^{i})$ .\",\n \"Each record tuple comprises a user $u \\\\in U$ , an activity $c_{u}^{i} \\\\in C_{u}$ , a location $l_{u}^{i} \\\\in L_{u}$ and the visit time $t_{u}^{i} \\\\in T_{u}$ .\",\n \"Definition 2 (Trajectory) A trajectory is a sequence of mobility records that belongs to a user $u \\\\in U$ , denoted by $R_{u}=\\\\lbrace r_{u}^{i}\\\\rbrace _{i=1}^{|R_{u}|}$ .\",\n \"Each trajectory $R_{u}$ can be divided into a history trajectory $R_{u}^{his}=\\\\lbrace r_{u}^{i}\\\\rbrace _{i=1}^{|R_{u}|-1}$ and the final record $r_{u}^{|R_{u}|}$ .\",\n \"In this way, we can obtain the user $u$ 's activity trajectories $R_{u}^{C}={\\\\lbrace c_{u}^{i}\\\\rbrace }_{i=1}^{|R_{u}|}$ , location trajectories $R_{u}^{L}={\\\\lbrace l_{u}^{i}\\\\rbrace }_{i=1}^{|R^{u}|}$ , and time trajectories $R_{u}^{T}={\\\\lbrace t_{u}^{i}\\\\rbrace }_{i=1}^{|R_{u}|}$ with their history trajectories $R_{u}^{C, his}$ , $R_{u}^{L, his}$ and $R_{u}^{T, his}$ .\",\n \"Given the definitions above, the next location prediction problem can be formulated as follows: Problem 1 (Next Location Prediction) Given a user $u \\\\in U$ 's history trajectory $R_{u}^{his}$ as input, we consider $u$ 's next record as its future state.\",\n \"The human mobility prediction task $\\\\mathcal {T}$ maps $u$ 's history trajectory $R_{u}^{his}$ to $u$ 's next location $l_{u}^{|R_{u}|}$ in the future.\",\n \"The above-described process can be summarized as follows: $R_{u}^{his}\\\\stackrel{\\\\mathcal {T}(\\\\cdot ;\\\\theta )}{\\\\longrightarrow }l_{u}^{|R_{u}|},$ where $\\\\theta $ is the parameters of mapping $\\\\mathcal {T}$ .\"\n ],\n [\n \"Methodology\",\n \"The Hgarn's workflow is demonstrated in Figure REF .\",\n \"The raw data is first encoded in the embedding layer and then input to the hierarchical graph attention module to model multi-dependencies.\",\n \"Finally, the user's personalized embeddings are fused with the learned hierarchical graph representations and input to the recurrent encoder-decoder module to make predictions.\",\n \"In the following sections, we will elaborate on the details of our Hgarn.\"\n ],\n [\n \"Embedding Layer\",\n \"We first introduce our embedding layer, which aims to assign trainable embeddings to each unique mobility record element, i.e., the user, activity, location and time.\",\n \"It is worth noting that all these elements are discrete values except for time, and it is necessary to split the continuous time into discrete intervals to facilitate the encoding of time.\",\n \"In this work we discretize the 24 hours of the day into $|T^{h}|$ time slots $T^{h}=\\\\lbrace h\\\\rbrace _{h=1}^{|T^{h}|}$ and another variable $T^{w}=\\\\lbrace w\\\\rbrace _{w=1}^{|T^{w}|}$ indicating the day of the week.\",\n \"Note that all $t \\\\in T$ can be written in the form of $t=(h, w)$ .\",\n \"We obtain the embeddings of each user, activity, location and time $e_{u} \\\\in \\\\mathbb {R}^{d^u}$ , $e_{c}, e_{l} \\\\in \\\\mathbb {R}^{d}$ , $e_{t} \\\\in \\\\mathbb {R}^{d^t}$ by multiplying their one-hot vectors with the corresponding trainable transformation matrices, where $d^u$ , $d$ and $d^t$ represent embedding dimensions.\",\n \"$e_{U} \\\\in \\\\mathbb {R}^{|U| \\\\times {d^u}}$ , $e_{C} \\\\in \\\\mathbb {R}^{|C| \\\\times {d}}$ , $e_{L} \\\\in \\\\mathbb {R}^{|L| \\\\times {d}}$ and $e_{T} \\\\in \\\\mathbb {R}^{|T| \\\\times {d^t}}$ are used to denote the embedding matrices for the user, activity, location and time, respectively.\",\n \"$e_{r_{u}^{i}}=\\\\left(e_{u}, e_{c_{u}^{i}}, e_{l_{u}^{i}}, e_{t_{u}^{i}}\\\\right)$ is used to denote a user $u$ 's embeddings of $i$ th record.\"\n ],\n [\n \"Hierarchical Graph Attention Module\",\n \"The hierarchical graph attention module consists of two parts: hierarchical graph construction and hierarchical graph attention (HGA) networks on the graph for multi-dependencies modeling.\",\n \"Figure: The design of hierarchical graph attention module.\"\n ],\n [\n \"Hierarchical Graph Construction\",\n \"Given that a spatial network is naturally a graph and recent advances in graph neural networks [23], [32], we choose to model the location-location, location-activity and activity-activity dependencies with a hierarchical graph.\",\n \"The constructed hierarchical graph is shown in Figure REF , which includes three layers: location layer, localized-activity layer, and activity layer, where the localized-activity layer is adopted to suppress noise aggregated from the location layer.\",\n \"We formally describe the hierarchical graph with notation $G=(V,E)$ , where $V$ $=$ $V^{L} \\\\cup V^{C} \\\\cup V^{C^\\\\prime }$ and $E=\\\\lbrace A^{L}$ ,$A^{C}$ ,$A^{LC^\\\\prime }$ ,$A^{CC^\\\\prime }\\\\rbrace $ .\",\n \"Specifically, $V^{L}$ and $V^{C}$ represents the location node-set and the activity node-set.\",\n \"$V^{C^\\\\prime }$ indicates the localized-activity node-set.\",\n \"$E$ comprises four adjacency matrices that denote connectivity between (1) two location nodes, (2) two activity nodes, (3) a location node and a localized-activity node, (4) an activity node and a localized-activity node.\",\n \"For the location adjacency matrix $A^{L}$ , we utilize the geographical distance to determine whether two nodes are linked.\",\n \"We employ the haversine formula to compute distances between locations.\",\n \"Given $l^{i}$ and $l^{j}$ and their GPS data, $A^{L} \\\\in \\\\mathbb {R}^{|L| \\\\times |L|}$ is as: $A^{L}_{l^{i}, l^{j}}={\\\\left\\\\lbrace \\\\begin{array}{ll}1, & Haversine \\\\left(\\\\text{GPS}^{i}, \\\\text{GPS}^{j} \\\\right) < D^{h}\\\\\\\\0, & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"},$ where $D^{h}$ is a hyperparameter to control the distance threshold to influence the connectivity between locations.\",\n \"The construction of $A^{C}$ is based on history trajectories of all users.\",\n \"Intuitively, the dependency between two activities can be measured by the frequency of co-occurrence in the same time interval $h \\\\in T^{h}$ .\",\n \"However, if we directly consider the activity co-occurrence frequency based on all trajectories (regardless of the user), it may lead to unrelated activities being interrelated (e.g., check-in at subway stations and gyms both often occur in the evening), due to the difference in user preferences.\",\n \"Instead, a more reasonable way is to learn the inter-activity dependencies based on activity co-occurrence within individual-level trajectory sets.\",\n \"Therefore, we can traverse the history trajectories of each user $u \\\\in U$ and calculate the co-occurrence matrix $M^{C} \\\\in \\\\mathbb {R}^{|C| \\\\times |C|}$ : $M^{C}_{c_{u}^{i}, c_{u}^{j}}=\\\\sum _{u \\\\in U}\\\\sum _{i=1}^{|R_{u}^{his}|}\\\\sum ^{|R_{u}^{his}|}_{j=i+1} \\\\mathbf {1}_{h}\\\\left(h_{u}^{i}, h_{u}^{j}\\\\right),$ $\\\\mathbf {1}_{h}\\\\left(h_{u}^{i}, h_{u}^{j}\\\\right)={\\\\left\\\\lbrace \\\\begin{array}{ll}1, & \\\\text{if } {h}_{u}^{i} = {h}_{u}^{j} \\\\\\\\0, & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"},$ where $\\\\mathbf {1}_{h}(\\\\cdot )$ is an indicator function.\",\n \"Larger elements in the $M^{C}$ indicate stronger dependencies between the corresponding two activities.\",\n \"For the sake of simplicity and suppression of correlation between some of the less relevant activities, we calculate the adjacency between each activity pair $(c^{i}, c^{j})$ to attain $A^{C} \\\\in \\\\mathbb {R}^{|C| \\\\times |C|}$ : $A^{C}_{c^{i}, c^{j}}={\\\\left\\\\lbrace \\\\begin{array}{ll}1, & \\\\text{if } M^{C}_{c^{i}, c^{j}} > \\\\text{mean}\\\\left(M^{C}\\\\right) \\\\\\\\0, & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"},$ The matrix $A^{LC^{\\\\prime }}$ defines the adjacency between nodes of the location layer and nodes of the localized-activity layer.\",\n \"Each node of $V^{L}$ is linked to only one node of $V^{C^\\\\prime }$ , representing the corresponding activity category at that location.\",\n \"In contrast, each node of $V^{C^\\\\prime }$ may be linked to multiple nodes of $V^{L}$ , as several locations can share the same activity.\",\n \"Formally, we define $A^{LC^{\\\\prime }}_{L} \\\\in \\\\mathbb {R}^{|L| \\\\times |C|}$ based on the affiliations of locations and activities, where each row corresponds to a location and each column an activity.\",\n \"Additionally, we construct the adjacency matrix $A^{LC^{\\\\prime }} \\\\in \\\\mathbb {R}^{(|L|+|C|) \\\\times (|L|+|C|)}$ based on $A^{LC^{\\\\prime }}_{L}$ in the following block matrix form: $A^{LC^{\\\\prime }}=\\\\begin{bmatrix}O_{L} & A^{LC^{\\\\prime }}_{L} \\\\\\\\{(A^{LC^{\\\\prime }}_{L})}^{\\\\top } & O_{C}\\\\end{bmatrix},$ where ${(A^{LC^{\\\\prime }})}^{\\\\top }$ is the transpose of $A^{LC^{\\\\prime }}$ , $O_{L} \\\\in \\\\mathbb {R}^{|L| \\\\times |L|}$ and $O_{C} \\\\in \\\\mathbb {R}^{|C| \\\\times |C|}$ are two zero matrices.\",\n \"The function of the localized-activity layer is to suppress noise from the location layer aggregated to the activity layer, so we simply define that each node in the localized-activity layer is connected to the node in the activity layer representing the same activity.\",\n \"Mathematically, similar to Equation REF , we have $A^{CC^\\\\prime } \\\\in \\\\mathbb {R}^{2|C| \\\\times 2|C|}$ : $A^{CC^\\\\prime }=\\\\begin{bmatrix}O_{C} & I_{C} \\\\\\\\I_{C} & O_{C}\\\\end{bmatrix},$ where $I_{C} \\\\in \\\\mathbb {R}^{|C| \\\\times |C|}$ is an identity matrix.\"\n ],\n [\n \"Hierarchical Graph Attention Networks\",\n \"Inspired by recent work conducted deep learning on hierarchical graphs [11], [34], [41], we design a hierarchical graph attention network (HGA) to model the hierarchical graph we just defined.\",\n \"Since the importance of locations within a certain distance is different for a specific location, we employ graph attention mechanisms to model spatial-view location-location dependencies: $H^{L}=\\\\textsc {\\\\textsc {Gat}}_{L}\\\\left(e_{L}, A^{L}\\\\right),$ where $\\\\textsc {\\\\textsc {Gat}}(\\\\cdot )$ is a standard implementation of [32], $H^{L} \\\\in \\\\mathbb {R}^{|L| \\\\times d^{g}}$ is the learned representations as the output of the $\\\\textsc {Gat}_{L}$ .\",\n \"To integrate location information into representation learning of activities and suppress the noise aggregated to the nodes of the activity layer, we introduce the localized-activity layer to pre-aggregate location embeddings.\",\n \"We first concatenate $e_L$ and $e_C$ to obtain the fused embedding matrix $e_{LC} \\\\in \\\\mathbb {R}^{(|L|+|C|) \\\\times d} = {\\\\begin{bmatrix}e_L & e_C\\\\end{bmatrix}}^{\\\\top }$ .\",\n \"Then the localized-activity process is implemented as: $H^{LC^{\\\\prime }}=\\\\textsc {Gat}_{LC}\\\\left(e_{LC}, A^{LC^{\\\\prime }}\\\\right),$ where $H^{LC^{\\\\prime }} \\\\in \\\\mathbb {R}^{(|L|+|C|) \\\\times d^{g}}$ is the output of $\\\\textsc {Gat}_{LC}$ , we further slice out the first $|L|$ rows of matrix $H^{LC^{\\\\prime }}$ to obtain the pre-aggregated representation matrix ${H^{C^{\\\\prime }}} \\\\in \\\\mathbb {R}^{|C| \\\\times d^{g}}$ .\",\n \"The learned representation $H^{C^{\\\\prime }}$ is again concatenated with activity embeddings $e_C$ as the $\\\\textsc {Gat}_C$ 's input $e_{CC}={\\\\begin{bmatrix}e_C & H^{C^{\\\\prime }}\\\\end{bmatrix}}^{\\\\top }$ .\",\n \"It is worth noting that for all nodes in the activity layer, we could simultaneously aggregate information from neighbors in the localized-activity layer and neighbors in the same layer by simply modifying the matrix $A^{CC^\\\\prime }$ to: $A^{CC^\\\\prime }_{new}=\\\\begin{bmatrix}A^{C} & I_{C} \\\\\\\\I_{C} & O_{C}\\\\end{bmatrix}.$ We still employ a similar strategy to update the representation of the nodes in the activity layer: ${H^{C}}^\\\\prime =\\\\textsc {Gat}_{C}\\\\left(e_{CC}, A^{CC^\\\\prime }_{new}\\\\right),$ where ${H^{C}}^\\\\prime \\\\in \\\\mathbb {R}^{2|C| \\\\times d^{g}}$ is the learned representation from $\\\\textsc {Gat}_{C}$ , we slice out the last $|C|$ rows of matrix ${H^{C}}^\\\\prime $ to obtain the updated activity node representation $H^{C} \\\\in \\\\mathbb {R}^{|C| \\\\times d^{g}}$ .\"\n ],\n [\n \"Recurrent Encoder-Decoder Module\",\n \"The recurrent encoder-decoder module 's structure is shown in Figure REF .\",\n \"Given a user $u \\\\in U$ 's history trajectory, the input of the activity and the location encoder at $i$ th iteration are implemented as: $X^{C,i}_{u}= {e_{u}} \\\\Vert e_{t_{u}^{i}} \\\\Vert e_{c_{u}^{i}} \\\\Vert H^{C}_{c_{u}^{i}},$ $X^{L,i}_{u}= {e_{u}} \\\\Vert e_{t_{u}^{i}} \\\\Vert e_{l_{u}^{i}} \\\\Vert H^{C}_{c_{u}^{i}} \\\\Vert H^{L}_{l_{u}^{i}},$ where $\\\\Vert $ is the concatenation operation, $H^{C}_{c_{u}^{i}}$ and $H^{L}_{l_{u}^{i}}$ are the learned hierarchical graph representation of the activity node $c_{u}^{i}$ and the location node $l_{u}^{i}$ , the concatenated $X^{C,i}_{u}$ and $X^{L,i}_{u}$ are the inputs of the two encoders at the $i$ th iteration.\",\n \"Figure: Recurrent encoder-decoder module's architecture.Both activity and location encoders are implemented with Lstm, where hidden states updating process between $h^{\\\\cdot ,i}_{u}$ and $h^{\\\\cdot ,i-1}_{u}$ can be illustrated as: $\\\\begin{aligned}p_{u}^{i}, f_{u}^{i}, g_{u}^{i}, o_{u}^{i} =\\\\text{split} ( ( & W_{x}X^{\\\\cdot ,i}_{u}+b_{x} ) + ( W_{h}h^{\\\\cdot ,i-1}_{u}+b_{h} ) ),\\\\end{aligned}$ $c^{\\\\cdot , i}_{u} = \\\\sigma \\\\left(f_{u}^{i}\\\\right) \\\\odot c^{\\\\cdot , i-1}_{u} + \\\\sigma \\\\left(p_{u}^{i}\\\\right) \\\\odot \\\\tanh \\\\left(g_{u}^{i}\\\\right),$ $h^{\\\\cdot , i}_{u} = \\\\sigma \\\\left(o_{u}^{i}\\\\right) \\\\odot \\\\tanh \\\\left(c^{\\\\cdot , i}_{u}\\\\right),$ where “$\\\\cdot $ ” can be either $C$ or $L$ .\",\n \"$p_{u}^{i}$ , $f_{u}^{i}$ , $g_{u}^{i}$ , $o_{u}^{i}$ are the input, forget, cell and output gates.\",\n \"$c^{\\\\cdot , i}_{u}$ is the cell state at iteration $i$ .\",\n \"$\\\\sigma (\\\\cdot )$ and $\\\\tanh (\\\\cdot )$ are the sigmoid and tanh activation functions.\",\n \"$W_{x}$ , $W_{h}$ , $b_{x}$ and $b_{h}$ are the trainable weights shared by inputs and $\\\\odot $ is the Hadamard product.\",\n \"Initial hidden states for $h^{\\\\cdot , 1}_{u}$ and $c^{\\\\cdot , 1}_{u}$ are set to zeros.\",\n \"After obtaining final hidden states of activity and location encoder as $h^{C}_{u}$ and $h^{L}_{u}$ , we implement our activity decoder as a multi-layer perceptron (Mlp) to get the next activity logits $\\\\widetilde{h^{C}_{u}} \\\\in \\\\mathbb {R}^{|C|}$ : $\\\\widetilde{h^{C}_{u}}=\\\\textsc {Mlp}_{C}\\\\left(h^{C}_{u}\\\\right).$ We finally residually combine the obtained activity logits with the encoded $h^{L}_{u}$ to learn the location logits $\\\\widetilde{h^{L}_{u}} \\\\in \\\\mathbb {R}^{|L|}$ : $\\\\begin{aligned}\\\\widetilde{h^{L}_{u}}=& \\\\lambda _{r}\\\\cdot \\\\textsc {Mlp}_{L}^{r}\\\\left(h^{L}_{u}\\\\right)+ (1-\\\\lambda _{r})\\\\cdot \\\\textsc {Mlp}_{L}\\\\left(\\\\textsc {Mlp}_{L}^{h}\\\\left(h^{L}_{u} \\\\Vert h^{C}_{u}\\\\right) \\\\Vert \\\\widetilde{h^{C}_{u}}\\\\right),\\\\end{aligned}$ where $\\\\lambda _{r}$ is a factor that controls the residual connections to trades off different features.\"\n ],\n [\n \"In this section we introduce the model-agnostic history-enhanced confidence label and explain the optimization process of Hgarn.\"\n ],\n [\n \"Model-Agnostic History-Enhanced Confidence Label Construction\",\n \"As human mobility trajectories exhibit repetitive nature (i.e., recurring mobility), existing models [9], [8] often try to capture periodicity by applying attention mechanisms to all user trajectories, have limited performances and interpretability.\",\n \"To overcome this, we modify the original label to get the model-agnostic history-enhanced confidence label (MaHec label, in short), which can guide our model to focus on relevant user trajectories.\",\n \"Specifically, for each location $l^{i}\\\\in L$ , we differentiate its confidence for a user $u$ 's next location in two types: ${\\\\textsc {MaHec}}_{l^{i}}^{u}={\\\\left\\\\lbrace \\\\begin{array}{ll}w^{c}, & \\\\text{if } l^{i}=l_{u}^{|R_{u}|} \\\\\\\\\\\\left(1-w^{c}\\\\right) \\\\cdot f_{l^{i}}^{u}, & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"},$ $f_{l^{i}}^{u}=\\\\frac{1}{|R_{u}^{his}|}\\\\sum _{{l^{i}}\\\\in L} \\\\mathbf {1}_{R^{L,his}_{u}}\\\\left(l^{i}\\\\right),\\\\text{ }\\\\text{ }\\\\mathbf {1}_{R^{L,his}_{u}}\\\\left(l^{i}\\\\right)={\\\\left\\\\lbrace \\\\begin{array}{ll}1, & \\\\text{if } l^{i} \\\\in R^{L,his}_{u} \\\\\\\\0, & \\\\text{if } l^{i} \\\\notin R^{L,his}_{u}\\\\end{array}\\\\right.\",\n \"},$ where $\\\\mathbf {1}_{R^{L,his}_{u}}(\\\\cdot )$ is an indicator function and $w^{c}\\\\in [0,1]$ is a hyperparameter that indicates the confidence of the $u$ 's ground truth label and $f_{l^{i}}^{u}$ denotes the user $u$ 's history visit frequency to $l^{i}$ .\",\n \"Then the MaHec label for the user $u$ 's next location is as: ${\\\\textsc {MaHec}}_{L}^{u}=\\\\left({\\\\textsc {MaHec}}_{l^{i}}^{u},\\\\right)_{i=1}^{|L|}\\\\in \\\\mathbb {R}^{|L|},$ where each element in ${\\\\textsc {MaHec}}_{L}^{u}$ represents the confidence that the user $u$ decides to pick as its next location.\",\n \"Similarly, we conduct the same operations for user activity trajectories to obtain ${\\\\textsc {MaHec}}_{C}^{u}$ .\"\n ],\n [\n \"Model Optimization\",\n \"Since next location prediction is a classification problem, we transform $\\\\widetilde{h^{L}_{u}}$ to the probability distribution $\\\\widehat{h^{L}_{u}} \\\\in \\\\mathbb {R}^{|L|}$ of all locations by $\\\\widehat{h^{L}_{u}}=\\\\text{Softmax}\\\\left(\\\\widetilde{h^{L}_{u}}\\\\right)$ .\",\n \"Given ${\\\\textsc {MaHec}}_{L}^{u}$ and $\\\\widehat{h^{L}_{u}}$ , we compute $\\\\mathcal {L}_L$ based on cross-entropy loss: $\\\\mathcal {L}_L=- \\\\frac{1}{|U|}\\\\sum _{u \\\\in U} \\\\sum _{i=1}^{|L|}{\\\\textsc {MaHec}}^{u}_{l^{i}}\\\\cdot \\\\log \\\\left(\\\\widehat{h^{L, i}_{u}}\\\\right),$ where $\\\\widehat{h^{L, i}_{u}}$ is the $i$ th element of $\\\\widehat{h^{L}_{u}}$ and we compute the next activity loss $\\\\mathcal {L}_{C}$ based on same above described operations.\",\n \"Finally, we could train our Hgarn end-to-end with a total loss function: $\\\\mathcal {L}=\\\\lambda _{L}\\\\cdot \\\\mathcal {L}_{L}+\\\\lambda _{C}\\\\cdot \\\\mathcal {L}_{C}$ $\\\\lambda _{L}$ and $\\\\lambda _{C}$ are hyperparameters that trade off different loss terms.\"\n ],\n [\n \"Experiments\",\n \"In this section, we compare our Hgarn with existing SOTAs on two real-world LBSN check-in datasets.\"\n ],\n [\n \"Datasets\",\n \"We adopt two LBSN datasets [38] containing Foursquare check-in records in New York City (NYC) and Tokyo (TKY) from April 12, 2012 to February 16, 2013, including 227,428 check-ins for NYC and 573,703 check-ins for TKY.\",\n \"The location distributions of NYC and TKY datasets are shown in Figure REF in the appendix.\",\n \"Users and locations with less than 10 records are removed following previous work, and after cleaning, NYC and TKY have 308 and 286 activities, respectively.\",\n \"We divide the data into a training and testing sets in a ratio of 8:2, following the setting of [42].\",\n \"Key data summary statistics are listed in Table REF .\",\n \"Table: Main performance comparisons of different methods in two real-world datasets.\"\n ],\n [\n \"Baselines & Experimental Details\",\n \"The following baseline methods are considered in experiments: [leftmargin=*] Mc [28]: Mc is a widely used sequential prediction approach which models transition patterns based on visited locations.\",\n \"Strnn [21]: Strnn is an Rnn-based model that incorporates the spatial-temporal contexts by leveraging transition matrices.\",\n \"DeepMove [9]: DeepMove uses attention mechanisms and an Rnn module to capture users' long- and shot-term preferences.\",\n \"Lstpm [30]: Lstpm introduces a non-local network and a geo-dilated Lstm to model users' long- and short-term preferences.\",\n \"Flashback [36]: Flashback is an Rnn-based model that leverages spatial and temporal intervals to compute an aggregated hidden state from past hidden states for prediction.\",\n \"Plspl [35]: Plspl incorporates activity information to learn user preferences and utilizes two Lstms to capture long-term and short-term preferences.\",\n \"PG2Net [16]: PG2Net learns users' group and personalized preferences with spatial-temporal dependencies and attention-based Bi-Lstm.\",\n \"Gcdan [8]: Gcdan leverages graph convolution to learn spatial-temporal representations and use dual-attention to model the sequential dependencies.\",\n \"Cslsl [14]: Cslsl employs multi-task learning to model decision logic and two Lstms to capture long- and short-term preferences.\",\n \"Graph-Flashback [27]: Graph-Flashback adds Gcn to Flashback to enrich learned transition graph representations constructed based on defined similarity functions over embeddings from the existing Knowledge Graph Embedding method.\",\n \"For all baselines and our method, we adopt two commonly-employed metrics in prior works: Rec@K (Recall) and NDCG@K (Normalized Discounted Cumulative Gain).\",\n \"Specific formulas of these two metrics are defined as: $Recall@K=\\\\frac{1}{|U|}\\\\sum _{u \\\\in U}\\\\frac{|l_{u}^{|R_{u}|} \\\\cap \\\\widehat{l_{u, K}^{|R_{u}|}}|}{|l_{u}^{|R_{u}|}|}$ and $NDCG@K=\\\\frac{1}{|U|}\\\\sum _{u \\\\in U}\\\\sum _{i=1}^{K}\\\\frac{|l_{u}^{|R_{u}|} \\\\cap \\\\widehat{l_{u, i}^{|R_{u}|}}|}{\\\\log (i+1)}$ , where $\\\\widehat{l^{\\\\cdot }_{u, k}}$ indicates the top $k$ predicted locations.\",\n \"We port all the baselines to our run time environment for fair comparisons based on their open-source codes.\",\n \"We carefully tuned their hyperparameters to get the best results.\",\n \"Additionally, unlike previous works that only evaluate overall model performances (under the main setting), we also conduct experiments under the recurring and explorative settings for more comprehensive performance evaluation against the existing SOTAs.\",\n \"For the main and recurring settings, we choose $K=\\\\lbrace 1, 5, 10\\\\rbrace $ for evaluation.\",\n \"As the performance is generally poorer under the explorative setting, we set $K=\\\\lbrace 10, 20\\\\rbrace $ .\",\n \"For the choice of hyperparameters, we set both $\\\\lambda _{L}$ and $\\\\lambda _{C}$ to 1, $\\\\lambda _{r}$ to 0.6 for both datasets.For embedding dimensions, we set $d=200$ , $d^{u}=10$ , $d^{t}=30$ , $d^{g}=50$ and the dimension of encoders' hidden states are set to 600.\",\n \"Detailed reproducibility information can be found in supplementary materials.\"\n ],\n [\n \"Main Results\",\n \"Table REF shows the main performance comparisons of different methods for the next location prediction.\",\n \"Our Hgarn achieves state-of-the-art performances on all metrics for both datasets.\",\n \"Specifically, Hgarn outperforms the best baseline approach by 12-19% on Recall@$K$ and NDCG@$K$ for NYC, and 11-20% for TKY.\",\n \"Its advantages become more significant as $K$ increases, validating the effectiveness of the hierarchical graph modeling and MaHec label for the next location prediction task.\",\n \"Table: Results under the recurring setting.In addition, we also evaluate different models separately in the recurring and explorative settings.\",\n \"In the recurring setting, based on results in Table REF , our model shows an improvement of (10.8%, 23.4%, 31.6%, 10.8%, 19.1%, 20.1%) and (9.4%, 19.2%, 19.7%, 9.4%, 16.5%, 16.9%) on all metrics (R@1, R@5, R@10, N@1, N@5, N@10) compared to the best baseline on the two datasets.\",\n \"Hgarn outperforms all baselines significantly in the recurring setting.\",\n \"Table: Results under the explorative setting.As shown in Table REFThe Mc's results are all zeros and thus deleted from the table., the overall results in the explorative setting are much lower than those in the main and recurring settings, which is intuitive because of the inherent difficulty of predicting unseen locations.\",\n \"A possible approach to improving the prediction in the explorative setting is to model the dependencies between locations.\",\n \"In addition, due to the larger number of locations in the TKY dataset, the hierarchical graph modeling may introduce noise, making our model less effective in ranking the predicted locations.\",\n \"The above hypotheses may also be why our model performs better in Recall but are not consistently better in NDCG.\"\n ],\n [\n \"Ablation Study\",\n \"To study each component of Hgarn, we conduct an ablation study considering the following four variants of Hgarn: (1) Hgarn w/o HGat contains only the recurrent encoder-decoder module for the next location prediction.\",\n \"(2) Hgarn w/o AGat: This variant's hierarchical graph attention module contains only the location layer and corresponding graph attention networks.\",\n \"(3) Hgarn w/o Res is the variant that removes the residual connection of the recurrent encoder-decoder module.\",\n \"(4) Hgarn w/o MaHec is the variant that leverages original labels to optimize our model.\",\n \"The ablation study results are shown in Figure REF .\",\n \"It is found that all metrics get improved as more components are included.\",\n \"The gradual increase in prediction results from Hgarn w/o HGat to Hgarn w/o AGat to Hgarn is a fine-grained demonstration of the effectiveness of Gat in each layer of the hierarchical graph.\",\n \"In addition, the model's performance improvements by adding the MaHec label is significant when $K$ are large, verifying its effectiveness under simplicity.\"\n ],\n [\n \"Hyperparameter Sensitivity Analysis\",\n \"We further study the sensitivity of a few key parameters by varying each parameter while keeping others constant.\",\n \"(1) $D^{h}$ affects the location dependencies, too high or too low will lead to decreased performances.\",\n \"The best results were obtained at $D^{h}=1$ km on NYC and optimal at $D^{h}=0.1$ km on TKY, this is probably because the geographic space and the distance between locations are larger in NYC than in TKY, as in Figure REF .\",\n \"(2) $w^{c}$ A affects the model's attention to history locations.\",\n \"The results show an upward and then downward trend as $w^{c}$ rises on both datasets, indicating that both too large and too small dependence on locations in the history trajectory is detrimental to the model's performance.\",\n \"(3) $\\\\lambda _{r}$ affects how much activity information is fused in predicting the next location.\",\n \"Intuitively, a $\\\\lambda _{r}$ too high would introduce noise, and too low may result in ineffective utilization of activity information.\",\n \"The results align with our conjecture that the model achieves optimal performance with $\\\\lambda _{r}$ at 0.6 for both NYC and TKY datasets.\"\n ],\n [\n \"How does \",\n \"To understand the mechanism of MaHec label, we randomly selected two human trajectories from two datasets and visualized the predicted probability difference of “Hgarn” and “Hgarn w/o MaHec”.\",\n \"Figure REF shows the probability change, where the purple line represents the locations in the current user's history trajectory.\",\n \"With MaHec labels, the probability distribution of the next location predicted by the model increases in most of the visited locations, demonstrating that our MaHec labels can effectively guide the model to pay attention to the user's history trajectory when predicting the next location.\",\n \"The mechanism of MaHec can also interpret, to some extent, why the prediction performances of our Hgarn can far exceed that of the baseline methods, especially under the recurring setting.\"\n ],\n [\n \"What the Hierarchical Graph learned?\",\n \"Unlike other methods that have difficulty interpreting learned higher-order spatial-temporal dependencies, our Hgarn can somewhat understand the dependencies between activities through the learned Hierarchical Graph.\",\n \"We visualize one attention head of $\\\\textsc {Gat}_{C}$ 's sliced attention matrix to analyze the learned activity-activity dependencies.\",\n \"In Figure REF , We select four activity pairs to show the related activities and their corresponding attention scores.\",\n \"These activity pairs are intuitive and consistent with common sense human travel, such as the high dependencies between Gyms and Stadiums, Bus Stops and Travel Lounges.\",\n \"These results have important implications for understanding human activity patterns and predicting the next location.\"\n ],\n [\n \"Conclusion\",\n \"Both travel behavior theories and empirical evidence suggest that human mobility patterns largely depend on the need to participate in activities at different times of the day.\",\n \"Therefore, it is crucial to consider the latter when modeling the former.\",\n \"In this paper, we propose a Hierarchical Graph Attention Recurrent Network (Hgarn) for activity-aware human mobility prediction.\",\n \"Specifically, Hgarn introduces hierarchical graph attention mechanisms on a constructed hierarchical graph to model time-activity-location dependencies, and employ next activity prediction as an auxiliary task to further improve the main task of the next location prediction.\",\n \"In addition, we propose a simple yet powerful MaHec label that can guide our model to flexibly tradeoff the importance of history locations when predicting future locations.\",\n \"Finally, we perform comprehensive experiments to demonstrate the superiority of Hgarn, this is the first work to evaluate existing models and our Hgarn under recurring and explorative settings to our best knowledge.\",\n \"We find that introducing activity information can effectively improve the model's prediction accuracy.\",\n \"In addition, our results show that the existing models have unsatisfactory prediction performances under the explorative setting.\",\n \"We hope our work could spark the attention of future work on explorative mobility prediction.\",\n \"For future work, we consider adding a ranking-based component to organize high-probability candidate locations and incorporate human travel decision logic modeling to make more interpretable predictions.\"\n ],\n [\n \"Appendix\",\n \"In this section, we provide more details, visualizations and experimental results.\"\n ],\n [\n \"Experimental Supplementary\",\n \"Figure REF demonstrated the location distribution of NYC and TKY datasets.\",\n \"Figure: Location distributions of NYC and TKY.Figure REF displayed two more probability distribution change cases with the effect of MaHec labels.\",\n \"Figure: More visualizations of predicted locations probaility distributions on two datasets.\"\n ],\n [\n \"Introduction's supplementary\",\n \"Given the recent emphasis on modeling contextual information (e.g., spatio-temporal) in mobility prediction approaches, we find little work analyzing changes in model prediction performances other than adding or removing relevant components (i.e., ablation studies) and parameter sensitivities.\",\n \"However, it is well-known that human mobility exhibits two different tendencies, i.e., returning to \\\"old\\\" places and exploring \\\"new\\\" ones [26].\",\n \"Inspired by this dichotomy, we can define recurring and explorative mobility based on whether a user' next location is seen before, as illustrated in Figure REF .\",\n \"Human mobility trajectories contain both recurring and explorative tendencies, although different users may have different mixtures of the two.\",\n \"When data is sparse, the latter may seem more pronounced, exacerbating the challenge of the next location prediction.\",\n \"Therefore, when evaluating those mobility prediction models, it is necessary to assess their predictive power under both recurring and explorative settings in addition to the main setting.\",\n \"Figure: Activity distribution statistics in New York and Tokyo at two different time periods.Figure REF demonstrated a strong temporal dependency between activities, and we remove TKY's top-1 activity Train Station for better visualization.\"\n ],\n [\n \"Reproducibility\",\n \"In this section, to address the reproducibility of our experiments, we provide the specific information about computing devices and detailed hyperparameter settings used in our experiments.\",\n \"All models (our Hgarn and other baselines) with learnable parameters are trained on a desktop with Intel Core i9-10990K CPU @3.7GHz $\\\\times $ 10, 2666MHz $\\\\times $ 2 $\\\\times $ 16GB RAM, GeForce RTX 3080 $\\\\times $ 2, 500GB SSD.\",\n \"We implement Hgarn based on Pytorch.\",\n \"Parameters of Hgarn are randomly initialized and optimized by the Adam optimizer with a learning rate of 2e-4, decaying by 0.8 with each epoch and we train Hgarn by 80 epochs.\",\n \"For hyperparameter settings, we set $\\\\lambda _{L}$ =$\\\\lambda _{C}$ =1, and dimensions $d^{g}=50$ , $d=200$ , $d^{u}=20$ , $d^{t}=30$ .\",\n \"Location and activity encoders have hidden states with dimensions of 600.\",\n \"The above settings remain the same in all experiment settings.\",\n \"For the main and recurring settings of all datasets, we employ 2 attention heads and set the dropout to 0.1, while for the explorative setting of all datasets, we let the number of attention heads be 1 and set the dropout to 0.6 to prevent overfitting.\",\n \"For the NYC main and recurring settings, we set $D^{h}$ to 1, $w^{c}$ to 0.8, and $\\\\lambda _{r}$ to 0.6, while for the explorative setting, $D^{h}$ , $w^{c}$ , and $\\\\lambda _{r}$ is set to 0.1, 0.9, and 1, respectively.\",\n \"for the TKY main and recurring settings, $D^{h}$ , $w^{c}$ , and $\\\\lambda _{r}$ are set to 0.1, 0.6 and 0.6 for TKY's main and recurring, respectively, and 0.1, 1, 0.5 for the explorative setting.\",\n \"(All notations mentioned above, their corresponding explanations could be found in the main text)\"\n ],\n [\n \"Full Numerical Results\",\n \"In this section, we show the complete numerical results for the figures presented in the main part.\",\n \"Table REF contains complete numerical results for $D^{h}$ 's sensitivity experiments.\",\n \"Table REF contains complete numerical results for $\\\\lambda _{r}$ 's sensitivity experiments.\",\n \"Table REF contains complete numerical results for $w^{c}$ 's sensitivity experiments.\",\n \"Table REF contains complete numerical results for our ablation study.\",\n \"Detailed analysis of the relevant experimental results can be found in the experimental section of the main part.\",\n \"Table: Full results of w c w^{c}'s sensitivity experiments.Table: Full results of D h D^{h}'s sensitivity experiments.Table: Full results of λ r \\\\lambda _{r}'s sensitivity experiments.Table: Full results of the ablation study.\"\n ]\n]"},"id":{"kind":"string","value":"2210.07765"}}},{"rowIdx":2994,"cells":{"text":{"kind":"list like","value":[["The Triangle Condition for the Marked Random Connection Model"],["Abstract We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $\\mathbb{R}^d$.","Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks.","Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely.","Our focus is on the transition between the low- and high-density phase, where the system is critical.","We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid.","This implies the triangle condition, thus indicating mean-field behaviour.","We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates."],["Phase transition in percolation","For percolation, each edge of the hypercubic lattice is retained independently with fixed probability $p\\in [0,1]$ .","It is well-known that the model undergoes a phase transition: in any dimension $d\\ge 2$ there is a $p_c\\in (0,1)$ such that for $pp_c$ there is a unique infinite cluster.","Behaviour of the model at (and near) the critical threshold $p_c$ is a major theme in probability theory.","Indeed, we have good control on the critical behaviour in two regimes: if $d=2$ , then planarity and the link with complex analysis allows a refined description of the critical behaviour (to some extent only on the triangular lattice), see [43], [44], [5] and references therein; if $d$ is large (at least $d>6$ ), then Hara and Slade [25] established the famous triangle condition, which implies that various critical exponents exist and take on their mean-field values [1], [3], [37], [30], [31], [20], [28], see also [23] for a corresponding result for site percolation.","While it is generally believed that $d>6$ is sufficient, current proof techniques only allow the proof for $d\\ge 11$ [11].","Already [25] show it for $d>6$ under an additional spread-out assumption.","The focus of the present paper is on the second case, which we coin the “mean-field regime”.","The triangle condition, and hence the mean-field value of various critical exponents, has also been established for long-range percolation in lower dimension provided that the edge occupation probability is decaying slowly enough [22], [6], [7].","The triangle condition for the random connection model, a continuum percolation model where vertices are given as a Poisson point process, was proven recently [21].","Similarly, it has been established for non-amenable graphs under varying assumptions [39], [40], [27].","All of the above papers crucially rely on the independence of edge occupation.","In the present work, we establish the triangle condition for dependent percolation models.","Mean-field bounds on some critical exponents were proven recently in the context of Boolean percolation [9], but a proof of the triangle condition in such situations is new."],["Complex networks","Random geometric graphs are versatile stochastic models for real networks.","As many real-world networks exhibit certain stylised facts such as scale-free degrees, short distances and geometric clustering, it is clear that mathematical models of complex networks must share such features.","While numerous different mathematical models for complex networks exist, it has been pointed out by [13] that the weight-dependent random connection model provides a unified approach that generalises many models for (infinite) geometric random graphs.","This is a continuum random graph model whose vertices are given as a Poisson point process on $\\mathbb {R}^d\\times (0,1)$ .","If $x,y\\in \\mathbb {R}^d$ and $s,t\\in (0,1)$ , where edges $\\big \\lbrace (x,s),(y,t)\\rbrace $ are inserted with probability given by a connection function $\\varphi ((x,s),(y,t))$ .","A standard assumption is that $\\varphi $ is decreasing in the spatial distance $|x-y|$ as well as in both marks $s$ and $t$ .","Relevance of this model comes from the fact that different choices for the connection function $\\varphi $ generalise various well-known random graph models such as Boolean percolation, scale-free Gilbert graph, scale-free percolation, soft random geometric graphs, long-range percolation, and spatial preferential attachment graphs.","Results proven in the generalised setup are thus valid in great generality.","For some choices of $\\varphi $ , these models are known to be robust, i.e., infinite components exist for all edge densities; see [15], [16].","Our interest is in regimes where these graphs are non-robust and thus a proper percolation phase transition exist: if $\\varphi $ is small (scaled by a certain parameter) then there is no infinite cluster, but if $\\varphi $ is large, then an infinite cluster does appear.","Investigation of such transitions is classical in percolation theory, and of great importance in modern probability.","The present paper is the first one to prove an infrared bound for an inhomogeneous spatial percolation model.","While such results are well-known for the homogeneous case (i.e., without marks), there are hardly any results for the percolation phase transition for inhomogeneous models.","A notable exception is Boolean percolation, where sharp phase transition is known [8] and the mean-field bounds on three critical exponents is identified [9].","The key challenge is therefore, to overcome the dependencies imposed by the vertex weights.","We prove the infrared bound under specific conditions, and verify these conditions for various concrete examples."],["The Marked Random Connection Model","In this paper we consider a generalisation of the weight-dependent random connection model, which we first describe informally and refer to Section REF for a formal construction.","It is for our purposes more convenient to generalise the setup by considering a general measurable mark space $\\mathcal {E}$ and thus define the model on the space $\\mathbb {X} = \\mathbb {R}^d\\times \\mathcal {E}$ .","Let $\\nu = {\\rm Leb}\\times \\mathcal {P}$ be a $\\sigma $ -finite measure on $\\mathbb {X}$ where ${\\rm Leb}$ denotes the Lebesgue measure on $\\mathbb {R}^d$ and $\\mathcal {P}$ is a probability measure on $\\mathcal {E}$ .","Let $\\lambda >0$ , and then $\\lambda \\nu $ is an intensity measure on $\\mathbb {X}$ .","The connection function is given by the measurable and symmetric $\\varphi \\colon \\mathbb {X}^2\\rightarrow \\left[0,1\\right]$ .","We furthermore require that $\\varphi $ is translation and reflection invariant in the position components.","That is, if we denote $x=\\left(\\overline{x},a\\right)$ and $y=\\left(\\overline{y},b\\right)$ for $\\overline{x},\\overline{y}\\in \\mathbb {R}^d$ and $a,b\\in \\mathcal {E}$ , we require that $\\varphi \\left(x,y\\right) = \\varphi (\\overline{x}-\\overline{y};a,b) = \\varphi (-\\overline{x}+\\overline{y};a,b)$ .","We denote by $\\eta $ the Poisson point process with intensity measure $\\lambda \\nu $ (which represents the vertex set of the random graph) and by $\\xi $ the full random graph including additional randomness for the formation of edges.","Once we have the concept of two vertices being adjacent, we can consider the concept of two vertices being connected.","We say $x,y\\in \\mathbb {X}$ are connected, and write $x\\longleftrightarrow y$ , if they are equal or there exists a path in $\\xi $ connecting the two vertices.","By this we mean that there exists a sequence of distinct vertices $x = u_0, u_1, \\ldots , u_k, u_{k+1} = y \\in \\eta $ (with $k \\in \\mathbb {N}_0$ ) such that $u_i \\sim u_{i+1}$ for all $0 \\le i \\le k$ .","This definition can also be extended to connectedness on the RCM augmented by the vertex $x\\in \\mathbb {X}$ (resp.","$x,y\\in \\mathbb {X}$ ), where $\\left(\\eta ,\\xi \\right)$ is replaced by $\\left(\\eta ^x,\\xi ^x\\right)$ (resp.","$\\left(\\eta ^{x,y},\\xi ^{x,y}\\right)$ ).","Therefore given a pair of points $x,y\\in \\mathbb {X}$ we can consider the event $\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ , and in turn the probability of this event.","We define the pair-connectedness (or two-point) function $\\tau _\\lambda \\colon \\mathbb {X}^2 \\rightarrow \\left[0,1\\right]$ to be $\\tau _\\lambda \\left(x,y\\right) := \\mathbb {P}_\\lambda \\left(x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right).$ Using the connection and pair-connectedness functions $\\varphi ,\\tau _\\lambda \\colon \\mathbb {X}^2 \\rightarrow \\left[0,1\\right]$ as kernel functions, we construct associated linear integral operators.","Where $L^2\\left(\\mathbb {X}\\right)$ is the Hilbert space of square integrable functions on $\\mathbb {X}$ , we define the connection operator, $\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ , and the pair-connectedness operator, $\\mathcal {T}_\\lambda \\colon L^2\\left(\\mathbb {X}\\right)\\rightarrow L^2\\left(\\mathbb {X}\\right)$ .","For $f\\in L^2\\left(\\mathbb {X}\\right)$ , we have $f \\left(x\\right) = \\int \\varphi (x,y)f(y)\\nu \\left(\\mathrm {d}y\\right), \\qquad \\mathcal {T}_\\lambda f \\left(x\\right) = \\int \\tau _\\lambda (x,y)f(y)\\nu \\left(\\mathrm {d}y\\right),$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .","Note that the composition of two of these operators can be seen as a “convolution over $\\mathbb {X}$ \" of the kernel functions.","Where we rarely use it, we reserve the $\\star $ notation for straight-forward convolution over $\\mathbb {R}^d$ only.","For example, given $\\overline{x}\\in \\mathbb {R}^d$ and $a_1,a_2,a_3,a_4\\in \\mathcal {E}$ , $\\left(\\varphi \\left(\\cdot ;a_1,a_2\\right)\\star \\varphi \\left(\\cdot ;a_3,a_4\\right)\\right)\\left(\\overline{x}\\right) = \\int _{\\mathbb {R}^d}\\varphi \\left(\\overline{x}-\\overline{y};a_1,a_2\\right)\\varphi \\left(\\overline{y};a_3,a_4\\right) \\mathrm {d}\\overline{y}.$ We will also consider Fourier transformed versions of these operators.","For a Lebesgue-integrable function $g\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ , we define the Fourier transform of $g$ to be $\\widehat{g}(k) = \\int \\text{e}^{ik\\cdot \\overline{x}}g\\left(\\overline{x}\\right) \\mathrm {d}\\overline{x}$ for $k\\in \\mathbb {R}^d$ where $k\\cdot \\overline{x}= \\sum ^d_{j=1}k_j\\overline{x}_j$ denotes the standard inner product on $\\mathbb {R}^d$ .","For given marks $a,b\\in \\mathcal {E}$ , we define $\\widehat{\\varphi }(k;a,b)$ and $\\widehat{\\tau }_\\lambda (k;a,b)$ as the Fourier transforms of $\\varphi (\\overline{x};a,b)$ and $\\tau _\\lambda (\\overline{x};a,b)$ respectively.","We use these $\\mathcal {E}^2\\rightarrow \\mathbb {R}$ functions to define linear operators on $L^2(\\mathcal {E})$ , the space of square integrable functions on $\\mathcal {E}$ .","For each $k\\in \\mathbb {R}^d$ we define the operators $\\widehat{}(k),\\widehat{\\mathcal {T}}_\\lambda (k)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ by their action on $f\\in L^2(\\mathcal {E})$ : $\\left[\\widehat{}(k) f\\right] \\left(a\\right) = \\int \\widehat{\\varphi }(k;a,b)f(b)\\mathcal {P}\\left(\\mathrm {d}b\\right), \\qquad \\left[\\widehat{\\mathcal {T}}_\\lambda (k) f\\right] \\left(a\\right) = \\int \\tau _\\lambda (k;a,b)f(b)\\mathcal {P}\\left(\\mathrm {d}b\\right),$ for $\\mathcal {P}$ -almost every $a$ .","Our main aims of this paper are to derive an infrared bound for the two-point function (via its integral operators), and to prove that the so-called triangle condition is satisfied by a large family of models in sufficiently high dimension.","For $\\lambda >0$ we define $\\triangle _\\lambda := \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int \\tau _\\lambda (x,u)\\tau _\\lambda (u,v)\\tau _\\lambda (v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right).$ It is relatively simple to show that $\\triangle _\\lambda $ is finite for sub-critical $\\lambda $ and infinite for super-critical $\\lambda $ , but a model is said to satisfy the triangle condition if $\\triangle _\\lambda $ is finite at criticality.","An aside: the essential supremum taken in (REF ) is with respect to the measure $\\nu $ on $\\mathbb {X}$ .","We will be taking the essential supremum many times in this paper over various spaces, and each space will naturally will have a different measure implicitly associated with it.","It should be clear from the context what that measure is."],["Overview","In Section we provide the main results of the paper.","We begin by making precise the idea of a critical intensity in Section REF .","These propositions are largely independent of the rest of the paper and their proofs can be found in Appendix .","The main concern of the paper is in proving the results of Section REF .","Given the set of assumption REF , REF , and REF , we prove that the triangle condition is satisfied and we provide an infrared bound on the two-point operator in terms of the adjacency operator.","Some examples of models that satisfy these assumptions are provided in Section REF , with the proofs that they satisfy them given in Appendix .","Before we begin with the proof proper, we give some preliminaries in Section .","First Section REF gives a formal construction of the RCM.","Then we have Sections REF and REF , which give standard but important results in probability and linear algebra respectively that we will frequently use in our arguments.","The probabilistic lemmas will be familiar to students of point process theory and percolation theory, with the caveat that we require them to apply with the augmented space $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ taking the place of the more usual $\\mathbb {R}^d$ .","This adds no complication.","The linear algebra results may well be foreign to a probabilistic audience, but will nevertheless be fundamental to our proof.","Here we provide a sufficient technical description of the linear operators we construct and the Hilbert spaces upon which they act.","Perhaps the most important result we call upon from this section is the Spectral Theorem (Theorem REF ), which allows us to relate bounded self-adjoint linear operators to a multiplication operator by some unitary map.","Multiplication operators are far simpler to work with, and we use the theorem many times towards the end of our proof.","For operators on finite dimensional vector spaces, the spectral theorem is equivalent to saying that a Hermitian matrix can be diagonalized over $\\mathbb {R}$ .","A slightly more unusual lemma in this section is Lemma REF , in which we relate the spectra of the raw operators $$ and $\\mathcal {T}_\\lambda $ to the spectra of the Fourier transformed operators $\\widehat{}(k)$ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ respectively.","We conclude Section with Section REF , in which we show that the two-point operator and its derived operators are differentiable with respect to $\\lambda $ in the sub-critical regime and we provide bounds on their derivatives.","This will be used to show equicontinuity in a step of the proof of Proposition REF , in which we use a forbidden-region argument to bound a bootstrap function.","At a broad description level of the proof, we begin by deriving a finite lace expansion for the sub-critical two-point operator and show that this converges to an Ornstein-Zernike equation (OZE).","Via a bootstrap argument this will also prove that various expressions - including $\\triangle _\\lambda $ - are bounded uniformly for subcritical $\\lambda $ , and therefore we will be able to extend these bounds and the OZE to critical $\\lambda $ by using monotone and dominated convergence-like arguments.","We will then be able to take full advantage of the operator description to solve the linear OZE to get the critical two-point operator in terms of the adjacency operator and a lace-expansion-coefficient operator.","Section deals with finding the finite lace expansion for the sub-critical $\\lambda $ .","Formulating this expansion in terms of the relevant functions is essentially the same as for the lace expansion argument appearing in [21] for the unmarked random connection model, except we replace $\\mathbb {R}^d$ with $\\mathbb {X}$ .","For this reason we give only a very brief description of the argument - presenting the important lemmas and ideas - and refer the reader to that previous work for the details.","The step of formulating the finite lace expansion in terms of operators (done in Section REF ) is then simple.","To show that the finite lace expansion converges, we require bounds on various operators arising from it.","We begin in Section by bounding the operator norms of these operators by fairly long and complicated integrals of $\\tau _\\lambda $ and $\\varphi $ (see Proposition REF ).","We are then able to divide these complicated integrals into smaller and more manageable elementary `diagrams' - it is here that $\\triangle _\\lambda $ first makes an appearance.","It is also here that we have our first significant detour from the analogous arguments for the proof for the unmarked models.","We not only reformulate the proof strategy to be more straight-forward, direct, and transparent than it as appeared previously, but also reduce the reliance on the assumption analogous to the bounds on the sets $B(x)$ defined in (REF ).","In [21] this was used multiple times to bound diagrams arising from both the displaced and non-displaced expansion coefficients, but for us this part of the assumption is only ever used to control one simple diagram in Proposition REF .","Unfortunately, it does indeed seem that this particular diagram would need to be controlled via a specific assumption, and the bounds on the sets $B(x)$ appear to be relatively easy to check for many models.","We also end up with more elementary diagrams than would appear in the unmarked case.","The term we describe here as $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ appearing in Propositions REF and REF would simplify into another diagram in the unmarked case, but the inhomogeneity that the marks introduce means that we cannot necessarily commute terms and it does not simplify in our case.","Providing bounds for these more elementary diagrams that are uniform in $\\lambda $ in the subcritical regime is now a much more realistic task, one we achieve in Section by a bootstrap argument.","Many of the elementary diagrams contain instances of the two-point function, which is not explicitly known.","In Section REF we therefore replace each of them with an explicitly known Green's function $G_{\\mu _\\lambda }$ at the cost of one factor of a bootstrap function $f(\\lambda )$ , and in Section REF we use a bootstrap argument with a forbidden-region argument to prove that $f$ is uniformly bounded for subcritical $\\lambda $ .","This then proves that the OZE is satisfied subcritically and that the elementary diagrams - including $\\triangle _\\lambda $ - are uniformly bounded for subcritical $\\lambda $ .","Now we have the OZE and uniform bounds for subcritical $\\lambda $ , we want to extend these to critical $\\lambda $ .","We begin Section by using monotone convergence to extend our uniform bound on $\\triangle _\\lambda $ to include critical $\\lambda $ .","Using similar monotone and dominated convergence ideas we are able to also show that OZEs for the Fourier transformed operators hold at criticality.","We are then finally able to find an infrared bound for the two-point operator at criticality by solving these OZEs using our linear operator formalism."],["Results","As we will be working with operators, we will require some concept of size for these operators.","This will take a number of forms, as some will have advantages in various scenarios.","Let $\\left(\\mathcal {X},\\mu \\right)\\in \\left\\lbrace \\left(\\mathbb {X},\\nu \\right),\\left(\\mathcal {E},\\mathcal {P}\\right)\\right\\rbrace $ , and let $L^2\\left(\\mathcal {X}\\right)\\in \\left\\lbrace L^2\\left(\\mathbb {X}\\right),L^2\\left(\\mathcal {E}\\right)\\right\\rbrace $ denote the corresponding space of square integrable functions.","In practice it will be clear which is being used.","We will require five various ideas of size.","First suppose $H\\colon L^2\\left(\\mathcal {X}\\right) \\rightarrow L^2\\left(\\mathcal {X}\\right)$ is a bounded linear operator.","Then the operator norm is defined by $\\left\\Vert H\\right\\Vert _{\\rm op} := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{*{Hf}_2}{*{f}_2},$ where $*{\\cdot }_2$ is the standard $L^2$ -norm on $L^2\\left(\\mathcal {X}\\right)$ .","In particular, we have $\\left\\Vert H\\right\\Vert _{\\rm op} = \\sup \\left\\lbrace *{z}:z\\in \\sigma (H)\\right\\rbrace $ , where $\\sigma \\left(H\\right)\\subset \\mathbb {C}$ denotes the spectrum of $H$ .","Let ${\\cdot }{\\cdot }$ be the inner product on $L^2\\left(\\mathcal {X}\\right)$ defined by (REF ).","If $H$ is also self-adjoint, then ${f}{Hf}\\in \\mathbb {R}$ and we define the spectral supremum $\\mathbb {S}\\left(H\\right) := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{{f}{Hf}}{{f}{f}}.$ In particular, the self-adjointness of $H$ implies that $\\sigma (H)\\subset \\mathbb {R}$ and $\\mathbb {S}\\left(H\\right) = \\sup \\left\\lbrace z:z\\in \\sigma (H)\\right\\rbrace $ .","Note that $\\mathbb {S}\\left(\\cdot \\right)$ is not a norm: it is not necessarily non-negative.","Now suppose $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ is a bounded linear integral operator with measurable kernel function $h\\colon \\mathcal {X}^2 \\rightarrow \\mathbb {C}$ .","Then we have the explicit norms: $\\left\\Vert H\\right\\Vert _{1,\\infty } &:= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}*{h\\left(\\cdot ,y\\right)}_1 = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}\\int *{h\\left(x,y\\right)}\\mu \\left(\\mathrm {d}x\\right), \\\\\\left\\Vert H\\right\\Vert _{2,\\infty } &:= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}*{h\\left(\\cdot ,y\\right)}_2 = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}\\left(\\int *{h\\left(x,y\\right)}^2\\mu \\left(\\mathrm {d}x\\right)\\right)^\\frac{1}{2},\\\\\\left\\Vert H\\right\\Vert _{\\infty ,\\infty } &:= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}*{h\\left(\\cdot ,y\\right)}_\\infty = \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathcal {X}}*{h\\left(x,y\\right)} .$"],["Critical Intensities","Before we move on to our results describing behaviour at and around criticality, we will first explore how we are defining criticality.","In our main results in Section REF we use the operator critical intensity $\\lambda _O$ , defined by $\\lambda _O := \\inf \\left\\lbrace \\lambda >0:\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}=\\infty \\right\\rbrace .$ Whilst the divergence of a two-point operator has been used previously to define a critical threshold in random graphs before (for example in [26]), at first glance it may seem to be quite far removed from the standard interpretations of a percolation transition.","A variety of transition thresholds for the continuum Boolean disc model are discussed in [38], [17], [8].","However, we are able to show that under certain conditions the operator critical intensity coincides with more familiar interpretations.","For $\\lambda \\ge 0$ , we define the two functions $\\theta _\\lambda \\colon \\mathcal {E}\\mapsto \\left[0,1\\right]$ and $\\chi _\\lambda \\colon \\mathcal {E}\\mapsto \\left[0,+\\infty \\right]$ as $\\theta _\\lambda (a) &:= \\mathbb {P}_\\lambda \\left(*{\\overline{0},a)}=\\infty \\right),\\\\\\chi _\\lambda (a) &:= \\mathbb {E}_\\lambda \\left[*{\\overline{0},a)}\\right].$ For each $p\\in \\left[1,+\\infty \\right]$ , we define corresponding functions $\\theta ^{(p)}\\colon \\lambda \\mapsto {\\theta _\\lambda }_p$ and $\\chi ^{(p)}\\colon \\lambda \\mapsto {\\chi _\\lambda }_p$ .","These suggest the critical intensities $\\lambda _c^{(p)} &:= \\inf \\left\\lbrace \\lambda \\ge 0: {\\theta _\\lambda }_p >0\\right\\rbrace ,\\\\\\lambda _T^{(p)} &:= \\inf \\left\\lbrace \\lambda \\ge 0: {\\chi _\\lambda }_p =\\infty \\right\\rbrace .$ Some relations between these intensities can be swiftly deduced without requiring the RCM to satisfy any special conditions.","Their definitions clearly show that $\\theta _\\lambda (a)>0$ implies $\\chi _\\lambda (a)=\\infty $ .","Therefore $\\lambda ^{(p)}_c \\ge \\lambda ^{(p)}_T$ for all $p\\in \\left[1,\\infty \\right]$ .","If $*{\\theta _\\lambda }_p>0$ for some $p\\in \\left[1,\\infty \\right]$ , then it is non-zero on some $\\mathcal {P}$ -positive set, and therefore $*{\\theta _\\lambda }_q>0$ for all $q\\in \\left[1,\\infty \\right]$ .","Therefore $\\lambda _c^{(p)} = \\lambda _c^{(1)}$ for all $p\\in \\left[1,+\\infty \\right]$ , and we denote this value simply by $\\lambda _c$ .","By Jensen's inequality, $1\\le p_1 \\le p_2 \\le \\infty $ implies the ordering $*{\\chi _\\lambda }_1 \\le *{\\chi _\\lambda }_{p_1} \\le *{\\chi _\\lambda }_{p_2} \\le *{\\chi _\\lambda }_\\infty $ .","Therefore we have $\\lambda _T^{(1)} \\ge \\lambda ^{(p_1)}_T \\ge \\lambda ^{(p_2)}_T \\ge \\lambda ^{(\\infty )}_T$ .","With some of the techniques developed in this paper it is possible to relate $\\lambda _O$ to the intensities $\\lambda ^{(1)}_T$ , $\\lambda ^{(\\infty )}_T$ , and $\\lambda _c$ described here.","Proposition 2.1 The critical intensities satisfy $\\lambda ^{(\\infty )}_T \\le \\lambda _O \\le \\lambda ^{(1)}_T \\le \\lambda _c$ .","If $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi (\\overline{x};a,b)\\mathrm {d}\\overline{x}<\\infty ,$ then we also have $\\lambda ^{(p)}_T = \\lambda _O$ for all $p\\in \\left[1,\\infty \\right]$ .","This proof of Proposition REF uses Mecke's equation, Lemma REF , Lemma REF , and some simple probabilistic ideas.","The details can be found in Appendix .","The coincidence of critical intensities for percolation in marked random connection models has also been considered by [35], [17], [8] in the specific case of a family of Boolean defect models, including Boolean disc models with random bounded radii.","In our terminology they introduced a new mark, $\\dagger $ , which forms connections as if it had radius 0, and proved that $\\theta _\\lambda \\left(\\dagger \\right) =0$ if and only if $\\chi _\\lambda \\left(\\dagger \\right) < \\infty $ .","Therefore if we define $\\lambda _c\\left(\\dagger \\right) := \\inf \\left\\lbrace \\lambda \\ge 0: \\theta _\\lambda \\left(\\dagger \\right) >0\\right\\rbrace , \\qquad \\lambda _T\\left(\\dagger \\right) := \\inf \\left\\lbrace \\lambda \\ge 0: \\chi _\\lambda \\left(\\dagger \\right) = \\infty \\right\\rbrace ,$ this gives $\\lambda _c\\left(\\dagger \\right) = \\lambda _T\\left(\\dagger \\right)$ .","The following Proposition uses this to prove that for the Boolean disc model with bounded radii, all the critical intensities we have considered are equal.","Proposition 2.2 In the case of the Boolean disc model with bounded radii, $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right) = \\lambda _c = \\lambda _O = \\lambda _T^{(p)},$ for all $p\\in \\left[1,\\infty \\right]$ .","To prove Proposition REF we show $\\lambda _T\\left(\\dagger \\right) \\le \\lambda ^{(p)}_T$ by considering points in $\\eta $ adjacent to the root point with mark $\\dagger $ .","The details can be found in Appendix ."],["Two-Point Operator","The majority of this paper is concerned with describing the two-point function and the associated two-point operator.","Our results will require Assumption REF described below to hold.","This can be considered to be a generalisation of the “finite variance\" models described by [21] - extending to cases with multiple marks.","HAssumption H We require $\\varphi $ to satisfy the following properties: For all dimensions $d$ we have $\\mathbb {S}\\left(\\widehat{}(0)\\right) < \\infty $ , and there exists a $d$ -independent constant $C>0$ such that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } &\\le C \\mathbb {S}\\left(\\widehat{}(0)\\right),\\\\\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } &\\le C \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ There exist $d$ -independent constants $C_1\\in \\left(0,1\\right)$ and $C_2>0$ such that $\\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(0\\right)\\right)} \\le C_1 \\vee \\left(1 - C_2{k}^2\\right).$ There exists a function $g\\colon \\mathbb {N} \\rightarrow \\mathbb {R}_{\\ge 0}$ with the following three properties.","Firstly, that $g(d)\\rightarrow 0$ as $d\\rightarrow \\infty $ .","Secondly, that $\\mathbb {S}\\left(\\widehat{}(0)\\right)^{-3}\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6)\\right)\\left(\\overline{x}\\right) \\le g(d).$ Thirdly, that the family of sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ given by $B\\left(x\\right) := \\left\\lbrace y\\in \\mathbb {X}: \\mathbb {S}\\left(\\widehat{}(0)\\right)^{-2}\\int \\varphi \\left(x,u\\right)\\varphi \\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right) > g\\left(d\\right)\\right\\rbrace $ satisfy $\\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\nu \\left(B\\left(x\\right)\\right) \\le g\\left(d\\right)$ .","Remark 2.3 We have some remarks regarding these assumptions.","By Jensen's inequality, the condition (REF ) in REF also implies that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty } \\le C \\mathbb {S}\\left(\\widehat{}(0)\\right).$ We will often use these bounds (the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ bound in particular) more directly than (REF ) itself.","From (REF ) and (REF ) we can use a supremum bound to deduce that analogous bounds hold for the convolution of $m$ adjacency functions (for $m\\ge 3$ ) with corresponding bound $C^{m-3}g(d)$ .","The sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ appearing in REF are all measurable - this follows from the measurability of $\\varphi $ .","Furthermore, the symmetry of $\\varphi $ implies that $y\\in B\\left(x\\right)\\iff x\\in B\\left(y\\right)$ .","In stating our theorems, we will use a parameter $\\beta =\\beta (\\varphi , d)$ that depends upon $g(d)$ .","The definition depends upon the asymptotic properties of $g(d)$ in that $\\beta (d) := {\\left\\lbrace \\begin{array}{ll}g(d)^{\\frac{1}{4} -\\frac{3}{2d}}d^{-\\frac{3}{2}} &: \\lim _{d\\rightarrow \\infty }g(d)\\rho ^{-d}\\Gamma \\left(\\frac{d}{2}+1\\right)^2 = 0 \\qquad \\forall \\rho >0,\\\\g(d)^\\frac{1}{4}&: \\text{otherwise}.\\end{array}\\right.","}$ That is, $\\beta (d) = g(d)^\\frac{1}{4}$ unless $g(d)$ approaches zero particularly quickly.","The parameter $\\beta $ will reappear many times in this paper - often in the case where $\\beta $ is `small.'","This corresponds to `large' dimension $d$ , and in particular, where the Landau notation $\\mathcal {O}\\left(\\beta \\right)$ is used the asymptotics are as $d\\rightarrow \\infty $ .","Theorem 2.4 Suppose $\\varphi $ and $\\nu $ satisfy the assumptions REF , REF , and REF .","Then there exists a $\\lambda $ -independent $d^*>6$ and $C>0$ such that for $\\lambda \\in \\left[0,\\lambda _O\\right]$ there exists an operator $_\\lambda $ with $\\left\\Vert _\\lambda \\right\\Vert _{\\rm op}\\le C\\beta $ and a family of operators $\\left\\lbrace \\widehat{}_{\\lambda }(k)\\right\\rbrace _{k\\in \\mathbb {R}^d}$ with $\\left\\Vert \\widehat{}_{\\lambda }(k)\\right\\Vert _{\\rm op}\\le C\\beta $ , such that for all $d\\ge d^*$ the following statements are satisfied.","For $\\lambda \\in \\left[0,\\lambda _O\\right)$ the bounded operator $\\mathcal {T}_\\lambda $ satisfies the Ornstein–Zernike equation: $\\mathcal {T}_\\lambda = + _\\lambda + \\lambda \\mathcal {T}_\\lambda \\left( + _\\lambda \\right).$ For $\\lambda \\in \\left[0,\\lambda _O\\right)$ and $k\\in \\mathbb {R}^d$ , or $\\lambda = \\lambda _O$ and $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ , the bounded operator $\\widehat{\\mathcal {T}}_\\lambda (k)$ satisfies the Ornstein–Zernike equation in Fourier form: $\\widehat{\\mathcal {T}}_\\lambda (k) = \\widehat{}(k) + \\widehat{}_{\\lambda }(k) + \\lambda \\widehat{\\mathcal {T}}_\\lambda (k) \\left( \\widehat{}(k) + \\widehat{}_{\\lambda }(k) \\right).$ Theorem 2.5 If $\\varphi $ and $\\nu $ satisfy the assumptions REF , REF , and REF , and $\\lambda \\in \\left[0,\\lambda _O\\right]$ , then there exist $\\lambda $ -independent $d^*>6$ and $C>0$ such that for all $d\\ge d^*$ $\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)+ C \\beta }{\\mathbb {S}\\left(\\widehat{}\\left(0\\right)\\right) - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\vee 1 \\qquad \\forall k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace .$ Furthermore, we have $\\triangle _\\lambda \\le C\\beta ^2$ and $*{\\theta _{\\lambda }}_p=0$ for all $p\\in \\left[1,\\infty \\right]$ and $\\lambda \\in \\left[0,\\lambda _O\\right]$ .","This infrared bound can also be related directly to the two-point function by Lemma REF via $\\int \\widehat{\\tau }_\\lambda \\left(k;a,b\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a, \\mathrm {d}b\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}.$ Remark 2.6 Note that in Theorem REF we say there exists some $d^*>6$ such that the results hold for all $d\\ge d^*$ .","If we were willing to say that there exists some $d^*>10$ instead (which would not change the strict logical content of the result), then the proof would be far simpler and Assumptions REF and REF could be weakened.","The $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ in (REF ) and () could be replaced with $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ and $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ respectively, and (REF ) could be replaced with $\\left\\Vert ^3\\right\\Vert _{\\infty ,\\infty }\\le g(d)\\mathbb {S}\\left(\\widehat{}(0)\\right)^3$ .","We do not pursue this approach because it is expected that a very similar argument to that presented here (with context appropriate changes to the assumptions) would prove the corresponding result for the spread-out model in $d>6$ .","For the simpler argument and weaker conditions, we would only get the spread-out result in $d>10$ , and this would be a logically weaker statement."],["Examples of Models","It is natural to ask if there are any models for which $\\varphi $ and $\\nu $ do indeed satisfy REF , REF , and REF .","All the models we describe here also satisfy the condition (REF ) for every dimension $d$ .","We prove in Appendix that all the models below do satisfy our assumptions."],["Single Mark Finite Variance Model.","Suppose $\\mathcal {E}$ is a singleton, so essentially $\\mathbb {X}=\\mathbb {R}^d$ and $\\nu = {\\rm Leb}$ .","Then the finite variance models described in [21] all satisfy our conditions.","These include the Poisson blob model (or the spherical Boolean model) in which $\\varphi (\\overline{x}) = {1}_{\\lbrace *{\\overline{x}} \\le r_d\\rbrace }$ where $r_d$ is the radius of the unit volume $d$ -ball, and the Gaussian connection model with $\\smash{\\varphi }(\\overline{x}) = \\left(2\\pi \\right)^{-d/2}\\exp \\big (-\\tfrac{1}{2}*{\\overline{x}}^2\\big )$ ."],["Double Mark Finite Variance Model.","Suppose $\\mathcal {E}= \\lbrace a_1,a_2\\rbrace $ has only two distinct elements with $\\mathcal {P}= p\\delta _{a_1} + q\\delta _{a_2}$ for some $p = 1-q \\in \\left(0,1\\right)$ , and that all the three independent functions $\\overline{x}\\mapsto \\varphi (\\overline{x};a,b)$ (for $a,b\\in \\mathcal {E}$ ) each satisfy the conditions required of a single mark finite variance model.","Further suppose that (without loss of generality) $p^2\\varphi (\\overline{x};a_1,a_1) \\ge q^2\\varphi (\\overline{x};a_2,a_2)$ for almost every $\\overline{x}\\in \\mathbb {R}^d$ .","Then our conditions are satisfied."],["Space-Mark Factorisation Model","Suppose now that the connection function admits the factorisation $\\varphi (\\overline{x};a,b) = \\overline{\\varphi }(\\overline{x})K(a,b)$ .","We require that $\\overline{\\varphi }$ satisfies the conditions of a single mark finite variance model, and that the ($d$ -independent) mark-kernel $K\\colon \\mathcal {E}^2 \\rightarrow \\left[0,1\\right]$ is symmetric and measurable.","Note that this means that the integral operator $\\mathcal {K}\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ with kernel function $K$ is a bounded self-adjoint operator with operator norm $\\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}\\le 1$ .","We further suppose that $\\mathbb {S}\\left(\\mathcal {K}\\right) = \\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}$ .","Then the connection operator is a bounded operator on the mark-space multiplied by a scalar function of spatial position.","This formulation places no constraint on the number of marks, permitting even models with uncountably many marks."],["Marked Multivariate Gaussian Model.","Let $\\Sigma \\colon \\mathcal {E}^2\\rightarrow \\mathbb {R}^{d\\times d}$ be a measurable map where for every $a,b\\in \\mathcal {E}$ , $\\Sigma \\left(a,b\\right)$ is itself a symmetric positive definite covariance matrix.","We further require that there exists $\\Sigma _{\\max }<\\infty $ and $\\Sigma _{\\min } = \\Sigma _{\\min }(d) > 0$ such that the spectrum $\\sigma (\\Sigma (a,b)) \\subset \\left[\\Sigma _{\\min },\\Sigma _{\\max }\\right]$ for all $a,b\\in \\mathcal {E}$ and dimension $d$ .","Then for $\\mathcal {A} = \\mathcal {A}(d)>0$ such that $\\mathcal {A}^2 \\le \\left(2\\pi \\right)^d\\det \\Sigma (a,b)$ for all $a,b\\in \\mathcal {E}$ and $\\limsup _{d\\rightarrow \\infty } \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2} = 0$ , let the connection function be given by $\\varphi (\\overline{x};a,b) = \\mathcal {A}\\left(2\\pi \\right)^{-d/2}\\left(\\det \\Sigma (a,b)\\right)^{-1/2}\\exp \\left(-\\frac{1}{2}\\overline{x}^{\\intercal }\\Sigma (a,b)^{-1}\\overline{x}\\right).$ This class of model includes examples of models with (uncountably) infinitely many marks, and for which the mark and spatial behaviours are truly coupled.","For a concrete example, consider $\\mathcal {E}=\\left[\\tfrac{1}{4\\pi },1\\right]$ with $\\Sigma (a,b) = (a+b)1$ and $\\mathcal {A} = 1$ ."],["Bounded-Volume Boolean Disc Model.","Let $\\left(\\left[0,1\\right], \\Sigma _{\\left[0,1\\right]}, \\mathcal {P}\\right)$ be a separable probability space and $\\left\\lbrace R_d\\right\\rbrace _{d\\ge 1}$ be a sequence of measurable functions $R_d\\colon \\left[0,1\\right]\\rightarrow \\mathbb {R}_+$ such that there exist $ 00$ such that $R^{({\\rm min})}_d:= c_1 d^{\\frac{1}{2}} \\le R_d\\left(a\\right) \\le \\frac{1}{2\\sqrt{\\pi }}\\left(V^{({\\rm max})}\\right)^{\\frac{1}{d}} \\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d} =: R^{({\\rm max})}_d,$ for all $a\\in [0,1]$ .","Note that as $d\\rightarrow \\infty $ , Stirling's formula gives that the upper bound $R^{({\\rm max})}_d\\sim \\sqrt{d/8\\pi \\text{e}}$ .","It is also required that there exist $\\epsilon >0$ and $c_2>0$ such that $\\mathcal {P}\\left(R_d > \\left(1-\\frac{c_2}{d}\\right)R^{({\\rm max})}_d\\right)>\\epsilon $ for all $d$ .","Then let the connection function be defined by $\\varphi _d(\\overline{x};a,b) = {1}\\left\\lbrace *{\\overline{x}} \\le R_d(a) + R_d(b)\\right\\rbrace .$ It is also perhaps worth mentioning a model that does not satisfy condition REF .","Such an example would be a Boolean disk model with random radii for which $\\mathcal {P}$ gives positive measure to arbitrarily large radii.","No matter the decay rate of $\\mathcal {P}$ for large radii, the presence of the suprema in (REF ) and () ensure that REF cannot be satisfied.","Similarly the condition (REF ) is also not satisfied for such a model."],["Outlook","Theorem REF establishes the triangle condition, which is a strong indication of mean-field behaviour.","The identification of mean-field critical exponents in the inhomogeneous setting is postponed to future work.","We believe that the infrared bound is valid beyond the restricted setting of Hypothesis REF , in particular for the Boolean model with unbounded radii as well as models for complex networks admitting power-law degree graphs (under suitable moment assumptions).","However, our current proof does not admit such generalisation, as explained above."],["Formal Construction of the RCM","We choose to construct our RCM in largely the same way as [21], with appropriate replacements of $\\mathbb {R}^d$ with the marked space $\\mathbb {X}$ .","As they did, we follow [34] in defining the RCM, $\\xi $ , as an independent edge-marking of a Poisson point process $\\eta $ .","Recall $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ .","We denote by $\\mathbf {N}\\left(\\mathbb {X}\\right)$ the set of all at most countably infinite subsets of $\\mathbb {X}$ and accompany it with the $\\sigma $ -algebra $\\mathcal {N}(\\mathbb {X})$ generated by the sets $\\lbrace A\\in \\mathbf {N}\\left(\\mathbb {X}\\right):|A \\cap B| = k, B \\in \\Sigma _\\mathbb {X}, k \\in \\mathbb {N}_0\\rbrace $ , where $*{A\\cap B}$ denotes the cardinality of $A\\cap B$ .","Then a point process on $\\mathbb {X}$ is a measurable mapping $\\zeta \\colon \\Omega \\rightarrow \\mathbf {N}\\left(\\mathbb {X}\\right)$ for some underlying probability space $\\left(\\Omega ,\\Sigma _\\Omega ,\\mathbb {P}\\right)$ .","The intensity measure of such a point process is then the measure on $\\mathbb {X}$ given by $B\\mapsto \\mathbb {E}\\left[*{\\zeta \\cap B}\\right]$ for $B\\in \\Sigma _\\mathbb {X}$ .","Let $\\zeta $ be a point process on $\\mathbb {X}$ with a $\\sigma $ -finite intensity measure.","Then there exist measurable $\\pi _i\\colon \\mathbf {N} \\left(\\mathbb {X}\\right) \\rightarrow \\mathbb {X}$ , $i\\in \\mathbb {N}$ such that almost surely $\\zeta = \\left\\lbrace \\pi _i\\left(\\zeta \\right): i\\le *{\\zeta \\cap \\mathbb {X}}\\right\\rbrace .$ For a $\\sigma $ -finite non-atomic measure on $\\mathbb {X}$ , say $\\nu $ , a Poisson point process (PPP) on $\\mathbb {X}$ with intensity measure $\\nu $ is a point process $\\zeta $ such that the number of points $*{\\zeta \\cap B}$ is $\\text{Poi}(\\nu (B))$ -distributed for each $B \\in \\Sigma _\\mathbb {X}$ , and the random variables $*{\\zeta \\cap B_1}, \\ldots , *{\\zeta \\cap B_m}$ are independent whenever $B_1, \\ldots , B_m\\in \\Sigma _\\mathbb {X}$ are pairwise disjoint.","In this paper we will be considering a PPP $\\eta $ in $\\mathbb {X}$ with intensity measure $\\lambda \\nu $ , where $\\nu ={\\rm Leb}\\otimes \\mathcal {P}$ and $\\lambda >0$ .","We write $\\eta =\\left\\lbrace X_i:i\\in \\mathbb {N}\\right\\rbrace $ , where $X_i=\\pi _i\\left(\\eta \\right)$ for $i\\in \\mathbb {N}$ .","We now construct the RCM as a (non-Poissionian) point process on the space $\\mathbb {X}^{[2]}\\times \\left[0,1\\right]$ .","We denote by $\\mathbb {X}^{[2]}$ the set of all subsets of $\\mathbb {X}$ containing precisely two elements, and each $e\\in \\mathbb {X}^{[2]}$ is then a potential edge of the RCM.","Also let $\\left\\lbrace U_{i,j}\\right\\rbrace _{i,j\\in \\mathbb {N}}$ be a double sequence of independent identically distributed random variables uniformly distributed on $\\left[0,1\\right]$ , such that the sequence is independent of the PPP $\\eta $ .","We define $\\xi = \\left\\lbrace \\left(\\left\\lbrace X_i,X_j\\right\\rbrace ,U_{i,j}\\right):1\\le i\\le j\\right\\rbrace ,$ where we recall $X_i=\\pi _i(\\eta )$ .","Thus $\\xi $ is a random point process on $\\mathbb {X}^{[2]}\\times \\left[0,1\\right]$ , which we interpret as a random marked graph and say that $\\xi $ is an independent edge-marking of $\\eta $ .","The independent edge-marking $\\xi $ of $\\eta $ can then be related by a deterministic functional, $\\Gamma _\\varphi $ , to a random graph on $\\mathbb {X}$ by defining its vertex and edge sets: $V(\\Gamma _\\varphi (\\xi )) = \\eta = \\lbrace X_i:i\\in \\mathbb {N}\\rbrace ,\\qquad E(\\Gamma _\\varphi (\\xi )) = \\lbrace \\lbrace X_i,X_j\\rbrace :U_{i,j} \\le \\varphi (X_i,X_j), 1\\le i\\le j\\rbrace .$ Since the sequence $U_{i,j}$ is independent of the Poisson process $\\eta $ , the ordering in (REF ) does not change the distribution of the resulting random graph.","This interpretation of the RCM also allows for the introduction of additional points.","This will be required many times in this paper, not least in our definition of the two-point function itself.","For additional points $x_1,x_2\\in \\mathbb {X}$ , the point process $\\eta $ is augmented to make $\\eta ^{x_1}:=\\eta \\cup \\lbrace x_1\\rbrace ,\\quad \\eta ^{x_1,x_2}:=\\eta \\cup \\lbrace x_1,x_2\\rbrace .$ To extend these point processes on $\\mathbb {X}$ to a RCM, we extend the sequence $\\left\\lbrace U_{m,n}\\right\\rbrace _{m,n \\ge 1}$ to a sequence $\\left\\lbrace U_{m,n}\\right\\rbrace _{m,n \\ge -1}$ of independent random variables uniformly distributed on $[0,1]$ , independent of the Poisson process $\\eta $ .","We then define $\\xi ^{x_1,x_2}$ on $\\mathbb {X}^{[2]}\\times [0,1]$ as $\\xi ^{x_1,x_2}:=\\lbrace (\\lbrace X_i,X_j\\rbrace ,U_{i,j}):-1\\le i\\le j\\rbrace ,$ where $X_0 = x_1$ and $X_{-1} = x_2$ .","It is straightforward to define $\\xi ^{x_1, \\ldots , x_m}$ for arbitrary $m \\ge 3$ by the same idea, and to define $\\xi ^{x_1}$ by removing all (marked) edges incident to $x_2$ from $\\xi ^{x_1,x_2}$ .","We assume implicitly that the augmented points are properly coupled such that, e.g., $\\xi ^{x_1}$ obtained in this way from $\\xi ^{x_1,x_2}$ (by removing all edges adjacent to $x_2$ ) is the same as if obtained from $\\xi ^{x_1,x_3}$ (by removing all edges adjacent to $x_3$ ); see [21] for further details."],["Probabilistic Lemmas","We introduce here a number of important lemmas that are standard in point process theory or percolation theory."],["The Mecke Equation","An important part of our analysis will be a version of the Mecke equation - for a discussion see [33].","This result allows us to make sums over the random sum of points in $\\eta $ more tractable.","Given $m\\in \\mathbb {N}$ and a measurable function $f\\colon \\mathbf {N}\\left((\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1]\\right) \\times \\mathbb {X}^m \\rightarrow \\mathbb {R}_{\\ge 0}$ , the Mecke equation for $\\xi $ states that $\\mathbb {E}_\\lambda \\left[ \\sum _{\\vec{x} \\in \\eta ^{(m)}} f(\\xi , \\vec{x})\\right] = \\lambda ^m \\int \\mathbb {E}_\\lambda \\left[ f\\left(\\xi ^{x_1, \\ldots , x_m}, \\vec{x}\\right)\\right] \\nu ^{\\otimes m}\\left(\\mathrm {d}\\vec{x}\\right), $ where $\\vec{x}=(x_1,\\ldots ,x_m)$ , $\\eta ^{(m)}=\\lbrace (x_1,\\ldots ,x_m): x_i \\in \\eta , x_i \\ne x_j \\text{ for } i \\ne j\\rbrace $ , and $\\nu ^{\\otimes m}$ is the $m$ -product measure of $\\nu $ on $\\mathbb {X}^m$ .","We will only need (REF ) for $m\\le 3$ , and largely only use it for $m=1$ ."],["The Margulis-Russo Formula","An often used and very important tool in (discrete) percolation theory, the Margulis-Russo formula will be widely used in our analysis as well.","The version we use follows from a slight adjustment to the more general result given in [34].","Writing $\\mathbf {N}:=\\mathbf {N}\\left((\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1]\\right)$ , let $\\Lambda \\subset \\mathbb {X}$ be $\\nu $ -finite, $\\zeta \\in \\mathbf {N}$ , and define $\\zeta _\\Lambda := \\lbrace (\\lbrace (x,v),(y,w)\\rbrace ,u)\\in \\zeta :\\lbrace x,y\\rbrace \\subseteq \\Lambda \\rbrace .","$ We call $\\zeta _\\Lambda $ the restriction of $\\zeta $ to $\\Lambda $ .","We say that $f\\colon \\mathbf {N}\\rightarrow \\mathbb {R}$ lives on $\\Lambda $ if $f(\\zeta ) = f(\\zeta _\\Lambda )$ for every $\\zeta \\in \\mathbf {N}$ .","Assume that there exists a $\\nu $ -finite set $\\Lambda \\subset \\mathbb {X}$ such that $f$ lives on $\\Lambda $ .","Moreover, assume that there exists $\\lambda _0>0$ such that $\\mathbb {E}_{\\lambda _0} \\left[*{f(\\xi )}\\right] < \\infty $ .","Then the Margulis-Russo formula states that, for all $\\lambda \\le \\lambda _0$ , $\\frac{\\partial }{\\partial \\lambda } \\mathbb {E}_\\lambda [f(\\xi )] = \\int _\\Lambda \\mathbb {E}_\\lambda \\left[f(\\xi ^x) - f(\\xi )\\right] \\nu \\left(\\mathrm {d}x\\right).","$ The result in [34] requires that $\\mathbb {X}=\\mathbb {R}^d$ and that $\\Lambda \\subset \\mathbb {R}^d$ is compact, but [32] allows us to make the necessary adjustments to permit $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ ."],["The BK Inequality","The idea of the BK inequality is that we acquire a simple upper bound for the probability that two increasing events happen, and that they do so on disjoint subsets of the space.","For example, we may want to bound the probability that there exist two disjoint paths (up to the start and end points) in $\\xi $ between two given points.","The BK inequality we present here is slightly more general than the usual BK inequality in that the point process $\\xi $ is augmented with independent random variables.","It differs only slightly from the version appearing in [21].","For notational clarity, we write $\\mathbf {N}:=\\mathbf {N}((\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1])$ and denote the $\\sigma $ -algebra on $\\mathbf {N}$ by $\\mathcal {N}$ .","We call a set $E\\subset \\mathbf {N}$ increasing if $\\mu \\in E$ implies $\\nu \\in E$ for each $\\nu \\in \\mathbf {N}$ with $\\mu \\subseteq \\nu $ .","Let $(\\mathbb {Y}_1,\\mathcal {Y}_1)$ , $(\\mathbb {Y}_2,\\mathcal {Y}_2)$ be two measurable spaces.","We say that a set $E_i\\subset \\mathbf {N}\\times \\mathbb {Y}_i$ is increasing if $\\mathbb {E}^z_i:=\\lbrace \\mu \\in \\mathbf {N}:(\\mu ,z)\\in E_i\\rbrace $ is increasing for each $z\\in \\mathbb {X}_i$ .","For Borel $\\Lambda \\in \\mathcal {B}\\left(\\mathbb {R}^d\\right)$ and $\\mu \\in \\mathbf {N}$ , we define $\\mu _\\Lambda $ , the restriction of $\\mu $ to all edges completely contained in $\\left(\\Lambda \\times \\mathcal {E}\\right)\\times \\mathbb {M}$ , analogously to (REF ).","Also let $\\mathcal {R}$ denote the ring of all finite unions of half-open $d$ -dimensional rectangles in $\\mathbb {R}^d$ with rational coordinates.","Then for increasing $E_i\\in \\mathcal {N}\\otimes \\mathcal {Y}_i$ , we define $E_1 \\circ E_2 := \\left\\lbrace (\\mu ,z_1,z_2)\\in \\mathbf {N}\\times \\mathbb {Y}_1\\times \\mathbb {Y}_2: \\exists K_1,K_2 \\in \\mathcal {R} \\text{ s.t.~} \\right.\\\\\\left.K_1\\cap K_2 = \\varnothing , (\\mu _{K_1},z_1)\\in E_1,(\\mu _{K_2},z_2)\\in E_2 \\right\\rbrace .","$ A set $E_i\\in \\mathcal {N}\\otimes \\mathcal {Y}_i$ lives on $\\Lambda $ if $1_{E_i}(\\mu ,z)=1_{E_i}(\\mu _\\Lambda ,z)$ for each $(\\mu ,z)\\in \\mathbf {N}\\times \\mathbb {Y}_i$ .","We consider random elements $W_1,W_2$ of $\\mathbb {Y}_1$ and $\\mathbb {Y}_2$ , respectively, and assume that $\\xi $ , $W_1$ , and $W_2$ are independent.","Theorem 3.1 (BK inequality) Let $E_1\\in \\mathcal {N}\\otimes \\mathcal {Y}_1$ and $E_2\\in \\mathcal {N}\\otimes \\mathcal {Y}_2$ be increasing events that live on $\\Lambda \\times \\mathcal {E}$ for some bounded set $\\Lambda \\in \\mathcal {B}\\left(\\mathbb {R}^d\\right)$ .","Then $\\mathbb {P}_\\lambda ((\\xi ,W_1,W_2)\\in E_1 \\circ E_2) \\le \\mathbb {P}_\\lambda ((\\xi ,W_1)\\in E_1)\\mathbb {P}_\\lambda ((\\xi ,W_2)\\in E_2).$ A proof for this theorem can be found in [21].","The presence of the marked space $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ rather than just $\\mathbb {R}^d$ makes no difference to the validity of their proof.","It is worth noting that the operation $\\circ $ is commutative and associative, which allows repeated application of the inequality."],["The FKG Inequality","In contrast to the BK inequality, the FKG inequality gives us a simple lower bound on the probability of two increasing events occurring.","Given two increasing and integrable functions $f,g$ on $(\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1]$ , we have $\\mathbb {E}_\\lambda \\left[f(\\xi ) g(\\xi )\\right] = \\mathbb {E}_\\lambda \\left[ \\mathbb {E}_\\lambda [f(\\xi )g(\\xi ) \\mid \\eta ] \\right] \\ge \\mathbb {E}_\\lambda \\left[ \\mathbb {E}_\\lambda \\left[f(\\xi )\\mid \\eta \\right] \\; \\mathbb {E}_\\lambda \\left[g(\\xi )\\mid \\eta \\right] \\right]\\ge \\mathbb {E}_\\lambda \\left[f(\\xi )\\right] \\; \\mathbb {E}_\\lambda \\left[g(\\xi )\\right].","$ The first inequality was obtained by applying FKG to the random graph conditioned to have $\\eta $ as its vertex set, the second inequality by applying FKG for point processes (see [33] for details)."],["Linear Operator Lemmas","We shall formulate our problem as one of linear operators acting on Hilbert spaces.","We will want to define analogous objects on the full marked space $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ and on the mark space $\\mathcal {E}$ itself.","Let $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)\\in \\left\\lbrace \\left(\\mathbb {X},\\Sigma _\\mathbb {X},\\nu \\right),\\left(\\mathcal {E},\\Sigma _\\mathcal {E},\\mathcal {P}\\right)\\right\\rbrace $ , and $L^2\\left(\\mathcal {X}\\right)\\in \\left\\lbrace L^2\\left(\\mathbb {X}\\right),L^2\\left(\\mathcal {E}\\right)\\right\\rbrace $ correspondingly.","Given the measure space $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ , we consider the Banach space of square-integrable functions $L^2(\\mathcal {X}) = L^2\\left(\\mathcal {X},\\Sigma _{\\mathcal {X}},\\mu \\right)$ .","We augment this with the inner product defined by ${f}{g} := \\int \\overline{f\\left(x\\right)}g\\left(x\\right) \\mu \\left(\\mathrm {d}x \\right),$ for all $f,g\\in L^2(\\mathcal {X})$ , where $\\overline{f(x)}$ is the complex conjugate of $f(x)$ .","With this inner product, $L^2(\\mathcal {X})$ is a Hilbert space.","The measure space $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ is separable if there exists a countable family $\\left\\lbrace E_k\\right\\rbrace _{k=1}^\\infty \\subset \\Sigma _\\mathcal {X}$ such that given $E\\in \\Sigma _\\mathcal {X}$ with $\\mu (E)<\\infty $ there exists an $E$ -dependent subsequence $\\lbrace n_k\\rbrace _{k=1}^\\infty $ such that $\\mu \\left(E\\Delta E_{n_k}\\right) \\rightarrow 0, \\qquad \\text{ as }k\\rightarrow \\infty ,$ where $\\Delta $ here is the symmetric difference.","A well-known example of a separable measure space is $\\left(\\mathbb {R}^d,\\mathfrak {B},{\\rm Leb}\\right)$ - where $\\mathfrak {B}$ is the Borel $\\sigma $ -algebra - and therefore if the probability measure space $\\left(\\mathcal {E},\\Sigma _{\\mathcal {E}},\\mathcal {P}\\right)$ is separable then so is the product measure space $\\left(\\mathbb {X},\\Sigma _\\mathbb {X},\\nu \\right)$ .","A topological space is called separable if there exists a countable dense subset.","If the measure space $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ is separable, then the function spaces $L^p\\left(\\mathcal {X}\\right) = L^p\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ with $p\\in \\left[1,\\infty \\right)$ are all separable (see, for example, [42]).","Since we have assumed that $\\left(\\mathcal {E},\\Sigma _\\mathcal {E},\\mathcal {P}\\right)$ is separable, we know that both $L^2(\\mathbb {X})$ and $L^2(\\mathcal {E})$ are separable Hilbert spaces.","The inner product for $L^2(\\mathcal {X})$ allows us to define the $L^2$ -norm for $f\\in L^2(\\mathcal {X})$ : $*{f}_2 = {f}{f}^{\\frac{1}{2}}.$ This allows us to define a norm on the space of bounded linear operators on $L^2(\\mathcal {X})$ .","The space of bounded linear operators on a Banach space is itself a Banach space when augmented with the operator norm defined by $\\left\\Vert H\\right\\Vert _{\\rm op} := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{*{Hf}_2}{*{f}_2},$ for a linear operator $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ .","Since $H$ is linear, the supremum can equivalently be taken over $f\\in L^2(\\mathcal {X})$ such that $*{f}_2=1$ .","Note that $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ is sub-multiplicative, in the sense that for linear operators $G,H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ we have $\\left\\Vert GH\\right\\Vert _{\\rm op} \\le \\left\\Vert G\\right\\Vert _{\\rm op}\\left\\Vert H\\right\\Vert _{\\rm op}.$ Given a linear operator $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ on Hilbert spaces, the adjoint of $H$ is the operator $H^\\dagger \\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ such that ${f}{Hg} = {H^\\dagger f}{g}$ for all $f,g\\in L^2(\\mathcal {X})$ .","An operator is self-adjoint if $H=H^\\dagger $ .","It is worth noting that a bounded integral operator $H$ with kernel function $h$ is self-adjoint if and only if $h(x,y) = \\overline{h(y,x)}$ for $\\mu $ -a.e.","$x,y\\in \\mathcal {X}$ .","If $h$ is real valued, this is equivalent to $h$ being symmetric on such a set of $x,y$ .","A simple calculation shows that if $H$ is bounded and self-adjoint, then ${f}{Hf}\\in \\mathbb {R}$ for all $f\\in L^2(\\mathcal {X})$ .","This allows us to define the following object.","Given a bounded and self-adjoint linear $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ , define $\\mathbb {S}\\left(H\\right) := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{{f}{Hf}}{{f}{f}}.$ Note that, in contrast to $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ , the spectral supremum $\\mathbb {S}\\left(\\cdot \\right)$ is not a norm - it is possible for it to be negative.","The objects $\\left\\Vert H\\right\\Vert _{\\rm op}$ and $\\mathbb {S}\\left(H\\right)$ can also be understood via the spectrum of $H$ .","The spectrum, denoted $\\sigma (H)$ , is the set of $z\\in \\mathbb {C}$ such that $z1- H$ does not have a bounded linear inverse.","For linear operators on finite dimensional vector spaces this coincides with the set of eigenvalues of $H$ .","If $H$ is self-adjoint, then $\\sigma (H)\\subset \\mathbb {R}$ .","We can then express $\\left\\Vert H\\right\\Vert _{\\rm op}$ and $\\mathbb {S}\\left(H\\right)$ in terms of the spectrum: $\\left\\Vert H\\right\\Vert _{\\rm op} = \\sup \\left\\lbrace *{z}:z\\in \\sigma (H)\\right\\rbrace ,\\qquad \\mathbb {S}\\left(H\\right) = \\sup \\left\\lbrace z:z\\in \\sigma (H)\\right\\rbrace .$ Note that for self-adjoint operators the operator norm is equal to the spectral radius of the operator.","While $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ and $\\mathbb {S}\\left(\\cdot \\right)$ are very important properties of the operators we will be considering, they are not necessarily simple to compute for an integral operator given its kernel function.","Recall that in (REF )-() we defined the three integral operator norms $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ .","These have the advantage that they are explicitly defined in terms of the kernel function.","We now provide some results regarding the four norms (plus $\\mathbb {S}\\left(\\cdot \\right)$ ), including relations between some of them.","Lemma 3.2 (Bounds on the Operator Norm) Let $H\\colon L^2\\left(\\mathcal {X}\\right)\\rightarrow L^2\\left(\\mathcal {X}\\right)$ be a bounded self-adjoint integral operator.","Then we can bound the operator norm: $*{\\mathbb {S}\\left(H\\right)} \\le \\left\\Vert H\\right\\Vert _{\\rm op} \\le \\left\\Vert H\\right\\Vert _{1,\\infty }.$ Furthermore, if $\\mathcal {X}=\\mathcal {E}$ and $H$ has kernel function $h$ , then $\\int h\\left(a,b\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\mathbb {S}\\left(H\\right)$ The first inequality in (REF ) holds from the expressions of $\\mathbb {S}\\left(\\cdot \\right)$ and $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ in terms of the spectrum (see (REF )).","The boundedness and self-adjointness of $H$ implies that the second inequality follows from an application of the Schur Test and the conjugate symmetry of its kernel function (see, for example [24]).","For the lower bound on $\\mathbb {S}\\left(\\cdot \\right)$ , we use the definition (REF ) and consider the test function $f(a) = 1$ .","Lemma 3.3 Suppose $h,g\\colon \\mathcal {X}^2\\rightarrow \\mathbb {C}$ are kernel functions for the self-adjoint linear operators $H,G\\colon L^2\\left(\\mathcal {X}\\right) \\rightarrow L^2\\left(\\mathcal {X}\\right)$ , such that $h\\left(x,y\\right)\\in \\mathbb {R}_{\\ge 0}$ and $h\\left(x,y\\right) \\ge *{g\\left(x,y\\right)}$ for $\\mu $ -a.e.","$x,y\\in \\mathcal {X}$ .","Then $\\mathbb {S}\\left(H\\right)\\ge \\mathbb {S}\\left(G\\right),\\quad \\left\\Vert H\\right\\Vert _{\\rm op}\\ge \\left\\Vert G\\right\\Vert _{\\rm op}, \\quad \\left\\Vert H\\right\\Vert _{1,\\infty }\\ge \\left\\Vert G\\right\\Vert _{1,\\infty }, \\quad \\left\\Vert H\\right\\Vert _{2,\\infty }\\ge \\left\\Vert G\\right\\Vert _{2,\\infty }, \\quad \\left\\Vert H\\right\\Vert _{\\infty ,\\infty }\\ge \\left\\Vert G\\right\\Vert _{\\infty ,\\infty }.$ For the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ norms the inequality is clear from the definitions.","For the $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ version, let $f\\in L^2\\left(\\mathcal {X}\\right)$ , and use $\\rm Abs$ to denote the operator $f\\mapsto *{f}$ .","While this operator is not linear, it is clear that $*{{\\rm Abs}f}_2 = *{f}_2$ .","With this setup, $*{Gf}^2_2 = \\int *{\\int g\\left(x,y\\right) f\\left(y\\right) \\nu \\left(\\mathrm {d}y\\right)}^2\\nu \\left(\\mathrm {d}x\\right) \\le \\int \\left(\\int *{g\\left(x,y\\right)} *{f\\left(y\\right)} \\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\\\\\le \\int \\left(\\int h\\left(x,y\\right) *{f\\left(y\\right)} \\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) = *{H{\\rm Abs}f}^2_2.$ Therefore $\\left\\Vert G\\right\\Vert _{\\rm op} = \\sup _f\\frac{*{Gf}_2}{*{f}_2} \\le \\sup _f\\frac{*{H{\\rm Abs}f}_2}{*{f}_2} = \\sup _f\\frac{*{H{\\rm Abs}f}_2}{*{{\\rm Abs}f}_2}\\le \\sup _f\\frac{*{Hf}_2}{*{f}_2} =\\left\\Vert H\\right\\Vert _{\\rm op}.$ Here we used that the image of $\\rm Abs$ is a subset of $L^2\\left(\\mathcal {X}\\right)$ .","A similar argument works for the $\\mathbb {S}\\left(\\cdot \\right)$ inequality.","Given $f\\in L^2(\\mathcal {X})$ , $*{\\left} \\le \\int *{f(x)}*{g(x,y)}*{f(y)}\\nu ^{\\otimes 2}\\left(\\mathrm {d}x,\\mathrm {d}y\\right) \\le \\left<{\\rm Abs}f,H{\\rm Abs}f\\right>.$ Therefore $\\mathbb {S}\\left(G\\right) = \\sup _f\\frac{\\left}{*{f}^2_2} \\le \\sup _f\\frac{\\left<{\\rm Abs}f,H{\\rm Abs}f\\right>}{*{f}^2_2} = \\sup _f\\frac{\\left<{\\rm Abs}f,H{\\rm Abs}f\\right>}{*{{\\rm Abs}f}^2_2}\\le \\sup _f\\frac{\\left}{*{f}^2_2} =\\mathbb {S}\\left(H\\right).$ Lemma 3.4 Let $G,H$ be bounded self-adjoint linear operators on a separable Hilbert space.","Then $*{\\mathbb {S}\\left(G+H\\right) - \\mathbb {S}\\left(G\\right)} \\le \\left\\Vert H\\right\\Vert _{\\rm op}.$ It is a well-known result (for example see [29]) that $\\sigma (G+H)$ is contained in the closed $\\left\\Vert H\\right\\Vert _{\\rm op}$ -neighbourhood of $\\sigma (G)$ .","Perturbing $G$ by $+H$ , and perturbing $G+H$ by $-H$ then implies the result.","The main advantage of the Hilbert space formulation for the operators is that it allows us to use the following theorem to better understand the operators.","Although the “Multiplication Operator\" version presented here may not provide as much information as the “Projection-Valued Measure\" or “Direct Integral\" versions (see [19]), it has the advantage of being simpler to state and is sufficient for our purposes.","Theorem 3.5 (Spectral Theorem - Multiplication Operator Version) Let $\\left\\lbrace H_i\\right\\rbrace _{i\\in I}$ be a commutative family of bounded, self-adjoint operators on a separable Hilbert space $\\mathbb {H}$ .","Then there exists a $\\sigma $ -finite measure space $\\left(\\Omega ,\\Sigma _\\Omega , \\mu \\right)$ , a family of bounded, measurable, real-valued functions $\\left\\lbrace h_i\\right\\rbrace _{i\\in I}$ on $\\Omega $ , and a unitary map $U \\colon \\mathbb {H}\\rightarrow L^2\\left(\\Omega ,\\Sigma _\\Omega , \\mu \\right)$ such that for $\\mu $ -a.e.","$\\omega $ , $\\left(UH_iU^{-1}f\\right)\\left(\\omega \\right) = h_i\\left(\\omega \\right)f\\left(\\omega \\right)$ for all $f\\in L^2\\left(\\Omega ,\\Sigma _\\Omega .","\\mu \\right)$ and $i\\in I$ .","For a proof in more generality, see [12].","In the context of a finite dimensional Hilbert space $\\mathbb {H}$ , the Spectral Theorem is effectively just the statement that Hermitian matrices can be diagonalized over $\\mathbb {R}$ .","Here $U$ is represented by a unitary matrix composed from the orthonormal eigenbasis of $\\left\\lbrace H_i\\right\\rbrace $ , and the entries of the resulting diagonal matrix corresponding to $H_i$ are the values of the function $h_i(\\omega )$ .","One famous application of the Spectral Theorem for infinite dimensional Hilbert spaces is in the context of convolution operators on $\\mathbb {R}^d$ with Lebesgue measure.","Suppose the operator $G\\colon L^2\\left(\\mathbb {R}^d\\right) \\rightarrow L^2\\left(\\mathbb {R}^d\\right)$ acts on the function $f$ by $Gf(\\overline{x}) = \\int g(\\overline{x}-\\overline{y})f(\\overline{y}) \\mathrm {d}\\overline{y}$ for some function $g\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ .","If $g$ is integrable then Schur's test proves that $G$ is bounded, and if $g$ is reflection invariant (that is, $g(\\overline{x})=g(-\\overline{x})$ ) then expanding the inner product proves that $G$ is self-adjoint.","Supposing that these conditions hold, then the Fourier transform of $g$ is given by the bounded real function $\\widehat{g}\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\widehat{g}(k) = \\int \\text{e}^{i\\overline{x}\\cdot k}g(\\overline{x})\\mathrm {d}\\overline{x}.$ Furthermore, the Plancherel Theorem states that for a function $f\\in L^2\\left(\\mathbb {R}^d\\right)\\cap L^1\\left(\\mathbb {R}^d\\right)$ then the $L^2$ -norm of $f$ is equal to the $L^2$ -norm of $\\widehat{f}$ .","This - along with the clear linearity of the Fourier transform map - shows that the Fourier transform map can be considered a bounded linear operator $L^2\\left(\\mathbb {R}^d\\right) \\rightarrow L^2\\left(\\mathbb {R}^d\\right)$ .","The Fourier Inversion Theorem also relates the Fourier transform map to its inverse and its adjoint, and this can be used to show that the Fourier transform map is in fact a unitary linear operator.","The famous Convolution Theorem says that the Fourier transform of the convolution of two functions is the pointwise product of their individual Fourier transforms.","This means that $\\widehat{Gf}(k) = \\widehat{g}(k)\\widehat{f}(k).$ This can then be understood as an instance of the Spectral Theorem.","In this case, both the Hilbert spaces are $L^2\\left(\\mathbb {R}^d\\right)$ , the unitary map is (the inverse of) the Fourier transform map, and the function $h$ takes the values of the Fourier function, $\\widehat{g}$ , on $\\mathbb {R}^d$ .","Part of the power of the Fourier transform lies in that fact that this single unitary map simultaneously “diagonalizes\" a large class of operators.","The translation and reflection invariance of the Lebesgue measure on $\\mathbb {R}^d$ means that for integrable functions $g,h\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ , $\\int g(\\overline{x}-\\overline{y})h(\\overline{y})\\mathrm {d}\\overline{y}= \\int h(\\overline{x}-\\overline{u})g(\\overline{u})\\mathrm {d}\\overline{u}$ (using the substitution $\\overline{u}=\\overline{x}-\\overline{y}$ ).","Therefore the family of convolution operators described here form a commutative family of operators.","We also present here a lemma that relates the operator norm and spectral supremum for operators on $L^2(\\mathbb {X})$ to operators on $L^2(\\mathcal {E})$ .","Given a family $\\left\\lbrace A_k\\right\\rbrace _{k\\in \\mathbb {R}^d}$ of subsets of $\\mathbb {C}$ , we define the essential union of these sets to be ${\\rm ess.","}\\bigcup _{k\\in \\mathbb {R}^d}A_k := \\left\\lbrace z\\in \\mathbb {C}:\\forall \\epsilon >0, {\\rm Leb}\\left(k\\in \\mathbb {R}^d:\\exists x\\in A_k \\text{ s.t.","}*{x-z}<\\epsilon \\right)>0\\right\\rbrace .$ This can be understood as the union of $A_k$ over $k$ neglecting null sets of $k$ .","Lemma 3.6 We have $\\sigma \\left(\\right) = {\\rm ess.","}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{}(k)\\right) , \\qquad \\sigma \\left(\\mathcal {T}_\\lambda \\right) = {\\rm ess.","}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right).$ It then follows that $\\left\\Vert \\right\\Vert _{\\rm op} = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\left\\Vert \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op}, \\qquad \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op},$ $\\mathbb {S}\\left(\\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\mathbb {S}\\left(\\widehat{}(k)\\right), \\qquad \\mathbb {S}\\left(\\mathcal {T}_\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right).$ We present the proof of (REF ) for the connection operator case.","The two-point operator case follows similarly.","First suppose $\\zeta \\in {\\rm ess.","}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{}(k)\\right)$ .","Therefore for all $\\epsilon >0$ there exists $P_\\epsilon \\subset \\mathbb {R}^d$ such that ${\\rm Leb}\\left(P_\\epsilon \\right)>0$ and for all $k\\in P_\\epsilon $ there exists $z\\in \\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ such that $*{\\zeta -z}<\\epsilon $ .","Furthermore, given $z\\in \\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ there exists $g\\in L^2\\left(\\mathcal {E}\\right)$ with $*{g}_2=1$ such that $*{\\left(z1- \\widehat{}\\left(k\\right)\\right)g}_2<\\epsilon $ .","For all $k\\in P_\\epsilon $ we let $z^{(k)}$ and $g^{(k)}$ denote such choices.","Now let $\\widehat{f}$ be any element of $L^2\\left(\\mathbb {R}^d\\right)$ supported on $P_\\epsilon $ with $*{\\widehat{f}}_2=1$ .","Then define $h\\left(\\overline{x},a\\right) := \\int _{P_\\epsilon } \\text{e}^{-i k\\cdot \\overline{x}}\\widehat{f}(k)g^{(k)}(a)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}.$ By the Fourier Inversion Theorem, the Fourier transform of $h$ is given by $\\widehat{h}\\left(k,a\\right) = \\widehat{f}(k)g^{(k)}(a)$ for $k\\in P_\\epsilon $ and vanishes elsewhere.","By Plancherel's Theorem (that is, the Fourier transform is unitary), $h$ and $\\widehat{h}$ have the same $L^2$ -norm: $*{h}^2_2= *{\\widehat{h}}^2_2 = \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\left(\\int *{g^{(k)}(a)}^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\\\= \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2*{g^{(k)}}^2_2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} = \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} = *{\\widehat{f}}^2_2 = 1.$ By applying Plancherel's Theorem once again, we can write $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\int *{\\zeta g^{(k)}(a) - \\int \\widehat{\\varphi }(k;a,b)g^{(k)}(b)\\mathcal {P}\\left(\\mathrm {d}b\\right)}^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\\\= \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2*{\\left(\\zeta 1- \\widehat{}(k)\\right)g^{(k)}}^2_2 \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} < \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\left(\\epsilon + \\underbrace{*{\\left(z(k)1- \\widehat{}(k)\\right)g^{(k)}}_2}_{<\\epsilon }\\right)^2 \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} < 4\\epsilon ^2.$ Here we use the triangle inequality and $*{g^{(k)}}_2=1$ to replace $\\zeta $ with $z(k)$ at the cost of an $\\epsilon $ .","Since for each $\\epsilon >0$ we have constructed a unit $h\\in L^2\\left(\\mathbb {X}\\right)$ such that $*{\\left(\\zeta 1- \\right)h}_2<2\\epsilon $ , we have proven that $\\left(\\zeta 1- \\right)$ has no bounded linear inverse and therefore $\\zeta \\in \\sigma \\left(\\right)$ .","We now show the converse.","Suppose $\\zeta \\in \\sigma \\left(\\right)$ .","Then for all $\\epsilon >0$ there exists $h \\in L^2\\left(\\mathbb {X}\\right)$ such that $*{h}_2=1$ and $*{\\left(\\zeta 1- \\right)h}_2 < \\epsilon $ .","By using Plancherel's Theorem to replace the original functions with their Fourier transforms, we get $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int *{\\zeta h\\left(\\overline{x},a\\right) - \\int \\varphi \\left(\\overline{x}-\\overline{y};a,b\\right)h\\left(\\overline{y},b\\right)\\mathrm {d}\\overline{y}\\mathcal {P}\\left(\\mathrm {d}b\\right)}^2\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\= \\int *{\\zeta \\widehat{h}\\left(k,a\\right) - \\int \\widehat{\\varphi }\\left(k;a,b\\right)\\widehat{h}\\left(k,b\\right)\\mathcal {P}\\left(\\mathrm {d}b\\right)}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\mathcal {P}\\left(\\mathrm {d}a\\right).$ Now let us define the family of functions $g^{(k)}\\colon \\mathcal {E}\\rightarrow \\mathbb {C}$ by $g^{(k)}\\left(a\\right) := \\widehat{h}\\left(k,a\\right)$ for all $k\\in \\mathbb {R}^d$ .","Then we can again re-write the equality above as $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int *{\\left(\\zeta 1- \\widehat{}\\left(k\\right)\\right)g^{(k)}}^2_2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ For each $k\\in \\mathbb {R}^d$ , $\\widehat{}(k)$ is self-adjoint and therefore by the Spectral Theorem (Theorem REF ) there exists a Hilbert space $L^2\\left(\\mathfrak {E}_k,\\mu _k\\right)$ , a unitary operator $U_k\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathfrak {E}_k\\right)$ , and a measurable function $e\\mapsto \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)$ such that $\\widehat{}(k)$ is unitarily equivalent to pointwise multiplication by $\\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)$ on $L^2\\left(\\mathfrak {E}_k\\right)$ .","If we define $f^{\\left(k\\right)\\left(e\\right)} := \\left(U_kg^{\\left(k\\right)}\\right)\\left(e\\right)\\in \\mathbb {C}$ , then (REF ) becomes $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int \\left(\\int *{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}^2*{f^{(k)(e)}}^2\\mu _k\\left(\\mathrm {d}e\\right)\\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ Note that since the Fourier transform and the maps $\\left\\lbrace U_k\\right\\rbrace _k$ are all unitary, the normalisation of $h$ passes on to $g^{(k)}$ and on to $f^{(k)(e)}$ so that $\\int \\left(\\int *{f^{(k)(e)}}^2\\mu _k\\left(\\mathrm {d}e\\right)\\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}=1.$ Now suppose for contradiction that $*{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}\\ge \\epsilon $ for $\\mu _k$ -a.e.","$e\\in \\mathfrak {E}_k$ , for ${\\rm Leb}$ -a.e.","$k\\in \\mathbb {R}^d$ .","Then the normalisation of $f^{(k)(e)}$ implies that $\\int \\left(\\int *{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}^2*{f^{(k)(e)}}^2\\mu _k\\left(\\mathrm {d}e\\right)\\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\ge \\epsilon ^2.$ However this contradicts the condition that $*{\\left(\\zeta 1- \\right)h}_2 < \\epsilon $ .","Instead, for each $\\epsilon >0$ there exists $P_\\epsilon \\subset \\mathbb {R}^d$ with ${\\rm Leb}\\left(P_\\epsilon \\right)>0$ such that for all $k\\in P_\\epsilon $ there exists $Q^{(k)}_\\epsilon \\subset \\mathfrak {E}_k$ with $\\mu _k\\left(Q^{(k)}_\\epsilon \\right)>0$ such that for all $e\\in Q^{(k)}_\\epsilon $ we have $*{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}< \\epsilon $ .","For each $k\\in \\mathbb {R}^d$ , the operator $U_k\\widehat{}(k)U_k^{-1}$ is a multiplication operator on $L^2\\left(\\mathfrak {E}_k\\right)$ with function $e\\mapsto \\widetilde{\\varphi }(k)(e)$ , and therefore it is easy to calculate its spectrum.","Since the operator is unchanged by changing $\\widetilde{\\varphi }(k)(e)$ on a $\\mu _k$ -null set, the spectrum is given by the essential image of $\\widetilde{\\varphi }(k)$ : ${\\rm ess.Im}\\left(\\widetilde{\\varphi }\\left(k\\right)\\right) = \\left\\lbrace z\\in \\mathbb {C}:\\forall \\epsilon >0,\\mu _k\\left(e\\in \\mathfrak {E}_k:*{\\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)-z}>\\epsilon \\right)>0\\right\\rbrace .$ Since the unitary equivalence preserves the spectrum, this gives us an expression for $\\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ .","In particular, we can write our previous conclusion as the following.","For all $\\epsilon >0$ there exists $P_\\epsilon \\subset \\mathbb {R}^d$ with ${\\rm Leb}\\left(P_\\epsilon \\right)>0$ such that for all $k\\in P_\\epsilon $ there exists $z\\in \\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ such that $*{\\zeta -z}<\\epsilon $ .","Equivalently, this means that $\\zeta \\in {\\rm ess.","}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{}(k)\\right)$ .","Since $\\left\\Vert A\\right\\Vert _{\\rm op} = \\sup \\left\\lbrace *{z}:z\\in \\sigma \\left(A\\right)\\right\\rbrace $ and $\\mathbb {S}\\left(A\\right) = \\sup \\left\\lbrace z:z\\in \\sigma \\left(A\\right)\\subset \\mathbb {R}\\right\\rbrace $ for self-adjoint operator $A$ , the remaining equalities follow from (REF )."],["Differentiating the Two-Point Operator","The ideas of this section can be found in [21], but we will need extra care to account for the fact that we are dealing with operators and marks and not just functions on $\\mathbb {R}^d$ .","Recall $\\tau _\\lambda \\left(x,y\\right) := \\mathbb {P}_\\lambda (x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y})$ .","Using the notation $x = \\left(\\overline{x},a\\right)\\in \\mathbb {R}^d\\times \\mathcal {E}$ , define $\\Lambda _n\\left(x\\right) = \\left(\\overline{x} + \\left[-n,n\\right)^d\\right)\\times \\mathcal {E}$ .","Then we can define the truncated two-point function $\\tau _\\lambda ^n\\left(x,y\\right) := \\mathbb {P}_\\lambda \\left(x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}_{\\Lambda _n\\left(y\\right)}\\right).$ We will want to give meaning to the event $\\left\\lbrace x \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^x_{\\Lambda _n\\left(y\\right)}\\right\\rbrace $ .","To do so we add a “ghost vertex” $g$ in the same way we added deterministic vertices, and add an edge between $v \\in \\xi _{\\Lambda _n\\left(y\\right)}$ and $g$ with probability $1-\\exp \\left(-\\int _{\\Lambda _n^c\\left(y\\right)} \\varphi (u,v) \\nu \\left(\\mathrm {d}u\\right)\\right)$ .","We now identify $\\Lambda _n^c\\left(y\\right)$ with $g$ .","Lemma 3.7 (Differentiability of $\\tau _\\lambda $ ) Let $x,y\\in \\mathbb {X}$ and $\\varepsilon >0$ be arbitrary.","The function $\\lambda \\mapsto \\tau _\\lambda ^n(x,y)$ is differentiable on $[0, \\lambda _O-\\varepsilon ]$ for any $n\\in \\mathbb {N}$ .","Furthermore, $ \\tau _\\lambda ^n(x,y)$ converges to $\\tau _\\lambda (x,y)$ uniformly in $\\lambda $ and $\\tfrac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y)$ converges to a limit uniformly in $\\lambda $ .","Consequently, $\\tau _\\lambda (x,y)$ is differentiable w.r.t.","$\\lambda $ on $[0,\\lambda _O)$ and $\\lim _{n\\rightarrow \\infty } \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) = \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\tau _\\lambda (x,y) =\\int \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y,u}, y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y}) \\nu \\left(\\mathrm {d}u\\right).$ For $S \\subseteq \\mathbb {X}$ , let $\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y} \\textrm { through } S \\rbrace $ be the event that $x$ is connected to $b$ in $\\xi ^{x,y}$ , and all such paths use a vertex from $S$ .","Also, let $\\lbrace x \\longleftrightarrow S\\textrm { in } \\xi ^x\\rbrace $ be the event that there exists $y \\in \\eta ^x\\cap S$ that is connected to $x$ .","The convergence $\\tau _\\lambda ^n(x,y) \\rightarrow \\tau _\\lambda (x,y)$ is uniform for $\\lambda \\in [0,\\lambda _O-\\varepsilon ]$ , as $|\\tau _\\lambda (x,y) - \\tau _\\lambda ^n(x,y)| &= \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\textrm { through } \\Lambda _n^c\\left(y\\right)) \\\\& \\le \\mathbb {P}_{\\lambda _O-\\varepsilon }(y \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{y})\\xrightarrow{} 0.$ We now go about showing $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) \\xrightarrow{} f_\\lambda (x,y) := \\int \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x,}, y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}) \\nu \\left(\\mathrm {d}u\\right) $ uniformly in $\\lambda $ .","Note that $\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}_{\\Lambda _n\\left(y\\right)}\\rbrace $ lives on the bounded set $\\Lambda _n\\left(y\\right)$ , and so the the Margulis-Russo formula gives us differentiability and the following explicit expression for the derivative $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) = \\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda \\left( y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x}_{\\Lambda _n\\left(y\\right)}, y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}_{\\Lambda _n\\left(y\\right)} \\right) \\nu \\left(\\mathrm {d}u\\right).","$ Using this we can write $\\left|\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) - f_\\lambda (x,y)\\right|&= \\left|\\int _{\\Lambda _n} \\Big ( \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x}) -\\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}) \\right.\\nonumber \\\\& \\qquad \\qquad -\\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x}_{\\Lambda _n\\left(y\\right)}) +\\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}_{\\Lambda _n\\left(y\\right)}) \\Big ) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& \\qquad \\qquad \\qquad \\left.+ \\int _{\\Lambda _n^c\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x,}, y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}) \\nu \\left(\\mathrm {d}u\\right) \\right|\\\\& \\le \\left|\\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x} \\textrm { through } \\Lambda _n^c\\left(y\\right))\\right.\\nonumber \\\\&\\qquad \\qquad - \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\textrm { through } \\Lambda _n^c\\left(y\\right)) \\nu \\left(\\mathrm {d}u\\right) \\bigg \\vert \\nonumber \\\\& \\qquad \\qquad \\qquad + \\int _{\\Lambda _n^c\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow u\\textrm { in } \\xi ^{y,u} ) \\nu \\left(\\mathrm {d}u\\right) \\\\& \\le \\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x} \\textrm { through } \\Lambda _n^c\\left(y\\right) \\text{ and through } u) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& \\qquad \\qquad + \\int _{\\Lambda _n^c\\left(y\\right)} \\tau _{\\lambda _O-\\varepsilon }(u,y) \\nu \\left(\\mathrm {d}u\\right) .$ We now have two main observations.","First, that the event $\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x} \\textrm { through } \\Lambda _n^c\\left(y\\right) \\text{ and through } u\\rbrace $ is contained in $& \\left( \\lbrace y \\longleftrightarrow u\\textrm { in } \\xi ^{y,u}\\rbrace \\circ \\lbrace u \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{u}\\rbrace \\circ \\lbrace \\Lambda _n^c\\left(y\\right) \\longleftrightarrow x\\textrm { in } \\xi ^{x}\\rbrace \\right) \\nonumber \\\\&\\qquad \\cup \\left( \\lbrace y \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{y}\\rbrace \\circ \\lbrace \\Lambda _n^c\\left(y\\right) \\longleftrightarrow u\\textrm { in } \\xi ^{u}\\rbrace \\circ \\lbrace u \\longleftrightarrow x\\textrm { in } \\xi ^{u,x}\\rbrace \\right).$ Secondly, from $\\lambda <\\lambda _O$ we have $\\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} <\\infty .$ This means that for $\\nu $ -almost every $y\\in \\mathbb {X}$ we have $\\int _{\\Lambda _n^c\\left(y\\right)} \\tau _{\\lambda _O-\\varepsilon }(u,y) \\nu \\left(\\mathrm {d}u\\right)\\rightarrow 0$ as $n\\rightarrow \\infty $ .","Applying the BK inequality then gives $\\left|\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) - f_\\lambda (x,y)\\right|& \\le \\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda (u \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^u) \\Big [\\tau _\\lambda (u,y)\\mathbb {P}_\\lambda (x \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^x) \\nonumber \\\\&\\qquad \\qquad + \\tau _\\lambda (x,u)\\mathbb {P}_\\lambda (y \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{y}) \\Big ] \\nu \\left(\\mathrm {d}u\\right) +o(1) \\\\& \\le 2 \\max _{z\\in \\lbrace y,x\\rbrace } \\mathbb {P}_{\\lambda _O-\\varepsilon }\\big (z \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^z\\big ) \\int \\tau _{\\lambda _O-\\varepsilon }(u,y) \\nu \\left(\\mathrm {d}u\\right) + o(1).$ The remaining integral is again finite, and the remaining probability tends to zero uniformly in $\\lambda $ .","The uniform convergence justifies the exchange of limit and derivative in (REF ).","A family of operators $\\left\\lbrace H_x\\right\\rbrace _{x\\in \\mathbb {R}}$ is differentiable at $x$ if there exists an operator $F_x$ such that $\\lim _{\\varepsilon \\rightarrow 0}\\left\\Vert \\frac{1}{\\varepsilon }\\left(H_{x+\\varepsilon }-H_x\\right) - F_x\\right\\Vert _{\\rm op} = 0.$ Using the notation $x = \\left(\\overline{x},a\\right)\\in \\mathbb {R}^d\\times \\mathcal {E}$ , recall that we require that the model has $\\mathbb {R}^d$ -translation invariance and so we can write $\\tau _\\lambda \\left(x,y\\right) = \\tau _\\lambda \\left(\\overline{x}-\\overline{y};a,b\\right)$ .","Also recall that we defined the Fourier transform $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel $\\widehat{\\tau }_\\lambda \\left(k;a,b\\right) = \\int \\cos \\left(k\\cdot \\overline{x}\\right)\\tau _\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}$ .","We now define the displaced Fourier transform $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel $\\widehat{\\tau }_{\\lambda ,k}\\left(l;a,b\\right) = \\int \\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\cos \\left(l\\cdot \\overline{x}\\right)\\tau _\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}$ .","Corollary 3.8 (Differentiability of $\\mathcal {T}_\\lambda $ ) The operators $\\mathcal {T}_\\lambda $ , $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)$ , and $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)$ are differentiable w.r.t.","$\\lambda $ on $\\left(0,\\lambda _O\\right)$ and their derivatives are the bounded linear operators $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ , $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ , and $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ with kernel functions $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\tau _\\lambda \\left(x,y\\right)$ , $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\tau }_\\lambda \\left(k;a,b\\right)$ , and $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\tau }_{\\lambda ,k}\\left(l;a,b\\right)$ respectively.","Specifically, we have the following bounds on the derivatives: $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} &\\le \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}^2,\\\\\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} &\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2,\\\\\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} &\\le 4\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}.","$ We use Lemma REF in conjunction with the Leibniz integral rule to differentiate the two-point operator.","Fix $f\\in L^2\\left(\\mathbb {X}\\right)$ and let $\\varepsilon \\in \\left(0,\\lambda _O\\right)$ .","From the reasoning of (REF ) we know that the subcritical integral $\\int \\tau _{\\lambda _O-\\varepsilon }(\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{(\\otimes 2)}\\left(\\mathrm {d}a,\\mathrm {d}b\\right)<\\infty $ , and therefore $\\int \\tau _{\\lambda _O-\\varepsilon }\\left(x,u\\right)^2\\nu \\left(\\mathrm {d}u\\right) \\le \\int \\tau _{\\lambda _O-\\varepsilon }\\left(x,u\\right)\\nu \\left(\\mathrm {d}u\\right) < \\infty ,$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .","Here we have also used that $\\tau _\\lambda (x,u)$ is $\\left[0,1\\right]$ -valued.","This implies that $\\tau _{\\lambda _O-\\varepsilon }\\left(x,\\cdot \\right)\\in L^2\\left(\\mathbb {X}\\right)$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .","In conjunction with $f\\in L^2\\left(\\mathbb {X}\\right)$ , Hölder's inequality implies that the function $\\Theta _{\\varepsilon ,x}\\colon y\\mapsto \\tau _{\\lambda _O-\\varepsilon }\\left(x,y\\right)*{f\\left(y\\right)}$ is in $L^1\\left(\\mathbb {X}\\right)$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .","Furthermore, $\\tau _\\lambda \\left(x,y\\right)$ is non-decreasing in $\\lambda $ and so for all $\\lambda <\\lambda _O$ there exists $\\varepsilon >0$ such that $\\Theta _{\\varepsilon ,x}$ dominates $y\\mapsto \\tau _\\lambda \\left(x,y\\right)f\\left(y\\right)$ .","This domination by an $L^1\\left(\\mathbb {X}\\right)$ function allows us to use the Leibniz integral rule to exchange the integral and derivative to get (for $\\nu $ -almost every $x\\in \\mathbb {X}$ ) $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\left(\\left[\\mathcal {T}_\\lambda f\\right]\\left(x\\right)\\right) = \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\int \\tau _\\lambda \\left(x,u\\right)f\\left(u\\right)\\nu \\left(\\mathrm {d}u\\right) = \\int \\left(\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda \\left(x,u\\right)\\right)f\\left(u\\right)\\nu \\left(\\mathrm {d}u\\right) = \\left[\\left(\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right)f\\right]\\left(x\\right),$ where Lemma REF gives the differentiability of the function $\\tau _\\lambda \\left(x,y\\right)$ .","We now prove that the operator $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda $ is bounded.","From Lemma REF and the BK inequality, we get the bound $0\\le \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) \\le \\int \\mathbb {P}_\\lambda \\left( \\lbrace y \\longleftrightarrow u\\textrm { in } \\xi ^{y,u}_{\\Lambda _n\\left(y\\right)} \\rbrace \\circ \\lbrace u \\longleftrightarrow x\\textrm { in } \\xi ^{u,x}_{\\Lambda _n\\left(y\\right)} \\rbrace \\right) \\nu \\left(\\mathrm {d}u\\right) \\\\\\le \\int \\tau _\\lambda ^n(u,y)\\mathbb {P}_\\lambda \\left(u \\longleftrightarrow x\\textrm { in } \\xi ^{u,x}_{\\Lambda _n\\left(y\\right)}\\right) \\nu \\left(\\mathrm {d}u\\right) \\le \\int \\tau _\\lambda \\left(u,y\\right)\\tau _\\lambda \\left(x,u\\right)\\nu \\left(\\mathrm {d}u\\right),$ where the last inequality holds because we are removing the $\\Lambda _n\\left(y\\right)$ restriction.","Note that the right hand side of this equation is the kernel function of the operator $\\mathcal {T}_\\lambda ^2$ .","Choose $f\\in L^2\\left(\\mathbb {X}\\right)$ , and use $\\rm Abs$ to denote the map $f\\mapsto *{f}$ .","Now the above inequality implies that $*{\\left[\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right] f}^2_2 & \\le \\int \\left(\\int *{\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\tau _\\lambda \\left(x,y\\right)}*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2 \\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\& \\le \\int \\left(\\int \\int \\tau _\\lambda \\left(x,u\\right)\\tau _\\lambda \\left(u,y\\right)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}u\\right)\\nu \\left(\\mathrm {d}y\\right)\\right)^2 \\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\int \\left(\\left[\\mathcal {T}_\\lambda ^2 {\\rm Abs}{f}\\right]\\left(x\\right)\\right)^2 \\nu \\left(\\mathrm {d}x\\right) = *{\\mathcal {T}_\\lambda ^2 {\\rm Abs}f}^2_2.$ Since $*{{\\rm Abs}f}_2 = *{f}_2$ , this then gives us $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} \\le \\sup _f\\frac{*{\\mathcal {T}_\\lambda ^2 {\\rm Abs}f}_2}{*{f}_2} = \\sup _f\\frac{*{\\mathcal {T}_\\lambda ^2 {\\rm Abs}f}_2}{*{{\\rm Abs}f}_2}\\le \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\rm op} = \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}^2 < \\infty .$ Now we address the Fourier transform.","Pick $\\varepsilon \\in (0,\\lambda _O-\\lambda )$ .","Recall $*{\\text{e}^{ik\\cdot \\overline{x}}\\tau _\\lambda \\left(\\overline{x};a,b\\right)}$ is bounded by $\\tau _{\\lambda _O-\\varepsilon }\\left(\\overline{x};a,b\\right)$ for all $\\overline{x}\\in \\mathbb {R}^d$ and $a,b\\in \\mathcal {E}$ .","From the reasoning of (REF ), we know that $\\tau _{\\lambda _O-\\varepsilon }\\left(\\cdot ;a,b\\right)$ is Lebesgue integrable for $\\mathcal {P}$ -a.e.","$a,b\\in \\mathcal {E}$ .","We are then justified in using the Leibniz integral rule to say that for every $b\\in \\mathcal {E}$ and $\\mathcal {P}$ -a.e.","$a\\in \\mathcal {E}$ , $\\tfrac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\int \\text{e}^{ik\\cdot \\overline{x}} \\tau _\\lambda (\\overline{x};a,b) \\mathrm {d}\\overline{x}= \\int \\text{e}^{ik\\cdot \\overline{x}}\\tfrac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda (\\overline{x};a,b) \\mathrm {d}\\overline{x}$ .","Applying Lemma REF as well as (REF ), we derive that for such $a,b$ we have $*{\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\widehat{\\tau }_\\lambda (k;a,b)} \\le \\int \\lim _{n\\rightarrow \\infty } \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(\\overline{x};a,b) \\mathrm {d}\\overline{x}\\le \\int \\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) \\,\\mathrm {d}\\overline{x}\\,\\mathrm {d}\\overline{y}\\,\\mathcal {P}\\left(\\mathrm {d}c \\right).$ Note that the right hand side of this equation is the kernel function of the operator $\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)^2$ .","By repeating the argument for $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda $ - with $\\mathbb {X}$ replaced by $\\mathcal {E}$ - we similarly arrive at $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2.$ For the displaced Fourier transform, note that $0\\le 1-\\cos \\left(k\\cdot \\overline{x}\\right)\\le 2$ , and so the same argument as for the Fourier transform allows us to use the Leibniz integral rule to exchange derivative and integral.","We can also improve on the immediate bound of $2\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2$ for the operator norm of the derivative.","Let $f\\in L^2\\left(\\mathcal {E}\\right)$ , $*{\\left[\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right]f}^2_2 &\\le \\int \\left(\\int *{\\text{e}^{il\\cdot \\overline{x}}}\\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) *{f\\left(b\\right)}\\mathrm {d}\\overline{x}\\,\\mathrm {d}\\overline{y}\\,\\mathcal {P}\\left(\\mathrm {d}c \\right)\\mathcal {P}\\left(\\mathrm {d}b\\right)\\right)^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\& \\le \\int \\left(\\int 2\\Big [\\left(1 - \\cos \\left(k\\cdot \\left(\\overline{x}-\\overline{y}\\right)\\right)\\right)\\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) \\right.","\\nonumber \\\\&\\qquad + \\left(1 - \\cos \\left(k\\cdot \\overline{y}\\right)\\right)\\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) \\Big ]*{f\\left(b\\right)}\\,\\mathrm {d}\\overline{x}\\,\\mathrm {d}\\overline{y}\\,\\mathcal {P}\\left(\\mathrm {d}c \\right)\\mathcal {P}\\left(\\mathrm {d}b\\right)\\Big )^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\& = 4 *{\\left[\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) + \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)\\right]{\\rm Abs} f}^2_2,$ where we have used the cosine-splitting lemma (Lemma REF ) to divide the displacement factor $\\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)$ into one spanning $\\overline{x}-\\overline{y}$ and one spanning $\\overline{y}$ .","Using the above approach of (REF ) and the triangle inequality then produces the required bound.","This is the cosine-splitting lemma referenced in the previous proof.","Lemma 3.9 (Split of cosines, [10]) Let $t_i \\in \\mathbb {R}$ for $i=1, \\ldots , m$ and $t = \\sum _{i=1}^{m} t_i$ .","Then $1-\\cos (t) \\le m \\sum _{i=1}^{m} [1- \\cos (t_i)].","$ We use the following lemma in the proof of Proposition REF to bound the displacement on the two-point operator using the displacement of the connection operator when working in the sub-critical regime away from the critical threshold.","Lemma 3.10 For $\\lambda <\\lambda _O$ , $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\text{e}^{4\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}}\\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op}.$ For the inequality removing $l$ , let $f\\in L^2\\left(\\mathcal {E}\\right)$ .","Then $*{\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right)f}^2_2 &\\le \\int \\left(\\int *{\\text{e}^{il\\cdot \\overline{x}}}\\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\tau _\\lambda (\\overline{x};a,b) *{f\\left(b\\right)}\\,\\mathrm {d}\\overline{x}\\, \\mathcal {P}\\left(\\mathrm {d}b\\right)\\right)^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\& = *{\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right){\\rm Abs} f}^2_2.$ Since $*{{\\rm Abs}f}_2=*{f}_2$ , we can use the approach of (REF ) to get the first inequality.","For the second inequality, we first note that $\\widehat{\\mathcal {T}}_0\\left(k\\right) = \\widehat{}\\left(k\\right)$ .","We then use the differential inequality () to extend to $\\lambda <\\lambda _O$ .","Using the reverse triangle inequality, we have $\\limsup _{\\varepsilon \\downarrow 0}\\frac{1}{\\varepsilon }*{\\left\\Vert \\widehat{\\mathcal {T}}_{s+\\varepsilon }\\left(0\\right) - \\widehat{\\mathcal {T}}_{s+\\varepsilon }\\left(k\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right) - \\widehat{\\mathcal {T}}_{s}\\left(k\\right)\\right\\Vert _{\\rm op}} &\\le \\left\\Vert \\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\left(\\widehat{\\mathcal {T}}_{\\lambda }\\left(0\\right) - \\widehat{\\mathcal {T}}_{\\lambda }\\left(k\\right)\\right)\\right|_{\\lambda =s}\\right\\Vert _{\\rm op} \\nonumber \\\\&\\le 4 \\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right) - \\widehat{\\mathcal {T}}_{s}\\left(k\\right)\\right\\Vert _{\\rm op}.$ Since $\\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right)\\right\\Vert _{\\rm op}$ is monotone increasing for $s \\in \\left[0,\\lambda \\right]$ , the right hand side of the inequality is bounded by $4 \\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda }\\left(0\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right) - \\widehat{\\mathcal {T}}_{s}\\left(k\\right)\\right\\Vert _{\\rm op}$ , giving the exponential factor in the result."],["The Expansion","There are two parts to this section.","First we perform an expansion of the two-point function.","This argument is essentially identical to the corresponding step in [21], so we refer to this reference for most of the details.","The novelty in this section is rather in the formulation of this expansion in terms of operators, contained in the second part."],["Function-level expansion","The expansion of the two-point function $\\tau _\\lambda $ proceeds in essentially the same way as in [21].","The similarity holds because the argument uses quite general properties of Poisson point processes and connection models - the change from $\\mathbb {R}^d$ to $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ adds no further complication.","We give here a very brief overview of the derivation - the details can be found in that reference.","We first introduce some notation.","It will be important to consider thinning events and pivotal points.","Definition 4.1 Let $x,y \\in \\mathbb {X}$ , and let $A \\subset \\mathbb {X}$ be locally finite and of cardinality ${A}$ .","(1) Set $\\bar{\\varphi }(A,x) :=\\prod _{y\\in A}(1-\\varphi (x,y)) $ and define $\\eta _{\\langle A \\rangle }$ as a $\\bar{\\varphi }(A, \\cdot )$ -thinning of $\\eta $ (or simply $A$ -thinning of $\\eta $) as follows.","We keep a point $w \\in \\eta $ as a point of $\\eta _{\\langle A \\rangle }$ with probability $\\bar{\\varphi }(A,w)$ independently of all other points of $\\eta $ .","We similarly define $\\eta _{\\langle A \\rangle }^x$ as a $\\bar{\\varphi }(A,\\cdot )$ -thinning of $\\eta ^x$ using the marks in $\\xi ^x$ .","(2) We write $x {\\,\\,A\\,\\,} y\\textrm { in } \\xi $ if both $\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi \\rbrace $ and $\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi [\\eta _{\\langle A \\rangle } \\cup \\lbrace x\\rbrace ] \\rbrace $ take place.","In words, $\\lbrace x {\\,\\,A\\,\\,} y\\textrm { in } \\xi \\rbrace $ is the event that $x,y\\in \\eta $ and $x$ is connected to $y$ in $\\xi $ , but that this connection is broken by an $A$ -thinning of $\\eta \\setminus \\lbrace x\\rbrace $ .","In particular, the connection does not survive if $y$ is thinned out.","(3) We define $\\tau _\\lambda ^{A}(x,y) = \\mathbb {P}_\\lambda \\left(y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y}[\\eta ^y_{\\langle A \\rangle }\\cup \\lbrace x\\rbrace ] \\right).","$ In words, $\\tau _\\lambda ^A(x,y)$ is the probability of the event that there exists an open path between $x$ and $y$ in an RCM driven by an $A$ -thinning of $\\eta ^y$ , where the point $x$ is fixed to be present (but $y$ is not).","(4) Given $x,y\\in \\mathbb {X}$ and edge-marking $\\xi $ , we say $u\\in \\mathbb {X}$ is pivotal and $u\\in \\textsf {Piv}(y,x,\\xi )$ if $\\left\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ and yet $\\left\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi \\left[\\eta \\setminus \\left\\lbrace u\\right\\rbrace \\right]\\right\\rbrace $ .","That is, every path on $\\xi ^{x,y}$ connecting $x$ and $y$ uses the vertex $u$ .","Note that the end points $x$ and $y$ are never pivotal.","(5) We will also use the set $E(x,y;A,\\xi ) := \\lbrace x {\\,\\,A\\,\\,} y\\textrm { in } \\xi \\rbrace \\cap \\lbrace \\nexists w \\in \\textsf {Piv}(x,y;\\xi ): x {\\,\\,A\\,\\,} w\\textrm { in } \\xi \\rbrace , $ for a locally finite set $A \\subset \\mathbb {X}$ , and $x,y\\in \\mathbb {R}^d$ .","If we consider the pivotal points of the $x$ to $y$ connection in $\\xi $ in sequence, then this is the event that an $A$ -thinning breaks the connection after the last pivotal point, but not before.","Now we can state a continuum version of a standard lemma often used in discrete models - see, for example, [25].","In bond percolation, it has the name “Cutting-bond lemma”.","The proof of this continuum version can be found in [21].","Lemma 4.2 (Cutting-point lemma) Let $\\lambda \\ge 0$ and let $v,u,x \\in \\mathbb {X}$ with $u \\ne x$ and let $A\\subset \\mathbb {X}$ be locally finite.","Then $\\mathbb {E}_\\lambda \\big [1_{E(v,u;A,\\xi ^{v,u,x})} 1_{\\lbrace u \\in \\textsf {\\textup {Piv}}(v,x;\\xi ^{v,u,x})\\rbrace } \\big ]= \\mathbb {E}_\\lambda \\left[1_{E(v,u;A,\\xi ^{v,u})} \\cdot \\tau _\\lambda ^{v,\\xi ^v) }(x,u) \\right].$ Moreover, $\\mathbb {P}_\\lambda \\left(y \\Longleftrightarrow u\\textrm { in } \\xi ^{y,u,x}, u \\in \\textsf {\\textup {Piv}}(y,x; \\xi ^{y,u,x})\\right) = \\mathbb {E}_\\lambda \\left[1_{\\lbrace y \\Longleftrightarrow u\\textrm { in } \\xi ^{y,u}\\rbrace } \\cdot \\tau _\\lambda ^{y,\\xi ^{y}) }(x,u) \\right].$ We now define the lace expansion coefficient function.","Definition 4.3 (Lace-expansion function coefficients) For $n\\in \\mathbb {N}$ and $x,y\\in \\mathbb {X}$ , we define $\\pi _\\lambda ^{(0)}(x,y) &:= \\mathbb {P}_\\lambda \\left(y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) - \\varphi (x,y), \\\\\\pi _\\lambda ^{(n)}(x,y) &:= \\lambda ^n \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y \\Longleftrightarrow u_0\\textrm { in } \\xi ^{y, u_0}_{0}\\right\\rbrace \\cap \\bigcap _{i=1}^{n} E\\left(u_{i-1},u_i; {i-1}, \\xi ^{u_{i-1}, u_i}_{i}\\right) \\right) \\nu ^{\\otimes n}\\left(\\mathrm {d}\\vec{u}_{[0,n-1]}\\right) , $ where $u_n=x$ and ${i} = u_{i-1}, \\xi ^{u_{i-1}}_{i})$ is the cluster of $u_{i-1}$ in $\\xi ^{u_{i-1}}_i$ .","Further define the remainder functions $r_{\\lambda , 0} (x,y) &:= - \\lambda \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y \\Longleftrightarrow u_0\\textrm { in } \\xi ^{y, u_0}_0\\right\\rbrace \\cap \\left\\lbrace u_0 {\\,\\,0\\,\\,} x\\textrm { in } \\xi ^{u_0,x}_1\\right\\rbrace \\right) \\nu \\left(\\mathrm {d}u_0\\right), \\\\r_{\\lambda , n}(x,y) &:= (-\\lambda )^{n+1} \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y \\Longleftrightarrow u_0\\textrm { in } \\xi ^{y, u_0}_0\\right\\rbrace \\cap \\bigcap _{i=1}^{n} E\\left(u_{i-1},u_i; {i-1}, \\xi ^{u_{i-1}, u_i}_{i}\\right)\\right.","\\nonumber \\\\& \\hspace{170.71652pt}\\left.\\cap \\left\\lbrace u_n {\\,\\,n\\,\\,} x\\textrm { in } \\xi ^{u_n,x}_{n+1} \\right\\rbrace \\right) \\nu ^{\\otimes \\left(n+1\\right)}\\left(\\mathrm {d}\\vec{u}_{[0,n]}\\right).$ Additionally, define $\\pi _{\\lambda , n}$ as the alternating partial sum $\\pi _{\\lambda , n}(x,y) := \\sum _{m=0}^{n} (-1)^m \\pi _\\lambda ^{(m)}(x,y).","$ We will also be requiring Fourier transforms of these functions.","Since the whole model is spatially translation invariant, so are the functions defined above.","Therefore for each pair of marks $(a,b)$ we perform the Fourier transform on the spatial displacement to get the functions $\\widehat{\\pi }^{(n)}_\\lambda (k;a,b)$ , $\\widehat{\\pi }_{\\lambda ,n}(k;a,b)$ , and $\\widehat{r}_{\\lambda ,n}(k;a,b)$ for $n\\ge 0$ .","Proposition 4.4 (Lace expansion: function-level) Let $\\lambda \\in [0,\\lambda _O)$ and $n\\ge 0$ .","Then for $x,y\\in \\mathbb {X}$ , and for $k\\in \\mathbb {R}^d$ and $a,b\\in \\mathcal {E}$ , $\\tau _\\lambda \\left(x,y\\right) &= \\varphi \\left(x,y\\right) + \\pi _{\\lambda , n}\\left(x,y\\right) + \\lambda \\int \\tau _\\lambda \\left(x,u\\right)\\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right) + r_{\\lambda , n}\\left(x,y\\right), \\\\\\widehat{\\tau }_\\lambda \\left(k;a,b\\right) &= \\widehat{\\varphi }\\left(k;a,b\\right) + \\widehat{\\pi }_{\\lambda , n}\\left(k;a,b\\right) + \\lambda \\int \\widehat{\\tau }_\\lambda \\left(k;a,c\\right)\\left(\\widehat{\\varphi }+ \\widehat{\\pi }_{\\lambda , n}\\right)\\left(k;c,b\\right)\\mathcal {P}\\left(\\mathrm {d}c\\right) + \\widehat{r}_{\\lambda , n}\\left(k;a,b\\right).","$ We outline the proof of (REF ) here.","The equation () then follows by applying the Fourier transform.","The details of the proof of (REF ) - with appropriate contextual changes - can be found in [21].","Firstly, the definitions give $\\tau _\\lambda (x,y) = \\varphi (x,y) + \\pi _\\lambda ^{(0)}(x,y) + \\mathbb {P}_\\lambda \\left(y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}, y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right).","$ With an appropriate event partition, Mecke's equation and the Cutting-point lemma allow us to re-write the probability term as $\\mathbb {P}_\\lambda \\left(y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}, y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) = \\lambda \\int \\mathbb {E}_\\lambda \\left[ 1_{\\lbrace y \\Longleftrightarrow u\\textrm { in } \\xi ^{y,u}\\rbrace } \\tau _\\lambda ^{0}(x,u) \\right] \\mathrm {d}u$ Now we use the identity $\\tau _\\lambda ^{A}(x,u) = \\tau _\\lambda (x,u) - \\mathbb {P}_\\lambda \\left( u {\\,\\,A\\,\\,} x\\textrm { in } \\xi ^{u,x} \\right), $ to extract the deterministic function $\\tau _\\lambda (x,u)$ from the expectation at the cost of an extra term.","The $\\tau _\\lambda $ term then gives us the second $\\pi ^{(0)}_\\lambda $ term and the extra term becomes the remainder $r_{\\lambda ,0}$ .","Repeated use of (REF ) with Mecke's equation and the Cutting-point lemma gives an alternating series that produces the various $\\pi _\\lambda ^{(n)}$ and $r_{\\lambda ,n}$ terms.","Operator-level expansion Definition 4.5 (Lace-expansion operator coefficients) For $n\\in \\mathbb {N}$ , we define $_\\lambda ^{(n)}\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator with kernel function $\\pi ^{(n)}_\\lambda $ , and $R_{\\lambda ,n}\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator with kernel function $r_{\\lambda ,n}$ .","We also define $_{\\lambda ,n}\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator with kernel function $\\pi _{\\lambda , n}$ .","It follows that $_{\\lambda , n} = \\sum _{m=0}^{n} (-1)^m _\\lambda ^{(m)}.$ We also use the kernel functions $\\widehat{\\pi }^{(n)}_\\lambda (k;a,b)$ , $\\widehat{\\pi }_{\\lambda ,n}(k;a,b)$ , and $\\widehat{r}_{\\lambda ,n}(k;a,b)$ to define the operators $\\widehat{}^{(n)}_{\\lambda }(k),\\widehat{}_{\\lambda ,n}(k),\\widehat{R}_{\\lambda ,n}(k)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ respectively for each $k\\in \\mathbb {R}^d$ .","Proposition 4.6 (Lace expansion: operator-level) Let $\\lambda \\in [0,\\lambda _O)$ .","Then for $n \\ge 0$ and $k\\in \\mathbb {R}^d$ , $\\mathcal {T}_\\lambda &= + _{\\lambda , n} + \\lambda \\mathcal {T}_\\lambda \\left(+ _{\\lambda , n}\\right) + R_{\\lambda , n},\\\\\\widehat{\\mathcal {T}}_\\lambda (k) &= \\left(\\widehat{}+ \\widehat{}_{\\lambda , n}\\right)(k) + \\lambda \\widehat{\\mathcal {T}}_\\lambda (k)\\left(\\widehat{}+ \\widehat{}_{\\lambda , n}\\right)(k) + \\widehat{R}_{\\lambda , n}(k).$ Let $f\\in L^2\\left(\\mathbb {X}\\right)$ be a test function.","Then Proposition REF gives for $x\\in \\mathbb {X}$ , $\\mathcal {T}_\\lambda f\\left(x\\right) &= \\int \\tau _\\lambda \\left(x,y\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right)\\nonumber \\\\& = \\int \\left(\\varphi \\left(x,y\\right) + \\pi _{\\lambda , n}\\left(x,y\\right) + \\lambda \\int \\tau _\\lambda \\left(x,u\\right)\\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right) + r_{\\lambda , n}\\left(x,y\\right)\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right)\\nonumber \\\\& = \\left(+ _{\\lambda , n} + \\lambda \\mathcal {T}_\\lambda \\left(+ _{\\lambda , n}\\right) + R_{\\lambda , n}\\right)f\\left(x\\right).$ Note that in this last equality we used $\\int \\int \\tau _\\lambda \\left(x,u\\right)\\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right) &= \\int \\tau _\\lambda \\left(x,u\\right)\\int \\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right)\\nu \\left(\\mathrm {d}u\\right)\\nonumber \\\\& = \\mathcal {T}_\\lambda \\left(+ _{\\lambda , n}\\right)f\\left(x\\right).$ This exchange of integrals is valid by Fubini's theorem because $\\lambda <\\lambda _O$ ensures the integrals are finite.","The same argument applies for the Fourier transformed operators to complete the proof.","Diagrammatic bounds For our argument, we will require bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}$ for all $n\\in \\mathbb {N}$ and $k\\in \\mathbb {R}^d$ .","However, it turns out that we can bound the last two remainder terms using others.","Lemma 5.1 For all $n\\in \\mathbb {N}$ , $k\\in \\mathbb {R}^d$ , and $\\lambda >0$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op},\\\\\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(0)\\right\\Vert _{\\rm op}.$ As $\\mathbb {P}_\\lambda (u_n {\\,\\,A\\,\\,} x\\textrm { in } \\xi ^{u_n,x}_{n+1} ) \\le \\tau _\\lambda (x,u_n)$ for an arbitrary locally finite set $A$ , the definition () gives $*{r_{\\lambda ,n}(x)} \\le \\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right).$ Let $f\\in L^2\\left(\\mathbb {X}\\right)$ .","Then this gives $*{R_{\\lambda ,n}f}^2_2 &\\le \\int \\left(\\int *{r_{\\lambda ,n}(x,y)}*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&\\le \\lambda ^2\\int \\left(\\int \\left(\\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right)\\right)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\lambda ^2\\int \\left(\\int \\tau _\\lambda (x,u)\\left(\\int \\pi _\\lambda ^{(n)}(u,y)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right) \\nu \\left(\\mathrm {d}u\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\& = \\lambda ^2 *{\\mathcal {T}_\\lambda _\\lambda ^{(n)}{\\rm Abs}f}^2_2.$ Note that we were able to use Tonelli's theorem to swap the $u$ and $y$ integrals because $\\tau _\\lambda $ and $\\pi ^{(n)}_\\lambda $ are both non-negative and measurable.","Finally, the approach of (REF ) implies the result.","For each $n\\ge 0$ , either $r_{\\lambda ,n}(x)\\ge 0$ for all $x\\in \\mathbb {X}$ or $r_{\\lambda ,n}(x)\\le 0$ for all $x\\in \\mathbb {X}$ .","Therefore $*{\\widehat{r}_{\\lambda ,n}(k;a,b)}\\le *{\\widehat{r}_{\\lambda ,n}(0;a,b)}$ for all $a,b\\in \\mathcal {E}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{R}_{\\lambda ,n}(0)\\right\\Vert _{\\rm op}$ (by Lemma REF ) for all $k\\in \\mathbb {R}^d$ .","From (REF ) we have $*{\\widehat{r}_{\\lambda ,n}(0;a,b)} \\le \\int \\widehat{\\tau }_\\lambda (0;a,c)\\widehat{\\pi }_\\lambda ^{(n)}(0;c,b) \\mathcal {P}\\left(\\mathrm {d}c\\right).$ Therefore the same argument as above gives the second bound in the result.","Because the $n=0$ cases for our bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ are so simple, we give the proof here before moving onto $n\\ge 1$ .","Recall $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right) = \\widehat{\\mathcal {T}}_\\lambda \\left(k+l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right)$ .","We will find it convenient to also define $\\widehat{}_k\\left(l\\right) = \\widehat{}\\left(k+l\\right) - \\widehat{}\\left(l\\right)$ .","Proposition 5.2 (Bounds for $n=0$ ) Let $\\lambda \\in [0,\\lambda _O)$ .","Then $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }.$ For $k\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op} &\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le \\lambda ^2\\left(\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } + \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\right).$ Our first comment relates to these bounds and every bound we perform hereafter in Section .","From the positivity of $\\pi ^{(0)}_{\\lambda }(x,y)$ and Lemma REF we have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op}$ , and by Lemma REF we have $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty }$ .","However from the definitions of the norm we also have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty } = \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ , and so we will be interested in bounding this last norm.","Now note that for the event $\\lbrace y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\rbrace $ to hold, either there is a direct edge between $y$ and $x$ , or there exist vertices $w,z$ in $\\eta $ that are direct neighbours of the origin and have respective disjoint paths to $x$ that both do not contain $y$ .","Hence, by the Mecke equation (REF ), $\\mathbb {P}_\\lambda ( y \\Longleftrightarrow x\\textrm { in } \\xi ^{\\mathbf {0},x}) &\\le \\varphi (x,y) + \\tfrac{1}{2} \\mathbb {E}_\\lambda \\Big [ \\sum _{(w,z) \\in \\eta ^{(2)}} 1_{(\\lbrace y\\sim w \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^x\\rbrace ) \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^x \\rbrace )} \\Big ] \\nonumber \\\\& = \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\int \\mathbb {P}_\\lambda \\big ( (\\lbrace y\\sim y \\text{ in } \\xi ^{y, w}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{x,w}\\rbrace )\\nonumber \\\\& \\hspace{99.58464pt} \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y, z}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^{x,z} \\rbrace ) \\big ) \\nu ^{\\otimes 2}\\left(\\mathrm {d}w,\\mathrm {d}z\\right).$ After applying the BK inequality to the above probability, $\\mathbb {P}_\\lambda \\left( y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) \\le \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left( \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y\\sim w \\text{ in } \\xi ^{y,w}\\right\\rbrace \\cap \\left\\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{w,x}\\right\\rbrace \\right) \\nu \\left(\\mathrm {d}w\\right) \\right)^2 \\\\= \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2.","$ Thus, recalling that $\\pi _\\lambda ^{(0)}(x,y) = \\mathbb {P}_\\lambda (y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x})-\\varphi (x,y) \\ge 0$ , and using the symmetry of $\\varphi $ and $\\tau _\\lambda $ , $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty } &= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\pi ^{(0)}_\\lambda \\left(x,y\\right)\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi \\left(y,u\\right)\\tau _\\lambda \\left(u,x\\right)\\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}w,\\mathrm {d}u\\right)\\nonumber \\\\&\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty }.$ We then use a supremum bound on this integral to split one of the $$ off from the others, and bound the other from above with $\\mathcal {T}_\\lambda $ to get the first two bounds of the result.","For the last bound of Proposition REF , we apply (REF ) and obtain $&\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{1,\\infty }\\nonumber \\\\&\\qquad = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(0)}_\\lambda \\left(\\overline{x};a,b\\right) \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\left(\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\mathrm {d}\\overline{u}\\mathcal {P}\\left(\\mathrm {d}c\\right)\\right)^2 \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b,b^\\prime \\in \\mathcal {E}}\\int \\Big [\\left(1 - \\cos \\left(k\\cdot \\left(\\overline{x}-\\overline{u}\\right)\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b^\\prime ,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{85.35826pt}+ \\left(1 - \\cos \\left(k\\cdot \\overline{u}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\Big ] \\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}+ \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty }.$ Here we have again used the symmetry of $\\varphi $ and $\\tau _\\lambda $ , and also used Lemma REF to split the cosine factor over $\\overline{x}-\\overline{u}$ and $\\overline{u}$ .","We can consider each term individually by using the triangle inequality.","By using a supremum bound, the symmetry of $\\varphi $ , and by bounding $\\varphi $ with $\\tau _\\lambda $ , we get the bound $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty }.$ Bounds on expansion coefficients In this section we aim to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ .","Lemmas REF and REF along with the positivity of $\\pi ^{(n)}_\\lambda (x,y)$ prove that it is sufficient to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty }$ to get bounds on both of these.","Much of the hard work for this has been done previously - for example in [21].","The novelty here is to see how the argument can be naturally written in terms of operators.","We inherit the following notation from [21].","Note that we make use of the Dirac delta function - specifically the one that holds with respect to the measure $\\nu $ .","They are used here primarily for convenience and to make the argument more readable.","In particular, they appear when applying the Mecke equation (REF ) to obtain $\\mathbb {E}\\left[ \\sum _{y\\in \\eta ^u} f(y, \\xi ^u) \\right] = \\int \\left(\\lambda +\\delta _{y,u}\\right) \\mathbb {E}_\\lambda \\left[ f\\left(y,\\xi ^{u,y}\\right)\\right] \\nu \\left(\\mathrm {d}y\\right),$ and so the factor $\\lambda +\\delta _{y,u}$ encodes a case distinction of whether point $y$ coincides with $u$ or not.","Definition 5.3 (The $\\psi $ functions) Let $r,s,u,w,x,y\\in \\mathbb {X}$ .","We first set $\\tau _\\lambda ^\\circ (x,y) := \\lambda ^{-1}\\delta _{x,y} + \\tau _\\lambda (x,y)$ .","Also define $\\psi _0^{(1)}(w,u,y) &:= \\lambda ^2\\tau _\\lambda (y,u)\\tau _\\lambda (u,w)\\tau _\\lambda (w,y),\\\\\\psi _0^{(2)}(w,u,y) &:=\\lambda ^2 \\delta _{w,y} \\tau _\\lambda (y,u)\\int \\tau _\\lambda (u,t)\\tau _\\lambda (t,y) \\nu \\left(\\mathrm {d}t\\right),\\\\\\psi _0^{(3)}(w,u,y) &:= \\lambda ^2\\varphi (u,y) \\left(\\lambda ^{-1}\\delta _{w,y}\\right),\\\\\\psi ^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda ^\\circ (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\psi ^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\left(\\lambda ^{-1}\\delta _{w,s}\\right)\\tau _\\lambda (u,r),\\\\\\psi _n^{(1)} (x,r,s) &:= \\lambda ^3\\int \\tau _\\lambda ^\\circ (t,s)\\tau _\\lambda (z,r)\\tau _\\lambda (t,z)\\tau _\\lambda (z,x)\\tau _\\lambda (x,t)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right),\\\\\\psi _n^{(2)}(x,r,s) &:=\\lambda \\tau _\\lambda (x,s)\\tau _\\lambda (x,r),$ and set $\\psi _0 := \\psi _0^{(1)}+\\psi _0^{(2)}+\\psi _0^{(3)}$ , $\\psi _n := \\psi _n^{(1)} + \\psi _n^{(2)}$ , and $\\psi := \\psi ^{(1)}+\\psi ^{(2)}+\\psi ^{(3)} + \\psi ^{(4)}$ .","Note that by using $\\lambda ^{-1}\\delta _{\\left(\\cdot ,\\cdot \\right)}$ in $\\tau _\\lambda ^\\circ $ and in $\\psi ^{(3)}_0$ , we ensure that each integral that will be truly performed (that is, not an integration over a Dirac delta function accounting for a coincidence of points) has a $\\lambda $ factor associated with it when the bound $\\pi ^{(n)}_\\lambda $ is calculated.","Diagrammatic representations of these functions can be found in Figure REF .","In these diagrams, the declared variables are represented by (0,0) circle (2pt) vertices whereas the variables that are integrated over are represented by (0,0) circle (2pt) vertices.","If $\\tau _\\lambda $ connects two variables then a standard edge (0,0) – (1,0); connects their vertices, if $\\tau _\\lambda ^\\circ $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; connects their vertices, and if $\\varphi $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\sim $; connects their vertices.","Figure: Diagrams of the ψ 0 \\psi _0, ψ\\psi , and ψ n \\psi _n functions.Definition 5.4 (The $\\Psi $ operators) Here we turn the functions of Definition REF into linear operators, with the $\\psi $ functions acting as their kernels.","For $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , we define $\\Psi ^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\Psi ^{(j)}_0 f\\left(w,u\\right) = \\int \\psi ^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right).$ For $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , we define $\\Psi ^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)} f\\left(w,u\\right) = \\int \\psi ^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ , we define $\\Psi ^{(j)}_n\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)}_n f\\left(x\\right) = \\int \\psi ^{(j)}_n(x,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ We can then naturally define $\\Psi _0\\colon L^2(\\mathbb {X})\\rightarrow L^2(\\mathbb {X}^2)$ , $\\Psi \\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X}^2)$ , and $\\Psi _n\\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X})$ as the sums of $\\Psi ^{(j)}_0$ , $\\Psi ^{(j)}$ , and $\\Psi ^{(j)}_n$ respectively, or equivalently as those operators having kernel functions $\\psi _0$ , $\\psi $ , and $\\psi _n$ respectively.","Proposition 5.5 (Bound in terms of $\\psi $ and $\\Psi $ ) Let $n \\ge 1$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in [0,\\lambda _O)$ .","Then $\\lambda \\pi _\\lambda ^{(n)}(x,y) \\le \\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes \\left(2n\\right)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right),$ where $\\vec{v}_i = (w_i,u_i,w_{i-1},u_{i-1})$ .","In operator terms, this means $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n \\Psi ^{n-1} \\Psi _0\\right\\Vert _{1,\\infty }.$ The argument for (REF ) is essentially identical to [21].","It uses only general properties of Poisson point process and connection models - for example thinning edges and Mecke's equation.","The inequality (REF ) then follows from the clear sequential structure of (REF ).","The inequality (REF ) suggests that we will bound the lace coefficient operators with some product of $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ terms.","This is nearly correct.","Unfortunately the norms $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ will not have the decay in $d$ that we will require.","We will have to consider the norms of pairs of operators.","Definition 5.6 Define $\\triangle _\\lambda &:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^\\circ _\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^{\\circ \\circ }_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda ^\\circ \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1,\\\\\\overline{\\triangle ^{\\circ \\circ }_\\lambda }&:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda ^\\circ \\left(\\overline{u}-\\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}.$ We can think of $\\triangle _\\lambda $ as an $\\mathbb {X}$ -convolution of three $\\tau _\\lambda $ functions with suprema taken over the end vertices.","Then $\\triangle ^\\circ _\\lambda $ and $\\triangle ^{\\circ \\circ }_\\lambda $ are produced by adding on the $\\mathbb {X}$ -convolution of two $\\tau _\\lambda $ functions and one $\\tau _\\lambda $ function (which will trivially take value 1 once the suprema are taken).","The object $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the marks at adjacent $\\tau _\\lambda $ functions need not be equal, and then we take the supremum over all the marks - not just the end ones.","We then use these elementary diagrams to produce composite objects: $U_\\lambda := \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }+ \\overline{\\triangle ^{\\circ \\circ }_\\lambda }^2 + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }, \\qquad V_\\lambda := \\left(\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }U_\\lambda \\right)^\\frac{1}{2}.$ We note a few relations.","Firstly, since all the terms in $U_\\lambda $ are non-negative we have $\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le U_\\lambda $ and thus $V_\\lambda \\le U_\\lambda $ .","Also note that $1\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ and thus $U_\\lambda \\ge \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .","In particular this means that $U_\\lambda \\ge 1$ and $U_\\lambda \\ge \\triangle ^\\circ _\\lambda $ .","The following lemma tells us that we can use $U_\\lambda $ to bound single operators, and that we can use $V_\\lambda ^2$ to bound pairs of operators.","To aid the reader's understanding, bear in mind that later (in Section ) we will prove that $U_\\lambda = \\mathcal {O}\\left(1\\right)$ , whilst $V_\\lambda $ is much smaller and is $\\mathcal {O}\\left(\\beta \\right)$ .","Lemma 5.7 For all $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j,j^{\\prime }\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $U_\\lambda &\\ge \\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\right\\Vert _{1,\\infty },\\\\V_\\lambda ^2 &\\ge \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j^{\\prime })}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }.$ We begin with the single operators.","For $\\Psi ^{(1)}$ we find $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } &= \\lambda ^4\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (w,u)\\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 4}\\left(\\mathrm {d}z, \\mathrm {d}t, \\mathrm {d}w, \\mathrm {d}u\\right), \\nonumber \\\\& \\le \\lambda ^4\\left(\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (r,z) \\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (t,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t \\right)\\right) \\nonumber \\\\&\\hspace{142.26378pt} \\times \\left(\\operatornamewithlimits{ess\\,sup}_{t^{\\prime },z^{\\prime }\\in \\mathbb {X}}\\int \\tau _\\lambda (t^{\\prime },w)\\tau _\\lambda (w,u)\\tau _\\lambda (u,z^{\\prime })\\nu ^{\\otimes 2}\\left(\\mathrm {d}w, \\mathrm {d}u\\right)\\right),\\nonumber \\\\& = \\triangle ^\\circ _\\lambda \\triangle _\\lambda ,$ where we have used a supremum bound on the $t$ and $z$ integrals to split them into $L^1$ and $L^\\infty $ bounds.","It is also clear that $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty }= \\triangle _\\lambda $ and $\\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\le \\triangle ^\\circ _\\lambda $ .","The $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ bound is more involved and we return to it in a moment.","We are able to represent these $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ norms and their bounds pictorially.","The vertices of the diagrams represent the variables $w,u,z,t,a,b$ , etc.","appearing in the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ expression.","The variables over which a $\\nu $ -integral is taken are represented by the (0,0) circle (2pt) vertices, and the variables over which a supremum is taken are represented by the (0,0) circle (2pt) vertices.","The presence of a $\\tau _\\lambda $ connecting two variables is then represented by a standard edge (0,0) – (1,0); between their vertices, and the presence of a $\\tau _\\lambda ^\\circ $ connecting two variables is correspondingly represented by an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; between their vertices.","When using a supremum bound to split the diagrams, an integrated vertex gets split into two new vertices, precisely one integrated vertex and one supremum vertex.","Each edge connected to such a split vertex can independently choose which of the new vertices to associate to.","The calculations producing the bounds on $\\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty }$ for $j=1,3,4$ can then be represented by: $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } \\quad &=\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\\\\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } + \\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad .$ Our strategy for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ is to use the spatial translation invariance to shift the origin to the bottom right vertex, and then use a supremum bound to split the diagram in two.","Unfortunately, this spatial shift decouples the spatial and mark components of the vertices so - for example - at some vertices we end up taking suprema over marks whilst integrating over space.","Let us introduce some more notation.","The vertex mark size=2pt] at (0,0) square* indicates that the spatial component is integrated over whilst the mark component is fixed and its supremum is taken after all the integrals.","We also introduce the edge-vertex combination [dashed] (0,0) – (1,0); [fill=white] (0,0) circle (2pt) which indicates that there is no term connecting the associated vertices, but there is a supremum taken over the spatial displacement.","Note that in principle we could use a vertex that had a spatial supremum and a mark integral, but we will always bound that probability integral by the supremum.","The bound for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ can therefore be expressed in the following: $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1);(0,-0.6) -- (1,-0.6);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (1.5,-0.1) circle (2pt);(1.5,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);[fill=white] (0,0) circle (2pt);\\end{tikzpicture}}.$ To clarify the calculation for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ , we write out the integral here: $&\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}-\\overline{s};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}- \\overline{w}; a_4,a_3)\\tau _\\lambda (\\overline{u}-\\overline{t};a_6,a_4)\\tau _\\lambda (\\overline{z}-\\overline{u};a_5,a_6) \\nonumber \\\\&\\hspace{142.26378pt}\\times \\tau _\\lambda (\\overline{t}- \\overline{z}; a_4,a_5)\\tau _\\lambda (\\overline{z}; a_5,a_2) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5, \\mathrm {d}a_6\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\left(\\operatornamewithlimits{ess\\,sup}_{\\overline{s},\\overline{t}^{\\prime },\\overline{z}^{\\prime }\\in \\mathbb {R}^d,a_1,a_2,a_4,a_5\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathcal {P}\\left(\\mathrm {d}a_3\\right)\\right)\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_6\\in \\mathcal {E}}\\int \\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6)\\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\right).$ Then the two components of the first factor that are joined by the $\\raisebox {2pt}{{[dashed] (0,0) -- (1,0); [fill=white] (0,0) circle (2pt)}}$ edge can be spatially translated together to form three consecutive edges.","Note however, that the marks on either side of this join are not necessarily equal.","We can therefore bound this first term by $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .","In summary, these manipulations then imply that $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\triangle _\\lambda $ .","For $\\Psi ^{(1)}_0$ and $\\Psi ^{(2)}_0$ the calculations are identical to each other, and we can once again represent the calculations pictorially: $\\left\\Vert \\Psi ^{(1)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_0\\right\\Vert _{1,\\infty } =\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- cycle;[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad = \\triangle _\\lambda ,$ where splitting the single supremum into two produces an upper bound.","For the third case we have $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) -- (1,0);(0.5,0) circle (0pt) node[above]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad = \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }.$ The diagrams representing the calculations for $\\Psi _n$ are as follows: $\\left\\Vert \\Psi _n^{(1)}\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(1,-0.6) -- (0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{113.81102pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\qquad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda , \\nonumber \\\\\\left\\Vert \\Psi ^{(2)}_n\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda .$ In the first case we expanded a $\\tau _\\lambda ^\\circ $ edge into a $\\tau _\\lambda $ edge and a contraction.","Then in the contracted case we bound $\\tau _\\lambda \\le 1$ on the diagonal edge.","We have now proven the bounds for the single operator norms.","For most of the operator pairs, we can then use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ to immediately get sufficient bounds.","The only pairs for which this strategy fails are: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ .","We now deal with these cases by hand: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- (0,0.6);(0,0) circle (0pt) node[above, rotate=90]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda , \\\\\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda .$ For the first bound we used $\\varphi \\le \\tau _\\lambda $ , and then split the single supremum into two.","For the second bound we used $\\tau _\\lambda \\le 1$ on the diagonal edge.","Proposition 5.8 For $n\\ge 1$ , $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le 6\\times 4^{n-1}U_\\lambda V_\\lambda ^n.$ Recall from Lemma REF that $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n\\Psi ^{n-1}\\Psi _0\\right\\Vert _{1,\\infty }$ .","We therefore aim to bound $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ for $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ .","We have two cases depending upon the parity of $n$ .","If $n$ is odd then we have an even number of operators.","We pair them off, use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , and Lemma REF to get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le \\left(V_\\lambda ^2\\right)^{\\frac{n+1}{2}} = V_\\lambda ^{n+1}.$ If $n$ is even, then we pair off the first $n$ operators and treat the last $\\Psi ^{(j_n)}_n$ operator by itself.","We then get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le U_\\lambda \\left(V_\\lambda ^2\\right)^{\\frac{n}{2}} = U_\\lambda V_\\lambda ^{n}.$ Since $U_\\lambda \\ge V_\\lambda $ , we have the bound $U_\\lambda V_\\lambda ^{n}$ for all $n\\ge 1$ .","Since there are 3 choices for $j_0$ , 4 choices for each of $j_1,\\ldots ,j_{n-1}$ , and 2 choices for $j_n$ , we get the pre-factor of $6\\times 4^{n-1}$ in our result by applying the triangle inequality.","Bounds on displaced expansion coefficients In this section we will be aiming to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ for $n\\ge 1$ .","Our strategy draws inspiration from the corresponding step in [21], whilst being adapted to account for inhomogeneous marks in a similar way to Section REF .","The central idea is to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(n)}_\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right),$ and bound $\\pi ^{(n)}_\\lambda $ using the function expression from Proposition REF .","This produces an integral (or diagram) composed from simpler segments with the displacement factor, $\\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)$ , spanning the whole length of the diagram.","We can then use the Cosine-Splitting result (Lemma REF ) to get a sum of diagrams where the displacement factor only spans a single segment of each diagram.","To demonstrate this, consider the diagram corresponding to $\\Psi ^{(2)}_n\\Psi ^{(1)}\\Psi ^{(1)}_0$ with the displacement factor: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8) -- (3,0.8) -- (4,0.2);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\le 3\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\right.","\\\\ \\left.\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (1,0.8) -- (3,0.8);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right).$ In these diagrams and hereafter we use [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; to denote that a displacement factor is connecting the two indicated vertices.","As above, the path the [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; takes will also suggest the route along which we will intend to use cosine-splitting.","For the sake of simplicity, we will be taking the displacement path across the `top' of the diagram from the perspective of the usual orientation of the $\\Psi _0$ segment.","However - as can be seen in the above example - the way the operators compose with each other means that the orientation of each segment inverts when compared to its neighbours.","Therefore the displacement may be across the `top' or the `bottom' of the $\\Psi $ and $\\Psi _n$ segments (in their usual orientation) depending upon the parity of its place in the sequence of segments.","Our general strategy will be to isolate the displaced segment (and perhaps a neighbouring segment) by splitting off `earlier' and `later' segments.","When we split off earlier segments (and sometimes later segments), we will use the $\\Psi $ structure we have been using so far.","However, we will sometimes want to group vertices and edges slightly differently when we split off `later' segments.","We use an observation from [21] to conveniently group these.","Let us define the kernel functions $\\overline{\\psi }^{(1)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\tau _\\lambda (y,r)\\tau _\\lambda (r,s)\\tau _\\lambda (s,x),\\\\\\overline{\\psi }^{(2)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\left(\\lambda ^{-1}\\delta _{y,r}\\right)\\left(\\lambda ^{-1}\\delta _{y,s}\\right),\\\\\\overline{\\psi }^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\overline{\\psi }^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\left(\\lambda ^{-1}\\delta _{u,r}\\right),$ and set $\\overline{\\psi }_0 := \\overline{\\psi }^{(1)}_0+\\overline{\\psi }^{(2)}_0$ and $\\overline{\\psi }:= \\overline{\\psi }^{(1)}+\\overline{\\psi }^{(2)}+\\overline{\\psi }^{(3)} + \\overline{\\psi }^{(4)}$ .","Diagrammatic representations of these functions can be found in Figure REF .","Note the similarity to the $\\psi $ functions, with the $\\tau _\\lambda ^\\circ $ edge appearing elsewhere in the integral.","Figure: Diagrams of the ψ ¯ 0 \\overline{\\psi }_0, and ψ ¯\\overline{\\psi } functions.For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\overline{\\Psi }^{(j)}_0 f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right),$ and for $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\overline{\\Psi }^{(j)} f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ In our terminology, the important observation of [21] was that for some $m\\ge 0$ the `later' segments of each diagram can be bounded using $\\big (\\overline{\\Psi }^m\\overline{\\Psi }_0\\big )^\\dagger $ , the adjoint of $\\overline{\\Psi }^m\\overline{\\Psi }_0$ .","In terms of the kernel functions, taking the adjoint in this case amounts to reflecting the `input' and `output' arguments, and for the diagrams this amounts to reflecting in the vertical plane.","Lemma 5.9 For $n\\ge 1$ , $m\\in \\left\\lbrace 0,\\ldots ,n\\right\\rbrace $ , $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^m,\\\\\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^{n-m},\\\\\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le {\\left\\lbrace \\begin{array}{ll}U_\\lambda &: m=0\\\\U_\\lambda ^2V_\\lambda ^{m-1} &: m\\ge 1.\\end{array}\\right.","}$ The argument for $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty }$ is the same as that in Lemma REF and Proposition REF .","We consider bounds for single operators and pairs of operators, and then use sub-multiplicity to get a bound for each diagram.","For $\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty }$ the argument is a little more complicated.","For $m=0,1$ we can easily check each diagram satisfies the bound.","For $m\\ge 2$ we have the issue that there exists one pair of operators which we can't bound by $V_\\lambda ^2$ : $\\left\\Vert \\overline{\\Psi }^{(4)}\\overline{\\Psi }^{(2)}_0\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) to [out=10,in=170] (1,0) to [out=190,in=350] (0,0);[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda \\le U_\\lambda ^2.$ If $m\\ge 2$ is even (an odd number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }_0$ term which we bound with $U_\\lambda $ .","We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda V_\\lambda ^m$ .","If $m\\ge 2$ is odd (an even number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }\\overline{\\Psi }_0$ pair which we bound with $U_\\lambda ^2$ .","We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda ^2V_\\lambda ^{m-1}$ .","Definition 5.10 In addition to the terms defined in Definition REF , we will now use $W_k&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int \\tau _{\\lambda ,k}(x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right) = \\lambda \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty },\\\\\\overline{W_k}&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _{\\lambda ,k}(\\overline{x}-\\overline{u};a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4)\\mathrm {d}u,\\\\\\overline{H_k}&:= \\lambda ^5 \\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,a_1,\\ldots ,a_{16}\\in \\mathcal {E}}\\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8)\\nonumber \\\\& \\hspace{42.67912pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\tau _{\\lambda ,k}(\\overline{t}-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u}.$ Note that we only use $\\overline{W_k}$ in Propositions REF and REF because $\\overline{W_k}\\ge W_k$ , but in the proof we will sometimes use $W_k$ to make the derivation easier to follow.","The diagrams for $\\overline{W_k}$ and $\\overline{H_k}$ are in Figure REF .","Also recall the sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ assumed to exist in Assumption REF .","Then define $\\mathbb {B} &:= \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\nu \\left(B\\left(x\\right)\\right),\\\\\\triangle ^{(B)}_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}: r\\notin {B\\left(s\\right)}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right).$ Note that these are only required to deal with one specific diagram in the following $n=1$ case.","Figure: Diagrams representing W k ¯\\overline{W_k} and H k ¯\\overline{H_k}.","In this figure, the marks on edges incident to filled square vertices need not be equal.Proposition 5.11 For $k\\in \\mathbb {R}^d$ and $n\\ge 2$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}\\left(n+1\\right)^24^{n-1}\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^4 V_\\lambda ^{n-4} &: n\\ge 4\\\\16\\times 16\\left(50\\overline{W_k}V_\\lambda ^2 + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=3\\\\9\\times 4\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=2.\\end{array}\\right.","}$ Proposition 5.12 For $k\\in \\mathbb {R}^d$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(1)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(1)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le 13\\overline{W_k}V_\\lambda U_\\lambda + \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\left(V_\\lambda + \\triangle ^{(B)}_\\lambda \\right) + \\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + 4\\lambda \\mathbb {B}\\overline{W_k}.$ [Proof of Proposition REF ] In bounding the diagrams with one displacement segment, we have three broad cases.","Let $i\\in \\left\\lbrace 0,1,\\ldots ,n\\right\\rbrace $ denote which segment the displacement lies upon.","Our three cases are then: the displacement lies on the $\\psi _n$ segment (i.e.","$i=n$ ), the displacement lies on the $\\psi _0$ segment (i.e.","$i=0$ ), the displacement lies on a $\\psi $ segment (i.e.","$i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ ).","We will aim to get a single bound for each diagram having a displacement crossing a single segment.","This will be uniform in the sequence $\\left(j_0,j_1,\\ldots ,j_n\\right)$ .","We then account for the number of possible choices of $\\left(j_0,j_1,\\ldots ,j_n\\right)$ , and finally include the factor of $(n+1)^2$ arising from using cosine-splitting (the explicit factor and the number of diagrams).","Case REF is the simplest.","We use a supremum bound to pull off the displaced $\\Psi _n$ term.","We then bound this by hand and use Lemma REF to bound the remainder.","In representing this scheme we use a shorthand $\\Psi ^{n-1}\\Psi _0$ to denote a particular sequence $\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0$ rather than actually the operators $\\Psi $ and $\\Psi _0$ defined above.","We also use labelled grey shapes to represent these groupings of diagrams, highlighting only end vertices or vertices that connect to neighbouring segments.","This scheme look like: $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8) -- (4,0.2);(2.8,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (2,0.8) -- (3,0.2);(2.5,0.5) circle (0pt) node[rotate = -30]{\\times };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{28.45274pt}\\le U_\\lambda V_\\lambda ^{n-1} \\left(2\\left( \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 2U_\\lambda V_\\lambda ^{n-1}\\left(\\overline{W_k}\\triangle ^{\\circ \\circ }_\\lambda + W_k\\triangle ^\\circ _\\lambda \\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 4\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that this bound also holds if the displacement runs along the bottom of the $\\Psi _n$ segment.","Since there are $3\\times 4^{n-1}$ possible sequences of indices for the $\\Psi ^{n-1}\\Psi _0$ diagram, this case contributes that as a pre-factor.","For Case REF , we need a few more tricks.","Since we are taking the displacement across the top of the diagram, the only term that will make a contribution is $\\Psi ^{(1)}_0$ .","To tackle this case we first use the observation of [21] that the diagram we want to bound can in turn be bounded by a diagram using $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ .","As above, we use $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ as a shorthand for a particular sequence $\\overline{\\Psi }^{(j_{n-1})}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0$ .","We are also imprecise in this notation over whether it is this operator or its adjoint.","Nevertheless the diagrams we use are clear on which vertices are integrated over and which have suprema, and this will avoid ambiguity.","We then spatially translate the supremum to the far end and pull off the $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram using a supremum bound.","This looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}\\\\\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (3,-0.6) to [out=100,in=260] (3,0.6) to [out=190,in=350] (2,0.6);\\node at (2.5,0) {\\overline{\\Psi }};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0};(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}.$ For $\\overline{\\Psi }^{(j)}$ with $j=1,2,3$ , a simple supremum bound shows that this starting diagram can be bounded by $\\overline{W_k}\\triangle ^\\circ _\\lambda \\left\\Vert \\overline{\\Psi }^{(j)}\\right\\Vert _{1,\\infty }\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2$ .","For $j=4$ a little more care is needed: $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ We therefore find that $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le {\\left\\lbrace \\begin{array}{ll}\\overline{W_k}V_\\lambda ^{n-1}U_\\lambda ^2 &: n\\ge 3\\\\\\overline{W_k}V_\\lambda ^2U_\\lambda &: n=2.\\end{array}\\right.","}$ Since there are 4 choices for the appended $\\overline{\\Psi }^{(j)}$ term, and $4^{n-2}\\times 2$ for the remaining $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram, this case contributes $2\\times 4^{n-1}$ as a pre-factor.","For Case REF , we shall find we have 3 sub-cases, depending not only on the $j$ -index of the displaced segment, but on which term we are looking at once it has been expanded.","Sub-cases $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ deal with $j=1,2,3$ .","In Appendix we expand out the various displaced segments using cosine-splitting and by expanding $\\tau _\\lambda ^\\circ $ edges into $\\tau _\\lambda $ edges and points, and label each of the resulting diagrams as sub-case $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ .","These sub-cases are distinguished by the strategy we use to bound them.","Sub-case $\\mathrm {\\left(III\\right)}$ then addresses the case with $j=4$ .","For sub-case $\\mathrm {\\left(I\\right)}$ , we just use the $\\Psi $ structure of the diagrams.","We first split off segments from the left of the displaced term like we did in Case REF .","If $i=n-1$ we can then bound the remaining two pair of segments with $\\overline{W_k}V_\\lambda ^2$ (Appendix explains how this bound is found).","This can be described diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda V_\\lambda ^{n}.$ If $i\\le n-2$ , then we also split off the segments to the right of this pair, and then bound the pair by $\\overline{W_k}V_\\lambda ^2$ again.","Diagrammatically this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (6,-0.6) to [out=40,in=200] (7,0) to [out=160,in=320] (6,0.6) to [out=190,in=350] (4,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };\\node at (5,0) {\\Psi _n\\Psi ^{n-i-2}};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(7,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda ^2V_\\lambda ^{n-3} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that the since $U_\\lambda \\ge V_\\lambda $ , the bound in (REF ) is greater than or equal to the bound in (REF ).","It will be convenient to find an $i$ -independent bound.","To this end, since the $i\\le n-2$ case only exists for $n\\ge 3$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(I\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}&: n\\ge 3\\\\\\overline{W_k}U_\\lambda V_\\lambda ^{2}&: n=2,\\end{array}\\right.","}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .","For sub-case $\\mathrm {\\left(II\\right)}$ , we use a different strategy.","First we split off the earlier (left) segments as we did for the first sub-case and in Case REF .","However we arrange the later terms differently - more like we did in Case REF .","We associate our displaced $\\Psi $ term with $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges from the subsequent term.","If the subsequent term would be $\\Psi ^{(2)}_n$ , these are actually two $\\tau _\\lambda $ edges, but in our schematic diagrams we will draw a $\\tau _\\lambda ^\\circ $ edge and bear this special case in mind when it is relevant.","While the displacement portrayed on the `top' of the segment could equally well be on the `bottom', the $\\tau _\\lambda ^\\circ $ edge will always be attached to the bottom in the usual orientation.","Everything to the right of these $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges can then be described as a sequence of $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ terms.","If $i=n-1$ , then we pair our augmented displaced segment with the subsequent $\\overline{\\Psi }_0$ segment.","We then perform a spatial translation to make the right-most vertex a supremum vertex like we did in (REF ).","The remaining diagram of a $\\Psi ^{(j)}$ segment and $\\overline{\\Psi }_0$ segment connected by a $\\tau _\\lambda ^\\circ $ and $\\tau _\\lambda $ edge can then be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ).","Recall that if $\\overline{\\Psi }_0$ is in fact $\\overline{\\Psi }^{(2)}_0$ then we have two $\\tau _\\lambda $ edges connecting the terms, not a $\\tau _\\lambda $ edge and a $\\tau _\\lambda ^\\circ $ edge.","We need to note this to get the $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ bound.","The scheme is then represented diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(6,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda V_\\lambda ^{n-2}.$ If $i\\le n-2$ , we also perform a spatial translation to move a supremum to the right-most vertex, but then also split off all $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ segments after the $\\overline{\\Psi }$ segment immediately following the displaced term.","Once again the remaining pair can be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ) and the split-off diagrams bounded using Lemma REF .","The scheme can be represented diagrammatically as $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);[fill=gray!50] (5,0.6) to [out=280,in=80] (5,-0.6) to [out=10,in=170] (7,-0.6) to [out=40,in=200] (8,0) to [out=160,in=320] (7,0.6) to [out=190,in=350] (5,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};\\node at (6,0) {\\overline{\\Psi }^{n-i-2}\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(5,0.6) circle (2pt);(5,-0.6) circle (2pt);(8,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\hspace{170.71652pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times {\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.}","\\nonumber \\\\&\\hspace{170.71652pt}\\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right){\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.","}$ Again for simplicity, we would like to get and $i$ -independent bound for such diagrams.","Since the $i = n-2$ case only exists for $n\\ge 3$ , and the $i\\le n-3$ case only exists for $n\\ge 4$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(II\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^3V_\\lambda ^{n-4}&: n\\ge 4\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^2&: n=3\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda &: n=2,\\end{array}\\right.","}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .","We now address sub-case $\\mathrm {\\left(III\\right)}$ , that is when the displacement crosses a $\\Psi ^{(4)}$ segment.","This is the simplest sub-case because when we take a displacement across $\\Psi ^{(4)}$ in isolation, we get exactly $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad =W_k.$ We will be able to use the $\\Psi $ -arrangement of segments here.","For $n\\ge 4$ we split off the earlier and later segments without pairing the displaced term with any of them.","With Lemma REF this produces the bound $\\left(U_\\lambda V_\\lambda ^{i-1}\\right)W_k\\left(U_\\lambda V_\\lambda ^{n-i-1}\\right) = \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-2} \\qquad :n\\ge 4.$ For $n=3$ we split off two segments from one side and one from the other.","This gives the finer bound $W_kU_\\lambda V_\\lambda ^2 \\quad :n=3.$ For $n=2$ , we use $j_0$ to indicate which $\\Psi _0$ term we have and $j_2$ to denote which $\\Psi _n$ term we have.","We note that if $j_0=1,2$ then $\\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ , and if $j_2 = 1$ then $\\left\\Vert \\Psi ^{(j_2)}_n\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ .","Therefore if $\\left(j_0,j_2\\right) \\ne \\left(3,2\\right)$ we have the bound $W_kU_\\lambda V_\\lambda $ .","If $\\left(j_0,j_2\\right) = \\left(3,2\\right)$ we proceed by hand.","We bound $\\varphi \\le \\tau _\\lambda $ and spatially shift one vertex to show $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };(1,0) circle (0pt) node[rotate=90, above]{\\sim };(0,0) circle (0pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (1,0.6) {{square*}};(1,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda .$ Carefully comparing the above bounds gives the required bounds for $n\\le 2$ .","Including the multiplicity of each diagram (including that arising from cosine-splitting), we find that we can bound the displaced and subsequent non-displaced pair by $\\left(24\\overline{W_k}V_\\lambda ^2 + 2\\overline{H_k}\\right)$ .","Note that this includes summing over the $j$ -index of the displaced term but not the index of the subsequent term.","Since there are $3\\times 4^{n-2}\\times 2$ options for the $j$ -indices of the other segments, this case contributes this to the counting pre-factors in our result.","[Proof of Proposition REF ] For $n=1$ , we can consider each of the six cases by hand.","Most of the diagrams can be dealt with routinely using cosine-splitting, supremum bounds, and spatial shifts.","We enumerate the diagrams by their $\\left(j_0,j_1\\right)$ values corresponding to the constituent segments.","$(1,1):& \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (0,-0.2) -- (1,-0.8) -- (2,-0.8) -- (3,-0.2);(1.5,-0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 3\\left(\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda + \\triangle _\\lambda \\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(1,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (0,0.2) -- (1,0.8) -- (2,0.2);(0.8,0.68) circle (0pt) node[rotate=30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda W_k\\right)\\\\(2,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (1,0.8) -- (2,0.8) -- (3,0.2);(1.8,0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\triangle _\\lambda \\overline{W_k}\\triangle ^\\circ _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(2,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (1,0.8) -- (2,0.2);(1.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle _\\lambda W_k\\\\(3,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6) -- cycle;(1,0.6) -- (1,-0.6);[<->] (0,0.8) -- (1,0.8) -- (2,0.2);(0.8,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(2,0) circle (2pt);(0,-0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\overline{W_k}\\triangle _\\lambda + \\overline{W_k}\\triangle _\\lambda + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\triangle _\\lambda W_k+ \\triangle _\\lambda W_k\\right)$ These terms sum to no more than $13\\overline{W_k}V_\\lambda U_\\lambda .$ For the $\\left(3,2\\right)$ diagram, we use a different `trick.'","In Lemma REF we will prove the bound $\\tau _\\lambda \\left(x,y\\right) \\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right)$ .","When applied to this diagram, this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (1.2,0.6) -- (1.2,-0.6);(1.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right),$ Here we have also used the cosine-splitting result on the second part.","For the first of these 4-vertex diagrams, we split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\lambda \\triangle _\\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }$ .","For the second of the 4-vertex diagrams, we also split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ .","For the 3-vertex diagram we condition on whether two of the vertices are in each other's $B$ -set (defined in Assumption REF ).","In the following diagrams [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\in B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\in B\\left(u\\right)\\right\\rbrace }$ relates two variables, and [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\notin B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\notin B\\left(u\\right)\\right\\rbrace }$ relates them: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.7) -- (0,-0.8) -- (1,-0.2);(-0.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-20pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,-0.8) -- (1,-0.2);(0.5,-0.5) circle (0pt) node[rotate=30]{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right) \\nonumber \\\\&\\le \\left(\\quad \\raisebox {-7pt}{\\begin{tikzpicture}(0,0) -- (1,0);[<->] (0,0.2) -- (1,0.2);(0.5,0.2) circle (0pt) node{\\times };(0.5,0) circle (0pt) node[below]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node[above, rotate=90]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) +4\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);[<->] (0,-0.2) -- (1,-0.2);(0.5,-0.2) circle (0pt) node[below]{\\scriptsize \\in B};\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right)\\nonumber \\\\& \\le \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\triangle ^{(B)}_\\lambda + 4\\lambda \\mathbb {B}W_k.$ Bootstrap analysis In this section we will be assuming that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = 1.$ By Lemma REF this also implies $\\mathbb {S}\\left(\\right)\\le 1$ .","A re-scaling argument similar to [36] explains how this can be done while maintaining generality.","First note that since $\\widehat{\\varphi }(0;a,b)>0$ for a positive $\\mathcal {P}$ -measure set of $a,b\\in \\mathcal {E}$ , we have $\\mathbb {S}\\left(\\widehat{}(0)\\right)>0$ .","Now suppose we scale $\\mathbb {R}^d$ by a factor of $q^{-1/d}$ (that is, the new unit radius ball is the previous ball of radius $q^{1/d}$ ), and leave the mark space $\\mathcal {E}$ unchanged.","Then we find that the system we are left with has the distribution of an RCM model with the new parameters $\\lambda ^* = \\mathbb {E}_\\lambda \\left[*{\\eta \\cap \\left[0, q^{\\frac{1}{d}}\\right]^d }\\right] = \\lambda q, \\qquad \\varphi ^*\\left(\\overline{x};a,b\\right) = \\varphi \\left(q^{\\frac{1}{d}}\\overline{x};a,b \\right).$ Therefore for all $a,b\\in \\mathcal {E}$ we have $\\widehat{\\varphi }^*(\\overline{x};a,b) = \\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\widehat{\\varphi }(\\overline{x};a,b),$ and $\\mathbb {S}\\left(\\widehat{}^*(0)\\right) = q^{-1} \\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","We can therefore achieve our assumption by choosing $q=\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","Furthermore, under this scaling we find that the old event $\\left\\lbrace \\left(\\overline{x},a\\right) \\longleftrightarrow \\left(\\overline{y},b\\right)\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ becomes the new event $\\left\\lbrace \\left(q^{-1/d}\\overline{x},a\\right) \\longleftrightarrow \\left(q^{-1/d}\\overline{y},b\\right)\\textrm { in } \\xi ^{\\left(q^{-1/d}\\overline{x},a\\right),\\left(q^{-1/d}\\overline{y},b\\right)} \\right\\rbrace $ , and we get $\\tau ^*_{\\lambda ^*}(\\overline{x};a,b) = \\tau _\\lambda \\left(q^\\frac{1}{d}\\overline{x};a,b\\right)$ where $\\tau ^*_{\\lambda ^*}$ is the two-point function in the RCM governed by the connection function $\\varphi ^*$ and intensity $\\lambda ^*$ .","It is also worth noting that the scaling choice to have $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ means that REF reduces to the condition that there exists a $d$ -independent constant $C>0$ such that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } &\\le C,\\\\\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } &\\le C \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Note that the inequalities relating $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ follow from Jensen's inequality and a supremum bound on a probability space.","The Bootstrap Function To show that the bounds arising from the diagrams in Section are sufficient for our purposes, we make use of a bootstrap argument.","Define $\\mu _\\lambda & = 1- \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)},\\\\\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) &= \\frac{1}{1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)},\\\\\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) &=\\left(1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\left(\\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right)\\right).$ From these we can then define the bootstrap function $f= f_1\\vee f_2\\vee f_3$ , where $f_1\\left(\\lambda \\right) = \\lambda , \\qquad f_2\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}, \\qquad f_3\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)}.$ Let us now outline some initial properties of $\\mu _\\lambda $ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .","Lemma 6.1 The map $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous and bounded below by $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ on $\\left[0,\\lambda _O\\right)$ .","Therefore the map $\\lambda \\mapsto \\mu _\\lambda $ is a continuous map from $\\left[0,\\lambda _O\\right)\\rightarrow \\left[0,1\\right)$ .","Furthermore, $\\widehat{G}_{\\mu _\\lambda }(k)$ is bounded below by $\\left(1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)^{-1}$ for all $k\\in \\mathbb {R}^d$ .","We first recall that $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (0)$ is differentiable with respect to the operator norm from Corollary REF , and is therefore continuous.","Lemma REF then implies that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous.","For the bounded below property, we use Lemma REF to get $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge \\mathbb {S}\\left(\\widehat{}(0)\\right) = 1$ .","The continuity of $\\mu _\\lambda $ follows directly from the continuity and positivity of $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ .","The bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ bounds $\\mu _\\lambda \\ge 0$ , and $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ shows that $\\mu _\\lambda <1$ for all $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","The case $\\mu _0 = 0$ follows immediately from $\\mathcal {T}_0 = $ .","Because $\\mu _\\lambda \\in \\left[0,1\\right)$ , to minimise $\\widehat{G}_{\\mu _\\lambda }(k)$ we want to find a lower bound for $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ .","The $k$ -uniform bound on $\\widehat{G}_{\\mu _\\lambda }(k)$ then follows from $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\ge -\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\rm op} \\ge -\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}$ .","Using the Bootstrap function to bound terms Bounding the Lace Operator We have managed to bound the lace operator and its displacement in terms of a variety of objects.","For the lace operator itself (and its associated Fourier transformed operators), these are: $\\lambda , \\triangle _\\lambda , \\overline{\\triangle ^{\\circ \\circ }_\\lambda }, \\left\\Vert \\right\\Vert _{1,\\infty }.$ As a guide, we expect $\\lambda $ , $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ , and $\\left\\Vert \\right\\Vert _{1,\\infty }$ to be bounded uniformly in $d$ and $\\lambda <\\lambda _O$ , whereas we expect $\\triangle _\\lambda $ to decay uniformly in $\\lambda <\\lambda _O$ as $d\\rightarrow \\infty $ .","For some, these properties are easy to prove.","The boundedness of $\\left\\Vert \\right\\Vert _{1,\\infty }$ follows from $\\left\\Vert \\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we have used REF via (REF ).","Regarding $\\lambda $ , since we are only interested in $\\lambda <\\lambda _O$ the question is perhaps more clearly written as “Is $\\lambda _O$ uniformly bounded in $d$ ?","\".","While there may be more elementary ways of proving this directly, we will get it for free from the bootstrap argument.","Since we have the bound $\\lambda \\le f(\\lambda )$ , proving the uniform boundedness of $f(\\lambda )$ with respect to $d$ and $\\lambda $ for $\\lambda <\\lambda _O$ will prove the same for $\\lambda _O$ .","This will leave us with only $\\triangle _\\lambda $ and $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .","Recall that we can write $\\triangle _\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }, \\quad \\triangle ^{\\circ \\circ }_\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1.$ Then $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the supremum is taken over all the marks, and adjacent $\\tau _\\lambda $ can have different marks.","To get the desired decay and boundedness behaviour, we will therefore only need to prove that $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ decays and to bound $&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v},\\\\&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}.$ We begin by proving the decay of $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ .","Lemma 6.2 Let $x,y \\in \\mathbb {X}$ .","Then $\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\varphi (x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right),\\\\\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right).$ Then for $n\\ge 2$ , $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le \\sum _{m=0}^n\\genfrac(){0.0pt}0{n}{m}\\lambda ^{m} \\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }.$ By combining Mecke's formula and the BK inequality, we obtain $\\tau _\\lambda (x,y) & \\le \\varphi (x,y) + \\mathbb {E}_\\lambda \\left[\\sum _{u \\in \\eta } 1_{\\lbrace x \\sim u \\text{ in } \\xi ^{x} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y}\\rbrace } \\right] \\nonumber \\\\& = \\varphi (x,y) + \\lambda \\int \\mathbb {P}_\\lambda \\left(\\lbrace x \\sim u \\text{ in } \\xi ^{x,u} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y,u}\\rbrace \\right) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& = \\varphi (x,y) +\\lambda \\int \\varphi (x,u) \\tau _\\lambda (u,y) \\nu \\left(\\mathrm {d}u\\right).$ In the last line, we have used that the two intersected events are independent.","This is due to the fact that $\\nu $ is non-atomic and $x\\notin \\eta ^{y,u}$ a.s.","The second inequality is then proven nearly identically, but with the adjacency to the intermediate point $u$ holding for $y$ instead of $x$ .","To get (REF ), we first use () to expand out the right-most $\\mathcal {T}_\\lambda $ .","This produces one term with $$ replacing the $\\mathcal {T}_\\lambda $ , and one term with $\\lambda \\mathcal {T}_\\lambda $ replacing it.","We then repeatedly use (REF ) to expand the left-most $\\mathcal {T}_\\lambda $ to get a $$ factor and a $\\lambda \\mathcal {T}_\\lambda $ factor until we have a total of $n$ factors of $$ in the expression.","Counting the number of ways to get to each possible end term then gives the binomial $\\genfrac(){0.0pt}1{n}{m}$ factor as required.","The following lemma will allow us to describe $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ using Hilbert space language.","Lemma 6.3 Let $H\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ be an integral linear operator with kernel function $h\\colon \\mathcal {E}^2\\rightarrow \\mathbb {R}$ .","Then for all $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ and $\\varepsilon >0$ , there exist $\\mathcal {P}$ -positive sets $E_1,E_2\\subset \\mathcal {E}$ such that $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ satisfy $*{h\\left(a,b\\right) - {g_1}{Hg_2}} \\le \\varepsilon .$ Since $h$ is measurable, the set $A=\\left\\lbrace c,c^{\\prime }\\in \\mathcal {E}:*{h\\left(c,c^{\\prime }\\right)-h\\left(a,b\\right)}\\le \\varepsilon \\right\\rbrace $ is measurable.","Since $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ , it has positive measure: $\\mathcal {P}\\left(A\\right)>0$ .","Furthermore, by considering the $\\pi $ -system of product sets $\\left\\lbrace E_1\\times E_2: E_1,E_2\\subset \\mathcal {E}\\text{ measurable}\\right\\rbrace $ that generates the $\\sigma $ -algebra on $\\mathcal {E}^2$ , we can find $B = E_1\\times E_2$ such that $E_1,E_2\\subset \\mathcal {E}$ are both measurable, $B\\subset A$ , and $\\mathcal {P}^{\\otimes 2}\\left(B\\right) = \\mathcal {P}\\left(E_1\\right)\\mathcal {P}\\left(E_2\\right)>0$ .","Then use $E_1,E_2$ to define $g_1,g_2$ as in the statement on the lemma.","Since $E_1,E_2$ are both measurable and have positive measure, both $g_1,g_2\\in L^2\\left(\\mathcal {E}\\right)$ .","We also have ${g_1}{Hg_2} = \\frac{1}{\\mathcal {P}^{\\otimes 2}\\left(B\\right)}\\int _B h\\left(c,c^{\\prime }\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right) \\in \\left[h(a,b) - \\varepsilon ,h(a,b) +\\varepsilon \\right],$ proving the result.","Let us define the function $\\overline{\\epsilon }\\colon \\mathbb {N}\\rightarrow \\mathbb {R}_+$ , where $\\overline{\\epsilon }(d) := {\\left\\lbrace \\begin{array}{ll}\\frac{1}{d} + \\frac{\\log d}{\\log g(d)} &: \\lim _{d\\rightarrow \\infty }g(d)\\rho ^{-d}\\Gamma \\left(\\frac{d}{2}+1\\right)^2 = 0 \\qquad \\forall \\rho >0,\\\\0&: \\text{otherwise}.\\end{array}\\right.","}$ Recall $g(d)$ is the function defined in REF .","In particular, $\\overline{\\epsilon }(d) = 0$ unless $g(d)$ approaches zero particularly quickly, and $\\beta (d) = g(d)^{\\frac{1}{4}-\\frac{3}{2}\\overline{\\epsilon }(d)}$ .","Lemma 6.4 Let $\\lambda <\\lambda _O$ , $n\\ge 2$ , and $d>2m$ .","Then there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=2\\\\c_fg(d)^{\\frac{1}{2}-m\\overline{\\epsilon }(d)} &: n\\ge 3.\\end{array}\\right.","}$ The condition $\\lambda < \\lambda _O$ here is required to allow us to swap integrals over the mark space with integrals over the Fourier space.","We are able to use Fubini's Theorem because sub-criticality implies that the relevant integrals are finite.","For clarity, let us introduce the notation $\\varphi ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\varphi (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),\\quad \\tau _\\lambda ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\tau _\\lambda (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),$ where $u_0=x$ and $u_n=y$ .","Furthermore, for each $\\overline{x}\\in \\mathbb {R}^d$ let us define the integral linear operator $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) \\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $(a,b)\\mapsto \\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) := \\int \\varphi ^{(n-1)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda ^{(m)}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime }\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ Given $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $a,b\\in \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right)\\in {\\rm ess.Im}\\left(\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};\\cdot ,\\cdot \\right)\\right)$ , we can use Lemma REF to get measurable and $\\mathcal {P}$ -positive sets $E^{(\\overline{x},a,b,\\delta )}_1,E^{(\\overline{x},a,b,\\delta )}_2\\subset \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) \\le *{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2} + \\delta ,$ where $g^{(\\overline{x},a,b,\\delta )}_i(c) = \\frac{1}{\\mathcal {P}\\left(E^{(\\overline{x},a,b,\\delta )}_i\\right)}1_{E^{(\\overline{x},a,b,\\delta )}_i}\\left(c\\right)$ for $i=1,2$ .","Now let $E_1,E_2\\subset \\mathcal {E}$ be arbitrary measurable $\\mathcal {P}$ -positive sets and $g_1,g_2$ be their associated functions.","We aim to produce bounds that are independent of the choice of $E_1,E_2$ .","We first use the Fourier inversion theorem to reformulate the position behaviour in terms of a $k$ -integral of the Fourier transforms, and use REF and $\\lambda <\\lambda _O$ (so the relevant integrals are finite) to allow us to swap the $k$ -integral and the mark integrals.","This produces $*{*{g_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g_2}} = *{\\int \\text{e}^{-i\\overline{x}\\cdot k}*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}}\\\\ \\le \\int *{*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ Recall that $\\widehat{}(k)$ is self-adjoint, so we can move $\\widehat{}(k)^{n-1}$ to the other side of the inner product, and then we can use Cauchy-Schwarz and the definition of the operator norm to get $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} = *{\\widehat{}(k)^{n-1}g_1}{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\\\\\le *{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}_2 \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}^m*{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{}(k)g_2}_2.$ We can then use the definition of the bootstrap function $f(\\lambda )$ to replace the $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ factors with factors of $f(\\lambda )$ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .","Also writing the norm $*{\\cdot }_2$ as the square root of an inner product and using the self-adjoint property of $\\widehat{}(k)$ gives $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\le f(\\lambda )^m\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}.$ We therefore want to derive bounds for the integral $\\int \\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ We first derive pointwise and integral bounds for the $\\widehat{}$ factors.","Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\widehat{\\varphi }^{(2N)}(k;c,c^{\\prime })\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right)^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2}.$ By using Cauchy-Schwarz and supremum bounds we can split this into norms of $\\widehat{}(k)$ only, then use Lemma REF to replace $k$ with 0, and finally use REF to bound these terms: $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2}\\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty }^{N-1} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }^{N-1} \\le C^N.$ For the integral bound, we apply Cauchy-Schwarz to the $k$ -integral and swap the $k$ -integrals with the mark integrals to un-do the Fourier transforms.","We get $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\left(\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2}\\left(\\int *{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2} \\\\= *{g_1}{^{(2n-2)}\\left(\\overline{0}\\right)g_1}^\\frac{1}{2}*{g_2}{^{(2)}\\left(\\overline{0}\\right)g_2}^\\frac{1}{2},$ where $^{(N)}\\left(\\overline{0}\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ is the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\varphi ^{(N)}\\left(\\overline{0};a,b\\right)$ .","Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{^{(2N)}\\left(\\overline{0}\\right)g} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\varphi ^{(2N)}\\left(\\overline{0};c,c^{\\prime }\\right)\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right) \\le \\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }.$ Here for $N\\ge 2$ we bound with $\\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert ^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le g(d)^\\frac{1}{2}$ from REF .","For $N=1$ we bound using $\\varphi \\left(\\overline{x};a,b\\right)\\in \\left[0,1\\right]$ and REF : $\\left\\Vert ^{(2)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,c\\right)\\varphi \\left(-\\overline{x};c,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}c\\right) \\le \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\\\= \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }\\le C.$ Therefore $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le {\\left\\lbrace \\begin{array}{ll}C &: n=2,\\\\C^\\frac{1}{2}g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.","}$ We now return to bounding (REF ).","To perform the $k$ -integral, we split the domain.","We first consider $B_\\varepsilon (0) = \\lbrace k\\in \\mathbb {R}^d: *{k} < \\varepsilon \\rbrace $ , where we choose $\\varepsilon >0$ such that $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1,C_2$ coming from the assumption REF .","From REF , we have $\\widehat{G}_{\\mu _\\lambda }(k) = \\left(1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)^{-1}\\le 1/C_2*{k}^2$ on $B_\\varepsilon (0)$ and $\\widehat{G}_{\\mu _\\lambda }(k)\\le 1/\\left(C_2\\varepsilon ^2\\right)$ on $B_\\varepsilon (0)^c$ .","Our pointwise bound on the $\\widehat{}$ factors then gives $\\int _{B_\\varepsilon (0)}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\int ^\\varepsilon _0\\frac{1}{r^{2m}}r^{d-1}\\mathrm {d}r = \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{d-2m}\\frac{\\varepsilon ^{d-2m}}{\\left(2\\pi \\right)^d},$ where $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.","Note that for all fixed $\\varepsilon >0$ this term vanishes in the $d\\rightarrow \\infty $ limit.","To deal with the integral over $B_\\varepsilon (0)^c$ , we find the upper bound $\\int _{B_\\varepsilon (0)^c}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{1}{C^m_2\\varepsilon ^{2m}}\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\frac{C^\\frac{1}{2}}{C^m_2\\varepsilon ^{2m}}{\\left\\lbrace \\begin{array}{ll}C^\\frac{1}{2} &: n=2\\\\g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.","}$ Note that the bounds we found were independent of the sets $E_1,E_2$ , and therefore these bounds also apply to $*{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2}$ uniformly over every $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $\\mathcal {P}$ -almost every $a,b\\in \\mathcal {E}$ .","Since we can take $\\delta \\rightarrow 0$ , our bound also applies to $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }$ .","For $n=2$ , the $B_\\varepsilon (0)^c$ integral dominates the $B_\\varepsilon (0)$ integral, and so the result is proven.","For $n\\ge 3$ , both parts of the integral approach 0 as $d\\rightarrow \\infty $ .","If both are of the same order or if the $B_\\varepsilon (0)^c$ integral dominates, then we can fix $\\varepsilon >0$ and get the result.","On the other hand, if $g(d)^\\frac{1}{2} \\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^{d}$ for all fixed $\\varepsilon >0$ and the $B_\\varepsilon (0)$ integral always dominates, we can improve the overall bound by letting $\\varepsilon =\\varepsilon (d)$ and having $\\varepsilon (d)\\rightarrow 0$ as $d\\rightarrow \\infty $ .","Having a smaller value of $\\varepsilon $ produces a smaller bound for the $B_\\varepsilon (0)$ integral, but a larger bound for the $B_\\varepsilon (0)^c$ integral.","To get an optimal $\\varepsilon $ we can set both terms to be of the same order - that is by having $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{2m}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .","For this choice, we have both integrals of the order $\\varepsilon ^{-2m}g(d)^\\frac{1}{2} = 2^{-2m}\\pi ^{-m}A^{-\\frac{2m}{d}} g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\left(1-\\frac{2m}{d}\\right)^{-\\frac{2m}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{2m}{d}} \\\\= \\left(\\frac{\\text{e}}{2\\pi }\\right)^m g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\frac{1}{d^m}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .","Proposition 6.5 (Bound for the $n$ -gon diagrams) Let $\\lambda < \\lambda _O$ .","Then for each $n$ such that $d>2n$ , there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=1,2,\\\\c_fg(d)^{\\frac{1}{2}-n\\overline{\\epsilon }(d)}&: n\\ge 3.\\end{array}\\right.","}$ The $n=1$ case is trivial because $\\tau _\\lambda \\left(\\overline{x},a,b\\right)\\in \\left[0,1\\right]$ .","For $n=2$ , we first bound $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty }$ .","We get $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int *{\\varphi (x,u)\\varphi (u,y)}\\nu (\\mathrm {d}u) \\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}^2\\nu (\\mathrm {d}u) \\\\\\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}\\nu (\\mathrm {d}u) = \\left\\Vert \\right\\Vert _{1,\\infty },$ by using Cauchy-Schwarz and symmetry of $\\varphi $ in the first inequality and $\\varphi (x,u)\\in \\left[0,1\\right]$ in the second.","We then use REF to bound this by the constant $C$ .","We use Lemma REF to get the required bound for the remaining terms arising from (REF ) in Lemma REF .","For $n\\ge 3$ , we once again bound the terms in the relevant expansion in (REF ).","For $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }$ we use REF to get $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }= O\\left(g(d)^\\frac{1}{2}\\right)$ .","We then use Lemma REF to bound the remaining terms.","For our proofs of Lemma REF and Lemma REF , we adopt the following notation.","Given $g(\\overline{x};a,b)$ and $h(\\overline{x};a,b)$ , we define $(gh)(\\overline{x};a,b) := \\int g(\\overline{x}-\\overline{y};a,c)h(\\overline{y};c,b)\\mathrm {d}\\overline{y}\\mathcal {P}(\\mathrm {d}c).$ This notational convention is associative and therefore can generalise to three or more terms unambiguously.","Lemma 6.6 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}\\le c_f,\\\\\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le c_f,$ and therefore $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le 1 + 2c_f.$ By applying (REF ) and (), we can get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ We will in fact prove the more general result for the convolution of $m\\ge 2$ $\\tau _\\lambda $ -functions in dimensions $d>2m$ .","By bounding $\\lambda \\le f(\\lambda )$ , and using Lemma REF and the Fourier inversion theorem we can realise that we only need to get further bounds for the following objects.","For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^m$ , we want to bound $\\int \\prod ^m_{i=1} P^{j_i}_{2i-1,2i}(k)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{p,q}(k) := {\\left\\lbrace \\begin{array}{ll}*{*{g_p}{\\widehat{}(k) g_q}} &:j=1\\\\*{*{g_p}{\\widehat{}(k)^2g_q}} &:j=2\\\\*{*{g_p}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_q}} &: j=3,\\end{array}\\right.","}$ where $\\left\\lbrace g_i\\right\\rbrace _{i=1}^{2m}$ are the functions of the form $g_i(a) = \\frac{1}{\\mathcal {P}(E_i)}1_{E_i}(a)$ arising from the application of Lemma REF .","We now bound the following terms in the same way as (REF ): $*{*{g_1}{\\widehat{}(k)^2g_2}} &\\le *{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}\\\\*{*{g_1}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_2}} &\\le f\\left(\\lambda \\right)\\widehat{G}_{\\mu _\\lambda }(k)*{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}.$ In terms of pointwise uniform bounds for the $\\widehat{}$ terms, we have $*{*{g_1}{\\widehat{}(k)g_2}}\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C$ and $*{g}{\\widehat{}(k)^2g}\\le C^2$ (the latter also having been used in the proof of Lemma REF ).","As in Lemma REF , we will require various integral bounds for these $\\widehat{}$ terms.","Suppose we have $r \\in \\left\\lbrace 1,\\ldots ,m-1\\right\\rbrace $ instances of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and $m-r$ instances of $P^{1}_{p,q}(k)$ .","Then by applying Cauchy-Schwarz and the definitions of the $g_i$ we get $&\\int \\prod ^{r}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\prod ^{m}_{i^{\\prime }=r+1}*{*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int \\prod ^{r}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\prod ^{m}_{i^{\\prime }=r+1}*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2r-1},a_{2r})\\right.\\right.\\nonumber \\\\&\\hspace{199.16928pt}\\left.\\left.\\star \\varphi (\\cdot ;a_{2r+1},a_{2r+2})\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m+r}.$ In this last inequality we have used $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\le C$ and $\\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }^2 \\le C^2$ to extract off $\\varphi $ and $\\varphi ^{(2)}$ via supremum bounds on the spatial position.","We will also require integral bounds where there are $m$ factors of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and where there are $m$ factors of $P^{1}_{p,q}(k)$ : $&\\int \\prod ^{m}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\left(\\int \\prod ^{m}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{2m}.\\\\&\\int \\prod ^{m}_{i=1}*{*{g_{2i-1}}{\\widehat{}(k)g_{2i}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)g_2}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\left(\\prod ^{m}_{i=2}*{g_{2i-1}}{\\widehat{}(k)g_{2i}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m}.$ Note that for $m\\ge 3$ , better bounds are available via REF , but will not be required here.","We can now proceed to bound each of the expressions of the form (REF ) in much the same way as we did in the proof of Lemma REF .","We set $\\varepsilon >0$ and partition the integral over $\\mathbb {R}^d$ into one over $B_\\varepsilon (0)$ and one over $B_\\varepsilon (0)^c$ .","For the $B_\\varepsilon (0)$ integral we use the uniform bounds on the $\\widehat{}$ terms and perform the integral of the $\\widehat{G}_{\\mu _\\lambda }$ terms (if there are any) as before.","Note that we require $d>6$ for all of these to be finite.","For the $B_\\varepsilon (0)^c$ integrals we uniformly bound $\\widehat{G}_{\\mu _\\lambda }(k) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and use the integral bounds we calculated above.","Since we are only asking for a constant bound, we don't need to worry about having $\\varepsilon \\rightarrow 0$ as $d\\rightarrow \\infty $ .","Bounding the Lace Operator Displacement We now consider the objects required to bound the displacement of the lace operator.","In addition to the objects used to bound the lace operator itself, we require bounds on $\\left\\Vert _k\\right\\Vert _{1,\\infty }, \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }, \\overline{W_k}, \\overline{H_k}, \\triangle ^{(B)}_\\lambda , \\mathbb {B}.$ The decay $\\mathbb {B}\\le g(d)$ follows from Assumption REF , and the following Observation REF demonstrates the decay of $\\triangle ^{(B)}_\\lambda $ .","Observation 6.7 Note that the inequality (REF ) allows us to bound $\\triangle ^{(B)}_\\lambda $ in terms of other terms.","By using the inequality twice and appropriately bounding $\\varphi $ with $\\tau _\\lambda $ , we get $\\int \\tau _\\lambda (r,u)\\tau _\\lambda (u,s)\\nu (\\mathrm {d}u) \\le \\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) + 2\\lambda \\int \\tau _\\lambda (r,v)\\tau _\\lambda (v,u)\\tau _\\lambda (u,s)\\nu ^{\\otimes 2}(\\mathrm {d}u, \\mathrm {d}v)$ for any $r,s\\in \\mathbb {X}$ .","Therefore $\\triangle ^{(B)}_\\lambda \\le 3\\triangle _\\lambda + \\lambda \\operatornamewithlimits{ess\\,sup}_{r\\notin B(s)}\\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) \\le 3\\triangle _\\lambda + \\lambda g(d),$ where we have used REF .","Therefore Proposition REF implies that for $d>6$ there exists $c_f$ such that $\\triangle ^{(B)}_\\lambda \\le c_f\\beta ^2.$ We are then left with $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ , $\\overline{W_k}$ , and $\\overline{H_k}$ , and we will deal them in this order.","First note that $\\left\\Vert _k\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi _k(x,y)\\nu \\left(\\mathrm {d}x\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int (1-\\cos \\left(k\\cdot \\overline{x}\\right))\\varphi (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}(\\mathrm {d}a) \\\\ = \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty }.$ The following lemma therefore also allows us to bound $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ .","Its extra generality will be required later.","Lemma 6.8 For all $k,l\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\infty ,\\infty } \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where $C>0$ is the $d$ -independent constant given in REF .","The first two inequalities in (REF ) holds from the general inequalities that holds for these norms (on a probability space for the second inequality).","Then we can remove the $l$ -dependence by performing the following calculation: $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+ k\\right)\\right\\Vert _{\\infty ,\\infty } \\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\text{e}^{il\\cdot \\overline{x}}}*{\\left(1-\\cos \\left( k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}\\\\= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}= \\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left( k\\right)\\right\\Vert _{\\infty ,\\infty }.$ The final step relating the $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ -norm to the difference of the spectral suprema is precisely that given by () in REF .","We now address $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ .","Lemma 6.9 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ First note that for $k=0$ the kernel function $\\tau _{\\lambda ,k}\\left(x,y\\right)=0$ for all $x,y\\in \\mathbb {X}$ , and so the required inequality holds trivially.","In this proof we hereafter may assume $k\\ne 0$ .","We use (REF ) to write $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }.$ We first get the bound on $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ , using a similar approach to Lemma REF .","For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda \\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}^{\\prime }-\\overline{u}^{\\prime \\prime };c^{\\prime },c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime \\prime };c^{\\prime \\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathrm {d}\\overline{u}^{\\prime \\prime }\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime },\\mathrm {d}c^{\\prime \\prime }\\right).$ Recall the notation $\\varphi ^{(2)}$ used in (REF ).","As in the proof of Lemma REF , we let $E_1,E_2\\subset \\mathcal {E}$ be measurable and $\\mathcal {P}$ -positive sets, and aim to bound ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ (where $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ ) independently of the choice of $E_1,E_2$ .","If we do this we will have proven the result.","As before, we use the Fourier inversion theorem to write ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ as the integral of an inner product of Fourier transformed operators over the Fourier argument.","Having $\\lambda <\\lambda _O$ allows us to swap the integrals in this step.","Then we can use Cauchy-Schwarz and the definitions of the operator norm and the bootstrap functions to extract factors of $\\widehat{G}_{\\mu _\\lambda }(l)$ and $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ from the inner product.","The net result is $*{g_1}{\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} &\\le \\int *{*{g_1}{\\widehat{}(l)^{2}\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l)g_2}}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\le f(\\lambda )^2\\int \\widehat{G}_{\\mu _\\lambda }(l) \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ The $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ We partition our integral using open $\\varepsilon $ -balls around the poles where $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1$ and $C_2$ coming from REF .","Let $B_{\\varepsilon }(p)$ denote the open $\\varepsilon $ -ball around $p\\in \\mathbb {R}^d$ .","For $n\\in \\left\\lbrace 0,1,2,3\\right\\rbrace $ , let $A_n := \\left\\lbrace l\\in \\mathbb {R}^d: l\\in B_\\varepsilon (p) \\text{ for precisely $n$ elements }p\\in \\left\\lbrace k,0,-k\\right\\rbrace \\right\\rbrace .$ Firstly it is possible for $A_3,A_2 = \\emptyset $ if $*{k}$ is sufficiently large compared to $\\varepsilon $ .","It is also easy to see that $A_3\\subset B_{\\varepsilon }(0)$ and $A_2\\subset B_{2\\varepsilon }(0)$ .","We will partition $\\mathbb {R}^d$ into $A_3$ , $A_2$ , $A_1$ , and $A_0$ and bound the $l$ -integral on each part.","Recall that REF implies that $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2*{l}^{-2}$ for $*{l}\\le \\varepsilon $ and $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2\\varepsilon ^2$ for $*{l}\\ge \\varepsilon $ .","Also recall the pointwise (REF ) and integral (REF ) bounds we derived for the $\\widehat{}$ factors.","We treat $A_3$ first: $&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)\\cap B_\\varepsilon (k)}\\frac{1}{*{l}^4*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\qquad \\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\\\&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{A_3}\\frac{1}{*{l+k}^2*{l}^2*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (-k)}\\frac{1}{*{l+k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ Recall $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.","In these calculations we used Hölder's inequality and increased the domain of integration to get upper bounds.","These integrals are finite for $d>6$ , and approach 0 if we take bounded $\\varepsilon $ .","For $A_2$ the counting of cases is the only extra complication.","A precise counting is entirely possible, but unnecessary for our purposes.","It is simple to see that the arguments outlined for $A_3$ can be applied to get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^6} + \\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a pair of two distinct points from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with three, two, and one factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on these two points.","This bound is finite for $d>6$ , and approaches 0 if we take bounded $\\varepsilon $ .","Repeating this for $A_1$ , we get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} + \\frac{1}{\\varepsilon ^6} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a single point from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with two, one, and zero factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on this single point.","This bound is finite for $d>4$ , and approaches 0 if we take bounded $\\varepsilon $ .","The $A_0$ integral is qualitatively different.","By taking the integral bound for the $\\widehat{}$ factors and a uniform (on $A_0$ ) bound for the factors of $\\widehat{G}_{\\mu _\\lambda }$ , we get the bound $\\frac{C^3}{C^3_2}\\varepsilon ^{-6}g(d)^\\frac{1}{2}.$ As in Lemma REF , if the $g(d)^\\frac{1}{2}$ term dominates, then we have a sufficient bound for our result.","The largest of the other terms (for $d$ sufficiently large and $\\varepsilon \\le 1$ ) is the $\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ term.","Like we did in Lemma REF , we can make these other terms smaller by decreasing $\\varepsilon $ at the cost of making the $g(d)^\\frac{1}{2}$ bound greater.","The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{6}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .","For this choice, we have both integrals of the order $\\varepsilon ^{-6}g(d)^\\frac{1}{2} = 2^{-6}\\pi ^{-3}A^{-\\frac{6}{d}}g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\left(1-\\frac{6}{d}\\right)^{-\\frac{6}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{6}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^3 g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\frac{1}{d^3}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .","Bounding the $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ term is similar.","For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ We once again take $\\mathcal {P}$ -positive sets $E_1,E_2$ and corresponding $g_1,g_2$ .","Then we use the Fourier inversion theorem and Cauchy-Schwarz to get $*{g_1}{\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} \\le f(\\lambda )\\int \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Then once again the $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Counting the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ centred on each of the poles $\\left\\lbrace k,0,-k\\right\\rbrace $ then tells us that there is a uniform bound that is some integer multiple of $\\frac{C^2}{C^2_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^{4}} + \\frac{1}{\\varepsilon ^2*{l}^{2}} + \\frac{1}{\\varepsilon ^4}\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} + \\varepsilon ^{-4}g(d)^\\frac{1}{2}\\right) \\\\= \\frac{C^2}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-4} + \\frac{C^2}{C^2_2}\\varepsilon ^{-4}g(d)^\\frac{1}{2}.$ Once again, if the $g(d)^\\frac{1}{2}$ term dominates the result is proven.","Otherwise, we let $\\varepsilon $ vary with $d$ .","The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{4}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .","For this choice, we have both integrals of the order $\\varepsilon ^{-4}\\beta ^2 = 2^{-4}\\pi ^{-2}A^{-\\frac{4}{d}} g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(1-\\frac{4}{d}\\right)^{-\\frac{4}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{4}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^2 g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\frac{1}{d^2}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .","For large $d$ , this is dominated by $g(d)^{\\frac{1}{2}-\\frac{3}{d}}d^{-3}$ and therefore $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ is dominated by $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ and we prove the result.","The following lemma allows us to deal with occurrences of $\\overline{W_k}$ .","Lemma 6.10 Let $x,y \\in \\mathbb {X}$ .","Then $\\tau _{\\lambda ,k}(x,y) \\le \\varphi _k(x,y) + 2\\lambda \\int \\left(\\tau _\\lambda (x,u)\\varphi _k(u,y) + \\tau _{\\lambda ,k}(x,u)\\varphi (u,y)\\right)\\nu \\left(\\mathrm {d}u\\right).","$ For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{W_k}\\le c_f\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The first inequality follows by applying the cosine-splitting result (Lemma REF ) to ().","For the $\\tau _\\lambda $ term in $\\overline{W_k}$ we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of (REF ), we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we can bound $\\overline{W_k}$ by bounding the following objects.","For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace \\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {W}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l)Q^{j_2}_{3,4}(l;k)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{1,2}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_1}{\\widehat{}(l) g_2}} &:j=1\\\\*{*{g_1}{\\widehat{}(l)^2g_2}} &:j=2\\\\*{*{g_1}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_2}} &: j=3,\\end{array}\\right.","}\\\\Q^{j}_{3,4}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} &:j=1\\\\*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} &:j=2\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{}_k(l) g_{4}}} &:j=3\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &:j=4\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} &:j=5,\\end{array}\\right.","}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{4}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .","For the $P^j_{1,2}(l)$ factor we can get bounds as we did in the proof of Lemma REF .","For $Q^{j}_{3,4}(l;k)$ we need to do a bit more processing.","For $j=1$ , $*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} \\le \\frac{1}{\\mathcal {P}(E_{3})\\mathcal {P}(E_{4})}\\int _{E_{3}\\times E_{4}}*{\\widehat{\\varphi }_k(l;a,b)}\\mathcal {P}^{\\otimes 2}(\\mathrm {d}a, \\mathrm {d}b) \\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right),$ where in the last inequality we used Lemma REF .","For $j=2$ , we use Cauchy-Schwarz to get $*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} \\le *{g_3}{\\widehat{}_k(l)^2 g_3}^{\\frac{1}{2}} *{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}} \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}.$ An identical argument gives the same bound for $j=3$ .","For $j=4$ , we can use the self-adjointness of $\\widehat{}(l)$ and $\\widehat{\\mathcal {T}}_\\lambda (l)$ and the Cauchy-Schwarz inequality to get $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &\\le *{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)^2\\widehat{}(l)g_{3}}^{\\frac{1}{2}} *{g_{4}}{\\widehat{}_k(l)^2g_{4}}^{\\frac{1}{2}} \\nonumber \\\\&\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}}\\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }\\nonumber \\\\&\\le Cf(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}},$ and similarly $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} \\le f(\\lambda )\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2} \\\\ \\le Cf(\\lambda )\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2},$ where we write for convenience $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k) = \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3-k) + \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k) + \\widehat{G}_{\\mu _\\lambda }(l_3-k)\\widehat{G}_{\\mu _\\lambda }(l_3+k).$ Note that all the bounds for the $Q^{j}_{3,4}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.","So we only need to prove that the remaining $l$ -integral of $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors is bounded.","As in our previous proofs, we partition $\\mathbb {R}^d$ into the sets $A_n$ defined earlier in (REF ).","For the $A_1$ , $A_2$ , and $A_3$ parts we then use the bounds $C$ and $C^2$ for the various $\\widehat{}$ factors and $1/\\left(C_2*{l}^2\\right)$ to bound the $\\widehat{G}_{\\mu _\\lambda }$ factors.","By using the techniques used previously in this section, we can show that these contributions is bounded by some $\\varepsilon $ and $d$ (only) dependent constant.","Since we don't need to show decay, we don't need to worry about taking $\\varepsilon $ to 0.","For the $A_0$ contribution, we bound $\\widehat{G}_{\\mu _\\lambda }$ factors by $1/\\left(C_2\\varepsilon ^2\\right)$ and show that the integral of the $\\widehat{}$ factors over all of $\\mathbb {R}^d$ is bounded.","For $j_2\\ne 1$ this proceeds similarly to the argument in the proof of Lemma REF .","For $j_2=1$ we need to take a step back.","These versions of (REF ) arise from trying to bound one particular type of term, which we now treat separately.","We bound $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\operatornamewithlimits{ess\\,sup}_{a_3,a_4\\in \\mathcal {E}}\\int \\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{\\infty ,\\infty }.$ Then Lemma REF gives the result.","We are now only left with bounding $\\overline{H_k}$ .","lemmaMartiniBound For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{H_k}\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The proof of the lemma uses the same techniques and tools as our previous lemmas, but is much more complicated in terms of keeping track of cases and components of integrals.","To avoid overburdening a reader on an early reading, the proof has been placed in Appendix .","The Bootstrap Argument In this section we bring together the diagrammatic bounds of Section with the estimates of Section REF to bound the relevant operators in such a way that the finite lace expansion of Proposition REF converges to an Ornstein-Zernike equation (OZE) for both the raw operators on $L^2\\left(\\mathbb {X}\\right)$ and the Fourier-transformed operators on $L^2\\left(\\mathcal {E}\\right)$ .","Proposition REF does this for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , with the caveat that there is a constant $c_f$ that is increasing in the bootstrap function $f$ (but independent of $d$ ) appearing in the bounds.","Observation REF then highlights that if $f$ is uniformly bounded on $\\left[0,\\lambda _O\\right)$ then so is $c_f$ and the associated bounds from Proposition REF .","The crucial step to the bootstrap argument is then Proposition REF .","Here a forbidden-region argument shows that $f$ is continuous, that there is an interval of finite values that it cannot take (the interval being independent of $\\lambda $ ), and that it starts below this interval.","This proves that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , and so are our bounds in Proposition REF .","Proposition 6.11 (Convergence of the operator lace expansion and OZE) Let $\\lambda \\in [0, \\lambda _O)$ and $d>6$ be sufficiently large.","Then there exists $c_f=c(f(\\lambda ))$ (which is increasing in $f$ and independent of $d$ ) such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f\\beta , &\\qquad \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}\\le c_f\\beta , \\\\ \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le c_f\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right)-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\beta , \\\\\\left\\Vert R_{\\lambda , n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} c_f\\beta ^{n}.","$ Furthermore, the limit $_\\lambda := \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}_\\lambda ^{(n)}$ exists and is a bounded operator with Fourier transform $\\widehat{}_{\\lambda }\\left(k\\right) = \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}\\widehat{}_{\\lambda }^{(n)}\\left(k\\right)$ for $k\\in \\mathbb {R}^d$ .","The operators $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ satisfy operator Ornstein-Zernike equations, taking the form $\\mathcal {T}_\\lambda = + _\\lambda + \\lambda \\mathcal {T}_\\lambda (+_\\lambda ), $ and $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right) + \\lambda \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\left(\\widehat{}\\left(k\\right)+\\widehat{}_{\\lambda }\\left(k\\right)\\right), $ for all $\\lambda < \\lambda _O$ and $k\\in \\mathbb {R}^d$ .","Before we prove this proposition, we make the following observation, which will guide the rest of the section.","Observation 6.12 (Uniform convergence of the lace expansion) Suppose that there exists finite $C>0$ such that $f \\le C$ on $[0,\\lambda _O)$ .","Then for $d>6$ sufficiently large there exits $c>0$ (independent of $\\lambda $ and $d$ ) such that the bounds (REF ), (), () hold with $c_f$ replaced by $c$ .","We now return to prove Proposition REF .","[Proof of Proposition REF ] So far in this section we have provided bounds for the building blocks of the bounds appearing in Propositions REF , REF , REF , and REF .","We still have to convert these into bounds for the composite terms $U_\\lambda $ and $V_\\lambda $ .","From their definitions and the bounds on the building blocks, it is simple to see that there exists $c_f^{\\prime }$ such that for $d>6$ $U_\\lambda \\le c_f^{\\prime }, \\qquad V_\\lambda \\le c_f^{\\prime }\\beta .$ This then implies that there exists $c_f^{\\prime \\prime }$ such that for $d>6$ $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime }\\beta ^2 &: n=0,\\\\c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n &: n\\ge 1.\\end{array}\\right.","}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime }\\left(\\beta ^2 + \\frac{c_f^{\\prime \\prime }\\beta }{1-c_f^{\\prime \\prime }\\beta }\\right) \\le c_f\\beta .$ When we want to consider the displaced terms, we also make use of Observation REF , and Lemmas REF , REF , REF , and REF to give us the existence of $c_f^{\\prime \\prime \\prime }$ such that for $d>6$ $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime \\prime }\\beta ^2\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=0,3,\\\\c_f^{\\prime \\prime \\prime }\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=1,2,\\\\\\left(n+1\\right)^2c_f^{\\prime \\prime \\prime }\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^{n-3}\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n\\ge 4.\\end{array}\\right.","}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime \\prime }\\left(2\\beta ^2 + 2\\beta + \\frac{16\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^3-39\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^2 + 25c_f^{\\prime \\prime \\prime }\\beta }{\\left(1-c_f^{\\prime \\prime \\prime }\\beta \\right)^3}\\right)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\\\\\le c_f\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Lemma REF then allows us to use (REF ) to bound $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ .","This immediately gives the required result for $n\\ge 1$ .","For $n=0$ , we simply additionally require $d$ to be large enough that $\\beta <1$ and therefore $\\beta ^2<\\beta ^0$ .","Note that the dual space of a Banach space (endowed with the operator norm) is also a Banach space.","In particular, it is complete.","Since $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n$ for $n\\ge 1$ , the sequence $_{\\lambda ,n}$ is a Cauchy sequence in the dual space of $L^2(\\mathbb {X})$ for sufficiently large $d$ .","Therefore the limit $_\\lambda $ exists and is a bounded linear operator on $L^2(\\mathbb {X})$ .","Since the Fourier transform is a unitary linear operator it is bounded and continuous, and therefore the same argument says that the limit $\\widehat{}_{\\lambda }(k)$ exists and is indeed the Fourier transform of $_\\lambda $ .","Since $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}$ is finite for $\\lambda <\\lambda _O$ , we have $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}\\rightarrow 0$ as $n\\rightarrow \\infty $ if $d$ is sufficiently large.","As a a consequence of this and equation (REF ) from Proposition REF , the $\\mathcal {T}_\\lambda $ operator satisfies the operator Ornstein-Zernike equation.","The following lemma has two main uses in what follows.","Firstly, it takes the OZEs from Proposition REF and uses it to write the two-point operators in terms of the sum of the adjacency operators and the lace expansion coefficient operators.","Secondly, it uses this and an intermediate value theorem argument to bound the spectral supremum of the sum of the adjacency operators and the lace expansion coefficient operators.","Lemma 6.13 Let $\\lambda \\in \\left[0,\\lambda _O\\right)$ and $d>6$ be sufficiently large.","Then the operators $\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)$ and $\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)$ all have bounded linear inverses and $\\mathcal {T}_\\lambda &= \\left(+ _\\lambda \\right)\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)^{-1}, \\\\\\widehat{\\mathcal {T}}_\\lambda (k) &= \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1}, \\qquad \\forall k\\in \\mathbb {R}^d.","$ Furthermore, $\\lambda \\mapsto _\\lambda $ and $\\lambda \\mapsto \\widehat{}_{\\lambda }(k)$ are all continuous in the operator norm topology and $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) &< 1,\\\\\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &< 1, \\qquad \\forall k\\in \\mathbb {R}^d.$ We first prove that $1- \\lambda \\left(+ _\\lambda \\right)$ has a bounded linear inverse.","This is clear for $\\lambda = 0$ , so we only need to consider $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","Suppose for contradiction that $1- \\lambda \\left(+ _\\lambda \\right)$ does not have a bounded linear inverse.","Then there exists a sequence $f_n\\in L^2(\\mathbb {X})$ such that $*{f_n}_2 = 1$ and $*{\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)f_n}_2\\rightarrow 0$ .","This also means that $\\lambda *{\\left(+ _\\lambda \\right)f_n}\\rightarrow 1$ .","Then from Lemma REF and sub-criticality we have $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ , and therefore the OZE equation (REF ) implies that $*{\\left(+ _\\lambda \\right)f_n}\\rightarrow 0$ , a contradiction.","A similar argument proves the corresponding statement for the Fourier transformed operators.","Once we know that the inverses exist, it is only a matter of rearranging the OZE equations to get the expressions for $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ in terms of the other operators.","Since $+ _\\lambda $ is a bounded operator, a similar argument proves that $1+ \\lambda \\mathcal {T}_\\lambda $ has a bounded inverse, and therefore $+ _\\lambda = \\left(1+ \\lambda \\mathcal {T}_\\lambda \\right)^{-1}\\mathcal {T}_\\lambda .$ From Corollary REF we have the continuity of $\\lambda \\mapsto \\mathcal {T}_\\lambda $ , and therefore the continuity $+ _\\lambda $ (and therefore $_\\lambda $ ).","We also have the continuity of $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (k)$ from Corollary REF , and we similarly get the continuity of $\\widehat{}_{\\lambda }(k)$ .","The continuity of $+ _\\lambda $ implies the continuity of $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right)$ (via Lemma REF ).","At $\\lambda = 0$ it is clear that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) = 0$ , and our above argument implies that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) \\ne 1$ .","Therefore an intermediate value theorem argument proves that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) < 1$ .","This argument also works for $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .","It now remains to show that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ .","We do this by performing a forbidden-region argument.","In Proposition REF we prove that $f(0)$ is bounded and $f$ is continuous on $\\left[0,\\lambda _O\\right)$ .","However we also prove that $f$ is never in the region $\\left(\\kappa ,\\kappa +1\\right]$ , where $\\kappa := 10 C \\left(1+C\\right)^2,$ $C$ being the constant appearing in REF .","Note that (REF ) in REF requires that $C\\ge 1$ .","The intermediate value theorem then implies that $\\kappa $ acts as an upper bound on the whole domain $\\left[0,\\lambda _O\\right)$ .","One should not read too much into the value of $\\kappa $ here.","It is sufficient for our purposes, but not remotely optimal.","Proposition 6.14 (The forbidden-region argument) The following three statements are all true: $f$ satisfies $f(0) \\le \\kappa $ .","$f(\\lambda ) \\notin (\\kappa ,\\kappa +1]$ for all $\\lambda \\in [0,\\lambda _O)$ provided that $d$ is sufficiently large.","$f$ is continuous on $[0, \\lambda _O)$ .","Therefore $f(\\lambda ) \\le \\kappa $ holds uniformly in $\\lambda <\\lambda _O$ for all $d$ sufficiently large.","We will deal with each of the three statements in Proposition REF in turn.","[Proof of Proposition REF -REF ] Firstly, it is clear $f_1\\left(0\\right) = 0$ .","Then note that $\\mathcal {T}_0 = $ , and hence $\\left\\Vert \\widehat{\\mathcal {T}_0}\\left(k\\right)\\right\\Vert _{\\rm op} = \\left\\Vert \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we use REF (via (REF )) in the last inequality.","Furthermore $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_0(0)\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) =1$ , and therefore $\\widehat{G}_{\\mu _{0}}\\left(k\\right) = \\widehat{G}_0\\left(k\\right) = 1$ and $f_2\\left(0\\right) \\le C$ .","Similarly we find that $f_3\\left(0\\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op}}{3\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]}.$ We use then Lemma REF to get the bound $\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Hence $f_3\\left(0\\right) \\le \\tfrac{2}{3}C$ .","[Proof of Proposition REF -REF ] We will assume that $f\\le \\kappa +1$ and show that this implies that $f\\le \\kappa $ .","Most crucially, the assumption $f\\le \\kappa +1$ allows us to apply Observation REF .","To highlight the $\\kappa $ -dependence, we write $c_{\\kappa +1}$ as the constant arising from this observation.","It will be convenient to introduce some temporary notation for this section.","Define $\\widehat{a}\\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as $\\widehat{a}\\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)$ for all $k\\in \\mathbb {R}^d$ .","We will also make use of the functions $\\delta _\\lambda \\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\delta _\\lambda \\left(k\\right) := \\mathbb {S}\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)\\right) - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ It is clear from the triangle inequality and Lemma REF that ${\\delta _\\lambda \\left(k\\right)}\\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k\\right)\\right\\Vert _{\\rm op}\\le c_{\\kappa +1}\\beta ,$ but we will also need the following inequality.","We will prove it at the end of this section.","Lemma 6.15 Let $k_1,k_2\\in \\mathbb {R}^d$ .","Then $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2 \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right) - \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ Figure: If 𝒯 ^ λ (k)\\widehat{\\mathcal {T}}_\\lambda (k) and a ^(k)\\widehat{a} (k) commute, then they are simultaneously diagonalizable in the sense of Theorem .","The arguments of their diagonal functions (τ ˜ λ (k)\\widetilde{\\tau }_\\lambda (k) and a ˜(k)\\widetilde{a}(k) respectively) are related by the monotone increasing function x↦x 1-λxx\\mapsto \\tfrac{x}{1-\\lambda x} depicted here.It will be important to note how $\\widehat{a}(k)$ relates to $\\widehat{\\mathcal {T}}_\\lambda (k)$ .","From Observation REF , the OZE gives us the following expression involving the commutator $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right] = \\widehat{\\mathcal {T}}_\\lambda (k) \\widehat{a}(k) - \\widehat{a}(k)\\widehat{\\mathcal {T}}_\\lambda (k)$ : $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right]\\left(1- \\lambda \\widehat{a}(k)\\right) = 0.$ Lemma REF implies that $\\left(1- \\lambda \\widehat{a}(k)\\right)$ has a bounded linear inverse and with (REF ) implies that $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ commute.","Therefore there exists a single unitary map that `diagonalizes' them both in the sense of Theorem REF .","By considering the diagonal functions, the OZE then implies that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\mathbb {S}\\left(\\widehat{a}(k)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)\\right)$ .","In particular, this relation allows us to write an alternative expression for $\\mu _\\lambda $ : $\\mu _\\lambda := 1 - \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} = \\lambda + \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)-1}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)} = \\lambda + \\frac{\\delta _\\lambda \\left(0\\right)}{1+\\delta _\\lambda \\left(0\\right)}.$ We first address $f_1$ .","From the above discussion relating $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ , we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\ge \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) = \\mathbb {S}\\left(\\widehat{a}(0)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(0)\\right)\\right)$ .","From Lemma REF and our choice of scaling we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right) \\ge 1 - \\left\\Vert \\widehat{}_{\\lambda }(0)\\right\\Vert _{\\rm op} \\ge 1 - c_{\\kappa +1}\\beta $ , and therefore we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right)>0$ for sufficiently large $d$ and $\\lambda \\le \\frac{1}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}\\left(1 - \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\right) \\le \\frac{1}{1-c_{\\kappa +1}\\beta } \\le 1 + 2c_{\\kappa +1}\\beta .$ Here we have used an idea that we will use very often in this part of the proof.","For $d$ large enough that $c_{\\kappa +1}\\beta <1$ we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1} = \\sum ^\\infty _{n=0}\\left(c_{\\kappa +1}\\beta \\right)^n$ , and for $d$ sufficiently large enough ($c_{\\kappa +1}\\beta \\le 1/2$ ) we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1}\\le 1 + 2c_{\\kappa +1}\\beta $ .","We also use a similar idea of neglecting higher order terms by increasing an earlier term's coefficient in other cases.","For $f_2$ we first replace $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ .","Since $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a} (k)$ commute, they are simultaneously diagonalizable in the sense of Theorem REF , and therefore their spectra are related by the function $x\\mapsto \\tfrac{x}{1-\\lambda x}$ (monotone increasing on $\\left(-\\infty ,1/\\lambda \\right)$ ).","Therefore we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op} = \\sup _{x\\in \\sigma \\left(\\widehat{a} (k)\\right)}*{\\frac{x}{1-\\lambda x}}$ .","Now (perhaps with the aid of Figure REF ) it is easy to see that $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\vee \\mathbb {S}\\left(-\\widehat{a}(k)\\right) = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) \\vee \\mathbb {S}\\left(-\\widehat{a} (k)\\right).$ Now since $\\mathbb {S}\\left(-\\widehat{a}(k)\\right) \\le \\left\\Vert \\widehat{a}(k)\\right\\Vert _{\\rm op}\\le 1 + c_{\\kappa +1}\\beta $ , and $\\widehat{G}_{\\mu _\\lambda }(k)^{-1} \\le 1+C$ (recall Lemma REF and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C$ from REF ), we only need to bound $\\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We now partition $k\\in \\mathbb {R}^d$ into two.","Define the subset of the Fourier space $A_{\\frac{1}{2}} := \\left\\lbrace k\\in \\mathbb {R}^d:\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\le \\frac{1}{2}\\right\\rbrace .$ Then we only need to bound the two functions $f_4,f_5$ , where $f_4\\left(\\lambda \\right) := \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)},\\qquad \\qquad f_5\\left(\\lambda \\right):= \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}^c} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We first deal with the simpler $f_4$ .","We consider $k\\in A_{\\frac{1}{2}}$ and choose $\\beta $ small enough that $1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right) \\ge 1 - \\frac{\\frac{1}{2}+c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta } \\ge \\frac{1}{2} - 2c_{\\kappa +1}\\beta > 0.$ Then $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\frac{1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}}{\\frac{1}{2} - 2c_{\\kappa +1}\\beta } \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right).","$ Since we have $\\mathbb {S}\\left(\\widehat{a}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{a}(0)\\right) \\le 1 + c_{\\kappa +1}\\beta $ , we then get $f_4(\\lambda ) \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right)\\left(1 + c_{\\kappa +1}\\beta \\right) \\le 2\\left(1+C\\right)\\left(1+7c_{\\kappa +1}\\beta \\right).$ To address $f_5$ we will have $k\\in A^c_\\rho $ .","We define: $\\widehat{N}(k) = \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{F}(k) = \\frac{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{Q}(k) = \\frac{1+\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}.$ Note that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\widehat{N}(k) / \\widehat{F}(k)$ .","Rearranging gets us to $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &= \\widehat{N}(k) \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)}\\nonumber \\\\&= \\widehat{Q}(k) + \\frac{\\widehat{N}(k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)} \\left(1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - \\frac{\\widehat{Q}(k)}{\\widehat{N}(k)}\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)\\right) .$ Since $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ (by our scaling choice), the extracted term $\\widehat{Q}(k)$ satisfies $*{\\widehat{Q}(k)} \\le 1 + 3c_{\\kappa +1}\\beta $ .","We further observe that $\\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{Q}(k)}{\\widehat{N}(k)} = \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\left(1+ \\delta _\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) +\\delta _\\lambda (k)} = 1 - \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right) \\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} =: 1- \\widehat{b}(k).$ Recalling identity (REF ) for $\\mu _\\lambda $ , we can rewrite the quantity $\\left[1- \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)- (1-\\widehat{b}(k)) \\widehat{F}(k)\\right]$ , appearing in (REF ), as $\\frac{1+\\delta _\\lambda (0) - \\left[\\lambda + \\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right] \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - 1 + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)+\\delta _\\lambda (k)\\right) + \\widehat{b}(k)\\left(1-\\lambda \\widehat{a}(k)\\right)}{1+\\delta _\\lambda (0)} \\\\= \\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}+ \\frac{\\widehat{b}(k)(1-\\lambda \\widehat{a}(k))}{1+\\delta _\\lambda (0)}.$ Noting that ${\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)} \\le 2c_{\\kappa +1}\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\beta $ by Lemma REF and Observation REF , the first term can be bounded $&*{\\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}} \\nonumber \\\\&\\qquad \\le \\frac{ c_{\\kappa +1}\\beta + \\left(1 + 2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta + 2\\left(1+2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta }{1 - c_{\\kappa +1}\\beta }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 5c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 6c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Using (REF ), the last term is $*{\\frac{\\widehat{b}(k)(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right))}{1+\\delta _\\lambda (0)}} &= *{ \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\delta _\\lambda (k)}{\\left(1+\\delta _\\lambda (0)\\right)\\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)\\right)}- \\frac{\\lambda \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\delta _\\lambda (k)}{1+\\delta _\\lambda (0)}} \\nonumber \\\\& = \\frac{*{\\delta _{\\lambda }(k)}}{*{1+\\delta _\\lambda (0)}}*{ \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} -\\lambda }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le \\frac{c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta }\\left( \\frac{1}{\\frac{1}{2}-c_{\\kappa +1}\\beta } + \\frac{1}{1-c_{\\kappa +1}\\beta }\\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\le c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left(3+8c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le 4c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Putting these bounds back into (REF ), we can find constants $M$ and $\\tilde{M}$ such that $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &\\le \\widehat{Q}(k) + 10c_{\\kappa +1}\\beta \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\frac{\\widehat{N}(k)}{\\widehat{F}(k)} \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\ \\\\& \\le 1+ 3c_{\\kappa +1}\\beta + 20c_{\\kappa +1}\\beta \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)} \\\\& \\le 1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta .","$ Note that we have used the bound $\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\le \\left[1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]$ (since $\\mu _\\lambda \\le 1$ and $\\mathbb {S}\\left(\\widehat{}(k)\\right)>\\frac{1}{2} >0$ ) to get from $\\widehat{G}_1(k)^{-1}$ to $\\widehat{G}_{\\mu _\\lambda }(k)^{-1}$ .","As $\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\in \\left(\\frac{1}{2}, 1\\right]$ , this concludes the improvement of $f_5$ and hence of $f_2$ .","Before we treat $f_3$ , we introduce $f_6$ given by $f_6(\\lambda ) := \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}\\left(k\\right)\\right)}.$ We show that $f(\\lambda ) \\le \\kappa +1$ implies $f_6(\\lambda ) \\le M$ uniformly for some $\\kappa $ -independent constant $M$ .","For $k\\in A_\\frac{1}{2}$ we showed in (REF ) that we can bound uniformly with $3\\left(1+C\\right)$ for sufficiently large small $\\beta $ .","For $k \\in A_{\\frac{1}{2}}^c$ , we have $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{G}_{\\mu _\\lambda }(k)} \\cdot \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) *{ 1- \\frac{\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}} \\\\ \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) \\left(1+\\frac{c_{\\kappa +1}\\beta }{\\frac{1}{2}- c_{\\kappa +1}\\beta } \\right) \\le 1+ \\left(20\\kappa + 27\\right)c_{\\kappa +1}\\beta .","$ Note that for the first bound in (REF ), we used the estimate established in (REF ), which is stronger than a bound on $f_5$ .","We therefore have the uniform bound $f_6(\\lambda ) \\le 3 + 3C$ for $\\beta $ sufficiently small.","Let us now improve the bound on $f_3$ .","It will be notationally convenient to denote $A\\left(k\\right) := \\left[1- \\lambda \\widehat{a}\\left(k\\right)\\right]^{-1}$ .","Since $\\widehat{a}\\left(k\\right)$ is self-adjoint, it can be “diagonalized\" in the sense of Theorem REF .","Therefore $A\\left(k\\right)$ can be “diagonalized\" over the same space and with the same unitary operator - it will have the diagonal function $\\widetilde{A}(k)(e) = 1/\\left(1-\\widetilde{a}\\left(k\\right)(e)\\right)$ .","It is then clear that $A\\left(k\\right)$ and $\\widehat{a}\\left(k\\right)$ commute (for each $k\\in \\mathbb {R}^d$ ).","Using Observation REF , we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} &= \\left\\Vert \\widehat{a}\\left(l\\right)A\\left(l\\right) - \\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op}\\nonumber \\\\&= \\left\\Vert A\\left(l\\right)\\widehat{a}\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l+k\\right)\\right]A\\left(l+k\\right) - A\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l\\right)\\right]\\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op} \\nonumber \\\\&= \\left\\Vert A(l)\\left(\\widehat{a}(l)-\\widehat{a}(l+k)\\right)A(l+k)\\right\\Vert _{\\rm op} \\nonumber \\\\&\\le \\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}\\left\\Vert A(l)\\right\\Vert _{\\rm op}\\left\\Vert A(l+k)\\right\\Vert _{\\rm op}.","$ For the $\\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}$ factor, we use the triangle inequality to separate this into a term of $\\widehat{}$ and $\\widehat{}_{\\lambda }$ .","Lemma REF immediately gives $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ for the $C>0$ arising from REF .","For the $\\widehat{}_{\\lambda }$ term, a similar argument to (REF ) gives $\\left\\Vert \\widehat{}_{\\lambda }\\left(l\\right) - \\widehat{}_{\\lambda }\\left(1+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(0\\right) - \\widehat{}_{\\lambda }\\left( k\\right)\\right\\Vert _{1,\\infty } \\le c_{\\kappa +1}\\beta \\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where the last inequality follows from Observation REF .","We then deal with the remaining factors of (REF ) by noting that for all $k\\in \\mathbb {R}^d$ , $\\left\\Vert A(k)\\right\\Vert _{\\rm op} = \\frac{1}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\widehat{G}_{\\mu _\\lambda }(k) \\le 3\\left(1+C\\right) \\widehat{G}_{\\mu _\\lambda }(k), $ employing our improved bound on $f_6$ .","The result of this is that we have the bound $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C + c_{\\kappa +1}\\beta \\right)\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right).$ Then since $\\widehat{G}_{\\mu _\\lambda }(k)\\ge 0$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C+c_{\\kappa +1}\\beta \\right)\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right).$ [Proof of Proposition REF -REF ] The continuity of $f_1$ is obvious.","For the other two functions we prove equicontinuity of a family of functions and use [41] to show the continuity of the desired functions.","The procedure is outlined here.","Suppose we wish to show that $H\\left(\\lambda \\right) := \\sup _{\\alpha \\in B}\\left\\Vert h_\\alpha \\left(\\lambda \\right)\\right\\Vert _{\\rm op}$ is continuous on $\\left[0,\\lambda _O\\right)$ .","For our purposes the parameter $\\alpha $ will be either $k$ or $\\left(k,l\\right)$ and thus $B=\\mathbb {R}^d$ or $B=\\mathbb {R}^{2d}$ .","The continuity on the half-open interval is implied by having continuity on the closed interval $\\left[0,\\lambda _O-\\rho \\right]$ for any $\\rho >0$ .","For closed intervals, [41] gives continuity if the family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous and $H\\left(\\lambda \\right)<+\\infty $ for all $\\lambda $ in the closed interval.","The family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous if for all $\\varepsilon >0$ there exists $\\delta >0$ such that $|s-t|<\\delta $ implies $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\varepsilon $ uniformly in $\\alpha \\in B$ .","In fact, the reverse triangle inequality implies that $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\left\\Vert h_\\alpha (s)-h_\\alpha (t)\\right\\Vert _{\\rm op},$ and therefore we only need to prove equicontinuity for the un-normed family $\\left\\lbrace h_\\alpha \\right\\rbrace _{\\alpha \\in B}$ .","We prove equicontinuity by bounding the `near-derivative' $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert h_\\alpha (\\lambda +\\varepsilon )-h_\\alpha (\\lambda )\\right\\Vert _{\\rm op}$ uniformly in $\\alpha \\in B$ for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ and arbitrary $\\rho >0$ .","For $f_2$ the operator-valued functions $h_\\alpha $ are $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)/\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)$ .","By using a variation on the chain rule, we get $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)$ is differentiable with the bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2$ .","We also have $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ from Lemma REF and $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ immediately from the definition.","It remains to deal with the $\\mu _\\lambda $ term.","Recall that $\\mu _\\lambda := 1 - \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^{-1}$ .","Therefore, using the reverse triangle inequality, $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda } = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)\\right)-\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} \\\\\\le \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)-\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2,$ where we have used $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ from Lemma REF in the last inequality.","Therefore $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + *{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2\\\\\\le \\left(1 + C\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + C\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^3.$ We therefore have a finite $k$ -independent bound.","This has proven the equicontinuity.","For the uniform boundedness, we note that for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}<\\infty $ .","In conjunction with $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ we have the required uniform boundedness and therefore the continuity of $f_2$ .","We repeat this approach for $f_3$ .","The corresponding step to (REF ) now reads $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda +\\varepsilon }}\\left(k,l\\right)} - \\frac{\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)$ is differentiable with the operator norm bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le 4\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}$ .","The operator itself has the similar bound $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}$ .","We would like to have the bound (REF ), but this was proven under the assumption $f\\le \\kappa +1$ , which we no longer assume.","Fortunately, we only require our bound for $\\lambda < \\lambda _O -\\rho $ .","Therefore we can use Lemma REF and Lemma REF to get $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\text{e}^{4\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\\\\\le C\\text{e}^{4\\left(\\lambda _O-\\rho \\right)\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}}\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ From this and $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge \\left(1+C\\right)^{-1}$ , there exists a constant $\\widetilde{M}>0$ such that both $\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M} \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}, \\qquad \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M},$ for sufficiently large $d$ .","Now recall $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ and $\\left(1+C\\right)^{-1}\\le \\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\le \\left(1-\\mu _\\lambda \\right)^{-1} = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}$ .","Therefore an application of the chain rule gives the bound $*{\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }} \\le 6\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^4\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op},$ uniformly in $k$ and $l$ .","These properties, along with the bound on $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }$ from above, are sufficient to prove equicontinuity and the uniform boundedness for the required $\\lambda $ .","Therefore we have proved that $f_3\\left(\\lambda \\right)$ is continuous.","It just remains to prove that we can indeed bound the difference of $\\delta _\\lambda (k)$ with the difference of $\\widehat{}_{\\lambda }(k)$ as we claimed.","[Proof of Lemma REF ] First note that there is nothing to prove if $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}=\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}=0$ , because the triangle inequality then forces $\\delta _\\lambda (k_1) = \\delta _\\lambda (k_2)=0$ .","Without loss of generality, assume $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ with $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}>0$ .","Then $\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2) = \\frac{\\delta _\\lambda (k_1)}{\\left\\Vert \\widehat{}_{\\lambda }(k_1)\\right\\Vert _{\\rm op}}\\left(\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}\\right) + \\left[\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)\\right]$ The second factor is the more troublesome: $*{\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)} = \\frac{1}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\delta _\\lambda (k_1)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op} - \\delta _\\lambda (k_2)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} \\\\\\le \\frac{*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}.$ Since $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ , we have $*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}$ .","Therefore $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}} \\le 2\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)-\\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op},$ where the last inequality holds because of the reverse triangle inequality.","Proof of Main Theorems Proposition 7.1 Let $d>6$ be sufficiently large.","Then for $\\lambda \\le \\lambda _O$ there exists $c>0$ such that $\\triangle _\\lambda < c\\beta ^2$ .","Proposition REF gives us that there exists $c_f$ (increasing in $f$ ) such that $\\triangle _\\lambda \\le c_f\\beta ^2$ for all $\\lambda <\\lambda _O$ .","Proposition REF then implies that there exists $c$ such that $c_f\\le c$ uniformly for all $\\lambda < \\lambda _O$ .","It remains to prove the assertion for $\\lambda =\\lambda _O$ .","Recall from Section REF the definition of $\\tau _\\lambda ^n(x,y)$ and $\\Lambda _n(x)$ .","Furthermore, recall that if $x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}$ then such a connection is achieved in finitely many steps and the bound in $\\ref {Assump:2ndMoment}$ then implies that $\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right\\rbrace = \\bigcup ^\\infty _{n=1}\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}_{\\Lambda _n(y)}\\right\\rbrace .$ Therefore by monotone convergence we have the pointwise limit $\\tau _\\lambda ^n(x,y) \\rightarrow \\tau _\\lambda (x,y)$ as $n\\rightarrow \\infty $ for all $\\lambda >0$ and $x,y\\in \\mathbb {X}$ .","Since $\\tau _\\lambda ^n(x,y)$ only depends upon the finite region $\\Lambda _n(y)$ , the functions $\\lambda \\mapsto \\tau _\\lambda ^n(x,y)$ are continuous for all $x,y\\in \\mathbb {X}$ .","This, with the monotonicity $\\tau _\\lambda ^n(x,y) \\le \\tau _\\lambda ^{n+1}(x,y)$ , implies that the function $\\lambda \\mapsto \\tau _\\lambda (x,y)$ is lower-semicontinuous.","Since $\\tau _\\lambda (x,y)$ is non-decreasing in $\\lambda $ , this lower semi-continuity implies that the pointwise limit $\\tau _\\lambda (x,y)\\uparrow \\tau _{\\lambda _O}(x,y)$ as $\\lambda \\uparrow \\lambda _O$ holds, and the limit holds monotonically.","This monotonic pointwise convergence implies that the integral $\\int \\tau _\\lambda (x,u)\\tau _\\lambda (u,v)\\tau _\\lambda (v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) \\rightarrow \\int \\tau _{\\lambda _O}(x,u)\\tau _{\\lambda _O}(u,v)\\tau _{\\lambda _O}(v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right)$ for all $x,y\\in \\mathbb {X}$ as $\\lambda \\uparrow \\lambda _O$ , and that this convergence is monotone increasing.","If we take the supremum over $x,y\\in \\mathbb {X}$ , we get that $\\lim _{\\lambda \\uparrow \\lambda _O}\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } = \\left\\Vert \\mathcal {T}_{\\lambda _O}^3\\right\\Vert _{\\infty ,\\infty }$ .","The uniform bound for $\\lambda <\\lambda _O$ then implies that this bound also holds at $\\lambda =\\lambda _O$ .","The following proposition uses the bound $\\triangle _{\\lambda _O} < c\\beta ^2$ to show that percolation does not occur at criticality.","Another element of the proof is that there is almost surely at most one infinite cluster.","There are general considerations that show that this is the case.","Indeed [14] establishes that on the discrete space $\\mathbb {Z}^d$ there is at most one infinite cluster if the edge occupation measure is stationary and obeys the ‘finite energy property’, and an analogous result for Poisson processes in the continuum applies in our case (see [4], [36]).","Proposition 7.2 The critical percolation function $\\theta _{\\lambda _O}(a)=0$ for $\\mathcal {P}$ -almost every $a\\in \\mathcal {E}$ .","Assume that $(\\overline{0},a),(\\overline{x},b)\\in \\mathbb {X}$ are in the (a.s. unique) infinite cluster, and are therefore connected.","Then the FKG inequality implies that $\\theta _{\\lambda _O}(a)\\theta _{\\lambda _O}(b)\\le \\mathbb {P}_{\\lambda _O}\\big (|\\overline{0},a)|=\\infty ,|\\overline{x},b)|=\\infty \\big )\\le \\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x},b)).$ Since $\\triangle _{\\lambda _O} \\le c\\beta ^2$ , for $\\mathcal {P}$ -a.e.","$a,b\\in \\mathcal {E}$ there exists a sequence $\\overline{x}^{(a,b)}_n$ such that $\\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x}^{(a,b)}_n,b))\\rightarrow 0$ .","Therefore for $\\mathcal {P}$ -a.e.","$a\\in \\mathcal {E}$ we have $\\theta _{\\lambda _O}(a)=0$ .","We can define the linear operator $\\mathcal {T}_{\\lambda _O}$ by its action on $f\\in L^2(\\mathbb {X})$ : $\\mathcal {T}_{\\lambda _O}f(x) = \\int \\tau _{\\lambda _O}(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right).$ Contrary to $\\mathcal {T}_\\lambda $ for $\\lambda <\\lambda _O$ , the operator $\\mathcal {T}_{\\lambda _O}$ may be an unbounded linear operator.","Proposition 7.3 (The operator OZE at the critical point) Let $d>6$ be sufficiently large.","The Ornstein-Zernike equation then extends to $\\lambda _O$ in the sense that the sum of (unbounded) linear operators vanishes: $\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) = 0.$ Furthermore, for $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ the linear operator $\\widehat{\\mathcal {T}}_{\\lambda _O}(k)$ is a bounded linear operator and the following equality holds: $\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right) - \\widehat{}\\left(k\\right) - \\widehat{}_{\\lambda _O}\\left(k\\right) - \\lambda _O\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right)\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda _O}\\left(k\\right)\\right) = 0.$ Finally, there exists $c>0$ such that for all $k\\in \\mathbb {R}^d$ $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le c\\beta , \\qquad \\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}\\le c\\beta .$ Let $f\\in L^2(\\mathbb {X})$ .","Then by the triangle inequality, for $\\lambda <\\lambda _O$ ${\\left(\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right)\\right)f}_2 \\le {\\left(\\mathcal {T}_\\lambda - - _\\lambda - \\lambda \\mathcal {T}_\\lambda \\left(+ _\\lambda \\right)\\right)f}_2 \\\\+ {\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda - _{\\lambda _O}+ _\\lambda - \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) + \\lambda \\mathcal {T}_\\lambda \\left(+_\\lambda \\right)\\right)f}_2.$ The first norm on the right hand side vanishes, since the OZE holds for $\\lambda <\\lambda _O$ .","To prove that the left hand side vanishes, we aim to show that the second norm on the right hand side vanishes as $\\lambda \\uparrow \\lambda _O$ .","To do this, the triangle inequality implies that we only need to show that the following limits hold as $\\lambda \\uparrow \\lambda _O$ : ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2 &\\rightarrow 0, \\\\{\\left(_{\\lambda _O}- _\\lambda \\right)f}_2 &\\rightarrow 0,\\\\{\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2 &\\rightarrow 0.$ We begin with (REF ).","Recall from the proof of Proposition REF that $\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y) := \\tau _{\\lambda _O}(x,y) - \\tau _\\lambda (x,y)$ converges monotonically and pointwise to zero.","From the definition of the ${\\cdot }_2$ norm, ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\\\&\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes (3)}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right).$ The monotone convergence of $\\left(\\tau _{\\lambda _O}-\\tau _\\lambda \\right)(x,y)\\downarrow 0$ then implies that the limit (REF ) holds.","For (), we use the functions $h^{(n)}_\\lambda \\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ defined by $h^{(n)}_\\lambda (x,y) := {\\left\\lbrace \\begin{array}{ll}\\lambda ^{-1}\\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes (2n)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right) &:n\\ge 1\\\\\\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2 &: n=0.\\end{array}\\right.","}$ From Proposition REF and the proof of Proposition REF , we know that $0\\le \\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_\\lambda (x,y)$ for all $n\\ge 0$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in \\left(0,\\lambda _O\\right)$ - the non-negativity follows from the definition of $\\pi ^{(n)}_\\lambda $ .","The functions $ h^{(n)}_\\lambda $ can also be defined for $\\lambda =\\lambda _O$ , and since they are monotone increasing in $\\lambda $ , we have the $\\lambda $ -independent bound $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda <\\lambda _O$ and $x,y\\in \\mathbb {X}$ .","For the single-mark version of the model it is proven in [21] that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.","This same argument works for our multi-mark version, and therefore $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda \\le \\lambda _O$ .","The triangle inequality then implies that $*{\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)} = *{\\sum ^\\infty _{n=0}\\left(-1\\right)^n\\left( \\pi ^{(n)}_{\\lambda _O}(x,y) - \\pi ^{(n)}_{\\lambda }(x,y)\\right)} \\le 2\\sum ^\\infty _{n=0}h^{(n)}_{\\lambda _O}(x,y).$ It will be convenient to define $h_{\\lambda }\\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ as $h_\\lambda (x,y) := \\sum ^\\infty _{n=0}h^{(n)}_{\\lambda }(x,y)$ .","Then ${\\left(_{\\lambda _O}- _\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,z)f(y)f(z)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right) \\\\& \\le 4\\int h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right)\\nonumber \\\\& = 4\\int \\left(\\int h_{\\lambda _O}(x,y)*{f(y)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right).","$ By Schur's test, this integral is finite for all $f\\in L^2(\\mathbb {X})$ if both the values $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right) \\qquad \\text{ and } \\qquad \\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ are finite.","To prove this for the former, we repeat the arguments of Proposition REF and Section REF to bound it in terms of $\\lambda _O$ , the triangle diagrams, and other $\\lambda $ -independent terms.","The argument of Proposition REF with the uniform bounds of Observation REF then proves that $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ is indeed finite.","A similar argument also holds for $\\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ .","We now have an integrable function $h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}$ that dominates the integrand of (REF ).","Recall that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.","Thus the dominated convergence theorem implies that the limit () holds.","For (), we can use $h_{\\lambda _O}$ to get the bound ${\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2^2 \\\\\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,u)h_{\\lambda _O}(u,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,v)h_{\\lambda _O}(v,z)*{f(y)}*{f(z)} \\nu ^{\\otimes 5}\\left(\\mathrm {d}u,\\mathrm {d}v, \\mathrm {d}x, \\mathrm {d}y, \\mathrm {d}z\\right).$ The integrand converges monotonically to zero, and therefore the integral vanishes in the limit.","We now consider the Fourier version of the OZE.","We use the notation that, given a bounded linear operator $A\\colon L^2(\\mathbb {X}) \\rightarrow L^2(\\mathbb {X})$ , the Fourier transform is given by $\\mathcal {F}\\left( A \\right)(k)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ .","By hypothesis, $$ is a bounded operator, and the finiteness of $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ implies that $_{\\lambda _O}$ is also bounded.","Then (REF ) implies that $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) = \\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)$ for all $k\\in \\mathbb {R}^d$ .","We now make the claim that $\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)$ has a bounded linear inverse for all $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ .","This is proven in Lemma REF below.","This claim then implies that for $k\\ne 0$ we have $&\\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) \\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)^{-1} \\right)(k) = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k).$ That is, $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k)$ is a bounded linear operator for $k\\ne 0$ .","The limit () also allows us to extend our sub-critical bound on $_\\lambda $ to apply at criticality.","The limit () may be phrased as saying that the function $\\lambda \\mapsto *{_\\lambda f}_2$ is continuous at $\\lambda =\\lambda _O$ (from the left) for all $f\\in L^2\\left(\\mathbb {X}\\right)$ .","Since $\\lambda \\mapsto \\left\\Vert _\\lambda \\right\\Vert _{\\rm op}$ can be written as a supremum of such functions, it is a lower semi-continuous function.","Our bound $\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta $ for $\\lambda < \\lambda _O$ (from Proposition REF ) then implies $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le \\liminf _{\\lambda \\uparrow \\lambda _O}\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta .$ The limit $*{\\left(\\widehat{}_{\\lambda _O}^{(n)}(0) - \\widehat{}_{\\lambda }^{(n)}(0)\\right)f}_2 \\rightarrow 0$ for each $f\\in L^2\\left(\\mathcal {E}\\right)$ can be derived in essentially the same way as () above.","The above lower semi-continuity argument can then be used to show that the bound from Proposition REF extends to also hold at $\\lambda =\\lambda _O$ .","From Lemma REF and the positivity of $\\pi ^{(n)}_{\\lambda _O}(x,y)$ this bound also holds for all $k\\in \\mathbb {R}^d$ .","By summing over $n$ , we then get the desired bound on $\\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}$ .","Lemma 7.4 For $\\lambda = \\lambda _O$ and $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ , the bounded linear operator $\\left(1- \\lambda _O\\left(\\widehat{}(k) + \\widehat{}_{\\lambda _O}(k)\\right)\\right)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ has a bounded linear inverse.","We first introduce some notation.","Let $A\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ be a self-adjoint linear operator.","Then we define the minimum modulus to be $\\left\\Vert A\\right\\Vert _{\\rm min} := \\inf _{0\\ne f\\in L^2(\\mathcal {E})}\\frac{*{Af}_2}{{f}_2}.$ Note that it follows from the Spectral Theorem (Theorem REF ) that $A$ has a bounded linear inverse if and only if $\\left\\Vert A\\right\\Vert _{\\rm min}>0$ .","Now suppose that $A=A_\\lambda $ , and that for all $f\\in L^2(\\mathcal {E})$ the map $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .","Then the map $\\lambda \\mapsto \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}$ is upper semicontinuous on $\\left[0,\\lambda _O\\right]$ .","That is, for any sequence $\\lambda _n$ in $\\left[0,\\lambda _O\\right]$ such that $\\lambda _n\\rightarrow \\lambda $ , $\\limsup _{n\\rightarrow \\infty }\\left\\Vert A_{\\lambda _n}\\right\\Vert _{\\rm min} \\le \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}.$ We now consider $A_\\lambda = 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .","A similar argument to the way we proved () shows that $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .","Our result is then proven if we can show that for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ we have an $\\varepsilon = \\varepsilon (k)>0$ such that $\\left\\Vert 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right\\Vert _{\\rm min} \\ge \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","We will in fact show the stronger result that $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","Recalling the definition of $\\delta _\\lambda (k)$ from (REF ), we have $1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &= 1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\lambda \\delta _{\\lambda }(k) \\nonumber \\\\&= \\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{>0 \\text{ from Lemma~\\ref {lem:Optlam-a}}} + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) + \\lambda \\left(\\delta _{\\lambda }(0) - \\delta _{\\lambda }(k)\\right) \\nonumber \\\\& > \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1+\\mathcal {O}\\left(\\beta \\right)\\right).$ Here we have used Lemma REF to bound the difference of $\\delta $ with the difference of $\\widehat{}_{\\lambda _O}$ .","For sufficiently large $d$ , REF then proves $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","Note that in conjunction, Proposition REF , Proposition REF and Proposition REF prove the OZE equations of Theorem REF .","In particular, the former two prove that the OZE holds in the subcritical regime, and the latter extends this to criticality where possible.","Furthermore, Proposition REF gives the bounds on $_\\lambda $ and $\\widehat{}_{\\lambda }(k)$ subcritically whilst Proposition REF extends this bound to criticality.","Regarding Theorem REF , the bound on $\\triangle _\\lambda $ and the lack of percolation at criticality were proven at the beginning of this section.","The infrared bound component of the theorem is proven below.","[Proof of Theorem REF ] First note that there is nothing to prove for $k=0$ .","From Proposition REF and Lemma REF , we can write $\\widehat{\\mathcal {T}}_\\lambda (k) = \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1},$ for all $k\\in \\mathbb {R}^d$ if $\\lambda <\\lambda _O$ , and for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ if $\\lambda = \\lambda _O$ .","If we use the Spectral Theorem (Theorem REF ) to `diagonalize' the operator $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , we get a multiplication operator on some Hilbert space $L^2(\\mathfrak {E}_k,\\mu _k)$ taking values $\\widetilde{a}(k)(e)$ for $e\\in \\mathfrak {E}_k$ .","Since, $\\widehat{\\mathcal {T}}_\\lambda (k)$ is formed from $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , it can also be diagonalized over $L^2(\\mathfrak {E}_k,\\mu _k)$ , and takes values $\\widetilde{\\tau }_\\lambda (k)(e) = \\frac{\\widetilde{a}(k)(e)}{1 - \\lambda \\widetilde{a}(k)(e)}.$ Since $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) < 1$ (from the proof of Lemma REF ), we know $\\lambda \\widetilde{a}(k)(e) >1$ and $\\lambda \\widetilde{\\tau }_\\lambda (k)(e) \\ge -1$ for $\\mu _k$ -almost every $e\\in \\mathfrak {E}_k$ .","It therefore only remains to bound $\\widetilde{\\tau }_\\lambda (k)(e)$ from above.","Equivalently, we only need to bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)$ from above.","We also note that the proof of () can be adapted to show that $\\lambda \\mapsto \\left$ is continuous on $\\left[0,\\lambda _O\\right]$ for all $f\\in L^2(\\mathcal {E})$ .","Then because $\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) = \\sup _{f}\\tfrac{\\left}{\\left}$ , this proves that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)$ is lower-semicontinuous on $\\left[0,\\lambda _O\\right]$ .","The bound $\\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) <1$ for $\\lambda < \\lambda _O$ (from Lemma REF ) then implies that $\\lambda _O\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda _O}(0)\\right)\\le 1$ .","From (REF ) the calculation proceeds: $\\lambda \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) &= \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& = \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{\\ge 0} + \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) + \\delta _{\\lambda }(0) - \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\delta _{\\lambda }(k)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1 + \\mathcal {O}\\left(\\beta \\right)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)},$ where we have bounded $\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)$ using Lemma REF and Observation REF .","Critical Intensities [Proof of Proposition REF ] By the Mecke equation (REF ), we find $*{\\chi _\\lambda }_1 & = 1 + \\lambda \\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) = 1 + \\lambda \\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right),\\\\*{\\chi _\\lambda }_\\infty & = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\widehat{\\tau }_\\lambda (0;a,b) \\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }.$ From Lemma REF we have $\\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }$ .","Therefore $\\lambda ^{(1)}_T \\ge \\lambda _O \\ge \\lambda _T^{(\\infty )}$ .","Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(1)}_T$ .","Therefore we only now need to prove that $\\lambda ^{(1)}_T = \\lambda ^{(\\infty )}_T$ .","We do this by proving that if $*{\\chi _\\lambda }_1<\\infty $ then $*{\\chi _\\lambda }_\\infty <\\infty $ .","Fix $b\\in \\mathcal {E}$ .","Then by considering the vertices in the cluster $\\overline{0},b)$ adjacent to $\\left(\\overline{0},b\\right)$ , we find $\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\mathbb {E}_\\lambda \\left[\\sum _{\\left(\\overline{x},a\\right)\\in \\eta :\\left(\\overline{x},a\\right)\\sim \\left(\\overline{0},b\\right)}*{\\overline{x},a)}\\right] = 1 + \\lambda \\int \\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right]\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right).$ In this equality we have used Mecke's equation (REF ).","From the spatial translation invariance of the model, $\\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right] = \\mathbb {E}_\\lambda \\left[*{\\overline{0},a)}\\right]$ for all $\\overline{x}\\in \\mathbb {R}^d$ .","Therefore an application of a supremum bound to the $a$ -integral gives $*{\\chi _\\lambda }_\\infty = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)\\int \\mathbb {E}_\\lambda \\left[*{\\overline{0},c)}\\right]\\mathcal {P}\\left(\\mathrm {d}c\\right)\\\\= 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)*{\\chi _\\lambda }_1.$ From the finiteness of the parenthesised factor, $*{\\chi _\\lambda }_1<\\infty $ implies $*{\\chi _\\lambda }_\\infty <\\infty $ and the result is proven.","[Proof of Proposition REF ] The equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ holds from [35].","The equality $\\lambda _O = \\lambda _T^{(p)}$ follows from Proposition REF because the condition (REF ) clearly holds for the bounded radii model.","Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(p)}_T$ for all $p\\in \\left[1,\\infty \\right]$ .","Now note that for all $a,b\\in \\mathcal {E}$ we have that if $\\left(\\overline{0},\\dagger \\right)\\sim \\left(\\overline{x},b\\right)$ then $\\left(\\overline{0},a\\right)\\sim \\left(\\overline{x},b\\right)$ .","Therefore $\\lambda _c\\left(\\dagger \\right) \\ge \\lambda _c$ , and the equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ then means we only need to prove $\\lambda _T\\left(\\dagger \\right) \\le \\lambda _T^{(p)}$ for some $p\\in \\left[1,\\infty \\right]$ .","We will do so for $p=1$ .","Suppose that $\\lambda >\\lambda _T^{(1)}$ , so that $\\int \\mathbb {E}_\\lambda \\left[(\\overline{0};a\\right)\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\infty $ .","That is, the expected cluster size of a vertex with a random ($\\mathcal {P}$ -distributed) mark is infinite.","Note that we have an ordering of the marks in that if two radii $a,b$ satisfy $a\\ge b$ , then $\\varphi \\left(\\overline{x};a,c\\right) \\ge \\varphi \\left(\\overline{x};b,c\\right)$ for all $\\overline{x}\\in \\mathbb {R}^d$ and $c\\in \\mathcal {E}$ .","This also implies that $a\\mapsto \\mathbb {E}_\\lambda \\left[*{(\\overline{0},a}\\right]$ is a non-decreasing function.","This in turn implies that if $\\lambda >\\lambda _T^{(1)}$ and $a$ satisfies $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , then $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right) = \\infty $ .","Given some radius $a\\in \\mathcal {E}$ such that $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we define a new mark, $\\star _a$ , that forms connections according to $\\varphi \\left(\\overline{x};\\star _a,b\\right) = 1\\left\\lbrace *{\\overline{x}}\\le a\\right\\rbrace ,$ for all $b\\in \\mathcal {E}$ .","Note that if $b\\ge a$ then $\\varphi \\left(\\overline{x};\\star _a,c\\right) \\le \\varphi \\left(\\overline{x};b,c\\right)$ for all $c\\in \\mathcal {E}$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\le \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]$ .","We now consider the expected cluster size of vertices adjacent to $\\left(\\overline{0},\\star _a\\right)$ with radius $\\ge a$ .","We also restrict to cases for which there is an unique neighbour, and that neighbour has radius $\\ge a$ .","Since the probability of a connection is independent of the radius of the proposed neighbour, the distribution of the unique neighbour is equal to $\\mathcal {P}$ - conditioned upon having radius $\\ge a$ .","Therefore we have the lower bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\ge 1 + \\mathbb {P}_\\lambda \\left(\\exists !","\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right).$ As noted above, $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)=\\infty $ .","Furthermore, as $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , the random variable $\\#\\left\\lbrace \\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right\\rbrace $ obeys a Poisson distribution with strictly positive mean.","Therefore $\\mathbb {P}_\\lambda \\left(\\exists !","\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)>0$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right]=\\infty $ .","This in turn implies that $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right] =\\infty $ for all $a>0$ .","To relate this to $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]$ , we perform a similar lower bound.","Given a radius $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we restrict to cases where there is an unique neighbour of $\\left(\\overline{0},\\dagger \\right)$ with radius $\\ge a$ and bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right] \\ge 1+ \\mathbb {P}_\\lambda \\left(\\exists !","\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\dagger \\right),b\\ge a\\right) \\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right].$ As before, the probability factor is strictly positive and the expected cluster size term is infinite.","Hence $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]=\\infty $ and $\\lambda \\ge \\lambda \\left(\\dagger \\right)$ .","Therefore $\\lambda _T\\left(\\dagger \\right)\\le \\lambda ^{(1)}_T$ as required.","Model Properties Here we will prove that the models outlined in Section REF do indeed satisfy Assumption REF .","First we consider the single mark “finite variance models” considered by [21].","As proven in that reference, this includes the Poisson blob and Gaussian connection models.","Lemma B.1 Single mark finite variance models satisfy the conditions of Assumption REF .","First note that since $\\mathcal {E}$ is a singleton, the operators $\\widehat{}(k)$ are simply scalars.","In particular, this means that the conditions (REF ), and () of REF hold trivially.","The finiteness of $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is also equivalent to the finiteness of $\\int \\varphi (x)\\mathrm {d}x$ required by [21].","Since $\\widehat{}(k)$ is simply a scalar, assumption REF follows directly from the third finite variance condition of [21].","The condition REF is similarly a generalisation of the second finite variance condition, and therefore follows.","Specifically, the sets $B(x)$ are the $\\varepsilon $ -balls around $x\\in \\mathbb {R}^d$ where $0<\\varepsilon = p^2\\widehat{\\varphi }_{1,1}(k)f_1^2 + 2pq\\widehat{\\varphi }_{1,2}(k)f_1f_2 + q^2\\widehat{\\varphi }_{2,2}(k)f_2^2,$ and therefore the spectrum of $\\widehat{}(k)$ is equal to the set of eigenvalues of the matrix $\\begin{pmatrix}p^2\\widehat{\\varphi }_{1,1}(k) & pq\\widehat{\\varphi }_{1,2}(k)\\\\pq\\widehat{\\varphi }_{1,2}(k) & q^2\\widehat{\\varphi }_{2,2}(k)\\end{pmatrix}.$ This $2\\times 2$ matrix has characteristic polynomial given by $z \\mapsto z^2 - \\left(p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)\\right) z + p^2q^2\\left( \\widehat{\\varphi }_{1,1}(k)\\widehat{\\varphi }_{2,2}(k) - \\widehat{\\varphi }_{1,2}(k)^2\\right).$ Solving for roots of this quadratic then gives $\\mathbb {S}\\left(\\widehat{}(k)\\right) = \\frac{p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)}{2} + \\sqrt{\\left(\\frac{p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)}{2}\\right)^2 + p^2q^2\\widehat{\\varphi }_{1,2}(k)^2},$ where the square root takes the positive value.","Note that by the definition of the double mark finite variance model $p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)$ is the Fourier transform of a non-negative (or non-positive) function, and therefore $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ is a sum of terms maximised at $k=0$ .","Furthermore, each approaches that maximum quadratically and at least one of $\\widehat{\\varphi }_{1,1}(k)$ , $\\widehat{\\varphi }_{2,2}(k)$ , and $\\widehat{\\varphi }_{1,2}(k)$ uniquely achieves its maximum at $k=0$ (otherwise we have $\\varphi = 0$ almost everywhere).","Therefore REF is satisfied.","Since we have only finitely many marks, there are only finitely many sequences of marks of a given length.","Therefore since each of the $\\varphi _{i,j}$ individually satisfy the 1-mark analog of REF , the 2-mark version is satisfied by $\\varphi $ too.","It is not inconceivable that the above argument could be generalised to 3 or 4 mark systems.","The degree 3 and 4 characteristic polynomials arising from the matrices have roots that can be solved for in radicals.","It may be then that a particular clever rearrangement of the expression allows the required properties to be deduced like above.","Such an investigation is currently beyond the authors' patience.","Of course, the famous Abel-Ruffini Theorem states that no such solution in radicals exists for arbitrary polynomials of degree 5 and higher, and so this approach seems doomed for systems with such families of marks.","Lemma B.3 Space-mark factorisation models satisfy the conditions of Assumption REF .","Note that $\\widehat{}(k) = \\widehat{\\overline{\\varphi }}(k)\\mathcal {K}$ , where $\\mathcal {K}\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ is the self-adjoint linear integral operator with kernel function $K$ .","Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b}\\int K(a,b)\\mathcal {P}(\\mathrm {d}a)\\le 1$ , $\\mathcal {K}$ is a bounded operator and $\\mathbb {S}\\left(\\widehat{}(k)\\right) = {\\left\\lbrace \\begin{array}{ll}\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) \\ge 0\\\\-\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(-\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) < 0.\\end{array}\\right.","}$ Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}\\le 1$ , the finiteness of $\\widehat{\\overline{\\varphi }}(0)$ implies that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite, and the $d$ -independence of $\\mathcal {K}$ implies that REF holds.","Since $\\overline{\\varphi }(\\overline{x})\\ge 0$ , we have $*{\\widehat{\\overline{\\varphi }}(k)} < \\widehat{\\overline{\\varphi }}(0)$ for all $k\\ne 0$ .","Furthermore, $\\mathbb {S}\\left(\\mathcal {K}\\right) = \\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}$ implies that $\\mathbb {S}\\left(\\mathcal {K}\\right)\\ge \\mathbb {S}\\left(-\\mathcal {K}\\right)$ .","Then $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ inherits the properties required by REF from $\\widehat{\\overline{\\varphi }}(k)$ .","For REF , we note that the $d$ -independence of $\\mathcal {K}$ means that required decay properties are again inherited from $\\widehat{\\overline{\\varphi }}(k)$ .","Lemma B.4 The marked multivariate Gaussian model satisfies the conditions of Assumption REF .","Firstly note that for each $\\left(a,b\\right)\\in \\mathcal {E}^2$ the Gaussian structure of $\\varphi $ means that it factorises over the $d$ eigenvector directions of $\\Sigma (a,b)$ .","This factorisation ensures that that the $d$ -dimensional Fourier transform is the product of the 1-dimensional Fourier transforms.","We have $\\widehat{\\varphi }(k;a,b) = \\mathcal {A} \\exp \\left(-\\frac{\\pi ^2}{2}k^{\\intercal }\\Sigma (a,b)^{-1}k\\right).$ In particular, when $k=0$ the Fourier transform is $a,b$ -independent.","This means $\\widehat{}(0) = \\mathcal {A}\\mathbf {1}$ where $\\mathbf {1}$ is the integral operator with constant kernel function 1.","For $f\\in L^2(\\mathcal {E})$ we have ${f}{\\mathbf {1}f} &= \\int f(a) \\left(\\int f(b) \\mathcal {P}\\left(\\mathrm {d}b\\right)\\right) \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f\\right]^2,\\\\{f}{f} &= \\int f(a)^2 \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f^2\\right].$ By Jensen's inequality we therefore have $\\mathbb {S}\\left(\\mathbf {1}\\right) = 1$ (equality follows by considering the test function $f(a)\\equiv 1$ ).","It therefore follows that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = \\mathcal {A}<\\infty $ .","From (REF ) we have $\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } = \\mathcal {A}\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\exp \\left(-\\pi ^2k^{\\intercal }\\Sigma (a,b)^{-1}k\\right) \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This bound then implies the condition (REF ) of REF holds with $C=1$ .","Before we address () of REF , we turn to REF .","From our above bounds we already have $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) \\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This exponential term ensures that REF is satisfied.","We now return to () of REF .","For all $x\\in \\mathbb {R}$ we have $1-\\cos (x) \\le \\frac{1}{2}x^2$ , and therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\frac{1}{2}*{k}^2 \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x},$ for some unit vector $e\\in \\mathbb {R}^d$ .","Since the vector $e$ picks out the second moment of $\\varphi $ in only one direction, a standard calculation gives $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\Sigma _{\\max }^2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }.$ Therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty }$ is bounded by a $d$ -independent quadratic function.","In conjunction with the quadratic bound of REF , this then proves () in the $k\\rightarrow 0$ regime.","The remaining regime then holds because the triangle inequality implies $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le 2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } = 2\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","To address REF , recall that the spatial convolution of two multivariate Gaussian functions with means $\\overline{0}$ and covariance matrices $\\Sigma _1$ and $\\Sigma _2$ is a multivariate Gaussian function with mean $\\overline{0}$ and covariance matrix $\\Sigma _1+\\Sigma _2$ .","Therefore for all $\\overline{x}\\in \\mathbb {R}^d$ we have $&\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\int \\varphi (\\overline{x}-\\overline{y};a_1,a_2)\\varphi (\\overline{y};a_3,a_4)\\mathrm {d}\\overline{y}\\nonumber \\\\&\\hspace{14.22636pt}= \\mathcal {A}^2\\left(2\\pi \\right)^{-d/2}\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\left(\\det \\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)\\right)^{-1/2}\\exp \\left(-\\frac{1}{2}\\overline{x}^{\\intercal }\\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)^{-1}\\overline{x}\\right) \\nonumber \\\\& \\hspace{14.22636pt} \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2}\\exp \\left(-\\frac{*{\\overline{x}}^2}{4\\Sigma _{\\min }}\\right) \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2} \\rightarrow 0.$ In particular, this bounds the integral appearing in the definition of the sets $B(x)$ (see (REF )) and therefore proves they are empty.","For the bound of the convolution of three $\\varphi $ functions, we can get this immediately from the above calculation and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","Specifically REF holds with $g(d) = \\left(4\\pi \\Sigma _{\\min }(d)\\right)^{-d/2}$ and the sets $B(x) = \\emptyset $ .","Lemma B.5 The bounded-volume Boolean disc model satisfies the conditions of Assumption REF .","Crucial to the conditions of REF holding is the upper bound on $R_d(a)$ in (REF ).","We use the shorthand $R_{a,b} := R_d(a) + R_d(b),\\qquad V_{a,b} := \\frac{\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R^d_{a,b},$ so $V_{a,b}$ equals the Lebesgue volume of the $d$ -dimensional Euclidean ball with radius $R_{a,b}$ .","Note that the upper bound $R^{({\\rm max})}_d$ on $R_d(a)$ in (REF ) ensures $V_{a,b} \\le V^{({\\rm max})}$ uniformly in $d$ .","We first find $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int V_{a,b}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\le V^{({\\rm max})}.$ This proves the statement that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite in REF .","From the condition (REF ) we also get $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\ge \\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a, \\mathrm {d}b\\right) \\ge \\epsilon ^2\\frac{d\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)} \\int ^{2\\left(1-\\frac{c_2}{d}\\right)R^{({\\rm max})}_d}_0 r^{d-1}\\mathrm {d}r \\\\= \\epsilon ^2V^{({\\rm max})}\\left(1-\\frac{c_2}{d}\\right)^d \\rightarrow \\epsilon ^2\\text{e}^{-c_2}V^{({\\rm max})},$ where we have used the test function $f(a) \\equiv 1$ to lower bound $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","We then prove (REF ) by bounding $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } =\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}V_{a,b} \\le V^{({\\rm max})}.$ We will return for () after addressing REF .","For REF , it will be useful to have an expression for $\\widehat{\\varphi }(k;a,b)$ for each $a,b\\in \\left[0,1\\right]$ .","Suppose we are able to find $C_1\\in \\left(0,1\\right)$ and $C_2 >0$ such that for all $a,b\\in \\left[0,1\\right]$ $*{\\widehat{\\varphi }(k;a,b)} \\le \\widehat{\\varphi }(0;a,b) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right).$ Then since the $k$ -dependent factor is $a,b$ -independent we can use Lemma REF to show that $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{}(0)\\right) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right),$ and therefore prove that REF is satisfied.","In finding an expression for $\\widehat{\\varphi }(k;a,b)$ we are assisted by the spherical symmetry of the connection function and follow [18].","It transpires that $\\widehat{\\varphi }(k;a,b) = \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}}\\left(R_{a,b}*{k}\\right),$ where $J_{\\frac{d}{2}}$ is the Bessel function of the first kind of order $\\frac{d}{2}$ .","This function has the expansion $J_{\\frac{d}{2}}\\left(r\\right) = \\sum ^\\infty _{m=0} \\frac{\\left(-1\\right)^{m}}{m!\\Gamma \\left(\\frac{d}{2}+m+1\\right)}\\left(\\frac{r}{2}\\right)^{\\frac{d}{2}+2m},$ converging for all $r\\ge 0$ .","We will consider three different regimes for $*{k}$ .","For $r \\ll \\sqrt{d}$ the expansion (REF ) is asymptotic, and since $R^{({\\rm max})}_d$ and $R^{({\\rm min})}_d$ are both asymptotically proportional to $\\sqrt{d}$ we have $\\widehat{\\varphi }(k;a,b) \\sim \\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R_{a,b}^d - \\frac{\\pi ^\\frac{d}{2}}{4\\Gamma \\left(\\frac{d}{2}+2\\right)}R_{a,b}^{d+2}*{k}^2 = V_{a,b}\\left(1 - \\frac{R^2_{a,b}}{2\\left(d+2\\right)}*{k}^2\\right)$ for $*{k}\\ll 1$ .","This proves the quadratic part of the desired bound.","The Bessel function $J_{\\frac{d}{2}}$ is bounded and achieves its global maximum (in absolute value) at its first non-zero stationary point, denoted $j^{\\prime }_{\\frac{d}{2},1}$ .","From [2], we have $j^{\\prime }_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _1 \\approx 0.81$ , and $*{J_{\\frac{d}{2}}\\left(j^{\\prime }_{\\frac{d}{2},1}\\right)} \\le M d^{-\\frac{1}{3}}$ for a $d$ -independent $M>0$ .","Since $R_{a,b} \\in \\left[2c_1,1/\\sqrt{2\\pi \\text{e}}+o(1)\\right]d^\\frac{1}{2}$ (for $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right]$ from (REF )) and $j^{\\prime }_{\\frac{d}{2},1} / R_{a,b} \\in \\left[\\sqrt{\\frac{\\pi \\text{e}}{2}}+o(1),\\frac{1}{4c_1}+o(1)\\right]d^{\\frac{1}{2}}$ we have $*{\\widehat{\\varphi }(k;a,b)} \\le \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\le \\left(2e^{-1} + o(1)\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\rightarrow 0$ for $*{k} \\ge j^{\\prime }_{\\frac{d}{2},1} / R_{a,b}$ .","We are now left with the intermediate range for $*{k}$ .","The first positive zero of $J_\\frac{d}{2}$ occurs at $j_{\\frac{d}{2},1}$ , where $j_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _2 \\approx 1.86$ (see [2]).","In particular, we will always have $j^{\\prime }_{\\frac{d}{2},1} < j_{\\frac{d}{2},1}$ .","From differential identities relating Bessel functions (see [18]), we have $\\frac{\\mathrm {d}}{\\mathrm {d}*{k}} \\widehat{\\varphi }(k;a,b) = -R_{a,b}\\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}+1}\\left(R_{a,b}*{k}\\right).$ The Bessel function $J_{\\frac{d}{2}+1}(r)$ is positive for $r>0$ until its first positive zero at $j_{\\frac{d}{2}+1,1}$ , and $j_{\\frac{d}{2}+1,1} > j_{\\frac{d}{2},1}$ .","Therefore $\\widehat{\\varphi }(k;a,b)$ is positive and decreasing on the whole region $*{k}\\in \\left(0,j_{\\frac{d}{2},1}/R_{a,b}\\right]$ .","Since this overlaps with the high $*{k}$ range, and we have the uniform quadratic behaviour near $k=0$ , the function $k\\mapsto *{\\widehat{\\varphi }(k;a,b)}$ can never increase and approach $\\widehat{\\varphi }(0;a,b)$ again after leaving a neighbourhood of $k=0$ .","We therefore have the bound (REF ) and thus have proven that REF is satisfied.","Figure: Sketch of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) against *k*{k}.","It approaches its maximum quadratically as *k→0*{k}\\rightarrow 0.","The first local maximum of J d 2 J_{\\frac{d}{2}} occurs at j d 2,1 ' ∼d 2+γ 1 d 2 1 3 j^{\\prime }_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3}.","The first zero of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) occurs at *kR a,b =j d 2,1 ∼d 2+γ 2 d 2 1 3 *{k}R_{a,b} = j_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} where γ 2 >γ 1 \\gamma _2>\\gamma _1.","Furthermore, ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) is strictly decreasing until *kR a,b =j d 2+1,1 ∼d 2+γ 2 d 2 1 3 +1 2*{k}R_{a,b} = j_{\\frac{d}{2}+1,1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\frac{1}{2}.We return to () in REF .","We first calculate the second moment: $\\int *{\\overline{x}}^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{d}{d+2}V_{a,b}R^2_{a,b}.$ From $*{\\overline{x}}^2 = \\overline{x}_1^2 + \\ldots + \\overline{x}^2_d$ (for an orthonormal basis) and the rotational symmetry, we have $\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{1}{d+2}V_{a,b}R^2_{a,b}$ for all unit vectors $e\\in \\mathbb {R}^d$ .","Therefore $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\frac{4}{d+2}V^{({\\rm max})}\\left(R^{({\\rm max})}_d\\right)^2.$ While $V^{({\\rm max})}$ is a constant, $R^{({\\rm max})}_d\\sim \\sqrt{d/8\\pi \\text{e}}$ and therefore we have a bound uniform in $d$ .","The argument now proceeds similarly to the corresponding part of the proof for the multivariate Gaussian model.","From (REF ) and REF we have both sides of () satisfying quadratic bounds around $k=0$ , and the triangle inequality with our other bounds prove that the bound also holds on the remainder.","For REF we first want to bound the convolution of three $\\varphi $ functions.","Given marks $a_1,\\ldots ,a_6\\in \\left[0,1\\right]$ and $\\overline{x}\\in \\mathbb {R}^d$ , let $\\mathbb {B}_{a,b}(\\overline{x})$ denote the ball of radius $R_{a,b}$ centred on $\\overline{x}$ , and let $1_{a,b}\\left(\\overline{x}\\right) = 1\\left\\lbrace *{\\overline{x}}\\le R_{a,b}\\right\\rbrace $ .","Then $\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6)\\right)\\left(\\overline{x}\\right)&= \\int 1_{a_1,a_2}(\\overline{y})1_{a_3,a_4}(\\overline{z}-\\overline{y})1_{a_5,a_6}(-\\overline{z})\\mathrm {d}\\overline{y}\\mathrm {d}\\overline{z}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)}\\left(\\int _{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)} 1_{a_3,a_4}(\\overline{z}-\\overline{y})\\mathrm {d}\\overline{z}\\right) \\mathrm {d}\\overline{y}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}.$ Then for $\\delta \\in \\left(0,1\\right)$ we have $&\\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}\\nonumber \\\\&\\qquad \\le \\int _{\\delta \\mathbb {B}_{a,b}\\left(\\overline{0}\\right)}*{\\mathbb {B}_{a_3,a_4}\\left(\\overline{0}\\right)}\\mathrm {d}\\overline{y}+ *{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} \\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} \\nonumber \\\\& \\qquad \\le \\left(V^{({\\rm max})}\\right)^2\\delta ^d + V^{({\\rm max})}\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}.$ Let $\\overline{e}_1\\in \\mathbb {R}^d$ be a unit vector.","Then by spherical symmetry we have $\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} = *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)}.$ We bound the volume of this intersection with the volume of the $d$ -ball with radius equal to the radius of the $d-1$ -sphere formed by the intersection of their boundaries.","This radius is equal to the length $h$ in Figure REF , which is maximised when $R_{a_5,a_6}$ and $R_{a_3,a_4}$ are maximised and $\\delta R_{a_1,a_2}$ is minimised.","Recall we defined the constant $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right)$ when we defined the model in Section REF .","Since we have $R_{a_5,a_6},R_{a_3,a_4} \\le 2R^{({\\rm max})}_d$ and $\\delta R_{a_1,a_2} \\ge \\delta c_1 d^\\frac{1}{2}$ , we have $h\\le \\sqrt{\\left(2 R^{({\\rm max})}_d\\right)^2 - \\delta ^2 c^2_1 d}$ - we have used the fact that the triangle is isosceles if both the $R_{a_5,a_6}$ and $R_{a_3,a_4}$ edges attain their maximum.","The ball with this radius then has the volume $\\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}h^d \\le V^{({\\rm max})}\\left(1 - \\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}}\\right)^\\frac{d}{2}.$ Since $\\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}} = 2\\pi \\text{e}\\delta ^2c_1^2\\left(1+o(1)\\right)$ and $c_1>0$ , this shows that $*{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)} \\le V^{({\\rm max})}\\left(1 - \\pi \\text{e}\\delta ^2c_1^2\\right)^\\frac{d}{2}$ for $d$ sufficiently large.","Sine we now have a mark independent bound with the required decay, we have proved the bound for the convolution of three $\\varphi $ functions.","Figure: For both the arguments, we bound the intersection of two dd-balls with a new dd-ball that has the radius hh in these diagrams.For finding the sets $B(x)$ in (REF ), we write $\\int \\varphi \\left(\\overline{x}- \\overline{y};a,c\\right)\\varphi \\left(\\overline{y};c,b\\right)\\mathrm {d}\\overline{y}\\mathcal {P}\\left(\\mathrm {d}c\\right) = \\int *{\\mathbb {B}_{a,c}\\left(\\overline{x}\\right)\\cap \\mathbb {B}_{c,b}\\left(\\overline{0}\\right)}\\mathcal {P}\\left(\\mathrm {d}c \\right).$ As before, we can bound the volume of this intersection independently of the mark $c$ .","We can bound it with the volume of the ball with radius $h$ where $h\\le \\sqrt{\\left(2R^{({\\rm max})}_d\\right)^2 - \\frac{1}{4}*{\\overline{x}}^2}$ (as can be seen in Figure REF ).","As before, if $*{\\overline{x}} \\ge \\kappa d^\\frac{1}{2}$ for some $\\kappa >0$ , then the volume of the ball with radius $h$ vanishes in the $d\\rightarrow \\infty $ limit.","Furthermore, if $\\kappa \\le \\frac{1}{\\sqrt{2\\pi \\text{e}}}$ then the volume of the ball with radius $\\kappa d^\\frac{1}{2}$ vanishes in the $d\\rightarrow \\infty $ limit.","Therefore assumption REF is satisfied with the sets $B(x)\\equiv B\\left(\\overline{x},a\\right)=\\left\\lbrace \\left(\\overline{y},b\\right)\\in \\mathbb {X}:*{\\overline{x}-\\overline{y}} \\le \\kappa d^\\frac{1}{2}\\right\\rbrace $ for any $\\kappa \\in \\left(0,\\frac{1}{\\sqrt{2\\pi \\text{e}}}\\right)$ .","Displacement Diagram Case Distinctions This section provides the details omitted from the proof of Proposition REF .","Here we show which diagrams arise from displacements crossing a $\\Psi $ segment, and whether they can be dealt with by the first sub-case, $\\mathrm {\\left(I\\right)}$ , or by the second sub-case, $\\mathrm {\\left(II\\right)}$ .","Recall that $\\mathrm {\\left(I\\right)}$ are those that use a $\\Psi $ decomposition of the larger diagram only (and the displacement is not crossing a $\\Psi ^{(4)}$ segment), while $\\mathrm {\\left(II\\right)}$ are those that will require a combination of a $\\Psi $ decomposition and a $\\overline{\\Psi }$ decomposition.","We first expand out the displaced $\\Psi $ terms and label which are in sub-case $\\mathrm {\\left(I\\right)}$ and which are in sub-case $\\mathrm {\\left(II\\right)}$ .","Note that the displacement may be across the `top' or the `bottom' of the segments in the usual orientation.","If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(1)}$ term, we use cosine-splitting and sometimes expand $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (2,0.8);(1,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[below]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad \\right)\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}\\quad \\right).$ Using the same ideas, if we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(2)}$ term we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };(0.75,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\right.\\nonumber \\\\&\\hspace{28.45274pt}\\left.\\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\quad \\right).$ If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(3)}$ term, then by symmetry in both cases we get $\\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}.$ We first show the bounds for the single segments in sub-case $\\mathrm {\\left(I\\right)}$ .","Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2.$ We first consider the displaced $\\Psi ^{(j)}$ term by itself.","By using supremum bounds and spatial translations we arrive at $\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , \\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda .$ For most of these, it is sufficient to separate the pair using a supremum bound and use the individual bound above for the displaced $\\Psi ^{(j)}$ term and the bound on the $\\Psi $ or $\\Psi _n$ term from Lemma REF .","The only pairs that need a more careful approach are those that have the bound $W_k\\triangle ^\\circ _\\lambda $ above for the displaced $\\Psi ^{(j)}$ , followed by a $\\Psi ^{(4)}$ or $\\Psi ^{(2)}_n$ term.","In these cases we consider the pair together and get the diagram bounds $\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2,\\qquad \\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2.$ In both of these cases we bounded a $\\tau _\\lambda $ edge by 1 (essentially omitting the edge) before applying a supremum bound.","In the first we omit the diagonal edge and split the upper supremum vertex in two.","In the second we omit the left-most vertical edge and apply a supremum bound at the bottom right vertex.","We now turn our attention to sub-case $\\mathrm {\\left(II\\right)}$ .","Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda .$ Also recall that if the $\\overline{\\Psi }_0$ term is actually a $\\overline{\\Psi }^{(2)}_0$ term, then the $\\tau _\\lambda ^\\circ $ edge connecting them is actually just a $\\tau _\\lambda $ edge.","We first investigate bounds for the displaced $\\Psi ^{(j)}$ term with the connecting $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges.","By using supremum bounds, spatial translations, splitting supremum vertices in two, and sometimes expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction, we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(3,0.6) -- (2,0.6) -- (2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);\\node [mark size=2pt] at (0,-0.6) {{square*}};[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(2,-0.6) -- (3,-0.6);(1.5,0.6) -- (3,0.6);[dashed] (3,0.6) -- (3,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (3,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(1.5,0.6) -- (3,0.6);[dashed] (0,0.6) -- (0,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\le \\overline{H_k}+ W_k\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda .$ For most of these, it is clear from a crude supremum bound that we have the bound $\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda $ for the pair of these with the subsequent $\\overline{\\Psi }$ or $\\overline{\\Psi }_0$ term (like we did for sub-case $\\mathrm {\\left(I\\right)}$ ).","The only pairs that need a more careful approach are those that have the bound $\\overline{W_k}\\triangle ^\\circ _\\lambda $ or $\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda $ above, followed by a $\\overline{\\Psi }^{(4)}$ or $\\overline{\\Psi }^{(2)}_0$ term.","We first consider the cases where they are followed by a $\\overline{\\Psi }^{(4)}$ term.","Here we use supremum bounds, spatial translations, and expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2.$ We now consider the cases where they are followed by a $\\overline{\\Psi }^{(2)}_0$ term.","Recall that for this case, the connecting $\\tau _\\lambda ^\\circ $ edge is in fact only a $\\tau _\\lambda $ edge.","We then get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ The remaining sub-case $\\mathrm {\\left(III\\right)}$ was fully dealt with in the main text.","Proof of Lemma REF Recall that we wish to prove the following result: * We begin by defining a sightly different object to $\\overline{H_k}$ .","Given $\\vec{a}=\\left(a_1,\\ldots ,a_{16}\\right)\\in \\mathcal {E}^{16}$ , define $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right) :=& \\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8) \\nonumber \\\\&\\hspace{28.45274pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _{\\lambda ,k}(\\overline{t}+ \\overline{x}_3-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u},$ so we have $\\overline{H_k}\\le f(\\lambda )^5\\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,\\vec{a}\\in \\mathcal {E}^{16}}H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{0};\\vec{a};k\\right).$ We then proceed to bound $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right)$ to get our result.","This form is preferred because it more easily indicates what the Fourier arguments should be when we take a Fourier transform.","A schematic Fourier diagram for this can be found in [21].","We then expand out the $\\tau _\\lambda $ and $\\tau _{\\lambda ,k}$ terms.","For each of the $\\tau _\\lambda $ terms we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of the cosine-splitting lemma, we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we only need to get appropriate bounds for the following objects.","For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^7\\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {H}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l_1)P^{j_2}_{3,4}(l_1)P^{j_3}_{5,6}(l_2)P^{j_4}_{7,8}(l_2)P^{j_5}_{9,10}(l_1-l_2)P^{j_6}_{11,12}(l_1-l_3)P^{j_7}_{13,14}(l_2-l_3) \\\\ \\times Q^{j_8}_{15,16}(l_3;k)\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}},$ where $P^{j}_{m,n}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_m}{\\widehat{}(l) g_n}} &:j=1\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &:j=2\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &: j=3,\\end{array}\\right.}","\\qquad \\text{for }m,n\\in \\left\\lbrace 1,2,\\ldots ,14\\right\\rbrace ,\\\\Q^{j}_{15,16}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &:j=1\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &:j=2\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{}_k(l) g_{16}}} &:j=3\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &:j=4\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &:j=5,\\end{array}\\right.","}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{16}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .","If we can bound these $\\mathcal {H}(k;\\vec{j})$ appropriately, then we can combine them to get the required bound.","We now recall pointwise bounds for the $P^{j}_{m,n}(l)$ and $Q^{j}_{15,16}(l)$ terms.","In the proof of Lemma REF we derived $*{*{g_m}{\\widehat{}(l)g_n}} &\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &\\le *{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}}\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}},$ and $*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{16}}{\\widehat{}(l)^2g_{16}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{15}}{\\widehat{}(l)^2g_{15}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(l)*{g_{15}}{\\widehat{}(l)^2g_{15}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k)*{g_{15}}{\\widehat{}(l)^2g_{15}}^\\frac{1}{2}*{g_{16}}{\\widehat{}(l)^2g_{16}}^\\frac{1}{2}$ where $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k)\\widehat{G}_{\\mu _\\lambda }(l+k).$ Also recall that we can bound $*{*{g_m}{\\widehat{}(l)g_n}}\\le C$ and $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ if required.","Note that all the bounds for the $Q^{j}_{15,16}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.","We therefore only need to show that the remaining integrals of $\\widehat{}$ and $\\widehat{G}_{\\mu _\\lambda }$ terms can all be bounded by some constant multiple of $\\beta ^2$ .","Also note that the bounds for $Q^{1}_{15,16}(l)$ , $Q^{2}_{15,16}(l)$ , and $Q^{3}_{15,16}(l)$ only differ by a uniform constant (after we bound $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ ), and so once we have dealt with one of these we have dealt with the other two.","Let us first consider $\\vec{j}= \\left(1,1,1,1,1,1,1,1\\right)$ .","After applying the uniform pointwise bound for $Q^{1}_{15,16}(l)$ , we use the Cauchy-Schwarz inequality to bound $&\\int *{*{g_1}{\\widehat{}(l_1)g_2}}*{*{g_3}{\\widehat{}(l_1)g_4}}*{*{g_5}{\\widehat{}(l_2)g_6}}*{*{g_7}{\\widehat{}(l_2)g_8}}*{*{g_9}{\\widehat{}(l_1-l_2)g_{10}}} \\nonumber \\\\&\\hspace{199.16928pt}\\times *{*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}}*{*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\&\\qquad \\le \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int *{g_7}{\\widehat{}(l_2)g_8}^2*{g_9}{\\widehat{}(l_1-l_2)g_{10}}^2*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.","$ To deal with these parentheses, we use a volume-preserving change of variables and factorise each into three integrals.","For example, $\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\\\= \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2\\frac{\\mathrm {d}l_1}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_5}{\\widehat{}(l_2)g_6}^2\\frac{\\mathrm {d}l_2}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_{13}}{\\widehat{}(l_3^{\\prime })g_{14}}^2\\frac{\\mathrm {d}l_3^{\\prime }}{\\left(2\\pi \\right)^{d}}\\right).$ There are now two types of integral we need to bound: $&\\int *{g_m}{\\widehat{}(l)g_n}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int \\left(\\int _{E_m\\times E_n \\times E_m \\times E_n} \\widehat{\\varphi }(l;a,b)\\widehat{\\varphi }(l;c,d) \\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\right) \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int _{E_m\\times E_n \\times E_m \\times E_n} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right)\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b)\\varphi (-\\overline{x};c,d)\\mathrm {d}\\overline{x}\\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b) \\mathrm {d}\\overline{x}\\le C, \\\\&\\int *{g_1}{\\widehat{}(l)g_2}^2 *{g_3}{\\widehat{}(l)g_4}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\le g(d).","$ In (REF ) we used $\\varphi \\left(-\\overline{x};c,d\\right)\\in \\left[0,1\\right]$ and REF .","In () we bounded the convolution of four adjacency functions using REF .","The result of this is that we can bound (REF ) by $C^{\\frac{5}{2}}g(d)^{\\frac{1}{2}}$ .","Since $g(d)\\le \\beta ^4$ , this satisfies the bound we require.","As noted above, this argument also proves the required bound for $\\vec{j}= \\left(1,1,1,1,1,1,1,2\\right)$ and $\\vec{j}= \\left(1,1,1,1,1,1,1,3\\right)$ .","Now suppose that we change some subset of $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace $ from having value 1 to having value 2.","The argument proceeds similarly, except that after we have factorised our integrals we now want to also bound integrals of the following types: $&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int \\left(\\int _{E_1\\times E_1\\times E_2\\times E_2}\\widehat{\\varphi }^{(2)}(l;a,b)\\widehat{\\varphi }^{(2)}(l;c,d)\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d)\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int _{E_1\\times E_1\\times E_2\\times E_2}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)(\\overline{0})\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d) \\nonumber \\\\& \\hspace{56.9055pt}\\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\le g(d), \\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)g_4}^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^2 g(d)\\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)^2g_3} *{g_4}{\\widehat{}(l)^2g_4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi ^{(2)}(\\cdot ;a_5,a_6) \\star \\varphi ^{(2)}(\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^4 g(d).$ Up to a constant, these are all less than or equal to the bounds we used for the $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace = \\left\\lbrace 1,\\ldots ,1\\right\\rbrace $ case, and therefore that case dominates.","These other cases would behave as constant multiple of one of $g(d)$ , $g(d)^{\\frac{3}{2}}$ , $g(d)^2$ , $g(d)^{\\frac{5}{2}}$ , and $g(d)^3$ .","We have now dealt with all the cases which have no factors of $\\widehat{G}_{\\mu _\\lambda }$ .","To demonstrate how we will deal with these, we first consider $\\vec{j}=\\left\\lbrace 3,3,3,3,3,3,3,5\\right\\rbrace $ .","For notational compactness, we define $\\left<\\widehat{}(l)^2\\right>_{I} := \\prod _{m\\in I}*{g_m}{\\widehat{}(l)^2g_m},$ where $I\\subset \\mathbb {N}$ and $l\\in \\mathbb {R}^d$ .","Since we have already extracted the factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ via the pointwise bound on $Q^{5}_{15,16}(l;k)$ , we are left trying to bound the following integral: $\\int \\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\\\ \\qquad \\times \\left<\\widehat{}(l_1)^2\\right>^{\\frac{1}{2}}_{1,2,3,4} \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_2)^2\\right>^{\\frac{1}{2}}_{9,10} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}.$ To bound this integral we need to partition our space.","Fix $\\varepsilon >0$ and define $B_1 &:= \\left\\lbrace *{l_1}< \\varepsilon \\right\\rbrace \\\\B_2 &:= \\left\\lbrace *{l_2}< \\varepsilon \\right\\rbrace \\\\B_3 &:= \\left\\lbrace *{l_1-l_3}< \\varepsilon \\right\\rbrace \\\\B_4 &:= \\left\\lbrace *{l_1-l_2}< \\varepsilon \\right\\rbrace \\\\B_5 &:= \\left\\lbrace *{l_2-l_3}< \\varepsilon \\right\\rbrace \\\\B_6 &:= \\left\\lbrace *{l_3}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3-k}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3+k}< \\varepsilon \\right\\rbrace ,$ and for $n=0,1,\\ldots ,6$ define $A_n := \\left\\lbrace (l_1,l_2,l_3)\\in \\left(\\mathbb {R}^{d}\\right)^3 \\colon \\#\\left\\lbrace m\\colon (l_1,l_2,l_3)\\in B_m \\right\\rbrace =n\\right\\rbrace .$ In words, $A_n$ is the set of points within $\\varepsilon $ (in Euclidean distance) of precisely $n$ of the hyperplanes on which the $\\widehat{G}_{\\mu _\\lambda }$ factors diverge.","If we are working with a factor of $\\widehat{G}_{\\mu _\\lambda }$ on the set $B_m$ associated with its relevant hyperplane, we use the bound $\\widehat{G}_{\\mu _\\lambda }(l)\\le 1/\\left(C_2*{l}^2\\right)$ , and if we are working off this set we use the bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ .","When we integrate over $A_0$ , we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ for all such factors and recycle the bound on the “only $\\widehat{}$ \" integrals from above.","The contribution from $A_0$ is therefore bounded by some constant multiple of $\\frac{1}{\\varepsilon ^{18}}g(d)^3.$ For $A_1$ , first suppose we are considering $A_1\\cap B_m$ for $m\\ne 6$ .","First we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\le 3/\\left(C_2\\varepsilon ^2\\right)$ for the “irrelevant\" directions, and use the pointwise bound for $\\left<\\widehat{}(l)^2\\right>_{m,n}$ in the “relevant\" direction.","Then we can use the Cauchy-Schwarz and factorise argument we used above and recycle the integral bounds for the $\\widehat{}$ factors as we did above.","However we are careful about the way we split apart terms in the Cauchy-Schwarz step.","We split the $\\widehat{G}_{\\mu _\\lambda }$ terms across both factors and arrange the remaining $\\widehat{}$ terms so that each factor has “directions\" spanning $\\left(\\mathbb {R}^d\\right)^3$ .","To demonstrate we consider $A_1\\cap B_1$ : $&\\int _{A_1\\cap B_1} \\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12} \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\& \\hspace{56.9055pt} \\le C^2\\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\left<\\widehat{}(l_2)^2\\right>_{5,6}\\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\& \\hspace{142.26378pt}\\times \\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>_{7,8} \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.$ After a change of variables, factorising and using our integral bounds of $\\widehat{}$ terms from above, we are only left needing to bound $\\int _{*{l}<\\varepsilon } \\widehat{G}_{\\mu _\\lambda }(l)^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\le \\frac{1}{C^2_2} \\int _{*{l}<\\varepsilon } \\frac{1}{*{l}^4} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-4}}{d-4}.$ Therefore the contribution from $A_1\\cap B_m$ for $m\\ne 6$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ .","For $A_1\\cap B_6$ we need to be slightly more careful because for $*{k}<2\\varepsilon $ the constituent parts of $B_6$ overlap - this is the same considerations we had to account for in the proof of Lemma REF .","We can overcome it in the same way by using Hölder's inequality (actually just Cauchy-Schwarz in this case) and then re-centring the integrals.","The net result is that the bound of a constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ also applies to $A_1\\cap B_6$ .","In summary, the contribution from $A_1$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2.$ The $A_2$ case proceeds similarly.","After partitioning the set according to which are the overlapping hyperplanes, we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions and one $\\widehat{}$ factor from a linearly independent direction, and pointwise bound all the other $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors.","The contribution from $A_2$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-4\\right)^2}g(d),$ arising from the $A_1\\cap B_1\\cap B_2$ , $A_1\\cap B_1\\cap B_6$ , and $A_1\\cap B_2\\cap B_6$ integrals.","The $A_3$ case is more complicated because there are two qualitatively different sub-cases to consider.","First suppose that the three normal vectors to the overlapping hyperplanes are linearly independent.","Then we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions, pointwise bound the other $\\widehat{G}_{\\mu _\\lambda }$ factors and all the $\\widehat{}$ factors.","The contribution from this sub-case of $A_3$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3}$ , arising from the $A_1\\cap B_1\\cap B_2\\cap B_6$ integral.","On the other hand, if the three normal vectors to the overlapping hyperplanes are linearly dependent, then we need to retain a $\\widehat{}$ from the remaining linearly independent direction.","To demonstrate, let us consider the $A_3\\cap B_1 \\cap B_2 \\cap B_4$ integral.","After removing unnecessary terms via uniform bounds, $&\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\& \\qquad \\le \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}.$ Applying a change of variables, factorising the integral, and applying Hölder's inequality allows us to bound these parentheses: $& \\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)\\left(\\int \\left<\\widehat{}(l)^2\\right>_{13,14}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^4}\\frac{1}{*{l_2}^4}\\frac{1}{*{l_1-l_2}^2} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right) g(d)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^6}\\frac{1}{*{l_2}^3} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{2}{3}}\\left(\\int _{\\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_2}^6}\\frac{1}{*{l_1-l_2}^6} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{1}{3}} g(d)\\nonumber \\\\& \\qquad = \\frac{1}{C^5_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{4}{3}}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-3}\\frac{\\varepsilon ^{d-3}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}g(d).$ For all of these linearly dependent cases, we can bound the integral by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d)$ .","In summary, the contribution from $A_3$ is bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3} + \\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d).$ For $A_4$ , $A_5$ , and $A_6$ we don't have the two linearly dependent/independent cases because any four of the directions spans $\\left(\\mathbb {R}^d\\right)^3$ .","We therefore don't need factors of $\\widehat{}$ to control the unbounded directions.","Instead, we need to be careful that when we split our integral at the Cauchy-Schwarz step the resulting parentheses have sufficiently few factors of $\\widehat{G}_{\\mu _\\lambda }$ that their bounds will be finite for $d>6$ .","The way we perform this split will be different for the different parts of $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k)$ .","Let us define $\\widehat{D}(l;k) := \\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k).$ We first consider $A_6$ .","By writing $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)$ , we have two integrals we want to bound.","First $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.","$ We then perform the $l_i$ integrals in specific orders.","For the first factor we integrate over $l_3$ (which factorises out already), then we integrate over $l_2$ for fixed $l_1$ , and finally integrate over $l_1$ .","For the second factor we integrate over $l_1$ for fixed $l_2$ and $l_3$ , then we integrate over $l_3$ for fixed $l_2$ , and finally we integrate over $l_2$ .","There are therefore three forms of integral we need to bound: $\\int _{*{l}<\\varepsilon }\\widehat{G}_{\\mu _\\lambda }(l)^3\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l}^2}\\frac{1}{*{l-l^{\\prime }}^4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\&\\le \\frac{1}{C^3_2}\\left(\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{1}{3}\\left(\\int _{*{l-l^{\\prime }}<\\varepsilon }\\frac{1}{*{l-l^{\\prime }}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{2}{3}\\nonumber \\\\& = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ In this last inequality, $C^{\\prime }>0$ is some uniform constant.","The last inequality is derived by applying Hölder's inequality in much the same way as for the second inequality, taking care to see when the three sets that are integrated over are overlapping.","The calculation uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .","The second integral we want to bound for $A_6$ is $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3-l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.","$ We then perform the $l_i$ integrals in specific orders.","For the first factor we integrate over $l_1$ (which factorises out already), then we integrate over $l_2$ for fixed $l_3$ , and finally integrate over $l_3$ .","For the second factor we integrate over $l_3$ for fixed $l_1$ and $l_2$ , then we integrate over $l_1$ for fixed $l_2$ , and finally we integrate over $l_2$ .","The new integrals we need to bound are: $\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}$ $\\int _{\\left(\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace \\right)\\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4}\\right)\\varepsilon ^{d-6}.$ The calculation of these two bounds uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .","The result of these bounds is that the integral in (REF ) is bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ For $A_4$ and $A_5$ we can repeat the argument for $A_6$ , but note that in some places the bound $1/\\left(C_2*{l}^2\\right)$ will be replaced by $1/\\left(C_2\\varepsilon ^2\\right)$ .","The net result of this is that the factor $\\left(d-6\\right)^3$ can be replaced by $\\left(d-6\\right)^{\\alpha _0}\\left(d-4\\right)^{\\alpha _1}\\left(d-2\\right)^{\\alpha _2} d^{\\alpha _3}$ for some $\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _3\\le 0$ such that $\\alpha _0+\\alpha _1+\\alpha _2+\\alpha _3 =3$ .","The net result is that the contribution from both $A_4$ and $A_5$ can also be bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ Above we have established bounds for the various parts of the integral when $\\vec{j}=\\left(3,3,3,3,3,3,3,5\\right)$ .","Having $j_8=4$ can also be dealt with by very similar arguments to that outlined above - the $l_3$ integrals are in fact simpler.","The above bounds still hold with the $\\varepsilon ^{-18}$ becoming a $\\varepsilon ^{-16}$ .","Since $\\varepsilon <1$ , this produces a larger bound.","Recall that up to a constant factor, having $j_m=1$ produces a larger bound than $j_m=2$ for $m=1,\\ldots ,7$ .","If we replace $j_m=3$ with $j_m=1$ , then after ignoring constant values we lose a factor of $\\varepsilon ^{-2}$ , but we may also lose a factor of $g(d)^\\frac{1}{2}$ unless that was for $m=1,2$ (corresponding to $l_1$ direction) or $m=3,4$ (corresponding to $l_2$ direction).","Note that doing for both directions will lose a $g(d)^\\frac{1}{2}$ factor.","If $\\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d= O\\left(g(d)^{\\frac{1}{2}}\\right)$ for all fixed $\\varepsilon >0$ , then it is clear that the dominating bound is $g(d)^\\frac{1}{2}$ .","This arises, for example from the bound for $\\vec{j}=\\left(1,1,1,1,1,1,1,1\\right)$ .","The situation is more complicated if $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , because we can choose to have $\\varepsilon =\\varepsilon (d)$ and take it to 0 as $d\\rightarrow \\infty $ .","It is easy to see that all the $A_1$ components (and $A_2,\\ldots ,A_6$ components) for each $\\vec{j}$ can be bounded by some constant multiple of $\\frac{1}{d}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\varepsilon ^d$ (provided $d>6$ ).","We therefore only need to compare this to the contributions from $A_0$ .","The bound we get from the above arguments for each $A_0$ contribution are of the form $\\varepsilon ^{-2m}g(d)^\\frac{n}{2}$ , where $m$ is the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ and $n$ is related to the multiplicity of $$ terms in each Fourier direction.","To find the terms with the largest bound, we want to have as many factors of $\\widehat{G}_{\\mu _\\lambda }$ as possible without producing extra factors of $g(d)^\\frac{1}{2}$ .","Since $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , increasing $n$ here instantly produces a smaller bound.","By changing $j_8$ from 1 to 4 or 5, or any other element from 1 to 3, we do gain a factor of $\\widehat{G}_{\\mu _\\lambda }$ but we also increase the number of $$ factors and therefore in most places we acquire extra factors of $g(d)^{\\frac{1}{2}}$ .","The exception to this is for $j_1$ and $j_2$ , which correspond to the $l_1$ directions, since they already produce a factor of $g(d)^{\\frac{1}{2}}$ .","The $A_0$ component of the $\\vec{j}=\\left(3,3,1,1,1,1,1,1\\right)$ term can be bounded by constant multiple of $\\varepsilon ^{-4}g(d)^{\\frac{1}{2}}$ .","We then optimise our choice of $\\varepsilon $ to have our two dominant bounds be of the same order.","This produces $\\varepsilon (d)^2 = g(d)^{\\frac{1}{d}}\\frac{2\\pi d}{\\text{e}}\\left(1+o(1)\\right)$ , and therefore our overall bound is given by $\\varepsilon ^{-4}g(d)^\\frac{1}{2} = g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(\\frac{\\text{e}}{2\\pi }\\right)^2\\frac{1}{d^2}\\left(1+o(1)\\right).$ Since $g(d)^{\\frac{1}{2}-\\frac{2}{d}} \\le g(d)^{\\frac{1}{2}-\\frac{3}{d}}$ for sufficiently large $d$ , this bound is of order $\\beta ^2$ and our bound is proven.","Acknowledgements.","This work is supported by Deutsche Forschungsgemeinschaft (project number 443880457) through priority program “Random Geometric Systems” (SPP 2265).","The authors thank the Centre de recherches mathématiques Montreal for hospitality during a research visit in spring 2022 through the Simons-CRM scholar-in-residence program."],["Diagrammatic bounds","For our argument, we will require bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}$ for all $n\\in \\mathbb {N}$ and $k\\in \\mathbb {R}^d$ .","However, it turns out that we can bound the last two remainder terms using others.","Lemma 5.1 For all $n\\in \\mathbb {N}$ , $k\\in \\mathbb {R}^d$ , and $\\lambda >0$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op},\\\\\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(0)\\right\\Vert _{\\rm op}.$ As $\\mathbb {P}_\\lambda (u_n {\\,\\,A\\,\\,} x\\textrm { in } \\xi ^{u_n,x}_{n+1} ) \\le \\tau _\\lambda (x,u_n)$ for an arbitrary locally finite set $A$ , the definition () gives $*{r_{\\lambda ,n}(x)} \\le \\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right).$ Let $f\\in L^2\\left(\\mathbb {X}\\right)$ .","Then this gives $*{R_{\\lambda ,n}f}^2_2 &\\le \\int \\left(\\int *{r_{\\lambda ,n}(x,y)}*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&\\le \\lambda ^2\\int \\left(\\int \\left(\\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right)\\right)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\lambda ^2\\int \\left(\\int \\tau _\\lambda (x,u)\\left(\\int \\pi _\\lambda ^{(n)}(u,y)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right) \\nu \\left(\\mathrm {d}u\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\& = \\lambda ^2 *{\\mathcal {T}_\\lambda _\\lambda ^{(n)}{\\rm Abs}f}^2_2.$ Note that we were able to use Tonelli's theorem to swap the $u$ and $y$ integrals because $\\tau _\\lambda $ and $\\pi ^{(n)}_\\lambda $ are both non-negative and measurable.","Finally, the approach of (REF ) implies the result.","For each $n\\ge 0$ , either $r_{\\lambda ,n}(x)\\ge 0$ for all $x\\in \\mathbb {X}$ or $r_{\\lambda ,n}(x)\\le 0$ for all $x\\in \\mathbb {X}$ .","Therefore $*{\\widehat{r}_{\\lambda ,n}(k;a,b)}\\le *{\\widehat{r}_{\\lambda ,n}(0;a,b)}$ for all $a,b\\in \\mathcal {E}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{R}_{\\lambda ,n}(0)\\right\\Vert _{\\rm op}$ (by Lemma REF ) for all $k\\in \\mathbb {R}^d$ .","From (REF ) we have $*{\\widehat{r}_{\\lambda ,n}(0;a,b)} \\le \\int \\widehat{\\tau }_\\lambda (0;a,c)\\widehat{\\pi }_\\lambda ^{(n)}(0;c,b) \\mathcal {P}\\left(\\mathrm {d}c\\right).$ Therefore the same argument as above gives the second bound in the result.","Because the $n=0$ cases for our bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ are so simple, we give the proof here before moving onto $n\\ge 1$ .","Recall $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right) = \\widehat{\\mathcal {T}}_\\lambda \\left(k+l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right)$ .","We will find it convenient to also define $\\widehat{}_k\\left(l\\right) = \\widehat{}\\left(k+l\\right) - \\widehat{}\\left(l\\right)$ .","Proposition 5.2 (Bounds for $n=0$ ) Let $\\lambda \\in [0,\\lambda _O)$ .","Then $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }.$ For $k\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op} &\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le \\lambda ^2\\left(\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } + \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\right).$ Our first comment relates to these bounds and every bound we perform hereafter in Section .","From the positivity of $\\pi ^{(0)}_{\\lambda }(x,y)$ and Lemma REF we have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op}$ , and by Lemma REF we have $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty }$ .","However from the definitions of the norm we also have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty } = \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ , and so we will be interested in bounding this last norm.","Now note that for the event $\\lbrace y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\rbrace $ to hold, either there is a direct edge between $y$ and $x$ , or there exist vertices $w,z$ in $\\eta $ that are direct neighbours of the origin and have respective disjoint paths to $x$ that both do not contain $y$ .","Hence, by the Mecke equation (REF ), $\\mathbb {P}_\\lambda ( y \\Longleftrightarrow x\\textrm { in } \\xi ^{\\mathbf {0},x}) &\\le \\varphi (x,y) + \\tfrac{1}{2} \\mathbb {E}_\\lambda \\Big [ \\sum _{(w,z) \\in \\eta ^{(2)}} 1_{(\\lbrace y\\sim w \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^x\\rbrace ) \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^x \\rbrace )} \\Big ] \\nonumber \\\\& = \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\int \\mathbb {P}_\\lambda \\big ( (\\lbrace y\\sim y \\text{ in } \\xi ^{y, w}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{x,w}\\rbrace )\\nonumber \\\\& \\hspace{99.58464pt} \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y, z}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^{x,z} \\rbrace ) \\big ) \\nu ^{\\otimes 2}\\left(\\mathrm {d}w,\\mathrm {d}z\\right).$ After applying the BK inequality to the above probability, $\\mathbb {P}_\\lambda \\left( y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) \\le \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left( \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y\\sim w \\text{ in } \\xi ^{y,w}\\right\\rbrace \\cap \\left\\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{w,x}\\right\\rbrace \\right) \\nu \\left(\\mathrm {d}w\\right) \\right)^2 \\\\= \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2.","$ Thus, recalling that $\\pi _\\lambda ^{(0)}(x,y) = \\mathbb {P}_\\lambda (y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x})-\\varphi (x,y) \\ge 0$ , and using the symmetry of $\\varphi $ and $\\tau _\\lambda $ , $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty } &= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\pi ^{(0)}_\\lambda \\left(x,y\\right)\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi \\left(y,u\\right)\\tau _\\lambda \\left(u,x\\right)\\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}w,\\mathrm {d}u\\right)\\nonumber \\\\&\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty }.$ We then use a supremum bound on this integral to split one of the $$ off from the others, and bound the other from above with $\\mathcal {T}_\\lambda $ to get the first two bounds of the result.","For the last bound of Proposition REF , we apply (REF ) and obtain $&\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{1,\\infty }\\nonumber \\\\&\\qquad = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(0)}_\\lambda \\left(\\overline{x};a,b\\right) \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\left(\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\mathrm {d}\\overline{u}\\mathcal {P}\\left(\\mathrm {d}c\\right)\\right)^2 \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b,b^\\prime \\in \\mathcal {E}}\\int \\Big [\\left(1 - \\cos \\left(k\\cdot \\left(\\overline{x}-\\overline{u}\\right)\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b^\\prime ,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{85.35826pt}+ \\left(1 - \\cos \\left(k\\cdot \\overline{u}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\Big ] \\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}+ \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty }.$ Here we have again used the symmetry of $\\varphi $ and $\\tau _\\lambda $ , and also used Lemma REF to split the cosine factor over $\\overline{x}-\\overline{u}$ and $\\overline{u}$ .","We can consider each term individually by using the triangle inequality.","By using a supremum bound, the symmetry of $\\varphi $ , and by bounding $\\varphi $ with $\\tau _\\lambda $ , we get the bound $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty }.$"],["Bounds on expansion coefficients","In this section we aim to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ .","Lemmas REF and REF along with the positivity of $\\pi ^{(n)}_\\lambda (x,y)$ prove that it is sufficient to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty }$ to get bounds on both of these.","Much of the hard work for this has been done previously - for example in [21].","The novelty here is to see how the argument can be naturally written in terms of operators.","We inherit the following notation from [21].","Note that we make use of the Dirac delta function - specifically the one that holds with respect to the measure $\\nu $ .","They are used here primarily for convenience and to make the argument more readable.","In particular, they appear when applying the Mecke equation (REF ) to obtain $\\mathbb {E}\\left[ \\sum _{y\\in \\eta ^u} f(y, \\xi ^u) \\right] = \\int \\left(\\lambda +\\delta _{y,u}\\right) \\mathbb {E}_\\lambda \\left[ f\\left(y,\\xi ^{u,y}\\right)\\right] \\nu \\left(\\mathrm {d}y\\right),$ and so the factor $\\lambda +\\delta _{y,u}$ encodes a case distinction of whether point $y$ coincides with $u$ or not.","Definition 5.3 (The $\\psi $ functions) Let $r,s,u,w,x,y\\in \\mathbb {X}$ .","We first set $\\tau _\\lambda ^\\circ (x,y) := \\lambda ^{-1}\\delta _{x,y} + \\tau _\\lambda (x,y)$ .","Also define $\\psi _0^{(1)}(w,u,y) &:= \\lambda ^2\\tau _\\lambda (y,u)\\tau _\\lambda (u,w)\\tau _\\lambda (w,y),\\\\\\psi _0^{(2)}(w,u,y) &:=\\lambda ^2 \\delta _{w,y} \\tau _\\lambda (y,u)\\int \\tau _\\lambda (u,t)\\tau _\\lambda (t,y) \\nu \\left(\\mathrm {d}t\\right),\\\\\\psi _0^{(3)}(w,u,y) &:= \\lambda ^2\\varphi (u,y) \\left(\\lambda ^{-1}\\delta _{w,y}\\right),\\\\\\psi ^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda ^\\circ (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\psi ^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\left(\\lambda ^{-1}\\delta _{w,s}\\right)\\tau _\\lambda (u,r),\\\\\\psi _n^{(1)} (x,r,s) &:= \\lambda ^3\\int \\tau _\\lambda ^\\circ (t,s)\\tau _\\lambda (z,r)\\tau _\\lambda (t,z)\\tau _\\lambda (z,x)\\tau _\\lambda (x,t)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right),\\\\\\psi _n^{(2)}(x,r,s) &:=\\lambda \\tau _\\lambda (x,s)\\tau _\\lambda (x,r),$ and set $\\psi _0 := \\psi _0^{(1)}+\\psi _0^{(2)}+\\psi _0^{(3)}$ , $\\psi _n := \\psi _n^{(1)} + \\psi _n^{(2)}$ , and $\\psi := \\psi ^{(1)}+\\psi ^{(2)}+\\psi ^{(3)} + \\psi ^{(4)}$ .","Note that by using $\\lambda ^{-1}\\delta _{\\left(\\cdot ,\\cdot \\right)}$ in $\\tau _\\lambda ^\\circ $ and in $\\psi ^{(3)}_0$ , we ensure that each integral that will be truly performed (that is, not an integration over a Dirac delta function accounting for a coincidence of points) has a $\\lambda $ factor associated with it when the bound $\\pi ^{(n)}_\\lambda $ is calculated.","Diagrammatic representations of these functions can be found in Figure REF .","In these diagrams, the declared variables are represented by (0,0) circle (2pt) vertices whereas the variables that are integrated over are represented by (0,0) circle (2pt) vertices.","If $\\tau _\\lambda $ connects two variables then a standard edge (0,0) – (1,0); connects their vertices, if $\\tau _\\lambda ^\\circ $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; connects their vertices, and if $\\varphi $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\sim $; connects their vertices.","Figure: Diagrams of the ψ 0 \\psi _0, ψ\\psi , and ψ n \\psi _n functions.Definition 5.4 (The $\\Psi $ operators) Here we turn the functions of Definition REF into linear operators, with the $\\psi $ functions acting as their kernels.","For $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , we define $\\Psi ^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\Psi ^{(j)}_0 f\\left(w,u\\right) = \\int \\psi ^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right).$ For $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , we define $\\Psi ^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)} f\\left(w,u\\right) = \\int \\psi ^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ , we define $\\Psi ^{(j)}_n\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)}_n f\\left(x\\right) = \\int \\psi ^{(j)}_n(x,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ We can then naturally define $\\Psi _0\\colon L^2(\\mathbb {X})\\rightarrow L^2(\\mathbb {X}^2)$ , $\\Psi \\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X}^2)$ , and $\\Psi _n\\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X})$ as the sums of $\\Psi ^{(j)}_0$ , $\\Psi ^{(j)}$ , and $\\Psi ^{(j)}_n$ respectively, or equivalently as those operators having kernel functions $\\psi _0$ , $\\psi $ , and $\\psi _n$ respectively.","Proposition 5.5 (Bound in terms of $\\psi $ and $\\Psi $ ) Let $n \\ge 1$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in [0,\\lambda _O)$ .","Then $\\lambda \\pi _\\lambda ^{(n)}(x,y) \\le \\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes \\left(2n\\right)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right),$ where $\\vec{v}_i = (w_i,u_i,w_{i-1},u_{i-1})$ .","In operator terms, this means $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n \\Psi ^{n-1} \\Psi _0\\right\\Vert _{1,\\infty }.$ The argument for (REF ) is essentially identical to [21].","It uses only general properties of Poisson point process and connection models - for example thinning edges and Mecke's equation.","The inequality (REF ) then follows from the clear sequential structure of (REF ).","The inequality (REF ) suggests that we will bound the lace coefficient operators with some product of $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ terms.","This is nearly correct.","Unfortunately the norms $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ will not have the decay in $d$ that we will require.","We will have to consider the norms of pairs of operators.","Definition 5.6 Define $\\triangle _\\lambda &:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^\\circ _\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^{\\circ \\circ }_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda ^\\circ \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1,\\\\\\overline{\\triangle ^{\\circ \\circ }_\\lambda }&:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda ^\\circ \\left(\\overline{u}-\\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}.$ We can think of $\\triangle _\\lambda $ as an $\\mathbb {X}$ -convolution of three $\\tau _\\lambda $ functions with suprema taken over the end vertices.","Then $\\triangle ^\\circ _\\lambda $ and $\\triangle ^{\\circ \\circ }_\\lambda $ are produced by adding on the $\\mathbb {X}$ -convolution of two $\\tau _\\lambda $ functions and one $\\tau _\\lambda $ function (which will trivially take value 1 once the suprema are taken).","The object $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the marks at adjacent $\\tau _\\lambda $ functions need not be equal, and then we take the supremum over all the marks - not just the end ones.","We then use these elementary diagrams to produce composite objects: $U_\\lambda := \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }+ \\overline{\\triangle ^{\\circ \\circ }_\\lambda }^2 + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }, \\qquad V_\\lambda := \\left(\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }U_\\lambda \\right)^\\frac{1}{2}.$ We note a few relations.","Firstly, since all the terms in $U_\\lambda $ are non-negative we have $\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le U_\\lambda $ and thus $V_\\lambda \\le U_\\lambda $ .","Also note that $1\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ and thus $U_\\lambda \\ge \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .","In particular this means that $U_\\lambda \\ge 1$ and $U_\\lambda \\ge \\triangle ^\\circ _\\lambda $ .","The following lemma tells us that we can use $U_\\lambda $ to bound single operators, and that we can use $V_\\lambda ^2$ to bound pairs of operators.","To aid the reader's understanding, bear in mind that later (in Section ) we will prove that $U_\\lambda = \\mathcal {O}\\left(1\\right)$ , whilst $V_\\lambda $ is much smaller and is $\\mathcal {O}\\left(\\beta \\right)$ .","Lemma 5.7 For all $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j,j^{\\prime }\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $U_\\lambda &\\ge \\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\right\\Vert _{1,\\infty },\\\\V_\\lambda ^2 &\\ge \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j^{\\prime })}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }.$ We begin with the single operators.","For $\\Psi ^{(1)}$ we find $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } &= \\lambda ^4\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (w,u)\\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 4}\\left(\\mathrm {d}z, \\mathrm {d}t, \\mathrm {d}w, \\mathrm {d}u\\right), \\nonumber \\\\& \\le \\lambda ^4\\left(\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (r,z) \\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (t,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t \\right)\\right) \\nonumber \\\\&\\hspace{142.26378pt} \\times \\left(\\operatornamewithlimits{ess\\,sup}_{t^{\\prime },z^{\\prime }\\in \\mathbb {X}}\\int \\tau _\\lambda (t^{\\prime },w)\\tau _\\lambda (w,u)\\tau _\\lambda (u,z^{\\prime })\\nu ^{\\otimes 2}\\left(\\mathrm {d}w, \\mathrm {d}u\\right)\\right),\\nonumber \\\\& = \\triangle ^\\circ _\\lambda \\triangle _\\lambda ,$ where we have used a supremum bound on the $t$ and $z$ integrals to split them into $L^1$ and $L^\\infty $ bounds.","It is also clear that $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty }= \\triangle _\\lambda $ and $\\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\le \\triangle ^\\circ _\\lambda $ .","The $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ bound is more involved and we return to it in a moment.","We are able to represent these $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ norms and their bounds pictorially.","The vertices of the diagrams represent the variables $w,u,z,t,a,b$ , etc.","appearing in the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ expression.","The variables over which a $\\nu $ -integral is taken are represented by the (0,0) circle (2pt) vertices, and the variables over which a supremum is taken are represented by the (0,0) circle (2pt) vertices.","The presence of a $\\tau _\\lambda $ connecting two variables is then represented by a standard edge (0,0) – (1,0); between their vertices, and the presence of a $\\tau _\\lambda ^\\circ $ connecting two variables is correspondingly represented by an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; between their vertices.","When using a supremum bound to split the diagrams, an integrated vertex gets split into two new vertices, precisely one integrated vertex and one supremum vertex.","Each edge connected to such a split vertex can independently choose which of the new vertices to associate to.","The calculations producing the bounds on $\\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty }$ for $j=1,3,4$ can then be represented by: $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } \\quad &=\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\\\\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } + \\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad .$ Our strategy for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ is to use the spatial translation invariance to shift the origin to the bottom right vertex, and then use a supremum bound to split the diagram in two.","Unfortunately, this spatial shift decouples the spatial and mark components of the vertices so - for example - at some vertices we end up taking suprema over marks whilst integrating over space.","Let us introduce some more notation.","The vertex mark size=2pt] at (0,0) square* indicates that the spatial component is integrated over whilst the mark component is fixed and its supremum is taken after all the integrals.","We also introduce the edge-vertex combination [dashed] (0,0) – (1,0); [fill=white] (0,0) circle (2pt) which indicates that there is no term connecting the associated vertices, but there is a supremum taken over the spatial displacement.","Note that in principle we could use a vertex that had a spatial supremum and a mark integral, but we will always bound that probability integral by the supremum.","The bound for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ can therefore be expressed in the following: $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1);(0,-0.6) -- (1,-0.6);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (1.5,-0.1) circle (2pt);(1.5,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);[fill=white] (0,0) circle (2pt);\\end{tikzpicture}}.$ To clarify the calculation for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ , we write out the integral here: $&\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}-\\overline{s};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}- \\overline{w}; a_4,a_3)\\tau _\\lambda (\\overline{u}-\\overline{t};a_6,a_4)\\tau _\\lambda (\\overline{z}-\\overline{u};a_5,a_6) \\nonumber \\\\&\\hspace{142.26378pt}\\times \\tau _\\lambda (\\overline{t}- \\overline{z}; a_4,a_5)\\tau _\\lambda (\\overline{z}; a_5,a_2) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5, \\mathrm {d}a_6\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\left(\\operatornamewithlimits{ess\\,sup}_{\\overline{s},\\overline{t}^{\\prime },\\overline{z}^{\\prime }\\in \\mathbb {R}^d,a_1,a_2,a_4,a_5\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathcal {P}\\left(\\mathrm {d}a_3\\right)\\right)\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_6\\in \\mathcal {E}}\\int \\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6)\\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\right).$ Then the two components of the first factor that are joined by the $\\raisebox {2pt}{{[dashed] (0,0) -- (1,0); [fill=white] (0,0) circle (2pt)}}$ edge can be spatially translated together to form three consecutive edges.","Note however, that the marks on either side of this join are not necessarily equal.","We can therefore bound this first term by $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .","In summary, these manipulations then imply that $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\triangle _\\lambda $ .","For $\\Psi ^{(1)}_0$ and $\\Psi ^{(2)}_0$ the calculations are identical to each other, and we can once again represent the calculations pictorially: $\\left\\Vert \\Psi ^{(1)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_0\\right\\Vert _{1,\\infty } =\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- cycle;[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad = \\triangle _\\lambda ,$ where splitting the single supremum into two produces an upper bound.","For the third case we have $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) -- (1,0);(0.5,0) circle (0pt) node[above]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad = \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }.$ The diagrams representing the calculations for $\\Psi _n$ are as follows: $\\left\\Vert \\Psi _n^{(1)}\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(1,-0.6) -- (0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{113.81102pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\qquad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda , \\nonumber \\\\\\left\\Vert \\Psi ^{(2)}_n\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda .$ In the first case we expanded a $\\tau _\\lambda ^\\circ $ edge into a $\\tau _\\lambda $ edge and a contraction.","Then in the contracted case we bound $\\tau _\\lambda \\le 1$ on the diagonal edge.","We have now proven the bounds for the single operator norms.","For most of the operator pairs, we can then use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ to immediately get sufficient bounds.","The only pairs for which this strategy fails are: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ .","We now deal with these cases by hand: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- (0,0.6);(0,0) circle (0pt) node[above, rotate=90]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda , \\\\\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda .$ For the first bound we used $\\varphi \\le \\tau _\\lambda $ , and then split the single supremum into two.","For the second bound we used $\\tau _\\lambda \\le 1$ on the diagonal edge.","Proposition 5.8 For $n\\ge 1$ , $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le 6\\times 4^{n-1}U_\\lambda V_\\lambda ^n.$ Recall from Lemma REF that $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n\\Psi ^{n-1}\\Psi _0\\right\\Vert _{1,\\infty }$ .","We therefore aim to bound $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ for $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ .","We have two cases depending upon the parity of $n$ .","If $n$ is odd then we have an even number of operators.","We pair them off, use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , and Lemma REF to get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le \\left(V_\\lambda ^2\\right)^{\\frac{n+1}{2}} = V_\\lambda ^{n+1}.$ If $n$ is even, then we pair off the first $n$ operators and treat the last $\\Psi ^{(j_n)}_n$ operator by itself.","We then get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le U_\\lambda \\left(V_\\lambda ^2\\right)^{\\frac{n}{2}} = U_\\lambda V_\\lambda ^{n}.$ Since $U_\\lambda \\ge V_\\lambda $ , we have the bound $U_\\lambda V_\\lambda ^{n}$ for all $n\\ge 1$ .","Since there are 3 choices for $j_0$ , 4 choices for each of $j_1,\\ldots ,j_{n-1}$ , and 2 choices for $j_n$ , we get the pre-factor of $6\\times 4^{n-1}$ in our result by applying the triangle inequality."],["Bounds on displaced expansion coefficients","In this section we will be aiming to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ for $n\\ge 1$ .","Our strategy draws inspiration from the corresponding step in [21], whilst being adapted to account for inhomogeneous marks in a similar way to Section REF .","The central idea is to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(n)}_\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right),$ and bound $\\pi ^{(n)}_\\lambda $ using the function expression from Proposition REF .","This produces an integral (or diagram) composed from simpler segments with the displacement factor, $\\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)$ , spanning the whole length of the diagram.","We can then use the Cosine-Splitting result (Lemma REF ) to get a sum of diagrams where the displacement factor only spans a single segment of each diagram.","To demonstrate this, consider the diagram corresponding to $\\Psi ^{(2)}_n\\Psi ^{(1)}\\Psi ^{(1)}_0$ with the displacement factor: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8) -- (3,0.8) -- (4,0.2);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\le 3\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\right.","\\\\ \\left.\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (1,0.8) -- (3,0.8);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right).$ In these diagrams and hereafter we use [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; to denote that a displacement factor is connecting the two indicated vertices.","As above, the path the [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; takes will also suggest the route along which we will intend to use cosine-splitting.","For the sake of simplicity, we will be taking the displacement path across the `top' of the diagram from the perspective of the usual orientation of the $\\Psi _0$ segment.","However - as can be seen in the above example - the way the operators compose with each other means that the orientation of each segment inverts when compared to its neighbours.","Therefore the displacement may be across the `top' or the `bottom' of the $\\Psi $ and $\\Psi _n$ segments (in their usual orientation) depending upon the parity of its place in the sequence of segments.","Our general strategy will be to isolate the displaced segment (and perhaps a neighbouring segment) by splitting off `earlier' and `later' segments.","When we split off earlier segments (and sometimes later segments), we will use the $\\Psi $ structure we have been using so far.","However, we will sometimes want to group vertices and edges slightly differently when we split off `later' segments.","We use an observation from [21] to conveniently group these.","Let us define the kernel functions $\\overline{\\psi }^{(1)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\tau _\\lambda (y,r)\\tau _\\lambda (r,s)\\tau _\\lambda (s,x),\\\\\\overline{\\psi }^{(2)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\left(\\lambda ^{-1}\\delta _{y,r}\\right)\\left(\\lambda ^{-1}\\delta _{y,s}\\right),\\\\\\overline{\\psi }^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\overline{\\psi }^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\left(\\lambda ^{-1}\\delta _{u,r}\\right),$ and set $\\overline{\\psi }_0 := \\overline{\\psi }^{(1)}_0+\\overline{\\psi }^{(2)}_0$ and $\\overline{\\psi }:= \\overline{\\psi }^{(1)}+\\overline{\\psi }^{(2)}+\\overline{\\psi }^{(3)} + \\overline{\\psi }^{(4)}$ .","Diagrammatic representations of these functions can be found in Figure REF .","Note the similarity to the $\\psi $ functions, with the $\\tau _\\lambda ^\\circ $ edge appearing elsewhere in the integral.","Figure: Diagrams of the ψ ¯ 0 \\overline{\\psi }_0, and ψ ¯\\overline{\\psi } functions.For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\overline{\\Psi }^{(j)}_0 f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right),$ and for $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\overline{\\Psi }^{(j)} f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ In our terminology, the important observation of [21] was that for some $m\\ge 0$ the `later' segments of each diagram can be bounded using $\\big (\\overline{\\Psi }^m\\overline{\\Psi }_0\\big )^\\dagger $ , the adjoint of $\\overline{\\Psi }^m\\overline{\\Psi }_0$ .","In terms of the kernel functions, taking the adjoint in this case amounts to reflecting the `input' and `output' arguments, and for the diagrams this amounts to reflecting in the vertical plane.","Lemma 5.9 For $n\\ge 1$ , $m\\in \\left\\lbrace 0,\\ldots ,n\\right\\rbrace $ , $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^m,\\\\\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^{n-m},\\\\\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le {\\left\\lbrace \\begin{array}{ll}U_\\lambda &: m=0\\\\U_\\lambda ^2V_\\lambda ^{m-1} &: m\\ge 1.\\end{array}\\right.","}$ The argument for $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty }$ is the same as that in Lemma REF and Proposition REF .","We consider bounds for single operators and pairs of operators, and then use sub-multiplicity to get a bound for each diagram.","For $\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty }$ the argument is a little more complicated.","For $m=0,1$ we can easily check each diagram satisfies the bound.","For $m\\ge 2$ we have the issue that there exists one pair of operators which we can't bound by $V_\\lambda ^2$ : $\\left\\Vert \\overline{\\Psi }^{(4)}\\overline{\\Psi }^{(2)}_0\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) to [out=10,in=170] (1,0) to [out=190,in=350] (0,0);[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda \\le U_\\lambda ^2.$ If $m\\ge 2$ is even (an odd number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }_0$ term which we bound with $U_\\lambda $ .","We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda V_\\lambda ^m$ .","If $m\\ge 2$ is odd (an even number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }\\overline{\\Psi }_0$ pair which we bound with $U_\\lambda ^2$ .","We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda ^2V_\\lambda ^{m-1}$ .","Definition 5.10 In addition to the terms defined in Definition REF , we will now use $W_k&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int \\tau _{\\lambda ,k}(x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right) = \\lambda \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty },\\\\\\overline{W_k}&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _{\\lambda ,k}(\\overline{x}-\\overline{u};a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4)\\mathrm {d}u,\\\\\\overline{H_k}&:= \\lambda ^5 \\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,a_1,\\ldots ,a_{16}\\in \\mathcal {E}}\\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8)\\nonumber \\\\& \\hspace{42.67912pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\tau _{\\lambda ,k}(\\overline{t}-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u}.$ Note that we only use $\\overline{W_k}$ in Propositions REF and REF because $\\overline{W_k}\\ge W_k$ , but in the proof we will sometimes use $W_k$ to make the derivation easier to follow.","The diagrams for $\\overline{W_k}$ and $\\overline{H_k}$ are in Figure REF .","Also recall the sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ assumed to exist in Assumption REF .","Then define $\\mathbb {B} &:= \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\nu \\left(B\\left(x\\right)\\right),\\\\\\triangle ^{(B)}_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}: r\\notin {B\\left(s\\right)}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right).$ Note that these are only required to deal with one specific diagram in the following $n=1$ case.","Figure: Diagrams representing W k ¯\\overline{W_k} and H k ¯\\overline{H_k}.","In this figure, the marks on edges incident to filled square vertices need not be equal.Proposition 5.11 For $k\\in \\mathbb {R}^d$ and $n\\ge 2$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}\\left(n+1\\right)^24^{n-1}\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^4 V_\\lambda ^{n-4} &: n\\ge 4\\\\16\\times 16\\left(50\\overline{W_k}V_\\lambda ^2 + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=3\\\\9\\times 4\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=2.\\end{array}\\right.","}$ Proposition 5.12 For $k\\in \\mathbb {R}^d$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(1)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(1)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le 13\\overline{W_k}V_\\lambda U_\\lambda + \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\left(V_\\lambda + \\triangle ^{(B)}_\\lambda \\right) + \\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + 4\\lambda \\mathbb {B}\\overline{W_k}.$ [Proof of Proposition REF ] In bounding the diagrams with one displacement segment, we have three broad cases.","Let $i\\in \\left\\lbrace 0,1,\\ldots ,n\\right\\rbrace $ denote which segment the displacement lies upon.","Our three cases are then: the displacement lies on the $\\psi _n$ segment (i.e.","$i=n$ ), the displacement lies on the $\\psi _0$ segment (i.e.","$i=0$ ), the displacement lies on a $\\psi $ segment (i.e.","$i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ ).","We will aim to get a single bound for each diagram having a displacement crossing a single segment.","This will be uniform in the sequence $\\left(j_0,j_1,\\ldots ,j_n\\right)$ .","We then account for the number of possible choices of $\\left(j_0,j_1,\\ldots ,j_n\\right)$ , and finally include the factor of $(n+1)^2$ arising from using cosine-splitting (the explicit factor and the number of diagrams).","Case REF is the simplest.","We use a supremum bound to pull off the displaced $\\Psi _n$ term.","We then bound this by hand and use Lemma REF to bound the remainder.","In representing this scheme we use a shorthand $\\Psi ^{n-1}\\Psi _0$ to denote a particular sequence $\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0$ rather than actually the operators $\\Psi $ and $\\Psi _0$ defined above.","We also use labelled grey shapes to represent these groupings of diagrams, highlighting only end vertices or vertices that connect to neighbouring segments.","This scheme look like: $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8) -- (4,0.2);(2.8,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (2,0.8) -- (3,0.2);(2.5,0.5) circle (0pt) node[rotate = -30]{\\times };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{28.45274pt}\\le U_\\lambda V_\\lambda ^{n-1} \\left(2\\left( \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 2U_\\lambda V_\\lambda ^{n-1}\\left(\\overline{W_k}\\triangle ^{\\circ \\circ }_\\lambda + W_k\\triangle ^\\circ _\\lambda \\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 4\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that this bound also holds if the displacement runs along the bottom of the $\\Psi _n$ segment.","Since there are $3\\times 4^{n-1}$ possible sequences of indices for the $\\Psi ^{n-1}\\Psi _0$ diagram, this case contributes that as a pre-factor.","For Case REF , we need a few more tricks.","Since we are taking the displacement across the top of the diagram, the only term that will make a contribution is $\\Psi ^{(1)}_0$ .","To tackle this case we first use the observation of [21] that the diagram we want to bound can in turn be bounded by a diagram using $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ .","As above, we use $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ as a shorthand for a particular sequence $\\overline{\\Psi }^{(j_{n-1})}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0$ .","We are also imprecise in this notation over whether it is this operator or its adjoint.","Nevertheless the diagrams we use are clear on which vertices are integrated over and which have suprema, and this will avoid ambiguity.","We then spatially translate the supremum to the far end and pull off the $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram using a supremum bound.","This looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}\\\\\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (3,-0.6) to [out=100,in=260] (3,0.6) to [out=190,in=350] (2,0.6);\\node at (2.5,0) {\\overline{\\Psi }};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0};(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}.$ For $\\overline{\\Psi }^{(j)}$ with $j=1,2,3$ , a simple supremum bound shows that this starting diagram can be bounded by $\\overline{W_k}\\triangle ^\\circ _\\lambda \\left\\Vert \\overline{\\Psi }^{(j)}\\right\\Vert _{1,\\infty }\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2$ .","For $j=4$ a little more care is needed: $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ We therefore find that $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le {\\left\\lbrace \\begin{array}{ll}\\overline{W_k}V_\\lambda ^{n-1}U_\\lambda ^2 &: n\\ge 3\\\\\\overline{W_k}V_\\lambda ^2U_\\lambda &: n=2.\\end{array}\\right.","}$ Since there are 4 choices for the appended $\\overline{\\Psi }^{(j)}$ term, and $4^{n-2}\\times 2$ for the remaining $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram, this case contributes $2\\times 4^{n-1}$ as a pre-factor.","For Case REF , we shall find we have 3 sub-cases, depending not only on the $j$ -index of the displaced segment, but on which term we are looking at once it has been expanded.","Sub-cases $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ deal with $j=1,2,3$ .","In Appendix we expand out the various displaced segments using cosine-splitting and by expanding $\\tau _\\lambda ^\\circ $ edges into $\\tau _\\lambda $ edges and points, and label each of the resulting diagrams as sub-case $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ .","These sub-cases are distinguished by the strategy we use to bound them.","Sub-case $\\mathrm {\\left(III\\right)}$ then addresses the case with $j=4$ .","For sub-case $\\mathrm {\\left(I\\right)}$ , we just use the $\\Psi $ structure of the diagrams.","We first split off segments from the left of the displaced term like we did in Case REF .","If $i=n-1$ we can then bound the remaining two pair of segments with $\\overline{W_k}V_\\lambda ^2$ (Appendix explains how this bound is found).","This can be described diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda V_\\lambda ^{n}.$ If $i\\le n-2$ , then we also split off the segments to the right of this pair, and then bound the pair by $\\overline{W_k}V_\\lambda ^2$ again.","Diagrammatically this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (6,-0.6) to [out=40,in=200] (7,0) to [out=160,in=320] (6,0.6) to [out=190,in=350] (4,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };\\node at (5,0) {\\Psi _n\\Psi ^{n-i-2}};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(7,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda ^2V_\\lambda ^{n-3} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that the since $U_\\lambda \\ge V_\\lambda $ , the bound in (REF ) is greater than or equal to the bound in (REF ).","It will be convenient to find an $i$ -independent bound.","To this end, since the $i\\le n-2$ case only exists for $n\\ge 3$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(I\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}&: n\\ge 3\\\\\\overline{W_k}U_\\lambda V_\\lambda ^{2}&: n=2,\\end{array}\\right.","}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .","For sub-case $\\mathrm {\\left(II\\right)}$ , we use a different strategy.","First we split off the earlier (left) segments as we did for the first sub-case and in Case REF .","However we arrange the later terms differently - more like we did in Case REF .","We associate our displaced $\\Psi $ term with $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges from the subsequent term.","If the subsequent term would be $\\Psi ^{(2)}_n$ , these are actually two $\\tau _\\lambda $ edges, but in our schematic diagrams we will draw a $\\tau _\\lambda ^\\circ $ edge and bear this special case in mind when it is relevant.","While the displacement portrayed on the `top' of the segment could equally well be on the `bottom', the $\\tau _\\lambda ^\\circ $ edge will always be attached to the bottom in the usual orientation.","Everything to the right of these $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges can then be described as a sequence of $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ terms.","If $i=n-1$ , then we pair our augmented displaced segment with the subsequent $\\overline{\\Psi }_0$ segment.","We then perform a spatial translation to make the right-most vertex a supremum vertex like we did in (REF ).","The remaining diagram of a $\\Psi ^{(j)}$ segment and $\\overline{\\Psi }_0$ segment connected by a $\\tau _\\lambda ^\\circ $ and $\\tau _\\lambda $ edge can then be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ).","Recall that if $\\overline{\\Psi }_0$ is in fact $\\overline{\\Psi }^{(2)}_0$ then we have two $\\tau _\\lambda $ edges connecting the terms, not a $\\tau _\\lambda $ edge and a $\\tau _\\lambda ^\\circ $ edge.","We need to note this to get the $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ bound.","The scheme is then represented diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(6,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda V_\\lambda ^{n-2}.$ If $i\\le n-2$ , we also perform a spatial translation to move a supremum to the right-most vertex, but then also split off all $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ segments after the $\\overline{\\Psi }$ segment immediately following the displaced term.","Once again the remaining pair can be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ) and the split-off diagrams bounded using Lemma REF .","The scheme can be represented diagrammatically as $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);[fill=gray!50] (5,0.6) to [out=280,in=80] (5,-0.6) to [out=10,in=170] (7,-0.6) to [out=40,in=200] (8,0) to [out=160,in=320] (7,0.6) to [out=190,in=350] (5,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};\\node at (6,0) {\\overline{\\Psi }^{n-i-2}\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(5,0.6) circle (2pt);(5,-0.6) circle (2pt);(8,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\hspace{170.71652pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times {\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.}","\\nonumber \\\\&\\hspace{170.71652pt}\\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right){\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.","}$ Again for simplicity, we would like to get and $i$ -independent bound for such diagrams.","Since the $i = n-2$ case only exists for $n\\ge 3$ , and the $i\\le n-3$ case only exists for $n\\ge 4$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(II\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^3V_\\lambda ^{n-4}&: n\\ge 4\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^2&: n=3\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda &: n=2,\\end{array}\\right.","}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .","We now address sub-case $\\mathrm {\\left(III\\right)}$ , that is when the displacement crosses a $\\Psi ^{(4)}$ segment.","This is the simplest sub-case because when we take a displacement across $\\Psi ^{(4)}$ in isolation, we get exactly $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad =W_k.$ We will be able to use the $\\Psi $ -arrangement of segments here.","For $n\\ge 4$ we split off the earlier and later segments without pairing the displaced term with any of them.","With Lemma REF this produces the bound $\\left(U_\\lambda V_\\lambda ^{i-1}\\right)W_k\\left(U_\\lambda V_\\lambda ^{n-i-1}\\right) = \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-2} \\qquad :n\\ge 4.$ For $n=3$ we split off two segments from one side and one from the other.","This gives the finer bound $W_kU_\\lambda V_\\lambda ^2 \\quad :n=3.$ For $n=2$ , we use $j_0$ to indicate which $\\Psi _0$ term we have and $j_2$ to denote which $\\Psi _n$ term we have.","We note that if $j_0=1,2$ then $\\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ , and if $j_2 = 1$ then $\\left\\Vert \\Psi ^{(j_2)}_n\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ .","Therefore if $\\left(j_0,j_2\\right) \\ne \\left(3,2\\right)$ we have the bound $W_kU_\\lambda V_\\lambda $ .","If $\\left(j_0,j_2\\right) = \\left(3,2\\right)$ we proceed by hand.","We bound $\\varphi \\le \\tau _\\lambda $ and spatially shift one vertex to show $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };(1,0) circle (0pt) node[rotate=90, above]{\\sim };(0,0) circle (0pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (1,0.6) {{square*}};(1,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda .$ Carefully comparing the above bounds gives the required bounds for $n\\le 2$ .","Including the multiplicity of each diagram (including that arising from cosine-splitting), we find that we can bound the displaced and subsequent non-displaced pair by $\\left(24\\overline{W_k}V_\\lambda ^2 + 2\\overline{H_k}\\right)$ .","Note that this includes summing over the $j$ -index of the displaced term but not the index of the subsequent term.","Since there are $3\\times 4^{n-2}\\times 2$ options for the $j$ -indices of the other segments, this case contributes this to the counting pre-factors in our result.","[Proof of Proposition REF ] For $n=1$ , we can consider each of the six cases by hand.","Most of the diagrams can be dealt with routinely using cosine-splitting, supremum bounds, and spatial shifts.","We enumerate the diagrams by their $\\left(j_0,j_1\\right)$ values corresponding to the constituent segments.","$(1,1):& \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (0,-0.2) -- (1,-0.8) -- (2,-0.8) -- (3,-0.2);(1.5,-0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 3\\left(\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda + \\triangle _\\lambda \\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(1,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (0,0.2) -- (1,0.8) -- (2,0.2);(0.8,0.68) circle (0pt) node[rotate=30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda W_k\\right)\\\\(2,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (1,0.8) -- (2,0.8) -- (3,0.2);(1.8,0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\triangle _\\lambda \\overline{W_k}\\triangle ^\\circ _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(2,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (1,0.8) -- (2,0.2);(1.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle _\\lambda W_k\\\\(3,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6) -- cycle;(1,0.6) -- (1,-0.6);[<->] (0,0.8) -- (1,0.8) -- (2,0.2);(0.8,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(2,0) circle (2pt);(0,-0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\overline{W_k}\\triangle _\\lambda + \\overline{W_k}\\triangle _\\lambda + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\triangle _\\lambda W_k+ \\triangle _\\lambda W_k\\right)$ These terms sum to no more than $13\\overline{W_k}V_\\lambda U_\\lambda .$ For the $\\left(3,2\\right)$ diagram, we use a different `trick.'","In Lemma REF we will prove the bound $\\tau _\\lambda \\left(x,y\\right) \\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right)$ .","When applied to this diagram, this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (1.2,0.6) -- (1.2,-0.6);(1.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right),$ Here we have also used the cosine-splitting result on the second part.","For the first of these 4-vertex diagrams, we split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\lambda \\triangle _\\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }$ .","For the second of the 4-vertex diagrams, we also split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ .","For the 3-vertex diagram we condition on whether two of the vertices are in each other's $B$ -set (defined in Assumption REF ).","In the following diagrams [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\in B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\in B\\left(u\\right)\\right\\rbrace }$ relates two variables, and [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\notin B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\notin B\\left(u\\right)\\right\\rbrace }$ relates them: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.7) -- (0,-0.8) -- (1,-0.2);(-0.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-20pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,-0.8) -- (1,-0.2);(0.5,-0.5) circle (0pt) node[rotate=30]{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right) \\nonumber \\\\&\\le \\left(\\quad \\raisebox {-7pt}{\\begin{tikzpicture}(0,0) -- (1,0);[<->] (0,0.2) -- (1,0.2);(0.5,0.2) circle (0pt) node{\\times };(0.5,0) circle (0pt) node[below]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node[above, rotate=90]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) +4\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);[<->] (0,-0.2) -- (1,-0.2);(0.5,-0.2) circle (0pt) node[below]{\\scriptsize \\in B};\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right)\\nonumber \\\\& \\le \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\triangle ^{(B)}_\\lambda + 4\\lambda \\mathbb {B}W_k.$"],["Bootstrap analysis","In this section we will be assuming that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = 1.$ By Lemma REF this also implies $\\mathbb {S}\\left(\\right)\\le 1$ .","A re-scaling argument similar to [36] explains how this can be done while maintaining generality.","First note that since $\\widehat{\\varphi }(0;a,b)>0$ for a positive $\\mathcal {P}$ -measure set of $a,b\\in \\mathcal {E}$ , we have $\\mathbb {S}\\left(\\widehat{}(0)\\right)>0$ .","Now suppose we scale $\\mathbb {R}^d$ by a factor of $q^{-1/d}$ (that is, the new unit radius ball is the previous ball of radius $q^{1/d}$ ), and leave the mark space $\\mathcal {E}$ unchanged.","Then we find that the system we are left with has the distribution of an RCM model with the new parameters $\\lambda ^* = \\mathbb {E}_\\lambda \\left[*{\\eta \\cap \\left[0, q^{\\frac{1}{d}}\\right]^d }\\right] = \\lambda q, \\qquad \\varphi ^*\\left(\\overline{x};a,b\\right) = \\varphi \\left(q^{\\frac{1}{d}}\\overline{x};a,b \\right).$ Therefore for all $a,b\\in \\mathcal {E}$ we have $\\widehat{\\varphi }^*(\\overline{x};a,b) = \\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\widehat{\\varphi }(\\overline{x};a,b),$ and $\\mathbb {S}\\left(\\widehat{}^*(0)\\right) = q^{-1} \\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","We can therefore achieve our assumption by choosing $q=\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","Furthermore, under this scaling we find that the old event $\\left\\lbrace \\left(\\overline{x},a\\right) \\longleftrightarrow \\left(\\overline{y},b\\right)\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ becomes the new event $\\left\\lbrace \\left(q^{-1/d}\\overline{x},a\\right) \\longleftrightarrow \\left(q^{-1/d}\\overline{y},b\\right)\\textrm { in } \\xi ^{\\left(q^{-1/d}\\overline{x},a\\right),\\left(q^{-1/d}\\overline{y},b\\right)} \\right\\rbrace $ , and we get $\\tau ^*_{\\lambda ^*}(\\overline{x};a,b) = \\tau _\\lambda \\left(q^\\frac{1}{d}\\overline{x};a,b\\right)$ where $\\tau ^*_{\\lambda ^*}$ is the two-point function in the RCM governed by the connection function $\\varphi ^*$ and intensity $\\lambda ^*$ .","It is also worth noting that the scaling choice to have $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ means that REF reduces to the condition that there exists a $d$ -independent constant $C>0$ such that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } &\\le C,\\\\\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } &\\le C \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Note that the inequalities relating $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ follow from Jensen's inequality and a supremum bound on a probability space."],["The Bootstrap Function","To show that the bounds arising from the diagrams in Section are sufficient for our purposes, we make use of a bootstrap argument.","Define $\\mu _\\lambda & = 1- \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)},\\\\\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) &= \\frac{1}{1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)},\\\\\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) &=\\left(1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\left(\\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right)\\right).$ From these we can then define the bootstrap function $f= f_1\\vee f_2\\vee f_3$ , where $f_1\\left(\\lambda \\right) = \\lambda , \\qquad f_2\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}, \\qquad f_3\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)}.$ Let us now outline some initial properties of $\\mu _\\lambda $ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .","Lemma 6.1 The map $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous and bounded below by $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ on $\\left[0,\\lambda _O\\right)$ .","Therefore the map $\\lambda \\mapsto \\mu _\\lambda $ is a continuous map from $\\left[0,\\lambda _O\\right)\\rightarrow \\left[0,1\\right)$ .","Furthermore, $\\widehat{G}_{\\mu _\\lambda }(k)$ is bounded below by $\\left(1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)^{-1}$ for all $k\\in \\mathbb {R}^d$ .","We first recall that $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (0)$ is differentiable with respect to the operator norm from Corollary REF , and is therefore continuous.","Lemma REF then implies that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous.","For the bounded below property, we use Lemma REF to get $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge \\mathbb {S}\\left(\\widehat{}(0)\\right) = 1$ .","The continuity of $\\mu _\\lambda $ follows directly from the continuity and positivity of $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ .","The bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ bounds $\\mu _\\lambda \\ge 0$ , and $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ shows that $\\mu _\\lambda <1$ for all $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","The case $\\mu _0 = 0$ follows immediately from $\\mathcal {T}_0 = $ .","Because $\\mu _\\lambda \\in \\left[0,1\\right)$ , to minimise $\\widehat{G}_{\\mu _\\lambda }(k)$ we want to find a lower bound for $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ .","The $k$ -uniform bound on $\\widehat{G}_{\\mu _\\lambda }(k)$ then follows from $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\ge -\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\rm op} \\ge -\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}$ ."],["Bounding the Lace Operator","We have managed to bound the lace operator and its displacement in terms of a variety of objects.","For the lace operator itself (and its associated Fourier transformed operators), these are: $\\lambda , \\triangle _\\lambda , \\overline{\\triangle ^{\\circ \\circ }_\\lambda }, \\left\\Vert \\right\\Vert _{1,\\infty }.$ As a guide, we expect $\\lambda $ , $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ , and $\\left\\Vert \\right\\Vert _{1,\\infty }$ to be bounded uniformly in $d$ and $\\lambda <\\lambda _O$ , whereas we expect $\\triangle _\\lambda $ to decay uniformly in $\\lambda <\\lambda _O$ as $d\\rightarrow \\infty $ .","For some, these properties are easy to prove.","The boundedness of $\\left\\Vert \\right\\Vert _{1,\\infty }$ follows from $\\left\\Vert \\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we have used REF via (REF ).","Regarding $\\lambda $ , since we are only interested in $\\lambda <\\lambda _O$ the question is perhaps more clearly written as “Is $\\lambda _O$ uniformly bounded in $d$ ?","\".","While there may be more elementary ways of proving this directly, we will get it for free from the bootstrap argument.","Since we have the bound $\\lambda \\le f(\\lambda )$ , proving the uniform boundedness of $f(\\lambda )$ with respect to $d$ and $\\lambda $ for $\\lambda <\\lambda _O$ will prove the same for $\\lambda _O$ .","This will leave us with only $\\triangle _\\lambda $ and $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .","Recall that we can write $\\triangle _\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }, \\quad \\triangle ^{\\circ \\circ }_\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1.$ Then $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the supremum is taken over all the marks, and adjacent $\\tau _\\lambda $ can have different marks.","To get the desired decay and boundedness behaviour, we will therefore only need to prove that $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ decays and to bound $&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v},\\\\&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}.$ We begin by proving the decay of $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ .","Lemma 6.2 Let $x,y \\in \\mathbb {X}$ .","Then $\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\varphi (x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right),\\\\\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right).$ Then for $n\\ge 2$ , $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le \\sum _{m=0}^n\\genfrac(){0.0pt}0{n}{m}\\lambda ^{m} \\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }.$ By combining Mecke's formula and the BK inequality, we obtain $\\tau _\\lambda (x,y) & \\le \\varphi (x,y) + \\mathbb {E}_\\lambda \\left[\\sum _{u \\in \\eta } 1_{\\lbrace x \\sim u \\text{ in } \\xi ^{x} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y}\\rbrace } \\right] \\nonumber \\\\& = \\varphi (x,y) + \\lambda \\int \\mathbb {P}_\\lambda \\left(\\lbrace x \\sim u \\text{ in } \\xi ^{x,u} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y,u}\\rbrace \\right) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& = \\varphi (x,y) +\\lambda \\int \\varphi (x,u) \\tau _\\lambda (u,y) \\nu \\left(\\mathrm {d}u\\right).$ In the last line, we have used that the two intersected events are independent.","This is due to the fact that $\\nu $ is non-atomic and $x\\notin \\eta ^{y,u}$ a.s.","The second inequality is then proven nearly identically, but with the adjacency to the intermediate point $u$ holding for $y$ instead of $x$ .","To get (REF ), we first use () to expand out the right-most $\\mathcal {T}_\\lambda $ .","This produces one term with $$ replacing the $\\mathcal {T}_\\lambda $ , and one term with $\\lambda \\mathcal {T}_\\lambda $ replacing it.","We then repeatedly use (REF ) to expand the left-most $\\mathcal {T}_\\lambda $ to get a $$ factor and a $\\lambda \\mathcal {T}_\\lambda $ factor until we have a total of $n$ factors of $$ in the expression.","Counting the number of ways to get to each possible end term then gives the binomial $\\genfrac(){0.0pt}1{n}{m}$ factor as required.","The following lemma will allow us to describe $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ using Hilbert space language.","Lemma 6.3 Let $H\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ be an integral linear operator with kernel function $h\\colon \\mathcal {E}^2\\rightarrow \\mathbb {R}$ .","Then for all $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ and $\\varepsilon >0$ , there exist $\\mathcal {P}$ -positive sets $E_1,E_2\\subset \\mathcal {E}$ such that $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ satisfy $*{h\\left(a,b\\right) - {g_1}{Hg_2}} \\le \\varepsilon .$ Since $h$ is measurable, the set $A=\\left\\lbrace c,c^{\\prime }\\in \\mathcal {E}:*{h\\left(c,c^{\\prime }\\right)-h\\left(a,b\\right)}\\le \\varepsilon \\right\\rbrace $ is measurable.","Since $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ , it has positive measure: $\\mathcal {P}\\left(A\\right)>0$ .","Furthermore, by considering the $\\pi $ -system of product sets $\\left\\lbrace E_1\\times E_2: E_1,E_2\\subset \\mathcal {E}\\text{ measurable}\\right\\rbrace $ that generates the $\\sigma $ -algebra on $\\mathcal {E}^2$ , we can find $B = E_1\\times E_2$ such that $E_1,E_2\\subset \\mathcal {E}$ are both measurable, $B\\subset A$ , and $\\mathcal {P}^{\\otimes 2}\\left(B\\right) = \\mathcal {P}\\left(E_1\\right)\\mathcal {P}\\left(E_2\\right)>0$ .","Then use $E_1,E_2$ to define $g_1,g_2$ as in the statement on the lemma.","Since $E_1,E_2$ are both measurable and have positive measure, both $g_1,g_2\\in L^2\\left(\\mathcal {E}\\right)$ .","We also have ${g_1}{Hg_2} = \\frac{1}{\\mathcal {P}^{\\otimes 2}\\left(B\\right)}\\int _B h\\left(c,c^{\\prime }\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right) \\in \\left[h(a,b) - \\varepsilon ,h(a,b) +\\varepsilon \\right],$ proving the result.","Let us define the function $\\overline{\\epsilon }\\colon \\mathbb {N}\\rightarrow \\mathbb {R}_+$ , where $\\overline{\\epsilon }(d) := {\\left\\lbrace \\begin{array}{ll}\\frac{1}{d} + \\frac{\\log d}{\\log g(d)} &: \\lim _{d\\rightarrow \\infty }g(d)\\rho ^{-d}\\Gamma \\left(\\frac{d}{2}+1\\right)^2 = 0 \\qquad \\forall \\rho >0,\\\\0&: \\text{otherwise}.\\end{array}\\right.","}$ Recall $g(d)$ is the function defined in REF .","In particular, $\\overline{\\epsilon }(d) = 0$ unless $g(d)$ approaches zero particularly quickly, and $\\beta (d) = g(d)^{\\frac{1}{4}-\\frac{3}{2}\\overline{\\epsilon }(d)}$ .","Lemma 6.4 Let $\\lambda <\\lambda _O$ , $n\\ge 2$ , and $d>2m$ .","Then there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=2\\\\c_fg(d)^{\\frac{1}{2}-m\\overline{\\epsilon }(d)} &: n\\ge 3.\\end{array}\\right.","}$ The condition $\\lambda < \\lambda _O$ here is required to allow us to swap integrals over the mark space with integrals over the Fourier space.","We are able to use Fubini's Theorem because sub-criticality implies that the relevant integrals are finite.","For clarity, let us introduce the notation $\\varphi ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\varphi (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),\\quad \\tau _\\lambda ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\tau _\\lambda (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),$ where $u_0=x$ and $u_n=y$ .","Furthermore, for each $\\overline{x}\\in \\mathbb {R}^d$ let us define the integral linear operator $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) \\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $(a,b)\\mapsto \\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) := \\int \\varphi ^{(n-1)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda ^{(m)}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime }\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ Given $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $a,b\\in \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right)\\in {\\rm ess.Im}\\left(\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};\\cdot ,\\cdot \\right)\\right)$ , we can use Lemma REF to get measurable and $\\mathcal {P}$ -positive sets $E^{(\\overline{x},a,b,\\delta )}_1,E^{(\\overline{x},a,b,\\delta )}_2\\subset \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) \\le *{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2} + \\delta ,$ where $g^{(\\overline{x},a,b,\\delta )}_i(c) = \\frac{1}{\\mathcal {P}\\left(E^{(\\overline{x},a,b,\\delta )}_i\\right)}1_{E^{(\\overline{x},a,b,\\delta )}_i}\\left(c\\right)$ for $i=1,2$ .","Now let $E_1,E_2\\subset \\mathcal {E}$ be arbitrary measurable $\\mathcal {P}$ -positive sets and $g_1,g_2$ be their associated functions.","We aim to produce bounds that are independent of the choice of $E_1,E_2$ .","We first use the Fourier inversion theorem to reformulate the position behaviour in terms of a $k$ -integral of the Fourier transforms, and use REF and $\\lambda <\\lambda _O$ (so the relevant integrals are finite) to allow us to swap the $k$ -integral and the mark integrals.","This produces $*{*{g_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g_2}} = *{\\int \\text{e}^{-i\\overline{x}\\cdot k}*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}}\\\\ \\le \\int *{*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ Recall that $\\widehat{}(k)$ is self-adjoint, so we can move $\\widehat{}(k)^{n-1}$ to the other side of the inner product, and then we can use Cauchy-Schwarz and the definition of the operator norm to get $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} = *{\\widehat{}(k)^{n-1}g_1}{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\\\\\le *{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}_2 \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}^m*{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{}(k)g_2}_2.$ We can then use the definition of the bootstrap function $f(\\lambda )$ to replace the $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ factors with factors of $f(\\lambda )$ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .","Also writing the norm $*{\\cdot }_2$ as the square root of an inner product and using the self-adjoint property of $\\widehat{}(k)$ gives $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\le f(\\lambda )^m\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}.$ We therefore want to derive bounds for the integral $\\int \\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ We first derive pointwise and integral bounds for the $\\widehat{}$ factors.","Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\widehat{\\varphi }^{(2N)}(k;c,c^{\\prime })\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right)^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2}.$ By using Cauchy-Schwarz and supremum bounds we can split this into norms of $\\widehat{}(k)$ only, then use Lemma REF to replace $k$ with 0, and finally use REF to bound these terms: $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2}\\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty }^{N-1} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }^{N-1} \\le C^N.$ For the integral bound, we apply Cauchy-Schwarz to the $k$ -integral and swap the $k$ -integrals with the mark integrals to un-do the Fourier transforms.","We get $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\left(\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2}\\left(\\int *{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2} \\\\= *{g_1}{^{(2n-2)}\\left(\\overline{0}\\right)g_1}^\\frac{1}{2}*{g_2}{^{(2)}\\left(\\overline{0}\\right)g_2}^\\frac{1}{2},$ where $^{(N)}\\left(\\overline{0}\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ is the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\varphi ^{(N)}\\left(\\overline{0};a,b\\right)$ .","Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{^{(2N)}\\left(\\overline{0}\\right)g} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\varphi ^{(2N)}\\left(\\overline{0};c,c^{\\prime }\\right)\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right) \\le \\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }.$ Here for $N\\ge 2$ we bound with $\\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert ^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le g(d)^\\frac{1}{2}$ from REF .","For $N=1$ we bound using $\\varphi \\left(\\overline{x};a,b\\right)\\in \\left[0,1\\right]$ and REF : $\\left\\Vert ^{(2)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,c\\right)\\varphi \\left(-\\overline{x};c,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}c\\right) \\le \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\\\= \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }\\le C.$ Therefore $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le {\\left\\lbrace \\begin{array}{ll}C &: n=2,\\\\C^\\frac{1}{2}g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.","}$ We now return to bounding (REF ).","To perform the $k$ -integral, we split the domain.","We first consider $B_\\varepsilon (0) = \\lbrace k\\in \\mathbb {R}^d: *{k} < \\varepsilon \\rbrace $ , where we choose $\\varepsilon >0$ such that $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1,C_2$ coming from the assumption REF .","From REF , we have $\\widehat{G}_{\\mu _\\lambda }(k) = \\left(1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)^{-1}\\le 1/C_2*{k}^2$ on $B_\\varepsilon (0)$ and $\\widehat{G}_{\\mu _\\lambda }(k)\\le 1/\\left(C_2\\varepsilon ^2\\right)$ on $B_\\varepsilon (0)^c$ .","Our pointwise bound on the $\\widehat{}$ factors then gives $\\int _{B_\\varepsilon (0)}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\int ^\\varepsilon _0\\frac{1}{r^{2m}}r^{d-1}\\mathrm {d}r = \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{d-2m}\\frac{\\varepsilon ^{d-2m}}{\\left(2\\pi \\right)^d},$ where $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.","Note that for all fixed $\\varepsilon >0$ this term vanishes in the $d\\rightarrow \\infty $ limit.","To deal with the integral over $B_\\varepsilon (0)^c$ , we find the upper bound $\\int _{B_\\varepsilon (0)^c}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{1}{C^m_2\\varepsilon ^{2m}}\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\frac{C^\\frac{1}{2}}{C^m_2\\varepsilon ^{2m}}{\\left\\lbrace \\begin{array}{ll}C^\\frac{1}{2} &: n=2\\\\g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.","}$ Note that the bounds we found were independent of the sets $E_1,E_2$ , and therefore these bounds also apply to $*{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2}$ uniformly over every $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $\\mathcal {P}$ -almost every $a,b\\in \\mathcal {E}$ .","Since we can take $\\delta \\rightarrow 0$ , our bound also applies to $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }$ .","For $n=2$ , the $B_\\varepsilon (0)^c$ integral dominates the $B_\\varepsilon (0)$ integral, and so the result is proven.","For $n\\ge 3$ , both parts of the integral approach 0 as $d\\rightarrow \\infty $ .","If both are of the same order or if the $B_\\varepsilon (0)^c$ integral dominates, then we can fix $\\varepsilon >0$ and get the result.","On the other hand, if $g(d)^\\frac{1}{2} \\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^{d}$ for all fixed $\\varepsilon >0$ and the $B_\\varepsilon (0)$ integral always dominates, we can improve the overall bound by letting $\\varepsilon =\\varepsilon (d)$ and having $\\varepsilon (d)\\rightarrow 0$ as $d\\rightarrow \\infty $ .","Having a smaller value of $\\varepsilon $ produces a smaller bound for the $B_\\varepsilon (0)$ integral, but a larger bound for the $B_\\varepsilon (0)^c$ integral.","To get an optimal $\\varepsilon $ we can set both terms to be of the same order - that is by having $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{2m}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .","For this choice, we have both integrals of the order $\\varepsilon ^{-2m}g(d)^\\frac{1}{2} = 2^{-2m}\\pi ^{-m}A^{-\\frac{2m}{d}} g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\left(1-\\frac{2m}{d}\\right)^{-\\frac{2m}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{2m}{d}} \\\\= \\left(\\frac{\\text{e}}{2\\pi }\\right)^m g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\frac{1}{d^m}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .","Proposition 6.5 (Bound for the $n$ -gon diagrams) Let $\\lambda < \\lambda _O$ .","Then for each $n$ such that $d>2n$ , there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=1,2,\\\\c_fg(d)^{\\frac{1}{2}-n\\overline{\\epsilon }(d)}&: n\\ge 3.\\end{array}\\right.","}$ The $n=1$ case is trivial because $\\tau _\\lambda \\left(\\overline{x},a,b\\right)\\in \\left[0,1\\right]$ .","For $n=2$ , we first bound $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty }$ .","We get $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int *{\\varphi (x,u)\\varphi (u,y)}\\nu (\\mathrm {d}u) \\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}^2\\nu (\\mathrm {d}u) \\\\\\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}\\nu (\\mathrm {d}u) = \\left\\Vert \\right\\Vert _{1,\\infty },$ by using Cauchy-Schwarz and symmetry of $\\varphi $ in the first inequality and $\\varphi (x,u)\\in \\left[0,1\\right]$ in the second.","We then use REF to bound this by the constant $C$ .","We use Lemma REF to get the required bound for the remaining terms arising from (REF ) in Lemma REF .","For $n\\ge 3$ , we once again bound the terms in the relevant expansion in (REF ).","For $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }$ we use REF to get $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }= O\\left(g(d)^\\frac{1}{2}\\right)$ .","We then use Lemma REF to bound the remaining terms.","For our proofs of Lemma REF and Lemma REF , we adopt the following notation.","Given $g(\\overline{x};a,b)$ and $h(\\overline{x};a,b)$ , we define $(gh)(\\overline{x};a,b) := \\int g(\\overline{x}-\\overline{y};a,c)h(\\overline{y};c,b)\\mathrm {d}\\overline{y}\\mathcal {P}(\\mathrm {d}c).$ This notational convention is associative and therefore can generalise to three or more terms unambiguously.","Lemma 6.6 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}\\le c_f,\\\\\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le c_f,$ and therefore $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le 1 + 2c_f.$ By applying (REF ) and (), we can get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ We will in fact prove the more general result for the convolution of $m\\ge 2$ $\\tau _\\lambda $ -functions in dimensions $d>2m$ .","By bounding $\\lambda \\le f(\\lambda )$ , and using Lemma REF and the Fourier inversion theorem we can realise that we only need to get further bounds for the following objects.","For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^m$ , we want to bound $\\int \\prod ^m_{i=1} P^{j_i}_{2i-1,2i}(k)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{p,q}(k) := {\\left\\lbrace \\begin{array}{ll}*{*{g_p}{\\widehat{}(k) g_q}} &:j=1\\\\*{*{g_p}{\\widehat{}(k)^2g_q}} &:j=2\\\\*{*{g_p}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_q}} &: j=3,\\end{array}\\right.","}$ where $\\left\\lbrace g_i\\right\\rbrace _{i=1}^{2m}$ are the functions of the form $g_i(a) = \\frac{1}{\\mathcal {P}(E_i)}1_{E_i}(a)$ arising from the application of Lemma REF .","We now bound the following terms in the same way as (REF ): $*{*{g_1}{\\widehat{}(k)^2g_2}} &\\le *{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}\\\\*{*{g_1}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_2}} &\\le f\\left(\\lambda \\right)\\widehat{G}_{\\mu _\\lambda }(k)*{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}.$ In terms of pointwise uniform bounds for the $\\widehat{}$ terms, we have $*{*{g_1}{\\widehat{}(k)g_2}}\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C$ and $*{g}{\\widehat{}(k)^2g}\\le C^2$ (the latter also having been used in the proof of Lemma REF ).","As in Lemma REF , we will require various integral bounds for these $\\widehat{}$ terms.","Suppose we have $r \\in \\left\\lbrace 1,\\ldots ,m-1\\right\\rbrace $ instances of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and $m-r$ instances of $P^{1}_{p,q}(k)$ .","Then by applying Cauchy-Schwarz and the definitions of the $g_i$ we get $&\\int \\prod ^{r}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\prod ^{m}_{i^{\\prime }=r+1}*{*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int \\prod ^{r}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\prod ^{m}_{i^{\\prime }=r+1}*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2r-1},a_{2r})\\right.\\right.\\nonumber \\\\&\\hspace{199.16928pt}\\left.\\left.\\star \\varphi (\\cdot ;a_{2r+1},a_{2r+2})\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m+r}.$ In this last inequality we have used $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\le C$ and $\\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }^2 \\le C^2$ to extract off $\\varphi $ and $\\varphi ^{(2)}$ via supremum bounds on the spatial position.","We will also require integral bounds where there are $m$ factors of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and where there are $m$ factors of $P^{1}_{p,q}(k)$ : $&\\int \\prod ^{m}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\left(\\int \\prod ^{m}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{2m}.\\\\&\\int \\prod ^{m}_{i=1}*{*{g_{2i-1}}{\\widehat{}(k)g_{2i}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)g_2}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\left(\\prod ^{m}_{i=2}*{g_{2i-1}}{\\widehat{}(k)g_{2i}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m}.$ Note that for $m\\ge 3$ , better bounds are available via REF , but will not be required here.","We can now proceed to bound each of the expressions of the form (REF ) in much the same way as we did in the proof of Lemma REF .","We set $\\varepsilon >0$ and partition the integral over $\\mathbb {R}^d$ into one over $B_\\varepsilon (0)$ and one over $B_\\varepsilon (0)^c$ .","For the $B_\\varepsilon (0)$ integral we use the uniform bounds on the $\\widehat{}$ terms and perform the integral of the $\\widehat{G}_{\\mu _\\lambda }$ terms (if there are any) as before.","Note that we require $d>6$ for all of these to be finite.","For the $B_\\varepsilon (0)^c$ integrals we uniformly bound $\\widehat{G}_{\\mu _\\lambda }(k) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and use the integral bounds we calculated above.","Since we are only asking for a constant bound, we don't need to worry about having $\\varepsilon \\rightarrow 0$ as $d\\rightarrow \\infty $ ."],["Bounding the Lace Operator Displacement","We now consider the objects required to bound the displacement of the lace operator.","In addition to the objects used to bound the lace operator itself, we require bounds on $\\left\\Vert _k\\right\\Vert _{1,\\infty }, \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }, \\overline{W_k}, \\overline{H_k}, \\triangle ^{(B)}_\\lambda , \\mathbb {B}.$ The decay $\\mathbb {B}\\le g(d)$ follows from Assumption REF , and the following Observation REF demonstrates the decay of $\\triangle ^{(B)}_\\lambda $ .","Observation 6.7 Note that the inequality (REF ) allows us to bound $\\triangle ^{(B)}_\\lambda $ in terms of other terms.","By using the inequality twice and appropriately bounding $\\varphi $ with $\\tau _\\lambda $ , we get $\\int \\tau _\\lambda (r,u)\\tau _\\lambda (u,s)\\nu (\\mathrm {d}u) \\le \\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) + 2\\lambda \\int \\tau _\\lambda (r,v)\\tau _\\lambda (v,u)\\tau _\\lambda (u,s)\\nu ^{\\otimes 2}(\\mathrm {d}u, \\mathrm {d}v)$ for any $r,s\\in \\mathbb {X}$ .","Therefore $\\triangle ^{(B)}_\\lambda \\le 3\\triangle _\\lambda + \\lambda \\operatornamewithlimits{ess\\,sup}_{r\\notin B(s)}\\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) \\le 3\\triangle _\\lambda + \\lambda g(d),$ where we have used REF .","Therefore Proposition REF implies that for $d>6$ there exists $c_f$ such that $\\triangle ^{(B)}_\\lambda \\le c_f\\beta ^2.$ We are then left with $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ , $\\overline{W_k}$ , and $\\overline{H_k}$ , and we will deal them in this order.","First note that $\\left\\Vert _k\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi _k(x,y)\\nu \\left(\\mathrm {d}x\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int (1-\\cos \\left(k\\cdot \\overline{x}\\right))\\varphi (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}(\\mathrm {d}a) \\\\ = \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty }.$ The following lemma therefore also allows us to bound $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ .","Its extra generality will be required later.","Lemma 6.8 For all $k,l\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\infty ,\\infty } \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where $C>0$ is the $d$ -independent constant given in REF .","The first two inequalities in (REF ) holds from the general inequalities that holds for these norms (on a probability space for the second inequality).","Then we can remove the $l$ -dependence by performing the following calculation: $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+ k\\right)\\right\\Vert _{\\infty ,\\infty } \\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\text{e}^{il\\cdot \\overline{x}}}*{\\left(1-\\cos \\left( k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}\\\\= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}= \\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left( k\\right)\\right\\Vert _{\\infty ,\\infty }.$ The final step relating the $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ -norm to the difference of the spectral suprema is precisely that given by () in REF .","We now address $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ .","Lemma 6.9 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ First note that for $k=0$ the kernel function $\\tau _{\\lambda ,k}\\left(x,y\\right)=0$ for all $x,y\\in \\mathbb {X}$ , and so the required inequality holds trivially.","In this proof we hereafter may assume $k\\ne 0$ .","We use (REF ) to write $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }.$ We first get the bound on $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ , using a similar approach to Lemma REF .","For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda \\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}^{\\prime }-\\overline{u}^{\\prime \\prime };c^{\\prime },c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime \\prime };c^{\\prime \\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathrm {d}\\overline{u}^{\\prime \\prime }\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime },\\mathrm {d}c^{\\prime \\prime }\\right).$ Recall the notation $\\varphi ^{(2)}$ used in (REF ).","As in the proof of Lemma REF , we let $E_1,E_2\\subset \\mathcal {E}$ be measurable and $\\mathcal {P}$ -positive sets, and aim to bound ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ (where $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ ) independently of the choice of $E_1,E_2$ .","If we do this we will have proven the result.","As before, we use the Fourier inversion theorem to write ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ as the integral of an inner product of Fourier transformed operators over the Fourier argument.","Having $\\lambda <\\lambda _O$ allows us to swap the integrals in this step.","Then we can use Cauchy-Schwarz and the definitions of the operator norm and the bootstrap functions to extract factors of $\\widehat{G}_{\\mu _\\lambda }(l)$ and $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ from the inner product.","The net result is $*{g_1}{\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} &\\le \\int *{*{g_1}{\\widehat{}(l)^{2}\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l)g_2}}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\le f(\\lambda )^2\\int \\widehat{G}_{\\mu _\\lambda }(l) \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ The $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ We partition our integral using open $\\varepsilon $ -balls around the poles where $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1$ and $C_2$ coming from REF .","Let $B_{\\varepsilon }(p)$ denote the open $\\varepsilon $ -ball around $p\\in \\mathbb {R}^d$ .","For $n\\in \\left\\lbrace 0,1,2,3\\right\\rbrace $ , let $A_n := \\left\\lbrace l\\in \\mathbb {R}^d: l\\in B_\\varepsilon (p) \\text{ for precisely $n$ elements }p\\in \\left\\lbrace k,0,-k\\right\\rbrace \\right\\rbrace .$ Firstly it is possible for $A_3,A_2 = \\emptyset $ if $*{k}$ is sufficiently large compared to $\\varepsilon $ .","It is also easy to see that $A_3\\subset B_{\\varepsilon }(0)$ and $A_2\\subset B_{2\\varepsilon }(0)$ .","We will partition $\\mathbb {R}^d$ into $A_3$ , $A_2$ , $A_1$ , and $A_0$ and bound the $l$ -integral on each part.","Recall that REF implies that $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2*{l}^{-2}$ for $*{l}\\le \\varepsilon $ and $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2\\varepsilon ^2$ for $*{l}\\ge \\varepsilon $ .","Also recall the pointwise (REF ) and integral (REF ) bounds we derived for the $\\widehat{}$ factors.","We treat $A_3$ first: $&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)\\cap B_\\varepsilon (k)}\\frac{1}{*{l}^4*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\qquad \\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\\\&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{A_3}\\frac{1}{*{l+k}^2*{l}^2*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (-k)}\\frac{1}{*{l+k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ Recall $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.","In these calculations we used Hölder's inequality and increased the domain of integration to get upper bounds.","These integrals are finite for $d>6$ , and approach 0 if we take bounded $\\varepsilon $ .","For $A_2$ the counting of cases is the only extra complication.","A precise counting is entirely possible, but unnecessary for our purposes.","It is simple to see that the arguments outlined for $A_3$ can be applied to get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^6} + \\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a pair of two distinct points from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with three, two, and one factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on these two points.","This bound is finite for $d>6$ , and approaches 0 if we take bounded $\\varepsilon $ .","Repeating this for $A_1$ , we get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} + \\frac{1}{\\varepsilon ^6} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a single point from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with two, one, and zero factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on this single point.","This bound is finite for $d>4$ , and approaches 0 if we take bounded $\\varepsilon $ .","The $A_0$ integral is qualitatively different.","By taking the integral bound for the $\\widehat{}$ factors and a uniform (on $A_0$ ) bound for the factors of $\\widehat{G}_{\\mu _\\lambda }$ , we get the bound $\\frac{C^3}{C^3_2}\\varepsilon ^{-6}g(d)^\\frac{1}{2}.$ As in Lemma REF , if the $g(d)^\\frac{1}{2}$ term dominates, then we have a sufficient bound for our result.","The largest of the other terms (for $d$ sufficiently large and $\\varepsilon \\le 1$ ) is the $\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ term.","Like we did in Lemma REF , we can make these other terms smaller by decreasing $\\varepsilon $ at the cost of making the $g(d)^\\frac{1}{2}$ bound greater.","The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{6}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .","For this choice, we have both integrals of the order $\\varepsilon ^{-6}g(d)^\\frac{1}{2} = 2^{-6}\\pi ^{-3}A^{-\\frac{6}{d}}g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\left(1-\\frac{6}{d}\\right)^{-\\frac{6}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{6}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^3 g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\frac{1}{d^3}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .","Bounding the $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ term is similar.","For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ We once again take $\\mathcal {P}$ -positive sets $E_1,E_2$ and corresponding $g_1,g_2$ .","Then we use the Fourier inversion theorem and Cauchy-Schwarz to get $*{g_1}{\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} \\le f(\\lambda )\\int \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Then once again the $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Counting the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ centred on each of the poles $\\left\\lbrace k,0,-k\\right\\rbrace $ then tells us that there is a uniform bound that is some integer multiple of $\\frac{C^2}{C^2_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^{4}} + \\frac{1}{\\varepsilon ^2*{l}^{2}} + \\frac{1}{\\varepsilon ^4}\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} + \\varepsilon ^{-4}g(d)^\\frac{1}{2}\\right) \\\\= \\frac{C^2}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-4} + \\frac{C^2}{C^2_2}\\varepsilon ^{-4}g(d)^\\frac{1}{2}.$ Once again, if the $g(d)^\\frac{1}{2}$ term dominates the result is proven.","Otherwise, we let $\\varepsilon $ vary with $d$ .","The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{4}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .","For this choice, we have both integrals of the order $\\varepsilon ^{-4}\\beta ^2 = 2^{-4}\\pi ^{-2}A^{-\\frac{4}{d}} g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(1-\\frac{4}{d}\\right)^{-\\frac{4}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{4}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^2 g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\frac{1}{d^2}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .","For large $d$ , this is dominated by $g(d)^{\\frac{1}{2}-\\frac{3}{d}}d^{-3}$ and therefore $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ is dominated by $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ and we prove the result.","The following lemma allows us to deal with occurrences of $\\overline{W_k}$ .","Lemma 6.10 Let $x,y \\in \\mathbb {X}$ .","Then $\\tau _{\\lambda ,k}(x,y) \\le \\varphi _k(x,y) + 2\\lambda \\int \\left(\\tau _\\lambda (x,u)\\varphi _k(u,y) + \\tau _{\\lambda ,k}(x,u)\\varphi (u,y)\\right)\\nu \\left(\\mathrm {d}u\\right).","$ For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{W_k}\\le c_f\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The first inequality follows by applying the cosine-splitting result (Lemma REF ) to ().","For the $\\tau _\\lambda $ term in $\\overline{W_k}$ we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of (REF ), we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we can bound $\\overline{W_k}$ by bounding the following objects.","For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace \\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {W}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l)Q^{j_2}_{3,4}(l;k)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{1,2}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_1}{\\widehat{}(l) g_2}} &:j=1\\\\*{*{g_1}{\\widehat{}(l)^2g_2}} &:j=2\\\\*{*{g_1}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_2}} &: j=3,\\end{array}\\right.","}\\\\Q^{j}_{3,4}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} &:j=1\\\\*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} &:j=2\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{}_k(l) g_{4}}} &:j=3\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &:j=4\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} &:j=5,\\end{array}\\right.","}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{4}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .","For the $P^j_{1,2}(l)$ factor we can get bounds as we did in the proof of Lemma REF .","For $Q^{j}_{3,4}(l;k)$ we need to do a bit more processing.","For $j=1$ , $*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} \\le \\frac{1}{\\mathcal {P}(E_{3})\\mathcal {P}(E_{4})}\\int _{E_{3}\\times E_{4}}*{\\widehat{\\varphi }_k(l;a,b)}\\mathcal {P}^{\\otimes 2}(\\mathrm {d}a, \\mathrm {d}b) \\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right),$ where in the last inequality we used Lemma REF .","For $j=2$ , we use Cauchy-Schwarz to get $*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} \\le *{g_3}{\\widehat{}_k(l)^2 g_3}^{\\frac{1}{2}} *{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}} \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}.$ An identical argument gives the same bound for $j=3$ .","For $j=4$ , we can use the self-adjointness of $\\widehat{}(l)$ and $\\widehat{\\mathcal {T}}_\\lambda (l)$ and the Cauchy-Schwarz inequality to get $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &\\le *{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)^2\\widehat{}(l)g_{3}}^{\\frac{1}{2}} *{g_{4}}{\\widehat{}_k(l)^2g_{4}}^{\\frac{1}{2}} \\nonumber \\\\&\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}}\\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }\\nonumber \\\\&\\le Cf(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}},$ and similarly $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} \\le f(\\lambda )\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2} \\\\ \\le Cf(\\lambda )\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2},$ where we write for convenience $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k) = \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3-k) + \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k) + \\widehat{G}_{\\mu _\\lambda }(l_3-k)\\widehat{G}_{\\mu _\\lambda }(l_3+k).$ Note that all the bounds for the $Q^{j}_{3,4}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.","So we only need to prove that the remaining $l$ -integral of $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors is bounded.","As in our previous proofs, we partition $\\mathbb {R}^d$ into the sets $A_n$ defined earlier in (REF ).","For the $A_1$ , $A_2$ , and $A_3$ parts we then use the bounds $C$ and $C^2$ for the various $\\widehat{}$ factors and $1/\\left(C_2*{l}^2\\right)$ to bound the $\\widehat{G}_{\\mu _\\lambda }$ factors.","By using the techniques used previously in this section, we can show that these contributions is bounded by some $\\varepsilon $ and $d$ (only) dependent constant.","Since we don't need to show decay, we don't need to worry about taking $\\varepsilon $ to 0.","For the $A_0$ contribution, we bound $\\widehat{G}_{\\mu _\\lambda }$ factors by $1/\\left(C_2\\varepsilon ^2\\right)$ and show that the integral of the $\\widehat{}$ factors over all of $\\mathbb {R}^d$ is bounded.","For $j_2\\ne 1$ this proceeds similarly to the argument in the proof of Lemma REF .","For $j_2=1$ we need to take a step back.","These versions of (REF ) arise from trying to bound one particular type of term, which we now treat separately.","We bound $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\operatornamewithlimits{ess\\,sup}_{a_3,a_4\\in \\mathcal {E}}\\int \\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{\\infty ,\\infty }.$ Then Lemma REF gives the result.","We are now only left with bounding $\\overline{H_k}$ .","lemmaMartiniBound For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{H_k}\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The proof of the lemma uses the same techniques and tools as our previous lemmas, but is much more complicated in terms of keeping track of cases and components of integrals.","To avoid overburdening a reader on an early reading, the proof has been placed in Appendix ."],["The Bootstrap Argument","In this section we bring together the diagrammatic bounds of Section with the estimates of Section REF to bound the relevant operators in such a way that the finite lace expansion of Proposition REF converges to an Ornstein-Zernike equation (OZE) for both the raw operators on $L^2\\left(\\mathbb {X}\\right)$ and the Fourier-transformed operators on $L^2\\left(\\mathcal {E}\\right)$ .","Proposition REF does this for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , with the caveat that there is a constant $c_f$ that is increasing in the bootstrap function $f$ (but independent of $d$ ) appearing in the bounds.","Observation REF then highlights that if $f$ is uniformly bounded on $\\left[0,\\lambda _O\\right)$ then so is $c_f$ and the associated bounds from Proposition REF .","The crucial step to the bootstrap argument is then Proposition REF .","Here a forbidden-region argument shows that $f$ is continuous, that there is an interval of finite values that it cannot take (the interval being independent of $\\lambda $ ), and that it starts below this interval.","This proves that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , and so are our bounds in Proposition REF .","Proposition 6.11 (Convergence of the operator lace expansion and OZE) Let $\\lambda \\in [0, \\lambda _O)$ and $d>6$ be sufficiently large.","Then there exists $c_f=c(f(\\lambda ))$ (which is increasing in $f$ and independent of $d$ ) such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f\\beta , &\\qquad \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}\\le c_f\\beta , \\\\ \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le c_f\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right)-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\beta , \\\\\\left\\Vert R_{\\lambda , n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} c_f\\beta ^{n}.","$ Furthermore, the limit $_\\lambda := \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}_\\lambda ^{(n)}$ exists and is a bounded operator with Fourier transform $\\widehat{}_{\\lambda }\\left(k\\right) = \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}\\widehat{}_{\\lambda }^{(n)}\\left(k\\right)$ for $k\\in \\mathbb {R}^d$ .","The operators $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ satisfy operator Ornstein-Zernike equations, taking the form $\\mathcal {T}_\\lambda = + _\\lambda + \\lambda \\mathcal {T}_\\lambda (+_\\lambda ), $ and $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right) + \\lambda \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\left(\\widehat{}\\left(k\\right)+\\widehat{}_{\\lambda }\\left(k\\right)\\right), $ for all $\\lambda < \\lambda _O$ and $k\\in \\mathbb {R}^d$ .","Before we prove this proposition, we make the following observation, which will guide the rest of the section.","Observation 6.12 (Uniform convergence of the lace expansion) Suppose that there exists finite $C>0$ such that $f \\le C$ on $[0,\\lambda _O)$ .","Then for $d>6$ sufficiently large there exits $c>0$ (independent of $\\lambda $ and $d$ ) such that the bounds (REF ), (), () hold with $c_f$ replaced by $c$ .","We now return to prove Proposition REF .","[Proof of Proposition REF ] So far in this section we have provided bounds for the building blocks of the bounds appearing in Propositions REF , REF , REF , and REF .","We still have to convert these into bounds for the composite terms $U_\\lambda $ and $V_\\lambda $ .","From their definitions and the bounds on the building blocks, it is simple to see that there exists $c_f^{\\prime }$ such that for $d>6$ $U_\\lambda \\le c_f^{\\prime }, \\qquad V_\\lambda \\le c_f^{\\prime }\\beta .$ This then implies that there exists $c_f^{\\prime \\prime }$ such that for $d>6$ $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime }\\beta ^2 &: n=0,\\\\c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n &: n\\ge 1.\\end{array}\\right.","}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime }\\left(\\beta ^2 + \\frac{c_f^{\\prime \\prime }\\beta }{1-c_f^{\\prime \\prime }\\beta }\\right) \\le c_f\\beta .$ When we want to consider the displaced terms, we also make use of Observation REF , and Lemmas REF , REF , REF , and REF to give us the existence of $c_f^{\\prime \\prime \\prime }$ such that for $d>6$ $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime \\prime }\\beta ^2\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=0,3,\\\\c_f^{\\prime \\prime \\prime }\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=1,2,\\\\\\left(n+1\\right)^2c_f^{\\prime \\prime \\prime }\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^{n-3}\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n\\ge 4.\\end{array}\\right.","}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime \\prime }\\left(2\\beta ^2 + 2\\beta + \\frac{16\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^3-39\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^2 + 25c_f^{\\prime \\prime \\prime }\\beta }{\\left(1-c_f^{\\prime \\prime \\prime }\\beta \\right)^3}\\right)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\\\\\le c_f\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Lemma REF then allows us to use (REF ) to bound $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ .","This immediately gives the required result for $n\\ge 1$ .","For $n=0$ , we simply additionally require $d$ to be large enough that $\\beta <1$ and therefore $\\beta ^2<\\beta ^0$ .","Note that the dual space of a Banach space (endowed with the operator norm) is also a Banach space.","In particular, it is complete.","Since $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n$ for $n\\ge 1$ , the sequence $_{\\lambda ,n}$ is a Cauchy sequence in the dual space of $L^2(\\mathbb {X})$ for sufficiently large $d$ .","Therefore the limit $_\\lambda $ exists and is a bounded linear operator on $L^2(\\mathbb {X})$ .","Since the Fourier transform is a unitary linear operator it is bounded and continuous, and therefore the same argument says that the limit $\\widehat{}_{\\lambda }(k)$ exists and is indeed the Fourier transform of $_\\lambda $ .","Since $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}$ is finite for $\\lambda <\\lambda _O$ , we have $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}\\rightarrow 0$ as $n\\rightarrow \\infty $ if $d$ is sufficiently large.","As a a consequence of this and equation (REF ) from Proposition REF , the $\\mathcal {T}_\\lambda $ operator satisfies the operator Ornstein-Zernike equation.","The following lemma has two main uses in what follows.","Firstly, it takes the OZEs from Proposition REF and uses it to write the two-point operators in terms of the sum of the adjacency operators and the lace expansion coefficient operators.","Secondly, it uses this and an intermediate value theorem argument to bound the spectral supremum of the sum of the adjacency operators and the lace expansion coefficient operators.","Lemma 6.13 Let $\\lambda \\in \\left[0,\\lambda _O\\right)$ and $d>6$ be sufficiently large.","Then the operators $\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)$ and $\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)$ all have bounded linear inverses and $\\mathcal {T}_\\lambda &= \\left(+ _\\lambda \\right)\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)^{-1}, \\\\\\widehat{\\mathcal {T}}_\\lambda (k) &= \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1}, \\qquad \\forall k\\in \\mathbb {R}^d.","$ Furthermore, $\\lambda \\mapsto _\\lambda $ and $\\lambda \\mapsto \\widehat{}_{\\lambda }(k)$ are all continuous in the operator norm topology and $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) &< 1,\\\\\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &< 1, \\qquad \\forall k\\in \\mathbb {R}^d.$ We first prove that $1- \\lambda \\left(+ _\\lambda \\right)$ has a bounded linear inverse.","This is clear for $\\lambda = 0$ , so we only need to consider $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","Suppose for contradiction that $1- \\lambda \\left(+ _\\lambda \\right)$ does not have a bounded linear inverse.","Then there exists a sequence $f_n\\in L^2(\\mathbb {X})$ such that $*{f_n}_2 = 1$ and $*{\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)f_n}_2\\rightarrow 0$ .","This also means that $\\lambda *{\\left(+ _\\lambda \\right)f_n}\\rightarrow 1$ .","Then from Lemma REF and sub-criticality we have $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ , and therefore the OZE equation (REF ) implies that $*{\\left(+ _\\lambda \\right)f_n}\\rightarrow 0$ , a contradiction.","A similar argument proves the corresponding statement for the Fourier transformed operators.","Once we know that the inverses exist, it is only a matter of rearranging the OZE equations to get the expressions for $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ in terms of the other operators.","Since $+ _\\lambda $ is a bounded operator, a similar argument proves that $1+ \\lambda \\mathcal {T}_\\lambda $ has a bounded inverse, and therefore $+ _\\lambda = \\left(1+ \\lambda \\mathcal {T}_\\lambda \\right)^{-1}\\mathcal {T}_\\lambda .$ From Corollary REF we have the continuity of $\\lambda \\mapsto \\mathcal {T}_\\lambda $ , and therefore the continuity $+ _\\lambda $ (and therefore $_\\lambda $ ).","We also have the continuity of $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (k)$ from Corollary REF , and we similarly get the continuity of $\\widehat{}_{\\lambda }(k)$ .","The continuity of $+ _\\lambda $ implies the continuity of $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right)$ (via Lemma REF ).","At $\\lambda = 0$ it is clear that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) = 0$ , and our above argument implies that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) \\ne 1$ .","Therefore an intermediate value theorem argument proves that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) < 1$ .","This argument also works for $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .","It now remains to show that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ .","We do this by performing a forbidden-region argument.","In Proposition REF we prove that $f(0)$ is bounded and $f$ is continuous on $\\left[0,\\lambda _O\\right)$ .","However we also prove that $f$ is never in the region $\\left(\\kappa ,\\kappa +1\\right]$ , where $\\kappa := 10 C \\left(1+C\\right)^2,$ $C$ being the constant appearing in REF .","Note that (REF ) in REF requires that $C\\ge 1$ .","The intermediate value theorem then implies that $\\kappa $ acts as an upper bound on the whole domain $\\left[0,\\lambda _O\\right)$ .","One should not read too much into the value of $\\kappa $ here.","It is sufficient for our purposes, but not remotely optimal.","Proposition 6.14 (The forbidden-region argument) The following three statements are all true: $f$ satisfies $f(0) \\le \\kappa $ .","$f(\\lambda ) \\notin (\\kappa ,\\kappa +1]$ for all $\\lambda \\in [0,\\lambda _O)$ provided that $d$ is sufficiently large.","$f$ is continuous on $[0, \\lambda _O)$ .","Therefore $f(\\lambda ) \\le \\kappa $ holds uniformly in $\\lambda <\\lambda _O$ for all $d$ sufficiently large.","We will deal with each of the three statements in Proposition REF in turn.","[Proof of Proposition REF -REF ] Firstly, it is clear $f_1\\left(0\\right) = 0$ .","Then note that $\\mathcal {T}_0 = $ , and hence $\\left\\Vert \\widehat{\\mathcal {T}_0}\\left(k\\right)\\right\\Vert _{\\rm op} = \\left\\Vert \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we use REF (via (REF )) in the last inequality.","Furthermore $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_0(0)\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) =1$ , and therefore $\\widehat{G}_{\\mu _{0}}\\left(k\\right) = \\widehat{G}_0\\left(k\\right) = 1$ and $f_2\\left(0\\right) \\le C$ .","Similarly we find that $f_3\\left(0\\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op}}{3\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]}.$ We use then Lemma REF to get the bound $\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Hence $f_3\\left(0\\right) \\le \\tfrac{2}{3}C$ .","[Proof of Proposition REF -REF ] We will assume that $f\\le \\kappa +1$ and show that this implies that $f\\le \\kappa $ .","Most crucially, the assumption $f\\le \\kappa +1$ allows us to apply Observation REF .","To highlight the $\\kappa $ -dependence, we write $c_{\\kappa +1}$ as the constant arising from this observation.","It will be convenient to introduce some temporary notation for this section.","Define $\\widehat{a}\\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as $\\widehat{a}\\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)$ for all $k\\in \\mathbb {R}^d$ .","We will also make use of the functions $\\delta _\\lambda \\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\delta _\\lambda \\left(k\\right) := \\mathbb {S}\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)\\right) - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ It is clear from the triangle inequality and Lemma REF that ${\\delta _\\lambda \\left(k\\right)}\\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k\\right)\\right\\Vert _{\\rm op}\\le c_{\\kappa +1}\\beta ,$ but we will also need the following inequality.","We will prove it at the end of this section.","Lemma 6.15 Let $k_1,k_2\\in \\mathbb {R}^d$ .","Then $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2 \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right) - \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ Figure: If 𝒯 ^ λ (k)\\widehat{\\mathcal {T}}_\\lambda (k) and a ^(k)\\widehat{a} (k) commute, then they are simultaneously diagonalizable in the sense of Theorem .","The arguments of their diagonal functions (τ ˜ λ (k)\\widetilde{\\tau }_\\lambda (k) and a ˜(k)\\widetilde{a}(k) respectively) are related by the monotone increasing function x↦x 1-λxx\\mapsto \\tfrac{x}{1-\\lambda x} depicted here.It will be important to note how $\\widehat{a}(k)$ relates to $\\widehat{\\mathcal {T}}_\\lambda (k)$ .","From Observation REF , the OZE gives us the following expression involving the commutator $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right] = \\widehat{\\mathcal {T}}_\\lambda (k) \\widehat{a}(k) - \\widehat{a}(k)\\widehat{\\mathcal {T}}_\\lambda (k)$ : $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right]\\left(1- \\lambda \\widehat{a}(k)\\right) = 0.$ Lemma REF implies that $\\left(1- \\lambda \\widehat{a}(k)\\right)$ has a bounded linear inverse and with (REF ) implies that $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ commute.","Therefore there exists a single unitary map that `diagonalizes' them both in the sense of Theorem REF .","By considering the diagonal functions, the OZE then implies that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\mathbb {S}\\left(\\widehat{a}(k)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)\\right)$ .","In particular, this relation allows us to write an alternative expression for $\\mu _\\lambda $ : $\\mu _\\lambda := 1 - \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} = \\lambda + \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)-1}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)} = \\lambda + \\frac{\\delta _\\lambda \\left(0\\right)}{1+\\delta _\\lambda \\left(0\\right)}.$ We first address $f_1$ .","From the above discussion relating $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ , we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\ge \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) = \\mathbb {S}\\left(\\widehat{a}(0)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(0)\\right)\\right)$ .","From Lemma REF and our choice of scaling we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right) \\ge 1 - \\left\\Vert \\widehat{}_{\\lambda }(0)\\right\\Vert _{\\rm op} \\ge 1 - c_{\\kappa +1}\\beta $ , and therefore we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right)>0$ for sufficiently large $d$ and $\\lambda \\le \\frac{1}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}\\left(1 - \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\right) \\le \\frac{1}{1-c_{\\kappa +1}\\beta } \\le 1 + 2c_{\\kappa +1}\\beta .$ Here we have used an idea that we will use very often in this part of the proof.","For $d$ large enough that $c_{\\kappa +1}\\beta <1$ we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1} = \\sum ^\\infty _{n=0}\\left(c_{\\kappa +1}\\beta \\right)^n$ , and for $d$ sufficiently large enough ($c_{\\kappa +1}\\beta \\le 1/2$ ) we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1}\\le 1 + 2c_{\\kappa +1}\\beta $ .","We also use a similar idea of neglecting higher order terms by increasing an earlier term's coefficient in other cases.","For $f_2$ we first replace $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ .","Since $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a} (k)$ commute, they are simultaneously diagonalizable in the sense of Theorem REF , and therefore their spectra are related by the function $x\\mapsto \\tfrac{x}{1-\\lambda x}$ (monotone increasing on $\\left(-\\infty ,1/\\lambda \\right)$ ).","Therefore we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op} = \\sup _{x\\in \\sigma \\left(\\widehat{a} (k)\\right)}*{\\frac{x}{1-\\lambda x}}$ .","Now (perhaps with the aid of Figure REF ) it is easy to see that $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\vee \\mathbb {S}\\left(-\\widehat{a}(k)\\right) = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) \\vee \\mathbb {S}\\left(-\\widehat{a} (k)\\right).$ Now since $\\mathbb {S}\\left(-\\widehat{a}(k)\\right) \\le \\left\\Vert \\widehat{a}(k)\\right\\Vert _{\\rm op}\\le 1 + c_{\\kappa +1}\\beta $ , and $\\widehat{G}_{\\mu _\\lambda }(k)^{-1} \\le 1+C$ (recall Lemma REF and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C$ from REF ), we only need to bound $\\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We now partition $k\\in \\mathbb {R}^d$ into two.","Define the subset of the Fourier space $A_{\\frac{1}{2}} := \\left\\lbrace k\\in \\mathbb {R}^d:\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\le \\frac{1}{2}\\right\\rbrace .$ Then we only need to bound the two functions $f_4,f_5$ , where $f_4\\left(\\lambda \\right) := \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)},\\qquad \\qquad f_5\\left(\\lambda \\right):= \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}^c} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We first deal with the simpler $f_4$ .","We consider $k\\in A_{\\frac{1}{2}}$ and choose $\\beta $ small enough that $1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right) \\ge 1 - \\frac{\\frac{1}{2}+c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta } \\ge \\frac{1}{2} - 2c_{\\kappa +1}\\beta > 0.$ Then $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\frac{1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}}{\\frac{1}{2} - 2c_{\\kappa +1}\\beta } \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right).","$ Since we have $\\mathbb {S}\\left(\\widehat{a}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{a}(0)\\right) \\le 1 + c_{\\kappa +1}\\beta $ , we then get $f_4(\\lambda ) \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right)\\left(1 + c_{\\kappa +1}\\beta \\right) \\le 2\\left(1+C\\right)\\left(1+7c_{\\kappa +1}\\beta \\right).$ To address $f_5$ we will have $k\\in A^c_\\rho $ .","We define: $\\widehat{N}(k) = \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{F}(k) = \\frac{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{Q}(k) = \\frac{1+\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}.$ Note that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\widehat{N}(k) / \\widehat{F}(k)$ .","Rearranging gets us to $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &= \\widehat{N}(k) \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)}\\nonumber \\\\&= \\widehat{Q}(k) + \\frac{\\widehat{N}(k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)} \\left(1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - \\frac{\\widehat{Q}(k)}{\\widehat{N}(k)}\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)\\right) .$ Since $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ (by our scaling choice), the extracted term $\\widehat{Q}(k)$ satisfies $*{\\widehat{Q}(k)} \\le 1 + 3c_{\\kappa +1}\\beta $ .","We further observe that $\\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{Q}(k)}{\\widehat{N}(k)} = \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\left(1+ \\delta _\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) +\\delta _\\lambda (k)} = 1 - \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right) \\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} =: 1- \\widehat{b}(k).$ Recalling identity (REF ) for $\\mu _\\lambda $ , we can rewrite the quantity $\\left[1- \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)- (1-\\widehat{b}(k)) \\widehat{F}(k)\\right]$ , appearing in (REF ), as $\\frac{1+\\delta _\\lambda (0) - \\left[\\lambda + \\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right] \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - 1 + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)+\\delta _\\lambda (k)\\right) + \\widehat{b}(k)\\left(1-\\lambda \\widehat{a}(k)\\right)}{1+\\delta _\\lambda (0)} \\\\= \\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}+ \\frac{\\widehat{b}(k)(1-\\lambda \\widehat{a}(k))}{1+\\delta _\\lambda (0)}.$ Noting that ${\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)} \\le 2c_{\\kappa +1}\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\beta $ by Lemma REF and Observation REF , the first term can be bounded $&*{\\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}} \\nonumber \\\\&\\qquad \\le \\frac{ c_{\\kappa +1}\\beta + \\left(1 + 2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta + 2\\left(1+2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta }{1 - c_{\\kappa +1}\\beta }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 5c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 6c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Using (REF ), the last term is $*{\\frac{\\widehat{b}(k)(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right))}{1+\\delta _\\lambda (0)}} &= *{ \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\delta _\\lambda (k)}{\\left(1+\\delta _\\lambda (0)\\right)\\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)\\right)}- \\frac{\\lambda \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\delta _\\lambda (k)}{1+\\delta _\\lambda (0)}} \\nonumber \\\\& = \\frac{*{\\delta _{\\lambda }(k)}}{*{1+\\delta _\\lambda (0)}}*{ \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} -\\lambda }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le \\frac{c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta }\\left( \\frac{1}{\\frac{1}{2}-c_{\\kappa +1}\\beta } + \\frac{1}{1-c_{\\kappa +1}\\beta }\\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\le c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left(3+8c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le 4c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Putting these bounds back into (REF ), we can find constants $M$ and $\\tilde{M}$ such that $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &\\le \\widehat{Q}(k) + 10c_{\\kappa +1}\\beta \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\frac{\\widehat{N}(k)}{\\widehat{F}(k)} \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\ \\\\& \\le 1+ 3c_{\\kappa +1}\\beta + 20c_{\\kappa +1}\\beta \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)} \\\\& \\le 1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta .","$ Note that we have used the bound $\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\le \\left[1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]$ (since $\\mu _\\lambda \\le 1$ and $\\mathbb {S}\\left(\\widehat{}(k)\\right)>\\frac{1}{2} >0$ ) to get from $\\widehat{G}_1(k)^{-1}$ to $\\widehat{G}_{\\mu _\\lambda }(k)^{-1}$ .","As $\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\in \\left(\\frac{1}{2}, 1\\right]$ , this concludes the improvement of $f_5$ and hence of $f_2$ .","Before we treat $f_3$ , we introduce $f_6$ given by $f_6(\\lambda ) := \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}\\left(k\\right)\\right)}.$ We show that $f(\\lambda ) \\le \\kappa +1$ implies $f_6(\\lambda ) \\le M$ uniformly for some $\\kappa $ -independent constant $M$ .","For $k\\in A_\\frac{1}{2}$ we showed in (REF ) that we can bound uniformly with $3\\left(1+C\\right)$ for sufficiently large small $\\beta $ .","For $k \\in A_{\\frac{1}{2}}^c$ , we have $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{G}_{\\mu _\\lambda }(k)} \\cdot \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) *{ 1- \\frac{\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}} \\\\ \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) \\left(1+\\frac{c_{\\kappa +1}\\beta }{\\frac{1}{2}- c_{\\kappa +1}\\beta } \\right) \\le 1+ \\left(20\\kappa + 27\\right)c_{\\kappa +1}\\beta .","$ Note that for the first bound in (REF ), we used the estimate established in (REF ), which is stronger than a bound on $f_5$ .","We therefore have the uniform bound $f_6(\\lambda ) \\le 3 + 3C$ for $\\beta $ sufficiently small.","Let us now improve the bound on $f_3$ .","It will be notationally convenient to denote $A\\left(k\\right) := \\left[1- \\lambda \\widehat{a}\\left(k\\right)\\right]^{-1}$ .","Since $\\widehat{a}\\left(k\\right)$ is self-adjoint, it can be “diagonalized\" in the sense of Theorem REF .","Therefore $A\\left(k\\right)$ can be “diagonalized\" over the same space and with the same unitary operator - it will have the diagonal function $\\widetilde{A}(k)(e) = 1/\\left(1-\\widetilde{a}\\left(k\\right)(e)\\right)$ .","It is then clear that $A\\left(k\\right)$ and $\\widehat{a}\\left(k\\right)$ commute (for each $k\\in \\mathbb {R}^d$ ).","Using Observation REF , we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} &= \\left\\Vert \\widehat{a}\\left(l\\right)A\\left(l\\right) - \\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op}\\nonumber \\\\&= \\left\\Vert A\\left(l\\right)\\widehat{a}\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l+k\\right)\\right]A\\left(l+k\\right) - A\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l\\right)\\right]\\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op} \\nonumber \\\\&= \\left\\Vert A(l)\\left(\\widehat{a}(l)-\\widehat{a}(l+k)\\right)A(l+k)\\right\\Vert _{\\rm op} \\nonumber \\\\&\\le \\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}\\left\\Vert A(l)\\right\\Vert _{\\rm op}\\left\\Vert A(l+k)\\right\\Vert _{\\rm op}.","$ For the $\\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}$ factor, we use the triangle inequality to separate this into a term of $\\widehat{}$ and $\\widehat{}_{\\lambda }$ .","Lemma REF immediately gives $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ for the $C>0$ arising from REF .","For the $\\widehat{}_{\\lambda }$ term, a similar argument to (REF ) gives $\\left\\Vert \\widehat{}_{\\lambda }\\left(l\\right) - \\widehat{}_{\\lambda }\\left(1+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(0\\right) - \\widehat{}_{\\lambda }\\left( k\\right)\\right\\Vert _{1,\\infty } \\le c_{\\kappa +1}\\beta \\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where the last inequality follows from Observation REF .","We then deal with the remaining factors of (REF ) by noting that for all $k\\in \\mathbb {R}^d$ , $\\left\\Vert A(k)\\right\\Vert _{\\rm op} = \\frac{1}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\widehat{G}_{\\mu _\\lambda }(k) \\le 3\\left(1+C\\right) \\widehat{G}_{\\mu _\\lambda }(k), $ employing our improved bound on $f_6$ .","The result of this is that we have the bound $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C + c_{\\kappa +1}\\beta \\right)\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right).$ Then since $\\widehat{G}_{\\mu _\\lambda }(k)\\ge 0$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C+c_{\\kappa +1}\\beta \\right)\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right).$ [Proof of Proposition REF -REF ] The continuity of $f_1$ is obvious.","For the other two functions we prove equicontinuity of a family of functions and use [41] to show the continuity of the desired functions.","The procedure is outlined here.","Suppose we wish to show that $H\\left(\\lambda \\right) := \\sup _{\\alpha \\in B}\\left\\Vert h_\\alpha \\left(\\lambda \\right)\\right\\Vert _{\\rm op}$ is continuous on $\\left[0,\\lambda _O\\right)$ .","For our purposes the parameter $\\alpha $ will be either $k$ or $\\left(k,l\\right)$ and thus $B=\\mathbb {R}^d$ or $B=\\mathbb {R}^{2d}$ .","The continuity on the half-open interval is implied by having continuity on the closed interval $\\left[0,\\lambda _O-\\rho \\right]$ for any $\\rho >0$ .","For closed intervals, [41] gives continuity if the family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous and $H\\left(\\lambda \\right)<+\\infty $ for all $\\lambda $ in the closed interval.","The family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous if for all $\\varepsilon >0$ there exists $\\delta >0$ such that $|s-t|<\\delta $ implies $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\varepsilon $ uniformly in $\\alpha \\in B$ .","In fact, the reverse triangle inequality implies that $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\left\\Vert h_\\alpha (s)-h_\\alpha (t)\\right\\Vert _{\\rm op},$ and therefore we only need to prove equicontinuity for the un-normed family $\\left\\lbrace h_\\alpha \\right\\rbrace _{\\alpha \\in B}$ .","We prove equicontinuity by bounding the `near-derivative' $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert h_\\alpha (\\lambda +\\varepsilon )-h_\\alpha (\\lambda )\\right\\Vert _{\\rm op}$ uniformly in $\\alpha \\in B$ for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ and arbitrary $\\rho >0$ .","For $f_2$ the operator-valued functions $h_\\alpha $ are $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)/\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)$ .","By using a variation on the chain rule, we get $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)$ is differentiable with the bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2$ .","We also have $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ from Lemma REF and $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ immediately from the definition.","It remains to deal with the $\\mu _\\lambda $ term.","Recall that $\\mu _\\lambda := 1 - \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^{-1}$ .","Therefore, using the reverse triangle inequality, $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda } = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)\\right)-\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} \\\\\\le \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)-\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2,$ where we have used $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ from Lemma REF in the last inequality.","Therefore $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + *{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2\\\\\\le \\left(1 + C\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + C\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^3.$ We therefore have a finite $k$ -independent bound.","This has proven the equicontinuity.","For the uniform boundedness, we note that for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}<\\infty $ .","In conjunction with $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ we have the required uniform boundedness and therefore the continuity of $f_2$ .","We repeat this approach for $f_3$ .","The corresponding step to (REF ) now reads $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda +\\varepsilon }}\\left(k,l\\right)} - \\frac{\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)$ is differentiable with the operator norm bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le 4\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}$ .","The operator itself has the similar bound $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}$ .","We would like to have the bound (REF ), but this was proven under the assumption $f\\le \\kappa +1$ , which we no longer assume.","Fortunately, we only require our bound for $\\lambda < \\lambda _O -\\rho $ .","Therefore we can use Lemma REF and Lemma REF to get $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\text{e}^{4\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\\\\\le C\\text{e}^{4\\left(\\lambda _O-\\rho \\right)\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}}\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ From this and $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge \\left(1+C\\right)^{-1}$ , there exists a constant $\\widetilde{M}>0$ such that both $\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M} \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}, \\qquad \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M},$ for sufficiently large $d$ .","Now recall $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ and $\\left(1+C\\right)^{-1}\\le \\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\le \\left(1-\\mu _\\lambda \\right)^{-1} = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}$ .","Therefore an application of the chain rule gives the bound $*{\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }} \\le 6\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^4\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op},$ uniformly in $k$ and $l$ .","These properties, along with the bound on $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }$ from above, are sufficient to prove equicontinuity and the uniform boundedness for the required $\\lambda $ .","Therefore we have proved that $f_3\\left(\\lambda \\right)$ is continuous.","It just remains to prove that we can indeed bound the difference of $\\delta _\\lambda (k)$ with the difference of $\\widehat{}_{\\lambda }(k)$ as we claimed.","[Proof of Lemma REF ] First note that there is nothing to prove if $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}=\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}=0$ , because the triangle inequality then forces $\\delta _\\lambda (k_1) = \\delta _\\lambda (k_2)=0$ .","Without loss of generality, assume $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ with $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}>0$ .","Then $\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2) = \\frac{\\delta _\\lambda (k_1)}{\\left\\Vert \\widehat{}_{\\lambda }(k_1)\\right\\Vert _{\\rm op}}\\left(\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}\\right) + \\left[\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)\\right]$ The second factor is the more troublesome: $*{\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)} = \\frac{1}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\delta _\\lambda (k_1)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op} - \\delta _\\lambda (k_2)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} \\\\\\le \\frac{*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}.$ Since $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ , we have $*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}$ .","Therefore $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}} \\le 2\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)-\\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op},$ where the last inequality holds because of the reverse triangle inequality."],["Proof of Main Theorems","Proposition 7.1 Let $d>6$ be sufficiently large.","Then for $\\lambda \\le \\lambda _O$ there exists $c>0$ such that $\\triangle _\\lambda < c\\beta ^2$ .","Proposition REF gives us that there exists $c_f$ (increasing in $f$ ) such that $\\triangle _\\lambda \\le c_f\\beta ^2$ for all $\\lambda <\\lambda _O$ .","Proposition REF then implies that there exists $c$ such that $c_f\\le c$ uniformly for all $\\lambda < \\lambda _O$ .","It remains to prove the assertion for $\\lambda =\\lambda _O$ .","Recall from Section REF the definition of $\\tau _\\lambda ^n(x,y)$ and $\\Lambda _n(x)$ .","Furthermore, recall that if $x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}$ then such a connection is achieved in finitely many steps and the bound in $\\ref {Assump:2ndMoment}$ then implies that $\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right\\rbrace = \\bigcup ^\\infty _{n=1}\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}_{\\Lambda _n(y)}\\right\\rbrace .$ Therefore by monotone convergence we have the pointwise limit $\\tau _\\lambda ^n(x,y) \\rightarrow \\tau _\\lambda (x,y)$ as $n\\rightarrow \\infty $ for all $\\lambda >0$ and $x,y\\in \\mathbb {X}$ .","Since $\\tau _\\lambda ^n(x,y)$ only depends upon the finite region $\\Lambda _n(y)$ , the functions $\\lambda \\mapsto \\tau _\\lambda ^n(x,y)$ are continuous for all $x,y\\in \\mathbb {X}$ .","This, with the monotonicity $\\tau _\\lambda ^n(x,y) \\le \\tau _\\lambda ^{n+1}(x,y)$ , implies that the function $\\lambda \\mapsto \\tau _\\lambda (x,y)$ is lower-semicontinuous.","Since $\\tau _\\lambda (x,y)$ is non-decreasing in $\\lambda $ , this lower semi-continuity implies that the pointwise limit $\\tau _\\lambda (x,y)\\uparrow \\tau _{\\lambda _O}(x,y)$ as $\\lambda \\uparrow \\lambda _O$ holds, and the limit holds monotonically.","This monotonic pointwise convergence implies that the integral $\\int \\tau _\\lambda (x,u)\\tau _\\lambda (u,v)\\tau _\\lambda (v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) \\rightarrow \\int \\tau _{\\lambda _O}(x,u)\\tau _{\\lambda _O}(u,v)\\tau _{\\lambda _O}(v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right)$ for all $x,y\\in \\mathbb {X}$ as $\\lambda \\uparrow \\lambda _O$ , and that this convergence is monotone increasing.","If we take the supremum over $x,y\\in \\mathbb {X}$ , we get that $\\lim _{\\lambda \\uparrow \\lambda _O}\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } = \\left\\Vert \\mathcal {T}_{\\lambda _O}^3\\right\\Vert _{\\infty ,\\infty }$ .","The uniform bound for $\\lambda <\\lambda _O$ then implies that this bound also holds at $\\lambda =\\lambda _O$ .","The following proposition uses the bound $\\triangle _{\\lambda _O} < c\\beta ^2$ to show that percolation does not occur at criticality.","Another element of the proof is that there is almost surely at most one infinite cluster.","There are general considerations that show that this is the case.","Indeed [14] establishes that on the discrete space $\\mathbb {Z}^d$ there is at most one infinite cluster if the edge occupation measure is stationary and obeys the ‘finite energy property’, and an analogous result for Poisson processes in the continuum applies in our case (see [4], [36]).","Proposition 7.2 The critical percolation function $\\theta _{\\lambda _O}(a)=0$ for $\\mathcal {P}$ -almost every $a\\in \\mathcal {E}$ .","Assume that $(\\overline{0},a),(\\overline{x},b)\\in \\mathbb {X}$ are in the (a.s. unique) infinite cluster, and are therefore connected.","Then the FKG inequality implies that $\\theta _{\\lambda _O}(a)\\theta _{\\lambda _O}(b)\\le \\mathbb {P}_{\\lambda _O}\\big (|\\overline{0},a)|=\\infty ,|\\overline{x},b)|=\\infty \\big )\\le \\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x},b)).$ Since $\\triangle _{\\lambda _O} \\le c\\beta ^2$ , for $\\mathcal {P}$ -a.e.","$a,b\\in \\mathcal {E}$ there exists a sequence $\\overline{x}^{(a,b)}_n$ such that $\\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x}^{(a,b)}_n,b))\\rightarrow 0$ .","Therefore for $\\mathcal {P}$ -a.e.","$a\\in \\mathcal {E}$ we have $\\theta _{\\lambda _O}(a)=0$ .","We can define the linear operator $\\mathcal {T}_{\\lambda _O}$ by its action on $f\\in L^2(\\mathbb {X})$ : $\\mathcal {T}_{\\lambda _O}f(x) = \\int \\tau _{\\lambda _O}(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right).$ Contrary to $\\mathcal {T}_\\lambda $ for $\\lambda <\\lambda _O$ , the operator $\\mathcal {T}_{\\lambda _O}$ may be an unbounded linear operator.","Proposition 7.3 (The operator OZE at the critical point) Let $d>6$ be sufficiently large.","The Ornstein-Zernike equation then extends to $\\lambda _O$ in the sense that the sum of (unbounded) linear operators vanishes: $\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) = 0.$ Furthermore, for $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ the linear operator $\\widehat{\\mathcal {T}}_{\\lambda _O}(k)$ is a bounded linear operator and the following equality holds: $\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right) - \\widehat{}\\left(k\\right) - \\widehat{}_{\\lambda _O}\\left(k\\right) - \\lambda _O\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right)\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda _O}\\left(k\\right)\\right) = 0.$ Finally, there exists $c>0$ such that for all $k\\in \\mathbb {R}^d$ $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le c\\beta , \\qquad \\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}\\le c\\beta .$ Let $f\\in L^2(\\mathbb {X})$ .","Then by the triangle inequality, for $\\lambda <\\lambda _O$ ${\\left(\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right)\\right)f}_2 \\le {\\left(\\mathcal {T}_\\lambda - - _\\lambda - \\lambda \\mathcal {T}_\\lambda \\left(+ _\\lambda \\right)\\right)f}_2 \\\\+ {\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda - _{\\lambda _O}+ _\\lambda - \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) + \\lambda \\mathcal {T}_\\lambda \\left(+_\\lambda \\right)\\right)f}_2.$ The first norm on the right hand side vanishes, since the OZE holds for $\\lambda <\\lambda _O$ .","To prove that the left hand side vanishes, we aim to show that the second norm on the right hand side vanishes as $\\lambda \\uparrow \\lambda _O$ .","To do this, the triangle inequality implies that we only need to show that the following limits hold as $\\lambda \\uparrow \\lambda _O$ : ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2 &\\rightarrow 0, \\\\{\\left(_{\\lambda _O}- _\\lambda \\right)f}_2 &\\rightarrow 0,\\\\{\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2 &\\rightarrow 0.$ We begin with (REF ).","Recall from the proof of Proposition REF that $\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y) := \\tau _{\\lambda _O}(x,y) - \\tau _\\lambda (x,y)$ converges monotonically and pointwise to zero.","From the definition of the ${\\cdot }_2$ norm, ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\\\&\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes (3)}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right).$ The monotone convergence of $\\left(\\tau _{\\lambda _O}-\\tau _\\lambda \\right)(x,y)\\downarrow 0$ then implies that the limit (REF ) holds.","For (), we use the functions $h^{(n)}_\\lambda \\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ defined by $h^{(n)}_\\lambda (x,y) := {\\left\\lbrace \\begin{array}{ll}\\lambda ^{-1}\\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes (2n)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right) &:n\\ge 1\\\\\\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2 &: n=0.\\end{array}\\right.","}$ From Proposition REF and the proof of Proposition REF , we know that $0\\le \\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_\\lambda (x,y)$ for all $n\\ge 0$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in \\left(0,\\lambda _O\\right)$ - the non-negativity follows from the definition of $\\pi ^{(n)}_\\lambda $ .","The functions $ h^{(n)}_\\lambda $ can also be defined for $\\lambda =\\lambda _O$ , and since they are monotone increasing in $\\lambda $ , we have the $\\lambda $ -independent bound $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda <\\lambda _O$ and $x,y\\in \\mathbb {X}$ .","For the single-mark version of the model it is proven in [21] that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.","This same argument works for our multi-mark version, and therefore $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda \\le \\lambda _O$ .","The triangle inequality then implies that $*{\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)} = *{\\sum ^\\infty _{n=0}\\left(-1\\right)^n\\left( \\pi ^{(n)}_{\\lambda _O}(x,y) - \\pi ^{(n)}_{\\lambda }(x,y)\\right)} \\le 2\\sum ^\\infty _{n=0}h^{(n)}_{\\lambda _O}(x,y).$ It will be convenient to define $h_{\\lambda }\\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ as $h_\\lambda (x,y) := \\sum ^\\infty _{n=0}h^{(n)}_{\\lambda }(x,y)$ .","Then ${\\left(_{\\lambda _O}- _\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,z)f(y)f(z)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right) \\\\& \\le 4\\int h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right)\\nonumber \\\\& = 4\\int \\left(\\int h_{\\lambda _O}(x,y)*{f(y)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right).","$ By Schur's test, this integral is finite for all $f\\in L^2(\\mathbb {X})$ if both the values $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right) \\qquad \\text{ and } \\qquad \\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ are finite.","To prove this for the former, we repeat the arguments of Proposition REF and Section REF to bound it in terms of $\\lambda _O$ , the triangle diagrams, and other $\\lambda $ -independent terms.","The argument of Proposition REF with the uniform bounds of Observation REF then proves that $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ is indeed finite.","A similar argument also holds for $\\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ .","We now have an integrable function $h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}$ that dominates the integrand of (REF ).","Recall that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.","Thus the dominated convergence theorem implies that the limit () holds.","For (), we can use $h_{\\lambda _O}$ to get the bound ${\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2^2 \\\\\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,u)h_{\\lambda _O}(u,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,v)h_{\\lambda _O}(v,z)*{f(y)}*{f(z)} \\nu ^{\\otimes 5}\\left(\\mathrm {d}u,\\mathrm {d}v, \\mathrm {d}x, \\mathrm {d}y, \\mathrm {d}z\\right).$ The integrand converges monotonically to zero, and therefore the integral vanishes in the limit.","We now consider the Fourier version of the OZE.","We use the notation that, given a bounded linear operator $A\\colon L^2(\\mathbb {X}) \\rightarrow L^2(\\mathbb {X})$ , the Fourier transform is given by $\\mathcal {F}\\left( A \\right)(k)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ .","By hypothesis, $$ is a bounded operator, and the finiteness of $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ implies that $_{\\lambda _O}$ is also bounded.","Then (REF ) implies that $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) = \\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)$ for all $k\\in \\mathbb {R}^d$ .","We now make the claim that $\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)$ has a bounded linear inverse for all $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ .","This is proven in Lemma REF below.","This claim then implies that for $k\\ne 0$ we have $&\\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) \\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)^{-1} \\right)(k) = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k).$ That is, $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k)$ is a bounded linear operator for $k\\ne 0$ .","The limit () also allows us to extend our sub-critical bound on $_\\lambda $ to apply at criticality.","The limit () may be phrased as saying that the function $\\lambda \\mapsto *{_\\lambda f}_2$ is continuous at $\\lambda =\\lambda _O$ (from the left) for all $f\\in L^2\\left(\\mathbb {X}\\right)$ .","Since $\\lambda \\mapsto \\left\\Vert _\\lambda \\right\\Vert _{\\rm op}$ can be written as a supremum of such functions, it is a lower semi-continuous function.","Our bound $\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta $ for $\\lambda < \\lambda _O$ (from Proposition REF ) then implies $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le \\liminf _{\\lambda \\uparrow \\lambda _O}\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta .$ The limit $*{\\left(\\widehat{}_{\\lambda _O}^{(n)}(0) - \\widehat{}_{\\lambda }^{(n)}(0)\\right)f}_2 \\rightarrow 0$ for each $f\\in L^2\\left(\\mathcal {E}\\right)$ can be derived in essentially the same way as () above.","The above lower semi-continuity argument can then be used to show that the bound from Proposition REF extends to also hold at $\\lambda =\\lambda _O$ .","From Lemma REF and the positivity of $\\pi ^{(n)}_{\\lambda _O}(x,y)$ this bound also holds for all $k\\in \\mathbb {R}^d$ .","By summing over $n$ , we then get the desired bound on $\\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}$ .","Lemma 7.4 For $\\lambda = \\lambda _O$ and $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ , the bounded linear operator $\\left(1- \\lambda _O\\left(\\widehat{}(k) + \\widehat{}_{\\lambda _O}(k)\\right)\\right)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ has a bounded linear inverse.","We first introduce some notation.","Let $A\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ be a self-adjoint linear operator.","Then we define the minimum modulus to be $\\left\\Vert A\\right\\Vert _{\\rm min} := \\inf _{0\\ne f\\in L^2(\\mathcal {E})}\\frac{*{Af}_2}{{f}_2}.$ Note that it follows from the Spectral Theorem (Theorem REF ) that $A$ has a bounded linear inverse if and only if $\\left\\Vert A\\right\\Vert _{\\rm min}>0$ .","Now suppose that $A=A_\\lambda $ , and that for all $f\\in L^2(\\mathcal {E})$ the map $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .","Then the map $\\lambda \\mapsto \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}$ is upper semicontinuous on $\\left[0,\\lambda _O\\right]$ .","That is, for any sequence $\\lambda _n$ in $\\left[0,\\lambda _O\\right]$ such that $\\lambda _n\\rightarrow \\lambda $ , $\\limsup _{n\\rightarrow \\infty }\\left\\Vert A_{\\lambda _n}\\right\\Vert _{\\rm min} \\le \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}.$ We now consider $A_\\lambda = 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .","A similar argument to the way we proved () shows that $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .","Our result is then proven if we can show that for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ we have an $\\varepsilon = \\varepsilon (k)>0$ such that $\\left\\Vert 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right\\Vert _{\\rm min} \\ge \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","We will in fact show the stronger result that $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","Recalling the definition of $\\delta _\\lambda (k)$ from (REF ), we have $1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &= 1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\lambda \\delta _{\\lambda }(k) \\nonumber \\\\&= \\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{>0 \\text{ from Lemma~\\ref {lem:Optlam-a}}} + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) + \\lambda \\left(\\delta _{\\lambda }(0) - \\delta _{\\lambda }(k)\\right) \\nonumber \\\\& > \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1+\\mathcal {O}\\left(\\beta \\right)\\right).$ Here we have used Lemma REF to bound the difference of $\\delta $ with the difference of $\\widehat{}_{\\lambda _O}$ .","For sufficiently large $d$ , REF then proves $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .","Note that in conjunction, Proposition REF , Proposition REF and Proposition REF prove the OZE equations of Theorem REF .","In particular, the former two prove that the OZE holds in the subcritical regime, and the latter extends this to criticality where possible.","Furthermore, Proposition REF gives the bounds on $_\\lambda $ and $\\widehat{}_{\\lambda }(k)$ subcritically whilst Proposition REF extends this bound to criticality.","Regarding Theorem REF , the bound on $\\triangle _\\lambda $ and the lack of percolation at criticality were proven at the beginning of this section.","The infrared bound component of the theorem is proven below.","[Proof of Theorem REF ] First note that there is nothing to prove for $k=0$ .","From Proposition REF and Lemma REF , we can write $\\widehat{\\mathcal {T}}_\\lambda (k) = \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1},$ for all $k\\in \\mathbb {R}^d$ if $\\lambda <\\lambda _O$ , and for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ if $\\lambda = \\lambda _O$ .","If we use the Spectral Theorem (Theorem REF ) to `diagonalize' the operator $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , we get a multiplication operator on some Hilbert space $L^2(\\mathfrak {E}_k,\\mu _k)$ taking values $\\widetilde{a}(k)(e)$ for $e\\in \\mathfrak {E}_k$ .","Since, $\\widehat{\\mathcal {T}}_\\lambda (k)$ is formed from $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , it can also be diagonalized over $L^2(\\mathfrak {E}_k,\\mu _k)$ , and takes values $\\widetilde{\\tau }_\\lambda (k)(e) = \\frac{\\widetilde{a}(k)(e)}{1 - \\lambda \\widetilde{a}(k)(e)}.$ Since $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) < 1$ (from the proof of Lemma REF ), we know $\\lambda \\widetilde{a}(k)(e) >1$ and $\\lambda \\widetilde{\\tau }_\\lambda (k)(e) \\ge -1$ for $\\mu _k$ -almost every $e\\in \\mathfrak {E}_k$ .","It therefore only remains to bound $\\widetilde{\\tau }_\\lambda (k)(e)$ from above.","Equivalently, we only need to bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)$ from above.","We also note that the proof of () can be adapted to show that $\\lambda \\mapsto \\left$ is continuous on $\\left[0,\\lambda _O\\right]$ for all $f\\in L^2(\\mathcal {E})$ .","Then because $\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) = \\sup _{f}\\tfrac{\\left}{\\left}$ , this proves that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)$ is lower-semicontinuous on $\\left[0,\\lambda _O\\right]$ .","The bound $\\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) <1$ for $\\lambda < \\lambda _O$ (from Lemma REF ) then implies that $\\lambda _O\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda _O}(0)\\right)\\le 1$ .","From (REF ) the calculation proceeds: $\\lambda \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) &= \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& = \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{\\ge 0} + \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) + \\delta _{\\lambda }(0) - \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\delta _{\\lambda }(k)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1 + \\mathcal {O}\\left(\\beta \\right)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)},$ where we have bounded $\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)$ using Lemma REF and Observation REF ."],["Critical Intensities","[Proof of Proposition REF ] By the Mecke equation (REF ), we find $*{\\chi _\\lambda }_1 & = 1 + \\lambda \\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) = 1 + \\lambda \\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right),\\\\*{\\chi _\\lambda }_\\infty & = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\widehat{\\tau }_\\lambda (0;a,b) \\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }.$ From Lemma REF we have $\\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }$ .","Therefore $\\lambda ^{(1)}_T \\ge \\lambda _O \\ge \\lambda _T^{(\\infty )}$ .","Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(1)}_T$ .","Therefore we only now need to prove that $\\lambda ^{(1)}_T = \\lambda ^{(\\infty )}_T$ .","We do this by proving that if $*{\\chi _\\lambda }_1<\\infty $ then $*{\\chi _\\lambda }_\\infty <\\infty $ .","Fix $b\\in \\mathcal {E}$ .","Then by considering the vertices in the cluster $\\overline{0},b)$ adjacent to $\\left(\\overline{0},b\\right)$ , we find $\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\mathbb {E}_\\lambda \\left[\\sum _{\\left(\\overline{x},a\\right)\\in \\eta :\\left(\\overline{x},a\\right)\\sim \\left(\\overline{0},b\\right)}*{\\overline{x},a)}\\right] = 1 + \\lambda \\int \\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right]\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right).$ In this equality we have used Mecke's equation (REF ).","From the spatial translation invariance of the model, $\\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right] = \\mathbb {E}_\\lambda \\left[*{\\overline{0},a)}\\right]$ for all $\\overline{x}\\in \\mathbb {R}^d$ .","Therefore an application of a supremum bound to the $a$ -integral gives $*{\\chi _\\lambda }_\\infty = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)\\int \\mathbb {E}_\\lambda \\left[*{\\overline{0},c)}\\right]\\mathcal {P}\\left(\\mathrm {d}c\\right)\\\\= 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)*{\\chi _\\lambda }_1.$ From the finiteness of the parenthesised factor, $*{\\chi _\\lambda }_1<\\infty $ implies $*{\\chi _\\lambda }_\\infty <\\infty $ and the result is proven.","[Proof of Proposition REF ] The equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ holds from [35].","The equality $\\lambda _O = \\lambda _T^{(p)}$ follows from Proposition REF because the condition (REF ) clearly holds for the bounded radii model.","Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(p)}_T$ for all $p\\in \\left[1,\\infty \\right]$ .","Now note that for all $a,b\\in \\mathcal {E}$ we have that if $\\left(\\overline{0},\\dagger \\right)\\sim \\left(\\overline{x},b\\right)$ then $\\left(\\overline{0},a\\right)\\sim \\left(\\overline{x},b\\right)$ .","Therefore $\\lambda _c\\left(\\dagger \\right) \\ge \\lambda _c$ , and the equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ then means we only need to prove $\\lambda _T\\left(\\dagger \\right) \\le \\lambda _T^{(p)}$ for some $p\\in \\left[1,\\infty \\right]$ .","We will do so for $p=1$ .","Suppose that $\\lambda >\\lambda _T^{(1)}$ , so that $\\int \\mathbb {E}_\\lambda \\left[(\\overline{0};a\\right)\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\infty $ .","That is, the expected cluster size of a vertex with a random ($\\mathcal {P}$ -distributed) mark is infinite.","Note that we have an ordering of the marks in that if two radii $a,b$ satisfy $a\\ge b$ , then $\\varphi \\left(\\overline{x};a,c\\right) \\ge \\varphi \\left(\\overline{x};b,c\\right)$ for all $\\overline{x}\\in \\mathbb {R}^d$ and $c\\in \\mathcal {E}$ .","This also implies that $a\\mapsto \\mathbb {E}_\\lambda \\left[*{(\\overline{0},a}\\right]$ is a non-decreasing function.","This in turn implies that if $\\lambda >\\lambda _T^{(1)}$ and $a$ satisfies $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , then $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right) = \\infty $ .","Given some radius $a\\in \\mathcal {E}$ such that $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we define a new mark, $\\star _a$ , that forms connections according to $\\varphi \\left(\\overline{x};\\star _a,b\\right) = 1\\left\\lbrace *{\\overline{x}}\\le a\\right\\rbrace ,$ for all $b\\in \\mathcal {E}$ .","Note that if $b\\ge a$ then $\\varphi \\left(\\overline{x};\\star _a,c\\right) \\le \\varphi \\left(\\overline{x};b,c\\right)$ for all $c\\in \\mathcal {E}$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\le \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]$ .","We now consider the expected cluster size of vertices adjacent to $\\left(\\overline{0},\\star _a\\right)$ with radius $\\ge a$ .","We also restrict to cases for which there is an unique neighbour, and that neighbour has radius $\\ge a$ .","Since the probability of a connection is independent of the radius of the proposed neighbour, the distribution of the unique neighbour is equal to $\\mathcal {P}$ - conditioned upon having radius $\\ge a$ .","Therefore we have the lower bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\ge 1 + \\mathbb {P}_\\lambda \\left(\\exists !","\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right).$ As noted above, $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)=\\infty $ .","Furthermore, as $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , the random variable $\\#\\left\\lbrace \\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right\\rbrace $ obeys a Poisson distribution with strictly positive mean.","Therefore $\\mathbb {P}_\\lambda \\left(\\exists !","\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)>0$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right]=\\infty $ .","This in turn implies that $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right] =\\infty $ for all $a>0$ .","To relate this to $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]$ , we perform a similar lower bound.","Given a radius $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we restrict to cases where there is an unique neighbour of $\\left(\\overline{0},\\dagger \\right)$ with radius $\\ge a$ and bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right] \\ge 1+ \\mathbb {P}_\\lambda \\left(\\exists !","\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\dagger \\right),b\\ge a\\right) \\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right].$ As before, the probability factor is strictly positive and the expected cluster size term is infinite.","Hence $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]=\\infty $ and $\\lambda \\ge \\lambda \\left(\\dagger \\right)$ .","Therefore $\\lambda _T\\left(\\dagger \\right)\\le \\lambda ^{(1)}_T$ as required."],["Model Properties","Here we will prove that the models outlined in Section REF do indeed satisfy Assumption REF .","First we consider the single mark “finite variance models” considered by [21].","As proven in that reference, this includes the Poisson blob and Gaussian connection models.","Lemma B.1 Single mark finite variance models satisfy the conditions of Assumption REF .","First note that since $\\mathcal {E}$ is a singleton, the operators $\\widehat{}(k)$ are simply scalars.","In particular, this means that the conditions (REF ), and () of REF hold trivially.","The finiteness of $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is also equivalent to the finiteness of $\\int \\varphi (x)\\mathrm {d}x$ required by [21].","Since $\\widehat{}(k)$ is simply a scalar, assumption REF follows directly from the third finite variance condition of [21].","The condition REF is similarly a generalisation of the second finite variance condition, and therefore follows.","Specifically, the sets $B(x)$ are the $\\varepsilon $ -balls around $x\\in \\mathbb {R}^d$ where $0<\\varepsilon = p^2\\widehat{\\varphi }_{1,1}(k)f_1^2 + 2pq\\widehat{\\varphi }_{1,2}(k)f_1f_2 + q^2\\widehat{\\varphi }_{2,2}(k)f_2^2,$ and therefore the spectrum of $\\widehat{}(k)$ is equal to the set of eigenvalues of the matrix $\\begin{pmatrix}p^2\\widehat{\\varphi }_{1,1}(k) & pq\\widehat{\\varphi }_{1,2}(k)\\\\pq\\widehat{\\varphi }_{1,2}(k) & q^2\\widehat{\\varphi }_{2,2}(k)\\end{pmatrix}.$ This $2\\times 2$ matrix has characteristic polynomial given by $z \\mapsto z^2 - \\left(p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)\\right) z + p^2q^2\\left( \\widehat{\\varphi }_{1,1}(k)\\widehat{\\varphi }_{2,2}(k) - \\widehat{\\varphi }_{1,2}(k)^2\\right).$ Solving for roots of this quadratic then gives $\\mathbb {S}\\left(\\widehat{}(k)\\right) = \\frac{p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)}{2} + \\sqrt{\\left(\\frac{p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)}{2}\\right)^2 + p^2q^2\\widehat{\\varphi }_{1,2}(k)^2},$ where the square root takes the positive value.","Note that by the definition of the double mark finite variance model $p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)$ is the Fourier transform of a non-negative (or non-positive) function, and therefore $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ is a sum of terms maximised at $k=0$ .","Furthermore, each approaches that maximum quadratically and at least one of $\\widehat{\\varphi }_{1,1}(k)$ , $\\widehat{\\varphi }_{2,2}(k)$ , and $\\widehat{\\varphi }_{1,2}(k)$ uniquely achieves its maximum at $k=0$ (otherwise we have $\\varphi = 0$ almost everywhere).","Therefore REF is satisfied.","Since we have only finitely many marks, there are only finitely many sequences of marks of a given length.","Therefore since each of the $\\varphi _{i,j}$ individually satisfy the 1-mark analog of REF , the 2-mark version is satisfied by $\\varphi $ too.","It is not inconceivable that the above argument could be generalised to 3 or 4 mark systems.","The degree 3 and 4 characteristic polynomials arising from the matrices have roots that can be solved for in radicals.","It may be then that a particular clever rearrangement of the expression allows the required properties to be deduced like above.","Such an investigation is currently beyond the authors' patience.","Of course, the famous Abel-Ruffini Theorem states that no such solution in radicals exists for arbitrary polynomials of degree 5 and higher, and so this approach seems doomed for systems with such families of marks.","Lemma B.3 Space-mark factorisation models satisfy the conditions of Assumption REF .","Note that $\\widehat{}(k) = \\widehat{\\overline{\\varphi }}(k)\\mathcal {K}$ , where $\\mathcal {K}\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ is the self-adjoint linear integral operator with kernel function $K$ .","Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b}\\int K(a,b)\\mathcal {P}(\\mathrm {d}a)\\le 1$ , $\\mathcal {K}$ is a bounded operator and $\\mathbb {S}\\left(\\widehat{}(k)\\right) = {\\left\\lbrace \\begin{array}{ll}\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) \\ge 0\\\\-\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(-\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) < 0.\\end{array}\\right.","}$ Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}\\le 1$ , the finiteness of $\\widehat{\\overline{\\varphi }}(0)$ implies that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite, and the $d$ -independence of $\\mathcal {K}$ implies that REF holds.","Since $\\overline{\\varphi }(\\overline{x})\\ge 0$ , we have $*{\\widehat{\\overline{\\varphi }}(k)} < \\widehat{\\overline{\\varphi }}(0)$ for all $k\\ne 0$ .","Furthermore, $\\mathbb {S}\\left(\\mathcal {K}\\right) = \\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}$ implies that $\\mathbb {S}\\left(\\mathcal {K}\\right)\\ge \\mathbb {S}\\left(-\\mathcal {K}\\right)$ .","Then $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ inherits the properties required by REF from $\\widehat{\\overline{\\varphi }}(k)$ .","For REF , we note that the $d$ -independence of $\\mathcal {K}$ means that required decay properties are again inherited from $\\widehat{\\overline{\\varphi }}(k)$ .","Lemma B.4 The marked multivariate Gaussian model satisfies the conditions of Assumption REF .","Firstly note that for each $\\left(a,b\\right)\\in \\mathcal {E}^2$ the Gaussian structure of $\\varphi $ means that it factorises over the $d$ eigenvector directions of $\\Sigma (a,b)$ .","This factorisation ensures that that the $d$ -dimensional Fourier transform is the product of the 1-dimensional Fourier transforms.","We have $\\widehat{\\varphi }(k;a,b) = \\mathcal {A} \\exp \\left(-\\frac{\\pi ^2}{2}k^{\\intercal }\\Sigma (a,b)^{-1}k\\right).$ In particular, when $k=0$ the Fourier transform is $a,b$ -independent.","This means $\\widehat{}(0) = \\mathcal {A}\\mathbf {1}$ where $\\mathbf {1}$ is the integral operator with constant kernel function 1.","For $f\\in L^2(\\mathcal {E})$ we have ${f}{\\mathbf {1}f} &= \\int f(a) \\left(\\int f(b) \\mathcal {P}\\left(\\mathrm {d}b\\right)\\right) \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f\\right]^2,\\\\{f}{f} &= \\int f(a)^2 \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f^2\\right].$ By Jensen's inequality we therefore have $\\mathbb {S}\\left(\\mathbf {1}\\right) = 1$ (equality follows by considering the test function $f(a)\\equiv 1$ ).","It therefore follows that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = \\mathcal {A}<\\infty $ .","From (REF ) we have $\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } = \\mathcal {A}\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\exp \\left(-\\pi ^2k^{\\intercal }\\Sigma (a,b)^{-1}k\\right) \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This bound then implies the condition (REF ) of REF holds with $C=1$ .","Before we address () of REF , we turn to REF .","From our above bounds we already have $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) \\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This exponential term ensures that REF is satisfied.","We now return to () of REF .","For all $x\\in \\mathbb {R}$ we have $1-\\cos (x) \\le \\frac{1}{2}x^2$ , and therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\frac{1}{2}*{k}^2 \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x},$ for some unit vector $e\\in \\mathbb {R}^d$ .","Since the vector $e$ picks out the second moment of $\\varphi $ in only one direction, a standard calculation gives $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\Sigma _{\\max }^2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }.$ Therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty }$ is bounded by a $d$ -independent quadratic function.","In conjunction with the quadratic bound of REF , this then proves () in the $k\\rightarrow 0$ regime.","The remaining regime then holds because the triangle inequality implies $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le 2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } = 2\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","To address REF , recall that the spatial convolution of two multivariate Gaussian functions with means $\\overline{0}$ and covariance matrices $\\Sigma _1$ and $\\Sigma _2$ is a multivariate Gaussian function with mean $\\overline{0}$ and covariance matrix $\\Sigma _1+\\Sigma _2$ .","Therefore for all $\\overline{x}\\in \\mathbb {R}^d$ we have $&\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\int \\varphi (\\overline{x}-\\overline{y};a_1,a_2)\\varphi (\\overline{y};a_3,a_4)\\mathrm {d}\\overline{y}\\nonumber \\\\&\\hspace{14.22636pt}= \\mathcal {A}^2\\left(2\\pi \\right)^{-d/2}\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\left(\\det \\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)\\right)^{-1/2}\\exp \\left(-\\frac{1}{2}\\overline{x}^{\\intercal }\\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)^{-1}\\overline{x}\\right) \\nonumber \\\\& \\hspace{14.22636pt} \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2}\\exp \\left(-\\frac{*{\\overline{x}}^2}{4\\Sigma _{\\min }}\\right) \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2} \\rightarrow 0.$ In particular, this bounds the integral appearing in the definition of the sets $B(x)$ (see (REF )) and therefore proves they are empty.","For the bound of the convolution of three $\\varphi $ functions, we can get this immediately from the above calculation and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","Specifically REF holds with $g(d) = \\left(4\\pi \\Sigma _{\\min }(d)\\right)^{-d/2}$ and the sets $B(x) = \\emptyset $ .","Lemma B.5 The bounded-volume Boolean disc model satisfies the conditions of Assumption REF .","Crucial to the conditions of REF holding is the upper bound on $R_d(a)$ in (REF ).","We use the shorthand $R_{a,b} := R_d(a) + R_d(b),\\qquad V_{a,b} := \\frac{\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R^d_{a,b},$ so $V_{a,b}$ equals the Lebesgue volume of the $d$ -dimensional Euclidean ball with radius $R_{a,b}$ .","Note that the upper bound $R^{({\\rm max})}_d$ on $R_d(a)$ in (REF ) ensures $V_{a,b} \\le V^{({\\rm max})}$ uniformly in $d$ .","We first find $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int V_{a,b}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\le V^{({\\rm max})}.$ This proves the statement that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite in REF .","From the condition (REF ) we also get $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\ge \\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a, \\mathrm {d}b\\right) \\ge \\epsilon ^2\\frac{d\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)} \\int ^{2\\left(1-\\frac{c_2}{d}\\right)R^{({\\rm max})}_d}_0 r^{d-1}\\mathrm {d}r \\\\= \\epsilon ^2V^{({\\rm max})}\\left(1-\\frac{c_2}{d}\\right)^d \\rightarrow \\epsilon ^2\\text{e}^{-c_2}V^{({\\rm max})},$ where we have used the test function $f(a) \\equiv 1$ to lower bound $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .","We then prove (REF ) by bounding $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } =\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}V_{a,b} \\le V^{({\\rm max})}.$ We will return for () after addressing REF .","For REF , it will be useful to have an expression for $\\widehat{\\varphi }(k;a,b)$ for each $a,b\\in \\left[0,1\\right]$ .","Suppose we are able to find $C_1\\in \\left(0,1\\right)$ and $C_2 >0$ such that for all $a,b\\in \\left[0,1\\right]$ $*{\\widehat{\\varphi }(k;a,b)} \\le \\widehat{\\varphi }(0;a,b) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right).$ Then since the $k$ -dependent factor is $a,b$ -independent we can use Lemma REF to show that $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{}(0)\\right) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right),$ and therefore prove that REF is satisfied.","In finding an expression for $\\widehat{\\varphi }(k;a,b)$ we are assisted by the spherical symmetry of the connection function and follow [18].","It transpires that $\\widehat{\\varphi }(k;a,b) = \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}}\\left(R_{a,b}*{k}\\right),$ where $J_{\\frac{d}{2}}$ is the Bessel function of the first kind of order $\\frac{d}{2}$ .","This function has the expansion $J_{\\frac{d}{2}}\\left(r\\right) = \\sum ^\\infty _{m=0} \\frac{\\left(-1\\right)^{m}}{m!\\Gamma \\left(\\frac{d}{2}+m+1\\right)}\\left(\\frac{r}{2}\\right)^{\\frac{d}{2}+2m},$ converging for all $r\\ge 0$ .","We will consider three different regimes for $*{k}$ .","For $r \\ll \\sqrt{d}$ the expansion (REF ) is asymptotic, and since $R^{({\\rm max})}_d$ and $R^{({\\rm min})}_d$ are both asymptotically proportional to $\\sqrt{d}$ we have $\\widehat{\\varphi }(k;a,b) \\sim \\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R_{a,b}^d - \\frac{\\pi ^\\frac{d}{2}}{4\\Gamma \\left(\\frac{d}{2}+2\\right)}R_{a,b}^{d+2}*{k}^2 = V_{a,b}\\left(1 - \\frac{R^2_{a,b}}{2\\left(d+2\\right)}*{k}^2\\right)$ for $*{k}\\ll 1$ .","This proves the quadratic part of the desired bound.","The Bessel function $J_{\\frac{d}{2}}$ is bounded and achieves its global maximum (in absolute value) at its first non-zero stationary point, denoted $j^{\\prime }_{\\frac{d}{2},1}$ .","From [2], we have $j^{\\prime }_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _1 \\approx 0.81$ , and $*{J_{\\frac{d}{2}}\\left(j^{\\prime }_{\\frac{d}{2},1}\\right)} \\le M d^{-\\frac{1}{3}}$ for a $d$ -independent $M>0$ .","Since $R_{a,b} \\in \\left[2c_1,1/\\sqrt{2\\pi \\text{e}}+o(1)\\right]d^\\frac{1}{2}$ (for $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right]$ from (REF )) and $j^{\\prime }_{\\frac{d}{2},1} / R_{a,b} \\in \\left[\\sqrt{\\frac{\\pi \\text{e}}{2}}+o(1),\\frac{1}{4c_1}+o(1)\\right]d^{\\frac{1}{2}}$ we have $*{\\widehat{\\varphi }(k;a,b)} \\le \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\le \\left(2e^{-1} + o(1)\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\rightarrow 0$ for $*{k} \\ge j^{\\prime }_{\\frac{d}{2},1} / R_{a,b}$ .","We are now left with the intermediate range for $*{k}$ .","The first positive zero of $J_\\frac{d}{2}$ occurs at $j_{\\frac{d}{2},1}$ , where $j_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _2 \\approx 1.86$ (see [2]).","In particular, we will always have $j^{\\prime }_{\\frac{d}{2},1} < j_{\\frac{d}{2},1}$ .","From differential identities relating Bessel functions (see [18]), we have $\\frac{\\mathrm {d}}{\\mathrm {d}*{k}} \\widehat{\\varphi }(k;a,b) = -R_{a,b}\\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}+1}\\left(R_{a,b}*{k}\\right).$ The Bessel function $J_{\\frac{d}{2}+1}(r)$ is positive for $r>0$ until its first positive zero at $j_{\\frac{d}{2}+1,1}$ , and $j_{\\frac{d}{2}+1,1} > j_{\\frac{d}{2},1}$ .","Therefore $\\widehat{\\varphi }(k;a,b)$ is positive and decreasing on the whole region $*{k}\\in \\left(0,j_{\\frac{d}{2},1}/R_{a,b}\\right]$ .","Since this overlaps with the high $*{k}$ range, and we have the uniform quadratic behaviour near $k=0$ , the function $k\\mapsto *{\\widehat{\\varphi }(k;a,b)}$ can never increase and approach $\\widehat{\\varphi }(0;a,b)$ again after leaving a neighbourhood of $k=0$ .","We therefore have the bound (REF ) and thus have proven that REF is satisfied.","Figure: Sketch of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) against *k*{k}.","It approaches its maximum quadratically as *k→0*{k}\\rightarrow 0.","The first local maximum of J d 2 J_{\\frac{d}{2}} occurs at j d 2,1 ' ∼d 2+γ 1 d 2 1 3 j^{\\prime }_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3}.","The first zero of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) occurs at *kR a,b =j d 2,1 ∼d 2+γ 2 d 2 1 3 *{k}R_{a,b} = j_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} where γ 2 >γ 1 \\gamma _2>\\gamma _1.","Furthermore, ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) is strictly decreasing until *kR a,b =j d 2+1,1 ∼d 2+γ 2 d 2 1 3 +1 2*{k}R_{a,b} = j_{\\frac{d}{2}+1,1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\frac{1}{2}.We return to () in REF .","We first calculate the second moment: $\\int *{\\overline{x}}^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{d}{d+2}V_{a,b}R^2_{a,b}.$ From $*{\\overline{x}}^2 = \\overline{x}_1^2 + \\ldots + \\overline{x}^2_d$ (for an orthonormal basis) and the rotational symmetry, we have $\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{1}{d+2}V_{a,b}R^2_{a,b}$ for all unit vectors $e\\in \\mathbb {R}^d$ .","Therefore $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\frac{4}{d+2}V^{({\\rm max})}\\left(R^{({\\rm max})}_d\\right)^2.$ While $V^{({\\rm max})}$ is a constant, $R^{({\\rm max})}_d\\sim \\sqrt{d/8\\pi \\text{e}}$ and therefore we have a bound uniform in $d$ .","The argument now proceeds similarly to the corresponding part of the proof for the multivariate Gaussian model.","From (REF ) and REF we have both sides of () satisfying quadratic bounds around $k=0$ , and the triangle inequality with our other bounds prove that the bound also holds on the remainder.","For REF we first want to bound the convolution of three $\\varphi $ functions.","Given marks $a_1,\\ldots ,a_6\\in \\left[0,1\\right]$ and $\\overline{x}\\in \\mathbb {R}^d$ , let $\\mathbb {B}_{a,b}(\\overline{x})$ denote the ball of radius $R_{a,b}$ centred on $\\overline{x}$ , and let $1_{a,b}\\left(\\overline{x}\\right) = 1\\left\\lbrace *{\\overline{x}}\\le R_{a,b}\\right\\rbrace $ .","Then $\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6)\\right)\\left(\\overline{x}\\right)&= \\int 1_{a_1,a_2}(\\overline{y})1_{a_3,a_4}(\\overline{z}-\\overline{y})1_{a_5,a_6}(-\\overline{z})\\mathrm {d}\\overline{y}\\mathrm {d}\\overline{z}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)}\\left(\\int _{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)} 1_{a_3,a_4}(\\overline{z}-\\overline{y})\\mathrm {d}\\overline{z}\\right) \\mathrm {d}\\overline{y}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}.$ Then for $\\delta \\in \\left(0,1\\right)$ we have $&\\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}\\nonumber \\\\&\\qquad \\le \\int _{\\delta \\mathbb {B}_{a,b}\\left(\\overline{0}\\right)}*{\\mathbb {B}_{a_3,a_4}\\left(\\overline{0}\\right)}\\mathrm {d}\\overline{y}+ *{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} \\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} \\nonumber \\\\& \\qquad \\le \\left(V^{({\\rm max})}\\right)^2\\delta ^d + V^{({\\rm max})}\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}.$ Let $\\overline{e}_1\\in \\mathbb {R}^d$ be a unit vector.","Then by spherical symmetry we have $\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} = *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)}.$ We bound the volume of this intersection with the volume of the $d$ -ball with radius equal to the radius of the $d-1$ -sphere formed by the intersection of their boundaries.","This radius is equal to the length $h$ in Figure REF , which is maximised when $R_{a_5,a_6}$ and $R_{a_3,a_4}$ are maximised and $\\delta R_{a_1,a_2}$ is minimised.","Recall we defined the constant $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right)$ when we defined the model in Section REF .","Since we have $R_{a_5,a_6},R_{a_3,a_4} \\le 2R^{({\\rm max})}_d$ and $\\delta R_{a_1,a_2} \\ge \\delta c_1 d^\\frac{1}{2}$ , we have $h\\le \\sqrt{\\left(2 R^{({\\rm max})}_d\\right)^2 - \\delta ^2 c^2_1 d}$ - we have used the fact that the triangle is isosceles if both the $R_{a_5,a_6}$ and $R_{a_3,a_4}$ edges attain their maximum.","The ball with this radius then has the volume $\\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}h^d \\le V^{({\\rm max})}\\left(1 - \\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}}\\right)^\\frac{d}{2}.$ Since $\\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}} = 2\\pi \\text{e}\\delta ^2c_1^2\\left(1+o(1)\\right)$ and $c_1>0$ , this shows that $*{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)} \\le V^{({\\rm max})}\\left(1 - \\pi \\text{e}\\delta ^2c_1^2\\right)^\\frac{d}{2}$ for $d$ sufficiently large.","Sine we now have a mark independent bound with the required decay, we have proved the bound for the convolution of three $\\varphi $ functions.","Figure: For both the arguments, we bound the intersection of two dd-balls with a new dd-ball that has the radius hh in these diagrams.For finding the sets $B(x)$ in (REF ), we write $\\int \\varphi \\left(\\overline{x}- \\overline{y};a,c\\right)\\varphi \\left(\\overline{y};c,b\\right)\\mathrm {d}\\overline{y}\\mathcal {P}\\left(\\mathrm {d}c\\right) = \\int *{\\mathbb {B}_{a,c}\\left(\\overline{x}\\right)\\cap \\mathbb {B}_{c,b}\\left(\\overline{0}\\right)}\\mathcal {P}\\left(\\mathrm {d}c \\right).$ As before, we can bound the volume of this intersection independently of the mark $c$ .","We can bound it with the volume of the ball with radius $h$ where $h\\le \\sqrt{\\left(2R^{({\\rm max})}_d\\right)^2 - \\frac{1}{4}*{\\overline{x}}^2}$ (as can be seen in Figure REF ).","As before, if $*{\\overline{x}} \\ge \\kappa d^\\frac{1}{2}$ for some $\\kappa >0$ , then the volume of the ball with radius $h$ vanishes in the $d\\rightarrow \\infty $ limit.","Furthermore, if $\\kappa \\le \\frac{1}{\\sqrt{2\\pi \\text{e}}}$ then the volume of the ball with radius $\\kappa d^\\frac{1}{2}$ vanishes in the $d\\rightarrow \\infty $ limit.","Therefore assumption REF is satisfied with the sets $B(x)\\equiv B\\left(\\overline{x},a\\right)=\\left\\lbrace \\left(\\overline{y},b\\right)\\in \\mathbb {X}:*{\\overline{x}-\\overline{y}} \\le \\kappa d^\\frac{1}{2}\\right\\rbrace $ for any $\\kappa \\in \\left(0,\\frac{1}{\\sqrt{2\\pi \\text{e}}}\\right)$ ."],["Displacement Diagram Case Distinctions","This section provides the details omitted from the proof of Proposition REF .","Here we show which diagrams arise from displacements crossing a $\\Psi $ segment, and whether they can be dealt with by the first sub-case, $\\mathrm {\\left(I\\right)}$ , or by the second sub-case, $\\mathrm {\\left(II\\right)}$ .","Recall that $\\mathrm {\\left(I\\right)}$ are those that use a $\\Psi $ decomposition of the larger diagram only (and the displacement is not crossing a $\\Psi ^{(4)}$ segment), while $\\mathrm {\\left(II\\right)}$ are those that will require a combination of a $\\Psi $ decomposition and a $\\overline{\\Psi }$ decomposition.","We first expand out the displaced $\\Psi $ terms and label which are in sub-case $\\mathrm {\\left(I\\right)}$ and which are in sub-case $\\mathrm {\\left(II\\right)}$ .","Note that the displacement may be across the `top' or the `bottom' of the segments in the usual orientation.","If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(1)}$ term, we use cosine-splitting and sometimes expand $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (2,0.8);(1,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[below]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad \\right)\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}\\quad \\right).$ Using the same ideas, if we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(2)}$ term we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };(0.75,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\right.\\nonumber \\\\&\\hspace{28.45274pt}\\left.\\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\quad \\right).$ If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(3)}$ term, then by symmetry in both cases we get $\\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}.$ We first show the bounds for the single segments in sub-case $\\mathrm {\\left(I\\right)}$ .","Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2.$ We first consider the displaced $\\Psi ^{(j)}$ term by itself.","By using supremum bounds and spatial translations we arrive at $\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , \\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda .$ For most of these, it is sufficient to separate the pair using a supremum bound and use the individual bound above for the displaced $\\Psi ^{(j)}$ term and the bound on the $\\Psi $ or $\\Psi _n$ term from Lemma REF .","The only pairs that need a more careful approach are those that have the bound $W_k\\triangle ^\\circ _\\lambda $ above for the displaced $\\Psi ^{(j)}$ , followed by a $\\Psi ^{(4)}$ or $\\Psi ^{(2)}_n$ term.","In these cases we consider the pair together and get the diagram bounds $\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2,\\qquad \\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2.$ In both of these cases we bounded a $\\tau _\\lambda $ edge by 1 (essentially omitting the edge) before applying a supremum bound.","In the first we omit the diagonal edge and split the upper supremum vertex in two.","In the second we omit the left-most vertical edge and apply a supremum bound at the bottom right vertex.","We now turn our attention to sub-case $\\mathrm {\\left(II\\right)}$ .","Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda .$ Also recall that if the $\\overline{\\Psi }_0$ term is actually a $\\overline{\\Psi }^{(2)}_0$ term, then the $\\tau _\\lambda ^\\circ $ edge connecting them is actually just a $\\tau _\\lambda $ edge.","We first investigate bounds for the displaced $\\Psi ^{(j)}$ term with the connecting $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges.","By using supremum bounds, spatial translations, splitting supremum vertices in two, and sometimes expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction, we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(3,0.6) -- (2,0.6) -- (2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);\\node [mark size=2pt] at (0,-0.6) {{square*}};[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(2,-0.6) -- (3,-0.6);(1.5,0.6) -- (3,0.6);[dashed] (3,0.6) -- (3,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (3,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(1.5,0.6) -- (3,0.6);[dashed] (0,0.6) -- (0,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\le \\overline{H_k}+ W_k\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda .$ For most of these, it is clear from a crude supremum bound that we have the bound $\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda $ for the pair of these with the subsequent $\\overline{\\Psi }$ or $\\overline{\\Psi }_0$ term (like we did for sub-case $\\mathrm {\\left(I\\right)}$ ).","The only pairs that need a more careful approach are those that have the bound $\\overline{W_k}\\triangle ^\\circ _\\lambda $ or $\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda $ above, followed by a $\\overline{\\Psi }^{(4)}$ or $\\overline{\\Psi }^{(2)}_0$ term.","We first consider the cases where they are followed by a $\\overline{\\Psi }^{(4)}$ term.","Here we use supremum bounds, spatial translations, and expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2.$ We now consider the cases where they are followed by a $\\overline{\\Psi }^{(2)}_0$ term.","Recall that for this case, the connecting $\\tau _\\lambda ^\\circ $ edge is in fact only a $\\tau _\\lambda $ edge.","We then get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ The remaining sub-case $\\mathrm {\\left(III\\right)}$ was fully dealt with in the main text."],["Proof of Lemma ","Recall that we wish to prove the following result: * We begin by defining a sightly different object to $\\overline{H_k}$ .","Given $\\vec{a}=\\left(a_1,\\ldots ,a_{16}\\right)\\in \\mathcal {E}^{16}$ , define $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right) :=& \\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8) \\nonumber \\\\&\\hspace{28.45274pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _{\\lambda ,k}(\\overline{t}+ \\overline{x}_3-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u},$ so we have $\\overline{H_k}\\le f(\\lambda )^5\\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,\\vec{a}\\in \\mathcal {E}^{16}}H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{0};\\vec{a};k\\right).$ We then proceed to bound $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right)$ to get our result.","This form is preferred because it more easily indicates what the Fourier arguments should be when we take a Fourier transform.","A schematic Fourier diagram for this can be found in [21].","We then expand out the $\\tau _\\lambda $ and $\\tau _{\\lambda ,k}$ terms.","For each of the $\\tau _\\lambda $ terms we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of the cosine-splitting lemma, we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we only need to get appropriate bounds for the following objects.","For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^7\\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {H}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l_1)P^{j_2}_{3,4}(l_1)P^{j_3}_{5,6}(l_2)P^{j_4}_{7,8}(l_2)P^{j_5}_{9,10}(l_1-l_2)P^{j_6}_{11,12}(l_1-l_3)P^{j_7}_{13,14}(l_2-l_3) \\\\ \\times Q^{j_8}_{15,16}(l_3;k)\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}},$ where $P^{j}_{m,n}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_m}{\\widehat{}(l) g_n}} &:j=1\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &:j=2\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &: j=3,\\end{array}\\right.}","\\qquad \\text{for }m,n\\in \\left\\lbrace 1,2,\\ldots ,14\\right\\rbrace ,\\\\Q^{j}_{15,16}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &:j=1\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &:j=2\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{}_k(l) g_{16}}} &:j=3\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &:j=4\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &:j=5,\\end{array}\\right.","}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{16}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .","If we can bound these $\\mathcal {H}(k;\\vec{j})$ appropriately, then we can combine them to get the required bound.","We now recall pointwise bounds for the $P^{j}_{m,n}(l)$ and $Q^{j}_{15,16}(l)$ terms.","In the proof of Lemma REF we derived $*{*{g_m}{\\widehat{}(l)g_n}} &\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &\\le *{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}}\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}},$ and $*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{16}}{\\widehat{}(l)^2g_{16}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{15}}{\\widehat{}(l)^2g_{15}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(l)*{g_{15}}{\\widehat{}(l)^2g_{15}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k)*{g_{15}}{\\widehat{}(l)^2g_{15}}^\\frac{1}{2}*{g_{16}}{\\widehat{}(l)^2g_{16}}^\\frac{1}{2}$ where $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k)\\widehat{G}_{\\mu _\\lambda }(l+k).$ Also recall that we can bound $*{*{g_m}{\\widehat{}(l)g_n}}\\le C$ and $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ if required.","Note that all the bounds for the $Q^{j}_{15,16}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.","We therefore only need to show that the remaining integrals of $\\widehat{}$ and $\\widehat{G}_{\\mu _\\lambda }$ terms can all be bounded by some constant multiple of $\\beta ^2$ .","Also note that the bounds for $Q^{1}_{15,16}(l)$ , $Q^{2}_{15,16}(l)$ , and $Q^{3}_{15,16}(l)$ only differ by a uniform constant (after we bound $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ ), and so once we have dealt with one of these we have dealt with the other two.","Let us first consider $\\vec{j}= \\left(1,1,1,1,1,1,1,1\\right)$ .","After applying the uniform pointwise bound for $Q^{1}_{15,16}(l)$ , we use the Cauchy-Schwarz inequality to bound $&\\int *{*{g_1}{\\widehat{}(l_1)g_2}}*{*{g_3}{\\widehat{}(l_1)g_4}}*{*{g_5}{\\widehat{}(l_2)g_6}}*{*{g_7}{\\widehat{}(l_2)g_8}}*{*{g_9}{\\widehat{}(l_1-l_2)g_{10}}} \\nonumber \\\\&\\hspace{199.16928pt}\\times *{*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}}*{*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\&\\qquad \\le \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int *{g_7}{\\widehat{}(l_2)g_8}^2*{g_9}{\\widehat{}(l_1-l_2)g_{10}}^2*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.","$ To deal with these parentheses, we use a volume-preserving change of variables and factorise each into three integrals.","For example, $\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\\\= \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2\\frac{\\mathrm {d}l_1}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_5}{\\widehat{}(l_2)g_6}^2\\frac{\\mathrm {d}l_2}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_{13}}{\\widehat{}(l_3^{\\prime })g_{14}}^2\\frac{\\mathrm {d}l_3^{\\prime }}{\\left(2\\pi \\right)^{d}}\\right).$ There are now two types of integral we need to bound: $&\\int *{g_m}{\\widehat{}(l)g_n}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int \\left(\\int _{E_m\\times E_n \\times E_m \\times E_n} \\widehat{\\varphi }(l;a,b)\\widehat{\\varphi }(l;c,d) \\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\right) \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int _{E_m\\times E_n \\times E_m \\times E_n} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right)\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b)\\varphi (-\\overline{x};c,d)\\mathrm {d}\\overline{x}\\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b) \\mathrm {d}\\overline{x}\\le C, \\\\&\\int *{g_1}{\\widehat{}(l)g_2}^2 *{g_3}{\\widehat{}(l)g_4}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\le g(d).","$ In (REF ) we used $\\varphi \\left(-\\overline{x};c,d\\right)\\in \\left[0,1\\right]$ and REF .","In () we bounded the convolution of four adjacency functions using REF .","The result of this is that we can bound (REF ) by $C^{\\frac{5}{2}}g(d)^{\\frac{1}{2}}$ .","Since $g(d)\\le \\beta ^4$ , this satisfies the bound we require.","As noted above, this argument also proves the required bound for $\\vec{j}= \\left(1,1,1,1,1,1,1,2\\right)$ and $\\vec{j}= \\left(1,1,1,1,1,1,1,3\\right)$ .","Now suppose that we change some subset of $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace $ from having value 1 to having value 2.","The argument proceeds similarly, except that after we have factorised our integrals we now want to also bound integrals of the following types: $&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int \\left(\\int _{E_1\\times E_1\\times E_2\\times E_2}\\widehat{\\varphi }^{(2)}(l;a,b)\\widehat{\\varphi }^{(2)}(l;c,d)\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d)\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int _{E_1\\times E_1\\times E_2\\times E_2}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)(\\overline{0})\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d) \\nonumber \\\\& \\hspace{56.9055pt}\\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\le g(d), \\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)g_4}^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^2 g(d)\\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)^2g_3} *{g_4}{\\widehat{}(l)^2g_4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi ^{(2)}(\\cdot ;a_5,a_6) \\star \\varphi ^{(2)}(\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^4 g(d).$ Up to a constant, these are all less than or equal to the bounds we used for the $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace = \\left\\lbrace 1,\\ldots ,1\\right\\rbrace $ case, and therefore that case dominates.","These other cases would behave as constant multiple of one of $g(d)$ , $g(d)^{\\frac{3}{2}}$ , $g(d)^2$ , $g(d)^{\\frac{5}{2}}$ , and $g(d)^3$ .","We have now dealt with all the cases which have no factors of $\\widehat{G}_{\\mu _\\lambda }$ .","To demonstrate how we will deal with these, we first consider $\\vec{j}=\\left\\lbrace 3,3,3,3,3,3,3,5\\right\\rbrace $ .","For notational compactness, we define $\\left<\\widehat{}(l)^2\\right>_{I} := \\prod _{m\\in I}*{g_m}{\\widehat{}(l)^2g_m},$ where $I\\subset \\mathbb {N}$ and $l\\in \\mathbb {R}^d$ .","Since we have already extracted the factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ via the pointwise bound on $Q^{5}_{15,16}(l;k)$ , we are left trying to bound the following integral: $\\int \\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\\\ \\qquad \\times \\left<\\widehat{}(l_1)^2\\right>^{\\frac{1}{2}}_{1,2,3,4} \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_2)^2\\right>^{\\frac{1}{2}}_{9,10} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}.$ To bound this integral we need to partition our space.","Fix $\\varepsilon >0$ and define $B_1 &:= \\left\\lbrace *{l_1}< \\varepsilon \\right\\rbrace \\\\B_2 &:= \\left\\lbrace *{l_2}< \\varepsilon \\right\\rbrace \\\\B_3 &:= \\left\\lbrace *{l_1-l_3}< \\varepsilon \\right\\rbrace \\\\B_4 &:= \\left\\lbrace *{l_1-l_2}< \\varepsilon \\right\\rbrace \\\\B_5 &:= \\left\\lbrace *{l_2-l_3}< \\varepsilon \\right\\rbrace \\\\B_6 &:= \\left\\lbrace *{l_3}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3-k}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3+k}< \\varepsilon \\right\\rbrace ,$ and for $n=0,1,\\ldots ,6$ define $A_n := \\left\\lbrace (l_1,l_2,l_3)\\in \\left(\\mathbb {R}^{d}\\right)^3 \\colon \\#\\left\\lbrace m\\colon (l_1,l_2,l_3)\\in B_m \\right\\rbrace =n\\right\\rbrace .$ In words, $A_n$ is the set of points within $\\varepsilon $ (in Euclidean distance) of precisely $n$ of the hyperplanes on which the $\\widehat{G}_{\\mu _\\lambda }$ factors diverge.","If we are working with a factor of $\\widehat{G}_{\\mu _\\lambda }$ on the set $B_m$ associated with its relevant hyperplane, we use the bound $\\widehat{G}_{\\mu _\\lambda }(l)\\le 1/\\left(C_2*{l}^2\\right)$ , and if we are working off this set we use the bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ .","When we integrate over $A_0$ , we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ for all such factors and recycle the bound on the “only $\\widehat{}$ \" integrals from above.","The contribution from $A_0$ is therefore bounded by some constant multiple of $\\frac{1}{\\varepsilon ^{18}}g(d)^3.$ For $A_1$ , first suppose we are considering $A_1\\cap B_m$ for $m\\ne 6$ .","First we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\le 3/\\left(C_2\\varepsilon ^2\\right)$ for the “irrelevant\" directions, and use the pointwise bound for $\\left<\\widehat{}(l)^2\\right>_{m,n}$ in the “relevant\" direction.","Then we can use the Cauchy-Schwarz and factorise argument we used above and recycle the integral bounds for the $\\widehat{}$ factors as we did above.","However we are careful about the way we split apart terms in the Cauchy-Schwarz step.","We split the $\\widehat{G}_{\\mu _\\lambda }$ terms across both factors and arrange the remaining $\\widehat{}$ terms so that each factor has “directions\" spanning $\\left(\\mathbb {R}^d\\right)^3$ .","To demonstrate we consider $A_1\\cap B_1$ : $&\\int _{A_1\\cap B_1} \\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12} \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\& \\hspace{56.9055pt} \\le C^2\\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\left<\\widehat{}(l_2)^2\\right>_{5,6}\\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\& \\hspace{142.26378pt}\\times \\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>_{7,8} \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.$ After a change of variables, factorising and using our integral bounds of $\\widehat{}$ terms from above, we are only left needing to bound $\\int _{*{l}<\\varepsilon } \\widehat{G}_{\\mu _\\lambda }(l)^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\le \\frac{1}{C^2_2} \\int _{*{l}<\\varepsilon } \\frac{1}{*{l}^4} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-4}}{d-4}.$ Therefore the contribution from $A_1\\cap B_m$ for $m\\ne 6$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ .","For $A_1\\cap B_6$ we need to be slightly more careful because for $*{k}<2\\varepsilon $ the constituent parts of $B_6$ overlap - this is the same considerations we had to account for in the proof of Lemma REF .","We can overcome it in the same way by using Hölder's inequality (actually just Cauchy-Schwarz in this case) and then re-centring the integrals.","The net result is that the bound of a constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ also applies to $A_1\\cap B_6$ .","In summary, the contribution from $A_1$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2.$ The $A_2$ case proceeds similarly.","After partitioning the set according to which are the overlapping hyperplanes, we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions and one $\\widehat{}$ factor from a linearly independent direction, and pointwise bound all the other $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors.","The contribution from $A_2$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-4\\right)^2}g(d),$ arising from the $A_1\\cap B_1\\cap B_2$ , $A_1\\cap B_1\\cap B_6$ , and $A_1\\cap B_2\\cap B_6$ integrals.","The $A_3$ case is more complicated because there are two qualitatively different sub-cases to consider.","First suppose that the three normal vectors to the overlapping hyperplanes are linearly independent.","Then we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions, pointwise bound the other $\\widehat{G}_{\\mu _\\lambda }$ factors and all the $\\widehat{}$ factors.","The contribution from this sub-case of $A_3$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3}$ , arising from the $A_1\\cap B_1\\cap B_2\\cap B_6$ integral.","On the other hand, if the three normal vectors to the overlapping hyperplanes are linearly dependent, then we need to retain a $\\widehat{}$ from the remaining linearly independent direction.","To demonstrate, let us consider the $A_3\\cap B_1 \\cap B_2 \\cap B_4$ integral.","After removing unnecessary terms via uniform bounds, $&\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\& \\qquad \\le \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}.$ Applying a change of variables, factorising the integral, and applying Hölder's inequality allows us to bound these parentheses: $& \\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)\\left(\\int \\left<\\widehat{}(l)^2\\right>_{13,14}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^4}\\frac{1}{*{l_2}^4}\\frac{1}{*{l_1-l_2}^2} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right) g(d)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^6}\\frac{1}{*{l_2}^3} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{2}{3}}\\left(\\int _{\\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_2}^6}\\frac{1}{*{l_1-l_2}^6} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{1}{3}} g(d)\\nonumber \\\\& \\qquad = \\frac{1}{C^5_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{4}{3}}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-3}\\frac{\\varepsilon ^{d-3}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}g(d).$ For all of these linearly dependent cases, we can bound the integral by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d)$ .","In summary, the contribution from $A_3$ is bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3} + \\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d).$ For $A_4$ , $A_5$ , and $A_6$ we don't have the two linearly dependent/independent cases because any four of the directions spans $\\left(\\mathbb {R}^d\\right)^3$ .","We therefore don't need factors of $\\widehat{}$ to control the unbounded directions.","Instead, we need to be careful that when we split our integral at the Cauchy-Schwarz step the resulting parentheses have sufficiently few factors of $\\widehat{G}_{\\mu _\\lambda }$ that their bounds will be finite for $d>6$ .","The way we perform this split will be different for the different parts of $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k)$ .","Let us define $\\widehat{D}(l;k) := \\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k).$ We first consider $A_6$ .","By writing $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)$ , we have two integrals we want to bound.","First $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.","$ We then perform the $l_i$ integrals in specific orders.","For the first factor we integrate over $l_3$ (which factorises out already), then we integrate over $l_2$ for fixed $l_1$ , and finally integrate over $l_1$ .","For the second factor we integrate over $l_1$ for fixed $l_2$ and $l_3$ , then we integrate over $l_3$ for fixed $l_2$ , and finally we integrate over $l_2$ .","There are therefore three forms of integral we need to bound: $\\int _{*{l}<\\varepsilon }\\widehat{G}_{\\mu _\\lambda }(l)^3\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l}^2}\\frac{1}{*{l-l^{\\prime }}^4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\&\\le \\frac{1}{C^3_2}\\left(\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{1}{3}\\left(\\int _{*{l-l^{\\prime }}<\\varepsilon }\\frac{1}{*{l-l^{\\prime }}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{2}{3}\\nonumber \\\\& = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ In this last inequality, $C^{\\prime }>0$ is some uniform constant.","The last inequality is derived by applying Hölder's inequality in much the same way as for the second inequality, taking care to see when the three sets that are integrated over are overlapping.","The calculation uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .","The second integral we want to bound for $A_6$ is $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3-l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.","$ We then perform the $l_i$ integrals in specific orders.","For the first factor we integrate over $l_1$ (which factorises out already), then we integrate over $l_2$ for fixed $l_3$ , and finally integrate over $l_3$ .","For the second factor we integrate over $l_3$ for fixed $l_1$ and $l_2$ , then we integrate over $l_1$ for fixed $l_2$ , and finally we integrate over $l_2$ .","The new integrals we need to bound are: $\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}$ $\\int _{\\left(\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace \\right)\\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4}\\right)\\varepsilon ^{d-6}.$ The calculation of these two bounds uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .","The result of these bounds is that the integral in (REF ) is bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ For $A_4$ and $A_5$ we can repeat the argument for $A_6$ , but note that in some places the bound $1/\\left(C_2*{l}^2\\right)$ will be replaced by $1/\\left(C_2\\varepsilon ^2\\right)$ .","The net result of this is that the factor $\\left(d-6\\right)^3$ can be replaced by $\\left(d-6\\right)^{\\alpha _0}\\left(d-4\\right)^{\\alpha _1}\\left(d-2\\right)^{\\alpha _2} d^{\\alpha _3}$ for some $\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _3\\le 0$ such that $\\alpha _0+\\alpha _1+\\alpha _2+\\alpha _3 =3$ .","The net result is that the contribution from both $A_4$ and $A_5$ can also be bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ Above we have established bounds for the various parts of the integral when $\\vec{j}=\\left(3,3,3,3,3,3,3,5\\right)$ .","Having $j_8=4$ can also be dealt with by very similar arguments to that outlined above - the $l_3$ integrals are in fact simpler.","The above bounds still hold with the $\\varepsilon ^{-18}$ becoming a $\\varepsilon ^{-16}$ .","Since $\\varepsilon <1$ , this produces a larger bound.","Recall that up to a constant factor, having $j_m=1$ produces a larger bound than $j_m=2$ for $m=1,\\ldots ,7$ .","If we replace $j_m=3$ with $j_m=1$ , then after ignoring constant values we lose a factor of $\\varepsilon ^{-2}$ , but we may also lose a factor of $g(d)^\\frac{1}{2}$ unless that was for $m=1,2$ (corresponding to $l_1$ direction) or $m=3,4$ (corresponding to $l_2$ direction).","Note that doing for both directions will lose a $g(d)^\\frac{1}{2}$ factor.","If $\\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d= O\\left(g(d)^{\\frac{1}{2}}\\right)$ for all fixed $\\varepsilon >0$ , then it is clear that the dominating bound is $g(d)^\\frac{1}{2}$ .","This arises, for example from the bound for $\\vec{j}=\\left(1,1,1,1,1,1,1,1\\right)$ .","The situation is more complicated if $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , because we can choose to have $\\varepsilon =\\varepsilon (d)$ and take it to 0 as $d\\rightarrow \\infty $ .","It is easy to see that all the $A_1$ components (and $A_2,\\ldots ,A_6$ components) for each $\\vec{j}$ can be bounded by some constant multiple of $\\frac{1}{d}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\varepsilon ^d$ (provided $d>6$ ).","We therefore only need to compare this to the contributions from $A_0$ .","The bound we get from the above arguments for each $A_0$ contribution are of the form $\\varepsilon ^{-2m}g(d)^\\frac{n}{2}$ , where $m$ is the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ and $n$ is related to the multiplicity of $$ terms in each Fourier direction.","To find the terms with the largest bound, we want to have as many factors of $\\widehat{G}_{\\mu _\\lambda }$ as possible without producing extra factors of $g(d)^\\frac{1}{2}$ .","Since $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , increasing $n$ here instantly produces a smaller bound.","By changing $j_8$ from 1 to 4 or 5, or any other element from 1 to 3, we do gain a factor of $\\widehat{G}_{\\mu _\\lambda }$ but we also increase the number of $$ factors and therefore in most places we acquire extra factors of $g(d)^{\\frac{1}{2}}$ .","The exception to this is for $j_1$ and $j_2$ , which correspond to the $l_1$ directions, since they already produce a factor of $g(d)^{\\frac{1}{2}}$ .","The $A_0$ component of the $\\vec{j}=\\left(3,3,1,1,1,1,1,1\\right)$ term can be bounded by constant multiple of $\\varepsilon ^{-4}g(d)^{\\frac{1}{2}}$ .","We then optimise our choice of $\\varepsilon $ to have our two dominant bounds be of the same order.","This produces $\\varepsilon (d)^2 = g(d)^{\\frac{1}{d}}\\frac{2\\pi d}{\\text{e}}\\left(1+o(1)\\right)$ , and therefore our overall bound is given by $\\varepsilon ^{-4}g(d)^\\frac{1}{2} = g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(\\frac{\\text{e}}{2\\pi }\\right)^2\\frac{1}{d^2}\\left(1+o(1)\\right).$ Since $g(d)^{\\frac{1}{2}-\\frac{2}{d}} \\le g(d)^{\\frac{1}{2}-\\frac{3}{d}}$ for sufficiently large $d$ , this bound is of order $\\beta ^2$ and our bound is proven."],["Acknowledgements.","This work is supported by Deutsche Forschungsgemeinschaft (project number 443880457) through priority program “Random Geometric Systems” (SPP 2265).","The authors thank the Centre de recherches mathématiques Montreal for hospitality during a research visit in spring 2022 through the Simons-CRM scholar-in-residence program."]],"string":"[\n [\n \"The Triangle Condition for the Marked Random Connection Model\"\n ],\n [\n \"Abstract We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $\\\\mathbb{R}^d$.\",\n \"Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks.\",\n \"Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely.\",\n \"Our focus is on the transition between the low- and high-density phase, where the system is critical.\",\n \"We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid.\",\n \"This implies the triangle condition, thus indicating mean-field behaviour.\",\n \"We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates.\"\n ],\n [\n \"Phase transition in percolation\",\n \"For percolation, each edge of the hypercubic lattice is retained independently with fixed probability $p\\\\in [0,1]$ .\",\n \"It is well-known that the model undergoes a phase transition: in any dimension $d\\\\ge 2$ there is a $p_c\\\\in (0,1)$ such that for $pp_c$ there is a unique infinite cluster.\",\n \"Behaviour of the model at (and near) the critical threshold $p_c$ is a major theme in probability theory.\",\n \"Indeed, we have good control on the critical behaviour in two regimes: if $d=2$ , then planarity and the link with complex analysis allows a refined description of the critical behaviour (to some extent only on the triangular lattice), see [43], [44], [5] and references therein; if $d$ is large (at least $d>6$ ), then Hara and Slade [25] established the famous triangle condition, which implies that various critical exponents exist and take on their mean-field values [1], [3], [37], [30], [31], [20], [28], see also [23] for a corresponding result for site percolation.\",\n \"While it is generally believed that $d>6$ is sufficient, current proof techniques only allow the proof for $d\\\\ge 11$ [11].\",\n \"Already [25] show it for $d>6$ under an additional spread-out assumption.\",\n \"The focus of the present paper is on the second case, which we coin the “mean-field regime”.\",\n \"The triangle condition, and hence the mean-field value of various critical exponents, has also been established for long-range percolation in lower dimension provided that the edge occupation probability is decaying slowly enough [22], [6], [7].\",\n \"The triangle condition for the random connection model, a continuum percolation model where vertices are given as a Poisson point process, was proven recently [21].\",\n \"Similarly, it has been established for non-amenable graphs under varying assumptions [39], [40], [27].\",\n \"All of the above papers crucially rely on the independence of edge occupation.\",\n \"In the present work, we establish the triangle condition for dependent percolation models.\",\n \"Mean-field bounds on some critical exponents were proven recently in the context of Boolean percolation [9], but a proof of the triangle condition in such situations is new.\"\n ],\n [\n \"Complex networks\",\n \"Random geometric graphs are versatile stochastic models for real networks.\",\n \"As many real-world networks exhibit certain stylised facts such as scale-free degrees, short distances and geometric clustering, it is clear that mathematical models of complex networks must share such features.\",\n \"While numerous different mathematical models for complex networks exist, it has been pointed out by [13] that the weight-dependent random connection model provides a unified approach that generalises many models for (infinite) geometric random graphs.\",\n \"This is a continuum random graph model whose vertices are given as a Poisson point process on $\\\\mathbb {R}^d\\\\times (0,1)$ .\",\n \"If $x,y\\\\in \\\\mathbb {R}^d$ and $s,t\\\\in (0,1)$ , where edges $\\\\big \\\\lbrace (x,s),(y,t)\\\\rbrace $ are inserted with probability given by a connection function $\\\\varphi ((x,s),(y,t))$ .\",\n \"A standard assumption is that $\\\\varphi $ is decreasing in the spatial distance $|x-y|$ as well as in both marks $s$ and $t$ .\",\n \"Relevance of this model comes from the fact that different choices for the connection function $\\\\varphi $ generalise various well-known random graph models such as Boolean percolation, scale-free Gilbert graph, scale-free percolation, soft random geometric graphs, long-range percolation, and spatial preferential attachment graphs.\",\n \"Results proven in the generalised setup are thus valid in great generality.\",\n \"For some choices of $\\\\varphi $ , these models are known to be robust, i.e., infinite components exist for all edge densities; see [15], [16].\",\n \"Our interest is in regimes where these graphs are non-robust and thus a proper percolation phase transition exist: if $\\\\varphi $ is small (scaled by a certain parameter) then there is no infinite cluster, but if $\\\\varphi $ is large, then an infinite cluster does appear.\",\n \"Investigation of such transitions is classical in percolation theory, and of great importance in modern probability.\",\n \"The present paper is the first one to prove an infrared bound for an inhomogeneous spatial percolation model.\",\n \"While such results are well-known for the homogeneous case (i.e., without marks), there are hardly any results for the percolation phase transition for inhomogeneous models.\",\n \"A notable exception is Boolean percolation, where sharp phase transition is known [8] and the mean-field bounds on three critical exponents is identified [9].\",\n \"The key challenge is therefore, to overcome the dependencies imposed by the vertex weights.\",\n \"We prove the infrared bound under specific conditions, and verify these conditions for various concrete examples.\"\n ],\n [\n \"The Marked Random Connection Model\",\n \"In this paper we consider a generalisation of the weight-dependent random connection model, which we first describe informally and refer to Section REF for a formal construction.\",\n \"It is for our purposes more convenient to generalise the setup by considering a general measurable mark space $\\\\mathcal {E}$ and thus define the model on the space $\\\\mathbb {X} = \\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ .\",\n \"Let $\\\\nu = {\\\\rm Leb}\\\\times \\\\mathcal {P}$ be a $\\\\sigma $ -finite measure on $\\\\mathbb {X}$ where ${\\\\rm Leb}$ denotes the Lebesgue measure on $\\\\mathbb {R}^d$ and $\\\\mathcal {P}$ is a probability measure on $\\\\mathcal {E}$ .\",\n \"Let $\\\\lambda >0$ , and then $\\\\lambda \\\\nu $ is an intensity measure on $\\\\mathbb {X}$ .\",\n \"The connection function is given by the measurable and symmetric $\\\\varphi \\\\colon \\\\mathbb {X}^2\\\\rightarrow \\\\left[0,1\\\\right]$ .\",\n \"We furthermore require that $\\\\varphi $ is translation and reflection invariant in the position components.\",\n \"That is, if we denote $x=\\\\left(\\\\overline{x},a\\\\right)$ and $y=\\\\left(\\\\overline{y},b\\\\right)$ for $\\\\overline{x},\\\\overline{y}\\\\in \\\\mathbb {R}^d$ and $a,b\\\\in \\\\mathcal {E}$ , we require that $\\\\varphi \\\\left(x,y\\\\right) = \\\\varphi (\\\\overline{x}-\\\\overline{y};a,b) = \\\\varphi (-\\\\overline{x}+\\\\overline{y};a,b)$ .\",\n \"We denote by $\\\\eta $ the Poisson point process with intensity measure $\\\\lambda \\\\nu $ (which represents the vertex set of the random graph) and by $\\\\xi $ the full random graph including additional randomness for the formation of edges.\",\n \"Once we have the concept of two vertices being adjacent, we can consider the concept of two vertices being connected.\",\n \"We say $x,y\\\\in \\\\mathbb {X}$ are connected, and write $x\\\\longleftrightarrow y$ , if they are equal or there exists a path in $\\\\xi $ connecting the two vertices.\",\n \"By this we mean that there exists a sequence of distinct vertices $x = u_0, u_1, \\\\ldots , u_k, u_{k+1} = y \\\\in \\\\eta $ (with $k \\\\in \\\\mathbb {N}_0$ ) such that $u_i \\\\sim u_{i+1}$ for all $0 \\\\le i \\\\le k$ .\",\n \"This definition can also be extended to connectedness on the RCM augmented by the vertex $x\\\\in \\\\mathbb {X}$ (resp.\",\n \"$x,y\\\\in \\\\mathbb {X}$ ), where $\\\\left(\\\\eta ,\\\\xi \\\\right)$ is replaced by $\\\\left(\\\\eta ^x,\\\\xi ^x\\\\right)$ (resp.\",\n \"$\\\\left(\\\\eta ^{x,y},\\\\xi ^{x,y}\\\\right)$ ).\",\n \"Therefore given a pair of points $x,y\\\\in \\\\mathbb {X}$ we can consider the event $\\\\left\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}\\\\right\\\\rbrace $ , and in turn the probability of this event.\",\n \"We define the pair-connectedness (or two-point) function $\\\\tau _\\\\lambda \\\\colon \\\\mathbb {X}^2 \\\\rightarrow \\\\left[0,1\\\\right]$ to be $\\\\tau _\\\\lambda \\\\left(x,y\\\\right) := \\\\mathbb {P}_\\\\lambda \\\\left(x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}\\\\right).$ Using the connection and pair-connectedness functions $\\\\varphi ,\\\\tau _\\\\lambda \\\\colon \\\\mathbb {X}^2 \\\\rightarrow \\\\left[0,1\\\\right]$ as kernel functions, we construct associated linear integral operators.\",\n \"Where $L^2\\\\left(\\\\mathbb {X}\\\\right)$ is the Hilbert space of square integrable functions on $\\\\mathbb {X}$ , we define the connection operator, $\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ , and the pair-connectedness operator, $\\\\mathcal {T}_\\\\lambda \\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ .\",\n \"For $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ , we have $f \\\\left(x\\\\right) = \\\\int \\\\varphi (x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right), \\\\qquad \\\\mathcal {T}_\\\\lambda f \\\\left(x\\\\right) = \\\\int \\\\tau _\\\\lambda (x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right),$ for $\\\\nu $ -almost every $x\\\\in \\\\mathbb {X}$ .\",\n \"Note that the composition of two of these operators can be seen as a “convolution over $\\\\mathbb {X}$ \\\" of the kernel functions.\",\n \"Where we rarely use it, we reserve the $\\\\star $ notation for straight-forward convolution over $\\\\mathbb {R}^d$ only.\",\n \"For example, given $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ and $a_1,a_2,a_3,a_4\\\\in \\\\mathcal {E}$ , $\\\\left(\\\\varphi \\\\left(\\\\cdot ;a_1,a_2\\\\right)\\\\star \\\\varphi \\\\left(\\\\cdot ;a_3,a_4\\\\right)\\\\right)\\\\left(\\\\overline{x}\\\\right) = \\\\int _{\\\\mathbb {R}^d}\\\\varphi \\\\left(\\\\overline{x}-\\\\overline{y};a_1,a_2\\\\right)\\\\varphi \\\\left(\\\\overline{y};a_3,a_4\\\\right) \\\\mathrm {d}\\\\overline{y}.$ We will also consider Fourier transformed versions of these operators.\",\n \"For a Lebesgue-integrable function $g\\\\colon \\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}$ , we define the Fourier transform of $g$ to be $\\\\widehat{g}(k) = \\\\int \\\\text{e}^{ik\\\\cdot \\\\overline{x}}g\\\\left(\\\\overline{x}\\\\right) \\\\mathrm {d}\\\\overline{x}$ for $k\\\\in \\\\mathbb {R}^d$ where $k\\\\cdot \\\\overline{x}= \\\\sum ^d_{j=1}k_j\\\\overline{x}_j$ denotes the standard inner product on $\\\\mathbb {R}^d$ .\",\n \"For given marks $a,b\\\\in \\\\mathcal {E}$ , we define $\\\\widehat{\\\\varphi }(k;a,b)$ and $\\\\widehat{\\\\tau }_\\\\lambda (k;a,b)$ as the Fourier transforms of $\\\\varphi (\\\\overline{x};a,b)$ and $\\\\tau _\\\\lambda (\\\\overline{x};a,b)$ respectively.\",\n \"We use these $\\\\mathcal {E}^2\\\\rightarrow \\\\mathbb {R}$ functions to define linear operators on $L^2(\\\\mathcal {E})$ , the space of square integrable functions on $\\\\mathcal {E}$ .\",\n \"For each $k\\\\in \\\\mathbb {R}^d$ we define the operators $\\\\widehat{}(k),\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ by their action on $f\\\\in L^2(\\\\mathcal {E})$ : $\\\\left[\\\\widehat{}(k) f\\\\right] \\\\left(a\\\\right) = \\\\int \\\\widehat{\\\\varphi }(k;a,b)f(b)\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right), \\\\qquad \\\\left[\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) f\\\\right] \\\\left(a\\\\right) = \\\\int \\\\tau _\\\\lambda (k;a,b)f(b)\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right),$ for $\\\\mathcal {P}$ -almost every $a$ .\",\n \"Our main aims of this paper are to derive an infrared bound for the two-point function (via its integral operators), and to prove that the so-called triangle condition is satisfied by a large family of models in sufficiently high dimension.\",\n \"For $\\\\lambda >0$ we define $\\\\triangle _\\\\lambda := \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{x,y\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (x,u)\\\\tau _\\\\lambda (u,v)\\\\tau _\\\\lambda (v,y)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right).$ It is relatively simple to show that $\\\\triangle _\\\\lambda $ is finite for sub-critical $\\\\lambda $ and infinite for super-critical $\\\\lambda $ , but a model is said to satisfy the triangle condition if $\\\\triangle _\\\\lambda $ is finite at criticality.\",\n \"An aside: the essential supremum taken in (REF ) is with respect to the measure $\\\\nu $ on $\\\\mathbb {X}$ .\",\n \"We will be taking the essential supremum many times in this paper over various spaces, and each space will naturally will have a different measure implicitly associated with it.\",\n \"It should be clear from the context what that measure is.\"\n ],\n [\n \"Overview\",\n \"In Section we provide the main results of the paper.\",\n \"We begin by making precise the idea of a critical intensity in Section REF .\",\n \"These propositions are largely independent of the rest of the paper and their proofs can be found in Appendix .\",\n \"The main concern of the paper is in proving the results of Section REF .\",\n \"Given the set of assumption REF , REF , and REF , we prove that the triangle condition is satisfied and we provide an infrared bound on the two-point operator in terms of the adjacency operator.\",\n \"Some examples of models that satisfy these assumptions are provided in Section REF , with the proofs that they satisfy them given in Appendix .\",\n \"Before we begin with the proof proper, we give some preliminaries in Section .\",\n \"First Section REF gives a formal construction of the RCM.\",\n \"Then we have Sections REF and REF , which give standard but important results in probability and linear algebra respectively that we will frequently use in our arguments.\",\n \"The probabilistic lemmas will be familiar to students of point process theory and percolation theory, with the caveat that we require them to apply with the augmented space $\\\\mathbb {X}=\\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ taking the place of the more usual $\\\\mathbb {R}^d$ .\",\n \"This adds no complication.\",\n \"The linear algebra results may well be foreign to a probabilistic audience, but will nevertheless be fundamental to our proof.\",\n \"Here we provide a sufficient technical description of the linear operators we construct and the Hilbert spaces upon which they act.\",\n \"Perhaps the most important result we call upon from this section is the Spectral Theorem (Theorem REF ), which allows us to relate bounded self-adjoint linear operators to a multiplication operator by some unitary map.\",\n \"Multiplication operators are far simpler to work with, and we use the theorem many times towards the end of our proof.\",\n \"For operators on finite dimensional vector spaces, the spectral theorem is equivalent to saying that a Hermitian matrix can be diagonalized over $\\\\mathbb {R}$ .\",\n \"A slightly more unusual lemma in this section is Lemma REF , in which we relate the spectra of the raw operators $$ and $\\\\mathcal {T}_\\\\lambda $ to the spectra of the Fourier transformed operators $\\\\widehat{}(k)$ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ respectively.\",\n \"We conclude Section with Section REF , in which we show that the two-point operator and its derived operators are differentiable with respect to $\\\\lambda $ in the sub-critical regime and we provide bounds on their derivatives.\",\n \"This will be used to show equicontinuity in a step of the proof of Proposition REF , in which we use a forbidden-region argument to bound a bootstrap function.\",\n \"At a broad description level of the proof, we begin by deriving a finite lace expansion for the sub-critical two-point operator and show that this converges to an Ornstein-Zernike equation (OZE).\",\n \"Via a bootstrap argument this will also prove that various expressions - including $\\\\triangle _\\\\lambda $ - are bounded uniformly for subcritical $\\\\lambda $ , and therefore we will be able to extend these bounds and the OZE to critical $\\\\lambda $ by using monotone and dominated convergence-like arguments.\",\n \"We will then be able to take full advantage of the operator description to solve the linear OZE to get the critical two-point operator in terms of the adjacency operator and a lace-expansion-coefficient operator.\",\n \"Section deals with finding the finite lace expansion for the sub-critical $\\\\lambda $ .\",\n \"Formulating this expansion in terms of the relevant functions is essentially the same as for the lace expansion argument appearing in [21] for the unmarked random connection model, except we replace $\\\\mathbb {R}^d$ with $\\\\mathbb {X}$ .\",\n \"For this reason we give only a very brief description of the argument - presenting the important lemmas and ideas - and refer the reader to that previous work for the details.\",\n \"The step of formulating the finite lace expansion in terms of operators (done in Section REF ) is then simple.\",\n \"To show that the finite lace expansion converges, we require bounds on various operators arising from it.\",\n \"We begin in Section by bounding the operator norms of these operators by fairly long and complicated integrals of $\\\\tau _\\\\lambda $ and $\\\\varphi $ (see Proposition REF ).\",\n \"We are then able to divide these complicated integrals into smaller and more manageable elementary `diagrams' - it is here that $\\\\triangle _\\\\lambda $ first makes an appearance.\",\n \"It is also here that we have our first significant detour from the analogous arguments for the proof for the unmarked models.\",\n \"We not only reformulate the proof strategy to be more straight-forward, direct, and transparent than it as appeared previously, but also reduce the reliance on the assumption analogous to the bounds on the sets $B(x)$ defined in (REF ).\",\n \"In [21] this was used multiple times to bound diagrams arising from both the displaced and non-displaced expansion coefficients, but for us this part of the assumption is only ever used to control one simple diagram in Proposition REF .\",\n \"Unfortunately, it does indeed seem that this particular diagram would need to be controlled via a specific assumption, and the bounds on the sets $B(x)$ appear to be relatively easy to check for many models.\",\n \"We also end up with more elementary diagrams than would appear in the unmarked case.\",\n \"The term we describe here as $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ appearing in Propositions REF and REF would simplify into another diagram in the unmarked case, but the inhomogeneity that the marks introduce means that we cannot necessarily commute terms and it does not simplify in our case.\",\n \"Providing bounds for these more elementary diagrams that are uniform in $\\\\lambda $ in the subcritical regime is now a much more realistic task, one we achieve in Section by a bootstrap argument.\",\n \"Many of the elementary diagrams contain instances of the two-point function, which is not explicitly known.\",\n \"In Section REF we therefore replace each of them with an explicitly known Green's function $G_{\\\\mu _\\\\lambda }$ at the cost of one factor of a bootstrap function $f(\\\\lambda )$ , and in Section REF we use a bootstrap argument with a forbidden-region argument to prove that $f$ is uniformly bounded for subcritical $\\\\lambda $ .\",\n \"This then proves that the OZE is satisfied subcritically and that the elementary diagrams - including $\\\\triangle _\\\\lambda $ - are uniformly bounded for subcritical $\\\\lambda $ .\",\n \"Now we have the OZE and uniform bounds for subcritical $\\\\lambda $ , we want to extend these to critical $\\\\lambda $ .\",\n \"We begin Section by using monotone convergence to extend our uniform bound on $\\\\triangle _\\\\lambda $ to include critical $\\\\lambda $ .\",\n \"Using similar monotone and dominated convergence ideas we are able to also show that OZEs for the Fourier transformed operators hold at criticality.\",\n \"We are then finally able to find an infrared bound for the two-point operator at criticality by solving these OZEs using our linear operator formalism.\"\n ],\n [\n \"Results\",\n \"As we will be working with operators, we will require some concept of size for these operators.\",\n \"This will take a number of forms, as some will have advantages in various scenarios.\",\n \"Let $\\\\left(\\\\mathcal {X},\\\\mu \\\\right)\\\\in \\\\left\\\\lbrace \\\\left(\\\\mathbb {X},\\\\nu \\\\right),\\\\left(\\\\mathcal {E},\\\\mathcal {P}\\\\right)\\\\right\\\\rbrace $ , and let $L^2\\\\left(\\\\mathcal {X}\\\\right)\\\\in \\\\left\\\\lbrace L^2\\\\left(\\\\mathbb {X}\\\\right),L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\right\\\\rbrace $ denote the corresponding space of square integrable functions.\",\n \"In practice it will be clear which is being used.\",\n \"We will require five various ideas of size.\",\n \"First suppose $H\\\\colon L^2\\\\left(\\\\mathcal {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {X}\\\\right)$ is a bounded linear operator.\",\n \"Then the operator norm is defined by $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op} := \\\\sup _{f\\\\in L^2(\\\\mathcal {X}):f\\\\ne 0}\\\\frac{*{Hf}_2}{*{f}_2},$ where $*{\\\\cdot }_2$ is the standard $L^2$ -norm on $L^2\\\\left(\\\\mathcal {X}\\\\right)$ .\",\n \"In particular, we have $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op} = \\\\sup \\\\left\\\\lbrace *{z}:z\\\\in \\\\sigma (H)\\\\right\\\\rbrace $ , where $\\\\sigma \\\\left(H\\\\right)\\\\subset \\\\mathbb {C}$ denotes the spectrum of $H$ .\",\n \"Let ${\\\\cdot }{\\\\cdot }$ be the inner product on $L^2\\\\left(\\\\mathcal {X}\\\\right)$ defined by (REF ).\",\n \"If $H$ is also self-adjoint, then ${f}{Hf}\\\\in \\\\mathbb {R}$ and we define the spectral supremum $\\\\mathbb {S}\\\\left(H\\\\right) := \\\\sup _{f\\\\in L^2(\\\\mathcal {X}):f\\\\ne 0}\\\\frac{{f}{Hf}}{{f}{f}}.$ In particular, the self-adjointness of $H$ implies that $\\\\sigma (H)\\\\subset \\\\mathbb {R}$ and $\\\\mathbb {S}\\\\left(H\\\\right) = \\\\sup \\\\left\\\\lbrace z:z\\\\in \\\\sigma (H)\\\\right\\\\rbrace $ .\",\n \"Note that $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ is not a norm: it is not necessarily non-negative.\",\n \"Now suppose $H\\\\colon L^2(\\\\mathcal {X})\\\\rightarrow L^2(\\\\mathcal {X})$ is a bounded linear integral operator with measurable kernel function $h\\\\colon \\\\mathcal {X}^2 \\\\rightarrow \\\\mathbb {C}$ .\",\n \"Then we have the explicit norms: $\\\\left\\\\Vert H\\\\right\\\\Vert _{1,\\\\infty } &:= \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathcal {X}}*{h\\\\left(\\\\cdot ,y\\\\right)}_1 = \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathcal {X}}\\\\int *{h\\\\left(x,y\\\\right)}\\\\mu \\\\left(\\\\mathrm {d}x\\\\right), \\\\\\\\\\\\left\\\\Vert H\\\\right\\\\Vert _{2,\\\\infty } &:= \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathcal {X}}*{h\\\\left(\\\\cdot ,y\\\\right)}_2 = \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathcal {X}}\\\\left(\\\\int *{h\\\\left(x,y\\\\right)}^2\\\\mu \\\\left(\\\\mathrm {d}x\\\\right)\\\\right)^\\\\frac{1}{2},\\\\\\\\\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &:= \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathcal {X}}*{h\\\\left(\\\\cdot ,y\\\\right)}_\\\\infty = \\\\operatornamewithlimits{ess\\\\,sup}_{x,y\\\\in \\\\mathcal {X}}*{h\\\\left(x,y\\\\right)} .$\"\n ],\n [\n \"Critical Intensities\",\n \"Before we move on to our results describing behaviour at and around criticality, we will first explore how we are defining criticality.\",\n \"In our main results in Section REF we use the operator critical intensity $\\\\lambda _O$ , defined by $\\\\lambda _O := \\\\inf \\\\left\\\\lbrace \\\\lambda >0:\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}=\\\\infty \\\\right\\\\rbrace .$ Whilst the divergence of a two-point operator has been used previously to define a critical threshold in random graphs before (for example in [26]), at first glance it may seem to be quite far removed from the standard interpretations of a percolation transition.\",\n \"A variety of transition thresholds for the continuum Boolean disc model are discussed in [38], [17], [8].\",\n \"However, we are able to show that under certain conditions the operator critical intensity coincides with more familiar interpretations.\",\n \"For $\\\\lambda \\\\ge 0$ , we define the two functions $\\\\theta _\\\\lambda \\\\colon \\\\mathcal {E}\\\\mapsto \\\\left[0,1\\\\right]$ and $\\\\chi _\\\\lambda \\\\colon \\\\mathcal {E}\\\\mapsto \\\\left[0,+\\\\infty \\\\right]$ as $\\\\theta _\\\\lambda (a) &:= \\\\mathbb {P}_\\\\lambda \\\\left(*{\\\\overline{0},a)}=\\\\infty \\\\right),\\\\\\\\\\\\chi _\\\\lambda (a) &:= \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},a)}\\\\right].$ For each $p\\\\in \\\\left[1,+\\\\infty \\\\right]$ , we define corresponding functions $\\\\theta ^{(p)}\\\\colon \\\\lambda \\\\mapsto {\\\\theta _\\\\lambda }_p$ and $\\\\chi ^{(p)}\\\\colon \\\\lambda \\\\mapsto {\\\\chi _\\\\lambda }_p$ .\",\n \"These suggest the critical intensities $\\\\lambda _c^{(p)} &:= \\\\inf \\\\left\\\\lbrace \\\\lambda \\\\ge 0: {\\\\theta _\\\\lambda }_p >0\\\\right\\\\rbrace ,\\\\\\\\\\\\lambda _T^{(p)} &:= \\\\inf \\\\left\\\\lbrace \\\\lambda \\\\ge 0: {\\\\chi _\\\\lambda }_p =\\\\infty \\\\right\\\\rbrace .$ Some relations between these intensities can be swiftly deduced without requiring the RCM to satisfy any special conditions.\",\n \"Their definitions clearly show that $\\\\theta _\\\\lambda (a)>0$ implies $\\\\chi _\\\\lambda (a)=\\\\infty $ .\",\n \"Therefore $\\\\lambda ^{(p)}_c \\\\ge \\\\lambda ^{(p)}_T$ for all $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"If $*{\\\\theta _\\\\lambda }_p>0$ for some $p\\\\in \\\\left[1,\\\\infty \\\\right]$ , then it is non-zero on some $\\\\mathcal {P}$ -positive set, and therefore $*{\\\\theta _\\\\lambda }_q>0$ for all $q\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"Therefore $\\\\lambda _c^{(p)} = \\\\lambda _c^{(1)}$ for all $p\\\\in \\\\left[1,+\\\\infty \\\\right]$ , and we denote this value simply by $\\\\lambda _c$ .\",\n \"By Jensen's inequality, $1\\\\le p_1 \\\\le p_2 \\\\le \\\\infty $ implies the ordering $*{\\\\chi _\\\\lambda }_1 \\\\le *{\\\\chi _\\\\lambda }_{p_1} \\\\le *{\\\\chi _\\\\lambda }_{p_2} \\\\le *{\\\\chi _\\\\lambda }_\\\\infty $ .\",\n \"Therefore we have $\\\\lambda _T^{(1)} \\\\ge \\\\lambda ^{(p_1)}_T \\\\ge \\\\lambda ^{(p_2)}_T \\\\ge \\\\lambda ^{(\\\\infty )}_T$ .\",\n \"With some of the techniques developed in this paper it is possible to relate $\\\\lambda _O$ to the intensities $\\\\lambda ^{(1)}_T$ , $\\\\lambda ^{(\\\\infty )}_T$ , and $\\\\lambda _c$ described here.\",\n \"Proposition 2.1 The critical intensities satisfy $\\\\lambda ^{(\\\\infty )}_T \\\\le \\\\lambda _O \\\\le \\\\lambda ^{(1)}_T \\\\le \\\\lambda _c$ .\",\n \"If $\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}<\\\\infty ,$ then we also have $\\\\lambda ^{(p)}_T = \\\\lambda _O$ for all $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"This proof of Proposition REF uses Mecke's equation, Lemma REF , Lemma REF , and some simple probabilistic ideas.\",\n \"The details can be found in Appendix .\",\n \"The coincidence of critical intensities for percolation in marked random connection models has also been considered by [35], [17], [8] in the specific case of a family of Boolean defect models, including Boolean disc models with random bounded radii.\",\n \"In our terminology they introduced a new mark, $\\\\dagger $ , which forms connections as if it had radius 0, and proved that $\\\\theta _\\\\lambda \\\\left(\\\\dagger \\\\right) =0$ if and only if $\\\\chi _\\\\lambda \\\\left(\\\\dagger \\\\right) < \\\\infty $ .\",\n \"Therefore if we define $\\\\lambda _c\\\\left(\\\\dagger \\\\right) := \\\\inf \\\\left\\\\lbrace \\\\lambda \\\\ge 0: \\\\theta _\\\\lambda \\\\left(\\\\dagger \\\\right) >0\\\\right\\\\rbrace , \\\\qquad \\\\lambda _T\\\\left(\\\\dagger \\\\right) := \\\\inf \\\\left\\\\lbrace \\\\lambda \\\\ge 0: \\\\chi _\\\\lambda \\\\left(\\\\dagger \\\\right) = \\\\infty \\\\right\\\\rbrace ,$ this gives $\\\\lambda _c\\\\left(\\\\dagger \\\\right) = \\\\lambda _T\\\\left(\\\\dagger \\\\right)$ .\",\n \"The following Proposition uses this to prove that for the Boolean disc model with bounded radii, all the critical intensities we have considered are equal.\",\n \"Proposition 2.2 In the case of the Boolean disc model with bounded radii, $\\\\lambda _c\\\\left(\\\\dagger \\\\right)= \\\\lambda _T\\\\left(\\\\dagger \\\\right) = \\\\lambda _c = \\\\lambda _O = \\\\lambda _T^{(p)},$ for all $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"To prove Proposition REF we show $\\\\lambda _T\\\\left(\\\\dagger \\\\right) \\\\le \\\\lambda ^{(p)}_T$ by considering points in $\\\\eta $ adjacent to the root point with mark $\\\\dagger $ .\",\n \"The details can be found in Appendix .\"\n ],\n [\n \"Two-Point Operator\",\n \"The majority of this paper is concerned with describing the two-point function and the associated two-point operator.\",\n \"Our results will require Assumption REF described below to hold.\",\n \"This can be considered to be a generalisation of the “finite variance\\\" models described by [21] - extending to cases with multiple marks.\",\n \"HAssumption H We require $\\\\varphi $ to satisfy the following properties: For all dimensions $d$ we have $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) < \\\\infty $ , and there exists a $d$ -independent constant $C>0$ such that $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &\\\\le C \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right),\\\\\\\\\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &\\\\le C \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ There exist $d$ -independent constants $C_1\\\\in \\\\left(0,1\\\\right)$ and $C_2>0$ such that $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(0\\\\right)\\\\right)} \\\\le C_1 \\\\vee \\\\left(1 - C_2{k}^2\\\\right).$ There exists a function $g\\\\colon \\\\mathbb {N} \\\\rightarrow \\\\mathbb {R}_{\\\\ge 0}$ with the following three properties.\",\n \"Firstly, that $g(d)\\\\rightarrow 0$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Secondly, that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)^{-3}\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6)\\\\right)\\\\left(\\\\overline{x}\\\\right) \\\\le g(d).$ Thirdly, that the family of sets $\\\\left\\\\lbrace B\\\\left(x\\\\right)\\\\right\\\\rbrace _{x\\\\in \\\\mathbb {X}}$ given by $B\\\\left(x\\\\right) := \\\\left\\\\lbrace y\\\\in \\\\mathbb {X}: \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)^{-2}\\\\int \\\\varphi \\\\left(x,u\\\\right)\\\\varphi \\\\left(u,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) > g\\\\left(d\\\\right)\\\\right\\\\rbrace $ satisfy $\\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\nu \\\\left(B\\\\left(x\\\\right)\\\\right) \\\\le g\\\\left(d\\\\right)$ .\",\n \"Remark 2.3 We have some remarks regarding these assumptions.\",\n \"By Jensen's inequality, the condition (REF ) in REF also implies that $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty } \\\\le C \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right).$ We will often use these bounds (the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ bound in particular) more directly than (REF ) itself.\",\n \"From (REF ) and (REF ) we can use a supremum bound to deduce that analogous bounds hold for the convolution of $m$ adjacency functions (for $m\\\\ge 3$ ) with corresponding bound $C^{m-3}g(d)$ .\",\n \"The sets $\\\\left\\\\lbrace B\\\\left(x\\\\right)\\\\right\\\\rbrace _{x\\\\in \\\\mathbb {X}}$ appearing in REF are all measurable - this follows from the measurability of $\\\\varphi $ .\",\n \"Furthermore, the symmetry of $\\\\varphi $ implies that $y\\\\in B\\\\left(x\\\\right)\\\\iff x\\\\in B\\\\left(y\\\\right)$ .\",\n \"In stating our theorems, we will use a parameter $\\\\beta =\\\\beta (\\\\varphi , d)$ that depends upon $g(d)$ .\",\n \"The definition depends upon the asymptotic properties of $g(d)$ in that $\\\\beta (d) := {\\\\left\\\\lbrace \\\\begin{array}{ll}g(d)^{\\\\frac{1}{4} -\\\\frac{3}{2d}}d^{-\\\\frac{3}{2}} &: \\\\lim _{d\\\\rightarrow \\\\infty }g(d)\\\\rho ^{-d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^2 = 0 \\\\qquad \\\\forall \\\\rho >0,\\\\\\\\g(d)^\\\\frac{1}{4}&: \\\\text{otherwise}.\\\\end{array}\\\\right.\",\n \"}$ That is, $\\\\beta (d) = g(d)^\\\\frac{1}{4}$ unless $g(d)$ approaches zero particularly quickly.\",\n \"The parameter $\\\\beta $ will reappear many times in this paper - often in the case where $\\\\beta $ is `small.'\",\n \"This corresponds to `large' dimension $d$ , and in particular, where the Landau notation $\\\\mathcal {O}\\\\left(\\\\beta \\\\right)$ is used the asymptotics are as $d\\\\rightarrow \\\\infty $ .\",\n \"Theorem 2.4 Suppose $\\\\varphi $ and $\\\\nu $ satisfy the assumptions REF , REF , and REF .\",\n \"Then there exists a $\\\\lambda $ -independent $d^*>6$ and $C>0$ such that for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right]$ there exists an operator $_\\\\lambda $ with $\\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op}\\\\le C\\\\beta $ and a family of operators $\\\\left\\\\lbrace \\\\widehat{}_{\\\\lambda }(k)\\\\right\\\\rbrace _{k\\\\in \\\\mathbb {R}^d}$ with $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le C\\\\beta $ , such that for all $d\\\\ge d^*$ the following statements are satisfied.\",\n \"For $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ the bounded operator $\\\\mathcal {T}_\\\\lambda $ satisfies the Ornstein–Zernike equation: $\\\\mathcal {T}_\\\\lambda = + _\\\\lambda + \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left( + _\\\\lambda \\\\right).$ For $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ and $k\\\\in \\\\mathbb {R}^d$ , or $\\\\lambda = \\\\lambda _O$ and $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ , the bounded operator $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ satisfies the Ornstein–Zernike equation in Fourier form: $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) = \\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k) + \\\\lambda \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) \\\\left( \\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k) \\\\right).$ Theorem 2.5 If $\\\\varphi $ and $\\\\nu $ satisfy the assumptions REF , REF , and REF , and $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right]$ , then there exist $\\\\lambda $ -independent $d^*>6$ and $C>0$ such that for all $d\\\\ge d^*$ $\\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)+ C \\\\beta }{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(0\\\\right)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}\\\\vee 1 \\\\qquad \\\\forall k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace .$ Furthermore, we have $\\\\triangle _\\\\lambda \\\\le C\\\\beta ^2$ and $*{\\\\theta _{\\\\lambda }}_p=0$ for all $p\\\\in \\\\left[1,\\\\infty \\\\right]$ and $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"This infrared bound can also be related directly to the two-point function by Lemma REF via $\\\\int \\\\widehat{\\\\tau }_\\\\lambda \\\\left(k;a,b\\\\right)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a, \\\\mathrm {d}b\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}.$ Remark 2.6 Note that in Theorem REF we say there exists some $d^*>6$ such that the results hold for all $d\\\\ge d^*$ .\",\n \"If we were willing to say that there exists some $d^*>10$ instead (which would not change the strict logical content of the result), then the proof would be far simpler and Assumptions REF and REF could be weakened.\",\n \"The $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ in (REF ) and () could be replaced with $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{2,\\\\infty }$ and $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ respectively, and (REF ) could be replaced with $\\\\left\\\\Vert ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\le g(d)\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)^3$ .\",\n \"We do not pursue this approach because it is expected that a very similar argument to that presented here (with context appropriate changes to the assumptions) would prove the corresponding result for the spread-out model in $d>6$ .\",\n \"For the simpler argument and weaker conditions, we would only get the spread-out result in $d>10$ , and this would be a logically weaker statement.\"\n ],\n [\n \"Examples of Models\",\n \"It is natural to ask if there are any models for which $\\\\varphi $ and $\\\\nu $ do indeed satisfy REF , REF , and REF .\",\n \"All the models we describe here also satisfy the condition (REF ) for every dimension $d$ .\",\n \"We prove in Appendix that all the models below do satisfy our assumptions.\"\n ],\n [\n \"Single Mark Finite Variance Model.\",\n \"Suppose $\\\\mathcal {E}$ is a singleton, so essentially $\\\\mathbb {X}=\\\\mathbb {R}^d$ and $\\\\nu = {\\\\rm Leb}$ .\",\n \"Then the finite variance models described in [21] all satisfy our conditions.\",\n \"These include the Poisson blob model (or the spherical Boolean model) in which $\\\\varphi (\\\\overline{x}) = {1}_{\\\\lbrace *{\\\\overline{x}} \\\\le r_d\\\\rbrace }$ where $r_d$ is the radius of the unit volume $d$ -ball, and the Gaussian connection model with $\\\\smash{\\\\varphi }(\\\\overline{x}) = \\\\left(2\\\\pi \\\\right)^{-d/2}\\\\exp \\\\big (-\\\\tfrac{1}{2}*{\\\\overline{x}}^2\\\\big )$ .\"\n ],\n [\n \"Double Mark Finite Variance Model.\",\n \"Suppose $\\\\mathcal {E}= \\\\lbrace a_1,a_2\\\\rbrace $ has only two distinct elements with $\\\\mathcal {P}= p\\\\delta _{a_1} + q\\\\delta _{a_2}$ for some $p = 1-q \\\\in \\\\left(0,1\\\\right)$ , and that all the three independent functions $\\\\overline{x}\\\\mapsto \\\\varphi (\\\\overline{x};a,b)$ (for $a,b\\\\in \\\\mathcal {E}$ ) each satisfy the conditions required of a single mark finite variance model.\",\n \"Further suppose that (without loss of generality) $p^2\\\\varphi (\\\\overline{x};a_1,a_1) \\\\ge q^2\\\\varphi (\\\\overline{x};a_2,a_2)$ for almost every $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ .\",\n \"Then our conditions are satisfied.\"\n ],\n [\n \"Space-Mark Factorisation Model\",\n \"Suppose now that the connection function admits the factorisation $\\\\varphi (\\\\overline{x};a,b) = \\\\overline{\\\\varphi }(\\\\overline{x})K(a,b)$ .\",\n \"We require that $\\\\overline{\\\\varphi }$ satisfies the conditions of a single mark finite variance model, and that the ($d$ -independent) mark-kernel $K\\\\colon \\\\mathcal {E}^2 \\\\rightarrow \\\\left[0,1\\\\right]$ is symmetric and measurable.\",\n \"Note that this means that the integral operator $\\\\mathcal {K}\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ with kernel function $K$ is a bounded self-adjoint operator with operator norm $\\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{\\\\rm op}\\\\le 1$ .\",\n \"We further suppose that $\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right) = \\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Then the connection operator is a bounded operator on the mark-space multiplied by a scalar function of spatial position.\",\n \"This formulation places no constraint on the number of marks, permitting even models with uncountably many marks.\"\n ],\n [\n \"Marked Multivariate Gaussian Model.\",\n \"Let $\\\\Sigma \\\\colon \\\\mathcal {E}^2\\\\rightarrow \\\\mathbb {R}^{d\\\\times d}$ be a measurable map where for every $a,b\\\\in \\\\mathcal {E}$ , $\\\\Sigma \\\\left(a,b\\\\right)$ is itself a symmetric positive definite covariance matrix.\",\n \"We further require that there exists $\\\\Sigma _{\\\\max }<\\\\infty $ and $\\\\Sigma _{\\\\min } = \\\\Sigma _{\\\\min }(d) > 0$ such that the spectrum $\\\\sigma (\\\\Sigma (a,b)) \\\\subset \\\\left[\\\\Sigma _{\\\\min },\\\\Sigma _{\\\\max }\\\\right]$ for all $a,b\\\\in \\\\mathcal {E}$ and dimension $d$ .\",\n \"Then for $\\\\mathcal {A} = \\\\mathcal {A}(d)>0$ such that $\\\\mathcal {A}^2 \\\\le \\\\left(2\\\\pi \\\\right)^d\\\\det \\\\Sigma (a,b)$ for all $a,b\\\\in \\\\mathcal {E}$ and $\\\\limsup _{d\\\\rightarrow \\\\infty } \\\\mathcal {A}^2\\\\left(4\\\\pi \\\\Sigma _{\\\\min }\\\\right)^{-d/2} = 0$ , let the connection function be given by $\\\\varphi (\\\\overline{x};a,b) = \\\\mathcal {A}\\\\left(2\\\\pi \\\\right)^{-d/2}\\\\left(\\\\det \\\\Sigma (a,b)\\\\right)^{-1/2}\\\\exp \\\\left(-\\\\frac{1}{2}\\\\overline{x}^{\\\\intercal }\\\\Sigma (a,b)^{-1}\\\\overline{x}\\\\right).$ This class of model includes examples of models with (uncountably) infinitely many marks, and for which the mark and spatial behaviours are truly coupled.\",\n \"For a concrete example, consider $\\\\mathcal {E}=\\\\left[\\\\tfrac{1}{4\\\\pi },1\\\\right]$ with $\\\\Sigma (a,b) = (a+b)1$ and $\\\\mathcal {A} = 1$ .\"\n ],\n [\n \"Bounded-Volume Boolean Disc Model.\",\n \"Let $\\\\left(\\\\left[0,1\\\\right], \\\\Sigma _{\\\\left[0,1\\\\right]}, \\\\mathcal {P}\\\\right)$ be a separable probability space and $\\\\left\\\\lbrace R_d\\\\right\\\\rbrace _{d\\\\ge 1}$ be a sequence of measurable functions $R_d\\\\colon \\\\left[0,1\\\\right]\\\\rightarrow \\\\mathbb {R}_+$ such that there exist $ 00$ such that $R^{({\\\\rm min})}_d:= c_1 d^{\\\\frac{1}{2}} \\\\le R_d\\\\left(a\\\\right) \\\\le \\\\frac{1}{2\\\\sqrt{\\\\pi }}\\\\left(V^{({\\\\rm max})}\\\\right)^{\\\\frac{1}{d}} \\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d} =: R^{({\\\\rm max})}_d,$ for all $a\\\\in [0,1]$ .\",\n \"Note that as $d\\\\rightarrow \\\\infty $ , Stirling's formula gives that the upper bound $R^{({\\\\rm max})}_d\\\\sim \\\\sqrt{d/8\\\\pi \\\\text{e}}$ .\",\n \"It is also required that there exist $\\\\epsilon >0$ and $c_2>0$ such that $\\\\mathcal {P}\\\\left(R_d > \\\\left(1-\\\\frac{c_2}{d}\\\\right)R^{({\\\\rm max})}_d\\\\right)>\\\\epsilon $ for all $d$ .\",\n \"Then let the connection function be defined by $\\\\varphi _d(\\\\overline{x};a,b) = {1}\\\\left\\\\lbrace *{\\\\overline{x}} \\\\le R_d(a) + R_d(b)\\\\right\\\\rbrace .$ It is also perhaps worth mentioning a model that does not satisfy condition REF .\",\n \"Such an example would be a Boolean disk model with random radii for which $\\\\mathcal {P}$ gives positive measure to arbitrarily large radii.\",\n \"No matter the decay rate of $\\\\mathcal {P}$ for large radii, the presence of the suprema in (REF ) and () ensure that REF cannot be satisfied.\",\n \"Similarly the condition (REF ) is also not satisfied for such a model.\"\n ],\n [\n \"Outlook\",\n \"Theorem REF establishes the triangle condition, which is a strong indication of mean-field behaviour.\",\n \"The identification of mean-field critical exponents in the inhomogeneous setting is postponed to future work.\",\n \"We believe that the infrared bound is valid beyond the restricted setting of Hypothesis REF , in particular for the Boolean model with unbounded radii as well as models for complex networks admitting power-law degree graphs (under suitable moment assumptions).\",\n \"However, our current proof does not admit such generalisation, as explained above.\"\n ],\n [\n \"Formal Construction of the RCM\",\n \"We choose to construct our RCM in largely the same way as [21], with appropriate replacements of $\\\\mathbb {R}^d$ with the marked space $\\\\mathbb {X}$ .\",\n \"As they did, we follow [34] in defining the RCM, $\\\\xi $ , as an independent edge-marking of a Poisson point process $\\\\eta $ .\",\n \"Recall $\\\\mathbb {X}=\\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ .\",\n \"We denote by $\\\\mathbf {N}\\\\left(\\\\mathbb {X}\\\\right)$ the set of all at most countably infinite subsets of $\\\\mathbb {X}$ and accompany it with the $\\\\sigma $ -algebra $\\\\mathcal {N}(\\\\mathbb {X})$ generated by the sets $\\\\lbrace A\\\\in \\\\mathbf {N}\\\\left(\\\\mathbb {X}\\\\right):|A \\\\cap B| = k, B \\\\in \\\\Sigma _\\\\mathbb {X}, k \\\\in \\\\mathbb {N}_0\\\\rbrace $ , where $*{A\\\\cap B}$ denotes the cardinality of $A\\\\cap B$ .\",\n \"Then a point process on $\\\\mathbb {X}$ is a measurable mapping $\\\\zeta \\\\colon \\\\Omega \\\\rightarrow \\\\mathbf {N}\\\\left(\\\\mathbb {X}\\\\right)$ for some underlying probability space $\\\\left(\\\\Omega ,\\\\Sigma _\\\\Omega ,\\\\mathbb {P}\\\\right)$ .\",\n \"The intensity measure of such a point process is then the measure on $\\\\mathbb {X}$ given by $B\\\\mapsto \\\\mathbb {E}\\\\left[*{\\\\zeta \\\\cap B}\\\\right]$ for $B\\\\in \\\\Sigma _\\\\mathbb {X}$ .\",\n \"Let $\\\\zeta $ be a point process on $\\\\mathbb {X}$ with a $\\\\sigma $ -finite intensity measure.\",\n \"Then there exist measurable $\\\\pi _i\\\\colon \\\\mathbf {N} \\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow \\\\mathbb {X}$ , $i\\\\in \\\\mathbb {N}$ such that almost surely $\\\\zeta = \\\\left\\\\lbrace \\\\pi _i\\\\left(\\\\zeta \\\\right): i\\\\le *{\\\\zeta \\\\cap \\\\mathbb {X}}\\\\right\\\\rbrace .$ For a $\\\\sigma $ -finite non-atomic measure on $\\\\mathbb {X}$ , say $\\\\nu $ , a Poisson point process (PPP) on $\\\\mathbb {X}$ with intensity measure $\\\\nu $ is a point process $\\\\zeta $ such that the number of points $*{\\\\zeta \\\\cap B}$ is $\\\\text{Poi}(\\\\nu (B))$ -distributed for each $B \\\\in \\\\Sigma _\\\\mathbb {X}$ , and the random variables $*{\\\\zeta \\\\cap B_1}, \\\\ldots , *{\\\\zeta \\\\cap B_m}$ are independent whenever $B_1, \\\\ldots , B_m\\\\in \\\\Sigma _\\\\mathbb {X}$ are pairwise disjoint.\",\n \"In this paper we will be considering a PPP $\\\\eta $ in $\\\\mathbb {X}$ with intensity measure $\\\\lambda \\\\nu $ , where $\\\\nu ={\\\\rm Leb}\\\\otimes \\\\mathcal {P}$ and $\\\\lambda >0$ .\",\n \"We write $\\\\eta =\\\\left\\\\lbrace X_i:i\\\\in \\\\mathbb {N}\\\\right\\\\rbrace $ , where $X_i=\\\\pi _i\\\\left(\\\\eta \\\\right)$ for $i\\\\in \\\\mathbb {N}$ .\",\n \"We now construct the RCM as a (non-Poissionian) point process on the space $\\\\mathbb {X}^{[2]}\\\\times \\\\left[0,1\\\\right]$ .\",\n \"We denote by $\\\\mathbb {X}^{[2]}$ the set of all subsets of $\\\\mathbb {X}$ containing precisely two elements, and each $e\\\\in \\\\mathbb {X}^{[2]}$ is then a potential edge of the RCM.\",\n \"Also let $\\\\left\\\\lbrace U_{i,j}\\\\right\\\\rbrace _{i,j\\\\in \\\\mathbb {N}}$ be a double sequence of independent identically distributed random variables uniformly distributed on $\\\\left[0,1\\\\right]$ , such that the sequence is independent of the PPP $\\\\eta $ .\",\n \"We define $\\\\xi = \\\\left\\\\lbrace \\\\left(\\\\left\\\\lbrace X_i,X_j\\\\right\\\\rbrace ,U_{i,j}\\\\right):1\\\\le i\\\\le j\\\\right\\\\rbrace ,$ where we recall $X_i=\\\\pi _i(\\\\eta )$ .\",\n \"Thus $\\\\xi $ is a random point process on $\\\\mathbb {X}^{[2]}\\\\times \\\\left[0,1\\\\right]$ , which we interpret as a random marked graph and say that $\\\\xi $ is an independent edge-marking of $\\\\eta $ .\",\n \"The independent edge-marking $\\\\xi $ of $\\\\eta $ can then be related by a deterministic functional, $\\\\Gamma _\\\\varphi $ , to a random graph on $\\\\mathbb {X}$ by defining its vertex and edge sets: $V(\\\\Gamma _\\\\varphi (\\\\xi )) = \\\\eta = \\\\lbrace X_i:i\\\\in \\\\mathbb {N}\\\\rbrace ,\\\\qquad E(\\\\Gamma _\\\\varphi (\\\\xi )) = \\\\lbrace \\\\lbrace X_i,X_j\\\\rbrace :U_{i,j} \\\\le \\\\varphi (X_i,X_j), 1\\\\le i\\\\le j\\\\rbrace .$ Since the sequence $U_{i,j}$ is independent of the Poisson process $\\\\eta $ , the ordering in (REF ) does not change the distribution of the resulting random graph.\",\n \"This interpretation of the RCM also allows for the introduction of additional points.\",\n \"This will be required many times in this paper, not least in our definition of the two-point function itself.\",\n \"For additional points $x_1,x_2\\\\in \\\\mathbb {X}$ , the point process $\\\\eta $ is augmented to make $\\\\eta ^{x_1}:=\\\\eta \\\\cup \\\\lbrace x_1\\\\rbrace ,\\\\quad \\\\eta ^{x_1,x_2}:=\\\\eta \\\\cup \\\\lbrace x_1,x_2\\\\rbrace .$ To extend these point processes on $\\\\mathbb {X}$ to a RCM, we extend the sequence $\\\\left\\\\lbrace U_{m,n}\\\\right\\\\rbrace _{m,n \\\\ge 1}$ to a sequence $\\\\left\\\\lbrace U_{m,n}\\\\right\\\\rbrace _{m,n \\\\ge -1}$ of independent random variables uniformly distributed on $[0,1]$ , independent of the Poisson process $\\\\eta $ .\",\n \"We then define $\\\\xi ^{x_1,x_2}$ on $\\\\mathbb {X}^{[2]}\\\\times [0,1]$ as $\\\\xi ^{x_1,x_2}:=\\\\lbrace (\\\\lbrace X_i,X_j\\\\rbrace ,U_{i,j}):-1\\\\le i\\\\le j\\\\rbrace ,$ where $X_0 = x_1$ and $X_{-1} = x_2$ .\",\n \"It is straightforward to define $\\\\xi ^{x_1, \\\\ldots , x_m}$ for arbitrary $m \\\\ge 3$ by the same idea, and to define $\\\\xi ^{x_1}$ by removing all (marked) edges incident to $x_2$ from $\\\\xi ^{x_1,x_2}$ .\",\n \"We assume implicitly that the augmented points are properly coupled such that, e.g., $\\\\xi ^{x_1}$ obtained in this way from $\\\\xi ^{x_1,x_2}$ (by removing all edges adjacent to $x_2$ ) is the same as if obtained from $\\\\xi ^{x_1,x_3}$ (by removing all edges adjacent to $x_3$ ); see [21] for further details.\"\n ],\n [\n \"Probabilistic Lemmas\",\n \"We introduce here a number of important lemmas that are standard in point process theory or percolation theory.\"\n ],\n [\n \"The Mecke Equation\",\n \"An important part of our analysis will be a version of the Mecke equation - for a discussion see [33].\",\n \"This result allows us to make sums over the random sum of points in $\\\\eta $ more tractable.\",\n \"Given $m\\\\in \\\\mathbb {N}$ and a measurable function $f\\\\colon \\\\mathbf {N}\\\\left((\\\\mathbb {X}\\\\times \\\\mathbb {M})^{[2]}\\\\times [0,1]\\\\right) \\\\times \\\\mathbb {X}^m \\\\rightarrow \\\\mathbb {R}_{\\\\ge 0}$ , the Mecke equation for $\\\\xi $ states that $\\\\mathbb {E}_\\\\lambda \\\\left[ \\\\sum _{\\\\vec{x} \\\\in \\\\eta ^{(m)}} f(\\\\xi , \\\\vec{x})\\\\right] = \\\\lambda ^m \\\\int \\\\mathbb {E}_\\\\lambda \\\\left[ f\\\\left(\\\\xi ^{x_1, \\\\ldots , x_m}, \\\\vec{x}\\\\right)\\\\right] \\\\nu ^{\\\\otimes m}\\\\left(\\\\mathrm {d}\\\\vec{x}\\\\right), $ where $\\\\vec{x}=(x_1,\\\\ldots ,x_m)$ , $\\\\eta ^{(m)}=\\\\lbrace (x_1,\\\\ldots ,x_m): x_i \\\\in \\\\eta , x_i \\\\ne x_j \\\\text{ for } i \\\\ne j\\\\rbrace $ , and $\\\\nu ^{\\\\otimes m}$ is the $m$ -product measure of $\\\\nu $ on $\\\\mathbb {X}^m$ .\",\n \"We will only need (REF ) for $m\\\\le 3$ , and largely only use it for $m=1$ .\"\n ],\n [\n \"The Margulis-Russo Formula\",\n \"An often used and very important tool in (discrete) percolation theory, the Margulis-Russo formula will be widely used in our analysis as well.\",\n \"The version we use follows from a slight adjustment to the more general result given in [34].\",\n \"Writing $\\\\mathbf {N}:=\\\\mathbf {N}\\\\left((\\\\mathbb {X}\\\\times \\\\mathbb {M})^{[2]}\\\\times [0,1]\\\\right)$ , let $\\\\Lambda \\\\subset \\\\mathbb {X}$ be $\\\\nu $ -finite, $\\\\zeta \\\\in \\\\mathbf {N}$ , and define $\\\\zeta _\\\\Lambda := \\\\lbrace (\\\\lbrace (x,v),(y,w)\\\\rbrace ,u)\\\\in \\\\zeta :\\\\lbrace x,y\\\\rbrace \\\\subseteq \\\\Lambda \\\\rbrace .\",\n \"$ We call $\\\\zeta _\\\\Lambda $ the restriction of $\\\\zeta $ to $\\\\Lambda $ .\",\n \"We say that $f\\\\colon \\\\mathbf {N}\\\\rightarrow \\\\mathbb {R}$ lives on $\\\\Lambda $ if $f(\\\\zeta ) = f(\\\\zeta _\\\\Lambda )$ for every $\\\\zeta \\\\in \\\\mathbf {N}$ .\",\n \"Assume that there exists a $\\\\nu $ -finite set $\\\\Lambda \\\\subset \\\\mathbb {X}$ such that $f$ lives on $\\\\Lambda $ .\",\n \"Moreover, assume that there exists $\\\\lambda _0>0$ such that $\\\\mathbb {E}_{\\\\lambda _0} \\\\left[*{f(\\\\xi )}\\\\right] < \\\\infty $ .\",\n \"Then the Margulis-Russo formula states that, for all $\\\\lambda \\\\le \\\\lambda _0$ , $\\\\frac{\\\\partial }{\\\\partial \\\\lambda } \\\\mathbb {E}_\\\\lambda [f(\\\\xi )] = \\\\int _\\\\Lambda \\\\mathbb {E}_\\\\lambda \\\\left[f(\\\\xi ^x) - f(\\\\xi )\\\\right] \\\\nu \\\\left(\\\\mathrm {d}x\\\\right).\",\n \"$ The result in [34] requires that $\\\\mathbb {X}=\\\\mathbb {R}^d$ and that $\\\\Lambda \\\\subset \\\\mathbb {R}^d$ is compact, but [32] allows us to make the necessary adjustments to permit $\\\\mathbb {X}=\\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ .\"\n ],\n [\n \"The BK Inequality\",\n \"The idea of the BK inequality is that we acquire a simple upper bound for the probability that two increasing events happen, and that they do so on disjoint subsets of the space.\",\n \"For example, we may want to bound the probability that there exist two disjoint paths (up to the start and end points) in $\\\\xi $ between two given points.\",\n \"The BK inequality we present here is slightly more general than the usual BK inequality in that the point process $\\\\xi $ is augmented with independent random variables.\",\n \"It differs only slightly from the version appearing in [21].\",\n \"For notational clarity, we write $\\\\mathbf {N}:=\\\\mathbf {N}((\\\\mathbb {X}\\\\times \\\\mathbb {M})^{[2]}\\\\times [0,1])$ and denote the $\\\\sigma $ -algebra on $\\\\mathbf {N}$ by $\\\\mathcal {N}$ .\",\n \"We call a set $E\\\\subset \\\\mathbf {N}$ increasing if $\\\\mu \\\\in E$ implies $\\\\nu \\\\in E$ for each $\\\\nu \\\\in \\\\mathbf {N}$ with $\\\\mu \\\\subseteq \\\\nu $ .\",\n \"Let $(\\\\mathbb {Y}_1,\\\\mathcal {Y}_1)$ , $(\\\\mathbb {Y}_2,\\\\mathcal {Y}_2)$ be two measurable spaces.\",\n \"We say that a set $E_i\\\\subset \\\\mathbf {N}\\\\times \\\\mathbb {Y}_i$ is increasing if $\\\\mathbb {E}^z_i:=\\\\lbrace \\\\mu \\\\in \\\\mathbf {N}:(\\\\mu ,z)\\\\in E_i\\\\rbrace $ is increasing for each $z\\\\in \\\\mathbb {X}_i$ .\",\n \"For Borel $\\\\Lambda \\\\in \\\\mathcal {B}\\\\left(\\\\mathbb {R}^d\\\\right)$ and $\\\\mu \\\\in \\\\mathbf {N}$ , we define $\\\\mu _\\\\Lambda $ , the restriction of $\\\\mu $ to all edges completely contained in $\\\\left(\\\\Lambda \\\\times \\\\mathcal {E}\\\\right)\\\\times \\\\mathbb {M}$ , analogously to (REF ).\",\n \"Also let $\\\\mathcal {R}$ denote the ring of all finite unions of half-open $d$ -dimensional rectangles in $\\\\mathbb {R}^d$ with rational coordinates.\",\n \"Then for increasing $E_i\\\\in \\\\mathcal {N}\\\\otimes \\\\mathcal {Y}_i$ , we define $E_1 \\\\circ E_2 := \\\\left\\\\lbrace (\\\\mu ,z_1,z_2)\\\\in \\\\mathbf {N}\\\\times \\\\mathbb {Y}_1\\\\times \\\\mathbb {Y}_2: \\\\exists K_1,K_2 \\\\in \\\\mathcal {R} \\\\text{ s.t.~} \\\\right.\\\\\\\\\\\\left.K_1\\\\cap K_2 = \\\\varnothing , (\\\\mu _{K_1},z_1)\\\\in E_1,(\\\\mu _{K_2},z_2)\\\\in E_2 \\\\right\\\\rbrace .\",\n \"$ A set $E_i\\\\in \\\\mathcal {N}\\\\otimes \\\\mathcal {Y}_i$ lives on $\\\\Lambda $ if $1_{E_i}(\\\\mu ,z)=1_{E_i}(\\\\mu _\\\\Lambda ,z)$ for each $(\\\\mu ,z)\\\\in \\\\mathbf {N}\\\\times \\\\mathbb {Y}_i$ .\",\n \"We consider random elements $W_1,W_2$ of $\\\\mathbb {Y}_1$ and $\\\\mathbb {Y}_2$ , respectively, and assume that $\\\\xi $ , $W_1$ , and $W_2$ are independent.\",\n \"Theorem 3.1 (BK inequality) Let $E_1\\\\in \\\\mathcal {N}\\\\otimes \\\\mathcal {Y}_1$ and $E_2\\\\in \\\\mathcal {N}\\\\otimes \\\\mathcal {Y}_2$ be increasing events that live on $\\\\Lambda \\\\times \\\\mathcal {E}$ for some bounded set $\\\\Lambda \\\\in \\\\mathcal {B}\\\\left(\\\\mathbb {R}^d\\\\right)$ .\",\n \"Then $\\\\mathbb {P}_\\\\lambda ((\\\\xi ,W_1,W_2)\\\\in E_1 \\\\circ E_2) \\\\le \\\\mathbb {P}_\\\\lambda ((\\\\xi ,W_1)\\\\in E_1)\\\\mathbb {P}_\\\\lambda ((\\\\xi ,W_2)\\\\in E_2).$ A proof for this theorem can be found in [21].\",\n \"The presence of the marked space $\\\\mathbb {X}=\\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ rather than just $\\\\mathbb {R}^d$ makes no difference to the validity of their proof.\",\n \"It is worth noting that the operation $\\\\circ $ is commutative and associative, which allows repeated application of the inequality.\"\n ],\n [\n \"The FKG Inequality\",\n \"In contrast to the BK inequality, the FKG inequality gives us a simple lower bound on the probability of two increasing events occurring.\",\n \"Given two increasing and integrable functions $f,g$ on $(\\\\mathbb {X}\\\\times \\\\mathbb {M})^{[2]}\\\\times [0,1]$ , we have $\\\\mathbb {E}_\\\\lambda \\\\left[f(\\\\xi ) g(\\\\xi )\\\\right] = \\\\mathbb {E}_\\\\lambda \\\\left[ \\\\mathbb {E}_\\\\lambda [f(\\\\xi )g(\\\\xi ) \\\\mid \\\\eta ] \\\\right] \\\\ge \\\\mathbb {E}_\\\\lambda \\\\left[ \\\\mathbb {E}_\\\\lambda \\\\left[f(\\\\xi )\\\\mid \\\\eta \\\\right] \\\\; \\\\mathbb {E}_\\\\lambda \\\\left[g(\\\\xi )\\\\mid \\\\eta \\\\right] \\\\right]\\\\ge \\\\mathbb {E}_\\\\lambda \\\\left[f(\\\\xi )\\\\right] \\\\; \\\\mathbb {E}_\\\\lambda \\\\left[g(\\\\xi )\\\\right].\",\n \"$ The first inequality was obtained by applying FKG to the random graph conditioned to have $\\\\eta $ as its vertex set, the second inequality by applying FKG for point processes (see [33] for details).\"\n ],\n [\n \"Linear Operator Lemmas\",\n \"We shall formulate our problem as one of linear operators acting on Hilbert spaces.\",\n \"We will want to define analogous objects on the full marked space $\\\\mathbb {X}=\\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ and on the mark space $\\\\mathcal {E}$ itself.\",\n \"Let $\\\\left(\\\\mathcal {X},\\\\Sigma _\\\\mathcal {X},\\\\mu \\\\right)\\\\in \\\\left\\\\lbrace \\\\left(\\\\mathbb {X},\\\\Sigma _\\\\mathbb {X},\\\\nu \\\\right),\\\\left(\\\\mathcal {E},\\\\Sigma _\\\\mathcal {E},\\\\mathcal {P}\\\\right)\\\\right\\\\rbrace $ , and $L^2\\\\left(\\\\mathcal {X}\\\\right)\\\\in \\\\left\\\\lbrace L^2\\\\left(\\\\mathbb {X}\\\\right),L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\right\\\\rbrace $ correspondingly.\",\n \"Given the measure space $\\\\left(\\\\mathcal {X},\\\\Sigma _\\\\mathcal {X},\\\\mu \\\\right)$ , we consider the Banach space of square-integrable functions $L^2(\\\\mathcal {X}) = L^2\\\\left(\\\\mathcal {X},\\\\Sigma _{\\\\mathcal {X}},\\\\mu \\\\right)$ .\",\n \"We augment this with the inner product defined by ${f}{g} := \\\\int \\\\overline{f\\\\left(x\\\\right)}g\\\\left(x\\\\right) \\\\mu \\\\left(\\\\mathrm {d}x \\\\right),$ for all $f,g\\\\in L^2(\\\\mathcal {X})$ , where $\\\\overline{f(x)}$ is the complex conjugate of $f(x)$ .\",\n \"With this inner product, $L^2(\\\\mathcal {X})$ is a Hilbert space.\",\n \"The measure space $\\\\left(\\\\mathcal {X},\\\\Sigma _\\\\mathcal {X},\\\\mu \\\\right)$ is separable if there exists a countable family $\\\\left\\\\lbrace E_k\\\\right\\\\rbrace _{k=1}^\\\\infty \\\\subset \\\\Sigma _\\\\mathcal {X}$ such that given $E\\\\in \\\\Sigma _\\\\mathcal {X}$ with $\\\\mu (E)<\\\\infty $ there exists an $E$ -dependent subsequence $\\\\lbrace n_k\\\\rbrace _{k=1}^\\\\infty $ such that $\\\\mu \\\\left(E\\\\Delta E_{n_k}\\\\right) \\\\rightarrow 0, \\\\qquad \\\\text{ as }k\\\\rightarrow \\\\infty ,$ where $\\\\Delta $ here is the symmetric difference.\",\n \"A well-known example of a separable measure space is $\\\\left(\\\\mathbb {R}^d,\\\\mathfrak {B},{\\\\rm Leb}\\\\right)$ - where $\\\\mathfrak {B}$ is the Borel $\\\\sigma $ -algebra - and therefore if the probability measure space $\\\\left(\\\\mathcal {E},\\\\Sigma _{\\\\mathcal {E}},\\\\mathcal {P}\\\\right)$ is separable then so is the product measure space $\\\\left(\\\\mathbb {X},\\\\Sigma _\\\\mathbb {X},\\\\nu \\\\right)$ .\",\n \"A topological space is called separable if there exists a countable dense subset.\",\n \"If the measure space $\\\\left(\\\\mathcal {X},\\\\Sigma _\\\\mathcal {X},\\\\mu \\\\right)$ is separable, then the function spaces $L^p\\\\left(\\\\mathcal {X}\\\\right) = L^p\\\\left(\\\\mathcal {X},\\\\Sigma _\\\\mathcal {X},\\\\mu \\\\right)$ with $p\\\\in \\\\left[1,\\\\infty \\\\right)$ are all separable (see, for example, [42]).\",\n \"Since we have assumed that $\\\\left(\\\\mathcal {E},\\\\Sigma _\\\\mathcal {E},\\\\mathcal {P}\\\\right)$ is separable, we know that both $L^2(\\\\mathbb {X})$ and $L^2(\\\\mathcal {E})$ are separable Hilbert spaces.\",\n \"The inner product for $L^2(\\\\mathcal {X})$ allows us to define the $L^2$ -norm for $f\\\\in L^2(\\\\mathcal {X})$ : $*{f}_2 = {f}{f}^{\\\\frac{1}{2}}.$ This allows us to define a norm on the space of bounded linear operators on $L^2(\\\\mathcal {X})$ .\",\n \"The space of bounded linear operators on a Banach space is itself a Banach space when augmented with the operator norm defined by $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op} := \\\\sup _{f\\\\in L^2(\\\\mathcal {X}):f\\\\ne 0}\\\\frac{*{Hf}_2}{*{f}_2},$ for a linear operator $H\\\\colon L^2(\\\\mathcal {X})\\\\rightarrow L^2(\\\\mathcal {X})$ .\",\n \"Since $H$ is linear, the supremum can equivalently be taken over $f\\\\in L^2(\\\\mathcal {X})$ such that $*{f}_2=1$ .\",\n \"Note that $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\rm op}$ is sub-multiplicative, in the sense that for linear operators $G,H\\\\colon L^2(\\\\mathcal {X})\\\\rightarrow L^2(\\\\mathcal {X})$ we have $\\\\left\\\\Vert GH\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert G\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}.$ Given a linear operator $H\\\\colon L^2(\\\\mathcal {X})\\\\rightarrow L^2(\\\\mathcal {X})$ on Hilbert spaces, the adjoint of $H$ is the operator $H^\\\\dagger \\\\colon L^2(\\\\mathcal {X})\\\\rightarrow L^2(\\\\mathcal {X})$ such that ${f}{Hg} = {H^\\\\dagger f}{g}$ for all $f,g\\\\in L^2(\\\\mathcal {X})$ .\",\n \"An operator is self-adjoint if $H=H^\\\\dagger $ .\",\n \"It is worth noting that a bounded integral operator $H$ with kernel function $h$ is self-adjoint if and only if $h(x,y) = \\\\overline{h(y,x)}$ for $\\\\mu $ -a.e.\",\n \"$x,y\\\\in \\\\mathcal {X}$ .\",\n \"If $h$ is real valued, this is equivalent to $h$ being symmetric on such a set of $x,y$ .\",\n \"A simple calculation shows that if $H$ is bounded and self-adjoint, then ${f}{Hf}\\\\in \\\\mathbb {R}$ for all $f\\\\in L^2(\\\\mathcal {X})$ .\",\n \"This allows us to define the following object.\",\n \"Given a bounded and self-adjoint linear $H\\\\colon L^2(\\\\mathcal {X})\\\\rightarrow L^2(\\\\mathcal {X})$ , define $\\\\mathbb {S}\\\\left(H\\\\right) := \\\\sup _{f\\\\in L^2(\\\\mathcal {X}):f\\\\ne 0}\\\\frac{{f}{Hf}}{{f}{f}}.$ Note that, in contrast to $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\rm op}$ , the spectral supremum $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ is not a norm - it is possible for it to be negative.\",\n \"The objects $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}$ and $\\\\mathbb {S}\\\\left(H\\\\right)$ can also be understood via the spectrum of $H$ .\",\n \"The spectrum, denoted $\\\\sigma (H)$ , is the set of $z\\\\in \\\\mathbb {C}$ such that $z1- H$ does not have a bounded linear inverse.\",\n \"For linear operators on finite dimensional vector spaces this coincides with the set of eigenvalues of $H$ .\",\n \"If $H$ is self-adjoint, then $\\\\sigma (H)\\\\subset \\\\mathbb {R}$ .\",\n \"We can then express $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}$ and $\\\\mathbb {S}\\\\left(H\\\\right)$ in terms of the spectrum: $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op} = \\\\sup \\\\left\\\\lbrace *{z}:z\\\\in \\\\sigma (H)\\\\right\\\\rbrace ,\\\\qquad \\\\mathbb {S}\\\\left(H\\\\right) = \\\\sup \\\\left\\\\lbrace z:z\\\\in \\\\sigma (H)\\\\right\\\\rbrace .$ Note that for self-adjoint operators the operator norm is equal to the spectral radius of the operator.\",\n \"While $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\rm op}$ and $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ are very important properties of the operators we will be considering, they are not necessarily simple to compute for an integral operator given its kernel function.\",\n \"Recall that in (REF )-() we defined the three integral operator norms $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{2,\\\\infty }$ , and $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"These have the advantage that they are explicitly defined in terms of the kernel function.\",\n \"We now provide some results regarding the four norms (plus $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ ), including relations between some of them.\",\n \"Lemma 3.2 (Bounds on the Operator Norm) Let $H\\\\colon L^2\\\\left(\\\\mathcal {X}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {X}\\\\right)$ be a bounded self-adjoint integral operator.\",\n \"Then we can bound the operator norm: $*{\\\\mathbb {S}\\\\left(H\\\\right)} \\\\le \\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert H\\\\right\\\\Vert _{1,\\\\infty }.$ Furthermore, if $\\\\mathcal {X}=\\\\mathcal {E}$ and $H$ has kernel function $h$ , then $\\\\int h\\\\left(a,b\\\\right)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right) \\\\le \\\\mathbb {S}\\\\left(H\\\\right)$ The first inequality in (REF ) holds from the expressions of $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ and $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\rm op}$ in terms of the spectrum (see (REF )).\",\n \"The boundedness and self-adjointness of $H$ implies that the second inequality follows from an application of the Schur Test and the conjugate symmetry of its kernel function (see, for example [24]).\",\n \"For the lower bound on $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ , we use the definition (REF ) and consider the test function $f(a) = 1$ .\",\n \"Lemma 3.3 Suppose $h,g\\\\colon \\\\mathcal {X}^2\\\\rightarrow \\\\mathbb {C}$ are kernel functions for the self-adjoint linear operators $H,G\\\\colon L^2\\\\left(\\\\mathcal {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {X}\\\\right)$ , such that $h\\\\left(x,y\\\\right)\\\\in \\\\mathbb {R}_{\\\\ge 0}$ and $h\\\\left(x,y\\\\right) \\\\ge *{g\\\\left(x,y\\\\right)}$ for $\\\\mu $ -a.e.\",\n \"$x,y\\\\in \\\\mathcal {X}$ .\",\n \"Then $\\\\mathbb {S}\\\\left(H\\\\right)\\\\ge \\\\mathbb {S}\\\\left(G\\\\right),\\\\quad \\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}\\\\ge \\\\left\\\\Vert G\\\\right\\\\Vert _{\\\\rm op}, \\\\quad \\\\left\\\\Vert H\\\\right\\\\Vert _{1,\\\\infty }\\\\ge \\\\left\\\\Vert G\\\\right\\\\Vert _{1,\\\\infty }, \\\\quad \\\\left\\\\Vert H\\\\right\\\\Vert _{2,\\\\infty }\\\\ge \\\\left\\\\Vert G\\\\right\\\\Vert _{2,\\\\infty }, \\\\quad \\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\ge \\\\left\\\\Vert G\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ For the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{2,\\\\infty }$ , and $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ norms the inequality is clear from the definitions.\",\n \"For the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\rm op}$ version, let $f\\\\in L^2\\\\left(\\\\mathcal {X}\\\\right)$ , and use $\\\\rm Abs$ to denote the operator $f\\\\mapsto *{f}$ .\",\n \"While this operator is not linear, it is clear that $*{{\\\\rm Abs}f}_2 = *{f}_2$ .\",\n \"With this setup, $*{Gf}^2_2 = \\\\int *{\\\\int g\\\\left(x,y\\\\right) f\\\\left(y\\\\right) \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)}^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\le \\\\int \\\\left(\\\\int *{g\\\\left(x,y\\\\right)} *{f\\\\left(y\\\\right)} \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\\\\\\\\\le \\\\int \\\\left(\\\\int h\\\\left(x,y\\\\right) *{f\\\\left(y\\\\right)} \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) = *{H{\\\\rm Abs}f}^2_2.$ Therefore $\\\\left\\\\Vert G\\\\right\\\\Vert _{\\\\rm op} = \\\\sup _f\\\\frac{*{Gf}_2}{*{f}_2} \\\\le \\\\sup _f\\\\frac{*{H{\\\\rm Abs}f}_2}{*{f}_2} = \\\\sup _f\\\\frac{*{H{\\\\rm Abs}f}_2}{*{{\\\\rm Abs}f}_2}\\\\le \\\\sup _f\\\\frac{*{Hf}_2}{*{f}_2} =\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}.$ Here we used that the image of $\\\\rm Abs$ is a subset of $L^2\\\\left(\\\\mathcal {X}\\\\right)$ .\",\n \"A similar argument works for the $\\\\mathbb {S}\\\\left(\\\\cdot \\\\right)$ inequality.\",\n \"Given $f\\\\in L^2(\\\\mathcal {X})$ , $*{\\\\left} \\\\le \\\\int *{f(x)}*{g(x,y)}*{f(y)}\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y\\\\right) \\\\le \\\\left<{\\\\rm Abs}f,H{\\\\rm Abs}f\\\\right>.$ Therefore $\\\\mathbb {S}\\\\left(G\\\\right) = \\\\sup _f\\\\frac{\\\\left}{*{f}^2_2} \\\\le \\\\sup _f\\\\frac{\\\\left<{\\\\rm Abs}f,H{\\\\rm Abs}f\\\\right>}{*{f}^2_2} = \\\\sup _f\\\\frac{\\\\left<{\\\\rm Abs}f,H{\\\\rm Abs}f\\\\right>}{*{{\\\\rm Abs}f}^2_2}\\\\le \\\\sup _f\\\\frac{\\\\left}{*{f}^2_2} =\\\\mathbb {S}\\\\left(H\\\\right).$ Lemma 3.4 Let $G,H$ be bounded self-adjoint linear operators on a separable Hilbert space.\",\n \"Then $*{\\\\mathbb {S}\\\\left(G+H\\\\right) - \\\\mathbb {S}\\\\left(G\\\\right)} \\\\le \\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}.$ It is a well-known result (for example see [29]) that $\\\\sigma (G+H)$ is contained in the closed $\\\\left\\\\Vert H\\\\right\\\\Vert _{\\\\rm op}$ -neighbourhood of $\\\\sigma (G)$ .\",\n \"Perturbing $G$ by $+H$ , and perturbing $G+H$ by $-H$ then implies the result.\",\n \"The main advantage of the Hilbert space formulation for the operators is that it allows us to use the following theorem to better understand the operators.\",\n \"Although the “Multiplication Operator\\\" version presented here may not provide as much information as the “Projection-Valued Measure\\\" or “Direct Integral\\\" versions (see [19]), it has the advantage of being simpler to state and is sufficient for our purposes.\",\n \"Theorem 3.5 (Spectral Theorem - Multiplication Operator Version) Let $\\\\left\\\\lbrace H_i\\\\right\\\\rbrace _{i\\\\in I}$ be a commutative family of bounded, self-adjoint operators on a separable Hilbert space $\\\\mathbb {H}$ .\",\n \"Then there exists a $\\\\sigma $ -finite measure space $\\\\left(\\\\Omega ,\\\\Sigma _\\\\Omega , \\\\mu \\\\right)$ , a family of bounded, measurable, real-valued functions $\\\\left\\\\lbrace h_i\\\\right\\\\rbrace _{i\\\\in I}$ on $\\\\Omega $ , and a unitary map $U \\\\colon \\\\mathbb {H}\\\\rightarrow L^2\\\\left(\\\\Omega ,\\\\Sigma _\\\\Omega , \\\\mu \\\\right)$ such that for $\\\\mu $ -a.e.\",\n \"$\\\\omega $ , $\\\\left(UH_iU^{-1}f\\\\right)\\\\left(\\\\omega \\\\right) = h_i\\\\left(\\\\omega \\\\right)f\\\\left(\\\\omega \\\\right)$ for all $f\\\\in L^2\\\\left(\\\\Omega ,\\\\Sigma _\\\\Omega .\",\n \"\\\\mu \\\\right)$ and $i\\\\in I$ .\",\n \"For a proof in more generality, see [12].\",\n \"In the context of a finite dimensional Hilbert space $\\\\mathbb {H}$ , the Spectral Theorem is effectively just the statement that Hermitian matrices can be diagonalized over $\\\\mathbb {R}$ .\",\n \"Here $U$ is represented by a unitary matrix composed from the orthonormal eigenbasis of $\\\\left\\\\lbrace H_i\\\\right\\\\rbrace $ , and the entries of the resulting diagonal matrix corresponding to $H_i$ are the values of the function $h_i(\\\\omega )$ .\",\n \"One famous application of the Spectral Theorem for infinite dimensional Hilbert spaces is in the context of convolution operators on $\\\\mathbb {R}^d$ with Lebesgue measure.\",\n \"Suppose the operator $G\\\\colon L^2\\\\left(\\\\mathbb {R}^d\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {R}^d\\\\right)$ acts on the function $f$ by $Gf(\\\\overline{x}) = \\\\int g(\\\\overline{x}-\\\\overline{y})f(\\\\overline{y}) \\\\mathrm {d}\\\\overline{y}$ for some function $g\\\\colon \\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}$ .\",\n \"If $g$ is integrable then Schur's test proves that $G$ is bounded, and if $g$ is reflection invariant (that is, $g(\\\\overline{x})=g(-\\\\overline{x})$ ) then expanding the inner product proves that $G$ is self-adjoint.\",\n \"Supposing that these conditions hold, then the Fourier transform of $g$ is given by the bounded real function $\\\\widehat{g}\\\\colon \\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}$ defined by $\\\\widehat{g}(k) = \\\\int \\\\text{e}^{i\\\\overline{x}\\\\cdot k}g(\\\\overline{x})\\\\mathrm {d}\\\\overline{x}.$ Furthermore, the Plancherel Theorem states that for a function $f\\\\in L^2\\\\left(\\\\mathbb {R}^d\\\\right)\\\\cap L^1\\\\left(\\\\mathbb {R}^d\\\\right)$ then the $L^2$ -norm of $f$ is equal to the $L^2$ -norm of $\\\\widehat{f}$ .\",\n \"This - along with the clear linearity of the Fourier transform map - shows that the Fourier transform map can be considered a bounded linear operator $L^2\\\\left(\\\\mathbb {R}^d\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {R}^d\\\\right)$ .\",\n \"The Fourier Inversion Theorem also relates the Fourier transform map to its inverse and its adjoint, and this can be used to show that the Fourier transform map is in fact a unitary linear operator.\",\n \"The famous Convolution Theorem says that the Fourier transform of the convolution of two functions is the pointwise product of their individual Fourier transforms.\",\n \"This means that $\\\\widehat{Gf}(k) = \\\\widehat{g}(k)\\\\widehat{f}(k).$ This can then be understood as an instance of the Spectral Theorem.\",\n \"In this case, both the Hilbert spaces are $L^2\\\\left(\\\\mathbb {R}^d\\\\right)$ , the unitary map is (the inverse of) the Fourier transform map, and the function $h$ takes the values of the Fourier function, $\\\\widehat{g}$ , on $\\\\mathbb {R}^d$ .\",\n \"Part of the power of the Fourier transform lies in that fact that this single unitary map simultaneously “diagonalizes\\\" a large class of operators.\",\n \"The translation and reflection invariance of the Lebesgue measure on $\\\\mathbb {R}^d$ means that for integrable functions $g,h\\\\colon \\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}$ , $\\\\int g(\\\\overline{x}-\\\\overline{y})h(\\\\overline{y})\\\\mathrm {d}\\\\overline{y}= \\\\int h(\\\\overline{x}-\\\\overline{u})g(\\\\overline{u})\\\\mathrm {d}\\\\overline{u}$ (using the substitution $\\\\overline{u}=\\\\overline{x}-\\\\overline{y}$ ).\",\n \"Therefore the family of convolution operators described here form a commutative family of operators.\",\n \"We also present here a lemma that relates the operator norm and spectral supremum for operators on $L^2(\\\\mathbb {X})$ to operators on $L^2(\\\\mathcal {E})$ .\",\n \"Given a family $\\\\left\\\\lbrace A_k\\\\right\\\\rbrace _{k\\\\in \\\\mathbb {R}^d}$ of subsets of $\\\\mathbb {C}$ , we define the essential union of these sets to be ${\\\\rm ess.\",\n \"}\\\\bigcup _{k\\\\in \\\\mathbb {R}^d}A_k := \\\\left\\\\lbrace z\\\\in \\\\mathbb {C}:\\\\forall \\\\epsilon >0, {\\\\rm Leb}\\\\left(k\\\\in \\\\mathbb {R}^d:\\\\exists x\\\\in A_k \\\\text{ s.t.\",\n \"}*{x-z}<\\\\epsilon \\\\right)>0\\\\right\\\\rbrace .$ This can be understood as the union of $A_k$ over $k$ neglecting null sets of $k$ .\",\n \"Lemma 3.6 We have $\\\\sigma \\\\left(\\\\right) = {\\\\rm ess.\",\n \"}\\\\bigcup _{k\\\\in \\\\mathbb {R}^d}\\\\sigma \\\\left(\\\\widehat{}(k)\\\\right) , \\\\qquad \\\\sigma \\\\left(\\\\mathcal {T}_\\\\lambda \\\\right) = {\\\\rm ess.\",\n \"}\\\\bigcup _{k\\\\in \\\\mathbb {R}^d}\\\\sigma \\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right).$ It then follows that $\\\\left\\\\Vert \\\\right\\\\Vert _{\\\\rm op} = \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d}\\\\left\\\\Vert \\\\widehat{}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}, \\\\qquad \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} = \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op},$ $\\\\mathbb {S}\\\\left(\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d}\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right), \\\\qquad \\\\mathbb {S}\\\\left(\\\\mathcal {T}_\\\\lambda \\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d}\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right).$ We present the proof of (REF ) for the connection operator case.\",\n \"The two-point operator case follows similarly.\",\n \"First suppose $\\\\zeta \\\\in {\\\\rm ess.\",\n \"}\\\\bigcup _{k\\\\in \\\\mathbb {R}^d}\\\\sigma \\\\left(\\\\widehat{}(k)\\\\right)$ .\",\n \"Therefore for all $\\\\epsilon >0$ there exists $P_\\\\epsilon \\\\subset \\\\mathbb {R}^d$ such that ${\\\\rm Leb}\\\\left(P_\\\\epsilon \\\\right)>0$ and for all $k\\\\in P_\\\\epsilon $ there exists $z\\\\in \\\\sigma \\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ such that $*{\\\\zeta -z}<\\\\epsilon $ .\",\n \"Furthermore, given $z\\\\in \\\\sigma \\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ there exists $g\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ with $*{g}_2=1$ such that $*{\\\\left(z1- \\\\widehat{}\\\\left(k\\\\right)\\\\right)g}_2<\\\\epsilon $ .\",\n \"For all $k\\\\in P_\\\\epsilon $ we let $z^{(k)}$ and $g^{(k)}$ denote such choices.\",\n \"Now let $\\\\widehat{f}$ be any element of $L^2\\\\left(\\\\mathbb {R}^d\\\\right)$ supported on $P_\\\\epsilon $ with $*{\\\\widehat{f}}_2=1$ .\",\n \"Then define $h\\\\left(\\\\overline{x},a\\\\right) := \\\\int _{P_\\\\epsilon } \\\\text{e}^{-i k\\\\cdot \\\\overline{x}}\\\\widehat{f}(k)g^{(k)}(a)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}.$ By the Fourier Inversion Theorem, the Fourier transform of $h$ is given by $\\\\widehat{h}\\\\left(k,a\\\\right) = \\\\widehat{f}(k)g^{(k)}(a)$ for $k\\\\in P_\\\\epsilon $ and vanishes elsewhere.\",\n \"By Plancherel's Theorem (that is, the Fourier transform is unitary), $h$ and $\\\\widehat{h}$ have the same $L^2$ -norm: $*{h}^2_2= *{\\\\widehat{h}}^2_2 = \\\\int _{P_\\\\epsilon }*{\\\\widehat{f}(k)}^2\\\\left(\\\\int *{g^{(k)}(a)}^2\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\right)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\\\\\= \\\\int _{P_\\\\epsilon }*{\\\\widehat{f}(k)}^2*{g^{(k)}}^2_2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} = \\\\int _{P_\\\\epsilon }*{\\\\widehat{f}(k)}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} = *{\\\\widehat{f}}^2_2 = 1.$ By applying Plancherel's Theorem once again, we can write $*{\\\\left(\\\\zeta 1- \\\\right)h}^2_2 = \\\\int _{P_\\\\epsilon }*{\\\\widehat{f}(k)}^2\\\\int *{\\\\zeta g^{(k)}(a) - \\\\int \\\\widehat{\\\\varphi }(k;a,b)g^{(k)}(b)\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)}^2\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\\\\\= \\\\int _{P_\\\\epsilon }*{\\\\widehat{f}(k)}^2*{\\\\left(\\\\zeta 1- \\\\widehat{}(k)\\\\right)g^{(k)}}^2_2 \\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} < \\\\int _{P_\\\\epsilon }*{\\\\widehat{f}(k)}^2\\\\left(\\\\epsilon + \\\\underbrace{*{\\\\left(z(k)1- \\\\widehat{}(k)\\\\right)g^{(k)}}_2}_{<\\\\epsilon }\\\\right)^2 \\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} < 4\\\\epsilon ^2.$ Here we use the triangle inequality and $*{g^{(k)}}_2=1$ to replace $\\\\zeta $ with $z(k)$ at the cost of an $\\\\epsilon $ .\",\n \"Since for each $\\\\epsilon >0$ we have constructed a unit $h\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ such that $*{\\\\left(\\\\zeta 1- \\\\right)h}_2<2\\\\epsilon $ , we have proven that $\\\\left(\\\\zeta 1- \\\\right)$ has no bounded linear inverse and therefore $\\\\zeta \\\\in \\\\sigma \\\\left(\\\\right)$ .\",\n \"We now show the converse.\",\n \"Suppose $\\\\zeta \\\\in \\\\sigma \\\\left(\\\\right)$ .\",\n \"Then for all $\\\\epsilon >0$ there exists $h \\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ such that $*{h}_2=1$ and $*{\\\\left(\\\\zeta 1- \\\\right)h}_2 < \\\\epsilon $ .\",\n \"By using Plancherel's Theorem to replace the original functions with their Fourier transforms, we get $*{\\\\left(\\\\zeta 1- \\\\right)h}^2_2 = \\\\int *{\\\\zeta h\\\\left(\\\\overline{x},a\\\\right) - \\\\int \\\\varphi \\\\left(\\\\overline{x}-\\\\overline{y};a,b\\\\right)h\\\\left(\\\\overline{y},b\\\\right)\\\\mathrm {d}\\\\overline{y}\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)}^2\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\nonumber \\\\\\\\= \\\\int *{\\\\zeta \\\\widehat{h}\\\\left(k,a\\\\right) - \\\\int \\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right)\\\\widehat{h}\\\\left(k,b\\\\right)\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right).$ Now let us define the family of functions $g^{(k)}\\\\colon \\\\mathcal {E}\\\\rightarrow \\\\mathbb {C}$ by $g^{(k)}\\\\left(a\\\\right) := \\\\widehat{h}\\\\left(k,a\\\\right)$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"Then we can again re-write the equality above as $*{\\\\left(\\\\zeta 1- \\\\right)h}^2_2 = \\\\int *{\\\\left(\\\\zeta 1- \\\\widehat{}\\\\left(k\\\\right)\\\\right)g^{(k)}}^2_2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}.$ For each $k\\\\in \\\\mathbb {R}^d$ , $\\\\widehat{}(k)$ is self-adjoint and therefore by the Spectral Theorem (Theorem REF ) there exists a Hilbert space $L^2\\\\left(\\\\mathfrak {E}_k,\\\\mu _k\\\\right)$ , a unitary operator $U_k\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathfrak {E}_k\\\\right)$ , and a measurable function $e\\\\mapsto \\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)$ such that $\\\\widehat{}(k)$ is unitarily equivalent to pointwise multiplication by $\\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)$ on $L^2\\\\left(\\\\mathfrak {E}_k\\\\right)$ .\",\n \"If we define $f^{\\\\left(k\\\\right)\\\\left(e\\\\right)} := \\\\left(U_kg^{\\\\left(k\\\\right)}\\\\right)\\\\left(e\\\\right)\\\\in \\\\mathbb {C}$ , then (REF ) becomes $*{\\\\left(\\\\zeta 1- \\\\right)h}^2_2 = \\\\int \\\\left(\\\\int *{\\\\zeta - \\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)}^2*{f^{(k)(e)}}^2\\\\mu _k\\\\left(\\\\mathrm {d}e\\\\right)\\\\right)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}.$ Note that since the Fourier transform and the maps $\\\\left\\\\lbrace U_k\\\\right\\\\rbrace _k$ are all unitary, the normalisation of $h$ passes on to $g^{(k)}$ and on to $f^{(k)(e)}$ so that $\\\\int \\\\left(\\\\int *{f^{(k)(e)}}^2\\\\mu _k\\\\left(\\\\mathrm {d}e\\\\right)\\\\right)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}=1.$ Now suppose for contradiction that $*{\\\\zeta - \\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)}\\\\ge \\\\epsilon $ for $\\\\mu _k$ -a.e.\",\n \"$e\\\\in \\\\mathfrak {E}_k$ , for ${\\\\rm Leb}$ -a.e.\",\n \"$k\\\\in \\\\mathbb {R}^d$ .\",\n \"Then the normalisation of $f^{(k)(e)}$ implies that $\\\\int \\\\left(\\\\int *{\\\\zeta - \\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)}^2*{f^{(k)(e)}}^2\\\\mu _k\\\\left(\\\\mathrm {d}e\\\\right)\\\\right)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\ge \\\\epsilon ^2.$ However this contradicts the condition that $*{\\\\left(\\\\zeta 1- \\\\right)h}_2 < \\\\epsilon $ .\",\n \"Instead, for each $\\\\epsilon >0$ there exists $P_\\\\epsilon \\\\subset \\\\mathbb {R}^d$ with ${\\\\rm Leb}\\\\left(P_\\\\epsilon \\\\right)>0$ such that for all $k\\\\in P_\\\\epsilon $ there exists $Q^{(k)}_\\\\epsilon \\\\subset \\\\mathfrak {E}_k$ with $\\\\mu _k\\\\left(Q^{(k)}_\\\\epsilon \\\\right)>0$ such that for all $e\\\\in Q^{(k)}_\\\\epsilon $ we have $*{\\\\zeta - \\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)}< \\\\epsilon $ .\",\n \"For each $k\\\\in \\\\mathbb {R}^d$ , the operator $U_k\\\\widehat{}(k)U_k^{-1}$ is a multiplication operator on $L^2\\\\left(\\\\mathfrak {E}_k\\\\right)$ with function $e\\\\mapsto \\\\widetilde{\\\\varphi }(k)(e)$ , and therefore it is easy to calculate its spectrum.\",\n \"Since the operator is unchanged by changing $\\\\widetilde{\\\\varphi }(k)(e)$ on a $\\\\mu _k$ -null set, the spectrum is given by the essential image of $\\\\widetilde{\\\\varphi }(k)$ : ${\\\\rm ess.Im}\\\\left(\\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\right) = \\\\left\\\\lbrace z\\\\in \\\\mathbb {C}:\\\\forall \\\\epsilon >0,\\\\mu _k\\\\left(e\\\\in \\\\mathfrak {E}_k:*{\\\\widetilde{\\\\varphi }\\\\left(k\\\\right)\\\\left(e\\\\right)-z}>\\\\epsilon \\\\right)>0\\\\right\\\\rbrace .$ Since the unitary equivalence preserves the spectrum, this gives us an expression for $\\\\sigma \\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ .\",\n \"In particular, we can write our previous conclusion as the following.\",\n \"For all $\\\\epsilon >0$ there exists $P_\\\\epsilon \\\\subset \\\\mathbb {R}^d$ with ${\\\\rm Leb}\\\\left(P_\\\\epsilon \\\\right)>0$ such that for all $k\\\\in P_\\\\epsilon $ there exists $z\\\\in \\\\sigma \\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ such that $*{\\\\zeta -z}<\\\\epsilon $ .\",\n \"Equivalently, this means that $\\\\zeta \\\\in {\\\\rm ess.\",\n \"}\\\\bigcup _{k\\\\in \\\\mathbb {R}^d}\\\\sigma \\\\left(\\\\widehat{}(k)\\\\right)$ .\",\n \"Since $\\\\left\\\\Vert A\\\\right\\\\Vert _{\\\\rm op} = \\\\sup \\\\left\\\\lbrace *{z}:z\\\\in \\\\sigma \\\\left(A\\\\right)\\\\right\\\\rbrace $ and $\\\\mathbb {S}\\\\left(A\\\\right) = \\\\sup \\\\left\\\\lbrace z:z\\\\in \\\\sigma \\\\left(A\\\\right)\\\\subset \\\\mathbb {R}\\\\right\\\\rbrace $ for self-adjoint operator $A$ , the remaining equalities follow from (REF ).\"\n ],\n [\n \"Differentiating the Two-Point Operator\",\n \"The ideas of this section can be found in [21], but we will need extra care to account for the fact that we are dealing with operators and marks and not just functions on $\\\\mathbb {R}^d$ .\",\n \"Recall $\\\\tau _\\\\lambda \\\\left(x,y\\\\right) := \\\\mathbb {P}_\\\\lambda (x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y})$ .\",\n \"Using the notation $x = \\\\left(\\\\overline{x},a\\\\right)\\\\in \\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ , define $\\\\Lambda _n\\\\left(x\\\\right) = \\\\left(\\\\overline{x} + \\\\left[-n,n\\\\right)^d\\\\right)\\\\times \\\\mathcal {E}$ .\",\n \"Then we can define the truncated two-point function $\\\\tau _\\\\lambda ^n\\\\left(x,y\\\\right) := \\\\mathbb {P}_\\\\lambda \\\\left(x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}_{\\\\Lambda _n\\\\left(y\\\\right)}\\\\right).$ We will want to give meaning to the event $\\\\left\\\\lbrace x \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^x_{\\\\Lambda _n\\\\left(y\\\\right)}\\\\right\\\\rbrace $ .\",\n \"To do so we add a “ghost vertex” $g$ in the same way we added deterministic vertices, and add an edge between $v \\\\in \\\\xi _{\\\\Lambda _n\\\\left(y\\\\right)}$ and $g$ with probability $1-\\\\exp \\\\left(-\\\\int _{\\\\Lambda _n^c\\\\left(y\\\\right)} \\\\varphi (u,v) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\right)$ .\",\n \"We now identify $\\\\Lambda _n^c\\\\left(y\\\\right)$ with $g$ .\",\n \"Lemma 3.7 (Differentiability of $\\\\tau _\\\\lambda $ ) Let $x,y\\\\in \\\\mathbb {X}$ and $\\\\varepsilon >0$ be arbitrary.\",\n \"The function $\\\\lambda \\\\mapsto \\\\tau _\\\\lambda ^n(x,y)$ is differentiable on $[0, \\\\lambda _O-\\\\varepsilon ]$ for any $n\\\\in \\\\mathbb {N}$ .\",\n \"Furthermore, $ \\\\tau _\\\\lambda ^n(x,y)$ converges to $\\\\tau _\\\\lambda (x,y)$ uniformly in $\\\\lambda $ and $\\\\tfrac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y)$ converges to a limit uniformly in $\\\\lambda $ .\",\n \"Consequently, $\\\\tau _\\\\lambda (x,y)$ is differentiable w.r.t.\",\n \"$\\\\lambda $ on $[0,\\\\lambda _O)$ and $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y) = \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\tau _\\\\lambda (x,y) =\\\\int \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,y,u}, y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,y}) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ For $S \\\\subseteq \\\\mathbb {X}$ , let $\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y} \\\\textrm { through } S \\\\rbrace $ be the event that $x$ is connected to $b$ in $\\\\xi ^{x,y}$ , and all such paths use a vertex from $S$ .\",\n \"Also, let $\\\\lbrace x \\\\longleftrightarrow S\\\\textrm { in } \\\\xi ^x\\\\rbrace $ be the event that there exists $y \\\\in \\\\eta ^x\\\\cap S$ that is connected to $x$ .\",\n \"The convergence $\\\\tau _\\\\lambda ^n(x,y) \\\\rightarrow \\\\tau _\\\\lambda (x,y)$ is uniform for $\\\\lambda \\\\in [0,\\\\lambda _O-\\\\varepsilon ]$ , as $|\\\\tau _\\\\lambda (x,y) - \\\\tau _\\\\lambda ^n(x,y)| &= \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x} \\\\textrm { through } \\\\Lambda _n^c\\\\left(y\\\\right)) \\\\\\\\& \\\\le \\\\mathbb {P}_{\\\\lambda _O-\\\\varepsilon }(y \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^{y})\\\\xrightarrow{} 0.$ We now go about showing $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y) \\\\xrightarrow{} f_\\\\lambda (x,y) := \\\\int \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x,}, y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) $ uniformly in $\\\\lambda $ .\",\n \"Note that $\\\\lbrace y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}_{\\\\Lambda _n\\\\left(y\\\\right)}\\\\rbrace $ lives on the bounded set $\\\\Lambda _n\\\\left(y\\\\right)$ , and so the the Margulis-Russo formula gives us differentiability and the following explicit expression for the derivative $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y) = \\\\int _{\\\\Lambda _n\\\\left(y\\\\right)} \\\\mathbb {P}_\\\\lambda \\\\left( y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x}_{\\\\Lambda _n\\\\left(y\\\\right)}, y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}_{\\\\Lambda _n\\\\left(y\\\\right)} \\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right).\",\n \"$ Using this we can write $\\\\left|\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y) - f_\\\\lambda (x,y)\\\\right|&= \\\\left|\\\\int _{\\\\Lambda _n} \\\\Big ( \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x}) -\\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}) \\\\right.\\\\nonumber \\\\\\\\& \\\\qquad \\\\qquad -\\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x}_{\\\\Lambda _n\\\\left(y\\\\right)}) +\\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}_{\\\\Lambda _n\\\\left(y\\\\right)}) \\\\Big ) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\nonumber \\\\\\\\& \\\\qquad \\\\qquad \\\\qquad \\\\left.+ \\\\int _{\\\\Lambda _n^c\\\\left(y\\\\right)} \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x,}, y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\right|\\\\\\\\& \\\\le \\\\left|\\\\int _{\\\\Lambda _n\\\\left(y\\\\right)} \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x} \\\\textrm { through } \\\\Lambda _n^c\\\\left(y\\\\right))\\\\right.\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad - \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x} \\\\textrm { through } \\\\Lambda _n^c\\\\left(y\\\\right)) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\bigg \\\\vert \\\\nonumber \\\\\\\\& \\\\qquad \\\\qquad \\\\qquad + \\\\int _{\\\\Lambda _n^c\\\\left(y\\\\right)} \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow u\\\\textrm { in } \\\\xi ^{y,u} ) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\\\\\& \\\\le \\\\int _{\\\\Lambda _n\\\\left(y\\\\right)} \\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x} \\\\textrm { through } \\\\Lambda _n^c\\\\left(y\\\\right) \\\\text{ and through } u) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\nonumber \\\\\\\\& \\\\qquad \\\\qquad + \\\\int _{\\\\Lambda _n^c\\\\left(y\\\\right)} \\\\tau _{\\\\lambda _O-\\\\varepsilon }(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) .$ We now have two main observations.\",\n \"First, that the event $\\\\lbrace y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,u,x} \\\\textrm { through } \\\\Lambda _n^c\\\\left(y\\\\right) \\\\text{ and through } u\\\\rbrace $ is contained in $& \\\\left( \\\\lbrace y \\\\longleftrightarrow u\\\\textrm { in } \\\\xi ^{y,u}\\\\rbrace \\\\circ \\\\lbrace u \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^{u}\\\\rbrace \\\\circ \\\\lbrace \\\\Lambda _n^c\\\\left(y\\\\right) \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x}\\\\rbrace \\\\right) \\\\nonumber \\\\\\\\&\\\\qquad \\\\cup \\\\left( \\\\lbrace y \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^{y}\\\\rbrace \\\\circ \\\\lbrace \\\\Lambda _n^c\\\\left(y\\\\right) \\\\longleftrightarrow u\\\\textrm { in } \\\\xi ^{u}\\\\rbrace \\\\circ \\\\lbrace u \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{u,x}\\\\rbrace \\\\right).$ Secondly, from $\\\\lambda <\\\\lambda _O$ we have $\\\\int \\\\tau _\\\\lambda (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right) \\\\le \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} <\\\\infty .$ This means that for $\\\\nu $ -almost every $y\\\\in \\\\mathbb {X}$ we have $\\\\int _{\\\\Lambda _n^c\\\\left(y\\\\right)} \\\\tau _{\\\\lambda _O-\\\\varepsilon }(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\rightarrow 0$ as $n\\\\rightarrow \\\\infty $ .\",\n \"Applying the BK inequality then gives $\\\\left|\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y) - f_\\\\lambda (x,y)\\\\right|& \\\\le \\\\int _{\\\\Lambda _n\\\\left(y\\\\right)} \\\\mathbb {P}_\\\\lambda (u \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^u) \\\\Big [\\\\tau _\\\\lambda (u,y)\\\\mathbb {P}_\\\\lambda (x \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^x) \\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad + \\\\tau _\\\\lambda (x,u)\\\\mathbb {P}_\\\\lambda (y \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^{y}) \\\\Big ] \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) +o(1) \\\\\\\\& \\\\le 2 \\\\max _{z\\\\in \\\\lbrace y,x\\\\rbrace } \\\\mathbb {P}_{\\\\lambda _O-\\\\varepsilon }\\\\big (z \\\\longleftrightarrow \\\\Lambda _n^c\\\\left(y\\\\right)\\\\textrm { in } \\\\xi ^z\\\\big ) \\\\int \\\\tau _{\\\\lambda _O-\\\\varepsilon }(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) + o(1).$ The remaining integral is again finite, and the remaining probability tends to zero uniformly in $\\\\lambda $ .\",\n \"The uniform convergence justifies the exchange of limit and derivative in (REF ).\",\n \"A family of operators $\\\\left\\\\lbrace H_x\\\\right\\\\rbrace _{x\\\\in \\\\mathbb {R}}$ is differentiable at $x$ if there exists an operator $F_x$ such that $\\\\lim _{\\\\varepsilon \\\\rightarrow 0}\\\\left\\\\Vert \\\\frac{1}{\\\\varepsilon }\\\\left(H_{x+\\\\varepsilon }-H_x\\\\right) - F_x\\\\right\\\\Vert _{\\\\rm op} = 0.$ Using the notation $x = \\\\left(\\\\overline{x},a\\\\right)\\\\in \\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ , recall that we require that the model has $\\\\mathbb {R}^d$ -translation invariance and so we can write $\\\\tau _\\\\lambda \\\\left(x,y\\\\right) = \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{y};a,b\\\\right)$ .\",\n \"Also recall that we defined the Fourier transform $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel $\\\\widehat{\\\\tau }_\\\\lambda \\\\left(k;a,b\\\\right) = \\\\int \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}$ .\",\n \"We now define the displaced Fourier transform $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel $\\\\widehat{\\\\tau }_{\\\\lambda ,k}\\\\left(l;a,b\\\\right) = \\\\int \\\\left(1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\cos \\\\left(l\\\\cdot \\\\overline{x}\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}$ .\",\n \"Corollary 3.8 (Differentiability of $\\\\mathcal {T}_\\\\lambda $ ) The operators $\\\\mathcal {T}_\\\\lambda $ , $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)$ , and $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)$ are differentiable w.r.t.\",\n \"$\\\\lambda $ on $\\\\left(0,\\\\lambda _O\\\\right)$ and their derivatives are the bounded linear operators $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda \\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ , $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ , and $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ with kernel functions $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\tau _\\\\lambda \\\\left(x,y\\\\right)$ , $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\tau }_\\\\lambda \\\\left(k;a,b\\\\right)$ , and $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\tau }_{\\\\lambda ,k}\\\\left(l;a,b\\\\right)$ respectively.\",\n \"Specifically, we have the following bounds on the derivatives: $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op}^2,\\\\\\\\\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2,\\\\\\\\\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} &\\\\le 4\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}.\",\n \"$ We use Lemma REF in conjunction with the Leibniz integral rule to differentiate the two-point operator.\",\n \"Fix $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ and let $\\\\varepsilon \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"From the reasoning of (REF ) we know that the subcritical integral $\\\\int \\\\tau _{\\\\lambda _O-\\\\varepsilon }(\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{(\\\\otimes 2)}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right)<\\\\infty $ , and therefore $\\\\int \\\\tau _{\\\\lambda _O-\\\\varepsilon }\\\\left(x,u\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\le \\\\int \\\\tau _{\\\\lambda _O-\\\\varepsilon }\\\\left(x,u\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) < \\\\infty ,$ for $\\\\nu $ -almost every $x\\\\in \\\\mathbb {X}$ .\",\n \"Here we have also used that $\\\\tau _\\\\lambda (x,u)$ is $\\\\left[0,1\\\\right]$ -valued.\",\n \"This implies that $\\\\tau _{\\\\lambda _O-\\\\varepsilon }\\\\left(x,\\\\cdot \\\\right)\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ for $\\\\nu $ -almost every $x\\\\in \\\\mathbb {X}$ .\",\n \"In conjunction with $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ , Hölder's inequality implies that the function $\\\\Theta _{\\\\varepsilon ,x}\\\\colon y\\\\mapsto \\\\tau _{\\\\lambda _O-\\\\varepsilon }\\\\left(x,y\\\\right)*{f\\\\left(y\\\\right)}$ is in $L^1\\\\left(\\\\mathbb {X}\\\\right)$ for $\\\\nu $ -almost every $x\\\\in \\\\mathbb {X}$ .\",\n \"Furthermore, $\\\\tau _\\\\lambda \\\\left(x,y\\\\right)$ is non-decreasing in $\\\\lambda $ and so for all $\\\\lambda <\\\\lambda _O$ there exists $\\\\varepsilon >0$ such that $\\\\Theta _{\\\\varepsilon ,x}$ dominates $y\\\\mapsto \\\\tau _\\\\lambda \\\\left(x,y\\\\right)f\\\\left(y\\\\right)$ .\",\n \"This domination by an $L^1\\\\left(\\\\mathbb {X}\\\\right)$ function allows us to use the Leibniz integral rule to exchange the integral and derivative to get (for $\\\\nu $ -almost every $x\\\\in \\\\mathbb {X}$ ) $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\left(\\\\left[\\\\mathcal {T}_\\\\lambda f\\\\right]\\\\left(x\\\\right)\\\\right) = \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\int \\\\tau _\\\\lambda \\\\left(x,u\\\\right)f\\\\left(u\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) = \\\\int \\\\left(\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\right)f\\\\left(u\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) = \\\\left[\\\\left(\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda \\\\right)f\\\\right]\\\\left(x\\\\right),$ where Lemma REF gives the differentiability of the function $\\\\tau _\\\\lambda \\\\left(x,y\\\\right)$ .\",\n \"We now prove that the operator $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda $ is bounded.\",\n \"From Lemma REF and the BK inequality, we get the bound $0\\\\le \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(x,y) \\\\le \\\\int \\\\mathbb {P}_\\\\lambda \\\\left( \\\\lbrace y \\\\longleftrightarrow u\\\\textrm { in } \\\\xi ^{y,u}_{\\\\Lambda _n\\\\left(y\\\\right)} \\\\rbrace \\\\circ \\\\lbrace u \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{u,x}_{\\\\Lambda _n\\\\left(y\\\\right)} \\\\rbrace \\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\\\\\\\\\le \\\\int \\\\tau _\\\\lambda ^n(u,y)\\\\mathbb {P}_\\\\lambda \\\\left(u \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{u,x}_{\\\\Lambda _n\\\\left(y\\\\right)}\\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\le \\\\int \\\\tau _\\\\lambda \\\\left(u,y\\\\right)\\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right),$ where the last inequality holds because we are removing the $\\\\Lambda _n\\\\left(y\\\\right)$ restriction.\",\n \"Note that the right hand side of this equation is the kernel function of the operator $\\\\mathcal {T}_\\\\lambda ^2$ .\",\n \"Choose $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ , and use $\\\\rm Abs$ to denote the map $f\\\\mapsto *{f}$ .\",\n \"Now the above inequality implies that $*{\\\\left[\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda \\\\right] f}^2_2 & \\\\le \\\\int \\\\left(\\\\int *{\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\tau _\\\\lambda \\\\left(x,y\\\\right)}*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2 \\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\& \\\\le \\\\int \\\\left(\\\\int \\\\int \\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,y\\\\right)*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2 \\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&= \\\\int \\\\left(\\\\left[\\\\mathcal {T}_\\\\lambda ^2 {\\\\rm Abs}{f}\\\\right]\\\\left(x\\\\right)\\\\right)^2 \\\\nu \\\\left(\\\\mathrm {d}x\\\\right) = *{\\\\mathcal {T}_\\\\lambda ^2 {\\\\rm Abs}f}^2_2.$ Since $*{{\\\\rm Abs}f}_2 = *{f}_2$ , this then gives us $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\sup _f\\\\frac{*{\\\\mathcal {T}_\\\\lambda ^2 {\\\\rm Abs}f}_2}{*{f}_2} = \\\\sup _f\\\\frac{*{\\\\mathcal {T}_\\\\lambda ^2 {\\\\rm Abs}f}_2}{*{{\\\\rm Abs}f}_2}\\\\le \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\rm op} = \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op}^2 < \\\\infty .$ Now we address the Fourier transform.\",\n \"Pick $\\\\varepsilon \\\\in (0,\\\\lambda _O-\\\\lambda )$ .\",\n \"Recall $*{\\\\text{e}^{ik\\\\cdot \\\\overline{x}}\\\\tau _\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right)}$ is bounded by $\\\\tau _{\\\\lambda _O-\\\\varepsilon }\\\\left(\\\\overline{x};a,b\\\\right)$ for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ and $a,b\\\\in \\\\mathcal {E}$ .\",\n \"From the reasoning of (REF ), we know that $\\\\tau _{\\\\lambda _O-\\\\varepsilon }\\\\left(\\\\cdot ;a,b\\\\right)$ is Lebesgue integrable for $\\\\mathcal {P}$ -a.e.\",\n \"$a,b\\\\in \\\\mathcal {E}$ .\",\n \"We are then justified in using the Leibniz integral rule to say that for every $b\\\\in \\\\mathcal {E}$ and $\\\\mathcal {P}$ -a.e.\",\n \"$a\\\\in \\\\mathcal {E}$ , $\\\\tfrac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\int \\\\text{e}^{ik\\\\cdot \\\\overline{x}} \\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}= \\\\int \\\\text{e}^{ik\\\\cdot \\\\overline{x}}\\\\tfrac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}$ .\",\n \"Applying Lemma REF as well as (REF ), we derive that for such $a,b$ we have $*{\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\widehat{\\\\tau }_\\\\lambda (k;a,b)} \\\\le \\\\int \\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda } \\\\tau _\\\\lambda ^n(\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}\\\\le \\\\int \\\\tau _\\\\lambda (\\\\overline{x}-\\\\overline{y};a,c) \\\\tau _\\\\lambda (\\\\overline{y};c,b) \\\\,\\\\mathrm {d}\\\\overline{x}\\\\,\\\\mathrm {d}\\\\overline{y}\\\\,\\\\mathcal {P}\\\\left(\\\\mathrm {d}c \\\\right).$ Note that the right hand side of this equation is the kernel function of the operator $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)^2$ .\",\n \"By repeating the argument for $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\mathcal {T}_\\\\lambda $ - with $\\\\mathbb {X}$ replaced by $\\\\mathcal {E}$ - we similarly arrive at $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2.$ For the displaced Fourier transform, note that $0\\\\le 1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\le 2$ , and so the same argument as for the Fourier transform allows us to use the Leibniz integral rule to exchange derivative and integral.\",\n \"We can also improve on the immediate bound of $2\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2$ for the operator norm of the derivative.\",\n \"Let $f\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ , $*{\\\\left[\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right]f}^2_2 &\\\\le \\\\int \\\\left(\\\\int *{\\\\text{e}^{il\\\\cdot \\\\overline{x}}}\\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\tau _\\\\lambda (\\\\overline{x}-\\\\overline{y};a,c) \\\\tau _\\\\lambda (\\\\overline{y};c,b) *{f\\\\left(b\\\\right)}\\\\mathrm {d}\\\\overline{x}\\\\,\\\\mathrm {d}\\\\overline{y}\\\\,\\\\mathcal {P}\\\\left(\\\\mathrm {d}c \\\\right)\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)\\\\right)^2\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\nonumber \\\\\\\\& \\\\le \\\\int \\\\left(\\\\int 2\\\\Big [\\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\left(\\\\overline{x}-\\\\overline{y}\\\\right)\\\\right)\\\\right)\\\\tau _\\\\lambda (\\\\overline{x}-\\\\overline{y};a,c) \\\\tau _\\\\lambda (\\\\overline{y};c,b) \\\\right.\",\n \"\\\\nonumber \\\\\\\\&\\\\qquad + \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{y}\\\\right)\\\\right)\\\\tau _\\\\lambda (\\\\overline{x}-\\\\overline{y};a,c) \\\\tau _\\\\lambda (\\\\overline{y};c,b) \\\\Big ]*{f\\\\left(b\\\\right)}\\\\,\\\\mathrm {d}\\\\overline{x}\\\\,\\\\mathrm {d}\\\\overline{y}\\\\,\\\\mathcal {P}\\\\left(\\\\mathrm {d}c \\\\right)\\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)\\\\Big )^2\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\nonumber \\\\\\\\& = 4 *{\\\\left[\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right)\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) + \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right)\\\\right]{\\\\rm Abs} f}^2_2,$ where we have used the cosine-splitting lemma (Lemma REF ) to divide the displacement factor $\\\\left(1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)$ into one spanning $\\\\overline{x}-\\\\overline{y}$ and one spanning $\\\\overline{y}$ .\",\n \"Using the above approach of (REF ) and the triangle inequality then produces the required bound.\",\n \"This is the cosine-splitting lemma referenced in the previous proof.\",\n \"Lemma 3.9 (Split of cosines, [10]) Let $t_i \\\\in \\\\mathbb {R}$ for $i=1, \\\\ldots , m$ and $t = \\\\sum _{i=1}^{m} t_i$ .\",\n \"Then $1-\\\\cos (t) \\\\le m \\\\sum _{i=1}^{m} [1- \\\\cos (t_i)].\",\n \"$ We use the following lemma in the proof of Proposition REF to bound the displacement on the two-point operator using the displacement of the connection operator when working in the sub-critical regime away from the critical threshold.\",\n \"Lemma 3.10 For $\\\\lambda <\\\\lambda _O$ , $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\text{e}^{4\\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}}\\\\left\\\\Vert \\\\widehat{}\\\\left(0\\\\right) - \\\\widehat{}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}.$ For the inequality removing $l$ , let $f\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ .\",\n \"Then $*{\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l+k\\\\right)\\\\right)f}^2_2 &\\\\le \\\\int \\\\left(\\\\int *{\\\\text{e}^{il\\\\cdot \\\\overline{x}}}\\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\tau _\\\\lambda (\\\\overline{x};a,b) *{f\\\\left(b\\\\right)}\\\\,\\\\mathrm {d}\\\\overline{x}\\\\, \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)\\\\right)^2\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\nonumber \\\\\\\\& = *{\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right){\\\\rm Abs} f}^2_2.$ Since $*{{\\\\rm Abs}f}_2=*{f}_2$ , we can use the approach of (REF ) to get the first inequality.\",\n \"For the second inequality, we first note that $\\\\widehat{\\\\mathcal {T}}_0\\\\left(k\\\\right) = \\\\widehat{}\\\\left(k\\\\right)$ .\",\n \"We then use the differential inequality () to extend to $\\\\lambda <\\\\lambda _O$ .\",\n \"Using the reverse triangle inequality, we have $\\\\limsup _{\\\\varepsilon \\\\downarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{s+\\\\varepsilon }\\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_{s+\\\\varepsilon }\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}} &\\\\le \\\\left\\\\Vert \\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\left(\\\\widehat{\\\\mathcal {T}}_{\\\\lambda }\\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_{\\\\lambda }\\\\left(k\\\\right)\\\\right)\\\\right|_{\\\\lambda =s}\\\\right\\\\Vert _{\\\\rm op} \\\\nonumber \\\\\\\\&\\\\le 4 \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}.$ Since $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ is monotone increasing for $s \\\\in \\\\left[0,\\\\lambda \\\\right]$ , the right hand side of the inequality is bounded by $4 \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda }\\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_{s}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ , giving the exponential factor in the result.\"\n ],\n [\n \"The Expansion\",\n \"There are two parts to this section.\",\n \"First we perform an expansion of the two-point function.\",\n \"This argument is essentially identical to the corresponding step in [21], so we refer to this reference for most of the details.\",\n \"The novelty in this section is rather in the formulation of this expansion in terms of operators, contained in the second part.\"\n ],\n [\n \"Function-level expansion\",\n \"The expansion of the two-point function $\\\\tau _\\\\lambda $ proceeds in essentially the same way as in [21].\",\n \"The similarity holds because the argument uses quite general properties of Poisson point processes and connection models - the change from $\\\\mathbb {R}^d$ to $\\\\mathbb {X}=\\\\mathbb {R}^d\\\\times \\\\mathcal {E}$ adds no further complication.\",\n \"We give here a very brief overview of the derivation - the details can be found in that reference.\",\n \"We first introduce some notation.\",\n \"It will be important to consider thinning events and pivotal points.\",\n \"Definition 4.1 Let $x,y \\\\in \\\\mathbb {X}$ , and let $A \\\\subset \\\\mathbb {X}$ be locally finite and of cardinality ${A}$ .\",\n \"(1) Set $\\\\bar{\\\\varphi }(A,x) :=\\\\prod _{y\\\\in A}(1-\\\\varphi (x,y)) $ and define $\\\\eta _{\\\\langle A \\\\rangle }$ as a $\\\\bar{\\\\varphi }(A, \\\\cdot )$ -thinning of $\\\\eta $ (or simply $A$ -thinning of $\\\\eta $) as follows.\",\n \"We keep a point $w \\\\in \\\\eta $ as a point of $\\\\eta _{\\\\langle A \\\\rangle }$ with probability $\\\\bar{\\\\varphi }(A,w)$ independently of all other points of $\\\\eta $ .\",\n \"We similarly define $\\\\eta _{\\\\langle A \\\\rangle }^x$ as a $\\\\bar{\\\\varphi }(A,\\\\cdot )$ -thinning of $\\\\eta ^x$ using the marks in $\\\\xi ^x$ .\",\n \"(2) We write $x {\\\\,\\\\,A\\\\,\\\\,} y\\\\textrm { in } \\\\xi $ if both $\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi \\\\rbrace $ and $\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi [\\\\eta _{\\\\langle A \\\\rangle } \\\\cup \\\\lbrace x\\\\rbrace ] \\\\rbrace $ take place.\",\n \"In words, $\\\\lbrace x {\\\\,\\\\,A\\\\,\\\\,} y\\\\textrm { in } \\\\xi \\\\rbrace $ is the event that $x,y\\\\in \\\\eta $ and $x$ is connected to $y$ in $\\\\xi $ , but that this connection is broken by an $A$ -thinning of $\\\\eta \\\\setminus \\\\lbrace x\\\\rbrace $ .\",\n \"In particular, the connection does not survive if $y$ is thinned out.\",\n \"(3) We define $\\\\tau _\\\\lambda ^{A}(x,y) = \\\\mathbb {P}_\\\\lambda \\\\left(y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,y}[\\\\eta ^y_{\\\\langle A \\\\rangle }\\\\cup \\\\lbrace x\\\\rbrace ] \\\\right).\",\n \"$ In words, $\\\\tau _\\\\lambda ^A(x,y)$ is the probability of the event that there exists an open path between $x$ and $y$ in an RCM driven by an $A$ -thinning of $\\\\eta ^y$ , where the point $x$ is fixed to be present (but $y$ is not).\",\n \"(4) Given $x,y\\\\in \\\\mathbb {X}$ and edge-marking $\\\\xi $ , we say $u\\\\in \\\\mathbb {X}$ is pivotal and $u\\\\in \\\\textsf {Piv}(y,x,\\\\xi )$ if $\\\\left\\\\lbrace y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,y}\\\\right\\\\rbrace $ and yet $\\\\left\\\\lbrace y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi \\\\left[\\\\eta \\\\setminus \\\\left\\\\lbrace u\\\\right\\\\rbrace \\\\right]\\\\right\\\\rbrace $ .\",\n \"That is, every path on $\\\\xi ^{x,y}$ connecting $x$ and $y$ uses the vertex $u$ .\",\n \"Note that the end points $x$ and $y$ are never pivotal.\",\n \"(5) We will also use the set $E(x,y;A,\\\\xi ) := \\\\lbrace x {\\\\,\\\\,A\\\\,\\\\,} y\\\\textrm { in } \\\\xi \\\\rbrace \\\\cap \\\\lbrace \\\\nexists w \\\\in \\\\textsf {Piv}(x,y;\\\\xi ): x {\\\\,\\\\,A\\\\,\\\\,} w\\\\textrm { in } \\\\xi \\\\rbrace , $ for a locally finite set $A \\\\subset \\\\mathbb {X}$ , and $x,y\\\\in \\\\mathbb {R}^d$ .\",\n \"If we consider the pivotal points of the $x$ to $y$ connection in $\\\\xi $ in sequence, then this is the event that an $A$ -thinning breaks the connection after the last pivotal point, but not before.\",\n \"Now we can state a continuum version of a standard lemma often used in discrete models - see, for example, [25].\",\n \"In bond percolation, it has the name “Cutting-bond lemma”.\",\n \"The proof of this continuum version can be found in [21].\",\n \"Lemma 4.2 (Cutting-point lemma) Let $\\\\lambda \\\\ge 0$ and let $v,u,x \\\\in \\\\mathbb {X}$ with $u \\\\ne x$ and let $A\\\\subset \\\\mathbb {X}$ be locally finite.\",\n \"Then $\\\\mathbb {E}_\\\\lambda \\\\big [1_{E(v,u;A,\\\\xi ^{v,u,x})} 1_{\\\\lbrace u \\\\in \\\\textsf {\\\\textup {Piv}}(v,x;\\\\xi ^{v,u,x})\\\\rbrace } \\\\big ]= \\\\mathbb {E}_\\\\lambda \\\\left[1_{E(v,u;A,\\\\xi ^{v,u})} \\\\cdot \\\\tau _\\\\lambda ^{v,\\\\xi ^v) }(x,u) \\\\right].$ Moreover, $\\\\mathbb {P}_\\\\lambda \\\\left(y \\\\Longleftrightarrow u\\\\textrm { in } \\\\xi ^{y,u,x}, u \\\\in \\\\textsf {\\\\textup {Piv}}(y,x; \\\\xi ^{y,u,x})\\\\right) = \\\\mathbb {E}_\\\\lambda \\\\left[1_{\\\\lbrace y \\\\Longleftrightarrow u\\\\textrm { in } \\\\xi ^{y,u}\\\\rbrace } \\\\cdot \\\\tau _\\\\lambda ^{y,\\\\xi ^{y}) }(x,u) \\\\right].$ We now define the lace expansion coefficient function.\",\n \"Definition 4.3 (Lace-expansion function coefficients) For $n\\\\in \\\\mathbb {N}$ and $x,y\\\\in \\\\mathbb {X}$ , we define $\\\\pi _\\\\lambda ^{(0)}(x,y) &:= \\\\mathbb {P}_\\\\lambda \\\\left(y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}\\\\right) - \\\\varphi (x,y), \\\\\\\\\\\\pi _\\\\lambda ^{(n)}(x,y) &:= \\\\lambda ^n \\\\int \\\\mathbb {P}_\\\\lambda \\\\left( \\\\left\\\\lbrace y \\\\Longleftrightarrow u_0\\\\textrm { in } \\\\xi ^{y, u_0}_{0}\\\\right\\\\rbrace \\\\cap \\\\bigcap _{i=1}^{n} E\\\\left(u_{i-1},u_i; {i-1}, \\\\xi ^{u_{i-1}, u_i}_{i}\\\\right) \\\\right) \\\\nu ^{\\\\otimes n}\\\\left(\\\\mathrm {d}\\\\vec{u}_{[0,n-1]}\\\\right) , $ where $u_n=x$ and ${i} = u_{i-1}, \\\\xi ^{u_{i-1}}_{i})$ is the cluster of $u_{i-1}$ in $\\\\xi ^{u_{i-1}}_i$ .\",\n \"Further define the remainder functions $r_{\\\\lambda , 0} (x,y) &:= - \\\\lambda \\\\int \\\\mathbb {P}_\\\\lambda \\\\left( \\\\left\\\\lbrace y \\\\Longleftrightarrow u_0\\\\textrm { in } \\\\xi ^{y, u_0}_0\\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace u_0 {\\\\,\\\\,0\\\\,\\\\,} x\\\\textrm { in } \\\\xi ^{u_0,x}_1\\\\right\\\\rbrace \\\\right) \\\\nu \\\\left(\\\\mathrm {d}u_0\\\\right), \\\\\\\\r_{\\\\lambda , n}(x,y) &:= (-\\\\lambda )^{n+1} \\\\int \\\\mathbb {P}_\\\\lambda \\\\left( \\\\left\\\\lbrace y \\\\Longleftrightarrow u_0\\\\textrm { in } \\\\xi ^{y, u_0}_0\\\\right\\\\rbrace \\\\cap \\\\bigcap _{i=1}^{n} E\\\\left(u_{i-1},u_i; {i-1}, \\\\xi ^{u_{i-1}, u_i}_{i}\\\\right)\\\\right.\",\n \"\\\\nonumber \\\\\\\\& \\\\hspace{170.71652pt}\\\\left.\\\\cap \\\\left\\\\lbrace u_n {\\\\,\\\\,n\\\\,\\\\,} x\\\\textrm { in } \\\\xi ^{u_n,x}_{n+1} \\\\right\\\\rbrace \\\\right) \\\\nu ^{\\\\otimes \\\\left(n+1\\\\right)}\\\\left(\\\\mathrm {d}\\\\vec{u}_{[0,n]}\\\\right).$ Additionally, define $\\\\pi _{\\\\lambda , n}$ as the alternating partial sum $\\\\pi _{\\\\lambda , n}(x,y) := \\\\sum _{m=0}^{n} (-1)^m \\\\pi _\\\\lambda ^{(m)}(x,y).\",\n \"$ We will also be requiring Fourier transforms of these functions.\",\n \"Since the whole model is spatially translation invariant, so are the functions defined above.\",\n \"Therefore for each pair of marks $(a,b)$ we perform the Fourier transform on the spatial displacement to get the functions $\\\\widehat{\\\\pi }^{(n)}_\\\\lambda (k;a,b)$ , $\\\\widehat{\\\\pi }_{\\\\lambda ,n}(k;a,b)$ , and $\\\\widehat{r}_{\\\\lambda ,n}(k;a,b)$ for $n\\\\ge 0$ .\",\n \"Proposition 4.4 (Lace expansion: function-level) Let $\\\\lambda \\\\in [0,\\\\lambda _O)$ and $n\\\\ge 0$ .\",\n \"Then for $x,y\\\\in \\\\mathbb {X}$ , and for $k\\\\in \\\\mathbb {R}^d$ and $a,b\\\\in \\\\mathcal {E}$ , $\\\\tau _\\\\lambda \\\\left(x,y\\\\right) &= \\\\varphi \\\\left(x,y\\\\right) + \\\\pi _{\\\\lambda , n}\\\\left(x,y\\\\right) + \\\\lambda \\\\int \\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\left(\\\\varphi + \\\\pi _{\\\\lambda , n}\\\\right)\\\\left(u,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) + r_{\\\\lambda , n}\\\\left(x,y\\\\right), \\\\\\\\\\\\widehat{\\\\tau }_\\\\lambda \\\\left(k;a,b\\\\right) &= \\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) + \\\\widehat{\\\\pi }_{\\\\lambda , n}\\\\left(k;a,b\\\\right) + \\\\lambda \\\\int \\\\widehat{\\\\tau }_\\\\lambda \\\\left(k;a,c\\\\right)\\\\left(\\\\widehat{\\\\varphi }+ \\\\widehat{\\\\pi }_{\\\\lambda , n}\\\\right)\\\\left(k;c,b\\\\right)\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right) + \\\\widehat{r}_{\\\\lambda , n}\\\\left(k;a,b\\\\right).\",\n \"$ We outline the proof of (REF ) here.\",\n \"The equation () then follows by applying the Fourier transform.\",\n \"The details of the proof of (REF ) - with appropriate contextual changes - can be found in [21].\",\n \"Firstly, the definitions give $\\\\tau _\\\\lambda (x,y) = \\\\varphi (x,y) + \\\\pi _\\\\lambda ^{(0)}(x,y) + \\\\mathbb {P}_\\\\lambda \\\\left(y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}, y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}\\\\right).\",\n \"$ With an appropriate event partition, Mecke's equation and the Cutting-point lemma allow us to re-write the probability term as $\\\\mathbb {P}_\\\\lambda \\\\left(y \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}, y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}\\\\right) = \\\\lambda \\\\int \\\\mathbb {E}_\\\\lambda \\\\left[ 1_{\\\\lbrace y \\\\Longleftrightarrow u\\\\textrm { in } \\\\xi ^{y,u}\\\\rbrace } \\\\tau _\\\\lambda ^{0}(x,u) \\\\right] \\\\mathrm {d}u$ Now we use the identity $\\\\tau _\\\\lambda ^{A}(x,u) = \\\\tau _\\\\lambda (x,u) - \\\\mathbb {P}_\\\\lambda \\\\left( u {\\\\,\\\\,A\\\\,\\\\,} x\\\\textrm { in } \\\\xi ^{u,x} \\\\right), $ to extract the deterministic function $\\\\tau _\\\\lambda (x,u)$ from the expectation at the cost of an extra term.\",\n \"The $\\\\tau _\\\\lambda $ term then gives us the second $\\\\pi ^{(0)}_\\\\lambda $ term and the extra term becomes the remainder $r_{\\\\lambda ,0}$ .\",\n \"Repeated use of (REF ) with Mecke's equation and the Cutting-point lemma gives an alternating series that produces the various $\\\\pi _\\\\lambda ^{(n)}$ and $r_{\\\\lambda ,n}$ terms.\",\n \"Operator-level expansion Definition 4.5 (Lace-expansion operator coefficients) For $n\\\\in \\\\mathbb {N}$ , we define $_\\\\lambda ^{(n)}\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ as the linear operator with kernel function $\\\\pi ^{(n)}_\\\\lambda $ , and $R_{\\\\lambda ,n}\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ as the linear operator with kernel function $r_{\\\\lambda ,n}$ .\",\n \"We also define $_{\\\\lambda ,n}\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ as the linear operator with kernel function $\\\\pi _{\\\\lambda , n}$ .\",\n \"It follows that $_{\\\\lambda , n} = \\\\sum _{m=0}^{n} (-1)^m _\\\\lambda ^{(m)}.$ We also use the kernel functions $\\\\widehat{\\\\pi }^{(n)}_\\\\lambda (k;a,b)$ , $\\\\widehat{\\\\pi }_{\\\\lambda ,n}(k;a,b)$ , and $\\\\widehat{r}_{\\\\lambda ,n}(k;a,b)$ to define the operators $\\\\widehat{}^{(n)}_{\\\\lambda }(k),\\\\widehat{}_{\\\\lambda ,n}(k),\\\\widehat{R}_{\\\\lambda ,n}(k)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ respectively for each $k\\\\in \\\\mathbb {R}^d$ .\",\n \"Proposition 4.6 (Lace expansion: operator-level) Let $\\\\lambda \\\\in [0,\\\\lambda _O)$ .\",\n \"Then for $n \\\\ge 0$ and $k\\\\in \\\\mathbb {R}^d$ , $\\\\mathcal {T}_\\\\lambda &= + _{\\\\lambda , n} + \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left(+ _{\\\\lambda , n}\\\\right) + R_{\\\\lambda , n},\\\\\\\\\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) &= \\\\left(\\\\widehat{}+ \\\\widehat{}_{\\\\lambda , n}\\\\right)(k) + \\\\lambda \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\left(\\\\widehat{}+ \\\\widehat{}_{\\\\lambda , n}\\\\right)(k) + \\\\widehat{R}_{\\\\lambda , n}(k).$ Let $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ be a test function.\",\n \"Then Proposition REF gives for $x\\\\in \\\\mathbb {X}$ , $\\\\mathcal {T}_\\\\lambda f\\\\left(x\\\\right) &= \\\\int \\\\tau _\\\\lambda \\\\left(x,y\\\\right)f\\\\left(y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\nonumber \\\\\\\\& = \\\\int \\\\left(\\\\varphi \\\\left(x,y\\\\right) + \\\\pi _{\\\\lambda , n}\\\\left(x,y\\\\right) + \\\\lambda \\\\int \\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\left(\\\\varphi + \\\\pi _{\\\\lambda , n}\\\\right)\\\\left(u,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) + r_{\\\\lambda , n}\\\\left(x,y\\\\right)\\\\right)f\\\\left(y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\nonumber \\\\\\\\& = \\\\left(+ _{\\\\lambda , n} + \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left(+ _{\\\\lambda , n}\\\\right) + R_{\\\\lambda , n}\\\\right)f\\\\left(x\\\\right).$ Note that in this last equality we used $\\\\int \\\\int \\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\left(\\\\varphi + \\\\pi _{\\\\lambda , n}\\\\right)\\\\left(u,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right)f\\\\left(y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right) &= \\\\int \\\\tau _\\\\lambda \\\\left(x,u\\\\right)\\\\int \\\\left(\\\\varphi + \\\\pi _{\\\\lambda , n}\\\\right)\\\\left(u,y\\\\right)f\\\\left(y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\nonumber \\\\\\\\& = \\\\mathcal {T}_\\\\lambda \\\\left(+ _{\\\\lambda , n}\\\\right)f\\\\left(x\\\\right).$ This exchange of integrals is valid by Fubini's theorem because $\\\\lambda <\\\\lambda _O$ ensures the integrals are finite.\",\n \"The same argument applies for the Fourier transformed operators to complete the proof.\",\n \"Diagrammatic bounds For our argument, we will require bounds on $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op}$ and $\\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(k)\\\\right\\\\Vert _{\\\\rm op}$ for all $n\\\\in \\\\mathbb {N}$ and $k\\\\in \\\\mathbb {R}^d$ .\",\n \"However, it turns out that we can bound the last two remainder terms using others.\",\n \"Lemma 5.1 For all $n\\\\in \\\\mathbb {N}$ , $k\\\\in \\\\mathbb {R}^d$ , and $\\\\lambda >0$ , $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op},\\\\\\\\\\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(k)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(0)\\\\right\\\\Vert _{\\\\rm op}.$ As $\\\\mathbb {P}_\\\\lambda (u_n {\\\\,\\\\,A\\\\,\\\\,} x\\\\textrm { in } \\\\xi ^{u_n,x}_{n+1} ) \\\\le \\\\tau _\\\\lambda (x,u_n)$ for an arbitrary locally finite set $A$ , the definition () gives $*{r_{\\\\lambda ,n}(x)} \\\\le \\\\int \\\\tau _\\\\lambda (x,u)\\\\pi _\\\\lambda ^{(n)}(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ Let $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ .\",\n \"Then this gives $*{R_{\\\\lambda ,n}f}^2_2 &\\\\le \\\\int \\\\left(\\\\int *{r_{\\\\lambda ,n}(x,y)}*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&\\\\le \\\\lambda ^2\\\\int \\\\left(\\\\int \\\\left(\\\\int \\\\tau _\\\\lambda (x,u)\\\\pi _\\\\lambda ^{(n)}(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\right)*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&= \\\\lambda ^2\\\\int \\\\left(\\\\int \\\\tau _\\\\lambda (x,u)\\\\left(\\\\int \\\\pi _\\\\lambda ^{(n)}(u,y)*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\& = \\\\lambda ^2 *{\\\\mathcal {T}_\\\\lambda _\\\\lambda ^{(n)}{\\\\rm Abs}f}^2_2.$ Note that we were able to use Tonelli's theorem to swap the $u$ and $y$ integrals because $\\\\tau _\\\\lambda $ and $\\\\pi ^{(n)}_\\\\lambda $ are both non-negative and measurable.\",\n \"Finally, the approach of (REF ) implies the result.\",\n \"For each $n\\\\ge 0$ , either $r_{\\\\lambda ,n}(x)\\\\ge 0$ for all $x\\\\in \\\\mathbb {X}$ or $r_{\\\\lambda ,n}(x)\\\\le 0$ for all $x\\\\in \\\\mathbb {X}$ .\",\n \"Therefore $*{\\\\widehat{r}_{\\\\lambda ,n}(k;a,b)}\\\\le *{\\\\widehat{r}_{\\\\lambda ,n}(0;a,b)}$ for all $a,b\\\\in \\\\mathcal {E}$ and $\\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(0)\\\\right\\\\Vert _{\\\\rm op}$ (by Lemma REF ) for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"From (REF ) we have $*{\\\\widehat{r}_{\\\\lambda ,n}(0;a,b)} \\\\le \\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,c)\\\\widehat{\\\\pi }_\\\\lambda ^{(n)}(0;c,b) \\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right).$ Therefore the same argument as above gives the second bound in the result.\",\n \"Because the $n=0$ cases for our bounds on $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}$ , and $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ are so simple, we give the proof here before moving onto $n\\\\ge 1$ .\",\n \"Recall $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right) = \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k+l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right)$ .\",\n \"We will find it convenient to also define $\\\\widehat{}_k\\\\left(l\\\\right) = \\\\widehat{}\\\\left(k+l\\\\right) - \\\\widehat{}\\\\left(l\\\\right)$ .\",\n \"Proposition 5.2 (Bounds for $n=0$ ) Let $\\\\lambda \\\\in [0,\\\\lambda _O)$ .\",\n \"Then $\\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ For $k\\\\in \\\\mathbb {R}^d$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(k)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda ^2\\\\left(\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty } + \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\right).$ Our first comment relates to these bounds and every bound we perform hereafter in Section .\",\n \"From the positivity of $\\\\pi ^{(0)}_{\\\\lambda }(x,y)$ and Lemma REF we have $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{\\\\rm op}$ , and by Lemma REF we have $\\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{1,\\\\infty }$ and $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"However from the definitions of the norm we also have $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{1,\\\\infty }$ , and so we will be interested in bounding this last norm.\",\n \"Now note that for the event $\\\\lbrace y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x} \\\\rbrace $ to hold, either there is a direct edge between $y$ and $x$ , or there exist vertices $w,z$ in $\\\\eta $ that are direct neighbours of the origin and have respective disjoint paths to $x$ that both do not contain $y$ .\",\n \"Hence, by the Mecke equation (REF ), $\\\\mathbb {P}_\\\\lambda ( y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{\\\\mathbf {0},x}) &\\\\le \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\mathbb {E}_\\\\lambda \\\\Big [ \\\\sum _{(w,z) \\\\in \\\\eta ^{(2)}} 1_{(\\\\lbrace y\\\\sim w \\\\text{ in } \\\\xi ^{y}\\\\rbrace \\\\cap \\\\lbrace w \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^x\\\\rbrace ) \\\\circ (\\\\lbrace y \\\\sim z \\\\text{ in } \\\\xi ^{y}\\\\rbrace \\\\cap \\\\lbrace z \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^x \\\\rbrace )} \\\\Big ] \\\\nonumber \\\\\\\\& = \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\lambda ^2 \\\\int \\\\mathbb {P}_\\\\lambda \\\\big ( (\\\\lbrace y\\\\sim y \\\\text{ in } \\\\xi ^{y, w}\\\\rbrace \\\\cap \\\\lbrace w \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,w}\\\\rbrace )\\\\nonumber \\\\\\\\& \\\\hspace{99.58464pt} \\\\circ (\\\\lbrace y \\\\sim z \\\\text{ in } \\\\xi ^{y, z}\\\\rbrace \\\\cap \\\\lbrace z \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,z} \\\\rbrace ) \\\\big ) \\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}w,\\\\mathrm {d}z\\\\right).$ After applying the BK inequality to the above probability, $\\\\mathbb {P}_\\\\lambda \\\\left( y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}\\\\right) \\\\le \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\lambda ^2 \\\\left( \\\\int \\\\mathbb {P}_\\\\lambda \\\\left( \\\\left\\\\lbrace y\\\\sim w \\\\text{ in } \\\\xi ^{y,w}\\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace w \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{w,x}\\\\right\\\\rbrace \\\\right) \\\\nu \\\\left(\\\\mathrm {d}w\\\\right) \\\\right)^2 \\\\\\\\= \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\lambda ^2 \\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}w\\\\right)\\\\right)^2.\",\n \"$ Thus, recalling that $\\\\pi _\\\\lambda ^{(0)}(x,y) = \\\\mathbb {P}_\\\\lambda (y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x})-\\\\varphi (x,y) \\\\ge 0$ , and using the symmetry of $\\\\varphi $ and $\\\\tau _\\\\lambda $ , $\\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{1,\\\\infty } &= \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\pi ^{(0)}_\\\\lambda \\\\left(x,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\nonumber \\\\\\\\& \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}w\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\nonumber \\\\\\\\& = \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\varphi \\\\left(y,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,x\\\\right)\\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu ^{\\\\otimes 3}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}w,\\\\mathrm {d}u\\\\right)\\\\nonumber \\\\\\\\&\\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ We then use a supremum bound on this integral to split one of the $$ off from the others, and bound the other from above with $\\\\mathcal {T}_\\\\lambda $ to get the first two bounds of the result.\",\n \"For the last bound of Proposition REF , we apply (REF ) and obtain $&\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(k\\\\right)\\\\right\\\\Vert _{1,\\\\infty }\\\\nonumber \\\\\\\\&\\\\qquad = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\pi ^{(0)}_\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right) \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right)\\\\right)^2 \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad = \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\varphi \\\\left(-\\\\overline{u}^\\\\prime ;b,c^\\\\prime \\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}^\\\\prime -\\\\overline{x};c^\\\\prime ,a\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\nonumber \\\\\\\\&\\\\qquad \\\\hspace{284.52756pt}\\\\times \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^\\\\prime \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^\\\\prime ,\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{b,b^\\\\prime \\\\in \\\\mathcal {E}}\\\\int \\\\Big [\\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\left(\\\\overline{x}-\\\\overline{u}\\\\right)\\\\right)\\\\right)\\\\varphi \\\\left(-\\\\overline{u}^\\\\prime ;b^\\\\prime ,c^\\\\prime \\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}^\\\\prime -\\\\overline{x};c^\\\\prime ,a\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\nonumber \\\\\\\\&\\\\qquad \\\\hspace{85.35826pt}+ \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{u}\\\\right)\\\\right)\\\\varphi \\\\left(-\\\\overline{u}^\\\\prime ;b,c^\\\\prime \\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}^\\\\prime -\\\\overline{x};c^\\\\prime ,a\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\Big ] \\\\nonumber \\\\\\\\&\\\\qquad \\\\hspace{284.52756pt}\\\\times \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^\\\\prime \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^\\\\prime ,\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}+ \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda _k\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Here we have again used the symmetry of $\\\\varphi $ and $\\\\tau _\\\\lambda $ , and also used Lemma REF to split the cosine factor over $\\\\overline{x}-\\\\overline{u}$ and $\\\\overline{u}$ .\",\n \"We can consider each term individually by using the triangle inequality.\",\n \"By using a supremum bound, the symmetry of $\\\\varphi $ , and by bounding $\\\\varphi $ with $\\\\tau _\\\\lambda $ , we get the bound $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda _k\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }.$ Bounds on expansion coefficients In this section we aim to bound $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}$ and $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Lemmas REF and REF along with the positivity of $\\\\pi ^{(n)}_\\\\lambda (x,y)$ prove that it is sufficient to bound $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty }$ to get bounds on both of these.\",\n \"Much of the hard work for this has been done previously - for example in [21].\",\n \"The novelty here is to see how the argument can be naturally written in terms of operators.\",\n \"We inherit the following notation from [21].\",\n \"Note that we make use of the Dirac delta function - specifically the one that holds with respect to the measure $\\\\nu $ .\",\n \"They are used here primarily for convenience and to make the argument more readable.\",\n \"In particular, they appear when applying the Mecke equation (REF ) to obtain $\\\\mathbb {E}\\\\left[ \\\\sum _{y\\\\in \\\\eta ^u} f(y, \\\\xi ^u) \\\\right] = \\\\int \\\\left(\\\\lambda +\\\\delta _{y,u}\\\\right) \\\\mathbb {E}_\\\\lambda \\\\left[ f\\\\left(y,\\\\xi ^{u,y}\\\\right)\\\\right] \\\\nu \\\\left(\\\\mathrm {d}y\\\\right),$ and so the factor $\\\\lambda +\\\\delta _{y,u}$ encodes a case distinction of whether point $y$ coincides with $u$ or not.\",\n \"Definition 5.3 (The $\\\\psi $ functions) Let $r,s,u,w,x,y\\\\in \\\\mathbb {X}$ .\",\n \"We first set $\\\\tau _\\\\lambda ^\\\\circ (x,y) := \\\\lambda ^{-1}\\\\delta _{x,y} + \\\\tau _\\\\lambda (x,y)$ .\",\n \"Also define $\\\\psi _0^{(1)}(w,u,y) &:= \\\\lambda ^2\\\\tau _\\\\lambda (y,u)\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,y),\\\\\\\\\\\\psi _0^{(2)}(w,u,y) &:=\\\\lambda ^2 \\\\delta _{w,y} \\\\tau _\\\\lambda (y,u)\\\\int \\\\tau _\\\\lambda (u,t)\\\\tau _\\\\lambda (t,y) \\\\nu \\\\left(\\\\mathrm {d}t\\\\right),\\\\\\\\\\\\psi _0^{(3)}(w,u,y) &:= \\\\lambda ^2\\\\varphi (u,y) \\\\left(\\\\lambda ^{-1}\\\\delta _{w,y}\\\\right),\\\\\\\\\\\\psi ^{(1)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda (w,u)\\\\int \\\\tau _\\\\lambda ^\\\\circ (t,s) \\\\tau _\\\\lambda (t,w)\\\\tau _\\\\lambda (u,z)\\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\psi ^{(2)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda ^\\\\circ (w,s)\\\\int \\\\tau _\\\\lambda (t,z)\\\\tau _\\\\lambda (z,u)\\\\tau _\\\\lambda (u,t)\\\\tau _\\\\lambda ^\\\\circ (t,w)\\\\tau _\\\\lambda (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\psi ^{(3)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,s)\\\\tau _\\\\lambda (u,r),\\\\\\\\\\\\psi ^{(4)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\left(\\\\lambda ^{-1}\\\\delta _{w,s}\\\\right)\\\\tau _\\\\lambda (u,r),\\\\\\\\\\\\psi _n^{(1)} (x,r,s) &:= \\\\lambda ^3\\\\int \\\\tau _\\\\lambda ^\\\\circ (t,s)\\\\tau _\\\\lambda (z,r)\\\\tau _\\\\lambda (t,z)\\\\tau _\\\\lambda (z,x)\\\\tau _\\\\lambda (x,t)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right),\\\\\\\\\\\\psi _n^{(2)}(x,r,s) &:=\\\\lambda \\\\tau _\\\\lambda (x,s)\\\\tau _\\\\lambda (x,r),$ and set $\\\\psi _0 := \\\\psi _0^{(1)}+\\\\psi _0^{(2)}+\\\\psi _0^{(3)}$ , $\\\\psi _n := \\\\psi _n^{(1)} + \\\\psi _n^{(2)}$ , and $\\\\psi := \\\\psi ^{(1)}+\\\\psi ^{(2)}+\\\\psi ^{(3)} + \\\\psi ^{(4)}$ .\",\n \"Note that by using $\\\\lambda ^{-1}\\\\delta _{\\\\left(\\\\cdot ,\\\\cdot \\\\right)}$ in $\\\\tau _\\\\lambda ^\\\\circ $ and in $\\\\psi ^{(3)}_0$ , we ensure that each integral that will be truly performed (that is, not an integration over a Dirac delta function accounting for a coincidence of points) has a $\\\\lambda $ factor associated with it when the bound $\\\\pi ^{(n)}_\\\\lambda $ is calculated.\",\n \"Diagrammatic representations of these functions can be found in Figure REF .\",\n \"In these diagrams, the declared variables are represented by (0,0) circle (2pt) vertices whereas the variables that are integrated over are represented by (0,0) circle (2pt) vertices.\",\n \"If $\\\\tau _\\\\lambda $ connects two variables then a standard edge (0,0) – (1,0); connects their vertices, if $\\\\tau _\\\\lambda ^\\\\circ $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\circ $; connects their vertices, and if $\\\\varphi $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\sim $; connects their vertices.\",\n \"Figure: Diagrams of the ψ 0 \\\\psi _0, ψ\\\\psi , and ψ n \\\\psi _n functions.Definition 5.4 (The $\\\\Psi $ operators) Here we turn the functions of Definition REF into linear operators, with the $\\\\psi $ functions acting as their kernels.\",\n \"For $j\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , we define $\\\\Psi ^{(j)}_0\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as the linear operator acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ as $\\\\Psi ^{(j)}_0 f\\\\left(w,u\\\\right) = \\\\int \\\\psi ^{(j)}_0(w,u,y) f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right).$ For $j\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , we define $\\\\Psi ^{(j)}\\\\colon L^2\\\\left(\\\\mathbb {X}^2\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as the linear operator acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as $\\\\Psi ^{(j)} f\\\\left(w,u\\\\right) = \\\\int \\\\psi ^{(j)}(w,u,r,s) f(r,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}r, \\\\mathrm {d}s\\\\right).$ For $j\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ , we define $\\\\Psi ^{(j)}_n\\\\colon L^2\\\\left(\\\\mathbb {X}^2\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ as the linear operator acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as $\\\\Psi ^{(j)}_n f\\\\left(x\\\\right) = \\\\int \\\\psi ^{(j)}_n(x,r,s) f(r,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}r, \\\\mathrm {d}s\\\\right).$ We can then naturally define $\\\\Psi _0\\\\colon L^2(\\\\mathbb {X})\\\\rightarrow L^2(\\\\mathbb {X}^2)$ , $\\\\Psi \\\\colon L^2(\\\\mathbb {X}^2)\\\\rightarrow L^2(\\\\mathbb {X}^2)$ , and $\\\\Psi _n\\\\colon L^2(\\\\mathbb {X}^2)\\\\rightarrow L^2(\\\\mathbb {X})$ as the sums of $\\\\Psi ^{(j)}_0$ , $\\\\Psi ^{(j)}$ , and $\\\\Psi ^{(j)}_n$ respectively, or equivalently as those operators having kernel functions $\\\\psi _0$ , $\\\\psi $ , and $\\\\psi _n$ respectively.\",\n \"Proposition 5.5 (Bound in terms of $\\\\psi $ and $\\\\Psi $ ) Let $n \\\\ge 1$ , $x,y\\\\in \\\\mathbb {X}$ , and $\\\\lambda \\\\in [0,\\\\lambda _O)$ .\",\n \"Then $\\\\lambda \\\\pi _\\\\lambda ^{(n)}(x,y) \\\\le \\\\int \\\\psi _n(x,w_{n-1},u_{n-1}) \\\\left( \\\\prod _{i=1}^{n-1} \\\\psi (\\\\vec{v}_i) \\\\right) \\\\psi _0(w_0,u_0,y) \\\\nu ^{\\\\otimes \\\\left(2n\\\\right)}\\\\left(\\\\mathrm {d}\\\\left( \\\\left(\\\\vec{w}, \\\\vec{u}\\\\right)_{[0,n-1]} \\\\right)\\\\right),$ where $\\\\vec{v}_i = (w_i,u_i,w_{i-1},u_{i-1})$ .\",\n \"In operator terms, this means $\\\\lambda \\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\Psi _n \\\\Psi ^{n-1} \\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }.$ The argument for (REF ) is essentially identical to [21].\",\n \"It uses only general properties of Poisson point process and connection models - for example thinning edges and Mecke's equation.\",\n \"The inequality (REF ) then follows from the clear sequential structure of (REF ).\",\n \"The inequality (REF ) suggests that we will bound the lace coefficient operators with some product of $\\\\left\\\\Vert \\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi \\\\right\\\\Vert _{1,\\\\infty }$ , and $\\\\left\\\\Vert \\\\Psi _n\\\\right\\\\Vert _{1,\\\\infty }$ terms.\",\n \"This is nearly correct.\",\n \"Unfortunately the norms $\\\\left\\\\Vert \\\\Psi _n\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi \\\\right\\\\Vert _{1,\\\\infty }$ and $\\\\left\\\\Vert \\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }$ will not have the decay in $d$ that we will require.\",\n \"We will have to consider the norms of pairs of operators.\",\n \"Definition 5.6 Define $\\\\triangle _\\\\lambda &:= \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda \\\\left(r,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\triangle ^\\\\circ _\\\\lambda &:= \\\\lambda ^2 \\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(r,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda &:= \\\\lambda ^2 \\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(r,u\\\\right)\\\\tau _\\\\lambda ^\\\\circ \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + 1,\\\\\\\\\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }&:= \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda ^\\\\circ \\\\left(\\\\overline{u}-\\\\overline{v};a_3,a_4\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{v};a_5,a_6\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{v}.$ We can think of $\\\\triangle _\\\\lambda $ as an $\\\\mathbb {X}$ -convolution of three $\\\\tau _\\\\lambda $ functions with suprema taken over the end vertices.\",\n \"Then $\\\\triangle ^\\\\circ _\\\\lambda $ and $\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda $ are produced by adding on the $\\\\mathbb {X}$ -convolution of two $\\\\tau _\\\\lambda $ functions and one $\\\\tau _\\\\lambda $ function (which will trivially take value 1 once the suprema are taken).\",\n \"The object $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ differs from $\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda $ in that now the marks at adjacent $\\\\tau _\\\\lambda $ functions need not be equal, and then we take the supremum over all the marks - not just the end ones.\",\n \"We then use these elementary diagrams to produce composite objects: $U_\\\\lambda := \\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }+ \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }^2 + \\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }, \\\\qquad V_\\\\lambda := \\\\left(\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }U_\\\\lambda \\\\right)^\\\\frac{1}{2}.$ We note a few relations.\",\n \"Firstly, since all the terms in $U_\\\\lambda $ are non-negative we have $\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le U_\\\\lambda $ and thus $V_\\\\lambda \\\\le U_\\\\lambda $ .\",\n \"Also note that $1\\\\le \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ and thus $U_\\\\lambda \\\\ge \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ .\",\n \"In particular this means that $U_\\\\lambda \\\\ge 1$ and $U_\\\\lambda \\\\ge \\\\triangle ^\\\\circ _\\\\lambda $ .\",\n \"The following lemma tells us that we can use $U_\\\\lambda $ to bound single operators, and that we can use $V_\\\\lambda ^2$ to bound pairs of operators.\",\n \"To aid the reader's understanding, bear in mind that later (in Section ) we will prove that $U_\\\\lambda = \\\\mathcal {O}\\\\left(1\\\\right)$ , whilst $V_\\\\lambda $ is much smaller and is $\\\\mathcal {O}\\\\left(\\\\beta \\\\right)$ .\",\n \"Lemma 5.7 For all $j_0\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , $j,j^{\\\\prime }\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , and $j_n\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ , $U_\\\\lambda &\\\\ge \\\\left\\\\Vert \\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j)}\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\right\\\\Vert _{1,\\\\infty },\\\\\\\\V_\\\\lambda ^2 &\\\\ge \\\\left\\\\Vert \\\\Psi ^{(j)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j)}\\\\Psi ^{(j^{\\\\prime })}\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j)}\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }.$ We begin with the single operators.\",\n \"For $\\\\Psi ^{(1)}$ we find $\\\\left\\\\Vert \\\\Psi ^{(1)}\\\\right\\\\Vert _{1,\\\\infty } &= \\\\lambda ^4\\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (w,u)\\\\tau _\\\\lambda ^\\\\circ (t,s) \\\\tau _\\\\lambda (t,w)\\\\tau _\\\\lambda (u,z)\\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda (z,r)\\\\nu ^{\\\\otimes 4}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t, \\\\mathrm {d}w, \\\\mathrm {d}u\\\\right), \\\\nonumber \\\\\\\\& \\\\le \\\\lambda ^4\\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (r,z) \\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda ^\\\\circ (t,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t \\\\right)\\\\right) \\\\nonumber \\\\\\\\&\\\\hspace{142.26378pt} \\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{t^{\\\\prime },z^{\\\\prime }\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (t^{\\\\prime },w)\\\\tau _\\\\lambda (w,u)\\\\tau _\\\\lambda (u,z^{\\\\prime })\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}w, \\\\mathrm {d}u\\\\right)\\\\right),\\\\nonumber \\\\\\\\& = \\\\triangle ^\\\\circ _\\\\lambda \\\\triangle _\\\\lambda ,$ where we have used a supremum bound on the $t$ and $z$ integrals to split them into $L^1$ and $L^\\\\infty $ bounds.\",\n \"It is also clear that $\\\\left\\\\Vert \\\\Psi ^{(3)}\\\\right\\\\Vert _{1,\\\\infty }= \\\\triangle _\\\\lambda $ and $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\triangle ^\\\\circ _\\\\lambda $ .\",\n \"The $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ bound is more involved and we return to it in a moment.\",\n \"We are able to represent these $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ norms and their bounds pictorially.\",\n \"The vertices of the diagrams represent the variables $w,u,z,t,a,b$ , etc.\",\n \"appearing in the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ expression.\",\n \"The variables over which a $\\\\nu $ -integral is taken are represented by the (0,0) circle (2pt) vertices, and the variables over which a supremum is taken are represented by the (0,0) circle (2pt) vertices.\",\n \"The presence of a $\\\\tau _\\\\lambda $ connecting two variables is then represented by a standard edge (0,0) – (1,0); between their vertices, and the presence of a $\\\\tau _\\\\lambda ^\\\\circ $ connecting two variables is correspondingly represented by an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\circ $; between their vertices.\",\n \"When using a supremum bound to split the diagrams, an integrated vertex gets split into two new vertices, precisely one integrated vertex and one supremum vertex.\",\n \"Each edge connected to such a split vertex can independently choose which of the new vertices to associate to.\",\n \"The calculations producing the bounds on $\\\\left\\\\Vert \\\\Psi ^{(j)}\\\\right\\\\Vert _{1,\\\\infty }$ for $j=1,3,4$ can then be represented by: $\\\\left\\\\Vert \\\\Psi ^{(1)}\\\\right\\\\Vert _{1,\\\\infty } \\\\quad &=\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(3)}\\\\right\\\\Vert _{1,\\\\infty } + \\\\left\\\\Vert \\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } \\\\quad &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad .$ Our strategy for $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ is to use the spatial translation invariance to shift the origin to the bottom right vertex, and then use a supremum bound to split the diagram in two.\",\n \"Unfortunately, this spatial shift decouples the spatial and mark components of the vertices so - for example - at some vertices we end up taking suprema over marks whilst integrating over space.\",\n \"Let us introduce some more notation.\",\n \"The vertex mark size=2pt] at (0,0) square* indicates that the spatial component is integrated over whilst the mark component is fixed and its supremum is taken after all the integrals.\",\n \"We also introduce the edge-vertex combination [dashed] (0,0) – (1,0); [fill=white] (0,0) circle (2pt) which indicates that there is no term connecting the associated vertices, but there is a supremum taken over the spatial displacement.\",\n \"Note that in principle we could use a vertex that had a spatial supremum and a mark integral, but we will always bound that probability integral by the supremum.\",\n \"The bound for $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ can therefore be expressed in the following: $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(0.75,0.6) circle (0pt) node[above]{\\\\circ };(1.5,0.25) circle (0pt) node[left]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}\\\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\\\circ };(1.5,0.25) circle (0pt) node[left]{\\\\circ };(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}\\\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1);(0,-0.6) -- (1,-0.6);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\\\circ };(1.5,0.25) circle (0pt) node[left]{\\\\circ };[fill=white] (1.5,-0.1) circle (2pt);(1.5,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);[fill=white] (0,0) circle (2pt);\\\\end{tikzpicture}}.$ To clarify the calculation for $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ , we write out the integral here: $&\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{s}\\\\in \\\\mathbb {R}^d,a_1,a_2\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ (\\\\overline{w}-\\\\overline{s};a_3,a_1)\\\\tau _\\\\lambda ^\\\\circ (\\\\overline{t}- \\\\overline{w}; a_4,a_3)\\\\tau _\\\\lambda (\\\\overline{u}-\\\\overline{t};a_6,a_4)\\\\tau _\\\\lambda (\\\\overline{z}-\\\\overline{u};a_5,a_6) \\\\nonumber \\\\\\\\&\\\\hspace{142.26378pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{t}- \\\\overline{z}; a_4,a_5)\\\\tau _\\\\lambda (\\\\overline{z}; a_5,a_2) \\\\mathrm {d}\\\\overline{t}\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{w}\\\\mathrm {d}\\\\overline{z}\\\\mathcal {P}^{\\\\otimes 4}\\\\left(\\\\mathrm {d}a_3, \\\\mathrm {d}a_4, \\\\mathrm {d}a_5, \\\\mathrm {d}a_6\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{s}\\\\in \\\\mathbb {R}^d,a_1,a_2,a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ (\\\\overline{w}^{\\\\prime }-\\\\overline{s}+\\\\overline{u};a_3,a_1)\\\\tau _\\\\lambda ^\\\\circ (\\\\overline{t}^{\\\\prime } - \\\\overline{w}^{\\\\prime }; a_4,a_3)\\\\tau _\\\\lambda (-\\\\overline{t}^{\\\\prime };a_6,a_4)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime };a_5,a_6) \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{t}^{\\\\prime } - \\\\overline{z}^{\\\\prime }; a_4,a_5)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime } + \\\\overline{u}; a_5,a_2) \\\\mathrm {d}\\\\overline{t}^{\\\\prime } \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{w}^{\\\\prime } \\\\mathrm {d}\\\\overline{z}^{\\\\prime } \\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}a_3, \\\\mathrm {d}a_4, \\\\mathrm {d}a_5\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}\\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{s},\\\\overline{t}^{\\\\prime },\\\\overline{z}^{\\\\prime }\\\\in \\\\mathbb {R}^d,a_1,a_2,a_4,a_5\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ (\\\\overline{w}^{\\\\prime }-\\\\overline{s}+\\\\overline{u};a_3,a_1)\\\\tau _\\\\lambda ^\\\\circ (\\\\overline{t}^{\\\\prime } - \\\\overline{w}^{\\\\prime }; a_4,a_3)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime } + \\\\overline{u}; a_5,a_2) \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{w}^{\\\\prime } \\\\mathcal {P}\\\\left(\\\\mathrm {d}a_3\\\\right)\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda (-\\\\overline{t}^{\\\\prime };a_6,a_4)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime };a_5,a_6)\\\\tau _\\\\lambda (\\\\overline{t}^{\\\\prime } - \\\\overline{z}^{\\\\prime }; a_4,a_5) \\\\mathrm {d}\\\\overline{t}^{\\\\prime } \\\\mathrm {d}\\\\overline{z}^{\\\\prime } \\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a_4, \\\\mathrm {d}a_5\\\\right)\\\\right).$ Then the two components of the first factor that are joined by the $\\\\raisebox {2pt}{{[dashed] (0,0) -- (1,0); [fill=white] (0,0) circle (2pt)}}$ edge can be spatially translated together to form three consecutive edges.\",\n \"Note however, that the marks on either side of this join are not necessarily equal.\",\n \"We can therefore bound this first term by $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ .\",\n \"In summary, these manipulations then imply that $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\triangle _\\\\lambda $ .\",\n \"For $\\\\Psi ^{(1)}_0$ and $\\\\Psi ^{(2)}_0$ the calculations are identical to each other, and we can once again represent the calculations pictorially: $\\\\left\\\\Vert \\\\Psi ^{(1)}_0\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\Psi ^{(2)}_0\\\\right\\\\Vert _{1,\\\\infty } =\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- cycle;[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad = \\\\triangle _\\\\lambda ,$ where splitting the single supremum into two produces an upper bound.\",\n \"For the third case we have $\\\\left\\\\Vert \\\\Psi ^{(3)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\quad \\\\raisebox {0pt}{\\\\begin{tikzpicture}(0,0) -- (1,0);(0.5,0) circle (0pt) node[above]{\\\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad = \\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }.$ The diagrams representing the calculations for $\\\\Psi _n$ are as follows: $\\\\left\\\\Vert \\\\Psi _n^{(1)}\\\\right\\\\Vert _{1,\\\\infty } & = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(1,-0.6) -- (0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\ &\\\\hspace{113.81102pt}\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\qquad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda , \\\\nonumber \\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\right\\\\Vert _{1,\\\\infty } & = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle ^\\\\circ _\\\\lambda .$ In the first case we expanded a $\\\\tau _\\\\lambda ^\\\\circ $ edge into a $\\\\tau _\\\\lambda $ edge and a contraction.\",\n \"Then in the contracted case we bound $\\\\tau _\\\\lambda \\\\le 1$ on the diagonal edge.\",\n \"We have now proven the bounds for the single operator norms.\",\n \"For most of the operator pairs, we can then use the sub-multiplicity of $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ to immediately get sufficient bounds.\",\n \"The only pairs for which this strategy fails are: $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(4)}_0\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(3)}_0\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty }$ , and $\\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"We now deal with these cases by hand: $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(3)}_0\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(3)}_0\\\\right\\\\Vert _{1,\\\\infty } &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- (0,0.6);(0,0) circle (0pt) node[above, rotate=90]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle _\\\\lambda , \\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle _\\\\lambda .$ For the first bound we used $\\\\varphi \\\\le \\\\tau _\\\\lambda $ , and then split the single supremum into two.\",\n \"For the second bound we used $\\\\tau _\\\\lambda \\\\le 1$ on the diagonal edge.\",\n \"Proposition 5.8 For $n\\\\ge 1$ , $\\\\lambda \\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le 6\\\\times 4^{n-1}U_\\\\lambda V_\\\\lambda ^n.$ Recall from Lemma REF that $\\\\lambda \\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\Psi _n\\\\Psi ^{n-1}\\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"We therefore aim to bound $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }$ for $j_0\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , $j_1,\\\\ldots ,j_{n-1}\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , and $j_n\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ .\",\n \"We have two cases depending upon the parity of $n$ .\",\n \"If $n$ is odd then we have an even number of operators.\",\n \"We pair them off, use the sub-multiplicity of $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ , and Lemma REF to get $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left(V_\\\\lambda ^2\\\\right)^{\\\\frac{n+1}{2}} = V_\\\\lambda ^{n+1}.$ If $n$ is even, then we pair off the first $n$ operators and treat the last $\\\\Psi ^{(j_n)}_n$ operator by itself.\",\n \"We then get $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\le U_\\\\lambda \\\\left(V_\\\\lambda ^2\\\\right)^{\\\\frac{n}{2}} = U_\\\\lambda V_\\\\lambda ^{n}.$ Since $U_\\\\lambda \\\\ge V_\\\\lambda $ , we have the bound $U_\\\\lambda V_\\\\lambda ^{n}$ for all $n\\\\ge 1$ .\",\n \"Since there are 3 choices for $j_0$ , 4 choices for each of $j_1,\\\\ldots ,j_{n-1}$ , and 2 choices for $j_n$ , we get the pre-factor of $6\\\\times 4^{n-1}$ in our result by applying the triangle inequality.\",\n \"Bounds on displaced expansion coefficients In this section we will be aiming to bound $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ for $n\\\\ge 1$ .\",\n \"Our strategy draws inspiration from the corresponding step in [21], whilst being adapted to account for inhomogeneous marks in a similar way to Section REF .\",\n \"The central idea is to bound $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1- \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\pi ^{(n)}_\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right),$ and bound $\\\\pi ^{(n)}_\\\\lambda $ using the function expression from Proposition REF .\",\n \"This produces an integral (or diagram) composed from simpler segments with the displacement factor, $\\\\left(1- \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)$ , spanning the whole length of the diagram.\",\n \"We can then use the Cosine-Splitting result (Lemma REF ) to get a sum of diagrams where the displacement factor only spans a single segment of each diagram.\",\n \"To demonstrate this, consider the diagram corresponding to $\\\\Psi ^{(2)}_n\\\\Psi ^{(1)}\\\\Psi ^{(1)}_0$ with the displacement factor: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (0,0.2) -- (1,0.8) -- (3,0.8) -- (4,0.2);(2,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le 3\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\right.\",\n \"\\\\\\\\ \\\\left.\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (1,0.8) -- (3,0.8);(2,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right).$ In these diagrams and hereafter we use [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; to denote that a displacement factor is connecting the two indicated vertices.\",\n \"As above, the path the [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; takes will also suggest the route along which we will intend to use cosine-splitting.\",\n \"For the sake of simplicity, we will be taking the displacement path across the `top' of the diagram from the perspective of the usual orientation of the $\\\\Psi _0$ segment.\",\n \"However - as can be seen in the above example - the way the operators compose with each other means that the orientation of each segment inverts when compared to its neighbours.\",\n \"Therefore the displacement may be across the `top' or the `bottom' of the $\\\\Psi $ and $\\\\Psi _n$ segments (in their usual orientation) depending upon the parity of its place in the sequence of segments.\",\n \"Our general strategy will be to isolate the displaced segment (and perhaps a neighbouring segment) by splitting off `earlier' and `later' segments.\",\n \"When we split off earlier segments (and sometimes later segments), we will use the $\\\\Psi $ structure we have been using so far.\",\n \"However, we will sometimes want to group vertices and edges slightly differently when we split off `later' segments.\",\n \"We use an observation from [21] to conveniently group these.\",\n \"Let us define the kernel functions $\\\\overline{\\\\psi }^{(1)}_0\\\\left(w,u,x\\\\right) &:= \\\\lambda ^2\\\\tau _\\\\lambda (y,r)\\\\tau _\\\\lambda (r,s)\\\\tau _\\\\lambda (s,x),\\\\\\\\\\\\overline{\\\\psi }^{(2)}_0\\\\left(w,u,x\\\\right) &:= \\\\lambda ^2\\\\left(\\\\lambda ^{-1}\\\\delta _{y,r}\\\\right)\\\\left(\\\\lambda ^{-1}\\\\delta _{y,s}\\\\right),\\\\\\\\\\\\overline{\\\\psi }^{(1)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda (w,u)\\\\int \\\\tau _\\\\lambda (t,s) \\\\tau _\\\\lambda (t,w)\\\\tau _\\\\lambda (u,z)\\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda ^\\\\circ (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\overline{\\\\psi }^{(2)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda (w,s)\\\\int \\\\tau _\\\\lambda (t,z)\\\\tau _\\\\lambda (z,u)\\\\tau _\\\\lambda (u,t)\\\\tau _\\\\lambda ^\\\\circ (t,w)\\\\tau _\\\\lambda ^\\\\circ (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\overline{\\\\psi }^{(3)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,s)\\\\tau _\\\\lambda (u,r),\\\\\\\\\\\\overline{\\\\psi }^{(4)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,s)\\\\left(\\\\lambda ^{-1}\\\\delta _{u,r}\\\\right),$ and set $\\\\overline{\\\\psi }_0 := \\\\overline{\\\\psi }^{(1)}_0+\\\\overline{\\\\psi }^{(2)}_0$ and $\\\\overline{\\\\psi }:= \\\\overline{\\\\psi }^{(1)}+\\\\overline{\\\\psi }^{(2)}+\\\\overline{\\\\psi }^{(3)} + \\\\overline{\\\\psi }^{(4)}$ .\",\n \"Diagrammatic representations of these functions can be found in Figure REF .\",\n \"Note the similarity to the $\\\\psi $ functions, with the $\\\\tau _\\\\lambda ^\\\\circ $ edge appearing elsewhere in the integral.\",\n \"Figure: Diagrams of the ψ ¯ 0 \\\\overline{\\\\psi }_0, and ψ ¯\\\\overline{\\\\psi } functions.For $j\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ we define $\\\\overline{\\\\Psi }^{(j)}_0\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ to be the linear operators acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ as $\\\\overline{\\\\Psi }^{(j)}_0 f\\\\left(w,u\\\\right) = \\\\int \\\\overline{\\\\psi }^{(j)}_0(w,u,y) f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right),$ and for $j\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ we define $\\\\overline{\\\\Psi }^{(j)}\\\\colon L^2\\\\left(\\\\mathbb {X}^2\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ to be the linear operators acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as $\\\\overline{\\\\Psi }^{(j)} f\\\\left(w,u\\\\right) = \\\\int \\\\overline{\\\\psi }^{(j)}(w,u,r,s) f(r,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}r, \\\\mathrm {d}s\\\\right).$ In our terminology, the important observation of [21] was that for some $m\\\\ge 0$ the `later' segments of each diagram can be bounded using $\\\\big (\\\\overline{\\\\Psi }^m\\\\overline{\\\\Psi }_0\\\\big )^\\\\dagger $ , the adjoint of $\\\\overline{\\\\Psi }^m\\\\overline{\\\\Psi }_0$ .\",\n \"In terms of the kernel functions, taking the adjoint in this case amounts to reflecting the `input' and `output' arguments, and for the diagrams this amounts to reflecting in the vertical plane.\",\n \"Lemma 5.9 For $n\\\\ge 1$ , $m\\\\in \\\\left\\\\lbrace 0,\\\\ldots ,n\\\\right\\\\rbrace $ , $j_0\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , $j_1,\\\\ldots ,j_{n-1}\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , and $j_n\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ , $\\\\left\\\\Vert \\\\Psi ^{(j_m)}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } &\\\\le U_\\\\lambda V_\\\\lambda ^m,\\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_{m})}\\\\right\\\\Vert _{1,\\\\infty } &\\\\le U_\\\\lambda V_\\\\lambda ^{n-m},\\\\\\\\\\\\left\\\\Vert \\\\overline{\\\\Psi }^{(j_m)}\\\\ldots \\\\overline{\\\\Psi }^{(j_1)}\\\\overline{\\\\Psi }^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } &\\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}U_\\\\lambda &: m=0\\\\\\\\U_\\\\lambda ^2V_\\\\lambda ^{m-1} &: m\\\\ge 1.\\\\end{array}\\\\right.\",\n \"}$ The argument for $\\\\left\\\\Vert \\\\Psi ^{(j_m)}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }$ and $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_{m})}\\\\right\\\\Vert _{1,\\\\infty }$ is the same as that in Lemma REF and Proposition REF .\",\n \"We consider bounds for single operators and pairs of operators, and then use sub-multiplicity to get a bound for each diagram.\",\n \"For $\\\\left\\\\Vert \\\\overline{\\\\Psi }^{(j_m)}\\\\ldots \\\\overline{\\\\Psi }^{(j_1)}\\\\overline{\\\\Psi }^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }$ the argument is a little more complicated.\",\n \"For $m=0,1$ we can easily check each diagram satisfies the bound.\",\n \"For $m\\\\ge 2$ we have the issue that there exists one pair of operators which we can't bound by $V_\\\\lambda ^2$ : $\\\\left\\\\Vert \\\\overline{\\\\Psi }^{(4)}\\\\overline{\\\\Psi }^{(2)}_0\\\\right\\\\Vert _{1,\\\\infty } = \\\\quad \\\\raisebox {0pt}{\\\\begin{tikzpicture}(0,0) to [out=10,in=170] (1,0) to [out=190,in=350] (0,0);[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle ^\\\\circ _\\\\lambda \\\\le U_\\\\lambda ^2.$ If $m\\\\ge 2$ is even (an odd number of operators) then we use sub-multiplicity to pull off the $\\\\overline{\\\\Psi }_0$ term which we bound with $U_\\\\lambda $ .\",\n \"We then pull off pairs which we can bound with $V_\\\\lambda ^2$ to get the bound $U_\\\\lambda V_\\\\lambda ^m$ .\",\n \"If $m\\\\ge 2$ is odd (an even number of operators) then we use sub-multiplicity to pull off the $\\\\overline{\\\\Psi }\\\\overline{\\\\Psi }_0$ pair which we bound with $U_\\\\lambda ^2$ .\",\n \"We then pull off pairs which we can bound with $V_\\\\lambda ^2$ to get the bound $U_\\\\lambda ^2V_\\\\lambda ^{m-1}$ .\",\n \"Definition 5.10 In addition to the terms defined in Definition REF , we will now use $W_k&:= \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{x,y\\\\in \\\\mathbb {X}}\\\\int \\\\tau _{\\\\lambda ,k}(x,u)\\\\tau _\\\\lambda (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) = \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\overline{W_k}&:= \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\tau _{\\\\lambda ,k}(\\\\overline{x}-\\\\overline{u};a_1,a_2)\\\\tau _\\\\lambda (\\\\overline{u};a_3,a_4)\\\\mathrm {d}u,\\\\\\\\\\\\overline{H_k}&:= \\\\lambda ^5 \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}_1,\\\\overline{x}_2\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_{16}\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{x}_1;a_1,a_2)\\\\tau _\\\\lambda (\\\\overline{u};a_3,a_4) \\\\tau _\\\\lambda (\\\\overline{v}-\\\\overline{s};a_5,a_6) \\\\tau _\\\\lambda (\\\\overline{v}+\\\\overline{x}_2-\\\\overline{t};a_7,a_8)\\\\nonumber \\\\\\\\& \\\\hspace{42.67912pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{w};a_9,a_{10})\\\\tau _\\\\lambda (\\\\overline{w}-\\\\overline{u};a_{11},a_{12}) \\\\tau _\\\\lambda (\\\\overline{t}-\\\\overline{w};a_{13},a_{14}) \\\\tau _{\\\\lambda ,k}(\\\\overline{t}-\\\\overline{u};a_{15},a_{16}) \\\\mathrm {d}\\\\overline{t}\\\\mathrm {d}\\\\overline{w}\\\\mathrm {d}\\\\overline{z}\\\\mathrm {d}\\\\overline{u}.$ Note that we only use $\\\\overline{W_k}$ in Propositions REF and REF because $\\\\overline{W_k}\\\\ge W_k$ , but in the proof we will sometimes use $W_k$ to make the derivation easier to follow.\",\n \"The diagrams for $\\\\overline{W_k}$ and $\\\\overline{H_k}$ are in Figure REF .\",\n \"Also recall the sets $\\\\left\\\\lbrace B\\\\left(x\\\\right)\\\\right\\\\rbrace _{x\\\\in \\\\mathbb {X}}$ assumed to exist in Assumption REF .\",\n \"Then define $\\\\mathbb {B} &:= \\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\nu \\\\left(B\\\\left(x\\\\right)\\\\right),\\\\\\\\\\\\triangle ^{(B)}_\\\\lambda &:= \\\\lambda ^2 \\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}: r\\\\notin {B\\\\left(s\\\\right)}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(r,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right).$ Note that these are only required to deal with one specific diagram in the following $n=1$ case.\",\n \"Figure: Diagrams representing W k ¯\\\\overline{W_k} and H k ¯\\\\overline{H_k}.\",\n \"In this figure, the marks on edges incident to filled square vertices need not be equal.Proposition 5.11 For $k\\\\in \\\\mathbb {R}^d$ and $n\\\\ge 2$ , $\\\\lambda \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(n+1\\\\right)^24^{n-1}\\\\left(50\\\\overline{W_k}V_\\\\lambda + 6\\\\overline{H_k}\\\\right) U_\\\\lambda ^4 V_\\\\lambda ^{n-4} &: n\\\\ge 4\\\\\\\\16\\\\times 16\\\\left(50\\\\overline{W_k}V_\\\\lambda ^2 + 6\\\\overline{H_k}\\\\right) U_\\\\lambda ^2 &: n=3\\\\\\\\9\\\\times 4\\\\left(50\\\\overline{W_k}V_\\\\lambda + 6\\\\overline{H_k}\\\\right) U_\\\\lambda ^2 &: n=2.\\\\end{array}\\\\right.\",\n \"}$ Proposition 5.12 For $k\\\\in \\\\mathbb {R}^d$ , $\\\\lambda \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(1)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(1)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le 13\\\\overline{W_k}V_\\\\lambda U_\\\\lambda + \\\\lambda \\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }\\\\left(V_\\\\lambda + \\\\triangle ^{(B)}_\\\\lambda \\\\right) + \\\\lambda ^3\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + 4\\\\lambda \\\\mathbb {B}\\\\overline{W_k}.$ [Proof of Proposition REF ] In bounding the diagrams with one displacement segment, we have three broad cases.\",\n \"Let $i\\\\in \\\\left\\\\lbrace 0,1,\\\\ldots ,n\\\\right\\\\rbrace $ denote which segment the displacement lies upon.\",\n \"Our three cases are then: the displacement lies on the $\\\\psi _n$ segment (i.e.\",\n \"$i=n$ ), the displacement lies on the $\\\\psi _0$ segment (i.e.\",\n \"$i=0$ ), the displacement lies on a $\\\\psi $ segment (i.e.\",\n \"$i\\\\in \\\\left\\\\lbrace 1,\\\\ldots ,n-1\\\\right\\\\rbrace $ ).\",\n \"We will aim to get a single bound for each diagram having a displacement crossing a single segment.\",\n \"This will be uniform in the sequence $\\\\left(j_0,j_1,\\\\ldots ,j_n\\\\right)$ .\",\n \"We then account for the number of possible choices of $\\\\left(j_0,j_1,\\\\ldots ,j_n\\\\right)$ , and finally include the factor of $(n+1)^2$ arising from using cosine-splitting (the explicit factor and the number of diagrams).\",\n \"Case REF is the simplest.\",\n \"We use a supremum bound to pull off the displaced $\\\\Psi _n$ term.\",\n \"We then bound this by hand and use Lemma REF to bound the remainder.\",\n \"In representing this scheme we use a shorthand $\\\\Psi ^{n-1}\\\\Psi _0$ to denote a particular sequence $\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0$ rather than actually the operators $\\\\Psi $ and $\\\\Psi _0$ defined above.\",\n \"We also use labelled grey shapes to represent these groupings of diagrams, highlighting only end vertices or vertices that connect to neighbouring segments.\",\n \"This scheme look like: $&\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8) -- (4,0.2);(2.8,0.8) circle (0pt) node[rotate = 0]{\\\\times };(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node at (1,0) {\\\\Psi ^{n-1}\\\\Psi _0};[fill=white] (-1,0) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad +\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (2,0.8) -- (3,0.2);(2.5,0.5) circle (0pt) node[rotate = -30]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{n-1}\\\\Psi _0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\ &\\\\hspace{28.45274pt}\\\\le U_\\\\lambda V_\\\\lambda ^{n-1} \\\\left(2\\\\left( \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\\\times };(2.5,-0.6) circle (0pt) node[above]{\\\\circ };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right) + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\mathbf {\\\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le 2U_\\\\lambda V_\\\\lambda ^{n-1}\\\\left(\\\\overline{W_k}\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda + W_k\\\\triangle ^\\\\circ _\\\\lambda \\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le 4\\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-1}.$ Note that this bound also holds if the displacement runs along the bottom of the $\\\\Psi _n$ segment.\",\n \"Since there are $3\\\\times 4^{n-1}$ possible sequences of indices for the $\\\\Psi ^{n-1}\\\\Psi _0$ diagram, this case contributes that as a pre-factor.\",\n \"For Case REF , we need a few more tricks.\",\n \"Since we are taking the displacement across the top of the diagram, the only term that will make a contribution is $\\\\Psi ^{(1)}_0$ .\",\n \"To tackle this case we first use the observation of [21] that the diagram we want to bound can in turn be bounded by a diagram using $\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0$ .\",\n \"As above, we use $\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0$ as a shorthand for a particular sequence $\\\\overline{\\\\Psi }^{(j_{n-1})}\\\\ldots \\\\overline{\\\\Psi }^{(j_1)}\\\\overline{\\\\Psi }^{(j_0)}_0$ .\",\n \"We are also imprecise in this notation over whether it is this operator or its adjoint.\",\n \"Nevertheless the diagrams we use are clear on which vertices are integrated over and which have suprema, and this will avoid ambiguity.\",\n \"We then spatially translate the supremum to the far end and pull off the $\\\\overline{\\\\Psi }^{n-2}\\\\overline{\\\\Psi }_0$ diagram using a supremum bound.\",\n \"This looks like $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0};\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (3,-0.6) to [out=100,in=260] (3,0.6) to [out=190,in=350] (2,0.6);\\\\node at (2.5,0) {\\\\overline{\\\\Psi }};\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-2}\\\\overline{\\\\Psi }_0};(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\\\end{tikzpicture}}.$ For $\\\\overline{\\\\Psi }^{(j)}$ with $j=1,2,3$ , a simple supremum bound shows that this starting diagram can be bounded by $\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\left\\\\Vert \\\\overline{\\\\Psi }^{(j)}\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le \\\\overline{W_k}V_\\\\lambda ^2$ .\",\n \"For $j=4$ a little more care is needed: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ We therefore find that $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\overline{W_k}V_\\\\lambda ^{n-1}U_\\\\lambda ^2 &: n\\\\ge 3\\\\\\\\\\\\overline{W_k}V_\\\\lambda ^2U_\\\\lambda &: n=2.\\\\end{array}\\\\right.\",\n \"}$ Since there are 4 choices for the appended $\\\\overline{\\\\Psi }^{(j)}$ term, and $4^{n-2}\\\\times 2$ for the remaining $\\\\overline{\\\\Psi }^{n-2}\\\\overline{\\\\Psi }_0$ diagram, this case contributes $2\\\\times 4^{n-1}$ as a pre-factor.\",\n \"For Case REF , we shall find we have 3 sub-cases, depending not only on the $j$ -index of the displaced segment, but on which term we are looking at once it has been expanded.\",\n \"Sub-cases $\\\\mathrm {\\\\left(I\\\\right)}$ and $\\\\mathrm {\\\\left(II\\\\right)}$ deal with $j=1,2,3$ .\",\n \"In Appendix we expand out the various displaced segments using cosine-splitting and by expanding $\\\\tau _\\\\lambda ^\\\\circ $ edges into $\\\\tau _\\\\lambda $ edges and points, and label each of the resulting diagrams as sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ and $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"These sub-cases are distinguished by the strategy we use to bound them.\",\n \"Sub-case $\\\\mathrm {\\\\left(III\\\\right)}$ then addresses the case with $j=4$ .\",\n \"For sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ , we just use the $\\\\Psi $ structure of the diagrams.\",\n \"We first split off segments from the left of the displaced term like we did in Case REF .\",\n \"If $i=n-1$ we can then bound the remaining two pair of segments with $\\\\overline{W_k}V_\\\\lambda ^2$ (Appendix explains how this bound is found).\",\n \"This can be described diagrammatically as $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{n-2}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi _n};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le U_\\\\lambda V_\\\\lambda ^{n-2} \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\ \\\\le \\\\overline{W_k}U_\\\\lambda V_\\\\lambda ^{n}.$ If $i\\\\le n-2$ , then we also split off the segments to the right of this pair, and then bound the pair by $\\\\overline{W_k}V_\\\\lambda ^2$ again.\",\n \"Diagrammatically this looks like $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (6,-0.6) to [out=40,in=200] (7,0) to [out=160,in=320] (6,0.6) to [out=190,in=350] (4,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{i-1}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi };\\\\node at (5,0) {\\\\Psi _n\\\\Psi ^{n-i-2}};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(7,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le U_\\\\lambda ^2V_\\\\lambda ^{n-3} \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\ \\\\le \\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-1}.$ Note that the since $U_\\\\lambda \\\\ge V_\\\\lambda $ , the bound in (REF ) is greater than or equal to the bound in (REF ).\",\n \"It will be convenient to find an $i$ -independent bound.\",\n \"To this end, since the $i\\\\le n-2$ case only exists for $n\\\\ge 3$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ with ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-1}&: n\\\\ge 3\\\\\\\\\\\\overline{W_k}U_\\\\lambda V_\\\\lambda ^{2}&: n=2,\\\\end{array}\\\\right.\",\n \"}$ uniformly in $i\\\\in \\\\left\\\\lbrace 1,\\\\ldots ,n-1\\\\right\\\\rbrace $ .\",\n \"For sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ , we use a different strategy.\",\n \"First we split off the earlier (left) segments as we did for the first sub-case and in Case REF .\",\n \"However we arrange the later terms differently - more like we did in Case REF .\",\n \"We associate our displaced $\\\\Psi $ term with $\\\\tau _\\\\lambda $ and $\\\\tau _\\\\lambda ^\\\\circ $ edges from the subsequent term.\",\n \"If the subsequent term would be $\\\\Psi ^{(2)}_n$ , these are actually two $\\\\tau _\\\\lambda $ edges, but in our schematic diagrams we will draw a $\\\\tau _\\\\lambda ^\\\\circ $ edge and bear this special case in mind when it is relevant.\",\n \"While the displacement portrayed on the `top' of the segment could equally well be on the `bottom', the $\\\\tau _\\\\lambda ^\\\\circ $ edge will always be attached to the bottom in the usual orientation.\",\n \"Everything to the right of these $\\\\tau _\\\\lambda $ and $\\\\tau _\\\\lambda ^\\\\circ $ edges can then be described as a sequence of $\\\\overline{\\\\Psi }$ and $\\\\overline{\\\\Psi }_0$ terms.\",\n \"If $i=n-1$ , then we pair our augmented displaced segment with the subsequent $\\\\overline{\\\\Psi }_0$ segment.\",\n \"We then perform a spatial translation to make the right-most vertex a supremum vertex like we did in (REF ).\",\n \"The remaining diagram of a $\\\\Psi ^{(j)}$ segment and $\\\\overline{\\\\Psi }_0$ segment connected by a $\\\\tau _\\\\lambda ^\\\\circ $ and $\\\\tau _\\\\lambda $ edge can then be bounded by $\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)$ (see Appendix ).\",\n \"Recall that if $\\\\overline{\\\\Psi }_0$ is in fact $\\\\overline{\\\\Psi }^{(2)}_0$ then we have two $\\\\tau _\\\\lambda $ edges connecting the terms, not a $\\\\tau _\\\\lambda $ edge and a $\\\\tau _\\\\lambda ^\\\\circ $ edge.\",\n \"We need to note this to get the $\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)$ bound.\",\n \"The scheme is then represented diagrammatically as $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{n-2}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(6,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le U_\\\\lambda V_\\\\lambda ^{n-2} \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\ \\\\le \\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda V_\\\\lambda ^{n-2}.$ If $i\\\\le n-2$ , we also perform a spatial translation to move a supremum to the right-most vertex, but then also split off all $\\\\overline{\\\\Psi }$ and $\\\\overline{\\\\Psi }_0$ segments after the $\\\\overline{\\\\Psi }$ segment immediately following the displaced term.\",\n \"Once again the remaining pair can be bounded by $\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)$ (see Appendix ) and the split-off diagrams bounded using Lemma REF .\",\n \"The scheme can be represented diagrammatically as $&\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);[fill=gray!50] (5,0.6) to [out=280,in=80] (5,-0.6) to [out=10,in=170] (7,-0.6) to [out=40,in=200] (8,0) to [out=160,in=320] (7,0.6) to [out=190,in=350] (5,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{i-1}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }};\\\\node at (6,0) {\\\\overline{\\\\Psi }^{n-i-2}\\\\overline{\\\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(5,0.6) circle (2pt);(5,-0.6) circle (2pt);(8,0) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times {\\\\left\\\\lbrace \\\\begin{array}{ll}U_\\\\lambda ^2 V_\\\\lambda ^{n-4}&: i\\\\le n-3 \\\\\\\\U_\\\\lambda V_\\\\lambda ^{n-3}&: i=n-2\\\\end{array}\\\\right.}\",\n \"\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\le \\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right){\\\\left\\\\lbrace \\\\begin{array}{ll}U_\\\\lambda ^2 V_\\\\lambda ^{n-4}&: i\\\\le n-3 \\\\\\\\U_\\\\lambda V_\\\\lambda ^{n-3}&: i=n-2\\\\end{array}\\\\right.\",\n \"}$ Again for simplicity, we would like to get and $i$ -independent bound for such diagrams.\",\n \"Since the $i = n-2$ case only exists for $n\\\\ge 3$ , and the $i\\\\le n-3$ case only exists for $n\\\\ge 4$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ with ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda ^3V_\\\\lambda ^{n-4}&: n\\\\ge 4\\\\\\\\\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda ^2&: n=3\\\\\\\\\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda &: n=2,\\\\end{array}\\\\right.\",\n \"}$ uniformly in $i\\\\in \\\\left\\\\lbrace 1,\\\\ldots ,n-1\\\\right\\\\rbrace $ .\",\n \"We now address sub-case $\\\\mathrm {\\\\left(III\\\\right)}$ , that is when the displacement crosses a $\\\\Psi ^{(4)}$ segment.\",\n \"This is the simplest sub-case because when we take a displacement across $\\\\Psi ^{(4)}$ in isolation, we get exactly $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad =W_k.$ We will be able to use the $\\\\Psi $ -arrangement of segments here.\",\n \"For $n\\\\ge 4$ we split off the earlier and later segments without pairing the displaced term with any of them.\",\n \"With Lemma REF this produces the bound $\\\\left(U_\\\\lambda V_\\\\lambda ^{i-1}\\\\right)W_k\\\\left(U_\\\\lambda V_\\\\lambda ^{n-i-1}\\\\right) = \\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-2} \\\\qquad :n\\\\ge 4.$ For $n=3$ we split off two segments from one side and one from the other.\",\n \"This gives the finer bound $W_kU_\\\\lambda V_\\\\lambda ^2 \\\\quad :n=3.$ For $n=2$ , we use $j_0$ to indicate which $\\\\Psi _0$ term we have and $j_2$ to denote which $\\\\Psi _n$ term we have.\",\n \"We note that if $j_0=1,2$ then $\\\\left\\\\Vert \\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le V_\\\\lambda $ , and if $j_2 = 1$ then $\\\\left\\\\Vert \\\\Psi ^{(j_2)}_n\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le V_\\\\lambda $ .\",\n \"Therefore if $\\\\left(j_0,j_2\\\\right) \\\\ne \\\\left(3,2\\\\right)$ we have the bound $W_kU_\\\\lambda V_\\\\lambda $ .\",\n \"If $\\\\left(j_0,j_2\\\\right) = \\\\left(3,2\\\\right)$ we proceed by hand.\",\n \"We bound $\\\\varphi \\\\le \\\\tau _\\\\lambda $ and spatially shift one vertex to show $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\\\times };(1,0) circle (0pt) node[rotate=90, above]{\\\\sim };(0,0) circle (0pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\\\times };\\\\node [mark size=2pt] at (1,0.6) {{square*}};(1,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda .$ Carefully comparing the above bounds gives the required bounds for $n\\\\le 2$ .\",\n \"Including the multiplicity of each diagram (including that arising from cosine-splitting), we find that we can bound the displaced and subsequent non-displaced pair by $\\\\left(24\\\\overline{W_k}V_\\\\lambda ^2 + 2\\\\overline{H_k}\\\\right)$ .\",\n \"Note that this includes summing over the $j$ -index of the displaced term but not the index of the subsequent term.\",\n \"Since there are $3\\\\times 4^{n-2}\\\\times 2$ options for the $j$ -indices of the other segments, this case contributes this to the counting pre-factors in our result.\",\n \"[Proof of Proposition REF ] For $n=1$ , we can consider each of the six cases by hand.\",\n \"Most of the diagrams can be dealt with routinely using cosine-splitting, supremum bounds, and spatial shifts.\",\n \"We enumerate the diagrams by their $\\\\left(j_0,j_1\\\\right)$ values corresponding to the constituent segments.\",\n \"$(1,1):& \\\\quad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (0,-0.2) -- (1,-0.8) -- (2,-0.8) -- (3,-0.2);(1.5,-0.8) circle (0pt) node{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 3\\\\left(\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle _\\\\lambda + \\\\triangle _\\\\lambda \\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda W_k\\\\right)\\\\\\\\(1,2):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (0,0.2) -- (1,0.8) -- (2,0.2);(0.8,0.68) circle (0pt) node[rotate=30]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\triangle _\\\\lambda W_k\\\\right)\\\\\\\\(2,1):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (1,0.8) -- (2,0.8) -- (3,0.2);(1.8,0.8) circle (0pt) node{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\triangle _\\\\lambda \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda + \\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda W_k\\\\right)\\\\\\\\(2,2):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (1,0.8) -- (2,0.2);(1.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\triangle _\\\\lambda W_k\\\\\\\\(3,1):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6) -- cycle;(1,0.6) -- (1,-0.6);[<->] (0,0.8) -- (1,0.8) -- (2,0.2);(0.8,0.8) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(2,0) circle (2pt);(0,-0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\triangle _\\\\lambda W_k+ \\\\triangle _\\\\lambda W_k\\\\right)$ These terms sum to no more than $13\\\\overline{W_k}V_\\\\lambda U_\\\\lambda .$ For the $\\\\left(3,2\\\\right)$ diagram, we use a different `trick.'\",\n \"In Lemma REF we will prove the bound $\\\\tau _\\\\lambda \\\\left(x,y\\\\right) \\\\le \\\\varphi (x,y) + \\\\lambda \\\\int \\\\tau _\\\\lambda (x,u)\\\\varphi (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right)$ .\",\n \"When applied to this diagram, this looks like $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad 2\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.6) circle (0pt) node[below]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (1.2,0.6) -- (1.2,-0.6);(1.2,0) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.6) circle (0pt) node[above]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right),$ Here we have also used the cosine-splitting result on the second part.\",\n \"For the first of these 4-vertex diagrams, we split the supremum in two and get the bound $\\\\lambda ^3\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda _k\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\lambda \\\\triangle _\\\\lambda \\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"For the second of the 4-vertex diagrams, we also split the supremum in two and get the bound $\\\\lambda ^3\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"For the 3-vertex diagram we condition on whether two of the vertices are in each other's $B$ -set (defined in Assumption REF ).\",\n \"In the following diagrams [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\in B$; indicates that an indicator of the form ${1}_{\\\\left\\\\lbrace w \\\\in B\\\\left(u\\\\right)\\\\right\\\\rbrace }$ relates two variables, and [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\notin B$; indicates that an indicator of the form ${1}_{\\\\left\\\\lbrace w \\\\notin B\\\\left(u\\\\right)\\\\right\\\\rbrace }$ relates them: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,below]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.7) -- (0,-0.8) -- (1,-0.2);(-0.2,0) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,below]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\ &\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + 2\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node{\\\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-20pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,-0.8) -- (1,-0.2);(0.5,-0.5) circle (0pt) node[rotate=30]{\\\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right) \\\\nonumber \\\\\\\\&\\\\le \\\\left(\\\\quad \\\\raisebox {-7pt}{\\\\begin{tikzpicture}(0,0) -- (1,0);[<->] (0,0.2) -- (1,0.2);(0.5,0.2) circle (0pt) node{\\\\times };(0.5,0) circle (0pt) node[below]{\\\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node[above, rotate=90]{\\\\scriptsize \\\\notin B};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right) +4\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);[<->] (0,-0.2) -- (1,-0.2);(0.5,-0.2) circle (0pt) node[below]{\\\\scriptsize \\\\in B};\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\mathbf {\\\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right)\\\\nonumber \\\\\\\\& \\\\le \\\\lambda \\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }\\\\triangle ^{(B)}_\\\\lambda + 4\\\\lambda \\\\mathbb {B}W_k.$ Bootstrap analysis In this section we will be assuming that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) = 1.$ By Lemma REF this also implies $\\\\mathbb {S}\\\\left(\\\\right)\\\\le 1$ .\",\n \"A re-scaling argument similar to [36] explains how this can be done while maintaining generality.\",\n \"First note that since $\\\\widehat{\\\\varphi }(0;a,b)>0$ for a positive $\\\\mathcal {P}$ -measure set of $a,b\\\\in \\\\mathcal {E}$ , we have $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)>0$ .\",\n \"Now suppose we scale $\\\\mathbb {R}^d$ by a factor of $q^{-1/d}$ (that is, the new unit radius ball is the previous ball of radius $q^{1/d}$ ), and leave the mark space $\\\\mathcal {E}$ unchanged.\",\n \"Then we find that the system we are left with has the distribution of an RCM model with the new parameters $\\\\lambda ^* = \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\eta \\\\cap \\\\left[0, q^{\\\\frac{1}{d}}\\\\right]^d }\\\\right] = \\\\lambda q, \\\\qquad \\\\varphi ^*\\\\left(\\\\overline{x};a,b\\\\right) = \\\\varphi \\\\left(q^{\\\\frac{1}{d}}\\\\overline{x};a,b \\\\right).$ Therefore for all $a,b\\\\in \\\\mathcal {E}$ we have $\\\\widehat{\\\\varphi }^*(\\\\overline{x};a,b) = \\\\int \\\\varphi ^*(\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}= q^{-1}\\\\int \\\\varphi ^*(\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}= q^{-1}\\\\widehat{\\\\varphi }(\\\\overline{x};a,b),$ and $\\\\mathbb {S}\\\\left(\\\\widehat{}^*(0)\\\\right) = q^{-1} \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"We can therefore achieve our assumption by choosing $q=\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"Furthermore, under this scaling we find that the old event $\\\\left\\\\lbrace \\\\left(\\\\overline{x},a\\\\right) \\\\longleftrightarrow \\\\left(\\\\overline{y},b\\\\right)\\\\textrm { in } \\\\xi ^{x,y}\\\\right\\\\rbrace $ becomes the new event $\\\\left\\\\lbrace \\\\left(q^{-1/d}\\\\overline{x},a\\\\right) \\\\longleftrightarrow \\\\left(q^{-1/d}\\\\overline{y},b\\\\right)\\\\textrm { in } \\\\xi ^{\\\\left(q^{-1/d}\\\\overline{x},a\\\\right),\\\\left(q^{-1/d}\\\\overline{y},b\\\\right)} \\\\right\\\\rbrace $ , and we get $\\\\tau ^*_{\\\\lambda ^*}(\\\\overline{x};a,b) = \\\\tau _\\\\lambda \\\\left(q^\\\\frac{1}{d}\\\\overline{x};a,b\\\\right)$ where $\\\\tau ^*_{\\\\lambda ^*}$ is the two-point function in the RCM governed by the connection function $\\\\varphi ^*$ and intensity $\\\\lambda ^*$ .\",\n \"It is also worth noting that the scaling choice to have $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)=1$ means that REF reduces to the condition that there exists a $d$ -independent constant $C>0$ such that $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &\\\\le C,\\\\\\\\\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &\\\\le C \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ Note that the inequalities relating $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{2,\\\\infty }$ , and $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ follow from Jensen's inequality and a supremum bound on a probability space.\",\n \"The Bootstrap Function To show that the bounds arising from the diagrams in Section are sufficient for our purposes, we make use of a bootstrap argument.\",\n \"Define $\\\\mu _\\\\lambda & = 1- \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)},\\\\\\\\\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) &= \\\\frac{1}{1 - \\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)},\\\\\\\\\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right) &=\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right)\\\\left(\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l-k\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l\\\\right) + \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l+k\\\\right) + \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l-k\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l+k\\\\right)\\\\right).$ From these we can then define the bootstrap function $f= f_1\\\\vee f_2\\\\vee f_3$ , where $f_1\\\\left(\\\\lambda \\\\right) = \\\\lambda , \\\\qquad f_2\\\\left(\\\\lambda \\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d} \\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}, \\\\qquad f_3\\\\left(\\\\lambda \\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k,l\\\\in \\\\mathbb {R}^d}\\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)}.$ Let us now outline some initial properties of $\\\\mu _\\\\lambda $ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ .\",\n \"Lemma 6.1 The map $\\\\lambda \\\\mapsto \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)$ is continuous and bounded below by $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ on $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"Therefore the map $\\\\lambda \\\\mapsto \\\\mu _\\\\lambda $ is a continuous map from $\\\\left[0,\\\\lambda _O\\\\right)\\\\rightarrow \\\\left[0,1\\\\right)$ .\",\n \"Furthermore, $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ is bounded below by $\\\\left(1+\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)^{-1}$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"We first recall that $\\\\lambda \\\\mapsto \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)$ is differentiable with respect to the operator norm from Corollary REF , and is therefore continuous.\",\n \"Lemma REF then implies that $\\\\lambda \\\\mapsto \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)$ is continuous.\",\n \"For the bounded below property, we use Lemma REF to get $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\ge \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) = 1$ .\",\n \"The continuity of $\\\\mu _\\\\lambda $ follows directly from the continuity and positivity of $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)$ .\",\n \"The bound $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\ge 1$ bounds $\\\\mu _\\\\lambda \\\\ge 0$ , and $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} < \\\\infty $ shows that $\\\\mu _\\\\lambda <1$ for all $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"The case $\\\\mu _0 = 0$ follows immediately from $\\\\mathcal {T}_0 = $ .\",\n \"Because $\\\\mu _\\\\lambda \\\\in \\\\left[0,1\\\\right)$ , to minimise $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ we want to find a lower bound for $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)$ .\",\n \"The $k$ -uniform bound on $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ then follows from $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) \\\\ge -\\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\rm op} \\\\ge -\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Using the Bootstrap function to bound terms Bounding the Lace Operator We have managed to bound the lace operator and its displacement in terms of a variety of objects.\",\n \"For the lace operator itself (and its associated Fourier transformed operators), these are: $\\\\lambda , \\\\triangle _\\\\lambda , \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }, \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }.$ As a guide, we expect $\\\\lambda $ , $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ , and $\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }$ to be bounded uniformly in $d$ and $\\\\lambda <\\\\lambda _O$ , whereas we expect $\\\\triangle _\\\\lambda $ to decay uniformly in $\\\\lambda <\\\\lambda _O$ as $d\\\\rightarrow \\\\infty $ .\",\n \"For some, these properties are easy to prove.\",\n \"The boundedness of $\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }$ follows from $\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le C,$ where we have used REF via (REF ).\",\n \"Regarding $\\\\lambda $ , since we are only interested in $\\\\lambda <\\\\lambda _O$ the question is perhaps more clearly written as “Is $\\\\lambda _O$ uniformly bounded in $d$ ?\",\n \"\\\".\",\n \"While there may be more elementary ways of proving this directly, we will get it for free from the bootstrap argument.\",\n \"Since we have the bound $\\\\lambda \\\\le f(\\\\lambda )$ , proving the uniform boundedness of $f(\\\\lambda )$ with respect to $d$ and $\\\\lambda $ for $\\\\lambda <\\\\lambda _O$ will prove the same for $\\\\lambda _O$ .\",\n \"This will leave us with only $\\\\triangle _\\\\lambda $ and $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ .\",\n \"Recall that we can write $\\\\triangle _\\\\lambda = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }, \\\\quad \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + 1.$ Then $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ differs from $\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda $ in that now the supremum is taken over all the marks, and adjacent $\\\\tau _\\\\lambda $ can have different marks.\",\n \"To get the desired decay and boundedness behaviour, we will therefore only need to prove that $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ decays and to bound $&\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}- \\\\overline{v};a_3,a_4\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{v};a_5,a_6\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{v},\\\\\\\\&\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}.$ We begin by proving the decay of $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"Lemma 6.2 Let $x,y \\\\in \\\\mathbb {X}$ .\",\n \"Then $\\\\tau _\\\\lambda (x,y) &\\\\le \\\\varphi (x,y) + \\\\lambda \\\\int \\\\varphi (x,u)\\\\tau _\\\\lambda (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right),\\\\\\\\\\\\tau _\\\\lambda (x,y) &\\\\le \\\\varphi (x,y) + \\\\lambda \\\\int \\\\tau _\\\\lambda (x,u)\\\\varphi (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ Then for $n\\\\ge 2$ , $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\sum _{m=0}^n\\\\genfrac(){0.0pt}0{n}{m}\\\\lambda ^{m} \\\\left\\\\Vert ^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ By combining Mecke's formula and the BK inequality, we obtain $\\\\tau _\\\\lambda (x,y) & \\\\le \\\\varphi (x,y) + \\\\mathbb {E}_\\\\lambda \\\\left[\\\\sum _{u \\\\in \\\\eta } 1_{\\\\lbrace x \\\\sim u \\\\text{ in } \\\\xi ^{x} \\\\rbrace \\\\cap \\\\lbrace u \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{y}\\\\rbrace } \\\\right] \\\\nonumber \\\\\\\\& = \\\\varphi (x,y) + \\\\lambda \\\\int \\\\mathbb {P}_\\\\lambda \\\\left(\\\\lbrace x \\\\sim u \\\\text{ in } \\\\xi ^{x,u} \\\\rbrace \\\\cap \\\\lbrace u \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{y,u}\\\\rbrace \\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\nonumber \\\\\\\\& = \\\\varphi (x,y) +\\\\lambda \\\\int \\\\varphi (x,u) \\\\tau _\\\\lambda (u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ In the last line, we have used that the two intersected events are independent.\",\n \"This is due to the fact that $\\\\nu $ is non-atomic and $x\\\\notin \\\\eta ^{y,u}$ a.s.\",\n \"The second inequality is then proven nearly identically, but with the adjacency to the intermediate point $u$ holding for $y$ instead of $x$ .\",\n \"To get (REF ), we first use () to expand out the right-most $\\\\mathcal {T}_\\\\lambda $ .\",\n \"This produces one term with $$ replacing the $\\\\mathcal {T}_\\\\lambda $ , and one term with $\\\\lambda \\\\mathcal {T}_\\\\lambda $ replacing it.\",\n \"We then repeatedly use (REF ) to expand the left-most $\\\\mathcal {T}_\\\\lambda $ to get a $$ factor and a $\\\\lambda \\\\mathcal {T}_\\\\lambda $ factor until we have a total of $n$ factors of $$ in the expression.\",\n \"Counting the number of ways to get to each possible end term then gives the binomial $\\\\genfrac(){0.0pt}1{n}{m}$ factor as required.\",\n \"The following lemma will allow us to describe $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ using Hilbert space language.\",\n \"Lemma 6.3 Let $H\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ be an integral linear operator with kernel function $h\\\\colon \\\\mathcal {E}^2\\\\rightarrow \\\\mathbb {R}$ .\",\n \"Then for all $h\\\\left(a,b\\\\right)\\\\in {\\\\rm ess.Im}\\\\left(h\\\\right)$ and $\\\\varepsilon >0$ , there exist $\\\\mathcal {P}$ -positive sets $E_1,E_2\\\\subset \\\\mathcal {E}$ such that $g_i(c) := \\\\frac{1}{\\\\mathcal {P}\\\\left(E_i\\\\right)}1_{E_i}\\\\left(c\\\\right)$ for $i=1,2$ satisfy $*{h\\\\left(a,b\\\\right) - {g_1}{Hg_2}} \\\\le \\\\varepsilon .$ Since $h$ is measurable, the set $A=\\\\left\\\\lbrace c,c^{\\\\prime }\\\\in \\\\mathcal {E}:*{h\\\\left(c,c^{\\\\prime }\\\\right)-h\\\\left(a,b\\\\right)}\\\\le \\\\varepsilon \\\\right\\\\rbrace $ is measurable.\",\n \"Since $h\\\\left(a,b\\\\right)\\\\in {\\\\rm ess.Im}\\\\left(h\\\\right)$ , it has positive measure: $\\\\mathcal {P}\\\\left(A\\\\right)>0$ .\",\n \"Furthermore, by considering the $\\\\pi $ -system of product sets $\\\\left\\\\lbrace E_1\\\\times E_2: E_1,E_2\\\\subset \\\\mathcal {E}\\\\text{ measurable}\\\\right\\\\rbrace $ that generates the $\\\\sigma $ -algebra on $\\\\mathcal {E}^2$ , we can find $B = E_1\\\\times E_2$ such that $E_1,E_2\\\\subset \\\\mathcal {E}$ are both measurable, $B\\\\subset A$ , and $\\\\mathcal {P}^{\\\\otimes 2}\\\\left(B\\\\right) = \\\\mathcal {P}\\\\left(E_1\\\\right)\\\\mathcal {P}\\\\left(E_2\\\\right)>0$ .\",\n \"Then use $E_1,E_2$ to define $g_1,g_2$ as in the statement on the lemma.\",\n \"Since $E_1,E_2$ are both measurable and have positive measure, both $g_1,g_2\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ .\",\n \"We also have ${g_1}{Hg_2} = \\\\frac{1}{\\\\mathcal {P}^{\\\\otimes 2}\\\\left(B\\\\right)}\\\\int _B h\\\\left(c,c^{\\\\prime }\\\\right)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime }\\\\right) \\\\in \\\\left[h(a,b) - \\\\varepsilon ,h(a,b) +\\\\varepsilon \\\\right],$ proving the result.\",\n \"Let us define the function $\\\\overline{\\\\epsilon }\\\\colon \\\\mathbb {N}\\\\rightarrow \\\\mathbb {R}_+$ , where $\\\\overline{\\\\epsilon }(d) := {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{1}{d} + \\\\frac{\\\\log d}{\\\\log g(d)} &: \\\\lim _{d\\\\rightarrow \\\\infty }g(d)\\\\rho ^{-d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^2 = 0 \\\\qquad \\\\forall \\\\rho >0,\\\\\\\\0&: \\\\text{otherwise}.\\\\end{array}\\\\right.\",\n \"}$ Recall $g(d)$ is the function defined in REF .\",\n \"In particular, $\\\\overline{\\\\epsilon }(d) = 0$ unless $g(d)$ approaches zero particularly quickly, and $\\\\beta (d) = g(d)^{\\\\frac{1}{4}-\\\\frac{3}{2}\\\\overline{\\\\epsilon }(d)}$ .\",\n \"Lemma 6.4 Let $\\\\lambda <\\\\lambda _O$ , $n\\\\ge 2$ , and $d>2m$ .\",\n \"Then there exists finite $c_f= c_f(m,n,f(\\\\lambda ))$ (increasing in $f(\\\\lambda )$ and independent of $d$ ) such that $\\\\left\\\\Vert ^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f&: n=2\\\\\\\\c_fg(d)^{\\\\frac{1}{2}-m\\\\overline{\\\\epsilon }(d)} &: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ The condition $\\\\lambda < \\\\lambda _O$ here is required to allow us to swap integrals over the mark space with integrals over the Fourier space.\",\n \"We are able to use Fubini's Theorem because sub-criticality implies that the relevant integrals are finite.\",\n \"For clarity, let us introduce the notation $\\\\varphi ^{(n)}(x,y) = \\\\int \\\\prod ^{n}_{j=1}\\\\varphi (u_{j-1},u_{j})\\\\nu ^{\\\\otimes (n-1)}\\\\left(\\\\mathrm {d}u_{[1,n-1]}\\\\right),\\\\quad \\\\tau _\\\\lambda ^{(n)}(x,y) = \\\\int \\\\prod ^{n}_{j=1}\\\\tau _\\\\lambda (u_{j-1},u_{j})\\\\nu ^{\\\\otimes (n-1)}\\\\left(\\\\mathrm {d}u_{[1,n-1]}\\\\right),$ where $u_0=x$ and $u_n=y$ .\",\n \"Furthermore, for each $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ let us define the integral linear operator $\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) \\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel function $(a,b)\\\\mapsto \\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) := \\\\int \\\\varphi ^{(n-1)}\\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\tau _\\\\lambda ^{(m)}\\\\left(\\\\overline{u}-\\\\overline{u}^{\\\\prime };c,c^{\\\\prime }\\\\right)\\\\varphi \\\\left(\\\\overline{u}^{\\\\prime };c^{\\\\prime },b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^{\\\\prime }\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime }\\\\right).$ Given $\\\\delta >0$ , $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ , and $a,b\\\\in \\\\mathcal {E}$ such that $\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};a,b\\\\right)\\\\in {\\\\rm ess.Im}\\\\left(\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};\\\\cdot ,\\\\cdot \\\\right)\\\\right)$ , we can use Lemma REF to get measurable and $\\\\mathcal {P}$ -positive sets $E^{(\\\\overline{x},a,b,\\\\delta )}_1,E^{(\\\\overline{x},a,b,\\\\delta )}_2\\\\subset \\\\mathcal {E}$ such that $\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) \\\\le *{g^{(\\\\overline{x},a,b,\\\\delta )}_1}{\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) g^{(\\\\overline{x},a,b,\\\\delta )}_2} + \\\\delta ,$ where $g^{(\\\\overline{x},a,b,\\\\delta )}_i(c) = \\\\frac{1}{\\\\mathcal {P}\\\\left(E^{(\\\\overline{x},a,b,\\\\delta )}_i\\\\right)}1_{E^{(\\\\overline{x},a,b,\\\\delta )}_i}\\\\left(c\\\\right)$ for $i=1,2$ .\",\n \"Now let $E_1,E_2\\\\subset \\\\mathcal {E}$ be arbitrary measurable $\\\\mathcal {P}$ -positive sets and $g_1,g_2$ be their associated functions.\",\n \"We aim to produce bounds that are independent of the choice of $E_1,E_2$ .\",\n \"We first use the Fourier inversion theorem to reformulate the position behaviour in terms of a $k$ -integral of the Fourier transforms, and use REF and $\\\\lambda <\\\\lambda _O$ (so the relevant integrals are finite) to allow us to swap the $k$ -integral and the mark integrals.\",\n \"This produces $*{*{g_1}{\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) g_2}} = *{\\\\int \\\\text{e}^{-i\\\\overline{x}\\\\cdot k}*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}}\\\\\\\\ \\\\le \\\\int *{*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}.$ Recall that $\\\\widehat{}(k)$ is self-adjoint, so we can move $\\\\widehat{}(k)^{n-1}$ to the other side of the inner product, and then we can use Cauchy-Schwarz and the definition of the operator norm to get $*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2} = *{\\\\widehat{}(k)^{n-1}g_1}{\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2} \\\\\\\\\\\\le *{\\\\widehat{}(k)^{n-1}g_1}_2*{\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2}_2 \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op}^m*{\\\\widehat{}(k)^{n-1}g_1}_2*{\\\\widehat{}(k)g_2}_2.$ We can then use the definition of the bootstrap function $f(\\\\lambda )$ to replace the $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op}$ factors with factors of $f(\\\\lambda )$ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ .\",\n \"Also writing the norm $*{\\\\cdot }_2$ as the square root of an inner product and using the self-adjoint property of $\\\\widehat{}(k)$ gives $*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2} \\\\le f(\\\\lambda )^m\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}.$ We therefore want to derive bounds for the integral $\\\\int \\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}.$ We first derive pointwise and integral bounds for the $\\\\widehat{}$ factors.\",\n \"Given $N\\\\ge 1$ , and $g(c)=\\\\frac{1}{\\\\mathcal {P}\\\\left(E\\\\right)}1_E(c)$ for some measurable and $\\\\mathcal {P}$ -positive set $E\\\\subset \\\\mathcal {E}$ , $*{g}{\\\\widehat{}(k)^{2N}g}^\\\\frac{1}{2} = \\\\left(\\\\frac{1}{\\\\mathcal {P}(E)^2}\\\\int _E\\\\int _E\\\\widehat{\\\\varphi }^{(2N)}(k;c,c^{\\\\prime })\\\\mathcal {P}(\\\\mathrm {d}c)\\\\mathcal {P}(\\\\mathrm {d}c^{\\\\prime })\\\\right)^\\\\frac{1}{2} \\\\le \\\\left\\\\Vert \\\\widehat{}(k)^{2N}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2}.$ By using Cauchy-Schwarz and supremum bounds we can split this into norms of $\\\\widehat{}(k)$ only, then use Lemma REF to replace $k$ with 0, and finally use REF to bound these terms: $*{g}{\\\\widehat{}(k)^{2N}g}^\\\\frac{1}{2}\\\\le \\\\left\\\\Vert \\\\widehat{}(k)^{2N}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2} \\\\le \\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{2,\\\\infty }\\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty }^{N-1} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty }\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty }^{N-1} \\\\le C^N.$ For the integral bound, we apply Cauchy-Schwarz to the $k$ -integral and swap the $k$ -integrals with the mark integrals to un-do the Fourier transforms.\",\n \"We get $\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^\\\\frac{1}{2}\\\\left(\\\\int *{g_2}{\\\\widehat{}(k)^2g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^\\\\frac{1}{2} \\\\\\\\= *{g_1}{^{(2n-2)}\\\\left(\\\\overline{0}\\\\right)g_1}^\\\\frac{1}{2}*{g_2}{^{(2)}\\\\left(\\\\overline{0}\\\\right)g_2}^\\\\frac{1}{2},$ where $^{(N)}\\\\left(\\\\overline{0}\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ is the integral operator with kernel function $\\\\left(a,b\\\\right)\\\\mapsto \\\\varphi ^{(N)}\\\\left(\\\\overline{0};a,b\\\\right)$ .\",\n \"Given $N\\\\ge 1$ , and $g(c)=\\\\frac{1}{\\\\mathcal {P}\\\\left(E\\\\right)}1_E(c)$ for some measurable and $\\\\mathcal {P}$ -positive set $E\\\\subset \\\\mathcal {E}$ , $*{g}{^{(2N)}\\\\left(\\\\overline{0}\\\\right)g} = \\\\left(\\\\frac{1}{\\\\mathcal {P}(E)^2}\\\\int _E\\\\int _E\\\\varphi ^{(2N)}\\\\left(\\\\overline{0};c,c^{\\\\prime }\\\\right)\\\\mathcal {P}(\\\\mathrm {d}c)\\\\mathcal {P}(\\\\mathrm {d}c^{\\\\prime })\\\\right) \\\\le \\\\left\\\\Vert ^{(2N)}\\\\left(\\\\overline{0}\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Here for $N\\\\ge 2$ we bound with $\\\\left\\\\Vert ^{(2N)}\\\\left(\\\\overline{0}\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2} \\\\le \\\\left\\\\Vert ^{2N}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2} \\\\le g(d)^\\\\frac{1}{2}$ from REF .\",\n \"For $N=1$ we bound using $\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\in \\\\left[0,1\\\\right]$ and REF : $\\\\left\\\\Vert ^{(2)}\\\\left(\\\\overline{0}\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,c\\\\right)\\\\varphi \\\\left(-\\\\overline{x};c,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right) \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\\\\\= \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty }\\\\le C.$ Therefore $\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}C &: n=2,\\\\\\\\C^\\\\frac{1}{2}g(d)^\\\\frac{1}{2} &: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ We now return to bounding (REF ).\",\n \"To perform the $k$ -integral, we split the domain.\",\n \"We first consider $B_\\\\varepsilon (0) = \\\\lbrace k\\\\in \\\\mathbb {R}^d: *{k} < \\\\varepsilon \\\\rbrace $ , where we choose $\\\\varepsilon >0$ such that $\\\\varepsilon ^2 \\\\le \\\\tfrac{1-C_1}{C_2}$ - the constants $C_1,C_2$ coming from the assumption REF .\",\n \"From REF , we have $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k) = \\\\left(1 - \\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)^{-1}\\\\le 1/C_2*{k}^2$ on $B_\\\\varepsilon (0)$ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)\\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ on $B_\\\\varepsilon (0)^c$ .\",\n \"Our pointwise bound on the $\\\\widehat{}$ factors then gives $\\\\int _{B_\\\\varepsilon (0)}\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\\\\\\\\\le \\\\frac{C^n}{C^m_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\int ^\\\\varepsilon _0\\\\frac{1}{r^{2m}}r^{d-1}\\\\mathrm {d}r = \\\\frac{C^n}{C^m_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{d-2m}\\\\frac{\\\\varepsilon ^{d-2m}}{\\\\left(2\\\\pi \\\\right)^d},$ where $\\\\mathfrak {S}_{d-1}= d\\\\pi ^\\\\frac{d}{2}/\\\\Gamma \\\\left(\\\\tfrac{d}{2}+1\\\\right)$ is the surface area of an unit $d$ -sphere.\",\n \"Note that for all fixed $\\\\varepsilon >0$ this term vanishes in the $d\\\\rightarrow \\\\infty $ limit.\",\n \"To deal with the integral over $B_\\\\varepsilon (0)^c$ , we find the upper bound $\\\\int _{B_\\\\varepsilon (0)^c}\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\\\\\\\\\le \\\\frac{1}{C^m_2\\\\varepsilon ^{2m}}\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\le \\\\frac{C^\\\\frac{1}{2}}{C^m_2\\\\varepsilon ^{2m}}{\\\\left\\\\lbrace \\\\begin{array}{ll}C^\\\\frac{1}{2} &: n=2\\\\\\\\g(d)^\\\\frac{1}{2} &: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ Note that the bounds we found were independent of the sets $E_1,E_2$ , and therefore these bounds also apply to $*{g^{(\\\\overline{x},a,b,\\\\delta )}_1}{\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) g^{(\\\\overline{x},a,b,\\\\delta )}_2}$ uniformly over every $\\\\delta >0$ , $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ , and $\\\\mathcal {P}$ -almost every $a,b\\\\in \\\\mathcal {E}$ .\",\n \"Since we can take $\\\\delta \\\\rightarrow 0$ , our bound also applies to $\\\\left\\\\Vert ^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"For $n=2$ , the $B_\\\\varepsilon (0)^c$ integral dominates the $B_\\\\varepsilon (0)$ integral, and so the result is proven.\",\n \"For $n\\\\ge 3$ , both parts of the integral approach 0 as $d\\\\rightarrow \\\\infty $ .\",\n \"If both are of the same order or if the $B_\\\\varepsilon (0)^c$ integral dominates, then we can fix $\\\\varepsilon >0$ and get the result.\",\n \"On the other hand, if $g(d)^\\\\frac{1}{2} \\\\ll \\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^{d}$ for all fixed $\\\\varepsilon >0$ and the $B_\\\\varepsilon (0)$ integral always dominates, we can improve the overall bound by letting $\\\\varepsilon =\\\\varepsilon (d)$ and having $\\\\varepsilon (d)\\\\rightarrow 0$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Having a smaller value of $\\\\varepsilon $ produces a smaller bound for the $B_\\\\varepsilon (0)$ integral, but a larger bound for the $B_\\\\varepsilon (0)^c$ integral.\",\n \"To get an optimal $\\\\varepsilon $ we can set both terms to be of the same order - that is by having $\\\\varepsilon (d) = 2\\\\pi ^{\\\\frac{1}{2}}A^\\\\frac{1}{d}g(d)^\\\\frac{1}{2d}\\\\left(1-\\\\frac{2m}{d}\\\\right)^\\\\frac{1}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d}$ for some fixed $A>0$ .\",\n \"For this choice, we have both integrals of the order $\\\\varepsilon ^{-2m}g(d)^\\\\frac{1}{2} = 2^{-2m}\\\\pi ^{-m}A^{-\\\\frac{2m}{d}} g(d)^{\\\\frac{1}{2}-\\\\frac{m}{d}}\\\\left(1-\\\\frac{2m}{d}\\\\right)^{-\\\\frac{2m}{d}}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^{-\\\\frac{2m}{d}} \\\\\\\\= \\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^m g(d)^{\\\\frac{1}{2}-\\\\frac{m}{d}}\\\\frac{1}{d^m}\\\\left(1+o\\\\left(1\\\\right)\\\\right),$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Proposition 6.5 (Bound for the $n$ -gon diagrams) Let $\\\\lambda < \\\\lambda _O$ .\",\n \"Then for each $n$ such that $d>2n$ , there exists finite $c_f= c_f(m,n,f(\\\\lambda ))$ (increasing in $f(\\\\lambda )$ and independent of $d$ ) such that $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f&: n=1,2,\\\\\\\\c_fg(d)^{\\\\frac{1}{2}-n\\\\overline{\\\\epsilon }(d)}&: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ The $n=1$ case is trivial because $\\\\tau _\\\\lambda \\\\left(\\\\overline{x},a,b\\\\right)\\\\in \\\\left[0,1\\\\right]$ .\",\n \"For $n=2$ , we first bound $\\\\left\\\\Vert ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"We get $\\\\left\\\\Vert ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{x,y\\\\in \\\\mathbb {X}}\\\\int *{\\\\varphi (x,u)\\\\varphi (u,y)}\\\\nu (\\\\mathrm {d}u) \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\int *{\\\\varphi (x,u)}^2\\\\nu (\\\\mathrm {d}u) \\\\\\\\\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\int *{\\\\varphi (x,u)}\\\\nu (\\\\mathrm {d}u) = \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty },$ by using Cauchy-Schwarz and symmetry of $\\\\varphi $ in the first inequality and $\\\\varphi (x,u)\\\\in \\\\left[0,1\\\\right]$ in the second.\",\n \"We then use REF to bound this by the constant $C$ .\",\n \"We use Lemma REF to get the required bound for the remaining terms arising from (REF ) in Lemma REF .\",\n \"For $n\\\\ge 3$ , we once again bound the terms in the relevant expansion in (REF ).\",\n \"For $\\\\left\\\\Vert ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ we use REF to get $\\\\left\\\\Vert ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty }= O\\\\left(g(d)^\\\\frac{1}{2}\\\\right)$ .\",\n \"We then use Lemma REF to bound the remaining terms.\",\n \"For our proofs of Lemma REF and Lemma REF , we adopt the following notation.\",\n \"Given $g(\\\\overline{x};a,b)$ and $h(\\\\overline{x};a,b)$ , we define $(gh)(\\\\overline{x};a,b) := \\\\int g(\\\\overline{x}-\\\\overline{y};a,c)h(\\\\overline{y};c,b)\\\\mathrm {d}\\\\overline{y}\\\\mathcal {P}(\\\\mathrm {d}c).$ This notational convention is associative and therefore can generalise to three or more terms unambiguously.\",\n \"Lemma 6.6 For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}&\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}- \\\\overline{v};a_3,a_4\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{v};a_5,a_6\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{v}\\\\le c_f,\\\\\\\\\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}&\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\le c_f,$ and therefore $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le 1 + 2c_f.$ By applying (REF ) and (), we can get $\\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\le \\\\varphi (\\\\overline{x};a,b) + \\\\lambda (\\\\varphi \\\\varphi )(\\\\overline{x};a,b)+ \\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi )(\\\\overline{x};a,b).$ We will in fact prove the more general result for the convolution of $m\\\\ge 2$ $\\\\tau _\\\\lambda $ -functions in dimensions $d>2m$ .\",\n \"By bounding $\\\\lambda \\\\le f(\\\\lambda )$ , and using Lemma REF and the Fourier inversion theorem we can realise that we only need to get further bounds for the following objects.\",\n \"For $\\\\vec{j}\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace ^m$ , we want to bound $\\\\int \\\\prod ^m_{i=1} P^{j_i}_{2i-1,2i}(k)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}},$ where $P^{j}_{p,q}(k) := {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_p}{\\\\widehat{}(k) g_q}} &:j=1\\\\\\\\*{*{g_p}{\\\\widehat{}(k)^2g_q}} &:j=2\\\\\\\\*{*{g_p}{\\\\widehat{}(k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\widehat{}(k)g_q}} &: j=3,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\left\\\\lbrace g_i\\\\right\\\\rbrace _{i=1}^{2m}$ are the functions of the form $g_i(a) = \\\\frac{1}{\\\\mathcal {P}(E_i)}1_{E_i}(a)$ arising from the application of Lemma REF .\",\n \"We now bound the following terms in the same way as (REF ): $*{*{g_1}{\\\\widehat{}(k)^2g_2}} &\\\\le *{g_1}{\\\\widehat{}(k)^2g_1}^{\\\\frac{1}{2}}*{g_2}{\\\\widehat{}(k)^2g_2}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_1}{\\\\widehat{}(k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\widehat{}(k)g_2}} &\\\\le f\\\\left(\\\\lambda \\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)*{g_1}{\\\\widehat{}(k)^2g_1}^{\\\\frac{1}{2}}*{g_2}{\\\\widehat{}(k)^2g_2}^{\\\\frac{1}{2}}.$ In terms of pointwise uniform bounds for the $\\\\widehat{}$ terms, we have $*{*{g_1}{\\\\widehat{}(k)g_2}}\\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C$ and $*{g}{\\\\widehat{}(k)^2g}\\\\le C^2$ (the latter also having been used in the proof of Lemma REF ).\",\n \"As in Lemma REF , we will require various integral bounds for these $\\\\widehat{}$ terms.\",\n \"Suppose we have $r \\\\in \\\\left\\\\lbrace 1,\\\\ldots ,m-1\\\\right\\\\rbrace $ instances of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and $m-r$ instances of $P^{1}_{p,q}(k)$ .\",\n \"Then by applying Cauchy-Schwarz and the definitions of the $g_i$ we get $&\\\\int \\\\prod ^{r}_{i=1} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}^{\\\\frac{1}{2}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}^{\\\\frac{1}{2}} \\\\prod ^{m}_{i^{\\\\prime }=r+1}*{*{g_{2i^{\\\\prime }-1}}{\\\\widehat{}(k)g_{2i^{\\\\prime }}}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)^2g_1}*{g_2}{\\\\widehat{}(k)^2g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\int \\\\prod ^{r}_{i=2} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}\\\\prod ^{m}_{i^{\\\\prime }=r+1}*{g_{2i^{\\\\prime }-1}}{\\\\widehat{}(k)g_{2i^{\\\\prime }}}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_5,\\\\ldots ,a_{2m}\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_5,a_6)\\\\star \\\\ldots \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_{2r-1},a_{2r})\\\\right.\\\\right.\\\\nonumber \\\\\\\\&\\\\hspace{199.16928pt}\\\\left.\\\\left.\\\\star \\\\varphi (\\\\cdot ;a_{2r+1},a_{2r+2})\\\\star \\\\ldots \\\\star \\\\varphi (\\\\cdot ;a_{2m-1},a_{2m})\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le C^{m+r}.$ In this last inequality we have used $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\le C$ and $\\\\left\\\\Vert \\\\widehat{}(0)^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty }^2 \\\\le C^2$ to extract off $\\\\varphi $ and $\\\\varphi ^{(2)}$ via supremum bounds on the spatial position.\",\n \"We will also require integral bounds where there are $m$ factors of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and where there are $m$ factors of $P^{1}_{p,q}(k)$ : $&\\\\int \\\\prod ^{m}_{i=1} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}^{\\\\frac{1}{2}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}^{\\\\frac{1}{2}} \\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)^2g_1}*{g_2}{\\\\widehat{}(k)^2g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}}\\\\left(\\\\int \\\\prod ^{m}_{i=2} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_5,\\\\ldots ,a_{2m}\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_5,a_6)\\\\star \\\\ldots \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_{2m-1},a_{2m})\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le C^{2m}.\\\\\\\\&\\\\int \\\\prod ^{m}_{i=1}*{*{g_{2i-1}}{\\\\widehat{}(k)g_{2i}}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)g_2}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\left(\\\\prod ^{m}_{i=2}*{g_{2i-1}}{\\\\widehat{}(k)g_{2i}}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_5,\\\\ldots ,a_{2m}\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_5,a_6)\\\\star \\\\ldots \\\\star \\\\varphi (\\\\cdot ;a_{2m-1},a_{2m})\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le C^{m}.$ Note that for $m\\\\ge 3$ , better bounds are available via REF , but will not be required here.\",\n \"We can now proceed to bound each of the expressions of the form (REF ) in much the same way as we did in the proof of Lemma REF .\",\n \"We set $\\\\varepsilon >0$ and partition the integral over $\\\\mathbb {R}^d$ into one over $B_\\\\varepsilon (0)$ and one over $B_\\\\varepsilon (0)^c$ .\",\n \"For the $B_\\\\varepsilon (0)$ integral we use the uniform bounds on the $\\\\widehat{}$ terms and perform the integral of the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ terms (if there are any) as before.\",\n \"Note that we require $d>6$ for all of these to be finite.\",\n \"For the $B_\\\\varepsilon (0)^c$ integrals we uniformly bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ and use the integral bounds we calculated above.\",\n \"Since we are only asking for a constant bound, we don't need to worry about having $\\\\varepsilon \\\\rightarrow 0$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Bounding the Lace Operator Displacement We now consider the objects required to bound the displacement of the lace operator.\",\n \"In addition to the objects used to bound the lace operator itself, we require bounds on $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }, \\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }, \\\\overline{W_k}, \\\\overline{H_k}, \\\\triangle ^{(B)}_\\\\lambda , \\\\mathbb {B}.$ The decay $\\\\mathbb {B}\\\\le g(d)$ follows from Assumption REF , and the following Observation REF demonstrates the decay of $\\\\triangle ^{(B)}_\\\\lambda $ .\",\n \"Observation 6.7 Note that the inequality (REF ) allows us to bound $\\\\triangle ^{(B)}_\\\\lambda $ in terms of other terms.\",\n \"By using the inequality twice and appropriately bounding $\\\\varphi $ with $\\\\tau _\\\\lambda $ , we get $\\\\int \\\\tau _\\\\lambda (r,u)\\\\tau _\\\\lambda (u,s)\\\\nu (\\\\mathrm {d}u) \\\\le \\\\int \\\\varphi (r,u)\\\\varphi (u,s)\\\\nu (\\\\mathrm {d}u) + 2\\\\lambda \\\\int \\\\tau _\\\\lambda (r,v)\\\\tau _\\\\lambda (v,u)\\\\tau _\\\\lambda (u,s)\\\\nu ^{\\\\otimes 2}(\\\\mathrm {d}u, \\\\mathrm {d}v)$ for any $r,s\\\\in \\\\mathbb {X}$ .\",\n \"Therefore $\\\\triangle ^{(B)}_\\\\lambda \\\\le 3\\\\triangle _\\\\lambda + \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{r\\\\notin B(s)}\\\\int \\\\varphi (r,u)\\\\varphi (u,s)\\\\nu (\\\\mathrm {d}u) \\\\le 3\\\\triangle _\\\\lambda + \\\\lambda g(d),$ where we have used REF .\",\n \"Therefore Proposition REF implies that for $d>6$ there exists $c_f$ such that $\\\\triangle ^{(B)}_\\\\lambda \\\\le c_f\\\\beta ^2.$ We are then left with $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ , $\\\\overline{W_k}$ , and $\\\\overline{H_k}$ , and we will deal them in this order.\",\n \"First note that $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\varphi _k(x,y)\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int (1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right))\\\\varphi (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}(\\\\mathrm {d}a) \\\\\\\\ = \\\\left\\\\Vert \\\\widehat{}_k(0)\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty }.$ The following lemma therefore also allows us to bound $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"Its extra generality will be required later.\",\n \"Lemma 6.8 For all $k,l\\\\in \\\\mathbb {R}^d$ , $\\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right],$ where $C>0$ is the $d$ -independent constant given in REF .\",\n \"The first two inequalities in (REF ) holds from the general inequalities that holds for these norms (on a probability space for the second inequality).\",\n \"Then we can remove the $l$ -dependence by performing the following calculation: $\\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+ k\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int *{\\\\text{e}^{il\\\\cdot \\\\overline{x}}}*{\\\\left(1-\\\\cos \\\\left( k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)}\\\\mathrm {d}\\\\overline{x}\\\\\\\\= \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int *{\\\\left(1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)}\\\\mathrm {d}\\\\overline{x}= \\\\left\\\\Vert \\\\widehat{}\\\\left(0\\\\right) - \\\\widehat{}\\\\left( k\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ The final step relating the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ -norm to the difference of the spectral suprema is precisely that given by () in REF .\",\n \"We now address $\\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"Lemma 6.9 For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\le c_f\\\\beta ^2\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ First note that for $k=0$ the kernel function $\\\\tau _{\\\\lambda ,k}\\\\left(x,y\\\\right)=0$ for all $x,y\\\\in \\\\mathbb {X}$ , and so the required inequality holds trivially.\",\n \"In this proof we hereafter may assume $k\\\\ne 0$ .\",\n \"We use (REF ) to write $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\left\\\\Vert ^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert ^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ We first get the bound on $\\\\left\\\\Vert ^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ , using a similar approach to Lemma REF .\",\n \"For each $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ we define the integral linear operator $\\\\left(^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel function $\\\\left(a,b\\\\right)\\\\mapsto \\\\left(^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) \\\\\\\\:= \\\\int \\\\varphi ^{(2)}\\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}-\\\\overline{u}^{\\\\prime };c,c^{\\\\prime }\\\\right)\\\\tau _{\\\\lambda ,k}\\\\left(\\\\overline{u}^{\\\\prime }-\\\\overline{u}^{\\\\prime \\\\prime };c^{\\\\prime },c^{\\\\prime \\\\prime }\\\\right)\\\\varphi \\\\left(\\\\overline{u}^{\\\\prime \\\\prime };c^{\\\\prime \\\\prime },b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^{\\\\prime } \\\\mathrm {d}\\\\overline{u}^{\\\\prime \\\\prime }\\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime },\\\\mathrm {d}c^{\\\\prime \\\\prime }\\\\right).$ Recall the notation $\\\\varphi ^{(2)}$ used in (REF ).\",\n \"As in the proof of Lemma REF , we let $E_1,E_2\\\\subset \\\\mathcal {E}$ be measurable and $\\\\mathcal {P}$ -positive sets, and aim to bound ${g_1}{\\\\left(\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2}$ (where $g_i(c) := \\\\frac{1}{\\\\mathcal {P}\\\\left(E_i\\\\right)}1_{E_i}\\\\left(c\\\\right)$ for $i=1,2$ ) independently of the choice of $E_1,E_2$ .\",\n \"If we do this we will have proven the result.\",\n \"As before, we use the Fourier inversion theorem to write ${g_1}{\\\\left(\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2}$ as the integral of an inner product of Fourier transformed operators over the Fourier argument.\",\n \"Having $\\\\lambda <\\\\lambda _O$ allows us to swap the integrals in this step.\",\n \"Then we can use Cauchy-Schwarz and the definitions of the operator norm and the bootstrap functions to extract factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)$ and $\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)$ from the inner product.\",\n \"The net result is $*{g_1}{\\\\left(^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2} &\\\\le \\\\int *{*{g_1}{\\\\widehat{}(l)^{2}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l)g_2}}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\&\\\\le f(\\\\lambda )^2\\\\int \\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right) *{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ The $\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)$ term produces the required factor of $\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\\\int \\\\left(2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)\\\\right)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ We partition our integral using open $\\\\varepsilon $ -balls around the poles where $\\\\varepsilon ^2 \\\\le \\\\tfrac{1-C_1}{C_2}$ - the constants $C_1$ and $C_2$ coming from REF .\",\n \"Let $B_{\\\\varepsilon }(p)$ denote the open $\\\\varepsilon $ -ball around $p\\\\in \\\\mathbb {R}^d$ .\",\n \"For $n\\\\in \\\\left\\\\lbrace 0,1,2,3\\\\right\\\\rbrace $ , let $A_n := \\\\left\\\\lbrace l\\\\in \\\\mathbb {R}^d: l\\\\in B_\\\\varepsilon (p) \\\\text{ for precisely $n$ elements }p\\\\in \\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace \\\\right\\\\rbrace .$ Firstly it is possible for $A_3,A_2 = \\\\emptyset $ if $*{k}$ is sufficiently large compared to $\\\\varepsilon $ .\",\n \"It is also easy to see that $A_3\\\\subset B_{\\\\varepsilon }(0)$ and $A_2\\\\subset B_{2\\\\varepsilon }(0)$ .\",\n \"We will partition $\\\\mathbb {R}^d$ into $A_3$ , $A_2$ , $A_1$ , and $A_0$ and bound the $l$ -integral on each part.\",\n \"Recall that REF implies that $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le C^{-1}_2*{l}^{-2}$ for $*{l}\\\\le \\\\varepsilon $ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le C^{-1}_2\\\\varepsilon ^2$ for $*{l}\\\\ge \\\\varepsilon $ .\",\n \"Also recall the pointwise (REF ) and integral (REF ) bounds we derived for the $\\\\widehat{}$ factors.\",\n \"We treat $A_3$ first: $&\\\\int _{A_3}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\le \\\\frac{C^3}{C^3_2}\\\\int _{B_\\\\varepsilon (0)\\\\cap B_\\\\varepsilon (k)}\\\\frac{1}{*{l}^4*{l-k}^2} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{2}{3}}\\\\left(\\\\int _{B_\\\\varepsilon (k)}\\\\frac{1}{*{l-k}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}} \\\\le \\\\frac{C^3}{C^3_2}\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}\\\\\\\\&\\\\int _{A_3}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\le \\\\frac{C^3}{C^3_2}\\\\int _{A_3}\\\\frac{1}{*{l+k}^2*{l}^2*{l-k}^2} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\le \\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (-k)}\\\\frac{1}{*{l+k}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}}\\\\left(\\\\int _{B_\\\\varepsilon (k)}\\\\frac{1}{*{l-k}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}} \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\le \\\\frac{C^3}{C^3_2}\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}$ Recall $\\\\mathfrak {S}_{d-1}= d\\\\pi ^\\\\frac{d}{2}/\\\\Gamma \\\\left(\\\\tfrac{d}{2}+1\\\\right)$ is the surface area of an unit $d$ -sphere.\",\n \"In these calculations we used Hölder's inequality and increased the domain of integration to get upper bounds.\",\n \"These integrals are finite for $d>6$ , and approach 0 if we take bounded $\\\\varepsilon $ .\",\n \"For $A_2$ the counting of cases is the only extra complication.\",\n \"A precise counting is entirely possible, but unnecessary for our purposes.\",\n \"It is simple to see that the arguments outlined for $A_3$ can be applied to get the bound as an integer multiple of $\\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\left(\\\\frac{1}{*{l}^6} + \\\\frac{1}{\\\\varepsilon ^2*{l}^4} + \\\\frac{1}{\\\\varepsilon ^4*{l}^2} \\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right) = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4} + \\\\frac{1}{d-2}\\\\right)\\\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a pair of two distinct points from $\\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace $ , there are terms with three, two, and one factor(s) of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ centred on these two points.\",\n \"This bound is finite for $d>6$ , and approaches 0 if we take bounded $\\\\varepsilon $ .\",\n \"Repeating this for $A_1$ , we get the bound as an integer multiple of $\\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\left(\\\\frac{1}{\\\\varepsilon ^2*{l}^4} + \\\\frac{1}{\\\\varepsilon ^4*{l}^2} + \\\\frac{1}{\\\\varepsilon ^6} \\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right) = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\left(\\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a single point from $\\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace $ , there are terms with two, one, and zero factor(s) of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ centred on this single point.\",\n \"This bound is finite for $d>4$ , and approaches 0 if we take bounded $\\\\varepsilon $ .\",\n \"The $A_0$ integral is qualitatively different.\",\n \"By taking the integral bound for the $\\\\widehat{}$ factors and a uniform (on $A_0$ ) bound for the factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ , we get the bound $\\\\frac{C^3}{C^3_2}\\\\varepsilon ^{-6}g(d)^\\\\frac{1}{2}.$ As in Lemma REF , if the $g(d)^\\\\frac{1}{2}$ term dominates, then we have a sufficient bound for our result.\",\n \"The largest of the other terms (for $d$ sufficiently large and $\\\\varepsilon \\\\le 1$ ) is the $\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}$ term.\",\n \"Like we did in Lemma REF , we can make these other terms smaller by decreasing $\\\\varepsilon $ at the cost of making the $g(d)^\\\\frac{1}{2}$ bound greater.\",\n \"The largest term can be minimised (up to a constant) by taking $\\\\varepsilon (d) = 2\\\\pi ^{\\\\frac{1}{2}}A^\\\\frac{1}{d}g(d)^\\\\frac{1}{2d}\\\\left(1-\\\\frac{6}{d}\\\\right)^\\\\frac{1}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d}$ for some fixed $A>0$ .\",\n \"For this choice, we have both integrals of the order $\\\\varepsilon ^{-6}g(d)^\\\\frac{1}{2} = 2^{-6}\\\\pi ^{-3}A^{-\\\\frac{6}{d}}g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}\\\\left(1-\\\\frac{6}{d}\\\\right)^{-\\\\frac{6}{d}}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^{-\\\\frac{6}{d}} = \\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^3 g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}\\\\frac{1}{d^3}\\\\left(1+o\\\\left(1\\\\right)\\\\right),$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Bounding the $\\\\left\\\\Vert ^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ term is similar.\",\n \"For each $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ we define the integral linear operator $\\\\left(^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel function $\\\\left(a,b\\\\right)\\\\mapsto \\\\left(^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) \\\\\\\\:= \\\\int \\\\varphi ^{(2)}\\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\tau _{\\\\lambda ,k}\\\\left(\\\\overline{u}-\\\\overline{u}^{\\\\prime };c,c^{\\\\prime \\\\prime }\\\\right)\\\\varphi \\\\left(\\\\overline{u}^{\\\\prime };c^{\\\\prime },b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^{\\\\prime } \\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime }\\\\right).$ We once again take $\\\\mathcal {P}$ -positive sets $E_1,E_2$ and corresponding $g_1,g_2$ .\",\n \"Then we use the Fourier inversion theorem and Cauchy-Schwarz to get $*{g_1}{\\\\left(^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2} \\\\le f(\\\\lambda )\\\\int \\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right) *{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ Then once again the $\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)$ term produces the required factor of $\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\\\int \\\\left(2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)\\\\right)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ Counting the number of factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ centred on each of the poles $\\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace $ then tells us that there is a uniform bound that is some integer multiple of $\\\\frac{C^2}{C^2_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\left(\\\\frac{1}{*{l}^{4}} + \\\\frac{1}{\\\\varepsilon ^2*{l}^{2}} + \\\\frac{1}{\\\\varepsilon ^4}\\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} + \\\\varepsilon ^{-4}g(d)^\\\\frac{1}{2}\\\\right) \\\\\\\\= \\\\frac{C^2}{C^2_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\left(\\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-4} + \\\\frac{C^2}{C^2_2}\\\\varepsilon ^{-4}g(d)^\\\\frac{1}{2}.$ Once again, if the $g(d)^\\\\frac{1}{2}$ term dominates the result is proven.\",\n \"Otherwise, we let $\\\\varepsilon $ vary with $d$ .\",\n \"The largest term can be minimised (up to a constant) by taking $\\\\varepsilon (d) = 2\\\\pi ^{\\\\frac{1}{2}}A^\\\\frac{1}{d}g(d)^\\\\frac{1}{2d}\\\\left(1-\\\\frac{4}{d}\\\\right)^\\\\frac{1}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d}$ for some fixed $A>0$ .\",\n \"For this choice, we have both integrals of the order $\\\\varepsilon ^{-4}\\\\beta ^2 = 2^{-4}\\\\pi ^{-2}A^{-\\\\frac{4}{d}} g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}}\\\\left(1-\\\\frac{4}{d}\\\\right)^{-\\\\frac{4}{d}}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^{-\\\\frac{4}{d}} = \\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^2 g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}}\\\\frac{1}{d^2}\\\\left(1+o\\\\left(1\\\\right)\\\\right),$ as $d\\\\rightarrow \\\\infty $ .\",\n \"For large $d$ , this is dominated by $g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}d^{-3}$ and therefore $\\\\left\\\\Vert ^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ is dominated by $\\\\left\\\\Vert ^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ and we prove the result.\",\n \"The following lemma allows us to deal with occurrences of $\\\\overline{W_k}$ .\",\n \"Lemma 6.10 Let $x,y \\\\in \\\\mathbb {X}$ .\",\n \"Then $\\\\tau _{\\\\lambda ,k}(x,y) \\\\le \\\\varphi _k(x,y) + 2\\\\lambda \\\\int \\\\left(\\\\tau _\\\\lambda (x,u)\\\\varphi _k(u,y) + \\\\tau _{\\\\lambda ,k}(x,u)\\\\varphi (u,y)\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right).\",\n \"$ For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\overline{W_k}\\\\le c_f\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ The first inequality follows by applying the cosine-splitting result (Lemma REF ) to ().\",\n \"For the $\\\\tau _\\\\lambda $ term in $\\\\overline{W_k}$ we apply (REF ) and () to get $\\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\le \\\\varphi (\\\\overline{x};a,b) + \\\\lambda (\\\\varphi \\\\varphi )(\\\\overline{x};a,b)+ \\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi )(\\\\overline{x};a,b).$ Similarly, with the additional use of (REF ), we get $\\\\tau _{\\\\lambda ,k}(\\\\overline{x};a,b) \\\\le \\\\varphi _k(\\\\overline{x};a,b) + 2\\\\lambda \\\\left( (\\\\varphi _k\\\\varphi )(\\\\overline{x};a,b) +(\\\\varphi \\\\varphi _k)(\\\\overline{x};a,b)\\\\right)+ 4\\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi _k)(\\\\overline{x};a,b) \\\\\\\\ + 4\\\\lambda ^2(\\\\varphi \\\\tau _{\\\\lambda ,k}\\\\varphi )(\\\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\\\lambda \\\\le f(\\\\lambda )$ , we can realise that we can bound $\\\\overline{W_k}$ by bounding the following objects.\",\n \"For $\\\\vec{j}\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace \\\\times \\\\left\\\\lbrace 1,2,3,4,5\\\\right\\\\rbrace $ , define $\\\\mathcal {W}(k;\\\\vec{j}) := \\\\int P^{j_1}_{1,2}(l)Q^{j_2}_{3,4}(l;k)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}},$ where $P^{j}_{1,2}(l) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_1}{\\\\widehat{}(l) g_2}} &:j=1\\\\\\\\*{*{g_1}{\\\\widehat{}(l)^2g_2}} &:j=2\\\\\\\\*{*{g_1}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}(l)g_2}} &: j=3,\\\\end{array}\\\\right.\",\n \"}\\\\\\\\Q^{j}_{3,4}(l;k) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_{3}}{\\\\widehat{}_k(l) g_{4}}} &:j=1\\\\\\\\*{*{g_{3}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{4}}} &:j=2\\\\\\\\*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{}_k(l) g_{4}}} &:j=3\\\\\\\\*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{4}}} &:j=4\\\\\\\\*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{4}}} &:j=5,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\left\\\\lbrace g_m\\\\right\\\\rbrace _{m=1}^{4}$ are the functions of the form $g_m(a) = \\\\frac{1}{\\\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .\",\n \"For the $P^j_{1,2}(l)$ factor we can get bounds as we did in the proof of Lemma REF .\",\n \"For $Q^{j}_{3,4}(l;k)$ we need to do a bit more processing.\",\n \"For $j=1$ , $*{*{g_{3}}{\\\\widehat{}_k(l) g_{4}}} \\\\le \\\\frac{1}{\\\\mathcal {P}(E_{3})\\\\mathcal {P}(E_{4})}\\\\int _{E_{3}\\\\times E_{4}}*{\\\\widehat{\\\\varphi }_k(l;a,b)}\\\\mathcal {P}^{\\\\otimes 2}(\\\\mathrm {d}a, \\\\mathrm {d}b) \\\\le \\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\\\\\\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right),$ where in the last inequality we used Lemma REF .\",\n \"For $j=2$ , we use Cauchy-Schwarz to get $*{*{g_{3}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{4}}} \\\\le *{g_3}{\\\\widehat{}_k(l)^2 g_3}^{\\\\frac{1}{2}} *{g_4}{\\\\widehat{}(l)^2 g_4}^{\\\\frac{1}{2}}\\\\le \\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }*{g_4}{\\\\widehat{}(l)^2 g_4}^{\\\\frac{1}{2}} \\\\\\\\\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)*{g_4}{\\\\widehat{}(l)^2 g_4}^{\\\\frac{1}{2}}.$ An identical argument gives the same bound for $j=3$ .\",\n \"For $j=4$ , we can use the self-adjointness of $\\\\widehat{}(l)$ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)$ and the Cauchy-Schwarz inequality to get $*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{4}}} &\\\\le *{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)^2\\\\widehat{}(l)g_{3}}^{\\\\frac{1}{2}} *{g_{4}}{\\\\widehat{}_k(l)^2g_{4}}^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\le f(\\\\lambda )\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)*{g_{3}}{\\\\widehat{}(l)^2g_{3}}^{\\\\frac{1}{2}}\\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\nonumber \\\\\\\\&\\\\le Cf(\\\\lambda )\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)*{g_{3}}{\\\\widehat{}(l)^2g_{3}}^{\\\\frac{1}{2}},$ and similarly $*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{4}}} \\\\le f(\\\\lambda )\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)*{g_3}{\\\\widehat{}(l)^2g_3}^\\\\frac{1}{2}*{g_4}{\\\\widehat{}(l)^2g_4}^\\\\frac{1}{2} \\\\\\\\ \\\\le Cf(\\\\lambda )\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)*{g_3}{\\\\widehat{}(l)^2g_3}^\\\\frac{1}{2}*{g_4}{\\\\widehat{}(l)^2g_4}^\\\\frac{1}{2},$ where we write for convenience $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l_3;k) = \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k).$ Note that all the bounds for the $Q^{j}_{3,4}(l)$ terms have a $\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ factor.\",\n \"So we only need to prove that the remaining $l$ -integral of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ and $\\\\widehat{}$ factors is bounded.\",\n \"As in our previous proofs, we partition $\\\\mathbb {R}^d$ into the sets $A_n$ defined earlier in (REF ).\",\n \"For the $A_1$ , $A_2$ , and $A_3$ parts we then use the bounds $C$ and $C^2$ for the various $\\\\widehat{}$ factors and $1/\\\\left(C_2*{l}^2\\\\right)$ to bound the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors.\",\n \"By using the techniques used previously in this section, we can show that these contributions is bounded by some $\\\\varepsilon $ and $d$ (only) dependent constant.\",\n \"Since we don't need to show decay, we don't need to worry about taking $\\\\varepsilon $ to 0.\",\n \"For the $A_0$ contribution, we bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors by $1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ and show that the integral of the $\\\\widehat{}$ factors over all of $\\\\mathbb {R}^d$ is bounded.\",\n \"For $j_2\\\\ne 1$ this proceeds similarly to the argument in the proof of Lemma REF .\",\n \"For $j_2=1$ we need to take a step back.\",\n \"These versions of (REF ) arise from trying to bound one particular type of term, which we now treat separately.\",\n \"We bound $\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\varphi _k\\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_3,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\varphi _k\\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\le \\\\left\\\\Vert \\\\widehat{}_k(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Then Lemma REF gives the result.\",\n \"We are now only left with bounding $\\\\overline{H_k}$ .\",\n \"lemmaMartiniBound For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\overline{H_k}\\\\le c_f\\\\beta ^2\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ The proof of the lemma uses the same techniques and tools as our previous lemmas, but is much more complicated in terms of keeping track of cases and components of integrals.\",\n \"To avoid overburdening a reader on an early reading, the proof has been placed in Appendix .\",\n \"The Bootstrap Argument In this section we bring together the diagrammatic bounds of Section with the estimates of Section REF to bound the relevant operators in such a way that the finite lace expansion of Proposition REF converges to an Ornstein-Zernike equation (OZE) for both the raw operators on $L^2\\\\left(\\\\mathbb {X}\\\\right)$ and the Fourier-transformed operators on $L^2\\\\left(\\\\mathcal {E}\\\\right)$ .\",\n \"Proposition REF does this for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ , with the caveat that there is a constant $c_f$ that is increasing in the bootstrap function $f$ (but independent of $d$ ) appearing in the bounds.\",\n \"Observation REF then highlights that if $f$ is uniformly bounded on $\\\\left[0,\\\\lambda _O\\\\right)$ then so is $c_f$ and the associated bounds from Proposition REF .\",\n \"The crucial step to the bootstrap argument is then Proposition REF .\",\n \"Here a forbidden-region argument shows that $f$ is continuous, that there is an interval of finite values that it cannot take (the interval being independent of $\\\\lambda $ ), and that it starts below this interval.\",\n \"This proves that $f$ is indeed uniformly bounded for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ , and so are our bounds in Proposition REF .\",\n \"Proposition 6.11 (Convergence of the operator lace expansion and OZE) Let $\\\\lambda \\\\in [0, \\\\lambda _O)$ and $d>6$ be sufficiently large.\",\n \"Then there exists $c_f=c(f(\\\\lambda ))$ (which is increasing in $f$ and independent of $d$ ) such that $\\\\sum _{n\\\\ge 0}\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}\\\\le c_f\\\\beta , &\\\\qquad \\\\sum _{n\\\\ge 0}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le c_f\\\\beta , \\\\\\\\ \\\\sum _{n\\\\ge 0}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} &\\\\le c_f\\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) \\\\beta , \\\\\\\\\\\\left\\\\Vert R_{\\\\lambda , n}\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} c_f\\\\beta ^{n}.\",\n \"$ Furthermore, the limit $_\\\\lambda := \\\\lim _{M\\\\rightarrow \\\\infty } \\\\sum ^M_{n=0}_\\\\lambda ^{(n)}$ exists and is a bounded operator with Fourier transform $\\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right) = \\\\lim _{M\\\\rightarrow \\\\infty } \\\\sum ^M_{n=0}\\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)$ for $k\\\\in \\\\mathbb {R}^d$ .\",\n \"The operators $\\\\mathcal {T}_\\\\lambda $ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ satisfy operator Ornstein-Zernike equations, taking the form $\\\\mathcal {T}_\\\\lambda = + _\\\\lambda + \\\\lambda \\\\mathcal {T}_\\\\lambda (+_\\\\lambda ), $ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right) = \\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right) + \\\\lambda \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\left(\\\\widehat{}\\\\left(k\\\\right)+\\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)\\\\right), $ for all $\\\\lambda < \\\\lambda _O$ and $k\\\\in \\\\mathbb {R}^d$ .\",\n \"Before we prove this proposition, we make the following observation, which will guide the rest of the section.\",\n \"Observation 6.12 (Uniform convergence of the lace expansion) Suppose that there exists finite $C>0$ such that $f \\\\le C$ on $[0,\\\\lambda _O)$ .\",\n \"Then for $d>6$ sufficiently large there exits $c>0$ (independent of $\\\\lambda $ and $d$ ) such that the bounds (REF ), (), () hold with $c_f$ replaced by $c$ .\",\n \"We now return to prove Proposition REF .\",\n \"[Proof of Proposition REF ] So far in this section we have provided bounds for the building blocks of the bounds appearing in Propositions REF , REF , REF , and REF .\",\n \"We still have to convert these into bounds for the composite terms $U_\\\\lambda $ and $V_\\\\lambda $ .\",\n \"From their definitions and the bounds on the building blocks, it is simple to see that there exists $c_f^{\\\\prime }$ such that for $d>6$ $U_\\\\lambda \\\\le c_f^{\\\\prime }, \\\\qquad V_\\\\lambda \\\\le c_f^{\\\\prime }\\\\beta .$ This then implies that there exists $c_f^{\\\\prime \\\\prime }$ such that for $d>6$ $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f^{\\\\prime \\\\prime }\\\\beta ^2 &: n=0,\\\\\\\\c_f^{\\\\prime \\\\prime }\\\\left(c_f^{\\\\prime \\\\prime }\\\\beta \\\\right)^n &: n\\\\ge 1.\\\\end{array}\\\\right.\",\n \"}$ If $d$ is sufficiently large we have $c_f^{\\\\prime \\\\prime }\\\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\\\sum _{n\\\\ge 0}\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op} \\\\le c_f^{\\\\prime \\\\prime }\\\\left(\\\\beta ^2 + \\\\frac{c_f^{\\\\prime \\\\prime }\\\\beta }{1-c_f^{\\\\prime \\\\prime }\\\\beta }\\\\right) \\\\le c_f\\\\beta .$ When we want to consider the displaced terms, we also make use of Observation REF , and Lemmas REF , REF , REF , and REF to give us the existence of $c_f^{\\\\prime \\\\prime \\\\prime }$ such that for $d>6$ $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta ^2\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) &: n=0,3,\\\\\\\\c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) &: n=1,2,\\\\\\\\\\\\left(n+1\\\\right)^2c_f^{\\\\prime \\\\prime \\\\prime }\\\\left(c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^{n-3}\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) &: n\\\\ge 4.\\\\end{array}\\\\right.\",\n \"}$ If $d$ is sufficiently large we have $c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\\\sum _{n\\\\ge 0}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le c_f^{\\\\prime \\\\prime \\\\prime }\\\\left(2\\\\beta ^2 + 2\\\\beta + \\\\frac{16\\\\left(c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^3-39\\\\left(c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^2 + 25c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta }{\\\\left(1-c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^3}\\\\right)\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) \\\\\\\\\\\\le c_f\\\\beta \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ Lemma REF then allows us to use (REF ) to bound $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"This immediately gives the required result for $n\\\\ge 1$ .\",\n \"For $n=0$ , we simply additionally require $d$ to be large enough that $\\\\beta <1$ and therefore $\\\\beta ^2<\\\\beta ^0$ .\",\n \"Note that the dual space of a Banach space (endowed with the operator norm) is also a Banach space.\",\n \"In particular, it is complete.\",\n \"Since $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}\\\\le c_f^{\\\\prime \\\\prime }\\\\left(c_f^{\\\\prime \\\\prime }\\\\beta \\\\right)^n$ for $n\\\\ge 1$ , the sequence $_{\\\\lambda ,n}$ is a Cauchy sequence in the dual space of $L^2(\\\\mathbb {X})$ for sufficiently large $d$ .\",\n \"Therefore the limit $_\\\\lambda $ exists and is a bounded linear operator on $L^2(\\\\mathbb {X})$ .\",\n \"Since the Fourier transform is a unitary linear operator it is bounded and continuous, and therefore the same argument says that the limit $\\\\widehat{}_{\\\\lambda }(k)$ exists and is indeed the Fourier transform of $_\\\\lambda $ .\",\n \"Since $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op}$ is finite for $\\\\lambda <\\\\lambda _O$ , we have $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op}\\\\rightarrow 0$ as $n\\\\rightarrow \\\\infty $ if $d$ is sufficiently large.\",\n \"As a a consequence of this and equation (REF ) from Proposition REF , the $\\\\mathcal {T}_\\\\lambda $ operator satisfies the operator Ornstein-Zernike equation.\",\n \"The following lemma has two main uses in what follows.\",\n \"Firstly, it takes the OZEs from Proposition REF and uses it to write the two-point operators in terms of the sum of the adjacency operators and the lace expansion coefficient operators.\",\n \"Secondly, it uses this and an intermediate value theorem argument to bound the spectral supremum of the sum of the adjacency operators and the lace expansion coefficient operators.\",\n \"Lemma 6.13 Let $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ and $d>6$ be sufficiently large.\",\n \"Then the operators $\\\\left(1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)$ and $\\\\left(1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right)$ all have bounded linear inverses and $\\\\mathcal {T}_\\\\lambda &= \\\\left(+ _\\\\lambda \\\\right)\\\\left(1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)^{-1}, \\\\\\\\\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) &= \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\left(1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right)^{-1}, \\\\qquad \\\\forall k\\\\in \\\\mathbb {R}^d.\",\n \"$ Furthermore, $\\\\lambda \\\\mapsto _\\\\lambda $ and $\\\\lambda \\\\mapsto \\\\widehat{}_{\\\\lambda }(k)$ are all continuous in the operator norm topology and $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) &< 1,\\\\\\\\\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) &< 1, \\\\qquad \\\\forall k\\\\in \\\\mathbb {R}^d.$ We first prove that $1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)$ has a bounded linear inverse.\",\n \"This is clear for $\\\\lambda = 0$ , so we only need to consider $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"Suppose for contradiction that $1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)$ does not have a bounded linear inverse.\",\n \"Then there exists a sequence $f_n\\\\in L^2(\\\\mathbb {X})$ such that $*{f_n}_2 = 1$ and $*{\\\\left(1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)f_n}_2\\\\rightarrow 0$ .\",\n \"This also means that $\\\\lambda *{\\\\left(+ _\\\\lambda \\\\right)f_n}\\\\rightarrow 1$ .\",\n \"Then from Lemma REF and sub-criticality we have $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} < \\\\infty $ , and therefore the OZE equation (REF ) implies that $*{\\\\left(+ _\\\\lambda \\\\right)f_n}\\\\rightarrow 0$ , a contradiction.\",\n \"A similar argument proves the corresponding statement for the Fourier transformed operators.\",\n \"Once we know that the inverses exist, it is only a matter of rearranging the OZE equations to get the expressions for $\\\\mathcal {T}_\\\\lambda $ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ in terms of the other operators.\",\n \"Since $+ _\\\\lambda $ is a bounded operator, a similar argument proves that $1+ \\\\lambda \\\\mathcal {T}_\\\\lambda $ has a bounded inverse, and therefore $+ _\\\\lambda = \\\\left(1+ \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\right)^{-1}\\\\mathcal {T}_\\\\lambda .$ From Corollary REF we have the continuity of $\\\\lambda \\\\mapsto \\\\mathcal {T}_\\\\lambda $ , and therefore the continuity $+ _\\\\lambda $ (and therefore $_\\\\lambda $ ).\",\n \"We also have the continuity of $\\\\lambda \\\\mapsto \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ from Corollary REF , and we similarly get the continuity of $\\\\widehat{}_{\\\\lambda }(k)$ .\",\n \"The continuity of $+ _\\\\lambda $ implies the continuity of $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right)$ (via Lemma REF ).\",\n \"At $\\\\lambda = 0$ it is clear that $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) = 0$ , and our above argument implies that $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) \\\\ne 1$ .\",\n \"Therefore an intermediate value theorem argument proves that $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) < 1$ .\",\n \"This argument also works for $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)$ .\",\n \"It now remains to show that $f$ is indeed uniformly bounded for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"We do this by performing a forbidden-region argument.\",\n \"In Proposition REF we prove that $f(0)$ is bounded and $f$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"However we also prove that $f$ is never in the region $\\\\left(\\\\kappa ,\\\\kappa +1\\\\right]$ , where $\\\\kappa := 10 C \\\\left(1+C\\\\right)^2,$ $C$ being the constant appearing in REF .\",\n \"Note that (REF ) in REF requires that $C\\\\ge 1$ .\",\n \"The intermediate value theorem then implies that $\\\\kappa $ acts as an upper bound on the whole domain $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"One should not read too much into the value of $\\\\kappa $ here.\",\n \"It is sufficient for our purposes, but not remotely optimal.\",\n \"Proposition 6.14 (The forbidden-region argument) The following three statements are all true: $f$ satisfies $f(0) \\\\le \\\\kappa $ .\",\n \"$f(\\\\lambda ) \\\\notin (\\\\kappa ,\\\\kappa +1]$ for all $\\\\lambda \\\\in [0,\\\\lambda _O)$ provided that $d$ is sufficiently large.\",\n \"$f$ is continuous on $[0, \\\\lambda _O)$ .\",\n \"Therefore $f(\\\\lambda ) \\\\le \\\\kappa $ holds uniformly in $\\\\lambda <\\\\lambda _O$ for all $d$ sufficiently large.\",\n \"We will deal with each of the three statements in Proposition REF in turn.\",\n \"[Proof of Proposition REF -REF ] Firstly, it is clear $f_1\\\\left(0\\\\right) = 0$ .\",\n \"Then note that $\\\\mathcal {T}_0 = $ , and hence $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}_0}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} = \\\\left\\\\Vert \\\\widehat{}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le C,$ where we use REF (via (REF )) in the last inequality.\",\n \"Furthermore $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_0(0)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) =1$ , and therefore $\\\\widehat{G}_{\\\\mu _{0}}\\\\left(k\\\\right) = \\\\widehat{G}_0\\\\left(k\\\\right) = 1$ and $f_2\\\\left(0\\\\right) \\\\le C$ .\",\n \"Similarly we find that $f_3\\\\left(0\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k,l\\\\in \\\\mathbb {R}^d}\\\\frac{\\\\left\\\\Vert \\\\widehat{}_k\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{3\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]}.$ We use then Lemma REF to get the bound $\\\\left\\\\Vert \\\\widehat{}_k\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le C\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ Hence $f_3\\\\left(0\\\\right) \\\\le \\\\tfrac{2}{3}C$ .\",\n \"[Proof of Proposition REF -REF ] We will assume that $f\\\\le \\\\kappa +1$ and show that this implies that $f\\\\le \\\\kappa $ .\",\n \"Most crucially, the assumption $f\\\\le \\\\kappa +1$ allows us to apply Observation REF .\",\n \"To highlight the $\\\\kappa $ -dependence, we write $c_{\\\\kappa +1}$ as the constant arising from this observation.\",\n \"It will be convenient to introduce some temporary notation for this section.\",\n \"Define $\\\\widehat{a}\\\\left(k\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as $\\\\widehat{a}\\\\left(k\\\\right) = \\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"We will also make use of the functions $\\\\delta _\\\\lambda \\\\colon \\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}$ defined by $\\\\delta _\\\\lambda \\\\left(k\\\\right) := \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ It is clear from the triangle inequality and Lemma REF that ${\\\\delta _\\\\lambda \\\\left(k\\\\right)}\\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le c_{\\\\kappa +1}\\\\beta ,$ but we will also need the following inequality.\",\n \"We will prove it at the end of this section.\",\n \"Lemma 6.15 Let $k_1,k_2\\\\in \\\\mathbb {R}^d$ .\",\n \"Then $*{\\\\delta _\\\\lambda (k_1)-\\\\delta _\\\\lambda (k_2)} \\\\le 2 \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right) - \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ Figure: If 𝒯 ^ λ (k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) and a ^(k)\\\\widehat{a} (k) commute, then they are simultaneously diagonalizable in the sense of Theorem .\",\n \"The arguments of their diagonal functions (τ ˜ λ (k)\\\\widetilde{\\\\tau }_\\\\lambda (k) and a ˜(k)\\\\widetilde{a}(k) respectively) are related by the monotone increasing function x↦x 1-λxx\\\\mapsto \\\\tfrac{x}{1-\\\\lambda x} depicted here.It will be important to note how $\\\\widehat{a}(k)$ relates to $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ .\",\n \"From Observation REF , the OZE gives us the following expression involving the commutator $\\\\left[\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k),\\\\widehat{a}(k)\\\\right] = \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) \\\\widehat{a}(k) - \\\\widehat{a}(k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ : $\\\\left[\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k),\\\\widehat{a}(k)\\\\right]\\\\left(1- \\\\lambda \\\\widehat{a}(k)\\\\right) = 0.$ Lemma REF implies that $\\\\left(1- \\\\lambda \\\\widehat{a}(k)\\\\right)$ has a bounded linear inverse and with (REF ) implies that $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ and $\\\\widehat{a}(k)$ commute.\",\n \"Therefore there exists a single unitary map that `diagonalizes' them both in the sense of Theorem REF .\",\n \"By considering the diagonal functions, the OZE then implies that $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)/\\\\left(1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)\\\\right)$ .\",\n \"In particular, this relation allows us to write an alternative expression for $\\\\mu _\\\\lambda $ : $\\\\mu _\\\\lambda := 1 - \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)} = \\\\lambda + \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)-1}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)} = \\\\lambda + \\\\frac{\\\\delta _\\\\lambda \\\\left(0\\\\right)}{1+\\\\delta _\\\\lambda \\\\left(0\\\\right)}.$ We first address $f_1$ .\",\n \"From the above discussion relating $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ and $\\\\widehat{a}(k)$ , we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} \\\\ge \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)/\\\\left(1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)\\\\right)$ .\",\n \"From Lemma REF and our choice of scaling we have $\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right) \\\\ge 1 - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }(0)\\\\right\\\\Vert _{\\\\rm op} \\\\ge 1 - c_{\\\\kappa +1}\\\\beta $ , and therefore we have $\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)>0$ for sufficiently large $d$ and $\\\\lambda \\\\le \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}\\\\left(1 - \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}}\\\\right) \\\\le \\\\frac{1}{1-c_{\\\\kappa +1}\\\\beta } \\\\le 1 + 2c_{\\\\kappa +1}\\\\beta .$ Here we have used an idea that we will use very often in this part of the proof.\",\n \"For $d$ large enough that $c_{\\\\kappa +1}\\\\beta <1$ we have $\\\\left(1-c_{\\\\kappa +1}\\\\beta \\\\right)^{-1} = \\\\sum ^\\\\infty _{n=0}\\\\left(c_{\\\\kappa +1}\\\\beta \\\\right)^n$ , and for $d$ sufficiently large enough ($c_{\\\\kappa +1}\\\\beta \\\\le 1/2$ ) we have $\\\\left(1-c_{\\\\kappa +1}\\\\beta \\\\right)^{-1}\\\\le 1 + 2c_{\\\\kappa +1}\\\\beta $ .\",\n \"We also use a similar idea of neglecting higher order terms by increasing an earlier term's coefficient in other cases.\",\n \"For $f_2$ we first replace $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Since $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ and $\\\\widehat{a} (k)$ commute, they are simultaneously diagonalizable in the sense of Theorem REF , and therefore their spectra are related by the function $x\\\\mapsto \\\\tfrac{x}{1-\\\\lambda x}$ (monotone increasing on $\\\\left(-\\\\infty ,1/\\\\lambda \\\\right)$ ).\",\n \"Therefore we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op} = \\\\sup _{x\\\\in \\\\sigma \\\\left(\\\\widehat{a} (k)\\\\right)}*{\\\\frac{x}{1-\\\\lambda x}}$ .\",\n \"Now (perhaps with the aid of Figure REF ) it is easy to see that $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}{1- \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\vee \\\\mathbb {S}\\\\left(-\\\\widehat{a}(k)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) \\\\vee \\\\mathbb {S}\\\\left(-\\\\widehat{a} (k)\\\\right).$ Now since $\\\\mathbb {S}\\\\left(-\\\\widehat{a}(k)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{a}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le 1 + c_{\\\\kappa +1}\\\\beta $ , and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^{-1} \\\\le 1+C$ (recall Lemma REF and $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le C$ from REF ), we only need to bound $\\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d}\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)}.$ We now partition $k\\\\in \\\\mathbb {R}^d$ into two.\",\n \"Define the subset of the Fourier space $A_{\\\\frac{1}{2}} := \\\\left\\\\lbrace k\\\\in \\\\mathbb {R}^d:\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\le \\\\frac{1}{2}\\\\right\\\\rbrace .$ Then we only need to bound the two functions $f_4,f_5$ , where $f_4\\\\left(\\\\lambda \\\\right) := \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in A_{\\\\frac{1}{2}}} \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)},\\\\qquad \\\\qquad f_5\\\\left(\\\\lambda \\\\right):= \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in A_{\\\\frac{1}{2}}^c} \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)}.$ We first deal with the simpler $f_4$ .\",\n \"We consider $k\\\\in A_{\\\\frac{1}{2}}$ and choose $\\\\beta $ small enough that $1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right) \\\\ge 1 - \\\\frac{\\\\frac{1}{2}+c_{\\\\kappa +1}\\\\beta }{1-c_{\\\\kappa +1}\\\\beta } \\\\ge \\\\frac{1}{2} - 2c_{\\\\kappa +1}\\\\beta > 0.$ Then $\\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\le \\\\frac{1+\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}}{\\\\frac{1}{2} - 2c_{\\\\kappa +1}\\\\beta } \\\\le 2\\\\left(1+C\\\\right)\\\\left(1+5c_{\\\\kappa +1}\\\\beta \\\\right).\",\n \"$ Since we have $\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right) \\\\le \\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right) \\\\le 1 + c_{\\\\kappa +1}\\\\beta $ , we then get $f_4(\\\\lambda ) \\\\le 2\\\\left(1+C\\\\right)\\\\left(1+5c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left(1 + c_{\\\\kappa +1}\\\\beta \\\\right) \\\\le 2\\\\left(1+C\\\\right)\\\\left(1+7c_{\\\\kappa +1}\\\\beta \\\\right).$ To address $f_5$ we will have $k\\\\in A^c_\\\\rho $ .\",\n \"We define: $\\\\widehat{N}(k) = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}, \\\\qquad \\\\widehat{F}(k) = \\\\frac{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}, \\\\qquad \\\\widehat{Q}(k) = \\\\frac{1+\\\\delta _\\\\lambda (k)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}.$ Note that $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) = \\\\widehat{N}(k) / \\\\widehat{F}(k)$ .\",\n \"Rearranging gets us to $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} &= \\\\widehat{N}(k) \\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{F}(k)}\\\\nonumber \\\\\\\\&= \\\\widehat{Q}(k) + \\\\frac{\\\\widehat{N}(k)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{F}(k)} \\\\left(1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) - \\\\frac{\\\\widehat{Q}(k)}{\\\\widehat{N}(k)}\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{F}(k)\\\\right) .$ Since $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)=1$ (by our scaling choice), the extracted term $\\\\widehat{Q}(k)$ satisfies $*{\\\\widehat{Q}(k)} \\\\le 1 + 3c_{\\\\kappa +1}\\\\beta $ .\",\n \"We further observe that $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\widehat{Q}(k)}{\\\\widehat{N}(k)} = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\left(1+ \\\\delta _\\\\lambda (k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) +\\\\delta _\\\\lambda (k)} = 1 - \\\\frac{\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right) \\\\delta _\\\\lambda (k)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) + \\\\delta _\\\\lambda (k)} =: 1- \\\\widehat{b}(k).$ Recalling identity (REF ) for $\\\\mu _\\\\lambda $ , we can rewrite the quantity $\\\\left[1- \\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)- (1-\\\\widehat{b}(k)) \\\\widehat{F}(k)\\\\right]$ , appearing in (REF ), as $\\\\frac{1+\\\\delta _\\\\lambda (0) - \\\\left[\\\\lambda + \\\\delta _\\\\lambda (0) + \\\\lambda \\\\delta _\\\\lambda (0)\\\\right] \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) - 1 + \\\\lambda \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)+\\\\delta _\\\\lambda (k)\\\\right) + \\\\widehat{b}(k)\\\\left(1-\\\\lambda \\\\widehat{a}(k)\\\\right)}{1+\\\\delta _\\\\lambda (0)} \\\\\\\\= \\\\frac{ \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\left(\\\\delta _\\\\lambda (0) + \\\\lambda \\\\delta _\\\\lambda (0)\\\\right) + \\\\lambda \\\\left[\\\\delta _\\\\lambda (k)-\\\\delta _\\\\lambda (0)\\\\right]}{1+\\\\delta _\\\\lambda (0)}+ \\\\frac{\\\\widehat{b}(k)(1-\\\\lambda \\\\widehat{a}(k))}{1+\\\\delta _\\\\lambda (0)}.$ Noting that ${\\\\delta _{\\\\lambda }(0)-\\\\delta _{\\\\lambda }(k)} \\\\le 2c_{\\\\kappa +1}\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\beta $ by Lemma REF and Observation REF , the first term can be bounded $&*{\\\\frac{ \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\left(\\\\delta _\\\\lambda (0) + \\\\lambda \\\\delta _\\\\lambda (0)\\\\right) + \\\\lambda \\\\left[\\\\delta _\\\\lambda (k)-\\\\delta _\\\\lambda (0)\\\\right]}{1+\\\\delta _\\\\lambda (0)}} \\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\frac{ c_{\\\\kappa +1}\\\\beta + \\\\left(1 + 2c_{\\\\kappa +1}\\\\beta \\\\right)c_{\\\\kappa +1}\\\\beta + 2\\\\left(1+2c_{\\\\kappa +1}\\\\beta \\\\right)c_{\\\\kappa +1}\\\\beta }{1 - c_{\\\\kappa +1}\\\\beta }\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\&\\\\qquad \\\\le 5c_{\\\\kappa +1}\\\\beta \\\\left(1+2c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\&\\\\qquad \\\\le 6c_{\\\\kappa +1}\\\\beta \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ Using (REF ), the last term is $*{\\\\frac{\\\\widehat{b}(k)(1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right))}{1+\\\\delta _\\\\lambda (0)}} &= *{ \\\\frac{\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right)\\\\delta _\\\\lambda (k)}{\\\\left(1+\\\\delta _\\\\lambda (0)\\\\right)\\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) + \\\\delta _\\\\lambda (k)\\\\right)}- \\\\frac{\\\\lambda \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\delta _\\\\lambda (k)}{1+\\\\delta _\\\\lambda (0)}} \\\\nonumber \\\\\\\\& = \\\\frac{*{\\\\delta _{\\\\lambda }(k)}}{*{1+\\\\delta _\\\\lambda (0)}}*{ \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) + \\\\delta _\\\\lambda (k)} -\\\\lambda }\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\& \\\\le \\\\frac{c_{\\\\kappa +1}\\\\beta }{1-c_{\\\\kappa +1}\\\\beta }\\\\left( \\\\frac{1}{\\\\frac{1}{2}-c_{\\\\kappa +1}\\\\beta } + \\\\frac{1}{1-c_{\\\\kappa +1}\\\\beta }\\\\right)\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\&\\\\le c_{\\\\kappa +1}\\\\beta \\\\left(1+2c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left(3+8c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\& \\\\le 4c_{\\\\kappa +1}\\\\beta \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ Putting these bounds back into (REF ), we can find constants $M$ and $\\\\tilde{M}$ such that $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} &\\\\le \\\\widehat{Q}(k) + 10c_{\\\\kappa +1}\\\\beta \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}\\\\frac{\\\\widehat{N}(k)}{\\\\widehat{F}(k)} \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\ \\\\\\\\& \\\\le 1+ 3c_{\\\\kappa +1}\\\\beta + 20c_{\\\\kappa +1}\\\\beta \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} \\\\\\\\& \\\\le 1+ \\\\left(20\\\\kappa + 23\\\\right)c_{\\\\kappa +1}\\\\beta .\",\n \"$ Note that we have used the bound $\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\le \\\\left[1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]$ (since $\\\\mu _\\\\lambda \\\\le 1$ and $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)>\\\\frac{1}{2} >0$ ) to get from $\\\\widehat{G}_1(k)^{-1}$ to $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^{-1}$ .\",\n \"As $\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\in \\\\left(\\\\frac{1}{2}, 1\\\\right]$ , this concludes the improvement of $f_5$ and hence of $f_2$ .\",\n \"Before we treat $f_3$ , we introduce $f_6$ given by $f_6(\\\\lambda ) := \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d} \\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{1- \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}\\\\left(k\\\\right)\\\\right)}.$ We show that $f(\\\\lambda ) \\\\le \\\\kappa +1$ implies $f_6(\\\\lambda ) \\\\le M$ uniformly for some $\\\\kappa $ -independent constant $M$ .\",\n \"For $k\\\\in A_\\\\frac{1}{2}$ we showed in (REF ) that we can bound uniformly with $3\\\\left(1+C\\\\right)$ for sufficiently large small $\\\\beta $ .\",\n \"For $k \\\\in A_{\\\\frac{1}{2}}^c$ , we have $\\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} \\\\cdot \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\le \\\\left(1+ \\\\left(20\\\\kappa + 23\\\\right)c_{\\\\kappa +1}\\\\beta \\\\right) *{ 1- \\\\frac{\\\\delta _\\\\lambda (k)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}} \\\\\\\\ \\\\le \\\\left(1+ \\\\left(20\\\\kappa + 23\\\\right)c_{\\\\kappa +1}\\\\beta \\\\right) \\\\left(1+\\\\frac{c_{\\\\kappa +1}\\\\beta }{\\\\frac{1}{2}- c_{\\\\kappa +1}\\\\beta } \\\\right) \\\\le 1+ \\\\left(20\\\\kappa + 27\\\\right)c_{\\\\kappa +1}\\\\beta .\",\n \"$ Note that for the first bound in (REF ), we used the estimate established in (REF ), which is stronger than a bound on $f_5$ .\",\n \"We therefore have the uniform bound $f_6(\\\\lambda ) \\\\le 3 + 3C$ for $\\\\beta $ sufficiently small.\",\n \"Let us now improve the bound on $f_3$ .\",\n \"It will be notationally convenient to denote $A\\\\left(k\\\\right) := \\\\left[1- \\\\lambda \\\\widehat{a}\\\\left(k\\\\right)\\\\right]^{-1}$ .\",\n \"Since $\\\\widehat{a}\\\\left(k\\\\right)$ is self-adjoint, it can be “diagonalized\\\" in the sense of Theorem REF .\",\n \"Therefore $A\\\\left(k\\\\right)$ can be “diagonalized\\\" over the same space and with the same unitary operator - it will have the diagonal function $\\\\widetilde{A}(k)(e) = 1/\\\\left(1-\\\\widetilde{a}\\\\left(k\\\\right)(e)\\\\right)$ .\",\n \"It is then clear that $A\\\\left(k\\\\right)$ and $\\\\widehat{a}\\\\left(k\\\\right)$ commute (for each $k\\\\in \\\\mathbb {R}^d$ ).\",\n \"Using Observation REF , we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} &= \\\\left\\\\Vert \\\\widehat{a}\\\\left(l\\\\right)A\\\\left(l\\\\right) - \\\\widehat{a}\\\\left(l+k\\\\right)A\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\nonumber \\\\\\\\&= \\\\left\\\\Vert A\\\\left(l\\\\right)\\\\widehat{a}\\\\left(l\\\\right)\\\\left[1- \\\\lambda \\\\widehat{a}\\\\left(l+k\\\\right)\\\\right]A\\\\left(l+k\\\\right) - A\\\\left(l\\\\right)\\\\left[1- \\\\lambda \\\\widehat{a}\\\\left(l\\\\right)\\\\right]\\\\widehat{a}\\\\left(l+k\\\\right)A\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\nonumber \\\\\\\\&= \\\\left\\\\Vert A(l)\\\\left(\\\\widehat{a}(l)-\\\\widehat{a}(l+k)\\\\right)A(l+k)\\\\right\\\\Vert _{\\\\rm op} \\\\nonumber \\\\\\\\&\\\\le \\\\left\\\\Vert \\\\widehat{a}(l)-\\\\widehat{a}(l+k)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert A(l)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert A(l+k)\\\\right\\\\Vert _{\\\\rm op}.\",\n \"$ For the $\\\\left\\\\Vert \\\\widehat{a}(l)-\\\\widehat{a}(l+k)\\\\right\\\\Vert _{\\\\rm op}$ factor, we use the triangle inequality to separate this into a term of $\\\\widehat{}$ and $\\\\widehat{}_{\\\\lambda }$ .\",\n \"Lemma REF immediately gives $\\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le C\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right],$ for the $C>0$ arising from REF .\",\n \"For the $\\\\widehat{}_{\\\\lambda }$ term, a similar argument to (REF ) gives $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(l\\\\right) - \\\\widehat{}_{\\\\lambda }\\\\left(1+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }\\\\left( k\\\\right)\\\\right\\\\Vert _{1,\\\\infty } \\\\le c_{\\\\kappa +1}\\\\beta \\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right],$ where the last inequality follows from Observation REF .\",\n \"We then deal with the remaining factors of (REF ) by noting that for all $k\\\\in \\\\mathbb {R}^d$ , $\\\\left\\\\Vert A(k)\\\\right\\\\Vert _{\\\\rm op} = \\\\frac{1}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} = \\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\widehat{G}_{\\\\mu _\\\\lambda }(k) \\\\le 3\\\\left(1+C\\\\right) \\\\widehat{G}_{\\\\mu _\\\\lambda }(k), $ employing our improved bound on $f_6$ .\",\n \"The result of this is that we have the bound $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le 9\\\\left(1+C\\\\right)^2\\\\left(C + c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l+k\\\\right).$ Then since $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)\\\\ge 0$ we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le 9\\\\left(1+C\\\\right)^2\\\\left(C+c_{\\\\kappa +1}\\\\beta \\\\right)\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right).$ [Proof of Proposition REF -REF ] The continuity of $f_1$ is obvious.\",\n \"For the other two functions we prove equicontinuity of a family of functions and use [41] to show the continuity of the desired functions.\",\n \"The procedure is outlined here.\",\n \"Suppose we wish to show that $H\\\\left(\\\\lambda \\\\right) := \\\\sup _{\\\\alpha \\\\in B}\\\\left\\\\Vert h_\\\\alpha \\\\left(\\\\lambda \\\\right)\\\\right\\\\Vert _{\\\\rm op}$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"For our purposes the parameter $\\\\alpha $ will be either $k$ or $\\\\left(k,l\\\\right)$ and thus $B=\\\\mathbb {R}^d$ or $B=\\\\mathbb {R}^{2d}$ .\",\n \"The continuity on the half-open interval is implied by having continuity on the closed interval $\\\\left[0,\\\\lambda _O-\\\\rho \\\\right]$ for any $\\\\rho >0$ .\",\n \"For closed intervals, [41] gives continuity if the family $\\\\left\\\\lbrace \\\\left\\\\Vert h_\\\\alpha \\\\right\\\\Vert _{\\\\rm op}\\\\right\\\\rbrace _{\\\\alpha \\\\in B}$ is equicontinuous and $H\\\\left(\\\\lambda \\\\right)<+\\\\infty $ for all $\\\\lambda $ in the closed interval.\",\n \"The family $\\\\left\\\\lbrace \\\\left\\\\Vert h_\\\\alpha \\\\right\\\\Vert _{\\\\rm op}\\\\right\\\\rbrace _{\\\\alpha \\\\in B}$ is equicontinuous if for all $\\\\varepsilon >0$ there exists $\\\\delta >0$ such that $|s-t|<\\\\delta $ implies $*{\\\\left\\\\Vert h_\\\\alpha (s)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert h_\\\\alpha (t)\\\\right\\\\Vert _{\\\\rm op}} \\\\le \\\\varepsilon $ uniformly in $\\\\alpha \\\\in B$ .\",\n \"In fact, the reverse triangle inequality implies that $*{\\\\left\\\\Vert h_\\\\alpha (s)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert h_\\\\alpha (t)\\\\right\\\\Vert _{\\\\rm op}} \\\\le \\\\left\\\\Vert h_\\\\alpha (s)-h_\\\\alpha (t)\\\\right\\\\Vert _{\\\\rm op},$ and therefore we only need to prove equicontinuity for the un-normed family $\\\\left\\\\lbrace h_\\\\alpha \\\\right\\\\rbrace _{\\\\alpha \\\\in B}$ .\",\n \"We prove equicontinuity by bounding the `near-derivative' $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert h_\\\\alpha (\\\\lambda +\\\\varepsilon )-h_\\\\alpha (\\\\lambda )\\\\right\\\\Vert _{\\\\rm op}$ uniformly in $\\\\alpha \\\\in B$ for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O-\\\\rho \\\\right]$ and arbitrary $\\\\rho >0$ .\",\n \"For $f_2$ the operator-valued functions $h_\\\\alpha $ are $\\\\lambda \\\\mapsto \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)/\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)$ .\",\n \"By using a variation on the chain rule, we get $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }\\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _{\\\\lambda +\\\\varepsilon }}\\\\left(k\\\\right)} - \\\\frac{\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{1}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\\\\\+ \\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)^2}*{\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{G}_\\\\mu \\\\left(k\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda }}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda }.$ Recall from Corollary REF that $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)$ is differentiable with the bound $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2$ .\",\n \"We also have $\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\ge 1/\\\\left(1 + \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)$ from Lemma REF and $\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{G}_\\\\mu \\\\left(k\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda } = \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)^2\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ immediately from the definition.\",\n \"It remains to deal with the $\\\\mu _\\\\lambda $ term.\",\n \"Recall that $\\\\mu _\\\\lambda := 1 - \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^{-1}$ .\",\n \"Therefore, using the reverse triangle inequality, $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda } = \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^2}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }(0)\\\\right)-\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)} \\\\\\\\\\\\le \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^2}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }(0)-\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} = \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^2}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^2,$ where we have used $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\ge 1$ from Lemma REF in the last inequality.\",\n \"Therefore $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }\\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _{\\\\lambda +\\\\varepsilon }}\\\\left(k\\\\right)} - \\\\frac{\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left(1 + \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2 + *{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^2\\\\\\\\\\\\le \\\\left(1 + C\\\\right)\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2 + C\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^3.$ We therefore have a finite $k$ -independent bound.\",\n \"This has proven the equicontinuity.\",\n \"For the uniform boundedness, we note that for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O-\\\\rho \\\\right]$ we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O-\\\\rho }(0)\\\\right\\\\Vert _{\\\\rm op}<\\\\infty $ .\",\n \"In conjunction with $\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\ge 1/\\\\left(1 + \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)$ we have the required uniform boundedness and therefore the continuity of $f_2$ .\",\n \"We repeat this approach for $f_3$ .\",\n \"The corresponding step to (REF ) now reads $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon ,k}\\\\left(l\\\\right)}{\\\\widehat{J}_{\\\\mu _{\\\\lambda +\\\\varepsilon }}\\\\left(k,l\\\\right)} - \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\\\\\+ \\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)^2}*{\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{J}_{\\\\mu }\\\\left(k,l\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda }}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda }.$ Recall from Corollary REF that $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)$ is differentiable with the operator norm bound $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le 4\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"The operator itself has the similar bound $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"We would like to have the bound (REF ), but this was proven under the assumption $f\\\\le \\\\kappa +1$ , which we no longer assume.\",\n \"Fortunately, we only require our bound for $\\\\lambda < \\\\lambda _O -\\\\rho $ .\",\n \"Therefore we can use Lemma REF and Lemma REF to get $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\text{e}^{4\\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}}\\\\left\\\\Vert \\\\widehat{}\\\\left(0\\\\right) - \\\\widehat{}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\\\\\\\\\le C\\\\text{e}^{4\\\\left(\\\\lambda _O-\\\\rho \\\\right)\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O-\\\\rho }(0)\\\\right\\\\Vert _{\\\\rm op}}\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ From this and $\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\ge \\\\left(1+C\\\\right)^{-1}$ , there exists a constant $\\\\widetilde{M}>0$ such that both $\\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\widetilde{M} \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}, \\\\qquad \\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\widetilde{M},$ for sufficiently large $d$ .\",\n \"Now recall $\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{G}_\\\\mu \\\\left(k\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda } = \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)^2\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ and $\\\\left(1+C\\\\right)^{-1}\\\\le \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\le \\\\left(1-\\\\mu _\\\\lambda \\\\right)^{-1} = \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Therefore an application of the chain rule gives the bound $*{\\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{J}_{\\\\mu }\\\\left(k,l\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda }} \\\\le 6\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^4\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op},$ uniformly in $k$ and $l$ .\",\n \"These properties, along with the bound on $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda }$ from above, are sufficient to prove equicontinuity and the uniform boundedness for the required $\\\\lambda $ .\",\n \"Therefore we have proved that $f_3\\\\left(\\\\lambda \\\\right)$ is continuous.\",\n \"It just remains to prove that we can indeed bound the difference of $\\\\delta _\\\\lambda (k)$ with the difference of $\\\\widehat{}_{\\\\lambda }(k)$ as we claimed.\",\n \"[Proof of Lemma REF ] First note that there is nothing to prove if $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}=\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}=0$ , because the triangle inequality then forces $\\\\delta _\\\\lambda (k_1) = \\\\delta _\\\\lambda (k_2)=0$ .\",\n \"Without loss of generality, assume $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\ge \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ with $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}>0$ .\",\n \"Then $\\\\delta _\\\\lambda (k_1)-\\\\delta _\\\\lambda (k_2) = \\\\frac{\\\\delta _\\\\lambda (k_1)}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }(k_1)\\\\right\\\\Vert _{\\\\rm op}}\\\\left(\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\right) + \\\\left[\\\\delta _\\\\lambda (k_1)\\\\frac{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}} - \\\\delta _\\\\lambda (k_2)\\\\right]$ The second factor is the more troublesome: $*{\\\\delta _\\\\lambda (k_1)\\\\frac{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}} - \\\\delta _\\\\lambda (k_2)} = \\\\frac{1}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}}*{\\\\delta _\\\\lambda (k_1)\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\delta _\\\\lambda (k_2)\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}} \\\\\\\\\\\\le \\\\frac{*{\\\\delta _\\\\lambda (k_1)} \\\\vee *{\\\\delta _\\\\lambda (k_2)}}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}}*{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}}.$ Since $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\ge \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ , we have $*{\\\\delta _\\\\lambda (k_1)} \\\\vee *{\\\\delta _\\\\lambda (k_2)} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Therefore $*{\\\\delta _\\\\lambda (k_1)-\\\\delta _\\\\lambda (k_2)} \\\\le 2*{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}} \\\\le 2\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)-\\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op},$ where the last inequality holds because of the reverse triangle inequality.\",\n \"Proof of Main Theorems Proposition 7.1 Let $d>6$ be sufficiently large.\",\n \"Then for $\\\\lambda \\\\le \\\\lambda _O$ there exists $c>0$ such that $\\\\triangle _\\\\lambda < c\\\\beta ^2$ .\",\n \"Proposition REF gives us that there exists $c_f$ (increasing in $f$ ) such that $\\\\triangle _\\\\lambda \\\\le c_f\\\\beta ^2$ for all $\\\\lambda <\\\\lambda _O$ .\",\n \"Proposition REF then implies that there exists $c$ such that $c_f\\\\le c$ uniformly for all $\\\\lambda < \\\\lambda _O$ .\",\n \"It remains to prove the assertion for $\\\\lambda =\\\\lambda _O$ .\",\n \"Recall from Section REF the definition of $\\\\tau _\\\\lambda ^n(x,y)$ and $\\\\Lambda _n(x)$ .\",\n \"Furthermore, recall that if $x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}$ then such a connection is achieved in finitely many steps and the bound in $\\\\ref {Assump:2ndMoment}$ then implies that $\\\\left\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}\\\\right\\\\rbrace = \\\\bigcup ^\\\\infty _{n=1}\\\\left\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}_{\\\\Lambda _n(y)}\\\\right\\\\rbrace .$ Therefore by monotone convergence we have the pointwise limit $\\\\tau _\\\\lambda ^n(x,y) \\\\rightarrow \\\\tau _\\\\lambda (x,y)$ as $n\\\\rightarrow \\\\infty $ for all $\\\\lambda >0$ and $x,y\\\\in \\\\mathbb {X}$ .\",\n \"Since $\\\\tau _\\\\lambda ^n(x,y)$ only depends upon the finite region $\\\\Lambda _n(y)$ , the functions $\\\\lambda \\\\mapsto \\\\tau _\\\\lambda ^n(x,y)$ are continuous for all $x,y\\\\in \\\\mathbb {X}$ .\",\n \"This, with the monotonicity $\\\\tau _\\\\lambda ^n(x,y) \\\\le \\\\tau _\\\\lambda ^{n+1}(x,y)$ , implies that the function $\\\\lambda \\\\mapsto \\\\tau _\\\\lambda (x,y)$ is lower-semicontinuous.\",\n \"Since $\\\\tau _\\\\lambda (x,y)$ is non-decreasing in $\\\\lambda $ , this lower semi-continuity implies that the pointwise limit $\\\\tau _\\\\lambda (x,y)\\\\uparrow \\\\tau _{\\\\lambda _O}(x,y)$ as $\\\\lambda \\\\uparrow \\\\lambda _O$ holds, and the limit holds monotonically.\",\n \"This monotonic pointwise convergence implies that the integral $\\\\int \\\\tau _\\\\lambda (x,u)\\\\tau _\\\\lambda (u,v)\\\\tau _\\\\lambda (v,y)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) \\\\rightarrow \\\\int \\\\tau _{\\\\lambda _O}(x,u)\\\\tau _{\\\\lambda _O}(u,v)\\\\tau _{\\\\lambda _O}(v,y)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right)$ for all $x,y\\\\in \\\\mathbb {X}$ as $\\\\lambda \\\\uparrow \\\\lambda _O$ , and that this convergence is monotone increasing.\",\n \"If we take the supremum over $x,y\\\\in \\\\mathbb {X}$ , we get that $\\\\lim _{\\\\lambda \\\\uparrow \\\\lambda _O}\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda _O}^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"The uniform bound for $\\\\lambda <\\\\lambda _O$ then implies that this bound also holds at $\\\\lambda =\\\\lambda _O$ .\",\n \"The following proposition uses the bound $\\\\triangle _{\\\\lambda _O} < c\\\\beta ^2$ to show that percolation does not occur at criticality.\",\n \"Another element of the proof is that there is almost surely at most one infinite cluster.\",\n \"There are general considerations that show that this is the case.\",\n \"Indeed [14] establishes that on the discrete space $\\\\mathbb {Z}^d$ there is at most one infinite cluster if the edge occupation measure is stationary and obeys the ‘finite energy property’, and an analogous result for Poisson processes in the continuum applies in our case (see [4], [36]).\",\n \"Proposition 7.2 The critical percolation function $\\\\theta _{\\\\lambda _O}(a)=0$ for $\\\\mathcal {P}$ -almost every $a\\\\in \\\\mathcal {E}$ .\",\n \"Assume that $(\\\\overline{0},a),(\\\\overline{x},b)\\\\in \\\\mathbb {X}$ are in the (a.s. unique) infinite cluster, and are therefore connected.\",\n \"Then the FKG inequality implies that $\\\\theta _{\\\\lambda _O}(a)\\\\theta _{\\\\lambda _O}(b)\\\\le \\\\mathbb {P}_{\\\\lambda _O}\\\\big (|\\\\overline{0},a)|=\\\\infty ,|\\\\overline{x},b)|=\\\\infty \\\\big )\\\\le \\\\tau _{\\\\lambda _O}((\\\\overline{0},a),(\\\\overline{x},b)).$ Since $\\\\triangle _{\\\\lambda _O} \\\\le c\\\\beta ^2$ , for $\\\\mathcal {P}$ -a.e.\",\n \"$a,b\\\\in \\\\mathcal {E}$ there exists a sequence $\\\\overline{x}^{(a,b)}_n$ such that $\\\\tau _{\\\\lambda _O}((\\\\overline{0},a),(\\\\overline{x}^{(a,b)}_n,b))\\\\rightarrow 0$ .\",\n \"Therefore for $\\\\mathcal {P}$ -a.e.\",\n \"$a\\\\in \\\\mathcal {E}$ we have $\\\\theta _{\\\\lambda _O}(a)=0$ .\",\n \"We can define the linear operator $\\\\mathcal {T}_{\\\\lambda _O}$ by its action on $f\\\\in L^2(\\\\mathbb {X})$ : $\\\\mathcal {T}_{\\\\lambda _O}f(x) = \\\\int \\\\tau _{\\\\lambda _O}(x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right).$ Contrary to $\\\\mathcal {T}_\\\\lambda $ for $\\\\lambda <\\\\lambda _O$ , the operator $\\\\mathcal {T}_{\\\\lambda _O}$ may be an unbounded linear operator.\",\n \"Proposition 7.3 (The operator OZE at the critical point) Let $d>6$ be sufficiently large.\",\n \"The Ornstein-Zernike equation then extends to $\\\\lambda _O$ in the sense that the sum of (unbounded) linear operators vanishes: $\\\\mathcal {T}_{\\\\lambda _O}- - _{\\\\lambda _O}- \\\\lambda _O\\\\mathcal {T}_{\\\\lambda _O}\\\\left(+ _{\\\\lambda _O}\\\\right) = 0.$ Furthermore, for $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ the linear operator $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O}(k)$ is a bounded linear operator and the following equality holds: $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O}\\\\left(k\\\\right) - \\\\widehat{}\\\\left(k\\\\right) - \\\\widehat{}_{\\\\lambda _O}\\\\left(k\\\\right) - \\\\lambda _O\\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O}\\\\left(k\\\\right)\\\\left(\\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda _O}\\\\left(k\\\\right)\\\\right) = 0.$ Finally, there exists $c>0$ such that for all $k\\\\in \\\\mathbb {R}^d$ $\\\\left\\\\Vert _{\\\\lambda _O}\\\\right\\\\Vert _{\\\\rm op} \\\\le c\\\\beta , \\\\qquad \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda _O}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le c\\\\beta .$ Let $f\\\\in L^2(\\\\mathbb {X})$ .\",\n \"Then by the triangle inequality, for $\\\\lambda <\\\\lambda _O$ ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- - _{\\\\lambda _O}- \\\\lambda _O\\\\mathcal {T}_{\\\\lambda _O}\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right)f}_2 \\\\le {\\\\left(\\\\mathcal {T}_\\\\lambda - - _\\\\lambda - \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)f}_2 \\\\\\\\+ {\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda - _{\\\\lambda _O}+ _\\\\lambda - \\\\lambda _O\\\\mathcal {T}_{\\\\lambda _O}\\\\left(+ _{\\\\lambda _O}\\\\right) + \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left(+_\\\\lambda \\\\right)\\\\right)f}_2.$ The first norm on the right hand side vanishes, since the OZE holds for $\\\\lambda <\\\\lambda _O$ .\",\n \"To prove that the left hand side vanishes, we aim to show that the second norm on the right hand side vanishes as $\\\\lambda \\\\uparrow \\\\lambda _O$ .\",\n \"To do this, the triangle inequality implies that we only need to show that the following limits hold as $\\\\lambda \\\\uparrow \\\\lambda _O$ : ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda \\\\right)f}_2 &\\\\rightarrow 0, \\\\\\\\{\\\\left(_{\\\\lambda _O}- _\\\\lambda \\\\right)f}_2 &\\\\rightarrow 0,\\\\\\\\{\\\\left(\\\\mathcal {T}_{\\\\lambda _O}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda _\\\\lambda \\\\right)f}_2 &\\\\rightarrow 0.$ We begin with (REF ).\",\n \"Recall from the proof of Proposition REF that $\\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,y) := \\\\tau _{\\\\lambda _O}(x,y) - \\\\tau _\\\\lambda (x,y)$ converges monotonically and pointwise to zero.\",\n \"From the definition of the ${\\\\cdot }_2$ norm, ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda \\\\right)f}_2^2 &= \\\\int \\\\left(\\\\int \\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\\\\\&\\\\le \\\\int \\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,y)\\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,z)*{f(y)}*{f(z)}\\\\nu ^{\\\\otimes (3)}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y, \\\\mathrm {d}z\\\\right).$ The monotone convergence of $\\\\left(\\\\tau _{\\\\lambda _O}-\\\\tau _\\\\lambda \\\\right)(x,y)\\\\downarrow 0$ then implies that the limit (REF ) holds.\",\n \"For (), we use the functions $h^{(n)}_\\\\lambda \\\\colon \\\\mathbb {X}^2\\\\rightarrow \\\\mathbb {R}_+$ defined by $h^{(n)}_\\\\lambda (x,y) := {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\lambda ^{-1}\\\\int \\\\psi _n(x,w_{n-1},u_{n-1}) \\\\left( \\\\prod _{i=1}^{n-1} \\\\psi (\\\\vec{v}_i) \\\\right) \\\\psi _0(w_0,u_0,y) \\\\nu ^{\\\\otimes (2n)}\\\\left(\\\\mathrm {d}\\\\left( \\\\left(\\\\vec{w}, \\\\vec{u}\\\\right)_{[0,n-1]} \\\\right)\\\\right) &:n\\\\ge 1\\\\\\\\\\\\tfrac{1}{2} \\\\lambda ^2 \\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}w\\\\right)\\\\right)^2 &: n=0.\\\\end{array}\\\\right.\",\n \"}$ From Proposition REF and the proof of Proposition REF , we know that $0\\\\le \\\\pi ^{(n)}_\\\\lambda (x,y) \\\\le h^{(n)}_\\\\lambda (x,y)$ for all $n\\\\ge 0$ , $x,y\\\\in \\\\mathbb {X}$ , and $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ - the non-negativity follows from the definition of $\\\\pi ^{(n)}_\\\\lambda $ .\",\n \"The functions $ h^{(n)}_\\\\lambda $ can also be defined for $\\\\lambda =\\\\lambda _O$ , and since they are monotone increasing in $\\\\lambda $ , we have the $\\\\lambda $ -independent bound $\\\\pi ^{(n)}_\\\\lambda (x,y) \\\\le h^{(n)}_{\\\\lambda _O}(x,y)$ for all $\\\\lambda <\\\\lambda _O$ and $x,y\\\\in \\\\mathbb {X}$ .\",\n \"For the single-mark version of the model it is proven in [21] that $\\\\pi _{\\\\lambda _O}-\\\\pi _{\\\\lambda }$ converges pointwise to zero.\",\n \"This same argument works for our multi-mark version, and therefore $\\\\pi ^{(n)}_\\\\lambda (x,y) \\\\le h^{(n)}_{\\\\lambda _O}(x,y)$ for all $\\\\lambda \\\\le \\\\lambda _O$ .\",\n \"The triangle inequality then implies that $*{\\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,y)} = *{\\\\sum ^\\\\infty _{n=0}\\\\left(-1\\\\right)^n\\\\left( \\\\pi ^{(n)}_{\\\\lambda _O}(x,y) - \\\\pi ^{(n)}_{\\\\lambda }(x,y)\\\\right)} \\\\le 2\\\\sum ^\\\\infty _{n=0}h^{(n)}_{\\\\lambda _O}(x,y).$ It will be convenient to define $h_{\\\\lambda }\\\\colon \\\\mathbb {X}^2\\\\rightarrow \\\\mathbb {R}_+$ as $h_\\\\lambda (x,y) := \\\\sum ^\\\\infty _{n=0}h^{(n)}_{\\\\lambda }(x,y)$ .\",\n \"Then ${\\\\left(_{\\\\lambda _O}- _\\\\lambda \\\\right)f}_2^2 &= \\\\int \\\\left(\\\\int \\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&= \\\\int \\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,y)\\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,z)f(y)f(z)\\\\nu ^{\\\\otimes 3}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y, \\\\mathrm {d}z\\\\right) \\\\\\\\& \\\\le 4\\\\int h_{\\\\lambda _O}(x,y)h_{\\\\lambda _O}(x,z)*{f(y)}*{f(z)}\\\\nu ^{\\\\otimes 3}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y, \\\\mathrm {d}z\\\\right)\\\\nonumber \\\\\\\\& = 4\\\\int \\\\left(\\\\int h_{\\\\lambda _O}(x,y)*{f(y)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right).\",\n \"$ By Schur's test, this integral is finite for all $f\\\\in L^2(\\\\mathbb {X})$ if both the values $\\\\operatornamewithlimits{ess\\\\,sup}_{y}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\qquad \\\\text{ and } \\\\qquad \\\\operatornamewithlimits{ess\\\\,sup}_{x}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)$ are finite.\",\n \"To prove this for the former, we repeat the arguments of Proposition REF and Section REF to bound it in terms of $\\\\lambda _O$ , the triangle diagrams, and other $\\\\lambda $ -independent terms.\",\n \"The argument of Proposition REF with the uniform bounds of Observation REF then proves that $\\\\operatornamewithlimits{ess\\\\,sup}_{y}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}x\\\\right)$ is indeed finite.\",\n \"A similar argument also holds for $\\\\operatornamewithlimits{ess\\\\,sup}_{x}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)$ .\",\n \"We now have an integrable function $h_{\\\\lambda _O}(x,y)h_{\\\\lambda _O}(x,z)*{f(y)}*{f(z)}$ that dominates the integrand of (REF ).\",\n \"Recall that $\\\\pi _{\\\\lambda _O}-\\\\pi _{\\\\lambda }$ converges pointwise to zero.\",\n \"Thus the dominated convergence theorem implies that the limit () holds.\",\n \"For (), we can use $h_{\\\\lambda _O}$ to get the bound ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda _\\\\lambda \\\\right)f}_2^2 \\\\\\\\\\\\le \\\\int \\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,u)h_{\\\\lambda _O}(u,y)\\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,v)h_{\\\\lambda _O}(v,z)*{f(y)}*{f(z)} \\\\nu ^{\\\\otimes 5}\\\\left(\\\\mathrm {d}u,\\\\mathrm {d}v, \\\\mathrm {d}x, \\\\mathrm {d}y, \\\\mathrm {d}z\\\\right).$ The integrand converges monotonically to zero, and therefore the integral vanishes in the limit.\",\n \"We now consider the Fourier version of the OZE.\",\n \"We use the notation that, given a bounded linear operator $A\\\\colon L^2(\\\\mathbb {X}) \\\\rightarrow L^2(\\\\mathbb {X})$ , the Fourier transform is given by $\\\\mathcal {F}\\\\left( A \\\\right)(k)\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ .\",\n \"By hypothesis, $$ is a bounded operator, and the finiteness of $\\\\operatornamewithlimits{ess\\\\,sup}_{y}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}x\\\\right)$ implies that $_{\\\\lambda _O}$ is also bounded.\",\n \"Then (REF ) implies that $\\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O}\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right) \\\\right)(k) = \\\\mathcal {F}\\\\left( + _{\\\\lambda _O} \\\\right)(k)$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"We now make the claim that $\\\\mathcal {F}\\\\left( 1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right) \\\\right)(k)$ has a bounded linear inverse for all $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ .\",\n \"This is proven in Lemma REF below.\",\n \"This claim then implies that for $k\\\\ne 0$ we have $&\\\\mathcal {F}\\\\left( + _{\\\\lambda _O} \\\\right)(k)\\\\mathcal {F}\\\\left( 1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right) \\\\right)(k)^{-1} \\\\nonumber \\\\\\\\& \\\\hspace{85.35826pt} = \\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O}\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right) \\\\right)(k) \\\\mathcal {F}\\\\left( 1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right) \\\\right)(k)^{-1} \\\\nonumber \\\\\\\\& \\\\hspace{85.35826pt} = \\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O}\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right)\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right)^{-1} \\\\right)(k) = \\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O} \\\\right)(k).$ That is, $\\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O} \\\\right)(k)$ is a bounded linear operator for $k\\\\ne 0$ .\",\n \"The limit () also allows us to extend our sub-critical bound on $_\\\\lambda $ to apply at criticality.\",\n \"The limit () may be phrased as saying that the function $\\\\lambda \\\\mapsto *{_\\\\lambda f}_2$ is continuous at $\\\\lambda =\\\\lambda _O$ (from the left) for all $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ .\",\n \"Since $\\\\lambda \\\\mapsto \\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op}$ can be written as a supremum of such functions, it is a lower semi-continuous function.\",\n \"Our bound $\\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le c\\\\beta $ for $\\\\lambda < \\\\lambda _O$ (from Proposition REF ) then implies $\\\\left\\\\Vert _{\\\\lambda _O}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\liminf _{\\\\lambda \\\\uparrow \\\\lambda _O}\\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le c\\\\beta .$ The limit $*{\\\\left(\\\\widehat{}_{\\\\lambda _O}^{(n)}(0) - \\\\widehat{}_{\\\\lambda }^{(n)}(0)\\\\right)f}_2 \\\\rightarrow 0$ for each $f\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ can be derived in essentially the same way as () above.\",\n \"The above lower semi-continuity argument can then be used to show that the bound from Proposition REF extends to also hold at $\\\\lambda =\\\\lambda _O$ .\",\n \"From Lemma REF and the positivity of $\\\\pi ^{(n)}_{\\\\lambda _O}(x,y)$ this bound also holds for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"By summing over $n$ , we then get the desired bound on $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda _O}(k)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Lemma 7.4 For $\\\\lambda = \\\\lambda _O$ and $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ , the bounded linear operator $\\\\left(1- \\\\lambda _O\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda _O}(k)\\\\right)\\\\right)\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ has a bounded linear inverse.\",\n \"We first introduce some notation.\",\n \"Let $A\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ be a self-adjoint linear operator.\",\n \"Then we define the minimum modulus to be $\\\\left\\\\Vert A\\\\right\\\\Vert _{\\\\rm min} := \\\\inf _{0\\\\ne f\\\\in L^2(\\\\mathcal {E})}\\\\frac{*{Af}_2}{{f}_2}.$ Note that it follows from the Spectral Theorem (Theorem REF ) that $A$ has a bounded linear inverse if and only if $\\\\left\\\\Vert A\\\\right\\\\Vert _{\\\\rm min}>0$ .\",\n \"Now suppose that $A=A_\\\\lambda $ , and that for all $f\\\\in L^2(\\\\mathcal {E})$ the map $\\\\lambda \\\\mapsto *{A_\\\\lambda f}$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"Then the map $\\\\lambda \\\\mapsto \\\\left\\\\Vert A_\\\\lambda \\\\right\\\\Vert _{\\\\rm min}$ is upper semicontinuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"That is, for any sequence $\\\\lambda _n$ in $\\\\left[0,\\\\lambda _O\\\\right]$ such that $\\\\lambda _n\\\\rightarrow \\\\lambda $ , $\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\left\\\\Vert A_{\\\\lambda _n}\\\\right\\\\Vert _{\\\\rm min} \\\\le \\\\left\\\\Vert A_\\\\lambda \\\\right\\\\Vert _{\\\\rm min}.$ We now consider $A_\\\\lambda = 1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)$ .\",\n \"A similar argument to the way we proved () shows that $\\\\lambda \\\\mapsto *{A_\\\\lambda f}$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"Our result is then proven if we can show that for all $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\lbrace 0\\\\rbrace $ we have an $\\\\varepsilon = \\\\varepsilon (k)>0$ such that $\\\\left\\\\Vert 1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right\\\\Vert _{\\\\rm min} \\\\ge \\\\varepsilon $ uniformly in $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"We will in fact show the stronger result that $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) \\\\le 1 - \\\\varepsilon $ uniformly in $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"Recalling the definition of $\\\\delta _\\\\lambda (k)$ from (REF ), we have $1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) &= 1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) - \\\\lambda \\\\delta _{\\\\lambda }(k) \\\\nonumber \\\\\\\\&= \\\\underbrace{1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right)}_{>0 \\\\text{ from Lemma~\\\\ref {lem:Optlam-a}}} + \\\\lambda \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) + \\\\lambda \\\\left(\\\\delta _{\\\\lambda }(0) - \\\\delta _{\\\\lambda }(k)\\\\right) \\\\nonumber \\\\\\\\& > \\\\lambda \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\left(1+\\\\mathcal {O}\\\\left(\\\\beta \\\\right)\\\\right).$ Here we have used Lemma REF to bound the difference of $\\\\delta $ with the difference of $\\\\widehat{}_{\\\\lambda _O}$ .\",\n \"For sufficiently large $d$ , REF then proves $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) \\\\le 1 - \\\\varepsilon $ uniformly in $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"Note that in conjunction, Proposition REF , Proposition REF and Proposition REF prove the OZE equations of Theorem REF .\",\n \"In particular, the former two prove that the OZE holds in the subcritical regime, and the latter extends this to criticality where possible.\",\n \"Furthermore, Proposition REF gives the bounds on $_\\\\lambda $ and $\\\\widehat{}_{\\\\lambda }(k)$ subcritically whilst Proposition REF extends this bound to criticality.\",\n \"Regarding Theorem REF , the bound on $\\\\triangle _\\\\lambda $ and the lack of percolation at criticality were proven at the beginning of this section.\",\n \"The infrared bound component of the theorem is proven below.\",\n \"[Proof of Theorem REF ] First note that there is nothing to prove for $k=0$ .\",\n \"From Proposition REF and Lemma REF , we can write $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) = \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\left(1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right)^{-1},$ for all $k\\\\in \\\\mathbb {R}^d$ if $\\\\lambda <\\\\lambda _O$ , and for all $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\lbrace 0\\\\rbrace $ if $\\\\lambda = \\\\lambda _O$ .\",\n \"If we use the Spectral Theorem (Theorem REF ) to `diagonalize' the operator $\\\\widehat{a}(k) =\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)$ , we get a multiplication operator on some Hilbert space $L^2(\\\\mathfrak {E}_k,\\\\mu _k)$ taking values $\\\\widetilde{a}(k)(e)$ for $e\\\\in \\\\mathfrak {E}_k$ .\",\n \"Since, $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ is formed from $\\\\widehat{a}(k) =\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)$ , it can also be diagonalized over $L^2(\\\\mathfrak {E}_k,\\\\mu _k)$ , and takes values $\\\\widetilde{\\\\tau }_\\\\lambda (k)(e) = \\\\frac{\\\\widetilde{a}(k)(e)}{1 - \\\\lambda \\\\widetilde{a}(k)(e)}.$ Since $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) < 1$ (from the proof of Lemma REF ), we know $\\\\lambda \\\\widetilde{a}(k)(e) >1$ and $\\\\lambda \\\\widetilde{\\\\tau }_\\\\lambda (k)(e) \\\\ge -1$ for $\\\\mu _k$ -almost every $e\\\\in \\\\mathfrak {E}_k$ .\",\n \"It therefore only remains to bound $\\\\widetilde{\\\\tau }_\\\\lambda (k)(e)$ from above.\",\n \"Equivalently, we only need to bound $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)$ from above.\",\n \"We also note that the proof of () can be adapted to show that $\\\\lambda \\\\mapsto \\\\left$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right]$ for all $f\\\\in L^2(\\\\mathcal {E})$ .\",\n \"Then because $\\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right) = \\\\sup _{f}\\\\tfrac{\\\\left}{\\\\left}$ , this proves that $\\\\lambda \\\\mapsto \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right)$ is lower-semicontinuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"The bound $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right) <1$ for $\\\\lambda < \\\\lambda _O$ (from Lemma REF ) then implies that $\\\\lambda _O\\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda _O}(0)\\\\right)\\\\le 1$ .\",\n \"From (REF ) the calculation proceeds: $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) &= \\\\frac{\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)}{1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)} \\\\nonumber \\\\\\\\& = \\\\frac{\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)}{\\\\underbrace{1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right)}_{\\\\ge 0} + \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right) - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)} \\\\nonumber \\\\\\\\& \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) + \\\\delta _{\\\\lambda }(0) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) - \\\\delta _{\\\\lambda }(k)} \\\\nonumber \\\\\\\\& \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) + \\\\mathcal {O}\\\\left(\\\\beta \\\\right)}{\\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\left(1 + \\\\mathcal {O}\\\\left(\\\\beta \\\\right)\\\\right)} = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) + \\\\mathcal {O}\\\\left(\\\\beta \\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)},$ where we have bounded $\\\\delta _{\\\\lambda }(0)-\\\\delta _{\\\\lambda }(k)$ using Lemma REF and Observation REF .\",\n \"Critical Intensities [Proof of Proposition REF ] By the Mecke equation (REF ), we find $*{\\\\chi _\\\\lambda }_1 & = 1 + \\\\lambda \\\\int \\\\tau _\\\\lambda (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right) = 1 + \\\\lambda \\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,b)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right),\\\\\\\\*{\\\\chi _\\\\lambda }_\\\\infty & = 1 + \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{b}\\\\int \\\\tau _\\\\lambda (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = 1 + \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{b}\\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,b) \\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = 1 + \\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{1,\\\\infty }.$ From Lemma REF we have $\\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,b)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"Therefore $\\\\lambda ^{(1)}_T \\\\ge \\\\lambda _O \\\\ge \\\\lambda _T^{(\\\\infty )}$ .\",\n \"Also recall from the discussion in Section REF that $\\\\lambda _c \\\\ge \\\\lambda ^{(1)}_T$ .\",\n \"Therefore we only now need to prove that $\\\\lambda ^{(1)}_T = \\\\lambda ^{(\\\\infty )}_T$ .\",\n \"We do this by proving that if $*{\\\\chi _\\\\lambda }_1<\\\\infty $ then $*{\\\\chi _\\\\lambda }_\\\\infty <\\\\infty $ .\",\n \"Fix $b\\\\in \\\\mathcal {E}$ .\",\n \"Then by considering the vertices in the cluster $\\\\overline{0},b)$ adjacent to $\\\\left(\\\\overline{0},b\\\\right)$ , we find $\\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},b)}\\\\right] \\\\le 1 + \\\\mathbb {E}_\\\\lambda \\\\left[\\\\sum _{\\\\left(\\\\overline{x},a\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},a\\\\right)\\\\sim \\\\left(\\\\overline{0},b\\\\right)}*{\\\\overline{x},a)}\\\\right] = 1 + \\\\lambda \\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{x},a)}\\\\right]\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right).$ In this equality we have used Mecke's equation (REF ).\",\n \"From the spatial translation invariance of the model, $\\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{x},a)}\\\\right] = \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},a)}\\\\right]$ for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ .\",\n \"Therefore an application of a supremum bound to the $a$ -integral gives $*{\\\\chi _\\\\lambda }_\\\\infty = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},b)}\\\\right] \\\\le 1 + \\\\lambda \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\right)\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},c)}\\\\right]\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right)\\\\\\\\= 1 + \\\\lambda \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\right)*{\\\\chi _\\\\lambda }_1.$ From the finiteness of the parenthesised factor, $*{\\\\chi _\\\\lambda }_1<\\\\infty $ implies $*{\\\\chi _\\\\lambda }_\\\\infty <\\\\infty $ and the result is proven.\",\n \"[Proof of Proposition REF ] The equality $\\\\lambda _c\\\\left(\\\\dagger \\\\right)= \\\\lambda _T\\\\left(\\\\dagger \\\\right)$ holds from [35].\",\n \"The equality $\\\\lambda _O = \\\\lambda _T^{(p)}$ follows from Proposition REF because the condition (REF ) clearly holds for the bounded radii model.\",\n \"Also recall from the discussion in Section REF that $\\\\lambda _c \\\\ge \\\\lambda ^{(p)}_T$ for all $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"Now note that for all $a,b\\\\in \\\\mathcal {E}$ we have that if $\\\\left(\\\\overline{0},\\\\dagger \\\\right)\\\\sim \\\\left(\\\\overline{x},b\\\\right)$ then $\\\\left(\\\\overline{0},a\\\\right)\\\\sim \\\\left(\\\\overline{x},b\\\\right)$ .\",\n \"Therefore $\\\\lambda _c\\\\left(\\\\dagger \\\\right) \\\\ge \\\\lambda _c$ , and the equality $\\\\lambda _c\\\\left(\\\\dagger \\\\right)= \\\\lambda _T\\\\left(\\\\dagger \\\\right)$ then means we only need to prove $\\\\lambda _T\\\\left(\\\\dagger \\\\right) \\\\le \\\\lambda _T^{(p)}$ for some $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"We will do so for $p=1$ .\",\n \"Suppose that $\\\\lambda >\\\\lambda _T^{(1)}$ , so that $\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[(\\\\overline{0};a\\\\right)\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\infty $ .\",\n \"That is, the expected cluster size of a vertex with a random ($\\\\mathcal {P}$ -distributed) mark is infinite.\",\n \"Note that we have an ordering of the marks in that if two radii $a,b$ satisfy $a\\\\ge b$ , then $\\\\varphi \\\\left(\\\\overline{x};a,c\\\\right) \\\\ge \\\\varphi \\\\left(\\\\overline{x};b,c\\\\right)$ for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ and $c\\\\in \\\\mathcal {E}$ .\",\n \"This also implies that $a\\\\mapsto \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},a}\\\\right]$ is a non-decreasing function.\",\n \"This in turn implies that if $\\\\lambda >\\\\lambda _T^{(1)}$ and $a$ satisfies $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , then $\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right) = \\\\infty $ .\",\n \"Given some radius $a\\\\in \\\\mathcal {E}$ such that $a>0$ and $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , we define a new mark, $\\\\star _a$ , that forms connections according to $\\\\varphi \\\\left(\\\\overline{x};\\\\star _a,b\\\\right) = 1\\\\left\\\\lbrace *{\\\\overline{x}}\\\\le a\\\\right\\\\rbrace ,$ for all $b\\\\in \\\\mathcal {E}$ .\",\n \"Note that if $b\\\\ge a$ then $\\\\varphi \\\\left(\\\\overline{x};\\\\star _a,c\\\\right) \\\\le \\\\varphi \\\\left(\\\\overline{x};b,c\\\\right)$ for all $c\\\\in \\\\mathcal {E}$ and $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};\\\\star _a}\\\\right] \\\\le \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]$ .\",\n \"We now consider the expected cluster size of vertices adjacent to $\\\\left(\\\\overline{0},\\\\star _a\\\\right)$ with radius $\\\\ge a$ .\",\n \"We also restrict to cases for which there is an unique neighbour, and that neighbour has radius $\\\\ge a$ .\",\n \"Since the probability of a connection is independent of the radius of the proposed neighbour, the distribution of the unique neighbour is equal to $\\\\mathcal {P}$ - conditioned upon having radius $\\\\ge a$ .\",\n \"Therefore we have the lower bound $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};\\\\star _a}\\\\right] \\\\ge 1 + \\\\mathbb {P}_\\\\lambda \\\\left(\\\\exists !\",\n \"\\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\star _a\\\\right),b\\\\ge a\\\\right)\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right).$ As noted above, $\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)=\\\\infty $ .\",\n \"Furthermore, as $a>0$ and $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , the random variable $\\\\#\\\\left\\\\lbrace \\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\star _a\\\\right),b\\\\ge a\\\\right\\\\rbrace $ obeys a Poisson distribution with strictly positive mean.\",\n \"Therefore $\\\\mathbb {P}_\\\\lambda \\\\left(\\\\exists !\",\n \"\\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\star _a\\\\right),b\\\\ge a\\\\right)>0$ and $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};\\\\star _a}\\\\right]=\\\\infty $ .\",\n \"This in turn implies that $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};a}\\\\right] =\\\\infty $ for all $a>0$ .\",\n \"To relate this to $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},\\\\dagger }\\\\right]$ , we perform a similar lower bound.\",\n \"Given a radius $a>0$ and $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , we restrict to cases where there is an unique neighbour of $\\\\left(\\\\overline{0},\\\\dagger \\\\right)$ with radius $\\\\ge a$ and bound $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},\\\\dagger }\\\\right] \\\\ge 1+ \\\\mathbb {P}_\\\\lambda \\\\left(\\\\exists !\",\n \"\\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\dagger \\\\right),b\\\\ge a\\\\right) \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};a}\\\\right].$ As before, the probability factor is strictly positive and the expected cluster size term is infinite.\",\n \"Hence $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},\\\\dagger }\\\\right]=\\\\infty $ and $\\\\lambda \\\\ge \\\\lambda \\\\left(\\\\dagger \\\\right)$ .\",\n \"Therefore $\\\\lambda _T\\\\left(\\\\dagger \\\\right)\\\\le \\\\lambda ^{(1)}_T$ as required.\",\n \"Model Properties Here we will prove that the models outlined in Section REF do indeed satisfy Assumption REF .\",\n \"First we consider the single mark “finite variance models” considered by [21].\",\n \"As proven in that reference, this includes the Poisson blob and Gaussian connection models.\",\n \"Lemma B.1 Single mark finite variance models satisfy the conditions of Assumption REF .\",\n \"First note that since $\\\\mathcal {E}$ is a singleton, the operators $\\\\widehat{}(k)$ are simply scalars.\",\n \"In particular, this means that the conditions (REF ), and () of REF hold trivially.\",\n \"The finiteness of $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ is also equivalent to the finiteness of $\\\\int \\\\varphi (x)\\\\mathrm {d}x$ required by [21].\",\n \"Since $\\\\widehat{}(k)$ is simply a scalar, assumption REF follows directly from the third finite variance condition of [21].\",\n \"The condition REF is similarly a generalisation of the second finite variance condition, and therefore follows.\",\n \"Specifically, the sets $B(x)$ are the $\\\\varepsilon $ -balls around $x\\\\in \\\\mathbb {R}^d$ where $0<\\\\varepsilon = p^2\\\\widehat{\\\\varphi }_{1,1}(k)f_1^2 + 2pq\\\\widehat{\\\\varphi }_{1,2}(k)f_1f_2 + q^2\\\\widehat{\\\\varphi }_{2,2}(k)f_2^2,$ and therefore the spectrum of $\\\\widehat{}(k)$ is equal to the set of eigenvalues of the matrix $\\\\begin{pmatrix}p^2\\\\widehat{\\\\varphi }_{1,1}(k) & pq\\\\widehat{\\\\varphi }_{1,2}(k)\\\\\\\\pq\\\\widehat{\\\\varphi }_{1,2}(k) & q^2\\\\widehat{\\\\varphi }_{2,2}(k)\\\\end{pmatrix}.$ This $2\\\\times 2$ matrix has characteristic polynomial given by $z \\\\mapsto z^2 - \\\\left(p^2\\\\widehat{\\\\varphi }_{1,1}(k) + q^2\\\\widehat{\\\\varphi }_{2,2}(k)\\\\right) z + p^2q^2\\\\left( \\\\widehat{\\\\varphi }_{1,1}(k)\\\\widehat{\\\\varphi }_{2,2}(k) - \\\\widehat{\\\\varphi }_{1,2}(k)^2\\\\right).$ Solving for roots of this quadratic then gives $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) = \\\\frac{p^2\\\\widehat{\\\\varphi }_{1,1}(k) + q^2\\\\widehat{\\\\varphi }_{2,2}(k)}{2} + \\\\sqrt{\\\\left(\\\\frac{p^2\\\\widehat{\\\\varphi }_{1,1}(k) - q^2\\\\widehat{\\\\varphi }_{2,2}(k)}{2}\\\\right)^2 + p^2q^2\\\\widehat{\\\\varphi }_{1,2}(k)^2},$ where the square root takes the positive value.\",\n \"Note that by the definition of the double mark finite variance model $p^2\\\\widehat{\\\\varphi }_{1,1}(k) - q^2\\\\widehat{\\\\varphi }_{2,2}(k)$ is the Fourier transform of a non-negative (or non-positive) function, and therefore $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)$ is a sum of terms maximised at $k=0$ .\",\n \"Furthermore, each approaches that maximum quadratically and at least one of $\\\\widehat{\\\\varphi }_{1,1}(k)$ , $\\\\widehat{\\\\varphi }_{2,2}(k)$ , and $\\\\widehat{\\\\varphi }_{1,2}(k)$ uniquely achieves its maximum at $k=0$ (otherwise we have $\\\\varphi = 0$ almost everywhere).\",\n \"Therefore REF is satisfied.\",\n \"Since we have only finitely many marks, there are only finitely many sequences of marks of a given length.\",\n \"Therefore since each of the $\\\\varphi _{i,j}$ individually satisfy the 1-mark analog of REF , the 2-mark version is satisfied by $\\\\varphi $ too.\",\n \"It is not inconceivable that the above argument could be generalised to 3 or 4 mark systems.\",\n \"The degree 3 and 4 characteristic polynomials arising from the matrices have roots that can be solved for in radicals.\",\n \"It may be then that a particular clever rearrangement of the expression allows the required properties to be deduced like above.\",\n \"Such an investigation is currently beyond the authors' patience.\",\n \"Of course, the famous Abel-Ruffini Theorem states that no such solution in radicals exists for arbitrary polynomials of degree 5 and higher, and so this approach seems doomed for systems with such families of marks.\",\n \"Lemma B.3 Space-mark factorisation models satisfy the conditions of Assumption REF .\",\n \"Note that $\\\\widehat{}(k) = \\\\widehat{\\\\overline{\\\\varphi }}(k)\\\\mathcal {K}$ , where $\\\\mathcal {K}\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ is the self-adjoint linear integral operator with kernel function $K$ .\",\n \"Since $\\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{b}\\\\int K(a,b)\\\\mathcal {P}(\\\\mathrm {d}a)\\\\le 1$ , $\\\\mathcal {K}$ is a bounded operator and $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) = {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widehat{\\\\overline{\\\\varphi }}(k)\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right) &: \\\\widehat{\\\\overline{\\\\varphi }}(k) \\\\ge 0\\\\\\\\-\\\\widehat{\\\\overline{\\\\varphi }}(k)\\\\mathbb {S}\\\\left(-\\\\mathcal {K}\\\\right) &: \\\\widehat{\\\\overline{\\\\varphi }}(k) < 0.\\\\end{array}\\\\right.\",\n \"}$ Since $\\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{\\\\rm op}\\\\le 1$ , the finiteness of $\\\\widehat{\\\\overline{\\\\varphi }}(0)$ implies that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ is finite, and the $d$ -independence of $\\\\mathcal {K}$ implies that REF holds.\",\n \"Since $\\\\overline{\\\\varphi }(\\\\overline{x})\\\\ge 0$ , we have $*{\\\\widehat{\\\\overline{\\\\varphi }}(k)} < \\\\widehat{\\\\overline{\\\\varphi }}(0)$ for all $k\\\\ne 0$ .\",\n \"Furthermore, $\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right) = \\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{\\\\rm op}$ implies that $\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right)\\\\ge \\\\mathbb {S}\\\\left(-\\\\mathcal {K}\\\\right)$ .\",\n \"Then $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)$ inherits the properties required by REF from $\\\\widehat{\\\\overline{\\\\varphi }}(k)$ .\",\n \"For REF , we note that the $d$ -independence of $\\\\mathcal {K}$ means that required decay properties are again inherited from $\\\\widehat{\\\\overline{\\\\varphi }}(k)$ .\",\n \"Lemma B.4 The marked multivariate Gaussian model satisfies the conditions of Assumption REF .\",\n \"Firstly note that for each $\\\\left(a,b\\\\right)\\\\in \\\\mathcal {E}^2$ the Gaussian structure of $\\\\varphi $ means that it factorises over the $d$ eigenvector directions of $\\\\Sigma (a,b)$ .\",\n \"This factorisation ensures that that the $d$ -dimensional Fourier transform is the product of the 1-dimensional Fourier transforms.\",\n \"We have $\\\\widehat{\\\\varphi }(k;a,b) = \\\\mathcal {A} \\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2}k^{\\\\intercal }\\\\Sigma (a,b)^{-1}k\\\\right).$ In particular, when $k=0$ the Fourier transform is $a,b$ -independent.\",\n \"This means $\\\\widehat{}(0) = \\\\mathcal {A}\\\\mathbf {1}$ where $\\\\mathbf {1}$ is the integral operator with constant kernel function 1.\",\n \"For $f\\\\in L^2(\\\\mathcal {E})$ we have ${f}{\\\\mathbf {1}f} &= \\\\int f(a) \\\\left(\\\\int f(b) \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)\\\\right) \\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\mathbb {E}_\\\\mathcal {P}\\\\left[f\\\\right]^2,\\\\\\\\{f}{f} &= \\\\int f(a)^2 \\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\mathbb {E}_\\\\mathcal {P}\\\\left[f^2\\\\right].$ By Jensen's inequality we therefore have $\\\\mathbb {S}\\\\left(\\\\mathbf {1}\\\\right) = 1$ (equality follows by considering the test function $f(a)\\\\equiv 1$ ).\",\n \"It therefore follows that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) = \\\\mathcal {A}<\\\\infty $ .\",\n \"From (REF ) we have $\\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\mathcal {A}\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}} \\\\exp \\\\left(-\\\\pi ^2k^{\\\\intercal }\\\\Sigma (a,b)^{-1}k\\\\right) \\\\le \\\\mathcal {A}\\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2\\\\Sigma _{\\\\max }}*{k}^2\\\\right).$ This bound then implies the condition (REF ) of REF holds with $C=1$ .\",\n \"Before we address () of REF , we turn to REF .\",\n \"From our above bounds we already have $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\mathcal {A}\\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2\\\\Sigma _{\\\\max }}*{k}^2\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2\\\\Sigma _{\\\\max }}*{k}^2\\\\right).$ This exponential term ensures that REF is satisfied.\",\n \"We now return to () of REF .\",\n \"For all $x\\\\in \\\\mathbb {R}$ we have $1-\\\\cos (x) \\\\le \\\\frac{1}{2}x^2$ , and therefore $\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\frac{1}{2}*{k}^2 \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x},$ for some unit vector $e\\\\in \\\\mathbb {R}^d$ .\",\n \"Since the vector $e$ picks out the second moment of $\\\\varphi $ in only one direction, a standard calculation gives $\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\le \\\\Sigma _{\\\\max }^2\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Therefore $\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ is bounded by a $d$ -independent quadratic function.\",\n \"In conjunction with the quadratic bound of REF , this then proves () in the $k\\\\rightarrow 0$ regime.\",\n \"The remaining regime then holds because the triangle inequality implies $\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le 2\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = 2\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"To address REF , recall that the spatial convolution of two multivariate Gaussian functions with means $\\\\overline{0}$ and covariance matrices $\\\\Sigma _1$ and $\\\\Sigma _2$ is a multivariate Gaussian function with mean $\\\\overline{0}$ and covariance matrix $\\\\Sigma _1+\\\\Sigma _2$ .\",\n \"Therefore for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ we have $&\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,a_2,a_3,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\varphi (\\\\overline{x}-\\\\overline{y};a_1,a_2)\\\\varphi (\\\\overline{y};a_3,a_4)\\\\mathrm {d}\\\\overline{y}\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}= \\\\mathcal {A}^2\\\\left(2\\\\pi \\\\right)^{-d/2}\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,a_2,a_3,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\det \\\\left(\\\\Sigma (a_1,a_2) + \\\\Sigma (a_3,a_4)\\\\right)\\\\right)^{-1/2}\\\\exp \\\\left(-\\\\frac{1}{2}\\\\overline{x}^{\\\\intercal }\\\\left(\\\\Sigma (a_1,a_2) + \\\\Sigma (a_3,a_4)\\\\right)^{-1}\\\\overline{x}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{14.22636pt} \\\\le \\\\mathcal {A}^2\\\\left(4\\\\pi \\\\Sigma _{\\\\min }\\\\right)^{-d/2}\\\\exp \\\\left(-\\\\frac{*{\\\\overline{x}}^2}{4\\\\Sigma _{\\\\min }}\\\\right) \\\\le \\\\mathcal {A}^2\\\\left(4\\\\pi \\\\Sigma _{\\\\min }\\\\right)^{-d/2} \\\\rightarrow 0.$ In particular, this bounds the integral appearing in the definition of the sets $B(x)$ (see (REF )) and therefore proves they are empty.\",\n \"For the bound of the convolution of three $\\\\varphi $ functions, we can get this immediately from the above calculation and $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"Specifically REF holds with $g(d) = \\\\left(4\\\\pi \\\\Sigma _{\\\\min }(d)\\\\right)^{-d/2}$ and the sets $B(x) = \\\\emptyset $ .\",\n \"Lemma B.5 The bounded-volume Boolean disc model satisfies the conditions of Assumption REF .\",\n \"Crucial to the conditions of REF holding is the upper bound on $R_d(a)$ in (REF ).\",\n \"We use the shorthand $R_{a,b} := R_d(a) + R_d(b),\\\\qquad V_{a,b} := \\\\frac{\\\\pi ^{\\\\frac{d}{2}}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)}R^d_{a,b},$ so $V_{a,b}$ equals the Lebesgue volume of the $d$ -dimensional Euclidean ball with radius $R_{a,b}$ .\",\n \"Note that the upper bound $R^{({\\\\rm max})}_d$ on $R_d(a)$ in (REF ) ensures $V_{a,b} \\\\le V^{({\\\\rm max})}$ uniformly in $d$ .\",\n \"We first find $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\left[0,1\\\\right]}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\left[0,1\\\\right]}\\\\int V_{a,b}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\le V^{({\\\\rm max})}.$ This proves the statement that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ is finite in REF .\",\n \"From the condition (REF ) we also get $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\ge \\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a, \\\\mathrm {d}b\\\\right) \\\\ge \\\\epsilon ^2\\\\frac{d\\\\pi ^\\\\frac{d}{2}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)} \\\\int ^{2\\\\left(1-\\\\frac{c_2}{d}\\\\right)R^{({\\\\rm max})}_d}_0 r^{d-1}\\\\mathrm {d}r \\\\\\\\= \\\\epsilon ^2V^{({\\\\rm max})}\\\\left(1-\\\\frac{c_2}{d}\\\\right)^d \\\\rightarrow \\\\epsilon ^2\\\\text{e}^{-c_2}V^{({\\\\rm max})},$ where we have used the test function $f(a) \\\\equiv 1$ to lower bound $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"We then prove (REF ) by bounding $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } =\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\left[0,1\\\\right]}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}= \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\left[0,1\\\\right]}V_{a,b} \\\\le V^{({\\\\rm max})}.$ We will return for () after addressing REF .\",\n \"For REF , it will be useful to have an expression for $\\\\widehat{\\\\varphi }(k;a,b)$ for each $a,b\\\\in \\\\left[0,1\\\\right]$ .\",\n \"Suppose we are able to find $C_1\\\\in \\\\left(0,1\\\\right)$ and $C_2 >0$ such that for all $a,b\\\\in \\\\left[0,1\\\\right]$ $*{\\\\widehat{\\\\varphi }(k;a,b)} \\\\le \\\\widehat{\\\\varphi }(0;a,b) \\\\left(C_1 \\\\vee \\\\left(1 - C_2*{k}^2\\\\right)\\\\right).$ Then since the $k$ -dependent factor is $a,b$ -independent we can use Lemma REF to show that $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) \\\\le \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\left(C_1 \\\\vee \\\\left(1 - C_2*{k}^2\\\\right)\\\\right),$ and therefore prove that REF is satisfied.\",\n \"In finding an expression for $\\\\widehat{\\\\varphi }(k;a,b)$ we are assisted by the spherical symmetry of the connection function and follow [18].\",\n \"It transpires that $\\\\widehat{\\\\varphi }(k;a,b) = \\\\left(\\\\frac{2\\\\pi R_{a,b}}{*{k}}\\\\right)^\\\\frac{d}{2}J_{\\\\frac{d}{2}}\\\\left(R_{a,b}*{k}\\\\right),$ where $J_{\\\\frac{d}{2}}$ is the Bessel function of the first kind of order $\\\\frac{d}{2}$ .\",\n \"This function has the expansion $J_{\\\\frac{d}{2}}\\\\left(r\\\\right) = \\\\sum ^\\\\infty _{m=0} \\\\frac{\\\\left(-1\\\\right)^{m}}{m!\\\\Gamma \\\\left(\\\\frac{d}{2}+m+1\\\\right)}\\\\left(\\\\frac{r}{2}\\\\right)^{\\\\frac{d}{2}+2m},$ converging for all $r\\\\ge 0$ .\",\n \"We will consider three different regimes for $*{k}$ .\",\n \"For $r \\\\ll \\\\sqrt{d}$ the expansion (REF ) is asymptotic, and since $R^{({\\\\rm max})}_d$ and $R^{({\\\\rm min})}_d$ are both asymptotically proportional to $\\\\sqrt{d}$ we have $\\\\widehat{\\\\varphi }(k;a,b) \\\\sim \\\\frac{\\\\pi ^\\\\frac{d}{2}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)}R_{a,b}^d - \\\\frac{\\\\pi ^\\\\frac{d}{2}}{4\\\\Gamma \\\\left(\\\\frac{d}{2}+2\\\\right)}R_{a,b}^{d+2}*{k}^2 = V_{a,b}\\\\left(1 - \\\\frac{R^2_{a,b}}{2\\\\left(d+2\\\\right)}*{k}^2\\\\right)$ for $*{k}\\\\ll 1$ .\",\n \"This proves the quadratic part of the desired bound.\",\n \"The Bessel function $J_{\\\\frac{d}{2}}$ is bounded and achieves its global maximum (in absolute value) at its first non-zero stationary point, denoted $j^{\\\\prime }_{\\\\frac{d}{2},1}$ .\",\n \"From [2], we have $j^{\\\\prime }_{\\\\frac{d}{2},1} = \\\\frac{d}{2} + \\\\gamma _1\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} + \\\\mathcal {O}\\\\left(d^{-\\\\frac{1}{3}}\\\\right)$ for a given $\\\\gamma _1 \\\\approx 0.81$ , and $*{J_{\\\\frac{d}{2}}\\\\left(j^{\\\\prime }_{\\\\frac{d}{2},1}\\\\right)} \\\\le M d^{-\\\\frac{1}{3}}$ for a $d$ -independent $M>0$ .\",\n \"Since $R_{a,b} \\\\in \\\\left[2c_1,1/\\\\sqrt{2\\\\pi \\\\text{e}}+o(1)\\\\right]d^\\\\frac{1}{2}$ (for $c_1\\\\in \\\\left(0,1/\\\\sqrt{8\\\\pi \\\\text{e}}\\\\right]$ from (REF )) and $j^{\\\\prime }_{\\\\frac{d}{2},1} / R_{a,b} \\\\in \\\\left[\\\\sqrt{\\\\frac{\\\\pi \\\\text{e}}{2}}+o(1),\\\\frac{1}{4c_1}+o(1)\\\\right]d^{\\\\frac{1}{2}}$ we have $*{\\\\widehat{\\\\varphi }(k;a,b)} \\\\le \\\\left(\\\\frac{2\\\\pi R_{a,b}}{*{k}}\\\\right)^\\\\frac{d}{2} M d^{-\\\\frac{1}{3}} \\\\le \\\\left(2e^{-1} + o(1)\\\\right)^\\\\frac{d}{2} M d^{-\\\\frac{1}{3}} \\\\rightarrow 0$ for $*{k} \\\\ge j^{\\\\prime }_{\\\\frac{d}{2},1} / R_{a,b}$ .\",\n \"We are now left with the intermediate range for $*{k}$ .\",\n \"The first positive zero of $J_\\\\frac{d}{2}$ occurs at $j_{\\\\frac{d}{2},1}$ , where $j_{\\\\frac{d}{2},1} = \\\\frac{d}{2} + \\\\gamma _2\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} + \\\\mathcal {O}\\\\left(d^{-\\\\frac{1}{3}}\\\\right)$ for a given $\\\\gamma _2 \\\\approx 1.86$ (see [2]).\",\n \"In particular, we will always have $j^{\\\\prime }_{\\\\frac{d}{2},1} < j_{\\\\frac{d}{2},1}$ .\",\n \"From differential identities relating Bessel functions (see [18]), we have $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}*{k}} \\\\widehat{\\\\varphi }(k;a,b) = -R_{a,b}\\\\left(\\\\frac{2\\\\pi R_{a,b}}{*{k}}\\\\right)^\\\\frac{d}{2}J_{\\\\frac{d}{2}+1}\\\\left(R_{a,b}*{k}\\\\right).$ The Bessel function $J_{\\\\frac{d}{2}+1}(r)$ is positive for $r>0$ until its first positive zero at $j_{\\\\frac{d}{2}+1,1}$ , and $j_{\\\\frac{d}{2}+1,1} > j_{\\\\frac{d}{2},1}$ .\",\n \"Therefore $\\\\widehat{\\\\varphi }(k;a,b)$ is positive and decreasing on the whole region $*{k}\\\\in \\\\left(0,j_{\\\\frac{d}{2},1}/R_{a,b}\\\\right]$ .\",\n \"Since this overlaps with the high $*{k}$ range, and we have the uniform quadratic behaviour near $k=0$ , the function $k\\\\mapsto *{\\\\widehat{\\\\varphi }(k;a,b)}$ can never increase and approach $\\\\widehat{\\\\varphi }(0;a,b)$ again after leaving a neighbourhood of $k=0$ .\",\n \"We therefore have the bound (REF ) and thus have proven that REF is satisfied.\",\n \"Figure: Sketch of ϕ ^k;a,b\\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) against *k*{k}.\",\n \"It approaches its maximum quadratically as *k→0*{k}\\\\rightarrow 0.\",\n \"The first local maximum of J d 2 J_{\\\\frac{d}{2}} occurs at j d 2,1 ' ∼d 2+γ 1 d 2 1 3 j^{\\\\prime }_{\\\\frac{d}{2},1}\\\\sim \\\\frac{d}{2}+\\\\gamma _1\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3}.\",\n \"The first zero of ϕ ^k;a,b\\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) occurs at *kR a,b =j d 2,1 ∼d 2+γ 2 d 2 1 3 *{k}R_{a,b} = j_{\\\\frac{d}{2},1}\\\\sim \\\\frac{d}{2}+\\\\gamma _2\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} where γ 2 >γ 1 \\\\gamma _2>\\\\gamma _1.\",\n \"Furthermore, ϕ ^k;a,b\\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) is strictly decreasing until *kR a,b =j d 2+1,1 ∼d 2+γ 2 d 2 1 3 +1 2*{k}R_{a,b} = j_{\\\\frac{d}{2}+1,1}\\\\sim \\\\frac{d}{2}+\\\\gamma _2\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} + \\\\frac{1}{2}.We return to () in REF .\",\n \"We first calculate the second moment: $\\\\int *{\\\\overline{x}}^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}= \\\\frac{d}{d+2}V_{a,b}R^2_{a,b}.$ From $*{\\\\overline{x}}^2 = \\\\overline{x}_1^2 + \\\\ldots + \\\\overline{x}^2_d$ (for an orthonormal basis) and the rotational symmetry, we have $\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}= \\\\frac{1}{d+2}V_{a,b}R^2_{a,b}$ for all unit vectors $e\\\\in \\\\mathbb {R}^d$ .\",\n \"Therefore $\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\le \\\\frac{4}{d+2}V^{({\\\\rm max})}\\\\left(R^{({\\\\rm max})}_d\\\\right)^2.$ While $V^{({\\\\rm max})}$ is a constant, $R^{({\\\\rm max})}_d\\\\sim \\\\sqrt{d/8\\\\pi \\\\text{e}}$ and therefore we have a bound uniform in $d$ .\",\n \"The argument now proceeds similarly to the corresponding part of the proof for the multivariate Gaussian model.\",\n \"From (REF ) and REF we have both sides of () satisfying quadratic bounds around $k=0$ , and the triangle inequality with our other bounds prove that the bound also holds on the remainder.\",\n \"For REF we first want to bound the convolution of three $\\\\varphi $ functions.\",\n \"Given marks $a_1,\\\\ldots ,a_6\\\\in \\\\left[0,1\\\\right]$ and $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ , let $\\\\mathbb {B}_{a,b}(\\\\overline{x})$ denote the ball of radius $R_{a,b}$ centred on $\\\\overline{x}$ , and let $1_{a,b}\\\\left(\\\\overline{x}\\\\right) = 1\\\\left\\\\lbrace *{\\\\overline{x}}\\\\le R_{a,b}\\\\right\\\\rbrace $ .\",\n \"Then $\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6)\\\\right)\\\\left(\\\\overline{x}\\\\right)&= \\\\int 1_{a_1,a_2}(\\\\overline{y})1_{a_3,a_4}(\\\\overline{z}-\\\\overline{y})1_{a_5,a_6}(-\\\\overline{z})\\\\mathrm {d}\\\\overline{y}\\\\mathrm {d}\\\\overline{z}\\\\nonumber \\\\\\\\&= \\\\int _{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)}\\\\left(\\\\int _{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)} 1_{a_3,a_4}(\\\\overline{z}-\\\\overline{y})\\\\mathrm {d}\\\\overline{z}\\\\right) \\\\mathrm {d}\\\\overline{y}\\\\nonumber \\\\\\\\&= \\\\int _{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)}\\\\mathrm {d}\\\\overline{y}.$ Then for $\\\\delta \\\\in \\\\left(0,1\\\\right)$ we have $&\\\\int _{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)}\\\\mathrm {d}\\\\overline{y}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\int _{\\\\delta \\\\mathbb {B}_{a,b}\\\\left(\\\\overline{0}\\\\right)}*{\\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{0}\\\\right)}\\\\mathrm {d}\\\\overline{y}+ *{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{y}\\\\in \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)} \\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\left(V^{({\\\\rm max})}\\\\right)^2\\\\delta ^d + V^{({\\\\rm max})}\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{y}\\\\in \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)}.$ Let $\\\\overline{e}_1\\\\in \\\\mathbb {R}^d$ be a unit vector.\",\n \"Then by spherical symmetry we have $\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{y}\\\\in \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)} = *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\delta R_{a_1,a_2}\\\\overline{e}_1\\\\right)}.$ We bound the volume of this intersection with the volume of the $d$ -ball with radius equal to the radius of the $d-1$ -sphere formed by the intersection of their boundaries.\",\n \"This radius is equal to the length $h$ in Figure REF , which is maximised when $R_{a_5,a_6}$ and $R_{a_3,a_4}$ are maximised and $\\\\delta R_{a_1,a_2}$ is minimised.\",\n \"Recall we defined the constant $c_1\\\\in \\\\left(0,1/\\\\sqrt{8\\\\pi \\\\text{e}}\\\\right)$ when we defined the model in Section REF .\",\n \"Since we have $R_{a_5,a_6},R_{a_3,a_4} \\\\le 2R^{({\\\\rm max})}_d$ and $\\\\delta R_{a_1,a_2} \\\\ge \\\\delta c_1 d^\\\\frac{1}{2}$ , we have $h\\\\le \\\\sqrt{\\\\left(2 R^{({\\\\rm max})}_d\\\\right)^2 - \\\\delta ^2 c^2_1 d}$ - we have used the fact that the triangle is isosceles if both the $R_{a_5,a_6}$ and $R_{a_3,a_4}$ edges attain their maximum.\",\n \"The ball with this radius then has the volume $\\\\frac{\\\\pi ^\\\\frac{d}{2}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)}h^d \\\\le V^{({\\\\rm max})}\\\\left(1 - \\\\frac{\\\\pi \\\\delta ^2 c^2_1 d}{\\\\left(V^{({\\\\rm max})}\\\\right)^\\\\frac{2}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{2}{d}}\\\\right)^\\\\frac{d}{2}.$ Since $\\\\frac{\\\\pi \\\\delta ^2 c^2_1 d}{\\\\left(V^{({\\\\rm max})}\\\\right)^\\\\frac{2}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{2}{d}} = 2\\\\pi \\\\text{e}\\\\delta ^2c_1^2\\\\left(1+o(1)\\\\right)$ and $c_1>0$ , this shows that $*{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\delta R_{a_1,a_2}\\\\overline{e}_1\\\\right)} \\\\le V^{({\\\\rm max})}\\\\left(1 - \\\\pi \\\\text{e}\\\\delta ^2c_1^2\\\\right)^\\\\frac{d}{2}$ for $d$ sufficiently large.\",\n \"Sine we now have a mark independent bound with the required decay, we have proved the bound for the convolution of three $\\\\varphi $ functions.\",\n \"Figure: For both the arguments, we bound the intersection of two dd-balls with a new dd-ball that has the radius hh in these diagrams.For finding the sets $B(x)$ in (REF ), we write $\\\\int \\\\varphi \\\\left(\\\\overline{x}- \\\\overline{y};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{y};c,b\\\\right)\\\\mathrm {d}\\\\overline{y}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right) = \\\\int *{\\\\mathbb {B}_{a,c}\\\\left(\\\\overline{x}\\\\right)\\\\cap \\\\mathbb {B}_{c,b}\\\\left(\\\\overline{0}\\\\right)}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c \\\\right).$ As before, we can bound the volume of this intersection independently of the mark $c$ .\",\n \"We can bound it with the volume of the ball with radius $h$ where $h\\\\le \\\\sqrt{\\\\left(2R^{({\\\\rm max})}_d\\\\right)^2 - \\\\frac{1}{4}*{\\\\overline{x}}^2}$ (as can be seen in Figure REF ).\",\n \"As before, if $*{\\\\overline{x}} \\\\ge \\\\kappa d^\\\\frac{1}{2}$ for some $\\\\kappa >0$ , then the volume of the ball with radius $h$ vanishes in the $d\\\\rightarrow \\\\infty $ limit.\",\n \"Furthermore, if $\\\\kappa \\\\le \\\\frac{1}{\\\\sqrt{2\\\\pi \\\\text{e}}}$ then the volume of the ball with radius $\\\\kappa d^\\\\frac{1}{2}$ vanishes in the $d\\\\rightarrow \\\\infty $ limit.\",\n \"Therefore assumption REF is satisfied with the sets $B(x)\\\\equiv B\\\\left(\\\\overline{x},a\\\\right)=\\\\left\\\\lbrace \\\\left(\\\\overline{y},b\\\\right)\\\\in \\\\mathbb {X}:*{\\\\overline{x}-\\\\overline{y}} \\\\le \\\\kappa d^\\\\frac{1}{2}\\\\right\\\\rbrace $ for any $\\\\kappa \\\\in \\\\left(0,\\\\frac{1}{\\\\sqrt{2\\\\pi \\\\text{e}}}\\\\right)$ .\",\n \"Displacement Diagram Case Distinctions This section provides the details omitted from the proof of Proposition REF .\",\n \"Here we show which diagrams arise from displacements crossing a $\\\\Psi $ segment, and whether they can be dealt with by the first sub-case, $\\\\mathrm {\\\\left(I\\\\right)}$ , or by the second sub-case, $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"Recall that $\\\\mathrm {\\\\left(I\\\\right)}$ are those that use a $\\\\Psi $ decomposition of the larger diagram only (and the displacement is not crossing a $\\\\Psi ^{(4)}$ segment), while $\\\\mathrm {\\\\left(II\\\\right)}$ are those that will require a combination of a $\\\\Psi $ decomposition and a $\\\\overline{\\\\Psi }$ decomposition.\",\n \"We first expand out the displaced $\\\\Psi $ terms and label which are in sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ and which are in sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"Note that the displacement may be across the `top' or the `bottom' of the segments in the usual orientation.\",\n \"If we consider a displacement across the `top' or the `bottom' of a $\\\\Psi ^{(1)}$ term, we use cosine-splitting and sometimes expand $\\\\tau _\\\\lambda ^\\\\circ $ edges into a $\\\\tau _\\\\lambda $ edge and a contraction to get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (2,0.8);(1,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[below]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad \\\\right)\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}}\\\\quad \\\\right).$ Using the same ideas, if we consider a displacement across the `top' or the `bottom' of a $\\\\Psi ^{(2)}$ term we get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.25) circle (0pt) node[right]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}}\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.25) circle (0pt) node[right]{\\\\circ };(0.75,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad +\\\\quad 2\\\\times \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}} \\\\quad +\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}}\\\\right.\\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\left.\\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad +\\\\quad 2\\\\times \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}} \\\\quad +\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}}\\\\quad \\\\right).$ If we consider a displacement across the `top' or the `bottom' of a $\\\\Psi ^{(3)}$ term, then by symmetry in both cases we get $\\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}}.$ We first show the bounds for the single segments in sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ .\",\n \"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\vee \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ We first consider the displaced $\\\\Psi ^{(j)}$ term by itself.\",\n \"By using supremum bounds and spatial translations we arrive at $\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda , & \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda , & \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda , \\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle ^\\\\circ _\\\\lambda , & \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda , & \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle ^\\\\circ _\\\\lambda .$ For most of these, it is sufficient to separate the pair using a supremum bound and use the individual bound above for the displaced $\\\\Psi ^{(j)}$ term and the bound on the $\\\\Psi $ or $\\\\Psi _n$ term from Lemma REF .\",\n \"The only pairs that need a more careful approach are those that have the bound $W_k\\\\triangle ^\\\\circ _\\\\lambda $ above for the displaced $\\\\Psi ^{(j)}$ , followed by a $\\\\Psi ^{(4)}$ or $\\\\Psi ^{(2)}_n$ term.\",\n \"In these cases we consider the pair together and get the diagram bounds $\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda \\\\le W_kV_\\\\lambda ^2,\\\\qquad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda \\\\le W_kV_\\\\lambda ^2.$ In both of these cases we bounded a $\\\\tau _\\\\lambda $ edge by 1 (essentially omitting the edge) before applying a supremum bound.\",\n \"In the first we omit the diagonal edge and split the upper supremum vertex in two.\",\n \"In the second we omit the left-most vertical edge and apply a supremum bound at the bottom right vertex.\",\n \"We now turn our attention to sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\vee \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda .$ Also recall that if the $\\\\overline{\\\\Psi }_0$ term is actually a $\\\\overline{\\\\Psi }^{(2)}_0$ term, then the $\\\\tau _\\\\lambda ^\\\\circ $ edge connecting them is actually just a $\\\\tau _\\\\lambda $ edge.\",\n \"We first investigate bounds for the displaced $\\\\Psi ^{(j)}$ term with the connecting $\\\\tau _\\\\lambda $ and $\\\\tau _\\\\lambda ^\\\\circ $ edges.\",\n \"By using supremum bounds, spatial translations, splitting supremum vertices in two, and sometimes expanding $\\\\tau _\\\\lambda ^\\\\circ $ edges into a $\\\\tau _\\\\lambda $ edge and a contraction, we get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(3,0.6) -- (2,0.6) -- (2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(2.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda },\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\quad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(2,-0.6) -- (3,-0.6);(1.5,0.6) -- (3,0.6);[dashed] (3,0.6) -- (3,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\\\times };\\\\node [mark size=2pt] at (3,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad +\\\\quad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(1.5,0.6) -- (3,0.6);[dashed] (0,0.6) -- (0,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\\\times };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\&\\\\le \\\\overline{H_k}+ W_k\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda },\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda .$ For most of these, it is clear from a crude supremum bound that we have the bound $\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda $ for the pair of these with the subsequent $\\\\overline{\\\\Psi }$ or $\\\\overline{\\\\Psi }_0$ term (like we did for sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ ).\",\n \"The only pairs that need a more careful approach are those that have the bound $\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda $ or $\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda $ above, followed by a $\\\\overline{\\\\Psi }^{(4)}$ or $\\\\overline{\\\\Psi }^{(2)}_0$ term.\",\n \"We first consider the cases where they are followed by a $\\\\overline{\\\\Psi }^{(4)}$ term.\",\n \"Here we use supremum bounds, spatial translations, and expanding $\\\\tau _\\\\lambda ^\\\\circ $ edges into a $\\\\tau _\\\\lambda $ edge and a contraction to get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\triangle ^\\\\circ _\\\\lambda \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda + \\\\triangle ^\\\\circ _\\\\lambda \\\\overline{W_k}\\\\triangle _\\\\lambda = \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda + \\\\overline{W_k}\\\\triangle _\\\\lambda = \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ We now consider the cases where they are followed by a $\\\\overline{\\\\Psi }^{(2)}_0$ term.\",\n \"Recall that for this case, the connecting $\\\\tau _\\\\lambda ^\\\\circ $ edge is in fact only a $\\\\tau _\\\\lambda $ edge.\",\n \"We then get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ The remaining sub-case $\\\\mathrm {\\\\left(III\\\\right)}$ was fully dealt with in the main text.\",\n \"Proof of Lemma REF Recall that we wish to prove the following result: * We begin by defining a sightly different object to $\\\\overline{H_k}$ .\",\n \"Given $\\\\vec{a}=\\\\left(a_1,\\\\ldots ,a_{16}\\\\right)\\\\in \\\\mathcal {E}^{16}$ , define $H^{\\\\prime }\\\\left(\\\\overline{x}_1,\\\\overline{x}_2,\\\\overline{x}_3;\\\\vec{a};k\\\\right) :=& \\\\int \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{x}_1;a_1,a_2)\\\\tau _\\\\lambda (\\\\overline{u};a_3,a_4) \\\\tau _\\\\lambda (\\\\overline{v}-\\\\overline{s};a_5,a_6) \\\\tau _\\\\lambda (\\\\overline{v}+\\\\overline{x}_2-\\\\overline{t};a_7,a_8) \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{w};a_9,a_{10})\\\\tau _\\\\lambda (\\\\overline{w}-\\\\overline{u};a_{11},a_{12}) \\\\tau _\\\\lambda (\\\\overline{t}-\\\\overline{w};a_{13},a_{14}) \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\tau _{\\\\lambda ,k}(\\\\overline{t}+ \\\\overline{x}_3-\\\\overline{u};a_{15},a_{16}) \\\\mathrm {d}\\\\overline{t}\\\\mathrm {d}\\\\overline{w}\\\\mathrm {d}\\\\overline{z}\\\\mathrm {d}\\\\overline{u},$ so we have $\\\\overline{H_k}\\\\le f(\\\\lambda )^5\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}_1,\\\\overline{x}_2\\\\in \\\\mathbb {R}^d,\\\\vec{a}\\\\in \\\\mathcal {E}^{16}}H^{\\\\prime }\\\\left(\\\\overline{x}_1,\\\\overline{x}_2,\\\\overline{0};\\\\vec{a};k\\\\right).$ We then proceed to bound $H^{\\\\prime }\\\\left(\\\\overline{x}_1,\\\\overline{x}_2,\\\\overline{x}_3;\\\\vec{a};k\\\\right)$ to get our result.\",\n \"This form is preferred because it more easily indicates what the Fourier arguments should be when we take a Fourier transform.\",\n \"A schematic Fourier diagram for this can be found in [21].\",\n \"We then expand out the $\\\\tau _\\\\lambda $ and $\\\\tau _{\\\\lambda ,k}$ terms.\",\n \"For each of the $\\\\tau _\\\\lambda $ terms we apply (REF ) and () to get $\\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\le \\\\varphi (\\\\overline{x};a,b) + \\\\lambda (\\\\varphi \\\\varphi )(\\\\overline{x};a,b)+ \\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi )(\\\\overline{x};a,b).$ Similarly, with the additional use of the cosine-splitting lemma, we get $\\\\tau _{\\\\lambda ,k}(\\\\overline{x};a,b) \\\\le \\\\varphi _k(\\\\overline{x};a,b) + 2\\\\lambda \\\\left( (\\\\varphi _k\\\\varphi )(\\\\overline{x};a,b) +(\\\\varphi \\\\varphi _k)(\\\\overline{x};a,b)\\\\right)+ 4\\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi _k)(\\\\overline{x};a,b) \\\\\\\\ + 4\\\\lambda ^2(\\\\varphi \\\\tau _{\\\\lambda ,k}\\\\varphi )(\\\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\\\lambda \\\\le f(\\\\lambda )$ , we can realise that we only need to get appropriate bounds for the following objects.\",\n \"For $\\\\vec{j}\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace ^7\\\\times \\\\left\\\\lbrace 1,2,3,4,5\\\\right\\\\rbrace $ , define $\\\\mathcal {H}(k;\\\\vec{j}) := \\\\int P^{j_1}_{1,2}(l_1)P^{j_2}_{3,4}(l_1)P^{j_3}_{5,6}(l_2)P^{j_4}_{7,8}(l_2)P^{j_5}_{9,10}(l_1-l_2)P^{j_6}_{11,12}(l_1-l_3)P^{j_7}_{13,14}(l_2-l_3) \\\\\\\\ \\\\times Q^{j_8}_{15,16}(l_3;k)\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}},$ where $P^{j}_{m,n}(l) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_m}{\\\\widehat{}(l) g_n}} &:j=1\\\\\\\\*{*{g_m}{\\\\widehat{}(l)^2g_n}} &:j=2\\\\\\\\*{*{g_m}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}(l)g_n}} &: j=3,\\\\end{array}\\\\right.}\",\n \"\\\\qquad \\\\text{for }m,n\\\\in \\\\left\\\\lbrace 1,2,\\\\ldots ,14\\\\right\\\\rbrace ,\\\\\\\\Q^{j}_{15,16}(l;k) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_{15}}{\\\\widehat{}_k(l) g_{16}}} &:j=1\\\\\\\\*{*{g_{15}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{16}}} &:j=2\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{}_k(l) g_{16}}} &:j=3\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{16}}} &:j=4\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{16}}} &:j=5,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\left\\\\lbrace g_m\\\\right\\\\rbrace _{m=1}^{16}$ are the functions of the form $g_m(a) = \\\\frac{1}{\\\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .\",\n \"If we can bound these $\\\\mathcal {H}(k;\\\\vec{j})$ appropriately, then we can combine them to get the required bound.\",\n \"We now recall pointwise bounds for the $P^{j}_{m,n}(l)$ and $Q^{j}_{15,16}(l)$ terms.\",\n \"In the proof of Lemma REF we derived $*{*{g_m}{\\\\widehat{}(l)g_n}} &\\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\\\\\*{*{g_m}{\\\\widehat{}(l)^2g_n}} &\\\\le *{g_m}{\\\\widehat{}(l)^2g_m}^{\\\\frac{1}{2}}*{g_n}{\\\\widehat{}(l)^2g_n}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_m}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}(l)g_n}} &\\\\le f(\\\\lambda )\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)*{g_m}{\\\\widehat{}(l)^2g_m}^{\\\\frac{1}{2}}*{g_n}{\\\\widehat{}(l)^2g_n}^{\\\\frac{1}{2}},$ and $*{*{g_{15}}{\\\\widehat{}_k(l) g_{16}}} &\\\\le \\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\\\\\*{*{g_{15}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{16}}} &\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) *{g_{16}}{\\\\widehat{}(l)^2g_{16}}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_{15}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{16}}} &\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) *{g_{15}}{\\\\widehat{}(l)^2g_{15}}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{16}}} &\\\\le Cf(\\\\lambda )\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)*{g_{15}}{\\\\widehat{}(l)^2g_{15}}\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{16}}} &\\\\le Cf(\\\\lambda )\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k)*{g_{15}}{\\\\widehat{}(l)^2g_{15}}^\\\\frac{1}{2}*{g_{16}}{\\\\widehat{}(l)^2g_{16}}^\\\\frac{1}{2}$ where $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) = \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k).$ Also recall that we can bound $*{*{g_m}{\\\\widehat{}(l)g_n}}\\\\le C$ and $*{g_m}{\\\\widehat{}(l)^2g_m}\\\\le C^2$ if required.\",\n \"Note that all the bounds for the $Q^{j}_{15,16}(l)$ terms have a $\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ factor.\",\n \"We therefore only need to show that the remaining integrals of $\\\\widehat{}$ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ terms can all be bounded by some constant multiple of $\\\\beta ^2$ .\",\n \"Also note that the bounds for $Q^{1}_{15,16}(l)$ , $Q^{2}_{15,16}(l)$ , and $Q^{3}_{15,16}(l)$ only differ by a uniform constant (after we bound $*{g_m}{\\\\widehat{}(l)^2g_m}\\\\le C^2$ ), and so once we have dealt with one of these we have dealt with the other two.\",\n \"Let us first consider $\\\\vec{j}= \\\\left(1,1,1,1,1,1,1,1\\\\right)$ .\",\n \"After applying the uniform pointwise bound for $Q^{1}_{15,16}(l)$ , we use the Cauchy-Schwarz inequality to bound $&\\\\int *{*{g_1}{\\\\widehat{}(l_1)g_2}}*{*{g_3}{\\\\widehat{}(l_1)g_4}}*{*{g_5}{\\\\widehat{}(l_2)g_6}}*{*{g_7}{\\\\widehat{}(l_2)g_8}}*{*{g_9}{\\\\widehat{}(l_1-l_2)g_{10}}} \\\\nonumber \\\\\\\\&\\\\hspace{199.16928pt}\\\\times *{*{g_{11}}{\\\\widehat{}(l_1-l_3)g_{12}}}*{*{g_{13}}{\\\\widehat{}(l_2-l_3)g_{14}}}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}} \\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(l_1)g_2}^2*{g_3}{\\\\widehat{}(l_1)g_4}^2*{g_5}{\\\\widehat{}(l_2)g_6}^2*{g_{13}}{\\\\widehat{}(l_2-l_3)g_{14}}^2\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\int *{g_7}{\\\\widehat{}(l_2)g_8}^2*{g_9}{\\\\widehat{}(l_1-l_2)g_{10}}^2*{g_{11}}{\\\\widehat{}(l_1-l_3)g_{12}}^2\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.\",\n \"$ To deal with these parentheses, we use a volume-preserving change of variables and factorise each into three integrals.\",\n \"For example, $\\\\int *{g_1}{\\\\widehat{}(l_1)g_2}^2*{g_3}{\\\\widehat{}(l_1)g_4}^2*{g_5}{\\\\widehat{}(l_2)g_6}^2*{g_{13}}{\\\\widehat{}(l_2-l_3)g_{14}}^2\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}} \\\\\\\\= \\\\left(\\\\int *{g_1}{\\\\widehat{}(l_1)g_2}^2*{g_3}{\\\\widehat{}(l_1)g_4}^2\\\\frac{\\\\mathrm {d}l_1}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)\\\\left(\\\\int *{g_5}{\\\\widehat{}(l_2)g_6}^2\\\\frac{\\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)\\\\left(\\\\int *{g_{13}}{\\\\widehat{}(l_3^{\\\\prime })g_{14}}^2\\\\frac{\\\\mathrm {d}l_3^{\\\\prime }}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right).$ There are now two types of integral we need to bound: $&\\\\int *{g_m}{\\\\widehat{}(l)g_n}^2 \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} = \\\\frac{1}{\\\\mathcal {P}\\\\left(E_m\\\\right)^2\\\\mathcal {P}\\\\left(E_n\\\\right)^2}\\\\int \\\\left(\\\\int _{E_m\\\\times E_n \\\\times E_m \\\\times E_n} \\\\widehat{\\\\varphi }(l;a,b)\\\\widehat{\\\\varphi }(l;c,d) \\\\mathcal {P}^{\\\\otimes 4}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b,\\\\mathrm {d}c,\\\\mathrm {d}d\\\\right)\\\\right) \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} = \\\\frac{1}{\\\\mathcal {P}\\\\left(E_m\\\\right)^2\\\\mathcal {P}\\\\left(E_n\\\\right)^2}\\\\int _{E_m\\\\times E_n \\\\times E_m \\\\times E_n} \\\\left(\\\\varphi (\\\\cdot ;a,b)\\\\star \\\\varphi (\\\\cdot ;c,d)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\mathcal {P}^{\\\\otimes 4}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b,\\\\mathrm {d}c,\\\\mathrm {d}d\\\\right)\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b,c,d\\\\in \\\\mathcal {E}} \\\\left(\\\\varphi (\\\\cdot ;a,b)\\\\star \\\\varphi (\\\\cdot ;c,d)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b,c,d\\\\in \\\\mathcal {E}} \\\\int \\\\varphi (\\\\overline{x};a,b)\\\\varphi (-\\\\overline{x};c,d)\\\\mathrm {d}\\\\overline{x}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}} \\\\int \\\\varphi (\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}\\\\le C, \\\\\\\\&\\\\int *{g_1}{\\\\widehat{}(l)g_2}^2 *{g_3}{\\\\widehat{}(l)g_4}^2 \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_8\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6) \\\\star \\\\varphi (\\\\cdot ;a_7,a_8)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\le g(d).\",\n \"$ In (REF ) we used $\\\\varphi \\\\left(-\\\\overline{x};c,d\\\\right)\\\\in \\\\left[0,1\\\\right]$ and REF .\",\n \"In () we bounded the convolution of four adjacency functions using REF .\",\n \"The result of this is that we can bound (REF ) by $C^{\\\\frac{5}{2}}g(d)^{\\\\frac{1}{2}}$ .\",\n \"Since $g(d)\\\\le \\\\beta ^4$ , this satisfies the bound we require.\",\n \"As noted above, this argument also proves the required bound for $\\\\vec{j}= \\\\left(1,1,1,1,1,1,1,2\\\\right)$ and $\\\\vec{j}= \\\\left(1,1,1,1,1,1,1,3\\\\right)$ .\",\n \"Now suppose that we change some subset of $\\\\left\\\\lbrace j_1,\\\\ldots ,j_7\\\\right\\\\rbrace $ from having value 1 to having value 2.\",\n \"The argument proceeds similarly, except that after we have factorised our integrals we now want to also bound integrals of the following types: $&\\\\int *{g_1}{\\\\widehat{}(l)^2g_1} *{g_2}{\\\\widehat{}(l)^2g_2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt}= \\\\frac{1}{\\\\mathcal {P}\\\\left(E_1\\\\right)^2\\\\mathcal {P}\\\\left(E_2\\\\right)^2}\\\\int \\\\left(\\\\int _{E_1\\\\times E_1\\\\times E_2\\\\times E_2}\\\\widehat{\\\\varphi }^{(2)}(l;a,b)\\\\widehat{\\\\varphi }^{(2)}(l;c,d)\\\\mathcal {P}^{\\\\otimes 4}(\\\\mathrm {d}a,\\\\mathrm {d}b, \\\\mathrm {d}c, \\\\mathrm {d}d)\\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt}= \\\\frac{1}{\\\\mathcal {P}\\\\left(E_1\\\\right)^2\\\\mathcal {P}\\\\left(E_2\\\\right)^2}\\\\int _{E_1\\\\times E_1\\\\times E_2\\\\times E_2}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a,b)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;c,d)\\\\right)(\\\\overline{0})\\\\mathcal {P}^{\\\\otimes 4}(\\\\mathrm {d}a,\\\\mathrm {d}b, \\\\mathrm {d}c, \\\\mathrm {d}d) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b,c,d\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a,b)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;c,d)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\le g(d), \\\\\\\\&\\\\int *{g_1}{\\\\widehat{}(l)^2g_1} *{g_2}{\\\\widehat{}(l)^2g_2}*{g_3}{\\\\widehat{}(l)g_4}^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_8\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6) \\\\star \\\\varphi (\\\\cdot ;a_7,a_8)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^2\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\le C^2 g(d)\\\\\\\\&\\\\int *{g_1}{\\\\widehat{}(l)^2g_1} *{g_2}{\\\\widehat{}(l)^2g_2}*{g_3}{\\\\widehat{}(l)^2g_3} *{g_4}{\\\\widehat{}(l)^2g_4}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_8\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4) \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_5,a_6) \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_7,a_8)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^2\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\le C^4 g(d).$ Up to a constant, these are all less than or equal to the bounds we used for the $\\\\left\\\\lbrace j_1,\\\\ldots ,j_7\\\\right\\\\rbrace = \\\\left\\\\lbrace 1,\\\\ldots ,1\\\\right\\\\rbrace $ case, and therefore that case dominates.\",\n \"These other cases would behave as constant multiple of one of $g(d)$ , $g(d)^{\\\\frac{3}{2}}$ , $g(d)^2$ , $g(d)^{\\\\frac{5}{2}}$ , and $g(d)^3$ .\",\n \"We have now dealt with all the cases which have no factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ .\",\n \"To demonstrate how we will deal with these, we first consider $\\\\vec{j}=\\\\left\\\\lbrace 3,3,3,3,3,3,3,5\\\\right\\\\rbrace $ .\",\n \"For notational compactness, we define $\\\\left<\\\\widehat{}(l)^2\\\\right>_{I} := \\\\prod _{m\\\\in I}*{g_m}{\\\\widehat{}(l)^2g_m},$ where $I\\\\subset \\\\mathbb {N}$ and $l\\\\in \\\\mathbb {R}^d$ .\",\n \"Since we have already extracted the factor of $\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ via the pointwise bound on $Q^{5}_{15,16}(l;k)$ , we are left trying to bound the following integral: $\\\\int \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) \\\\\\\\ \\\\qquad \\\\times \\\\left<\\\\widehat{}(l_1)^2\\\\right>^{\\\\frac{1}{2}}_{1,2,3,4} \\\\left<\\\\widehat{}(l_2)^2\\\\right>^{\\\\frac{1}{2}}_{5,6,7,8} \\\\left<\\\\widehat{}(l_1-l_2)^2\\\\right>^{\\\\frac{1}{2}}_{9,10} \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{11,12}\\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}.$ To bound this integral we need to partition our space.\",\n \"Fix $\\\\varepsilon >0$ and define $B_1 &:= \\\\left\\\\lbrace *{l_1}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_2 &:= \\\\left\\\\lbrace *{l_2}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_3 &:= \\\\left\\\\lbrace *{l_1-l_3}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_4 &:= \\\\left\\\\lbrace *{l_1-l_2}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_5 &:= \\\\left\\\\lbrace *{l_2-l_3}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_6 &:= \\\\left\\\\lbrace *{l_3}< \\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l_3-k}< \\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l_3+k}< \\\\varepsilon \\\\right\\\\rbrace ,$ and for $n=0,1,\\\\ldots ,6$ define $A_n := \\\\left\\\\lbrace (l_1,l_2,l_3)\\\\in \\\\left(\\\\mathbb {R}^{d}\\\\right)^3 \\\\colon \\\\#\\\\left\\\\lbrace m\\\\colon (l_1,l_2,l_3)\\\\in B_m \\\\right\\\\rbrace =n\\\\right\\\\rbrace .$ In words, $A_n$ is the set of points within $\\\\varepsilon $ (in Euclidean distance) of precisely $n$ of the hyperplanes on which the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors diverge.\",\n \"If we are working with a factor of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ on the set $B_m$ associated with its relevant hyperplane, we use the bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\le 1/\\\\left(C_2*{l}^2\\\\right)$ , and if we are working off this set we use the bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ .\",\n \"When we integrate over $A_0$ , we bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ for all such factors and recycle the bound on the “only $\\\\widehat{}$ \\\" integrals from above.\",\n \"The contribution from $A_0$ is therefore bounded by some constant multiple of $\\\\frac{1}{\\\\varepsilon ^{18}}g(d)^3.$ For $A_1$ , first suppose we are considering $A_1\\\\cap B_m$ for $m\\\\ne 6$ .\",\n \"First we bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ and $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) \\\\le 3/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ for the “irrelevant\\\" directions, and use the pointwise bound for $\\\\left<\\\\widehat{}(l)^2\\\\right>_{m,n}$ in the “relevant\\\" direction.\",\n \"Then we can use the Cauchy-Schwarz and factorise argument we used above and recycle the integral bounds for the $\\\\widehat{}$ factors as we did above.\",\n \"However we are careful about the way we split apart terms in the Cauchy-Schwarz step.\",\n \"We split the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ terms across both factors and arrange the remaining $\\\\widehat{}$ terms so that each factor has “directions\\\" spanning $\\\\left(\\\\mathbb {R}^d\\\\right)^3$ .\",\n \"To demonstrate we consider $A_1\\\\cap B_1$ : $&\\\\int _{A_1\\\\cap B_1} \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2 \\\\left<\\\\widehat{}(l_2)^2\\\\right>^{\\\\frac{1}{2}}_{5,6,7,8} \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{11,12} \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le C^2\\\\left(\\\\int _{A_1\\\\cap B_1}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\left<\\\\widehat{}(l_2)^2\\\\right>_{5,6}\\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>_{11,12}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\& \\\\hspace{142.26378pt}\\\\times \\\\left(\\\\int _{A_1\\\\cap B_1}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2 \\\\left<\\\\widehat{}(l_2)^2\\\\right>_{7,8} \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>_{13,14}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.$ After a change of variables, factorising and using our integral bounds of $\\\\widehat{}$ terms from above, we are only left needing to bound $\\\\int _{*{l}<\\\\varepsilon } \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^2 \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\le \\\\frac{1}{C^2_2} \\\\int _{*{l}<\\\\varepsilon } \\\\frac{1}{*{l}^4} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} = \\\\frac{1}{C^2_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-4}}{d-4}.$ Therefore the contribution from $A_1\\\\cap B_m$ for $m\\\\ne 6$ is bounded by some constant multiple of $\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-18}}{d-4}g(d)^2$ .\",\n \"For $A_1\\\\cap B_6$ we need to be slightly more careful because for $*{k}<2\\\\varepsilon $ the constituent parts of $B_6$ overlap - this is the same considerations we had to account for in the proof of Lemma REF .\",\n \"We can overcome it in the same way by using Hölder's inequality (actually just Cauchy-Schwarz in this case) and then re-centring the integrals.\",\n \"The net result is that the bound of a constant multiple of $\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-18}}{d-4}g(d)^2$ also applies to $A_1\\\\cap B_6$ .\",\n \"In summary, the contribution from $A_1$ is bounded by some constant multiple of $\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-18}}{d-4}g(d)^2.$ The $A_2$ case proceeds similarly.\",\n \"After partitioning the set according to which are the overlapping hyperplanes, we take the relevant $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors from these directions and one $\\\\widehat{}$ factor from a linearly independent direction, and pointwise bound all the other $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ and $\\\\widehat{}$ factors.\",\n \"The contribution from $A_2$ is then bounded by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^2\\\\frac{\\\\varepsilon ^{2d-18}}{\\\\left(d-4\\\\right)^2}g(d),$ arising from the $A_1\\\\cap B_1\\\\cap B_2$ , $A_1\\\\cap B_1\\\\cap B_6$ , and $A_1\\\\cap B_2\\\\cap B_6$ integrals.\",\n \"The $A_3$ case is more complicated because there are two qualitatively different sub-cases to consider.\",\n \"First suppose that the three normal vectors to the overlapping hyperplanes are linearly independent.\",\n \"Then we take the relevant $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors from these directions, pointwise bound the other $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors and all the $\\\\widehat{}$ factors.\",\n \"The contribution from this sub-case of $A_3$ is then bounded by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\frac{\\\\varepsilon ^{3d-18}}{\\\\left(d-4\\\\right)^3}$ , arising from the $A_1\\\\cap B_1\\\\cap B_2\\\\cap B_6$ integral.\",\n \"On the other hand, if the three normal vectors to the overlapping hyperplanes are linearly dependent, then we need to retain a $\\\\widehat{}$ from the remaining linearly independent direction.\",\n \"To demonstrate, let us consider the $A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4$ integral.\",\n \"After removing unnecessary terms via uniform bounds, $&\\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{13,14} \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{11,12}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\left(\\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^\\\\frac{1}{2}\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad \\\\times \\\\left(\\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>_{11,12} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^\\\\frac{1}{2}.$ Applying a change of variables, factorising the integral, and applying Hölder's inequality allows us to bound these parentheses: $& \\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int _{\\\\left\\\\lbrace *{l_1}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_1-l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right)\\\\left(\\\\int \\\\left<\\\\widehat{}(l)^2\\\\right>_{13,14}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\frac{1}{C^5_2}\\\\left(\\\\int _{\\\\left\\\\lbrace *{l_1}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_1-l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l_1}^4}\\\\frac{1}{*{l_2}^4}\\\\frac{1}{*{l_1-l_2}^2} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right) g(d)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\frac{1}{C^5_2}\\\\left(\\\\int _{\\\\left\\\\lbrace *{l_1}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l_1}^6}\\\\frac{1}{*{l_2}^3} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right)^{\\\\frac{2}{3}}\\\\left(\\\\int _{\\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_1-l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l_2}^6}\\\\frac{1}{*{l_1-l_2}^6} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right)^{\\\\frac{1}{3}} g(d)\\\\nonumber \\\\\\\\& \\\\qquad = \\\\frac{1}{C^5_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{4}{3}}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{d-3}\\\\frac{\\\\varepsilon ^{d-3}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{2}{3}}g(d).$ For all of these linearly dependent cases, we can bound the integral by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^2\\\\frac{\\\\varepsilon ^{2d-18}}{\\\\left(d-6\\\\right)^2}g(d)$ .\",\n \"In summary, the contribution from $A_3$ is bounded by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\frac{\\\\varepsilon ^{3d-18}}{\\\\left(d-4\\\\right)^3} + \\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^2\\\\frac{\\\\varepsilon ^{2d-18}}{\\\\left(d-6\\\\right)^2}g(d).$ For $A_4$ , $A_5$ , and $A_6$ we don't have the two linearly dependent/independent cases because any four of the directions spans $\\\\left(\\\\mathbb {R}^d\\\\right)^3$ .\",\n \"We therefore don't need factors of $\\\\widehat{}$ to control the unbounded directions.\",\n \"Instead, we need to be careful that when we split our integral at the Cauchy-Schwarz step the resulting parentheses have sufficiently few factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ that their bounds will be finite for $d>6$ .\",\n \"The way we perform this split will be different for the different parts of $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l_3;k)$ .\",\n \"Let us define $\\\\widehat{D}(l;k) := \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k).$ We first consider $A_6$ .\",\n \"By writing $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) = \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{D}(l;k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)$ , we have two integrals we want to bound.\",\n \"First $&\\\\int _{A_6}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{D}(l_3;k) \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^3\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{D}(l_3;k)^2\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad \\\\times \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^3\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)^2\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.\",\n \"$ We then perform the $l_i$ integrals in specific orders.\",\n \"For the first factor we integrate over $l_3$ (which factorises out already), then we integrate over $l_2$ for fixed $l_1$ , and finally integrate over $l_1$ .\",\n \"For the second factor we integrate over $l_1$ for fixed $l_2$ and $l_3$ , then we integrate over $l_3$ for fixed $l_2$ , and finally we integrate over $l_2$ .\",\n \"There are therefore three forms of integral we need to bound: $\\\\int _{*{l}<\\\\varepsilon }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^3\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} &\\\\le \\\\frac{1}{C^3_2}\\\\int _{*{l}<\\\\varepsilon }\\\\frac{1}{*{l}^6}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} = \\\\frac{1}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\frac{\\\\varepsilon ^{d-6}}{d-6}\\\\\\\\\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l-l^{\\\\prime }}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-l^{\\\\prime })^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} &\\\\le \\\\frac{1}{C^3_2}\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l-l^{\\\\prime }}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l}^2}\\\\frac{1}{*{l-l^{\\\\prime }}^4}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\&\\\\le \\\\frac{1}{C^3_2}\\\\left(\\\\int _{*{l}<\\\\varepsilon }\\\\frac{1}{*{l}^6}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^\\\\frac{1}{3}\\\\left(\\\\int _{*{l-l^{\\\\prime }}<\\\\varepsilon }\\\\frac{1}{*{l-l^{\\\\prime }}^6}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^\\\\frac{2}{3}\\\\nonumber \\\\\\\\& = \\\\frac{1}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\frac{\\\\varepsilon ^{d-6}}{d-6}\\\\\\\\\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l-k}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l+k}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{D}(l;k)^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} &\\\\le \\\\frac{C^{\\\\prime }}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-6}.$ In this last inequality, $C^{\\\\prime }>0$ is some uniform constant.\",\n \"The last inequality is derived by applying Hölder's inequality in much the same way as for the second inequality, taking care to see when the three sets that are integrated over are overlapping.\",\n \"The calculation uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .\",\n \"The second integral we want to bound for $A_6$ is $&\\\\int _{A_6}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k) \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^3\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad \\\\times \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^3\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k)\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.\",\n \"$ We then perform the $l_i$ integrals in specific orders.\",\n \"For the first factor we integrate over $l_1$ (which factorises out already), then we integrate over $l_2$ for fixed $l_3$ , and finally integrate over $l_3$ .\",\n \"For the second factor we integrate over $l_3$ for fixed $l_1$ and $l_2$ , then we integrate over $l_1$ for fixed $l_2$ , and finally we integrate over $l_2$ .\",\n \"The new integrals we need to bound are: $\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l-k}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l+k}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\\\\\ \\\\le \\\\frac{C^{\\\\prime }}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-6}$ $\\\\int _{\\\\left(\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l-k}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l+k}<\\\\varepsilon \\\\right\\\\rbrace \\\\right)\\\\cap \\\\left\\\\lbrace *{l-l^{\\\\prime }}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-l^{\\\\prime })^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\\\\\ \\\\le \\\\frac{C^{\\\\prime }}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4}\\\\right)\\\\varepsilon ^{d-6}.$ The calculation of these two bounds uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .\",\n \"The result of these bounds is that the integral in (REF ) is bounded by some constant multiple of $\\\\frac{1}{\\\\left(d-6\\\\right)^3}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\varepsilon ^{3d-18}.$ For $A_4$ and $A_5$ we can repeat the argument for $A_6$ , but note that in some places the bound $1/\\\\left(C_2*{l}^2\\\\right)$ will be replaced by $1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ .\",\n \"The net result of this is that the factor $\\\\left(d-6\\\\right)^3$ can be replaced by $\\\\left(d-6\\\\right)^{\\\\alpha _0}\\\\left(d-4\\\\right)^{\\\\alpha _1}\\\\left(d-2\\\\right)^{\\\\alpha _2} d^{\\\\alpha _3}$ for some $\\\\alpha _0,\\\\alpha _1,\\\\alpha _2,\\\\alpha _3\\\\le 0$ such that $\\\\alpha _0+\\\\alpha _1+\\\\alpha _2+\\\\alpha _3 =3$ .\",\n \"The net result is that the contribution from both $A_4$ and $A_5$ can also be bounded by some constant multiple of $\\\\frac{1}{\\\\left(d-6\\\\right)^3}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\varepsilon ^{3d-18}.$ Above we have established bounds for the various parts of the integral when $\\\\vec{j}=\\\\left(3,3,3,3,3,3,3,5\\\\right)$ .\",\n \"Having $j_8=4$ can also be dealt with by very similar arguments to that outlined above - the $l_3$ integrals are in fact simpler.\",\n \"The above bounds still hold with the $\\\\varepsilon ^{-18}$ becoming a $\\\\varepsilon ^{-16}$ .\",\n \"Since $\\\\varepsilon <1$ , this produces a larger bound.\",\n \"Recall that up to a constant factor, having $j_m=1$ produces a larger bound than $j_m=2$ for $m=1,\\\\ldots ,7$ .\",\n \"If we replace $j_m=3$ with $j_m=1$ , then after ignoring constant values we lose a factor of $\\\\varepsilon ^{-2}$ , but we may also lose a factor of $g(d)^\\\\frac{1}{2}$ unless that was for $m=1,2$ (corresponding to $l_1$ direction) or $m=3,4$ (corresponding to $l_2$ direction).\",\n \"Note that doing for both directions will lose a $g(d)^\\\\frac{1}{2}$ factor.\",\n \"If $\\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^d= O\\\\left(g(d)^{\\\\frac{1}{2}}\\\\right)$ for all fixed $\\\\varepsilon >0$ , then it is clear that the dominating bound is $g(d)^\\\\frac{1}{2}$ .\",\n \"This arises, for example from the bound for $\\\\vec{j}=\\\\left(1,1,1,1,1,1,1,1\\\\right)$ .\",\n \"The situation is more complicated if $g(d)^{\\\\frac{1}{2}}\\\\ll \\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^d$ for fixed $\\\\varepsilon $ , because we can choose to have $\\\\varepsilon =\\\\varepsilon (d)$ and take it to 0 as $d\\\\rightarrow \\\\infty $ .\",\n \"It is easy to see that all the $A_1$ components (and $A_2,\\\\ldots ,A_6$ components) for each $\\\\vec{j}$ can be bounded by some constant multiple of $\\\\frac{1}{d}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\varepsilon ^d$ (provided $d>6$ ).\",\n \"We therefore only need to compare this to the contributions from $A_0$ .\",\n \"The bound we get from the above arguments for each $A_0$ contribution are of the form $\\\\varepsilon ^{-2m}g(d)^\\\\frac{n}{2}$ , where $m$ is the number of factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ and $n$ is related to the multiplicity of $$ terms in each Fourier direction.\",\n \"To find the terms with the largest bound, we want to have as many factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ as possible without producing extra factors of $g(d)^\\\\frac{1}{2}$ .\",\n \"Since $g(d)^{\\\\frac{1}{2}}\\\\ll \\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^d$ for fixed $\\\\varepsilon $ , increasing $n$ here instantly produces a smaller bound.\",\n \"By changing $j_8$ from 1 to 4 or 5, or any other element from 1 to 3, we do gain a factor of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ but we also increase the number of $$ factors and therefore in most places we acquire extra factors of $g(d)^{\\\\frac{1}{2}}$ .\",\n \"The exception to this is for $j_1$ and $j_2$ , which correspond to the $l_1$ directions, since they already produce a factor of $g(d)^{\\\\frac{1}{2}}$ .\",\n \"The $A_0$ component of the $\\\\vec{j}=\\\\left(3,3,1,1,1,1,1,1\\\\right)$ term can be bounded by constant multiple of $\\\\varepsilon ^{-4}g(d)^{\\\\frac{1}{2}}$ .\",\n \"We then optimise our choice of $\\\\varepsilon $ to have our two dominant bounds be of the same order.\",\n \"This produces $\\\\varepsilon (d)^2 = g(d)^{\\\\frac{1}{d}}\\\\frac{2\\\\pi d}{\\\\text{e}}\\\\left(1+o(1)\\\\right)$ , and therefore our overall bound is given by $\\\\varepsilon ^{-4}g(d)^\\\\frac{1}{2} = g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}}\\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^2\\\\frac{1}{d^2}\\\\left(1+o(1)\\\\right).$ Since $g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}} \\\\le g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}$ for sufficiently large $d$ , this bound is of order $\\\\beta ^2$ and our bound is proven.\",\n \"Acknowledgements.\",\n \"This work is supported by Deutsche Forschungsgemeinschaft (project number 443880457) through priority program “Random Geometric Systems” (SPP 2265).\",\n \"The authors thank the Centre de recherches mathématiques Montreal for hospitality during a research visit in spring 2022 through the Simons-CRM scholar-in-residence program.\"\n ],\n [\n \"Diagrammatic bounds\",\n \"For our argument, we will require bounds on $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op}$ and $\\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(k)\\\\right\\\\Vert _{\\\\rm op}$ for all $n\\\\in \\\\mathbb {N}$ and $k\\\\in \\\\mathbb {R}^d$ .\",\n \"However, it turns out that we can bound the last two remainder terms using others.\",\n \"Lemma 5.1 For all $n\\\\in \\\\mathbb {N}$ , $k\\\\in \\\\mathbb {R}^d$ , and $\\\\lambda >0$ , $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op},\\\\\\\\\\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(k)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(0)\\\\right\\\\Vert _{\\\\rm op}.$ As $\\\\mathbb {P}_\\\\lambda (u_n {\\\\,\\\\,A\\\\,\\\\,} x\\\\textrm { in } \\\\xi ^{u_n,x}_{n+1} ) \\\\le \\\\tau _\\\\lambda (x,u_n)$ for an arbitrary locally finite set $A$ , the definition () gives $*{r_{\\\\lambda ,n}(x)} \\\\le \\\\int \\\\tau _\\\\lambda (x,u)\\\\pi _\\\\lambda ^{(n)}(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ Let $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ .\",\n \"Then this gives $*{R_{\\\\lambda ,n}f}^2_2 &\\\\le \\\\int \\\\left(\\\\int *{r_{\\\\lambda ,n}(x,y)}*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&\\\\le \\\\lambda ^2\\\\int \\\\left(\\\\int \\\\left(\\\\int \\\\tau _\\\\lambda (x,u)\\\\pi _\\\\lambda ^{(n)}(u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\right)*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&= \\\\lambda ^2\\\\int \\\\left(\\\\int \\\\tau _\\\\lambda (x,u)\\\\left(\\\\int \\\\pi _\\\\lambda ^{(n)}(u,y)*{f\\\\left(y\\\\right)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\& = \\\\lambda ^2 *{\\\\mathcal {T}_\\\\lambda _\\\\lambda ^{(n)}{\\\\rm Abs}f}^2_2.$ Note that we were able to use Tonelli's theorem to swap the $u$ and $y$ integrals because $\\\\tau _\\\\lambda $ and $\\\\pi ^{(n)}_\\\\lambda $ are both non-negative and measurable.\",\n \"Finally, the approach of (REF ) implies the result.\",\n \"For each $n\\\\ge 0$ , either $r_{\\\\lambda ,n}(x)\\\\ge 0$ for all $x\\\\in \\\\mathbb {X}$ or $r_{\\\\lambda ,n}(x)\\\\le 0$ for all $x\\\\in \\\\mathbb {X}$ .\",\n \"Therefore $*{\\\\widehat{r}_{\\\\lambda ,n}(k;a,b)}\\\\le *{\\\\widehat{r}_{\\\\lambda ,n}(0;a,b)}$ for all $a,b\\\\in \\\\mathcal {E}$ and $\\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{R}_{\\\\lambda ,n}(0)\\\\right\\\\Vert _{\\\\rm op}$ (by Lemma REF ) for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"From (REF ) we have $*{\\\\widehat{r}_{\\\\lambda ,n}(0;a,b)} \\\\le \\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,c)\\\\widehat{\\\\pi }_\\\\lambda ^{(n)}(0;c,b) \\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right).$ Therefore the same argument as above gives the second bound in the result.\",\n \"Because the $n=0$ cases for our bounds on $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}$ , and $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ are so simple, we give the proof here before moving onto $n\\\\ge 1$ .\",\n \"Recall $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right) = \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k+l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right)$ .\",\n \"We will find it convenient to also define $\\\\widehat{}_k\\\\left(l\\\\right) = \\\\widehat{}\\\\left(k+l\\\\right) - \\\\widehat{}\\\\left(l\\\\right)$ .\",\n \"Proposition 5.2 (Bounds for $n=0$ ) Let $\\\\lambda \\\\in [0,\\\\lambda _O)$ .\",\n \"Then $\\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ For $k\\\\in \\\\mathbb {R}^d$ , $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(k)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda ^2\\\\left(\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty } + \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\right).$ Our first comment relates to these bounds and every bound we perform hereafter in Section .\",\n \"From the positivity of $\\\\pi ^{(0)}_{\\\\lambda }(x,y)$ and Lemma REF we have $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{\\\\rm op}$ , and by Lemma REF we have $\\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{1,\\\\infty }$ and $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"However from the definitions of the norm we also have $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}(0)\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{1,\\\\infty }$ , and so we will be interested in bounding this last norm.\",\n \"Now note that for the event $\\\\lbrace y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x} \\\\rbrace $ to hold, either there is a direct edge between $y$ and $x$ , or there exist vertices $w,z$ in $\\\\eta $ that are direct neighbours of the origin and have respective disjoint paths to $x$ that both do not contain $y$ .\",\n \"Hence, by the Mecke equation (REF ), $\\\\mathbb {P}_\\\\lambda ( y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{\\\\mathbf {0},x}) &\\\\le \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\mathbb {E}_\\\\lambda \\\\Big [ \\\\sum _{(w,z) \\\\in \\\\eta ^{(2)}} 1_{(\\\\lbrace y\\\\sim w \\\\text{ in } \\\\xi ^{y}\\\\rbrace \\\\cap \\\\lbrace w \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^x\\\\rbrace ) \\\\circ (\\\\lbrace y \\\\sim z \\\\text{ in } \\\\xi ^{y}\\\\rbrace \\\\cap \\\\lbrace z \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^x \\\\rbrace )} \\\\Big ] \\\\nonumber \\\\\\\\& = \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\lambda ^2 \\\\int \\\\mathbb {P}_\\\\lambda \\\\big ( (\\\\lbrace y\\\\sim y \\\\text{ in } \\\\xi ^{y, w}\\\\rbrace \\\\cap \\\\lbrace w \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,w}\\\\rbrace )\\\\nonumber \\\\\\\\& \\\\hspace{99.58464pt} \\\\circ (\\\\lbrace y \\\\sim z \\\\text{ in } \\\\xi ^{y, z}\\\\rbrace \\\\cap \\\\lbrace z \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{x,z} \\\\rbrace ) \\\\big ) \\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}w,\\\\mathrm {d}z\\\\right).$ After applying the BK inequality to the above probability, $\\\\mathbb {P}_\\\\lambda \\\\left( y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x}\\\\right) \\\\le \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\lambda ^2 \\\\left( \\\\int \\\\mathbb {P}_\\\\lambda \\\\left( \\\\left\\\\lbrace y\\\\sim w \\\\text{ in } \\\\xi ^{y,w}\\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace w \\\\longleftrightarrow x\\\\textrm { in } \\\\xi ^{w,x}\\\\right\\\\rbrace \\\\right) \\\\nu \\\\left(\\\\mathrm {d}w\\\\right) \\\\right)^2 \\\\\\\\= \\\\varphi (x,y) + \\\\tfrac{1}{2} \\\\lambda ^2 \\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}w\\\\right)\\\\right)^2.\",\n \"$ Thus, recalling that $\\\\pi _\\\\lambda ^{(0)}(x,y) = \\\\mathbb {P}_\\\\lambda (y \\\\Longleftrightarrow x\\\\textrm { in } \\\\xi ^{y,x})-\\\\varphi (x,y) \\\\ge 0$ , and using the symmetry of $\\\\varphi $ and $\\\\tau _\\\\lambda $ , $\\\\left\\\\Vert _\\\\lambda ^{(0)}\\\\right\\\\Vert _{1,\\\\infty } &= \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\pi ^{(0)}_\\\\lambda \\\\left(x,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\nonumber \\\\\\\\& \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}w\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\nonumber \\\\\\\\& = \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\varphi \\\\left(y,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,x\\\\right)\\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu ^{\\\\otimes 3}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}w,\\\\mathrm {d}u\\\\right)\\\\nonumber \\\\\\\\&\\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ We then use a supremum bound on this integral to split one of the $$ off from the others, and bound the other from above with $\\\\mathcal {T}_\\\\lambda $ to get the first two bounds of the result.\",\n \"For the last bound of Proposition REF , we apply (REF ) and obtain $&\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(0)}\\\\left(k\\\\right)\\\\right\\\\Vert _{1,\\\\infty }\\\\nonumber \\\\\\\\&\\\\qquad = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\pi ^{(0)}_\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right) \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right)\\\\right)^2 \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad = \\\\tfrac{1}{2}\\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\varphi \\\\left(-\\\\overline{u}^\\\\prime ;b,c^\\\\prime \\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}^\\\\prime -\\\\overline{x};c^\\\\prime ,a\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\nonumber \\\\\\\\&\\\\qquad \\\\hspace{284.52756pt}\\\\times \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^\\\\prime \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^\\\\prime ,\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{b,b^\\\\prime \\\\in \\\\mathcal {E}}\\\\int \\\\Big [\\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\left(\\\\overline{x}-\\\\overline{u}\\\\right)\\\\right)\\\\right)\\\\varphi \\\\left(-\\\\overline{u}^\\\\prime ;b^\\\\prime ,c^\\\\prime \\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}^\\\\prime -\\\\overline{x};c^\\\\prime ,a\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\nonumber \\\\\\\\&\\\\qquad \\\\hspace{85.35826pt}+ \\\\left(1 - \\\\cos \\\\left(k\\\\cdot \\\\overline{u}\\\\right)\\\\right)\\\\varphi \\\\left(-\\\\overline{u}^\\\\prime ;b,c^\\\\prime \\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}^\\\\prime -\\\\overline{x};c^\\\\prime ,a\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{u};c,b\\\\right)\\\\Big ] \\\\nonumber \\\\\\\\&\\\\qquad \\\\hspace{284.52756pt}\\\\times \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^\\\\prime \\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^\\\\prime ,\\\\mathrm {d}a\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}+ \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda _k\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Here we have again used the symmetry of $\\\\varphi $ and $\\\\tau _\\\\lambda $ , and also used Lemma REF to split the cosine factor over $\\\\overline{x}-\\\\overline{u}$ and $\\\\overline{u}$ .\",\n \"We can consider each term individually by using the triangle inequality.\",\n \"By using a supremum bound, the symmetry of $\\\\varphi $ , and by bounding $\\\\varphi $ with $\\\\tau _\\\\lambda $ , we get the bound $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda _k\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }.$\"\n ],\n [\n \"Bounds on expansion coefficients\",\n \"In this section we aim to bound $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}$ and $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Lemmas REF and REF along with the positivity of $\\\\pi ^{(n)}_\\\\lambda (x,y)$ prove that it is sufficient to bound $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty }$ to get bounds on both of these.\",\n \"Much of the hard work for this has been done previously - for example in [21].\",\n \"The novelty here is to see how the argument can be naturally written in terms of operators.\",\n \"We inherit the following notation from [21].\",\n \"Note that we make use of the Dirac delta function - specifically the one that holds with respect to the measure $\\\\nu $ .\",\n \"They are used here primarily for convenience and to make the argument more readable.\",\n \"In particular, they appear when applying the Mecke equation (REF ) to obtain $\\\\mathbb {E}\\\\left[ \\\\sum _{y\\\\in \\\\eta ^u} f(y, \\\\xi ^u) \\\\right] = \\\\int \\\\left(\\\\lambda +\\\\delta _{y,u}\\\\right) \\\\mathbb {E}_\\\\lambda \\\\left[ f\\\\left(y,\\\\xi ^{u,y}\\\\right)\\\\right] \\\\nu \\\\left(\\\\mathrm {d}y\\\\right),$ and so the factor $\\\\lambda +\\\\delta _{y,u}$ encodes a case distinction of whether point $y$ coincides with $u$ or not.\",\n \"Definition 5.3 (The $\\\\psi $ functions) Let $r,s,u,w,x,y\\\\in \\\\mathbb {X}$ .\",\n \"We first set $\\\\tau _\\\\lambda ^\\\\circ (x,y) := \\\\lambda ^{-1}\\\\delta _{x,y} + \\\\tau _\\\\lambda (x,y)$ .\",\n \"Also define $\\\\psi _0^{(1)}(w,u,y) &:= \\\\lambda ^2\\\\tau _\\\\lambda (y,u)\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,y),\\\\\\\\\\\\psi _0^{(2)}(w,u,y) &:=\\\\lambda ^2 \\\\delta _{w,y} \\\\tau _\\\\lambda (y,u)\\\\int \\\\tau _\\\\lambda (u,t)\\\\tau _\\\\lambda (t,y) \\\\nu \\\\left(\\\\mathrm {d}t\\\\right),\\\\\\\\\\\\psi _0^{(3)}(w,u,y) &:= \\\\lambda ^2\\\\varphi (u,y) \\\\left(\\\\lambda ^{-1}\\\\delta _{w,y}\\\\right),\\\\\\\\\\\\psi ^{(1)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda (w,u)\\\\int \\\\tau _\\\\lambda ^\\\\circ (t,s) \\\\tau _\\\\lambda (t,w)\\\\tau _\\\\lambda (u,z)\\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\psi ^{(2)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda ^\\\\circ (w,s)\\\\int \\\\tau _\\\\lambda (t,z)\\\\tau _\\\\lambda (z,u)\\\\tau _\\\\lambda (u,t)\\\\tau _\\\\lambda ^\\\\circ (t,w)\\\\tau _\\\\lambda (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\psi ^{(3)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,s)\\\\tau _\\\\lambda (u,r),\\\\\\\\\\\\psi ^{(4)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\left(\\\\lambda ^{-1}\\\\delta _{w,s}\\\\right)\\\\tau _\\\\lambda (u,r),\\\\\\\\\\\\psi _n^{(1)} (x,r,s) &:= \\\\lambda ^3\\\\int \\\\tau _\\\\lambda ^\\\\circ (t,s)\\\\tau _\\\\lambda (z,r)\\\\tau _\\\\lambda (t,z)\\\\tau _\\\\lambda (z,x)\\\\tau _\\\\lambda (x,t)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right),\\\\\\\\\\\\psi _n^{(2)}(x,r,s) &:=\\\\lambda \\\\tau _\\\\lambda (x,s)\\\\tau _\\\\lambda (x,r),$ and set $\\\\psi _0 := \\\\psi _0^{(1)}+\\\\psi _0^{(2)}+\\\\psi _0^{(3)}$ , $\\\\psi _n := \\\\psi _n^{(1)} + \\\\psi _n^{(2)}$ , and $\\\\psi := \\\\psi ^{(1)}+\\\\psi ^{(2)}+\\\\psi ^{(3)} + \\\\psi ^{(4)}$ .\",\n \"Note that by using $\\\\lambda ^{-1}\\\\delta _{\\\\left(\\\\cdot ,\\\\cdot \\\\right)}$ in $\\\\tau _\\\\lambda ^\\\\circ $ and in $\\\\psi ^{(3)}_0$ , we ensure that each integral that will be truly performed (that is, not an integration over a Dirac delta function accounting for a coincidence of points) has a $\\\\lambda $ factor associated with it when the bound $\\\\pi ^{(n)}_\\\\lambda $ is calculated.\",\n \"Diagrammatic representations of these functions can be found in Figure REF .\",\n \"In these diagrams, the declared variables are represented by (0,0) circle (2pt) vertices whereas the variables that are integrated over are represented by (0,0) circle (2pt) vertices.\",\n \"If $\\\\tau _\\\\lambda $ connects two variables then a standard edge (0,0) – (1,0); connects their vertices, if $\\\\tau _\\\\lambda ^\\\\circ $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\circ $; connects their vertices, and if $\\\\varphi $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\sim $; connects their vertices.\",\n \"Figure: Diagrams of the ψ 0 \\\\psi _0, ψ\\\\psi , and ψ n \\\\psi _n functions.Definition 5.4 (The $\\\\Psi $ operators) Here we turn the functions of Definition REF into linear operators, with the $\\\\psi $ functions acting as their kernels.\",\n \"For $j\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , we define $\\\\Psi ^{(j)}_0\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as the linear operator acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ as $\\\\Psi ^{(j)}_0 f\\\\left(w,u\\\\right) = \\\\int \\\\psi ^{(j)}_0(w,u,y) f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right).$ For $j\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , we define $\\\\Psi ^{(j)}\\\\colon L^2\\\\left(\\\\mathbb {X}^2\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as the linear operator acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as $\\\\Psi ^{(j)} f\\\\left(w,u\\\\right) = \\\\int \\\\psi ^{(j)}(w,u,r,s) f(r,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}r, \\\\mathrm {d}s\\\\right).$ For $j\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ , we define $\\\\Psi ^{(j)}_n\\\\colon L^2\\\\left(\\\\mathbb {X}^2\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}\\\\right)$ as the linear operator acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as $\\\\Psi ^{(j)}_n f\\\\left(x\\\\right) = \\\\int \\\\psi ^{(j)}_n(x,r,s) f(r,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}r, \\\\mathrm {d}s\\\\right).$ We can then naturally define $\\\\Psi _0\\\\colon L^2(\\\\mathbb {X})\\\\rightarrow L^2(\\\\mathbb {X}^2)$ , $\\\\Psi \\\\colon L^2(\\\\mathbb {X}^2)\\\\rightarrow L^2(\\\\mathbb {X}^2)$ , and $\\\\Psi _n\\\\colon L^2(\\\\mathbb {X}^2)\\\\rightarrow L^2(\\\\mathbb {X})$ as the sums of $\\\\Psi ^{(j)}_0$ , $\\\\Psi ^{(j)}$ , and $\\\\Psi ^{(j)}_n$ respectively, or equivalently as those operators having kernel functions $\\\\psi _0$ , $\\\\psi $ , and $\\\\psi _n$ respectively.\",\n \"Proposition 5.5 (Bound in terms of $\\\\psi $ and $\\\\Psi $ ) Let $n \\\\ge 1$ , $x,y\\\\in \\\\mathbb {X}$ , and $\\\\lambda \\\\in [0,\\\\lambda _O)$ .\",\n \"Then $\\\\lambda \\\\pi _\\\\lambda ^{(n)}(x,y) \\\\le \\\\int \\\\psi _n(x,w_{n-1},u_{n-1}) \\\\left( \\\\prod _{i=1}^{n-1} \\\\psi (\\\\vec{v}_i) \\\\right) \\\\psi _0(w_0,u_0,y) \\\\nu ^{\\\\otimes \\\\left(2n\\\\right)}\\\\left(\\\\mathrm {d}\\\\left( \\\\left(\\\\vec{w}, \\\\vec{u}\\\\right)_{[0,n-1]} \\\\right)\\\\right),$ where $\\\\vec{v}_i = (w_i,u_i,w_{i-1},u_{i-1})$ .\",\n \"In operator terms, this means $\\\\lambda \\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\Psi _n \\\\Psi ^{n-1} \\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }.$ The argument for (REF ) is essentially identical to [21].\",\n \"It uses only general properties of Poisson point process and connection models - for example thinning edges and Mecke's equation.\",\n \"The inequality (REF ) then follows from the clear sequential structure of (REF ).\",\n \"The inequality (REF ) suggests that we will bound the lace coefficient operators with some product of $\\\\left\\\\Vert \\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi \\\\right\\\\Vert _{1,\\\\infty }$ , and $\\\\left\\\\Vert \\\\Psi _n\\\\right\\\\Vert _{1,\\\\infty }$ terms.\",\n \"This is nearly correct.\",\n \"Unfortunately the norms $\\\\left\\\\Vert \\\\Psi _n\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi \\\\right\\\\Vert _{1,\\\\infty }$ and $\\\\left\\\\Vert \\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }$ will not have the decay in $d$ that we will require.\",\n \"We will have to consider the norms of pairs of operators.\",\n \"Definition 5.6 Define $\\\\triangle _\\\\lambda &:= \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda \\\\left(r,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\triangle ^\\\\circ _\\\\lambda &:= \\\\lambda ^2 \\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(r,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda &:= \\\\lambda ^2 \\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(r,u\\\\right)\\\\tau _\\\\lambda ^\\\\circ \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + 1,\\\\\\\\\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }&:= \\\\lambda ^2\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda ^\\\\circ \\\\left(\\\\overline{u}-\\\\overline{v};a_3,a_4\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{v};a_5,a_6\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{v}.$ We can think of $\\\\triangle _\\\\lambda $ as an $\\\\mathbb {X}$ -convolution of three $\\\\tau _\\\\lambda $ functions with suprema taken over the end vertices.\",\n \"Then $\\\\triangle ^\\\\circ _\\\\lambda $ and $\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda $ are produced by adding on the $\\\\mathbb {X}$ -convolution of two $\\\\tau _\\\\lambda $ functions and one $\\\\tau _\\\\lambda $ function (which will trivially take value 1 once the suprema are taken).\",\n \"The object $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ differs from $\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda $ in that now the marks at adjacent $\\\\tau _\\\\lambda $ functions need not be equal, and then we take the supremum over all the marks - not just the end ones.\",\n \"We then use these elementary diagrams to produce composite objects: $U_\\\\lambda := \\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }+ \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }^2 + \\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }, \\\\qquad V_\\\\lambda := \\\\left(\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }U_\\\\lambda \\\\right)^\\\\frac{1}{2}.$ We note a few relations.\",\n \"Firstly, since all the terms in $U_\\\\lambda $ are non-negative we have $\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le U_\\\\lambda $ and thus $V_\\\\lambda \\\\le U_\\\\lambda $ .\",\n \"Also note that $1\\\\le \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ and thus $U_\\\\lambda \\\\ge \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ .\",\n \"In particular this means that $U_\\\\lambda \\\\ge 1$ and $U_\\\\lambda \\\\ge \\\\triangle ^\\\\circ _\\\\lambda $ .\",\n \"The following lemma tells us that we can use $U_\\\\lambda $ to bound single operators, and that we can use $V_\\\\lambda ^2$ to bound pairs of operators.\",\n \"To aid the reader's understanding, bear in mind that later (in Section ) we will prove that $U_\\\\lambda = \\\\mathcal {O}\\\\left(1\\\\right)$ , whilst $V_\\\\lambda $ is much smaller and is $\\\\mathcal {O}\\\\left(\\\\beta \\\\right)$ .\",\n \"Lemma 5.7 For all $j_0\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , $j,j^{\\\\prime }\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , and $j_n\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ , $U_\\\\lambda &\\\\ge \\\\left\\\\Vert \\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j)}\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\right\\\\Vert _{1,\\\\infty },\\\\\\\\V_\\\\lambda ^2 &\\\\ge \\\\left\\\\Vert \\\\Psi ^{(j)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j)}\\\\Psi ^{(j^{\\\\prime })}\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j)}\\\\right\\\\Vert _{1,\\\\infty } \\\\vee \\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }.$ We begin with the single operators.\",\n \"For $\\\\Psi ^{(1)}$ we find $\\\\left\\\\Vert \\\\Psi ^{(1)}\\\\right\\\\Vert _{1,\\\\infty } &= \\\\lambda ^4\\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (w,u)\\\\tau _\\\\lambda ^\\\\circ (t,s) \\\\tau _\\\\lambda (t,w)\\\\tau _\\\\lambda (u,z)\\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda (z,r)\\\\nu ^{\\\\otimes 4}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t, \\\\mathrm {d}w, \\\\mathrm {d}u\\\\right), \\\\nonumber \\\\\\\\& \\\\le \\\\lambda ^4\\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (r,z) \\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda ^\\\\circ (t,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t \\\\right)\\\\right) \\\\nonumber \\\\\\\\&\\\\hspace{142.26378pt} \\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{t^{\\\\prime },z^{\\\\prime }\\\\in \\\\mathbb {X}}\\\\int \\\\tau _\\\\lambda (t^{\\\\prime },w)\\\\tau _\\\\lambda (w,u)\\\\tau _\\\\lambda (u,z^{\\\\prime })\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}w, \\\\mathrm {d}u\\\\right)\\\\right),\\\\nonumber \\\\\\\\& = \\\\triangle ^\\\\circ _\\\\lambda \\\\triangle _\\\\lambda ,$ where we have used a supremum bound on the $t$ and $z$ integrals to split them into $L^1$ and $L^\\\\infty $ bounds.\",\n \"It is also clear that $\\\\left\\\\Vert \\\\Psi ^{(3)}\\\\right\\\\Vert _{1,\\\\infty }= \\\\triangle _\\\\lambda $ and $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\triangle ^\\\\circ _\\\\lambda $ .\",\n \"The $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ bound is more involved and we return to it in a moment.\",\n \"We are able to represent these $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ norms and their bounds pictorially.\",\n \"The vertices of the diagrams represent the variables $w,u,z,t,a,b$ , etc.\",\n \"appearing in the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ expression.\",\n \"The variables over which a $\\\\nu $ -integral is taken are represented by the (0,0) circle (2pt) vertices, and the variables over which a supremum is taken are represented by the (0,0) circle (2pt) vertices.\",\n \"The presence of a $\\\\tau _\\\\lambda $ connecting two variables is then represented by a standard edge (0,0) – (1,0); between their vertices, and the presence of a $\\\\tau _\\\\lambda ^\\\\circ $ connecting two variables is correspondingly represented by an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\circ $; between their vertices.\",\n \"When using a supremum bound to split the diagrams, an integrated vertex gets split into two new vertices, precisely one integrated vertex and one supremum vertex.\",\n \"Each edge connected to such a split vertex can independently choose which of the new vertices to associate to.\",\n \"The calculations producing the bounds on $\\\\left\\\\Vert \\\\Psi ^{(j)}\\\\right\\\\Vert _{1,\\\\infty }$ for $j=1,3,4$ can then be represented by: $\\\\left\\\\Vert \\\\Psi ^{(1)}\\\\right\\\\Vert _{1,\\\\infty } \\\\quad &=\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(3)}\\\\right\\\\Vert _{1,\\\\infty } + \\\\left\\\\Vert \\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } \\\\quad &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad .$ Our strategy for $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ is to use the spatial translation invariance to shift the origin to the bottom right vertex, and then use a supremum bound to split the diagram in two.\",\n \"Unfortunately, this spatial shift decouples the spatial and mark components of the vertices so - for example - at some vertices we end up taking suprema over marks whilst integrating over space.\",\n \"Let us introduce some more notation.\",\n \"The vertex mark size=2pt] at (0,0) square* indicates that the spatial component is integrated over whilst the mark component is fixed and its supremum is taken after all the integrals.\",\n \"We also introduce the edge-vertex combination [dashed] (0,0) – (1,0); [fill=white] (0,0) circle (2pt) which indicates that there is no term connecting the associated vertices, but there is a supremum taken over the spatial displacement.\",\n \"Note that in principle we could use a vertex that had a spatial supremum and a mark integral, but we will always bound that probability integral by the supremum.\",\n \"The bound for $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ can therefore be expressed in the following: $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(0.75,0.6) circle (0pt) node[above]{\\\\circ };(1.5,0.25) circle (0pt) node[left]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}\\\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\\\circ };(1.5,0.25) circle (0pt) node[left]{\\\\circ };(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}\\\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1);(0,-0.6) -- (1,-0.6);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\\\circ };(1.5,0.25) circle (0pt) node[left]{\\\\circ };[fill=white] (1.5,-0.1) circle (2pt);(1.5,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);[fill=white] (0,0) circle (2pt);\\\\end{tikzpicture}}.$ To clarify the calculation for $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }$ , we write out the integral here: $&\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{s}\\\\in \\\\mathbb {R}^d,a_1,a_2\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ (\\\\overline{w}-\\\\overline{s};a_3,a_1)\\\\tau _\\\\lambda ^\\\\circ (\\\\overline{t}- \\\\overline{w}; a_4,a_3)\\\\tau _\\\\lambda (\\\\overline{u}-\\\\overline{t};a_6,a_4)\\\\tau _\\\\lambda (\\\\overline{z}-\\\\overline{u};a_5,a_6) \\\\nonumber \\\\\\\\&\\\\hspace{142.26378pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{t}- \\\\overline{z}; a_4,a_5)\\\\tau _\\\\lambda (\\\\overline{z}; a_5,a_2) \\\\mathrm {d}\\\\overline{t}\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{w}\\\\mathrm {d}\\\\overline{z}\\\\mathcal {P}^{\\\\otimes 4}\\\\left(\\\\mathrm {d}a_3, \\\\mathrm {d}a_4, \\\\mathrm {d}a_5, \\\\mathrm {d}a_6\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{s}\\\\in \\\\mathbb {R}^d,a_1,a_2,a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ (\\\\overline{w}^{\\\\prime }-\\\\overline{s}+\\\\overline{u};a_3,a_1)\\\\tau _\\\\lambda ^\\\\circ (\\\\overline{t}^{\\\\prime } - \\\\overline{w}^{\\\\prime }; a_4,a_3)\\\\tau _\\\\lambda (-\\\\overline{t}^{\\\\prime };a_6,a_4)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime };a_5,a_6) \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{t}^{\\\\prime } - \\\\overline{z}^{\\\\prime }; a_4,a_5)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime } + \\\\overline{u}; a_5,a_2) \\\\mathrm {d}\\\\overline{t}^{\\\\prime } \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{w}^{\\\\prime } \\\\mathrm {d}\\\\overline{z}^{\\\\prime } \\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}a_3, \\\\mathrm {d}a_4, \\\\mathrm {d}a_5\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}\\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{s},\\\\overline{t}^{\\\\prime },\\\\overline{z}^{\\\\prime }\\\\in \\\\mathbb {R}^d,a_1,a_2,a_4,a_5\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda ^\\\\circ (\\\\overline{w}^{\\\\prime }-\\\\overline{s}+\\\\overline{u};a_3,a_1)\\\\tau _\\\\lambda ^\\\\circ (\\\\overline{t}^{\\\\prime } - \\\\overline{w}^{\\\\prime }; a_4,a_3)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime } + \\\\overline{u}; a_5,a_2) \\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{w}^{\\\\prime } \\\\mathcal {P}\\\\left(\\\\mathrm {d}a_3\\\\right)\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda (-\\\\overline{t}^{\\\\prime };a_6,a_4)\\\\tau _\\\\lambda (\\\\overline{z}^{\\\\prime };a_5,a_6)\\\\tau _\\\\lambda (\\\\overline{t}^{\\\\prime } - \\\\overline{z}^{\\\\prime }; a_4,a_5) \\\\mathrm {d}\\\\overline{t}^{\\\\prime } \\\\mathrm {d}\\\\overline{z}^{\\\\prime } \\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a_4, \\\\mathrm {d}a_5\\\\right)\\\\right).$ Then the two components of the first factor that are joined by the $\\\\raisebox {2pt}{{[dashed] (0,0) -- (1,0); [fill=white] (0,0) circle (2pt)}}$ edge can be spatially translated together to form three consecutive edges.\",\n \"Note however, that the marks on either side of this join are not necessarily equal.\",\n \"We can therefore bound this first term by $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ .\",\n \"In summary, these manipulations then imply that $\\\\left\\\\Vert \\\\Psi ^{(2)}\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\triangle _\\\\lambda $ .\",\n \"For $\\\\Psi ^{(1)}_0$ and $\\\\Psi ^{(2)}_0$ the calculations are identical to each other, and we can once again represent the calculations pictorially: $\\\\left\\\\Vert \\\\Psi ^{(1)}_0\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\Psi ^{(2)}_0\\\\right\\\\Vert _{1,\\\\infty } =\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- cycle;[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad = \\\\triangle _\\\\lambda ,$ where splitting the single supremum into two produces an upper bound.\",\n \"For the third case we have $\\\\left\\\\Vert \\\\Psi ^{(3)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\quad \\\\raisebox {0pt}{\\\\begin{tikzpicture}(0,0) -- (1,0);(0.5,0) circle (0pt) node[above]{\\\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad = \\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }.$ The diagrams representing the calculations for $\\\\Psi _n$ are as follows: $\\\\left\\\\Vert \\\\Psi _n^{(1)}\\\\right\\\\Vert _{1,\\\\infty } & = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(0.5,0.6) circle (0pt) node[above]{\\\\circ };(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(1,-0.6) -- (0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\ &\\\\hspace{113.81102pt}\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\qquad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda , \\\\nonumber \\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\right\\\\Vert _{1,\\\\infty } & = \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle ^\\\\circ _\\\\lambda .$ In the first case we expanded a $\\\\tau _\\\\lambda ^\\\\circ $ edge into a $\\\\tau _\\\\lambda $ edge and a contraction.\",\n \"Then in the contracted case we bound $\\\\tau _\\\\lambda \\\\le 1$ on the diagonal edge.\",\n \"We have now proven the bounds for the single operator norms.\",\n \"For most of the operator pairs, we can then use the sub-multiplicity of $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ to immediately get sufficient bounds.\",\n \"The only pairs for which this strategy fails are: $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(4)}_0\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(3)}_0\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty }$ , and $\\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"We now deal with these cases by hand: $\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(3)}_0\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(3)}_0\\\\right\\\\Vert _{1,\\\\infty } &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- (0,0.6);(0,0) circle (0pt) node[above, rotate=90]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle _\\\\lambda , \\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(4)}\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\Psi ^{(2)}_n\\\\Psi ^{(4)}\\\\right\\\\Vert _{1,\\\\infty } &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle _\\\\lambda .$ For the first bound we used $\\\\varphi \\\\le \\\\tau _\\\\lambda $ , and then split the single supremum into two.\",\n \"For the second bound we used $\\\\tau _\\\\lambda \\\\le 1$ on the diagonal edge.\",\n \"Proposition 5.8 For $n\\\\ge 1$ , $\\\\lambda \\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le 6\\\\times 4^{n-1}U_\\\\lambda V_\\\\lambda ^n.$ Recall from Lemma REF that $\\\\lambda \\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\Psi _n\\\\Psi ^{n-1}\\\\Psi _0\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"We therefore aim to bound $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }$ for $j_0\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , $j_1,\\\\ldots ,j_{n-1}\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , and $j_n\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ .\",\n \"We have two cases depending upon the parity of $n$ .\",\n \"If $n$ is odd then we have an even number of operators.\",\n \"We pair them off, use the sub-multiplicity of $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ , and Lemma REF to get $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left(V_\\\\lambda ^2\\\\right)^{\\\\frac{n+1}{2}} = V_\\\\lambda ^{n+1}.$ If $n$ is even, then we pair off the first $n$ operators and treat the last $\\\\Psi ^{(j_n)}_n$ operator by itself.\",\n \"We then get $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } \\\\le U_\\\\lambda \\\\left(V_\\\\lambda ^2\\\\right)^{\\\\frac{n}{2}} = U_\\\\lambda V_\\\\lambda ^{n}.$ Since $U_\\\\lambda \\\\ge V_\\\\lambda $ , we have the bound $U_\\\\lambda V_\\\\lambda ^{n}$ for all $n\\\\ge 1$ .\",\n \"Since there are 3 choices for $j_0$ , 4 choices for each of $j_1,\\\\ldots ,j_{n-1}$ , and 2 choices for $j_n$ , we get the pre-factor of $6\\\\times 4^{n-1}$ in our result by applying the triangle inequality.\"\n ],\n [\n \"Bounds on displaced expansion coefficients\",\n \"In this section we will be aiming to bound $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ for $n\\\\ge 1$ .\",\n \"Our strategy draws inspiration from the corresponding step in [21], whilst being adapted to account for inhomogeneous marks in a similar way to Section REF .\",\n \"The central idea is to bound $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\left(1- \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\pi ^{(n)}_\\\\lambda \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right),$ and bound $\\\\pi ^{(n)}_\\\\lambda $ using the function expression from Proposition REF .\",\n \"This produces an integral (or diagram) composed from simpler segments with the displacement factor, $\\\\left(1- \\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)$ , spanning the whole length of the diagram.\",\n \"We can then use the Cosine-Splitting result (Lemma REF ) to get a sum of diagrams where the displacement factor only spans a single segment of each diagram.\",\n \"To demonstrate this, consider the diagram corresponding to $\\\\Psi ^{(2)}_n\\\\Psi ^{(1)}\\\\Psi ^{(1)}_0$ with the displacement factor: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (0,0.2) -- (1,0.8) -- (3,0.8) -- (4,0.2);(2,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le 3\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\right.\",\n \"\\\\\\\\ \\\\left.\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (1,0.8) -- (3,0.8);(2,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right).$ In these diagrams and hereafter we use [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; to denote that a displacement factor is connecting the two indicated vertices.\",\n \"As above, the path the [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; takes will also suggest the route along which we will intend to use cosine-splitting.\",\n \"For the sake of simplicity, we will be taking the displacement path across the `top' of the diagram from the perspective of the usual orientation of the $\\\\Psi _0$ segment.\",\n \"However - as can be seen in the above example - the way the operators compose with each other means that the orientation of each segment inverts when compared to its neighbours.\",\n \"Therefore the displacement may be across the `top' or the `bottom' of the $\\\\Psi $ and $\\\\Psi _n$ segments (in their usual orientation) depending upon the parity of its place in the sequence of segments.\",\n \"Our general strategy will be to isolate the displaced segment (and perhaps a neighbouring segment) by splitting off `earlier' and `later' segments.\",\n \"When we split off earlier segments (and sometimes later segments), we will use the $\\\\Psi $ structure we have been using so far.\",\n \"However, we will sometimes want to group vertices and edges slightly differently when we split off `later' segments.\",\n \"We use an observation from [21] to conveniently group these.\",\n \"Let us define the kernel functions $\\\\overline{\\\\psi }^{(1)}_0\\\\left(w,u,x\\\\right) &:= \\\\lambda ^2\\\\tau _\\\\lambda (y,r)\\\\tau _\\\\lambda (r,s)\\\\tau _\\\\lambda (s,x),\\\\\\\\\\\\overline{\\\\psi }^{(2)}_0\\\\left(w,u,x\\\\right) &:= \\\\lambda ^2\\\\left(\\\\lambda ^{-1}\\\\delta _{y,r}\\\\right)\\\\left(\\\\lambda ^{-1}\\\\delta _{y,s}\\\\right),\\\\\\\\\\\\overline{\\\\psi }^{(1)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda (w,u)\\\\int \\\\tau _\\\\lambda (t,s) \\\\tau _\\\\lambda (t,w)\\\\tau _\\\\lambda (u,z)\\\\tau _\\\\lambda (z,t)\\\\tau _\\\\lambda ^\\\\circ (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\overline{\\\\psi }^{(2)}(w,u,r,s) &:= \\\\lambda ^4\\\\tau _\\\\lambda (w,s)\\\\int \\\\tau _\\\\lambda (t,z)\\\\tau _\\\\lambda (z,u)\\\\tau _\\\\lambda (u,t)\\\\tau _\\\\lambda ^\\\\circ (t,w)\\\\tau _\\\\lambda ^\\\\circ (z,r)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}z, \\\\mathrm {d}t\\\\right), \\\\\\\\\\\\overline{\\\\psi }^{(3)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,s)\\\\tau _\\\\lambda (u,r),\\\\\\\\\\\\overline{\\\\psi }^{(4)}(w,u,r,s) &:= \\\\lambda ^2\\\\tau _\\\\lambda (u,w)\\\\tau _\\\\lambda (w,s)\\\\left(\\\\lambda ^{-1}\\\\delta _{u,r}\\\\right),$ and set $\\\\overline{\\\\psi }_0 := \\\\overline{\\\\psi }^{(1)}_0+\\\\overline{\\\\psi }^{(2)}_0$ and $\\\\overline{\\\\psi }:= \\\\overline{\\\\psi }^{(1)}+\\\\overline{\\\\psi }^{(2)}+\\\\overline{\\\\psi }^{(3)} + \\\\overline{\\\\psi }^{(4)}$ .\",\n \"Diagrammatic representations of these functions can be found in Figure REF .\",\n \"Note the similarity to the $\\\\psi $ functions, with the $\\\\tau _\\\\lambda ^\\\\circ $ edge appearing elsewhere in the integral.\",\n \"Figure: Diagrams of the ψ ¯ 0 \\\\overline{\\\\psi }_0, and ψ ¯\\\\overline{\\\\psi } functions.For $j\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ we define $\\\\overline{\\\\Psi }^{(j)}_0\\\\colon L^2\\\\left(\\\\mathbb {X}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ to be the linear operators acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ as $\\\\overline{\\\\Psi }^{(j)}_0 f\\\\left(w,u\\\\right) = \\\\int \\\\overline{\\\\psi }^{(j)}_0(w,u,y) f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right),$ and for $j\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ we define $\\\\overline{\\\\Psi }^{(j)}\\\\colon L^2\\\\left(\\\\mathbb {X}^2\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ to be the linear operators acting on $f\\\\in L^2\\\\left(\\\\mathbb {X}^2\\\\right)$ as $\\\\overline{\\\\Psi }^{(j)} f\\\\left(w,u\\\\right) = \\\\int \\\\overline{\\\\psi }^{(j)}(w,u,r,s) f(r,s)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}r, \\\\mathrm {d}s\\\\right).$ In our terminology, the important observation of [21] was that for some $m\\\\ge 0$ the `later' segments of each diagram can be bounded using $\\\\big (\\\\overline{\\\\Psi }^m\\\\overline{\\\\Psi }_0\\\\big )^\\\\dagger $ , the adjoint of $\\\\overline{\\\\Psi }^m\\\\overline{\\\\Psi }_0$ .\",\n \"In terms of the kernel functions, taking the adjoint in this case amounts to reflecting the `input' and `output' arguments, and for the diagrams this amounts to reflecting in the vertical plane.\",\n \"Lemma 5.9 For $n\\\\ge 1$ , $m\\\\in \\\\left\\\\lbrace 0,\\\\ldots ,n\\\\right\\\\rbrace $ , $j_0\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace $ , $j_1,\\\\ldots ,j_{n-1}\\\\in \\\\left\\\\lbrace 1,2,3,4\\\\right\\\\rbrace $ , and $j_n\\\\in \\\\left\\\\lbrace 1,2\\\\right\\\\rbrace $ , $\\\\left\\\\Vert \\\\Psi ^{(j_m)}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } &\\\\le U_\\\\lambda V_\\\\lambda ^m,\\\\\\\\\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_{m})}\\\\right\\\\Vert _{1,\\\\infty } &\\\\le U_\\\\lambda V_\\\\lambda ^{n-m},\\\\\\\\\\\\left\\\\Vert \\\\overline{\\\\Psi }^{(j_m)}\\\\ldots \\\\overline{\\\\Psi }^{(j_1)}\\\\overline{\\\\Psi }^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty } &\\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}U_\\\\lambda &: m=0\\\\\\\\U_\\\\lambda ^2V_\\\\lambda ^{m-1} &: m\\\\ge 1.\\\\end{array}\\\\right.\",\n \"}$ The argument for $\\\\left\\\\Vert \\\\Psi ^{(j_m)}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }$ and $\\\\left\\\\Vert \\\\Psi ^{(j_n)}_n\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_{m})}\\\\right\\\\Vert _{1,\\\\infty }$ is the same as that in Lemma REF and Proposition REF .\",\n \"We consider bounds for single operators and pairs of operators, and then use sub-multiplicity to get a bound for each diagram.\",\n \"For $\\\\left\\\\Vert \\\\overline{\\\\Psi }^{(j_m)}\\\\ldots \\\\overline{\\\\Psi }^{(j_1)}\\\\overline{\\\\Psi }^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }$ the argument is a little more complicated.\",\n \"For $m=0,1$ we can easily check each diagram satisfies the bound.\",\n \"For $m\\\\ge 2$ we have the issue that there exists one pair of operators which we can't bound by $V_\\\\lambda ^2$ : $\\\\left\\\\Vert \\\\overline{\\\\Psi }^{(4)}\\\\overline{\\\\Psi }^{(2)}_0\\\\right\\\\Vert _{1,\\\\infty } = \\\\quad \\\\raisebox {0pt}{\\\\begin{tikzpicture}(0,0) to [out=10,in=170] (1,0) to [out=190,in=350] (0,0);[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\triangle ^\\\\circ _\\\\lambda \\\\le U_\\\\lambda ^2.$ If $m\\\\ge 2$ is even (an odd number of operators) then we use sub-multiplicity to pull off the $\\\\overline{\\\\Psi }_0$ term which we bound with $U_\\\\lambda $ .\",\n \"We then pull off pairs which we can bound with $V_\\\\lambda ^2$ to get the bound $U_\\\\lambda V_\\\\lambda ^m$ .\",\n \"If $m\\\\ge 2$ is odd (an even number of operators) then we use sub-multiplicity to pull off the $\\\\overline{\\\\Psi }\\\\overline{\\\\Psi }_0$ pair which we bound with $U_\\\\lambda ^2$ .\",\n \"We then pull off pairs which we can bound with $V_\\\\lambda ^2$ to get the bound $U_\\\\lambda ^2V_\\\\lambda ^{m-1}$ .\",\n \"Definition 5.10 In addition to the terms defined in Definition REF , we will now use $W_k&:= \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{x,y\\\\in \\\\mathbb {X}}\\\\int \\\\tau _{\\\\lambda ,k}(x,u)\\\\tau _\\\\lambda (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right) = \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty },\\\\\\\\\\\\overline{W_k}&:= \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\tau _{\\\\lambda ,k}(\\\\overline{x}-\\\\overline{u};a_1,a_2)\\\\tau _\\\\lambda (\\\\overline{u};a_3,a_4)\\\\mathrm {d}u,\\\\\\\\\\\\overline{H_k}&:= \\\\lambda ^5 \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}_1,\\\\overline{x}_2\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_{16}\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{x}_1;a_1,a_2)\\\\tau _\\\\lambda (\\\\overline{u};a_3,a_4) \\\\tau _\\\\lambda (\\\\overline{v}-\\\\overline{s};a_5,a_6) \\\\tau _\\\\lambda (\\\\overline{v}+\\\\overline{x}_2-\\\\overline{t};a_7,a_8)\\\\nonumber \\\\\\\\& \\\\hspace{42.67912pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{w};a_9,a_{10})\\\\tau _\\\\lambda (\\\\overline{w}-\\\\overline{u};a_{11},a_{12}) \\\\tau _\\\\lambda (\\\\overline{t}-\\\\overline{w};a_{13},a_{14}) \\\\tau _{\\\\lambda ,k}(\\\\overline{t}-\\\\overline{u};a_{15},a_{16}) \\\\mathrm {d}\\\\overline{t}\\\\mathrm {d}\\\\overline{w}\\\\mathrm {d}\\\\overline{z}\\\\mathrm {d}\\\\overline{u}.$ Note that we only use $\\\\overline{W_k}$ in Propositions REF and REF because $\\\\overline{W_k}\\\\ge W_k$ , but in the proof we will sometimes use $W_k$ to make the derivation easier to follow.\",\n \"The diagrams for $\\\\overline{W_k}$ and $\\\\overline{H_k}$ are in Figure REF .\",\n \"Also recall the sets $\\\\left\\\\lbrace B\\\\left(x\\\\right)\\\\right\\\\rbrace _{x\\\\in \\\\mathbb {X}}$ assumed to exist in Assumption REF .\",\n \"Then define $\\\\mathbb {B} &:= \\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\nu \\\\left(B\\\\left(x\\\\right)\\\\right),\\\\\\\\\\\\triangle ^{(B)}_\\\\lambda &:= \\\\lambda ^2 \\\\operatornamewithlimits{ess\\\\,sup}_{r,s\\\\in \\\\mathbb {X}: r\\\\notin {B\\\\left(s\\\\right)}}\\\\int \\\\tau _\\\\lambda ^\\\\circ \\\\left(r,u\\\\right)\\\\tau _\\\\lambda \\\\left(u,v\\\\right)\\\\tau _\\\\lambda \\\\left(v,s\\\\right)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right).$ Note that these are only required to deal with one specific diagram in the following $n=1$ case.\",\n \"Figure: Diagrams representing W k ¯\\\\overline{W_k} and H k ¯\\\\overline{H_k}.\",\n \"In this figure, the marks on edges incident to filled square vertices need not be equal.Proposition 5.11 For $k\\\\in \\\\mathbb {R}^d$ and $n\\\\ge 2$ , $\\\\lambda \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(n+1\\\\right)^24^{n-1}\\\\left(50\\\\overline{W_k}V_\\\\lambda + 6\\\\overline{H_k}\\\\right) U_\\\\lambda ^4 V_\\\\lambda ^{n-4} &: n\\\\ge 4\\\\\\\\16\\\\times 16\\\\left(50\\\\overline{W_k}V_\\\\lambda ^2 + 6\\\\overline{H_k}\\\\right) U_\\\\lambda ^2 &: n=3\\\\\\\\9\\\\times 4\\\\left(50\\\\overline{W_k}V_\\\\lambda + 6\\\\overline{H_k}\\\\right) U_\\\\lambda ^2 &: n=2.\\\\end{array}\\\\right.\",\n \"}$ Proposition 5.12 For $k\\\\in \\\\mathbb {R}^d$ , $\\\\lambda \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(1)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(1)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le 13\\\\overline{W_k}V_\\\\lambda U_\\\\lambda + \\\\lambda \\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }\\\\left(V_\\\\lambda + \\\\triangle ^{(B)}_\\\\lambda \\\\right) + \\\\lambda ^3\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + 4\\\\lambda \\\\mathbb {B}\\\\overline{W_k}.$ [Proof of Proposition REF ] In bounding the diagrams with one displacement segment, we have three broad cases.\",\n \"Let $i\\\\in \\\\left\\\\lbrace 0,1,\\\\ldots ,n\\\\right\\\\rbrace $ denote which segment the displacement lies upon.\",\n \"Our three cases are then: the displacement lies on the $\\\\psi _n$ segment (i.e.\",\n \"$i=n$ ), the displacement lies on the $\\\\psi _0$ segment (i.e.\",\n \"$i=0$ ), the displacement lies on a $\\\\psi $ segment (i.e.\",\n \"$i\\\\in \\\\left\\\\lbrace 1,\\\\ldots ,n-1\\\\right\\\\rbrace $ ).\",\n \"We will aim to get a single bound for each diagram having a displacement crossing a single segment.\",\n \"This will be uniform in the sequence $\\\\left(j_0,j_1,\\\\ldots ,j_n\\\\right)$ .\",\n \"We then account for the number of possible choices of $\\\\left(j_0,j_1,\\\\ldots ,j_n\\\\right)$ , and finally include the factor of $(n+1)^2$ arising from using cosine-splitting (the explicit factor and the number of diagrams).\",\n \"Case REF is the simplest.\",\n \"We use a supremum bound to pull off the displaced $\\\\Psi _n$ term.\",\n \"We then bound this by hand and use Lemma REF to bound the remainder.\",\n \"In representing this scheme we use a shorthand $\\\\Psi ^{n-1}\\\\Psi _0$ to denote a particular sequence $\\\\Psi ^{(j_{n-1})}\\\\ldots \\\\Psi ^{(j_1)}\\\\Psi ^{(j_0)}_0$ rather than actually the operators $\\\\Psi $ and $\\\\Psi _0$ defined above.\",\n \"We also use labelled grey shapes to represent these groupings of diagrams, highlighting only end vertices or vertices that connect to neighbouring segments.\",\n \"This scheme look like: $&\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8) -- (4,0.2);(2.8,0.8) circle (0pt) node[rotate = 0]{\\\\times };(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node at (1,0) {\\\\Psi ^{n-1}\\\\Psi _0};[fill=white] (-1,0) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad +\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (2,0.8) -- (3,0.2);(2.5,0.5) circle (0pt) node[rotate = -30]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{n-1}\\\\Psi _0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\ &\\\\hspace{28.45274pt}\\\\le U_\\\\lambda V_\\\\lambda ^{n-1} \\\\left(2\\\\left( \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\\\times };(2.5,-0.6) circle (0pt) node[above]{\\\\circ };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right) + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\mathbf {\\\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le 2U_\\\\lambda V_\\\\lambda ^{n-1}\\\\left(\\\\overline{W_k}\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda + W_k\\\\triangle ^\\\\circ _\\\\lambda \\\\right)\\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le 4\\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-1}.$ Note that this bound also holds if the displacement runs along the bottom of the $\\\\Psi _n$ segment.\",\n \"Since there are $3\\\\times 4^{n-1}$ possible sequences of indices for the $\\\\Psi ^{n-1}\\\\Psi _0$ diagram, this case contributes that as a pre-factor.\",\n \"For Case REF , we need a few more tricks.\",\n \"Since we are taking the displacement across the top of the diagram, the only term that will make a contribution is $\\\\Psi ^{(1)}_0$ .\",\n \"To tackle this case we first use the observation of [21] that the diagram we want to bound can in turn be bounded by a diagram using $\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0$ .\",\n \"As above, we use $\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0$ as a shorthand for a particular sequence $\\\\overline{\\\\Psi }^{(j_{n-1})}\\\\ldots \\\\overline{\\\\Psi }^{(j_1)}\\\\overline{\\\\Psi }^{(j_0)}_0$ .\",\n \"We are also imprecise in this notation over whether it is this operator or its adjoint.\",\n \"Nevertheless the diagrams we use are clear on which vertices are integrated over and which have suprema, and this will avoid ambiguity.\",\n \"We then spatially translate the supremum to the far end and pull off the $\\\\overline{\\\\Psi }^{n-2}\\\\overline{\\\\Psi }_0$ diagram using a supremum bound.\",\n \"This looks like $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0};\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (3,-0.6) to [out=100,in=260] (3,0.6) to [out=190,in=350] (2,0.6);\\\\node at (2.5,0) {\\\\overline{\\\\Psi }};\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-2}\\\\overline{\\\\Psi }_0};(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\\\end{tikzpicture}}.$ For $\\\\overline{\\\\Psi }^{(j)}$ with $j=1,2,3$ , a simple supremum bound shows that this starting diagram can be bounded by $\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\left\\\\Vert \\\\overline{\\\\Psi }^{(j)}\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le \\\\overline{W_k}V_\\\\lambda ^2$ .\",\n \"For $j=4$ a little more care is needed: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };\\\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ We therefore find that $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\\\node at (3,0) {\\\\overline{\\\\Psi }^{n-1}\\\\overline{\\\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\overline{W_k}V_\\\\lambda ^{n-1}U_\\\\lambda ^2 &: n\\\\ge 3\\\\\\\\\\\\overline{W_k}V_\\\\lambda ^2U_\\\\lambda &: n=2.\\\\end{array}\\\\right.\",\n \"}$ Since there are 4 choices for the appended $\\\\overline{\\\\Psi }^{(j)}$ term, and $4^{n-2}\\\\times 2$ for the remaining $\\\\overline{\\\\Psi }^{n-2}\\\\overline{\\\\Psi }_0$ diagram, this case contributes $2\\\\times 4^{n-1}$ as a pre-factor.\",\n \"For Case REF , we shall find we have 3 sub-cases, depending not only on the $j$ -index of the displaced segment, but on which term we are looking at once it has been expanded.\",\n \"Sub-cases $\\\\mathrm {\\\\left(I\\\\right)}$ and $\\\\mathrm {\\\\left(II\\\\right)}$ deal with $j=1,2,3$ .\",\n \"In Appendix we expand out the various displaced segments using cosine-splitting and by expanding $\\\\tau _\\\\lambda ^\\\\circ $ edges into $\\\\tau _\\\\lambda $ edges and points, and label each of the resulting diagrams as sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ and $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"These sub-cases are distinguished by the strategy we use to bound them.\",\n \"Sub-case $\\\\mathrm {\\\\left(III\\\\right)}$ then addresses the case with $j=4$ .\",\n \"For sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ , we just use the $\\\\Psi $ structure of the diagrams.\",\n \"We first split off segments from the left of the displaced term like we did in Case REF .\",\n \"If $i=n-1$ we can then bound the remaining two pair of segments with $\\\\overline{W_k}V_\\\\lambda ^2$ (Appendix explains how this bound is found).\",\n \"This can be described diagrammatically as $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{n-2}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi _n};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le U_\\\\lambda V_\\\\lambda ^{n-2} \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\ \\\\le \\\\overline{W_k}U_\\\\lambda V_\\\\lambda ^{n}.$ If $i\\\\le n-2$ , then we also split off the segments to the right of this pair, and then bound the pair by $\\\\overline{W_k}V_\\\\lambda ^2$ again.\",\n \"Diagrammatically this looks like $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (6,-0.6) to [out=40,in=200] (7,0) to [out=160,in=320] (6,0.6) to [out=190,in=350] (4,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{i-1}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi };\\\\node at (5,0) {\\\\Psi _n\\\\Psi ^{n-i-2}};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(7,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le U_\\\\lambda ^2V_\\\\lambda ^{n-3} \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\ \\\\le \\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-1}.$ Note that the since $U_\\\\lambda \\\\ge V_\\\\lambda $ , the bound in (REF ) is greater than or equal to the bound in (REF ).\",\n \"It will be convenient to find an $i$ -independent bound.\",\n \"To this end, since the $i\\\\le n-2$ case only exists for $n\\\\ge 3$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ with ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-1}&: n\\\\ge 3\\\\\\\\\\\\overline{W_k}U_\\\\lambda V_\\\\lambda ^{2}&: n=2,\\\\end{array}\\\\right.\",\n \"}$ uniformly in $i\\\\in \\\\left\\\\lbrace 1,\\\\ldots ,n-1\\\\right\\\\rbrace $ .\",\n \"For sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ , we use a different strategy.\",\n \"First we split off the earlier (left) segments as we did for the first sub-case and in Case REF .\",\n \"However we arrange the later terms differently - more like we did in Case REF .\",\n \"We associate our displaced $\\\\Psi $ term with $\\\\tau _\\\\lambda $ and $\\\\tau _\\\\lambda ^\\\\circ $ edges from the subsequent term.\",\n \"If the subsequent term would be $\\\\Psi ^{(2)}_n$ , these are actually two $\\\\tau _\\\\lambda $ edges, but in our schematic diagrams we will draw a $\\\\tau _\\\\lambda ^\\\\circ $ edge and bear this special case in mind when it is relevant.\",\n \"While the displacement portrayed on the `top' of the segment could equally well be on the `bottom', the $\\\\tau _\\\\lambda ^\\\\circ $ edge will always be attached to the bottom in the usual orientation.\",\n \"Everything to the right of these $\\\\tau _\\\\lambda $ and $\\\\tau _\\\\lambda ^\\\\circ $ edges can then be described as a sequence of $\\\\overline{\\\\Psi }$ and $\\\\overline{\\\\Psi }_0$ terms.\",\n \"If $i=n-1$ , then we pair our augmented displaced segment with the subsequent $\\\\overline{\\\\Psi }_0$ segment.\",\n \"We then perform a spatial translation to make the right-most vertex a supremum vertex like we did in (REF ).\",\n \"The remaining diagram of a $\\\\Psi ^{(j)}$ segment and $\\\\overline{\\\\Psi }_0$ segment connected by a $\\\\tau _\\\\lambda ^\\\\circ $ and $\\\\tau _\\\\lambda $ edge can then be bounded by $\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)$ (see Appendix ).\",\n \"Recall that if $\\\\overline{\\\\Psi }_0$ is in fact $\\\\overline{\\\\Psi }^{(2)}_0$ then we have two $\\\\tau _\\\\lambda $ edges connecting the terms, not a $\\\\tau _\\\\lambda $ edge and a $\\\\tau _\\\\lambda ^\\\\circ $ edge.\",\n \"We need to note this to get the $\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)$ bound.\",\n \"The scheme is then represented diagrammatically as $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{n-2}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(6,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le U_\\\\lambda V_\\\\lambda ^{n-2} \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\\\end{tikzpicture}}\\\\\\\\ \\\\le \\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda V_\\\\lambda ^{n-2}.$ If $i\\\\le n-2$ , we also perform a spatial translation to move a supremum to the right-most vertex, but then also split off all $\\\\overline{\\\\Psi }$ and $\\\\overline{\\\\Psi }_0$ segments after the $\\\\overline{\\\\Psi }$ segment immediately following the displaced term.\",\n \"Once again the remaining pair can be bounded by $\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)$ (see Appendix ) and the split-off diagrams bounded using Lemma REF .\",\n \"The scheme can be represented diagrammatically as $&\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);[fill=gray!50] (5,0.6) to [out=280,in=80] (5,-0.6) to [out=10,in=170] (7,-0.6) to [out=40,in=200] (8,0) to [out=160,in=320] (7,0.6) to [out=190,in=350] (5,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (1,0) {\\\\Psi ^{i-1}\\\\Psi _0};\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }};\\\\node at (6,0) {\\\\overline{\\\\Psi }^{n-i-2}\\\\overline{\\\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(5,0.6) circle (2pt);(5,-0.6) circle (2pt);(8,0) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times {\\\\left\\\\lbrace \\\\begin{array}{ll}U_\\\\lambda ^2 V_\\\\lambda ^{n-4}&: i\\\\le n-3 \\\\\\\\U_\\\\lambda V_\\\\lambda ^{n-3}&: i=n-2\\\\end{array}\\\\right.}\",\n \"\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\le \\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right){\\\\left\\\\lbrace \\\\begin{array}{ll}U_\\\\lambda ^2 V_\\\\lambda ^{n-4}&: i\\\\le n-3 \\\\\\\\U_\\\\lambda V_\\\\lambda ^{n-3}&: i=n-2\\\\end{array}\\\\right.\",\n \"}$ Again for simplicity, we would like to get and $i$ -independent bound for such diagrams.\",\n \"Since the $i = n-2$ case only exists for $n\\\\ge 3$ , and the $i\\\\le n-3$ case only exists for $n\\\\ge 4$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ with ${\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda ^3V_\\\\lambda ^{n-4}&: n\\\\ge 4\\\\\\\\\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda ^2&: n=3\\\\\\\\\\\\left(\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda \\\\right)U_\\\\lambda &: n=2,\\\\end{array}\\\\right.\",\n \"}$ uniformly in $i\\\\in \\\\left\\\\lbrace 1,\\\\ldots ,n-1\\\\right\\\\rbrace $ .\",\n \"We now address sub-case $\\\\mathrm {\\\\left(III\\\\right)}$ , that is when the displacement crosses a $\\\\Psi ^{(4)}$ segment.\",\n \"This is the simplest sub-case because when we take a displacement across $\\\\Psi ^{(4)}$ in isolation, we get exactly $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad =W_k.$ We will be able to use the $\\\\Psi $ -arrangement of segments here.\",\n \"For $n\\\\ge 4$ we split off the earlier and later segments without pairing the displaced term with any of them.\",\n \"With Lemma REF this produces the bound $\\\\left(U_\\\\lambda V_\\\\lambda ^{i-1}\\\\right)W_k\\\\left(U_\\\\lambda V_\\\\lambda ^{n-i-1}\\\\right) = \\\\overline{W_k}U_\\\\lambda ^2V_\\\\lambda ^{n-2} \\\\qquad :n\\\\ge 4.$ For $n=3$ we split off two segments from one side and one from the other.\",\n \"This gives the finer bound $W_kU_\\\\lambda V_\\\\lambda ^2 \\\\quad :n=3.$ For $n=2$ , we use $j_0$ to indicate which $\\\\Psi _0$ term we have and $j_2$ to denote which $\\\\Psi _n$ term we have.\",\n \"We note that if $j_0=1,2$ then $\\\\left\\\\Vert \\\\Psi ^{(j_0)}_0\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le V_\\\\lambda $ , and if $j_2 = 1$ then $\\\\left\\\\Vert \\\\Psi ^{(j_2)}_n\\\\right\\\\Vert _{1,\\\\infty }\\\\le \\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le V_\\\\lambda $ .\",\n \"Therefore if $\\\\left(j_0,j_2\\\\right) \\\\ne \\\\left(3,2\\\\right)$ we have the bound $W_kU_\\\\lambda V_\\\\lambda $ .\",\n \"If $\\\\left(j_0,j_2\\\\right) = \\\\left(3,2\\\\right)$ we proceed by hand.\",\n \"We bound $\\\\varphi \\\\le \\\\tau _\\\\lambda $ and spatially shift one vertex to show $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\\\times };(1,0) circle (0pt) node[rotate=90, above]{\\\\sim };(0,0) circle (0pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\\\times };\\\\node [mark size=2pt] at (1,0.6) {{square*}};(1,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda .$ Carefully comparing the above bounds gives the required bounds for $n\\\\le 2$ .\",\n \"Including the multiplicity of each diagram (including that arising from cosine-splitting), we find that we can bound the displaced and subsequent non-displaced pair by $\\\\left(24\\\\overline{W_k}V_\\\\lambda ^2 + 2\\\\overline{H_k}\\\\right)$ .\",\n \"Note that this includes summing over the $j$ -index of the displaced term but not the index of the subsequent term.\",\n \"Since there are $3\\\\times 4^{n-2}\\\\times 2$ options for the $j$ -indices of the other segments, this case contributes this to the counting pre-factors in our result.\",\n \"[Proof of Proposition REF ] For $n=1$ , we can consider each of the six cases by hand.\",\n \"Most of the diagrams can be dealt with routinely using cosine-splitting, supremum bounds, and spatial shifts.\",\n \"We enumerate the diagrams by their $\\\\left(j_0,j_1\\\\right)$ values corresponding to the constituent segments.\",\n \"$(1,1):& \\\\quad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (0,-0.2) -- (1,-0.8) -- (2,-0.8) -- (3,-0.2);(1.5,-0.8) circle (0pt) node{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 3\\\\left(\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle _\\\\lambda + \\\\triangle _\\\\lambda \\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda W_k\\\\right)\\\\\\\\(1,2):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (0,0.2) -- (1,0.8) -- (2,0.2);(0.8,0.68) circle (0pt) node[rotate=30]{\\\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\triangle _\\\\lambda W_k\\\\right)\\\\\\\\(2,1):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (1,0.8) -- (2,0.8) -- (3,0.2);(1.8,0.8) circle (0pt) node{\\\\times };(1.5,-0.6) circle (0pt) node[above]{\\\\circ };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\triangle _\\\\lambda \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda + \\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda W_k\\\\right)\\\\\\\\(2,2):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (1,0.8) -- (2,0.2);(1.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\triangle _\\\\lambda W_k\\\\\\\\(3,1):& \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6) -- cycle;(1,0.6) -- (1,-0.6);[<->] (0,0.8) -- (1,0.8) -- (2,0.2);(0.8,0.8) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(2,0) circle (2pt);(0,-0.6) circle (2pt);(1,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\overline{W_k}\\\\triangle _\\\\lambda + \\\\lambda \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }\\\\triangle _\\\\lambda W_k+ \\\\triangle _\\\\lambda W_k\\\\right)$ These terms sum to no more than $13\\\\overline{W_k}V_\\\\lambda U_\\\\lambda .$ For the $\\\\left(3,2\\\\right)$ diagram, we use a different `trick.'\",\n \"In Lemma REF we will prove the bound $\\\\tau _\\\\lambda \\\\left(x,y\\\\right) \\\\le \\\\varphi (x,y) + \\\\lambda \\\\int \\\\tau _\\\\lambda (x,u)\\\\varphi (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right)$ .\",\n \"When applied to this diagram, this looks like $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad 2\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.6) circle (0pt) node[below]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (1.2,0.6) -- (1.2,-0.6);(1.2,0) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.6) circle (0pt) node[above]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right),$ Here we have also used the cosine-splitting result on the second part.\",\n \"For the first of these 4-vertex diagrams, we split the supremum in two and get the bound $\\\\lambda ^3\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_\\\\lambda _k\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\lambda \\\\triangle _\\\\lambda \\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"For the second of the 4-vertex diagrams, we also split the supremum in two and get the bound $\\\\lambda ^3\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"For the 3-vertex diagram we condition on whether two of the vertices are in each other's $B$ -set (defined in Assumption REF ).\",\n \"In the following diagrams [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\in B$; indicates that an indicator of the form ${1}_{\\\\left\\\\lbrace w \\\\in B\\\\left(u\\\\right)\\\\right\\\\rbrace }$ relates two variables, and [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\\\notin B$; indicates that an indicator of the form ${1}_{\\\\left\\\\lbrace w \\\\notin B\\\\left(u\\\\right)\\\\right\\\\rbrace }$ relates them: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,above]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &= \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0,0) circle (0pt) node[rotate=90,below]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.7) -- (0,-0.8) -- (1,-0.2);(-0.2,0) circle (0pt) node{\\\\times };(0,0) circle (0pt) node[rotate=90,below]{\\\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\ &\\\\le \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\times };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + 2\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node{\\\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad + \\\\quad \\\\raisebox {-20pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,-0.8) -- (1,-0.2);(0.5,-0.5) circle (0pt) node[rotate=30]{\\\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\\\scriptsize \\\\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right) \\\\nonumber \\\\\\\\&\\\\le \\\\left(\\\\quad \\\\raisebox {-7pt}{\\\\begin{tikzpicture}(0,0) -- (1,0);[<->] (0,0.2) -- (1,0.2);(0.5,0.2) circle (0pt) node{\\\\times };(0.5,0) circle (0pt) node[below]{\\\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node[above, rotate=90]{\\\\scriptsize \\\\notin B};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right) +4\\\\left(\\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);[<->] (0,-0.2) -- (1,-0.2);(0.5,-0.2) circle (0pt) node[below]{\\\\scriptsize \\\\in B};\\\\end{tikzpicture}}\\\\quad \\\\times \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\\\mathbf {\\\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\right)\\\\nonumber \\\\\\\\& \\\\le \\\\lambda \\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }\\\\triangle ^{(B)}_\\\\lambda + 4\\\\lambda \\\\mathbb {B}W_k.$\"\n ],\n [\n \"Bootstrap analysis\",\n \"In this section we will be assuming that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) = 1.$ By Lemma REF this also implies $\\\\mathbb {S}\\\\left(\\\\right)\\\\le 1$ .\",\n \"A re-scaling argument similar to [36] explains how this can be done while maintaining generality.\",\n \"First note that since $\\\\widehat{\\\\varphi }(0;a,b)>0$ for a positive $\\\\mathcal {P}$ -measure set of $a,b\\\\in \\\\mathcal {E}$ , we have $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)>0$ .\",\n \"Now suppose we scale $\\\\mathbb {R}^d$ by a factor of $q^{-1/d}$ (that is, the new unit radius ball is the previous ball of radius $q^{1/d}$ ), and leave the mark space $\\\\mathcal {E}$ unchanged.\",\n \"Then we find that the system we are left with has the distribution of an RCM model with the new parameters $\\\\lambda ^* = \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\eta \\\\cap \\\\left[0, q^{\\\\frac{1}{d}}\\\\right]^d }\\\\right] = \\\\lambda q, \\\\qquad \\\\varphi ^*\\\\left(\\\\overline{x};a,b\\\\right) = \\\\varphi \\\\left(q^{\\\\frac{1}{d}}\\\\overline{x};a,b \\\\right).$ Therefore for all $a,b\\\\in \\\\mathcal {E}$ we have $\\\\widehat{\\\\varphi }^*(\\\\overline{x};a,b) = \\\\int \\\\varphi ^*(\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}= q^{-1}\\\\int \\\\varphi ^*(\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}= q^{-1}\\\\widehat{\\\\varphi }(\\\\overline{x};a,b),$ and $\\\\mathbb {S}\\\\left(\\\\widehat{}^*(0)\\\\right) = q^{-1} \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"We can therefore achieve our assumption by choosing $q=\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"Furthermore, under this scaling we find that the old event $\\\\left\\\\lbrace \\\\left(\\\\overline{x},a\\\\right) \\\\longleftrightarrow \\\\left(\\\\overline{y},b\\\\right)\\\\textrm { in } \\\\xi ^{x,y}\\\\right\\\\rbrace $ becomes the new event $\\\\left\\\\lbrace \\\\left(q^{-1/d}\\\\overline{x},a\\\\right) \\\\longleftrightarrow \\\\left(q^{-1/d}\\\\overline{y},b\\\\right)\\\\textrm { in } \\\\xi ^{\\\\left(q^{-1/d}\\\\overline{x},a\\\\right),\\\\left(q^{-1/d}\\\\overline{y},b\\\\right)} \\\\right\\\\rbrace $ , and we get $\\\\tau ^*_{\\\\lambda ^*}(\\\\overline{x};a,b) = \\\\tau _\\\\lambda \\\\left(q^\\\\frac{1}{d}\\\\overline{x};a,b\\\\right)$ where $\\\\tau ^*_{\\\\lambda ^*}$ is the two-point function in the RCM governed by the connection function $\\\\varphi ^*$ and intensity $\\\\lambda ^*$ .\",\n \"It is also worth noting that the scaling choice to have $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)=1$ means that REF reduces to the condition that there exists a $d$ -independent constant $C>0$ such that $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &\\\\le C,\\\\\\\\\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } &\\\\le C \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ Note that the inequalities relating $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{2,\\\\infty }$ , and $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ follow from Jensen's inequality and a supremum bound on a probability space.\"\n ],\n [\n \"The Bootstrap Function\",\n \"To show that the bounds arising from the diagrams in Section are sufficient for our purposes, we make use of a bootstrap argument.\",\n \"Define $\\\\mu _\\\\lambda & = 1- \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)},\\\\\\\\\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) &= \\\\frac{1}{1 - \\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)},\\\\\\\\\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right) &=\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right)\\\\left(\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l-k\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l\\\\right) + \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l+k\\\\right) + \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l-k\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l+k\\\\right)\\\\right).$ From these we can then define the bootstrap function $f= f_1\\\\vee f_2\\\\vee f_3$ , where $f_1\\\\left(\\\\lambda \\\\right) = \\\\lambda , \\\\qquad f_2\\\\left(\\\\lambda \\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d} \\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}, \\\\qquad f_3\\\\left(\\\\lambda \\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k,l\\\\in \\\\mathbb {R}^d}\\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)}.$ Let us now outline some initial properties of $\\\\mu _\\\\lambda $ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ .\",\n \"Lemma 6.1 The map $\\\\lambda \\\\mapsto \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)$ is continuous and bounded below by $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ on $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"Therefore the map $\\\\lambda \\\\mapsto \\\\mu _\\\\lambda $ is a continuous map from $\\\\left[0,\\\\lambda _O\\\\right)\\\\rightarrow \\\\left[0,1\\\\right)$ .\",\n \"Furthermore, $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ is bounded below by $\\\\left(1+\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)^{-1}$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"We first recall that $\\\\lambda \\\\mapsto \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)$ is differentiable with respect to the operator norm from Corollary REF , and is therefore continuous.\",\n \"Lemma REF then implies that $\\\\lambda \\\\mapsto \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)$ is continuous.\",\n \"For the bounded below property, we use Lemma REF to get $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\ge \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) = 1$ .\",\n \"The continuity of $\\\\mu _\\\\lambda $ follows directly from the continuity and positivity of $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)$ .\",\n \"The bound $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\ge 1$ bounds $\\\\mu _\\\\lambda \\\\ge 0$ , and $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} < \\\\infty $ shows that $\\\\mu _\\\\lambda <1$ for all $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"The case $\\\\mu _0 = 0$ follows immediately from $\\\\mathcal {T}_0 = $ .\",\n \"Because $\\\\mu _\\\\lambda \\\\in \\\\left[0,1\\\\right)$ , to minimise $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ we want to find a lower bound for $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)$ .\",\n \"The $k$ -uniform bound on $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ then follows from $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) \\\\ge -\\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\rm op} \\\\ge -\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}$ .\"\n ],\n [\n \"Bounding the Lace Operator\",\n \"We have managed to bound the lace operator and its displacement in terms of a variety of objects.\",\n \"For the lace operator itself (and its associated Fourier transformed operators), these are: $\\\\lambda , \\\\triangle _\\\\lambda , \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }, \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }.$ As a guide, we expect $\\\\lambda $ , $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ , and $\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }$ to be bounded uniformly in $d$ and $\\\\lambda <\\\\lambda _O$ , whereas we expect $\\\\triangle _\\\\lambda $ to decay uniformly in $\\\\lambda <\\\\lambda _O$ as $d\\\\rightarrow \\\\infty $ .\",\n \"For some, these properties are easy to prove.\",\n \"The boundedness of $\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty }$ follows from $\\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le C,$ where we have used REF via (REF ).\",\n \"Regarding $\\\\lambda $ , since we are only interested in $\\\\lambda <\\\\lambda _O$ the question is perhaps more clearly written as “Is $\\\\lambda _O$ uniformly bounded in $d$ ?\",\n \"\\\".\",\n \"While there may be more elementary ways of proving this directly, we will get it for free from the bootstrap argument.\",\n \"Since we have the bound $\\\\lambda \\\\le f(\\\\lambda )$ , proving the uniform boundedness of $f(\\\\lambda )$ with respect to $d$ and $\\\\lambda $ for $\\\\lambda <\\\\lambda _O$ will prove the same for $\\\\lambda _O$ .\",\n \"This will leave us with only $\\\\triangle _\\\\lambda $ and $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ .\",\n \"Recall that we can write $\\\\triangle _\\\\lambda = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }, \\\\quad \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda = \\\\lambda ^2\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + 1.$ Then $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }$ differs from $\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda $ in that now the supremum is taken over all the marks, and adjacent $\\\\tau _\\\\lambda $ can have different marks.\",\n \"To get the desired decay and boundedness behaviour, we will therefore only need to prove that $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ decays and to bound $&\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}- \\\\overline{v};a_3,a_4\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{v};a_5,a_6\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{v},\\\\\\\\&\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}.$ We begin by proving the decay of $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"Lemma 6.2 Let $x,y \\\\in \\\\mathbb {X}$ .\",\n \"Then $\\\\tau _\\\\lambda (x,y) &\\\\le \\\\varphi (x,y) + \\\\lambda \\\\int \\\\varphi (x,u)\\\\tau _\\\\lambda (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right),\\\\\\\\\\\\tau _\\\\lambda (x,y) &\\\\le \\\\varphi (x,y) + \\\\lambda \\\\int \\\\tau _\\\\lambda (x,u)\\\\varphi (u,y)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ Then for $n\\\\ge 2$ , $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\sum _{m=0}^n\\\\genfrac(){0.0pt}0{n}{m}\\\\lambda ^{m} \\\\left\\\\Vert ^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ By combining Mecke's formula and the BK inequality, we obtain $\\\\tau _\\\\lambda (x,y) & \\\\le \\\\varphi (x,y) + \\\\mathbb {E}_\\\\lambda \\\\left[\\\\sum _{u \\\\in \\\\eta } 1_{\\\\lbrace x \\\\sim u \\\\text{ in } \\\\xi ^{x} \\\\rbrace \\\\cap \\\\lbrace u \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{y}\\\\rbrace } \\\\right] \\\\nonumber \\\\\\\\& = \\\\varphi (x,y) + \\\\lambda \\\\int \\\\mathbb {P}_\\\\lambda \\\\left(\\\\lbrace x \\\\sim u \\\\text{ in } \\\\xi ^{x,u} \\\\rbrace \\\\cap \\\\lbrace u \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{y,u}\\\\rbrace \\\\right) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right) \\\\nonumber \\\\\\\\& = \\\\varphi (x,y) +\\\\lambda \\\\int \\\\varphi (x,u) \\\\tau _\\\\lambda (u,y) \\\\nu \\\\left(\\\\mathrm {d}u\\\\right).$ In the last line, we have used that the two intersected events are independent.\",\n \"This is due to the fact that $\\\\nu $ is non-atomic and $x\\\\notin \\\\eta ^{y,u}$ a.s.\",\n \"The second inequality is then proven nearly identically, but with the adjacency to the intermediate point $u$ holding for $y$ instead of $x$ .\",\n \"To get (REF ), we first use () to expand out the right-most $\\\\mathcal {T}_\\\\lambda $ .\",\n \"This produces one term with $$ replacing the $\\\\mathcal {T}_\\\\lambda $ , and one term with $\\\\lambda \\\\mathcal {T}_\\\\lambda $ replacing it.\",\n \"We then repeatedly use (REF ) to expand the left-most $\\\\mathcal {T}_\\\\lambda $ to get a $$ factor and a $\\\\lambda \\\\mathcal {T}_\\\\lambda $ factor until we have a total of $n$ factors of $$ in the expression.\",\n \"Counting the number of ways to get to each possible end term then gives the binomial $\\\\genfrac(){0.0pt}1{n}{m}$ factor as required.\",\n \"The following lemma will allow us to describe $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ using Hilbert space language.\",\n \"Lemma 6.3 Let $H\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ be an integral linear operator with kernel function $h\\\\colon \\\\mathcal {E}^2\\\\rightarrow \\\\mathbb {R}$ .\",\n \"Then for all $h\\\\left(a,b\\\\right)\\\\in {\\\\rm ess.Im}\\\\left(h\\\\right)$ and $\\\\varepsilon >0$ , there exist $\\\\mathcal {P}$ -positive sets $E_1,E_2\\\\subset \\\\mathcal {E}$ such that $g_i(c) := \\\\frac{1}{\\\\mathcal {P}\\\\left(E_i\\\\right)}1_{E_i}\\\\left(c\\\\right)$ for $i=1,2$ satisfy $*{h\\\\left(a,b\\\\right) - {g_1}{Hg_2}} \\\\le \\\\varepsilon .$ Since $h$ is measurable, the set $A=\\\\left\\\\lbrace c,c^{\\\\prime }\\\\in \\\\mathcal {E}:*{h\\\\left(c,c^{\\\\prime }\\\\right)-h\\\\left(a,b\\\\right)}\\\\le \\\\varepsilon \\\\right\\\\rbrace $ is measurable.\",\n \"Since $h\\\\left(a,b\\\\right)\\\\in {\\\\rm ess.Im}\\\\left(h\\\\right)$ , it has positive measure: $\\\\mathcal {P}\\\\left(A\\\\right)>0$ .\",\n \"Furthermore, by considering the $\\\\pi $ -system of product sets $\\\\left\\\\lbrace E_1\\\\times E_2: E_1,E_2\\\\subset \\\\mathcal {E}\\\\text{ measurable}\\\\right\\\\rbrace $ that generates the $\\\\sigma $ -algebra on $\\\\mathcal {E}^2$ , we can find $B = E_1\\\\times E_2$ such that $E_1,E_2\\\\subset \\\\mathcal {E}$ are both measurable, $B\\\\subset A$ , and $\\\\mathcal {P}^{\\\\otimes 2}\\\\left(B\\\\right) = \\\\mathcal {P}\\\\left(E_1\\\\right)\\\\mathcal {P}\\\\left(E_2\\\\right)>0$ .\",\n \"Then use $E_1,E_2$ to define $g_1,g_2$ as in the statement on the lemma.\",\n \"Since $E_1,E_2$ are both measurable and have positive measure, both $g_1,g_2\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ .\",\n \"We also have ${g_1}{Hg_2} = \\\\frac{1}{\\\\mathcal {P}^{\\\\otimes 2}\\\\left(B\\\\right)}\\\\int _B h\\\\left(c,c^{\\\\prime }\\\\right)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime }\\\\right) \\\\in \\\\left[h(a,b) - \\\\varepsilon ,h(a,b) +\\\\varepsilon \\\\right],$ proving the result.\",\n \"Let us define the function $\\\\overline{\\\\epsilon }\\\\colon \\\\mathbb {N}\\\\rightarrow \\\\mathbb {R}_+$ , where $\\\\overline{\\\\epsilon }(d) := {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{1}{d} + \\\\frac{\\\\log d}{\\\\log g(d)} &: \\\\lim _{d\\\\rightarrow \\\\infty }g(d)\\\\rho ^{-d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^2 = 0 \\\\qquad \\\\forall \\\\rho >0,\\\\\\\\0&: \\\\text{otherwise}.\\\\end{array}\\\\right.\",\n \"}$ Recall $g(d)$ is the function defined in REF .\",\n \"In particular, $\\\\overline{\\\\epsilon }(d) = 0$ unless $g(d)$ approaches zero particularly quickly, and $\\\\beta (d) = g(d)^{\\\\frac{1}{4}-\\\\frac{3}{2}\\\\overline{\\\\epsilon }(d)}$ .\",\n \"Lemma 6.4 Let $\\\\lambda <\\\\lambda _O$ , $n\\\\ge 2$ , and $d>2m$ .\",\n \"Then there exists finite $c_f= c_f(m,n,f(\\\\lambda ))$ (increasing in $f(\\\\lambda )$ and independent of $d$ ) such that $\\\\left\\\\Vert ^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f&: n=2\\\\\\\\c_fg(d)^{\\\\frac{1}{2}-m\\\\overline{\\\\epsilon }(d)} &: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ The condition $\\\\lambda < \\\\lambda _O$ here is required to allow us to swap integrals over the mark space with integrals over the Fourier space.\",\n \"We are able to use Fubini's Theorem because sub-criticality implies that the relevant integrals are finite.\",\n \"For clarity, let us introduce the notation $\\\\varphi ^{(n)}(x,y) = \\\\int \\\\prod ^{n}_{j=1}\\\\varphi (u_{j-1},u_{j})\\\\nu ^{\\\\otimes (n-1)}\\\\left(\\\\mathrm {d}u_{[1,n-1]}\\\\right),\\\\quad \\\\tau _\\\\lambda ^{(n)}(x,y) = \\\\int \\\\prod ^{n}_{j=1}\\\\tau _\\\\lambda (u_{j-1},u_{j})\\\\nu ^{\\\\otimes (n-1)}\\\\left(\\\\mathrm {d}u_{[1,n-1]}\\\\right),$ where $u_0=x$ and $u_n=y$ .\",\n \"Furthermore, for each $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ let us define the integral linear operator $\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) \\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel function $(a,b)\\\\mapsto \\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) := \\\\int \\\\varphi ^{(n-1)}\\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\tau _\\\\lambda ^{(m)}\\\\left(\\\\overline{u}-\\\\overline{u}^{\\\\prime };c,c^{\\\\prime }\\\\right)\\\\varphi \\\\left(\\\\overline{u}^{\\\\prime };c^{\\\\prime },b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^{\\\\prime }\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime }\\\\right).$ Given $\\\\delta >0$ , $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ , and $a,b\\\\in \\\\mathcal {E}$ such that $\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};a,b\\\\right)\\\\in {\\\\rm ess.Im}\\\\left(\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};\\\\cdot ,\\\\cdot \\\\right)\\\\right)$ , we can use Lemma REF to get measurable and $\\\\mathcal {P}$ -positive sets $E^{(\\\\overline{x},a,b,\\\\delta )}_1,E^{(\\\\overline{x},a,b,\\\\delta )}_2\\\\subset \\\\mathcal {E}$ such that $\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) \\\\le *{g^{(\\\\overline{x},a,b,\\\\delta )}_1}{\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) g^{(\\\\overline{x},a,b,\\\\delta )}_2} + \\\\delta ,$ where $g^{(\\\\overline{x},a,b,\\\\delta )}_i(c) = \\\\frac{1}{\\\\mathcal {P}\\\\left(E^{(\\\\overline{x},a,b,\\\\delta )}_i\\\\right)}1_{E^{(\\\\overline{x},a,b,\\\\delta )}_i}\\\\left(c\\\\right)$ for $i=1,2$ .\",\n \"Now let $E_1,E_2\\\\subset \\\\mathcal {E}$ be arbitrary measurable $\\\\mathcal {P}$ -positive sets and $g_1,g_2$ be their associated functions.\",\n \"We aim to produce bounds that are independent of the choice of $E_1,E_2$ .\",\n \"We first use the Fourier inversion theorem to reformulate the position behaviour in terms of a $k$ -integral of the Fourier transforms, and use REF and $\\\\lambda <\\\\lambda _O$ (so the relevant integrals are finite) to allow us to swap the $k$ -integral and the mark integrals.\",\n \"This produces $*{*{g_1}{\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) g_2}} = *{\\\\int \\\\text{e}^{-i\\\\overline{x}\\\\cdot k}*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}}\\\\\\\\ \\\\le \\\\int *{*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}.$ Recall that $\\\\widehat{}(k)$ is self-adjoint, so we can move $\\\\widehat{}(k)^{n-1}$ to the other side of the inner product, and then we can use Cauchy-Schwarz and the definition of the operator norm to get $*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2} = *{\\\\widehat{}(k)^{n-1}g_1}{\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2} \\\\\\\\\\\\le *{\\\\widehat{}(k)^{n-1}g_1}_2*{\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2}_2 \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op}^m*{\\\\widehat{}(k)^{n-1}g_1}_2*{\\\\widehat{}(k)g_2}_2.$ We can then use the definition of the bootstrap function $f(\\\\lambda )$ to replace the $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op}$ factors with factors of $f(\\\\lambda )$ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)$ .\",\n \"Also writing the norm $*{\\\\cdot }_2$ as the square root of an inner product and using the self-adjoint property of $\\\\widehat{}(k)$ gives $*{g_1}{\\\\widehat{}(k)^{n-1}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)^m\\\\widehat{}(k)g_2} \\\\le f(\\\\lambda )^m\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}.$ We therefore want to derive bounds for the integral $\\\\int \\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}.$ We first derive pointwise and integral bounds for the $\\\\widehat{}$ factors.\",\n \"Given $N\\\\ge 1$ , and $g(c)=\\\\frac{1}{\\\\mathcal {P}\\\\left(E\\\\right)}1_E(c)$ for some measurable and $\\\\mathcal {P}$ -positive set $E\\\\subset \\\\mathcal {E}$ , $*{g}{\\\\widehat{}(k)^{2N}g}^\\\\frac{1}{2} = \\\\left(\\\\frac{1}{\\\\mathcal {P}(E)^2}\\\\int _E\\\\int _E\\\\widehat{\\\\varphi }^{(2N)}(k;c,c^{\\\\prime })\\\\mathcal {P}(\\\\mathrm {d}c)\\\\mathcal {P}(\\\\mathrm {d}c^{\\\\prime })\\\\right)^\\\\frac{1}{2} \\\\le \\\\left\\\\Vert \\\\widehat{}(k)^{2N}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2}.$ By using Cauchy-Schwarz and supremum bounds we can split this into norms of $\\\\widehat{}(k)$ only, then use Lemma REF to replace $k$ with 0, and finally use REF to bound these terms: $*{g}{\\\\widehat{}(k)^{2N}g}^\\\\frac{1}{2}\\\\le \\\\left\\\\Vert \\\\widehat{}(k)^{2N}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2} \\\\le \\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{2,\\\\infty }\\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty }^{N-1} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty }\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty }^{N-1} \\\\le C^N.$ For the integral bound, we apply Cauchy-Schwarz to the $k$ -integral and swap the $k$ -integrals with the mark integrals to un-do the Fourier transforms.\",\n \"We get $\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^\\\\frac{1}{2}\\\\left(\\\\int *{g_2}{\\\\widehat{}(k)^2g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^\\\\frac{1}{2} \\\\\\\\= *{g_1}{^{(2n-2)}\\\\left(\\\\overline{0}\\\\right)g_1}^\\\\frac{1}{2}*{g_2}{^{(2)}\\\\left(\\\\overline{0}\\\\right)g_2}^\\\\frac{1}{2},$ where $^{(N)}\\\\left(\\\\overline{0}\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right) \\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ is the integral operator with kernel function $\\\\left(a,b\\\\right)\\\\mapsto \\\\varphi ^{(N)}\\\\left(\\\\overline{0};a,b\\\\right)$ .\",\n \"Given $N\\\\ge 1$ , and $g(c)=\\\\frac{1}{\\\\mathcal {P}\\\\left(E\\\\right)}1_E(c)$ for some measurable and $\\\\mathcal {P}$ -positive set $E\\\\subset \\\\mathcal {E}$ , $*{g}{^{(2N)}\\\\left(\\\\overline{0}\\\\right)g} = \\\\left(\\\\frac{1}{\\\\mathcal {P}(E)^2}\\\\int _E\\\\int _E\\\\varphi ^{(2N)}\\\\left(\\\\overline{0};c,c^{\\\\prime }\\\\right)\\\\mathcal {P}(\\\\mathrm {d}c)\\\\mathcal {P}(\\\\mathrm {d}c^{\\\\prime })\\\\right) \\\\le \\\\left\\\\Vert ^{(2N)}\\\\left(\\\\overline{0}\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Here for $N\\\\ge 2$ we bound with $\\\\left\\\\Vert ^{(2N)}\\\\left(\\\\overline{0}\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2} \\\\le \\\\left\\\\Vert ^{2N}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^\\\\frac{1}{2} \\\\le g(d)^\\\\frac{1}{2}$ from REF .\",\n \"For $N=1$ we bound using $\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\in \\\\left[0,1\\\\right]$ and REF : $\\\\left\\\\Vert ^{(2)}\\\\left(\\\\overline{0}\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,c\\\\right)\\\\varphi \\\\left(-\\\\overline{x};c,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right) \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\\\\\= \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty }\\\\le C.$ Therefore $\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}C &: n=2,\\\\\\\\C^\\\\frac{1}{2}g(d)^\\\\frac{1}{2} &: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ We now return to bounding (REF ).\",\n \"To perform the $k$ -integral, we split the domain.\",\n \"We first consider $B_\\\\varepsilon (0) = \\\\lbrace k\\\\in \\\\mathbb {R}^d: *{k} < \\\\varepsilon \\\\rbrace $ , where we choose $\\\\varepsilon >0$ such that $\\\\varepsilon ^2 \\\\le \\\\tfrac{1-C_1}{C_2}$ - the constants $C_1,C_2$ coming from the assumption REF .\",\n \"From REF , we have $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k) = \\\\left(1 - \\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)^{-1}\\\\le 1/C_2*{k}^2$ on $B_\\\\varepsilon (0)$ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)\\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ on $B_\\\\varepsilon (0)^c$ .\",\n \"Our pointwise bound on the $\\\\widehat{}$ factors then gives $\\\\int _{B_\\\\varepsilon (0)}\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\\\\\\\\\le \\\\frac{C^n}{C^m_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\int ^\\\\varepsilon _0\\\\frac{1}{r^{2m}}r^{d-1}\\\\mathrm {d}r = \\\\frac{C^n}{C^m_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{d-2m}\\\\frac{\\\\varepsilon ^{d-2m}}{\\\\left(2\\\\pi \\\\right)^d},$ where $\\\\mathfrak {S}_{d-1}= d\\\\pi ^\\\\frac{d}{2}/\\\\Gamma \\\\left(\\\\tfrac{d}{2}+1\\\\right)$ is the surface area of an unit $d$ -sphere.\",\n \"Note that for all fixed $\\\\varepsilon >0$ this term vanishes in the $d\\\\rightarrow \\\\infty $ limit.\",\n \"To deal with the integral over $B_\\\\varepsilon (0)^c$ , we find the upper bound $\\\\int _{B_\\\\varepsilon (0)^c}\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^m*{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\\\\\\\\\le \\\\frac{1}{C^m_2\\\\varepsilon ^{2m}}\\\\int *{g_1}{\\\\widehat{}(k)^{2n-2}g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(k)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^d} \\\\le \\\\frac{C^\\\\frac{1}{2}}{C^m_2\\\\varepsilon ^{2m}}{\\\\left\\\\lbrace \\\\begin{array}{ll}C^\\\\frac{1}{2} &: n=2\\\\\\\\g(d)^\\\\frac{1}{2} &: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ Note that the bounds we found were independent of the sets $E_1,E_2$ , and therefore these bounds also apply to $*{g^{(\\\\overline{x},a,b,\\\\delta )}_1}{\\\\left(^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right)\\\\left(\\\\overline{x}\\\\right) g^{(\\\\overline{x},a,b,\\\\delta )}_2}$ uniformly over every $\\\\delta >0$ , $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ , and $\\\\mathcal {P}$ -almost every $a,b\\\\in \\\\mathcal {E}$ .\",\n \"Since we can take $\\\\delta \\\\rightarrow 0$ , our bound also applies to $\\\\left\\\\Vert ^{n-1}\\\\mathcal {T}_\\\\lambda ^m\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"For $n=2$ , the $B_\\\\varepsilon (0)^c$ integral dominates the $B_\\\\varepsilon (0)$ integral, and so the result is proven.\",\n \"For $n\\\\ge 3$ , both parts of the integral approach 0 as $d\\\\rightarrow \\\\infty $ .\",\n \"If both are of the same order or if the $B_\\\\varepsilon (0)^c$ integral dominates, then we can fix $\\\\varepsilon >0$ and get the result.\",\n \"On the other hand, if $g(d)^\\\\frac{1}{2} \\\\ll \\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^{d}$ for all fixed $\\\\varepsilon >0$ and the $B_\\\\varepsilon (0)$ integral always dominates, we can improve the overall bound by letting $\\\\varepsilon =\\\\varepsilon (d)$ and having $\\\\varepsilon (d)\\\\rightarrow 0$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Having a smaller value of $\\\\varepsilon $ produces a smaller bound for the $B_\\\\varepsilon (0)$ integral, but a larger bound for the $B_\\\\varepsilon (0)^c$ integral.\",\n \"To get an optimal $\\\\varepsilon $ we can set both terms to be of the same order - that is by having $\\\\varepsilon (d) = 2\\\\pi ^{\\\\frac{1}{2}}A^\\\\frac{1}{d}g(d)^\\\\frac{1}{2d}\\\\left(1-\\\\frac{2m}{d}\\\\right)^\\\\frac{1}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d}$ for some fixed $A>0$ .\",\n \"For this choice, we have both integrals of the order $\\\\varepsilon ^{-2m}g(d)^\\\\frac{1}{2} = 2^{-2m}\\\\pi ^{-m}A^{-\\\\frac{2m}{d}} g(d)^{\\\\frac{1}{2}-\\\\frac{m}{d}}\\\\left(1-\\\\frac{2m}{d}\\\\right)^{-\\\\frac{2m}{d}}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^{-\\\\frac{2m}{d}} \\\\\\\\= \\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^m g(d)^{\\\\frac{1}{2}-\\\\frac{m}{d}}\\\\frac{1}{d^m}\\\\left(1+o\\\\left(1\\\\right)\\\\right),$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Proposition 6.5 (Bound for the $n$ -gon diagrams) Let $\\\\lambda < \\\\lambda _O$ .\",\n \"Then for each $n$ such that $d>2n$ , there exists finite $c_f= c_f(m,n,f(\\\\lambda ))$ (increasing in $f(\\\\lambda )$ and independent of $d$ ) such that $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f&: n=1,2,\\\\\\\\c_fg(d)^{\\\\frac{1}{2}-n\\\\overline{\\\\epsilon }(d)}&: n\\\\ge 3.\\\\end{array}\\\\right.\",\n \"}$ The $n=1$ case is trivial because $\\\\tau _\\\\lambda \\\\left(\\\\overline{x},a,b\\\\right)\\\\in \\\\left[0,1\\\\right]$ .\",\n \"For $n=2$ , we first bound $\\\\left\\\\Vert ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"We get $\\\\left\\\\Vert ^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{x,y\\\\in \\\\mathbb {X}}\\\\int *{\\\\varphi (x,u)\\\\varphi (u,y)}\\\\nu (\\\\mathrm {d}u) \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\int *{\\\\varphi (x,u)}^2\\\\nu (\\\\mathrm {d}u) \\\\\\\\\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{x\\\\in \\\\mathbb {X}}\\\\int *{\\\\varphi (x,u)}\\\\nu (\\\\mathrm {d}u) = \\\\left\\\\Vert \\\\right\\\\Vert _{1,\\\\infty },$ by using Cauchy-Schwarz and symmetry of $\\\\varphi $ in the first inequality and $\\\\varphi (x,u)\\\\in \\\\left[0,1\\\\right]$ in the second.\",\n \"We then use REF to bound this by the constant $C$ .\",\n \"We use Lemma REF to get the required bound for the remaining terms arising from (REF ) in Lemma REF .\",\n \"For $n\\\\ge 3$ , we once again bound the terms in the relevant expansion in (REF ).\",\n \"For $\\\\left\\\\Vert ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ we use REF to get $\\\\left\\\\Vert ^n\\\\right\\\\Vert _{\\\\infty ,\\\\infty }= O\\\\left(g(d)^\\\\frac{1}{2}\\\\right)$ .\",\n \"We then use Lemma REF to bound the remaining terms.\",\n \"For our proofs of Lemma REF and Lemma REF , we adopt the following notation.\",\n \"Given $g(\\\\overline{x};a,b)$ and $h(\\\\overline{x};a,b)$ , we define $(gh)(\\\\overline{x};a,b) := \\\\int g(\\\\overline{x}-\\\\overline{y};a,c)h(\\\\overline{y};c,b)\\\\mathrm {d}\\\\overline{y}\\\\mathcal {P}(\\\\mathrm {d}c).$ This notational convention is associative and therefore can generalise to three or more terms unambiguously.\",\n \"Lemma 6.6 For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_6\\\\in \\\\mathcal {E}}&\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}- \\\\overline{v};a_3,a_4\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{v};a_5,a_6\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{v}\\\\le c_f,\\\\\\\\\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}&\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\le c_f,$ and therefore $\\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le 1 + 2c_f.$ By applying (REF ) and (), we can get $\\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\le \\\\varphi (\\\\overline{x};a,b) + \\\\lambda (\\\\varphi \\\\varphi )(\\\\overline{x};a,b)+ \\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi )(\\\\overline{x};a,b).$ We will in fact prove the more general result for the convolution of $m\\\\ge 2$ $\\\\tau _\\\\lambda $ -functions in dimensions $d>2m$ .\",\n \"By bounding $\\\\lambda \\\\le f(\\\\lambda )$ , and using Lemma REF and the Fourier inversion theorem we can realise that we only need to get further bounds for the following objects.\",\n \"For $\\\\vec{j}\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace ^m$ , we want to bound $\\\\int \\\\prod ^m_{i=1} P^{j_i}_{2i-1,2i}(k)\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}},$ where $P^{j}_{p,q}(k) := {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_p}{\\\\widehat{}(k) g_q}} &:j=1\\\\\\\\*{*{g_p}{\\\\widehat{}(k)^2g_q}} &:j=2\\\\\\\\*{*{g_p}{\\\\widehat{}(k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\widehat{}(k)g_q}} &: j=3,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\left\\\\lbrace g_i\\\\right\\\\rbrace _{i=1}^{2m}$ are the functions of the form $g_i(a) = \\\\frac{1}{\\\\mathcal {P}(E_i)}1_{E_i}(a)$ arising from the application of Lemma REF .\",\n \"We now bound the following terms in the same way as (REF ): $*{*{g_1}{\\\\widehat{}(k)^2g_2}} &\\\\le *{g_1}{\\\\widehat{}(k)^2g_1}^{\\\\frac{1}{2}}*{g_2}{\\\\widehat{}(k)^2g_2}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_1}{\\\\widehat{}(k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\widehat{}(k)g_2}} &\\\\le f\\\\left(\\\\lambda \\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)*{g_1}{\\\\widehat{}(k)^2g_1}^{\\\\frac{1}{2}}*{g_2}{\\\\widehat{}(k)^2g_2}^{\\\\frac{1}{2}}.$ In terms of pointwise uniform bounds for the $\\\\widehat{}$ terms, we have $*{*{g_1}{\\\\widehat{}(k)g_2}}\\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C$ and $*{g}{\\\\widehat{}(k)^2g}\\\\le C^2$ (the latter also having been used in the proof of Lemma REF ).\",\n \"As in Lemma REF , we will require various integral bounds for these $\\\\widehat{}$ terms.\",\n \"Suppose we have $r \\\\in \\\\left\\\\lbrace 1,\\\\ldots ,m-1\\\\right\\\\rbrace $ instances of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and $m-r$ instances of $P^{1}_{p,q}(k)$ .\",\n \"Then by applying Cauchy-Schwarz and the definitions of the $g_i$ we get $&\\\\int \\\\prod ^{r}_{i=1} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}^{\\\\frac{1}{2}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}^{\\\\frac{1}{2}} \\\\prod ^{m}_{i^{\\\\prime }=r+1}*{*{g_{2i^{\\\\prime }-1}}{\\\\widehat{}(k)g_{2i^{\\\\prime }}}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)^2g_1}*{g_2}{\\\\widehat{}(k)^2g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\int \\\\prod ^{r}_{i=2} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}\\\\prod ^{m}_{i^{\\\\prime }=r+1}*{g_{2i^{\\\\prime }-1}}{\\\\widehat{}(k)g_{2i^{\\\\prime }}}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_5,\\\\ldots ,a_{2m}\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_5,a_6)\\\\star \\\\ldots \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_{2r-1},a_{2r})\\\\right.\\\\right.\\\\nonumber \\\\\\\\&\\\\hspace{199.16928pt}\\\\left.\\\\left.\\\\star \\\\varphi (\\\\cdot ;a_{2r+1},a_{2r+2})\\\\star \\\\ldots \\\\star \\\\varphi (\\\\cdot ;a_{2m-1},a_{2m})\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le C^{m+r}.$ In this last inequality we have used $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\le C$ and $\\\\left\\\\Vert \\\\widehat{}(0)^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{2,\\\\infty }^2 \\\\le C^2$ to extract off $\\\\varphi $ and $\\\\varphi ^{(2)}$ via supremum bounds on the spatial position.\",\n \"We will also require integral bounds where there are $m$ factors of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and where there are $m$ factors of $P^{1}_{p,q}(k)$ : $&\\\\int \\\\prod ^{m}_{i=1} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}^{\\\\frac{1}{2}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}^{\\\\frac{1}{2}} \\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)^2g_1}*{g_2}{\\\\widehat{}(k)^2g_2}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}}\\\\left(\\\\int \\\\prod ^{m}_{i=2} *{g_{2i-1}}{\\\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\\\widehat{}(k)^2g_{2i}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_5,\\\\ldots ,a_{2m}\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_5,a_6)\\\\star \\\\ldots \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_{2m-1},a_{2m})\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le C^{2m}.\\\\\\\\&\\\\int \\\\prod ^{m}_{i=1}*{*{g_{2i-1}}{\\\\widehat{}(k)g_{2i}}}\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(k)g_2}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\left(\\\\prod ^{m}_{i=2}*{g_{2i-1}}{\\\\widehat{}(k)g_{2i}}^2\\\\frac{\\\\mathrm {d}k}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{170.71652pt}\\\\times \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a_5,\\\\ldots ,a_{2m}\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_5,a_6)\\\\star \\\\ldots \\\\star \\\\varphi (\\\\cdot ;a_{2m-1},a_{2m})\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\right)^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt} \\\\le C^{m}.$ Note that for $m\\\\ge 3$ , better bounds are available via REF , but will not be required here.\",\n \"We can now proceed to bound each of the expressions of the form (REF ) in much the same way as we did in the proof of Lemma REF .\",\n \"We set $\\\\varepsilon >0$ and partition the integral over $\\\\mathbb {R}^d$ into one over $B_\\\\varepsilon (0)$ and one over $B_\\\\varepsilon (0)^c$ .\",\n \"For the $B_\\\\varepsilon (0)$ integral we use the uniform bounds on the $\\\\widehat{}$ terms and perform the integral of the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ terms (if there are any) as before.\",\n \"Note that we require $d>6$ for all of these to be finite.\",\n \"For the $B_\\\\varepsilon (0)^c$ integrals we uniformly bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ and use the integral bounds we calculated above.\",\n \"Since we are only asking for a constant bound, we don't need to worry about having $\\\\varepsilon \\\\rightarrow 0$ as $d\\\\rightarrow \\\\infty $ .\"\n ],\n [\n \"Bounding the Lace Operator Displacement\",\n \"We now consider the objects required to bound the displacement of the lace operator.\",\n \"In addition to the objects used to bound the lace operator itself, we require bounds on $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }, \\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }, \\\\overline{W_k}, \\\\overline{H_k}, \\\\triangle ^{(B)}_\\\\lambda , \\\\mathbb {B}.$ The decay $\\\\mathbb {B}\\\\le g(d)$ follows from Assumption REF , and the following Observation REF demonstrates the decay of $\\\\triangle ^{(B)}_\\\\lambda $ .\",\n \"Observation 6.7 Note that the inequality (REF ) allows us to bound $\\\\triangle ^{(B)}_\\\\lambda $ in terms of other terms.\",\n \"By using the inequality twice and appropriately bounding $\\\\varphi $ with $\\\\tau _\\\\lambda $ , we get $\\\\int \\\\tau _\\\\lambda (r,u)\\\\tau _\\\\lambda (u,s)\\\\nu (\\\\mathrm {d}u) \\\\le \\\\int \\\\varphi (r,u)\\\\varphi (u,s)\\\\nu (\\\\mathrm {d}u) + 2\\\\lambda \\\\int \\\\tau _\\\\lambda (r,v)\\\\tau _\\\\lambda (v,u)\\\\tau _\\\\lambda (u,s)\\\\nu ^{\\\\otimes 2}(\\\\mathrm {d}u, \\\\mathrm {d}v)$ for any $r,s\\\\in \\\\mathbb {X}$ .\",\n \"Therefore $\\\\triangle ^{(B)}_\\\\lambda \\\\le 3\\\\triangle _\\\\lambda + \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{r\\\\notin B(s)}\\\\int \\\\varphi (r,u)\\\\varphi (u,s)\\\\nu (\\\\mathrm {d}u) \\\\le 3\\\\triangle _\\\\lambda + \\\\lambda g(d),$ where we have used REF .\",\n \"Therefore Proposition REF implies that for $d>6$ there exists $c_f$ such that $\\\\triangle ^{(B)}_\\\\lambda \\\\le c_f\\\\beta ^2.$ We are then left with $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }$ , $\\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ , $\\\\overline{W_k}$ , and $\\\\overline{H_k}$ , and we will deal them in this order.\",\n \"First note that $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{y\\\\in \\\\mathbb {X}}\\\\int \\\\varphi _k(x,y)\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\int (1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right))\\\\varphi (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}(\\\\mathrm {d}a) \\\\\\\\ = \\\\left\\\\Vert \\\\widehat{}_k(0)\\\\right\\\\Vert _{1,\\\\infty } = \\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty }.$ The following lemma therefore also allows us to bound $\\\\left\\\\Vert _k\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"Its extra generality will be required later.\",\n \"Lemma 6.8 For all $k,l\\\\in \\\\mathbb {R}^d$ , $\\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right],$ where $C>0$ is the $d$ -independent constant given in REF .\",\n \"The first two inequalities in (REF ) holds from the general inequalities that holds for these norms (on a probability space for the second inequality).\",\n \"Then we can remove the $l$ -dependence by performing the following calculation: $\\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+ k\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int *{\\\\text{e}^{il\\\\cdot \\\\overline{x}}}*{\\\\left(1-\\\\cos \\\\left( k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)}\\\\mathrm {d}\\\\overline{x}\\\\\\\\= \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int *{\\\\left(1-\\\\cos \\\\left(k\\\\cdot \\\\overline{x}\\\\right)\\\\right)\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)}\\\\mathrm {d}\\\\overline{x}= \\\\left\\\\Vert \\\\widehat{}\\\\left(0\\\\right) - \\\\widehat{}\\\\left( k\\\\right)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ The final step relating the $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ -norm to the difference of the spectral suprema is precisely that given by () in REF .\",\n \"We now address $\\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda ,k}\\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"Lemma 6.9 For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\le c_f\\\\beta ^2\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ First note that for $k=0$ the kernel function $\\\\tau _{\\\\lambda ,k}\\\\left(x,y\\\\right)=0$ for all $x,y\\\\in \\\\mathbb {X}$ , and so the required inequality holds trivially.\",\n \"In this proof we hereafter may assume $k\\\\ne 0$ .\",\n \"We use (REF ) to write $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\left\\\\Vert ^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty } + \\\\lambda \\\\left\\\\Vert ^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ We first get the bound on $\\\\left\\\\Vert ^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ , using a similar approach to Lemma REF .\",\n \"For each $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ we define the integral linear operator $\\\\left(^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel function $\\\\left(a,b\\\\right)\\\\mapsto \\\\left(^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) \\\\\\\\:= \\\\int \\\\varphi ^{(2)}\\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\tau _\\\\lambda \\\\left(\\\\overline{u}-\\\\overline{u}^{\\\\prime };c,c^{\\\\prime }\\\\right)\\\\tau _{\\\\lambda ,k}\\\\left(\\\\overline{u}^{\\\\prime }-\\\\overline{u}^{\\\\prime \\\\prime };c^{\\\\prime },c^{\\\\prime \\\\prime }\\\\right)\\\\varphi \\\\left(\\\\overline{u}^{\\\\prime \\\\prime };c^{\\\\prime \\\\prime },b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^{\\\\prime } \\\\mathrm {d}\\\\overline{u}^{\\\\prime \\\\prime }\\\\mathcal {P}^{\\\\otimes 3}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime },\\\\mathrm {d}c^{\\\\prime \\\\prime }\\\\right).$ Recall the notation $\\\\varphi ^{(2)}$ used in (REF ).\",\n \"As in the proof of Lemma REF , we let $E_1,E_2\\\\subset \\\\mathcal {E}$ be measurable and $\\\\mathcal {P}$ -positive sets, and aim to bound ${g_1}{\\\\left(\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2}$ (where $g_i(c) := \\\\frac{1}{\\\\mathcal {P}\\\\left(E_i\\\\right)}1_{E_i}\\\\left(c\\\\right)$ for $i=1,2$ ) independently of the choice of $E_1,E_2$ .\",\n \"If we do this we will have proven the result.\",\n \"As before, we use the Fourier inversion theorem to write ${g_1}{\\\\left(\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2}$ as the integral of an inner product of Fourier transformed operators over the Fourier argument.\",\n \"Having $\\\\lambda <\\\\lambda _O$ allows us to swap the integrals in this step.\",\n \"Then we can use Cauchy-Schwarz and the definitions of the operator norm and the bootstrap functions to extract factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)$ and $\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)$ from the inner product.\",\n \"The net result is $*{g_1}{\\\\left(^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2} &\\\\le \\\\int *{*{g_1}{\\\\widehat{}(l)^{2}\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l)g_2}}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\&\\\\le f(\\\\lambda )^2\\\\int \\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right) *{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ The $\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)$ term produces the required factor of $\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\\\int \\\\left(2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)\\\\right)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ We partition our integral using open $\\\\varepsilon $ -balls around the poles where $\\\\varepsilon ^2 \\\\le \\\\tfrac{1-C_1}{C_2}$ - the constants $C_1$ and $C_2$ coming from REF .\",\n \"Let $B_{\\\\varepsilon }(p)$ denote the open $\\\\varepsilon $ -ball around $p\\\\in \\\\mathbb {R}^d$ .\",\n \"For $n\\\\in \\\\left\\\\lbrace 0,1,2,3\\\\right\\\\rbrace $ , let $A_n := \\\\left\\\\lbrace l\\\\in \\\\mathbb {R}^d: l\\\\in B_\\\\varepsilon (p) \\\\text{ for precisely $n$ elements }p\\\\in \\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace \\\\right\\\\rbrace .$ Firstly it is possible for $A_3,A_2 = \\\\emptyset $ if $*{k}$ is sufficiently large compared to $\\\\varepsilon $ .\",\n \"It is also easy to see that $A_3\\\\subset B_{\\\\varepsilon }(0)$ and $A_2\\\\subset B_{2\\\\varepsilon }(0)$ .\",\n \"We will partition $\\\\mathbb {R}^d$ into $A_3$ , $A_2$ , $A_1$ , and $A_0$ and bound the $l$ -integral on each part.\",\n \"Recall that REF implies that $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le C^{-1}_2*{l}^{-2}$ for $*{l}\\\\le \\\\varepsilon $ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le C^{-1}_2\\\\varepsilon ^2$ for $*{l}\\\\ge \\\\varepsilon $ .\",\n \"Also recall the pointwise (REF ) and integral (REF ) bounds we derived for the $\\\\widehat{}$ factors.\",\n \"We treat $A_3$ first: $&\\\\int _{A_3}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\le \\\\frac{C^3}{C^3_2}\\\\int _{B_\\\\varepsilon (0)\\\\cap B_\\\\varepsilon (k)}\\\\frac{1}{*{l}^4*{l-k}^2} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{2}{3}}\\\\left(\\\\int _{B_\\\\varepsilon (k)}\\\\frac{1}{*{l-k}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}} \\\\le \\\\frac{C^3}{C^3_2}\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}\\\\\\\\&\\\\int _{A_3}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\le \\\\frac{C^3}{C^3_2}\\\\int _{A_3}\\\\frac{1}{*{l+k}^2*{l}^2*{l-k}^2} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\le \\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (-k)}\\\\frac{1}{*{l+k}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}}\\\\left(\\\\int _{B_\\\\varepsilon (k)}\\\\frac{1}{*{l-k}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{1}{3}} \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\le \\\\frac{C^3}{C^3_2}\\\\int _{B_\\\\varepsilon (0)}\\\\frac{1}{*{l}^6} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}$ Recall $\\\\mathfrak {S}_{d-1}= d\\\\pi ^\\\\frac{d}{2}/\\\\Gamma \\\\left(\\\\tfrac{d}{2}+1\\\\right)$ is the surface area of an unit $d$ -sphere.\",\n \"In these calculations we used Hölder's inequality and increased the domain of integration to get upper bounds.\",\n \"These integrals are finite for $d>6$ , and approach 0 if we take bounded $\\\\varepsilon $ .\",\n \"For $A_2$ the counting of cases is the only extra complication.\",\n \"A precise counting is entirely possible, but unnecessary for our purposes.\",\n \"It is simple to see that the arguments outlined for $A_3$ can be applied to get the bound as an integer multiple of $\\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\left(\\\\frac{1}{*{l}^6} + \\\\frac{1}{\\\\varepsilon ^2*{l}^4} + \\\\frac{1}{\\\\varepsilon ^4*{l}^2} \\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right) = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4} + \\\\frac{1}{d-2}\\\\right)\\\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a pair of two distinct points from $\\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace $ , there are terms with three, two, and one factor(s) of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ centred on these two points.\",\n \"This bound is finite for $d>6$ , and approaches 0 if we take bounded $\\\\varepsilon $ .\",\n \"Repeating this for $A_1$ , we get the bound as an integer multiple of $\\\\frac{C^3}{C^3_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\left(\\\\frac{1}{\\\\varepsilon ^2*{l}^4} + \\\\frac{1}{\\\\varepsilon ^4*{l}^2} + \\\\frac{1}{\\\\varepsilon ^6} \\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\right) = \\\\frac{C^3}{C^3_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\left(\\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a single point from $\\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace $ , there are terms with two, one, and zero factor(s) of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ centred on this single point.\",\n \"This bound is finite for $d>4$ , and approaches 0 if we take bounded $\\\\varepsilon $ .\",\n \"The $A_0$ integral is qualitatively different.\",\n \"By taking the integral bound for the $\\\\widehat{}$ factors and a uniform (on $A_0$ ) bound for the factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ , we get the bound $\\\\frac{C^3}{C^3_2}\\\\varepsilon ^{-6}g(d)^\\\\frac{1}{2}.$ As in Lemma REF , if the $g(d)^\\\\frac{1}{2}$ term dominates, then we have a sufficient bound for our result.\",\n \"The largest of the other terms (for $d$ sufficiently large and $\\\\varepsilon \\\\le 1$ ) is the $\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}$ term.\",\n \"Like we did in Lemma REF , we can make these other terms smaller by decreasing $\\\\varepsilon $ at the cost of making the $g(d)^\\\\frac{1}{2}$ bound greater.\",\n \"The largest term can be minimised (up to a constant) by taking $\\\\varepsilon (d) = 2\\\\pi ^{\\\\frac{1}{2}}A^\\\\frac{1}{d}g(d)^\\\\frac{1}{2d}\\\\left(1-\\\\frac{6}{d}\\\\right)^\\\\frac{1}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d}$ for some fixed $A>0$ .\",\n \"For this choice, we have both integrals of the order $\\\\varepsilon ^{-6}g(d)^\\\\frac{1}{2} = 2^{-6}\\\\pi ^{-3}A^{-\\\\frac{6}{d}}g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}\\\\left(1-\\\\frac{6}{d}\\\\right)^{-\\\\frac{6}{d}}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^{-\\\\frac{6}{d}} = \\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^3 g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}\\\\frac{1}{d^3}\\\\left(1+o\\\\left(1\\\\right)\\\\right),$ as $d\\\\rightarrow \\\\infty $ .\",\n \"Bounding the $\\\\left\\\\Vert ^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ term is similar.\",\n \"For each $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ we define the integral linear operator $\\\\left(^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as the integral operator with kernel function $\\\\left(a,b\\\\right)\\\\mapsto \\\\left(^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x};a,b\\\\right) \\\\\\\\:= \\\\int \\\\varphi ^{(2)}\\\\left(\\\\overline{x}-\\\\overline{u};a,c\\\\right)\\\\tau _{\\\\lambda ,k}\\\\left(\\\\overline{u}-\\\\overline{u}^{\\\\prime };c,c^{\\\\prime \\\\prime }\\\\right)\\\\varphi \\\\left(\\\\overline{u}^{\\\\prime };c^{\\\\prime },b\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\mathrm {d}\\\\overline{u}^{\\\\prime } \\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}c,\\\\mathrm {d}c^{\\\\prime }\\\\right).$ We once again take $\\\\mathcal {P}$ -positive sets $E_1,E_2$ and corresponding $g_1,g_2$ .\",\n \"Then we use the Fourier inversion theorem and Cauchy-Schwarz to get $*{g_1}{\\\\left(^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right)\\\\left(\\\\overline{x}\\\\right)g_2} \\\\le f(\\\\lambda )\\\\int \\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right) *{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ Then once again the $\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)$ term produces the required factor of $\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\\\int \\\\left(2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)\\\\right)*{g_1}{\\\\widehat{}(l)^4g_1}^\\\\frac{1}{2}*{g_2}{\\\\widehat{}(l)^2g_2}^\\\\frac{1}{2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}.$ Counting the number of factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ centred on each of the poles $\\\\left\\\\lbrace k,0,-k\\\\right\\\\rbrace $ then tells us that there is a uniform bound that is some integer multiple of $\\\\frac{C^2}{C^2_2}\\\\left(\\\\int _{B_\\\\varepsilon (0)}\\\\left(\\\\frac{1}{*{l}^{4}} + \\\\frac{1}{\\\\varepsilon ^2*{l}^{2}} + \\\\frac{1}{\\\\varepsilon ^4}\\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} + \\\\varepsilon ^{-4}g(d)^\\\\frac{1}{2}\\\\right) \\\\\\\\= \\\\frac{C^2}{C^2_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\left(\\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-4} + \\\\frac{C^2}{C^2_2}\\\\varepsilon ^{-4}g(d)^\\\\frac{1}{2}.$ Once again, if the $g(d)^\\\\frac{1}{2}$ term dominates the result is proven.\",\n \"Otherwise, we let $\\\\varepsilon $ vary with $d$ .\",\n \"The largest term can be minimised (up to a constant) by taking $\\\\varepsilon (d) = 2\\\\pi ^{\\\\frac{1}{2}}A^\\\\frac{1}{d}g(d)^\\\\frac{1}{2d}\\\\left(1-\\\\frac{4}{d}\\\\right)^\\\\frac{1}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{1}{d}$ for some fixed $A>0$ .\",\n \"For this choice, we have both integrals of the order $\\\\varepsilon ^{-4}\\\\beta ^2 = 2^{-4}\\\\pi ^{-2}A^{-\\\\frac{4}{d}} g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}}\\\\left(1-\\\\frac{4}{d}\\\\right)^{-\\\\frac{4}{d}}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^{-\\\\frac{4}{d}} = \\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^2 g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}}\\\\frac{1}{d^2}\\\\left(1+o\\\\left(1\\\\right)\\\\right),$ as $d\\\\rightarrow \\\\infty $ .\",\n \"For large $d$ , this is dominated by $g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}d^{-3}$ and therefore $\\\\left\\\\Vert ^2\\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ is dominated by $\\\\left\\\\Vert ^2\\\\mathcal {T}_\\\\lambda \\\\mathcal {T}_{\\\\lambda ,k}\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ and we prove the result.\",\n \"The following lemma allows us to deal with occurrences of $\\\\overline{W_k}$ .\",\n \"Lemma 6.10 Let $x,y \\\\in \\\\mathbb {X}$ .\",\n \"Then $\\\\tau _{\\\\lambda ,k}(x,y) \\\\le \\\\varphi _k(x,y) + 2\\\\lambda \\\\int \\\\left(\\\\tau _\\\\lambda (x,u)\\\\varphi _k(u,y) + \\\\tau _{\\\\lambda ,k}(x,u)\\\\varphi (u,y)\\\\right)\\\\nu \\\\left(\\\\mathrm {d}u\\\\right).\",\n \"$ For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\overline{W_k}\\\\le c_f\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ The first inequality follows by applying the cosine-splitting result (Lemma REF ) to ().\",\n \"For the $\\\\tau _\\\\lambda $ term in $\\\\overline{W_k}$ we apply (REF ) and () to get $\\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\le \\\\varphi (\\\\overline{x};a,b) + \\\\lambda (\\\\varphi \\\\varphi )(\\\\overline{x};a,b)+ \\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi )(\\\\overline{x};a,b).$ Similarly, with the additional use of (REF ), we get $\\\\tau _{\\\\lambda ,k}(\\\\overline{x};a,b) \\\\le \\\\varphi _k(\\\\overline{x};a,b) + 2\\\\lambda \\\\left( (\\\\varphi _k\\\\varphi )(\\\\overline{x};a,b) +(\\\\varphi \\\\varphi _k)(\\\\overline{x};a,b)\\\\right)+ 4\\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi _k)(\\\\overline{x};a,b) \\\\\\\\ + 4\\\\lambda ^2(\\\\varphi \\\\tau _{\\\\lambda ,k}\\\\varphi )(\\\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\\\lambda \\\\le f(\\\\lambda )$ , we can realise that we can bound $\\\\overline{W_k}$ by bounding the following objects.\",\n \"For $\\\\vec{j}\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace \\\\times \\\\left\\\\lbrace 1,2,3,4,5\\\\right\\\\rbrace $ , define $\\\\mathcal {W}(k;\\\\vec{j}) := \\\\int P^{j_1}_{1,2}(l)Q^{j_2}_{3,4}(l;k)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}},$ where $P^{j}_{1,2}(l) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_1}{\\\\widehat{}(l) g_2}} &:j=1\\\\\\\\*{*{g_1}{\\\\widehat{}(l)^2g_2}} &:j=2\\\\\\\\*{*{g_1}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}(l)g_2}} &: j=3,\\\\end{array}\\\\right.\",\n \"}\\\\\\\\Q^{j}_{3,4}(l;k) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_{3}}{\\\\widehat{}_k(l) g_{4}}} &:j=1\\\\\\\\*{*{g_{3}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{4}}} &:j=2\\\\\\\\*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{}_k(l) g_{4}}} &:j=3\\\\\\\\*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{4}}} &:j=4\\\\\\\\*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{4}}} &:j=5,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\left\\\\lbrace g_m\\\\right\\\\rbrace _{m=1}^{4}$ are the functions of the form $g_m(a) = \\\\frac{1}{\\\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .\",\n \"For the $P^j_{1,2}(l)$ factor we can get bounds as we did in the proof of Lemma REF .\",\n \"For $Q^{j}_{3,4}(l;k)$ we need to do a bit more processing.\",\n \"For $j=1$ , $*{*{g_{3}}{\\\\widehat{}_k(l) g_{4}}} \\\\le \\\\frac{1}{\\\\mathcal {P}(E_{3})\\\\mathcal {P}(E_{4})}\\\\int _{E_{3}\\\\times E_{4}}*{\\\\widehat{\\\\varphi }_k(l;a,b)}\\\\mathcal {P}^{\\\\otimes 2}(\\\\mathrm {d}a, \\\\mathrm {d}b) \\\\le \\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\\\\\\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right),$ where in the last inequality we used Lemma REF .\",\n \"For $j=2$ , we use Cauchy-Schwarz to get $*{*{g_{3}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{4}}} \\\\le *{g_3}{\\\\widehat{}_k(l)^2 g_3}^{\\\\frac{1}{2}} *{g_4}{\\\\widehat{}(l)^2 g_4}^{\\\\frac{1}{2}}\\\\le \\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }*{g_4}{\\\\widehat{}(l)^2 g_4}^{\\\\frac{1}{2}} \\\\\\\\\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)*{g_4}{\\\\widehat{}(l)^2 g_4}^{\\\\frac{1}{2}}.$ An identical argument gives the same bound for $j=3$ .\",\n \"For $j=4$ , we can use the self-adjointness of $\\\\widehat{}(l)$ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)$ and the Cauchy-Schwarz inequality to get $*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{4}}} &\\\\le *{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)^2\\\\widehat{}(l)g_{3}}^{\\\\frac{1}{2}} *{g_{4}}{\\\\widehat{}_k(l)^2g_{4}}^{\\\\frac{1}{2}} \\\\nonumber \\\\\\\\&\\\\le f(\\\\lambda )\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)*{g_{3}}{\\\\widehat{}(l)^2g_{3}}^{\\\\frac{1}{2}}\\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\nonumber \\\\\\\\&\\\\le Cf(\\\\lambda )\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)*{g_{3}}{\\\\widehat{}(l)^2g_{3}}^{\\\\frac{1}{2}},$ and similarly $*{*{g_{3}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{4}}} \\\\le f(\\\\lambda )\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right)*{g_3}{\\\\widehat{}(l)^2g_3}^\\\\frac{1}{2}*{g_4}{\\\\widehat{}(l)^2g_4}^\\\\frac{1}{2} \\\\\\\\ \\\\le Cf(\\\\lambda )\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)*{g_3}{\\\\widehat{}(l)^2g_3}^\\\\frac{1}{2}*{g_4}{\\\\widehat{}(l)^2g_4}^\\\\frac{1}{2},$ where we write for convenience $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l_3;k) = \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k).$ Note that all the bounds for the $Q^{j}_{3,4}(l)$ terms have a $\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ factor.\",\n \"So we only need to prove that the remaining $l$ -integral of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ and $\\\\widehat{}$ factors is bounded.\",\n \"As in our previous proofs, we partition $\\\\mathbb {R}^d$ into the sets $A_n$ defined earlier in (REF ).\",\n \"For the $A_1$ , $A_2$ , and $A_3$ parts we then use the bounds $C$ and $C^2$ for the various $\\\\widehat{}$ factors and $1/\\\\left(C_2*{l}^2\\\\right)$ to bound the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors.\",\n \"By using the techniques used previously in this section, we can show that these contributions is bounded by some $\\\\varepsilon $ and $d$ (only) dependent constant.\",\n \"Since we don't need to show decay, we don't need to worry about taking $\\\\varepsilon $ to 0.\",\n \"For the $A_0$ contribution, we bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors by $1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ and show that the integral of the $\\\\widehat{}$ factors over all of $\\\\mathbb {R}^d$ is bounded.\",\n \"For $j_2\\\\ne 1$ this proceeds similarly to the argument in the proof of Lemma REF .\",\n \"For $j_2=1$ we need to take a step back.\",\n \"These versions of (REF ) arise from trying to bound one particular type of term, which we now treat separately.\",\n \"We bound $\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\tau _\\\\lambda \\\\left(\\\\overline{x}-\\\\overline{u};a_1,a_2\\\\right)\\\\varphi _k\\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_3,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\varphi _k\\\\left(\\\\overline{u};a_3,a_4\\\\right)\\\\mathrm {d}\\\\overline{u}\\\\le \\\\left\\\\Vert \\\\widehat{}_k(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Then Lemma REF gives the result.\",\n \"We are now only left with bounding $\\\\overline{H_k}$ .\",\n \"lemmaMartiniBound For $\\\\lambda <\\\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\\\overline{H_k}\\\\le c_f\\\\beta ^2\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ The proof of the lemma uses the same techniques and tools as our previous lemmas, but is much more complicated in terms of keeping track of cases and components of integrals.\",\n \"To avoid overburdening a reader on an early reading, the proof has been placed in Appendix .\"\n ],\n [\n \"The Bootstrap Argument\",\n \"In this section we bring together the diagrammatic bounds of Section with the estimates of Section REF to bound the relevant operators in such a way that the finite lace expansion of Proposition REF converges to an Ornstein-Zernike equation (OZE) for both the raw operators on $L^2\\\\left(\\\\mathbb {X}\\\\right)$ and the Fourier-transformed operators on $L^2\\\\left(\\\\mathcal {E}\\\\right)$ .\",\n \"Proposition REF does this for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ , with the caveat that there is a constant $c_f$ that is increasing in the bootstrap function $f$ (but independent of $d$ ) appearing in the bounds.\",\n \"Observation REF then highlights that if $f$ is uniformly bounded on $\\\\left[0,\\\\lambda _O\\\\right)$ then so is $c_f$ and the associated bounds from Proposition REF .\",\n \"The crucial step to the bootstrap argument is then Proposition REF .\",\n \"Here a forbidden-region argument shows that $f$ is continuous, that there is an interval of finite values that it cannot take (the interval being independent of $\\\\lambda $ ), and that it starts below this interval.\",\n \"This proves that $f$ is indeed uniformly bounded for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ , and so are our bounds in Proposition REF .\",\n \"Proposition 6.11 (Convergence of the operator lace expansion and OZE) Let $\\\\lambda \\\\in [0, \\\\lambda _O)$ and $d>6$ be sufficiently large.\",\n \"Then there exists $c_f=c(f(\\\\lambda ))$ (which is increasing in $f$ and independent of $d$ ) such that $\\\\sum _{n\\\\ge 0}\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}\\\\le c_f\\\\beta , &\\\\qquad \\\\sum _{n\\\\ge 0}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le c_f\\\\beta , \\\\\\\\ \\\\sum _{n\\\\ge 0}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} &\\\\le c_f\\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) \\\\beta , \\\\\\\\\\\\left\\\\Vert R_{\\\\lambda , n}\\\\right\\\\Vert _{\\\\rm op} &\\\\le \\\\lambda \\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} c_f\\\\beta ^{n}.\",\n \"$ Furthermore, the limit $_\\\\lambda := \\\\lim _{M\\\\rightarrow \\\\infty } \\\\sum ^M_{n=0}_\\\\lambda ^{(n)}$ exists and is a bounded operator with Fourier transform $\\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right) = \\\\lim _{M\\\\rightarrow \\\\infty } \\\\sum ^M_{n=0}\\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)$ for $k\\\\in \\\\mathbb {R}^d$ .\",\n \"The operators $\\\\mathcal {T}_\\\\lambda $ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ satisfy operator Ornstein-Zernike equations, taking the form $\\\\mathcal {T}_\\\\lambda = + _\\\\lambda + \\\\lambda \\\\mathcal {T}_\\\\lambda (+_\\\\lambda ), $ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right) = \\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right) + \\\\lambda \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\left(\\\\widehat{}\\\\left(k\\\\right)+\\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)\\\\right), $ for all $\\\\lambda < \\\\lambda _O$ and $k\\\\in \\\\mathbb {R}^d$ .\",\n \"Before we prove this proposition, we make the following observation, which will guide the rest of the section.\",\n \"Observation 6.12 (Uniform convergence of the lace expansion) Suppose that there exists finite $C>0$ such that $f \\\\le C$ on $[0,\\\\lambda _O)$ .\",\n \"Then for $d>6$ sufficiently large there exits $c>0$ (independent of $\\\\lambda $ and $d$ ) such that the bounds (REF ), (), () hold with $c_f$ replaced by $c$ .\",\n \"We now return to prove Proposition REF .\",\n \"[Proof of Proposition REF ] So far in this section we have provided bounds for the building blocks of the bounds appearing in Propositions REF , REF , REF , and REF .\",\n \"We still have to convert these into bounds for the composite terms $U_\\\\lambda $ and $V_\\\\lambda $ .\",\n \"From their definitions and the bounds on the building blocks, it is simple to see that there exists $c_f^{\\\\prime }$ such that for $d>6$ $U_\\\\lambda \\\\le c_f^{\\\\prime }, \\\\qquad V_\\\\lambda \\\\le c_f^{\\\\prime }\\\\beta .$ This then implies that there exists $c_f^{\\\\prime \\\\prime }$ such that for $d>6$ $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f^{\\\\prime \\\\prime }\\\\beta ^2 &: n=0,\\\\\\\\c_f^{\\\\prime \\\\prime }\\\\left(c_f^{\\\\prime \\\\prime }\\\\beta \\\\right)^n &: n\\\\ge 1.\\\\end{array}\\\\right.\",\n \"}$ If $d$ is sufficiently large we have $c_f^{\\\\prime \\\\prime }\\\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\\\sum _{n\\\\ge 0}\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op} \\\\le c_f^{\\\\prime \\\\prime }\\\\left(\\\\beta ^2 + \\\\frac{c_f^{\\\\prime \\\\prime }\\\\beta }{1-c_f^{\\\\prime \\\\prime }\\\\beta }\\\\right) \\\\le c_f\\\\beta .$ When we want to consider the displaced terms, we also make use of Observation REF , and Lemmas REF , REF , REF , and REF to give us the existence of $c_f^{\\\\prime \\\\prime \\\\prime }$ such that for $d>6$ $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le {\\\\left\\\\lbrace \\\\begin{array}{ll}c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta ^2\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) &: n=0,3,\\\\\\\\c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) &: n=1,2,\\\\\\\\\\\\left(n+1\\\\right)^2c_f^{\\\\prime \\\\prime \\\\prime }\\\\left(c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^{n-3}\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) &: n\\\\ge 4.\\\\end{array}\\\\right.\",\n \"}$ If $d$ is sufficiently large we have $c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\\\sum _{n\\\\ge 0}\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }^{(n)}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le c_f^{\\\\prime \\\\prime \\\\prime }\\\\left(2\\\\beta ^2 + 2\\\\beta + \\\\frac{16\\\\left(c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^3-39\\\\left(c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^2 + 25c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta }{\\\\left(1-c_f^{\\\\prime \\\\prime \\\\prime }\\\\beta \\\\right)^3}\\\\right)\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) \\\\\\\\\\\\le c_f\\\\beta \\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right).$ Lemma REF then allows us to use (REF ) to bound $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"This immediately gives the required result for $n\\\\ge 1$ .\",\n \"For $n=0$ , we simply additionally require $d$ to be large enough that $\\\\beta <1$ and therefore $\\\\beta ^2<\\\\beta ^0$ .\",\n \"Note that the dual space of a Banach space (endowed with the operator norm) is also a Banach space.\",\n \"In particular, it is complete.\",\n \"Since $\\\\left\\\\Vert _\\\\lambda ^{(n)}\\\\right\\\\Vert _{\\\\rm op}\\\\le c_f^{\\\\prime \\\\prime }\\\\left(c_f^{\\\\prime \\\\prime }\\\\beta \\\\right)^n$ for $n\\\\ge 1$ , the sequence $_{\\\\lambda ,n}$ is a Cauchy sequence in the dual space of $L^2(\\\\mathbb {X})$ for sufficiently large $d$ .\",\n \"Therefore the limit $_\\\\lambda $ exists and is a bounded linear operator on $L^2(\\\\mathbb {X})$ .\",\n \"Since the Fourier transform is a unitary linear operator it is bounded and continuous, and therefore the same argument says that the limit $\\\\widehat{}_{\\\\lambda }(k)$ exists and is indeed the Fourier transform of $_\\\\lambda $ .\",\n \"Since $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op}$ is finite for $\\\\lambda <\\\\lambda _O$ , we have $\\\\left\\\\Vert R_{\\\\lambda ,n}\\\\right\\\\Vert _{\\\\rm op}\\\\rightarrow 0$ as $n\\\\rightarrow \\\\infty $ if $d$ is sufficiently large.\",\n \"As a a consequence of this and equation (REF ) from Proposition REF , the $\\\\mathcal {T}_\\\\lambda $ operator satisfies the operator Ornstein-Zernike equation.\",\n \"The following lemma has two main uses in what follows.\",\n \"Firstly, it takes the OZEs from Proposition REF and uses it to write the two-point operators in terms of the sum of the adjacency operators and the lace expansion coefficient operators.\",\n \"Secondly, it uses this and an intermediate value theorem argument to bound the spectral supremum of the sum of the adjacency operators and the lace expansion coefficient operators.\",\n \"Lemma 6.13 Let $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ and $d>6$ be sufficiently large.\",\n \"Then the operators $\\\\left(1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)$ and $\\\\left(1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right)$ all have bounded linear inverses and $\\\\mathcal {T}_\\\\lambda &= \\\\left(+ _\\\\lambda \\\\right)\\\\left(1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)^{-1}, \\\\\\\\\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) &= \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\left(1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right)^{-1}, \\\\qquad \\\\forall k\\\\in \\\\mathbb {R}^d.\",\n \"$ Furthermore, $\\\\lambda \\\\mapsto _\\\\lambda $ and $\\\\lambda \\\\mapsto \\\\widehat{}_{\\\\lambda }(k)$ are all continuous in the operator norm topology and $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) &< 1,\\\\\\\\\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) &< 1, \\\\qquad \\\\forall k\\\\in \\\\mathbb {R}^d.$ We first prove that $1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)$ has a bounded linear inverse.\",\n \"This is clear for $\\\\lambda = 0$ , so we only need to consider $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"Suppose for contradiction that $1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)$ does not have a bounded linear inverse.\",\n \"Then there exists a sequence $f_n\\\\in L^2(\\\\mathbb {X})$ such that $*{f_n}_2 = 1$ and $*{\\\\left(1- \\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)f_n}_2\\\\rightarrow 0$ .\",\n \"This also means that $\\\\lambda *{\\\\left(+ _\\\\lambda \\\\right)f_n}\\\\rightarrow 1$ .\",\n \"Then from Lemma REF and sub-criticality we have $\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} < \\\\infty $ , and therefore the OZE equation (REF ) implies that $*{\\\\left(+ _\\\\lambda \\\\right)f_n}\\\\rightarrow 0$ , a contradiction.\",\n \"A similar argument proves the corresponding statement for the Fourier transformed operators.\",\n \"Once we know that the inverses exist, it is only a matter of rearranging the OZE equations to get the expressions for $\\\\mathcal {T}_\\\\lambda $ and $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ in terms of the other operators.\",\n \"Since $+ _\\\\lambda $ is a bounded operator, a similar argument proves that $1+ \\\\lambda \\\\mathcal {T}_\\\\lambda $ has a bounded inverse, and therefore $+ _\\\\lambda = \\\\left(1+ \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\right)^{-1}\\\\mathcal {T}_\\\\lambda .$ From Corollary REF we have the continuity of $\\\\lambda \\\\mapsto \\\\mathcal {T}_\\\\lambda $ , and therefore the continuity $+ _\\\\lambda $ (and therefore $_\\\\lambda $ ).\",\n \"We also have the continuity of $\\\\lambda \\\\mapsto \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ from Corollary REF , and we similarly get the continuity of $\\\\widehat{}_{\\\\lambda }(k)$ .\",\n \"The continuity of $+ _\\\\lambda $ implies the continuity of $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right)$ (via Lemma REF ).\",\n \"At $\\\\lambda = 0$ it is clear that $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) = 0$ , and our above argument implies that $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) \\\\ne 1$ .\",\n \"Therefore an intermediate value theorem argument proves that $\\\\lambda \\\\mathbb {S}\\\\left(+ _\\\\lambda \\\\right) < 1$ .\",\n \"This argument also works for $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)$ .\",\n \"It now remains to show that $f$ is indeed uniformly bounded for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"We do this by performing a forbidden-region argument.\",\n \"In Proposition REF we prove that $f(0)$ is bounded and $f$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"However we also prove that $f$ is never in the region $\\\\left(\\\\kappa ,\\\\kappa +1\\\\right]$ , where $\\\\kappa := 10 C \\\\left(1+C\\\\right)^2,$ $C$ being the constant appearing in REF .\",\n \"Note that (REF ) in REF requires that $C\\\\ge 1$ .\",\n \"The intermediate value theorem then implies that $\\\\kappa $ acts as an upper bound on the whole domain $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"One should not read too much into the value of $\\\\kappa $ here.\",\n \"It is sufficient for our purposes, but not remotely optimal.\",\n \"Proposition 6.14 (The forbidden-region argument) The following three statements are all true: $f$ satisfies $f(0) \\\\le \\\\kappa $ .\",\n \"$f(\\\\lambda ) \\\\notin (\\\\kappa ,\\\\kappa +1]$ for all $\\\\lambda \\\\in [0,\\\\lambda _O)$ provided that $d$ is sufficiently large.\",\n \"$f$ is continuous on $[0, \\\\lambda _O)$ .\",\n \"Therefore $f(\\\\lambda ) \\\\le \\\\kappa $ holds uniformly in $\\\\lambda <\\\\lambda _O$ for all $d$ sufficiently large.\",\n \"We will deal with each of the three statements in Proposition REF in turn.\",\n \"[Proof of Proposition REF -REF ] Firstly, it is clear $f_1\\\\left(0\\\\right) = 0$ .\",\n \"Then note that $\\\\mathcal {T}_0 = $ , and hence $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}_0}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} = \\\\left\\\\Vert \\\\widehat{}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le C,$ where we use REF (via (REF )) in the last inequality.\",\n \"Furthermore $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_0(0)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) =1$ , and therefore $\\\\widehat{G}_{\\\\mu _{0}}\\\\left(k\\\\right) = \\\\widehat{G}_0\\\\left(k\\\\right) = 1$ and $f_2\\\\left(0\\\\right) \\\\le C$ .\",\n \"Similarly we find that $f_3\\\\left(0\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{k,l\\\\in \\\\mathbb {R}^d}\\\\frac{\\\\left\\\\Vert \\\\widehat{}_k\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{3\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]}.$ We use then Lemma REF to get the bound $\\\\left\\\\Vert \\\\widehat{}_k\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le C\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ Hence $f_3\\\\left(0\\\\right) \\\\le \\\\tfrac{2}{3}C$ .\",\n \"[Proof of Proposition REF -REF ] We will assume that $f\\\\le \\\\kappa +1$ and show that this implies that $f\\\\le \\\\kappa $ .\",\n \"Most crucially, the assumption $f\\\\le \\\\kappa +1$ allows us to apply Observation REF .\",\n \"To highlight the $\\\\kappa $ -dependence, we write $c_{\\\\kappa +1}$ as the constant arising from this observation.\",\n \"It will be convenient to introduce some temporary notation for this section.\",\n \"Define $\\\\widehat{a}\\\\left(k\\\\right)\\\\colon L^2\\\\left(\\\\mathcal {E}\\\\right)\\\\rightarrow L^2\\\\left(\\\\mathcal {E}\\\\right)$ as $\\\\widehat{a}\\\\left(k\\\\right) = \\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"We will also make use of the functions $\\\\delta _\\\\lambda \\\\colon \\\\mathbb {R}^d\\\\rightarrow \\\\mathbb {R}$ defined by $\\\\delta _\\\\lambda \\\\left(k\\\\right) := \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ It is clear from the triangle inequality and Lemma REF that ${\\\\delta _\\\\lambda \\\\left(k\\\\right)}\\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le c_{\\\\kappa +1}\\\\beta ,$ but we will also need the following inequality.\",\n \"We will prove it at the end of this section.\",\n \"Lemma 6.15 Let $k_1,k_2\\\\in \\\\mathbb {R}^d$ .\",\n \"Then $*{\\\\delta _\\\\lambda (k_1)-\\\\delta _\\\\lambda (k_2)} \\\\le 2 \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right) - \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ Figure: If 𝒯 ^ λ (k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) and a ^(k)\\\\widehat{a} (k) commute, then they are simultaneously diagonalizable in the sense of Theorem .\",\n \"The arguments of their diagonal functions (τ ˜ λ (k)\\\\widetilde{\\\\tau }_\\\\lambda (k) and a ˜(k)\\\\widetilde{a}(k) respectively) are related by the monotone increasing function x↦x 1-λxx\\\\mapsto \\\\tfrac{x}{1-\\\\lambda x} depicted here.It will be important to note how $\\\\widehat{a}(k)$ relates to $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ .\",\n \"From Observation REF , the OZE gives us the following expression involving the commutator $\\\\left[\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k),\\\\widehat{a}(k)\\\\right] = \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) \\\\widehat{a}(k) - \\\\widehat{a}(k)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ : $\\\\left[\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k),\\\\widehat{a}(k)\\\\right]\\\\left(1- \\\\lambda \\\\widehat{a}(k)\\\\right) = 0.$ Lemma REF implies that $\\\\left(1- \\\\lambda \\\\widehat{a}(k)\\\\right)$ has a bounded linear inverse and with (REF ) implies that $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ and $\\\\widehat{a}(k)$ commute.\",\n \"Therefore there exists a single unitary map that `diagonalizes' them both in the sense of Theorem REF .\",\n \"By considering the diagonal functions, the OZE then implies that $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)/\\\\left(1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)\\\\right)$ .\",\n \"In particular, this relation allows us to write an alternative expression for $\\\\mu _\\\\lambda $ : $\\\\mu _\\\\lambda := 1 - \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)} = \\\\lambda + \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)-1}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)} = \\\\lambda + \\\\frac{\\\\delta _\\\\lambda \\\\left(0\\\\right)}{1+\\\\delta _\\\\lambda \\\\left(0\\\\right)}.$ We first address $f_1$ .\",\n \"From the above discussion relating $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ and $\\\\widehat{a}(k)$ , we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} \\\\ge \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)/\\\\left(1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)\\\\right)$ .\",\n \"From Lemma REF and our choice of scaling we have $\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right) \\\\ge 1 - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }(0)\\\\right\\\\Vert _{\\\\rm op} \\\\ge 1 - c_{\\\\kappa +1}\\\\beta $ , and therefore we have $\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)>0$ for sufficiently large $d$ and $\\\\lambda \\\\le \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}\\\\left(1 - \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}}\\\\right) \\\\le \\\\frac{1}{1-c_{\\\\kappa +1}\\\\beta } \\\\le 1 + 2c_{\\\\kappa +1}\\\\beta .$ Here we have used an idea that we will use very often in this part of the proof.\",\n \"For $d$ large enough that $c_{\\\\kappa +1}\\\\beta <1$ we have $\\\\left(1-c_{\\\\kappa +1}\\\\beta \\\\right)^{-1} = \\\\sum ^\\\\infty _{n=0}\\\\left(c_{\\\\kappa +1}\\\\beta \\\\right)^n$ , and for $d$ sufficiently large enough ($c_{\\\\kappa +1}\\\\beta \\\\le 1/2$ ) we have $\\\\left(1-c_{\\\\kappa +1}\\\\beta \\\\right)^{-1}\\\\le 1 + 2c_{\\\\kappa +1}\\\\beta $ .\",\n \"We also use a similar idea of neglecting higher order terms by increasing an earlier term's coefficient in other cases.\",\n \"For $f_2$ we first replace $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Since $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ and $\\\\widehat{a} (k)$ commute, they are simultaneously diagonalizable in the sense of Theorem REF , and therefore their spectra are related by the function $x\\\\mapsto \\\\tfrac{x}{1-\\\\lambda x}$ (monotone increasing on $\\\\left(-\\\\infty ,1/\\\\lambda \\\\right)$ ).\",\n \"Therefore we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right\\\\Vert _{\\\\rm op} = \\\\sup _{x\\\\in \\\\sigma \\\\left(\\\\widehat{a} (k)\\\\right)}*{\\\\frac{x}{1-\\\\lambda x}}$ .\",\n \"Now (perhaps with the aid of Figure REF ) it is easy to see that $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}{1- \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\vee \\\\mathbb {S}\\\\left(-\\\\widehat{a}(k)\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) \\\\vee \\\\mathbb {S}\\\\left(-\\\\widehat{a} (k)\\\\right).$ Now since $\\\\mathbb {S}\\\\left(-\\\\widehat{a}(k)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{a}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le 1 + c_{\\\\kappa +1}\\\\beta $ , and $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^{-1} \\\\le 1+C$ (recall Lemma REF and $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } \\\\le C$ from REF ), we only need to bound $\\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d}\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)}.$ We now partition $k\\\\in \\\\mathbb {R}^d$ into two.\",\n \"Define the subset of the Fourier space $A_{\\\\frac{1}{2}} := \\\\left\\\\lbrace k\\\\in \\\\mathbb {R}^d:\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\le \\\\frac{1}{2}\\\\right\\\\rbrace .$ Then we only need to bound the two functions $f_4,f_5$ , where $f_4\\\\left(\\\\lambda \\\\right) := \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in A_{\\\\frac{1}{2}}} \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)},\\\\qquad \\\\qquad f_5\\\\left(\\\\lambda \\\\right):= \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in A_{\\\\frac{1}{2}}^c} \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)}.$ We first deal with the simpler $f_4$ .\",\n \"We consider $k\\\\in A_{\\\\frac{1}{2}}$ and choose $\\\\beta $ small enough that $1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right) \\\\ge 1 - \\\\frac{\\\\frac{1}{2}+c_{\\\\kappa +1}\\\\beta }{1-c_{\\\\kappa +1}\\\\beta } \\\\ge \\\\frac{1}{2} - 2c_{\\\\kappa +1}\\\\beta > 0.$ Then $\\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\le \\\\frac{1+\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}}{\\\\frac{1}{2} - 2c_{\\\\kappa +1}\\\\beta } \\\\le 2\\\\left(1+C\\\\right)\\\\left(1+5c_{\\\\kappa +1}\\\\beta \\\\right).\",\n \"$ Since we have $\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right) \\\\le \\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right) \\\\le 1 + c_{\\\\kappa +1}\\\\beta $ , we then get $f_4(\\\\lambda ) \\\\le 2\\\\left(1+C\\\\right)\\\\left(1+5c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left(1 + c_{\\\\kappa +1}\\\\beta \\\\right) \\\\le 2\\\\left(1+C\\\\right)\\\\left(1+7c_{\\\\kappa +1}\\\\beta \\\\right).$ To address $f_5$ we will have $k\\\\in A^c_\\\\rho $ .\",\n \"We define: $\\\\widehat{N}(k) = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}, \\\\qquad \\\\widehat{F}(k) = \\\\frac{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}, \\\\qquad \\\\widehat{Q}(k) = \\\\frac{1+\\\\delta _\\\\lambda (k)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(0)\\\\right)}.$ Note that $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) = \\\\widehat{N}(k) / \\\\widehat{F}(k)$ .\",\n \"Rearranging gets us to $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} &= \\\\widehat{N}(k) \\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{F}(k)}\\\\nonumber \\\\\\\\&= \\\\widehat{Q}(k) + \\\\frac{\\\\widehat{N}(k)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{F}(k)} \\\\left(1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) - \\\\frac{\\\\widehat{Q}(k)}{\\\\widehat{N}(k)}\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{F}(k)\\\\right) .$ Since $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)=1$ (by our scaling choice), the extracted term $\\\\widehat{Q}(k)$ satisfies $*{\\\\widehat{Q}(k)} \\\\le 1 + 3c_{\\\\kappa +1}\\\\beta $ .\",\n \"We further observe that $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\widehat{Q}(k)}{\\\\widehat{N}(k)} = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\left(1+ \\\\delta _\\\\lambda (k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) +\\\\delta _\\\\lambda (k)} = 1 - \\\\frac{\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right) \\\\delta _\\\\lambda (k)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) + \\\\delta _\\\\lambda (k)} =: 1- \\\\widehat{b}(k).$ Recalling identity (REF ) for $\\\\mu _\\\\lambda $ , we can rewrite the quantity $\\\\left[1- \\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)- (1-\\\\widehat{b}(k)) \\\\widehat{F}(k)\\\\right]$ , appearing in (REF ), as $\\\\frac{1+\\\\delta _\\\\lambda (0) - \\\\left[\\\\lambda + \\\\delta _\\\\lambda (0) + \\\\lambda \\\\delta _\\\\lambda (0)\\\\right] \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) - 1 + \\\\lambda \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)+\\\\delta _\\\\lambda (k)\\\\right) + \\\\widehat{b}(k)\\\\left(1-\\\\lambda \\\\widehat{a}(k)\\\\right)}{1+\\\\delta _\\\\lambda (0)} \\\\\\\\= \\\\frac{ \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\left(\\\\delta _\\\\lambda (0) + \\\\lambda \\\\delta _\\\\lambda (0)\\\\right) + \\\\lambda \\\\left[\\\\delta _\\\\lambda (k)-\\\\delta _\\\\lambda (0)\\\\right]}{1+\\\\delta _\\\\lambda (0)}+ \\\\frac{\\\\widehat{b}(k)(1-\\\\lambda \\\\widehat{a}(k))}{1+\\\\delta _\\\\lambda (0)}.$ Noting that ${\\\\delta _{\\\\lambda }(0)-\\\\delta _{\\\\lambda }(k)} \\\\le 2c_{\\\\kappa +1}\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\beta $ by Lemma REF and Observation REF , the first term can be bounded $&*{\\\\frac{ \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\left(\\\\delta _\\\\lambda (0) + \\\\lambda \\\\delta _\\\\lambda (0)\\\\right) + \\\\lambda \\\\left[\\\\delta _\\\\lambda (k)-\\\\delta _\\\\lambda (0)\\\\right]}{1+\\\\delta _\\\\lambda (0)}} \\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\frac{ c_{\\\\kappa +1}\\\\beta + \\\\left(1 + 2c_{\\\\kappa +1}\\\\beta \\\\right)c_{\\\\kappa +1}\\\\beta + 2\\\\left(1+2c_{\\\\kappa +1}\\\\beta \\\\right)c_{\\\\kappa +1}\\\\beta }{1 - c_{\\\\kappa +1}\\\\beta }\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\&\\\\qquad \\\\le 5c_{\\\\kappa +1}\\\\beta \\\\left(1+2c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\&\\\\qquad \\\\le 6c_{\\\\kappa +1}\\\\beta \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ Using (REF ), the last term is $*{\\\\frac{\\\\widehat{b}(k)(1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right))}{1+\\\\delta _\\\\lambda (0)}} &= *{ \\\\frac{\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right)\\\\delta _\\\\lambda (k)}{\\\\left(1+\\\\delta _\\\\lambda (0)\\\\right)\\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) + \\\\delta _\\\\lambda (k)\\\\right)}- \\\\frac{\\\\lambda \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\delta _\\\\lambda (k)}{1+\\\\delta _\\\\lambda (0)}} \\\\nonumber \\\\\\\\& = \\\\frac{*{\\\\delta _{\\\\lambda }(k)}}{*{1+\\\\delta _\\\\lambda (0)}}*{ \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) + \\\\delta _\\\\lambda (k)} -\\\\lambda }\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\& \\\\le \\\\frac{c_{\\\\kappa +1}\\\\beta }{1-c_{\\\\kappa +1}\\\\beta }\\\\left( \\\\frac{1}{\\\\frac{1}{2}-c_{\\\\kappa +1}\\\\beta } + \\\\frac{1}{1-c_{\\\\kappa +1}\\\\beta }\\\\right)\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\&\\\\le c_{\\\\kappa +1}\\\\beta \\\\left(1+2c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left(3+8c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\nonumber \\\\\\\\& \\\\le 4c_{\\\\kappa +1}\\\\beta \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ Putting these bounds back into (REF ), we can find constants $M$ and $\\\\tilde{M}$ such that $\\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} &\\\\le \\\\widehat{Q}(k) + 10c_{\\\\kappa +1}\\\\beta \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}\\\\frac{\\\\widehat{N}(k)}{\\\\widehat{F}(k)} \\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\ \\\\\\\\& \\\\le 1+ 3c_{\\\\kappa +1}\\\\beta + 20c_{\\\\kappa +1}\\\\beta \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} \\\\\\\\& \\\\le 1+ \\\\left(20\\\\kappa + 23\\\\right)c_{\\\\kappa +1}\\\\beta .\",\n \"$ Note that we have used the bound $\\\\left[1-\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right] \\\\le \\\\left[1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]$ (since $\\\\mu _\\\\lambda \\\\le 1$ and $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)>\\\\frac{1}{2} >0$ ) to get from $\\\\widehat{G}_1(k)^{-1}$ to $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)^{-1}$ .\",\n \"As $\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\in \\\\left(\\\\frac{1}{2}, 1\\\\right]$ , this concludes the improvement of $f_5$ and hence of $f_2$ .\",\n \"Before we treat $f_3$ , we introduce $f_6$ given by $f_6(\\\\lambda ) := \\\\operatornamewithlimits{ess\\\\,sup}_{k\\\\in \\\\mathbb {R}^d} \\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{1- \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}\\\\left(k\\\\right)\\\\right)}.$ We show that $f(\\\\lambda ) \\\\le \\\\kappa +1$ implies $f_6(\\\\lambda ) \\\\le M$ uniformly for some $\\\\kappa $ -independent constant $M$ .\",\n \"For $k\\\\in A_\\\\frac{1}{2}$ we showed in (REF ) that we can bound uniformly with $3\\\\left(1+C\\\\right)$ for sufficiently large small $\\\\beta $ .\",\n \"For $k \\\\in A_{\\\\frac{1}{2}}^c$ , we have $\\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right) \\\\widehat{G}_{\\\\mu _\\\\lambda }(k)} \\\\cdot \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\le \\\\left(1+ \\\\left(20\\\\kappa + 23\\\\right)c_{\\\\kappa +1}\\\\beta \\\\right) *{ 1- \\\\frac{\\\\delta _\\\\lambda (k)}{\\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)}} \\\\\\\\ \\\\le \\\\left(1+ \\\\left(20\\\\kappa + 23\\\\right)c_{\\\\kappa +1}\\\\beta \\\\right) \\\\left(1+\\\\frac{c_{\\\\kappa +1}\\\\beta }{\\\\frac{1}{2}- c_{\\\\kappa +1}\\\\beta } \\\\right) \\\\le 1+ \\\\left(20\\\\kappa + 27\\\\right)c_{\\\\kappa +1}\\\\beta .\",\n \"$ Note that for the first bound in (REF ), we used the estimate established in (REF ), which is stronger than a bound on $f_5$ .\",\n \"We therefore have the uniform bound $f_6(\\\\lambda ) \\\\le 3 + 3C$ for $\\\\beta $ sufficiently small.\",\n \"Let us now improve the bound on $f_3$ .\",\n \"It will be notationally convenient to denote $A\\\\left(k\\\\right) := \\\\left[1- \\\\lambda \\\\widehat{a}\\\\left(k\\\\right)\\\\right]^{-1}$ .\",\n \"Since $\\\\widehat{a}\\\\left(k\\\\right)$ is self-adjoint, it can be “diagonalized\\\" in the sense of Theorem REF .\",\n \"Therefore $A\\\\left(k\\\\right)$ can be “diagonalized\\\" over the same space and with the same unitary operator - it will have the diagonal function $\\\\widetilde{A}(k)(e) = 1/\\\\left(1-\\\\widetilde{a}\\\\left(k\\\\right)(e)\\\\right)$ .\",\n \"It is then clear that $A\\\\left(k\\\\right)$ and $\\\\widehat{a}\\\\left(k\\\\right)$ commute (for each $k\\\\in \\\\mathbb {R}^d$ ).\",\n \"Using Observation REF , we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} &= \\\\left\\\\Vert \\\\widehat{a}\\\\left(l\\\\right)A\\\\left(l\\\\right) - \\\\widehat{a}\\\\left(l+k\\\\right)A\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\nonumber \\\\\\\\&= \\\\left\\\\Vert A\\\\left(l\\\\right)\\\\widehat{a}\\\\left(l\\\\right)\\\\left[1- \\\\lambda \\\\widehat{a}\\\\left(l+k\\\\right)\\\\right]A\\\\left(l+k\\\\right) - A\\\\left(l\\\\right)\\\\left[1- \\\\lambda \\\\widehat{a}\\\\left(l\\\\right)\\\\right]\\\\widehat{a}\\\\left(l+k\\\\right)A\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\nonumber \\\\\\\\&= \\\\left\\\\Vert A(l)\\\\left(\\\\widehat{a}(l)-\\\\widehat{a}(l+k)\\\\right)A(l+k)\\\\right\\\\Vert _{\\\\rm op} \\\\nonumber \\\\\\\\&\\\\le \\\\left\\\\Vert \\\\widehat{a}(l)-\\\\widehat{a}(l+k)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert A(l)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert A(l+k)\\\\right\\\\Vert _{\\\\rm op}.\",\n \"$ For the $\\\\left\\\\Vert \\\\widehat{a}(l)-\\\\widehat{a}(l+k)\\\\right\\\\Vert _{\\\\rm op}$ factor, we use the triangle inequality to separate this into a term of $\\\\widehat{}$ and $\\\\widehat{}_{\\\\lambda }$ .\",\n \"Lemma REF immediately gives $\\\\left\\\\Vert \\\\widehat{}\\\\left(l\\\\right) - \\\\widehat{}\\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le C\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right],$ for the $C>0$ arising from REF .\",\n \"For the $\\\\widehat{}_{\\\\lambda }$ term, a similar argument to (REF ) gives $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(l\\\\right) - \\\\widehat{}_{\\\\lambda }\\\\left(1+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(0\\\\right) - \\\\widehat{}_{\\\\lambda }\\\\left( k\\\\right)\\\\right\\\\Vert _{1,\\\\infty } \\\\le c_{\\\\kappa +1}\\\\beta \\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right],$ where the last inequality follows from Observation REF .\",\n \"We then deal with the remaining factors of (REF ) by noting that for all $k\\\\in \\\\mathbb {R}^d$ , $\\\\left\\\\Vert A(k)\\\\right\\\\Vert _{\\\\rm op} = \\\\frac{1}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} = \\\\frac{1-\\\\mu _\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)}{1-\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{a}(k)\\\\right)} \\\\widehat{G}_{\\\\mu _\\\\lambda }(k) \\\\le 3\\\\left(1+C\\\\right) \\\\widehat{G}_{\\\\mu _\\\\lambda }(k), $ employing our improved bound on $f_6$ .\",\n \"The result of this is that we have the bound $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le 9\\\\left(1+C\\\\right)^2\\\\left(C + c_{\\\\kappa +1}\\\\beta \\\\right)\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right]\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(l+k\\\\right).$ Then since $\\\\widehat{G}_{\\\\mu _\\\\lambda }(k)\\\\ge 0$ we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le 9\\\\left(1+C\\\\right)^2\\\\left(C+c_{\\\\kappa +1}\\\\beta \\\\right)\\\\widehat{J}_{\\\\mu _\\\\lambda }\\\\left(k,l\\\\right).$ [Proof of Proposition REF -REF ] The continuity of $f_1$ is obvious.\",\n \"For the other two functions we prove equicontinuity of a family of functions and use [41] to show the continuity of the desired functions.\",\n \"The procedure is outlined here.\",\n \"Suppose we wish to show that $H\\\\left(\\\\lambda \\\\right) := \\\\sup _{\\\\alpha \\\\in B}\\\\left\\\\Vert h_\\\\alpha \\\\left(\\\\lambda \\\\right)\\\\right\\\\Vert _{\\\\rm op}$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right)$ .\",\n \"For our purposes the parameter $\\\\alpha $ will be either $k$ or $\\\\left(k,l\\\\right)$ and thus $B=\\\\mathbb {R}^d$ or $B=\\\\mathbb {R}^{2d}$ .\",\n \"The continuity on the half-open interval is implied by having continuity on the closed interval $\\\\left[0,\\\\lambda _O-\\\\rho \\\\right]$ for any $\\\\rho >0$ .\",\n \"For closed intervals, [41] gives continuity if the family $\\\\left\\\\lbrace \\\\left\\\\Vert h_\\\\alpha \\\\right\\\\Vert _{\\\\rm op}\\\\right\\\\rbrace _{\\\\alpha \\\\in B}$ is equicontinuous and $H\\\\left(\\\\lambda \\\\right)<+\\\\infty $ for all $\\\\lambda $ in the closed interval.\",\n \"The family $\\\\left\\\\lbrace \\\\left\\\\Vert h_\\\\alpha \\\\right\\\\Vert _{\\\\rm op}\\\\right\\\\rbrace _{\\\\alpha \\\\in B}$ is equicontinuous if for all $\\\\varepsilon >0$ there exists $\\\\delta >0$ such that $|s-t|<\\\\delta $ implies $*{\\\\left\\\\Vert h_\\\\alpha (s)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert h_\\\\alpha (t)\\\\right\\\\Vert _{\\\\rm op}} \\\\le \\\\varepsilon $ uniformly in $\\\\alpha \\\\in B$ .\",\n \"In fact, the reverse triangle inequality implies that $*{\\\\left\\\\Vert h_\\\\alpha (s)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert h_\\\\alpha (t)\\\\right\\\\Vert _{\\\\rm op}} \\\\le \\\\left\\\\Vert h_\\\\alpha (s)-h_\\\\alpha (t)\\\\right\\\\Vert _{\\\\rm op},$ and therefore we only need to prove equicontinuity for the un-normed family $\\\\left\\\\lbrace h_\\\\alpha \\\\right\\\\rbrace _{\\\\alpha \\\\in B}$ .\",\n \"We prove equicontinuity by bounding the `near-derivative' $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert h_\\\\alpha (\\\\lambda +\\\\varepsilon )-h_\\\\alpha (\\\\lambda )\\\\right\\\\Vert _{\\\\rm op}$ uniformly in $\\\\alpha \\\\in B$ for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O-\\\\rho \\\\right]$ and arbitrary $\\\\rho >0$ .\",\n \"For $f_2$ the operator-valued functions $h_\\\\alpha $ are $\\\\lambda \\\\mapsto \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)/\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)$ .\",\n \"By using a variation on the chain rule, we get $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }\\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _{\\\\lambda +\\\\varepsilon }}\\\\left(k\\\\right)} - \\\\frac{\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{1}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\\\\\+ \\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)^2}*{\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{G}_\\\\mu \\\\left(k\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda }}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda }.$ Recall from Corollary REF that $\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)$ is differentiable with the bound $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2$ .\",\n \"We also have $\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\ge 1/\\\\left(1 + \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)$ from Lemma REF and $\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{G}_\\\\mu \\\\left(k\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda } = \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)^2\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ immediately from the definition.\",\n \"It remains to deal with the $\\\\mu _\\\\lambda $ term.\",\n \"Recall that $\\\\mu _\\\\lambda := 1 - \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^{-1}$ .\",\n \"Therefore, using the reverse triangle inequality, $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda } = \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^2}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }(0)\\\\right)-\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)} \\\\\\\\\\\\le \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^2}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }(0)-\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} = \\\\frac{1}{\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right)^2}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^2,$ where we have used $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\ge 1$ from Lemma REF in the last inequality.\",\n \"Therefore $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon }\\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _{\\\\lambda +\\\\varepsilon }}\\\\left(k\\\\right)} - \\\\frac{\\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)}{\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left(1 + \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2 + *{\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^2\\\\\\\\\\\\le \\\\left(1 + C\\\\right)\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}^2 + C\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^3.$ We therefore have a finite $k$ -independent bound.\",\n \"This has proven the equicontinuity.\",\n \"For the uniform boundedness, we note that for $\\\\lambda \\\\in \\\\left[0,\\\\lambda _O-\\\\rho \\\\right]$ we have $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O-\\\\rho }(0)\\\\right\\\\Vert _{\\\\rm op}<\\\\infty $ .\",\n \"In conjunction with $\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\ge 1/\\\\left(1 + \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op}\\\\right)$ we have the required uniform boundedness and therefore the continuity of $f_2$ .\",\n \"We repeat this approach for $f_3$ .\",\n \"The corresponding step to (REF ) now reads $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }\\\\left\\\\Vert \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda +\\\\varepsilon ,k}\\\\left(l\\\\right)}{\\\\widehat{J}_{\\\\mu _{\\\\lambda +\\\\varepsilon }}\\\\left(k,l\\\\right)} - \\\\frac{\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\\\\\+ \\\\frac{\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)^2}*{\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{J}_{\\\\mu }\\\\left(k,l\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda }}\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda }.$ Recall from Corollary REF that $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)$ is differentiable with the operator norm bound $\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le 4\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"The operator itself has the similar bound $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"We would like to have the bound (REF ), but this was proven under the assumption $f\\\\le \\\\kappa +1$ , which we no longer assume.\",\n \"Fortunately, we only require our bound for $\\\\lambda < \\\\lambda _O -\\\\rho $ .\",\n \"Therefore we can use Lemma REF and Lemma REF to get $\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l\\\\right) - \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(l+k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\text{e}^{4\\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}}\\\\left\\\\Vert \\\\widehat{}\\\\left(0\\\\right) - \\\\widehat{}\\\\left(k\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\\\\\\\\\le C\\\\text{e}^{4\\\\left(\\\\lambda _O-\\\\rho \\\\right)\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O-\\\\rho }(0)\\\\right\\\\Vert _{\\\\rm op}}\\\\left[1 - \\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)\\\\right].$ From this and $\\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\ge \\\\left(1+C\\\\right)^{-1}$ , there exists a constant $\\\\widetilde{M}>0$ such that both $\\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left\\\\Vert \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\lambda }\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\widetilde{M} \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda \\\\left(0\\\\right)\\\\right\\\\Vert _{\\\\rm op}, \\\\qquad \\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}\\\\left(l\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\widetilde{M},$ for sufficiently large $d$ .\",\n \"Now recall $\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{G}_\\\\mu \\\\left(k\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda } = \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right)^2\\\\mathbb {S}\\\\left(\\\\widehat{}\\\\left(k\\\\right)\\\\right)$ and $\\\\left(1+C\\\\right)^{-1}\\\\le \\\\widehat{G}_{\\\\mu _\\\\lambda }\\\\left(k\\\\right) \\\\le \\\\left(1-\\\\mu _\\\\lambda \\\\right)^{-1} = \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Therefore an application of the chain rule gives the bound $*{\\\\frac{1}{\\\\widehat{J}_{\\\\mu _{\\\\lambda }}\\\\left(k,l\\\\right)}\\\\left.\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}\\\\mu }\\\\widehat{J}_{\\\\mu }\\\\left(k,l\\\\right)\\\\right|_{\\\\mu =\\\\mu _\\\\lambda }} \\\\le 6\\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op}^4\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\rm op},$ uniformly in $k$ and $l$ .\",\n \"These properties, along with the bound on $\\\\limsup _{\\\\varepsilon \\\\rightarrow 0}\\\\frac{1}{\\\\varepsilon }*{\\\\mu _{\\\\lambda +\\\\varepsilon } - \\\\mu _\\\\lambda }$ from above, are sufficient to prove equicontinuity and the uniform boundedness for the required $\\\\lambda $ .\",\n \"Therefore we have proved that $f_3\\\\left(\\\\lambda \\\\right)$ is continuous.\",\n \"It just remains to prove that we can indeed bound the difference of $\\\\delta _\\\\lambda (k)$ with the difference of $\\\\widehat{}_{\\\\lambda }(k)$ as we claimed.\",\n \"[Proof of Lemma REF ] First note that there is nothing to prove if $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}=\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}=0$ , because the triangle inequality then forces $\\\\delta _\\\\lambda (k_1) = \\\\delta _\\\\lambda (k_2)=0$ .\",\n \"Without loss of generality, assume $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\ge \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ with $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}>0$ .\",\n \"Then $\\\\delta _\\\\lambda (k_1)-\\\\delta _\\\\lambda (k_2) = \\\\frac{\\\\delta _\\\\lambda (k_1)}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }(k_1)\\\\right\\\\Vert _{\\\\rm op}}\\\\left(\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}\\\\right) + \\\\left[\\\\delta _\\\\lambda (k_1)\\\\frac{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}} - \\\\delta _\\\\lambda (k_2)\\\\right]$ The second factor is the more troublesome: $*{\\\\delta _\\\\lambda (k_1)\\\\frac{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}} - \\\\delta _\\\\lambda (k_2)} = \\\\frac{1}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}}*{\\\\delta _\\\\lambda (k_1)\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\delta _\\\\lambda (k_2)\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}} \\\\\\\\\\\\le \\\\frac{*{\\\\delta _\\\\lambda (k_1)} \\\\vee *{\\\\delta _\\\\lambda (k_2)}}{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}}*{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}}.$ Since $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} \\\\ge \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ , we have $*{\\\\delta _\\\\lambda (k_1)} \\\\vee *{\\\\delta _\\\\lambda (k_2)} \\\\le \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Therefore $*{\\\\delta _\\\\lambda (k_1)-\\\\delta _\\\\lambda (k_2)} \\\\le 2*{\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op} - \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_2\\\\right)\\\\right\\\\Vert _{\\\\rm op}} \\\\le 2\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)-\\\\widehat{}_{\\\\lambda }\\\\left(k_1\\\\right)\\\\right\\\\Vert _{\\\\rm op},$ where the last inequality holds because of the reverse triangle inequality.\"\n ],\n [\n \"Proof of Main Theorems\",\n \"Proposition 7.1 Let $d>6$ be sufficiently large.\",\n \"Then for $\\\\lambda \\\\le \\\\lambda _O$ there exists $c>0$ such that $\\\\triangle _\\\\lambda < c\\\\beta ^2$ .\",\n \"Proposition REF gives us that there exists $c_f$ (increasing in $f$ ) such that $\\\\triangle _\\\\lambda \\\\le c_f\\\\beta ^2$ for all $\\\\lambda <\\\\lambda _O$ .\",\n \"Proposition REF then implies that there exists $c$ such that $c_f\\\\le c$ uniformly for all $\\\\lambda < \\\\lambda _O$ .\",\n \"It remains to prove the assertion for $\\\\lambda =\\\\lambda _O$ .\",\n \"Recall from Section REF the definition of $\\\\tau _\\\\lambda ^n(x,y)$ and $\\\\Lambda _n(x)$ .\",\n \"Furthermore, recall that if $x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}$ then such a connection is achieved in finitely many steps and the bound in $\\\\ref {Assump:2ndMoment}$ then implies that $\\\\left\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}\\\\right\\\\rbrace = \\\\bigcup ^\\\\infty _{n=1}\\\\left\\\\lbrace x \\\\longleftrightarrow y\\\\textrm { in } \\\\xi ^{x,y}_{\\\\Lambda _n(y)}\\\\right\\\\rbrace .$ Therefore by monotone convergence we have the pointwise limit $\\\\tau _\\\\lambda ^n(x,y) \\\\rightarrow \\\\tau _\\\\lambda (x,y)$ as $n\\\\rightarrow \\\\infty $ for all $\\\\lambda >0$ and $x,y\\\\in \\\\mathbb {X}$ .\",\n \"Since $\\\\tau _\\\\lambda ^n(x,y)$ only depends upon the finite region $\\\\Lambda _n(y)$ , the functions $\\\\lambda \\\\mapsto \\\\tau _\\\\lambda ^n(x,y)$ are continuous for all $x,y\\\\in \\\\mathbb {X}$ .\",\n \"This, with the monotonicity $\\\\tau _\\\\lambda ^n(x,y) \\\\le \\\\tau _\\\\lambda ^{n+1}(x,y)$ , implies that the function $\\\\lambda \\\\mapsto \\\\tau _\\\\lambda (x,y)$ is lower-semicontinuous.\",\n \"Since $\\\\tau _\\\\lambda (x,y)$ is non-decreasing in $\\\\lambda $ , this lower semi-continuity implies that the pointwise limit $\\\\tau _\\\\lambda (x,y)\\\\uparrow \\\\tau _{\\\\lambda _O}(x,y)$ as $\\\\lambda \\\\uparrow \\\\lambda _O$ holds, and the limit holds monotonically.\",\n \"This monotonic pointwise convergence implies that the integral $\\\\int \\\\tau _\\\\lambda (x,u)\\\\tau _\\\\lambda (u,v)\\\\tau _\\\\lambda (v,y)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right) \\\\rightarrow \\\\int \\\\tau _{\\\\lambda _O}(x,u)\\\\tau _{\\\\lambda _O}(u,v)\\\\tau _{\\\\lambda _O}(v,y)\\\\nu ^{\\\\otimes 2}\\\\left(\\\\mathrm {d}u, \\\\mathrm {d}v\\\\right)$ for all $x,y\\\\in \\\\mathbb {X}$ as $\\\\lambda \\\\uparrow \\\\lambda _O$ , and that this convergence is monotone increasing.\",\n \"If we take the supremum over $x,y\\\\in \\\\mathbb {X}$ , we get that $\\\\lim _{\\\\lambda \\\\uparrow \\\\lambda _O}\\\\left\\\\Vert \\\\mathcal {T}_\\\\lambda ^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\left\\\\Vert \\\\mathcal {T}_{\\\\lambda _O}^3\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ .\",\n \"The uniform bound for $\\\\lambda <\\\\lambda _O$ then implies that this bound also holds at $\\\\lambda =\\\\lambda _O$ .\",\n \"The following proposition uses the bound $\\\\triangle _{\\\\lambda _O} < c\\\\beta ^2$ to show that percolation does not occur at criticality.\",\n \"Another element of the proof is that there is almost surely at most one infinite cluster.\",\n \"There are general considerations that show that this is the case.\",\n \"Indeed [14] establishes that on the discrete space $\\\\mathbb {Z}^d$ there is at most one infinite cluster if the edge occupation measure is stationary and obeys the ‘finite energy property’, and an analogous result for Poisson processes in the continuum applies in our case (see [4], [36]).\",\n \"Proposition 7.2 The critical percolation function $\\\\theta _{\\\\lambda _O}(a)=0$ for $\\\\mathcal {P}$ -almost every $a\\\\in \\\\mathcal {E}$ .\",\n \"Assume that $(\\\\overline{0},a),(\\\\overline{x},b)\\\\in \\\\mathbb {X}$ are in the (a.s. unique) infinite cluster, and are therefore connected.\",\n \"Then the FKG inequality implies that $\\\\theta _{\\\\lambda _O}(a)\\\\theta _{\\\\lambda _O}(b)\\\\le \\\\mathbb {P}_{\\\\lambda _O}\\\\big (|\\\\overline{0},a)|=\\\\infty ,|\\\\overline{x},b)|=\\\\infty \\\\big )\\\\le \\\\tau _{\\\\lambda _O}((\\\\overline{0},a),(\\\\overline{x},b)).$ Since $\\\\triangle _{\\\\lambda _O} \\\\le c\\\\beta ^2$ , for $\\\\mathcal {P}$ -a.e.\",\n \"$a,b\\\\in \\\\mathcal {E}$ there exists a sequence $\\\\overline{x}^{(a,b)}_n$ such that $\\\\tau _{\\\\lambda _O}((\\\\overline{0},a),(\\\\overline{x}^{(a,b)}_n,b))\\\\rightarrow 0$ .\",\n \"Therefore for $\\\\mathcal {P}$ -a.e.\",\n \"$a\\\\in \\\\mathcal {E}$ we have $\\\\theta _{\\\\lambda _O}(a)=0$ .\",\n \"We can define the linear operator $\\\\mathcal {T}_{\\\\lambda _O}$ by its action on $f\\\\in L^2(\\\\mathbb {X})$ : $\\\\mathcal {T}_{\\\\lambda _O}f(x) = \\\\int \\\\tau _{\\\\lambda _O}(x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right).$ Contrary to $\\\\mathcal {T}_\\\\lambda $ for $\\\\lambda <\\\\lambda _O$ , the operator $\\\\mathcal {T}_{\\\\lambda _O}$ may be an unbounded linear operator.\",\n \"Proposition 7.3 (The operator OZE at the critical point) Let $d>6$ be sufficiently large.\",\n \"The Ornstein-Zernike equation then extends to $\\\\lambda _O$ in the sense that the sum of (unbounded) linear operators vanishes: $\\\\mathcal {T}_{\\\\lambda _O}- - _{\\\\lambda _O}- \\\\lambda _O\\\\mathcal {T}_{\\\\lambda _O}\\\\left(+ _{\\\\lambda _O}\\\\right) = 0.$ Furthermore, for $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ the linear operator $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O}(k)$ is a bounded linear operator and the following equality holds: $\\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O}\\\\left(k\\\\right) - \\\\widehat{}\\\\left(k\\\\right) - \\\\widehat{}_{\\\\lambda _O}\\\\left(k\\\\right) - \\\\lambda _O\\\\widehat{\\\\mathcal {T}}_{\\\\lambda _O}\\\\left(k\\\\right)\\\\left(\\\\widehat{}\\\\left(k\\\\right) + \\\\widehat{}_{\\\\lambda _O}\\\\left(k\\\\right)\\\\right) = 0.$ Finally, there exists $c>0$ such that for all $k\\\\in \\\\mathbb {R}^d$ $\\\\left\\\\Vert _{\\\\lambda _O}\\\\right\\\\Vert _{\\\\rm op} \\\\le c\\\\beta , \\\\qquad \\\\left\\\\Vert \\\\widehat{}_{\\\\lambda _O}(k)\\\\right\\\\Vert _{\\\\rm op}\\\\le c\\\\beta .$ Let $f\\\\in L^2(\\\\mathbb {X})$ .\",\n \"Then by the triangle inequality, for $\\\\lambda <\\\\lambda _O$ ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- - _{\\\\lambda _O}- \\\\lambda _O\\\\mathcal {T}_{\\\\lambda _O}\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right)f}_2 \\\\le {\\\\left(\\\\mathcal {T}_\\\\lambda - - _\\\\lambda - \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left(+ _\\\\lambda \\\\right)\\\\right)f}_2 \\\\\\\\+ {\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda - _{\\\\lambda _O}+ _\\\\lambda - \\\\lambda _O\\\\mathcal {T}_{\\\\lambda _O}\\\\left(+ _{\\\\lambda _O}\\\\right) + \\\\lambda \\\\mathcal {T}_\\\\lambda \\\\left(+_\\\\lambda \\\\right)\\\\right)f}_2.$ The first norm on the right hand side vanishes, since the OZE holds for $\\\\lambda <\\\\lambda _O$ .\",\n \"To prove that the left hand side vanishes, we aim to show that the second norm on the right hand side vanishes as $\\\\lambda \\\\uparrow \\\\lambda _O$ .\",\n \"To do this, the triangle inequality implies that we only need to show that the following limits hold as $\\\\lambda \\\\uparrow \\\\lambda _O$ : ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda \\\\right)f}_2 &\\\\rightarrow 0, \\\\\\\\{\\\\left(_{\\\\lambda _O}- _\\\\lambda \\\\right)f}_2 &\\\\rightarrow 0,\\\\\\\\{\\\\left(\\\\mathcal {T}_{\\\\lambda _O}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda _\\\\lambda \\\\right)f}_2 &\\\\rightarrow 0.$ We begin with (REF ).\",\n \"Recall from the proof of Proposition REF that $\\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,y) := \\\\tau _{\\\\lambda _O}(x,y) - \\\\tau _\\\\lambda (x,y)$ converges monotonically and pointwise to zero.\",\n \"From the definition of the ${\\\\cdot }_2$ norm, ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda \\\\right)f}_2^2 &= \\\\int \\\\left(\\\\int \\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right)\\\\\\\\&\\\\le \\\\int \\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,y)\\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,z)*{f(y)}*{f(z)}\\\\nu ^{\\\\otimes (3)}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y, \\\\mathrm {d}z\\\\right).$ The monotone convergence of $\\\\left(\\\\tau _{\\\\lambda _O}-\\\\tau _\\\\lambda \\\\right)(x,y)\\\\downarrow 0$ then implies that the limit (REF ) holds.\",\n \"For (), we use the functions $h^{(n)}_\\\\lambda \\\\colon \\\\mathbb {X}^2\\\\rightarrow \\\\mathbb {R}_+$ defined by $h^{(n)}_\\\\lambda (x,y) := {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\lambda ^{-1}\\\\int \\\\psi _n(x,w_{n-1},u_{n-1}) \\\\left( \\\\prod _{i=1}^{n-1} \\\\psi (\\\\vec{v}_i) \\\\right) \\\\psi _0(w_0,u_0,y) \\\\nu ^{\\\\otimes (2n)}\\\\left(\\\\mathrm {d}\\\\left( \\\\left(\\\\vec{w}, \\\\vec{u}\\\\right)_{[0,n-1]} \\\\right)\\\\right) &:n\\\\ge 1\\\\\\\\\\\\tfrac{1}{2} \\\\lambda ^2 \\\\left(\\\\int \\\\tau _\\\\lambda \\\\left(x,w\\\\right)\\\\varphi \\\\left(w,y\\\\right)\\\\nu \\\\left(\\\\mathrm {d}w\\\\right)\\\\right)^2 &: n=0.\\\\end{array}\\\\right.\",\n \"}$ From Proposition REF and the proof of Proposition REF , we know that $0\\\\le \\\\pi ^{(n)}_\\\\lambda (x,y) \\\\le h^{(n)}_\\\\lambda (x,y)$ for all $n\\\\ge 0$ , $x,y\\\\in \\\\mathbb {X}$ , and $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ - the non-negativity follows from the definition of $\\\\pi ^{(n)}_\\\\lambda $ .\",\n \"The functions $ h^{(n)}_\\\\lambda $ can also be defined for $\\\\lambda =\\\\lambda _O$ , and since they are monotone increasing in $\\\\lambda $ , we have the $\\\\lambda $ -independent bound $\\\\pi ^{(n)}_\\\\lambda (x,y) \\\\le h^{(n)}_{\\\\lambda _O}(x,y)$ for all $\\\\lambda <\\\\lambda _O$ and $x,y\\\\in \\\\mathbb {X}$ .\",\n \"For the single-mark version of the model it is proven in [21] that $\\\\pi _{\\\\lambda _O}-\\\\pi _{\\\\lambda }$ converges pointwise to zero.\",\n \"This same argument works for our multi-mark version, and therefore $\\\\pi ^{(n)}_\\\\lambda (x,y) \\\\le h^{(n)}_{\\\\lambda _O}(x,y)$ for all $\\\\lambda \\\\le \\\\lambda _O$ .\",\n \"The triangle inequality then implies that $*{\\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,y)} = *{\\\\sum ^\\\\infty _{n=0}\\\\left(-1\\\\right)^n\\\\left( \\\\pi ^{(n)}_{\\\\lambda _O}(x,y) - \\\\pi ^{(n)}_{\\\\lambda }(x,y)\\\\right)} \\\\le 2\\\\sum ^\\\\infty _{n=0}h^{(n)}_{\\\\lambda _O}(x,y).$ It will be convenient to define $h_{\\\\lambda }\\\\colon \\\\mathbb {X}^2\\\\rightarrow \\\\mathbb {R}_+$ as $h_\\\\lambda (x,y) := \\\\sum ^\\\\infty _{n=0}h^{(n)}_{\\\\lambda }(x,y)$ .\",\n \"Then ${\\\\left(_{\\\\lambda _O}- _\\\\lambda \\\\right)f}_2^2 &= \\\\int \\\\left(\\\\int \\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,y)f(y)\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\nonumber \\\\\\\\&= \\\\int \\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,y)\\\\left(\\\\pi _{\\\\lambda _O} - \\\\pi _{\\\\lambda }\\\\right)(x,z)f(y)f(z)\\\\nu ^{\\\\otimes 3}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y, \\\\mathrm {d}z\\\\right) \\\\\\\\& \\\\le 4\\\\int h_{\\\\lambda _O}(x,y)h_{\\\\lambda _O}(x,z)*{f(y)}*{f(z)}\\\\nu ^{\\\\otimes 3}\\\\left(\\\\mathrm {d}x,\\\\mathrm {d}y, \\\\mathrm {d}z\\\\right)\\\\nonumber \\\\\\\\& = 4\\\\int \\\\left(\\\\int h_{\\\\lambda _O}(x,y)*{f(y)}\\\\nu \\\\left(\\\\mathrm {d}y\\\\right)\\\\right)^2\\\\nu \\\\left(\\\\mathrm {d}x\\\\right).\",\n \"$ By Schur's test, this integral is finite for all $f\\\\in L^2(\\\\mathbb {X})$ if both the values $\\\\operatornamewithlimits{ess\\\\,sup}_{y}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}x\\\\right) \\\\qquad \\\\text{ and } \\\\qquad \\\\operatornamewithlimits{ess\\\\,sup}_{x}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)$ are finite.\",\n \"To prove this for the former, we repeat the arguments of Proposition REF and Section REF to bound it in terms of $\\\\lambda _O$ , the triangle diagrams, and other $\\\\lambda $ -independent terms.\",\n \"The argument of Proposition REF with the uniform bounds of Observation REF then proves that $\\\\operatornamewithlimits{ess\\\\,sup}_{y}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}x\\\\right)$ is indeed finite.\",\n \"A similar argument also holds for $\\\\operatornamewithlimits{ess\\\\,sup}_{x}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}y\\\\right)$ .\",\n \"We now have an integrable function $h_{\\\\lambda _O}(x,y)h_{\\\\lambda _O}(x,z)*{f(y)}*{f(z)}$ that dominates the integrand of (REF ).\",\n \"Recall that $\\\\pi _{\\\\lambda _O}-\\\\pi _{\\\\lambda }$ converges pointwise to zero.\",\n \"Thus the dominated convergence theorem implies that the limit () holds.\",\n \"For (), we can use $h_{\\\\lambda _O}$ to get the bound ${\\\\left(\\\\mathcal {T}_{\\\\lambda _O}_{\\\\lambda _O}- \\\\mathcal {T}_\\\\lambda _\\\\lambda \\\\right)f}_2^2 \\\\\\\\\\\\le \\\\int \\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,u)h_{\\\\lambda _O}(u,y)\\\\left(\\\\tau _{\\\\lambda _O} - \\\\tau _\\\\lambda \\\\right)(x,v)h_{\\\\lambda _O}(v,z)*{f(y)}*{f(z)} \\\\nu ^{\\\\otimes 5}\\\\left(\\\\mathrm {d}u,\\\\mathrm {d}v, \\\\mathrm {d}x, \\\\mathrm {d}y, \\\\mathrm {d}z\\\\right).$ The integrand converges monotonically to zero, and therefore the integral vanishes in the limit.\",\n \"We now consider the Fourier version of the OZE.\",\n \"We use the notation that, given a bounded linear operator $A\\\\colon L^2(\\\\mathbb {X}) \\\\rightarrow L^2(\\\\mathbb {X})$ , the Fourier transform is given by $\\\\mathcal {F}\\\\left( A \\\\right)(k)\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ .\",\n \"By hypothesis, $$ is a bounded operator, and the finiteness of $\\\\operatornamewithlimits{ess\\\\,sup}_{y}\\\\int h_{\\\\lambda _O}(x,y) \\\\nu \\\\left(\\\\mathrm {d}x\\\\right)$ implies that $_{\\\\lambda _O}$ is also bounded.\",\n \"Then (REF ) implies that $\\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O}\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right) \\\\right)(k) = \\\\mathcal {F}\\\\left( + _{\\\\lambda _O} \\\\right)(k)$ for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"We now make the claim that $\\\\mathcal {F}\\\\left( 1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right) \\\\right)(k)$ has a bounded linear inverse for all $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ .\",\n \"This is proven in Lemma REF below.\",\n \"This claim then implies that for $k\\\\ne 0$ we have $&\\\\mathcal {F}\\\\left( + _{\\\\lambda _O} \\\\right)(k)\\\\mathcal {F}\\\\left( 1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right) \\\\right)(k)^{-1} \\\\nonumber \\\\\\\\& \\\\hspace{85.35826pt} = \\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O}\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right) \\\\right)(k) \\\\mathcal {F}\\\\left( 1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right) \\\\right)(k)^{-1} \\\\nonumber \\\\\\\\& \\\\hspace{85.35826pt} = \\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O}\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right)\\\\left(1- \\\\lambda _O\\\\left(+ _{\\\\lambda _O}\\\\right)\\\\right)^{-1} \\\\right)(k) = \\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O} \\\\right)(k).$ That is, $\\\\mathcal {F}\\\\left( \\\\mathcal {T}_{\\\\lambda _O} \\\\right)(k)$ is a bounded linear operator for $k\\\\ne 0$ .\",\n \"The limit () also allows us to extend our sub-critical bound on $_\\\\lambda $ to apply at criticality.\",\n \"The limit () may be phrased as saying that the function $\\\\lambda \\\\mapsto *{_\\\\lambda f}_2$ is continuous at $\\\\lambda =\\\\lambda _O$ (from the left) for all $f\\\\in L^2\\\\left(\\\\mathbb {X}\\\\right)$ .\",\n \"Since $\\\\lambda \\\\mapsto \\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op}$ can be written as a supremum of such functions, it is a lower semi-continuous function.\",\n \"Our bound $\\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le c\\\\beta $ for $\\\\lambda < \\\\lambda _O$ (from Proposition REF ) then implies $\\\\left\\\\Vert _{\\\\lambda _O}\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\liminf _{\\\\lambda \\\\uparrow \\\\lambda _O}\\\\left\\\\Vert _\\\\lambda \\\\right\\\\Vert _{\\\\rm op} \\\\le c\\\\beta .$ The limit $*{\\\\left(\\\\widehat{}_{\\\\lambda _O}^{(n)}(0) - \\\\widehat{}_{\\\\lambda }^{(n)}(0)\\\\right)f}_2 \\\\rightarrow 0$ for each $f\\\\in L^2\\\\left(\\\\mathcal {E}\\\\right)$ can be derived in essentially the same way as () above.\",\n \"The above lower semi-continuity argument can then be used to show that the bound from Proposition REF extends to also hold at $\\\\lambda =\\\\lambda _O$ .\",\n \"From Lemma REF and the positivity of $\\\\pi ^{(n)}_{\\\\lambda _O}(x,y)$ this bound also holds for all $k\\\\in \\\\mathbb {R}^d$ .\",\n \"By summing over $n$ , we then get the desired bound on $\\\\left\\\\Vert \\\\widehat{}_{\\\\lambda _O}(k)\\\\right\\\\Vert _{\\\\rm op}$ .\",\n \"Lemma 7.4 For $\\\\lambda = \\\\lambda _O$ and $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\left\\\\lbrace 0\\\\right\\\\rbrace $ , the bounded linear operator $\\\\left(1- \\\\lambda _O\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda _O}(k)\\\\right)\\\\right)\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ has a bounded linear inverse.\",\n \"We first introduce some notation.\",\n \"Let $A\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ be a self-adjoint linear operator.\",\n \"Then we define the minimum modulus to be $\\\\left\\\\Vert A\\\\right\\\\Vert _{\\\\rm min} := \\\\inf _{0\\\\ne f\\\\in L^2(\\\\mathcal {E})}\\\\frac{*{Af}_2}{{f}_2}.$ Note that it follows from the Spectral Theorem (Theorem REF ) that $A$ has a bounded linear inverse if and only if $\\\\left\\\\Vert A\\\\right\\\\Vert _{\\\\rm min}>0$ .\",\n \"Now suppose that $A=A_\\\\lambda $ , and that for all $f\\\\in L^2(\\\\mathcal {E})$ the map $\\\\lambda \\\\mapsto *{A_\\\\lambda f}$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"Then the map $\\\\lambda \\\\mapsto \\\\left\\\\Vert A_\\\\lambda \\\\right\\\\Vert _{\\\\rm min}$ is upper semicontinuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"That is, for any sequence $\\\\lambda _n$ in $\\\\left[0,\\\\lambda _O\\\\right]$ such that $\\\\lambda _n\\\\rightarrow \\\\lambda $ , $\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\left\\\\Vert A_{\\\\lambda _n}\\\\right\\\\Vert _{\\\\rm min} \\\\le \\\\left\\\\Vert A_\\\\lambda \\\\right\\\\Vert _{\\\\rm min}.$ We now consider $A_\\\\lambda = 1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)$ .\",\n \"A similar argument to the way we proved () shows that $\\\\lambda \\\\mapsto *{A_\\\\lambda f}$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"Our result is then proven if we can show that for all $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\lbrace 0\\\\rbrace $ we have an $\\\\varepsilon = \\\\varepsilon (k)>0$ such that $\\\\left\\\\Vert 1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right\\\\Vert _{\\\\rm min} \\\\ge \\\\varepsilon $ uniformly in $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"We will in fact show the stronger result that $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) \\\\le 1 - \\\\varepsilon $ uniformly in $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"Recalling the definition of $\\\\delta _\\\\lambda (k)$ from (REF ), we have $1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) &= 1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) - \\\\lambda \\\\delta _{\\\\lambda }(k) \\\\nonumber \\\\\\\\&= \\\\underbrace{1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right)}_{>0 \\\\text{ from Lemma~\\\\ref {lem:Optlam-a}}} + \\\\lambda \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) + \\\\lambda \\\\left(\\\\delta _{\\\\lambda }(0) - \\\\delta _{\\\\lambda }(k)\\\\right) \\\\nonumber \\\\\\\\& > \\\\lambda \\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\left(1+\\\\mathcal {O}\\\\left(\\\\beta \\\\right)\\\\right).$ Here we have used Lemma REF to bound the difference of $\\\\delta $ with the difference of $\\\\widehat{}_{\\\\lambda _O}$ .\",\n \"For sufficiently large $d$ , REF then proves $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) \\\\le 1 - \\\\varepsilon $ uniformly in $\\\\lambda \\\\in \\\\left(0,\\\\lambda _O\\\\right)$ .\",\n \"Note that in conjunction, Proposition REF , Proposition REF and Proposition REF prove the OZE equations of Theorem REF .\",\n \"In particular, the former two prove that the OZE holds in the subcritical regime, and the latter extends this to criticality where possible.\",\n \"Furthermore, Proposition REF gives the bounds on $_\\\\lambda $ and $\\\\widehat{}_{\\\\lambda }(k)$ subcritically whilst Proposition REF extends this bound to criticality.\",\n \"Regarding Theorem REF , the bound on $\\\\triangle _\\\\lambda $ and the lack of percolation at criticality were proven at the beginning of this section.\",\n \"The infrared bound component of the theorem is proven below.\",\n \"[Proof of Theorem REF ] First note that there is nothing to prove for $k=0$ .\",\n \"From Proposition REF and Lemma REF , we can write $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k) = \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\left(1- \\\\lambda \\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)\\\\right)^{-1},$ for all $k\\\\in \\\\mathbb {R}^d$ if $\\\\lambda <\\\\lambda _O$ , and for all $k\\\\in \\\\mathbb {R}^d\\\\setminus \\\\lbrace 0\\\\rbrace $ if $\\\\lambda = \\\\lambda _O$ .\",\n \"If we use the Spectral Theorem (Theorem REF ) to `diagonalize' the operator $\\\\widehat{a}(k) =\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)$ , we get a multiplication operator on some Hilbert space $L^2(\\\\mathfrak {E}_k,\\\\mu _k)$ taking values $\\\\widetilde{a}(k)(e)$ for $e\\\\in \\\\mathfrak {E}_k$ .\",\n \"Since, $\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)$ is formed from $\\\\widehat{a}(k) =\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)$ , it can also be diagonalized over $L^2(\\\\mathfrak {E}_k,\\\\mu _k)$ , and takes values $\\\\widetilde{\\\\tau }_\\\\lambda (k)(e) = \\\\frac{\\\\widetilde{a}(k)(e)}{1 - \\\\lambda \\\\widetilde{a}(k)(e)}.$ Since $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right) < 1$ (from the proof of Lemma REF ), we know $\\\\lambda \\\\widetilde{a}(k)(e) >1$ and $\\\\lambda \\\\widetilde{\\\\tau }_\\\\lambda (k)(e) \\\\ge -1$ for $\\\\mu _k$ -almost every $e\\\\in \\\\mathfrak {E}_k$ .\",\n \"It therefore only remains to bound $\\\\widetilde{\\\\tau }_\\\\lambda (k)(e)$ from above.\",\n \"Equivalently, we only need to bound $\\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right)$ from above.\",\n \"We also note that the proof of () can be adapted to show that $\\\\lambda \\\\mapsto \\\\left$ is continuous on $\\\\left[0,\\\\lambda _O\\\\right]$ for all $f\\\\in L^2(\\\\mathcal {E})$ .\",\n \"Then because $\\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right) = \\\\sup _{f}\\\\tfrac{\\\\left}{\\\\left}$ , this proves that $\\\\lambda \\\\mapsto \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right)$ is lower-semicontinuous on $\\\\left[0,\\\\lambda _O\\\\right]$ .\",\n \"The bound $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right) <1$ for $\\\\lambda < \\\\lambda _O$ (from Lemma REF ) then implies that $\\\\lambda _O\\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda _O}(0)\\\\right)\\\\le 1$ .\",\n \"From (REF ) the calculation proceeds: $\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{\\\\mathcal {T}}_\\\\lambda (k)\\\\right) &= \\\\frac{\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)}{1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)} \\\\nonumber \\\\\\\\& = \\\\frac{\\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)}{\\\\underbrace{1 - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right)}_{\\\\ge 0} + \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(0) + \\\\widehat{}_{\\\\lambda }(0)\\\\right) - \\\\lambda \\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)} \\\\nonumber \\\\\\\\& \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}(k) + \\\\widehat{}_{\\\\lambda }(k)\\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) + \\\\delta _{\\\\lambda }(0) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) - \\\\delta _{\\\\lambda }(k)} \\\\nonumber \\\\\\\\& \\\\le \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) + \\\\mathcal {O}\\\\left(\\\\beta \\\\right)}{\\\\left(\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\left(1 + \\\\mathcal {O}\\\\left(\\\\beta \\\\right)\\\\right)} = \\\\frac{\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) + \\\\mathcal {O}\\\\left(\\\\beta \\\\right)}{\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)},$ where we have bounded $\\\\delta _{\\\\lambda }(0)-\\\\delta _{\\\\lambda }(k)$ using Lemma REF and Observation REF .\"\n ],\n [\n \"Critical Intensities\",\n \"[Proof of Proposition REF ] By the Mecke equation (REF ), we find $*{\\\\chi _\\\\lambda }_1 & = 1 + \\\\lambda \\\\int \\\\tau _\\\\lambda (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right) = 1 + \\\\lambda \\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,b)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right),\\\\\\\\*{\\\\chi _\\\\lambda }_\\\\infty & = 1 + \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{b}\\\\int \\\\tau _\\\\lambda (\\\\overline{x};a,b)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = 1 + \\\\lambda \\\\operatornamewithlimits{ess\\\\,sup}_{b}\\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,b) \\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = 1 + \\\\lambda \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{1,\\\\infty }.$ From Lemma REF we have $\\\\int \\\\widehat{\\\\tau }_\\\\lambda (0;a,b)\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{\\\\rm op} \\\\le \\\\left\\\\Vert \\\\widehat{\\\\mathcal {T}}_\\\\lambda (0)\\\\right\\\\Vert _{1,\\\\infty }$ .\",\n \"Therefore $\\\\lambda ^{(1)}_T \\\\ge \\\\lambda _O \\\\ge \\\\lambda _T^{(\\\\infty )}$ .\",\n \"Also recall from the discussion in Section REF that $\\\\lambda _c \\\\ge \\\\lambda ^{(1)}_T$ .\",\n \"Therefore we only now need to prove that $\\\\lambda ^{(1)}_T = \\\\lambda ^{(\\\\infty )}_T$ .\",\n \"We do this by proving that if $*{\\\\chi _\\\\lambda }_1<\\\\infty $ then $*{\\\\chi _\\\\lambda }_\\\\infty <\\\\infty $ .\",\n \"Fix $b\\\\in \\\\mathcal {E}$ .\",\n \"Then by considering the vertices in the cluster $\\\\overline{0},b)$ adjacent to $\\\\left(\\\\overline{0},b\\\\right)$ , we find $\\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},b)}\\\\right] \\\\le 1 + \\\\mathbb {E}_\\\\lambda \\\\left[\\\\sum _{\\\\left(\\\\overline{x},a\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},a\\\\right)\\\\sim \\\\left(\\\\overline{0},b\\\\right)}*{\\\\overline{x},a)}\\\\right] = 1 + \\\\lambda \\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{x},a)}\\\\right]\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right).$ In this equality we have used Mecke's equation (REF ).\",\n \"From the spatial translation invariance of the model, $\\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{x},a)}\\\\right] = \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},a)}\\\\right]$ for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ .\",\n \"Therefore an application of a supremum bound to the $a$ -integral gives $*{\\\\chi _\\\\lambda }_\\\\infty = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\mathcal {E}}\\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},b)}\\\\right] \\\\le 1 + \\\\lambda \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\right)\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{\\\\overline{0},c)}\\\\right]\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right)\\\\\\\\= 1 + \\\\lambda \\\\left(\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\right)*{\\\\chi _\\\\lambda }_1.$ From the finiteness of the parenthesised factor, $*{\\\\chi _\\\\lambda }_1<\\\\infty $ implies $*{\\\\chi _\\\\lambda }_\\\\infty <\\\\infty $ and the result is proven.\",\n \"[Proof of Proposition REF ] The equality $\\\\lambda _c\\\\left(\\\\dagger \\\\right)= \\\\lambda _T\\\\left(\\\\dagger \\\\right)$ holds from [35].\",\n \"The equality $\\\\lambda _O = \\\\lambda _T^{(p)}$ follows from Proposition REF because the condition (REF ) clearly holds for the bounded radii model.\",\n \"Also recall from the discussion in Section REF that $\\\\lambda _c \\\\ge \\\\lambda ^{(p)}_T$ for all $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"Now note that for all $a,b\\\\in \\\\mathcal {E}$ we have that if $\\\\left(\\\\overline{0},\\\\dagger \\\\right)\\\\sim \\\\left(\\\\overline{x},b\\\\right)$ then $\\\\left(\\\\overline{0},a\\\\right)\\\\sim \\\\left(\\\\overline{x},b\\\\right)$ .\",\n \"Therefore $\\\\lambda _c\\\\left(\\\\dagger \\\\right) \\\\ge \\\\lambda _c$ , and the equality $\\\\lambda _c\\\\left(\\\\dagger \\\\right)= \\\\lambda _T\\\\left(\\\\dagger \\\\right)$ then means we only need to prove $\\\\lambda _T\\\\left(\\\\dagger \\\\right) \\\\le \\\\lambda _T^{(p)}$ for some $p\\\\in \\\\left[1,\\\\infty \\\\right]$ .\",\n \"We will do so for $p=1$ .\",\n \"Suppose that $\\\\lambda >\\\\lambda _T^{(1)}$ , so that $\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[(\\\\overline{0};a\\\\right)\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\infty $ .\",\n \"That is, the expected cluster size of a vertex with a random ($\\\\mathcal {P}$ -distributed) mark is infinite.\",\n \"Note that we have an ordering of the marks in that if two radii $a,b$ satisfy $a\\\\ge b$ , then $\\\\varphi \\\\left(\\\\overline{x};a,c\\\\right) \\\\ge \\\\varphi \\\\left(\\\\overline{x};b,c\\\\right)$ for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ and $c\\\\in \\\\mathcal {E}$ .\",\n \"This also implies that $a\\\\mapsto \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},a}\\\\right]$ is a non-decreasing function.\",\n \"This in turn implies that if $\\\\lambda >\\\\lambda _T^{(1)}$ and $a$ satisfies $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , then $\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right) = \\\\infty $ .\",\n \"Given some radius $a\\\\in \\\\mathcal {E}$ such that $a>0$ and $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , we define a new mark, $\\\\star _a$ , that forms connections according to $\\\\varphi \\\\left(\\\\overline{x};\\\\star _a,b\\\\right) = 1\\\\left\\\\lbrace *{\\\\overline{x}}\\\\le a\\\\right\\\\rbrace ,$ for all $b\\\\in \\\\mathcal {E}$ .\",\n \"Note that if $b\\\\ge a$ then $\\\\varphi \\\\left(\\\\overline{x};\\\\star _a,c\\\\right) \\\\le \\\\varphi \\\\left(\\\\overline{x};b,c\\\\right)$ for all $c\\\\in \\\\mathcal {E}$ and $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};\\\\star _a}\\\\right] \\\\le \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]$ .\",\n \"We now consider the expected cluster size of vertices adjacent to $\\\\left(\\\\overline{0},\\\\star _a\\\\right)$ with radius $\\\\ge a$ .\",\n \"We also restrict to cases for which there is an unique neighbour, and that neighbour has radius $\\\\ge a$ .\",\n \"Since the probability of a connection is independent of the radius of the proposed neighbour, the distribution of the unique neighbour is equal to $\\\\mathcal {P}$ - conditioned upon having radius $\\\\ge a$ .\",\n \"Therefore we have the lower bound $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};\\\\star _a}\\\\right] \\\\ge 1 + \\\\mathbb {P}_\\\\lambda \\\\left(\\\\exists !\",\n \"\\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\star _a\\\\right),b\\\\ge a\\\\right)\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right).$ As noted above, $\\\\int \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};b}\\\\right]1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)=\\\\infty $ .\",\n \"Furthermore, as $a>0$ and $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , the random variable $\\\\#\\\\left\\\\lbrace \\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\star _a\\\\right),b\\\\ge a\\\\right\\\\rbrace $ obeys a Poisson distribution with strictly positive mean.\",\n \"Therefore $\\\\mathbb {P}_\\\\lambda \\\\left(\\\\exists !\",\n \"\\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\star _a\\\\right),b\\\\ge a\\\\right)>0$ and $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};\\\\star _a}\\\\right]=\\\\infty $ .\",\n \"This in turn implies that $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};a}\\\\right] =\\\\infty $ for all $a>0$ .\",\n \"To relate this to $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},\\\\dagger }\\\\right]$ , we perform a similar lower bound.\",\n \"Given a radius $a>0$ and $\\\\int 1\\\\left\\\\lbrace b\\\\ge a\\\\right\\\\rbrace \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)>0$ , we restrict to cases where there is an unique neighbour of $\\\\left(\\\\overline{0},\\\\dagger \\\\right)$ with radius $\\\\ge a$ and bound $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},\\\\dagger }\\\\right] \\\\ge 1+ \\\\mathbb {P}_\\\\lambda \\\\left(\\\\exists !\",\n \"\\\\left(\\\\overline{x},b\\\\right)\\\\in \\\\eta :\\\\left(\\\\overline{x},b\\\\right)\\\\sim \\\\left(\\\\overline{0},\\\\dagger \\\\right),b\\\\ge a\\\\right) \\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0};a}\\\\right].$ As before, the probability factor is strictly positive and the expected cluster size term is infinite.\",\n \"Hence $\\\\mathbb {E}_\\\\lambda \\\\left[*{(\\\\overline{0},\\\\dagger }\\\\right]=\\\\infty $ and $\\\\lambda \\\\ge \\\\lambda \\\\left(\\\\dagger \\\\right)$ .\",\n \"Therefore $\\\\lambda _T\\\\left(\\\\dagger \\\\right)\\\\le \\\\lambda ^{(1)}_T$ as required.\"\n ],\n [\n \"Model Properties\",\n \"Here we will prove that the models outlined in Section REF do indeed satisfy Assumption REF .\",\n \"First we consider the single mark “finite variance models” considered by [21].\",\n \"As proven in that reference, this includes the Poisson blob and Gaussian connection models.\",\n \"Lemma B.1 Single mark finite variance models satisfy the conditions of Assumption REF .\",\n \"First note that since $\\\\mathcal {E}$ is a singleton, the operators $\\\\widehat{}(k)$ are simply scalars.\",\n \"In particular, this means that the conditions (REF ), and () of REF hold trivially.\",\n \"The finiteness of $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ is also equivalent to the finiteness of $\\\\int \\\\varphi (x)\\\\mathrm {d}x$ required by [21].\",\n \"Since $\\\\widehat{}(k)$ is simply a scalar, assumption REF follows directly from the third finite variance condition of [21].\",\n \"The condition REF is similarly a generalisation of the second finite variance condition, and therefore follows.\",\n \"Specifically, the sets $B(x)$ are the $\\\\varepsilon $ -balls around $x\\\\in \\\\mathbb {R}^d$ where $0<\\\\varepsilon = p^2\\\\widehat{\\\\varphi }_{1,1}(k)f_1^2 + 2pq\\\\widehat{\\\\varphi }_{1,2}(k)f_1f_2 + q^2\\\\widehat{\\\\varphi }_{2,2}(k)f_2^2,$ and therefore the spectrum of $\\\\widehat{}(k)$ is equal to the set of eigenvalues of the matrix $\\\\begin{pmatrix}p^2\\\\widehat{\\\\varphi }_{1,1}(k) & pq\\\\widehat{\\\\varphi }_{1,2}(k)\\\\\\\\pq\\\\widehat{\\\\varphi }_{1,2}(k) & q^2\\\\widehat{\\\\varphi }_{2,2}(k)\\\\end{pmatrix}.$ This $2\\\\times 2$ matrix has characteristic polynomial given by $z \\\\mapsto z^2 - \\\\left(p^2\\\\widehat{\\\\varphi }_{1,1}(k) + q^2\\\\widehat{\\\\varphi }_{2,2}(k)\\\\right) z + p^2q^2\\\\left( \\\\widehat{\\\\varphi }_{1,1}(k)\\\\widehat{\\\\varphi }_{2,2}(k) - \\\\widehat{\\\\varphi }_{1,2}(k)^2\\\\right).$ Solving for roots of this quadratic then gives $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) = \\\\frac{p^2\\\\widehat{\\\\varphi }_{1,1}(k) + q^2\\\\widehat{\\\\varphi }_{2,2}(k)}{2} + \\\\sqrt{\\\\left(\\\\frac{p^2\\\\widehat{\\\\varphi }_{1,1}(k) - q^2\\\\widehat{\\\\varphi }_{2,2}(k)}{2}\\\\right)^2 + p^2q^2\\\\widehat{\\\\varphi }_{1,2}(k)^2},$ where the square root takes the positive value.\",\n \"Note that by the definition of the double mark finite variance model $p^2\\\\widehat{\\\\varphi }_{1,1}(k) - q^2\\\\widehat{\\\\varphi }_{2,2}(k)$ is the Fourier transform of a non-negative (or non-positive) function, and therefore $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)$ is a sum of terms maximised at $k=0$ .\",\n \"Furthermore, each approaches that maximum quadratically and at least one of $\\\\widehat{\\\\varphi }_{1,1}(k)$ , $\\\\widehat{\\\\varphi }_{2,2}(k)$ , and $\\\\widehat{\\\\varphi }_{1,2}(k)$ uniquely achieves its maximum at $k=0$ (otherwise we have $\\\\varphi = 0$ almost everywhere).\",\n \"Therefore REF is satisfied.\",\n \"Since we have only finitely many marks, there are only finitely many sequences of marks of a given length.\",\n \"Therefore since each of the $\\\\varphi _{i,j}$ individually satisfy the 1-mark analog of REF , the 2-mark version is satisfied by $\\\\varphi $ too.\",\n \"It is not inconceivable that the above argument could be generalised to 3 or 4 mark systems.\",\n \"The degree 3 and 4 characteristic polynomials arising from the matrices have roots that can be solved for in radicals.\",\n \"It may be then that a particular clever rearrangement of the expression allows the required properties to be deduced like above.\",\n \"Such an investigation is currently beyond the authors' patience.\",\n \"Of course, the famous Abel-Ruffini Theorem states that no such solution in radicals exists for arbitrary polynomials of degree 5 and higher, and so this approach seems doomed for systems with such families of marks.\",\n \"Lemma B.3 Space-mark factorisation models satisfy the conditions of Assumption REF .\",\n \"Note that $\\\\widehat{}(k) = \\\\widehat{\\\\overline{\\\\varphi }}(k)\\\\mathcal {K}$ , where $\\\\mathcal {K}\\\\colon L^2(\\\\mathcal {E}) \\\\rightarrow L^2(\\\\mathcal {E})$ is the self-adjoint linear integral operator with kernel function $K$ .\",\n \"Since $\\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{b}\\\\int K(a,b)\\\\mathcal {P}(\\\\mathrm {d}a)\\\\le 1$ , $\\\\mathcal {K}$ is a bounded operator and $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) = {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\widehat{\\\\overline{\\\\varphi }}(k)\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right) &: \\\\widehat{\\\\overline{\\\\varphi }}(k) \\\\ge 0\\\\\\\\-\\\\widehat{\\\\overline{\\\\varphi }}(k)\\\\mathbb {S}\\\\left(-\\\\mathcal {K}\\\\right) &: \\\\widehat{\\\\overline{\\\\varphi }}(k) < 0.\\\\end{array}\\\\right.\",\n \"}$ Since $\\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{\\\\rm op}\\\\le 1$ , the finiteness of $\\\\widehat{\\\\overline{\\\\varphi }}(0)$ implies that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ is finite, and the $d$ -independence of $\\\\mathcal {K}$ implies that REF holds.\",\n \"Since $\\\\overline{\\\\varphi }(\\\\overline{x})\\\\ge 0$ , we have $*{\\\\widehat{\\\\overline{\\\\varphi }}(k)} < \\\\widehat{\\\\overline{\\\\varphi }}(0)$ for all $k\\\\ne 0$ .\",\n \"Furthermore, $\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right) = \\\\left\\\\Vert \\\\mathcal {K}\\\\right\\\\Vert _{\\\\rm op}$ implies that $\\\\mathbb {S}\\\\left(\\\\mathcal {K}\\\\right)\\\\ge \\\\mathbb {S}\\\\left(-\\\\mathcal {K}\\\\right)$ .\",\n \"Then $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)$ inherits the properties required by REF from $\\\\widehat{\\\\overline{\\\\varphi }}(k)$ .\",\n \"For REF , we note that the $d$ -independence of $\\\\mathcal {K}$ means that required decay properties are again inherited from $\\\\widehat{\\\\overline{\\\\varphi }}(k)$ .\",\n \"Lemma B.4 The marked multivariate Gaussian model satisfies the conditions of Assumption REF .\",\n \"Firstly note that for each $\\\\left(a,b\\\\right)\\\\in \\\\mathcal {E}^2$ the Gaussian structure of $\\\\varphi $ means that it factorises over the $d$ eigenvector directions of $\\\\Sigma (a,b)$ .\",\n \"This factorisation ensures that that the $d$ -dimensional Fourier transform is the product of the 1-dimensional Fourier transforms.\",\n \"We have $\\\\widehat{\\\\varphi }(k;a,b) = \\\\mathcal {A} \\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2}k^{\\\\intercal }\\\\Sigma (a,b)^{-1}k\\\\right).$ In particular, when $k=0$ the Fourier transform is $a,b$ -independent.\",\n \"This means $\\\\widehat{}(0) = \\\\mathcal {A}\\\\mathbf {1}$ where $\\\\mathbf {1}$ is the integral operator with constant kernel function 1.\",\n \"For $f\\\\in L^2(\\\\mathcal {E})$ we have ${f}{\\\\mathbf {1}f} &= \\\\int f(a) \\\\left(\\\\int f(b) \\\\mathcal {P}\\\\left(\\\\mathrm {d}b\\\\right)\\\\right) \\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\mathbb {E}_\\\\mathcal {P}\\\\left[f\\\\right]^2,\\\\\\\\{f}{f} &= \\\\int f(a)^2 \\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\mathbb {E}_\\\\mathcal {P}\\\\left[f^2\\\\right].$ By Jensen's inequality we therefore have $\\\\mathbb {S}\\\\left(\\\\mathbf {1}\\\\right) = 1$ (equality follows by considering the test function $f(a)\\\\equiv 1$ ).\",\n \"It therefore follows that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) = \\\\mathcal {A}<\\\\infty $ .\",\n \"From (REF ) we have $\\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = \\\\mathcal {A}\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}} \\\\exp \\\\left(-\\\\pi ^2k^{\\\\intercal }\\\\Sigma (a,b)^{-1}k\\\\right) \\\\le \\\\mathcal {A}\\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2\\\\Sigma _{\\\\max }}*{k}^2\\\\right).$ This bound then implies the condition (REF ) of REF holds with $C=1$ .\",\n \"Before we address () of REF , we turn to REF .\",\n \"From our above bounds we already have $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{1,\\\\infty } \\\\le \\\\left\\\\Vert \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\mathcal {A}\\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2\\\\Sigma _{\\\\max }}*{k}^2\\\\right) = \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\exp \\\\left(-\\\\frac{\\\\pi ^2}{2\\\\Sigma _{\\\\max }}*{k}^2\\\\right).$ This exponential term ensures that REF is satisfied.\",\n \"We now return to () of REF .\",\n \"For all $x\\\\in \\\\mathbb {R}$ we have $1-\\\\cos (x) \\\\le \\\\frac{1}{2}x^2$ , and therefore $\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le \\\\frac{1}{2}*{k}^2 \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x},$ for some unit vector $e\\\\in \\\\mathbb {R}^d$ .\",\n \"Since the vector $e$ picks out the second moment of $\\\\varphi $ in only one direction, a standard calculation gives $\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\le \\\\Sigma _{\\\\max }^2\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }.$ Therefore $\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }$ is bounded by a $d$ -independent quadratic function.\",\n \"In conjunction with the quadratic bound of REF , this then proves () in the $k\\\\rightarrow 0$ regime.\",\n \"The remaining regime then holds because the triangle inequality implies $\\\\left\\\\Vert \\\\widehat{}(0) - \\\\widehat{}(k)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le 2\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } = 2\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"To address REF , recall that the spatial convolution of two multivariate Gaussian functions with means $\\\\overline{0}$ and covariance matrices $\\\\Sigma _1$ and $\\\\Sigma _2$ is a multivariate Gaussian function with mean $\\\\overline{0}$ and covariance matrix $\\\\Sigma _1+\\\\Sigma _2$ .\",\n \"Therefore for all $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ we have $&\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,a_2,a_3,a_4\\\\in \\\\mathcal {E}}\\\\int \\\\varphi (\\\\overline{x}-\\\\overline{y};a_1,a_2)\\\\varphi (\\\\overline{y};a_3,a_4)\\\\mathrm {d}\\\\overline{y}\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}= \\\\mathcal {A}^2\\\\left(2\\\\pi \\\\right)^{-d/2}\\\\operatornamewithlimits{ess\\\\,sup}_{a_1,a_2,a_3,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\det \\\\left(\\\\Sigma (a_1,a_2) + \\\\Sigma (a_3,a_4)\\\\right)\\\\right)^{-1/2}\\\\exp \\\\left(-\\\\frac{1}{2}\\\\overline{x}^{\\\\intercal }\\\\left(\\\\Sigma (a_1,a_2) + \\\\Sigma (a_3,a_4)\\\\right)^{-1}\\\\overline{x}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{14.22636pt} \\\\le \\\\mathcal {A}^2\\\\left(4\\\\pi \\\\Sigma _{\\\\min }\\\\right)^{-d/2}\\\\exp \\\\left(-\\\\frac{*{\\\\overline{x}}^2}{4\\\\Sigma _{\\\\min }}\\\\right) \\\\le \\\\mathcal {A}^2\\\\left(4\\\\pi \\\\Sigma _{\\\\min }\\\\right)^{-d/2} \\\\rightarrow 0.$ In particular, this bounds the integral appearing in the definition of the sets $B(x)$ (see (REF )) and therefore proves they are empty.\",\n \"For the bound of the convolution of three $\\\\varphi $ functions, we can get this immediately from the above calculation and $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"Specifically REF holds with $g(d) = \\\\left(4\\\\pi \\\\Sigma _{\\\\min }(d)\\\\right)^{-d/2}$ and the sets $B(x) = \\\\emptyset $ .\",\n \"Lemma B.5 The bounded-volume Boolean disc model satisfies the conditions of Assumption REF .\",\n \"Crucial to the conditions of REF holding is the upper bound on $R_d(a)$ in (REF ).\",\n \"We use the shorthand $R_{a,b} := R_d(a) + R_d(b),\\\\qquad V_{a,b} := \\\\frac{\\\\pi ^{\\\\frac{d}{2}}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)}R^d_{a,b},$ so $V_{a,b}$ equals the Lebesgue volume of the $d$ -dimensional Euclidean ball with radius $R_{a,b}$ .\",\n \"Note that the upper bound $R^{({\\\\rm max})}_d$ on $R_d(a)$ in (REF ) ensures $V_{a,b} \\\\le V^{({\\\\rm max})}$ uniformly in $d$ .\",\n \"We first find $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{1,\\\\infty } = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\left[0,1\\\\right]}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) = \\\\operatornamewithlimits{ess\\\\,sup}_{b\\\\in \\\\left[0,1\\\\right]}\\\\int V_{a,b}\\\\mathcal {P}\\\\left(\\\\mathrm {d}a\\\\right) \\\\le V^{({\\\\rm max})}.$ This proves the statement that $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ is finite in REF .\",\n \"From the condition (REF ) we also get $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\ge \\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\mathcal {P}^{\\\\otimes 2}\\\\left(\\\\mathrm {d}a, \\\\mathrm {d}b\\\\right) \\\\ge \\\\epsilon ^2\\\\frac{d\\\\pi ^\\\\frac{d}{2}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)} \\\\int ^{2\\\\left(1-\\\\frac{c_2}{d}\\\\right)R^{({\\\\rm max})}_d}_0 r^{d-1}\\\\mathrm {d}r \\\\\\\\= \\\\epsilon ^2V^{({\\\\rm max})}\\\\left(1-\\\\frac{c_2}{d}\\\\right)^d \\\\rightarrow \\\\epsilon ^2\\\\text{e}^{-c_2}V^{({\\\\rm max})},$ where we have used the test function $f(a) \\\\equiv 1$ to lower bound $\\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right)$ .\",\n \"We then prove (REF ) by bounding $\\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } =\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\left[0,1\\\\right]}\\\\int \\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}= \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\left[0,1\\\\right]}V_{a,b} \\\\le V^{({\\\\rm max})}.$ We will return for () after addressing REF .\",\n \"For REF , it will be useful to have an expression for $\\\\widehat{\\\\varphi }(k;a,b)$ for each $a,b\\\\in \\\\left[0,1\\\\right]$ .\",\n \"Suppose we are able to find $C_1\\\\in \\\\left(0,1\\\\right)$ and $C_2 >0$ such that for all $a,b\\\\in \\\\left[0,1\\\\right]$ $*{\\\\widehat{\\\\varphi }(k;a,b)} \\\\le \\\\widehat{\\\\varphi }(0;a,b) \\\\left(C_1 \\\\vee \\\\left(1 - C_2*{k}^2\\\\right)\\\\right).$ Then since the $k$ -dependent factor is $a,b$ -independent we can use Lemma REF to show that $\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right) \\\\le \\\\mathbb {S}\\\\left(\\\\widehat{}(0)\\\\right) \\\\left(C_1 \\\\vee \\\\left(1 - C_2*{k}^2\\\\right)\\\\right),$ and therefore prove that REF is satisfied.\",\n \"In finding an expression for $\\\\widehat{\\\\varphi }(k;a,b)$ we are assisted by the spherical symmetry of the connection function and follow [18].\",\n \"It transpires that $\\\\widehat{\\\\varphi }(k;a,b) = \\\\left(\\\\frac{2\\\\pi R_{a,b}}{*{k}}\\\\right)^\\\\frac{d}{2}J_{\\\\frac{d}{2}}\\\\left(R_{a,b}*{k}\\\\right),$ where $J_{\\\\frac{d}{2}}$ is the Bessel function of the first kind of order $\\\\frac{d}{2}$ .\",\n \"This function has the expansion $J_{\\\\frac{d}{2}}\\\\left(r\\\\right) = \\\\sum ^\\\\infty _{m=0} \\\\frac{\\\\left(-1\\\\right)^{m}}{m!\\\\Gamma \\\\left(\\\\frac{d}{2}+m+1\\\\right)}\\\\left(\\\\frac{r}{2}\\\\right)^{\\\\frac{d}{2}+2m},$ converging for all $r\\\\ge 0$ .\",\n \"We will consider three different regimes for $*{k}$ .\",\n \"For $r \\\\ll \\\\sqrt{d}$ the expansion (REF ) is asymptotic, and since $R^{({\\\\rm max})}_d$ and $R^{({\\\\rm min})}_d$ are both asymptotically proportional to $\\\\sqrt{d}$ we have $\\\\widehat{\\\\varphi }(k;a,b) \\\\sim \\\\frac{\\\\pi ^\\\\frac{d}{2}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)}R_{a,b}^d - \\\\frac{\\\\pi ^\\\\frac{d}{2}}{4\\\\Gamma \\\\left(\\\\frac{d}{2}+2\\\\right)}R_{a,b}^{d+2}*{k}^2 = V_{a,b}\\\\left(1 - \\\\frac{R^2_{a,b}}{2\\\\left(d+2\\\\right)}*{k}^2\\\\right)$ for $*{k}\\\\ll 1$ .\",\n \"This proves the quadratic part of the desired bound.\",\n \"The Bessel function $J_{\\\\frac{d}{2}}$ is bounded and achieves its global maximum (in absolute value) at its first non-zero stationary point, denoted $j^{\\\\prime }_{\\\\frac{d}{2},1}$ .\",\n \"From [2], we have $j^{\\\\prime }_{\\\\frac{d}{2},1} = \\\\frac{d}{2} + \\\\gamma _1\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} + \\\\mathcal {O}\\\\left(d^{-\\\\frac{1}{3}}\\\\right)$ for a given $\\\\gamma _1 \\\\approx 0.81$ , and $*{J_{\\\\frac{d}{2}}\\\\left(j^{\\\\prime }_{\\\\frac{d}{2},1}\\\\right)} \\\\le M d^{-\\\\frac{1}{3}}$ for a $d$ -independent $M>0$ .\",\n \"Since $R_{a,b} \\\\in \\\\left[2c_1,1/\\\\sqrt{2\\\\pi \\\\text{e}}+o(1)\\\\right]d^\\\\frac{1}{2}$ (for $c_1\\\\in \\\\left(0,1/\\\\sqrt{8\\\\pi \\\\text{e}}\\\\right]$ from (REF )) and $j^{\\\\prime }_{\\\\frac{d}{2},1} / R_{a,b} \\\\in \\\\left[\\\\sqrt{\\\\frac{\\\\pi \\\\text{e}}{2}}+o(1),\\\\frac{1}{4c_1}+o(1)\\\\right]d^{\\\\frac{1}{2}}$ we have $*{\\\\widehat{\\\\varphi }(k;a,b)} \\\\le \\\\left(\\\\frac{2\\\\pi R_{a,b}}{*{k}}\\\\right)^\\\\frac{d}{2} M d^{-\\\\frac{1}{3}} \\\\le \\\\left(2e^{-1} + o(1)\\\\right)^\\\\frac{d}{2} M d^{-\\\\frac{1}{3}} \\\\rightarrow 0$ for $*{k} \\\\ge j^{\\\\prime }_{\\\\frac{d}{2},1} / R_{a,b}$ .\",\n \"We are now left with the intermediate range for $*{k}$ .\",\n \"The first positive zero of $J_\\\\frac{d}{2}$ occurs at $j_{\\\\frac{d}{2},1}$ , where $j_{\\\\frac{d}{2},1} = \\\\frac{d}{2} + \\\\gamma _2\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} + \\\\mathcal {O}\\\\left(d^{-\\\\frac{1}{3}}\\\\right)$ for a given $\\\\gamma _2 \\\\approx 1.86$ (see [2]).\",\n \"In particular, we will always have $j^{\\\\prime }_{\\\\frac{d}{2},1} < j_{\\\\frac{d}{2},1}$ .\",\n \"From differential identities relating Bessel functions (see [18]), we have $\\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}*{k}} \\\\widehat{\\\\varphi }(k;a,b) = -R_{a,b}\\\\left(\\\\frac{2\\\\pi R_{a,b}}{*{k}}\\\\right)^\\\\frac{d}{2}J_{\\\\frac{d}{2}+1}\\\\left(R_{a,b}*{k}\\\\right).$ The Bessel function $J_{\\\\frac{d}{2}+1}(r)$ is positive for $r>0$ until its first positive zero at $j_{\\\\frac{d}{2}+1,1}$ , and $j_{\\\\frac{d}{2}+1,1} > j_{\\\\frac{d}{2},1}$ .\",\n \"Therefore $\\\\widehat{\\\\varphi }(k;a,b)$ is positive and decreasing on the whole region $*{k}\\\\in \\\\left(0,j_{\\\\frac{d}{2},1}/R_{a,b}\\\\right]$ .\",\n \"Since this overlaps with the high $*{k}$ range, and we have the uniform quadratic behaviour near $k=0$ , the function $k\\\\mapsto *{\\\\widehat{\\\\varphi }(k;a,b)}$ can never increase and approach $\\\\widehat{\\\\varphi }(0;a,b)$ again after leaving a neighbourhood of $k=0$ .\",\n \"We therefore have the bound (REF ) and thus have proven that REF is satisfied.\",\n \"Figure: Sketch of ϕ ^k;a,b\\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) against *k*{k}.\",\n \"It approaches its maximum quadratically as *k→0*{k}\\\\rightarrow 0.\",\n \"The first local maximum of J d 2 J_{\\\\frac{d}{2}} occurs at j d 2,1 ' ∼d 2+γ 1 d 2 1 3 j^{\\\\prime }_{\\\\frac{d}{2},1}\\\\sim \\\\frac{d}{2}+\\\\gamma _1\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3}.\",\n \"The first zero of ϕ ^k;a,b\\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) occurs at *kR a,b =j d 2,1 ∼d 2+γ 2 d 2 1 3 *{k}R_{a,b} = j_{\\\\frac{d}{2},1}\\\\sim \\\\frac{d}{2}+\\\\gamma _2\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} where γ 2 >γ 1 \\\\gamma _2>\\\\gamma _1.\",\n \"Furthermore, ϕ ^k;a,b\\\\widehat{\\\\varphi }\\\\left(k;a,b\\\\right) is strictly decreasing until *kR a,b =j d 2+1,1 ∼d 2+γ 2 d 2 1 3 +1 2*{k}R_{a,b} = j_{\\\\frac{d}{2}+1,1}\\\\sim \\\\frac{d}{2}+\\\\gamma _2\\\\left(\\\\frac{d}{2}\\\\right)^\\\\frac{1}{3} + \\\\frac{1}{2}.We return to () in REF .\",\n \"We first calculate the second moment: $\\\\int *{\\\\overline{x}}^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}= \\\\frac{d}{d+2}V_{a,b}R^2_{a,b}.$ From $*{\\\\overline{x}}^2 = \\\\overline{x}_1^2 + \\\\ldots + \\\\overline{x}^2_d$ (for an orthonormal basis) and the rotational symmetry, we have $\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}= \\\\frac{1}{d+2}V_{a,b}R^2_{a,b}$ for all unit vectors $e\\\\in \\\\mathbb {R}^d$ .\",\n \"Therefore $\\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}}\\\\int \\\\left(e\\\\cdot \\\\overline{x}\\\\right)^2\\\\varphi \\\\left(\\\\overline{x};a,b\\\\right)\\\\mathrm {d}\\\\overline{x}\\\\le \\\\frac{4}{d+2}V^{({\\\\rm max})}\\\\left(R^{({\\\\rm max})}_d\\\\right)^2.$ While $V^{({\\\\rm max})}$ is a constant, $R^{({\\\\rm max})}_d\\\\sim \\\\sqrt{d/8\\\\pi \\\\text{e}}$ and therefore we have a bound uniform in $d$ .\",\n \"The argument now proceeds similarly to the corresponding part of the proof for the multivariate Gaussian model.\",\n \"From (REF ) and REF we have both sides of () satisfying quadratic bounds around $k=0$ , and the triangle inequality with our other bounds prove that the bound also holds on the remainder.\",\n \"For REF we first want to bound the convolution of three $\\\\varphi $ functions.\",\n \"Given marks $a_1,\\\\ldots ,a_6\\\\in \\\\left[0,1\\\\right]$ and $\\\\overline{x}\\\\in \\\\mathbb {R}^d$ , let $\\\\mathbb {B}_{a,b}(\\\\overline{x})$ denote the ball of radius $R_{a,b}$ centred on $\\\\overline{x}$ , and let $1_{a,b}\\\\left(\\\\overline{x}\\\\right) = 1\\\\left\\\\lbrace *{\\\\overline{x}}\\\\le R_{a,b}\\\\right\\\\rbrace $ .\",\n \"Then $\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6)\\\\right)\\\\left(\\\\overline{x}\\\\right)&= \\\\int 1_{a_1,a_2}(\\\\overline{y})1_{a_3,a_4}(\\\\overline{z}-\\\\overline{y})1_{a_5,a_6}(-\\\\overline{z})\\\\mathrm {d}\\\\overline{y}\\\\mathrm {d}\\\\overline{z}\\\\nonumber \\\\\\\\&= \\\\int _{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)}\\\\left(\\\\int _{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)} 1_{a_3,a_4}(\\\\overline{z}-\\\\overline{y})\\\\mathrm {d}\\\\overline{z}\\\\right) \\\\mathrm {d}\\\\overline{y}\\\\nonumber \\\\\\\\&= \\\\int _{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)}\\\\mathrm {d}\\\\overline{y}.$ Then for $\\\\delta \\\\in \\\\left(0,1\\\\right)$ we have $&\\\\int _{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)}\\\\mathrm {d}\\\\overline{y}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\int _{\\\\delta \\\\mathbb {B}_{a,b}\\\\left(\\\\overline{0}\\\\right)}*{\\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{0}\\\\right)}\\\\mathrm {d}\\\\overline{y}+ *{\\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} \\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{y}\\\\in \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)} \\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\left(V^{({\\\\rm max})}\\\\right)^2\\\\delta ^d + V^{({\\\\rm max})}\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{y}\\\\in \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)}.$ Let $\\\\overline{e}_1\\\\in \\\\mathbb {R}^d$ be a unit vector.\",\n \"Then by spherical symmetry we have $\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{y}\\\\in \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)\\\\setminus \\\\delta \\\\mathbb {B}_{a_1,a_2}\\\\left(\\\\overline{0}\\\\right)} *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\overline{y}\\\\right)} = *{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\delta R_{a_1,a_2}\\\\overline{e}_1\\\\right)}.$ We bound the volume of this intersection with the volume of the $d$ -ball with radius equal to the radius of the $d-1$ -sphere formed by the intersection of their boundaries.\",\n \"This radius is equal to the length $h$ in Figure REF , which is maximised when $R_{a_5,a_6}$ and $R_{a_3,a_4}$ are maximised and $\\\\delta R_{a_1,a_2}$ is minimised.\",\n \"Recall we defined the constant $c_1\\\\in \\\\left(0,1/\\\\sqrt{8\\\\pi \\\\text{e}}\\\\right)$ when we defined the model in Section REF .\",\n \"Since we have $R_{a_5,a_6},R_{a_3,a_4} \\\\le 2R^{({\\\\rm max})}_d$ and $\\\\delta R_{a_1,a_2} \\\\ge \\\\delta c_1 d^\\\\frac{1}{2}$ , we have $h\\\\le \\\\sqrt{\\\\left(2 R^{({\\\\rm max})}_d\\\\right)^2 - \\\\delta ^2 c^2_1 d}$ - we have used the fact that the triangle is isosceles if both the $R_{a_5,a_6}$ and $R_{a_3,a_4}$ edges attain their maximum.\",\n \"The ball with this radius then has the volume $\\\\frac{\\\\pi ^\\\\frac{d}{2}}{\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)}h^d \\\\le V^{({\\\\rm max})}\\\\left(1 - \\\\frac{\\\\pi \\\\delta ^2 c^2_1 d}{\\\\left(V^{({\\\\rm max})}\\\\right)^\\\\frac{2}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{2}{d}}\\\\right)^\\\\frac{d}{2}.$ Since $\\\\frac{\\\\pi \\\\delta ^2 c^2_1 d}{\\\\left(V^{({\\\\rm max})}\\\\right)^\\\\frac{2}{d}\\\\Gamma \\\\left(\\\\frac{d}{2}+1\\\\right)^\\\\frac{2}{d}} = 2\\\\pi \\\\text{e}\\\\delta ^2c_1^2\\\\left(1+o(1)\\\\right)$ and $c_1>0$ , this shows that $*{\\\\mathbb {B}_{a_5,a_6}\\\\left(\\\\overline{0}\\\\right)\\\\cap \\\\mathbb {B}_{a_3,a_4}\\\\left(\\\\delta R_{a_1,a_2}\\\\overline{e}_1\\\\right)} \\\\le V^{({\\\\rm max})}\\\\left(1 - \\\\pi \\\\text{e}\\\\delta ^2c_1^2\\\\right)^\\\\frac{d}{2}$ for $d$ sufficiently large.\",\n \"Sine we now have a mark independent bound with the required decay, we have proved the bound for the convolution of three $\\\\varphi $ functions.\",\n \"Figure: For both the arguments, we bound the intersection of two dd-balls with a new dd-ball that has the radius hh in these diagrams.For finding the sets $B(x)$ in (REF ), we write $\\\\int \\\\varphi \\\\left(\\\\overline{x}- \\\\overline{y};a,c\\\\right)\\\\varphi \\\\left(\\\\overline{y};c,b\\\\right)\\\\mathrm {d}\\\\overline{y}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c\\\\right) = \\\\int *{\\\\mathbb {B}_{a,c}\\\\left(\\\\overline{x}\\\\right)\\\\cap \\\\mathbb {B}_{c,b}\\\\left(\\\\overline{0}\\\\right)}\\\\mathcal {P}\\\\left(\\\\mathrm {d}c \\\\right).$ As before, we can bound the volume of this intersection independently of the mark $c$ .\",\n \"We can bound it with the volume of the ball with radius $h$ where $h\\\\le \\\\sqrt{\\\\left(2R^{({\\\\rm max})}_d\\\\right)^2 - \\\\frac{1}{4}*{\\\\overline{x}}^2}$ (as can be seen in Figure REF ).\",\n \"As before, if $*{\\\\overline{x}} \\\\ge \\\\kappa d^\\\\frac{1}{2}$ for some $\\\\kappa >0$ , then the volume of the ball with radius $h$ vanishes in the $d\\\\rightarrow \\\\infty $ limit.\",\n \"Furthermore, if $\\\\kappa \\\\le \\\\frac{1}{\\\\sqrt{2\\\\pi \\\\text{e}}}$ then the volume of the ball with radius $\\\\kappa d^\\\\frac{1}{2}$ vanishes in the $d\\\\rightarrow \\\\infty $ limit.\",\n \"Therefore assumption REF is satisfied with the sets $B(x)\\\\equiv B\\\\left(\\\\overline{x},a\\\\right)=\\\\left\\\\lbrace \\\\left(\\\\overline{y},b\\\\right)\\\\in \\\\mathbb {X}:*{\\\\overline{x}-\\\\overline{y}} \\\\le \\\\kappa d^\\\\frac{1}{2}\\\\right\\\\rbrace $ for any $\\\\kappa \\\\in \\\\left(0,\\\\frac{1}{\\\\sqrt{2\\\\pi \\\\text{e}}}\\\\right)$ .\"\n ],\n [\n \"Displacement Diagram Case Distinctions\",\n \"This section provides the details omitted from the proof of Proposition REF .\",\n \"Here we show which diagrams arise from displacements crossing a $\\\\Psi $ segment, and whether they can be dealt with by the first sub-case, $\\\\mathrm {\\\\left(I\\\\right)}$ , or by the second sub-case, $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"Recall that $\\\\mathrm {\\\\left(I\\\\right)}$ are those that use a $\\\\Psi $ decomposition of the larger diagram only (and the displacement is not crossing a $\\\\Psi ^{(4)}$ segment), while $\\\\mathrm {\\\\left(II\\\\right)}$ are those that will require a combination of a $\\\\Psi $ decomposition and a $\\\\overline{\\\\Psi }$ decomposition.\",\n \"We first expand out the displaced $\\\\Psi $ terms and label which are in sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ and which are in sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"Note that the displacement may be across the `top' or the `bottom' of the segments in the usual orientation.\",\n \"If we consider a displacement across the `top' or the `bottom' of a $\\\\Psi ^{(1)}$ term, we use cosine-splitting and sometimes expand $\\\\tau _\\\\lambda ^\\\\circ $ edges into a $\\\\tau _\\\\lambda $ edge and a contraction to get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (2,0.8);(1,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[below]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad \\\\right)\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}}\\\\quad \\\\right).$ Using the same ideas, if we consider a displacement across the `top' or the `bottom' of a $\\\\Psi ^{(2)}$ term we get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.25) circle (0pt) node[right]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad + \\\\quad \\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}}\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.25) circle (0pt) node[right]{\\\\circ };(0.75,0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le 2\\\\left(\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad +\\\\quad 2\\\\times \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}} \\\\quad +\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}}\\\\right.\\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\left.\\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}} \\\\quad +\\\\quad 2\\\\times \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}} \\\\quad +\\\\quad \\\\underbrace{\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(I\\\\right)}}\\\\quad \\\\right).$ If we consider a displacement across the `top' or the `bottom' of a $\\\\Psi ^{(3)}$ term, then by symmetry in both cases we get $\\\\underbrace{\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}}_{\\\\mathrm {\\\\left(II\\\\right)}}.$ We first show the bounds for the single segments in sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ .\",\n \"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\vee \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (3.5,0) {\\\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ We first consider the displaced $\\\\Psi ^{(j)}$ term by itself.\",\n \"By using supremum bounds and spatial translations we arrive at $\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda , & \\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda , & \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda , \\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle ^\\\\circ _\\\\lambda , & \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda , & \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle ^\\\\circ _\\\\lambda .$ For most of these, it is sufficient to separate the pair using a supremum bound and use the individual bound above for the displaced $\\\\Psi ^{(j)}$ term and the bound on the $\\\\Psi $ or $\\\\Psi _n$ term from Lemma REF .\",\n \"The only pairs that need a more careful approach are those that have the bound $W_k\\\\triangle ^\\\\circ _\\\\lambda $ above for the displaced $\\\\Psi ^{(j)}$ , followed by a $\\\\Psi ^{(4)}$ or $\\\\Psi ^{(2)}_n$ term.\",\n \"In these cases we consider the pair together and get the diagram bounds $\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda \\\\le W_kV_\\\\lambda ^2,\\\\qquad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le W_k\\\\triangle _\\\\lambda \\\\le W_kV_\\\\lambda ^2.$ In both of these cases we bounded a $\\\\tau _\\\\lambda $ edge by 1 (essentially omitting the edge) before applying a supremum bound.\",\n \"In the first we omit the diagonal edge and split the upper supremum vertex in two.\",\n \"In the second we omit the left-most vertical edge and apply a supremum bound at the bottom right vertex.\",\n \"We now turn our attention to sub-case $\\\\mathrm {\\\\left(II\\\\right)}$ .\",\n \"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\vee \\\\quad \\\\raisebox {-15pt}{\\\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };\\\\node at (2.5,0) {\\\\Psi ^{(j)}};\\\\node at (4.5,0) {\\\\overline{\\\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad \\\\le \\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda .$ Also recall that if the $\\\\overline{\\\\Psi }_0$ term is actually a $\\\\overline{\\\\Psi }^{(2)}_0$ term, then the $\\\\tau _\\\\lambda ^\\\\circ $ edge connecting them is actually just a $\\\\tau _\\\\lambda $ edge.\",\n \"We first investigate bounds for the displaced $\\\\Psi ^{(j)}$ term with the connecting $\\\\tau _\\\\lambda $ and $\\\\tau _\\\\lambda ^\\\\circ $ edges.\",\n \"By using supremum bounds, spatial translations, splitting supremum vertices in two, and sometimes expanding $\\\\tau _\\\\lambda ^\\\\circ $ edges into a $\\\\tau _\\\\lambda $ edge and a contraction, we get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(3,0.6) -- (2,0.6) -- (2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(2.5,-0.6) circle (0pt) node[above]{\\\\circ };[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda },\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\quad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(2,-0.6) -- (3,-0.6);(1.5,0.6) -- (3,0.6);[dashed] (3,0.6) -- (3,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\\\times };\\\\node [mark size=2pt] at (3,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad +\\\\quad \\\\raisebox {-25pt}{\\\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(1.5,0.6) -- (3,0.6);[dashed] (0,0.6) -- (0,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\\\times };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\nonumber \\\\\\\\&\\\\le \\\\overline{H_k}+ W_k\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda },\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda ,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda .$ For most of these, it is clear from a crude supremum bound that we have the bound $\\\\overline{W_k}V_\\\\lambda ^2 + \\\\overline{H_k}U_\\\\lambda $ for the pair of these with the subsequent $\\\\overline{\\\\Psi }$ or $\\\\overline{\\\\Psi }_0$ term (like we did for sub-case $\\\\mathrm {\\\\left(I\\\\right)}$ ).\",\n \"The only pairs that need a more careful approach are those that have the bound $\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda $ or $\\\\overline{W_k}\\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda $ above, followed by a $\\\\overline{\\\\Psi }^{(4)}$ or $\\\\overline{\\\\Psi }^{(2)}_0$ term.\",\n \"We first consider the cases where they are followed by a $\\\\overline{\\\\Psi }^{(4)}$ term.\",\n \"Here we use supremum bounds, spatial translations, and expanding $\\\\tau _\\\\lambda ^\\\\circ $ edges into a $\\\\tau _\\\\lambda $ edge and a contraction to get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\triangle ^\\\\circ _\\\\lambda \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda + \\\\triangle ^\\\\circ _\\\\lambda \\\\overline{W_k}\\\\triangle _\\\\lambda = \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda + \\\\overline{W_k}\\\\triangle _\\\\lambda = \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\overline{\\\\triangle ^{\\\\circ \\\\circ }_\\\\lambda }\\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ We now consider the cases where they are followed by a $\\\\overline{\\\\Psi }^{(2)}_0$ term.\",\n \"Recall that for this case, the connecting $\\\\tau _\\\\lambda ^\\\\circ $ edge is in fact only a $\\\\tau _\\\\lambda $ edge.\",\n \"We then get $\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(0.5,0.6) circle (0pt) node[above]{\\\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\triangle ^\\\\circ _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-15pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2,\\\\\\\\\\\\raisebox {-25pt}{\\\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\\\end{tikzpicture}}\\\\quad &\\\\le \\\\overline{W_k}\\\\triangle _\\\\lambda \\\\le \\\\overline{W_k}V_\\\\lambda ^2.$ The remaining sub-case $\\\\mathrm {\\\\left(III\\\\right)}$ was fully dealt with in the main text.\"\n ],\n [\n \"Proof of Lemma \",\n \"Recall that we wish to prove the following result: * We begin by defining a sightly different object to $\\\\overline{H_k}$ .\",\n \"Given $\\\\vec{a}=\\\\left(a_1,\\\\ldots ,a_{16}\\\\right)\\\\in \\\\mathcal {E}^{16}$ , define $H^{\\\\prime }\\\\left(\\\\overline{x}_1,\\\\overline{x}_2,\\\\overline{x}_3;\\\\vec{a};k\\\\right) :=& \\\\int \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{x}_1;a_1,a_2)\\\\tau _\\\\lambda (\\\\overline{u};a_3,a_4) \\\\tau _\\\\lambda (\\\\overline{v}-\\\\overline{s};a_5,a_6) \\\\tau _\\\\lambda (\\\\overline{v}+\\\\overline{x}_2-\\\\overline{t};a_7,a_8) \\\\nonumber \\\\\\\\&\\\\hspace{28.45274pt}\\\\times \\\\tau _\\\\lambda (\\\\overline{s}-\\\\overline{w};a_9,a_{10})\\\\tau _\\\\lambda (\\\\overline{w}-\\\\overline{u};a_{11},a_{12}) \\\\tau _\\\\lambda (\\\\overline{t}-\\\\overline{w};a_{13},a_{14}) \\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\tau _{\\\\lambda ,k}(\\\\overline{t}+ \\\\overline{x}_3-\\\\overline{u};a_{15},a_{16}) \\\\mathrm {d}\\\\overline{t}\\\\mathrm {d}\\\\overline{w}\\\\mathrm {d}\\\\overline{z}\\\\mathrm {d}\\\\overline{u},$ so we have $\\\\overline{H_k}\\\\le f(\\\\lambda )^5\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}_1,\\\\overline{x}_2\\\\in \\\\mathbb {R}^d,\\\\vec{a}\\\\in \\\\mathcal {E}^{16}}H^{\\\\prime }\\\\left(\\\\overline{x}_1,\\\\overline{x}_2,\\\\overline{0};\\\\vec{a};k\\\\right).$ We then proceed to bound $H^{\\\\prime }\\\\left(\\\\overline{x}_1,\\\\overline{x}_2,\\\\overline{x}_3;\\\\vec{a};k\\\\right)$ to get our result.\",\n \"This form is preferred because it more easily indicates what the Fourier arguments should be when we take a Fourier transform.\",\n \"A schematic Fourier diagram for this can be found in [21].\",\n \"We then expand out the $\\\\tau _\\\\lambda $ and $\\\\tau _{\\\\lambda ,k}$ terms.\",\n \"For each of the $\\\\tau _\\\\lambda $ terms we apply (REF ) and () to get $\\\\tau _\\\\lambda (\\\\overline{x};a,b) \\\\le \\\\varphi (\\\\overline{x};a,b) + \\\\lambda (\\\\varphi \\\\varphi )(\\\\overline{x};a,b)+ \\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi )(\\\\overline{x};a,b).$ Similarly, with the additional use of the cosine-splitting lemma, we get $\\\\tau _{\\\\lambda ,k}(\\\\overline{x};a,b) \\\\le \\\\varphi _k(\\\\overline{x};a,b) + 2\\\\lambda \\\\left( (\\\\varphi _k\\\\varphi )(\\\\overline{x};a,b) +(\\\\varphi \\\\varphi _k)(\\\\overline{x};a,b)\\\\right)+ 4\\\\lambda ^2(\\\\varphi \\\\tau _\\\\lambda \\\\varphi _k)(\\\\overline{x};a,b) \\\\\\\\ + 4\\\\lambda ^2(\\\\varphi \\\\tau _{\\\\lambda ,k}\\\\varphi )(\\\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\\\lambda \\\\le f(\\\\lambda )$ , we can realise that we only need to get appropriate bounds for the following objects.\",\n \"For $\\\\vec{j}\\\\in \\\\left\\\\lbrace 1,2,3\\\\right\\\\rbrace ^7\\\\times \\\\left\\\\lbrace 1,2,3,4,5\\\\right\\\\rbrace $ , define $\\\\mathcal {H}(k;\\\\vec{j}) := \\\\int P^{j_1}_{1,2}(l_1)P^{j_2}_{3,4}(l_1)P^{j_3}_{5,6}(l_2)P^{j_4}_{7,8}(l_2)P^{j_5}_{9,10}(l_1-l_2)P^{j_6}_{11,12}(l_1-l_3)P^{j_7}_{13,14}(l_2-l_3) \\\\\\\\ \\\\times Q^{j_8}_{15,16}(l_3;k)\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}},$ where $P^{j}_{m,n}(l) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_m}{\\\\widehat{}(l) g_n}} &:j=1\\\\\\\\*{*{g_m}{\\\\widehat{}(l)^2g_n}} &:j=2\\\\\\\\*{*{g_m}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}(l)g_n}} &: j=3,\\\\end{array}\\\\right.}\",\n \"\\\\qquad \\\\text{for }m,n\\\\in \\\\left\\\\lbrace 1,2,\\\\ldots ,14\\\\right\\\\rbrace ,\\\\\\\\Q^{j}_{15,16}(l;k) &:= {\\\\left\\\\lbrace \\\\begin{array}{ll}*{*{g_{15}}{\\\\widehat{}_k(l) g_{16}}} &:j=1\\\\\\\\*{*{g_{15}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{16}}} &:j=2\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{}_k(l) g_{16}}} &:j=3\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{16}}} &:j=4\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{16}}} &:j=5,\\\\end{array}\\\\right.\",\n \"}$ where $\\\\left\\\\lbrace g_m\\\\right\\\\rbrace _{m=1}^{16}$ are the functions of the form $g_m(a) = \\\\frac{1}{\\\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .\",\n \"If we can bound these $\\\\mathcal {H}(k;\\\\vec{j})$ appropriately, then we can combine them to get the required bound.\",\n \"We now recall pointwise bounds for the $P^{j}_{m,n}(l)$ and $Q^{j}_{15,16}(l)$ terms.\",\n \"In the proof of Lemma REF we derived $*{*{g_m}{\\\\widehat{}(l)g_n}} &\\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }\\\\\\\\*{*{g_m}{\\\\widehat{}(l)^2g_n}} &\\\\le *{g_m}{\\\\widehat{}(l)^2g_m}^{\\\\frac{1}{2}}*{g_n}{\\\\widehat{}(l)^2g_n}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_m}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}(l)g_n}} &\\\\le f(\\\\lambda )\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)*{g_m}{\\\\widehat{}(l)^2g_m}^{\\\\frac{1}{2}}*{g_n}{\\\\widehat{}(l)^2g_n}^{\\\\frac{1}{2}},$ and $*{*{g_{15}}{\\\\widehat{}_k(l) g_{16}}} &\\\\le \\\\left\\\\Vert \\\\widehat{}_k(l)\\\\right\\\\Vert _{\\\\infty ,\\\\infty } \\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\\\\\*{*{g_{15}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{16}}} &\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) *{g_{16}}{\\\\widehat{}(l)^2g_{16}}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_{15}}{\\\\widehat{}_k(l)\\\\widehat{}(l) g_{16}}} &\\\\le C\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right) *{g_{15}}{\\\\widehat{}(l)^2g_{15}}^{\\\\frac{1}{2}}\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_\\\\lambda (l)\\\\widehat{}_k(l) g_{16}}} &\\\\le Cf(\\\\lambda )\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)*{g_{15}}{\\\\widehat{}(l)^2g_{15}}\\\\\\\\*{*{g_{15}}{\\\\widehat{}(l)\\\\widehat{\\\\mathcal {T}}_{\\\\lambda ,k}(l)\\\\widehat{}(l) g_{16}}} &\\\\le Cf(\\\\lambda )\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k)*{g_{15}}{\\\\widehat{}(l)^2g_{15}}^\\\\frac{1}{2}*{g_{16}}{\\\\widehat{}(l)^2g_{16}}^\\\\frac{1}{2}$ where $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) = \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k).$ Also recall that we can bound $*{*{g_m}{\\\\widehat{}(l)g_n}}\\\\le C$ and $*{g_m}{\\\\widehat{}(l)^2g_m}\\\\le C^2$ if required.\",\n \"Note that all the bounds for the $Q^{j}_{15,16}(l)$ terms have a $\\\\left(1 - \\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ factor.\",\n \"We therefore only need to show that the remaining integrals of $\\\\widehat{}$ and $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ terms can all be bounded by some constant multiple of $\\\\beta ^2$ .\",\n \"Also note that the bounds for $Q^{1}_{15,16}(l)$ , $Q^{2}_{15,16}(l)$ , and $Q^{3}_{15,16}(l)$ only differ by a uniform constant (after we bound $*{g_m}{\\\\widehat{}(l)^2g_m}\\\\le C^2$ ), and so once we have dealt with one of these we have dealt with the other two.\",\n \"Let us first consider $\\\\vec{j}= \\\\left(1,1,1,1,1,1,1,1\\\\right)$ .\",\n \"After applying the uniform pointwise bound for $Q^{1}_{15,16}(l)$ , we use the Cauchy-Schwarz inequality to bound $&\\\\int *{*{g_1}{\\\\widehat{}(l_1)g_2}}*{*{g_3}{\\\\widehat{}(l_1)g_4}}*{*{g_5}{\\\\widehat{}(l_2)g_6}}*{*{g_7}{\\\\widehat{}(l_2)g_8}}*{*{g_9}{\\\\widehat{}(l_1-l_2)g_{10}}} \\\\nonumber \\\\\\\\&\\\\hspace{199.16928pt}\\\\times *{*{g_{11}}{\\\\widehat{}(l_1-l_3)g_{12}}}*{*{g_{13}}{\\\\widehat{}(l_2-l_3)g_{14}}}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}} \\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int *{g_1}{\\\\widehat{}(l_1)g_2}^2*{g_3}{\\\\widehat{}(l_1)g_4}^2*{g_5}{\\\\widehat{}(l_2)g_6}^2*{g_{13}}{\\\\widehat{}(l_2-l_3)g_{14}}^2\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\hspace{56.9055pt}\\\\times \\\\left(\\\\int *{g_7}{\\\\widehat{}(l_2)g_8}^2*{g_9}{\\\\widehat{}(l_1-l_2)g_{10}}^2*{g_{11}}{\\\\widehat{}(l_1-l_3)g_{12}}^2\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.\",\n \"$ To deal with these parentheses, we use a volume-preserving change of variables and factorise each into three integrals.\",\n \"For example, $\\\\int *{g_1}{\\\\widehat{}(l_1)g_2}^2*{g_3}{\\\\widehat{}(l_1)g_4}^2*{g_5}{\\\\widehat{}(l_2)g_6}^2*{g_{13}}{\\\\widehat{}(l_2-l_3)g_{14}}^2\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}} \\\\\\\\= \\\\left(\\\\int *{g_1}{\\\\widehat{}(l_1)g_2}^2*{g_3}{\\\\widehat{}(l_1)g_4}^2\\\\frac{\\\\mathrm {d}l_1}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)\\\\left(\\\\int *{g_5}{\\\\widehat{}(l_2)g_6}^2\\\\frac{\\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)\\\\left(\\\\int *{g_{13}}{\\\\widehat{}(l_3^{\\\\prime })g_{14}}^2\\\\frac{\\\\mathrm {d}l_3^{\\\\prime }}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right).$ There are now two types of integral we need to bound: $&\\\\int *{g_m}{\\\\widehat{}(l)g_n}^2 \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} = \\\\frac{1}{\\\\mathcal {P}\\\\left(E_m\\\\right)^2\\\\mathcal {P}\\\\left(E_n\\\\right)^2}\\\\int \\\\left(\\\\int _{E_m\\\\times E_n \\\\times E_m \\\\times E_n} \\\\widehat{\\\\varphi }(l;a,b)\\\\widehat{\\\\varphi }(l;c,d) \\\\mathcal {P}^{\\\\otimes 4}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b,\\\\mathrm {d}c,\\\\mathrm {d}d\\\\right)\\\\right) \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} = \\\\frac{1}{\\\\mathcal {P}\\\\left(E_m\\\\right)^2\\\\mathcal {P}\\\\left(E_n\\\\right)^2}\\\\int _{E_m\\\\times E_n \\\\times E_m \\\\times E_n} \\\\left(\\\\varphi (\\\\cdot ;a,b)\\\\star \\\\varphi (\\\\cdot ;c,d)\\\\right)\\\\left(\\\\overline{0}\\\\right)\\\\mathcal {P}^{\\\\otimes 4}\\\\left(\\\\mathrm {d}a,\\\\mathrm {d}b,\\\\mathrm {d}c,\\\\mathrm {d}d\\\\right)\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b,c,d\\\\in \\\\mathcal {E}} \\\\left(\\\\varphi (\\\\cdot ;a,b)\\\\star \\\\varphi (\\\\cdot ;c,d)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b,c,d\\\\in \\\\mathcal {E}} \\\\int \\\\varphi (\\\\overline{x};a,b)\\\\varphi (-\\\\overline{x};c,d)\\\\mathrm {d}\\\\overline{x}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b\\\\in \\\\mathcal {E}} \\\\int \\\\varphi (\\\\overline{x};a,b) \\\\mathrm {d}\\\\overline{x}\\\\le C, \\\\\\\\&\\\\int *{g_1}{\\\\widehat{}(l)g_2}^2 *{g_3}{\\\\widehat{}(l)g_4}^2 \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^d} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_8\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi (\\\\cdot ;a_1,a_2)\\\\star \\\\varphi (\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6) \\\\star \\\\varphi (\\\\cdot ;a_7,a_8)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\le g(d).\",\n \"$ In (REF ) we used $\\\\varphi \\\\left(-\\\\overline{x};c,d\\\\right)\\\\in \\\\left[0,1\\\\right]$ and REF .\",\n \"In () we bounded the convolution of four adjacency functions using REF .\",\n \"The result of this is that we can bound (REF ) by $C^{\\\\frac{5}{2}}g(d)^{\\\\frac{1}{2}}$ .\",\n \"Since $g(d)\\\\le \\\\beta ^4$ , this satisfies the bound we require.\",\n \"As noted above, this argument also proves the required bound for $\\\\vec{j}= \\\\left(1,1,1,1,1,1,1,2\\\\right)$ and $\\\\vec{j}= \\\\left(1,1,1,1,1,1,1,3\\\\right)$ .\",\n \"Now suppose that we change some subset of $\\\\left\\\\lbrace j_1,\\\\ldots ,j_7\\\\right\\\\rbrace $ from having value 1 to having value 2.\",\n \"The argument proceeds similarly, except that after we have factorised our integrals we now want to also bound integrals of the following types: $&\\\\int *{g_1}{\\\\widehat{}(l)^2g_1} *{g_2}{\\\\widehat{}(l)^2g_2}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt}= \\\\frac{1}{\\\\mathcal {P}\\\\left(E_1\\\\right)^2\\\\mathcal {P}\\\\left(E_2\\\\right)^2}\\\\int \\\\left(\\\\int _{E_1\\\\times E_1\\\\times E_2\\\\times E_2}\\\\widehat{\\\\varphi }^{(2)}(l;a,b)\\\\widehat{\\\\varphi }^{(2)}(l;c,d)\\\\mathcal {P}^{\\\\otimes 4}(\\\\mathrm {d}a,\\\\mathrm {d}b, \\\\mathrm {d}c, \\\\mathrm {d}d)\\\\right)\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt}= \\\\frac{1}{\\\\mathcal {P}\\\\left(E_1\\\\right)^2\\\\mathcal {P}\\\\left(E_2\\\\right)^2}\\\\int _{E_1\\\\times E_1\\\\times E_2\\\\times E_2}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a,b)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;c,d)\\\\right)(\\\\overline{0})\\\\mathcal {P}^{\\\\otimes 4}(\\\\mathrm {d}a,\\\\mathrm {d}b, \\\\mathrm {d}c, \\\\mathrm {d}d) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt}\\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a,b,c,d\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a,b)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;c,d)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\le g(d), \\\\\\\\&\\\\int *{g_1}{\\\\widehat{}(l)^2g_1} *{g_2}{\\\\widehat{}(l)^2g_2}*{g_3}{\\\\widehat{}(l)g_4}^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_8\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4) \\\\star \\\\varphi (\\\\cdot ;a_5,a_6) \\\\star \\\\varphi (\\\\cdot ;a_7,a_8)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^2\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\le C^2 g(d)\\\\\\\\&\\\\int *{g_1}{\\\\widehat{}(l)^2g_1} *{g_2}{\\\\widehat{}(l)^2g_2}*{g_3}{\\\\widehat{}(l)^2g_3} *{g_4}{\\\\widehat{}(l)^2g_4}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\operatornamewithlimits{ess\\\\,sup}_{a_1,\\\\ldots ,a_8\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4) \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_5,a_6) \\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_7,a_8)\\\\right)\\\\left(\\\\overline{0}\\\\right) \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le \\\\left\\\\Vert \\\\widehat{}(0)^2\\\\right\\\\Vert _{\\\\infty ,\\\\infty }^2\\\\operatornamewithlimits{ess\\\\,sup}_{\\\\overline{x}\\\\in \\\\mathbb {R}^d,a_1,\\\\ldots ,a_4\\\\in \\\\mathcal {E}}\\\\left(\\\\varphi ^{(2)}(\\\\cdot ;a_1,a_2)\\\\star \\\\varphi ^{(2)}(\\\\cdot ;a_3,a_4)\\\\right)\\\\left(\\\\overline{x}\\\\right)\\\\le C^4 g(d).$ Up to a constant, these are all less than or equal to the bounds we used for the $\\\\left\\\\lbrace j_1,\\\\ldots ,j_7\\\\right\\\\rbrace = \\\\left\\\\lbrace 1,\\\\ldots ,1\\\\right\\\\rbrace $ case, and therefore that case dominates.\",\n \"These other cases would behave as constant multiple of one of $g(d)$ , $g(d)^{\\\\frac{3}{2}}$ , $g(d)^2$ , $g(d)^{\\\\frac{5}{2}}$ , and $g(d)^3$ .\",\n \"We have now dealt with all the cases which have no factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ .\",\n \"To demonstrate how we will deal with these, we first consider $\\\\vec{j}=\\\\left\\\\lbrace 3,3,3,3,3,3,3,5\\\\right\\\\rbrace $ .\",\n \"For notational compactness, we define $\\\\left<\\\\widehat{}(l)^2\\\\right>_{I} := \\\\prod _{m\\\\in I}*{g_m}{\\\\widehat{}(l)^2g_m},$ where $I\\\\subset \\\\mathbb {N}$ and $l\\\\in \\\\mathbb {R}^d$ .\",\n \"Since we have already extracted the factor of $\\\\left(1-\\\\mathbb {S}\\\\left(\\\\widehat{}(k)\\\\right)\\\\right)$ via the pointwise bound on $Q^{5}_{15,16}(l;k)$ , we are left trying to bound the following integral: $\\\\int \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) \\\\\\\\ \\\\qquad \\\\times \\\\left<\\\\widehat{}(l_1)^2\\\\right>^{\\\\frac{1}{2}}_{1,2,3,4} \\\\left<\\\\widehat{}(l_2)^2\\\\right>^{\\\\frac{1}{2}}_{5,6,7,8} \\\\left<\\\\widehat{}(l_1-l_2)^2\\\\right>^{\\\\frac{1}{2}}_{9,10} \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{11,12}\\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}.$ To bound this integral we need to partition our space.\",\n \"Fix $\\\\varepsilon >0$ and define $B_1 &:= \\\\left\\\\lbrace *{l_1}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_2 &:= \\\\left\\\\lbrace *{l_2}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_3 &:= \\\\left\\\\lbrace *{l_1-l_3}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_4 &:= \\\\left\\\\lbrace *{l_1-l_2}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_5 &:= \\\\left\\\\lbrace *{l_2-l_3}< \\\\varepsilon \\\\right\\\\rbrace \\\\\\\\B_6 &:= \\\\left\\\\lbrace *{l_3}< \\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l_3-k}< \\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l_3+k}< \\\\varepsilon \\\\right\\\\rbrace ,$ and for $n=0,1,\\\\ldots ,6$ define $A_n := \\\\left\\\\lbrace (l_1,l_2,l_3)\\\\in \\\\left(\\\\mathbb {R}^{d}\\\\right)^3 \\\\colon \\\\#\\\\left\\\\lbrace m\\\\colon (l_1,l_2,l_3)\\\\in B_m \\\\right\\\\rbrace =n\\\\right\\\\rbrace .$ In words, $A_n$ is the set of points within $\\\\varepsilon $ (in Euclidean distance) of precisely $n$ of the hyperplanes on which the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors diverge.\",\n \"If we are working with a factor of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ on the set $B_m$ associated with its relevant hyperplane, we use the bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\le 1/\\\\left(C_2*{l}^2\\\\right)$ , and if we are working off this set we use the bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ .\",\n \"When we integrate over $A_0$ , we bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ for all such factors and recycle the bound on the “only $\\\\widehat{}$ \\\" integrals from above.\",\n \"The contribution from $A_0$ is therefore bounded by some constant multiple of $\\\\frac{1}{\\\\varepsilon ^{18}}g(d)^3.$ For $A_1$ , first suppose we are considering $A_1\\\\cap B_m$ for $m\\\\ne 6$ .\",\n \"First we bound $\\\\widehat{G}_{\\\\mu _\\\\lambda }(l) \\\\le 1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ and $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) \\\\le 3/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ for the “irrelevant\\\" directions, and use the pointwise bound for $\\\\left<\\\\widehat{}(l)^2\\\\right>_{m,n}$ in the “relevant\\\" direction.\",\n \"Then we can use the Cauchy-Schwarz and factorise argument we used above and recycle the integral bounds for the $\\\\widehat{}$ factors as we did above.\",\n \"However we are careful about the way we split apart terms in the Cauchy-Schwarz step.\",\n \"We split the $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ terms across both factors and arrange the remaining $\\\\widehat{}$ terms so that each factor has “directions\\\" spanning $\\\\left(\\\\mathbb {R}^d\\\\right)^3$ .\",\n \"To demonstrate we consider $A_1\\\\cap B_1$ : $&\\\\int _{A_1\\\\cap B_1} \\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2 \\\\left<\\\\widehat{}(l_2)^2\\\\right>^{\\\\frac{1}{2}}_{5,6,7,8} \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{11,12} \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}} \\\\nonumber \\\\\\\\& \\\\hspace{56.9055pt} \\\\le C^2\\\\left(\\\\int _{A_1\\\\cap B_1}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\left<\\\\widehat{}(l_2)^2\\\\right>_{5,6}\\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>_{11,12}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\& \\\\hspace{142.26378pt}\\\\times \\\\left(\\\\int _{A_1\\\\cap B_1}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2 \\\\left<\\\\widehat{}(l_2)^2\\\\right>_{7,8} \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>_{13,14}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.$ After a change of variables, factorising and using our integral bounds of $\\\\widehat{}$ terms from above, we are only left needing to bound $\\\\int _{*{l}<\\\\varepsilon } \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^2 \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\le \\\\frac{1}{C^2_2} \\\\int _{*{l}<\\\\varepsilon } \\\\frac{1}{*{l}^4} \\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} = \\\\frac{1}{C^2_2}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-4}}{d-4}.$ Therefore the contribution from $A_1\\\\cap B_m$ for $m\\\\ne 6$ is bounded by some constant multiple of $\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-18}}{d-4}g(d)^2$ .\",\n \"For $A_1\\\\cap B_6$ we need to be slightly more careful because for $*{k}<2\\\\varepsilon $ the constituent parts of $B_6$ overlap - this is the same considerations we had to account for in the proof of Lemma REF .\",\n \"We can overcome it in the same way by using Hölder's inequality (actually just Cauchy-Schwarz in this case) and then re-centring the integrals.\",\n \"The net result is that the bound of a constant multiple of $\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-18}}{d-4}g(d)^2$ also applies to $A_1\\\\cap B_6$ .\",\n \"In summary, the contribution from $A_1$ is bounded by some constant multiple of $\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\frac{\\\\varepsilon ^{d-18}}{d-4}g(d)^2.$ The $A_2$ case proceeds similarly.\",\n \"After partitioning the set according to which are the overlapping hyperplanes, we take the relevant $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors from these directions and one $\\\\widehat{}$ factor from a linearly independent direction, and pointwise bound all the other $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ and $\\\\widehat{}$ factors.\",\n \"The contribution from $A_2$ is then bounded by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^2\\\\frac{\\\\varepsilon ^{2d-18}}{\\\\left(d-4\\\\right)^2}g(d),$ arising from the $A_1\\\\cap B_1\\\\cap B_2$ , $A_1\\\\cap B_1\\\\cap B_6$ , and $A_1\\\\cap B_2\\\\cap B_6$ integrals.\",\n \"The $A_3$ case is more complicated because there are two qualitatively different sub-cases to consider.\",\n \"First suppose that the three normal vectors to the overlapping hyperplanes are linearly independent.\",\n \"Then we take the relevant $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors from these directions, pointwise bound the other $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ factors and all the $\\\\widehat{}$ factors.\",\n \"The contribution from this sub-case of $A_3$ is then bounded by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\frac{\\\\varepsilon ^{3d-18}}{\\\\left(d-4\\\\right)^3}$ , arising from the $A_1\\\\cap B_1\\\\cap B_2\\\\cap B_6$ integral.\",\n \"On the other hand, if the three normal vectors to the overlapping hyperplanes are linearly dependent, then we need to retain a $\\\\widehat{}$ from the remaining linearly independent direction.\",\n \"To demonstrate, let us consider the $A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4$ integral.\",\n \"After removing unnecessary terms via uniform bounds, $&\\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{13,14} \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>^{\\\\frac{1}{2}}_{11,12}\\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\left(\\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^\\\\frac{1}{2}\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad \\\\times \\\\left(\\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_1-l_3)^2\\\\right>_{11,12} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^\\\\frac{1}{2}.$ Applying a change of variables, factorising the integral, and applying Hölder's inequality allows us to bound these parentheses: $& \\\\int _{A_3\\\\cap B_1 \\\\cap B_2 \\\\cap B_4}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\left<\\\\widehat{}(l_2-l_3)^2\\\\right>_{13,14} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int _{\\\\left\\\\lbrace *{l_1}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_1-l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2) \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right)\\\\left(\\\\int \\\\left<\\\\widehat{}(l)^2\\\\right>_{13,14}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\frac{1}{C^5_2}\\\\left(\\\\int _{\\\\left\\\\lbrace *{l_1}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_1-l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l_1}^4}\\\\frac{1}{*{l_2}^4}\\\\frac{1}{*{l_1-l_2}^2} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right) g(d)\\\\nonumber \\\\\\\\& \\\\qquad \\\\le \\\\frac{1}{C^5_2}\\\\left(\\\\int _{\\\\left\\\\lbrace *{l_1}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l_1}^6}\\\\frac{1}{*{l_2}^3} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right)^{\\\\frac{2}{3}}\\\\left(\\\\int _{\\\\left\\\\lbrace *{l_2}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l_1-l_2}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l_2}^6}\\\\frac{1}{*{l_1-l_2}^6} \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2}{\\\\left(2\\\\pi \\\\right)^{2d}}\\\\right)^{\\\\frac{1}{3}} g(d)\\\\nonumber \\\\\\\\& \\\\qquad = \\\\frac{1}{C^5_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{d-6}\\\\frac{\\\\varepsilon ^{d-6}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{4}{3}}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{d-3}\\\\frac{\\\\varepsilon ^{d-3}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^{\\\\frac{2}{3}}g(d).$ For all of these linearly dependent cases, we can bound the integral by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^2\\\\frac{\\\\varepsilon ^{2d-18}}{\\\\left(d-6\\\\right)^2}g(d)$ .\",\n \"In summary, the contribution from $A_3$ is bounded by some constant multiple of $\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\frac{\\\\varepsilon ^{3d-18}}{\\\\left(d-4\\\\right)^3} + \\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^2\\\\frac{\\\\varepsilon ^{2d-18}}{\\\\left(d-6\\\\right)^2}g(d).$ For $A_4$ , $A_5$ , and $A_6$ we don't have the two linearly dependent/independent cases because any four of the directions spans $\\\\left(\\\\mathbb {R}^d\\\\right)^3$ .\",\n \"We therefore don't need factors of $\\\\widehat{}$ to control the unbounded directions.\",\n \"Instead, we need to be careful that when we split our integral at the Cauchy-Schwarz step the resulting parentheses have sufficiently few factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ that their bounds will be finite for $d>6$ .\",\n \"The way we perform this split will be different for the different parts of $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l_3;k)$ .\",\n \"Let us define $\\\\widehat{D}(l;k) := \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k).$ We first consider $A_6$ .\",\n \"By writing $\\\\widehat{\\\\mathcal {G}}_{\\\\mu _\\\\lambda }(l;k) = \\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{D}(l;k) + \\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)$ , we have two integrals we want to bound.\",\n \"First $&\\\\int _{A_6}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{D}(l_3;k) \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^3\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{D}(l_3;k)^2\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad \\\\times \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^3\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)^2\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.\",\n \"$ We then perform the $l_i$ integrals in specific orders.\",\n \"For the first factor we integrate over $l_3$ (which factorises out already), then we integrate over $l_2$ for fixed $l_1$ , and finally integrate over $l_1$ .\",\n \"For the second factor we integrate over $l_1$ for fixed $l_2$ and $l_3$ , then we integrate over $l_3$ for fixed $l_2$ , and finally we integrate over $l_2$ .\",\n \"There are therefore three forms of integral we need to bound: $\\\\int _{*{l}<\\\\varepsilon }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)^3\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} &\\\\le \\\\frac{1}{C^3_2}\\\\int _{*{l}<\\\\varepsilon }\\\\frac{1}{*{l}^6}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} = \\\\frac{1}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\frac{\\\\varepsilon ^{d-6}}{d-6}\\\\\\\\\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l-l^{\\\\prime }}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-l^{\\\\prime })^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} &\\\\le \\\\frac{1}{C^3_2}\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cap \\\\left\\\\lbrace *{l-l^{\\\\prime }}<\\\\varepsilon \\\\right\\\\rbrace }\\\\frac{1}{*{l}^2}\\\\frac{1}{*{l-l^{\\\\prime }}^4}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\nonumber \\\\\\\\&\\\\le \\\\frac{1}{C^3_2}\\\\left(\\\\int _{*{l}<\\\\varepsilon }\\\\frac{1}{*{l}^6}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^\\\\frac{1}{3}\\\\left(\\\\int _{*{l-l^{\\\\prime }}<\\\\varepsilon }\\\\frac{1}{*{l-l^{\\\\prime }}^6}\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}}\\\\right)^\\\\frac{2}{3}\\\\nonumber \\\\\\\\& = \\\\frac{1}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\frac{\\\\varepsilon ^{d-6}}{d-6}\\\\\\\\\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l-k}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l+k}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l)\\\\widehat{D}(l;k)^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} &\\\\le \\\\frac{C^{\\\\prime }}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-6}.$ In this last inequality, $C^{\\\\prime }>0$ is some uniform constant.\",\n \"The last inequality is derived by applying Hölder's inequality in much the same way as for the second inequality, taking care to see when the three sets that are integrated over are overlapping.\",\n \"The calculation uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .\",\n \"The second integral we want to bound for $A_6$ is $&\\\\int _{A_6}\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k) \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\le \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-k)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2-l_3)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)^3\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}\\\\nonumber \\\\\\\\&\\\\qquad \\\\qquad \\\\times \\\\left(\\\\int _{A_6}\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_2)^3\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_1-l_2)^2\\\\right]\\\\left[\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3-l_1)^2\\\\widehat{G}_{\\\\mu _\\\\lambda }(l_3+k)\\\\right] \\\\frac{\\\\mathrm {d}l_1 \\\\mathrm {d}l_2 \\\\mathrm {d}l_3}{\\\\left(2\\\\pi \\\\right)^{3d}}\\\\right)^{\\\\frac{1}{2}}.\",\n \"$ We then perform the $l_i$ integrals in specific orders.\",\n \"For the first factor we integrate over $l_1$ (which factorises out already), then we integrate over $l_2$ for fixed $l_3$ , and finally integrate over $l_3$ .\",\n \"For the second factor we integrate over $l_3$ for fixed $l_1$ and $l_2$ , then we integrate over $l_1$ for fixed $l_2$ , and finally we integrate over $l_2$ .\",\n \"The new integrals we need to bound are: $\\\\int _{\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l-k}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l+k}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-k)^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\\\\\ \\\\le \\\\frac{C^{\\\\prime }}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4} + \\\\frac{1}{d-2} + \\\\frac{1}{d}\\\\right)\\\\varepsilon ^{d-6}$ $\\\\int _{\\\\left(\\\\left\\\\lbrace *{l}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l-k}<\\\\varepsilon \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace *{l+k}<\\\\varepsilon \\\\right\\\\rbrace \\\\right)\\\\cap \\\\left\\\\lbrace *{l-l^{\\\\prime }}<\\\\varepsilon \\\\right\\\\rbrace }\\\\widehat{G}_{\\\\mu _\\\\lambda }(l+k)\\\\widehat{G}_{\\\\mu _\\\\lambda }(l-l^{\\\\prime })^2\\\\frac{\\\\mathrm {d}l}{\\\\left(2\\\\pi \\\\right)^{d}} \\\\\\\\ \\\\le \\\\frac{C^{\\\\prime }}{C^3_2}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)\\\\left(\\\\frac{1}{d-6} + \\\\frac{1}{d-4}\\\\right)\\\\varepsilon ^{d-6}.$ The calculation of these two bounds uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .\",\n \"The result of these bounds is that the integral in (REF ) is bounded by some constant multiple of $\\\\frac{1}{\\\\left(d-6\\\\right)^3}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\varepsilon ^{3d-18}.$ For $A_4$ and $A_5$ we can repeat the argument for $A_6$ , but note that in some places the bound $1/\\\\left(C_2*{l}^2\\\\right)$ will be replaced by $1/\\\\left(C_2\\\\varepsilon ^2\\\\right)$ .\",\n \"The net result of this is that the factor $\\\\left(d-6\\\\right)^3$ can be replaced by $\\\\left(d-6\\\\right)^{\\\\alpha _0}\\\\left(d-4\\\\right)^{\\\\alpha _1}\\\\left(d-2\\\\right)^{\\\\alpha _2} d^{\\\\alpha _3}$ for some $\\\\alpha _0,\\\\alpha _1,\\\\alpha _2,\\\\alpha _3\\\\le 0$ such that $\\\\alpha _0+\\\\alpha _1+\\\\alpha _2+\\\\alpha _3 =3$ .\",\n \"The net result is that the contribution from both $A_4$ and $A_5$ can also be bounded by some constant multiple of $\\\\frac{1}{\\\\left(d-6\\\\right)^3}\\\\left(\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\right)^3\\\\varepsilon ^{3d-18}.$ Above we have established bounds for the various parts of the integral when $\\\\vec{j}=\\\\left(3,3,3,3,3,3,3,5\\\\right)$ .\",\n \"Having $j_8=4$ can also be dealt with by very similar arguments to that outlined above - the $l_3$ integrals are in fact simpler.\",\n \"The above bounds still hold with the $\\\\varepsilon ^{-18}$ becoming a $\\\\varepsilon ^{-16}$ .\",\n \"Since $\\\\varepsilon <1$ , this produces a larger bound.\",\n \"Recall that up to a constant factor, having $j_m=1$ produces a larger bound than $j_m=2$ for $m=1,\\\\ldots ,7$ .\",\n \"If we replace $j_m=3$ with $j_m=1$ , then after ignoring constant values we lose a factor of $\\\\varepsilon ^{-2}$ , but we may also lose a factor of $g(d)^\\\\frac{1}{2}$ unless that was for $m=1,2$ (corresponding to $l_1$ direction) or $m=3,4$ (corresponding to $l_2$ direction).\",\n \"Note that doing for both directions will lose a $g(d)^\\\\frac{1}{2}$ factor.\",\n \"If $\\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^d= O\\\\left(g(d)^{\\\\frac{1}{2}}\\\\right)$ for all fixed $\\\\varepsilon >0$ , then it is clear that the dominating bound is $g(d)^\\\\frac{1}{2}$ .\",\n \"This arises, for example from the bound for $\\\\vec{j}=\\\\left(1,1,1,1,1,1,1,1\\\\right)$ .\",\n \"The situation is more complicated if $g(d)^{\\\\frac{1}{2}}\\\\ll \\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^d$ for fixed $\\\\varepsilon $ , because we can choose to have $\\\\varepsilon =\\\\varepsilon (d)$ and take it to 0 as $d\\\\rightarrow \\\\infty $ .\",\n \"It is easy to see that all the $A_1$ components (and $A_2,\\\\ldots ,A_6$ components) for each $\\\\vec{j}$ can be bounded by some constant multiple of $\\\\frac{1}{d}\\\\frac{\\\\mathfrak {S}_{d-1}}{\\\\left(2\\\\pi \\\\right)^d}\\\\varepsilon ^d$ (provided $d>6$ ).\",\n \"We therefore only need to compare this to the contributions from $A_0$ .\",\n \"The bound we get from the above arguments for each $A_0$ contribution are of the form $\\\\varepsilon ^{-2m}g(d)^\\\\frac{n}{2}$ , where $m$ is the number of factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ and $n$ is related to the multiplicity of $$ terms in each Fourier direction.\",\n \"To find the terms with the largest bound, we want to have as many factors of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ as possible without producing extra factors of $g(d)^\\\\frac{1}{2}$ .\",\n \"Since $g(d)^{\\\\frac{1}{2}}\\\\ll \\\\frac{1}{d}\\\\mathfrak {S}_{d-1}\\\\varepsilon ^d$ for fixed $\\\\varepsilon $ , increasing $n$ here instantly produces a smaller bound.\",\n \"By changing $j_8$ from 1 to 4 or 5, or any other element from 1 to 3, we do gain a factor of $\\\\widehat{G}_{\\\\mu _\\\\lambda }$ but we also increase the number of $$ factors and therefore in most places we acquire extra factors of $g(d)^{\\\\frac{1}{2}}$ .\",\n \"The exception to this is for $j_1$ and $j_2$ , which correspond to the $l_1$ directions, since they already produce a factor of $g(d)^{\\\\frac{1}{2}}$ .\",\n \"The $A_0$ component of the $\\\\vec{j}=\\\\left(3,3,1,1,1,1,1,1\\\\right)$ term can be bounded by constant multiple of $\\\\varepsilon ^{-4}g(d)^{\\\\frac{1}{2}}$ .\",\n \"We then optimise our choice of $\\\\varepsilon $ to have our two dominant bounds be of the same order.\",\n \"This produces $\\\\varepsilon (d)^2 = g(d)^{\\\\frac{1}{d}}\\\\frac{2\\\\pi d}{\\\\text{e}}\\\\left(1+o(1)\\\\right)$ , and therefore our overall bound is given by $\\\\varepsilon ^{-4}g(d)^\\\\frac{1}{2} = g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}}\\\\left(\\\\frac{\\\\text{e}}{2\\\\pi }\\\\right)^2\\\\frac{1}{d^2}\\\\left(1+o(1)\\\\right).$ Since $g(d)^{\\\\frac{1}{2}-\\\\frac{2}{d}} \\\\le g(d)^{\\\\frac{1}{2}-\\\\frac{3}{d}}$ for sufficiently large $d$ , this bound is of order $\\\\beta ^2$ and our bound is proven.\"\n ],\n [\n \"Acknowledgements.\",\n \"This work is supported by Deutsche Forschungsgemeinschaft (project number 443880457) through priority program “Random Geometric Systems” (SPP 2265).\",\n \"The authors thank the Centre de recherches mathématiques Montreal for hospitality during a research visit in spring 2022 through the Simons-CRM scholar-in-residence program.\"\n ]\n]"},"id":{"kind":"string","value":"2210.07727"}}},{"rowIdx":2995,"cells":{"text":{"kind":"list like","value":[["Object-Category Aware Reinforcement Learning"],["Abstract Object-oriented reinforcement learning (OORL) is a promising way to improve the sample efficiency and generalization ability over standard RL.","Recent works that try to solve OORL tasks without additional feature engineering mainly focus on learning the object representations and then solving tasks via reasoning based on these object representations.","However, none of these works tries to explicitly model the inherent similarity between different object instances of the same category.","Objects of the same category should share similar functionalities; therefore, the category is the most critical property of an object.","Following this insight, we propose a novel framework named Object-Category Aware Reinforcement Learning (OCARL), which utilizes the category information of objects to facilitate both perception and reasoning.","OCARL consists of three parts: (1) Category-Aware Unsupervised Object Discovery (UOD), which discovers the objects as well as their corresponding categories; (2) Object-Category Aware Perception, which encodes the category information and is also robust to the incompleteness of (1) at the same time; (3) Object-Centric Modular Reasoning, which adopts multiple independent and object-category-specific networks when reasoning based on objects.","Our experiments show that OCARL can improve both the sample efficiency and generalization in the OORL domain."],["Introduction","Reinforcement Learning (RL) has achieved impressive progress in recent years, such as results in Atari [atari] and Go [go] in which RL agents even perform better than human beings.","Despite its successes, conventional RL is also known to be of low sample efficiency [rlintro] and poor generalization ability [procgen].","Object-oriented RL (OORL) [rmdp, oomdp, d-oomdp,schema,rrl] is a promising way to deal with these limitations.","Inspired by studies in cognitive science [piaget] that objects are the basic units of recognizing the world, OORL focuses on learning the invariance of objects' functionalities in different scenarios to achieve better generalization ability.","In OORL, the agent's observation is a set of object representations, and the task can be solved via reasoning based on these objects.","Most previous works [rmdp, schema, oomdp] in OORL employ hand-crafted object features given in advance, which require human expertise and result in low generality, limiting the applications of these methods.","Recent works [rrl, smorl, ics] try to broaden the scope of OORL's applications by avoiding using additional feature engineering.","Their works can be divided into either top-down approaches [rrl] which purely rely on reward signals to ground representations to objects, or bottom-up approaches [smorl, ics] which utilize unsupervised object discovery (UOD) technologies to provide structural object representations.","However, none of these works [rrl, smorl, ics] tries to explicitly model the inherent similarity between different object instances of the same category when reasoning based on objects.","In most cases, objects of the same category should share the same functionalities, such as obeying similar behaviour patterns or having the same effects when interacting with other objects.","Therefore, an intelligent agent should recognize an object's category to be aware of its functionalities.","Since objects from different categories differ in their functionalities, the objects' functionalities can also be separately modelled (according to their categories), leading to modularity which is beneficial for generalization [modularity, modularity2,naturetaskcluster] in general.","Following the above insight, we propose a framework named Object-Category Aware Reinforcement Learning (OCARL), which utilizes the category information of objects to facilitate both perception and reasoning.","OCARL consists of three parts: (1) category-aware UOD, which can automatically discover the objects as well as their corresponding categories via UOD and unsupervised clustering.","(2) Object-Category Aware Perception (OCAP), a perception module that takes the object category information obtained in (1) as an additional supervision signal.","OCAP can encode the category information into representations and is also robust to the incompleteness of UOD at the same time.","(3) Object-Centric Modular Reasoning (OCMR), a reasoning module that adopts multiple independent and object-category-specific networks, each of which processes the object features of the same corresponding category.","Such a modular mechanism can enhance further the generalization ability of OCARL.","Our experiments show that OCARL can improve the sample efficiency on several tasks and also enable the agent to generalize to unseen combinations of objects where other baselines fail."],["Unsupervised Object Discovery","Unsupervised object discovery (UOD) methods try to automatically discover objects without additional supervision.","Roughly speaking, there are two main categories in this area: spatial attention models and spatial mixture models.","Spatial attention models such as SCALOR [scalor] and TBA [tba] explicitly factorize the scene into a set of object properties such as position, scale and presence.","Then these properties are utilized by a spatial transformer network to select small patches in the original image, which constitute a set of object proposals.","These methods can deal with a flexible number of objects and thus are prominent in the UOD tasks.","Spacial mixture models such as Slot Attention [slotatten], IODINE [idonet] and MONet [monet] decompose scenes via clustering pixels that belong to the same object (often in an iterative manner).","However, they often assume a fixed maximum number of objects and thus cannot deal with a large number of objects.","SPACE [space], a UOD method used in this paper, combines both the spatial attention model and spatial mixture model.","The spatial attention model extracts objects from the scene, whereas the spacial mixture model is responsible for decomposing the remaining background.","Such a combination enables SPACE to distinguish salient objects from relative complex backgrounds."],["Object-Oriented Reinforcement Learning","Object-oriented reinforcement learning (OORL) has been widely studied in RL community.","There are many different structural assumptions on the underlying MDP in OORL, such as relational MDP [rmdp], propositional object-oriented MDPs [oomdp], and deictic object-oriented MDP [d-oomdp].","Although these assumptions differ in detail, they all share a common spirit that the state-space of MDPs can be represented in terms of objects.","Following these assumptions, [objq] proposes object-focused Q-learning to improve sample efficiency.","[schema] learns an object-based casual model for regression planning.","[newobj] improves the generalization ability to novel objects by leveraging relation types between objects that are given in advance.","Although these works have been demonstrated to be useful in their corresponding domain, they require explicit encodings of object representations, which limits their applications.","There are other lines of work in OORL such as COBRA [cobra], OODP [oodp], and OP3 [op3] that try to solve OORL tasks in an end-to-end fashion.","Most of these works fall into a model-based paradigm in which an object-centric model is trained and then utilized to plan.","Although these methods are effective in their corresponding domains, in this paper, we restrict ourselves to the model-free setting, which can generally achieve better asymptotic performance than model-based methods [rlsurvey, imagine].","The most related works should be RRL [rrl] and SMORL [smorl].","RRL proposes an attention-based neural network to introduce a relational inductive bias into RL agents.","However, since RRL relies purely on this bias, it may fail to capture the objects from the observations.","On the other hand, SMORL utilizes an advanced UOD method to extract object representations from raw images, and then these representations are directly used as the whole observations.","However, SMORL adopts a relatively simple reasoning module, which limits its applications in multi-object scenarios."],["Method","OCARL consists of three parts: (1) category-aware unsupervised object discovery (category-aware UOD) module, (2) object-category aware perception (OCAP), and (3) object-centric modular reasoning (OCMR) module.","The overall architecture is shown in Figure REF .","We will introduce (1)(2)(3) in Section REF , Section REF , and Section REF respectively."],["Unsupervised Object Discovery","In this work, SPACE [space] is utilized to discover objects from raw images.","In SPACE, an image $\\mathbf {x}$ is decomposed into two latent representations: foreground $\\mathbf {z}^{fg}$ and background $\\mathbf {z}^{bg}$ .","For simplicity, we only introduce the inference model of the foreground latent $\\mathbf {z}^{fg}$ , which is a set of object representations.","We encourage the readers to refer to [space] for more details.","To obtain $\\mathbf {z}^{fg}$ , $\\mathbf {x}$ is treated as if it were divided into $H\\times W$ cells and each cell is tasked with modelling at most one (nearby) object.","Therefore, $\\mathbf {z}^{fg}$ consists of a set of $H\\times W$ object representations (i.e.","$\\mathbf {z}^{fg} = \\lbrace \\mathbf {z}_{ij}^{fg}\\rbrace _{i=1}^H{}_{j=1}^W$ ), each of which is a 3-tuplewe omit $\\mathbf {z}_{ij}^{depth}$ for simplicity, because we only consider 2-D environment without object occlusions in this paper.","$\\mathbf {z}_{ij}^{fg} = (z_{ij}^{pres}, \\mathbf {z}_{ij}^{where}, \\mathbf {z}_{ij}^{what})$ .","$z_{ij}^{pres}\\in \\lbrace 0,1\\rbrace $ is a 1-D variable that indicates the presence of any object in cell $(i,j)$ , $\\mathbf {z}_{ij}^{where}$ encodes $\\mathbf {z}_{ij}^{fg}$ 's size and location, and $\\mathbf {z}_{ij}^{what}$ is a latent vector that identifies $\\mathbf {z}_{ij}^{fg}$ itself.","The whole inference model of $\\mathbf {z}^{fg}$ can be written as: $\\begin{aligned}q(\\mathbf {z}^{fg}|\\mathbf {x}) = \\prod _{i=1}^{H}\\prod _{j=1}^W q(z_{ij}^{pres}|\\mathbf {x}) (q(\\mathbf {z}_{ij}^{where}|\\mathbf {x}) q(\\mathbf {x}_{ij} | \\mathbf {z}_{ij}^{where}, \\mathbf {x}) q(\\mathbf {z}_{ij}^{what}|\\mathbf {x}_{ij}))^{z_{ij}^{pres}},\\end{aligned}$ where $\\mathbf {x}_{ij}$ is a small patch from the $\\mathbf {x}$ that is obtained by the proposal bounding box identified by $\\mathbf {z}_{ij}^{where}$ , which should contain exactly one object if $z_{ij}^{pres}=1$ ."],["Unsupervised Clustering","After applying SPACE on the randomly collected dataset, we can obtain a set of object representations: $D=\\lbrace z_{nij}^{pres}, \\mathbf {z}_{nij}^{where}, \\mathbf {z}_{nij}^{what}, \\mathbf {x}_{nij}\\rbrace _{n=1}^N{}_{i=1}^H{}_{j=1}^W$ , where $N$ is the size of the dataset, $z_{nij}^{pres}, \\mathbf {z}_{nij}^{where}, \\mathbf {z}_{nij}^{what}, \\mathbf {x}_{nij}$ are defined in Eq.","(REF ).","We first select object patches from $D$ that is of high object-presence probability by a threshold $\\tau $ : $D_\\tau = \\lbrace \\mathbf {x}_{nij}\\in D: p(z_{nij}^{pres}=1) > \\tau \\rbrace $ .","Roughly speaking, each patch $\\mathbf {x}_{nij}\\in D_\\tau $ should contain exactly one object.","Therefore, we first project $\\mathbf {x}_{nij}\\in D_\\tau $ into a low-dimensional latent space $\\mathbf {R}^d$ by running IncrementalPCA [ipca] on $D_\\tau $ , and then adopt KMeans clustering with a given cluster number $C$ upon the projected latent representations.","The object-category predictor $q(\\mathbf {z}_{ij}^{cat}|\\mathbf {x}_{ij}) = \\Psi _\\texttt {KMeans}\\circ \\Psi _\\texttt {IncrementalPCA}$ is the composition of IncrementalPCA and KMeans: $\\begin{aligned}\\mathbf {z}_{ij}^{cat} = \\Psi _{\\texttt {KMeans}}\\circ \\Psi _{\\texttt {IcreamentalPCA}} (\\mathbf {x}_{ij})\\end{aligned}$ where $\\mathbf {z}_{ij}^{cat}\\in \\mathbf {R}^C$ is an one-hot latent which indicates the category of $\\mathbf {z}_{ij}$ .","For simplicity, we regard the background (i.e.","$z_{ij}^{pres}=0$ ) as special 'object', thus $\\mathbf {z}_{ij}^{cat}$ can be written as $\\mathbf {\\hat{z}}_{ij}^{cat}=\\Psi _{\\texttt {concatenate}}([1-z^{pres}_{ij}, z^{pres}_{ij}\\cdot \\mathbf {z}_{ij}^{cat}])$ .","Note that $\\hat{\\mathbf {z}}_{ij}^{cat}$ is an one-hot latent $\\in \\mathbf {R}^{C+1}$ .","Although the object-category predictor Eq.","(REF ) is very simple, we find it works well in our experiments.","In practice, the cluster number $C$ of KMeans can be set by leveraging prior knowledge about ground-truth category number, or by analysing some clustering quality metrics such as silhouette coefficients.","In more complex environments, we can also rely on other advanced unsupervised clustering methods such as NVISA [cluster1], CC [cluster2], etc.","Figure: OCARL consists of 3 parts: (1) category-aware UOD, (2) OCAP, and (3) OCMR.","(1) In category-aware UOD, the objects are firstly discovered by a UOD method (SPACE) and then assigned an (extended) category label by unsupervised clustering (IcreamentalPCA + KMeans).","(2) In OCAP, the observation is first encoded into a set of features 𝐙\\mathbf {Z} by convolution networks.","Each vector in 𝐳 i =𝐙 :,i,j ∈𝐙\\mathbf {z}_i=\\mathbf {Z}_{:,i,j}\\in \\mathbf {Z} is given a task by the OCAP module to predict the corresponding (extended) object category, which is given by (1).","𝐙\\mathbf {Z} is then fed to the OCMR module.","(3) In OCMR, x-y coordinate information is added into 𝐙\\mathbf {Z}, which gives us 𝐙 ^\\mathbf {\\widehat{Z}}.","Then we apply self-attention on 𝐙 ^\\mathbf {\\widehat{Z}} to query related information 𝐐 ^\\mathbf {\\widehat{Q}} for each object in 𝐙 ^\\mathbf {\\widehat{Z}}.","𝐐 ^\\mathbf {\\widehat{Q}} and 𝐙 ^\\mathbf {\\widehat{Z}} are concatenated to form 𝐙 ˜\\mathbf {\\widetilde{Z}}.","Each entity in 𝐙 ˜\\mathbf {\\widetilde{Z}} will be passed through different independent neural networks according to its corresponding object category, which will give us 𝐙 out \\mathbf {Z}^{out}.","Finally, we perform a max operation along the H,WH,W dimension of 𝐙 out +𝐙 ^\\mathbf {Z}^{out} + \\mathbf {\\widehat{Z}}, and use the resulting feature vector to predict the value function and action probabilities."],["OCAP: Object-Category Aware Perception","The OCAP module is designed to robustly incorporate the object knowledge of the (category-aware) UOD model defined in Section REF .","In a word, OCAP is a plain convolution encoder that accepts the category information from the UOD model as additional supervision signals.","Such a design makes OCARL robust to the incompleteness of the UOD model: in extreme cases in which the UOD model fails to discover any objects, OCAP degenerates into a plain convolution encoder which is still able to extract useful information from the raw images with the help of reward signals.","Suppose the convolution encoder in OCAP maps the raw image observation $\\mathbf {X} \\in \\mathbf {R}^{3\\times H_{img}\\times W_{img}}$ to a latent representation $\\mathbf {Z}\\in \\mathbf {R}^{F\\times H\\times W}$ , where $H, W$ are the same with those in the UOD model (see Eq.","(REF )).","On the other hand, $\\mathbf {X}$ is also fed into the UOD model to get the (extended) object category information $\\hat{\\mathbf {z}}^{cat} = \\lbrace \\hat{\\mathbf {z}}_{ij}^{cat}\\rbrace _{i=1}^H{}_{j=1}^W$ via Eq.","(REF ).","The OCAP module forces the latent representation $\\mathbf {Z}$ to encode the object category information, which is implemented by training a additional category predictor $f_{cat}: \\mathbf {R}^F\\rightarrow \\texttt {Categorical}(C+1)$ .","$f_{cat}$ predicts $\\hat{\\mathbf {z}}_{ij}^{cat}$ given $\\mathbf {Z}_{:,i,j}$ which is a single channel in $\\mathbf {Z}$ .","Therefore, the additional supervision signal that incorporates the object knowledge of the UOD model is given as: $\\begin{aligned}L_{cat} = \\sum _{i=1}^H\\sum _{j=1}^W \\texttt {CrossEntropyLoss}(f_{cat}(\\mathbf {Z}_{:,i,j}); \\hat{\\mathbf {z}}_{ij}^{cat}).\\end{aligned}$ $L_{cat}$ is used for training the convolution encoder in OCAP.","In practice, $L_{cat}$ can be used as an auxiliary loss to any RL algorithm: $\\begin{aligned}L_{total} = L_{RL} + \\lambda _{cat} L_{cat},\\end{aligned}$ where $\\lambda _{cat}\\in \\mathbf {R}^+$ is a coefficient."],["OCMR: Object-Centric Modular Reasoning","OCMR is a module that takes $\\mathbf {Z}\\in \\mathbf {R}^{F\\times H\\times W}$ from the OCAP module as input and outputs a feature vector that summarizes $\\mathbf {Z}$ .","The key design philosophy of OCMR is to adopt multiple independent and object-category-specific networks, each of which focuses on processing the object features of the same corresponding category.","Compared with using a universal category-agnostic network, the processing logic of each independent network in OCMR is much simpler and therefore easier to master, which in turn allows for improved generalization as we will show in Section REF .","This design philosophy also agrees with the recent discovery from [naturesepnet,modularity] which says that neural modules of specialization can lead to better generalization ability.","Given $\\mathbf {Z}\\in \\mathbf {R}^{F\\times H\\times W}$ from OCAP, we first encode the x-y coordinates information (corresponding to the $H\\times W$ grid) to each channel, and then map the resulting tensor into $\\widehat{\\mathbf {Z}}\\in \\mathbf {R}^{HW\\times F}$ .","$\\widehat{\\mathbf {Z}}$ is treated as $HW$ objects with x-y coordinate information encoded.","To model the relations between objects, we apply a self-attention [attention] module on $\\widehat{\\mathbf {Z}}$ to query information for each object from its related objects: $\\begin{aligned}\\widehat{\\mathbf {Q}} = \\texttt {softmax}(\\widehat{\\mathbf {Z}}\\mathbf {W}_q(\\widehat{\\mathbf {Z}}\\mathbf {W}_k)^T)\\widehat{\\mathbf {Z}}\\mathbf {W}_v \\in \\mathbf {R}^{HW\\times F},\\end{aligned}$ where $\\mathbf {W}_q,\\mathbf {W}_k,\\mathbf {W}_v\\in \\mathbf {R}^{F\\times F}$ are trainable parameters.","Modeling the relations is important, because one object's high-level semantic feature can be derived from other objects.","For example, whether a door can be opened is determined by the existence of the key; therefore, the door should query other objects to check whether it is openable.","After the attention module, $\\widehat{\\mathbf {Q}}$ and $\\widehat{\\mathbf {Z}}$ are concatenated together to get $\\widetilde{\\mathbf {Z}} = \\Psi _\\texttt {concatenate}([\\widehat{\\mathbf {Q}},\\widehat{\\mathbf {Z}}])\\in \\mathbf {R}^{HW\\times 2F}$ .","Each $\\widetilde{\\mathbf {Z}}_{i,j,:}$ in $\\widetilde{\\mathbf {Z}}$ is then fed into different independent neural networks according to its corresponding object category $\\hat{\\mathbf {z}}_{ij}^{cat}$ .","In practice, this can be implemented as: $\\begin{aligned}\\mathbf {Z}^{out}_{i,j,:} = \\sum _{c=1}^{C+1} f_c(\\widetilde{\\mathbf {Z}}_{i,j,:}) \\cdot \\hat{z}_{ij;c}^{cat},\\end{aligned}$ where $f_c:\\mathbf {R}^{2F} \\rightarrow \\mathbf {R}^F$ , $[\\hat{z}_{ij;1}^{cat},...,\\hat{z}_{ij;C+1}^{cat}] = \\hat{\\mathbf {z}}_{ij}^{cat}$ .","Eq.","(REF ) can be computed in parallel, resulting a tensor $\\mathbf {Z}^{out}\\in \\mathbf {R}^{H W\\times F}$ .","Finally, the $\\mathbf {Z}^{out}$ is added to $\\widehat{\\mathbf {Z}}$ (i.e.","a residual connection).","We perform a max operation along the $H,W$ dimension, and using the resulting vector $\\in \\mathbf {R}^F$ to predict the value function and action probabilities: $\\begin{aligned}\\texttt {action\\_probs, value} = f_{ac}(\\max _{H,W}(\\mathbf {Z}^{out} + \\widehat{\\mathbf {Z}})).\\end{aligned}$"],["Task Description","In this work, we consider two task domains: Crafter and Hunter.","The observations on both tasks are raw images of shape $64\\times 64$ ."],["Crafter","[crafter] is a complex 2-D MineCraft-style RL task, where complex behaviors are necessary for the agent' survival.","The environment is procedurally generated by arranging various resources, terrain types, and objects (18 in total).","The agent is rewarded if it can craft new items and accomplish achievements (22 possible achievements in total)."],["Hunter","is much simpler than Crafter.","It is also procedurally generated but only contains 4 types of objects: Hunter, Cow, Zombie and Wall The images to render these objects come from Crafter [crafter], which is under MIT license..","The agent can control Hunter and can get positive reward (=1) if Hunter catches a Cow or kill a Zombie.","The agent will be given a high reward (=5) if it can accomplish this task by catching/killing all Cows and Zombies.","However, once Hunter is caught by a Zombie, the agent will receive a negative reward (=-1), and the episode ends.","In a word, the agent in Hunter should master two different behavior patterns: chase & catch the Cow, and avoid & shoot at Zombie.","Although Hunter is simple, it is a typical OORL task.","We can derive different environment instances from Hunter by setting the number of Zombies and Cows.","We use Hunter-ZmCn to denote an environment that spawns (m Zombies + n Cows) at the beginning of each episode, and Hunter-ZmCn/ZnCm an environment that spawns (m Zombies + n Cows) or (n Zombies + m Cows)."],["Evaluation","In this section, we evaluate OCARL's effect both on sample efficiency and generalization ability.","We consider the following baselines: PPO [ppo], RRL [rrl], and SMORL [smorl].","PPO is a general-purpose on-policy RL algorithm with a plain neural network (i.e.","Convolution + MLP).","RRL proposes an attention-based neural network to introduce relational inductive biases and iterated relational reasoning into RL agent, which aims to solve object-oriented RL tasks without any other external supervision signals.","SMORL utilizes the UOD model to decompose the observation into a set of object representations which are directly used as the agent's new observation.","All algorithms (RRL, SMORL, and OCARL) are (re-)implemented upon PPO to ensure a fair comparison.","Note that OCARL is orthogonal to the backbone RL algorithm, therefore other advanced RL methods such as [rainbow,hvf] are also applicable.","For more implementation details, please refer to the Appendix."],["Sample Efficiency","The results on Hunter domain are shown in Figure REF .","We can conclude three facts from this figure: (1) OCARL performs better and is more stable than other baselines on all tasks.","(2) RRL is competitive with OCARL on Hunter-Z4C4 but significantly worse on other simper environments.","This is because RRL relies solely on the reward signals to ground the visual images into objects; therefore, it requires enough (rewarded) interactions between objects, which simpler environments such as Hunter-Z1C1 fail to supply.","(3) SMORL performs better than other baselines (PPO, RRL) on simple environments such as Hunter-Z1C0/Z0C1, Hunter-Z1C1 and Hunter-Z4C0/Z0C4, but unable to make progress in Hunter-Z4C4 which consists of more objects.","This is because SMORL adopts a simple reasoning module that may not be well-suited to multi-object reasoning.","The results on Crafter domain are shown in Table REF and Figure REF .","Crafter is a complex environment in that it features sparse (rewarded) interactions between objects and large object category numbers.","Besides, some objects are omitted by the UOD model (see Appendix) because they almost do not appear in the training dataset due to the insufficient exploration of the random policy.","Due to these features, both RRL and SMORL fail to provide noteworthy results (with scores of 4.22 and 3.94, respectively).","On the other hand, OCARL is able to present more meaningful behaviours (with a score of 12.31), such as collecting coal, defeating zombies, making wood pickaxes and so on, as shown in Figure REF .","The learning curves and detailed achievement success rates are provided in the Appendix.","Figure: Success rates (averaged across 12 seeds) on 22 achievements in Crafter.","These success rates are calculated using the final 1M environment steps during training.","OCARL presents more meaningful behaviours than other baselines, such as collecting coal, defeating zombies, making wood pickaxes and so on."],["Generalization","To evaluate the generalization ability, we train agents on Hunter-Z1C0/Z0C1, Hunter-Z1C1 and Hunter-Z4C0/Z0C4 separately, and then observe their test performance on Hunter-Z4C4.","In Hunter-Z1C1, the agent needs to generalize from a few objects to more objects.","In Hunter-Z4C0/Z0C4, the agent never observes the coincidence of Cow and Zombie.","The generalization from Hunter-Z1C0/Z0C1 to Hunter-Z4C4 is the most difficult.","Such a paradigm actually follows the out-of-distribution (OOD) setting, in that the object combination (4 Zombies + 4 Cows) is never seen in the training environments.","Therefore, although such generalization is possible for humans, it is much harder for RL algorithms.","As shown in Figure REF , OCARL is the only algorithm that make significant progress on the test environment.","Although other baselines can improve their training performance (Figure REF ), they are unable to generalize to the test environment.","The generalization performance of OCARL is most impressive on Hunter-Z1C1, because agent trained on Hunter-Z1C1 even achieves slightly better than that on Hunter-Z4C4 (i.e.","directly trained on the test environment, see the last figure in Figure REF .",")."],["Robustness to the Incompleteness of Object Discovery Methods","OCARL relies on UOD methods to discover different objects from given observations, which are not guaranteed to discover all objects, especially in complex environments [benchmarkoom].","Therefore, one may naturally ask to what extent OCARL's performance is affected by such incompleteness.","To answer this question, we randomly mask out each object (by setting $z_{ij}^{pres}$ to 0) discovered by the UOD model with a given probability $p\\in [0,1]$ , and evaluate OCARL's performance after training 5M environment steps on Hunter.","Such random confusion of the UOD model will affect both the OCAP and OCMR modules of OCARL.","Note that OCARL($p=0$ ) is OCARL itself, and OCARL($p=1$ ) is exactly RRL This is because Eq.","(REF ) takes no effect and Eq.","(REF ) also degenerates into a single universal network because all object features are passed through the same network $f_{bg}$ which is corresponding to the 'background'..","The results are shown in Figure REF .","Although increasing $p$ does hinder OCARL's performance, OCARL can still benefit from the imperfect UOD model in that its performance is still better than RRL in general.","Such robustness should be attributed to the fact that OCARL does not directly use the output of the UOD model as observation but instead treats it as an additional supervision signal that help the encoder to capture object-category information, which gives OCARL a second chance to find other useful information from the raw image."],["OCMR Improves Generalization","One key advantage of OCARL is that it can generalize to unseen combinations of objects by leveraging the invariant relations between objects learnt during training.","We argue that such an advantage is established via OCMR, which adopts object-category-specific networks to deal with different objects.","To confirm our argument, we consider a variant of OCARL (i.e.","OCARL w/o OCMR) in which the OCMR module is removed and replaced by an RRL-like reasoning module which adopts a universal network (instead of multiple object-category-specific networks as OCARL does).","The comparison between OCARL w/ and w/o OCMR is shown in Figure REF .","OCARL w/o OCMR (almost completely) fails to generalize from Hunter-Z1C0/Z0C1, Hunter-Z4C0/Z0C4 to Hunter-Z4C4, while OCARL w/ OCMR does.","In Figure REF , we also consider a variant of OCARL (i.e.","OCARL w/o OCAP) in which the OCAP module is disabled by setting $\\lambda _{cat}=0$ in Eq.","(REF ).","In this paradigm, only OCMR is in effect.","As shown in Figure REF , OCMR alone can still bring benefits without the help of OCAP (although not as consistently as OCMR+OCAP ).","According to these results, we can conclude that OCMR is crucial in OCARL's success.","Table: The Modularity of OCMR.","We first train agent on Hunter-Z4C4 and then disable the module f Z f_Z (corresponding to Zombie) or f C f_C (corresponding to Cow) in OCMR (see Eq.()).","Each agent is evaluated on both Hunter-Z4C0 and Hunter-Z0C4, and the mean and standard deviation of 12 seeds are reported .","The results show that the behaviour patterns (chase & catch the Cow) and (avoid & shoot at Zombie) are encoded in f C f_C and f Z f_Z respectively."],["The Modularity of OCMR","OCMR adopts different modules to deal with object features from different categories.","In this section, we will show that each module in OCMR actually encodes an object-category-specific behaviour pattern.","As stated in Section REF , the agent should master two useful behaviour patterns in Hunter: (1) chase & catch the Cow, and (2) avoid & shoot at Zombie.","The existence of these patterns can be observed by evaluating the agent in Hunter-Z0C4 and Hunter-Z4C0 respectively, because agents that master pattern (1) (pattern (2)) should achieve better performance on Hunter-Z0C4 (Hunter-Z4C0).","In Table REF , we first train agent on Hunter-Z4C4 and then disable the module $f_Z$ (corresponding to Zombie) or $f_C$ (corresponding to Cow) by replacing $f_C$ or $f_E$ with $f_{bg}$ (corresponding to the 'background') in Eq.","(REF ).","The results shows that disabling $f_Z$ does not affect the performance on Hunter-Z0C4 ($8.29\\rightarrow 8.13$ ), which means that $f_Z$ is not relative to behaviour pattern (1).","Instead, $f_Z$ is corresponding to behaviour pattern (2) because we can observe a significant performance decline ($8.27\\rightarrow 1.17$ ) on Hunter-Z4C0 when we disable $f_Z$ .","Similar phenomenon can also be observed on $f_C$ ."],["Conclusion","In this paper, we propose OCARL, which utilize the category information of objects to facilitate both perception.","For the perception, we propose the OCAP, which enables the encoder to distinguish the categories of objects.","For the reasoning, we propose the OCMR, which adopts object-category-specific networks to deal with different objects.","Our experiments are carried out on Crafter and Hunter.","Experiments show that OCARL outperforms other baselines both on sample efficiency and generalization ability.","We also perform several ablation studies to show that (1) OCARL is robust to the incompleteness of UOD methods, (2) OCMR is critical to improving generalization, and (3) each module in OCMR actually encodes an object-category-specific behaviour pattern in Hunter."],["Limitation","The main limitation is that we only test OCARL's generalization ability to unseen object combinations, but not novel object instances.","In order to generalize to novel object instances that looks very different from old ones, the agent should interact with the novel objects first and then infer the their underlying categories according to these interactions.","Such a problem is likely to be in the meta-RL domain and needs more efforts to solve, which we would like to leave for future work."],["Acknowledgements","This work is partially supported by the National Key Research and Development Program of China(under Grant 2018AAA0103300), the NSF of China(under Grants 61925208, 62102399, 62002338, U19B2019, 61906179, 61732020), Strategic Priority Research Program of Chinese Academy of Science (XDB32050200), Beijing Academy of Artificial Intelligence (BAAI) and Beijing Nova Program of Science and Technology (Z191100001119093) , CAS Project for Young Scientists in Basic Research(YSBR-029), Youth Innovation Promotion Association CAS and Xplore Prize.","numbers Checklist For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?","Did you describe the limitations of your work?","Did you discuss any potential negative societal impacts of your work?","Have you read the ethics review guidelines and ensured that your paper conforms to them?","If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?","Did you include complete proofs of all theoretical results?","If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?","Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?","see Appendix.","Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?","Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?","If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...","If your work uses existing assets, did you cite the creators?","Did you mention the license of the assets?","Partially.","Did you include any new assets either in the supplemental material or as a URL?","Did you discuss whether and how consent was obtained from people whose data you're using/curating?","Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?","If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?","Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?","Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?","Implementation Details The implementation of OCARL is available at https://anonymous.4open.science/r/OCARL-51BF.","Hyper-parameters for SPACE In this paper, we use SPACE for object discovery.","Most hyper-parameters are the same as the default parameters for Atari in SPACE, except that (1) we use a smaller network to deal with $64\\times 64$ images; (2) For Hunter, we disable the background module in SPACE because the observations in Hunter is quite simple.","(3) We set the dimension of object representations $\\mathbf {z}^{what}_{ij}$ to be 16, and (in Crafter) $\\textbf {z}^{fg}$ to be 8.","The training data for SPACE is obtained via running a random policy on Hunter-Z4C4 and Crafter for 100000 environment steps.","Hyper-parameters for PPO Our PPO implementation is based on Tianshou [tianshou] which is purely based on PyTorch.","We adopt the default hyper-parameters in Tianshou, which are shown in Table REF .","Hyper-parameters for OCARL OCARL only introduces 2 hyper-parameter $\\lambda _{cat}$ (the coefficient for $L_{cat}$ defined in Eq.","(REF )) and $C$ (the number of object categories).","We set $\\lambda _{cat}=0.01$ for all experiments, $C=4,8$ for Hunter and Crafter respectively.","Network Architecture In OCAP, we use the convolution encoder from IMPALA [impla], which consists of 3 residual blocks with 16, 32, and 32 channels, respectively.","By the encoder, the $3\\times 64\\times 64$ image observation is encoded into $32\\times 8\\times 8$ .","The category predictor $f_{cat}$ is a simple 2-layer MLP with hidden size 32.","The object-category-specific networks $f_1,...,f_{C+1}$ in the OCMR module consists of $C+1$ MLPs, each of which is of 2-layer with hidden size 64.","Implementation for RRL The main implementation differences between RRL and OCARL are that (1) RRL does not use the supervision signals from the UOD model (i.e.","$\\lambda _{cat}=0$ in Eq.","(REF )); (2) RRL adopts a different (although very similar) network architecture.","In RRL, we use the same convolution encoder as OCARL.","The reasoning module in RRL is similar to OCARL, except that it utilizes a non-linear universe transformation instead of multiple object-category-specific networks.","We also use only one relational block in RRL as OCARL does, because we found more blocks may hinder the performance in our early experiments.","Implementation for SMORL We use SPACE [space] instead of SCALOR [scalor] (as suggested in the SMORL paper) to obtain object representations for SMORL.","Since SMORL is originally designed for goal-based RL, it uses an attention model to gather information from the set of object representations, in which the goal serves as a query vector.","In our settings, we do not consider goal-based RL; therefore, we use $L$ learnable vectors as queries, which are also used in the original SMORL.","We search over $L=[1,2,4,8]$ and find that $L=4$ works best in our experiments.","We also find that it is better to apply an MLP to the object representations before they are fed into the attention module.","To be more specific, we first run SPACE to get the object representations, sharing the same procedure as OCARL.","The information of these objects is first processed by a 2-layer MLP with hidden size 32 and then gathered through an attention module in which the query consists of $L=4$ learnable vectors.","The resulting tensor (of shape $L\\times 32$ ) is flattened and then fed into a 2-layer MLP with hidden size 64, which finally output the value function and action probabilities.","Table: PPO hyper-parameters.","Details of SPACE OCARL utilizes SPACE to discovery objects from raw objects.","In Section REF , we have introduced the inference model of $\\mathbf {z}_{}^{fg}$ .","For completeness, we would like to introduce the remaining parts of SPACE in this section: (1) inference model of $\\mathbf {z}_{}^{bg}$ ; (2) generative model of the image $\\mathbf {x}$ ; and (3) the ELBO to train the model.","Inference model of $\\mathbf {z}_{}^{bg}$ $\\mathbf {z}_{}^{bg}$ consists of several components $\\mathbf {z}_{k}^{bg}$ , and these components are inferred from the image $\\mathbf {x}$ in an iterative manner: $q(\\mathbf {z}_{k}^{bg}|\\mathbf {x})=\\prod _{k=1}^{K}q(\\mathbf {z}_{k}^{bg}|\\mathbf {z}_{ \\\\tau \\\\rbrace $ .\",\n \"Roughly speaking, each patch $\\\\mathbf {x}_{nij}\\\\in D_\\\\tau $ should contain exactly one object.\",\n \"Therefore, we first project $\\\\mathbf {x}_{nij}\\\\in D_\\\\tau $ into a low-dimensional latent space $\\\\mathbf {R}^d$ by running IncrementalPCA [ipca] on $D_\\\\tau $ , and then adopt KMeans clustering with a given cluster number $C$ upon the projected latent representations.\",\n \"The object-category predictor $q(\\\\mathbf {z}_{ij}^{cat}|\\\\mathbf {x}_{ij}) = \\\\Psi _\\\\texttt {KMeans}\\\\circ \\\\Psi _\\\\texttt {IncrementalPCA}$ is the composition of IncrementalPCA and KMeans: $\\\\begin{aligned}\\\\mathbf {z}_{ij}^{cat} = \\\\Psi _{\\\\texttt {KMeans}}\\\\circ \\\\Psi _{\\\\texttt {IcreamentalPCA}} (\\\\mathbf {x}_{ij})\\\\end{aligned}$ where $\\\\mathbf {z}_{ij}^{cat}\\\\in \\\\mathbf {R}^C$ is an one-hot latent which indicates the category of $\\\\mathbf {z}_{ij}$ .\",\n \"For simplicity, we regard the background (i.e.\",\n \"$z_{ij}^{pres}=0$ ) as special 'object', thus $\\\\mathbf {z}_{ij}^{cat}$ can be written as $\\\\mathbf {\\\\hat{z}}_{ij}^{cat}=\\\\Psi _{\\\\texttt {concatenate}}([1-z^{pres}_{ij}, z^{pres}_{ij}\\\\cdot \\\\mathbf {z}_{ij}^{cat}])$ .\",\n \"Note that $\\\\hat{\\\\mathbf {z}}_{ij}^{cat}$ is an one-hot latent $\\\\in \\\\mathbf {R}^{C+1}$ .\",\n \"Although the object-category predictor Eq.\",\n \"(REF ) is very simple, we find it works well in our experiments.\",\n \"In practice, the cluster number $C$ of KMeans can be set by leveraging prior knowledge about ground-truth category number, or by analysing some clustering quality metrics such as silhouette coefficients.\",\n \"In more complex environments, we can also rely on other advanced unsupervised clustering methods such as NVISA [cluster1], CC [cluster2], etc.\",\n \"Figure: OCARL consists of 3 parts: (1) category-aware UOD, (2) OCAP, and (3) OCMR.\",\n \"(1) In category-aware UOD, the objects are firstly discovered by a UOD method (SPACE) and then assigned an (extended) category label by unsupervised clustering (IcreamentalPCA + KMeans).\",\n \"(2) In OCAP, the observation is first encoded into a set of features 𝐙\\\\mathbf {Z} by convolution networks.\",\n \"Each vector in 𝐳 i =𝐙 :,i,j ∈𝐙\\\\mathbf {z}_i=\\\\mathbf {Z}_{:,i,j}\\\\in \\\\mathbf {Z} is given a task by the OCAP module to predict the corresponding (extended) object category, which is given by (1).\",\n \"𝐙\\\\mathbf {Z} is then fed to the OCMR module.\",\n \"(3) In OCMR, x-y coordinate information is added into 𝐙\\\\mathbf {Z}, which gives us 𝐙 ^\\\\mathbf {\\\\widehat{Z}}.\",\n \"Then we apply self-attention on 𝐙 ^\\\\mathbf {\\\\widehat{Z}} to query related information 𝐐 ^\\\\mathbf {\\\\widehat{Q}} for each object in 𝐙 ^\\\\mathbf {\\\\widehat{Z}}.\",\n \"𝐐 ^\\\\mathbf {\\\\widehat{Q}} and 𝐙 ^\\\\mathbf {\\\\widehat{Z}} are concatenated to form 𝐙 ˜\\\\mathbf {\\\\widetilde{Z}}.\",\n \"Each entity in 𝐙 ˜\\\\mathbf {\\\\widetilde{Z}} will be passed through different independent neural networks according to its corresponding object category, which will give us 𝐙 out \\\\mathbf {Z}^{out}.\",\n \"Finally, we perform a max operation along the H,WH,W dimension of 𝐙 out +𝐙 ^\\\\mathbf {Z}^{out} + \\\\mathbf {\\\\widehat{Z}}, and use the resulting feature vector to predict the value function and action probabilities.\"\n ],\n [\n \"OCAP: Object-Category Aware Perception\",\n \"The OCAP module is designed to robustly incorporate the object knowledge of the (category-aware) UOD model defined in Section REF .\",\n \"In a word, OCAP is a plain convolution encoder that accepts the category information from the UOD model as additional supervision signals.\",\n \"Such a design makes OCARL robust to the incompleteness of the UOD model: in extreme cases in which the UOD model fails to discover any objects, OCAP degenerates into a plain convolution encoder which is still able to extract useful information from the raw images with the help of reward signals.\",\n \"Suppose the convolution encoder in OCAP maps the raw image observation $\\\\mathbf {X} \\\\in \\\\mathbf {R}^{3\\\\times H_{img}\\\\times W_{img}}$ to a latent representation $\\\\mathbf {Z}\\\\in \\\\mathbf {R}^{F\\\\times H\\\\times W}$ , where $H, W$ are the same with those in the UOD model (see Eq.\",\n \"(REF )).\",\n \"On the other hand, $\\\\mathbf {X}$ is also fed into the UOD model to get the (extended) object category information $\\\\hat{\\\\mathbf {z}}^{cat} = \\\\lbrace \\\\hat{\\\\mathbf {z}}_{ij}^{cat}\\\\rbrace _{i=1}^H{}_{j=1}^W$ via Eq.\",\n \"(REF ).\",\n \"The OCAP module forces the latent representation $\\\\mathbf {Z}$ to encode the object category information, which is implemented by training a additional category predictor $f_{cat}: \\\\mathbf {R}^F\\\\rightarrow \\\\texttt {Categorical}(C+1)$ .\",\n \"$f_{cat}$ predicts $\\\\hat{\\\\mathbf {z}}_{ij}^{cat}$ given $\\\\mathbf {Z}_{:,i,j}$ which is a single channel in $\\\\mathbf {Z}$ .\",\n \"Therefore, the additional supervision signal that incorporates the object knowledge of the UOD model is given as: $\\\\begin{aligned}L_{cat} = \\\\sum _{i=1}^H\\\\sum _{j=1}^W \\\\texttt {CrossEntropyLoss}(f_{cat}(\\\\mathbf {Z}_{:,i,j}); \\\\hat{\\\\mathbf {z}}_{ij}^{cat}).\\\\end{aligned}$ $L_{cat}$ is used for training the convolution encoder in OCAP.\",\n \"In practice, $L_{cat}$ can be used as an auxiliary loss to any RL algorithm: $\\\\begin{aligned}L_{total} = L_{RL} + \\\\lambda _{cat} L_{cat},\\\\end{aligned}$ where $\\\\lambda _{cat}\\\\in \\\\mathbf {R}^+$ is a coefficient.\"\n ],\n [\n \"OCMR: Object-Centric Modular Reasoning\",\n \"OCMR is a module that takes $\\\\mathbf {Z}\\\\in \\\\mathbf {R}^{F\\\\times H\\\\times W}$ from the OCAP module as input and outputs a feature vector that summarizes $\\\\mathbf {Z}$ .\",\n \"The key design philosophy of OCMR is to adopt multiple independent and object-category-specific networks, each of which focuses on processing the object features of the same corresponding category.\",\n \"Compared with using a universal category-agnostic network, the processing logic of each independent network in OCMR is much simpler and therefore easier to master, which in turn allows for improved generalization as we will show in Section REF .\",\n \"This design philosophy also agrees with the recent discovery from [naturesepnet,modularity] which says that neural modules of specialization can lead to better generalization ability.\",\n \"Given $\\\\mathbf {Z}\\\\in \\\\mathbf {R}^{F\\\\times H\\\\times W}$ from OCAP, we first encode the x-y coordinates information (corresponding to the $H\\\\times W$ grid) to each channel, and then map the resulting tensor into $\\\\widehat{\\\\mathbf {Z}}\\\\in \\\\mathbf {R}^{HW\\\\times F}$ .\",\n \"$\\\\widehat{\\\\mathbf {Z}}$ is treated as $HW$ objects with x-y coordinate information encoded.\",\n \"To model the relations between objects, we apply a self-attention [attention] module on $\\\\widehat{\\\\mathbf {Z}}$ to query information for each object from its related objects: $\\\\begin{aligned}\\\\widehat{\\\\mathbf {Q}} = \\\\texttt {softmax}(\\\\widehat{\\\\mathbf {Z}}\\\\mathbf {W}_q(\\\\widehat{\\\\mathbf {Z}}\\\\mathbf {W}_k)^T)\\\\widehat{\\\\mathbf {Z}}\\\\mathbf {W}_v \\\\in \\\\mathbf {R}^{HW\\\\times F},\\\\end{aligned}$ where $\\\\mathbf {W}_q,\\\\mathbf {W}_k,\\\\mathbf {W}_v\\\\in \\\\mathbf {R}^{F\\\\times F}$ are trainable parameters.\",\n \"Modeling the relations is important, because one object's high-level semantic feature can be derived from other objects.\",\n \"For example, whether a door can be opened is determined by the existence of the key; therefore, the door should query other objects to check whether it is openable.\",\n \"After the attention module, $\\\\widehat{\\\\mathbf {Q}}$ and $\\\\widehat{\\\\mathbf {Z}}$ are concatenated together to get $\\\\widetilde{\\\\mathbf {Z}} = \\\\Psi _\\\\texttt {concatenate}([\\\\widehat{\\\\mathbf {Q}},\\\\widehat{\\\\mathbf {Z}}])\\\\in \\\\mathbf {R}^{HW\\\\times 2F}$ .\",\n \"Each $\\\\widetilde{\\\\mathbf {Z}}_{i,j,:}$ in $\\\\widetilde{\\\\mathbf {Z}}$ is then fed into different independent neural networks according to its corresponding object category $\\\\hat{\\\\mathbf {z}}_{ij}^{cat}$ .\",\n \"In practice, this can be implemented as: $\\\\begin{aligned}\\\\mathbf {Z}^{out}_{i,j,:} = \\\\sum _{c=1}^{C+1} f_c(\\\\widetilde{\\\\mathbf {Z}}_{i,j,:}) \\\\cdot \\\\hat{z}_{ij;c}^{cat},\\\\end{aligned}$ where $f_c:\\\\mathbf {R}^{2F} \\\\rightarrow \\\\mathbf {R}^F$ , $[\\\\hat{z}_{ij;1}^{cat},...,\\\\hat{z}_{ij;C+1}^{cat}] = \\\\hat{\\\\mathbf {z}}_{ij}^{cat}$ .\",\n \"Eq.\",\n \"(REF ) can be computed in parallel, resulting a tensor $\\\\mathbf {Z}^{out}\\\\in \\\\mathbf {R}^{H W\\\\times F}$ .\",\n \"Finally, the $\\\\mathbf {Z}^{out}$ is added to $\\\\widehat{\\\\mathbf {Z}}$ (i.e.\",\n \"a residual connection).\",\n \"We perform a max operation along the $H,W$ dimension, and using the resulting vector $\\\\in \\\\mathbf {R}^F$ to predict the value function and action probabilities: $\\\\begin{aligned}\\\\texttt {action\\\\_probs, value} = f_{ac}(\\\\max _{H,W}(\\\\mathbf {Z}^{out} + \\\\widehat{\\\\mathbf {Z}})).\\\\end{aligned}$\"\n ],\n [\n \"Task Description\",\n \"In this work, we consider two task domains: Crafter and Hunter.\",\n \"The observations on both tasks are raw images of shape $64\\\\times 64$ .\"\n ],\n [\n \"Crafter\",\n \"[crafter] is a complex 2-D MineCraft-style RL task, where complex behaviors are necessary for the agent' survival.\",\n \"The environment is procedurally generated by arranging various resources, terrain types, and objects (18 in total).\",\n \"The agent is rewarded if it can craft new items and accomplish achievements (22 possible achievements in total).\"\n ],\n [\n \"Hunter\",\n \"is much simpler than Crafter.\",\n \"It is also procedurally generated but only contains 4 types of objects: Hunter, Cow, Zombie and Wall The images to render these objects come from Crafter [crafter], which is under MIT license..\",\n \"The agent can control Hunter and can get positive reward (=1) if Hunter catches a Cow or kill a Zombie.\",\n \"The agent will be given a high reward (=5) if it can accomplish this task by catching/killing all Cows and Zombies.\",\n \"However, once Hunter is caught by a Zombie, the agent will receive a negative reward (=-1), and the episode ends.\",\n \"In a word, the agent in Hunter should master two different behavior patterns: chase & catch the Cow, and avoid & shoot at Zombie.\",\n \"Although Hunter is simple, it is a typical OORL task.\",\n \"We can derive different environment instances from Hunter by setting the number of Zombies and Cows.\",\n \"We use Hunter-ZmCn to denote an environment that spawns (m Zombies + n Cows) at the beginning of each episode, and Hunter-ZmCn/ZnCm an environment that spawns (m Zombies + n Cows) or (n Zombies + m Cows).\"\n ],\n [\n \"Evaluation\",\n \"In this section, we evaluate OCARL's effect both on sample efficiency and generalization ability.\",\n \"We consider the following baselines: PPO [ppo], RRL [rrl], and SMORL [smorl].\",\n \"PPO is a general-purpose on-policy RL algorithm with a plain neural network (i.e.\",\n \"Convolution + MLP).\",\n \"RRL proposes an attention-based neural network to introduce relational inductive biases and iterated relational reasoning into RL agent, which aims to solve object-oriented RL tasks without any other external supervision signals.\",\n \"SMORL utilizes the UOD model to decompose the observation into a set of object representations which are directly used as the agent's new observation.\",\n \"All algorithms (RRL, SMORL, and OCARL) are (re-)implemented upon PPO to ensure a fair comparison.\",\n \"Note that OCARL is orthogonal to the backbone RL algorithm, therefore other advanced RL methods such as [rainbow,hvf] are also applicable.\",\n \"For more implementation details, please refer to the Appendix.\"\n ],\n [\n \"Sample Efficiency\",\n \"The results on Hunter domain are shown in Figure REF .\",\n \"We can conclude three facts from this figure: (1) OCARL performs better and is more stable than other baselines on all tasks.\",\n \"(2) RRL is competitive with OCARL on Hunter-Z4C4 but significantly worse on other simper environments.\",\n \"This is because RRL relies solely on the reward signals to ground the visual images into objects; therefore, it requires enough (rewarded) interactions between objects, which simpler environments such as Hunter-Z1C1 fail to supply.\",\n \"(3) SMORL performs better than other baselines (PPO, RRL) on simple environments such as Hunter-Z1C0/Z0C1, Hunter-Z1C1 and Hunter-Z4C0/Z0C4, but unable to make progress in Hunter-Z4C4 which consists of more objects.\",\n \"This is because SMORL adopts a simple reasoning module that may not be well-suited to multi-object reasoning.\",\n \"The results on Crafter domain are shown in Table REF and Figure REF .\",\n \"Crafter is a complex environment in that it features sparse (rewarded) interactions between objects and large object category numbers.\",\n \"Besides, some objects are omitted by the UOD model (see Appendix) because they almost do not appear in the training dataset due to the insufficient exploration of the random policy.\",\n \"Due to these features, both RRL and SMORL fail to provide noteworthy results (with scores of 4.22 and 3.94, respectively).\",\n \"On the other hand, OCARL is able to present more meaningful behaviours (with a score of 12.31), such as collecting coal, defeating zombies, making wood pickaxes and so on, as shown in Figure REF .\",\n \"The learning curves and detailed achievement success rates are provided in the Appendix.\",\n \"Figure: Success rates (averaged across 12 seeds) on 22 achievements in Crafter.\",\n \"These success rates are calculated using the final 1M environment steps during training.\",\n \"OCARL presents more meaningful behaviours than other baselines, such as collecting coal, defeating zombies, making wood pickaxes and so on.\"\n ],\n [\n \"Generalization\",\n \"To evaluate the generalization ability, we train agents on Hunter-Z1C0/Z0C1, Hunter-Z1C1 and Hunter-Z4C0/Z0C4 separately, and then observe their test performance on Hunter-Z4C4.\",\n \"In Hunter-Z1C1, the agent needs to generalize from a few objects to more objects.\",\n \"In Hunter-Z4C0/Z0C4, the agent never observes the coincidence of Cow and Zombie.\",\n \"The generalization from Hunter-Z1C0/Z0C1 to Hunter-Z4C4 is the most difficult.\",\n \"Such a paradigm actually follows the out-of-distribution (OOD) setting, in that the object combination (4 Zombies + 4 Cows) is never seen in the training environments.\",\n \"Therefore, although such generalization is possible for humans, it is much harder for RL algorithms.\",\n \"As shown in Figure REF , OCARL is the only algorithm that make significant progress on the test environment.\",\n \"Although other baselines can improve their training performance (Figure REF ), they are unable to generalize to the test environment.\",\n \"The generalization performance of OCARL is most impressive on Hunter-Z1C1, because agent trained on Hunter-Z1C1 even achieves slightly better than that on Hunter-Z4C4 (i.e.\",\n \"directly trained on the test environment, see the last figure in Figure REF .\",\n \").\"\n ],\n [\n \"Robustness to the Incompleteness of Object Discovery Methods\",\n \"OCARL relies on UOD methods to discover different objects from given observations, which are not guaranteed to discover all objects, especially in complex environments [benchmarkoom].\",\n \"Therefore, one may naturally ask to what extent OCARL's performance is affected by such incompleteness.\",\n \"To answer this question, we randomly mask out each object (by setting $z_{ij}^{pres}$ to 0) discovered by the UOD model with a given probability $p\\\\in [0,1]$ , and evaluate OCARL's performance after training 5M environment steps on Hunter.\",\n \"Such random confusion of the UOD model will affect both the OCAP and OCMR modules of OCARL.\",\n \"Note that OCARL($p=0$ ) is OCARL itself, and OCARL($p=1$ ) is exactly RRL This is because Eq.\",\n \"(REF ) takes no effect and Eq.\",\n \"(REF ) also degenerates into a single universal network because all object features are passed through the same network $f_{bg}$ which is corresponding to the 'background'..\",\n \"The results are shown in Figure REF .\",\n \"Although increasing $p$ does hinder OCARL's performance, OCARL can still benefit from the imperfect UOD model in that its performance is still better than RRL in general.\",\n \"Such robustness should be attributed to the fact that OCARL does not directly use the output of the UOD model as observation but instead treats it as an additional supervision signal that help the encoder to capture object-category information, which gives OCARL a second chance to find other useful information from the raw image.\"\n ],\n [\n \"OCMR Improves Generalization\",\n \"One key advantage of OCARL is that it can generalize to unseen combinations of objects by leveraging the invariant relations between objects learnt during training.\",\n \"We argue that such an advantage is established via OCMR, which adopts object-category-specific networks to deal with different objects.\",\n \"To confirm our argument, we consider a variant of OCARL (i.e.\",\n \"OCARL w/o OCMR) in which the OCMR module is removed and replaced by an RRL-like reasoning module which adopts a universal network (instead of multiple object-category-specific networks as OCARL does).\",\n \"The comparison between OCARL w/ and w/o OCMR is shown in Figure REF .\",\n \"OCARL w/o OCMR (almost completely) fails to generalize from Hunter-Z1C0/Z0C1, Hunter-Z4C0/Z0C4 to Hunter-Z4C4, while OCARL w/ OCMR does.\",\n \"In Figure REF , we also consider a variant of OCARL (i.e.\",\n \"OCARL w/o OCAP) in which the OCAP module is disabled by setting $\\\\lambda _{cat}=0$ in Eq.\",\n \"(REF ).\",\n \"In this paradigm, only OCMR is in effect.\",\n \"As shown in Figure REF , OCMR alone can still bring benefits without the help of OCAP (although not as consistently as OCMR+OCAP ).\",\n \"According to these results, we can conclude that OCMR is crucial in OCARL's success.\",\n \"Table: The Modularity of OCMR.\",\n \"We first train agent on Hunter-Z4C4 and then disable the module f Z f_Z (corresponding to Zombie) or f C f_C (corresponding to Cow) in OCMR (see Eq.()).\",\n \"Each agent is evaluated on both Hunter-Z4C0 and Hunter-Z0C4, and the mean and standard deviation of 12 seeds are reported .\",\n \"The results show that the behaviour patterns (chase & catch the Cow) and (avoid & shoot at Zombie) are encoded in f C f_C and f Z f_Z respectively.\"\n ],\n [\n \"The Modularity of OCMR\",\n \"OCMR adopts different modules to deal with object features from different categories.\",\n \"In this section, we will show that each module in OCMR actually encodes an object-category-specific behaviour pattern.\",\n \"As stated in Section REF , the agent should master two useful behaviour patterns in Hunter: (1) chase & catch the Cow, and (2) avoid & shoot at Zombie.\",\n \"The existence of these patterns can be observed by evaluating the agent in Hunter-Z0C4 and Hunter-Z4C0 respectively, because agents that master pattern (1) (pattern (2)) should achieve better performance on Hunter-Z0C4 (Hunter-Z4C0).\",\n \"In Table REF , we first train agent on Hunter-Z4C4 and then disable the module $f_Z$ (corresponding to Zombie) or $f_C$ (corresponding to Cow) by replacing $f_C$ or $f_E$ with $f_{bg}$ (corresponding to the 'background') in Eq.\",\n \"(REF ).\",\n \"The results shows that disabling $f_Z$ does not affect the performance on Hunter-Z0C4 ($8.29\\\\rightarrow 8.13$ ), which means that $f_Z$ is not relative to behaviour pattern (1).\",\n \"Instead, $f_Z$ is corresponding to behaviour pattern (2) because we can observe a significant performance decline ($8.27\\\\rightarrow 1.17$ ) on Hunter-Z4C0 when we disable $f_Z$ .\",\n \"Similar phenomenon can also be observed on $f_C$ .\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we propose OCARL, which utilize the category information of objects to facilitate both perception.\",\n \"For the perception, we propose the OCAP, which enables the encoder to distinguish the categories of objects.\",\n \"For the reasoning, we propose the OCMR, which adopts object-category-specific networks to deal with different objects.\",\n \"Our experiments are carried out on Crafter and Hunter.\",\n \"Experiments show that OCARL outperforms other baselines both on sample efficiency and generalization ability.\",\n \"We also perform several ablation studies to show that (1) OCARL is robust to the incompleteness of UOD methods, (2) OCMR is critical to improving generalization, and (3) each module in OCMR actually encodes an object-category-specific behaviour pattern in Hunter.\"\n ],\n [\n \"Limitation\",\n \"The main limitation is that we only test OCARL's generalization ability to unseen object combinations, but not novel object instances.\",\n \"In order to generalize to novel object instances that looks very different from old ones, the agent should interact with the novel objects first and then infer the their underlying categories according to these interactions.\",\n \"Such a problem is likely to be in the meta-RL domain and needs more efforts to solve, which we would like to leave for future work.\"\n ],\n [\n \"Acknowledgements\",\n \"This work is partially supported by the National Key Research and Development Program of China(under Grant 2018AAA0103300), the NSF of China(under Grants 61925208, 62102399, 62002338, U19B2019, 61906179, 61732020), Strategic Priority Research Program of Chinese Academy of Science (XDB32050200), Beijing Academy of Artificial Intelligence (BAAI) and Beijing Nova Program of Science and Technology (Z191100001119093) , CAS Project for Young Scientists in Basic Research(YSBR-029), Youth Innovation Promotion Association CAS and Xplore Prize.\",\n \"numbers Checklist For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\",\n \"Did you describe the limitations of your work?\",\n \"Did you discuss any potential negative societal impacts of your work?\",\n \"Have you read the ethics review guidelines and ensured that your paper conforms to them?\",\n \"If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?\",\n \"Did you include complete proofs of all theoretical results?\",\n \"If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\",\n \"Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\",\n \"see Appendix.\",\n \"Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\",\n \"Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\",\n \"If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...\",\n \"If your work uses existing assets, did you cite the creators?\",\n \"Did you mention the license of the assets?\",\n \"Partially.\",\n \"Did you include any new assets either in the supplemental material or as a URL?\",\n \"Did you discuss whether and how consent was obtained from people whose data you're using/curating?\",\n \"Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?\",\n \"If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?\",\n \"Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\",\n \"Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\",\n \"Implementation Details The implementation of OCARL is available at https://anonymous.4open.science/r/OCARL-51BF.\",\n \"Hyper-parameters for SPACE In this paper, we use SPACE for object discovery.\",\n \"Most hyper-parameters are the same as the default parameters for Atari in SPACE, except that (1) we use a smaller network to deal with $64\\\\times 64$ images; (2) For Hunter, we disable the background module in SPACE because the observations in Hunter is quite simple.\",\n \"(3) We set the dimension of object representations $\\\\mathbf {z}^{what}_{ij}$ to be 16, and (in Crafter) $\\\\textbf {z}^{fg}$ to be 8.\",\n \"The training data for SPACE is obtained via running a random policy on Hunter-Z4C4 and Crafter for 100000 environment steps.\",\n \"Hyper-parameters for PPO Our PPO implementation is based on Tianshou [tianshou] which is purely based on PyTorch.\",\n \"We adopt the default hyper-parameters in Tianshou, which are shown in Table REF .\",\n \"Hyper-parameters for OCARL OCARL only introduces 2 hyper-parameter $\\\\lambda _{cat}$ (the coefficient for $L_{cat}$ defined in Eq.\",\n \"(REF )) and $C$ (the number of object categories).\",\n \"We set $\\\\lambda _{cat}=0.01$ for all experiments, $C=4,8$ for Hunter and Crafter respectively.\",\n \"Network Architecture In OCAP, we use the convolution encoder from IMPALA [impla], which consists of 3 residual blocks with 16, 32, and 32 channels, respectively.\",\n \"By the encoder, the $3\\\\times 64\\\\times 64$ image observation is encoded into $32\\\\times 8\\\\times 8$ .\",\n \"The category predictor $f_{cat}$ is a simple 2-layer MLP with hidden size 32.\",\n \"The object-category-specific networks $f_1,...,f_{C+1}$ in the OCMR module consists of $C+1$ MLPs, each of which is of 2-layer with hidden size 64.\",\n \"Implementation for RRL The main implementation differences between RRL and OCARL are that (1) RRL does not use the supervision signals from the UOD model (i.e.\",\n \"$\\\\lambda _{cat}=0$ in Eq.\",\n \"(REF )); (2) RRL adopts a different (although very similar) network architecture.\",\n \"In RRL, we use the same convolution encoder as OCARL.\",\n \"The reasoning module in RRL is similar to OCARL, except that it utilizes a non-linear universe transformation instead of multiple object-category-specific networks.\",\n \"We also use only one relational block in RRL as OCARL does, because we found more blocks may hinder the performance in our early experiments.\",\n \"Implementation for SMORL We use SPACE [space] instead of SCALOR [scalor] (as suggested in the SMORL paper) to obtain object representations for SMORL.\",\n \"Since SMORL is originally designed for goal-based RL, it uses an attention model to gather information from the set of object representations, in which the goal serves as a query vector.\",\n \"In our settings, we do not consider goal-based RL; therefore, we use $L$ learnable vectors as queries, which are also used in the original SMORL.\",\n \"We search over $L=[1,2,4,8]$ and find that $L=4$ works best in our experiments.\",\n \"We also find that it is better to apply an MLP to the object representations before they are fed into the attention module.\",\n \"To be more specific, we first run SPACE to get the object representations, sharing the same procedure as OCARL.\",\n \"The information of these objects is first processed by a 2-layer MLP with hidden size 32 and then gathered through an attention module in which the query consists of $L=4$ learnable vectors.\",\n \"The resulting tensor (of shape $L\\\\times 32$ ) is flattened and then fed into a 2-layer MLP with hidden size 64, which finally output the value function and action probabilities.\",\n \"Table: PPO hyper-parameters.\",\n \"Details of SPACE OCARL utilizes SPACE to discovery objects from raw objects.\",\n \"In Section REF , we have introduced the inference model of $\\\\mathbf {z}_{}^{fg}$ .\",\n \"For completeness, we would like to introduce the remaining parts of SPACE in this section: (1) inference model of $\\\\mathbf {z}_{}^{bg}$ ; (2) generative model of the image $\\\\mathbf {x}$ ; and (3) the ELBO to train the model.\",\n \"Inference model of $\\\\mathbf {z}_{}^{bg}$ $\\\\mathbf {z}_{}^{bg}$ consists of several components $\\\\mathbf {z}_{k}^{bg}$ , and these components are inferred from the image $\\\\mathbf {x}$ in an iterative manner: $q(\\\\mathbf {z}_{k}^{bg}|\\\\mathbf {x})=\\\\prod _{k=1}^{K}q(\\\\mathbf {z}_{k}^{bg}|\\\\mathbf {z}_{ 0$ , denote $\\ell \\lambda \\vec{\\ell }$ and set for $e = [x,y] \\in E(\\mathbb {Z}^{d})$ , $c(e) = c_{*}(e)e^{(x+y)\\cdot \\ell }.$ We focus on the regime where $\\mathbf {P}\\left(c_{*}(e) \\ge x \\right) = \\mu ([x,+\\infty [) = L(x) x^{-\\gamma } \\quad \\textnormal {for any }x \\ge 0,$ with $\\gamma \\in (0,1)$ and $L(\\cdot )$ is a slowly-varying function.","For a fixed environment $\\omega $ , we consider the random walk $(X_n)_{n \\ge 0}$ with conductances $(c^\\omega (e))_{e \\in E(\\mathbb {Z}^d)}$ .","We will call $P_{x}^{\\omega }$ the quenched law of the random walk on the environment starting from $x$ and with transition probabilities $p^\\omega (x, y)$ for $x, y \\in \\mathbb {Z}^d$ $p^\\omega (x, y) = \\frac{c^\\omega ([x, y])}{\\sum \\limits _{y \\sim x} c^\\omega ([x, y])}.$ Let $\\mathbb {P}_{x} = \\mathbf {E}[P_{x}^{\\omega }(.",")]$ be the annealed law of this random walk ($\\mathbf {E}[\\cdot ]$ being the expectation with respect to $\\mathbf {P}$ and $\\mathbf {Var}$ the corresponding variance).","The main theorem of this work is the following.","Theorem 1.1 Let $d \\ge 5$ and fix some $T > 0$ , the following statements hold for $\\mathbf {P}$ -almost every environment $\\omega \\in \\Omega $ .","There exists a deterministic $v_0 \\in \\mathbb {S}^{d-1}$ with $v_0 \\cdot \\vec{\\ell } > 0$ such that, under the quenched law $P^\\omega _0(\\cdot )$ , we have the following weak convergences holding $\\left( \\frac{X_{{n t}}}{n^\\gamma /L(n)} \\right)_{0 \\le t \\le T} \\overset{(\\mathrm {d})}{\\rightarrow } \\left( v_0 C_\\infty ^{-\\gamma } \\mathcal {S}_\\gamma ^{-1}(t) \\right)_{0 \\le t \\le T},$ $\\left( \\frac{X_{{n t}} - (X_{{n t}} \\cdot v_0) v_0}{\\sqrt{n^\\gamma /L(n)}} \\right)_{0 \\le t \\le T} \\overset{(\\mathrm {d})}{\\rightarrow }\\left( M_d B_{\\mathcal {S}_\\gamma ^{-1}(t)} \\right)_{0 \\le t \\le T}.$ For a deterministic constant $C_\\infty >0$ , a deterministic $d \\times d$ matrix $M_d$ of rank $d-1$ , a standard Brownian motion $B$ and a stable subordinator of index $\\gamma $ , independent of $B$ , $\\mathcal {S}_\\gamma $ .","The first convergence holds in the uniform topology while the second holds in the Skorohod's $J_1$ topology."],["Quenched convergence via series convergence","Let us explain the main strategy that we will follow.","In one can find the definition of a regeneration structure with suitable independence properties, in particular let $(\\tau _k)_{k \\ge 0}$ be the regeneration times.","See Section REF where we recall the rigorous definition of $(\\tau _k)_{k \\ge 0}$ .","Call $v \\mathbb {E}[X_{\\tau _2} - X_{\\tau _1} ]$ , then we introduce, for all $t \\ge 0$ $Y_n(t) = \\frac{X_{\\tau _{{tn}}}}{n}, \\quad Z_n(t) = \\frac{X_{\\tau _{{tn}}} - v n t}{n^{1/2}},\\quad \\textnormal {and} \\quad S_n(t) = \\frac{\\tau _{{tn}}}{\\mathrm {Inv}(n)},$ where $\\mathrm {Inv}(t) \\inf \\left\\lbrace s: \\mathbf {P}[c_{*} > s]\\le 1/t \\right\\rbrace ,$ is the generalised right-continuous inverse of the tail of $c_{*}$ , let us point out that $\\mathrm {Inv}(n) \\approx n^{1/\\gamma }$ .","Our initial aim is to prove that for $\\mathbf {P}$ -almost all environments $\\omega \\in \\Omega $ these three quantities converge weakly.","Then we will use this to prove Theorem REF .","We get the quenched convergence of $Y_n(t)$ for free from .","For technical reasons let us define the process $S^*_n(t) = \\frac{\\tau _{{tn}+1} - \\tau _{1}}{\\mathrm {Inv}(n)}.$ To prove quenched convergence of $S^*_n(t)$ and $Z_n(t)$ we will use alternative formulations considered in and to rephrase the problem.","These conditions are stated in the following theorem.","Let us define $F(h) = \\exp \\left(-\\lambda _1 h(t_1) - \\lambda _2(h(t_2) - h(t_1)) -... - \\lambda _m(h(t_m) - h(t_{m-1}))\\right), $ with $\\lambda _1,\\cdots ,\\lambda _m \\ge 0$ and $0 < t_1 < \\cdots < t_m \\le T$ .","Theorem 1.2 Let $d \\ge 5$ , then the following holds for any $F_1$ as in (REF ) $ \\sum _{n = 1 }^{+\\infty }\\mathbf {Var}\\left(E_{0}^{\\omega }\\left[F_1(S^*_{b^n})\\right]\\right)< +\\infty ,$ with $b \\in (1,2)$ .","Moreover, by denoting with $Z_n$ the polygonal interpolant of the quantity denoted with the same letter in (REF ), we have $ \\sum _{n = 1 }^{+\\infty }\\mathbf {Var}\\left(E_{0}^{\\omega }[F_2(Z_{b^n})]\\right)< +\\infty ,$ where $F_2$ is any bounded $K$ -Lipschitz function from $\\mathcal {C}([0, T],\\mathbb {R}^{d}) \\rightarrow \\mathbb {R}$ ."],["Sketch of proof","To prove Theorem REF , we can consider under each environment two independent random walks $X^1$ and $X^2$ which have the same law as $X$ under $P_0^{\\omega }$ .","Then the term in the second sum above is equal to $\\mathbb {E}[F_2(Z_{b^n}^1)F_2(Z_{b^n}^2)] - \\mathbb {E}[F_2(Z_{b^n}^1)]\\mathbb {E}[F_2(Z_{b^n}^2)].$ We will show that the two walks do not meet each other after reaching a distance not too far from the origin, then deduce that the quantity above goes quickly to 0.","Indeed, when the two walks do not meet, they behave almost as if they evolved in independent environments.","The important ideas can be grouped in two main parts.","The first one deals with joint regeneration levels while the second consists in proving the asymptotic separation using joint regeneration levels.","Joint regeneration levels: Section will be devoted to extend the notion of regeneration times of to the case where we have two walks.","We will construct joint regeneration levels: a sequence of levels such that the hitting time of these are regeneration times for both walks.","Asymptotic separation: Once we have constructed joint regeneration levels the idea is the following: at each regeneration level, both walks have a decent chance of staying far apart for a long time, after which it is unlikely that they will ever meet again.","As joint regeneration levels are common, the walks will decouple relatively quickly.","In this part we use a method which is inspired from .","However, we cannot directly use their results because our model lacks both the uniform ellipticity property of the jump probabilities and it is not ballistic."],["Notations","We fix an orthonormal basis $(e_1,...,e_d)$ of $\\mathbb {R}^d$ with elements of $\\mathbb {Z}^d$ such that $e_1\\cdot \\vec{\\ell } \\ge e_2\\cdot \\vec{\\ell } \\ge ... \\ge e_d\\cdot \\vec{\\ell }$ .","In particular we have $e_1\\cdot \\vec{\\ell } \\ge 1/\\sqrt{d}$ .","We also fix an orthonormal basis of $\\mathbb {R}^d$ , $(f_1,...,f_d)$ such that $f_1 = \\vec{\\ell }$ .","For any $L \\in \\mathbb {R}$ we set $\\mathcal {H}^{+}(L) = \\lbrace z \\in \\mathbb {Z}^d ; \\, z\\cdot \\vec{\\ell } > L\\rbrace \\,\\, \\mathrm {and} \\,\\, \\mathcal {H}^{-}(L) = \\lbrace z \\in \\mathbb {Z}^d ; \\, z\\cdot \\vec{\\ell } \\le L\\rbrace .$ Similarly, for $z\\in \\mathbb {Z}^d$ $\\mathcal {H}^{+}_{z} = \\mathcal {H}^{+}(z\\cdot \\vec{\\ell }) \\,\\, \\mathrm {and} \\,\\, \\mathcal {H}^{-}_{z} = \\mathcal {H}^{-}(z\\cdot \\vec{\\ell }).$ For any $x \\in \\mathbb {Z}^d$ we define $ \\mathcal {V}_x = \\lbrace x,x+e_1,x-e_1,\\cdots ,x+e_d,x-e_d\\rbrace .$ For any $x,y \\in \\mathbb {Z}^d$ we say that $x$ and $y$ are adjacent and write $x \\sim y$ if $x = y \\pm e_i$ where $i \\in \\lbrace 1,\\cdots ,d\\rbrace $ .","For $x$ in $\\mathbb {Z}^d$ and $e = [e^{+}, e^{-}]$ and edge, we write $x \\sim e$ if $x \\sim e^{-}$ or $x \\sim e^{+}$ and $x \\notin \\lbrace e^{+},e^{-}\\rbrace $ .","Two distinct edges $e$ and $e^{\\prime }$ are said to be adjacent and we write $e\\sim e^{\\prime }$ if they share one endpoint.","For $x \\in \\mathbb {Z}^d$ we define $\\mathcal {E}_x \\lbrace [x,y] \\, | \\, y\\sim x\\rbrace , \\,\\, \\text{ and for }A\\subset \\mathbb {Z}^d \\text{, }\\mathcal {E}_A = \\bigcup _{x\\in A}\\mathcal {E}_x.$ $\\mathcal {L}^x \\lbrace [y,z] \\,\\, | \\,\\, y \\cdot \\vec{\\ell } \\le x \\cdot \\vec{\\ell } \\text{ and } z \\cdot \\vec{\\ell } \\le x \\cdot \\vec{\\ell }\\rbrace \\cup \\mathcal {E}_x.$ $\\mathcal {R}^x \\lbrace [y,z] \\,\\, | \\,\\, y\\cdot \\vec{\\ell } > x\\cdot \\vec{\\ell } \\text{ or } z\\cdot \\vec{\\ell } > x\\cdot \\vec{\\ell }\\rbrace \\cup \\mathcal {E}_x.$ For any subset $V$ of $\\mathbb {Z}^d$ we define its edge-set $E(V) \\lbrace [x,y] \\text{ ; } x \\in V \\text{ and } y \\in V\\rbrace ,$ its vertex-boundary $\\partial V \\lbrace x \\notin V \\text{ ; }\\exists y \\in V, x \\sim y \\rbrace ,$ and its edge-boundary $\\partial _{E} V \\lbrace [x,y]\\in E(\\mathbb {Z}^d)\\text{ ; } y \\in V, x \\notin V \\rbrace .$ For any subset $A \\subset \\mathbb {Z}^d$ we define its width $W(A)= \\max _{1\\le i \\le d}\\left(\\max _{y \\in A}y \\cdot e_i - \\min _{y \\in A}y \\cdot e_i\\right).$ For any $L,L^{\\prime } > 0$ and $y \\in \\mathbb {Z}^d$ we define the box $\\mathcal {B}_y(L,L^{\\prime }) = \\lbrace x \\in \\mathbb {Z}^d \\text{ ; }|(x-y)\\cdot \\vec{\\ell }| \\le L \\text{ and } |(x-y)\\cdot f_i| \\le L^{\\prime } \\text{ for all } i \\in \\lbrace 2,...,d\\rbrace \\rbrace ,$ and its positive boundary $\\partial ^{+}\\mathcal {B}_y(L,L^{\\prime }) = \\lbrace x \\in \\partial \\mathcal {B}_y(L,L^{\\prime }) \\text{ s.t.","}|(x-y)\\cdot \\vec{\\ell }| > L\\rbrace .$ When $y = 0$ we just use the notation $\\mathcal {B}(L,L^{\\prime })$ .","For a general random walk on a graph $(V,E)$ we define for $B \\subset V$ the hitting times of $B$ , $T_B = \\inf \\lbrace n \\ge 0 \\text{ : } X_n \\in B\\rbrace \\text{ and } T^{+}_B = \\inf \\lbrace n \\ge 1 \\text{ : } X_n \\in B\\rbrace ,$ and the exit time of $B$ $T^{\\mathrm {ex}}_B = \\inf \\lbrace n \\ge 0 \\text{ ; } X_n \\notin B\\rbrace .$ When $B$ is a singleton $\\lbrace x\\rbrace $ we use the notation $T_x$ .","For $R \\in \\mathbb {R}$ we define $T_{R} = T_{\\mathcal {H}^{+}(R)}.$ We denote by $\\theta _n$ the translation of $n$ units of time.","For $\\mathbb {P}$ a probability measure, $A_1,\\cdots ,A_n$ events and $X$ a random variable the notation $\\mathbb {E}[A_1,\\cdots ,A_n,X]$ denotes $\\mathbb {E}[{1}_{A_1}\\cdots {1}_{A_n}X]$ .","In most of the paper we will add an index $i \\in \\lbrace 1,2\\rbrace $ as superscript when the event depends on $X^i$ , where $X^i$ are two realisations of the random walk (independent with respect to the quenched distribution).","For example, $T^{i}_R$ will be the hitting time of $\\mathcal {H}^{+}(R)$ by the $i$ -th random walk.","In the following we will use the notations $c$ , $C$ , $a$ to denote constants which only depend on the dimension $d$ and the strength of the bias $\\lambda $ , its direction $\\vec{\\ell }$ and a parameter $K > 0$ ."],["Probability measures.","Let us recall that $P^\\omega $ denotes the quenched law while $\\mathbb {P}$ denotes the annealed law of the walk.","As we already mentioned we will often consider two walks $X^1_{\\cdot }$ and $X^2_{\\cdot }$ , for these walks it is useful to introduce two probability measures.","The first is for two walks that evolve independently (according to the law defined in (REF )) in the same environment.","It is characterised by the formula: $\\mathbb {P}_{U_1, U_2}[X^1_{\\cdot } \\in A_1, X^2_{\\cdot } \\in A_2] \\mathbf {E}\\left[ P_{U_1}^\\omega \\left( X^1_{\\cdot } \\in A_1 \\right) \\times P_{U_2}^\\omega \\left( X^2_{\\cdot } \\in A_2 \\right) \\right].$ It is also possible to define a measure according to which the two walks evolve with the same quenched law in two independent environments $\\omega _1, \\omega _2$ sampled according to $\\mathbf {P}$ .","In formulas, $Q_{U_1, U_2}[X^1_{\\cdot } \\in A_1, X^2_{\\cdot } \\in A_2] &\\mathbf {E}\\left[ P_{U_1}^{\\omega _1}\\left( X^1_{\\cdot } \\in A_1 \\right) \\times P_{U_2}^{\\omega _2}\\left( X^2_{\\cdot } \\in A_2 \\right) \\right] \\\\ &= \\mathbb {P}_{U_1}\\left( X^1_{\\cdot } \\in A_1 \\right) \\times \\mathbb {P}_{U_1}\\left( X^2_{\\cdot } \\in A_2 \\right).\\nonumber $ We denote by $\\mathbb {E}_{U_1,U_2}[\\cdot ]$ the expectation with respect to the measure (REF ).","This is the same notation as the annealed law of one walk (with the exception of the two starting points), this because by considering the marginal of $X^1_{\\cdot }$ that is the same as the law of the single walk.","$\\mathbb {E}^Q[\\cdot ]$ is the expectation associated to (REF ).","For $a \\in \\mathbb {R}^{2d}$ and $x \\in \\mathbb {Z}^d$ we write $\\mathbb {P}_x^{a}$ (resp.","$\\mathbb {P}_{x,y}^{a}$ ) for the annealed law where we set the conductances in $\\mathcal {E}_x$ (resp.","in $\\mathcal {E}_x$ and $\\mathcal {E}_y$ ) to the value $a$ .","A special case is when $a = (K,\\ldots ,K)$ with $K > 0$ , in that case we write $\\mathbb {P}_x^{K}$ (resp.","$\\mathbb {P}_{x,y}^{K}$ ).","The notation $Q_{x,y}^{a}$ means the law $\\mathbb {P}_x^{a}\\otimes \\mathbb {P}_y^{a}$ .","In accordance with the notation of (that we will recall in Section REF ) we write $\\mathbb {P}_x^{K}(\\cdot |D=+\\infty )$ for the law of the future of the walk after a regeneration time at $x$ .","We also write $Q_{x,y}^K(\\cdot |D^{\\otimes }=+\\infty ) = \\mathbb {P}_x^{K}(\\cdot |D=+\\infty )\\otimes \\mathbb {P}_y^{K}(\\cdot |D=+\\infty )$ ."],["Joint regeneration levels","For this model a regeneration structure was introduced in .","It was further improved in to obtain the independence of trajectories between regeneration times.","For our purpose it will be more convenient to have joint regeneration levels.","Let us now introduce a joint regeneration structure for two walks in the same environment.","Figure: Two trajectories and the first joint regeneration level ℒ 1 \\mathcal {L}_1.In principle a regeneration time is simply a new maximum which will be a minimum for the future of the trajectory.","However, for technical reasons we want to work with a subset of those regeneration times: those happening on points with bounded conductances.","This is why we will adapt the construction of which uses $K$ -open ladder points.","The first step is to introduce the notion of joint $K$ -open ladder point and we do this in the next subsection."],["Notations and preliminary results","A point $z \\in \\mathbb {Z}^d$ is said to be $K$ -open if for all $e \\in \\mathcal {E}_x$ we have $c_{*}(e) \\in [1/K,K]$ (we call such edges $K$ -normal).","Joint $K$ -open ladder points are new maxima in the bias direction for both walks such that the points are $K$ -open and the maxima are close in the sense that they are both at roughly the same distance, say $\\approx R$ , from the origin in the bias direction.","Note that the distance cannot be exactly the same due to the lattice structure.","The main goal of this subsection is to prove that the first joint $K$ -open ladder point is close enough to the origin in the bias direction.","The main result, that will be used in the rest of the paper is Proposition REF which follows from Lemma REF .","Lemma REF which is related to Proposition REF will also be instrumental for the rest of this work.","The proofs and the other results are purely technical and will not be used in the rest of the paper.","Let us first introduce some useful quantities and results from .","We introduce the inner positive boundary of $\\mathcal {B}_{x}(L, L^{\\prime })$ $\\partial _{\\mathrm {in}}^+ \\mathcal {B}_{x}(L, L^{\\prime }) \\lbrace z \\in \\mathcal {B}_{x}(L, L^{\\prime }), \\text{ where } z \\sim y \\text{ with } y \\in \\partial ^+ \\mathcal {B}_{x}(L, L^{\\prime }) \\rbrace .$ We consider two walks $(X^i_n)_{n\\ge 0}$ for $i\\in \\lbrace 1,2\\rbrace $ and define the events $ A^{i}(L) = \\lbrace T^i_{\\partial \\mathcal {B}_{X^{i}_{0}}(L, L^\\alpha )} \\ge T^i_{\\partial ^+_{\\mathrm {in}} \\mathcal {B}_{X^{i}_{0}}(L, L^\\alpha )}\\rbrace , \\quad \\mathrm {and} \\quad A(L) A^{1}(L) \\cap A^{2}(L).$ In all the following we write $\\mathcal {B}_{i} = \\mathcal {B}_{X_{0}^{i}}$ and $\\mathcal {B} = \\mathcal {B}_{0}$ .","Lemma 3.1 (Analog of ) For $\\alpha > d + 3$ and any $L > 0$ we have $\\sup _{U_1, U_2 \\in \\mathbb {Z}^d} \\mathbb {P}_{U_1, U_2}\\left( A(L)^c \\right) \\le C e^{-c L}.$ where the constants $C,c$ only depends on $\\lambda ,\\vec{\\ell },d$ .","Fix any $U_1, U_2 \\in \\mathbb {Z}^d$ .","We aim to bound $\\mathbb {P}_{U_1, U_2}\\left( A(L)^c \\right) &= \\mathbb {P}_{U_1, U_2}\\left( A^1(L)^c \\cup A^2(L)^c \\right) \\le \\mathbb {P}_{U_1}\\left( A^1(L)^c \\right) + \\mathbb {P}_{U_2}\\left( A^2(L)^c \\right),$ where the inequality is due to a union bound.","By we have that $\\mathbb {P}_{0}\\left[ A^{i}(L)^c \\right] \\le C \\exp (-c L)$ , thanks to translation invariance of the environment we get $\\mathbb {P}_{U_1, U_2}\\left( A(L)^c \\right) \\le C e^{-c L},$ as desired.","We say that a vertex $x \\in \\mathcal {B}(L, L^\\alpha )$ is $K$ -$L$ -closed if there exists a neighbour $y \\notin \\mathcal {H}^+(L)$ of $x$ such that $c^*([x, y]) \\notin [1/K, K]$ .","Let $\\bar{K}_x(L)$ be the $K$ -$L$ -closed connected component of $x$ (set it to $\\lbrace x\\rbrace $ in case it is empty) and let us define the event $ B(L) = \\left\\lbrace \\textnormal {for all } x \\in \\partial _{\\mathrm {in}}^+ \\mathcal {B}_1(L, L^\\alpha ) \\cup \\partial _{\\mathrm {in}}^+ \\mathcal {B}_2(L, L^\\alpha ), \\textnormal { } |\\bar{K}_x(L)| \\le \\log (L) \\right\\rbrace .$ Lemma 3.2 (Analog of ) For any $M<\\infty $ , we can find $K_0$ such that for all $K \\ge K_0$ $\\mathbf {P}\\left( B(L)^c \\right) \\le C L^{-M}.$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","Thanks to we have the bound for the event $\\widehat{B}(L) = \\lbrace \\textnormal {for all } x \\in \\partial _{\\mathrm {in}}^+ \\mathcal {B}(L, L^\\alpha ), \\textnormal { we have } |\\bar{K}_x(L)| \\le \\log (L) \\rbrace ,$ $\\mathbf {P}\\left( \\widehat{B}(L)^c \\right) \\le C L^{-M}.$ Just as in the proof of the previous Lemma, we get the result by union bound and translation invariance of the the measure $\\mathbf {P}$ .","We also recall the following useful lemma.","Lemma 3.3 Take $G \\ne \\varnothing $ to be a finite connected subset of $\\mathbb {Z}^d$ .","Assume each edge $e$ of $\\mathbb {Z}^d$ is assigned a positive conductance $c(e)$ and that there exists a constant $c_1>0$ , $x \\in \\partial G$ and $y \\in G$ such that $x \\sim y$ and $c([x, y]) \\ge c_1 c(e)$ for any $e \\in \\partial _{E} G$ .","We have $P_y\\left( T_x \\le T_{\\partial G} \\right) \\ge \\frac{c_1}{4d}|G|^{-1},$ where $P_y$ is the law of the random walk on the family of conductances $\\lbrace c(e)\\rbrace _{e \\in E(\\mathbb {Z}^d)}$ , starting from $y$ ."],["Properties of joint $K$ -open ladder points","For $i \\in \\lbrace 1,2\\rbrace $ we recall the notation $T^i_R$ defined in (REF ).","We introduce $\\mathcal {M}^{\\bullet } = \\mathcal {M}^{\\bullet (K)} = \\inf \\Big \\lbrace R > &\\max \\lbrace |X^{1}_0\\cdot \\vec{\\ell }|, |X^{2}_0\\cdot \\vec{\\ell }|\\rbrace \\text{ ; for } i\\in \\lbrace 1,2\\rbrace ,\\text{ } X^i_{T^i_{R}} \\text{ is } K\\text{-open, } \\nonumber \\\\ &X^i_{j} \\cdot \\vec{\\ell } < X^i_{T^i_{R}-2} \\cdot \\vec{\\ell } \\text{ for } j < T^i_{R}-2,\\\\& \\nonumber X^i_{T^i_{R}} = X^i_{T^i_{R}-1} + e_1,X^i_{T^i_{R}} = X^i_{T^i_{R}-2} + 2e_1 \\Big \\rbrace .$ In general, we will put a bullet when referring to quantities related to two walks evolving in the same environment to avoid confusion with similar quantities for a single walk or two walks in independent environments.","Since the definition of $\\mathcal {M}^{\\bullet }$ corresponds to a level, the joint $K$ -open ladder point is then $(X^1_{T^1_{\\mathcal {M}^{\\bullet }}},X^2_{T^2_{\\mathcal {M}^{\\bullet }}})$ .","However, as we will show in the rest of the article, it is much more convenient to work with levels rather than time or points when we consider two walks.","The main proposition of this subsection is the following.","Proposition 3.4 For any $M > 0$ there exists $K_0 > 0$ such that for $K\\ge K_0$ we have $\\sup _{U_1, U_2 \\in \\mathcal {U}}\\mathbb {P}_{U_1,U_2}(\\mathcal {M}^{\\bullet (K)} > n) \\le Cn^{-M},$ where we have $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","To prove this result we require a technical lemma, for $1 \\le k \\le n$ , let us define the events $R^{(K)}(n k) \\left\\lbrace \\mathcal {M}^{\\bullet (K)} > kn + 3 \\right\\rbrace .$ Lemma 3.5 For any $\\varepsilon ^{\\prime } > 0$ and $M > 0$ , we can find $K_0 = K_0(\\varepsilon ^{\\prime }, M)$ large enough such that for any $U_1,U_2 \\in \\mathbb {Z}^d$ with $|U_1\\cdot \\vec{\\ell }| \\le 1$ and $|U_2\\cdot \\vec{\\ell }| \\le 1$ , all $K > K_0$ , $1< k \\le n$ we have $\\displaystyle \\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n k)\\right)\\le \\left(1-cn^{-\\varepsilon ^{\\prime }}\\right)\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n (k-1))\\right) + Cn^{-M},$ where the constants $C,c$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We postpone the proof of Lemma REF after the proof of the main result.","[Proof of Proposition REF ] This Proposition is the analogous of .","In particular, one can iterate Lemma REF to get a tail estimate on the quantity $\\mathcal {M}^{\\bullet (K)}$ .","Indeed, $\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n k)\\right)\\le \\left(1-cn^{-1/2}\\right)\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n (k-1))\\right) + Cn^{-M},$ implies, by induction, that $\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n^2)\\right)\\le \\left(1-cn^{-1/2}\\right)^n + Cn^{-M + 1} \\le 2Cn^{-M + 1}.$ In the end we get $\\mathbb {P}_{U_1,U_2}\\left(\\mathcal {M}^{\\bullet (K)} > n^2 + 3\\right) \\le 2Cn^{-M + 1},$ or equivalently $\\mathbb {P}_{U_1,U_2}\\left(\\mathcal {M}^{\\bullet (K)} > n\\right) \\le Cn^{-(M + 1)/2},$ which is enough to conclude since $M$ is arbitrary.","Figure: Black points represent the KK-open points.","We look at the successive attempts to reach a joint KK-open ladder point when the walks exit boxes.","[Proof of Lemma REF ] The proof essentially goes the same way as .","Fix $U_1,U_2 \\in \\mathcal {U}$ as in the hypothesis of the Lemma.","For $i = 1, 2$ let us denote as $\\widetilde{\\mathcal {B}}^i_n$ the largest box of the form $\\mathcal {B}_i(m, m^\\alpha )$ with $m \\in \\mathbb {R}_+$ contained in $\\mathcal {H}^{-}(n)$ .","Note that $ \\mathcal {B}_i((n-1), (n-1)^\\alpha ) \\subset \\widetilde{\\mathcal {B}}^i_n \\subset \\mathcal {B}_i(n+1, (n+1)^\\alpha )$ .","Recall that $\\bar{K}_x(n)$ denotes the $K$ -$n$ -closed connected component of $x$ defined just before (REF ).","We introduce $\\mathcal {K}^i(n) \\bar{K}_{X_{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}}} (n) \\subseteq \\widetilde{\\mathcal {B}}^i_n, \\quad i = 1, 2.$ We then set $\\mathcal {K}(n) = \\mathcal {K}^1(n) \\cup \\mathcal {K}^2(n)$ .","In case $T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n} = + \\infty $ we set $\\mathcal {K}^i(n) = \\varnothing $ and $\\partial \\mathcal {K}^i(n) = \\varnothing $ .","We recall the definition of $A(L)$ , $B(L)$ in (REF ),(REF ).","We slightly modify them and define, with a little abuse of the notation, $A^{i}(n) = \\left\\lbrace T^i_{\\partial \\widetilde{\\mathcal {B}}_{X^{i}_{0}}(n, n^\\alpha )} \\ge T^i_{\\partial ^+_{\\mathrm {in}} \\widetilde{\\mathcal {B}}_{X^{i}_{0}}(n, n^\\alpha )}\\right\\rbrace , \\quad \\mathrm {and} \\quad A(n) A^{1}(n) \\cap A^{2}(n);$ $B(n) = \\left\\lbrace \\textnormal {for all } x \\in \\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}_{X^{1}_{0}}(n, n^\\alpha )\\cup \\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}_{X^{2}_{0}}(n, n^\\alpha ), \\textnormal { we have } |\\bar{K}_x(n)| \\le \\log (n) \\right\\rbrace .$ Notice that the estimates derived in Lemmas REF REF would still hold thanks to the fact that $ \\mathcal {B}_i((n-1), (n-1)^\\alpha ) \\subset \\widetilde{\\mathcal {B}}^i_n \\subset \\mathcal {B}_i(n+1, (n+1)^\\alpha )$ .","Moreover, we introduce the events: $&\\bullet \\hspace{3.00003pt}C(n) \\left\\lbrace \\text{for all } x \\in \\partial \\mathcal {K}(n)\\cap \\mathcal {H}^+(n), \\text{ the vertex } x \\text{ is } K\\textnormal {-open} \\right\\rbrace .","\\\\\\vspace{2.125pt}&\\bullet \\hspace{3.00003pt} D^{i}(n) \\left\\lbrace T^i_{\\partial \\mathcal {K}^{i}(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}} = T^i_{\\partial \\mathcal {K}^{i}(n) \\cap \\mathcal {H}^+(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}} \\right\\rbrace .","\\\\\\vspace{2.125pt}&\\bullet \\hspace{3.00003pt} E^{i}(n) \\left\\lbrace X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 2} = X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 1} + e_1 = X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n}} + 2 e_1 \\textnormal { and } X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 2}, X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 1} \\textnormal { are \\right.K-open}.$ Furthermore, define $D(n) = D^{1}(n) \\cap D^{2}(n)$ and $E(n) = E^{1}(n) \\cap E^{2}(n)$ .","Let us consider the event $A(n) \\cap C(n) \\cap D(n) \\cap E(n)$ then it is possible to notice the following: On $A(n)$ , defined in (REF ), for $i = 1, 2$ , we have $T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} \\ge T^i_{\\partial ^+_{\\mathrm {in}} \\widetilde{\\mathcal {B}}^i_n}$ so that $T^i_{\\partial ^+ \\widetilde{\\mathcal {B}}^i_n} \\ge T^i_{\\partial \\mathcal {K}(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}} \\widetilde{\\mathcal {B}}^i_n}}.$ On $D(n)$ , for $i = 1, 2$ , we have $T_{\\partial \\mathcal {K}^{i}(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}} = T_{\\partial \\mathcal {K}^{i}(n) \\cap \\mathcal {H}^+(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}}.$ On $C(n)$ , for $i = 1, 2$ , we have that all points in $\\partial \\mathcal {K}^i(n) \\cap \\mathcal {H}^+(n)$ are $K$ -open.","Putting these observations together one can see that, for $i \\in \\lbrace 1,2\\rbrace $ , $X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n}}$ is a new maximum in the direction $\\vec{\\ell }$ which is also $K$ -open and the next two steps are in the direction $e_1$ on $K$ -open points, so that $\\lbrace A(n) \\cap C(n) \\cap D(n) \\cap E(n)\\rbrace \\subset \\lbrace \\mathcal {M} \\le n + 3 \\rbrace .$ By following the steps of from Equation $(6.6)$ to $(6.10)$ we get the following $\\mathbb {P}\\left( R(kn) \\right) &\\le \\mathbb {P}\\left( A(kn)^c \\right) + \\mathbb {P}\\left( B(kn)^c \\right) + \\cdots \\\\&+ \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn)^c \\right) + \\cdots \\\\&+ \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) + \\cdots \\\\&+ \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) .$ The first two terms in (REF ) are controlled, respectively by Lemma REF and Lemma REF .","In order to finish the proof we need to control the terms in (), () and ().","Step 1: Control of the term ().","Recall the definition of $\\mathcal {K}(kn)$ at the start of this proof.","For $k \\le n$ , on the event $A(kn) \\cap B(kn)$ it holds that $|\\mathcal {K}(kn)| \\le 4 \\log (n),$ since for both $i = 1, 2$ it holds that $|\\mathcal {K}^i(kn)| \\le 2 \\log (n)$ .","We deduce that the term $\\mathbb {P}( R((k-1)n), A(kn), B(kn), C(kn)^c )$ can be rewritten as $&\\sum _{i =1, 2} \\sum _{\\begin{array}{c}F^i \\subset \\mathbb {Z}^d, \\\\ |F^i|\\le 2 \\log (n)\\end{array}} \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn)^c, \\mathcal {K}(kn) = F^1 \\cup F^2 \\right) \\\\& = \\sum _{\\begin{array}{c}i =1, 2, \\\\F^i \\subset \\mathbb {Z}^d, \\\\ |F^i|\\le 2 \\log (n)\\end{array}} \\mathbf {E} \\left[ P^\\omega \\left( R((k-1)n), A(kn), B(kn), \\mathcal {K}(kn) = F^1 \\cup F^2 \\right) {1}_{\\begin{array}{c}\\lbrace \\text{some }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\\\ \\text{ is closed} \\rbrace \\end{array}} \\right].$ We may notice that: $P^\\omega \\left( R((k-1)n), A(kn), B(kn), \\mathcal {K}(kn) = F^1 \\cup F^2 \\right)$ is a $\\mathbf {P}$ -random variable measurable with respect to the sigma algebra $\\sigma \\left(c([x, y]), \\,\\, x, y \\notin \\mathcal {H}^+(kn) \\right)$ .","The event $\\left\\lbrace \\text{some }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\text{ is closed} \\right\\rbrace $ is measurable with respect to the sigma algebra $\\sigma \\left(c([x, y]), \\,\\, x, y \\in \\mathcal {H}^+(kn)\\right)$ .","Hence we have $\\mathbf {P}$ -independence between these two random objects.","Hence, after some standard manipulation last sum can be rewritten as $\\sum _{i =1, 2} \\sum _{\\begin{array}{c}F^i \\subset \\mathbb {Z}^d,\\\\ |F^i|\\le 2 \\log (n)\\end{array}} &\\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), \\mathcal {K}(kn) = F^1 \\cup F^2 \\right) \\\\& \\times \\left( 1 - \\mathbf {P}\\left( \\text{all }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\text{ are open} \\right) \\right)$ Since $|F^{i}| \\le 2\\log (n)$ and that $|F^1 \\cup F^2| \\le |F^1| + |F^2|$ we have $\\mathbf {P}\\left( \\text{all }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\text{ are open} \\right) & \\ge \\left(1 - \\varepsilon (K)\\right)^{2d |F^1 \\cup F^2|}\\\\& \\ge \\left(1 - \\varepsilon (K)\\right)^{8d\\log (n)} \\\\& = n^{8d \\log (1 - \\varepsilon (K))},$ where $\\varepsilon (K) \\mathbf {P}\\left( c_* \\notin [1/K, K] \\right) $ .","For any $\\varepsilon ^{\\prime } > 0$ it is possible to fix $K$ large enough such that $8d \\log (1 - \\varepsilon (K)) \\ge - \\varepsilon ^{\\prime }$ .","Then we have $\\displaystyle \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn)^c \\right) \\le ( 1 - n^{-\\varepsilon ^{\\prime }}) \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn)\\right),$ which implies $n^{-\\varepsilon ^{\\prime }} \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn)\\right) \\le \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn) \\right).","$ Step 2: Control of the term ().","Notice that, on the event $A(n)$ we have that, for $i = 1, 2$ , $X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}}} \\in \\partial ^+_{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}$ and by definition of $\\mathcal {K}^{i}(kn)$ , $X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}}} \\in \\mathcal {K}^{i}(kn)$ .","Moreover on $B(kn)$ we also have that $|\\mathcal {K}^{i}(kn)| \\le 2 \\log (n)$ .","Hence $&\\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) \\\\&\\le \\sum _{i =1, 2}\\sum _{z^i} \\sum _{F^i} \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), \\bigcap _{i = 1, 2} \\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\rbrace , C(kn), D(kn)^c \\right),$ where $\\displaystyle \\sum _{z^i} \\sum _{F^i}$ is a shortcut for $\\displaystyle \\sum _{z^i \\in \\partial ^+_{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}} \\sum _{\\begin{array}{c}F^i \\subset \\mathbb {Z}^d \\\\ |F^i| \\le 2 \\log (n)\\\\ z^i \\in F^i\\end{array}}$ .","For $\\omega $ fixed, the events $\\displaystyle R((k-1) n ), A(kn), B(kn), \\bigcap _{i = 1, 2} \\left\\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\right\\rbrace $ are $P^\\omega $ -measurable with respect to $\\sigma \\Big (\\lbrace X^1_j, j \\le T^1_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^1_{kn}}\\rbrace , \\lbrace X^2_k, k \\le T^2_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^2_{kn}} \\rbrace \\Big )$ .","Thus, thanks to Lemma REF and the quenched independence of the two walks, we can write $\\mathbb {P}&\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) \\nonumber \\\\&\\le \\sum _{i =1, 2} \\sum _{z^i} \\sum _{F^i} \\mathbf {E}\\bigg [ P^\\omega \\Big ( R((k-1)n), A(kn), B(kn), \\bigcap _{i = 1, 2} \\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\rbrace \\Big ) \\nonumber \\\\& \\times P_{z^1, z^2}^\\omega \\left(T^1_{\\partial F^1} < T^1_{\\partial F^1 \\cap \\mathcal {H}^+(kn)} \\textnormal { or } T^2_{\\partial F^2} < T^2_{\\partial F^2 \\cap \\mathcal {H}^+(kn)}\\right) {1}_{\\lbrace x \\in \\partial F \\cap \\mathcal {H}^+(kn) \\text{ is open} \\rbrace } \\bigg ].","$ For both $F^1$ and $F^2$ we can notice thanks to the definition of $\\mathcal {K}(kn)$ and the fact that we are considering the event $\\lbrace x \\in \\partial F \\cap \\mathcal {H}^+(kn) \\text{ is } K\\textnormal {-open} \\rbrace $ we get that all the edges $e \\in \\partial _E F$ are $K$ -normal.","Just as a remark also notice that $F^1$ and $F^2$ are either the same set or disjoint sets, so overall we get that $\\partial _E F^1$ and $\\partial _E F^2$ are two (possibly one) sets of normal edges.","We obtain a bound for $P_{z^1}^\\omega (T^1_{\\partial F^1} \\ge T^1_{\\partial F^1 \\cap \\mathcal {H}^+(kn)})$ , the case $P_{z^2}^\\omega (T^2_{\\partial F^2} \\ge T^2_{\\partial F^2 \\cap \\mathcal {H}^+(kn)})$ can be treated by following the same reasoning.","For any $z^1 \\in F^1 \\cap \\partial ^+_{\\text{in}} \\widetilde{\\mathcal {B}}^1_{kn}$ , there exists a neighbour $y^1 \\in \\mathcal {H}^+(kn)$ .","Thanks to the fact that $F^1 \\subset \\widetilde{\\mathcal {B}}_{kn}$ we have that, for any $x \\in \\partial F^1 \\cup F^1$ , $(y^1 - x) \\cdot \\vec{\\ell } \\ge - 1$ and $(z^1 - x) \\cdot \\vec{\\ell } \\ge - 2$ .","Then, thanks to this observation and the definition of $K$ -normal edges, we see that $c^\\omega (e) \\le K^2 e^{3 \\lambda } c^{\\omega }([z^1, y^1]), \\quad \\textnormal {for } e \\in \\partial _E F^1.$ Thus we can apply Lemma REF with $G= F^1$ (resp.","$F^2$ ) and obtain, thanks to quenched independence $P_{z^1, z^2}^\\omega &\\left(T^1_{\\partial F^1} < T^1_{\\partial F^1 \\cap \\mathcal {H}^+(kn)} \\textnormal { or } T^2_{\\partial F^2} < T^2_{\\partial F^2 \\cap \\mathcal {H}^+(kn)}\\right) \\\\ &\\le P_{z^1, z^2}^\\omega \\left(T^1_{\\partial F^1} < T^1_{y^1} \\textnormal { or } T^2_{\\partial F^2} < T^2_{y^2}\\right) \\\\& = \\left( 1 - P_{z^1}^\\omega \\left( T^1_{y^1} \\le T^1_{\\partial F^1}\\right) P_{z^2}^\\omega \\left( T^2_{y^2} \\le T^2_{\\partial F^2}\\right) \\right) \\\\&\\le (1 - c\\log (n)^{-2}),$ using $|F^1| \\le 2 \\log (n)$ .","By plugging it in Equation (REF ) we get $\\mathbb {P}&\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) \\\\&\\le \\sum _{i =1, 2} \\sum _{z^i} \\sum _{F^i} \\mathbf {E}\\bigg [ P^\\omega \\bigg ( R((k-1)n), A(kn), B(kn), \\bigcap _{i = 1, 2} \\Big \\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\Big \\rbrace \\bigg ) \\\\& \\times {1}_{\\lbrace x \\in \\partial F \\cap \\mathcal {H}^+(kn) \\text{ is open} \\rbrace } \\bigg ] (1 - c\\log (n)^{-2})\\\\&\\le (1 - c\\log (n)^{-2}) \\mathbb {P}\\left( R((k - 1)n ), A(n), B(n), C(n) \\right).$ Which readily implies that for some positive constant $c>0$ we get $c\\log (n)^{-2} & \\mathbb {P}\\left( R((k - 1)n ), A(n), B(n), C(n) \\right) \\nonumber \\\\&\\le \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn) \\right).","$ Step 3: Control of the term ().","On the event $A(n)\\cap B(n) \\cap C(n) \\cap D(n)$ we know that both $X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} \\in \\partial ^+ \\widetilde{\\mathcal {B}}^1_{kn}$ and $X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} \\in \\partial ^+ \\widetilde{\\mathcal {B}}^2_{kn}$ are $K$ -open new maxima in the direction $\\vec{\\ell }$ .","Moreover, the event $E(kn)$ does not depend on what happens in the “double box” $\\widetilde{\\mathcal {B}}^1_{kn} \\cup \\widetilde{\\mathcal {B}}^2_{kn}$ .","Let us also introduce the concept of $x, y$ -open vertex, we say that $z$ is $x, y$ -open if all the edges in the set $\\mathcal {E}_{z} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{y})$ are $K$ -normal.","Let us also define $R^{\\prime }((k - 1)n) = R((k - 1)n) \\cap A(n)\\cap B(n) \\cap C(n) \\cap D(n).$ Recall (see (REF ) and considerations above) that on this event we have that both walks, when they hit level $kn$ , are at new $K$ -open maxima.","We can write $\\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) \\\\&\\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbb {P}\\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2, \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}, E(kn)^c \\right)$ $&\\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbf {E}\\bigg [ P^\\omega \\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\right) {1}_{\\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace } \\\\&\\big ( {1}_{\\lbrace \\textnormal {A vertex }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed}\\rbrace } + {1}_{\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace } \\\\&\\times P^\\omega _{x^1, x^2}\\left( X^1_1 \\ne x^1 + e_1 \\text{ or } X^1_2 \\ne x^1 + 2e_1 \\text{ or } X^2_1 \\ne x^2 + e_1 \\text{ or } X^2_2 \\ne x^2 + 2e_1 \\right) \\big ) \\bigg ].$ On the event $\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace \\cap \\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace $ one has that $P^\\omega _{x^i}\\left( X^i_1 = x^i + e_1, X^i_2 = x^i + 2e_1 \\right) \\ge c > 0$ for both $i = 1, 2$ .","Then $\\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) \\\\&\\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbf {E}\\bigg [ P^\\omega \\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\right) {1}_{\\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace } \\\\& \\big ( {1}_{\\lbrace \\textnormal {A vertex in }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed}\\rbrace } \\\\ &+ (1-c^2){1}_{\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace } \\big ) \\bigg ].$ The $\\mathbf {P}$ -random variables $P^\\omega \\Big ( R^{\\prime }((k - 1)n), X^1_{T_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\Big )$ and ${1}_{\\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace }$ are measurable with respect to $\\sigma ( e \\in E(\\widetilde{\\mathcal {B}}^1_{kn} \\cup \\widetilde{\\mathcal {B}}^2_{kn})\\cup \\mathcal {E}_{x^1} \\cup \\mathcal {E}_{x^2})$ , we recall that $\\mathcal {E}_x, x \\in \\mathbb {Z}^d$ is defined in (REF ).","On the other hand the events $\\lbrace \\textnormal {A vertex in }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed}\\rbrace $ and $\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace $ are measurable with respect to $\\sigma ( e \\notin E(\\widetilde{\\mathcal {B}}^1_{kn} \\cup \\widetilde{\\mathcal {B}}^2_{kn})\\cup \\mathcal {E}_{x^1} \\cup \\mathcal {E}_{x^2} )$ , thanks to the fact that we are summing over $x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}$ .","This implies $\\mathbf {P}$ -independence between these two groups of random variables.","This yields $\\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) \\\\& \\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbf {E}\\left[ P^\\omega \\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\right) \\right] \\times \\\\& \\times \\bigg ( \\mathbf {P}\\left( \\textnormal {A vertex in }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed} \\right) + \\\\&+ (1 - c^{\\prime }) \\mathbf {P}\\left( \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open} \\right) \\bigg ) \\\\& \\le (1 - c^{\\prime }) \\mathbb {P}\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn) \\right),$ since $\\mathbf {P}\\left( \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open} \\right) > 0$ and its value does not depend on $x^1,x^2$ .","This implies that there exists $c>0$ such that $c \\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn) \\right) \\nonumber \\\\ &\\le \\mathbb {P}\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)\\right).","$ Step 4: Conclusion.","For any $\\varepsilon ^{\\prime } > 0$ , $M > 0$ , one can find a $K$ chosen large enough such that, putting together (REF ) with Lemma REF and Lemma REF and the estimates attained in (REF ), (REF ) and (REF ), implies that, for all $k \\in \\lbrace 2, \\dots , n \\rbrace $ $\\mathbb {P}\\left( R(kn) \\right) \\le \\mathbb {P}\\left( R((k-1)n) \\right)\\left( 1 - c\\log (n)^{-2} n^{-\\varepsilon ^{\\prime }} \\right) + n^{-M},$ which implies the result.","For the construction of joint regeneration times we will need for the joint $K$ -open ladder points to be close.","We introduce for $R > 0$ the variable $\\mathcal {M}^{\\bullet (K)}_R = \\mathcal {M^\\bullet }_R \\mathcal {M}^{\\bullet (K)} \\circ \\theta _{T_R} - R$ where we recall that $\\mathcal {M}^{\\bullet (K)}$ is defined in (REF ) and $\\theta _{T_R}$ means that the first walk is shifted by $T^{1}_R$ and the second one by $T^{2}_R$ .","We introduce $ \\Upsilon (n) = \\bigcap _{R = 0}^{n}\\left\\lbrace \\mathcal {M^\\bullet }_R\\le n^{\\frac{1}{2}}\\right\\rbrace ,$ under this event before reaching $\\mathcal {H}^{+}(n)$ the successive joint $K$ -open ladder points are always at distance less than $n^{\\frac{1}{2}}$ in the direction $\\vec{\\ell }$ .","Proposition 3.6 For any $M > 0$ there exists $K > 0$ $\\sup _{U_1, U_2 \\in \\mathcal {U}}\\mathbb {P}_{U_1, U_2}(\\Upsilon (n)^{c}) \\le Cn^{-M}.$ where we recall $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and $C$ only depends on $d,\\lambda $ .","The proof is similar to the proof of .","Let us consider the events, for $j \\in \\lbrace 0, \\dots , n\\rbrace $ $\\Upsilon _j(n) = \\bigcap _{R = 0}^{j - 1}\\left\\lbrace \\mathcal {M^\\bullet }_R\\le n^{\\frac{1}{2}}\\right\\rbrace \\quad \\textnormal {and} \\quad \\Theta _j(n) = \\left\\lbrace \\mathcal {M^\\bullet }_j > n^{\\frac{1}{2}}\\right\\rbrace .$ We can split the event over the minimal $j \\in \\lbrace 0, \\dots , n\\rbrace $ such that $\\Theta _j(n)$ is true and get $ \\mathbb {P}_{U_1, U_2}(\\Upsilon (n)^{c}) \\le \\sum _{j = 0}^n \\mathbb {P}_{U_1, U_2} \\left( \\Upsilon _j(n), \\Theta _j(n), T^1_{\\partial \\widetilde{\\mathcal {B}}^1_n} = T^1_{\\partial ^+ \\widetilde{\\mathcal {B}}^1_n} , T^2_{\\partial \\widetilde{\\mathcal {B}}^2_n} = T^2_{\\partial ^+ \\widetilde{\\mathcal {B}}^2_n} \\right) + C e^{-c n}.$ On the event $\\Upsilon _j(n) \\cap \\Theta _j(n) \\cap \\lbrace T^1_{\\partial \\widetilde{\\mathcal {B}}^1_n} = T^1_{\\partial ^+ \\widetilde{\\mathcal {B}}^1_n} \\rbrace \\cap \\lbrace T^2_{\\partial \\widetilde{\\mathcal {B}}^2_n} = T^2_{\\partial ^+ \\widetilde{\\mathcal {B}}^2_n} \\rbrace $ one is sure that $X^i_{T_{j}} \\in \\widetilde{\\mathcal {B}}^i_n$ for $i = 1, 2$ then by union bound argument $ \\nonumber \\mathbb {P}_{U_1, U_2} &\\left( \\Upsilon _j(n), \\Theta _j(n), T^1_{\\partial \\widetilde{\\mathcal {B}}^1_n} = T^1_{\\partial ^+ \\widetilde{\\mathcal {B}}^1_n}, T^2_{\\partial \\widetilde{\\mathcal {B}}^2_n} = T^2_{\\partial ^+ \\widetilde{\\mathcal {B}}^2_n} \\right) \\\\&\\nonumber \\le \\sum _{x_1, x_2 \\in \\widetilde{\\mathcal {B}}^1_n, \\widetilde{\\mathcal {B}}^2_n} \\mathbb {P}_{U_1, U_2} \\left( X^1_{T_j} = x_1, X^2_{T_j} = x_2, \\mathcal {M}_{j}^{\\bullet (K)} > n^{1/2} \\right) \\\\&\\le |\\mathcal {B}_{n+1}|^2 \\sup _{x_1, x_2}\\mathbb {P}_{x_1,x_2}\\left(\\mathcal {M}^{\\bullet (K)} - j \\ge n^{1/2}\\right),$ where we used Lemma REF in the last inequality and the $\\sup $ is over all $x_1,x_2 \\in \\widetilde{\\mathcal {B}}^1_n, \\widetilde{\\mathcal {B}}^2_n$ such that $x_1,x_2$ are compatible with $\\lbrace X^1_{T_j} = x_1, X^2_{T_j} = x_2\\rbrace $ .","Now we conclude using the fact that $\\mathbb {P}_{x_1,x_2}\\left(\\mathcal {M}^{\\bullet (K)} - j \\ge n^{1/2}\\right) = \\mathbb {P}_{0,x_2-x_1}\\left(\\mathcal {M}^{\\bullet (K)} - j+ x_1\\cdot \\vec{\\ell } \\ge n^{1/2}\\right).$ Where we used translation invariance of the environment, now using the fact that $|x_1\\cdot \\vec{\\ell } - j| \\le 1$ and $|(x_2-x_1)\\cdot \\vec{\\ell }| \\le 1$ , the assumptions of Proposition REF are verified.","We conclude that $\\sup _{x_1, x_2}\\mathbb {P}_{x_1,x_2}\\left(\\mathcal {M}^{\\bullet (K)} - j \\ge n^{1/2}\\right) \\le Cn^{-M},$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","We then conclude the domination of (REF ) using the fact that $|\\mathcal {B}_{n+1}| \\le Cn^{2d\\alpha }$ ."],["Enhanced walks","The traditional definition of regeneration times has the drawback that it doesn't guarantee independence since the knowledge of the conductances at $x$ influences the walk at any time it reaches a neighbour of $x$ (even when that neighbour $y$ is such that $y\\cdot \\vec{\\ell }> x \\cdot \\vec{\\ell }$ ).","In order to obtain stronger independence properties, which we need for our purpose, we need to introduce a variant in the construction of regeneration times.","Such a task was already done in using a so-called “enhanced-walk”.","We recall the definition of the enhanced walk which is the same as in .","For any environment $\\omega $ we define two transition probabilities for $x \\sim y$ $p^{\\omega }(x,y) = \\frac{c_{*}(x,y)e^{(x+y)\\cdot \\ell }}{\\displaystyle \\sum _{z \\sim x}^{} c_{*}(x,z)e^{(x+z)\\cdot \\ell }},$ and $p_K^{\\omega }(x,y) = \\frac{(c_{*}(x,y)\\wedge K^{-1})e^{(x+y)\\cdot \\ell }}{\\displaystyle \\sum _{ z \\sim x}^{} (c_{*}(x,z)\\vee K)e^{(x+z)\\cdot \\ell }},$ and 0 in both cases if $x$ and $y$ are not adjacent.","Note that, for any $x,y \\in \\mathbb {Z}^d$ , we have that $p^{\\omega }_{K}(x,y) \\le p^{\\omega }(x,y)$ .","For any $(x,z) \\in \\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ we define the Markov chain $(\\tilde{X}_{n})_{n \\ge 0} = (X_{n}, Z_{n})_{n \\ge 0}$ with law $P^\\omega _{(x,z)}$ with transitions probabilities $p^{\\omega }((x_1,z_1),(x_2,z_2))$ defined by $&1.","\\,\\, \\tilde{X}_{0} = (x,z), P^{\\omega }_{(x,z)}\\text{-a.s},\\\\&2.\\,\\, p^{\\omega }((x,z),(y,1)) = p_K^{\\omega }(x,y),\\\\&3.\\,\\, p^{\\omega }((x,z),(y,0)) = p^{\\omega }(x,y) - p_K^{\\omega }(x,y).$ This definition ensures that the first coordinate has the same law that our classic walk in $\\omega $ .","When we are talking of the annealed law for the enhanced walk we will also write $\\mathbb {P}_{(x,z)}$ .","We observe that the evolution of the first coordinate does not depend on the second coordinate, thus when we are talking about only the first coordinate, we will simply write $X$ instead of $\\tilde{X}$ and $P_{x}^{\\omega }$ instead of $P^{\\omega }_{(x,z)}$ ."],["Regeneration times of a single walk","In this subsection we recall the rigorous definition of regeneration times introduced .","This definition will be used at the end of the paper and we highlight the fact that the definition of joint regeneration levels in Section REF is very similar to the definition of regeneration times.","First we introduce $\\mathcal {M}^{(K)}= \\inf \\Big \\lbrace &i \\ge 2 \\text{ ; }X_i \\text{ is }K\\text{-open, }X_j\\cdot \\vec{\\ell } < X_{i-2} \\cdot \\vec{\\ell } \\text{ for any } j < i-2 \\\\& \\text{ and } X_i = X_{i-1} + e_1 = X_{i-2} + 2e_1 \\Big \\rbrace .$ We also introduce $& \\mathbf {BACK} = \\inf \\left\\lbrace n \\ge 1 \\text{ ; } X_n \\cdot \\vec{\\ell } \\le X_0 \\cdot \\vec{\\ell }\\right\\rbrace \\\\& \\mathbf {ORI} = \\inf \\left\\lbrace n \\ge 1 \\text{ ; }X_{n-1} \\in \\mathcal {V}_{X_0} \\text{ and } Z_n = 0\\right\\rbrace ,$ and finally $D = \\mathbf {BACK} \\wedge \\mathbf {ORI}.$ The event $\\lbrace \\mathbf {BACK} > n\\rbrace $ means that the walk $X$ does not backtrack below $X_0$ before time $n$ .","The event $\\lbrace \\mathbf {ORI} > n\\rbrace $ is essential to get independence between the regeneration times.","In particular, a coupling argument shows that, as long as $\\lbrace \\mathbf {ORI} > n\\rbrace $ occurs, the conductances around the starting point do not affect the future of the walk if the point is $K$ -open.","That is why we will be interested in the event $\\lbrace D = +\\infty \\rbrace $ to regenerate.","Now we introduce the sequence of dependent stopping times $(S_n)_{n \\ge 0}$ and the levels $(M_n)_{n \\ge 0}$ $&\\hspace{40.0pt}S_0 = 0\\text{, }M_0 = X_{0} \\cdot \\vec{\\ell } \\text{ and }\\\\& \\text{ for } n \\ge 0\\text{, } S_{n+1} = \\mathcal {M}^{(K)} \\circ \\theta _{T_{\\mathcal {H}^{+}(M_n)}} + T_{\\mathcal {H}^{+}(M_n)},$ where $M_n = \\sup \\lbrace X_k \\cdot \\vec{\\ell } \\text{ with } 0 \\le k \\le R_n \\rbrace ,$ with $R_n = S_n + D \\circ \\theta _{S_n}.$ Then, we define the first regeneration time as $\\tau _1^{(K)} = S_N \\text{ with } N = \\inf \\lbrace n \\ge 1 \\text{ ; }S_n < +\\infty \\text{ and } M_n = +\\infty \\rbrace .$ Remark 3.7 Most of the time we will forego the superscript $K$ in $\\tau _1^{(K)}$ since it will be fixed large enough.","Using we obtain that $\\tau _1$ is $\\mathbb {P}_0$ -almost surely finite and we can define inductively finite regeneration times by $\\tau _{k+1} = \\tau _1 + \\tau _{k}((X_{\\tau _1+\\cdot } - X_{\\tau _1}),\\omega ( \\cdot +X_{\\tau _1})).$ The key result states that the $\\tau _k$ are $\\mathbb {P}_0$ -almost surely finite and that $(X_{\\tau _{k}+\\cdot } - X_{\\tau _k})$ is independent of $(X_i)_{0\\le i \\le \\tau _k}$ and distributed as a random walk under $\\mathbb {P}_0^K( \\, \\cdot \\text{ | }D=+\\infty )$ .","In the next section we will generalize these notions to two walks.","As mentioned before, regeneration times will turn into regeneration levels and the independence property will become a Markov-type property.","Remark 3.8 The reader should be careful in the following since we are going to define joint regeneration levels and we will use the notation $\\mathcal {D}^{^{\\bullet }}$ which is not the same as $D$ and $D^{\\otimes }$ , which are the natural counterparts for a single walk or for two independent walks under the annealed measure."],["Enhanced walks and joint regeneration","We consider two enhanced random walks $\\tilde{X}^1$ and $\\tilde{X}^2$ independent under the quenched law.","We recall the notation set in Section REF .","For $i \\in \\lbrace 1,2\\rbrace $ we introduce the random variables: $&\\mathbf {BACK}^i = \\inf \\left\\lbrace n > 0 \\text{ ; } X^i_n \\cdot \\vec{\\ell } \\le X^i_0 \\cdot \\vec{\\ell } \\right\\rbrace , \\\\& \\mathbf {ORI}^i = \\inf \\left\\lbrace n > 0 \\text{ ; } X^i_{n-1} \\in \\mathcal {V}_{X^1_0} \\cup \\mathcal {V}_{X^2_0} \\text{ and } Z^i_n = 0 \\right\\rbrace .","$ Then we introduce $\\mathcal {D}^{\\bullet i} = \\mathbf {BACK}^i \\wedge \\mathbf {ORI}^i.$ Finally we set $\\mathcal {D}^{\\bullet } = \\mathcal {D}^{\\bullet 1} \\wedge \\mathcal {D}^{\\bullet 2}.$ Remark 3.9 The notation $\\mathbf {BACK}^i$ refers to the first time the walk $X^i$ backtracks.","The notation $\\mathbf {ORI}^i$ encapsulates information around the starting points in such a way that Proposition REF holds.","In the following we will be interested in the case $\\mathcal {D}^{\\bullet } = +\\infty $ and $X_0^1,X_0^2$ are $K$ -open.","In that case the walks never backtrack and they behave as if for all $e \\in \\mathcal {E}_{X_0^{1}} \\cup \\mathcal {E}_{X_0^{2}}$ , $c_{*}(e) = K$ .","We notice that $\\mathcal {D}^{\\bullet 1}$ is not measurable with respect to $\\sigma (X_0^{1},(\\tilde{X}^{1}_{n})_{n \\ge 1})$ but with respect to $\\sigma (X_0^{1},X_0^{2},(\\tilde{X}^{1}_{n})_{n \\ge 1})$ .","Remark 3.10 We can notice that under $P^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}$ , $\\mathcal {D}^{\\bullet 1}$ and $\\mathcal {D}^{\\bullet 2}$ are independent.","Indeed, the variables $X^1_0$ and $X^2_0$ are almost surely constant under $P^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}$ and then we use the quenched independence of the two walks.","The following proposition shows that under $\\mathcal {D}^{\\bullet } = +\\infty $ the walks behave as if the environment around the starting points is deterministic.","For $\\omega $ an environment and $(x_1,z_1),(x_2,z_2) \\in \\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ we define $P^{\\omega _K }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}$ the quenched probability such that the conductances $c_{*}(e)$ with $e \\in \\mathcal {E}_{x_1}\\cup \\mathcal {E}_{x_2}$ are set to $K$ .","Proposition 3.11 Fix $(x_1,z_1),(x_2,z_2) \\in \\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ .","If $x_1$ and $x_2$ are both $K$ open points then for any bounded $ \\sigma \\big (\\tilde{X}^{1}_{n},\\tilde{X}^{2}_{n}\\big )$ -measurable function $f(\\cdot )$ , we have the equality: $E^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right] = E^{\\omega _K }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right].$ To lighten the notation in the proof we will write $P^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}} = P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}.$ The proof goes the same way as the proof of that uses a coupling argument.","The idea of the proof is to construct a couple $(U^1,U^2)$ with the same law as $(\\tilde{X}^1,\\tilde{X}^2)$ under $P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}$ and a couple $(V^1,V^2)$ with the same law as $(\\tilde{X}^1,\\tilde{X}^2)$ under $P^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ .","In addition we will construct the random variables in such a way that $\\displaystyle (U_i^1,U_i^2)_{0\\le i \\le n} = (V_i^1,V_i^2)_{0 \\le i \\le n}$ as soon as $\\mathcal {D}^{\\bullet }(U) > n$ where $\\mathcal {D}^{\\bullet }(U)$ denotes the quantity $\\mathcal {D}$ defined in (REF ) associated to $(U^1,U^2)$ .","We give the construction of $(U^1_{\\cdot },V^{1}_{\\cdot })$ under a probability measure $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ .","We can then build $(U^2_{\\cdot },V^{2}_{\\cdot })$ independently in the same way under $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ .","To simplify the notations and highlight the fact that $U^1$ has the same law as $\\tilde{X}^1 = (X^1,Z^1)$ we will write $U^1 = U = (X_n,Z_n)_{n \\ge 0}$ and $V^1 = V = (Y_n,W_n)_{n\\ge 0}$ .","We also call $\\mathcal {D}^{\\bullet }_U = \\mathcal {D}^{\\bullet 1}(U)$ and $\\mathcal {D}^{\\bullet }_V = \\mathcal {D}^{\\bullet 1}(V)$ .","We set $((X_0,Z_0),(Y_0,W_0)) = ((x_1,z_1),(x_1,z_1))$ $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.","Given the trajectory up to time $n \\ge 0$ the conditional law of $(U_{n+1},V_{n+1})$ is given by the following rules: $1.$ If $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace $ and if $U_n = V_n = x \\in \\mathbb {Z}^d$ .","Then $U$ makes a step according to $P^{\\omega }$ and (a) If $x \\notin \\lbrace x_1,x_2\\rbrace $ and $x \\notin \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ then $V_{n+1}= U_{n+1}$ .","(b) If $x \\in \\lbrace x_1,x_2\\rbrace $ or $x \\in \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ and if $Z_{n+1} = 1$ then $V_{n+1}= U_{n+1}$ .","(c) If $x \\in \\lbrace x_1,x_2\\rbrace $ or $x \\in \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ and if $Z_{n+1} = 0$ then for any $j \\in \\lbrace 1,...,2d\\rbrace $ , $(Y_{n+1},W_{n+1}) = (x + e_j,0)$ with probability $\\frac{p^{\\omega _K}(x,x+e_j)-p_K^{\\omega _K}(x,x+e_j)}{1- \\displaystyle \\sum _{1 \\le j \\le 2d}^{} p_{K}^{\\omega _K}(x,x+e_j)}.$ $2.$ if $\\lbrace \\mathcal {D}^{\\bullet }_U \\le n\\rbrace \\cup \\lbrace \\mathcal {D}^{\\bullet }_V \\le n\\rbrace $ then $U$ and $V$ move independentely according to $P^{\\omega }$ and $P^{\\omega _K}$ respectively.","We have to show that $(U,V)$ is well defined following this construction.","By construction, if $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace \\cap \\lbrace U_n = V_n\\rbrace $ then $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely either $\\lbrace \\mathcal {D}^{\\bullet }_U > n+1\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n+1\\rbrace \\cap \\lbrace U_{n+1} = V_{n+1}\\rbrace $ or $\\lbrace \\mathcal {D}^{\\bullet }_U = \\mathcal {D}^{\\bullet }_V = n+1\\rbrace $ .","By induction it follows that: The event $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace \\cap \\lbrace U_n \\ne V_n\\rbrace $ never occurs $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.","Thus $U$ and $V$ are well defined.","Under $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace $ we have $X_k = Y_k$ for all $k \\in \\lbrace 0,...,n\\rbrace $ $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.","We have $\\mathcal {D}^{\\bullet }_U = \\mathcal {D}^{\\bullet }_V$ $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.","Thus we have $&P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left(\\lbrace \\mathcal {D}^{\\bullet }_U = +\\infty \\rbrace \\cap \\left\\lbrace \\lbrace \\exists n \\ge 0; U_n \\ne V_n\\rbrace \\cup \\lbrace \\mathcal {D}^{\\bullet }_V < +\\infty \\rbrace \\right\\rbrace \\right) \\\\&\\le P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left(\\mathcal {D}^{\\bullet }_U \\ne \\mathcal {D}^{\\bullet }_V\\right) + \\sum _{n = 0}^{+\\infty } P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left(\\mathcal {D}^{\\bullet }_U > n+1, \\mathcal {D}^{\\bullet }_V >n+1, X_{n+1} \\ne Y_{n+1}\\right) = 0.","\\nonumber $ We notice that $p_{K}^{\\omega }(x,\\cdot ) = p_{K}^{\\omega _K}(x,\\cdot )$ for any $x \\in \\mathbb {Z}^d$ and $p^{\\omega }(x,\\cdot ) = p^{\\omega _K}(x,\\cdot )$ as soon as $x \\notin \\lbrace x_1,x_2\\rbrace \\cup \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ , it is easy to check that the law of $U$ (resp.","$V$ ) under $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ is the law of $\\tilde{X}^1$ under $P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}$ (resp.","the law of $\\tilde{X}^1$ under $P^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ ).","Then take $A_0,\\cdots ,A_n$ subsets of $\\mathbb {Z}^d\\times \\lbrace 0,1\\rbrace $ where $A_i$ are in the $\\sigma $ -algebra generated by the subsets of $\\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ and call $A = A_0 \\times \\cdots \\times A_n$ .","Thus we have: $&E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (\\tilde{X}^1_{0},\\cdots ,\\tilde{X}^1_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet 1} = +\\infty } \\right] \\\\&= E^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (U_{0},...,U_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet }_U = +\\infty } \\right]\\\\&= E^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (V_{0},...,V_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet }_V = +\\infty } \\right] \\\\& = E^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (\\tilde{X}^1_{0},...,\\tilde{X}^1_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet 1} = +\\infty } \\right],$ where the second equality follows from (REF ).","By a Monotone Class Theorem, for any bounded $\\sigma \\big (\\tilde{X}^{1}_{n}\\big )$ -measurable function we have $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right] = E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right].$ Then, for bounded measurable functions $f$ and $g$ we have: $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right]&= E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }{1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right] \\\\& = E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right] E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right]\\\\&= E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right] E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right] \\\\&=E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }{1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right] \\\\&=E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right].$ To extend the previous result to bounded measurable functions of the form $f(\\cdot ,\\cdot )$ we use once again the Monotone Class Lemma."],["$K$ -good points","We recall the definition of a $K$ -good point taken from .","A point $z \\in \\mathbb {Z}^d$ is said to be $K$ -good if there exists an infinite path $(x_0 = z, x_1,x_2,\\cdots )$ such that all $x_i$ are $K$ -open and $x_{2i+1}-x_{2i} = e_1$ and $x_{2i+2}-x_{2i+1} = \\lbrace e_1,e_2,\\cdots ,e_d\\rbrace $ .","Proposition 3.12 There exists a constant $K_0>0$ and $\\eta > 0$ such that for $K > K_0$ we have for any $U_1,U_2 \\in \\mathbb {Z}^d$ we have $\\mathbf {P}(U_1 \\text{ and }U_2\\text{ are }K\\text{-good points}) > \\eta > 0$ .","For $e_1,\\cdots ,e_n \\in E(\\mathbb {Z}^d)$ distinct edges, the events $(c^{*}_{e_i} \\in [1/K,K])_{1\\le i \\le n}$ are independent and have the same probability.","Thus $\\displaystyle ({1}_{c^{*}_{e} \\in [1/K,K]})_{e\\in E(\\mathbb {Z}^d)}$ is a percolation on $\\mathbb {Z}^d$ .","Since for $x \\in \\mathbb {Z}^d$ the event $\\lbrace x \\text{ is }K\\text{-good}\\rbrace $ is an increasing event we can apply FKG inequality and obtain $\\mathbf {P}(U_1 \\text{ and }U_2\\text{ are }K\\text{-good points}) \\ge \\mathbf {P}(U_1 \\text{ is a }K\\text{-good point})\\mathbf {P}(U_2 \\text{ is a }K\\text{-good point}),$ then we conclude using ."],["Construction of joint regeneration levels and main result","We recall the notation $\\mathcal {M}^{\\bullet }$ defined in (REF ).","We set $L_0 = 0, \\quad M_{0} = X^1_0 \\cdot \\vec{\\ell } \\vee X^{2}_{0} \\cdot \\vec{\\ell },$ and, for $k\\ge 0$ , $ L_{k+1} = \\mathcal {M}^{\\bullet }\\circ \\theta _{T_{M_k}}.$ Where $ M_k = \\sup \\bigg \\lbrace \\lbrace X^1_m\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le m \\le R_k\\rbrace \\cup \\lbrace X^2_m\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le m \\le R_k\\rbrace \\bigg \\rbrace ,$ with $R_k = \\mathcal {D}^{\\bullet } \\circ \\theta _{T_{L_k}}.$ Here we set $N = \\inf \\lbrace k \\ge 0 \\text{ ; } L_k < +\\infty \\text{ and } R_k =+\\infty \\rbrace $ we define the first joint regeneration level as $\\mathcal {L}_1 = L_{N}$ .","With this definition the associated joint regeneration times (respectively points) for $X^1$ and $X^2$ are $T^{1}_{\\mathcal {L}_1}$ and $T^{2}_{\\mathcal {L}_1}$ (respectively $X^{1}_{T^{1}_{\\mathcal {L}_1}}$ and $X^{2}_{T^{2}_{\\mathcal {L}_1}}$ ).","We observe that the definition is very similar to the definition in Section REF but instead of defining a sequence of stopping times $(S_k)$ we define a sequence of levels $(L_k)$ .","The first regeneration level is the first level $L_k$ such that the walks do not backtrack and properly encodes the information in variables $Z^1, Z^2$ around the starting points.","The first thing to show is that the quantity $\\mathcal {L}_1$ , as defined above, is $\\mathbb {P}_0$ -almost surely finite, Proposition REF below shows that the first joint regeneration level happens pretty quickly.","Then we will focus in proving the Markov property in Theorem REF .","Proposition 3.13 For any $M \\in (0,+\\infty )$ there exists $K_0 >0$ such that for $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ we have $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}(\\mathcal {L}_1 \\ge n) \\le Cn^{-M},$ where the constant $C$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","The proof will be showed later, as we need several lemmas in order to be able to prove this result.","We recall the notation $\\mathbb {P}_{U_1,U_2}^{K}$ which is the usual annealed measure with the restriction that the conductances around $U_1,U_2$ are set to have value $K$ ."],["Proof of Proposition ","Lemma 3.14 There exists $K_0 > 0$ and $ \\eta > 0$ such that for $K \\ge K_0$ , for $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ we have $& \\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty ) > \\eta > 0, \\\\& \\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}^{K}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty ) > \\eta > 0,$ where $\\eta $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","We follow the lines of .","We only prove the result for $\\mathbb {P}_{U_1,U_2}$ since the proof for $\\mathbb {P}_{U_1,U_2}^{K}$ is similar.","We have $\\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty ) > c \\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}),$ where $c = \\mathbb {P}(U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}) > \\eta > 0$ using Proposition REF and where $\\eta $ only depends on $d$ .","On event $\\lbrace U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}\\rbrace $ we fix $\\mathcal {P}^i$ a path given by the fact that $U_i$ is a $K$ -good point.","We necessarily have $\\mathcal {P}^{i}(1) = e_1$ and $\\mathcal {P}^{i}(2) = e_1 + e_{j_2}$ with $e_{j_2} \\in \\lbrace e_1,\\cdots ,e_d\\rbrace $ .","We introduce $L^i_{\\partial ^{+}\\mathcal {B}(n,n^2)+U_i} = \\inf \\lbrace k \\in \\mathbb {N} \\text{ ; }\\mathcal {P}^{i}(k) \\in \\partial ^{+}\\mathcal {B}(n,n^2)+U_i \\rbrace $ and $x^i = \\mathcal {P}^{i}(L^{i}_{\\partial ^{+}\\mathcal {B}(n,n^2)+ U_i})$ .","We introduce $&A_1^{i} = \\lbrace (X^{i}_1,Z^{i}_1) = (e_1,1)\\rbrace ,\\\\&A_2^{i} = \\lbrace (X^{i}_2,Z^{i}_2) = (e_1+e_{j_2},1)\\rbrace ,\\\\&A_3^{i} = \\lbrace X^{i}_{k} = \\mathcal {P}^{i}(k) \\text{ for } 3\\le k \\le L_{\\partial ^{+}\\mathcal {B}(n,n^2)+U_i}\\rbrace ,\\\\&A_4^{i} = \\lbrace T^{i}_{\\mathcal {H}^{-}(2)} \\circ \\theta _{T_{x^i}} = +\\infty \\rbrace .$ and define $A_k = A^1_k \\cap A^2_k$ and $\\displaystyle A = \\bigcap _{k=1}^{4}A_{k}$ .","We see that $A \\subset \\lbrace \\mathcal {D}^{\\bullet } =+\\infty \\rbrace $ .","As $U_1$ and $U_2$ are $K$ -good we have $L^{i}_{\\partial ^{+}\\mathcal {B}(n,n^2)+U_i} \\le Cn$ and since all points in $\\mathcal {P}^{i}$ are $K$ -open, we have $P_{U_1,U_2}^{\\omega }(A_1,A_2,A_3) \\ge c^{n},$ with $c > 0$ which only depend on $K$ , $d$ and $\\ell $ .","Thus, by Markov property we have $&\\mathbf {E}[P_{U_1,U_2}^{\\omega }(A_1,A_2,A_3,A_4) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}]\\\\&= \\mathbf {E}[P_{U_1,U_2}^{\\omega }(A_1,A_2,A_3)P^{\\omega }_{x^1,x^2}(A^1_4,A^2_4) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}]\\\\&\\ge c^n\\mathbf {E}[P^{\\omega }_{x^1,x^2}(A_4) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}]\\\\& = c^n \\mathbf {E}\\left[P^{\\omega }_{x^1,x^2}(T^{1}_{\\mathcal {H}^{-}(2)} = +\\infty ,T^{2}_{\\mathcal {H}^{-}(2)} = +\\infty ) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}\\right].$ We also have $&\\mathbf {E}\\left[P^{\\omega }_{x^1,x^2}( \\lbrace T^1_{\\mathcal {H}^{-}(2)} < +\\infty \\rbrace \\cup \\lbrace T^2_{\\mathcal {H}^{-}(2)} < +\\infty \\rbrace ) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}\\right]\\\\&\\le C\\mathbb {P}_0\\left(T_{\\mathcal {H}^{-}(-n+2)} < +\\infty \\right) \\\\&\\le C\\exp (-cn).$ Where we used a union bound, translation invariance of the law of the environment and the fact $\\mathbf {P}(U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}) > \\eta > 0$ and .","Hence we see that there exists $n_0$ big enough which only depend on $\\lambda ,\\vec{\\ell },d,K$ such that this quantity is smaller than $1/2$ for all $n \\ge n_0$ and it concludes the proof.","The following lemmas correspond to the ones in .","We recall the definitions of $L_k$ and $M_k$ in (REF ), (REF ), then we introduce the events: $&M^{K}(n) = \\left\\lbrace \\text{for }k \\text{ such that }M_k < n \\text{ we have }L_{k+1} - M_k < n^{1/2} \\right\\rbrace , \\\\&S(n) = \\left\\lbrace \\text{for } k \\text{ such that }L_k < n \\text{ and } M_k < +\\infty \\text{ we have } M_k - L_k < n^{1/2}\\right\\rbrace .$ In the following we may drop the superscript $K$ to lighten the notations.","Lemma 3.15 For $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and for any $M > 0$ there exists $K > 0$ and a constant $C > 0$ such that $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}\\left(M^{K}(n)^{c}\\right) \\le Cn^{-M},$ where the constant $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","It follows from Proposition REF .","Indeed, $M^{K}(n)^{c}$ implies that there exists $0 \\le R \\le n$ such that $\\mathcal {M}^{\\bullet }_R \\ge n^{\\frac{1}{2}}$ .","We introduce $\\bar{L} = \\sup \\left\\lbrace \\lbrace X^1_n\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le n \\le \\mathcal {D}^{\\bullet }\\rbrace \\cup \\lbrace X^2_n\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le n \\le \\mathcal {D}^{\\bullet }\\rbrace \\right\\rbrace .$ The following proposition shows that under $\\mathcal {D} < +\\infty $ , $\\bar{L}$ cannot be too big.","Lemma 3.16 Let $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ , then there exists constants $C,c$ such that $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}\\left(\\bar{L} \\ge n \\text{ }|\\text{ }\\mathcal {D}^{\\bullet }<+\\infty \\right) \\le C\\exp (-cn),$ where $C,c$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","First thing to say is that $\\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet } < +\\infty ) > \\eta > 0$ with $\\eta $ which does not depends on $U_1,U_2$ .","Indeed $0 < \\mathbb {P}_{U_1,U_2}(X_1^1 = U_1+e_1,X^1_2 = U_1) = \\mathbb {P}_{0}(X_1 = e_1 , X_2 = 0) < \\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet } < +\\infty ),$ where we used translation invariance of the environment for the equality.","Thus, we only have to dominate $\\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }<+\\infty ,\\bar{L} \\ge n)$ .","By union bound and symmetry, we only have to dominate $\\mathbb {P}_{U_1,U_2}\\left(\\mathcal {D}^{\\bullet }<+\\infty ,\\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } \\ge n\\right).$ Without loss of generality we can suppose $U_1= U_2 =0$ , indeed the proof is exactly the same in the general case but instead of considering the boxes $\\mathcal {B}(2^k,2^{\\alpha k})$ we should consider the boxes $\\mathcal {B}(2^k,2^{\\alpha k}) + U_i$ .","First we have $\\mathbb {P}_{0}\\left(2^k \\le \\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } < 2^{k+1} \\right) \\le &\\mathbb {P}_{0}\\left(T^1_{\\partial \\mathcal {B}(2^k,2^{\\alpha k})} \\ne T^1_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}\\right) +\\\\&\\mathbb {P}_{0}\\Big (T^1_{\\partial \\mathcal {B}(2^k,2^{\\alpha k})} = T^1_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})},\\\\&T^{+}_{\\mathcal {H}^{-}(0)}\\circ \\theta _{T_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}}< T^{+}_{\\mathcal {H}^{+}(2^{k+1})}\\circ \\theta _{T_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}} \\Big ).$ The first term is smaller than $C\\exp (-c2^k)$ using .","Then using a union bound and translation invariance of the environment the second term is dominated by $(2^{k})^{\\alpha d}\\mathbb {P}_{0}\\left(T^1_{\\partial \\mathcal {B}(2^k,2^{\\alpha k})} \\ne T^1_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}\\right) \\le C2^{\\alpha kd}\\exp (-c2^k),$ where we have used once again Theorem $5.1$ of .","Then we conclude using the following domination and the previous inequality $\\mathbb {P}_{0}\\left(\\mathcal {D}^{\\bullet }<+\\infty ,\\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } \\ge n\\right) \\le C\\sum _{k,2^{k}\\ge n}^{} \\mathbb {P}_{0}\\left(2^k \\le \\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } < 2^{k+1}\\right).$ We recall the definition of $S(n)$ in ().","Lemma 3.17 Let $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ , then there exists constants $C,c > 0$ such that $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}\\left(S(n)^{c}\\right) \\le C\\exp (-cn^{\\frac{1}{2}}).$ It is an adaptation of .","First we notice that $\\mathbb {P}_{0}$ -almost surely, $\\# \\lbrace k \\ge 0 \\text{ such that } L_k < n\\rbrace $ is smaller than $d n$ .","Hence we can write $\\displaystyle \\mathbb {P}_{U_1,U_2}\\left(S(n)^{c}\\right) \\le &\\mathbb {P}_{U_1}\\left(T^1_{\\partial \\mathcal {B}_1(n,n^{\\alpha })} \\ne T^1_{ \\partial ^{+} \\mathcal {B}_1(n,n^{\\alpha })} \\right) + \\mathbb {P}_{U_2}\\left(T^2_{\\partial \\mathcal {B}_2(n,n^{\\alpha })} \\ne T^2_{ \\partial ^{+} \\mathcal {B}_2(n,n^{\\alpha })} \\right) \\\\&+ \\sum _{k \\le d n}^{} \\sum _{\\begin{array}{c}x \\in \\mathcal {B}_1(n,n^{\\alpha })\\\\y \\in \\mathcal {B}_2(n,n^{\\alpha })\\end{array}}^{} \\mathbb {P}_{U_1,U_2}\\left(M_k < +\\infty , M_k - L_k \\ge n^{\\frac{1}{2}},X^{1}_{T_{L_k}} = x, X^{2}_{T_{L_k}} = y\\right).$ The first two terms are controlled by Theorem 5.1 of and using translation invariance of the environment.","We can dominate the term in the sum by $& \\mathbb {P}_{U_1,U_2}\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet } \\circ \\theta _{T^1_{x}} + T^1_{x}}(X^{1}_j - x)\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet }\\circ \\theta _{T_{x,y}} < +\\infty ,X^{1}_{T_{L_k}} = x,X^{2}_{T_{L_k}} = y\\right) +\\\\&\\mathbb {P}_{U_1,U_2}\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet } \\circ \\theta _{T^2_{y}} + T^2_{y}}(X^{2}_j - y)\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet }\\circ \\theta _{T_{x,y}} < +\\infty ,,X^{1}_{T_{L_k}} = x,X^{2}_{T_{L_k}} = y\\right).$ Let us dominate the first term, the second is dominated the same way by symmetry.","Using Lemma REF and the invariance of the environment it is bounded by $\\mathbf {E}\\left[P_{x,y}^{\\omega }\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet }}(X^{1}_j -x)\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet } < +\\infty \\right) \\right]&\\le \\mathbf {E}\\left[P_{0,y-x}^{\\omega }\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet }} X^{1}_j\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet } < +\\infty \\right) \\right] \\\\&\\le \\mathbb {P}_{0,y-x}\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet }} X^{1}_j\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}}\\text{ }|\\text{ }\\mathcal {D}^{\\bullet }<+\\infty \\right)\\\\& \\le \\mathbb {P}_{0,y-x}\\left(\\bar{L} > n^{\\frac{1}{2}}\\text{ }|\\text{ }\\mathcal {D}^{\\bullet }<+\\infty \\right).$ The last term is dominated using Lemma REF since we have $ |(y-x) \\cdot \\vec{\\ell }| \\le 1$ by definition of $L_k$ .","Taking the sum over $k,x,y$ we conclude that we have $\\displaystyle \\mathbb {P}_{U_1,U_2}(S(n)^{c}) \\le Cn^{\\beta }\\exp (-cn^{\\frac{1}{2}}).$ where $\\beta $ only depends on $d,\\alpha $ .","It concludes the proof by taking $C$ large enough and $c$ small enough.","The following lemma is the equivalent of .","Recall that we defined $N = \\inf \\lbrace k \\ge 0 \\text{ ; } L_k < +\\infty \\text{ and } R_k =+\\infty \\rbrace $ , and the definition of $\\mathcal {E}_x, \\mathcal {L}^x, \\mathcal {R}^x$ as in Equations (REF )-(REF ).","Furthermore, we define $\\mathcal {E}^{2}_x \\mathcal {E}_x \\cup \\mathcal {E}_{x-e_1}\\cup \\mathcal {E}_{x-2e_1}\\,\\, \\textnormal {the set of all edges which touch} \\,\\, \\lbrace x,x-e_1,x-2e_1\\rbrace .$ Lemma 3.18 There exists $K_0 > 0$ such that for $K \\ge K_0$ $\\sup _{U_1, U_2 \\in \\mathcal {U}}\\mathbb {P}_{U_1,U_2}(N \\ge n) \\le C\\exp (-cn).$ Where $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and $C,c > 0$ only depends on $\\lambda ,\\vec{\\ell },d$ .","We follow the lines of the proof of .","The idea of the proof is simple, each time we reach a level $L_k$ there is a positive probability to regenerate and using the way we have defined $\\mathcal {D}$ we should have good independence properties, hence $N$ should decay at least like a geometric random variable.","We introduce event $C(n) = \\lbrace \\text{for all }k \\le n \\text{ such that }L_k <+\\infty \\text{ we have } \\mathcal {D}^{\\bullet } \\circ \\theta _{T_{L_k}} < +\\infty \\rbrace .$ We have $\\lbrace N \\ge n\\rbrace \\subset C(n)$ , hence we only have to dominate $C(n)$ .","We can write $\\mathbb {P}_{U_1,U_2}(C(n+1)) \\le \\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n),\\mathcal {D}^{\\bullet }\\circ \\theta _{T_{L_k}} < +\\infty \\right)\\right].$ The event $\\lbrace X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\rbrace $ is measurable with respect to $\\sigma \\lbrace (X^{1}_n,Z_{n}^1),(X^{2}_k,Z_{k}^2)\\text{ for }0 \\le n \\le T^1_x \\text{ and }0\\le k \\le T^2_y\\rbrace $ .","Thus we can use Lemma REF and get $&P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n),\\mathcal {D}^{\\bullet } \\circ \\theta _{T_{L_k}} < +\\infty \\right) \\\\=& P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)P_{x,y}^{\\omega }(\\mathcal {D}^{\\bullet } < +\\infty ).$ Then, the sum is also equal to $&\\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)P_{x,y}^{\\omega }(\\mathcal {D}^{\\bullet } < +\\infty )\\right]\\\\&= \\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)P_{x,y}^{\\omega _{K}}(\\mathcal {D}^{\\bullet } < +\\infty )\\right],$ where we used the fact that both $x$ and $y$ are $K$ -open and Proposition REF .","Since $P_{U_1,U_2}^{\\omega }\\big (X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\big )$ is measurable with respect to $\\sigma \\big (c_{*}(e) \\text{ with } e \\in \\mathcal {L}^{x-2e_1} \\cup \\mathcal {L}^{y-2e_1} \\cup \\mathcal {E}^2_{x} \\cup \\mathcal {E}^2_{y}\\big )$ and $P_{x,y}^{\\omega _{K}}(\\mathcal {D}^{\\bullet } < +\\infty )$ is measurable with respect to $\\sigma \\big (c_{*}(e) \\text{ with } e \\in \\mathcal {R}^{x} \\cup \\mathcal {R}^{y} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{y})\\big )$ and using the fact that those sets of edges are disjoint (see Figure REF ) we can write by $\\mathbf {P}$ independence that the sum is equal to $\\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)\\right]\\mathbf {E}\\left[P_{x,y}^{\\omega _{K}}(\\mathcal {D}^{\\bullet } < +\\infty )\\right].$ Using Lemma REF with $K_0$ large enough we have $\\mathbb {P}_{x,y}^{K}(\\mathcal {D}^{\\bullet }<+\\infty ) \\le \\eta < 1$ with $\\eta $ a constant which only depends on $\\lambda ,\\vec{\\ell },d$ .","It follows that $\\mathbb {P}_{U_1,U_2}(C(n+1)) \\le \\eta \\mathbb {P}_{U_1,U_2}(C(n)).$ The result follows by induction.","We now have all the tools to prove Proposition REF .","Indeed combining Lemmas REF , REF and REF , it is unlikely to have the $n$ first $L_k$ finite and $L_{k+1} - L_{k} \\ge n^{\\frac{1}{2}}$ for $k \\le n$ .","[Proof of Proposition REF ] On $\\lbrace \\mathcal {L}_1 \\ge n\\rbrace \\cap S(n)\\cap M(n)$ we have $N \\ge n^{1/3}$ for $n$ large enough.","Indeed, on $\\lbrace \\mathcal {L}_1 \\ge n\\rbrace \\cap S(n)\\cap M(n)$ for $k < n$ such that $L_k < n $ and $M_k < n $ , we have $L_{k+1} - L_k < 2n^{1/2}$ , and since $\\mathcal {L}_1 \\ge n$ , as long as $L_{k+1} < n$ we have $M_{k+1} < +\\infty $ .","Hence it follows that we must have $N \\ge n^{\\frac{1}{3}}$ since in the opposite case we would have $\\displaystyle \\mathcal {L}_1 = L_{N} = \\sum _{k=0}^{N-1} L_{k+1}-L_k < N2n^{1/2} < 2n^{5/6} 0$ there exists $K_0 > 0$ such that for $K > K_0$ $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}^{K}(\\mathcal {L}_1 > n) \\le Cn^{-M}.$ Where $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and the constant $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","For the same reasons as in previous proofs we restrict to the case where $U_1 = U_2 = 0$ without loss of generality, indeed we simply have to replaces boxes $\\mathcal {B}(n,n^{\\alpha })$ with $\\widetilde{\\mathcal {B}}_i(n,n^{\\alpha })$ .","We write: $&\\mathbb {P}_{0}^{K}(\\mathcal {L}_1 > n) \\le \\mathbb {P}^{K}_{0}\\left(T^1_{\\partial B(n/2,(n/2)^{\\alpha })} \\ne T^1_{\\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })} \\right) + \\mathbb {P}^{K}_{0}\\left(T^2_{\\partial \\mathcal {B}(n/2,(n/2)^{\\alpha })} \\ne T^2_{\\partial ^{+} \\mathcal {B}(n/2,(n/2)^{\\alpha })} \\right)\\\\&+\\sum _{x,y \\in \\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })}^{} \\mathbb {P}^{K}_{0}(X^1_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = y,T^1_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T_x} < +\\infty )\\\\&+ \\sum _{x,y \\in \\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })}^{} \\mathbb {P}^{K}_{0}(X^1_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = y,T^2_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T^2_y} < +\\infty )\\\\&+\\sum _{x,y \\in \\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })}^{} \\mathbb {P}^{K}_{0}(X^1_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = y,\\\\&\\hspace{128.0374pt} T^1_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T^1_x} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T^2_y} = +\\infty ,\\mathcal {L}_1 \\circ \\theta _{T_{x,y}} \\ge n/2),$ where in the last sum $\\theta _{T_{x,y}}$ means that the first walk is shifted by $T_x$ and the second by $T_y$ .","The first two terms are dominated thanks to .","Let us dominate the third term.","The fourth can be dominated in the same way by symmetry.","The term in the sum can be written using Markov property $\\mathbf {E}&\\left[P^{\\omega _K}_{0}\\left(X^1_{T_{\\partial B((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial B((n/2),(n/2)^{\\alpha })}} = y\\right)P_{x}^{\\omega ^{0}_K}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty \\right)\\right] \\\\ &\\le \\mathbf {E}\\left[P_{x}^{\\omega ^{0}_K}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty \\right)\\right],$ where the notation $\\omega ^{0}_{K}$ means that we have set the conductances of $\\mathcal {E}_0$ to $K$ .","Then we notice that $P_{x}^{\\omega ^{0}_K}(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty )$ does not depend on the conductances of $\\mathcal {E}_0$ , so it is also equal to $P_{x}^{\\omega }(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty )$ .","Using translation invariance of the environment and the sum is bounded by $Cn^{2d\\alpha }\\exp (-cn)$ .","Let us dominate the fifth term.","Using Lemma REF the term in the sum can be written $&\\mathbf {E}\\left[P_{0}^{\\omega _{K}}\\left(X^1_{T_{\\partial B((n/2)),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial B((n/2),(n/2)^{\\alpha })}} = y\\right)P_{x,y}^{\\omega ^{0}_{K}}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,\\mathcal {L}_1\\ge n\\right)\\right]\\\\&\\le \\mathbf {E}\\left[P_{x,y}^{\\omega ^{0}_{K}}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,\\mathcal {L}_1\\ge n\\right)\\right].$ Under $\\lbrace T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty \\rbrace $ both walks never meet conductances in $\\mathcal {E}_{0}$ and so we can replace $\\omega ^{0}_{K}$ by $\\omega $ .","Then we write $\\mathbf {E}\\left[P_{x,y}^{\\omega ^{0}_{K}}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,\\mathcal {L}_1\\ge n\\right)\\right]&\\le \\mathbb {P}_{x,y}(\\mathcal {L}_1\\ge n) \\\\&\\le \\mathbb {P}_{0,y-x}(\\mathcal {L}_1\\ge n/4),$ where we used Proposition REF since we have $|(y-x) \\cdot \\vec{\\ell }| \\le 1$ because $x,y \\in \\partial ^{+} B((n/2),(n/2)^{\\alpha })$ , translation invariance of the environment.","Then we see that the sum is bounded by $ C n^{2d\\alpha } n^{-(M+2d\\alpha )} $ by taking $K$ large enough.","This concludes the proof.","We can define successive joint regeneration levels as $\\mathcal {L}_{k+1} = \\mathcal {L}_{k} \\circ \\theta _{T_{\\mathcal {L}_1}}.$ Let us define $\\mathcal {G}_k = &\\sigma \\big (\\mathcal {L}_0,\\dots ,\\mathcal {L}_k,(\\tilde{X}^1_{T^1_{\\mathcal {L}_k}\\wedge m})_{m \\ge 0},(\\tilde{X}^2_{T^2_{\\mathcal {L}_k}\\wedge m})_{m \\ge 0}, \\nonumber \\\\ & c_{*}(e) \\text{ with } e \\in E(\\mathcal {H}^{-}(\\mathcal {L}_k-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{X^1_{T^1_{\\mathcal {L}_k}}} \\cup \\mathcal {E}^{2}_{X^2_{T^2_{\\mathcal {L}_k}}}\\big ).$ The definition of $\\mathcal {G}_k$ may seem a bit complicated at first sight.","It represents the information that the walkers see before reaching level $\\mathcal {L}_k$ and also the whole environment on the left of $\\mathcal {L}_k-2e_1\\cdot \\vec{\\ell }$ .","Figure: The sigma algebra 𝒢 k \\mathcal {G}_k: In red and blue the trajectories before reaching ℒ k \\mathcal {L}_k and in orange the edges considered.Now we want to show that when we consider our walks shifted of $T_{\\mathcal {L}_k}$ they behave like new walks starting from respectively $X^1_{T^1_{\\mathcal {L}_k}}$ and $X^2_{T^2_{\\mathcal {L}_k}}$ conditionned to never backtrack and such that $c_{*}(e) = K$ for $e \\in \\mathcal {E}_{X^1_{T^1_{\\mathcal {L}_k}}} \\cup \\mathcal {E}_{X^2_{T^2_{\\mathcal {L}_k}}}$ .","This is the purpose of the next theorem.","Theorem 3.20 Fix $K$ large enough.","For any $k \\ge 1$ , we have $\\mathcal {L}_k < +\\infty $ $\\mathbb {P}_0$ -almost surely.","Now let $f, g, h_k$ be bounded functions which are measurable with respect to $\\sigma \\left( (X^1_0,X^2_0), (\\tilde{X}^1_n,\\tilde{X}^2_n)_{n \\ge 1}\\right)$ , $\\sigma \\left(c_{*}(e) \\text{ ; } e \\in \\mathcal {R}^{X^1_0} \\cup \\mathcal {R}^{X^2_0} \\backslash (\\mathcal {E}_{X^1_0} \\cup \\mathcal {E}_{X^2_0})\\right)$ and $\\mathcal {G}_k$ , respectively.","We have the following equality $\\mathbb {E}_{\\begin{array}{c}(0,z_1)\\\\(0,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{\\mathcal {L}_k}+\\cdot },\\tilde{X}^{2}_{T^{2}_{\\mathcal {L}_k}+\\cdot })g\\circ \\theta _{T_{\\mathcal {L}_k}}h_k\\right] = \\mathbb {E}_{\\begin{array}{c}(0,z_1)\\\\(0,z_2)\\end{array}}\\left[h_k\\mathbb {E}^K_{X^1_{T^{1}_{\\mathcal {L}_k}},X^2_{T^{2}_{\\mathcal {L}_k}}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot })g \\text{ } | \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ This proof can be done by induction.","We can restrict to the case $k = 1$ , but we have to prove the result with initial points $|U_1\\cdot \\vec{\\ell }| \\le 1$ and $|U_2 \\cdot \\vec{\\ell }| \\le 1$ and also when the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ are fixed to a certain value $a \\in [1/K,K]^{\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}}$ $\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{\\mathcal {L}_1}+\\cdot },\\tilde{X}^{2}_{T^{2}_{\\mathcal {L}_1}+\\cdot })g\\circ \\theta _{T_{\\mathcal {L}_1}}h_1\\right] = \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1\\mathbb {E}^K_{X^1_{T^{1}_{\\mathcal {L}_1}},X^2_{T^{2}_{\\mathcal {L}_1}}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot })g \\text{ } | \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ Here the notation $\\mathbb {E}^{a}$ means that we fix the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ to $a$ .","The proof with $\\mathbb {E}$ is the same.","By definition of $\\mathcal {L}_1$ , we have $\\displaystyle &\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{\\mathcal {L}_1}+\\cdot },\\tilde{X}^{2}_{T^{2}_{\\mathcal {L}_1}+\\cdot })g\\circ \\theta _{T_{\\mathcal {L}_1}}h_1\\right] \\\\&= \\sum _{k \\ge 0}^{} \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{L_k}+\\cdot },\\tilde{X}^{2}_{T^{2}_{L_k}+\\cdot })g\\circ \\theta _{T_{L_k}}h_1,L_k <+\\infty , R_k = +\\infty \\right]\\\\&= \\sum _{\\begin{array}{c}k \\ge 0 \\\\ p \\ge 0\\end{array}, \\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}^{} \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{p}+\\cdot },\\tilde{X}^{2}_{T^{2}_{p}+\\cdot })g\\circ \\theta _{T_p}h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_{p}} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right].$ For $p,(x,z),(x^{\\prime },z^{\\prime })$ in the sum and following the arguments of there exists $h_1^{\\prime }$ measurable with respect to $\\sigma \\left((\\tilde{X}^1_{T^1_{p}\\wedge m})_{m \\ge 0},(\\tilde{X}^2_{T^2_{p}\\wedge m})_{m \\ge 0}, c_{*}(e) \\text{ with } e \\in \\mathcal {L}^{x- 2e_1} \\cup \\mathcal {L}^{x^{\\prime }- 2e_1} \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }}\\right),$ such that on the event $\\lbrace \\tilde{X}^1_{T^{1}_p} = (x,z)\\rbrace \\cap \\lbrace \\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\rbrace \\cap \\lbrace \\mathcal {L}_1 = p\\rbrace $ the functions $h_1$ and $h^{^{\\prime }}_1$ coincide.","Fix $k,p \\ge 0$ and $(x,z),(x^{\\prime },z^{\\prime })$ , in the sum.","We can replace $h_1$ by $h_1^{\\prime }$ but to simplify notations we will still write $h_1$ and suppose it is measurable with respect to the above $\\sigma $ -algebra.","We can do the same with $g \\circ \\theta _{T_p}$ , indeed we can consider $g$ as a function of $(X^1_0,X^2_0,\\omega )$ and then under $\\lbrace \\mathcal {L}_1 = p,X^1_{T^1_{\\mathcal {L}_1}} = x,X^2_{T^2_{\\mathcal {L}_1}} = x^{\\prime } \\rbrace $ we can replace $g$ by a function which is measurable with respect to $\\sigma \\left(c_{*}(e) \\text{ ; } e \\in \\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }})\\right).$ The term in the sum can then be written $&\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{p}+\\cdot },\\tilde{X}^{2}_{T^{2}_{p}+\\cdot })g^{(x,x^{\\prime })}h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\\\&=\\mathbf {E}\\left[E^{\\omega _a}_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\times g^{(x,x^{\\prime })}E^{\\omega _a}_{\\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}\\left[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ In the following we will use the shortcut $E^{\\omega _a}_{\\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}\\left[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty \\right] = E^{\\omega _a}_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty \\right].$ Where we have used Lemma REF with $\\tilde{X}^1$ and $\\tilde{X}^2$ and with the stopping times $T^1_{p}$ and $T^2_{p}$ .","We recall the notation $\\omega _a$ which means we fix the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ to $a$ .","We want to stress that in the expression $E^{\\omega _a}_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ , with a little abuse of the notation, we fix to the value $a$ the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ and not the ones in $\\mathcal {E}_{x}\\cup \\mathcal {E}_{x^{\\prime }}$ as one might expect.","Moreover, thanks to the fact that $P^{\\omega _a}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely the walks never backtrack below $x$ and $x^{\\prime }$ on $\\lbrace \\mathcal {D}^{\\bullet } = +\\infty \\rbrace $ , the value of the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ does not matter in the expectation so we can replace $\\omega _a$ with $\\omega $ and write $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ .","Then, since under $\\lbrace L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\rbrace $ the points $x$ and $x^{\\prime }$ are $K$ -open and using Proposition REF , we can replace $\\omega $ by $\\omega _K$ in $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ .","Notice that $f$ and $\\mathcal {D}$ are measurable with respect to $\\sigma \\big ( X^1_0,X^2_0,(\\tilde{X}^1_{n})_{n \\ge 1},(\\tilde{X}^2_{n})_{n\\ge 1} \\big )$ and thus can be written as $\\phi \\left(X^1_0,X^2_0,(\\tilde{X}^1_{n})_{n \\ge 1},(\\tilde{X}^2_{n})_{n\\ge 1}\\right)$ with $\\phi $ measurable.","Since the law of $\\left( X^1_0,X^2_0,(\\tilde{X}^1_{n})_{n \\ge 1},(\\tilde{X}^2_{n})_{n\\ge 1}\\right)$ is the same under $P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}$ and under $P^{\\omega }_{(\\mathbf {x}, 0)}$ , we replace $z,z^{\\prime }$ by 0 in $E^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ .","We observe that $E^{\\omega _K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ is measurable with respect to $\\sigma \\left(c_{*}(e) \\text{ ; } e \\in \\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }})\\right).$ Moreover, $E^{\\omega _a}_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })]$ is measurable with respect to $\\sigma \\left(c_{*}(e) \\text{ with } e \\in E(\\mathcal {H}^{-}(p-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }}\\right).$ In addition, $\\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }})$ and $E(\\mathcal {H}^{-}(p-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }}$ can be proved to be disjoint in an elementary way as illustrated in the figure below.","Figure: In orange E(ℋ - (p-2e 1 ·ℓ →))∪ℰ x 2 ∪ℰ x ' 2 E(\\mathcal {H}^{-}(p-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }} and in green ℛ x ∪ℛ x ' ∖(ℰ x ∪ℰ x ' )\\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }}).By independence of the two $\\sigma $ -algebra we deduce that $&\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{p}+\\cdot },\\tilde{X}^{2}_{T^{2}_{p}+\\cdot })g^{(x,x^{\\prime })}h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\\\&=\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.","}),\\mathcal {D}^{\\bullet } =+\\infty \\right].$ Last equality used in the particular case where $f = 1$ and $g = 1$ gives $&\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\\\&=\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {P}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}(\\mathcal {D}^{\\bullet } =+\\infty ).$ Hence we can write $& \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.","}),\\mathcal {D}^{\\bullet } =+\\infty \\right] \\\\&= \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.})","\\text{ }| \\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right].$ Then taking the sum we obtain $&\\sum _{\\begin{array}{c}k \\ge 0 \\\\ p \\ge 0\\end{array}, \\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}^{} \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.})","\\text{ }| \\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right]\\\\&= \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1\\mathbb {E}^K_{X^1_{T^{1}_{\\mathcal {L}_1}},X^2_{T^{2}_{\\mathcal {L}_1}}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot })g \\text{ } | \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ Then for the induction, if the formula has been proved for $k$ , from Proposition REF we get that $\\mathcal {L}_{k+1}$ is almost surely finite.","The proof works in a similar way to go from $k$ to $k+1$ and the details of the adaptation can be found in .","A consequence of this theorem is the analogous of Proposition REF but in a more general setting.","Proposition 3.21 For any $M > 0$ there exists a constant $c$ such that for any $k > 0$ $\\mathbb {P}_{0}(\\mathcal {L}_{k+1} - \\mathcal {L}_k \\ge n) \\le cn^{-M}.$ We write $\\mathbb {P}_{0}(\\mathcal {L}_{k+1} - \\mathcal {L}_k \\ge n) &= \\sum _{p,x,x^{\\prime }}^{} \\mathbb {E}_{0}\\left[\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\prime }, \\mathcal {L}_{k+1} - \\mathcal {L}_k \\ge n\\right]\\\\& = \\sum _{p,x,x^{\\prime }}^{} \\mathbb {E}_{0}\\left[\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\prime }, \\mathcal {L}_1 \\circ \\theta _{T_{x,x^{^{\\prime }}}} - p \\ge n\\right]\\\\& = \\sum _{p,x,x^{\\prime }}^{} \\mathbb {E}_{0}\\left[\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\prime },\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n \\text{ } | \\text{ }\\mathcal {D}^{\\bullet } =+\\infty )\\right].$ where the third equality follows from Theorem REF .","We only have to dominate $\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n \\text{ } | \\text{ }\\mathcal {D}^{\\bullet } =+\\infty )$ to conclude.","To do so, using the fact that $\\mathbb {P}_{x,x^{\\prime }}^{K}(\\mathcal {D}^{\\bullet } =+\\infty ) > \\alpha > 0$ with $\\alpha $ which does not depend on $x$ and $x^{\\prime }$ , we can see it suffices to bound $\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n )$ .","Set $y = x^{\\prime } - x$ , by the invariance in law of the environment by translation we have $\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n ) = \\mathbb {P}^{K}_{0,y}(\\mathcal {L}_1 -p + x\\cdot \\vec{\\ell } \\ge n ) $ .","Notice that we have $|x\\cdot \\vec{\\ell } - p| \\le 1$ , which follows from the fact that $X^1_{T^{1}_p} = x$ .","Thus the result follows from Proposition REF ."],["Asymptotic separation","The aim of this section is to prove the following result: the trajectories of the two walks do not cross anymore in $\\mathcal {H}^{+}_N$ with high probability.","We recall the notation $\\mathcal {H}^{+}_{N} = \\lbrace x \\in \\mathbb {Z}^d \\text{ }|\\text{ } x\\cdot \\vec{\\ell } > N\\rbrace $ .","We introduce the event that the traces of the walks in $\\mathcal {H}^{+}_N$ get close $ \\mathbf {M}_N \\bigg \\lbrace \\inf _{\\begin{array}{c}x \\in \\mathcal {H}^{+}_{N}\\cap \\lbrace X_n^1\\rbrace \\\\y \\in \\mathcal {H}^{+}_{N}\\cap \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\le 1 \\bigg \\rbrace $ Proposition 4.1 For any $N \\ge 0$ , the following inequality holds $\\mathbb {P}_{0}(\\mathbf {M}_N ) \\le CN^{-c},$ where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d$ .","In this part we will adapt methods from .","Firstly we will deal with the case when the two walks $X^1$ and $X^2$ live in independent environments.","Then, when the two walks live in the same environment, we will try use the fact that when they do not cross, they behave like if they were in independent environments."],["independent environments","First we recall some notations: The law $Q_{U_1,U_2}$ is defined as $Q_{U_1,U_2} = \\mathbb {P}_{U_1} \\otimes \\mathbb {P}_{U_2}$ .","Hence $X^1$ and $X^2$ are independent under $Q_{U_1,U_2}$ .","We also have $Q^{K}_{U_1,U_2} =\\mathbb {P}^{K}_{U_1} \\otimes \\mathbb {P}^{K}_{U_2}$ .","We write $Q_{U_1,U_2}(\\cdot \\, | \\, D^{\\otimes }=+\\infty ) = \\mathbb {P}_{U_1}(\\cdot \\, | \\, D=+\\infty ) \\otimes \\mathbb {P}_{U_2}(\\cdot \\, | \\, D=+\\infty )$ where $D$ is defined in in the definition of regeneration times.","Proposition 4.2 There exists $K_0 > 0$ such that for any $U_1, U_2 \\in \\mathbb {Z}^d$ and $K > K_0$ we have $Q^{K}_{U_1, U_2}\\bigg (\\inf _{\\begin{array}{c}x \\in \\lbrace X_n^1\\rbrace \\\\y \\in \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\le 1 \\, \\bigg | \\, \\textnormal {}D^{\\otimes }=+\\infty \\bigg ) \\le C\\Vert U_1 - U_2\\Vert ^{-c},$ where the constants $C,c > 0 $ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Remark 4.3 We observe that if $U_1 = U_2$ the above inequality remains true with the convention $\\Vert U_1-U_2\\Vert ^{-c} = +\\infty $ .","For $z \\in \\mathbb {Z}^d$ we define $F_i(z) = \\mathbb {P}^{K}_{U_i}(\\exists k, X_k^{i} = z \\, | \\, \\text{}D=+\\infty ),$ and $F_i^{(R)}(z) = F_i(z){1}_{\\lbrace \\Vert z-U_i\\Vert > R\\rbrace }.$"],["Step 1: bounding our quantity using $\\Vert F_i^{(R)}\\Vert _2$ and {{formula:7505ed72-501c-44cd-9caa-ec40deaa1136}} .","Fix $R = \\Vert U_1-U_2\\Vert /3$ .","If $\\Vert X^{1}_p-X^2_q\\Vert \\le 1$ then necessarily either $\\Vert U_1-X^1_p\\Vert \\ge R$ or $\\Vert U_2-X^2_q\\Vert \\ge R$ , otherwise we would have $\\Vert U_1-U_2\\Vert \\le 2/3\\Vert U_1-U_2\\Vert + 2 < \\Vert U_1-U_2\\Vert $ (as soon as $\\Vert U_1-U_2\\Vert \\ge 7$ ).","It follows $\\nonumber Q^{K}_{U_1,U_2} &\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\&\\le Q^{K}_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\text{ and } \\Vert U_1-X^1_p\\Vert \\ge R \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\& + Q^{K}_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\text{ and } \\Vert U_2-X^2_q\\Vert \\ge R \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right).$ We write: $Q^{K}_{U_1,U_2} &\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\text{ and } \\Vert U_1-X^1_p\\Vert \\ge R \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\&\\le \\sum _{\\Vert z-U_1\\Vert \\ge R}^{} Q^{K}_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } X^1_p = z, X^2_q \\in \\mathcal {V}_z\\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\& \\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{\\Vert z-U_1\\Vert \\ge R}^{} F_1^{(R)}(z)F_2^{}(z+e) \\\\&\\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{z \\in \\mathbb {Z}^d}^{} F_1^{(R)}(z)F_2^{}(z+e) \\\\& \\le C\\Vert F^{(R)}_1\\Vert _2\\Vert F_2\\Vert _2,$ where the third inequality comes from Cauchy-Schwarz inequality and where the constant $C > 0$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","Symmetrically, we can do the same computation for (REF ).","It follows that if we show that $\\Vert F_i\\Vert _2 < +\\infty $ and $\\Vert F_i^{(R)}\\Vert _2 \\le CR^{-c}$ with $C,c > 0$ constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ , the proof is finished."],["Step 2: bound on $\\Vert F_i\\Vert _2$ and {{formula:23b48c57-5a69-4a5d-8b96-9d7b1d444caf}} .","For $n \\in \\mathbb {N}$ we introduce $H_i(z,n) = \\mathbb {P}^{K}_{U_i}(\\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\text{, } X_k^{i} = z \\, | \\, \\text{}D=+\\infty ),$ and $H_i^{(R)}(z,n) = H_i(z,n){1}_{\\lbrace \\Vert z-U_i \\Vert > R \\rbrace }.$ Using the triangle inequality we write $\\Vert F_i\\Vert _{2} \\le \\sum _{n=1}^{+\\infty } \\Vert H_i(\\cdot ,n)\\Vert _{2},$ and $\\Vert F_i^{(R)}\\Vert _{2} \\le \\sum _{n = 1}^{+\\infty } \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2}.$ We want to bound $\\Vert H_i(\\cdot ,n)\\Vert _2$ and $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _2$ .","The quantity $H_i(\\cdot ,n)$ can be rewritten as a convolution because $X_{\\tau _n^{i}}^{i}$ and $\\lbrace X_{\\tau _n^{i} + \\, \\cdot }^{i} - X_{\\tau _n^{i}}^{i} \\rbrace $ are independent by .","Thus $H_i(\\cdot ,n) = T_i^{n} \\ast J,$ where $T_i^{n}(z) = \\mathbb {P}_{U_i}^{K}(X^{i}_{\\tau ^{i}_n} = z),$ and $J(z) = \\mathbb {P}_{0}^{K}\\left(\\exists 0 \\le k \\le \\tau _{1}^{i} \\text{, } X_k^{i} = z \\, | \\, \\text{}D=+\\infty \\right).$ Young's inequality gives $\\Vert H_i(\\cdot ,n)\\Vert _{2} \\le \\Vert T_i^{n}\\Vert _{2} \\Vert J\\Vert _{1}.$ We have $\\Gamma := \\Vert J\\Vert _{1} < +\\infty $ .","Indeed $\\sum _{z \\in \\mathbb {Z}^d}^{} \\mathbb {P}_{0}^{K}\\left(\\exists 0 \\le k \\le \\tau _{1}^{i} \\text{, } X_k^{i} = z \\, | \\, \\text{}D=+\\infty \\right) \\le C\\mathbb {E}_{0}^{K}\\bigg [\\sum _{z \\in \\mathbb {Z}^d}^{} {1}_{\\exists 0 \\le k \\le \\tau ^i_{1} \\text{, } X_k^{i} = z} \\bigg ],$ where $C$ is a constant which only depend on $\\lambda ,\\vec{\\ell },d,K$ using .","The term in the last expectation is equal to $\\#\\lbrace X_k^{i} \\text{ ; } 0\\le k \\le \\tau _1^{i}\\rbrace $ , now recall the notation $\\chi ^i = \\inf \\lbrace m \\ge 0 \\text{ ; }\\lbrace X_k^{i} \\text{ ; } 0\\le k \\le \\tau _1^{i}\\rbrace \\subset X^i_0 + \\mathcal {B}(m,m^{\\alpha })\\rbrace $ , then we clearly have $\\#\\lbrace X_k^{i} \\text{ ; } 0\\le k \\le \\tau _1^{i}\\rbrace \\le (\\chi ^i)^{\\beta }$ for $\\beta $ large enough which only depends on $d,\\alpha $ .","By , for $K$ large enough, $\\chi ^i$ has finite moment of order $\\beta $ .","It follows that $||J||_1 < +\\infty $ .","For the term $\\Vert T_i^{n}\\Vert _2$ , using Proposition REF we have for $K$ high enough $\\displaystyle \\sup _{z \\in \\mathbb {Z}^d}\\mathbb {P}_{U_i}^{K}(X_{\\tau _n^{i}}^{i} = z) \\le Cn^{-\\frac{d}{2}+\\varepsilon },$ where $C$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","It gives the bound $\\Vert H_i^{}(\\cdot ,n)\\Vert _{2} \\le Cn^{-\\frac{d}{4}+\\frac{\\varepsilon }{2}}.$ In dimension $d \\ge 5$ we have $\\Vert F_i\\Vert _{2} < +\\infty .$ We still have to bound $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2}$ .","A first bound is given by (REF ), however for $n$ small this bound is not precise enough, thus we compute another bound which is useful for small $n$ .","Let us bound $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1}$ and then use the inequality $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2} \\le \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1}^{1/2}$ .","We can rewrite $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1} &= \\sum _{||z-U_{i}|| \\ge R}\\mathbb {E}^{K}_{U_i}\\left[ {1}_{ \\left\\lbrace \\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\, ; \\, X_k^{i} = z \\right\\rbrace } \\, \\big | \\, \\text{}D=+\\infty \\right] \\nonumber \\\\ &= \\mathbb {E}^{K}_{U_i}\\left[ \\sum _{||z - U_i|| \\ge R}{1}_{ \\left\\lbrace \\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\, ; \\, X_k^{i} = z \\right\\rbrace } \\, \\big | \\, \\text{}D=+\\infty \\right].","$ Define $A(n,R) \\lbrace \\exists \\, \\tau _n \\le k \\le \\tau _{n+1} \\textnormal { such that } |X^i_k-X^i_0| \\ge R\\rbrace $ .","We introduce the notation $\\chi ^i_n = \\chi ^i(X^{i}_{\\tau _n + \\cdot })$ .","The term $\\displaystyle \\sum _{||z - U_i|| \\ge R}{1}_{ \\left\\lbrace \\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\, ; \\, X_k^{i} = z \\right\\rbrace }$ in the expectation (REF ) is bounded by ${1}_{A(n,R)}\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })$ , indeed it is clearly bounded by $\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })$ and under $A(n,R)^{c}$ it is equal to 0.","Using the fact that $\\mathrm {diam}(\\mathcal {B}(L,L^{\\alpha })) \\le CL^{\\alpha }$ where $C$ is a constant which only depends on $\\vec{\\ell },d$ , we have the following inclusion: $A(n,R) \\subset \\bigg \\lbrace \\sum _{k=0}^{n} (C\\chi ^{i}_k)^{\\alpha } \\ge R \\bigg \\rbrace .$ Since $((\\chi ^{i}_{k})^{\\alpha })_{k \\ge 0}$ is a family of independent identically distributed random variables and for $K$ large enough $(\\chi ^{i}_0)^{\\alpha }$ has a finite first moment under $\\mathbb {P}^K_{U_i}(\\cdot \\text{ | }D=+\\infty )$ by , we have $\\mathbb {P}^{K}_{U_i}\\left(A(n,R) \\text{ | }D=+\\infty \\right) &\\le \\mathbb {P}^{K}_{U_i}\\left(\\sum _{k=0}^{n} C(\\chi ^{i}_k)^{\\alpha } \\ge R\\text{ | }D=+\\infty \\right)\\\\&\\le C\\frac{(n+1)\\, \\mathbb {E}_{U_i}^{K}\\left[(\\chi ^{i}_{1})^{\\alpha }\\text{ | }D=+\\infty \\right]}{R} = \\frac{Cn}{R},$ where $C > 0$ is a constant which only depends on $\\lambda ,\\vec{\\ell },d,K$ .","$\\bullet $ Now using , for any $M > 0$ there exists $K_0 > 0$ such that for $K \\ge K_0$ the variable $\\chi ^i_n$ has a moment of order $M$ .","By chosing $K_0$ big enough, $\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })$ has a moment of order 2.","Indeed $\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha }) \\le C(\\chi ^i_n)^{d\\alpha }$ with $C$ a constant which only depends on $d$ .","Using Cauchy-Schwarz inequality we have $\\nonumber \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1} &\\le \\mathbb {P}_{U_i}^{K}\\left( A(n,R)\\text{ | }D=+\\infty \\right)^{\\frac{1}{2}}\\mathbb {E}^{K}_{U_i}\\left[\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })^2 \\text{ | }D=+\\infty \\right]^{\\frac{1}{2}}\\\\& \\le Cn^{\\frac{1}{2}}R^{-\\frac{1}{2}},$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","It follows that we have $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2} \\le \\Vert H_i^{(R)}(\\cdot ,n)\\Vert ^{\\frac{1}{2}}_{1} \\le Cn^{\\frac{1}{4}}R^{-\\frac{1}{4}}.$ Pick $a > 0$ arbitrarily small.","We bound $\\Vert F_i^{(R)}\\Vert _{2}$ by $\\Vert F_i^{(R)}\\Vert _{2} &\\le \\sum _{n=1}^{\\lfloor R^{a} \\rfloor } \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2} +\\sum _{n=\\lfloor R^{a} \\rfloor +1}^{+\\infty } \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2}\\\\&\\underset{(\\ref {first_bound}),(\\ref {second_bound})}{\\le } C \\left(\\displaystyle \\sum _{n = 1}^{\\lfloor R^{a} \\rfloor }n^{\\frac{1}{4}}R^{-\\frac{1}{4}} + \\sum _{n = \\lfloor R^{a} \\rfloor + 1}^{+\\infty }n^{-\\frac{d}{4}+\\frac{\\varepsilon }{2}} \\right)\\\\&\\le CR^{-\\frac{1}{4}}R^{\\frac{5}{4}a} + C(R^{a})^{-\\frac{d}{4}+1+\\frac{\\varepsilon }{2}}.$ By taking $a$ small enough such that $\\frac{5}{4}a-\\frac{1}{4} < 0$ and since $-\\frac{d}{4}+1+\\frac{\\varepsilon }{2} < 0$ we have $\\Vert F_i^{(R)}\\Vert _{2} &\\le CR^{-c},$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We introduce for $R \\ge 0$ the event $\\nonumber F(R) = &\\hspace{10.0pt}\\left\\lbrace \\exists p,q \\ge 2 \\text{, }\\Vert X^1_p- X^2_q\\Vert \\le 1\\right\\rbrace \\\\ \\nonumber & \\cap \\left\\lbrace \\forall p,q \\ge 2 \\text{, }\\Vert X^1_p-X^2_q\\Vert \\le 1 \\Rightarrow \\Vert X^1_p-X^1_0\\Vert \\wedge \\Vert X^2_q - X^1_0\\Vert \\ge R \\right\\rbrace \\\\&\\cap \\left\\lbrace \\forall p,q \\ge 2 \\text{, }\\Vert X^1_p-X^2_q\\Vert \\le 1 \\Rightarrow \\Vert X^1_p-X^2_0\\Vert \\wedge \\Vert X^2_q - X^2_0\\Vert \\ge R \\right\\rbrace .$ Roughly speaking, on the event $F(R)$ the two walks almost intersect (actually get at distance less than 1) but do not intersect at a point at distance less than $R$ from $U_1$ or $U_2$ .","The last proof immediately gives the following corollary.","Corollary 4.4 There exists $K_0 > 0$ such that, for any $U_1,U_2 \\in \\mathbb {Z}^d$ and for any $K \\ge K_0$ , we have that $Q^{K}_{U_1,U_2}\\left(F(R)\\, | \\,\\text{}D^{\\otimes }=+\\infty \\right) \\le CR^{-c},$ where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Taking the same notations as in last proof we have: $\\displaystyle Q^{K}_{U_1,U_2}\\left(F(R)\\, | \\,\\text{}D^{\\otimes }=+\\infty \\right) &\\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{\\begin{array}{c}z \\in \\mathbb {Z}^d\\\\ \\Vert z-U_1\\Vert \\ge R \\\\ \\Vert z-U_2\\Vert \\ge R\\end{array}}^{} F_1^{(R)}(z)F_2^{(R)}(z + e)\\\\&\\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{\\begin{array}{c}z \\in \\mathbb {Z}^d\\end{array}}^{} F_1^{(R)}(z)F_2^{(R - 1)}(z + e).$ We conclude as in last proof by using Cauchy-Schwarz inequality.","The last corollary shows that there is little probability that the two walks only intersect on points which are far from the starting points.","In other words, with high probability, if the two walks do intersect, then it has to be close to one of the starting points.","We need a last proposition before moving on to the case where both walks are in the same environment.","We recall the notation $T^i_{R}$ denotes the first hitting time of the set $\\mathcal {H}^{+}(R)$ for the walk $X^i$ .","We introduce the notation: $R_0 \\max (X^{1}_0\\cdot \\vec{\\ell },X^{2}_0 \\cdot \\vec{\\ell }).$ For $R > 0$ we define the event $ A(R) {\\left\\lbrace \\begin{array}{ll}X_p^{1}\\cdot \\vec{\\ell } > X^1_0\\cdot \\vec{\\ell } \\quad & \\forall 0 < p \\le T^1_{R+R_0} + 2,\\\\X_q^{2}\\cdot \\vec{\\ell } > X^2_0\\cdot \\vec{\\ell } \\quad &\\forall 0 < q \\le T^2_{R+R_0} + 2,\\\\\\Vert X_p^1 - X_q^2\\Vert \\ge 2 \\quad &\\forall 1 < p \\le T^1_{R+R_0} + 2 \\,\\, \\textnormal {and} \\,\\, \\forall 1 < q \\le T^2_{R+R_0} + 2.\\end{array}\\right.","}$ Proposition 4.5 Let $R_0 = \\max (X_0^1 \\cdot \\vec{\\ell },X_0^2 \\cdot \\vec{\\ell }) $ and $R > 0$ .","There exists $\\Lambda > 0$ and $K_0 > 0$ such that, for $K > K_0$ , $U_1,U_2 \\in \\mathbb {Z}^d$ such that $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $\\Vert U_1-U_2\\Vert \\ge \\Lambda $ , $Q^{K}_{U_1,U_2}(A(R)) \\ge \\rho > 0,$ where the constant $\\rho $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","We have $Q^{K}_{U_1,U_2}(A(R)) \\ge cQ^{K}_{U_1,U_2}(A(R) \\, | \\, \\text{}D=+\\infty )$ with $c = \\mathbb {P}^{K}_{0}(D=+\\infty )^2$ .","Thus, under the law $Q^{K}_{U_1,U_2}( \\cdot \\, | \\, \\text{}D^{\\otimes }=+\\infty )$ the event $A(R)^{c}$ implies $ \\left\\lbrace \\exists \\, p,q \\in \\mathbb {N}, \\,\\, \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\right\\rbrace $ .","It follows from Proposition REF that one can bound $Q^{K}_{U_1,U_2}(A(R)^{c} \\, | \\, \\text{}D^{\\otimes }=+\\infty ) &\\le Q^K_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N}, \\,\\, \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right) \\\\ &\\le C\\Vert U_1 - U_2\\Vert ^{-c}.$ The result comes by choosing $\\Lambda $ large enough."],["Identical environments","We will try to exploit the results attained in the last part in order to deduce estimates for two walks in the same environment.","When the two walks are far enough they live in independent environments and so, by using the previous subsection, we will be able to show that they have a positive probability to not hit each other anymore.","If they intersect, then we wait for the walks to get far enough and we iterate this way.","First we state the analogous of Proposition REF .","We recall the definition of $A(R)$ in (REF ) and the notations $\\mathcal {E}_x, \\mathcal {E}_A$ in (REF ).","Proposition 4.6 There exists $\\Lambda > 0$ and $K_0 > 0$ such that, for any $K > K_0$ , $R > 0$ and $U_1,U_2 \\in \\mathbb {Z}^d$ with $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $||U_1-U_2|| \\ge \\Lambda $ , $\\mathbb {P}_{U_1,U_2}^{K}(A(R) \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty )\\ge \\rho ,$ the constant $\\rho $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","We only have to bound $\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c} \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty )$ .","Hence, using Lemma REF we only have to bound $\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c})$ .","Since under $A(R)$ the paths of the two walks stay at distance more than 2 we have $\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c}) = Q_{U_1,U_2}^{K}(A(R)^{c})$ .","Indeed we can write $\\mathbb {P}_{U_1,U_2}^{K}(A(R)) & = \\sum _{(x_i),(y_j)}^{} \\mathbf {E}\\bigg [P^{\\omega _K}_{U_1}\\bigg ((X^1_i) = (x_i)\\bigg )P^{\\omega _K}_{U_2}\\bigg ((X^2_j) = (y_j)\\bigg )\\bigg ],$ where the paths $(x_i),(y_j)$ are chosen under $A(R)$ .","We highlight the fact that in $P^{\\omega _K}_{U_1}$ the conductances in $\\mathcal {E}_{U_2}$ are not set to $K$ , while in $\\mathbb {P}^{K}_{U_1,U_2}$ they are.","Indeed $P^{\\omega _{K}}_{U_1}\\left((X^1_i) = (x_i)\\right)$ depends on the values $c^{*}(e)$ where $e \\in \\mathcal {E}_{(x_i)}$ , using the fact that $(x_i)$ and $(y_j)$ stay at distance more than 2 we have $\\mathcal {E}_{(x_i)} \\cap \\mathcal {E}_{U_2} = \\emptyset $ .","Thus modifying the value of $c^{*}(e)$ where $e \\in \\mathcal {E}_{U_2}$ does not modify the value of $P^{\\omega _{K}}_{U_1}\\left((X^1_i) = (x_i)\\right)$ .","The term $P^{\\omega _K}_{U_1}\\left((X^1_i) = (x_i)\\right)$ is measurable with respect to $ \\sigma (c^{*}(e) \\text{ ; } e\\in \\mathcal {E}_{(x_i)})$ and $P^{\\omega _K}_{U_2}((X^2_j) = (y_j))$ is measurable with respect to $ \\sigma (c^{*}(e) \\text{ ; } e\\in \\mathcal {E}_{(y_j)})$ .","The two sigma algebras are independent since for all $i,j$ we have $\\Vert x_i-y_j\\Vert \\ge 2$ .","We conclude that we have $\\mathbb {P}_{U_1,U_2}^{K}(A(R)) & = \\sum _{(x_i),(y_j)}^{} \\mathbf {E}\\left[P^{\\omega _K}_{U_1}\\left((X^1_i) = (x_i) \\right) \\right] \\mathbf {E} \\left[P^{\\omega _K}_{U_2}\\left((X^2_j) = (y_j)\\right)\\right]\\\\& = Q_{U_1,U_2}^{K}(A(R)).$ We conclude using Proposition REF .","We want to extend the last result to the case when $U_1$ and $U_2$ may be close.","Proposition 4.7 There exists $K_0 > 0$ such that for any $K > K_0$ and any $U_1,U_2 \\in \\mathbb {Z}^d$ with $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $R > 0$ : $\\mathbb {P}_{U_1,U_2}^{K}(A(R) \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty )\\ge \\rho ,$ where the constant $\\rho > 0$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","First with the same arguments as in last proposition we see that we only have to find a lower bound for $\\mathbb {P}_{U_1,U_2}^{K}(A(R))$ .","Fix $\\Lambda $ as in Proposition REF .","We may suppose $\\Vert U_1-U_2\\Vert \\le \\Lambda $ otherwise it is just applying Proposition REF .","We recall the notation $R_0 = \\min (U_1\\cdot \\vec{\\ell },U_2\\cdot \\vec{\\ell })$ .","We build paths that start form $U_1$ and $U_2$ in the following way: Fix $V_1$ and $V_2$ such that $\\Lambda \\le ||V_1-V_2|| < 2\\Lambda $ and $ R_0+5< \\lfloor V_1\\cdot \\vec{\\ell } \\rfloor ,\\lfloor V_2\\cdot \\vec{\\ell } \\rfloor < R_0+6$ .","Build two finite paths $(x_i)$ and $(y_j)$ starting from $U_1$ , $U_2$ respectively and ending at $V_1$ , $V_2$ , which are: Compatible with having joint regeneration points on $(V_1,V_2)$ .","Of length lower than $C\\Lambda $ where $C$ is a large enough constant.","Compatible with $A(R)$ , i.e.","on those paths the walks get at distance more than 2 after the first 2 steps and stay on the right side (in terms of inner product with $\\vec{\\ell }$ ) of their starting point.","It is always possible to find such paths, see the figure below for a visual explanation of why these paths exist.","Figure: On the left, the general construction of the paths described above, the distance between U 1 U_1 (resp U 2 U_2) and V 1 V_1 (resp.","V 2 V_2), in the direction ℓ →\\vec{\\ell }, is between 5 and 6 because of the lattice geometry.","On the right, a “worst case scenario” when the two starting points coincide and the bias is in the direction of the vector e 1 e_1.","This is why we allow, in the definition of A(R)A(R) (), that the walks might be close after one step (in orange).We have the following inequality: $\\mathbb {P}_{U_1,U_2}^{K}\\left(A(R)\\right) \\ge \\mathbb {P}_{U_1,U_2}^{K}\\left(\\begin{array}{c}(X^1_i) = (x_i),(X^2_j)=(y_j),(x_i) \\text{ and }(y_j)\\text{ are all }K\\text{-open points}\\\\A(R) \\circ \\theta _{T_{V_1},T_{V_2}}, \\mathcal {D}^{\\bullet } \\circ \\theta _{T_{V_1},T_{V_2}}\\end{array}\\right).$ Using Theorem REF last term can be rewritten as $&\\mathbb {P}_{U_1,U_2}^{K}\\left((X^1_i) = (x_i),(X^2_j)=(y_j),(x_i) \\text{ and }(y_j)\\text{ are all }K\\text{-open points}\\right)\\\\&\\times \\mathbb {P}_{V_1,V_2}^{K}\\left(A(R) ,\\mathcal {D}^{\\bullet } = +\\infty \\right).$ The left term of the product is bounded below by a constant which only depends on $\\lambda ,\\vec{\\ell },d,K$ and the right term is bounded by a $\\rho > 0$ using Proposition REF and Lemma REF .","It concludes the proof.","We show that under $\\mathbb {P}_0$ , on $A(R)$ and when the two trajectories reach a joint regeneration level before reaching level $R$ , then it is unlikely that they meet in the future.","We recall the definition of $\\mathbf {M}_N$ in (REF ).","Proposition 4.8 We recall the notation $R_0 = \\max (U_1\\cdot \\vec{\\ell },U_2\\cdot \\vec{\\ell })$ .","There exists $K_0 > 0$ such that for $K > K_0$ , for $U_1,U_2 \\in \\mathbb {Z}^d$ such that $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $R > 0$ we have $\\mathbb {P}_{U_1,U_2}^{K}\\left( A(R) \\cap \\mathrm {\\bf {M}}_{R+R_0} \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right) \\le CR^{-c},$ where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We want to use Corollary REF .","First notice that it suffices to bound $\\mathbb {P}_{U_1,U_2}^{K}\\big ( A(R) \\cap \\mathrm {\\bf {M}}_{R+R_0}\\cap \\mathcal {D}^{\\bullet } = +\\infty \\big )$ , indeed it follows from Lemma REF that $\\mathbb {P}_{U_1,U_2}^{K}\\big (\\mathcal {D}^{\\bullet } = +\\infty \\big )$ is bounded below by a constant $\\eta > 0$ which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Let us consider the set $\\mathcal {P}^{\\bullet }$ of all finite paths $((\\tilde{x}_i),(\\tilde{y}_j))$ for the enhanced walks $(\\tilde{X}^1_{\\cdot },\\tilde{X}^2_{\\cdot })$ (defined in Section REF ) such that $(x_i)$ and $(y_j)$ are at distance 1 at their ending points and always more than 2 (except at the starting point) otherwise and which are compatible with $\\lbrace \\mathcal {D}^{\\bullet } = +\\infty \\rbrace \\cap A(R)$ .","We then only have to bound $\\mathbb {P}^{K}_{U_1,U_2}\\left(\\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace \\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j) \\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\right).$ Under all paths considered, the environments each walk encounters are independent.","Then, the quantity above can be bounded in the following way $\\nonumber \\mathbb {P}^{K}_{U_1,U_2}\\Bigg (\\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace &\\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\Bigg ) \\\\ \\nonumber &= \\sum _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\mathbb {P}^{K}_{U_1,U_2}\\left( \\lbrace \\tilde{X}^1_i\\rbrace = \\lbrace \\tilde{x}_i\\rbrace ,\\lbrace \\tilde{X}^2_j\\rbrace = \\lbrace \\tilde{y}_j\\rbrace \\right)\\\\&\\nonumber \\le C \\sum _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} Q^{K}_{U_1,U_2}\\left( \\lbrace \\tilde{X}^1_i\\rbrace = \\lbrace \\tilde{x}_i\\rbrace ,\\lbrace \\tilde{X}^2_j\\rbrace = \\lbrace \\tilde{y}_j\\rbrace \\right)\\\\& = C Q^{K}_{U_1,U_2}\\left(\\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace \\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\right).$ Where the inequality is using the fact that the term $P^{\\omega _K}_{U_1,U_2}\\big (\\lbrace \\tilde{X}^1_i\\rbrace = \\lbrace \\tilde{x}_i\\rbrace \\big )$ is measurable with respect to $\\sigma \\big (c^{*}(e) \\text{ ; }e \\in \\mathcal {E}_{\\lbrace \\tilde{x}_i\\rbrace _{i\\ge 0}\\backslash \\lbrace \\tilde{x}_{\\infty }\\rbrace }\\big )$ and the term $P^{\\omega _K}_{U_1,U_2}\\big (\\lbrace \\tilde{X}^2_j\\rbrace = \\lbrace \\tilde{y}_j\\rbrace \\big )$ is measurable with respect to $\\sigma \\big (c^{*}(e) \\text{ ; }e \\in \\mathcal {E}_{\\lbrace \\tilde{y}_j\\rbrace _{j\\ge 0}\\backslash \\lbrace \\tilde{y}_{\\infty }\\rbrace }\\big )$ where $\\tilde{x}_{\\infty },\\tilde{y}_{\\infty }$ denotes the ending point of the paths.","In most cases, we simply use the fact that the two set of edges considered are disjoint and the independence of the conductances would give an equality.","We only have an inequality because when $\\Vert U_1 - U_2 \\Vert _{1} \\le 1$ and $\\vec{\\ell } = e_1$ , we might be forced to take the first step in the direction $e_1$ by the definition of $A(R)$ .","In this last case we just lose a constant because there is a positive chance that the second site visited is $K$ -open under both $\\mathbb {P}^{K}$ and $Q^{K}$ .","Note that we could also have an inequality in the opposite direction with a different constant if needed.","See the right of Figure REF for a visualisation of the “worst case scenario”.","Let us introduce the notation: $\\mathbf {TRAJ} = \\Big \\lbrace \\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace \\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\Big \\rbrace .$ We can rewrite (REF ) as $\\displaystyle & Q^{K}_{U_1,U_2}\\left(\\mathbf {TRAJ},D^{\\otimes } < +\\infty \\right) \\\\& + Q^{K}_{U_1,U_2}\\left(\\mathbf {TRAJ},D^{\\otimes }=+\\infty \\right).$ Using the fact that $\\mathbf {TRAJ}$ is compatible with $\\mathcal {D}^{\\bullet } = +\\infty $ and that the trajectories go beyond level $R+R_0$ , if $D^1 < +\\infty $ (resp.","$D^2 < +\\infty )$ then we have $X^1_{\\tau _1}\\cdot \\vec{\\ell } > R + R_0$ (resp.","$X^2_{\\tau _1}\\cdot \\vec{\\ell } > R + R_0$ ).","Using , we dominate the first term by $CR^{-c}$ .","It remains to bound the term (REF ).","To do so, we define $C_i, i=1, 2$ : the walk $X^i$ regenerates between the levels $R_0 + R/2$ and $R_0 + R$ .","Using once again of we have $ Q^{K}_{U_1,U_2}\\left(C_1^{c} \\cup C_2^{c}\\text{ | }D^{\\otimes }=+\\infty \\right) \\le CR^{-c}.$ On the event $\\displaystyle C_1\\cap C_2 \\cap \\mathbf {TRAJ}$ , all points $(x,y)$ in the two trajectories (except with the starting points) which are at distance less than $R/2$ from $\\lbrace U_1,U_2\\rbrace $ we have $\\Vert x-y\\Vert _1 \\ge 2$ .","Thus $\\displaystyle C_1\\cap C_2 \\cap \\mathbf {TRAJ}$ implies $F(R/2)$ where $F(R/2)$ is defined in (REF ).","Using Corollary REF we conclude that $ Q^{K}_{U_1,U_2}\\left(C_1 \\cap C_2 \\cap \\mathbf {TRAJ} \\text{ | }D^{\\otimes }=+\\infty \\right) \\le Q^{K}_{U_1,U_2}\\left(F(R/2)\\right) \\le CR^{-c}.$ Finally we dominate (REF ) by $& Q^{K}_{U_1,U_2}\\left(C_1^{c} \\cup C_2^{c}\\text{ | }D^{\\otimes }=+\\infty \\right)+ Q^{K}_{U_1,U_2}\\left(C_1 \\cap C_2 \\cap \\mathbf {TRAJ} \\text{ | }D^{\\otimes }=+\\infty \\right),$ and we conclude with the estimates (REF ) and (REF )."],["Asymptotic separation proved","We have everything needed to prove the asymptotic separation stated in Proposition REF .","We fix $\\psi \\in (0, 1)$ arbitrarily small.","We also introduce $\\mathbf {d} = N^{1-\\psi }$ .","We set $\\gamma _1 = \\mathcal {L}_1$ and for $ k \\ge 2 $ we call $\\gamma _k$ the first joint regeneration level in $\\mathcal {H}^{+}(\\gamma _{k-1}+2\\mathbf {d})$ .","Recall the notation $A(R)$ for $R > 0$ defined in (REF ).","We introduce the event $S_k$ which is just event $A(\\mathbf {d})$ where the walks start respectively from $X_{T^1_{\\gamma _k}}$ and $X_{T^2_{\\gamma _k}}$ .","We set: $ \\forall k \\ge 0, \\text{ }S_k = A(\\mathbf {d}) \\circ \\theta _{T_{\\gamma _k}} \\text{ and }S(N) = \\bigcup _{k=1}^{\\lfloor N^{\\psi }/4 \\rfloor } S_k.$ The idea of the proof is the following: each time our walks reach a level $\\gamma _k$ we check whether $S_k$ occurs or not.","If it does, using Proposition REF , with high probability the walks will never meet after that that.","If they do meet, we wait for the walks to reach $\\gamma _{k+1}$ and repeat the procedure.","Using the independence property of joint regeneration levels (see Theorem REF ) we expect the first success of this process to happen very quickly (See Figure REF ).","Figure: Successive attempts for events S k S_k.Proposition 4.9 There exists $K_0 > 0$ such that for any $K > K_0$ , there exists $\\rho > 0$ such that for any $k \\ge 1$ we have $\\displaystyle \\mathbb {P}_{0}(S_k \\, | \\, S_0^{c},\\cdots ,S_{k-1}^{c} ) \\ge \\rho ,$ where $\\rho $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","This is a direct application of Theorem REF and Proposition REF .","Indeed, using Theorem REF we have $\\mathbb {P}_0(S_k,S_{k-1}^{c},\\cdots ,S_0^{c}) &=\\sum _{U_1,U_2}^{} \\mathbb {P}_{0}(S_{k-1}^{c},\\cdots ,S_0^{c},X^{1}_{T_{\\gamma _k}} = U_1,X^{2}_{T_{\\gamma _k}} = U_2)\\mathbb {P}_{U_1,U_2}^{K}(A(w) \\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty ).$ Applying Proposition REF gives the result.","We introduce $ \\textbf {Reg}(N) = \\bigcap _{k=1}^{N^2}\\lbrace \\mathcal {L}_{k} - \\mathcal {L}_{k-1} \\le \\mathbf {d}\\rbrace .$ which is the event under which the $N^2$ first joint regeneration levels are close enough.","We also recall the definition of $\\mathbf {M}_N$ in (REF ).","We have the following proposition.","Proposition 4.10 There exists $K_0 > 0$ such that for any $K > K_0$ and any $0 \\le k \\le \\lfloor N^{\\psi }/4 \\rfloor $ , we have the following inequality: $\\mathbb {P}_0(\\textnormal {\\textbf {Reg}}(N)\\cap \\mathrm {\\bf {M}}_N \\cap S_k) \\le CN^{-c(1-\\psi )}$ , where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We sum over all possible values for $(X^1_{T_{\\gamma _k}},X^2_{T_{\\gamma _k}})$ on the event $\\mathbf {Reg}(N)$ $\\mathbb {P}_0(\\textbf {Reg}(N)\\cap \\mathrm {\\bf {M}}_N \\cap S_k) &= \\sum _{U_1,U_2}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2,\\textbf {Reg}(N)\\cap \\mathrm {\\bf {M}}_N \\cap S_k\\right) \\\\&\\le \\sum _{\\begin{array}{c}U_1,U_2\\\\\\text{compatible with }\\mathbf {Reg}(N)\\end{array}}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2, \\mathrm {\\bf {M}}_N \\cap S_k\\right).$ Since $U_1,U_2$ where chosen on event $\\textbf {Reg}(N)$ we have $U_1\\cdot \\vec{\\ell } < 3N/4$ and $U_2\\cdot \\vec{\\ell } < 3N/4$ .","The worst case clearly happens for $k = \\lfloor N^{\\psi }/4 \\rfloor $ .","We claim that for every $0 \\le j \\le \\lfloor N^{\\psi }/4 \\rfloor $ on the event $\\mathbf {Reg}(N)$ , we have $\\gamma _{j+1} \\le \\gamma _j + 3\\mathbf {d}$ .","Indeed, between $\\gamma _j$ and $\\gamma _{j+1}$ there are at most $C \\mathbf {d}$ joint regeneration levels, for some constant depending on the dimension $d$ and $\\vec{\\ell }$ .","The claim follows from the fact that, for $0 \\le j \\le \\lfloor N^{\\psi }/4 \\rfloor $ , we are choosing all the $\\gamma _{j}$ between the first $N^2$ regeneration levels.","Then, from the claim we deduce that for $k = \\lfloor N^{\\psi }/4 \\rfloor $ we have $\\gamma _k \\le 3\\mathbf {d}N^{\\psi }/4 \\le 3N/4$ .","We recall $S_k = A(\\mathbf {d}) \\circ \\theta _{T_{\\gamma _k}}$ and since $\\gamma _k \\le 3N/4$ on $\\mathbf {Reg}(N)$ , $\\mathrm {\\bf {M}}_N$ implies $\\mathrm {\\bf {M}}_N \\circ \\theta _{T_{\\gamma _k}}$ .","We deduce that the last sum is bounded by $\\sum _{\\begin{array}{c}U_1,U_2\\\\\\text{compatible with }\\mathbf {Reg}(N)\\end{array}}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2, \\mathrm {\\bf {M}}_N \\circ \\theta _{T_{\\gamma _k}}, A(\\mathbf {d}) \\circ \\theta _{T_{\\gamma _k}}\\right).$ Using Theorem REF this sum can be rewritten as $ \\sum _{U_1,U_2}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2\\right)\\mathbb {P}_{U_1,U_2}^{K}\\left(\\mathrm {\\bf {M}}_N\\cap A(\\mathbf {d})\\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right).$ We recall $R_0 = \\max (U_1 \\cdot \\vec{\\ell },U_2 \\cdot \\vec{\\ell })$ .","Now, $\\mathbb {P}_{U_1,U_2}^{K}$ -almost surely $\\mathrm {\\bf {M}}_{N}$ implies $\\mathbf {M}_{R_0+\\mathbf {d}}$ .","Indeed, $\\displaystyle U_1\\cdot \\vec{\\ell } < 3N/4$ and $\\displaystyle U_2\\cdot \\vec{\\ell } < 3N/4$ , hence $\\mathbb {P}^{K}_{U_1,U_2}$ -almost surely $\\displaystyle R_0 + \\mathbf {d} \\le 3N/4 + N^{1-\\psi } \\le N$ which gives $\\mathbf {M}_N$ implies $\\mathbf {M}_{R_0+\\mathbf {d}}$ .","It follows that (REF ) is bounded by $\\sum _{U_1,U_2}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2\\right)\\mathbb {P}_{U_1,U_2}^{K}\\left(\\mathrm {\\bf {M}}_{R_0+\\mathbf {d}}\\cap A(\\mathbf {d})\\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right).$ We conclude using Proposition REF .","Finally, we prove the asymptotic separation.","[Proof of Proposition REF ] We recall the definition of $S(N)$ in (REF ) and of $\\mathbf {Reg}(N)$ in (REF ).","Using Proposition REF we have $\\mathbb {P}_{0}\\left(S(N)^c\\right) \\le (1-\\rho )^{N^{\\psi }} $ and by Proposition REF for any $M > 0$ we can take $K$ large enough such that we have $\\mathbb {P}_{0}(\\mathbf {Reg}(N)^{c}) \\le CN^{-M}$ for any $M > 0$ where the constants $\\rho ,C>0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Hence, $\\mathbb {P}_0(\\mathrm {\\bf {M}}_N) &= \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right) + \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N \\cap (\\mathbf {Reg}(N)^c \\cup S(N)^c)\\right) \\\\ &\\le \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right) + CN^{-M} + (1-\\rho )^{N^{\\psi }}.$ We only need to bound $\\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right)$ , by the definition of $S(N)$ and union bound we get $\\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right) &= \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N \\cap \\mathbf {Reg}(N) \\cap \\bigg ( \\bigcup _{k=1}^{\\lfloor N^{\\psi }/4 \\rfloor } S_k\\bigg )\\right)\\\\& \\le \\sum _{k=1}^{\\lfloor N^{\\psi }/4 \\rfloor } \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S_k\\right) \\\\&\\le CN^{\\psi }N^{-c(1-\\psi )},$ where the last step is a consequence of Proposition REF .","We conclude by choosing $\\psi $ small enough, we notice that this choice depends on $c$ which only depend on $\\lambda ,\\vec{\\ell },d,K$ ."],["For the trajectory","In this part we aim to prove that $\\mathbb {E}[F(Z_n^1)F(Z_n^2)] - \\mathbb {E}[F(Z_n^1)]\\mathbb {E}[F(Z_n^2)]$ converges polynomially fast to 0 as $n \\rightarrow +\\infty $ .","Where $F$ is a bounded $L$ -Lipschitz function from $\\mathcal {C}([0,1],\\mathbb {R}^{d}) \\rightarrow \\mathbb {R}$ and $Z^{i}_n(t) = \\displaystyle \\frac{X^{i}_{\\tau _{\\lfloor nt \\rfloor }} - nvt}{n^{1/2}}$ for $t \\in [0,1]$ .","Actually, with a slight abuse of the notation, in all this section we will refer with $Z^{i}_n(t)$ to the polygonal interpolant of $\\displaystyle \\frac{X^{i}_{\\tau _{\\lfloor nt \\rfloor }} - nvt}{n^{1/2}}$ , which is needed for technical reasons.","Throughout this proof we assume that the function $F$ is bounded by 1 and has Lipschitz constant $L=1$ as an extension to other constants follows with the same reasoning.","We take $t \\in [0,1]$ as it is the same for $t \\in [0,T]$ .","We recall the notation: $ \\chi ^{i}_k = \\inf \\left\\lbrace m \\ge 0 \\text{ ; } \\lbrace X^i_{j}-X^i_{\\tau ^{i}_k}\\rbrace _{\\tau ^{i}_k \\le j \\le \\tau ^{i}_{k+1}} \\subset \\mathcal {B}(m,m^{\\alpha })\\right\\rbrace .$ where we recall that $\\alpha $ is a constant chosen greater than $d+3$ .","Let us state the main theorem of this part.","Theorem 5.1 There exists $K > 0$ such that the following inequality holds: $\\Big |\\mathbb {E}[F(Z_n^1)F(Z_n^2)] - \\mathbb {E}[F(Z_n^1)]\\mathbb {E}[F(Z_n^2)]\\Big | \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We postpone the proof to the end of the section but let us give some intuition.","Roughly speaking, on the event $\\mathbf {M}_{n^\\theta }$ (see (REF )) and after few regeneration times, the two walks do not intersect anymore.","It follows that they should behave like two regenerating random walks in two independent environment.","This is the point of Propositions REF -REF which are the main steps to prove Theorem REF .","The rest of the proof basically consists in proving that the beginning of the trajectories does not matter in the re-scaled trajectory $Z_n$ .","Fix $0 < \\theta < 1/2$ .","Let us start with an easy estimate which will be useful later on.","Proposition 5.2 The following inequality holds: $\\left|\\mathbb {E}_0\\left[F(Z_n)\\right] - \\mathbb {E}_0\\left[F(Z_n)\\text{ }|\\text{ }D=+\\infty \\right]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We introduce the scaled position of the walk starting after the first regeneration time and re-centered $\\bar{Z}_n(t) = Z_n(t+1/n) - Z_n(1/n).$ We define $B= \\big \\lbrace \\forall k \\in \\lbrace 0,\\cdots ,n+1\\rbrace \\text{ ; }\\chi _k \\le n^{\\varepsilon }\\big \\rbrace $ where $\\varepsilon > 0$ is a constant that we choose arbitrarily small.","Then, under $B$ , we have $\\Vert \\bar{Z}_n-Z_n\\Vert _{\\infty } \\le Cn^{-\\frac{1}{2}}n^{\\alpha \\varepsilon } \\le Cn^{-c},$ where $c = \\alpha \\varepsilon - \\frac{1}{2} < 0$ when choosing $\\varepsilon $ small enough.","It follows that under $B$ and using the fact that $F$ is 1-Lipschitz we get $|F(Z_n)-F(\\bar{Z}_n)| \\le Cn^{-c},$ By , for $K$ large enough we have $\\mathbb {P}_0(B_1^{c} \\cup B_2^{c}) \\le Cn^{-c}$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Using this inequality and the fact that $F$ is bounded by 1 we get that $\\left|\\mathbb {E}_0[F(Z_n)]-\\mathbb {E}_0[F(\\bar{Z}_n)]\\right| \\le Cn^{-c}.$ We conclude the proof observing that using we have that: $\\mathbb {E}_0[F(\\bar{Z}_n)] = \\mathbb {E}_0[F(Z_n)\\text{ | }D=+\\infty ].$ Let us introduce some events and estimates on them: We recall $\\mathrm {\\bf {M}}_{n^\\theta } = \\bigg \\lbrace \\displaystyle \\inf _{\\begin{array}{c}x \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^1\\rbrace \\\\y \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\ge 2 \\bigg \\rbrace .$ Using Proposition REF we have $\\mathbb {P}_{0}(\\mathrm {\\bf {M}}_{n^\\theta }^{c}) \\le Cn^{-c}$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We define $\\mathcal {R}_0$ the first joint regeneration level in $ \\mathcal {H}^{+}_{n^{\\theta }}$ .","For $i \\in \\lbrace 1,2\\rbrace $ we introduce $\\rho _0^{i} = T^{i}_{\\mathcal {R}_0}$ .","In particular, $T^i_{\\mathcal {R}_0}$ is a regeneration time for the walk $X^i$ .","We call $B \\lbrace \\mathcal {R}_0 \\le 2n^{\\theta }\\rbrace $ .","Using , there exists $K > 0$ such that $\\mathbb {P}_{0}(B^{c}) \\le Cn^{-c}$ , where $C,c$ depends on $\\theta , \\lambda ,d,\\vec{\\ell },K$ .","Indeed, under the event $\\lbrace \\forall i \\in \\lbrace 1,\\cdots ,dn^{\\theta }\\rbrace , \\text{ }\\mathcal {L}_{k+1}-\\mathcal {L}_{k} \\le n^{\\theta }\\rbrace $ with the proviso $\\mathcal {L}_{0} = 0$ (and using the fact that we always have $\\mathcal {L}_{k+1}-\\mathcal {L}_k \\ge \\frac{2}{\\sqrt{d}}$ ) we have $\\mathcal {R}_0 \\le 2n^{\\theta }$ .","The following inequality follows: $ \\mathbb {P}_{0}(B^{c}) & \\le \\mathbb {P}_{0}\\bigg (\\exists k \\in \\lbrace 1,\\cdots ,dn^{\\theta }\\rbrace , \\text{ }\\mathcal {L}_{k+1}-\\mathcal {L}_{k} \\ge n^{\\theta }\\bigg ).$ Using Proposition REF for $K$ large enough the last term is bounded above, for $M$ arbitrarily big, by $Cn^{\\theta }n^{-M\\theta }.$ We call $\\rho ^i_{n}$ the $n^{\\text{th}}$ (not joint) regeneration time after $\\rho _{0}^{i}$ for $X^{i}$ .","We recall the notation (REF ).","Then define $F_i = \\big \\lbrace \\forall k \\in \\lbrace 1,\\cdots ,n^2\\rbrace \\text{, }\\chi ^{i}_k \\le n^{\\theta }\\big \\rbrace $ .","Using , there exists $K > 0$ such that $\\mathbb {P}(F^{c}_{i}) \\le Cn^{-M}$ , where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","Define $E:= \\mathrm {\\bf {M}}_{n^\\theta } \\cap B \\cap F_1 \\cap F_2$ .","The estimates above can be summarized in the following proposition.","Proposition 5.3 There exists $K > 0$ such that we have: $\\mathbb {P}_{0}(E^{c}) \\le Cn^{-c},$ where the constants $C,c >0$ only depend on $\\theta , \\lambda ,\\vec{\\ell },d,K$ .","Let us introduce $\\tilde{X}^i_{\\cdot } = X^{i}_{\\rho _{0}^{i}+\\cdot } - X^{i}_{\\rho ^i_0}$ which is just the walk $X^i$ shifted by $\\rho _0^{i}$ and recentered, call $\\tilde{Z}^{i}_{n}(t) = (\\tilde{X}^{i}_{\\tau _{\\lfloor nt \\rfloor }} - nvt)/n^{1/2}$ .","Let us show that under $E$ , the trajectories $\\tilde{Z}^{i}_n$ and $Z_n^{i}$ stay close.","Proposition 5.4 The following inequality holds almost surely under the event $E$ $\\forall i \\in \\lbrace 1,2\\rbrace , \\text{ } \\sup _{t \\in [0, 1]}||\\tilde{Z}^{i}_n(t) - Z^{i}_n(t)||_{\\infty } \\le Cn^{-c},$ where $C,c$ depend on $\\lambda ,\\vec{\\ell }, d, \\theta $ .","Fix $i \\in \\lbrace 1,2\\rbrace $ and $k \\in \\lbrace 0,\\cdots ,n\\rbrace $ , let us show that $X^{i}_{\\tau _k}$ is close to $X^{i}_{\\rho _k}$ after the proper rescaling.","We have, $\\mathcal {R}_0 \\le 2n^{\\theta }$ and since $\\rho _{0}^{i}$ is a regeneration time, it can be written as $\\tau ^{i}_{k_0}$ with $k_0 \\in \\lbrace 0,\\cdots ,d\\lfloor n^{2\\theta } \\rfloor \\rbrace $ .","Indeed, at each regeneration time the walk has increased at least $2/\\sqrt{d}$ in the direction $\\vec{\\ell }$ .","Using the fact that under $E$ , $\\chi ^{i}_k \\le n^{\\theta }$ for all $k \\in \\lbrace 0,\\cdots ,n^2\\rbrace $ we have $\\Vert X^{i}_{\\rho _{k}} - X^i_{\\rho _{0}}- X^{i}_{\\tau _k}\\Vert _\\infty =\\Vert X^{i}_{\\tau _{k+k_0}} - X^{i}_{\\tau _{k_0}} - X^{i}_{\\tau _k}\\Vert _\\infty & \\le Ck_0n^{\\alpha \\theta }\\\\& \\le Cn^{(\\alpha +2)\\theta },$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d$ .","The first inequality comes from the fact that the diameter of $\\mathcal {B}(m,m^{\\alpha })$ is bounded by $Cm^{\\alpha }$ .","Then, we only have to chose $\\theta $ such that $(\\alpha + 2)\\theta \\le 1/4$ to conclude, indeed for all $k \\in \\lbrace 1, \\cdots , n\\rbrace $ : $\\Vert Z^{i}_n(k/n) - \\tilde{Z}^{i}_n(k/n)\\Vert _\\infty &= \\bigg \\Vert \\frac{X^{i}_{\\tau _k} +X^{i}_{\\rho _0} - {X}_{\\rho _k}^{i}}{n^{1/2}}\\bigg \\Vert _\\infty \\le Cn^{-1/4},$ we conclude by taking $c = 1/4$ .","We give the following corollary of the last proposition.","Corollary 5.5 The following inequality holds: $\\left|\\mathbb {E}_0\\left[F(Z^1_n)F(Z^2_n)\\right] - \\mathbb {E}_0\\left[F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","It follows from last Proposition REF , Proposition REF and the fact that $F$ is a bounded Lipschitz function.","We introduce $I = \\bigg \\lbrace \\displaystyle \\inf _{\\begin{array}{c}p,q \\ge 0\\end{array}}\\Vert X^1_{ p}-X^2_{q}\\Vert _{1}\\ge 2 \\bigg \\rbrace $ the event that the two walks never “see” the same edge.","The following proposition is crucial and shows that when shifted and non-intersecting anymore, we can approximate the behavior of our walks as if they lived in independent environments.","We also, highlight the fact that the following proposition gives an inequality.","One can prove it is an equality with an error term of order $Cn^{-c}$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Proposition 5.6 For any bounded positive function $f$ measurable with respect to $\\sigma \\left(\\big \\lbrace X^1_0,X^2_0,(X^1_{p},Z^1_{p}),(X^2_{q},Z^2_{q}) \\text{ }|\\text{ }p,q \\ge 1 \\big \\rbrace \\cup \\big \\lbrace c^{*}(e) \\text{ } |\\text{ }e \\in \\mathcal {E}_{(X^1_n)_{n\\ge 0}} \\cup \\mathcal {E}_{(X^2_n)_{n\\ge 0}} \\backslash (\\mathcal {E}_{X^1_0}\\cup \\mathcal {E}_{X^2_0})\\big \\rbrace \\right),$ we have the following inequality: $\\mathbb {E}_0[f \\circ \\theta _{\\rho _0} {1}_{I}\\circ \\theta _{\\rho _0}] \\le \\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }I, D^{\\otimes } = +\\infty \\bigg ]\\bigg ].$ Remark 5.7 Looking at this inequality one may be surprised.","Indeed, on the left term $\\rho _0$ is using the notion of joint regeneration levels while on the right term, $\\lbrace D^{\\otimes } = \\infty \\rbrace = \\lbrace D^1=+\\infty ,D^2=+\\infty \\rbrace $ refers to the classical notion of regeneration time for a single walk.","However, thinking about it carefully, we observe that under event $I$ , the event $\\lbrace \\mathcal {D}^{\\bullet } = +\\infty \\rbrace $ coincide with $\\lbrace D^{\\otimes } = + \\infty \\rbrace $ .","First using Theorem REF we can write $\\mathbb {E}_0\\left[f \\circ \\theta _{\\rho _0} {1}_{I} \\circ \\theta _{\\rho _0}\\right] = \\mathbb {E}_{0}\\left[\\mathbb {E}^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left[f{1}_{I} \\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right]\\right].$ Indeed, we can first restrict to functions $f$ of type ${1}_{X^1 = (a_0,\\cdots ,a_k)}{1}_{X^2 = (b_0,\\cdots ,b_k)}g$ where $g$ is measurable with respect to $\\sigma (c^{*}(e) \\text{ }|\\text{ } e \\in \\mathcal {R}_{X^1_0} \\cup \\mathcal {R}_{X^2_0} \\backslash (\\mathcal {E}_{X^1_0 \\cup X^2_0}))$ .","In addition, ${1}_{X^1 = (a_0,\\cdots ,a_k)}{1}_{X^2 = (b_0,\\cdots ,b_k)}{1}_{I}$ is measurable with respect to $\\sigma ((X_0^1,X_0^2),(\\tilde{X}^1_n,\\tilde{X}^2_n)_{n \\ge 1})$ .","Then for the general case we use the fact that $f$ can be written as a limit of an increasing sequence of functions which are a linear combination of the above type and a monotone convergence theorem.","Now fix $a,b \\in \\mathbb {Z}^d$ such that $\\mathbb {P}_{0}\\left(X^1_{\\rho ^1_0}=a,X^2_{\\rho ^2_0} = b\\right) > 0$ .","First notice that $\\mathbb {P}^{K}_{a,b}$ -almost surely, under $I$ the event $\\mathcal {D}^{\\bullet } =+\\infty $ coincide with $\\lbrace D^{\\otimes }=+\\infty \\rbrace $ .","Moreover, for any bounded function $g(\\cdot )$ measurable with respect to $\\sigma \\left(\\left\\lbrace X^1_0,X^2_0,(X^1_{p},Z^1_{p}),(X^2_{q},Z^2_{q}) \\text{ }|\\text{ }p,q \\ge 1 \\right\\rbrace \\cup \\left\\lbrace c^{*}(e) \\text{ } |\\text{ }e \\in \\mathcal {E}_{(X^1_n)_{n\\ge 0}} \\cup \\mathcal {E}_{(X^2_n)_{n\\ge 0}} \\backslash (\\mathcal {E}_{X^1_0}\\cup \\mathcal {E}_{X^2_0})\\right\\rbrace \\right),$ we have that $\\mathbb {E}^K_{a,b}\\left[g{1}_{I}\\right] = \\mathbb {E}^{Q^{K}_{a,b}}\\left[g{1}_{I}\\right].$ Indeed, it follows by restricting to simple functions with disjoint set of edges for both walks and using the fact $g{1}_{I}$ can be written as an increasing limit of linear combination of functions of this type.","In particular, we have $\\mathbb {E}^K_{a,b}\\left[f{1}_{I}{1}_{D^1=+\\infty }{1}_{D^2=+\\infty }\\right] = \\mathbb {E}^{Q^{K}_{a,b}}\\left[f{1}_{I}{1}_{D^1=+\\infty }{1}_{D^2=+\\infty }\\right].$ We conclude the proof using the fact that $\\mathbb {P}_{a,b}^{K}\\left(I,D^{\\otimes }=+\\infty \\right)\\le \\mathbb {P}_{a,b}^{K}\\left(\\mathcal {D}^{\\bullet } =+\\infty \\right).$ The next lemma will enable us to erase the conditioning with the event $I$ .","Lemma 5.8 We have the following inequality: $\\mathbb {E}_{0}\\left[Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c} \\text{ }|\\text{ }D^{\\otimes }=+\\infty \\right)\\right] \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","By we have $Q^K_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(D^{\\otimes }=+\\infty \\right) = \\mathbb {P}_0(D=+\\infty )^2 > \\eta > 0$ where $\\eta $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","Hence we only have to dominate $\\mathbb {E}_{0}\\left[Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c},D^{\\otimes }=+\\infty \\right)\\right].$ By the same argument as in last proof, it is equal to $\\mathbb {E}_{0}\\left[\\mathbb {P}^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c},\\mathcal {D}^{\\bullet } =+\\infty \\right)\\right].$ Using Lemma REF , last term is bounded above by $C\\mathbb {E}_{0}\\left[\\mathbb {P}^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c}\\text{ | }\\mathcal {D}^{\\bullet } =+\\infty \\right)\\right],$ where $C > 0$ is a constant which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","By Proposition REF , this expectation is equal to $\\mathbb {P}_{0}\\left(I^{c} \\circ \\theta _{\\rho _0}\\right) \\le \\mathbb {P}_0\\bigg (\\bigg \\lbrace \\displaystyle \\inf _{\\begin{array}{c}x \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^1\\rbrace \\\\y \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\le 1 \\bigg \\rbrace \\bigg ) \\le Cn^{-c},$ where last inequality comes by using Proposition REF .","Proposition 5.9 For any bounded positive function $f$ we have the following inequality: $&\\left|\\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ } I,D^{\\otimes }=+\\infty \\bigg ]\\bigg ] - \\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }D^{\\otimes }=+\\infty \\bigg ]\\bigg ]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","We can rewrite the left term $\\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }D^{\\otimes }=+\\infty \\bigg ]\\bigg ]$ as $& \\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }I,D^{\\otimes }=+\\infty \\bigg ]Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\bigg (I\\text{ | }D^{\\otimes }=+\\infty \\bigg )\\bigg ] +\\\\&\\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }I^{c},D^{\\otimes }=+\\infty \\bigg ]Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\bigg (I^{c}\\text{ | }D^{\\otimes }=+\\infty \\bigg )\\bigg ].$ Using the fact that $f$ is bounded, it follows that the term in the statement of the proposition is bounded above by $C\\mathbb {E}_{0}\\bigg [Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\bigg (I^{c}\\text{ | }D^{\\otimes }=+\\infty \\bigg )\\bigg ].$ We conclude using Proposition REF .","Now we can conclude with the proof of Theorem REF .","[Proof of Theorem REF ] We recall that we want to dominate $\\mathbb {E}_0\\left[F(Z^1_n)F(Z^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right].$ First, using Corollary REF we dominate this by $Cn^{-c}+\\mathbb {E}_0\\left[F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right].$ Thanks to the fact that $\\mathbb {P}_{0}(I^{c} \\circ \\theta _{\\rho _0}) \\le Cn^{-c}$ by Proposition REF and the fact that $F$ is bounded we dominate it by $Cn^{-c}+\\mathbb {E}_0\\left[{1}_{I} \\circ \\theta _{\\rho _0} F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right].$ Let us bound above the term $\\mathbb {E}_0\\left[{1}_{I} \\circ \\theta _{\\rho _0} F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right].$ Let us introduce the function $f = F\\left(Z^1_n-Z^1_n(0)\\right)F\\left(Z^2_n-Z^2_n(0)\\right)$ , then the last term can be rewritten as $\\mathbb {E}_0\\left[{1}_{I} \\circ \\theta _{\\rho _0} f\\circ \\theta _{\\rho _0} \\right].$ Furthermore, we observe that $f$ satisfies the assumptions of Proposition REF and Proposition REF , hence last term is bounded above by $Cn^{-c} + \\mathbb {E}_0\\bigg [\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\left[f\\text{ | }D^{\\otimes }=+\\infty \\right]\\bigg ].$ Using and the form of the function $f$ we can rewrite $\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\left[F\\left(Z^1_{n}-Z^1_n(0)\\right)F\\left(Z^2_{n}-Z^2_n(0)\\right)\\text{ | }D^{\\otimes }=+\\infty \\right],$ as $\\mathbb {E}^K_0\\left[F(Z_n) \\text{ | }D=+\\infty \\right]^2.$ To summarise we have dominated the term $\\mathbb {E}_0\\left[F(Z^1_n)F(Z^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right],$ by $Cn^{-c} + \\mathbb {E}^K_0\\left[F(Z_n) \\text{ | }D=+\\infty \\right]^2 - \\mathbb {E}_0\\left[F(Z_n)\\right]^2.$ We conclude by using the fact that $F$ is bounded and Proposition REF ."],["For the clock process","We recall the notation $S^*_n(t) =\\displaystyle \\frac{\\tau _{{tn} + 1} - \\tau _1}{\\mathrm {Inv}(n)}$ where $\\mathrm {Inv}(n) = \\inf \\lbrace t \\ge 0 \\text{ ; }\\mathbf {P}[c^{*} > t] \\le 1/n\\rbrace $ .","In this part we aim to prove the following main result.","Proposition 5.10 There exists $K > 0$ such that the following inequality holds: $|\\mathbb {E}[F(S^{*1}_n)F(S^{*2}_n)] - \\mathbb {E}[F(S^{*1}_n)]\\mathbb {E}[F(S^{*2}_n)]| \\le Cn^{-c},$ where $F$ is as in (REF ), $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Before proving this result we will need to state and prove Lemma REF .","Let us also admit the following lemma which is crucial in the proof of Lemma REF .","Lemma 5.11 For all $\\eta \\in (0,1)$ and $\\rho < \\eta / \\gamma $ , the probability $\\mathbb {P}_{0} \\left( S^*_n(n^{-\\eta }) > n^{-\\rho } \\right) = \\mathbb {P}_{0} \\left( S_n(n^{-\\eta }) > n^{-\\rho } | D = + \\infty \\right)$ decays polynomially fast to 0 as $n \\rightarrow \\infty $ .","We let $\\rho ^i_0$ be $T^i_{\\mathcal {R}_0}$ , where $\\mathcal {R}_0$ is the first joint regeneration level in $\\mathcal {H}^+_{n^\\theta }$ .","Let us also introduce the random integer $k_n^{i} \\ge 1$ , which is the one such that $\\tau ^{i}_{k_n^{i}} = \\rho ^{i}_0$ .","Then let us introduce, for $i \\in \\lbrace 1,2\\rbrace $ $ \\widetilde{S}^{*i}_n(t) = \\mathrm {Inv}(n)^{-1}\\sum _{j = k^i_n+1}^{k^i_n + {nt}} (\\tau ^{i}_{j+1} - \\tau ^{i}_j) = S^{*i}_n\\left(\\frac{k^{i}_n}{n}+t\\right) - S^{*i}_n\\left(\\frac{k^{i}_n}{n}\\right),$ which is the value taken by $S^{*i}_n$ for the walk shifted by $\\tau _{k^{i}_n}$ .","Lemma 5.12 The following inequality holds: $\\left|\\mathbb {E}_0\\left[F(S^{*1}_n)F(S^{*2}_n)\\right] - \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .","Let us compute the difference $|F(S^{*i}_n)-F(\\widetilde{S}^{*i}_n(t))|$ .","Let us write the two terms properly: $& F(S^{*i}_n) = \\exp \\left(-\\lambda _1 S^{*i}_n(t_1) - \\cdots - \\lambda _m (S^{*i}_n(t_{m}) - S^{*i}_n(t_{m-1}))\\right)\\\\& F(\\tilde{S}^{*i}_n) = \\exp \\left(-\\lambda _1 \\tilde{S}^{*i}_n(t_1) - \\cdots - \\lambda _m (\\tilde{S}^{*i}_n(t_{m}) - \\tilde{S}^{*i}_n(t_{m-1}))\\right).$ Then using (REF ), and factorising by $F(S^{*i}_n)$ , the difference is dominated by $ F(S^{*i}_n) \\left[1-\\exp \\bigg (CS^{*i}_n( k^{i}_n / n)\\bigg )\\prod _{j=1}^{m}\\exp \\bigg (C\\bigg (S^{*i}_n( t_j + k^{i}_n / n) - S^{*i}_n( t_j )\\bigg )\\bigg )\\right],$ where $C = \\max \\left\\lbrace \\lambda _1,\\cdots ,\\lambda _m\\right\\rbrace $ .","Let us introduce the following event $ R \\mathcal {R}_0 \\le 2n^{\\theta },$ for $\\mathcal {R}_0$ being the first joint regeneration level inside $\\mathcal {H}^+_{n^\\theta }$ .","We already proved that $\\mathbb {P}_0(R^c)\\le Cn^{-c}$ in (REF ).","Let us put ourselves on the event $R$ and fix $1 > 1 - \\eta > \\theta > 0$ .","Then for $i \\in \\lbrace 1,2\\rbrace $ , $X^i_{\\tau ^i_{{n^{1-\\eta }}}}$ is $\\mathbb {P}_{0}$ -almost surely in $\\mathcal {H}^+_{2 n^\\theta }$ , hence $\\tau ^i_{k^i_n} < \\tau ^i_{n^{1-\\eta }}$ since $\\mathcal {R}_0 \\le 2n^{\\theta }$ .","It follows that event $R$ implies $C_n^{i} \\lbrace k_n^{i} \\le n^{1 - \\eta }\\rbrace $ for $i \\in \\lbrace 1,2\\rbrace $ .","In addition, under event $C^{i}_n$ the term $S^{*i}_n\\left(t_{j} + (k_n^i)/n)\\right) - S^{*i}_n\\left(t_{j}\\right)$ is dominated by $S^{*i}_n(t_{j} + n^{-\\eta }) - S^{*i}_n(t_{j})$ which is distributed as $S^{*i}_n(n^{-\\eta })$ under $\\mathbb {P}_0$ .","We define (fixing $t_0 = 0$ ): $A^i_n = \\left\\lbrace \\forall j \\in \\lbrace 0,\\cdots ,m-1\\rbrace \\text{, } S^{*i}_n\\left(t_{j} + (k_n^i)/n)\\right) - S^{*i}_n\\left(t_{j}\\right) \\le n^{\\rho }\\right\\rbrace .$ Then, using our estimate (REF ), Lemma REF , union bound and the i.i.d structure of regeneration times one gets $ &\\mathbb {P}_0\\left( (A^{i}_n)^{c} \\right) \\le m\\mathbb {P}\\left(S^{*i}_n(n^{-\\eta }) \\ge n^{\\rho } \\text{ | }D=+\\infty \\right) + \\mathbb {P}_0\\left(R^{c}\\right) \\le Cn^{-c},$ where $\\rho < \\eta / \\gamma < 1/\\gamma $ .","Therefore, on the event $A^{i}_n$ we have that $|F(S^{*i}_n)-F(\\widetilde{S}^{*i}_n(t))| \\le Cn^{-c},$ where $C,c$ only depend on $\\lambda ,\\vec{\\ell },d,K,\\lambda _1,\\cdots ,\\lambda _m$ .","Putting everything together we obtain $&\\left|\\mathbb {E}_0\\left[F(S^{*1}_n)F(S^{*2}_n)\\right] - \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right]\\right|&&\\\\ & \\hspace{100.0pt} \\le \\left|\\mathbb {E}_0\\left[F(S^{*1}_n)F(S^{*2}_n) {1}_{\\lbrace A^i_n\\rbrace }\\right] - \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n){1}_{\\lbrace A^i_n\\rbrace }\\right] \\right| + Cn^{-c} && \\\\& \\hspace{100.0pt} \\le Cn^{-c}.&&$ In the first inequality we used (REF ) and in the second one the fact that $F$ is bounded, (REF ) and the expression (REF ).","We are now able to prove the key result.","[Proof of Proposition REF ] Let us fix again $1 > 1 - \\eta > \\theta > 0$ and recall that $I = \\Big \\lbrace \\displaystyle \\inf _{\\begin{array}{c}p,q \\ge 0\\end{array}}(\\Vert X^1_{ p}-X^2_{q}\\Vert _{1})\\ge 2 \\Big \\rbrace $ .","Thanks to Lemma REF we get that $\\mathbb {E}[F(S^{*1}_n)F(S^{*2}_n)] - \\mathbb {E}[F(S^{*1}_n)]\\mathbb {E}[F(S^{*2}_n)] \\le Cn^{-c} + \\mathbb {E}[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)] - \\mathbb {E}[F(S^{*1}_n)]\\mathbb {E}[F(S^{*2}_n)]$ Moreover, we also get that $\\left| \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right] - \\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace }F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right] \\right| &\\le \\mathbb {P}_{0}\\left(I^{c} \\circ \\theta _{\\rho _0} \\right)\\nonumber \\\\&\\le \\mathbb {P}_0\\left(\\mathbf {M}_{n^{\\theta }}\\right)\\nonumber \\\\ &\\le C n^{-c}.$ where we recall the definition of $\\mathbf {M}_N$ in (REF ) and we used Proposition REF .","We introduce $f = F(S^{*1}_n)F(S^{*2}_n)$ and note that $\\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace }F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right] = \\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace } f \\circ \\theta _{\\rho _0}\\right].$ Furthermore, $f$ satisfies the assumptions of Proposition REF and REF , hence we get $\\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace } f \\circ \\theta _{\\rho _0} \\right] \\le Cn^{-c} + \\mathbb {E}_0\\bigg [\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\Big [f\\text{ | }D^{\\otimes }=+\\infty \\Big ]\\bigg ].$ We can rewrite the last term in the following way $\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\Big [f\\text{ | }D^{\\otimes }=+\\infty \\Big ] &= \\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\Big [F(S^{*1}_n)F(S^{*2}_n)\\text{ | }D^{\\otimes }=+\\infty \\Big ]\\nonumber \\\\&= \\mathbb {E}\\left[F(S^{*}_n)\\text{ | }D=+\\infty \\right]^2 = \\mathbb {E}\\left[F(S^{*}_n)\\right]^2.$ Putting together the estimates (REF ) and (REF ) we have shown that $\\big | \\mathbb {E} \\left[ F(S^{*1}_n)F(S^{*2}_n) \\right] - \\mathbb {E} \\left[ F(S^{*1}_n)\\right] \\mathbb {E} \\left[ F(S^{*2}_n)\\right] \\big | \\le Cn^{-c},$ which finishes the proof."],["Small times do not matter","Lemma 5.13 (Fuk-Nagaev inequality, ) Let $(X_i)_{i \\ge 1}$ be a family of independent identically distributed random variables with regularly varying tails and the tail parameter $\\displaystyle \\gamma \\in (0, 1)$ .","Let $\\displaystyle {S}(n) \\sum _{i = 1}^n X_i$ and $\\displaystyle {M}(n) \\max _{i \\in \\lbrace 1,\\cdots ,n\\rbrace } X_i$ , then there exists a constant $c>0$ such that, for all $y \\le x$ $P \\left( {S}(n) > x, \\, {M}(n) < y \\right) \\stackrel{\\textnormal {F-N}}{\\le } \\left( c n \\frac{y}{x} L(y) y^{-\\gamma }\\right)^{x/y}.$ Where $L(y)$ is the slowly varying function associated to the tail of $X_1$ .","Let us consider $\\lbrace \\tau _1^{(i)}\\rbrace _{i \\ge 1}$ , a sequence of independent identically distributed regeneration times under the measure $\\mathbb {P}^K_{0}(\\cdot \\text{ | }D=+\\infty )$ .","In it is proved that the sum ${S}(n) \\sum _{i = 1}^{n} \\tau _1^{(i)}$ is in the domain of attraction of a completely asymmetric stable law of index $\\gamma $ .","This implies, thanks to , that $\\mathbb {P}_0^K(\\tau _1 > t \\text{ | }D=+\\infty ) = \\psi (t)t^{-\\gamma }$ for some slowly varying function $\\psi (\\cdot )$ .","Lemma 5.14 For all $\\eta \\in (0,1)$ set $\\displaystyle {M}(n^{1-\\eta }) = \\max _{1 \\le i \\le {n^{1-\\eta }}} \\tau _1^{(i)}$ then for all $\\rho < \\eta /\\gamma $ and all $\\varepsilon < (\\eta - \\rho \\gamma )/3\\gamma $ there exists $\\nu >0$ such that $\\mathbb {P}\\left( {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) \\le Cn^{-\\nu }.$ $\\mathbb {P}\\left( {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) &\\le n^{1 - \\eta } \\mathbb {P}_{0}^{K}\\left( \\tau _1 > n^{1/\\gamma - \\rho - 2\\varepsilon } | D=\\infty \\right) \\\\&\\le Cn^{1-\\eta } n^{-1+\\gamma \\rho + 2\\varepsilon \\gamma } \\psi (n^{1/\\gamma - \\rho - 2\\varepsilon }) \\\\&\\le Cn^{-\\eta + \\gamma \\rho + 3\\varepsilon \\gamma }\\le n^{-\\nu },$ where we used the fact that a slowly varying function can always be bounded above by any polynomial so that $\\psi (n^{1/\\gamma }) \\le C n^{\\varepsilon /\\gamma }$ (Potter's bound ).","Finally, it holds that $\\eta - \\gamma \\rho - 3\\varepsilon \\gamma > 0$ thanks to the choice of $\\varepsilon $ , one has that the last inequality holds for some $\\nu $ small enough.","Lemma 5.15 For all $\\eta \\in (0, 1)$ recall $\\displaystyle {M}(n^{1-\\eta }) = \\max _{1 \\le i \\le {n^{1-\\eta }}}^{} \\tau _1^{(i)}$ and let $ \\displaystyle {S}(n^{1-\\eta }) \\displaystyle \\sum _{i = 1}^{{n^{1-\\eta }}} \\tau _1^{(i)}$ then for all $\\rho < \\eta /\\gamma $ and all $\\varepsilon < (\\eta - \\rho \\gamma )/3\\gamma $ there exists $\\nu >0$ such that $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho } \\right) \\le n^{-\\nu }.$ First notice that, by total probability $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho } \\right) &= \\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho }, {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) \\\\ & +\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho }, {M}(n^{1-\\eta }) \\le n^{1/\\gamma - \\rho - 2\\varepsilon } \\right),$ Thanks to Lemma REF $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma -\\epsilon - \\rho }, {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) \\le Cn^{-\\nu },$ for some $\\nu >0$ .","So we just need to bound the second term of the sum.","We can appeal to Lemma REF with the couple $\\quad \\quad x = n^{1/\\gamma - \\varepsilon - \\rho }, \\quad \\quad y=n^{1/\\gamma - \\rho - 2\\varepsilon }.$ Plugging these quantities into the formula gives $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho }, {M} \\le n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) &\\le \\left( C n^{1-\\eta } n^{-\\varepsilon } n^{-1 + \\gamma \\rho + 2\\gamma \\varepsilon } \\psi (n^{1/\\gamma - \\rho - 2\\varepsilon }) \\right)^{n^\\varepsilon }.$ Since the exponent is large we just need to check that what is inside the brackets is small.","This is immediate as $C n^{1-\\eta } n^{-\\varepsilon } n^{-1 + \\gamma \\rho + 2\\gamma \\varepsilon } \\psi (n^{1/\\gamma - \\rho - 2\\varepsilon })\\le C n^{2\\varepsilon \\gamma - \\eta + \\gamma \\rho } \\le e^{-1},$ as $\\psi (n^{1/\\gamma }) \\le Cn^{\\varepsilon }$ thanks to Potter's bound and $2\\varepsilon \\gamma - \\eta + \\gamma \\rho < 0$ by our choice of $\\varepsilon $ , this gives a super-polynomial bound for the desired quantity.","[Proof of Lemma REF ] Let us start with the first statement.","First let us note that, under $\\mathbb {P}_0$ , $\\lbrace \\tau _{i + 1} - \\tau _i\\rbrace _{i \\ge 1}$ are independent identically distributed random variables distributed as $\\tau _1$ under $\\mathbb {P}^K_{0}(.\\text{ | }D=+\\infty )$ .","Thanks to this fact we can re-write $S^*_n(n^{-\\eta }) \\stackrel{(\\mathrm {d})}{=} \\mathrm {Inv}(n)^{-1}\\sum _{i = 1}^{n^{1 - \\eta }} \\tau _{1}^{(i)},$ where $\\lbrace \\tau _{1}^{(i)}\\rbrace _{i \\ge 1}$ are i.i.d.","copies of $\\tau _1$ under the measure $\\mathbb {P}^{K}_{0}[ \\cdot | D = \\infty ]$ .","Then Lemma REF is enough to conclude together with the fact that for all $\\varepsilon >0$ $\\frac{n^{1/\\gamma - \\varepsilon }}{\\mathrm {Inv}(n)} \\rightarrow 0.$"],["Quenched convergence","We start this section by proving Theorem REF .","[Proof of Theorem REF ] The result is a straightforward consequence of Theorem REF and Proposition REF .","In particular one just needs to notice that $\\mathbf {Var}\\left(E_{0}^{\\omega }\\left[F_2(Z_{b^n})\\right]\\right) = \\mathbb {E}[F_2(Z_{b^n}^1)F_2(Z_{b^n}^2)] - \\mathbb {E}[F_2(Z_{b^n}^1)]\\mathbb {E}[F_2(Z_{b^n}^2)].$ The same decomposition holds for the process $S^*$ and the function $F_1$ .","Recall $D(\\mathbb {R})$ (resp.","$D(\\mathbb {R}^d)$ ) is the space of $\\mathbb {R}$ -valued (resp.","$\\mathbb {R}^d$ -valued) càdlàg functions.","In the following we consider $U$ the uniform topology and $J_1$ , $M_1$ two classical Skorohod topologies.","Let us recall that the vector $v$ and the matrix $\\Sigma $ have been introduced in .","In particular $v \\mathbb {E}_{0}^K[X_{\\tau _1}| D = \\infty ]$ while $\\Sigma $ is the covariance matrix of $X_{\\tau _1}$ under $\\mathbb {P}_{0}^{K}[ \\cdot | D = \\infty ]$ .","Moreover, let us introduce $v_0 = v / \\Vert v\\Vert $ , $I_d$ as the $d$ -dimensional identity matrix and $P_{v_0}$ as the projection matrix on $v_0$ .","Finally let $M_d C_\\infty ^{-\\gamma /2}(I_d - P_{v_0}) \\sqrt{\\Sigma }.$ Proposition 6.1 Fix any $T > 0$ , the followings holds for almost every environment $\\omega \\in \\Omega $ .","The law of $(Y_n(t))_{0 \\le t \\le T}$ converges under $P^\\omega _0(\\cdot )$ for the $U$ topology on $D^d$ towards the distribution of $(v t)_{0 \\le t \\le T}$ .","Moreover, the law of $(Z_n(t), S_n(t))_{0 \\le t \\le T}$ under $P^\\omega _0(\\cdot )$ converges in $D(\\mathbb {R}^d) \\times D(\\mathbb {R})$ for the $J_1 \\times M_1$ topology as $n \\rightarrow \\infty $ towards the law of $(\\sqrt{\\Sigma } B_t, C_\\infty \\mathcal {S}_\\gamma (t))_{0 \\le t \\le T}$ where $B$ is a standard Brownian motion and $\\mathcal {S}_\\gamma $ is a stable Subordinator of index $\\gamma $ independent of $B$ .","The convergence of $(Y_n(t))_{0 \\le t \\le T}$ is a consequence of .","We start by proving that Theorem REF together with imply the fact that $S_n$ converges in the process $M_1$ sense under the quenched law towards $\\mathcal {S}_\\gamma $ , we follow the proof of .","First we observe that $S_n(t) = \\frac{1}{\\mathrm {Inv}(n)} \\left( \\tau _1 + \\sum _{i = 1}^{{nt} - 1} (\\tau _{i + 1} - \\tau _i) \\right).$ We have that $\\tau _1 < + \\infty $ $\\mathbb {P}$ -a.s. thanks to , this means that for $\\mathbf {P}$ -almost all realisation of the environment $\\tau _1< + \\infty $ $P^{\\omega }$ -a.s..","But then, applying Slutsky's theorem, we just need to prove the quenched convergence of $S^*_n$ towards $\\mathcal {S}_\\gamma $ .","In order to prove convergence of the finite dimensional distributions we can restrict to prove convergence of the Laplace functionals $F$ of the form $F(h) = \\exp \\Big (-\\lambda _1h(t_1)-\\lambda _2(h(t_2)-h(t_1)) -\\cdots -\\lambda _n(h(t_n)-h(t_{n-1}))\\Big )$ Thanks to we have annealed convergence, so that Laplace functionals converge, in formulas $\\mathbb {E}\\left[F(S_n^*)\\right] = \\mathbf {E}\\left[ E^\\omega \\left[ F(S_n^*) \\right]\\right] \\underset{n \\rightarrow +\\infty }{\\rightarrow } \\mathbb {E}\\left[ F(\\mathcal {S}_\\gamma ) \\right] = \\exp \\left\\lbrace - t_1 \\psi (\\lambda _1) - \\dots - (t_m - t_{m -1})\\psi (\\lambda _m) \\right\\rbrace ,$ where $\\psi $ is the Laplace exponent of the stable Subordinator.","Moreover thanks to Theorem REF we know that for $b \\in (1, 2)$ $\\sum _{k = 1 }^{+\\infty }\\mathbf {Var}\\left[E_{0}^{\\omega }(F(S^*_{b^k}))\\right]< +\\infty .$ Therefore, by applying Borel-Cantelli and Chebyshev inequality to the sequence of events $|E_{0}^{\\omega }(F(S^*_{b^k})) - \\mathbb {E}_{0}\\left[ F(S^*_{b^k}) \\right]| > \\varepsilon $ and using the annealed convergence we get that $E_{0}^{\\omega }\\left[ F(S^*_{b^k}) \\right] \\rightarrow \\mathbb {E}\\left[ F(\\mathcal {S}_\\gamma ) \\right],$ almost surely along $b^k$ .","Moreover, the convergence holds jointly on a set of full measure for all functions $F$ of the form above for $b$ rational.","On the set of full measure where the joint convergence holds, we want to show that the promised convergence holds.","Let $k_n$ be the largest integer such that $b^{k_n} \\le n$ .","Thanks to the definition of $F$ , for any two increasing processes $h$ and $h^{\\prime }$ starting from 0 one gets $|F(h) - F(h^{\\prime })| \\le C \\max _{1 \\le i \\le m} |h(t_i) - h^{\\prime }(t_i)| \\wedge 1.$ Observe that, $S^*_n(t_i) = \\left( \\frac{b^{k_n}}{n} \\right)^{1/\\gamma } S^*_{b^{k_n}} \\left( \\frac{n}{b^{k_n}} t_i\\right).$ By the definition of $k_n$ and the monotonicity of $S^*_{b^{k_n}}(t)$ the latter is smaller than $S^*_{b^{k_n}}(b t_i)$ , as a consequence $(b^{-\\frac{1}{\\gamma }}-1)S^*_{b^{k_n}}(t_i) \\le S^*_n(t_i) - S^*_{b^{k_n}}(t_i) \\le S^*_{b^{k_n}}(b t_i) - S^*_{b^{k_n}}(t_i).$ The quantity $\\limsup _{n \\rightarrow \\infty } \\left| E^\\omega \\left[ F(S^*_n(t_i)) \\right] - E^\\omega \\left[ F(S^*_{b^{k_n}}(t_i)) \\right] \\right|,$ is thus bounded from above, up to a multiplicative constant, by $\\limsup _{n \\rightarrow \\infty } E^\\omega \\left[ \\max _{1 \\le i \\le m} \\bigg (|S^*_n(t_i) - S^*_{b^{k_n}}(t_i)|,|(b^{-\\frac{1}{\\gamma }}-1)S^*_{b^{k_n}}(t_i)|\\bigg ) \\wedge 1 \\right].$ Thanks (REF ) we get that this last quantity is bounded by $&\\limsup _{n \\rightarrow \\infty } E^\\omega \\left[ \\max _{1 \\le i \\le m} \\bigg (|S^*_{b^{k_n}}(b t_i) - S^*_{b^{k_n}}(t_i)|,|(b^{-\\frac{1}{\\gamma }}-1)S^*_{b^{k_n}}(t_i)|\\bigg ) \\wedge 1 \\right] \\\\&= \\mathbb {E} \\left[ \\max _{1 \\le i \\le m} \\bigg (|\\mathcal {S}_\\gamma (b t_i) - \\mathcal {S}_\\gamma (t_i)|,|(b^{-\\frac{1}{\\gamma }}-1)\\mathcal {S}_\\gamma (t_i)| \\bigg ) \\wedge 1 \\right].$ By the fact that $\\mathcal {S}_\\gamma $ is almost surely continuous at deterministic times we get that the quantity on the right converges to 0 as $b \\rightarrow 1$ along rationals by dominated convergence theorem since the quantity is bounded by 1.","The tightness of the process in $M_1$ comes from this f.d.d.","distributions convergence we just proved, thanks to the fact that all processes involved are increasing.","The convergence of the position $Z_n(t)$ (both seen as the polygonal interpolation and in the canonical way) is proved with the same hypotheses and thesis in .","We now need to prove joint convergence of $S_n$ and $Z_n$ .","First notice that, thanks to the previous two points and , tightness of the marginals is enough to have tightness in the product space.","Therefore, we have that the law $(S_n, Z_n)$ is tight on $D(\\mathbb {R}^d) \\times D(\\mathbb {R})$ endowed with the $J_1 \\times M_1$ -topology on a set of the environment $\\Omega _1 \\subset \\Omega $ of full $\\mathbf {P}$ -measure.","We need to prove that on a set of full measure $\\Omega _2$ , the law of the increments of the process $W_n = (S_n, Z_n)$ converge to the one of $W = (\\mathcal {S}_\\gamma , \\sqrt{\\Sigma } B)$ .","The proof follows the lines of the proof of the convergence of $S_n$ and the argument of .","Let $\\lambda _1, \\dots , \\lambda _m \\in \\mathbb {R}^{d+1}$ and $0 < t_0 < \\dots < t_m$ as it is sufficient to check on intervals that do not contain 0.","Let us define the function $G(w) = \\exp \\Big ( i \\lambda _1 \\cdot \\left( w(t_1) - w(t_0) \\right) + \\dots + i \\lambda _m \\cdot \\left( w(t_m) - w(t_{m - 1}) \\right) \\Big ).$ From we have that $\\mathbb {E}\\left[ G(W_n) \\right] \\underset{n \\rightarrow +\\infty }{\\longrightarrow } \\mathbb {E}\\left[ G(W) \\right].$ By proceeding in the same way as we did for Theorem REF (using all the instrumental lemmas and propositions that we proved in Section ) it is possible to prove that for all $b \\in (1, 2)$ $\\sum _{k = 1 }^{+\\infty }\\mathbf {Var}\\left[E_{0}^{\\omega }(G(W_{b^k}))\\right]< +\\infty .$ As a consequence we get that $E_{0}^{\\omega }\\left[ G(W_{b^k}) \\right] \\rightarrow \\mathbb {E}\\left[ G(W) \\right].$ Then, by following the argument that we did before, it is possible to show that for all $G$ of the form of (REF ) with $\\lambda _1, \\dots , \\lambda _m$ and $0 < t_0 < \\dots < t_m$ rationals on a set $\\Omega _2$ of full measure one gets $E_{0}^{\\omega }\\left[ G(W_{n}) \\right] \\rightarrow \\mathbb {E}\\left[ G(W) \\right].$ Finally, by right continuity of $W_n$ and $W$ and continuity of the Fourier functional it is possible to show that the limit holds for all $0 \\le t_0 \\le \\dots \\le t_m$ and for all $\\lambda _1, \\dots , \\lambda _m \\in \\mathbb {R}^{d+1}$ .","This finishes the proof as we showed that on a set of full measure $\\Omega _1 \\cap \\Omega _2$ the sequence $W_n$ is tight and has just one possible limit $W$ .","We are now able to prove the main result, the method is the same as the proof of .","[Proof of Theorem REF ] We fix a realisation of the environment $\\omega \\in \\Omega $ , the following convergences in distribution are meant to be understood under the quenched law $P^\\omega $ , we assume the result of Proposition REF .","Recall that $D(\\mathbb {R})$ and $D(\\mathbb {R}^d)$ denote the sets of càdlàg functions in dimension one (resp.","$d$ ).","Let $D_{\\uparrow }$ be the set of non-deacreasing càdlàg functions, $D_{\\uparrow \\uparrow }$ the set of strictly increasing càdlàg functions and $D_u$ the set of unbounded càdlàg functions.","Let $\\mathcal {C}$ be the set of continuous functions and $U$ the uniform topology.","It is a known fact that $\\mathcal {S}_\\gamma ^{-1}$ is almost surely continuous and increasing.","Firstly we get that $\\left(Y_{n^\\gamma /L(n)}\\left( S_{n^\\gamma /L(n)}^{-1}\\left(\\frac{nt}{\\mathrm {Inv}\\left(n^\\gamma /L(n)\\right)}\\right) \\right) \\right)_{t \\in [0, T]} \\rightarrow \\left( v C_\\infty ^{-\\gamma } \\mathcal {S}_\\gamma ^{-1}(t) \\right)_{t \\in [0, T]},$ in the uniform topology.","This thanks to the fact the right continuous inverse is continuous from $(D_{u, \\uparrow \\uparrow }, M_1)$ to $(\\mathcal {C}, U)$ thanks to , and if $(x_n, y_n) \\rightarrow (x, y)$ in $D^d \\times D_{\\uparrow }$ with $(x, y) \\in \\mathcal {C}^d \\times \\mathcal {C}_{\\uparrow }$ , then $x_n \\circ y_n \\rightarrow x \\circ y$ in the $U$ topology (see ).","Lastly, note that $(nt/\\mathrm {Inv}(n^\\gamma /L(n)))_{t \\in [0, T]}$ converges uniformly to $(t)_{t \\in [0, T]}$ .","This proves the first result.","Moreover, thanks to the same results it holds that $\\left(Z_{n^\\gamma /L(n)}\\left( S_{n^\\gamma /L(n)}^{-1}\\left(\\frac{nt}{\\mathrm {Inv}\\left(n^\\gamma /L(n)\\right)}\\right) \\right) \\right)_{t \\in [0, T]} \\stackrel{(\\mathrm {d})}{\\rightarrow } \\left( C_\\infty ^{-\\gamma /2} B_{\\mathcal {S}_\\gamma ^{-1}(t)} \\sqrt{\\Sigma } \\right)_{t \\in [0, T]},$ in the $J_1$ topology.","We now need to ensure that $\\mathbf {P}$ -a.s., with high $P^\\omega $ -probability $X_{{nt}}$ is close to $X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }}$ .","It is immediate to prove that $\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }$ is the smallest $\\tau _i$ such that $\\tau _i > tn$ .","Then we can bound the distance $X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }} - X_{{nt}}$ by a constant times the size of a regeneration block.","$M_n &\\max _{t \\in [0, T]} \\left\\Vert X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }} - X_{{nt}} \\right\\Vert _\\infty \\\\& \\le C \\max _{k = 1, \\dots , {nT} + 1} \\left\\Vert X_{\\tau _k} - X_{\\tau _{k - 1}} \\right\\Vert _\\infty .$ By the fact that $\\Vert X_{\\tau _k} - X_{\\tau _{k - 1}}\\Vert _\\infty \\le \\chi _k^\\alpha $ defined in (REF ), applying and using a union bound one gets that, for all $M>0$ and $\\delta >0$ $\\mathbb {P}_0\\left( M_n > \\delta n^{\\gamma /4} \\right) = \\mathbf {E}\\left[ P_0^\\omega (M_n > \\delta n^{\\gamma /4}) \\right] \\le n^{-M}.$ Markov inequality gives that $\\mathbf {P}\\left( P_0^\\omega (M_n > \\delta n^{\\gamma /4}) > n^{-1} \\right) \\le n^{-2}.$ Then Borel-Cantelli implies that $\\mathbf {P}$ -almost surely $P_0^\\omega [M_n > \\delta n^{\\gamma /4}] \\le 1/n$ for all $n$ large enough, which leads us to conclude that $X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }} - X_{{nt}} \\le \\delta n^{\\gamma /4}$ in the uniform sense with high $P^\\omega $ -probability.","This implies, together with (REF ) and (REF ) and Slutsky's theorem, that $\\mathbf {P}$ -a.s., under the quenched law $P^\\omega $ the statement (REF ) holds and that $\\left(\\frac{X_{{nt}} - v \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}\\left(\\frac{nt}{\\mathrm {Inv}\\left(n^\\gamma /L(n)\\right)}\\right)}{\\sqrt{n^\\gamma /L(n)}} \\right)_{t \\in [0, T]} \\stackrel{(\\mathrm {d})}{\\rightarrow } \\left( C_\\infty ^{-\\gamma /2} B_{\\mathcal {S}_\\gamma ^{-1}(t)} \\sqrt{\\Sigma } \\right)_{t \\in [0, T]}.$ Finally, notice that (REF ) follows by applying the linear transformation $(I_d - P_{v_0})$ to (REF ) as $(I_d - P_{v_0}) v = 0$ .","All linear combinations are continuous in the $J_1$ -topology at continuous functions."],["Extension of Markov property","The appendix gives some results which are useful in the paper but considered as tools.","Since in the whole paper we will consider two random walks we extend the Markov property when we have two Markov chains with the general following lemma.","Lemma A.1 Consider two Markov chains $(X_n)_{n \\ge 0}$ and $(Y_n)_{n \\ge 0}$ which take values in a countable set $E$ .","Consider the product measure $P_{(x,y)} = P_x \\otimes P_y$ .","We consider the canonical markov chain on $E^2$ .","We define for $k,k^{\\prime } \\in \\mathbb {N}$ the translation operator defined by $\\begin{array}{l|rcl}\\end{array}\\theta _{k,k^{\\prime }}: & (E^2)^{\\mathbb {N}} & \\longrightarrow & (E^2)^{\\mathbb {N}} \\\\& ((x_n),(y_n)) & \\longmapsto & ((x_{n+k}),(y_{n+k^{\\prime }})) $ .$$ Consider also $P_{x,y} = P_x \\otimes P_y$ almost surely finite stopping times $\\tau $ and $\\rho $ for $X$ and $Y$ .","Consider two bounded and measurable functions $f$ and $g$ which a respectively measurable with respect to $\\sigma \\Big (X_0,...,X_{\\tau },Y_0,...,Y_{\\rho }\\Big ) \\text{ and }\\sigma \\Big ((X_n)_{n \\ge 0},(Y_n)_{n \\ge 0}\\Big ).$ Then we have the following equality: $E_{x,y}\\left[f g \\circ \\theta _{\\tau ,\\rho }\\right] = E_{x,y}\\left[fE_{X_{\\tau },Y_{\\rho }}[g]\\right].$ The proof is straightforward using strong Markov property and a Monotone Class Theorem.","Indeed, if $f$ is of the form $f((x_n),(y_n)) = f_1((x_n))f_2((y_n))$ with $f_1$ measurable with respect to $\\sigma \\big (X_0,...,X_{\\tau }\\big )$ and $f_2$ measurable with respect to $\\sigma \\big (Y_0,...,Y_{\\rho }\\big )$ and $g((x_n),(y_n)) = g_1((x_n))g_2((y_n))$ with $g_1$ measurable with respect to $\\sigma \\big ((X_n)_{n \\ge 0}\\big )$ and $g_2$ measurable with respect to $\\sigma \\big ((Y_n)_{n \\ge 0}\\big )$ .","Then we have that $&E_{x,y}\\left[f g \\circ \\theta _{\\tau ,\\rho }\\right]\\\\&= E_{x,y}\\left[f_1 g_1 \\theta _{\\tau } \\cdot f_2g_2 \\circ \\theta _{\\rho }\\right]\\\\& = E_{x}\\left[f_1 g_1 \\circ \\theta _{\\tau }\\right]\\cdot E_{y}\\left[f_2 g_2 \\circ \\theta _{\\rho }\\right],$ where the last equality comes from the independence of the two walks.","Then using the Strong Markov property we get $&E_{x}\\left[f_1 E_{X_{\\tau }}[g_1]\\right]\\cdot E_{y}\\left[f_2 E_{Y_{\\rho }}[g_2]\\right] \\\\& = E_{x,y}\\left[f_1f_2 E_{X_{\\tau }}[g_1]E_{Y_{\\rho }}[g_2]\\right]\\\\& = E_{x,y}\\left[f E_{X_{\\tau },Y_{\\rho }}[g]\\right].$ We conclude using a Monotone Class Theorem argument."],["Polynomial decay","Proposition B.1 For any $\\varepsilon > 0$ , there exists $K_0 > 0$ such that for any $K > K_0$ , any $n \\in \\mathbb {N}^{*}$ and any $z \\in \\mathbb {Z}^d$ we have $\\mathbb {P}_0(X_{\\tau _n} = z) \\le Cn^{\\varepsilon -\\frac{d}{2}},$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","Remark B.2 We recall that the notion of regeneration time introduced in (see the definition in Section REF ) depends on the value of $K$ , but we always take $K$ large enough.","First we have the following equality: $\\mathbb {P}_0(X_{\\tau _n} = z)& =\\mathbb {P}_0\\left(X_{\\tau _1} +\\sum _{i=1}^{n-1} (X_{\\tau _{i+1}}-X_{\\tau _i}) = z\\right)\\\\&\\nonumber =\\mathbb {P}_0\\left(\\sum _{i=1}^{n-1} (X_{\\tau _{i+1}}-X_{\\tau _i} - v ) = z - X_{\\tau _1} - (n-1)v\\right),$ where $v = \\mathbb {E}[X_{\\tau _{2}}-X_{\\tau _1}]$ , we introduce for $i\\ge 1$ , $\\xi _i = X_{\\tau _{i+1}}-X_{\\tau _i} - v $ .","The family $(\\xi _i)_{i \\ge 1}$ is an independent family of random variables identically distributed and independent of $X_{\\tau _1}$ .","By , for any $M > 0$ there exists $K_0 > 0$ such that for any $K > K_0$ we have $\\mathbb {E}_{0}\\left[\\Vert \\xi _1\\Vert ^{M}\\right] < +\\infty $ .","Thus we can write (REF ) as $\\sum _{y \\in \\mathbb {Z}^d}^{} \\mathbb {P}_0(X_{\\tau _1} = y)\\mathbb {P}_0\\bigg (\\sum _{i=1}^{n-1} \\xi _i = e - y - (n-1)v\\bigg ).$ Then we only need a bound for the value of $\\mathbb {P}_0\\bigg (\\displaystyle \\sum _{i=1}^{n-1} \\xi _i = x\\bigg )$ with $x = e - y - (n-1)v$ uniformly for $y \\in \\mathbb {Z}^d$ .","We define $\\begin{array}{l|rcl}\\end{array}H: & \\mathbb {R} & \\longrightarrow & \\mathbb {R} \\\\& x & \\longmapsto & x^{M} $$, then we have $ E0(H(||i||)) < +$.","$ H$ increases faster than $ x x2+$ and slower than $ x ex$.","Using \\cite [Theorem 3]{Zaitsev} we can construct a probability space and two family of random variables $ (Yi)$ and $ (Ni)$ such that:\\begin{itemize}\\item [\\bullet ] (Y_i)_{i \\ge 1} has the same law as (\\xi _i)_{i \\ge 1} under \\mathbb {P}_0.\\item [\\bullet ] (N_i)_{i \\ge 1} is a family of independent identically distributed centered gaussian vectors with covariance matrix \\Sigma where \\Sigma is the covariance matrix of \\xi _1.\\end{itemize}Moreover, they satisfy\\begin{equation*}\\mathbb {P}\\left(\\max _{\\begin{array}{c}1 \\le j \\le n\\end{array}}(|\\sum _{i=1}^{j} (Y_i - N_i)|) \\ge cn^{\\varepsilon }\\right) \\le C\\frac{n}{n^{\\varepsilon M}},\\end{equation*}where the constants $ C,c > 0$ only depend on $ ,,d,K$.","We write{\\begin{@align}{1}{-1}\\nonumber \\mathbb {P}_0\\bigg (\\sum _{i=1}^{n-1} \\xi _i = x\\bigg ) &= \\mathbb {P}\\bigg (\\sum _{i=1}^{n-1} Y_i = x\\bigg ) \\\\&\\nonumber \\le Cn^{1-\\varepsilon M} + \\mathbb {P}\\bigg (\\sum _{i=1}^{n-1} N_i \\in B_{\\Vert \\cdot \\Vert _{1}}(x,cn^{\\varepsilon })\\bigg ) \\\\& \\le Cn^{1-\\varepsilon M} + \\mathbb {P}\\bigg ( \\frac{\\sum _{i=1}^{n-1} N_i}{\\sqrt{n-1}} \\in B_{\\Vert \\cdot \\Vert _{1}}\\bigg (\\frac{x}{\\sqrt{n-1}},\\frac{cn^{\\varepsilon }}{\\sqrt{n-1}}\\bigg ) \\bigg ).\\end{@align}}Since $ i=1n-1 Nin-1$ is a centered Gaussian vector with covariance matrix $$, the right term in (\\ref {Gaussian}) is bounded above by{\\begin{@align*}{1}{-1}C\\mu \\left(B_{\\Vert \\cdot \\Vert _{1}}\\bigg (\\frac{x}{\\sqrt{n-1}},\\frac{cn^{\\varepsilon }}{\\sqrt{n-1}}\\bigg )\\right),\\end{@align*}}where $$ denotes the Lebesgue measure on $ Rd$ and $ C$ is a constant which depends on $$, hence on $ ,,d,K$.","We deduce that last term is dominated by{\\begin{@align*}{1}{-1}Cn^{d\\varepsilon -\\frac{d}{2}},\\end{@align*}}where $ C$ is a constant which only depends on $ ,,d,K$.$ Choosing $M$ large enough such that $1-\\varepsilon M < - \\frac{d}{2}$ we obtain that (REF ) is bounded above by $Cn^{1-\\varepsilon M} + Cn^{d\\varepsilon -\\frac{d}{2}} \\le C n^{\\varepsilon d-\\frac{d}{2}},$ where the constant $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .","We conclude since the result is true for all $\\varepsilon > 0$ ."],["Acknowledgments","We would like to thank Daniel Kious for encouraging this collaboration and useful comments at various stages of the project.","We would like also to thank Christophe Sabot and Noam Berger for insightful discussions.","AF acknowledges the support of an NSERC Discovery Grant.","CS is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/S022945/1."]],"string":"[\n [\n \"Quenched invariance principle for biased random walks in random\\n conductances in the sub-ballistic regime\"\n ],\n [\n \"Abstract We consider a biased random walk in positive random conductances on $\\\\mathbb{Z}^d$ for $d\\\\geq 5$.\",\n \"In the sub-ballistic regime, we prove the quenched convergence of the properly rescaled random walk towards a Fractional Kinetics.\"\n ],\n [\n \"Introduction\",\n \"Random walks in random environment (RWRE) have been extensively studied since the mid seventies and especially so in the last two decades.\",\n \"We refer the reader to , , , and for different surveys of the field.\",\n \"A lot of effort has been directed towards proving invariance principles in various models.\",\n \"The majority of those results have been obtained in isotropic settings.\",\n \"We could mention as an example the numerous works done for the simple random walk on supercritical percolation clusters in $\\\\mathbb {Z}^d$ .\",\n \"This started with results in the annealed setting in the 80s, see and (based on works of and ) which was later generalized into a quenched result, initially for $d\\\\ge 4$ in and later in dimensions 2 and 3 in and .\",\n \"In models where trapping occurs, the classical invariance principle is not expected to hold.\",\n \"However, using a different time scaling, it is still possible to obtain a convergence result where the limiting Brownian motion is replaced by a Fractional Kinetics (a Brownian motion time-changed by a stable subordinator).\",\n \"For example, Barlow and Černý prove in subdiffusive scaling limit in both the Bouchaud trap model (BTM) and the random conductance model.\",\n \"Mourrat gives a different proof for the result of for the Bouchaud trap model in .\",\n \"All the results mentioned above are for isotropic models.\",\n \"Other cases, such as models with directional transience, have attracted a lot of interest as well.\",\n \"Annealed invariance principles have been obtained for such models on $\\\\mathbb {Z}$ (see ), trees (see , ) and on $\\\\mathbb {Z}^d$ with $d \\\\ge 2$ (see , ).\",\n \"Moreover, there are a few results in the quenched setting as well on $\\\\mathbb {Z}$ (see ), trees (see ) and $\\\\mathbb {Z}^d$ (see ).\",\n \"The models with directional transience of interest are typically studied because several of them exhibit slowdown effects due to a trapping phenomenon, such as , , , , , , , , .\",\n \"The most precise results for an anisotropic model on $\\\\mathbb {Z}^d$ with $d \\\\ge 2$ were obtained by Fribergh and Kious in , where it is shown that under heavy tail condition for conductances we have an aging phenomenon for the walk and an annealed scaling limit related to Fractional Kinetics.\",\n \"In light of Mourrat's result in it is natural to expect that the annealed scaling limit obtained in should hold in the quenched setting as well.\",\n \"This is the main result of this article.\",\n \"It provides a first example for a directionally transient RWRE which converges to a Fractional Kinetics under the quenched setting.\"\n ],\n [\n \"Definition of the model and main result\",\n \"Fix $d \\\\ge 5$ .\",\n \"Let us consider the following random walk in random environment.\",\n \"Let $\\\\mathbf {P}$ be a measure on the set of nearest neighbour edges of the square lattice $E(\\\\mathbb {Z}^d)$ , i.e.\",\n \"$e \\\\in E(\\\\mathbb {Z}^d)$ if $e = [x, y]$ for $x \\\\sim y$ (adjacent vertices in $\\\\mathbb {Z}^d$ ).\",\n \"In particular, let $\\\\mathbf {P}(\\\\cdot ) = \\\\mu ^{\\\\otimes E(\\\\mathbb {Z}^d)}$ , where $\\\\mu $ is a probability measure on $(0,+\\\\infty )$ .\",\n \"For $e \\\\in E(\\\\mathbb {Z}^d)$ , we denote by $c_{*}(e)$ the random variable of the marginal of the conductance of the edge $e$ under $\\\\mathbf {P}$ .\",\n \"Let us call the space of environments $\\\\Omega $ and a single realisation of the environment $\\\\omega \\\\in \\\\Omega $ .\",\n \"To introduce the bias consider a unit vector $\\\\vec{\\\\ell } \\\\in \\\\mathbb {S}^{d-1}$ and $\\\\lambda > 0$ , denote $\\\\ell \\\\lambda \\\\vec{\\\\ell }$ and set for $e = [x,y] \\\\in E(\\\\mathbb {Z}^{d})$ , $c(e) = c_{*}(e)e^{(x+y)\\\\cdot \\\\ell }.$ We focus on the regime where $\\\\mathbf {P}\\\\left(c_{*}(e) \\\\ge x \\\\right) = \\\\mu ([x,+\\\\infty [) = L(x) x^{-\\\\gamma } \\\\quad \\\\textnormal {for any }x \\\\ge 0,$ with $\\\\gamma \\\\in (0,1)$ and $L(\\\\cdot )$ is a slowly-varying function.\",\n \"For a fixed environment $\\\\omega $ , we consider the random walk $(X_n)_{n \\\\ge 0}$ with conductances $(c^\\\\omega (e))_{e \\\\in E(\\\\mathbb {Z}^d)}$ .\",\n \"We will call $P_{x}^{\\\\omega }$ the quenched law of the random walk on the environment starting from $x$ and with transition probabilities $p^\\\\omega (x, y)$ for $x, y \\\\in \\\\mathbb {Z}^d$ $p^\\\\omega (x, y) = \\\\frac{c^\\\\omega ([x, y])}{\\\\sum \\\\limits _{y \\\\sim x} c^\\\\omega ([x, y])}.$ Let $\\\\mathbb {P}_{x} = \\\\mathbf {E}[P_{x}^{\\\\omega }(.\",\n \")]$ be the annealed law of this random walk ($\\\\mathbf {E}[\\\\cdot ]$ being the expectation with respect to $\\\\mathbf {P}$ and $\\\\mathbf {Var}$ the corresponding variance).\",\n \"The main theorem of this work is the following.\",\n \"Theorem 1.1 Let $d \\\\ge 5$ and fix some $T > 0$ , the following statements hold for $\\\\mathbf {P}$ -almost every environment $\\\\omega \\\\in \\\\Omega $ .\",\n \"There exists a deterministic $v_0 \\\\in \\\\mathbb {S}^{d-1}$ with $v_0 \\\\cdot \\\\vec{\\\\ell } > 0$ such that, under the quenched law $P^\\\\omega _0(\\\\cdot )$ , we have the following weak convergences holding $\\\\left( \\\\frac{X_{{n t}}}{n^\\\\gamma /L(n)} \\\\right)_{0 \\\\le t \\\\le T} \\\\overset{(\\\\mathrm {d})}{\\\\rightarrow } \\\\left( v_0 C_\\\\infty ^{-\\\\gamma } \\\\mathcal {S}_\\\\gamma ^{-1}(t) \\\\right)_{0 \\\\le t \\\\le T},$ $\\\\left( \\\\frac{X_{{n t}} - (X_{{n t}} \\\\cdot v_0) v_0}{\\\\sqrt{n^\\\\gamma /L(n)}} \\\\right)_{0 \\\\le t \\\\le T} \\\\overset{(\\\\mathrm {d})}{\\\\rightarrow }\\\\left( M_d B_{\\\\mathcal {S}_\\\\gamma ^{-1}(t)} \\\\right)_{0 \\\\le t \\\\le T}.$ For a deterministic constant $C_\\\\infty >0$ , a deterministic $d \\\\times d$ matrix $M_d$ of rank $d-1$ , a standard Brownian motion $B$ and a stable subordinator of index $\\\\gamma $ , independent of $B$ , $\\\\mathcal {S}_\\\\gamma $ .\",\n \"The first convergence holds in the uniform topology while the second holds in the Skorohod's $J_1$ topology.\"\n ],\n [\n \"Quenched convergence via series convergence\",\n \"Let us explain the main strategy that we will follow.\",\n \"In one can find the definition of a regeneration structure with suitable independence properties, in particular let $(\\\\tau _k)_{k \\\\ge 0}$ be the regeneration times.\",\n \"See Section REF where we recall the rigorous definition of $(\\\\tau _k)_{k \\\\ge 0}$ .\",\n \"Call $v \\\\mathbb {E}[X_{\\\\tau _2} - X_{\\\\tau _1} ]$ , then we introduce, for all $t \\\\ge 0$ $Y_n(t) = \\\\frac{X_{\\\\tau _{{tn}}}}{n}, \\\\quad Z_n(t) = \\\\frac{X_{\\\\tau _{{tn}}} - v n t}{n^{1/2}},\\\\quad \\\\textnormal {and} \\\\quad S_n(t) = \\\\frac{\\\\tau _{{tn}}}{\\\\mathrm {Inv}(n)},$ where $\\\\mathrm {Inv}(t) \\\\inf \\\\left\\\\lbrace s: \\\\mathbf {P}[c_{*} > s]\\\\le 1/t \\\\right\\\\rbrace ,$ is the generalised right-continuous inverse of the tail of $c_{*}$ , let us point out that $\\\\mathrm {Inv}(n) \\\\approx n^{1/\\\\gamma }$ .\",\n \"Our initial aim is to prove that for $\\\\mathbf {P}$ -almost all environments $\\\\omega \\\\in \\\\Omega $ these three quantities converge weakly.\",\n \"Then we will use this to prove Theorem REF .\",\n \"We get the quenched convergence of $Y_n(t)$ for free from .\",\n \"For technical reasons let us define the process $S^*_n(t) = \\\\frac{\\\\tau _{{tn}+1} - \\\\tau _{1}}{\\\\mathrm {Inv}(n)}.$ To prove quenched convergence of $S^*_n(t)$ and $Z_n(t)$ we will use alternative formulations considered in and to rephrase the problem.\",\n \"These conditions are stated in the following theorem.\",\n \"Let us define $F(h) = \\\\exp \\\\left(-\\\\lambda _1 h(t_1) - \\\\lambda _2(h(t_2) - h(t_1)) -... - \\\\lambda _m(h(t_m) - h(t_{m-1}))\\\\right), $ with $\\\\lambda _1,\\\\cdots ,\\\\lambda _m \\\\ge 0$ and $0 < t_1 < \\\\cdots < t_m \\\\le T$ .\",\n \"Theorem 1.2 Let $d \\\\ge 5$ , then the following holds for any $F_1$ as in (REF ) $ \\\\sum _{n = 1 }^{+\\\\infty }\\\\mathbf {Var}\\\\left(E_{0}^{\\\\omega }\\\\left[F_1(S^*_{b^n})\\\\right]\\\\right)< +\\\\infty ,$ with $b \\\\in (1,2)$ .\",\n \"Moreover, by denoting with $Z_n$ the polygonal interpolant of the quantity denoted with the same letter in (REF ), we have $ \\\\sum _{n = 1 }^{+\\\\infty }\\\\mathbf {Var}\\\\left(E_{0}^{\\\\omega }[F_2(Z_{b^n})]\\\\right)< +\\\\infty ,$ where $F_2$ is any bounded $K$ -Lipschitz function from $\\\\mathcal {C}([0, T],\\\\mathbb {R}^{d}) \\\\rightarrow \\\\mathbb {R}$ .\"\n ],\n [\n \"Sketch of proof\",\n \"To prove Theorem REF , we can consider under each environment two independent random walks $X^1$ and $X^2$ which have the same law as $X$ under $P_0^{\\\\omega }$ .\",\n \"Then the term in the second sum above is equal to $\\\\mathbb {E}[F_2(Z_{b^n}^1)F_2(Z_{b^n}^2)] - \\\\mathbb {E}[F_2(Z_{b^n}^1)]\\\\mathbb {E}[F_2(Z_{b^n}^2)].$ We will show that the two walks do not meet each other after reaching a distance not too far from the origin, then deduce that the quantity above goes quickly to 0.\",\n \"Indeed, when the two walks do not meet, they behave almost as if they evolved in independent environments.\",\n \"The important ideas can be grouped in two main parts.\",\n \"The first one deals with joint regeneration levels while the second consists in proving the asymptotic separation using joint regeneration levels.\",\n \"Joint regeneration levels: Section will be devoted to extend the notion of regeneration times of to the case where we have two walks.\",\n \"We will construct joint regeneration levels: a sequence of levels such that the hitting time of these are regeneration times for both walks.\",\n \"Asymptotic separation: Once we have constructed joint regeneration levels the idea is the following: at each regeneration level, both walks have a decent chance of staying far apart for a long time, after which it is unlikely that they will ever meet again.\",\n \"As joint regeneration levels are common, the walks will decouple relatively quickly.\",\n \"In this part we use a method which is inspired from .\",\n \"However, we cannot directly use their results because our model lacks both the uniform ellipticity property of the jump probabilities and it is not ballistic.\"\n ],\n [\n \"Notations\",\n \"We fix an orthonormal basis $(e_1,...,e_d)$ of $\\\\mathbb {R}^d$ with elements of $\\\\mathbb {Z}^d$ such that $e_1\\\\cdot \\\\vec{\\\\ell } \\\\ge e_2\\\\cdot \\\\vec{\\\\ell } \\\\ge ... \\\\ge e_d\\\\cdot \\\\vec{\\\\ell }$ .\",\n \"In particular we have $e_1\\\\cdot \\\\vec{\\\\ell } \\\\ge 1/\\\\sqrt{d}$ .\",\n \"We also fix an orthonormal basis of $\\\\mathbb {R}^d$ , $(f_1,...,f_d)$ such that $f_1 = \\\\vec{\\\\ell }$ .\",\n \"For any $L \\\\in \\\\mathbb {R}$ we set $\\\\mathcal {H}^{+}(L) = \\\\lbrace z \\\\in \\\\mathbb {Z}^d ; \\\\, z\\\\cdot \\\\vec{\\\\ell } > L\\\\rbrace \\\\,\\\\, \\\\mathrm {and} \\\\,\\\\, \\\\mathcal {H}^{-}(L) = \\\\lbrace z \\\\in \\\\mathbb {Z}^d ; \\\\, z\\\\cdot \\\\vec{\\\\ell } \\\\le L\\\\rbrace .$ Similarly, for $z\\\\in \\\\mathbb {Z}^d$ $\\\\mathcal {H}^{+}_{z} = \\\\mathcal {H}^{+}(z\\\\cdot \\\\vec{\\\\ell }) \\\\,\\\\, \\\\mathrm {and} \\\\,\\\\, \\\\mathcal {H}^{-}_{z} = \\\\mathcal {H}^{-}(z\\\\cdot \\\\vec{\\\\ell }).$ For any $x \\\\in \\\\mathbb {Z}^d$ we define $ \\\\mathcal {V}_x = \\\\lbrace x,x+e_1,x-e_1,\\\\cdots ,x+e_d,x-e_d\\\\rbrace .$ For any $x,y \\\\in \\\\mathbb {Z}^d$ we say that $x$ and $y$ are adjacent and write $x \\\\sim y$ if $x = y \\\\pm e_i$ where $i \\\\in \\\\lbrace 1,\\\\cdots ,d\\\\rbrace $ .\",\n \"For $x$ in $\\\\mathbb {Z}^d$ and $e = [e^{+}, e^{-}]$ and edge, we write $x \\\\sim e$ if $x \\\\sim e^{-}$ or $x \\\\sim e^{+}$ and $x \\\\notin \\\\lbrace e^{+},e^{-}\\\\rbrace $ .\",\n \"Two distinct edges $e$ and $e^{\\\\prime }$ are said to be adjacent and we write $e\\\\sim e^{\\\\prime }$ if they share one endpoint.\",\n \"For $x \\\\in \\\\mathbb {Z}^d$ we define $\\\\mathcal {E}_x \\\\lbrace [x,y] \\\\, | \\\\, y\\\\sim x\\\\rbrace , \\\\,\\\\, \\\\text{ and for }A\\\\subset \\\\mathbb {Z}^d \\\\text{, }\\\\mathcal {E}_A = \\\\bigcup _{x\\\\in A}\\\\mathcal {E}_x.$ $\\\\mathcal {L}^x \\\\lbrace [y,z] \\\\,\\\\, | \\\\,\\\\, y \\\\cdot \\\\vec{\\\\ell } \\\\le x \\\\cdot \\\\vec{\\\\ell } \\\\text{ and } z \\\\cdot \\\\vec{\\\\ell } \\\\le x \\\\cdot \\\\vec{\\\\ell }\\\\rbrace \\\\cup \\\\mathcal {E}_x.$ $\\\\mathcal {R}^x \\\\lbrace [y,z] \\\\,\\\\, | \\\\,\\\\, y\\\\cdot \\\\vec{\\\\ell } > x\\\\cdot \\\\vec{\\\\ell } \\\\text{ or } z\\\\cdot \\\\vec{\\\\ell } > x\\\\cdot \\\\vec{\\\\ell }\\\\rbrace \\\\cup \\\\mathcal {E}_x.$ For any subset $V$ of $\\\\mathbb {Z}^d$ we define its edge-set $E(V) \\\\lbrace [x,y] \\\\text{ ; } x \\\\in V \\\\text{ and } y \\\\in V\\\\rbrace ,$ its vertex-boundary $\\\\partial V \\\\lbrace x \\\\notin V \\\\text{ ; }\\\\exists y \\\\in V, x \\\\sim y \\\\rbrace ,$ and its edge-boundary $\\\\partial _{E} V \\\\lbrace [x,y]\\\\in E(\\\\mathbb {Z}^d)\\\\text{ ; } y \\\\in V, x \\\\notin V \\\\rbrace .$ For any subset $A \\\\subset \\\\mathbb {Z}^d$ we define its width $W(A)= \\\\max _{1\\\\le i \\\\le d}\\\\left(\\\\max _{y \\\\in A}y \\\\cdot e_i - \\\\min _{y \\\\in A}y \\\\cdot e_i\\\\right).$ For any $L,L^{\\\\prime } > 0$ and $y \\\\in \\\\mathbb {Z}^d$ we define the box $\\\\mathcal {B}_y(L,L^{\\\\prime }) = \\\\lbrace x \\\\in \\\\mathbb {Z}^d \\\\text{ ; }|(x-y)\\\\cdot \\\\vec{\\\\ell }| \\\\le L \\\\text{ and } |(x-y)\\\\cdot f_i| \\\\le L^{\\\\prime } \\\\text{ for all } i \\\\in \\\\lbrace 2,...,d\\\\rbrace \\\\rbrace ,$ and its positive boundary $\\\\partial ^{+}\\\\mathcal {B}_y(L,L^{\\\\prime }) = \\\\lbrace x \\\\in \\\\partial \\\\mathcal {B}_y(L,L^{\\\\prime }) \\\\text{ s.t.\",\n \"}|(x-y)\\\\cdot \\\\vec{\\\\ell }| > L\\\\rbrace .$ When $y = 0$ we just use the notation $\\\\mathcal {B}(L,L^{\\\\prime })$ .\",\n \"For a general random walk on a graph $(V,E)$ we define for $B \\\\subset V$ the hitting times of $B$ , $T_B = \\\\inf \\\\lbrace n \\\\ge 0 \\\\text{ : } X_n \\\\in B\\\\rbrace \\\\text{ and } T^{+}_B = \\\\inf \\\\lbrace n \\\\ge 1 \\\\text{ : } X_n \\\\in B\\\\rbrace ,$ and the exit time of $B$ $T^{\\\\mathrm {ex}}_B = \\\\inf \\\\lbrace n \\\\ge 0 \\\\text{ ; } X_n \\\\notin B\\\\rbrace .$ When $B$ is a singleton $\\\\lbrace x\\\\rbrace $ we use the notation $T_x$ .\",\n \"For $R \\\\in \\\\mathbb {R}$ we define $T_{R} = T_{\\\\mathcal {H}^{+}(R)}.$ We denote by $\\\\theta _n$ the translation of $n$ units of time.\",\n \"For $\\\\mathbb {P}$ a probability measure, $A_1,\\\\cdots ,A_n$ events and $X$ a random variable the notation $\\\\mathbb {E}[A_1,\\\\cdots ,A_n,X]$ denotes $\\\\mathbb {E}[{1}_{A_1}\\\\cdots {1}_{A_n}X]$ .\",\n \"In most of the paper we will add an index $i \\\\in \\\\lbrace 1,2\\\\rbrace $ as superscript when the event depends on $X^i$ , where $X^i$ are two realisations of the random walk (independent with respect to the quenched distribution).\",\n \"For example, $T^{i}_R$ will be the hitting time of $\\\\mathcal {H}^{+}(R)$ by the $i$ -th random walk.\",\n \"In the following we will use the notations $c$ , $C$ , $a$ to denote constants which only depend on the dimension $d$ and the strength of the bias $\\\\lambda $ , its direction $\\\\vec{\\\\ell }$ and a parameter $K > 0$ .\"\n ],\n [\n \"Probability measures.\",\n \"Let us recall that $P^\\\\omega $ denotes the quenched law while $\\\\mathbb {P}$ denotes the annealed law of the walk.\",\n \"As we already mentioned we will often consider two walks $X^1_{\\\\cdot }$ and $X^2_{\\\\cdot }$ , for these walks it is useful to introduce two probability measures.\",\n \"The first is for two walks that evolve independently (according to the law defined in (REF )) in the same environment.\",\n \"It is characterised by the formula: $\\\\mathbb {P}_{U_1, U_2}[X^1_{\\\\cdot } \\\\in A_1, X^2_{\\\\cdot } \\\\in A_2] \\\\mathbf {E}\\\\left[ P_{U_1}^\\\\omega \\\\left( X^1_{\\\\cdot } \\\\in A_1 \\\\right) \\\\times P_{U_2}^\\\\omega \\\\left( X^2_{\\\\cdot } \\\\in A_2 \\\\right) \\\\right].$ It is also possible to define a measure according to which the two walks evolve with the same quenched law in two independent environments $\\\\omega _1, \\\\omega _2$ sampled according to $\\\\mathbf {P}$ .\",\n \"In formulas, $Q_{U_1, U_2}[X^1_{\\\\cdot } \\\\in A_1, X^2_{\\\\cdot } \\\\in A_2] &\\\\mathbf {E}\\\\left[ P_{U_1}^{\\\\omega _1}\\\\left( X^1_{\\\\cdot } \\\\in A_1 \\\\right) \\\\times P_{U_2}^{\\\\omega _2}\\\\left( X^2_{\\\\cdot } \\\\in A_2 \\\\right) \\\\right] \\\\\\\\ &= \\\\mathbb {P}_{U_1}\\\\left( X^1_{\\\\cdot } \\\\in A_1 \\\\right) \\\\times \\\\mathbb {P}_{U_1}\\\\left( X^2_{\\\\cdot } \\\\in A_2 \\\\right).\\\\nonumber $ We denote by $\\\\mathbb {E}_{U_1,U_2}[\\\\cdot ]$ the expectation with respect to the measure (REF ).\",\n \"This is the same notation as the annealed law of one walk (with the exception of the two starting points), this because by considering the marginal of $X^1_{\\\\cdot }$ that is the same as the law of the single walk.\",\n \"$\\\\mathbb {E}^Q[\\\\cdot ]$ is the expectation associated to (REF ).\",\n \"For $a \\\\in \\\\mathbb {R}^{2d}$ and $x \\\\in \\\\mathbb {Z}^d$ we write $\\\\mathbb {P}_x^{a}$ (resp.\",\n \"$\\\\mathbb {P}_{x,y}^{a}$ ) for the annealed law where we set the conductances in $\\\\mathcal {E}_x$ (resp.\",\n \"in $\\\\mathcal {E}_x$ and $\\\\mathcal {E}_y$ ) to the value $a$ .\",\n \"A special case is when $a = (K,\\\\ldots ,K)$ with $K > 0$ , in that case we write $\\\\mathbb {P}_x^{K}$ (resp.\",\n \"$\\\\mathbb {P}_{x,y}^{K}$ ).\",\n \"The notation $Q_{x,y}^{a}$ means the law $\\\\mathbb {P}_x^{a}\\\\otimes \\\\mathbb {P}_y^{a}$ .\",\n \"In accordance with the notation of (that we will recall in Section REF ) we write $\\\\mathbb {P}_x^{K}(\\\\cdot |D=+\\\\infty )$ for the law of the future of the walk after a regeneration time at $x$ .\",\n \"We also write $Q_{x,y}^K(\\\\cdot |D^{\\\\otimes }=+\\\\infty ) = \\\\mathbb {P}_x^{K}(\\\\cdot |D=+\\\\infty )\\\\otimes \\\\mathbb {P}_y^{K}(\\\\cdot |D=+\\\\infty )$ .\"\n ],\n [\n \"Joint regeneration levels\",\n \"For this model a regeneration structure was introduced in .\",\n \"It was further improved in to obtain the independence of trajectories between regeneration times.\",\n \"For our purpose it will be more convenient to have joint regeneration levels.\",\n \"Let us now introduce a joint regeneration structure for two walks in the same environment.\",\n \"Figure: Two trajectories and the first joint regeneration level ℒ 1 \\\\mathcal {L}_1.In principle a regeneration time is simply a new maximum which will be a minimum for the future of the trajectory.\",\n \"However, for technical reasons we want to work with a subset of those regeneration times: those happening on points with bounded conductances.\",\n \"This is why we will adapt the construction of which uses $K$ -open ladder points.\",\n \"The first step is to introduce the notion of joint $K$ -open ladder point and we do this in the next subsection.\"\n ],\n [\n \"Notations and preliminary results\",\n \"A point $z \\\\in \\\\mathbb {Z}^d$ is said to be $K$ -open if for all $e \\\\in \\\\mathcal {E}_x$ we have $c_{*}(e) \\\\in [1/K,K]$ (we call such edges $K$ -normal).\",\n \"Joint $K$ -open ladder points are new maxima in the bias direction for both walks such that the points are $K$ -open and the maxima are close in the sense that they are both at roughly the same distance, say $\\\\approx R$ , from the origin in the bias direction.\",\n \"Note that the distance cannot be exactly the same due to the lattice structure.\",\n \"The main goal of this subsection is to prove that the first joint $K$ -open ladder point is close enough to the origin in the bias direction.\",\n \"The main result, that will be used in the rest of the paper is Proposition REF which follows from Lemma REF .\",\n \"Lemma REF which is related to Proposition REF will also be instrumental for the rest of this work.\",\n \"The proofs and the other results are purely technical and will not be used in the rest of the paper.\",\n \"Let us first introduce some useful quantities and results from .\",\n \"We introduce the inner positive boundary of $\\\\mathcal {B}_{x}(L, L^{\\\\prime })$ $\\\\partial _{\\\\mathrm {in}}^+ \\\\mathcal {B}_{x}(L, L^{\\\\prime }) \\\\lbrace z \\\\in \\\\mathcal {B}_{x}(L, L^{\\\\prime }), \\\\text{ where } z \\\\sim y \\\\text{ with } y \\\\in \\\\partial ^+ \\\\mathcal {B}_{x}(L, L^{\\\\prime }) \\\\rbrace .$ We consider two walks $(X^i_n)_{n\\\\ge 0}$ for $i\\\\in \\\\lbrace 1,2\\\\rbrace $ and define the events $ A^{i}(L) = \\\\lbrace T^i_{\\\\partial \\\\mathcal {B}_{X^{i}_{0}}(L, L^\\\\alpha )} \\\\ge T^i_{\\\\partial ^+_{\\\\mathrm {in}} \\\\mathcal {B}_{X^{i}_{0}}(L, L^\\\\alpha )}\\\\rbrace , \\\\quad \\\\mathrm {and} \\\\quad A(L) A^{1}(L) \\\\cap A^{2}(L).$ In all the following we write $\\\\mathcal {B}_{i} = \\\\mathcal {B}_{X_{0}^{i}}$ and $\\\\mathcal {B} = \\\\mathcal {B}_{0}$ .\",\n \"Lemma 3.1 (Analog of ) For $\\\\alpha > d + 3$ and any $L > 0$ we have $\\\\sup _{U_1, U_2 \\\\in \\\\mathbb {Z}^d} \\\\mathbb {P}_{U_1, U_2}\\\\left( A(L)^c \\\\right) \\\\le C e^{-c L}.$ where the constants $C,c$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d$ .\",\n \"Fix any $U_1, U_2 \\\\in \\\\mathbb {Z}^d$ .\",\n \"We aim to bound $\\\\mathbb {P}_{U_1, U_2}\\\\left( A(L)^c \\\\right) &= \\\\mathbb {P}_{U_1, U_2}\\\\left( A^1(L)^c \\\\cup A^2(L)^c \\\\right) \\\\le \\\\mathbb {P}_{U_1}\\\\left( A^1(L)^c \\\\right) + \\\\mathbb {P}_{U_2}\\\\left( A^2(L)^c \\\\right),$ where the inequality is due to a union bound.\",\n \"By we have that $\\\\mathbb {P}_{0}\\\\left[ A^{i}(L)^c \\\\right] \\\\le C \\\\exp (-c L)$ , thanks to translation invariance of the environment we get $\\\\mathbb {P}_{U_1, U_2}\\\\left( A(L)^c \\\\right) \\\\le C e^{-c L},$ as desired.\",\n \"We say that a vertex $x \\\\in \\\\mathcal {B}(L, L^\\\\alpha )$ is $K$ -$L$ -closed if there exists a neighbour $y \\\\notin \\\\mathcal {H}^+(L)$ of $x$ such that $c^*([x, y]) \\\\notin [1/K, K]$ .\",\n \"Let $\\\\bar{K}_x(L)$ be the $K$ -$L$ -closed connected component of $x$ (set it to $\\\\lbrace x\\\\rbrace $ in case it is empty) and let us define the event $ B(L) = \\\\left\\\\lbrace \\\\textnormal {for all } x \\\\in \\\\partial _{\\\\mathrm {in}}^+ \\\\mathcal {B}_1(L, L^\\\\alpha ) \\\\cup \\\\partial _{\\\\mathrm {in}}^+ \\\\mathcal {B}_2(L, L^\\\\alpha ), \\\\textnormal { } |\\\\bar{K}_x(L)| \\\\le \\\\log (L) \\\\right\\\\rbrace .$ Lemma 3.2 (Analog of ) For any $M<\\\\infty $ , we can find $K_0$ such that for all $K \\\\ge K_0$ $\\\\mathbf {P}\\\\left( B(L)^c \\\\right) \\\\le C L^{-M}.$ where $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Thanks to we have the bound for the event $\\\\widehat{B}(L) = \\\\lbrace \\\\textnormal {for all } x \\\\in \\\\partial _{\\\\mathrm {in}}^+ \\\\mathcal {B}(L, L^\\\\alpha ), \\\\textnormal { we have } |\\\\bar{K}_x(L)| \\\\le \\\\log (L) \\\\rbrace ,$ $\\\\mathbf {P}\\\\left( \\\\widehat{B}(L)^c \\\\right) \\\\le C L^{-M}.$ Just as in the proof of the previous Lemma, we get the result by union bound and translation invariance of the the measure $\\\\mathbf {P}$ .\",\n \"We also recall the following useful lemma.\",\n \"Lemma 3.3 Take $G \\\\ne \\\\varnothing $ to be a finite connected subset of $\\\\mathbb {Z}^d$ .\",\n \"Assume each edge $e$ of $\\\\mathbb {Z}^d$ is assigned a positive conductance $c(e)$ and that there exists a constant $c_1>0$ , $x \\\\in \\\\partial G$ and $y \\\\in G$ such that $x \\\\sim y$ and $c([x, y]) \\\\ge c_1 c(e)$ for any $e \\\\in \\\\partial _{E} G$ .\",\n \"We have $P_y\\\\left( T_x \\\\le T_{\\\\partial G} \\\\right) \\\\ge \\\\frac{c_1}{4d}|G|^{-1},$ where $P_y$ is the law of the random walk on the family of conductances $\\\\lbrace c(e)\\\\rbrace _{e \\\\in E(\\\\mathbb {Z}^d)}$ , starting from $y$ .\"\n ],\n [\n \"Properties of joint $K$ -open ladder points\",\n \"For $i \\\\in \\\\lbrace 1,2\\\\rbrace $ we recall the notation $T^i_R$ defined in (REF ).\",\n \"We introduce $\\\\mathcal {M}^{\\\\bullet } = \\\\mathcal {M}^{\\\\bullet (K)} = \\\\inf \\\\Big \\\\lbrace R > &\\\\max \\\\lbrace |X^{1}_0\\\\cdot \\\\vec{\\\\ell }|, |X^{2}_0\\\\cdot \\\\vec{\\\\ell }|\\\\rbrace \\\\text{ ; for } i\\\\in \\\\lbrace 1,2\\\\rbrace ,\\\\text{ } X^i_{T^i_{R}} \\\\text{ is } K\\\\text{-open, } \\\\nonumber \\\\\\\\ &X^i_{j} \\\\cdot \\\\vec{\\\\ell } < X^i_{T^i_{R}-2} \\\\cdot \\\\vec{\\\\ell } \\\\text{ for } j < T^i_{R}-2,\\\\\\\\& \\\\nonumber X^i_{T^i_{R}} = X^i_{T^i_{R}-1} + e_1,X^i_{T^i_{R}} = X^i_{T^i_{R}-2} + 2e_1 \\\\Big \\\\rbrace .$ In general, we will put a bullet when referring to quantities related to two walks evolving in the same environment to avoid confusion with similar quantities for a single walk or two walks in independent environments.\",\n \"Since the definition of $\\\\mathcal {M}^{\\\\bullet }$ corresponds to a level, the joint $K$ -open ladder point is then $(X^1_{T^1_{\\\\mathcal {M}^{\\\\bullet }}},X^2_{T^2_{\\\\mathcal {M}^{\\\\bullet }}})$ .\",\n \"However, as we will show in the rest of the article, it is much more convenient to work with levels rather than time or points when we consider two walks.\",\n \"The main proposition of this subsection is the following.\",\n \"Proposition 3.4 For any $M > 0$ there exists $K_0 > 0$ such that for $K\\\\ge K_0$ we have $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}}\\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {M}^{\\\\bullet (K)} > n) \\\\le Cn^{-M},$ where we have $\\\\mathcal {U} \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ and $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"To prove this result we require a technical lemma, for $1 \\\\le k \\\\le n$ , let us define the events $R^{(K)}(n k) \\\\left\\\\lbrace \\\\mathcal {M}^{\\\\bullet (K)} > kn + 3 \\\\right\\\\rbrace .$ Lemma 3.5 For any $\\\\varepsilon ^{\\\\prime } > 0$ and $M > 0$ , we can find $K_0 = K_0(\\\\varepsilon ^{\\\\prime }, M)$ large enough such that for any $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ with $|U_1\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and $|U_2\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ , all $K > K_0$ , $1< k \\\\le n$ we have $\\\\displaystyle \\\\mathbb {P}_{U_1,U_2}\\\\left(R^{(K)}(n k)\\\\right)\\\\le \\\\left(1-cn^{-\\\\varepsilon ^{\\\\prime }}\\\\right)\\\\mathbb {P}_{U_1,U_2}\\\\left(R^{(K)}(n (k-1))\\\\right) + Cn^{-M},$ where the constants $C,c$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We postpone the proof of Lemma REF after the proof of the main result.\",\n \"[Proof of Proposition REF ] This Proposition is the analogous of .\",\n \"In particular, one can iterate Lemma REF to get a tail estimate on the quantity $\\\\mathcal {M}^{\\\\bullet (K)}$ .\",\n \"Indeed, $\\\\mathbb {P}_{U_1,U_2}\\\\left(R^{(K)}(n k)\\\\right)\\\\le \\\\left(1-cn^{-1/2}\\\\right)\\\\mathbb {P}_{U_1,U_2}\\\\left(R^{(K)}(n (k-1))\\\\right) + Cn^{-M},$ implies, by induction, that $\\\\mathbb {P}_{U_1,U_2}\\\\left(R^{(K)}(n^2)\\\\right)\\\\le \\\\left(1-cn^{-1/2}\\\\right)^n + Cn^{-M + 1} \\\\le 2Cn^{-M + 1}.$ In the end we get $\\\\mathbb {P}_{U_1,U_2}\\\\left(\\\\mathcal {M}^{\\\\bullet (K)} > n^2 + 3\\\\right) \\\\le 2Cn^{-M + 1},$ or equivalently $\\\\mathbb {P}_{U_1,U_2}\\\\left(\\\\mathcal {M}^{\\\\bullet (K)} > n\\\\right) \\\\le Cn^{-(M + 1)/2},$ which is enough to conclude since $M$ is arbitrary.\",\n \"Figure: Black points represent the KK-open points.\",\n \"We look at the successive attempts to reach a joint KK-open ladder point when the walks exit boxes.\",\n \"[Proof of Lemma REF ] The proof essentially goes the same way as .\",\n \"Fix $U_1,U_2 \\\\in \\\\mathcal {U}$ as in the hypothesis of the Lemma.\",\n \"For $i = 1, 2$ let us denote as $\\\\widetilde{\\\\mathcal {B}}^i_n$ the largest box of the form $\\\\mathcal {B}_i(m, m^\\\\alpha )$ with $m \\\\in \\\\mathbb {R}_+$ contained in $\\\\mathcal {H}^{-}(n)$ .\",\n \"Note that $ \\\\mathcal {B}_i((n-1), (n-1)^\\\\alpha ) \\\\subset \\\\widetilde{\\\\mathcal {B}}^i_n \\\\subset \\\\mathcal {B}_i(n+1, (n+1)^\\\\alpha )$ .\",\n \"Recall that $\\\\bar{K}_x(n)$ denotes the $K$ -$n$ -closed connected component of $x$ defined just before (REF ).\",\n \"We introduce $\\\\mathcal {K}^i(n) \\\\bar{K}_{X_{T^i_{\\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}^i_n}}} (n) \\\\subseteq \\\\widetilde{\\\\mathcal {B}}^i_n, \\\\quad i = 1, 2.$ We then set $\\\\mathcal {K}(n) = \\\\mathcal {K}^1(n) \\\\cup \\\\mathcal {K}^2(n)$ .\",\n \"In case $T^i_{\\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}^i_n} = + \\\\infty $ we set $\\\\mathcal {K}^i(n) = \\\\varnothing $ and $\\\\partial \\\\mathcal {K}^i(n) = \\\\varnothing $ .\",\n \"We recall the definition of $A(L)$ , $B(L)$ in (REF ),(REF ).\",\n \"We slightly modify them and define, with a little abuse of the notation, $A^{i}(n) = \\\\left\\\\lbrace T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}_{X^{i}_{0}}(n, n^\\\\alpha )} \\\\ge T^i_{\\\\partial ^+_{\\\\mathrm {in}} \\\\widetilde{\\\\mathcal {B}}_{X^{i}_{0}}(n, n^\\\\alpha )}\\\\right\\\\rbrace , \\\\quad \\\\mathrm {and} \\\\quad A(n) A^{1}(n) \\\\cap A^{2}(n);$ $B(n) = \\\\left\\\\lbrace \\\\textnormal {for all } x \\\\in \\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}_{X^{1}_{0}}(n, n^\\\\alpha )\\\\cup \\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}_{X^{2}_{0}}(n, n^\\\\alpha ), \\\\textnormal { we have } |\\\\bar{K}_x(n)| \\\\le \\\\log (n) \\\\right\\\\rbrace .$ Notice that the estimates derived in Lemmas REF REF would still hold thanks to the fact that $ \\\\mathcal {B}_i((n-1), (n-1)^\\\\alpha ) \\\\subset \\\\widetilde{\\\\mathcal {B}}^i_n \\\\subset \\\\mathcal {B}_i(n+1, (n+1)^\\\\alpha )$ .\",\n \"Moreover, we introduce the events: $&\\\\bullet \\\\hspace{3.00003pt}C(n) \\\\left\\\\lbrace \\\\text{for all } x \\\\in \\\\partial \\\\mathcal {K}(n)\\\\cap \\\\mathcal {H}^+(n), \\\\text{ the vertex } x \\\\text{ is } K\\\\textnormal {-open} \\\\right\\\\rbrace .\",\n \"\\\\\\\\\\\\vspace{2.125pt}&\\\\bullet \\\\hspace{3.00003pt} D^{i}(n) \\\\left\\\\lbrace T^i_{\\\\partial \\\\mathcal {K}^{i}(n)} \\\\circ \\\\theta _{T^i_{\\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}^i_n}} = T^i_{\\\\partial \\\\mathcal {K}^{i}(n) \\\\cap \\\\mathcal {H}^+(n)} \\\\circ \\\\theta _{T^i_{\\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}^i_n}} \\\\right\\\\rbrace .\",\n \"\\\\\\\\\\\\vspace{2.125pt}&\\\\bullet \\\\hspace{3.00003pt} E^{i}(n) \\\\left\\\\lbrace X^i_{T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n} + 2} = X^i_{T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n} + 1} + e_1 = X^i_{T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n}} + 2 e_1 \\\\textnormal { and } X^i_{T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n} + 2}, X^i_{T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n} + 1} \\\\textnormal { are \\\\right.K-open}.$ Furthermore, define $D(n) = D^{1}(n) \\\\cap D^{2}(n)$ and $E(n) = E^{1}(n) \\\\cap E^{2}(n)$ .\",\n \"Let us consider the event $A(n) \\\\cap C(n) \\\\cap D(n) \\\\cap E(n)$ then it is possible to notice the following: On $A(n)$ , defined in (REF ), for $i = 1, 2$ , we have $T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n} \\\\ge T^i_{\\\\partial ^+_{\\\\mathrm {in}} \\\\widetilde{\\\\mathcal {B}}^i_n}$ so that $T^i_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^i_n} \\\\ge T^i_{\\\\partial \\\\mathcal {K}(n)} \\\\circ \\\\theta _{T^i_{\\\\partial _{\\\\mathrm {in}} \\\\widetilde{\\\\mathcal {B}}^i_n}}.$ On $D(n)$ , for $i = 1, 2$ , we have $T_{\\\\partial \\\\mathcal {K}^{i}(n)} \\\\circ \\\\theta _{T^i_{\\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}^i_n}} = T_{\\\\partial \\\\mathcal {K}^{i}(n) \\\\cap \\\\mathcal {H}^+(n)} \\\\circ \\\\theta _{T^i_{\\\\partial _{\\\\mathrm {in}}^+ \\\\widetilde{\\\\mathcal {B}}^i_n}}.$ On $C(n)$ , for $i = 1, 2$ , we have that all points in $\\\\partial \\\\mathcal {K}^i(n) \\\\cap \\\\mathcal {H}^+(n)$ are $K$ -open.\",\n \"Putting these observations together one can see that, for $i \\\\in \\\\lbrace 1,2\\\\rbrace $ , $X^i_{T^i_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^i_n}}$ is a new maximum in the direction $\\\\vec{\\\\ell }$ which is also $K$ -open and the next two steps are in the direction $e_1$ on $K$ -open points, so that $\\\\lbrace A(n) \\\\cap C(n) \\\\cap D(n) \\\\cap E(n)\\\\rbrace \\\\subset \\\\lbrace \\\\mathcal {M} \\\\le n + 3 \\\\rbrace .$ By following the steps of from Equation $(6.6)$ to $(6.10)$ we get the following $\\\\mathbb {P}\\\\left( R(kn) \\\\right) &\\\\le \\\\mathbb {P}\\\\left( A(kn)^c \\\\right) + \\\\mathbb {P}\\\\left( B(kn)^c \\\\right) + \\\\cdots \\\\\\\\&+ \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn)^c \\\\right) + \\\\cdots \\\\\\\\&+ \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\\\right) + \\\\cdots \\\\\\\\&+ \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\\\right) .$ The first two terms in (REF ) are controlled, respectively by Lemma REF and Lemma REF .\",\n \"In order to finish the proof we need to control the terms in (), () and ().\",\n \"Step 1: Control of the term ().\",\n \"Recall the definition of $\\\\mathcal {K}(kn)$ at the start of this proof.\",\n \"For $k \\\\le n$ , on the event $A(kn) \\\\cap B(kn)$ it holds that $|\\\\mathcal {K}(kn)| \\\\le 4 \\\\log (n),$ since for both $i = 1, 2$ it holds that $|\\\\mathcal {K}^i(kn)| \\\\le 2 \\\\log (n)$ .\",\n \"We deduce that the term $\\\\mathbb {P}( R((k-1)n), A(kn), B(kn), C(kn)^c )$ can be rewritten as $&\\\\sum _{i =1, 2} \\\\sum _{\\\\begin{array}{c}F^i \\\\subset \\\\mathbb {Z}^d, \\\\\\\\ |F^i|\\\\le 2 \\\\log (n)\\\\end{array}} \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn)^c, \\\\mathcal {K}(kn) = F^1 \\\\cup F^2 \\\\right) \\\\\\\\& = \\\\sum _{\\\\begin{array}{c}i =1, 2, \\\\\\\\F^i \\\\subset \\\\mathbb {Z}^d, \\\\\\\\ |F^i|\\\\le 2 \\\\log (n)\\\\end{array}} \\\\mathbf {E} \\\\left[ P^\\\\omega \\\\left( R((k-1)n), A(kn), B(kn), \\\\mathcal {K}(kn) = F^1 \\\\cup F^2 \\\\right) {1}_{\\\\begin{array}{c}\\\\lbrace \\\\text{some }x \\\\in \\\\partial (F^1 \\\\cup F^2) \\\\cap \\\\mathcal {H}^+(kn) \\\\\\\\ \\\\text{ is closed} \\\\rbrace \\\\end{array}} \\\\right].$ We may notice that: $P^\\\\omega \\\\left( R((k-1)n), A(kn), B(kn), \\\\mathcal {K}(kn) = F^1 \\\\cup F^2 \\\\right)$ is a $\\\\mathbf {P}$ -random variable measurable with respect to the sigma algebra $\\\\sigma \\\\left(c([x, y]), \\\\,\\\\, x, y \\\\notin \\\\mathcal {H}^+(kn) \\\\right)$ .\",\n \"The event $\\\\left\\\\lbrace \\\\text{some }x \\\\in \\\\partial (F^1 \\\\cup F^2) \\\\cap \\\\mathcal {H}^+(kn) \\\\text{ is closed} \\\\right\\\\rbrace $ is measurable with respect to the sigma algebra $\\\\sigma \\\\left(c([x, y]), \\\\,\\\\, x, y \\\\in \\\\mathcal {H}^+(kn)\\\\right)$ .\",\n \"Hence we have $\\\\mathbf {P}$ -independence between these two random objects.\",\n \"Hence, after some standard manipulation last sum can be rewritten as $\\\\sum _{i =1, 2} \\\\sum _{\\\\begin{array}{c}F^i \\\\subset \\\\mathbb {Z}^d,\\\\\\\\ |F^i|\\\\le 2 \\\\log (n)\\\\end{array}} &\\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), \\\\mathcal {K}(kn) = F^1 \\\\cup F^2 \\\\right) \\\\\\\\& \\\\times \\\\left( 1 - \\\\mathbf {P}\\\\left( \\\\text{all }x \\\\in \\\\partial (F^1 \\\\cup F^2) \\\\cap \\\\mathcal {H}^+(kn) \\\\text{ are open} \\\\right) \\\\right)$ Since $|F^{i}| \\\\le 2\\\\log (n)$ and that $|F^1 \\\\cup F^2| \\\\le |F^1| + |F^2|$ we have $\\\\mathbf {P}\\\\left( \\\\text{all }x \\\\in \\\\partial (F^1 \\\\cup F^2) \\\\cap \\\\mathcal {H}^+(kn) \\\\text{ are open} \\\\right) & \\\\ge \\\\left(1 - \\\\varepsilon (K)\\\\right)^{2d |F^1 \\\\cup F^2|}\\\\\\\\& \\\\ge \\\\left(1 - \\\\varepsilon (K)\\\\right)^{8d\\\\log (n)} \\\\\\\\& = n^{8d \\\\log (1 - \\\\varepsilon (K))},$ where $\\\\varepsilon (K) \\\\mathbf {P}\\\\left( c_* \\\\notin [1/K, K] \\\\right) $ .\",\n \"For any $\\\\varepsilon ^{\\\\prime } > 0$ it is possible to fix $K$ large enough such that $8d \\\\log (1 - \\\\varepsilon (K)) \\\\ge - \\\\varepsilon ^{\\\\prime }$ .\",\n \"Then we have $\\\\displaystyle \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn)^c \\\\right) \\\\le ( 1 - n^{-\\\\varepsilon ^{\\\\prime }}) \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn)\\\\right),$ which implies $n^{-\\\\varepsilon ^{\\\\prime }} \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn)\\\\right) \\\\le \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn) \\\\right).\",\n \"$ Step 2: Control of the term ().\",\n \"Notice that, on the event $A(n)$ we have that, for $i = 1, 2$ , $X^{i}_{T^i_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^i_{kn}}} \\\\in \\\\partial ^+_{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^i_{kn}$ and by definition of $\\\\mathcal {K}^{i}(kn)$ , $X^{i}_{T^i_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^i_{kn}}} \\\\in \\\\mathcal {K}^{i}(kn)$ .\",\n \"Moreover on $B(kn)$ we also have that $|\\\\mathcal {K}^{i}(kn)| \\\\le 2 \\\\log (n)$ .\",\n \"Hence $&\\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\\\right) \\\\\\\\&\\\\le \\\\sum _{i =1, 2}\\\\sum _{z^i} \\\\sum _{F^i} \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), \\\\bigcap _{i = 1, 2} \\\\lbrace X^{i}_{T^i_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}_{kn}}} = z^i, \\\\mathcal {K}^{i}(kn) = F^i \\\\rbrace , C(kn), D(kn)^c \\\\right),$ where $\\\\displaystyle \\\\sum _{z^i} \\\\sum _{F^i}$ is a shortcut for $\\\\displaystyle \\\\sum _{z^i \\\\in \\\\partial ^+_{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^i_{kn}} \\\\sum _{\\\\begin{array}{c}F^i \\\\subset \\\\mathbb {Z}^d \\\\\\\\ |F^i| \\\\le 2 \\\\log (n)\\\\\\\\ z^i \\\\in F^i\\\\end{array}}$ .\",\n \"For $\\\\omega $ fixed, the events $\\\\displaystyle R((k-1) n ), A(kn), B(kn), \\\\bigcap _{i = 1, 2} \\\\left\\\\lbrace X^{i}_{T^i_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}_{kn}}} = z^i, \\\\mathcal {K}^{i}(kn) = F^i \\\\right\\\\rbrace $ are $P^\\\\omega $ -measurable with respect to $\\\\sigma \\\\Big (\\\\lbrace X^1_j, j \\\\le T^1_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^1_{kn}}\\\\rbrace , \\\\lbrace X^2_k, k \\\\le T^2_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^2_{kn}} \\\\rbrace \\\\Big )$ .\",\n \"Thus, thanks to Lemma REF and the quenched independence of the two walks, we can write $\\\\mathbb {P}&\\\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\\\right) \\\\nonumber \\\\\\\\&\\\\le \\\\sum _{i =1, 2} \\\\sum _{z^i} \\\\sum _{F^i} \\\\mathbf {E}\\\\bigg [ P^\\\\omega \\\\Big ( R((k-1)n), A(kn), B(kn), \\\\bigcap _{i = 1, 2} \\\\lbrace X^{i}_{T^i_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^i_{kn}}} = z^i, \\\\mathcal {K}^{i}(kn) = F^i \\\\rbrace \\\\Big ) \\\\nonumber \\\\\\\\& \\\\times P_{z^1, z^2}^\\\\omega \\\\left(T^1_{\\\\partial F^1} < T^1_{\\\\partial F^1 \\\\cap \\\\mathcal {H}^+(kn)} \\\\textnormal { or } T^2_{\\\\partial F^2} < T^2_{\\\\partial F^2 \\\\cap \\\\mathcal {H}^+(kn)}\\\\right) {1}_{\\\\lbrace x \\\\in \\\\partial F \\\\cap \\\\mathcal {H}^+(kn) \\\\text{ is open} \\\\rbrace } \\\\bigg ].\",\n \"$ For both $F^1$ and $F^2$ we can notice thanks to the definition of $\\\\mathcal {K}(kn)$ and the fact that we are considering the event $\\\\lbrace x \\\\in \\\\partial F \\\\cap \\\\mathcal {H}^+(kn) \\\\text{ is } K\\\\textnormal {-open} \\\\rbrace $ we get that all the edges $e \\\\in \\\\partial _E F$ are $K$ -normal.\",\n \"Just as a remark also notice that $F^1$ and $F^2$ are either the same set or disjoint sets, so overall we get that $\\\\partial _E F^1$ and $\\\\partial _E F^2$ are two (possibly one) sets of normal edges.\",\n \"We obtain a bound for $P_{z^1}^\\\\omega (T^1_{\\\\partial F^1} \\\\ge T^1_{\\\\partial F^1 \\\\cap \\\\mathcal {H}^+(kn)})$ , the case $P_{z^2}^\\\\omega (T^2_{\\\\partial F^2} \\\\ge T^2_{\\\\partial F^2 \\\\cap \\\\mathcal {H}^+(kn)})$ can be treated by following the same reasoning.\",\n \"For any $z^1 \\\\in F^1 \\\\cap \\\\partial ^+_{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}^1_{kn}$ , there exists a neighbour $y^1 \\\\in \\\\mathcal {H}^+(kn)$ .\",\n \"Thanks to the fact that $F^1 \\\\subset \\\\widetilde{\\\\mathcal {B}}_{kn}$ we have that, for any $x \\\\in \\\\partial F^1 \\\\cup F^1$ , $(y^1 - x) \\\\cdot \\\\vec{\\\\ell } \\\\ge - 1$ and $(z^1 - x) \\\\cdot \\\\vec{\\\\ell } \\\\ge - 2$ .\",\n \"Then, thanks to this observation and the definition of $K$ -normal edges, we see that $c^\\\\omega (e) \\\\le K^2 e^{3 \\\\lambda } c^{\\\\omega }([z^1, y^1]), \\\\quad \\\\textnormal {for } e \\\\in \\\\partial _E F^1.$ Thus we can apply Lemma REF with $G= F^1$ (resp.\",\n \"$F^2$ ) and obtain, thanks to quenched independence $P_{z^1, z^2}^\\\\omega &\\\\left(T^1_{\\\\partial F^1} < T^1_{\\\\partial F^1 \\\\cap \\\\mathcal {H}^+(kn)} \\\\textnormal { or } T^2_{\\\\partial F^2} < T^2_{\\\\partial F^2 \\\\cap \\\\mathcal {H}^+(kn)}\\\\right) \\\\\\\\ &\\\\le P_{z^1, z^2}^\\\\omega \\\\left(T^1_{\\\\partial F^1} < T^1_{y^1} \\\\textnormal { or } T^2_{\\\\partial F^2} < T^2_{y^2}\\\\right) \\\\\\\\& = \\\\left( 1 - P_{z^1}^\\\\omega \\\\left( T^1_{y^1} \\\\le T^1_{\\\\partial F^1}\\\\right) P_{z^2}^\\\\omega \\\\left( T^2_{y^2} \\\\le T^2_{\\\\partial F^2}\\\\right) \\\\right) \\\\\\\\&\\\\le (1 - c\\\\log (n)^{-2}),$ using $|F^1| \\\\le 2 \\\\log (n)$ .\",\n \"By plugging it in Equation (REF ) we get $\\\\mathbb {P}&\\\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\\\right) \\\\\\\\&\\\\le \\\\sum _{i =1, 2} \\\\sum _{z^i} \\\\sum _{F^i} \\\\mathbf {E}\\\\bigg [ P^\\\\omega \\\\bigg ( R((k-1)n), A(kn), B(kn), \\\\bigcap _{i = 1, 2} \\\\Big \\\\lbrace X^{i}_{T^i_{\\\\partial _{\\\\text{in}} \\\\widetilde{\\\\mathcal {B}}_{kn}}} = z^i, \\\\mathcal {K}^{i}(kn) = F^i \\\\Big \\\\rbrace \\\\bigg ) \\\\\\\\& \\\\times {1}_{\\\\lbrace x \\\\in \\\\partial F \\\\cap \\\\mathcal {H}^+(kn) \\\\text{ is open} \\\\rbrace } \\\\bigg ] (1 - c\\\\log (n)^{-2})\\\\\\\\&\\\\le (1 - c\\\\log (n)^{-2}) \\\\mathbb {P}\\\\left( R((k - 1)n ), A(n), B(n), C(n) \\\\right).$ Which readily implies that for some positive constant $c>0$ we get $c\\\\log (n)^{-2} & \\\\mathbb {P}\\\\left( R((k - 1)n ), A(n), B(n), C(n) \\\\right) \\\\nonumber \\\\\\\\&\\\\le \\\\mathbb {P}\\\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn) \\\\right).\",\n \"$ Step 3: Control of the term ().\",\n \"On the event $A(n)\\\\cap B(n) \\\\cap C(n) \\\\cap D(n)$ we know that both $X^1_{T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_{kn}}} \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^1_{kn}$ and $X^2_{T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_{kn}}} \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^2_{kn}$ are $K$ -open new maxima in the direction $\\\\vec{\\\\ell }$ .\",\n \"Moreover, the event $E(kn)$ does not depend on what happens in the “double box” $\\\\widetilde{\\\\mathcal {B}}^1_{kn} \\\\cup \\\\widetilde{\\\\mathcal {B}}^2_{kn}$ .\",\n \"Let us also introduce the concept of $x, y$ -open vertex, we say that $z$ is $x, y$ -open if all the edges in the set $\\\\mathcal {E}_{z} \\\\backslash (\\\\mathcal {E}_{x} \\\\cup \\\\mathcal {E}_{y})$ are $K$ -normal.\",\n \"Let us also define $R^{\\\\prime }((k - 1)n) = R((k - 1)n) \\\\cap A(n)\\\\cap B(n) \\\\cap C(n) \\\\cap D(n).$ Recall (see (REF ) and considerations above) that on this event we have that both walks, when they hit level $kn$ , are at new $K$ -open maxima.\",\n \"We can write $\\\\mathbb {P}&\\\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\\\right) \\\\\\\\&\\\\le \\\\sum _{i = 1, 2} \\\\sum _{x^i \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^i_{kn}} \\\\mathbb {P}\\\\left( R^{\\\\prime }((k - 1)n), X^1_{T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_{kn}}} = x^2, \\\\lbrace x^1, x^2\\\\rbrace \\\\textnormal { are }K\\\\textnormal {-open}, E(kn)^c \\\\right)$ $&\\\\le \\\\sum _{i = 1, 2} \\\\sum _{x^i \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^i_{kn}} \\\\mathbf {E}\\\\bigg [ P^\\\\omega \\\\left( R^{\\\\prime }((k - 1)n), X^1_{T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_{kn}}} = x^2 \\\\right) {1}_{\\\\lbrace \\\\lbrace x^1, x^2\\\\rbrace \\\\textnormal { are }K\\\\textnormal {-open}\\\\rbrace } \\\\\\\\&\\\\big ( {1}_{\\\\lbrace \\\\textnormal {A vertex }\\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { is }x^1, x^2\\\\textnormal {-closed}\\\\rbrace } + {1}_{\\\\lbrace \\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { are }x^1, x^2\\\\textnormal {-open}\\\\rbrace } \\\\\\\\&\\\\times P^\\\\omega _{x^1, x^2}\\\\left( X^1_1 \\\\ne x^1 + e_1 \\\\text{ or } X^1_2 \\\\ne x^1 + 2e_1 \\\\text{ or } X^2_1 \\\\ne x^2 + e_1 \\\\text{ or } X^2_2 \\\\ne x^2 + 2e_1 \\\\right) \\\\big ) \\\\bigg ].$ On the event $\\\\lbrace \\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { are }x^1, x^2\\\\textnormal {-open}\\\\rbrace \\\\cap \\\\lbrace \\\\lbrace x^1, x^2\\\\rbrace \\\\textnormal { are }K\\\\textnormal {-open}\\\\rbrace $ one has that $P^\\\\omega _{x^i}\\\\left( X^i_1 = x^i + e_1, X^i_2 = x^i + 2e_1 \\\\right) \\\\ge c > 0$ for both $i = 1, 2$ .\",\n \"Then $\\\\mathbb {P}&\\\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\\\right) \\\\\\\\&\\\\le \\\\sum _{i = 1, 2} \\\\sum _{x^i \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^i_{kn}} \\\\mathbf {E}\\\\bigg [ P^\\\\omega \\\\left( R^{\\\\prime }((k - 1)n), X^1_{T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_{kn}}} = x^2 \\\\right) {1}_{\\\\lbrace \\\\lbrace x^1, x^2\\\\rbrace \\\\textnormal { are }K\\\\textnormal {-open}\\\\rbrace } \\\\\\\\& \\\\big ( {1}_{\\\\lbrace \\\\textnormal {A vertex in }\\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { is }x^1, x^2\\\\textnormal {-closed}\\\\rbrace } \\\\\\\\ &+ (1-c^2){1}_{\\\\lbrace \\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { are }x^1, x^2\\\\textnormal {-open}\\\\rbrace } \\\\big ) \\\\bigg ].$ The $\\\\mathbf {P}$ -random variables $P^\\\\omega \\\\Big ( R^{\\\\prime }((k - 1)n), X^1_{T_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_{kn}}} = x^2 \\\\Big )$ and ${1}_{\\\\lbrace \\\\lbrace x^1, x^2\\\\rbrace \\\\textnormal { are }K\\\\textnormal {-open}\\\\rbrace }$ are measurable with respect to $\\\\sigma ( e \\\\in E(\\\\widetilde{\\\\mathcal {B}}^1_{kn} \\\\cup \\\\widetilde{\\\\mathcal {B}}^2_{kn})\\\\cup \\\\mathcal {E}_{x^1} \\\\cup \\\\mathcal {E}_{x^2})$ , we recall that $\\\\mathcal {E}_x, x \\\\in \\\\mathbb {Z}^d$ is defined in (REF ).\",\n \"On the other hand the events $\\\\lbrace \\\\textnormal {A vertex in }\\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { is }x^1, x^2\\\\textnormal {-closed}\\\\rbrace $ and $\\\\lbrace \\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { are }x^1, x^2\\\\textnormal {-open}\\\\rbrace $ are measurable with respect to $\\\\sigma ( e \\\\notin E(\\\\widetilde{\\\\mathcal {B}}^1_{kn} \\\\cup \\\\widetilde{\\\\mathcal {B}}^2_{kn})\\\\cup \\\\mathcal {E}_{x^1} \\\\cup \\\\mathcal {E}_{x^2} )$ , thanks to the fact that we are summing over $x^i \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^i_{kn}$ .\",\n \"This implies $\\\\mathbf {P}$ -independence between these two groups of random variables.\",\n \"This yields $\\\\mathbb {P}&\\\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\\\right) \\\\\\\\& \\\\le \\\\sum _{i = 1, 2} \\\\sum _{x^i \\\\in \\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^i_{kn}} \\\\mathbf {E}\\\\left[ P^\\\\omega \\\\left( R^{\\\\prime }((k - 1)n), X^1_{T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_{kn}}} = x^2 \\\\right) \\\\right] \\\\times \\\\\\\\& \\\\times \\\\bigg ( \\\\mathbf {P}\\\\left( \\\\textnormal {A vertex in }\\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { is }x^1, x^2\\\\textnormal {-closed} \\\\right) + \\\\\\\\&+ (1 - c^{\\\\prime }) \\\\mathbf {P}\\\\left( \\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { are }x^1, x^2\\\\textnormal {-open} \\\\right) \\\\bigg ) \\\\\\\\& \\\\le (1 - c^{\\\\prime }) \\\\mathbb {P}\\\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn) \\\\right),$ since $\\\\mathbf {P}\\\\left( \\\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\\\rbrace \\\\textnormal { are }x^1, x^2\\\\textnormal {-open} \\\\right) > 0$ and its value does not depend on $x^1,x^2$ .\",\n \"This implies that there exists $c>0$ such that $c \\\\mathbb {P}&\\\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn) \\\\right) \\\\nonumber \\\\\\\\ &\\\\le \\\\mathbb {P}\\\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)\\\\right).\",\n \"$ Step 4: Conclusion.\",\n \"For any $\\\\varepsilon ^{\\\\prime } > 0$ , $M > 0$ , one can find a $K$ chosen large enough such that, putting together (REF ) with Lemma REF and Lemma REF and the estimates attained in (REF ), (REF ) and (REF ), implies that, for all $k \\\\in \\\\lbrace 2, \\\\dots , n \\\\rbrace $ $\\\\mathbb {P}\\\\left( R(kn) \\\\right) \\\\le \\\\mathbb {P}\\\\left( R((k-1)n) \\\\right)\\\\left( 1 - c\\\\log (n)^{-2} n^{-\\\\varepsilon ^{\\\\prime }} \\\\right) + n^{-M},$ which implies the result.\",\n \"For the construction of joint regeneration times we will need for the joint $K$ -open ladder points to be close.\",\n \"We introduce for $R > 0$ the variable $\\\\mathcal {M}^{\\\\bullet (K)}_R = \\\\mathcal {M^\\\\bullet }_R \\\\mathcal {M}^{\\\\bullet (K)} \\\\circ \\\\theta _{T_R} - R$ where we recall that $\\\\mathcal {M}^{\\\\bullet (K)}$ is defined in (REF ) and $\\\\theta _{T_R}$ means that the first walk is shifted by $T^{1}_R$ and the second one by $T^{2}_R$ .\",\n \"We introduce $ \\\\Upsilon (n) = \\\\bigcap _{R = 0}^{n}\\\\left\\\\lbrace \\\\mathcal {M^\\\\bullet }_R\\\\le n^{\\\\frac{1}{2}}\\\\right\\\\rbrace ,$ under this event before reaching $\\\\mathcal {H}^{+}(n)$ the successive joint $K$ -open ladder points are always at distance less than $n^{\\\\frac{1}{2}}$ in the direction $\\\\vec{\\\\ell }$ .\",\n \"Proposition 3.6 For any $M > 0$ there exists $K > 0$ $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}}\\\\mathbb {P}_{U_1, U_2}(\\\\Upsilon (n)^{c}) \\\\le Cn^{-M}.$ where we recall $\\\\mathcal {U} = \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ and $C$ only depends on $d,\\\\lambda $ .\",\n \"The proof is similar to the proof of .\",\n \"Let us consider the events, for $j \\\\in \\\\lbrace 0, \\\\dots , n\\\\rbrace $ $\\\\Upsilon _j(n) = \\\\bigcap _{R = 0}^{j - 1}\\\\left\\\\lbrace \\\\mathcal {M^\\\\bullet }_R\\\\le n^{\\\\frac{1}{2}}\\\\right\\\\rbrace \\\\quad \\\\textnormal {and} \\\\quad \\\\Theta _j(n) = \\\\left\\\\lbrace \\\\mathcal {M^\\\\bullet }_j > n^{\\\\frac{1}{2}}\\\\right\\\\rbrace .$ We can split the event over the minimal $j \\\\in \\\\lbrace 0, \\\\dots , n\\\\rbrace $ such that $\\\\Theta _j(n)$ is true and get $ \\\\mathbb {P}_{U_1, U_2}(\\\\Upsilon (n)^{c}) \\\\le \\\\sum _{j = 0}^n \\\\mathbb {P}_{U_1, U_2} \\\\left( \\\\Upsilon _j(n), \\\\Theta _j(n), T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_n} = T^1_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^1_n} , T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_n} = T^2_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^2_n} \\\\right) + C e^{-c n}.$ On the event $\\\\Upsilon _j(n) \\\\cap \\\\Theta _j(n) \\\\cap \\\\lbrace T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_n} = T^1_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^1_n} \\\\rbrace \\\\cap \\\\lbrace T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_n} = T^2_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^2_n} \\\\rbrace $ one is sure that $X^i_{T_{j}} \\\\in \\\\widetilde{\\\\mathcal {B}}^i_n$ for $i = 1, 2$ then by union bound argument $ \\\\nonumber \\\\mathbb {P}_{U_1, U_2} &\\\\left( \\\\Upsilon _j(n), \\\\Theta _j(n), T^1_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^1_n} = T^1_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^1_n}, T^2_{\\\\partial \\\\widetilde{\\\\mathcal {B}}^2_n} = T^2_{\\\\partial ^+ \\\\widetilde{\\\\mathcal {B}}^2_n} \\\\right) \\\\\\\\&\\\\nonumber \\\\le \\\\sum _{x_1, x_2 \\\\in \\\\widetilde{\\\\mathcal {B}}^1_n, \\\\widetilde{\\\\mathcal {B}}^2_n} \\\\mathbb {P}_{U_1, U_2} \\\\left( X^1_{T_j} = x_1, X^2_{T_j} = x_2, \\\\mathcal {M}_{j}^{\\\\bullet (K)} > n^{1/2} \\\\right) \\\\\\\\&\\\\le |\\\\mathcal {B}_{n+1}|^2 \\\\sup _{x_1, x_2}\\\\mathbb {P}_{x_1,x_2}\\\\left(\\\\mathcal {M}^{\\\\bullet (K)} - j \\\\ge n^{1/2}\\\\right),$ where we used Lemma REF in the last inequality and the $\\\\sup $ is over all $x_1,x_2 \\\\in \\\\widetilde{\\\\mathcal {B}}^1_n, \\\\widetilde{\\\\mathcal {B}}^2_n$ such that $x_1,x_2$ are compatible with $\\\\lbrace X^1_{T_j} = x_1, X^2_{T_j} = x_2\\\\rbrace $ .\",\n \"Now we conclude using the fact that $\\\\mathbb {P}_{x_1,x_2}\\\\left(\\\\mathcal {M}^{\\\\bullet (K)} - j \\\\ge n^{1/2}\\\\right) = \\\\mathbb {P}_{0,x_2-x_1}\\\\left(\\\\mathcal {M}^{\\\\bullet (K)} - j+ x_1\\\\cdot \\\\vec{\\\\ell } \\\\ge n^{1/2}\\\\right).$ Where we used translation invariance of the environment, now using the fact that $|x_1\\\\cdot \\\\vec{\\\\ell } - j| \\\\le 1$ and $|(x_2-x_1)\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ , the assumptions of Proposition REF are verified.\",\n \"We conclude that $\\\\sup _{x_1, x_2}\\\\mathbb {P}_{x_1,x_2}\\\\left(\\\\mathcal {M}^{\\\\bullet (K)} - j \\\\ge n^{1/2}\\\\right) \\\\le Cn^{-M},$ where $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We then conclude the domination of (REF ) using the fact that $|\\\\mathcal {B}_{n+1}| \\\\le Cn^{2d\\\\alpha }$ .\"\n ],\n [\n \"Enhanced walks\",\n \"The traditional definition of regeneration times has the drawback that it doesn't guarantee independence since the knowledge of the conductances at $x$ influences the walk at any time it reaches a neighbour of $x$ (even when that neighbour $y$ is such that $y\\\\cdot \\\\vec{\\\\ell }> x \\\\cdot \\\\vec{\\\\ell }$ ).\",\n \"In order to obtain stronger independence properties, which we need for our purpose, we need to introduce a variant in the construction of regeneration times.\",\n \"Such a task was already done in using a so-called “enhanced-walk”.\",\n \"We recall the definition of the enhanced walk which is the same as in .\",\n \"For any environment $\\\\omega $ we define two transition probabilities for $x \\\\sim y$ $p^{\\\\omega }(x,y) = \\\\frac{c_{*}(x,y)e^{(x+y)\\\\cdot \\\\ell }}{\\\\displaystyle \\\\sum _{z \\\\sim x}^{} c_{*}(x,z)e^{(x+z)\\\\cdot \\\\ell }},$ and $p_K^{\\\\omega }(x,y) = \\\\frac{(c_{*}(x,y)\\\\wedge K^{-1})e^{(x+y)\\\\cdot \\\\ell }}{\\\\displaystyle \\\\sum _{ z \\\\sim x}^{} (c_{*}(x,z)\\\\vee K)e^{(x+z)\\\\cdot \\\\ell }},$ and 0 in both cases if $x$ and $y$ are not adjacent.\",\n \"Note that, for any $x,y \\\\in \\\\mathbb {Z}^d$ , we have that $p^{\\\\omega }_{K}(x,y) \\\\le p^{\\\\omega }(x,y)$ .\",\n \"For any $(x,z) \\\\in \\\\mathbb {Z}^d \\\\times \\\\lbrace 0,1\\\\rbrace $ we define the Markov chain $(\\\\tilde{X}_{n})_{n \\\\ge 0} = (X_{n}, Z_{n})_{n \\\\ge 0}$ with law $P^\\\\omega _{(x,z)}$ with transitions probabilities $p^{\\\\omega }((x_1,z_1),(x_2,z_2))$ defined by $&1.\",\n \"\\\\,\\\\, \\\\tilde{X}_{0} = (x,z), P^{\\\\omega }_{(x,z)}\\\\text{-a.s},\\\\\\\\&2.\\\\,\\\\, p^{\\\\omega }((x,z),(y,1)) = p_K^{\\\\omega }(x,y),\\\\\\\\&3.\\\\,\\\\, p^{\\\\omega }((x,z),(y,0)) = p^{\\\\omega }(x,y) - p_K^{\\\\omega }(x,y).$ This definition ensures that the first coordinate has the same law that our classic walk in $\\\\omega $ .\",\n \"When we are talking of the annealed law for the enhanced walk we will also write $\\\\mathbb {P}_{(x,z)}$ .\",\n \"We observe that the evolution of the first coordinate does not depend on the second coordinate, thus when we are talking about only the first coordinate, we will simply write $X$ instead of $\\\\tilde{X}$ and $P_{x}^{\\\\omega }$ instead of $P^{\\\\omega }_{(x,z)}$ .\"\n ],\n [\n \"Regeneration times of a single walk\",\n \"In this subsection we recall the rigorous definition of regeneration times introduced .\",\n \"This definition will be used at the end of the paper and we highlight the fact that the definition of joint regeneration levels in Section REF is very similar to the definition of regeneration times.\",\n \"First we introduce $\\\\mathcal {M}^{(K)}= \\\\inf \\\\Big \\\\lbrace &i \\\\ge 2 \\\\text{ ; }X_i \\\\text{ is }K\\\\text{-open, }X_j\\\\cdot \\\\vec{\\\\ell } < X_{i-2} \\\\cdot \\\\vec{\\\\ell } \\\\text{ for any } j < i-2 \\\\\\\\& \\\\text{ and } X_i = X_{i-1} + e_1 = X_{i-2} + 2e_1 \\\\Big \\\\rbrace .$ We also introduce $& \\\\mathbf {BACK} = \\\\inf \\\\left\\\\lbrace n \\\\ge 1 \\\\text{ ; } X_n \\\\cdot \\\\vec{\\\\ell } \\\\le X_0 \\\\cdot \\\\vec{\\\\ell }\\\\right\\\\rbrace \\\\\\\\& \\\\mathbf {ORI} = \\\\inf \\\\left\\\\lbrace n \\\\ge 1 \\\\text{ ; }X_{n-1} \\\\in \\\\mathcal {V}_{X_0} \\\\text{ and } Z_n = 0\\\\right\\\\rbrace ,$ and finally $D = \\\\mathbf {BACK} \\\\wedge \\\\mathbf {ORI}.$ The event $\\\\lbrace \\\\mathbf {BACK} > n\\\\rbrace $ means that the walk $X$ does not backtrack below $X_0$ before time $n$ .\",\n \"The event $\\\\lbrace \\\\mathbf {ORI} > n\\\\rbrace $ is essential to get independence between the regeneration times.\",\n \"In particular, a coupling argument shows that, as long as $\\\\lbrace \\\\mathbf {ORI} > n\\\\rbrace $ occurs, the conductances around the starting point do not affect the future of the walk if the point is $K$ -open.\",\n \"That is why we will be interested in the event $\\\\lbrace D = +\\\\infty \\\\rbrace $ to regenerate.\",\n \"Now we introduce the sequence of dependent stopping times $(S_n)_{n \\\\ge 0}$ and the levels $(M_n)_{n \\\\ge 0}$ $&\\\\hspace{40.0pt}S_0 = 0\\\\text{, }M_0 = X_{0} \\\\cdot \\\\vec{\\\\ell } \\\\text{ and }\\\\\\\\& \\\\text{ for } n \\\\ge 0\\\\text{, } S_{n+1} = \\\\mathcal {M}^{(K)} \\\\circ \\\\theta _{T_{\\\\mathcal {H}^{+}(M_n)}} + T_{\\\\mathcal {H}^{+}(M_n)},$ where $M_n = \\\\sup \\\\lbrace X_k \\\\cdot \\\\vec{\\\\ell } \\\\text{ with } 0 \\\\le k \\\\le R_n \\\\rbrace ,$ with $R_n = S_n + D \\\\circ \\\\theta _{S_n}.$ Then, we define the first regeneration time as $\\\\tau _1^{(K)} = S_N \\\\text{ with } N = \\\\inf \\\\lbrace n \\\\ge 1 \\\\text{ ; }S_n < +\\\\infty \\\\text{ and } M_n = +\\\\infty \\\\rbrace .$ Remark 3.7 Most of the time we will forego the superscript $K$ in $\\\\tau _1^{(K)}$ since it will be fixed large enough.\",\n \"Using we obtain that $\\\\tau _1$ is $\\\\mathbb {P}_0$ -almost surely finite and we can define inductively finite regeneration times by $\\\\tau _{k+1} = \\\\tau _1 + \\\\tau _{k}((X_{\\\\tau _1+\\\\cdot } - X_{\\\\tau _1}),\\\\omega ( \\\\cdot +X_{\\\\tau _1})).$ The key result states that the $\\\\tau _k$ are $\\\\mathbb {P}_0$ -almost surely finite and that $(X_{\\\\tau _{k}+\\\\cdot } - X_{\\\\tau _k})$ is independent of $(X_i)_{0\\\\le i \\\\le \\\\tau _k}$ and distributed as a random walk under $\\\\mathbb {P}_0^K( \\\\, \\\\cdot \\\\text{ | }D=+\\\\infty )$ .\",\n \"In the next section we will generalize these notions to two walks.\",\n \"As mentioned before, regeneration times will turn into regeneration levels and the independence property will become a Markov-type property.\",\n \"Remark 3.8 The reader should be careful in the following since we are going to define joint regeneration levels and we will use the notation $\\\\mathcal {D}^{^{\\\\bullet }}$ which is not the same as $D$ and $D^{\\\\otimes }$ , which are the natural counterparts for a single walk or for two independent walks under the annealed measure.\"\n ],\n [\n \"Enhanced walks and joint regeneration\",\n \"We consider two enhanced random walks $\\\\tilde{X}^1$ and $\\\\tilde{X}^2$ independent under the quenched law.\",\n \"We recall the notation set in Section REF .\",\n \"For $i \\\\in \\\\lbrace 1,2\\\\rbrace $ we introduce the random variables: $&\\\\mathbf {BACK}^i = \\\\inf \\\\left\\\\lbrace n > 0 \\\\text{ ; } X^i_n \\\\cdot \\\\vec{\\\\ell } \\\\le X^i_0 \\\\cdot \\\\vec{\\\\ell } \\\\right\\\\rbrace , \\\\\\\\& \\\\mathbf {ORI}^i = \\\\inf \\\\left\\\\lbrace n > 0 \\\\text{ ; } X^i_{n-1} \\\\in \\\\mathcal {V}_{X^1_0} \\\\cup \\\\mathcal {V}_{X^2_0} \\\\text{ and } Z^i_n = 0 \\\\right\\\\rbrace .\",\n \"$ Then we introduce $\\\\mathcal {D}^{\\\\bullet i} = \\\\mathbf {BACK}^i \\\\wedge \\\\mathbf {ORI}^i.$ Finally we set $\\\\mathcal {D}^{\\\\bullet } = \\\\mathcal {D}^{\\\\bullet 1} \\\\wedge \\\\mathcal {D}^{\\\\bullet 2}.$ Remark 3.9 The notation $\\\\mathbf {BACK}^i$ refers to the first time the walk $X^i$ backtracks.\",\n \"The notation $\\\\mathbf {ORI}^i$ encapsulates information around the starting points in such a way that Proposition REF holds.\",\n \"In the following we will be interested in the case $\\\\mathcal {D}^{\\\\bullet } = +\\\\infty $ and $X_0^1,X_0^2$ are $K$ -open.\",\n \"In that case the walks never backtrack and they behave as if for all $e \\\\in \\\\mathcal {E}_{X_0^{1}} \\\\cup \\\\mathcal {E}_{X_0^{2}}$ , $c_{*}(e) = K$ .\",\n \"We notice that $\\\\mathcal {D}^{\\\\bullet 1}$ is not measurable with respect to $\\\\sigma (X_0^{1},(\\\\tilde{X}^{1}_{n})_{n \\\\ge 1})$ but with respect to $\\\\sigma (X_0^{1},X_0^{2},(\\\\tilde{X}^{1}_{n})_{n \\\\ge 1})$ .\",\n \"Remark 3.10 We can notice that under $P^{\\\\omega }_{\\\\begin{array}{c}(x_1,z_1)\\\\\\\\(x_2,z_2)\\\\end{array}}$ , $\\\\mathcal {D}^{\\\\bullet 1}$ and $\\\\mathcal {D}^{\\\\bullet 2}$ are independent.\",\n \"Indeed, the variables $X^1_0$ and $X^2_0$ are almost surely constant under $P^{\\\\omega }_{\\\\begin{array}{c}(x_1,z_1)\\\\\\\\(x_2,z_2)\\\\end{array}}$ and then we use the quenched independence of the two walks.\",\n \"The following proposition shows that under $\\\\mathcal {D}^{\\\\bullet } = +\\\\infty $ the walks behave as if the environment around the starting points is deterministic.\",\n \"For $\\\\omega $ an environment and $(x_1,z_1),(x_2,z_2) \\\\in \\\\mathbb {Z}^d \\\\times \\\\lbrace 0,1\\\\rbrace $ we define $P^{\\\\omega _K }_{\\\\begin{array}{c}(x_1,z_1)\\\\\\\\(x_2,z_2)\\\\end{array}}$ the quenched probability such that the conductances $c_{*}(e)$ with $e \\\\in \\\\mathcal {E}_{x_1}\\\\cup \\\\mathcal {E}_{x_2}$ are set to $K$ .\",\n \"Proposition 3.11 Fix $(x_1,z_1),(x_2,z_2) \\\\in \\\\mathbb {Z}^d \\\\times \\\\lbrace 0,1\\\\rbrace $ .\",\n \"If $x_1$ and $x_2$ are both $K$ open points then for any bounded $ \\\\sigma \\\\big (\\\\tilde{X}^{1}_{n},\\\\tilde{X}^{2}_{n}\\\\big )$ -measurable function $f(\\\\cdot )$ , we have the equality: $E^{\\\\omega }_{\\\\begin{array}{c}(x_1,z_1)\\\\\\\\(x_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet } = +\\\\infty }\\\\right] = E^{\\\\omega _K }_{\\\\begin{array}{c}(x_1,z_1)\\\\\\\\(x_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet } = +\\\\infty }\\\\right].$ To lighten the notation in the proof we will write $P^{\\\\omega }_{\\\\begin{array}{c}(x_1,z_1)\\\\\\\\(x_2,z_2)\\\\end{array}} = P^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}.$ The proof goes the same way as the proof of that uses a coupling argument.\",\n \"The idea of the proof is to construct a couple $(U^1,U^2)$ with the same law as $(\\\\tilde{X}^1,\\\\tilde{X}^2)$ under $P^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}$ and a couple $(V^1,V^2)$ with the same law as $(\\\\tilde{X}^1,\\\\tilde{X}^2)$ under $P^{\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ .\",\n \"In addition we will construct the random variables in such a way that $\\\\displaystyle (U_i^1,U_i^2)_{0\\\\le i \\\\le n} = (V_i^1,V_i^2)_{0 \\\\le i \\\\le n}$ as soon as $\\\\mathcal {D}^{\\\\bullet }(U) > n$ where $\\\\mathcal {D}^{\\\\bullet }(U)$ denotes the quantity $\\\\mathcal {D}$ defined in (REF ) associated to $(U^1,U^2)$ .\",\n \"We give the construction of $(U^1_{\\\\cdot },V^{1}_{\\\\cdot })$ under a probability measure $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ .\",\n \"We can then build $(U^2_{\\\\cdot },V^{2}_{\\\\cdot })$ independently in the same way under $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ .\",\n \"To simplify the notations and highlight the fact that $U^1$ has the same law as $\\\\tilde{X}^1 = (X^1,Z^1)$ we will write $U^1 = U = (X_n,Z_n)_{n \\\\ge 0}$ and $V^1 = V = (Y_n,W_n)_{n\\\\ge 0}$ .\",\n \"We also call $\\\\mathcal {D}^{\\\\bullet }_U = \\\\mathcal {D}^{\\\\bullet 1}(U)$ and $\\\\mathcal {D}^{\\\\bullet }_V = \\\\mathcal {D}^{\\\\bullet 1}(V)$ .\",\n \"We set $((X_0,Z_0),(Y_0,W_0)) = ((x_1,z_1),(x_1,z_1))$ $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ -almost surely.\",\n \"Given the trajectory up to time $n \\\\ge 0$ the conditional law of $(U_{n+1},V_{n+1})$ is given by the following rules: $1.$ If $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U > n\\\\rbrace \\\\cap \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V > n\\\\rbrace $ and if $U_n = V_n = x \\\\in \\\\mathbb {Z}^d$ .\",\n \"Then $U$ makes a step according to $P^{\\\\omega }$ and (a) If $x \\\\notin \\\\lbrace x_1,x_2\\\\rbrace $ and $x \\\\notin \\\\lbrace x_i + e_j \\\\text{ | } j \\\\in \\\\lbrace 1,...,2d\\\\rbrace \\\\text{ and }i \\\\in \\\\lbrace 1,2\\\\rbrace \\\\rbrace $ then $V_{n+1}= U_{n+1}$ .\",\n \"(b) If $x \\\\in \\\\lbrace x_1,x_2\\\\rbrace $ or $x \\\\in \\\\lbrace x_i + e_j \\\\text{ | } j \\\\in \\\\lbrace 1,...,2d\\\\rbrace \\\\text{ and }i \\\\in \\\\lbrace 1,2\\\\rbrace \\\\rbrace $ and if $Z_{n+1} = 1$ then $V_{n+1}= U_{n+1}$ .\",\n \"(c) If $x \\\\in \\\\lbrace x_1,x_2\\\\rbrace $ or $x \\\\in \\\\lbrace x_i + e_j \\\\text{ | } j \\\\in \\\\lbrace 1,...,2d\\\\rbrace \\\\text{ and }i \\\\in \\\\lbrace 1,2\\\\rbrace \\\\rbrace $ and if $Z_{n+1} = 0$ then for any $j \\\\in \\\\lbrace 1,...,2d\\\\rbrace $ , $(Y_{n+1},W_{n+1}) = (x + e_j,0)$ with probability $\\\\frac{p^{\\\\omega _K}(x,x+e_j)-p_K^{\\\\omega _K}(x,x+e_j)}{1- \\\\displaystyle \\\\sum _{1 \\\\le j \\\\le 2d}^{} p_{K}^{\\\\omega _K}(x,x+e_j)}.$ $2.$ if $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U \\\\le n\\\\rbrace \\\\cup \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V \\\\le n\\\\rbrace $ then $U$ and $V$ move independentely according to $P^{\\\\omega }$ and $P^{\\\\omega _K}$ respectively.\",\n \"We have to show that $(U,V)$ is well defined following this construction.\",\n \"By construction, if $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U > n\\\\rbrace \\\\cap \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V > n\\\\rbrace \\\\cap \\\\lbrace U_n = V_n\\\\rbrace $ then $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ -almost surely either $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U > n+1\\\\rbrace \\\\cap \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V > n+1\\\\rbrace \\\\cap \\\\lbrace U_{n+1} = V_{n+1}\\\\rbrace $ or $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U = \\\\mathcal {D}^{\\\\bullet }_V = n+1\\\\rbrace $ .\",\n \"By induction it follows that: The event $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U > n\\\\rbrace \\\\cap \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V > n\\\\rbrace \\\\cap \\\\lbrace U_n \\\\ne V_n\\\\rbrace $ never occurs $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ -almost surely.\",\n \"Thus $U$ and $V$ are well defined.\",\n \"Under $\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U > n\\\\rbrace \\\\cap \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V > n\\\\rbrace $ we have $X_k = Y_k$ for all $k \\\\in \\\\lbrace 0,...,n\\\\rbrace $ $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ -almost surely.\",\n \"We have $\\\\mathcal {D}^{\\\\bullet }_U = \\\\mathcal {D}^{\\\\bullet }_V$ $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ -almost surely.\",\n \"Thus we have $&P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left(\\\\lbrace \\\\mathcal {D}^{\\\\bullet }_U = +\\\\infty \\\\rbrace \\\\cap \\\\left\\\\lbrace \\\\lbrace \\\\exists n \\\\ge 0; U_n \\\\ne V_n\\\\rbrace \\\\cup \\\\lbrace \\\\mathcal {D}^{\\\\bullet }_V < +\\\\infty \\\\rbrace \\\\right\\\\rbrace \\\\right) \\\\\\\\&\\\\le P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left(\\\\mathcal {D}^{\\\\bullet }_U \\\\ne \\\\mathcal {D}^{\\\\bullet }_V\\\\right) + \\\\sum _{n = 0}^{+\\\\infty } P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left(\\\\mathcal {D}^{\\\\bullet }_U > n+1, \\\\mathcal {D}^{\\\\bullet }_V >n+1, X_{n+1} \\\\ne Y_{n+1}\\\\right) = 0.\",\n \"\\\\nonumber $ We notice that $p_{K}^{\\\\omega }(x,\\\\cdot ) = p_{K}^{\\\\omega _K}(x,\\\\cdot )$ for any $x \\\\in \\\\mathbb {Z}^d$ and $p^{\\\\omega }(x,\\\\cdot ) = p^{\\\\omega _K}(x,\\\\cdot )$ as soon as $x \\\\notin \\\\lbrace x_1,x_2\\\\rbrace \\\\cup \\\\lbrace x_i + e_j \\\\text{ | } j \\\\in \\\\lbrace 1,...,2d\\\\rbrace \\\\text{ and }i \\\\in \\\\lbrace 1,2\\\\rbrace \\\\rbrace $ , it is easy to check that the law of $U$ (resp.\",\n \"$V$ ) under $P^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ is the law of $\\\\tilde{X}^1$ under $P^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}$ (resp.\",\n \"the law of $\\\\tilde{X}^1$ under $P^{\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ ).\",\n \"Then take $A_0,\\\\cdots ,A_n$ subsets of $\\\\mathbb {Z}^d\\\\times \\\\lbrace 0,1\\\\rbrace $ where $A_i$ are in the $\\\\sigma $ -algebra generated by the subsets of $\\\\mathbb {Z}^d \\\\times \\\\lbrace 0,1\\\\rbrace $ and call $A = A_0 \\\\times \\\\cdots \\\\times A_n$ .\",\n \"Thus we have: $&E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[{1}_{}\\\\lbrace (\\\\tilde{X}^1_{0},\\\\cdots ,\\\\tilde{X}^1_{n}) \\\\in A\\\\rbrace {1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty } \\\\right] \\\\\\\\&= E^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[{1}_{}\\\\lbrace (U_{0},...,U_{n}) \\\\in A\\\\rbrace {1}_{\\\\mathcal {D}^{\\\\bullet }_U = +\\\\infty } \\\\right]\\\\\\\\&= E^{\\\\omega ,\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[{1}_{}\\\\lbrace (V_{0},...,V_{n}) \\\\in A\\\\rbrace {1}_{\\\\mathcal {D}^{\\\\bullet }_V = +\\\\infty } \\\\right] \\\\\\\\& = E^{\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[{1}_{}\\\\lbrace (\\\\tilde{X}^1_{0},...,\\\\tilde{X}^1_{n}) \\\\in A\\\\rbrace {1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty } \\\\right],$ where the second equality follows from (REF ).\",\n \"By a Monotone Class Theorem, for any bounded $\\\\sigma \\\\big (\\\\tilde{X}^{1}_{n}\\\\big )$ -measurable function we have $E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty }\\\\right] = E^{\\\\omega _K }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty }\\\\right].$ Then, for bounded measurable functions $f$ and $g$ we have: $E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot })g(\\\\tilde{X}^2_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet } = +\\\\infty }\\\\right]&= E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot })g(\\\\tilde{X}^2_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty }{1}_{\\\\mathcal {D}^{\\\\bullet 2} = +\\\\infty }\\\\right] \\\\\\\\& = E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty }\\\\right] E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[g(\\\\tilde{X}^2_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 2} = +\\\\infty }\\\\right]\\\\\\\\&= E^{\\\\omega _K }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty }\\\\right] E^{\\\\omega _K }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[g(\\\\tilde{X}^2_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 2} = +\\\\infty }\\\\right] \\\\\\\\&=E^{\\\\omega _K }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot })g(\\\\tilde{X}^2_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet 1} = +\\\\infty }{1}_{\\\\mathcal {D}^{\\\\bullet 2} = +\\\\infty }\\\\right] \\\\\\\\&=E^{\\\\omega _K }_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot })g(\\\\tilde{X}^2_{\\\\cdot }){1}_{\\\\mathcal {D}^{\\\\bullet } = +\\\\infty }\\\\right].$ To extend the previous result to bounded measurable functions of the form $f(\\\\cdot ,\\\\cdot )$ we use once again the Monotone Class Lemma.\"\n ],\n [\n \"$K$ -good points\",\n \"We recall the definition of a $K$ -good point taken from .\",\n \"A point $z \\\\in \\\\mathbb {Z}^d$ is said to be $K$ -good if there exists an infinite path $(x_0 = z, x_1,x_2,\\\\cdots )$ such that all $x_i$ are $K$ -open and $x_{2i+1}-x_{2i} = e_1$ and $x_{2i+2}-x_{2i+1} = \\\\lbrace e_1,e_2,\\\\cdots ,e_d\\\\rbrace $ .\",\n \"Proposition 3.12 There exists a constant $K_0>0$ and $\\\\eta > 0$ such that for $K > K_0$ we have for any $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ we have $\\\\mathbf {P}(U_1 \\\\text{ and }U_2\\\\text{ are }K\\\\text{-good points}) > \\\\eta > 0$ .\",\n \"For $e_1,\\\\cdots ,e_n \\\\in E(\\\\mathbb {Z}^d)$ distinct edges, the events $(c^{*}_{e_i} \\\\in [1/K,K])_{1\\\\le i \\\\le n}$ are independent and have the same probability.\",\n \"Thus $\\\\displaystyle ({1}_{c^{*}_{e} \\\\in [1/K,K]})_{e\\\\in E(\\\\mathbb {Z}^d)}$ is a percolation on $\\\\mathbb {Z}^d$ .\",\n \"Since for $x \\\\in \\\\mathbb {Z}^d$ the event $\\\\lbrace x \\\\text{ is }K\\\\text{-good}\\\\rbrace $ is an increasing event we can apply FKG inequality and obtain $\\\\mathbf {P}(U_1 \\\\text{ and }U_2\\\\text{ are }K\\\\text{-good points}) \\\\ge \\\\mathbf {P}(U_1 \\\\text{ is a }K\\\\text{-good point})\\\\mathbf {P}(U_2 \\\\text{ is a }K\\\\text{-good point}),$ then we conclude using .\"\n ],\n [\n \"Construction of joint regeneration levels and main result\",\n \"We recall the notation $\\\\mathcal {M}^{\\\\bullet }$ defined in (REF ).\",\n \"We set $L_0 = 0, \\\\quad M_{0} = X^1_0 \\\\cdot \\\\vec{\\\\ell } \\\\vee X^{2}_{0} \\\\cdot \\\\vec{\\\\ell },$ and, for $k\\\\ge 0$ , $ L_{k+1} = \\\\mathcal {M}^{\\\\bullet }\\\\circ \\\\theta _{T_{M_k}}.$ Where $ M_k = \\\\sup \\\\bigg \\\\lbrace \\\\lbrace X^1_m\\\\cdot \\\\vec{\\\\ell } \\\\text{ ; with } 0 \\\\le m \\\\le R_k\\\\rbrace \\\\cup \\\\lbrace X^2_m\\\\cdot \\\\vec{\\\\ell } \\\\text{ ; with } 0 \\\\le m \\\\le R_k\\\\rbrace \\\\bigg \\\\rbrace ,$ with $R_k = \\\\mathcal {D}^{\\\\bullet } \\\\circ \\\\theta _{T_{L_k}}.$ Here we set $N = \\\\inf \\\\lbrace k \\\\ge 0 \\\\text{ ; } L_k < +\\\\infty \\\\text{ and } R_k =+\\\\infty \\\\rbrace $ we define the first joint regeneration level as $\\\\mathcal {L}_1 = L_{N}$ .\",\n \"With this definition the associated joint regeneration times (respectively points) for $X^1$ and $X^2$ are $T^{1}_{\\\\mathcal {L}_1}$ and $T^{2}_{\\\\mathcal {L}_1}$ (respectively $X^{1}_{T^{1}_{\\\\mathcal {L}_1}}$ and $X^{2}_{T^{2}_{\\\\mathcal {L}_1}}$ ).\",\n \"We observe that the definition is very similar to the definition in Section REF but instead of defining a sequence of stopping times $(S_k)$ we define a sequence of levels $(L_k)$ .\",\n \"The first regeneration level is the first level $L_k$ such that the walks do not backtrack and properly encodes the information in variables $Z^1, Z^2$ around the starting points.\",\n \"The first thing to show is that the quantity $\\\\mathcal {L}_1$ , as defined above, is $\\\\mathbb {P}_0$ -almost surely finite, Proposition REF below shows that the first joint regeneration level happens pretty quickly.\",\n \"Then we will focus in proving the Markov property in Theorem REF .\",\n \"Proposition 3.13 For any $M \\\\in (0,+\\\\infty )$ there exists $K_0 >0$ such that for $\\\\mathcal {U} \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ we have $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {L}_1 \\\\ge n) \\\\le Cn^{-M},$ where the constant $C$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"The proof will be showed later, as we need several lemmas in order to be able to prove this result.\",\n \"We recall the notation $\\\\mathbb {P}_{U_1,U_2}^{K}$ which is the usual annealed measure with the restriction that the conductances around $U_1,U_2$ are set to have value $K$ .\"\n ],\n [\n \"Proof of Proposition \",\n \"Lemma 3.14 There exists $K_0 > 0$ and $ \\\\eta > 0$ such that for $K \\\\ge K_0$ , for $\\\\mathcal {U} = \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ we have $& \\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet }=+\\\\infty ) > \\\\eta > 0, \\\\\\\\& \\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}^{K}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet }=+\\\\infty ) > \\\\eta > 0,$ where $\\\\eta $ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We follow the lines of .\",\n \"We only prove the result for $\\\\mathbb {P}_{U_1,U_2}$ since the proof for $\\\\mathbb {P}_{U_1,U_2}^{K}$ is similar.\",\n \"We have $\\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet }=+\\\\infty ) > c \\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet }=+\\\\infty \\\\text{ | }U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}),$ where $c = \\\\mathbb {P}(U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}) > \\\\eta > 0$ using Proposition REF and where $\\\\eta $ only depends on $d$ .\",\n \"On event $\\\\lbrace U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}\\\\rbrace $ we fix $\\\\mathcal {P}^i$ a path given by the fact that $U_i$ is a $K$ -good point.\",\n \"We necessarily have $\\\\mathcal {P}^{i}(1) = e_1$ and $\\\\mathcal {P}^{i}(2) = e_1 + e_{j_2}$ with $e_{j_2} \\\\in \\\\lbrace e_1,\\\\cdots ,e_d\\\\rbrace $ .\",\n \"We introduce $L^i_{\\\\partial ^{+}\\\\mathcal {B}(n,n^2)+U_i} = \\\\inf \\\\lbrace k \\\\in \\\\mathbb {N} \\\\text{ ; }\\\\mathcal {P}^{i}(k) \\\\in \\\\partial ^{+}\\\\mathcal {B}(n,n^2)+U_i \\\\rbrace $ and $x^i = \\\\mathcal {P}^{i}(L^{i}_{\\\\partial ^{+}\\\\mathcal {B}(n,n^2)+ U_i})$ .\",\n \"We introduce $&A_1^{i} = \\\\lbrace (X^{i}_1,Z^{i}_1) = (e_1,1)\\\\rbrace ,\\\\\\\\&A_2^{i} = \\\\lbrace (X^{i}_2,Z^{i}_2) = (e_1+e_{j_2},1)\\\\rbrace ,\\\\\\\\&A_3^{i} = \\\\lbrace X^{i}_{k} = \\\\mathcal {P}^{i}(k) \\\\text{ for } 3\\\\le k \\\\le L_{\\\\partial ^{+}\\\\mathcal {B}(n,n^2)+U_i}\\\\rbrace ,\\\\\\\\&A_4^{i} = \\\\lbrace T^{i}_{\\\\mathcal {H}^{-}(2)} \\\\circ \\\\theta _{T_{x^i}} = +\\\\infty \\\\rbrace .$ and define $A_k = A^1_k \\\\cap A^2_k$ and $\\\\displaystyle A = \\\\bigcap _{k=1}^{4}A_{k}$ .\",\n \"We see that $A \\\\subset \\\\lbrace \\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\rbrace $ .\",\n \"As $U_1$ and $U_2$ are $K$ -good we have $L^{i}_{\\\\partial ^{+}\\\\mathcal {B}(n,n^2)+U_i} \\\\le Cn$ and since all points in $\\\\mathcal {P}^{i}$ are $K$ -open, we have $P_{U_1,U_2}^{\\\\omega }(A_1,A_2,A_3) \\\\ge c^{n},$ with $c > 0$ which only depend on $K$ , $d$ and $\\\\ell $ .\",\n \"Thus, by Markov property we have $&\\\\mathbf {E}[P_{U_1,U_2}^{\\\\omega }(A_1,A_2,A_3,A_4) \\\\text{ | }U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}]\\\\\\\\&= \\\\mathbf {E}[P_{U_1,U_2}^{\\\\omega }(A_1,A_2,A_3)P^{\\\\omega }_{x^1,x^2}(A^1_4,A^2_4) \\\\text{ | }U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}]\\\\\\\\&\\\\ge c^n\\\\mathbf {E}[P^{\\\\omega }_{x^1,x^2}(A_4) \\\\text{ | }U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}]\\\\\\\\& = c^n \\\\mathbf {E}\\\\left[P^{\\\\omega }_{x^1,x^2}(T^{1}_{\\\\mathcal {H}^{-}(2)} = +\\\\infty ,T^{2}_{\\\\mathcal {H}^{-}(2)} = +\\\\infty ) \\\\text{ | }U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}\\\\right].$ We also have $&\\\\mathbf {E}\\\\left[P^{\\\\omega }_{x^1,x^2}( \\\\lbrace T^1_{\\\\mathcal {H}^{-}(2)} < +\\\\infty \\\\rbrace \\\\cup \\\\lbrace T^2_{\\\\mathcal {H}^{-}(2)} < +\\\\infty \\\\rbrace ) \\\\text{ | }U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}\\\\right]\\\\\\\\&\\\\le C\\\\mathbb {P}_0\\\\left(T_{\\\\mathcal {H}^{-}(-n+2)} < +\\\\infty \\\\right) \\\\\\\\&\\\\le C\\\\exp (-cn).$ Where we used a union bound, translation invariance of the law of the environment and the fact $\\\\mathbf {P}(U_1 \\\\text{ and } U_2 \\\\text{ are } K\\\\text{-good}) > \\\\eta > 0$ and .\",\n \"Hence we see that there exists $n_0$ big enough which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ such that this quantity is smaller than $1/2$ for all $n \\\\ge n_0$ and it concludes the proof.\",\n \"The following lemmas correspond to the ones in .\",\n \"We recall the definitions of $L_k$ and $M_k$ in (REF ), (REF ), then we introduce the events: $&M^{K}(n) = \\\\left\\\\lbrace \\\\text{for }k \\\\text{ such that }M_k < n \\\\text{ we have }L_{k+1} - M_k < n^{1/2} \\\\right\\\\rbrace , \\\\\\\\&S(n) = \\\\left\\\\lbrace \\\\text{for } k \\\\text{ such that }L_k < n \\\\text{ and } M_k < +\\\\infty \\\\text{ we have } M_k - L_k < n^{1/2}\\\\right\\\\rbrace .$ In the following we may drop the superscript $K$ to lighten the notations.\",\n \"Lemma 3.15 For $\\\\mathcal {U} = \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ and for any $M > 0$ there exists $K > 0$ and a constant $C > 0$ such that $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}_{U_1,U_2}\\\\left(M^{K}(n)^{c}\\\\right) \\\\le Cn^{-M},$ where the constant $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"It follows from Proposition REF .\",\n \"Indeed, $M^{K}(n)^{c}$ implies that there exists $0 \\\\le R \\\\le n$ such that $\\\\mathcal {M}^{\\\\bullet }_R \\\\ge n^{\\\\frac{1}{2}}$ .\",\n \"We introduce $\\\\bar{L} = \\\\sup \\\\left\\\\lbrace \\\\lbrace X^1_n\\\\cdot \\\\vec{\\\\ell } \\\\text{ ; with } 0 \\\\le n \\\\le \\\\mathcal {D}^{\\\\bullet }\\\\rbrace \\\\cup \\\\lbrace X^2_n\\\\cdot \\\\vec{\\\\ell } \\\\text{ ; with } 0 \\\\le n \\\\le \\\\mathcal {D}^{\\\\bullet }\\\\rbrace \\\\right\\\\rbrace .$ The following proposition shows that under $\\\\mathcal {D} < +\\\\infty $ , $\\\\bar{L}$ cannot be too big.\",\n \"Lemma 3.16 Let $\\\\mathcal {U} = \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ , then there exists constants $C,c$ such that $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}_{U_1,U_2}\\\\left(\\\\bar{L} \\\\ge n \\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet }<+\\\\infty \\\\right) \\\\le C\\\\exp (-cn),$ where $C,c$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"First thing to say is that $\\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty ) > \\\\eta > 0$ with $\\\\eta $ which does not depends on $U_1,U_2$ .\",\n \"Indeed $0 < \\\\mathbb {P}_{U_1,U_2}(X_1^1 = U_1+e_1,X^1_2 = U_1) = \\\\mathbb {P}_{0}(X_1 = e_1 , X_2 = 0) < \\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty ),$ where we used translation invariance of the environment for the equality.\",\n \"Thus, we only have to dominate $\\\\mathbb {P}_{U_1,U_2}(\\\\mathcal {D}^{\\\\bullet }<+\\\\infty ,\\\\bar{L} \\\\ge n)$ .\",\n \"By union bound and symmetry, we only have to dominate $\\\\mathbb {P}_{U_1,U_2}\\\\left(\\\\mathcal {D}^{\\\\bullet }<+\\\\infty ,\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }}X^1_{j}\\\\cdot \\\\vec{\\\\ell } \\\\ge n\\\\right).$ Without loss of generality we can suppose $U_1= U_2 =0$ , indeed the proof is exactly the same in the general case but instead of considering the boxes $\\\\mathcal {B}(2^k,2^{\\\\alpha k})$ we should consider the boxes $\\\\mathcal {B}(2^k,2^{\\\\alpha k}) + U_i$ .\",\n \"First we have $\\\\mathbb {P}_{0}\\\\left(2^k \\\\le \\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }}X^1_{j}\\\\cdot \\\\vec{\\\\ell } < 2^{k+1} \\\\right) \\\\le &\\\\mathbb {P}_{0}\\\\left(T^1_{\\\\partial \\\\mathcal {B}(2^k,2^{\\\\alpha k})} \\\\ne T^1_{\\\\partial ^{+} \\\\mathcal {B}(2^k,2^{\\\\alpha k})}\\\\right) +\\\\\\\\&\\\\mathbb {P}_{0}\\\\Big (T^1_{\\\\partial \\\\mathcal {B}(2^k,2^{\\\\alpha k})} = T^1_{\\\\partial ^{+} \\\\mathcal {B}(2^k,2^{\\\\alpha k})},\\\\\\\\&T^{+}_{\\\\mathcal {H}^{-}(0)}\\\\circ \\\\theta _{T_{\\\\partial ^{+} \\\\mathcal {B}(2^k,2^{\\\\alpha k})}}< T^{+}_{\\\\mathcal {H}^{+}(2^{k+1})}\\\\circ \\\\theta _{T_{\\\\partial ^{+} \\\\mathcal {B}(2^k,2^{\\\\alpha k})}} \\\\Big ).$ The first term is smaller than $C\\\\exp (-c2^k)$ using .\",\n \"Then using a union bound and translation invariance of the environment the second term is dominated by $(2^{k})^{\\\\alpha d}\\\\mathbb {P}_{0}\\\\left(T^1_{\\\\partial \\\\mathcal {B}(2^k,2^{\\\\alpha k})} \\\\ne T^1_{\\\\partial ^{+} \\\\mathcal {B}(2^k,2^{\\\\alpha k})}\\\\right) \\\\le C2^{\\\\alpha kd}\\\\exp (-c2^k),$ where we have used once again Theorem $5.1$ of .\",\n \"Then we conclude using the following domination and the previous inequality $\\\\mathbb {P}_{0}\\\\left(\\\\mathcal {D}^{\\\\bullet }<+\\\\infty ,\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }}X^1_{j}\\\\cdot \\\\vec{\\\\ell } \\\\ge n\\\\right) \\\\le C\\\\sum _{k,2^{k}\\\\ge n}^{} \\\\mathbb {P}_{0}\\\\left(2^k \\\\le \\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }}X^1_{j}\\\\cdot \\\\vec{\\\\ell } < 2^{k+1}\\\\right).$ We recall the definition of $S(n)$ in ().\",\n \"Lemma 3.17 Let $\\\\mathcal {U} \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ , then there exists constants $C,c > 0$ such that $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}_{U_1,U_2}\\\\left(S(n)^{c}\\\\right) \\\\le C\\\\exp (-cn^{\\\\frac{1}{2}}).$ It is an adaptation of .\",\n \"First we notice that $\\\\mathbb {P}_{0}$ -almost surely, $\\\\# \\\\lbrace k \\\\ge 0 \\\\text{ such that } L_k < n\\\\rbrace $ is smaller than $d n$ .\",\n \"Hence we can write $\\\\displaystyle \\\\mathbb {P}_{U_1,U_2}\\\\left(S(n)^{c}\\\\right) \\\\le &\\\\mathbb {P}_{U_1}\\\\left(T^1_{\\\\partial \\\\mathcal {B}_1(n,n^{\\\\alpha })} \\\\ne T^1_{ \\\\partial ^{+} \\\\mathcal {B}_1(n,n^{\\\\alpha })} \\\\right) + \\\\mathbb {P}_{U_2}\\\\left(T^2_{\\\\partial \\\\mathcal {B}_2(n,n^{\\\\alpha })} \\\\ne T^2_{ \\\\partial ^{+} \\\\mathcal {B}_2(n,n^{\\\\alpha })} \\\\right) \\\\\\\\&+ \\\\sum _{k \\\\le d n}^{} \\\\sum _{\\\\begin{array}{c}x \\\\in \\\\mathcal {B}_1(n,n^{\\\\alpha })\\\\\\\\y \\\\in \\\\mathcal {B}_2(n,n^{\\\\alpha })\\\\end{array}}^{} \\\\mathbb {P}_{U_1,U_2}\\\\left(M_k < +\\\\infty , M_k - L_k \\\\ge n^{\\\\frac{1}{2}},X^{1}_{T_{L_k}} = x, X^{2}_{T_{L_k}} = y\\\\right).$ The first two terms are controlled by Theorem 5.1 of and using translation invariance of the environment.\",\n \"We can dominate the term in the sum by $& \\\\mathbb {P}_{U_1,U_2}\\\\left(\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet } \\\\circ \\\\theta _{T^1_{x}} + T^1_{x}}(X^{1}_j - x)\\\\cdot \\\\vec{\\\\ell } \\\\ge n^{\\\\frac{1}{2}},\\\\mathcal {D}^{\\\\bullet }\\\\circ \\\\theta _{T_{x,y}} < +\\\\infty ,X^{1}_{T_{L_k}} = x,X^{2}_{T_{L_k}} = y\\\\right) +\\\\\\\\&\\\\mathbb {P}_{U_1,U_2}\\\\left(\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet } \\\\circ \\\\theta _{T^2_{y}} + T^2_{y}}(X^{2}_j - y)\\\\cdot \\\\vec{\\\\ell } \\\\ge n^{\\\\frac{1}{2}},\\\\mathcal {D}^{\\\\bullet }\\\\circ \\\\theta _{T_{x,y}} < +\\\\infty ,,X^{1}_{T_{L_k}} = x,X^{2}_{T_{L_k}} = y\\\\right).$ Let us dominate the first term, the second is dominated the same way by symmetry.\",\n \"Using Lemma REF and the invariance of the environment it is bounded by $\\\\mathbf {E}\\\\left[P_{x,y}^{\\\\omega }\\\\left(\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }}(X^{1}_j -x)\\\\cdot \\\\vec{\\\\ell } \\\\ge n^{\\\\frac{1}{2}},\\\\mathcal {D}^{\\\\bullet } < +\\\\infty \\\\right) \\\\right]&\\\\le \\\\mathbf {E}\\\\left[P_{0,y-x}^{\\\\omega }\\\\left(\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }} X^{1}_j\\\\cdot \\\\vec{\\\\ell } \\\\ge n^{\\\\frac{1}{2}},\\\\mathcal {D}^{\\\\bullet } < +\\\\infty \\\\right) \\\\right] \\\\\\\\&\\\\le \\\\mathbb {P}_{0,y-x}\\\\left(\\\\sup _{j \\\\le \\\\mathcal {D}^{\\\\bullet }} X^{1}_j\\\\cdot \\\\vec{\\\\ell } \\\\ge n^{\\\\frac{1}{2}}\\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet }<+\\\\infty \\\\right)\\\\\\\\& \\\\le \\\\mathbb {P}_{0,y-x}\\\\left(\\\\bar{L} > n^{\\\\frac{1}{2}}\\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet }<+\\\\infty \\\\right).$ The last term is dominated using Lemma REF since we have $ |(y-x) \\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ by definition of $L_k$ .\",\n \"Taking the sum over $k,x,y$ we conclude that we have $\\\\displaystyle \\\\mathbb {P}_{U_1,U_2}(S(n)^{c}) \\\\le Cn^{\\\\beta }\\\\exp (-cn^{\\\\frac{1}{2}}).$ where $\\\\beta $ only depends on $d,\\\\alpha $ .\",\n \"It concludes the proof by taking $C$ large enough and $c$ small enough.\",\n \"The following lemma is the equivalent of .\",\n \"Recall that we defined $N = \\\\inf \\\\lbrace k \\\\ge 0 \\\\text{ ; } L_k < +\\\\infty \\\\text{ and } R_k =+\\\\infty \\\\rbrace $ , and the definition of $\\\\mathcal {E}_x, \\\\mathcal {L}^x, \\\\mathcal {R}^x$ as in Equations (REF )-(REF ).\",\n \"Furthermore, we define $\\\\mathcal {E}^{2}_x \\\\mathcal {E}_x \\\\cup \\\\mathcal {E}_{x-e_1}\\\\cup \\\\mathcal {E}_{x-2e_1}\\\\,\\\\, \\\\textnormal {the set of all edges which touch} \\\\,\\\\, \\\\lbrace x,x-e_1,x-2e_1\\\\rbrace .$ Lemma 3.18 There exists $K_0 > 0$ such that for $K \\\\ge K_0$ $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}}\\\\mathbb {P}_{U_1,U_2}(N \\\\ge n) \\\\le C\\\\exp (-cn).$ Where $\\\\mathcal {U} \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ and $C,c > 0$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d$ .\",\n \"We follow the lines of the proof of .\",\n \"The idea of the proof is simple, each time we reach a level $L_k$ there is a positive probability to regenerate and using the way we have defined $\\\\mathcal {D}$ we should have good independence properties, hence $N$ should decay at least like a geometric random variable.\",\n \"We introduce event $C(n) = \\\\lbrace \\\\text{for all }k \\\\le n \\\\text{ such that }L_k <+\\\\infty \\\\text{ we have } \\\\mathcal {D}^{\\\\bullet } \\\\circ \\\\theta _{T_{L_k}} < +\\\\infty \\\\rbrace .$ We have $\\\\lbrace N \\\\ge n\\\\rbrace \\\\subset C(n)$ , hence we only have to dominate $C(n)$ .\",\n \"We can write $\\\\mathbb {P}_{U_1,U_2}(C(n+1)) \\\\le \\\\sum _{x,y}^{} \\\\mathbf {E}\\\\left[P_{U_1,U_2}^{\\\\omega }\\\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n),\\\\mathcal {D}^{\\\\bullet }\\\\circ \\\\theta _{T_{L_k}} < +\\\\infty \\\\right)\\\\right].$ The event $\\\\lbrace X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\\\rbrace $ is measurable with respect to $\\\\sigma \\\\lbrace (X^{1}_n,Z_{n}^1),(X^{2}_k,Z_{k}^2)\\\\text{ for }0 \\\\le n \\\\le T^1_x \\\\text{ and }0\\\\le k \\\\le T^2_y\\\\rbrace $ .\",\n \"Thus we can use Lemma REF and get $&P_{U_1,U_2}^{\\\\omega }\\\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n),\\\\mathcal {D}^{\\\\bullet } \\\\circ \\\\theta _{T_{L_k}} < +\\\\infty \\\\right) \\\\\\\\=& P_{U_1,U_2}^{\\\\omega }\\\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\\\right)P_{x,y}^{\\\\omega }(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty ).$ Then, the sum is also equal to $&\\\\sum _{x,y}^{} \\\\mathbf {E}\\\\left[P_{U_1,U_2}^{\\\\omega }\\\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\\\right)P_{x,y}^{\\\\omega }(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty )\\\\right]\\\\\\\\&= \\\\sum _{x,y}^{} \\\\mathbf {E}\\\\left[P_{U_1,U_2}^{\\\\omega }\\\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\\\right)P_{x,y}^{\\\\omega _{K}}(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty )\\\\right],$ where we used the fact that both $x$ and $y$ are $K$ -open and Proposition REF .\",\n \"Since $P_{U_1,U_2}^{\\\\omega }\\\\big (X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\\\big )$ is measurable with respect to $\\\\sigma \\\\big (c_{*}(e) \\\\text{ with } e \\\\in \\\\mathcal {L}^{x-2e_1} \\\\cup \\\\mathcal {L}^{y-2e_1} \\\\cup \\\\mathcal {E}^2_{x} \\\\cup \\\\mathcal {E}^2_{y}\\\\big )$ and $P_{x,y}^{\\\\omega _{K}}(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty )$ is measurable with respect to $\\\\sigma \\\\big (c_{*}(e) \\\\text{ with } e \\\\in \\\\mathcal {R}^{x} \\\\cup \\\\mathcal {R}^{y} \\\\backslash (\\\\mathcal {E}_{x} \\\\cup \\\\mathcal {E}_{y})\\\\big )$ and using the fact that those sets of edges are disjoint (see Figure REF ) we can write by $\\\\mathbf {P}$ independence that the sum is equal to $\\\\sum _{x,y}^{} \\\\mathbf {E}\\\\left[P_{U_1,U_2}^{\\\\omega }\\\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\\\right)\\\\right]\\\\mathbf {E}\\\\left[P_{x,y}^{\\\\omega _{K}}(\\\\mathcal {D}^{\\\\bullet } < +\\\\infty )\\\\right].$ Using Lemma REF with $K_0$ large enough we have $\\\\mathbb {P}_{x,y}^{K}(\\\\mathcal {D}^{\\\\bullet }<+\\\\infty ) \\\\le \\\\eta < 1$ with $\\\\eta $ a constant which only depends on $\\\\lambda ,\\\\vec{\\\\ell },d$ .\",\n \"It follows that $\\\\mathbb {P}_{U_1,U_2}(C(n+1)) \\\\le \\\\eta \\\\mathbb {P}_{U_1,U_2}(C(n)).$ The result follows by induction.\",\n \"We now have all the tools to prove Proposition REF .\",\n \"Indeed combining Lemmas REF , REF and REF , it is unlikely to have the $n$ first $L_k$ finite and $L_{k+1} - L_{k} \\\\ge n^{\\\\frac{1}{2}}$ for $k \\\\le n$ .\",\n \"[Proof of Proposition REF ] On $\\\\lbrace \\\\mathcal {L}_1 \\\\ge n\\\\rbrace \\\\cap S(n)\\\\cap M(n)$ we have $N \\\\ge n^{1/3}$ for $n$ large enough.\",\n \"Indeed, on $\\\\lbrace \\\\mathcal {L}_1 \\\\ge n\\\\rbrace \\\\cap S(n)\\\\cap M(n)$ for $k < n$ such that $L_k < n $ and $M_k < n $ , we have $L_{k+1} - L_k < 2n^{1/2}$ , and since $\\\\mathcal {L}_1 \\\\ge n$ , as long as $L_{k+1} < n$ we have $M_{k+1} < +\\\\infty $ .\",\n \"Hence it follows that we must have $N \\\\ge n^{\\\\frac{1}{3}}$ since in the opposite case we would have $\\\\displaystyle \\\\mathcal {L}_1 = L_{N} = \\\\sum _{k=0}^{N-1} L_{k+1}-L_k < N2n^{1/2} < 2n^{5/6} 0$ there exists $K_0 > 0$ such that for $K > K_0$ $\\\\sup _{U_1, U_2 \\\\in \\\\mathcal {U}} \\\\mathbb {P}_{U_1,U_2}^{K}(\\\\mathcal {L}_1 > n) \\\\le Cn^{-M}.$ Where $\\\\mathcal {U} \\\\lbrace U \\\\in \\\\mathbb {Z}^d : |U\\\\cdot \\\\vec{\\\\ell }|\\\\le 1 \\\\rbrace $ and the constant $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"For the same reasons as in previous proofs we restrict to the case where $U_1 = U_2 = 0$ without loss of generality, indeed we simply have to replaces boxes $\\\\mathcal {B}(n,n^{\\\\alpha })$ with $\\\\widetilde{\\\\mathcal {B}}_i(n,n^{\\\\alpha })$ .\",\n \"We write: $&\\\\mathbb {P}_{0}^{K}(\\\\mathcal {L}_1 > n) \\\\le \\\\mathbb {P}^{K}_{0}\\\\left(T^1_{\\\\partial B(n/2,(n/2)^{\\\\alpha })} \\\\ne T^1_{\\\\partial ^{+} \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })} \\\\right) + \\\\mathbb {P}^{K}_{0}\\\\left(T^2_{\\\\partial \\\\mathcal {B}(n/2,(n/2)^{\\\\alpha })} \\\\ne T^2_{\\\\partial ^{+} \\\\mathcal {B}(n/2,(n/2)^{\\\\alpha })} \\\\right)\\\\\\\\&+\\\\sum _{x,y \\\\in \\\\partial ^{+} \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}^{} \\\\mathbb {P}^{K}_{0}(X^1_{T_{\\\\partial \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}} = x,X^2_{T_{\\\\partial \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}} = y,T^1_{\\\\mathcal {H}^{-}_{e_1}} \\\\circ \\\\theta _{T_x} < +\\\\infty )\\\\\\\\&+ \\\\sum _{x,y \\\\in \\\\partial ^{+} \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}^{} \\\\mathbb {P}^{K}_{0}(X^1_{T_{\\\\partial \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}} = x,X^2_{T_{\\\\partial \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}} = y,T^2_{\\\\mathcal {H}^{-}_{e_1}} \\\\circ \\\\theta _{T^2_y} < +\\\\infty )\\\\\\\\&+\\\\sum _{x,y \\\\in \\\\partial ^{+} \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}^{} \\\\mathbb {P}^{K}_{0}(X^1_{T_{\\\\partial \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}} = x,X^2_{T_{\\\\partial \\\\mathcal {B}((n/2),(n/2)^{\\\\alpha })}} = y,\\\\\\\\&\\\\hspace{128.0374pt} T^1_{\\\\mathcal {H}^{-}_{e_1}} \\\\circ \\\\theta _{T^1_x} = +\\\\infty ,T^2_{\\\\mathcal {H}^{-}_{e_1}} \\\\circ \\\\theta _{T^2_y} = +\\\\infty ,\\\\mathcal {L}_1 \\\\circ \\\\theta _{T_{x,y}} \\\\ge n/2),$ where in the last sum $\\\\theta _{T_{x,y}}$ means that the first walk is shifted by $T_x$ and the second by $T_y$ .\",\n \"The first two terms are dominated thanks to .\",\n \"Let us dominate the third term.\",\n \"The fourth can be dominated in the same way by symmetry.\",\n \"The term in the sum can be written using Markov property $\\\\mathbf {E}&\\\\left[P^{\\\\omega _K}_{0}\\\\left(X^1_{T_{\\\\partial B((n/2),(n/2)^{\\\\alpha })}} = x,X^2_{T_{\\\\partial B((n/2),(n/2)^{\\\\alpha })}} = y\\\\right)P_{x}^{\\\\omega ^{0}_K}\\\\left(T^1_{\\\\mathcal {H}^{-}_{e_1}} < +\\\\infty \\\\right)\\\\right] \\\\\\\\ &\\\\le \\\\mathbf {E}\\\\left[P_{x}^{\\\\omega ^{0}_K}\\\\left(T^1_{\\\\mathcal {H}^{-}_{e_1}} < +\\\\infty \\\\right)\\\\right],$ where the notation $\\\\omega ^{0}_{K}$ means that we have set the conductances of $\\\\mathcal {E}_0$ to $K$ .\",\n \"Then we notice that $P_{x}^{\\\\omega ^{0}_K}(T^1_{\\\\mathcal {H}^{-}_{e_1}} < +\\\\infty )$ does not depend on the conductances of $\\\\mathcal {E}_0$ , so it is also equal to $P_{x}^{\\\\omega }(T^1_{\\\\mathcal {H}^{-}_{e_1}} < +\\\\infty )$ .\",\n \"Using translation invariance of the environment and the sum is bounded by $Cn^{2d\\\\alpha }\\\\exp (-cn)$ .\",\n \"Let us dominate the fifth term.\",\n \"Using Lemma REF the term in the sum can be written $&\\\\mathbf {E}\\\\left[P_{0}^{\\\\omega _{K}}\\\\left(X^1_{T_{\\\\partial B((n/2)),(n/2)^{\\\\alpha })}} = x,X^2_{T_{\\\\partial B((n/2),(n/2)^{\\\\alpha })}} = y\\\\right)P_{x,y}^{\\\\omega ^{0}_{K}}\\\\left(T^1_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,T^2_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,\\\\mathcal {L}_1\\\\ge n\\\\right)\\\\right]\\\\\\\\&\\\\le \\\\mathbf {E}\\\\left[P_{x,y}^{\\\\omega ^{0}_{K}}\\\\left(T^1_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,T^2_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,\\\\mathcal {L}_1\\\\ge n\\\\right)\\\\right].$ Under $\\\\lbrace T^1_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,T^2_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty \\\\rbrace $ both walks never meet conductances in $\\\\mathcal {E}_{0}$ and so we can replace $\\\\omega ^{0}_{K}$ by $\\\\omega $ .\",\n \"Then we write $\\\\mathbf {E}\\\\left[P_{x,y}^{\\\\omega ^{0}_{K}}\\\\left(T^1_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,T^2_{\\\\mathcal {H}^{-}_{e_1}} = +\\\\infty ,\\\\mathcal {L}_1\\\\ge n\\\\right)\\\\right]&\\\\le \\\\mathbb {P}_{x,y}(\\\\mathcal {L}_1\\\\ge n) \\\\\\\\&\\\\le \\\\mathbb {P}_{0,y-x}(\\\\mathcal {L}_1\\\\ge n/4),$ where we used Proposition REF since we have $|(y-x) \\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ because $x,y \\\\in \\\\partial ^{+} B((n/2),(n/2)^{\\\\alpha })$ , translation invariance of the environment.\",\n \"Then we see that the sum is bounded by $ C n^{2d\\\\alpha } n^{-(M+2d\\\\alpha )} $ by taking $K$ large enough.\",\n \"This concludes the proof.\",\n \"We can define successive joint regeneration levels as $\\\\mathcal {L}_{k+1} = \\\\mathcal {L}_{k} \\\\circ \\\\theta _{T_{\\\\mathcal {L}_1}}.$ Let us define $\\\\mathcal {G}_k = &\\\\sigma \\\\big (\\\\mathcal {L}_0,\\\\dots ,\\\\mathcal {L}_k,(\\\\tilde{X}^1_{T^1_{\\\\mathcal {L}_k}\\\\wedge m})_{m \\\\ge 0},(\\\\tilde{X}^2_{T^2_{\\\\mathcal {L}_k}\\\\wedge m})_{m \\\\ge 0}, \\\\nonumber \\\\\\\\ & c_{*}(e) \\\\text{ with } e \\\\in E(\\\\mathcal {H}^{-}(\\\\mathcal {L}_k-2e_1\\\\cdot \\\\vec{\\\\ell })) \\\\cup \\\\mathcal {E}^{2}_{X^1_{T^1_{\\\\mathcal {L}_k}}} \\\\cup \\\\mathcal {E}^{2}_{X^2_{T^2_{\\\\mathcal {L}_k}}}\\\\big ).$ The definition of $\\\\mathcal {G}_k$ may seem a bit complicated at first sight.\",\n \"It represents the information that the walkers see before reaching level $\\\\mathcal {L}_k$ and also the whole environment on the left of $\\\\mathcal {L}_k-2e_1\\\\cdot \\\\vec{\\\\ell }$ .\",\n \"Figure: The sigma algebra 𝒢 k \\\\mathcal {G}_k: In red and blue the trajectories before reaching ℒ k \\\\mathcal {L}_k and in orange the edges considered.Now we want to show that when we consider our walks shifted of $T_{\\\\mathcal {L}_k}$ they behave like new walks starting from respectively $X^1_{T^1_{\\\\mathcal {L}_k}}$ and $X^2_{T^2_{\\\\mathcal {L}_k}}$ conditionned to never backtrack and such that $c_{*}(e) = K$ for $e \\\\in \\\\mathcal {E}_{X^1_{T^1_{\\\\mathcal {L}_k}}} \\\\cup \\\\mathcal {E}_{X^2_{T^2_{\\\\mathcal {L}_k}}}$ .\",\n \"This is the purpose of the next theorem.\",\n \"Theorem 3.20 Fix $K$ large enough.\",\n \"For any $k \\\\ge 1$ , we have $\\\\mathcal {L}_k < +\\\\infty $ $\\\\mathbb {P}_0$ -almost surely.\",\n \"Now let $f, g, h_k$ be bounded functions which are measurable with respect to $\\\\sigma \\\\left( (X^1_0,X^2_0), (\\\\tilde{X}^1_n,\\\\tilde{X}^2_n)_{n \\\\ge 1}\\\\right)$ , $\\\\sigma \\\\left(c_{*}(e) \\\\text{ ; } e \\\\in \\\\mathcal {R}^{X^1_0} \\\\cup \\\\mathcal {R}^{X^2_0} \\\\backslash (\\\\mathcal {E}_{X^1_0} \\\\cup \\\\mathcal {E}_{X^2_0})\\\\right)$ and $\\\\mathcal {G}_k$ , respectively.\",\n \"We have the following equality $\\\\mathbb {E}_{\\\\begin{array}{c}(0,z_1)\\\\\\\\(0,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{\\\\mathcal {L}_k}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{\\\\mathcal {L}_k}+\\\\cdot })g\\\\circ \\\\theta _{T_{\\\\mathcal {L}_k}}h_k\\\\right] = \\\\mathbb {E}_{\\\\begin{array}{c}(0,z_1)\\\\\\\\(0,z_2)\\\\end{array}}\\\\left[h_k\\\\mathbb {E}^K_{X^1_{T^{1}_{\\\\mathcal {L}_k}},X^2_{T^{2}_{\\\\mathcal {L}_k}}}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot })g \\\\text{ } | \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right]\\\\right].$ This proof can be done by induction.\",\n \"We can restrict to the case $k = 1$ , but we have to prove the result with initial points $|U_1\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and $|U_2 \\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and also when the conductances in $\\\\mathcal {E}_{U_1}\\\\cup \\\\mathcal {E}_{U_2}$ are fixed to a certain value $a \\\\in [1/K,K]^{\\\\mathcal {E}_{U_1}\\\\cup \\\\mathcal {E}_{U_2}}$ $\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{\\\\mathcal {L}_1}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{\\\\mathcal {L}_1}+\\\\cdot })g\\\\circ \\\\theta _{T_{\\\\mathcal {L}_1}}h_1\\\\right] = \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1\\\\mathbb {E}^K_{X^1_{T^{1}_{\\\\mathcal {L}_1}},X^2_{T^{2}_{\\\\mathcal {L}_1}}}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot })g \\\\text{ } | \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right]\\\\right].$ Here the notation $\\\\mathbb {E}^{a}$ means that we fix the conductances in $\\\\mathcal {E}_{U_1}\\\\cup \\\\mathcal {E}_{U_2}$ to $a$ .\",\n \"The proof with $\\\\mathbb {E}$ is the same.\",\n \"By definition of $\\\\mathcal {L}_1$ , we have $\\\\displaystyle &\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{\\\\mathcal {L}_1}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{\\\\mathcal {L}_1}+\\\\cdot })g\\\\circ \\\\theta _{T_{\\\\mathcal {L}_1}}h_1\\\\right] \\\\\\\\&= \\\\sum _{k \\\\ge 0}^{} \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{L_k}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{L_k}+\\\\cdot })g\\\\circ \\\\theta _{T_{L_k}}h_1,L_k <+\\\\infty , R_k = +\\\\infty \\\\right]\\\\\\\\&= \\\\sum _{\\\\begin{array}{c}k \\\\ge 0 \\\\\\\\ p \\\\ge 0\\\\end{array}, \\\\begin{array}{c}(x,z)\\\\\\\\(x^{\\\\prime },z^{\\\\prime })\\\\end{array}}^{} \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{p}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{p}+\\\\cdot })g\\\\circ \\\\theta _{T_p}h_1,L_k =p, R_k = +\\\\infty ,\\\\tilde{X}^1_{T^{1}_{p}} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right].$ For $p,(x,z),(x^{\\\\prime },z^{\\\\prime })$ in the sum and following the arguments of there exists $h_1^{\\\\prime }$ measurable with respect to $\\\\sigma \\\\left((\\\\tilde{X}^1_{T^1_{p}\\\\wedge m})_{m \\\\ge 0},(\\\\tilde{X}^2_{T^2_{p}\\\\wedge m})_{m \\\\ge 0}, c_{*}(e) \\\\text{ with } e \\\\in \\\\mathcal {L}^{x- 2e_1} \\\\cup \\\\mathcal {L}^{x^{\\\\prime }- 2e_1} \\\\cup \\\\mathcal {E}^{2}_{x} \\\\cup \\\\mathcal {E}^{2}_{x^{\\\\prime }}\\\\right),$ such that on the event $\\\\lbrace \\\\tilde{X}^1_{T^{1}_p} = (x,z)\\\\rbrace \\\\cap \\\\lbrace \\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\rbrace \\\\cap \\\\lbrace \\\\mathcal {L}_1 = p\\\\rbrace $ the functions $h_1$ and $h^{^{\\\\prime }}_1$ coincide.\",\n \"Fix $k,p \\\\ge 0$ and $(x,z),(x^{\\\\prime },z^{\\\\prime })$ , in the sum.\",\n \"We can replace $h_1$ by $h_1^{\\\\prime }$ but to simplify notations we will still write $h_1$ and suppose it is measurable with respect to the above $\\\\sigma $ -algebra.\",\n \"We can do the same with $g \\\\circ \\\\theta _{T_p}$ , indeed we can consider $g$ as a function of $(X^1_0,X^2_0,\\\\omega )$ and then under $\\\\lbrace \\\\mathcal {L}_1 = p,X^1_{T^1_{\\\\mathcal {L}_1}} = x,X^2_{T^2_{\\\\mathcal {L}_1}} = x^{\\\\prime } \\\\rbrace $ we can replace $g$ by a function which is measurable with respect to $\\\\sigma \\\\left(c_{*}(e) \\\\text{ ; } e \\\\in \\\\mathcal {R}^{x} \\\\cup \\\\mathcal {R}^{x^{\\\\prime }} \\\\backslash (\\\\mathcal {E}_{x} \\\\cup \\\\mathcal {E}_{x^{\\\\prime }})\\\\right).$ The term in the sum can then be written $&\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{p}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{p}+\\\\cdot })g^{(x,x^{\\\\prime })}h_1,L_k =p, R_k = +\\\\infty ,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\\\\\&=\\\\mathbf {E}\\\\left[E^{\\\\omega _a}_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\times g^{(x,x^{\\\\prime })}E^{\\\\omega _a}_{\\\\begin{array}{c}(x,z)\\\\\\\\(x^{\\\\prime },z^{\\\\prime })\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right]\\\\right].$ In the following we will use the shortcut $E^{\\\\omega _a}_{\\\\begin{array}{c}(x,z)\\\\\\\\(x^{\\\\prime },z^{\\\\prime })\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right] = E^{\\\\omega _a}_{(\\\\mathbf {x}, \\\\mathbf {z})}\\\\left[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right].$ Where we have used Lemma REF with $\\\\tilde{X}^1$ and $\\\\tilde{X}^2$ and with the stopping times $T^1_{p}$ and $T^2_{p}$ .\",\n \"We recall the notation $\\\\omega _a$ which means we fix the conductances in $\\\\mathcal {E}_{U_1}\\\\cup \\\\mathcal {E}_{U_2}$ to $a$ .\",\n \"We want to stress that in the expression $E^{\\\\omega _a}_{(\\\\mathbf {x}, \\\\mathbf {z})}[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty ]$ , with a little abuse of the notation, we fix to the value $a$ the conductances in $\\\\mathcal {E}_{U_1}\\\\cup \\\\mathcal {E}_{U_2}$ and not the ones in $\\\\mathcal {E}_{x}\\\\cup \\\\mathcal {E}_{x^{\\\\prime }}$ as one might expect.\",\n \"Moreover, thanks to the fact that $P^{\\\\omega _a}_{(\\\\mathbf {x}, \\\\mathbf {z})}$ -almost surely the walks never backtrack below $x$ and $x^{\\\\prime }$ on $\\\\lbrace \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\rbrace $ , the value of the conductances in $\\\\mathcal {E}_{U_1}\\\\cup \\\\mathcal {E}_{U_2}$ does not matter in the expectation so we can replace $\\\\omega _a$ with $\\\\omega $ and write $E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty ]$ .\",\n \"Then, since under $\\\\lbrace L_k =p,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\rbrace $ the points $x$ and $x^{\\\\prime }$ are $K$ -open and using Proposition REF , we can replace $\\\\omega $ by $\\\\omega _K$ in $E^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty ]$ .\",\n \"Notice that $f$ and $\\\\mathcal {D}$ are measurable with respect to $\\\\sigma \\\\big ( X^1_0,X^2_0,(\\\\tilde{X}^1_{n})_{n \\\\ge 1},(\\\\tilde{X}^2_{n})_{n\\\\ge 1} \\\\big )$ and thus can be written as $\\\\phi \\\\left(X^1_0,X^2_0,(\\\\tilde{X}^1_{n})_{n \\\\ge 1},(\\\\tilde{X}^2_{n})_{n\\\\ge 1}\\\\right)$ with $\\\\phi $ measurable.\",\n \"Since the law of $\\\\left( X^1_0,X^2_0,(\\\\tilde{X}^1_{n})_{n \\\\ge 1},(\\\\tilde{X}^2_{n})_{n\\\\ge 1}\\\\right)$ is the same under $P^{\\\\omega }_{(\\\\mathbf {x}, \\\\mathbf {z})}$ and under $P^{\\\\omega }_{(\\\\mathbf {x}, 0)}$ , we replace $z,z^{\\\\prime }$ by 0 in $E^{\\\\omega _K}_{(\\\\mathbf {x}, \\\\mathbf {z})}[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty ]$ .\",\n \"We observe that $E^{\\\\omega _K}_{\\\\begin{array}{c}x,x^{\\\\prime }\\\\end{array}}[f(\\\\tilde{X}^{1}_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot }), \\\\mathcal {D}^{\\\\bullet } = +\\\\infty ]$ is measurable with respect to $\\\\sigma \\\\left(c_{*}(e) \\\\text{ ; } e \\\\in \\\\mathcal {R}^{x} \\\\cup \\\\mathcal {R}^{x^{\\\\prime }} \\\\backslash (\\\\mathcal {E}_{x} \\\\cup \\\\mathcal {E}_{x^{\\\\prime }})\\\\right).$ Moreover, $E^{\\\\omega _a}_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}[h_1,L_k =p,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })]$ is measurable with respect to $\\\\sigma \\\\left(c_{*}(e) \\\\text{ with } e \\\\in E(\\\\mathcal {H}^{-}(p-2e_1\\\\cdot \\\\vec{\\\\ell })) \\\\cup \\\\mathcal {E}^{2}_{x} \\\\cup \\\\mathcal {E}^{2}_{x^{\\\\prime }}\\\\right).$ In addition, $\\\\mathcal {R}^{x} \\\\cup \\\\mathcal {R}^{x^{\\\\prime }} \\\\backslash (\\\\mathcal {E}_{x} \\\\cup \\\\mathcal {E}_{x^{\\\\prime }})$ and $E(\\\\mathcal {H}^{-}(p-2e_1\\\\cdot \\\\vec{\\\\ell })) \\\\cup \\\\mathcal {E}^{2}_{x} \\\\cup \\\\mathcal {E}^{2}_{x^{\\\\prime }}$ can be proved to be disjoint in an elementary way as illustrated in the figure below.\",\n \"Figure: In orange E(ℋ - (p-2e 1 ·ℓ →))∪ℰ x 2 ∪ℰ x ' 2 E(\\\\mathcal {H}^{-}(p-2e_1\\\\cdot \\\\vec{\\\\ell })) \\\\cup \\\\mathcal {E}^{2}_{x} \\\\cup \\\\mathcal {E}^{2}_{x^{\\\\prime }} and in green ℛ x ∪ℛ x ' ∖(ℰ x ∪ℰ x ' )\\\\mathcal {R}^{x} \\\\cup \\\\mathcal {R}^{x^{\\\\prime }} \\\\backslash (\\\\mathcal {E}_{x} \\\\cup \\\\mathcal {E}_{x^{\\\\prime }}).By independence of the two $\\\\sigma $ -algebra we deduce that $&\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[f(\\\\tilde{X}^{1}_{T^{1}_{p}+\\\\cdot },\\\\tilde{X}^{2}_{T^{2}_{p}+\\\\cdot })g^{(x,x^{\\\\prime })}h_1,L_k =p, R_k = +\\\\infty ,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\\\\\&=\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\mathbb {E}^{K}_{\\\\begin{array}{c}x,x^{\\\\prime }\\\\end{array}}\\\\left[gf(\\\\tilde{X}^{1}_{.},\\\\tilde{X}^{2}_{.\",\n \"}),\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right].$ Last equality used in the particular case where $f = 1$ and $g = 1$ gives $&\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p, R_k = +\\\\infty ,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\\\\\&=\\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\mathbb {P}^{K}_{\\\\begin{array}{c}x,x^{\\\\prime }\\\\end{array}}(\\\\mathcal {D}^{\\\\bullet } =+\\\\infty ).$ Hence we can write $& \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\mathbb {E}^{K}_{\\\\begin{array}{c}x,x^{\\\\prime }\\\\end{array}}\\\\left[gf(\\\\tilde{X}^{1}_{.},\\\\tilde{X}^{2}_{.\",\n \"}),\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right] \\\\\\\\&= \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p, R_k = +\\\\infty ,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\mathbb {E}^{K}_{\\\\begin{array}{c}x,x^{\\\\prime }\\\\end{array}}\\\\left[gf(\\\\tilde{X}^{1}_{.},\\\\tilde{X}^{2}_{.})\",\n \"\\\\text{ }| \\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right].$ Then taking the sum we obtain $&\\\\sum _{\\\\begin{array}{c}k \\\\ge 0 \\\\\\\\ p \\\\ge 0\\\\end{array}, \\\\begin{array}{c}(x,z)\\\\\\\\(x^{\\\\prime },z^{\\\\prime })\\\\end{array}}^{} \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1,L_k =p, R_k = +\\\\infty ,\\\\tilde{X}^1_{T^{1}_p} = (x,z),\\\\tilde{X}^2_{T^{2}_p} = (x^{\\\\prime },z^{\\\\prime })\\\\right]\\\\mathbb {E}^{K}_{\\\\begin{array}{c}x,x^{\\\\prime }\\\\end{array}}\\\\left[gf(\\\\tilde{X}^{1}_{.},\\\\tilde{X}^{2}_{.})\",\n \"\\\\text{ }| \\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right]\\\\\\\\&= \\\\mathbb {E}^a_{\\\\begin{array}{c}(U_1,z_1)\\\\\\\\(U_2,z_2)\\\\end{array}}\\\\left[h_1\\\\mathbb {E}^K_{X^1_{T^{1}_{\\\\mathcal {L}_1}},X^2_{T^{2}_{\\\\mathcal {L}_1}}}\\\\left[f(\\\\tilde{X}^1_{\\\\cdot },\\\\tilde{X}^{2}_{\\\\cdot })g \\\\text{ } | \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right]\\\\right].$ Then for the induction, if the formula has been proved for $k$ , from Proposition REF we get that $\\\\mathcal {L}_{k+1}$ is almost surely finite.\",\n \"The proof works in a similar way to go from $k$ to $k+1$ and the details of the adaptation can be found in .\",\n \"A consequence of this theorem is the analogous of Proposition REF but in a more general setting.\",\n \"Proposition 3.21 For any $M > 0$ there exists a constant $c$ such that for any $k > 0$ $\\\\mathbb {P}_{0}(\\\\mathcal {L}_{k+1} - \\\\mathcal {L}_k \\\\ge n) \\\\le cn^{-M}.$ We write $\\\\mathbb {P}_{0}(\\\\mathcal {L}_{k+1} - \\\\mathcal {L}_k \\\\ge n) &= \\\\sum _{p,x,x^{\\\\prime }}^{} \\\\mathbb {E}_{0}\\\\left[\\\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\\\prime }, \\\\mathcal {L}_{k+1} - \\\\mathcal {L}_k \\\\ge n\\\\right]\\\\\\\\& = \\\\sum _{p,x,x^{\\\\prime }}^{} \\\\mathbb {E}_{0}\\\\left[\\\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\\\prime }, \\\\mathcal {L}_1 \\\\circ \\\\theta _{T_{x,x^{^{\\\\prime }}}} - p \\\\ge n\\\\right]\\\\\\\\& = \\\\sum _{p,x,x^{\\\\prime }}^{} \\\\mathbb {E}_{0}\\\\left[\\\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\\\prime },\\\\mathbb {P}^{K}_{x,x^{\\\\prime }}(\\\\mathcal {L}_1 - p \\\\ge n \\\\text{ } | \\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty )\\\\right].$ where the third equality follows from Theorem REF .\",\n \"We only have to dominate $\\\\mathbb {P}^{K}_{x,x^{\\\\prime }}(\\\\mathcal {L}_1 - p \\\\ge n \\\\text{ } | \\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty )$ to conclude.\",\n \"To do so, using the fact that $\\\\mathbb {P}_{x,x^{\\\\prime }}^{K}(\\\\mathcal {D}^{\\\\bullet } =+\\\\infty ) > \\\\alpha > 0$ with $\\\\alpha $ which does not depend on $x$ and $x^{\\\\prime }$ , we can see it suffices to bound $\\\\mathbb {P}^{K}_{x,x^{\\\\prime }}(\\\\mathcal {L}_1 - p \\\\ge n )$ .\",\n \"Set $y = x^{\\\\prime } - x$ , by the invariance in law of the environment by translation we have $\\\\mathbb {P}^{K}_{x,x^{\\\\prime }}(\\\\mathcal {L}_1 - p \\\\ge n ) = \\\\mathbb {P}^{K}_{0,y}(\\\\mathcal {L}_1 -p + x\\\\cdot \\\\vec{\\\\ell } \\\\ge n ) $ .\",\n \"Notice that we have $|x\\\\cdot \\\\vec{\\\\ell } - p| \\\\le 1$ , which follows from the fact that $X^1_{T^{1}_p} = x$ .\",\n \"Thus the result follows from Proposition REF .\"\n ],\n [\n \"Asymptotic separation\",\n \"The aim of this section is to prove the following result: the trajectories of the two walks do not cross anymore in $\\\\mathcal {H}^{+}_N$ with high probability.\",\n \"We recall the notation $\\\\mathcal {H}^{+}_{N} = \\\\lbrace x \\\\in \\\\mathbb {Z}^d \\\\text{ }|\\\\text{ } x\\\\cdot \\\\vec{\\\\ell } > N\\\\rbrace $ .\",\n \"We introduce the event that the traces of the walks in $\\\\mathcal {H}^{+}_N$ get close $ \\\\mathbf {M}_N \\\\bigg \\\\lbrace \\\\inf _{\\\\begin{array}{c}x \\\\in \\\\mathcal {H}^{+}_{N}\\\\cap \\\\lbrace X_n^1\\\\rbrace \\\\\\\\y \\\\in \\\\mathcal {H}^{+}_{N}\\\\cap \\\\lbrace X_n^2\\\\rbrace \\\\end{array}}\\\\Vert x-y\\\\Vert _{1}\\\\le 1 \\\\bigg \\\\rbrace $ Proposition 4.1 For any $N \\\\ge 0$ , the following inequality holds $\\\\mathbb {P}_{0}(\\\\mathbf {M}_N ) \\\\le CN^{-c},$ where the constants $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d$ .\",\n \"In this part we will adapt methods from .\",\n \"Firstly we will deal with the case when the two walks $X^1$ and $X^2$ live in independent environments.\",\n \"Then, when the two walks live in the same environment, we will try use the fact that when they do not cross, they behave like if they were in independent environments.\"\n ],\n [\n \"independent environments\",\n \"First we recall some notations: The law $Q_{U_1,U_2}$ is defined as $Q_{U_1,U_2} = \\\\mathbb {P}_{U_1} \\\\otimes \\\\mathbb {P}_{U_2}$ .\",\n \"Hence $X^1$ and $X^2$ are independent under $Q_{U_1,U_2}$ .\",\n \"We also have $Q^{K}_{U_1,U_2} =\\\\mathbb {P}^{K}_{U_1} \\\\otimes \\\\mathbb {P}^{K}_{U_2}$ .\",\n \"We write $Q_{U_1,U_2}(\\\\cdot \\\\, | \\\\, D^{\\\\otimes }=+\\\\infty ) = \\\\mathbb {P}_{U_1}(\\\\cdot \\\\, | \\\\, D=+\\\\infty ) \\\\otimes \\\\mathbb {P}_{U_2}(\\\\cdot \\\\, | \\\\, D=+\\\\infty )$ where $D$ is defined in in the definition of regeneration times.\",\n \"Proposition 4.2 There exists $K_0 > 0$ such that for any $U_1, U_2 \\\\in \\\\mathbb {Z}^d$ and $K > K_0$ we have $Q^{K}_{U_1, U_2}\\\\bigg (\\\\inf _{\\\\begin{array}{c}x \\\\in \\\\lbrace X_n^1\\\\rbrace \\\\\\\\y \\\\in \\\\lbrace X_n^2\\\\rbrace \\\\end{array}}\\\\Vert x-y\\\\Vert _{1}\\\\le 1 \\\\, \\\\bigg | \\\\, \\\\textnormal {}D^{\\\\otimes }=+\\\\infty \\\\bigg ) \\\\le C\\\\Vert U_1 - U_2\\\\Vert ^{-c},$ where the constants $C,c > 0 $ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Remark 4.3 We observe that if $U_1 = U_2$ the above inequality remains true with the convention $\\\\Vert U_1-U_2\\\\Vert ^{-c} = +\\\\infty $ .\",\n \"For $z \\\\in \\\\mathbb {Z}^d$ we define $F_i(z) = \\\\mathbb {P}^{K}_{U_i}(\\\\exists k, X_k^{i} = z \\\\, | \\\\, \\\\text{}D=+\\\\infty ),$ and $F_i^{(R)}(z) = F_i(z){1}_{\\\\lbrace \\\\Vert z-U_i\\\\Vert > R\\\\rbrace }.$\"\n ],\n [\n \"Step 1: bounding our quantity using $\\\\Vert F_i^{(R)}\\\\Vert _2$ and {{formula:7505ed72-501c-44cd-9caa-ec40deaa1136}} .\",\n \"Fix $R = \\\\Vert U_1-U_2\\\\Vert /3$ .\",\n \"If $\\\\Vert X^{1}_p-X^2_q\\\\Vert \\\\le 1$ then necessarily either $\\\\Vert U_1-X^1_p\\\\Vert \\\\ge R$ or $\\\\Vert U_2-X^2_q\\\\Vert \\\\ge R$ , otherwise we would have $\\\\Vert U_1-U_2\\\\Vert \\\\le 2/3\\\\Vert U_1-U_2\\\\Vert + 2 < \\\\Vert U_1-U_2\\\\Vert $ (as soon as $\\\\Vert U_1-U_2\\\\Vert \\\\ge 7$ ).\",\n \"It follows $\\\\nonumber Q^{K}_{U_1,U_2} &\\\\left(\\\\exists p,q \\\\in \\\\mathbb {N} \\\\text{ ; } \\\\Vert X_p^1 - X_q^2\\\\Vert \\\\le 1 \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right)\\\\\\\\&\\\\le Q^{K}_{U_1,U_2}\\\\left(\\\\exists p,q \\\\in \\\\mathbb {N} \\\\text{ ; } \\\\Vert X_p^1 - X_q^2\\\\Vert \\\\le 1 \\\\text{ and } \\\\Vert U_1-X^1_p\\\\Vert \\\\ge R \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right)\\\\\\\\& + Q^{K}_{U_1,U_2}\\\\left(\\\\exists p,q \\\\in \\\\mathbb {N} \\\\text{ ; } \\\\Vert X_p^1 - X_q^2\\\\Vert \\\\le 1 \\\\text{ and } \\\\Vert U_2-X^2_q\\\\Vert \\\\ge R \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right).$ We write: $Q^{K}_{U_1,U_2} &\\\\left(\\\\exists p,q \\\\in \\\\mathbb {N} \\\\text{ ; } \\\\Vert X_p^1 - X_q^2\\\\Vert \\\\le 1 \\\\text{ and } \\\\Vert U_1-X^1_p\\\\Vert \\\\ge R \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right)\\\\\\\\&\\\\le \\\\sum _{\\\\Vert z-U_1\\\\Vert \\\\ge R}^{} Q^{K}_{U_1,U_2}\\\\left(\\\\exists p,q \\\\in \\\\mathbb {N} \\\\text{ ; } X^1_p = z, X^2_q \\\\in \\\\mathcal {V}_z\\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right)\\\\\\\\& \\\\le \\\\sum _{e \\\\in \\\\mathcal {V}_{0}}^{} \\\\sum _{\\\\Vert z-U_1\\\\Vert \\\\ge R}^{} F_1^{(R)}(z)F_2^{}(z+e) \\\\\\\\&\\\\le \\\\sum _{e \\\\in \\\\mathcal {V}_{0}}^{} \\\\sum _{z \\\\in \\\\mathbb {Z}^d}^{} F_1^{(R)}(z)F_2^{}(z+e) \\\\\\\\& \\\\le C\\\\Vert F^{(R)}_1\\\\Vert _2\\\\Vert F_2\\\\Vert _2,$ where the third inequality comes from Cauchy-Schwarz inequality and where the constant $C > 0$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Symmetrically, we can do the same computation for (REF ).\",\n \"It follows that if we show that $\\\\Vert F_i\\\\Vert _2 < +\\\\infty $ and $\\\\Vert F_i^{(R)}\\\\Vert _2 \\\\le CR^{-c}$ with $C,c > 0$ constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ , the proof is finished.\"\n ],\n [\n \"Step 2: bound on $\\\\Vert F_i\\\\Vert _2$ and {{formula:23b48c57-5a69-4a5d-8b96-9d7b1d444caf}} .\",\n \"For $n \\\\in \\\\mathbb {N}$ we introduce $H_i(z,n) = \\\\mathbb {P}^{K}_{U_i}(\\\\exists \\\\tau _n^{i} \\\\le k \\\\le \\\\tau _{n+1}^{i} \\\\text{, } X_k^{i} = z \\\\, | \\\\, \\\\text{}D=+\\\\infty ),$ and $H_i^{(R)}(z,n) = H_i(z,n){1}_{\\\\lbrace \\\\Vert z-U_i \\\\Vert > R \\\\rbrace }.$ Using the triangle inequality we write $\\\\Vert F_i\\\\Vert _{2} \\\\le \\\\sum _{n=1}^{+\\\\infty } \\\\Vert H_i(\\\\cdot ,n)\\\\Vert _{2},$ and $\\\\Vert F_i^{(R)}\\\\Vert _{2} \\\\le \\\\sum _{n = 1}^{+\\\\infty } \\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{2}.$ We want to bound $\\\\Vert H_i(\\\\cdot ,n)\\\\Vert _2$ and $\\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _2$ .\",\n \"The quantity $H_i(\\\\cdot ,n)$ can be rewritten as a convolution because $X_{\\\\tau _n^{i}}^{i}$ and $\\\\lbrace X_{\\\\tau _n^{i} + \\\\, \\\\cdot }^{i} - X_{\\\\tau _n^{i}}^{i} \\\\rbrace $ are independent by .\",\n \"Thus $H_i(\\\\cdot ,n) = T_i^{n} \\\\ast J,$ where $T_i^{n}(z) = \\\\mathbb {P}_{U_i}^{K}(X^{i}_{\\\\tau ^{i}_n} = z),$ and $J(z) = \\\\mathbb {P}_{0}^{K}\\\\left(\\\\exists 0 \\\\le k \\\\le \\\\tau _{1}^{i} \\\\text{, } X_k^{i} = z \\\\, | \\\\, \\\\text{}D=+\\\\infty \\\\right).$ Young's inequality gives $\\\\Vert H_i(\\\\cdot ,n)\\\\Vert _{2} \\\\le \\\\Vert T_i^{n}\\\\Vert _{2} \\\\Vert J\\\\Vert _{1}.$ We have $\\\\Gamma := \\\\Vert J\\\\Vert _{1} < +\\\\infty $ .\",\n \"Indeed $\\\\sum _{z \\\\in \\\\mathbb {Z}^d}^{} \\\\mathbb {P}_{0}^{K}\\\\left(\\\\exists 0 \\\\le k \\\\le \\\\tau _{1}^{i} \\\\text{, } X_k^{i} = z \\\\, | \\\\, \\\\text{}D=+\\\\infty \\\\right) \\\\le C\\\\mathbb {E}_{0}^{K}\\\\bigg [\\\\sum _{z \\\\in \\\\mathbb {Z}^d}^{} {1}_{\\\\exists 0 \\\\le k \\\\le \\\\tau ^i_{1} \\\\text{, } X_k^{i} = z} \\\\bigg ],$ where $C$ is a constant which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ using .\",\n \"The term in the last expectation is equal to $\\\\#\\\\lbrace X_k^{i} \\\\text{ ; } 0\\\\le k \\\\le \\\\tau _1^{i}\\\\rbrace $ , now recall the notation $\\\\chi ^i = \\\\inf \\\\lbrace m \\\\ge 0 \\\\text{ ; }\\\\lbrace X_k^{i} \\\\text{ ; } 0\\\\le k \\\\le \\\\tau _1^{i}\\\\rbrace \\\\subset X^i_0 + \\\\mathcal {B}(m,m^{\\\\alpha })\\\\rbrace $ , then we clearly have $\\\\#\\\\lbrace X_k^{i} \\\\text{ ; } 0\\\\le k \\\\le \\\\tau _1^{i}\\\\rbrace \\\\le (\\\\chi ^i)^{\\\\beta }$ for $\\\\beta $ large enough which only depends on $d,\\\\alpha $ .\",\n \"By , for $K$ large enough, $\\\\chi ^i$ has finite moment of order $\\\\beta $ .\",\n \"It follows that $||J||_1 < +\\\\infty $ .\",\n \"For the term $\\\\Vert T_i^{n}\\\\Vert _2$ , using Proposition REF we have for $K$ high enough $\\\\displaystyle \\\\sup _{z \\\\in \\\\mathbb {Z}^d}\\\\mathbb {P}_{U_i}^{K}(X_{\\\\tau _n^{i}}^{i} = z) \\\\le Cn^{-\\\\frac{d}{2}+\\\\varepsilon },$ where $C$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"It gives the bound $\\\\Vert H_i^{}(\\\\cdot ,n)\\\\Vert _{2} \\\\le Cn^{-\\\\frac{d}{4}+\\\\frac{\\\\varepsilon }{2}}.$ In dimension $d \\\\ge 5$ we have $\\\\Vert F_i\\\\Vert _{2} < +\\\\infty .$ We still have to bound $\\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{2}$ .\",\n \"A first bound is given by (REF ), however for $n$ small this bound is not precise enough, thus we compute another bound which is useful for small $n$ .\",\n \"Let us bound $\\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{1}$ and then use the inequality $\\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{2} \\\\le \\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{1}^{1/2}$ .\",\n \"We can rewrite $\\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{1} &= \\\\sum _{||z-U_{i}|| \\\\ge R}\\\\mathbb {E}^{K}_{U_i}\\\\left[ {1}_{ \\\\left\\\\lbrace \\\\exists \\\\tau _n^{i} \\\\le k \\\\le \\\\tau _{n+1}^{i} \\\\, ; \\\\, X_k^{i} = z \\\\right\\\\rbrace } \\\\, \\\\big | \\\\, \\\\text{}D=+\\\\infty \\\\right] \\\\nonumber \\\\\\\\ &= \\\\mathbb {E}^{K}_{U_i}\\\\left[ \\\\sum _{||z - U_i|| \\\\ge R}{1}_{ \\\\left\\\\lbrace \\\\exists \\\\tau _n^{i} \\\\le k \\\\le \\\\tau _{n+1}^{i} \\\\, ; \\\\, X_k^{i} = z \\\\right\\\\rbrace } \\\\, \\\\big | \\\\, \\\\text{}D=+\\\\infty \\\\right].\",\n \"$ Define $A(n,R) \\\\lbrace \\\\exists \\\\, \\\\tau _n \\\\le k \\\\le \\\\tau _{n+1} \\\\textnormal { such that } |X^i_k-X^i_0| \\\\ge R\\\\rbrace $ .\",\n \"We introduce the notation $\\\\chi ^i_n = \\\\chi ^i(X^{i}_{\\\\tau _n + \\\\cdot })$ .\",\n \"The term $\\\\displaystyle \\\\sum _{||z - U_i|| \\\\ge R}{1}_{ \\\\left\\\\lbrace \\\\exists \\\\tau _n^{i} \\\\le k \\\\le \\\\tau _{n+1}^{i} \\\\, ; \\\\, X_k^{i} = z \\\\right\\\\rbrace }$ in the expectation (REF ) is bounded by ${1}_{A(n,R)}\\\\#\\\\mathcal {B}(\\\\chi ^{i}_n,(\\\\chi ^{i}_n)^{\\\\alpha })$ , indeed it is clearly bounded by $\\\\#\\\\mathcal {B}(\\\\chi ^{i}_n,(\\\\chi ^{i}_n)^{\\\\alpha })$ and under $A(n,R)^{c}$ it is equal to 0.\",\n \"Using the fact that $\\\\mathrm {diam}(\\\\mathcal {B}(L,L^{\\\\alpha })) \\\\le CL^{\\\\alpha }$ where $C$ is a constant which only depends on $\\\\vec{\\\\ell },d$ , we have the following inclusion: $A(n,R) \\\\subset \\\\bigg \\\\lbrace \\\\sum _{k=0}^{n} (C\\\\chi ^{i}_k)^{\\\\alpha } \\\\ge R \\\\bigg \\\\rbrace .$ Since $((\\\\chi ^{i}_{k})^{\\\\alpha })_{k \\\\ge 0}$ is a family of independent identically distributed random variables and for $K$ large enough $(\\\\chi ^{i}_0)^{\\\\alpha }$ has a finite first moment under $\\\\mathbb {P}^K_{U_i}(\\\\cdot \\\\text{ | }D=+\\\\infty )$ by , we have $\\\\mathbb {P}^{K}_{U_i}\\\\left(A(n,R) \\\\text{ | }D=+\\\\infty \\\\right) &\\\\le \\\\mathbb {P}^{K}_{U_i}\\\\left(\\\\sum _{k=0}^{n} C(\\\\chi ^{i}_k)^{\\\\alpha } \\\\ge R\\\\text{ | }D=+\\\\infty \\\\right)\\\\\\\\&\\\\le C\\\\frac{(n+1)\\\\, \\\\mathbb {E}_{U_i}^{K}\\\\left[(\\\\chi ^{i}_{1})^{\\\\alpha }\\\\text{ | }D=+\\\\infty \\\\right]}{R} = \\\\frac{Cn}{R},$ where $C > 0$ is a constant which only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"$\\\\bullet $ Now using , for any $M > 0$ there exists $K_0 > 0$ such that for $K \\\\ge K_0$ the variable $\\\\chi ^i_n$ has a moment of order $M$ .\",\n \"By chosing $K_0$ big enough, $\\\\#\\\\mathcal {B}(\\\\chi ^{i}_n,(\\\\chi ^{i}_n)^{\\\\alpha })$ has a moment of order 2.\",\n \"Indeed $\\\\#\\\\mathcal {B}(\\\\chi ^{i}_n,(\\\\chi ^{i}_n)^{\\\\alpha }) \\\\le C(\\\\chi ^i_n)^{d\\\\alpha }$ with $C$ a constant which only depends on $d$ .\",\n \"Using Cauchy-Schwarz inequality we have $\\\\nonumber \\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{1} &\\\\le \\\\mathbb {P}_{U_i}^{K}\\\\left( A(n,R)\\\\text{ | }D=+\\\\infty \\\\right)^{\\\\frac{1}{2}}\\\\mathbb {E}^{K}_{U_i}\\\\left[\\\\#\\\\mathcal {B}(\\\\chi ^{i}_n,(\\\\chi ^{i}_n)^{\\\\alpha })^2 \\\\text{ | }D=+\\\\infty \\\\right]^{\\\\frac{1}{2}}\\\\\\\\& \\\\le Cn^{\\\\frac{1}{2}}R^{-\\\\frac{1}{2}},$ where $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"It follows that we have $\\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{2} \\\\le \\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert ^{\\\\frac{1}{2}}_{1} \\\\le Cn^{\\\\frac{1}{4}}R^{-\\\\frac{1}{4}}.$ Pick $a > 0$ arbitrarily small.\",\n \"We bound $\\\\Vert F_i^{(R)}\\\\Vert _{2}$ by $\\\\Vert F_i^{(R)}\\\\Vert _{2} &\\\\le \\\\sum _{n=1}^{\\\\lfloor R^{a} \\\\rfloor } \\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{2} +\\\\sum _{n=\\\\lfloor R^{a} \\\\rfloor +1}^{+\\\\infty } \\\\Vert H_i^{(R)}(\\\\cdot ,n)\\\\Vert _{2}\\\\\\\\&\\\\underset{(\\\\ref {first_bound}),(\\\\ref {second_bound})}{\\\\le } C \\\\left(\\\\displaystyle \\\\sum _{n = 1}^{\\\\lfloor R^{a} \\\\rfloor }n^{\\\\frac{1}{4}}R^{-\\\\frac{1}{4}} + \\\\sum _{n = \\\\lfloor R^{a} \\\\rfloor + 1}^{+\\\\infty }n^{-\\\\frac{d}{4}+\\\\frac{\\\\varepsilon }{2}} \\\\right)\\\\\\\\&\\\\le CR^{-\\\\frac{1}{4}}R^{\\\\frac{5}{4}a} + C(R^{a})^{-\\\\frac{d}{4}+1+\\\\frac{\\\\varepsilon }{2}}.$ By taking $a$ small enough such that $\\\\frac{5}{4}a-\\\\frac{1}{4} < 0$ and since $-\\\\frac{d}{4}+1+\\\\frac{\\\\varepsilon }{2} < 0$ we have $\\\\Vert F_i^{(R)}\\\\Vert _{2} &\\\\le CR^{-c},$ where $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We introduce for $R \\\\ge 0$ the event $\\\\nonumber F(R) = &\\\\hspace{10.0pt}\\\\left\\\\lbrace \\\\exists p,q \\\\ge 2 \\\\text{, }\\\\Vert X^1_p- X^2_q\\\\Vert \\\\le 1\\\\right\\\\rbrace \\\\\\\\ \\\\nonumber & \\\\cap \\\\left\\\\lbrace \\\\forall p,q \\\\ge 2 \\\\text{, }\\\\Vert X^1_p-X^2_q\\\\Vert \\\\le 1 \\\\Rightarrow \\\\Vert X^1_p-X^1_0\\\\Vert \\\\wedge \\\\Vert X^2_q - X^1_0\\\\Vert \\\\ge R \\\\right\\\\rbrace \\\\\\\\&\\\\cap \\\\left\\\\lbrace \\\\forall p,q \\\\ge 2 \\\\text{, }\\\\Vert X^1_p-X^2_q\\\\Vert \\\\le 1 \\\\Rightarrow \\\\Vert X^1_p-X^2_0\\\\Vert \\\\wedge \\\\Vert X^2_q - X^2_0\\\\Vert \\\\ge R \\\\right\\\\rbrace .$ Roughly speaking, on the event $F(R)$ the two walks almost intersect (actually get at distance less than 1) but do not intersect at a point at distance less than $R$ from $U_1$ or $U_2$ .\",\n \"The last proof immediately gives the following corollary.\",\n \"Corollary 4.4 There exists $K_0 > 0$ such that, for any $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ and for any $K \\\\ge K_0$ , we have that $Q^{K}_{U_1,U_2}\\\\left(F(R)\\\\, | \\\\,\\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right) \\\\le CR^{-c},$ where the constants $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Taking the same notations as in last proof we have: $\\\\displaystyle Q^{K}_{U_1,U_2}\\\\left(F(R)\\\\, | \\\\,\\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right) &\\\\le \\\\sum _{e \\\\in \\\\mathcal {V}_{0}}^{} \\\\sum _{\\\\begin{array}{c}z \\\\in \\\\mathbb {Z}^d\\\\\\\\ \\\\Vert z-U_1\\\\Vert \\\\ge R \\\\\\\\ \\\\Vert z-U_2\\\\Vert \\\\ge R\\\\end{array}}^{} F_1^{(R)}(z)F_2^{(R)}(z + e)\\\\\\\\&\\\\le \\\\sum _{e \\\\in \\\\mathcal {V}_{0}}^{} \\\\sum _{\\\\begin{array}{c}z \\\\in \\\\mathbb {Z}^d\\\\end{array}}^{} F_1^{(R)}(z)F_2^{(R - 1)}(z + e).$ We conclude as in last proof by using Cauchy-Schwarz inequality.\",\n \"The last corollary shows that there is little probability that the two walks only intersect on points which are far from the starting points.\",\n \"In other words, with high probability, if the two walks do intersect, then it has to be close to one of the starting points.\",\n \"We need a last proposition before moving on to the case where both walks are in the same environment.\",\n \"We recall the notation $T^i_{R}$ denotes the first hitting time of the set $\\\\mathcal {H}^{+}(R)$ for the walk $X^i$ .\",\n \"We introduce the notation: $R_0 \\\\max (X^{1}_0\\\\cdot \\\\vec{\\\\ell },X^{2}_0 \\\\cdot \\\\vec{\\\\ell }).$ For $R > 0$ we define the event $ A(R) {\\\\left\\\\lbrace \\\\begin{array}{ll}X_p^{1}\\\\cdot \\\\vec{\\\\ell } > X^1_0\\\\cdot \\\\vec{\\\\ell } \\\\quad & \\\\forall 0 < p \\\\le T^1_{R+R_0} + 2,\\\\\\\\X_q^{2}\\\\cdot \\\\vec{\\\\ell } > X^2_0\\\\cdot \\\\vec{\\\\ell } \\\\quad &\\\\forall 0 < q \\\\le T^2_{R+R_0} + 2,\\\\\\\\\\\\Vert X_p^1 - X_q^2\\\\Vert \\\\ge 2 \\\\quad &\\\\forall 1 < p \\\\le T^1_{R+R_0} + 2 \\\\,\\\\, \\\\textnormal {and} \\\\,\\\\, \\\\forall 1 < q \\\\le T^2_{R+R_0} + 2.\\\\end{array}\\\\right.\",\n \"}$ Proposition 4.5 Let $R_0 = \\\\max (X_0^1 \\\\cdot \\\\vec{\\\\ell },X_0^2 \\\\cdot \\\\vec{\\\\ell }) $ and $R > 0$ .\",\n \"There exists $\\\\Lambda > 0$ and $K_0 > 0$ such that, for $K > K_0$ , $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ such that $|(U_1-U_2)\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and $\\\\Vert U_1-U_2\\\\Vert \\\\ge \\\\Lambda $ , $Q^{K}_{U_1,U_2}(A(R)) \\\\ge \\\\rho > 0,$ where the constant $\\\\rho $ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We have $Q^{K}_{U_1,U_2}(A(R)) \\\\ge cQ^{K}_{U_1,U_2}(A(R) \\\\, | \\\\, \\\\text{}D=+\\\\infty )$ with $c = \\\\mathbb {P}^{K}_{0}(D=+\\\\infty )^2$ .\",\n \"Thus, under the law $Q^{K}_{U_1,U_2}( \\\\cdot \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty )$ the event $A(R)^{c}$ implies $ \\\\left\\\\lbrace \\\\exists \\\\, p,q \\\\in \\\\mathbb {N}, \\\\,\\\\, \\\\Vert X_p^1 - X_q^2\\\\Vert \\\\le 1 \\\\right\\\\rbrace $ .\",\n \"It follows from Proposition REF that one can bound $Q^{K}_{U_1,U_2}(A(R)^{c} \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty ) &\\\\le Q^K_{U_1,U_2}\\\\left(\\\\exists p,q \\\\in \\\\mathbb {N}, \\\\,\\\\, \\\\Vert X_p^1 - X_q^2\\\\Vert \\\\le 1 \\\\, | \\\\, \\\\text{}D^{\\\\otimes }=+\\\\infty \\\\right) \\\\\\\\ &\\\\le C\\\\Vert U_1 - U_2\\\\Vert ^{-c}.$ The result comes by choosing $\\\\Lambda $ large enough.\"\n ],\n [\n \"Identical environments\",\n \"We will try to exploit the results attained in the last part in order to deduce estimates for two walks in the same environment.\",\n \"When the two walks are far enough they live in independent environments and so, by using the previous subsection, we will be able to show that they have a positive probability to not hit each other anymore.\",\n \"If they intersect, then we wait for the walks to get far enough and we iterate this way.\",\n \"First we state the analogous of Proposition REF .\",\n \"We recall the definition of $A(R)$ in (REF ) and the notations $\\\\mathcal {E}_x, \\\\mathcal {E}_A$ in (REF ).\",\n \"Proposition 4.6 There exists $\\\\Lambda > 0$ and $K_0 > 0$ such that, for any $K > K_0$ , $R > 0$ and $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ with $|(U_1-U_2)\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and $||U_1-U_2|| \\\\ge \\\\Lambda $ , $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R) \\\\text{ }| \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty )\\\\ge \\\\rho ,$ the constant $\\\\rho $ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We only have to bound $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c} \\\\text{ }| \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty )$ .\",\n \"Hence, using Lemma REF we only have to bound $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c})$ .\",\n \"Since under $A(R)$ the paths of the two walks stay at distance more than 2 we have $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c}) = Q_{U_1,U_2}^{K}(A(R)^{c})$ .\",\n \"Indeed we can write $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R)) & = \\\\sum _{(x_i),(y_j)}^{} \\\\mathbf {E}\\\\bigg [P^{\\\\omega _K}_{U_1}\\\\bigg ((X^1_i) = (x_i)\\\\bigg )P^{\\\\omega _K}_{U_2}\\\\bigg ((X^2_j) = (y_j)\\\\bigg )\\\\bigg ],$ where the paths $(x_i),(y_j)$ are chosen under $A(R)$ .\",\n \"We highlight the fact that in $P^{\\\\omega _K}_{U_1}$ the conductances in $\\\\mathcal {E}_{U_2}$ are not set to $K$ , while in $\\\\mathbb {P}^{K}_{U_1,U_2}$ they are.\",\n \"Indeed $P^{\\\\omega _{K}}_{U_1}\\\\left((X^1_i) = (x_i)\\\\right)$ depends on the values $c^{*}(e)$ where $e \\\\in \\\\mathcal {E}_{(x_i)}$ , using the fact that $(x_i)$ and $(y_j)$ stay at distance more than 2 we have $\\\\mathcal {E}_{(x_i)} \\\\cap \\\\mathcal {E}_{U_2} = \\\\emptyset $ .\",\n \"Thus modifying the value of $c^{*}(e)$ where $e \\\\in \\\\mathcal {E}_{U_2}$ does not modify the value of $P^{\\\\omega _{K}}_{U_1}\\\\left((X^1_i) = (x_i)\\\\right)$ .\",\n \"The term $P^{\\\\omega _K}_{U_1}\\\\left((X^1_i) = (x_i)\\\\right)$ is measurable with respect to $ \\\\sigma (c^{*}(e) \\\\text{ ; } e\\\\in \\\\mathcal {E}_{(x_i)})$ and $P^{\\\\omega _K}_{U_2}((X^2_j) = (y_j))$ is measurable with respect to $ \\\\sigma (c^{*}(e) \\\\text{ ; } e\\\\in \\\\mathcal {E}_{(y_j)})$ .\",\n \"The two sigma algebras are independent since for all $i,j$ we have $\\\\Vert x_i-y_j\\\\Vert \\\\ge 2$ .\",\n \"We conclude that we have $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R)) & = \\\\sum _{(x_i),(y_j)}^{} \\\\mathbf {E}\\\\left[P^{\\\\omega _K}_{U_1}\\\\left((X^1_i) = (x_i) \\\\right) \\\\right] \\\\mathbf {E} \\\\left[P^{\\\\omega _K}_{U_2}\\\\left((X^2_j) = (y_j)\\\\right)\\\\right]\\\\\\\\& = Q_{U_1,U_2}^{K}(A(R)).$ We conclude using Proposition REF .\",\n \"We want to extend the last result to the case when $U_1$ and $U_2$ may be close.\",\n \"Proposition 4.7 There exists $K_0 > 0$ such that for any $K > K_0$ and any $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ with $|(U_1-U_2)\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and $R > 0$ : $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R) \\\\text{ }| \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty )\\\\ge \\\\rho ,$ where the constant $\\\\rho > 0$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"First with the same arguments as in last proposition we see that we only have to find a lower bound for $\\\\mathbb {P}_{U_1,U_2}^{K}(A(R))$ .\",\n \"Fix $\\\\Lambda $ as in Proposition REF .\",\n \"We may suppose $\\\\Vert U_1-U_2\\\\Vert \\\\le \\\\Lambda $ otherwise it is just applying Proposition REF .\",\n \"We recall the notation $R_0 = \\\\min (U_1\\\\cdot \\\\vec{\\\\ell },U_2\\\\cdot \\\\vec{\\\\ell })$ .\",\n \"We build paths that start form $U_1$ and $U_2$ in the following way: Fix $V_1$ and $V_2$ such that $\\\\Lambda \\\\le ||V_1-V_2|| < 2\\\\Lambda $ and $ R_0+5< \\\\lfloor V_1\\\\cdot \\\\vec{\\\\ell } \\\\rfloor ,\\\\lfloor V_2\\\\cdot \\\\vec{\\\\ell } \\\\rfloor < R_0+6$ .\",\n \"Build two finite paths $(x_i)$ and $(y_j)$ starting from $U_1$ , $U_2$ respectively and ending at $V_1$ , $V_2$ , which are: Compatible with having joint regeneration points on $(V_1,V_2)$ .\",\n \"Of length lower than $C\\\\Lambda $ where $C$ is a large enough constant.\",\n \"Compatible with $A(R)$ , i.e.\",\n \"on those paths the walks get at distance more than 2 after the first 2 steps and stay on the right side (in terms of inner product with $\\\\vec{\\\\ell }$ ) of their starting point.\",\n \"It is always possible to find such paths, see the figure below for a visual explanation of why these paths exist.\",\n \"Figure: On the left, the general construction of the paths described above, the distance between U 1 U_1 (resp U 2 U_2) and V 1 V_1 (resp.\",\n \"V 2 V_2), in the direction ℓ →\\\\vec{\\\\ell }, is between 5 and 6 because of the lattice geometry.\",\n \"On the right, a “worst case scenario” when the two starting points coincide and the bias is in the direction of the vector e 1 e_1.\",\n \"This is why we allow, in the definition of A(R)A(R) (), that the walks might be close after one step (in orange).We have the following inequality: $\\\\mathbb {P}_{U_1,U_2}^{K}\\\\left(A(R)\\\\right) \\\\ge \\\\mathbb {P}_{U_1,U_2}^{K}\\\\left(\\\\begin{array}{c}(X^1_i) = (x_i),(X^2_j)=(y_j),(x_i) \\\\text{ and }(y_j)\\\\text{ are all }K\\\\text{-open points}\\\\\\\\A(R) \\\\circ \\\\theta _{T_{V_1},T_{V_2}}, \\\\mathcal {D}^{\\\\bullet } \\\\circ \\\\theta _{T_{V_1},T_{V_2}}\\\\end{array}\\\\right).$ Using Theorem REF last term can be rewritten as $&\\\\mathbb {P}_{U_1,U_2}^{K}\\\\left((X^1_i) = (x_i),(X^2_j)=(y_j),(x_i) \\\\text{ and }(y_j)\\\\text{ are all }K\\\\text{-open points}\\\\right)\\\\\\\\&\\\\times \\\\mathbb {P}_{V_1,V_2}^{K}\\\\left(A(R) ,\\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right).$ The left term of the product is bounded below by a constant which only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ and the right term is bounded by a $\\\\rho > 0$ using Proposition REF and Lemma REF .\",\n \"It concludes the proof.\",\n \"We show that under $\\\\mathbb {P}_0$ , on $A(R)$ and when the two trajectories reach a joint regeneration level before reaching level $R$ , then it is unlikely that they meet in the future.\",\n \"We recall the definition of $\\\\mathbf {M}_N$ in (REF ).\",\n \"Proposition 4.8 We recall the notation $R_0 = \\\\max (U_1\\\\cdot \\\\vec{\\\\ell },U_2\\\\cdot \\\\vec{\\\\ell })$ .\",\n \"There exists $K_0 > 0$ such that for $K > K_0$ , for $U_1,U_2 \\\\in \\\\mathbb {Z}^d$ such that $|(U_1-U_2)\\\\cdot \\\\vec{\\\\ell }| \\\\le 1$ and $R > 0$ we have $\\\\mathbb {P}_{U_1,U_2}^{K}\\\\left( A(R) \\\\cap \\\\mathrm {\\\\bf {M}}_{R+R_0} \\\\text{ }| \\\\text{ } \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\right) \\\\le CR^{-c},$ where the constants $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We want to use Corollary REF .\",\n \"First notice that it suffices to bound $\\\\mathbb {P}_{U_1,U_2}^{K}\\\\big ( A(R) \\\\cap \\\\mathrm {\\\\bf {M}}_{R+R_0}\\\\cap \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\big )$ , indeed it follows from Lemma REF that $\\\\mathbb {P}_{U_1,U_2}^{K}\\\\big (\\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\big )$ is bounded below by a constant $\\\\eta > 0$ which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Let us consider the set $\\\\mathcal {P}^{\\\\bullet }$ of all finite paths $((\\\\tilde{x}_i),(\\\\tilde{y}_j))$ for the enhanced walks $(\\\\tilde{X}^1_{\\\\cdot },\\\\tilde{X}^2_{\\\\cdot })$ (defined in Section REF ) such that $(x_i)$ and $(y_j)$ are at distance 1 at their ending points and always more than 2 (except at the starting point) otherwise and which are compatible with $\\\\lbrace \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\rbrace \\\\cap A(R)$ .\",\n \"We then only have to bound $\\\\mathbb {P}^{K}_{U_1,U_2}\\\\left(\\\\lbrace \\\\tilde{X}^1_{i}\\\\rbrace \\\\times \\\\lbrace \\\\tilde{X}^2_{j}\\\\rbrace \\\\in \\\\bigcup _{(\\\\tilde{x}_i),(\\\\tilde{y}_j) \\\\in \\\\mathcal {P}^{\\\\bullet }}^{} \\\\lbrace \\\\tilde{x}_i\\\\rbrace \\\\times \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\right).$ Under all paths considered, the environments each walk encounters are independent.\",\n \"Then, the quantity above can be bounded in the following way $\\\\nonumber \\\\mathbb {P}^{K}_{U_1,U_2}\\\\Bigg (\\\\lbrace \\\\tilde{X}^1_{i}\\\\rbrace \\\\times \\\\lbrace \\\\tilde{X}^2_{j}\\\\rbrace &\\\\in \\\\bigcup _{(\\\\tilde{x}_i),(\\\\tilde{y}_j)\\\\in \\\\mathcal {P}^{\\\\bullet }}^{} \\\\lbrace \\\\tilde{x}_i\\\\rbrace \\\\times \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\Bigg ) \\\\\\\\ \\\\nonumber &= \\\\sum _{(\\\\tilde{x}_i),(\\\\tilde{y}_j)\\\\in \\\\mathcal {P}^{\\\\bullet }}^{} \\\\mathbb {P}^{K}_{U_1,U_2}\\\\left( \\\\lbrace \\\\tilde{X}^1_i\\\\rbrace = \\\\lbrace \\\\tilde{x}_i\\\\rbrace ,\\\\lbrace \\\\tilde{X}^2_j\\\\rbrace = \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\right)\\\\\\\\&\\\\nonumber \\\\le C \\\\sum _{(\\\\tilde{x}_i),(\\\\tilde{y}_j)\\\\in \\\\mathcal {P}^{\\\\bullet }}^{} Q^{K}_{U_1,U_2}\\\\left( \\\\lbrace \\\\tilde{X}^1_i\\\\rbrace = \\\\lbrace \\\\tilde{x}_i\\\\rbrace ,\\\\lbrace \\\\tilde{X}^2_j\\\\rbrace = \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\right)\\\\\\\\& = C Q^{K}_{U_1,U_2}\\\\left(\\\\lbrace \\\\tilde{X}^1_{i}\\\\rbrace \\\\times \\\\lbrace \\\\tilde{X}^2_{j}\\\\rbrace \\\\in \\\\bigcup _{(\\\\tilde{x}_i),(\\\\tilde{y}_j)\\\\in \\\\mathcal {P}^{\\\\bullet }}^{} \\\\lbrace \\\\tilde{x}_i\\\\rbrace \\\\times \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\right).$ Where the inequality is using the fact that the term $P^{\\\\omega _K}_{U_1,U_2}\\\\big (\\\\lbrace \\\\tilde{X}^1_i\\\\rbrace = \\\\lbrace \\\\tilde{x}_i\\\\rbrace \\\\big )$ is measurable with respect to $\\\\sigma \\\\big (c^{*}(e) \\\\text{ ; }e \\\\in \\\\mathcal {E}_{\\\\lbrace \\\\tilde{x}_i\\\\rbrace _{i\\\\ge 0}\\\\backslash \\\\lbrace \\\\tilde{x}_{\\\\infty }\\\\rbrace }\\\\big )$ and the term $P^{\\\\omega _K}_{U_1,U_2}\\\\big (\\\\lbrace \\\\tilde{X}^2_j\\\\rbrace = \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\big )$ is measurable with respect to $\\\\sigma \\\\big (c^{*}(e) \\\\text{ ; }e \\\\in \\\\mathcal {E}_{\\\\lbrace \\\\tilde{y}_j\\\\rbrace _{j\\\\ge 0}\\\\backslash \\\\lbrace \\\\tilde{y}_{\\\\infty }\\\\rbrace }\\\\big )$ where $\\\\tilde{x}_{\\\\infty },\\\\tilde{y}_{\\\\infty }$ denotes the ending point of the paths.\",\n \"In most cases, we simply use the fact that the two set of edges considered are disjoint and the independence of the conductances would give an equality.\",\n \"We only have an inequality because when $\\\\Vert U_1 - U_2 \\\\Vert _{1} \\\\le 1$ and $\\\\vec{\\\\ell } = e_1$ , we might be forced to take the first step in the direction $e_1$ by the definition of $A(R)$ .\",\n \"In this last case we just lose a constant because there is a positive chance that the second site visited is $K$ -open under both $\\\\mathbb {P}^{K}$ and $Q^{K}$ .\",\n \"Note that we could also have an inequality in the opposite direction with a different constant if needed.\",\n \"See the right of Figure REF for a visualisation of the “worst case scenario”.\",\n \"Let us introduce the notation: $\\\\mathbf {TRAJ} = \\\\Big \\\\lbrace \\\\lbrace \\\\tilde{X}^1_{i}\\\\rbrace \\\\times \\\\lbrace \\\\tilde{X}^2_{j}\\\\rbrace \\\\in \\\\bigcup _{(\\\\tilde{x}_i),(\\\\tilde{y}_j)\\\\in \\\\mathcal {P}^{\\\\bullet }}^{} \\\\lbrace \\\\tilde{x}_i\\\\rbrace \\\\times \\\\lbrace \\\\tilde{y}_j\\\\rbrace \\\\Big \\\\rbrace .$ We can rewrite (REF ) as $\\\\displaystyle & Q^{K}_{U_1,U_2}\\\\left(\\\\mathbf {TRAJ},D^{\\\\otimes } < +\\\\infty \\\\right) \\\\\\\\& + Q^{K}_{U_1,U_2}\\\\left(\\\\mathbf {TRAJ},D^{\\\\otimes }=+\\\\infty \\\\right).$ Using the fact that $\\\\mathbf {TRAJ}$ is compatible with $\\\\mathcal {D}^{\\\\bullet } = +\\\\infty $ and that the trajectories go beyond level $R+R_0$ , if $D^1 < +\\\\infty $ (resp.\",\n \"$D^2 < +\\\\infty )$ then we have $X^1_{\\\\tau _1}\\\\cdot \\\\vec{\\\\ell } > R + R_0$ (resp.\",\n \"$X^2_{\\\\tau _1}\\\\cdot \\\\vec{\\\\ell } > R + R_0$ ).\",\n \"Using , we dominate the first term by $CR^{-c}$ .\",\n \"It remains to bound the term (REF ).\",\n \"To do so, we define $C_i, i=1, 2$ : the walk $X^i$ regenerates between the levels $R_0 + R/2$ and $R_0 + R$ .\",\n \"Using once again of we have $ Q^{K}_{U_1,U_2}\\\\left(C_1^{c} \\\\cup C_2^{c}\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\right) \\\\le CR^{-c}.$ On the event $\\\\displaystyle C_1\\\\cap C_2 \\\\cap \\\\mathbf {TRAJ}$ , all points $(x,y)$ in the two trajectories (except with the starting points) which are at distance less than $R/2$ from $\\\\lbrace U_1,U_2\\\\rbrace $ we have $\\\\Vert x-y\\\\Vert _1 \\\\ge 2$ .\",\n \"Thus $\\\\displaystyle C_1\\\\cap C_2 \\\\cap \\\\mathbf {TRAJ}$ implies $F(R/2)$ where $F(R/2)$ is defined in (REF ).\",\n \"Using Corollary REF we conclude that $ Q^{K}_{U_1,U_2}\\\\left(C_1 \\\\cap C_2 \\\\cap \\\\mathbf {TRAJ} \\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\right) \\\\le Q^{K}_{U_1,U_2}\\\\left(F(R/2)\\\\right) \\\\le CR^{-c}.$ Finally we dominate (REF ) by $& Q^{K}_{U_1,U_2}\\\\left(C_1^{c} \\\\cup C_2^{c}\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\right)+ Q^{K}_{U_1,U_2}\\\\left(C_1 \\\\cap C_2 \\\\cap \\\\mathbf {TRAJ} \\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\right),$ and we conclude with the estimates (REF ) and (REF ).\"\n ],\n [\n \"Asymptotic separation proved\",\n \"We have everything needed to prove the asymptotic separation stated in Proposition REF .\",\n \"We fix $\\\\psi \\\\in (0, 1)$ arbitrarily small.\",\n \"We also introduce $\\\\mathbf {d} = N^{1-\\\\psi }$ .\",\n \"We set $\\\\gamma _1 = \\\\mathcal {L}_1$ and for $ k \\\\ge 2 $ we call $\\\\gamma _k$ the first joint regeneration level in $\\\\mathcal {H}^{+}(\\\\gamma _{k-1}+2\\\\mathbf {d})$ .\",\n \"Recall the notation $A(R)$ for $R > 0$ defined in (REF ).\",\n \"We introduce the event $S_k$ which is just event $A(\\\\mathbf {d})$ where the walks start respectively from $X_{T^1_{\\\\gamma _k}}$ and $X_{T^2_{\\\\gamma _k}}$ .\",\n \"We set: $ \\\\forall k \\\\ge 0, \\\\text{ }S_k = A(\\\\mathbf {d}) \\\\circ \\\\theta _{T_{\\\\gamma _k}} \\\\text{ and }S(N) = \\\\bigcup _{k=1}^{\\\\lfloor N^{\\\\psi }/4 \\\\rfloor } S_k.$ The idea of the proof is the following: each time our walks reach a level $\\\\gamma _k$ we check whether $S_k$ occurs or not.\",\n \"If it does, using Proposition REF , with high probability the walks will never meet after that that.\",\n \"If they do meet, we wait for the walks to reach $\\\\gamma _{k+1}$ and repeat the procedure.\",\n \"Using the independence property of joint regeneration levels (see Theorem REF ) we expect the first success of this process to happen very quickly (See Figure REF ).\",\n \"Figure: Successive attempts for events S k S_k.Proposition 4.9 There exists $K_0 > 0$ such that for any $K > K_0$ , there exists $\\\\rho > 0$ such that for any $k \\\\ge 1$ we have $\\\\displaystyle \\\\mathbb {P}_{0}(S_k \\\\, | \\\\, S_0^{c},\\\\cdots ,S_{k-1}^{c} ) \\\\ge \\\\rho ,$ where $\\\\rho $ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"This is a direct application of Theorem REF and Proposition REF .\",\n \"Indeed, using Theorem REF we have $\\\\mathbb {P}_0(S_k,S_{k-1}^{c},\\\\cdots ,S_0^{c}) &=\\\\sum _{U_1,U_2}^{} \\\\mathbb {P}_{0}(S_{k-1}^{c},\\\\cdots ,S_0^{c},X^{1}_{T_{\\\\gamma _k}} = U_1,X^{2}_{T_{\\\\gamma _k}} = U_2)\\\\mathbb {P}_{U_1,U_2}^{K}(A(w) \\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty ).$ Applying Proposition REF gives the result.\",\n \"We introduce $ \\\\textbf {Reg}(N) = \\\\bigcap _{k=1}^{N^2}\\\\lbrace \\\\mathcal {L}_{k} - \\\\mathcal {L}_{k-1} \\\\le \\\\mathbf {d}\\\\rbrace .$ which is the event under which the $N^2$ first joint regeneration levels are close enough.\",\n \"We also recall the definition of $\\\\mathbf {M}_N$ in (REF ).\",\n \"We have the following proposition.\",\n \"Proposition 4.10 There exists $K_0 > 0$ such that for any $K > K_0$ and any $0 \\\\le k \\\\le \\\\lfloor N^{\\\\psi }/4 \\\\rfloor $ , we have the following inequality: $\\\\mathbb {P}_0(\\\\textnormal {\\\\textbf {Reg}}(N)\\\\cap \\\\mathrm {\\\\bf {M}}_N \\\\cap S_k) \\\\le CN^{-c(1-\\\\psi )}$ , where the constants $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We sum over all possible values for $(X^1_{T_{\\\\gamma _k}},X^2_{T_{\\\\gamma _k}})$ on the event $\\\\mathbf {Reg}(N)$ $\\\\mathbb {P}_0(\\\\textbf {Reg}(N)\\\\cap \\\\mathrm {\\\\bf {M}}_N \\\\cap S_k) &= \\\\sum _{U_1,U_2}^{} \\\\mathbb {P}_0\\\\left(X^1_{T_{\\\\gamma _k}} = U_1,X^2_{T_{\\\\gamma _k}} = U_2,\\\\textbf {Reg}(N)\\\\cap \\\\mathrm {\\\\bf {M}}_N \\\\cap S_k\\\\right) \\\\\\\\&\\\\le \\\\sum _{\\\\begin{array}{c}U_1,U_2\\\\\\\\\\\\text{compatible with }\\\\mathbf {Reg}(N)\\\\end{array}}^{} \\\\mathbb {P}_0\\\\left(X^1_{T_{\\\\gamma _k}} = U_1,X^2_{T_{\\\\gamma _k}} = U_2, \\\\mathrm {\\\\bf {M}}_N \\\\cap S_k\\\\right).$ Since $U_1,U_2$ where chosen on event $\\\\textbf {Reg}(N)$ we have $U_1\\\\cdot \\\\vec{\\\\ell } < 3N/4$ and $U_2\\\\cdot \\\\vec{\\\\ell } < 3N/4$ .\",\n \"The worst case clearly happens for $k = \\\\lfloor N^{\\\\psi }/4 \\\\rfloor $ .\",\n \"We claim that for every $0 \\\\le j \\\\le \\\\lfloor N^{\\\\psi }/4 \\\\rfloor $ on the event $\\\\mathbf {Reg}(N)$ , we have $\\\\gamma _{j+1} \\\\le \\\\gamma _j + 3\\\\mathbf {d}$ .\",\n \"Indeed, between $\\\\gamma _j$ and $\\\\gamma _{j+1}$ there are at most $C \\\\mathbf {d}$ joint regeneration levels, for some constant depending on the dimension $d$ and $\\\\vec{\\\\ell }$ .\",\n \"The claim follows from the fact that, for $0 \\\\le j \\\\le \\\\lfloor N^{\\\\psi }/4 \\\\rfloor $ , we are choosing all the $\\\\gamma _{j}$ between the first $N^2$ regeneration levels.\",\n \"Then, from the claim we deduce that for $k = \\\\lfloor N^{\\\\psi }/4 \\\\rfloor $ we have $\\\\gamma _k \\\\le 3\\\\mathbf {d}N^{\\\\psi }/4 \\\\le 3N/4$ .\",\n \"We recall $S_k = A(\\\\mathbf {d}) \\\\circ \\\\theta _{T_{\\\\gamma _k}}$ and since $\\\\gamma _k \\\\le 3N/4$ on $\\\\mathbf {Reg}(N)$ , $\\\\mathrm {\\\\bf {M}}_N$ implies $\\\\mathrm {\\\\bf {M}}_N \\\\circ \\\\theta _{T_{\\\\gamma _k}}$ .\",\n \"We deduce that the last sum is bounded by $\\\\sum _{\\\\begin{array}{c}U_1,U_2\\\\\\\\\\\\text{compatible with }\\\\mathbf {Reg}(N)\\\\end{array}}^{} \\\\mathbb {P}_0\\\\left(X^1_{T_{\\\\gamma _k}} = U_1,X^2_{T_{\\\\gamma _k}} = U_2, \\\\mathrm {\\\\bf {M}}_N \\\\circ \\\\theta _{T_{\\\\gamma _k}}, A(\\\\mathbf {d}) \\\\circ \\\\theta _{T_{\\\\gamma _k}}\\\\right).$ Using Theorem REF this sum can be rewritten as $ \\\\sum _{U_1,U_2}^{} \\\\mathbb {P}_0\\\\left(X^1_{T_{\\\\gamma _k}} = U_1,X^2_{T_{\\\\gamma _k}} = U_2\\\\right)\\\\mathbb {P}_{U_1,U_2}^{K}\\\\left(\\\\mathrm {\\\\bf {M}}_N\\\\cap A(\\\\mathbf {d})\\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right).$ We recall $R_0 = \\\\max (U_1 \\\\cdot \\\\vec{\\\\ell },U_2 \\\\cdot \\\\vec{\\\\ell })$ .\",\n \"Now, $\\\\mathbb {P}_{U_1,U_2}^{K}$ -almost surely $\\\\mathrm {\\\\bf {M}}_{N}$ implies $\\\\mathbf {M}_{R_0+\\\\mathbf {d}}$ .\",\n \"Indeed, $\\\\displaystyle U_1\\\\cdot \\\\vec{\\\\ell } < 3N/4$ and $\\\\displaystyle U_2\\\\cdot \\\\vec{\\\\ell } < 3N/4$ , hence $\\\\mathbb {P}^{K}_{U_1,U_2}$ -almost surely $\\\\displaystyle R_0 + \\\\mathbf {d} \\\\le 3N/4 + N^{1-\\\\psi } \\\\le N$ which gives $\\\\mathbf {M}_N$ implies $\\\\mathbf {M}_{R_0+\\\\mathbf {d}}$ .\",\n \"It follows that (REF ) is bounded by $\\\\sum _{U_1,U_2}^{} \\\\mathbb {P}_0\\\\left(X^1_{T_{\\\\gamma _k}} = U_1,X^2_{T_{\\\\gamma _k}} = U_2\\\\right)\\\\mathbb {P}_{U_1,U_2}^{K}\\\\left(\\\\mathrm {\\\\bf {M}}_{R_0+\\\\mathbf {d}}\\\\cap A(\\\\mathbf {d})\\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right).$ We conclude using Proposition REF .\",\n \"Finally, we prove the asymptotic separation.\",\n \"[Proof of Proposition REF ] We recall the definition of $S(N)$ in (REF ) and of $\\\\mathbf {Reg}(N)$ in (REF ).\",\n \"Using Proposition REF we have $\\\\mathbb {P}_{0}\\\\left(S(N)^c\\\\right) \\\\le (1-\\\\rho )^{N^{\\\\psi }} $ and by Proposition REF for any $M > 0$ we can take $K$ large enough such that we have $\\\\mathbb {P}_{0}(\\\\mathbf {Reg}(N)^{c}) \\\\le CN^{-M}$ for any $M > 0$ where the constants $\\\\rho ,C>0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Hence, $\\\\mathbb {P}_0(\\\\mathrm {\\\\bf {M}}_N) &= \\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N\\\\cap \\\\mathbf {Reg}(N) \\\\cap S(N)\\\\right) + \\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N \\\\cap (\\\\mathbf {Reg}(N)^c \\\\cup S(N)^c)\\\\right) \\\\\\\\ &\\\\le \\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N\\\\cap \\\\mathbf {Reg}(N) \\\\cap S(N)\\\\right) + CN^{-M} + (1-\\\\rho )^{N^{\\\\psi }}.$ We only need to bound $\\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N\\\\cap \\\\mathbf {Reg}(N) \\\\cap S(N)\\\\right)$ , by the definition of $S(N)$ and union bound we get $\\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N\\\\cap \\\\mathbf {Reg}(N) \\\\cap S(N)\\\\right) &= \\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N \\\\cap \\\\mathbf {Reg}(N) \\\\cap \\\\bigg ( \\\\bigcup _{k=1}^{\\\\lfloor N^{\\\\psi }/4 \\\\rfloor } S_k\\\\bigg )\\\\right)\\\\\\\\& \\\\le \\\\sum _{k=1}^{\\\\lfloor N^{\\\\psi }/4 \\\\rfloor } \\\\mathbb {P}_0\\\\left(\\\\mathrm {\\\\bf {M}}_N\\\\cap \\\\mathbf {Reg}(N) \\\\cap S_k\\\\right) \\\\\\\\&\\\\le CN^{\\\\psi }N^{-c(1-\\\\psi )},$ where the last step is a consequence of Proposition REF .\",\n \"We conclude by choosing $\\\\psi $ small enough, we notice that this choice depends on $c$ which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\"\n ],\n [\n \"For the trajectory\",\n \"In this part we aim to prove that $\\\\mathbb {E}[F(Z_n^1)F(Z_n^2)] - \\\\mathbb {E}[F(Z_n^1)]\\\\mathbb {E}[F(Z_n^2)]$ converges polynomially fast to 0 as $n \\\\rightarrow +\\\\infty $ .\",\n \"Where $F$ is a bounded $L$ -Lipschitz function from $\\\\mathcal {C}([0,1],\\\\mathbb {R}^{d}) \\\\rightarrow \\\\mathbb {R}$ and $Z^{i}_n(t) = \\\\displaystyle \\\\frac{X^{i}_{\\\\tau _{\\\\lfloor nt \\\\rfloor }} - nvt}{n^{1/2}}$ for $t \\\\in [0,1]$ .\",\n \"Actually, with a slight abuse of the notation, in all this section we will refer with $Z^{i}_n(t)$ to the polygonal interpolant of $\\\\displaystyle \\\\frac{X^{i}_{\\\\tau _{\\\\lfloor nt \\\\rfloor }} - nvt}{n^{1/2}}$ , which is needed for technical reasons.\",\n \"Throughout this proof we assume that the function $F$ is bounded by 1 and has Lipschitz constant $L=1$ as an extension to other constants follows with the same reasoning.\",\n \"We take $t \\\\in [0,1]$ as it is the same for $t \\\\in [0,T]$ .\",\n \"We recall the notation: $ \\\\chi ^{i}_k = \\\\inf \\\\left\\\\lbrace m \\\\ge 0 \\\\text{ ; } \\\\lbrace X^i_{j}-X^i_{\\\\tau ^{i}_k}\\\\rbrace _{\\\\tau ^{i}_k \\\\le j \\\\le \\\\tau ^{i}_{k+1}} \\\\subset \\\\mathcal {B}(m,m^{\\\\alpha })\\\\right\\\\rbrace .$ where we recall that $\\\\alpha $ is a constant chosen greater than $d+3$ .\",\n \"Let us state the main theorem of this part.\",\n \"Theorem 5.1 There exists $K > 0$ such that the following inequality holds: $\\\\Big |\\\\mathbb {E}[F(Z_n^1)F(Z_n^2)] - \\\\mathbb {E}[F(Z_n^1)]\\\\mathbb {E}[F(Z_n^2)]\\\\Big | \\\\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We postpone the proof to the end of the section but let us give some intuition.\",\n \"Roughly speaking, on the event $\\\\mathbf {M}_{n^\\\\theta }$ (see (REF )) and after few regeneration times, the two walks do not intersect anymore.\",\n \"It follows that they should behave like two regenerating random walks in two independent environment.\",\n \"This is the point of Propositions REF -REF which are the main steps to prove Theorem REF .\",\n \"The rest of the proof basically consists in proving that the beginning of the trajectories does not matter in the re-scaled trajectory $Z_n$ .\",\n \"Fix $0 < \\\\theta < 1/2$ .\",\n \"Let us start with an easy estimate which will be useful later on.\",\n \"Proposition 5.2 The following inequality holds: $\\\\left|\\\\mathbb {E}_0\\\\left[F(Z_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(Z_n)\\\\text{ }|\\\\text{ }D=+\\\\infty \\\\right]\\\\right| \\\\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We introduce the scaled position of the walk starting after the first regeneration time and re-centered $\\\\bar{Z}_n(t) = Z_n(t+1/n) - Z_n(1/n).$ We define $B= \\\\big \\\\lbrace \\\\forall k \\\\in \\\\lbrace 0,\\\\cdots ,n+1\\\\rbrace \\\\text{ ; }\\\\chi _k \\\\le n^{\\\\varepsilon }\\\\big \\\\rbrace $ where $\\\\varepsilon > 0$ is a constant that we choose arbitrarily small.\",\n \"Then, under $B$ , we have $\\\\Vert \\\\bar{Z}_n-Z_n\\\\Vert _{\\\\infty } \\\\le Cn^{-\\\\frac{1}{2}}n^{\\\\alpha \\\\varepsilon } \\\\le Cn^{-c},$ where $c = \\\\alpha \\\\varepsilon - \\\\frac{1}{2} < 0$ when choosing $\\\\varepsilon $ small enough.\",\n \"It follows that under $B$ and using the fact that $F$ is 1-Lipschitz we get $|F(Z_n)-F(\\\\bar{Z}_n)| \\\\le Cn^{-c},$ By , for $K$ large enough we have $\\\\mathbb {P}_0(B_1^{c} \\\\cup B_2^{c}) \\\\le Cn^{-c}$ where $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Using this inequality and the fact that $F$ is bounded by 1 we get that $\\\\left|\\\\mathbb {E}_0[F(Z_n)]-\\\\mathbb {E}_0[F(\\\\bar{Z}_n)]\\\\right| \\\\le Cn^{-c}.$ We conclude the proof observing that using we have that: $\\\\mathbb {E}_0[F(\\\\bar{Z}_n)] = \\\\mathbb {E}_0[F(Z_n)\\\\text{ | }D=+\\\\infty ].$ Let us introduce some events and estimates on them: We recall $\\\\mathrm {\\\\bf {M}}_{n^\\\\theta } = \\\\bigg \\\\lbrace \\\\displaystyle \\\\inf _{\\\\begin{array}{c}x \\\\in \\\\mathcal {H}^{+}_{n^{\\\\theta }}\\\\cap \\\\lbrace X_n^1\\\\rbrace \\\\\\\\y \\\\in \\\\mathcal {H}^{+}_{n^{\\\\theta }}\\\\cap \\\\lbrace X_n^2\\\\rbrace \\\\end{array}}\\\\Vert x-y\\\\Vert _{1}\\\\ge 2 \\\\bigg \\\\rbrace .$ Using Proposition REF we have $\\\\mathbb {P}_{0}(\\\\mathrm {\\\\bf {M}}_{n^\\\\theta }^{c}) \\\\le Cn^{-c}$ where $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We define $\\\\mathcal {R}_0$ the first joint regeneration level in $ \\\\mathcal {H}^{+}_{n^{\\\\theta }}$ .\",\n \"For $i \\\\in \\\\lbrace 1,2\\\\rbrace $ we introduce $\\\\rho _0^{i} = T^{i}_{\\\\mathcal {R}_0}$ .\",\n \"In particular, $T^i_{\\\\mathcal {R}_0}$ is a regeneration time for the walk $X^i$ .\",\n \"We call $B \\\\lbrace \\\\mathcal {R}_0 \\\\le 2n^{\\\\theta }\\\\rbrace $ .\",\n \"Using , there exists $K > 0$ such that $\\\\mathbb {P}_{0}(B^{c}) \\\\le Cn^{-c}$ , where $C,c$ depends on $\\\\theta , \\\\lambda ,d,\\\\vec{\\\\ell },K$ .\",\n \"Indeed, under the event $\\\\lbrace \\\\forall i \\\\in \\\\lbrace 1,\\\\cdots ,dn^{\\\\theta }\\\\rbrace , \\\\text{ }\\\\mathcal {L}_{k+1}-\\\\mathcal {L}_{k} \\\\le n^{\\\\theta }\\\\rbrace $ with the proviso $\\\\mathcal {L}_{0} = 0$ (and using the fact that we always have $\\\\mathcal {L}_{k+1}-\\\\mathcal {L}_k \\\\ge \\\\frac{2}{\\\\sqrt{d}}$ ) we have $\\\\mathcal {R}_0 \\\\le 2n^{\\\\theta }$ .\",\n \"The following inequality follows: $ \\\\mathbb {P}_{0}(B^{c}) & \\\\le \\\\mathbb {P}_{0}\\\\bigg (\\\\exists k \\\\in \\\\lbrace 1,\\\\cdots ,dn^{\\\\theta }\\\\rbrace , \\\\text{ }\\\\mathcal {L}_{k+1}-\\\\mathcal {L}_{k} \\\\ge n^{\\\\theta }\\\\bigg ).$ Using Proposition REF for $K$ large enough the last term is bounded above, for $M$ arbitrarily big, by $Cn^{\\\\theta }n^{-M\\\\theta }.$ We call $\\\\rho ^i_{n}$ the $n^{\\\\text{th}}$ (not joint) regeneration time after $\\\\rho _{0}^{i}$ for $X^{i}$ .\",\n \"We recall the notation (REF ).\",\n \"Then define $F_i = \\\\big \\\\lbrace \\\\forall k \\\\in \\\\lbrace 1,\\\\cdots ,n^2\\\\rbrace \\\\text{, }\\\\chi ^{i}_k \\\\le n^{\\\\theta }\\\\big \\\\rbrace $ .\",\n \"Using , there exists $K > 0$ such that $\\\\mathbb {P}(F^{c}_{i}) \\\\le Cn^{-M}$ , where $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Define $E:= \\\\mathrm {\\\\bf {M}}_{n^\\\\theta } \\\\cap B \\\\cap F_1 \\\\cap F_2$ .\",\n \"The estimates above can be summarized in the following proposition.\",\n \"Proposition 5.3 There exists $K > 0$ such that we have: $\\\\mathbb {P}_{0}(E^{c}) \\\\le Cn^{-c},$ where the constants $C,c >0$ only depend on $\\\\theta , \\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Let us introduce $\\\\tilde{X}^i_{\\\\cdot } = X^{i}_{\\\\rho _{0}^{i}+\\\\cdot } - X^{i}_{\\\\rho ^i_0}$ which is just the walk $X^i$ shifted by $\\\\rho _0^{i}$ and recentered, call $\\\\tilde{Z}^{i}_{n}(t) = (\\\\tilde{X}^{i}_{\\\\tau _{\\\\lfloor nt \\\\rfloor }} - nvt)/n^{1/2}$ .\",\n \"Let us show that under $E$ , the trajectories $\\\\tilde{Z}^{i}_n$ and $Z_n^{i}$ stay close.\",\n \"Proposition 5.4 The following inequality holds almost surely under the event $E$ $\\\\forall i \\\\in \\\\lbrace 1,2\\\\rbrace , \\\\text{ } \\\\sup _{t \\\\in [0, 1]}||\\\\tilde{Z}^{i}_n(t) - Z^{i}_n(t)||_{\\\\infty } \\\\le Cn^{-c},$ where $C,c$ depend on $\\\\lambda ,\\\\vec{\\\\ell }, d, \\\\theta $ .\",\n \"Fix $i \\\\in \\\\lbrace 1,2\\\\rbrace $ and $k \\\\in \\\\lbrace 0,\\\\cdots ,n\\\\rbrace $ , let us show that $X^{i}_{\\\\tau _k}$ is close to $X^{i}_{\\\\rho _k}$ after the proper rescaling.\",\n \"We have, $\\\\mathcal {R}_0 \\\\le 2n^{\\\\theta }$ and since $\\\\rho _{0}^{i}$ is a regeneration time, it can be written as $\\\\tau ^{i}_{k_0}$ with $k_0 \\\\in \\\\lbrace 0,\\\\cdots ,d\\\\lfloor n^{2\\\\theta } \\\\rfloor \\\\rbrace $ .\",\n \"Indeed, at each regeneration time the walk has increased at least $2/\\\\sqrt{d}$ in the direction $\\\\vec{\\\\ell }$ .\",\n \"Using the fact that under $E$ , $\\\\chi ^{i}_k \\\\le n^{\\\\theta }$ for all $k \\\\in \\\\lbrace 0,\\\\cdots ,n^2\\\\rbrace $ we have $\\\\Vert X^{i}_{\\\\rho _{k}} - X^i_{\\\\rho _{0}}- X^{i}_{\\\\tau _k}\\\\Vert _\\\\infty =\\\\Vert X^{i}_{\\\\tau _{k+k_0}} - X^{i}_{\\\\tau _{k_0}} - X^{i}_{\\\\tau _k}\\\\Vert _\\\\infty & \\\\le Ck_0n^{\\\\alpha \\\\theta }\\\\\\\\& \\\\le Cn^{(\\\\alpha +2)\\\\theta },$ where $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d$ .\",\n \"The first inequality comes from the fact that the diameter of $\\\\mathcal {B}(m,m^{\\\\alpha })$ is bounded by $Cm^{\\\\alpha }$ .\",\n \"Then, we only have to chose $\\\\theta $ such that $(\\\\alpha + 2)\\\\theta \\\\le 1/4$ to conclude, indeed for all $k \\\\in \\\\lbrace 1, \\\\cdots , n\\\\rbrace $ : $\\\\Vert Z^{i}_n(k/n) - \\\\tilde{Z}^{i}_n(k/n)\\\\Vert _\\\\infty &= \\\\bigg \\\\Vert \\\\frac{X^{i}_{\\\\tau _k} +X^{i}_{\\\\rho _0} - {X}_{\\\\rho _k}^{i}}{n^{1/2}}\\\\bigg \\\\Vert _\\\\infty \\\\le Cn^{-1/4},$ we conclude by taking $c = 1/4$ .\",\n \"We give the following corollary of the last proposition.\",\n \"Corollary 5.5 The following inequality holds: $\\\\left|\\\\mathbb {E}_0\\\\left[F(Z^1_n)F(Z^2_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(\\\\tilde{Z}^1_n)F(\\\\tilde{Z}^2_n)\\\\right]\\\\right| \\\\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"It follows from last Proposition REF , Proposition REF and the fact that $F$ is a bounded Lipschitz function.\",\n \"We introduce $I = \\\\bigg \\\\lbrace \\\\displaystyle \\\\inf _{\\\\begin{array}{c}p,q \\\\ge 0\\\\end{array}}\\\\Vert X^1_{ p}-X^2_{q}\\\\Vert _{1}\\\\ge 2 \\\\bigg \\\\rbrace $ the event that the two walks never “see” the same edge.\",\n \"The following proposition is crucial and shows that when shifted and non-intersecting anymore, we can approximate the behavior of our walks as if they lived in independent environments.\",\n \"We also, highlight the fact that the following proposition gives an inequality.\",\n \"One can prove it is an equality with an error term of order $Cn^{-c}$ where $C,c > 0$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Proposition 5.6 For any bounded positive function $f$ measurable with respect to $\\\\sigma \\\\left(\\\\big \\\\lbrace X^1_0,X^2_0,(X^1_{p},Z^1_{p}),(X^2_{q},Z^2_{q}) \\\\text{ }|\\\\text{ }p,q \\\\ge 1 \\\\big \\\\rbrace \\\\cup \\\\big \\\\lbrace c^{*}(e) \\\\text{ } |\\\\text{ }e \\\\in \\\\mathcal {E}_{(X^1_n)_{n\\\\ge 0}} \\\\cup \\\\mathcal {E}_{(X^2_n)_{n\\\\ge 0}} \\\\backslash (\\\\mathcal {E}_{X^1_0}\\\\cup \\\\mathcal {E}_{X^2_0})\\\\big \\\\rbrace \\\\right),$ we have the following inequality: $\\\\mathbb {E}_0[f \\\\circ \\\\theta _{\\\\rho _0} {1}_{I}\\\\circ \\\\theta _{\\\\rho _0}] \\\\le \\\\mathbb {E}_{0}\\\\bigg [\\\\mathbb {E}^{Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}}\\\\bigg [f \\\\text{ }|\\\\text{ }I, D^{\\\\otimes } = +\\\\infty \\\\bigg ]\\\\bigg ].$ Remark 5.7 Looking at this inequality one may be surprised.\",\n \"Indeed, on the left term $\\\\rho _0$ is using the notion of joint regeneration levels while on the right term, $\\\\lbrace D^{\\\\otimes } = \\\\infty \\\\rbrace = \\\\lbrace D^1=+\\\\infty ,D^2=+\\\\infty \\\\rbrace $ refers to the classical notion of regeneration time for a single walk.\",\n \"However, thinking about it carefully, we observe that under event $I$ , the event $\\\\lbrace \\\\mathcal {D}^{\\\\bullet } = +\\\\infty \\\\rbrace $ coincide with $\\\\lbrace D^{\\\\otimes } = + \\\\infty \\\\rbrace $ .\",\n \"First using Theorem REF we can write $\\\\mathbb {E}_0\\\\left[f \\\\circ \\\\theta _{\\\\rho _0} {1}_{I} \\\\circ \\\\theta _{\\\\rho _0}\\\\right] = \\\\mathbb {E}_{0}\\\\left[\\\\mathbb {E}^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\left[f{1}_{I} \\\\text{ }|\\\\text{ }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right]\\\\right].$ Indeed, we can first restrict to functions $f$ of type ${1}_{X^1 = (a_0,\\\\cdots ,a_k)}{1}_{X^2 = (b_0,\\\\cdots ,b_k)}g$ where $g$ is measurable with respect to $\\\\sigma (c^{*}(e) \\\\text{ }|\\\\text{ } e \\\\in \\\\mathcal {R}_{X^1_0} \\\\cup \\\\mathcal {R}_{X^2_0} \\\\backslash (\\\\mathcal {E}_{X^1_0 \\\\cup X^2_0}))$ .\",\n \"In addition, ${1}_{X^1 = (a_0,\\\\cdots ,a_k)}{1}_{X^2 = (b_0,\\\\cdots ,b_k)}{1}_{I}$ is measurable with respect to $\\\\sigma ((X_0^1,X_0^2),(\\\\tilde{X}^1_n,\\\\tilde{X}^2_n)_{n \\\\ge 1})$ .\",\n \"Then for the general case we use the fact that $f$ can be written as a limit of an increasing sequence of functions which are a linear combination of the above type and a monotone convergence theorem.\",\n \"Now fix $a,b \\\\in \\\\mathbb {Z}^d$ such that $\\\\mathbb {P}_{0}\\\\left(X^1_{\\\\rho ^1_0}=a,X^2_{\\\\rho ^2_0} = b\\\\right) > 0$ .\",\n \"First notice that $\\\\mathbb {P}^{K}_{a,b}$ -almost surely, under $I$ the event $\\\\mathcal {D}^{\\\\bullet } =+\\\\infty $ coincide with $\\\\lbrace D^{\\\\otimes }=+\\\\infty \\\\rbrace $ .\",\n \"Moreover, for any bounded function $g(\\\\cdot )$ measurable with respect to $\\\\sigma \\\\left(\\\\left\\\\lbrace X^1_0,X^2_0,(X^1_{p},Z^1_{p}),(X^2_{q},Z^2_{q}) \\\\text{ }|\\\\text{ }p,q \\\\ge 1 \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace c^{*}(e) \\\\text{ } |\\\\text{ }e \\\\in \\\\mathcal {E}_{(X^1_n)_{n\\\\ge 0}} \\\\cup \\\\mathcal {E}_{(X^2_n)_{n\\\\ge 0}} \\\\backslash (\\\\mathcal {E}_{X^1_0}\\\\cup \\\\mathcal {E}_{X^2_0})\\\\right\\\\rbrace \\\\right),$ we have that $\\\\mathbb {E}^K_{a,b}\\\\left[g{1}_{I}\\\\right] = \\\\mathbb {E}^{Q^{K}_{a,b}}\\\\left[g{1}_{I}\\\\right].$ Indeed, it follows by restricting to simple functions with disjoint set of edges for both walks and using the fact $g{1}_{I}$ can be written as an increasing limit of linear combination of functions of this type.\",\n \"In particular, we have $\\\\mathbb {E}^K_{a,b}\\\\left[f{1}_{I}{1}_{D^1=+\\\\infty }{1}_{D^2=+\\\\infty }\\\\right] = \\\\mathbb {E}^{Q^{K}_{a,b}}\\\\left[f{1}_{I}{1}_{D^1=+\\\\infty }{1}_{D^2=+\\\\infty }\\\\right].$ We conclude the proof using the fact that $\\\\mathbb {P}_{a,b}^{K}\\\\left(I,D^{\\\\otimes }=+\\\\infty \\\\right)\\\\le \\\\mathbb {P}_{a,b}^{K}\\\\left(\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right).$ The next lemma will enable us to erase the conditioning with the event $I$ .\",\n \"Lemma 5.8 We have the following inequality: $\\\\mathbb {E}_{0}\\\\left[Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\left(I^{c} \\\\text{ }|\\\\text{ }D^{\\\\otimes }=+\\\\infty \\\\right)\\\\right] \\\\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"By we have $Q^K_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\left(D^{\\\\otimes }=+\\\\infty \\\\right) = \\\\mathbb {P}_0(D=+\\\\infty )^2 > \\\\eta > 0$ where $\\\\eta $ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Hence we only have to dominate $\\\\mathbb {E}_{0}\\\\left[Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\left(I^{c},D^{\\\\otimes }=+\\\\infty \\\\right)\\\\right].$ By the same argument as in last proof, it is equal to $\\\\mathbb {E}_{0}\\\\left[\\\\mathbb {P}^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\left(I^{c},\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right)\\\\right].$ Using Lemma REF , last term is bounded above by $C\\\\mathbb {E}_{0}\\\\left[\\\\mathbb {P}^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\left(I^{c}\\\\text{ | }\\\\mathcal {D}^{\\\\bullet } =+\\\\infty \\\\right)\\\\right],$ where $C > 0$ is a constant which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"By Proposition REF , this expectation is equal to $\\\\mathbb {P}_{0}\\\\left(I^{c} \\\\circ \\\\theta _{\\\\rho _0}\\\\right) \\\\le \\\\mathbb {P}_0\\\\bigg (\\\\bigg \\\\lbrace \\\\displaystyle \\\\inf _{\\\\begin{array}{c}x \\\\in \\\\mathcal {H}^{+}_{n^{\\\\theta }}\\\\cap \\\\lbrace X_n^1\\\\rbrace \\\\\\\\y \\\\in \\\\mathcal {H}^{+}_{n^{\\\\theta }}\\\\cap \\\\lbrace X_n^2\\\\rbrace \\\\end{array}}\\\\Vert x-y\\\\Vert _{1}\\\\le 1 \\\\bigg \\\\rbrace \\\\bigg ) \\\\le Cn^{-c},$ where last inequality comes by using Proposition REF .\",\n \"Proposition 5.9 For any bounded positive function $f$ we have the following inequality: $&\\\\left|\\\\mathbb {E}_{0}\\\\bigg [\\\\mathbb {E}^{Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}}\\\\bigg [f \\\\text{ }|\\\\text{ } I,D^{\\\\otimes }=+\\\\infty \\\\bigg ]\\\\bigg ] - \\\\mathbb {E}_{0}\\\\bigg [\\\\mathbb {E}^{Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}}\\\\bigg [f \\\\text{ }|\\\\text{ }D^{\\\\otimes }=+\\\\infty \\\\bigg ]\\\\bigg ]\\\\right| \\\\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We can rewrite the left term $\\\\mathbb {E}_{0}\\\\bigg [\\\\mathbb {E}^{Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}}\\\\bigg [f \\\\text{ }|\\\\text{ }D^{\\\\otimes }=+\\\\infty \\\\bigg ]\\\\bigg ]$ as $& \\\\mathbb {E}_{0}\\\\bigg [\\\\mathbb {E}^{Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}}\\\\bigg [f \\\\text{ }|\\\\text{ }I,D^{\\\\otimes }=+\\\\infty \\\\bigg ]Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\bigg (I\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\bigg )\\\\bigg ] +\\\\\\\\&\\\\mathbb {E}_{0}\\\\bigg [\\\\mathbb {E}^{Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}}\\\\bigg [f \\\\text{ }|\\\\text{ }I^{c},D^{\\\\otimes }=+\\\\infty \\\\bigg ]Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\bigg (I^{c}\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\bigg )\\\\bigg ].$ Using the fact that $f$ is bounded, it follows that the term in the statement of the proposition is bounded above by $C\\\\mathbb {E}_{0}\\\\bigg [Q^{K}_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}\\\\bigg (I^{c}\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\bigg )\\\\bigg ].$ We conclude using Proposition REF .\",\n \"Now we can conclude with the proof of Theorem REF .\",\n \"[Proof of Theorem REF ] We recall that we want to dominate $\\\\mathbb {E}_0\\\\left[F(Z^1_n)F(Z^2_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(Z^1_n)\\\\right]\\\\mathbb {E}_0\\\\left[F(Z^2_n)\\\\right].$ First, using Corollary REF we dominate this by $Cn^{-c}+\\\\mathbb {E}_0\\\\left[F(\\\\tilde{Z}^1_n)F(\\\\tilde{Z}^2_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(Z^1_n)\\\\right]\\\\mathbb {E}_0\\\\left[F(Z^2_n)\\\\right].$ Thanks to the fact that $\\\\mathbb {P}_{0}(I^{c} \\\\circ \\\\theta _{\\\\rho _0}) \\\\le Cn^{-c}$ by Proposition REF and the fact that $F$ is bounded we dominate it by $Cn^{-c}+\\\\mathbb {E}_0\\\\left[{1}_{I} \\\\circ \\\\theta _{\\\\rho _0} F(\\\\tilde{Z}^1_n)F(\\\\tilde{Z}^2_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(Z^1_n)\\\\right]\\\\mathbb {E}_0\\\\left[F(Z^2_n)\\\\right].$ Let us bound above the term $\\\\mathbb {E}_0\\\\left[{1}_{I} \\\\circ \\\\theta _{\\\\rho _0} F(\\\\tilde{Z}^1_n)F(\\\\tilde{Z}^2_n)\\\\right].$ Let us introduce the function $f = F\\\\left(Z^1_n-Z^1_n(0)\\\\right)F\\\\left(Z^2_n-Z^2_n(0)\\\\right)$ , then the last term can be rewritten as $\\\\mathbb {E}_0\\\\left[{1}_{I} \\\\circ \\\\theta _{\\\\rho _0} f\\\\circ \\\\theta _{\\\\rho _0} \\\\right].$ Furthermore, we observe that $f$ satisfies the assumptions of Proposition REF and Proposition REF , hence last term is bounded above by $Cn^{-c} + \\\\mathbb {E}_0\\\\bigg [\\\\mathbb {E}^{Q_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}^{K}}\\\\left[f\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\right]\\\\bigg ].$ Using and the form of the function $f$ we can rewrite $\\\\mathbb {E}^{Q_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}^{K}}\\\\left[F\\\\left(Z^1_{n}-Z^1_n(0)\\\\right)F\\\\left(Z^2_{n}-Z^2_n(0)\\\\right)\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\right],$ as $\\\\mathbb {E}^K_0\\\\left[F(Z_n) \\\\text{ | }D=+\\\\infty \\\\right]^2.$ To summarise we have dominated the term $\\\\mathbb {E}_0\\\\left[F(Z^1_n)F(Z^2_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(Z^1_n)\\\\right]\\\\mathbb {E}_0\\\\left[F(Z^2_n)\\\\right],$ by $Cn^{-c} + \\\\mathbb {E}^K_0\\\\left[F(Z_n) \\\\text{ | }D=+\\\\infty \\\\right]^2 - \\\\mathbb {E}_0\\\\left[F(Z_n)\\\\right]^2.$ We conclude by using the fact that $F$ is bounded and Proposition REF .\"\n ],\n [\n \"For the clock process\",\n \"We recall the notation $S^*_n(t) =\\\\displaystyle \\\\frac{\\\\tau _{{tn} + 1} - \\\\tau _1}{\\\\mathrm {Inv}(n)}$ where $\\\\mathrm {Inv}(n) = \\\\inf \\\\lbrace t \\\\ge 0 \\\\text{ ; }\\\\mathbf {P}[c^{*} > t] \\\\le 1/n\\\\rbrace $ .\",\n \"In this part we aim to prove the following main result.\",\n \"Proposition 5.10 There exists $K > 0$ such that the following inequality holds: $|\\\\mathbb {E}[F(S^{*1}_n)F(S^{*2}_n)] - \\\\mathbb {E}[F(S^{*1}_n)]\\\\mathbb {E}[F(S^{*2}_n)]| \\\\le Cn^{-c},$ where $F$ is as in (REF ), $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Before proving this result we will need to state and prove Lemma REF .\",\n \"Let us also admit the following lemma which is crucial in the proof of Lemma REF .\",\n \"Lemma 5.11 For all $\\\\eta \\\\in (0,1)$ and $\\\\rho < \\\\eta / \\\\gamma $ , the probability $\\\\mathbb {P}_{0} \\\\left( S^*_n(n^{-\\\\eta }) > n^{-\\\\rho } \\\\right) = \\\\mathbb {P}_{0} \\\\left( S_n(n^{-\\\\eta }) > n^{-\\\\rho } | D = + \\\\infty \\\\right)$ decays polynomially fast to 0 as $n \\\\rightarrow \\\\infty $ .\",\n \"We let $\\\\rho ^i_0$ be $T^i_{\\\\mathcal {R}_0}$ , where $\\\\mathcal {R}_0$ is the first joint regeneration level in $\\\\mathcal {H}^+_{n^\\\\theta }$ .\",\n \"Let us also introduce the random integer $k_n^{i} \\\\ge 1$ , which is the one such that $\\\\tau ^{i}_{k_n^{i}} = \\\\rho ^{i}_0$ .\",\n \"Then let us introduce, for $i \\\\in \\\\lbrace 1,2\\\\rbrace $ $ \\\\widetilde{S}^{*i}_n(t) = \\\\mathrm {Inv}(n)^{-1}\\\\sum _{j = k^i_n+1}^{k^i_n + {nt}} (\\\\tau ^{i}_{j+1} - \\\\tau ^{i}_j) = S^{*i}_n\\\\left(\\\\frac{k^{i}_n}{n}+t\\\\right) - S^{*i}_n\\\\left(\\\\frac{k^{i}_n}{n}\\\\right),$ which is the value taken by $S^{*i}_n$ for the walk shifted by $\\\\tau _{k^{i}_n}$ .\",\n \"Lemma 5.12 The following inequality holds: $\\\\left|\\\\mathbb {E}_0\\\\left[F(S^{*1}_n)F(S^{*2}_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n)\\\\right]\\\\right| \\\\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Let us compute the difference $|F(S^{*i}_n)-F(\\\\widetilde{S}^{*i}_n(t))|$ .\",\n \"Let us write the two terms properly: $& F(S^{*i}_n) = \\\\exp \\\\left(-\\\\lambda _1 S^{*i}_n(t_1) - \\\\cdots - \\\\lambda _m (S^{*i}_n(t_{m}) - S^{*i}_n(t_{m-1}))\\\\right)\\\\\\\\& F(\\\\tilde{S}^{*i}_n) = \\\\exp \\\\left(-\\\\lambda _1 \\\\tilde{S}^{*i}_n(t_1) - \\\\cdots - \\\\lambda _m (\\\\tilde{S}^{*i}_n(t_{m}) - \\\\tilde{S}^{*i}_n(t_{m-1}))\\\\right).$ Then using (REF ), and factorising by $F(S^{*i}_n)$ , the difference is dominated by $ F(S^{*i}_n) \\\\left[1-\\\\exp \\\\bigg (CS^{*i}_n( k^{i}_n / n)\\\\bigg )\\\\prod _{j=1}^{m}\\\\exp \\\\bigg (C\\\\bigg (S^{*i}_n( t_j + k^{i}_n / n) - S^{*i}_n( t_j )\\\\bigg )\\\\bigg )\\\\right],$ where $C = \\\\max \\\\left\\\\lbrace \\\\lambda _1,\\\\cdots ,\\\\lambda _m\\\\right\\\\rbrace $ .\",\n \"Let us introduce the following event $ R \\\\mathcal {R}_0 \\\\le 2n^{\\\\theta },$ for $\\\\mathcal {R}_0$ being the first joint regeneration level inside $\\\\mathcal {H}^+_{n^\\\\theta }$ .\",\n \"We already proved that $\\\\mathbb {P}_0(R^c)\\\\le Cn^{-c}$ in (REF ).\",\n \"Let us put ourselves on the event $R$ and fix $1 > 1 - \\\\eta > \\\\theta > 0$ .\",\n \"Then for $i \\\\in \\\\lbrace 1,2\\\\rbrace $ , $X^i_{\\\\tau ^i_{{n^{1-\\\\eta }}}}$ is $\\\\mathbb {P}_{0}$ -almost surely in $\\\\mathcal {H}^+_{2 n^\\\\theta }$ , hence $\\\\tau ^i_{k^i_n} < \\\\tau ^i_{n^{1-\\\\eta }}$ since $\\\\mathcal {R}_0 \\\\le 2n^{\\\\theta }$ .\",\n \"It follows that event $R$ implies $C_n^{i} \\\\lbrace k_n^{i} \\\\le n^{1 - \\\\eta }\\\\rbrace $ for $i \\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"In addition, under event $C^{i}_n$ the term $S^{*i}_n\\\\left(t_{j} + (k_n^i)/n)\\\\right) - S^{*i}_n\\\\left(t_{j}\\\\right)$ is dominated by $S^{*i}_n(t_{j} + n^{-\\\\eta }) - S^{*i}_n(t_{j})$ which is distributed as $S^{*i}_n(n^{-\\\\eta })$ under $\\\\mathbb {P}_0$ .\",\n \"We define (fixing $t_0 = 0$ ): $A^i_n = \\\\left\\\\lbrace \\\\forall j \\\\in \\\\lbrace 0,\\\\cdots ,m-1\\\\rbrace \\\\text{, } S^{*i}_n\\\\left(t_{j} + (k_n^i)/n)\\\\right) - S^{*i}_n\\\\left(t_{j}\\\\right) \\\\le n^{\\\\rho }\\\\right\\\\rbrace .$ Then, using our estimate (REF ), Lemma REF , union bound and the i.i.d structure of regeneration times one gets $ &\\\\mathbb {P}_0\\\\left( (A^{i}_n)^{c} \\\\right) \\\\le m\\\\mathbb {P}\\\\left(S^{*i}_n(n^{-\\\\eta }) \\\\ge n^{\\\\rho } \\\\text{ | }D=+\\\\infty \\\\right) + \\\\mathbb {P}_0\\\\left(R^{c}\\\\right) \\\\le Cn^{-c},$ where $\\\\rho < \\\\eta / \\\\gamma < 1/\\\\gamma $ .\",\n \"Therefore, on the event $A^{i}_n$ we have that $|F(S^{*i}_n)-F(\\\\widetilde{S}^{*i}_n(t))| \\\\le Cn^{-c},$ where $C,c$ only depend on $\\\\lambda ,\\\\vec{\\\\ell },d,K,\\\\lambda _1,\\\\cdots ,\\\\lambda _m$ .\",\n \"Putting everything together we obtain $&\\\\left|\\\\mathbb {E}_0\\\\left[F(S^{*1}_n)F(S^{*2}_n)\\\\right] - \\\\mathbb {E}_0\\\\left[F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n)\\\\right]\\\\right|&&\\\\\\\\ & \\\\hspace{100.0pt} \\\\le \\\\left|\\\\mathbb {E}_0\\\\left[F(S^{*1}_n)F(S^{*2}_n) {1}_{\\\\lbrace A^i_n\\\\rbrace }\\\\right] - \\\\mathbb {E}_0\\\\left[F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n){1}_{\\\\lbrace A^i_n\\\\rbrace }\\\\right] \\\\right| + Cn^{-c} && \\\\\\\\& \\\\hspace{100.0pt} \\\\le Cn^{-c}.&&$ In the first inequality we used (REF ) and in the second one the fact that $F$ is bounded, (REF ) and the expression (REF ).\",\n \"We are now able to prove the key result.\",\n \"[Proof of Proposition REF ] Let us fix again $1 > 1 - \\\\eta > \\\\theta > 0$ and recall that $I = \\\\Big \\\\lbrace \\\\displaystyle \\\\inf _{\\\\begin{array}{c}p,q \\\\ge 0\\\\end{array}}(\\\\Vert X^1_{ p}-X^2_{q}\\\\Vert _{1})\\\\ge 2 \\\\Big \\\\rbrace $ .\",\n \"Thanks to Lemma REF we get that $\\\\mathbb {E}[F(S^{*1}_n)F(S^{*2}_n)] - \\\\mathbb {E}[F(S^{*1}_n)]\\\\mathbb {E}[F(S^{*2}_n)] \\\\le Cn^{-c} + \\\\mathbb {E}[F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n)] - \\\\mathbb {E}[F(S^{*1}_n)]\\\\mathbb {E}[F(S^{*2}_n)]$ Moreover, we also get that $\\\\left| \\\\mathbb {E}_0\\\\left[F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n)\\\\right] - \\\\mathbb {E}_0\\\\left[{1}_{\\\\lbrace I \\\\circ \\\\theta _{\\\\rho _0}\\\\rbrace }F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n)\\\\right] \\\\right| &\\\\le \\\\mathbb {P}_{0}\\\\left(I^{c} \\\\circ \\\\theta _{\\\\rho _0} \\\\right)\\\\nonumber \\\\\\\\&\\\\le \\\\mathbb {P}_0\\\\left(\\\\mathbf {M}_{n^{\\\\theta }}\\\\right)\\\\nonumber \\\\\\\\ &\\\\le C n^{-c}.$ where we recall the definition of $\\\\mathbf {M}_N$ in (REF ) and we used Proposition REF .\",\n \"We introduce $f = F(S^{*1}_n)F(S^{*2}_n)$ and note that $\\\\mathbb {E}_0\\\\left[{1}_{\\\\lbrace I \\\\circ \\\\theta _{\\\\rho _0}\\\\rbrace }F(\\\\widetilde{S}^{*1}_n)F(\\\\widetilde{S}^{*2}_n)\\\\right] = \\\\mathbb {E}_0\\\\left[{1}_{\\\\lbrace I \\\\circ \\\\theta _{\\\\rho _0}\\\\rbrace } f \\\\circ \\\\theta _{\\\\rho _0}\\\\right].$ Furthermore, $f$ satisfies the assumptions of Proposition REF and REF , hence we get $\\\\mathbb {E}_0\\\\left[{1}_{\\\\lbrace I \\\\circ \\\\theta _{\\\\rho _0}\\\\rbrace } f \\\\circ \\\\theta _{\\\\rho _0} \\\\right] \\\\le Cn^{-c} + \\\\mathbb {E}_0\\\\bigg [\\\\mathbb {E}^{Q_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}^{K}}\\\\Big [f\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\Big ]\\\\bigg ].$ We can rewrite the last term in the following way $\\\\mathbb {E}^{Q_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}^{K}}\\\\Big [f\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\Big ] &= \\\\mathbb {E}^{Q_{X^1_{\\\\rho ^1_0},X^2_{\\\\rho ^2_0}}^{K}}\\\\Big [F(S^{*1}_n)F(S^{*2}_n)\\\\text{ | }D^{\\\\otimes }=+\\\\infty \\\\Big ]\\\\nonumber \\\\\\\\&= \\\\mathbb {E}\\\\left[F(S^{*}_n)\\\\text{ | }D=+\\\\infty \\\\right]^2 = \\\\mathbb {E}\\\\left[F(S^{*}_n)\\\\right]^2.$ Putting together the estimates (REF ) and (REF ) we have shown that $\\\\big | \\\\mathbb {E} \\\\left[ F(S^{*1}_n)F(S^{*2}_n) \\\\right] - \\\\mathbb {E} \\\\left[ F(S^{*1}_n)\\\\right] \\\\mathbb {E} \\\\left[ F(S^{*2}_n)\\\\right] \\\\big | \\\\le Cn^{-c},$ which finishes the proof.\"\n ],\n [\n \"Small times do not matter\",\n \"Lemma 5.13 (Fuk-Nagaev inequality, ) Let $(X_i)_{i \\\\ge 1}$ be a family of independent identically distributed random variables with regularly varying tails and the tail parameter $\\\\displaystyle \\\\gamma \\\\in (0, 1)$ .\",\n \"Let $\\\\displaystyle {S}(n) \\\\sum _{i = 1}^n X_i$ and $\\\\displaystyle {M}(n) \\\\max _{i \\\\in \\\\lbrace 1,\\\\cdots ,n\\\\rbrace } X_i$ , then there exists a constant $c>0$ such that, for all $y \\\\le x$ $P \\\\left( {S}(n) > x, \\\\, {M}(n) < y \\\\right) \\\\stackrel{\\\\textnormal {F-N}}{\\\\le } \\\\left( c n \\\\frac{y}{x} L(y) y^{-\\\\gamma }\\\\right)^{x/y}.$ Where $L(y)$ is the slowly varying function associated to the tail of $X_1$ .\",\n \"Let us consider $\\\\lbrace \\\\tau _1^{(i)}\\\\rbrace _{i \\\\ge 1}$ , a sequence of independent identically distributed regeneration times under the measure $\\\\mathbb {P}^K_{0}(\\\\cdot \\\\text{ | }D=+\\\\infty )$ .\",\n \"In it is proved that the sum ${S}(n) \\\\sum _{i = 1}^{n} \\\\tau _1^{(i)}$ is in the domain of attraction of a completely asymmetric stable law of index $\\\\gamma $ .\",\n \"This implies, thanks to , that $\\\\mathbb {P}_0^K(\\\\tau _1 > t \\\\text{ | }D=+\\\\infty ) = \\\\psi (t)t^{-\\\\gamma }$ for some slowly varying function $\\\\psi (\\\\cdot )$ .\",\n \"Lemma 5.14 For all $\\\\eta \\\\in (0,1)$ set $\\\\displaystyle {M}(n^{1-\\\\eta }) = \\\\max _{1 \\\\le i \\\\le {n^{1-\\\\eta }}} \\\\tau _1^{(i)}$ then for all $\\\\rho < \\\\eta /\\\\gamma $ and all $\\\\varepsilon < (\\\\eta - \\\\rho \\\\gamma )/3\\\\gamma $ there exists $\\\\nu >0$ such that $\\\\mathbb {P}\\\\left( {M}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } \\\\right) \\\\le Cn^{-\\\\nu }.$ $\\\\mathbb {P}\\\\left( {M}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } \\\\right) &\\\\le n^{1 - \\\\eta } \\\\mathbb {P}_{0}^{K}\\\\left( \\\\tau _1 > n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } | D=\\\\infty \\\\right) \\\\\\\\&\\\\le Cn^{1-\\\\eta } n^{-1+\\\\gamma \\\\rho + 2\\\\varepsilon \\\\gamma } \\\\psi (n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon }) \\\\\\\\&\\\\le Cn^{-\\\\eta + \\\\gamma \\\\rho + 3\\\\varepsilon \\\\gamma }\\\\le n^{-\\\\nu },$ where we used the fact that a slowly varying function can always be bounded above by any polynomial so that $\\\\psi (n^{1/\\\\gamma }) \\\\le C n^{\\\\varepsilon /\\\\gamma }$ (Potter's bound ).\",\n \"Finally, it holds that $\\\\eta - \\\\gamma \\\\rho - 3\\\\varepsilon \\\\gamma > 0$ thanks to the choice of $\\\\varepsilon $ , one has that the last inequality holds for some $\\\\nu $ small enough.\",\n \"Lemma 5.15 For all $\\\\eta \\\\in (0, 1)$ recall $\\\\displaystyle {M}(n^{1-\\\\eta }) = \\\\max _{1 \\\\le i \\\\le {n^{1-\\\\eta }}}^{} \\\\tau _1^{(i)}$ and let $ \\\\displaystyle {S}(n^{1-\\\\eta }) \\\\displaystyle \\\\sum _{i = 1}^{{n^{1-\\\\eta }}} \\\\tau _1^{(i)}$ then for all $\\\\rho < \\\\eta /\\\\gamma $ and all $\\\\varepsilon < (\\\\eta - \\\\rho \\\\gamma )/3\\\\gamma $ there exists $\\\\nu >0$ such that $\\\\mathbb {P}\\\\left( {S}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\varepsilon - \\\\rho } \\\\right) \\\\le n^{-\\\\nu }.$ First notice that, by total probability $\\\\mathbb {P}\\\\left( {S}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\varepsilon - \\\\rho } \\\\right) &= \\\\mathbb {P}\\\\left( {S}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\varepsilon - \\\\rho }, {M}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } \\\\right) \\\\\\\\ & +\\\\mathbb {P}\\\\left( {S}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\varepsilon - \\\\rho }, {M}(n^{1-\\\\eta }) \\\\le n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } \\\\right),$ Thanks to Lemma REF $\\\\mathbb {P}\\\\left( {S}(n^{1-\\\\eta }) > n^{1/\\\\gamma -\\\\epsilon - \\\\rho }, {M}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } \\\\right) \\\\le Cn^{-\\\\nu },$ for some $\\\\nu >0$ .\",\n \"So we just need to bound the second term of the sum.\",\n \"We can appeal to Lemma REF with the couple $\\\\quad \\\\quad x = n^{1/\\\\gamma - \\\\varepsilon - \\\\rho }, \\\\quad \\\\quad y=n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon }.$ Plugging these quantities into the formula gives $\\\\mathbb {P}\\\\left( {S}(n^{1-\\\\eta }) > n^{1/\\\\gamma - \\\\varepsilon - \\\\rho }, {M} \\\\le n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon } \\\\right) &\\\\le \\\\left( C n^{1-\\\\eta } n^{-\\\\varepsilon } n^{-1 + \\\\gamma \\\\rho + 2\\\\gamma \\\\varepsilon } \\\\psi (n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon }) \\\\right)^{n^\\\\varepsilon }.$ Since the exponent is large we just need to check that what is inside the brackets is small.\",\n \"This is immediate as $C n^{1-\\\\eta } n^{-\\\\varepsilon } n^{-1 + \\\\gamma \\\\rho + 2\\\\gamma \\\\varepsilon } \\\\psi (n^{1/\\\\gamma - \\\\rho - 2\\\\varepsilon })\\\\le C n^{2\\\\varepsilon \\\\gamma - \\\\eta + \\\\gamma \\\\rho } \\\\le e^{-1},$ as $\\\\psi (n^{1/\\\\gamma }) \\\\le Cn^{\\\\varepsilon }$ thanks to Potter's bound and $2\\\\varepsilon \\\\gamma - \\\\eta + \\\\gamma \\\\rho < 0$ by our choice of $\\\\varepsilon $ , this gives a super-polynomial bound for the desired quantity.\",\n \"[Proof of Lemma REF ] Let us start with the first statement.\",\n \"First let us note that, under $\\\\mathbb {P}_0$ , $\\\\lbrace \\\\tau _{i + 1} - \\\\tau _i\\\\rbrace _{i \\\\ge 1}$ are independent identically distributed random variables distributed as $\\\\tau _1$ under $\\\\mathbb {P}^K_{0}(.\\\\text{ | }D=+\\\\infty )$ .\",\n \"Thanks to this fact we can re-write $S^*_n(n^{-\\\\eta }) \\\\stackrel{(\\\\mathrm {d})}{=} \\\\mathrm {Inv}(n)^{-1}\\\\sum _{i = 1}^{n^{1 - \\\\eta }} \\\\tau _{1}^{(i)},$ where $\\\\lbrace \\\\tau _{1}^{(i)}\\\\rbrace _{i \\\\ge 1}$ are i.i.d.\",\n \"copies of $\\\\tau _1$ under the measure $\\\\mathbb {P}^{K}_{0}[ \\\\cdot | D = \\\\infty ]$ .\",\n \"Then Lemma REF is enough to conclude together with the fact that for all $\\\\varepsilon >0$ $\\\\frac{n^{1/\\\\gamma - \\\\varepsilon }}{\\\\mathrm {Inv}(n)} \\\\rightarrow 0.$\"\n ],\n [\n \"Quenched convergence\",\n \"We start this section by proving Theorem REF .\",\n \"[Proof of Theorem REF ] The result is a straightforward consequence of Theorem REF and Proposition REF .\",\n \"In particular one just needs to notice that $\\\\mathbf {Var}\\\\left(E_{0}^{\\\\omega }\\\\left[F_2(Z_{b^n})\\\\right]\\\\right) = \\\\mathbb {E}[F_2(Z_{b^n}^1)F_2(Z_{b^n}^2)] - \\\\mathbb {E}[F_2(Z_{b^n}^1)]\\\\mathbb {E}[F_2(Z_{b^n}^2)].$ The same decomposition holds for the process $S^*$ and the function $F_1$ .\",\n \"Recall $D(\\\\mathbb {R})$ (resp.\",\n \"$D(\\\\mathbb {R}^d)$ ) is the space of $\\\\mathbb {R}$ -valued (resp.\",\n \"$\\\\mathbb {R}^d$ -valued) càdlàg functions.\",\n \"In the following we consider $U$ the uniform topology and $J_1$ , $M_1$ two classical Skorohod topologies.\",\n \"Let us recall that the vector $v$ and the matrix $\\\\Sigma $ have been introduced in .\",\n \"In particular $v \\\\mathbb {E}_{0}^K[X_{\\\\tau _1}| D = \\\\infty ]$ while $\\\\Sigma $ is the covariance matrix of $X_{\\\\tau _1}$ under $\\\\mathbb {P}_{0}^{K}[ \\\\cdot | D = \\\\infty ]$ .\",\n \"Moreover, let us introduce $v_0 = v / \\\\Vert v\\\\Vert $ , $I_d$ as the $d$ -dimensional identity matrix and $P_{v_0}$ as the projection matrix on $v_0$ .\",\n \"Finally let $M_d C_\\\\infty ^{-\\\\gamma /2}(I_d - P_{v_0}) \\\\sqrt{\\\\Sigma }.$ Proposition 6.1 Fix any $T > 0$ , the followings holds for almost every environment $\\\\omega \\\\in \\\\Omega $ .\",\n \"The law of $(Y_n(t))_{0 \\\\le t \\\\le T}$ converges under $P^\\\\omega _0(\\\\cdot )$ for the $U$ topology on $D^d$ towards the distribution of $(v t)_{0 \\\\le t \\\\le T}$ .\",\n \"Moreover, the law of $(Z_n(t), S_n(t))_{0 \\\\le t \\\\le T}$ under $P^\\\\omega _0(\\\\cdot )$ converges in $D(\\\\mathbb {R}^d) \\\\times D(\\\\mathbb {R})$ for the $J_1 \\\\times M_1$ topology as $n \\\\rightarrow \\\\infty $ towards the law of $(\\\\sqrt{\\\\Sigma } B_t, C_\\\\infty \\\\mathcal {S}_\\\\gamma (t))_{0 \\\\le t \\\\le T}$ where $B$ is a standard Brownian motion and $\\\\mathcal {S}_\\\\gamma $ is a stable Subordinator of index $\\\\gamma $ independent of $B$ .\",\n \"The convergence of $(Y_n(t))_{0 \\\\le t \\\\le T}$ is a consequence of .\",\n \"We start by proving that Theorem REF together with imply the fact that $S_n$ converges in the process $M_1$ sense under the quenched law towards $\\\\mathcal {S}_\\\\gamma $ , we follow the proof of .\",\n \"First we observe that $S_n(t) = \\\\frac{1}{\\\\mathrm {Inv}(n)} \\\\left( \\\\tau _1 + \\\\sum _{i = 1}^{{nt} - 1} (\\\\tau _{i + 1} - \\\\tau _i) \\\\right).$ We have that $\\\\tau _1 < + \\\\infty $ $\\\\mathbb {P}$ -a.s. thanks to , this means that for $\\\\mathbf {P}$ -almost all realisation of the environment $\\\\tau _1< + \\\\infty $ $P^{\\\\omega }$ -a.s..\",\n \"But then, applying Slutsky's theorem, we just need to prove the quenched convergence of $S^*_n$ towards $\\\\mathcal {S}_\\\\gamma $ .\",\n \"In order to prove convergence of the finite dimensional distributions we can restrict to prove convergence of the Laplace functionals $F$ of the form $F(h) = \\\\exp \\\\Big (-\\\\lambda _1h(t_1)-\\\\lambda _2(h(t_2)-h(t_1)) -\\\\cdots -\\\\lambda _n(h(t_n)-h(t_{n-1}))\\\\Big )$ Thanks to we have annealed convergence, so that Laplace functionals converge, in formulas $\\\\mathbb {E}\\\\left[F(S_n^*)\\\\right] = \\\\mathbf {E}\\\\left[ E^\\\\omega \\\\left[ F(S_n^*) \\\\right]\\\\right] \\\\underset{n \\\\rightarrow +\\\\infty }{\\\\rightarrow } \\\\mathbb {E}\\\\left[ F(\\\\mathcal {S}_\\\\gamma ) \\\\right] = \\\\exp \\\\left\\\\lbrace - t_1 \\\\psi (\\\\lambda _1) - \\\\dots - (t_m - t_{m -1})\\\\psi (\\\\lambda _m) \\\\right\\\\rbrace ,$ where $\\\\psi $ is the Laplace exponent of the stable Subordinator.\",\n \"Moreover thanks to Theorem REF we know that for $b \\\\in (1, 2)$ $\\\\sum _{k = 1 }^{+\\\\infty }\\\\mathbf {Var}\\\\left[E_{0}^{\\\\omega }(F(S^*_{b^k}))\\\\right]< +\\\\infty .$ Therefore, by applying Borel-Cantelli and Chebyshev inequality to the sequence of events $|E_{0}^{\\\\omega }(F(S^*_{b^k})) - \\\\mathbb {E}_{0}\\\\left[ F(S^*_{b^k}) \\\\right]| > \\\\varepsilon $ and using the annealed convergence we get that $E_{0}^{\\\\omega }\\\\left[ F(S^*_{b^k}) \\\\right] \\\\rightarrow \\\\mathbb {E}\\\\left[ F(\\\\mathcal {S}_\\\\gamma ) \\\\right],$ almost surely along $b^k$ .\",\n \"Moreover, the convergence holds jointly on a set of full measure for all functions $F$ of the form above for $b$ rational.\",\n \"On the set of full measure where the joint convergence holds, we want to show that the promised convergence holds.\",\n \"Let $k_n$ be the largest integer such that $b^{k_n} \\\\le n$ .\",\n \"Thanks to the definition of $F$ , for any two increasing processes $h$ and $h^{\\\\prime }$ starting from 0 one gets $|F(h) - F(h^{\\\\prime })| \\\\le C \\\\max _{1 \\\\le i \\\\le m} |h(t_i) - h^{\\\\prime }(t_i)| \\\\wedge 1.$ Observe that, $S^*_n(t_i) = \\\\left( \\\\frac{b^{k_n}}{n} \\\\right)^{1/\\\\gamma } S^*_{b^{k_n}} \\\\left( \\\\frac{n}{b^{k_n}} t_i\\\\right).$ By the definition of $k_n$ and the monotonicity of $S^*_{b^{k_n}}(t)$ the latter is smaller than $S^*_{b^{k_n}}(b t_i)$ , as a consequence $(b^{-\\\\frac{1}{\\\\gamma }}-1)S^*_{b^{k_n}}(t_i) \\\\le S^*_n(t_i) - S^*_{b^{k_n}}(t_i) \\\\le S^*_{b^{k_n}}(b t_i) - S^*_{b^{k_n}}(t_i).$ The quantity $\\\\limsup _{n \\\\rightarrow \\\\infty } \\\\left| E^\\\\omega \\\\left[ F(S^*_n(t_i)) \\\\right] - E^\\\\omega \\\\left[ F(S^*_{b^{k_n}}(t_i)) \\\\right] \\\\right|,$ is thus bounded from above, up to a multiplicative constant, by $\\\\limsup _{n \\\\rightarrow \\\\infty } E^\\\\omega \\\\left[ \\\\max _{1 \\\\le i \\\\le m} \\\\bigg (|S^*_n(t_i) - S^*_{b^{k_n}}(t_i)|,|(b^{-\\\\frac{1}{\\\\gamma }}-1)S^*_{b^{k_n}}(t_i)|\\\\bigg ) \\\\wedge 1 \\\\right].$ Thanks (REF ) we get that this last quantity is bounded by $&\\\\limsup _{n \\\\rightarrow \\\\infty } E^\\\\omega \\\\left[ \\\\max _{1 \\\\le i \\\\le m} \\\\bigg (|S^*_{b^{k_n}}(b t_i) - S^*_{b^{k_n}}(t_i)|,|(b^{-\\\\frac{1}{\\\\gamma }}-1)S^*_{b^{k_n}}(t_i)|\\\\bigg ) \\\\wedge 1 \\\\right] \\\\\\\\&= \\\\mathbb {E} \\\\left[ \\\\max _{1 \\\\le i \\\\le m} \\\\bigg (|\\\\mathcal {S}_\\\\gamma (b t_i) - \\\\mathcal {S}_\\\\gamma (t_i)|,|(b^{-\\\\frac{1}{\\\\gamma }}-1)\\\\mathcal {S}_\\\\gamma (t_i)| \\\\bigg ) \\\\wedge 1 \\\\right].$ By the fact that $\\\\mathcal {S}_\\\\gamma $ is almost surely continuous at deterministic times we get that the quantity on the right converges to 0 as $b \\\\rightarrow 1$ along rationals by dominated convergence theorem since the quantity is bounded by 1.\",\n \"The tightness of the process in $M_1$ comes from this f.d.d.\",\n \"distributions convergence we just proved, thanks to the fact that all processes involved are increasing.\",\n \"The convergence of the position $Z_n(t)$ (both seen as the polygonal interpolation and in the canonical way) is proved with the same hypotheses and thesis in .\",\n \"We now need to prove joint convergence of $S_n$ and $Z_n$ .\",\n \"First notice that, thanks to the previous two points and , tightness of the marginals is enough to have tightness in the product space.\",\n \"Therefore, we have that the law $(S_n, Z_n)$ is tight on $D(\\\\mathbb {R}^d) \\\\times D(\\\\mathbb {R})$ endowed with the $J_1 \\\\times M_1$ -topology on a set of the environment $\\\\Omega _1 \\\\subset \\\\Omega $ of full $\\\\mathbf {P}$ -measure.\",\n \"We need to prove that on a set of full measure $\\\\Omega _2$ , the law of the increments of the process $W_n = (S_n, Z_n)$ converge to the one of $W = (\\\\mathcal {S}_\\\\gamma , \\\\sqrt{\\\\Sigma } B)$ .\",\n \"The proof follows the lines of the proof of the convergence of $S_n$ and the argument of .\",\n \"Let $\\\\lambda _1, \\\\dots , \\\\lambda _m \\\\in \\\\mathbb {R}^{d+1}$ and $0 < t_0 < \\\\dots < t_m$ as it is sufficient to check on intervals that do not contain 0.\",\n \"Let us define the function $G(w) = \\\\exp \\\\Big ( i \\\\lambda _1 \\\\cdot \\\\left( w(t_1) - w(t_0) \\\\right) + \\\\dots + i \\\\lambda _m \\\\cdot \\\\left( w(t_m) - w(t_{m - 1}) \\\\right) \\\\Big ).$ From we have that $\\\\mathbb {E}\\\\left[ G(W_n) \\\\right] \\\\underset{n \\\\rightarrow +\\\\infty }{\\\\longrightarrow } \\\\mathbb {E}\\\\left[ G(W) \\\\right].$ By proceeding in the same way as we did for Theorem REF (using all the instrumental lemmas and propositions that we proved in Section ) it is possible to prove that for all $b \\\\in (1, 2)$ $\\\\sum _{k = 1 }^{+\\\\infty }\\\\mathbf {Var}\\\\left[E_{0}^{\\\\omega }(G(W_{b^k}))\\\\right]< +\\\\infty .$ As a consequence we get that $E_{0}^{\\\\omega }\\\\left[ G(W_{b^k}) \\\\right] \\\\rightarrow \\\\mathbb {E}\\\\left[ G(W) \\\\right].$ Then, by following the argument that we did before, it is possible to show that for all $G$ of the form of (REF ) with $\\\\lambda _1, \\\\dots , \\\\lambda _m$ and $0 < t_0 < \\\\dots < t_m$ rationals on a set $\\\\Omega _2$ of full measure one gets $E_{0}^{\\\\omega }\\\\left[ G(W_{n}) \\\\right] \\\\rightarrow \\\\mathbb {E}\\\\left[ G(W) \\\\right].$ Finally, by right continuity of $W_n$ and $W$ and continuity of the Fourier functional it is possible to show that the limit holds for all $0 \\\\le t_0 \\\\le \\\\dots \\\\le t_m$ and for all $\\\\lambda _1, \\\\dots , \\\\lambda _m \\\\in \\\\mathbb {R}^{d+1}$ .\",\n \"This finishes the proof as we showed that on a set of full measure $\\\\Omega _1 \\\\cap \\\\Omega _2$ the sequence $W_n$ is tight and has just one possible limit $W$ .\",\n \"We are now able to prove the main result, the method is the same as the proof of .\",\n \"[Proof of Theorem REF ] We fix a realisation of the environment $\\\\omega \\\\in \\\\Omega $ , the following convergences in distribution are meant to be understood under the quenched law $P^\\\\omega $ , we assume the result of Proposition REF .\",\n \"Recall that $D(\\\\mathbb {R})$ and $D(\\\\mathbb {R}^d)$ denote the sets of càdlàg functions in dimension one (resp.\",\n \"$d$ ).\",\n \"Let $D_{\\\\uparrow }$ be the set of non-deacreasing càdlàg functions, $D_{\\\\uparrow \\\\uparrow }$ the set of strictly increasing càdlàg functions and $D_u$ the set of unbounded càdlàg functions.\",\n \"Let $\\\\mathcal {C}$ be the set of continuous functions and $U$ the uniform topology.\",\n \"It is a known fact that $\\\\mathcal {S}_\\\\gamma ^{-1}$ is almost surely continuous and increasing.\",\n \"Firstly we get that $\\\\left(Y_{n^\\\\gamma /L(n)}\\\\left( S_{n^\\\\gamma /L(n)}^{-1}\\\\left(\\\\frac{nt}{\\\\mathrm {Inv}\\\\left(n^\\\\gamma /L(n)\\\\right)}\\\\right) \\\\right) \\\\right)_{t \\\\in [0, T]} \\\\rightarrow \\\\left( v C_\\\\infty ^{-\\\\gamma } \\\\mathcal {S}_\\\\gamma ^{-1}(t) \\\\right)_{t \\\\in [0, T]},$ in the uniform topology.\",\n \"This thanks to the fact the right continuous inverse is continuous from $(D_{u, \\\\uparrow \\\\uparrow }, M_1)$ to $(\\\\mathcal {C}, U)$ thanks to , and if $(x_n, y_n) \\\\rightarrow (x, y)$ in $D^d \\\\times D_{\\\\uparrow }$ with $(x, y) \\\\in \\\\mathcal {C}^d \\\\times \\\\mathcal {C}_{\\\\uparrow }$ , then $x_n \\\\circ y_n \\\\rightarrow x \\\\circ y$ in the $U$ topology (see ).\",\n \"Lastly, note that $(nt/\\\\mathrm {Inv}(n^\\\\gamma /L(n)))_{t \\\\in [0, T]}$ converges uniformly to $(t)_{t \\\\in [0, T]}$ .\",\n \"This proves the first result.\",\n \"Moreover, thanks to the same results it holds that $\\\\left(Z_{n^\\\\gamma /L(n)}\\\\left( S_{n^\\\\gamma /L(n)}^{-1}\\\\left(\\\\frac{nt}{\\\\mathrm {Inv}\\\\left(n^\\\\gamma /L(n)\\\\right)}\\\\right) \\\\right) \\\\right)_{t \\\\in [0, T]} \\\\stackrel{(\\\\mathrm {d})}{\\\\rightarrow } \\\\left( C_\\\\infty ^{-\\\\gamma /2} B_{\\\\mathcal {S}_\\\\gamma ^{-1}(t)} \\\\sqrt{\\\\Sigma } \\\\right)_{t \\\\in [0, T]},$ in the $J_1$ topology.\",\n \"We now need to ensure that $\\\\mathbf {P}$ -a.s., with high $P^\\\\omega $ -probability $X_{{nt}}$ is close to $X_{\\\\tau _{\\\\left\\\\lfloor \\\\frac{n^\\\\gamma }{L(n)} S_{n^\\\\gamma /L(n)}^{-1}(nt/\\\\mathrm {Inv}(n^\\\\gamma /L(n)))\\\\right\\\\rfloor }}$ .\",\n \"It is immediate to prove that $\\\\tau _{\\\\left\\\\lfloor \\\\frac{n^\\\\gamma }{L(n)} S_{n^\\\\gamma /L(n)}^{-1}(nt/\\\\mathrm {Inv}(n^\\\\gamma /L(n)))\\\\right\\\\rfloor }$ is the smallest $\\\\tau _i$ such that $\\\\tau _i > tn$ .\",\n \"Then we can bound the distance $X_{\\\\tau _{\\\\left\\\\lfloor \\\\frac{n^\\\\gamma }{L(n)} S_{n^\\\\gamma /L(n)}^{-1}(nt/\\\\mathrm {Inv}(n^\\\\gamma /L(n)))\\\\right\\\\rfloor }} - X_{{nt}}$ by a constant times the size of a regeneration block.\",\n \"$M_n &\\\\max _{t \\\\in [0, T]} \\\\left\\\\Vert X_{\\\\tau _{\\\\left\\\\lfloor \\\\frac{n^\\\\gamma }{L(n)} S_{n^\\\\gamma /L(n)}^{-1}(nt/\\\\mathrm {Inv}(n^\\\\gamma /L(n)))\\\\right\\\\rfloor }} - X_{{nt}} \\\\right\\\\Vert _\\\\infty \\\\\\\\& \\\\le C \\\\max _{k = 1, \\\\dots , {nT} + 1} \\\\left\\\\Vert X_{\\\\tau _k} - X_{\\\\tau _{k - 1}} \\\\right\\\\Vert _\\\\infty .$ By the fact that $\\\\Vert X_{\\\\tau _k} - X_{\\\\tau _{k - 1}}\\\\Vert _\\\\infty \\\\le \\\\chi _k^\\\\alpha $ defined in (REF ), applying and using a union bound one gets that, for all $M>0$ and $\\\\delta >0$ $\\\\mathbb {P}_0\\\\left( M_n > \\\\delta n^{\\\\gamma /4} \\\\right) = \\\\mathbf {E}\\\\left[ P_0^\\\\omega (M_n > \\\\delta n^{\\\\gamma /4}) \\\\right] \\\\le n^{-M}.$ Markov inequality gives that $\\\\mathbf {P}\\\\left( P_0^\\\\omega (M_n > \\\\delta n^{\\\\gamma /4}) > n^{-1} \\\\right) \\\\le n^{-2}.$ Then Borel-Cantelli implies that $\\\\mathbf {P}$ -almost surely $P_0^\\\\omega [M_n > \\\\delta n^{\\\\gamma /4}] \\\\le 1/n$ for all $n$ large enough, which leads us to conclude that $X_{\\\\tau _{\\\\left\\\\lfloor \\\\frac{n^\\\\gamma }{L(n)} S_{n^\\\\gamma /L(n)}^{-1}(nt/\\\\mathrm {Inv}(n^\\\\gamma /L(n)))\\\\right\\\\rfloor }} - X_{{nt}} \\\\le \\\\delta n^{\\\\gamma /4}$ in the uniform sense with high $P^\\\\omega $ -probability.\",\n \"This implies, together with (REF ) and (REF ) and Slutsky's theorem, that $\\\\mathbf {P}$ -a.s., under the quenched law $P^\\\\omega $ the statement (REF ) holds and that $\\\\left(\\\\frac{X_{{nt}} - v \\\\frac{n^\\\\gamma }{L(n)} S_{n^\\\\gamma /L(n)}^{-1}\\\\left(\\\\frac{nt}{\\\\mathrm {Inv}\\\\left(n^\\\\gamma /L(n)\\\\right)}\\\\right)}{\\\\sqrt{n^\\\\gamma /L(n)}} \\\\right)_{t \\\\in [0, T]} \\\\stackrel{(\\\\mathrm {d})}{\\\\rightarrow } \\\\left( C_\\\\infty ^{-\\\\gamma /2} B_{\\\\mathcal {S}_\\\\gamma ^{-1}(t)} \\\\sqrt{\\\\Sigma } \\\\right)_{t \\\\in [0, T]}.$ Finally, notice that (REF ) follows by applying the linear transformation $(I_d - P_{v_0})$ to (REF ) as $(I_d - P_{v_0}) v = 0$ .\",\n \"All linear combinations are continuous in the $J_1$ -topology at continuous functions.\"\n ],\n [\n \"Extension of Markov property\",\n \"The appendix gives some results which are useful in the paper but considered as tools.\",\n \"Since in the whole paper we will consider two random walks we extend the Markov property when we have two Markov chains with the general following lemma.\",\n \"Lemma A.1 Consider two Markov chains $(X_n)_{n \\\\ge 0}$ and $(Y_n)_{n \\\\ge 0}$ which take values in a countable set $E$ .\",\n \"Consider the product measure $P_{(x,y)} = P_x \\\\otimes P_y$ .\",\n \"We consider the canonical markov chain on $E^2$ .\",\n \"We define for $k,k^{\\\\prime } \\\\in \\\\mathbb {N}$ the translation operator defined by $\\\\begin{array}{l|rcl}\\\\end{array}\\\\theta _{k,k^{\\\\prime }}: & (E^2)^{\\\\mathbb {N}} & \\\\longrightarrow & (E^2)^{\\\\mathbb {N}} \\\\\\\\& ((x_n),(y_n)) & \\\\longmapsto & ((x_{n+k}),(y_{n+k^{\\\\prime }})) $ .$$ Consider also $P_{x,y} = P_x \\\\otimes P_y$ almost surely finite stopping times $\\\\tau $ and $\\\\rho $ for $X$ and $Y$ .\",\n \"Consider two bounded and measurable functions $f$ and $g$ which a respectively measurable with respect to $\\\\sigma \\\\Big (X_0,...,X_{\\\\tau },Y_0,...,Y_{\\\\rho }\\\\Big ) \\\\text{ and }\\\\sigma \\\\Big ((X_n)_{n \\\\ge 0},(Y_n)_{n \\\\ge 0}\\\\Big ).$ Then we have the following equality: $E_{x,y}\\\\left[f g \\\\circ \\\\theta _{\\\\tau ,\\\\rho }\\\\right] = E_{x,y}\\\\left[fE_{X_{\\\\tau },Y_{\\\\rho }}[g]\\\\right].$ The proof is straightforward using strong Markov property and a Monotone Class Theorem.\",\n \"Indeed, if $f$ is of the form $f((x_n),(y_n)) = f_1((x_n))f_2((y_n))$ with $f_1$ measurable with respect to $\\\\sigma \\\\big (X_0,...,X_{\\\\tau }\\\\big )$ and $f_2$ measurable with respect to $\\\\sigma \\\\big (Y_0,...,Y_{\\\\rho }\\\\big )$ and $g((x_n),(y_n)) = g_1((x_n))g_2((y_n))$ with $g_1$ measurable with respect to $\\\\sigma \\\\big ((X_n)_{n \\\\ge 0}\\\\big )$ and $g_2$ measurable with respect to $\\\\sigma \\\\big ((Y_n)_{n \\\\ge 0}\\\\big )$ .\",\n \"Then we have that $&E_{x,y}\\\\left[f g \\\\circ \\\\theta _{\\\\tau ,\\\\rho }\\\\right]\\\\\\\\&= E_{x,y}\\\\left[f_1 g_1 \\\\theta _{\\\\tau } \\\\cdot f_2g_2 \\\\circ \\\\theta _{\\\\rho }\\\\right]\\\\\\\\& = E_{x}\\\\left[f_1 g_1 \\\\circ \\\\theta _{\\\\tau }\\\\right]\\\\cdot E_{y}\\\\left[f_2 g_2 \\\\circ \\\\theta _{\\\\rho }\\\\right],$ where the last equality comes from the independence of the two walks.\",\n \"Then using the Strong Markov property we get $&E_{x}\\\\left[f_1 E_{X_{\\\\tau }}[g_1]\\\\right]\\\\cdot E_{y}\\\\left[f_2 E_{Y_{\\\\rho }}[g_2]\\\\right] \\\\\\\\& = E_{x,y}\\\\left[f_1f_2 E_{X_{\\\\tau }}[g_1]E_{Y_{\\\\rho }}[g_2]\\\\right]\\\\\\\\& = E_{x,y}\\\\left[f E_{X_{\\\\tau },Y_{\\\\rho }}[g]\\\\right].$ We conclude using a Monotone Class Theorem argument.\"\n ],\n [\n \"Polynomial decay\",\n \"Proposition B.1 For any $\\\\varepsilon > 0$ , there exists $K_0 > 0$ such that for any $K > K_0$ , any $n \\\\in \\\\mathbb {N}^{*}$ and any $z \\\\in \\\\mathbb {Z}^d$ we have $\\\\mathbb {P}_0(X_{\\\\tau _n} = z) \\\\le Cn^{\\\\varepsilon -\\\\frac{d}{2}},$ where $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"Remark B.2 We recall that the notion of regeneration time introduced in (see the definition in Section REF ) depends on the value of $K$ , but we always take $K$ large enough.\",\n \"First we have the following equality: $\\\\mathbb {P}_0(X_{\\\\tau _n} = z)& =\\\\mathbb {P}_0\\\\left(X_{\\\\tau _1} +\\\\sum _{i=1}^{n-1} (X_{\\\\tau _{i+1}}-X_{\\\\tau _i}) = z\\\\right)\\\\\\\\&\\\\nonumber =\\\\mathbb {P}_0\\\\left(\\\\sum _{i=1}^{n-1} (X_{\\\\tau _{i+1}}-X_{\\\\tau _i} - v ) = z - X_{\\\\tau _1} - (n-1)v\\\\right),$ where $v = \\\\mathbb {E}[X_{\\\\tau _{2}}-X_{\\\\tau _1}]$ , we introduce for $i\\\\ge 1$ , $\\\\xi _i = X_{\\\\tau _{i+1}}-X_{\\\\tau _i} - v $ .\",\n \"The family $(\\\\xi _i)_{i \\\\ge 1}$ is an independent family of random variables identically distributed and independent of $X_{\\\\tau _1}$ .\",\n \"By , for any $M > 0$ there exists $K_0 > 0$ such that for any $K > K_0$ we have $\\\\mathbb {E}_{0}\\\\left[\\\\Vert \\\\xi _1\\\\Vert ^{M}\\\\right] < +\\\\infty $ .\",\n \"Thus we can write (REF ) as $\\\\sum _{y \\\\in \\\\mathbb {Z}^d}^{} \\\\mathbb {P}_0(X_{\\\\tau _1} = y)\\\\mathbb {P}_0\\\\bigg (\\\\sum _{i=1}^{n-1} \\\\xi _i = e - y - (n-1)v\\\\bigg ).$ Then we only need a bound for the value of $\\\\mathbb {P}_0\\\\bigg (\\\\displaystyle \\\\sum _{i=1}^{n-1} \\\\xi _i = x\\\\bigg )$ with $x = e - y - (n-1)v$ uniformly for $y \\\\in \\\\mathbb {Z}^d$ .\",\n \"We define $\\\\begin{array}{l|rcl}\\\\end{array}H: & \\\\mathbb {R} & \\\\longrightarrow & \\\\mathbb {R} \\\\\\\\& x & \\\\longmapsto & x^{M} $$, then we have $ E0(H(||i||)) < +$.\",\n \"$ H$ increases faster than $ x x2+$ and slower than $ x ex$.\",\n \"Using \\\\cite [Theorem 3]{Zaitsev} we can construct a probability space and two family of random variables $ (Yi)$ and $ (Ni)$ such that:\\\\begin{itemize}\\\\item [\\\\bullet ] (Y_i)_{i \\\\ge 1} has the same law as (\\\\xi _i)_{i \\\\ge 1} under \\\\mathbb {P}_0.\\\\item [\\\\bullet ] (N_i)_{i \\\\ge 1} is a family of independent identically distributed centered gaussian vectors with covariance matrix \\\\Sigma where \\\\Sigma is the covariance matrix of \\\\xi _1.\\\\end{itemize}Moreover, they satisfy\\\\begin{equation*}\\\\mathbb {P}\\\\left(\\\\max _{\\\\begin{array}{c}1 \\\\le j \\\\le n\\\\end{array}}(|\\\\sum _{i=1}^{j} (Y_i - N_i)|) \\\\ge cn^{\\\\varepsilon }\\\\right) \\\\le C\\\\frac{n}{n^{\\\\varepsilon M}},\\\\end{equation*}where the constants $ C,c > 0$ only depend on $ ,,d,K$.\",\n \"We write{\\\\begin{@align}{1}{-1}\\\\nonumber \\\\mathbb {P}_0\\\\bigg (\\\\sum _{i=1}^{n-1} \\\\xi _i = x\\\\bigg ) &= \\\\mathbb {P}\\\\bigg (\\\\sum _{i=1}^{n-1} Y_i = x\\\\bigg ) \\\\\\\\&\\\\nonumber \\\\le Cn^{1-\\\\varepsilon M} + \\\\mathbb {P}\\\\bigg (\\\\sum _{i=1}^{n-1} N_i \\\\in B_{\\\\Vert \\\\cdot \\\\Vert _{1}}(x,cn^{\\\\varepsilon })\\\\bigg ) \\\\\\\\& \\\\le Cn^{1-\\\\varepsilon M} + \\\\mathbb {P}\\\\bigg ( \\\\frac{\\\\sum _{i=1}^{n-1} N_i}{\\\\sqrt{n-1}} \\\\in B_{\\\\Vert \\\\cdot \\\\Vert _{1}}\\\\bigg (\\\\frac{x}{\\\\sqrt{n-1}},\\\\frac{cn^{\\\\varepsilon }}{\\\\sqrt{n-1}}\\\\bigg ) \\\\bigg ).\\\\end{@align}}Since $ i=1n-1 Nin-1$ is a centered Gaussian vector with covariance matrix $$, the right term in (\\\\ref {Gaussian}) is bounded above by{\\\\begin{@align*}{1}{-1}C\\\\mu \\\\left(B_{\\\\Vert \\\\cdot \\\\Vert _{1}}\\\\bigg (\\\\frac{x}{\\\\sqrt{n-1}},\\\\frac{cn^{\\\\varepsilon }}{\\\\sqrt{n-1}}\\\\bigg )\\\\right),\\\\end{@align*}}where $$ denotes the Lebesgue measure on $ Rd$ and $ C$ is a constant which depends on $$, hence on $ ,,d,K$.\",\n \"We deduce that last term is dominated by{\\\\begin{@align*}{1}{-1}Cn^{d\\\\varepsilon -\\\\frac{d}{2}},\\\\end{@align*}}where $ C$ is a constant which only depends on $ ,,d,K$.$ Choosing $M$ large enough such that $1-\\\\varepsilon M < - \\\\frac{d}{2}$ we obtain that (REF ) is bounded above by $Cn^{1-\\\\varepsilon M} + Cn^{d\\\\varepsilon -\\\\frac{d}{2}} \\\\le C n^{\\\\varepsilon d-\\\\frac{d}{2}},$ where the constant $C$ only depends on $\\\\lambda ,\\\\vec{\\\\ell },d,K$ .\",\n \"We conclude since the result is true for all $\\\\varepsilon > 0$ .\"\n ],\n [\n \"Acknowledgments\",\n \"We would like to thank Daniel Kious for encouraging this collaboration and useful comments at various stages of the project.\",\n \"We would like also to thank Christophe Sabot and Noam Berger for insightful discussions.\",\n \"AF acknowledges the support of an NSERC Discovery Grant.\",\n \"CS is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/S022945/1.\"\n ]\n]"},"id":{"kind":"string","value":"2210.07825"}}},{"rowIdx":2997,"cells":{"text":{"kind":"list like","value":[["(1,1)-Cluster Editing is Polynomial-time Solvable"],["Abstract A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques.","Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\\ V(G)\\rightarrow \\{0,1,\\dots, a\\}$ and $d^*{}:\\ V(G)\\rightarrow \\{0,1,\\dots, d\\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\\in V(G)$.","Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3."],["Introduction","A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques.","Addition or deletion of an edge to a graph is called an edge edition.","Given a graph $G$ and an integer $k \\ge 0,$ the Cluster Editing problem asks whether $k$ or less edge editions can transform $G$ into a clique graph.","Cluster Editing is NP-Complete [11], [12], but it is fixed-parameter tractable if parameterized by $k$ [5].","The parameterization has received considerable attention [2], [3], [4], [6], [7], [8], [9].","In particular, an $O^*(1.618^k)$ -time algorithm was designed in [2] and a kernel with at most $2k$ vertices in [7].","Unfortunately, $k$ is often not small for real-world instances.","For example, in a protein similarity data set that has been frequently used for evaluating Cluster Editing algorithms, the instances with $n \\ge 30,$ where $n$ is the number of vertices, have an average number of edge editions between $2n$ and $4n$ [3].","This has led to introduction of different parameterizations in [1], [10].","We will consider the following variation of Cluster Editing problem introduced by Abu-Khzam [1].","In $(a,d)$ -Cluster Editing, for fixed natural numbers $a$ and $d$ , given a graph $G$ and vertex-weights $a^*_{}:\\ V(G)\\rightarrow \\lbrace 0,1,\\dots , a\\rbrace $ and $d^*_{}:\\ V(G)\\rightarrow \\lbrace 0,1,\\dots , d\\rbrace $ , decide whether $G$ can be turned into a cluster graph by deleting at most $d^*_{}(v)$ edges incident to every $v\\in V(G)$ and adding at most $a^*_{}(v)$ edges incident to every $v\\in V(G)$ .","Note that there is no upper bound on the total number of additions and deletions, see Paragraph 7 of Section 2 and Paragraph 1 of Section 5.1 in [1].","$(a,d)$ -Cluster Editing is similar to a version of Cluster Editing studied by Komusiewicz and Uhlmann [10].","In particular, they proved that $(0,d)$ -Cluster Editing is NP-hard for every fixed integer $d\\ge 2.$ Abu-Khzam [1] proved additional results on $(a,d)$ -Cluster Editing and established computational complexity of $(a,d)$ -Cluster Editing for every value of $a$ and $d$ apart from $a=d=1$ : $(a,d)$ -Cluster Editing is in P when $(a,d)\\in \\lbrace (a,0), (0,1):\\ a\\in \\lbrace 0,1,2\\rbrace \\rbrace $ and NP-complete, otherwise, apart from possibly the case of $a=d=1$ .","In fact, Abu-Khzam [1] conjectured that $(1,1)$ -Cluster Editing is polynomial-time solvable.","In this paper we resolve Abu-Khzam's conjecture in affirmative by (i) providing, in Section , a serious of five polynomial-time reductions to $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3, and (ii) designing, in Section , a polynomial-time algorithm for solving $(1,1)$ -Cluster Editing on $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3.","An edge $uv$ of $G$ is deletable (in $G$ ) if $d^*_{}(u)=d^*_{}(v)=1$ and non-deletable, otherwise.","An edge $uv$ of the complement $\\overline{G}$ of $G$ is addable (to $G$ ) if $a^*_{}(u)=a^*_{}(v)=1$ and non-addable, otherwise.","Now we reformulate $(1,1)$ -Cluster Editing as follows: Given a graph $G$ , decide whether $G$ has a matching $D$ of edges deletable in $G$ and $\\overline{G}$ has a matching $A$ of edges addable to $G$ such that $G-D+A$ is a clique graph.","We call it the matching formulation of $(1,1)$ -Cluster Editing and the desirable clique graph $G-D+A$ a solution.","Note that we will often write a solution in the form $G-D+A$ , which defines the matchings $D$ and $A$ rather than the resulting cliques.","We will denote by $P_n$ ($C_n$ , respectively) a path (cycle, respectively) on $n$ vertices, by $K_n$ a complete graph on $n$ vertices, and by $K_{n,m}$ a complete bipartite graph with partite sets of sizes $n$ and $m.$ For a positive integer $p$ , $[p]=\\lbrace 1,2,\\dots , p\\rbrace .$"],["Five polynomial-time reductions","For two sets ${\\cal G}$ and ${\\cal H}$ of graphs, we say that $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}$ to ${\\cal H}$ if for all $G\\in {\\cal G}$ and its vertex-weights $a^*_{},d^*_{}$ , we can either decide in polynomial time in $|V(G)|$ whether $(G,a^*_{},d^*_{})$ is a YES-instance or NO-instance, or construct $H \\in {\\cal H}$ with vertex-weights $a^+_{},d^+_{}$ such that $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if $(H,a^+_{},d^+_{})$ is a YES-instance of $(1,1)$ -Cluster Editing.","Note that as the reduction is a polynomial-time algorithm, the size of $H$ is also bounded above by a polynomial in the size of $G$ .","When the vertex-weights are clear from the context, we will often say that $G$ is an instance of the problem (i.e., of $(1,1)$ -Cluster Editing).","We say that a graph $G$ is triangle-free if it does not contain a cycle $C_3$ as a subgraph and is quadrilateral-free if it does not contain a cycle $C_4$ as a subgraph.","We will now prove a number of lemmas which will culminate in a final polynomial-time algorithm for $(1,1)$ -Cluster Editing.","Let ${\\cal G}_1$ denote all graphs and let ${\\cal G}_2$ denote all triangle-free graphs of maximum degree at most 3.","Lemma 1 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_1$ to ${\\cal G}_2$ .","Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing.","Suppose that $G$ has a triangle $T\\cong C_3$ .","We can delete at most one edge from $T$ and add at most one edge to $T$ (after an edge was deleted).","Thus, if $(G,a^*_{},d^*_{})$ has a solution, it must contain $T$ as (part of) a clique.","In the algorithm below to shorten its description, once a deletable edge $uv$ is deleted from $G$ , we immediately set $d^*_{}(u)=d^*_{}(v)=0$ and if an addable edge is added to $G$ , we immediately set $a^*_{}(u)=a^*_{}(v)=0.$ Let $Q:=T$ and execute the following loop.","While there is a vertex outside $Q$ adjacent to a vertex in $Q$ , for every such vertex $v$ do the following cases in turn.","If there is only one edge $e$ between $v$ and $Q$ then if $e$ is an deletable edge, then $e$ is deleted and the loop is continued.","If there are exactly $|V(Q)|-1$ edges between $v$ and $V(Q)$ in $G$ and the edge in $\\overline{G}$ between $v$ and $V(Q)$ is addable, then add the addable edge to $G$ and continue the loop.","If there are $|V(Q)|$ edges between $v$ and $V(Q)$ in $G$ , then add vertex $v$ and the edges between $v$ and $Q$ to $Q$ .","Otherwise, $v$ has between two and $|V(Q)|-2$ edges to $Q$ and we conclude that $(G,a^*_{},d^*_{})$ is a NO-instance and stop the loop.","If the loop stops without concluding that $(G,a^*_{},d^*_{})$ is a NO-instance, then delete $Q$ from $G.$ We can continue as above and either eliminate all triangles from $G$ or conclude that $(G,a^*_{},d^*_{})$ is a NO-instance.","In the former case, we obtain a triangle-free instance $(G,a^*_{},d^*_{})$ which is a YES-instance if and only if the initial instance is.","Note that it takes a polynomial time to compute the triangle-free instance or conclude that the initial instance is a NO-instance.","Now suppose that $(G,a^*_{},d^*_{})$ is a triangle-free instance and $G$ has a vertex $v$ of degree at least four.","Since we can delete at most one edge incident to $v$ and $G$ is triangle-free, we cannot make a clique including $v$ without adding at least two edges incident to a neighbour of $v$ .","Thus, $(G,a^*_{},d^*_{})$ is a NO-instance.","It is not hard to verify that the algorithm runs in polynomial time.","Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing and let $Q \\subseteq V(G)$ be arbitrary.","Consider the graph $G-Q$ , where we delete $Q$ (and all edges incident with $Q$ ).","We define $d^*_{Q}$ such that $d^*_{Q}(v) = d^*_{}(v) - |N(v) \\cap Q|$ for all $v \\in V(G-Q)$ .","We define $a^*_{Q}$ to be the function $a^*_{}$ restricted to $V(G-Q)$ .","Note that by definition of vertex-weights, if $d^*_{Q}(v) <0$ for any $v \\in V(G-Q)$ then $(G-Q,a^*_{Q},d^*_{Q})$ is a NO-instance.","Similarly, if $F\\subseteq E(G)$ , then $d^*_{F}(v) = d^*_{}(v) - |E(v) \\cap F|$ , where $E(v)$ is the set of edges incident to $v$ .","Also, $a^*_{F}=a^*_{}.$ Again, if $d^*_{F}(v) <0$ for any $v \\in V(G)$ then $(G-F,a^*_{F},d^*_{F})$ is a NO-instance.","We say that a vertex set $X$ of $G$ be R-deletable if $X$ has constant size and every solution to $(G,a^*,d^*)$ (if any) puts all edges between $X$ and $V(G)-X$ in $D.$ Note that vacuously this implies that if $G$ is a NO-instance then any set $X$ of constant size is an R-deletable set.","Lemma 2 Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing.","If a vertex set $X$ of $G$ is R-deletable, then we can either solve $(G,a^*_{},d^*_{})$ is constant time or reduce the instance to $G-X$ .","Note that if $(G,a^*_{},d^*_{})$ is a NO-instance, then any arbitrary vertex set $X$ of constant size is R-deletable and therefore satisfies the above.","If $X=V(G)$ then we can solve the problem in constant time, so assume that $X \\ne V(G).$ Let $F$ be all edges between $X$ and $V(D)-X$ and let $G^{\\prime } =G-F.$ If any $d^*_{F}$ value drops below 0 we have a NO-instance by definition of vertex-weights.","Otherwise, $G$ is a YES-instance if and only if $G^{\\prime }$ is a YES-instance.","And as $X$ has constant size we can determine if there is a solution to $G^{\\prime }[X]$ in constant time.","If there is no solution to $G^{\\prime }[X]$ then $G$ is a NO-instance and if there is a solution to $G^{\\prime }[X]$ , then we can reduce $G$ by deleting $X$ , as $G$ is a YES-instance if and only if $G^{\\prime }-X=G-X$ is a YES-instance.","Note that the above also holds when $(G,a^*_{},d^*_{})$ is a NO-instance as then either $G[X]$ is a NO-instance (after adjustment of $d^*$ ), which can be decided in constant time, or $G-X$ is a NO-instance (after adjustment of $d^*$ ).","Lemma 3 Let $G$ be a triangle-free graph with maximum degree at most 3.","Then $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if there is a matching $D$ of deletable edges from $G$ such that every connected component $C$ of $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ or $C_4$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ and if $C\\cong C_4$ then the two chords of $C$ , which lie in $\\overline{G}$ , are addable to $G$ .","Let $(G,a^*_{},d^*_{})$ be a YES-instance of $(1,1)$ -Cluster Editing and $G-D+A$ is a matching solution.","Since $G$ is a triangle-free graph with maximum degree at most 3, each connected component of $G-D+A$ is isomorphic to $K_1$ or $K_2$ or $K_3$ or $K_4$ .","Now the lemma follows from the fact that $G$ is a triangle-free graph.","Lemma 4 Let $G$ be a triangle-free graph with maximum degree at most 3 containing a $K_{2,3}$ as a subgraph.","Then $(G,a^*_{},d^*_{})$ is a NO-instance of $(1,1)$ -Cluster Editing for every $(a^*,d^*)$ .","Let $G$ be a triangle-free graph with maximum degree at most 3 containing a $K_{2,3}$ subgraph, $R$ , where $V(R)=\\lbrace x_1,x_2,y_1,y_2,y_3\\rbrace $ and $E(R)=\\lbrace x_i y_j \\; | \\; i=1,2 \\mbox{ and } j=1,2,3\\rbrace $ .","Assume that $(G,a^*_{},d^*_{})$ is a YES-instance with solution $G-D+A$ .","By Lemma REF , no vertex in $G-D$ has degree 3.","Thus, we must have $x_1 y_i \\in D$ and $x_2 y_j \\in D$ for some $i \\ne j$ (as $D$ is a matching) and these are the only edges in $E(R)$ that belong to $D$ .","However this implies that $R-D$ is a $P_5$ , which is a contradiction by Lemma REF .","So, $(G,a^*_{},d^*_{})$ is a NO-instance.","Let ${\\cal G}_3$ denote all triangle-free graphs of maximum degree at most 3 which contain no 4-cycle, all of whose vertices have degree 3.","Lemma 5 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_2$ to ${\\cal G}_3$ in polynomial time.","Let $G \\in {\\cal G}_2$ be arbitrary with vertex-weights $(a^*,d^*)$ .","If there is no 4-cycle in $G$ where all vertices have degree three then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be a 4-cycle in $G$ with $d(v_i)=3$ for all $i=1,2,3,4$ .","Let $w_i$ be the neighbour of $v_i$ in $G$ which does not lie on $C$ .","Let $V_8=\\lbrace v_1,v_2,v_3,v_4,w_1,w_2,w_3,w_4\\rbrace $ .","We will show that $|V_8|=8$ (or we have a NO-instance), so assume for the sake of contradiction that this is not the case.","As $G$ is triangle-free, we must have $w_1=w_3$ or $w_2=w_4$ .","In either case $K_{2,3}$ is a subgraph of $G$ and we have a NO-instance by Lemma REF .","Therefore we may assume that $|V_8|=8$ .","We now prove the following claims.","Claim A: If $(G,a^*_{},d^*_{})$ is a YES-instance and $G-D+A$ is a solution, then at least one of the following options holds.","(a): $v_i w_i \\in D$ for all $i=1,2,3,4$ and no edge on $C$ belongs to $D$ .","(b): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_1 v_2, v_3 v_4 \\in D$ and $v_2 v_3, v_4 v_1 \\notin D$ .","(c): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_2 v_3, v_4 v_1 \\in D$ and $v_1 v_2, v_3 v_4 \\notin D$ .","Furthermore, for every such 4-cycle in $G$ either zero or two edges of the cycle belong to $D$ .","Proof of Claim A: Let $C^{\\prime }$ be any 4-cycle in $G$ .","If one edge $e \\in E(C^{\\prime })$ belongs to $D$ , then at least two edges of $C^{\\prime }$ must belong to $D$ , as otherwise $C^{\\prime } - e$ is a $P_4$ in $G-D$ , which is not part of a 4-cycle in $G-D$ , a contradiction by Lemma REF .","As each vertex is adjacent to at most one edge from $D$ , we note that we cannot have more than two edges from $C^{\\prime }$ in $D$ .","Option (a) now corresponds to the case when no edge from $C$ belongs to $D$ and Options (b) and (c) correspond to the two possible ways that we can add two (non-adjacent) edges of $C$ to $D$ .","This completes the proof of Claim A.","($\\diamond $ ) Claim B: We may assume that $d^*_{}(v_i)=a^*_{}(v_i)=1$ for all $i=1,2,3,4,$ as otherwise we can reduce our instance or solve it in polynomial time.","Proof of Claim B: If $d^*_{}(v_i)=0$ for some $i\\in [4]$ , then none of the options in Claim A is possible, so we have a NO-instance.","So we may assume that $d^*_{}(v_i)=1$ for all $i\\in [4]$ .","For the sake of contradiction, assume that $a^*_{}(v_1)=0$ and $(G,a^*_{},d^*_{})$ is a YES-instance with a solution $G-D+A$ .","In this case Option (a) of Claim A cannot hold as we cannot add the edge $v_1 v_3$ to $C$ .","Option (b) of Claim A cannot hold as we cannot add the edge $w_1 v_4$ to $G-D$ .","Also, Option (c) of Claim A cannot hold as we cannot add the edge $w_1 v_2$ to $G-D$ .","So in this case we have a NO-instance.","Analogously, we may assume that $a^*_{}(v_i)=1$ for all $i\\in [4]$ , which completes the proof of Claim B.","($\\diamond $ ) Let $e_1 = w_1 w_2$ , $e_2 = w_2 w_3$ , $e_3 = w_3 w_4$ and $e_4 = w_4 w_1$ be edges that may or may not exist in $G$ .","We now prove the following claims.","Claim C: We may assume that $e_1 \\notin E(G)$ and that Option (c) in Claim A cannot occur in any solution of $(G,a^*,d^*)$ , as otherwise we can reduce our instance or solve it in polynomial time.","Furthermore, if we know that there is no solution of $(G,a^*,d^*)$ for which Option (b) of Claim A occurs, then we can reduce or solve our instance in polynomial time.","Also, if we know that there is no solution of $(G,a^*,d^*)$ for which Option (a) of Claim A occurs, then we can reduce or solve our instance in polynomial time.","Proof of Claim C: We may assume that $G$ is connected; otherwise consider the connected component containing $V_8.$ By renaming vertices we may assume that $e_1 \\notin E(G)$ , unless $e_i \\in E(G)$ for all $i \\in [4]$ .","So assume that $e_i \\in E(G)$ for all $i \\in [4]$ , which, as $G$ is connected and of maximum degree three, implies that $V(G)=V_8$ .","In this case we can determine if we have a YES-instance (which is the case if and only if $d^*_{}(v)=a^*_{}(v)=1$ for all $v \\in V(G)$ ) or NO-instance in constant time, and so we would be done.","So we may indeed assume that $e_1 \\notin E(G)$ .","If Option (c) in Claim A occurs, then $w_1 v_1 v_2 w_2$ is a $P_4$ in $G-D$ where $w_1w_2 \\notin E(G)$ , which is not possible by Lemma REF , so Option (c) in Claim A cannot occur.","By Lemma REF , we may assume that, in the rest of the proof, $(G,a^*,d^*)$ is a YES-instance.","As we may assume that Option (c) in Claim A cannot occur, either Option (a) or Option (b) of Claim A must occur for every solution of $(G,a^*_{},d^*_{})$ .","First assume that we can show that Option (b) does not occur, and therefore Option (a) must occur in all solutions.","In this case, $w_iv_i \\in D$ for all $i \\in [4]$ and therefore $V(C)$ is a R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .","Now assume that Option (a) cannot occur which implies that Option (b) must occur in all solutions.","This implies that all edges between $V_8$ and $V(G) \\setminus V_8$ must belong to $D$ .","Thus, either $V(G)=V_8$ in which case we are done or $V_8$ is an R-deletable set, so we can reduce $G$ to $G-V_8$ by Lemma REF .","This completes the proof of Claim C.($\\diamond $ ) Figure: Graph (a) is an illustration of Claim D in the proof of Lemma , while Graph (b) is an illustration of Claim F.Claim D: We may assume that $e_2,e_4 \\in E(G)$ and $e_1,e_3 \\notin E(G),$ as otherwise we can reduce our instance or solve it in polynomial time.","(See Figure REF (a) for an illustration.)","We may also assume that $d^*_{}(w_i)=a^*_{}(w_i)=1$ for all $i \\in \\lbrace 1,2,3,4\\rbrace $ .","Proof of Claim D: By Claim C we may assume that $e_1 \\notin E(G)$ .","For the sake of contradiction assume that $e_2 \\notin E(G)$ .","By Claim C we note that part (c) in Claim A cannot occur.","Analogously, if Option (b) in Claim A occurs, then $w_2 v_2 v_3 w_3$ is a $P_4$ in $G-D$ where $w_2w_3 \\notin E(G)$ , which is not possible.","So, Option (b) in Claim A can also not occur and we are done by Claim C. Hence, we may assume that $e_2 \\in E(G)$ .","The fact that $e_4 \\in E(G)$ can be proved analogously, so we may assume that $e_2,e_4 \\in E(G)$ .","For the sake of contradiction assume that $e_3 \\in E(G)$ .","In this case Option (a) in Claim A cannot occur, as $w_2 w_3 w_4 w_1$ would then be a $P_4$ in $G-D$ where $w_1 w_2 \\notin E(G-D)$ , a contradiction.","So, again we are done by Claim C. So we may assume that $e_3 \\notin E(G)$ , as otherwise we either reduce our instance or solve it in constant time.","This completes the proof of the first part of Claim D. In order to prove the second part we consider the following two cases.","First assume that $d^*_{}(w_i)=0$ for some $i \\in [4]$ .","Then Option (a) in Claim A cannot occur, so by Claim C, Option (b) must occur.","As above we now see that we either can solve our instance in constant time or $V_8$ is a R-deletable set, so we can reduce $G$ to $G-V_8$ by Lemma REF .","Hence, we may assume that $d^*_{}(w_i)=1$ for all $i \\in [4]$ .","Now assume that $a^*_{}(w_i)=0$ for some $i \\in [4]$ .","In this case Option (b) in Claim A cannot occur, so by Claim C, Option (a) must occur.","In this case $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .","So we may assume that $a^*_{}(w_i)=1$ for all $i \\in [4]$ .","This completes the proof of Claim D.($\\diamond $ ) Definition: If $d(w_1)=3$ then let $z_1$ be defined such that $N(w_1)=\\lbrace z_1,v_1,w_4\\rbrace $ .","If $d(w_2)=3$ then let $z_2$ be defined such that $N(w_2)=\\lbrace z_2,v_2,w_3\\rbrace $ .","If $d(w_3)=3$ then let $z_3$ be defined such that $N(w_3)=\\lbrace z_3,v_3,w_2\\rbrace $ .","If $d(w_4)=3$ then let $z_4$ be defined such that $N(w_4)=\\lbrace z_4,v_4,w_1\\rbrace $ .","Note that the $z_i$ 's may not be distinct and we may have $z_i=w_j$ for some $i$ and $j$ .","But, by Claim D we have $z_1 \\ne w_2$ , $z_2 \\ne w_1$ , $z_3 \\ne w_4$ and $z_4 \\ne w_3$ .","Claim E: We may assume that if $z_i$ exists then $z_i \\notin V_8$ for all $i=1,2,3,4$ and if $z_i$ and $z_j$ both exist with $i \\ne j$ , then $z_i \\ne z_j$ .","Otherwise, we can reduce our instance or solve it in polynomial time.","Proof of Claim E: For the sake of contradiction assume that $z_1$ exists and $z_1 \\in V_8$ .","By definition there exists a $j \\in [4]$ such that $z_i = w_j$ .","By Claim D, we must have $j=3$ (and therefore $z_1=w_3$ and $z_3=w_1$ ).","First consider the case when $z_2$ exists and $z_2=w_4$ .","In this case, as $G$ is connected, we have $V(G)=V_8$ and we can determine if $(G, a^*,d^*)$ is a YES-instance or a NO-instance in constant time.","So we may assume that $w_2w_4 \\notin E(G)$ .","Option (a) in Claim A cannot occur, as $w_4 w_1 w_3 w_2$ would then be a $P_4$ in $G-D$ where $w_2 w_4 \\notin E(G-D)$ , a contradiction.","So, by Claim C, we are done in this case.","Analogously we can show that $z_i \\notin V_8$ for all $i=1,2,3,4$ (or we can reduce our instance or solve it in polynomial time).","Now for the sake of contradiction assume that $z_i$ and $z_j$ both exist and $i \\ne j$ but $z_i=z_j$ .","Option (b) in Claim A cannot occur, as in this case $w_i z_i \\in D$ and $w_j z_j \\in D$ , but $z_i$ can be incident with at most one edge from $D$ .","So, we are done by Claim C. This completes the proof of Claim E.($\\diamond $ ) Figure: The reduction in Claim G of Lemma .Claim F: If $z_1$ and $z_4$ both exist then $z_1 z_4 \\in E(G)$ .","Otherwise, we can reduce our instance or solve it in polynomial time.","Analogously, we may assume that if $z_2$ and $z_3$ both exist then $z_2 z_3 \\in E(G)$ .","Proof of Claim F: For the sake of contradiction assume that $z_1$ and $z_4$ both exist, but $z_1 z_4 \\notin E(G)$ (see Figure REF (b)).","By Claim C, either Option (a) or Option (b) in Claim A must occur.","Assume that Option (a) in Claim A occurs.","This implies that $v_i w_i \\in D$ for all $i \\in [4]$ .","Hence, $z_1 w_1 w_4 z_4$ is a $P_4$ in $G-D$ where $z_1 z_4 \\notin E(G)$ , a contradiction.","So Option (b) in Claim A must occur.","We are now done by Claim C. This completes the proof of Claim F.($\\diamond $ ) Claim G: If $z_1$ , $z_3$ and $z_4$ all exist, then we may assume that $z_3 z_4 z_1$ is a path in $G$ .","Otherwise, we can reduce our instance or solve it in polynomial time.","Analogously if $z_1$ , $z_2$ and $z_4$ all exist, then we may assume that $z_4 z_1 z_2$ is a path in $G$ .","And if $z_2$ , $z_3$ and $z_4$ all exist, then that $z_2 z_3 z_4$ is a path in $G$ .","And finally if $z_1$ , $z_2$ and $z_3$ all exist, then that $z_1 z_2 z_3$ is a path in $G$ .","Proof of Claim G: Assume that $z_1$ , $z_3$ and $z_4$ exist.","By Claim F we may assume that $z_1 z_4 \\in E(G)$ .","For the sake of contradiction assume that $z_3 z_4 \\notin E(G)$ .","Let $G^+$ be obtained from $G$ by deleting the vertices $v_1,v_2,v_3,v_4$ and adding the edges $w_1 w_2$ and $w_3 w_4$ (the vertex weights $d^*_{}$ and $a^*_{}$ remain unchanged).","See Figure REF for an illustration.","We will now show that $G^+$ is a YES-instance if and only if $G$ is a YES-instance.","First assume that $(G,a^*,d^*)$ is a YES-instance.","By Claim C we note that Option (a) or Option (b) in Claim A must occur.","If Option (a) occurs then $v_i w_i \\in D$ for all $i \\in [4]$ in $G$ .","In $G^+$ we use exactly the same $D$ , except we remove $v_1w_1, v_2w_2, v_3w_3, v_4w_4$ and add $w_1w_2$ and $w_3 w_4$ .","This way we get exactly the same components in $G$ and $G^+$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G$ .","Now assume that Option (b) occurs in Claim A.","In this case we use the same $D$ in $G^+$ and in $G$ , except we remove the edges $v_1 v_2$ and $v_3 v_4$ .","Now we get exactly the same components in $G$ and $G^+$ , except the 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ have been replaced by $w_1 w_2 w_3 w_4 w_1$ in $G^+$ .","So in all cases $G^+$ is a YES-instance.","Conversely assume that $G^+$ is a YES-instance.","We will now show that $G$ is a YES-instance.","Let $D^+$ be the matching in $G^+$ such that all components in $G^+-D^+$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).","Let $C^+$ be the 4-cycle $w_1 w_2 w_3 w_4 w_1$ in $G^+$ .","As in the proof of Claim A we note that $D^+$ contains no edge from $C^+$ or it contains exactly two edges from $C^+$ , which are non-adjacent.","Note that $w_2w_3, w_4w_1 \\in D^+$ is not possible as the path $z_4 w_4 w_3 z_3$ would be a $P_4$ in $G^+ - D^+$ and $z_3 z_4 \\notin E(G)$ .","So either $w_1w_2, w_3w_4 \\in D^+$ or no edge from $C^+$ is in $D^+$ .","First assume that $w_1w_2, w_3w_4 \\in D^+$ .","Then $z_1 w_1 w_4 z_4 z_1$ is a $C_4$ in $G^+ - D^+$ and $w_2 w_3 z_3$ is a $P_2$ in $G^+ - D^+$ (which may be part of a $C_4$ if $z_2$ exists).","In this case let $D = D^+ \\cup \\lbrace v_1 w_1, v_2w_2, v_3w_3, v_4w_4\\rbrace \\setminus \\lbrace w_1w_2, w_3w_4\\rbrace $ and note that we obtain exactly the same components in $G$ and $G^+$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G$ .","So we now consider the case when no edge from $C^+$ is in $D^+$ .","In this case $w_1 z_1, w_3 z_3, w_4 z_4 \\in D^+$ .","Let $D = D^+ \\cup \\lbrace v_1 v_2, v_3 v_4\\rbrace $ and note that we obtain exactly the same components in $G$ and $G^+$ , except the 4-cycle $w_1 w_2 w_3 w_4 w_1$ in $G^+$ has been replaced by the 4-cycles $w_1 v_1 v_4 w_4 w_1$ and $w_2 v_2 v_3 w_3 w_2$ in $G$ .","We have now proven that $G^+$ is a YES-instance if and only if $G$ is a YES-instance.","This completes the proof of Claim G.($\\diamond $ ) Claim H: We may assume that at most one of $z_1$ and $z_4$ exist.","Otherwise, we can reduce our instance or solve it in polynomial time.","Similarly, at most one of $z_2$ and $z_3$ exist.","Proof of Claim H: Assume that both $z_1$ and $z_4$ exist.","We first consider the case when both $z_2$ and $z_3$ also exist.","By Claim G, we have that $z_1 z_2 z_3 z_4 z_1$ is a 4-cycle in $G$ and therefore $G$ has 12 vertices, and we can determine if it is a YES-instance or a NO-instance in constant time.","So we may assume that either none of $z_2$ and $z_3$ exist or exactly one of them exist.","First assume that $z_3$ exists, but $z_2$ does not exist.","By Claim G, $z_1 z_4, z_3 z_4 \\in E(G)$ .","Let $V_{11}=V_8 \\cup \\lbrace z_1,z_3,z_4\\rbrace $ .","If $V(G)=V_{11}$ then we can solve our instance in constant time so assume that this is not the case.","We consider the cases when there exists a $w \\in V(G) \\setminus V_{11}$ with $\\lbrace z_1,z_3\\rbrace \\subseteq N(w)$ and when no such $w$ exists.","If such a $w$ exists then Option (a) in Claim A cannot occur, as if it did both the edges $wz_1$ and $wz_3$ would need to be in $D$ (as $z_1 w_1 w_4 z_4 z_1$ would be a $C_4$ in $G-D$ and $w_2 w_3 z_3$ would be a $P_3$ in $G-D$ ), a contradiction.","So, we are done by Claim C. We may therefore now assume that no such $w$ exists.","As $G$ is connected and $V(G) \\ne V_{11}$ we must have a vertex $r \\in V(G) \\setminus V_{11}$ which is adjacent to either $z_1$ or to $z_3$ .","Now Option (b) of Claim A cannot occur, as if it did then the edges $w_1z_1$ , $w_3z_3$ and $w_4z_4$ would need to be in $D$ , which implies that either $r z_1 z_4 z_3$ or $z_1 z_4 z_3 r$ is a $P_4$ in $G-D$ where the end-points of the path are not adjacent in $G$ (as no $w$ exists in $V(G) \\setminus V_{11}$ that is adjacent to both $z_1$ and $z_3$ ).","So, again we are done by Claim C. This completes all cases and implies that we can reduce when $z_3$ exists, but $z_2$ does not exist.","Figure: The graph when z 2 z_2 and z 3 z_3 do not exist.Analogously we can also reduce if $z_2$ exists, but $z_3$ does not exist.","So the last case to consider is when neither $z_2$ nor $z_3$ exist.","In this case the graph looks like in Figure REF .","If we had considered the cycle $w_1 v_1 v_4 w_4 w_1$ instead of $v_1 v_2 v_3 v_4 v_1$ , then we would be in the case where $G$ has 10 vertices (when $d(z_1)=d(z_4)=2$ ) or we would have been in a case where at least three $z$ -vertices would exist (namely $\\lbrace w_2,w_3\\rbrace \\cup {N(z_1)\\cup N(z_4)}\\setminus \\lbrace z_1,w_1,z_4,w_4\\rbrace $ ), which means we can reduce as seen above.","This completes the proof of Claim H.($\\diamond $ ) We now return to the proof of Lemma REF .","By Claim H we may assume that at most one of $z_1$ and $z_4$ exist and at most one of $z_2$ and $z_3$ exist.","If no $z_i$ exist then $G$ has 8 vertices, so we can solve our problem in constant time.","Therefore we may, without loss of generality assume that $z_1$ exists.","If $z_2$ exists then let $z=z_2$ and if $z_3$ exists then let $z=z_3$ and if neither $z_2$ or $z_3$ exist then let $z$ be undefined.","We now create $G^*$ from $G$ by removing the vertices $w_1, w_2, w_3, w_4$ and adding the edge $v_1 z_1$ and if $z$ exists also the edge $v_3 z$ .","We will show that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.","First assume that $G$ is a YES-instance.","By Claim C we note that Option (a) or Option (b) in Claim A must occur.","If Option (a) occurs then $v_i w_i \\in D$ for all $i \\in [4]$ in $G$ .","Let $D^* = D \\cup \\lbrace v_1 v_2, v_3 v_4\\rbrace \\setminus \\lbrace v_1w_1, v_2w_2, v_3w_3, v_4w_4 \\rbrace $ .","This way we get exactly the same components in $G$ and $G^*$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ , the $P_3=z_1w_1w_4$ and the $P_2$ or $P_3$ on the vertex set $\\lbrace w_2, w_3\\rbrace $ or $\\lbrace w_2, w_3, z\\rbrace $ in $G$ get replaced by the $P_3=z_1 v_1 v_4$ and either the $P_2=v_2 v_3$ (if $z$ does not exist) or the $P_3=v_2 v_3 z$ in $G^*$ .","Now assume that Option (b) occurs in Claim A.","In this case $z_1 w_1 \\in D$ and if $z$ exists then $w_2 z$ or $w_3z$ is in $D$ .","Also, we let $D^* = D - \\lbrace v_1 v_2, v_3 v_4\\rbrace $ , where we also replace $z_1w_1$ by $z_1v_1$ and if $z$ exists then the edge $w_2 z$ or $w_3z$ gets replaced by $v_3z$ .","Now we get exactly the same components in $G-D$ and $G^* - D^*$ , except the 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ have been replaced by $v_1 v_2 v_3 v_4 v_1$ in $G^*$ .","So in all cases $G^*$ is a YES-instance.","Conversely assume that $G^*$ is a YES-instance.","We will now show that $G$ is a YES-instance.","Let $D^*$ be the matching in $G^*$ such that all components in $G^* - D^*$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).","As in the proof of Claim A we note that $D^*$ contains no edge from $C$ or it contains exactly two edges from $C$ , which are non-adjacent.","First assume that $D^*$ contains no edge from $C$ .","Then $z_1 v_1 \\in D^*$ and if $z$ exists then $z v_3 \\in D$ .","In this case, let $D$ be obtained from $D^* + \\lbrace v_1v_2, v_3v_4,z_1 w_1\\rbrace - \\lbrace z_1 v_1\\rbrace $ by replacing $v_3 z$ by the edge from $\\lbrace w_2,w_3\\rbrace $ to $z$ if $z$ exists.","Again, we obtain exactly the same components in $G-D$ and $G^* - D^*$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G^*$ has been replaced by the two 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ .","So we now consider the case when $D^*$ contains exactly two edges from $C$ .","In this case $G^* - D^*$ contains a $P_3$ containing the edge $z_1 v_1$ and either the edge $v_1 v_2$ or $v_1 v_4$ .","Similarly, if $z$ exists, then $G^* - D^*$ contains a $P_3$ containing the edge $z v_3$ and either the edge $v_3 v_4$ or $v_3 v_2$ .","In this case let $D$ be obtained from $D^*$ by deleting these two edges on $C$ that belong to $D^*$ and adding the edges $\\lbrace v_1w_1, v_2w_2, v_3w_3, v_4w_4 \\rbrace $ instead.","This way we obtain exactly the same components in $G-D$ and $G^* - D^*$ , except for the following: The $P_3$ in $G^* - D^*$ containing $v_1$ now becomes the $P_2$ $z_1 w_1 w_4$ and the $P_3$ containing $z$ (if it exists) now becomes a $P_3$ containing the edge $w_2 w_3$ as well as the vertex $z$ .","Furthermore $G-D$ will contain the 4-cycle $C$ , which is not in $G^* - D^*$ .","This proves that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.","Let ${\\cal G}_4$ denote all triangle-free graphs of maximum degree at most 3 which contain no 4-cycle, with at least three vertices of degree three.","Lemma 6 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_3$ to ${\\cal G}_4$ in polynomial time.","Let $G \\in {\\cal G}_3$ be arbitrary with vertex-weights $(a^*,d^*)$ .","If there is no 4-cycle in $G$ where three of the vertices have degree three then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be a 4-cycle in $G$ with $d(v_i)=3$ for all $i=1,2,3$ and $d(v_4)=2$ .","Let $w_i$ be the neighbour of $v_i$ in $G$ which does not lie on $C$ for $i=1,2,3$ .","Let $V_7=\\lbrace v_1,v_2,v_3,v_4,w_1,w_2,w_3\\rbrace $ .","We will show that $|V_7|=7$ (or we have a NO-instance), so assume for the sake of contradiction that this is not the case.","As $G$ is triangle-free, we must have $w_1=w_3$ , which implies that $K_{2,3}$ is a subgraph of $G$ and we have a NO-instance by Lemma REF .","Therefore we may assume that $|V_7|=7$ .","The following claim can be proved analogously to Claim A in Lemma REF and thus its proof is omitted.","Claim A: If $(G,a^*_{},d^*_{})$ is a YES-instance and $G-D+A$ is a solution.","Then one of the following options hold.","(a): $v_i w_i \\in D$ for all $i=1,2,3$ and no edge on $C$ belongs to $D$ .","(b): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_1 v_2, v_3 v_4 \\in D$ and $v_2 v_3, v_4 v_1 \\notin D$ .","(c): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_2 v_3, v_4 v_1 \\in D$ and $v_1 v_2, v_3 v_4 \\notin D$ .","Furthermore, for every such 4-cycle in $G$ either zero or two edges of the cycle belong to $D$ .","Claim B: We may assume that $d^*_{}(v_i)=1$ for all $i=1,2,3,4$ .","Otherwise, we have a NO-instance.","Proof of Claim B: If $d^*_{}(v_i)=0$ for some $i\\in [4]$ , then none of the options in Claim A is possible, so we have a NO-instance.","This proves Claim B.","($\\diamond $ ) Claim C: If Option (b) in Claim A occurs, then we must have $w_2 w_3 \\in E(G)$ .","If Option (c) occurs in Claim A, then we must have $w_1 w_2 \\in E(G)$ .","Proof of Claim C: For the sake of contradiction, assume that Option (b) occurs in Claim A, but $w_2 w_3 \\notin E(G)$ .","In this case $w_2 v_2 v_3 w_3$ is a $P_4$ in $G-D$ where $w_2 w_3 \\notin E(G)$ , a contradiction.","The fact that if Option (c) occurs in Claim A, then we must have $w_1 w_2 \\in E(G)$ is proved analogously.","($\\diamond $ ) Claim D: We may assume that exactly one of the edges $w_1w_2$ and $w_2w_3$ belongs to $G$ .","Proof of Claim D: If $w_1w_2 \\notin E(G)$ and $w_2w_3 \\notin E(G)$ then, by Claim C, neither Option (b) nor Option (c) occurs in Claim A.","Therefore Option (a) must occur in Claim A.","This implies that $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .","We may therefore assume that at least one of $w_1w_2$ and $w_2w_3$ belong to $G$ .","Assume that $w_1w_2 \\in E(G)$ and $w_2w_3 \\in E(G)$ .","As $G \\in {\\cal G}_3$ we note that the two 4-cycles $w_1 v_1 v_2 w_2 w_1$ and $w_2 v_2 v_3 w_3 w_2$ both must contain a degree-2 vertex.","This implies that $d(w_1)=d(w_3)=2$ and $V(G)=V_7$ .","We can therefore solve the problem in constant time.","We may therefore assume that one of the edges $w_1w_2$ and $w_2w_3$ do not belong to $G$ .","This proves Claim D.($\\diamond $ ) By Claim D, we may without loss of generality assume that $w_1w_2 \\in E(G)$ and $w_2w_3 \\notin E(G)$ .","Claim E: Option (b) in Claim A cannot occur.","Furthermore, we may assume that $d(w_1) \\le 2$ or $d(w_2) \\le 2$ .","We may also assume that $d(w_3) \\le 2$ as otherwise we can either solve our problem in polynomial time or reduce the instance.","Proof of Claim E: Since $w_2w_3 \\notin E(G)$ by Claim C, Option (b) in Claim A cannot occur.","Note that $d(w_1)=d(w_2)=3$ is not possible as $G \\in {\\cal G}_3$ and $d(v_1)=d(v_2)=3$ .","For the sake of contradiction assume that $d(w_3)=3$ and that $N(w_3)=\\lbrace v_3,a,b\\rbrace $ .","Recall that Option (b) in Claim A cannot occur.","Assume that Option (c) in Claim A occurs.","This implies that either $v_4 v_3 w_3 a$ or $v_4 v_3 w_3 b$ is a $P_4$ in $G-D$ where the end-points are not adjacent in $G$ , a contradiction.","So Option (c) in Claim A cannot occur, which implies that Option (a) in Claim A must occur.","This implies that $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .","This proves Claim E.($\\diamond $ ) We now return to the proof of Lemma REF .","If $d(w_1)=3$ then let $s=1$ and otherwise let $s=2$ .","By Claim E we note that $d(w_{3-s})=2$ .","Let $G^*$ be obtained from $G-\\lbrace v_1,v_2,v_3,v_4,w_{3-s}\\rbrace $ by adding a vertex $x$ and the edge $w_s w_3$ and $w_s x$ and setting $d^*_{}(x)=0$ and $a^*_{}(x)=1$ .","See Figure REF for an illustration.","We will now show that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.","Figure: An illustration of G * G^* in the proof of Lemma when s=2s=2.First assume that $G$ is a YES-instance.","By Claim E we note that Option (a) or Option (c) in Claim A must occur.","If Option (a) occurs then $v_i w_i \\in D$ for all $i \\in [3]$ in $G$ .","Let $D^* = D \\cup \\lbrace w_s w_3 \\rbrace - \\lbrace v_1 w_1, v_2 w_2, v_3 w_3\\rbrace $ .","Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G-D$ and the component containing $w_1 w_2$ have been replaced by a component containing $x w_s$ in $G^* - D^*$ .","However the component containing $w_1 w_2$ in $G-D$ is isomorphic with the component containing $x w_s$ in $G^*-D^*$ .","So we get a solution for $G^*$ .","See Figure REF for an illustration.","Figure: An illustration of G-DG-D and G * -D * G^* - D^* when part (a) of Claim A occurs.If Option (c) occurs then $v_2 v_3, v_4 v_1 \\in D$ .","Let $D^* = D - \\lbrace v_2 v_3, v_4 v_1\\rbrace $ .","Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 w_1 w_2 v_2 v_1$ and the path $v_4 v_3 w_3$ in $G-D$ have been replaced by the path $x w_s w_3$ in $G^* - D^*$ .","So we get a solution for $G^*$ .","See Figure REF for an illustration.","Figure: An illustration of G-DG-D and G * -D * G^* - D^* when part (b) of Claim A occurs.Conversely assume that $G^*$ is a YES-instance.","We will now show that $G$ is a YES-instance.","Let $D^*$ be the matching in $G^*$ such that all components in $G^*-D^*$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).","As $d^*_{}(x)=0$ in $G^*$ we note that $x w_s \\notin D^*$ .","We now consider the case when $w_s w_3$ belongs to $D^*$ and then the case when it does not.","So first assume that $w_s w_3 \\in D^*$ .","In this case let $D = D^* + \\lbrace v_1w_1, v_2 w_2, v_3 w_3 \\rbrace - \\lbrace w_s w_3\\rbrace $ .","Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 v_4 w_4 w_1 v_1$ is in $G-D$ and the component containing $x w_s$ have been replaced by an isomorphic component containing $w_1 w_2$ in $G^* - D^*$ .","So we get a solution for $G$ (this corresponds to the reverse process to that illustrated in Figure REF ).","Now assume that $w_s w_3 \\notin D^*$ .","This implies that any edges incident with $w_s$ and $w_3$ , different from $w_s w_3$ and $x w_s$ must belong to $D^*$ (due to the path $x w_s w_3$ in $G^* - D^*$ ).","Let $D = D^* + \\lbrace v_2v_3, v_4 v_1 \\rbrace $ .","Now $G-D$ and $G^* - D^*$ have exactly the same components except the path $x w_s w_3$ in $G^* - D^*$ has been replaced by the 4-cycle $v_1 w_1 w_2 v_2 v_1$ and the path $v_4 v_3 w_3$ in $G-D$ .","So we get a solution for $G$ (this corresponds to the reverse process to that illustrated in Figure REF ).","The above reduction completes the proof of the lemma.","Let ${\\cal G}_5$ denote all triangle-free and quadrilateral-free graphs of maximum degree at most 3.","Lemma 7 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_4$ to ${\\cal G}_5$ in polynomial time.","Let $G \\in {\\cal G}_4$ be arbitrary with vertex-weights $(a^*,d^*)$ .","We may assume that $G$ is connected.","If there is no 4-cycle in $G$ then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be any 4-cycle in $G$ .","As $G \\in {\\cal G}_4$ , we know that at most two of the vertices in $C$ have degree three.","If no vertex of $G$ has degree three then $V(G)=V(C)$ and we can solve our problem in constant time.","So we may assume that at least one vertex on $C$ has degree three.","If $d^*_{}(v_i)=0$ for any $i \\in [4]$ then any solution $G-D+A$ to our instance must have all edges between $C$ and $V(G)-V(C)$ in $D$ , which implies that $V(C)$ is an R-deletable set.","So by Lemma REF , we can reduce the instance in this case.","Thus, we may assume that $d^*_{}(v_i)=1$ for all $i \\in [4]$ .","We first consider the case when exactly one vertex on $C$ has degree three.","Without loss of generality assume that $d(v_1)=3$ and $d(v_2)=d(v_3)=d(v_4)=2$ .","Let $e$ be the edge incident with $v_1$ that does not belong to $C$ .","Let $G^*$ be obtained from $G^*=G-\\lbrace v_3,v_4\\rbrace $ by letting $d^*_{}(v_2)=0$ .","Now $G^*$ is a YES-instance if and only if $G$ is a YES-instance, due to the following.","Deleting $e$ from $G$ corresponds to deleting $e$ from $G^*$ and vice versa.","Not deleting $e$ from $G$ corresponds to deleting two edges from $C$ in $G$ which corresponds to not deleting $e$ in $G^*$ .","So $G$ is a YES-instance if and only if $G^*$ is a YES-instance.","We may therefore assume that exactly two vertices on $C$ have degree 3.","Let $x$ and $y$ be the two vertices on $C$ of degree 3.","Let $x^{\\prime }$ be the neighbour of $x$ not on $C$ and let $y^{\\prime }$ be the neighbour of $y$ not on $C$ .","Analogously to Claim A in both Lemma REF and Lemma REF we note that if our instance has a solution and $D$ is the set of edges that are deleted from $G$ in the solution, then one of the following holds.","(a): $x x^{\\prime }, y y^{\\prime } \\in D$ and no edge on $C$ belongs to $D$ .","(b): $x x^{\\prime }, y y^{\\prime } \\notin D$ and $v_1 v_2, v_3 v_4 \\in D$ and $v_2 v_3, v_4 v_1 \\notin D$ .","(c): $x x^{\\prime }, y y^{\\prime } \\notin D$ and $v_2 v_3, v_4 v_1 \\in D$ and $v_1 v_2, v_3 v_4 \\notin D$ .","First consider the case when $x$ and $y$ are adjacent vertices on $C$ .","Then without loss of generality assume that $x=v_1$ and $y=v_4$ .","As $G$ is triangle-free we have $x^{\\prime } \\ne y^{\\prime }$ .","First assume that $x^{\\prime } y^{\\prime } \\in E(G)$ .","Then $V(G)=\\lbrace v_1, v_2, v_3, v_4, x^{\\prime }, y^{\\prime }\\rbrace $ as $G \\in {\\cal G}_4$ , which implies that the cycle $x^{\\prime } x y y^{\\prime } x^{\\prime }$ ($=x^{\\prime } v_1 v_4 y^{\\prime } x^{\\prime }$ ) has at most two vertices of degree three, which are $x$ and $y$ .","So in this case we can solve our problem in constant time.","Therefore, we may assume that $x^{\\prime } y^{\\prime } \\notin E(G)$ .","This also implies that part (b) above is not possible, as if part (b) occurs then $x^{\\prime } v_1 v_4 y^{\\prime }$ is a $P_4$ in $G-D$ where $x^{\\prime } y^{\\prime } \\notin E(G)$ , a contradiction.","So either part (a) or part (c) occurs.","This is illustrated in Figure REF .","Figure: Possibble options for G-DG-D when xx and yy are adjacent.If $x$ and $y$ are not adjacent, then without loss of generality assume that $x=v_1$ and $y=v_3$ .","If $x^{\\prime }=y^{\\prime }$ then $G$ contains a $K_{2,3}$ -subgraph, which implies that our instance is a NO-instance by Lemma REF .","Thus, we may assume that $x^{\\prime } \\ne y^{\\prime }$ .","Now part (a), part (b) and part (c) are illustrated in Figure REF .","Figure: Possibble options for G-DG-D when xx and yy are not adjacent.So whether $x$ and $y$ is adjacent or not, we will in $G-D$ either have both edges $xx^{\\prime }$ and $yy^{\\prime }$ in $D$ or neither in $D$ .","And if neither is in $D$ then each of $x^{\\prime }$ and $y^{\\prime }$ is the endpoint of a $P_3$ containing two vertices from $C$ .","So in both cases we will remove $V(C)$ from $G$ and add the gadget $G(x,y)$ , illustrated in Figure REF , to $G$ instead.","Note that $V(G(x,y))=\\lbrace q_1, x, q_3, q_4, y, q_6 \\rbrace $ and $E(G(x,y))=\\lbrace q_1 x, x q_3, q_3 q_4, q_4 y, y q_5 \\rbrace $ and we add the edges $x x^{\\prime }$ and $y y^{\\prime }$ .","Furthermore $d^*_{}(q_1)=d^*_{}(q_6)=0$ and all other $d^*_{}$ -values and $a^*_{}$ -values are one.","Figure: The gadget G(x,y)G(x,y).It is not difficult to check that if the modified graph, $G^{\\prime }$ , is a YES-instance and $D^{\\prime }$ the matching that gets removed from $G^{\\prime }$ , then we must have one of the two cases illustrated in Figure REF .","Figure: The two options, (I) and (II), for G ' -D ' G^{\\prime }-D^{\\prime }.So $G^{\\prime \\prime }$ has either both edges $xx^{\\prime }$ and $yy^{\\prime }$ in $D^{\\prime }$ or neither in $D^{\\prime }$ .","And if neither are in $D^{\\prime }$ then both $x^{\\prime }$ and $y^{\\prime }$ are endpoints of a $P_3$ containing two vertices from the gadget $G(x,y)$ .","This was exactly the same property we had in $G$ , so $G$ is a YES-instance if and only if $G^{\\prime }$ is.","Furthermore, this reduction deletes a 4-cycle.","So even though the reduction adds an edge (that is, $|E(G^{\\prime })|=|E(G)|+1$ ) the reduction can be carried out at most as many times as there are 4-cycles in $G$ which is at most a polynomial number.","Therefore repeatadly performing the above reduction creates a graph in ${\\cal G}_5$ in polynomial time, as desired."],["Polynomial-time algorithm for triangle-free and quadrilateral-free graphs of maximum degree at most 3","Lemma REF implies the following: Lemma 8 Let $G$ be a triangle-free and quadrilateral-free graph with maximum degree at most 3.","Then $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if there is a matching $D$ of deletable edges from $G$ such that every connected component $C$ of $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .","Let ${\\cal G}_6$ denote all triangle-free and quadrilateral-free graphs of maximum degree at most 3, such that the following holds.","All vertices $v$ of degree at least 2 have $d^*_{}(v)=1$ .","No vertex $v$ of degree 3 is adjacent to a vertex $w$ with $a^*_{}(w)=0$ .","The vertices of degree 3 form an independent set.","Lemma 9 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_5$ to ${\\cal G}_6$ in polynomial time.","Let $G \\in {\\cal G}_5$ be arbitrary with vertex-weights $(a^*,d^*)$ .","Note that if some vertex $v$ has $d(v)=3$ and $d^*_{}(v)=0$ then $(G,a^*_{},d^*_{})$ is a NO-instance, so we may assume that $d^*_{}(v)=1$ for all vertices, $v$ , of degree 3.","If $d(v)=2$ and $d^*_{}(v)=0$ , then $(G,a^*_{},d^*_{})$ is a YES-instance if and only if both edges incident to $v$ have to remain in $G-D$ and therefore form a $P_3.$ Let $N(x)=\\lbrace u,v\\rbrace ,$ and note that $\\lbrace x,u,v\\rbrace $ is an R-deletable set.","So by Lemma REF , we can reduce the instance.","Assume that there exists a $uw \\in E(G)$ with $d(u)=3$ and $a^*_{}(w)=0$ .","Assume that $G$ is a YES-instance and $G-D+A$ is a cluster graph.","If $uw \\notin D$ , then $w$ is the endpoint of a $P_3$ in $G-D$ which is a contradiction as $a^*_{}(w)=0$ .","So, we must have $uw \\in D$ .","Since $uw \\in D$ , $d^*_{}(u)=d^*_{}({w})=1$ or $(G,a^*,d^*)$ is a NO-instance.","Form a new instance with $G^{\\prime }=G- uw$ and $d^*_{}(u)=d^*_{}({w})=0$ .","Then $G$ is a YES-instance if and only if $G^{\\prime }$ is a YES-instance.","This reduction allows us to get rid of all cases where a degree three vertex is adjacent to a vertex with $a^*_{}$ -value zero.","Now assume that there exists a $uw \\in E(G)$ with $d(u)=d(w)=3$ .","Assume that $G$ is a YES-instance and $G-D+A$ is a cluster graph.","If $uw \\notin D$ , then we obtain a $P_4$ in $G-D$ containing the edge $uw$ , a contradiction.","So, we must have $uw \\in D$ .","As $d(u)=d(w)=3$ , we have $d^*(u)=d^*(w)=1$ .","Now we form a new instance with $G^{\\prime }=G- uw$ and $d^*_{}(u)=d^*_{}(v)=0$ as above and note that $G$ is a YES-instance if and only if $G^{\\prime }$ is a YES-instance.","Thus, we may assume that $uw \\notin E(G)$ if $d(u)=d(w)=3$ .","This completes the proof.","In the proof of the next lemma, we will use the well-known result that given a graph $G$ in polynomial time either a perfect matching can be found in $G$ or we can decide that $G$ has no perfect matching.","Lemma 10 Let $G$ be a graph and let $Y$ and $Z$ be disjoint sets of vertices in $G$ .","In polynomial time, we can decide whether $G$ contains a matching covering $Y$ , but not covering $Z$ .","Let $G^{\\prime }=G-Z$ .","Observe that $G$ contains a matching covering $Y$ and not covering $Z$ if and only if $G^{\\prime }$ contains a matching covering $Y$ .","Let $U= V(G^{\\prime })-Y$ and if $G^{\\prime } $ has odd number of vertices then add an extra (isolated) vertex to $G^{\\prime }$ which is not in $Y$ (and therefore will be added to $U$ ).","Finally add edges between all pairs of non-adjacent vertices in $U$ .","Let the resulting graph be $G^{\\prime \\prime }.$ Now we will show that $G^{\\prime \\prime }$ has a perfect matching if and only if there is a matching in $G ^{\\prime }$ covering $Y.$ If $M^{\\prime \\prime }$ is a perfect matching in $G^{\\prime \\prime }$ then by deleting all edge with both end-points in $U$ we get the desired matching in $G^{\\prime }.$ And if $M^{\\prime }$ is the desired matching in $G^{\\prime }$ then adding as many edges as possible with both endpoints in $U$ gives us a perfect matching in $G^{\\prime \\prime }.$ Lemma 11 $(1,1)$ -Cluster Editing can be solved in polynomial time for all graphs in ${\\cal G}_6$ in polynomial time.","Let $G \\in {\\cal G}_6$ be arbitrary with vertex-weights $(a^*,d^*)$ .","Let $X_i = \\lbrace v \\; | \\; d_G(v)=i \\rbrace $ for all $i=0,1,2,3$ .","We may assume that $G$ is connected as otherwise we can just consider each connected component separately (it will never be advantages to add edges between different components).","We may clearly also assume that $G$ has at least two vertices, which implies that $X_0 = \\emptyset $ .","Let $X_2^{\\prime }$ denote all vertices in $X_2$ with an edge to a vertex in $X_3$ and let $X_2^{\\prime \\prime } = X_2 \\setminus X_2^{\\prime }$ .","Let $Z$ contain all vertices, $z$ , in $X_2^{\\prime \\prime }$ , such that $z$ is adjacent to a vertex, $w$ , with $a^*_{}(w)=0$ .","In other words, $Z$ contains all vertices from $X_2^{\\prime \\prime }$ which are neighbours of a vertex which we are not allowed to add edges to.","Define $Y$ as follows: $Y = Z \\cup X_2^{\\prime } \\cup X_3.$ We will show that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, if and only if there exists a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .","This will give us the desired polynomial algorithm by Lemma REF .","First assume that $D$ is a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .","We may assume that $D$ is maximal, as otherwise we just keep adding edges to $D$ (where both end-points have $d^*$ -value 1) until it becomes maximal (which is not necessarily maximum).","Let $G^{\\prime } = G-D$ .","By Lemma REF it suffices to show that every component, $C$ , in $G^{\\prime }$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .","As every vertex in $X_3$ belongs to $Y$ we note that $\\Delta (G^{\\prime }) \\le 2$ .","For the sake of contradiction assume that $uv \\in E(G^{\\prime })$ and $d_{G^{\\prime }}(u)=d_{G^{\\prime }}(v)=2$ .","As all vertices in $X_2^{\\prime }$ will have degree 1 in $G^{\\prime }$ we note that neither $u$ nor $v$ belong to $X_2^{\\prime }$ .","As all vertices in $X_3$ are only adjacent to vertices in $X_2^{\\prime }$ , by the definition of ${\\cal G}_6$ and $X_2^{\\prime }$ , we note that neither $u$ nor $v$ belong to $X_3$ .","So $u$ and $v$ must both belong to $X_2^{\\prime \\prime }$ .","By the definition of ${\\cal G}_6$ we note that $d^*_{}(u)=d^*_{}(v)=1$ .","However, this contradicts the fact that $D$ is maximal, as we could have added the edge $uv$ to $D$ .","So, no $uv \\in E(G^{\\prime })$ has $d_{G^{\\prime }}(u)=d_{G^{\\prime }}(v)=2$ .","This implies that all components in $G^{\\prime }$ are isomorphic to $P_1$ or $P_2$ or $P_3$ .","Assume that $C$ is a component in $G^{\\prime }$ isomorphic to $P_3$ and let $v_1 v_2 v_3$ be the 3-path in $C$ .","By the construction of $G^{\\prime }$ we note that $v_2 \\in X_3$ or $v_2 \\in X_2^{\\prime \\prime }$ .","If $v_2 \\in X_3$ then, by the definition of ${\\cal G}_6$ , we note that $a^*_{}(v_1)=a^*_{}(v_3)=1$ .","And if $v_2 \\in X_2^{\\prime \\prime }$ , then $v_2 \\notin Y$ which by the definition of $Z$ implies that $a^*_{}(v_1)=a^*_{}(v_3)=1$ .","So in both cases $a^*_{}(v_1)=a^*_{}(v_3)=1$ , which by Lemma REF , implies that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, as desired.","Now conversely assume that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing and that $D$ denotes the edges deleted from $G$ in the solution.","By definition $D$ is a matching in $G$ and by Lemma REF every component, $C$ , in $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .","Let $G^* = G-D$ .","As $\\Delta (G^*) \\le 2$ we note that $D$ covers all vertices in $X_3$ .","Let $u \\in X_2^{\\prime }$ be arbitrary and let $v$ be a neighbour of $u$ in $G$ such that $v \\in X_3$ (which exists by the definition of $X_2^{\\prime }$ ).","Let $N(u)=\\lbrace x,v\\rbrace $ and let $N(v)=\\lbrace u,s,t\\rbrace $ , where as $G$ is triangle-free and quadrilateral-free implies that $x,v,u,s,t$ are all distinct.","If no edge incident with $u$ belongs to $D$ , then either $xuvs$ or $xuvt$ will be a $P_4$ in $G^*$ , a contradiction, so $u$ is covered by $D$ .","So all vertices in $X_2^{\\prime }$ are covered by $D$ and previously we showed that all vertices in $X_3$ are also covered by $D$ .","Let $z \\in Z$ be arbitrary.","That is $z \\in X_2^{\\prime \\prime }$ and $z$ is adjacent to a vertex, $w$ , with $a^*_{}(w)=0$ .","Let $N(z)=\\lbrace w,q\\rbrace $ and note that if $z$ is not covered by $D$ then $wzq$ is a $P_2$ in $G^*$ , but as $a^*_{}(w)=0$ , the edge $wq \\in \\overline{G}$ is not addable to $G$ .","So $z$ must be covered by $D$ .","This implies that $D$ is a matching covering $Y$ in $G$ .","Furthermore, by definition, $D$ does not cover any vertex with $d$ -value zero.","We have therefore shown that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, if and only if there exists a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .","Corollary 12 $(1,1)$ -Cluster Editing can be solved in polynomial time.","This follows immediately from Lemmas REF , REF , REF , REF , REF and REF ."],["Discussion","The main result of our paper concludes a complete dichotomy of complexity of $(a,d)$ -Cluster Editing.","We proved that $(1,1)$ -Cluster Editing can be solved in polynomial time.","Our proof consists of two stages: (i) providing a serious of five polynomial-time reductions to $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$ -Cluster Editing on $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3.","While Stage 2 is relatively short, Stage 1 is not.","Moreover, while our reduction from all graphs to $C_3$ -free graphs of maximum degree at most 3 is not hard, getting rid of 4-cycles required a lengthy and non-trivial series of four polynomial-time reductions.","It would be interesting to see whether the four reductions can be replaced by a shorter and simpler series of reductions."]],"string":"[\n [\n \"(1,1)-Cluster Editing is Polynomial-time Solvable\"\n ],\n [\n \"Abstract A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques.\",\n \"Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\\\\ V(G)\\\\rightarrow \\\\{0,1,\\\\dots, a\\\\}$ and $d^*{}:\\\\ V(G)\\\\rightarrow \\\\{0,1,\\\\dots, d\\\\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\\\\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\\\\in V(G)$.\",\n \"Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3.\"\n ],\n [\n \"Introduction\",\n \"A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques.\",\n \"Addition or deletion of an edge to a graph is called an edge edition.\",\n \"Given a graph $G$ and an integer $k \\\\ge 0,$ the Cluster Editing problem asks whether $k$ or less edge editions can transform $G$ into a clique graph.\",\n \"Cluster Editing is NP-Complete [11], [12], but it is fixed-parameter tractable if parameterized by $k$ [5].\",\n \"The parameterization has received considerable attention [2], [3], [4], [6], [7], [8], [9].\",\n \"In particular, an $O^*(1.618^k)$ -time algorithm was designed in [2] and a kernel with at most $2k$ vertices in [7].\",\n \"Unfortunately, $k$ is often not small for real-world instances.\",\n \"For example, in a protein similarity data set that has been frequently used for evaluating Cluster Editing algorithms, the instances with $n \\\\ge 30,$ where $n$ is the number of vertices, have an average number of edge editions between $2n$ and $4n$ [3].\",\n \"This has led to introduction of different parameterizations in [1], [10].\",\n \"We will consider the following variation of Cluster Editing problem introduced by Abu-Khzam [1].\",\n \"In $(a,d)$ -Cluster Editing, for fixed natural numbers $a$ and $d$ , given a graph $G$ and vertex-weights $a^*_{}:\\\\ V(G)\\\\rightarrow \\\\lbrace 0,1,\\\\dots , a\\\\rbrace $ and $d^*_{}:\\\\ V(G)\\\\rightarrow \\\\lbrace 0,1,\\\\dots , d\\\\rbrace $ , decide whether $G$ can be turned into a cluster graph by deleting at most $d^*_{}(v)$ edges incident to every $v\\\\in V(G)$ and adding at most $a^*_{}(v)$ edges incident to every $v\\\\in V(G)$ .\",\n \"Note that there is no upper bound on the total number of additions and deletions, see Paragraph 7 of Section 2 and Paragraph 1 of Section 5.1 in [1].\",\n \"$(a,d)$ -Cluster Editing is similar to a version of Cluster Editing studied by Komusiewicz and Uhlmann [10].\",\n \"In particular, they proved that $(0,d)$ -Cluster Editing is NP-hard for every fixed integer $d\\\\ge 2.$ Abu-Khzam [1] proved additional results on $(a,d)$ -Cluster Editing and established computational complexity of $(a,d)$ -Cluster Editing for every value of $a$ and $d$ apart from $a=d=1$ : $(a,d)$ -Cluster Editing is in P when $(a,d)\\\\in \\\\lbrace (a,0), (0,1):\\\\ a\\\\in \\\\lbrace 0,1,2\\\\rbrace \\\\rbrace $ and NP-complete, otherwise, apart from possibly the case of $a=d=1$ .\",\n \"In fact, Abu-Khzam [1] conjectured that $(1,1)$ -Cluster Editing is polynomial-time solvable.\",\n \"In this paper we resolve Abu-Khzam's conjecture in affirmative by (i) providing, in Section , a serious of five polynomial-time reductions to $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3, and (ii) designing, in Section , a polynomial-time algorithm for solving $(1,1)$ -Cluster Editing on $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3.\",\n \"An edge $uv$ of $G$ is deletable (in $G$ ) if $d^*_{}(u)=d^*_{}(v)=1$ and non-deletable, otherwise.\",\n \"An edge $uv$ of the complement $\\\\overline{G}$ of $G$ is addable (to $G$ ) if $a^*_{}(u)=a^*_{}(v)=1$ and non-addable, otherwise.\",\n \"Now we reformulate $(1,1)$ -Cluster Editing as follows: Given a graph $G$ , decide whether $G$ has a matching $D$ of edges deletable in $G$ and $\\\\overline{G}$ has a matching $A$ of edges addable to $G$ such that $G-D+A$ is a clique graph.\",\n \"We call it the matching formulation of $(1,1)$ -Cluster Editing and the desirable clique graph $G-D+A$ a solution.\",\n \"Note that we will often write a solution in the form $G-D+A$ , which defines the matchings $D$ and $A$ rather than the resulting cliques.\",\n \"We will denote by $P_n$ ($C_n$ , respectively) a path (cycle, respectively) on $n$ vertices, by $K_n$ a complete graph on $n$ vertices, and by $K_{n,m}$ a complete bipartite graph with partite sets of sizes $n$ and $m.$ For a positive integer $p$ , $[p]=\\\\lbrace 1,2,\\\\dots , p\\\\rbrace .$\"\n ],\n [\n \"Five polynomial-time reductions\",\n \"For two sets ${\\\\cal G}$ and ${\\\\cal H}$ of graphs, we say that $(1,1)$ -Cluster Editing can be reduced from ${\\\\cal G}$ to ${\\\\cal H}$ if for all $G\\\\in {\\\\cal G}$ and its vertex-weights $a^*_{},d^*_{}$ , we can either decide in polynomial time in $|V(G)|$ whether $(G,a^*_{},d^*_{})$ is a YES-instance or NO-instance, or construct $H \\\\in {\\\\cal H}$ with vertex-weights $a^+_{},d^+_{}$ such that $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if $(H,a^+_{},d^+_{})$ is a YES-instance of $(1,1)$ -Cluster Editing.\",\n \"Note that as the reduction is a polynomial-time algorithm, the size of $H$ is also bounded above by a polynomial in the size of $G$ .\",\n \"When the vertex-weights are clear from the context, we will often say that $G$ is an instance of the problem (i.e., of $(1,1)$ -Cluster Editing).\",\n \"We say that a graph $G$ is triangle-free if it does not contain a cycle $C_3$ as a subgraph and is quadrilateral-free if it does not contain a cycle $C_4$ as a subgraph.\",\n \"We will now prove a number of lemmas which will culminate in a final polynomial-time algorithm for $(1,1)$ -Cluster Editing.\",\n \"Let ${\\\\cal G}_1$ denote all graphs and let ${\\\\cal G}_2$ denote all triangle-free graphs of maximum degree at most 3.\",\n \"Lemma 1 $(1,1)$ -Cluster Editing can be reduced from ${\\\\cal G}_1$ to ${\\\\cal G}_2$ .\",\n \"Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing.\",\n \"Suppose that $G$ has a triangle $T\\\\cong C_3$ .\",\n \"We can delete at most one edge from $T$ and add at most one edge to $T$ (after an edge was deleted).\",\n \"Thus, if $(G,a^*_{},d^*_{})$ has a solution, it must contain $T$ as (part of) a clique.\",\n \"In the algorithm below to shorten its description, once a deletable edge $uv$ is deleted from $G$ , we immediately set $d^*_{}(u)=d^*_{}(v)=0$ and if an addable edge is added to $G$ , we immediately set $a^*_{}(u)=a^*_{}(v)=0.$ Let $Q:=T$ and execute the following loop.\",\n \"While there is a vertex outside $Q$ adjacent to a vertex in $Q$ , for every such vertex $v$ do the following cases in turn.\",\n \"If there is only one edge $e$ between $v$ and $Q$ then if $e$ is an deletable edge, then $e$ is deleted and the loop is continued.\",\n \"If there are exactly $|V(Q)|-1$ edges between $v$ and $V(Q)$ in $G$ and the edge in $\\\\overline{G}$ between $v$ and $V(Q)$ is addable, then add the addable edge to $G$ and continue the loop.\",\n \"If there are $|V(Q)|$ edges between $v$ and $V(Q)$ in $G$ , then add vertex $v$ and the edges between $v$ and $Q$ to $Q$ .\",\n \"Otherwise, $v$ has between two and $|V(Q)|-2$ edges to $Q$ and we conclude that $(G,a^*_{},d^*_{})$ is a NO-instance and stop the loop.\",\n \"If the loop stops without concluding that $(G,a^*_{},d^*_{})$ is a NO-instance, then delete $Q$ from $G.$ We can continue as above and either eliminate all triangles from $G$ or conclude that $(G,a^*_{},d^*_{})$ is a NO-instance.\",\n \"In the former case, we obtain a triangle-free instance $(G,a^*_{},d^*_{})$ which is a YES-instance if and only if the initial instance is.\",\n \"Note that it takes a polynomial time to compute the triangle-free instance or conclude that the initial instance is a NO-instance.\",\n \"Now suppose that $(G,a^*_{},d^*_{})$ is a triangle-free instance and $G$ has a vertex $v$ of degree at least four.\",\n \"Since we can delete at most one edge incident to $v$ and $G$ is triangle-free, we cannot make a clique including $v$ without adding at least two edges incident to a neighbour of $v$ .\",\n \"Thus, $(G,a^*_{},d^*_{})$ is a NO-instance.\",\n \"It is not hard to verify that the algorithm runs in polynomial time.\",\n \"Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing and let $Q \\\\subseteq V(G)$ be arbitrary.\",\n \"Consider the graph $G-Q$ , where we delete $Q$ (and all edges incident with $Q$ ).\",\n \"We define $d^*_{Q}$ such that $d^*_{Q}(v) = d^*_{}(v) - |N(v) \\\\cap Q|$ for all $v \\\\in V(G-Q)$ .\",\n \"We define $a^*_{Q}$ to be the function $a^*_{}$ restricted to $V(G-Q)$ .\",\n \"Note that by definition of vertex-weights, if $d^*_{Q}(v) <0$ for any $v \\\\in V(G-Q)$ then $(G-Q,a^*_{Q},d^*_{Q})$ is a NO-instance.\",\n \"Similarly, if $F\\\\subseteq E(G)$ , then $d^*_{F}(v) = d^*_{}(v) - |E(v) \\\\cap F|$ , where $E(v)$ is the set of edges incident to $v$ .\",\n \"Also, $a^*_{F}=a^*_{}.$ Again, if $d^*_{F}(v) <0$ for any $v \\\\in V(G)$ then $(G-F,a^*_{F},d^*_{F})$ is a NO-instance.\",\n \"We say that a vertex set $X$ of $G$ be R-deletable if $X$ has constant size and every solution to $(G,a^*,d^*)$ (if any) puts all edges between $X$ and $V(G)-X$ in $D.$ Note that vacuously this implies that if $G$ is a NO-instance then any set $X$ of constant size is an R-deletable set.\",\n \"Lemma 2 Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing.\",\n \"If a vertex set $X$ of $G$ is R-deletable, then we can either solve $(G,a^*_{},d^*_{})$ is constant time or reduce the instance to $G-X$ .\",\n \"Note that if $(G,a^*_{},d^*_{})$ is a NO-instance, then any arbitrary vertex set $X$ of constant size is R-deletable and therefore satisfies the above.\",\n \"If $X=V(G)$ then we can solve the problem in constant time, so assume that $X \\\\ne V(G).$ Let $F$ be all edges between $X$ and $V(D)-X$ and let $G^{\\\\prime } =G-F.$ If any $d^*_{F}$ value drops below 0 we have a NO-instance by definition of vertex-weights.\",\n \"Otherwise, $G$ is a YES-instance if and only if $G^{\\\\prime }$ is a YES-instance.\",\n \"And as $X$ has constant size we can determine if there is a solution to $G^{\\\\prime }[X]$ in constant time.\",\n \"If there is no solution to $G^{\\\\prime }[X]$ then $G$ is a NO-instance and if there is a solution to $G^{\\\\prime }[X]$ , then we can reduce $G$ by deleting $X$ , as $G$ is a YES-instance if and only if $G^{\\\\prime }-X=G-X$ is a YES-instance.\",\n \"Note that the above also holds when $(G,a^*_{},d^*_{})$ is a NO-instance as then either $G[X]$ is a NO-instance (after adjustment of $d^*$ ), which can be decided in constant time, or $G-X$ is a NO-instance (after adjustment of $d^*$ ).\",\n \"Lemma 3 Let $G$ be a triangle-free graph with maximum degree at most 3.\",\n \"Then $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if there is a matching $D$ of deletable edges from $G$ such that every connected component $C$ of $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ or $C_4$ and if $C\\\\cong P_3$ then the edge of $\\\\overline{G}$ between the end-vertices of $C$ is addable to $G$ and if $C\\\\cong C_4$ then the two chords of $C$ , which lie in $\\\\overline{G}$ , are addable to $G$ .\",\n \"Let $(G,a^*_{},d^*_{})$ be a YES-instance of $(1,1)$ -Cluster Editing and $G-D+A$ is a matching solution.\",\n \"Since $G$ is a triangle-free graph with maximum degree at most 3, each connected component of $G-D+A$ is isomorphic to $K_1$ or $K_2$ or $K_3$ or $K_4$ .\",\n \"Now the lemma follows from the fact that $G$ is a triangle-free graph.\",\n \"Lemma 4 Let $G$ be a triangle-free graph with maximum degree at most 3 containing a $K_{2,3}$ as a subgraph.\",\n \"Then $(G,a^*_{},d^*_{})$ is a NO-instance of $(1,1)$ -Cluster Editing for every $(a^*,d^*)$ .\",\n \"Let $G$ be a triangle-free graph with maximum degree at most 3 containing a $K_{2,3}$ subgraph, $R$ , where $V(R)=\\\\lbrace x_1,x_2,y_1,y_2,y_3\\\\rbrace $ and $E(R)=\\\\lbrace x_i y_j \\\\; | \\\\; i=1,2 \\\\mbox{ and } j=1,2,3\\\\rbrace $ .\",\n \"Assume that $(G,a^*_{},d^*_{})$ is a YES-instance with solution $G-D+A$ .\",\n \"By Lemma REF , no vertex in $G-D$ has degree 3.\",\n \"Thus, we must have $x_1 y_i \\\\in D$ and $x_2 y_j \\\\in D$ for some $i \\\\ne j$ (as $D$ is a matching) and these are the only edges in $E(R)$ that belong to $D$ .\",\n \"However this implies that $R-D$ is a $P_5$ , which is a contradiction by Lemma REF .\",\n \"So, $(G,a^*_{},d^*_{})$ is a NO-instance.\",\n \"Let ${\\\\cal G}_3$ denote all triangle-free graphs of maximum degree at most 3 which contain no 4-cycle, all of whose vertices have degree 3.\",\n \"Lemma 5 $(1,1)$ -Cluster Editing can be reduced from ${\\\\cal G}_2$ to ${\\\\cal G}_3$ in polynomial time.\",\n \"Let $G \\\\in {\\\\cal G}_2$ be arbitrary with vertex-weights $(a^*,d^*)$ .\",\n \"If there is no 4-cycle in $G$ where all vertices have degree three then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be a 4-cycle in $G$ with $d(v_i)=3$ for all $i=1,2,3,4$ .\",\n \"Let $w_i$ be the neighbour of $v_i$ in $G$ which does not lie on $C$ .\",\n \"Let $V_8=\\\\lbrace v_1,v_2,v_3,v_4,w_1,w_2,w_3,w_4\\\\rbrace $ .\",\n \"We will show that $|V_8|=8$ (or we have a NO-instance), so assume for the sake of contradiction that this is not the case.\",\n \"As $G$ is triangle-free, we must have $w_1=w_3$ or $w_2=w_4$ .\",\n \"In either case $K_{2,3}$ is a subgraph of $G$ and we have a NO-instance by Lemma REF .\",\n \"Therefore we may assume that $|V_8|=8$ .\",\n \"We now prove the following claims.\",\n \"Claim A: If $(G,a^*_{},d^*_{})$ is a YES-instance and $G-D+A$ is a solution, then at least one of the following options holds.\",\n \"(a): $v_i w_i \\\\in D$ for all $i=1,2,3,4$ and no edge on $C$ belongs to $D$ .\",\n \"(b): $v_i w_i \\\\notin D$ for all $i=1,2,3,4$ and $v_1 v_2, v_3 v_4 \\\\in D$ and $v_2 v_3, v_4 v_1 \\\\notin D$ .\",\n \"(c): $v_i w_i \\\\notin D$ for all $i=1,2,3,4$ and $v_2 v_3, v_4 v_1 \\\\in D$ and $v_1 v_2, v_3 v_4 \\\\notin D$ .\",\n \"Furthermore, for every such 4-cycle in $G$ either zero or two edges of the cycle belong to $D$ .\",\n \"Proof of Claim A: Let $C^{\\\\prime }$ be any 4-cycle in $G$ .\",\n \"If one edge $e \\\\in E(C^{\\\\prime })$ belongs to $D$ , then at least two edges of $C^{\\\\prime }$ must belong to $D$ , as otherwise $C^{\\\\prime } - e$ is a $P_4$ in $G-D$ , which is not part of a 4-cycle in $G-D$ , a contradiction by Lemma REF .\",\n \"As each vertex is adjacent to at most one edge from $D$ , we note that we cannot have more than two edges from $C^{\\\\prime }$ in $D$ .\",\n \"Option (a) now corresponds to the case when no edge from $C$ belongs to $D$ and Options (b) and (c) correspond to the two possible ways that we can add two (non-adjacent) edges of $C$ to $D$ .\",\n \"This completes the proof of Claim A.\",\n \"($\\\\diamond $ ) Claim B: We may assume that $d^*_{}(v_i)=a^*_{}(v_i)=1$ for all $i=1,2,3,4,$ as otherwise we can reduce our instance or solve it in polynomial time.\",\n \"Proof of Claim B: If $d^*_{}(v_i)=0$ for some $i\\\\in [4]$ , then none of the options in Claim A is possible, so we have a NO-instance.\",\n \"So we may assume that $d^*_{}(v_i)=1$ for all $i\\\\in [4]$ .\",\n \"For the sake of contradiction, assume that $a^*_{}(v_1)=0$ and $(G,a^*_{},d^*_{})$ is a YES-instance with a solution $G-D+A$ .\",\n \"In this case Option (a) of Claim A cannot hold as we cannot add the edge $v_1 v_3$ to $C$ .\",\n \"Option (b) of Claim A cannot hold as we cannot add the edge $w_1 v_4$ to $G-D$ .\",\n \"Also, Option (c) of Claim A cannot hold as we cannot add the edge $w_1 v_2$ to $G-D$ .\",\n \"So in this case we have a NO-instance.\",\n \"Analogously, we may assume that $a^*_{}(v_i)=1$ for all $i\\\\in [4]$ , which completes the proof of Claim B.\",\n \"($\\\\diamond $ ) Let $e_1 = w_1 w_2$ , $e_2 = w_2 w_3$ , $e_3 = w_3 w_4$ and $e_4 = w_4 w_1$ be edges that may or may not exist in $G$ .\",\n \"We now prove the following claims.\",\n \"Claim C: We may assume that $e_1 \\\\notin E(G)$ and that Option (c) in Claim A cannot occur in any solution of $(G,a^*,d^*)$ , as otherwise we can reduce our instance or solve it in polynomial time.\",\n \"Furthermore, if we know that there is no solution of $(G,a^*,d^*)$ for which Option (b) of Claim A occurs, then we can reduce or solve our instance in polynomial time.\",\n \"Also, if we know that there is no solution of $(G,a^*,d^*)$ for which Option (a) of Claim A occurs, then we can reduce or solve our instance in polynomial time.\",\n \"Proof of Claim C: We may assume that $G$ is connected; otherwise consider the connected component containing $V_8.$ By renaming vertices we may assume that $e_1 \\\\notin E(G)$ , unless $e_i \\\\in E(G)$ for all $i \\\\in [4]$ .\",\n \"So assume that $e_i \\\\in E(G)$ for all $i \\\\in [4]$ , which, as $G$ is connected and of maximum degree three, implies that $V(G)=V_8$ .\",\n \"In this case we can determine if we have a YES-instance (which is the case if and only if $d^*_{}(v)=a^*_{}(v)=1$ for all $v \\\\in V(G)$ ) or NO-instance in constant time, and so we would be done.\",\n \"So we may indeed assume that $e_1 \\\\notin E(G)$ .\",\n \"If Option (c) in Claim A occurs, then $w_1 v_1 v_2 w_2$ is a $P_4$ in $G-D$ where $w_1w_2 \\\\notin E(G)$ , which is not possible by Lemma REF , so Option (c) in Claim A cannot occur.\",\n \"By Lemma REF , we may assume that, in the rest of the proof, $(G,a^*,d^*)$ is a YES-instance.\",\n \"As we may assume that Option (c) in Claim A cannot occur, either Option (a) or Option (b) of Claim A must occur for every solution of $(G,a^*_{},d^*_{})$ .\",\n \"First assume that we can show that Option (b) does not occur, and therefore Option (a) must occur in all solutions.\",\n \"In this case, $w_iv_i \\\\in D$ for all $i \\\\in [4]$ and therefore $V(C)$ is a R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .\",\n \"Now assume that Option (a) cannot occur which implies that Option (b) must occur in all solutions.\",\n \"This implies that all edges between $V_8$ and $V(G) \\\\setminus V_8$ must belong to $D$ .\",\n \"Thus, either $V(G)=V_8$ in which case we are done or $V_8$ is an R-deletable set, so we can reduce $G$ to $G-V_8$ by Lemma REF .\",\n \"This completes the proof of Claim C.($\\\\diamond $ ) Figure: Graph (a) is an illustration of Claim D in the proof of Lemma , while Graph (b) is an illustration of Claim F.Claim D: We may assume that $e_2,e_4 \\\\in E(G)$ and $e_1,e_3 \\\\notin E(G),$ as otherwise we can reduce our instance or solve it in polynomial time.\",\n \"(See Figure REF (a) for an illustration.)\",\n \"We may also assume that $d^*_{}(w_i)=a^*_{}(w_i)=1$ for all $i \\\\in \\\\lbrace 1,2,3,4\\\\rbrace $ .\",\n \"Proof of Claim D: By Claim C we may assume that $e_1 \\\\notin E(G)$ .\",\n \"For the sake of contradiction assume that $e_2 \\\\notin E(G)$ .\",\n \"By Claim C we note that part (c) in Claim A cannot occur.\",\n \"Analogously, if Option (b) in Claim A occurs, then $w_2 v_2 v_3 w_3$ is a $P_4$ in $G-D$ where $w_2w_3 \\\\notin E(G)$ , which is not possible.\",\n \"So, Option (b) in Claim A can also not occur and we are done by Claim C. Hence, we may assume that $e_2 \\\\in E(G)$ .\",\n \"The fact that $e_4 \\\\in E(G)$ can be proved analogously, so we may assume that $e_2,e_4 \\\\in E(G)$ .\",\n \"For the sake of contradiction assume that $e_3 \\\\in E(G)$ .\",\n \"In this case Option (a) in Claim A cannot occur, as $w_2 w_3 w_4 w_1$ would then be a $P_4$ in $G-D$ where $w_1 w_2 \\\\notin E(G-D)$ , a contradiction.\",\n \"So, again we are done by Claim C. So we may assume that $e_3 \\\\notin E(G)$ , as otherwise we either reduce our instance or solve it in constant time.\",\n \"This completes the proof of the first part of Claim D. In order to prove the second part we consider the following two cases.\",\n \"First assume that $d^*_{}(w_i)=0$ for some $i \\\\in [4]$ .\",\n \"Then Option (a) in Claim A cannot occur, so by Claim C, Option (b) must occur.\",\n \"As above we now see that we either can solve our instance in constant time or $V_8$ is a R-deletable set, so we can reduce $G$ to $G-V_8$ by Lemma REF .\",\n \"Hence, we may assume that $d^*_{}(w_i)=1$ for all $i \\\\in [4]$ .\",\n \"Now assume that $a^*_{}(w_i)=0$ for some $i \\\\in [4]$ .\",\n \"In this case Option (b) in Claim A cannot occur, so by Claim C, Option (a) must occur.\",\n \"In this case $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .\",\n \"So we may assume that $a^*_{}(w_i)=1$ for all $i \\\\in [4]$ .\",\n \"This completes the proof of Claim D.($\\\\diamond $ ) Definition: If $d(w_1)=3$ then let $z_1$ be defined such that $N(w_1)=\\\\lbrace z_1,v_1,w_4\\\\rbrace $ .\",\n \"If $d(w_2)=3$ then let $z_2$ be defined such that $N(w_2)=\\\\lbrace z_2,v_2,w_3\\\\rbrace $ .\",\n \"If $d(w_3)=3$ then let $z_3$ be defined such that $N(w_3)=\\\\lbrace z_3,v_3,w_2\\\\rbrace $ .\",\n \"If $d(w_4)=3$ then let $z_4$ be defined such that $N(w_4)=\\\\lbrace z_4,v_4,w_1\\\\rbrace $ .\",\n \"Note that the $z_i$ 's may not be distinct and we may have $z_i=w_j$ for some $i$ and $j$ .\",\n \"But, by Claim D we have $z_1 \\\\ne w_2$ , $z_2 \\\\ne w_1$ , $z_3 \\\\ne w_4$ and $z_4 \\\\ne w_3$ .\",\n \"Claim E: We may assume that if $z_i$ exists then $z_i \\\\notin V_8$ for all $i=1,2,3,4$ and if $z_i$ and $z_j$ both exist with $i \\\\ne j$ , then $z_i \\\\ne z_j$ .\",\n \"Otherwise, we can reduce our instance or solve it in polynomial time.\",\n \"Proof of Claim E: For the sake of contradiction assume that $z_1$ exists and $z_1 \\\\in V_8$ .\",\n \"By definition there exists a $j \\\\in [4]$ such that $z_i = w_j$ .\",\n \"By Claim D, we must have $j=3$ (and therefore $z_1=w_3$ and $z_3=w_1$ ).\",\n \"First consider the case when $z_2$ exists and $z_2=w_4$ .\",\n \"In this case, as $G$ is connected, we have $V(G)=V_8$ and we can determine if $(G, a^*,d^*)$ is a YES-instance or a NO-instance in constant time.\",\n \"So we may assume that $w_2w_4 \\\\notin E(G)$ .\",\n \"Option (a) in Claim A cannot occur, as $w_4 w_1 w_3 w_2$ would then be a $P_4$ in $G-D$ where $w_2 w_4 \\\\notin E(G-D)$ , a contradiction.\",\n \"So, by Claim C, we are done in this case.\",\n \"Analogously we can show that $z_i \\\\notin V_8$ for all $i=1,2,3,4$ (or we can reduce our instance or solve it in polynomial time).\",\n \"Now for the sake of contradiction assume that $z_i$ and $z_j$ both exist and $i \\\\ne j$ but $z_i=z_j$ .\",\n \"Option (b) in Claim A cannot occur, as in this case $w_i z_i \\\\in D$ and $w_j z_j \\\\in D$ , but $z_i$ can be incident with at most one edge from $D$ .\",\n \"So, we are done by Claim C. This completes the proof of Claim E.($\\\\diamond $ ) Figure: The reduction in Claim G of Lemma .Claim F: If $z_1$ and $z_4$ both exist then $z_1 z_4 \\\\in E(G)$ .\",\n \"Otherwise, we can reduce our instance or solve it in polynomial time.\",\n \"Analogously, we may assume that if $z_2$ and $z_3$ both exist then $z_2 z_3 \\\\in E(G)$ .\",\n \"Proof of Claim F: For the sake of contradiction assume that $z_1$ and $z_4$ both exist, but $z_1 z_4 \\\\notin E(G)$ (see Figure REF (b)).\",\n \"By Claim C, either Option (a) or Option (b) in Claim A must occur.\",\n \"Assume that Option (a) in Claim A occurs.\",\n \"This implies that $v_i w_i \\\\in D$ for all $i \\\\in [4]$ .\",\n \"Hence, $z_1 w_1 w_4 z_4$ is a $P_4$ in $G-D$ where $z_1 z_4 \\\\notin E(G)$ , a contradiction.\",\n \"So Option (b) in Claim A must occur.\",\n \"We are now done by Claim C. This completes the proof of Claim F.($\\\\diamond $ ) Claim G: If $z_1$ , $z_3$ and $z_4$ all exist, then we may assume that $z_3 z_4 z_1$ is a path in $G$ .\",\n \"Otherwise, we can reduce our instance or solve it in polynomial time.\",\n \"Analogously if $z_1$ , $z_2$ and $z_4$ all exist, then we may assume that $z_4 z_1 z_2$ is a path in $G$ .\",\n \"And if $z_2$ , $z_3$ and $z_4$ all exist, then that $z_2 z_3 z_4$ is a path in $G$ .\",\n \"And finally if $z_1$ , $z_2$ and $z_3$ all exist, then that $z_1 z_2 z_3$ is a path in $G$ .\",\n \"Proof of Claim G: Assume that $z_1$ , $z_3$ and $z_4$ exist.\",\n \"By Claim F we may assume that $z_1 z_4 \\\\in E(G)$ .\",\n \"For the sake of contradiction assume that $z_3 z_4 \\\\notin E(G)$ .\",\n \"Let $G^+$ be obtained from $G$ by deleting the vertices $v_1,v_2,v_3,v_4$ and adding the edges $w_1 w_2$ and $w_3 w_4$ (the vertex weights $d^*_{}$ and $a^*_{}$ remain unchanged).\",\n \"See Figure REF for an illustration.\",\n \"We will now show that $G^+$ is a YES-instance if and only if $G$ is a YES-instance.\",\n \"First assume that $(G,a^*,d^*)$ is a YES-instance.\",\n \"By Claim C we note that Option (a) or Option (b) in Claim A must occur.\",\n \"If Option (a) occurs then $v_i w_i \\\\in D$ for all $i \\\\in [4]$ in $G$ .\",\n \"In $G^+$ we use exactly the same $D$ , except we remove $v_1w_1, v_2w_2, v_3w_3, v_4w_4$ and add $w_1w_2$ and $w_3 w_4$ .\",\n \"This way we get exactly the same components in $G$ and $G^+$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G$ .\",\n \"Now assume that Option (b) occurs in Claim A.\",\n \"In this case we use the same $D$ in $G^+$ and in $G$ , except we remove the edges $v_1 v_2$ and $v_3 v_4$ .\",\n \"Now we get exactly the same components in $G$ and $G^+$ , except the 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ have been replaced by $w_1 w_2 w_3 w_4 w_1$ in $G^+$ .\",\n \"So in all cases $G^+$ is a YES-instance.\",\n \"Conversely assume that $G^+$ is a YES-instance.\",\n \"We will now show that $G$ is a YES-instance.\",\n \"Let $D^+$ be the matching in $G^+$ such that all components in $G^+-D^+$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).\",\n \"Let $C^+$ be the 4-cycle $w_1 w_2 w_3 w_4 w_1$ in $G^+$ .\",\n \"As in the proof of Claim A we note that $D^+$ contains no edge from $C^+$ or it contains exactly two edges from $C^+$ , which are non-adjacent.\",\n \"Note that $w_2w_3, w_4w_1 \\\\in D^+$ is not possible as the path $z_4 w_4 w_3 z_3$ would be a $P_4$ in $G^+ - D^+$ and $z_3 z_4 \\\\notin E(G)$ .\",\n \"So either $w_1w_2, w_3w_4 \\\\in D^+$ or no edge from $C^+$ is in $D^+$ .\",\n \"First assume that $w_1w_2, w_3w_4 \\\\in D^+$ .\",\n \"Then $z_1 w_1 w_4 z_4 z_1$ is a $C_4$ in $G^+ - D^+$ and $w_2 w_3 z_3$ is a $P_2$ in $G^+ - D^+$ (which may be part of a $C_4$ if $z_2$ exists).\",\n \"In this case let $D = D^+ \\\\cup \\\\lbrace v_1 w_1, v_2w_2, v_3w_3, v_4w_4\\\\rbrace \\\\setminus \\\\lbrace w_1w_2, w_3w_4\\\\rbrace $ and note that we obtain exactly the same components in $G$ and $G^+$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G$ .\",\n \"So we now consider the case when no edge from $C^+$ is in $D^+$ .\",\n \"In this case $w_1 z_1, w_3 z_3, w_4 z_4 \\\\in D^+$ .\",\n \"Let $D = D^+ \\\\cup \\\\lbrace v_1 v_2, v_3 v_4\\\\rbrace $ and note that we obtain exactly the same components in $G$ and $G^+$ , except the 4-cycle $w_1 w_2 w_3 w_4 w_1$ in $G^+$ has been replaced by the 4-cycles $w_1 v_1 v_4 w_4 w_1$ and $w_2 v_2 v_3 w_3 w_2$ in $G$ .\",\n \"We have now proven that $G^+$ is a YES-instance if and only if $G$ is a YES-instance.\",\n \"This completes the proof of Claim G.($\\\\diamond $ ) Claim H: We may assume that at most one of $z_1$ and $z_4$ exist.\",\n \"Otherwise, we can reduce our instance or solve it in polynomial time.\",\n \"Similarly, at most one of $z_2$ and $z_3$ exist.\",\n \"Proof of Claim H: Assume that both $z_1$ and $z_4$ exist.\",\n \"We first consider the case when both $z_2$ and $z_3$ also exist.\",\n \"By Claim G, we have that $z_1 z_2 z_3 z_4 z_1$ is a 4-cycle in $G$ and therefore $G$ has 12 vertices, and we can determine if it is a YES-instance or a NO-instance in constant time.\",\n \"So we may assume that either none of $z_2$ and $z_3$ exist or exactly one of them exist.\",\n \"First assume that $z_3$ exists, but $z_2$ does not exist.\",\n \"By Claim G, $z_1 z_4, z_3 z_4 \\\\in E(G)$ .\",\n \"Let $V_{11}=V_8 \\\\cup \\\\lbrace z_1,z_3,z_4\\\\rbrace $ .\",\n \"If $V(G)=V_{11}$ then we can solve our instance in constant time so assume that this is not the case.\",\n \"We consider the cases when there exists a $w \\\\in V(G) \\\\setminus V_{11}$ with $\\\\lbrace z_1,z_3\\\\rbrace \\\\subseteq N(w)$ and when no such $w$ exists.\",\n \"If such a $w$ exists then Option (a) in Claim A cannot occur, as if it did both the edges $wz_1$ and $wz_3$ would need to be in $D$ (as $z_1 w_1 w_4 z_4 z_1$ would be a $C_4$ in $G-D$ and $w_2 w_3 z_3$ would be a $P_3$ in $G-D$ ), a contradiction.\",\n \"So, we are done by Claim C. We may therefore now assume that no such $w$ exists.\",\n \"As $G$ is connected and $V(G) \\\\ne V_{11}$ we must have a vertex $r \\\\in V(G) \\\\setminus V_{11}$ which is adjacent to either $z_1$ or to $z_3$ .\",\n \"Now Option (b) of Claim A cannot occur, as if it did then the edges $w_1z_1$ , $w_3z_3$ and $w_4z_4$ would need to be in $D$ , which implies that either $r z_1 z_4 z_3$ or $z_1 z_4 z_3 r$ is a $P_4$ in $G-D$ where the end-points of the path are not adjacent in $G$ (as no $w$ exists in $V(G) \\\\setminus V_{11}$ that is adjacent to both $z_1$ and $z_3$ ).\",\n \"So, again we are done by Claim C. This completes all cases and implies that we can reduce when $z_3$ exists, but $z_2$ does not exist.\",\n \"Figure: The graph when z 2 z_2 and z 3 z_3 do not exist.Analogously we can also reduce if $z_2$ exists, but $z_3$ does not exist.\",\n \"So the last case to consider is when neither $z_2$ nor $z_3$ exist.\",\n \"In this case the graph looks like in Figure REF .\",\n \"If we had considered the cycle $w_1 v_1 v_4 w_4 w_1$ instead of $v_1 v_2 v_3 v_4 v_1$ , then we would be in the case where $G$ has 10 vertices (when $d(z_1)=d(z_4)=2$ ) or we would have been in a case where at least three $z$ -vertices would exist (namely $\\\\lbrace w_2,w_3\\\\rbrace \\\\cup {N(z_1)\\\\cup N(z_4)}\\\\setminus \\\\lbrace z_1,w_1,z_4,w_4\\\\rbrace $ ), which means we can reduce as seen above.\",\n \"This completes the proof of Claim H.($\\\\diamond $ ) We now return to the proof of Lemma REF .\",\n \"By Claim H we may assume that at most one of $z_1$ and $z_4$ exist and at most one of $z_2$ and $z_3$ exist.\",\n \"If no $z_i$ exist then $G$ has 8 vertices, so we can solve our problem in constant time.\",\n \"Therefore we may, without loss of generality assume that $z_1$ exists.\",\n \"If $z_2$ exists then let $z=z_2$ and if $z_3$ exists then let $z=z_3$ and if neither $z_2$ or $z_3$ exist then let $z$ be undefined.\",\n \"We now create $G^*$ from $G$ by removing the vertices $w_1, w_2, w_3, w_4$ and adding the edge $v_1 z_1$ and if $z$ exists also the edge $v_3 z$ .\",\n \"We will show that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.\",\n \"First assume that $G$ is a YES-instance.\",\n \"By Claim C we note that Option (a) or Option (b) in Claim A must occur.\",\n \"If Option (a) occurs then $v_i w_i \\\\in D$ for all $i \\\\in [4]$ in $G$ .\",\n \"Let $D^* = D \\\\cup \\\\lbrace v_1 v_2, v_3 v_4\\\\rbrace \\\\setminus \\\\lbrace v_1w_1, v_2w_2, v_3w_3, v_4w_4 \\\\rbrace $ .\",\n \"This way we get exactly the same components in $G$ and $G^*$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ , the $P_3=z_1w_1w_4$ and the $P_2$ or $P_3$ on the vertex set $\\\\lbrace w_2, w_3\\\\rbrace $ or $\\\\lbrace w_2, w_3, z\\\\rbrace $ in $G$ get replaced by the $P_3=z_1 v_1 v_4$ and either the $P_2=v_2 v_3$ (if $z$ does not exist) or the $P_3=v_2 v_3 z$ in $G^*$ .\",\n \"Now assume that Option (b) occurs in Claim A.\",\n \"In this case $z_1 w_1 \\\\in D$ and if $z$ exists then $w_2 z$ or $w_3z$ is in $D$ .\",\n \"Also, we let $D^* = D - \\\\lbrace v_1 v_2, v_3 v_4\\\\rbrace $ , where we also replace $z_1w_1$ by $z_1v_1$ and if $z$ exists then the edge $w_2 z$ or $w_3z$ gets replaced by $v_3z$ .\",\n \"Now we get exactly the same components in $G-D$ and $G^* - D^*$ , except the 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ have been replaced by $v_1 v_2 v_3 v_4 v_1$ in $G^*$ .\",\n \"So in all cases $G^*$ is a YES-instance.\",\n \"Conversely assume that $G^*$ is a YES-instance.\",\n \"We will now show that $G$ is a YES-instance.\",\n \"Let $D^*$ be the matching in $G^*$ such that all components in $G^* - D^*$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).\",\n \"As in the proof of Claim A we note that $D^*$ contains no edge from $C$ or it contains exactly two edges from $C$ , which are non-adjacent.\",\n \"First assume that $D^*$ contains no edge from $C$ .\",\n \"Then $z_1 v_1 \\\\in D^*$ and if $z$ exists then $z v_3 \\\\in D$ .\",\n \"In this case, let $D$ be obtained from $D^* + \\\\lbrace v_1v_2, v_3v_4,z_1 w_1\\\\rbrace - \\\\lbrace z_1 v_1\\\\rbrace $ by replacing $v_3 z$ by the edge from $\\\\lbrace w_2,w_3\\\\rbrace $ to $z$ if $z$ exists.\",\n \"Again, we obtain exactly the same components in $G-D$ and $G^* - D^*$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G^*$ has been replaced by the two 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ .\",\n \"So we now consider the case when $D^*$ contains exactly two edges from $C$ .\",\n \"In this case $G^* - D^*$ contains a $P_3$ containing the edge $z_1 v_1$ and either the edge $v_1 v_2$ or $v_1 v_4$ .\",\n \"Similarly, if $z$ exists, then $G^* - D^*$ contains a $P_3$ containing the edge $z v_3$ and either the edge $v_3 v_4$ or $v_3 v_2$ .\",\n \"In this case let $D$ be obtained from $D^*$ by deleting these two edges on $C$ that belong to $D^*$ and adding the edges $\\\\lbrace v_1w_1, v_2w_2, v_3w_3, v_4w_4 \\\\rbrace $ instead.\",\n \"This way we obtain exactly the same components in $G-D$ and $G^* - D^*$ , except for the following: The $P_3$ in $G^* - D^*$ containing $v_1$ now becomes the $P_2$ $z_1 w_1 w_4$ and the $P_3$ containing $z$ (if it exists) now becomes a $P_3$ containing the edge $w_2 w_3$ as well as the vertex $z$ .\",\n \"Furthermore $G-D$ will contain the 4-cycle $C$ , which is not in $G^* - D^*$ .\",\n \"This proves that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.\",\n \"Let ${\\\\cal G}_4$ denote all triangle-free graphs of maximum degree at most 3 which contain no 4-cycle, with at least three vertices of degree three.\",\n \"Lemma 6 $(1,1)$ -Cluster Editing can be reduced from ${\\\\cal G}_3$ to ${\\\\cal G}_4$ in polynomial time.\",\n \"Let $G \\\\in {\\\\cal G}_3$ be arbitrary with vertex-weights $(a^*,d^*)$ .\",\n \"If there is no 4-cycle in $G$ where three of the vertices have degree three then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be a 4-cycle in $G$ with $d(v_i)=3$ for all $i=1,2,3$ and $d(v_4)=2$ .\",\n \"Let $w_i$ be the neighbour of $v_i$ in $G$ which does not lie on $C$ for $i=1,2,3$ .\",\n \"Let $V_7=\\\\lbrace v_1,v_2,v_3,v_4,w_1,w_2,w_3\\\\rbrace $ .\",\n \"We will show that $|V_7|=7$ (or we have a NO-instance), so assume for the sake of contradiction that this is not the case.\",\n \"As $G$ is triangle-free, we must have $w_1=w_3$ , which implies that $K_{2,3}$ is a subgraph of $G$ and we have a NO-instance by Lemma REF .\",\n \"Therefore we may assume that $|V_7|=7$ .\",\n \"The following claim can be proved analogously to Claim A in Lemma REF and thus its proof is omitted.\",\n \"Claim A: If $(G,a^*_{},d^*_{})$ is a YES-instance and $G-D+A$ is a solution.\",\n \"Then one of the following options hold.\",\n \"(a): $v_i w_i \\\\in D$ for all $i=1,2,3$ and no edge on $C$ belongs to $D$ .\",\n \"(b): $v_i w_i \\\\notin D$ for all $i=1,2,3,4$ and $v_1 v_2, v_3 v_4 \\\\in D$ and $v_2 v_3, v_4 v_1 \\\\notin D$ .\",\n \"(c): $v_i w_i \\\\notin D$ for all $i=1,2,3,4$ and $v_2 v_3, v_4 v_1 \\\\in D$ and $v_1 v_2, v_3 v_4 \\\\notin D$ .\",\n \"Furthermore, for every such 4-cycle in $G$ either zero or two edges of the cycle belong to $D$ .\",\n \"Claim B: We may assume that $d^*_{}(v_i)=1$ for all $i=1,2,3,4$ .\",\n \"Otherwise, we have a NO-instance.\",\n \"Proof of Claim B: If $d^*_{}(v_i)=0$ for some $i\\\\in [4]$ , then none of the options in Claim A is possible, so we have a NO-instance.\",\n \"This proves Claim B.\",\n \"($\\\\diamond $ ) Claim C: If Option (b) in Claim A occurs, then we must have $w_2 w_3 \\\\in E(G)$ .\",\n \"If Option (c) occurs in Claim A, then we must have $w_1 w_2 \\\\in E(G)$ .\",\n \"Proof of Claim C: For the sake of contradiction, assume that Option (b) occurs in Claim A, but $w_2 w_3 \\\\notin E(G)$ .\",\n \"In this case $w_2 v_2 v_3 w_3$ is a $P_4$ in $G-D$ where $w_2 w_3 \\\\notin E(G)$ , a contradiction.\",\n \"The fact that if Option (c) occurs in Claim A, then we must have $w_1 w_2 \\\\in E(G)$ is proved analogously.\",\n \"($\\\\diamond $ ) Claim D: We may assume that exactly one of the edges $w_1w_2$ and $w_2w_3$ belongs to $G$ .\",\n \"Proof of Claim D: If $w_1w_2 \\\\notin E(G)$ and $w_2w_3 \\\\notin E(G)$ then, by Claim C, neither Option (b) nor Option (c) occurs in Claim A.\",\n \"Therefore Option (a) must occur in Claim A.\",\n \"This implies that $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .\",\n \"We may therefore assume that at least one of $w_1w_2$ and $w_2w_3$ belong to $G$ .\",\n \"Assume that $w_1w_2 \\\\in E(G)$ and $w_2w_3 \\\\in E(G)$ .\",\n \"As $G \\\\in {\\\\cal G}_3$ we note that the two 4-cycles $w_1 v_1 v_2 w_2 w_1$ and $w_2 v_2 v_3 w_3 w_2$ both must contain a degree-2 vertex.\",\n \"This implies that $d(w_1)=d(w_3)=2$ and $V(G)=V_7$ .\",\n \"We can therefore solve the problem in constant time.\",\n \"We may therefore assume that one of the edges $w_1w_2$ and $w_2w_3$ do not belong to $G$ .\",\n \"This proves Claim D.($\\\\diamond $ ) By Claim D, we may without loss of generality assume that $w_1w_2 \\\\in E(G)$ and $w_2w_3 \\\\notin E(G)$ .\",\n \"Claim E: Option (b) in Claim A cannot occur.\",\n \"Furthermore, we may assume that $d(w_1) \\\\le 2$ or $d(w_2) \\\\le 2$ .\",\n \"We may also assume that $d(w_3) \\\\le 2$ as otherwise we can either solve our problem in polynomial time or reduce the instance.\",\n \"Proof of Claim E: Since $w_2w_3 \\\\notin E(G)$ by Claim C, Option (b) in Claim A cannot occur.\",\n \"Note that $d(w_1)=d(w_2)=3$ is not possible as $G \\\\in {\\\\cal G}_3$ and $d(v_1)=d(v_2)=3$ .\",\n \"For the sake of contradiction assume that $d(w_3)=3$ and that $N(w_3)=\\\\lbrace v_3,a,b\\\\rbrace $ .\",\n \"Recall that Option (b) in Claim A cannot occur.\",\n \"Assume that Option (c) in Claim A occurs.\",\n \"This implies that either $v_4 v_3 w_3 a$ or $v_4 v_3 w_3 b$ is a $P_4$ in $G-D$ where the end-points are not adjacent in $G$ , a contradiction.\",\n \"So Option (c) in Claim A cannot occur, which implies that Option (a) in Claim A must occur.\",\n \"This implies that $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .\",\n \"This proves Claim E.($\\\\diamond $ ) We now return to the proof of Lemma REF .\",\n \"If $d(w_1)=3$ then let $s=1$ and otherwise let $s=2$ .\",\n \"By Claim E we note that $d(w_{3-s})=2$ .\",\n \"Let $G^*$ be obtained from $G-\\\\lbrace v_1,v_2,v_3,v_4,w_{3-s}\\\\rbrace $ by adding a vertex $x$ and the edge $w_s w_3$ and $w_s x$ and setting $d^*_{}(x)=0$ and $a^*_{}(x)=1$ .\",\n \"See Figure REF for an illustration.\",\n \"We will now show that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.\",\n \"Figure: An illustration of G * G^* in the proof of Lemma when s=2s=2.First assume that $G$ is a YES-instance.\",\n \"By Claim E we note that Option (a) or Option (c) in Claim A must occur.\",\n \"If Option (a) occurs then $v_i w_i \\\\in D$ for all $i \\\\in [3]$ in $G$ .\",\n \"Let $D^* = D \\\\cup \\\\lbrace w_s w_3 \\\\rbrace - \\\\lbrace v_1 w_1, v_2 w_2, v_3 w_3\\\\rbrace $ .\",\n \"Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G-D$ and the component containing $w_1 w_2$ have been replaced by a component containing $x w_s$ in $G^* - D^*$ .\",\n \"However the component containing $w_1 w_2$ in $G-D$ is isomorphic with the component containing $x w_s$ in $G^*-D^*$ .\",\n \"So we get a solution for $G^*$ .\",\n \"See Figure REF for an illustration.\",\n \"Figure: An illustration of G-DG-D and G * -D * G^* - D^* when part (a) of Claim A occurs.If Option (c) occurs then $v_2 v_3, v_4 v_1 \\\\in D$ .\",\n \"Let $D^* = D - \\\\lbrace v_2 v_3, v_4 v_1\\\\rbrace $ .\",\n \"Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 w_1 w_2 v_2 v_1$ and the path $v_4 v_3 w_3$ in $G-D$ have been replaced by the path $x w_s w_3$ in $G^* - D^*$ .\",\n \"So we get a solution for $G^*$ .\",\n \"See Figure REF for an illustration.\",\n \"Figure: An illustration of G-DG-D and G * -D * G^* - D^* when part (b) of Claim A occurs.Conversely assume that $G^*$ is a YES-instance.\",\n \"We will now show that $G$ is a YES-instance.\",\n \"Let $D^*$ be the matching in $G^*$ such that all components in $G^*-D^*$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).\",\n \"As $d^*_{}(x)=0$ in $G^*$ we note that $x w_s \\\\notin D^*$ .\",\n \"We now consider the case when $w_s w_3$ belongs to $D^*$ and then the case when it does not.\",\n \"So first assume that $w_s w_3 \\\\in D^*$ .\",\n \"In this case let $D = D^* + \\\\lbrace v_1w_1, v_2 w_2, v_3 w_3 \\\\rbrace - \\\\lbrace w_s w_3\\\\rbrace $ .\",\n \"Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 v_4 w_4 w_1 v_1$ is in $G-D$ and the component containing $x w_s$ have been replaced by an isomorphic component containing $w_1 w_2$ in $G^* - D^*$ .\",\n \"So we get a solution for $G$ (this corresponds to the reverse process to that illustrated in Figure REF ).\",\n \"Now assume that $w_s w_3 \\\\notin D^*$ .\",\n \"This implies that any edges incident with $w_s$ and $w_3$ , different from $w_s w_3$ and $x w_s$ must belong to $D^*$ (due to the path $x w_s w_3$ in $G^* - D^*$ ).\",\n \"Let $D = D^* + \\\\lbrace v_2v_3, v_4 v_1 \\\\rbrace $ .\",\n \"Now $G-D$ and $G^* - D^*$ have exactly the same components except the path $x w_s w_3$ in $G^* - D^*$ has been replaced by the 4-cycle $v_1 w_1 w_2 v_2 v_1$ and the path $v_4 v_3 w_3$ in $G-D$ .\",\n \"So we get a solution for $G$ (this corresponds to the reverse process to that illustrated in Figure REF ).\",\n \"The above reduction completes the proof of the lemma.\",\n \"Let ${\\\\cal G}_5$ denote all triangle-free and quadrilateral-free graphs of maximum degree at most 3.\",\n \"Lemma 7 $(1,1)$ -Cluster Editing can be reduced from ${\\\\cal G}_4$ to ${\\\\cal G}_5$ in polynomial time.\",\n \"Let $G \\\\in {\\\\cal G}_4$ be arbitrary with vertex-weights $(a^*,d^*)$ .\",\n \"We may assume that $G$ is connected.\",\n \"If there is no 4-cycle in $G$ then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be any 4-cycle in $G$ .\",\n \"As $G \\\\in {\\\\cal G}_4$ , we know that at most two of the vertices in $C$ have degree three.\",\n \"If no vertex of $G$ has degree three then $V(G)=V(C)$ and we can solve our problem in constant time.\",\n \"So we may assume that at least one vertex on $C$ has degree three.\",\n \"If $d^*_{}(v_i)=0$ for any $i \\\\in [4]$ then any solution $G-D+A$ to our instance must have all edges between $C$ and $V(G)-V(C)$ in $D$ , which implies that $V(C)$ is an R-deletable set.\",\n \"So by Lemma REF , we can reduce the instance in this case.\",\n \"Thus, we may assume that $d^*_{}(v_i)=1$ for all $i \\\\in [4]$ .\",\n \"We first consider the case when exactly one vertex on $C$ has degree three.\",\n \"Without loss of generality assume that $d(v_1)=3$ and $d(v_2)=d(v_3)=d(v_4)=2$ .\",\n \"Let $e$ be the edge incident with $v_1$ that does not belong to $C$ .\",\n \"Let $G^*$ be obtained from $G^*=G-\\\\lbrace v_3,v_4\\\\rbrace $ by letting $d^*_{}(v_2)=0$ .\",\n \"Now $G^*$ is a YES-instance if and only if $G$ is a YES-instance, due to the following.\",\n \"Deleting $e$ from $G$ corresponds to deleting $e$ from $G^*$ and vice versa.\",\n \"Not deleting $e$ from $G$ corresponds to deleting two edges from $C$ in $G$ which corresponds to not deleting $e$ in $G^*$ .\",\n \"So $G$ is a YES-instance if and only if $G^*$ is a YES-instance.\",\n \"We may therefore assume that exactly two vertices on $C$ have degree 3.\",\n \"Let $x$ and $y$ be the two vertices on $C$ of degree 3.\",\n \"Let $x^{\\\\prime }$ be the neighbour of $x$ not on $C$ and let $y^{\\\\prime }$ be the neighbour of $y$ not on $C$ .\",\n \"Analogously to Claim A in both Lemma REF and Lemma REF we note that if our instance has a solution and $D$ is the set of edges that are deleted from $G$ in the solution, then one of the following holds.\",\n \"(a): $x x^{\\\\prime }, y y^{\\\\prime } \\\\in D$ and no edge on $C$ belongs to $D$ .\",\n \"(b): $x x^{\\\\prime }, y y^{\\\\prime } \\\\notin D$ and $v_1 v_2, v_3 v_4 \\\\in D$ and $v_2 v_3, v_4 v_1 \\\\notin D$ .\",\n \"(c): $x x^{\\\\prime }, y y^{\\\\prime } \\\\notin D$ and $v_2 v_3, v_4 v_1 \\\\in D$ and $v_1 v_2, v_3 v_4 \\\\notin D$ .\",\n \"First consider the case when $x$ and $y$ are adjacent vertices on $C$ .\",\n \"Then without loss of generality assume that $x=v_1$ and $y=v_4$ .\",\n \"As $G$ is triangle-free we have $x^{\\\\prime } \\\\ne y^{\\\\prime }$ .\",\n \"First assume that $x^{\\\\prime } y^{\\\\prime } \\\\in E(G)$ .\",\n \"Then $V(G)=\\\\lbrace v_1, v_2, v_3, v_4, x^{\\\\prime }, y^{\\\\prime }\\\\rbrace $ as $G \\\\in {\\\\cal G}_4$ , which implies that the cycle $x^{\\\\prime } x y y^{\\\\prime } x^{\\\\prime }$ ($=x^{\\\\prime } v_1 v_4 y^{\\\\prime } x^{\\\\prime }$ ) has at most two vertices of degree three, which are $x$ and $y$ .\",\n \"So in this case we can solve our problem in constant time.\",\n \"Therefore, we may assume that $x^{\\\\prime } y^{\\\\prime } \\\\notin E(G)$ .\",\n \"This also implies that part (b) above is not possible, as if part (b) occurs then $x^{\\\\prime } v_1 v_4 y^{\\\\prime }$ is a $P_4$ in $G-D$ where $x^{\\\\prime } y^{\\\\prime } \\\\notin E(G)$ , a contradiction.\",\n \"So either part (a) or part (c) occurs.\",\n \"This is illustrated in Figure REF .\",\n \"Figure: Possibble options for G-DG-D when xx and yy are adjacent.If $x$ and $y$ are not adjacent, then without loss of generality assume that $x=v_1$ and $y=v_3$ .\",\n \"If $x^{\\\\prime }=y^{\\\\prime }$ then $G$ contains a $K_{2,3}$ -subgraph, which implies that our instance is a NO-instance by Lemma REF .\",\n \"Thus, we may assume that $x^{\\\\prime } \\\\ne y^{\\\\prime }$ .\",\n \"Now part (a), part (b) and part (c) are illustrated in Figure REF .\",\n \"Figure: Possibble options for G-DG-D when xx and yy are not adjacent.So whether $x$ and $y$ is adjacent or not, we will in $G-D$ either have both edges $xx^{\\\\prime }$ and $yy^{\\\\prime }$ in $D$ or neither in $D$ .\",\n \"And if neither is in $D$ then each of $x^{\\\\prime }$ and $y^{\\\\prime }$ is the endpoint of a $P_3$ containing two vertices from $C$ .\",\n \"So in both cases we will remove $V(C)$ from $G$ and add the gadget $G(x,y)$ , illustrated in Figure REF , to $G$ instead.\",\n \"Note that $V(G(x,y))=\\\\lbrace q_1, x, q_3, q_4, y, q_6 \\\\rbrace $ and $E(G(x,y))=\\\\lbrace q_1 x, x q_3, q_3 q_4, q_4 y, y q_5 \\\\rbrace $ and we add the edges $x x^{\\\\prime }$ and $y y^{\\\\prime }$ .\",\n \"Furthermore $d^*_{}(q_1)=d^*_{}(q_6)=0$ and all other $d^*_{}$ -values and $a^*_{}$ -values are one.\",\n \"Figure: The gadget G(x,y)G(x,y).It is not difficult to check that if the modified graph, $G^{\\\\prime }$ , is a YES-instance and $D^{\\\\prime }$ the matching that gets removed from $G^{\\\\prime }$ , then we must have one of the two cases illustrated in Figure REF .\",\n \"Figure: The two options, (I) and (II), for G ' -D ' G^{\\\\prime }-D^{\\\\prime }.So $G^{\\\\prime \\\\prime }$ has either both edges $xx^{\\\\prime }$ and $yy^{\\\\prime }$ in $D^{\\\\prime }$ or neither in $D^{\\\\prime }$ .\",\n \"And if neither are in $D^{\\\\prime }$ then both $x^{\\\\prime }$ and $y^{\\\\prime }$ are endpoints of a $P_3$ containing two vertices from the gadget $G(x,y)$ .\",\n \"This was exactly the same property we had in $G$ , so $G$ is a YES-instance if and only if $G^{\\\\prime }$ is.\",\n \"Furthermore, this reduction deletes a 4-cycle.\",\n \"So even though the reduction adds an edge (that is, $|E(G^{\\\\prime })|=|E(G)|+1$ ) the reduction can be carried out at most as many times as there are 4-cycles in $G$ which is at most a polynomial number.\",\n \"Therefore repeatadly performing the above reduction creates a graph in ${\\\\cal G}_5$ in polynomial time, as desired.\"\n ],\n [\n \"Polynomial-time algorithm for triangle-free and quadrilateral-free graphs of maximum degree at most 3\",\n \"Lemma REF implies the following: Lemma 8 Let $G$ be a triangle-free and quadrilateral-free graph with maximum degree at most 3.\",\n \"Then $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if there is a matching $D$ of deletable edges from $G$ such that every connected component $C$ of $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\\\cong P_3$ then the edge of $\\\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .\",\n \"Let ${\\\\cal G}_6$ denote all triangle-free and quadrilateral-free graphs of maximum degree at most 3, such that the following holds.\",\n \"All vertices $v$ of degree at least 2 have $d^*_{}(v)=1$ .\",\n \"No vertex $v$ of degree 3 is adjacent to a vertex $w$ with $a^*_{}(w)=0$ .\",\n \"The vertices of degree 3 form an independent set.\",\n \"Lemma 9 $(1,1)$ -Cluster Editing can be reduced from ${\\\\cal G}_5$ to ${\\\\cal G}_6$ in polynomial time.\",\n \"Let $G \\\\in {\\\\cal G}_5$ be arbitrary with vertex-weights $(a^*,d^*)$ .\",\n \"Note that if some vertex $v$ has $d(v)=3$ and $d^*_{}(v)=0$ then $(G,a^*_{},d^*_{})$ is a NO-instance, so we may assume that $d^*_{}(v)=1$ for all vertices, $v$ , of degree 3.\",\n \"If $d(v)=2$ and $d^*_{}(v)=0$ , then $(G,a^*_{},d^*_{})$ is a YES-instance if and only if both edges incident to $v$ have to remain in $G-D$ and therefore form a $P_3.$ Let $N(x)=\\\\lbrace u,v\\\\rbrace ,$ and note that $\\\\lbrace x,u,v\\\\rbrace $ is an R-deletable set.\",\n \"So by Lemma REF , we can reduce the instance.\",\n \"Assume that there exists a $uw \\\\in E(G)$ with $d(u)=3$ and $a^*_{}(w)=0$ .\",\n \"Assume that $G$ is a YES-instance and $G-D+A$ is a cluster graph.\",\n \"If $uw \\\\notin D$ , then $w$ is the endpoint of a $P_3$ in $G-D$ which is a contradiction as $a^*_{}(w)=0$ .\",\n \"So, we must have $uw \\\\in D$ .\",\n \"Since $uw \\\\in D$ , $d^*_{}(u)=d^*_{}({w})=1$ or $(G,a^*,d^*)$ is a NO-instance.\",\n \"Form a new instance with $G^{\\\\prime }=G- uw$ and $d^*_{}(u)=d^*_{}({w})=0$ .\",\n \"Then $G$ is a YES-instance if and only if $G^{\\\\prime }$ is a YES-instance.\",\n \"This reduction allows us to get rid of all cases where a degree three vertex is adjacent to a vertex with $a^*_{}$ -value zero.\",\n \"Now assume that there exists a $uw \\\\in E(G)$ with $d(u)=d(w)=3$ .\",\n \"Assume that $G$ is a YES-instance and $G-D+A$ is a cluster graph.\",\n \"If $uw \\\\notin D$ , then we obtain a $P_4$ in $G-D$ containing the edge $uw$ , a contradiction.\",\n \"So, we must have $uw \\\\in D$ .\",\n \"As $d(u)=d(w)=3$ , we have $d^*(u)=d^*(w)=1$ .\",\n \"Now we form a new instance with $G^{\\\\prime }=G- uw$ and $d^*_{}(u)=d^*_{}(v)=0$ as above and note that $G$ is a YES-instance if and only if $G^{\\\\prime }$ is a YES-instance.\",\n \"Thus, we may assume that $uw \\\\notin E(G)$ if $d(u)=d(w)=3$ .\",\n \"This completes the proof.\",\n \"In the proof of the next lemma, we will use the well-known result that given a graph $G$ in polynomial time either a perfect matching can be found in $G$ or we can decide that $G$ has no perfect matching.\",\n \"Lemma 10 Let $G$ be a graph and let $Y$ and $Z$ be disjoint sets of vertices in $G$ .\",\n \"In polynomial time, we can decide whether $G$ contains a matching covering $Y$ , but not covering $Z$ .\",\n \"Let $G^{\\\\prime }=G-Z$ .\",\n \"Observe that $G$ contains a matching covering $Y$ and not covering $Z$ if and only if $G^{\\\\prime }$ contains a matching covering $Y$ .\",\n \"Let $U= V(G^{\\\\prime })-Y$ and if $G^{\\\\prime } $ has odd number of vertices then add an extra (isolated) vertex to $G^{\\\\prime }$ which is not in $Y$ (and therefore will be added to $U$ ).\",\n \"Finally add edges between all pairs of non-adjacent vertices in $U$ .\",\n \"Let the resulting graph be $G^{\\\\prime \\\\prime }.$ Now we will show that $G^{\\\\prime \\\\prime }$ has a perfect matching if and only if there is a matching in $G ^{\\\\prime }$ covering $Y.$ If $M^{\\\\prime \\\\prime }$ is a perfect matching in $G^{\\\\prime \\\\prime }$ then by deleting all edge with both end-points in $U$ we get the desired matching in $G^{\\\\prime }.$ And if $M^{\\\\prime }$ is the desired matching in $G^{\\\\prime }$ then adding as many edges as possible with both endpoints in $U$ gives us a perfect matching in $G^{\\\\prime \\\\prime }.$ Lemma 11 $(1,1)$ -Cluster Editing can be solved in polynomial time for all graphs in ${\\\\cal G}_6$ in polynomial time.\",\n \"Let $G \\\\in {\\\\cal G}_6$ be arbitrary with vertex-weights $(a^*,d^*)$ .\",\n \"Let $X_i = \\\\lbrace v \\\\; | \\\\; d_G(v)=i \\\\rbrace $ for all $i=0,1,2,3$ .\",\n \"We may assume that $G$ is connected as otherwise we can just consider each connected component separately (it will never be advantages to add edges between different components).\",\n \"We may clearly also assume that $G$ has at least two vertices, which implies that $X_0 = \\\\emptyset $ .\",\n \"Let $X_2^{\\\\prime }$ denote all vertices in $X_2$ with an edge to a vertex in $X_3$ and let $X_2^{\\\\prime \\\\prime } = X_2 \\\\setminus X_2^{\\\\prime }$ .\",\n \"Let $Z$ contain all vertices, $z$ , in $X_2^{\\\\prime \\\\prime }$ , such that $z$ is adjacent to a vertex, $w$ , with $a^*_{}(w)=0$ .\",\n \"In other words, $Z$ contains all vertices from $X_2^{\\\\prime \\\\prime }$ which are neighbours of a vertex which we are not allowed to add edges to.\",\n \"Define $Y$ as follows: $Y = Z \\\\cup X_2^{\\\\prime } \\\\cup X_3.$ We will show that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, if and only if there exists a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .\",\n \"This will give us the desired polynomial algorithm by Lemma REF .\",\n \"First assume that $D$ is a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .\",\n \"We may assume that $D$ is maximal, as otherwise we just keep adding edges to $D$ (where both end-points have $d^*$ -value 1) until it becomes maximal (which is not necessarily maximum).\",\n \"Let $G^{\\\\prime } = G-D$ .\",\n \"By Lemma REF it suffices to show that every component, $C$ , in $G^{\\\\prime }$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\\\cong P_3$ then the edge of $\\\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .\",\n \"As every vertex in $X_3$ belongs to $Y$ we note that $\\\\Delta (G^{\\\\prime }) \\\\le 2$ .\",\n \"For the sake of contradiction assume that $uv \\\\in E(G^{\\\\prime })$ and $d_{G^{\\\\prime }}(u)=d_{G^{\\\\prime }}(v)=2$ .\",\n \"As all vertices in $X_2^{\\\\prime }$ will have degree 1 in $G^{\\\\prime }$ we note that neither $u$ nor $v$ belong to $X_2^{\\\\prime }$ .\",\n \"As all vertices in $X_3$ are only adjacent to vertices in $X_2^{\\\\prime }$ , by the definition of ${\\\\cal G}_6$ and $X_2^{\\\\prime }$ , we note that neither $u$ nor $v$ belong to $X_3$ .\",\n \"So $u$ and $v$ must both belong to $X_2^{\\\\prime \\\\prime }$ .\",\n \"By the definition of ${\\\\cal G}_6$ we note that $d^*_{}(u)=d^*_{}(v)=1$ .\",\n \"However, this contradicts the fact that $D$ is maximal, as we could have added the edge $uv$ to $D$ .\",\n \"So, no $uv \\\\in E(G^{\\\\prime })$ has $d_{G^{\\\\prime }}(u)=d_{G^{\\\\prime }}(v)=2$ .\",\n \"This implies that all components in $G^{\\\\prime }$ are isomorphic to $P_1$ or $P_2$ or $P_3$ .\",\n \"Assume that $C$ is a component in $G^{\\\\prime }$ isomorphic to $P_3$ and let $v_1 v_2 v_3$ be the 3-path in $C$ .\",\n \"By the construction of $G^{\\\\prime }$ we note that $v_2 \\\\in X_3$ or $v_2 \\\\in X_2^{\\\\prime \\\\prime }$ .\",\n \"If $v_2 \\\\in X_3$ then, by the definition of ${\\\\cal G}_6$ , we note that $a^*_{}(v_1)=a^*_{}(v_3)=1$ .\",\n \"And if $v_2 \\\\in X_2^{\\\\prime \\\\prime }$ , then $v_2 \\\\notin Y$ which by the definition of $Z$ implies that $a^*_{}(v_1)=a^*_{}(v_3)=1$ .\",\n \"So in both cases $a^*_{}(v_1)=a^*_{}(v_3)=1$ , which by Lemma REF , implies that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, as desired.\",\n \"Now conversely assume that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing and that $D$ denotes the edges deleted from $G$ in the solution.\",\n \"By definition $D$ is a matching in $G$ and by Lemma REF every component, $C$ , in $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\\\cong P_3$ then the edge of $\\\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .\",\n \"Let $G^* = G-D$ .\",\n \"As $\\\\Delta (G^*) \\\\le 2$ we note that $D$ covers all vertices in $X_3$ .\",\n \"Let $u \\\\in X_2^{\\\\prime }$ be arbitrary and let $v$ be a neighbour of $u$ in $G$ such that $v \\\\in X_3$ (which exists by the definition of $X_2^{\\\\prime }$ ).\",\n \"Let $N(u)=\\\\lbrace x,v\\\\rbrace $ and let $N(v)=\\\\lbrace u,s,t\\\\rbrace $ , where as $G$ is triangle-free and quadrilateral-free implies that $x,v,u,s,t$ are all distinct.\",\n \"If no edge incident with $u$ belongs to $D$ , then either $xuvs$ or $xuvt$ will be a $P_4$ in $G^*$ , a contradiction, so $u$ is covered by $D$ .\",\n \"So all vertices in $X_2^{\\\\prime }$ are covered by $D$ and previously we showed that all vertices in $X_3$ are also covered by $D$ .\",\n \"Let $z \\\\in Z$ be arbitrary.\",\n \"That is $z \\\\in X_2^{\\\\prime \\\\prime }$ and $z$ is adjacent to a vertex, $w$ , with $a^*_{}(w)=0$ .\",\n \"Let $N(z)=\\\\lbrace w,q\\\\rbrace $ and note that if $z$ is not covered by $D$ then $wzq$ is a $P_2$ in $G^*$ , but as $a^*_{}(w)=0$ , the edge $wq \\\\in \\\\overline{G}$ is not addable to $G$ .\",\n \"So $z$ must be covered by $D$ .\",\n \"This implies that $D$ is a matching covering $Y$ in $G$ .\",\n \"Furthermore, by definition, $D$ does not cover any vertex with $d$ -value zero.\",\n \"We have therefore shown that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, if and only if there exists a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .\",\n \"Corollary 12 $(1,1)$ -Cluster Editing can be solved in polynomial time.\",\n \"This follows immediately from Lemmas REF , REF , REF , REF , REF and REF .\"\n ],\n [\n \"Discussion\",\n \"The main result of our paper concludes a complete dichotomy of complexity of $(a,d)$ -Cluster Editing.\",\n \"We proved that $(1,1)$ -Cluster Editing can be solved in polynomial time.\",\n \"Our proof consists of two stages: (i) providing a serious of five polynomial-time reductions to $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$ -Cluster Editing on $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3.\",\n \"While Stage 2 is relatively short, Stage 1 is not.\",\n \"Moreover, while our reduction from all graphs to $C_3$ -free graphs of maximum degree at most 3 is not hard, getting rid of 4-cycles required a lengthy and non-trivial series of four polynomial-time reductions.\",\n \"It would be interesting to see whether the four reductions can be replaced by a shorter and simpler series of reductions.\"\n ]\n]"},"id":{"kind":"string","value":"2210.07722"}}},{"rowIdx":2998,"cells":{"text":{"kind":"list like","value":[["Theory and Approximate Solvers for Branched Optimal Transport with\n Multiple Sources"],["Abstract Branched Optimal Transport (BOT) is a generalization of optimal transport in which transportation costs along an edge are subadditive.","This subadditivity models an increase in transport efficiency when shipping mass along the same route, favoring branched transportation networks.","We here study the NP-hard optimization of BOT networks connecting a finite number of sources and sinks in $\\mathbb{R}^2$.","First, we show how to efficiently find the best geometry of a BOT network for many sources and sinks, given a topology.","Second, we argue that a topology with more than three edges meeting at a branching point is never optimal.","Third, we show that the results obtained for the Euclidean plane generalize directly to optimal transportation networks on two-dimensional Riemannian manifolds.","Finally, we present a simple but effective approximate BOT solver combining geometric optimization with a combinatorial optimization of the network topology."],["Introduction","Optimal transport (OT) [27], [6], [23] stipulates transportation costs that increase linearly with the transported mass.","However, in many systems of practical and theoretical interest, a diminishing cost property is more realistic: it is more economic to jointly transport two loads with nearby destinations along the same route.","The optimal transportation networks under diminishing costs exhibit branching; and indeed, nature and societies are using branched networks, e.g.","in blood circulation, gas supply or mail delivery.","In this paper, we study the theory and practice of finding good or even optimal solutions in branched optimal transport (BOT).","More formally, we consider a finite set of sources $S$ with supplies $\\mu _S > 0$ and sinks $T$ with demands $\\mu _T > 0$ , located at fixed positions $x_S$ and $x_T$ in $\\mathbb {R}^2$ .","A possible transportation network is represented as a directed, edge-weighted graph $G(V,E)$ with nodes $V = S \\cup T \\cup B$ .","The edges $E \\subset V \\times V$ interconnect the terminals $S$ and $T$ with the help of a set of additional nodes $B$ , so-called branching points (BPs), with coordinates $x_B$ .","The edge direction indicates the direction of mass flow.","The edge weights, denoted by $m_e$ , specify the absolute flows.","Gilbert first proposed the BOT problem [9] in which the objective is to solve for $ \\underset{B, E, x_B, m_E}{\\arg \\min } \\ \\sum _{(i,j) \\, \\in \\, E} m_{ij}^\\alpha &\\left\\Vert x_i - x_j \\right\\Vert _2 , \\text{ subject to }\\\\ \\nonumber \\text{supply } \\ \\mu _s = \\sum \\nolimits _{k} m_{sk} &- \\sum \\nolimits _{k} m_{ks} \\ \\text{ at each source } s,\\\\ \\nonumber \\text{demand } \\ \\mu _t = \\sum \\nolimits _{k} m_{kt} &- \\sum \\nolimits _{k} m_{tk} \\ \\text{ at each sink } t,\\\\ \\nonumber \\text{conservation } \\ \\sum \\nolimits _{k} m_{kb} &= \\sum \\nolimits _{k} m_{bk} \\ \\text{ at each BP } b,$ given a single parameter $\\alpha \\in [0,1]$ .","The problem of BOT is interesting in that it combines combinatorial optimization (over $B$ , $E$ ) with continuous optimization (over $x_B$ , $m_E$ ).","Figure: Branched optimal transport (Eq. )","interpolates between optimal transport and the Euclidean Steiner tree problem.","On a toy example, shown are good BOT solutions (found by our approximate solver for α≠1\\alpha \\ne 1, see Sect. )","for the same set of sources (red) and sinks (blue).","The disk sizes indicate the demands and supplies, the edge widths the mass transported along each edge.For $\\alpha = 1$ , the BOT problem is the discrete version of the famous optimal transport problem for which optimal solutions can be found efficiently [6], [23].","However, due to the linearity of the cost function, OT solutions do not exhibit any branching but consist of straight lines between sources and sinks, see Fig.","REF .","In contrast, for $\\alpha \\in [0,1)$ , the subadditivity of $m \\mapsto m^\\alpha $ reflects the increased efficiency of transporting loads together, i.e.","$(m_1 + m_2)^\\alpha < m_1^\\alpha + m_2^\\alpha $ .","Thus, for $\\alpha \\in [0,1)$ , BOT solutions show a branched structure, see Fig.","REF -al0.","Unlike OT, the optimization problem of BOT is NP-hard [11].","In the special case of $\\alpha = 0$, BOT turns into the well-studied Euclidean Steiner tree problem (ESTP) [28], [13].","In the ESTP, the objective is to find the overall shortest network that interconnects all terminals (with the help of BPs), independently of the edge flows, since $m^0 = 1$ .","For different values of $\\alpha $ , BOT interpolates between these two optimization problems, see Fig.","REF ."],["Connection of BOT to machine learning.","Optimal transport has emerged as an important tool in machine learning [1], [5], [23].","BOT is a strict generalization, describing a more versatile concept and more challenging optimization problem.","BOT offers a mathematical formalism that is deceivingly simple (cf. Eq.","(REF )) and yet engenders non-trivial structure.","Many machine learning problems such as tracking of divisible targets (computer vision), skeletonization (image analysis), trajectory inference (bioinformatics) come with input that is essentially continuous (images, distributions) and require structured output that is discrete, e.g. graphs.","Arguably, this transition from continuous to discrete is one of the most interesting aspects (and an unsolved problem) in current machine learning research.","It is also a problem that cannot be solved by a mere upscaling of standard deep learning architectures.","In addition, routing problems have become a popular problem to challenge machine learning and amortized optimization algorithms with difficult optimization problems [17], [3], [15].","Combining combinatorial and continuous optimization, BOT is a highly instructive target for new machine learning approaches.","In Sect.","we address the generalization of BOT to higher-dimensional Euclidean space, particularly relevant for applications in data science.","In this paper, we make the following contributions: We generalize an existing method for constructing BOT solutions with optimal geometry to the case of multiple sources.","Based on this generalization, we present an analytical and numerical scheme to rule out $n$ -degree branchings with $n > 3$ .","Further, we demonstrate how to extend geometric and topological properties of optimal BOT solutions to two-dimensional Riemannian manifolds.","Lastly, we propose a more practical numerical algorithm for the geometry optimization together with a simple but compelling heuristic, addressing the optimization of the BOT topology.","To the best of our knowledge, no readily accessible code for finding BOT solutions is publicly available.","By making our code available at https://github.com/hci-unihd/BranchedOT we hope to aid the evolution of the field."],["Topology and geometry of BOT solutions","A BOT problem can be divided into the combinatorial optimization of the network topology, specified by the set of BPs $B$ and edges $E$ (see Sect.","), and the geometric optimization of the BP positions $x_B$ .","[2] showed that optimal BOT solutions can be assumed to be acyclic, which restricts the search for the optimal topology to trees.","Given $n$ terminals, WLOG, the topology can be represented as a so-called full tree topology, which has $n-2$ BPs, each of degree three.","Higher-degree branchings may effectively form during the geometry optimization if multiple BPs settle at the same position.","A set of such BPs is referred to as coupled BP, cf. Fig.","REF .","The union set of all neighbors of the individual BPs (not including the BPs themselves) is referred to as set of effective neighbors.","Conversely, a BP configuration in which all BPs are uncoupled and located away from the terminals is called non-degenerate, see Fig.","REF .","The number of distinct full tree topologies interconnecting $n$ terminals is given by $(2n - 5)!!","= (2n - 5) \\cdot (2n - 7) \\cdot ... \\cdot 3 \\cdot 1$ and hence increases super-exponentially with the number of terminals [25].","Given 100 terminals, one would have to consider more than $10^{18}$ possible full tree topologies, making an exhaustive search computationally intractable already for problems of modest size.","Fortunately, given a tree topology, the geometric optimization of the BP positions reduces to a convex optimization problem, as all edge flows $m_{ij}$ are already uniquely determined by the flow constraints in Eq.","(REF ).","The corresponding linear system can be solved in linear time by dynamic programming, called “elimination on leaves of a tree” in [26].","Since the Euclidean norm, like any norm, is convex, given a fixed tree topology, the cost function in Eq.","(REF ) becomes a convex function of the BP positions.","Together with the independence of the individual BPs, this implies the following lemma on the optimal substructure of BOT solutions (see App. ).","Definition 2.1 For a chosen topology $T$ , a BOT solution is called a relatively optimal solution (ROS of $T$ ) if its BP configuration has minimal cost.","The overall best BOT solution, given by the optimal topology together with its ROS, is called the globally optimal solution (GOS).","Lemma 2.1 (a) For a given tree topology, a BOT solution is relatively optimal if and only if every (coupled) BP connects its (effective) neighbors at minimal cost.","(b) In a globally optimal solution, every subsolution restricted to a connected subset of nodes solves its respective subproblem (cf. App. )","globally optimally.","Figure: Two BP configurations for the same full tree topology: A higher-degree branching may effectively form by coupling the two BPs at the same position."],["Geometric optimization of BOT solutions","Although the BP optimization for a given tree topology is a convex problem, as argued above, it is non-trivial, since the objective function is not everywhere differentiable.","Here, we present a principled geometric approach, which was first suggested by Gilbert in [9] and previously developed in the context of the ESTP [20].","More recently, this approach was discussed in the comprehensive work by [2], where it was applied exclusively to BOT problems with a single source.","A generalization to the case of multiple sources was posed as an open problem by the authors (see Problem 15.11), for which we give the solution in this section."],["Geometric solution for one source and two sinks","Motivated by Lem.","REF , we start by considering a single BP $b$ in isolation (cf. Fig.","REF ), following [2].","Given a source at positionWe will often use the node label, e.g., $a_0$ , to denote also the position of the node, instead of writing $x_{a_0}$ .","$a_0$ and two sinks at positions $a_1$ and $a_2$ , we aim to find the optimal position $b^*$ for the BP connecting the three terminals, i.e., the minimizer of $ \\mathcal {C}(b) = m_1^\\alpha \\vert a_1 - b \\vert + m_2^\\alpha \\vert a_2 - b \\vert + (m_1 + m_2)^\\alpha \\vert a_0 - b \\vert ,$ where $m_1$ and $m_2$ are the respective demands of the two sinks.","Due to the convexity of $\\mathcal {C}(b)$ , the minimum must lie either at a stationary point at which $\\nabla _{\\!","\\!","b} \\, \\mathcal {C} = 0$ or at a non-differentiable point, where $b$ coincides with one of the $a_i$ .","[2] showed that the gradient is equal to zero if and only if the branching angles $\\theta _i$ , see Fig.","REF , are given by $&\\theta ^*_1 = \\arccos \\bigg ( \\frac{k^{2\\alpha } + 1 - (1-k)^{2 \\alpha }}{2 k^\\alpha } \\bigg ) =: f(\\alpha , k), \\nonumber \\\\&\\theta ^*_2 = \\arccos \\bigg ( \\frac{(1 \\!","- \\!","k)^{2\\alpha } + 1 - k^{2 \\alpha }}{2 (1 \\!","- \\!","k)^\\alpha } \\bigg ) = f(\\alpha , 1 \\!","-k), \\\\&\\theta ^*_1 + \\theta ^*_2 = \\arccos \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) =: h(\\alpha , k) \\nonumber ,$ where we have defined the flow fraction $k:= m_1/(m_1 + m_2)$ and the two functions $f$ and $h$ , related via $h(\\alpha , k) = f(\\alpha , k) + f(\\alpha , 1-k)$ .","If a BP exists that realizes the branching angles $\\theta _i^*$ , it can be constructed geometrically based on the central angle property (see App. ).","It states that, given a circle through $a_1$ and $a_2$ , the angle $\\angle a_1 o a_2$ at the center $o$ is twice the angle enclosed with a point anywhere on the opposite circle arc, cf. Fig.","REF .","In particular, let us construct the so-called pivot circle with central angle $\\angle a_1 o a_2 = 2\\theta ^*_1 + 2\\theta ^*_2$ and pivot point $p$ as in Fig.","REF .","Applying the central angle property twice (once for $\\theta _1^*$ and once for $\\theta _2^*$ ), a BP located at the intersection of the lower circle arc and the connection line $\\overline{a_0p}$ realizes both angles $\\theta ^*_i$ and is therefore optimal.","Figure: (a) Branching point bb connecting one source and two sinks with branching angles θ 1 \\theta _1 and θ 2 \\theta _2.","(b) Construction of the optimal BP b * b^* applying twice the central angle property illustrated in (c).","(d) shows the relation of the optimal branching angles θ i * \\theta ^*_i and ϑ i * \\vartheta ^*_i, relevant for the case of asymmetric branchings (see Sect.",").However, given the pivot point and pivot circle, $\\overline{a_0 p}$ may not intersect the lower circle arc, depending on the position of $a_0$ .","Accordingly, the lower half plane can be partitioned into a region for which the described construction yields an optimal Y-shaped branching and three other regions, see Fig.","REF .","For $a_0$ located in one of these regions, the optimal BP position coincides with one of the terminals, resulting in a V-shaped branching ($b^* = a_0$ ) or an L-shaped branching ($b^* \\in \\lbrace a_1,a_2\\rbrace $ ), cf. Fig.","REF [2]."],["Geometric construction of BOT solutions for a given topology","Applying the geometric construction from above in a recursive manner, one can construct the ROS (see Def.","REF ) for larger BOT problems, as illustrated in Fig.","REF .","Given a full tree topology $T$ , first, we determine all edge flows (see Sect. )","and consequently the optimal branching angles.","Then, a root node is chosen, arbitrarily (here $a_0$ ), and all other nodes are sorted based on the number of edges to $a_0$ (ignoring edge directions and resolving ties arbitrarily).","Starting from the furthest nodes and working towards the root, two nodes are recursively summarized by a pivot point, constructed from the optimal branching angles, see Fig.","REF -recur2.","Afterwards, in reversed order, the optimal BPs are placed iteratively, each as in the 1-to-2 case, see Fig.","REF -recur4.","In this manner, the optimal branching angles are realized at every BP and the resulting solution is a ROS of $T$ by Lem.","REF .","The choice of the root node induces a node ordering as described above.","Given this ordering, consider any BP $b$ and denote its children by $a_1$ and $a_2$ and its parent node by $a_0$ .","The construction of the pivot point now requires the positions of $a_1$ and $a_2$ and the optimal branching angles enclosed by the children edges $(b,a_1)$ and $(b,a_2)$ .","However, the branching angles do not only depend on the absolute flows $m_1$ and $m_2$ of the respective edges but also on the flow directions.","Given that both flows point towards $b$ or given that both flows point away from $b$ , as in Fig.","REF , the branching is referred to as symmetric and the optimal branching angles of interest are given by $\\theta _i^*$ , cf. Eq.","(REF ).","Note that BOT problems and their solutions are fully symmetric under complete exchange of sinks and sources (up to reversal of all flow directions).","On the contrary, in case of one flow pointing towards $b$ and one pointing away from $b$ , referred to as asymmetric branching, the optimal branching angles $\\vartheta _i^*$ enclosed by the children edges are calculated differently, see Fig.","REF .","However, the branching angles $\\vartheta _i^*$ can be related geometrically to the known $\\theta _i^*$ .","Using the functions $f$ and $h$ from Eq.","(REF ), we find that $\\begin{aligned}\\vartheta ^*_1 &= \\pi - \\theta ^*_1 - \\theta ^*_2 = \\pi - h\\Big (\\alpha , k=\\frac{m_2 - m_1}{m_2}\\Big ) \\\\\\vartheta ^*_2 &= \\theta ^*_1 = f\\Big (\\alpha , k = \\frac{m_2 - m_1}{m_2}\\Big ).\\end{aligned}$ After determining the two angles $\\vartheta _1^*$ and $\\vartheta _2^*$ from the flows $m_1$ and $m_2$ , the BP construction based on the central angle property works analogously to the symmetric case.","Crucially, this distinction of symmetric and asymmetric branching makes the recursive construction applicable also to problems with multiple sources, where asymmetric branchings may be unavoidable, consider e.g. Fig.","REF with $a_0$ and $a_3$ as sources and $a_1$ and $a_2$ as sinks (see App.","REF ).","Further, note that the known conditions for optimal V- and L-branching can be transferred to the asymmetric case simply by relabelling $a_0 \\rightarrow a_2$ , $a_1 \\rightarrow a_0$ and $a_2 \\rightarrow a_1$ , cf. Fig.","REF and Fig.","REF .","In terms of angular inequalities (derived in App.","), these conditions, for both branching types, are summarized in Table REF .","Figure: Recursive geometric construction of a relatively optimal solution, using one pivot point and pivot circle per branching point to collectively realize the optimal branching angles.Figure: Regions of optimal Y-, V- and L-branching.","Table: NO_CAPTION tableRelations between the L- and V-branching conditions.","In principle, given a full tree topology, the described method efficiently constructs the ROS in linear time.","However, as already pointed out by [9], the approach has some practical limitations, even after our generalization.","Figure REF shows how the pivot point is constructed only from the positions of two children $a_1$ and $a_2$ and the corresponding optimal branching angles.","However, a priori there are two possible pivot point locations, one in the upper and one in the lower half plane with respect to $\\overline{a_1a_2}$ .","Hence, the construction relies on knowing in which half plane the third node $a_0$ lies.","For larger trees, the topological parent $a_0$ may itself be a BP whose position is not yet determined.","In the worst case, one would thus have to try all $2^{n-2}$ possible pivot point combinations to find the ROS.","This pivot point degeneracy gets substantially worse in higher dimensions, making the recursive construction applicable only in $\\mathbb {R}^2$ .","Secondly, the geometric construction only produces solutions which are non-degenerate, i.e., solutions without edge contractions.","For now, the geometric construction is therefore primarily of theoretical interest; and indeed, it forms the basis of our following arguments.","Note that both of the aforementioned problems could be solved elegantly in the special case of $\\alpha = 0$ [12], [14]."],["Properties of optimal BOT topologies","Let us now consider topological modifications in order to improve the transportation cost of a BOT solution.","In particular, we intend to show that a topology $T$ can be improved if its ROS contains coupled BPs.","Let us start by considering a general BOT solution which contains a coupled 4-BP, i.e., a coupled BP with four effective neighbors, as in Fig.","REF .","Lemma REF states that a solution is not globally optimal if any subsolution is not globally optimal.","It will therefore suffice to study the coupled BP as an isolated subproblem."],["Non-optimality of coupled branching points","Given two sources and two sinks, there are two possible configurations in which the terminals can be arranged, cf. Fig.","REF ,fork2.","First we address the case in which the two sources are at opposite corners of the terminal quadrilateral, as in Fig.","REF .","Based on Lem.","REF , a necessary condition for the existence of a globally optimal 4-BP is that all four V-branchings between neighboring terminals are optimal.","This puts a lower bound on each of the angles $\\gamma _i$ , see Tab.","REF .","The general idea, also regarding the other 4-branching scenarios, is to show that the angular sum of these lower bounds already exceeds $2 \\pi $ .","This will immediately imply that not all V-branchings can be optimal simultaneously and thus a coupled 4-BP cannot be globally optimal.","Given a 4-BP as in Fig.","REF , all V-branchings are asymmetric (i.e.","neighboring flows point in opposite directions).","Hence, all four lower bounds (in Tab.","REF ) are of the form $\\gamma _i \\ge \\pi - \\theta ^*_2 = \\pi - f(\\alpha , 1 - k)$ and indeed $\\pi - f(\\alpha , 1 - k) > \\pi / 2$ , see Lem.","REF , so that their sum exceeds $2 \\pi $ .","Next, let us consider the scenario in Fig.","REF with two sources at neighboring corners.","WLOG, we use the normalization $m_1 + m_2 = 1 = m_3 + m_4$ and assume that $m_1 > m_3$ and $m_2 < m_4$ .","In this case, the four conditions for optimal V-branching in Tab.","REF read: $\\gamma _1 \\ge \\pi - f\\Big (\\alpha , 1-\\frac{m_1 - m_3}{m_1} \\Big ) = \\pi - f\\Big (\\alpha , \\frac{m_3}{m_1} \\Big )$ , $\\gamma _2 \\ge h\\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) = h(\\alpha , m_1)$ , $\\gamma _3 \\ge \\!","\\pi - f\\Big (\\alpha , 1 \\!","- \\frac{m_4 - m_2}{m_4}\\Big ) \\!","= \\pi - f\\Big (\\alpha , \\frac{1- m_1}{1 - m_3} \\Big )$ , $\\gamma _4 \\ge h\\Big (\\alpha , \\frac{m_3}{m_3 + m_4} \\Big ) = h (\\alpha , m_3)$ , Figure: Different scenarios of coupled 4-BPs with symmetric branching angles in blue, asymmetric ones in orange.","where the expressions (REF ) were plugged into the V-branching conditions in Tab.","REF for symmetric and asymmetric branching respectively, as indicated by the colors in Fig.","REF .","Let us show that in fact for all combinations of $\\alpha $ , $m_1$ and $m_3$ the sum of the lower bounds already exceeds $2 \\pi $ .","Indeed, summing the lower bounds and subtracting $2\\pi $ yields $&\\underbrace{h(m_1)}_{= \\ h(1-m_1)} + \\ h(m_3) - f\\Big (\\underbrace{\\frac{m_3}{m_1}}_{> \\, m_3} \\Big ) - f\\Big (\\underbrace{ \\frac{1 - m_1}{1 - m_3} }_{> \\, 1 - m_1} \\Big ) >h(1-m_1) + h(m_3) - f(m_3) - f(1-m_1) \\\\&= f(m_1) + f(1-m_3) > 0,$ using $h(\\alpha ,k) = f(\\alpha , k) + f(\\alpha , 1-k)$ and the fact that $f(\\alpha , k)$ is strictly decreasing with respect to $k$, see Lem.","REF .","To summarize, we have arrived at the following lemma: Lemma 4.1 A coupled 4-BP not coincident with a terminal connecting two sources and two sinks is never globally optimal.","Exactly the same logic applies for a coupled 4-BP connecting one source and three sinks (or equivalently 3 sources and 1 sink), as in Fig.","REF .","WLOG, in the following, we normalize the flows so that $m_1 + m_2 + m_3 =1$ .","We then determine the necessary conditions under which all V-branchings are optimal.","We again intend to show that such a 4-BP can never be globally optimal by showing that for any combination of $\\alpha $ and ${m_i}$ the sum of the lower bounds exceeds $2 \\pi $ .","This is equivalent to proving the following inequality (see App.","REF ): $h \\Big (\\frac{m_1}{m_1 + m_2} \\Big ) &- f( m_1) + h \\Big (\\frac{m_3}{m_3 + m_2} \\Big ) - f(m_3) > 0.$ Assuming a globally optimal 4-BP existed, one could continuously displace a terminal in a way such that for the resulting BOT problem a coupled 4-BP is still globally optimal.","Choosing different such displacements four additional inequalities can be derived (see App.","REF ): Proposition 4.2 Given a BOT problem with one source and three sinks, with demands $m_1, m_2, m_3$ as in Fig.","REF , a coupled 4-BP away from the terminals cannot be globally optimal if at least one of the following inequalities holds true: $\\Gamma &= h \\Big (\\frac{m_1}{m_1 + m_2} \\Big ) - f(m_1) + h \\Big (\\frac{m_3}{m_3 + m_2} \\Big ) - f( m_3) > 0, \\\\\\Gamma _{1,*} &= f(1 -m_*) + f\\Big (1 - \\frac{m_2}{1-m_*} \\Big ) - f(1-m_*-m_2) > 0, \\\\\\Gamma _{2,*} &= h \\Big (\\frac{m_*}{m_* + m_2} \\Big ) + f\\Big (\\frac{m_2}{1 - m_*} \\Big ) - h(m_*) > 0$ where $*=1,3$ .","Note that $\\Gamma = \\Gamma _{1,1} + \\Gamma _{2,1} = \\Gamma _{1,3} + \\Gamma _{2,3}$ .","In App.","REF , we prove the inequalities analytically for a large subset of the parameter space.","For the remainder we present a numerical argument (see App.","REF ).","In addition, we show by induction how, given that coupled 4-BPs are never globally optimal, one can further rule out coupled $n$ -BPs (with $n$ effective neighbors) for all $n > 4$ .","Theorem 4.3 Given a BOT problem in the Euclidean plane and assuming that coupled 4-BPs are never globally optimal, in a globally optimal BOT solution each branching point not coincident with a terminal must have degree three."],["Generalization of BOT to Riemannian manifolds","In this section, we extend the BOT problem together with many of the previous results to two-dimensional Riemannian manifolds $\\mathcal {M}$ embedded into $\\mathbb {R}^3$ [18].","This includes the sphere as important special case, particularly relevant for global transportation networks.","In the generalized BOT cost function (REF ) we replace the Euclidean metric by the geodesic distance $d: \\mathcal {M} \\times \\mathcal {M} \\rightarrow \\mathbb {R}^+$ , i.e.","$ \\mathcal {C}_M = \\sum _{(i,j) \\, \\in \\, E} m_{ij}^\\alpha \\, d(x_i,x_j).$ As we assume the manifold to be embedded, the length of a geodesics can be measured in $\\mathbb {R}^3$ .","First, we generalize the non-optimality of cyclic solutions.","The corresponding proof in [2] readily applies also to two- and higher-dimensional manifolds.","As before, solving a BOT problem on a curved surface can thus be separated into the combinatorial topology optimization and the continuous optimization of the BP configuration."],["Linear approximation of BOT solutions on manifolds","Intuitively speaking, a two-dimensional Riemannian manifold locally looks like the Euclidean plane.","If we zoom in on a sufficiently small region, geodesics again resemble straight lines and the geodesic distance approaches the Euclidean one.","This can be used to show that the branching angles which were optimal for Y-branchings in the Euclidean plane are also optimal on Riemannian manifolds.","Below, we summarize the main steps of the proof.","All details can be found in App. .","Given a Y-branching on a manifold, we measure the angles between the three geodesics in the tangent space $T_b \\mathcal {M}$ at the BP $b$ .","We now zoom in on a small neighborhood $U$ around $b$ and consider only the subsolution in $\\mathcal {M} \\cap U$ .","The terminals of the corresponding subproblem are projected orthogonally onto the tangent space, more specifically onto a small disk of radius $r$ , denoted by $D(r)$ , see Fig.","REF .","Let us denote the cost of the subsolution on the manifold by $\\mathcal {C}_M(b)$ and the cost of the corresponding subproblem in the flat disk by $\\mathcal {C}(b)$ .","Now, assuming that the angles between the geodesics deviate from the optimal branching angles, the same holds true for the projected subsolution.","Consequently, there exists an alternative BP $b^*$ in the disk with cheaper cost $\\mathcal {C}(b^*)$ .","Note that the radius of this disk becomes smaller the smaller we choose the region $\\mathcal {M} \\cap U$ of the subproblem.","Figure: On the flat disk D(r)⊂T b ℳD(r) \\subset T_b \\mathcal {M}, the BP bb does not solve the problem with terminals a ^ i \\hat{a}_i optimally if the angles between the dotted geodesics are not optimal.Crucially, the cost difference between a subsolution on the manifold and its projection onto the plane tends to zero quadratically in the limit of $r \\rightarrow 0$ .","The intuitive reason for this is that the tangent space $T_b \\mathcal {M}$ locally approximates the manifold to linear order.","On the contrary, the costs $\\mathcal {C}(b)$ and $\\mathcal {C}(b^*)$ in the disk scale linearly in $r$ and so does the cost improvement $\\mathcal {C}(b) - \\mathcal {C}(b^*) = M \\, r$, for some fixed $M > 0$ .","To conclude the proof, one projects $b^*$ onto the manifold and evaluates the cost difference of the two subsolutions there.","The difference is of the form $M \\, r + O(r^2)$ , with second order differences due to the projection from $D(r)$ to the manifold.","Consequently, a finite radius $r > 0$ must exist for which the cost difference is truly positive.","A BOT solution on the manifold for which the Y-branching angles deviate from the optimal branching angles can thus be improved and is not relatively optimal.","The logic of the proof outlined here can easily be extended to the V- and L-branching conditions as well as our results regarding the non-optimality of coupled BPs.","Again, improving the BOT solution locally in the tangent plane (w.r.t.","its geometry or topology) and projecting back to the manifold results in an improved solution on the manifold (see App.","REF ).","Theorem 5.1 Consider the solution to a generalized BOT problem on a two-dimensional Riemannian manifold embedded into $\\mathbb {R}^3$ .","For the solution to be relatively optimal, it is a necessary condition that each BP satisfies the optimal angle conditions for Y-, V- and L-branching, which apply for BOT in the Euclidean plane.","For it to be globally optimal, assuming that coupled 4-BPs are not optimal in the plane, it is a necessary condition that BPs not coincident with a terminal have degree three.","Though there is no readily available algorithm to solve BOT on embedded surfaces, we discuss some possible approaches in App.","REF ."],["Heuristics and numerical optimization","In this section, we present a simple but effective algorithm for the geometry optimization, followed by a compelling heuristic for the topology optimization.","As pointed out earlier, the difficulty of solving a BOT problem stems from the super-exponentially growing number of possible full tree topologies.","Obtaining an exact solution by brute-force is almost always computationally infeasible and hence fast heuristic solvers are needed.","For BOT problems with a single source, a branch-and-bound method is applicable [31], enabling exact solutions for up to 16 nodes.","However, this method does not generalize directly to the case of multiple sources.","While some literature exists on heuristics for BOT problems with a single source [29], we are not aware of heuristics for multiple sources, except [24].","The authors of [24] present a simulated annealing based optimization strategy for BOT, based on hand-crafted geometrical and topological modifications, which may require user supervision.","Furthermore, continuous approaches to solve BOT exist which do not rely on a subdivision into geometry and topology optimization.","The authors of [22] phrase BOT as a limit of functional minimization problems.","Since their algorithm discretizes the plane and the BOT cost function, their output is however not sparse but a discretized function."],["Numerical branching point optimization for a given topology","Brute-force and heuristic BOT solvers alike typically rely on the geometry optimization of many different topologies.","A fast and reliable BP optimization routine is therefore essential, as it determines the computational bottleneck of these algorithms.","For a given tree topology $T$ , all edge flows $m_{ij}$ are known (see Sect. ).","The objective is thus to minimize the following convex cost function: $ \\mathcal {C}(\\lbrace x_i\\rbrace ) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\left\\Vert x_i - x_j \\right\\Vert _2,$ where, for $1 \\le i \\le n$ , the $x_i$ hold the fixed coordinates of the terminals and, for $n+1 \\le i \\le n+m$ , the variable BP positions.","Since the cost function is not everywhere differentiable, we suggest the following generalization of Smith's algorithm developed for geometry optimization in the ESTP [26].","It is an effective algorithm specifically for minimizing the sum of Euclidean norms in two- and higher-dimensional Euclidean space.","Unlike the geometric construction in Section REF , it is applicable to all (not necessarily full) tree topologies.","Starting from a non-optimal, non-degenerate BP configuration, e.g.","from a random initialization, the gradient with respect to each BP position $x_i$ is set to zero for $n+1 \\le i \\le n+m$ , resulting in the following non-linear system of $m$ equations: $x_i = \\sum _{j \\, : \\, (i,j) \\in T} m_{ij}^\\alpha \\frac{x_j}{\\vert x_i - x_j \\vert }\\ \\Bigg / \\sum _{j \\, : \\, (i,j) \\in T} \\frac{m_{ij}^\\alpha }{\\vert x_i - x_j \\vert }.$ This system can be solved approximately, by iteratively solving the following linearized system $ &x_i^{(k + 1)} = \\!","\\!","\\sum _{j \\, : \\, (i,j) \\in T} m_{ij}^\\alpha \\frac{x_j^{(k+1)}}{\\vert x_i^{(k)} - x_j^{(k)} \\vert }\\ \\Bigg / \\sum _{j \\, : \\, (i,j) \\in T} \\frac{m_{ij}^\\alpha }{\\vert x_i^{(k)} - x_j^{(k)} \\vert }, \\hspace{11.38092pt}\\text{for } n+1 \\le i \\le n+m.$ Note that $x_i^{(k)} = x_i$ is fixed for $1 \\le i \\le n$ .","For each iteration, the solution can be found in linear time, again by “elimination on leaves of a tree”, similar to determining all edge flows from the flow constraints.","The algorithm is easily parallelized over $d$ spatial dimensions of a BOT problem so that a single iteration is of order $O(nd)$ .","In essence, this is an iteratively reweighted least squares (IRLS) approach [4].","The connection is made explicit in App.","REF .","Details on the proof of convergence, the empirical runtime of the algorithm and suitable convergence criteria can be found in App.","REF and in [26].","The arguments in [26] readily apply to our generalization.","Besides our method, other techniques may be used for the geometry optimization, for instance the interior point method presented in [30]."],["A greedy randomized algorithm for the topology optimization","Our heuristic for the optimization of the BOT topology is inspired by the idea of simulated annealing [16], which has been applied in different variants to combinatorial problems such as the Traveling Salesman Problem [19] or the ESTP [10].","In our heuristic, the BOT topology is iteratively modified by randomly deleting an edge and replacing it with a new one.","At each step, the new solution is accepted according to a criterion, which typically depends on the cost difference between the solutions and a user-chosen hyperparameter, the temperature, used to mimic a physical cooling process.","However, because in practice it works already sufficiently well (see Fig.","REF ), we refrained from designing an elaborate cooling scheme.","Instead, we apply the heuristic most greedily, i.e., in the zero-temperature limit, where a new state is accepted only if it decreases the cost.","Starting from an initial tree topology $T$ , e.g., the minimum spanning tree (mST) or the OT solutionIn particular in the regime $\\alpha \\approx 1$ , our BOT solver benefits from existing efficient OT solvers by using their solution as initial guess., we uniformly sample an edge $\\hat{e} \\in E$ and remove it from $T$ .","Let the incident node of $\\hat{e}$ which ended up in the smaller connected component be $\\ell $ .","Then, one calculates the distance $d(e, \\ell )$ between $\\ell $ and every edge $e=(i,j)$ in the larger component and samples one of these edges with probability $p(e) \\propto \\exp (-d(e, \\ell )^2 / d_{min}^2)$ , where $d_{min}$ is the distance to the closest considered edge.","The node $\\ell $ is then connected to the sampled edge via a new BP to produce a new tree topology.","For this topology, we optimize the geometry (as described in Sect.","REF ) and compare costs with the previous solution.","If the new state is rejected, start the next iteration by sampling $\\hat{e}$ without replacement until either a move is accepted and all above steps are repeated; or until no accepted move is found, upon which the search terminates.","Experiments for small BOT problems suggest that even in the greedy zero-temperature limit the algorithm often finds the globally optimal solution, after comparatively few iterations.","For this, the greedy heuristic (using the mST as initialization) was compared against exact solutions with up to nine terminals, obtained by brute-force.","For each $n$ , 100 BOT problems were sampled uniformly with respect to $\\alpha $ , the terminal positions and demands and supplies, cf. Alg.","REF .","The ratios of the heuristic's cost divided by the cost of the exact solution are plotted in Fig.","REF .","On average the heuristic solution is less than 0.5% worse than the brute-force solution.","This is impressive, considering the fact that for $n=9$ the brute-force solver requires over $10^5$ BP optimizations, whereas the simulated annealing heuristic on average required $29 \\pm 10$ iterations to converge.","Additional experiments (also for larger BOT problems) suggest that the number of BP optimizations until convergence scales better than $O(n^2)$ , see App.","REF .","Further, the cost ratios in Fig.","REF stay roughly constant as $n$ increases.","Additional experiments for BOT in higher dimensions (see Fig.","REF ) indicate that the average quality of the heuristic solution decreases only very slightly with $n$ .","Unfortunately, one can only speculate how this trend extends to larger BOT problems, where brute-force solutions are no longer feasible.","Figure REF shows heuristic solutions of a larger example problem for different values of $\\alpha $ .","In particular, we find that the greedy heuristic is very effective at removing higher-degree branchings and undesirable edge crossings."],["Generalization to higher-dimensional BOT","Optimal BOT solutions are acyclic also in $\\mathbb {R}^d$ [2].","Thus, for a given topology, the edge flows are known, the optimal substructure property of Lemma REF generalizes and the convex geometry optimization can be separated from the combinatorial topology optimization.","Though, the optimal angle conditions for Y-, V- and L-branching (see Sect. )","hold also in $\\mathbb {R}^d$ , the results on the degree limitation do not generalize, as the arguments rely on the fact that the angles between edges meeting at a higher-degree branching point sum up to $2\\pi $ (cf.","Sect.","REF ).","The numerical geometry optimization as well as the greedy algorithm for the topology optimization presented in Sect.","are readily applicable to BOT problems in $\\mathbb {R}^d$ (see also App.","REF and App.","REF )."],["Conclusions","We have studied branched optimal transport in $\\mathbb {R}^2$ from a theoretical and practical perspective.","First, we have tackled the geometric optimization of BOT solutions, given a tree topology.","We generalized the existing exact method presented in [2], [9] to the case of multiple sources.","Based on theory developed in the process of this generalization, we formulated a catalog of necessary and sufficient conditions for optimal BOT solutions and argued that $n$ -degree branching points for $n>3$ are never optimal.","Moreover, we showed that these conditions also apply for BOT on two-dimensional manifolds.","Lastly, we presented a greedy randomized algorithm, which optimizes the tree topology, combined with an efficient numerical branching point optimization method.","We compared our algorithm to the optimal solution for small examples, obtaining compelling results.","BOT provides a unifying framework for optimal transport and the Euclidean Steiner tree problem and is itself of great theoretical and practical interest.","The emergent branching in BOT can be used to simulate and study the myriad of efficient transportation systems which exhibit subadditive costs.","Moreover, BOT combines both combinatorial and convex optimization and could be an inspiring problem to be solved by machine learning techniques.","The number of optimality criteria derived in this paper can guide further research in this area and the presented approximate solvers may serve as competitive baseline for new ML-based approaches.","We would like to thank Jarosław Piersa for sharing his code with us for a comparison to his work.","Further, we thank Edouard Oudet for helpful hints on the comparison to his related work and Fabian Egersdoerfer for his improved C++ implementation of the geometry optimization.","This work is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), by Informatics for Life and by SIMPLAIX funded by the Klaus Tschira Foundation."],["Appendix overview","The appendix is subdivided into the following seven topics: app-central: Quick proof of the central angle property, used in Sect.","REF in the recursive construction of relatively optimal solutions based on the optimal branching angles.","secLV: Derivation of the conditions listed in Tab.","REF under which V- or L-branching are optimal.","sec:subopt: Formal proof of Lemma REF on the optimal substructure of BOT solutions.","appfh: Collection of small lemmas on the monotonicity and other properties of the analytical expression for the branching angles (cf. Eq.","(REF )).","appsec:nonopt: Technical proofs and numerical scheme to show the non-optimality of higher-degree branchings discussed in Sect.","REF .","app:mf: Formal proof of Theorem REF , which generalizes the optimal branching conditions and other properties from the Euclidean plane to embedded surfaces.","A sketch of the proof can be found in Sect.","REF .","appsec:algo: Additional details and experiments for the different algorithms presented in the main paper.","Section REF focuses on the numerical geometry optimization and Sect.","REF on the greedy algorithm for the topology optimization.","Section REF holds a few examples of the recursive geometric construction of relatively optimal solutions for BOT problems with multiple sources."],["Central angle property","In this section, we present a geometric proof of the central angle property used in the geometric construction of relatively optimal solutions for a given full tree topology (see Sect.","REF ).","It states that for a circle, as in Fig.","REF , the central angle $\\angle a_1 o a_2$ is twice the angle $\\angle a_1 q a_2 = \\theta $ for all $q$ on the lower circle arc.","Figure: (a) Illustration of the central angle theorem.","(b) Isosceles triangles used in the geometric proof.Let us start by constructing the three isosceles triangles $\\triangle o q a_1$ , $\\triangle o q a_2$ and $\\triangle o a_1 a_2$ with angles as denoted in Fig.","REF .","Now consider the angular sums in the following triangles: $\\triangle q a_1 a_2 : \\ 2 \\delta _1 + 2 \\delta _2 + 2 \\tau = 180^\\circ \\ \\text{ and } \\ \\triangle o a_1 a_2 : \\gamma + 2 \\tau = 180^\\circ \\ .$ Subtracting the two equations immediately reveals that $\\gamma = 2 \\delta _1 + 2 \\delta _2 = 2 \\theta $ and the proof is complete."],["Optimal L- and V-shaped branching","Below, we formally derive the conditions listed in Tab.","REF under which V- or L-branching provide the optimal solution to a BOT problem with one source and two sinks.","The proof is inspired by the approach in [21], where subdifferentials are applied to the related Fermat-Torricelli problem.","Definition B.1 (Subgradient and subdifferential) A vector $v \\in \\mathbb {R}^n$ is called a subgradient of a convex scalar function $g: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ at a specific point $y$ if for all $x \\in \\mathbb {R}^n$ it satisfies $g(x) \\ge g(y) + \\langle v, x - y \\rangle $ The set of all subgradients of the function $g$ at a given point $y$ is called the subdifferential of $g$ at $y$ and is denoted by $\\partial g(y)$ .","From a geometric point of view, the subdifferential of $g$ at $y$ is the set of gradients of all straight lines which cross $g(y)$ and lie below the image of $g$ .","The subdifferential rule of Fermat follows immediately from the definition and states that $g$ achieves an absolute minimum at $y$ if and only if $0 \\in \\partial g(y)$ .","Now, for $g(b) = c \\cdot \\left\\Vert b - a \\right\\Vert _2$ with $c \\in \\mathbb {R}$ and $a \\in \\mathbb {R}^n$ the subdifferential is given by $\\partial g(b) = {\\left\\lbrace \\begin{array}{ll}\\mathbb {B}_{c} \\, , \\quad \\quad \\quad \\ \\text{ for } b = a \\\\\\bigg \\lbrace c \\frac{b - a}{\\left\\Vert b - a \\right\\Vert _2} \\bigg \\rbrace \\, , \\text{ elsewhere, }\\end{array}\\right.","}$ with $B_r = \\lbrace v \\in \\mathbb {R}^n : \\left\\Vert v \\right\\Vert _2 \\le r \\rbrace $ , the ball of radius $r$ .","Furthermore, it can be easily shown that for $g(x) = g_1(x) + g_2(x)$, one has $\\partial g(y) = \\partial g_1(y) + \\nabla g_2(y)$ , given that both $g_1$ and $g_2$ are convex functions and $g_2$ is differentiable.The sum of a vector and a set of vectors, as in $\\partial g_1(y) + \\nabla g_2(y)$ , is known as the Minkowski sum.","Using this, we calculate the subdifferentials of the cost function of the 1-to-2 branching in Eq.","(REF ).","The subdifferentials at $b = a_i$ are of the form $\\partial \\mathcal {C}(a_i) = \\mathbb {B}_{m_i^\\alpha } + \\sum _{j \\ne i}m_j^\\alpha \\frac{ a_i - a_j}{ \\left\\Vert a_i - a_j \\right\\Vert } \\ .$ Based on the rule of Fermat, the cost function achieves an absolute minimum at $b = a_i$ if and only if $ \\left\\Vert \\ \\sum _{j \\ne i} m_j^\\alpha \\frac{ a_i - a_j}{\\left\\Vert a_i - a_j \\right\\Vert } \\ \\right\\Vert \\le m_i^\\alpha \\ .","\\\\[-0.4cm]\\nonumber $"],["V-branching.","We square condition (REF ) and evaluate it for $i=0$ in order to determine under which condition a V-shaped branching with $b^* = a_0$ is optimal: $\\left\\Vert \\ m_1^\\alpha \\frac{a_0 - a_1}{\\left\\Vert a_0 - a_1 \\right\\Vert }+ m_2^\\alpha \\frac{a_0 - a_2}{\\left\\Vert a_0 - a_2 \\right\\Vert }\\ \\right\\Vert ^2 =m_1^{2 \\alpha } + m_2^{2 \\alpha } + 2 m_1^\\alpha m_2^\\alpha \\cos (\\psi )\\le m_0^{2\\alpha } = (m_1 + m_2)^{2 \\alpha } \\,$ where $\\psi $ denotes the angle of the terminal triangle at $a_0$ , i.e., $\\psi = \\angle a_1a_0a_2$.","The condition in terms of $\\psi $ reads $ \\psi \\ge \\arccos \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) = h(\\alpha , k) = \\theta ^*_1 + \\theta ^*_2 \\ ,$ where we have used the flow fraction $k = m_1/(m_1 + m_2)$ .","We immediately recognize the expression for the optimal branching angle $\\theta ^*_1 + \\theta ^*_2$ , cf. Eq.","(REF ).","We already know that $\\psi = \\theta ^*_1 + \\theta ^*_2$ on the lower circle arc of the pivot circle by construction.","And indeed one can easily check that $\\psi > \\theta ^*_1 + \\theta ^*_2$ if an only if the source $a_0$ lies inside the lower half of the pivot circle, as in Fig.","REF .","For that, we construct a line through $a_1$ and $a_0$ and the intersection of $\\overline{a_1a_0}$ with the lower pivot circle we denote by $q$ .","By construction of the pivot circle, $\\angle a_1 q a_2 = \\theta ^*_1 + \\theta ^*_2$ .","Using the angular sum in the triangle $\\triangle a_0 q a_2$ , one immediately obtains: $180^\\circ = \\delta + ( 180^\\circ - \\psi ) + (\\theta ^*_1 + \\theta ^*_2) \\ \\rightarrow \\ \\psi - (\\theta ^*_1 + \\theta ^*_2) = \\delta > 0 \\ .$ Figure: Sketch to show that ψ>θ 1 * +θ 2 * \\psi > \\theta ^*_1 + \\theta ^*_2.Hence, for $a_0$ located inside the lower half of the pivot circle indeed $\\psi > \\theta ^*_1 + \\theta ^*_2$ .","A similar argument can show that for any point outside the pivot circle the optimal V-branching condition is not fulfilled."],["L-branching.","Analogous to the above steps, one obtains conditions for optimal L1- and L2-branching, where $b^*= a_1$ and $b^*= a_2$ respectively.","Again squaring the general condition (REF ), now for $i=1,2$ , one eventually finds that $\\varphi &\\ge \\pi - f(\\alpha , 1-k) = \\pi - \\theta ^*_2 \\ , \\\\\\varrho &\\ge \\pi - f(\\alpha , k) = \\pi - \\theta ^*_1 \\ ,$ where the angle $\\varphi $ and $\\varrho $ denote the angles of the terminal triangle located at $a_1$ and $a_2$ , i.e.","$\\varphi = \\angle a_2a_1a_0$ and $\\varrho = \\angle a_0a_2a_1$ (see also Fig.","REF ,vll4).","Let us now demonstrate that these conditions are indeed fulfilled if and only if the source is located in the L1- and L2-region, as marked also in Fig.","REF .","The pivot point $p$ is constructed such that $\\angle a_1 o p = 2 \\theta ^*_1$ and $\\angle p o a_2 = 2 \\theta ^*_2$ , cf. Fig.","REF .","Besides that, by construction, we have that $\\angle p a_1 a_2 = \\theta ^*_2$ and $\\angle a_1 a_2 p = \\theta ^*_1$ (as shown in Fig.","REF ,vll4).","Then, looking at Fig.","REF , it is evident that indeed for any source $a_0$ inside the L1-branching sector the condition $\\varphi = \\angle a_2a_1a_0 \\ge \\pi - \\theta ^*_2$ holds.","The respective condition (REF ) for L2-branching holds true exactly inside the highlighted L2-region.","Definition B.2 (Transient and strict V- and L-branchings) A V- or L-branching for which the inequality conditions (REF )-(REF ) hold as equality is referred to as transient V- or L-branching.","The reason for this is that, in such a case, one of the terminal positions may be perturbed infinitesimally, so that the condition is no longer fulfilled and the optimal solution transitions to a Y-shaped branching.","On the contrary, if an L- or V-branching condition is fulfilled as strict inequality, we call the L- or V-branching strict.","Figure: After constructing the pivot point the position of the source a 0 a_0 determines whether a Y-,V- or L-shaped branching is optimal."],["Relation of V- and L-conditions between symmetric and asymmetric branchings","Let us briefly show that the corner cases of Y-, V- and L-branching work analogously for both flow scenarios (described in Fig.","REF ,case2a).","The conditions under which L- and V-branching are optimal in the asymmetric branching case could be again determined straightforwardly by plugging into the subdifferential condition (REF ), as before.","However, looking at Fig.","REF ,case2a, we notice that the well-known symmetric branching case can be transformed into the asymmetric case by relabeling $a_0 \\rightarrow a_2$ , $a_1 \\rightarrow a_0$ and $a_2 \\rightarrow a_1$ .","This relation provides a direct correspondence of the L- and V-branching conditions.","The conditions for the corner case are transferred according to Table REF .","Note that the L- and V-branching conditions for both cases are of the exact same form, only that for the asymmetric case the stationary branching angles are $\\vartheta ^*_i$ instead of $\\theta ^*_i$ .","Moreover, for fixed children positions $a_1$ and $a_2$ , the position of the source again distinguishes between optimal Y-, V- and L-branching.","The partitioning of the lower half plane into the respective regions is completely analogous to Fig.","REF .","Table: Relations between the L- and V-branching conditions (see Fig. )","for symmetric and asymmetric branchings."],["Optimal substructure of BOT solutions (Proof of Lem. ","In this section, we provide the formal proof to Lemma REF , repeated below for completeness: Lemma C.1 (a) For a given tree topology, a BOT solutions is a relatively optimal if and only if every (coupled) BP connects its (effective) neighbors at minimal cost.","(b) In a globally optimal solution, every subsolution restricted to a connected subset of nodes solves its respective subproblem globally optimally.","Let us start with the following definition, which divides a BOT problem and its possible solutions into subproblems and corresponding subsolutions.","Definition C.1 (Subproblems and subsolutions) A given BOT solution may be split into two subsolutions, by choosing a number of edges $\\lbrace e_i\\rbrace $ and cutting them at points $\\lbrace x_i\\rbrace $ , so that the topology is split into two connected components.","This procedure induces two subproblems and two subtopologies.","Each subproblem consist of the terminals contained in the respective component plus additional terminals at the positions $x_i$ .","The demands or supplies of the additional terminals at $x_i$ are equal to the amount of flow through the corresponding edge $e_i$ that was cut.","The terminal becomes a sink in one subproblem and a source in the other according to the direction of flow through $e_i$ .","The two subtopologies are given by the induced subgraph on all terminals contained in one component.","The subsolutions to the created subproblems are given by the subtopologies and the BP configurations of the respective subsets of branching points contained in each subproblem.","Note that using a number of such splits a given solution may be divided into several subsolutions, each solving their respective subproblem.","An illustrative example can be found in Fig REF .","Figure: A BOT solution (left) is split into two subsolutions (right) by cutting an edge as indicated by the dashed line."],["Proof of the optimal substructure as necessary condition for optimality.","The optimal substructure as necessary optimality condition follows immediately from the independence of the subproblems.","Given that a solution is minimal in cost (in the relative or global sense), each subproblems itself must be minimal in that sense.","Otherwise the cost of this subproblem could be decreased by (a) improving the BP configuration or (b) the topology.","In that case, also the full solution could be improved in cost and could thus not be optimal."],["Proof of the optimal substructure as sufficient condition for optimality.","The optimal substructure as sufficient optimality condition stems from the fact that, given a tree topology, the BOT cost function is convex with respect to the BP positions.","As a first step, let us prove the following lemmata: Lemma C.2 Let $b$ be a branching point with neighbors $\\lbrace a_i \\rbrace $ .","Let the optimal position of $b$ be at one of its neighbors with the condition (REF ) fulfilled as equality, i.e.","the branching is transient.","Then, for every $\\delta >0 $ there exists a BP location $b_\\delta \\in \\mathbb {R}^2$ away from all neighbors with $\\left\\Vert b - b_\\delta \\right\\Vert _2 < \\delta $ and a continuous function $f:\\mathbb {R^+}\\rightarrow \\mathbb {R}$ with $\\lim _{x\\rightarrow 0}f(x)=0$ , such that the gradient $\\left\\Vert \\nabla _{\\!","\\!","b} \\, \\mathcal {C}(b_\\delta ) \\right\\Vert _2 < \\epsilon $ for $0<\\epsilon 0$ , such that $b_\\delta $ is located away from all terminals.","The gradient with respect to the branching point position is well defined at $b_\\delta $ and reads: $\\nabla _{\\!","\\!","b} \\, \\mathcal {C}(b_\\delta ) = - m_0^\\alpha \\frac{v}{\\left\\Vert v \\right\\Vert } +\\sum _{i \\ne 0} m_i^\\alpha \\frac{ a_0 - a_i - \\delta v}{\\left\\Vert a_0 - a_i -\\delta v \\right\\Vert } \\ .$ Clearly, $\\left\\Vert \\nabla _{\\!","\\!","b} \\, \\mathcal {C}(b_\\delta ) \\right\\Vert $ can be brought arbitrarily close to zero as $\\delta \\rightarrow 0$ , since $\\left\\Vert \\nabla _{\\!","\\!","b} \\, \\mathcal {C}(b_\\delta ) \\right\\Vert ^2 = \\underbrace{m_0^{2 \\alpha } + m_0^{2 \\alpha } - 2 m_0^\\alpha \\Big \\langle v, \\frac{v}{\\left\\Vert v \\right\\Vert } \\Big \\rangle }_{= \\, 0} + O(\\delta ) \\ ,$ comprising the terms which go to zero as $\\delta \\rightarrow 0$ in $O(\\delta )$ .","The function absorbed in $O(\\delta )$ therefore provides the function $f$ and $b_\\delta $ is the desired branching point with arbitrarily small gradient.","The argument works for both coupled and uncoupled branching points.","$$ Lemma C.3 For a given full tree topology $T$ , let $B=\\lbrace b_i\\rbrace $ be the ROS of a BOT problem with terminals $A = \\lbrace a_i\\rbrace $ .","Let $B$ contain Y-branchings and transient V- or L-branchings.","Then, for every $\\delta >0 $ there exists a non-degenerate BP configuration $B_\\delta $ and a continuous function $f:\\mathbb {R^+}\\rightarrow \\mathbb {R}$ with $\\lim _{x\\rightarrow 0}f(x)=0$ , such that $\\left\\Vert B - B_\\delta \\right\\Vert _2 < \\delta $ and the gradient norm fulfills $\\left\\Vert \\nabla _{\\!","\\!","B} \\, \\mathcal {C}(B_\\delta ) \\right\\Vert _2 < \\epsilon $ for $0<\\epsilon 0$ .","Lemma REF states that we can find an arbitrarily small $\\delta $ so that $B^*_{f} + \\delta $ is non-degenerate and the gradient $\\nabla _{\\!","\\!","B_f} \\, \\mathcal {C}(B^*_{f} + \\delta )$ is arbitrarily small.","Due to the convexity of $\\mathcal {C}$ , the following inequality holds: $\\mathcal {C}(B_{f,0}) &\\ge \\mathcal {C}(B^*_{f} + \\delta ) + \\langle \\nabla _{\\!","\\!","B_f} \\, \\mathcal {C}(B^*_{f} + \\delta ), B_{f,0} - (B^*_{f} + \\delta ) \\rangle \\\\&= \\mathcal {C}(B^*_{f}) + \\underbrace{O( \\left\\Vert \\delta \\right\\Vert _2) + \\langle \\nabla _{\\!","\\!","B_f} \\, \\mathcal {C}(B^*_{f} + \\delta ), B_{f,0} - (B^*_{f} + \\delta ) \\rangle }_{=: \\ K(\\delta )}$ where we have used that $\\mathcal {C}(B^*_{f} + \\delta ) = \\mathcal {C}(B^*_{f}) + O( \\left\\Vert \\delta \\right\\Vert _2)$ , using Big-O notation.","All terms which tend to zero as $\\delta \\rightarrow 0$ have been summarized in $K(\\delta )$ .","$K(\\delta )$ can be brought arbitrarily close to zero by choosing a sufficiently small $\\delta $ .","In particular, there exists a $\\delta > 0$ so that $\\vert K(\\delta ) \\vert < \\Delta / 2$ .","Using the assumption $\\mathcal {C}(B^*_{f}) = \\mathcal {C}(B_{f,0}) + \\Delta $ , this leads to: $\\mathcal {C}(B_{f,0}) &\\ge \\mathcal {C}(B^*_{f}) + K(\\delta ) = \\mathcal {C}(B_{f,0}) + \\Delta + K(\\delta ) \\ge \\mathcal {C}(B_{f,0}) + \\frac{\\Delta }{2} \\ ,$ which is a contradiction.","Hence $B^*$ , is again a local minimum.","$$"],["Properties of the functions $f$ and {{formula:ae7b64eb-edf7-4367-a415-b0559ec00a0b}} describing the optimal branching angles","The optimal branching angles are expressed in terms of the following functions $f(\\alpha , k)$ and $h(\\alpha , k)$ , defined for $\\alpha \\in [0,1]$ and $k \\in (0,1)$ , cf. Eq.","(REF ): $f(\\alpha , k) &= \\arccos \\bigg ( \\frac{k^{2\\alpha } + 1 - (1-k)^{2 \\alpha }}{2 k^\\alpha } \\bigg ) \\ , \\\\h(\\alpha , k) &= \\arccos \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) \\ .$ Figure REF shows $f$ and $h$ as functions of $k$ for a number of different values of $\\alpha $ .","The two functions are related by $h(\\alpha , k) = f(\\alpha , k ) + f(\\alpha , 1-k)$ , so that $h$ is symmetric around $k=1/2$ .","Both functions are defined for inputs $\\alpha \\in [0,1]$ and $k \\in (0,1)$ .","For $\\alpha = 1$ , we have $f(\\alpha =1, k) = 0 = h(\\alpha =1, k)$ for all $k$ , reflecting the fact that Y-shaped branchings are never optimal in the optimal transport case.","On the other hand, we have $f(\\alpha =0, k) = \\pi /3$ and $h(\\alpha =0, k)= 2 \\pi / 3$ for all $k$ .","This limit corresponds to the Euclidean Steiner tree problem where in an optimal Y-branching all edges meet at $120^\\circ $ .","Moreover, considering the limits of $k \\rightarrow 0$ and $k \\rightarrow 1$ , one finds that $f(\\alpha , k \\rightarrow 0) \\rightarrow \\pi /2$ and $f(\\alpha , k \\rightarrow 1) \\rightarrow 0$ .","Consequently, $h(\\alpha , k) \\rightarrow \\pi /2$ for both $k \\rightarrow 0$ and for $k \\rightarrow 1$ .","In the following three lemmas, we investigate the monotonicities of the two functions.","Figure: The functions ff and hh vs. kk for different values of α\\alpha .Lemma D.1 $f(\\alpha , k)$ is strictly decreasing in $k$ for all $\\alpha \\in (0,1)$ and $f(\\alpha ,k) < \\pi /2$ for all $\\alpha \\in [0,1]$ and $k \\in (0,1)$ .","Proof.","Since the inverse cosine function decreases monotonically, it is sufficient to show that the argument is monotonically increasing.","Let us consider its derivative $\\frac{\\partial }{\\partial k} \\bigg ( \\frac{k^{2\\alpha } + 1 - (1-k)^{2 \\alpha }}{2 k^\\alpha } \\bigg )\\sim k^{2 \\alpha } + (1-k)^{2 \\alpha } + 2 k (1-k)^{2 \\alpha - 1} - 1 =: s(\\alpha , k) \\ .$ The $\\sim $ indicates that we have dropped overall factors which were clearly positive, as we are only interested in the sign of the derivative.","First, note that $s(\\alpha = 0, k) = \\frac{1+k}{1-k} > 0$ and $s(\\alpha = 1, k) = 0$ for all $k$ .","Secondly, $s(\\alpha , k)$ is strictly decreasing with respect to $\\alpha $ : $\\frac{\\partial s}{\\partial \\alpha } =2 k^{2\\alpha } \\log (k) + 2(1-k)^{2\\alpha } \\log (1-k) + 4k (1-k)^{2\\alpha -1} \\log (1-k) < 0 \\ .$ Each of the above terms are negative due to the fact that $k, (1-k) \\in (0,1)$ .","Together it follows that $s(\\alpha , k)$ is positive for any combination of $\\alpha $ and $k$ and the proof of the monotonicity is complete.","The fact that $f(\\alpha , k \\rightarrow 0) \\rightarrow \\pi /2$ for all $\\alpha $ then immediately implies that $f(\\alpha ,k) < \\pi /2$ .","$$ Lemma D.2 For $\\alpha \\le 0.5$ , $h(\\alpha , k)$ increases monotonically in $k$ for $k \\in (0,1/2]$ and decreases for larger values of $k$ .","Conversely, for $\\alpha \\ge 0.5$ , $h(\\alpha , k)$ increases first until in reaches a maximum at $k=1/2$ and decreases afterwards.","Proof.","The argument here follows the proof of Lemma 12.14 in [2].","Due to the monotonicity of the inverse cosine function, it is again sufficient to investigate the expression in the argument.","We rewrite this expression trivially, so that it becomes a function of the fraction $r := \\frac{k}{1-k}$ and consider the following derivative: $\\frac{\\partial }{\\partial r} \\bigg ( \\frac{(r+1 )^{2\\alpha } - r^{2 \\alpha } - 1}{2 r^\\alpha } \\bigg )\\sim -(r+1)^{2\\alpha } - r^{2\\alpha } + 1 + 2 r (r+1)^{2\\alpha - 1} =: t(\\alpha , r)$ We first note that $t(\\alpha = 0, r) = \\frac{r-1}{r+1} \\le 0$ and that $t(\\alpha = 0.5, r) = 0 $ as well as $t(\\alpha = 1, r) = 0$ for all $r$ .","Next, we show that $t(\\alpha , r)$ is a concave function with respect to $r$ for $r \\le 1$ , which corresponds to $k \\in (0,1/2]$ : $\\frac{\\partial ^2 t}{\\partial \\alpha ^2} = 4 (r -1) (r+1)^{2\\alpha -1} \\log ^2(r+1) - 4 r^{2\\alpha } \\log ^2(r) \\le 0 \\ ,$ since $r \\le 1$ .","Taken together, we conclude that $t(\\alpha ,r) \\le 0$ for all $\\alpha \\in [0,0.5]$ and that $t(\\alpha ,r) \\ge 0$ for all $\\alpha \\in [0.5,1]$ .","Moreover, since $r$ increases monotonically w.r.t.","$k$ , the monotonicity of $h(\\alpha , k)$ holds for $k \\in (0,1/2]$ .","Due to the symmetry of $h$ around $k = 1/2$ , the proof is complete.","$$ Lemma D.3 $f(\\alpha , k)$ decreases monotonically in $\\alpha $ for $\\alpha \\in [0.5,1]$ and $h(\\alpha , k)$ decreases monotonically in $\\alpha $ for all $\\alpha \\in [0,1]$ .","Proof.","As the inverse cosine function decreases monotonically, it is again sufficient to investigate the derivative of the function in the $\\arccos $ -argument.","For $f(\\alpha , k)$ , we consider $\\frac{\\partial }{\\partial \\alpha } \\bigg ( \\frac{k^{2\\alpha } + 1 - (1 \\!","- \\!","k)^{2\\alpha }}{2 k^a} \\bigg ) \\!= \\frac{1}{2} k^{-\\alpha } \\Big ( \\underbrace{ \\log (k)}_{< \\ 0} [ \\,\\underbrace{ k^{2 \\alpha } + (1 \\!","- \\!","k)^{2 \\alpha } - 1 }_{\\le \\ 0} \\, ] - 2 (1 \\!","- \\!","k)^{2 \\alpha } \\underbrace{\\log (1 \\!- \\!k)}_{< \\ 0} \\Big ) > 0 .$ To see that in fact the expression in the square bracket is smaller or equal to zero, we exploit that for $\\alpha \\ge 0.5$ the function $k \\mapsto k^{2\\alpha }$ is superadditive, so that $1 = 1^{2 \\alpha } = ( k + (1-k))^{2 \\alpha } \\ge k^{2 \\alpha } + (1-k)^{2 \\alpha }$ .","For $h(\\alpha , k)$ , we consider $\\frac{\\partial }{\\partial \\alpha } \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) = - \\frac{1}{2} (1-k)^{-\\alpha } k^{-\\alpha } \\Big (&\\underbrace{\\log (k)}_{< \\ 0} \\, [ k^{2 \\alpha } + 1 - (1-k)^{2 \\alpha } ] + \\\\&\\underbrace{ \\log (1- k)}_{< \\ 0} \\, [ (1-k)^{2 \\alpha } + 1 - k^{2 \\alpha } ]\\Big ) > 0 \\ .$ Since $k, (1-k) \\in (0,1)$ clearly the expressions in square brackets are positive so that the overall expression is positive too.","In the respective regions, $f(\\alpha , k)$ and $h(\\alpha , k)$ are thus monotonically decreasing in $\\alpha $ .","$$"],["Non-optimality of higher-degree branchings","This section supplements Sect.","REF of the paper.","First, we address the third and most involved 4-branching scenario in which a coupled 4-BP connects one source and three sinks (or equivalently 3 sources and 1 sink), see Fig.","REF .","We derive the inequalities listed in Proposition REF and prove them analytically for a large subset of the parameter space.","For the remainder we present a numerical argument (App.","REF ).","Lastly, we show by induction how, given that coupled 4-BPs are never globally optimal, one can further rule out coupled $n$ -BPs (with $n$ effective neighbors) for all $n > 4$ (App.","REF )."],["Non-optimality of coupled 4-BPs between one source and three sinks","Let us start by providing the derivation of Proposition REF , which we repeat here for completeness: Proposition E.1 Given a BOT problem with one source and three sinks, with demands $m_1, m_2, m_3$ as in Fig.","REF , a coupled 4-BP away from the terminals cannot be globally optimal if at least one of the following inequalities holds true: $\\Gamma &= h \\Big (\\frac{m_1}{m_1 + m_2} \\Big ) - f(m_1) + h \\Big (\\frac{m_3}{m_3 + m_2} \\Big ) - f( m_3) > 0, \\\\\\Gamma _{1,*} &= f(1 -m_*) + f\\Big (1 - \\frac{m_2}{1-m_*} \\Big ) - f(1-m_*-m_2) > 0, \\\\\\Gamma _{2,*} &= h \\Big (\\frac{m_*}{m_* + m_2} \\Big ) + f\\Big (\\frac{m_2}{1 - m_*} \\Big ) - h(m_*) > 0$ where $*=1,3$ .","Note that $\\Gamma = \\Gamma _{1,1} + \\Gamma _{1,3} = \\Gamma _{2,1} + \\Gamma _{2,3}$ .","Note that it is an important specification that we consider coupled 4-BPs away from the terminals.","For instance, in the OT case where $\\alpha = 1$ , all BPs are located at the terminals and coupled BPs with arbitrary number of neighbors may be globally optimal.","For all following considerations, we assume that $\\alpha \\in [0,1)$ .","Figure: Coupled 4-BP between one source and three sinks."],["Derivation of the $\\Gamma $ -inequality","WLOG, let us normalize the masses so that $m_1 + m_2 + m_3 =1$ and determine the necessary conditions under which all V-branchings are optimal: $\\gamma _1 &\\ge \\pi - f\\Big (\\alpha , 1- \\frac{1-m_1}{1}\\Big ) = \\pi - f(\\alpha , m_1) \\ , \\nonumber \\\\\\gamma _2 &\\ge \\pi - f \\Big ( \\alpha , 1 - \\frac{1-m_3}{1}\\Big ) = \\pi - f(\\alpha , m_3) \\ , \\nonumber \\\\\\gamma _3 &\\ge h \\Big (\\alpha , \\frac{m_3}{m_3 + m_2} \\Big ) \\ , \\nonumber \\\\\\gamma _4 &\\ge h \\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) \\ .$ We intend to show that such a 4-BP can never be globally optimal by showing that for any combination of $\\alpha $ and the masses $m_i$ the sum of the lower bounds is already larger than $2 \\pi $ .","This is equivalent to proving that the following inequality holds true for all parameter combinations: $\\Gamma := h \\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) - f(\\alpha , m_1) + h \\Big (\\alpha , \\frac{m_3}{m_3 + m_2} \\Big ) - f(\\alpha , m_3) > 0 \\ .$ The inequality reflects that the problem setup is inherently symmetric under exchange of $m_1$ and $m_3$ .","WLOG, we assume that $m_1 \\le m_3$ ."],["Derivation of the $\\Gamma _{1,*}$ - and {{formula:2319981d-25b2-4794-a10a-81724537d3b3}} -inequalities","For a BOT solution to be globally optimal it means that it is the cheapest relatively optimal solution of all possible full tree topologies.","In our case of four terminals, there are three distinct topologies, see Fig.","REF .","Let us assume that a globally optimal 4-BP away from the terminals exists and denote the terminal positions by $a_i$ .","Then, for all three topologies $T_1$ , $T_2$ and $T_3$ , this branching point configuration is the ROS, since clearly for all $T_i$ a coupled 4-BP configuration can be realized by coupling the two branching points.","Figure: The three distinct full tree topologies connecting four terminals.Let us investigate graphically under which conditions a coupled 4-BP provides the relatively optimal solution for the different topologies.","Figure REF shows the pivot circle and pivot point construction for topology $T_1$ , where $a_0$ has been chosen as root node.","Let us refer to the line through $p_2$ and the intersection of the two pivot circles as transition line, for the following reason: If $a_0$ was positioned to the left of the transition line, the ROS of $T_1$ would be non-degenerate, as shown in Fig.","REF for instance.","For $a_0$ to the right of the transition line, the ROS of $T_1$ is given by a coupled 4-BP.","Hence, the transition line marks the transition between a non-generate ROS of $T_1$ and a coupled 4-BP solution.","Consequently, $a_0$ must lie to the right of the transition line of $T_1$ .","But, $a_0$ must simultaneously also lie on the appropriate side of the transition lines of the two other topologies $T_2$ and $T_3$ .","One can now argue that the root node $a_0$ can be moved along a continuous path onto the transition line of topology $T_1$ , without crossing any of the other transition lines.","In doing so, the coupled 4-branching stays relatively optimal for all three topologies.","Most importantly, it thereby stays globally optimal.","During this procedure the terminals $a_1$ , $a_2$ and $a_3$ stay fixed so that the pivot points and pivot circles as well as the transition lines stay exactly the same.","For topology $T_1$ we have now arrived at a special case of coupled 4-branching, in which the V- and L-branchings are transient (see Def.","REF ).","Let us refer to such a BP configuration as transient 4-branching.","As a consequence, the angle $\\gamma _4$ may be expressed in terms of the following branching angles (see Fig.","REF ) $ \\gamma _4 = \\beta _1 + \\beta _2 - \\theta _1 = h(\\alpha , m_1) - f\\Big (\\alpha , \\frac{m_2}{1-m_1} \\Big ) \\ .$ Figure: Transient coupled 4-BP with a 0 a_0 placed on the transition line of topology T 1 T_1.All in all, the above argument shows that the existence of a globally optimal coupled 4-BP necessarily implies the existence of a globally optimal transient 4-branching, which can be constructed by changing only the coordinates of one of the terminals.","Clearly, for the globally optimal and transient 4-BP of topology $T_1$ all four necessary conditions for optimal V-branching must still apply.","In order to generally rule out globally optimal coupled 4-BPs, it is therefore sufficient to show that at least one of the following two conditions is fulfilled for all parameter combinations $\\alpha $ and $m_i$ : the sum of the lower bounds on $\\gamma _i$ always exceeds $2 \\pi $ , condition (REF ) is always incompatible with Eq.","(REF ).","The first condition is equivalent to the following inequality, obtained by substituting the lower bound for $\\gamma _4$ in (REF ) by (REF ): $ \\Gamma _{1,1}(\\alpha , m_1,m_2) := f(\\alpha , 1 -m_1) + f\\Big ( \\alpha , 1 - \\frac{m_2}{1-m_1} \\Big ) - f(\\alpha , 1-m_1-m_2) > 0 \\ .$ The second condition can be expressed as inequality simply by combining (REF ) and (REF ): $\\Gamma _{2,1}(\\alpha , m_1, m_2) := h \\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) + f\\Big ( \\alpha , \\frac{m_2}{1 - m_1} \\Big ) - h(\\alpha , m_1) > 0 \\ .$ Proving one of these two inequalities already suffices to rule out globally optimal 4-branching between one source and three sinks.","Note that the three inequalities presented so far are not completely independent but are related via $\\Gamma = \\Gamma _{1,1} + \\Gamma _{2,1}$ .","The above procedure of moving the root node onto the transition line of topology $T_1$ can be repeated exactly analogously for $T_2$ .","This results in the inequalities $\\Gamma _{1,3}$ and $\\Gamma _{2,3}$ , which are of the exact same form except that $m_3$ appears in all places instead of $m_1$ .","Note that, in both cases, we have used that $a_0$ can be moved onto the transition lines of $T_1$ and $T_2$ without crossing the transition line of topology $T_3$ .","A justification for this is given in form of the following lemma.","Lemma E.2 Starting from a globally optimal coupled 4-BP connecting the terminals $\\lbrace a_i\\rbrace $ and located away from all terminals, one terminal node may be moved along a continuous path onto the transition lines of topology $T_1$ and $T_2$ without crossing the transition line of topology $T_3$ first.","Proof.","Let us give a proof by contradiction.","Assume that that $a_0$ could actually be moved along a continuous path onto the transition line of topology $T_3$ without touching any of the other two transition lines.","Then, one may also move $a_0$ infinitesimally further across the transition line of topology $T_3$ , such that the ROS of $T_3$ becomes non-degenerate.","At the same time, for topology $T_1$ and $T_2$ the coupled 4-branching configuration stays relatively optimal.","Now, since the ROS of $T_3$ deviates from the coupled 4-branching configuration, the 4-branching can no longer be globally optimal.","Note that a coupled 4-BP can only be globally optimal if all three topologies agree.","Consequently, the ROS of $T_3$ must be globally optimal.","However, the non-degenerate solution of $T_3$ will necessarily contain a cycle.","To see this, let $b_1$ be the branching point connected to $a_0$ and $b_2$ the other branching point in $T_3$ .","After crossing the transition line of $T_3$ , the edge $(b_1, b_2)$ at first has finite but infinitesimal length, so that either one of the edges $(b_2,a_1)$ or $(b_2,a_3)$ must intersect with $(b_1,a_2)$ , thereby creating a cycle in the ROS of $T_3$ (cf.","Fig.","REF ).","However, it was proven in [2] that for $\\alpha \\in [0,1)$ cyclic BOT solution cannot be globally optimal, so that we have arrived at a contradiction.","$$ To summarize, we have shown the following lemma: Lemma E.3 The existence of any globally optimal 4-BP connecting one source and three sinks and located away from the terminals implies the existence of two globally optimal transient 4-branchings, one for topology $T_1$ and one for $T_2$ ."],["$\\Gamma > 0 \\ $ for {{formula:b6d5d064-a7aa-4620-b08e-8d52134ebf39}} :","For $\\alpha \\le 0.5$ we have $h \\ge \\pi /2$ and $f < \\pi /2$ for all combinations of $m_i$ (see App. )","and thus clearly $\\Gamma > 0$ if $\\alpha \\le 0.5$ .","For all following arguments, we therefore assume that $\\alpha >0.5$ .","The following analytical arguments all rely on the properties of $h$ and $f$ , which are listed and proven in App.","."],["$\\Gamma > 0 \\ $ for {{formula:4340b88c-d10b-4164-adc8-fcac7451b1ad}} :","Using that $f(\\alpha , k)$ is monotonically decreasing in $k$ , the following loose lower bound suffices to demonstrate that $\\Gamma > 0$ if $m_1 \\ge 1/4$ : $\\Gamma &\\ge 2 \\ \\min _k \\, h(\\alpha , k) - 2 f(\\alpha , m_1) \\\\&= 2 \\arccos (2^{2 \\alpha - 1} - 1) - 2 \\arccos \\bigg ( \\frac{m_1^{2\\alpha } + 1 - (1-m_1)^{2 \\alpha }}{2 m_1^\\alpha } \\bigg ) \\ ,$ where we have used that $f(m_1) \\ge f(m_3)$ due to our assumption $m_1 \\le m_3$ and that $h(\\alpha , k)$ forms a minimum at $k=1/2$ .","For $m_1 = 1/4$ , one finds that this expression is truly positive if and only if $9^\\alpha < 2 \\cdot 4^\\alpha + 1 \\ ,$ which is fulfilled for all $\\alpha \\in [0,1)$ due to the subadditivity of the function $m \\mapsto m^\\alpha $ , namely $9^\\alpha = (4 + 4 + 1)^\\alpha < 4^\\alpha + 4^\\alpha +1$.","It follows that $\\Gamma > 0$ also for all $m_1 > 1/4$ due to the monotonicity of $f$ .","Figure: m 1 m_1-m 2 m_2 parameter space largely ruled out analytically.","In the white region the inequalities are checked numerically."],["$\\Gamma _{2,3} > 0 \\ $ for {{formula:1550a853-7ad7-41ff-84d6-e6ee54b09d65}} :","Inequalities $\\Gamma _{2,3}$ is fulfilled if $m_3 \\ge 1/2$ , since $\\Gamma _{2,3}(\\alpha , m_3, m_2) > h \\Big (\\alpha , \\underbrace{\\frac{m_3}{m_3 + m_2}}_{> \\ m_3} \\Big ) - h(\\alpha , m_3) \\ge 0 \\ \\text{ for } \\ m_3 \\ge 1/2 \\ ,$ where we have used that for $\\alpha > 0.5$ , $h(\\alpha , k)$ is monotonically increasing in $k$ for $k \\in [0.5,1)$ , as shown in Lem.","REF .","The remaining parameter region, for which none of the inequalities have been shown yet, can be characterized by the following conditions: $ \\alpha > 0.5 \\, , \\ \\ m_1 < 0.25 \\ \\text{ and } \\ m_2 \\in [0.5 - m_1, 1 - 2 m_1] \\ .$ The constraints for $m_3$ are implicitly represented, using that the normalization was chosen such that $m_1 +m_2 +m_3 = 1$ .","A visualization of the remaining region can be found in Fig.","REF .","Finally, we propose a simple and watertight numerical scheme, which can be used to rule out almost the entire remaining volume, already with little numerical effort."],["Numerical treatment of the remaining parameter space","For the remaining parameter region characterized by the conditions in (REF ), we propose a numerical scheme which checks the inequality $\\Gamma _{2,1}$ for all practically relevant parameter combinations of $m_1$ , $m_2$ and $\\alpha $ .","For this, we split the remaining volume into cuboids $ I = [\\check{\\alpha }, \\hat{\\alpha }] \\times [\\check{m}_1, \\hat{m}_1] \\times [\\check{m}_2, \\hat{m}_2]$ which are divided recursively into smaller cuboids based on the following octree scheme.","For each cuboid, we determine a lower bound of $\\Gamma _{2,1}$ .","This lower bound becomes tighter the smaller the cuboid $I$ .","If, for a cuboid, this lower bound is not yet positive, it is divided further into eight new cuboids by splitting each of the three intervals in half.","This procedure is iterated until for all cuboids the lower bound is truly positive.","For all $m_1, m_2, \\alpha $ in a cuboid $I=[\\check{\\alpha }, \\hat{\\alpha }] \\times [\\check{m}_1, \\hat{m}_1] \\times [\\check{m}_2, \\hat{m}_2]$ , the lower bound of $\\Gamma _{2,1}$ is obtained by minimizing each summand individually: $\\Gamma _{2,1}(\\alpha , m_1, m_2) &\\ge \\underset{I}{\\min } \\ h \\big (\\alpha , \\frac{m_1}{m_1 + m_2} \\big ) + \\underset{I}{\\min } \\ f\\big ( \\alpha , \\frac{m_2}{1 - m_1} \\big ) - \\underset{I}{\\max } \\ h(\\alpha , m_1) \\nonumber \\\\&= h \\Big (\\hat{\\alpha }, \\frac{\\hat{m}_1}{\\hat{m}_1 + \\check{m}_2} \\Big )+ f \\Big ( \\hat{\\alpha }, \\frac{ \\hat{m}_2}{1 - \\hat{m}_1} \\Big )- h(\\check{\\alpha }, \\check{m}_1)$ where we have used the crucial fact that for the remaining volume, characterized by (REF ), the functions $h(\\alpha , k)$ and $f(\\alpha , k)$ are monotonically decreasing in both arguments.","The proof of these monotonicities can be found in App. .","Note that the described procedure allows to rigorously confirm the inequality across a continuous region with finitely many evaluations.","For $\\alpha \\rightarrow 1$ or $m_1 \\rightarrow 0$ , the value of $\\Gamma _{2,1}$ approaches zero, so that the above scheme cannot be used to proof the inequality for values arbitrarily close to these limits.","However, if we restrict us to $m_1 > \\delta $ and $\\alpha < 1 - \\epsilon $ for finite $\\epsilon , \\delta > 0$ , the inequality $\\Gamma _{2,1}$ can be shown for practically all parameter combinations with little numerical effort.","For $\\epsilon = \\delta = 10^{-3}$ , using the proposed scheme, it was checked in only a few minutes that the lower bound in Eq.","(REF ) is larger than $10^{-4}$ everywhere in the remaining volume.","To be numerically on the safe side, we have stopped splitting a cuboid not if the lower bound exceeded zero but set a suitable finite threshold, in this case $10^{-4}$ .","In other words, we have stopped splitting a cuboid if its respective lower bound was $>10^{-4}$ .","The smallest terms which occur during the arithmetic operations inside the functions $h$ and $f$ are of the order $\\delta ^2$ , the largest terms are of order one.","It is therefore safe to say that numerical errors at the order of the machine accuracy are negligibly small against the margin of $10^{-4}$ and we may say that all together globally optimal 4-BPs are ruled out, for all practical parameter combinations.","In principle, the presented scheme can be used to check the inequality $\\Gamma _{2,1}$ up to even smaller $\\epsilon $ and $\\delta $ .","The Python code of the numerical scheme is made available at https://github.com/hci-unihd/BranchedOT."],["Non-optimality of five- and higher-degree branchings","In this section, we formally prove that globally optimal coupled $n$ -BPs not coincident with a terminal can be ruled out in general, given that 4-BPs are not globally optimal.","Lemma REF states that a solution is not globally optimal if any subsolution is not globally optimal.","It will therefore suffice to study the coupled BP as an isolated subproblem.","Let us start by proving the following corollary about the preservation of relative optimality under edge extensions for transient V- and L-branchings (see Def.","REF ): Corollary E.4 Consider a BOT problem with terminals $A=\\lbrace a_i\\rbrace $ and let the BP configuration $B = \\lbrace b_i \\rbrace $ be relatively optimal for a given topology $T$ .","Let $B$ contain a transient V-branching between two terminals, say $a_0$ and $a_1$ and denote the branching point connected to $a_0$ and $a_1$ by $b_1$ .","Otherwise, let $B$ not contain any strict L-or V-branchings.","Let us further denote the branching point to which $b_1$ is coupled in a transient V-branching by $b_2$ , as illustrated on the left side of Fig.","REF .","Then, there exists a direction in which the zero length edge $(b_1, b_2)$ can be extended to finite length $l>0$ (cf.","right side of Fig.","REF ) such that the new BOT problem (with shifted terminal positions) is solved relatively optimally by the new BP configuration (with shifted $b_1$ ).","Proof.","By assumption, the BP configuration of interest $B = \\lbrace b_i \\rbrace $ contains only Y-branchings and transient V- and L-branchings.","Then, according to Lemma REF , there exists a set of arbitrarily small displacements $\\delta _i$ , one for each branching point $b_i$ , so that $B + \\delta = \\lbrace b_i + \\delta _i \\rbrace $ is a non-degenerate BP configuration with arbitrarily small gradients $\\left\\Vert \\nabla _{\\!","\\!","b_i} \\, \\mathcal {C}(B+\\delta , A) \\right\\Vert $ .","Note that the notation $C(B,A)$ emphasizes that the cost function also depends on the terminal positions.","Since the BP configuration $B + \\delta $ is non-degenerate, any edge, in our case $(b_1,b_2)$ , can be easily expanded in length without changing any of the branching angles, assuming that the extension preserves the direction of the edge and that the BPs and terminals are moved along correspondingly.","As the gradient $\\nabla _{\\!","\\!","b_i} \\, \\mathcal {C}$ depends only on the directions of the edges meeting at $b_i$ (i.e.","the branching angles), the gradient is not changed by this procedure.","Let us summarize the shifted BPs by $B_{shift}$ and the shifted terminals by $A_{shift}$ .","Then, $\\left\\Vert \\nabla _{\\!","\\!","b_i} \\, \\mathcal {C}(B + \\delta , A) \\right\\Vert = \\left\\Vert \\nabla _{\\!","\\!","b_i} \\, \\mathcal {C}(B_{shift} + \\delta , A_{shift}) \\right\\Vert $ can also be made arbitrarily small as $\\delta \\rightarrow 0$ .","Let us prove by contradiction that $B_{shift}$ is the ROS of the BOT problem with terminals $A_{shift}$ : Assuming that a different BP configuration $B_0 \\ne B_{shift}$ was the ROS, there would exist a constant $\\Delta >0$ , so that $\\mathcal {C}(B_{shift} , A_{shift}) = \\mathcal {C}(B_0, A_{shift}) + \\Delta $ .","However, since the cost is a convex function w.r.t.","the BPs, we have $\\mathcal {C}(B_0, A_{shift}) \\ge \\mathcal {C}(B_{shift} + \\delta , A_{shift})+ \\langle \\nabla _{\\!","\\!","b_i} \\, \\mathcal {C}(B_{shift} + \\delta , A_{shift}), B_0 - (B_{shift} + \\delta ) \\rangle \\\\= \\mathcal {C}(B_{shift} , A_{shift}) + \\underbrace{O(\\delta ) + \\langle \\nabla _{\\!","\\!","b_i} \\, \\mathcal {C}(B_{shift} + \\delta , A_{shift}), B_0 - (B_{shift} + \\delta ) \\rangle }_{=:K} \\, .$ As $\\delta \\rightarrow 0$ , the latter two terms summarized by $K$ can clearly be made arbitrarily small.","In particular, there exists an $\\delta > 0$ , such that $\\vert K \\vert < \\Delta /2$ , which together with $\\mathcal {C}(B_{shift} , A_{shift}) = \\mathcal {C}(B_0, A_{shift}) + \\Delta $ implies that $\\mathcal {C}(B_0, A_{shift}) \\ge \\mathcal {C}(B_{shift}, A_{shift}) + K > \\mathcal {C}(B_0, A_{shift}) + \\Delta /2$ and we have thereby arrived at a contradiction, similarly to the reasoning in part (c) of the proof in App. .","$$ Figure: Extension of the edge (b 1 ,b 2 )(b_1,b_2) in the direction of the pivot point p 1 p_1, preserving relative optimality.Coming back to the non-optimality of coupled $n$ -BPs, let us, for concreteness, consider a globally optimal coupled 5-BP not coincident with a terminal.","By induction, repeating the presented argument one can then rule out all globally optimal $n$ -BPs.","The proof is by contradiction, so let us assume a globally optimal 5-BP existed, between terminals $a_0$ , $a_1$ , $a_2$ , $a_3$ and $a_4$ .","In terms of enclosed angles $\\gamma _i$ (see Fig.","REF ), it is a necessary condition that all $\\gamma _i$ must exceed their respective lower bound, specified by the optimal V-branching conditions in Table REF (Section REF ).","Again, for a 5-BP configuration to be globally optimal all possible full tree topologies must agree on the 5-BP configuration as their relative optimal solution.","Starting from this configuration, one may continuously move one of the terminals, e.g.","$a_0$ , such that the globally optimal coupled 5-BP starts to decouple, meaning that $a_0$ is moved until for (at least) one of the possible topologies, say $\\tilde{T}$ , the 5-BP configuration no longer provides the ROS.","This can always be achieved, for instance by bringing $a_0$ sufficiently close to $a_1$ so that a Y-branching between the two terminals becomes optimal.","Similar to the 4-branching case, the ROS of such a topology $\\tilde{T}$ in this moment becomes the globally optimal solution, as the other topologies are still in the 5-branching configuration which can only be globally optimal if all topologies agree on it.","The globally optimal solution of topology $\\tilde{T}$ must exhibit one of the following two properties it contains one Y-branching and a coupled 4-BP or it is non-degenerate and contains only Y-branchings.","In the first case, it would mean that a coupled 4-BP exists which is globally optimal as subgraph of a globally optimal solution (using the necessarily optimal substructure of Lem.","REF ).","This contradicts our assumption that 4-BPs are not globally optimal.","Figure: Edge extension to finite length l>0l > 0 in a globally optimal and transient 5-branching with transient V-branching between a 0 a_0 and a 1 a_1.","The resulting ROS is split into two subsolutions, as indicted by the red dashed line.Regarding the second option, we proceed as follows.","Let us move $a_0$ back to the point in which the coupled 5-BP configuration was still globally optimal but the ROS of $\\tilde{T}$ is on the verge of decoupling into a non-degenerate branching configuration.","Denote this special location of the terminals by $\\lbrace a_i^* \\rbrace $ .","In this configuration an infinitesimal movement of $a_0$ away from $a_0^*$ can cause the ROS of $\\tilde{T}$ to transition from coupled 5-branching to a non-degenerate BP configuration, very much analogous to the case of the transient 4-branching illustrated in Fig.","REF .","This means that in this configuration all V- and L-branchings are transient in the ROS of $T$ and WLOG we choose the labeling of the terminals such that the V-branching between $a_0^*$ and $a_1^*$ is transient, cf. Fig.","REF .","At this point, let us split the set of all possible full tree topologies $\\mathcal {T}$ into the following subsets.","The subset in which the terminals $a_0$ and $a_1$ are connected to a common branching point, say $b_1$ , is denoted by $\\mathcal {T}^{(0,1)}$ .","Let us label the BP to which $b_1$ is connected in these topologies by $b_2$ .","The ROS of all topologies $T \\in \\mathcal {T}^{(0,1)}$ for the current BOT problem contains a transient V-branching at branching point $b_1$ .","Moreover, let us single out a specific subset in $\\mathcal {T}^{(0,1)}$ , defined by the following condition: $\\mathcal {T}^{(0,1)}_{trans} = \\lbrace T \\in \\mathcal {T}^{(0,1)} : \\text{ROS of $T$ is transient if the terminals are located at $a_i^*$} \\rbrace \\ .$ Visibly, the branching at $b_1$ appears as V-branching but it may also be seen as a Y-branching with a zero length stub.","Let us now extend this zero length edge between $b_1$ and $b_2$ to finite length $l>0$ into the direction of the pivot point between $a_0^*$ and $a_1^*$ , as explained in the proof of Corollary REF .","The two terminals $a_1$ and $a_2$ are shifted from $a_0^*$ and $a_1^*$ to $a_0(l)$ and $a_1(l)$ and Corollary REF guarantees that the resulting BP configuration for all $T \\in \\mathcal {T}^{(0,1)}_{trans}$ solves the new BOT problem relatively optimally.","For an illustration of the edge extension see Fig.","REF .","Note that for all $T \\in \\mathcal {T}^{(0,1)}_{trans}$ this ROS is the same.","Now split this ROS into two subsolutions as indicted in Fig.","REF .","This induces two subproblems and subtopologies, as described in Def.","REF .","We focus on the upper right subproblem, consisting of four terminals.","Note that any topology $T^{(4)}$ on this four terminal subproblem, may be induced as subtopology by a topology $T \\in \\mathcal {T}^{(0,1)}$ .","Let us distinguish the following two cases: a) On the four terminal subproblem the topology of the GOS, denoted by $T^{*(4)}$ , is induced by a topology $T \\in \\mathcal {T}^{(0,1)}_{trans}$ .","Or b) $T^{*(4)}$ is induced by a topology $T \\in \\mathcal {T}^{(0,1)} \\setminus \\mathcal {T}^{(0,1)}_{trans}$ .","In case a) the ROS of $T^{*(4)}$ is given by the right subsolution in Fig.","REF , as the edge extension preserved the relative optimality for all $T \\in \\mathcal {T}^{(0,1)}_{trans}$ .","Hence, the globally optimal solution on the four terminal subproblem is given by a coupled 4-BP and we have arrived at a contradiction.","Otherwise, in case b), we do not know the ROS of $T^{*(4)}$ a priori, but it cannot be a coupled 4-branching configuration and must hence be non-degenerate.","Crucially, it must be non-degenerate for any finite length extension $l > 0$ , even if we consider the limit of $l \\rightarrow 0$ .","But this means that the ROS of $T^{*(4)}$ already transitions from a coupled 4-BP into a non-degenerate ROS, if the terminals $a_0$ and $a_1$ are perturbed infinitesimally ($l > 0$ but infinitesimal).","Consequently, for $l=0$ the ROS of $T^{*(4)}$ is transient.","But this means that $T^{*(4)}$ can be induced as a subtopology of a transient topology $T \\in \\mathcal {T}^{(0,1)}_{trans}$ and we have arrived yet at another contradiction.","$$"],["BOT on two-dimensional Riemannian manifolds","In the following section, we prove that the optimal branching angles for Y-shaped branchings on two-dimensional manifolds are the same as the optimal branching angles derived for BOT in the Euclidean plane.","An outline of the proof was given in Section REF .","The same strategy, presented below for the optimal branching angles, can be used to generalize other necessary conditions for optimal BOT solutions to manifolds, as explained in Sect.","REF ."],["Optimal Y-branching on two-dimensional Riemannian manifolds","For a Y-shaped branching at BP $b$ connecting the terminals $a_0$ , $a_1$ and $a_2$ in Euclidean plane the cost function of BOT is given by $\\mathcal {C}(b) = \\sum _i m_i^\\alpha \\left\\Vert a_i - b \\right\\Vert _2 \\ ,$ where the $m_i$ are the known edge flows.","Let us consider a two-dimensional Riemannian manifold $\\mathcal {M}$ embedded into $\\mathbb {R}^3$ , for which the metric is induced by the standard Euclidean inner product in $\\mathbb {R}^3$ .","Let the geodesic distance be denoted by $d(x,y)$ .","The generalized cost function for 1-to-2 branching then reads $\\mathcal {C}_M(b) = \\sum _i m_i^\\alpha \\, d( a_i, b) \\ .$ All points $b$ and $a_i$ now lie on the manifold and are assumed to have differing positions.","In a solution which minimizes $\\mathcal {C}_M$ the terminals $a_i$ are connected to $b$ via geodesics.","For $b$ to be a valid solution to the BOT problem on the manifold, these geodesics must exist.","We denote the geodesic which connects $b$ and $a_i$ by $v_i(\\lambda ) \\in \\mathcal {M}$ , parametrized by the length $\\lambda $ .","The tangent space at $b$ is denoted by $T_b \\mathcal {M}$ .","Furthermore, let $\\hat{n}_i$ be the unit tangent vectors of the geodesics $v_i(\\lambda )$ at the branching point $b$ , i.e.","$\\hat{n}_i = \\partial _\\lambda v_i(\\lambda ) \\big \\vert _{\\lambda = 0}$ .","WLOG, for all following considerations let us rotate and translate the manifold so that $b \\in \\mathbb {R}^3$ lies at the origin, i.e.","$b=0$ , and that the tangent space $T_b \\mathcal {M}$ is equal to the $x_1$ -$x_2$ -plane of $\\mathcal {R}^3$ , i.e.","$T_b \\mathcal {M} = \\mathbb {R}^2 \\times \\lbrace 0\\rbrace $ ."],["Restriction to a local subsolution on the manifold.","Since $\\mathcal {M}$ is embedded into $\\mathbb {R}^3$ , there exists an $r > 0$ and an environment $U(r) \\subset \\mathbb {R}^3$ around $b=0$ such that the manifold $\\mathcal {M} \\cap U(r)$ can be represented as the graph of a function: $\\mathcal {M} \\cap U(r) = \\lbrace (x, u(x)): x \\in D(r) \\subset \\mathbb {R}^2 \\rbrace \\ ,$ where $u$ is a smooth, scalar function $u: D(r) \\rightarrow \\mathbb {R}$ , defined on the disk of radius $r$ , denoted by $D(r) := \\lbrace (x_1, x_2) \\in \\mathbb {R}^2 : \\left\\Vert (x_1,x_2)^T \\right\\Vert _2 \\le r \\rbrace $ .","Note that due to the mentioned rotation and translation of $\\mathcal {M}$ , we have $u(0) = 0$ and $\\nabla u(0) = 0$ , where $\\nabla u$ denotes the gradient of $u$ .","The existence of such a function $u$ is guaranteed by the implicit function theorem.","A formal proof can be found in John M. Lee's book [18] in Proposition 8.24.","Further, let us define the orthogonal projection $\\sigma $ from the manifold onto the $x_1$ -$x_2$ -plane as $\\sigma : \\mathcal {M} \\cap U(r) \\rightarrow D(r) \\times \\lbrace 0\\rbrace , \\ \\begin{pmatrix} x_1 \\\\ x_2 \\\\ u(x_1,x_2) \\end{pmatrix} \\mapsto \\begin{pmatrix} x_1 \\\\ x_2 \\\\ 0 \\end{pmatrix} \\ .$ WLOG, $r$ is chosen sufficiently small so that this projection is bijective.","Now, Taylor's theorem states that $u$ can be approximated by the following expansion around $x = 0$ : $ u(x) = \\underbrace{u(0)}_{= \\ 0} + \\langle \\underbrace{\\nabla u(0)}_{= \\ 0}, x \\rangle + O(\\left\\Vert x \\right\\Vert _2^2) \\in O(r^2) \\ ,$ where $\\langle \\cdot , \\cdot \\rangle $ is the standard Euclidean inner product and we have introduced the Big-O notation.","A term is of order $O(r^2)$ if it goes to zero for $r \\rightarrow 0$ at least as fast as $r^2$ , or more formally: $p(r) \\in O(q(r)) \\ \\Leftrightarrow \\ \\lim _{r \\rightarrow 0} \\frac{p(r)}{q(r)} = c$ for some finite constant $c$ .","Consequently, a point $a = (x,u(x))$ in $\\mathcal {M} \\cap U(r)$ and its projection onto the plane $\\sigma (a)$ agree to first order, i.e.","$ \\left\\Vert \\sigma (a) - a \\right\\Vert _2 = u(x) \\in O(r^2) \\ .$ One of the key ingredients when transferring BOT problems from two-dimensional surfaces to the tangent plane $T_b \\mathcal {M}$ is the following Lemma about the difference between the Euclidean distance and the geodesic distance: Lemma F.1 (Relation between geodesic and Euclidean distance) Let $r$ be a small radius, which characterizes the environment $U(r)$ around the origin $b = 0$ on a two-dimensional Riemannian manifold as described above.","Let $a \\in \\mathcal {M} \\cap U$ be a point in this environment, located at $a = (x,u(x)) \\in \\mathbb {R}^3$ for some $x \\in D(r)$ and $u$ as above.","Then, the geodesic distance can be expressed through the Euclidean distance as $d(a,b) = \\left\\Vert a - b \\right\\Vert _2 + O(r^3) \\ .$ Proof.","Let us first note that $d(a,b) \\ge \\left\\Vert a - b \\right\\Vert _2$ .","It is hence sufficient to show that $d(a,b) \\le \\left\\Vert a - b \\right\\Vert _2 + O(r^3)$ .","Let us consider the following curve $\\gamma (t) = (tx, u(tx))$ on the manifold $\\mathcal {M}$ , which for $t \\in [0,1]$ connects $a = (x,u(x))$ and $b = 0$ .","In general, $\\gamma (t)$ is not a geodesic between $a$ and $b$ .","Thus, the length of $\\gamma (t)$ provides an upper bound to $d(a,b)$ .","For the calculation of the length we use that $\\gamma \\hspace{0.85355pt} ^{\\prime }(t) = (x, \\langle \\nabla u(tx), x \\rangle )$ and simply integrate $\\left\\Vert \\gamma \\hspace{0.85355pt} ^{\\prime }(t) \\right\\Vert _2$ along the curve in $\\mathbb {R}^3$ , since the metric of the embedding is induced by the standard Euclidean inner product in $\\mathbb {R}^3$ : $\\frac{d(a,b)}{\\left\\Vert a - b \\right\\Vert _2} &\\le \\frac{1}{\\sqrt{\\left\\Vert x \\right\\Vert ^2_2 + u^2(x)}} \\int _0^1 \\left\\Vert \\gamma \\hspace{0.85355pt} ^{\\prime }(t) \\right\\Vert _2 \\mathrm {d} t \\\\&\\le \\frac{1}{\\left\\Vert x \\right\\Vert _2} \\int _0^1 \\sqrt{\\left\\Vert x \\right\\Vert ^2_2 + \\vert \\langle \\nabla u(tx), x \\rangle \\vert ^2} \\ \\mathrm {d} t \\\\&= \\int _0^1 \\sqrt{1 + \\left| \\Big \\langle \\nabla u(tx), \\frac{x}{\\left\\Vert x \\right\\Vert _2} \\Big \\rangle \\right|^2} \\ \\mathrm {d}t \\\\&\\le 1 + \\frac{1}{2} \\int _0^1 \\left| \\Big \\langle \\nabla u(tx), \\frac{x}{\\left\\Vert x \\right\\Vert _2} \\Big \\rangle \\right|^2 \\mathrm {d}t \\\\&\\le 1 + \\frac{1}{2} \\int _0^1 \\left\\Vert \\nabla u(tx) \\right\\Vert _2^2 \\ \\mathrm {d}t \\ .$ For the last step we have used the Cauchy-Schwarz inequality.","Furthermore, we have used that $\\sqrt{1 + x^2} \\le 1 + x^2 / 2$ .","The only thing left to show is that the remaining integral is $O(r^2)$ , as $\\left\\Vert a - b \\right\\Vert = ( \\left\\Vert x \\right\\Vert ^2_2 + u^2(x) )^{1/2} \\in O(r)$ , since $\\left\\Vert x \\right\\Vert _2 \\le r$ and $u(x) \\in O(r^2)$, cf. Eq.","(REF ).","For that, let us consider the Taylor expansion of the $i$ -th component of $\\nabla u(tx)$ around 0.","For $i=1,2$ one has $\\partial _i \\, u(tx) = \\underbrace{\\partial _i \\, u(0)}_{= \\ 0 } + \\partial _j \\partial _i \\, u(0) \\ t x_j + O(r^2) \\ ,$ where the sum in $j$ is implicit and runs over both indices $j=1,2$ .","And thus: $\\frac{1}{2} \\int _0^1 \\left\\Vert \\nabla u(tx) \\right\\Vert _2^2 \\ \\mathrm {d}t = \\frac{1}{6} \\, \\partial _j \\partial _i \\, u(0) \\ x_j \\ \\partial _k \\partial _i \\, u(0) \\ x_k + O(r^2) \\ ,$ where again the sum over all index pairs is implied.","Since for all components $x_i$ we have $x_i \\le r$ , this completes the proof.","$$"],["Projection of a subproblem onto the tangent plane.","Coming back to the 1-to-2 BOT problem on the manifold, let us project the geodesics $v_i$ onto the $x_1$ -$x_2$ -plane using the orthogonal projection $\\sigma $ from above, in order to transfer the BOT problem from the manifold onto the tangent plane.","WLOG, we choose the orientation of the $x_1$ and $x_2$ axis such that $\\hat{n}_0$ points along the $x_1$ -axis.","Then, one may easily check that the implicit function theorem guarantees that for sufficiently small $r$ the projected geodesic $\\sigma (v_0)$ can be represented by a graph $(x_1, w(x_1))$ with smooth $w: (-\\epsilon ,r+\\epsilon ) \\rightarrow \\mathbb {R}$ .","Note, that the projected geodesic $\\sigma (v_0)$ may be considered on the open interval with $\\epsilon >0$ such that all derivatives with respect to $x_1 \\in [0,r]$ are well-defined.","Due to our special orientation of the $x_1$ -$x_2$ -plane, we have $w(0)=0$ and $w^{\\prime }(0) = 0$ .","Let us now define the following two points inside the tangent plane: Define $\\hat{a}_0$ as the point which lies in the direction of $\\hat{n}_0$ at a distance $r$ away from the origin, i.e.","$\\hat{a}_0 = r \\cdot \\hat{n}_0 = (r,0)$ .","Similarly, define $\\hat{a}_i = r \\cdot \\hat{n}_i$ .","These will be the three terminals of the BOT subproblem of interest in the Euclidean plane.","For an illustration see Fig.","REF .","Define $a_{\\perp ,0} = (r, w(r))$ as special point on the projected geodesic $\\sigma (v_0)$ .","From an analogous construction using the projected geodesics $\\sigma (v_1)$ and $\\sigma (v_2)$ , we obtain the $a_{\\perp ,i}$ also for $i=1,2$ .","No explicit representation of these will be necessary.","What is however important is that the two defined points agree up to linear order, in the sense that $\\left\\Vert \\hat{a}_i - a_{\\perp ,i} \\right\\Vert _2 \\in O(r^2)$ for all $i=0,1,2$ .","This is shown for $i=0$ , by Taylor expansion of $w(x_1)$ around $x_1 = 0$ , but holds of course equally true for $i=1,2$ : $ \\left\\Vert \\hat{a}_0 - a_{\\perp ,0} \\right\\Vert _2 = w(r)= \\underbrace{w(0)}_{= \\ 0} + \\underbrace{w^{\\prime }(0)}_{= \\ 0} \\, r + O(r^2) \\in O(r^2) .$ Note that since the points $a_{\\perp ,i}$ lie on the projected geodesics, after an inverse projection, the $\\sigma ^{-1}(a_{\\perp ,i})$ mark special points on the unprojected geodesics $v_i \\in \\mathcal {M}$ .","These will be the terminals of the subproblem of interest on the manifold.","Combining Eq.","(REF ) and (REF ), we can relate $\\sigma ^{-1}(a_{\\perp , i}) \\in \\mathcal {M}$ to the terminals $\\hat{a}_i$ of the BOT problem of interest in the plane by $ \\left\\Vert \\sigma ^{-1}(a_{\\perp , i}) - \\hat{a}_i \\right\\Vert _2 \\in O(r^2) \\ ,$ Note that all steps above can be repeated for radii smaller than the chosen $r$ .","In the following we will decrease the scale $r$ of the two problems and therefore indicate the $r$ -dependence in $\\sigma ^{-1}(a_{\\perp , i}(r))$ and $\\hat{a}_i(r)$ explicitly.","Figure: The tangent vectors of the geodesics (dotted) in bb define the position of the terminals a ^ i \\hat{a}_i of the subproblem on the flat disk D(r)⊂T b ℳD(r) \\subset T_b \\mathcal {M}."],["Improved subsolutions on the manifold by projection from the tangent space.","Now, follows the main line of arguments to show that $b$ does not connect $a_i \\in \\mathcal {M}$ at minimal cost if the geodesic angles in the tangent space $T_b \\mathcal {M}$ are different from the optimal angles in the Euclidean plane.","We start by considering the subproblem on the manifold with terminals $\\sigma ^{-1}(a_{\\perp , i}) \\in \\mathcal {M}$ .","The generalized cost function of this subproblem reads: $\\mathcal {C}_M(b) &= \\sum _i m_i^\\alpha \\, d(\\sigma ^{-1}(a_{\\perp ,i}(r)), b) \\ .$ We can now use Eq.","(REF ) to express the terminals on the manifold through the terminals $\\hat{a}_i$ in the tangent plane and express the geodesic distance in terms of the Euclidean distance, based on Lem.","REF .","All additional terms are at least of order $O(r^2)$ and we have $ \\mathcal {C}_M(b) = \\underbrace{\\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b \\right\\Vert _2}_{=: \\ \\mathcal {C}_b(r)} + \\ O(r^2) \\ .$ Crucially, with $\\mathcal {C}_b(r)$ we have arrived at the cost of the BOT problem which purely lives on the tangent space.","It is the cost of the solution in which $b$ is connected via straight lines to the terminals $\\hat{a}_i(r) = r \\cdot \\hat{n}_i$ at a distance $r$ away from $b$ , as shown in Fig.","REF .","By assumption, this solution is not relatively optimally since the angles (or equivalently the directions $\\hat{n}_i$ ) are assumed to deviate from the optimal angles in the Euclidean plane.","This means that a branching point $b^* \\in D(r)\\times \\lbrace 0 \\rbrace $ exists which provides a better solution with cost denoted by $\\mathcal {C}_{b^*}(r) < \\mathcal {C}_b(r)$ .","From the definition of $\\hat{a}_i = r \\cdot \\hat{n}_i$ , we see that two BOT problems within disks of different radii $D(r)$ and $D(\\nu \\cdot r)$ are related simply by rescaling the coordinates.","Under rescaling of coordinates the Euclidean distance between points and consequently also the cost functions changes proportionally, i.e.","$\\mathcal {C} \\rightarrow \\nu \\mathcal {C}$ .","Thus, there exist non-negative constants $M_1$ and $M_2$ , so that $\\mathcal {C}_b(r) = M_1 \\cdot r$ and $\\mathcal {C}_{b^*}(r) = M_2 \\cdot r$ .","Let us distinguish the following two cases: a) $\\ \\ \\ \\mathcal {C}_{b^*}(r) = \\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b^* \\right\\Vert _2 = 0$ .","This special case looks unusual, but is in principle possible, e.g.","if all terminals $a_i$ and $b$ lie on a common geodesic.","In the case, where $C_{b^*}(r) = 0$ , we project $b^*$ onto the manifold using $\\sigma ^{-1}$ and go backwards in the above steps.","We find that $0 = \\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b^* \\right\\Vert _2 =\\underbrace{\\sum _i m_i^\\alpha \\, d( \\sigma ^{-1}(a_{\\perp ,i}(r)) , \\sigma ^{-1}(b^*))}_{= \\ \\mathcal {C}_M(\\sigma ^{-1}(b^*)))} + O(r^2) \\, .$ Clearly, a sufficiently small $r > 0$ exists so that the terms contained in $O(r^2)$ are much smaller than $M_1 \\cdot r$ , so that $\\mathcal {C}_M(\\sigma ^{-1}(b^*))) = 0 + O(r^2) < M_1 \\cdot r + O(r^2) = \\mathcal {C}_M(b) \\ .$ This proves that the projection of $b^*$ onto the manifold provides a cheaper cost solution on the manifold than $b$ .","b) $\\ \\ \\ \\mathcal {C}_{b^*}(r) > 0$ , and thus $M_2 >0$ .","In this case, let us write the ratio of the two costs as $ \\mathcal {C}_b(r)/ \\mathcal {C}_{b^*}(r) = M_1 / M_2 =: 1 + \\kappa $ for some $\\kappa > 0$ .","Again, $b^*$ and its projection onto the manifold agree to first order, that is $\\left\\Vert \\sigma ^{-1}(b^*) - b^* \\right\\Vert \\in O(r^2)$ .","We now show that $\\sigma ^{-1}(b^*)$ provides a better solution to the BOT problem on the manifold than $b$ .","We proceed from Eq.","(REF ), using that $\\mathcal {C}_{b^*}(r)$ is linear in $r$ : $\\mathcal {C}_M(b) &= C_b(r) + O(r^2) = (1 + \\kappa ) \\, \\mathcal {C}_{b^*}(r) + O(r^2) \\nonumber \\\\&= \\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b^* \\right\\Vert _2 + \\kappa M_2 \\cdot r + O(r^2) \\nonumber \\\\&= \\underbrace{\\sum _i m_i^\\alpha \\, d( \\sigma ^{-1}(a_{\\perp , i}(r)) , \\sigma ^{-1}(b^*))}_{= \\ \\mathcal {C}_M(\\sigma ^{-1}(b^*))} + \\kappa M_2 \\cdot r + O(r^2) \\ .$ In the last step, we have projected inversely onto the manifold and replaced the Euclidean distance by the geodesic distance (all with differences at least of $O(r^2)$ ).","We have therefore arrived at the cost of the new solution on the manifold.","Since the term $\\kappa M_2 \\cdot r$ is positive, we conclude that $r$ may always be chosen sufficiently small, so that the linear term dominates over higher-order terms and we have: $\\mathcal {C}_M(\\sigma ^{-1}(b^*)) < \\mathcal {C}_M(b) \\ .$ Together with Lem.","REF on the necessarily optimal substructure of all subsolutions, this concludes the proof that any optimally placed branching point on a two-dimensional embedded Riemannian manifold must exhibit the same optimal branching angles as in the Euclidean plane.$$"],["Other local properties of optimal BOT solutions on manifolds","The logic of the proof outlined above can be easily extended to the conditions for optimal V- and L-branching.","For BOT problems in the Euclidean plane, Table REF lists these conditions under which the optimal BP position in a 1-to-2 branching coincides with one of the terminals.","Let us now consider a 1-to-2 BOT solution on a two-dimensional manifold in which $b$ is located at the terminal $a_0$ and is connected to the terminals $a_1$ and $a_2$ via the geodesics $v_1$ and $v_2$ respectively.","Imagine that in the tangent plane $T_b \\mathcal {M}$ the angle enclosed by these geodesics does not fulfill the optimal V-branching criterion.","In this case, one may again consider a sufficiently small disk of radius $r$ around $b = a_0$ in the tangent space $T_b \\mathcal {M}$ and project the corresponding subproblem from the manifold onto this disk.","In the Euclidean plane, an improved BP location $b^*$ must exists, since $b = a_0$ is the optimal solution if and only if the V-branching condition is fulfilled.","All arguments about the scaling of the cost improvement apply as described above and the projection $b^*$ back to the manifold will provide an improved solution to the subproblem on the manifold if $r$ is chosen sufficiently small.","Even more so, the same reasoning can also be applied to generalize our results regarding the non-optimality of higher-degree branchings.","Given a coupled $n$ -BP at position $b$ not coincident with a terminal, one may again consider a sufficiently small region around $b$ on the manifold and project the corresponding subsolution onto $T_b \\mathcal {M}$ .","Exactly analogous to the previous arguments, improving the topology locally in the plane and projecting back to the manifold eventually results in an improved solution on the manifold."],["The practical side of BOT on manifolds","Although much of the theory generalizes nicely to Riemannian manifolds, the generalization of the practical algorithms is highly non-trivial.","Unlike in Euclidean space, realizing the optimal branching angles is a necessary but no longer sufficient condition for relatively optimal solutions.","For instance, on the sphere the meridians of three terminals located in the southern hemisphere at longitudes $0^\\circ $ , $120^\\circ $ , $240^\\circ $ will intersect at both poles at angles of $120^\\circ $ , which is the optimal angle for $\\alpha =0$ .","Nonetheless, only the south pole is the optimal branching point.","In essence, the geometry optimization aims to assign simultaneously to each branching point the coordinates of the weighted geometric median of its neighbors, a problem that is considered in [8].","The topology optimization presented in Sect.","REF could be easily generalized to manifolds if the geodesic distance can be computed.","Due to these obstacles, previous works of the Steiner Tree problem ($\\alpha = 1$ ) have focused mostly on the sphere as important special case [7]."],["Hardware and code availability.","Python code for all experiments can be found at https://github.com/hci-unihd/BranchedOT.","For the different experiments, a single machine with 56 CPUs (Intel(R) Xeon(R) CPU E5-2660 v4 @ 2.00GHz) and 256GB RAM was used.","Execution times for all described experiments lie (at most) in the order of hours.","A more detailed estimate can be obtained from the performance statistics reported in Fig.","REF and Fig.","REF ."],["Geometric construction of relatively optimal solutions for BOT with multiple sources","In Sect.","REF we have presented the the exact geometric construction of relatively optimal solutions based on [2], [9] and its generalization to the case of BOT with multiple sources.","We have thereby solved the open problem 15.11 in [2], for which an example is illustrated in Fig.","REF .","Figure REF shows a simple BOT problem with a chosen topology for which no root node can be chosen such that all branchings are symmetric (see Sect.","REF ).","Choosing for instance the left source as a root node, the branching at $b_1$ is symmetric whereas the one at $b_2$ is asymmetric.","However, after having solved the asymmetric branching case analogously to the symmetric one, the relatively optimal solution can now be constructed geometrically as shown in Fig.","REF .","More examples of the geometric construction for a given topology applied BOT problems with multiple sources are shown in Fig.","REF .","Figure: Simple setup with symmetric branching and asymmetric branching, see Sect.",".Figure: Examples for geometric construction of relatively optimal solutions with fixed topology for BOT problems with multiple sources.","The construction is based on the optimal branching angles and uses one pivot circle and pivot point (light blue) per branching point.","Sources are shown in red, sinks in blue."],["Continuity of the optimal BP configuration.","The limitations of the geometric construction above are discussed in Sect.","REF .","Still, it forms the basis for our theoretical arguments in the paper.","Moreover, from the construction of a single branching point based on the branching angles (see Fig.","REF ), it can be seen that its optimal position changes continuously w.r.t.","the neighbor positions, the edge flows or $\\alpha $ .","Namely, the Y-, V- and L- branching change continuously as the source $a_0$ is moved around.","In addition, the optimal branching angles are given by continuous functions of $k$ and $\\alpha $ (cf. Eq.","(REF )), making the construction of the pivot point and pivot circle continuous.","By transitivity and based on the optimal substructure in Lem.","REF , the continuity generalizes also to the construction of larger ROS for a given topology."],["Numerical algorithm for geometry optimization","In this section, we provide theoretical details and practical experiments for the numerical geometry optimization presented in Sect.","REF .","We have generalized this approached from the context of the ESTP [26] to BOT.","For a given tree topology $T$ , the BP configuration is optimized by minimizing the following cost function: $\\mathcal {C}(X) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\left\\Vert x_i - x_j \\right\\Vert _2 ,$ where the $x_i$ for $1 \\le i \\le n$ are fixed terminals and the $x_i$ for $n+1 \\le i \\le n+m$ denote the variable branching point positions.","All coordinates are collectively summarized by $X$ .","Starting from a non-optimal, non-degenerate BP configuration denoted $X^{(0)}$ , e.g.","from a random initialization of the branching points, the algorithm iteratively solves the following linear system of equations $ x_i^{(k + 1)} = \\!","\\!","\\sum _{j \\, : \\, (i,j) \\in T} m_{ij}^\\alpha \\frac{x_j^{(k+1)}}{\\vert x_i^{(k)} - x_j^{(k)} \\vert }\\ \\Bigg / \\sum _{j \\, : \\, (i,j) \\in T} \\frac{m_{ij}^\\alpha }{\\vert x_i^{(k)} - x_j^{(k)} \\vert }, \\hspace{11.38092pt}\\text{for } n+1 \\le i \\le n+m.$ In essence, this is an iteratively reweighted least squares (IRLS) approach [4].","To see this, let us rewrite the cost function in Eq.","(REF ) into pseudo-quadratic from: $\\mathcal {C}(X) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\vert x_i - x_j \\vert ^{(\\beta -2) + 2} = \\sum _{(i,j) \\, \\in \\, T} \\underbrace{m_{ij}^\\alpha \\vert x_i - x_j \\vert ^{\\beta -2}}_{=: w_{ij}(X)} \\vert x_i - x_j \\vert ^2 \\, ,$ with $\\beta = 1$ for BOT.","Clearly, the weights $w_{ij}$ depend on the branching point positions.","However, the idea of IRLS is to insert the coordinates $X^{(k)} = \\lbrace x_i^{(k)} \\rbrace $ at $k$ -th iteration into $w_{ij}$ to obtain a truly quadratic form: $Q^{(k)}(X) &= w_{ij}(X^{(k)}) \\vert x_i - x_j \\vert ^2 =\\sum _{ (i,j) \\, \\in \\, T} m_{ij}^\\alpha \\frac{ \\vert x_i - x_j \\vert ^2}{ \\vert x_i^{(k)} - x_j^{(k)} \\vert } \\, ,$ where we have plugged in $w_{ij}(X^{(k)})$ and $\\beta = 1$ .","Indeed, minimizing this quadratic form yields $X^{(k+1)}$ as in Eq.","(REF ).","The updated coordinates $X^{(k+1)}$ are then plugged into $w_{ij}(X)$ during the next iteration.","This iterative updating of the weights $w_{ij}$ gives IRLS its name.","[26] proved in detail that this iterative solver converges to the minimum cost BP configuration.","The reasoning in [26] is based on the following key argument: As $X^{(k+1)}$ is the minimizer of $Q^{(k)}(X)$ , surely $Q^{(k)}(X^{(k)}) \\ge Q^{(k)}(X^{(k+1)})$ .","Together with the fact that $\\mathcal {C} (X^{(k)}) = Q^{(k)}(X^{(k)})$ , we have $\\mathcal {C}(X^{(k)}) &= Q^{(k)}(X^{(k)}) \\ge Q^{(k)}(X^{(k+1)}) \\\\&= \\sum _{ (i,j) \\, \\in \\, T: } m_{ij}^\\alpha \\frac{\\big ( \\vert x_i^{(k)} - x_j^{(k)} \\vert +\\vert x_i^{(k+1)} - x_j^{(k+1)} \\vert - \\vert x_i^{(k)} - x_j^{(k)} \\vert \\big )^2}{ \\vert x_i^{(k)} - x_j^{(k)} \\vert } \\\\&= \\mathcal {C}(X^{(k)}) + 2 [\\mathcal {C}(X^{(k+1)}) - \\mathcal {C}(X^{(k)}) ] \\\\&\\hspace{54.06006pt} + \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\frac{ \\big ( \\vert x_i^{(k+1)} - x_j^{(k+1)} \\vert - \\vert x_i^{(k)} - x_j^{(k)} \\vert \\big )^2}{ \\vert x_i^{(k)} - x_j^{(k)} \\vert } \\ .$ Since the sum in the last expression is clearly non-negative, the inequality above implies that $\\mathcal {C}(X^{(k)}) \\ge \\mathcal {C}(X^{(k+1)})$ .","This means that the cost of the BP configuration decreases with each iteration which, as shown in [26], implies that the iterations defined by Eq.","(REF ) converge to an absolute minimum of $\\mathcal {C}(X)$ .","As can be seen from the above derivation, the edge flows $m_{ij}^\\alpha $ are constant coefficients which do not complicate the considerations in [26].","Consequently, all arguments presented there can be directly transferred from ESTP to BOT.","For a detailed discussion on the complexity of the algorithm and suitable convergence criteria, we refer the reader to the work of Smith and only briefly state the results here.","Based on an analytically tractable example, Smith claims that the algorithm requires at most $O(N/\\epsilon )$ iterations to converge to a solution of the ESTP whose cost is within $\\epsilon $ of the optimal cost.","As a convergence criterion, Smith suggests to stop the iteration when the angles at all branching points are sufficiently close to the optimal angle conditions.","In order to be able to apply the algorithm also to trees with higher-degree branching points where the optimal angle conditions are a necessary but not sufficient condition for the cost minimum, the experiments in this paper use a different criterion.","The algorithm is considered to have converged if from one iteration to the next the relative cost improvement $\\big (\\mathcal {C}(X^{(k)}) - \\mathcal {C}(X^{(k+1)}) \\big ) / \\mathcal {C}(X^{(k+1)})$ has dropped below a certain threshold.","The BP optimization routine for a given tree topology is summarized in Alg.","REF .","Note that, in practice, the denominators $\\vert x_i^{(k)} - x_j^{(k)} \\vert $ in Eq.","(REF ) are clipped to $10^{-7}$ to avoid numerical instabilities.","BP optimization routine Input: threshold $\\eta $ , tree topology $T$ , BOT problem (terminal positions $x_{1:n}$ , supplies/demands $\\mu $ , $\\alpha $ ) Output: numerical minimal cost BP configuration [1] $F \\leftarrow $ get_all_edge_flows($T,\\mu $ ) $\\triangleright \\ $ uniquely determined from flow constraints $x_{n+1:n+m} \\leftarrow $ randomly initialized.","$C_{old} \\leftarrow \\infty $ $C \\leftarrow $ BOT_cost$(T,x_{1:n+m},F, \\alpha )$ $\\frac{\\mathcal {C}_{old} - \\mathcal {C}}{\\mathcal {C}} > \\eta $ $C_{old} \\leftarrow C$ $x_{1:n+m} \\leftarrow $ update_BPs($x_{1:n+m},F, \\alpha $ ) $\\triangleright \\ $ solves linear system in Eq.","(REF ) $C \\leftarrow $ BOT_cost$(T,x_{1:n+m},F, \\alpha )$ return $x_{1:n+m}$"],["BOT with costs scaling non-linearly with the edge length.","Let us briefly consider BOT with a modified cost function of $\\mathcal {C}(X) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\left\\Vert x_i - x_j \\right\\Vert ^\\beta _2 \\,$ including one additional parameter $\\beta $ , regulating the scaling of the cost function w.r.t.","edge lengths.","For $\\beta \\ge 1$ the cost function as a function of the branching points is still convex, as $x \\mapsto x^\\beta $ is convex and increasing and the Euclidean norm is convex.","Thus, it has a unique minimum.","As can be seen from Eq.","(REF ), the IRLS scheme completely absorbs the $\\beta $ into the weights $w_{ij}$ .","Consequently, the geometry optimization as presented here is readily applicable also to the modified cost in Eq.","(REF ).","However, for $\\beta >1$ we are not aware of any theoretical convergence guarantee to the minimum, therefore, further investigation is required.","Though, in the special case of $\\beta =2$ , it is clear that the geometry optimization can be solved to global optimality in just a single iteration."],["Performance test of numerical BP optimization.","In order to evaluate how long the BP optimization for a given topology takes, we have randomly generated 1000 BOT problems for different number of terminals using Alg.","REF .","For each of these problems, we have applied the BP optimization routine, given a uniformly sampled full tree topology.","Regarding the generation of problem setups, note that BOT solutions and problems are invariant under rescaling of the total demand and supply as well as under global rescaling of the coordinates, see Eq.","(REF ).","WLOG, these scales are chosen to be 1.","Figure REF shows the average runtime in seconds plotted against the number of terminals.","For BOT problems with 10 to 1000 terminals our efficient C++ implementation of the geometry optimization takes just a fraction of a second.","The relatively large error bars, indicating the standard deviation, are not due to an insufficient sample size but due to the natural run time variability for different problems that exists independent of the $n$ -dependence.","For instance, plotting the average runtime of all problems with $\\alpha \\le 0.5$ and $\\alpha > 0.5$ separately reveals that the optimization on average requires more time for problems with higher $\\alpha $ .","The reason for this was not investigated further, but we suspect that for larger $\\alpha $ on average more V- and L-branchings occur which may require more iterations than Y-branchings for convergence.","For this and all following experiments we chose the convergence threshold $\\eta $ to be $10^{-6}$ .","Figure: Average runtime of BP optimization routine applied to each 1000 random BOT problems of different size.","Average of all problems with α≤0.5\\alpha \\le 0.5 in orange (dotted) and α>0.5\\alpha > 0.5 in green (dashed)."],["Scaling of the geometry optimization","A single iteration of the geometry optimization, i.e.","solving the linear system defined by Eq.","(REF ) once, takes $O(nd)$ operations for a problem with $n$ terminals in $d$ spatial dimensions.","The elimination scheme used in our efficient C++ implementation is based on the “elimination on leaves of a tree” found in [26].","From Eq.","(REF ), one can easily see that the geometry optimization parallelizes over the spatial dimensions, as the linear system of the form $Ax = b$ shares the same matrix $A$ across the different dimensions and only the $b$ is different.","Although a single iteration is of order $O(nd)$ , it is a priori not clear how many iterations are required until convergence is reached.","Paralleling the setup of Fig.","REF , we have conducted an experiment where we report the number of iterations, for a batch of BOT problems.","We performed the same experiment in Euclidean space of different dimensions ($d \\in \\lbrace 3,4,5,10,30,100 \\rbrace $ ).","Figure REF shows the number of iterations until convergence plotted against the number of terminals.","The plot suggests that, for the investigated regime of $n$ and $d$ , the number of iterations on average scales like $\\log (n)$ .","However, this is merely an empirical observation and theoretical investigations are an interesting subject for future research.","Figure: Average number of iterations required by the BP optimization routine until convergence for each 1000 random BOT problems of different sizes nn and different dimensionality dd.","The x-axis is in log-scale.","The plot suggests that, for the investigated regime of nn and dd, the number of iterations on average scales like log(n)\\log (n), though one cannot generalize this statement.Random BOT problem generation Input: number of terminals $n$ Output: BOT problem with $n$ terminals located in $[0,1]\\times [0,1]$ , total supply and demand equal to 1 [1] $\\alpha \\sim $ Uniform($[0,1]$ ) $\\triangleright \\ $ sample uniformly from $[0,1]$ $n^+ \\sim $ Uniform($\\lbrace 1,2,...,n-1\\rbrace $ ) $\\triangleright \\ $ number of sources $n^- \\leftarrow n - n^+$ $\\mu ^+_{1:n^+} \\sim $ Uniform($[0,1]$ , size=$n^+$ ) $\\triangleright \\ $ array of supplies $\\mu ^-_{1:n^-} \\sim $ Uniform($[0,1]$ , size=$n^-$ ) $\\triangleright \\ $ array of demands $a_{1:n} \\sim $ Uniform($[0,1]$ , size=($n$ ,2)) $\\triangleright \\ $ terminal coordinates return $a_{1:n}, \\ \\mu ^+/\\mathrm {sum}(\\mu ^+), \\ \\mu ^-/\\mathrm {sum}(\\mu ^-), \\ \\alpha $"],["Greedy randomized heuristic for topology optimization","In this section, we give the details of the greedy randomized heuristic presented in Section REF and show the results of some additional experiments conducted with it.","Starting from a tree topology $T$ , a uniformly sampled edge $\\hat{e}$ is removed.","The node in the smaller connected component $\\ell $ is connected via a new branching point to an edge in the other component.","This edge is sampled according to $p(e) \\propto \\exp (-d(e, \\ell )^2 / d_{min}^2)$ , where $d(e, \\ell )$ is the distance between an edge $e=(i,j)$ and node $\\ell $ , defined by $ d(e, \\ell ) = \\underset{\\lambda \\in [0,1]}{\\min } \\ \\left\\Vert [x_i + \\lambda (x_j - x_i) ] - \\ell \\right\\Vert _2 \\ $ and $d_{min}$ is the distance to the closest considered edge.","The resulting new tree topology $T_{new}$ is accepted if it decreases the cost compared to the previous topology $T$ and the above procedure is iterated.","This greedy optimization strategy is summarized in Alg.","REF below.","Greedy randomized heuristic (zero temperature limit) Input: BOT problem $P$ , tree $T$ interconnecting terminals $a_{1:n}$ (with help of BPs) Output: heuristic BOT solution [1] $B \\leftarrow $ optimize_BP_configuration($T, P$ ) $\\triangleright \\ $ returns coordinates of BPs and terminals, see Alg.","REF $\\mathcal {C} \\leftarrow $ BOT_cost$(T,B,P)$ $E \\leftarrow $ list(T.edges()) $E$ not empty $S \\leftarrow $ Store current state $(T,B)$ $\\mathcal {C}_{old} \\leftarrow \\mathcal {C}$ $\\hat{e} \\sim $ Uniform($E$ ) $\\triangleright \\ $ sample uniformly from $E$ $E$ .remove($e$ ) $T$ .remove_edge($\\hat{e}$ ) $C_b \\leftarrow $ subgraph component of $T$ with more nodes $E_b \\leftarrow $ list($C_b$ .edges) $b \\leftarrow $ node of $\\hat{e}$ in $C_b$ $\\ell \\leftarrow $ node of $\\hat{e}$ not in $C_b$ degree($b$ ) == 2 $n_1,n_2 \\leftarrow $ neighbors($T, b$ ) $T$ .remove_node($b$ ) $\\triangleright \\ $ remove unnecessary BPs with degree 2 $T$ .add_edge($n_1,n_2$ ) $E_b$ .remove(($n_1,b$ ), ($n_2,b$ )) $d \\leftarrow $ Initalize array of size length($E_b$ ) edge $e \\in E_b$ $d[e] \\leftarrow $ get_distance($e,\\ell $ ) $\\triangleright \\ $ calculate distance defined by Eq.","(REF ) $d_{min} \\leftarrow $ min($d$ ) $e_c \\sim \\exp (-d^2/d_{min}^2)$ $\\triangleright \\ $ sample edge according to distance kernel $T$ .add_node($b_{new}$ ) $\\triangleright \\ $ Initialize a new BP $T$ .add_edges($(\\ell ,b_{new}),(b_{new}, e_c[0]),(b_{new}, e_c[1])$ ) $B \\leftarrow $ optimize_BP_configuration($T, P$ ) $\\mathcal {C} \\leftarrow $ BOT_cost$(T,X,P)$ $ \\mathcal {C} < \\mathcal {C}_{old} $ $E \\leftarrow $ list(T.edges()) $\\triangleright \\ $ accept new state $T,B \\leftarrow S$ $\\triangleright \\ $ Restore old state $(T,B)$ $\\mathcal {C} \\leftarrow \\mathcal {C}_{old}$ return $T, B$ Figure REF shows an example BOT problem with 100 terminals to which our greedy heuristic has been applied.","As starting point, the topology with the least possible structure was used, a star-like tree centred around a single BP of degree 100, cf. Fig.","REF .","It can be seen how the topology evolves over less than 2000 iterations to the final solution.","Figure REF shows the decreasing transportation cost $\\mathcal {C}$ plotted against the number of iterations."],["Performance test of heuristic topology optimization.","Finally, in a large experiment, we have investigated how many iterations the greedy heuristic requires on average to converge.","For that, the greedy heuristic was applied to a number of BOT problems with $n$ terminals and the number of iterations until convergence were counted.","The mean and standard deviation of the required iterations are plotted in Fig.","REF .","The reason for the relatively large standard deviations are due to intrinsic variation of the sampled problems.","To illustrate, for instance, the influence of the $\\alpha $ -value, the average number of iterations until convergence were plotted separately for all BOT problems with $\\alpha \\le 0.5$ and $\\alpha > 0.5$ .","We find that the greedy heuristic systematically needs more iterations to converge for $\\alpha > 0.5$ .","Figure: The greedy topology optimization applied to each 150 problems with nn terminals: Mean and standard deviation of required iterations plotted vs. nn in blue.","Average for problems with α≤0.5\\alpha \\le 0.5 in green (dashed) and for problems with α>0.5\\alpha > 0.5 in orange (dotted)."],["Influence of the edge sampling kernel.","Our heuristic topology optimization involves a number of design choices which may affect its performance.","A systematic investigation is beyond the scope of this work and left for future research.","However, one hyperparameter of particular interest is the kernel (chosen to be Gaussian) and its width (chosen to be $d_{min}$ ), which together define the replacement probability of the edges, see l. REF in Alg.","REF .","To study its influence on the performance we varied the width of the Gaussian kernel $\\exp (-d^2/(\\omega d_{min})^2)$ by tuning the parameter $\\omega $ .","Based on 150 random problems of various sizes $n$ , we calculated the average cost ratio of the heuristic with different $\\omega $ to the default of $\\omega = 1$ (cf.","Fig.","REF ).","Indeed, Fig.","REF shows that the default choice of $\\omega = 1$ is quite strong and relatively robust given that $\\omega = 0.5$ or $\\omega = 0.1$ work similarly well.","Clearly, wider kernels lead to larger (i.e.","less local) changes of the topology.","At later stages of the algorithm most of these topology changes will be unfavorable, explaining why the algorithm for wider kernels terminates with comparatively less optimal solutions.","Furthermore, we have investigated the influence of the kernel width on the number of iterations required (cf.","Fig.","REF ).","Qualitatively, Fig.","REF confirms that for wider kernels, which encourage exploration, more iterations are required.","Fitting a power law of the form $x \\mapsto ax^b$ to the curves in Fig.","REF , one finds that, depending on the kernel, the greedy heuristic on average scales between $O(n^{1.5})$ and $O(n^{1.7})$ .","Again, this is a purely empirical statement.","A careful theoretical analysis to obtain guarantees, also for problem sizes not covered in our experiments, is beyond the scope of this work.","Figure: Influence of kernel width in topology optimization based on 150 random problems of various sizes nn: (a) Ratio of the average cost of the heuristic solution with kernel width factor ω\\omega to the default of ω=1\\omega =1.","(b) Average number of iterations until convergence for different kernel width factors ω\\omega .","Depending on the kernel width the number of iterations on average scales between O(n 1.5 )O(n^{1.5}) and O(n 1.7 )O(n^{1.7}).Figure: Average cost ratios of our greedy heuristic and brute-force solutions (the closer to 1 the better) for different number of terminals nn and dimensions dd.","For each nn, we uniformly sampled 100 different BOT problems.","Though the average cost ratio increases slightly with nn, our approximate BOT solver compares very well against the ground truth solutions, independently of the dimensionality dd."],["Greedy topology optimization vs. brute-force search for higher-dimensional BOT.","Both the numerical geometry optimization and the greedy algorithm for the topology optimization presented in Sect.","are readily applicable to BOT problems in $\\mathbb {R}^d$ .","Paralleling the experiment presented in Fig.","REF , we have compared our heuristic topology optimization against brute-forced solutions for each 100 problems of size $n=5$ to $n=9$ and spatial dimension $d$ .","Although Fig.","REF suggests that the average cost ratio increases slightly with $n$ , our approximate BOT solver again compares very well against the ground truth solutions, independently of the dimensionality $d$ ."]],"string":"[\n [\n \"Theory and Approximate Solvers for Branched Optimal Transport with\\n Multiple Sources\"\n ],\n [\n \"Abstract Branched Optimal Transport (BOT) is a generalization of optimal transport in which transportation costs along an edge are subadditive.\",\n \"This subadditivity models an increase in transport efficiency when shipping mass along the same route, favoring branched transportation networks.\",\n \"We here study the NP-hard optimization of BOT networks connecting a finite number of sources and sinks in $\\\\mathbb{R}^2$.\",\n \"First, we show how to efficiently find the best geometry of a BOT network for many sources and sinks, given a topology.\",\n \"Second, we argue that a topology with more than three edges meeting at a branching point is never optimal.\",\n \"Third, we show that the results obtained for the Euclidean plane generalize directly to optimal transportation networks on two-dimensional Riemannian manifolds.\",\n \"Finally, we present a simple but effective approximate BOT solver combining geometric optimization with a combinatorial optimization of the network topology.\"\n ],\n [\n \"Introduction\",\n \"Optimal transport (OT) [27], [6], [23] stipulates transportation costs that increase linearly with the transported mass.\",\n \"However, in many systems of practical and theoretical interest, a diminishing cost property is more realistic: it is more economic to jointly transport two loads with nearby destinations along the same route.\",\n \"The optimal transportation networks under diminishing costs exhibit branching; and indeed, nature and societies are using branched networks, e.g.\",\n \"in blood circulation, gas supply or mail delivery.\",\n \"In this paper, we study the theory and practice of finding good or even optimal solutions in branched optimal transport (BOT).\",\n \"More formally, we consider a finite set of sources $S$ with supplies $\\\\mu _S > 0$ and sinks $T$ with demands $\\\\mu _T > 0$ , located at fixed positions $x_S$ and $x_T$ in $\\\\mathbb {R}^2$ .\",\n \"A possible transportation network is represented as a directed, edge-weighted graph $G(V,E)$ with nodes $V = S \\\\cup T \\\\cup B$ .\",\n \"The edges $E \\\\subset V \\\\times V$ interconnect the terminals $S$ and $T$ with the help of a set of additional nodes $B$ , so-called branching points (BPs), with coordinates $x_B$ .\",\n \"The edge direction indicates the direction of mass flow.\",\n \"The edge weights, denoted by $m_e$ , specify the absolute flows.\",\n \"Gilbert first proposed the BOT problem [9] in which the objective is to solve for $ \\\\underset{B, E, x_B, m_E}{\\\\arg \\\\min } \\\\ \\\\sum _{(i,j) \\\\, \\\\in \\\\, E} m_{ij}^\\\\alpha &\\\\left\\\\Vert x_i - x_j \\\\right\\\\Vert _2 , \\\\text{ subject to }\\\\\\\\ \\\\nonumber \\\\text{supply } \\\\ \\\\mu _s = \\\\sum \\\\nolimits _{k} m_{sk} &- \\\\sum \\\\nolimits _{k} m_{ks} \\\\ \\\\text{ at each source } s,\\\\\\\\ \\\\nonumber \\\\text{demand } \\\\ \\\\mu _t = \\\\sum \\\\nolimits _{k} m_{kt} &- \\\\sum \\\\nolimits _{k} m_{tk} \\\\ \\\\text{ at each sink } t,\\\\\\\\ \\\\nonumber \\\\text{conservation } \\\\ \\\\sum \\\\nolimits _{k} m_{kb} &= \\\\sum \\\\nolimits _{k} m_{bk} \\\\ \\\\text{ at each BP } b,$ given a single parameter $\\\\alpha \\\\in [0,1]$ .\",\n \"The problem of BOT is interesting in that it combines combinatorial optimization (over $B$ , $E$ ) with continuous optimization (over $x_B$ , $m_E$ ).\",\n \"Figure: Branched optimal transport (Eq. )\",\n \"interpolates between optimal transport and the Euclidean Steiner tree problem.\",\n \"On a toy example, shown are good BOT solutions (found by our approximate solver for α≠1\\\\alpha \\\\ne 1, see Sect. )\",\n \"for the same set of sources (red) and sinks (blue).\",\n \"The disk sizes indicate the demands and supplies, the edge widths the mass transported along each edge.For $\\\\alpha = 1$ , the BOT problem is the discrete version of the famous optimal transport problem for which optimal solutions can be found efficiently [6], [23].\",\n \"However, due to the linearity of the cost function, OT solutions do not exhibit any branching but consist of straight lines between sources and sinks, see Fig.\",\n \"REF .\",\n \"In contrast, for $\\\\alpha \\\\in [0,1)$ , the subadditivity of $m \\\\mapsto m^\\\\alpha $ reflects the increased efficiency of transporting loads together, i.e.\",\n \"$(m_1 + m_2)^\\\\alpha < m_1^\\\\alpha + m_2^\\\\alpha $ .\",\n \"Thus, for $\\\\alpha \\\\in [0,1)$ , BOT solutions show a branched structure, see Fig.\",\n \"REF -al0.\",\n \"Unlike OT, the optimization problem of BOT is NP-hard [11].\",\n \"In the special case of $\\\\alpha = 0$, BOT turns into the well-studied Euclidean Steiner tree problem (ESTP) [28], [13].\",\n \"In the ESTP, the objective is to find the overall shortest network that interconnects all terminals (with the help of BPs), independently of the edge flows, since $m^0 = 1$ .\",\n \"For different values of $\\\\alpha $ , BOT interpolates between these two optimization problems, see Fig.\",\n \"REF .\"\n ],\n [\n \"Connection of BOT to machine learning.\",\n \"Optimal transport has emerged as an important tool in machine learning [1], [5], [23].\",\n \"BOT is a strict generalization, describing a more versatile concept and more challenging optimization problem.\",\n \"BOT offers a mathematical formalism that is deceivingly simple (cf. Eq.\",\n \"(REF )) and yet engenders non-trivial structure.\",\n \"Many machine learning problems such as tracking of divisible targets (computer vision), skeletonization (image analysis), trajectory inference (bioinformatics) come with input that is essentially continuous (images, distributions) and require structured output that is discrete, e.g. graphs.\",\n \"Arguably, this transition from continuous to discrete is one of the most interesting aspects (and an unsolved problem) in current machine learning research.\",\n \"It is also a problem that cannot be solved by a mere upscaling of standard deep learning architectures.\",\n \"In addition, routing problems have become a popular problem to challenge machine learning and amortized optimization algorithms with difficult optimization problems [17], [3], [15].\",\n \"Combining combinatorial and continuous optimization, BOT is a highly instructive target for new machine learning approaches.\",\n \"In Sect.\",\n \"we address the generalization of BOT to higher-dimensional Euclidean space, particularly relevant for applications in data science.\",\n \"In this paper, we make the following contributions: We generalize an existing method for constructing BOT solutions with optimal geometry to the case of multiple sources.\",\n \"Based on this generalization, we present an analytical and numerical scheme to rule out $n$ -degree branchings with $n > 3$ .\",\n \"Further, we demonstrate how to extend geometric and topological properties of optimal BOT solutions to two-dimensional Riemannian manifolds.\",\n \"Lastly, we propose a more practical numerical algorithm for the geometry optimization together with a simple but compelling heuristic, addressing the optimization of the BOT topology.\",\n \"To the best of our knowledge, no readily accessible code for finding BOT solutions is publicly available.\",\n \"By making our code available at https://github.com/hci-unihd/BranchedOT we hope to aid the evolution of the field.\"\n ],\n [\n \"Topology and geometry of BOT solutions\",\n \"A BOT problem can be divided into the combinatorial optimization of the network topology, specified by the set of BPs $B$ and edges $E$ (see Sect.\",\n \"), and the geometric optimization of the BP positions $x_B$ .\",\n \"[2] showed that optimal BOT solutions can be assumed to be acyclic, which restricts the search for the optimal topology to trees.\",\n \"Given $n$ terminals, WLOG, the topology can be represented as a so-called full tree topology, which has $n-2$ BPs, each of degree three.\",\n \"Higher-degree branchings may effectively form during the geometry optimization if multiple BPs settle at the same position.\",\n \"A set of such BPs is referred to as coupled BP, cf. Fig.\",\n \"REF .\",\n \"The union set of all neighbors of the individual BPs (not including the BPs themselves) is referred to as set of effective neighbors.\",\n \"Conversely, a BP configuration in which all BPs are uncoupled and located away from the terminals is called non-degenerate, see Fig.\",\n \"REF .\",\n \"The number of distinct full tree topologies interconnecting $n$ terminals is given by $(2n - 5)!!\",\n \"= (2n - 5) \\\\cdot (2n - 7) \\\\cdot ... \\\\cdot 3 \\\\cdot 1$ and hence increases super-exponentially with the number of terminals [25].\",\n \"Given 100 terminals, one would have to consider more than $10^{18}$ possible full tree topologies, making an exhaustive search computationally intractable already for problems of modest size.\",\n \"Fortunately, given a tree topology, the geometric optimization of the BP positions reduces to a convex optimization problem, as all edge flows $m_{ij}$ are already uniquely determined by the flow constraints in Eq.\",\n \"(REF ).\",\n \"The corresponding linear system can be solved in linear time by dynamic programming, called “elimination on leaves of a tree” in [26].\",\n \"Since the Euclidean norm, like any norm, is convex, given a fixed tree topology, the cost function in Eq.\",\n \"(REF ) becomes a convex function of the BP positions.\",\n \"Together with the independence of the individual BPs, this implies the following lemma on the optimal substructure of BOT solutions (see App. ).\",\n \"Definition 2.1 For a chosen topology $T$ , a BOT solution is called a relatively optimal solution (ROS of $T$ ) if its BP configuration has minimal cost.\",\n \"The overall best BOT solution, given by the optimal topology together with its ROS, is called the globally optimal solution (GOS).\",\n \"Lemma 2.1 (a) For a given tree topology, a BOT solution is relatively optimal if and only if every (coupled) BP connects its (effective) neighbors at minimal cost.\",\n \"(b) In a globally optimal solution, every subsolution restricted to a connected subset of nodes solves its respective subproblem (cf. App. )\",\n \"globally optimally.\",\n \"Figure: Two BP configurations for the same full tree topology: A higher-degree branching may effectively form by coupling the two BPs at the same position.\"\n ],\n [\n \"Geometric optimization of BOT solutions\",\n \"Although the BP optimization for a given tree topology is a convex problem, as argued above, it is non-trivial, since the objective function is not everywhere differentiable.\",\n \"Here, we present a principled geometric approach, which was first suggested by Gilbert in [9] and previously developed in the context of the ESTP [20].\",\n \"More recently, this approach was discussed in the comprehensive work by [2], where it was applied exclusively to BOT problems with a single source.\",\n \"A generalization to the case of multiple sources was posed as an open problem by the authors (see Problem 15.11), for which we give the solution in this section.\"\n ],\n [\n \"Geometric solution for one source and two sinks\",\n \"Motivated by Lem.\",\n \"REF , we start by considering a single BP $b$ in isolation (cf. Fig.\",\n \"REF ), following [2].\",\n \"Given a source at positionWe will often use the node label, e.g., $a_0$ , to denote also the position of the node, instead of writing $x_{a_0}$ .\",\n \"$a_0$ and two sinks at positions $a_1$ and $a_2$ , we aim to find the optimal position $b^*$ for the BP connecting the three terminals, i.e., the minimizer of $ \\\\mathcal {C}(b) = m_1^\\\\alpha \\\\vert a_1 - b \\\\vert + m_2^\\\\alpha \\\\vert a_2 - b \\\\vert + (m_1 + m_2)^\\\\alpha \\\\vert a_0 - b \\\\vert ,$ where $m_1$ and $m_2$ are the respective demands of the two sinks.\",\n \"Due to the convexity of $\\\\mathcal {C}(b)$ , the minimum must lie either at a stationary point at which $\\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b} \\\\, \\\\mathcal {C} = 0$ or at a non-differentiable point, where $b$ coincides with one of the $a_i$ .\",\n \"[2] showed that the gradient is equal to zero if and only if the branching angles $\\\\theta _i$ , see Fig.\",\n \"REF , are given by $&\\\\theta ^*_1 = \\\\arccos \\\\bigg ( \\\\frac{k^{2\\\\alpha } + 1 - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha } \\\\bigg ) =: f(\\\\alpha , k), \\\\nonumber \\\\\\\\&\\\\theta ^*_2 = \\\\arccos \\\\bigg ( \\\\frac{(1 \\\\!\",\n \"- \\\\!\",\n \"k)^{2\\\\alpha } + 1 - k^{2 \\\\alpha }}{2 (1 \\\\!\",\n \"- \\\\!\",\n \"k)^\\\\alpha } \\\\bigg ) = f(\\\\alpha , 1 \\\\!\",\n \"-k), \\\\\\\\&\\\\theta ^*_1 + \\\\theta ^*_2 = \\\\arccos \\\\bigg ( \\\\frac{1 - k^{2 \\\\alpha } - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha (1-k)^\\\\alpha } \\\\bigg ) =: h(\\\\alpha , k) \\\\nonumber ,$ where we have defined the flow fraction $k:= m_1/(m_1 + m_2)$ and the two functions $f$ and $h$ , related via $h(\\\\alpha , k) = f(\\\\alpha , k) + f(\\\\alpha , 1-k)$ .\",\n \"If a BP exists that realizes the branching angles $\\\\theta _i^*$ , it can be constructed geometrically based on the central angle property (see App. ).\",\n \"It states that, given a circle through $a_1$ and $a_2$ , the angle $\\\\angle a_1 o a_2$ at the center $o$ is twice the angle enclosed with a point anywhere on the opposite circle arc, cf. Fig.\",\n \"REF .\",\n \"In particular, let us construct the so-called pivot circle with central angle $\\\\angle a_1 o a_2 = 2\\\\theta ^*_1 + 2\\\\theta ^*_2$ and pivot point $p$ as in Fig.\",\n \"REF .\",\n \"Applying the central angle property twice (once for $\\\\theta _1^*$ and once for $\\\\theta _2^*$ ), a BP located at the intersection of the lower circle arc and the connection line $\\\\overline{a_0p}$ realizes both angles $\\\\theta ^*_i$ and is therefore optimal.\",\n \"Figure: (a) Branching point bb connecting one source and two sinks with branching angles θ 1 \\\\theta _1 and θ 2 \\\\theta _2.\",\n \"(b) Construction of the optimal BP b * b^* applying twice the central angle property illustrated in (c).\",\n \"(d) shows the relation of the optimal branching angles θ i * \\\\theta ^*_i and ϑ i * \\\\vartheta ^*_i, relevant for the case of asymmetric branchings (see Sect.\",\n \").However, given the pivot point and pivot circle, $\\\\overline{a_0 p}$ may not intersect the lower circle arc, depending on the position of $a_0$ .\",\n \"Accordingly, the lower half plane can be partitioned into a region for which the described construction yields an optimal Y-shaped branching and three other regions, see Fig.\",\n \"REF .\",\n \"For $a_0$ located in one of these regions, the optimal BP position coincides with one of the terminals, resulting in a V-shaped branching ($b^* = a_0$ ) or an L-shaped branching ($b^* \\\\in \\\\lbrace a_1,a_2\\\\rbrace $ ), cf. Fig.\",\n \"REF [2].\"\n ],\n [\n \"Geometric construction of BOT solutions for a given topology\",\n \"Applying the geometric construction from above in a recursive manner, one can construct the ROS (see Def.\",\n \"REF ) for larger BOT problems, as illustrated in Fig.\",\n \"REF .\",\n \"Given a full tree topology $T$ , first, we determine all edge flows (see Sect. )\",\n \"and consequently the optimal branching angles.\",\n \"Then, a root node is chosen, arbitrarily (here $a_0$ ), and all other nodes are sorted based on the number of edges to $a_0$ (ignoring edge directions and resolving ties arbitrarily).\",\n \"Starting from the furthest nodes and working towards the root, two nodes are recursively summarized by a pivot point, constructed from the optimal branching angles, see Fig.\",\n \"REF -recur2.\",\n \"Afterwards, in reversed order, the optimal BPs are placed iteratively, each as in the 1-to-2 case, see Fig.\",\n \"REF -recur4.\",\n \"In this manner, the optimal branching angles are realized at every BP and the resulting solution is a ROS of $T$ by Lem.\",\n \"REF .\",\n \"The choice of the root node induces a node ordering as described above.\",\n \"Given this ordering, consider any BP $b$ and denote its children by $a_1$ and $a_2$ and its parent node by $a_0$ .\",\n \"The construction of the pivot point now requires the positions of $a_1$ and $a_2$ and the optimal branching angles enclosed by the children edges $(b,a_1)$ and $(b,a_2)$ .\",\n \"However, the branching angles do not only depend on the absolute flows $m_1$ and $m_2$ of the respective edges but also on the flow directions.\",\n \"Given that both flows point towards $b$ or given that both flows point away from $b$ , as in Fig.\",\n \"REF , the branching is referred to as symmetric and the optimal branching angles of interest are given by $\\\\theta _i^*$ , cf. Eq.\",\n \"(REF ).\",\n \"Note that BOT problems and their solutions are fully symmetric under complete exchange of sinks and sources (up to reversal of all flow directions).\",\n \"On the contrary, in case of one flow pointing towards $b$ and one pointing away from $b$ , referred to as asymmetric branching, the optimal branching angles $\\\\vartheta _i^*$ enclosed by the children edges are calculated differently, see Fig.\",\n \"REF .\",\n \"However, the branching angles $\\\\vartheta _i^*$ can be related geometrically to the known $\\\\theta _i^*$ .\",\n \"Using the functions $f$ and $h$ from Eq.\",\n \"(REF ), we find that $\\\\begin{aligned}\\\\vartheta ^*_1 &= \\\\pi - \\\\theta ^*_1 - \\\\theta ^*_2 = \\\\pi - h\\\\Big (\\\\alpha , k=\\\\frac{m_2 - m_1}{m_2}\\\\Big ) \\\\\\\\\\\\vartheta ^*_2 &= \\\\theta ^*_1 = f\\\\Big (\\\\alpha , k = \\\\frac{m_2 - m_1}{m_2}\\\\Big ).\\\\end{aligned}$ After determining the two angles $\\\\vartheta _1^*$ and $\\\\vartheta _2^*$ from the flows $m_1$ and $m_2$ , the BP construction based on the central angle property works analogously to the symmetric case.\",\n \"Crucially, this distinction of symmetric and asymmetric branching makes the recursive construction applicable also to problems with multiple sources, where asymmetric branchings may be unavoidable, consider e.g. Fig.\",\n \"REF with $a_0$ and $a_3$ as sources and $a_1$ and $a_2$ as sinks (see App.\",\n \"REF ).\",\n \"Further, note that the known conditions for optimal V- and L-branching can be transferred to the asymmetric case simply by relabelling $a_0 \\\\rightarrow a_2$ , $a_1 \\\\rightarrow a_0$ and $a_2 \\\\rightarrow a_1$ , cf. Fig.\",\n \"REF and Fig.\",\n \"REF .\",\n \"In terms of angular inequalities (derived in App.\",\n \"), these conditions, for both branching types, are summarized in Table REF .\",\n \"Figure: Recursive geometric construction of a relatively optimal solution, using one pivot point and pivot circle per branching point to collectively realize the optimal branching angles.Figure: Regions of optimal Y-, V- and L-branching.\",\n \"Table: NO_CAPTION tableRelations between the L- and V-branching conditions.\",\n \"In principle, given a full tree topology, the described method efficiently constructs the ROS in linear time.\",\n \"However, as already pointed out by [9], the approach has some practical limitations, even after our generalization.\",\n \"Figure REF shows how the pivot point is constructed only from the positions of two children $a_1$ and $a_2$ and the corresponding optimal branching angles.\",\n \"However, a priori there are two possible pivot point locations, one in the upper and one in the lower half plane with respect to $\\\\overline{a_1a_2}$ .\",\n \"Hence, the construction relies on knowing in which half plane the third node $a_0$ lies.\",\n \"For larger trees, the topological parent $a_0$ may itself be a BP whose position is not yet determined.\",\n \"In the worst case, one would thus have to try all $2^{n-2}$ possible pivot point combinations to find the ROS.\",\n \"This pivot point degeneracy gets substantially worse in higher dimensions, making the recursive construction applicable only in $\\\\mathbb {R}^2$ .\",\n \"Secondly, the geometric construction only produces solutions which are non-degenerate, i.e., solutions without edge contractions.\",\n \"For now, the geometric construction is therefore primarily of theoretical interest; and indeed, it forms the basis of our following arguments.\",\n \"Note that both of the aforementioned problems could be solved elegantly in the special case of $\\\\alpha = 0$ [12], [14].\"\n ],\n [\n \"Properties of optimal BOT topologies\",\n \"Let us now consider topological modifications in order to improve the transportation cost of a BOT solution.\",\n \"In particular, we intend to show that a topology $T$ can be improved if its ROS contains coupled BPs.\",\n \"Let us start by considering a general BOT solution which contains a coupled 4-BP, i.e., a coupled BP with four effective neighbors, as in Fig.\",\n \"REF .\",\n \"Lemma REF states that a solution is not globally optimal if any subsolution is not globally optimal.\",\n \"It will therefore suffice to study the coupled BP as an isolated subproblem.\"\n ],\n [\n \"Non-optimality of coupled branching points\",\n \"Given two sources and two sinks, there are two possible configurations in which the terminals can be arranged, cf. Fig.\",\n \"REF ,fork2.\",\n \"First we address the case in which the two sources are at opposite corners of the terminal quadrilateral, as in Fig.\",\n \"REF .\",\n \"Based on Lem.\",\n \"REF , a necessary condition for the existence of a globally optimal 4-BP is that all four V-branchings between neighboring terminals are optimal.\",\n \"This puts a lower bound on each of the angles $\\\\gamma _i$ , see Tab.\",\n \"REF .\",\n \"The general idea, also regarding the other 4-branching scenarios, is to show that the angular sum of these lower bounds already exceeds $2 \\\\pi $ .\",\n \"This will immediately imply that not all V-branchings can be optimal simultaneously and thus a coupled 4-BP cannot be globally optimal.\",\n \"Given a 4-BP as in Fig.\",\n \"REF , all V-branchings are asymmetric (i.e.\",\n \"neighboring flows point in opposite directions).\",\n \"Hence, all four lower bounds (in Tab.\",\n \"REF ) are of the form $\\\\gamma _i \\\\ge \\\\pi - \\\\theta ^*_2 = \\\\pi - f(\\\\alpha , 1 - k)$ and indeed $\\\\pi - f(\\\\alpha , 1 - k) > \\\\pi / 2$ , see Lem.\",\n \"REF , so that their sum exceeds $2 \\\\pi $ .\",\n \"Next, let us consider the scenario in Fig.\",\n \"REF with two sources at neighboring corners.\",\n \"WLOG, we use the normalization $m_1 + m_2 = 1 = m_3 + m_4$ and assume that $m_1 > m_3$ and $m_2 < m_4$ .\",\n \"In this case, the four conditions for optimal V-branching in Tab.\",\n \"REF read: $\\\\gamma _1 \\\\ge \\\\pi - f\\\\Big (\\\\alpha , 1-\\\\frac{m_1 - m_3}{m_1} \\\\Big ) = \\\\pi - f\\\\Big (\\\\alpha , \\\\frac{m_3}{m_1} \\\\Big )$ , $\\\\gamma _2 \\\\ge h\\\\Big (\\\\alpha , \\\\frac{m_1}{m_1 + m_2} \\\\Big ) = h(\\\\alpha , m_1)$ , $\\\\gamma _3 \\\\ge \\\\!\",\n \"\\\\pi - f\\\\Big (\\\\alpha , 1 \\\\!\",\n \"- \\\\frac{m_4 - m_2}{m_4}\\\\Big ) \\\\!\",\n \"= \\\\pi - f\\\\Big (\\\\alpha , \\\\frac{1- m_1}{1 - m_3} \\\\Big )$ , $\\\\gamma _4 \\\\ge h\\\\Big (\\\\alpha , \\\\frac{m_3}{m_3 + m_4} \\\\Big ) = h (\\\\alpha , m_3)$ , Figure: Different scenarios of coupled 4-BPs with symmetric branching angles in blue, asymmetric ones in orange.\",\n \"where the expressions (REF ) were plugged into the V-branching conditions in Tab.\",\n \"REF for symmetric and asymmetric branching respectively, as indicated by the colors in Fig.\",\n \"REF .\",\n \"Let us show that in fact for all combinations of $\\\\alpha $ , $m_1$ and $m_3$ the sum of the lower bounds already exceeds $2 \\\\pi $ .\",\n \"Indeed, summing the lower bounds and subtracting $2\\\\pi $ yields $&\\\\underbrace{h(m_1)}_{= \\\\ h(1-m_1)} + \\\\ h(m_3) - f\\\\Big (\\\\underbrace{\\\\frac{m_3}{m_1}}_{> \\\\, m_3} \\\\Big ) - f\\\\Big (\\\\underbrace{ \\\\frac{1 - m_1}{1 - m_3} }_{> \\\\, 1 - m_1} \\\\Big ) >h(1-m_1) + h(m_3) - f(m_3) - f(1-m_1) \\\\\\\\&= f(m_1) + f(1-m_3) > 0,$ using $h(\\\\alpha ,k) = f(\\\\alpha , k) + f(\\\\alpha , 1-k)$ and the fact that $f(\\\\alpha , k)$ is strictly decreasing with respect to $k$, see Lem.\",\n \"REF .\",\n \"To summarize, we have arrived at the following lemma: Lemma 4.1 A coupled 4-BP not coincident with a terminal connecting two sources and two sinks is never globally optimal.\",\n \"Exactly the same logic applies for a coupled 4-BP connecting one source and three sinks (or equivalently 3 sources and 1 sink), as in Fig.\",\n \"REF .\",\n \"WLOG, in the following, we normalize the flows so that $m_1 + m_2 + m_3 =1$ .\",\n \"We then determine the necessary conditions under which all V-branchings are optimal.\",\n \"We again intend to show that such a 4-BP can never be globally optimal by showing that for any combination of $\\\\alpha $ and ${m_i}$ the sum of the lower bounds exceeds $2 \\\\pi $ .\",\n \"This is equivalent to proving the following inequality (see App.\",\n \"REF ): $h \\\\Big (\\\\frac{m_1}{m_1 + m_2} \\\\Big ) &- f( m_1) + h \\\\Big (\\\\frac{m_3}{m_3 + m_2} \\\\Big ) - f(m_3) > 0.$ Assuming a globally optimal 4-BP existed, one could continuously displace a terminal in a way such that for the resulting BOT problem a coupled 4-BP is still globally optimal.\",\n \"Choosing different such displacements four additional inequalities can be derived (see App.\",\n \"REF ): Proposition 4.2 Given a BOT problem with one source and three sinks, with demands $m_1, m_2, m_3$ as in Fig.\",\n \"REF , a coupled 4-BP away from the terminals cannot be globally optimal if at least one of the following inequalities holds true: $\\\\Gamma &= h \\\\Big (\\\\frac{m_1}{m_1 + m_2} \\\\Big ) - f(m_1) + h \\\\Big (\\\\frac{m_3}{m_3 + m_2} \\\\Big ) - f( m_3) > 0, \\\\\\\\\\\\Gamma _{1,*} &= f(1 -m_*) + f\\\\Big (1 - \\\\frac{m_2}{1-m_*} \\\\Big ) - f(1-m_*-m_2) > 0, \\\\\\\\\\\\Gamma _{2,*} &= h \\\\Big (\\\\frac{m_*}{m_* + m_2} \\\\Big ) + f\\\\Big (\\\\frac{m_2}{1 - m_*} \\\\Big ) - h(m_*) > 0$ where $*=1,3$ .\",\n \"Note that $\\\\Gamma = \\\\Gamma _{1,1} + \\\\Gamma _{2,1} = \\\\Gamma _{1,3} + \\\\Gamma _{2,3}$ .\",\n \"In App.\",\n \"REF , we prove the inequalities analytically for a large subset of the parameter space.\",\n \"For the remainder we present a numerical argument (see App.\",\n \"REF ).\",\n \"In addition, we show by induction how, given that coupled 4-BPs are never globally optimal, one can further rule out coupled $n$ -BPs (with $n$ effective neighbors) for all $n > 4$ .\",\n \"Theorem 4.3 Given a BOT problem in the Euclidean plane and assuming that coupled 4-BPs are never globally optimal, in a globally optimal BOT solution each branching point not coincident with a terminal must have degree three.\"\n ],\n [\n \"Generalization of BOT to Riemannian manifolds\",\n \"In this section, we extend the BOT problem together with many of the previous results to two-dimensional Riemannian manifolds $\\\\mathcal {M}$ embedded into $\\\\mathbb {R}^3$ [18].\",\n \"This includes the sphere as important special case, particularly relevant for global transportation networks.\",\n \"In the generalized BOT cost function (REF ) we replace the Euclidean metric by the geodesic distance $d: \\\\mathcal {M} \\\\times \\\\mathcal {M} \\\\rightarrow \\\\mathbb {R}^+$ , i.e.\",\n \"$ \\\\mathcal {C}_M = \\\\sum _{(i,j) \\\\, \\\\in \\\\, E} m_{ij}^\\\\alpha \\\\, d(x_i,x_j).$ As we assume the manifold to be embedded, the length of a geodesics can be measured in $\\\\mathbb {R}^3$ .\",\n \"First, we generalize the non-optimality of cyclic solutions.\",\n \"The corresponding proof in [2] readily applies also to two- and higher-dimensional manifolds.\",\n \"As before, solving a BOT problem on a curved surface can thus be separated into the combinatorial topology optimization and the continuous optimization of the BP configuration.\"\n ],\n [\n \"Linear approximation of BOT solutions on manifolds\",\n \"Intuitively speaking, a two-dimensional Riemannian manifold locally looks like the Euclidean plane.\",\n \"If we zoom in on a sufficiently small region, geodesics again resemble straight lines and the geodesic distance approaches the Euclidean one.\",\n \"This can be used to show that the branching angles which were optimal for Y-branchings in the Euclidean plane are also optimal on Riemannian manifolds.\",\n \"Below, we summarize the main steps of the proof.\",\n \"All details can be found in App. .\",\n \"Given a Y-branching on a manifold, we measure the angles between the three geodesics in the tangent space $T_b \\\\mathcal {M}$ at the BP $b$ .\",\n \"We now zoom in on a small neighborhood $U$ around $b$ and consider only the subsolution in $\\\\mathcal {M} \\\\cap U$ .\",\n \"The terminals of the corresponding subproblem are projected orthogonally onto the tangent space, more specifically onto a small disk of radius $r$ , denoted by $D(r)$ , see Fig.\",\n \"REF .\",\n \"Let us denote the cost of the subsolution on the manifold by $\\\\mathcal {C}_M(b)$ and the cost of the corresponding subproblem in the flat disk by $\\\\mathcal {C}(b)$ .\",\n \"Now, assuming that the angles between the geodesics deviate from the optimal branching angles, the same holds true for the projected subsolution.\",\n \"Consequently, there exists an alternative BP $b^*$ in the disk with cheaper cost $\\\\mathcal {C}(b^*)$ .\",\n \"Note that the radius of this disk becomes smaller the smaller we choose the region $\\\\mathcal {M} \\\\cap U$ of the subproblem.\",\n \"Figure: On the flat disk D(r)⊂T b ℳD(r) \\\\subset T_b \\\\mathcal {M}, the BP bb does not solve the problem with terminals a ^ i \\\\hat{a}_i optimally if the angles between the dotted geodesics are not optimal.Crucially, the cost difference between a subsolution on the manifold and its projection onto the plane tends to zero quadratically in the limit of $r \\\\rightarrow 0$ .\",\n \"The intuitive reason for this is that the tangent space $T_b \\\\mathcal {M}$ locally approximates the manifold to linear order.\",\n \"On the contrary, the costs $\\\\mathcal {C}(b)$ and $\\\\mathcal {C}(b^*)$ in the disk scale linearly in $r$ and so does the cost improvement $\\\\mathcal {C}(b) - \\\\mathcal {C}(b^*) = M \\\\, r$, for some fixed $M > 0$ .\",\n \"To conclude the proof, one projects $b^*$ onto the manifold and evaluates the cost difference of the two subsolutions there.\",\n \"The difference is of the form $M \\\\, r + O(r^2)$ , with second order differences due to the projection from $D(r)$ to the manifold.\",\n \"Consequently, a finite radius $r > 0$ must exist for which the cost difference is truly positive.\",\n \"A BOT solution on the manifold for which the Y-branching angles deviate from the optimal branching angles can thus be improved and is not relatively optimal.\",\n \"The logic of the proof outlined here can easily be extended to the V- and L-branching conditions as well as our results regarding the non-optimality of coupled BPs.\",\n \"Again, improving the BOT solution locally in the tangent plane (w.r.t.\",\n \"its geometry or topology) and projecting back to the manifold results in an improved solution on the manifold (see App.\",\n \"REF ).\",\n \"Theorem 5.1 Consider the solution to a generalized BOT problem on a two-dimensional Riemannian manifold embedded into $\\\\mathbb {R}^3$ .\",\n \"For the solution to be relatively optimal, it is a necessary condition that each BP satisfies the optimal angle conditions for Y-, V- and L-branching, which apply for BOT in the Euclidean plane.\",\n \"For it to be globally optimal, assuming that coupled 4-BPs are not optimal in the plane, it is a necessary condition that BPs not coincident with a terminal have degree three.\",\n \"Though there is no readily available algorithm to solve BOT on embedded surfaces, we discuss some possible approaches in App.\",\n \"REF .\"\n ],\n [\n \"Heuristics and numerical optimization\",\n \"In this section, we present a simple but effective algorithm for the geometry optimization, followed by a compelling heuristic for the topology optimization.\",\n \"As pointed out earlier, the difficulty of solving a BOT problem stems from the super-exponentially growing number of possible full tree topologies.\",\n \"Obtaining an exact solution by brute-force is almost always computationally infeasible and hence fast heuristic solvers are needed.\",\n \"For BOT problems with a single source, a branch-and-bound method is applicable [31], enabling exact solutions for up to 16 nodes.\",\n \"However, this method does not generalize directly to the case of multiple sources.\",\n \"While some literature exists on heuristics for BOT problems with a single source [29], we are not aware of heuristics for multiple sources, except [24].\",\n \"The authors of [24] present a simulated annealing based optimization strategy for BOT, based on hand-crafted geometrical and topological modifications, which may require user supervision.\",\n \"Furthermore, continuous approaches to solve BOT exist which do not rely on a subdivision into geometry and topology optimization.\",\n \"The authors of [22] phrase BOT as a limit of functional minimization problems.\",\n \"Since their algorithm discretizes the plane and the BOT cost function, their output is however not sparse but a discretized function.\"\n ],\n [\n \"Numerical branching point optimization for a given topology\",\n \"Brute-force and heuristic BOT solvers alike typically rely on the geometry optimization of many different topologies.\",\n \"A fast and reliable BP optimization routine is therefore essential, as it determines the computational bottleneck of these algorithms.\",\n \"For a given tree topology $T$ , all edge flows $m_{ij}$ are known (see Sect. ).\",\n \"The objective is thus to minimize the following convex cost function: $ \\\\mathcal {C}(\\\\lbrace x_i\\\\rbrace ) = \\\\sum _{(i,j) \\\\, \\\\in \\\\, T} m_{ij}^\\\\alpha \\\\left\\\\Vert x_i - x_j \\\\right\\\\Vert _2,$ where, for $1 \\\\le i \\\\le n$ , the $x_i$ hold the fixed coordinates of the terminals and, for $n+1 \\\\le i \\\\le n+m$ , the variable BP positions.\",\n \"Since the cost function is not everywhere differentiable, we suggest the following generalization of Smith's algorithm developed for geometry optimization in the ESTP [26].\",\n \"It is an effective algorithm specifically for minimizing the sum of Euclidean norms in two- and higher-dimensional Euclidean space.\",\n \"Unlike the geometric construction in Section REF , it is applicable to all (not necessarily full) tree topologies.\",\n \"Starting from a non-optimal, non-degenerate BP configuration, e.g.\",\n \"from a random initialization, the gradient with respect to each BP position $x_i$ is set to zero for $n+1 \\\\le i \\\\le n+m$ , resulting in the following non-linear system of $m$ equations: $x_i = \\\\sum _{j \\\\, : \\\\, (i,j) \\\\in T} m_{ij}^\\\\alpha \\\\frac{x_j}{\\\\vert x_i - x_j \\\\vert }\\\\ \\\\Bigg / \\\\sum _{j \\\\, : \\\\, (i,j) \\\\in T} \\\\frac{m_{ij}^\\\\alpha }{\\\\vert x_i - x_j \\\\vert }.$ This system can be solved approximately, by iteratively solving the following linearized system $ &x_i^{(k + 1)} = \\\\!\",\n \"\\\\!\",\n \"\\\\sum _{j \\\\, : \\\\, (i,j) \\\\in T} m_{ij}^\\\\alpha \\\\frac{x_j^{(k+1)}}{\\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert }\\\\ \\\\Bigg / \\\\sum _{j \\\\, : \\\\, (i,j) \\\\in T} \\\\frac{m_{ij}^\\\\alpha }{\\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert }, \\\\hspace{11.38092pt}\\\\text{for } n+1 \\\\le i \\\\le n+m.$ Note that $x_i^{(k)} = x_i$ is fixed for $1 \\\\le i \\\\le n$ .\",\n \"For each iteration, the solution can be found in linear time, again by “elimination on leaves of a tree”, similar to determining all edge flows from the flow constraints.\",\n \"The algorithm is easily parallelized over $d$ spatial dimensions of a BOT problem so that a single iteration is of order $O(nd)$ .\",\n \"In essence, this is an iteratively reweighted least squares (IRLS) approach [4].\",\n \"The connection is made explicit in App.\",\n \"REF .\",\n \"Details on the proof of convergence, the empirical runtime of the algorithm and suitable convergence criteria can be found in App.\",\n \"REF and in [26].\",\n \"The arguments in [26] readily apply to our generalization.\",\n \"Besides our method, other techniques may be used for the geometry optimization, for instance the interior point method presented in [30].\"\n ],\n [\n \"A greedy randomized algorithm for the topology optimization\",\n \"Our heuristic for the optimization of the BOT topology is inspired by the idea of simulated annealing [16], which has been applied in different variants to combinatorial problems such as the Traveling Salesman Problem [19] or the ESTP [10].\",\n \"In our heuristic, the BOT topology is iteratively modified by randomly deleting an edge and replacing it with a new one.\",\n \"At each step, the new solution is accepted according to a criterion, which typically depends on the cost difference between the solutions and a user-chosen hyperparameter, the temperature, used to mimic a physical cooling process.\",\n \"However, because in practice it works already sufficiently well (see Fig.\",\n \"REF ), we refrained from designing an elaborate cooling scheme.\",\n \"Instead, we apply the heuristic most greedily, i.e., in the zero-temperature limit, where a new state is accepted only if it decreases the cost.\",\n \"Starting from an initial tree topology $T$ , e.g., the minimum spanning tree (mST) or the OT solutionIn particular in the regime $\\\\alpha \\\\approx 1$ , our BOT solver benefits from existing efficient OT solvers by using their solution as initial guess., we uniformly sample an edge $\\\\hat{e} \\\\in E$ and remove it from $T$ .\",\n \"Let the incident node of $\\\\hat{e}$ which ended up in the smaller connected component be $\\\\ell $ .\",\n \"Then, one calculates the distance $d(e, \\\\ell )$ between $\\\\ell $ and every edge $e=(i,j)$ in the larger component and samples one of these edges with probability $p(e) \\\\propto \\\\exp (-d(e, \\\\ell )^2 / d_{min}^2)$ , where $d_{min}$ is the distance to the closest considered edge.\",\n \"The node $\\\\ell $ is then connected to the sampled edge via a new BP to produce a new tree topology.\",\n \"For this topology, we optimize the geometry (as described in Sect.\",\n \"REF ) and compare costs with the previous solution.\",\n \"If the new state is rejected, start the next iteration by sampling $\\\\hat{e}$ without replacement until either a move is accepted and all above steps are repeated; or until no accepted move is found, upon which the search terminates.\",\n \"Experiments for small BOT problems suggest that even in the greedy zero-temperature limit the algorithm often finds the globally optimal solution, after comparatively few iterations.\",\n \"For this, the greedy heuristic (using the mST as initialization) was compared against exact solutions with up to nine terminals, obtained by brute-force.\",\n \"For each $n$ , 100 BOT problems were sampled uniformly with respect to $\\\\alpha $ , the terminal positions and demands and supplies, cf. Alg.\",\n \"REF .\",\n \"The ratios of the heuristic's cost divided by the cost of the exact solution are plotted in Fig.\",\n \"REF .\",\n \"On average the heuristic solution is less than 0.5% worse than the brute-force solution.\",\n \"This is impressive, considering the fact that for $n=9$ the brute-force solver requires over $10^5$ BP optimizations, whereas the simulated annealing heuristic on average required $29 \\\\pm 10$ iterations to converge.\",\n \"Additional experiments (also for larger BOT problems) suggest that the number of BP optimizations until convergence scales better than $O(n^2)$ , see App.\",\n \"REF .\",\n \"Further, the cost ratios in Fig.\",\n \"REF stay roughly constant as $n$ increases.\",\n \"Additional experiments for BOT in higher dimensions (see Fig.\",\n \"REF ) indicate that the average quality of the heuristic solution decreases only very slightly with $n$ .\",\n \"Unfortunately, one can only speculate how this trend extends to larger BOT problems, where brute-force solutions are no longer feasible.\",\n \"Figure REF shows heuristic solutions of a larger example problem for different values of $\\\\alpha $ .\",\n \"In particular, we find that the greedy heuristic is very effective at removing higher-degree branchings and undesirable edge crossings.\"\n ],\n [\n \"Generalization to higher-dimensional BOT\",\n \"Optimal BOT solutions are acyclic also in $\\\\mathbb {R}^d$ [2].\",\n \"Thus, for a given topology, the edge flows are known, the optimal substructure property of Lemma REF generalizes and the convex geometry optimization can be separated from the combinatorial topology optimization.\",\n \"Though, the optimal angle conditions for Y-, V- and L-branching (see Sect. )\",\n \"hold also in $\\\\mathbb {R}^d$ , the results on the degree limitation do not generalize, as the arguments rely on the fact that the angles between edges meeting at a higher-degree branching point sum up to $2\\\\pi $ (cf.\",\n \"Sect.\",\n \"REF ).\",\n \"The numerical geometry optimization as well as the greedy algorithm for the topology optimization presented in Sect.\",\n \"are readily applicable to BOT problems in $\\\\mathbb {R}^d$ (see also App.\",\n \"REF and App.\",\n \"REF ).\"\n ],\n [\n \"Conclusions\",\n \"We have studied branched optimal transport in $\\\\mathbb {R}^2$ from a theoretical and practical perspective.\",\n \"First, we have tackled the geometric optimization of BOT solutions, given a tree topology.\",\n \"We generalized the existing exact method presented in [2], [9] to the case of multiple sources.\",\n \"Based on theory developed in the process of this generalization, we formulated a catalog of necessary and sufficient conditions for optimal BOT solutions and argued that $n$ -degree branching points for $n>3$ are never optimal.\",\n \"Moreover, we showed that these conditions also apply for BOT on two-dimensional manifolds.\",\n \"Lastly, we presented a greedy randomized algorithm, which optimizes the tree topology, combined with an efficient numerical branching point optimization method.\",\n \"We compared our algorithm to the optimal solution for small examples, obtaining compelling results.\",\n \"BOT provides a unifying framework for optimal transport and the Euclidean Steiner tree problem and is itself of great theoretical and practical interest.\",\n \"The emergent branching in BOT can be used to simulate and study the myriad of efficient transportation systems which exhibit subadditive costs.\",\n \"Moreover, BOT combines both combinatorial and convex optimization and could be an inspiring problem to be solved by machine learning techniques.\",\n \"The number of optimality criteria derived in this paper can guide further research in this area and the presented approximate solvers may serve as competitive baseline for new ML-based approaches.\",\n \"We would like to thank Jarosław Piersa for sharing his code with us for a comparison to his work.\",\n \"Further, we thank Edouard Oudet for helpful hints on the comparison to his related work and Fabian Egersdoerfer for his improved C++ implementation of the geometry optimization.\",\n \"This work is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), by Informatics for Life and by SIMPLAIX funded by the Klaus Tschira Foundation.\"\n ],\n [\n \"Appendix overview\",\n \"The appendix is subdivided into the following seven topics: app-central: Quick proof of the central angle property, used in Sect.\",\n \"REF in the recursive construction of relatively optimal solutions based on the optimal branching angles.\",\n \"secLV: Derivation of the conditions listed in Tab.\",\n \"REF under which V- or L-branching are optimal.\",\n \"sec:subopt: Formal proof of Lemma REF on the optimal substructure of BOT solutions.\",\n \"appfh: Collection of small lemmas on the monotonicity and other properties of the analytical expression for the branching angles (cf. Eq.\",\n \"(REF )).\",\n \"appsec:nonopt: Technical proofs and numerical scheme to show the non-optimality of higher-degree branchings discussed in Sect.\",\n \"REF .\",\n \"app:mf: Formal proof of Theorem REF , which generalizes the optimal branching conditions and other properties from the Euclidean plane to embedded surfaces.\",\n \"A sketch of the proof can be found in Sect.\",\n \"REF .\",\n \"appsec:algo: Additional details and experiments for the different algorithms presented in the main paper.\",\n \"Section REF focuses on the numerical geometry optimization and Sect.\",\n \"REF on the greedy algorithm for the topology optimization.\",\n \"Section REF holds a few examples of the recursive geometric construction of relatively optimal solutions for BOT problems with multiple sources.\"\n ],\n [\n \"Central angle property\",\n \"In this section, we present a geometric proof of the central angle property used in the geometric construction of relatively optimal solutions for a given full tree topology (see Sect.\",\n \"REF ).\",\n \"It states that for a circle, as in Fig.\",\n \"REF , the central angle $\\\\angle a_1 o a_2$ is twice the angle $\\\\angle a_1 q a_2 = \\\\theta $ for all $q$ on the lower circle arc.\",\n \"Figure: (a) Illustration of the central angle theorem.\",\n \"(b) Isosceles triangles used in the geometric proof.Let us start by constructing the three isosceles triangles $\\\\triangle o q a_1$ , $\\\\triangle o q a_2$ and $\\\\triangle o a_1 a_2$ with angles as denoted in Fig.\",\n \"REF .\",\n \"Now consider the angular sums in the following triangles: $\\\\triangle q a_1 a_2 : \\\\ 2 \\\\delta _1 + 2 \\\\delta _2 + 2 \\\\tau = 180^\\\\circ \\\\ \\\\text{ and } \\\\ \\\\triangle o a_1 a_2 : \\\\gamma + 2 \\\\tau = 180^\\\\circ \\\\ .$ Subtracting the two equations immediately reveals that $\\\\gamma = 2 \\\\delta _1 + 2 \\\\delta _2 = 2 \\\\theta $ and the proof is complete.\"\n ],\n [\n \"Optimal L- and V-shaped branching\",\n \"Below, we formally derive the conditions listed in Tab.\",\n \"REF under which V- or L-branching provide the optimal solution to a BOT problem with one source and two sinks.\",\n \"The proof is inspired by the approach in [21], where subdifferentials are applied to the related Fermat-Torricelli problem.\",\n \"Definition B.1 (Subgradient and subdifferential) A vector $v \\\\in \\\\mathbb {R}^n$ is called a subgradient of a convex scalar function $g: \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ at a specific point $y$ if for all $x \\\\in \\\\mathbb {R}^n$ it satisfies $g(x) \\\\ge g(y) + \\\\langle v, x - y \\\\rangle $ The set of all subgradients of the function $g$ at a given point $y$ is called the subdifferential of $g$ at $y$ and is denoted by $\\\\partial g(y)$ .\",\n \"From a geometric point of view, the subdifferential of $g$ at $y$ is the set of gradients of all straight lines which cross $g(y)$ and lie below the image of $g$ .\",\n \"The subdifferential rule of Fermat follows immediately from the definition and states that $g$ achieves an absolute minimum at $y$ if and only if $0 \\\\in \\\\partial g(y)$ .\",\n \"Now, for $g(b) = c \\\\cdot \\\\left\\\\Vert b - a \\\\right\\\\Vert _2$ with $c \\\\in \\\\mathbb {R}$ and $a \\\\in \\\\mathbb {R}^n$ the subdifferential is given by $\\\\partial g(b) = {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathbb {B}_{c} \\\\, , \\\\quad \\\\quad \\\\quad \\\\ \\\\text{ for } b = a \\\\\\\\\\\\bigg \\\\lbrace c \\\\frac{b - a}{\\\\left\\\\Vert b - a \\\\right\\\\Vert _2} \\\\bigg \\\\rbrace \\\\, , \\\\text{ elsewhere, }\\\\end{array}\\\\right.\",\n \"}$ with $B_r = \\\\lbrace v \\\\in \\\\mathbb {R}^n : \\\\left\\\\Vert v \\\\right\\\\Vert _2 \\\\le r \\\\rbrace $ , the ball of radius $r$ .\",\n \"Furthermore, it can be easily shown that for $g(x) = g_1(x) + g_2(x)$, one has $\\\\partial g(y) = \\\\partial g_1(y) + \\\\nabla g_2(y)$ , given that both $g_1$ and $g_2$ are convex functions and $g_2$ is differentiable.The sum of a vector and a set of vectors, as in $\\\\partial g_1(y) + \\\\nabla g_2(y)$ , is known as the Minkowski sum.\",\n \"Using this, we calculate the subdifferentials of the cost function of the 1-to-2 branching in Eq.\",\n \"(REF ).\",\n \"The subdifferentials at $b = a_i$ are of the form $\\\\partial \\\\mathcal {C}(a_i) = \\\\mathbb {B}_{m_i^\\\\alpha } + \\\\sum _{j \\\\ne i}m_j^\\\\alpha \\\\frac{ a_i - a_j}{ \\\\left\\\\Vert a_i - a_j \\\\right\\\\Vert } \\\\ .$ Based on the rule of Fermat, the cost function achieves an absolute minimum at $b = a_i$ if and only if $ \\\\left\\\\Vert \\\\ \\\\sum _{j \\\\ne i} m_j^\\\\alpha \\\\frac{ a_i - a_j}{\\\\left\\\\Vert a_i - a_j \\\\right\\\\Vert } \\\\ \\\\right\\\\Vert \\\\le m_i^\\\\alpha \\\\ .\",\n \"\\\\\\\\[-0.4cm]\\\\nonumber $\"\n ],\n [\n \"V-branching.\",\n \"We square condition (REF ) and evaluate it for $i=0$ in order to determine under which condition a V-shaped branching with $b^* = a_0$ is optimal: $\\\\left\\\\Vert \\\\ m_1^\\\\alpha \\\\frac{a_0 - a_1}{\\\\left\\\\Vert a_0 - a_1 \\\\right\\\\Vert }+ m_2^\\\\alpha \\\\frac{a_0 - a_2}{\\\\left\\\\Vert a_0 - a_2 \\\\right\\\\Vert }\\\\ \\\\right\\\\Vert ^2 =m_1^{2 \\\\alpha } + m_2^{2 \\\\alpha } + 2 m_1^\\\\alpha m_2^\\\\alpha \\\\cos (\\\\psi )\\\\le m_0^{2\\\\alpha } = (m_1 + m_2)^{2 \\\\alpha } \\\\,$ where $\\\\psi $ denotes the angle of the terminal triangle at $a_0$ , i.e., $\\\\psi = \\\\angle a_1a_0a_2$.\",\n \"The condition in terms of $\\\\psi $ reads $ \\\\psi \\\\ge \\\\arccos \\\\bigg ( \\\\frac{1 - k^{2 \\\\alpha } - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha (1-k)^\\\\alpha } \\\\bigg ) = h(\\\\alpha , k) = \\\\theta ^*_1 + \\\\theta ^*_2 \\\\ ,$ where we have used the flow fraction $k = m_1/(m_1 + m_2)$ .\",\n \"We immediately recognize the expression for the optimal branching angle $\\\\theta ^*_1 + \\\\theta ^*_2$ , cf. Eq.\",\n \"(REF ).\",\n \"We already know that $\\\\psi = \\\\theta ^*_1 + \\\\theta ^*_2$ on the lower circle arc of the pivot circle by construction.\",\n \"And indeed one can easily check that $\\\\psi > \\\\theta ^*_1 + \\\\theta ^*_2$ if an only if the source $a_0$ lies inside the lower half of the pivot circle, as in Fig.\",\n \"REF .\",\n \"For that, we construct a line through $a_1$ and $a_0$ and the intersection of $\\\\overline{a_1a_0}$ with the lower pivot circle we denote by $q$ .\",\n \"By construction of the pivot circle, $\\\\angle a_1 q a_2 = \\\\theta ^*_1 + \\\\theta ^*_2$ .\",\n \"Using the angular sum in the triangle $\\\\triangle a_0 q a_2$ , one immediately obtains: $180^\\\\circ = \\\\delta + ( 180^\\\\circ - \\\\psi ) + (\\\\theta ^*_1 + \\\\theta ^*_2) \\\\ \\\\rightarrow \\\\ \\\\psi - (\\\\theta ^*_1 + \\\\theta ^*_2) = \\\\delta > 0 \\\\ .$ Figure: Sketch to show that ψ>θ 1 * +θ 2 * \\\\psi > \\\\theta ^*_1 + \\\\theta ^*_2.Hence, for $a_0$ located inside the lower half of the pivot circle indeed $\\\\psi > \\\\theta ^*_1 + \\\\theta ^*_2$ .\",\n \"A similar argument can show that for any point outside the pivot circle the optimal V-branching condition is not fulfilled.\"\n ],\n [\n \"L-branching.\",\n \"Analogous to the above steps, one obtains conditions for optimal L1- and L2-branching, where $b^*= a_1$ and $b^*= a_2$ respectively.\",\n \"Again squaring the general condition (REF ), now for $i=1,2$ , one eventually finds that $\\\\varphi &\\\\ge \\\\pi - f(\\\\alpha , 1-k) = \\\\pi - \\\\theta ^*_2 \\\\ , \\\\\\\\\\\\varrho &\\\\ge \\\\pi - f(\\\\alpha , k) = \\\\pi - \\\\theta ^*_1 \\\\ ,$ where the angle $\\\\varphi $ and $\\\\varrho $ denote the angles of the terminal triangle located at $a_1$ and $a_2$ , i.e.\",\n \"$\\\\varphi = \\\\angle a_2a_1a_0$ and $\\\\varrho = \\\\angle a_0a_2a_1$ (see also Fig.\",\n \"REF ,vll4).\",\n \"Let us now demonstrate that these conditions are indeed fulfilled if and only if the source is located in the L1- and L2-region, as marked also in Fig.\",\n \"REF .\",\n \"The pivot point $p$ is constructed such that $\\\\angle a_1 o p = 2 \\\\theta ^*_1$ and $\\\\angle p o a_2 = 2 \\\\theta ^*_2$ , cf. Fig.\",\n \"REF .\",\n \"Besides that, by construction, we have that $\\\\angle p a_1 a_2 = \\\\theta ^*_2$ and $\\\\angle a_1 a_2 p = \\\\theta ^*_1$ (as shown in Fig.\",\n \"REF ,vll4).\",\n \"Then, looking at Fig.\",\n \"REF , it is evident that indeed for any source $a_0$ inside the L1-branching sector the condition $\\\\varphi = \\\\angle a_2a_1a_0 \\\\ge \\\\pi - \\\\theta ^*_2$ holds.\",\n \"The respective condition (REF ) for L2-branching holds true exactly inside the highlighted L2-region.\",\n \"Definition B.2 (Transient and strict V- and L-branchings) A V- or L-branching for which the inequality conditions (REF )-(REF ) hold as equality is referred to as transient V- or L-branching.\",\n \"The reason for this is that, in such a case, one of the terminal positions may be perturbed infinitesimally, so that the condition is no longer fulfilled and the optimal solution transitions to a Y-shaped branching.\",\n \"On the contrary, if an L- or V-branching condition is fulfilled as strict inequality, we call the L- or V-branching strict.\",\n \"Figure: After constructing the pivot point the position of the source a 0 a_0 determines whether a Y-,V- or L-shaped branching is optimal.\"\n ],\n [\n \"Relation of V- and L-conditions between symmetric and asymmetric branchings\",\n \"Let us briefly show that the corner cases of Y-, V- and L-branching work analogously for both flow scenarios (described in Fig.\",\n \"REF ,case2a).\",\n \"The conditions under which L- and V-branching are optimal in the asymmetric branching case could be again determined straightforwardly by plugging into the subdifferential condition (REF ), as before.\",\n \"However, looking at Fig.\",\n \"REF ,case2a, we notice that the well-known symmetric branching case can be transformed into the asymmetric case by relabeling $a_0 \\\\rightarrow a_2$ , $a_1 \\\\rightarrow a_0$ and $a_2 \\\\rightarrow a_1$ .\",\n \"This relation provides a direct correspondence of the L- and V-branching conditions.\",\n \"The conditions for the corner case are transferred according to Table REF .\",\n \"Note that the L- and V-branching conditions for both cases are of the exact same form, only that for the asymmetric case the stationary branching angles are $\\\\vartheta ^*_i$ instead of $\\\\theta ^*_i$ .\",\n \"Moreover, for fixed children positions $a_1$ and $a_2$ , the position of the source again distinguishes between optimal Y-, V- and L-branching.\",\n \"The partitioning of the lower half plane into the respective regions is completely analogous to Fig.\",\n \"REF .\",\n \"Table: Relations between the L- and V-branching conditions (see Fig. )\",\n \"for symmetric and asymmetric branchings.\"\n ],\n [\n \"Optimal substructure of BOT solutions (Proof of Lem. \",\n \"In this section, we provide the formal proof to Lemma REF , repeated below for completeness: Lemma C.1 (a) For a given tree topology, a BOT solutions is a relatively optimal if and only if every (coupled) BP connects its (effective) neighbors at minimal cost.\",\n \"(b) In a globally optimal solution, every subsolution restricted to a connected subset of nodes solves its respective subproblem globally optimally.\",\n \"Let us start with the following definition, which divides a BOT problem and its possible solutions into subproblems and corresponding subsolutions.\",\n \"Definition C.1 (Subproblems and subsolutions) A given BOT solution may be split into two subsolutions, by choosing a number of edges $\\\\lbrace e_i\\\\rbrace $ and cutting them at points $\\\\lbrace x_i\\\\rbrace $ , so that the topology is split into two connected components.\",\n \"This procedure induces two subproblems and two subtopologies.\",\n \"Each subproblem consist of the terminals contained in the respective component plus additional terminals at the positions $x_i$ .\",\n \"The demands or supplies of the additional terminals at $x_i$ are equal to the amount of flow through the corresponding edge $e_i$ that was cut.\",\n \"The terminal becomes a sink in one subproblem and a source in the other according to the direction of flow through $e_i$ .\",\n \"The two subtopologies are given by the induced subgraph on all terminals contained in one component.\",\n \"The subsolutions to the created subproblems are given by the subtopologies and the BP configurations of the respective subsets of branching points contained in each subproblem.\",\n \"Note that using a number of such splits a given solution may be divided into several subsolutions, each solving their respective subproblem.\",\n \"An illustrative example can be found in Fig REF .\",\n \"Figure: A BOT solution (left) is split into two subsolutions (right) by cutting an edge as indicated by the dashed line.\"\n ],\n [\n \"Proof of the optimal substructure as necessary condition for optimality.\",\n \"The optimal substructure as necessary optimality condition follows immediately from the independence of the subproblems.\",\n \"Given that a solution is minimal in cost (in the relative or global sense), each subproblems itself must be minimal in that sense.\",\n \"Otherwise the cost of this subproblem could be decreased by (a) improving the BP configuration or (b) the topology.\",\n \"In that case, also the full solution could be improved in cost and could thus not be optimal.\"\n ],\n [\n \"Proof of the optimal substructure as sufficient condition for optimality.\",\n \"The optimal substructure as sufficient optimality condition stems from the fact that, given a tree topology, the BOT cost function is convex with respect to the BP positions.\",\n \"As a first step, let us prove the following lemmata: Lemma C.2 Let $b$ be a branching point with neighbors $\\\\lbrace a_i \\\\rbrace $ .\",\n \"Let the optimal position of $b$ be at one of its neighbors with the condition (REF ) fulfilled as equality, i.e.\",\n \"the branching is transient.\",\n \"Then, for every $\\\\delta >0 $ there exists a BP location $b_\\\\delta \\\\in \\\\mathbb {R}^2$ away from all neighbors with $\\\\left\\\\Vert b - b_\\\\delta \\\\right\\\\Vert _2 < \\\\delta $ and a continuous function $f:\\\\mathbb {R^+}\\\\rightarrow \\\\mathbb {R}$ with $\\\\lim _{x\\\\rightarrow 0}f(x)=0$ , such that the gradient $\\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b} \\\\, \\\\mathcal {C}(b_\\\\delta ) \\\\right\\\\Vert _2 < \\\\epsilon $ for $0<\\\\epsilon 0$ , such that $b_\\\\delta $ is located away from all terminals.\",\n \"The gradient with respect to the branching point position is well defined at $b_\\\\delta $ and reads: $\\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b} \\\\, \\\\mathcal {C}(b_\\\\delta ) = - m_0^\\\\alpha \\\\frac{v}{\\\\left\\\\Vert v \\\\right\\\\Vert } +\\\\sum _{i \\\\ne 0} m_i^\\\\alpha \\\\frac{ a_0 - a_i - \\\\delta v}{\\\\left\\\\Vert a_0 - a_i -\\\\delta v \\\\right\\\\Vert } \\\\ .$ Clearly, $\\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b} \\\\, \\\\mathcal {C}(b_\\\\delta ) \\\\right\\\\Vert $ can be brought arbitrarily close to zero as $\\\\delta \\\\rightarrow 0$ , since $\\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b} \\\\, \\\\mathcal {C}(b_\\\\delta ) \\\\right\\\\Vert ^2 = \\\\underbrace{m_0^{2 \\\\alpha } + m_0^{2 \\\\alpha } - 2 m_0^\\\\alpha \\\\Big \\\\langle v, \\\\frac{v}{\\\\left\\\\Vert v \\\\right\\\\Vert } \\\\Big \\\\rangle }_{= \\\\, 0} + O(\\\\delta ) \\\\ ,$ comprising the terms which go to zero as $\\\\delta \\\\rightarrow 0$ in $O(\\\\delta )$ .\",\n \"The function absorbed in $O(\\\\delta )$ therefore provides the function $f$ and $b_\\\\delta $ is the desired branching point with arbitrarily small gradient.\",\n \"The argument works for both coupled and uncoupled branching points.\",\n \"$$ Lemma C.3 For a given full tree topology $T$ , let $B=\\\\lbrace b_i\\\\rbrace $ be the ROS of a BOT problem with terminals $A = \\\\lbrace a_i\\\\rbrace $ .\",\n \"Let $B$ contain Y-branchings and transient V- or L-branchings.\",\n \"Then, for every $\\\\delta >0 $ there exists a non-degenerate BP configuration $B_\\\\delta $ and a continuous function $f:\\\\mathbb {R^+}\\\\rightarrow \\\\mathbb {R}$ with $\\\\lim _{x\\\\rightarrow 0}f(x)=0$ , such that $\\\\left\\\\Vert B - B_\\\\delta \\\\right\\\\Vert _2 < \\\\delta $ and the gradient norm fulfills $\\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"B} \\\\, \\\\mathcal {C}(B_\\\\delta ) \\\\right\\\\Vert _2 < \\\\epsilon $ for $0<\\\\epsilon 0$ .\",\n \"Lemma REF states that we can find an arbitrarily small $\\\\delta $ so that $B^*_{f} + \\\\delta $ is non-degenerate and the gradient $\\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"B_f} \\\\, \\\\mathcal {C}(B^*_{f} + \\\\delta )$ is arbitrarily small.\",\n \"Due to the convexity of $\\\\mathcal {C}$ , the following inequality holds: $\\\\mathcal {C}(B_{f,0}) &\\\\ge \\\\mathcal {C}(B^*_{f} + \\\\delta ) + \\\\langle \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"B_f} \\\\, \\\\mathcal {C}(B^*_{f} + \\\\delta ), B_{f,0} - (B^*_{f} + \\\\delta ) \\\\rangle \\\\\\\\&= \\\\mathcal {C}(B^*_{f}) + \\\\underbrace{O( \\\\left\\\\Vert \\\\delta \\\\right\\\\Vert _2) + \\\\langle \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"B_f} \\\\, \\\\mathcal {C}(B^*_{f} + \\\\delta ), B_{f,0} - (B^*_{f} + \\\\delta ) \\\\rangle }_{=: \\\\ K(\\\\delta )}$ where we have used that $\\\\mathcal {C}(B^*_{f} + \\\\delta ) = \\\\mathcal {C}(B^*_{f}) + O( \\\\left\\\\Vert \\\\delta \\\\right\\\\Vert _2)$ , using Big-O notation.\",\n \"All terms which tend to zero as $\\\\delta \\\\rightarrow 0$ have been summarized in $K(\\\\delta )$ .\",\n \"$K(\\\\delta )$ can be brought arbitrarily close to zero by choosing a sufficiently small $\\\\delta $ .\",\n \"In particular, there exists a $\\\\delta > 0$ so that $\\\\vert K(\\\\delta ) \\\\vert < \\\\Delta / 2$ .\",\n \"Using the assumption $\\\\mathcal {C}(B^*_{f}) = \\\\mathcal {C}(B_{f,0}) + \\\\Delta $ , this leads to: $\\\\mathcal {C}(B_{f,0}) &\\\\ge \\\\mathcal {C}(B^*_{f}) + K(\\\\delta ) = \\\\mathcal {C}(B_{f,0}) + \\\\Delta + K(\\\\delta ) \\\\ge \\\\mathcal {C}(B_{f,0}) + \\\\frac{\\\\Delta }{2} \\\\ ,$ which is a contradiction.\",\n \"Hence $B^*$ , is again a local minimum.\",\n \"$$\"\n ],\n [\n \"Properties of the functions $f$ and {{formula:ae7b64eb-edf7-4367-a415-b0559ec00a0b}} describing the optimal branching angles\",\n \"The optimal branching angles are expressed in terms of the following functions $f(\\\\alpha , k)$ and $h(\\\\alpha , k)$ , defined for $\\\\alpha \\\\in [0,1]$ and $k \\\\in (0,1)$ , cf. Eq.\",\n \"(REF ): $f(\\\\alpha , k) &= \\\\arccos \\\\bigg ( \\\\frac{k^{2\\\\alpha } + 1 - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha } \\\\bigg ) \\\\ , \\\\\\\\h(\\\\alpha , k) &= \\\\arccos \\\\bigg ( \\\\frac{1 - k^{2 \\\\alpha } - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha (1-k)^\\\\alpha } \\\\bigg ) \\\\ .$ Figure REF shows $f$ and $h$ as functions of $k$ for a number of different values of $\\\\alpha $ .\",\n \"The two functions are related by $h(\\\\alpha , k) = f(\\\\alpha , k ) + f(\\\\alpha , 1-k)$ , so that $h$ is symmetric around $k=1/2$ .\",\n \"Both functions are defined for inputs $\\\\alpha \\\\in [0,1]$ and $k \\\\in (0,1)$ .\",\n \"For $\\\\alpha = 1$ , we have $f(\\\\alpha =1, k) = 0 = h(\\\\alpha =1, k)$ for all $k$ , reflecting the fact that Y-shaped branchings are never optimal in the optimal transport case.\",\n \"On the other hand, we have $f(\\\\alpha =0, k) = \\\\pi /3$ and $h(\\\\alpha =0, k)= 2 \\\\pi / 3$ for all $k$ .\",\n \"This limit corresponds to the Euclidean Steiner tree problem where in an optimal Y-branching all edges meet at $120^\\\\circ $ .\",\n \"Moreover, considering the limits of $k \\\\rightarrow 0$ and $k \\\\rightarrow 1$ , one finds that $f(\\\\alpha , k \\\\rightarrow 0) \\\\rightarrow \\\\pi /2$ and $f(\\\\alpha , k \\\\rightarrow 1) \\\\rightarrow 0$ .\",\n \"Consequently, $h(\\\\alpha , k) \\\\rightarrow \\\\pi /2$ for both $k \\\\rightarrow 0$ and for $k \\\\rightarrow 1$ .\",\n \"In the following three lemmas, we investigate the monotonicities of the two functions.\",\n \"Figure: The functions ff and hh vs. kk for different values of α\\\\alpha .Lemma D.1 $f(\\\\alpha , k)$ is strictly decreasing in $k$ for all $\\\\alpha \\\\in (0,1)$ and $f(\\\\alpha ,k) < \\\\pi /2$ for all $\\\\alpha \\\\in [0,1]$ and $k \\\\in (0,1)$ .\",\n \"Proof.\",\n \"Since the inverse cosine function decreases monotonically, it is sufficient to show that the argument is monotonically increasing.\",\n \"Let us consider its derivative $\\\\frac{\\\\partial }{\\\\partial k} \\\\bigg ( \\\\frac{k^{2\\\\alpha } + 1 - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha } \\\\bigg )\\\\sim k^{2 \\\\alpha } + (1-k)^{2 \\\\alpha } + 2 k (1-k)^{2 \\\\alpha - 1} - 1 =: s(\\\\alpha , k) \\\\ .$ The $\\\\sim $ indicates that we have dropped overall factors which were clearly positive, as we are only interested in the sign of the derivative.\",\n \"First, note that $s(\\\\alpha = 0, k) = \\\\frac{1+k}{1-k} > 0$ and $s(\\\\alpha = 1, k) = 0$ for all $k$ .\",\n \"Secondly, $s(\\\\alpha , k)$ is strictly decreasing with respect to $\\\\alpha $ : $\\\\frac{\\\\partial s}{\\\\partial \\\\alpha } =2 k^{2\\\\alpha } \\\\log (k) + 2(1-k)^{2\\\\alpha } \\\\log (1-k) + 4k (1-k)^{2\\\\alpha -1} \\\\log (1-k) < 0 \\\\ .$ Each of the above terms are negative due to the fact that $k, (1-k) \\\\in (0,1)$ .\",\n \"Together it follows that $s(\\\\alpha , k)$ is positive for any combination of $\\\\alpha $ and $k$ and the proof of the monotonicity is complete.\",\n \"The fact that $f(\\\\alpha , k \\\\rightarrow 0) \\\\rightarrow \\\\pi /2$ for all $\\\\alpha $ then immediately implies that $f(\\\\alpha ,k) < \\\\pi /2$ .\",\n \"$$ Lemma D.2 For $\\\\alpha \\\\le 0.5$ , $h(\\\\alpha , k)$ increases monotonically in $k$ for $k \\\\in (0,1/2]$ and decreases for larger values of $k$ .\",\n \"Conversely, for $\\\\alpha \\\\ge 0.5$ , $h(\\\\alpha , k)$ increases first until in reaches a maximum at $k=1/2$ and decreases afterwards.\",\n \"Proof.\",\n \"The argument here follows the proof of Lemma 12.14 in [2].\",\n \"Due to the monotonicity of the inverse cosine function, it is again sufficient to investigate the expression in the argument.\",\n \"We rewrite this expression trivially, so that it becomes a function of the fraction $r := \\\\frac{k}{1-k}$ and consider the following derivative: $\\\\frac{\\\\partial }{\\\\partial r} \\\\bigg ( \\\\frac{(r+1 )^{2\\\\alpha } - r^{2 \\\\alpha } - 1}{2 r^\\\\alpha } \\\\bigg )\\\\sim -(r+1)^{2\\\\alpha } - r^{2\\\\alpha } + 1 + 2 r (r+1)^{2\\\\alpha - 1} =: t(\\\\alpha , r)$ We first note that $t(\\\\alpha = 0, r) = \\\\frac{r-1}{r+1} \\\\le 0$ and that $t(\\\\alpha = 0.5, r) = 0 $ as well as $t(\\\\alpha = 1, r) = 0$ for all $r$ .\",\n \"Next, we show that $t(\\\\alpha , r)$ is a concave function with respect to $r$ for $r \\\\le 1$ , which corresponds to $k \\\\in (0,1/2]$ : $\\\\frac{\\\\partial ^2 t}{\\\\partial \\\\alpha ^2} = 4 (r -1) (r+1)^{2\\\\alpha -1} \\\\log ^2(r+1) - 4 r^{2\\\\alpha } \\\\log ^2(r) \\\\le 0 \\\\ ,$ since $r \\\\le 1$ .\",\n \"Taken together, we conclude that $t(\\\\alpha ,r) \\\\le 0$ for all $\\\\alpha \\\\in [0,0.5]$ and that $t(\\\\alpha ,r) \\\\ge 0$ for all $\\\\alpha \\\\in [0.5,1]$ .\",\n \"Moreover, since $r$ increases monotonically w.r.t.\",\n \"$k$ , the monotonicity of $h(\\\\alpha , k)$ holds for $k \\\\in (0,1/2]$ .\",\n \"Due to the symmetry of $h$ around $k = 1/2$ , the proof is complete.\",\n \"$$ Lemma D.3 $f(\\\\alpha , k)$ decreases monotonically in $\\\\alpha $ for $\\\\alpha \\\\in [0.5,1]$ and $h(\\\\alpha , k)$ decreases monotonically in $\\\\alpha $ for all $\\\\alpha \\\\in [0,1]$ .\",\n \"Proof.\",\n \"As the inverse cosine function decreases monotonically, it is again sufficient to investigate the derivative of the function in the $\\\\arccos $ -argument.\",\n \"For $f(\\\\alpha , k)$ , we consider $\\\\frac{\\\\partial }{\\\\partial \\\\alpha } \\\\bigg ( \\\\frac{k^{2\\\\alpha } + 1 - (1 \\\\!\",\n \"- \\\\!\",\n \"k)^{2\\\\alpha }}{2 k^a} \\\\bigg ) \\\\!= \\\\frac{1}{2} k^{-\\\\alpha } \\\\Big ( \\\\underbrace{ \\\\log (k)}_{< \\\\ 0} [ \\\\,\\\\underbrace{ k^{2 \\\\alpha } + (1 \\\\!\",\n \"- \\\\!\",\n \"k)^{2 \\\\alpha } - 1 }_{\\\\le \\\\ 0} \\\\, ] - 2 (1 \\\\!\",\n \"- \\\\!\",\n \"k)^{2 \\\\alpha } \\\\underbrace{\\\\log (1 \\\\!- \\\\!k)}_{< \\\\ 0} \\\\Big ) > 0 .$ To see that in fact the expression in the square bracket is smaller or equal to zero, we exploit that for $\\\\alpha \\\\ge 0.5$ the function $k \\\\mapsto k^{2\\\\alpha }$ is superadditive, so that $1 = 1^{2 \\\\alpha } = ( k + (1-k))^{2 \\\\alpha } \\\\ge k^{2 \\\\alpha } + (1-k)^{2 \\\\alpha }$ .\",\n \"For $h(\\\\alpha , k)$ , we consider $\\\\frac{\\\\partial }{\\\\partial \\\\alpha } \\\\bigg ( \\\\frac{1 - k^{2 \\\\alpha } - (1-k)^{2 \\\\alpha }}{2 k^\\\\alpha (1-k)^\\\\alpha } \\\\bigg ) = - \\\\frac{1}{2} (1-k)^{-\\\\alpha } k^{-\\\\alpha } \\\\Big (&\\\\underbrace{\\\\log (k)}_{< \\\\ 0} \\\\, [ k^{2 \\\\alpha } + 1 - (1-k)^{2 \\\\alpha } ] + \\\\\\\\&\\\\underbrace{ \\\\log (1- k)}_{< \\\\ 0} \\\\, [ (1-k)^{2 \\\\alpha } + 1 - k^{2 \\\\alpha } ]\\\\Big ) > 0 \\\\ .$ Since $k, (1-k) \\\\in (0,1)$ clearly the expressions in square brackets are positive so that the overall expression is positive too.\",\n \"In the respective regions, $f(\\\\alpha , k)$ and $h(\\\\alpha , k)$ are thus monotonically decreasing in $\\\\alpha $ .\",\n \"$$\"\n ],\n [\n \"Non-optimality of higher-degree branchings\",\n \"This section supplements Sect.\",\n \"REF of the paper.\",\n \"First, we address the third and most involved 4-branching scenario in which a coupled 4-BP connects one source and three sinks (or equivalently 3 sources and 1 sink), see Fig.\",\n \"REF .\",\n \"We derive the inequalities listed in Proposition REF and prove them analytically for a large subset of the parameter space.\",\n \"For the remainder we present a numerical argument (App.\",\n \"REF ).\",\n \"Lastly, we show by induction how, given that coupled 4-BPs are never globally optimal, one can further rule out coupled $n$ -BPs (with $n$ effective neighbors) for all $n > 4$ (App.\",\n \"REF ).\"\n ],\n [\n \"Non-optimality of coupled 4-BPs between one source and three sinks\",\n \"Let us start by providing the derivation of Proposition REF , which we repeat here for completeness: Proposition E.1 Given a BOT problem with one source and three sinks, with demands $m_1, m_2, m_3$ as in Fig.\",\n \"REF , a coupled 4-BP away from the terminals cannot be globally optimal if at least one of the following inequalities holds true: $\\\\Gamma &= h \\\\Big (\\\\frac{m_1}{m_1 + m_2} \\\\Big ) - f(m_1) + h \\\\Big (\\\\frac{m_3}{m_3 + m_2} \\\\Big ) - f( m_3) > 0, \\\\\\\\\\\\Gamma _{1,*} &= f(1 -m_*) + f\\\\Big (1 - \\\\frac{m_2}{1-m_*} \\\\Big ) - f(1-m_*-m_2) > 0, \\\\\\\\\\\\Gamma _{2,*} &= h \\\\Big (\\\\frac{m_*}{m_* + m_2} \\\\Big ) + f\\\\Big (\\\\frac{m_2}{1 - m_*} \\\\Big ) - h(m_*) > 0$ where $*=1,3$ .\",\n \"Note that $\\\\Gamma = \\\\Gamma _{1,1} + \\\\Gamma _{1,3} = \\\\Gamma _{2,1} + \\\\Gamma _{2,3}$ .\",\n \"Note that it is an important specification that we consider coupled 4-BPs away from the terminals.\",\n \"For instance, in the OT case where $\\\\alpha = 1$ , all BPs are located at the terminals and coupled BPs with arbitrary number of neighbors may be globally optimal.\",\n \"For all following considerations, we assume that $\\\\alpha \\\\in [0,1)$ .\",\n \"Figure: Coupled 4-BP between one source and three sinks.\"\n ],\n [\n \"Derivation of the $\\\\Gamma $ -inequality\",\n \"WLOG, let us normalize the masses so that $m_1 + m_2 + m_3 =1$ and determine the necessary conditions under which all V-branchings are optimal: $\\\\gamma _1 &\\\\ge \\\\pi - f\\\\Big (\\\\alpha , 1- \\\\frac{1-m_1}{1}\\\\Big ) = \\\\pi - f(\\\\alpha , m_1) \\\\ , \\\\nonumber \\\\\\\\\\\\gamma _2 &\\\\ge \\\\pi - f \\\\Big ( \\\\alpha , 1 - \\\\frac{1-m_3}{1}\\\\Big ) = \\\\pi - f(\\\\alpha , m_3) \\\\ , \\\\nonumber \\\\\\\\\\\\gamma _3 &\\\\ge h \\\\Big (\\\\alpha , \\\\frac{m_3}{m_3 + m_2} \\\\Big ) \\\\ , \\\\nonumber \\\\\\\\\\\\gamma _4 &\\\\ge h \\\\Big (\\\\alpha , \\\\frac{m_1}{m_1 + m_2} \\\\Big ) \\\\ .$ We intend to show that such a 4-BP can never be globally optimal by showing that for any combination of $\\\\alpha $ and the masses $m_i$ the sum of the lower bounds is already larger than $2 \\\\pi $ .\",\n \"This is equivalent to proving that the following inequality holds true for all parameter combinations: $\\\\Gamma := h \\\\Big (\\\\alpha , \\\\frac{m_1}{m_1 + m_2} \\\\Big ) - f(\\\\alpha , m_1) + h \\\\Big (\\\\alpha , \\\\frac{m_3}{m_3 + m_2} \\\\Big ) - f(\\\\alpha , m_3) > 0 \\\\ .$ The inequality reflects that the problem setup is inherently symmetric under exchange of $m_1$ and $m_3$ .\",\n \"WLOG, we assume that $m_1 \\\\le m_3$ .\"\n ],\n [\n \"Derivation of the $\\\\Gamma _{1,*}$ - and {{formula:2319981d-25b2-4794-a10a-81724537d3b3}} -inequalities\",\n \"For a BOT solution to be globally optimal it means that it is the cheapest relatively optimal solution of all possible full tree topologies.\",\n \"In our case of four terminals, there are three distinct topologies, see Fig.\",\n \"REF .\",\n \"Let us assume that a globally optimal 4-BP away from the terminals exists and denote the terminal positions by $a_i$ .\",\n \"Then, for all three topologies $T_1$ , $T_2$ and $T_3$ , this branching point configuration is the ROS, since clearly for all $T_i$ a coupled 4-BP configuration can be realized by coupling the two branching points.\",\n \"Figure: The three distinct full tree topologies connecting four terminals.Let us investigate graphically under which conditions a coupled 4-BP provides the relatively optimal solution for the different topologies.\",\n \"Figure REF shows the pivot circle and pivot point construction for topology $T_1$ , where $a_0$ has been chosen as root node.\",\n \"Let us refer to the line through $p_2$ and the intersection of the two pivot circles as transition line, for the following reason: If $a_0$ was positioned to the left of the transition line, the ROS of $T_1$ would be non-degenerate, as shown in Fig.\",\n \"REF for instance.\",\n \"For $a_0$ to the right of the transition line, the ROS of $T_1$ is given by a coupled 4-BP.\",\n \"Hence, the transition line marks the transition between a non-generate ROS of $T_1$ and a coupled 4-BP solution.\",\n \"Consequently, $a_0$ must lie to the right of the transition line of $T_1$ .\",\n \"But, $a_0$ must simultaneously also lie on the appropriate side of the transition lines of the two other topologies $T_2$ and $T_3$ .\",\n \"One can now argue that the root node $a_0$ can be moved along a continuous path onto the transition line of topology $T_1$ , without crossing any of the other transition lines.\",\n \"In doing so, the coupled 4-branching stays relatively optimal for all three topologies.\",\n \"Most importantly, it thereby stays globally optimal.\",\n \"During this procedure the terminals $a_1$ , $a_2$ and $a_3$ stay fixed so that the pivot points and pivot circles as well as the transition lines stay exactly the same.\",\n \"For topology $T_1$ we have now arrived at a special case of coupled 4-branching, in which the V- and L-branchings are transient (see Def.\",\n \"REF ).\",\n \"Let us refer to such a BP configuration as transient 4-branching.\",\n \"As a consequence, the angle $\\\\gamma _4$ may be expressed in terms of the following branching angles (see Fig.\",\n \"REF ) $ \\\\gamma _4 = \\\\beta _1 + \\\\beta _2 - \\\\theta _1 = h(\\\\alpha , m_1) - f\\\\Big (\\\\alpha , \\\\frac{m_2}{1-m_1} \\\\Big ) \\\\ .$ Figure: Transient coupled 4-BP with a 0 a_0 placed on the transition line of topology T 1 T_1.All in all, the above argument shows that the existence of a globally optimal coupled 4-BP necessarily implies the existence of a globally optimal transient 4-branching, which can be constructed by changing only the coordinates of one of the terminals.\",\n \"Clearly, for the globally optimal and transient 4-BP of topology $T_1$ all four necessary conditions for optimal V-branching must still apply.\",\n \"In order to generally rule out globally optimal coupled 4-BPs, it is therefore sufficient to show that at least one of the following two conditions is fulfilled for all parameter combinations $\\\\alpha $ and $m_i$ : the sum of the lower bounds on $\\\\gamma _i$ always exceeds $2 \\\\pi $ , condition (REF ) is always incompatible with Eq.\",\n \"(REF ).\",\n \"The first condition is equivalent to the following inequality, obtained by substituting the lower bound for $\\\\gamma _4$ in (REF ) by (REF ): $ \\\\Gamma _{1,1}(\\\\alpha , m_1,m_2) := f(\\\\alpha , 1 -m_1) + f\\\\Big ( \\\\alpha , 1 - \\\\frac{m_2}{1-m_1} \\\\Big ) - f(\\\\alpha , 1-m_1-m_2) > 0 \\\\ .$ The second condition can be expressed as inequality simply by combining (REF ) and (REF ): $\\\\Gamma _{2,1}(\\\\alpha , m_1, m_2) := h \\\\Big (\\\\alpha , \\\\frac{m_1}{m_1 + m_2} \\\\Big ) + f\\\\Big ( \\\\alpha , \\\\frac{m_2}{1 - m_1} \\\\Big ) - h(\\\\alpha , m_1) > 0 \\\\ .$ Proving one of these two inequalities already suffices to rule out globally optimal 4-branching between one source and three sinks.\",\n \"Note that the three inequalities presented so far are not completely independent but are related via $\\\\Gamma = \\\\Gamma _{1,1} + \\\\Gamma _{2,1}$ .\",\n \"The above procedure of moving the root node onto the transition line of topology $T_1$ can be repeated exactly analogously for $T_2$ .\",\n \"This results in the inequalities $\\\\Gamma _{1,3}$ and $\\\\Gamma _{2,3}$ , which are of the exact same form except that $m_3$ appears in all places instead of $m_1$ .\",\n \"Note that, in both cases, we have used that $a_0$ can be moved onto the transition lines of $T_1$ and $T_2$ without crossing the transition line of topology $T_3$ .\",\n \"A justification for this is given in form of the following lemma.\",\n \"Lemma E.2 Starting from a globally optimal coupled 4-BP connecting the terminals $\\\\lbrace a_i\\\\rbrace $ and located away from all terminals, one terminal node may be moved along a continuous path onto the transition lines of topology $T_1$ and $T_2$ without crossing the transition line of topology $T_3$ first.\",\n \"Proof.\",\n \"Let us give a proof by contradiction.\",\n \"Assume that that $a_0$ could actually be moved along a continuous path onto the transition line of topology $T_3$ without touching any of the other two transition lines.\",\n \"Then, one may also move $a_0$ infinitesimally further across the transition line of topology $T_3$ , such that the ROS of $T_3$ becomes non-degenerate.\",\n \"At the same time, for topology $T_1$ and $T_2$ the coupled 4-branching configuration stays relatively optimal.\",\n \"Now, since the ROS of $T_3$ deviates from the coupled 4-branching configuration, the 4-branching can no longer be globally optimal.\",\n \"Note that a coupled 4-BP can only be globally optimal if all three topologies agree.\",\n \"Consequently, the ROS of $T_3$ must be globally optimal.\",\n \"However, the non-degenerate solution of $T_3$ will necessarily contain a cycle.\",\n \"To see this, let $b_1$ be the branching point connected to $a_0$ and $b_2$ the other branching point in $T_3$ .\",\n \"After crossing the transition line of $T_3$ , the edge $(b_1, b_2)$ at first has finite but infinitesimal length, so that either one of the edges $(b_2,a_1)$ or $(b_2,a_3)$ must intersect with $(b_1,a_2)$ , thereby creating a cycle in the ROS of $T_3$ (cf.\",\n \"Fig.\",\n \"REF ).\",\n \"However, it was proven in [2] that for $\\\\alpha \\\\in [0,1)$ cyclic BOT solution cannot be globally optimal, so that we have arrived at a contradiction.\",\n \"$$ To summarize, we have shown the following lemma: Lemma E.3 The existence of any globally optimal 4-BP connecting one source and three sinks and located away from the terminals implies the existence of two globally optimal transient 4-branchings, one for topology $T_1$ and one for $T_2$ .\"\n ],\n [\n \"$\\\\Gamma > 0 \\\\ $ for {{formula:b6d5d064-a7aa-4620-b08e-8d52134ebf39}} :\",\n \"For $\\\\alpha \\\\le 0.5$ we have $h \\\\ge \\\\pi /2$ and $f < \\\\pi /2$ for all combinations of $m_i$ (see App. )\",\n \"and thus clearly $\\\\Gamma > 0$ if $\\\\alpha \\\\le 0.5$ .\",\n \"For all following arguments, we therefore assume that $\\\\alpha >0.5$ .\",\n \"The following analytical arguments all rely on the properties of $h$ and $f$ , which are listed and proven in App.\",\n \".\"\n ],\n [\n \"$\\\\Gamma > 0 \\\\ $ for {{formula:4340b88c-d10b-4164-adc8-fcac7451b1ad}} :\",\n \"Using that $f(\\\\alpha , k)$ is monotonically decreasing in $k$ , the following loose lower bound suffices to demonstrate that $\\\\Gamma > 0$ if $m_1 \\\\ge 1/4$ : $\\\\Gamma &\\\\ge 2 \\\\ \\\\min _k \\\\, h(\\\\alpha , k) - 2 f(\\\\alpha , m_1) \\\\\\\\&= 2 \\\\arccos (2^{2 \\\\alpha - 1} - 1) - 2 \\\\arccos \\\\bigg ( \\\\frac{m_1^{2\\\\alpha } + 1 - (1-m_1)^{2 \\\\alpha }}{2 m_1^\\\\alpha } \\\\bigg ) \\\\ ,$ where we have used that $f(m_1) \\\\ge f(m_3)$ due to our assumption $m_1 \\\\le m_3$ and that $h(\\\\alpha , k)$ forms a minimum at $k=1/2$ .\",\n \"For $m_1 = 1/4$ , one finds that this expression is truly positive if and only if $9^\\\\alpha < 2 \\\\cdot 4^\\\\alpha + 1 \\\\ ,$ which is fulfilled for all $\\\\alpha \\\\in [0,1)$ due to the subadditivity of the function $m \\\\mapsto m^\\\\alpha $ , namely $9^\\\\alpha = (4 + 4 + 1)^\\\\alpha < 4^\\\\alpha + 4^\\\\alpha +1$.\",\n \"It follows that $\\\\Gamma > 0$ also for all $m_1 > 1/4$ due to the monotonicity of $f$ .\",\n \"Figure: m 1 m_1-m 2 m_2 parameter space largely ruled out analytically.\",\n \"In the white region the inequalities are checked numerically.\"\n ],\n [\n \"$\\\\Gamma _{2,3} > 0 \\\\ $ for {{formula:1550a853-7ad7-41ff-84d6-e6ee54b09d65}} :\",\n \"Inequalities $\\\\Gamma _{2,3}$ is fulfilled if $m_3 \\\\ge 1/2$ , since $\\\\Gamma _{2,3}(\\\\alpha , m_3, m_2) > h \\\\Big (\\\\alpha , \\\\underbrace{\\\\frac{m_3}{m_3 + m_2}}_{> \\\\ m_3} \\\\Big ) - h(\\\\alpha , m_3) \\\\ge 0 \\\\ \\\\text{ for } \\\\ m_3 \\\\ge 1/2 \\\\ ,$ where we have used that for $\\\\alpha > 0.5$ , $h(\\\\alpha , k)$ is monotonically increasing in $k$ for $k \\\\in [0.5,1)$ , as shown in Lem.\",\n \"REF .\",\n \"The remaining parameter region, for which none of the inequalities have been shown yet, can be characterized by the following conditions: $ \\\\alpha > 0.5 \\\\, , \\\\ \\\\ m_1 < 0.25 \\\\ \\\\text{ and } \\\\ m_2 \\\\in [0.5 - m_1, 1 - 2 m_1] \\\\ .$ The constraints for $m_3$ are implicitly represented, using that the normalization was chosen such that $m_1 +m_2 +m_3 = 1$ .\",\n \"A visualization of the remaining region can be found in Fig.\",\n \"REF .\",\n \"Finally, we propose a simple and watertight numerical scheme, which can be used to rule out almost the entire remaining volume, already with little numerical effort.\"\n ],\n [\n \"Numerical treatment of the remaining parameter space\",\n \"For the remaining parameter region characterized by the conditions in (REF ), we propose a numerical scheme which checks the inequality $\\\\Gamma _{2,1}$ for all practically relevant parameter combinations of $m_1$ , $m_2$ and $\\\\alpha $ .\",\n \"For this, we split the remaining volume into cuboids $ I = [\\\\check{\\\\alpha }, \\\\hat{\\\\alpha }] \\\\times [\\\\check{m}_1, \\\\hat{m}_1] \\\\times [\\\\check{m}_2, \\\\hat{m}_2]$ which are divided recursively into smaller cuboids based on the following octree scheme.\",\n \"For each cuboid, we determine a lower bound of $\\\\Gamma _{2,1}$ .\",\n \"This lower bound becomes tighter the smaller the cuboid $I$ .\",\n \"If, for a cuboid, this lower bound is not yet positive, it is divided further into eight new cuboids by splitting each of the three intervals in half.\",\n \"This procedure is iterated until for all cuboids the lower bound is truly positive.\",\n \"For all $m_1, m_2, \\\\alpha $ in a cuboid $I=[\\\\check{\\\\alpha }, \\\\hat{\\\\alpha }] \\\\times [\\\\check{m}_1, \\\\hat{m}_1] \\\\times [\\\\check{m}_2, \\\\hat{m}_2]$ , the lower bound of $\\\\Gamma _{2,1}$ is obtained by minimizing each summand individually: $\\\\Gamma _{2,1}(\\\\alpha , m_1, m_2) &\\\\ge \\\\underset{I}{\\\\min } \\\\ h \\\\big (\\\\alpha , \\\\frac{m_1}{m_1 + m_2} \\\\big ) + \\\\underset{I}{\\\\min } \\\\ f\\\\big ( \\\\alpha , \\\\frac{m_2}{1 - m_1} \\\\big ) - \\\\underset{I}{\\\\max } \\\\ h(\\\\alpha , m_1) \\\\nonumber \\\\\\\\&= h \\\\Big (\\\\hat{\\\\alpha }, \\\\frac{\\\\hat{m}_1}{\\\\hat{m}_1 + \\\\check{m}_2} \\\\Big )+ f \\\\Big ( \\\\hat{\\\\alpha }, \\\\frac{ \\\\hat{m}_2}{1 - \\\\hat{m}_1} \\\\Big )- h(\\\\check{\\\\alpha }, \\\\check{m}_1)$ where we have used the crucial fact that for the remaining volume, characterized by (REF ), the functions $h(\\\\alpha , k)$ and $f(\\\\alpha , k)$ are monotonically decreasing in both arguments.\",\n \"The proof of these monotonicities can be found in App. .\",\n \"Note that the described procedure allows to rigorously confirm the inequality across a continuous region with finitely many evaluations.\",\n \"For $\\\\alpha \\\\rightarrow 1$ or $m_1 \\\\rightarrow 0$ , the value of $\\\\Gamma _{2,1}$ approaches zero, so that the above scheme cannot be used to proof the inequality for values arbitrarily close to these limits.\",\n \"However, if we restrict us to $m_1 > \\\\delta $ and $\\\\alpha < 1 - \\\\epsilon $ for finite $\\\\epsilon , \\\\delta > 0$ , the inequality $\\\\Gamma _{2,1}$ can be shown for practically all parameter combinations with little numerical effort.\",\n \"For $\\\\epsilon = \\\\delta = 10^{-3}$ , using the proposed scheme, it was checked in only a few minutes that the lower bound in Eq.\",\n \"(REF ) is larger than $10^{-4}$ everywhere in the remaining volume.\",\n \"To be numerically on the safe side, we have stopped splitting a cuboid not if the lower bound exceeded zero but set a suitable finite threshold, in this case $10^{-4}$ .\",\n \"In other words, we have stopped splitting a cuboid if its respective lower bound was $>10^{-4}$ .\",\n \"The smallest terms which occur during the arithmetic operations inside the functions $h$ and $f$ are of the order $\\\\delta ^2$ , the largest terms are of order one.\",\n \"It is therefore safe to say that numerical errors at the order of the machine accuracy are negligibly small against the margin of $10^{-4}$ and we may say that all together globally optimal 4-BPs are ruled out, for all practical parameter combinations.\",\n \"In principle, the presented scheme can be used to check the inequality $\\\\Gamma _{2,1}$ up to even smaller $\\\\epsilon $ and $\\\\delta $ .\",\n \"The Python code of the numerical scheme is made available at https://github.com/hci-unihd/BranchedOT.\"\n ],\n [\n \"Non-optimality of five- and higher-degree branchings\",\n \"In this section, we formally prove that globally optimal coupled $n$ -BPs not coincident with a terminal can be ruled out in general, given that 4-BPs are not globally optimal.\",\n \"Lemma REF states that a solution is not globally optimal if any subsolution is not globally optimal.\",\n \"It will therefore suffice to study the coupled BP as an isolated subproblem.\",\n \"Let us start by proving the following corollary about the preservation of relative optimality under edge extensions for transient V- and L-branchings (see Def.\",\n \"REF ): Corollary E.4 Consider a BOT problem with terminals $A=\\\\lbrace a_i\\\\rbrace $ and let the BP configuration $B = \\\\lbrace b_i \\\\rbrace $ be relatively optimal for a given topology $T$ .\",\n \"Let $B$ contain a transient V-branching between two terminals, say $a_0$ and $a_1$ and denote the branching point connected to $a_0$ and $a_1$ by $b_1$ .\",\n \"Otherwise, let $B$ not contain any strict L-or V-branchings.\",\n \"Let us further denote the branching point to which $b_1$ is coupled in a transient V-branching by $b_2$ , as illustrated on the left side of Fig.\",\n \"REF .\",\n \"Then, there exists a direction in which the zero length edge $(b_1, b_2)$ can be extended to finite length $l>0$ (cf.\",\n \"right side of Fig.\",\n \"REF ) such that the new BOT problem (with shifted terminal positions) is solved relatively optimally by the new BP configuration (with shifted $b_1$ ).\",\n \"Proof.\",\n \"By assumption, the BP configuration of interest $B = \\\\lbrace b_i \\\\rbrace $ contains only Y-branchings and transient V- and L-branchings.\",\n \"Then, according to Lemma REF , there exists a set of arbitrarily small displacements $\\\\delta _i$ , one for each branching point $b_i$ , so that $B + \\\\delta = \\\\lbrace b_i + \\\\delta _i \\\\rbrace $ is a non-degenerate BP configuration with arbitrarily small gradients $\\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b_i} \\\\, \\\\mathcal {C}(B+\\\\delta , A) \\\\right\\\\Vert $ .\",\n \"Note that the notation $C(B,A)$ emphasizes that the cost function also depends on the terminal positions.\",\n \"Since the BP configuration $B + \\\\delta $ is non-degenerate, any edge, in our case $(b_1,b_2)$ , can be easily expanded in length without changing any of the branching angles, assuming that the extension preserves the direction of the edge and that the BPs and terminals are moved along correspondingly.\",\n \"As the gradient $\\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b_i} \\\\, \\\\mathcal {C}$ depends only on the directions of the edges meeting at $b_i$ (i.e.\",\n \"the branching angles), the gradient is not changed by this procedure.\",\n \"Let us summarize the shifted BPs by $B_{shift}$ and the shifted terminals by $A_{shift}$ .\",\n \"Then, $\\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b_i} \\\\, \\\\mathcal {C}(B + \\\\delta , A) \\\\right\\\\Vert = \\\\left\\\\Vert \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b_i} \\\\, \\\\mathcal {C}(B_{shift} + \\\\delta , A_{shift}) \\\\right\\\\Vert $ can also be made arbitrarily small as $\\\\delta \\\\rightarrow 0$ .\",\n \"Let us prove by contradiction that $B_{shift}$ is the ROS of the BOT problem with terminals $A_{shift}$ : Assuming that a different BP configuration $B_0 \\\\ne B_{shift}$ was the ROS, there would exist a constant $\\\\Delta >0$ , so that $\\\\mathcal {C}(B_{shift} , A_{shift}) = \\\\mathcal {C}(B_0, A_{shift}) + \\\\Delta $ .\",\n \"However, since the cost is a convex function w.r.t.\",\n \"the BPs, we have $\\\\mathcal {C}(B_0, A_{shift}) \\\\ge \\\\mathcal {C}(B_{shift} + \\\\delta , A_{shift})+ \\\\langle \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b_i} \\\\, \\\\mathcal {C}(B_{shift} + \\\\delta , A_{shift}), B_0 - (B_{shift} + \\\\delta ) \\\\rangle \\\\\\\\= \\\\mathcal {C}(B_{shift} , A_{shift}) + \\\\underbrace{O(\\\\delta ) + \\\\langle \\\\nabla _{\\\\!\",\n \"\\\\!\",\n \"b_i} \\\\, \\\\mathcal {C}(B_{shift} + \\\\delta , A_{shift}), B_0 - (B_{shift} + \\\\delta ) \\\\rangle }_{=:K} \\\\, .$ As $\\\\delta \\\\rightarrow 0$ , the latter two terms summarized by $K$ can clearly be made arbitrarily small.\",\n \"In particular, there exists an $\\\\delta > 0$ , such that $\\\\vert K \\\\vert < \\\\Delta /2$ , which together with $\\\\mathcal {C}(B_{shift} , A_{shift}) = \\\\mathcal {C}(B_0, A_{shift}) + \\\\Delta $ implies that $\\\\mathcal {C}(B_0, A_{shift}) \\\\ge \\\\mathcal {C}(B_{shift}, A_{shift}) + K > \\\\mathcal {C}(B_0, A_{shift}) + \\\\Delta /2$ and we have thereby arrived at a contradiction, similarly to the reasoning in part (c) of the proof in App. .\",\n \"$$ Figure: Extension of the edge (b 1 ,b 2 )(b_1,b_2) in the direction of the pivot point p 1 p_1, preserving relative optimality.Coming back to the non-optimality of coupled $n$ -BPs, let us, for concreteness, consider a globally optimal coupled 5-BP not coincident with a terminal.\",\n \"By induction, repeating the presented argument one can then rule out all globally optimal $n$ -BPs.\",\n \"The proof is by contradiction, so let us assume a globally optimal 5-BP existed, between terminals $a_0$ , $a_1$ , $a_2$ , $a_3$ and $a_4$ .\",\n \"In terms of enclosed angles $\\\\gamma _i$ (see Fig.\",\n \"REF ), it is a necessary condition that all $\\\\gamma _i$ must exceed their respective lower bound, specified by the optimal V-branching conditions in Table REF (Section REF ).\",\n \"Again, for a 5-BP configuration to be globally optimal all possible full tree topologies must agree on the 5-BP configuration as their relative optimal solution.\",\n \"Starting from this configuration, one may continuously move one of the terminals, e.g.\",\n \"$a_0$ , such that the globally optimal coupled 5-BP starts to decouple, meaning that $a_0$ is moved until for (at least) one of the possible topologies, say $\\\\tilde{T}$ , the 5-BP configuration no longer provides the ROS.\",\n \"This can always be achieved, for instance by bringing $a_0$ sufficiently close to $a_1$ so that a Y-branching between the two terminals becomes optimal.\",\n \"Similar to the 4-branching case, the ROS of such a topology $\\\\tilde{T}$ in this moment becomes the globally optimal solution, as the other topologies are still in the 5-branching configuration which can only be globally optimal if all topologies agree on it.\",\n \"The globally optimal solution of topology $\\\\tilde{T}$ must exhibit one of the following two properties it contains one Y-branching and a coupled 4-BP or it is non-degenerate and contains only Y-branchings.\",\n \"In the first case, it would mean that a coupled 4-BP exists which is globally optimal as subgraph of a globally optimal solution (using the necessarily optimal substructure of Lem.\",\n \"REF ).\",\n \"This contradicts our assumption that 4-BPs are not globally optimal.\",\n \"Figure: Edge extension to finite length l>0l > 0 in a globally optimal and transient 5-branching with transient V-branching between a 0 a_0 and a 1 a_1.\",\n \"The resulting ROS is split into two subsolutions, as indicted by the red dashed line.Regarding the second option, we proceed as follows.\",\n \"Let us move $a_0$ back to the point in which the coupled 5-BP configuration was still globally optimal but the ROS of $\\\\tilde{T}$ is on the verge of decoupling into a non-degenerate branching configuration.\",\n \"Denote this special location of the terminals by $\\\\lbrace a_i^* \\\\rbrace $ .\",\n \"In this configuration an infinitesimal movement of $a_0$ away from $a_0^*$ can cause the ROS of $\\\\tilde{T}$ to transition from coupled 5-branching to a non-degenerate BP configuration, very much analogous to the case of the transient 4-branching illustrated in Fig.\",\n \"REF .\",\n \"This means that in this configuration all V- and L-branchings are transient in the ROS of $T$ and WLOG we choose the labeling of the terminals such that the V-branching between $a_0^*$ and $a_1^*$ is transient, cf. Fig.\",\n \"REF .\",\n \"At this point, let us split the set of all possible full tree topologies $\\\\mathcal {T}$ into the following subsets.\",\n \"The subset in which the terminals $a_0$ and $a_1$ are connected to a common branching point, say $b_1$ , is denoted by $\\\\mathcal {T}^{(0,1)}$ .\",\n \"Let us label the BP to which $b_1$ is connected in these topologies by $b_2$ .\",\n \"The ROS of all topologies $T \\\\in \\\\mathcal {T}^{(0,1)}$ for the current BOT problem contains a transient V-branching at branching point $b_1$ .\",\n \"Moreover, let us single out a specific subset in $\\\\mathcal {T}^{(0,1)}$ , defined by the following condition: $\\\\mathcal {T}^{(0,1)}_{trans} = \\\\lbrace T \\\\in \\\\mathcal {T}^{(0,1)} : \\\\text{ROS of $T$ is transient if the terminals are located at $a_i^*$} \\\\rbrace \\\\ .$ Visibly, the branching at $b_1$ appears as V-branching but it may also be seen as a Y-branching with a zero length stub.\",\n \"Let us now extend this zero length edge between $b_1$ and $b_2$ to finite length $l>0$ into the direction of the pivot point between $a_0^*$ and $a_1^*$ , as explained in the proof of Corollary REF .\",\n \"The two terminals $a_1$ and $a_2$ are shifted from $a_0^*$ and $a_1^*$ to $a_0(l)$ and $a_1(l)$ and Corollary REF guarantees that the resulting BP configuration for all $T \\\\in \\\\mathcal {T}^{(0,1)}_{trans}$ solves the new BOT problem relatively optimally.\",\n \"For an illustration of the edge extension see Fig.\",\n \"REF .\",\n \"Note that for all $T \\\\in \\\\mathcal {T}^{(0,1)}_{trans}$ this ROS is the same.\",\n \"Now split this ROS into two subsolutions as indicted in Fig.\",\n \"REF .\",\n \"This induces two subproblems and subtopologies, as described in Def.\",\n \"REF .\",\n \"We focus on the upper right subproblem, consisting of four terminals.\",\n \"Note that any topology $T^{(4)}$ on this four terminal subproblem, may be induced as subtopology by a topology $T \\\\in \\\\mathcal {T}^{(0,1)}$ .\",\n \"Let us distinguish the following two cases: a) On the four terminal subproblem the topology of the GOS, denoted by $T^{*(4)}$ , is induced by a topology $T \\\\in \\\\mathcal {T}^{(0,1)}_{trans}$ .\",\n \"Or b) $T^{*(4)}$ is induced by a topology $T \\\\in \\\\mathcal {T}^{(0,1)} \\\\setminus \\\\mathcal {T}^{(0,1)}_{trans}$ .\",\n \"In case a) the ROS of $T^{*(4)}$ is given by the right subsolution in Fig.\",\n \"REF , as the edge extension preserved the relative optimality for all $T \\\\in \\\\mathcal {T}^{(0,1)}_{trans}$ .\",\n \"Hence, the globally optimal solution on the four terminal subproblem is given by a coupled 4-BP and we have arrived at a contradiction.\",\n \"Otherwise, in case b), we do not know the ROS of $T^{*(4)}$ a priori, but it cannot be a coupled 4-branching configuration and must hence be non-degenerate.\",\n \"Crucially, it must be non-degenerate for any finite length extension $l > 0$ , even if we consider the limit of $l \\\\rightarrow 0$ .\",\n \"But this means that the ROS of $T^{*(4)}$ already transitions from a coupled 4-BP into a non-degenerate ROS, if the terminals $a_0$ and $a_1$ are perturbed infinitesimally ($l > 0$ but infinitesimal).\",\n \"Consequently, for $l=0$ the ROS of $T^{*(4)}$ is transient.\",\n \"But this means that $T^{*(4)}$ can be induced as a subtopology of a transient topology $T \\\\in \\\\mathcal {T}^{(0,1)}_{trans}$ and we have arrived yet at another contradiction.\",\n \"$$\"\n ],\n [\n \"BOT on two-dimensional Riemannian manifolds\",\n \"In the following section, we prove that the optimal branching angles for Y-shaped branchings on two-dimensional manifolds are the same as the optimal branching angles derived for BOT in the Euclidean plane.\",\n \"An outline of the proof was given in Section REF .\",\n \"The same strategy, presented below for the optimal branching angles, can be used to generalize other necessary conditions for optimal BOT solutions to manifolds, as explained in Sect.\",\n \"REF .\"\n ],\n [\n \"Optimal Y-branching on two-dimensional Riemannian manifolds\",\n \"For a Y-shaped branching at BP $b$ connecting the terminals $a_0$ , $a_1$ and $a_2$ in Euclidean plane the cost function of BOT is given by $\\\\mathcal {C}(b) = \\\\sum _i m_i^\\\\alpha \\\\left\\\\Vert a_i - b \\\\right\\\\Vert _2 \\\\ ,$ where the $m_i$ are the known edge flows.\",\n \"Let us consider a two-dimensional Riemannian manifold $\\\\mathcal {M}$ embedded into $\\\\mathbb {R}^3$ , for which the metric is induced by the standard Euclidean inner product in $\\\\mathbb {R}^3$ .\",\n \"Let the geodesic distance be denoted by $d(x,y)$ .\",\n \"The generalized cost function for 1-to-2 branching then reads $\\\\mathcal {C}_M(b) = \\\\sum _i m_i^\\\\alpha \\\\, d( a_i, b) \\\\ .$ All points $b$ and $a_i$ now lie on the manifold and are assumed to have differing positions.\",\n \"In a solution which minimizes $\\\\mathcal {C}_M$ the terminals $a_i$ are connected to $b$ via geodesics.\",\n \"For $b$ to be a valid solution to the BOT problem on the manifold, these geodesics must exist.\",\n \"We denote the geodesic which connects $b$ and $a_i$ by $v_i(\\\\lambda ) \\\\in \\\\mathcal {M}$ , parametrized by the length $\\\\lambda $ .\",\n \"The tangent space at $b$ is denoted by $T_b \\\\mathcal {M}$ .\",\n \"Furthermore, let $\\\\hat{n}_i$ be the unit tangent vectors of the geodesics $v_i(\\\\lambda )$ at the branching point $b$ , i.e.\",\n \"$\\\\hat{n}_i = \\\\partial _\\\\lambda v_i(\\\\lambda ) \\\\big \\\\vert _{\\\\lambda = 0}$ .\",\n \"WLOG, for all following considerations let us rotate and translate the manifold so that $b \\\\in \\\\mathbb {R}^3$ lies at the origin, i.e.\",\n \"$b=0$ , and that the tangent space $T_b \\\\mathcal {M}$ is equal to the $x_1$ -$x_2$ -plane of $\\\\mathcal {R}^3$ , i.e.\",\n \"$T_b \\\\mathcal {M} = \\\\mathbb {R}^2 \\\\times \\\\lbrace 0\\\\rbrace $ .\"\n ],\n [\n \"Restriction to a local subsolution on the manifold.\",\n \"Since $\\\\mathcal {M}$ is embedded into $\\\\mathbb {R}^3$ , there exists an $r > 0$ and an environment $U(r) \\\\subset \\\\mathbb {R}^3$ around $b=0$ such that the manifold $\\\\mathcal {M} \\\\cap U(r)$ can be represented as the graph of a function: $\\\\mathcal {M} \\\\cap U(r) = \\\\lbrace (x, u(x)): x \\\\in D(r) \\\\subset \\\\mathbb {R}^2 \\\\rbrace \\\\ ,$ where $u$ is a smooth, scalar function $u: D(r) \\\\rightarrow \\\\mathbb {R}$ , defined on the disk of radius $r$ , denoted by $D(r) := \\\\lbrace (x_1, x_2) \\\\in \\\\mathbb {R}^2 : \\\\left\\\\Vert (x_1,x_2)^T \\\\right\\\\Vert _2 \\\\le r \\\\rbrace $ .\",\n \"Note that due to the mentioned rotation and translation of $\\\\mathcal {M}$ , we have $u(0) = 0$ and $\\\\nabla u(0) = 0$ , where $\\\\nabla u$ denotes the gradient of $u$ .\",\n \"The existence of such a function $u$ is guaranteed by the implicit function theorem.\",\n \"A formal proof can be found in John M. Lee's book [18] in Proposition 8.24.\",\n \"Further, let us define the orthogonal projection $\\\\sigma $ from the manifold onto the $x_1$ -$x_2$ -plane as $\\\\sigma : \\\\mathcal {M} \\\\cap U(r) \\\\rightarrow D(r) \\\\times \\\\lbrace 0\\\\rbrace , \\\\ \\\\begin{pmatrix} x_1 \\\\\\\\ x_2 \\\\\\\\ u(x_1,x_2) \\\\end{pmatrix} \\\\mapsto \\\\begin{pmatrix} x_1 \\\\\\\\ x_2 \\\\\\\\ 0 \\\\end{pmatrix} \\\\ .$ WLOG, $r$ is chosen sufficiently small so that this projection is bijective.\",\n \"Now, Taylor's theorem states that $u$ can be approximated by the following expansion around $x = 0$ : $ u(x) = \\\\underbrace{u(0)}_{= \\\\ 0} + \\\\langle \\\\underbrace{\\\\nabla u(0)}_{= \\\\ 0}, x \\\\rangle + O(\\\\left\\\\Vert x \\\\right\\\\Vert _2^2) \\\\in O(r^2) \\\\ ,$ where $\\\\langle \\\\cdot , \\\\cdot \\\\rangle $ is the standard Euclidean inner product and we have introduced the Big-O notation.\",\n \"A term is of order $O(r^2)$ if it goes to zero for $r \\\\rightarrow 0$ at least as fast as $r^2$ , or more formally: $p(r) \\\\in O(q(r)) \\\\ \\\\Leftrightarrow \\\\ \\\\lim _{r \\\\rightarrow 0} \\\\frac{p(r)}{q(r)} = c$ for some finite constant $c$ .\",\n \"Consequently, a point $a = (x,u(x))$ in $\\\\mathcal {M} \\\\cap U(r)$ and its projection onto the plane $\\\\sigma (a)$ agree to first order, i.e.\",\n \"$ \\\\left\\\\Vert \\\\sigma (a) - a \\\\right\\\\Vert _2 = u(x) \\\\in O(r^2) \\\\ .$ One of the key ingredients when transferring BOT problems from two-dimensional surfaces to the tangent plane $T_b \\\\mathcal {M}$ is the following Lemma about the difference between the Euclidean distance and the geodesic distance: Lemma F.1 (Relation between geodesic and Euclidean distance) Let $r$ be a small radius, which characterizes the environment $U(r)$ around the origin $b = 0$ on a two-dimensional Riemannian manifold as described above.\",\n \"Let $a \\\\in \\\\mathcal {M} \\\\cap U$ be a point in this environment, located at $a = (x,u(x)) \\\\in \\\\mathbb {R}^3$ for some $x \\\\in D(r)$ and $u$ as above.\",\n \"Then, the geodesic distance can be expressed through the Euclidean distance as $d(a,b) = \\\\left\\\\Vert a - b \\\\right\\\\Vert _2 + O(r^3) \\\\ .$ Proof.\",\n \"Let us first note that $d(a,b) \\\\ge \\\\left\\\\Vert a - b \\\\right\\\\Vert _2$ .\",\n \"It is hence sufficient to show that $d(a,b) \\\\le \\\\left\\\\Vert a - b \\\\right\\\\Vert _2 + O(r^3)$ .\",\n \"Let us consider the following curve $\\\\gamma (t) = (tx, u(tx))$ on the manifold $\\\\mathcal {M}$ , which for $t \\\\in [0,1]$ connects $a = (x,u(x))$ and $b = 0$ .\",\n \"In general, $\\\\gamma (t)$ is not a geodesic between $a$ and $b$ .\",\n \"Thus, the length of $\\\\gamma (t)$ provides an upper bound to $d(a,b)$ .\",\n \"For the calculation of the length we use that $\\\\gamma \\\\hspace{0.85355pt} ^{\\\\prime }(t) = (x, \\\\langle \\\\nabla u(tx), x \\\\rangle )$ and simply integrate $\\\\left\\\\Vert \\\\gamma \\\\hspace{0.85355pt} ^{\\\\prime }(t) \\\\right\\\\Vert _2$ along the curve in $\\\\mathbb {R}^3$ , since the metric of the embedding is induced by the standard Euclidean inner product in $\\\\mathbb {R}^3$ : $\\\\frac{d(a,b)}{\\\\left\\\\Vert a - b \\\\right\\\\Vert _2} &\\\\le \\\\frac{1}{\\\\sqrt{\\\\left\\\\Vert x \\\\right\\\\Vert ^2_2 + u^2(x)}} \\\\int _0^1 \\\\left\\\\Vert \\\\gamma \\\\hspace{0.85355pt} ^{\\\\prime }(t) \\\\right\\\\Vert _2 \\\\mathrm {d} t \\\\\\\\&\\\\le \\\\frac{1}{\\\\left\\\\Vert x \\\\right\\\\Vert _2} \\\\int _0^1 \\\\sqrt{\\\\left\\\\Vert x \\\\right\\\\Vert ^2_2 + \\\\vert \\\\langle \\\\nabla u(tx), x \\\\rangle \\\\vert ^2} \\\\ \\\\mathrm {d} t \\\\\\\\&= \\\\int _0^1 \\\\sqrt{1 + \\\\left| \\\\Big \\\\langle \\\\nabla u(tx), \\\\frac{x}{\\\\left\\\\Vert x \\\\right\\\\Vert _2} \\\\Big \\\\rangle \\\\right|^2} \\\\ \\\\mathrm {d}t \\\\\\\\&\\\\le 1 + \\\\frac{1}{2} \\\\int _0^1 \\\\left| \\\\Big \\\\langle \\\\nabla u(tx), \\\\frac{x}{\\\\left\\\\Vert x \\\\right\\\\Vert _2} \\\\Big \\\\rangle \\\\right|^2 \\\\mathrm {d}t \\\\\\\\&\\\\le 1 + \\\\frac{1}{2} \\\\int _0^1 \\\\left\\\\Vert \\\\nabla u(tx) \\\\right\\\\Vert _2^2 \\\\ \\\\mathrm {d}t \\\\ .$ For the last step we have used the Cauchy-Schwarz inequality.\",\n \"Furthermore, we have used that $\\\\sqrt{1 + x^2} \\\\le 1 + x^2 / 2$ .\",\n \"The only thing left to show is that the remaining integral is $O(r^2)$ , as $\\\\left\\\\Vert a - b \\\\right\\\\Vert = ( \\\\left\\\\Vert x \\\\right\\\\Vert ^2_2 + u^2(x) )^{1/2} \\\\in O(r)$ , since $\\\\left\\\\Vert x \\\\right\\\\Vert _2 \\\\le r$ and $u(x) \\\\in O(r^2)$, cf. Eq.\",\n \"(REF ).\",\n \"For that, let us consider the Taylor expansion of the $i$ -th component of $\\\\nabla u(tx)$ around 0.\",\n \"For $i=1,2$ one has $\\\\partial _i \\\\, u(tx) = \\\\underbrace{\\\\partial _i \\\\, u(0)}_{= \\\\ 0 } + \\\\partial _j \\\\partial _i \\\\, u(0) \\\\ t x_j + O(r^2) \\\\ ,$ where the sum in $j$ is implicit and runs over both indices $j=1,2$ .\",\n \"And thus: $\\\\frac{1}{2} \\\\int _0^1 \\\\left\\\\Vert \\\\nabla u(tx) \\\\right\\\\Vert _2^2 \\\\ \\\\mathrm {d}t = \\\\frac{1}{6} \\\\, \\\\partial _j \\\\partial _i \\\\, u(0) \\\\ x_j \\\\ \\\\partial _k \\\\partial _i \\\\, u(0) \\\\ x_k + O(r^2) \\\\ ,$ where again the sum over all index pairs is implied.\",\n \"Since for all components $x_i$ we have $x_i \\\\le r$ , this completes the proof.\",\n \"$$\"\n ],\n [\n \"Projection of a subproblem onto the tangent plane.\",\n \"Coming back to the 1-to-2 BOT problem on the manifold, let us project the geodesics $v_i$ onto the $x_1$ -$x_2$ -plane using the orthogonal projection $\\\\sigma $ from above, in order to transfer the BOT problem from the manifold onto the tangent plane.\",\n \"WLOG, we choose the orientation of the $x_1$ and $x_2$ axis such that $\\\\hat{n}_0$ points along the $x_1$ -axis.\",\n \"Then, one may easily check that the implicit function theorem guarantees that for sufficiently small $r$ the projected geodesic $\\\\sigma (v_0)$ can be represented by a graph $(x_1, w(x_1))$ with smooth $w: (-\\\\epsilon ,r+\\\\epsilon ) \\\\rightarrow \\\\mathbb {R}$ .\",\n \"Note, that the projected geodesic $\\\\sigma (v_0)$ may be considered on the open interval with $\\\\epsilon >0$ such that all derivatives with respect to $x_1 \\\\in [0,r]$ are well-defined.\",\n \"Due to our special orientation of the $x_1$ -$x_2$ -plane, we have $w(0)=0$ and $w^{\\\\prime }(0) = 0$ .\",\n \"Let us now define the following two points inside the tangent plane: Define $\\\\hat{a}_0$ as the point which lies in the direction of $\\\\hat{n}_0$ at a distance $r$ away from the origin, i.e.\",\n \"$\\\\hat{a}_0 = r \\\\cdot \\\\hat{n}_0 = (r,0)$ .\",\n \"Similarly, define $\\\\hat{a}_i = r \\\\cdot \\\\hat{n}_i$ .\",\n \"These will be the three terminals of the BOT subproblem of interest in the Euclidean plane.\",\n \"For an illustration see Fig.\",\n \"REF .\",\n \"Define $a_{\\\\perp ,0} = (r, w(r))$ as special point on the projected geodesic $\\\\sigma (v_0)$ .\",\n \"From an analogous construction using the projected geodesics $\\\\sigma (v_1)$ and $\\\\sigma (v_2)$ , we obtain the $a_{\\\\perp ,i}$ also for $i=1,2$ .\",\n \"No explicit representation of these will be necessary.\",\n \"What is however important is that the two defined points agree up to linear order, in the sense that $\\\\left\\\\Vert \\\\hat{a}_i - a_{\\\\perp ,i} \\\\right\\\\Vert _2 \\\\in O(r^2)$ for all $i=0,1,2$ .\",\n \"This is shown for $i=0$ , by Taylor expansion of $w(x_1)$ around $x_1 = 0$ , but holds of course equally true for $i=1,2$ : $ \\\\left\\\\Vert \\\\hat{a}_0 - a_{\\\\perp ,0} \\\\right\\\\Vert _2 = w(r)= \\\\underbrace{w(0)}_{= \\\\ 0} + \\\\underbrace{w^{\\\\prime }(0)}_{= \\\\ 0} \\\\, r + O(r^2) \\\\in O(r^2) .$ Note that since the points $a_{\\\\perp ,i}$ lie on the projected geodesics, after an inverse projection, the $\\\\sigma ^{-1}(a_{\\\\perp ,i})$ mark special points on the unprojected geodesics $v_i \\\\in \\\\mathcal {M}$ .\",\n \"These will be the terminals of the subproblem of interest on the manifold.\",\n \"Combining Eq.\",\n \"(REF ) and (REF ), we can relate $\\\\sigma ^{-1}(a_{\\\\perp , i}) \\\\in \\\\mathcal {M}$ to the terminals $\\\\hat{a}_i$ of the BOT problem of interest in the plane by $ \\\\left\\\\Vert \\\\sigma ^{-1}(a_{\\\\perp , i}) - \\\\hat{a}_i \\\\right\\\\Vert _2 \\\\in O(r^2) \\\\ ,$ Note that all steps above can be repeated for radii smaller than the chosen $r$ .\",\n \"In the following we will decrease the scale $r$ of the two problems and therefore indicate the $r$ -dependence in $\\\\sigma ^{-1}(a_{\\\\perp , i}(r))$ and $\\\\hat{a}_i(r)$ explicitly.\",\n \"Figure: The tangent vectors of the geodesics (dotted) in bb define the position of the terminals a ^ i \\\\hat{a}_i of the subproblem on the flat disk D(r)⊂T b ℳD(r) \\\\subset T_b \\\\mathcal {M}.\"\n ],\n [\n \"Improved subsolutions on the manifold by projection from the tangent space.\",\n \"Now, follows the main line of arguments to show that $b$ does not connect $a_i \\\\in \\\\mathcal {M}$ at minimal cost if the geodesic angles in the tangent space $T_b \\\\mathcal {M}$ are different from the optimal angles in the Euclidean plane.\",\n \"We start by considering the subproblem on the manifold with terminals $\\\\sigma ^{-1}(a_{\\\\perp , i}) \\\\in \\\\mathcal {M}$ .\",\n \"The generalized cost function of this subproblem reads: $\\\\mathcal {C}_M(b) &= \\\\sum _i m_i^\\\\alpha \\\\, d(\\\\sigma ^{-1}(a_{\\\\perp ,i}(r)), b) \\\\ .$ We can now use Eq.\",\n \"(REF ) to express the terminals on the manifold through the terminals $\\\\hat{a}_i$ in the tangent plane and express the geodesic distance in terms of the Euclidean distance, based on Lem.\",\n \"REF .\",\n \"All additional terms are at least of order $O(r^2)$ and we have $ \\\\mathcal {C}_M(b) = \\\\underbrace{\\\\sum _i m_i^\\\\alpha \\\\left\\\\Vert \\\\hat{a}_i(r) - b \\\\right\\\\Vert _2}_{=: \\\\ \\\\mathcal {C}_b(r)} + \\\\ O(r^2) \\\\ .$ Crucially, with $\\\\mathcal {C}_b(r)$ we have arrived at the cost of the BOT problem which purely lives on the tangent space.\",\n \"It is the cost of the solution in which $b$ is connected via straight lines to the terminals $\\\\hat{a}_i(r) = r \\\\cdot \\\\hat{n}_i$ at a distance $r$ away from $b$ , as shown in Fig.\",\n \"REF .\",\n \"By assumption, this solution is not relatively optimally since the angles (or equivalently the directions $\\\\hat{n}_i$ ) are assumed to deviate from the optimal angles in the Euclidean plane.\",\n \"This means that a branching point $b^* \\\\in D(r)\\\\times \\\\lbrace 0 \\\\rbrace $ exists which provides a better solution with cost denoted by $\\\\mathcal {C}_{b^*}(r) < \\\\mathcal {C}_b(r)$ .\",\n \"From the definition of $\\\\hat{a}_i = r \\\\cdot \\\\hat{n}_i$ , we see that two BOT problems within disks of different radii $D(r)$ and $D(\\\\nu \\\\cdot r)$ are related simply by rescaling the coordinates.\",\n \"Under rescaling of coordinates the Euclidean distance between points and consequently also the cost functions changes proportionally, i.e.\",\n \"$\\\\mathcal {C} \\\\rightarrow \\\\nu \\\\mathcal {C}$ .\",\n \"Thus, there exist non-negative constants $M_1$ and $M_2$ , so that $\\\\mathcal {C}_b(r) = M_1 \\\\cdot r$ and $\\\\mathcal {C}_{b^*}(r) = M_2 \\\\cdot r$ .\",\n \"Let us distinguish the following two cases: a) $\\\\ \\\\ \\\\ \\\\mathcal {C}_{b^*}(r) = \\\\sum _i m_i^\\\\alpha \\\\left\\\\Vert \\\\hat{a}_i(r) - b^* \\\\right\\\\Vert _2 = 0$ .\",\n \"This special case looks unusual, but is in principle possible, e.g.\",\n \"if all terminals $a_i$ and $b$ lie on a common geodesic.\",\n \"In the case, where $C_{b^*}(r) = 0$ , we project $b^*$ onto the manifold using $\\\\sigma ^{-1}$ and go backwards in the above steps.\",\n \"We find that $0 = \\\\sum _i m_i^\\\\alpha \\\\left\\\\Vert \\\\hat{a}_i(r) - b^* \\\\right\\\\Vert _2 =\\\\underbrace{\\\\sum _i m_i^\\\\alpha \\\\, d( \\\\sigma ^{-1}(a_{\\\\perp ,i}(r)) , \\\\sigma ^{-1}(b^*))}_{= \\\\ \\\\mathcal {C}_M(\\\\sigma ^{-1}(b^*)))} + O(r^2) \\\\, .$ Clearly, a sufficiently small $r > 0$ exists so that the terms contained in $O(r^2)$ are much smaller than $M_1 \\\\cdot r$ , so that $\\\\mathcal {C}_M(\\\\sigma ^{-1}(b^*))) = 0 + O(r^2) < M_1 \\\\cdot r + O(r^2) = \\\\mathcal {C}_M(b) \\\\ .$ This proves that the projection of $b^*$ onto the manifold provides a cheaper cost solution on the manifold than $b$ .\",\n \"b) $\\\\ \\\\ \\\\ \\\\mathcal {C}_{b^*}(r) > 0$ , and thus $M_2 >0$ .\",\n \"In this case, let us write the ratio of the two costs as $ \\\\mathcal {C}_b(r)/ \\\\mathcal {C}_{b^*}(r) = M_1 / M_2 =: 1 + \\\\kappa $ for some $\\\\kappa > 0$ .\",\n \"Again, $b^*$ and its projection onto the manifold agree to first order, that is $\\\\left\\\\Vert \\\\sigma ^{-1}(b^*) - b^* \\\\right\\\\Vert \\\\in O(r^2)$ .\",\n \"We now show that $\\\\sigma ^{-1}(b^*)$ provides a better solution to the BOT problem on the manifold than $b$ .\",\n \"We proceed from Eq.\",\n \"(REF ), using that $\\\\mathcal {C}_{b^*}(r)$ is linear in $r$ : $\\\\mathcal {C}_M(b) &= C_b(r) + O(r^2) = (1 + \\\\kappa ) \\\\, \\\\mathcal {C}_{b^*}(r) + O(r^2) \\\\nonumber \\\\\\\\&= \\\\sum _i m_i^\\\\alpha \\\\left\\\\Vert \\\\hat{a}_i(r) - b^* \\\\right\\\\Vert _2 + \\\\kappa M_2 \\\\cdot r + O(r^2) \\\\nonumber \\\\\\\\&= \\\\underbrace{\\\\sum _i m_i^\\\\alpha \\\\, d( \\\\sigma ^{-1}(a_{\\\\perp , i}(r)) , \\\\sigma ^{-1}(b^*))}_{= \\\\ \\\\mathcal {C}_M(\\\\sigma ^{-1}(b^*))} + \\\\kappa M_2 \\\\cdot r + O(r^2) \\\\ .$ In the last step, we have projected inversely onto the manifold and replaced the Euclidean distance by the geodesic distance (all with differences at least of $O(r^2)$ ).\",\n \"We have therefore arrived at the cost of the new solution on the manifold.\",\n \"Since the term $\\\\kappa M_2 \\\\cdot r$ is positive, we conclude that $r$ may always be chosen sufficiently small, so that the linear term dominates over higher-order terms and we have: $\\\\mathcal {C}_M(\\\\sigma ^{-1}(b^*)) < \\\\mathcal {C}_M(b) \\\\ .$ Together with Lem.\",\n \"REF on the necessarily optimal substructure of all subsolutions, this concludes the proof that any optimally placed branching point on a two-dimensional embedded Riemannian manifold must exhibit the same optimal branching angles as in the Euclidean plane.$$\"\n ],\n [\n \"Other local properties of optimal BOT solutions on manifolds\",\n \"The logic of the proof outlined above can be easily extended to the conditions for optimal V- and L-branching.\",\n \"For BOT problems in the Euclidean plane, Table REF lists these conditions under which the optimal BP position in a 1-to-2 branching coincides with one of the terminals.\",\n \"Let us now consider a 1-to-2 BOT solution on a two-dimensional manifold in which $b$ is located at the terminal $a_0$ and is connected to the terminals $a_1$ and $a_2$ via the geodesics $v_1$ and $v_2$ respectively.\",\n \"Imagine that in the tangent plane $T_b \\\\mathcal {M}$ the angle enclosed by these geodesics does not fulfill the optimal V-branching criterion.\",\n \"In this case, one may again consider a sufficiently small disk of radius $r$ around $b = a_0$ in the tangent space $T_b \\\\mathcal {M}$ and project the corresponding subproblem from the manifold onto this disk.\",\n \"In the Euclidean plane, an improved BP location $b^*$ must exists, since $b = a_0$ is the optimal solution if and only if the V-branching condition is fulfilled.\",\n \"All arguments about the scaling of the cost improvement apply as described above and the projection $b^*$ back to the manifold will provide an improved solution to the subproblem on the manifold if $r$ is chosen sufficiently small.\",\n \"Even more so, the same reasoning can also be applied to generalize our results regarding the non-optimality of higher-degree branchings.\",\n \"Given a coupled $n$ -BP at position $b$ not coincident with a terminal, one may again consider a sufficiently small region around $b$ on the manifold and project the corresponding subsolution onto $T_b \\\\mathcal {M}$ .\",\n \"Exactly analogous to the previous arguments, improving the topology locally in the plane and projecting back to the manifold eventually results in an improved solution on the manifold.\"\n ],\n [\n \"The practical side of BOT on manifolds\",\n \"Although much of the theory generalizes nicely to Riemannian manifolds, the generalization of the practical algorithms is highly non-trivial.\",\n \"Unlike in Euclidean space, realizing the optimal branching angles is a necessary but no longer sufficient condition for relatively optimal solutions.\",\n \"For instance, on the sphere the meridians of three terminals located in the southern hemisphere at longitudes $0^\\\\circ $ , $120^\\\\circ $ , $240^\\\\circ $ will intersect at both poles at angles of $120^\\\\circ $ , which is the optimal angle for $\\\\alpha =0$ .\",\n \"Nonetheless, only the south pole is the optimal branching point.\",\n \"In essence, the geometry optimization aims to assign simultaneously to each branching point the coordinates of the weighted geometric median of its neighbors, a problem that is considered in [8].\",\n \"The topology optimization presented in Sect.\",\n \"REF could be easily generalized to manifolds if the geodesic distance can be computed.\",\n \"Due to these obstacles, previous works of the Steiner Tree problem ($\\\\alpha = 1$ ) have focused mostly on the sphere as important special case [7].\"\n ],\n [\n \"Hardware and code availability.\",\n \"Python code for all experiments can be found at https://github.com/hci-unihd/BranchedOT.\",\n \"For the different experiments, a single machine with 56 CPUs (Intel(R) Xeon(R) CPU E5-2660 v4 @ 2.00GHz) and 256GB RAM was used.\",\n \"Execution times for all described experiments lie (at most) in the order of hours.\",\n \"A more detailed estimate can be obtained from the performance statistics reported in Fig.\",\n \"REF and Fig.\",\n \"REF .\"\n ],\n [\n \"Geometric construction of relatively optimal solutions for BOT with multiple sources\",\n \"In Sect.\",\n \"REF we have presented the the exact geometric construction of relatively optimal solutions based on [2], [9] and its generalization to the case of BOT with multiple sources.\",\n \"We have thereby solved the open problem 15.11 in [2], for which an example is illustrated in Fig.\",\n \"REF .\",\n \"Figure REF shows a simple BOT problem with a chosen topology for which no root node can be chosen such that all branchings are symmetric (see Sect.\",\n \"REF ).\",\n \"Choosing for instance the left source as a root node, the branching at $b_1$ is symmetric whereas the one at $b_2$ is asymmetric.\",\n \"However, after having solved the asymmetric branching case analogously to the symmetric one, the relatively optimal solution can now be constructed geometrically as shown in Fig.\",\n \"REF .\",\n \"More examples of the geometric construction for a given topology applied BOT problems with multiple sources are shown in Fig.\",\n \"REF .\",\n \"Figure: Simple setup with symmetric branching and asymmetric branching, see Sect.\",\n \".Figure: Examples for geometric construction of relatively optimal solutions with fixed topology for BOT problems with multiple sources.\",\n \"The construction is based on the optimal branching angles and uses one pivot circle and pivot point (light blue) per branching point.\",\n \"Sources are shown in red, sinks in blue.\"\n ],\n [\n \"Continuity of the optimal BP configuration.\",\n \"The limitations of the geometric construction above are discussed in Sect.\",\n \"REF .\",\n \"Still, it forms the basis for our theoretical arguments in the paper.\",\n \"Moreover, from the construction of a single branching point based on the branching angles (see Fig.\",\n \"REF ), it can be seen that its optimal position changes continuously w.r.t.\",\n \"the neighbor positions, the edge flows or $\\\\alpha $ .\",\n \"Namely, the Y-, V- and L- branching change continuously as the source $a_0$ is moved around.\",\n \"In addition, the optimal branching angles are given by continuous functions of $k$ and $\\\\alpha $ (cf. Eq.\",\n \"(REF )), making the construction of the pivot point and pivot circle continuous.\",\n \"By transitivity and based on the optimal substructure in Lem.\",\n \"REF , the continuity generalizes also to the construction of larger ROS for a given topology.\"\n ],\n [\n \"Numerical algorithm for geometry optimization\",\n \"In this section, we provide theoretical details and practical experiments for the numerical geometry optimization presented in Sect.\",\n \"REF .\",\n \"We have generalized this approached from the context of the ESTP [26] to BOT.\",\n \"For a given tree topology $T$ , the BP configuration is optimized by minimizing the following cost function: $\\\\mathcal {C}(X) = \\\\sum _{(i,j) \\\\, \\\\in \\\\, T} m_{ij}^\\\\alpha \\\\left\\\\Vert x_i - x_j \\\\right\\\\Vert _2 ,$ where the $x_i$ for $1 \\\\le i \\\\le n$ are fixed terminals and the $x_i$ for $n+1 \\\\le i \\\\le n+m$ denote the variable branching point positions.\",\n \"All coordinates are collectively summarized by $X$ .\",\n \"Starting from a non-optimal, non-degenerate BP configuration denoted $X^{(0)}$ , e.g.\",\n \"from a random initialization of the branching points, the algorithm iteratively solves the following linear system of equations $ x_i^{(k + 1)} = \\\\!\",\n \"\\\\!\",\n \"\\\\sum _{j \\\\, : \\\\, (i,j) \\\\in T} m_{ij}^\\\\alpha \\\\frac{x_j^{(k+1)}}{\\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert }\\\\ \\\\Bigg / \\\\sum _{j \\\\, : \\\\, (i,j) \\\\in T} \\\\frac{m_{ij}^\\\\alpha }{\\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert }, \\\\hspace{11.38092pt}\\\\text{for } n+1 \\\\le i \\\\le n+m.$ In essence, this is an iteratively reweighted least squares (IRLS) approach [4].\",\n \"To see this, let us rewrite the cost function in Eq.\",\n \"(REF ) into pseudo-quadratic from: $\\\\mathcal {C}(X) = \\\\sum _{(i,j) \\\\, \\\\in \\\\, T} m_{ij}^\\\\alpha \\\\vert x_i - x_j \\\\vert ^{(\\\\beta -2) + 2} = \\\\sum _{(i,j) \\\\, \\\\in \\\\, T} \\\\underbrace{m_{ij}^\\\\alpha \\\\vert x_i - x_j \\\\vert ^{\\\\beta -2}}_{=: w_{ij}(X)} \\\\vert x_i - x_j \\\\vert ^2 \\\\, ,$ with $\\\\beta = 1$ for BOT.\",\n \"Clearly, the weights $w_{ij}$ depend on the branching point positions.\",\n \"However, the idea of IRLS is to insert the coordinates $X^{(k)} = \\\\lbrace x_i^{(k)} \\\\rbrace $ at $k$ -th iteration into $w_{ij}$ to obtain a truly quadratic form: $Q^{(k)}(X) &= w_{ij}(X^{(k)}) \\\\vert x_i - x_j \\\\vert ^2 =\\\\sum _{ (i,j) \\\\, \\\\in \\\\, T} m_{ij}^\\\\alpha \\\\frac{ \\\\vert x_i - x_j \\\\vert ^2}{ \\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert } \\\\, ,$ where we have plugged in $w_{ij}(X^{(k)})$ and $\\\\beta = 1$ .\",\n \"Indeed, minimizing this quadratic form yields $X^{(k+1)}$ as in Eq.\",\n \"(REF ).\",\n \"The updated coordinates $X^{(k+1)}$ are then plugged into $w_{ij}(X)$ during the next iteration.\",\n \"This iterative updating of the weights $w_{ij}$ gives IRLS its name.\",\n \"[26] proved in detail that this iterative solver converges to the minimum cost BP configuration.\",\n \"The reasoning in [26] is based on the following key argument: As $X^{(k+1)}$ is the minimizer of $Q^{(k)}(X)$ , surely $Q^{(k)}(X^{(k)}) \\\\ge Q^{(k)}(X^{(k+1)})$ .\",\n \"Together with the fact that $\\\\mathcal {C} (X^{(k)}) = Q^{(k)}(X^{(k)})$ , we have $\\\\mathcal {C}(X^{(k)}) &= Q^{(k)}(X^{(k)}) \\\\ge Q^{(k)}(X^{(k+1)}) \\\\\\\\&= \\\\sum _{ (i,j) \\\\, \\\\in \\\\, T: } m_{ij}^\\\\alpha \\\\frac{\\\\big ( \\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert +\\\\vert x_i^{(k+1)} - x_j^{(k+1)} \\\\vert - \\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert \\\\big )^2}{ \\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert } \\\\\\\\&= \\\\mathcal {C}(X^{(k)}) + 2 [\\\\mathcal {C}(X^{(k+1)}) - \\\\mathcal {C}(X^{(k)}) ] \\\\\\\\&\\\\hspace{54.06006pt} + \\\\sum _{(i,j) \\\\, \\\\in \\\\, T} m_{ij}^\\\\alpha \\\\frac{ \\\\big ( \\\\vert x_i^{(k+1)} - x_j^{(k+1)} \\\\vert - \\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert \\\\big )^2}{ \\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert } \\\\ .$ Since the sum in the last expression is clearly non-negative, the inequality above implies that $\\\\mathcal {C}(X^{(k)}) \\\\ge \\\\mathcal {C}(X^{(k+1)})$ .\",\n \"This means that the cost of the BP configuration decreases with each iteration which, as shown in [26], implies that the iterations defined by Eq.\",\n \"(REF ) converge to an absolute minimum of $\\\\mathcal {C}(X)$ .\",\n \"As can be seen from the above derivation, the edge flows $m_{ij}^\\\\alpha $ are constant coefficients which do not complicate the considerations in [26].\",\n \"Consequently, all arguments presented there can be directly transferred from ESTP to BOT.\",\n \"For a detailed discussion on the complexity of the algorithm and suitable convergence criteria, we refer the reader to the work of Smith and only briefly state the results here.\",\n \"Based on an analytically tractable example, Smith claims that the algorithm requires at most $O(N/\\\\epsilon )$ iterations to converge to a solution of the ESTP whose cost is within $\\\\epsilon $ of the optimal cost.\",\n \"As a convergence criterion, Smith suggests to stop the iteration when the angles at all branching points are sufficiently close to the optimal angle conditions.\",\n \"In order to be able to apply the algorithm also to trees with higher-degree branching points where the optimal angle conditions are a necessary but not sufficient condition for the cost minimum, the experiments in this paper use a different criterion.\",\n \"The algorithm is considered to have converged if from one iteration to the next the relative cost improvement $\\\\big (\\\\mathcal {C}(X^{(k)}) - \\\\mathcal {C}(X^{(k+1)}) \\\\big ) / \\\\mathcal {C}(X^{(k+1)})$ has dropped below a certain threshold.\",\n \"The BP optimization routine for a given tree topology is summarized in Alg.\",\n \"REF .\",\n \"Note that, in practice, the denominators $\\\\vert x_i^{(k)} - x_j^{(k)} \\\\vert $ in Eq.\",\n \"(REF ) are clipped to $10^{-7}$ to avoid numerical instabilities.\",\n \"BP optimization routine Input: threshold $\\\\eta $ , tree topology $T$ , BOT problem (terminal positions $x_{1:n}$ , supplies/demands $\\\\mu $ , $\\\\alpha $ ) Output: numerical minimal cost BP configuration [1] $F \\\\leftarrow $ get_all_edge_flows($T,\\\\mu $ ) $\\\\triangleright \\\\ $ uniquely determined from flow constraints $x_{n+1:n+m} \\\\leftarrow $ randomly initialized.\",\n \"$C_{old} \\\\leftarrow \\\\infty $ $C \\\\leftarrow $ BOT_cost$(T,x_{1:n+m},F, \\\\alpha )$ $\\\\frac{\\\\mathcal {C}_{old} - \\\\mathcal {C}}{\\\\mathcal {C}} > \\\\eta $ $C_{old} \\\\leftarrow C$ $x_{1:n+m} \\\\leftarrow $ update_BPs($x_{1:n+m},F, \\\\alpha $ ) $\\\\triangleright \\\\ $ solves linear system in Eq.\",\n \"(REF ) $C \\\\leftarrow $ BOT_cost$(T,x_{1:n+m},F, \\\\alpha )$ return $x_{1:n+m}$\"\n ],\n [\n \"BOT with costs scaling non-linearly with the edge length.\",\n \"Let us briefly consider BOT with a modified cost function of $\\\\mathcal {C}(X) = \\\\sum _{(i,j) \\\\, \\\\in \\\\, T} m_{ij}^\\\\alpha \\\\left\\\\Vert x_i - x_j \\\\right\\\\Vert ^\\\\beta _2 \\\\,$ including one additional parameter $\\\\beta $ , regulating the scaling of the cost function w.r.t.\",\n \"edge lengths.\",\n \"For $\\\\beta \\\\ge 1$ the cost function as a function of the branching points is still convex, as $x \\\\mapsto x^\\\\beta $ is convex and increasing and the Euclidean norm is convex.\",\n \"Thus, it has a unique minimum.\",\n \"As can be seen from Eq.\",\n \"(REF ), the IRLS scheme completely absorbs the $\\\\beta $ into the weights $w_{ij}$ .\",\n \"Consequently, the geometry optimization as presented here is readily applicable also to the modified cost in Eq.\",\n \"(REF ).\",\n \"However, for $\\\\beta >1$ we are not aware of any theoretical convergence guarantee to the minimum, therefore, further investigation is required.\",\n \"Though, in the special case of $\\\\beta =2$ , it is clear that the geometry optimization can be solved to global optimality in just a single iteration.\"\n ],\n [\n \"Performance test of numerical BP optimization.\",\n \"In order to evaluate how long the BP optimization for a given topology takes, we have randomly generated 1000 BOT problems for different number of terminals using Alg.\",\n \"REF .\",\n \"For each of these problems, we have applied the BP optimization routine, given a uniformly sampled full tree topology.\",\n \"Regarding the generation of problem setups, note that BOT solutions and problems are invariant under rescaling of the total demand and supply as well as under global rescaling of the coordinates, see Eq.\",\n \"(REF ).\",\n \"WLOG, these scales are chosen to be 1.\",\n \"Figure REF shows the average runtime in seconds plotted against the number of terminals.\",\n \"For BOT problems with 10 to 1000 terminals our efficient C++ implementation of the geometry optimization takes just a fraction of a second.\",\n \"The relatively large error bars, indicating the standard deviation, are not due to an insufficient sample size but due to the natural run time variability for different problems that exists independent of the $n$ -dependence.\",\n \"For instance, plotting the average runtime of all problems with $\\\\alpha \\\\le 0.5$ and $\\\\alpha > 0.5$ separately reveals that the optimization on average requires more time for problems with higher $\\\\alpha $ .\",\n \"The reason for this was not investigated further, but we suspect that for larger $\\\\alpha $ on average more V- and L-branchings occur which may require more iterations than Y-branchings for convergence.\",\n \"For this and all following experiments we chose the convergence threshold $\\\\eta $ to be $10^{-6}$ .\",\n \"Figure: Average runtime of BP optimization routine applied to each 1000 random BOT problems of different size.\",\n \"Average of all problems with α≤0.5\\\\alpha \\\\le 0.5 in orange (dotted) and α>0.5\\\\alpha > 0.5 in green (dashed).\"\n ],\n [\n \"Scaling of the geometry optimization\",\n \"A single iteration of the geometry optimization, i.e.\",\n \"solving the linear system defined by Eq.\",\n \"(REF ) once, takes $O(nd)$ operations for a problem with $n$ terminals in $d$ spatial dimensions.\",\n \"The elimination scheme used in our efficient C++ implementation is based on the “elimination on leaves of a tree” found in [26].\",\n \"From Eq.\",\n \"(REF ), one can easily see that the geometry optimization parallelizes over the spatial dimensions, as the linear system of the form $Ax = b$ shares the same matrix $A$ across the different dimensions and only the $b$ is different.\",\n \"Although a single iteration is of order $O(nd)$ , it is a priori not clear how many iterations are required until convergence is reached.\",\n \"Paralleling the setup of Fig.\",\n \"REF , we have conducted an experiment where we report the number of iterations, for a batch of BOT problems.\",\n \"We performed the same experiment in Euclidean space of different dimensions ($d \\\\in \\\\lbrace 3,4,5,10,30,100 \\\\rbrace $ ).\",\n \"Figure REF shows the number of iterations until convergence plotted against the number of terminals.\",\n \"The plot suggests that, for the investigated regime of $n$ and $d$ , the number of iterations on average scales like $\\\\log (n)$ .\",\n \"However, this is merely an empirical observation and theoretical investigations are an interesting subject for future research.\",\n \"Figure: Average number of iterations required by the BP optimization routine until convergence for each 1000 random BOT problems of different sizes nn and different dimensionality dd.\",\n \"The x-axis is in log-scale.\",\n \"The plot suggests that, for the investigated regime of nn and dd, the number of iterations on average scales like log(n)\\\\log (n), though one cannot generalize this statement.Random BOT problem generation Input: number of terminals $n$ Output: BOT problem with $n$ terminals located in $[0,1]\\\\times [0,1]$ , total supply and demand equal to 1 [1] $\\\\alpha \\\\sim $ Uniform($[0,1]$ ) $\\\\triangleright \\\\ $ sample uniformly from $[0,1]$ $n^+ \\\\sim $ Uniform($\\\\lbrace 1,2,...,n-1\\\\rbrace $ ) $\\\\triangleright \\\\ $ number of sources $n^- \\\\leftarrow n - n^+$ $\\\\mu ^+_{1:n^+} \\\\sim $ Uniform($[0,1]$ , size=$n^+$ ) $\\\\triangleright \\\\ $ array of supplies $\\\\mu ^-_{1:n^-} \\\\sim $ Uniform($[0,1]$ , size=$n^-$ ) $\\\\triangleright \\\\ $ array of demands $a_{1:n} \\\\sim $ Uniform($[0,1]$ , size=($n$ ,2)) $\\\\triangleright \\\\ $ terminal coordinates return $a_{1:n}, \\\\ \\\\mu ^+/\\\\mathrm {sum}(\\\\mu ^+), \\\\ \\\\mu ^-/\\\\mathrm {sum}(\\\\mu ^-), \\\\ \\\\alpha $\"\n ],\n [\n \"Greedy randomized heuristic for topology optimization\",\n \"In this section, we give the details of the greedy randomized heuristic presented in Section REF and show the results of some additional experiments conducted with it.\",\n \"Starting from a tree topology $T$ , a uniformly sampled edge $\\\\hat{e}$ is removed.\",\n \"The node in the smaller connected component $\\\\ell $ is connected via a new branching point to an edge in the other component.\",\n \"This edge is sampled according to $p(e) \\\\propto \\\\exp (-d(e, \\\\ell )^2 / d_{min}^2)$ , where $d(e, \\\\ell )$ is the distance between an edge $e=(i,j)$ and node $\\\\ell $ , defined by $ d(e, \\\\ell ) = \\\\underset{\\\\lambda \\\\in [0,1]}{\\\\min } \\\\ \\\\left\\\\Vert [x_i + \\\\lambda (x_j - x_i) ] - \\\\ell \\\\right\\\\Vert _2 \\\\ $ and $d_{min}$ is the distance to the closest considered edge.\",\n \"The resulting new tree topology $T_{new}$ is accepted if it decreases the cost compared to the previous topology $T$ and the above procedure is iterated.\",\n \"This greedy optimization strategy is summarized in Alg.\",\n \"REF below.\",\n \"Greedy randomized heuristic (zero temperature limit) Input: BOT problem $P$ , tree $T$ interconnecting terminals $a_{1:n}$ (with help of BPs) Output: heuristic BOT solution [1] $B \\\\leftarrow $ optimize_BP_configuration($T, P$ ) $\\\\triangleright \\\\ $ returns coordinates of BPs and terminals, see Alg.\",\n \"REF $\\\\mathcal {C} \\\\leftarrow $ BOT_cost$(T,B,P)$ $E \\\\leftarrow $ list(T.edges()) $E$ not empty $S \\\\leftarrow $ Store current state $(T,B)$ $\\\\mathcal {C}_{old} \\\\leftarrow \\\\mathcal {C}$ $\\\\hat{e} \\\\sim $ Uniform($E$ ) $\\\\triangleright \\\\ $ sample uniformly from $E$ $E$ .remove($e$ ) $T$ .remove_edge($\\\\hat{e}$ ) $C_b \\\\leftarrow $ subgraph component of $T$ with more nodes $E_b \\\\leftarrow $ list($C_b$ .edges) $b \\\\leftarrow $ node of $\\\\hat{e}$ in $C_b$ $\\\\ell \\\\leftarrow $ node of $\\\\hat{e}$ not in $C_b$ degree($b$ ) == 2 $n_1,n_2 \\\\leftarrow $ neighbors($T, b$ ) $T$ .remove_node($b$ ) $\\\\triangleright \\\\ $ remove unnecessary BPs with degree 2 $T$ .add_edge($n_1,n_2$ ) $E_b$ .remove(($n_1,b$ ), ($n_2,b$ )) $d \\\\leftarrow $ Initalize array of size length($E_b$ ) edge $e \\\\in E_b$ $d[e] \\\\leftarrow $ get_distance($e,\\\\ell $ ) $\\\\triangleright \\\\ $ calculate distance defined by Eq.\",\n \"(REF ) $d_{min} \\\\leftarrow $ min($d$ ) $e_c \\\\sim \\\\exp (-d^2/d_{min}^2)$ $\\\\triangleright \\\\ $ sample edge according to distance kernel $T$ .add_node($b_{new}$ ) $\\\\triangleright \\\\ $ Initialize a new BP $T$ .add_edges($(\\\\ell ,b_{new}),(b_{new}, e_c[0]),(b_{new}, e_c[1])$ ) $B \\\\leftarrow $ optimize_BP_configuration($T, P$ ) $\\\\mathcal {C} \\\\leftarrow $ BOT_cost$(T,X,P)$ $ \\\\mathcal {C} < \\\\mathcal {C}_{old} $ $E \\\\leftarrow $ list(T.edges()) $\\\\triangleright \\\\ $ accept new state $T,B \\\\leftarrow S$ $\\\\triangleright \\\\ $ Restore old state $(T,B)$ $\\\\mathcal {C} \\\\leftarrow \\\\mathcal {C}_{old}$ return $T, B$ Figure REF shows an example BOT problem with 100 terminals to which our greedy heuristic has been applied.\",\n \"As starting point, the topology with the least possible structure was used, a star-like tree centred around a single BP of degree 100, cf. Fig.\",\n \"REF .\",\n \"It can be seen how the topology evolves over less than 2000 iterations to the final solution.\",\n \"Figure REF shows the decreasing transportation cost $\\\\mathcal {C}$ plotted against the number of iterations.\"\n ],\n [\n \"Performance test of heuristic topology optimization.\",\n \"Finally, in a large experiment, we have investigated how many iterations the greedy heuristic requires on average to converge.\",\n \"For that, the greedy heuristic was applied to a number of BOT problems with $n$ terminals and the number of iterations until convergence were counted.\",\n \"The mean and standard deviation of the required iterations are plotted in Fig.\",\n \"REF .\",\n \"The reason for the relatively large standard deviations are due to intrinsic variation of the sampled problems.\",\n \"To illustrate, for instance, the influence of the $\\\\alpha $ -value, the average number of iterations until convergence were plotted separately for all BOT problems with $\\\\alpha \\\\le 0.5$ and $\\\\alpha > 0.5$ .\",\n \"We find that the greedy heuristic systematically needs more iterations to converge for $\\\\alpha > 0.5$ .\",\n \"Figure: The greedy topology optimization applied to each 150 problems with nn terminals: Mean and standard deviation of required iterations plotted vs. nn in blue.\",\n \"Average for problems with α≤0.5\\\\alpha \\\\le 0.5 in green (dashed) and for problems with α>0.5\\\\alpha > 0.5 in orange (dotted).\"\n ],\n [\n \"Influence of the edge sampling kernel.\",\n \"Our heuristic topology optimization involves a number of design choices which may affect its performance.\",\n \"A systematic investigation is beyond the scope of this work and left for future research.\",\n \"However, one hyperparameter of particular interest is the kernel (chosen to be Gaussian) and its width (chosen to be $d_{min}$ ), which together define the replacement probability of the edges, see l. REF in Alg.\",\n \"REF .\",\n \"To study its influence on the performance we varied the width of the Gaussian kernel $\\\\exp (-d^2/(\\\\omega d_{min})^2)$ by tuning the parameter $\\\\omega $ .\",\n \"Based on 150 random problems of various sizes $n$ , we calculated the average cost ratio of the heuristic with different $\\\\omega $ to the default of $\\\\omega = 1$ (cf.\",\n \"Fig.\",\n \"REF ).\",\n \"Indeed, Fig.\",\n \"REF shows that the default choice of $\\\\omega = 1$ is quite strong and relatively robust given that $\\\\omega = 0.5$ or $\\\\omega = 0.1$ work similarly well.\",\n \"Clearly, wider kernels lead to larger (i.e.\",\n \"less local) changes of the topology.\",\n \"At later stages of the algorithm most of these topology changes will be unfavorable, explaining why the algorithm for wider kernels terminates with comparatively less optimal solutions.\",\n \"Furthermore, we have investigated the influence of the kernel width on the number of iterations required (cf.\",\n \"Fig.\",\n \"REF ).\",\n \"Qualitatively, Fig.\",\n \"REF confirms that for wider kernels, which encourage exploration, more iterations are required.\",\n \"Fitting a power law of the form $x \\\\mapsto ax^b$ to the curves in Fig.\",\n \"REF , one finds that, depending on the kernel, the greedy heuristic on average scales between $O(n^{1.5})$ and $O(n^{1.7})$ .\",\n \"Again, this is a purely empirical statement.\",\n \"A careful theoretical analysis to obtain guarantees, also for problem sizes not covered in our experiments, is beyond the scope of this work.\",\n \"Figure: Influence of kernel width in topology optimization based on 150 random problems of various sizes nn: (a) Ratio of the average cost of the heuristic solution with kernel width factor ω\\\\omega to the default of ω=1\\\\omega =1.\",\n \"(b) Average number of iterations until convergence for different kernel width factors ω\\\\omega .\",\n \"Depending on the kernel width the number of iterations on average scales between O(n 1.5 )O(n^{1.5}) and O(n 1.7 )O(n^{1.7}).Figure: Average cost ratios of our greedy heuristic and brute-force solutions (the closer to 1 the better) for different number of terminals nn and dimensions dd.\",\n \"For each nn, we uniformly sampled 100 different BOT problems.\",\n \"Though the average cost ratio increases slightly with nn, our approximate BOT solver compares very well against the ground truth solutions, independently of the dimensionality dd.\"\n ],\n [\n \"Greedy topology optimization vs. brute-force search for higher-dimensional BOT.\",\n \"Both the numerical geometry optimization and the greedy algorithm for the topology optimization presented in Sect.\",\n \"are readily applicable to BOT problems in $\\\\mathbb {R}^d$ .\",\n \"Paralleling the experiment presented in Fig.\",\n \"REF , we have compared our heuristic topology optimization against brute-forced solutions for each 100 problems of size $n=5$ to $n=9$ and spatial dimension $d$ .\",\n \"Although Fig.\",\n \"REF suggests that the average cost ratio increases slightly with $n$ , our approximate BOT solver again compares very well against the ground truth solutions, independently of the dimensionality $d$ .\"\n ]\n]"},"id":{"kind":"string","value":"2210.07702"}}},{"rowIdx":2999,"cells":{"text":{"kind":"list like","value":[["An atom's worth of anonymity"],["Abstract Anonymity has gained notoriety in modern times as data about our actions and choices accumulates in the internet partly unbeknownst to us and partly by our own choice.","Usually people wish some data about themselves were private while some other date may be public or is even wanted to be public for publicity reasons.","There are different criteria which characterize the degree of anonymity of data.","Given data can also be anonymized by different techniques in order to increase its degree of anonymity.","In this paper we take a very simple \"atomic\" degree of anonymity as our starting place.","We axiomatize these atoms and propose the investigation of first order logic based on these atoms.","Considering the vast literature and the huge importance of anonymity our investigation may seem quite modest.","However, it is about the logic of anonymity, not about how to secure, create or break anonymity."],["I contribute this paper on anonymity to honor the birthday of John Crossley.","I am not only John's friend but also his grandson in the academic sense, as my doctoral supervisor was Peter Aczel, a student of John's.","Of course, it is my hope that this attempt to find the logic of anonymity amuses John.","Suppose we have data about some individuals and in the data the values of some attributes are publicly known while the values of some other attributes are sensitive personal data that should be protected from becoming public.","Anonymity in this context means the impossibility to derive the values of the sensitive attributes from the values of the publicly known attributes, and this impossibility should be true of each individual.","Anonymity has gained notoriety in modern times as data about our actions and choices accumulates in the internet partly unbeknownst to us and partly by our own choice.","Usually people wish some data about themselves were private while some other date may be public or is even wanted to be public for publicity reasons.","There are different criteria which characterize the degree of anonymity of data.","Given data can also be anonymized by different techniques in order to increase its degree of anonymity.","In this paper we take a very simple “atomic\" degree of anonymity as our starting place.","We axiomatize these atoms and propose the investigation of first order logic based on these atoms.","Considering the vast literature and the huge importance of anonymity our investigation may seem quite modest.","However, it is about the logic of anonymity, not about how to secure, create or break anonymity.","In this paper, as in database theory, variables denote attributes.","Thus our $x$ , $y$ , $z$ , etc stand for such attributes as “date\", “time\", “surname\", “first name\", “hometown\", “year of birth\", “gender\", “salary\", “blood type\", “systolic blood pressure\", “diastolic blood pressure\", etc.","It is natural to think of an assignment of values to such variables as a database record.","We call any set of such assignments a team.","Calling them teams carries no special meaning but is just a practice I adopted in my book [6], with teams of robots performing some task in mind.","Table: A team.In the team of Table REF the attributes hometown and salary do not reveal 100% the attribute surname because for all combinations of hometown and salary there are two different values of surname.","So, if we know hometown and salary we are still a little uncertain who is in question.","Of course, this is an extremely weak form of anonymity, and that is why we call it “atomic\" anonymity.","On the other hand, the attribute surname completely determines hometown and salary, so no anonymity or uncertainty there.","We formalize the concept of keeping part of data anonymous while publishing some other part.","We use $x,y,z,u,v,$ etc to denote finite sequences of attributes.","The concatenation of $x$ and $y$ is denoted $xy$ .","If $s$ is an assignment and $x=\\langle x_1,\\ldots , x_n\\rangle $ , then $\\langle s(x_1),\\ldots ,s(x_n)\\rangle $ is denoted $s(x)$ .","Definition 1 (Anonymity atom) A team $X$ satisfies “$x$ leaves $y$ anonymous” in the model $M$ , in symbols $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ , if $\\forall s\\in X\\exists s^{\\prime }\\in X(s(x)=s^{\\prime }(x)\\wedge s(y)\\ne s^{\\prime }(y)).$ The intuition of $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ is the following: We can publish $X\\upharpoonright x$ and still keep the anonymity of $X\\upharpoonright y$ as there is no row $s$ of $X$ (i.e.","no individual $s$ ) for which we can infer $s(y)$ from $s(x)$ .","A better anonymity is provided by the $k$ -anonymity atom which demands in Definition REF not just one but $k$ different values $s^{\\prime }(y)\\ne s(y)$ for each $s(y)$ .","Note that the full team (in a domain with at least two elements) and the empty team always satisfy $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .","Unlike ${=\\hspace{-0.66666pt}}(x,y)$ (as in [6]), the atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ is not closed downwards (or upwards).","We can regard $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ as a strong (maximal) denial of ${=\\hspace{-0.66666pt}}(x,y)$ .","But note that the independence atom $x\\ \\bot \\ y$ (as in [4]) does not imply $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ , because $x\\ \\bot \\ y$ may be true because $y$ is constant.","The anonymity atom is clearly closed under unions and it is first order.","Hence it is definable in inclusion logic [3], in fact, if $u$ is a sequence of new variables, as many as in $y$ , then $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\iff \\exists u(\\lnot u=y\\wedge xu\\subseteq xy).$ Thus `anonymity logic' is a sublogic on inclusion logic.","In fact, the two are equivalent [2].","See also [5].","We now ask the question, what are the general principles that propositions of the form $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ obey?","If we know a bunch of such propositions, what can we infer from them?","For example, is it true that if $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and $y\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ , then $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ ?","That this is not so can be seen by investigating the team of Table REF .","This team satisfies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and $y\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ but fails to satisfy $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ because it actually satisfies $x=z$ .","Table: Failure of transitivityDefinition 2 The axioms of anonymity are A1 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ implies $yx\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ and $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}uz$ .","(Permutation) A2 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ .","(Monotonicity) A3 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zy$ implies $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ .","(Cancellation) A4 $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}$ implies $\\bot $ .","$xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ implies $yx\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ and $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}uz$ .","(Permutation) $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ .","(Monotonicity) $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zy$ implies $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ .","(Cancellation) $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}$ implies $\\bot $ .","From the axioms (or rules) of Definition REF we get a notion $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ of derivability of an atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ from a set $\\Sigma $ of such atoms.","Theorem 3 The following are equivalent for any set $\\Sigma \\cup \\lbrace x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\rbrace $ of anonymity atoms: $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .","$M\\models _X\\Sigma $ implies $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ for every model $M$ and team $X$ .","$\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .","$M\\models _X\\Sigma $ implies $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ for every model $M$ and team $X$ .","Suppose first $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and $M\\models _X\\Sigma $ .","All the rules (A1)-(A4) preserve truth.","Hence $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .","Suppose then $\\Sigma \\nvdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .","By the validity of (A3) we may assume $x\\cap y=\\emptyset $ .","By the validity of (A4) we may assume $y\\ne \\emptyset $ .","Let $x=\\langle x_1,\\ldots ,x_n\\rangle $ and $y=\\langle y_1,\\ldots ,y_m\\rangle $ .","Let $W$ be the set of all variables occurring in $\\Sigma \\cup \\lbrace x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\rbrace $ .","Let $M$ be a model with at least three elements $0,1,2$ and $\\begin{array}{lcl}X&=&\\lbrace s\\in \\lbrace 0,1,2\\rbrace ^W : \\bigvee _{1\\le i\\le n}(s(x_i)\\ne 0)\\vee \\bigwedge _{1\\le i\\le m}(s(y_i)= 0)\\rbrace \\end{array}$ First of all, $M\\lnot \\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ for if $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and we let $s(w)=0$ for all $w\\in W$ , then there would be $s^{\\prime }\\in X$ such that $s^{\\prime }(x_i)=s(x_i)=0$ for all $i\\in \\lbrace 1,\\ldots ,n\\rbrace $ and $s^{\\prime }(y_i)\\ne s(y_i)=0$ for some $i\\in \\lbrace 1,\\ldots ,m\\rbrace $ , a contradiction.","We are left with proving $M\\models _X \\Sigma $ .","To see this, let ${u\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}v}\\in \\Sigma $ .","We may assume, w.l.o.g., $u\\cap v=\\emptyset $ .","Let $s\\in X$ .","We shall construct $s^{\\prime }\\in X$ such that $s^{\\prime }(u)=s(u)$ and $s^{\\prime }(v)\\ne s(v)$ .","Let us assume first: ${x\\lnot \\subseteq u.","}$ Since $x\\cap (W\\setminus u)\\ne \\emptyset $ , there is some $x_i\\notin u$ .","Either $x_i\\in v$ or $x_i\\notin v$ .","Case 1.1: $x_i\\in v$ , i.e.","$x_i=v_j$ for some $j\\in \\lbrace 1,\\ldots ,k\\rbrace $ .","Since there are three possible values for $s^{\\prime }(v_j)$ , we can make sure both $s^{\\prime }(v_j)\\ne s(v_j)$ and $s^{\\prime }(v_j)\\ne 0$ .","Then $s^{\\prime }(x)\\ne 0$ , since $x_i=v_j$ .","Thus $s^{\\prime }\\in X$ , whichever way we otherwise define $s^{\\prime }$ .","By construction $s^{\\prime }(v)\\ne s(v)$ .","Since $u\\cap v=\\emptyset $ , we are free to let $s^{\\prime }(u)=s(u)$ , and we are done.","Case 1.2: $x_i\\notin v$ .","Let $s^{\\prime }(u)=s(u)$ and let $s^{\\prime }(v)$ be such that $s^{\\prime }(v)\\ne s(v)$ .","Since now $x_i\\notin u\\cup v$ , we can let $s^{\\prime }(x_i)\\ne 0$ , and this guarantees $s^{\\prime }\\in X$ .","We are done again.","Let us then assume ${x\\subseteq u.","}$ Since $s\\in X$ , we know that $s(x)$ is not all zero or $s(y)$ is all zero.","Case 2.1: $s(x)$ is not all zero.","We let $s^{\\prime }(u)=s(u)$ .","Since now $s^{\\prime }(x)$ is not all zero, we have $s^{\\prime }\\in X$ whatever $s^{\\prime }$ is otherwise.","So we choose $s^{\\prime }(v)\\ne s(v)$ in an arbitrary way, and we are done.","So let us assume Case 2.2: $s(y)$ is all zero.","If $v\\subseteq y$ , then ${u\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}v}\\in \\Sigma $ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\in \\Sigma $ , by Axiom (A2), contrary to our assumption.","So we may pick $v_i\\in v\\setminus y$ .","Let $s^{\\prime }(v_i)\\ne s(v_i)$ .","Now we know $s^{\\prime }(v)\\ne s(v)$ .","Since $u\\cap v=\\emptyset $ , we can let $s^{\\prime }(u)=s(u)$ .","Since $v_i\\notin y$ we can let $s^{\\prime }(y)=s(y)=0$ .","Now $s^{\\prime }\\in X$ , whichever way $s^{\\prime }$ is otherwise defined.","We may combine $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ with logical operations as is done in dependence logic [6].","Here is a practical example: Example 4 Suppose we collect some data about individuals and we need to publish this data together with the addresses of the individuals.","Some individuals may, for privacy reasons, not want their address to be associated with other data about them.","For this purpose we allow any individual to declare their address private by setting ${\\tt publicity}={\\tt private}$ .","A team satisfies, according to the semantics of implication given in [6]: ${\\tt publicity}={\\tt private}\\rightarrow {\\tt data}\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address},$ iff the anonymity constraint ${\\tt data}\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address}$ holds (at least) in the part of the team that satisfies ${\\tt publicity}={\\tt private}$ .","Example 5 We continue the previous example.","A team satisfies $\\exists x (\\phi (x)\\wedge ({\\tt data}\\ x)\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address}),$ iff (possibly multiple) values can be assigned to $x$ so that the new values of $x$ satisfy $\\phi (x)$ and, in addition, the anonymity constraint $({\\tt data}\\ x)\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address}$ (still) holds.","In other words, data about the individuals can be updated with data about the attributes $x$ without violating the modest anonymity provided by the $\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}$ -atom.","Our anonymity atom provides the most basic form of anonymity, not sufficient for practical applications.","Therefore we strengthen the atom to $k$ -anonymity atom.","The bigger the $k$ , the more anonymity it provides.","The idea is that an attribute $y$ is $k$ -anonymous with respect to some other attributes $x$ if any data about $x$ in the team is associated with at least $k$ values for $y$ .","Intuitively, my name does not reveal me if there are $k$ different persons with the same name as me, and $k$ is relatively large.","In September 9, 2022, there were 4703 people alive with the last name Väänänen in Finland.","Thus I can remain relatively anonymous in Finland even if I reveal that my last name is Väänänen.","At the same moment of time there were only 7 people in Finland's population information system with the last name Crossley, providing much less anonymity if the last name was revealed.","Definition 6 ($k$ -anonymity atom) A team $X$ satisfies “publishing $x$ keeps $y$ $k$ -anonymous” in a model $M$ , in symbols $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ , if $\\forall s\\in X(|\\lbrace s^{\\prime }(y) : s^{\\prime }\\in X \\wedge s(x)=s^{\\prime }(x)\\rbrace |\\ge k).$ The atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ is called simple, if $|y|=1$ .","Note that the criterion is different from $\\forall s\\in X(|\\lbrace s\\in X : s(x)=s^{\\prime }(x)\\wedge s(y)\\ne s^{\\prime }(y)|\\ge k)$ but equivalent to $\\forall s\\in X\\exists s^{\\prime }_1\\ldots \\exists s^{\\prime }_k\\in X\\bigwedge _{1\\le i
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[
"MIINet: An Image Quality Improvement Framework for Supporting Medical\n Diagnosis"
],
[
"Abstract Medical images have been indispensable and useful tools for supporting medical experts in making diagnostic decisions.",
"However, taken medical images especially throat and endoscopy images are normally hazy, lack of focus, or uneven illumination.",
"Thus, these could difficult the diagnosis process for doctors.",
"In this paper, we propose MIINet, a novel image-to-image translation network for improving quality of medical images by unsupervised translating low-quality images to the high-quality clean version.",
"Our MIINet is not only capable of generating high-resolution clean images, but also preserving the attributes of original images, making the diagnostic more favorable for doctors.",
"Experiments on dehazing 100 practical throat images show that our MIINet largely improves the mean doctor opinion score (MDOS), which assesses the quality and the reproducibility of the images from the baseline of 2.36 to 4.11, while dehazed images by CycleGAN got lower score of 3.83.",
"The MIINet is confirmed by three physicians to be satisfying in supporting throat disease diagnostic from original low-quality images."
],
[
"Introduction",
"Medical images provide a lot of useful information and visual insight into different hidden body organs.",
"They are very effective for helping doctors in making correct diagnoses or can be used as valuable reference resources for better treatment.",
"Moreover, with the rapid development of artificial intelligence (AI), many breakthrough applications have been built on top of medical images data [27], [13], [24], [11], [32].",
"However, obtaining medical images especially endoscopic or throat images is never an easy task.",
"In practice, those images normally contain noise, hazy, uneven illumination, lack of focus, etc., due to many difficult shooting conditions inside the body.",
"Thus, these could greatly affect the medical diagnostic process.",
"Several studies applying machine learning techniques for diagnosing endoscopic and throat images have been reported that their systems are highly sensitive to the image conditions [1], [30], [14], [15], [29].",
"Poor image quality could easily lead to a misdetection, making it a very challenging task.",
"We are developing a special camera device for supporting doctors in diagnosis oral and throat diseases.",
"We also experienced that the inside environment of patient’s palate contains many negative factors that reduce the quality of images such as the hazy caused by patient’s breath on camera or the lack of focus.",
"Fig.",
"REF illustrates examples of throat images with undesirably quality and this is an obstacle for doctors from making medical decisions.",
"Therefore, a method to improve the quality of medical image to support the diagnosis is essential.",
"We believe that this problem can be addressed by applying image dehazing technique.",
"Figure: Two examples of throat images with undesirably quality (out of focus (left) or hazy (right)).Recent works have been utilizing a deep learning method called convolutional neural networks (CNNs) and shown tremendous success for recovering image quality from very dense haze and noise.",
"Those dehazing techniques can be divided into two major approaches: the supervised approach [4], [20], [26], [33], [35] and the unsupervised approach [10], [34], [9], [16].",
"The former normally achieves compelling results thanks to the modeling power of CNNs.",
"However, they require a large amount of paired ground-truth images for supervision which is almost unavailable to obtain in reality.",
"The latter offers more practical settings for image dehazing by removing the need of paired label training data.",
"They are all built on the success of CycleGAN [37], which is a generative adversarial network (GAN) [12] based image-to-image translation method.",
"CycleGAN introduced the cycle-consistency constraint that generated image from a domain should be identical to its original form when transforming it back.",
"Despite their impressive results, there are two main problems of these methods when applying to our practical throat images data.",
"Firstly, supervised and several unsupervised studies were still built based on the assumption that hazy images (training data) have unique haze and are generated by the atmosphere scattering model [22], [23].",
"For this reason, they may not be practical in scenarios when the disturbance deviates from prior assumptions (e.g., when shooting environment changed, such as differences in equipment, camera-setting or protocols).",
"Secondly, naïve CycleGAN is reported to not work well on high-resolution data [21] and it does not generate sufficient resolution output for our purposes.",
"We should note that the literature [10] suggested to use the Laplacian upscaling for the output of CycleGAN to obtain higher resolution results.",
"However, the obtained images are normally overly smooth and sometimes fails to accurately represent detailed structures.",
"Therefore, these abovementioned problems will make it difficult for doctors to diagnose through throat images.",
"A framework that generates clean throat images with high-resolution from original low-quality (LQ) images could be a great tool for supporting doctors in making medical decisions.",
"In this paper, we propose a medical image improvement framework named MIINet for helping doctors to make medical diagnostic decisions.",
"Our MIINet consists of two modules: the image dehazing module (IDM) and the image super-resolution (ISR) module.",
"The IDM is developed based on the CycleGAN [37] model with the aims of translating images from LQ domain to high-quality (HQ) domain.",
"In this work, we introduce a new loss term based on the perceptual loss function [17] with the aims to preserve original input image attributes such as structure, color, texture.",
"This function is essential since that original information is crucial in medical diagnosis.",
"Besides the IDM, we introduce a CNN-based image super-resolution (ISR) module to enlarge the output from our IDM, obtaining high-resolution results.",
"The ISR module acts as an optional module when doctors need to enlarge images for more diagnosis details.",
"Our contributions can be summarized as follows: We propose the MIINet that improves the quality of practical LQ throat images while preserving the structure of the involved areas.",
"Our MIINet with the introduction of the ISR module is able to produce high-resolution throat images, making the disease diagnosis more favorable for doctors.",
"The dehazed throat images obtained by our MIINet shows a significantly higher of the mean doctor opinion score (MDOS) of 4.11 compared to the original LQ images of 2.36, in assessing the quality and the reproducibility of the images."
],
[
"Proposed Method – MIINet",
"The proposed MIINet consists of two modules: (1) image dehazing module (IDM), and (2) image super-resolution (ISR) module.",
"Fig.",
"REF shows the schematic of our framework.",
"Given an input of original LQ throat image, our IDM will calibrate and convert that image into a HQ clean image.",
"Since the output of our IDM is relatively small size, it will be fed to the ISR to enlarge into higher resolution with $4\\times $ upscaling, this will help doctors to have a better visual inspection."
],
[
"The Image Dehazing Module – IDM",
"Our IDM is an improved version of CycleGAN [37] for unpaired throat image improvement.",
"It consists of a mapping function $G:X{\\rightarrow }Y$ that translates image from source domain ($X$ ) to target domain ($Y$ ), and an invert mapping function $F:Y{\\rightarrow }X$ to enforce the cycle-consistency.",
"Corresponding to two generators are the two adversarial discriminators $D_X$ and $D_Y$ , where $D_X$ is trying to discriminate the real image $x \\in X$ from the generated image $F(y)$ with $y \\in Y$ .",
"Similarly, $D_Y$ distinguishes the real image $y$ from the generated image $G(x)$ .",
"In this work, we assume $X$ is the LQ image domain while $Y$ is the HQ image domain.",
"Fig.",
"REF shows the dataflow of the translation from $X{\\rightarrow }Y$ .",
"Given a LQ image $x$ , the generator $G$ will transform it into a HQ clean image $x^{\\prime }$ .",
"Then, the image $y \\in Y$ and $x^{\\prime }$ are then fed into the discriminator $D_Y$ .",
"Note that the translation $Y{\\rightarrow }X$ is symmetric to the translation $X{\\rightarrow }Y$ .",
"Figure: Dataflow of the translation X→YX{\\rightarrow }Y from low-quality image (domain XX) to high-quality image (domain YY).Based on the GAN literature [12], the adversarial losses for both mapping functions $G:X{\\rightarrow }Y$ and $F:Y{\\rightarrow }X$ are $\\mathcal {L}_\\mathrm {adv}(G,D_Y)$ and $\\mathcal {L}_\\mathrm {adv}(F,D_X)$ , respectively.",
"Where: $\\mathcal {L}_\\mathrm {adv}(G,D_Y)=\\mathbb {E}_{y{\\sim }p_\\mathrm {data}(y)}[(D_Y(y)-1)^2]+\\\\\\mathbb {E}_{x{\\sim }p_\\mathrm {data}(x)}[(D_Y(x^{\\prime }))^2],$ and $\\mathcal {L}_\\mathrm {adv}(F,D_X)=\\mathbb {E}_{x{\\sim }p_\\mathrm {data}(x)}[(D_X(x)-1)^2]+\\\\\\mathbb {E}_{y{\\sim }p_\\mathrm {data}(y)}[(D_X(y^{\\prime }))^2].$ Note here that $x^{\\prime }=G(x)$ and $y^{\\prime }=F(y)$ .",
"The cycle consistency loss $\\mathcal {L}_\\mathrm {cyc}(G,F)$ is formulated as follows: $\\mathcal {L}_\\mathrm {cyc}(G,F)=\\mathbb {E}_{x{\\sim }p_\\mathrm {data}(x)}[|F(G(x))-x|_1]+\\\\\\mathbb {E}_{y{\\sim }p_\\mathrm {data}(y)}[|G(F(y))-y|_1].$ As we mentioned before that preserving the original attributes of input images (i.e., structure, texture, color) is crucial in medical diagnosis.",
"Therefore, we introduce a new loss term based on the perceptual loss $\\mathcal {L}_\\mathrm {percep}$ [17].",
"To ensure that the attributes of original input and output are as similar as possible, we minimize the L1 distance between the features extracted by a CNN model of both input and generated image.",
"Based on our preliminary experiments, we use the 2nd pooling layer of the ImageNet [7] pre-trained VGG16 [28] model to extract the features.",
"The $\\mathcal {L}_\\mathrm {percep}$ will be defined as: $\\mathcal {L}_\\mathrm {percep}(G,F)=\\mathbb {E}_{x{\\sim }p_\\mathrm {data}(x)}[|\\phi (G(x))-\\phi (x)|_1]+\\\\\\mathbb {E}_{y{\\sim }p_\\mathrm {data}(y)}[|\\phi (F(y))-\\phi (y)|_1],$ where $\\phi (\\cdot )$ is the features extracted from the VGG16 model.",
"Finally, our final objective function can be summed up as: $\\mathcal {L}(G,F,D_X,D_X)=\\mathcal {L}_\\mathrm {adv}(G,D_Y)+\\mathcal {L}_\\mathrm {adv}(F,D_X)+\\\\\\lambda \\mathcal {L}_\\mathrm {cyc}(G,F)+\\beta \\mathcal {L}_\\mathrm {percep}(G,F),$ where $\\lambda ,\\beta $ are the coefficient to control the balance of different loss terms."
],
[
"The Image Super-Resolution Module - ISR",
"Our ISR module is a GAN-based single image super-resolution (SISR) model, which aims to learn an end-to-end mapping function to recover a high-resolution (HR) image from a single low-resolution (LR) image [8], [19], [31].",
"Many SISR models have also been proposed and widely used in many practical applications ranging from medical imaging [6], [36], security and surveillance [3], satellite imaging [25], to agriculture [5].",
"In this work, we propose an SISR module namely throat image super-resolution (ISR) for enlarging the resolution of the clean throat image output from our IDM.",
"Our ISR module is built based on an excellent SR model so-called ESRGAN [31] which generates realistic perceptual quality results and achieved impressive performances in many benchmarks [2].",
"Similar to ESRGAN, our ISR module consists of two networks: a generator $S$ which generates super-resolved images from LR images and a discriminator $D_{SR}$ that discriminates the HR image from the super-resolved ones.",
"We use the architecture of the generator $S$ , the loss functions, and the hyperparameters as same as in ESRGAN literature.",
"For the discriminator $D_{SR}$ , we design our network to take the input of $224 \\times 224$ instead of the original $128 \\times 128$ as in ESRGAN since this setting helps our model gains slightly better performance based on our preliminary experiments.",
"The two networks are then trained together in an alternating manner to solve a minimax problem [12].",
"For more technical training details, please refer to the original ESRGAN article [31]."
],
[
"Throat Image Dataset",
"In this work, we collected 1,600 throat images from over 160 patients in which contain both ill-conditioned images and clean images.",
"They were taken by a special camera designed for throat diagnosis and each of which has the size of $1920 \\times 1080$ pixels.",
"Experts were asked to manually inspect and carefully select 200 images with hazy and lack of focus (see Fig.",
"REF ) as low-quality images and we refer it as the LQ Throat dataset.",
"Note that those images are the most difficult cases for physicians to diagnose.",
"From this LQ Throat dataset, 100 images are used for training and the others 100 are for testing.",
"The rest 1,400 images are clean and high-quality.",
"We refer it as the HQ Throat dataset."
],
[
"Training The IDM",
"Since the number of images between the two datasets LQ Throat and HQ Throat is quite different from each other.",
"We randomly selected 100 images from HQ Throat dataset (i.e., same amount as the LQ Throat test dataset) to train our IDM.",
"We then combined and applied different data augmentation techniques such as horizontal flip, random scale, random resize on both datasets beforehand.",
"Since the IDM (or other image-to-image translation GAN models such as CycleGAN) cannot handle high-resolution data due to the limitation of available GPU memory, we resized input images to the size of $480 \\times 270$ pixels before training.",
"As a result, each dataset has 2,300 images after data augmentation.",
"We applied the same training procedures as described in CycleGAN [37] to train our MIINet.",
"The Adam optimizer [18] was used to train the network.",
"We set the $\\lambda $ and $\\beta $ in Eq.",
"(5) equal to 10.0 and 1.5, respectively.",
"The training process finished after 400 epochs.",
"Please refer to [37] for more training details."
],
[
"Training The ISR Module",
"In this paper, we built our ISR module for super-resolving the output from the IDM.",
"The scaling factor of $\\times 4$ was used for enlarging HR from LR throat images.",
"We used the HQ Throat dataset described in section 3.1 to train our ISR model.",
"During the training, the HR images were obtained by randomly cropping from training images with the size of $224 \\times 224$ .",
"The LR images are $1/4 \\times $ down-sampled from HR images using bicubic interpolation.",
"We randomly applied Gaussian blur to the LR images with the standard deviation $\\sigma =5$ as we observed this helped our ISR module to generate better visual results.",
"Note here again that this ISR module acts as an optional module when doctors need to enlarge images for more diagnosis details.",
"Since the HR (clean) version of the input LR throat images is unavailable, we do not report the numerical results of our ISR module in this paper.",
"The training details are the same as in the ESRGAN literature [31] and was completed after 400 epochs.",
"Figure: Visual comparison among original LQ throat images, generated images by CycleGAN and MIINet, and the HQ images evaluated with MDOS.",
"The above HQ images are unrelated with the rest of the images, and we evaluated them as a reference in our experiment."
],
[
"The Mean Doctor Opinion Score",
"Since there are no quantitative metrics for assessing the throat image quality for diagnostic purposes, we introduce a new evaluation criteria called mean doctor opinion score (MDOS) based on the mean opinion score to evaluate the quality of throat images.",
"Specifically, only experienced doctors were requested to give the scores.",
"We asked each doctor to assess a given image under two aspects: the quality (i.e., how good is this image for diagnosis?)",
"and the reproducibility (i.e., how good is this image in preserving the structure, texture, color from the original throat image?).",
"We should note that scores for original LQ and HQ throat images are given based on the quality aspect only.",
"We asked three specialized doctors to assign a score from 1 (bad quality) to 5 (excellent quality) to the throat images.",
"The doctors rated three versions of each image on 100 test images from the LQ Throat dataset (i.e., original LQ throat images, generated images by CycleGAN and MIINet, respectively) and an addition 100 HQ images.",
"Each doctor thus rated 400 instances.",
"Table: The mean doctor opinion score (MDOS) results among original and generated throat images (100 images each; ranging from 1 to 5; higher is better)Figure: Line distributions of the mean doctor opinion score (MDOS) among original LQ throat images, generated images by CycleGAN and MIINet, and the HQ images.",
"Each line represents the scores distribution from 100 images."
],
[
"Results",
"For comparison purposes, we also trained a CycleGAN model and evaluated its dehazed images.",
"Comparisons of original LQ images, generated images by CycleGAN and MIINet, and HQ images are shown in Fig.",
"REF .",
"Our proposed MIINet successfully generates clean versions from original LQ images and have a much better capability of preserving the original attributes (i.e., structure, color, texture) than the CycleGAN model.",
"Note here that the HQ images provided in the examples have no association with the rest of the images, and we evaluated them as a reference for a better intuitive understanding about the MDOS in our experiment.",
"Our MIINet also significantly improved the MDOS from the LQ images and is better than CycleGAN as shown in Table REF and Fig.",
"REF ."
],
[
"Discussion",
"We confirm the effectiveness of our MIINet for supporting doctors in image-based throat diagnosis by using the MDOS testing.",
"From the results in Figs.",
"REF , REF and Table REF , it is clear that the original LQ images yield the lowest MDOS since they are affected by negative factors such as hazy, uneven illumination, lack of focus etc., making it difficult for doctors to make their decisions.",
"As for the result of CycleGAN, even it improves a much better visual quality than original LQ images, there is a significant difference in scores distribution in comparison with our MIINet (see Fig.",
"REF ) since the CycleGAN could not preserve the original attributes (i.e., structure, color, texture) of LQ throat images.",
"Visual results from Fig.",
"REF show that CycleGAN either changes the color or generates much different structure and shape from input images.",
"This is because the original CycleGAN only learns to generate images that look close to the samples from the target domain but has no mechanisms to preserve those original attributes.",
"We should note that the color distribution of the HQ Throat dataset is quite different from the LQ Throat dataset.",
"Thus, CycleGAN generated outputs that have similar color as the target domain.",
"Keeping the similar structure and color is very important for doctors to make their decisions and therefore, the generated images from CycleGAN are not preferable.",
"From doctor’s feedback, generated images by MIINet are recommended to support throat diagnosis.",
"Thanks to the introduction of the perceptual loss, our MIINet not only learns to generate compelling quality images but also helps preserving the originality from inputs, significantly improved the MDOS from original LQ images from 2.36 to 4.11.",
"Although our system has achieved a promising result, there are several cases when CycleGAN generates slightly better visual focus images than our MIINet as shown in Fig.",
"6.",
"This is the trade-off of adding the perceptual loss into the objective function of CycleGAN.",
"MIINet is forced to keep the characteristics of original inputs while CycleGAN has more freedom to generate close outputs to the HQ Throat dataset.",
"Despite that fact, it is worth to mention that the MDOS of MIINet generated images in most cases are higher than CycleGAN since the original attributes have been preserved.",
"For better medical decisions, doctors recommend utilizing both results from CycleGAN and MIINet when diagnosing throat images if necessary.",
"Moreover, proposing more objective quantitative evaluations beside the MDOS metric for our framework could be useful and we intend to develop it in future works."
],
[
"Conclusion",
"In this paper, we proposed the medical image improvement method (MIINet) to improve the quality of throat images for supporting in making medical diagnosis.",
"With the introduction of the simple but effective perceptual loss, our MIINet largely improved the quality of original LQ throat images and achieved a promising result on the real-world throat images dataset.",
"From the results, we believe that our proposed MIINet method could be a useful tool for supporting doctors in making medical decisions and has a potential impact on different types of medical images."
],
[
"Acknowledgment",
"This work was done while the first author did a research internship at Aillis Inc., Japan.",
"We would like to thank all researchers, specially doctor Sho Okiyama, Memori Fukuda, Kazutaka Okuda for their valuable comments and feedback."
]
] | 2011.14132 |
[
[
"Privacy-Preserving Federated Learning for UAV-Enabled Networks:\n Learning-Based Joint Scheduling and Resource Management"
],
[
"Abstract Unmanned aerial vehicles (UAVs) are capable of serving as flying base stations (BSs) for supporting data collection, artificial intelligence (AI) model training, and wireless communications.",
"However, due to the privacy concerns of devices and limited computation or communication resource of UAVs, it is impractical to send raw data of devices to UAV servers for model training.",
"Moreover, due to the dynamic channel condition and heterogeneous computing capacity of devices in UAV-enabled networks, the reliability and efficiency of data sharing require to be further improved.",
"In this paper, we develop an asynchronous federated learning (AFL) framework for multi-UAV-enabled networks, which can provide asynchronous distributed computing by enabling model training locally without transmitting raw sensitive data to UAV servers.",
"The device selection strategy is also introduced into the AFL framework to keep the low-quality devices from affecting the learning efficiency and accuracy.",
"Moreover, we propose an asynchronous advantage actor-critic (A3C) based joint device selection, UAVs placement, and resource management algorithm to enhance the federated convergence speed and accuracy.",
"Simulation results demonstrate that our proposed framework and algorithm achieve higher learning accuracy and faster federated execution time compared to other existing solutions."
],
[
"Introduction",
"With the rapid deployment of fifth-generation (5G) and beyond wireless networks, unmanned aerial vehicles (UAVs) have been emerged as a promising candidate paradigm to provide communication and computation services for ground mobile devices in sport stadiums, outdoor events, hot spots, and remote areas [1]-[3], where aerial base stations (BSs) or servers can be mounted on UAVs.",
"Given the benefits of agility, flexibility, mobility, and beneficial line-of-sight (LoS) propagation, UAV-enabled networks have been widely applied in quick-response wireless communications, data collection, artificial intelligence (AI) model training, and coverage enhancement [1].",
"Correspondingly, UAV-assisted computation and communication tasks have attracted significant interest recently in 5G and beyond wireless networks [3].",
"Again, deploying UAVs as flight BSs is able to flexibly to collect data and implement AI model training for ground devices, but it is impractical for a large number of devices to transmit their raw data to UAV servers due to the privacy concern and limited communication resources for data transmission [4].",
"Moreover, as the energy capacity, storage, and computational ability of UAVs are limited, it is still challenging for UAVs to process/training a large amount of raw data [1], [2], [4].",
"In face of the challenges, federated learning (FL) [5], [6] emerges as a promising paradigm aiming to protect device privacy by enabling devices to train AI model locally without sending their raw data to a server.",
"Instead of training AI model at the data server, FL enables devices to execute local training on their own data, which generally uses the gradient descent optimization method [7].",
"By using FL, UAVs can perform distributed AI tasks for ground mobile devices without relying on any centralized BS, and devices also do not need to send any raw data to UAVs during the training [4].",
"In particular, wireless devices use their respective local datasets to train AI models, and upload the local model parameters to an FL UAV server for model aggregation.",
"After collecting the local model parameters from devices, the UAV server then aggregates the updated model parameters before broadcasting the parameters to associated devices for another round of local model training.",
"During the process, keeping raw data at devices not only preserves privacy but also reduces network traffic congestion.",
"A number of rounds are performed until a target learning accuracy is obtained.",
"In essence, FL allows UAV-enabled wireless networks to train AI models in an efficient way, compared with centralized cloud-centric frameworks.",
"However, since the parameters of AI models in FL need to be exchanged between UAV servers and devices, the FL convergence and task consensus for UAV servers will inevitably be affected by transmission latency [4].",
"In addition, due to the mobility of UAVs and devices, dynamic channel conditions can negatively affect the FL convergence [8].",
"Moreover, UAV servers are generally limited in terms of computation and energy resources, resulting in the necessity of design for efficient scheduling and management approaches to minimize the FL execution time [1], [4], [9].",
"Hence, the aforementioned challenges pertaining to FL specifically that need to be investigated for large scale implementation of efficient FL in UAV-enabled wireless networks.",
"In recent years, implementing FL in wireless networks has attracted many research efforts, and lots of studies have presented their studies that how to adopt FL to improve the learning efficiency [10]-[17].",
"Tran $et~al.$ [10] formulated an FL framework over a wireless network as an optimization problem that minimizes the sum of FL aggregation latency and total device energy consumption.",
"In addition, the frequency of global aggregation under different resource constraints (i.e., device central processing unit (CPU), transmission delay, and model accuracy) has been optimized in [11], [12].",
"However, in [10]-[12], due to the limited wireless spectrum, it is difficult to implement practical wireless FL applications when all mobile devices are involved in each aggregation iteration.",
"Hence, some studies proposed to apply the device selection/scheduling scheme to improve the convergence speed of FL [13]-[16].",
"The authors in [13] studied the relationship between the number of rounds and different device scheduling policies (i.e., random scheduling, round-robin, and proportional fair), and compared their performances by simulations.",
"A deep reinforcement learning (DRL)-based device selection algorithm was proposed to enhance the reliability and efficiency in an asynchronous federated learning manner [14].",
"Shi $et~al.$ [9] formulated a joint channel allocation and device scheduling problem to study the convergence speed of FL, but the solution to the problem is not globally optimized.",
"In [15], the authors introduced a novel hierarchical federated edge learning framework, and a joint device and resource scheduling algorithm was proposed to minimize both the system energy and FL execution delay cost.",
"A greedy scheduling scheme was proposed in [16] that manages a part of clients to participate in the global FL aggregation according to their resource conditions.",
"However, the work [16] failed to verify the effectiveness of the scheme in the presence of dynamic channels and computation capabilities.",
"To the best of our knowledge, there are several studies [4], [8], [9], [18]-[22] that investigated how to apply FL to improve AI model learning efficiency in UAV-enabled communication scenarios.",
"For instance, the authors in [8], [18] developed a novel framework to enable FL within UAV swarms, and a joint power allocation and flying trajectory scheme was proposed to improve the FL learning efficiency [8].",
"Lim $et~al.$ [19] proposed an FL-based sensing and collaborative learning approach for UAV-enabled internet of vehicles (IoVs), where UAVs collect data and train AI model for IoVs.",
"In addition, Ng $et~al.$ [20] presented the use of UAVs as flight relays to support the wireless communications between IoVs and the FL server, hence enhancing the accuracy of FL.",
"Shiri $et~al.$ [9] adopted FL and mean-field game (MFG) theory to address the online path design problem of massive UAVs, where each UAV can share its own model parameters of AI with other UAVs in a federated manner.",
"The work in [4] provided discussions of several possible applications of FL in UAV-enabled wireless networks, and introduced the key challenges, open issues, and future research directions therein.",
"To preserve the privacy of devices, a two-stage FL algorithm among the devices, UAVs, and heterogeneous computing platform was proposed to collaboratively predict the content caching placement in a heterogeneous computing architecture [21].",
"Furthermore, the authors in [22] introduced a blockchain-based collaborative scheme in the proposed secure FL framework for UAV-assisted mobile crowdsensing, which effectively prevents potential security and privacy threats from affecting secure communication performance.",
"However, most of the studies [4], [8], [9], [18]-[22] did not jointly consider the importance of the tasks (i.e., learning accuracy and execution time) of FL that UAVs perform, and the optimal dynamic scheduling and resource management of UAVs to complete the tasks, subject to the resource constraints of UAVs and different computational capabilities of devices.",
"Joint management of both UAVs trajectory (or UAVs placement) and resource management has been widely studied to optimize the network performance, such as in [23]–[28].",
"In [23] and [24], a joint optimization for user association, resource allocation, and UAV placement was studied in multi-UAV wireless networks, to guarantee the quality of services (QoS) of mobile devices.",
"However, the conventional optimization methods adopted in [23], [24] may be not effective as the UAVs-enabled environment are dynamic and complex.",
"To address this issue, reinforcement learning (RL) or DRL has been adopted to learn the decision-making strategy [25]-[28].",
"The authors in [25], [26] applied the multi-agent DRL algorithm to address the joint UAVs trajectory and transmit power optimization problem, and also demonstrated the effectiveness of multi-agent DRL algorithms compared with baseline schemes.",
"Considering the fact that UAVs have continuous action spaces (i.e., trajectory, location, and transmit power), Zhu $et~al.$ [27] proposed an actor-critic (AC)-based algorithm to jointly adjust UAV's transmission control and three -dimensional (3D) flight to maximize the network throughput over continuous action spaces.",
"Similarly, in [28], the authors adopted the AC-based algorithm to optimize the device association, resource allocation, and trajectory of UAVs in dynamic environments.",
"However, to the best of our knowledge, there are no studies applying RL or DRL to achieve the implementation of FL for the UAVs-enabled wireless network, and optimize the FL convergence or accuracy by jointly designing scheduling and resource management."
],
[
"Contributions and Organization",
"Motivated by the aforementioned challenges, this paper first develops an asynchronous federated learning (AFL) to achieve fast convergence speed and high learning accuracy for multi-UAV-enabled wireless networks.",
"The proposed framework can enable mobile devices to train their AI models locally and asynchronously upload the model parameters to UAV servers for model aggregation without uploading the raw private data, which reduces communication cost and privacy threats, as well as improves the round efficiency and model accuracy.",
"Considering the dynamic environment and different computational capabilities of devices, we also propose an asynchronous advantage actor-critic (A3C)-based joint device selection, UAVs placement, and resource management algorithm to further enhance the overall federated execution efficiency and model accuracy.",
"The main contributions of this paper are summarized as follows.",
"We develop a novel privacy-preserving AFL framework for a multi-UAV-enabled wireless network to aggregate and update the AI model parameters in an asynchronous manner, where the mobile devices with high communication and computation capabilities are selected to participate in global aggregation instead of waiting for all associated devices to accomplish their local model update.",
"We propose an A3C-based joint device selection, UAVs placement, and resource management algorithm to further minimize the federated execution time and learning accuracy loss under dynamic environments as well as large-scale continuous action spaces.",
"We conducte extensive simulations to demonstrate that the proposed AFL framework and A3C-based algorithm can significantly improve the FL model accuracy, convergence speed, and federated execution efficiency compared to other baseline solutions under different scenarios.",
"Organization: The remainder of this paper is organized as follows.",
"The system model and problem formulation are provided in Section II.",
"Section III proposes the AFL framework and A3C-based algorithm for UAV-enabled wireless networks.",
"Simulation results and analysis are provided in Section IV, and the paper is concluded in Section V. We consider an FL instance consisting of a number of ground devices, which are associated with several parameter servers residing at multiple UAVs in the sky.",
"As shown in Fig.",
"1, the multi-UAV-enabled network consists of $N$ UAVs and $K$ single-antenna devices, denoted by $\\mathcal {N} = \\lbrace 1,...,N\\rbrace $ and $\\mathcal {K} = \\lbrace 1,...,K\\rbrace $ , respectively.",
"In practical wireless networks, the ground BS may face congestion due to a BS malfunction, or a temporary festival or a big sport event, and even ground BS may fail to provide coverage in some remote areas.",
"In this case, the deployment of UAVs has been emerged as a potential technique for providing wireless coverage for ground devices.",
"As illustrated in Fig.",
"1, mobile devices are randomly located on the ground, and multiple UAVs fly in the sky to provide wireless services for them by using frequency division multiple access (FDMA).",
"Here, the unity frequency reuse (UFR) design is employed in multi-UAV networks to enhance the spectrum utilization, where the communication band is reused across all cells with each cell being associated with one UAV [23]-[26].",
"Figure: Federated learning-based UAV-enabled wireless networks.In the FL instance, each device has its own personal dataset and it is willing to upload a part of inflammation (i.e., AI model parameters) to its associated UAV server in a privacy-preserving manner.",
"In addition, devices involved in a common computing task (e.g., training a classification model) are more likely to work with others together to finish the task collaboratively, by adopting AI techniques.",
"These devices will update their own local AI model parameters to associated UAV servers for global model aggregation.",
"Each device processes the local model training based on its local raw dataset without sharing the raw data with other devices, to protect the privacy of data providers.",
"After completing local model training, each device will send its local model parameters to its associated UAV server by the uplink communication channel, and the corresponding UAV server aggregates the local parameters from the selected devices before broadcasting the aggregated global parameters to the devices by the downlink communication channel.",
"For the FL aggregation, communication and computation resource optimization (related to both the computation capability and communication capability) is necessary to realize the implementation of model aggregation, such as local computation, communication [5], [6], and global computation at both the devices and the UAV servers.",
"The computation capacity of each device or each UAV server is captured by the CPU capability, learning time, and learning accuracy.",
"The communication capacity can be characterized in the forms of the available transmission rate and transmission latency.",
"It is worth noting that both the computation and communication capabilities may vary at different time slots due to different computing tasks and mobility of UAVs or devices.",
"Hence, this paper also focuses on joint device selection, UAVs placement design, and resource management to enhance the leaning efficiency and accuracy of FL."
],
[
"FL Model",
"This subsection briefly investigates the basics of FL in multi-UAV-enabled networks.",
"Hereinafter, the considered AI model that is trained on each device's training dataset is called local FL model, while the FL model that is built by each UAV server employing local model parameter inputs from its selected devices is called global FL model.",
"Let ${{\\bf {w}}_n}$ denote the model parameters which is related to the global model of the $n$ -th UAV server, let ${{\\bf {w}}_k}$ denote the local model parameters of the $k$ -th device, and let ${\\mathcal {D}_k}$ denote the set of training dataset used at the $k$ -th device.",
"Accordingly, if the $k$ -th device is associated with the $n$ -th UAV server, we introduce the loss function $f\\left({{{\\bf {w}}_n};{{\\bf {s}}_{k,i}},{z_{k,i}}} \\right)$ to quantify the FL performance error over the input data sample vector ${{\\bf {s}}_{k,i}}$ on the learning model ${\\bf {w}}$ and the desired output scalar ${z_{k,i}}$ for each input sample $i$ at the $k$ -th device.",
"For the $k$ -th device, the sum loss function on its training dataset ${\\mathcal {D}_k}$ can be expressed as [5], [6] $\\begin{split}{F_k}\\left( {{{\\bf {w}}_n}} \\right) = \\frac{1}{{\\left|{{{\\mathcal {D}_k}}} \\right|}}\\sum \\limits _{i \\in {{\\mathcal {D}_k}}}{f\\left( {{{\\bf {w}}_n};{{\\bf {s}}_{k,i}},{z_{k,i}}} \\right)},\\;\\;\\forall k \\in \\mathcal {K},\\end{split}$ where $\\left| {{\\mathcal {D}_k}} \\right|$ is the cardinality of set ${\\mathcal {D}_k}$ .",
"Accordingly, at the $n$ -th UAV server, the average global loss function with the distributed local datasets of all selected devices is defined as [5], [6] $\\begin{split}F\\left( {{{\\bf {w}}_n}} \\right) \\Delta \\over =\\sum \\limits _{k \\in {\\mathcal {K}_n}} {\\frac{{\\left|{{\\mathcal {D}_k}} \\right|{F_k}({{\\bf {w}}_n})}}{{\\left|{{\\mathcal {D}_n}} \\right|}}} = \\frac{1}{{\\left| {{\\mathcal {D}_n}}\\right|}}\\sum \\limits _{k \\in {\\mathcal {K}_n}} {\\sum \\limits _{i \\in {\\mathcal {D}_k}} {f\\left({{{\\bf {w}}_n};{{\\bf {s}}_{k,i}},{z_{k,i}}} \\right)} } ,\\end{split}$ where $\\left| {{\\mathcal {D}_n}} \\right| = \\sum \\nolimits _{k \\in {{\\mathcal {K}}_n}} {\\left| {{\\mathcal {D}_k}} \\right|} $ is the sum data samples from all selected devices at the $n$ -th UAV-enabled cell, and ${\\mathcal {K}_n}$ is the set of the devices associated with the $n$ -th UAV server with ${K_n} = |{\\mathcal {K}_n}|$ being the number of the selected devices.",
"The objective of the FL task is to search for the optimal model parameters at the $n$ -th UAV server that minimizes the global loss function as [5], [6] $\\begin{split}{\\bf {w}}_n^ * = \\arg \\min F\\left( {{{\\bf {w}}_n}}\\right),\\;\\forall n \\in \\mathcal {N}.\\end{split}$ Note that $F\\left( {{{\\bf {w}}_n}} \\right)$ cannot be directly computed by employing the raw datasets from selected devices, for the sake of protecting the information privacy and security of each device.",
"Hence, the problem in (3) can be addressed in a distributed manner.",
"We pay attention to the FL model-averaging implementation in the subsequent exposition while the same principle is generally adopted to realize alternative implementation according to the gradient-averaging method [5], [6]."
],
[
"Communication Model",
"Different from the propagation of ground communications, the air-to-ground or ground-to-air channel mainly depends on the propagation environments, transmission distance, and elevation angle [1]-[3].",
"Similar to existing works [23]-[28], the UAV-to-device or device-to-UAV communication channel link is modeled by a probabilistic path loss model, where both the LoS and non-LoS (NLoS) links are taken into account.",
"The LoS and NLoS path loss in dB from the $n$ -th UAV server to the $k$ -th device can be given by $\\begin{split}l_{n,k}^{_{{\\rm {LoS}}}} = 20\\log \\left( {{{4\\pi {f_c}{d_{n,k}}}\\mathord {\\left\\bad.",
"{\\vphantom{{4\\pi {f_c}{d_{n,k}}} c}} \\right.\\hspace{0.0pt}} c}} \\right) + {\\eta ^{{\\rm {LoS}}}},\\end{split}$ $\\begin{split}l_{n,k}^{_{{\\rm {NLoS}}}} = 20\\log \\left( {{{4\\pi {f_c}{d_{n,k}}}\\mathord {\\left\\bad.",
"{\\vphantom{{4\\pi {f_c}{d_{n,k}}} c}} \\right.\\hspace{0.0pt}} c}} \\right) + {\\eta ^{{\\rm {NLoS}}}},\\end{split}$ respectively, where ${f_c}$ and $c$ denote the carrier frequency and the speed of light, respectively.",
"${d_{n,k}}$ is the transmission distance from the $n$ -th UAV server to the $k$ -th device, given by ${d_{n,k}} = {\\left[ {{{({x_n} - {x_k})}^2} + {{({y_n} -{y_k})}^2} + h_n^2} \\right]^{{1 \\mathord {\\left\\bad.",
"{\\vphantom{1 2}} \\right.\\hspace{0.0pt}} 2}}}$ , where ${\\Theta _n} = ({x_n},{y_n},{h_n})$ denotes the location of the $n$ -th UAV server, ${\\Theta _k} = ({x_k},{y_k})$ is the location of the $k$ -th device, and $h$ is the height of UAV servers in the sky, and we assume that all UAVs have the same height.",
"${\\eta ^{{\\rm {LoS}}}}$ and ${\\eta ^{{\\rm {NLoS}}}}$ are the mean additional losses for the LoS and NLoS links due to the free space propagation loss, respectively, as defined in [29].",
"In the communication model, the probability of LoS connection between the $n$ -th UAV server and the $k$ -th device is expressed as $\\begin{split}{\\rm {P}}_{n,k}^{_{{\\rm {LoS}}}} = \\frac{1}{{1 + {\\xi _1}\\exp \\left({ - {\\xi _2}({\\theta _{n,k}} - {\\xi _1})} \\right)}},\\end{split}$ where ${\\xi _1}$ and ${\\xi _2}$ are constant values which depend on the carrier frequency and UAV network environment, and ${\\theta _{n,k}}$ is the elevation angle from UAV $n$ to device $k$ (in degree): ${\\theta _{n,k}} = {\\textstyle {{180} \\over {\\rm {\\pi }}}}\\cdot {\\sin ^{ - 1}}({\\textstyle {{{h_n}} \\over {{d_{n,k}}}}})$ .",
"Furthermore, the probability of NLoS is ${\\rm {P}}_{n,k}^{_{{\\rm {NLoS}}}} = 1 -{\\rm {P}}_{n,k}^{_{{\\rm {LoS}}}}$ .",
"In this context, the probabilistic path loss between the $n$ -th UAV and the $k$ -th device is given by $\\begin{split}{l_{n,k}} = {\\rm {P}}_{n,k}^{_{{\\rm {LoS}}}} \\cdot l_{n,k}^{_{{\\rm {LoS}}}} + {\\rm {P}}_{n,k}^{_{{\\rm {NLoS}}}} \\cdot l_{n,k}^{_{{\\rm {NLoS}}}}.\\end{split}$ We assume that orthogonal frequency-division multiple access (OFDMA) technique is used for uplink channel access, where each UAV-enabled cell has $M$ orthogonal uplink subchannels and these subchannels are reused across all cells.",
"Note that the total uplink bandwidth is divided into $M$ orthogonal subchannels, denoted by $\\mathcal {M} =\\lbrace 1,...,M\\rbrace $ .",
"In this case, each UAV server will suffer inter-cell interference (ICI) from other nearby devices associated by other cells on the same spectrum band.",
"Hence, according to the path loss model, when the $k$ -th device is associated with the $n$ -th UAV server, the received signal-to-interference plus-noise-ratio (SINR) over the allocated subchannel $m$ at the $n$ -th UAV server in the uplink is characterized as $\\begin{split}SINR_{n,k,m}^{\\rm {U}} = \\frac{{P_{k,m}^{\\rm {U}}{{10}^{ -{l_{n,k}}/10}}}}{{\\sum \\limits _{k^{\\prime } \\in \\mathcal {K},k^{\\prime } \\ne k}{P_{k^{\\prime },m}^{\\rm {U}}{{10}^{ - {l_{n,k}}/10}}} + {\\sigma ^2}}},\\end{split}$ where $P_{k,m}^{\\rm {U}}$ is the transmit power of the $k$ -th device allocated on the $m$ -th subchannel, and ${\\sigma ^2}$ denotes the power of the Gaussian noise.",
"Furthermore, $\\sum \\limits _{k^{\\prime } \\in \\mathcal {K},k^{\\prime } \\ne k} {P_{k^{\\prime },m}^{\\rm {U}}{{10}^{ - {l_{n,k}}/10}}}$ is the ICI received at UAV server $n$ over the $m$ -th subchannel which is generated from nearby devices associated by other cells.",
"Accordingly, the achievable uplink data rates (in bit/s) for the $k$ -th device over the allocated subchannels can be expressed as $\\begin{split}R_k^{\\rm {U}} = {B_{{\\rm {sub}}}}\\sum \\limits _{m = 1}^M {\\left({{\\chi _{n,k,m}}{{\\log }_2}(1 + SINR_{n,k,m}^{\\rm {U}})} \\right)},\\end{split}$ where ${B_{{\\rm {sub}}}} = {B^{\\rm {U}}}/M$ is the bandwidth of each uplink subchannel with ${B^{\\rm {U}}}$ being the total bandwidth in uplink, and ${\\chi _{n,k,m}}$ is the uplink subchannel allocation indicator, ${\\chi _{n,k,m}} \\in \\lbrace 0,1\\rbrace $ ; ${\\chi _{n,k,m}} =1$ shows that the $k$ -th device is associated with the $n$ -th UAV server on the $m$ -th subchannel ; otherwise, ${\\chi _{n,k,m}} =0$ .",
"For the downlink channel, we assume that each UAV server occupies a given downlink channel to broadcast the global model parameters to its associated devices.",
"As the deployment of multiple UAVs in the sky, devices located in the overlapped areas will also suffer ICI from other nearby UAV servers on the same spectrum band.",
"Then, when the $k$ -th device is associated with the $n$ -th UAV server, its received SINR in the downlink can be expressed as $\\begin{split}SINR_{n,k}^{\\rm {D}} = \\frac{{P_n^{\\rm {D}}{{10}^{ -{l_{n,k}}/10}}}}{{\\sum \\limits _{n^{\\prime } \\in \\mathcal {N}, n^{\\prime } \\ne n}{P_{n^{\\prime }}^{\\rm {D}}{{10}^{ - {l_{n,k}}/10}}} + {\\sigma ^2}}}.\\end{split}$ where $P_n^{\\rm {D}}$ is the transmit power of the $n$ -th UAV server.",
"Accordingly, the achievable downlink data rate at the $k$ -th device is given by $\\begin{split}R_k^{\\rm {D}} = {B^{\\rm {D}}}{\\log _2}(1 + SINR_{n,k}^{\\rm {D}}),\\end{split}$ where ${B^{\\rm {D}}}$ is the bandwidth in the downlink."
],
[
"FL Model Update Latency Analysis",
"Local/Global Model Update Latency: Define ${C_k}$ and ${C_n}$ as the number of CPU cycles used for training model on one sample data at the $k$ -th device and $n$ -th UAV server, respectively.",
"Let ${f_k}$ and ${f_n}$ denote the computation capability (CPU cycles/s) of device $k$ and UAV server $n$ , respectively, where ${f_k} \\in (f_k^{\\min },f_k^{\\max })$ with $f_k^{\\min }$ and $f_k^{\\max }$ being the minimum and maximum CPU computation capabilities of device $k$ , respectively.",
"Accordingly, the local model computation latency of device $k$ and the global model aggregation latency of UAV server $n$ at the $t$ -th time slot are respectively given by $\\begin{split}T_k^{{\\rm {Loc}},t} = \\left| {{\\mathcal {D}_k}}\\right|{C_k}/{f_k},\\;\\forall k,\\end{split}$ $\\begin{split}T_n^{{\\rm {Glo}},t} = \\left| {{\\mathcal {D}_n}}\\right|{C_n}/{f_n},\\;\\forall n.\\end{split}$ Global FL Model Broadcast Latency: Let ${L_n}$ denote the number of bits required for each UAV server $n$ to broadcast the global model parameters to the associated devices.",
"For the $n$ -th UAV server, the global model parameters broadcast latency is expressed as $\\begin{split}T_n^{{\\rm {D}},t} = {L_n}/R_n^{{\\rm {D}},t},\\;\\forall n.\\end{split}$ Local FL Model Upload Latency: Let ${L_k}$ denote the number of bits needed for each device $k$ to upload its local model parameters to its associated UAV server.",
"For the $k$ -th device, the local model parameters upload latency can be given by $\\begin{split}T_k^{{\\rm {U}},t} = {L_k}/R_k^{{\\rm {U}},t},\\;\\forall k.\\end{split}$ The one round time for scheduling the $k$ -th device is comprised of local model update latency, uplink local model upload latency, global model aggregation latency, and downlink global model broadcast latency.",
"Therefore, the total time cost for scheduling the FL-model of the $k$ -th device at one round can be given by $\\begin{split}T_k^t = T_k^{{\\rm {Loc}},t} + T_n^{{\\rm {Glo}},t}+ T_k^{{\\rm {D}},t}+ T_k^{{\\rm {U}},t},\\;\\;\\forall k.\\end{split}$"
],
[
"Problem Formulation",
"In UAV-enabled networks, the various computation capacities of devices and the time-varying communication channel conditions play an important role on the implementation of FL global aggregation.",
"It is desirable to select the devices with high computation capability, communication capability, and accurate learned models.",
"In addition, it is necessary to schedule UAVs' locations to provide the best channel gains for communication services, and manage the radio and power resources to improve the communication data rate for AI model parameters upload and broadcast.",
"Thus, we aim to select a subset of devices, design UAVs' locations, manage subchannel and transmit power resources to minimize the FL model execution time and the learning accuracy loss.",
"In the $n$ -th UAV-enabled cell, we define the execution time cost as $\\begin{split}c_n^{{\\rm {Time}}}(t) = \\frac{1}{{{K_n}}}\\sum \\limits _{k =1}^{{K_n}} {T_k^t} ,\\;\\;\\forall n,\\end{split}$ where ${K_n}$ is the number of devices selected by the $n$ -th UAV server for the federated model aggregation.",
"In addition, the learning accuracy loss can be defined as $\\begin{split}c_n^{{\\rm {Loss}}}(t) = \\frac{1}{{\\left| {{\\mathcal {D}_n}}\\right|}}\\sum \\limits _{k \\in {\\mathcal {K}_n}} {\\sum \\limits _{i \\in {\\mathcal {D}_k}} {f\\left({{{\\bf {w}}_n};{{\\bf {s}}_{k,i}},{z_{k,i}}} \\right)} } ,\\;\\;\\forall n.\\end{split}$ In the network, the learning accuracy loss is measured at the end of each time slot.",
"Given the aforementioned system model, our objective is to minimize the weighted sum of one-round FL model execution time and learning accuracy loss.",
"Targeting at learning acceleration and efficiency, it is desirable to select a subset of devices with high computation capability, place the UAVs' locations with best channel quality, as well as manage both subchannel and power resources.",
"Hence, the optimization problem can be formulated as follows $\\begin{split}\\begin{array}{l}\\mathop {\\min }\\limits _{{\\Theta _{n,}}{\\mathbf {\\rho }},{\\mathbf {\\chi }},P_n^{\\rm {D}}} \\left( {\\lambda c_n^{{\\rm {Time}}}(t) + (1 - \\lambda )c_n^{{\\rm {Loss}}}(t)} \\right),\\;\\;\\forall n\\\\s.t.\\;\\;{\\rm {a}})\\;{\\rho _{n,k}} \\in \\lbrace 0,1\\rbrace ,\\;{\\chi _{n,k,m}} \\in \\lbrace 0,1\\rbrace ,\\forall k,m,\\\\\\;\\;\\;\\;\\;\\;\\;{\\rm {b}})\\sum \\limits _{n \\in \\mathcal {N}} {{\\rho _{n,k}} \\le 1,\\forall k} ,\\\\\\;\\;\\;\\;\\;\\;\\;{\\rm {c}})\\sum \\limits _{k \\in \\mathcal {K}} {\\sum \\limits _{m \\in M} {{\\chi _{n,k,m}} \\le M,} } \\\\\\;\\;\\;\\;\\;\\;\\;{\\rm {d}})\\;0 \\le P_n^{\\rm {D}} \\le P_n^{{\\rm {max}}},\\\\\\;\\;\\;\\;\\;\\;\\;{\\rm {e}})\\;0 \\le {f_k} \\le f_k^{\\max },\\;\\forall k,\\end{array}\\end{split}$ where ${\\mathbf {\\rho }} = {[{\\rho _{n,1}}, \\ldots ,{\\rho _{n,K}}]^T}$ and ${\\mathbf {\\chi }} = [{\\chi _{n,1,1}}, \\ldots ,{\\chi _{n,1,M}},{\\chi _{n,2,1}}, \\ldots ,{\\chi _{n,2,M}}, \\ldots ,{\\chi _{n,K,M}}]$ , $P_n^{{\\rm {max}}}$ is the maximum transmit power of UAV server $n$ , and $\\lambda \\in [0,1]$ is a constant weight parameter which is used to balance the one-round execution time $c_n^{{\\rm {Time}}}$ and the learning accuracy loss $c_n^{{\\rm {Loss}}}$ .",
"Constraint (19b) indicates that each device can only associate one UAV server to preform model aggregation.",
"Constraint (19c) is used to ensure the maximum number of available subchannels of each UAV-enabled cell.",
"Constraint (19d) ensures the maximum transmit power of UAV server $n$ in the downlink.",
"Constraint (19d) represents the computation capacity range of devices.",
"We would like to mention that for the resource management issue, we focus our study on the uplink subchannel allocation since the global model parameters at each UAV server are broadcast by a given downlink band.",
"In addition, we also investigate the power allocation at each UAV server by assuming that the transmit power of each device is given."
],
[
"Asynchronous Federated Reinforcement Learning Solution",
"The optimization problem formulated in (19) is challenging to tackle as it is a non-convex combination and NP-hard problem.",
"In addition, the time-varying channel condition and different computation capacities of devices result in dynamic open and uncertain characteristics, which increases the difficulty of addressing the optimization problem.",
"Model-free RL is one of the dynamic programming technique which is capable of tackling the decision-making problem by learning an optimized policy in dynamic environments [30].",
"Thus, RL is introduced to implement the self-scheduling based FL aggregation process in multi-UAV-enable networks.",
"Moreover, traditional FL models mostly use a synchronous learning framework to update the model parameters between the UAV servers and client devices, which inherently have several key challenges.",
"Firstly, due to the mobility and different computation capacities of mobile devices, it is difficult to maintain continuous synchronized communication between UAV servers and client devices.",
"Secondly, each UAV server needs to wait for all selected devices to finish their local results before aggregating the model parameters, which consequently increases the global learning delay with low round efficiency.",
"Furthermore, a part of the action (i.e., transmission power and horizontal locations of UAVs) in our joint placement design and resource management optimization problem have continuous spaces.",
"Therefore, we propose an asynchronous advantage actor-critic-based asynchronous federated learning algorithm (called A3C-AFL) to address the aforementioned challenges, and the corresponding framework is provided with the following extensive details in this section.",
"Figure: The framework of A3C-AFL in UAV-enabled networks.The proposed A3C-AFL framework consists of three phases: device selection, UAVs placement as well as resource management; local training; and global aggregation, as illustrated in Fig.",
"2.",
"The strategy of the device selection, UAV placement design, and resource management can be achieved by using the A3C algorithm.",
"Then, the selected devices in each UAV-enabled cell perform local training and periodically upload their local model parameters to their corresponding associated UAV server for global aggregation.",
"Finally, each UAV server broadcasts the updated global model parameters to the associated devices.",
"Device Selection, UAVs Placement, and Resource Management: To improve the leaning efficiency and accuracy, each UAV server selects the devices with high computation and communication capacities to perform federated model parameters update.",
"Then, the joint UAVs placement design and resource management are implemented to minimize the execution time.",
"This phase is implemented by adopting the A3C-based algorithm [31], which will be elaborated in Section III.B.",
"Local Training: At the $t$ -th global communication round, after receiving the global model parameters ${{\\bf {w}}_n}(t - 1)$ from the associated UAV server, each selected device $k$ trains its local model parameters ${{\\bf {w}}_k}(t)$ based on its dataset ${\\mathcal {D}_k}$ by calculating the local stochastic gradient descent $\\nabla {F_k}\\left( {{{\\bf {w}}_n}(t - 1)} \\right)$ , i.e., $\\begin{split}{{\\bf {w}}_k}(t) = {{\\bf {w}}_n}(t - 1) - \\eta \\nabla {F_k}\\left({{{\\bf {w}}_n}(t - 1)} \\right),\\end{split}$ where $\\eta > 0$ denotes the learning step size and $\\nabla $ is the gradient operator.",
"After updating the parameters ${{\\bf {w}}_k}(t)$ , each device $k$ uploads its trained local model parameters ${{\\bf {w}}_k}(t)$ to its associated UAV server for further FL model aggregation.",
"Global Aggregation: Each UAV server $n$ (aggregator) retrieves the uploaded local FL model parameters from its selected devices and implements the global model aggregation through averaging and updating the global model parameters ${{\\bf {w}}_n}(t)$ as follows $\\begin{split}{{\\bf {w}}_n}(t) = \\frac{{\\sum \\nolimits _{k \\in {K_n}}{|{\\mathcal {D}_k}|{{\\bf {w}}_k}(t)} }}{{\\left| {{\\mathcal {D}_n}} \\right|}}.\\end{split}$ The mode parameters update process involves the iteration between (20) and (21) until the FL model coverages."
],
[
"Modeling of Reinforcement Learning Environment",
"We formulate the combinatorial optimization problem (19) as a Markov decision process (MDP), i.e., $\\left\\langle {\\mathcal {S},\\mathcal {A},\\mathcal {P},r} \\right\\rangle $ , where $\\mathcal {A}$ , $\\mathcal {S}$ , $\\mathcal {P}$ , and $r$ are network state space, action space, state transition probability, and reward function, respectively.",
"In multi-UAV-enabled networks, UAV servers are considered as multiple agents to observe the environment and try to maximize the expected sum reward.",
"Then, we use RL to tackle the device selection, UAVs placement, and resource management problem (19) in the federated learning scenario.",
"All of the multiple agents iteratively update their policies based on their observations by interacting with the network environment.",
"According to the aforementioned MDP, i.e., $\\left\\langle {\\mathcal {S},\\mathcal {A},\\mathcal {P},r} \\right\\rangle $ , the corresponding elements in MDP are described as below: State Space: At each time slot $t$ , the network state ${s_n}(t) \\in \\mathcal {S}$ of each UAV server $n$ for characterizing the environment is comprised of the following several parts: horizontal location of the $n$ -th UAV server ${\\Theta _n}(t) =\\left( {{x_n}(t),{y_n}(t)} \\right)$ ; device selection indicators at the $n$ -th UAV-enabled cell, ${\\left\\lbrace {{\\rho _{n,k}}(t- 1)} \\right\\rbrace _{k \\in {\\mathcal {K}_n}}}$ ; the subchannel allocation state ${\\left\\lbrace {{\\chi _{n,k,m}}(t - 1)} \\right\\rbrace _{k\\in {\\mathcal {K}_n}}}$ ; locations of selected devices, ${\\left\\lbrace {{\\Theta _k}(t) = \\left( {{x_k}(t),{y_k}(t)} \\right)} \\right\\rbrace _{k\\in {\\mathcal {K}_n}}}$ ; the remaining payload needs to be transmitted, ${\\left\\lbrace {{L_k}(t - 1)} \\right\\rbrace _{k \\in {K_n}}}$ and ${L_n}(t - 1)$ .",
"According to the above definitions, in the multi-agent RL model, the network state of the $n$ -th UAV server at time slot $t$ can be expressed by $\\begin{split}{s_n}(t) = \\left\\lbrace {{\\Theta _n}(t),{L_n}(t - 1),{{\\left\\lbrace {{\\Theta _k}(t),{\\rho _{n,k}}(t - 1),{\\chi _{n,k,m}}(t - 1),{L_k}(t)}\\right\\rbrace }_{k \\in {\\mathcal {K}_n}}}} \\right\\rbrace ,\\end{split}$ and the network state of all UAV servers is given by ${s_t} =\\left\\lbrace {{{\\lbrace {s_n}(t)\\rbrace }_{n \\in \\mathcal {N}}}} \\right\\rbrace $ .",
"Action Space: At the $t$ -th time slot, each agent $n$ (i.e., UAV server $n$ ) selects its corresponding action ${a_n}(t)\\in \\mathcal {A}$ according to the observed state ${s_n}(t)$ , where ${a_n}(t)$ consists of the horizontal position ${\\Theta _n}(t) =\\left( {{x_n}(t),{y_n}(t)} \\right)$ , the device selection indicators ${\\left\\lbrace {{\\rho _{n,k}}(t)} \\right\\rbrace _{k \\in {\\mathcal {K}_n}}}$ , the subchannel allocation indicators ${\\left\\lbrace {{\\chi _{n,k,m}}(t)} \\right\\rbrace _{k \\in {\\mathcal {K}_n}}}$ , and the transmit power allocation level $P_n^{\\rm {D}}(t)$ , that is $\\begin{split}{a_n}(t) = \\left\\lbrace {{\\Theta _n}(t),P_n^{\\rm {D}}(t),{{\\left\\lbrace {{\\rho _{n,k}}(t),{\\chi _{n,k,m}}(t)} \\right\\rbrace }_{k \\in {\\mathcal {K}_n}}}} \\right\\rbrace ,\\end{split}$ and the action of all UAV servers is given by ${a_t} = \\left\\lbrace {{{\\lbrace {a_n}(t)\\rbrace }_{n \\in \\mathcal {N}}}} \\right\\rbrace $ .",
"At the end of each time slot $t$ , each UAV server moves to an updated horizontal position, updates the device association indicators, allocates subchannel and power resources to devices, respectively.",
"State Transition Function: Let $\\mathcal {P}({s_{t +1}}|{s_t},{a_t})$ denote the transition probability of one UAV server entering a new state ${s_{t + 1}}$ , after executing an action ${a_t}$ at the current state ${s_t}$ .",
"Policy: Let $\\pi $ denote the policy function, which is a mapping from perceived states with the probability distribution over actions that the agent can select in those states, $\\pi (a|s)=\\mathcal {P}(a|s)$ .",
"Reward Function: In the context of minimizing the one round FL model execution time and learning accuracy loss, the reward function ${r_t}$ is designed to evaluate the quality of a learning policy under the current state-action pair $({s_t},{a_t})$ .",
"In this paper, we design a reward function that can capture the federated execution time and learning accuracy loss.",
"According to the objective function in (19), the presented reward function of the $n$ -th agent (i.e., UAV server $n$ ) at one time step is expressed as $\\begin{split}{r_n}({s_t},{a_t}) = - \\left( {\\lambda c_n^{{\\rm {Time}}}(t) + (1- \\lambda )c_n^{{\\rm {Loss}}}(t)} \\right).\\end{split}$ The objective of the UAV-enabled network is to minimize the execution time and learning accuracy loss in FL.",
"For the MDP model, the objective is to search for an action $a$ which is capable of maximizing the cumulative reward (minimize the total cumulative execution time and learning accuracy loss), given by $\\begin{split}{U_t} = \\sum \\limits _{\\tau = 0}^\\infty {{\\gamma _\\tau }r({s_{t +\\tau }},{a_{t + \\tau }})},\\end{split}$ where $\\gamma $ denotes the discount factor."
],
[
"Multi-Agent A3C Algorithm",
"In multi-UAV-enabled networks, the spaces of both state and action are large, and hence combining deep learning and RL, i.e., DRL algorithms (e.g., deep Q-learning and deep deterministic policy gradient), are effective for handling the large-scale decisions making problems [30].",
"These algorithms generally adopt experience replay to improve learning efficiency.",
"However, experience replay requires enough memory space and computation resources to guarantee the learning accuracy, and it only uses the data generated through old policy to update the learning process [31].",
"The negative issues of the aforementioned DRL algorithms motivate us to search for a better algorithm, A3C (asynchronous advantage actor-critic).",
"Different from a classical AC algorithm with only one learning agent, A3C is capable of enabling asynchronous multiple agents (i.e., UAV servers) to parallelly interact with their environments and achieve different exploration policies.",
"In A3C, the actor network generally adopts policy gradient schemes to select actions under a given parameterized policy $\\pi (a|s;{{\\mathbf {\\theta }}_{\\rm {a}}}) = \\mathcal {P}(a|s,{{\\mathbf {\\theta }}_{\\rm {a}}})$ with a set of actor parameters ${{\\mathbf {\\theta }}_{\\rm {a}}}$ , and then updates the parameters by the gradient-descent methods.",
"The critic network uses an estimator of the state value function ${V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})$ to qualify the expected return under a certain state $s$ with a set of critic parameters ${{\\mathbf {\\theta }}_{\\rm {c}}}$ .",
"Each UAV server acts as one agent to evaluate and optimize its policy based on the value function, which is defined as the expected long-term cumulative reward achieved over the entire learning process.",
"Here, we defined two value functions, called state value function and state-action value function, where the former one is the expected return under a given policy $\\pi $ while the latter one is the expected return under a given policy $\\pi $ after executing action $a$ in state $s$ .",
"These two functions are respectively expressed as $\\begin{split}{V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}}) = \\mathbb {E}\\left\\lbrace {{U_t}|{s_t} = s,\\pi } \\right\\rbrace ,\\end{split}$ $\\begin{split}{Q^\\pi }({s_t},{a_t}) = \\mathbb {E}\\left\\lbrace {{U_t}|{s_t} = s,{a_t} =a,\\pi } \\right\\rbrace ,\\end{split}$ where $\\mathbb {E}\\lbrace \\cdot \\rbrace $ is the expectation.",
"A3C adopts the multi-step reward to update the parameters of the policy in the actor network and the value function in the critic network [31].",
"Here, the $i$ -step reward is defined as [31] $\\begin{split}{U_t} = \\sum \\limits _{i = 0}^{\\tau - 1} {{\\gamma _i}r({s_{t +i}},{a_{t + i}}) + {\\gamma _\\tau }{V^\\pi }({s_{t + \\tau }};{{\\mathbf {\\theta }}_{\\rm {c}}})} .\\end{split}$ Similar to the AC framework, A3C also adopts policy gradient schemes to perform parameters update which may cause high variance in the critic network.",
"In order to address this issue, an advantage function $A(s,a) = {Q^\\pi }(s,a) - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})$ is employed to replace ${Q^\\pi }(s,a)$ in the policy gradient process.",
"Since ${Q^\\pi }(s,a)$ cannot be determined in A3C [31], we use ${U_t} - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})$ as an estimator for the advantage function $A(s,a) = {Q^\\pi }(s,a) - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})$ .",
"As a result, the advantage function is given by $\\begin{split}\\begin{array}{l}A({s_t},{a_t}) = {U_t} - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})\\\\= \\sum \\limits _{i = 0}^{\\tau - 1} {{\\gamma _i}r({s_{t + i}},{a_{t + i}}) + {\\gamma _\\tau }{V^\\pi }({s_{t + \\tau }};{{\\mathbf {\\theta }}_{\\rm {c}}}) - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})}.\\end{array}\\end{split}$ In the A3C framework, two loss functions are associated with the two deep neural network outputs of the actor network and the critic network.",
"According to the advantage function (29), the actor loss function [31] under a given policy $\\pi $ is defined as $\\begin{split}{f_\\pi }({{\\mathbf {\\theta }}_{\\rm {a}}}) = \\log \\pi ({a_t}|{s_t};{{\\mathbf {\\theta }}_{\\rm {a}}})\\left( {{U_t} - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})} \\right) + \\vartheta G\\left({\\pi ({s_t};{{\\mathbf {\\theta }}_{\\rm {a}}})} \\right),\\end{split}$ where $\\vartheta $ is a hyperparameter that controls the strength of the entropy regularization term, such as the exploration and exploitation management during the training process, and $G\\left({\\pi ({s_t};{{\\mathbf {\\theta }}_{\\rm {a}}})} \\right)$ is the entropy which is employed to favor exploration in training.",
"From (30), the accumulated gradient of the actor loss function ${f_\\pi }({{\\mathbf {\\theta }}_{\\rm {a}}})$ is expressed as $\\begin{split}d{{\\mathbf {\\theta }}_{\\rm {a}}} = d{{\\mathbf {\\theta }}_{\\rm {a}}} + {\\nabla _{{{{\\mathbf {\\theta ^{\\prime }}}}_{\\rm {a}}}}}\\log \\pi ({a_t}|{s_t};{{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}})\\left( {{U_t} - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})} \\right) + \\vartheta {\\nabla _{{{{\\mathbf {\\theta ^{\\prime }}}}_{\\rm {a}}}}}G\\left( {\\pi ({s_t};{{{\\mathbf {\\theta ^{\\prime }}}}_{\\rm {a}}})}\\right),\\end{split}$ where ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}}$ is the thread-specific actor network parameters.",
"In addition, the critic loss function of the estimated value function is defined as $\\begin{split}f({{\\mathbf {\\theta }}_{\\rm {c}}}) = {\\mathop {\\rm }\\nolimits } {\\left({{U_t} - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})} \\right)^2},\\end{split}$ and the accumulated gradient of the critic loss function $f({{\\mathbf {\\theta }}_{\\rm {c}}})$ in the critic network is calculated by $\\begin{split}d{{\\mathbf {\\theta }}_{\\rm {c}}} = d{{\\mathbf {\\theta }}_{\\rm {c}}} +\\frac{{\\partial {{\\left( {{U_t} - {V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}})} \\right)}^2}}}{{\\partial {{{\\mathbf {\\theta ^{\\prime }}}}_{\\rm {c}}}}},\\end{split}$ where ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {c}}}$ is the thread-specific critic network parameters.",
"After updating the accumulated gradients shown in (31) and (33), we adopt the standard non-centered RMSProp algorithm [32] to perform training for the loss function minimization in our presented A3C framework.",
"The estimated gradient with the RMSProp algorithm of the actor or critic network is expressed as $\\begin{split}g = \\alpha g + (1 - \\alpha ){(\\Delta {\\mathbf {\\theta }})^2},\\end{split}$ where $\\alpha $ denotes the momentum, and $\\Delta {\\mathbf {\\theta }}$ is the accumulated gradient ($d{{\\mathbf {\\theta }}_{\\rm {a}}}$ or $d{{\\mathbf {\\theta }}_{\\rm {c}}}$ ) of the actor loss function or the critic loss function.",
"Note that the estimated gradient $g$ can be either shared or separated across agent threads, but the shared mode tends to be more robust [31], [32].",
"The estimated gradient $g$ is used to update the parameters of both the actor and critic networks as $\\begin{split}{\\mathbf {\\theta }} \\leftarrow {\\mathbf {\\theta }} - \\beta \\frac{{\\Delta {\\mathbf {\\theta }}}}{{\\sqrt{g + \\varepsilon } }},\\end{split}$ where $\\beta $ is the learning rate, and $\\varepsilon $ is a small positive step."
],
[
"Training and Execution of A3C-AFL",
"Like most of machine learning algorithms, there are two stages in the proposed A3C-AFL algorithm, i.e., the training procedure and the execution procedure.",
"Both the training and execution datasets are generated from their interaction of a federated learning environment conducted by the multi-UAV-enabled wireless network.",
"1.1 [t] A3C-based Device Selection, UAVs Placement, and Resource Management 1:$~$ Initialization: Initialize the global parameters ${{\\mathbf {\\theta }}_{\\rm {a}}}$ and ${{\\mathbf {\\theta }}_{\\rm {c}}}$ of the actor and critic networks in the global network; $~~~~$ Initialize the thread-specific parameters ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}}$ and ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {c}}}$ in the local networks; $~~~$ Initialize the global shared counter $T = 0$ and thread counter $t = 1$ with the maximum counters $T_{\\max }^{{\\rm {A3C}}}$ and $t_{\\max }^{{\\rm {A3C}}}$ , respectively; 2:$~$ Set $\\alpha ,\\beta ,\\gamma ,\\varepsilon $ , $T_{\\max }^{{\\rm {A3C}}}$ , and $t_{\\max }^{{\\rm {A3C}}}$ , respectively; 3:$~$ while $T < T_{\\max }^{{\\rm {A3C}}}$ do 4:$~~~$ for each learning agent do 5:$~~~$ Reset two accumulated gradients: $d{{\\mathbf {\\theta }}_{\\rm {a}}} \\leftarrow 0$ and $d{{\\mathbf {\\theta }}_{\\rm {c}}}\\leftarrow 0$ ; 6:$~~~$ Synchronize thread-specific parameters ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}} = {{\\mathbf {\\theta }}_{\\rm {a}}}$ and ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {c}}} = {{\\mathbf {\\theta }}_{\\rm {c}}}$ ; 7:$~~~$ Set $t = {t_0}$ and observe a network state ${s_t}$ ; 8:$~~~$ for $t \\le t_{\\max }^{{\\rm {A3C}}}$ do 9:$~~~~~$ Select an action ${a_t}$ based on the policy $\\pi ({a_t}|{s_t};{{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}})$ ; 10:$~~~~~$ Receive an immediate reward ${r_t}$ and observe a new reward ${s_{t + 1}}$ ; 11:$~~~~~$$t = t + 1$ ; 12:$~~~$end for 13:$~$ Update the received return by $~~~~~~~~~~~$${R_t} = \\left\\lbrace \\begin{array}{l}0,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{\\rm {if~}}{s_t}\\;{\\rm {is~a~terminal~state,}}\\\\{V^\\pi }({s_t};{{\\mathbf {\\theta }}_{\\rm {c}}}),\\;\\;{\\rm {if~}}{s_t}\\;{\\rm {is~a~non - terminal~state}}{\\rm {.",
"}}\\end{array} \\right.$ 14:$~$ for $i = t - 1$ to ${t_0}$ do 15:$~~~$ $R = {r_i} + \\gamma {R_t}$ ; 16:$~~~$ Update the actor accumulative gradient $d{{\\mathbf {\\theta }}_{\\rm {a}}}$ by using (31); 17:$~~~$ Update the critic accumulative gradient $d{{\\mathbf {\\theta }}_{\\rm {c}}}$ by using (33); 18:$~~~$ Perform an asynchronous update of global parameters ${{\\mathbf {\\theta }}_{\\rm {a}}}$ and ${{\\mathbf {\\theta }}_{\\rm {c}}}$ based on (35), respectively; 19:$~~~$ $T = T + 1$ ; 20:$~~$ end for 21:$~$ end while 1.",
"The training procedure of A3C-AFL performs in an asynchronous way: The A3C Procedure: The A3C algorithm is adopted for device selection, UAVs placement, and resource management in UAV-enabled networks, which is presented in Algorithm 1.",
"The detailed processes of Algorithm 1 are shown as follows.",
"1) Before training the A3C model, we load the real-world UAV-enabled network dataset and mobile devices' information, which generate a simulated environment for the federated learning scenario.",
"2) At the $t$ -th global counter, the global A3C network parameters ${{\\mathbf {\\theta }}_{\\rm {a}}}$ and ${{\\mathbf {\\theta }}_{\\rm {c}}}$ as well as thread-specific parameters ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}}$ and ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {c}}}$ are initialized.",
"3) Each UAV server acts as a learning agent to observe a network state ${s_t}$ by interacting with the environment.",
"4) Each learning agent selects an action ${a_t}$ (i.e., device selection, UAVs placement, subchannel allocation, and power allocation) according to the policy probability distribution $\\pi ({a_t}|{s_t};{{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}})$ in the actor network, and receives an immediate reward ${r_t}$ as well as a new state ${s_{t + 1}}$ after executing the action ${a_t}$ .",
"5).",
"The state value function ${V^\\pi }({s_t};{{\\mathbf {\\theta ^{\\prime }}}_{\\rm {c}}})$ and the estimation of advantage function $A({s_t},{a_t})$ are updated in the critic network, and the probability distribution $\\pi ({a_t}|{s_t};{{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}})$ is updated in actor network.",
"6) At each thread counter $i$ , the accumulative gradients of the thread parameters ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {a}}}$ and ${{\\mathbf {\\theta ^{\\prime }}}_{\\rm {c}}}$ are updated according to (31) and (33), respectively.",
"7) The global network collects the parameters $d{{\\mathbf {\\theta }}_{\\rm {a}}}$ and $d{{\\mathbf {\\theta }}_{\\rm {c}}}$ , then perform asynchronous update of the global parameters ${{\\mathbf {\\theta }}_{\\rm {a}}}$ and ${{\\mathbf {\\theta }}_{\\rm {c}}}$ by (35), before broadcasting them to each thread separately.",
"8) The A3C algorithm repeats the above training steps until the number of iterations gets the maximum global shared counter $T_{\\max }^{{\\rm {A3C}}}$ .",
"Finally, the trained A3C model can be loaded to perform device selection, UAVs placement, and resource management for federated learning model updating.",
"1.1 [t] Asynchronous Federated Learning (AFL) Algorithm 1:$~$ Input: The maximum number of communication rounds ${T^{{\\rm {AFL}}}}$ , client devices set $\\mathcal {K}$ , number of local iteration ${E^{{\\rm {AFL}}}}$ per communication round, and learning rate $\\eta $ ; Server process: // running at each UAV server 2:$~$ Input: Execute joint device selection, UAVs placement, and resource management by running Algorithm 1.",
"3:$~$ Initializes global model parameters ${{\\bf {w}}_n}(0)$ at each UAV server $n \\in \\mathcal {N}$ ; 4:$~$ for each global round $t$ $=$ 0, 1, 2, ..., ${T^{{\\rm {AFL}}}}$ do 5:$~~~$ for each UAV server $n$ $=$ 0, 1, 2, ..., $N$ do in parallel Device process: // running at each selected device 6:$~~~~~$ for each selected device $k \\in {\\mathcal {K}_n}$ in parallel do 7:$~~~~~~~$ Initialize ${{\\bf {w}}_k}(t) = {{\\bf {w}}_n}(t)$ ; 8:$~~~~~~~$ for $j$ $=$ 0, 1, 2, ..., ${E^{{\\rm {AFL}}}}$ do 9:$~~~~~~~~~$ Sample $i \\in {\\mathcal {D}_k}$ uniformly at random and update the local parameters ${{\\bf {w}}_k}(t)$ as follows $~~~~~~~~~~~~~$ ${{\\bf {w}}_k}(t) = {{\\bf {w}}_n}(t - 1) - \\eta \\nabla {F_k}\\left({{{\\bf {w}}_n}(t - 1)}\\right)$ ; 10:$~~~~~~$ end for 11:$~~~~~~$ UAV server $n$ collects the parameters $\\lbrace {{\\bf {w}}_k}(t)\\rbrace _{k = 1}^{{K_n}}$ from selected devices, and updates the global parameters $~~~~~~~~~~~~$ ${{\\bf {w}}_n}(t + 1) = {{\\sum \\nolimits _{k \\in {\\mathcal {K}_n}}{|{\\mathcal {D}_k}|{{\\bf {w}}_k}(t)} } \\mathord {\\left\\bad.",
"{\\vphantom{{\\sum \\nolimits _{k \\in {\\mathcal {K}_n}} {|{\\mathcal {D}_k}|{{\\bf {w}}_k}(t)} } {\\left| {{\\mathcal {D}_n}} \\right|}}} \\right.\\hspace{0.0pt}} {\\left| {{\\mathcal {D}_n}} \\right|}}$ ; 12:$~~~~~$ end for 13:$~~~$ end for 14:$~$ Output: Finalized global FL model parameters ${{\\bf {w}}_n}$ of each UAV server $n \\in \\mathcal {N}$ .",
"The AFL Procedure: This procedure consists of two phases, i.e., local training and global aggregation, which is provided in Algorithm 2.",
"After performing Algorithm 1, the following procedure is implemented at each global communication round $t$ for a federated learning system.",
"1) Each UAV sever $n$ broadcasts its global model parameters ${{\\bf {w}}_n}(t - 1)$ to its associated devices, where the local model parameters at each participated device $k$ is set as ${{\\bf {w}}_k}(t) ={{\\bf {w}}_n}(t - 1)$ , $k \\in {\\mathcal {K}_n}$ .",
"2) In the $n$ -th UAV-enabled cell, each selected device $k \\in {\\mathcal {K}_n}$ updates its local model parameters in an iterative manner according to the gradient of its loss function ${F_k}\\left( {{{\\mathbf {w}}_n}(t - 1)}\\right)$ .",
"At each local iteration $j$ , the local parameters ${{\\bf {w}}_k}(t)$ are calculated by (20).",
"3) The ${K_n}$ selected devices upload their updated local model-parameters $\\lbrace {{\\bf {w}}_k}(t)\\rbrace _{k = 1}^{{K_n}}$ to its associated UAV server $n$ .",
"4) Each UAV server $n$ aggregates the uploaded local model parameters from the ${K_n}$ selected devices, and updates the global model parameters ${{\\bf {w}}_n}(t)$ by (21) before broadcasting them to the associated devices.",
"2.",
"Asynchronous Implementation of A3C-AFL: We load the trained A3C model (i.e., Algorithm 1) to perform device selection, UAVs placement, and resource management in multi-UAV-enabled wireless networks.",
"Then, the selected devices perform local training, and upload their local model-parameters to its associated UAV server over the allocated uplink subchannels.",
"Each UAV server aggregates the collected local model parameters, and broadcasts the updated global model parameters to each associated device in the downlink.",
"It is worth noting that the multiple agents (i.e., UAV servers) load their trained models (i.e., Algorithm 1) to asynchronously search for different exploration policies by interacting with their environment.",
"In addition, the local model training is asynchronously executed among a range of participated devices, in order to enhance the efficiency of trained local models.",
"Thus, to improve the federated aggregation efficiency, some model parameters update process of the associated devices may not be used for the global aggregation sometimes (i.e., Algorithm 2) ."
],
[
"Complexity and Convergence Analysis",
"This subsection provides the computational complexity and convergence analysis of the proposed A3C algorithm for federated learning systems.",
"Let us define $J$ and $V$ as the number of DNN layers of the actor network and the critic network, respectively.",
"Both the actor network and the critic network are also two fully connected networks.",
"Define ${J_j}$ as the number of neurons of the $j$ -th layer in the actor network, and define ${V_i}$ as the number of neurons of the $i$ -th layer in the critic network.",
"In the actor network, the computational complexity of the $j$ -th layer is $O\\left( {{J_{j -1}}{J_j} + {J_j}{J_{j + 1}}} \\right)$ , and the total computational complexity with $J$ layers is $O\\left( {\\sum \\nolimits _{j = 2}^{J - 1}({{J_{j - 1}}{J_j} + {J_j}{J_{j + 1}}}) } \\right)$ .",
"Similarly, in the critic network, the total computational complexity with $V$ layers is $O\\left( {\\sum \\nolimits _{i = 2}^{V - 1} ({{V_{i -1}}{V_i} + {V_i}{V_{i + 1}}}) } \\right)$ .",
"As the proposed A3C algorithm is comprised of both the actor network and the critic network, the computational complexity of each training iteration is $O\\left( {\\sum \\nolimits _{j = 2}^{J - 1} ({{J_{j - 1}}{J_j} +{J_j}{J_{j + 1}}}) + \\sum \\nolimits _{i = 2}^{V - 1} ({{V_{i -1}}{V_i} + {V_i}{V_{i + 1}}}) } \\right)$ .",
"We set that there have $E$ episodes in the training phase, and each episode has $T$ time steps .",
"Thus, the overall computational complexity of the proposed A3C algorithm in the training process is $O\\left({ET(\\sum \\nolimits _{j = 2}^{J - 1} ({{J_{j - 1}}{J_j} + {J_j}{J_{j +1}}}) + \\sum \\nolimits _{i = 2}^{V - 1} ({{V_{i - 1}}{V_i} +{V_i}{V_{i + 1}}} ))} \\right)$ .",
"Theorem 1: The A3C algorithm can reach convergence by using policy evaluation in the critic network and policy enhancement (the actor network) alternatively, i.e., $\\pi ({a_t}|{s_t};{{\\mathbf {\\theta }}_{\\rm {a}}}) \\in \\prod $ will converge to a policy ${\\pi ^ * }({a_t}|{s_t};{{\\mathbf {\\theta }}_{\\rm {a}}})$ which guarantees ${Q^{{\\pi ^ * }}}({s_t},{a_t})> {Q^\\pi }({s_t},{a_t})$ for $\\pi ({a_t}|{s_t};{{\\mathbf {\\theta }}_{\\rm {a}}}) \\in \\prod $ and $({a_t},{s_t}) \\in \\mathcal {S} \\times \\mathcal {A}$ , assuming $|\\mathcal {A}| < \\infty $ .",
"At the same time, it also requires to satisfy the following conditions [33], [34]: 1) the learning rates ${\\beta _a}(t)$ and ${\\beta _c}(t)$ of the actor network and critic network admit: $\\sum \\nolimits _{t = 0}^\\infty {{\\beta _a}} (t) = \\infty ,\\;\\sum \\nolimits _{t = 0}^\\infty {\\beta _a^2} (t) = \\infty $ , $\\sum \\nolimits _{t = 0}^\\infty {{\\beta _c}}(t) = \\infty ,\\;\\sum \\nolimits _{t = 0}^\\infty {\\beta _c^2} (t) =\\infty $ , and $\\mathop {\\lim }\\limits _{t \\rightarrow \\infty } {{{\\beta _a}(t)} \\mathord {\\left\\bad.",
"{\\vphantom{{{\\beta _a}(t)} {{\\beta _c}(t)}}} \\right.\\hspace{0.0pt}} {{\\beta _c}(t)}} = 0$ ; 2) The instantaneous reward ${\\rm {Var}}\\left\\lbrace {{r_t}} \\right\\rbrace $ is bounded; 3) The policy function $\\pi (a|s;{{\\mathbf {\\theta }}_{\\rm {a}}})$ is continuously differentiable in ${{\\mathbf {\\theta }}_{\\rm {a}}}$ ; 4) The sequence $({a_t},{s_t},{r_t})$ is Independent and identically distributed (i.i.d.",
"), and has uniformly bounded second moments [33], [34].",
"Proof: See Appendix A.",
"In the following, we discuss the convergence property of the FL algorithm.",
"Let us define the upper bond of the divergence between the federated loss function $F({\\bf {w}}) $ and the global optimal loss function $F({{\\bf {w}}^ * }) $ as $\\left| {F({\\bf {w}}) - F({{\\bf {w}}^ * })} \\right| $ , where ${{\\bf {w}}^ * } $ is the global optimal parameters.",
"Definition 1: The FL algorithm can achieve the global optimal convergence if it satisfies [10], [11] $\\begin{split}\\left| {F({\\bf {w}}) - F({{\\bf {w}}^ * })} \\right| \\le \\varepsilon ,\\end{split}$ where $\\varepsilon $ is a small positive constant $\\varepsilon > 0 $ .",
"Theorem 2: When $F({\\bf {w}}) $ is a $\\eta - $ convex and $\\sigma - $ smooth function, the upper bond of $\\left[ {F({\\bf {w}}) - F({{\\bf {w}}^ * })} \\right] $ can be expressed $\\begin{split}F({\\bf {w}}) - F({{\\bf {w}}^ * }) \\le \\varepsilon \\left( {F({\\bf {w}}(0)) - F({{\\bf {w}}^ * })} \\right).\\end{split}$ Proof: The details of the proof can be seen in [10], [11].",
"For appropriate selections of the iteration numbers, i.e., the global iterations ${T^{{\\rm {AFL}}}}$ and the local iterations ${E^{{\\rm {AFL}}}}$ , the FL algorithm will finally coverage to the global optimality (36), the more proof analysis can be found in [10], [11].",
"In this section, we evaluate the performance of our proposed A3C-AFL algorithm under different parameter settings.",
"In our simulations, the performance is evaluated in the Python 3 environment on a PC with Intel (R) Core(TM) i7-6700 CPU @ 3.40 GHz, 16 RAM, and the operating system is Windows 10 Ultimate 64 bits.",
"We also compare the performance of the following different algorithms/approaches: 1) AFL with device selection: Proposed asynchronous federated learning framework with adopting device selection strategy (i.e., high communication and computation capabilities) to perform model aggregation.",
"2) AFL without device selection: Proposed asynchronous federated learning framework with adopting random device selection strategy to perform model aggregation [17].",
"3) SFL with device selection: Synchronous federated learning framework with adopting device selection strategy to perform model aggregation [20].",
"4) A3C-AFL: Adopting proposed A3C to perform device selection, UAVs placement, and resource management in the asynchronous federated learning framework.",
"5) A3C-SFL: Adopting proposed A3C to perform device selection, UAVs placement, and resource management in the synchronous federated learning framework.",
"6) Gradient-AFL: Adopting gradient-based benchmark to perform device selection, UAVs placement, and resource management in the asynchronous federated learning framework.",
"For our simulations, we consider a multi-UAV-enabled network where four UAVs are deployed in the sky to support the coverage of a square area of 400 m $\\times $ 400 m. At the beginning, four UAVs are uniformly located in the sky at the height of 150 m. The device training data of each device $k$ follows uniform distribution [5,10] Mbits, and the CPU computation capacity of the devices range from 1.0 GHz to 2.0 GHz.",
"The transmit power of each device is set to be equal to 50 mW, and the maximum transmit power of each UAV is 150 mW.",
"The transmit data size of model parameters is 200 kbits, and the weight parameter is $\\lambda = 0.4$ in (19).",
"Both the actor network and the critic network are conducted with deep neural networks (DNNs), and they have three hidden fully-connected layers.",
"Each of the layers in the actor network or the critic network contains 256 neurons, 256 neurons, and 128 neurons, respectively.",
"The actor network is trained with the earning rate 0.0001, and the critic network is trained with the learning rate 0.001 [33], [34].",
"The discount factor is $\\gamma = 0.98$ .",
"We evaluate the proposed AFL on the MNIST dataset [15], [16], and the Convolutional Neural Network (CNN) tool is used to train the local model.",
"In each global communication round, FL has one global aggregation and 10 iterations for local training.",
"The relevant simulation parameters are provided in Table I."
],
[
"Convergence Comparison of Algorithms",
"We first evaluate the convergence of the proposed A3C-based learning algorithm with the different learning rates of the actor network, i.e., ${\\beta _a} = \\lbrace 0.001,0.0001,0.00001\\rbrace $ , and also compare it with the gradient-based benchmark algorithm.",
"Note that the system cost is the objective function in (19), which includes the model aggregation time and the learning accuracy loss.",
"The learning rate of DNN plays a key role on the convergence speed and system cost.",
"From Fig.",
"3, we can observe that a large value of learning rate (i.e., ${\\beta _a} = 0.001$ ) may cause oscillations while a small value of the learning rate (i.e., ${\\beta _a} =0.00001$ ) produces slow convergence.",
"Hence, we select a suitable learning rate, neither too large nor too small, and the value can be set around 0.0001 in the training model, which can guarantee the fast convergence speed and the low system cost.",
"In addition, it is interesting to note that both the proposed A3C-based learning algorithm and the gradient-based benchmark algorithm have the comparable convergence speed, and when the training episode approximately reaches 500, the system cost gradually converges despite some fluctuations due to dynamic environment characteristic and policy exploration.",
"However, our proposed algorithm achieves the lower system cost than that of the gradient-based benchmark algorithm.",
"In addition, we also carry out a test and trials for the critic's learning rate selection, where the learning rate of the critic network is selected at 0.001 which has high convergence speed and low system cost [33], [34]."
],
[
" UAVs Placement and Device Selection Evaluation",
"Figure 4 captures the 2D deployment of the UAVs (in a horizontal plane) and selected devices distribution in one time slot, where a number of $K$ =150 devices are randomly located over the coverage areas of their associated UAVs.",
"UAVs adaptively update their locations according to the number of associated devices and devices' distributions, in order to provide the best channel gain and minimize the communication delay between UAVs and the devices.",
"It is worth noting that a part of devices with high communication and computation capabilities (solid dots) are selected to participate the FL model aggregation, to minimize the aggregation time and learning accuracy loss, while the remaining low-quality devices (hollow dots) don't participate in the FL model aggregation in this time slot.",
"Figure: Two dimensional distribution of UAVs, ground devices andparticipated devices."
],
[
"Accuracy Comparison with Different Global Rounds and Implementation Time",
"In Fig.",
"5, we compare the accuracy performance versus the number of global rounds and wall-clock time for different FL approaches, where several devices are set as low-quality participants.",
"The low-quality participants are with low communication and computation capabilities, and even have low-quality training parameters [14].",
"From Fig.",
"5(a), we can observe that both the AFL and SFL approaches with device selection requires about 25 global rounds to achieve an accuracy of 90.0%, and both of them have similar convergence speed and accuracy performance.",
"However, at each global round, SFL has to wait until all the selected devices response, while AFL only needs a number of selected devices' response to move on to the next round which decreases the aggregation completion time during the learning process as shown in Fig.",
"5(b).",
"In addition, both the AFL and SFL approaches with device selection achieve higher accuracy and convergence speed than those of the AFL approach without device selection.",
"The results illustrate that the proposed device selection scheme can prevent low-quality devices from affecting the learning accuracy, and enhances the system performance significantly.",
"From Fig.",
"5(b), we can see that the accuracy of all approaches improves with the increase of the wall-clock time.",
"However, by comparing different approaches, the proposed AFL approach with device selection outperform the other two approaches in terms of both the convergence speed and accuracy.",
"The reason lies on the fact that the AFL approach without device selection may enable the low-quality devices to participate in the FL model aggregation, where the low-quality model parameters decrease the overall accuracy, as well as devices with low communication and computation capacities need more time to complete the model aggregation.",
"In addition, the SFL approach with device selection has to wait for all selected devices to complete their local model parameters update, among which there may require long computation time due to low computation capability.",
"Consequently, each global communication round of the SFL approach with device selection requires more time to finish model aggregation, and hence its accuracy performance slowly improves with the increase of wall-clock time.",
"The results from Fig.",
"5 demonstrate that the proposed AFL approach with device selection is capable of improving both the convergence speed and aggregation accuracy performance."
],
[
"Performance Comparison Versus Number of Devices",
"Figure 6 presents the aggregation accuracy and completion time achieved by different algorithms with various numbers of devices.",
"As we can see from Fig.",
"6(a), the proposed A3C-AFL and A3C-SFL algorithms achieve the superior accuracy compared to the AFL without device selection algorithm under different numbers of devices, and the advantage gap becomes large with the increase of the devices.",
"The reason is that the devices with low-quality model parameters in the AFL without device algorithm compromise the accuracy after model aggregation, while the proposed A3C algorithm selects the devices with high-quality model parameters for model aggregation which can significantly improve the overall accuracy.",
"Furthermore, the accuracy of the A3C algorithm increases with the increase of devices, because the network has the more probability of searching participated devices having high-quality model parameters to enhance the aggregation accuracy.",
"However, the accuracy of the AFL without device selection algorithm maintains at a horizontal level (i.e., about 78.0%) with some fluctuation during this process, since it doesn't keep the low-quality devices from decreasing the aggregation accuracy.",
"In addition, the accuracy performance of the gradient-AFL algorithm increases with $K$ , but it has a lower accuracy than that of the proposed A3C-AFL algorithm.",
"Fig.",
"6(b) illustrates that the aggregation completion time of the A3C-SFL and AFL without device selection algorithms increases with the increase of devices, while the A3C-AFL and gradient-AFL algorithms decrease slightly during this process.",
"Generally, the more devices there are, the more completion time is required to learn the optimal solution for the A3C-SFL algorithm.",
"The reason is that SFL needs to wait for all selected devices to complete their local parameters update before aggregating the local models at each communication round.",
"In addition, for the AFL without device selection algorithm, as more devices participate in the model aggregation, the participated devices with low-quality computation and communication capabilities consume longer time for global model aggregation.",
"Even though the gradient-AFL algorithm prevents low-quality devices from increasing the completion time, it needs more time to complete the decision making compared to the A3C-AFL algorithm.",
"The results show that the proposed A3C algorithm can carry out the better decision making for the device selection, UAVs placement, and resource management than that of the gradient-based benchmark."
],
[
"Conclusions",
"This paper has investigated how to minimize the execution time and learning accuracy loss of privacy-preserving federated learning in multi-UAV-enabled wireless networks.",
"Specifically, an AFL framework was proposed to provide asynchronous distributed computing by enabling model training locally without transmitting raw sensitive data to UAV servers.",
"The device selection strategy was also introduced into the AFL framework to select the mobile devices with high communication and computation capabilities to improve the learning efficiency and accuracy.",
"Moreover, we also proposed an A3C-based joint device selection, UAVs placement, and resource management algorithm to enhance the learning convergence speed and accuracy.",
"Simulation results have demonstrated that the proposed AFL framework with A3C-based algorithm outperform the existing solutions in terms of learning accuracy and execution time under different settings."
],
[
"Proof of Theorem 1",
"For all agents (i.e., UAV servers) in the environment, the network aims to search a joint policy in the coordinated multi-agent RL scenario, where the joint policy can be expressed as $\\begin{split}\\pi (s{\\rm {|}}a) = \\left[ \\begin{array}{l}{\\pi ^1}({s^1}{\\rm {|}}{a^1})\\\\\\;\\;\\;\\;\\; \\vdots \\\\{\\pi ^N}({s^N}{\\rm {|}}{a^N})\\end{array} \\right] = \\left[ \\begin{array}{l}\\;\\;\\;\\;\\arg \\;\\mathop {\\max }\\limits _{{a^1} \\in {\\mathcal {A}_1}} {Q^1}({s^1}|{a^1})\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\vdots \\\\\\arg \\;\\mathop {\\max }\\limits _{{a^N} \\in {\\mathcal {A}_N}} {Q^N}({s^N}{\\rm {|}}{a^N})\\end{array} \\right],\\end{split}$ where ${s^n},{a^n} $ , and ${\\mathcal {A}_n} $ denote the individual state, action and action space of the $n$ -th agent.",
"As A3C uses the policy evaluation in the critic network and policy enhancement in the actor network alternatively, we adopt the two-time-scale stochastic approximation [33], [34] to provide the convergence proof analysis of A3C.",
"In detail, the convergence of the critic network is first analyzed with the joint policy $\\pi (s|a) $ being fixed.",
"Then, we provide the convergence analysis of the policy parameter ${{\\mathbf {\\theta }}_{\\rm {a}}} $ upon the convergence of the actor network.",
"Here, let us define the transition probability of state-action pair as $\\mathcal {T}(s^{\\prime },a^{\\prime }|s,a) = \\mathcal {T}(s^{\\prime }|s,a)\\pi (s^{\\prime }|a^{\\prime }) $ and the stationary distribution of MDP as $D(s,a) = {\\rm {diag}}[d(s) \\cdot \\pi (s|a),s \\in \\mathcal {S},a \\in \\mathcal {A}] $ , where $d(s) $ denotes the stationary distribution of the Markov chain induced by policy $\\pi (s|a) $ .",
"The sum cumulative reward of all agents is expressed as $\\bar{U}(s,a) = \\sum \\nolimits _{n \\in \\mathcal {N}} {{U_n}(s,a)} $ , and the overall available cumulative reward set in learning process is defined as $\\hat{U}(s,a)\\bar{=} \\left[ {\\bar{U}(s,a),s \\in \\mathcal {S},a \\in \\mathcal {A}} \\right] $ .",
"The joint long-term return under the joint policy $\\pi (s|a) $ is expressed as $\\begin{split}J({\\mathbf {\\theta }}) = \\sum \\limits _{s \\in \\mathcal {S}} {d(s)} \\sum \\limits _{a \\in \\mathcal {A}} {\\pi (s|a)\\hat{U}(s,a)}.\\end{split}$ In addition, the operator ${W^Q} $ for any action-value function $Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) $ is defined as [33], [34] In A3C, the action-value function can be expressed by using the linear functions [33], [34], i.e., $Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) = {\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a)$ , where ${\\mathbf {\\Psi }}(s,a) = {\\left( {\\psi {}_1(s,a),\\psi {}_2(s,a), \\ldots ,\\psi {}_{|\\mathcal {S}|}(s,a)} \\right)^T} $ is the feature vector (basis function vector) at the state $s$ .",
"The critic network aims to find a unique solution ${{\\mathbf {\\theta }}_{\\rm {c}}} $ by satisfying $\\begin{split}{\\mathbf {\\Psi }}(s,a) \\cdot D(s,a)\\left[ {{W^Q}({\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a)) - {\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a)} \\right] = 0.\\end{split}$ As the solution to (41) is a limiting point of the $TD(0) $ method, and thus we approximate the action-value function $Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) $ instead of the state-value function $V(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}})$ .",
"This solution can be achieved by minimizing the Mean Square Projected Bellman Error, i.e., $\\begin{split}\\mathop {\\min }\\limits _{{{\\mathbf {\\theta }}_{\\rm {c}}}} \\left\\Vert {{\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a) - \\Pi \\left( {W^Q}\\left( {{\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a)} \\right) \\right)} \\right\\Vert _{D(s,a)}^2,\\end{split}$ where $\\Pi ( \\cdot ) $ denotes the operator that projects a vector to the space, and $\\left\\Vert \\cdot \\right\\Vert _{D(s,a)}^2 $ is the Euclidean norm.",
"In the coordinated scenario, each agent exchanges its decisions with each other, and all of them will achieve a copy of the estimation of the jointly averaged action-value function, i.e., ${\\mathbf {\\theta }}_{\\rm {c}}^n \\rightarrow {{\\mathbf {\\theta }}_{\\rm {c}}} $ for all $n \\in \\mathcal {N} $ .",
"The joint action $a = \\left\\langle {{a_1}, \\ldots ,{a_N}} \\right\\rangle $ of A3C in state $s$ is in a global point that all agents coordinately achieve the sum highest return from the environment, i.e., $a = \\mathop {\\arg \\max }\\limits _{a \\in \\mathcal {A}} Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) $ .",
"In other words, the final critic parameter ${{\\mathbf {\\theta }}_{\\rm {c}}} $ is achieved by iteratively minimizing (42) , and the $Q$ value will converge to the final point ${Q^ * } $ with probability 1 [33], [34].",
"To analyze the convergence of the actor network, the advantage function of the $n$ -th agent in (29) is rewritten as $\\begin{split}A_t^n(s,a) = {\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a) - \\sum \\limits _{{a^n} \\in \\mathcal {A}^n} {{\\pi ^n}({s_t},{a^n}) \\cdot {\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a)} .\\end{split}$ Using Assumptions 2.2 and 4.1-4.5 [34], for the $n$ -th agent, the policy parameter ${\\mathbf {\\theta }}_{\\rm {a}}^n $ of the actor network in (35) will converge to a point from the following set of asymptotically stable equilibria of $\\begin{split}{\\mathbf {\\theta }}_{\\rm {a}}^n = {\\Gamma ^n}\\left[ {{\\mathbb {E}_{{s_{t \\sim d(s),at \\sim \\pi }}}}\\left( {A_t^n(s,a) \\cdot {\\nabla _{{\\mathbf {\\theta }}_{\\rm {a}}^n}}\\log {\\pi ^n}(s|{a^n})} \\right)} \\right], {\\rm {for}}~n \\in \\mathcal {N} ,\\end{split}$ where $\\Gamma [ \\cdot ] $ denotes an operator that projects any vector onto the compact set.",
"The estimation of the policy gradient $A_t^n(s,a) \\cdot {\\nabla _{{\\mathbf {\\theta }}_{\\rm {a}}^n}}\\log {\\pi ^n}(s|{a^n}) $ satisfies $\\begin{split}\\begin{array}{l}{\\mathbb {E}_{{s_{t \\sim d(s),at \\sim \\pi }}}}\\left( {A_t^n(s,a) \\cdot {\\nabla _{{\\mathbf {\\theta }}_{\\rm {a}}^n}}\\log {\\pi ^n}(s|{a^n})} \\right)\\\\= {\\nabla _{{\\mathbf {\\theta }}_{\\rm {c}}^n}}J({\\mathbf {\\theta }}) + {\\mathbb {E}_{{s_{t \\sim d(s),at \\sim \\pi }}}}\\left( {\\left( {Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) - {\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a)} \\right) \\cdot {\\nabla _{{\\mathbf {\\theta }}_{\\rm {a}}^n}}\\log {\\pi ^n}(s|{a^n})} \\right).\\end{array}\\end{split}$ As the linear features here are not limited by the compatible features, we can get the convergence to the stationary point of ${\\mathbb {E}_{{s_{t \\sim d(s),at \\sim \\pi }}}}\\left( {A_t^n(s,a) \\cdot {\\nabla _{{\\mathbf {\\theta }}_{\\rm {a}}^n}}\\log {\\pi ^n}(s|{a^n})} \\right) = 0 $ in the set of the policy parameters.",
"In this case, when the long-term averaged return $J({\\mathbf {\\theta }}) $ satisfies ${\\nabla _{{\\mathbf {\\theta }}_{\\rm {c}}^n}}J({\\mathbf {\\theta }}) = 0 $ , the error between the approximation function ${\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a) $ and the action value function $Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) $ is small, i.e., $Q(s,a;{{\\mathbf {\\theta }}_{\\rm {c}}}) - {\\mathbf {\\theta }}_{\\rm {c}}^T \\cdot {\\mathbf {\\Psi }}(s,a) \\approx 0 $ .",
"Thus, we can achieve the best solution for A3C with general linear function approximation [33], [34].",
"Due to the update rule in (45) and the coordination nature of the coordinated multi-agent A3C, a joint policy $\\pi $ implied by multiple policies is updated by each agent, eventually converges to the final point.",
"The more details of the proof can be seen in [33], [34]."
]
] | 2011.14197 |
[
[
"Thin-suspended 2D Materials: Facile, Versatile, and Deterministic\n Transfer Assembly"
],
[
"Abstract We report a deterministic 2D material (2DM) transfer method to assemble any-stacking-order heterostructures incorporating suspended ultra-thin 2D materials, such as single-layer graphene (SLG) and bilayer graphene (BLG).",
"The transfer procedure relies on a single-step preparation nitrocellulose micro-stamp, which combines both outstanding adhesion and softness.",
"It permits the dry pick-up of naked 2D crystals (graphene, MoS$_2$, and hBN) directly from a SiO$_2$ substrate, and to precisely transfer them on substrates or trenches.",
"Optical and Raman data show that no significant defect is introduced upon transfer, even in suspended SLG and BLG.",
"The areas transferred range up $\\sim$ 1000 $\\mu$m$^2$ on substrate.",
"High-yield transfer of suspended ultra-thin 2DM does not require critical point drying for areas up to 15 $\\mu$m$^2$ or suspension heights down to 160 nm.",
"To demonstrate the method's capabilities, we assembled on-substrate and suspended optical cavities tuning BLG's Raman scattering intensity by factors of 19 and 4, respectively.",
"This resilient and rapid 2DM transfer procedure will accelerate the fabrication of many heterostructures and permit versatile suspension of 2DMs for research in twistronics, straintronics, and nano-opto-electro-mechanical systems (NOEMS)."
],
[
"toc"
],
[
"Deterministic any-stacking-order and suspended 2DM transfer procedure",
"The stamping method is separated into five steps as shown in Figure 1a-e of the main text.",
"The first step consists in the preparation of a nitrocellulose micro-stamp and finding a desired 2DM crystal on a SiO2 substrate.",
"Secondly, the micro-stamp is aligned above the 2DM and brought into contact.",
"Later, the 2DM is picked-up from the SiO2 substrate by carefully retracting the micro-stamp.",
"The micro-stamp/2DM assembly is subsequently aligned above the new substrate and brought into contact.",
"The transfer is completed with a facile microliter-volume solvent dissolution of the micro-stamp.",
"We summarize the key details of each of these five stages below, and we first briefly describe our stamping apparatus."
],
[
"Stamping apparatus with $x, y, z, \\theta $ alignment",
"The stamping setup used is shown in fig:S1.",
"It consists of a rotating stage with a vacuum system to hold a substrate in place, a three-axis micro manipulator ($x, y$ and $z$ -axis) that holds a glass slide with the micro-stamp (nitrocellulose droplet), a long working distance optical assembly, and a high-resolution CCD camera which is connected to a monitor for live viewing.",
"This setup is based on previous stamping techniques, often called deterministic transfer [1], [2], [3], [4].",
"Most steps of our micro-stamp transfer process are all-dry, and only the last step requires micro-liter amounts of mild solvents.",
"Figure: Stamping apparatus.",
"The substrate with the 2DM-to-be-transferred is placed on a rotating stage and held by vacuum, while a glass slide with a custom micro-stamp is attached to the x-y-z manipulator.",
"A long working distance optical assembly is connected to a digital camera which displays a live view of the transfer process."
],
[
"Micro-stamp preparation and 2DM exfoliation",
"Micro-stamp preparation – Previous deterministic 2DM transfer methods have used stamps based on materials such as polydimethylsiloxane (PDMS) [1], [3], polymethylmethacrylate (PMMA) [5], thermal release tape (TRP) [6], polyvinyl alcohol (PVA) [2], and polypropylene carbonate (PPC) [7], [8].",
"The novelty of our stamping method lies in the use of a nitrocellulose-based stamp in the shape of an ellipsoidal micro-droplet (see fig:S2).",
"The stamp is made with a commercially available product (Extra Life$^{\\mathrm {TM}}$ No Chip Top Coat – Revlon).",
"The size and shape of the micro-stamp determine the contact area between the stamp and 2DMs.",
"The optimal stamp size for our work was around 400 $\\mu $ m $\\times $ 600 $\\mu $ m $\\times $ 400 $\\mu $ m (see fig:S2), with a contact area (top of stamp) approximately 200 $\\mu $ m by 200 $\\mu $ m. To achieve these micro-stamp dimensions, we first submerge the tip of a 27 gauge needle into a small drop of the solution deposited on a glass slide (fig:S2a).",
"Due to capillary forces, a small volume of solution stays on needle when pulled away from it.",
"When the needle barely touches the target clean glass slide, it transfers a very smal droplet on its surface (see fig:S2b).",
"The resulting micro-stamp is inspected by optical microscopy to make sure that it has the wanted shape and size.",
"The narrow apex of the stamp appears as a bright spot in fig:S2c.",
"This process only takes a few minutes and is easily reproducible.",
"Figure: Nitrocellulose micro-stamp preparation.",
"(a) A sharp needle is submerged in a drop of nitrocellulose-solution (Extra Life TM ^{\\mathrm {TM}} No Chip Top Coat – Revlon), and then pulled up so that a small droplet remains attached to the needed by capillary forces.",
"(b) A slight contact of the droplet with the target glass slide leaves a reproducible micro-stamp.",
"(c) Top-view optical image of a typical micro-stamp, with a 200 μ\\mu m ×\\times 200 μ\\mu m contact area for 2DM transfer.",
"Inset: Illustration of the micro-stamp profile.2DM exfoliation – To exfoliate graphene crystals, we start with high quality HOPG graphite flakes (Kish Graphite B from Covalent Materials Corporation).",
"To exfoliate hexagonal boron-nitrate (hBN) 2D crystals, we start with high quality hBN crystals (PT110 Powder CTR from Momentive Performance Materials).",
"To exfoliate molybdenum disulfide (MoS2) crystals, we start with a synthetic MoS2 crystal from 2D Semiconductors.",
"Our wafers are made of 500 $\\mu $ m-thick Si with a 310 nm-thick SiO2 film grown on them.",
"We transferred a photolithography patterned coordinate grid on the SiO$_2$ /Si wafers, to easily record the location of the candidate 2DM crystals.",
"To exfoliate 2DMs, we first do a coarse mechanical cleavage with a razor blade to generate the thinnest flakes possible.",
"We then place these flakes on a piece of Scotch$^{\\mathrm {TM}}$ tape (exfoliation of hBN is done using Nitto$^{\\mathrm {TM}}$ tape) and further cleavage is done by repeatedly folding the tape [9].",
"Before transferring the exfoliated 2DM from the tape to the SiO2/Si substrate, we lightly etch the substrate with a dilute hydrochloric acid (HCl) and hydrogen peroxide (H$_2$ O$_2$ ) solution (18:1:1) at 75 C$^0$ for 5 minutes, to get rid of any surface residues.",
"This etching minimizes the substrate contamination and promotes adhesion between 2DM crystals and the oxide.",
"The substrate is carefully rinsed with DI water, blown dry with nitrogen and baked at 120 $^0$ C for 2 minutes to evaporate any water left.",
"The tape containing the 2DM flakes is gently pressed with a finger on the substrate, and we wait for about 10 minutes before slowly peeling off the tape (0.1 mm/s) with the help of tweezers.",
"The scotch tape itself leaves organic residues that need to be cleaned before the deterministic transfer.",
"The substrate with exfoliated 2DM is submerged in a warm bath of acetone at 75 $^0$ C for 5 minutes, rinsed with IPA, then with DI water and baked at 120 $^0$ C for 2 minutes.",
"The substrate is then mounted on the vacuum stage of the stamping apparatus (fig:S1) and the lens and camera system is focused on the target crystal."
],
[
"Making contact between the micro-stamp and 2D crystal",
"The glass slide with the micro-stamp is mounted on the micro-manipulator, the apex of the stamp is centred 1 mm above the crystal selected for pick-up (the micro-stamp is highly transparent and acts as a lens due to its ellipsoidal shape).",
"The stamp is left to dry for 10 $\\pm $ 3 min.",
"During this time the stamp's surface hardens.",
"It is then lowered slowly ($\\sim $ 50 $\\mu $ m/s) to contact the selected crystal and its immediate surrounding area (4000 $\\mu $ m$^2$ ).",
"There is a sudden change of colour when contact is made (see Figure 1b in the main text).",
"The contact must be done on the first trial, otherwise the micro-stamp deforms, and the pick-up process may induce crystal folding or work unreliably.",
"The stamp-2DM contact is maintained constant for 20 $\\pm $ 5 min to promote strong adhesion.",
"These parameters were the same for all of the 2D crystals we transferred: SLG, BLG, FLG, hBN and MoS2 crystals."
],
[
"Direct pick-up from SiO2",
"The pick-up speed of the micro-stamp/2DM assembly away from the original substrate is controlled with the $z$ -axis of the micro-manipulator.",
"The optimal speed depends on the thickness of the crystal.",
"We found that the nitrocellulose-based stamp acts as a hard surface at higher speeds and as a more flexible one at lower speeds.",
"The vertical pick-up speed used for atomically thin crystals (SLG, BLG and few layers) is $\\sim $ 500 $\\mu $ m/s, while for thicker crystals it is reduced to $\\sim $ 250 $\\mu $ m/s.",
"The live view option of the camera allows us to observe when the crystal is completely picked-up and if the process induces crystal folds.",
"Once the crystal is picked up (see fig:S3), we raise the stamp by an additional 500 $\\mu $ m, and exchange the old SiO2/Si substrate with the new target substrate.",
"Figure: Schematic of the 2DM pick-up process from a SiO2 substrate.",
"A micro-manipulator holds the glass slide to which the micro-stamp is attached.",
"By raising the slide, the 2DM is cleanly picked-up."
],
[
"Alignment and stamping of a 2D crystal on a target substrate",
"We often use hBN as the target substrate for transfer, but also successfully used SiO2, graphene, or aluminum.",
"We note that exfoliated hBN crystals on SiO2/Si substrates often exhibit “U” shape edges which can serve as naturally occurring trenches to suspend 2DMs.",
"We first characterize the substrate via AFM (thickness, shape, flatness).",
"Then, we place it on the stamping stage, and focus the optical assembly on the substrate.",
"We mark (trace) the substrate crystal (e.g.",
"hBN) contour on the video screen for future reference.",
"The 2DM-substrate alignment is done in successive steps.",
"First, the micro-stamp is centred and positioned about 0.5 mm above the target, and the substrate is aligned as desired relative to the 2D crystal orientation on the micro-stamp.",
"In a second stage, we focus the image right above the substrate and lower progressively the 2D crystal at about 20 $\\mu $ m/s until both the substrate and 2D crystal are clearly in focus.",
"In fig:S4 we show a final alignment done while making gentle contact at about 5 $\\mu $ m/s.",
"We stop lowering the micro-stamp when the 2D crystal contacts the target, as shown in fig:S4b.",
"We ensure that the micro-stamp is not pressed hard enough to deform significantly near its apex (stamping area), and does not contact the SiO$_2$ immediately surrounding the hBN substrate.",
"As visible in fig:S4c, there should be a spacing (a no-contact zone) of $\\sim $ 3 $\\mu $ m between the stamp (in contact with SiO2) and the hBN substrate.",
"A tilted SEM image of a suspended TLG structure can be seen in the inset of fig:S4c.",
"Figure: Alignment and stamping of a trilayer (TLG) graphene crystal on a hBN trench.",
"(a) The micro-stamp starts to make contact with the target substrate.",
"(b) TLG crystal makes slight contact with hBN substrate.",
"(c) The micro-stamp holding the TLG crystal is in contact with the SiO2 and hBN, but does not contact a 3 μ\\mu m-wide zone surrounding the hBN support.",
"Inset: Tilted SEM image of the final heterostructure."
],
[
"Transfer: Solvent dissolution of micro-stamp",
"A clean micro-pipette is used to inject one drop of acetone ($<$ 500 $\\mu $ L) in the spacing between the micro-stamp and the substrate (fig:S5a).",
"The acetone rapidly dissolves the stamp (this can be monitored in real time on the screen) and releases the 2D crystal.",
"Once the stamp has been dissolved, we raise the glass slide by 500 $\\mu $ m and do a local rinsing with IPA using the same micro-pipette.",
"We repeat a few times this rinsing with IPA to completely flush the acetone and polymer residues (fig:S5b).",
"At this point the 2D crystal has been transferred on the new substrate and is surrounded by the IPA solution (fig:S5c).",
"To transfer suspended crystals, we control the evaporation rate of IPA to avoid their collapse due to capillary forces.",
"The evaporation rate is easily tuned by raising or lowering the glass slide to tailor the exposure of the IPA solution to the atmosphere (fig:S5d).",
"This procedure removes the need for critical point drying of our suspended 2DMs.",
"It enables the stamping of defect-free suspended crystals with a simple table-top apparatus.",
"Figure: 2DM transfer via solvent dissolution of the micro-stamp (a) A micro-pipette is used to introduce a drop of acetone.",
"(b) The acetone starts to dissolve the stamp and the 2D crystal is released.",
"A few IPA drops are injected to flush the acetone and polymer residues.",
"(c) The stamp is completely dissolved, and fast evaporation of the solvent starts to occur.",
"(d) By the lowering of the glass slide we can decrease the rate of evaporation, minimizing the risk of collapse for suspended crystals."
],
[
"On-substrate heterostructures",
"In Figure 2 of the main text, we describe the capabilities of our method to transfer naked (not encapsulated) 2DM crystals.",
"We fabricated on-substrate heterostructures with precise alignment of large 2D crystals.",
"Below, we present in Table REF a detailed list of our 21 attempts at transferring 2DM crystals from one substrate to another following the recipe presented in Section and Figure 1 of the main text.",
"Figure: Transferring large bilayer graphene (BLG) crystals from SiO2 substrates to hBN substrates.",
"(a) Optical image of a large BLG crystal on SiO2 before the transfer procedure, and (b) the same BLG crystal on hBN after the transfer.",
"The BLG was picked up by the micro-stamp and transferred, without any tearing or encapsulation of the BLG.",
"The red lines show the location of Raman spectra acquisitions.",
"(c)-(d) Same information as in (a)-(b) for another large area BLG crystal transfer.For each device listed in Table REF , we acquired high quality optical images of the 2DM before and after its transfer, as shown in fig:S6 for two large area BLG crystals.",
"These optical images were used to extract the surface area of the samples listed in Table REF , and to verify that no significant macroscopic damage (tearing, folding, and bubbles) was introduced during the transfer.",
"Before and after transfer, we acquired Raman scans (along the red lines in fig:S6) and AFM images of the devices.",
"The comparison of the Raman spectra (Figure 2) before and after transfer confirmed that no microscopic disorder was introduced during the transfer.",
"The alignment accuracy of the transfer is $\\approx $ 1 $\\mu $ m when great care is taken.",
"We successfully transferred all of the 2DMs we attempted to.",
"They include various thicknesses of graphene (MLG, BLG, TLG, FLG), MoS2, and hBN.",
"tab:S1 details the thickness of the materials transferred and the final heterostructure geometries created.",
"T Table: List of all 21 on-substrate 2DM transfers done via our nitrocellulose micro-stamp method.The transfer procedure was very reproducible, and its success rate was very high.",
"Out of the 21 attempts, there was only 1 failure (no transfer) and 2 partial successes (some tearing of the crystal), and 18 fully successful transfers.",
"In terms of producing the desired planar heterostructures, we thus find the success rate to be around 95%."
],
[
"Suspended heterostructures",
"In Figure 3 of the main text, we describe our capability to transfer and align suspended ultra-thin 2DM crystals on hBN trenches.",
"We created suspended SLG, BLG, and FLG heterostructures with various suspension heights, including some structures with a variable (tilted) suspension height as shown in Figure 4.",
"Below, we present in tab:S2 a detailed list of our 15 attempts at transferring 2DM crystals from SiO2 substrates onto hBN trenches for suspension.",
"We followed the recipe presented in sec:S1 and Figure 1 of the main text.",
"Pristine graphene crystals feature two main Raman resonances, a first order G-peak and higher order 2D-peak.",
"In defected graphene crystals, there is a third Raman resonance called D-peak which appears.",
"All of these resonances are laser wavelength dependent.",
"With a 532-nm laser wavelength, the location of the D peak is around 1320 cm$^{-1}$ [10], G around 1580 cm$^{-1}$ [10], and 2D around 2700 cm$^{-1}$ .",
"Our heterostructures often include hBN crystals beneath the graphene crystals, and this material is also Raman active.",
"It features a resonance called G$_{\\mathrm {hBN}}$ located around 1370 cm$^{-1}$ [11].",
"It is possible to clearly distinguish between a graphene D-peak, and a G$_{\\mathrm {hBN}}$ .",
"In Figure 3 of the main text, we show suspended graphene crystals on top of “U” shaped hBN trenches.",
"There is a small portion of incoming laser photons being scattered by the edge of the hBN trench, and this leads to a very small intensity G$_{\\mathrm {hBN}}$ peak.",
"In fig:S7, we compare the intensities of the G$_{\\mathrm {hBN}}$ peak at different laser positions in the device shown in Figure 3c.",
"Figure: Comparing the G hBN _{\\mathrm {hBN}} peak intensities at different laser positions in a suspended BLG heterostructure.",
"(a) Top view image with a red-labeled and a black-labeled laser positions, respectively located on the hBN-supported and suspended BLG regions.",
"(b) BLG Raman data acquired at the location where the crystal is suspended (black data) and where it is supported by a hBN substrate (red data).",
"The inset shows a zoom-in on the data.None of our fabricated samples featured a defect-related D peak.",
"In fig:S8, we show two additional transferred suspended BLG devices and their Raman spectra.",
"A Raman D-peak is not visible, and neither is the G$_{\\mathrm {hBN}}$ peak.",
"This complete absence of G$_{\\mathrm {hBN}}$ is because the laser beam was farther away from the hBN edge during the Raman acquisition than for the device in fig:S7.",
"Figure: Transferred suspended BLG.",
"(a) SEM image of a transferred suspended BLG crystal with a uniform height.",
"(b) Raman spectra acquired at the red-marker location in (a), before (black data) and after (red data) the BLG transfer.",
"(c) Optical image of a transferred tilted-suspended BLG crystal.",
"(d) Raman spectra acquired at the red-marker location in (c), before (black data) and after (red data) the BLG transfer.For each device listed in tab:S2, we acquired high quality optical images of the 2DM before and after the transfer.",
"We also acquired SEM images after suspension in some devices, as shown in fig:S8 for two suspended BLG heterostructures.",
"One of these two devices has a uniform suspension height (fig:S8a) while the other has a titled-suspension (fig:S8b) resulting in a continuous variation of its suspension height.",
"These images, and others, were used to extract the surface area of the crystals listed in tab:S2, and to verify that no significant macroscopic damage (tearing, folding, and bubbles) was introduced during the transfer.",
"Before and after the transfer we also acquired multiple Raman scans of the samples.",
"The Raman spectra, see fig:S8 and Figure 3, confirmed that no microscopic disorder was introduced during the stamping.",
"The translational and rotational alignment accuracy of the deterministic transfer were down to 1 $\\mu $ m and less than 1 degree.",
"We successfully transferred the materials we attempted to, including various thicknesses of graphene (SLG, BLG, TLG, FLG).",
"The suspended heterostructures created were graphene/air/SiO2/Si with hBN trenches.",
"In order to determine the suspension height of graphene, and verify the absence of wrinkles introduced during transfer, we used tilted-SEM imaging and AFM imaging, as shown in fig:S9.",
"The results show uniform suspension heights (except in the tilted devices), and these precise suspension heights were confirmed by the quantitative measurements and modeling of Raman interferences presented in Figure 4.",
"Indeed, the Raman factors are highly sensitive to the suspension height of the graphene [10], as discussed in the next section.",
"Figure: Transferred suspended TLG heterostructure.",
"(a) Tilted-SEM image showing a uniform and wrinkle-free suspension.",
"Inset: diagram of the heterostructure geometry.",
"(b) AFM image of the same suspended TLG device.",
"The inset shows the AFM data along the dashed-red line cut in the main panel.The suspension transfer procedure is very reproducible.",
"Out of our 16 transfer attempts, there was only 1 failure (transferred, but no suspension) and 15 successes.",
"In terms of producing the desired suspended heterostructures, we thus find the success rate to be around 93%.",
"T Table: List of all 16 suspended 2DM transfers done via our nitrocellulose micro-stamp method."
],
[
"Tuning the Raman factors and light absorption in BLG",
"There is a strong interest in enhancing the light absorption of graphene [12], [13] and 2D transition metal dichalcogenides (TMDs)[14] to optimize their great potential for light harvesting applications, and also to develop new tools for NOEMS research [15].",
"The Raman scattering intensity in an isolated (i.e.",
"surrounded by vacuum) 2D crystal is linearly proportional to light absorption since only a tiny fraction of photons undergo Raman scattering [16].",
"In Figure 4 of the main text and fig:S10, we observe that BLG Raman scattering intensity in planar heterostructures can be enhanced via constructive and destructive interferences at the interfaces between the various 2D layers.",
"Here we first present in sec:S4.1 a quantitative model based on Fresnel’s equations to calculate the exclusive light absorption in BLG, $A_{BLG}$ , and then how this method can be extended to calculate the Raman Factors (Raman relative intensities) $F_{BLG-G}$ and $F_{BLG-2D}$ .",
"In sec:S4.2 we detail how we analyze and calibrate our experimental Raman data to establish a quantitative comparison between the data and the theoretical model.",
"Figure: Interference in a tilted-suspended BLG/air/SiO2/Si heterostructure.",
"(a) Optical image of tilted-suspended structure with tree distinct regions labeled I, II and III.",
"Inset: diagram of the heterostructure geometry.",
"(b) Raman spectra acquired at the numbered circles location in the optical picture."
],
[
"Calculating the BLG exclusive light absorption and Raman factors in planar heterostructures",
"Exclusive light absorption in BLG – We use a simple theoretical model based on Fresnel’s equations [17] and derived in previous work [13], [14].",
"This exclusive light absorption model predicts an absorption in the visible range of about 2.3% for a SLG (4.6% for BLG) when surrounded by vacuum.",
"This was confirmed experimentally [18].",
"Figure: Suspended-2DM optical cavity and optical interferences leading to absorbtion.",
"(a) Schematic of the enhanced circulating power inside the optical cavity formed by a suspended BLG and substrate surfaces (mirrors).",
"(b)Series of Fresnel reflections, r ij r_{ij}, and transmissions, t ij t_{ij}, at the various interfaces between media ii and jj.fig:S11a is a representation of the enhanced circulating power inside the optical cavity formed by a suspended BLG and back-plan substrate (Si) due to interferences.",
"This can lead to increased light absorption when the mirror spacing creates constructive interferences.",
"To describe this quantitatively, in fig:S11b we show the series of Fresnel reflections, $r_{ij}$ , and transmissions, $t_{ij}$ , at the various interfaces between media $i$ and $j$ .",
"The reflection (transmission) coefficient values are the ratio of the amplitude of the reflected (transmitted) light's electric field to the incident light.",
"They are given numerically by $r_{ij} = (n_i-n_j)/(n_i+n_j)$ and $t_{ij} = (2n_i)/(n_i+n_j))$ , where $n_{i}$ and $n_{j}$ are the complex indices of refraction of material, and they are wavelength dependent.",
"The real part of the index of refraction conserves energy, while the imaginary part absorbs energy (dissipative currents).",
"The term $\\beta _i$ is the complex phase shift introduced while light travels in medium $i$ , and depends on $n_i$ , the medium's thickness $h_i$ , and incident light wavelength $\\lambda $ , as per Equation REF .",
"tab:S3 below summarizes the indices of refraction we used in our calculations.",
"$ \\beta _i = \\frac{ 2\\pi n_i h_i}{\\lambda }$ T Table: List of all indices of refraction for different wavelength and material.We first show how we calculate a $N$ -layer heterostructure's total reflection coefficient, $r_{N}$ , and transmission coefficient, $t_{N}$ , starting from the individual interfacial coefficients $r_{ij}$ and $t_{ij}$ .",
"Then we show how $r_{N}$ and $t_{N}$ leads to an expression for $A_{BLG}$ , the exclusive light absorption of BLG when inserted in a planar heterostructure.",
"For a system with $N=$ 3 media (layers), we find that [17], $r_{3} &= \\frac{ r_{01} + r_{12} e^{2i\\beta _1} }{ 1+r_{01} r_{12} e^{2i\\beta _1} } \\\\[10pt]t_{3} &= \\frac{ t_{01} t_{12} e^{i\\beta _1} }{ 1+r_{01} r_{12} e^{2i\\beta _1} }$ where $r_{01}, r_{12}, t_{01}$ and $t_{12}$ are the reflection and transmission coefficients (Fresnel’s) for light rays at the interfaces of media 0 and 1, and 1 and 2 respectively.",
"We can simply use a recursive method to obtain all of the $r_{N}$ and $t_{N}$ , when $N > 3$ .",
"The recursive expression to derive the $N$ coefficients from the $N-1$ ones, is given by Equation REF .",
"Note that in Equation REF , the newly added medium is placed on top of the previous $N-1$ media (see fig:S11b), and is now assigned the label “0” while the other media’s labels are increased by 1.",
"$r_{N} &= \\frac{ r_{01} + r_{N-1} e^{2i\\beta _1} }{ 1+r_{01} r_{N-1} e^{2i\\beta _1} } \\\\[10pt]t_{N} &= \\frac{ t_{01} t_{12} e^{i\\beta _1} }{ 1+r_{01} r_{N-1} e^{2i\\beta _1} } \\ , $ Thus, in order to derive the coefficients for a $N=$ 4 media system, we simply plug the coefficients for $N=$ 3 from Equations REF and into Equations REF and , and find: $r_{4} &= \\frac{ r_{01} + r_{12} e^{2i\\beta _1}+(r_{01} r_{12} + e^{2i\\beta _1} ) r_{23} e^{2i\\beta _2} }{ 1+r_{01} r_{12} e^{2i\\beta _1} +(r_{12} + r_{01} e^{2i\\beta _1}) r_{23} e^{2i\\beta _2} } \\\\[10pt]t_{4} &= \\frac{ t_{01} t_{12} t_{23} e^{i (\\beta _1+\\beta _2)} }{ 1+r_{01} r_{12} e^{2i\\beta _1} +(r_{12} + r_{01} e^{2i\\beta _1}) r_{23} e^{2i\\beta _2} } \\ .",
"$ The fraction of power reflected, $R_{N}$ , by a $N$ media heterostructure is given by squaring the complex amplitude, $R_{N} = {r_{N}}^2$ .",
"The back plane substrate is a semi-infinite (very thick) medium and there is no transmission across it[14].",
"Thus, $A_{N}= 1 - R_{N}$ Our objective is to calculate specifically (exclusively) the light absorbed by the BLG (or another specific 2DM replacing it).",
"In the heterostructures (optical cavities) we study, the 2D layers used as spacers between the top BLG and bottom mirror (substrate) are air, hBN, or SiO$_2$ .",
"They all have purely real indices of refractions, and do not lead to any absorption.",
"The only material dissipating power, in addition to the BLG, is the bottom substrate/mirror, made of either Si or Al.",
"We therefore, calculate the exclusive light absorption of BLG as: $A_{\\mathrm {BLG}}=1-R_{N}-A_{substrate}$ where $A_{\\mathrm {substrate}} = A_{\\mathrm {Si}}$ in our BLG/air/SiO$_2$ /Si devices, and $A_{\\mathrm {substrate}} = A_{\\mathrm {Al}}$ in our BLG/hBN/Al devices.",
"Based on fig:S11b, we can identify $A_{\\mathrm {substrate}} = T_{N} = \\frac{n_N}{n_0} {t_{N}}^2$ , which can readily be calculated using Equation .",
"We mention that the wavelength used in Equation REF is the incident laser wavelength.",
"Raman factors in BLG – While it is simple to calculate $A_{\\mathrm {BLG}}$ , experimentally it is rather challenging to isolate it from other absorption processes.",
"To achieve an optical measurement which contains a unique BLG fingerprint, we use Raman spectroscopy.",
"While we cannot measure directly light absorption with Raman, we can measure closely related quantities called the Raman Factors, $F_{\\mathrm {BLG-G}}$ for the G-peak and $F_{\\mathrm {BLG-2D}}$ for the 2D-peak.",
"The Raman factor's meaning is the ration between the Raman intensity observed in a material (integrated Raman count) inside an heterostructure (i.e.",
"including interferences) and the Raman intensity in the same material surrounding by only vacuum.",
"Thus experimentally, $F_{\\mathrm {BLG}}$ = (BLG-in-heterostructure Raman count) / (BLG-in-vacuum Raman count).",
"The $F_{\\mathrm {BLG}}$ 's are calculated with the same Fresnel coefficients as $A_{\\mathrm {BLG}}$ , as described below.",
"Thus their experimental measurement can confirm that values necessary to extract the BLG absorption, in addition to demonstrating the tunability of the Raman scattering intensities.",
"Moreover, the ability to measure simultaneously both of the Raman Factors (same sample, same time, same laser, same systematic errors) in several devices, provides a very robust experimental verification of the calculations.",
"Figure: Schematic of reflection interferences both before (a) and after (b) Raman scattering.",
"(a) 𝐄 0 \\mathbf {E}_0 is the incident laser electric field, 𝐄 z \\mathbf {E}_z is the field inside medium 1.",
"The Fresnel equations are used to calculate the transmission and reflection amplitudes.",
"(b) We start with the 𝐄 Ram \\mathbf {E}_{\\mathrm {Ram}}, Raman scattered E-field produced inside medium 1 (BLG), to calculate the reflections and transmissions coefficients following the scattering event.",
"This Raman-shifted light also goes through multiple reflections and transmissions before exiting as 𝐄 out \\mathbf {E}_{\\mathrm {out}}.To calculate the $F_{\\mathrm {BLG}}$ 's, we must first calculate the absorption amplitude for $N$ -media structure, $F_{ab-N}$ , from the interferences due to multiple reflection/transmission paths at the incident laser light (see fig:S12a) as well the amplitudes, $F_{sc-N-G}$ or $F_{sc-N-2D}$ , due to the multiple reflection/transmission paths of the Raman-shifted light after scattering, as shown in fig:S12b [26].",
"We follow Yoon et al.",
"[12], and find for a $N =$ 3 media system, $F_{ab\\mathrm {-3}}(\\lambda = \\lambda _{laser}) &=t_{01} \\frac{ e^{-i \\beta _z} + r_{12} e^{-i (2\\beta _1-\\beta _z)} }{ 1+ r_{12} r_{01} e^{-2 i \\beta _1} } \\\\[10pt]F_{sc\\mathrm {-3}}(\\lambda = \\lambda _{Raman-shifted}) &=t_{10} \\frac{ e^{-i \\beta _z} + r_{12} e^{-i (2\\beta _1-\\beta _z)} }{ 1+ r_{12} r_{01} e^{-2 i \\beta _1} } \\ , $ where $\\beta _z$ is the complex phase shift introduced along the BLG thickness at position $z$ .",
"It is important to notice that Equation includes $t_{10}=2n_1/(n_1+n_0) \\ne t_{01}$ , and that the wavelength used is the one after the Raman shift, while the original (laser) wavelength is used in Equation REF .",
"Our incident laser has a wavelength of 532 nm, the shifted laser wavelength for the G peak is 581 nm (1580 cm$^{-1}$ ), and for the 2D peak it is 621 nm (2700 cm$^{-1}$ ).",
"Equations REF can also be extended to planar heterostructures made of $N$ -media as follows, $F_{ab\\mathrm {-N}}(\\lambda = \\lambda _{laser}) &=t_{01} \\frac{ e^{-i \\beta _z} + r_{\\mathrm {N-1}} e^{-i (2\\beta _1-\\beta _z)} }{ 1+ r_{\\mathrm {N-1}} r_{01} e^{-2 i \\beta _1} } \\\\[10pt]F_{sc\\mathrm {-N}}(\\lambda = \\lambda _{Raman-shifted}) &=t_{10} \\frac{ e^{-i \\beta _z} + r_{\\mathrm {N-1}} e^{-i (2\\beta _1-\\beta _z)} }{ 1+ r_{\\mathrm {N-1}} r_{01} e^{-2 i \\beta _1} } \\ .",
"$ For instance, when $N=4$ we find, $F_{ab\\mathrm {-4}}(\\lambda = \\lambda _{laser}) &=t_{01} \\frac{ (1+r_{12} r_{23} e^{-2i\\beta _2}) e^{-i \\beta _z} + (r_{12}+r_{23} e^{-2i\\beta _2} ) e^{-i (2\\beta _1-\\beta _z)} }{ 1+ r_{12} r_{23} e^{-2i\\beta _2}+(r_{12} + r_{23} e^{-2i\\beta _2}) r_{01} e^{-2 i \\beta _1} } \\\\[10pt]F_{sc\\mathrm {-4}}(\\lambda = \\lambda _{Raman-shifted}) &=t_{10} \\frac{ (1+r_{12} r_{23} e^{-2i\\beta _2}) e^{-i \\beta _z} + (r_{12}+r_{23} e^{-2i\\beta _2} ) e^{-i (2\\beta _1-\\beta _z)} }{ 1+ r_{12} r_{23} e^{-2i\\beta _2}+(r_{12} + r_{23} e^{-2i\\beta _2}) r_{01} e^{-2 i \\beta _1} } \\ .",
"$ Finally, to calculate the desired Raman Factors, $F_{BLG-G}$ and $F_{BLG-2D}$ , we use the above amplitudes following Yoon et al.",
"[12], $F_{BLG-G}=\\mathcal {N}_{G} \\int _0^{h_{BLG}} {F_{ab}\\cdot F_{sc-G} }^2 dz \\\\[10pt]F_{BLG-2D}=\\mathcal {N}_{2D} \\int _0^{h_{BLG}} {F_{ab}\\cdot F_{sc-2D} }^2 dz \\ , $ and $\\mathcal {N}$ is a normalization constant (1 divided by the integral calculated for BLG surrounded by vacuum only).",
"The integration is over the thickness of the BLG crystal, where z is a dummy variable representing the vertical position inside the graphene, see fig:S12.",
"Figure: Raman Factors and exclusive light absorption of BLG in Air/BLG/SiO2/Si heterostructure.",
"The left axis shows F BLG - no shift F_{\\mathrm {BLG-no shift}} (black), F BLG -G F_{\\mathrm {BLG-G}} (blue), and F BLG -2D F_{\\mathrm {BLG-2D}} (gold) versus h SiO 2 h_{\\mathrm {SiO_{2}}}, and the right axis display A BLG A_{\\mathrm {BLG}}.",
"The inset shows a diagram of the heterostructure geometry.fig:S13 shows Raman factors and absorption calculated in BLG/SiO$_2$ /Si heterostructures (inset) as a function of the thickness of the SiO$_2$ film.",
"The solid black trace shows $F_{\\mathrm {BLG-no shift}}$ , is the Raman Factor calculated under the approximation that there is zero Raman wavelength shift.",
"This is to be compared with the $F_{\\mathrm {BLG-G}}$ (blue trace) and the $F_{\\mathrm {BLG-2D}}$ , which clarify the impact of the wavelength shift on the predicted Raman intensity.",
"Lastly, the red trace shows the calculated $A_{\\mathrm {BLG}}$ (right axis), whose maxima align well with $F_{\\mathrm {BLG-no shift}}$ .",
"As an example, for our BLG/310$\\pm $ 3nm-SiO$_2$ /Si heterostructures, we can read from fig:S13 the calculated $F_{\\mathrm {BLG-G}}=$ 1.48 $\\pm $ 0.18 and $F_{\\mathrm {BLG-2D}}=$ 1.53 $\\pm $ 0.12, where the errors are systematic ones (i.e.",
"a rigid shift of the whole data sets depending on the exact SiO$_2$ thickness on the wafers).",
"The corresponding exclusive BLG light absorption in fig:S13 is 4.25 %.",
"We emphasize that these calculations do not include any fitting or free parameter."
],
[
"Experimental Raman intensity measurements, and calibration of the Raman factors",
"To measure experimentally Raman factors and make a quantitative comparison with the calculated ones, we first explain how to extract the raw experimental Raman counts, and then we discuss how to convert these Raman counts into Raman factors based on reliable calibration devices.",
"Experimental Raman intensity measurements – The Raman counts for the G and 2D peaks in BLG are measured by integrating the areas under the measured peaks.",
"In order to extract these numbers, we first do a fit of the experimental curves using a single Lorentzian (G-peak) or a combination of four Lorentzian functions (2D-peak).",
"Examples of these fits are shown respectively in fig:S14a-b for data from a BLG/310nm-SiO$_2$ /Si device.",
"Using these fits, we remove the background signal (vertical offset and linear background) from the raw data, and then integrate the area under the peaks as shown in fig:S14c-d.",
"The integrated counts shown in fig:S14c-d are typical and have very low uncertainties, and scale linearly with both laser power and integration time.",
"The range of integration is set such that, increasing its range does not change the extracted number of counts.",
"Figure: Curve fitting to extract the integrated Raman intensities in a BLG/310nm-SiO 2 _2/Si device.",
"(a) Fitting the G-peak and (b) the 2D-peak.",
"The open black circles are the raw data and the red traces are the fit functions for G and 2D peak.",
"(c) Integrated Raman count under the G-peak fit, and (d) under the 2D-peak fit.fig:S15a show the measured, and fig:S15b the fitted, laser intensity profile of our laser beam.",
"The width of our focused laser is 0.8$\\pm $ 0.05 $\\mu $ m and determines our spatial resolution.",
"Figure: Laser beam intensity profile.",
"(a) The measured intensity profile of our focused laser beam at 532 nm.",
"(b) The fitted laser intensity profile.",
"It gives beam width of 0.8 μ\\mu m, which sets the spatial resolution of our Raman measurements.Calibrating the experimental Raman factors – The theoretical calculations of $F_{\\mathrm {BLG}}$ discussed above do not involve any fitting parameter, but we need one single reference calibration (conversion factor) to relate the many measured Raman counts to the calculated Raman Factors.",
"For this calibration, we used a series of 5 identically prepared calibration samples air/BLG/310 nm-SiO2/Si.",
"Using the above calculated Raman Factors for such heterostructures ($F_{\\mathrm {BLG-G}}=$ 1.48 and $F_{\\mathrm {BLG-2D}}=$ 1.53), and the measured Raman counts in Figure fig:S14 we can establish a calibration factor relating experiment and theory.",
"The details of this calibration are shown in fig:S16.",
"In panel (a) we show one of the five air/BLG/310 nm-SiO2/Si reference samples and indicate in red the path along which many independent Raman measurements were made.",
"The normalized Raman counts $I/P_{L}$ , from dividing the integrated counts by the (laser power/area $\\times $ exposure time) used during the acquisitions, is shown in fig:S16b versus the position along the red line shown in fig:S16(a).",
"Each set of symbols in (b) represents the data from one for the five devices.",
"The laser powers and acquisition times used were 1 – 5 mW/$\\mu \\mathrm {m}^2$ , and 10 s respectively.",
"We found a very high consistency in the measured $I/P_{L}$ in different samples and at different locations, for both the G-peak and 2D-peak.",
"We used the average values of $I/P_{L}$ for the G and 2D resonances, and assign them the calculated Raman factors.",
"This gives us a the following conversion constant of $F_{\\mathrm {BLG-G}}=$ 1.0 $=$ 38000 $\\pm $ 1000, and $F_{\\mathrm {BLG-2D}}=$ 125000 $\\pm $ 4000 counts for a laser power of 1 ± 0.02 mW/$\\mu \\mathrm {m}^2$ and 10 second of exposure.",
"We verified carefully that the calibration ratio is unchanged by the experimental laser power in a range well beyond the values used in our data acquisition (1 – 20 mW/$\\mu \\mathrm {m}^2$ ).",
"We used this calibration for all reported experimental $F_{\\mathrm {BLG}}$ 's.",
"Figure: Calibration of the integrated Raman counts to the calculated Raman factors.",
"(a) Top view of a calibration BLG/310nm-SiO 2 _2/Si heterostructure.",
"The red line shows the locations of the Raman data acquisitions.",
"(b) The normalized integrated Raman counts I/P L I/P_{L}, for 10s exposures, measured in five devices similar to the one in (a) at many different positions X L X_{L}, for both the G-peak and 2D-peak.Error bars in Figure 4 of the main text The vertical uncertainties reported for the experimental data in Figure 4b-c, e-f, and h-i, stem primarily from the uncertainty of $\\pm $ 3 nm on the thickness of the SiO$_2$ film on our Raman factor calibration devices.",
"As explained in the paragraphs above, a change in SiO$_2$ thickness leads to a change in the calculated Raman factors, and thus in the conversion (calibration) constant for the experimental $F_{\\mathrm {BLG-G}}$ and $F_{\\mathrm {BLG-2D}}$ .",
"The horizontal uncertainties in the data reported in Figure 4 stems from the uncertainties on thickness of the optical cavity spacers ($h_{hBN}$ or $h_{air}$ .",
"This uncertainty itself stems from various sources including, the thickness variations of the hBN crystals or suspension heights at various laser positions, the precision of laser beam positions (around $\\pm $ 0.5 $\\mu $ m), AFM data noise, and AFM calibration limitations.",
"Specifically for Figures 4h-i, which refer to data from a tilted-suspended device, an additional source of horizontal-axis uncertainty is the cross-section profile (slope) of the suspended device.",
"Based on both the optical images and Raman data, we found that the suspension slope $dz/dx$ was roughly linear.",
"To establish a realistic uncertainty for this approximation, we compared (took the difference between) the assumed linear slope positions and the AFM measured positions in a similar tilted-suspended device.",
"This explains why the horizontal error bars in Figures 4h-i vary along the x-axis.",
"toctoc toc"
]
] | 2011.14166 |
[
[
"Model-independently calibrating the luminosity correlations of gamma-ray\n bursts using deep learning"
],
[
"Abstract Gamma-ray bursts (GRBs) detected at high redshift can be used to trace the Hubble diagram of the Universe.",
"However, the distance calibration of GRBs is not as easily as that of type Ia supernovae (SNe Ia).",
"For the calibrating method based on the empirical luminosity correlations, there is an underlying assumption that the correlations should be universal over the whole redshift range.",
"In this paper, we investigate the possible redshift dependence of six luminosity correlations with a completely model-independent deep learning method.",
"We construct a network combining the Recurrent Neural Networks (RNN) and the Bayesian Neural Networks (BNN), where RNN is used to reconstruct the distance-redshift relation by training the network with the Pantheon compilation, and BNN is used to calculate the uncertainty of the reconstruction.",
"Using the reconstructed distance-redshift relation of Pantheon, we test the redshift dependence of six luminosity correlations by dividing the full GRB sample into two subsamples (low-$z$ and high-$z$ subsamples), and find that only the $E_p-E_{\\gamma}$ relation has no evidence for redshift dependence.",
"We use the $E_p-E_{\\gamma}$ relation to calibrate GRBs, and the calibrated GRBs give tight constraint on the flat $\\Lambda$CDM model, with the best-fitting parameter $\\Omega_{\\rm M}$=0.307$^{+0.065}_{-0.073}$."
],
[
"Introduction",
"The accelerating expansion of the universe is first found from the fact that the luminosity of type Ia supernovae (SNe Ia) is dimmer than expected [44], [43].",
"Since then, many SNe Ia datasets have been compiled to cosmological researches [47], [4], [46].",
"However, due to the limited luminosity, most observable SNe Ia are at redshift $z\\lesssim 2$ .",
"With these low redshift data, cosmological models cannot be unambiguously distinguished [57], [17], [48], [53].",
"Among these SNe Ia samples, the most up-to-date Pantheon compilation [46] is the largest sample and the redshift of the furthest SNe reaches up to $z\\sim $ 2.3, but the number of SNe whose redshift is larger than 1.4 is only six.",
"Actually, the subtle difference between cosmological models in low redshift range would be remarkable in high redshift range.",
"For various cosmological models, such as $\\Lambda $ CDM model, wCDM model, Chevallier-Polarsky-Linder (CPL) model, holographic dark energy (HDE) model and generalised chaplygin gas (GCG) model, the evolutions of dark energy equation of state are discrepant in high redshift [10], [51], [42], [18].",
"Thus, it is important to extend the Hubble diagram to high redshift range.",
"Gamma-ray bursts (GRBs), as the most energetic explosions in the universe, is detectable up to very high redshift [12].",
"Therefore, it is possible to use GRBs as standard candles to trace the Hubble diagram at high redshift.",
"Combining GRBs with other standard candles, the cosmological parameters can be tightly constrained [21], [49], [28], [33], [5], [55], [52], [15], [7], [9], [35].",
"However, it is not easy to calibrate the distance of GRBs due to the lack of knowledge on the explosion mechanism.",
"Based on the correlations between various observables of the prompt or afterglow emission, several methods have been proposed to calibrate the distance of GRBs [14], [25], [20], [45], [32], [54], [37].",
"Most calibrating methods rely on one of the following 2-dimensional empirical luminosity calibrations found in long GRBs: the correlation between spectrum lag and isotropic peak luminosity ($\\tau _{\\rm lag} - L$ relation) [41], the correlation between time variability and isotropic peak luminosity ($V - L$ relation) [19], the correlation between peak energy and isotropic peak luminosity ($E_p - L$ relation) [56], the correlation between peak energy and collimation-corrected energy ($E_p - E_{\\gamma }$ relation) [25], the correlation between minimum rise time of light curve and isotropic peak luminosity ($\\tau _{\\rm RT}- L$ relation) [45] and the correlation between the peak energy of $\\nu F_{\\nu }$ spectrum and isotropic equivalent energy ($E_p - E_{\\rm iso}$ relation) [1].",
"What is noteworthy is that, all of the above correlations depend on a certain cosmological model, thus leading to the circularity problem when use the calibrated GRBs to constrain cosmological models.",
"Several model-independent methods have been proposed by using distance ladders to calibrate GRBs [32], [52], [54], [37].",
"Using cubic interpolation or other approximations from SNe Ia dataset, one can first calculates the distance for low-$z$ ($z<$ 1.4) GRBs to derive the empirical luminosity correlations.",
"Then by extrapolating the correlations to high-$z$ ($z>$ 1.4) GRBs, one can inversely derive the distance for high-$z$ GRBs.",
"However, in addition to the dependency of a certain approximation form, there is an underlying assumption in this method that the luminosity correlations are universal over all redshifts.",
"Many works have been devoted to test the possible redshift dependence of luminosity correlations [30], [3], [50], [34], [35].",
"Assuming a flat $\\Lambda $ CDM model with different cosmological parameters, [3] and [50] investigated the above six empirical luminosity correlations in four redshift bins, and found no statistically significant evidence for redshift evolution.",
"While [35] rechecked these correlation in two redshift bins, and found moderate evidence ($>3\\sigma $ ) for the redshift evolution in four out of six correlations.",
"All the above works to check the redshift dependence of GRB luminosity correlations still depend on cosmological model.",
"In this paper, we will investigate the redshift dependence of these correlations with a model-independent method, i.e.",
"the deep learning, which is one of the most exciting areas in machine learning.",
"Deep learning is tackling large and highly complex machine learning tasks by training deep neural networks constructed with an input layer to receive the features, several hidden layers to transform the information from the previous layer, and an output layer to export the target, where each layer contains hundreds of nonlinear processing neurons [2].",
"Recently, deep learning method has been widely employed in cosmological researches, such as predicting galaxy morphology [16], learning the universe at scale [40], constraining cosmological dark energy [18], and so on.",
"In this paper, we proposed a method to calibrate the luminosity correlations of GRBs using deep learning method, which is completely independent on the cosmological model.",
"We first combine the Recurrent Neural Networks (RNN) and the Bayesian Neural Networks (BNN) to reconstruct the distance-redshift relation from the Pantheon sample up to the highest redshift of GRB dataset, then we test the redshift dependence of GRB luminosity correlations by dividing the full GRB sample into low-$z$ ($z\\le 1.4$ ) and high-$z$ ($z>1.4$ ) subsamples.",
"Comparing with previous works [3], [50], [35], our method is model-independent and only relies on the training data, i.e.",
"the Pantheon compilation.",
"Without any assumption about the cosmological model or about the specific form of distance-redshift relation of SNe Ia, we directly calibrate the distance of GRBs from the Pantheon sample with deep learning.",
"The rest of the paper is organized as follows: In Section , we introduce the architecture of the RNN+BNN network.",
"In Section , we use the network to reconstruct the distance-redshift relation from Pantheon supernovae data set.",
"In Section , we use the reconstructed distance-redshift relation to test the possible redshift dependence of luminosity correlations of GRBs.",
"Finally, discussions and conclusion are given in Section ."
],
[
"The architecture of the neural network",
"Before testing the redshift dependence of luminosity correlations of GRBs, we firstly introduce the process of reconstructing the luminosity distance of SNe with deep learning.",
"Based on the training Artificial Neural Networks (ANN), such as Convolutional Neural Networks (CNN), Recurrent Neural Networks (RNN) and Bayesian Neural Networks (BNN), deep learning is versatile, powerful and scalable in tackling complex problems such as classifying billions of images, recognizing speech, detecting subtle patterns in data, etc [2].",
"Recently, the application of deep learning in cosmological research is very extensive and successful [16], [40], [18].",
"Following the work of [18], we reconstruct the distance moduli from the Pantheon compilation [46] with RNN+BNN.",
"In this process, the reconstruction of distance only depends on the Pantheon dataset, and without any assumption on the cosmological model.",
"RNN is a class of nets which can predict the future from the complex sequential information without any model assumption, but is incapable of estimating the uncertainty of target.",
"This shortcoming can be fixed up with BNN.",
"Therefore, our neural network is composed of RNN and BNN, the details of which are described bellow.",
"RNN is one of the supervised learning algorithm by training the neural networks with the real data, reaching an ideal network characterizing the relationship between the target and the feature by minimizing the loss function [2].",
"In our work, the Pantheon data set is used as the training data, in which the redshift is the feature and the distance module is the target.",
"In RNN, the activation not only flows from the input layer to the output layer, but also has connections pointing backward.",
"The architecture of RNN is shown in Figure REF .",
"In the unrolled RNN, the neurons at each time step $t$ receive the inputs as well as the outputs from the previous time step [2].",
"In the neural network, the loss function is used to depict the difference between the targets and the predicts.",
"We adopt the Mean Squared Error (MSE) function as the loss function and find the minimum with the Adam optimizer.",
"Figure: The architecture of our network with one hidden layer (left), unrolled through time step tt=4 (right).",
"In the unrolled network, each column is one of the tt time steps, while the three rows from bottom to top represent input layer, hidden layer and output layer, respectively.",
"The first two layers with tanh activation function consist of LSTM cell containing 100 neurons, while the output layer is a fully-connected (dense) layer.",
"To avoid overfitting, the dropout technique is employed between LSTM and its next layers, and we set the dropout rate to 0.2.Handling long sequences, the training of RNN will take a long time and the information of initial inputs will gradually fades away [2].",
"Thus, we adopt the time step $t=4$ to alleviate the long training time, and use the most popular basic cell called Long Short-Term Memory (LSTM) cell to solve the problem of information loss.",
"RNN with LSTM cell is aware of what to store, throw away and read.",
"The computations of LSTM are $i^{<t>}=\\sigma \\left(W_{xi}^T \\cdot x^{<t>}+W_{hi}^T \\cdot h^{<t-1>}+b_i\\right)$ $f^{<t>}=\\sigma \\left(W_{xf}^T \\cdot x^{<t>}+W_{hf}^T \\cdot h^{<t-1>}+b_f\\right),$ $o^{<t>}=\\sigma \\left(W_{xo}^T \\cdot x^{<t>}+W_{ho}^T \\cdot h^{<t-1>}+b_o\\right),$ $g^{<t>}=A_f\\left(W_{xg}^T \\cdot x^{<t>}+W_{hg}^T \\cdot h^{<t-1>}+b_g\\right),$ $c^{<t>}=f^{<t>}\\otimes c^{<t-1>}+i^{<t>}\\otimes g^{<t>},$ $y^{<t>}=h^{<t>}=o^{<t>}\\otimes A_f\\left(c^{<t>}\\right),$ where $\\sigma $ is the sigmoid function that outputs a value between 0 and 1, $t$ is the time step referring to the current sequence (for example $t=1$ for the first redshift).",
"The superscript $<t>$ indicates a vector of steps $t$ , the superscript $T$ is the transpose of the matrix, the dot is matrix product and $\\otimes $ is direct product.",
"$x^{<t>}$ and $y^{<t>}$ are respectively the current input and output vectors.",
"$h^{<t>}$ and $c^{<t>}$ are respectively the short-term state and long-term state of LSTM cells.",
"$A_f$ is an activation function to make the network be capable of solving complex tasks by introducing the non-linearity to network.",
"In our work, we use the tanh activation function, which is defined as $A_{f_{\\rm Tanh}}={\\rm tanh}(x)=\\frac{e^x-e^{-x}}{e^x+e^{-x}}.$ There are four connected layers playing different roles, where the main layer that outputs $g^{<t>}$ analyzes the current inputs $x^{<t>}$ and the previous state $h^{<t-1>}$ , the rest three layers are $gate \\ controllers$ : (a) $Input \\ gate$ controlled by $i^{<t>}$ determines which parts of $g^{<t>}$ should be added to $c^{<t>}$ , (b) $Forget \\ gate$ controlled by $f^{<t>}$ determines which parts of $c^{<t>}$ should be abandoned, (c) $Output \\ gate$ controlled by $o^{<t>}$ determines which parts of $c^{<t>}$ should be output.",
"It can be easily found that, these gate controllers are related to the logistic activation function $\\sigma $ , thus they would close the gate if output 0 and open it if output 1.",
"$W_{xi}$ , $W_{xf}$ , $W_{xo}$ and $W_{xg}$ are the weight matrices of each of above four layers connecting to the input vector.",
"$W_{hi}$ , $W_{hf}$ , $W_{ho}$ , and $W_{hg}$ are the weight matrices of each of layers connecting to the previous short-term state.",
"$b_i$ , $b_f$ , $b_o$ , and $b_g$ are the bias terms for each of layer.",
"In a deep neural network, the training may suffer from overfitting due to a large number of its own hyperparameters.",
"We can use the method called $regularization$ to prevent it from overfitting.",
"$Dropout$ is one of the most popular $regularization$ techniques, applying in some layers to reduce the overfitting risk [2].",
"In this way, some neurons has a probability of being ignored at every step controlled by $dropout \\ rate$ .",
"Besides, it is also of benefit to estimate the confidence of the training in BNN.",
"BNN is a supplementary of RNN for calculating the uncertainty of the prediction.",
"BNN is defined in terms of a prior distribution with parameters over the wights $p(\\omega )$ , which manifests a prior belief about parameters generating the observations.",
"With a given dataset $\\lbrace \\textbf {X},\\textbf {Y}\\rbrace $ , we can achieve the posterior distribution of the parameters space $p(\\omega |\\textbf {X},\\textbf {Y})$ .",
"Thus the output of a new input point $x$ can be anticipated by the integration $p(y^*|x^*,\\textbf {X},\\textbf {Y})=\\int p(y^*|x^*,\\omega )p(\\omega |\\textbf {X},\\textbf {Y})\\textrm {d}\\omega .$ A full BNN is extremely complex.",
"Several works had shown that a network with a dropout is approximately equivalent to the Bayesian model [22], [23], [24], [38].",
"Introducing the Bayesian machinery into the deep learning framework, [22], [23], [24] developed a new framework casting dropout training in deep neural network as approximate Bayesian inference in deep Gaussian processes and successfully applied in RNN.",
"Their results offer a Bayesian interpretation of the dropout technique, and verify that a network with a dropout is mathematically equivalent to the Bayesian model.",
"When the RNN is well-trained and executed $n$ times, the network is equivalent to BNN.",
"Therefore, we employ the $dropout$ in the training and call the trained network $n$ times to estimate the uncertainty of outputs, where the $dropout$ is an approximation of the Gaussian processes and cooperates with the activation function to determine the confidence regions of prediction."
],
[
"Reconstructing the distance-redshift relation from Pantheon",
"In order to reconstruct the Hubble diagram to high redshift range with our network, we use the latest Pantheon compilation [46] of SNe Ia as the training data.",
"The Pantheon compilation consists of 1048 well-calibrated SNe Ia in the redshift range $0.01<z<2.3$ .",
"The distance modulus of SNe Ia is given by $\\mu _B(z;\\alpha ,\\beta ,M_b)=m-M_b+\\alpha x(z)-\\beta c(z),$ where $m$ is the apparent magnitude, $M_b$ is the absolute magnitude, $x$ and $c$ are the stretch factor and color parameter respectively, $\\alpha $ and $\\beta $ are the nuisance parameter.",
"In the Pantheon dataset, the presented apparent magnitude has already been corrected for stretch and color, thus the stretch and color corrections are vanishing in eq.",
"(REF ).",
"The absolute magnitude is fixed to $M_b=-19.36$ .",
"We obtain the distance moduli $\\mu (z)$ from eq.",
"(REF ) and sort the data points according to the redshift from low to high.",
"We construct the RNN+BNN network and train it with the package TensorFlowhttps://www.tensorflow.org.",
"For clarity, we present the corresponding hyperparameters in Figure REF and list the steps to reconstruct data with our network as follow: (a) Data processing.",
"The scale of data has an effect on training.",
"Hence, we normalize the distance moduli of the sorted Pantheon data and re-arrange $\\mu -z$ as sequences with the step number $t=4$ .",
"(b) Building RNN.",
"We build RNN with three layers, i.e.",
"an input layer, a hidden layer and an output layer as described in Figure REF .",
"The first two layers are constructed with the LSTM cells of 100 neurons.",
"The redshifts $z^{<t>}$ and the corresponding distance moduli $\\mu ^{<t>}$ are the input and output vectors, respectively.",
"We employ the Adam optimizer to minimize the cost function MSE and train the network 1000 times.",
"(c) Building BNN.",
"We set the dropout rate to 0 in the input layer to avoid the lost of information, and to 0.2 in the second layer as well as the output layer [6], [39].",
"We execute the trained network 1000 times to obtain the distribution of distance moduli.",
"The highest redshift of GRB sample [50] is 8.2.",
"Therefore, we reconstruct the distance moduli by training RNN+BNN up to $z=8.5$ .",
"The result is shown in Figure REF .",
"The red dots with error bars and the light-blue dots are the observational data points of Pantheon and the central values of reconstruction, respectively.",
"The shaded regions are the 1$\\sigma $ and 2$\\sigma $ uncertainties of the reconstruction.",
"The inside plot at the lower-right corner of Figure REF is the MSE loss over the training epochs for both the train (blue) and test (orange) sets, which approach their minimum value when epoch near 400.",
"For comparison, we also plot the best-fitting curve of the $\\Lambda $ CDM model (black line).",
"The result shows that, our reconstruction is excellently consistent with the $\\Lambda $ CDM model within 1$\\sigma $ confidence level.",
"We want to emphasize again that our reconstruction of distance moduli is neither dependent on the cosmological model, nor has any assumption about the concrete form of the reconstructing function.",
"It is the architecture itself makes the neural network become a universal approximator, not the specific activation function [27].",
"We note that the Gaussian processes can also reconstruct the distance moduli without involving any model assumption [36].",
"However, the reconstructed uncertainty of Gaussian processes is very large in the region where the data points are sparse.",
"The advantage of RNN+BNN compared with the Gaussian processes is that, the former can reconstruct the curve precisely even far beyond the data points.",
"Figure: The reconstruction of distance moduli from Pantheon data set.",
"The red dots with 1σ1\\sigma error bars are the Pantheon data points.",
"The light-blue dots are the central values of reconstruction.",
"The shaded regions are the 1σ\\sigma and 2σ\\sigma uncertainties.",
"The black curve is the best-fitting curve of flat Λ\\Lambda CDM model.",
"Inside plot: the Mean Squared Error (MSE) loss over the training epochs for both the train (blue) and test (orange) sets.To investigate the influence of different choices of activation function on the reconstruction, we use Monte Carlo simulations.",
"We first simulate a set of Pantheon-like sample, whose redshifts and distance uncertainties are the same to that of Pantheon sample, and the distance moduli are sampled from Gaussian distribution $\\mu \\sim G(\\bar{\\mu },\\sigma _\\mu )$ , where $\\bar{\\mu }$ is calculated using the fiducial $\\Lambda $ CDM model with $\\Omega _M=0.3$ and $H_0=70~{\\rm km/s/Mpc}$ , and $\\sigma _\\mu $ is the uncertainty of the observed distance modulus of Pantheon sample.",
"Then we replace the Pantheon data with the mock data and retrain the network, and then reconstruct the distance-redshift relation using the trained network.",
"We also consider other three activation functions as a comparison with tanh function: $A_{f_{\\rm Relu}}={\\left\\lbrace \\begin{array}{ll}0,& x\\le 0\\\\x,& x>0\\end{array}\\right.",
"}$ $A_{f_{\\rm Elu}}={\\left\\lbrace \\begin{array}{ll}\\alpha \\left(e^x-1\\right),& x\\le 0\\\\x,& x>0\\end{array}\\right.",
"}$ $A_{f_{\\rm Selu}}={\\left\\lbrace \\begin{array}{ll}\\alpha \\lambda \\left(e^x-1\\right),& x\\le 0\\\\\\lambda x,& x>0\\end{array}\\right.",
"}$ where in Elu function $\\alpha =1$ [11], and in Selu function $\\alpha $ $\\approx $ 1.673 and $\\lambda $ $\\approx $ 1.051 [29].",
"The reconstructed results with four different activation functions are presented in Figure REF .",
"It is shown that only with the tanh function our network can correctly reconstruct the distance-redshift relation up to $z\\approx 8.5$ within $1\\sigma $ uncertainty.",
"For the rest three activation functions, the network couldn't correctly reconstruct the curve at high redshift.",
"Therefore, we choose the tanh function rather than the rest three in our work.",
"Note that the similarity between the elu and selu functions makes the reconstructed curves be similar to each other.",
"Figure: The reconstructions of distance-redshift relation from Pantheon-like sample (generated from fiducial Λ\\Lambda CDM model) with four different activation functions.",
"Top-left: tanh; top-right: relu; bottom-left: elu; bottom-right: selu.",
"This figure shows that the reconstruction depends on the activation functions.We fix the activation function to tanh to investigate the influence of other hyperparameters on our network.",
"Firstly, we fix the number of layers to 3 and choose three different number of neurons {50, 100 150} for comparison.",
"We also fix the number of neurons to 100 and choose three different number of layers {2,3,4} for comparison.",
"The training losses of different cases are presented in Figure REF .",
"For the same number of layers, the training loss of 50 neurons is distinctly higher than that of 100 and 150 neurons.",
"The increase of neurons from 100 to 150 does not significantly reduce the training loss.",
"For the same number of neurons, the training loss of 2 layers is much higher than that of 3 and 4 layers, but the training losses of 3 layers and 4 layers are not significantly distinguishable.",
"Therefore, our choice of 100 neurons and 3 layers is appropriate to construct a well-trained network.",
"Figure: The training loss in the cases of different numbers of neurons and layers with the activation function fixed to tanh."
],
[
"Testing the redshift dependence of luminosity correlations",
"After reconstructing the distance-redshift relation from SNe, we can use it to calibrate the luminosity correlations of GRBs.",
"The luminosity correlations of GRB can be expressed with the exponential form $R=AQ^b$ , and further be re-expressed with the linear form by taking logarithms, $y=a+bx \\ \\ \\ (y\\equiv \\log R, x\\equiv \\log Q, a\\equiv \\log A),$ in which `log' denotes the logarithm of base 10.",
"We definitely write the six luminosity correlations as follows $\\log \\frac{L}{\\textrm {erg} \\ \\textrm {s}^{-1}}=a_1+b_1\\log \\frac{\\tau _{\\textrm {lag},i}}{0.1s},$ $\\log \\frac{L}{\\textrm {erg} \\ \\textrm {s}^{-1}}=a_2+b_2\\log \\frac{V_i}{0.02},$ $\\log \\frac{L}{\\textrm {erg} \\ \\textrm {s}^{-1}}=a_3+b_3\\log \\frac{E_{p,i}}{300\\textrm {keV}},$ $\\log \\frac{E_{\\gamma }}{\\textrm {erg} }=a_4+b_4\\log \\frac{E_{p,i}}{300\\textrm {keV}},$ $\\log \\frac{L}{\\textrm {erg} \\ \\textrm {s}^{-1}}=a_5+b_5\\log \\frac{\\tau _{\\textrm {RT},i}}{0.1\\textrm {s}},$ $\\log \\frac{E_{\\textrm {iso}}}{\\textrm {erg}}=a_6+b_6\\log \\frac{E_{p,i}}{300\\textrm {keV}}.$ Here, quantities with a subscript `i' indicate that they are measured in the comoving frame, which are related to the quantities in the observer frame by $\\tau _{\\textrm {lag},i}=\\tau _{\\textrm {lag}}(1+z)^{-1}$ , $\\tau _{\\textrm {RT},i}=\\tau _{\\textrm {RT}}(1+z)^{-1}$ , $V_i=V(1+z)$ and $E_{p,i}=E_p(1+z)$ , where ${\\tau _{\\textrm {lag}},\\tau _{\\textrm {RT}},V,E_p}$ can be directly derived from the observations of spectrum or light curve of GRBs.",
"Assuming that GRBs radiate isotropically, the isotropic equivalent luminosity can be derived from the bolometric peak flux $P_{\\textrm {bolo}}$ by [45] $L=4\\pi d^2_L P_{\\textrm {bolo}},$ where $d_L$ is the luminosity distance of GRB, which can be obtained from the reconstructed distance moduli of Pantheon presented in section with the relation $\\mu =5\\log \\frac{d_L}{\\textrm {Mpc}}+25.$ Hence, the uncertainty of $L$ propagates from the uncertainties of $P_{\\rm bolo}$ and $d_L$ .",
"The isotropic equivalent energy $E_{\\rm iso}$ can be obtained from the bolometric fluence $S_{\\rm bolo}$ by $E_{\\rm iso}=4\\pi d^2_L S_{\\textrm {bolo}} (1+z)^{-1},$ the uncertainty of $E_{\\rm iso}$ propagates from the uncertainties of $S_{\\rm bolo}$ and $d_L$ .",
"If, on the other hand, GRBs radiate in two symmetric beams, then we can define the collimation-corrected energy $E_{\\gamma }$ as $E_{\\gamma }\\equiv E_{\\textrm {iso}}F_{\\textrm {beam}},$ where $F_{\\rm beam}\\equiv 1-\\cos \\theta _{\\rm jet}$ is the beaming factor, $\\theta _{\\rm jet}$ is the jet opening angle.",
"The uncertainty of $E_{\\gamma }$ propagates from the uncertainties of $E_{\\rm iso}$ and $F_{\\rm beam}$ .",
"Our GRBs sample is taken from [50], which consists of 116 long GRBs in redshift range $z\\in [0.17,8.2]$ .",
"Following [35], we divide GRBs into two subsamples, i.e.",
"the low-$z$ sample ($z\\le 1.4$ ) which consists of 50 GRBs, and the high-$z$ sample ($z> 1.4$ ) which consists of 66 GRBs.",
"We investigate the redshift dependence of luminosity correlations for this two subsamples, as well as for the full GRBs sample.",
"To fit the six luminosity correlations, we apply the D'Agostini's liklihood [13] $\\mathcal {L}\\left(\\sigma _{\\rm int},a,b\\right)\\propto \\prod _i\\frac{1}{\\sqrt{\\sigma ^2_{\\rm int}+\\sigma ^2_{yi}+b^2\\sigma ^2_{xi}}}\\times \\exp \\left[-\\frac{(y_i-a-bx_i)^2}{2(\\sigma ^2_{\\rm int}+\\sigma ^2_{yi}+b^2\\sigma ^2_{xi})}\\right].$ By maximizing this joint likelihood function, we can derive the best-fitting parameters $(a,b,\\sigma _{\\rm int})$ , where the intrinsic scatter $\\sigma _{\\rm int}$ denotes any other unknown errors except for the measurement errors.",
"Table: The best-fitting parameters of GRB luminosity correlations.",
"NN is the number of GRBs in each subsample.Figure: The luminosity correlations for low-zz (blue dots) and high-zz (red dots) GRBs.",
"Error bars denote the 1σ\\sigma uncertainties.",
"The lines are the best-fitting results, blue line for low-zz GRBs, red line for high-zz GRBs and black line for all-zz GRBs.Figure: The confidence contours and marginalized PDFs for the parameters.We perform a Markov Chain Monte Carlo analysis to calculate the posterior probability density function (PDF) of parameter space.",
"We assume a flat prior on all the free parameters and limit $\\sigma _{\\rm int}>0$ .",
"Note that not all GRBs can be used to analyze each luminosity correlation, because not all the necessary quantities are measurable for some GRBs.",
"For example, GRBs without measurement of the spectrum lag can not used in the $\\tau _{\\rm lag}-L$ analysis.",
"Hence, we present the best-fitting parameters, together with the number of available GRBs in each fitting in Table REF .",
"In Figure REF , we plot all the six luminosity correlations in logarithmic coordinates.",
"Low-$z$ and high-$z$ GRBs are represented by blue and red dots with the error bars denoting 1$\\sigma $ uncertainties.",
"The blue line, red line and black line stand for the best-fitting results for low-$z$ GRBs, high-$z$ GRBs and all-$z$ GRBs, respectively.",
"The 1$\\sigma $ and 2$\\sigma $ contours and the PDFs for parameter space are plotted in Figure REF .",
"As shown in Table REF , low-$z$ GRBs have a smaller intercept, but a sharper slope than high-$z$ GRBs for all the six luminosity correlations.",
"All-$z$ GRBs have the parameter values between that of low-$z$ and high-$z$ subsamples.",
"For the intrinsic scatter, low-$z$ GRBs have larger value than high-$z$ GRBs, and the $E_p-E_{\\gamma }$ relation has the smallest intrinsic scatter hence we can only obtain its upper limit.",
"The $V-L$ relation has the largest intrinsic scatter, thus it can not be fitted well with a simple line, which is legible in Figure REF .",
"In Figure REF , the contours in the ($a,b$ ) plane indicate that the $E_p-E_{\\gamma }$ relation of low-$z$ GRBs is consistent with that of high-$z$ GRBs at 1$\\sigma $ confidence level.",
"For the rest luminosity correlations, however, the intercepts and slopes for low-$z$ GRBs differ from that of high-$z$ GRBs at more than 2$\\sigma $ confidence level.",
"Having luminosity correlations calibrated, we can conversely using these correlations to calibrate the distance of GRBs, and further use GRBs to constrain cosmological models.",
"Since our calibration of luminosity correlations is independent of cosmological model, the circularity problem is avoided.",
"As we have seen, the $E_p-E_{\\gamma }$ relation is not significantly evolving with redshift, so we use this relation to calibrate the distance of GRBs.",
"Due to that the parameters of $E_p-E_{\\gamma }$ relation in three samples (low-$z$ , high-$z$ and full-$z$ samples) are consistent with each other, we directly apply the parameters obtained with all-$z$ sample.",
"The distance of 24 GRBs calibrated using $E_p-E_{\\gamma }$ are shown in the Hubble diagram in Figure REF .",
"With the Pantheon dataset, the matter density of the flat $\\Lambda $ CDM model is constrained to be $\\Omega _{\\rm M}$ =0.278$^{+0.007}_{-0.008}$ .",
"With 24 long GRBs alone, the matter density is constrained to be $\\Omega _{\\rm M}$ =0.307$^{+0.065}_{-0.073}$ .",
"It indicates that the Hubble diagram in high redshift is consistent with the $\\Lambda $ CDM model.",
"Figure: The Hubble diagram of GRBs.",
"The red dots with 1σ\\sigma error bars are the Pantheon data points.",
"The blue dots with with 1σ\\sigma error bars are 24 long GRBs calibrated using the E p -E γ E_p-E_{\\gamma } relation.",
"The lines are the best-fitting results to Λ\\Lambda CDM model with Pantheon (red line) and GRBs (blue dot-dashed line), respectively.",
"Note: the two best-fitting lines seem to overlap with each other."
],
[
"Discussions and conclusion",
"We have investigated the redshift dependence of six luminosity correlations in long GRBs using deep learning.",
"We first reconstructed the distance-redshift relation to high redshift from the Pantheon compilation using RNN, and derived the uncertainty using BNN.",
"Then the luminosity distance of GRBs is obtained without any assumption about the cosmological model.",
"To test the possible redshift dependence of luminosity correlations, we divided GRBs into low-$z$ and high-$z$ subsamples and investigate the correlations for each subsample separately.",
"It is found that, for all six luminosity correlations, low-$z$ subsample has a smaller intercept, but a sharper slope than high-$z$ subsample.",
"In four out of six correlations ($\\tau _{\\rm lag} - L$ , $E_p - L$ , $\\tau _{\\rm RT} - L$ and $E_p - E_{\\rm iso}$ ), the intercept and slope for low-$z$ subsample and high-$z$ subsample differ at more than $2\\sigma $ .",
"For the $V-L$ relation, the intrinsic scatter is too large to make a reliable conclusion.",
"For the $E_p-E_{\\gamma }$ relation, there is no evidence for redshift evolution, and the intrinsic scatter is the smallest among the six correlations.",
"However, the number of available GRBs for the $E_p-E_{\\gamma }$ relation is small, because most long GRBs lack the measurement of jet opening angle which is necessary to calculate the beaming factor.",
"The constraint on flat $\\Lambda $ CDM model from 24 GRBs calibrated using $E_p-E_{\\gamma }$ relation gives $\\Omega _{\\rm M}$ =0.307$^{+0.065}_{-0.073}$ , which is well consistent with the result constrained from Pantheon.",
"Our network is trained using Pantheon sample, so GRBs calibrated using the network should be consistent with Pantheon sample.",
"In the RNN+BNN network, the network is trained using the Pantheon data to extend the Hubble diagram to high redshift range without any assumption about the cosmological model.",
"In the deep learning method, the kernel is achieving the most efficient expression to denote the relationship between the features and the targets of Pantheon data, where the validity is reflected in the architecture of network.",
"In the cooperation of the Monte Carlo dropout and the activation function, our network not only predicts the central value, but also the corresponding confidence region.",
"Because the number of SNe Ia whose redshift beyond 1.4 is small, the uncertainty in the poor observational data range would be large.",
"Therefore, we choose a small dropout rate 0.2 with the specific activation function $A_{f_{\\rm Tanh}}$ to acquire a relatively small uncertainty.",
"The method proposed here has some advantages compared with other methods already existing in literatures.",
"[35] have tested the redshift-dependence of six luminosity correlations by dividing GRBs into low-$z$ and high-$z$ subsamples.",
"In their work, the slope and intercept parameters $(a,b)$ are calculated based on a specific cosmological model.",
"In contrast, our method proposed is completely independent on cosmological model.",
"Several works have been devoted to calibrating the distance of GRBs in a model-independent way [20], [31], [8], [54], [37].",
"All of these methods rely on the assumption that the luminosity correlations have no redshift evolution, which couldn't be tested by the method itself.",
"To calibrate high-$z$ GRBs, one must first obtain the luminosity correlation coefficients $(a,b)$ from the low-$z$ GRB, then directly extrapolate the correlation to high-$z$ GRBs.",
"proposed to calibrate GRBs by fitting the coefficients $(a,b)$ and the cosmological parameters simultaneously to a specific cosmological model, so the circularity problem can be avoided.",
"This calibrating method is still model-dependent, and the GRB calibrated in one cosmological model couldn't be used to constrain other models.",
"To constrain other cosmological models, the coefficients $(a,b)$ should be refitted again.",
"In comparison, our method can be used to test the redshift evolution of luminosity correlations model-independently.",
"If a correlation is redshift-independent, low-$z$ and high-$z$ GRBs can be calibrated simultaneously using this correlation.",
"GRBs calibrated in this way can be directly used to constrain cosmological models.",
"This work has been supported by the National Natural Science Fund of China Grant Nos.",
"11603005, 11775038 and 11947406, and the Fundamental Research Funds for the Central Universities Grant No.",
"2020CQJQY-Z003."
]
] | 2011.14040 |
[
[
"Contact Tracing Enhances the Efficiency of COVID-19 Group Testing"
],
[
"Abstract Group testing can save testing resources in the context of the ongoing COVID-19 pandemic.",
"In group testing, we are given $n$ samples, one per individual, and arrange them into $m < n$ pooled samples, where each pool is obtained by mixing a subset of the $n$ individual samples.",
"Infected individuals are then identified using a group testing algorithm.",
"In this paper, we use side information (SI) collected from contact tracing (CT) within non-adaptive/single-stage group testing algorithms.",
"We generate data by incorporating CT SI and characteristics of disease spread between individuals.",
"These data are fed into two signal and measurement models for group testing, where numerical results show that our algorithms provide improved sensitivity and specificity.",
"While Nikolopoulos et al.",
"utilized family structure to improve non-adaptive group testing, ours is the first work to explore and demonstrate how CT SI can further improve group testing performance."
],
[
"Introduction",
" Widespread testing has been promoted for combating the ongoing COVID-19 pandemic.",
"Samples are typically collected from nasal or oropharyngeal swabs, and then processed by a reverse transcription polymerase chain reaction (RT-PCR) machine.",
"However, widespread testing is hindered by supply chain constraints and long testing times.",
"Pooled or group testing has been suggested for improving testing efficiencies [2].",
"Group testing involves mixing a subset of $n$ individual samples into $m < n$ pools.",
"The measurement process can be expressed as $y = \\mathfrak {N}(Ax)$ , where $x$ is a vector that quantifies the health status of the $n$ individuals, $A$ is an $m \\times n$ binary pooling matrix with $A_{ij} = 1$ if the $j$ th individual contributes to the $i$ th pool, else $A_{ij} = 0$ , $y$ is a vector of $m$ noisy measurements or tests, and $\\mathfrak {N}$ represents a probabilistic noise model that relates the noiseless pooled results, $Ax$ , to $y$ .",
"We consider two signal and noise models.",
"Model M1: A binary noise model used by Zhu et al.",
"[3], where $x$ is binary, $w = Ax$ is an auxiliary vector, and the measurement $y_i\\in \\lbrace 0,1\\rbrace $ depends probabilistically on $w_i$ , where $\\Pr (y_i=1|w_i=0)$ and $\\Pr (y_i=0|w_i>0)$ are probabilities of erroneous tests.",
"Model M2: A multiplicative noise model of the form $y = Ax \\circ z$ as used in Ghosh et al.",
"[4], where $\\circ $ represents element-wise multiplication, $z$ is a vector of $m$ noisy elements defined as $z_i = (1+q)^{\\eta _i}$ , $q \\in (0, 1]$ is a known amplification factor for RT-PCR, $\\eta _i \\sim \\mathcal {N}(0,\\sigma ^2)$ , and $\\sigma ^2 \\ll 1$ is a known parameter controlling the strength of the noise in RT-PCR.",
"Under model M2, $x$ and $y$ represent viral loads in the $n$ individuals and $m$ pools, respectively.",
"Assuming reasonably high viral loads in $x$ , Poisson effects in $y$ can be ignored [4].",
"For both models, the objective is to estimate $x$ from $y$ and $A$ .",
"We use single-stage non-adaptive algorithms as in [3], [4], rather than two-stage algorithms, which employ a second stage of tests depending on results from the first stage, as in Heidarzadeh and Narayanan [5] or the classical approach by Dorfman [6].",
"The main advantage of non-adaptive algorithms is that they save on testing time, which is quite high for RT-PCR.",
"Algorithms for estimation of $x$ from $y$ and $A$ [4], [7] rely primarily on the sparsity of $x$ , which is a valid assumption for COVID-19 due to low prevalence rates [8].",
"Zhu et al.",
"[3] also exploit probabilistic information such as the prevalence rate, structure in $x$ , and side information (SI).",
"Finally, Nikolopoulos et al.",
"[1] showed how SI about family-style structure can reduce the number of required tests, $m$ .",
"In this paper, we show how to estimate $x$ while utilizing contact tracing (CT) SI, which allows one to analyze the spread of the pandemic [9].",
"Our contributions are twofold.",
"First, we propose a generative model for a population of $n$ individuals that characterizes the spread of COVID-19 by explicitly using CT SI.",
"Second, we show that CT SI, when used appropriately, can help algorithms such as generalized approximate message passing (GAMP) [10] or LASSO variants [11], [12] better estimate $x$ from $y$ and $A$ .",
"Our work uses more SI than Nikolopoulos et al.",
"[1], who only considered family-style structure in binary group testing."
],
[
"Data Generation Model",
" In this section, we present a generative infection model incorporating CT SI, which we later use to prepare simulated data for algorithmic evaluation.",
"We model a population of $n$ individuals using a dynamical or time-varying graphical model that contains nodes $\\lbrace v_i\\rbrace _{i=1}^n$ and undirected edges $\\big \\lbrace e_{ij}^{(t)}\\big \\rbrace _{i,j=1}^n$ .",
"On a given day $t$ , an edge $e_{ij}^{(t)}$ between nodes $v_i$ and $v_j$ encodes CT SI $\\big (\\tau ^{(t)}_{ij}, d^{(t)}_{ij}\\big )$ , which can be acquired via Bluetooth-based CT applications [13].",
"Here, $\\tau ^{(t)}_{ij}$ represents the contact duration and $d^{(t)}_{ij}$ represents a measure of the physical proximity between two individuals.",
"On day $t$ , a node can be in one of the following states: susceptible, infected, infectious, and recovered.",
"To keep the model simple, we assume that there are no reinfections, i.e., recovered is a terminal state, despite some reports of reinfection[14].",
"Figure: State transition diagram for a node.",
"A node is infectious only between days k 1 k_1 and k 2 k_2 (both inclusive)after getting infected.",
"In our work, we set (k 1 ,k 2 )=(3,7)(k_1,k_2) = (3,7).We adopt a simplified infection dynamic wherein the infectious period is preceded and followed by the infected state.",
"We propose the following design parameters for the infection dynamics based on a World Health Organization report on COVID-19 [15].",
"Specifically, a node $v_i$ remains infected but noninfectious for $k_1 = 3$ days.",
"On day $t+k_1$ , the node becomes infectious and may transmit the disease to a susceptible neighboring node $v_j$ with probability $p_{i,j}^{(t+k_1)}$ whose construction is described below.",
"An infectious node can potentially transmit the infection until $k_2 = 7$ days after getting infected, and becomes noninfectious afterward.",
"We also model a small fraction of stray infections that may occur, for example, due to sporadic contact with contaminated surfaces.",
"Such infections only affect nodes in the susceptible state with a probability $p_1 = 2 \\times 10^{-4}$ of our choice.",
"A state diagram for a general node is illustrated in Fig.",
"REF .",
"Regarding the viral load $x_i^{(t)}$ for node $i$ on day $t$ , we assume $x_i^{(t)} = 0$ if the node is susceptible or recovered.",
"For an infected or infectious node, we make a simplified assumption that its viral load $x_i^{(t)} \\sim \\textrm {Uniform}(1,32768)$ ,The cycle threshold for RT-PCR commonly ranges from 19 to 34 cycles [16], where 34 cycles corresponds to a low initial viral load of a few molecules, and each cycle roughly doubles the viral density.",
"Therefore, we estimate the largest possible viral load as $2^{34-19} = 2^{15} = 32768$ .",
"once drawn, remains constant throughout the combined 14-day period of infection.",
"Next, we model the probability $p_{i,j}^{(t)}$ that the disease is transmitted from node $v_i$ to $v_j$ on day $t$ .",
"We view the infection times of the population throughout the pandemic as a nonhomogeneous Poisson process with time-varying rate function $\\lambda (t)$ .",
"Consider a $\\tau ^{(t)}_{ij}$ -hour contact on day $t$ when susceptible node $v_j$ is exposed to infectious node $v_i$ .",
"The average infection rate $\\lambda _{ij}(t)$ for day $t$ is assumed to be proportional to both the viral load $x^{(t)}_i$ and the physical proximity $d^{(t)}_{ij}$ , namely, $\\lambda _{ij}(t) = \\lambda _0 \\, x^{(t)}_i \\, d^{(t)}_{ij}$ , where $\\lambda _0$ is a tunable, baseline Poisson rate.",
"The probability that $v_j$ is infected by the end of contact period $\\tau ^{(t)}_{ij}$ is therefore $p^{(t)}_{i,j} = 1 - \\exp \\left( -\\lambda _0 \\, x^{(t)}_i \\, d^{(t)}_{ij} \\, \\tau ^{(t)}_{ij} \\right)$ for $t \\in [k_1, k_2] + t_i$ .",
"From the standpoint of susceptible node $v_j$ , all its neighbors $v_k$ that are infectious contribute to its probability of getting infected on day $t$ , namely, $1-\\prod _{k} \\big ( 1 - p^{(t)}_{k,j} \\big )$ .",
"Figure: (a) The number of active infections, and (b) cumulative infections at different inter-clique contact levels α\\alpha .",
"We chose 50-day windows for testing proposed algorithms.While generating our data, we considered $n = 1000$ nodes divided into cliques based on the distribution of family sizes in India [17], for a duration of $t_{\\text{max}} = 250$ days.",
"Fig.",
"REF shows the number of active infections and the cumulative number of infections at the end of each day.",
"The clique structures were kept constant throughout the $t_{\\text{max}}$ days, whereas inter-clique contacts corresponding to sporadic contacts between people were dynamically added and removed.",
"The varying inter-clique contact level $\\alpha $ affects the sparsity of the underlying vector $x$ as it brings infections to new cliques/families.",
"Pooling of samples is performed at the beginning of each day from $t_{\\text{peak}}-24$ to $t_{\\text{peak}}+25$ , where $t_{\\text{peak}}$ is the day with the maximum number of active infections."
],
[
"Proposed Group Testing Algorithms",
" This section describes two classes of group testing algorithms for reconstructing the health status vector $x$ from the pooled tests, $y$ , and pooling matrix, $A$ .",
"Algorithms for binary noise.",
"For model M1, Zhu et al.",
"[3] use generalized approximate message passing (GAMP) [10] for group testing estimation.",
"GAMP is comprised of two components.",
"The first component consists of an input channel that relates a prior for $n$ individuals' viral loads, $x=(x_i)_{i=1}^n$ , and pseudo data, $v= x+ q\\in \\mathbb {R}^n$ , where the $n$ coordinates of $x$ are correlated, and $q$ is additive white Gaussian noise with $q_i \\sim \\mathcal {N}(0,\\Delta )$ .",
"We estimate $x$ from $v$ using a denoising function, often called a denoiser, $\\widehat{x}_{i}=g_{\\text{in}} \\left(v\\right)= \\mathbb {E}\\left[X_{i} \\mid V=v\\right] ,\\vspace*{-5.69054pt}$ where we use the convention that when both the upper and lower case versions of a symbol appear, the upper case is a random variable and the lower case its realization, and $\\mathbb {E}\\left[X_{i} | v\\right]$ represents $\\mathbb {E}\\left[X_{i} | V=v\\right]$ when the context is clear.",
"The second component of GAMP consists of an output channel relating the auxiliary vector $w$ to the noisy measurements $y$ as reviewed in Sec. .",
"We adopt the output channel denoiser of Zhu et al.",
"[3], $h_{i}=g_{\\text{out}}\\left(y_{i}; k_{i}, \\theta _{i}\\right) = ( \\mathbb {E}\\left[W_{i} \\mid y_{i}, k_{i}, \\theta _{i}\\right]-k_{i} ) / \\theta _i$ , where $\\theta _{i}$ is the estimated variance of $h_i$ , and $k_{i}$ is the mean of our estimate for $w_i$ .",
"Since $y_i$ depends probabilistically on $w_i$ , we have $f \\left(w_{i} \\mid y_{i}, k_{i}, \\theta _{i}\\right) \\propto \\operatorname{Pr}\\left(y_{i} \\mid w_{i}\\right) \\,\\exp \\left[-\\frac{\\left(w_{i}-k_{i}\\right)^{2}}{2 \\theta _{i}}\\right]$ , where $W_i$ is approximated as Gaussian in the derivation of GAMP.",
"While Zhu et al.",
"[3] considered Bernoulli $x$ , which implies a scalar separable denoiser $g_{\\text{in}}$ for the input channel, this paper accounts for probabilistic dependencies within $x$ .",
"Our first probabilistic model considers groups of people, for example, members of a family.",
"Each family is modeled as entirely healthy with probability $1-\\pi _{\\text{vf}}$ , else each individual within the family is infected with probability $\\pi _{\\text{ind}}$ .",
"This model relates to our generative model of Sec.",
"by using family structure as SI.",
"Denoting the pseudo data of family $\\mathcal {F}$ by $v_{\\mathcal {F}}$ , the denoiser for the $i$ th individual of family $\\mathcal {F}$ is given by $\\small g_{\\text{in}}^\\text{family}(v_{\\mathcal {F}} )= \\mathbb {E}\\big [ X_i | \\mathcal {F}\\text{ viral},v_{\\mathcal {F}} \\big ] \\, \\Pr \\big (\\mathcal {F}\\text{ viral}|v_{\\mathcal {F}} \\big ),\\normalsize \\vspace*{-2.84526pt}$ where $\\mathbb {E}\\big [X_i|\\mathcal {F}\\text{ viral},v_{\\mathcal {F}} \\big ]$ and $\\Pr \\big ( \\mathcal {F}\\text{ viral}|v_{\\mathcal {F}} \\big )$ are parameterized by $\\pi _{\\text{vf}}$ , $\\pi _{\\text{ind}}$ , and $\\Delta $ .",
"For detailed expressions, we refer readers to Sec.",
"1.1 of the supplemental document.",
"Our second probabilistic model uses CT.",
"Consider a hypothetical widespread testing program that relies on CT SI, where all individuals are tested 8 days before the group testing program begins resulting in a good estimate of their ground-truth health status.",
"After the program begins, probability estimates from the previous group test are used as priors for the $n$ individuals when performing the current group test.",
"We provide detailed analysis in Secs.",
"2.2–2.3 of the supplemental document on the use of prior infection status.",
"The final form of the denoiser for the CT model is as follows: $\\small g_{\\text{in}}^\\text{CT}(v_i)\\!=\\!",
"\\left\\lbrace 1 \\!+\\!",
"\\big [\\Pr (X_i\\!=\\!1)^{-1} \\!",
"- \\!",
"1\\big ] \\exp \\Big [ \\big (v_i\\!-\\!\\tfrac{1}{2}\\big ) \\big / \\Delta \\Big ] \\!\\right\\rbrace ^{-1}\\!.\\vspace*{-2.84526pt}$ Here, $\\Pr (X_i\\!=\\!1)$ for day $k+1$ can be estimated by aggregating CT information of individual $i$ over the past 8 days, namely, $\\widehat{\\Pr }^{(k+1)}\\!",
"(X_{i}\\!=\\!1) = 1 \\!-\\!",
"\\prod _{d=k-7}^k{\\prod _{j=1}^n{{ \\!\\big ( 1-\\widehat{p}^{(d)}_{i,j} \\big ) }}}$ , where $\\widehat{p}_{i,j}^{(d)}$ is the estimated probability of infection of individual $i$ due to contact with individual $j$ .",
"This probability, $\\widehat{p}_{i,j}^{(d)}$ , can be determined by the CT information ($\\tau _{ij}^{(d)}, d_{ij}^{(d)})$ , as well as their infection status as follows: $\\widehat{p}_{i,j}^{(d)}=\\exp \\left(-\\big (\\lambda \\, \\tau _{ij}^{(d)} \\, d_{ij}^{(d)} \\, \\Psi _{ij}^{(d)}+\\epsilon \\big )^{-1}\\right),\\vspace*{-2.84526pt}$ where $\\Psi _{ij}^{(d)} =1 - \\widehat{\\Pr }^{(d)}\\!",
"(X_i\\!=\\!0) \\, \\widehat{\\Pr }^{(d)}\\!",
"(X_j\\!=\\!0){\\color {orange}}$ , $\\lambda $ is a Poisson rate parameter, and $\\epsilon $ is used to avoid division by zero.",
"Note that $\\widehat{p}_{i,j}^{(t)}$ depends on $\\lambda $ , which is unknown in practice.",
"We estimate it using a plug-in approach by Ma et al. [18].",
"More details are given in Sec.",
"1.2 of the supplemental document.",
"Algorithms for multiplicative noise.",
"For model M2, recall that $x$ and $y$ represent viral loads of individual samples and pools, respectively.",
"The core algorithm presented in [4] uses the well-known Lasso estimator, $\\widehat{x}^{\\textsc {Lasso}{}} =\\text{arg}\\min _{x} \\Vert y-Ax\\Vert ^2_2 + \\rho \\Vert x\\Vert _1$ [19], where $\\rho $ is a smoothness parameter.",
"Lasso exploits the sparsity of $x$ but uses no SI.",
"Despite the multiplicative nature of the noise, Lasso yields good estimation performance [4] in terms of three commonly used measures: (i) relative root mean squared error (RRMSE) $= \\Vert x-\\widehat{x}\\Vert _2/ \\Vert x\\Vert _2$ ; (ii) false negative rate (FNR) = $\\#$ incorrectly detected negatives$\\big / \\#$ true positives; and (iii) false positive rate (FPR) = $\\#$ incorrectly detected positives$\\big / \\#$ true negatives.",
"Note that FNR $=1-$ sensitivity and FPR $=1-$ specificity.",
"In some cases, the $n$ individuals in $x$ can be partitioned into $n_1 \\ll n$ disjoint groups of people, for example family members, who interact closely with each other and are thus likely to pass the virus between group members.",
"This family-style structure leads to a situation where either all members of the group are uninfected, or a majority of members are infected.",
"Note that the family-style structure also includes groups of coworkers, students taking a course together, and people sharing common accommodation.",
"If reliable SI about how the $n$ individuals are partitioned into families is available, and only a small portion of families, $n_2 \\ll n_1$ , are infected, then Lasso can be replaced by group square-root Lasso (Sqrt-Glasso) [11].We observed that Sqrt-Glasso, which has an $\\ell _2$ data fidelity term instead of a squared $\\ell _2$ one [20], outperformed Glasso.",
"In contrast, conventional Lasso outperformed Sqrt-Lasso.",
"The latter is defined as $\\widehat{x}^\\textsc {Sqrt-Glasso}{} = \\operatorname{arg}\\min _{x} \\Vert y-Ax\\Vert _2 + \\rho \\sum _{g=1}^{n_1}\\Vert x_g\\Vert _2,\\vspace{-5.69054pt}$ where $x_g$ consists of viral loads of people from the $g$ th family.",
"Figure: Performance of the proposed group testing methods M1 (top row) with binary noise and M2 (bottom row) with multiplicative noise at four averaged sparsity levels and three measurement levels for a population of n=1000n = 1000 individuals.In other cases, accurate family SI may be unavailable or unreliable.",
"Moreover, family SI does not account for contact between members of different families.",
"In such cases, we use CT SI commonly available via Bluetooth [13] to directly infer family-style structure using clique detection algorithms; contacts between members of different families can also be considered to be small cliques.",
"In particular, we use the Bron–Kerbosch algorithm [21] to find maximal cliques in the CT graph, and label each clique as a family.",
"Note that one could generate these groups differently [22], for example, decomposition into $k$ -clique communities [23].",
"However, such a decomposition may partition the $n$ individuals into $n_3 \\ll n$ family structures that overlap with each other, unlike the earlier case of disjoint families.",
"In a scenario with overlapping families, we use the overlapping group square-root Lasso (Comp-Sqrt-Oglasso) estimator [22], $\\widehat{x}^\\textsc {Sqrt-Oglasso}{} = \\operatorname{arg}\\min _{x} \\Vert y-Ax\\Vert _2 + \\rho \\, \\Omega _{\\text{overlap}}(x),\\vspace{-5.69054pt}$ where $\\Omega _{\\text{overlap}}(x) = \\inf _{v \\in \\mathcal {V}_G, \\sum _{g \\in G} v_g = x} \\sum _{g \\in G} \\Vert v_g\\Vert _2$ , $G$ denotes a set of possibly overlapping groups each containing a subset of the $n$ individuals in $x$ , $\\mathcal {V}_G$ is a set of $|G|$ -tuples of vectors $v = (v_g)_{g \\in G}$ , $g$ is an index for the groups in $G$ , and $v_g \\in \\mathbb {R}^n$ is a vector whose support is a subset of $g$ .",
"Advantages of Oglasso over Glasso for overlapping groups are summarized in Fig.",
"1 and Sec.",
"3 of [22].",
"In all three algorithms, we imposed a non-negativity constraint on $x$ .",
"Moreover, all algorithms were preceded by a step that executed combinatorial orthogonal matching pursuit (Comp).",
"Comp declares all samples that contributed to a pool with zero viral load to be negative.",
"This Comp preprocessing step reduces the problem size and improves all three algorithms' performance, as well as speed.",
"We henceforth refer to our algorithms as Comp-Lasso, Comp-Sqrt-Glasso and Comp-Sqrt-Oglasso."
],
[
"Numerical Results",
"We now present numerical results obtained for the two models, M1 and M2.",
"The data was generated based on Sec.",
", and group testing inference was performed using the algorithms proposed in Sec. .",
"We generated datasets using four levels of cross-clique contacts, leading to four averaged sparsity levels, $2.12\\%$ , $3.98\\%$ , $6.01\\%$ , and $8.86\\%$ , for $x$ .",
"At each sparsity level, we perform pooling experiments using Kirkman triple matrices as proposed in [4] for three measurement setups, $m \\in \\lbrace 150, 300, 375\\rbrace $ .",
"Measurement vectors $y$ for M1 were generated using probabilities for erroneous binary tests, $\\Pr (y_i=1|w_i=0)=0.001$ and $\\Pr (y_i=0|w_i>0)=0.02$ , per Hanel and Thurner [24].",
"Vectors $y$ for M2 were generated by setting the parameter reflecting the strength of noise in RT-PCR to $\\sigma ^2 = 0.01$ .",
"Fig.",
"REF shows the performance of the proposed algorithms in terms of FNR and FPRFor M1, we chose to report the FPR and FNR pair such that the sum of the two error rates is minimized.",
"The complete ROC curves are shown in Sec.",
"2.1 of the supplemental document.",
"For M2, we reported the error rates by thresholding the estimated viral load using $\\tau = 0.2$ .",
"We noticed that the error rates do not change much when $\\tau $ varies between 0 and 1. averaged across the inference results obtained for the time window of 50 days described in Sec. .",
"For model M1, we tested the family denoiser (REF ) and the CT denoiser (REF ).",
"Fig.",
"REF shows that the CT denoiser outperforms the family denoiser in all settings.",
"Both algorithms yield lower (better) FNR and FPR as the number of measurements, $m$ , increases.",
"Moreover, the CT denoiser's error rates are below $0.05$ , except for the challenging cases where the sparsity level is $8.86\\%$ and $m\\in \\lbrace 150,300\\rbrace $ .",
"For model M2, we tested four algorithms: Comp, Comp-Lasso, Comp-Sqrt-Glasso, and Comp-Sqrt-Oglasso.",
"The results show that both Comp-Sqrt-Glasso and Comp-Sqrt-Oglasso outperform Comp-Lasso in terms of FNR and FPR, which shows the benefit of using CT SI.",
"Note that Comp-Sqrt-Oglasso performs on par with Comp-Sqrt-Glasso, even though the former infers everything on the fly from CT SI without explicit access to family SI.",
"The Comp algorithm by itself produces no false negatives (corresponding to $\\text{FNR} = 0$ ), but many false positives.",
"Further, all four algorithms yield lower (better) FNR and FPR as $m$ increases or the averaged sparsity level decreases.",
"Finally, we remark that Comp-Sqrt-Oglasso outperforms Comp-Sqrt-Glasso for more general CT graphs consisting of slightly incomplete cliques.",
"We refer the readers to Sec.",
"3 of the supplemental document for details.",
"Our algorithms, when presented with SI, reduce the FNR and FPR, despite not knowing which individuals are infected within each infected family (around $70\\%$ of the individuals are infected within an infected family on average).",
"Note that none of the algorithms for model M2 make use of previous inference results, whereas the CT denoiser for model M1 uses such information.",
"This distinction makes the two approaches applicable in different scenarios, namely, the CT denoiser can be used for a CT and testing program where the same population is tested at regular intervals, e.g., warehouse employees, whereas Comp-Sqrt-Oglasso is useful when a population has to be tested only once.",
"Furthermore, while model M1 performs well in the presence of erroneous binary tests, it does not yield viral load estimates as Comp-Sqrt-Oglasso does.",
"Viral load estimates could prove to be useful, since there is a positive correlation between mortality and viral loads [25], [26].",
"Acknowledgment: The authors thank Junan Zhu for allowing them to use his implementation of GAMP with SI in their implementation of the family and CT denoisers.",
"*0.970.5Note: This is a supplemental document for “Contact tracing enhances the efficiency of COVID-19 group testing,” submitted to 2021 IEEE International Conference on Acoustics, Speech and Signal Processing by Ritesh Goenka,$^{\\star }$ Shu-Jie Cao,$^{\\star }$ Chau-Wai Wong, Ajit Rajwade, and Dror Baron.",
"RG and SJC have made equal contributions to the paper."
],
[
"Family Denoiser",
"We now formalize a family-based infection mechanism that can be used in designing group testing algorithms for improving the detection accuracy.",
"We define $\\mathcal {M}_{\\mathcal {F}}$ as the set of indices of all members of family $\\mathcal {F}$ .",
"We say that $\\mathcal {F}$ is viral when there exists viral material in the family.",
"Next, define the infection probability of individual $i$ within viral family $\\mathcal {F}$ as $\\pi _{\\text{ind}}= \\Pr (X_i = 1 \\mid \\mathcal {F}\\text{ viral})$ , for all $i \\in \\mathcal {M}_{\\mathcal {F}}$ , and $\\pi _{\\text{vf}}= \\Pr (\\mathcal {F}\\text{ viral})$ .",
"Note that the infection status of individuals in a viral family are conditionally independent and identically distributed (i.i.d.).",
"Under our definition, family $\\mathcal {F}$ being viral need not be attributed to any individual $i \\in \\mathcal {M}_{\\mathcal {F}}$ .",
"After all, viral material can be on an infected pet or contaminated surface.",
"For this model, once the family is viral, the virus spreads independently with a fixed probability $\\pi _{\\text{ind}}$ .",
"Of course, our simplified model may not accurately reflect reality.",
"That said, without a consensus in the literature on how coronavirus spreads, it is unrealistic to create a more accurate model.",
"On the other hand, our model is plausible, and we will see that it is mathematically tractable.",
"We further assume that individuals cannot be infected unless the family is viral, i.e., $\\Pr (X_i = 1 \\mid \\mathcal {F}\\text{ not viral}) = 0$ .",
"The family structure serves as SI and allows the group testing algorithm to impose the constraint that people living together have strongly correlated health status.",
"Next, we derive the exact form of the denoiser (REF ) by incorporating the family-based infection mechanism.",
"Denote the pseudodata of the members of family $\\mathcal {F}$ as $v_{\\mathcal {F}} = (v_i)_{i \\in \\mathcal {M}_{\\mathcal {F}}}$ , the family-based denoiser for $i$ th individual can be decomposed as follows: $&g_{\\text{in}}^\\text{family}(v_\\mathcal {F}) \\\\=& \\mathbb {E}\\left[ X_{i} \\mid v_{\\mathcal {F}} \\right] \\\\=& \\Pr (X_i = 1 \\mid v_{\\mathcal {F}}) \\\\=& \\Pr (X_i = 1, \\mathcal {F}\\text{ viral}\\mid v_{\\mathcal {F}}) \\\\=& \\Pr (\\mathcal {F}\\text{ viral}\\mid v_{\\mathcal {F}}) \\Pr (X_i = 1 \\mid v_{\\mathcal {F}},\\mathcal {F}\\text{ viral}),$ where the first term of (REF ) is $&\\Pr (\\mathcal {F}\\text{ viral}\\mid v_{\\mathcal {F}}) \\nonumber \\\\=& \\, \\frac{f(v_{\\mathcal {F}},\\mathcal {F}\\text{ viral})}{f(v_{\\mathcal {F}},\\mathcal {F}\\text{ viral}) + f(v_{\\mathcal {F}},\\mathcal {F}\\text{ not viral}) }.$ The two quantities in (REF ) can be further expanded as $&f(v_{\\mathcal {F}},\\mathcal {F}\\text{ not viral}) \\\\= &(1-\\pi _{\\text{vf}}) \\, f(v_{\\mathcal {F}} \\mid \\mathcal {F}\\text{ not viral}) \\\\= &(1-\\pi _{\\text{vf}}) \\prod _{i\\in \\mathcal {M}_{\\mathcal {F}}}\\mathcal {N}({v_i}; {0}, {\\Delta }),$ and $&f(v_{\\mathcal {F}},\\ \\mathcal {F}\\text{ viral}) = \\pi _{\\text{vf}}\\, f(v_{\\mathcal {F}} \\mid \\mathcal {F}\\text{ viral})\\\\\\begin{split}=\\, &\\pi _{\\text{vf}}\\sum _{x_k \\in \\Omega _{\\mathcal {F}}} \\prod _{i \\in \\mathcal {M}_{\\mathcal {F}}} \\\\& \\Big [f(v_{i}|X_{i}=x_{k,i})\\Pr ( X_i = x_{k,i} | \\mathcal {F}\\text{ viral})\\Big ],\\end{split}\\normalsize $ where $\\mathcal {N}({x}; {\\mu }, {\\sigma ^2}) := \\frac{1}{\\sqrt{2 \\pi \\sigma ^2}} \\exp \\left( \\frac{(x-\\mu )^2}{2 \\sigma ^2} \\right)$ , and $\\Omega _{\\mathcal {F}} = \\lbrace 0...00,\\ 0...10,\\ \\dots ,\\ 1...11\\rbrace $ is a power set comprised of $2^{|\\mathcal {M}_{\\mathcal {F}}|}$ distinct infection patterns for family $\\mathcal {F}$ .",
"The second term of (REF ) can be simplified as follows: $&\\Pr (X_i = 1 \\mid v_{\\mathcal {F}},\\mathcal {F}\\text{ viral}) \\\\= & \\Pr (X_i = 1 \\mid v_i,\\mathcal {F}\\text{ viral}) \\\\=&\\Pr (X_i = 1, v_i \\mid \\mathcal {F}\\text{ viral}) \\, / \\, \\Pr (v_i \\mid \\mathcal {F}\\text{ viral}) \\\\=&\\dfrac{\\pi _{\\text{ind}}\\, \\mathcal {N}({v_i}; {1}, {\\Delta })}{\\pi _{\\text{ind}}\\, \\mathcal {N}({v_i}; {1}, {\\Delta }) + \\left(1-\\pi _{\\text{ind}}\\right) \\, \\mathcal {N}({v_i}; {0}, {\\Delta })} \\\\=& \\left( 1 + \\frac{1-\\pi _{\\text{ind}}}{\\pi _{\\text{ind}}} \\cdot \\frac{\\mathcal {N}({v_i}; {0}, {\\Delta })}{\\mathcal {N}({v_i}; {1}, {\\Delta })} \\right)^{-1} \\\\=& \\left( 1 + \\left( \\pi _{\\text{ind}}^{-1} - 1 \\right)\\exp \\Big [ \\big (v_i-\\tfrac{1}{2}\\big ) \\big / \\Delta \\Big ] \\right)^{-1}.$"
],
[
"Contact Tracing Denoiser",
"While family structure SI characterizes part of the spread of the disease, individual members of a family will presumably all come in close contact with each other, hence CT SI will include cliques for these individuals.",
"Additionally, CT SI describes inter-family contacts.",
"Therefore, CT SI can characterize the spread of the disease more comprehensively than family SI.",
"To exploit the CT SI, we encode it for each individual $i$ into the prior probability of infection, $\\Pr (X_i=1)$ , and use the following scalar denoiser: $&g_{\\text{in}}^\\text{CT}(v_i) \\nonumber \\\\=& \\mathbb {E}\\left[ X_i|v_i \\right] = \\Pr (X_i = 1|v_i) \\\\=& f(v_i|X_i=1)\\Pr (X_i=1)/f(v_i) \\\\=& \\left\\lbrace 1 \\!+\\!",
"\\big [\\Pr (X_i\\!=\\!1)^{-1} \\!",
"- \\!",
"1\\big ]\\exp \\Big [ \\big (v_i-\\tfrac{1}{2}\\big ) \\big / \\Delta \\Big ] \\!\\right\\rbrace ^{-1}.$ Here, $\\Pr (X_i\\!=\\!1)$ for day $k+1$ can be estimated by aggregating CT information of individual $i$ over a so-called SI period from day $k-7$ to day $k$ as follows $\\widehat{\\Pr }^{(k+1)}(X_{i}=1) = 1 - \\prod _{d=k-7}^k{\\prod _{j=1}^n{{ \\!\\big ( 1-\\widehat{p}^{(d)}_{i,j} \\big ) }}},$ where $\\widehat{p}_{i,j}^{(d)}$ is the estimated probability of infection of individual $i$ due to contact with individual $j$ .",
"This probability, $\\widehat{p}_{i,j}^{(d)}$ , can be determined by the CT information ($\\tau _{ij}^{(d)}, d_{ij}^{(d)})$ , as well as their infection status as follows: $\\widehat{p}_{i,j}^{(d)}=\\exp \\left(-\\big (\\lambda \\, \\tau _{ij}^{(d)} \\, d_{ij}^{(d)} \\, \\Psi _{ij}^{(d)}+\\epsilon \\big )^{-1}\\right),$ where $\\Psi _{ij}^{(d)} =1 - \\widehat{\\Pr }^{(d)}\\!",
"(X_i\\!=\\!0) \\, \\widehat{\\Pr }^{(d)}\\!",
"(X_j\\!=\\!0){\\color {orange}}$ , $\\lambda $ is an unknown Poisson rate parameter, and $\\epsilon $ is used to avoid division by zero.",
"We estimate $\\lambda $ with maximum likelihood (ML) using the pseudodata of all individuals, i.e., $\\widehat{\\lambda }^{\\text{ML}} = \\arg \\max _{\\lambda }\\ \\prod _{i=1}^n f(v_i|\\lambda ),$ where $f(v_i|\\lambda ) = f(v_i|X_{i}=1) \\, \\Pr (X_{i}=1|\\lambda )+f(v_i|X_{i}=0) \\, \\Pr (X_{i}=0|\\lambda )$ .",
"Once $\\widehat{\\lambda }^{\\text{ML}}$ is obtained, it is plugged into (REF ) for calculating the prior probability[18].",
"Note that this plug-in strategy is also used for two other denoisers, namely, $\\lambda =\\rho $ for $g_{\\text{in}}^\\text{Bernoulli}(v_i)$ and $\\lambda = (\\pi _{\\text{vf}}, \\pi _{\\text{ind}})$ for $g_{\\text{in}}^\\text{family}(v)$ ."
],
[
"Additional Results for M1",
"In Sec.",
"4 of the main paper, we reported the performance of M1 in a compact way, due to space limitations, by choosing a representative operating point on an ROC curve instead of using the whole curve.",
"In this section, we provide complete ROC curves that correspond to the top row of Fig.",
"3 of the main paper.",
"Fig.",
"REF illustrates M1's performance for family and CT denoisers at different measurement and sparsity levels.",
"The dot on each curve corresponds to the operating point that minimizes the total error rate, i.e., the sum of FPR and FNR, as reported in Sec.",
"4 of the main paper.",
"The closer a dot is to the origin of the FPR–FNR plane, the better the performance it reflects.",
"Comparing the ROC curves in the top row to those in the bottom row, we note that the CT denoiser significantly outperforms the family denoiser at all sparsity levels.",
"The CT denoiser, with most of its FNR and FPR $< 5\\%$ , can achieve as low as $15\\%$ of the total error rate of the family denoiser.",
"Across different sparsity levels, the algorithm performs less accurately as the sparsity level increases.",
"In each plot, lower measurement rates make it more challenging for the group testing algorithm.",
"We also examine the stability of the thresholds corresponding to the operating points we selected to report results in Fig.",
"3 of the main paper.",
"Our empirical results reveal that at a particular sparsity level, the variation of the threshold due to different design matrices or denoisers is less than $0.003$ .",
"As the sparsity level increases from $2.12\\%$ to $8.86\\%$ , the threshold only drops from $0.160$ to $0.137$ .",
"Hence, the threshold for minimizing the total error rate is insensitive to the testing conditions."
],
[
"Using Prior Knowledge of the Infection Status",
"In this subsection, we examine the advantage that prior knowledge of the population's infection status in the startup phase provides our proposed algorithm for the M1 binary model.",
"As stated in Sec.",
"3 of the main paper and in (REF ), our algorithm iteratively uses the updated probability of infection, $\\widehat{\\Pr }(X_i=1)$ , estimated from an SI period of 8 immediately preceding days.",
"Note that for days $k < 8$ , we had to use the ground-truth infection status of each individual in the startup phase to generate the results reported in Sec.",
"4 of the main paper.",
"However, ground-truth infection data from the startup phase may provide our approach an unfair advantage over the algorithms proposed for M2.",
"Below, we investigate whether this advantage is significant.",
"We examine how varying the amount of startup information impacts our algorithm's quality.",
"Specifically, we randomly replace a portion, $p_{\\text{excluded}} \\in \\lbrace 0, 0.1, 0.5, 0.75, 1\\rbrace $ , of the population's infection status by an estimated probability of infection, e.g., $5\\%$ , for a setup that has a true averaged sparsity level of $7.2\\%$ .",
"Using a probability instead of a binary value, 0 or 1, gives the algorithm soft probabilistic information instead of hard ground-truth style information.",
"Fig.",
"REF shows that even with 50% prior knowledge of the infection status of individuals, our detection accuracy for M1 is close to that when using complete prior information after ramping up for 8 days.",
"The averages of the total error rates across time for increasing $p_{\\text{excluded}}$ are $0.038$ , $0.039$ , $0.046$ , $0.148$ , and $0.407$ , respectively.",
"We also tried to replace the startup infection status by an estimated probability of infection of $10\\%$ , but only observed negligible performance differences.",
"The results show that the CT algorithm is robust to the absence of up to $50\\%$ of startup infection information.",
"Figure: Performance of M1 when a proportion, p excluded p_{\\text{excluded}}, of the population's health states in the startup phase is unknown.The curves reveal that in the absence of up to 50%50\\% prior knowledge of the infection status of the population,the accuracy of M1 is close to that when complete startup information is available."
],
[
"Duration of Startup SI Period",
"We investigated the impact of the duration of the startup SI on estimation performance.",
"In principle, the longer the SI duration, the more accurate we expect the results to be.",
"There is a trade-off between the accuracy of our algorithm and the startup SI infection status information that needs to be pre-collected before the initialization of the testing algorithm.",
"In our experiment, we tested three startup SI durations, namely, 4 days, 8 days, and 12 days.",
"Our experimental results (omitted for brevity) show that the estimation accuracy is somewhat insensitive to the duration of the SI period.",
"Hence, for the experiments conducted for this paper, we chose 8 days as the SI period."
],
[
"Data Generation",
"For this experiment, we use a different and slightly more general contact tracing graph to simulate the spread of infection.",
"Recall that the adjacency matrix of the contact graph has a block diagonal structure with sizes of cliques coming from the distribution of family sizes in India [17].",
"However, in this case, we allow two consecutive (according to the order in which cliques appear along the diagonal of the contact matrix) non-trivial cliques (i.e., cliques with more than one node) to have an overlap of one node with probability half.",
"This assumption is reasonable since the concept of family encompasses more general groups such as people at the same workplace, students studying in the same classroom, etc.",
"Furthermore, we remove $\\alpha = 5\\%$ of the edges from this block diagonal structure, thus converting the existing cliques into “almost-cliques.” This modified block diagonal structure is kept constant over time while the cross-clique contacts are updated every day.",
"Except for the changes in the underlying contact tracing graph, the rest of the data generation method is the same as that described in Sec.",
"2 of the main paper."
],
[
"Inference",
"We use the four algorithms (including Comp) for multiplicative noise described in Sec.",
"3 of the main paper.",
"However, instead of using maximal cliques as groups in Comp-sqrt-oglasso, we use the decomposition of the contact tracing graph into overlapping 3-clique communities [23].",
"An algorithm for detecting $k$ -clique communities can be found in Sec.",
"1 of [23].",
"The first step of this algorithm involves finding the maximal cliques in the contact graph, for which we use the Bron-Kerbosch algorithm [21].",
"In the next step, we detect 3-clique communities and label each of those as groups.",
"Further, we also label as groups the maximal cliques that are not part of any of these communities, in order to ensure that every contact is taken into account.",
"The advantage of using 3-clique communities over just maximal cliques is that the former is able to capture “almost cliques,\" i.e., cliques with a small fraction of absent pairwise contacts."
],
[
"Numerical Results",
"We present the results in a format similar to that in Sec.",
"4 of the main paper, but for the contact graph described in Sec.",
"REF .",
"Fig.",
"REF shows the mean values (across 50 signals) of the false negative rate (FNR) and false positive rate (FPR) obtained for four different sparsity levels.",
"The sparsity levels were obtained by varying the amount of cross-clique contacts.",
"We remark that the length of each bar in Fig.",
"REF , $\\text{FNR} + \\text{FPR}$ , is equal to $1 - \\text{Youden's Index}$ .",
"Further, we plot heat maps (Fig.",
"REF ) to compare the performance of the four algorithms under consideration—the intensity of gray corresponds to the mean value (across 50 signals) of the Matthews Correlation Coefficient (MCC).",
"The MCC is defined as $\\text{MCC} = \\dfrac{\\text{TP} \\times \\text{TN} - \\text{FP} \\times \\text{FN}}{\\sqrt{(\\text{TP}\\!+\\!\\text{FP})(\\text{TP}\\!+\\!\\text{FN})(\\text{TN}\\!+\\!\\text{FP})(\\text{TN}\\!+\\!\\text{FN})}},$ and has been proposed as a comprehensive metric to evaluate the performance of binary classification algorithms [27].",
"Its values range from $-1$ to $+1$ , where a value closer toward $+1$ is desirable.",
"The RRMSE values can be seen from the heat maps in Fig.",
"REF (we do not provide a heat map for Comp since it does not estimate viral loads).",
"Figure: Figure showing mean FNR and FPR values for the contact graph from Section , for mean sparsity levels of 3.20%, 4.84%, 6.25%, 8.66% (from left to right).Figure: Figure showing mean MCC values obtained using Comp, Comp-Lasso, Comp-Sqrt-Glasso, Comp-Sqrt-Oglasso (from left to right).Figure: Figure showing mean RRMSE values obtained using Comp-Lasso, Comp-Sqrt-Glasso, Comp-Sqrt-Oglasso (from left to right)."
],
[
"Additional Results for M2",
"For model M2, we present a comparison of the four algorithms for the experiment described in Sec.",
"4 of the main paper.",
"Fig.",
"REF and Fig.",
"REF show a comparison of the performance of the algorithms under consideration in terms of mean MCC and mean RRMSE values, respectively.",
"Further, we remark that the true viral loads of the false negatives yielded by Comp-Lasso variants are concentrated toward lower values.",
"For instance, only about 29% of the false negatives given by Comp-Sqrt-Oglasso had viral load values greater than $2^{12} = 4096$ .",
"Figure: Figure showing mean MCC values obtained using Comp, Comp-Lasso, Comp-Sqrt-Glasso, Comp-Sqrt-Oglasso (from left to right).Figure: Figure showing mean RRMSE values obtained using Comp-Lasso, Comp-Sqrt-Glasso, Comp-Sqrt-Oglasso (from left to right)."
],
[
"Sensing Matrix Design",
"As mentioned in the main paper, we use Kirkman triple matrices as sensing matrices for performing pooling.",
"A Kirkman triple (binary) matrix $A$ can be partitioned into $3n/m$ sub-matrices of dimensions $m \\times m/3$ , each of which contains exactly one nonzero entry in each row and three nonzero entries in each column.",
"Further, the dot product of any two columns of the matrix $A$ should not exceed 1.",
"For a given value of $n$ , $m (< n)$ must satisfy the following conditions: $m$ must be of the form $3n_1$ , where $n_1$ divides $n$ , since the number of sub-matrices and the number of columns in each sub-matrix must be integers.",
"$\\binom{3}{2} \\cdot n \\le m(m-1)/2$ since a triple contains $\\binom{3}{2}$ pairs and a pair must belong to at most one triple.",
"For $n = 1000$ , the only values of $m$ which satisfy the above constraints are 120, 150, 300, 375, 600, and 750.",
"We construct Kirkman triple matrices with $m = 150, 300, 375$ and use them in our experiments.",
"The matrices are constructed based on a few simple rules: The indices of ones in each column form an arithmetic progression (AP).",
"The matrix has a block structure and the common difference of the AP remains constant throughout each block.",
"Furthermore, the sum of all columns in a block yields the vector consisting of all ones.",
"The common difference values $\\lbrace d_B : B\\text{ is a block}\\rbrace $ are chosen such that the multi-set $\\lbrace r \\cdot d_B: r \\in \\lbrace n: n \\in \\mathbb {N}, n < 3\\rbrace , B \\text{ is a block}\\rbrace $ has no duplicate values.",
"Figure: A 375×1000375 \\times 1000 Kirkman triple matrix obtained using our approach.",
"The number written within each block is equal to the corresponding common difference value and the blocks without number markings are zero matrices.",
"The blocks having less than m=375m = 375 rows have dimensions equal to 3d B ×d B 3d_B \\times d_B.Fig.",
"REF shows the structure of a $375 \\times 1000$ Kirkman triple matrix obtained using the above approach.",
"Let $B$ be any block and let $d_B$ denote the common difference of the AP for block $B$ as indicated in the figure.",
"Then, the $i$ th column of $B$ is given by $B_i = \\sum _{j=1}^{3-1} e_{\\beta _i + j d_B},\\quad \\beta _i = \\text{mod}(i,d_B) + 3d_B[i/d_B],$ where $[\\cdot ]$ denoted the greatest integer function and $e_j$ denotes the $j$ th standard basis vector.",
"Clearly, any block $B$ must have dimensions $3 n_B \\times n_B$ , where $d_B$ divides $n_B$ .",
"As $m$ decreases, it becomes harder to design matrices satisfying all three rules specified earlier.",
"However, it is possible to relax the third rule in such cases and still obtain a matrix satisfying the required constraints.",
"For example, our $150 \\times 1000$ Kirkman triple matrix does not obey the third rule.",
"We further remark that one may design balanced matrices with a different number of (say $k$ ) ones in each column such that the dot product of every pair of columns is bounded by 1, using the above approach.",
"Such matrices would arise from the Steiner systems $S(2,k,m)$ [just as Kirkman matrices arise from $S(2,3,m)$ ].",
"For example, it is straightforward to design a $400 \\times 1000$ matrix with $k = 4$ using our approach."
]
] | 2011.14186 |
[
[
"Uncertainty-Aware Physically-Guided Proxy Tasks for Unseen Domain Face\n Anti-spoofing"
],
[
"Abstract Face anti-spoofing (FAS) seeks to discriminate genuine faces from fake ones arising from any type of spoofing attack.",
"Due to the wide varieties of attacks, it is implausible to obtain training data that spans all attack types.",
"We propose to leverage physical cues to attain better generalization on unseen domains.",
"As a specific demonstration, we use physically guided proxy cues such as depth, reflection, and material to complement our main anti-spoofing (a.k.a liveness detection) task, with the intuition that genuine faces across domains have consistent face-like geometry, minimal reflection, and skin material.",
"We introduce a novel uncertainty-aware attention scheme that independently learns to weigh the relative contributions of the main and proxy tasks, preventing the over-confident issue with traditional attention modules.",
"Further, we propose attribute-assisted hard negative mining to disentangle liveness-irrelevant features with liveness features during learning.",
"We evaluate extensively on public benchmarks with intra-dataset and inter-dataset protocols.",
"Our method achieves the superior performance especially in unseen domain generalization for FAS."
],
[
"Introduction",
"With growing prevalence of face recognition, it is increasingly subject to a wide variety of spoofing attacks.",
"Thus, face anti-spoofing or liveness detection is emerging as an essential precursor, with the key need being robust to various attacks that are possibly unseen previously and drastically different in appearance from training data.",
"This is an extremely challenging problem due to the fact that sophisticated spoofs might arise from similar camera and lighting setups as genuine inputs, leading to only subtle differences in appearance.",
"On the other hand, types of attacks range from printed photos to facial masks, which makes it laborious to obtain exhaustive training data for anti-spoofing task.",
"Figure: Traditional methods only focus on real/spoofing binary classification, which results in sensitive prediction.",
"By introducing the physical cues of depth, material and reflection as proxy tasks, our method largely close the domain gap from the training data to the unseen testing data and thus boost the performance reliably.In this paper, we aim to address the face anti-spoofing problem on unseen domains or attack types, where neither definition of the attack types nor training data under supervised or unsupervised condition is available.",
"To achieve this, we derive inspiration from physical cues that establish a commonality for genuine inputs and distinction from fake ones.",
"We refer to the estimation of these cues as proxy tasks, performed in conjunction with the main task, a.k.a appearance-based liveness detection.",
"While the proposed formulation is general and physical cues can be arbitrary, we focus on depth estimation, reflection detection and material classification as our proxy tasks.",
"Intuitively, we expect genuine faces to constitute face-like geometry and present skin as the material, while several presentation attacks might violate at least one of those conditions.",
"As a consequence, incorporating such proxy tasks enables to generalize the shared cues to unseen domains or attack types.",
"We bring the insights from single-image based face reconstruction using 3D morphable models (3DMM) [7] for depth proxy, single-image based material recognition trained on large-scale datasets [5] for material proxy, and a single image reflection separation model [58] to provide the pseudo labels for the reflection proxy.",
"A shared encoder is trained across the main and proxy tasks to transfer the insights into our deep appearance-based liveness detection problem.",
"In contrast to existing work [34] that incorporates depth cue to regularize sensitive binary liveness task, our proxy tasks are more general in the sense of considering more physical cues such as material, to gear towards a physically meaningful way to handle unseen domains.",
"Meanwhile, we organize the proxy tasks into a multi-channel learning framework to provide a more robust detection with an attention aggregation.",
"Note that domain adaptation is not applicable in our setting, since we assume that even unlabeled training data is not available, which cannot define the target domain.",
"Besides proxy tasks, we also leverage a pretext task in the form of face recognition, which is usually regularized by large scale labeled datasets and expected to provide high-level shared face analysis feature representation for liveness detection.",
"We thereby provide recipes for pre-training on face recognition that allows better generalization to unseen domains for the liveness task, as well as multi-channel training with liveness and proxy tasks.",
"We conduct extensive experiments on five publicly available benchmarks.",
"In each case, we demonstrate not only state-of-the-art results, but also that judicious use of pretext and proxy tasks allows better generalization of liveness detection to unseen domains.",
"Besides, the multi-task learning can result in channel conflict as the liveness feature is ideally invariant to identity information where the pretext task and our liveness task share the same network.",
"To this end, we leverage the attribute information in the proxy data to conduct a triplet metric learning based mining, expecting to better disentangle the non-liveness information from the learned feature and thus boosts the liveness detection.",
"To better exploit multiple physically meaningful resources, we further holistically weigh the relative contributions of the main task and various proxy tasks with an uncertainty-aware attention module.",
"Traditional attention modules are jointly optimized with all tasks and might cause the notorious over-confident issue due to training data bias.",
"While our uncertainty-aware attention is designed to independently estimate the tasks' variance, which does not capture feature fitness to the task but rather focusing on its deviation to the estimated mean or termed uncertainty of the feature estimation.",
"This property ensures less bias in uncertainty-aware attention module thus captures the property of input images better.",
"In summary, we propose the following contributions: We propose three physical-cue guided proxy tasks including depth, material and reflection, which share the commonality across domains to enable the unseen domain anti-spoofing.",
"We leverage an uncertainty-aware attention module to effectively combine the main and proxy tasks and boost the performance.",
"We propose a attribute-assisted mining scheme to make sure liveness-irrelevant features are properly disentangled and only liveness features are learned.",
"We conduct an extensive evaluation with both intra-dataset and inter-dataset protocols including the latest attribute-rich CelebA-Spoof dataset, highlighting our framework's better performance in unseen domain generalization for FAS."
],
[
"Related Work",
"We categorize face anti-spoofing literature into physical cue based, feature learning based methods, and whether they address the unseen spoofing attacks.",
"Physical Cue based Anti-spoofing: Early research on anti-spoofing leverages the physical cues, e.g.",
"head movement [27] and eye blinking [42], to indicate the genuineness.",
"These methods can be simply spoofed by printing faces with eye region cut, or wearing a facial mask and moving head.",
"By analyzing the lighting cooperated from different reflection, the remote Photoplethysmography (r-PPG) [15], [8], [33], [41] is proposed to identify spoofing attacks with material information.",
"However, this type of methods require the imaging quality to be high as the lighting measurement is less tolerated to noise.",
"Combining with CNN, depth is proposed [34], [46] as an auxiliary task that enables less sensitive and more explainable training.",
"[26] introduced reflection map as a supplement to depth for bipartite auxiliary supervision.",
"This method is limited in generalizing to other spoofing resources since their depth, reflection and r-PPG are trained on a single dataset.",
"Instead, our depth, reflection and material channels are guided by models trained on large scale datasets, i.e., a 3DMM based depth regression model [17], a single image reflection separation model [58] and a material classification model [5], which substantially improves the generalization.",
"Table: Unseen domain anti-spoofing methods comparison with robust feature , anomaly detection , binary or auxiliary supervision , deep tree learning (DTL) and MADDG .",
"“” means applying and “×\\times ” means not applying.Learning based Anti-spoofing: The handcrafted features, e.g., HoG [28], [23], SIFT [43] and LBP [9], [45], [39], [52], [57] are explored in early literature.",
"Such binary classification achieves good performance but is restricted to some defined domains.",
"Meanwhile, those methods do not consider environment variations, i.e., lighting, color tone or pose change.",
"To this end, the HSV and YCbCr [9], Fourier transform [31] and image low-rank decomposition [51] methods are also explored.",
"Some other works [1], [4], [49], [16], [61] utilize the temporal information assuming videos are available.",
"Later, deep learning based features [56], [16], [32], [3], [22], [34], [44] are utilized [56], [16], [32], [3], [22], [34], [44] and achieve better performance.",
"Notice that both [3] and [34] leverage texture and depth, which seem to be close to our setting.",
"However, instead of directly exploring texture, we formulate the texture into a more physically consistent cue, the material, and set up the material classification task to avoid rPPG calculation.",
"Moreover, our method is single image based which does not require the temporal information, thus reducing the run-time and model complexity.",
"Unseen Domain Anti-spoofing: Methods that explore handcrafted features can deal with unseen spoofing attacks as these features are independent from attack types.",
"However, due to the limitation of feature representation power, they cannot generalize well.",
"A method comparison is listed in Table REF .",
"Patel et al.",
"[44] propose to combine deep features with eye blinking cues for cross-dataset spoofing detection.",
"There are works [2], [54] formulating the anti-spoofing task into an anormaly or outlier detection, which highly rely on the definition of genuine samples.",
"To alleviate this, Liu et al.",
"[35] propose a zero-shot learning solution, whereas the unseen attacks are assigned to the most similar attacks predefined in the database.",
"These unseen attacks are wildly variant and leave the chance that the attacks are heavy outliers.",
"Shao et al.",
"[46] formulate a domain generalization approach to improve the generalization ability, which depends on the number and diversity of the seen domains, i.e., biased or long-tailed observed domains would degrade the performance.",
"Different from [35], [46], we propose the physical cue based proxy tasks that are less dependent on the data distribution, which generally could be more stable and consistent across seen and unseen attacks.",
"Uncertainty Analysis: Uncertainty provides an effective measurement for model/data reliability [30], [50], [24], [38].",
"It has been widely applied in many vision tasks such as classification [21], semantic segmentation [25] and face recognition [6].",
"Our method follows the setting of [25] by leveraging multiple tasks, but in a complete different problem as face anti-spoofing rather than segmentation.",
"We consider our proxy tasks are orthogonal to each other, whereas in [25] those semantic tasks are strongly correlated.",
"To the best of our knowledge, we are the first to leverage uncertainty in face anti-spoofing tasks.",
"Figure: The proposed framework consists of the pretext task “face recognition” (Sec.",
"), the proxy tasks (Sec. )",
"“depth estimation”,“material prediction”, “reflection detection” and the main task “liveness detection”.",
"A novel triplet mining regularization (Sec. )",
"is proposed to better disentangle the liveness feature and an uncertainty-aware attention module (Sec. )",
"aggregates the channel-wise results for boosted performance."
],
[
"Proposed Approach",
"In this section, we firstly introduce the shared feature extractor learning by incorporating the pretext task face recognition.",
"Then, the physical cue based proxy tasks, i.e., depth estimation, reflection detection and material classification, are introduced as the spoofing attack detection anchors.",
"Finally, an uncertainty-aware attention module is proposed to aggregate the proxy channels for optimal performance."
],
[
"Shared Feature Representation Learning",
"As shown in Figure REF , our framework consists of multiple channels of pretext and proxy tasks.",
"Separating each single task with independent CNNs results in network redundancy.",
"Moreover, the separated CNNs cannot leverage the rich information from the other tasks, where hyper-column [18] and deeply-supervised net [29] have shown a highly integrated framework for multiple tasks is beneficial.",
"To this end, we propose to use a single feature extractor $\\Phi $ to provide the shared feature for all the downstream tasks.",
"The shared features should provide high-level task-specific yet general information for downstream tasks such that we neither drift away from original tasks nor learn only task-driven representations.",
"Among the face analysis applications, face recognition is a promising pretext as it is usually trained with large-scale data including millions of identities, which guarantees the robustness as well as the discriminality.",
"Other candidates such as facial attribute classification, expression recognition or spoofing detection are not general or robust, as each of the tasks conduct a 10-way or 2-way classification, which can be sensitive or easily overfitting [34].",
"Thus, to initialize the feature extractor $\\Phi $ , we apply face recognition as our pretext task.",
"Denoting input image as $\\mathbf {x}_r, \\mathbf {x}_v$ and $\\mathbf {x}_m$ for recognition, spoofing and material data respectively.",
"After the shared feature extractor $\\Phi $ , the pretext task applies a filter $\\Psi _r$ to refine the face identity feature.",
"The loss is: $\\mathcal {L}_{r}&=-\\sum _i {1}(y_{i})\\log \\frac{\\exp (\\mathbf {w}_{i} \\Psi _r\\circ \\Phi (\\mathbf {x}_{r}))}{\\sum _j \\exp (\\mathbf {w}_{j} \\Psi _r\\circ \\Phi (\\mathbf {x}_{r}))}$ where $y_i$ is the ground truth label for identity $i$ .",
"$j$ varies across the whole number of identities.",
"$\\mathbf {w}_i$ is the $i_{th}$ separation hyper-plane of the classifier.",
"$\\circ $ denotes the sequential network flow."
],
[
"Multi-channel Proxy Task Learning",
"We introduce a transfer module $\\Gamma $ to adapt the rich feature extracted by $\\Phi $ into the spoofing detection related tasks.",
"Directly utilizing the shared feature leads to sub-optimal prediction as it incorporates unrelated face recognition cues, which may serve as noise.",
"Similar to the pretext task, we set up multiple channels for our proxy tasks, i.e., liveness detection $\\Psi _{v}$ , depth estimation $\\Psi _{d}$ , reflection detection $\\Psi _{r}$ and material prediction $\\Psi _{m}$ .",
"Liveness Detection Main Task: The spoofing detection is a well-known binary classification task.",
"The input is spoofing face $\\mathbf {x}_{v}$ .",
"After shared extractor and feature transfer module, we set up the spoofing detection channel filter $\\Psi _{v}$ to conduct the binary classification task, in which we adopt binary cross entropy loss as the objective: $\\mathcal {L}_{v} &= -y_v\\log (\\mathbf {p}(\\mathbf {z}))- (1-y_v)\\log (1-\\mathbf {p}(\\mathbf {z}))$ $\\mathbf {p}(\\mathbf {z}) = \\frac{\\exp (\\tilde{\\mathbf {w}}_0\\mathbf {z})}{\\exp (\\tilde{\\mathbf {w}}_0\\mathbf {z}) + \\exp (\\tilde{\\mathbf {w}}_1\\mathbf {z})}$ where $\\mathbf {y}_v$ is the ground truth of spoofing or genuine, $\\mathbf {z}=\\Psi _v\\circ \\Phi (\\mathbf {x}_v)$ denotes the spoofing detection feature after the spoofing detection filter $\\Psi _{v}$ .",
"$\\tilde{\\mathbf {w}}_0$ and $\\tilde{\\mathbf {w}}_1$ are the separation hyper-planes of the binary classifier.",
"Likelihood of being spoofing sample $\\mathbf {p}(\\mathbf {z})$ is estimated via a softmax operation in Equation REF .",
"Depth Proxy Task: We believe the physical cues should share similar characteristics for genuine faces across different attack types or spoofing datasets.",
"Thus the depth prediction should also be consistent.",
"We aim to predict the per-pixel depth map given the input face image.",
"We leverage an hourglass network structure to conduct this regression problem, which has been proved effective in key point detection [40] and image segmentation [12].",
"To prepare the ground truth depth map $\\mathbf {d}_{GT}$ , we apply a 3D face shape reconstruction algorithm [17] offline to estimate the dense point cloud for the face images.",
"As for genuine face image, we utilize the predicted depth as its ground truth depth map, where background is set as 0.",
"For 2D spoofing face images, according to their attack types, i.e., display screen or paper, we know that the actual depth is from a flat plane of either screen or paper.",
"Thus, we manually set the spoofing ground truth depth to be all 0.",
"The absolute depth is unnecessary since we only focus on the relative face geometry.",
"We show some examples of generated depth map results in Figure REF .",
"During training, an $l_1$ -based reconstruction loss is applied: $\\mathcal {L}_{d} = \\Vert \\Psi _d\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_{v}) - \\mathbf {d}_{GT}\\Vert _1$ where $\\Psi _d$ is the hourglass net depth estimation module, $\\Gamma $ is the feature transfer module and $\\mathbf {d}_{GT}$ is the ground truth depth.",
"Notice that for depth estimation, we input the spoofing data $\\mathbf {x}_v$ with the augmented ground truth depth map.",
"We do not utilize extra depth data for this channel.",
"Reflection Proxy Task: Reflection is another useful physical cue that indicates the genuine faces, as non-skin materials inevitably show abnormal reflection compared to skin.",
"As a result, for spoofing face, we use a single image reflection separation model [58] to generate the reflection map, while for genuine face, we set the it to zero denoting no reflection is present.",
"Visual examples of generated reflection maps are in Figure REF .",
"During training, we push the predicted reflection map to be close to the pseudo ground truth under $l_1$ -based constraint: $\\mathcal {L}_{r} = \\Vert \\Psi _r\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_{v}) - \\mathbf {R}_{GT}\\Vert _1$ where $\\Psi _r$ is the hourglass net reflection estimation moduleand $\\mathbf {r}_{GT}$ is the ground truth reflection map.",
"Material Proxy Task: Though reflection in some way indicates the material information, we explicitly introduce material as another proxy task to leverage the correlation among the multiple tasks, expecting to benefit from the multi-task learning.",
"The physical insight for material in liveness detection is that skin across different spoofing attacks or spoofing datasets should remain similar RGB information.",
"We automatically obtain the material type for face spoofing data according to its attack type.",
"For instance, we denote the material class of screen display and paper print as “glass” and “paper” respectively.",
"In this way, we actually unify the material type towards the general material recognition [5].",
"Notice that the number of material types in spoofing data can be limited, which may encounter the same sensitivity issue as the binary spoofing detection task.",
"To this end, we introduce the general material recognition data to anchor the material feature space from being collapsed.",
"Specifically, the general material recognition and our spoofing data material recognition share all the network structures except the last classifier layer.",
"As in general material recognition, there are 23 defined categories [5], such as brick, metal, plastic, skin, glass, etc.",
"We set up a 23-way classifier $\\mathbf {C}_g$ for the general material recognition and a 3-way classifier $\\mathbf {C}_v$ for our spoofing data material recognition.",
"A multi-source scheme is proposed to train the modules of $\\Phi $ , $\\Gamma $ and $\\Psi _{m}$ jointly.",
"Denoting the feature $\\mathbf {f}=\\Psi _m\\circ \\Gamma \\circ \\Phi (\\mathbf {x})$ , a combined multi-class softmax loss is applied to train $\\mathbf {C}_g$ and $\\mathbf {C}_v$ : $\\mathcal {L}_{m}&=-\\sum _{i=1}^{23} {1}(l_i)\\log \\frac{\\exp (\\mathbf {\\omega }_i \\Psi _m\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_m))}{\\sum _j \\exp (\\mathbf {\\omega }_j \\Psi _m\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_m))} - \\\\&\\sum _{i=1}^{3} {1}(l_i)\\log \\frac{\\exp (\\tilde{\\mathbf {\\omega }}_i \\Psi _m\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_v))}{\\sum _j \\exp (\\tilde{\\mathbf {\\omega }}_j \\Psi _m\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_v))}$ where $l_i$ is the material ground truth label, $\\omega _i$ and $\\omega _j$ , $\\tilde{\\mathbf {\\omega }}_i$ and $\\tilde{\\mathbf {\\omega }}_j$ are the separation hyper-planes for $\\mathbf {C}_g$ and $\\mathbf {C}_v$ respectively.",
"By alternatively feeding the material and spoofing data, we guarantee that $\\mathbf {f}$ is generalized for not only the standard material recognition, but also the material recognition for face spoofing data."
],
[
"Attributed-assisted Triplet Mining",
"To better disentangle the liveness feature apart from identity information and other facial attributes information, we leverage the metric learning to regularize the feature representation learning.",
"Specifically, given the input $\\mathbf {x}_v^{i}$ , we would expect the following loss to be minimized such that the identity information can be decoupled from the liveness feature.",
"$\\mathcal {L}_{tid} = \\lfloor \\Vert \\Phi (x_v^{i,j}) &- \\Phi (x_v^{i,k})\\Vert ^2 - \\\\&\\Vert \\Phi (x_v^{i,j}) - \\Phi (x_v^{h})\\Vert ^2 + m_1\\rfloor _{+}$ $x_v^{i,j}$ means the $j^{th}$ liveness sample from identity $i$ , while $x_v^{h}$ simply means the liveness sample from other identities as a negative sample.",
"Similarly for other face attributes introduced in CelebA [36], [59], we believe the orthogonality can be preserved if those attribute information is disentangled from the liveness information.",
"$\\mathcal {L}_{ta} = \\lfloor \\Vert \\Phi (x_v^{a_i,j}) &- \\Phi (x_v^{a_i,k})\\Vert ^2 - \\\\&\\Vert \\Phi (x_v^{a_i,j}) - \\Phi (x_v^{a_h})\\Vert ^2 + m_2\\rfloor _{+}$ $a_i$ indicates an attribute label and $a_h$ indicates a different attribute label for the negative sample.",
"$m_1$ and $m_2$ here are the margin hyper-parameter set to squeeze the classification boundary for better feature learning."
],
[
"Uncertainty-aware Attention Modeling",
"As each of the channels looks into different aspects of the spoofing characteristics, we seek to combine those independent channels adaptively to boost the final spoofing detection performance.",
"We introduce an uncertainty-driven attention module that is orthogonal to each of the main and proxy tasks, which thus effectively overcomes the over-confident issue of the traditional attention modules.",
"Given an input $\\mathbf {x}_v$ , the joint likelihood $p(y|\\mathbf {x}_v)=p(\\mathbf {z}|\\mathbf {x}_v)p(\\mathbf {d}|\\mathbf {x}_v)p(\\mathbf {r}|\\mathbf {x}_v)p(\\mathbf {f}|\\mathbf {x}_v)$ according to the channel independence assumption, where $\\mathbf {z}$ is from Equation REF as the main task feature, $\\mathbf {d}=\\Psi _d\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_{v})$ is from Equation REF as the reflection feature, $\\mathbf {r}=\\Psi _r\\circ \\Gamma \\circ \\Phi (\\mathbf {x}_{v})$ is from Equation REF as the depth feature, $\\mathbf {f}=\\Psi _m\\circ \\Gamma \\circ \\Phi (\\mathbf {x})$ is from Equation REF as the material feature.",
"Maximizing the joint likelihood leads to maximizing the summation of each likelihood: $\\operatornamewithlimits{arg\\,min}-\\log (p(y|\\mathbf {x}_v)= -\\sum _{\\mathbf {u}=\\mathbf {z},\\mathbf {d},\\mathbf {r},\\mathbf {f}}\\log (p(\\mathbf {u}|\\mathbf {x}_{v})$ For each channel, we assume a Gaussian distribution $p(\\mathbf {u}|\\mathbf {x}_v) \\sim \\mathcal {N}(\\mu _{\\mathbf {u}}, \\sigma _{\\mathbf {u}})$ to capture the uncertainty, where $\\mu _{\\mathbf {u}}$ is the corresponding channel $\\mathbf {u}$ classifier's separation hyper-plane vector or the mean depth map for the depth channel.",
"Under the probabilistic setting, such $\\mu _{\\mathbf {u}}$ conforms to another Gaussian distribution $\\mathcal {N}(\\mu _{\\mathbf {u}}, \\sigma _{\\mu _{\\mathbf {u}}})$ , where $\\sigma _{\\mu _{\\mathbf {u}}}$ is estimated upon sampling from multiple rounds of training.",
"$\\sigma _{\\mathbf {u}}$ is independently learned via a two FC-layer network structure in parallel to the feature $\\mathbf {u}$ .",
"$\\mu _{\\mathbf {u}}$ is jointly learned with $\\mathbf {u}$ during training and is fixed during the uncertainty training.",
"The objective to learn $\\sigma _{\\mathbf {u}}$ is defined in the following: $\\mathcal {L}_{\\sigma _{\\mathbf {u}}}=\\sum _{\\mathbf {u}=\\mathbf {z},\\mathbf {d},\\mathbf {r},\\mathbf {f}}\\left(\\frac{\\Vert \\mathbf {u} - \\mu _{\\mathbf {u}}\\Vert ^{2}}{2(\\sigma _{\\mathbf {u}}^{2}+\\sigma _{\\mu _{\\mathbf {u}}}^{2})}+\\frac{D}{2}\\log (\\sigma _{\\mathbf {u}}^{2}+\\sigma _{\\mu _{\\mathbf {u}}}^{2})\\right)$ where $D$ is the feature dimension.",
"It is independently optimized after the network is converged.",
"During inference, the network outputs $\\mathbf {u}$ and $\\sigma _{\\mathbf {u}}$ simultaneously.",
"Given $\\mu _{\\mathbf {u}}$ and $\\sigma _{\\mu _{\\mathbf {u}}}$ , we then obtain the uncertainty estimate for each channel with Equation REF .",
"To sum with, we propose a two-stage training procedure.",
"The first stage consists of the training of liveness main task, proxy tasks and pretext task.",
"And the loss is defined as following: $\\mathcal {L}=\\lambda _v\\mathcal {L}_v + \\lambda _d\\mathcal {L}_d + \\lambda _r\\mathcal {L}_r + \\lambda _m\\mathcal {L}_m + \\lambda _{t} (\\mathcal {L}_{tid} + \\mathcal {L}_{ta})$ Then in the second stage, the uncertainty attention module is trained with Equation REF .",
"In our implementation, we leverage a pre-trained face recognition engine and re-utilize the encoder as our shared feature extractor $\\Phi $ .",
"Then, we keep the face recognition as our pretext task and equip the main and proxy tasks to form a multi-source multi-channel training.",
"As illustrated in the methodology section, a two-stage training is conducted.",
"For the first stage joint training of pretext, main and proxy tasks, we apply random cropping and horizontal flipping as data augmentation.",
"We adopt Adam solver and the initial learning rate is set 0.0001.",
"The momentum and weight decay are fixed as 0.9 and 0, respectively.",
"Hyper-parameters in Equation REF is empirically searched via some hold-out validation as $\\lambda _r=1,\\lambda _d=1,\\lambda _v=1,\\lambda _m=0.1$ , $\\lambda _t=0.1$ for triplet need to add here together with $m_1$ and $m_2$ as in Equation REF and REF .",
"respectively.",
"For the second stage, when training the uncertainty-aware attention module, we re-use well-trained modules from the first stage, and only fine-tune the two-layer fully connected layers for each of the main and proxy tasks to estimate the variance.",
"Table: Intra-dataset evaluation on SiW dataset.",
"L: main spoofing/liveness detection.",
"L+D: spoofing and depth channels.",
"L+M: spoofing and material channels, L+D+M: spoofing, depth and material channels.",
"L+D+R+M: spoofing, depth, reflection and material channels.",
"L+D+A: spoofing and depth with attention.",
"L+D+M+A: spoofing, depth and material with attention.",
"L+D+R+M+A: our overall model with attention.Table: Inter-dataset evaluation on CASIA, Replay Attack and MSU, AUC(%) is reported.",
"We follow the “Leave one dataset & attack-type out” protocol in , where the attack types in testing set is unseen in the training set.",
"We abbreviate V, C-P, W-P, D-P, P-P, H-V and M-V for Video, Cut Photo, Warped Photo, Digital Photo, Printed Photo, HR Video and Mobile video, respectively.",
"*: retrained by their released codes."
],
[
"Datasets",
"CASIA [60]: A video based 2D spoofing attack database, consists of 600 videos from 50 people, 240 videos from 20 people for training and 360 videos from 30 people for testing.",
"Each people contains 12 videos with video re-display and photo print attacks, of which 8 are normal resolution videos and 4 are high resolution videos.",
"The photo attacks are further categorized into cut photo by cutting holes around eyes, noise, mouth, and warped photos by warping photos with different curvature.",
"Replay-Attack [13]: A 2D face spoofing attack database consists of $1,300$ video clips of photo and video attack attempts from 50 clients, under different lighting conditions.",
"To produce the attacks, high-resolution photos and videos from each client were taken under the same conditions as in their authentication step.",
"MSU-MFSD [53]: it consists of 280 video clips of photo and video attack attempts from 35 clients.",
"Mobile phones are used to capture both genuine faces and spoofing attacks.",
"Printed photos are generated from high quality color printers for another attack type.",
"It also provides replay video attacks with high resolution of $2048\\times 1536$ from iPad air screen.",
"Oulu-NPU [11]: A large-scale 2D spoofing attack dataset, consists of 4950 genuine and attack videos from 55 people.",
"They are recorded using the front cameras of different mobile devices with variant illuminations and backgrounds.",
"The attack types are print and video replay.",
"There are four protocols designed to consider generalization on cross attack types and capturing sensor types.",
"SiW [34]: Spoofing in the Wild dataset provides 4,478 genuine and spoofing videos from 165 subjects.",
"For each subject, 8 genuine and up to 20 spoofing videos are captured.",
"It systematically considers variations from subjects, camera sensors, spoofing attack types, lighting conditions, image resolution, and different sessions for capturing.",
"There are three evaluation protocols emphasizing the generalization on face PAD, cross attack types, and unknown attack types."
],
[
"Evaluation Metrics",
"The evaluation is focused on testing the generalization of cross attack types within one dataset, termed intra-dataset evaluation, and cross dataset spoofing detection, termed inter-dataset evaluation following [2].",
"To be consistent with most of the previous spoofing detection works, we apply the evaluation metrics as: Attack Presentation Classification Error Rate (APCER[19]), Bona Fide Presentation Classification Error Rate (BPCER[19]), ACER = 0.5(APCER+BPCER), Area Under Curve (AUC) ratio and Half-Total Error Rate (HTER).",
"Further following [34] in SiW protocol settings, we use Equal Error Rate (EER) as validation metric for all models to search the threshold to report performance.",
"Figure: Visual examples of our Uncertainty-aware Attention Modeling.",
"We show example of Spoof and Live Image on SiW dataset.",
"The first two rows shows examples is from SiW dataset while the bottom row is from CelebA-Spoof dataset ."
],
[
"Intra-Dataset Evaluation",
"We evaluate on a recent large scale spoofing dataset SiW with carefully designed cross attack type testing protocols.",
"We refer another intra-dataset evaluation on Replay-Attack to supplementary due to space limit.",
"SiW Evaluation: There are 3 protocols in SiW.",
"Protocol 1 focus on evaluating the performance of variations in face pose and expression.",
"Protocol 2 focuses on the unseen medium of replay attack.",
"It chooses 3 out of 4 display attacks, as training and leaving the remaining one as testing, which is iteratively conducted 4 times and averaged.",
"Protocol 3 evaluates cross presentation attack detection, i.e., from print attack to replay attack and vice versa.",
"Averaging over the two is reported.",
"In Table REF , our method consistently outperforms the other methods with significant margin, i.e., on Protocol 1, we achieve $\\textbf {0.36}$ ACER while FAS-TD-SF [52] is $1.05$ .",
"On Protocol 2, ours is $\\textbf {0.20}$ while the best compared method is 0.31 from FAS-TD-SF.",
"On Protocol 3, ours is $\\textbf {7.32}$ while the best compared method is 7.73.",
"Similar to Replay-Attack, we apply a gradually increasing module way to highlight effectiveness of the proposed modules.",
"The ablation over our proposed modules suggests: (1) Depth, reflection and material are beneficial proxy tasks.",
"(2) putting more proxy tasks together boosts the performance.",
"(3) Our uncertainty-aware attention on top of the baselines can further achieve performance gain with significant margin.",
"We also show an ablation contrasting w/ or w/o using attributed-assisted triplet mining, it shows that by adding triplet constraint, there is continuous margin gain over the other baselines."
],
[
"Inter-Dataset Evaluation",
"The inter-dataset setting mimics the real setting for unseen attack across types and datasets.",
"We consider two protocols.",
"One follows the traditional rule [2], a “Leave one dataset & attack-type out” protocol, taking CASIA, Replay-Attack and MSU-MFSD as our datasets.",
"Each of the three datasets contains three attack types.",
"When evaluating one attack type of one dataset, we pick the other two datasets for training and excluding the testing attack type from training.",
"The other less-strict setting is the “Leave one dataset out“ protocol used in MADDG[46], the difference to the former is that in this protocol, training and testing sets would have overlapping attack types.",
"Leave one dataset & attack-type out In Table REF , we evaluate each of the three attack types from three datasets.",
"Both traditional feature learning based methods and most recent deep learning based methods [34], [46] are compared.",
"Overall we achieve consistently stronger results than the other methods.",
"In CASIA, video attack is significantly better than other methods while cut photo and warped photo are among the top.",
"In Replay-Attack, we achieve clear better performance.",
"In MSU-MFSD, we observe 1% to 6% performance improvement over the compared methods.",
"Table: Inter-dataset evaluation on CASIA, Replay Attack, MSU and Oulu-NPU dataset.",
"AUC (%) is reported.",
"We follow the “Leave one dataset out” protocol in , where training and testing sets share attack types.",
"DD denotes disentangling source domains.Leave one dataset out In Table REF .",
"Since our physically-guided proxy task does not require any domain priors, we compare methods w and w/o source domains disentanglement (DD).",
"Our method surpass all methods without domain disentanglement including [34], while still achieve comparable performance compare to methods [47][48][20] that utilize extra source domains information."
],
[
"Analysis of Uncertainty-aware Attention",
"We visualize our Uncertainty-aware Attention Module in Figure REF .",
"Specifically, we visualize the last feature map of the last FC layer in the Attention Module across Liveness, Depth, Reflection and Spoof channel alongside with the corresponding Input Image, GT/Predicted Depth map and GT/Predicted Reflection map.",
"In Figure REF , we can see that, in most cases, those proxy tasks agree with each other like the examples from top two rows, while in some cases, those proxy tasks does not agree with each other.",
"for example, the figure in the bottom row, depth channel made the wrong prediction and disagree with other channels, However, out uncertainty-aware attention module are able to correct his by give depth a lower confidence and voted for prediction from other channels, thus correcting the final decision."
],
[
"Conclusion",
"In this work, we propose depth, reflection and material guided proxy tasks for unseen spoofing attacks.",
"We propose a multi-source multi-channel training scheme for model optimization.",
"Due to the consistency of depth, reflection and skin material across different spoofing scenario on genuine faces, by harnessing those physical proxy tasks, we expect the proposed method to deal with unseen spoofing attacks.",
"Finally, an uncertainty-aware attention module is introduced to aggregate the multiple channels for boosted performance.",
"Experiments across intra- and inter-dataset protocols show our method achieves consistently better performance and is effective for unseen spoofing detection."
]
] | 2011.14054 |
[
[
"Human-Agent Cooperation in Bridge Bidding"
],
[
"Abstract We introduce a human-compatible reinforcement-learning approach to a cooperative game, making use of a third-party hand-coded human-compatible bot to generate initial training data and to perform initial evaluation.",
"Our learning approach consists of imitation learning, search, and policy iteration.",
"Our trained agents achieve a new state-of-the-art for bridge bidding in three settings: an agent playing in partnership with a copy of itself; an agent partnering a pre-existing bot; and an agent partnering a human player."
],
[
"Introduction",
"Deep reinforcement learning has had great success in two-player, zero-sum games.",
"Key to the success of the algorithms used for these games is the fact that if neither player can unilaterally improve, the policies must be globally optimal [1], and therefore considering only unilateral policy deviations is sufficient to achieve optimal play.",
"On the other hand, collaborative multi-agent environments have many local optima – agents must coordinate their strategies with each other to perform well.",
"We are particularly interested in human-compatible agents for such domains, because of the general importance of agents that can cooperate with humans.",
"A type of environment where the need for coordination is particularly stark is a communication game; if one agent unilaterally diverges from an agreed “language”, the results are likely to be very poor.",
"The bidding phase of bridge is one such domain, involving grounded communication and competition between two teams of two.",
"It has been extensively studied by human players and has coordination and cooperation with a partner as key features.",
"There are many fundamentally different equilibria in this domain; in particular, humans have devised many mutually-incompatible communication systems.",
"We address the problem of training an agent to learn a human-compatible policy for bridge bidding.",
"Using imitation learning, search, and policy iteration we train an agent to cooperate with a strong hand-coded human-compatible bot (WBridge5), and evaluate it in play with a human expert.",
"Our approach starts by imitation-learning from a dataset of deals played by WBridge5 to obtain an initial human-compatible policy.",
"We then use a search algorithm that aims to improve a given policy, combining the prior policy with the results of rollouts.",
"Starting with our learned model of WBridge5, we perform several (up to 16) rounds of policy iteration.",
"In each round we first generate a dataset using the policy and search, and then update the policy from this dataset.",
"If we are aiming for human-compatibility, we use the imitation-learned model as a partner; if we are aiming for maximum team performance, we use ourself as our partner.",
"Finally, at test time we optionally apply a further larger search to improve our policy."
],
[
"Bridge Bidding",
"Contract bridge is a zero-sum game played between two teams of two players.",
"We summarize key aspects of the rules here; see the Laws of Duplicate Bridge [2] for more detail.",
"The four players are conventionally labelled North, East, South, and West, and the two partnerships are North-South and East-West.",
"Each player is dealt a private hand of 13 cards.",
"The game then proceeds in two phases: first bidding, then play.",
"We describe the play first, since this grounds the bidding.",
"In the play phase, 13 tricks are played.",
"The first player to a trick may play any of their remaining cards.",
"Subsequent players must play a card of the same suit as the first player if they have any, or else any card from their hand.",
"The trick is won by the highest trump if any were played, or otherwise the highest card of the suit led.",
"Scoring is based on the division of tricks between the two sides.",
"In the bidding phase, players take turns to make a call which is one of Pass, Double, Redouble, or a bid.",
"Bids specify a trick target and a trump suit (or no trump).",
"The collective sequence of bids made must be ascending, with bids ordered first by number of tricks and then by trump suit.",
"The final bid, or contract sets the declarer, trump suit and trick target for the play phase.",
"At the end of the play, the declarer's side receives a positive score if they met or exceeded the trick target; otherwise their score is negative.",
"Higher contracts receive significant bonuses if made; this incentivizes players to use low-level bids conventionally to exchange information and determine whether a high contract is feasible These conventions may be extremely intricate in expert play, for example enabling one player to ask their partner whether they have a single specific card.",
"Some partnerships have hundreds of pages of detailed notes.. Conversely, a player with a weak hand may make a high-level bid to restrict the ability of the opponents to communicate in this way.",
"Bids are thus grounded communication actions, serving multiple purposes.",
"Each bid communicates information about the bidder’s cards, and restricts the later bids of other players, as well as being a possible contract.",
"The rules require that players be fully informed of the bidding system used by their opponents, both so that they are able to draw the correct inferences from the actions taken and so that they are able to adapt their subsequent system.",
"In tournament play, this is achieved through a combination of disclosing agreements to the opponents in advance (allowing preparation), disclosing agreements during the course of the deal (at which point no discussion is permitted), and restricting the nature of permissible agreements, hence reducing the adaptations that may be necessary.",
"Following previous work [3], [4], [5], [6] we consider only the bidding phase of the game; after the bidding, we assign rewards based on double-dummy play; that is, the result of the play phase computed assuming perfect play for all players given perfect information.",
"This bidding-only game is a slightly different game from bidding in the full game of bridge for two reasons.",
"Firstly, because there is a cost during the play phase from having revealed information during the bidding phase, which may affect the best bidding strategy.",
"Secondly, because the expected result of the imperfect-information card play phase may be different from the result of optimal perfect-information play.",
"However, the average score will be close, and a statistical study by a world-class player concluded that “actual [expert-level] play versus double-dummy is pretty close, so analyses based on double-dummy play should generally be on track”[7]."
],
[
"WBridge5",
"WBridge5 [8] by Yves Costel is a strong computer bridge program which has won the World Computer Bridge Championship six times, most recently in 2018, and has been runner up seven times [9].",
"The format of this competition is the same as our team play setting.",
"WBridge5 is the standard comparator for machine-learning agents because it is free software, although not open source.",
"Some information on the algorithms used by the program may be gleaned from [10].",
"WBridge5 can be programmatically interacted with using the Blue Chip Bridge protocol [11].",
"For training purposes, we used a public dataset of one million deals played by WBridge5 [12].",
"In this dataset, WBridge5 is configured to play according to the Standard American Yellow Card system [13], which was devised by the American Contract Bridge League in the 1980s and has become popular in online play as a simple system for casual partnerships, which makes it a good starting point for human-agent cooperation."
],
[
"Tasks",
"We consider two distinct tasks: human-compatible and partnership play.",
"For human-compatible play, the aim is to produce agents which can play well in a partnership with humans.",
"Since WBridge5 is designed to play a human-like strategy, we use playing with WBridge5 as a proxy for training and evaluation of this task.",
"We then evaluate the best agent in this setting as a partner to a human expert.",
"For partnership play, the aim is produce agents that can play well in a partnership with themselves.",
"This is the task which has been addressed in previous work [3], [4], [5], [6], with the usual evaluation metric being performance against WBridge5.",
"In line with prior work, we ignore the system restrictions and disclosure requirements of bridge.",
"However evaluating our performance as a partner to WBridge5 implicitly ties us to the Standard American Yellow Card system, which does meet these requirements."
],
[
"Prior Work",
"Ginsberg [14] introduced the Borel search algorithm to improve a bridge bidding policy expressed as a set of rules in a domain-specific language: To select a bid from a candidate set $B$ , given a database $Z$ that suggests bids in various situations: Construct a set $D$ of deals consistent with the bidding thus far.",
"For each bid $b \\in B$ and each deal $d \\in D$ , use the database $Z$ to project how the auction will continue if the bid $b$ is made.",
"(If no bid is suggested by the database, the player in question is assumed to pass.)",
"Compute the double dummy result of the eventual contract, denoting it $s(b,d)$ .",
"Return that $b$ for which $s(b,d)$ is maximal.",
"Ginsberg used Borel search in the bridge-playing program GIB at test-time to potentially override the action selection given by the rules.",
"The paper describes a number of heuristics to attempt to avoid failures when there was a weakness in the original policy.",
"We did not find these to be necessary in our approach, we believe both because our starting policy was stronger, and also because our policy iteration would eventually remove such weaknesses.",
"Ginsberg mentions two specific classes of problems.",
"First: Suppose that the database $Z$ is somewhat conservative in its actions.",
"The projection in step 2 leads each player to assume his partner bids conservatively, and therefore to bid somewhat aggressively to compensate.",
"The partnership as a whole ends up overcompensating.",
"In our partnership policy iteration approach, both our policy and our partner's policy will gradually get more aggressive, and the iteration should converge at a locally optimal level of aggression.",
"Second: Worse still, suppose that there is an omission of some kind in $Z$ ; perhaps every time someone bids $7\\diamondsuit $ the database suggests a foolish action.",
"Since $7\\diamondsuit $ is a rare bid, a bidding system that matches its bids directly to the database will encounter this problem infrequently.",
"GIB, however, will be much more aggressive, bidding $7\\diamondsuit $ often on the grounds that doing so will cause the opponents to make a mistake.",
"In practice, of course, the bug in the database is unlikely to be replicated in the opponents' minds, and GIB's attempts to exploit the gap will be unrewarded or worse.",
"Our policy iteration approaches use our learned policy as an opponent model, so that if the starting policy does have a weakness like this, it will be repeatedly triggered in training data; subsequent search should find better countermeasures to the $7\\diamondsuit $ bid, which will be incorporated into the learned policy, eventually making the $7\\diamondsuit $ action less attractive.",
"While the proposed combination of imitation learning, search, and policy iteration is novel, very similar components have been used in past successful game-playing programs.",
"Combining a deep neural network policy with a search procedure at training time to generate improved training data, and at test time to improve upon the raw neural network policy was introduced in [15] and has been successfully used in a number of games, including Hex [16], Go [17], Chess and Shogi [18].",
"As in [17], we start the policy iteration process by learning from a dataset of games played by strong players, with human-compatibility as an additional motivation.",
"Recent work has shown that standard reinforcement learning self-play techniques can lead to good performance on bridge bidding [5].",
"Following this work, we use standard neural network architectures without any special auxiliary losses.",
"Improving a prior policy using search has achieved state-of-the-art results in the fully-cooperative game of Hanabi [19].",
"We extend this approach by sampling possible world states (as considering them all would be infeasible There are $6e18$ distinct possible distributions of the hidden cards from the point of view of a searching player.",
"For comparison, the search algorithm in [19] is only used when there are fewer than $1e4$ possible states.",
"), and by embedding the search in a policy iteration loop.",
"Other recent work examined approaches for exploring the policy space in cooperative games including bridge bidding [6].",
"Our work does not require this exploration, as we start from a fairly strong policy.",
"Improving upon a policy while maintaining the ability to cooperate with a human, or a bot as a proxy for a human, is also relevant in the natural language domain, where there is recent work on preventing “language drift”.",
"In [20], episodes of supervised learning from human demonstrations are interleaved with self-play to reduce divergence from the human policy.",
"In [21], a single student learns from data generated by successive generations of fine-tuned teachers.",
"One approach discussed in [22] is using rewards from partnering with a fixed listener to prevent drift in the policy of a speaker, which is analogous to our use of an imitation-learned model as a partner."
],
[
"Imitation learning",
"The first stage of our method is to learn a model which can predict the bids made by WBridge5.",
"We learn a policy network $P_\\Phi (a|f)$ which gives probabilities for each action $a$ given features $f$ describing the state of the game.",
"We use the OpenSpiel implementation of bridge [23] to track the game state and generate a 480-element input feature vector as follows: Table: NO_CAPTIONThis representation has perfect recall, i.e.",
"it is possible to reconstruct the entire sequence of actions and observations from the current observation.",
"Our policy network is a 4-layer MLP with 1024 neurons per layer, ReLU non-linearities, and a softmax output, giving probabilities for each of the 38 actions (35 bids, Pass, Double, and Redouble).",
"We train on a dataset of WBridge5 self-play [12], selecting decision points uniformly at random, with a minibatch size of 16.",
"We used 200,000 steps of the Adam optimizer with learning rate 0.0003 to optimize a cross-entropy loss; we ran experiments with various auxiliary losses, such as predicting the cards in partner's hand, but found that these did not improve performance."
],
[
"Search",
"We extend the Borel search discussed in REF to use neural policies and a soft update.",
"We use the term particle for a single determinization of the hidden information, i.e.",
"a complete deal which is consistent with the information possessed by one player.",
"Our extended Borel search, shown in Algorithm REF , uses the rollout evaluation in Algorithm REF for several particles to evaluate a small candidate set of actions.",
"There are several differences between Ginsberg's Borel search and our search algorithm.",
"Firstly, we represent policies by neural nets, which may be different for the different players.",
"We then select candidate particles at random, based on our estimated probability that they would result in the actual actions observed.",
"When performing rollouts, we select actions according to a stochastic policy, allowing for more diversity of possible outcomes.",
"Finally, our prior policy is adjusted towards the search results, not replaced by them.",
"The policies used to filter particles and generate rollouts are approximations when partnering WBridge5 or a human, because we don't have access to the true policy in these cases.",
"We optionally use a search to improve the policy at test time, which results in a divergence between the policy we follow and the policy we use to filter particles and perform rollouts.",
"It is infeasible to use the search-augmented policy here because to do so would require search-within-search, when the search process is already expensive.",
"[!ht] Inputinput Outputoutput Rollout Value $h^{\\prime } \\leftarrow h + a$ $h^{\\prime }$ is not terminal $j \\leftarrow \\operatorname{ActingPlayer}(h^{\\prime })$ sample action $a^{\\prime }$ from $\\pi ^j(h^{\\prime }, p_j)$ $h^{\\prime } \\leftarrow h^{\\prime } + a^{\\prime }$ reward for $\\operatorname{ActingPlayer}(h)$ of the final contract $h^{\\prime }$ , using double-dummy analysis [!ht] Inputinput Outputoutput Parameterparameter $t$ — temperature $R_{min}$ — minimum rollouts to use the result of the search $R_{max}$ — maximum rollouts for the search $P_{max}$ — maximum particles to consider $k$ — maximum actions to consider $p_{min}$ — minimum action probability to consider $\\pi _{posterior}$ — posterior policy for the searching player Borel Search with Non-Deterministic Model $s \\leftarrow \\operatorname{ActingPlayer}(h)$ $A \\leftarrow $ top-$k$ actions from $\\pi _{prior}$ with probability at least $p_{min}$ $V(a) \\leftarrow 0$ for $a \\in A$ $R \\leftarrow 0$ $P \\leftarrow 0$ $P < P_{max}$ and $R < R_{max}$ jointly sample private information $\\left\\lbrace p_{-s}\\right\\rbrace $ for players other than $s$ uniformly, consistent with $p_s$ form the particle $\\left\\lbrace p_i \\right\\rbrace = \\left\\lbrace p_s \\right\\rbrace \\cup \\left\\lbrace p_{-s} \\right\\rbrace $ $P \\leftarrow P + 1$ each $h^{\\prime }a^{\\prime } \\sqsubseteq h$ $j \\leftarrow \\operatorname{ActingPlayer}(h^{\\prime })$ with probability $1 - \\pi ^{j}(a^{\\prime } | h^{\\prime }, p_{j})$ , skip to next particle $a \\in A$ $V(a) \\leftarrow V(a) + \\operatorname{RolloutValue}(h, a, \\left\\lbrace p_i \\right\\rbrace , \\left\\lbrace \\pi ^i \\right\\rbrace )$ $R \\leftarrow R + 1$ $R > R_{min}$ $\\pi _{posterior} \\propto \\pi _{prior} \\times \\exp \\left( \\frac{V(a)}{t\\sqrt{R}} \\right)$ $\\pi _{posterior} = \\pi _{prior}$"
],
[
"Policy Iteration",
"We implement a policy iteration loop by generating experience from a search policy as set out in REF .",
"We denote the current reinforcement learning policy $\\pi _l$ ; it is initially set to the WBridge5 imitation policy $\\pi _b$ , as learned in REF , using the same neural network architecture.",
"At each iteration, we learn a new updated policy $\\pi _{l^{\\prime }}$ by imitating the policy produced by the search.",
"The search configuration is slightly different depending on the aim of the learning task (that is, learning to play with an existing agent or in self-play).",
"For both tasks, we use $\\pi _l$ for the prior policy, and to rollout our own and the opponents' actions during search.",
"For learning to play a policy compatible with WBridge5, we use $\\pi _b$ for our partner's policy.",
"For learning to play in partnership with ourselves, we use $\\pi _l$ for our partner's policy.",
"Note that this choice of opponent policies is slightly counter-intuitive: in our evaluations, the opponent agents are WBridge5, so the imitation policy $\\pi _b$ might be a better model than $\\pi _l$ .",
"Empirically, we found that using $\\pi _b$ in search produces policies which perform well against $\\pi _b$ , but not against WBridge5.",
"We conclude that the search policy exploits weaknesses in $\\pi _b$ which are not present in WBridge5 and which the policy iteration loop is able to remove.",
"Finally, we perform a search at test time using a larger search budget, to improve our policy further.",
"For this search, we use greedy versions of the policies for rollouts, selecting the highest-probability action for each player.",
"The search parameters during policy iteration and at test time were as follows: Table: NO_CAPTIONThe acting network was updated every hour, which is approximately every 10 million learner observations.",
"The experiments were stopped when performance of $\\pi _l$ against $\\pi _b$ appeared to be levelling off; this was after 6 network updates and 50 million learner observations for compatible learning, and 16 network updates and 170 million learner observations for partnership learning."
],
[
"Imitation learning",
"Our imitation model achieved 93.9% accuracy in predicting the actions of WBridge5 on a held-out test set."
],
[
"Bot evaluation",
"We evaluate each of our agents on the two tasks described in REF .",
"In both cases, we use IMPs[2] to rescale raw score differences; this is consistent with prior work and is the usual practice in tournament bridge play.",
"Let $S(pqrs)$ represent the score obtained by the North-South partnership when the deal is played with agent $p$ as North, agent $q$ as East, agent $r$ as South, and agent $s$ as West; and let $w$ represent WBridge5 and $a$ the agent under evaluation.",
"To evaluate WBridge5-compatible play, we compare deals played by four WBridge5 bots to the same deals played by three WBridge5 bots and one of the agent being tested.",
"Specifically, our metric for a single deal is: $&\\operatorname{IMP}\\left(S(awww) - S(wwww)\\right)- \\operatorname{IMP}\\left(S(waww) - S(wwww)\\right) \\\\+ &\\operatorname{IMP}\\left(S(wwaw) - S(wwww)\\right)- \\operatorname{IMP}\\left(S(wwwa) - S(wwww)\\right)$ This is positive for agents which are better partners to WBridge5 than WBridge5 itself.",
"To evaluate team play, each deal is played twice, once with our agent playing the North-South hands, and once with our agent playing the East-West hands.",
"Specifically, our metric for a single deal is $ \\operatorname{IMP}\\left(S(awaw) - S(wawa)\\right) $ .",
"This is positive if the agent achieves better results than WBridge5.",
"In both cases, we express our results in average IMPs per deal over a set of 10,000 held-out deals.",
"This is considerably more evaluation deals than prior work, which accounts for at least some of our tighter error bounds.",
"Table: IMPs per deal for learned agents.",
"The errors shown are the standard error of the mean.As we expect, the agents which learn to partner WBridge5 and agents which learn to partner themselves each do better in their respective evaluation settings.",
"Also as expected, test-time search improves performance, especially where the rollout policy $\\pi ^i$ used for our partner in the search aligns closely with the player we are partnering.",
"We note that for teams play, there are important differences between our algorithm and previous work which affect this comparison.",
"Our agents are initialized to a similar policy to WBridge5, and remain somewhat similar throughout training, as evidenced by the fact that all the agents from policy iteration are better partners to WBridge5 than another copy of WBridge5.",
"In previous work, learning has been without reference to human policies, and so this does not hold.",
"This means our agents are less likely to gain points by “confusing” WBridge5 with unexpected conventions (which would be illegal in the full game, where conventions must be disclosed to the opponents either in advance or during the course of play).",
"Our agents have explicitly trained to beat a model of WBridge5's play, whereas previous work has not.",
"It is possible that the agent is exploiting weaknesses in WBridge5 rather than exhibiting strong play itself.",
"Note however that any weakness would have to be present both in WBridge5 and also in our model of it, but not be removed by our policy iteration loop.",
"As a partial test of this, we evaluate our agent's performance on the subset of deals where one of the two partnerships does not make a bid with any configuration of agents.",
"Roughly 20% of deals fall into this category.",
"We can be sure that performance on these deals does not arise from exploiting weaknesses in WBridge5's policy, although there is generally less scope for interesting bidding in these cases, so we should expect relative scores to be smaller.",
"Our best scores on this subset of deals are +0.37 IMPs/deal in the partnership setting and +0.24 IMPs/deal in the compatible setting.",
"This confirms that our agents are genuinely improving on WBridge5 and not solely exploiting its weaknesses.",
"Duplicate bridge has four possible scoring tables, known as vulnerabilities [2].",
"The deals in the WBridge5 dataset were all played with “neither side vulnerable” in order to allow statistical analysis across the whole dataset.",
"We therefore used this single scoring table throughout our experiments.",
"Since this vulnerability gives the smallest absolute scores, this choice is likely to have reduced our reported performance compared to prior work which uses a mixture of all four scoring tables."
],
[
"Human Evaluation",
"We take the final network from compatible policy iteration and evaluate it as a partner to a human expert (one of the authors), with no test time search.",
"The human plays each deal eight times, once in each seat with WBridge5 as a partner, and once in each seat with our agent as a partner; the opponents are WBridge5 in both cases.",
"The order in which the human partners the bots is randomized for each deal independently so that the human does not know at the start of the deal which of the two possible bots they are partnering.",
"Representing the human as $h$ , our scoring metric is: $&\\operatorname{IMP}\\left(S(hwaw) - S(hwww)\\right)- \\operatorname{IMP}\\left(S(whwa) - S(whww)\\right) \\\\+ &\\operatorname{IMP}\\left(S(awhw) - S(wwhw)\\right)- \\operatorname{IMP}\\left(S(wawh) - S(wwwh)\\right)$ Instructions for the human expert were as follows: Bid each hand according to Standard American Yellow Card, selecting the calls you would make with an unfamiliar expert human partner playing the same system.",
"Do not take advantage of information gained from earlier plays of the same deal.",
"The second time you play a deal in a particular seat, make the same call as the first time if the situation is identical.",
"If the situation is merely similar, be consistent with the previous play-through unless there is a clear reason to judge differently.",
"This is a slightly artificial setting, but allows a meaningful comparison with relatively few deals played.",
"Over an evaluation set of 32 deals, the agent outperforms WBridge5 by 0.97 IMPs/deal, with standard error of the mean 0.76.",
"Qualitative observations from the human expert, reviewing the hands after play: Our agent conforms to the Standard American Yellow Card system; the differences to WBridge5 are of judgement rather than system.",
"Our agent is slightly more aggressive in competitive auctions, matching modern expert practice.",
"Our agent generally prefers simpler, more direct auctions, which are more robust to slight differences in interpretation.",
"Little, if any, of the agent's improved performance was due to differences from the full game of bridge; i.e.",
"the agent did not gain points because WBridge5 was concealing information in anticipation of the play phase, nor because of the double-dummy play assumption."
],
[
"Summary",
"We introduced the problem of learning human-compatible policies for bridge bidding.",
"This is an interesting task to tackle because it requires learning to communicate and collaborate with a human in a challenging domain – one which is played competitively and has been extensively studied.",
"Using a combination of imitation learning, policy iteration and search, we trained agents to improve on a hand-coded bot playing a human-like strategy.",
"Our agents improved on this baseline bot, both when playing with the bot itself and also with an expert human.",
"We believe that the strong collaborative performance is owing to our combination of a good starting policy and searching with the assumption that partner sticks to that starting policy, thereby penalizing incompatible policy innovations.",
"Our partnership learning approach achieves new state of the art performance when playing as a pair against WBridge5, which is the problem addressed in previous work.",
"Perhaps surprisingly, these agents also maintain compatibility with WBridge5, performing better as a partner than WBridge5 itself does.",
"We believe this compatibility is owing to the strong starting policy and the soft policy updates, which will result in conservative policy exploration rather than finding a radically different local optimum."
],
[
"Broader Impact",
"Agents that cooperate with humans are an important long-term goal of AI research, with the potential for significant societal benefit.",
"Approaches along the lines that we describe here could eventually be used in a wide range of environments to improve on existing cooperative agents without losing human-compatibility, or to learn an initial policy by imitation learning from human data and then improve on it, again without losing human-compatibility.",
"Humans could in turn learn from the improvements that the agent finds.",
"Our agent learns to collaborate only with one particular human policy.",
"The requirement for an existing agent or a large corpus of data on which to train means that our approach may not be readily extensible to conventions used by minority groups where this prior work or large datasets do not exist.",
"On the other hand, the possibility to generate a high-quality human-compatible agent through demonstration and self-play may be more feasible than hand-engineering for such communities.",
"Our imitation learning approach requires a homogeneous dataset, in which the same conventions are used throughout.",
"This means our approach is not directly applicable to situations where a group of people with diverse behaviour interact."
]
] | 2011.14124 |
[
[
"Short-Term Load Forecasting using Bi-directional Sequential Models and\n Feature Engineering for Small Datasets"
],
[
"Abstract Electricity load forecasting enables the grid operators to optimally implement the smart grid's most essential features such as demand response and energy efficiency.",
"Electricity demand profiles can vary drastically from one region to another on diurnal, seasonal and yearly scale.",
"Hence to devise a load forecasting technique that can yield the best estimates on diverse datasets, specially when the training data is limited, is a big challenge.",
"This paper presents a deep learning architecture for short-term load forecasting based on bidirectional sequential models in conjunction with feature engineering that extracts the hand-crafted derived features in order to aid the model for better learning and predictions.",
"In the proposed architecture, named as Deep Derived Feature Fusion (DeepDeFF), the raw input and hand-crafted features are trained at separate levels and then their respective outputs are combined to make the final prediction.",
"The efficacy of the proposed methodology is evaluated on datasets from five countries with completely different patterns.",
"The results demonstrate that the proposed technique is superior to the existing state of the art."
],
[
"Introduction",
"Smart grid, in simple terms, implies monitoring and control of the power system's assets in the generation, transmission, distribution, and utilization, to achieve high efficiency and reliability at low operational costs.",
"Its cardinal feature – demand response, can only be fully realized through accurate forecasting of various variables, the most important of which is the forecasting of electrical load [1].",
"Artificial intelligence is fast becoming cardinal for data analytics and enhanced control of modern power systems.",
"One of its most desirable application in recent times is the load forecasting through machine learning for predicting the trends in energy demand, so that the control decisions can be proactively optimized.",
"Long-term [2], mid-term [3], and short-term [4] are the different types of load forecasting found in the literature based on their duration of prediction from years to minutes.",
"Short-term load forecasting (SLF) is more difficult than the mid and long term forecasting because of the greater variance in the respective energy consumption patterns [5].",
"The advantage of SLF is that it provides better insight into the electricity consumption patterns and a greater degree of freedom for demand-side management.",
"Also, SLF can be aggregated to get mid-term and long-term forecasts.",
"Therefore this paper focuses on SLF.",
"One of the most challenging problems in SLF is posed by small datasets; particularly in the case of individual households which usually exhibit wide variations in energy consumption in short intervals, thus making it harder for the deep learning models to learn the underlying patterns [5].",
"Before the deep learning era, lot of research went into the engineering of hand-crafted features which were required as the inputs to machine learning algorithms.",
"The capability of deep learning to extract implicit features, removed the need of such complicated pre-processing of raw data.",
"However deep learning models require large training data to extract useful features, more so for the datasets with high variances.",
"This paper presents a novel deep learning architecture that combines the use of hand-crafted features with raw data, such that the deep learning model can work well for SLF of small datasets.",
"The results demonstrate significant improvements in the performance, specially for small datasets by using the proposed architecture."
],
[
"Literature Review",
"A significant amount of research has been carried out in recent times to develop SLF as the enabling tool for efficient monitoring and control of power system.",
"Before the deep learning era, hand-crafted features were used to be fed to a machine learning model for making predictions.",
"In [6] feature engineering was done to design a feature vector by performing entropy analysis with a specific tolerance band and auto-correlation function.",
"The designed feature vector was then passed through an artificial neural network (ANN) for prediction.",
"Ferreira and da Silva used a Bayesian based approach to solve the complexity of neural network and variable selection [7].",
"The approach has theoretical ground but relies on various assumptions regarding the network parameters distribution, requires three relevance thresholds and is computationally expensive.",
"Phase-space embedding method was used for the selection of input variable which allowed to include the preference of the past values of prediction quantity in the input vector [8].",
"A neural networks based approach to forecast next 24 hour load on medium and low voltage substations was presented in [9].",
"The use of separate models each for daily average power and for intraday variation in power, improved the accuracy of prediction compared to the model based on time series.",
"Learning the daily routine of the usage of various appliances can help in better forecasting of an household's load profile.",
"It was shown in [10], [11] that using the consumption data of the appliances together with the aggregated data of the whole house as the input to the long short term memory (LSTM) models gave better results than using the whole house readings alone.",
"Recently recurrent neural networks (RNN) have become the popular choice for load forecasting.",
"In [12] machine learning models were used for predicting the energy demand on publicly available RTE dataset [13].",
"The performances of RNN and support vector machine (SVM) models were compared using different input features.",
"The models were evaluated on a test set of 10 days of year 2017.",
"The results demonstrated that RNN performed better, with a MAPE of 3.52%, compared to SVM with a MAPE of 14.00%.",
"A recent study [5] demonstrated how the individual household level load forecasting can be challenging because of different patterns of energy consumption of individual consumers [14].",
"A two layers based LSTM model was proposed and compared with other models based on back-propagation neural network (BPNN), k-nearest neighbour (KNN), extreme learning machine (ELM) and input scheme combined with a hybrid forecasting framework (IS-HF).",
"Individual models for each household were trained and the best average MAPE of 44.06% was achieved through LSTM.",
"Also it was demonstrated that aggregating these individual forecasts resulted into quite the same net MAPE of $\\sim $ 8% that was yielded when the aggregated data of the consumers was trained and tested on a single model.",
"This difference in the MAPE of individual versus aggregated forecast established that the individual SLF is harder compared to aggregated one; however the advantage of individual SLF is that it provides better insight into the trend of each constituent customer and can easily be aggregated together to provide the net trend.",
"Electricity demand is influenced by weather, holiday, time of day etc.",
"Time dependant convolution neural network (TD-CNN) and cycle based long short term memory (C-LSTM) for short- and medium-term load forecasting was presented in [15].",
"Electric load on weekly basis was arranged in image format on which TD-CNN was run.",
"C-LSTM helped to extract time dependencies between sequences.",
"The models performed better than the traditional LSTM model while reducing the training time.",
"Another important application of SLF is in energy trading which is a complex process due to non-periodic variations in energy consumption.",
"Accurate forecasting for hourly spot price is the key to achieve the best trading decision which is vital for investors and retailers in electricity market.",
"A model based on a hybrid approach comprising of ARIMA, multiple linear regression (MLR), and Holt-Winter model was proposed in [16].",
"The hybrid model was tested for Iberian electricity market dataset to forecast hourly spot prices for various numbers of days.",
"A hybrid model based on non-linear regression and SVM was proposed in [17], that was tested on ERCOT data [18].",
"This hybrid model achieved MAPE of 7.30% compared to the individual models with 8.99% and 8.63% MAPE respectively.",
"Improvement of forecasting accuracy using standard LSTM model by feeding it processed features rather than raw data was proposed in [19].",
"The power load sequence was decomposed by complementary ensemble empirical mode decomposition (CEEMD), then the approximate entropy (AE) values of the obtained subsequences were calculated.",
"The subsequences with similar AE values were merged into new sequence to form the inputs of the load forecasting model.",
"This reduced the complexity of the power load sequence and improved the accuracy of load forecasting.",
"The vanilla LSTM network was improved in [20] by cleaning and processing the raw load data using isolated forest algorithm.",
"Electric load forecasting requires training of large number of neurons in hidden layer, which increases the size of the network and slows overall training process.",
"To reduce this overhead, a multi-column radial bias function (MCRF) with error correction algorithm designed to reduce the number of hidden neurons in a network, was proposed in [21].",
"It was shown that MCRF with only 50 neurons in hidden layer took only 10 minutes to train and achieved the MAPE of 4.59% compared to other models with more than 150 neurons that achieved better MAPE of 1.77% but took hours to train.",
"Accuracy of SLF can be improved through careful analysis of the load data to find the effectiveness of selected features.",
"A technique was proposed in [22] for features selection where the bisecting K-means algorithm was used to cluster the load data with high similarity for a forecast date.",
"The ensemble empirical mode decomposition (EEMD) helped to combine components with similar entropy.",
"A bidirectional recurrent neural network (BRNN) model was proposed to forecast the load of the network.",
"The model was verified on two datasets including a dataset from load forecasting competition.",
"The results showed that BRNN model performed better even than the winner of the competition.",
"The literature survey therefore implies that a better load forecasting technique with reduced statistical error is a hot topic of research for modern power systems."
],
[
"Proposed Methodology",
"Recently [15], [5], [10] deep learning solutions, particularly sequential models such as RNN and LSTM models are becoming popular choices for load forecasting.",
"LSTM [23] has become a state of the art tool for time series problems owing to its ability to learn temporal patterns in sequential data.",
"This paper presents a novel architecture named as deep derived feature fusion (DeepDeFF), comprising of a bidirectional sequential model with feature engineering for realizing a more accurate SLF technique."
],
[
"Bidirectional Sequential Models",
"Bidirectional model trains forward and reverse nodes using respectively: input in positive time, i.e.",
"the given input as it is, input in negative time, i.e.",
"a time-reversed copy of the original input.",
"The advantage of bidirectional model compared to conventional ANN models is that it observes the input in both forward and reverse directions to extract more information from the input sequence.",
"This technique of negative time and bidirectional layer was first discussed in [24].",
"This paper implements LSTM, RNN, gated recurrent unit (GRU) as well as their bidirectional counterparts (BLSTM) [25], BRNN and (BGRU) on several datasets for a comprehensive comparison presented in the results.",
"Derived Features The aim of derived features is to enrich the training data with useful features for more accurate predictions.",
"A deep learning model with enough computation time and data may extract such features on its own, but this cannot be guaranteed within the constraints of time and resource.",
"Thus providing these derived features explicitly as inputs can enable the model to learn more from the data and converge quickly, specially for small datasets.",
"Generally the performance of deep learning models improve by increasing the number of relevant input features unless it starts to over fit.",
"The basic features used for generation of derived features and as input for the DeepDeFF model are: Energy load consumption $E$ .",
"Time-stamp of the day $T$ , divided into 30 minutes interval each.",
"The feature is converted into One-hot encoding.",
"Current day of the week $W$ , converted into one-hot encoding.",
"Holidays represented by a binary label $H$ .",
"At the moment only weekends are marked as holidays, but in future work this can be expanded and synced with other public and national holidays.",
"Derived features are calculated for each record in the input sequence (1, $K$ , $f$ ), where $K$ represents the number of past records used for creating the input sequence and $f$ represents the basic features.",
"Following are the derived features that are calculated and used as input to the DeepDeFF model.",
"Average load consumption of $K$ time-steps.",
"Standard deviation of load consumption of $K$ time-steps.",
"Average load consumption of the time-stamp $t$ that is to be predicted, for past $K$ days.",
"Standard deviation of load consumption of the time-stamp $t$ that is to be predicted, for past $K$ days.",
"Proposed Architecture This paper proposes a two-layer bidirectional sequential model architecture DeepDeFF, which inputs the raw and derived input features into separate layers to extract learned features.",
"The idea behind using separate input layers for basic and derived sequences is to allow the sequential layers to learn from the two input sequences independently.",
"The goal is to exploit the relevance of basic and derived sequences with the predictions individually.",
"The learned representation from the individual sequential layers is then merged and fed to a dense layer followed by a linear activation output layer to make the final prediction of the load at the next time interval.",
"Fig.",
"REF shows the schematics of the DeepDeFF architecture.",
"The hyper parameter settings consists of: 20 nodes sequential layer a dropout of 0.2 Adam optimizer MAPE as loss function Learning rate: 0.01 Figure: System Architecture: The input data is first pre-processed to achieve the derived input features.",
"The raw input features TT, WW, HH are converted to one-hot encoding.",
"The raw and derived input features are then fed to two individual bi-directional sequential layers.",
"The information extracted from these separate layers is then merged and used as input to a dense layer followed by a final feed forward layer with LinearLinear activation function.",
"The Datasets The proposed methodology for SLF has been evaluated on five energy load datasets from different sources.",
"This section provides the salient parameters of the dataset and presents the pre-processing technique adopted for each.",
"Smart Grid Smart City (SGSC) dataset SGSC project was initiated by the Australian Government in year 2010 [14].",
"It gathered smart-meter data from around 78,000 customers for a period of 4-years.",
"In [5], individual models for each customer was proposed.",
"However since it is not feasible to train individual models for $\\sim $ 78,000 customers, therefore 69 customers having “hot water system” were selected.",
"The same subset is extracted here to evaluate the DeepDeFF architecture.",
"The Almanac of Minutely Power dataset (AMPds) AMPds [26] contains electricity, water and natural gas measurements of a single Canadian household with 19 appliances, recorded for 1 year with 1 minute resolution, which is down-sampled to 30 minutes resolution [10].",
"The variables for raw features used here are the same as for SGSC except that $E$ here is assigned to the Ampere reading.",
"Réseau de Transport d'Électricité (RTE) France dataset RTE dataset [13] is also used here to evaluate the proposed technique.",
"The dataset used spans from year 2013 to 2016 with the sampling interval of 30 minutes.",
"The raw inputs are programmed with same variables as for SGSC above.",
"The Electric Reliability Council of Texas (ERCOT) dataset ERCOT dataset [18] provides real time and historical statistics surrounding independent system operator (ISO) operations of the Texas region for a period of $\\sim $ 5 years recorded every 1 hour.",
"The raw features variables used here are the same as for SGSC except that the time $T$ here ranges 1-24 since the resolution is 1 hour.",
"Pakistan Residential Electricity Consumption (PRECON) PRECON dataset [27] records the electricity consumption patterns in a developing country for 42 households of varying financial status, daily routine and load profile.",
"The data is collected with 1 minute interval from 01-06-2018 to 31-09-2019.",
"The amount of data varies for each household due to different number and types of appliances that are selected for monitoring.",
"This dataset also captures the problem of power outages rampant in developing countries.",
"This is evident from several long 0KW data intervals.",
"For raw features, same variables as in SGSC are used here except that $E$ here refers to the KW usage.",
"Experiments & Results The proposed framework for SLF is achieved through an evolutionary process after numerous rigorous experiments on all five datasets.",
"This section discusses these experiments in sufficient detail and infers the results obtained.",
"The results from the DeepDeFF architecture are compared with the results of simple two layer sequential models trained on basic features and MAE as loss as proposed in [5].",
"SGSC Dataset Train & Test Setting The same settings provided in [5] are used to extract the subset of SGSC data for fair comparison on the same test set.",
"The data spanning the whole winter season of New South Wales Australia is subdivided into a split ratio of 0.7/0.2/0.1 as: Training set (01-Jun-2013 to 05-Aug-2013) Validation set (06-Aug-2013 to 22-Aug-2013) Test set (23-Aug-2013 to 31-Aug-2013) The first set is to train the DeepDeFF model, validation set is used to select the best model weights based on performance on validation set, while the test set is for the evaluation of the DeepDeFF model.",
"The data is spaced between 30 minutes interval; so for 69 customers the 9 days of evaluation implies the forecasting of 29,808 time points.",
"Figure: A comparison of train and test load data for two types of customers from SGSC dataset is shown, (a) shows a customer whose train and test data patterns have high correlation, (b) shows a customer who has very different load patterns for train and test data.Fig.",
"REF provides some insight into the diversity of customers by showing the similarity between their train and test data.",
"Test data is plotted over training data with matching numeric dates.",
"Fig.",
"REF shows the data of a customer with similarity between train and test data patterns, whereas Fig.",
"REF shows no similarity for another customer.",
"This indeed effects the results of DeepDeFF architecture which is reflected in their respective MAPE of 26.04% and 50.78% using BLSTM layer; thus the DeepDeFF model has been able to learn the underlying patterns and temporal relations for Fig.",
"REF but not as good for Fig.",
"REF .",
"Results Table-REF shows the comparison of results from rigorous experiments that are performed on SGSC dataset using the proposed DeepDeFF method in contrast with the implementation of the model proposed in [5].",
"The addition of derived features in the proposed architecture along with MAPE as loss function, outperforms the state of the art on the SGSC dataset as evident from the average MAPE computed in Table-REF .",
"Table: Results achieved on SGSC datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers for the two customers shown in Fig.Fig.",
"REF confirms that the DeepDeFF model indeed predicts the actual load very well for customer 8804804.",
"However, Fig.",
"REF shows that the model under performs for customer 8655993 due to uncorrelated train and test data, owing to disjoint customer behavior during training and testing days.",
"AMPds Train & Test Setting The AMPds data is converted from 1 minute resolution to 30 minutes, yielding 17,483 data points [10].",
"The data is subdivided with a split ratio of 0.7/0.2/0.1 into: Training set (01-Apr-2012 to 17-Dec-2012) Validation set (18-Dec-2012 to 23-Feb-2013) Test set (24-Feb-2013 to 01-Apr-2013) Figure: A comparison of train and test data, AMPdsFig.",
"REF shows that the train and test data have some common pattern and that there are no abrupt changes like in Fig.REF .",
"Even though, the training and testing data is not available for same dates of different years, there is a general trend that is being followed in the test data.",
"Results Table-REF shows a comparison of the results produced by the simple two layer sequential model and the DeepDeFF architecture with derived features.",
"The proposed architecture beats the benchmark of 26.23% achieved in [10] for 6 time-steps.",
"Table: Results achieved on AMP datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers.Fig.",
"REF shows that the DeepDeFF architecture performs well in predicting the general load and suffers in case of outliers.",
"This is because the model was able to learn the underlying general pattern from the training data, and gave it more importance than to outliers.",
"This problem occurred because the training data was not enough and does not cover all the months; so the test data is of a month that was never seen during training.",
"RTE Dataset Train & Test Setting RTE data is subdivided into three subsets with a split ratio of 0.7/0.2/0.1 as: Training set (01-Jan-2013 to 18-Nov-2015) Validation set (19-Nov-2015 to 07-Aug-2016) Test set (08-Aug-2016 to 31-Dec-2016) Figure: Training and testing data comparison, RTE datasetFig.",
"REF show the subsets of training and testing data for the dates mentioned in the figures' legends.",
"Such close resemblance in the test and train data helps the model to make accurate predictions as evident from the results.",
"Results It is observed from the results for SGSC and AMPds datasets that the experiments with 2 time-steps mostly yield the best results.",
"Henceforth 2 time steps is used for the experiments on other datasets.",
"Table-REF shows the results for RTE dataset.",
"The proposed model with GRU and derived features performed best with average MAPE of 0.81%.",
"Fig.",
"REF shows the prediction results against the actual system load which further confirms the excellent performance of the DeepDeFF architecture.",
"Table: Results achieved on RTE datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers.",
"ERCOT Dataset Train & Test Setting ERCOT data is subdivided into three subsets with a split ratio of 0.5/0.1/0.4 as: Training set (01-Jan-2011 to 26-May-2013) Validation set (27-May-2013 to 31-Dec-2013) Test set (01-Jan-2014 to 31-Dec-2015) Similar to RTE, ERCOT is also the accumulated load consumption data of Texas.",
"The train and test data for ERCOT also has close resemblance similar to Fig.",
"REF Results Table-IV shows the results for ERCOT dataset where the DeepDeFF model with BGRU performed best with average MAPE of 0.91%.",
"Figure-REF shows the results that establishes the effectiveness of the DeepDeFF architecture.",
"Table: Results achieved on ERCOT datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers.",
"PRECON Train & Test Setting Owing to the peculiar nature of the PRECON dataset, it is pre-processed in two steps in this research.",
"First, the data is converted from 1-minute interval to 30-minute intervals by taking the average over the 30 consecutive load readings.",
"The second step is to take care of close to zero values in the data that are mostly due to power outages.",
"Otherwise these values cause divide-by-zero problem when using MAPE function for evaluation, resulting into unrealistically high MAPE and adversely effecting the performance of the machine learning algorithm.",
"This is countered simply by adding a small offset of 0.1 KW to all the readings.",
"The offset is small enough to makes no significant change in the nominal values of the load and takes effect only for the near zero data.",
"This simplest pre-processing has shown remarkable impact on the performance of the DeepDeFF algorithm as evident from the results.",
"The data splitting is done in a unique way here due to the reason that it spanned over a period of only one year with no repeated data for any month.",
"So instead of using an overall split of data, as done in previous datasets, a month-wise split is proposed.",
"Here the training, validation, and testing data is taken from days 1 – 21, 22 – 26, 27 – 30/31 respectively for each month.",
"This corresponds roughly to an overall split of 0.7/0.2/0.1.",
"Results Table V shows the comparison of results obtained for PRECON dataset.",
"DeepDeFF models have consistently outperformed the basic models on all the houses, achieving on average 8.9% lower MAPE than basic models.",
"The value of the MAPE achieved by DeepDeFF models ranged from 7.67% on House 3 to 37.61% on House 29.",
"The graphs of predicted versus actual load of these two houses are shown in Fig.",
"REF and Fig.",
"REF respectively.",
"Table: Results achieved on PRECON datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers.The above mentioned results on five public datasets infer that SLF for individual households: SGSC, AMPDs and PRECON is more difficult than aggregated load forecasting of a country or state wide dataset because of high variances in load consumption patterns of the former ones.",
"However the proposed DeepDeFF architecture has been able to forecast better than the previously published techniques.",
"Conclusion Load forecasting is of critical importance to optimally schedule and reliably manage the operations of power systems.",
"This manuscript presented a deep learning architecture based on sequential layers, and a pre-processing method for introducing hand-crafted features into the end-to-end learning pipeline of the deep learning model, for short-term load forecasting.",
"It is demonstrated with rigorous experimentation that the inclusion of hand-crafted features has improved the learning and predictions of the model, specially for smaller datasets.",
"The proposed DeepDeFF architecture has been comprehensively tested on five different datasets – two country/state wide datasets and three household datasets.",
"The results achieved from the proposed methodology beat the current benchmark of these datasets for SLF."
],
[
"The Datasets",
"The proposed methodology for SLF has been evaluated on five energy load datasets from different sources.",
"This section provides the salient parameters of the dataset and presents the pre-processing technique adopted for each."
],
[
"Smart Grid Smart City (SGSC) dataset",
"SGSC project was initiated by the Australian Government in year 2010 [14].",
"It gathered smart-meter data from around 78,000 customers for a period of 4-years.",
"In [5], individual models for each customer was proposed.",
"However since it is not feasible to train individual models for $\\sim $ 78,000 customers, therefore 69 customers having “hot water system” were selected.",
"The same subset is extracted here to evaluate the DeepDeFF architecture."
],
[
"The Almanac of Minutely Power dataset (AMPds)",
"AMPds [26] contains electricity, water and natural gas measurements of a single Canadian household with 19 appliances, recorded for 1 year with 1 minute resolution, which is down-sampled to 30 minutes resolution [10].",
"The variables for raw features used here are the same as for SGSC except that $E$ here is assigned to the Ampere reading."
],
[
"Réseau de Transport d'Électricité (RTE) France dataset",
"RTE dataset [13] is also used here to evaluate the proposed technique.",
"The dataset used spans from year 2013 to 2016 with the sampling interval of 30 minutes.",
"The raw inputs are programmed with same variables as for SGSC above."
],
[
"The Electric Reliability Council of Texas (ERCOT) dataset",
"ERCOT dataset [18] provides real time and historical statistics surrounding independent system operator (ISO) operations of the Texas region for a period of $\\sim $ 5 years recorded every 1 hour.",
"The raw features variables used here are the same as for SGSC except that the time $T$ here ranges 1-24 since the resolution is 1 hour."
],
[
"Pakistan Residential Electricity Consumption (PRECON)",
"PRECON dataset [27] records the electricity consumption patterns in a developing country for 42 households of varying financial status, daily routine and load profile.",
"The data is collected with 1 minute interval from 01-06-2018 to 31-09-2019.",
"The amount of data varies for each household due to different number and types of appliances that are selected for monitoring.",
"This dataset also captures the problem of power outages rampant in developing countries.",
"This is evident from several long 0KW data intervals.",
"For raw features, same variables as in SGSC are used here except that $E$ here refers to the KW usage.",
"The proposed framework for SLF is achieved through an evolutionary process after numerous rigorous experiments on all five datasets.",
"This section discusses these experiments in sufficient detail and infers the results obtained.",
"The results from the DeepDeFF architecture are compared with the results of simple two layer sequential models trained on basic features and MAE as loss as proposed in [5]."
],
[
"Train & Test Setting",
"The same settings provided in [5] are used to extract the subset of SGSC data for fair comparison on the same test set.",
"The data spanning the whole winter season of New South Wales Australia is subdivided into a split ratio of 0.7/0.2/0.1 as: Training set (01-Jun-2013 to 05-Aug-2013) Validation set (06-Aug-2013 to 22-Aug-2013) Test set (23-Aug-2013 to 31-Aug-2013) The first set is to train the DeepDeFF model, validation set is used to select the best model weights based on performance on validation set, while the test set is for the evaluation of the DeepDeFF model.",
"The data is spaced between 30 minutes interval; so for 69 customers the 9 days of evaluation implies the forecasting of 29,808 time points.",
"Figure: A comparison of train and test load data for two types of customers from SGSC dataset is shown, (a) shows a customer whose train and test data patterns have high correlation, (b) shows a customer who has very different load patterns for train and test data.Fig.",
"REF provides some insight into the diversity of customers by showing the similarity between their train and test data.",
"Test data is plotted over training data with matching numeric dates.",
"Fig.",
"REF shows the data of a customer with similarity between train and test data patterns, whereas Fig.",
"REF shows no similarity for another customer.",
"This indeed effects the results of DeepDeFF architecture which is reflected in their respective MAPE of 26.04% and 50.78% using BLSTM layer; thus the DeepDeFF model has been able to learn the underlying patterns and temporal relations for Fig.",
"REF but not as good for Fig.",
"REF ."
],
[
"Results",
"Table-REF shows the comparison of results from rigorous experiments that are performed on SGSC dataset using the proposed DeepDeFF method in contrast with the implementation of the model proposed in [5].",
"The addition of derived features in the proposed architecture along with MAPE as loss function, outperforms the state of the art on the SGSC dataset as evident from the average MAPE computed in Table-REF .",
"Table: Results achieved on SGSC datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers for the two customers shown in Fig.Fig.",
"REF confirms that the DeepDeFF model indeed predicts the actual load very well for customer 8804804.",
"However, Fig.",
"REF shows that the model under performs for customer 8655993 due to uncorrelated train and test data, owing to disjoint customer behavior during training and testing days."
],
[
"Train & Test Setting",
"The AMPds data is converted from 1 minute resolution to 30 minutes, yielding 17,483 data points [10].",
"The data is subdivided with a split ratio of 0.7/0.2/0.1 into: Training set (01-Apr-2012 to 17-Dec-2012) Validation set (18-Dec-2012 to 23-Feb-2013) Test set (24-Feb-2013 to 01-Apr-2013) Figure: A comparison of train and test data, AMPdsFig.",
"REF shows that the train and test data have some common pattern and that there are no abrupt changes like in Fig.REF .",
"Even though, the training and testing data is not available for same dates of different years, there is a general trend that is being followed in the test data."
],
[
"Results",
"Table-REF shows a comparison of the results produced by the simple two layer sequential model and the DeepDeFF architecture with derived features.",
"The proposed architecture beats the benchmark of 26.23% achieved in [10] for 6 time-steps.",
"Table: Results achieved on AMP datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers.Fig.",
"REF shows that the DeepDeFF architecture performs well in predicting the general load and suffers in case of outliers.",
"This is because the model was able to learn the underlying general pattern from the training data, and gave it more importance than to outliers.",
"This problem occurred because the training data was not enough and does not cover all the months; so the test data is of a month that was never seen during training."
],
[
"Train & Test Setting",
"RTE data is subdivided into three subsets with a split ratio of 0.7/0.2/0.1 as: Training set (01-Jan-2013 to 18-Nov-2015) Validation set (19-Nov-2015 to 07-Aug-2016) Test set (08-Aug-2016 to 31-Dec-2016) Figure: Training and testing data comparison, RTE datasetFig.",
"REF show the subsets of training and testing data for the dates mentioned in the figures' legends.",
"Such close resemblance in the test and train data helps the model to make accurate predictions as evident from the results."
],
[
"Results",
"It is observed from the results for SGSC and AMPds datasets that the experiments with 2 time-steps mostly yield the best results.",
"Henceforth 2 time steps is used for the experiments on other datasets.",
"Table-REF shows the results for RTE dataset.",
"The proposed model with GRU and derived features performed best with average MAPE of 0.81%.",
"Fig.",
"REF shows the prediction results against the actual system load which further confirms the excellent performance of the DeepDeFF architecture.",
"Table: Results achieved on RTE datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers."
],
[
"Train & Test Setting",
"ERCOT data is subdivided into three subsets with a split ratio of 0.5/0.1/0.4 as: Training set (01-Jan-2011 to 26-May-2013) Validation set (27-May-2013 to 31-Dec-2013) Test set (01-Jan-2014 to 31-Dec-2015) Similar to RTE, ERCOT is also the accumulated load consumption data of Texas.",
"The train and test data for ERCOT also has close resemblance similar to Fig.",
"REF"
],
[
"Results",
"Table-IV shows the results for ERCOT dataset where the DeepDeFF model with BGRU performed best with average MAPE of 0.91%.",
"Figure-REF shows the results that establishes the effectiveness of the DeepDeFF architecture.",
"Table: Results achieved on ERCOT datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers."
],
[
"Train & Test Setting",
"Owing to the peculiar nature of the PRECON dataset, it is pre-processed in two steps in this research.",
"First, the data is converted from 1-minute interval to 30-minute intervals by taking the average over the 30 consecutive load readings.",
"The second step is to take care of close to zero values in the data that are mostly due to power outages.",
"Otherwise these values cause divide-by-zero problem when using MAPE function for evaluation, resulting into unrealistically high MAPE and adversely effecting the performance of the machine learning algorithm.",
"This is countered simply by adding a small offset of 0.1 KW to all the readings.",
"The offset is small enough to makes no significant change in the nominal values of the load and takes effect only for the near zero data.",
"This simplest pre-processing has shown remarkable impact on the performance of the DeepDeFF algorithm as evident from the results.",
"The data splitting is done in a unique way here due to the reason that it spanned over a period of only one year with no repeated data for any month.",
"So instead of using an overall split of data, as done in previous datasets, a month-wise split is proposed.",
"Here the training, validation, and testing data is taken from days 1 – 21, 22 – 26, 27 – 30/31 respectively for each month.",
"This corresponds roughly to an overall split of 0.7/0.2/0.1."
],
[
"Results",
"Table V shows the comparison of results obtained for PRECON dataset.",
"DeepDeFF models have consistently outperformed the basic models on all the houses, achieving on average 8.9% lower MAPE than basic models.",
"The value of the MAPE achieved by DeepDeFF models ranged from 7.67% on House 3 to 37.61% on House 29.",
"The graphs of predicted versus actual load of these two houses are shown in Fig.",
"REF and Fig.",
"REF respectively.",
"Table: Results achieved on PRECON datasetFigure: Actual load versus load predicted by DeepDeFF architecture utilizing different sequential layers.The above mentioned results on five public datasets infer that SLF for individual households: SGSC, AMPDs and PRECON is more difficult than aggregated load forecasting of a country or state wide dataset because of high variances in load consumption patterns of the former ones.",
"However the proposed DeepDeFF architecture has been able to forecast better than the previously published techniques."
],
[
"Conclusion",
"Load forecasting is of critical importance to optimally schedule and reliably manage the operations of power systems.",
"This manuscript presented a deep learning architecture based on sequential layers, and a pre-processing method for introducing hand-crafted features into the end-to-end learning pipeline of the deep learning model, for short-term load forecasting.",
"It is demonstrated with rigorous experimentation that the inclusion of hand-crafted features has improved the learning and predictions of the model, specially for smaller datasets.",
"The proposed DeepDeFF architecture has been comprehensively tested on five different datasets – two country/state wide datasets and three household datasets.",
"The results achieved from the proposed methodology beat the current benchmark of these datasets for SLF."
]
] | 2011.14137 |
[
[
"Cyberbiosecurity: DNA Injection Attack in Synthetic Biology"
],
[
"Abstract Today arbitrary synthetic DNA can be ordered online and delivered within several days.",
"In order to regulate both intentional and unintentional generation of dangerous substances, most synthetic gene providers screen DNA orders.",
"A weakness in the Screening Framework Guidance for Providers of Synthetic Double-Stranded DNA allows screening protocols based on this guidance to be circumvented using a generic obfuscation procedure inspired by early malware obfuscation techniques.",
"Furthermore, accessibility and automation of the synthetic gene engineering workflow, combined with insufficient cybersecurity controls, allow malware to interfere with biological processes within the victim's lab, closing the loop with the possibility of an exploit written into a DNA molecule presented by Ney et al.",
"in USENIX Security'17.",
"Here we present an end-to-end cyberbiological attack, in which unwitting biologists may be tricked into generating dangerous substances within their labs.",
"Consequently, despite common biosecurity assumptions, the attacker does not need to have physical contact with the generated substance.",
"The most challenging part of the attack, decoding of the obfuscated DNA, is executed within living cells while using primitive biological operations commonly employed by biologists during in-vivo gene editing.",
"This attack scenario underlines the need to harden the synthetic DNA supply chain with protections against cyberbiological threats.",
"To address these threats we propose an improved screening protocol that takes into account in-vivo gene editing."
],
[
"Introduction",
"Synthetic biology is an emerging bioengineering technology that plays a significant role in personalized medicine, pharmaceutical manufacturing, etc.",
"[45], [41].",
"Rapid development of biological systems is supported by large online libraries of genes [34], [60], [5], as well as integrated development environments (IDEs) and gene compilers for efficient gene coding [18].",
"Currently, the software stack used to develop synthetic genes is loosely secured (Section REF ), potentially allowing the injection of rogue genetic information into biological systems by a malware.",
"DNA synthesis companies, which produce and ship the DNA sequences, are an important element of the growing synthetic biology market.",
"Synthetic DNA is available in multiple ready-to-use forms, such as a circular DNA molecule called a plasmid (Section REF ).",
"A synthesized plasmid can be inserted into an organism by following a simple biological protocol, after which it can start producing proteins [44].",
"Many bioengineering tools are now easily accessible by do-it-yourself (DIY) biology enthusiasts.",
"The products of biological systems can be extremely dangerous substances, such as toxins or synthetic viruses [13], [36].",
"DNA sequences producing dangerous viruses may be as small as 1.7kbp [26]; natural and synthetic toxins may be constructed from less than 100bp [42].",
"The dual use of synthetic biology as a powerful technology for the benefit of mankind, and as a potential weapon, is a long-standing issue [50].",
"The dangers of synthetic biology are manifold, and they require rigorous security controls [24], [59].",
"One such control is the U.S. Department of Health and Human Services' (HSS) Screening Framework Guidance for Providers of Synthetic Double-Stranded DNA (HHS guidelines) [49].",
"We discuss state-of-the-art DNA screening in Section REF .",
"Biosecurity researchers agree that constant improvement of DNA screening methodologies is required to prevent both bioterrorists, and careless enthusiasts, from generating dangerous substances in their labs [50], [37].",
"Legislation is keeping pace.",
"Recently California obliged all customers to order synthetic genes from companies that perform gene screening [58].",
"Understanding the impact of cyber threats on biosecurity is extremely important.",
"Cyber attacks are considered a potential threat to the security and privacy of genomic data [27], [12] and the analysis of genetic material [22].",
"A recent biodefense report [50] mentions cyber threats with respect to identifying potential targets and the development of bioweapons.",
"Diggans and Leproust [20] highlight the value of cybersecurity methods applied in the domain of biosecurity.",
"Ney at el.",
"[47] have even suggested an attack where an exploit written in a DNA molecule may subvert a vulnerable sequencing machine.",
"However, the potential of delivering a harmful biological agent through cyberspace has not yet been considered.",
"It is currently believed that a criminal needs to have physical contact with the substance to produce and deliver it.",
"However, malware could easily replace a short sub-string of the DNA on a bioengineer's computer with a toxin producing sequence.",
"Order screening by synthetic DNA providers is the most effective line of defense against such attacks.",
"Unfortunately, the screening guidelines have not been adapted to reflect recent developments in synthetic biology and cyberwarfare.",
"In this paper we discuss two main attack vectors: (1) A traditional biosecurity vector where an adversary orders a dangerous synthetic DNA: We show that a weakness in the HHS guidelines can be exploited by the attacker to avoid detection by obfuscating the malicious DNA (Section REF ).",
"The proposed Gene Edit Distance (Section REF ) can efficiently detect sequences that can be decoded into malicious DNA within living cells.",
"A benchmark dataset of obfuscated DNA sequences demonstrates the superiority of Gene Edit Distance over a rigorous implementation of the HHS guidelines (Section REF ).",
"(2) A cyberbiological attack vector where malware on the biologist's computer interferes with biological processes: We discuss the attack surface of a synthetic biology pipeline (Section REF ) and highlight the overlooked impacts malware may have on biological processes (Section REF ).",
"We demonstrate an end-to-end scenario where a malware within a biological lab tricks the biologist victim into producing a substance of the attacker's choice.",
"This scenario closes the loop with a potential attack propagating from DNA to cyberspace [47] and stresses the importance of designing adversary-resilient biological protocols.",
"Simple mitigation steps applied by DNA providers (Section REF ) may significantly reduce the threat of a cyber attack propagating to the biological space."
],
[
"Background and related work",
" This section provides the most important information required in subsequent sections, where we describe potential cyberbiological threats and their mitigation.",
"Readers familiar with the basics of protein synthesis and the CRISPR-Cas system are encouraged to skip Section REF .",
"Important terms used later in the paper are highlighted in bold face."
],
[
"An introduction to DNA editing for cybersecurity experts",
"Genetic information in living cells is encoded in sequences of nucleotides called DNA.",
"Nucleotides are commonly denoted by four letters, C,G,A, and T, corresponding to four nucleobases; cytosine, guanine, adenine, and thymine, respectively.",
"In a double-stranded DNA (dsDNA) molecule, nucleobases on the opposite strands are bound together, C with G and A with T. Thus, dsDNA is sometimes seen as a sequence of base pairs (bp).",
"The fees charged by gene synthesis companies for synthetic DNA orders are typically based on the number of base pairs.",
"The orders are commonly delivered in the form of cyclic dsDNA molecules called plasmids which are very stable and can be replicated within a living cell.",
"The price for a plasmid can be as low as five cents per base pair.",
"Once provided with a plasmid, one can employ a sequencing procedure to get the string representation of the DNA molecule.",
"Companies that produce synthetic DNA usually also provide sequencing services.",
"A complement of a DNA sequence is formed by replacing every occurrence character with its pair ($C\\leftrightarrow G, A\\leftrightarrow T$ ).",
"A reverse complement of a DNA sequence, an operation commonly used in gene design, is formed by reversing the complement of a DNA sequence.",
"For example, the complement of GGCA is CCGT and its reverse complement is TGCC.",
"Along with other cell functions, DNA encodes proteins.",
"First, an RNA sequence is generated from a DNA region surrounded by special sequences called promoters and terminators.",
"An RNA molecule contains the same information as the respective DNA, but it is short-lived.",
"In computer terms it can be compared to volatile memory, whereas DNA would be considered persistent storage.",
"RNA containing special sequences called ribosome binding sites can be transformed by a ribosome into a sequence of amino acids.",
"Every three nucleotides form one amino acid, but different triplets (a codon) may form the same amino acid.",
"Translating amino acids back to triplets of nucleotides is called reverse translation.",
"The choice of which triplets are optimal depends on the organism in which the DNA is expressed.",
"There are 20 amino acids.",
"Short sequences of amino acids are called peptides, while long sequences are called proteins.",
"Both perform various functions within a living cell.",
"The clustered regularly interspaced short palindromic repeats (CRISPR) complex is a part of the bacterial immune system that was adapted by bioengineers to perform precise DNA editing in live biological systems.",
"The most common DNA editing system consists of a Cas9 protein and a guide RNA sequence (gRNA).",
"In this article we will use the term CRISPR to refer to the DNA system consisting of Cas9 and gRNA.",
"The Cas9 protein performs a cut in a dsDNA molecule at specific locations called protospacer adjacent motifs (PAMs).",
"The gRNA contains a short replica of the region following the PAM that needs to be cut by Cas9.",
"In computer terms, gRNAs can be regarded as pointers in an associative memory.",
"For gRNA creation, the DNA should contain a promoter, a copy of the gRNA target site, and a terminator; collectively these are referred to as a gRNA scaffold.",
"A dsDNA that was cut by a CRISPR can repair itself.",
"Such a repair process is error prone and can produce mutations at the cut point.",
"If such a mutation results in production of a different amino acid during protein formation, the protein may become non-functional (a.k.a.",
"a gene knockout).",
"Precise repairs of the cut DNA can be performed using a process known as homology directed repair (HDR).",
"To activate HDR, the cell should contain a DNA sequence that repeats the sequence of nucleotides to the left and right of the cut point (left and right arms of the HDR template respectively) and a small number of nucleotides that can be inserted between them at the cut point.",
"HDR can also correct a few small mutations close to the cut point.",
"Using CRISPR and HDR it is possible to remove and replace long DNA fragments, a process known as knock-in [57], [51].",
"The success rate of knock-in may vary significantly, from 2-3% up to almost 80% [28], [40]."
],
[
"Sequence alignment",
"Next we discuss sequence alignment, which plays a significant role in bioengineering and in biosecurity.",
"The Basic Local Alignment Search Tool () [3] is the first of a long series of algorithms developed for aligning sequences of nucleotides or amino acids.",
"algorithms are optimized both for speed and for searching large databases of sequences.",
"Let $q$ denote a query sequence and $t$ denote a subject sequence (also called a target sequence).",
"Let $q[i]$ denote the $i$ 'th character in $q$ .",
"operates by matching n-grams—short sequences of letters (words)—and extending these matches to form local alignments between the sequences.",
"Provided a query sequence, returns a set of target sequences similar to the query sequence and a set of alignments (also called ranges) for each target sequence.",
"Let $\\mathcal {A}_{q,t}$ be a set of such alignments between $q$ and $t$ found by .",
"Every alignment in $\\alpha \\in \\mathcal {A}_{q,t}$ maps a range of character positions (a substring) in $q$ to a range of character positions (a substring) in $t$ , such that any two successfully aligned character positions $i>j$ , $\\alpha (i)>\\alpha (j)$ .",
"Although $\\alpha $ is not a function, we use the terms domain ($dom(\\alpha )$ ) and image ($img(\\alpha )=a(dom(\\alpha ))$ ) to denote the respective substrings.",
"Let $\\alpha ^{-1}$ denote the inverse alignment, such that $dom(\\alpha ^{-1})=img(\\alpha )$ and $img(\\alpha ^{-1})=dom(\\alpha )$ .",
"The score of an alignment is computed based on the number of matched characters ($M=|\\lbrace i:q[i]=t[\\alpha (i)]\\rbrace |$ ); the number of mismatched characters ($MM=|\\lbrace i:q[i]\\ne t[\\alpha (i)]\\rbrace |$ ); the number of gaps opened in both the query and the target sequences $G=|\\lbrace i:\\alpha (i-1)\\ne \\bot \\wedge \\alpha (i)=\\bot \\rbrace |+|\\lbrace i:\\alpha ^{-1}(i-1)\\ne \\bot \\wedge \\alpha ^{-1}(i)=\\bot \\rbrace |$ ; and the total extent of the gaps ($GX=|\\lbrace i:\\alpha (i)=\\bot \\rbrace |+|\\lbrace i:\\alpha ^{-1}(i)=\\bot \\rbrace |$ ), where $\\bot $ means that the argument character is not aligned.",
"There is a reward ($rm$ ) for every matching character and penalties for mismatching characters ($pmm$ ), gap opening ($pgo$ ), and gap extension ($pgx$ ).",
"The reward and penalties are configurable.",
"The score of an alignment is: $Score = rm\\cdot M - pmm\\cdot MM - pgo\\cdot G - pgx\\cdot GX$ The fraction of $q$ characters successfully aligned is called query coverage: $ QC(\\alpha ) = \\frac{MM}{\\left|q\\right|}$ .",
"The percent identity is computed from the sizes of the domain, the image, and the number of successfully mapped characters: $ PI(\\alpha ) = 2\\cdot MM/(\\left|dom(\\alpha )\\right|+\\left|img(\\alpha )\\right|) $ Example 1 Consider the following alignment: $\\alpha =\\lbrace (1,1), (2,2), (3,\\bot ), (4,3), (\\bot ,4), (\\bot ,5), (5,6), (6,7)\\rbrace $ .",
"$\\begin{matrix}dom(\\alpha )& \\texttt {1234~~56}\\\\q= & \\texttt {TAGT—CA}\\\\\\alpha = & \\texttt {||~~~~||}\\\\t= &\\texttt {TA-CGGCA} \\\\img(\\alpha ) & \\texttt {12~34567}\\end{matrix}$ The first two characters and the last two characters are identical, the fourth is a mismatch, and there are three gaps.",
"The query coverage of $\\alpha $ is $QC(\\alpha )=4/6$ , the percent identity is $PI(\\alpha )=\\frac{2\\cdot 4}{6+7}$ , and the fraction of gaps is $Gaps(\\alpha )=3/8$ ."
],
[
"Related work on DNA screening",
"Some DNA sequences may encode extremely dangerous products, such as toxic peptides.",
"The Screening Framework Guidance for Providers of Synthetic Double-Stranded DNA (HHS guidelines), published by the United States Department of Health and Human Services, suggests methods to minimize the risk of unauthorized distribution of select agents [49].",
"Highly related to the HHS guidelines are the Harmonized Screening Protocol v2.0 (HSP), employed by the International Gene Synthesis Consortium (IGSC) [15], and the International Association Synthetic Biology (IASB) Code of Conduct for Best Practices in Gene Synthesis [29].",
"The HHS guidelines, IASB Code of Conduct, and HSP outline standards and practices aimed at preventing the misuse of synthetic genes.",
"They define procedures for customer screening and synthetic gene order screening for the presence of possible toxins, pathogens, and other biological agents that pose a significant threat to public health and safety, which are collectively referred to as sequences of concern (SoC).",
"US regulation also defines items on the Commerce Control List as sequences of concern.",
"While the HHS guidelines recommend screening specifically for the presence of sequences unique to sequences of concern, the HSP recommends identifying sequences derived from or encoding a sequence of concern.",
"It is generally advised to use a sequence alignment tool, such as BLAST (the Basic Local Alignment Search Tool) [3], to compare gene orders with known sequences in the GenBank database [5].",
"HHS guidelines recommend the Best Match approach to determine the legitimacy of an order based on the classification of the most similar sequence in the database.",
"Specifically, every fragment of 200bp in the order is searched within the database using sequence alignment.",
"If the Best Match of any fragment is a sequence of concern, the order is deemed a hit, and it is forwarded for further investigation.",
"False hit rate is a major issue during screening.",
"GenoTHREAT [1] software implements the HHS guidelines.",
"The query sequence is partitioned into 200bp fragments.",
"We evaluate the disclosed attack and defence against GenoTHREATand our own rigorous implementation of the HHS Guidelines based on .",
"Another software relying on sequence alignment for order screening is BlackWatch [32].",
"Early versions of both GenoTHREAT and BlackWatch were released as open source but were retracted later due to biosecurity concerns.",
"Note that general security practice advises against security by obscurity, striving instead for rapid identification and elimination of security weaknesses by the public.",
"A recently published NNTox [30] screening tool employs Gene Ontology (GO) Terms and machine learning for predicting the toxicity of a sequence.",
"Unfortunately, the published NNTox models failed to identify conotoxins and other short toxic peptides during preliminary tests and thus was not included in the evaluation.",
"In addition to biosecurity tools, there are multiple systems such as InterProScan[31], SeqScreen[2], and various machine learning approaches [39], [62], [21] that can be used to predict the function of DNA and protein sequences.",
"These approaches provide detailed information for the human analyst to investigate a hit including both suspicious and legitimate elements found within the DNA.",
"In this article, we primarily argue against the Best Match principle in DNA screening which allows an adversary to evade detection.",
"Similar to the cyber warfare, the penetration of machine learning and adversarial learning into biosecurity domain should increase."
],
[
"HHS guidelines criticism and responses",
" There are multiple concerns regarding the HHS guidelines [19], including the possibility of assembling a SoC using new bioengineering tools.",
"For example, using Gibson assembly, oligos (short DNA molecules) may be assembled to construct larger fragments [23].",
"Since the HHS guidelines do not address oligonucleotide screening, one might order small DNA fragments which are not screened [37] and can be assembled to create pathogens.",
"The HHS guidelines do not specify a database to use for screening, but it does suggest the GenBank as an example of such a database.",
"The lack of a formal database of SoCs is a problem, since it may lead to inconsistent screening protocols between companies, false positives due to housekeeping genes shared between pathogenic and non-pathogenic organisms, and increased cost of overall screening [37].",
"Housekeeping genes are required for basic cellular functions but do not produce toxins or other dangerous products.",
"Nevertheless, housekeeping genes have long been known to cause false hits [6].",
"A curated database of SoCs may reduce the cost of sequence screening [11], [2].",
"Other concerns are the possibility of poisoning public databases with adversarial sequences that have been misclassified as benign, whether mistakenly or with malicious intent [12].",
"The HHS guidelines recommend screening all sequences ordered, regardless of their length.",
"Some argue that screening DNA fragments that are shorter than 200bp may lead to false positives and increased cost [1], while others argue that the 200bp cutoff is not scientifically justifiable [55].",
"We provide justification for a 60bp cutoff in Section REF ."
],
[
"Related cyber attacks",
" Cyberbiological attacks are reminiscent of attacks on physical assets, such as critical infrastructure or industrial control systems, from cyberspace [46], [38].",
"They are perhaps most similar to attacks on additive manufacturing [61], where malware on the victim's computer manipulates blueprints of parts being produced on a 3D printer.",
"Those parts may function (or malfunction) far away from both the cybercriminal and the victim, extending the reach of the former.",
"Similarly, here the adversary affects a biological substance without direct contact with it.",
"The fundamental difference between cyberbiological attacks and cyberphysical attacks is that, DNA can and should be considered as an executable code, although, executable in a non-digital environment.",
"A defect inserted by a malware into a 3D printed rotor may cause a drone to crash.",
"But malicious gene injected into a DNA order by a malicious browser plugin may potentially subvert a DNA sequencing machine propagating back to the cyberspace.",
"This potential bio-cyber attack presented by Ney et al.",
"in USENIX Security'17 [47] closes the loop with the cyber-bio attack discussed here."
],
[
"The biosecurity scenario",
" Consider a case where a bioterrorist in California wants to build a synthetic virus or produce dangerous toxins, and distribute them locally.",
"Synthetic DNA providers, members of IGSC, are the first line of defense.",
"Following is a list of working assumptions considered in this section regarding the attacker and the defender: (1) The attacker orders long DNA sequences.",
"We consider oligos out of scope for this paper.",
"(2) The attacker places an order with one of the synthetic DNA providers that screen orders.",
"There are no other assumptions regarding the attacker.",
"(3) The defender automatically screens every order with a sliding window of fixed size (200bp but may change) and hits are determined according to the Best Match approach.",
"This assumption is well aligned with the HHS guidelines.",
"(4) Only when there is a hit during automatic screening is the order forwarded for further screening by a human analyst or by using DNA function prediction tools.",
"This assumption stems from the large number of orders, the large size of DNA databases used for screening, and the significant computational resources that are therefore needed [19], [21].",
"(5) We assume that all the malicious DNA sequences that the attacker tries to order are found in the defender's database.",
"It is a worst-case assumption that makes the obfuscation process described next more challenging.",
"[t] ${SoC}$ – a Sequence of Concern – a black-box screening algorithm $O$ obfuscated Sequence of Concern @post-condition: Expressing $O$ in Cas9 containing environment results in an assembly of $P$ Partition SoC into 64bp fragments ${SoC}=f_1+f_2+\\ldots +f_n$ $f_1\\leftarrow $ promoter and ribosome binding site $+f_1$ $f_n\\leftarrow f_n+$ terminator Let $m$ be a 23bp long efficient Cas9 cutting site [17] $m_{RC} \\leftarrow $ reverse complement of $m$ $\\forall _{1\\le i<n},f_i\\leftarrow f_i+m$ $\\forall _{1< i\\le n},f_i\\leftarrow m_{RC}+f_i$ $Body = \\sum _{i=1}^{n} $$f_i$ $\\forall _{1\\le i<n},hdr_i\\leftarrow $ 32bp suffix of $f_i+$ 32bp prefix of $f_{i+1}$ Let $grs$ be a gRNA scaffold targeting $m$ $grs$ also targets $m_{RC}$ on the opposite strand.",
"$Decoder = grs+\\sum _{i=1}^{n-1} $$hdr_i$ Assemble and return the obfuscated sequence $O=Body + Decoder$ $x$ Find benign gene $c$ that is most similar to $x$ .",
"Find best scored $(c,\\alpha ) \\in $ $x$ ,no gaps such that $img(\\alpha )$ is at least 200pb from $c$ 's ends $cx\\leftarrow $ replace $\\alpha (x)$ in $c$ with $x$ $cx$ =hit error obfuscation failed cx SoC Obfuscation 2 (SoCO2)"
],
[
"Below the DNA screening radar",
"At a glance, a sequence of concern (SoC) that needs to be obfuscated is split into small fragments.",
"The fragments are interleaved with legitimate DNA sequences which are as similar as possible to the SoC fragments.",
"As a result, for every 200bp window the Best Match is always a legitimate sequence.",
"This may sound like a trivial obfuscation against pattern-based detectors that long ago became obsolete in cyberwarfare, but its biological implementation poses several challenges, as discussed below.",
"We note that GenoTHREAT can easily be evaded due to its strict conditions for identifying a hit.",
"We omit the details hereSee Appendix for details on SoCO1.",
"because it is an attack against a specific vulnerable algorithm.",
"Later the specific attack against GenoTHREAT is referred to as level 1 SoC obfuscation (SoCO1) and the generic attack described next as level 2 SoC obfuscation (SoCO2).",
"Level 2 obfuscation allows a SoC to remain below the screening radar of protocols implemented according to the HHS guidelines.",
"Specifically, we exploit the specification stating that if for every 200bp fragment of a query sequence the Best Match is NSoC, then the query is not a hit.",
"The general obfuscation process is depicted in Figure REF a-f and Algorithm presents its pseudocode.",
"We will refer to the figure and the pseudocode while presenting SoCO2.",
"Figure: Level 2 SoC obfuscation.",
"Schematics of the SoC obfuscation process (a-f); a DNA containing obfuscated α\\alpha -conotoxin PeAI and a decoder sequence that facilitates its reconstruction (g); blast scores when scanning an obfuscated sequence with a 200bp sliding window (h).",
"Colors correspond to the schematic on the left: green and blue lines are alignment scores with camouflage genes, and the red line is the alignment score with the SoC subject.The main challenges an attacker faces in order to successfully evade detection are: (1) Finding legitimate DNA that can successfully camouflage the SoC fragments.",
"(2) Decoding the obfuscated DNA with a minimal set of standard biological primitives commonly used in experiments.",
"The camouflage genes.",
"Given a SoC, we split it into small fragments ($SoC=f_1+\\ldots +f_n$ ).",
"Plus $+$ denotes string concatenation.",
"We will refer to the length of these fragments later in this subsection.",
"In order to hide a SoC fragment we search for a NSoC gene that is most similar to the fragment.",
"We will refer to such NSoC genes as camouflage genes.",
"The search for camouflage genes can be performed within standard nucleotide and protein databases using https://blast.ncbi.nlm.nih.gov/Blast.cgi.",
"In order to efficiently find camouflage genes we search for each SoC fragment ($f_i$ ) while excluding organisms that are known to contain the SoC or similar compounds.",
"The search is performed with the lowest mismatch penalties and the highest gap penalties.",
"In the following discussion, let $c$ denote the appropriate camouflage gene and $\\alpha $ denote an alignment between $c$ and $f_i$ .",
"In order to hide the SoC fragment within the camouflage gene we replace the respective nucleotide sub-sequence ($img(\\alpha )$ ) with the SoC fragment $f_i$ .",
"This process is summarized in Algorithm function .",
"See also Figure REF a, c, and d. The SoC fragments embedded within the camouflage genes constitute the main body of the obfuscated DNA.",
"In order to properly function within living cells after decoding, the body also needs a promoter, a terminator, and a ribosom binding site.",
"See lines 1-3, 8 in Algorithm .",
"The target organism chosen can be one of the most prevalent organisms used in DIY biology, for example, E. Coli.",
"The decoder.",
"Similar to the old well-known encrypted or oligomorphic malware [52], building a decoder is the most challenging part in designing obfuscated malicious DNA.",
"Here the decoder should operate inside living cells rather than in cyberspace.",
"Decoding is facilitated by the maturity and prevalence of CRISPR-based DNA editing methods.",
"In general, the decoder needs to perform two tasks when reconstructing the malicious DNA: (1) It should remove the parts of the camouflage genes between consecutive SoC fragments.",
"(2) It should stitch consecutive SoC fragments together, forming operational DNA.",
"In order to cut out parts of the camouflage genes during decoding, we add cut points before and after each SoC fragment ($f_i\\text{\\ding {34}}camouflage\\text{\\ding {34}}f_{i+1}$ ).",
"Cut points are marked using a 23bp sequence $m$ which includes PAM and gRNA target site.",
"The following example of such a sequence is cut at the nucleotide marked with ✂.",
"$m=\\scriptsize \\begin{matrix}\\texttt {CCTTCC} & \\texttt {ACAAGCTCGCCGAGGTG}\\\\\\texttt {GGAAGG} & \\texttt {TGCTCGAGCGGCTCCAC}\\\\\\underbrace{\\texttt {~~\\ding {34}~~~}} & \\underbrace{\\texttt {~~~~~~~~~~~~~~~~~}} \\\\\\text{\\scriptsize PAM} & \\text{gRNA target sequence}\\end{matrix}$ It is important to note that CRISPR may bind to either side of the double-strand DNA molecule to perform the cut.",
"Therefore, a reverse complement of $m$ (denoted $m_{RC}$ ) is cut from the opposite strand, as shown in Equation REF : $m_{RC}=\\scriptsize \\begin{matrix}\\text{gRNA target sequence} & \\text{ PAM} \\\\\\overbrace{\\texttt {~~~~~~~~~~~~~~~~~}} & \\overbrace{\\texttt {~~~\\ding {34}~~}} \\\\\\texttt {CACCTCGGCGAGCTCGT}& \\texttt {GGAAGG}\\\\\\texttt {GTGGAGCCGCTCGAGCA}& \\texttt {CCTTCC}\\end{matrix}$ The gRNA target sites within the main body point to the locations that need to be cut (see Algorithm lines 6-7 and Figure REF e and f).",
"In order to actually perform the cuts the decoder should include the respective gRNA scaffold targeting the cut points (see Algorithm line 10 and Figure REF g).",
"Cas9 that is also required to perform the cuts is a part of standard gene editing instrumentation which can easily be purchased online (Figure REF ).",
"Next, there is a need to repair the DNA cut made by CRISPR when removing the camouflage and to repair the residue PAM nucleotidesFor example, the leftmost GG on the lower strand in Equation REF ..",
"Such a repair can be performed using the HDR process which requires a 64bp long template ($hdr_i$ ) – a DNA sequence covering both the last 32 nucleotides of $f_i$ and the first 32 nucleotides of $f_{i+1}$ (see Algorithm line 9 and Figure REF b).",
"A 64bp long HDR template is the shortest HDR template shown to perform well in practice [51].",
"The longer the HDR templates are, and the more replicas of these templates are found within the cell, the higher the HDR efficiency is.",
"Since $\\lbrace hdr_i\\rbrace $ are also fragments of the SoC sequence, it is important to camouflage them in order to remain below the screening radar.",
"Obfuscation variants.",
"There could be many variants of this general obfuscation scheme, including variations of the gRNA binding sites.",
"On the one hand, these sites may be designed individually for each cut point in order to improve the stealthiness of the obfuscation.",
"But in this case they will require multiple gRNA scaffolds, each targeting a different cut point, potentially reducing the decoding's effectiveness.",
"On the other hand, repeated gRNA binding sites may increase the likelihood of obfuscation failure.",
"The size of the SoC fragments and the HDR templates may also vary.",
"Reducing the size of the SoC fragments increases their likelihood of blending within the camouflage genes.",
"Yet it also increases the number of such fragments, which leads to a larger number of cuts and repairs, consequently reducing the effectiveness of the decoder.",
"Since 64bp is the currently known lower bound on the size of HDR templates, there is no point in using shorter SoC fragments if they are ordered together with the HDR templates.",
"The adversary may also choose to order the main body and the HDR templates separately and reduce the size of the SoC fragments."
],
[
"Gene Edit Distance",
" In order to harden synthetic DNA order screening and reduce the non-regulated distribution of select agents and toxins, we propose a new DNA screening algorithm termed Gene Edit Distance (GED).",
"The algorithm is designed to assess the difficulty of SoC assembly from a DNA sequence.",
"In order to do so, GED screens the query sequence to find all substrings which are similar to fragments of a SoC.",
"Then, GED quantifies the effort of assembling a SoC from these fragments.",
"Although designed with a focus on SoC, GED can quantify the effort required to assemble any target sequence $t$ from a query sequence $q$ using a standard CRISPR system.",
"More specifically, we count the number of cuts and repairs required for constructing the target sequence from the query sequence.",
"In a standard biological sequence alignment, a typical objective is to identify genes conserved in different genomes.",
"To achieve this objective, the match reward ($rm$ ), mismatch penalty ($pmm$ ), and gap penalties ($gpo$ , $gpx$ ) must be well balanced when computing the alignment score.",
"GED's objective is more complex.",
"On the one hand, we need to find short conserved regions with minimal gaps within the query sequence.",
"On the other hand, we want to concatenate the short conserved regions regardless of the gap length between them.",
"Example 2 A default configuration when aligning two sequences with [4] is $rm=2$ , $pmm=3$ , $pgo=5$ , and $pgx=2$ .",
"Figure REF presents the results of aligning the obfuscated $\\alpha $ -conotoxin PeIAA short toxic peptide.",
"from Figure REF to the PeIA sequence using default configuration.",
"returns three ranges (alignments).",
"As long as $pgx>0$ , these ranges will not be merged by due to the length of the gap that would be opened in the target sequence.",
"However, when removing a sequence between two consecutive SoC fragments using the CRISPR system and HDR templates, the distance between them does not play a critical role.",
"Gap removal penalty.",
"In order to achieve the objective of GED, we introduce a new gap removal penalty ($prm$ ) that substitutes some gap opening and extension penalties within the target sequence.Note that variation of gap penalties for query and target sequences has successfully been used in the past for other use cases [54].",
"Let $g=[a,b]$ be a gap in the target sequence ($\\forall _{i\\in g}, \\alpha (i)=\\bot $ ).",
"Removal of $[a,b]$ from the query sequence requires two cuts, at $a$ and at $b$ respectively, followed by an HDR (e.g., Figures REF and REF .10).",
"These operations may fail.",
"Let $\\gamma $ be the probability that a gap is successfully removed.",
"To define GED we neglect some biological constraints of the CRISPR system, such as specific PAM sites targeted by Cas9, assuming that cuts can occur at any point in a DNA sequence.",
"This is a worst-case assumption overestimating the potential risk.",
"Future versions of GED may take biological constraints into account to reduce false positives.",
"Currently, in the absence of advanced adversarial techniques, GED detects obfuscated SoC with 100% accuracy, as shown in Section REF .",
"If the gap $g=[a,b]$ is removed, the score of the alignment will increase by $pgo+pgx\\cdot (b-a+1)$ and reduce by $prm$ .",
"Let $\\mathcal {G}_{q}(\\alpha )$ and $\\mathcal {G}_{t}(\\alpha )$ be sets of all gaps in the query and target sequences respectively according to the alignment $\\alpha $ .",
"Let $\\mathcal {G}^k_t\\subseteq \\mathcal {G}_t$ be a subset of the $k$ longest gaps in the target sequence to be removed.",
"The probability that all gaps in $\\mathcal {G}^k_t$ will be successfully removed is $\\gamma ^{k}$ .",
"Next, we adjust the alignment score as a result of designating $\\mathcal {G}^k_t$ for removal.",
"Definition 1 (Adjusted alignment score) Given an alignment $\\alpha $ ; a subset of gaps $\\mathcal {G}^k_t$ in the target sequence of $\\alpha $ ; gap opening and gap extension penalties $pgo$ and $pgx$ respectively; gap removal penalty $prm$ ; and gap removal probability $\\gamma $ ; we define the change in the alignment score as the result of the removal of the $k$ longest gaps in the target sequence $t$ as: $Score^k(\\alpha )= Score + \\gamma ^{|\\mathcal {G}^k_t|}\\cdot \\sum _{g\\in \\mathcal {G}^k_t} \\left(pgo+pgx\\cdot |g|-prm\\right).$ Next, we examine the choice of gaps to be removed ($\\mathcal {G}_{trm}$ ) and the parametrization of $Score^k$ .",
"We assume that the match reward ($rm$ ), as well as the mismatch, gap opening, and gap extension penalties ($pmm,pgo,pgx$ ) are set at their default values or according to the biological considerations (which are beyond the scope of this article).",
"$Score^k$ approaches $Score$ when the number of gaps to be removed $k$ increases and $\\gamma <1$ .",
"The choice of $\\gamma $ depends on the assumptions made regarding the expected biological effectiveness of the attacks given typical bioengineering tools used by potential victims today.",
"$\\gamma =0$ signifies that no attacker could ever rely on residue Cas9 protein in the cells and the SoC obfuscation attack described in Section REF is impossible.",
"$\\gamma =1$ signifies 100% success of gene editing, which leads to the successful decoding of SoCO2 sequences regardless of the number of SoC fragments.",
"The gap removal penalty $prm$ should be set to a value that would justify gap removal using bioengineering tools.",
"For example, if only the removal of gaps larger than $x$ bp is justified, then the gap removal penalty should be set to: $prm = pgo + pgx\\cdot x.$ Obviously, the longer a gap is, the more worthwhile its removal is.",
"Thus, for the sake of computing $Score^k$ , only the longest gaps are selected.",
"The number of gaps $k$ that should be included in $\\mathcal {G}^k_t$ for the maximal $Score^k$ depends on the parameters and can be selected using a grid search or simple hill climbing algorithm.",
"Figure: Example of an alignment score correction using the GED ΔScore\\Delta Score.",
"(a) The gap length distribution of an example alignment.",
"(b) The corrected alignment score (Score+ΔScore k Score+ \\Delta Score^k) for various choices of kk.Example 3 Assume, for example, that we examine a 10Kpb long query sequence, comparing it to a 2Kpb long SoC.",
"Also assume a negatively scored optimal alignment between the two sequences which contains 2K matching base pairs, a few mismatches, and 50 gaps whose length is exponentially distributed, as depicted in Figure REF a.",
"Although, such alignment exists, it will never be returned by the search engine, because its score is worse than random.",
"Assuming default configuration, the specific alignment score could be lower than $-4,500$ .",
"Assume a gap removal penalty of $prm=20$ .",
"Replacing gap opening and gap extension penalties with $prm$ starting from the longest gaps would increase the score, as shown in Figure REF b.",
"With $\\gamma =0.98$ , setting $k=10$ results in the highest $Score^k$ value.",
"With $\\gamma =0.99$ , the adjusted alignment score can reach $2,000$ when $k=13$ in this example.",
"Example 4 Further following Example REF , consider the alignment of obfuscated $\\alpha $ -conotoxin PeIA with the unobfuscated peptide, as shown in Figure REF .",
"The blue alignment in Figure REF c, was not identified by , because it contains very long gaps in the target sequence.",
"Nevertheless, an adjusted score with the gap removal penalty of $prm=20$ and $\\gamma =0.99$ , according to definition REF , would result in a total score of 193 which is very close to a best match.",
"GED definition.",
"Next, we define the gene edit distance as the optimal number of gaps ($k$ ) to remove in a query sequence such that the adjusted alignment score is maximized.",
"Definition 2 (Unidirectional gene edit distance) The gene edit distance (GED) from $q$ to $t$ is: $GED(q,t) = \\texttt {ARGMAX}_k \\texttt {MAX}_{\\alpha } Score^k(\\alpha ).$ According to this definition $Score^{GED(q,t)}(\\alpha )$ has the maximal adjusted score.",
"Although we call GED a distance, it is not a valid mathematical distance metric, first and foremost because it is asymmetric.",
"GED quantifies the effort required to transform $q$ into $t$ but not vice versa.",
"As an academic exercise, one can define a true gene edit metric, but such definition is beyond the scope of this security article.",
"GED computation.",
"Current implementations will not return suitable alignments which minimize the small gaps but neglect the long ones.",
"Thus, in order to reduce time-to-market and maintain backward compatibility with existing engines, we implement a GED heuristic as postprocessing of standard outputs.",
"The GED heuristic pseudocode is presented in Algorithm REF .",
"[t] q – a query sequence t – a target sequence $k$ – the number of cut and repair operations $Score^k$ – adjusted alignment score List local alignments $\\mathcal {A}=\\lbrace \\alpha \\rbrace $ sorted by $dom(\\alpha ).start$ $\\mathcal {A}_{q,t} = {query=q,subject=t}$ each subset $P\\subseteq \\mathcal {A}_{q,t}$ of local alignments with disjoint domains $\\alpha _P = \\merge {P}$ $P^*=\\texttt {argmax}_{P} Score^{|P|-1}(\\alpha _{P})$ $k=|P^*|, Score^k(\\alpha _{P^*})$ $\\alpha _1,\\ldots ,\\alpha _k$ Find global alignment between unified domains and $t$ $\\mathcal {A}=(query=q[\\bigcup _i dom(\\alpha _i)],subject=t)$ $\\alpha =$ cleaned global alignment unifying $\\mathcal {A}$ Reintroduce gaps to form a continuous alignment $\\alpha \\cup =\\lbrace (i,\\bot ) : i\\in [dom(\\alpha _j).end,dom(\\alpha _{j+1}).start], 1\\le j<k \\rbrace $ $\\alpha $ Gene Edit Distance Standard algorithms compute a large set of small local alignments and extend these alignments when doing so increases the alignment score.",
"We take a similar approach when computing a set of alignments using standard configuration and merging them to maximize $Score^k$ .",
"Recall that $\\mathcal {A}_{q,t}$ is a set of local alignments between $q$ and $t$ returned by (Algorithm REF , line 1).",
"Let $.start$ and $.end$ denote the first and the last position respectively, in the domain or the image of an alignment $\\alpha \\in \\mathcal {A}_{q,t}$ .",
"Let $P=\\alpha _1,\\ldots ,\\alpha _k$ be a set of alignments whose domains are disjoint, $dom(\\alpha _i).end<dom(\\alpha _{i+1}).start$ (Algorithm REF , line 1).",
"Their images may overlap, as depicted in Figure REF .",
"can find a global alignment between the unified domains of the alignments and the target sequence i.e.",
"when sequence fragments between the alignment domains are removed.",
"Such unified alignments are called cleaned alignments according to .",
"Figure: Two local alignments with disjoint domains and overlapping images.The output of Algorithm REF is the number of cut and repair actions required to reconstruct the target sequence from the query sequence and the adjusted alignment score.",
"The latter quantifies both the similarity of the body of an obfuscated SoC and the effort required to decode it within the cells.",
"Unlike in the case of encrypted or oligomorphic malware where decoders are the easiest to detect, here we concentrate on detecting the main body of the obfuscated SoC, because the gRNA scaffolds and HDR templates comprising the decoder may be distributed among different plasmids or even different orders.",
"In order to return a final assessment of the risk a query sequence presents, the GED of the query sequence is evaluated against all SoC sequences in the database.",
"The final judgement is made based on the maximal adjusted score of the query sequence for any of the SoC sequences.",
"This allows the identification of seemingly benign sequences that can easily be transformed into malicious sequences that produce dangerous products.",
"In addition, such an approach is more resilient to an attack in which a public gene database is poisoned with legitimate sequences, although poisoning with legitimate sequences maliciously marked as a SoC may result in false hits and require human attention."
],
[
"Evaluation",
"In this section we evaluate different screening algorthms vs. the SoCO1 and SoCO2obfuscation algorithms.",
"Benchmark dataset.",
"For the purpose of evaluation we selected 50 SoCs from the UniProt database [16], all of which are marked as toxins, are manually reviewed, and contain between 33 and 100 amino acids.",
"These protein sequences were reverse translated to sequences of nucleotides using an EMBOSS online tool [43].",
"We ran queries using the nucleotide SoCs to ensure that they are well detected within the GenBank database using .",
"Doing so, we identified seven benign sequences with various levels of similarity to the SoCs.",
"We selected an additional 43 benign sequences from UniProt, non of which are toxic and all of which are manually reviewed.",
"The 50 SoC sequences were obfuscated using Algorithms and .",
"To create the SoCO2 sequences, we selected the camouflage genes from the 50 NSoC sequences described above.",
"Finally, we generated 50 random nucleotide sequences (denoted as Rnd) as a reference point.",
"There are a total of 250 nucleotide sequences in the benchmark dataset, equally split among NSoC, SoC, SoCO1, SoCO2, and Rnd.",
"Baseline screening algorithms.",
"GenoTHREAT can be easily circumvented, because it scans non-overlapping 200bp fragments of the query sequence and employs very strict match constraints (see Appendix for details).",
"In order to objectively assess the threat of DNA obfuscation, we implement a screening method, referred to as GenoTHREAT plus (GTP), that strictly implements HHS guidelines.",
"While we don't elaborate on the algorithm here due to space constraints, we provide some highlights below.",
"GTP scans the query sequence with a sliding window of length 200bp and step size of 1bp.",
"Every such 200bp nucleotide sequence is searched in the database.",
"We use a set of keywords and anti-keywords to identify dangerous substances in the returned results.",
"GTP records the score of the best matching SoC as well as the score of the best matching NSoC.",
"Later, we derive a confidence value from the difference between them.",
"Querying both nucleotide and protein databases for each sequence of 200bp is a highly inefficient, yet accurate approach that strictly follows the HHS guidelines.",
"In addition to GenoTHREAT and GTP, we employ a heuristic screener based on InterProScan 5 [31], a state of the art DNA and protein functional annotation framework.",
"The heuristic referred to as IPC, produces a hit if at least one of the tools within InterProScan annotates at least some part of the evaluated DNA as a toxin.",
"Results.",
"Benchmark sequence screening was performed using local copies of the complete GenBank NT and NR and UniProt databases.",
"While GenoTHREAT, IPC, and GTP require the entire database for accurate screening, GED only requires the SoCs to compare with the query sequences.",
"Figure: GTP, IPC, and GED confidence levels for five types of sequences in the benchmark dataset.Table: Number of hits produced by different screening approaches.We expect a good screening algorithm to report a hit when screening sequences in the SoC, SoCO1, or SoCO2.",
"We expect a non-hit when screening sequences in the NSoC or Rnd.",
"Similar to studies focusing on malware detection, we consider the true positive rate (TPR), the false negative rate (FNR), and the false positive rate (FPR).",
"High FNR (low FPR) indicate that the screening method can be evaded using obfuscation.",
"Since some malicious sequences are easier to detect than others, we the hit counts for each group of sequences.",
"In order to analyze the performance of the screening algorithms, we also inspect their confidence levels.",
"While GenoTHREAT only provides a binary decision on a query sequence, GTP can provide a confidence level along with the decision.",
"We compute the GTP confidence as follows.",
"Let $q$ be some 200bp fragment of the screened sequence.",
"Let $Max{SoC}(q)$ and $Max{NSoC}(q)$ be the highest score of an alignment of $q$ with a SoC and NSoC respectively.",
"If did not return alignments with SoC or NSoC, we set the respective score to zero.",
"We define GTP confidence as $GTPConf = \\max _q \\left\\lbrace \\frac{Max{SoC}(q)-Max{NSoC}(q)}{\\max \\lbrace Max{SoC}(q),Max{NSoC}(q)\\rbrace }\\right\\rbrace $ $GTPConf>0$ means a hit, because there is at least one 200bp fragment that is more similar to a SoC than to an NSoC.",
"The confident of IPC is the fraction of toxin annotations out of all successful annotations produced by InterProScan when screening $q$ .",
"$IPCConf>0$ means a hit.",
"The confidence of GED ($GEDConf$ ) is simply the maximal adjusted score it returns when screening $q$ .",
"The value of $GEDConf$ is between zero and one, where $GEDConf=1$ means that $q$ is definitely a SoC.",
"Table REF summarizes the performance of GenoTHREAT, GTP, and GED on the benchmark dataset.",
"Confidence levels of GTP and GED when screening the 250 benchmark sequences are presented in Figure REF .",
"As expected, all SoCs are the closest to being malicious (right part of the chart) and all NSoCs are the closest to being benign (left part of the chart).",
"Most random sequences are classified by GTP as benign (low negative confidence), because there are many more benign sequences than toxic sequences in the GenBank database.",
"Nevertheless, some random sequences contain 200bp fragments that are more similar to a SoC than to an NSoC.",
"One such fragment is sufficient to produce a hit and require human attention.",
"We consider such statistical errors as an inherent deficiency of the Best Match approach.",
"According to GED, random sequences are similar to benign sequences with respect to the effort required to transform them into something dangerous.",
"Next, we observe a high variance of the confidence levels of obfuscated sequences in Figure REF for all screening methods.",
"While SoCO1 sequences are always detected, some SoCO2 sequences remain below the radar of the Best Match approach, represented here by GTPadn below the radar of InterProScan functional annotation.",
"This is due, of course, to the camouflage genes which contribute the most to the PI value in each 200bp fragment.",
"The partitioning into fragments makes it harder for InterProScan to detect functional segments.",
"Nevertheless, some SoCO2 sequences are detected, because no sufficiently similar camouflage gene was found during obfuscation.",
"An important conclusion from these results is that some sequences are easier to obfuscate than others.",
"GED successfully detects all obfuscated malicious sequences.",
"Moreover the large confidence gap between the most suspicious benign sequence and the least suspicious malicious sequence ensures GED's robustness as a screening algorithm.",
"We also note that GED is an order of magnitude faster than GenoTHREAT, because it compares the query sequence with a small database of SoC and does not need to query for every 200bp fragment."
],
[
"DNA injection as a cyberbiological attack",
"In this section we introduce an attack pattern called DNA Injection following the common template of Attack Pattern Enumeration and Classification (CAPEC) descriptions.https://capec.mitre.org/ In contrast to classical biosecurity scenarios, the attack pattern discussed here assumes that the attack originates from cyberspace as do all cyberphysical attacks.",
"In the discussed attack scenario, a victim might be a do it yourself (DIY) biology enthusiast or a small bioengineering company that develops their own DNA sequences or combines existing genes to produce fuel, medical components, or resilient plants.",
"We assume that the victim does not use their own facilities to produce the DNA but prefers ordering synthetic DNA strands from synthetic gene providers.",
"Figure REF (steps 1-9) depicts a common workflow of a biologist experimenting with gene editing.",
"First, the biologist designs a system for adding, removing, or replacing a gene in some target organism (e.g.",
"E. Coli) using CRISPR, and then places a synthetic gene order with one of the providers, who screens the order for possible inclusion of SoC.",
"During the production a provider typically sends the biologist updates on production status and quality control.",
"Once the DNA is produced, a tube is delivered to the biologist.",
"Large labs prefer sequencing the order they received to ensure its quality, but DIY biologists and many small companies would trust the quality reports sent by the provider.",
"Finally the synthetic genes are applied to the organism of interest and their biological effects are assessed.",
"The attacker is a cybercriminal who wants to trick the biologist victim into producing dangerous biological components inside the victim's lab.",
"For this purpose the attacker may design an obfuscated malicious gene as described in Section REF .",
"Figure: A typical synthetic biology workflow (1-9), weaknesses (a-e red underlined), and attack components (a,c,d,e purble bold).course of an RDI attack.",
"The cell transformation schematic (9) depicts the decoding of the malicious DNA within a cell."
],
[
"Related weaknesses",
"Endpoint security.",
"DIY biologists are well trained in biosafety protocols and aware of biosecurity screening and the dangers of engineering pathogens, similar to most bioenginers [25].",
"However, like most computer users, they cannot be trusted to properly secure their end devices and mitigate social engineering attacks.",
"It is a common assumption that the end customer may be infected with malware (Figure REF a) and providers as well as software vendors must be alert to that.",
"Data at rest.",
"The first line of defense should have been provided by the integrated development environments (IDEs) for DNA coding.",
"IDEs provide the ability to create and edit DNA sequences.",
"We inspected the electronic integrity features provided by typical DNA IDEs, such as SnapGene, Serial Cloner, ApE (A plasmid Editor), and Genome Compiler, most of which support one or more common DNA file types, such as .genbank, .fasta, and .dna.",
"Some of the file formats are binary, but they do not contain electronic signatures or other means of integrity protection (Figure REF .b).",
"This allows a malicious attacker to change the sequences within DNA files without the user's consent.",
"Communication.",
"Most communication with gene synthesis companies, including gene orders, takes place through a company's website or email.",
"All synthetic gene orders are validated prior to purchase and during production.",
"Unfortunately, most validation reports are delivered through the same channel which, in the case of an attack, is presumably controlled by the attacker (Figure REF c).",
"Standard end-to-end encryption provided by HTTPS does not help when the data is corrupted, for example, by a malicious browser plugin.",
"None of the gene libraries, such as GeneBank, NCBI etc., provide electronic signatures for data download.",
"Some projects, such as InterProScan provide MD5 checksum for large downloads but this is insufficient.",
"None of the providers request a validation record for submitted orders.",
"Biosecurity screening.",
"Although gene orders are screened by providers, screening algorithms are aimed at computational performance and achieving zero false hits.",
"Current official screening recommendations have multiple weaknesses (Figure REF .d); we elaborated on one of the weaknesses, which enables an attacker to evade detection by a common screening protocol, in Section REF .",
"Additionally, small orders of less than 200bp might not be screened; neither are orders for short single-strand DNA (oligos).",
"Fortunately, constructing a pathogen from oligos requires expert knowledge and dedicated effort, and thus the biologist victim is unlikely to be tricked into constructing a pathogen without being aware of it.",
"Yet obfuscated malicious genes, capable of reconstructing and activating themselves, pass unnoticed.",
"Biosecurity protocols for customer screening won't help in this case since the malicious genes would be ordered by a legitimate customer.",
"Biological protocols.",
"Bioengineering best practices focus on safety and efficiency.",
"Most biological protocols today do not take into account cyber threats, or any adversarial manipulation of the genetic material, relying instead on physical perimeter security when biosecurity is a concern (Figure REF .e).",
"One example of a biological-grade vulnerability is negligence of Cas9 expressing DNA after performing the intended cuts in the DNA.",
"CRISPR best practice recommends discharging of gRNA in order to avoid unintended cuts during subsequent phases of the experiment.",
"However, the Cas9 expressing DNA is usually left intact within the cells."
],
[
"Prerequisites",
" In order to carry out a DNA injection the attacker must be able to interfere with the victim's communication with the synthetic DNA provider, which can be achieved using a malicious browser plugin or malware.",
"Minimally, the attacker needs the ability to modify the DNA order, either while it is stored on the victim's computer or when it is submitted to the provider.",
"The ability to modify provider reports will help to conceal the attack from the victim.",
"A malicious browser plugin is sufficient for all attack objectives if the victim's communication with the DNA providers takes place via the Web, including via a webmail client.",
"This is a valid assumption for most DYI biologists and small bioengineering companies."
],
[
"Skills and resources",
" An attacker need only possess the resources of an average individual, and an intermediate level of sophistication [53].",
"For example, they must be able to write a trojan plugin for a browser; intercept, parse, and modify a pdf file found as a webmail attachment or downloaded from the Web; and successfully execute a man-in-the-browser attack technique [33].",
"The attacker is, however, unable to penetrate the security premises of synthetic DNA providers.",
"The attacker must know the basics of synthetic biology and should be able to target DIY biologists or small bioengineering companies working with CRISPR.",
"Targeting may be performed via synthetic biology forums where the attacker promotes the attack agent."
],
[
"Execution flow",
" CAPEC defines three attack phases: Explore, Experiment, and Exploit.",
"The first two phases include the reconnaissance and weaponization activities performed by the attacker.",
"The last one is the actual execution of the attack pattern.",
"Explore.",
"In order to collect information the attacker places a synthetic DNA order with the targeted providers.",
"During the production of the DNA and its delivery the attacker inspects the provider's website pages related to the order including progress and quality reports, for all information concerning the specific DNA sequence ordered.",
"When the attack objective is the injection of select agents or toxins, the attacker identifies suitable short malicious DNA sequences that can be successfully obfuscated, i.e.",
"sequences for which camouflage genes can be found in common NCBI databases using .",
"This activity is similar to the evaluation of SoCO2 described in Section REF .",
"In order to better target the attack, an adversary may use his malware (a trojan plugin) to monitor the DNA orders of the victims, infer the target organism from the most common codons in the orders, and tailor the obfuscated DNA to the organism used by the victim in his experiments.",
"Experiment.",
"The attacker may test the DNA injection technique by placing additional orders replacing the submitted DNA with another legitimate DNA.",
"Further, the attacker experiments with the providers, placing small fragments of obfuscated SoCs within his orders.",
"Such orders may be placed with multiple providers in order to better target the attack.",
"Detection of a SoC results in the order being cancelled due to biosecurity concerns.",
"But this does not prevent the attacker from placing additional legitimate orders, up to some number of repeated violations, in accordance with the client screening guidance and internal company policies.",
"As a result, during this phase an attacker will start with the shortest SoC fragments gradually increasing their length.",
"Exploit.",
"The attacker injects obfuscated malicious DNA into the orders of biologist victims.",
"Please refer to Appendices and for an example instance of this attack and sufficient DIY biology instrumentation."
],
[
"Possible attack impacts",
"Biosecurity exploration.",
"Customer screening and information sharing among synthetic DNA providers may limit the ability of the attacker to experiment with actual DNA orders.",
"The attacker may circumvent customer screening by injecting obfuscated malicious DNA into orders placed by the biologist victims.",
"Orders whose production was completed indicate that the obfuscated SoC successfully bypassed the provider's screening.",
"Biosafety violation.",
"Similar to cyberphysical attacks here the attack results an actual toxic substance handled by an actual person.",
"For example, conotoxins are short peptides many of which are less than 30 amino acids long, i.e.",
"encoded by less than 90bp DNA.",
"They are found within the venom naturally produced by cone snails and can cause serious injury if inhaled, ingested, or even absorbed through skin.",
"Working with conotoxins in a biological lab requires biosafety level 2 [14], a level at which personnel must be properly trained in handling pathogenic agents and decontamination.",
"Biological grade bacteria such as E.Coli DH5a are safe even if inhaled, digested, or contact skin.",
"However, conotoxins produced by such a bacteria are released when the bacteria is destroyed by the human immune system or simply when transferred from one plate to another.",
"E.Coli is capable of producing custom complex protein at levels higher than $1mg/mL$ [7].",
"A lethal dose of conotoxins is $DL50=5\\mu g/Kg$ [56] making $0.5mL$ of a conotoxin producing E.Coli dangerous to a human weighting $100kg$ .",
"Denial of service.",
"The attacker may significantly slow down experiments by replacing one nucleotide.",
"Consider an angry developer that adds a rogue jump command after the 200th character in source code of some C program.",
"He can cause huge headache to his coworkers.",
"Now imagine that you can only use print debug with at most 3-4 print commands and every program execution takes 2-3 days.",
"Debugging DNA is much harder than debugging electronic hardware.",
"This variant of the attack is likely to target large biological labs rather than DIY biologists.",
"A possible goal of an attacker would be to gain competitive advantage by slowing the progress of labs developing similar biological systems."
],
[
"Mitigation",
" The digital parts of the DNA injection attack can be mitigated by additional levels of integrity control.",
"First, all reports sent during and after DNA production should be password protected following best practices of paperless communication in the banking and insurance domains.",
"Second, a hard-copy of the quality report including the DNA sequence should be delivered with the tube containing the synthetic DNA.",
"Finally, a sticker on the tube indicating the most important information from the security perspective would be highly valuable; for example, are there gRNA scaffolds or Cas9 encoding genes within the tube?",
"At the level of biosecurity, the hardened DNA screening method based on GED that we present in this article accurately identifies obfuscated malicious DNA.",
"At the biological level, any use of Cas9 should be considered a critical stage in the biological protocol.",
"A security-aware biologist should use existing methods to inhibit the expression of Cas9 within the cells [8].",
"Currently such methods are used only to achieve the desired biological effects, but not as safety or security measures."
],
[
"Conclusions",
" In this paper we demonstrated a potential attack where a remote attacker injects malicious DNA that produces a dangerous substance into the workflow of a biologist victim.",
"We demonstrated a new obfuscation technique that circumvents current synthetic DNA screening guidelines by utilizing gene editing using CRISPR-Cas9 for decoding of the obfuscated DNA.",
"Experiments demonstrate that 16 out of 50 obfuscated DNA samples are not detected when screened according to the HHS guidelines.",
"We further proposed a hardened screening algorithm termed Gene Edit Distance (GED) that successfully detects all obfuscated DNA samples.",
"Future enhancements to DNA screening may rely on machine learning for sequence analysis and DNA function prediction.",
"Adversarial learning techniques can be used to further increase the resilience of screening algorithms against malicious DNA sequences that are not yet on the SoC list.",
"The DNA injection attack discussed in this paper demonstrates a significant new threat of malicious code altering biological processes.",
"Although simpler attacks that may harm biological experiments exist, we've chosen to demonstrate the a scenario that makes use of multiple weaknesses at three levels of the bioengineering workflow: software, biosecurity screening, and biological protocols.",
"This scenario highlights the opportunities for applying cybersecurity know-how in new contexts such as biosecurity and gene coding.",
"A conceptual attack where malicious DNA exploits a vulnerability in a DNA sequencing machine propagating to cyberspace was presented in USENIX Security'17 [47].",
"Current work closes the loop by showing that code may propagate from cyberspace to DNA.",
"Future cyberbiological attacks may directly exploit desktop DNA assembly machines, diminishing the border between digital and biological.",
"DNA screening is not globally enforced yet.",
"So, if a bioterrorist would like to buy dangerous synthetic DNA today it is possible to do so, although only outside of California [58].",
"Current article is timely since it will cause any immediate harm, but hopefully, it sets the stage for robust adversary-resilient sequence screening and cybersecurity-hardened synthetic gene production services for the time when biosecurity screening will be enforced by local regulation around the world."
],
[
"Ethics and responsible disclosure",
" This paper comes at a time when dangerous DNA can still be purchased online without screening.",
"It's vital to preemptively enlist cybersecurity specialists for reviewing and hardening biosecurity protocols.",
"The defense-evasion technique was disclosed to the IGSC consortium during their monthly meeting and multiple partners expressed their interest in internal evaluation of the benchmark data.",
"The results of this evaluation are not yet available and their publication depends on the goodwill of the involved companies.",
"We will make all the code and the benchmark data set publicly available following feedback from IGSC.",
"Blinded."
],
[
"Example attack instance",
"The attack agent may be a trojan browser plugin (Figure REF a) that provides a useful function for biologists.",
"Legitimate plugins, e.g.",
"[35], are already used by synthetic biologists.",
"In this example the trojan plugin adds an annotated schematic adequately visualizing DNA sequences found on the Web pages or submitted in forms.",
"See Figure REF in the Appendix for an example of such a plugin.",
"Figure: DIY biology instrumentation, including a CRISPR kit from the-odin.com (left), and a microcentrifuge and TE buffer from eBay (right).A DIY biologist obtains a CRISPR kit from the-odin.com (Figure REF left).",
"The biologist also places an order with a DNA provider for a plasmid containing Green Fluorescent Protein (GFP) to make the E.Coli glow.",
"The malicious payload of the plugin manipulates the DNA during form submission (Figure REF c and 3) and substitutes the DNA a user entered for a successfully obfuscated conotoxinAn attacker would use a conotoxin with high number of disulfide bonds to resist standard bleach which is commonly used for bacteria disposal [9].",
"(Sections REF REF and Figure REF d).",
"Consequently the plugin substitutes the malicious DNA for the one the user entered in all provider's reports (Figure REF c and 7).",
"Since most gene synthesis companies charge by the number of base pairs [10], the attacker retains the length of the originally ordered DNA in order to avoid detection resulting from unexpected costs.",
"Once all orders arrive, the DIY biologist applies the protocol attached to the CRIPSR kit [48] allowing the bacteria to survive on a plate with a Strep antibiotic.",
"Then he inserts the GFP plasmid, which does not glow because it was replaced by a toxin-encoding plasmid.",
"Frustrated, the biologist will then dispose of the bacteria.",
"Since the bacteria is not pathogenic, the protocol requires simply putting 5% bleach on the plate and putting the plate in the trash [48].",
"However, some conotoxins resist bleach and disposing of them in this way may release the toxins to the air and potentially cause serious injury."
],
[
"DIY biology instrumentation",
"We note the wide applicability of the DNA injection threat in terms of biological instrumentation.",
"Everything that is required for a DIY biology enthusiast to develop her own genes and apply them to cells is available for purchase online.",
"Moreover, no special expertise or sophisticated equipment is required.",
"For instance, the DIY Bacterial Gene Engineering CRISPR Kit can be purchased from the-odin.com (Figure REF left).",
"The kit contains E. coli cells (that were stripped of any pathogens and are considered safe), Cas9 coding plasmids, growth media, petri plates, and other instrumentation, including detailed instructions on the CRISPR protocol.",
"When a plasmid (such as in Figure REF .g) is ordered online, it comes in a dry form and needs to be resuspended in a TE buffer, which is a commonly-used buffer solution.",
"Most resuspension protocols suggest centrifuging the synthetic DNA upon receipt, and the required microcentrifuge and TE buffer are available for purchase on eBay (Figure REF right)."
],
[
"Level 1 SoC Obfuscation (SoCO1)",
"As an implementation of the HHS guidelines, the designers of GenoTHREAT [1] chose to split the query sequence into consecutive 200bp fragments rather than using a sliding window of 200bp.",
"This is much more efficient in terms of the number of queries.",
"Yet, such a design is also susceptible to short DNA inserts.",
"According to GenoTHREAT an alignment between the query sequence and a target sequence is a Best Match if $QC=100\\%$ and $PI$ is the maximal over all other alignments of the query sequence.",
"Note that more than one alignment may match these conditions.",
"Further, if either the beginning or end of a 200bp fragment is aligned to a SoC and $QC>50\\%$ , then the fragment is extended in order to identify a possible alignment of the preceding or the following fragment, with a SoC.",
"Next we discuss the simplest DNA obfuscation method which is specifically targeted at circumventing GenoTHREAT.",
"We will refer to this method as Level 1 Sequence of Concern Obfuscation (SoCO1).",
"In Section REF we generalize this approach to withstand arbitrary Best Match screening.",
"The pseudocode of SoCO1 is presented in Algorithm .",
"In order to hide a sequence of concern (SoC) from GenoTHREAT, we split it into small fragments (${SoC}=f_1+\\ldots +f_n$ ) of 154bp each, padded with NSoC fillers.",
"Plus $+$ denotes string concatenation.",
"Similar to the old well-known encrypted or oligomorphic malware [52], building the decoder is the most challenging part in designing obfuscated malicious DNA.",
"Here the decoder should operate within living cells rather than in cyberspace.",
"In general, the decoder needs to perform two tasks when reconstructing the malicious DNA: (1) it should cut out the NSoC fillers between the SoC fragments, and (2) it should stitch the SoC fragments forming operational DNA.",
"We use $m+m_{RC}$ (Equations REF and REF ) as the NSoC fillers between SoC fragments ($f_i$ ), such that the length of $m_{RC}+f_i+m$ is exactly 200bp.",
"The DNA code block $f_1+m+m_{RC}+f_2+\\ldots +m_{RC}+f_n$ is not detected by GenoTHREAT as a SoC even if SoC exists within the SoC database, because there are no best matches with a query coverage of 100%.",
"We show how GTP mitigates this problem in Section REF .",
"We assume that Cas9 protein is available within the cells.",
"The decoder block of the malicious DNA should contain the gRNA scaffold targeting $m$ in order to form a CRISPR system that will cut out $m+m_{RC}$ between consecutive SoC fragments ($f_i\\text{\\ding {34}}m+m_{RC}\\text{\\ding {34}}f_{i+1}$ ).",
"Next, there is a need to repair the DNA cut made by CRISPR when removing the residue PAM nucleotides.",
"This process is the same for both SoCO1 and SoCO2.",
"Overall, the malicious DNA sequence injected into an online synthetic DNA order, should contain the split SoC, the gRNA scaffold, and a set of HDR templates, as shown in Algorithm .",
"We refer to such sequences as SoCO1 sequences.",
"Generating SoCO1 sequences requires minimal computing time, but they are detected easily as shown in Section REF .",
"In particular, SoCO1 sequences can be detected by relaxing GenoTHREAT 's 100% query coverage constraint and returning a hit if a sequence with the highest score is a SoC.",
"[t] $P$ – a Sequence of Concern $O$ – an obfuscated Sequence of Concern @post-condition: Expressing $O$ in Cas9 containing environment results in assembly of $P$ Partition $P$ into 154bp fragments $P=p_1+p_2+\\ldots +p_n$ Let $r$ be a promoter and a ribosome binding site Let $t$ be a terminator Let $f$ be a 23bp long efficient Cas9 cutting site [17] $f_{RC} \\leftarrow $ reverse complement of $F$ $body_1\\leftarrow r+p_1+f$ $\\forall _{i=2}^{n-1} body_i \\leftarrow f_{RC}+p_i+f$ $body_n\\leftarrow f_{RC}+p_n+t$ $Body\\leftarrow r+p_1+f+\\left(\\sum _{i=2}^{n-1} f_{RC}+p_i+f\\right)+f_{RC}+p_n+t$ Let $grs$ be a gRNA scaffold targeting $f$ $grs$ also targets $f_{RC}$ on the opposite strand.",
"each $p_{i},p_{i+1}$ $hdr_i\\leftarrow $ 32bp suffix of $p_i+$ 32bp prefix of $p_{i+1}$ Let $nonce$ be a 40bp long DNA sequence, which does not contain $f$ or $f_{RC}$ $Decoder\\leftarrow nonce+grs+hdr_1+nonce+\\ldots +hdr_{n-1}$ Assemble and return the obfuscated sequence $osoc=Body + Decoder$ SoC Obfuscation 1 (SoCO1) Figure: Sample trojan plugin.",
"Synthetic DNA order form (a).",
"A browser plugin (b) that adds a visualization of DNA sequences within form text fields (c).Code snippet of a malicious payload replacing the submitted DNA order with predefined attack DNA (d).Figure: Excerpts from alignment reports generated using BLASTX for the obfuscated α\\alpha -conotoxin PeIA.",
"(a) The alignment domains for 29 alignments to 10 target sequences specific to the conus family, a species that produces α\\alpha -conotoxin PeIA.",
"(b) The images (as defined in Section ) of three alignments to the α\\alpha -conotoxin PeIA peptide.The alignments are arranged according to the image start positions, and the respective domain ranges appear in braces [..] at the alignments' ends.The alignment markers α 1 \\alpha _1,α 2 \\alpha _2,α 3 \\alpha _3 and the domain ranges were added to theNCBI report.The scores of the three alignments range from 47.3 to 54.6.",
"(c) A 2D plot presenting the alignments between obfuscated and non-obfuscated α\\alpha -conotoxin PeIA, found by Blast 2 sequences .",
"The x-axis represents the obfuscated DNA, and the y-axis represents the DNA that encodes α\\alpha -conotoxin PeIA.",
"The red and blue lines represent extended alignments, with zero penalties for long gaps in the target sequence (deletion).",
"(d) Alignments of the unified domains of α 1 \\alpha _1 and α 3 \\alpha _3 with their unified images.",
"Two local alignments found by Blast 2 sequences are successfully merged into a cleaned alignment with 100 Percent Identity."
]
] | 2011.14224 |
[
[
"Optimal Semi-supervised Estimation and Inference for High-dimensional\n Linear Regression"
],
[
"Abstract There are many scenarios such as the electronic health records where the outcome is much more difficult to collect than the covariates.",
"In this paper, we consider the linear regression problem with such a data structure under the high dimensionality.",
"Our goal is to investigate when and how the unlabeled data can be exploited to improve the estimation and inference of the regression parameters in linear models, especially in light of the fact that such linear models may be misspecified in data analysis.",
"In particular, we address the following two important questions.",
"(1) Can we use the labeled data as well as the unlabeled data to construct a semi-supervised estimator such that its convergence rate is faster than the supervised estimators?",
"(2) Can we construct confidence intervals or hypothesis tests that are guaranteed to be more efficient or powerful than the supervised estimators?",
"To address the first question, we establish the minimax lower bound for parameter estimation in the semi-supervised setting.",
"We show that the upper bound from the supervised estimators that only use the labeled data cannot attain this lower bound.",
"We close this gap by proposing a new semi-supervised estimator which attains the lower bound.",
"To address the second question, based on our proposed semi-supervised estimator, we propose two additional estimators for semi-supervised inference, the efficient estimator and the safe estimator.",
"The former is fully efficient if the unknown conditional mean function is estimated consistently, but may not be more efficient than the supervised approach otherwise.",
"The latter usually does not aim to provide fully efficient inference, but is guaranteed to be no worse than the supervised approach, no matter whether the linear model is correctly specified or the conditional mean function is consistently estimated."
],
[
"Introduction",
"Thanks to the development of modern technology, big datasets are routinely collected in many areas including health care, business, epidemiology and social science.",
"Not surprisingly, the availability of large datasets increases the chance of complications and heterogeneity.",
"For example, the response variable(s) may be missing due to various reasons.",
"In Electronic Health Records (EHR) based association studies, one major challenge is the lack of gold-standard health outcomes or phenotypes.",
"The validated phenotypes are often obtained by manual chart reviews, which are often prohibitively expensive.",
"In reality, only a very small subset of patients can be annotated by experts.",
"For the rest of the patients, we only observe their covariate information, which is often high-dimensional.",
"Developing efficient statistical methods to analyze such data is a timely and important problem.",
"We consider the following so-called semi-supervised setting.",
"Let $Y$ denote the outcome variable and $X$ denote the $p$ -dimensional covariates.",
"In addition to $n$ i.i.d.",
"samples $(Y_1, X_1),...(Y_n, X_n)\\sim (Y, X)$ , we also observe $N$ i.i.d.",
"data consisting of only covariates, $X_{n+1},...,X_{N+n}\\sim X$ .",
"Following the convention, the former is referred to as labeled data and the latter is called unlabeled data.",
"For notation simplicity, we denote by $=(Y_1,...,Y_n)^T\\in ^n$ , $=(X_1,...,X_n)^T \\in ^{n\\times p}$ the outcomes and covariates from the labeled data, respectively, and $\\tilde{}=(X_1,...,X_{N+n})^T \\in ^{(N+n)\\times p}$ the covariates from the labeled data as well as the unlabeled data.",
"In this work, we focus on high-dimensional problems; namely, $p$ can be much larger than $n$ .",
"The size of the unlabeled data $N$ is allowed but not required to be larger than $n$ .",
"We consider the so-called assumption lean regression framework [9], [5], $ Y=f(X)+\\epsilon ,$ where $f(X)=E(Y|X)$ is the unknown conditional mean function, $\\epsilon $ is the random error independent of $X\\in ^p$ with $(\\epsilon )=0$ , $(\\epsilon ^2)=\\sigma ^2$ , and $\\sigma ^2$ is an unknown parameter.",
"We assume that $X$ and $Y$ are centered with $(X)=0$ and $(f(X))=0$ .",
"On the one hand, we would like to put as fewer model assumptions on $f(X)$ as possible to make the model more flexible.",
"On the other hand, for the purpose of interpretability, we often fit simple parametric models such as linear regression to explain the association between $Y$ and $X$ in many practical data analysis.",
"To meet both ends, we focus on the linear regression as a working model where the true data generating process follows (REF ).",
"Since $[(Y-X^T)^2]=[(f(X)-X^T)^2]+\\sigma ^2$ , the regression coefficients in a linear model correspond to the $L_2()$ projection of $f(X)$ onto the linear space spanned by $X$ , i.e., $^*=\\arg \\min _{\\theta \\in \\mathbb {R}^p} [(f(X)-X^T)^2]\\in ^p$ which describes the linear dependence between $Y$ and $X$ .",
"We note that since $(X)=0$ and $(f(X))=0$ , the intercept parameter in the working linear regression is 0.",
"Thus, we do not include the intercept parameter in the definition of $^*$ .",
"Our goal is to estimate the high dimensional parameter $^*$ and further construct confidence intervals or hypothesis tests for some linear functional of $^*$ , i.e., $^T^*$ for some given $\\in ^p$ .",
"In the supervised setting with $n$ labeled observations, a number of penalized estimators have been proposed to estimate $^*$ , such as lasso [39] and Dantzig selector [13].",
"For statistical inference, there has been some recent research on debiased lasso estimators for hypothesis tests and confidence intervals, for example, [46], [42], [22], [12], [29], [28], a list that is far from exhaustive.",
"While significant progress has been made towards understanding the estimation and inference in the fully supervised setting, there is very limited research in the semi-supervised setting.",
"It is important to observe that under (REF ), since the linear regression is the working model, the covariate $X$ is no longer the ancillary statistic for the regression parameter $^*$ .",
"Therefore, the information of $X$ in the unlabeled data may improve the estimation and inference of $^*$ .",
"In particular, we answer two critical questions the presence of the unlabeled data poses.",
"Q1: Can we use the labeled data as well as the unlabeled data to construct a semi-supervised estimator for $^*$ such that its convergence rate is faster than the supervised estimators (e.g., lasso and Dantzig selector)?",
"In the literature, [3] proposed a modified lasso estimator of $^*$ in the semi-supervised setting and showed that the excess risk of their estimator in prediction has the same rate of convergence as the standard lasso estimator.",
"Their results neither confirm nor deny the existence of estimators with improved statistical rate when unlabeled data are available.",
"To the best of our knowledge, this question remains an open problem.",
"In Theorem REF , we establish the minimax lower bound for the estimation of $^*$ in the semi-supervised setting.",
"Based on the lower bound, when $N$ is sufficiently large, the error term due to the model misspecification or equivalently the nonlinearity of $f(X)$ becomes ignorable.",
"This reveals one potential benefit from using the unlabeled data in that the estimation of $^*$ can be more robust to the model misspecification.",
"Moreover, we show that the fully supervised estimators (e.g., lasso and Dantzig selector) do not attain this lower bound.",
"To demonstrate how to achieve this lower bound, in Section REF we propose an estimator $\\hat{}_{D}$ under the assumption that the unknown conditional mean function $f(X)$ can be consistently estimated.",
"We show in Theorem REF that the proposed estimator $\\hat{}_{D}$ attains the minimax lower bound (up to the $\\log s$ factor).",
"For this reason, we refer $\\hat{}_{D}$ as the optimal semi-supervised estimator.",
"As a byproduct of this theorem, our estimator $\\hat{}_{D}$ has a faster convergence rate than the supervised estimators.",
"Q2: Does there exist an estimator of the linear functional of $^*$ that is asymptotically normal and is guaranteed to be more efficient than the supervised estimator (e.g., debiased lasso)?",
"In the literature, several methods have been proposed to address this question in low-dimensional setting with $p$ fixed.",
"[2] proposed an adjusted linear regression approach and [15] proposed an efficient and adaptive semi-supervised estimator (EASE) based on non-parametric imputation.",
"Both methods produce more efficient estimators for each component of $^*$ than the least square estimator (LSE) using only labeled data.",
"However, they do not guarantee the efficiency improvement if the parameter of interest is the linear combination of $^*$ (e.g., $\\theta ^*_1+\\theta ^*_2$ ).",
"Therefore, none of the existing methods provides a satisfactory answer to this question, even when $p$ is fixed.",
"Following our answer to Q1, while the estimator $\\hat{}_{D}$ is rate optimal, the limiting distribution of $\\hat{}_{D}$ is intractable due to the regularization.",
"To construct confidence intervals and hypothesis tests, we further propose two estimators based on the existing debiased lasso approach [46], [42], [22], [12], [29], [28].",
"In particular, using $\\hat{}_{D}$ as an initial estimator, we construct a one-step estimator $\\hat{}^d$ .",
"In Theorem REF , we show that the estimator $^T\\hat{}^d$ is asymptotically normal and attains the semi-parametric efficiency bound when $n\\ll N$ .",
"For this reason, we call $\\hat{}^d$ efficient semi-supervised estimator.",
"However, this estimator depends on the consistency of the estimate of $f(X)$ .",
"If $f(X)$ cannot be consistently estimated, the estimator $^T\\hat{}^d$ is not guaranteed to be more efficient than the supervised debiased lasso estimators [42].",
"To address the question Q2, we further propose a safe semi-supervised estimator which does not require the estimate of $f(X)$ .",
"The main idea is to construct a set of unbiased estimating functions and decorrelate the score function to reduce the variability.",
"In Theorem REF , we show that the corresponding estimator $^T\\hat{}_{S,\\psi }^d$ is asymptotically normal, where $\\psi $ is a tuning parameter.",
"When the linear model is misspecified and $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=\\rho $ for some $0\\le \\rho <1$ , the estimator $^T\\hat{}_{S,\\psi }^d$ with $0<\\psi <2$ is strictly more efficient than the debiased lasso, leading to more powerful hypothesis tests and shorter confidence intervals.",
"We attain the maximum variance reduction by choosing $\\psi =1$ .",
"In addition, if either the linear model is correctly specified $f(X)=X^T^*$ or the size of the unlabeled data is small $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=1$ (i.e., $N\\ll n$ ), the estimator $^T\\hat{}_{S,\\psi }^d$ is asymptotically equivalent to the debiased lasso estimator.",
"In summary, the estimator $^T\\hat{}_{S,\\psi }^d$ provides a safe use of the unlabeled data, since it is always no worse than the supervised estimators, no matter whether the linear model is correctly specified or the conditional mean function is consistently estimated."
],
[
"Other related work",
"In computer science, a large number of classification algorithms have been developed under semi-supervised setting, which mainly focus on the data with discrete labels; see [49], [16] for some surveys.",
"Common assumptions such as manifold assumption and cluster assumption were made in the literature in order to obtain fast rate of convergence in classification [33].",
"In non-parametric regression problem, [44] showed that unlabeled data do not always help to improve the rate of the mean squared error, but with semi-supervised smoothness assumption the estimator with faster rate was developed.",
"Recently, there are progress considering how to safely make use of the unlabeled data to achieve an estimator with smaller asymptotic variance for inference.",
"[45] proposed a general semi-supervised inference framework to improve the estimation of the population mean $(Y)$ without specific distributional assumptions relating $Y$ and $X$ .",
"They allowed the dimension of $X$ to grow but no faster than $n^{1/2}$ .",
"Therefore, the results cannot be used in high-dimensional setting.",
"With high-dimensional data, [47] proposed semi-supervised estimators of population mean and variance and established their asymptotic distributions.",
"By fitting a working linear regression, one can rewrite $(Y)$ as a linear combination of $^*$ , $(Y)=(X^T^*)$ .",
"The way they utilized the unlabeled data is to estimate $(X)$ by $\\sum _{i=n+1}^{n+N}X_i/N$ .",
"It is of interest to note that, they required $\\lim _{n\\rightarrow \\infty } \\frac{n}{n+N}<1/2$ to guarantee the safe inference on $(Y)$ (i.e., more efficient than the sample mean of $Y$ in the labeled data).",
"In comparison, our proposed safe semi-supervised inference requires a weaker condition $\\lim _{n\\rightarrow \\infty } \\frac{n}{n+N}<1$ .",
"In high dimensional regime, [11] considered how to estimate the explained variance $^{*T}^*$ in the semi-supervised setting.",
"Their estimator achieved the optimal rate of convergence and was asymptotically normal.",
"However, their results were established under the assumption that the working linear model is correctly specified, which differed from our assumption lean framework."
],
[
"Organization of the Paper",
"The rest of this paper is organized as follows.",
"In Section , we first give the minimax lower bound for semi-supervised estimation, then introduce the proposed estimator and its corresponding upper bound.",
"In Section we devote ourselves into semi-supervised inference, where we propose an efficient procedure provided that the conditional mean function is consistently estimated, and a safe procedure which is guaranteed to be no worse than the one using only the labeled data.",
"Numerical experiments and a real data application are in Sections and , respectively.",
"The paper is concluded with a discussion section.",
"All the technical proofs are contained in the appendix."
],
[
"Notations",
"The following notation is adopted throughout this paper.",
"For $v=(v^{(1)},...,v^{(p)})^{T} \\in \\mathbb {R}^p$ , and $1 \\le q \\le \\infty $ , we define $\\Vert v\\Vert _q=(\\sum _{i=1}^p |v^{(i)}|^q)^{1/q}$ , $\\Vert v\\Vert _0=|\\textrm {supp}(v)|$ , where $\\textrm {supp}(v)=\\lbrace i: v^{(i)}\\ne 0\\rbrace $ and $|A|$ is the cardinality of a set $A$ .",
"Denote $\\Vert v\\Vert _{\\infty }=\\max _{1\\le i \\le p} |v^{(i)}|$ and $v^{\\otimes 2}=v v^T$ .",
"For a matrix $=[M_{ij}]$ , $_{i\\cdot }$ and $_{\\cdot j}$ denote the $i$ -th row and $j$ -th column respectively.",
"Define $\\Vert \\Vert _{\\max }=\\max _{ij}|M_{ij}|$ , $\\Vert \\Vert _1=\\max _{j}\\sum _{i}|M_{ij}|$ , $\\Vert \\Vert _{\\infty }=\\max _{i}\\sum _{j}|M_{ij}|$ .",
"If the matrix $$ is symmetric, then $\\Lambda _{\\min }()$ and $\\Lambda _{\\max }()$ are the minimal and maximal eigenvalues of $$ .",
"We denote $_p$ by the $p\\times p$ identity matrix.",
"For $S\\subseteq \\lbrace 1,...,p\\rbrace $ , let $v_S=\\lbrace v^{(k)}: k\\in S\\rbrace $ and $ S^c$ be the complement of $S$ .",
"For matrix $\\in ^{n\\times p}$ and index set $D\\subseteq \\lbrace 1,...,n\\rbrace $ , $_D=\\lbrace X_i:i\\in D\\rbrace ^T\\in ^{|D|\\times p}$ .",
"For a function $f$ , let $\\Vert f\\Vert _2^2=[f(X)^2]$ denote the $L_2()$ norm of $f$ .",
"For two positive sequences $a_n$ and $b_n$ , we write $a_n\\asymp b_n$ if $C\\le a_n/b_n\\le C^{\\prime }$ for some $C,C^{\\prime }>0$ .",
"Similarly, we use $a\\lesssim b$ to denote $a\\le Cb$ for some constant $C>0$ .",
"Given $a,b\\in $ , let $a\\vee b$ and $a\\wedge b$ denote the maximum and minimum of $a$ and $b$ .",
"For notational simplicity, we use $C, C^{\\prime }, C^{\\prime \\prime }$ to denote generic constants, whose values can change from line to line.",
"To characterize the tail behavior of random variables, we introduce the following definition.",
"[Sub-Gaussian variable and vector] A random variable $X$ is called sub-Gaussian if there exists some positive constant $K_2$ such that $(|X|>t)\\le \\exp (1-t^2/K^2_2)$ for all $t\\ge 0$ .",
"The sub-Gaussian norm of $X$ is defined as $\\Vert X\\Vert _{\\psi _2}=\\sup _{p\\ge 1}p^{-1/2}(|X|^p)^{1/p}$ .",
"A vector $\\in ^p$ is a sub-Gaussian vector if the one-dimensional marginals $^T$ are sub-Gaussian for all $\\in ^p$ , and its sub-Gaussian norm is defined as $\\Vert \\Vert _{\\psi _2}=\\sup _{\\Vert \\Vert _2=1}\\Vert ^T\\Vert _{\\psi _2}$ ."
],
[
"Optimal Rate in Semi-supervised Estimation",
"In Section REF , we present the minimax lower bound for the estimation of $^*$ in the semi-supervised setting.",
"A key message from the lower bound is that the benefit of using the unlabeled data is to mitigate the error due to model misspecification.",
"In Section REF , we propose a new estimator that nearly attains the lower bound.",
"Throughout the paper, we consider the random design."
],
[
"Minimax lower bound for semi-supervised estimation",
"Recall that in the semi-supervised setting we observe $n$ i.i.d.",
"labeled data $(Y_1, X_1),...(Y_n, X_n)\\sim (X,Y)$ and additional $N$ unlabeled data $X_{n+1},...,X_{N+n}\\sim X$ .",
"Let $P_{X,Y}$ and $P_X$ denote the joint distribution of $(X,Y)$ and the marginal distribution of $X$ , respectively.",
"To establish the minimax lower bound, we consider the following class of distributions $\\mathcal {P}_{\\Phi ,\\sigma }=\\lbrace P_{X,Y}|~ Y = f(X)+\\epsilon ,~ \\Vert ^*\\Vert _0\\le s, ~\\textrm {and}~ (\\epsilon )=\\sigma ^2, (f(X)-X^T^*)^2 \\le \\Phi ^2 , P_X\\in \\mathcal {P}_X \\rbrace ,$ where $f(X)=(Y|X)$ , $^*=\\arg \\min _{\\in \\mathbb {R}^p}[(f(X)-X^T)^2]$ depends on the distribution $P_{X,Y}$ implicitly, and $\\mathcal {P}_X=\\lbrace P_X|~ [X]=0, (X_j)=1 ~\\textrm {and}~ \\lambda _{\\min }((X))\\ge C_{\\min }>0$ } with some constant $C_{\\min }$ .",
"For notational simplicity, we write $(\\cdot )$ for $_{P_{X,Y}}(\\cdot )$ .",
"We note that, $\\mathcal {P}_{\\Phi ,\\sigma }$ is indexed by two non-negative parameters $\\Phi ^2$ and $\\sigma ^2$ , where the former controls the magnitude of model misspecification $f(X)-X^T^*$ or equivalently the nonlinearity of $f(X)$ in the second moment and the latter is the variance of $\\epsilon $ .",
"In particular, we allow $\\Phi ^2$ to grow with $n$ in our framework.",
"The following theorem offers the lower bound for the convergence rate of any estimator of $^*$ over the class of distributions $\\mathcal {P}_{\\Phi ,\\sigma }$ .",
"If $s\\log (p/s)\\le C n$ , $(s-1)c_1^{\\prime }\\log (p/s)\\le 8(n+N)$ for some absolute constants $C, c_1^{\\prime }$ and $2\\le s\\le (n-1)/4$ , we have that for any $1\\le q\\le \\infty $ , $ \\inf _{\\hat{}} \\sup _{P_{X,Y}\\in \\mathcal {P}_{\\Phi ,\\sigma }} _{P_{X,Y}} \\Big [\\Vert \\hat{}-^*\\Vert _q\\ge c_1s^{1/q}\\Big (\\Phi \\sqrt{\\frac{\\log (p/s)}{n+N}}+\\sigma \\sqrt{\\frac{\\log (p/s)}{n}}\\Big )\\Big ]>c_2,$ where $\\inf _{\\hat{}}$ denotes the infimum over all estimators of $^*$ , $c_1$ and $c_2$ are some positive constants and we simply denote $s^{1/\\infty }=1$ .",
"(1) The lower bound (REF ) consists of two components.",
"Up to some absolute constants, the first term $s^{1/q}\\Phi \\sqrt{{\\log (p/s)}/(n+N)}$ corresponds to the error due to potential model misspecification and the second term $s^{1/q}\\sigma \\sqrt{{\\log (p/s)}/{n}}$ comes from the uncertainty inherited from the randomness of the error $\\epsilon $ , which always exists even if the regression function is linear $f(X)=X^T^*$ .",
"In this case, we have $\\Phi =0$ and the lower bound agrees with the existing result for sparse linear regression [43], [4].",
"(2) The sample size of the unlabeled data $N$ plays an important role in the lower bound (REF ).",
"We first consider the case $\\frac{\\Phi }{\\sigma }\\sqrt{\\frac{n}{n+N}}\\rightarrow \\infty $ , which may happen if $\\Phi \\rightarrow \\infty $ .",
"In Appendix REF , we construct examples in which we have $\\Phi \\asymp s^{1/2}$ so that $\\Phi $ tends to infinity as the sparsity grows.",
"In this case, the dominating term in the lower bound is $s^{1/q}\\Phi \\sqrt{{\\log (p/s)}/(n+N)}$ , which can be reduced as $N$ increases.",
"If $N$ is sufficiently large such that $\\frac{\\Phi }{\\sigma }\\sqrt{\\frac{n}{n+N}}\\rightarrow c<\\infty $ , the lower bound attains its minimum $s^{1/q}\\sigma \\sqrt{{\\log (p/s)}/{n}}$ , which can be viewed as the irreducible error in the semi-supervised setting since a further increase of $N$ would no longer decrease the lower bound.",
"As an illustration, we plot the lower bound in Figure REF .",
"(3) In the following, we compare the lower bound with the upper bound from the following supervised Dantzig selector, which only uses the labeled data, $\\hat{}_L=\\arg \\min \\Vert \\Vert _1,~~\\textrm {s.t.",
"}~~ \\Big \\Vert \\frac{1}{n}\\sum _{i=1}^n (Y_i-X_i^T)X_i\\Big \\Vert _\\infty \\le \\lambda _L,$ where $\\lambda _L$ is a tuning parameter.",
"Under the same assumption lean framework, we show in Lemma REF that the upper bound for $\\Vert \\hat{}_L-^*\\Vert _1$ is $s(\\Phi +\\sigma )\\sqrt{\\frac{\\log p}{n}},$ which differs from the lower bound in (REF ) with $q=1$ (ignoring the $\\log s$ factor); see Figure REF .",
"Moreover, under the condition $N\\gg n$ , it can be easily shown that the lower bound in (REF ) is strictly smaller in order than the upper bound of $\\hat{}_L$ if and only if $\\Phi /\\sigma \\rightarrow \\infty $ .",
"In this case, the fully supervised estimator does not attain the lower bound and thus is sub-optimal in the minimax sense; see Figure REF .",
"Figure: Plot of the lower bound with q=1q=1 in Theorem (the solid curve) and the upper bound from Dantzig in () (the dashed line) against the value of (N+n)/n(N+n)/n.",
"In the plot, we fix nn and vary the value of NN.",
"The region between these two lines corresponds to the gap between the lower bound and the upper bound for the supervised estimator."
],
[
"Optimal semi-supervised estimator",
"Motivation.",
"To motivate our estimator, we first briefly explain how the convergence rate of $\\hat{}_L$ in (REF ) is derived.",
"Following the standard argument in [8], the Dantizig selector satisfies $\\Vert \\hat{}_L-^*\\Vert _1=O_p(s\\lambda _L)$ , where the tuning parameter $\\lambda _L \\gtrsim \\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)\\Vert _\\infty $ .",
"In the proof of Lemma REF , we further show that $\\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)\\Vert _\\infty \\lesssim \\sqrt{\\frac{\\log p}{n}}\\lbrace (Y_i-X_i^T^*)^2\\rbrace ^{1/2}$ with high probability.",
"The desired bound (REF ) is obtained by noting that $(Y_i-X_i^T^*)^2=(Y_i-f(X_i))^2+(f(X_i)-X_i^T^*)^2\\le \\sigma ^2+\\Phi ^2,$ as $(\\epsilon |X)=0$ , where $(\\epsilon ^2)=\\sigma ^2$ and $(f(X)-X^T^*)^2\\le \\Phi ^2$ .",
"In view of (REF ) and Remark REF (3), we see that the slow rate of $\\hat{}_L$ is driven by the $(f(X)-X^T^*)^2$ term in (REF ).",
"Key Step.",
"To reduce the effect of $\\Phi ^2$ in the upper bound, our key idea is to decompose the score function $\\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)=\\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-f(X_i))+\\frac{1}{n}\\sum _{i=1}^nX_i(f(X_i)-X_i^T^*),$ and replace the last term with $\\frac{1}{n+N}\\sum _{i=1}^{n+N}X_i(f(X_i)-X_i^T^*)$ , the sample average over both labeled and unlabeled data.",
"This leads to the following modified score function $\\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-f(X_i))+\\frac{1}{n+N}\\sum _{i=1}^{n+N}X_i(f(X_i)-X_i^T^*)=\\bar{}-\\hat{}_{n+N}^*,$ where $\\hat{}_{n+N}=\\frac{1}{n+N}\\sum _{i=1}^{n+N}X_i^{\\otimes 2}$ and $\\bar{}= \\frac{1}{n}\\sum _{i=1}^n X_iY_i-\\frac{1}{n}\\sum _{i=1}^n X_if(X_i)+\\frac{1}{n+N}\\sum _{i=1}^{n+N}X_if(X_i).$ Computation of $\\bar{}$ .",
"To compute $\\bar{}$ , we need to find an estimator for $f(\\cdot )$ , the unknown conditional mean function.",
"For model flexibility, we can use semiparametric/nonparametric techniques to estimate $f(\\cdot )$ and will discuss some examples of $\\hat{f}(\\cdot )$ later, such as in Remark REF .",
"A serious challenge may arise from deriving the theoretical property of our proposed optimal semi-supervised estimator if we use all data to obtain $\\hat{f}(\\cdot )$ due to the dependence between the estimator $\\hat{f}(\\cdot )$ and the data $(X_i,Y_i)$ in the sample average from $\\bar{}$ .",
"To bypass this challenge, we adopt the popular cross-fitting technique that was devised for semiparametric estimation problems [7], [37] as well as for high-dimensional data [35], [18].",
"For notational simplicity, we denote by $D^*$ the labeled data and $D$ the full dataset with both labeled and unlabeled data.",
"Without loss of generality, we split the labeled data $D^*$ into two folds $D^*_1$ and $D_2^*$ with size $n_1=n_2=n/2$ .",
"Similarly, we split the unlabeled data into two folds $U_1$ and $U_2$ with size $N_1=N_2=N/2$ .",
"Merging with $D^*_1$ and $D_2^*$ respectively, we obtain two independent data sets $D_1=D_1^*\\cup U_1$ and $D_2=D_2^*\\cup U_2$ .",
"Next, for $j=\\lbrace 1,2\\rbrace $ , we train the estimator $\\hat{f}^{-j}$ using the data $D^*\\backslash D^*_j$ and then construct $ \\hat{}_j= \\frac{1}{n_j}\\sum _{i\\in D_j^*} X_iY_i-\\frac{1}{n_j}\\sum _{i\\in D_j^*} X_i\\hat{f}^{-j}(X_i)+\\frac{1}{n_j+N_j}\\sum _{i\\in D_j}X_i\\hat{f}^{-j}(X_i).$ In view of the modified score function (REF ), replacing $\\bar{}$ with $\\hat{}=(\\hat{}_1+\\hat{}_2)/2$ , we propose the following estimator $ \\hat{}_{D}=\\arg \\min \\Vert \\Vert _1,~~\\textrm {s.t.",
"}~~ \\Vert \\hat{}_{n+N}-\\hat{}\\Vert _\\infty \\le \\lambda _D.$"
],
[
"Upper bound of the proposed estimator",
"Now we develop the theoretical property of the proposed estimator $\\hat{}_{D}$ .",
"We make the following assumptions: $^{-1/2}X$ is a zero mean sub-gaussian vector with bounded sub-gaussian norm and $(X)=$ has smallest eigenvalue $\\Lambda _{\\min }()\\ge C_{\\min }>0$ for some positive constant $C_{\\min }$ .",
"Moreover, $\\max _{1\\le j\\le p}\\Sigma _{jj}=O(1)$ .",
"$\\max _{1\\le i\\le n+N}\\Vert X_i\\Vert _\\infty \\le K_1$ where we allow $K_1$ to diverge with $(n,N,p)$ .",
"$(\\epsilon ^2)=\\sigma ^2$ and $[(f(X)-X^T^*)^2]\\le \\Phi ^2$ .",
"$^*$ is $s$ -sparse with $\\Vert ^*\\Vert _0=s$ , and $\\frac{s\\log p}{n+N}=O(1)$ .",
"Assumption (A1) is a standard technical condition for $X$ in order to verify the restricted eigenvalue (RE) condition [8].",
"Assumption (A2) imposes the boundedness of the covariates, which simplifies the analysis when the linear model is misspecified [10].",
"In particular, when $X_i$ is uniformly bounded, $K_1$ becomes a constant.",
"If each component of $X_i$ is Gaussian or sub-Gaussian, Assumption (A2) still holds with high probability with $K_1=\\sqrt{\\log [p(n+N)]}$ .",
"Assumption (A3) only requires the existence of the second moment of $\\epsilon $ and $f(X)-X^T^*$ .",
"We note that, unlike most of the work in high-dimensional statistics including [8], we do not assume the residual $Y-X^T^*$ is sub-Gaussian.",
"This is because the misspecified model $Y-X^T^*=\\epsilon +(f(X)-X^T^*)$ contains the nonlinearity term $f(X)-X^T^*$ which can be large.",
"While we only assume the moment condition in Assumption (A3), the boundedness in Assumption (A2) enables us to apply the Nemirovski moment inequality (Lemma REF ) to control the deviation of the sample estimates from their population.",
"Assumption (A4) is the sparsity condition.",
"In particular, [10] provided some sufficient conditions on $f(X)$ and the distribution of $X$ under which $^*$ is sparse in the misspecified model.",
"We further require $\\frac{s\\log p}{n+N}=O(1)$ to verify the RE condition under the random design; see Lemma REF .",
"Given Assumption REF , the following theorem shows that the proposed estimator $\\hat{}_D$ in (REF ) achieves the near optimal rate for estimating $^*$ .",
"For the purpose of inference, we only present the asymptotic results, where $n,p\\rightarrow \\infty $ and $N$ can be either fixed or tends to infinity as well.",
"Suppose Assumption REF holds and the estimator $\\hat{f}^{-j}(X)$ satisfies $\\Vert \\hat{f}^{-j}-f\\Vert _2=O_p(b_n),$ for $j=1,2$ , where $b_n$ is a deterministic sequence.",
"With tuning parameter $\\lambda _D\\asymp K_1(\\Phi \\sqrt{\\frac{\\log p}{n+N}}+\\sigma \\sqrt{\\frac{\\log p}{n}}+b_n\\sqrt{\\frac{\\log p}{n}})$ , the estimator $\\hat{}_D$ in (REF ) achieves the following error bounds $\\Vert \\hat{}_D-^*\\Vert _1=O_p\\Big (K_1s\\Big \\lbrace \\Phi \\sqrt{\\frac{\\log p}{n+N}}+ (\\sigma +b_n)\\sqrt{\\frac{\\log p}{n}}\\Big \\rbrace \\Big ),$ $\\Vert \\hat{}_D-^*\\Vert _2=O_p\\Big (K_1s^{1/2}\\Big \\lbrace \\Phi \\sqrt{\\frac{\\log p}{n+N}}+ (\\sigma +b_n)\\sqrt{\\frac{\\log p}{n}}\\Big \\rbrace \\Big ).$ Moreover, if $b_n/\\sigma =o(1)$ and $K_1=O(1)$ , we obtain $\\Vert \\hat{}_D-^*\\Vert _1=O_p\\Big (s\\Big \\lbrace \\Phi \\sqrt{\\frac{\\log p}{n+N}}+ \\sigma \\sqrt{\\frac{\\log p}{n}}\\Big \\rbrace \\Big ),$ $\\Vert \\hat{}_D-^*\\Vert _2=O_p\\Big (s^{1/2}\\Big \\lbrace \\Phi \\sqrt{\\frac{\\log p}{n+N}}+ \\sigma \\sqrt{\\frac{\\log p}{n}}\\Big \\rbrace \\Big ),$ which match the minimax lower bound with $q=1, 2$ in Theorem REF up to a $\\log s$ factor.",
"Assume that $\\sigma ^2$ is bounded away from 0 by a constant.",
"In order to attain the near minimax optimal rate, we only need a very mild condition on $\\hat{f}^{-j}$ , that is $\\hat{f}^{-j}$ is consistent in the $L_2()$ norm.",
"If the knowledge of $f(X)$ is available to some extent, we can leverage this information to construct estimators of $f(X)$ .",
"For example, if $f(X_i)$ can be well approximated by sparse additive models $f(X_i)=\\sum _{k\\in S} f_k(X_{ik})$ for some set $S\\subseteq \\lbrace 1,...,p\\rbrace $ with small cardinality, then we can directly apply the existing estimators in the literature, see [25], [26], [21], [32] among many others.",
"In particular, Corollary 2 in [21] implies that their adaptive group lasso estimator $\\hat{f}$ satisfies $\\Vert \\hat{f}-f\\Vert _2=O_p(n^{-d/(2d+1)})$ , where $d$ is the smoothness of the function $f_k(\\cdot )$ .",
"This meets the condition $b_n=o(1)$ .",
"In a recent work, [3] proposed a modified lasso estimator for prediction in the semi-supervised setting, which can be reformatted as the following Dantzig selector $ \\hat{}_{U}=\\arg \\min \\Vert \\Vert _1,~~\\textrm {s.t.",
"}~~ \\Vert \\hat{}_{n+N}-\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i \\Vert _\\infty \\le \\lambda _U,$ where $\\hat{}_{n+N}=\\frac{1}{n+N}\\sum _{i=1}^{n+N}X_i^{\\otimes 2}$ .",
"They showed that if a large number of unlabeled data are used to compute $\\hat{}_{n+N}$ , it becomes more plausible to assume that the compatibility (or RE) constant is bounded away from zero.",
"Moreover, they proved that the error bound for the excess risk in prediction remains $O_p(s \\log p/n)$ under certain conditions, including $|Y|\\le C$ for some constant $C>0$ which indeed implies $\\Phi $ and $\\sigma =O(1)$ by their proof of Theorem 7.",
"To make a fair comparison of $\\hat{}_{U}$ with our estimator $\\hat{}_{D}$ , we show that under the same conditions in our Theorem REF , $\\Vert \\hat{}_U-^*\\Vert _1=O_p\\Big (s(\\Phi +\\sigma +(^{*T}^*)^{1/2})\\sqrt{\\frac{\\log p}{n}}\\Big ).$ The proof is deferred to Appendix REF .",
"It is seen that $\\hat{}_U$ has a slower rate than our estimator $\\hat{}_{D}$ if $\\Phi /\\sigma \\rightarrow \\infty $ or $^{*T}^*/\\sigma ^2\\rightarrow \\infty $ .",
"Perhaps, a more surprising fact is that the convergence rate of $\\hat{}_U$ can be even slower than the fully supervised estimator $\\hat{}_L$ in (REF ) if $^{*T}^*/(\\sigma ^2+\\Phi ^2)\\rightarrow \\infty $ .",
"Indeed, our simulation studies confirm that the estimator $\\hat{}_{U}$ often produces larger estimation error than the lasso/Dantzig selector that only use the labeled data.",
"Finally, we note that the estimator $\\hat{}_{U}$ can be considered as an extreme case of our estimator $\\hat{}_D$ that does not account for the estimated conditional mean function in (REF ) (i.e., set $\\hat{f}^{-j}=0$ in $\\hat{}_j$ ).",
"This remark shows that, if $f(\\cdot )$ is poorly estimated, it may not be beneficial to estimate $$ by $\\hat{}_{n+N}$ as in (REF ), a common procedure to incorporate the information from unlabeled data."
],
[
"Semi-supervised Inference",
"In this section, we address the inference problem for a linear combination of $^*$ under the semi-supervised setting.",
"In Section REF , we consider the case where there exists a proper estimator of the unknown conditional mean function $f(X)$ .",
"An efficient semi-supervised inference procedure is proposed.",
"We further extend our method in Section REF to a more general case where we do not require the estimation of $f(X)$ .",
"We propose a safe semi-supervised inference approach that guarantees the efficiency improvement over the supervised debiased estimators."
],
[
"Efficient semi-supervised inference",
"Motivated by the formulation of the regularized estimator $\\hat{}_{D}$ in (REF ), we can view $h(\\tilde{},Y;)=\\hat{}_{n+N}-\\hat{}$ as an estimating function for $$ .",
"Borrowing the idea from the classical one-step estimator and the debiased lasso, we construct the following estimator, $\\hat{}^d =\\hat{}_{D}-\\hat{}h(\\tilde{},Y;\\hat{}_D)=\\hat{}_{D}+\\hat{}(\\hat{}-\\hat{}_{n+N}\\hat{}_{D}),$ where $\\hat{}$ is an estimator of the precision matrix $=^{-1}$ .",
"To be specific, we consider the following node-wise lasso estimator [27] based on both labeled and unlabeled data $\\tilde{}$ .",
"For $k \\in [p]$ , we define the vector $\\hat{}_k=\\lbrace \\hat{\\gamma }_{k,j}:j\\in [p] ~\\textrm {and}~ j\\ne k\\rbrace $ as $\\hat{}_k=_{\\gamma \\in \\mathbb {R}^p}\\left\\lbrace \\frac{1}{n+N}||\\tilde{}_{\\cdot k}-\\tilde{}_{\\cdot -k}||_2^2+2\\lambda _k\\Vert \\Vert _1\\right\\rbrace .$ Denote by $\\hat{}=\\begin{bmatrix}1& -\\hat{\\gamma }_{1,2}&\\dots & -\\hat{\\gamma }_{1,p} \\\\-\\hat{\\gamma }_{2,1}& 1 & \\dots & -\\hat{\\gamma }_{2,p} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\-\\hat{\\gamma }_{p,1}&-\\hat{\\gamma }_{p,2} & \\dots &1\\end{bmatrix}$ and let $ \\hat{}^2=(\\hat{\\tau }_1^2,...,\\hat{\\tau }_p^2), ~\\textrm {where}~\\hat{\\tau }_k^2=\\frac{1}{n+N}(\\tilde{}_{\\cdot k}-\\tilde{}_{\\cdot -k}\\hat{}_k)^T\\tilde{}_{\\cdot k}.$ The node-wise lasso estimator is defined as $ \\hat{}=\\hat{}^{-2}\\hat{}.$ Denote $=^{-1}$ .",
"Assume $\\max _{1\\le i\\le n+N}\\Vert X_i\\Vert _\\infty \\le K_2$ , and $\\max _{1\\le k\\le p}\\Vert _{k\\cdot }\\Vert _0\\le s_{\\Omega }$ satisfies $K^2s_{\\Omega }\\sqrt{\\log p/(n+N)}=o(1)$ , where $K=K_1\\vee K_2$ with $K_1$ defined in Assumption REF .",
"Assumption REF and (A2) in Assumption REF together imply the strong boundedness condition and $\\max _{1\\le i\\le n+N}\\max _{1\\le k\\le p}|X_{i,-k}^T_k|=O(K)$ in [42] which further guarantees the rate of $\\hat{}$ in the matrix $L_{\\infty }$ norm.",
"While it is possible to relax the sparsity assumption $\\max _{1\\le k\\le p}\\Vert _{k\\cdot }\\Vert _0\\le s_{\\Omega }$ [22], we make this assumption in order to show the proposed estimator is regular and asymptotically linear, which facilitates the comparison with other competing estimators in terms of asymptotic efficiency.",
"Finally, we note that Assumptions REF and REF do not impose or imply any upper bound on $\\Lambda _{\\max }()$ .",
"For example, we allow $$ to be an equicorrelation matrix, whose largest eigenvalue is proportional to the dimension $p$ .",
"Given these assumptions, the following theorem shows that $^T\\hat{}^d$ is asymptotically normal for a linear functional $^T^*$ .",
"Suppose Assumptions REF and REF hold.",
"By choosing $\\lambda _D\\asymp K_1(\\Phi \\sqrt{\\frac{\\log p}{n+N}}+\\sigma \\sqrt{\\frac{\\log p}{n}}+b_n\\sqrt{\\frac{\\log p}{n}})$ and $\\lambda _k\\asymp K\\sqrt{\\frac{\\log p}{n+N}}$ uniformly over $k$ , we obtain that for any $\\ne \\mathbf {0}\\in ^p$ , $^T(\\hat{}^d-^*)&=\\frac{1}{n}\\sum _{i=1}^n^T_i(Y_i-f(X_i))+\\frac{1}{n+N}\\sum _{i=1}^{n+N} ^T_i(f(X_i)-X_i^T^*)+O_p(\\delta _n),$ where $_i=X_i$ and $\\delta _n=\\Vert \\Vert _1(R_1+R_2)$ with $R_1=K_1K(s\\vee s_\\Omega )\\Big ({\\frac{\\Phi \\log p}{n+N}}+{\\frac{(\\sigma +b_n)\\log p}{\\sqrt{n(n+N)}}}\\Big ),~~R_2=K_2b_n\\sqrt{\\frac{\\log p}{n}}$ and $b_n$ is defined in Theorem REF .",
"In addition, if $\\frac{n^{1/2}\\delta _n}{(^T(\\sigma ^2+\\frac{n}{n+N}))^{1/2}}=o(1)$ with $=(_i^{\\otimes 2}(f(X_i)-X_i^T^*)^2)$ , $\\epsilon $ and $\\eta (X)=f(X)-X^T^*$ satisfy $\\Vert \\Vert _1^{2+\\delta }K_2^{2+\\delta }\\Big [\\frac{ |\\epsilon |^{2+\\delta }}{n^{\\delta /2}(\\sigma ^2 ^T)^{1+\\delta /2}}+\\frac{|\\eta (X)|^{2+\\delta }}{(n+N)^{\\delta /2}(^T)^{1+\\delta /2}}\\Big ]=o(1),$ for some $\\delta >0$ , then $\\frac{n^{1/2}^T(\\hat{}^d-^*)}{(^T(\\sigma ^2+\\frac{n}{n+N}))^{1/2}} \\stackrel{d}{\\longrightarrow } \\mathcal {N}(0,1).$ The asymptotic expansion of $^T(\\hat{}^d-^*)$ is presented in (REF ), where the remainder term $\\delta _n$ consists of two components $R_1$ and $R_2$ , which come from the cross product of the estimation errors of $\\hat{}$ and $\\hat{}_D$ in Theorem REF and the plug-in error of $\\hat{f}^{-j}$ in $\\hat{}$ , respectively.",
"To establish the asymptotic normality of $^T(\\hat{}^d-^*)$ , we further need to assume that $\\delta _n$ is sufficiently small and the Lyapunov condition holds so that one can apply the central limit theorem to the leading terms in (REF ).",
"These two conditions are rigorously formulated in (REF ) and (REF ).",
"To further simplify (REF ) and (REF ), assume that $\\sigma ^2 ^T\\ge C\\Vert \\Vert _2^2$ and $^T\\ge C\\Vert \\Vert _2^2$ for some constant $C$ , $|\\epsilon |^{2+\\delta }, |\\eta (X)|^{2+\\delta }, K$ are all $O(1)$ and $b_n=o(1)$ .",
"Under these mild conditions, (REF ) and (REF ) are implied by $\\frac{\\Vert \\Vert _1}{\\Vert \\Vert _2}\\Big [\\frac{(s\\vee s_\\Omega )\\log p}{\\sqrt{n+N}}+b_n\\sqrt{\\log p}+n^{-\\frac{\\delta }{2(2+\\delta )}}\\Big ]=o(1).$ (1) The bound (REF ) requires the ratio $\\Vert \\Vert _1/\\Vert \\Vert _2$ cannot be too large which excludes the case that $$ has many large entries (e.g., $=(1,1,...,1)^T$ ).",
"This observation agrees with the theoretical results in [12], as the debiased estimator does not yield optimal confidence intervals for $^T^*$ when $$ is a dense vector.",
"To see some concrete examples that our results are applicable, we first note that if $=_j$ the $j$ th basis vector in $^p$ , then $^T\\hat{}^d=\\hat{\\theta }^d_j$ reduces to the estimate of $\\theta _j$ .",
"Our condition (REF ) becomes $(s\\vee s_\\Omega )\\log p=o(\\sqrt{n+N})$ and $b_n=o(1/\\sqrt{\\log p})$ .",
"The former is a standard condition for debiased inference adapted to the semi-supervised setting and the latter is slightly stronger than the consistency of $\\hat{f}^{-j}$ required in Theorem REF ; see Remark REF for details.",
"The same comments are applicable if the parameter of interest $^T^*$ is a linear combination of $^*$ with $\\Vert \\Vert _0$ fixed.",
"Indeed, the set of vector $$ in $^p$ satisfying (REF ) forms a cone $\\lbrace \\frac{\\Vert \\Vert _1}{\\Vert \\Vert _2}\\le t_n[\\frac{(s\\vee s_\\Omega )\\log p}{\\sqrt{n+N}}+b_n\\sqrt{\\log p}+n^{-\\frac{\\delta }{2(2+\\delta )}}]^{-1}\\rbrace $ for some $t_n=o(1)$ .",
"Compared to [12] who proposed the debiased estimator for $^T^*$ with sparse $$ , the cone condition (REF ) may still hold if $$ is approximately sparse with many small but nonzero entries.",
"Our results are still applicable in this case.",
"(2) Assuming $\\Vert \\Vert _1/\\Vert \\Vert _2$ is a constant and $N\\gg n$ , we can see from (REF ) that in the semi-supervised setting we need $(s\\vee s_\\Omega )\\log p=o(\\sqrt{N})$ , which is much weaker than the similar condition $(s\\vee s_\\Omega )\\log p=o(\\sqrt{n})$ for the supervised estimators (up to some logarithmic factors).",
"Thus, with a large amount of unlabeled data, our inference results may still hold for models with large $s$ .",
"[Efficiency improvement and semi-parametric efficiency bound] We first note that, when the linear model is correctly specified i.e.",
"$f(X)=X^T^*$ , we have $=0$ and the asymptotic variance of $^T\\hat{}^d$ reduces to $\\sigma ^2^T$ , which agrees with the asymptotic variance of the debiased estimator in fully supervised setting and also matches the semi-parametric efficiency bound.",
"In this case, the information of $X$ contained in the unlabeled data is ancillary and does not contribute to the inference on $$ ; see also [2], [15].",
"In the following, we assume $$ is strictly positive definite.",
"Recall that our asymptotic analysis requires $n,p\\rightarrow \\infty $ and allows $N$ to be either fixed or grow with $n$ .",
"In the following, we discuss the asymptotic variance of $^T\\hat{}^d$ in (REF ) according to the magnitude of $N$ .",
"(1) $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=1$ .",
"Denote $=[X^{\\otimes 2}(Y-X^T^*)^2]$ .",
"It is seen that $=\\sigma ^2+$ .",
"In this case, the asymptotic variance of $^T\\hat{}^d$ reduces to $^T(\\sigma ^2+)=^T$ , which is the asymptotic variance of the debiased estimator in the fully supervised setting; see [10], [29].",
"As expected, when $N\\ll n$ , the amount of unlabeled data is not sufficiently large to improve the asymptotic efficiency of the estimator.",
"(2) $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=\\rho $ for some $0<\\rho <1$ .",
"In this case, the asymptotic variance $^T(\\sigma ^2+\\rho )$ is strictly smaller than $^T=^T(\\sigma ^2+)$ .",
"Thus, the unlabeled data can be used to improve the asymptotic efficiency for inference.",
"(3) $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=0$ .",
"In the case, the asymptotic variance becomes $\\sigma ^2^T$ .",
"Indeed, if the distribution of $X$ is known, the semi-parametric efficiency bound for estimating $^T^*$ is exactly $\\sigma ^2^T$ as well; see [15] and the reference therein.",
"Thus, when $N\\gg n$ , our estimator attains the semi-parametric efficiency bound.",
"In the following, we consider how to estimate the asymptotic variance of $^T\\hat{}^d$ .",
"To estimate $\\sigma ^2$ , we apply the cross-fitting technique.",
"Specifically, for $j=\\lbrace 1,2\\rbrace $ , define $\\hat{\\sigma }^2_j=\\frac{1}{n_j}\\sum _{i\\in D^*_j}(Y_i-\\hat{f}^{-j}(X_i))^2.$ We estimate $\\sigma ^2$ by $\\hat{\\sigma }^2=(\\hat{\\sigma }^2_1+\\hat{\\sigma }^2_2)/2$ .",
"Similarly, define $\\hat{}_j=\\frac{1}{n_j+N_j}\\sum _{i\\in D_j} (\\hat{\\eta }_i^{-j})^2\\hat{}X_iX_i^T\\hat{},$ where $\\hat{\\eta }_i^{-j}=\\hat{f}^{-j}(X_i)-\\hat{}_D^TX_i$ and $\\hat{}$ is defined in (REF ).",
"We then estimate $$ by $\\hat{}=(\\hat{}_1+\\hat{}_2)/2$ .",
"The following Proposition shows that the asymptotic variance of $^T\\hat{}^d$ can be consistently estimated by the plug-in estimator $^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{})$ .",
"Suppose the conditions in Theorems REF and REF hold.",
"To simplify the presentation, we further assume $(\\epsilon ^4)=O(1), (\\eta ^4(X))=O(1)$ and $K \\sqrt{\\frac{s\\log p}{n+N}}=o(1)$ .",
"Then $\\Big |^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{})-^T(\\sigma ^2+\\frac{n}{n+N})\\Big |=O_p\\Big (\\Vert \\Vert _2^2(\\frac{1}{\\sqrt{n}}+b_n^2)+\\textrm {Rem}_N\\Big ),$ where $\\textrm {Rem}_N=\\frac{n}{n+N}K^2\\Vert \\Vert _1^2b_n+K^3\\Vert \\Vert _1^2(s\\vee s_\\Omega )\\sqrt{\\frac{\\log p}{n+N}}.$ Under the additional assumptions $\\sigma ^2^T\\ge C\\Vert \\Vert _2^2$ and $\\textrm {Rem}_N/\\Vert \\Vert _2^2=o(1)$ , we have $\\frac{n^{1/2}^T(\\hat{}^d-^*)}{(^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{}))^{1/2}} \\stackrel{d}{\\longrightarrow } \\mathcal {N}(0,1).$ To better understand the convergence rate of the estimated asymptotic variance, we decompose the error in (REF ) into two terms, $\\Vert \\Vert _2^2(\\frac{1}{\\sqrt{n}}+b_n^2)$ and $\\textrm {Rem}_N$ .",
"The former is due to the estimation error of $\\hat{\\sigma }^2$ and the latter comes from the error of $\\hat{}$ and $\\hat{}$ .",
"It is of interest to note that, if $N\\gg n$ , the error term $\\textrm {Rem}_N$ may vanish to 0 fast enough, so that the convergence rate of the estimated asymptotic variance in (REF ) is dominated by $\\Vert \\Vert _2^2(\\frac{1}{\\sqrt{n}}+b_n^2)$ .",
"In addition, for many practical estimators $\\hat{f}^{-j}$ , such as the group lasso estimator for sparse additive models in Remark REF , its convergence rate in $L_2()$ norm is no slower than $n^{-1/4}$ , that is $b_n=o(n^{-1/4})$ .",
"In this case, the rate in (REF ) further reduces to $\\Vert \\Vert ^2_2/\\sqrt{n}$ , which is the best possible rate for estimating the variance even if $$ , $$ and $f(X)$ are known.",
"Thus, the unlabeled data lead to a more accurate estimate of the asymptotic variance.",
"Finally, from (REF ) we can construct the $(1-\\alpha )$ confidence interval for $^T^*$ as $[^T\\hat{}^d-z_{1-\\alpha /2}n^{-1/2}sd, ^T\\hat{}^d+z_{1-\\alpha /2}n^{-1/2}sd]$ , where $z_{1-\\alpha /2}$ is the $1-\\alpha /2$ quantile of a standard normal distribution and $sd=(^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{}))^{1/2}$ .",
"Similarly, if one is interested in testing the hypothesis $H_0: ^T^*=0$ , we can construct the test statistic $n^{1/2}^T\\hat{}^d/(^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{}))^{1/2}$ based on (REF )."
],
[
"Safe semi-supervised inference",
"To construct the estimator $\\hat{}^d$ in the previous section, one assumption we make is that the conditional mean function $f(X)$ is consistently estimated with the desired rate.",
"If this assumption fails, there is no guarantee that the estimator $\\hat{}^d$ attains the semi-parametric efficiency bound or outperforms the supervised estimator.",
"To tackle this problem, in this section we propose a new safe semi-supervised inference approach, which does not rely on the estimation of the conditional mean function $f(X)$ but guarantees the efficiency improvement.",
"Given any $p$ -dimensional function $m(X) \\colon ^p\\rightarrow $ , we first construct a set of unbiased estimating functions $Xm(X)-$ , where $=[X m(X)]$ .",
"While these functions do not directly involve the unknown parameter $^*$ , they play an important role in the safe semi-supervised inference approach.",
"Using $Xm(X)-$ as covariates, we postulate a $p$ -variate working regression model with response variables $X(Y-X^T^*)$ , i.e.",
"$X(Y-X^T^*)=^T(X m(X)-)+,$ where $\\in ^p$ is the error vector and the coefficient matrix $\\in ^{p\\times p}$ is $=\\lbrace [(X m(X)-)^{\\otimes 2}]\\rbrace ^{-1}[(X^{\\otimes 2}m(X)(Y-X^T^*))].$ Since (REF ) is only a working model, the error $$ and covariates $Xm(X)-$ are not necessarily independent.",
"Recall that the response variable $X(Y-X^T^*)$ corresponds to the score function of $^*$ in the linear regression model, and can be rewritten as $X(\\epsilon +\\eta (X))$ , where $\\epsilon =Y-f(X)$ and $\\eta (X)=f(X)-X^T^*$ is the nonlinear effect.",
"Since $\\epsilon $ and $X m(X)-$ are independent, the goal of model (REF ) is to explain the nonlinear effect $X\\eta (X)$ by the covariates $Xm(X)-$ .",
"Indeed, we show in Remark REF that the optimal choice of $m(X)$ is $m(X)=f(X)-X^T^*=\\eta (X)$ and in this case the nonlinear effect $X\\eta (X)$ can be perfectly explained by $Xm(X)-$ .",
"Given $Xm(X)-$ and the coefficient matrix $$ , we define a class of unbiased estimating functions for $^*$ as $h_{\\psi }(X,Y; )=\\bar{}_\\psi -XX^T=X(Y-X^T)-\\psi ^T(Xm(X)-)$ , where $\\bar{}_\\psi =XY-\\psi ^T(Xm(X)-),$ with $\\psi \\in $ being a tuning parameter that balances two unbiased functions $Xm(X)-$ and $X(Y-X^T^*)$ .",
"In particular, we have $(h_{\\psi }(X,Y; ^*))=0$ for any $\\psi $ .",
"Indeed, we show in Remark REF that the optimal choice of $\\psi $ is $\\psi =1$ , which implies $h_{\\psi }(X,Y; ^*)=$ in view of (REF ).",
"Thus, from a geometric perspective, $h_{\\psi }(X,Y; )$ is the residual by projecting the score function $X(Y-X^T)$ onto the set of unbiased estimating functions $Xm(X)-$ in the $L_2()$ norm.",
"Following the insight from the above geometric interpretation, we now propose the safe semi-supervised inference approach.",
"To formulate the inference procedure, we first consider how to estimate the coefficient matrix $$ .",
"In view of (REF ) and the followup discussion, to estimate $$ , we can either pre-specify a nonlinear function $m(X)$ or perhaps use a more flexible approach to estimate $m(X)$ from the data.",
"To see this, we can define $m(X)=_{g\\in } (Y-g(X))^2$ , where $$ is a pre-specified class of functions of $X$ .",
"For example, $=\\lbrace \\sum _{j=1}^p \\alpha _jX_j+\\sum _{1\\le k<\\ell \\le p} \\beta _{k\\ell }X_\\ell X_k: \\alpha _j, \\beta _{k\\ell }\\in \\rbrace $ corresponds to the class of functions with main effects and the second-order interactions.",
"By fitting a penalized interaction model as in [48], we can construct an estimator $\\hat{m}(X)$ .",
"In the rest of the paper, we assume an estimator $\\hat{m}(X)$ of $m(X)$ is available.",
"The detailed technical conditions on $\\hat{m}(X)$ are shown in Assumption REF and Theorem REF .",
"Similar to Section REF , we apply a cross-fitting approach to estimate $$ .",
"Recall that we use $D^*_j$ to denote the $j$ th fold of labeled data and $D_j$ to denote the $j$ th fold of labeled and unlabeled data.",
"Given the estimator $\\hat{m}^{-j}(\\cdot )$ obtained from the labeled data $D^*\\backslash D_j^*$ for $j=1,2$ , we can estimate the $k$ th column of $$ by $ \\hat{}^j_{\\cdot k}=\\arg \\min _{\\in \\mathbb {R}^p} \\frac{1}{n_j}\\sum _{i\\in D_j^*} \\Big \\lbrace X_{ik} (Y_i- X_i^T\\hat{}_L)-^T (X_i\\hat{m}^{-j}(X_i) -\\hat{}^j)\\Big \\rbrace ^2+\\tilde{\\lambda }_k\\Vert \\Vert _1,$ where $\\hat{}_L$ is the Dantzig estimator in (REF ), $\\hat{}^j=\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\hat{m}^{-j}(X_i)X_i$ and $\\tilde{\\lambda }_k$ is a tuning parameter.",
"We note that it is possible to estimate $=[X m(X)]$ by using both labeled and unlabeled data $D_j$ .",
"However, the rate of the estimator $\\hat{}^j_{\\cdot k}$ remains the same.",
"The final estimator of $_{\\cdot k}$ is $\\hat{}_{\\cdot k}=(\\hat{}^1_{\\cdot k}+\\hat{}^2_{\\cdot k})/2$ , and this leads to $\\hat{}=(\\hat{}_{\\cdot 1},...,\\hat{}_{\\cdot p})$ .",
"Motivated by the form of $\\bar{}_\\psi $ in (REF ), we construct the following estimate of $=(XY)$ , $ \\hat{}_{S,\\psi }= \\frac{\\sum _{i=1}^nX_i Y_i}{n}-\\frac{\\psi }{2}\\hat{}^T\\sum _{j=1}^2\\Big (\\frac{\\sum _{i\\in D^*_{j}}X_i \\hat{m}^{-j}(X_i)}{n_j}-\\frac{\\sum _{i\\in D_j} X_i \\hat{m}^{-j}(X_i)}{n_j+N_j}\\Big ),$ where we apply the cross-fitting technique again, and $n_j=|D^*_j|$ and $n_j+N_j=|D_j|$ .",
"We note that different from the estimator $\\hat{}^j$ used in $\\hat{}^j_{\\cdot k}$ , we estimate $$ by $\\frac{1}{n_j+N_j}\\sum _{i\\in D_j} X_i \\hat{m}^{-j}(X_i)$ in (REF ), which incorporates the information from the unlabeled data.",
"Similar to the estimator $\\hat{}^d$ in (REF ), we propose the following safe semi-supervised estimator $\\hat{}^d_{S,\\psi }=\\hat{}_L+\\hat{}(\\hat{}_{S,\\psi }-\\hat{}_n\\hat{}),$ where $\\hat{}_L$ is the supervised Dantzig estimator in (REF ), $\\hat{}_{S,\\psi }$ is defined in (REF ), $\\hat{}_n=\\frac{1}{n}\\sum _{i=1}^nX_i^{\\otimes 2}$ and $\\hat{}$ is the node-wise lasso estimator in (REF ).",
"It is worthwhile to note that we estimate $$ by $\\hat{}_n$ in (REF ), whereas we use $\\hat{}_{n+N}=\\frac{1}{n+N}\\sum _{i=1}^{n+N}X_i^{\\otimes 2}$ in the estimator $\\hat{}^d$ .",
"Indeed, this is a critical difference as replacing $\\hat{}_n$ with $\\hat{}_{n+N}$ in (REF ) corresponds to an estimating function different from $h_{\\psi }(X,Y; )$ and therefore no longer leads to a more efficient estimator.",
"To show the theoretical properties of $\\hat{}^d_{S,\\psi }$ , we make the following assumptions: The smallest eigenvalue of $[(X m(X)-)^{\\otimes 2}]$ is lower bounded by a positive constant.",
"Assume that the second moment of the residual $E_j$ in (REF ) is less than $C$ , and $|m(X_i)|\\le C$ for some constant $C$ .",
"The estimator $\\hat{m}^{-j}(\\cdot )$ satisfies $\\Vert \\hat{m}^{-j}-m\\Vert _2=O_p(c_n)$ for a deterministic sequence $c_n$ .",
"We require $s_BK_1^2(c_n+\\sqrt{\\frac{\\log p}{n}})=o(1)$ and $\\sqrt{\\frac{s\\log p}{n}}=O(1)$ .",
"The columns of the matrix $$ are sparse with $\\max _{1\\le k\\le p}\\Vert _{\\cdot k}\\Vert _0 = s_B$ and $\\max _{1\\le k\\le p}\\Vert _{\\cdot k}\\Vert _1\\le L_B$ for some $L_B$ that may grow with $n$ .",
"$\\Vert \\Vert _\\infty \\le L_{\\Omega }$ and $(K_1L_\\Omega )^2s_{\\Omega }\\sqrt{\\log p/(n+N)}=o(1)$ , where $s_{\\Omega }$ is the maximum rowwise sparsity of $$ defined in Assumption REF .",
"$|\\epsilon |^{2+\\delta }=O(1)$ and $|\\eta |^{2+\\delta }=O(1)$ , where $\\epsilon =Y-f(X)$ and $\\eta =f(X)-X^T^*$ .",
"Assumption (E1) guarantees that the RE condition holds for the estimation of $_{\\cdot k}$ in (REF ).",
"Assumption (E2) is the sparsity assumption of $_{\\cdot k}$ .",
"For example, in the ideal case we can choose $m(X)=f(X)-X^T^*$ and then $=_p$ is sparse.",
"We note that there are other practical settings that the sparsity assumption of $_{\\cdot k}$ is reasonable (e.g., $X$ is blockwise independent).",
"We defer the detailed discussion to Appendix REF .",
"In Assumption (E2), we further require that the matrix $L_\\infty $ norm of $$ is bounded by $L_B$ , which is used to establish the rate of $\\hat{}$ .",
"Assumption (E3) is similar to Assumption REF .",
"In particular, to control the remainder term in the asymptotic expansion of $\\hat{}^d_{S,\\psi }$ , we need $\\Vert \\Vert _\\infty \\le L_{\\Omega }$ .",
"Together with the boundedness assumption $\\Vert X_i\\Vert _\\infty \\le K_1$ in Assumption REF , it implies $\\Vert X_i\\Vert _\\infty \\le \\Vert \\Vert _\\infty \\Vert X_i\\Vert _\\infty \\le K_1 L_{\\Omega }$ .",
"Replacing $K$ in Assumption REF with $K_1 L_{\\Omega }$ , (E3) is equivalent to Assumption REF .",
"Assumption (E4) assumes that $\\epsilon $ and $\\eta $ has bounded $(2+\\delta )$ moment, which is used to simplify the Lyapunov condition.",
"Denote $ _{\\psi }=(_{i1}^{\\otimes 2})-\\frac{N(2\\psi -\\psi ^2)}{n+N}(_{i2}_{i1}^T)^T\\lbrace (_{i2}^{\\otimes 2})\\rbrace ^{-1}(_{i2}_{i1}^T),$ where $_{i1}=X_i(Y_i-X_i^T^*)$ and $_{i2}=X_im(X_i)-$ .",
"Suppose Assumptions REF and REF hold.",
"We choose $\\lambda _L\\asymp K_1\\sqrt{\\frac{(\\sigma ^2+\\Phi ^2)\\log p}{n}}$ in (REF ), $\\lambda _k\\asymp K\\sqrt{\\frac{\\log p}{n+N}}$ in (REF ) and $\\tilde{\\lambda }_k=\\tilde{\\lambda }_{opt}$ in (REF ), where $\\tilde{\\lambda }_{opt}$ is defined in Proposition REF .",
"Then for any $\\ne \\mathbf {0}\\in ^p$ , $^T (\\hat{}^d_{S,\\psi }-^*)&=^T\\Big (\\frac{^T (-^*)}{n}-\\psi ^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n} -\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Big )+O_p(\\bar{\\delta }_n),$ where $\\bar{\\delta }_n&=\\Vert \\Vert _1 L_{\\Omega } K_1^2\\sqrt{\\frac{\\log p}{n}} \\Big [L_Bs_\\Omega \\sqrt{\\frac{\\log p}{n+N}}+s_B^{1/2}L_B(\\sqrt{\\frac{\\log p}{n}}+c_n)+K_1(s\\vee s_B)\\sqrt{\\frac{\\log p}{n}}\\Big ]$ with $c_n$ defined in Assumption REF (E1).",
"In addition, if $^T_\\psi \\ge C\\Vert \\Vert _2^2$ for some constant $C$ , $\\bar{\\delta }_n/\\Vert \\Vert _2=o(n^{-1/2})$ and $\\Big (\\frac{\\Vert \\Vert _1L_\\Omega K_1}{\\Vert \\Vert _2}\\Big )^{2+\\delta } \\frac{1}{n^{\\delta /2}}\\Big (1+\\frac{L_B^{2+\\delta }N}{n+N}\\Big )=o(1),$ then $\\frac{n^{1/2}^T(\\hat{}^d_{S,\\psi }-^*)}{(^T_\\psi )^{1/2}} \\stackrel{d}{\\longrightarrow } \\mathcal {N}(0,1).$ In a similar spirit to Theorem REF , the remainder term $\\bar{\\delta }_n$ in the asymptotic expansion of $^T (\\hat{}^d_{S,\\psi }-^*)$ characterizes the effect of the plug-in estimators $\\hat{}$ , $\\hat{}$ and $\\hat{}_L$ and (REF ) is the Lyapunov condition under Assumption (E4).",
"To further simplify the conditions in Theorem REF , we assume $\\Vert \\Vert _1/\\Vert \\Vert _2$ , $K_1, L_{\\Omega }, L_B$ are all of order $O(1)$ .",
"As a result, (REF ) always holds and the condition $\\bar{\\delta }_n/\\Vert \\Vert _2=o(n^{-1/2})$ is implied by $s_{\\Omega }{\\frac{\\log p}{\\sqrt{n+N}}}+(s\\vee s_B){\\frac{\\log p}{\\sqrt{n}}}+s_B^{1/2}c_n\\sqrt{\\log p}=o(1).$ Compared to (REF ) for Theorem REF , we need a slightly stronger condition $s({\\log p}/{\\sqrt{n}})=o(1)$ in (REF ).",
"This is because $\\hat{}_L$ is not only used as an initial estimator when constructing the one-step estimator $\\hat{}^d_{S,\\psi }$ but also used as a plug-in estimator to estimate $$ in (REF ).",
"The error of $\\hat{}_L$ accumulates in the asymptotic expansion of $^T (\\hat{}^d_{S,\\psi }-^*)$ , leading to the slow order $s({\\log p}/{\\sqrt{n}})$ in (REF ).",
"In terms of the rate of $\\hat{m}^{-j}$ in the $L_2()$ norm, (REF ) requires $c_n=o((s_B\\log p)^{-1/2})$ , which is also slightly stronger than the condition for $\\hat{f}^{-j}$ in (REF ).",
"[Efficiency improvement and optimality] Similar to Remark REF , we first note that when the linear model is correctly specified i.e.",
"$f(X)=X^T^*$ , we have $(_{i2}_{i1}^T)=0$ and $_\\psi =\\sigma ^2$ .",
"Thus, the asymptotic variance of $^T\\hat{}^d_{S,\\psi }$ reduces to $\\sigma ^2^T$ , which agrees with the asymptotic variance of the supervised debiased estimator.",
"In the following, we assume $(_{i2}_{i1}^T)=(X^{\\otimes 2}m(X)\\eta )$ is of full rank.",
"Since $(_{i2}^{\\otimes 2})$ is strictly positive definite by Assumption (E1), it implies that $(_{i2}_{i1}^T)^T\\lbrace (_{i2}^{\\otimes 2})\\rbrace ^{-1}(_{i2}_{i1}^T)$ is strictly positive definite.",
"Similar to Remark REF , we consider the following two cases.",
"(1) $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=1$ .",
"Recall that the asymptotic variance of the supervised debiased estimator is $^T$ , where $=(_{i1}^{\\otimes 2})$ with $_{i1}=X_i(Y_i-X_i^T^*)$ .",
"In this case, the asymptotic variance of $^T\\hat{}^d_{S,\\psi }$ is identical to $^T$ .",
"There is no efficiency improvement when $n\\gg N$ .",
"(2) $\\lim _{n\\rightarrow \\infty }\\frac{n}{n+N}=\\rho $ for some $0\\le \\rho <1$ .",
"In this case, the asymptotic variance $^T_\\psi $ is strictly smaller than $^T$ provided $2\\psi -\\psi ^2>0$ , i.e.",
"$0<\\psi <2$ .",
"Thus, we conclude that our estimator $^T\\hat{}^d_{S,\\psi }$ with $0<\\psi <2$ is more efficient than the supervised estimator.",
"In view of the form of $_\\psi $ , the variance reduction becomes more evident as $\\rho $ goes to 0 (i.e., $N$ increases).",
"Another interesting fact is that the asymptotic variance $^T_\\psi $ is minimized when taking $\\psi =1$ .",
"Thus, the estimator $^T\\hat{}^d_{S,\\psi =1}$ is optimal within the following class of estimators $\\lbrace ^T\\hat{}^d_{S,\\psi }: \\psi \\in \\rbrace $ in terms of asymptotic efficiency.",
"[Comparison with the estimator $\\hat{}^d$ ] To see the connection of the two estimators $\\hat{}^d$ and $\\hat{}^d_{S,\\psi }$ in (REF ) and (REF ), consider the ideal case with $m(X)=f(X)-X^T^*$ .",
"From (REF ), we can show that with $\\psi =1$ $_{\\psi }=\\sigma ^2+(X^{\\otimes 2}\\eta ^2)-\\frac{N}{n+N}(X^{\\otimes 2}\\eta ^2)=\\sigma ^2+\\frac{n}{n+N}(X^{\\otimes 2}\\eta ^2)$ where $\\eta =f(X)-X^T^*$ .",
"The asymptotic variances of $^T\\hat{}^d$ and $^T\\hat{}^d_{S,\\psi }$ are identical, since $^T_\\psi =^T(\\sigma ^2+\\frac{n}{n+N})$ , where $=(X^{\\otimes 2}\\eta ^2)$ .",
"Thus, in the ideal case when $f(X)$ is known, using $\\hat{}^d_{S,\\psi }$ with $m(X)=f(X)-X^T^*$ would not suffer efficiency loss compared to $\\hat{}^d$ and both estimators improve the efficiency of the debiased estimator (and attains the semi-parametric efficiency bound under certain conditions); see Remark REF and REF .",
"However, if there is no sufficient information for us to estimate $f(X)$ consistently, the estimator $^T\\hat{}^d$ may not improve the efficiency of the debiased estimator, whereas $^T\\hat{}^d_{S,\\psi }$ does not rely on the estimation of $f(X)$ and guarantees the efficiency improvement for any $m(X)$ that satisfies the conditions in Theorem REF .",
"We note that the amount of efficiency improvement of $^T\\hat{}^d_{S,\\psi }$ depends on the choice of $m(X)$ .",
"Unless we choose $m(X)=f(X)-X^T^*$ , the estimator $^T\\hat{}^d_{S,\\psi }$ in general would not attain the semi-parametric efficiency bound.",
"From Remarks REF and REF , we can see that the estimator $^T\\hat{}_{S,\\psi }^d$ provides a safe use of the unlabeled data, since it is no worse than the supervised approach, no matter whether the linear model is correctly specified or the conditional mean function is consistently estimated.",
"As mentioned in the introduction, when the dimension $p$ is fixed, [2] and [15] investigated how to incorporate the unlabeled data to improve the estimation efficiency for $\\theta _j^*$ .",
"In addition to the technical challenges arise from the high dimensionality, the way we construct our estimator $\\hat{}^d_{S,\\psi }$ is different from theirs.",
"Unlike $\\hat{}^d_{S,\\psi }$ , their estimators can not guarantee the efficiency improvement if the parameter of interest is the linear combination of $^*$ (e.g., $\\theta ^*_1+\\theta ^*_2$ ).",
"We refer to Appendix REF for more detailed discussions.",
"Finally, we consider how to estimate the asymptotic variance of $^T\\hat{}^d_S$ .",
"Recall that $\\hat{}$ is an estimator of $$ defined in (REF ).",
"To estimate $_\\psi $ in (REF ), we note that $\\hat{}^T$ is an estimate of $(_{i2}_{i1}^T)^T\\lbrace (_{i2}^{\\otimes 2})\\rbrace ^{-1}$ .",
"We can further estimate $_1=(_{i1}^{\\otimes 2})$ and $_2=(_{i2}_{i1}^T)$ by $\\hat{}_1=\\frac{1}{n}\\sum _{i=1}^n (Y_i-X_i^T\\hat{}_L)^2X_i^{\\otimes 2}$ and $\\hat{}_2=(\\hat{}^1_2+\\hat{}^2_2)/2$ , where $\\hat{}_2^j=\\frac{1}{n_j}\\sum _{i\\in D^*_j} (Y_i-X_i^T\\hat{}_L)\\hat{m}^{-j}(X_i)X_i^{\\otimes 2}.$ Given these estimates, an estimator of $_\\psi $ is defined as $\\hat{}_\\psi =\\hat{}_1-\\frac{N(2\\psi -\\psi ^2)}{n+N}\\hat{}^T\\hat{}_2.$ Suppose the conditions in Theorem REF hold, and assume $(\\epsilon ^4)=O(1), (\\eta ^4)=O(1)$ and $Rem=o(1)$ , where $Rem=K_1(s_B+s_B^{1/2}L_B)(\\sqrt{\\frac{\\log p}{n}}+c_n)+K_1^2\\sqrt{\\frac{ss_B\\log p}{n}}+K_1L_B\\sqrt{\\frac{s\\log p}{n}}.$ Under these assumptions, we can show that $\\Big |^T\\hat{}\\hat{}_\\psi \\hat{}-^T_\\psi \\Big |=O_p\\Big (\\Vert \\Vert _1^2(R_1+R_2+R_3)\\Big ),$ where $R_1=K_1L_{\\Omega }^2s_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}} \\Vert _\\psi \\Vert _{\\max }, ~~R_2=K_1^3L_{\\Omega }^2\\sqrt{\\frac{s\\log p}{n}},~~~R_3=\\frac{NK_1^2L_{\\Omega }^2}{n+N}Rem,$ with $Rem$ defined in (REF ).",
"Thus, if $\\Vert \\Vert _1^2(R_1+R_2+R_3)/\\Vert \\Vert _2^2=o(1)$ , we have $\\frac{n^{1/2}^T(\\hat{}^d_{S,\\psi }-^*)}{(^T\\hat{}\\hat{}_\\psi \\hat{})^{1/2}} \\stackrel{d}{\\longrightarrow } \\mathcal {N}(0,1).$ We note that the three terms $R_1, R_2$ and $R_3$ in the convergence rate (REF ) are inherited from the estimation errors of $\\hat{}$ , $\\hat{}_L$ and $\\hat{}$ , respectively.",
"To further simplify the conditions in Proposition REF , let us consider the case that $\\Vert \\Vert _1/\\Vert \\Vert _2$ , $K_1, L_{\\Omega }, L_B$ and $\\Vert _\\psi \\Vert _{\\max }$ are all of order $O(1)$ .",
"In this case, the asymptotic normality in (REF ) is valid provided $(s\\vee s_B)\\sqrt{\\frac{\\log p}{n}}=o(1)$ , $s_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}=o(1)$ and $s_Bc_n=o(1)$ ."
],
[
"Data generating models and practical implementation",
"We first generate a $p$ -dimensional multivariate normal random vector $U \\sim \\mathcal {N}(0,\\Sigma )$ with $\\Sigma _{jk}=0.3^{|j-k|}$ .",
"We set the covariate $X=(X_1,...,X_p)$ to be $X_1=|U_1|$ and $X_j=U_j ~\\mathrm {for}~ 1<j\\le p$ .",
"The reason we take $X_1=|U_1|$ is that this transformation implies $(X_1^k X_j)=0$ for $j\\ne 1$ but the parameter $\\theta ^*_1$ for centered $X_1$ is nonzero.",
"We consider the following two data generating models for $Y$ .",
"For model 1, we consider an additive model $Y=0.5X_1^2+0.8X_3^3-(X_4-2)^2+2(X_5+1)^2+2X_6+\\epsilon ,$ where $\\epsilon \\sim \\mathcal {N}(0,1)$ .",
"To calculate the corresponding regression parameter $^*$ under the working linear model, we first center $Y$ and $X_1$ so that their means are 0.",
"By Proposition 4 in [10], we know that the support of $^*$ is $S=\\lbrace 1,3,4,5,6\\rbrace $ and $\\theta _j^*$ for any $j\\in S$ is given by the $L_2()$ projection in the sub-model only with the variable $X_j$ (e.g, $\\theta _3^*=\\arg \\min (0.8X_3^2-\\theta _3 X_3)^2$ ).",
"After some calculation, we obtain $^*=(1.1,0,2.4,4,4,2,0,...,0)$ , which is sparse.",
"For model 2, we have $Y=0.6(X_1+X_2)^2+0.4X_4^3-X_5+2X_6+\\epsilon ,$ where $\\epsilon \\sim \\mathcal {N}(0,1)$ .",
"The model is non-additive since it includes an interaction term between $X_1$ and $X_2$ .",
"We can show that the corresponding regression parameter $^*$ is $(1.48,1.04,0,1.2,-1,2,0,...,0)$ .",
"Under each data generating model, we consider several combinations of $(n,p)$ and vary the ratio $N/n$ from 1 to 8.",
"We repeat the simulation 100 times.",
"Before we proceed to illustrate the results, we discuss several practical implementation issues for the proposed methods.",
"To compute our optimal semi-supervised estimator $\\hat{}_D$ in (REF ), we apply the group lasso with spline basis to estimate a sparse additive regression function $\\hat{f}$ [21].",
"To be specific, we use the cubic spline basis with degree of freedom $df=5$ .",
"To select the penalty parameter in group lasso and make computation easier, the BIC criterion is used; see Section 4 in [21] for the definition.",
"After we derive the estimator $\\hat{f}$ and subsequently $\\hat{}$ , we modify the source code in the flare package to compute the Dantzig type estimator $\\hat{}_D$ , where the tuning parameter $\\lambda _D$ is selected by 5 fold cross-validation.",
"Given the estimator $\\hat{}_D$ , we can compute the one-step estimator $\\hat{}^d$ in (REF ) for inference, where $\\hat{}$ is obtained by the node-wise lasso using the glmnet package with tuning parameter selected by 5 fold cross-validation.",
"To implement the safe semi-supervised method, we choose $\\hat{m}(\\cdot )=\\hat{f}(\\cdot )$ the estimated sparse additive function obtained previously.",
"We estimate each column of the coefficient matrix $$ by (REF ) using lasso with tuning parameters selected by cross-validation.",
"With the optimal choice $\\psi =1$ (see Remark REF ), we can compute the safe semi-supervised estimator $\\hat{}^d_{S,\\psi =1}$ in (REF ), where $\\hat{}$ is obtained previously and the Dantzig selector $\\hat{}_L$ is computed using the flare package.",
"To evaluate the estimation performance of the safe semi-supervised method, we can also compute the following sparse estimator $ \\hat{}_{S}=\\arg \\min \\Vert \\Vert _1,~~\\textrm {s.t.",
"}~~ \\Vert \\hat{}_{n}-\\hat{}_{S,\\psi }\\Vert _\\infty \\le \\lambda _S,$ where $\\hat{}_{S,\\psi }$ is defined in (REF ).",
"Similarly, the tuning parameter $\\lambda _S$ is selected by 5 fold cross-validation."
],
[
"Numerical results",
"We first compare the estimation error of several sparse estimators $\\hat{}_D$ in (REF ) (O-SSL), $\\hat{}_S$ in (REF ) (S-SSL), $\\hat{}_L$ in (REF ) (Dantzig) and $\\hat{}_U$ in (REF ) (U-Dantzig).",
"The $L_2$ and $L_1$ estimation error under Model 1 with $p=200$ and $n=100$ is shown in Figure REF .",
"Since the true data generating model is additive, O-SSL has the smallest error among the four estimators, which agrees with our results in Theorem REF .",
"While S-SSL is sub-optimal, it still outperforms Dantzig and U-Dantzig by a large margin.",
"One interesting observation is that U-Dantzig performs much worse than the fully supervised estimator Dantzig.",
"Thus, using the sample covariance $\\hat{}_{n+N}$ from both labeled and unlabeled data in the Dantizg selector may not provide any empirical improvement; see Remark REF for the theoretical justification.",
"In addition, as the size of unlabeled data $N$ increases, the improvement of our estimators O-SSL and S-SSL is more overwhelming, whereas the performance of U-Dantzig tends to deteriorate.",
"The simulation results with $p=500$ and $n=200$ demonstrate the same patterns and are deferred to Appendix .",
"The $L_1$ and $L_2$ estimation error under Model 2 with $p=500$ and $n=200$ is shown in Figure REF .",
"Since Model 2 includes an interaction term between $X_1$ and $X_2$ , the sparse additive model is inconsistent for the true regression function.",
"Thus our O-SSL estimator is no longer optimal.",
"This is validated from the first panel of Figure REF , where the $L_2$ estimation error of O-SSL is slightly larger than Dantzig.",
"In contrast, our S-SSL estimator leads to the smallest estimation error and provides a safe use of unlabeled data even if the imposed conditional mean model is incorrect.",
"Figure: The L 2 L_2 and L 1 L_1 estimation error under Model 1 with p=200p=200 and n=100n=100.",
"The length of the vertical bar represents the magnitude of the sample standard deviations.Figure: The L 2 L_2 and L 1 L_1 estimation error under Model 2 with p=500p=500 and n=200n=200.",
"The length of the vertical bar represents the magnitude of the sample standard deviations.To compare the inference results, we also consider two versions of debiased lasso estimators, $ \\hat{}_1^d =\\hat{}_{lasso}+ \\bar{}\\Big (\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i-\\hat{}_{n}\\hat{}_{lasso}\\Big ),~~\\hat{}_2^d =\\hat{}_{lasso}+\\hat{}\\Big (\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i-\\hat{}_{n}\\hat{}_{lasso}\\Big ),$ where $\\hat{}_{lasso}$ and $\\bar{}$ are the standard lasso and node-wise lasso estimator applied to the labeled data.",
"The only difference between $\\hat{}_1^d$ and $\\hat{}_2^d$ is the way of estimating the precision matrix $$ .",
"The two estimators $\\hat{}_1^d, \\hat{}_2^d$ and the associated confidence intervals can be computed using the hdi package with Robust option.",
"In Figure REF , we compare the performance of $95\\%$ confidence intervals (CIs) associated with $\\hat{}^d$ in (REF ) (O-SSL), $\\hat{}^d_{S,\\psi =1}$ in (REF ) (S-SSL), $\\hat{}^d_1$ (D-Lasso1) and $\\hat{}^d_2$ in (REF ) (D-Lasso2) under Model 1 with $p=200,n=100$ .",
"For simplicity, we still call our estimators as O-SSL and S-SSL.",
"Recall that under Model 1, the support set is $S=\\lbrace 1,3,4,5,6\\rbrace $ .",
"The top left panel contains the length and empirical coverage rate (shown in the number) of $95\\%$ CIs averaged over all the covariates not in the support set (denoted by $X0$ ).",
"Similarly, the rest of panels compare the CIs for the coefficients of three covariates $X_1$ , $X_5$ and $X_6$ in the support set.",
"It is seen that the coverage rates of all the methods are very close to the desired level $0.95$ .",
"Our O-SSL method as guaranteed in Theorem REF , produces the shortest CIs among all four methods.",
"The CIs from the S-SSL method are always shorter than those from D-Lasso1 and D-Lasso2, which agrees with Theorem REF .",
"As expected, the CIs from both O-SSL and S-SSL become shorter as the size of unlabeled data $N$ increases.",
"Finally, we note that the length of CIs from D-Lasso1 and D-Lasso2 is very similar, which shows that the way of estimating $$ has little effect on debiased lasso estimators.",
"The former is slightly shorter for the covariates not in the support set ($X0$ ) but longer for $X_1$ , $X_5$ and $X_6$ .",
"Additional simulation results with $p=500,n=200$ are deferred to Appendix .",
"Figure: The length and empirical coverage rate (shown in the number) of 95%95\\% CIs under Model 1 with p=200p=200 and n=100n=100.",
"The top left panel is drawn by averaging over all the covariates not in the support set.",
"The rest are for the coefficients of X 1 X_1, X 5 X_5 and X 6 X_6 in the support set.We illustrate the inference results under Model 2 with $p=500,n=200$ in Figure REF .",
"Although the CI from O-SSL is still shorter than D-Lasso1, it is no longer valid for the inference, as we can see a relatively low coverage rate for $X_1$ .",
"This results from a poor estimation of conditional mean as the true model is no longer additive.",
"However, S-SSL is still reliable in this scenario and shows a plausible improvement in efficiency, which becomes more significant as $N$ increases.",
"Additional simulation results are deferred to Appendix .",
"In summary, O-SSL outperforms the other methods when the conditional mean function is estimated consistently.",
"However, in practice, estimation of the conditional mean function can be difficult especially with high-dimensional covariates.",
"In this case, we recommend S-SSL as it provides a safe use of unlabeled data even if the imposed conditional mean model is incorrect.",
"Figure: The length and empirical coverage rate (shown in the number) of 95%95\\% CIs under Model 2 with p=500p=500 and n=200n=200.",
"The top left panel is drawn by averaging over all the covariates not in the support set.",
"The rest are for the coefficients of X 1 X_1, X 2 X_2 and X 6 X_6 in the support set."
],
[
"Real Data Application",
"In this section, we apply our proposed method to a real data example, where the data set is from the Medical Information Mart for Intensive Care III (MIMIC-III) database [23].",
"MIMIC-III is an openly available electronic health records system developed by the MIT Lab for Computational Physiology.",
"It comprises deidentified health-related data associated with intensive care unit patients with rich information including demographics, vital signs, laboratory test, medications, and so on.",
"Our initial motivation for this data analysis is the association study for the albumin level in the blood sample, a very indicative biomarker correlated with the phenotypes of different types of diseases [31].",
"We focus on a subset with around 4800 patients of the whole database that the albumin level is available.",
"Some data cleaning strategy is inevitable for handling electronic health records database.",
"In our situation, around 54% covariates contain missing values.",
"Among these covariates with missing values, the missingness proportions are 9.4% on average and the range is from 0.2% to 30.8%.",
"For those missing values, we simply impute them using the mean of observed samples, the so-called mean imputation.",
"For many clinical markers with continuous scale, the database collects the minimum, the maximum, as well as the mean, values across a certain period of time.",
"To alleviate the potential collinearity among these variables but also to maintain as much information as possible, we decide to only include the maximum and the mean values in our analysis.",
"Additionally, we convert the categorical variables, such as gender and marital status, to dummy variables.",
"The number of features after data pre-processing is $p=160$ .",
"We randomly sample 900 observations out of 4811 patients and divide them into $n=100$ labeled data and $N=800$ unlabeled data, where the value of the outcome variable albumin is removed for the 800 unlabeled data.",
"We also vary the size of the labeled and unlabeled data in this analysis.",
"Firstly, same as the simulations, we apply the hdi package to compute the debiased lasso estimator D-Lasso1 with robust option using the labeled data.",
"Due to multiple testing, the p-value for each covariate is corrected using the default holm approach.",
"The three variables iron binding capacity, calcium, and alkaline phosphatase are significant with corrected p-values smaller than $0.05$ .",
"We also applied the method D-Lasso2.",
"Its result is quite similar therefore omitted.",
"Next we apply our O-SSL and S-SSL methods.",
"The O-SSL estimator is implemented in the same way as the simulation studies.",
"In the S-SSL method implementation, we consider the working model $\\hat{m}$ as the linear model with lasso where the explanatory variables also include the squared terms and interaction terms of the three identified variables iron binding capacity, calcium, and alkaline phosphatase.",
"It shows that, in this working model, the coefficients of the interaction terms between iron binding capacity and calcium, and between iron binding capacity and alkaline phosphatase are nonzero; therefore, this working model indeed exhibits some nonlinearity between the outcome and the covariate variables.",
"In either O-SSL or S-SSL, the p-values are corrected using the same holm method.",
"The results are summarized in Table .",
"All the three methods D-Lasso1, O-SSL and S-SSL identify the three variables iron binding capacity, calcium, and alkaline phosphatase as significant at the FWER $5\\%$ level, with the average confidence interval length 0.11, 0.08, and 0.09, respectively.",
"Besides, the O-SSL method selects nine additional variables, and the S-SSL method provides a more parsimonious model with three additional variables.",
"The variables hemoglobin and lymphocytes are selected by both of the two methods.",
"The S-SSL method selects the variable white blood cells as well.",
"In the medical literature, there are various evidence documenting the associations between the albumin level in the blood sample with our selected biomarkers here, such as with the hemoglobin level [19], with the lymphocytes level [1], and with the count of the white blood cells [14].",
"More interestingly, researchers recently started to use the so-called HALP score, a combination of hemoglobin, albumin, lymphocyte, and platelet levels in the blood sample [17], as a prognostic factor for various disease types [30], [20], [38], [40].",
"Our statistical results are consistent with the medical literature.",
"Table also shows that, while the successful implementation of the O-SSL method needs the correct specification of the conditional mean model, both O-SSL and S-SSL achieve more efficiency gains compared to the D-Lasso1 method, in the sense that they have shorter confidence intervals and would be more powerful for conducting hypothesis tests.",
"Table: NO_CAPTION The point estimates, confidence intervals (and the length) and corrected p-values from the D-Lasso1, S-SSL and O-SSL methods when $n=100$ , $N=800$ ."
],
[
"Discussion",
"In this paper, we first establish the minimax lower bound for the estimation of $^*$ in the semi-supervised setting.",
"We show that the simple supervised estimator cannot attain this lower bound and thus there is a gap between the two.",
"We close this gap by proposing a new semi-supervised estimator $\\hat{}_D$ which attains the lower bound.",
"To construct confidence intervals and hypothesis tests, we further propose two semi-supervised estimators $^T\\hat{}^d$ and $^T\\hat{}^d_{S,\\psi }$ for $^T^*$ .",
"The former is fully efficient if $f(X)$ is estimated consistently, but may not be more efficient than the supervised approach otherwise.",
"The latter does not aim to provide fully efficient inference, but is guaranteed to be no worse than the supervised approach, no matter whether $f(X)=X^T^*$ or $f(X)$ is estimated consistently.",
"Thus, it provides a safe use of the unlabeled data for inference.",
"There are several future directions that warrant further investigation.",
"First, in many biomedical studies, the outcome variable of interest is often binary or discrete.",
"it is of both practical and theoretical importance to generalize the method and theory to models that can deal with discrete data, such as generalized linear models.",
"Second, in some applications, the labeled and unlabeled data are collected under different conditions or from different populations.",
"In this case, it is more appropriate to assume that the marginal distribution of $X$ in the unlabeled data differs from that in the labeled data [24].",
"With different marginal distributions of $X$ , it is unclear whether and how the unlabeled data can be used to improve the estimation and inference.",
"We plan to study this problem in the future."
],
[
"Acknowledgement",
"Ning is supported in part by NSF CAREER award DMS-1941945 and NSF award DMS-1854637.",
"Zhao is supported in part by NSF award DMS-1953526 and a start-up grant from the University of Wisconsin-Madison.",
"Zhang is supported in part by U.S. National Institutes of Health (R01HG010171 and R01MH116527).",
"We appreciate Dr. Abhishek Chakrabortty from Texas A&M University for some insightful discussions on the high-dimensional semi-supervised inference problem."
],
[
"Preliminary Lemmas",
"We start with several basic lemmas that we will apply in our proofs.",
"[Lemma B.1 in [18]] Let $\\lbrace X_n\\rbrace ,\\lbrace Y_n\\rbrace $ be sequences of random variables.",
"If for any $c>0$ , $(|X_n|>c|Y_n)=o_p(1)$ .",
"Then $X_n=o_p(1)$ .",
"[Nemirovski moment inequality, Lemma 14.24 in [6]] For $m\\ge 1$ and $p>e^{m-1}$ , we have $\\Big [\\max _{1\\le k\\le p}\\Big |\\sum _{i=1}^n(\\gamma _k(Z_i)-[\\gamma _k(Z_i)])\\Big |^m\\Big ]\\le (8\\log 2p)^{\\frac{m}{2}}\\Big [(\\max _{1\\le k\\le p}\\sum _{i=1}^n\\gamma _k^2(Z_i))^{m/2}\\Big ]$ [Theorem 3.1 in [36]] Assume that $\\in ^{n\\times p}$ has zero mean and covariance $$ .",
"Furthermore, assume that the rows of $^{-1/2}\\in ^{n\\times p}$ are independent sub-gaussian random vector with a bounded sub-gaussian constant and $\\Lambda _{\\min }()>C_{\\min }>0$ , $\\max _{1\\le j\\le p}\\Sigma _{jj}=O(1)$ .",
"Set $0<\\delta <1$ , $0<s_0<p$ , and $L>0$ .",
"Define the following event, $\\mathcal {B}_\\delta (n,s_0,L)=\\lbrace \\in ^{n\\times p}:(1-\\delta )\\sqrt{C_{\\min }} \\le \\frac{\\Vert \\Vert _2}{\\sqrt{n}\\Vert \\Vert _2},\\forall \\in \\mathcal {C}(s_0,L)~\\textrm {s.t.",
"}~\\ne 0\\rbrace .$ and $\\mathcal {C}(s_0,L)=\\lbrace \\in ^p:\\exists S\\subseteq \\lbrace 1,...,p\\rbrace ,|S|=s_0, \\Vert _{S^c}\\Vert _1\\le L\\Vert _S\\Vert _1\\rbrace $ .",
"Then, there exists a constant $c_1=c(L,\\delta )$ such that, for sample size $n\\ge c_1s_0\\log (p/s_0)$ , we have $(\\mathcal {B}_\\delta (n,s_0,L))\\ge 1-e^{-\\delta ^2n}.$"
],
[
"Proof of Theorem ",
"The lower bound consists of two parts.",
"By taking $f(X)=X^T^*$ , that is the linear model is correctly specified, the proof in Proposition 6.4 of [43] with conditions $s\\log (p/s)\\le C^{\\prime }n$ and $2\\le s\\le (n-1)/4$ directly implies $\\inf _{\\hat{\\theta }} \\sup _{P_{X,Y}\\in \\mathcal {P}_{\\Phi ,\\sigma }} _{P_{X,Y}} \\Big [\\Vert \\hat{}-^*\\Vert _q\\gtrsim c_1s^{1/q}\\sigma \\sqrt{\\frac{\\log (p/s)}{n}}\\Big ]>c_2.$ To establish the lower bound $s^{1/q}\\Phi \\sqrt{\\log (p/s)/(n+N)}$ , we first construct a set of hypotheses and then apply Theorem 2.7 in [41].",
"Define the set $\\mathcal {M}=\\lbrace x\\in \\lbrace 0,1\\rbrace ^{p-1}: \\Vert x\\Vert _0=s\\rbrace $ .",
"It follows from the Varshamov-Gilbert bound (e.g., Lemma 2.9 in [41]) and Lemma A.3 in [34] that there exists a subset $\\mathcal {M}^{\\prime }$ of $\\mathcal {M}$ such that for any $x,x^{\\prime }$ in $\\mathcal {M}^{\\prime }$ with $x\\ne x^{\\prime }$ , we have $\\rho _H(x,x^{\\prime })>\\frac{s}{16},~~\\textrm {and}~~\\log |\\mathcal {M}^{\\prime }|\\ge c_1^{\\prime }s\\log (\\frac{p}{s}),$ where $\\rho _H$ denotes the Hamming distance and $c_1^{\\prime }>0$ is an absolute constant.",
"Denote the element of the finite set $\\mathcal {M^{\\prime }}$ by $^j$ and the index set for the nonzero entries by $[j]$ .",
"For each $^j\\in ^{p-1}$ , we add a zero as the first entry to obtain a $p$ -dimensional vector $(0,^j)$ , and for notational simplicity, we still call it $^j$ .",
"Next, we construct a finite set of hypotheses by perturbing the distribution of $X$ .",
"Denote $N=(N_1,...,N_p)\\sim \\mathcal {N}(0,I_p)$ .",
"Let us consider the following hypotheses $\\begin{aligned}H_0&: X=(X_1,...,X_p) ~\\textrm {with}~ X_{\\ell }=N_{\\ell },\\\\H_j&: X=(X_1,...,X_p) ~\\textrm {with}~ X_{\\ell }={\\left\\lbrace \\begin{array}{ll}\\rho (N_1^2-1)+\\sqrt{1-2\\rho ^2}N_{\\ell }, &\\mbox{ if } ^j_\\ell =1,\\\\N_\\ell , &\\mbox{ if } ^j_\\ell =0,\\end{array}\\right.",
"}\\end{aligned}$ for $j=1,...,|^{\\prime }|$ , where $\\rho >0$ is a quantity to be chosen later.",
"Let $_j$ denote the expectation under $H_j$ .",
"Clearly, $_j(X_\\ell )=0$ .",
"After some simple calculation, we can verify that for $j=1,...,|^{\\prime }|$ , $_j[X_\\ell X_m]={\\left\\lbrace \\begin{array}{ll}2\\rho ^2 & \\mbox{if}~ ^j_\\ell ~\\mbox{and}~ ^j_m \\ne 0 ~\\mbox{and}~ \\ell \\ne m, \\\\1 & \\mbox{if}~ \\ell =m,\\\\0 & \\mbox{otherwise.}\\end{array}\\right.",
"}$ Denote by $=_j(XX^T)$ the covariance matrix of $X$ under $H_j$ .",
"From (REF ), there exists a permutation matrix $$ such that $=^T$ , where $=\\textrm {diag}(,I_{p-s})$ and $$ is a $s$ -dimensional equicorrelation matrix with the off-diagonal entry $2\\rho ^2$ and the diagonal entry 1.",
"Assume that the $(\\lambda ,)$ are the eigenvalue and corresponding eigenvector of $$ .",
"Following the definition of eigenvalues, $(\\lambda ,)$ are the eigenvalue and eigenvector of $$ .",
"As a result, $\\lambda _{\\min }()=\\lambda _{\\min }()=\\lambda _{\\min }()=1-2\\rho ^2$ , where the last two equalities follow from the property of the block diagonal matrix and equicorrelation matrix.",
"With the choice of $\\rho $ as specified in (REF ), we can derive that $\\lambda _{\\min }()>1/2$ and therefore $$ is positive definite.",
"We further choose $f(X)=\\frac{\\Phi }{\\sqrt{3}} X_1^2$ .",
"Our next step is to verify that the corresponding estimand $^j=\\lbrace _j(XX^T)\\rbrace ^{-1}_j(Xf(X))$ is $s$ -sparse.",
"To this end, we first note that $_j[X f(X)]=\\frac{2}{\\sqrt{3}} \\Phi \\rho ^j$ .",
"To calculate $^{-1}$ , we first rewrite the equicorrelation matrix $$ as $=(1-2\\rho ^2)I_s+2\\rho ^2\\mathbf {1}_s\\mathbf {1}_s^T$ , where $\\mathbf {1}_s$ is a $s$ -dimensional vector of 1.",
"The Woodbury formula implies $^{-1}=\\frac{1}{1-2\\rho ^2}I_s-\\frac{2\\rho ^2}{(1-2\\rho ^2)(1+2(s-1)\\rho ^2)}\\mathbf {1}_s\\mathbf {1}_s^T.$ Thus, we obtain $^{-1}=^{-1}^{T}$ with $^{-1}=\\textrm {diag}(^{-1}, I_{p-s})$ .",
"Let $_t$ denote the canonical basis in $^p$ with the $t$ th entry being 1 and the rest being 0.",
"Note that the permutation matrix $$ can be written as $=(_{j_1},...,_{j_s}, _{j_{s+1}},...,_{j_p})$ , where the indexes $j_1,...,j_s$ belong to $[j]$ and $j_{s+1},...,j_p$ are not in $[j]$ .",
"Combining the above argument, the estimand $^j$ under hypothesis $H_j$ is given by $^j&=\\frac{2\\Phi \\rho }{\\sqrt{3}}^{-1}^{T}^j=\\frac{2\\Phi \\rho }{\\sqrt{3}}^{-1}\\begin{pmatrix}\\mathbf {1}_s\\\\\\mathbf {0}_{p-s}\\end{pmatrix}=\\frac{2\\Phi \\rho }{\\sqrt{3}}\\begin{pmatrix}^{-1}{\\mathbf {1}_s}\\\\\\mathbf {0}_{p-s}\\end{pmatrix}\\nonumber \\\\&=\\frac{2\\Phi \\rho }{\\sqrt{3}(1+2(s-1)\\rho ^2)} ^j, $ from which we know $^j$ is $s$ -sparse.",
"It is easily seen that under $H_0$ the corresponding estimand is $^0=\\mathbf {0}$ .",
"In the sequel, we will verify $_j(f(X)-X^T^j)^2\\le \\Phi ^2$ holds.",
"Recall that $N=(N_1,...,N_p)\\sim \\mathcal {N}(0,I_p)$ .",
"By (REF ), we have $&_j(f(X)-X^T^j)^2\\\\& = \\Big [(\\frac{\\Phi }{\\sqrt{3}}N_1^2-\\frac{2s\\Phi \\rho ^2}{\\sqrt{3}(1+2(s-1)\\rho ^2)}(N_1^2-1)-\\frac{2\\Phi \\rho \\sqrt{1-2\\rho ^2}}{\\sqrt{3}(1+2(s-1)\\rho ^2)}\\sum _{k\\in [j]}N_k)^2\\Big ] \\nonumber \\\\&=\\frac{\\Phi ^2}{3}(3-\\frac{4s\\rho ^2}{1+2(s-1)\\rho ^2})\\le \\Phi ^2.$ Obviously, under $H_0$ we have $_0(f(X)-X^T^0)^2= \\Phi ^2$ .",
"Therefore, we have shown that the distribution of $(X,Y)$ under the hypotheses $H_j$ for $j=0,...,|^{\\prime }|$ belongs to the class of distributions $\\mathcal {P}_{\\Phi ,\\sigma }$ .",
"To apply Theorem 2.7 in [41], we need to (1) lower bound $\\Vert ^j-^{j^{\\prime }}\\Vert _q$ for $0\\le j<j^{\\prime }\\le |^{\\prime }|$ and (2) upper bound the Kullback-Leibler divergence between the probability measure of the data denoted by $\\mathcal {P}_j$ and $\\mathcal {P}_0$ under $H_j$ and $H_0$ .",
"For (1), we have from (REF ) that for $1\\le j<j^{\\prime }\\le |^{\\prime }|$ $\\Vert ^j-^{j^{\\prime }}\\Vert _q=\\frac{2\\Phi \\rho }{\\sqrt{3}+2\\sqrt{3}(s-1)\\rho ^2}\\rho _H^{1/q}(^j,^{j^{\\prime }}) \\ge \\frac{2s^{1/q}\\Phi \\rho }{16^{1/q}\\sqrt{3}(1+2(s-1)\\rho ^2)},$ where the last step follows from (REF ).",
"Since $^0=\\mathbf {0}$ , for $j=0$ and $j^{\\prime }\\ge 1$ we have $\\Vert ^j-^{j^{\\prime }}\\Vert _q=\\frac{2\\Phi \\rho s^{1/q}}{\\sqrt{3}(1+2(s-1)\\rho ^2)}.$ To quantify the Kullback-Leibler divergence, recall that the data in matrix form can be written as $(,\\tilde{})$ , where $\\tilde{}=(X_1,...,X_{n+N})^T$ and $=(Y_1,...,Y_{n})^T$ .",
"With a slight change of notation, we use $X_{i\\ell }$ to denote the $\\ell $ th component of $X_i$ for $1\\le \\ell \\le p$ and $1\\le i\\le n+N$ .",
"Under $H_j$ , the data distribution can be decomposed as $p_j(,\\tilde{})&=p(|)p_j(\\tilde{})=p(|)\\prod _{i=1}^{n+N} p_j(X_i)\\\\&=p(|)\\prod _{i=1}^{n+N} p_j(X_{i,-\\lbrace [j],1\\rbrace })p_j(X_{i,[j]}|X_{i1})p_j(X_{i1}),$ where we note that the p.d.f.",
"$p(|)$ remains the same across $j$ and $X_{i,-\\lbrace [j],1\\rbrace }$ stands for the subvector of $X_i$ by excluding the indexes in $\\lbrace [j],1\\rbrace $ .",
"From the above decomposition, the Kullback-Leibler divergence is given by $\\begin{aligned}\\mathcal {K}(\\mathcal {P}_j,\\mathcal {P}_0)&=_{j}\\Big [\\log \\frac{p(|)p_j(\\tilde{})}{p(|)p_0(\\tilde{})}\\Big ]=(n+N)_j\\Big [\\log \\frac{p_j(X_{i,[j]}| X_{i1})}{p_0(X_{i,[j]}| X_{i1})}\\Big ].\\end{aligned}$ Furthermore, notice that $p_j(X_{i,[j]}|X_{i1})=\\prod _{k\\in [j]}p_j(X_{ik}|X_{i1})=\\prod _{k\\in [j]}\\Big (\\frac{1}{\\sqrt{2\\pi (1-2\\rho ^2)}}\\exp \\Big \\lbrace -\\frac{(X_{ik}-\\rho (X_{i1}^2-1))^2}{2(1-2\\rho ^2)}\\Big \\rbrace \\Big ),$ $p_0(X_{i,[j]}|X_{i1})=\\prod _{k\\in [j]}p_0(X_{ik}|X_{i1})=\\prod _{k\\in [j]}\\Big (\\frac{1}{\\sqrt{2\\pi }}\\exp \\Big \\lbrace -\\frac{X_{ik}^2}{2}\\Big \\rbrace \\Big ).$ Hence, with some index $k\\in [j]$ we obtain $\\mathcal {K}(\\mathcal {P}_j,\\mathcal {P}_0)&=\\frac{(n+N)s}{2}\\Big (_j[-\\frac{(X_{ik}-\\rho (X_{i1}^2-1))^2}{(1-2\\rho ^2)}]-\\log (1-2\\rho ^2)+_j[X_{ik}^2]\\Big )\\nonumber \\\\&=\\frac{(n+N)s}{2}\\Big (1-\\log (1-2\\rho ^2)-\\frac{1}{(1-2\\rho ^2)}_j\\big [X_{ik}^2+\\rho ^2((X_{i1})^2-1)^2-2\\rho X_{ik}(X_{i1}^2-1)\\big ]\\Big )\\nonumber \\\\&=\\frac{(n+N)s}{2}\\log \\Big (1+\\frac{2\\rho ^2}{1-2\\rho ^2}\\Big )\\nonumber \\\\&\\le \\frac{s(n+N)\\rho ^2}{1-2\\rho ^2}.$ We set $\\rho =\\frac{1}{4\\sqrt{2}}\\sqrt{\\frac{c_1^{\\prime }\\log (p/s)}{n+N}},$ where $c_1^{\\prime }$ is specified in (REF ).",
"Given the condition $\\frac{(s-1)c_1^{\\prime }\\log (p/s)}{8(n+N)}\\le 1$ and $s\\ge 2$ , we have $1-2\\rho ^2\\ge \\frac{1}{2}$ .",
"Then from (REF ) the Kullback-Leibler divergence can be bounded as follows $\\mathcal {K}(\\mathcal {P}_j,\\mathcal {P}_0)\\le \\frac{s(n+N)\\rho ^2}{1-2\\rho ^2}\\le \\frac{ c_1^{\\prime }s \\log (p/s)}{16}\\le \\frac{1}{16}\\log |\\mathcal {M^{\\prime }}|,$ where the last inequality follows from (REF ).",
"Finally, with the choice of $\\rho $ in (REF ), we obtain from (REF ) and (REF ) that $\\Vert ^j-^{j^{\\prime }}\\Vert _q\\ge \\frac{s^{1/q}\\Phi }{3\\sqrt{6}16^{1/q}}\\sqrt{\\frac{c_1^{\\prime }\\log (p/s)}{n+N}}.$ where we use the inequality $1+2(s-1)\\rho ^2\\le 3/2$ .",
"We complete the proof by applying Theorem 2.7 in [41]."
],
[
"Proof of Theorem ",
"We can rewrite that $\\hat{}_j&= \\frac{1}{n_j} \\sum _{i\\in D^*_j} \\Big \\lbrace X_iY_i- (\\hat{f}^{-j}(X_i)X_i-\\sum _{k\\in D_j}\\frac{\\hat{f}^{-j}(X_k)X_k}{n_j+N_j})\\Big \\rbrace \\\\&=\\frac{1}{n_j} \\sum _{i\\in D^*_j}\\Big \\lbrace X_iY_i- (f(X_i)X_i-\\sum _{k\\in D_j}\\frac{f(X_k)X_k}{n_j+N_j})\\\\&+ (f(X_i)X_i-\\sum _{k\\in D_j}\\frac{f(X_k)X_k}{n_j+N_j})-(\\hat{f}^{-j}(X_i)X_i-\\sum _{k\\in D_j}\\frac{\\hat{f}^{-j}(X_k)X_k}{n_j+N_j})\\Big \\rbrace .$ Denote $I_1=\\frac{1}{n_j} \\sum _{i\\in D^*_j}\\Big \\lbrace f(X_i)X_i-\\sum _{k\\in D_j}\\frac{f(X_k)X_k}{n_j+N_j}-(\\hat{f}^{-j}(X_i)X_i-\\sum _{k\\in D_j}\\frac{\\hat{f}^{-j}(X_k)X_k}{n_j+N_j})\\Big \\rbrace .$ Next, we aim to show that $\\Vert I_!\\Vert _\\infty =O_p(K_1b_n\\sqrt{\\frac{\\log p}{n}})$ .",
"To this end, we further decompose $I$ as $\\Vert I_1\\Vert _\\infty &= \\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D^*_j} \\lbrace f(X_i)X_i-\\hat{f}^{-j}(X_i)X_i\\rbrace -\\frac{1}{n_j+N_j}\\sum _{i\\in D_j}\\lbrace f(X_i)X_i-\\hat{f}^{-j}(X_i)X_i\\rbrace \\Big \\Vert _\\infty \\nonumber \\\\&\\le \\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D^*_j}\\lbrace ( f(X_i)-\\hat{f}^{-j}(X_i))X_i\\rbrace -_{D^*_{-j}}\\big [(f(X)-\\hat{f}^{-j}(X))X\\big ]\\Big \\Vert _\\infty \\nonumber \\\\&~~+\\Big \\Vert \\frac{1}{n_j+N_j}\\sum _{i\\in D_j}\\lbrace (f(X_i)-\\hat{f}^{-j}(X_i))X_i\\rbrace -_{D^*_{-j}}\\big [(f(X)-\\hat{f}^{-j}(X))X\\big ]\\Big \\Vert _\\infty ,$ where $_{D^*_{-j}}$ denotes the conditional expectation given the data in $D^*_{-j}=D^*\\backslash D^*_j$ .",
"Let us denote $g_k(X)=(f(X)-\\hat{f}^{-j}(X))X_k$ and $\\gamma _k(X)=g_k(X)-_{D^*_{-j}}[g_k(X)]$ .",
"From (REF ), it suffices to upper bound $\\max _{1\\le k\\le p}\\frac{1}{n_j}\\sum _{i\\in D^*_j} \\gamma _k(X_i)$ and $\\max _{1\\le k\\le p}\\frac{1}{n_j+N_j}\\sum _{i\\in D_j} \\gamma _k(X_i)$ , respectively.",
"We know $_{D^*_{-j}}\\Big [\\max _{1\\le k\\le p}\\sum _{i\\in D^*_j} g_k^2(X_i)\\Big ]\\le n_jK_1^2\\Vert \\hat{f}^{-j}-f\\Vert _2^2,$ which follows from $\\Vert X\\Vert _\\infty \\le K_1$ in Assumption REF .",
"Therefore, with the application of lemma REF by choosing $m=2$ , we can show $_{D^*_{-j}}\\Big [\\max _{1\\le k\\le p}\\Big |\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\gamma _k(X_i)\\Big |^2\\Big ]\\le K_1^2\\frac{8\\log (2p)}{n_j}\\Vert \\hat{f}^{-j}-f\\Vert _2^2.$ Furthermore, the Markov inequality implies for any $c>0$ $&\\Big (\\max _{1\\le k\\le p}|\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\gamma _k(X_i)|\\ge cK_1b_n\\sqrt{\\frac{\\log 2p}{n_j}}\\Big | D^*_{-j}\\Big )\\nonumber \\\\&\\le \\Big (\\frac{_{D^*_{-j}}[\\max _{1\\le k\\le p}|\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\gamma _k(X_i)|^2]}{c^2K_1^2b_n^2}{\\frac{n_j}{\\log (2p)}}\\Big )\\wedge 1\\nonumber \\\\&\\le \\Big (\\frac{8\\Vert \\hat{f}^{-j}-f\\Vert _2^2}{c^2b_n^2}\\Big )\\wedge 1.$ For any $\\epsilon >0$ , let $c^{\\prime }$ be a sufficiently large constant such that the event $=\\lbrace \\Vert \\hat{f}^{-j}-f\\Vert _2^2\\le c^{\\prime }b_n^2\\rbrace $ holds with probability at least $1-\\epsilon $ .",
"From (REF ), we know that $&\\Big (\\max _{1\\le k\\le p}|\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\gamma _k(X_i)|\\ge cK_1b_n\\sqrt{\\frac{\\log 2p}{n_j}}\\Big )\\\\&=\\Big [\\Big (\\max _{1\\le k\\le p}|\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\gamma _k(X_i)|\\ge cK_1b_n\\sqrt{\\frac{\\log 2p}{n_j}}\\Big | D^*_{-j}\\Big )\\Big ]\\\\&\\le \\Big [\\frac{8\\Vert \\hat{f}^{-j}-f\\Vert _2^2}{c^2b_n^2}\\wedge 1\\Big ] I()+\\Big [\\frac{8\\Vert \\hat{f}^{-j}-f\\Vert _2^2}{c^2b_n^2}\\wedge 1\\Big ] I(^c)\\\\&\\le \\frac{8c^{\\prime }}{c^2}+(^c)\\le 2\\epsilon .$ where the last step holds by taking $c^2=8c^{\\prime }/\\epsilon $ and the definition $(^c)\\le \\epsilon $ .",
"This implies $\\max _{1\\le k\\le p}\\Big |\\frac{1}{n_j}\\sum _{i\\in D^*_j} \\gamma _k(X_i)\\Big |=O_p(K_1b_n\\sqrt{\\frac{\\log p}{n_j}}).$ Following the same argument, the following probability bound holds $\\max _{1\\le k\\le p}\\Big |\\frac{1}{n_j+N_j}\\sum _{i\\in D_j} \\gamma _k(X_i)\\Big |=O_p(K_1b_n\\sqrt{\\frac{\\log p}{n_j+N_j}}).$ The rest of the proof follows the same line as in the proof of Theorem 7.1 in [8].",
"Recall that we assume for $j=\\lbrace 1,2\\rbrace $ , $n_j=n/2$ and $N_j=N/2$ .",
"We can show that $ &\\Big \\Vert \\frac{\\sum _{i =1 }^{n+N}X_iX_i^T}{(n+N)}^*-\\frac{\\hat{}_1+\\hat{}_2}{2}\\Big \\Vert _\\infty \\nonumber \\\\&=\\Big \\Vert \\frac{\\sum _{i =1 }^{n+N}X_i(X_i^T^*- f(X_i))}{(n+N)}+\\frac{\\sum _{i=1}^{n}X_i^T(Y_i- f(X_i))}{n}\\Big \\Vert _\\infty +O_p(K_1b_n\\sqrt{\\frac{\\log p}{n}})\\nonumber \\\\& \\le \\Big \\Vert \\frac{\\sum _{i =1 }^{n+N}X_i(X_i^T^*- f(X_i))}{(n+N)}\\Big \\Vert _\\infty +\\Big \\Vert \\frac{\\sum _{i=1}^{n}X_i^T(Y_i- f(X_i))}{n}\\Big \\Vert _\\infty +O_p(K_1b_n\\sqrt{\\frac{\\log p}{n}})\\nonumber \\\\&= O_p(K_1\\Phi \\sqrt{\\frac{\\log p}{n+N}})+O_p(K_1\\sigma \\sqrt{\\frac{\\log p}{n}})+O_p(K_1b_n\\sqrt{\\frac{\\log p}{n}}).$ where the last probability bound holds by the same argument in the proof of example 14.3 in [6].",
"Then, if we set $\\lambda _D= C^{\\prime }K_1(\\Phi \\sqrt{\\frac{\\log p}{n+N}}+\\sigma \\sqrt{\\frac{\\log p}{n}}+b_n\\sqrt{\\frac{\\log p}{n}})$ for sufficiently large $C^{\\prime }$ , (REF ) implies $\\lambda _D\\ge \\Big \\Vert \\frac{\\sum _{i =1 }^{n+N}X_iX_i^T}{(n+N)}^*-\\frac{\\hat{}_1+\\hat{}_2}{2}\\Big \\Vert _\\infty $ holds with probability tending to 1.",
"Let $=\\hat{}_D-^*$ .",
"By the construction of Dantzig estimator, when (REF ) holds, we have $\\Vert _{T^c}\\Vert _1\\le \\Vert _T\\Vert _1$ where $T$ denotes the support of $^*$ and $\\begin{aligned}\\frac{1}{n+N}\\Vert \\tilde{}^T\\tilde{}\\Vert _\\infty \\le \\Vert \\frac{\\tilde{}^T\\tilde{}\\hat{}_D}{n+N}-\\frac{\\hat{}_1+\\hat{}_2}{2}\\Vert _\\infty +\\Vert \\frac{\\tilde{}^T\\tilde{}^*}{n+N}-\\frac{\\hat{}_1+\\hat{}_2}{2}\\Vert _\\infty \\le 2\\lambda _D.\\end{aligned}$ Therefore, $\\begin{aligned}\\frac{1}{n+N}\\Vert \\tilde{}\\Vert ^2_2&=\\frac{1}{n+N}^T\\tilde{}^T\\tilde{}\\\\&\\le \\frac{1}{n+N}\\Vert ^T\\tilde{}^T\\tilde{}\\Vert _\\infty \\Vert \\Vert _1\\\\&\\le 2\\lambda _D\\times 2\\Vert _T\\Vert _1\\le 4\\lambda _D\\sqrt{s}\\Vert _T\\Vert _2.\\end{aligned}$ With smallest eigenvalue condition (A1) and Lemma REF , we know on the event of $\\mathcal {B}_{c_1}(n+N,s,1)$ , $\\frac{1}{n+N}\\Vert \\tilde{}\\Vert _2^2\\ge (1-c_1)^2C_{\\min } \\Vert _T\\Vert _2^2$ .",
"Therefore, $\\Vert _T\\Vert _2\\le \\frac{4\\lambda _D\\sqrt{s}}{(1-c_1)^2C_{\\min }}.$ Above all, we know $\\Vert \\Vert _1\\le 2\\Vert _T\\Vert _1\\le 2\\sqrt{s}\\Vert _T\\Vert _2\\le \\frac{8\\lambda _D s}{(1-c_1)^2C_{\\min }}$ .",
"As a byproduct, we can show that $\\Vert \\Vert _2\\lesssim \\lambda _D\\sqrt{s}$ .",
"This completes the proof."
],
[
"Proof of Theorem ",
"We will first derive some preliminary probability bounds that will be used later in the proof.",
"With (A1)-(A5) in Assumptions (REF ) and (REF ), we can verify the assumptions (B1)-(B4) for strongly bounded case in Theorem 2.4 of [42] holds with $K=K_1\\vee K_2$ .",
"In particular, we have $|X^{(-k)T}\\gamma _k|&=|X^{(-k)T}_{-k,-k}^{-1}_{-k,k}|=|X^{(-k)T}_{-k,k}||_{kk}^{-1}|\\\\&=|X^{(-k)T}_{-k,k}|(_{kk}-_{k,-k}_{-k,-k}^{-1}_{-k,k})=O(K_2),$ uniformly over $1\\le k\\le p$ .",
"Under the the strongly bounded case with $s_\\Omega =o(\\frac{n+N}{\\log p})$ , and $\\max _k_{k,k}=O(1)$ , we can apply Theorem 2.4 and Lemma 5.3 in [42] and claim that the nodewise lasso estimator satisfies $\\Vert \\hat{}-\\Vert _\\infty =O_p(Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}),~~\\Vert _p-\\hat{}\\hat{}_{n+N}\\Vert _{\\max }=O_p(K\\sqrt{\\frac{\\log p}{n+N}}).$ The first probability bound in (REF ) is directly from Theorem 2.4 of [42].",
"To see the second probability bound, with the formulation of $\\hat{}$ and notation from nodewise Lasso (REF ), we know for each row of $\\hat{}$ , $\\Vert \\hat{}_{n+N}_{k\\cdot }^T-e_k\\Vert _\\infty \\le \\lambda _k/\\hat{\\tau }_k^2,$ where $e_k$ is the unit vector.",
"Furthermore, invoking Lemma 5.3 in [42], we know when we choose a suitable tuning parameter $\\lambda _k\\asymp K\\sqrt{\\frac{\\log p}{n+N}}$ uniformly over $k$ , we have $\\max _k 1/\\hat{\\tau }_k^2=O_p(1)$ .",
"Hence, $\\Vert _p-\\hat{}\\hat{}_{n+N}\\Vert _{\\max }\\le \\max _k (\\lambda _k/\\hat{\\tau }_k^2)=O_p(K\\sqrt{\\frac{\\log p}{n+N}})$ .",
"In addition, recalling from the derivation of (REF ) we have $\\Vert \\frac{^T(- _n)}{n}+\\frac{\\tilde{}^T( _{n+N}-\\tilde{}^*)}{n+N}\\Vert _\\infty =O_p(K_1\\sigma \\sqrt{\\frac{\\log p}{n}}+K_1\\Phi \\sqrt{\\frac{\\log p}{n+N}}),$ where $_n=(f(X_1),....,f(X_n))^T$ and $_{n+N}$ is defined similarly.",
"Given the above preliminary results, we focus on deriving the limiting distribution of $^T(\\hat{}^d-^*)$ .",
"Recall that we use the following notation $\\hat{}^{-j}_{D^*_j}=\\lbrace \\hat{f}^{-j}(X_i):i\\in D_j^*\\rbrace $ , $\\hat{}^{-j}_{D_j}=\\lbrace \\hat{f}^{-j}(X_i):i\\in D_j\\rbrace $ , and $_{D^*_j}$ and $_{D_j}$ are defined similarly.",
"We decompose the term $^T(\\hat{}^d-^*)$ as $^T(\\hat{}^d-^*)&=^T\\Big \\lbrace (_p-\\hat{}\\hat{}_{n+N})(\\hat{}_D-^*)+\\hat{}\\sum _{j=1}^2(\\frac{_{D^*_j}^T(_{D^*_j}-\\hat{}^{-j}_{D^*_j})}{2n_j}+\\frac{\\tilde{}_{D_j}^T( \\hat{}^{-j}_{D_j}-\\tilde{}_{D_j}^*)}{2n_j+2N_j})\\Big \\rbrace \\nonumber \\\\&=^T\\Big \\lbrace (_p-\\hat{}\\hat{}_{n+N})(\\hat{}_D-^*)+(\\hat{}-)(\\frac{^T(- _{n})}{n}+\\frac{\\tilde{}^T( _{n+N}-\\tilde{}^*)}{n+N})\\nonumber \\\\&+(\\frac{^T(-_{n})}{n}+\\frac{\\tilde{}^T(_{n+N}-\\tilde{}^*)}{n+N})-\\sum _{j=1}^2(\\frac{_{D^*_j}^T(\\hat{}^{-j}_{D^*_j}- _{D^*_j})}{2n_j}+\\frac{\\tilde{}_{D_j}^T( _{D_j}-\\hat{}^{-j}_{D_j})}{2n_j+2N_j})\\nonumber \\\\&+(-\\hat{}) \\sum _{j=1}^2(\\frac{_{D^*_j}^T(\\hat{}^{-j}_{D^*_j}- _{D^*_j})}{2n_j}+\\frac{\\tilde{}_{D_j}^T( _{D_j}-\\hat{}^{-j}_{D_j})}{2n_j+2N_j})\\Big \\rbrace .$ Therefore, with the preliminary results above in hand, we can show that $\\Vert (_p-\\hat{}\\hat{}_{n+N})(\\hat{}_D-^*)\\Vert _\\infty &\\le \\Vert _p-\\hat{}\\hat{}_{n+N}\\Vert _{\\max }\\Vert \\hat{}_D-^*\\Vert _1\\\\&=O_p\\Big (K_1Ks(\\Phi {\\frac{\\log p}{n+N}}+\\sigma {\\frac{\\log p}{\\sqrt{n(n+N)}}}+b_n{\\frac{\\log p}{\\sqrt{n(n+N)}}})\\Big )$ from (REF ) and Theorem REF .",
"Similarly, $&\\Vert (\\hat{}-)(\\frac{^T(- _{n})}{n}+\\frac{\\tilde{}^T( _{n+N}-\\tilde{}^*)}{n+N})\\Vert _\\infty \\\\&\\le \\Vert \\hat{}-\\Vert _\\infty \\Vert \\frac{^T(- _{n})}{n}+\\frac{\\tilde{}^T( _{n+N}-\\tilde{}^*)}{n+N}\\Vert _\\infty \\nonumber \\\\&=O_p\\Big (K_1Ks_\\Omega (\\Phi {\\frac{\\log p}{n+N}}+\\sigma {\\frac{\\log p}{\\sqrt{n(n+N)}}})\\Big ),$ from (REF ) and (REF ).",
"Following the similar argument in the analysis of $I_1$ in (REF ) together with the assumption that $\\Vert \\Vert _\\infty =\\Vert X\\Vert _\\infty \\le K_2$ , we obtain $\\Vert \\sum _{j=1}^2(\\frac{_{D^*_j}^T(\\hat{}^{-j}_{D^*_j}- _{D^*_j})}{2n_j}+\\frac{\\tilde{}_{D_j}^T( _{D_j}-\\hat{}^{-j}_{D_j})}{2n_j+2N_j})\\Vert _\\infty =O_p(K_2b_n\\sqrt{\\frac{\\log p}{n}}).$ Similarly, we have $&\\Vert (-\\hat{}) \\sum _{j=1}^2(\\frac{_{D^*_j}^T(\\hat{}^{-j}_{D^*_j}- _{D^*_j})}{2n_j}+\\frac{\\tilde{}_{D_j}^T( _{D_j}-\\hat{}^{-j}_{D_j})}{2n_j+2N_j})\\Vert _\\infty \\\\&\\le \\Vert -\\hat{}\\Vert _{\\infty } \\Vert \\sum _{j=1}^2(\\frac{_{D^*_j}^T(\\hat{}^{-j}_{D^*_j}- _{D^*_j})}{2n_j}+\\frac{\\tilde{}_{D_j}^T( _{D_j}-\\hat{}^{-j}_{D_j})}{2n_j+2N_j})\\Vert _\\infty \\\\&=O_p\\Big (K_1Kb_ns_\\Omega \\frac{\\log p}{\\sqrt{n(n+N)}}\\Big ).$ Collecting the above probability bounds and plugging into (REF ), we obtain $^T(\\hat{}^d-^*)&=^T(\\frac{^T(-_n)}{n}+\\frac{\\tilde{}^T(_{n+N}-\\tilde{}^*)}{n+N})+O_p(\\delta _n)\\nonumber \\\\&=\\frac{1}{n}\\sum _{i=1}^n^T_i(Y_i-f(X_i))+\\frac{1}{n+N}\\sum _{i=1}^{n+N} ^T_i(f(X_i)-X_i^T^*)+O_p(\\delta _n)\\nonumber \\\\&=\\sum _{i=1}^{n+N} \\xi _i+O_p(\\delta _n).$ where $\\xi _i={\\left\\lbrace \\begin{array}{ll}\\frac{1}{n}^T_i(Y_i-f(X_i)+\\frac{n}{n+N}(f(X_i)-X_i^T^*)) & \\mbox{for}~ 1\\le i\\le n, \\\\\\frac{1}{n+N}^T_i(f(X_i)-X_i^T^*) & \\mbox{for}~ n+1\\le i\\le n+N\\end{array}\\right.",
"}$ and $\\delta _n=\\Vert \\Vert _1\\Big [K_1K(s\\vee s_\\Omega )\\Big (\\Phi {\\frac{\\log p}{n+N}}+\\sigma {\\frac{\\log p}{\\sqrt{n(n+N)}}}+b_n{\\frac{\\log p}{\\sqrt{n(n+N)}}}\\Big )+K_2b_n\\sqrt{\\frac{\\log p}{n}}\\Big ].$ In the following, we will apply the Lindeberg-Feller Central Limit Theorem to (REF ).",
"First, we note that $_i(Y_i-f(X_i))=0$ and $_i(f(X_i)-X_i^T^*)=0$ .",
"Denote $\\eta _i=f(X_i)-X_i^T^*$ .",
"We have $\\sum _{i=1}^{n+N}(\\xi _i^2)&=\\sum _{i=1}^n\\frac{1}{n^2}\\lbrace ^T_i(\\epsilon _i+\\frac{n}{n+N}\\eta _i)\\rbrace ^2+\\sum _{i=n+1}^{n+N}\\frac{1}{(n+N)^2}(^T_i\\eta _i)^2\\\\&=\\frac{1}{n}^T(\\sigma ^2+\\frac{n}{n+N}):=t_n^2,$ where $=((f(X)-X^T^*))$ .",
"The Lyapunov condition holds as follows $\\frac{\\sum _{i=1}^{n+N}|\\xi _i|^{2+\\delta }}{t_n^{2+\\delta }}&\\le \\frac{\\Vert \\Vert _1^{2+\\delta } K_2^{2+\\delta }(\\sum _{i=1}^n(\\epsilon _i+\\frac{n}{n+N}\\eta _i)^{2+\\delta }/n^{2+\\delta }+\\sum _{i=n+1}^{n+N}\\eta _i^{2+\\delta }/(n+N)^{2+\\delta })}{t_n^{2+\\delta }}\\\\&\\le \\frac{\\Vert \\Vert _1^{2+\\delta } K_2^{2+\\delta }[2^{1+\\delta }\\lbrace \\epsilon _i^{2+\\delta }+(\\frac{n}{n+N})^{2+\\delta }\\eta _i^{2+\\delta }\\rbrace /n^{1+\\delta }+\\eta _i^{2+\\delta }N/(n+N)^{2+\\delta }]}{t_n^{2+\\delta }}\\\\&\\le \\frac{2^{1+\\delta } \\Vert \\Vert _1^{2+\\delta } K_2^{2+\\delta }\\epsilon _i^{2+\\delta }}{n^{\\delta /2}(\\sigma ^2 ^T)^{1+\\delta /2}}+\\frac{ \\Vert \\Vert _1^{2+\\delta } K_2^{2+\\delta }(2^{1+\\delta }n+N)\\eta _i^{2+\\delta }}{(n+N)^{1+\\delta /2}(^T)^{1+\\delta /2}}\\rightarrow 0,$ where the first inequality follows from $\\Vert \\Vert _\\infty =\\Vert X\\Vert _\\infty \\le K_2$ and the second one is due to the convexity of the function $x^{2+\\delta }$ for $x>0$ .",
"Therefore, the Lindeberg-Feller Central Limit Theorem leads to $\\sum _{i=1}^{n+N} \\xi _i/t_n\\stackrel{d}{\\longrightarrow } \\mathcal {N}(0,1).$ From (REF ) we obtain $\\frac{^T(\\hat{}^d-^*)}{t_n}&=\\sum _{i=1}^{n+N} \\xi _i/t_n+O_p(\\delta _n/t_n)\\stackrel{d}{\\longrightarrow } \\mathcal {N}(0,1),$ as $\\delta _n/t_n=o(1)$ .",
"This completes the proof."
],
[
"Proof of Proposition ",
"Under the same conditions in Proposition REF , we have $|\\hat{\\sigma }^2-\\sigma ^2|=O_p(n^{-1/2}+b^2_n),$ and $\\Big |^T(\\hat{}-)\\Big |=O_p\\Big (K^2\\Vert \\Vert _1^2(b_n+K\\sqrt{\\frac{s\\log p}{n}}+\\frac{K^2s\\log p}{n}+Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}})\\Big ).$ [Proof of Lemma REF ] To show (REF ), it suffices to upper bound $\\hat{\\sigma }^2_j-\\sigma ^2$ , that is $\\hat{\\sigma }^2_j-\\sigma ^2&=\\frac{1}{n_j}\\sum _{i\\in D^*_j}(Y_i-f(X_i)+f(X_i)-\\hat{f}^{-j}(X_i))^2-\\sigma ^2\\nonumber \\\\&=\\frac{1}{n_j}\\sum _{i\\in D^*_j} (\\epsilon _i^2-\\sigma ^2)+\\frac{2}{n_j}\\sum _{i\\in D^*_j}\\epsilon _i(f(X_i)-\\hat{f}^{-j}(X_i))+\\frac{1}{n_j}\\sum _{i\\in D^*_j}(f(X_i)-\\hat{f}^{-j}(X_i))^2.$ Chebyshev's inequality together with the assumption $(\\epsilon ^4)\\le C$ implies $\\frac{1}{n_j}\\sum _{i\\in D^*_j} (\\epsilon _i^2-\\sigma ^2)=O_p(n^{-1/2})$ .",
"As in the derivation of (REF ) we have $\\Big (\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\epsilon _i(f(X_i)-\\hat{f}^{-j}(X_i))>c\\sigma b_n/n_j^{1/2}|D_j^*\\Big )\\le \\frac{\\Vert \\hat{f}^{-j}-f\\Vert _2^2}{c^2b_n^2}\\wedge 1.$ As a result, we have $\\frac{1}{n_j}\\sum _{i\\in D^*_j}\\epsilon _i(f(X_i)-\\hat{f}^{-j}(X_i))=O_p(\\frac{b_n}{n^{1/2}}).$ Similarly, $\\frac{1}{n_j}\\sum _{i\\in D^*_j}(f(X_i)-\\hat{f}^{-j}(X_i))^2\\lesssim \\Vert \\hat{f}^{-j}-f\\Vert _2^2=O_p(b^2_n)$ .",
"Plugging into (REF ), we have $|\\hat{\\sigma }^2_j-\\sigma ^2|=O_p(n^{-1/2}+b^2_n)$ , which further implies (REF ).",
"To show (REF ), we decompose $^T(\\hat{}_j-)$ as follows $&^T(\\hat{}_j-)\\nonumber \\\\&=\\underbrace{^T\\hat{}\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}X_iX_i^T [(\\hat{\\eta }_i^{-j})^2-\\eta _i^2]\\hat{}}_{T_1}+\\underbrace{^T\\hat{}\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}\\lbrace X_iX_i^T \\eta _i^2-(X_iX_i^T \\eta _i^2)\\rbrace \\hat{}}_{T_2}\\nonumber \\\\&~~~~~+\\underbrace{^T(\\hat{}-) (X_iX_i^T \\eta _i^2)\\hat{}}_{T_3}+\\underbrace{^T(X_iX_i^T \\eta _i^2)(\\hat{}-)}_{T_4}$ Let us first consider $T_1$ , which can be rewritten as $T_1&=\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}(^T\\hat{}X_i)^2[\\hat{\\eta }_i^{-j}-\\eta _i]^2+\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}(^T\\hat{}X_i)^2\\eta _i[\\hat{\\eta }_i^{-j}-\\eta _i].$ We know that $|^T\\hat{}X_i|\\le \\Vert \\Vert _1\\Vert \\hat{}X_i\\Vert _\\infty \\le \\Vert \\Vert _1(\\Vert X_i\\Vert _\\infty +\\Vert \\hat{}-\\Vert _\\infty \\Vert X_i\\Vert _\\infty )\\lesssim K\\Vert \\Vert _1$ , since $K s_{\\Omega }(\\log p/(n+N))^{1/2}=o(1)$ .",
"In addition, $(\\hat{\\eta }_i^{-j}-\\eta _i)^2\\le 2(\\hat{f}^{-j}(X_i)-f(X_i))^2+2(X_i^T(\\hat{}_D-^*))^2$ .",
"Combining these results, Theorem REF and $b_n=o(1)$ , we derive $\\Big |\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}(^T\\hat{}X_i)^2[\\hat{\\eta }_i^{-j}-\\eta _i]^2\\Big |=O_p\\Big (K^2\\Vert \\Vert _1^2(b_n^2+\\frac{K_1^2s\\log p}{n})\\Big ),$ and $&\\Big |\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}(^T\\hat{}X_i)^2\\eta _i[\\hat{\\eta }_i^{-j}-\\eta _i]\\Big |\\\\&\\le \\Big |\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}(^T\\hat{}X_i)^2\\eta _i^2\\Big |^{1/2} \\Big |\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}(^T\\hat{}X_i)^2[\\hat{\\eta }_i^{-j}-\\eta _i]^2\\Big |^{1/2}\\\\&=O_p\\Big (K^2\\Vert \\Vert _1^2(b_n+K\\sqrt{\\frac{s\\log p}{n}})\\Big ),$ where the first step holds by Cauchy–Schwarz inequality.",
"This implies the following rate for $T_1$ : $|T_1|=O_p\\Big (K^2\\Vert \\Vert _1^2(b_n+K\\sqrt{\\frac{s\\log p}{n}}+\\frac{K^2s\\log p}{n})\\Big ).$ For $T_2$ , we can show that $T_2&= ^T(\\hat{}-) _n(\\hat{}-)+2 ^T(\\hat{}-)_n+^T_n$ where $_n=\\frac{1}{n_i+N_j}\\sum _{i\\in D_j}\\lbrace X_iX_i^T \\eta _i^2-(X_iX_i^T \\eta _i^2)\\rbrace $ .",
"We can bound the three terms in the right hand side of the above equation separately.",
"As an illustration, we have $^T(\\hat{}-) _n(\\hat{}-)\\le \\Vert \\Vert _1^2\\Vert \\hat{}-\\Vert ^2_\\infty \\Vert _n\\Vert _{\\max }=O_p\\Big (\\Vert \\Vert _1^2\\frac{K^2s^2_{\\Omega }\\log p}{n+N}K^2\\sqrt{\\frac{\\log p}{n+N}}\\Big ),$ where in the last step holds we plugin the rate of $\\hat{}$ in (REF ) and apply Lemma REF to upper bound $\\Vert _n\\Vert _{\\max }$ (as $X_{i}$ is uniformly bounded by $K_1\\le K$ and $(\\eta _i^2)$ is bounded as well).",
"Using a similar argument, one can derive $| ^T(\\hat{}-)_n|\\le \\Vert \\Vert _1\\Vert \\hat{}-\\Vert _\\infty \\Vert _n\\Vert _\\infty =O_p\\Big (\\Vert \\Vert _1^2Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}K^2\\sqrt{\\frac{\\log p}{n+N}}\\Big ),$ and $^T_n=O_p\\Big (\\Vert \\Vert _1^2K^2\\sqrt{\\frac{\\log p}{n+N}}\\Big ).$ Under the additional assumption that $K s_{\\Omega }(\\log p/(n+N))^{1/2}=o(1)$ , we can simplify the rate of $T_2$ as $|T_2|=O_p\\Big (\\Vert \\Vert _1^2K^2\\sqrt{\\frac{\\log p}{n+N}}\\Big ).$ Finally, let us consider $T_3$ and $T_4$ .",
"For $T_3$ , we have $T_3=^T(\\hat{}-) (X_iX_i^T \\eta _i^2)+^T(\\hat{}-) (X_iX_i^T \\eta _i^2)(\\hat{}-),$ where the first term is identical to $T_4$ and therefore it suffices to only consider the rate of $T_3$ .",
"Since $\\Vert (X_iX_i^T \\eta _i^2)\\Vert _\\infty \\le \\Vert \\Vert _1K^2(\\eta _i^2)\\lesssim \\Vert \\Vert _1K^2$ and $\\Vert (X_iX_i^T \\eta _i^2)\\Vert _{\\max }\\lesssim K^2$ , we have $|T_3|\\lesssim \\Vert \\Vert _1Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}} \\Vert \\Vert _1K^2+\\Vert \\Vert _1^2\\frac{K^2s^2_{\\Omega }\\log p}{n+N}K^2=O_p\\Big (\\Vert \\Vert _1^2K^3s_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}\\Big ).$ Collecting the upper bounds for $T_1,...T_4$ , from (REF ) we obtain the rate in (REF ).",
"[Proof of Proposition REF ] Note that $^T\\le \\Vert \\Vert _2^2\\lambda _{\\max }()\\lesssim \\Vert \\Vert _2^2$ since $\\lambda _{\\max }()=1/\\lambda _{\\min }()\\le 1/C$ .",
"From Lemma REF , we can show that $&^T\\hat{\\sigma }^2\\hat{}-^T\\sigma ^2\\\\&=(\\hat{\\sigma }^2-\\sigma ^2)^T+(\\hat{\\sigma }^2-\\sigma ^2)^T(\\hat{}-)+\\sigma ^2^T(\\hat{}-)\\\\&\\lesssim \\Vert \\Vert _2^2(n^{-1/2}+b_n^2)+ (n^{-1/2}+b_n^2)\\Vert \\Vert _1\\Vert \\Vert _\\infty Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}+\\Vert \\Vert _1\\Vert \\Vert _\\infty Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}\\\\&=O_p\\Big ( \\Vert \\Vert _2^2(n^{-1/2}+b_n^2)+\\Vert \\Vert _1\\Vert \\Vert _\\infty Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}\\Big ).$ This implies $&\\Big |^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{})-^T(\\sigma ^2+\\frac{n}{n+N})\\Big |\\\\&=O_p\\Big ( \\Vert \\Vert _2^2(n^{-1/2}+b_n^2)+\\Vert \\Vert _1\\Vert \\Vert _\\infty Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}\\\\&~~~~+K^2\\Vert \\Vert _1^2(\\frac{nb_n}{n+N}+K\\sqrt{\\frac{n}{n+N}}\\sqrt{\\frac{s\\log p}{n+N}}+\\frac{K^2s\\log p}{n+N}+Ks_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}\\frac{n}{n+N})\\Big ).$ By applying the condition $K (s\\log p/(n+N))^{1/2}=o(1)$ and $n/(n+N)\\le 1$ , we can further simplify the above rate and derive (REF ).",
"Note that $^T(\\sigma ^2+\\frac{n}{n+N})\\ge C\\Vert \\Vert _2^2$ which together with (REF ) leads to $\\Big |^T(\\hat{\\sigma }^2\\hat{}+\\frac{n}{n+N}\\hat{})/^T(\\sigma ^2+\\frac{n}{n+N})-1\\Big |=O_p(n^{-1/2}+b_n^2+Rem_N/\\Vert \\Vert _2^2).$ Finally, (REF ) holds by Theorem REF and the Slutsky Theorem.",
"This completes the proof."
],
[
"Proof of Theorem ",
"We first state several propositions and lemmas which are used in the proof.",
"Assume that Assumption REF holds.",
"Consider the Dantzig selector $\\hat{}_L$ in (REF ) with $\\lambda _L\\asymp K_1\\sqrt{\\frac{(\\sigma ^2+\\Phi ^2)\\log p}{n}}$ .",
"We have $\\Vert \\hat{}_L-^*\\Vert _1=O_p(s\\lambda _L),~~\\textrm {and}~~\\frac{1}{n}\\sum _{i=1}^n[X_i^T(\\hat{}_L-^*)]^2=O_p(s\\lambda _L^2).$ Moreover, we have $\\Vert \\hat{}_n(\\hat{}_L-^*)\\Vert _{\\infty }=O_p(\\lambda _L),$ where $\\hat{}_n=\\frac{1}{n}\\sum _{i=1}^nX_i^{\\otimes 2}$ .",
"The proof of the convergence rate of $\\hat{}_L$ in (REF ) is similar to Theorem 7.1 in [8].",
"The key step is to derive $\\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)\\Vert _\\infty \\lesssim K_1(\\sigma ^2+\\Phi ^2)^{1/2}\\sqrt{\\frac{\\log p}{n}},$ which is implied by Lemma REF together with $(Y_i-X_i^T^*)^2=\\sigma ^2+\\Phi ^2$ and $\\Vert X_i\\Vert _\\infty \\le K_1$ .",
"The rest of the proof is omitted.",
"To show the rate of $\\Vert \\hat{}_n(\\hat{}_L-^*)\\Vert _{\\infty }$ , we note that, with $\\lambda _L=C K_1\\sqrt{\\frac{(\\sigma ^2+\\Phi ^2)\\log p}{n}}$ for some sufficiently large $C$ , we have $\\Vert \\hat{}_n(\\hat{}_L-^*)\\Vert _{\\infty }&\\le \\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T\\hat{}_L)\\Vert _{\\infty }+\\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)\\Vert _{\\infty }\\le 2\\bar{\\lambda },$ where we invoke the KKT condition of $\\hat{}_L$ in the last step.",
"Under the same conditions in Theorem REF , for any $\\tilde{\\lambda }_k=\\tilde{\\lambda }\\ge CK_1(c_n+\\sqrt{\\log p/n})$ , we obtain that $\\Vert \\hat{}_{\\cdot k}- _{\\cdot k}\\Vert _1\\lesssim s_B\\tilde{\\lambda }+\\tilde{\\lambda }^{-1}\\Big [K_1L_B(c_n+\\sqrt{\\frac{\\log p}{n}})+K_1^2\\sqrt{\\frac{s(\\sigma ^2+\\Phi ^2)\\log p}{n}}\\Big ]^2.$ Define $\\tilde{\\lambda }_{opt}=_{\\tilde{\\lambda }\\ge CK_1(c_n+\\sqrt{\\log p/n})}s_B\\tilde{\\lambda }+\\tilde{\\lambda }^{-1}\\Big [K_1L_B(c_n+\\sqrt{\\frac{\\log p}{n}})+K_1^2\\sqrt{\\frac{s(\\sigma ^2+\\Phi ^2)\\log p}{n}}\\Big ]^2.$ By choosing $\\tilde{\\lambda }_k=\\tilde{\\lambda }_{opt}$ , we obtain $\\Vert \\hat{}_{\\cdot k}- _{\\cdot k}\\Vert _1\\lesssim K_1(s_B+s_B^{1/2}L_B)(\\sqrt{\\frac{\\log p}{n}}+c_n)+K_1^2\\sqrt{\\frac{ss_B(\\sigma ^2+\\Phi ^2)\\log p}{n}},$ which holds uniformly over $1\\le k \\le p$ The proof is deferred to Appendix REF .",
"Now we are ready to prove Theorem REF .",
"For notational simplicity, we use $\\hat{}^d_S$ for $\\hat{}^d_{S,\\psi }$ , $\\hat{}_S$ for $\\hat{}_{S,\\psi }$ and $\\hat{}$ for $\\hat{}_L$ .",
"We can rewrite $^T(\\hat{}^d_S-^*)&=v^T\\lbrace \\hat{}-^*-\\hat{}\\hat{}_n(\\hat{}-^*)+\\hat{}(\\hat{}_S-\\hat{}_n ^*)\\rbrace \\nonumber \\\\&=^T\\Big \\lbrace (_p-\\hat{}\\hat{}_n)(\\hat{}-^*)+J\\Big \\rbrace \\nonumber \\\\&=^T\\Big \\lbrace (_p-\\hat{}_n)(\\hat{}-^*)+(-\\hat{})\\hat{}_n(\\hat{}-^*)+J\\Big \\rbrace ,$ where $J=\\hat{}\\Big (\\frac{^T (-^*)}{n}-\\frac{\\psi }{2}\\hat{}^T\\sum _{j=1}^2(\\frac{\\sum _{i\\in D^*_{j}}X_i \\hat{m}^{-j}(X_i)}{n_j}-\\frac{\\sum _{i\\in D_j} X_i \\hat{m}^{-j}(X_i)}{n_j+N_j})\\Big ).$ We notice that $\\Vert X_i\\Vert _\\infty \\le \\Vert \\Vert _\\infty \\Vert X_i\\Vert _\\infty \\le L_{\\Omega }K_1$ , applying Hoeffding inequality, $\\Vert I_p-\\hat{}_n\\Vert _{\\max }=O_p(K_1^2L_{\\Omega }\\sqrt{\\frac{\\log p}{n}})$ .",
"Hence, $\\Vert (I_p-\\hat{}_n)(\\hat{}-^*)\\Vert _\\infty \\le \\Vert I_p-\\hat{}_n\\Vert _{\\max }\\Vert \\hat{}-^*\\Vert _1\\lesssim K_1^3L_{\\Omega }(\\sigma ^2+\\Phi ^2)^{1/2}\\frac{s\\log p}{n},$ where we use the convergence rate of $\\hat{}$ in Lemma REF .",
"Moreover, we know $\\Vert (-\\hat{})\\hat{}_n(\\hat{}-^*)\\Vert _\\infty \\le \\Vert -\\hat{}\\Vert _\\infty \\Vert \\hat{}_n(\\hat{}-^*)\\Vert _\\infty \\lesssim K_1^2L_{\\Omega }s_\\Omega \\frac{(\\sigma ^2+\\Phi ^2)^{1/2}\\log p}{\\sqrt{(n+N)n}},$ followed by (REF ) (we replace $K$ with $K_1L_{\\Omega }$ ) and again Lemma REF .",
"Now, we focus on the term $J$ .",
"We rewrite $J$ as $J&=\\Big (\\frac{^T (-^*)}{n}-\\psi ^T(\\frac{\\sum _{i=1}^{n} X_im(X_i)}{n} -\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Big )\\\\&+\\underbrace{(\\hat{}-)\\Big (\\frac{^T (-^*)}{n}-\\psi ^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n}-\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Big )}_{J_1}\\\\&+\\underbrace{(\\hat{}-)(\\hat{}_S-_0)}_{J_2}+\\underbrace{(\\hat{}_S-_0)}_{J_3},$ where $_0=\\frac{^T }{n}- \\psi ^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n}-\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N}).$ In the following, we will show that the three terms $\\Vert J_k\\Vert _\\infty $ for $k=1,2,3$ are sufficiently small.",
"We first recall that $\\Vert \\frac{1}{n}\\sum _{i=1}^n X_im(X_i)-\\Vert _\\infty \\lesssim K_1\\sqrt{\\frac{\\log p}{n}},~~~\\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)\\Vert _\\infty \\lesssim K_1(\\sigma ^2+\\Phi ^2)^{1/2}\\sqrt{\\frac{\\log p}{n}},$ by Hoeffding inequality and Lemma REF .",
"For $J_1$ , it holds that $\\Vert J_1\\Vert _\\infty &\\le \\Vert \\hat{}-\\Vert _\\infty \\Vert \\frac{^T (-^*)}{n}-\\psi ^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n}-\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N}))\\Vert _\\infty \\nonumber \\\\&\\le \\Vert \\hat{}-\\Vert _\\infty \\Big [\\Vert \\frac{1}{n}\\sum _{i=1}^n X_i(Y_i-X_i^T^*)\\Vert _\\infty +|\\psi | L_B\\Vert \\frac{1}{n}\\sum _{i=1}^n X_im(X_i)-\\Vert _\\infty \\nonumber \\\\&~~~~~~~+|\\psi | L_B\\Vert \\frac{1}{n+N}\\sum _{i=1}^{n+N} X_im(X_i)-\\Vert _\\infty \\Big ]\\nonumber \\\\&\\lesssim K_1^2L_{\\Omega }((\\sigma ^2+\\Phi ^2)^{1/2}+L_B)s_\\Omega \\frac{\\log p}{\\sqrt{(n+N)n}}.$ To upper bound the supnorm of $J_2$ and $J_3$ , we need the following bounds.",
"First, by Proposition REF , we have $&\\Vert (\\hat{}- )^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n}-\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Vert _\\infty \\\\&\\le \\max _k\\Vert \\hat{}_k- _k\\Vert _1 \\Big [\\Vert \\frac{1}{n}\\sum _{i=1}^n X_im(X_i)-\\Vert _\\infty +\\Vert \\frac{1}{n+N}\\sum _{i=1}^{n+N} X_im(X_i)-\\Vert _\\infty \\Big ]\\\\&\\lesssim K_1^2(s_B+s_B^{1/2}L_B)(\\frac{\\log p}{n}+c_n\\sqrt{\\frac{\\log p}{n}})+K_1^3{\\frac{\\sqrt{ss_B(\\sigma ^2+\\Phi ^2)}\\log p}{n}}.$ Furthermore, using a similar argument in the analysis of (REF ) in the proof of Theorem REF , we have $&\\Vert ^T\\frac{1}{2}\\sum _{j=1}^2(\\frac{\\sum _{i\\in D^*_{j}} X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j}-\\frac{\\sum _{i\\in D_j}X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j+N_j})\\Vert _\\infty \\\\&\\le L_B\\max _{j=1,2}\\Vert \\frac{\\sum _{i\\in D^*_{j}} X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j}-\\frac{\\sum _{i\\in D_j}X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j+N_j}\\Vert _\\infty \\\\&\\lesssim K_1L_Bc_n\\sqrt{\\frac{\\log p}{n}}.$ Under the condition $s_BK_1^2(c_n+\\sqrt{\\frac{\\log p}{n}})=o(1)$ , one can easily verify that $\\max _k\\Vert \\hat{}_k- _k\\Vert _1\\lesssim L_B+K_1^2\\sqrt{\\frac{ss_B(\\sigma ^2+\\Phi ^2)\\log p}{n}}$ , and therefore, $&\\Vert (\\hat{}- )^T\\frac{1}{2}\\sum _{j=1}^2(\\frac{\\sum _{i\\in D^*_{j}} X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j}-\\frac{\\sum _{i\\in D_j}X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j+N_j})\\Vert _\\infty \\\\&\\lesssim K_1L_Bc_n\\sqrt{\\frac{\\log p}{n}}+c_nK_1^3{\\frac{\\sqrt{ss_B(\\sigma ^2+\\Phi ^2)}\\log p}{n}},$ which is of smaller order than the previous two terms.",
"Given these bounds, we can decompose and bound $\\Vert _0- \\hat{}_S\\Vert _\\infty $ as $\\Vert _0- \\hat{}_S\\Vert _\\infty &=|\\psi |\\times \\Vert \\frac{1}{2}\\hat{}^T\\sum _{j=1}^2(\\frac{\\sum _{i\\in D^*_{j}}X_i \\hat{m}^{-j}(X_i)}{n_j}-\\frac{\\sum _{i\\in D_j} X_i \\hat{m}^{-j}(X_i)}{n_j+N_j})\\\\&~~~-^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n}-\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Vert _\\infty \\\\&\\le \\Vert (\\hat{}- )^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n}-\\frac{\\sum _{k=1}^{n+N} X_i m(X_i)}{n+N})\\Vert _\\infty \\\\&~~~+\\Vert (\\hat{}- )^T\\frac{1}{2}\\sum _{j=1}^2(\\frac{\\sum _{i\\in D^*_{j}}X_i (\\hat{m}^{-j}(X_i)-m(X_i)}{n_j}-\\frac{\\sum _{i\\in D_j} X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j+N_j})\\Vert _\\infty \\\\&~~~+\\Vert ^T\\frac{1}{2}\\sum _{j=1}^2(\\frac{\\sum _{i\\in D^*_{j}} X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j}-\\frac{\\sum _{i\\in D_j}X_i (\\hat{m}^{-j}(X_i)-m(X_i))}{n_j+N_j})\\Vert _\\infty \\\\&\\lesssim K_1^2(s_B+s_B^{1/2}L_B)(\\frac{\\log p}{n}+c_n\\sqrt{\\frac{\\log p}{n}})+K_1^3{\\frac{\\sqrt{ss_B(\\sigma ^2+\\Phi ^2)}\\log p}{n}}.$ Thus, for $J_2$ and $J_3$ we have $\\Vert J_3\\Vert _\\infty \\lesssim L_{\\Omega } \\Big [K_1^2(s_B+s_B^{1/2}L_B)(\\frac{\\log p}{n}+c_n\\sqrt{\\frac{\\log p}{n}})+K_1^3{\\frac{\\sqrt{ss_B(\\sigma ^2+\\Phi ^2)}\\log p}{n}}\\Big ].$ Since we have $K_1L_{\\Omega }s_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}}=o(1)$ , $\\Vert J_2\\Vert _\\infty $ is of smaller order than that of $\\Vert J_3\\Vert _\\infty $ .",
"Thus, from (REF ) and (REF ), we have $J= \\Big (\\frac{^T (-^*)}{n}-\\psi ^T(\\frac{\\sum _{i=1}^{n} X_im(X_i)}{n} -\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Big )+Rem,$ where $\\Vert Rem\\Vert _\\infty &\\lesssim L_{\\Omega } K_1^2\\Big [((\\sigma ^2+\\Phi ^2)^{1/2}+L_B)s_\\Omega \\frac{\\log p}{\\sqrt{(n+N)n}}\\\\&~~~~~+(s_B+s_B^{1/2}L_B)(\\frac{\\log p}{n}+c_n\\sqrt{\\frac{\\log p}{n}})+K_1{\\frac{\\sqrt{ss_B(\\sigma ^2+\\Phi ^2)}\\log p}{n}}\\Big ].$ Finally, combining with (REF ) and (REF ), we obtain from (REF ) that $^T (^d_S-^*)&=^T\\Big (\\frac{^T (-^*)}{n}-\\psi ^T(\\frac{\\sum _{i=1}^{n} X_i m(X_i)}{n} -\\frac{\\sum _{i=1}^{n+N} X_i m(X_i)}{n+N})\\Big )+O_p(\\bar{\\delta }_n),$ where $\\bar{\\delta }_n&=\\Vert \\Vert _1 L_{\\Omega } K_1^2\\Big [((\\sigma ^2+\\Phi ^2)^{1/2}+L_B)s_\\Omega \\frac{\\log p}{\\sqrt{(n+N)n}}\\\\&~~~~~~~~~~~+(s_B+s_B^{1/2}L_B)(\\frac{\\log p}{n}+c_n\\sqrt{\\frac{\\log p}{n}})+K_1(s\\vee s_B){\\frac{\\sqrt{(\\sigma ^2+\\Phi ^2)}\\log p}{n}}\\Big ].$ To show the asymptotic normality of $^T (^d_S-^*)$ , we denote $\\xi _i={\\left\\lbrace \\begin{array}{ll}\\frac{1}{n}^T\\lbrace _{i1}-\\frac{N\\psi }{n+N}^T_{i2}\\rbrace & \\mbox{for}~ 1\\le i\\le n, \\\\\\frac{\\psi }{n+N}^T^T_{i2}& \\mbox{for}~ n+1\\le i\\le n+N,\\end{array}\\right.",
"}$ where $_{i1}=X_i(Y_i-X_i^T^*)$ and $_{i2}=X_im(X_i)-$ .",
"One can rewrite (REF ) as $^T (^d_S-^*)=\\sum _{i=1}^{n+N}\\xi _i+O_p(\\bar{\\delta }_n).$ To apply the Lindeberg-Feller Central Limit Theorem, we first note that $(\\xi _i)=0$ .",
"Furthermore, $\\sum _{i=1}^{n+N}(\\xi _i^2)&=\\frac{1}{n}^T\\Big ((_{i1}^{\\otimes 2})-\\frac{2N\\psi }{n+N}^T(_{i2}_{i1}^T)+\\frac{N^2\\psi ^2}{(n+N)^2}^T(_{i2}^{\\otimes 2})\\Big )\\\\&~~~~~~~~~~+\\frac{\\psi ^2N}{(n+N)^2}^T^T(_{i2}^{\\otimes 2})\\\\&=\\frac{1}{n}^T\\Big ((_{i1}^{\\otimes 2})-\\frac{2N\\psi }{n+N}^T(_{i2}_{i1}^T)+\\frac{N\\psi ^2}{n+N}^T(_{i2}^{\\otimes 2})\\Big ).$ Recall that $=\\lbrace (_{i2}^{\\otimes 2})\\rbrace ^{-1}(_{i2}_{i1}^T)$ .",
"Thus, we have $\\sum _{i=1}^{n+N}(\\xi _i^2)=\\frac{1}{n}^T\\Big ((_{i1}^{\\otimes 2})-\\frac{N(2\\psi -\\psi ^2)}{n+N}(_{i2}_{i1}^T)^T\\lbrace (_{i2}^{\\otimes 2})\\rbrace ^{-1}(_{i2}_{i1}^T)\\Big ):=t_n^2.$ Note that $\\Vert _{i1}\\Vert _\\infty ^{2+\\delta }\\le K_1^{2+\\delta }|\\epsilon _i+\\eta _i|^{2+\\delta }\\lesssim K_1^{2+\\delta }$ and $\\Vert _{i2}\\Vert _\\infty ^{2+\\delta }\\lesssim K_1^{2+\\delta }$ .",
"In addition, our assumption implies $t_n^{2+\\delta }\\ge C\\Vert \\Vert _2^{2+\\delta }/n^{1+\\delta /2}$ .",
"Finally, we can verify that the Lyapunov condition holds $\\frac{\\sum _{i=1}^{n+N}|\\xi _i|^{2+\\delta }}{t_n^{2+\\delta }}&\\le \\frac{\\Vert \\Vert _1^{2+\\delta } L_{\\Omega }^{2+\\delta }}{t_n^{2+\\delta }}\\Big [\\frac{1}{n^{1+\\delta }}\\Vert _{i1}-\\frac{N\\psi }{n+N}^T_{i2}\\Vert ^{2+\\delta }_\\infty +\\frac{\\psi ^{2+\\delta }N}{(n+N)^{2+\\delta }}\\Vert ^T_{i2}\\Vert _\\infty ^{2+\\delta }\\Big ]\\\\&\\lesssim (\\frac{\\Vert \\Vert _1L_\\Omega K_1}{\\Vert \\Vert _2})^{2+\\delta } \\frac{1}{n^{\\delta /2}}(1+\\frac{L_B^{2+\\delta }N}{n+N})\\rightarrow 0.$ Therefore, we obtain the desired result by applying the Lindeberg-Feller Central Limit Theorem and Slutsky Theorem.",
"This completes the proof."
],
[
"Proof of Proposition ",
"We first establish the rate of $\\hat{}_1$ and $\\hat{}_2$ in the elementwise supnorm.",
"Define $_1=((Y_i-X_i^T^*)^2X_i^{\\otimes 2})$ .",
"We note that $\\Vert \\hat{}_1-_1\\Vert _{\\max }&\\le \\Big \\Vert \\frac{1}{n}\\sum _{i=1}^n(Y_i-X_i^T^*)^2X_i^{\\otimes 2}-_1\\Big \\Vert _{\\max }+\\Big \\Vert \\frac{2}{n}\\sum _{i=1}^n(Y_i-X_i^T^*)[X_i^T(\\hat{}_L-^*)]X_i^{\\otimes 2}\\Big \\Vert _{\\max }\\\\&~~~~~+\\Big \\Vert \\frac{1}{n}\\sum _{i=1}^n[X_i^T(\\hat{}_L-^*)]^2X_i^{\\otimes 2}\\Big \\Vert _{\\max }\\lesssim K_1^3\\sqrt{\\frac{s\\log p}{n}},$ where we use the moment inequality in Lemma REF and Lemma REF .",
"Similarly, define $_2=(_{i2}_{i1}^T)$ .",
"Since $\\hat{}_2$ is constructed by sample splitting, it suffices to derive the rate of $\\hat{}_2^j-_2$ .",
"Following the same type of argument, we can show that $&\\Vert \\hat{}_2^j-_2\\Vert _{\\max }\\\\&\\le \\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*}(Y_i-X_i^T\\hat{}_L)(\\hat{m}^{-j}(X_i)-m(X_i))X_i^{\\otimes 2}\\Big \\Vert _{\\max }+\\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*}X_i^T(\\hat{}_L-^*)m(X_i)X_i^{\\otimes 2}\\Big \\Vert _{\\max }\\\\&~~~~~~+\\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*}(Y_i-X_i^T^*)m(X_i)X_i^{\\otimes 2}-_2\\Big \\Vert _{\\max }\\\\&\\lesssim K_1^2 c_n+K_1^3\\sqrt{\\frac{s\\log p}{n}}.$ Together with Proposition REF , we have $&\\Vert \\hat{}^T\\hat{}_2-^T_2\\Vert _{\\max }\\\\&\\le \\Vert \\hat{}-\\Vert _\\infty \\Vert \\hat{}_2-_2\\Vert _{\\max }+\\Vert \\hat{}-\\Vert _\\infty \\Vert _2\\Vert _{\\max }+\\Vert \\Vert _\\infty \\Vert \\hat{}_2-_2\\Vert _{\\max }\\\\&\\lesssim K_1^2 Rem,$ where $Rem=K_1(s_B+s_B^{1/2}L_B)(\\sqrt{\\frac{\\log p}{n}}+c_n)+K_1^2\\sqrt{\\frac{ss_B\\log p}{n}}+K_1L_B\\sqrt{\\frac{s\\log p}{n}}$ As a result, $\\Vert \\hat{}_\\psi -_\\psi \\Vert _{\\max }\\lesssim K_1^3\\sqrt{\\frac{s\\log p}{n}}+\\frac{NK_1^2}{n+N}Rem.$ One can easily show that $\\Vert _\\psi \\Vert _{\\max }\\lesssim K_1^2+\\frac{N}{n+N}L_BK_1^2$ , which implies $\\Vert \\hat{}_\\psi \\Vert _{\\max }\\le \\Vert _\\psi \\Vert _{\\max }+ \\Vert \\hat{}_\\psi -_\\psi \\Vert _{\\max }\\lesssim K_1^2+\\frac{N}{n+N}L_BK_1^2$ under the assumption that $Rem=o(1)$ and $K_1\\sqrt{\\frac{s\\log p}{n}}=o(1)$ .",
"Similarly, $\\Vert \\hat{}\\Vert _\\infty \\lesssim L_{\\Omega }$ since $\\Vert \\hat{}-\\Vert _\\infty =o(1)$ .",
"Finally, we can establish the rate of convergence of the estimated variance $&|^T\\hat{}\\hat{}_\\psi \\hat{}-^T_\\psi |\\\\&\\le |^T(\\hat{}-)\\hat{}_\\psi \\hat{}|+|^T\\hat{}_\\psi (\\hat{}-)|+|^T(\\hat{}_\\psi -_\\psi )|\\\\&\\lesssim \\Vert \\Vert _1^2(L_{\\Omega }\\Vert \\hat{}-\\Vert _\\infty \\Vert \\hat{}_\\psi \\Vert _{\\max }+L_{\\Omega }^2 \\Vert \\hat{}_\\psi -_\\psi \\Vert _{\\max })\\\\&\\lesssim \\Vert \\Vert _1^2\\Big [K_1L_{\\Omega }^2s_{\\Omega }\\sqrt{\\frac{\\log p}{n+N}} \\Vert _\\psi \\Vert _{\\max }+K_1^3L_{\\Omega }^2\\sqrt{\\frac{s\\log p}{n}}+\\frac{NK_1^2L_{\\Omega }^2}{n+N}Rem\\Big ],$ which proves (REF ).",
"The proof of (REF ) is immediate by the Slutsky Theorem."
],
[
"Proof of Proposition ",
"To show Proposition REF , it suffices to show that the same rate of convergence holds for $\\hat{}^j_{\\cdot k}$ .",
"For simplicity of presentation, we use the notation $\\hat{}_L=\\hat{}$ , $\\hat{}_{\\cdot k}=\\hat{}$ , $_{\\cdot k}=$ , $\\hat{}=\\hat{}-$ and $\\lambda =\\tilde{\\lambda }_k$ , $\\hat{}_i=X_i\\hat{m}^{-j}(X_i) -\\hat{}^j$ and $ _i=X_i m(X_i) - $ .",
"We start from the inequality $&\\frac{1}{n_j}\\sum _{i\\in D_j^*} \\lbrace X_{ik} (Y_i- X_i^T\\hat{})-\\hat{}^T (X_i\\hat{m}^{-j}(X_i) -\\hat{}^j)\\rbrace ^2+\\lambda \\Vert \\hat{}\\Vert _1\\\\&\\le \\frac{1}{n_j}\\sum _{i\\in D_j^*} \\lbrace X_{ik} (Y_i- X_i^T\\hat{})-^T (X_i\\hat{m}^{-j}(X_i) -\\hat{}^j)\\rbrace ^2+\\lambda \\Vert \\Vert _1.$ Following the standard argument in the analysis of Lasso (e.g., the proof of Theorem 7.1 in [8]), the above inequality reduces to $\\hat{}^T\\hat{}\\hat{}&\\le \\lambda \\Vert \\Vert _1-\\lambda \\Vert \\hat{}\\Vert _1+\\frac{2}{n_j}\\sum _{i\\in D_j^*} \\lbrace X_{ik} (Y_i- X_i^T\\hat{})-^T\\hat{}_i\\rbrace \\hat{}_i^T\\hat{}\\nonumber \\\\&=\\lambda \\Vert \\Vert _1-\\lambda \\Vert \\hat{}\\Vert _1+I_1+I_2+I_3+I_4,$ where $I_1&=\\frac{2}{n_j}\\sum _{i\\in D_j^*} \\lbrace X_{ik} (Y_i- X_i^T ^*)-^T _i\\rbrace _i^T\\hat{},\\\\I_2&=\\frac{2}{n_j}\\sum _{i\\in D_j^*} \\lbrace X_{ik} (Y_i- X_i^T ^*)-^T _i\\rbrace (\\hat{}_i-_i)^T\\hat{},\\\\I_3&=-\\frac{2}{n_j}\\sum _{i\\in D_j^*} X_{ik} X_i^T (\\hat{}-^*) \\hat{}_i^T\\hat{},~~\\textrm {and}~~~I_4=-\\frac{2}{n_j}\\sum _{i\\in D_j^*} ^T(\\hat{}_i-_i)\\hat{}_i^T\\hat{}.$ To bound $I_1$ , we note that $\\lbrace X_{ik} (Y_i- X_i^T ^*)-^T _i\\rbrace _i=0$ by the definition of $$ .",
"To invoke Lemma REF , we control the second moment as $\\Big [\\max _{1\\le j\\le p}\\lbrace X_{ik} (Y_i- X_i^T ^*)-^T _i\\rbrace ^2 _{ij}^2\\Big ]&\\lesssim K_1^2\\Big [\\lbrace X_{ik} (Y_i- X_i^T ^*)-^T _i\\rbrace ^2 \\Big ]\\lesssim K_1^2,$ where first step $|_{ij}|=|X_{ij} m(X_i) - _j|\\lesssim K_1$ holds as $|X_{ij}|\\le K_1$ and $|m(X_i)|\\le C$ and second step relies on the assumption that the second moment of $X_{ik} (Y_i- X_i^T ^*)-^T _i$ is bounded.",
"Therefore, applying Holder inequality and Lemma REF with $m=2$ , we have $|I_1|\\lesssim \\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*} \\lbrace X_{ik} (Y_i- X_i^T ^*)-^T _i\\rbrace _i\\Big \\Vert _\\infty \\Vert \\hat{}\\Vert _1\\lesssim K_1\\sqrt{\\frac{\\log p}{n}}\\Vert \\hat{}\\Vert _1.$ Similarly, we have $|I_2|\\lesssim \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik} (\\hat{}_i-_i)\\Vert _\\infty \\Vert \\hat{}\\Vert _1$ , where $_{ik}=X_{ik} (Y_i- X_i^T ^*)-^T _i$ .",
"Furthermore, we notice that $\\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik} (\\hat{}_i-_i)&=\\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik} X_i(\\hat{m}^{-j}(X_i)-m(X_i))\\\\&~~~-(\\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik}) \\frac{1}{n_j}\\sum _{i\\in D_j^*} \\Big [X_i\\hat{m}^{-j}(X_i)-(X_i m(X_i))\\Big ].$ For the first term, we can apply Cauchy–Schwarz inequality to show that $&\\max _{1\\le l\\le p}\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik} X_{il}(\\hat{m}^{-j}(X_i)-m(X_i))\\Big |\\nonumber \\\\&\\le \\max _{1\\le l\\le p}\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} ^2_{ik} X^2_{il}\\Big |^{1/2}\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} (\\hat{m}^{-j}(X_i)-m(X_i))^2\\Big |^{1/2}\\lesssim K_1c_n,$ where in the last step we invoke Lemma REF again to show $\\max _{1\\le l\\le p}|\\frac{1}{n_j}\\sum _{i\\in D_j^*} ^2_{ik} X^2_{il}-(^2_{ik} X^2_{il})|\\lesssim K_1^2\\sqrt{\\frac{\\log p}{n}}$ and $(^2_{ik} X^2_{il})\\lesssim K_1^2$ together with triangle inequality imply the desired bound.",
"For the second term, we first notice that $\\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik}=O_p(1)$ , and then a similar argument leads to $&\\max _{1\\le l\\le p} \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} \\Big [X_{il}\\hat{m}^{-j}(X_i)-(X_{il}m(X_i))\\Big ]\\Big |\\nonumber \\\\&\\le \\max _{1\\le l\\le p} \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} X_{il}(\\hat{m}^{-j}(X_i) -m(X_i))\\Big |+\\max _{1\\le l\\le p} \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} (X_{il}m(X_i)-(X_{il}m(X_i)))\\Big |\\nonumber \\\\&\\lesssim K_1c_n+K_1\\sqrt{\\frac{\\log p}{n}},$ where we apply the Hoeffding inequality as $|X_{il}m(X_i)|\\le CK_1$ .",
"Combining (REF ) and (REF ), we have $|I_2|\\lesssim \\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*} _{ik} (\\hat{}_i-_i)\\Big \\Vert _\\infty \\Vert \\hat{}\\Vert _1\\lesssim (K_1c_n+K_1\\sqrt{\\frac{\\log p}{n}})\\Vert \\hat{}\\Vert _1.$ We now consider $I_3$ .",
"Recall that Lemma REF implies $\\frac{1}{n}\\sum _{i=1}^n[X_i^T(\\hat{}-^*)]^2=O_p(K_1^2s\\frac{(\\sigma ^2+\\Phi ^2)\\log p}{n})$ .",
"Thus, $|I_3|&\\lesssim \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} [X_i^T(\\hat{}-^*)]^2 X^2_{ik}\\Big |^{1/2}(\\hat{}^T\\hat{}\\hat{})^{1/2}\\lesssim K_1^2\\sqrt{\\frac{s(\\sigma ^2+\\Phi ^2)\\log p}{n}}(\\hat{}^T\\hat{}\\hat{})^{1/2}.$ Similarly, the Cauchy–Schwarz inequality yields $|I_4|\\lesssim (\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T(\\hat{}_i-_i)]^2)^{1/2}(\\hat{}^T\\hat{}\\hat{})^{1/2}$ .",
"Moreover, $\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T (\\hat{}_i-_i)]^2&\\le \\frac{2}{n_j}\\sum _{i\\in D_j^*} (^TX_i)^2(\\hat{m}^{-j}(X_i)-m(X_i))^2\\\\&~~~+2\\Big (\\frac{1}{n_j}\\sum _{i\\in D_j^*} \\Big [^TX_i m(X_i)-(^TX_i m(X_i))\\Big ]\\Big )^2\\\\&\\lesssim K_1^2L_B^2c_n^2+(K_1L_Bc_n+K_1L_B\\sqrt{\\frac{\\log p}{n}})^2$ where the last step holds by using a similar argument as in (REF ) and (REF ) together with the fact that $|^TX_i|\\le \\Vert \\Vert _1\\Vert X_i\\Vert _\\infty \\le K_1L_B$ .",
"As a result, we have $|I_4|\\lesssim (K_1L_Bc_n+K_1L_B\\sqrt{\\frac{\\log p}{n}})(\\hat{}^T\\hat{}\\hat{})^{1/2}.$ Collecting the bounds in (REF ), (REF ) and (REF ), we obtain from (REF ) that $\\hat{}^T\\hat{}\\hat{}&\\le \\lambda \\Vert \\Vert _1-\\lambda \\Vert \\hat{}\\Vert _1+t_1\\Vert \\hat{}\\Vert _1+t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}\\\\&\\le \\lambda \\Vert \\hat{}_S\\Vert _1-\\lambda \\Vert \\hat{}_{S^c}\\Vert _1+t_1\\Vert \\hat{}\\Vert _1+t_2(\\hat{}^T\\hat{}\\hat{})^{1/2},$ where $t_1=C(K_1c_n+K_1\\sqrt{\\frac{\\log p}{n}})$ and $t_2=C[K_1L_B(c_n+\\sqrt{\\frac{\\log p}{n}})+K_1^2\\sqrt{\\frac{s(\\sigma ^2+\\Phi ^2)\\log p}{n}}]$ for some sufficiently large constant $C$ .",
"By taking $\\lambda \\ge 2t_1$ , we have $\\hat{}^T\\hat{}\\hat{}\\le \\frac{3}{2}\\lambda \\Vert \\hat{}_S\\Vert _1-\\frac{1}{2}\\lambda \\Vert \\hat{}_{S^c}\\Vert _1+t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}.$ In the following, we consider two cases.",
"In case (1): $t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}\\le \\lambda \\Vert \\hat{}_S\\Vert _1$ , (REF ) further implies $\\hat{}^T\\hat{}\\hat{}\\le \\frac{5}{2}\\lambda \\Vert \\hat{}_S\\Vert _1-\\frac{1}{2}\\lambda \\Vert \\hat{}_{S^c}\\Vert _1,$ which leads to the standard cone condition $\\Vert \\hat{}_{S^c}\\Vert _1\\le 5\\Vert \\hat{}_S\\Vert _1$ .",
"With lemma REF , we can show that $\\Vert \\hat{}\\Vert _2^2\\lesssim \\lambda \\Vert \\hat{}_S\\Vert _1\\le \\lambda s_B^{1/2}\\Vert \\hat{}_S\\Vert _2$ and therefore $\\Vert \\hat{}\\Vert _2\\lesssim \\lambda s_B^{1/2}$ .",
"Similarly, we can derive $\\Vert \\hat{}\\Vert _1\\lesssim \\lambda s_B$ .",
"In case (2): $t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}> \\lambda \\Vert \\hat{}_S\\Vert _1$ , (REF ) implies $\\hat{}^T\\hat{}\\hat{}&\\le \\frac{3}{2}\\lambda \\Vert \\hat{}_S\\Vert _1+t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}\\le \\frac{5}{2}t_2(\\hat{}^T\\hat{}\\hat{})^{1/2},$ and therefore $\\hat{}^T\\hat{}\\hat{}\\le \\frac{25}{4}t_2^2$ .",
"Since $t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}> \\lambda \\Vert \\hat{}_S\\Vert _1$ holds in case (2), we immediately obtain $\\Vert \\hat{}_S\\Vert _1\\le \\frac{5t_2^2}{2\\lambda }$ .",
"To control $\\Vert \\hat{}_{S^c}\\Vert _1$ , we rely on (REF ) again, which is $\\frac{1}{2}\\lambda \\Vert \\hat{}_{S^c}\\Vert _1\\le \\frac{3}{2}\\lambda \\Vert \\hat{}_S\\Vert _1+t_2(\\hat{}^T\\hat{}\\hat{})^{1/2}.$ This leads to $\\Vert \\hat{}_{S^c}\\Vert _1\\le 3\\Vert \\hat{}_S\\Vert _1+\\frac{5t_2^2}{\\lambda }\\le \\frac{25t_2^2}{2\\lambda }$ , such that $\\Vert \\hat{}\\Vert _1\\lesssim t_2^2/\\lambda $ .",
"Combining the bounds in these two cases, we derive $\\Vert \\hat{}\\Vert _1\\lesssim \\lambda s_B+t_2^2/\\lambda ,$ where $\\lambda $ is subject to the constraint that $\\lambda \\ge 2t_1$ .",
"To establish a sharp rate of $\\Vert \\hat{}\\Vert _1$ , we can further minimize $f(\\lambda )=\\lambda s_B+t_2^2/\\lambda $ subject to the constraint $\\lambda \\ge 2t_1$ .",
"Define $\\lambda _{opt}=t_2/s_B^{1/2}$ .",
"When $\\lambda _{opt}\\ge 2t_1$ , the minimizer of $f(\\lambda )$ is $\\lambda _{opt}$ and the resulting minimal value is $f(\\lambda _{opt})=s_B^{1/2}t_2$ .",
"However, when $\\lambda _{opt}<2t_1$ , by the monotonicity of $f(\\lambda )$ the minimal is given by $f(2t_1)\\asymp t_1s_B$ .",
"Combining these two cases, finally, we obtain the desired rate $\\Vert \\hat{}\\Vert _1\\lesssim t_1s_B+s_B^{1/2}t_2.$ With a slight modification of the proof (e.g., $|I_1|\\lesssim K_1\\sqrt{\\frac{\\log p}{n}}\\Vert \\hat{}\\Vert _1$ still holds uniformly over $1\\le k\\le p$ ), we obtain the same rate for $\\Vert \\hat{}_{\\cdot k}-_{\\cdot k}\\Vert _1$ uniformly over $1\\le k\\le p$ .",
"This concludes the proof.",
"Recall that $\\hat{}_i=X_i\\hat{m}^{-j}(X_i) -\\hat{}^j$ , $ _i=X_i m(X_i) - $ and $\\hat{}=\\frac{1}{n_j}\\sum _{i\\in D_j^*} \\hat{}_i^{\\otimes 2}$ [RE condition for $\\hat{}_k$ ] Assume that the same conditions in Theorem REF hold.",
"Then with probability tending to 1, $\\inf _{\\in \\mathcal {C},\\ne 0}\\frac{^T\\hat{}}{\\Vert \\Vert _2^2}\\ge C,$ where $\\mathcal {C}=\\lbrace \\in ^p:\\exists S\\subseteq \\lbrace 1,...,p\\rbrace ,|S|=s_B, \\Vert _{S^c}\\Vert _1\\le \\xi \\Vert _S\\Vert _1\\rbrace $ for some constants $C,\\xi >0$ .",
"We define $=\\frac{1}{n_j}\\sum _{i\\in D_j^*} _i^{\\otimes 2}$ .",
"It holds that $^T\\hat{}&=^T(\\hat{}-)+^T(-())+^T()\\\\&\\ge ^T()-|^T(\\hat{}-)|-|^T(-())|.$ In what follows, we will bound the three terms in the last line one by one.",
"Clearly, $^T()\\ge C\\Vert \\Vert _2^2$ uniformly over $$ , as $()$ has bounded smallest eigenvalues.",
"For the last term, $|^T(-())|\\le \\Vert \\Vert _1^2\\Vert -()\\Vert _{\\max }\\le s_B(\\xi +1)^2\\Vert \\Vert _2^2\\Vert -()\\Vert _{\\max }\\lesssim s_B\\Vert \\Vert _2^2K_1^2\\sqrt{\\frac{\\log p}{n}},$ where the second step holds as $\\Vert \\Vert _1\\le (\\xi +1)\\Vert _S\\Vert _1\\le (\\xi +1)s_B^{1/2}\\Vert _S\\Vert _2\\le (\\xi +1)s_B^{1/2}\\Vert \\Vert _2$ and the last step is obtained by the Hoeffding inequality together with the bound $\\Vert X_im(X_i)\\Vert _\\infty \\le CK_1$ .",
"Under the condition $s_BK_1^2\\sqrt{\\frac{\\log p}{n}}=o(1)$ , we have $\\sup _{\\in \\mathcal {C},\\ne 0}\\frac{|^T(-())|}{\\Vert \\Vert _2^2}=o(1).$ Now, we focus on the second term $|^T(\\hat{}-)|$ .",
"To this end, we first note that $|^T(\\hat{}^j-)|&\\le \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} ^T X_i(\\hat{m}^{-j}(X_i)-m(X_i))\\Big |+\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} ^T (X_im(X_i)-(X_im(X_i)))\\Big |\\nonumber \\\\&\\le \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} (^T X_i)^2\\Big |^{1/2}\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} (\\hat{m}^{-j}(X_i)-m(X_i))^2\\Big |^{1/2}\\nonumber \\\\&~~~~~~+\\Vert \\Vert _1\\Big \\Vert \\frac{1}{n_j}\\sum _{i\\in D_j^*} (X_im(X_i)-(X_im(X_i)))\\Big \\Vert _\\infty \\nonumber \\\\&\\lesssim \\Vert \\Vert _1K_1(c_n+\\sqrt{\\frac{\\log p}{n}}),$ which is implied by the Hoeffding inequality in the last step and $\\Vert \\hat{m}^{-j}-m\\Vert _2\\lesssim c_n$ .",
"In addition, $\\frac{1}{n_j}\\sum _{i\\in D_j^*} (^T X_i)^2(\\hat{m}^{-j}(X_i)-m(X_i))^2\\lesssim \\Vert \\Vert _1^2K_1^2c_n^2.$ Combined with (REF ), we have $&\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)-m(X_i))-^T(\\hat{}^j-)]^2\\nonumber \\\\&\\le \\frac{2}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)-m(X_i))]^2+\\frac{2}{n_j}\\sum _{i\\in D_j^*} [^T(\\hat{}^j-)]^2\\nonumber \\\\&\\lesssim \\Vert \\Vert _1^2K_1^2(c_n+\\sqrt{\\frac{\\log p}{n}})^2.$ An implication of (REF ) is the following inequality $&\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)+m(X_i))-^T(\\hat{}^j+)]^2\\nonumber \\\\&\\le \\frac{2}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)-m(X_i))-^T(\\hat{}^j-)]^2+\\frac{2}{n_j}\\sum _{i\\in D_j^*} [2^T X_i m(X_i)-2^T)]^2\\nonumber \\\\&\\lesssim \\Vert \\Vert _1^2K_1^2(c_n+\\sqrt{\\frac{\\log p}{n}})^2+\\Vert \\Vert _1^2K_1^2 \\lesssim \\Vert \\Vert _1^2K_1^2.$ Finally, applying Cauchy–Schwarz inequality we can show that $&|^T(\\hat{}-)|\\nonumber \\\\&= \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} \\Big [(^T X_i \\hat{m}^{-j}(X_i)-^T\\hat{})^2- (^T X_i m(X_i)-^T)^2\\Big ]\\Big |\\nonumber \\\\&=\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)-m(X_i))-^T(\\hat{}-)][^T X_i (\\hat{m}^{-j}(X_i)+m(X_i))-^T(\\hat{}+)]\\Big |\\nonumber \\\\&\\le \\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)-m(X_i))-^T(\\hat{}^j-)]^2\\Big |^{1/2}\\nonumber \\\\&~~~~~~~~~~~~~~\\Big |\\frac{1}{n_j}\\sum _{i\\in D_j^*} [^T X_i (\\hat{m}^{-j}(X_i)+m(X_i))-^T(\\hat{}^j+)]^2\\Big |^{1/2}\\nonumber \\\\&\\lesssim \\Vert \\Vert _1^2K_1^2(c_n+\\sqrt{\\frac{\\log p}{n}})\\lesssim \\Vert \\Vert _2^2s_BK_1^2(c_n+\\sqrt{\\frac{\\log p}{n}}),$ where we use (REF ) and (REF ).",
"Therefore, from (REF ), (REF ) and (REF ), we obtain $\\inf _{\\in \\mathcal {C},\\ne 0}\\frac{^T\\hat{}}{\\Vert \\Vert _2^2}\\ge C-o(1).$"
],
[
"Examples of $(f(X)-X^T\\theta ^*)^2\\asymp s$",
"Assume that the true conditional mean function $f(X)$ has an additive form $f(X)=\\sum _{k\\in S}f_k(X_k)$ with $(f_k(X_k))=0$ , where $X_k$ is the $k$ th component of $X$ and $S$ is a subset of $\\lbrace 1,...,p\\rbrace $ with $|S|=s$ .",
"We further assume that all the covariates $X_1,...,X_p$ are mutually independent.",
"We have $(f(X)-X^T^*)^2&=\\Big \\lbrace \\sum _{k\\in S}(f_k(X_k)-X_k\\theta _k^*)-\\sum _{k\\notin S}X_k\\theta _k^*\\Big \\rbrace ^2\\\\&=\\Big \\lbrace \\sum _{k\\in S}(f_k(X_k)-X_k\\theta _k^*)\\Big \\rbrace ^2+\\Big \\lbrace \\sum _{k\\notin S}X_k\\theta _k^*\\Big \\rbrace ^2\\\\&=\\sum _{k\\in S}(f_k(X_k)-X_k\\theta ^*_k)^2+\\sum _{k\\notin S} \\Sigma _{kk}(\\theta ^*_k)^2,$ where we use the fact that $(f_k(X_k))=0$ and $(X_k)=0$ .",
"By the definition of $^*$ , we know that for $k\\notin S$ , $\\theta ^*_k=0$ .",
"For $k\\in S$ , $\\theta ^*_k=_{\\theta } (f_k(X_k)-X_k\\theta )^2=(f_k(X_k)X_k)/\\Sigma _{kk}$ and $(f_k(X_k)-X_k\\theta ^*_k)^2=(f_k^2(X_k))-[(f_k(X_k)X_k)]^2/\\Sigma _{kk}$ which is typically a constant.",
"For example, if $X_k\\sim N(0,1)$ and $f_k(X_k)=X_k^3$ , then $(f_k(X_k)-X_k\\theta ^*_k)^2=6$ .",
"Thus, we have $(f(X)-X^T^*)^2=6s$ ."
],
[
"Rate of $\\hat{\\theta }_{U}$",
"Recall that the modified Dantzig selector $\\hat{\\theta }_{U}$ is defined as $\\hat{}_{U}=\\arg \\min \\Vert \\Vert _1,~~\\textrm {s.t.",
"}~~ \\Vert \\hat{}_{n+N}-\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i \\Vert _\\infty \\le \\lambda _U.$ Assume that Assumption REF holds.",
"By choosing $\\lambda _U\\asymp K_1\\sqrt{\\frac{(\\sigma ^2+\\Phi ^2+^{*T}^*)\\log p}{n}}$ , we obtain that $\\Vert \\hat{}_U-^*\\Vert _1=O_p\\Big (sK_1(\\Phi +\\sigma +(^{*T}^*)^{1/2})\\sqrt{\\frac{\\log p}{n}}\\Big ).$ Under the conditions in our Theorem REF , we have $K_1=O(1)$ .",
"This implies the bound (REF ).",
"The proof follows from the same argument as in Lemma REF .",
"The only nontrivial step is to bound $\\Vert \\hat{}_{n+N}-\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i \\Vert _\\infty $ .",
"Using the triangle inequality, we have $\\Vert \\hat{}_{n+N}-\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i \\Vert _\\infty &\\le \\Vert \\frac{1}{n}\\sum _{i=1}^{n}X_i(Y_i-f(X_i)) \\Vert _\\infty + \\Vert \\frac{1}{n}\\sum _{i=1}^{n}X_i(f(X_i)-X_i^T^*) \\Vert _\\infty \\\\&~~~+ \\Vert (\\hat{}_n-)^* \\Vert _\\infty + \\Vert (\\hat{}_{n+N}-)^* \\Vert _\\infty .$ We have already derived in (REF ) that, $\\Vert \\frac{1}{n}\\sum _{i=1}^{n}X_i(Y_i-f(X_i)) \\Vert _\\infty \\lesssim K_1\\sigma \\sqrt{\\frac{\\log p}{n}},~~~ \\Vert \\frac{1}{n}\\sum _{i=1}^{n}X_i(f(X_i)-X_i^T^*) \\Vert _\\infty \\lesssim K_1\\Phi \\sqrt{\\frac{\\log p}{n}}.$ To control $\\Vert (\\hat{}_n-)^* \\Vert _\\infty $ , we note that $\\Vert X_i\\Vert _\\infty \\le K_1$ and $(X_i^T^*)^2=^{*T}^*$ .",
"We obtain $\\Vert (\\hat{}_n-)^* \\Vert _\\infty \\lesssim K_1(^{*T}^*)^{1/2}\\sqrt{\\frac{\\log p}{n}},$ by the Nemirovski moment inequality in Lemma REF and Markov inequality.",
"The last term $\\Vert (\\hat{}_{n+N}-)^* \\Vert _\\infty $ is dominated by $\\Vert (\\hat{}_n-)^* \\Vert _\\infty $ and can be ignored.",
"Thus, we obtain $\\Vert \\hat{}_{n+N}-\\frac{1}{n}\\sum _{i=1}^{n}X_iY_i \\Vert _\\infty \\lesssim K_1(\\Phi +\\sigma +(^{*T}^*)^{1/2})\\sqrt{\\frac{\\log p}{n}}.$"
],
[
"Sparsity assumption on $B$",
"We show here the blockwise independence structure in $X$ ensures the sparsity of $$ .",
"We denote the support of function $m$ by $S_m\\subseteq \\lbrace 1, .",
".",
".",
", p\\rbrace $ which is the index set of all the variables present in $m(\\cdot )$ .",
"Assume that the predictor variables exhibit block independence with blocks corresponding to the block-diagonal covariance matrix $$ , and the maximal block-size is equal to $b_{\\max }$ .",
"Under this assumption, we firstly know that for $1\\le k\\le p$ , if $k \\notin S_m $ $\\Vert [(X m(X)-)X^k m(X)]\\Vert _0\\le b_{\\max },$ where we use $X^k$ to denote the $k$ th component of $X$ .",
"Otherwise, $\\Vert [(X m(X)-)X^k m(X)]\\Vert _0\\le b_{\\max }|S_m|.$ Therefore, we can view the covariance matrix of $Xm(X)$ as a different block-diagonal matrix with the same blocks as in $$ for those variables not in working model $m$ .",
"Moreover, we can claim that $\\lbrace [(X m(X)-)^{\\otimes 2}]\\rbrace ^{-1}$ presents the same block structure as in covariance matrix of $Xm(X)$ .",
"On the other hand, for $k \\notin S_m\\cup S_\\eta $ , where $\\eta (X)=f(X)-X^T^*$ , $\\Vert [(XX^km(X)\\eta (X))]]\\Vert _0\\le b_{\\max },$ and the non-zero elements are within the dependence block of $X^k$ .",
"Therefore, given that $_{\\cdot k}=\\lbrace [(X m(X)-)^{\\otimes 2}]\\rbrace ^{-1}[(XX^km(X)\\eta (X))],$ we know $\\Vert _{\\cdot k}\\Vert _0\\le b_{\\max }$ since $B_{jk}$ is nonzero only when $j$ is in the corresponding dependence block of $X^k$ .",
"For $k \\in S_m\\cup S_\\eta $ , $\\Vert [(XX^km(X)\\eta (X))]]\\Vert _0\\le b_{\\max }|S_m\\cup S_\\eta |,$ and by equation REF and the block structure of $\\lbrace [(X m(X)-)^{\\otimes 2}]\\rbrace ^{-1}$ , we know $\\Vert _{\\cdot k}\\Vert _0\\le b_{\\max }|S_m\\cup S_\\eta |.$ Above all, if the supports of working model $m(\\cdot )$ and $\\eta (\\cdot )$ , the nonlinear part in $f$ are sparse, the blockwise independence structure of $X$ assumption is sufficient to guarantee the sparsity of $$ ."
],
[
"Comparison with related work in Section ",
"When the dimension $p$ is fixed and small, [2] and [15] investigated how to incorporate the unlabeled data to improve the estimation efficiency for regression coefficients in a working linear regression.",
"In addition to the technical challenges arise from the high dimensionality (e.g., regularization and one-step update), a key difference from the previous works is that our safe semi-supervised approach leads to a more efficient estimator for any linear combination of $^*$ .",
"In the following, we briefly summarize their methodologies and explain the differences.",
"To improve the estimation efficiency for $\\theta _j^*$ , [2] considered the following adjusted linear regression, for any $j\\in [p]$ $\\tilde{Y}_{ij}=\\theta ^*_j+_j^T _{ij}+\\tilde{\\delta }_{ij},$ where $\\tilde{Y}_{ij}=Y_i\\tilde{X}_{ij}$ , $_{ij}=(U_{ij1},...,U_{ijp})^T$ with $U_{ijk}=X_{ik}\\tilde{X}_{ij}$ for $k\\ne j$ and $U_{ijj}=X_{ij}\\tilde{X}_{ij}-1$ and $\\tilde{\\delta }_{ij}$ is a mean 0 random variable.",
"We use the notation $\\tilde{X}_{ij}=(X_{ij}-_j^TX_{i,-j})/[(X_{ij}-_j^TX_{i,-j})^2]$ where $_j$ is the estimand for the nodewise lasso (REF ).",
"One interesting property of the adjusted linear regression is that the parameter of interest $\\theta _j^*$ becomes the intercept parameter, because $(_{ij})=0$ and $\\theta _j^*=(Y_i\\tilde{X}_{ij})=(\\tilde{Y}_{ij})$ by the definition of $\\tilde{Y}_{ij}$ .",
"Thus, when $p$ is fixed and small, $\\theta ^*_j$ can be estimated by $\\hat{\\theta }_j^{A}$ the LSE from the adjusted linear regression, where the unlabeled data can help the estimation of $_j$ and $[(X_{ij}-_j^TX_{i,-j})^2]$ .",
"Thanks to the orthogonality of $\\tilde{\\delta }_{ij}$ and $_{ij}$ , the asymptotic variance of $\\hat{\\theta }_j^{A}$ is shown to be no greater than $\\hat{\\theta }_j^{LSE}$ , the $j$ th component of the standard LSE $\\hat{}^{LSE}=(^T)^{-1}^T$ ; see their Theorem 2.",
"As a result, if the parameter of interest is any component of $^*$ , their estimator provides the safe semi-supervised inference.",
"However, since the adjusted linear regression is estimated for each $j\\in [p]$ separately, their procedure does not guarantee the orthogonality of $\\tilde{\\delta }_{ij}$ and $_{ij^{\\prime }}$ for any $j^{\\prime }\\ne j$ when the true regression function $f(X)$ is nonlinear.",
"Therefore, the linear combination of their estimators such as $\\hat{\\theta }_j^{A}+\\hat{\\theta }_{j^{\\prime }}^{A}$ may not be more efficient than the standard LSE $\\hat{\\theta }_j^{LSE}+\\hat{\\theta }_{j^{\\prime }}^{LSE}$ .",
"Unlike their approach, our estimator is constructed based on the geometric interpretation of estimating functions.",
"The projection theory from estimating functions motivates us to consider the working regression model (REF ), which is different from the adjusted linear regression in [2].",
"[15] proposed a class of Efficient and Adaptive Semi-Supervised Estimators (EASE) which exploit the unlabeled data based on a semi-non-parametric smoothing and refitting estimate of a target imputation function $\\mu (X)$ .",
"They mainly focused on the context that $N$ is much larger than $n$ and $p$ is fixed.",
"With an estimated imputation function $\\hat{\\mu }(X)$ , they derived an initial semi-supervised estimator $\\hat{}_r$ through the estimating equation $\\frac{1}{N}\\sum _{i=n+1}^{n+N} X_i(\\hat{\\mu }(X_i)-X_i^T)=0.$ The estimator $\\hat{}_r$ attains the semi-parametric efficiency bound when the imputation is sufficient (i.e., the imputation function equals the conditional mean function $\\mu (X)=f(X)$ ) or the conditional mean function $f(X)$ is linear $f(X)=X^T^*$ .",
"As seen in Remark REF , these properties also hold for our efficient semi-supervised estimator $\\hat{}^d$ in (REF ).",
"To ensure the improved efficiency of EASE, they considered a further step of calibration which searched an optimal linear combination of $\\hat{}_r$ and the LSE $\\hat{}^{LSE}$ .",
"Their adaptive estimator is defined as $\\hat{}^E=\\hat{}^{LSE}+(\\hat{}_r-\\hat{}^{LSE})$ , where $$ is a diagonal matrix that minimizes the asymptotic variance of $\\hat{}^E_j$ for each $j\\in [p]$ .",
"When $$ is consistently estimated, $\\hat{\\theta }^E_j$ is always no less efficient than the LSE no matter whether the imputation is sufficient or $f(X)$ is linear.",
"However, by the construction of $\\hat{}^E$ , the efficiency improvement is not guaranteed if a linear combination of $^*$ is considered."
],
[
"Supplementary Plots",
"This appendix collects additional simulation results.",
"The $L_2$ and $L_1$ estimation error under Model 1 with $p=500$ and $n=200$ is shown in Figure REF .",
"The length and empirical coverage rate (shown in the number) of $95\\%$ CIs under Model 1 with $p=500$ and $n=200$ is shown in Figure REF .",
"The length and empirical coverage rate (shown in the number) of $95\\%$ CIs under Model 2 with $p=200$ and $n=200$ is shown in Figure REF .",
"Figure: The L 2 L_2 and L 1 L_1 estimation error under Model 1 with p=500p=500 and n=200n=200Figure: The length and empirical coverage rate (shown in the number) of 95%95\\% CIs under Model 1 with p=500p=500 and n=200n=200.",
"The top left panel is drawn by averaging over all the covariates not in the support set.",
"The rest are for the coefficients of X 1 X_1, X 5 X_5 and X 6 X_6 in the support set.Figure: The length and empirical coverage rate (shown in the number) of 95%95\\% CIs under Model 2 with p=200p=200 and n=200n=200.",
"The top left panel is drawn by averaging over all the covariates not in the support set.",
"The rest are for the coefficients of X 1 X_1, X 2 X_2 and X 6 X_6 in the support set.The length and empirical coverage rate (shown in the number) of $95\\%$ CIs under Model 2 with $p=200$ and $n=100$ is shown in Figure REF .",
"As Figure REF shows, O-SSL does not outperform Dantzig any more, but S-SSL still performs better.",
"This agrees with what we expect since the wrong additive model does not fit the conditional mean better than the original linear model.",
"On the other hand, the size of labeled data affects the coverage rate greatly.",
"As we can see with Dantzig, the inference result for $X_5$ is unreliable because of a low coverage rate.",
"Although the leverage of unlabeled data helps to alleviate the problem, U-Dantzig results in slightly larger CIs as well.",
"With S-SSL, we can incorporate the unlabeled data without losing efficiency.",
"Figure: The length and empirical coverage rate (shown in the number) of 95%95\\% CIs under Model 2 with p=200p=200 and n=100n=100.",
"The top left panel is drawn by averaging over all the covariates not in the support set.",
"The rest are for the coefficients of X 2 X_2, X 5 X_5 and X 6 X_6 in the support set."
]
] | 2011.14185 |
[
[
"Two-coil Wireless Power Transfer System Configured in Series-Series\n Topology: Fundamental Dynamics and Limitations on Transmitted Power"
],
[
"Abstract The dynamics and performance of a two-coil resonant coupled wireless power transfer system are investigated.",
"At high coupling, the frequency-splitting phenomenon occurs, in which the power transferred to the load attains its maximum at two frequencies away from the resonance frequency.",
"However, this behavior is not a universal property; there exist certain regions of resonator intrinsic parameters in which it is not present for any coupling strength.",
"Therefore, in order to suppress such a phenomenon, there is no need to constrain the coupling of the transmitter and receiver below a certain level as widely reported in the literature.",
"For low-power applications, optimizing the received power is essential.",
"We derive a rigorous asymptotic upper bound for the power that can be delivered to an arbitrary load from a generic source.",
"Our results quantitatively reveal the direct impacts of the unloaded $Q-$factors of the two resonators and the coupling between them on the actual output power.",
"We discuss that, in contrast to the often employed operating power gain, the transducer power gain constitutes a more suitable metric to optimize system efficiency.",
"Once the transferred power reaches its physical bound, the two gains collapse to a unique global optimal solution."
],
[
"Introduction",
"Wireless power transfer is an effective means for charging consumer electronic devices such as wireless sensor nodes in machinery and on human body [1], [2].",
"The feasibility of a wireless power transfer system (WPTS) utilizing magnetically coupled resonators was demonstrated both theoretically and experimentally [3], [4].",
"The wireless power transfer mechanism in this configuration has been extensively investigated in the literature based on either coupled-mode theory [4] or equivalent circuit theory [5].",
"The mechanism behind the transmission process can be also thoroughly explained in the electromagnetic domain [6].",
"These studies so far reveal a frequency splitting phenomenon occurring when the resonators are over-coupled, in which the power generated at the load reaches its peaks not at the original resonance frequency but at two adjacent frequencies.",
"Although it has been widely studied over the last decade, exact analytical solutions describing the position of these maxima and minimum as well as fundamental conditions defining the presence of this phenomenon are not available in the literature and therefore are of great interest.",
"Frequency splitting phenomena require the practical WPTS to either track the optimal frequency [7], [3], [8] or vary the distance between the transmitter and receiver [9] in order to maintain a sufficient power delivered to a load.",
"Several authors have attempted to eliminate the influences of the frequency splitting behavior by using tunable impedance matching networks [8], [10] or through improving designs of the resonance coils [11], [12], [13], [14].",
"All of these analyses were specific to those particular systems.",
"A generalized solution for frequency splitting suppression, which is independent of system structure, has not been fully developed yet.",
"A well-known technique to analyze the frequency splitting characteristics is to consider the input impedance, $Z_\\mathrm {in}$ .",
"The splitting frequencies are assumed to coincide with the resonance frequencies that are determined by setting the imaginary part of $Z_\\mathrm {in}$ equal to zero, $\\Im \\lbrace Z_\\mathrm {in}\\rbrace = 0$ .",
"We note that the resonance frequencies are also referred to as zero-phase angle or bifurcation frequencies.",
"Following that, the frequency splitting appears if the coupling coefficient $k$ of the two resonators goes beyond a critical value $k_\\mathrm {c}$ characterized by the condition such that there exist three resonance frequencies.",
"Therefore, in order to avoid the frequency splitting, $k$ should be selected to be less than $k_\\mathrm {c}$ , which in practice can be done by restricting the minimum distance between the transmitter and receiver.",
"This method has been extensively used even in the most recent works [15], [16].",
"However, we believe that the exact solutions of the splitting frequencies must be obtained by solving the equation $P_\\mathrm {L}/\\omega = 0$ instead of $\\Im \\lbrace Z_\\mathrm {in}\\rbrace = 0$ , where $P_\\mathrm {L}$ is the power transferred to the load and $\\omega $ is the driving angular frequency.",
"This different point of view could shed new light on the frequency splitting problem.",
"Most of the investigations on WPTSs are focused on the operating power gain (OPG), which is defined by the ratio between the power transferred to the load and the power input to the network [16], [17].",
"The OPG is referred to as the link efficiency $\\eta $ in the literature.",
"The input power is dependent on the input impedance of the system and is a function of the resonator characteristics, the load resistance, the coupling strength and the driving frequency.",
"It changes with variation of any of these parameters.",
"However, optimizing OPG does not always ensure maximum output power delivered to the load [18] and thus may not be the key factor for many low-power applications, such as biomedical sensing systems.",
"The studies in [16], [17] do not take into account the role of the power available from the source, $P_\\mathrm {avs}$ , which can be considered as the strict upper-bound on the output power of a WPTS.",
"For a realistic device, it is essential to determine the maximum transferable power of an arbitrarily given WPTS subject to mild restrictions on the physical parameters.",
"Different from other published works, we treat the actual transferred power $P_\\mathrm {L}$ as the central objective of the investigation rather than the transfer efficiency $\\eta $ .",
"From this distinguished perspective, we expect to provide an additional understanding of the performance of the WPTS.",
"The frequency splitting issue was partially discussed in [19], in which the problem was formulated for a system with matched resonance frequencies.",
"Several possible scenarios were addressed, in which two dimensionless parameters – the loaded $Q$ –factors of the two resonators – were chosen as the objectives of the study.",
"However, analytical solutions to those equations for a particular dimensional parameter such as capacitance have not been found yet.",
"Moreover, the relationship between the resonance and splitting frequencies under different conditions has not been fully exploited and is worthwhile to clarify.",
"It is essential to emphasize that the frequency splitting phenomenon can occur even when the resonance frequencies of the transmitter and receiver are not identical.",
"Therefore, a more thorough analysis and a method to solve the general case with arbitrary parameters of the two resonators are of great interest to further explore.",
"In this work, we theoretically investigate and tackle all the questions raised above with two major issues, (i) frequency splitting and (ii) an upper bound on power transferred.",
"We focus on (but not entirely limited to) a two-coil WPTS configured in series, which is perhaps the most widely used in practice.",
"In particular, we provide a universal comprehensive picture of the fundamental dynamics of the considered system.",
"We propose an efficient approach to prevent the frequency splitting property regardless of how the two coils are designed and how strong the coupling between them is.",
"A simple implementation is to adjust the nominal resonance frequency by changing the added capacitance $C_{1/2}$ ; an explicit solution of $C_{1/2}$ is given.",
"We also analytically determine the asymptotic rigorous upper bound on the optimum achievable output power, showing a general framework for designing and optimizing the system performance.",
"Several case studies are experimentally validated to support and demonstrate our theoretical hypothesis.",
"Note that, although the main target of the paper is low-power applications, all presented arguments hold true independently of any particular devices.",
"The article is organized as follows.",
"First of all, in Section , we introduce an equivalent circuit model of a two-coil WPTS and establish the analytical solution of the power transferred to the load.",
"Section is devoted to investigating the frequency splitting phenomenon and the performance of the output power with respect to the drive frequency.",
"Then, Section focuses on examining the general conditions under which the frequency splitting behavior is suppressed for any coupling strength.",
"In Section , a counterexample is provided to prove that there are scenarios where the delivered power has a unique maximum while exhibiting three distinguished resonance frequencies.",
"In addition, we also show that the frequency splitting phenomenon can occur even when there only exists a single zero-phase angle frequency.",
"Therefore, studying the imaginary part of the input impedance is not sufficient to describe the physical insights of the frequency splitting characteristics.",
"These findings return the problem to its true nature and essence.",
"The principle of the power optimization problem is presented in Section .",
"A unique asymptotic property of a perfect coupled system is discussed in Section .",
"In the last section, the experimental validations of some essential findings are carried out."
],
[
"Equivalent circuit model",
"In an inductive-based WPTS, a transmitter $L_1$ driven by an electric power source generates a time-varying electromagnetic field traveling through space, or some other medium, across a receiver $L_2$ where the electromagnetic energy is extracted and then supplied to an electrical load.",
"The magnetic induction between the transmitter and receiver is modeled by a mutual inductance $M$ , which depends on the two coil geometries and their separation distance.",
"A widely dimensionless figure of merit is the coupling coefficient defined by $k = M/\\sqrt{L_1 L_2}$ , which represents the fraction of magnetic flux density generated by $L_1$ that passes through $L_2$ when $L_2$ is in open circuit condition [20].",
"Resonant inductive coupling (or strongly coupled magnetic resonance) WPTSs are one of the most commonly used structures in which each transmitting/receiving coil is connected with a tunable capacitor in order to form a resonator.",
"Figure: Equivalent circuit model of two-coil WPTS configured in series and its representation in terms of Z-parameters.Figure REF shows an equivalent circuit model of a two-coil series-series compensated resonant WPTS, where $(L_1R_1C_1)$ and $(L_2R_2C_2)$ are the inductance, parasitic resistance and added capacitance of the transmitter and receiver coils, respectively.",
"The source is represented by a series circuit with a source voltage $V_\\mathrm {s}$ and a resistor $R_\\mathrm {s}$ ; $R_\\mathrm {L}$ is the load resistance.",
"The parasitic capacitances of the two coils are considered to be small and therefore neglected.",
"This model was verified and used in the literature (see [3], [5] for example), which is reliable for further analysis.",
"The T-network equivalent-circuit impedances of the inductive link are $\\small Z_{11} &= R_1 + j \\big (\\omega L_1 - 1/(\\omega C_1)\\big ), \\\\Z_{22} &= R_2 + j \\big (\\omega L_2 - 1/(\\omega C_2)\\big ), \\\\Z_{12} &= Z_{21} = j\\omega M $ where $\\omega $ is the angular driving frequency [18].",
"Therefore, $\\small \\begin{split}Z_\\mathrm {in} &= R_1 + j \\big (\\omega L_1 - 1/(\\omega C_1) \\big ) \\\\& + (\\omega M)^2 \\Big {/} \\big [ R_2 + \\displaystyle j \\big (\\omega L_2 - 1/(\\omega C_2) \\big ) + R_\\mathrm {L} \\big ], \\end{split}\\\\\\begin{split}Z_\\mathrm {out} &= R_2 + j \\big (\\omega L_2 - 1/(\\omega C_2) \\big ) \\\\& + (\\omega M)^2 \\Big {/} \\big [ R_1 + \\displaystyle j \\big (\\omega L_1 - 1/(\\omega C_1) \\big ) + R_\\mathrm {s} \\big ].",
"\\end{split}$ The general explicit form of the output power is derived as follows [18] $\\small \\begin{split}& P_\\mathrm {L} = \\frac{1}{2} {V_\\mathrm {s}}^2 \\frac{{Z_{21}}^2 R_\\mathrm {L}}{{(Z_{11} + Z_\\mathrm {s}) (Z_\\mathrm {out} + Z_\\mathrm {L})}^2} \\\\&= \\frac{1}{2} {V_\\mathrm {s}}^2 (\\omega M)^2 R_\\mathrm {L} \\kappa \\Big {/} \\!",
"\\left\\lbrace \\displaystyle \\big [ \\kappa (R_1 + R_\\mathrm {s}) (\\omega M)^2 + R_\\mathrm {L} + R_2 \\big ]^2 \\right.",
"\\\\& \\left.",
"+ \\big [ \\omega L_2 - 1/(\\omega C_2) - (\\omega M)^2 \\big (\\omega L_1 - 1/(\\omega C_1) \\big ) \\kappa \\big ]^2 \\right\\rbrace \\end{split} \\\\&\\text{where } \\kappa = \\big [ \\big (\\omega L_1 - 1/(\\omega C_1) \\big )^2 + (R_1 + R_\\mathrm {s})^2 \\big ]^{-1}.",
"$ Formula (REF ) is the central objective throughout the paper.",
"In this work, we choose to utilize two-port theory to investigate the performance of the two-coil WPTS since it is more familiar to electrical and electronics engineers than coupled mode theory, and therefore may be a better choice to bridge the gap between physics and engineering.",
"Another advantage of the former approach is its ability to express the coupled system in explicit algebraic forms in the frequency domain instead of describing the physical coupling by differential equations.",
"The equivalence of coupled mode and circuit theory was analyzed in, e.g., [21], [5], [22]."
],
[
"The frequency splitting phenomenon and fundamental dynamics of the series-configured Wireless Power Transfer System",
"In this section, we examine the changes on the transferred power (REF ) with respect to the driving angular frequency and reveal fundamental dynamics of the two-coil WPTS in series-series configuration.",
"We consider a general case for an arbitrary set of parameters, regardless of matched or unmatched resonance conditions.",
"The stationary point(s) are determined by equation $P_\\mathrm {L}/\\omega = 0$ , which can be written as $\\small f(\\Omega ) & = \\alpha \\Omega ^4 + \\beta \\Omega ^2 + \\gamma \\Omega + \\lambda = 0.",
"$ The coefficients $\\alpha , \\, \\beta , \\, \\gamma $ and $\\lambda $ can be normalized and expressed as functions of architecture-independent parameters, such as the electrical time constants $(\\tau _\\mathrm {s}, \\, \\tau _\\mathrm {L})$ and the angular resonance frequencies $(\\omega _1, \\, \\omega _2)$ of the transmitter and receiver coils, as follows $\\small \\alpha &= -\\frac{(k^2 - 1)^2}{(\\omega _1 \\omega _2)^4}, \\\\\\begin{split}\\beta &= \\frac{1}{(\\omega _1 \\omega _2)^4} \\bigg [ \\frac{1}{(\\tau _\\mathrm {s} \\tau _\\mathrm {L})^2} + 2(1-k^2) (\\omega _1 \\omega _2)^2 \\\\& + (\\omega _1^2 + \\omega _2^2)^2 - 2 \\Big (\\frac{\\omega _2}{\\tau _\\mathrm {s}} \\Big )^2 - 2 \\Big (\\frac{\\omega _1}{\\tau _\\mathrm {L}} \\Big )^2 \\bigg ],\\end{split} \\\\\\gamma &= \\frac{2}{(\\omega _1 \\omega _2)^4} \\bigg [ \\Big (\\frac{\\omega _1^2}{\\tau _\\mathrm {L}} \\Big )^2 \\!+\\!",
"\\Big (\\frac{\\omega _2^2}{\\tau _\\mathrm {s}} \\Big )^2\\!-\\!",
"2 (\\omega _1 \\omega _2)^2 (\\omega _1^2 \\!+\\!",
"\\omega _2^2) \\bigg ], \\\\\\lambda &= 3, \\\\\\tau _\\mathrm {s} &= L_1/(R_1 + R_\\mathrm {s}), \\,\\tau _\\mathrm {L} = L_2/(R_2 + R_\\mathrm {L}), \\\\\\omega _1 &= 1/\\sqrt{L_1 C_1}, \\,\\omega _2 = 1/\\sqrt{L_2 C_2}.$ The function $f(\\Omega )$ is a real coefficient polynomial of degree four without third degree term, in which all physical parameters such as resistance, inductance and capacitance are positive.",
"Without loss of generality, we assume that $k^2 \\ne 1$ (otherwise, the quartic polynomial $f(\\Omega )$ reduces to a quadratic function), therefore $\\alpha < 0$ .",
"By introducing intermediate parameters $p = \\beta /\\alpha , \\,\\, q = \\gamma /\\alpha $ and $r = \\lambda /\\alpha $ , we can write (REF ) in the standard form $\\small f_\\mathrm {s}(\\Omega ) = \\Omega ^4 + p \\Omega ^2 + q \\Omega + r = 0.",
"$ $f_\\mathrm {s}(\\Omega )$ collapses to the form shown in [19] when the two resonance frequencies are identical, $\\omega _1 = \\omega _2$ .",
"The cubic resolvent of the quartic equation (REF ) is defined as [23], [24] $\\small t^3 - p t^2 - 4 r t + (4 p r - q^2) = 0.$ As proven in [24], a third degree polynomial with real coefficients always has at least one real root.",
"Let $u$ be a real root of (REF ) (a complete set of solutions can be found in [25] and was presented in Appendix A of our previous paper [19]), then the four roots of the original quartic (REF ) are given by [23], [24] $\\small \\Omega _1 &= (G + H)/2, \\\\\\Omega _2 &= (G - H)/2, \\\\\\Omega _3 &= (-G + I)/2, \\\\\\Omega _4 &= (-G - I)/2 $ $\\text{where }\\small G &= \\sqrt{u - p}, \\\\H &= {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sqrt{-G^2 - 2 \\big (p + q/G \\big ) }, & G \\ne 0 \\\\\\displaystyle \\sqrt{- 2p + 2\\sqrt{u^2 - 4 r} }, & G = 0\\end{array}\\right.}",
", \\\\I &= {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sqrt{-G^2 - 2 \\big (p - q/G \\big ) }, & G \\ne 0 \\\\\\displaystyle \\sqrt{- 2p - 2\\sqrt{u^2 - 4 r} }, & G = 0\\end{array}\\right.",
"}.$ It should be noted that the notation in [(REF )-()] is the same as in [(15–(18)] in [19].",
"However, [(REF )-()] are the exact solutions for the general case where the transmitter and receiver parameters are arbitrary.",
"On the contrary, the corresponding formulas derived in [19] are only applicable when the resonance frequencies are matched.",
"In [19], we showed that equation $P_\\mathrm {L}/\\omega = 0$ (or equivalently, equations (REF ) and (REF )) must have at least one positive solution.",
"We now note the facts that, (i) there must exist an interval $(0, \\, \\epsilon _1)$ for some $\\epsilon _1 > 0$ on which $P_\\mathrm {L}$ increases and an interval $(\\epsilon _2, \\, +\\infty )$ for some $\\epsilon _2 > 0$ on which $P_\\mathrm {L}$ decreases, and (ii) (REF ) has at most four solutions [(REF )-()].",
"Thus, equation $P_\\mathrm {L}/\\omega = 0$ can only have either one or three positive solution(s); if it has two or four positive solutions, the conditions in (i) are violated.",
"As a summary, there are only two fundamental behaviors of $P_\\mathrm {L}$ with respect to $\\omega $ : the output power has either a unique maximum or two maxima and one minimum.",
"No other possibilities exist.",
"This is a unique property of the two-coil configuration in comparison with three- or four-coil systems, in which more than two power peaks can be observed under the right circumstances.",
"The latter results in the well-known frequency splitting phenomenon that has been experimentally reported in several physical systems.",
"A visual summary of these two characteristics are shown in Appendix , where signs of $\\mathrm {d}P_\\mathrm {L}/\\mathrm {d} \\omega $ and variations of $P_\\mathrm {L}$ are provided in detail.",
"All these findings complete the analysis of the dynamic performance of the series-configured WPTS that we partially discussed in [18].",
"Let us consider the necessary condition(s) such that (REF ) has three distinguishable positive solutions and where therefore the frequency splitting takes place.",
"Firstly, $G, \\,\\, H$ and $I$ must be real and non-negative.",
"If either of them is complex and not real, (REF ) only has at most two real solutions.",
"In the case $\\lbrace G, \\, H, \\, I\\rbrace \\in {\\mathbb {R}}_{\\ge 0}$ and (REF ) has four real solutions, since $\\Omega _4 \\le 0$ , the three possible non-negative solutions are $\\lbrace \\Omega _1,\\, \\Omega _2,\\, \\Omega _3\\rbrace $ .",
"Then accordingly, the extreme frequencies at which $P_\\mathrm {L}$ obtains its maximum and minimum values are $^\\mathrm {i}\\omega _\\mathrm {e} = \\sqrt{\\Omega _\\mathrm {i}}$ where $\\mathrm {i}\\in \\lbrace 1, \\,2, \\,3 \\rbrace $ .",
"However, $\\Omega = 0$ is not a solution of equation (REF ), hence $G, \\, H$ and $I$ are not simultaneously 0 and $\\lbrace I \\ne G, \\, G \\ne H \\rbrace $ .",
"In addition, $\\lbrace \\Omega _1,\\, \\Omega _2,\\, \\Omega _3\\rbrace $ are different, therefore the complete set of necessary conditions is {$I > G > H > 0$ and $(I \\pm H) \\ne 2 G$ }.",
"The frequency splitting is not present if any of these conditions is not fulfilled."
],
[
"General conditions under which the frequency splitting phenomenon does not occur.",
"We have shown that when the frequency splitting behavior occurs, there are exactly two local maxima and one minimum.",
"In general, this phenomenon is not a universal property of a WPTS.",
"The aim of this section is to point out scenarios in which (REF ) has a unique positive root, meaning that the frequency splitting phenomenon is eliminated for any coupling strength.",
"One approach is to negate the necessary conditions shown in Section and solve for a chosen variable.",
"However, we realize that it leads to cumbersome relations among system parameters which are difficult to interpret further.",
"Thus, we propose a more efficient alternative method below.",
"We proved that (REF ) must have at least one positive solution [19].",
"Therefore, if (REF ) has at most one positive solution in a set of system parameters, as a consequence, it is simultaneously the unique positive solution in that set.",
"We now apply the Descartes' sign rule to determine the maximum number of positive real roots of a polynomial, denoted by $n$ .",
"Here, $n$ is the number of sign changes, as we proceed from the lowest to the highest order (ignoring orders that do not appear).",
"Considering the polynomial $f(\\Omega )$ , since $\\alpha < 0$ and $\\lambda > 0$ , $n = 1$ if either of the following conditions is satisfied: (i) $\\beta \\le 0$ , (ii) $\\gamma \\ge 0$ .",
"The complete Descartes' sign rule table is shown in Appendix .",
"The sign of a real number does not change if we divide or multiply it by another positive number.",
"Letting $\\beta _\\mathrm {s} = \\beta (\\omega _1 \\omega _2)^2/2$ and $\\gamma _\\mathrm {s} = \\gamma \\omega _2^2 /2 $ , we obtain $\\operatorname{sign}(\\beta ) = \\operatorname{sign}(\\beta _\\mathrm {s})$ and $\\operatorname{sign}(\\gamma ) = \\operatorname{sign}(\\gamma _\\mathrm {s})$ .",
"The parameters $\\beta _\\mathrm {s}$ and $\\gamma _\\mathrm {s}$ are given by dimensionless forms as follows $\\small \\beta _\\mathrm {s} & \\!=\\!",
"1 \\!",
"+ \\!",
"\\frac{1}{2} \\frac{1}{(Q_\\mathrm {s} Q_\\mathrm {L})^2} \\!",
"+ \\!",
"\\frac{1}{2} \\big ( \\frac{\\omega _1}{\\omega _2} \\!",
"+ \\!",
"\\frac{\\omega _2}{\\omega _1} \\big )^2 \\!",
"- \\!",
"\\big ( \\frac{1}{Q_\\mathrm {s}^2} \\!",
"+ \\!",
"\\frac{1}{Q_\\mathrm {L}^2} \\big ) \\!",
"- \\!",
"k^2, \\\\\\gamma _\\mathrm {s} &= \\big ( \\frac{\\omega _2}{\\omega _1} \\big )^2 \\frac{1}{Q_\\mathrm {s}^2} + \\frac{1}{Q_\\mathrm {L}^2} - 2 \\Big [1 + \\big ( \\frac{\\omega _2}{\\omega _1} \\big )^2 \\Big ]$ where the loaded quality factors of the two coils are $\\small Q_\\mathrm {s} = \\frac{\\omega _1 L_1}{R_1 + R_\\mathrm {s}} = \\omega _1 \\tau _\\mathrm {s}, \\,Q_\\mathrm {L} = \\frac{\\omega _2 L_2}{R_2 + R_\\mathrm {L}} = \\omega _2 \\tau _\\mathrm {L}.$ The two conditions (i) and (ii) now become (I) $\\beta _\\mathrm {s} \\le 0$ and (II) $\\gamma _\\mathrm {s} \\ge 0$ respectively.",
"This analysis reveals that, in addition to the coupling coefficient, the loaded quality factors and the resonance frequencies of the transmitter and the receiver are decisive parameters that directly affect the existence of the frequency splitting phenomenon.",
"If $(k, \\, Q_\\mathrm {s}, \\, Q_\\mathrm {L}, \\, \\omega _{1}, \\, \\omega _{2})$ satisfy (I) or (II), the frequency splitting behavior is not present.",
"In particular, for a given coupling coefficient, lower resonance frequencies (and therefore lower coil quality factors) lead to suppression of this dynamical phenomenon.",
"We can utilize (I) and (II) as a general framework to determine relations between the key parameters of the two resonators in order to eliminate such a phenomenon.",
"An example is presented below.",
"We choose to treat the added capacitors $C_\\mathrm {i}$ as dependent variables instead of the coil inductances $L_\\mathrm {i}$ or parasitic resistances $R_\\mathrm {i}$ ($i = \\lbrace 1, \\, 2\\rbrace $ ).",
"This would also seem to be more common from a practical standpoint since $C_\\mathrm {i}$ can be adjusted easier than other parameters.",
"To reduce the complexity of the problem, we consider the case in which the two resonators are identical, meaning that $L_\\mathrm {i} = L, \\,\\, C_\\mathrm {i} = C, \\,\\, R_\\mathrm {i} = R$ and $\\omega _\\mathrm {i} = \\omega _0 = 1/\\sqrt{LC}$ .",
"The expressions of $\\beta _\\mathrm {s}$ and $\\gamma _\\mathrm {s}$ reduce to $\\small \\beta _\\mathrm {s} &= 3 + \\frac{1}{2} \\frac{1}{(Q_\\mathrm {s} Q_\\mathrm {L})^2} - \\Big ( \\frac{1}{Q_\\mathrm {s}^2} + \\frac{1}{Q_\\mathrm {L}^2} \\Big ) - k^2, \\\\\\gamma _\\mathrm {s} &= \\frac{1}{Q_\\mathrm {s}^2} + \\frac{1}{Q_\\mathrm {L}^2} - 4.$ The inequality equations are then solved for $C > 0$ .",
"Case (I): $\\small \\beta _\\mathrm {s} & \\le 0 \\Longleftrightarrow {\\left\\lbrace \\begin{array}{ll}\\mu \\ge 0 \\\\\\displaystyle C \\in \\mathbb {R}_1 = [\\max \\lbrace 0, \\, \\xi _1\\rbrace , \\,\\xi _2] \\setminus \\lbrace 0\\rbrace \\end{array}\\right.",
"}\\\\\\text{where }\\sigma _1 &= (R + R_\\mathrm {s})^2 (R + R_\\mathrm {L})^2 > 0, \\\\\\sigma _2 &= (R + R_\\mathrm {s})^2 + (R + R_\\mathrm {L})^2 > 0, \\\\\\mu &= \\sigma _2 ^2 - 2 (3-k^2) \\sigma _1,\\\\\\xi _1 &= \\frac{L}{\\sigma _1} (\\sigma _2 - \\sqrt{\\mu }) \\le \\xi _2, \\\\\\xi _2 &= \\frac{L }{\\sigma _1} (\\sigma _2 + \\sqrt{\\mu }) > 0.$ Case (II): $\\small \\gamma _\\mathrm {s} \\ge 0 \\iff C \\in \\mathbb {R}_2 = [{\\, C_\\gamma , +\\infty })\\text{ where } C_{\\gamma } = \\frac{4 L}{\\sigma _2}.$ Then the compact-form solution of $C$ is $ \\small {\\left\\lbrace \\begin{array}{ll}C \\in \\mathbb {R}_{\\mu -} = \\varnothing \\cup \\mathbb {R}_2 = \\mathbb {R}_2, & \\text{ if } \\mu < 0 \\\\C \\in \\mathbb {R}_{\\mu +} = \\mathbb {R}_1 \\cap \\mathbb {R}_2, & \\text{ if } \\mu \\ge 0\\end{array}\\right.}",
".$ Table: Solution of CC such that the frequency splitting phenomenon does not occur.We see that, for any values of $0 \\le k^2 < 1$ , there always exists $C$ in one or more region(s) (defined by other parameters) in which $P_\\mathrm {L}$ has a unique stationary point.",
"The solution of $C$ is summarized in Table REF , extracted from (REF ).",
"Note that, if $R_\\mathrm {L} = R_\\mathrm {s} = R^\\ast $ , we have $\\mu < 0$ for all $0 \\le k^2 < 1$ , so any $C \\ge 2L/(R+R^\\ast )^2$ results in a unique positive solution of (REF ).",
"In contrast to the well-known argument in the literature (i.e., to select the squared coupling coefficient $k^2$ to be less than a critical value), we have proven that it is possible to prohibit the frequency splitting phenomenon without compromising or inhibiting the coupling strength.",
"Figure REF presents an example where the frequency splitting is eliminated by choosing appropriate external capacitances, $C = 7$ nF and $C = 10$ nF.",
"The two resonators are set to be identical and an exemplary set of system parameters is as follows {$L = 9.13$ $\\mu $ H, $R = 2.25$ $\\Omega $ , $R_\\mathrm {s} = 50$ $\\Omega $ }, adapted from [7].",
"The load resistance and the coupling coefficient are (arbitrarily) kept fixed at $R_\\mathrm {L} = 50 \\, \\Omega $ and $k = 0.9$ .",
"The power available from the source is $P_\\mathrm {avs} = {V_\\mathrm {s}}^2/(8R_\\mathrm {s})$ and the primary resonance frequency of the two coils is $f_0 = \\omega _0/(2\\pi )$ .",
"Since $\\mu < 0$ , the condition of $C$ such that the frequency splitting phenomenon does not exist is $C \\ge C_\\gamma \\approx 6.7$ nF.",
"The capacitances below that value such as $C = 444$ pF and $C = 1$ nF lead to the frequency splitting behavior.",
"In a general trend, higher value of $C$ yields lower resonance frequency and (therefore) lower loaded quality factors, which reduces $\\beta _\\mathrm {s}$ and increases $\\gamma _\\mathrm {s}$ .",
"Until $\\beta _\\mathrm {s}$ is non-positive and/or $\\gamma _\\mathrm {s}$ is non-negative, the frequency splitting phenomenon is completely eliminated.",
"Figure: Frequency responses of the transferred power with a coupling coefficient of k=0.9k=0.9, a load resistance of R L =50ΩR_\\mathrm {L} = 50 \\, \\Omega and different added capacitances, parameterized by two ratios P L /P avs P_\\mathrm {L}/P_\\mathrm {avs} and f/f 0 f/f_0.",
"Solid circle: maximum/minimum power at extreme frequencies given by (), () and ().",
"The power is computed using ().This finding relaxes the requirement of, for instance, sophisticated designs [12], [13], [14], nonlinear resonant circuits [15], frequency/impedance tuning [8], coupling/mutual inductance adaptation [9], [16] or sensing, feedback and control units [10], among others.",
"The proposed approach is simpler and easier to implement while maintaining its effectiveness, compared to the conventional methods developed in previous works.",
"Up to this point, we are able to assert that all the experimental observations of such a frequency splitting reported in the literature are just one of two possibilities that can be performed by a WPTS.",
"If the transmitter and receiver coils are not identical but the parameters $(L_1, \\, R_1)$ and $(L_2, \\, R_2)$ are close, the same procedure can be applied to approximate the regions of $C_1$ and/or $C_2$ in which the delivered power has only one stationary point.",
"The general problem with arbitrary values of $(R_{1}, \\, R_{2}, \\, L_{1}, \\, L_{2}, \\, C_{1}, \\, C_{2})$ is much more complicated and its analytical solution is open for future study.",
"However, a numerical solution is possible and is demonstrated through the following example.",
"Without loss of generality, we assume that $C_2$ is a dependent variable and is given by $C_2 = L_1 C_1/L_2$ to match the resonance frequencies of the two resonators.",
"The numerical problem is formulated as $\\small C_\\mathrm {m} = \\min \\lbrace C_1\\rbrace $ such that there exists a unique positive solution among the four $\\lbrace \\Omega _1,\\, \\Omega _2,\\, \\Omega _3, \\, \\Omega _4\\rbrace $ .",
"Instead of using the same transmitter parameters, we now consider the case where $L_1 \\!=\\!",
"18.26$ $\\mu $ H and $R_1 \\!=\\!",
"4.50$ $\\Omega $ , which are double of those used in the previous example.",
"The electrical characteristics of the receiver coil are taken from [7], {$L_2 = 8.92$ $\\mu $ H, $R_2 = 2.5$ $\\Omega $ }.",
"Solving (REF ) with $k = 0.9$ and $R_\\mathrm {L} = 10$ $\\Omega $ yields $C_\\mathrm {m} \\approx 8.01$ nF, while $C_\\mathrm {m} \\approx 0.55$ nF with $R_\\mathrm {L} = 100$ $\\Omega $ .",
"The frequency splitting is present with any $C_1 < C_\\mathrm {m}$ and is suppressed for all $C_1 \\ge C_\\mathrm {m}$ .",
"As a general principle, lowering both or either of the two $Q$ –factors $(Q_\\mathrm {s}, \\, Q_\\mathrm {L})$ to a certain level could assure the suppression of the frequency splitting property [19].",
"Other system parameters can then be designed accordingly based on the required values of $Q_\\mathrm {s}$ and $Q_\\mathrm {L}$ .",
"However, the maximum possible power delivered to the load and the optimum inductive-link efficiency heavily depend on the coil quality factors.",
"Therefore, $Q_\\mathrm {s}$ and $Q_\\mathrm {L}$ should not be chosen too low.",
"Furthermore, even in circumstances where there is not much room for tuning $C_{1/2}$ , for instance, ($L_{1/2}, \\, R_{1/2}$ ) are given and the operating frequency is predetermined, $Q_\\mathrm {s}$ ($Q_\\mathrm {L}$ ) can still be adjusted by adding an external resistor in series with the transmitter (receiver).",
"It is important to note that, if a pair of $(C_1, \\, C_2)$ satisfies conditions such that the frequency splitting phenomenon is eliminated for a given load resistance $R_\\mathrm {L}$ , this conclusion also holds for all $R_\\mathrm {X} \\ge R_\\mathrm {L}$ .",
"It is a consequence of the general trend of reducing $Q$ –factors since a higher resistance leads to a lower quality factor.",
"Therefore, such behavior can be avoided across a wide range of $R_\\mathrm {L}$ .",
"In a dynamic operating condition where the electrical load varies from $R_\\mathrm {m}>0$ to $R_\\mathrm {n} > R_\\mathrm {m}$ , i.e., $R_\\mathrm {L} \\in [R_\\mathrm {m}, \\, R_\\mathrm {n}]$ , choosing suitable values of $(C_1, \\, C_2)$ only for $R_\\mathrm {m}$ is simultaneously sufficient for the whole range.",
"While the proposed method (i.e., increasing capacitance to reduce the resonance frequency and quality factor) can be utilized for various applications, it is perhaps the most suitable for biomedical wearable and implantable devices.",
"In these systems, a lower operating frequency is desired as it allows higher permissible external magnetic flux density that can be applied to the human body, according to IEEE safety standards and regulations [26]."
],
[
"Relations between Resonance ($\\omega _\\mathrm {r}$ ) and Extreme ({{formula:5413c6a9-0d81-4fc1-a49e-6ad210094f1a}} ) Frequencies",
"The aim of this section is to address the question on the effects of the resonance frequencies $^\\mathrm {i}\\omega _\\mathrm {r}$ on the dynamics of the output power.",
"As we have analyzed in [18] and recapped in [19], the extreme points (i.e., maxima and minimum) of $P_\\mathrm {L}$ are achieved at frequencies $^\\mathrm {i}\\omega _\\mathrm {e}$ that are close to $^\\mathrm {i}\\omega _\\mathrm {r}$ .",
"This can lead to a misunderstanding on that the resonance frequencies always cause or relate to the frequency splitting phenomenon.",
"However, we find that this might not be necessarily hold true.",
"One simple counterexample can be found in [19] and a more in-depth analysis is to be presented.",
"Taking the case when the two coils are identical as an example, if the following conditions are simultaneously fulfilled, equation $\\Im \\lbrace Z_\\mathrm {in} \\rbrace = 0$ has three positive solutions but the frequency splitting behavior does not occur: (a) $C \\in \\mathbb {R}_{\\mu +/\\mu -}$ , (b) The loaded quality factor of the receiver coil at the primary resonance frequency $^{0}\\omega _\\mathrm {r} = \\omega _0$ satisfies $Q_0 = \\omega _0 L /(R + R_\\mathrm {L}) > 1/\\sqrt{2}$ , or equivalently $C \\in {\\mathbb {R}}_\\mathrm {q} = (0, \\, C_\\mathrm {q})$ where $C_\\mathrm {q} = 2L/(R+R_\\mathrm {L})^2$ , and (c) $k^2 > k_\\mathrm {r}^2 = \\Theta (1-\\Theta /4)$ where $\\Theta = 1/Q_0^2$ .",
"The analytical expressions of all the resonance frequencies $^\\mathrm {i}\\omega _\\mathrm {r}$ were presented in Table 1 of [18].",
"In summary, if $C \\in \\lbrace \\mathbb {R}_{\\mu +/\\mu -} \\cap {\\mathbb {R}}_\\mathrm {q} \\ne \\varnothing \\rbrace $ and $k \\in (k_\\mathrm {r}, 1)$ , the system has a unique global maximum output power while exhibiting three resonance frequencies.",
"Considering the particular example shown in Figure REF of Section , we have {$L = 9.13$ $\\mu $ H, $R = 2.25$ $\\Omega $ , $R_\\mathrm {s} = 50$ $\\Omega $ , $k=0.9$ }.",
"With $R_\\mathrm {L} = 10 \\, \\Omega $ , due to the inequality $\\xi _1 < C_\\mathrm {q}$ , any $C \\in (\\xi _1, \\, C_\\mathrm {q})$ (for instance, $C = 9$ nF) leads to the behavior that is addressed above.",
"In contrast, if $R_\\mathrm {L} = 90 \\, \\Omega $ , we get $\\xi _1 > C_\\mathrm {q}$ and $\\mathbb {R}_{\\mu +/\\mu -} \\cap {\\mathbb {R}}_\\mathrm {q} = \\varnothing $ , the system has only one resonance frequency $\\omega _0 = 1/\\sqrt{LC}$ and $P_\\mathrm {L}$ attains a unique peak for all $C > \\xi _1$ (such as, $C = 5$ nF).",
"This same dynamics is observed if $R_\\mathrm {L} = R_\\mathrm {s} = R^\\ast $ since the two conditions (a) and (b) become $C > 2L/(R+ R^\\ast )^2$ and $C < 2L/(R+ R^\\ast )^2$ respectively, which cannot happen at the same time.",
"SPICE simulations are performed in order to substantiate these arguments.",
"More interestingly, there also exist circumstances in which equation $\\Im \\lbrace Z_\\mathrm {in} \\rbrace = 0$ exhibits a unique zero-phase angle frequency, however, the frequency splitting phenomenon still takes place.",
"An example is shown in Figure REF with an arbitrary system where the two coupled resonators have different resonance frequencies.",
"The parameter set is as follows: {$L_1 = 18.26$ $\\mu $ H, $R_1 = 4.50$ $\\Omega $ , $C_1 = 0.444$ nF, $L_2 = 8.92$ $\\mu $ H, $R_2 = 2.50$ $\\Omega $ , $C_2 = 1.362$ nF, $R_\\mathrm {s} = 50$ $\\Omega $ , $R_\\mathrm {L} = 30$ $\\Omega $ }.",
"These results prove that, although analyzing the input impedance and the resonance frequencies might be useful to roughly anticipate the frequency splitting behavior in some circumstances, it is not an appropriate approach to correctly comprehend this phenomenon or adequately describe the dynamics of the power transfer to the load.",
"And that, the multiple-resonance and frequency splitting phenomena should be considered independent of each other.",
"This clarification could provide a different perspective on the analysis established in the literature, and prevent any possible further misunderstanding.",
"Dynamic simulations with SPICE are executed to ensure the accuracy of the derivation and verify the results presented in Figures REF and REF .",
"It is worthwhile to note that by operating the series-series WPTS at either secondary resonance frequencies $^{1}\\omega _\\mathrm {r}$ or $^{2}\\omega _\\mathrm {r} \\ne \\, ^{0}\\omega _\\mathrm {r}$ , a constant output power is obtained regardless of coupling coefficient changes (e.g., due to distance variations or coil misalignment), which is then referred to as coupling-independent transmission regime [27].",
"In general, this constant transferred power is far below the maximum achievable value [27], and therefore utilizing $^{1,2}\\omega _\\mathrm {r}$ may be not suitable for the purpose of power transfer optimization.",
"Figure: Frequency responses of the system input impedance and the power delivered to the load with k=0.9k \\!=\\!",
"0.9 and f 1 =(2πL 1 C 1 ) -1 f_1 \\!=\\!",
"(2\\pi \\sqrt{L_1 C_1})^{-1}.",
"Solid circle: maximum/minimum power at extreme frequencies."
],
[
"Power optimization and physical bound",
"Although the two-coil WPTS has been widely investigated in the literature, most of the previous work considered the transmission efficiency as the key factor in evaluating the system performance.",
"In this paper, we choose to approach the problem from a different perspective and consider the actual power transferred to the load as the primary objective of our study.",
"Various aspects of the power optimization are to be investigated."
],
[
"Analytical general-solution to the optimal load",
"Determining the optimal load is a powerful method that has been widely used for maximizing the output power of vibration energy harvesters.",
"Since the energy-harvesting concept was proposed as an alternative solution in order to power wireless sensing systems [28], [29], there has been significant effort focusing on two fundamental issues: how much power can be obtained and how to approach it.",
"A method to achieve performance close to the maximum possible output power (that includes displacement-constrained operation) is to adapt the electrical load only [30], [31].",
"The two concerns raised above are also solid in the context of WPTS.",
"A closed-form analytical solution for the optimum load that achieves the maximum possible power efficiency under arbitrary input impedance conditions was presented in [32].",
"However, it was also shown that a different arrangement is necessary for maximizing the output power of a four-coil WPTS [33].",
"Motivated by these observations, we aim to investigate the optimum power with respect to the load using the conventional gradient descent method.",
"The stationary points of $P_\\mathrm {L}$ are evaluated by setting the derivative $P_\\mathrm {L}/R_\\mathrm {L}$ equal to zero, resulting in $\\small ^\\mathrm {opt}\\tau \\!=\\!",
"\\frac{L_2}{^\\mathrm {opt}R_\\mathrm {L}} \\!=\\!",
"\\bigg [ \\frac{m_6 \\omega ^6 + m_4 \\omega ^4 + m_2 \\omega ^2}{n_8 \\omega ^8 \\!",
"+ \\!",
"n_6 \\omega ^6 \\!",
"+ \\!",
"n_4 \\omega ^4 \\!",
"+ \\!",
"n_2 \\omega ^2 \\!",
"+ \\!",
"n_0} \\displaystyle \\bigg ]^{1/2} $ where $\\tau _1 = L_1/R_1, \\,\\tau _2 = L_2/R_2$ , $\\small m_2 &= \\omega _1^4, \\,m_4 = \\tau _\\mathrm {s}^{-1} - 2 \\omega _1^2 , \\,m_6 = 1, \\\\n_0 &= \\big (\\omega _1 \\omega _2 \\big )^4 , \\\\n_2 &= \\big (\\omega _2^2/\\tau _\\mathrm {s}\\big )^2 + \\big (\\omega _1^2/\\tau _2\\big )^2 - 2 \\big (\\omega _1 \\omega _2 \\big )^2 \\big (\\omega _1^2 + \\omega _2^2 \\big ) , \\\\\\begin{split}n_4 &= \\tau _\\mathrm {s}^{-2} \\big ( \\tau _2^{-2} - 2 \\omega _2^2 \\big ) + \\big (\\omega _1^2 + \\omega _2^2 \\big )^2 \\\\&- 2 \\omega _1^2 \\big [ \\tau _2^{-2} + (k^2 - 1) \\omega _2^2 \\big ],\\end{split} \\\\\\begin{split}n_6 &= \\big (\\tau _2^{-1} + \\tau _\\mathrm {s}^{-1} \\big )^2 \\\\&+ 2 (k^2 - 1) \\big [ (\\tau _2 \\tau _\\mathrm {s})^{-1} + \\big (\\omega _1^2 + \\omega _2^2 \\big ) \\big ] ,\\end{split} \\\\n_8 &= (k^2 - 1)^2.$ Formula (REF ) is the general closed form expression of the optimal load $^\\mathrm {opt} R_\\mathrm {L}$ derived as a function of the other system parameters.",
"Figure: Optimal load, characterized by the electrical time constant opt τ^\\mathrm {opt}\\tau , and optimum output power P L (k,ω, opt τ(k,ω))P_\\mathrm {L} (k,\\, \\omega , \\, ^\\mathrm {opt}\\tau (k,\\, \\omega )).Figure REF shows the changes of $^\\mathrm {opt} R_\\mathrm {L}$ with respect to $(k,\\, \\omega )$ characterized by the optimal electrical time scale $^\\mathrm {opt}\\tau = L_2/^\\mathrm {opt} R_\\mathrm {L}$ and the corresponding output power as the function of $(k,\\, \\omega , \\, ^\\mathrm {opt} R_\\mathrm {L})$ .",
"Other parameters are taken from the example presented in Figure REF .",
"In a general trend, increasing the coupling strength results in decrease of $n_8$ and, therefore, lower (higher) values of $^\\mathrm {opt}R_\\mathrm {L}$ ($^\\mathrm {opt}\\tau $ ).",
"Meanwhile, changes of $^\\mathrm {opt} R_\\mathrm {L}$ with $\\omega $ are much more complicated.",
"Note that, the dynamics of the perfect coupled system where $k = 1$ that makes $n_8 = 0$ is a different scenario and is considered in the next Section.",
"Optimizing the load produces multiple local optimum solutions of the output power with three distinguished branches as shown in Figure REF .",
"The optimal load at or nearby the resonance frequency $(f/f_0 \\approx 1)$ is perhaps the most convenient option in practice.",
"This observation is discussed in detail in the next Section.",
"Formula (REF ) can also provide a means for designing an optimized WPTS.",
"Given an equivalent load resistance and a transmission distance, (REF ) allows us to estimate appropriate(s) range or value(s) of key parameters such as the transmitter/receiver coil inductance and the selection of a suitable operating frequency.",
"See [34] as an example of rigorous design for a WPTS where the input impedance, the electrical load and the resonance frequency are specified."
],
[
"Limitations of transferred power",
"In order to evaluate the maximum achievable power for a given system, we first consider the conditions for optimizing the power delivered to a load from the two resonators, based on the general theorem presented in [35].",
"Here, we consider that the source and load resistances ($R_\\mathrm {s}$ and $R_\\mathrm {L}$ respectively) are objective variables.",
"A similar technique was used for optimizing the power or OPG of multiple receiver systems in [36], [37].",
"The model shown in Figure REF is equivalent to a simple AC circuit in Figure REF a, where $V_\\mathrm {out}$ is the Th$\\acute{e}$ venin equivalent voltage and the output impedance $Z_\\mathrm {out}$ is given by ().",
"The conditions for maximum power are summarized as follows $\\small \\begin{split}\\Im \\lbrace Z_\\mathrm {out}\\rbrace &= \\omega L_2 - 1/(\\omega C_2) \\\\&- \\kappa (\\omega M)^2 \\big ( \\omega L_1 - 1/(\\omega C_1) \\big ) = 0,\\end{split} \\\\\\Re \\lbrace Z_\\mathrm {out}\\rbrace &= R_2 + \\kappa (\\omega M)^2 (R_1 + R_\\mathrm {s}) = R_\\mathrm {L} $ where $\\kappa $ is defined by () and $\\kappa > 0$ .",
"One of practical solutions is to adjust the added capacitor $C_2$ on the receiving side and keep other parameters fixed.",
"$C_2$ is then computed by $\\small C_2 = \\frac{1}{\\omega } \\Big [\\omega L_2 - \\kappa (\\omega M)^2 \\Big ( \\omega L_1 - \\frac{1}{\\omega C_1} \\Big ) \\Big ]^{-1}.",
"$ Substitute the optimal load and secondary added capacitance in (REF ) and (REF ) back into (REF ), the transferred power is derived as $\\small P_\\mathrm {L} = \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{(R_1 + R_\\mathrm {s}) + R_2/\\big (\\kappa (\\omega M)^2\\big )}.",
"$ Note that $1/\\kappa \\ge (R_1 + R_\\mathrm {s})^2$ , therefore $\\small \\begin{split}P_\\mathrm {L} & \\le \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{(R_1 + R_\\mathrm {s}) \\Big [ 1 + R_2 (R_1 + R_\\mathrm {s})/(\\omega M)^2 \\Big ]} \\\\& < P_\\ast = \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{R_1 + R_\\mathrm {s}} < P_\\mathrm {avs} = \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{R_\\mathrm {s}}.",
"\\end{split}$ The equality on the left side of (REF ) can be attained if and only if $\\omega = \\omega _1 = 1/\\sqrt{L_1 C_1}$ .",
"The power available from the source $P_\\mathrm {avs}$ is a strict upper bound, however, it does not describe the system performance comprehensively.",
"We further maximize the output power attained in (REF ) with respect to the driving frequency.",
"The stationary point is determined by setting $P_\\mathrm {L}/\\omega = 0$ , which yields $\\small ^\\mathrm {opt}\\omega = \\frac{1}{\\sqrt{L_1 C_1 - \\displaystyle \\frac{1}{2} \\big [C_1(R_1 + R_\\mathrm {s}) \\big ]^2}} = \\frac{\\omega _1}{\\sqrt{1 - \\displaystyle \\frac{1}{2 Q_\\mathrm {s}^2}}}$ where $Q_\\mathrm {s}$ is defined by (REF ).",
"Since the desired self-resonance frequency of the transmitter coil $f_1 = \\omega _1/2\\pi $ is in the range of MHz, the quality factor $Q_\\mathrm {s}$ is significantly larger than unity and the optimal frequency $^\\mathrm {opt}\\omega $ can be approximated by $\\omega _1$ .",
"Hence, $P_\\mathrm {L} \\big |_{\\omega _1}$ is asymptotic to $P_\\mathrm {L} \\big |_{^\\mathrm {opt}\\omega }$ and is utilized in following derivations as an alternative for the sake of simplicity.",
"When $\\omega = \\omega _1$ , from (REF ) and (REF ), we get $\\small R_\\mathrm {L} &= R_2 + (\\omega _0 M)^2 \\big {/} (R_1 + R_\\mathrm {s}), \\\\\\omega _1 &= 1 \\big {/} \\sqrt{L_2 C_2} = \\omega _2.",
"$ Denoting $\\omega _1 = \\omega _2$ by $\\omega _0$ , the output power $P_\\mathrm {L} \\big |_{\\omega _0}$ reads as $\\small \\begin{split}P_0 &= \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{(R_1 + R_\\mathrm {s}) \\big [ 1 + R_2 (R_1 + R_\\mathrm {s}) \\big {/}(\\omega _0 M)^2 \\big ]} \\\\&= \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{(R_1 + R_\\mathrm {s}) \\big [ 1 + 1/(k^2 Q_2 Q_\\mathrm {s}) \\big ]} \\\\&= \\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{R_\\mathrm {s}} \\Big (1 - \\frac{Q_\\mathrm {s}}{Q_1} \\Big ) \\frac{k^2 Q_2 Q_\\mathrm {s}}{1 + k^2 Q_2 Q_\\mathrm {s}} \\end{split}$ where the unloaded quality factors of the transmitter and receiver coils at the resonance frequency $\\omega _0$ are defined by $\\small Q_1 = \\omega _0 L_1/R_1, \\,Q_2 = \\omega _0 L_2/R_2$ and $Q_1 > Q_\\mathrm {s} = \\omega _0 L_1/(R_1 + R_\\mathrm {s})$ .",
"For given coil parameters and a given distance between the two coils (i.e., $L_1, \\, R_1, \\, L_2, \\, R_2$ and $k$ are known), the first expression in (REF ) shows that higher resonance frequency $\\omega _0$ results in higher maximum output power.",
"In practice, this can be executed by reducing the added capacitances $C_1$ and $C_2$ .",
"In the case when only the load resistance is adapted as in (REF ), the relative relationship of $Q_\\mathrm {s}$ and $Q_1$ plays a critical role.",
"Once the ratio $Q_\\mathrm {s}/Q_1$ approaches unity, $P_0$ is dramatically reduced.",
"A smaller source resistance is preferable in general.",
"However, for this particular circumstance, if $R_\\mathrm {s} \\ll R_1$ , the ratio $P_0/P_\\mathrm {avs}$ is negligibly small, meaning that a tiny amount of power from the source is delivered to the load.",
"For a typical WPTS described in previous examples, the inductances $(L_1, \\, L_2)$ , the capacitances $(C_1, \\, C_2)$ and the resistances $(R_1, \\, R_2)$ are in the range of $\\mu $ H, nF/pF and (a few) $\\Omega $ respectively, therefore $Q_2 Q_\\mathrm {s} \\gg 1$ .",
"In the strong coupling regime, $P_0 \\xrightarrow{} P_\\ast $ , and if $Q_1$ is much larger than $Q_\\mathrm {s}$ , then $P_0 \\rightarrow P_\\mathrm {avs}$ .",
"Figure: The ´\\acute{e}venin equivalent circuits.While some other authors only accounted for the roles of the input power and OPG [36], [37], we also further investigate the conditions to maximize the power input to the network from a given source.",
"Similarly, the model in Figure REF is equivalent to the circuit depicted in Figure REF b, where the input impedance $Z_\\mathrm {in}$ is defined as (REF ).",
"Due to (), the imaginary part of the input impedance is eliminated, $\\Im \\lbrace Z_\\mathrm {in}\\rbrace = 0$ .",
"The other condition for maximum delivered power is $\\Re \\lbrace Z_\\mathrm {in}\\rbrace = R_\\mathrm {s}$ , which leads to $\\small R_\\mathrm {s} &= R_1 + (\\omega _0 M)^2 \\big {/} (R_2 + R_\\mathrm {L}).",
"$ In order to obtain the maximum transferable power from the source, two conditions (REF ) and (REF ) need to be satisfied simultaneously.",
"Considering other parameters as dependent variables, the solutions of $R_\\mathrm {s}$ and $R_\\mathrm {L}$ are expressed as $\\small R_\\mathrm {s} &= R_1 \\sqrt{1 + (\\omega _0 M)^2 \\big {/} (R_1 R_2)}, \\\\R_\\mathrm {L} &= R_2 \\sqrt{1 + (\\omega _0 M)^2 \\big {/} (R_1 R_2)} = R_\\mathrm {s} R_2 / R_1 .",
"$ Substituting (), (REF ) and () into (REF ), we obtain $\\small \\begin{split}P_\\mathrm {A} & \\!=\\!",
"\\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{R_\\mathrm {s}} \\frac{\\displaystyle \\sqrt{1 \\!+\\!",
"\\frac{(\\omega _0 M)^2}{R_1 R_2}} \\!-\\!",
"1}{\\displaystyle \\sqrt{1 \\!+\\!",
"\\frac{(\\omega _0 M)^2}{R_1 R_2}} \\!+\\!",
"1} \\!=\\!",
"\\frac{1}{8} \\frac{{V_\\mathrm {s}}^2}{R_\\mathrm {s}} \\frac{\\displaystyle \\sqrt{1 \\!+\\!",
"k^2 Q_1 Q_2} \\!-\\!",
"1}{\\displaystyle \\sqrt{1 \\!+\\!",
"k^2 Q_1 Q_2} \\!+\\!",
"1} \\\\&= P_\\mathrm {avs} \\frac{k^2 Q_1 Q_2}{\\displaystyle \\big ( \\sqrt{1 + k^2 Q_1 Q_2} + 1 \\big )^2}.",
"\\end{split}$ Here, $P_\\mathrm {A}$ can be (asymptotically) considered as a tight upper bound of the power that can be delivered to the load with a given power source (characterized by $P_\\mathrm {avs}$ ).",
"The result also quantitatively reveals the crucial role of $Q_1$ and $Q_2$ , in which both of them in addition to the coupling strength frankly determine the performance of a WPTS.",
"The asymptotic power bound approaches the power available from the source once $k^2 Q_1 Q_2 \\gg 1$ .",
"Figure: Comparison of: (i) The maximum output power presented in Figure at each coupling strength, (ii) The asymptotic power bound given by () and (iii) Numerical solution of the upper power bound ().For given resonators where $(R_{1}, \\, R_{2}, \\, L_{1}, \\, L_{2}, \\, C_{1}, \\, C_{2})$ are know, the rigorous upper bound of the transferable power is determined by $ \\max _{R_\\mathrm {L}, \\,R_\\mathrm {s}, \\, \\omega > 0} P_\\mathrm {B}$ where $P_\\mathrm {B} = P_\\mathrm {L}/P_\\mathrm {avs}$ .",
"We choose to numerically solve this problem due to its high complexity and nonlinearity with inequality constraints of the three variables, using the nonlinear Interior Point (or Sequential Quadratic) Programming methods [38].",
"Figure REF presents a comparison of the rigorous upper bound of the output power and the asymptotic closed-form solution.",
"Since they are nearly identical, (REF ) provides a very convenient and compact method to evaluate the performance of a WPTS.",
"In the case where we are able to arbitrarily adjust the source and load resistances, the maximum transferable power (the physical-bound performance) is attained if three conditions (), (REF ) and () are fulfilled.",
"In addition, the optimum power at each coupling coefficient, $^\\mathrm {opt}P_{k}$ , extracted from Figure REF is included.",
"$^\\mathrm {opt}P_{k}$ is verified by SPICE simulations, in which both $\\omega $ and $R_\\mathrm {L}$ are varied to determine their optimal values at each $k$ .",
"In the high coupling regime $(k > 0.2)$ , $^\\mathrm {opt}P_\\mathrm {k} \\approx P_\\mathrm {A} \\approx P_\\mathrm {B}$ , however there is a significant difference between $^\\mathrm {opt}P_\\mathrm {k}$ and $\\lbrace P_\\mathrm {A}, \\, P_\\mathrm {B}\\rbrace $ in the lower regions of $k$ .",
"Therefore, only adapting the electrical load at the resonance frequency is sufficient to approach the power bound at strong coupling strength.",
"This can be realized by using asynchronously switched electronic interfaces such as buck-boost converters since they can appear as an effective load resistance determined by the duty cycle of the switching circuit [39], [40].",
"Otherwise, adjusting the source impedance is required in a loosely coupled system.",
"Figure: Bi-conjugate impedance matching at source and load with two Π-\\Pi -networks."
],
[
"Power transfer efficiency: An additional discussion",
"Dionigi and Costanzo et.",
"al.",
"[33], [41] reported that maximizing power transfer efficiency (i.e., or operating power gain, defined by the ratio between the generated power and the power input to the network) and maximizing power delivered to the load lead to two different solutions.",
"This raises a question on the efficiency of the investigated WPTS under the optimum-power operation presented in Section REF .",
"Two widely used definitions for the power gain of the two-port network are the transducer power gain (TPG), $\\eta _\\mathrm {t} =$ power delivered to the load/power available from the source, and the operating power gain, $\\eta _\\mathrm {p} =$ power delivered to the load/power input to the network.",
"$\\eta _\\mathrm {p}$ is the most common coefficient used in context of wireless power transfer as the transmission (or inductive link) efficiency.",
"At the resonance frequency $\\omega = \\omega _0$ , $\\small \\eta _\\mathrm {t} &= \\frac{P_\\mathrm {L}}{P_\\mathrm {avs}} \\bigg |_{\\omega _0} = \\frac{4 R_\\mathrm {s} R_\\mathrm {L} (\\omega _0 M)^2}{\\big [(\\omega _0 M)^2 + (R_1 + R_\\mathrm {s})(R_\\mathrm {L} + R_2) \\big ]^2}, \\\\\\eta _\\mathrm {p} & \\!=\\!",
"\\frac{P_\\mathrm {L}}{P_\\mathrm {in}} \\bigg |_{\\omega _0} = \\frac{R_\\mathrm {L} (\\omega _0 M)^2}{\\big [(\\omega _0 M)^2 + R_1 (R_\\mathrm {L} + R_2) \\big ](R_\\mathrm {L} + R_2)}.",
"$ Substituting (REF ) and () into (REF ) and (), we get $\\small \\eta _\\mathrm {t} = \\eta _\\mathrm {p} = \\frac{k^2 Q_1 Q_2}{\\displaystyle \\big ( \\sqrt{1 + k^2 Q_1 Q_2} + 1 \\big )^2}.",
"$ We have revealed that, as the output power reaches its physical bound, $P_\\mathrm {in} = P_\\mathrm {avs}$ and the two power gains collapse to a unique solution simultaneously.",
"This value is also the maximum possible transmission efficiency of the near-field inductive link [32].",
"Therefore, the global solution of the power optimization problem and that of the efficiency maximization are essentially identical.",
"This interpretation has been extensively studied and validated in the microwave technology society.",
"However, it is still debating in the field of WPT.",
"For example, see [33], [41], in which the authors concluded that optimizing $\\eta _\\mathrm {t}$ and $\\eta _\\mathrm {p}$ do not give the same result.",
"This statement is the consequence of the fact that the authors only considered optimizing the load resistance or conjugate matching at the receiver, which results in local optimum point(s).",
"By explicitly deriving equation (REF ), we aim to bury the difference between the two optimization schemes and confirm that, as long as we seek for the global maximum, the obtained solutions are equivalent.",
"Some authors prefer to quantify efficiency instead through the overall/system efficiency, $\\small \\eta _\\mathrm {s} = P_\\mathrm {L} \\big {/} (P_\\mathrm {L} + P_\\mathrm {loss})$ where $P_\\mathrm {loss}$ is the total power dissipated in $R_\\mathrm {s}, \\, R_1$ and $R_2$ .",
"Under the optimum-power operation condition, $P_\\mathrm {loss} = P_\\mathrm {avs} + (P_\\mathrm {in} - P_\\mathrm {L}) = 2 P_\\mathrm {avs} - P_\\mathrm {L}$ , therefore $\\eta _\\mathrm {s}$ becomes $\\small \\eta _\\mathrm {s} = \\frac{1}{2} \\frac{P_\\mathrm {L}}{P_\\mathrm {avs}} = \\frac{1}{2} \\frac{k^2 Q_1 Q_2}{\\displaystyle \\big ( \\sqrt{1 + k^2 Q_1 Q_2} + 1 \\big )^2} < \\frac{1}{2}.$ This result collides with the general theorem discussed in [35], in which the overall efficiency can never go beyond 50 % as the transferred power is fully optimized."
],
[
"Possible method to obtain maximum output power: bi-conjugate impedance matching circuits",
"In the circumstance where the source, input and load impedances are specified, Monti et.",
"al.",
"presented closed-form formulas for the optimum design of a near-field magnetically coupled WPTS, with the main aim is to maximize the transferring efficiency [34].",
"The coil inductance $L_\\mathrm {i}$ , parasitic resistance $R_\\mathrm {i}$ and the lumped capacitance $C_\\mathrm {i}$ are then determined as functions of desired coil unloaded quality factor $Q_\\mathrm {i}$ ($i = \\lbrace 1, \\, 2\\rbrace $ ), given resonance frequency $\\omega _0$ and coupling coefficient $k$ .",
"This method is probably not appropriate in practice since $L_\\mathrm {i}$ , $R_\\mathrm {i}$ and $k$ are not independent of each other as considered in [34].",
"In realistic application scenarios, it is common that the source impedance and the electronic load are predetermined.",
"This section aims to realize the general power optimization technique presented in Section REF under such a situation.",
"Two additional $\\Pi -$ networks are used as impedance matching circuits at both transmitter and receiver sides, as shown in Figure REF .",
"The power transferred to the load is derived as follows $\\small P_\\mathrm {L} = \\frac{1}{2} {V_\\mathrm {th}}^2 \\frac{{Z_{21}}^2 \\Re \\lbrace Z_\\mathrm {L}\\rbrace }{{(Z_{11} + Z_\\mathrm {s}) (Z_\\mathrm {out} + Z_\\mathrm {L})}^2} $ where $\\small V_\\mathrm {th} &= V_\\mathrm {s} \\frac{\\displaystyle Z_\\mathrm {t} \\Big [ j \\omega C_5 \\Big (j \\omega L_4 + \\big (j \\omega C_5 \\big )^{-1} \\Big ) \\Big ]^{-1}}{\\displaystyle R_\\mathrm {s} + Z_\\mathrm {t}}, \\\\Z_\\mathrm {t} &= \\Big [ j \\omega C_6 + \\Big (j \\omega L_4 + \\big (j \\omega C_5\\big )^{-1} \\Big )^{-1} \\Big ]^{-1}, \\\\Z_\\mathrm {s} &= \\Big [ j \\omega C_5 + \\Big ( j \\omega L_4 + \\big ( j \\omega C_6 + R_\\mathrm {s}^{-1} \\big )^{-1} \\Big )^{-1} \\Big ]^{-1}, \\\\Z_\\mathrm {L} &= \\Big [ j \\omega C_3 + \\Big ( j \\omega L_3 + \\big ( j \\omega C_4 + R_\\mathrm {L}^{-1} \\big )^{-1} \\Big )^{-1} \\Big ]^{-1}, \\\\\\begin{split}Z_\\mathrm {out} &= R_2 + j \\big (\\omega L_2 - 1/(\\omega C_2) \\big ) \\\\& + (\\omega M)^2 \\Big [ R_1 + \\displaystyle j \\big (\\omega L_1 - 1/(\\omega C_1) \\big ) + Z_\\mathrm {s} \\Big ]^{-1}.",
"\\end{split}$ $Z_{11}$ and $Z_{12}$ are calculated as formulas (REF ) and ().",
"Note that (REF ) has the same explicit form and collapses to (REF ) when replacing $Z_\\mathrm {s}, \\, Z_\\mathrm {L}$ and $V_\\mathrm {th}$ by $R_\\mathrm {s}, \\, R_\\mathrm {L}$ and $V_\\mathrm {s}$ respectively.",
"The bi-conjugate matching conditions can be equivalently formulated as a power optimization problem as $ \\small \\max _{C_3, \\, L_3, \\, L_4, \\, C_5 > 0} P_\\mathrm {L}$ where two capacitances $C_4$ and $C_6$ are arbitrarily set equal to 10 nF.",
"The driving frequency is $f = f_0$ and the source resistance is $R_\\mathrm {s} = 50 \\, \\Omega $ .",
"Figure REF a shows that the output power obtained by using bi-conjugate impedance matching with two $\\Pi -$ networks is identical to that resulted from the asymptotic solution, regardless of the load resistance.",
"In other words, formula (REF ) is able to exactly predict the optimal performance of the series-series WPTS when the two-side impedance matching technique is utilized.",
"The statements presented in Section REF still holds true under this circumstance.",
"In addition, a few numerical solutions of $C_3, \\, L_3, \\, L_4$ and $C_5$ are imported to SPICE simulator and the corresponding power outputs are compared to verify the reliability of the optimization formulated in (REF ).",
"In Figure REF b, we also reveal that the optimum performance of other typical configurations, such as series-parallel (SP), parallel-parallel (PP) and parallel-series (PS) resonators [42], is close (nearly indistinguishable) to that of series-series structure.",
"Therefore, (REF ) can be used as an estimated physical power bound for all those three types as well.",
"The power bound of each structure is obtained from solving (REF ) numerically.",
"Here, Series and Parallel denote the way the coil and added capacitor connect to each other.",
"The question on how to design a particular system (e.g., [43], [44]) depends on specific applications and is out of scope of the paper.",
"Although the analysis is applied to a load resistance, all findings in this study are valid for a general complex load (i.e., $Z_\\mathrm {L} = R_\\mathrm {L} + j X_\\mathrm {L}$ )."
],
[
"Discussion on dynamics of a perfectly coupled system",
"We can re-write $P_\\mathrm {L}$ in (REF ) as $\\small P_\\mathrm {L} = a_6 \\omega ^6 \\big {/} (b_8 \\omega ^8 + b_6 \\omega ^6 + b_4 \\omega ^4 + b_2 \\omega ^2 + b_0) $ where $a_6 \\!=\\!",
"(C_1 C_2)^2 k^2 L_1 L_2 {V_\\mathrm {s}}^2 R_\\mathrm {L} \\!>\\!",
"0$ , $b_8 \\!=\\!",
"2 [C_1 L_1 C_2 L_2 (1 \\!-\\!",
"k^2)]^2 \\!>\\!",
"0$ , $b_6 \\!=\\!",
"2 (C_1 C_2)^2 \\big [ L_1(R_2 \\!+\\!",
"R_\\mathrm {L}) \\!+\\!",
"L_2 (R_1 \\!+\\!",
"R_\\mathrm {s})\\big ]^2 \\!>\\!",
"0$ and $b_0 \\!=\\!",
"2$ (other constants can also be extracted from (REF ), but are not necessary).",
"Consider an extreme case when the coupling factor between the two resonators is unity, $k = 1$ , the coefficient $b_8$ in (REF ) disappears, $b_8 = 0$ .",
"This leads to a special property of the system, where $\\small \\displaystyle ^{\\infty }P_\\mathrm {L} \\!=\\!",
"\\lim _{\\omega \\rightarrow +\\infty } \\!",
"P_\\mathrm {L} \\!=\\!",
"\\frac{a_6}{b_6}\\!=\\!",
"\\frac{1}{2} \\frac{{V_\\mathrm {s}}^2}{R_\\mathrm {s}} \\frac{\\big ( \\tau _\\mathrm {s}^{-1} \\!-\\!",
"\\tau _1^{-1} \\big ) \\big ( \\tau _\\mathrm {L}^{-1} \\!-\\!",
"\\tau _2^{-1} \\big )}{\\big ( \\tau _\\mathrm {s}^{-1} \\!+\\!",
"\\tau _\\mathrm {L}^{-1} \\big )^2} > 0 $ and is independent of any added capacitances.",
"If the transmitter and receiver are identical, $^{\\infty }P_\\mathrm {L}$ no longer depends on coil inductances, as follows $\\small ^{\\infty }P_\\mathrm {L} = {V_\\mathrm {s}}^2 R_\\mathrm {L} \\big {/} [2 (R_\\mathrm {s} + R_\\mathrm {L} + 2 R)^2].",
"$ It should be noted that the coupling coefficient is unity if and only if all the lines of the magnetic flux density of the transmitter coil cuts all of the turns of the receiver coil [20].",
"However, this condition rarely occurs in a finite-dimensional WPTS.",
"Therefore, the considered behavior is difficult to observe in practice."
],
[
"Experimental validations: Case studies",
"This section is devoted to validating the two following hypotheses.",
"(i) The frequency splitting behavior is not present even at a high coupling strength with the choice of an appropriate external capacitance, and (ii) At a moderate- or strong-coupling regime, the power bound can be approached by optimizing the load resistance at the nominal resonance frequency."
],
[
"Frequency splitting elimination",
"An experimental demonstration of frequency splitting suppression was reported [19].",
"However, the study presented in this Section aims to provide a more in-depth understanding to complete a comprehensive picture of the equivalent circuit model and the proposed approach.",
"Another goal is to verify an important property in Section , that the quality factors should not be chosen too low, or in other words, the added capacitances should not be too large, in order to keep the transferred power at an appropriate level.",
"Figure: Experimental setup of a WPTS in series-series configuration.Figure REF shows a schematic illustration of the setup for experiments, in which a Tektronix function generator is simultaneously utilized as a power source and a control unit to drive the transmitter coil.",
"The output voltage induced in the load is measured and recorded by a Tektronix oscilloscope.",
"The source voltage amplitude is set at $V_\\mathrm {s} = 10$ V, and the internal source impedance is specified as $R_\\mathrm {s} = 50$ $\\Omega $ .",
"Each transmitter/receiver coil has 3 turns with a diameter of 15 cm and the distance between them is 0.5 cm, approximately.",
"The inductance and resistance of the two coils are $(L_1, \\, L_2) = (3.90, \\, 3.81)$ $\\mu $ H and $(R_1, \\, R_2) = (0.10, \\, 0.71)$ $\\Omega $ , respectively.",
"The load resistance is arbitrarily chosen as $R_\\mathrm {L} = 25$ $\\Omega $ .",
"In order to analyze the frequency response of the output power, a linearly swept sinusoidal signal is used, with a duration of 90 s. The average power transferred to the load is computed as $P_\\mathrm {L} = \\frac{1}{t_2 - t_1} \\int _{t_1}^{t_2} \\frac{(V_\\mathrm {L}(t))^2}{R_\\mathrm {L}} t$ where $[t_1, \\, t_2]$ is the integral time interval.",
"Figure: Frequency responses of the delivered power with a coupling coefficient of k=52.16×10 -2 k = 52.16 \\, \\times 10^{-2}, a load resistance of R L =25R_\\mathrm {L} = 25 Ω\\Omega and added capacitances of C 1 =C 2 =77.08C_1 = C_2 = 77.08 pF.In the first examination, we select a small capacitance of $C_1 = C_2 = 77.08$ pF to ensure the presence of the frequency splitting characteristic, as presented in Figure REF .",
"Here, the equivalent series resistance (ESR) of these ceramic capacitors are small and neglected.",
"The measured coupling coefficient between the transmitter and receiver is $k = 52.16 \\, \\times 10^{-2}$ (or $k^2 = 272.06 \\times 10^{-3}$ ); this value is kept fixed while validating all cases.",
"The experimental data and the predictions by the model show a good agreement.",
"Figure: Frequency responses of the delivered power with a coupling coefficient of k=52.16×10 -2 k = 52.16 \\, \\times 10^{-2}, a load resistance of R L =25R_\\mathrm {L} = 25 Ω\\Omega and added capacitances of C 1 =C 2 =69C_1 = C_2 = 69 nF.Figure: Normalized output power (TPG) with respect to load resistance: Comparison between experimental data and simulation results.In order to prove the method proposed in Section is effective and feasible, we now seek a different value of $C_1$ and $C_2$ such that the frequency splitting phenomenon is eliminated, while keeping the other parameters unchanged.",
"For the system under consideration, we have $\\mu > 0, \\, \\xi _1 > 0$ and $C_\\gamma < \\xi _1$ .",
"Here, we use $L \\!=\\!",
"(L_1 \\!+\\!",
"L_2)/2$ and $R \\!=\\!",
"(R_1 \\!+\\!",
"R_2)/2$ to roughly calculate $\\mu , \\, \\xi , \\, C_\\gamma $ and other related parameters.",
"Based on Table REF , we (arbitrarily) choose $C_1 \\!=\\!",
"C_2 \\!=\\!",
"69$ nF that satisfies the condition $C_{1/2} \\in [{C_\\gamma , +\\infty })$ .",
"As a consequence, a single maximum of the power transferred to the load is obtained, as shown in Figure REF .",
"In this case, the two capacitors in use are made of Tantalum and the ERS cannot be negligible.",
"In particular, $R_{C_1} = 4.9 \\, \\Omega $ and $R_{C_2} = 6.45 \\, \\Omega $ , which are also in series with $R_1$ and $R_2$ , respectively.",
"Since we can always take the effects of ERS into consideration by replacing $R_1$ and $R_2$ by effective resistances $R_\\mathrm {E_1} = R_1 + R_{C_1}$ and $R_\\mathrm {E_2} = R_2 + R_{C_2}$ , the disregard of ERS in the original model does not compromise its generality.",
"Again, the experiment and simulation results are in good agreement.",
"It is important to note that the decrease of $P_\\mathrm {L}$ observed in Figure REF is only a particular case.",
"This behavior is not a universal consequence of the method.",
"As we depicted in Figure REF and in [19], it is possible to select appropriate values of $(C_1, \\, C_2)$ so that the frequency splitting property is suppressed while the power transferred to the load is not compromised."
],
[
"Approaching the power bound at strong coupling",
"As the actual transferred power is the central objective of the paper, this subsection aims to verify that, at a strong coupling regime, the power limit can be reached by only tuning the load resistance at (or nearby) the nominal resonance frequency of the two resonators.",
"The setup corresponding to Figure REF with $C_1 = C_2 = 77.08$ pF is utilized for this purpose.",
"Figure REF shows the variation of the output power with respect to the load resistance, characterized by the TPG, $\\eta _\\mathrm {t} = P_\\mathrm {L}/P_\\mathrm {avs}$ .",
"An optimal load of $^\\mathrm {opt}R_\\mathrm {L} = 0.26$ k$\\Omega $ is predicted by (REF ), while that of the experiment is $0.34$ k$\\Omega $ ; the corresponding maximum TPGs are 0.94 and 0.86.",
"At that given coupling coefficient, the limit of the TPG is $\\lim \\lbrace \\eta _\\mathrm {t}\\rbrace = 99.5 \\times 10^{-2}$ , obtained from (REF ).",
"The drive frequency is set at $f = 9$ MHz, which is close to the resonance frequencies of the transmitter and receiver ($f_1 = 9.18$ MHz and $f_2 = 9.29$ MHz, respectively).",
"Despite the small discrepancy between the measurements and simulations, the general trend is well captured by the model.",
"This difference can be explained by some factors that are not taken into account, such as the parasitic capacitances of the coils and load, the equivalent series inductance of the added capacitors, or the power loss due to contact resistance.",
"Finally, the validity of some of the important theoretical analyses in Sections [-] is justified, including (i) an effective mechanism to eliminate the frequency splitting phenomenon and (ii) approaching the power bound with optimizing the load resistance at moderate/high coupling regimes."
],
[
"Concluding remarks",
"We analyzed the fundamental frequency response and dynamics of a WPTS configured in series, with a focus on the transferred power.",
"We comprehensively revealed the physical insight of the frequency splitting phenomenon by examining the rate of change of the power with respect to the drive frequency rather than considering the characteristics of the input impedance.",
"We also provided a simple but effective method to eliminate this behavior.",
"In particular, high coil quality factors and strong coupling between the two resonators result in the frequency splitting, which can be avoided by adjusting the resonance frequency (or the added capacitors, equivalently).",
"This technique is independent of the coupling strength, meaning that it is not necessary to select the coupling coefficient less than a specific value or limit the minimum distance between the two coils, as suggested by other authors.",
"The optimal load resistance (i.e., positive stationary point(s) of the equation $P_\\mathrm {L}/R_\\mathrm {L} = 0$ ) was derived in a general form and expressed as a function of the other parameters.",
"Numerical computation of the rigorous power bound and analytical asymptotic formula were discussed and compared, showing nearly identical results.",
"The asymptotic solution is therefore an efficient and effective approach to examine the performance of a given WPTS, regardless of configuration (SS, PS, SP or PP).",
"For the completeness of the analysis, a distinctive dynamics of the perfect coupled system, where the coupling coefficient equals unity, is discussed.",
"In this circumstance, it is shown that the output power tends to saturate at a frequency-independent constant value when the driving frequency is relatively large (approaches infinity).",
"All important findings in this paper were verified by dynamic simulations using SPICE, and in addition, two theoretical analyses were validated with rigorous experiments.",
"Therefore, the mathematical model is reliable and can be used as a framework for designing an optimal system.",
"Although the study is initially motivated by low-power applications, in which the central objective is to optimize the power transferred to the load instead of maximizing the link efficiency, no specific assumptions were made regarding the particular physical nature of the system.",
"Thus, the obtained results and discussion remain valid independently of low- or high-power systems."
],
[
"Acknowledgment",
"The authors would like to thank the ISS Lab, University of Utah, for providing resources."
],
[
"Tables of signs and variations",
"Figure REF shows the signs of $\\mathrm {d}P_\\mathrm {L}/\\mathrm {d} \\omega $ and the variations of $P_\\mathrm {L}$ when (REF ) has unique positive solution $\\Omega _\\mathrm {u}$ .",
"Note that $\\alpha < 0$ and $P_\\mathrm {L}/\\omega \\xrightarrow{} -\\infty $ .",
"The signs of $\\mathrm {d}P_\\mathrm {L}/\\mathrm {d} \\omega $ and the variations of $P_\\mathrm {L}$ for the case in which (REF ) has three positive solutions $0 < \\Omega _\\mathrm {i} < \\Omega _\\mathrm {j} < \\Omega _\\mathrm {k}$ are shown in Figure REF .",
"Here $\\lbrace i, \\, j, \\, k\\rbrace $ takes one of three values $\\lbrace 1, \\, 2, \\, 3\\rbrace $ depending on the relations of $\\lbrace \\Omega _1, \\, \\Omega _2, \\, \\Omega _3\\rbrace $ in (REF ), () and ().",
"Two maxima and one minimum of $P_\\mathrm {L}$ are denoted by $P_\\mathrm {M-1}, \\,\\, P_\\mathrm {M-2}$ and $P_\\mathrm {m}$ respectively."
],
[
"Descartes' sign rule",
"Table REF shows all possibilities of the signs of $\\alpha , \\,\\, \\beta , \\,\\, \\gamma $ and $\\lambda $ which are the coefficients of the polynomial $f(\\Omega )$ .",
"When $k \\ne 1$ , $\\alpha $ is always negative and $\\lambda > 0$ for all positive-physical parameters of the system."
]
] | 2011.14215 |
[
[
"On periodic boundary solutions for cylindrical and spherical KdV-Burgers\n equations"
],
[
"Abstract For the KdV-Burgers equations for cylindrical and spherical waves the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary is studied.",
"The equation describes a medium which is both dissipative and dispersive.",
"For an appropriate combination of dispersion and dissipation the asymptotic profile looks like a periodical chain of shock fronts with a decreasing amplitude (sawtooth waves).",
"The development of such a profile is preceded by a head shock of a constant height and equal velocity which depends on spatial dimension as well as on integral characteristics of boundary condition; an explicit asymptotic for this head shock is found."
],
[
"Introduction",
"The behavior of solutions of the KdV and KdV - Burgers equations is well studied, yet they remain a subject of various recent research, [3]–[5] where these equations describe flat waves in one spatial dimension.",
"But cylindrical and spherical waves also have a variety applications (eg, waves generated by a downhole vibrator).",
"The paper is a continuation of the previous research of the author, [6] – [10].",
"The well known KdV-Burgers equation for flat waves is of the form $u_t=-2uu_x+\\varepsilon ^2 u_{xx}+\\delta u_{xxx}.$ Its cylindrical and spherical analogues are $u_t+\\frac{1}{2t}u=-2uu_x+\\varepsilon ^2 u_{xx}+\\delta u_{xxx}.$ and $u_t+\\frac{1}{t}u=-2uu_x+\\varepsilon ^2 u_{xx}+\\delta u_{xxx}.$ correspondingly, [1] – [2].",
"We consider the initial value - boundary problem (IVBP) for the KdV-Burgers equation on a finite interval: $u(x,0) =f(x), \\; u(a,t) = l(t),\\; u(b,t) =L(t), \\; u_x(b,t) =R(t), \\; x\\in [a,b].$ In the case $\\delta =0$ (that is, for Burgers equation), it comes to $u(x,0) =f(x), \\; u(a,t) = l(t),\\; u(b,t) =R(t), \\; x\\in [a,b].$ The case of the boundary conditions $u(a,t) = A\\sin (\\omega t),\\; u(b,t) =0$ and the related asymptotics are of a special interest here.",
"For numerical modelling we use $ x\\in [0,b]$ for appropriately large $b$ instead of $\\mathbb {R}^+$ ."
],
[
"Flat case: travelling waves",
"For $t\\gg 1$ equations (REF ) and (REF ) tend to (REF ) as well as their solutions.",
"In particular, the explicit form of traveling wave solutions for the flat KdV-Burgers (REF ) is as follows $u_{\\mathrm {tws}}(x,t)=\\frac{3\\varepsilon ^4\\tanh ^2(\\frac{\\varepsilon ^2(x-Vt-s)}{10\\delta })}{50\\delta } -\\frac{3\\varepsilon ^4\\tanh (\\frac{\\varepsilon ^2(x-Vt-s)}{10\\delta })}{25\\delta }+\\frac{V}{2}-\\frac{3\\varepsilon ^4}{50\\delta }$ Our IVBP requires $u|_{x=+\\infty }=0$ ; so the sole such travelling wave has a velocity $V=\\frac{6\\varepsilon ^4}{25\\delta }.$ Note that the height of the wave (REF ), $u|_{x=-\\infty }-u|_{x=+\\infty }=H-h=6\\varepsilon ^4/25\\delta $ does not depend on its velocity and is completely defined by the ratio $\\varepsilon ^4/\\delta $ which depends on the coefficients $\\varepsilon ,\\;\\lambda $ related to dispersion and dissipation.",
"Also note that the equations (REF )–(REF ) may be readily put in the form $w_t+\\frac{n}{2t}w=\\gamma w_{xx}-2ww_x+w_{xxx}$ by the change of variables $t\\rightarrow t\\sqrt{\\delta }$ , $x\\rightarrow x\\sqrt{\\delta }$ , $u\\rightarrow -\\frac{u}{2}$ .",
"Here $\\gamma =\\frac{\\varepsilon ^2}{\\sqrt{\\delta }}$ is the important parameter that defines a character of solutions; $n=0,1/2,1$ for flat, cylindrical and spherical waves correspondingly.",
"In the case $\\delta =0$ , the Burgers equation also has a variety of travelling wave solutions, vanishing at $x\\rightarrow +\\infty $ .",
"They are given by the formula $u_{\\mathrm {Btws}}(x,t)=\\frac{V}{2}\\left[1-\\tanh \\left(\\frac{V}{2\\varepsilon ^2}(x-Vt+s)\\right)\\right]$ We demonstrate that in the case of the above IVBP the perturbation of the equilibrium state (), () ultimately becomes very similar to the form of this shock."
],
[
"Burgers.",
"Here we demonstrate typical graphs for cylindrical and spherical Burgers waves, figure REF , REF .",
"Figure: Cylindrical Burgers, ε=0.1,\\varepsilon =0.1, Left: u 0 =sint,t=150.u_0=\\sin t, t=150.",
"Right: u 0 =sin10t,t=200.u_0=\\sin 10t, t=200.Figure: Spherical Burgers, u 0 =sintu_0=\\sin t, Left: ε=0.1,t=150\\varepsilon =0.1, t=150 Right: ε 2 =0.3,t=150\\varepsilon ^2=0.3, t=150"
],
[
"KdV-Burgers.",
"Typical graphs for cylindrical and spherical KdV-Burgers, figure REF , REF .",
"Figure: Cylindrical KdV-Burgars, Left: u 0 =sint,t=300,ε=0.1,δ=0.001u_0=\\sin t, t=300, \\varepsilon =0.1, \\delta =0.001.Right: u 0 =3sint,t=100,ε=0.1,δ=0.001.u_0=3\\sin t, t=100, \\varepsilon =0.1, \\delta =0.001.Figure: Spherical KdV-Burgers, u 0 =sintu_0=\\sin t, Left: t=300,ε=0.1,δ=0.001.t=300, \\varepsilon =0.1, \\delta =0.001.",
"Right: u↔-u,t=300,ε 2 =0.02,δ=0.001u\\leftrightarrow -u, t=300, \\varepsilon ^2=0.02, \\delta =0.001 ε 2 =0.2\\varepsilon ^2=0.2"
],
[
"Overview.",
"Stronger viscosity effectively damps oscillation and may result in absence of sawtooth effects.",
"Greater frequencies of initial perturbation decay much faster.",
"A signal of a greater amplitude results in increase of velocity and amplitude of travelling signal.",
"After the decay of initial oscillations, graphs become monotonic declining convex lines, terminating by a shock."
],
[
"Symmetries",
"Since cylindrical and spherical equations explicitly depend on time, their stock of symmetries is scarce.",
"The algebras of classical symmetries are generated by vector fields: $X &=& \\frac{\\partial }{\\partial x},\\ \\\\Y &=& x\\frac{\\partial }{\\partial x}+2t\\frac{\\partial }{\\partial t}-u\\frac{\\partial }{\\partial u}, \\\\Z &=& \\sqrt{t}\\frac{\\partial }{\\partial x}+\\frac{1}{4\\sqrt{t}}\\frac{\\partial }{\\partial u}, \\\\W &=& \\ln (t)\\frac{\\partial }{\\partial x}+\\frac{1}{2t}\\frac{\\partial }{\\partial u}.$ Table: NO_CAPTION"
],
[
"Conservation laws",
"First rewrite equations (REF ) – (REF ) into an appropriate, conservation law form $[t^n\\cdot u]_t=[t^n\\cdot (-u^2+\\varepsilon ^2 u_{x}+\\delta u_{xx})]_x,$ $n=0,\\; 1/2,\\; 1$ for flat, cylindrical and spherical cases correspondingly.",
"Hence for solutions of the above equations we have $\\oint \\limits _{\\partial \\mathcal { D}} t^n\\cdot [u\\,dx+(\\varepsilon ^2 u_x-u^2+\\delta u_{xx})\\,dt] =0,$ where $\\mathcal {D}$ is a rectangle $\\lbrace 0\\le x\\le L,\\;0\\le t\\le T\\rbrace .$ Bearing in mind the initial value/boundary conditions $u(x,0)=u(+\\infty ,t)=0$ , for $L=+\\infty $ the integrals read $\\int \\limits _{+\\infty }^0 T^n u(x,T)\\,dx+\\int \\limits _T^0 t^n(\\varepsilon ^2u_x(0,t)-u^2(0,t)+\\delta u_{xx}(0,t))\\,dt=0.$ Thus $\\int \\limits _0^{+\\infty } u(x,T)\\,dx=\\frac{1}{T^n}\\int \\limits _0^T t^n(-\\varepsilon ^2u_x(0,t)+u^2(0,t)-\\delta u_{xx}(0,t))\\,dt.$ Subsequently $\\frac{1}{T}\\int \\limits _0^{+\\infty } u(x,T)\\,dx=\\frac{1}{T}\\int \\limits _0^T \\frac{1}{T^n}t^n(-\\varepsilon ^2u_x(0,t)+u^2(0,t)-\\delta u_{xx}(0,t))\\,dt.$ The right-hand side of (REF ) is the mean value right-hand side of (REF ).",
"It can be computed in some simple cases or estimated."
],
[
"Constant boundary conditions",
"Consider boundary condition $u(0,t)=M$ .",
"The graphs of solution are shown on figure REF , left (compare the rate of decay caused solely by the spaces dimensions.)",
"Figure: Constant boundary solutions to Burgers equation, ε=0.1,t=200.\\varepsilon =0.1, t=200.",
"Left: Solid line — cylindrical, dots line — spherical.",
"Right: A trace of movement to the right of the spherical solution at moments t=37.5·k,k=1⋯6t=37.5\\cdot k, k=1\\dots 6For the resulting compression wave $u_x(0,t)=0$ and the right-hand side of (REF ) equals $\\frac{1}{T}\\int \\limits _0^T \\frac{M^2}{T^n}t^n\\,dt=\\frac{M^2}{n+1}$ As the figures REF — REF show, for periodic boundary condition, after the decay of initial oscillations, graphs become monotonic convex lines that begin approximately at the hight $A/2$ and broking at $x=V\\cdot T$ and at the height $V$ .",
"These monotonic lines are very similar to the graphs or constant-boundary solutions."
],
[
"\"Homothetic\" solutions",
"Looking at the solution's graph animation one can clearly see (eg, on figure REF , right) that the monotonic part and its head shock develops as a homothetic transformation of the initial configuration.",
"So we seek solutions of the form $u(x,t)=y(\\frac{x}{t})$ .",
"Substituting it into equations (REF ) – (REF ) we get the equation $-y^{\\prime }\\frac{x}{t^2}+\\frac{ny}{t}=\\frac{2yy^{\\prime }}{t}+\\frac{\\varepsilon ^2 y^{\\prime \\prime }}{t^2}+\\frac{\\delta y^{\\prime \\prime \\prime }}{t^3},$ or $-\\xi y^{\\prime }+ny=2yy^{\\prime }+\\frac{\\varepsilon ^2 y^{\\prime \\prime }}{t}+\\frac{\\delta y^{\\prime \\prime \\prime }}{t^2},$ for $y=y(\\xi )$ and $n=0,\\;1/2,\\;1$ .",
"For $t$ large enough we may omit last two summands.",
"It follows that appropriate solutions of the above ordinary differential equations are $u_1(x,t)=C_1,\\; C_1\\in \\mathbb {R},\\; n=0,\\mbox{ for flat waves equation;}$ $u_2(x,t)=-\\frac{2+\\sqrt{C_2\\xi +4}}{C_2},\\; C_2\\in \\mathbb {R},\\; n=\\frac{1}{2},\\; \\mbox{ for cylindrical and}$ $u_3(x,t)=\\exp \\left(\\mathop {\\mathrm {LambertW}}\\nolimits \\left(-\\frac{\\xi }{2}e^{-\\frac{C_3}{2}}\\right)+\\frac{C_3}{2}\\right),\\; C_3\\in \\mathbb {R},\\; n=1$ for spherical equation.",
"Let $V$ is the velocity of the signal propagation in the medium.",
"Since at the head shock $x=Vt$ and $u=V$ we obtain the condition for finding $C_i$ .",
"It is $y(V)=V$ .",
"It follows then that $C_1=V,\\; C_2=-\\frac{3}{V},\\; C_3=\\ln (V)+\\frac{1}{2}$ .",
"For flat waves it corresponds to a travelling wave solution of the classical Burgers equation.",
"For the cylindrical waves the monotonic part is given by $u_2=\\frac{1}{3}\\left(2V+V\\sqrt{4 -\\frac{3x}{Vt}}\\right);$ for spherical waves $u_3=V\\sqrt{e}\\exp \\left(\\mathop {\\mathrm {LambertW}}\\nolimits \\left(-\\frac{x}{2 V t\\sqrt{e}}\\right)\\right).$ Note that $u_2|_{x=0}=\\frac{4V}{3}\\mbox{ and }u_3|_{x=0}=V\\sqrt{e}\\approx 1.65 V.$ These formulas show that the velocity is proportional to the amplitude at the start of oscillation.",
"And it does not depend on frequency that together with amplitude define the oscillating part of solutions; more on that below.",
"The corresponding graphs ideally coincide with the graphs obtained by numerical modelling; for instance see comparison to the solution at ($t=100$ ) for the problem $u_t=0.01u_{xx}-2uu_x-u/t,\\; u(0,t) = 1, u(75,t) = 0, u(x,0) = 0$ on figure REF , left.",
"Figure: Left: Solid line — solution to (), dots line — its u 2 u_2 approximation.",
"Right: Solid line— solution to (), dots line — its u ˜ 2 \\tilde{u}_2 approximation; both at t=20t=20.Yet the smooth part of the periodic boundary solution ends with a break, which travels with a constant velocity and amplitude, very much like a head of the Burgers' travelling wave (TWS) solution (REF ).",
"A rather natural idea is to truncate a homothetic solution, multiplying it by a (normalized) Burgers TWS.",
"Namely, put For the cylindrical waves $\\tilde{u}_2=\\frac{1}{2}[1-\\tanh (\\frac{V}{\\varepsilon ^2}(x-Vt))]\\cdot \\frac{1}{3}\\left(2V+V\\sqrt{4 -\\frac{3x}{Vt}}\\right);$ for spherical waves $\\tilde{u}_3=\\frac{1}{2}[1-\\tanh (\\frac{V}{\\varepsilon ^2}(x-Vt))]\\cdot V\\sqrt{e}\\exp \\left(\\mathop {\\mathrm {LambertW}}\\nolimits \\left(-\\frac{x}{2 V t\\sqrt{e}}\\right)\\right).$ This construction produces an approximation of an astonishing accuracy, see figure REF , right; this figure corresponds to the cylindrical Burgers problem $u_t=0.01u_{xx}-2uu_x-u/2t,\\; u(0,t) =\\sin 10t , u(10,t) = 0, u(x,0) = 0.$ Moreover, it is evident that the graphs of $\\tilde{u}_2,\\, \\tilde{u}_3$ neatly represent the median lines of the approximated solutions on their whole range.",
"Recall that these medians may be evaluated independently via the right-hand side of (REF ).",
"Now evaluate the trapezoid area under $\\tilde{u}_2,\\, \\tilde{u}_3$ graphs: For cylindrical equation $\\int _{0}^{Vt}\\left[\\frac{[1-\\tanh (\\frac{V}{\\varepsilon ^2}(x-Vt))]}{2} \\frac{1}{3}\\left(2V+V\\sqrt{4 -\\frac{3x}{Vt}}\\right) \\right]dx=\\frac{32}{27}V^2t;$ for spherical equation $\\int _{0}^{Vt}\\left[\\frac{[1-\\tanh (\\frac{V}{\\varepsilon ^2}(x-Vt))]}{2} V\\sqrt{e}\\exp \\left(\\mathop {\\mathrm {LambertW}}\\nolimits \\left(\\frac{-x}{2 V t\\sqrt{e}}\\right)\\right) \\right]dx\\\\=\\frac{V^2t\\cdot e}{2}.$ Hence the mean value of the left-hand side of (REF ) can be estimated as follows.",
"Since the signal from $x=0$ spreads, after decay of oscillations, to the right with a constant speed $V$ and the same constant amplitude $V$ at the shock, and it is very well approximated by an appropriate homothetic solution, we get $\\frac{1}{T}\\int \\limits _0^{+\\infty } u(x,T)\\,dx=\\frac{1}{T}\\int \\limits _0^{VT} u(x,T)\\,dx\\approx \\left\\lbrace \\begin{array}{l}\\frac{32}{27}V^2 \\mbox{ in cylindrical case} ;\\\\[3pt]\\frac{V^2\\cdot e}{2} \\mbox{ in spherical case},\\end{array}\\right.$ This mean value can be also evaluated numerically.",
"In the case illustrated by figure REF the direct numerical evaluation differs from the estimation (REF ) by $1\\%$ .",
"For constant-boundary waves, it follows from (REF ) that $\\frac{M^2}{n+1}=\\left\\lbrace \\begin{array}{l}\\frac{32}{27}V^2 \\mbox{ in cylindrical case} ;\\\\[3pt]\\frac{V^2\\cdot e}{2} \\mbox{ in spherical case},\\end{array}\\right.$ see (REF ); of course this result coincides with (REF ).",
"So the mean value $M$ (see it on figure REF ) of arbitrary solution at the start of oscillations (or in a vicinity of the oscillator) is linearly linked to the velocity of the head shock.",
"But to find this mean value for an arbitrary border condition is a tricky task, because the integrands $u_x$ and $u_{xx}$ of the right-hand side of (REF ) have numerous breaks.",
"Yet numerical experiments show (eg, see figure REF ), that for the $u|_{x=0}=A\\sin (t)$ boundary condition such a value is $M\\approx A\\cdot a$ , where $a\\approx 0.467$ is the mean value for $1\\cdot \\sin (t)$ condition.",
"That is, $M$ depends on $A$ almost linearly.",
"Note, that this value may be obtained via the velocity of the head shock, which, in its turn, can be measured with great accuracy by the distance passed by the head shock after a sufficiently long time."
],
[
"Conclusion",
"We obtained a way to foretell the form of the head shock with a great accuracy: the links in the chain of causations are as follows.",
"First, using the boundary conditions we find, if approximately, the initial mean value of the solution by the formulae (REF ), (REF ).",
"This value defines the form of the declining homothetic part of the solution, in particular its velocity and median line.",
"When the amplitude of this declining part reaches the value if velocity, the solution jumps to zero value by the scenario of the Burgers travelling wave and becomes a part of a homothetic head shock.",
"In vicinity of the boundary oscillations occur around this homothetic line; their longevity, both in time and space, and whether they have a sawtooth form, is defined by the relations between the amplitude and frequency of forcing oscillations and viscosity and dispersive characteristics of the media.",
"The exact dependencies are now investigated; results will be published elsewhere.",
"This work was partially supported by the Russian Basic Research Foundation grant 18-29-10013."
]
] | 2011.14189 |
[
[
"Structure formation and the matter power-spectrum in the R_h=ct universe"
],
[
"Abstract Inflation drives quantum fluctuations beyond the Hubble horizon, freezing them out before the small-scale modes re-enter during the radiation dominated epoch, and subsequently decay, while large-scale modes re-enter later during the matter dominated epoch and grow.",
"This distinction shapes the matter power spectrum and provides observational evidence in support of the standard model.",
"In this paper, we demonstrate that another mechanism, based on the fluctuation growth in the R_h=ct universe, itself an FLRW cosmology with the added constraint of zero active mass (i.e., rho+3p=0), also accounts very well for the observed matter power spectrum, so this feature is not unique to LambdaCDM.",
"In R_h=ct, the shape of the matter power spectrum is set by the interplay between the more rapid decay of the gravitational potential for the smaller mode wavelengths and the longer dynamical timescale for the larger wavelengths.",
"This combination produces a characteristic peak that grows in both amplitude and mode number as a function of time.",
"Today, that peak lies at k approx 0.02 Mpc^-1, in agreement with the Ly-alpha and Planck data.",
"But there is no need of an inflationary expansion, and a complicated epoch dependence as one finds in LambdaCDM."
],
[
"Introduction",
"Our current view of cosmic evolution holds that large-scale structure originated from primordial quantum fluctuations, believed to have been seeded during inflation in the early Universe [1], [2], [3], [4], [5], [6].",
"According to this theory, quantum fluctuations classicalized and grew due to self-gravity to form the inhomogeneous Universe we see today.",
"The existence of these inhomogeneities in the CMB and matter power spectrum [7], [8], [9] has become quite evident with the advent of high precision measurements.",
"Moreover, the observation of galaxy rotation curves [10], [11], [12] and weak lensing measurements [13], [14], [15], [16], [17] have indicated a necessity for the existence of dark matter.",
"Hence primordial fluctuations must include dark and baryonic matter, and radiation.",
"Dark matter is assumed to be collisionless and non-interacting, so its fluctuations grew due to self-gravity only, without the suppression from radiation pressure.",
"Once dark matter perturbations grew beyond a critical limit, they formed bound objects [18].",
"In contrast, baryons could not form bound objects in the early Universe because their growth was suppressed by the radiation to which they were coupled.",
"As the baryons gradually decoupled from the radiation, however, they accreted into the potential wells created by dark matter to form the large-scale structure we see in the Universe today.",
"This scenario—in which dark matter participated in the formation of structure in the early Universe—is indispensable to $\\Lambda $ CDM.",
"Without the dark matter, structure formation would have been delayed by the baryon-radiation coupling, creating an inconsistency with the observations.",
"But the high precision data coming down the pipeline over the past decade have created a conflict with the timeline of structure formation in $\\Lambda $ CDM, even with the contribution due to dark matter.",
"In particular, the discovery of SDSS010013.02+280225.8, an ultraluminous quasar at redshift $z=6.3$ [19], has exacerbated the problem of supermassive black-hole growth and evolution in the early Universe [20].",
"Almost all of the 122 previously discovered quasars at redshifts $z\\approx 6$ [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] have a black hole with mass $\\sim 10^9\\;M_{\\odot }$ , challenging the standard model's predicted timeline, which would have afforded them less than 900 Myr to grow after the big bang, but likely even less than $\\sim $ 500 Myr since the onset of Population II star formation.",
"In the context of $\\Lambda $ CDM, the formation of Pop III stars might have occurred by redshift $z\\approx 20$ [38], [39], [40], [41], corresponding to a time of 200 Myrs.",
"Then the subsequent transition to Pop II stars would have taken at least 100 Myrs because the gas expelled by the first generation stars had to cool and recollapse [42], [43].",
"These Pop II stars would have acted as seeds for black holes.",
"If the early black holes grew according to the Eddington rate, it would have taken at least 930 Myrs for them to reach their observed mass.",
"The observed quasar at redshift $z = 6.3$ , however, would have been 880 Myrs old in $\\Lambda $ CDM, which is not consistent with how these objects grew.",
"Two possibilities have been proposed to reconcile this problem: 1) the black holes might have grown with highly anomalous accretion rates, but this phenomenon is not observed anywhere in the Universe; 2) Pop III stars might have formed earlier than expected, thus partially reconciling the problem.",
"But this too doesn't appear to be reasonable according to various simulations [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55].",
"In addition, the structure formation timeline in $\\Lambda $ CDM predicts that at redshifts $z = 4 - 8$ most massive galaxies should have been transitioning from an initial halo assembly to baryonic evolution.",
"But the recent Cosmic Assembly Near-infrared Deep Extragalactic Survey (CANDELS)[56], [57] and Spitzer Large Area Survey with Hyper-Suprime-Cam (SPLASH) surveys have found that these massive halos formed much earlier than predicted by $\\Lambda $ CDM, giving to rise to what some have called `The Impossibly Early Galaxy Problem' [58], [59], [60].",
"Attempts to reconcile this problem have indicated that a $\\sim 0.8$ dex change is required in the halo to stellar mass ratio over this redshift range, though such a drastic change could come about only with a complete absence of dark matter at redshift 8 or with essentially $100\\%$ of the baryons condensing into stars at higher redshifts [58].",
"Both of these scenarios constitute implausible physics, such as the need to convert all of the baryons into stars instantly upon halo virialization (see ref.",
"[58] and references cited therein).",
"Other attempted remedies, such as an evolution of the halo mass to light ratio, could reconcile the problem, but could only happen if the initial mass function were top-heavy.",
"Studies have shown that the ratio of halo mass to light should remain the same, at least within the redshift range ($4\\le z \\le 8$ ).",
"In this paper, we present the formation of structure in $R_{\\rm h}=ct$ [61], [62], [63], [64], [65], [66], an alternative Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology with zero active mass [65], [66].",
"The linear expansion in this model afforded sufficient time for supermassive objects to form, thus solving the `too early' appearance of massive quasars [20].",
"Even more importantly, we recently showed that the growth rate in this model accounts very well for the (otherwise too) early appearance of massive halos and galaxies [59], [60].",
"In § 2, we summarize the basis for the two cosmological models we consider here (i.e., $\\Lambda $ CDM and $R_{\\rm h}=ct$ ).",
"Then, in §§ 3–5, we shall derive the necessary mathematical formalism for the formation of structure in the $R_{\\rm h}=ct$ universe, and describe the evolution of the matter power spectrum in §§ 6–8.",
"Our conclusions will be presented in § 9.",
"The standard model contains dark energy, dark matter, radiation and baryons as the primary constituents.",
"The existence of dark energy is inferred from the distance-redshift relation of Type Ia SNe [67], [68], [69], while the presence of dark matter is inferred from the galaxy rotation [10], [11], [12] and weak lensing [13], [14], [15], [16] measurements, as noted earlier.",
"Its current energy budget is dominated by dark energy ($\\sim 70\\%$ ; see ref.",
"[70]), possibly in the form of a cosmological constant with an equation-of-state $w_{\\rm de}\\equiv p_{\\rm de}/\\rho _{\\rm de}=-1$ .",
"The dark-matter component contributes an additional $\\sim 27\\%$ of the energy budget, and is primarily responsible for the formation and growth of large-scale structures.",
"The rest of the energy density ($\\sim 3\\%$ ) is in the form of baryons, with a negligible contribution from radiation.",
"The early $\\Lambda $ CDM universe was radiation dominated, producing a phase of decelerated expansion, leading to the well-known temperature horizon problem.",
"The standard model is therefore strongly dependent on an early inflationary expansion, lasting from $10^{-35}$ to $10^{-32}$ s, in order to mitigate such internal inconsistencies [1].",
"In more recent times, however, the principal benefit of inflation has been viewed as the mechanism it fosters for the creation and growth of quantum fluctuations [2], [3], [4], [5] that might haved subsequently grown into large-scale inhomogeneities.",
"As stated earlier, the currently held view is that the classicalised version of these quantum fluctuations produced large-scale structure.",
"Although, the actual mechanism of classicalisation has remained elusive [71], [6], [72], [73], [74], [75], it is thought that primordial modes were driven beyond the Hubble horizon by inflation, frozen and turned into classical modes.",
"They re-entered the horizon once the Hubble radius grew sufficiently.",
"This exit and entry of modes across the horizon is quite critical, playing a vital role in producing the matter power spectrum that we shall describe in more detail in § 7."
],
[
"The $R_{\\rm h}=ct$ Universe",
"Our primary focus in this paper is to demonstrate how the $R_{\\rm h}=ct$ universe also provides a mechanism for producing the observed matter power spectrum.",
"This is an FLRW cosmology constrained by the zero active mass condition, which results in an expansion with an apparent (or gravitational) radius always equal to $ct$ [61], [62], [76], [64], [65], [66], [77].",
"It too assumes the presence of dark energy, radiation and baryonic and dark matter.",
"The primary difference between $\\Lambda $ CDM and $R_{\\rm h}=ct$ is that the latter is constrained by the equation-of-state $\\rho +3p=0$ [65], [66], where $p$ is the total pressure and $\\rho $ is the total energy density.",
"Numerous comparative tests examining which of the two models, $\\Lambda $ CDM or $R_{\\rm h}=ct$ , is favoured by the data have been carried out over the past decade.",
"These include high-$z$ quasars [78], [64], [79], [80], [81], [82], gamma ray bursts [83], [84], [85], cosmic chronometers [78], [82], [86], [87], [88], Type Ia SNe [76], [67], [68], [69], [89], Type Ic superluminous SNe [90], [91], and the age measurements of passively evolving galaxies [92], [93], [94], and strong gravitational lensing [95], [96], [97].",
"In all such one-on-one comparisons completed thus far, model selection tools show that the data favour $R_{\\rm h}=ct$ over $\\Lambda $ CDM [98], [81], [88], [20], [78], [85], [91], [89], [94].",
"What is particularly attractive about this alternative FLRW cosmology is that it did not have an early decelerated or accelerated phase, so it has no horizon problem [98].",
"An inflationary phase is not needed in the $R_{\\rm h}=ct$ universe, removing the need to find a self-consistent theory of inflation, which has thus far eluded us for over three decades.",
"The early $R_{\\rm h}=ct$ universe was dominated by dark energy and radiation, in the fractions $\\sim 80\\%$ and $\\sim 20\\%$ of the total energy density, respectively [101].",
"The late $R_{\\rm h}=ct$ universe is dominated by dark energy and matter, comprising 2/3 and 1/3 of the energy density, respectively.",
"In addition, since the ratio of horizon size to proper mode size always remains constant in the $R_{\\rm h}=ct$ universe, modes never cross the horizon.",
"Thus, the mechanism for generating the matter power spectrum in $R_{\\rm h}=ct$ is completely different than that in $\\Lambda $ CDM."
],
[
"Perturbed Einstein Equations in FLRW",
"We begin by deriving the perturbed Einstein equations for an arbitrary FLRW metric and then use them in the $R_{\\rm h}=ct$ universe to obtain the growth rate and power spectrum at various epochs.",
"All the modes remain sub-horizon in $R_{\\rm h}=ct$ , but one has to use relativistic perturbation theory rather than Newtonian theory because the cosmic fluid in $R_{\\rm h}=ct$ is always dominated by dark energy.",
"We proceed by perturbing the general FLRW metric [102], [103], [104], [105], [106], [107], [108], [109], [110], [111], which we write in the form $ds^2=a^2(\\eta )\\bigg [-(1-\\bar{h}_{00})\\,d\\eta ^2+2\\bar{h}_{0\\alpha }\\,d\\eta \\,dx^\\alpha +(\\delta _{\\alpha \\beta }+2\\bar{h}_{\\alpha \\beta })\\,dx^\\alpha \\,dx^\\beta \\bigg ]\\;,$ where $a(\\eta )$ is the scale factor in terms of the conformal time $\\eta $ .",
"The components of the tensor $\\bar{h}$ are the perturbations to the homogeneous FLRW metric.",
"The indices $\\alpha $ and $\\beta $ run from 1 to 3.",
"Using the conventional scalar-vector-tensor decomposition, we write $\\bar{h}_{00}$ , $\\bar{h}_{0\\alpha }$ and $\\bar{h}_{\\alpha \\beta }$ as follows $\\bar{h}_{00}=2\\Phi \\;,$ $\\bar{h}_{0\\alpha }=w_\\alpha =w_\\alpha ^\\perp +\\partial _\\alpha \\phi ^\\parallel \\;,$ and $\\bar{h}_{\\alpha \\beta }=\\bigg [ -\\psi \\delta _{\\alpha \\beta }+\\bigg (\\nabla _{\\alpha }U_{\\beta }^{\\perp }+\\nabla _{\\beta }U_{\\alpha }^{\\perp }\\bigg )+\\bigg (\\nabla _{\\alpha }\\nabla _{\\beta }-\\frac{1}{3}\\delta _{\\alpha \\beta }\\nabla ^2\\bigg )\\Phi _1+h_{\\alpha \\beta }^{\\perp \\perp }\\bigg ]\\;,$ with $\\psi ={h^\\alpha }_\\alpha $ and $h_\\alpha ^{\\perp \\perp {\\alpha }}=0$ .",
"All these metric perturbations are highly coordinate dependent, so one cannot automatically attribute any physical meaning to the perturbed components of the metric.",
"For infinitesimal coordinate transformations, such as $\\bar{x}^i=x^i+\\xi ^i$ , the decomposed perturbed metric components transform as follows $\\bar{\\Phi }=\\Phi -\\frac{1}{a}\\frac{da}{d\\eta }\\xi ^0\\;,$ $\\bar{\\phi }^\\parallel =\\phi ^\\parallel +\\xi ^0-\\frac{d\\xi }{d\\eta }\\;,$ $\\bar{w}^\\perp _\\alpha =w^\\perp _\\alpha -\\frac{d\\xi ^{\\perp }_\\alpha }{d\\eta }\\;,$ $\\bar{\\psi }=\\psi +\\frac{1}{a}\\frac{da}{d\\eta }\\xi ^0+\\frac{1}{3}\\nabla ^2\\xi \\;,$ $\\bar{\\Phi }_1=\\Phi _1-\\xi \\;,$ $\\bar{U}_\\alpha ^\\perp =U_\\alpha ^\\perp -\\frac{1}{2}\\xi _\\alpha ^\\perp \\;,$ and $\\bar{h}^{\\perp \\perp }_{\\alpha \\beta }=h^{\\perp \\perp }_{\\alpha \\beta }\\;.$ To use these perturbed components, one may either choose a particular observer (i.e., a particular set of coordinates), resort to a particular gauge, or work with Bardeen's four gauge independent variables [112].",
"It is evident from Equations (5)–(11) that there are seven independent metric perturbations, implying seven degrees of freedom, but there are only four gauge-invariant variables.",
"So when choosing a particular gauge, one must fix 3 degrees of freedom.",
"In this work, we follow the first approach by choosing the Newtonian gauge and fix three independent metric perturbations as follows (with $\\phi ^{\\parallel }=\\Phi _1=U_{\\alpha }^\\perp =0$ ): $\\Phi _A=\\Phi +\\frac{1}{a}\\frac{\\partial [a(\\phi ^\\parallel -\\frac{d\\Phi _1}{d\\eta })]}{\\partial \\eta }\\;,$ $\\Phi _H=\\frac{1}{a}\\frac{da}{d\\eta }(\\phi ^\\parallel -\\frac{d\\Phi _1}{d\\eta })-\\psi -\\frac{1}{3}\\partial ^\\alpha \\partial _\\alpha \\Phi _1\\;,$ $\\psi _\\alpha =w_\\alpha ^\\perp -2\\frac{dU_\\alpha ^{\\perp }}{d\\eta }\\;,$ and $h_{\\alpha \\beta }^\\perp =h_{\\alpha \\beta }^\\perp \\;.$ After setting three metric perturbations to zero (in the Newtonian gauge), one ends up with $\\Phi _A=\\Phi \\;,$ $\\Phi _H=-\\psi \\;,$ $\\psi _\\alpha =w_\\alpha ^\\perp \\;,$ and $\\bar{h}_{\\alpha \\beta }^\\perp =h_{\\alpha \\beta }^\\perp \\;.$ It is quite evident from the above equations that, in the Newtonian gauge, the independent metric perturbations are in fact the Bardeen variables.",
"This is the primary reason for choosing this gauge.",
"Then the perturbed metric is given as $\\qquad ds^2&=&a^2(\\eta )\\left[-(1+2\\Phi )\\,d\\eta ^2+2w_\\alpha ^\\perp \\,d\\eta \\,dx^\\alpha +\\right.\\nonumber \\\\&\\hbox{}&\\qquad \\qquad \\left.\\left\\lbrace (1-2\\psi )\\delta _{\\alpha \\beta }+2h_{\\alpha \\beta }^{\\perp \\perp }\\right\\rbrace \\,dx^\\alpha \\,dx^\\beta \\right]\\;.$ It is well known that the vector perturbations die away as the Universe expands, and since our primary motivation in this paper is to compute the power spectrum and evolution of the perturbed dark-matter density field, we shall ignore both the vector and tensor perturbations for this work.",
"The metric with only the scalar perturbations is then given as $ds^2=a^2(\\eta )\\bigg [-(1+2\\Phi )\\,d\\eta ^2+(1-2\\psi )\\delta _{\\alpha \\beta }\\,dx^\\alpha \\,dx^\\beta \\bigg ]\\;,$ which may be used to compute the Ricci tensor with the following components: $R_{00}=\\nabla ^2\\Phi +3\\frac{d^2\\psi }{d\\eta ^2}+3{\\mathcal {H}}\\bigg [\\frac{d\\Phi }{d\\eta }+\\frac{d\\psi }{d\\eta }\\bigg ]-3\\frac{d{\\mathcal {H}}}{d\\eta }\\;,$ $R_{0\\alpha }=2\\partial _\\alpha \\frac{d\\psi }{d\\eta }+2{\\mathcal {H}}\\partial _\\alpha \\Phi \\;,$ $R_{\\alpha \\beta }&=&\\left[\\frac{d{\\mathcal {H}}}{d\\eta }+2{\\mathcal {H}}^2-\\frac{d^2\\psi }{d\\eta ^2}+\\nabla ^2\\psi -2\\left(\\frac{d{\\mathcal {H}}}{d\\eta }+2{\\mathcal {H}}^2\\right)(\\psi +\\Phi )-\\right.\\nonumber \\\\&\\hbox{}&\\left.",
"{\\mathcal {H}}\\frac{d\\Phi }{d\\eta }-5{\\mathcal {H}}\\frac{d\\psi }{d\\eta }\\right]\\delta _{\\alpha \\beta }+\\partial _\\alpha \\partial _\\beta (\\psi +\\Phi )\\;,$ the Ricci scalar, $a^2{\\mathcal {R}}&=&6\\left(\\frac{d{\\mathcal {H}}}{d\\eta }+{\\mathcal {H}}^2\\right)-2\\nabla ^2\\Phi +4\\nabla ^2\\psi -12\\left(\\frac{{\\mathcal {H}}}{d\\eta }+{\\mathcal {H}}^2\\right)\\Phi -\\nonumber \\\\&\\hbox{}&6\\frac{d^2\\psi }{d\\eta ^2}-6{\\mathcal {H}}\\left(\\frac{d\\Phi }{d\\eta }+3\\frac{d^2\\psi }{d\\eta ^2}\\right)\\;,\\;\\;$ and also the Einstein tensor, $G_{00}=3{\\mathcal {H}}^2+2\\nabla ^2\\psi -6{\\mathcal {H}}\\frac{d\\psi }{d\\eta }\\;,$ $G_{0\\alpha }=2\\partial _\\alpha \\frac{d\\psi }{d\\eta }+2{\\mathcal {H}}\\partial _\\alpha \\Phi \\;,$ $G_{\\alpha \\beta }&\\hspace{-14.45377pt}=\\hspace{-14.45377pt}&-\\left(2\\frac{d{\\mathcal {H}}}{d\\eta }+{\\mathcal {H}}^2\\right)\\delta _{\\alpha \\beta }+\\left[\\nabla ^2(\\Phi -\\psi )+2\\frac{d^2\\psi }{d\\eta ^2}+2\\left(2\\frac{d{\\mathcal {H}}}{d\\eta }+\\right.\\right.\\nonumber \\\\&\\hbox{}&\\left.\\left.\\hspace{-21.68121pt}{\\mathcal {H}}^2\\right)(\\psi +\\Phi )+2{\\mathcal {H}}\\frac{d\\Phi }{d\\eta }+4{\\mathcal {H}}\\frac{d\\psi }{d\\eta }\\right]\\delta _{\\alpha \\beta }+\\partial _\\alpha \\partial _\\beta (\\psi +\\Phi )\\;.\\;$ In these expressions, ${\\mathcal {H}}$ is the Hubble parameter written as ${a^{^{\\prime }}}/{a}$ ."
],
[
"The Perturbed Stress-Energy Tensor",
"The perturbed stress-energy tensor for a perfect fluid may be written ${T^a}_b=(\\rho +\\delta \\rho +p+\\delta p)(u^a+\\delta u^a)(u_b+\\delta u_b)+ (p+\\delta p)\\delta ^a_b\\;,$ where $\\rho $ and $\\delta \\rho $ are the total, and perturbed, energy density of the Universe contributed by all the species, $p$ and $\\delta p$ are the total, and perturbed, pressure contributed by all the species, and $u_a$ and $\\delta u_a$ are the total, and perturbed, four-velocity.",
"Then, using $g_{ab}\\,u^au^b=-1$ , the perturbed velocity components are $\\delta u^0=-\\frac{\\Phi }{a}\\;,$ and $\\delta u^\\alpha =\\frac{v^\\alpha }{a}\\;,$ and we use the scalar-vector-decomposition to put $v_\\alpha =\\partial _\\alpha v+v_\\alpha ^\\perp $ .",
"Working out the spatial components of Einstein's equation, i.e., $G_{\\alpha \\beta }=8\\pi \\,Gg_{\\alpha c}T^c_\\beta $ , we get $\\left[\\nabla ^2(\\Phi -\\psi )\\right.&\\hspace{-21.68121pt}+\\hspace{-21.68121pt}&\\left.2\\frac{d^2\\psi }{d\\eta ^2}+2\\left(2\\frac{d{\\mathcal {H}}}{d\\eta }+{\\mathcal {H}}^2\\right)(\\psi +\\Phi )+2{\\mathcal {H}}\\frac{d\\Phi }{d\\eta }+\\right.\\nonumber \\\\&\\hbox{}&\\left.\\hspace{-36.135pt}4{\\mathcal {H}}\\frac{d\\psi }{d\\eta }\\right]\\delta _{\\alpha \\beta }+\\partial _\\alpha \\partial _\\beta (\\psi +\\Phi )=8\\pi Ga^2\\left[\\delta p-2\\psi p\\right]\\delta _{\\alpha \\beta }\\;.$ When $\\alpha \\ne \\beta $ , we get $\\psi =-\\Phi $ , and using this condition for the rest of the components in Einstein's equation, we find that $2\\frac{d^2\\Phi }{d\\eta ^2}+4\\bigg (2\\frac{d{\\mathcal {H}}}{d\\eta }+{\\mathcal {H}}^2\\bigg )\\Phi +6{\\mathcal {H}}\\frac{d\\Phi }{d\\eta }= 8\\pi Ga^2(\\delta p-2p\\Phi )\\;,$ $2\\nabla ^2\\Phi -6{\\mathcal {H}}\\frac{d\\Phi }{d\\eta }=8\\pi Ga^2(\\delta \\rho +2\\rho \\Phi )\\;,$ and $2\\partial _\\alpha \\frac{d\\Phi }{d\\eta }+2{\\mathcal {H}}\\partial _\\alpha \\Phi =8\\pi Ga^2(\\rho +p)v_\\alpha \\;,$ where $v_\\alpha =\\partial _\\alpha v+v_\\alpha ^\\perp $ .",
"Finally, using a wave decomposition with a Fourier series, defined for an arbitrary function as $F_k=\\frac{1}{(2\\pi )^3}\\int F(x)\\,e^{i\\vec{k}\\cdot \\vec{x}}\\,d^3x\\;,$ we get $k^2\\Phi _k+3{\\mathcal {H}}\\bigg (\\frac{d\\Phi _k}{d\\eta }+{\\mathcal {H}}\\Phi _k\\bigg )=4\\pi \\, Ga^2\\delta \\rho _k\\;.$ $\\delta \\rho _k$ in the above equation comprises of the components which deviate from ideal cosmic fluid.",
"In the case of $\\Lambda $ CDM $\\delta \\rho _k$ comprises of dark matter, baryons, neutrinos and radiation, whereas in the case of $R_{\\rm h}=ct$ Universe, this term comprises of dark matter, dark energy, baryons, neutrinos and radiation."
],
[
"The Perturbed Boltzmann Equations in $\\Lambda $ CDM",
"Equation (37) describes the evolution of the perturbed gravitational potential $\\Phi $ with a source term $\\delta \\rho $ driving its growth.",
"To solve it, one must determine the evolution equation for $\\delta \\rho $ as well.",
"In principle $\\delta \\rho $ represents the total perturbed energy density of the universe, comprised of perturbations of individual species that clump above the smooth background.",
"In this section, we shall obtain the evolution equations for individual species by perturbing their corresponding Boltzmann equations.",
"The Boltzmann equation for a given species (i.e., dark matter, baryons, radiation, etc.)",
"describes the evolution of its distribution function in the 8-dimensional phase space with 4-dimensions representing the four space-time coordinates, $x^{\\,i}$ , and the other four comprising the 4-momentum, $p^{\\,i}$ .",
"Using the additional constraint $g_{ij}p^{\\,i} p^{\\,j}=-m^2$ , this 8-dimensional phase space is reduced to 7-dimensions.",
"In this work, we choose the four spacetime coordinates, $x^{\\,i}$ , the magnitude of the 3-momentum, $p\\equiv |\\vec{p}|$ , and its direction, $\\hat{p}^{\\,\\mu }$ , as the independent variables.",
"From Liouville's theorem, one obtains the following equation for the evolution of the distribution function $f_s(x^{\\,i},p,\\hat{p}^{\\,\\mu })$ (for species `s'): $\\frac{df_s}{d\\lambda }&=&\\frac{\\partial f_s}{\\partial x^0}\\frac{\\partial x^0}{\\partial \\lambda }+\\frac{\\partial f_s}{\\partial x^{\\,\\mu }}\\frac{\\partial x^{\\,\\mu }}{\\partial \\lambda }+\\frac{\\partial f_s}{\\partial p}\\frac{\\partial p}{\\partial \\lambda }+\\frac{\\partial f_s}{\\partial \\hat{p}^{\\,\\mu }}\\frac{\\partial \\hat{p}^{\\,\\mu }}{\\partial \\lambda }\\nonumber \\\\&=&C[f_s]\\;,$ where $\\lambda $ and $C[f_s]$ are the affine parameter and the source/collision term for this species, respectively.",
"We define $P^{\\,i}={dx^{\\,i}}/{d\\lambda }$ , so that ${dx^{\\,0}}/{d\\lambda }=P^{\\,0}$ .",
"Thus, dividing the above equation by $P^{\\,0}$ , and neglecting the fourth term that is of second order, one gets $\\frac{df_s}{d\\eta }=\\frac{\\partial f_s}{\\partial \\eta }+\\frac{\\partial f_s}{\\partial x^{\\,\\mu }}\\frac{P^{\\,\\mu }}{P^{\\,0}}+\\frac{\\partial f_s}{\\partial p}\\frac{\\partial p}{\\partial \\eta }=\\frac{C[f_s]}{P^{\\,0}}\\;,$ where $\\eta $ is the conformal time, in terms of the expansion factor $a(t)$ and cosmic time $t$ .",
"Using ${P^{\\,\\mu }}/{P^{\\,0}}=({p}/{E})\\hat{p}^{\\,\\mu }$ (where $E$ is the energy) and geodesic equation, we may write the above equation as follows: $\\frac{dp}{d\\eta }=-{\\mathcal {H}}p+E\\hat{p}^{\\,\\mu }\\partial _{\\,\\mu }\\frac{h_{00}}{2}-\\frac{p}{2}\\frac{dh_{\\,\\mu \\nu }}{d\\eta }\\hat{p}^{\\,\\mu }\\hat{p}^{\\,\\nu }\\;,$ where $h_{\\mu \\nu }$ are the perturbed metric coefficients.",
"Then substituting Equation (40) into Equation (39), we get $\\frac{df_s}{d\\eta }+\\frac{p\\hat{p}^{\\,\\mu }}{E}\\frac{\\partial f_s}{\\partial x^{\\,\\mu }}+p\\bigg (-{\\mathcal {H}}+\\frac{E}{p}\\hat{p}^{\\,\\mu }\\partial _{\\,\\mu }\\frac{h_{00}}{2}-\\frac{1}{2}h_{\\,\\mu \\nu }^{^{\\prime }}\\hat{p}^\\mu \\hat{p}^\\nu \\bigg )\\frac{\\partial f_s}{\\partial p}=\\frac{a}{E}(1-\\Phi )C[f_s]\\;.$ In arriving at this equation, we have substituted $P_0=\\frac{E}{a}(1+\\Phi )$ , in terms of the perturbed gravitational potential, $\\Phi $ .",
"Next, we partition the distribution function into its unperturbed, $\\bar{f}_s$ , and perturbed, ${\\mathcal {F}}_s$ , components: $f_s(\\eta , x^i, p, \\hat{p}^i)=\\bar{f}_s(\\eta , x^i, p, \\hat{p}^i)+{\\mathcal {F}}_s(\\eta , x^i, p, \\hat{p}^i)\\;.$ Thus, multiplying Equation (41) by the energy and integrating over momentum space, and collecting the zeroth-order terms, one gets $\\int \\frac{d^3p}{(2\\pi )^3}E(p)\\frac{d\\bar{f}_s}{d\\eta }-\\int \\frac{d^3p}{(2\\pi )^3}{\\mathcal {H}}pE(p)\\frac{\\partial \\bar{f}_s}{\\partial p}=\\int \\frac{d^3p}{(2\\pi )^3}aC[f_s]\\;.$ In the context of $\\Lambda $ CDM, the quantity $C[f_s]$ is zero because the particle number is conserved during the phase of fluctuation growth.",
"Therefore, integrating the second term on the left-hand side by parts and neglecting the boundary term, we arrive at the expression $\\int \\frac{d^3p}{(2\\pi )^3}E(p)\\frac{d\\bar{f}_s}{d\\eta }+3{\\mathcal {H}}\\int \\frac{d^3p}{(2\\pi )^3}\\bigg (E+\\frac{p^2}{3E}\\bigg )\\,\\bar{f}_s=0\\;.$ This equation may be reduced further by using the following definitions of density and pressure: $\\rho _s=\\int \\frac{d^3p}{(2\\pi )^3}E(p)\\bar{f}_s\\;,$ and ${\\mathcal {P}}_s=\\int \\frac{d^3p}{(2\\pi )^3}\\frac{p^2}{3E}\\bar{f}_s\\;.$ If we now apply Equations (44-46) to the dark matter fluctuation, we arrive at the expression $\\frac{d\\rho _{\\rm dm}}{d\\eta }+3{\\mathcal {H}}(\\rho _{\\rm dm}+{\\mathcal {P}}_{\\rm dm})= 0\\;.$ We multiply Equation (41) by $E(p)$ and again integrate over momentum space using the definition of $\\rho $ and $\\mathcal {P}$ , but now collecting the first order terms: $\\frac{d(\\delta \\rho _{\\rm dm})}{d\\eta }+(\\rho _{\\rm dm}+{\\mathcal {P}}_{\\rm dm})\\partial _{\\,\\mu }v^{\\,\\mu }_{\\rm dm}+3{\\mathcal {H}}(\\delta \\rho _{\\rm dm}+\\delta {\\mathcal {P}}_{\\rm dm})+3(\\rho _{\\rm dm}+ {\\mathcal {P}}_{\\rm dm})\\frac{d\\Phi }{d\\eta }=0\\;.$ We define $\\delta _{\\rm dm}\\equiv {\\delta \\rho _{\\rm dm}}/{\\rho _{\\rm dm}}$ and set both ${\\mathcal {P}}_{\\rm dm}$ and $\\delta {\\mathcal {P}}_{\\rm dm}$ equal to zero.",
"Then, $\\frac{d\\delta _{\\rm dm}}{d\\eta }=\\frac{1}{\\rho _{\\rm dm}}\\frac{d(\\delta \\rho _{\\rm dm})}{d\\eta }-\\frac{\\delta _{\\rm dm}}{\\rho _{\\rm dm}}\\frac{d\\rho _{\\rm dm}}{d\\eta }\\;.$ Substituting this expression into Equation (48) and decomposing them into Fourier modes, we find that $\\frac{d\\delta _{{\\rm dm},\\,k}}{d\\eta } =-ku_{dm,k} -3\\frac{d\\Phi _k}{d\\eta }\\;.$ In arriving at this equation, we have also used $\\partial _{\\,\\mu }v^{\\,\\mu }_{k}=ku_{dm,k}$ , where $u_{dm,k}$ is the velocity perturbation of dark matter.",
"Now, taking the second moment of Equation (41) by multiplying it with $p\\hat{p}^{\\,\\mu }$ and contracting with $i\\hat{k}_{\\,\\mu }$ , and then integrating it over momentum, we get $\\frac{d(\\rho _{\\rm dm}u_{{\\rm dm},\\,k})}{d\\eta }+4{\\mathcal {H}}\\rho _{\\rm dm}u_{{\\rm dm},\\,k}+k\\rho _{\\rm dm}\\Phi _k=0\\;,$ where $u_{{\\rm dm},\\,k}$ is the $k^{\\rm th}$ velocity perturbation of dark matter.",
"Again substituting for ${d\\rho _{\\rm dm}}/{d\\eta }$ in this equation, we get $\\frac{du_{{\\rm dm},\\,k}}{d\\eta }=-\\frac{1}{a}\\frac{da}{d\\eta }u_{{\\rm dm},\\,k}-k\\Phi _k\\;.$"
],
[
"The Perturbed Boltzmann Equations in $R_{\\rm h}=ct$",
"The early $R_{\\rm h}=ct$ universe consisted of $\\approx 80\\%$ dark energy and $\\approx 20\\%$ radiation, with a small contamination of matter [101].",
"At late times, these fractions change to $\\approx 67\\%$ dark energy and $33\\%$ matter.",
"A coupling between dark matter and dark energy is therefore unavoidable in this model, since particle number is not conserved.",
"The Boltzmann equations in this model will therefore be considerably different from those in $\\Lambda $ CDM.",
"The pertinent equations were derived earlier in [60], but we summarize the procedure here for consistency and convenience.",
"Starting with Equation (41), an integration over momentum space yields $\\int \\frac{d^3p}{(2\\pi )^3}E(p)\\frac{d\\bar{f}_s}{d\\eta }+3{\\mathcal {H}}\\int \\frac{d^3p}{(2\\pi )^3}\\bigg (E+\\frac{p^2}{3E}\\bigg )\\bar{f}_s =\\int \\frac{d^3p}{(2\\pi )^3}aC[f_s]\\;,$ where the principal difference between this result and Equation (44) is that $C[f_s]$ on the righthand side is zero for $\\Lambda $ CDM, but not for $R_{\\rm h}=ct$ .",
"The interaction term cannot be zero in the latter due to the coupling between dark energy and dark matter.",
"Then, using the definition of density and pressure for dark matter (from Equations 45 and 46), one gets $\\frac{d\\rho _{\\rm dm}}{d\\eta }+3{\\mathcal {H}}\\rho _{\\rm dm}=\\int \\frac{d^3p}{(2\\pi )^3}aC[f_{\\rm dm}]\\;.$ The aforementioned transition in densities, e.g., $\\rho _{\\rm de}\\approx 0.8\\rho _{\\rm c}$ after the Big Bang to $\\approx 2\\rho _{\\rm c}/3$ more recently, and analogously for radiation and matter, may be modeled simply using a straightforward empirical expression, $\\rho _{\\rm dm}=({\\rho _{\\rm c}}/{3a^2}) \\exp \\bigg [-\\frac{a_*}{a}\\frac{(1-a)}{(1-a_*)}\\bigg ]\\;,$ to reflect the fact that, in $R_{\\rm h}=ct$ , the Universe transitioned from a radiation-dark-energy dominated early phase to a matter/dark-energy dominated one at late times.",
"In the above expression, $\\rho _{\\rm c}$ is the critical density today, and $a_*$ represents the scale factor at matter radiation equality.",
"Using this expression, one may easily evaluate the collision/source term, $C[\\bar{f}_{\\rm dm}]=\\frac{{\\mathcal {H}}E}{a}\\bar{f}_{\\rm dm}+\\frac{{\\mathcal {H}}E}{a^2}\\bar{f}_{\\rm dm}\\bigg (\\frac{a_*}{1-a_*}\\bigg )\\;,$ which describes the interaction between dark energy and dark matter.",
"A fraction of dark energy must be transformed (possibly via decay) into dark matter.",
"Thus, the corresponding collision term for dark energy is $C[\\bar{f}_{\\rm de}]=-\\frac{{\\mathcal {H}}E}{a}\\bar{f}_{\\rm dm}-\\frac{{\\mathcal {H}}E}{a^2}\\bar{f}_{\\rm dm}\\bigg (\\frac{a_*}{1-a_*}\\bigg )\\;.$ Returning now to Equation (41), with the use of Equation (56), we find for dark matter that $\\frac{df_{\\rm dm}}{d\\eta }&\\hspace{-21.68121pt}+\\hspace{-21.68121pt}&\\frac{p\\hat{p}^{\\,\\mu }}{E}\\frac{\\partial f_{\\rm dm}}{\\partial x^{\\,\\mu }}+p\\left(-{\\mathcal {H}}+\\frac{E}{p}\\hat{p}^{\\,\\mu }\\partial _{\\,\\mu }\\frac{h_{00}}{2}-\\frac{1}{2}h_{\\mu \\nu }^{^{\\prime }}\\hat{p}^{\\,\\mu }\\hat{p}^{\\,\\nu }\\right) \\frac{\\partial f_{\\rm dm}}{\\partial p}=\\nonumber \\\\&\\hbox{}&\\left[{\\mathcal {H}}f_{\\rm dm}+\\frac{{\\mathcal {H}}}{a}f_{\\rm dm} \\left(\\frac{a_*}{1-a_*}\\right)\\right](1-\\Phi )\\;.$ We take the first moment of this equation by multiplying it with $E(p)$ and integrating over momentum space.",
"Collecting the first order terms yields $\\frac{d(\\delta \\rho _{\\rm dm})}{d\\eta }&\\hspace{-21.68121pt}+\\hspace{-21.68121pt}&(\\rho _{\\rm dm}+{\\mathcal {P}}_{\\rm dm})\\partial _{\\,\\mu }v^{\\,\\mu }_{\\rm dm}+3{\\mathcal {H}}(\\delta \\rho _{\\rm dm}+\\delta {\\mathcal {P}}_{\\rm dm})+\\nonumber \\\\&\\hbox{}&\\hspace{-36.135pt}3(\\rho _{\\rm dm}+{\\mathcal {P}}_{\\rm dm})\\frac{d\\Phi }{d\\eta }=\\left[{\\mathcal {H}}+\\frac{{\\mathcal {H}}}{a}\\left(\\frac{a_*}{1-a_*}\\right)\\right](\\delta \\rho _{\\rm dm}-\\rho _{\\rm dm}\\Phi )\\;.$ We also have, from the definition of $\\delta _{\\rm dm}$ and ${\\mathcal {P}}_{\\rm dm}=\\delta {\\mathcal {P}}_{\\rm dm}=0$ for dark matter, that $\\frac{d\\delta _{\\rm dm}}{d\\eta }=\\frac{1}{\\rho _{\\rm dm}}\\frac{d(\\delta \\rho _{\\rm dm})}{d\\eta }-\\frac{\\delta _{\\rm dm}}{\\rho _{\\rm dm}}\\frac{d\\rho _{\\rm dm}}{d\\eta }\\;.$ And substituting for $d(\\delta \\rho _{\\rm dm})/d\\eta $ in this equation, and isolating the Fourier mode $k$ , we find that $\\frac{d\\delta _{{\\rm dm},\\,k}}{d\\eta } =-ku_{dm,k} -3\\frac{d\\Phi _k}{d\\eta }-{\\mathcal {H}}\\bigg [1+\\frac{a_*}{a(1-a_*)}\\bigg ]\\Phi _k\\;.$ Then, taking the second moment of Equation (58) by multiplying it with $p\\hat{p}^{\\,\\mu }$ , contracting it with $i\\hat{k}_{\\,\\mu }$ , and integrating over momentum space, we find $\\frac{d(\\rho _{\\rm dm}u_{{\\rm dm},\\,k})}{d\\eta }+4{\\mathcal {H}}\\rho _{\\rm dm}u_{{\\rm dm},\\,k}+k\\Phi _k\\rho _{\\rm dm}={\\mathcal {H}}\\bigg [1+\\frac{a_*}{a(1-a_*)}\\bigg ]\\rho _{\\rm dm}u_{{\\rm dm},\\,k}\\;.$ Substituting for ${d\\rho _{\\rm dm}}/{d\\eta }$ we thus get $\\frac{du_{{\\rm dm},\\,k}}{d\\eta }=-\\frac{1}{a}\\frac{da}{d\\eta }u_{{\\rm dm},\\,k}-k\\Phi _k\\;.$ Figure: Numerical solution of the perturbed potential, Φ\\Phi , in Λ\\Lambda CDM, for modesk=10 -4 k=10^{-4}, 10 -2 10^{-2} and 10 -1 10^{-1} Mpc -1 ^{-1}.A similar procedure allows to derive the analogous equation for dark energy.",
"Using the interaction term in Equation (57), we may write $\\frac{df_{\\rm de}}{d\\eta }&\\hspace{-21.68121pt}+\\hspace{-21.68121pt}&\\frac{p\\hat{p}^{\\,\\mu }}{E}\\frac{\\partial f_{\\rm de}}{\\partial x^{\\,\\mu }}+p\\left(-{\\mathcal {H}}+\\frac{E}{p}\\hat{p}^{\\,\\mu }\\partial _{\\,\\mu }\\frac{h_{00}}{2}-\\frac{1}{2}h_{\\,\\mu \\nu }^{^{\\prime }} \\hat{p}^{\\,\\mu }\\hat{p}^{\\,\\nu }\\right)\\frac{\\partial f_{\\rm de}}{\\partial p}=\\nonumber \\\\&\\hbox{}&-{\\mathcal {H}}(1-\\Phi )\\left[1+\\frac{a_*}{a(1-a_*)}\\right]f_{\\rm dm}\\;,$ where $f_{\\rm de}$ is the distribution function for dark energy.",
"Then, partitioning $f_{\\rm de}$ into its unperturbed ($\\bar{f_{de}}$ ) and perturbed (${\\mathcal {F}}_{de}$ ) components, we find $\\frac{d{\\mathcal {F}}_{\\rm de}}{d\\eta }&\\hspace{-21.68121pt}+\\hspace{-21.68121pt}&\\frac{p\\hat{p}^{\\,\\mu }}{E}\\frac{\\partial {\\mathcal {F}}_{\\rm de}}{\\partial x^{\\,\\mu }}-{\\mathcal {H}}p\\frac{\\partial {\\mathcal {F}}_{\\rm de}}{\\partial p}+p\\left(\\frac{E}{p}\\hat{p}^{\\,\\mu }\\partial _{\\,\\mu }\\frac{h_{00}}{2}-\\frac{1}{2}h_{\\,\\mu \\nu }^{^{\\prime }}\\hat{p}^{\\,\\mu } \\hat{p}^{\\,\\nu }\\right)\\frac{\\partial \\bar{f}_{\\rm de}}{\\partial p}=\\nonumber \\\\&\\hbox{}&-{\\mathcal {H}}\\left[1+\\frac{a_*}{a(1-a_*)}\\right]\\left({\\mathcal {F}}_{\\rm dm}-\\Phi \\bar{f}_{\\rm dm}\\right)\\;.$ Taking the first momentum of this equation and isolating them into Fourier modes gives $\\frac{d \\delta \\rho _{\\rm de}}{d\\eta }&\\hspace{-28.90755pt}+\\hspace{-28.90755pt}&\\rho _{\\rm de}(1+w_{\\rm de})\\partial _{\\mu }v^{\\ \\mu }_{de}+3{\\mathcal {H}}\\delta \\rho _{\\rm de}\\left(1+\\frac{\\delta {\\mathcal {P}}_{\\rm de}}{\\delta \\rho _{\\rm de}}\\right)+\\nonumber \\\\&\\hbox{}&\\hspace{-36.135pt}3\\frac{d\\Phi }{d\\eta }\\rho _{\\rm de}(1+w_{\\rm de})=-{\\mathcal {H}}\\left[1+\\frac{a_*}{a(1-a_*)}\\right](-\\rho _{\\rm dm}\\Phi +\\delta \\rho _{\\rm dm})\\;$ From the definition of $\\delta _{\\rm de}={\\delta \\rho _{\\rm de}}/{\\rho _{\\rm de}}$ and the inferred equation-of-state ${\\mathcal {P}}_{\\rm de}=-\\rho _{\\rm de}/2$ [77] for dark energy, we may write $\\frac{d\\delta _{\\rm de}}{d\\eta }=\\frac{1}{\\rho _{\\rm de}}\\frac{d \\delta \\rho _{\\rm de}}{d\\eta }-\\frac{\\delta \\rho _{\\rm de}}{\\rho _{\\rm de}^2}\\frac{d\\rho _{\\rm de}}{d\\eta }$ so that, combining Equations (66) and (67), we get $\\frac{d\\delta _{\\rm de}}{d\\eta }&\\hspace{-21.68121pt}=\\hspace{-21.68121pt}&-\\frac{1}{2}\\partial _{\\mu }v^{\\ \\mu }_{de}-3{\\mathcal {H}}\\delta _{\\rm de}\\left(\\frac{1}{2}+\\frac{\\delta {\\mathcal {P}}_{\\rm de}}{\\delta \\rho _{\\rm de}}\\right)-\\frac{3}{2}\\frac{d\\Phi }{d\\eta }+\\nonumber \\\\&\\hbox{}&{\\mathcal {H}}\\left[1+\\frac{a_*}{a(1-a_*)}\\right]\\frac{\\rho _{\\rm dm}}{\\rho _{\\rm de}}(\\delta _{\\rm de}-\\delta _{\\rm dm}+\\Phi )\\;.$ We do not yet know the sound speed for the coupled dark matter/dark energy fluid, so we characterize it as follows: $c_s^2\\equiv \\frac{\\delta {\\mathcal {P}}}{\\delta \\rho }=\\frac{\\delta {\\mathcal {P}}_{\\rm de}}{\\delta \\rho _{\\rm dm}+\\delta \\rho _{\\rm de}}=\\frac{\\delta {\\mathcal {P}}_{\\rm de}/\\delta \\rho _{\\rm de}}{(1+\\delta \\rho _{\\rm dm}/\\delta \\rho _{\\rm de})}$ and, assuming adiabatic fluctuations, we shall put $\\frac{\\delta {\\mathcal {P}}_{\\rm de}}{\\delta \\rho _{\\rm de}}=c_s^2\\bigg [1+\\frac{2\\rho _{\\rm dm}}{\\rho _{\\rm de}}\\bigg ]\\;.$ For the sake of simplicity, we assume the sound speed to be a constant limited to the range $0<(c_s/c)^2<1$ .",
"We have found that the actual value of this constant has a negligible impact on the solutions to the above equations since the ratio of dark matter density to dark energy is always much smaller than 1 in the $R_{\\rm h}=ct$ universe, and we therefore adopt the simple fraction $c_s^2=c^2/2$ throughout this work.",
"Thus, using Equations (68) and (70), we get $\\frac{d\\delta _{\\rm de,k}}{d\\eta } &\\hspace{-21.68121pt}=\\hspace{-21.68121pt}& -\\frac{k}{2}u_{{\\rm de},\\,k}-\\delta _{\\rm de,k}\\left(\\frac{3{\\mathcal {H}}}{2}+3{\\mathcal {H}}c_s^2+\\frac{6{\\mathcal {H}}c_s^2\\rho _{\\rm dm}}{\\rho _{\\rm de}}\\right)-\\frac{3}{2}\\frac{d\\Phi _k}{d\\eta }+\\nonumber \\\\&\\hbox{}&{\\mathcal {H}}\\left[1+\\frac{a_*}{a(1-a_*)}\\right]\\frac{\\rho _{\\rm dm}}{\\rho _{\\rm de}}(\\delta _{\\rm de,k}-\\delta _{\\rm dm,k}+\\Phi _k)\\;.$ Finally, taking the second moment of Equation (65), we find that $\\frac{du_{{\\rm de},\\,k}}{d\\eta }&\\hspace{-21.68121pt}=\\hspace{-21.68121pt}&-\\frac{5{\\mathcal {H}}}{2}u_{{\\rm de},\\,k}-k\\Phi _k+2kc_s^2\\left[1+\\frac{2\\rho _{\\rm dm}}{\\rho _{\\rm de}}\\right]\\delta _{\\rm de,k}+\\nonumber \\\\&\\hbox{}&{\\mathcal {H}}\\left[1+\\frac{a_*}{a(1-a_*)}\\right]\\frac{\\rho _{\\rm dm}}{\\rho _{\\rm de}}(u_{{\\rm de},\\,k}-2u_{{\\rm dm},\\,k})\\;.$"
],
[
"The Observed Matter Power Spectrum",
"The matter power spectrum is obtained using several steps and a combination of data from the CMB, and the Ly-$\\alpha $ forest, together with a model-dependent transfer function, that must be calculated individually for each cosmology.",
"For the sake of clarity, we begin by summarizing these key factors in the process.",
"Figure: Numerical solution of the matter fluctuation δ dm ,k \\delta _{{\\rm dm},\\,k} inΛ\\Lambda CDM, for modes k=10 -4 k=10^{-4}, 10 -2 10^{-2} and 10 -1 10^{-1} Mpc -1 ^{-1}."
],
[
"The Cosmic Microwave Background",
"First and foremost, all of the data are model dependent, and must be recalibrated when changing background cosmologies.",
"For the concordance model parameter values, the CMB measurements are shown as blue, orange and black circular dots in Figure 3.",
"The actual CMB observations are converted into a power spectrum using the approach of Tegmark & Zaldarriaga (see ref.",
"[113]).",
"The CMB angular power for multipole $\\ell $ may be written $C_\\ell =\\int _{-\\infty }^{\\infty }W_\\ell (k)P_*(k)\\,d\\ln k\\;,$ where $W_\\ell (k)$ is the transfer function that depends on the cosmic matter budget and the reionisation optical depth, and $P_*(k)$ is the primordial power spectrum, assumed for simplicity, to be $P_*(k)\\propto k$ .",
"These CMB measurements are mapped into the $k$ -space of the matter power spectrum using the approach of Tegmark & Zaldarriaga (see ref.",
"[113]), the measured CMB data points $d_i$ (in the case of CMB they are $C_\\ell $ ) are mapped into the probability distribution using the following equation $d_i=\\int _{-\\infty }^{\\infty }\\mathcal {P}_i(k)d\\ln k\\;,$ Figure: The matter power spectrum observed in the CMB (orange, blue and black dots), and theLy-α\\alpha survey (red triangles), compared with the power spectrum predicted byΛ\\Lambda CDM (solid black curve), where hh is the parameter H 0 /(100 km s -1 Mpc -1 )H_0/(100\\;{\\rm km}\\;{\\rm s}^{-1}\\;{\\rm Mpc}^{-1}).",
"Throughout this work, we assume the value h=0.6732h=0.6732, consistent with thelatest Planck measurements (Planck Collaboration 2016).where $\\mathcal {P}_i$ is the probability distribution (see ref.",
"[113]), in Figure 3, the CMB data points are placed horizontally at the $k$ -value corresponding to the median of this distribution, with a horizontal error bar ranging from the 20th to the 80th percentile of this distribution, representing a 1 $\\sigma $ range.",
"They are then plotted at a vertical position given by $P_{\\Lambda {\\rm CDM}}(k)\\simeq T^2_{\\Lambda {\\rm CDM}}(k_{\\rm eff})P_*(k)\\;,$ where $k_{\\rm eff}$ is the horizontal location of the median of the window function and $T_{\\Lambda {\\rm CDM}}(k)$ is the matter transfer function in $\\Lambda $ CDM.",
"In order to use these data for the $R_{\\rm h}=ct$ universe, they must be recalibrated using the appropriate matter transfer function, according to the expression $P_{R_{\\rm h}=ct}(k)=P_{\\Lambda {\\rm CDM}}(k)\\frac{T^2_{R_{\\rm h}=ct}(k)}{T^2_{\\Lambda {\\rm CDM}}(k)}\\;.$ The CMB data points recalibrated for $R_{\\rm h}=ct$ are shown (also as blue, orange and black circular dots) in Figure 6."
],
[
"The Ly-$\\alpha $ Forest",
"The Ly-$\\alpha $ forest (red triangles in Figs.",
"3 and 6) is produced by absorption seen in high redshift quasar spectra, associated with neutral hydrogen present in a continuously fluctuating photo-ionized intergalactic medium.",
"Simulations show that the underlying mass density is related to the optical depth of the Ly-$\\alpha $ absorption, which allows the Ly-$\\alpha $ forest to be used as a proxy for the matter power spectrum.",
"The gas traced by Ly-$\\alpha $ is modestly overdense compared to the cosmic mean, so if all the relevant physics can be simulated, one may construct the underlying matter power spectrum from the observed spectrum.",
"The TreeSPH hydrodynamical simulation that simulates the observed Ly-$\\alpha $ forest indicates that the optical depth $\\tau $ is proportional to the density of neutral hydrogen, according to the expression $\\tau (x)=A\\rho _b(x)^\\beta \\;,$ where $x$ is the line of sight distance towards the quasar and $A$ is the amplitude that depends on the cosmology and physical state of the gas.",
"This amplitude $A$ is obtained by matching the simulated and observed Ly-$\\alpha $ forests.",
"But given that the simulation depends on the assumed background cosmology, the value of $A$ is itself model dependent.",
"There are several caveats in obtaining the matter power spectrum from these data, however.",
"In particular, the hydro-simulations with a dark matter only prescription may not have included all of the relevant physics, and it is not clear how the uncertainties in the reionisation history, the ionizing background and its fluctuations propagate into the reconstruction of $P(k)$ .",
"These data may therefore not be as reliable as the others for model selection purposes.",
"Nevertheless, we here attempt to overcome this model dependence as much as possible.",
"The model dependence of the amplitude in the power spectrum arises from the process of matching the observed and simulated spectra.",
"Thus, in order to properly compare the data with the predicted matter power spectrum in $R_{\\rm h}=ct$ , one must carry out similar simulations with this model as the background cosmology.",
"These simulations have not been performed yet, however, so we shall rely on a statistical argument to calibrate the matter power spectrum obtained from the Ly-$\\alpha $ forest.",
"The power spectrum obtained from the CMB and the Ly-$\\alpha $ forest belong to the same sample, so one can use the $t$ -test to examine whether the matter power spectrum obtained from the Ly-$\\alpha $ forest through simulations is consistent with that obtained from the CMB.",
"One may then also use the $t$ -test to calibrate the matter power spectrum obtained from Ly-$\\alpha $ by ensuring consistency with the spectrum obtained from CMB.",
"The matter power spectrum from the Ly-$\\alpha $ forest may be calibrated using a two-step process.",
"First, we optimize the polynomial function $f(k)= a_1k^{-1}+b_1k^{-2}+c_1k^{-3}+D_1$ , where $a_1$ , $b_1$ , $c_1$ and $D_1$ are constants, by minimizing the $\\chi ^2$ in fitting the matter power spectrum from CMB.",
"A second optimization procedure is followed using a polynomial $g(k)=a_2k^{-1}+b_2k^{-2}+c_2k^{-3}+D_{2}$ to fit the matter power spectrum from Ly-$\\alpha $ , yielding the constants $a_2$ , $b_2$ , $c_2$ and $D_2$ via $\\chi ^2$ minimization.",
"A relative calibration between these two is obtained by varying the normalization constant $D_2$ until the p-value obtained from the $t$ -test lies above the $95\\%$ confidence level.",
"Using this approach, we find that the p-values for $\\Lambda $ CDM and $R_{\\rm h}=ct$ are, respectively, $\\approx 99.8\\%$ and $\\approx 98.7\\%$ ."
],
[
"The Matter Power Spectrum in $\\Lambda $ CDM",
"In $\\Lambda $ CDM, dark energy is a cosmological constant, contributing to the smooth background but not the fluctuations.",
"Thus, $\\delta \\rho $ in Equation (37) is comprised primarily of matter, though radiation ($\\delta \\rho _{\\gamma }$ ) may contribute as well.",
"But in this paper, we follow convention and consider only dark matter growth, so the radiation does not condense into the fluctuations.",
"Like dark energy, it contributes only to the smooth background.",
"The equations describing the growth of dark matter fluctuations in $\\Lambda $ CDM may therefore be written (see Equations 37, 50 and 52) $\\frac{d\\Phi _k}{d\\eta }=-\\bigg (1+\\frac{k^2}{3{\\mathcal {H}}^2}\\bigg ){\\mathcal {H}}\\Phi _k+\\frac{4\\pi G a^2 \\rho _{\\rm m}}{3{\\mathcal {H}}}\\delta _{\\rm m}\\;,$ $\\frac{d\\delta _{{\\rm dm},\\,k}}{d\\eta } =-ku_k -3\\frac{d\\Phi _k}{d\\eta }\\;,$ and $\\frac{du_{{\\rm dm},\\,k}}{d\\eta }=-\\frac{1}{a}\\frac{da}{d\\eta }u_{{\\rm dm},\\,k}-k\\Phi _k\\;.$ Equation (78) describes the evolution of the gravitational potential.",
"The second term on the right-hand side in this expression represents the self-gravity driving the fluctuation growth, while the first is due to the so-called Hubble friction that suppresses growth.",
"Equations (79) and (80) describe the evolution of the matter fluctuations and their velocity.",
"We solve these coupled first order differential equations starting with $\\delta _{m}=\\frac{3}{2}\\Phi $ and $u=\\frac{1}{2}k\\eta \\Phi $ , following the initial conditions used in the Cosmological Initial Conditions and Microwave Anisotropy Codes (COSMICS; see ref.",
"[114]).",
"As noted earlier, the expansion history in $\\Lambda $ CDM includes an exponential acceleration that drives the modes across the Hubble horizon.",
"They then freeze and later re-enter as the Universe continues to expand.",
"The small-scale modes re-enter during the radiation-dominated phase, while the large-scale modes re-enter when matter dominates.",
"The modes re-enter across the Hubble horizon at different times, so the starting time when classical growth begins depends on their proper wavelength (or, equivalently, on their comoving wavenumber $k$ ).",
"Hence we choose the initial (conformal) time at the beginning of the fluctuation growth to be $\\eta ={\\rm min}[10^{-3}k^{-1},10^{-1}h^{-1}Mpc]$ (COSMICS; see ref.",
"[114]).",
"The solution to these equations is plotted in Figures 1 and 2.",
"Figure 1 shows the evolution of the gravitational potential $\\Phi _k$ , consistent with (1) that all the modes are frozen outside the horizon, (2) that the small-scale modes that re-enter the horizon during the radiation-domination expansion decay at first, and grow only when matter starts to dominate, and (3) that the large-scale modes remain frozen outside the horizon during the radiation-dominated phase, and re-enter when matter dominates and grow continuously thereafter.",
"Using the transfer function defined as $T(k,a)=\\Phi _k(a)/\\Phi _k(a_i)$ (see ref.",
"[115]), we then compute the matter power spectrum $P(k)=AP_*(k)T^2(k,a)$ , in terms of the primordial power spectrum $P_*(k)$ generated by inflation, and the normalization factor $A$ .",
"The computed matter power spectrum is shown in Figure 3, along with the power spectrum observed in the CMB, and Ly-$\\alpha $ .",
"As noted earlier, the small-scale modes, $k\\gtrsim 0.02$ Mpc$^{-1}$ , re-enter the horizon during the radiation-dominated expansion and then decay, as one may see to the right in Figure 3.",
"The large-scale modes, $k\\lesssim 0.02$ Mpc$^{-1}$ , re-enter the horizon when matter is dominant and then grow.",
"To appreciate the behavior to the right of the peak in this plot, consider two typical modes $k_1$ and $k_2$ with $k_1>k_2$ .",
"The first mode has a shorter wavelength and therefore re-enters before the second one.",
"Mode $k_1$ thus suffers a greater decay due to the large Hubble friction.",
"Hence mode $k_1$ has less power than mode $k_2$ .",
"Starting from the small modes and progressing towards the larger ones, the power increases until $k\\approx 0.02$ Mpc$^{-1}$ , which corresponds to the mode that re-enters the Hubble horizon exactly at matter-radiation equality.",
"Modes with $k\\lesssim 0.02$ Mpc$^{-1}$ don't decay and instead grow during the matter-dominant era.",
"In principle, these modes should all have the same power if the primordial power spectrum is completely scale invariant, but Figure 3 indicates that the power declines as we progress towards even larger mode wavelengths, to the left of $k\\approx 0.02$ Mpc$^{-1}$ .",
"This happens because the primordial power spectrum $P_*=Ak^{n_s}$ is not exactly scale invariant; it has a scalar spectral index $n_s=0.967$ .",
"Clearly, the turning point in the power spectrum is highly sensitive to the energy content in the form of matter and radiation.",
"Any adjustment to these quantities will alter the location of the matter-radiation equality and therefore change the shape of the matter power spectrum."
],
[
"A Possible Failure of the Mechanism Generating the Matter Power Spectrum",
"The exit and re-entry of modes across the Hubble horizon are paramount for generating the observed matter power spectrum in $\\Lambda $ CDM.",
"This mechanism, however, relies on the existence of an inflationary epoch.",
"Should inflation eventually be disfavoured by the observations, $\\Lambda $ CDM would be unable to account for the observed matter power spectrum.",
"It is worth mentioning in this regard that the angular correlation function measured by the Cosmic Background Explorer (COBE)[116], [117], Wilkinson Microwave Anisotropy Probe (WMAP)[7], [8]) and particularly Planck (Planck Collaboration 2016)[70], have indicated a rather strong tension with the prediction of $\\Lambda $ CDM.",
"Slow-roll inflation predicts significant large-angle correlations in the CMB fluctuations which, however, are not seen by these instruments.",
"Large-angle correlations are disfavoured by the data at a confidence level exceeding $8\\sigma $ [118].",
"Cosmic variance could mitigate this problem only partially, so a resolution to this flaw in the model is still elusive.",
"Should this failure persist, it would herald, not only an internal inconsistency of $\\Lambda $ CDM, but also its inability to account for the observed matter power spectrum.",
"Figure: Numerical solution of the perturbed potential Φ k \\Phi _k in R h =ctR_{\\rm h}=ct, for modesk=0.0001k=0.0001, 0.0030.003 and 0.0010.001 Mpc -1 ^{-1}."
],
[
"The Matter Power Spectrum in $R_{\\rm h}=ct$",
"The fluctuations in $R_{\\rm h}=ct$ involve coupled species.",
"The density $\\delta \\rho $ in Equation (37) includes both dark energy and dark matter, i.e., $\\delta \\rho =\\delta \\rho _{\\rm dm}+\\delta \\rho _{\\rm de}$ .",
"Radiation and baryons are not directly coupled to dark matter, so we ignore their contribution in this paper.",
"The relevant equations describing the growth of dark-matter fluctuations in $R_{\\rm h}=ct$ may be derived as follows.",
"We convert the $\\eta $ derivatives in the coupled first-order differential Equations (37), (61), (63), (71) and (72) into derivatives with respect to the expansion factor $a$ , and evolve them starting from $a_i=10^{-12}$ to $a=1$ .",
"The evolution of $\\Phi _k$ over this range is shown for several modes in Figure 4.",
"This plot is to be compared with the corresponding potential growth for $\\Lambda $ CDM in Figure 1.",
"The $\\Lambda $ CDM and $R_{\\rm h}=ct$ models differ both in terms of how the perturbations are generated, and how they evolve.",
"The perturbations in $\\Lambda $ CDM are seeded in the inflaton field, and then exit the horizon and freeze out during the quasi-exponential expansion.",
"These frozen modes later re-enter through the horizon as the Hubble radius continues to grow, though at different times depending on the wavelength of the fluctuation.",
"In $R_{\\rm hl}=ct$ , on the other hand, the primordial fluctuations are generated within an incipient scalar field with an equation-of-state $p=-\\rho /3$ .",
"Since this model has no horizon problem [98], it also does not experience any inflated expansion, so none of the modes criss-cross the horizon as the Universe expands.",
"Instead, all of the modes are born at the Planck scale, though at different times corresponding to their wavelength.",
"The primordial power spectrum is produced from this birth mechanism, first proposed in a less well-defined manner by Hollands and Wald [99], [100].",
"The incipient scalar field should not be confused with the dark energy component, however, since the latter emerged after the former decayed into standard model particles and other fields seen in extensions to the standard model of particle physics, e.g., representing dark energy.",
"The incipient scalar field and dark energy have different equations-of-state.",
"As described earlier, modes born during inflation freeze upon exiting the horizon and remain coherent upon re-entry, a mechanism that is resposible for generating a near scale-invariant power spectrum.",
"In the case of the incipient field in $R_{\\rm h}=ct$ , a similar outcome ensues upon exiting the Planck regime, but these fluctuations then remain within the semi-classical Universe and never criss-cross the horizon.",
"They oscillate and grow until the Universe cools down to the GUT scale, and then presumably decay, as described above.",
"The second difference between the two models arises during the evolution of the fluctuations, which may be understood via an inspection of Equation (37).",
"After introducing the model dependence through ${\\mathcal {H}}$ , $\\rho _{\\rm dm}$ and $\\rho _{\\rm de}$ in $R_{\\rm h}=ct$ , this expression may be written $\\frac{d\\Phi _k}{da}=-\\bigg (1+\\frac{k^2c^2}{3H_0^2}\\bigg )\\frac{\\Phi _k}{a}+\\frac{1}{2a}\\bigg (\\varpi (a) \\delta _{\\rm dm,k}+ \\Upsilon (a)\\delta _{\\rm de,k}\\bigg )\\;,$ where $\\varpi (a)=0.3\\exp \\left[-\\frac{a_*}{a}\\frac{(1-a)}{(1-a_*)}\\right]$ and $\\Upsilon (a)=1-\\varpi (a)$ .",
"The corresponding expression in $\\Lambda $ CDM is $\\frac{d\\Phi _k}{da}=-\\bigg (1+\\frac{k^2c^2}{3a^2H^2}\\bigg )\\frac{\\Phi _k}{a}+\\frac{3H_0^2}{2H^2a^2}\\frac{\\Omega _{\\rm m}}{a^3}\\delta _{\\rm dm}\\;,$ Figure: Numerical solution of the perturbed δ dm,k \\delta _{dm,k} in R h =ctR_{\\rm h}=ct for modesk=0.0001k=0.0001, 0.0030.003 and 0.0010.001 Mpc -1 ^{-1}.where $H=H_0\\sqrt{\\Omega _{\\rm m}a^{\\,-3}+\\Omega _{\\rm r}a^{\\,-4}+\\Omega _{\\Lambda }}$ .",
"The Hubble friction term ${k^2c^2}/{H_0^2}$ in $R_{\\rm h}=ct$ may be written as ${4\\pi ^2R_{\\rm h}^2}/{\\lambda ^2}$ , where $R_{\\rm h}$ is the Hubble horizon and $\\lambda $ is the proper wavelength of the mode.",
"The primary difference between Equations (81) and (82) enters via the ratio $R_{\\rm h}/\\lambda $ .",
"This factor is constant throughout the history of the Universe in $R_{\\rm h}=ct$ for each given mode, while it evolves as a function of $a(t)$ in $\\Lambda $ CDM for every mode.",
"Figure: The matter power spectrum observed in the CMB (orange, blue and black dots),and the Ly-α\\alpha survey (red triangles), compared with the power spectrum predicted by R h =ctR_{\\rm h}=ct.The solution for $\\delta _{dm,k}$ as a function of $a(t)$ in $R_{\\rm h}=ct$ is shown in Figure 5, which is to be compared with Figure 2 for $\\Lambda $ CDM.",
"The behavior of these solutions may be understood quite easily because the only `forces' acting on the modes are: (i) the inward pull of gravity, and (ii) the outwardly directed Hubble friction due to the expansion of the Universe.",
"As one can see in Figure 4, the gravitational potential decays more rapidly for the smaller mode wavelengths, i.e., the larger $k$ 's.",
"As a result, the growth rate of the modes varies according to their wavelength—the larger ones have a longer dynamical timescale, and therefore grow more slowly than the smaller ones.",
"Another factor responsible for the difference between $R_{\\rm h}=ct$ and the standard model is that, at early times, the energy density in the former is a blend of dark energy and radiation, comprising $\\sim 80\\%$ and $20\\%$ , respectively.",
"As the Universe expands, some of the dark energy decays into dark matter and standard model particles, eventually reaching a fractional representation of 2/3 dark energy and 1/3 matter (dark plus baryonic).",
"The overall equation-of-state, however, always remains $p = -\\rho /3$ .",
"Thus, both gravity and the pressure always act inwards together, making strucuture formation more efficient and more rapid than in $\\Lambda $ CDM.",
"This is the principal reason why the growth rate in $R_{\\rm h}=ct$ is larger than in $\\Lambda $ CDM, and why it ends up matching the data much better than the latter.",
"For example, this strong growth rate is responsible for the early appearance of supermassive black holes and massive galaxies [60].",
"The dynamical equations show that, as matter forms from dark-energy decay, the small-$k$ modes (i.e., the large-size fluctuations) overcome the Hubble friction due to the expansion and form bound structures.",
"The large-$k$ modes (i.e., the small-size fluctuations), on the other hand, decay.",
"This maybe seen more quantitatively in Fig. 5.",
"Together with the transfer function $T(k,a)$ for $R_{\\rm h}=ct$ , we compute the power spectrum $P(k)$ using $P(k)=k^{n_s}T^2(k,a)$ , where $n_s=0.967$ .",
"The combination of different growth rates in $\\delta _{dm,k}$ and the $k$ -dependent decay rate of the potential, produces the shape for the power spectrum seen in Figure 6.",
"The computed power spectrum $P(k)$ in $R_{\\rm h}=ct$ is shown in Figure 6, together with the data recalibrated for this cosmological model (see § 6 above).",
"Above the peak, the mode decay rate increases with $k$ .",
"As before, let us compare the behaviour of two particular modes, $k_1$ and $k_2$ , with $k_1$ less than $0.001$ Mpc$^{-1}$ and the second mode $k_2$ greater than $k_1$ and $0.001$ Mpc$^{-1}$ .",
"The decay rate of $\\Phi _{k_1}$ is smaller than that of $\\Phi _{k_2}$ and the growth rate of $k_1$ is less than that of $k_2$ , but mode $k_2$ did not have sufficient time to cross over $k_1$ .",
"This may be inferred by comparing Figures 4 and 5.",
"Hence mode $k_1$ acquires more power than $k_2$ .",
"Now consider two modes $k_3$ and $k_4$ , with $k_3$ less than $0.001$ Mpc$^{-1}$ and $k_4$ greater than $k_3$ , though still less than $0.001$ Mpc$^{-1}$ .",
"As $k_3$ is smaller than $k_4$ , $\\Phi _{k_3}$ undergoes less decay than $\\Phi _{k_4}$ , so the growth of $k_3$ is less than that of $k_4$ , because the growth rate of $k_4$ exceeds that of $k_3$ and it had enough time to cross over.",
"It ends up with more power than $k_3$ .",
"These effects all combine to produce the particular shape of the power spectrum seen in Figure 6, particularly the height and location of the peak.",
"Figure: The matter power spectrum in R h =ctR_{\\rm h}=ct calculated at different valuesof the expansion factor, specifically a=10a=10, 1.01.0 and 0.0010.001.",
"Though the shaperemains qualitatively the same, the peak of the distribution shifts slowly towardshigher values of kk as the Universe ages.An inspection of Figure 7 demonstrates another important difference between the matter power spectrum produced in $R_{\\rm h}=ct$ versus its counterpart in $\\Lambda $ CDM.",
"The location of its peak is time dependent.",
"The qualitative shape of this distribution remains unchanged, but its turning point shifts as the Universe ages.",
"The peak shifts slowly towards higher $k$ with time.",
"Its predicted location at $a(t_0)=1$ (i.e., today) matches the data very well.",
"When viewed at $a=10$ (which in $R_{\\rm h}=ct$ corresponds to a Universe ten times older than today), however, the peak will have shifted from its current $k$ by a factor of almost 2.",
"In contrast, the peak's location does not change as the $\\Lambda $ CDM Universe ages.",
"A second important factor to be noted in the case of $R_{\\rm h}=ct$ is that the initial scale factor $a_i$ for evolving the modes is arbitrary in this approach; it does not pertain to any particular epoch in the Universe's early expansion, such as the time at which the modes re-enter the horizon, which does not happen in this model.",
"In reality, the initial value of $a(t)$ should correspond to the epoch at which the modes transitioned from the quantum realm to the classical realm.",
"But the `classicalization' of these quantum fluctuations is still far from being completely understood.",
"We have chosen the value $a_i=10^{-12}$ to ensure that it does in fact lie beyond this transition.",
"Of course, there is a wide range of possible values that could reflect this condition.",
"To examine how sensitive our results are to the choice of $a_i$ , we have therefore repeated the calculation varying the starting point, using $a_i=10^{-14}$ and $a_i=10^{-9}$ .",
"The results of this comparison are summarized in Figure 8.",
"Though the amplitude does change with $a_i$ , the shape is only very minimally dependent on it.",
"But the amplitude is normalized to obtain the measured $\\sigma _8$ today, so this difference cannot be distinguished from one case to another.",
"As one may see from this figure, the predicted result is not at all dependent on the choice of $a_i$ .",
"The principal physical reason behind this is that the transition from dark energy/radiation domination to dark energy/matter domination was optimized for $a_*=10^{-9}$ in this work.",
"The growth of the modes is therefore dominated by what takes place after this point.",
"One may turn this approach around and view the `fitting' of the power spectrum as a way of identifing the transition point $a_*$ .",
"Figure: A comparision of the computed matter power spectrum for different valuesof the starting point a i a_i in R h =ctR_{\\rm h}=ct."
],
[
"Conclusion",
"The mechanism responsible for the matter power spectrum in $R_{\\rm h}=ct$ is quite different from that developed for $\\Lambda $ CDM, primarily because the fluctuations in this cosmology include a coupled dark matter–dark energy fluid.",
"The formation of this spectrum in $R_{\\rm h}=ct$ is quite simple, largely influenced by the strong dependence of the perturbed gravitational potential's ($\\Phi _k$ ) decay rate on the size of the modes: the smaller the scale size, the larger the decay rate.",
"Coupled to the $k$ dependence of the mode growth rate, this trend shapes the matter power spectrum, producing a distribution consistent with the observations.",
"It is important to stress that this mechanism does not require inflation at all, even to generate quantum fluctuations in the first place, since the presence of any scalar field at the Planck scale would suffice [100].",
"In contrast, the formation of the matter power spectrum in $\\Lambda $ CDM requires the modes to exit and re-enter the Hubble horizon at specific times in order to produce the necessary growth and decay rates as functions of $k$ .",
"The physics of inflation is still unknown, however, so this mechanism is arguably more speculative than its counterpart in $R_{\\rm h}=ct$ .",
"If the inflationary paradigm turns out to be correct, then the principal effect responsible for shaping the matter power spectrum is the $k$ dependence of the time at which modes re-enter the horizon, with the smaller ones returning first, during the radiation-dominated epoch, and the larger ones re-entering later, when matter was dominant.",
"The different growth rates during these epochs is the explanation for the peak and shape of the distribution.",
"The fact that such a complicated process is not needed in $R_{\\rm h}=ct$ is comforting to see, particularly since inflation is also not needed to solve the temperature horizon problem, which does not exist in this model.",
"This distinction is important in view of the growing tension between $\\Lambda $ CDM and the data in other areas.",
"For example, with the discovery of the Higgs particle, the standard model is now facing a second major horizon problem, this time having to do with the electroweak phase transition [82].",
"While the inflationary (GUT) scale expansion may fix the temperature horizon problem, it cannot under any known circumstance fix the subsequent electroweak horizon inconsistency.",
"Coupled to this, is the problem with the timeline in $\\Lambda $ CDM, which has significant difficulty accounting for the `too early' appearance of massive halos and supermassive black holes.",
"With its much simpler structure formation process, the $R_{\\rm h}=ct$ universe is able to solve “the impossibly early galaxy problem\" and the too early appearance of high-$z$ quasars.",
"The fact that the cosmic fluid is dominated by dark energy throughout the Universe's history in $R_{\\rm h}=ct$ , comprising $\\approx 80\\%$ of $\\rho $ at early times and $2/3$ of $\\rho $ towards the present, means that a coupling of dark matter and dark energy is unavoidable in this cosmology.",
"But though these fractions are constrained quite tightly, the redshift at which this transition took place, gradual or otherwise, is not known.",
"In this paper, we have optimized the fit to the matter power spectrum by treating the time at which this transition occurred as an adjustable parameter, reflected in the value of $a_*$ in Equation (55).",
"Eventually, higher precision high-$z$ observations will be required to better constrain this process.",
"Gamma-ray bursts (GRB) from Pop III stars [121], [119], [120] may provide such a diagnostic.",
"In particular, the rate of gamma-ray bursts to be observed is strongly dependent on the growth rate, which itself depends on the transition redshift (from dark energy/radiation to dark energy/matter dominated epochs).",
"The observation of GRB's by the James Webb Space Telescope (JWST) may prove to be critical for a proper comparison and evaluation of these two models."
],
[
"Acknowledgments",
"We are grateful to Andreu Font-Ribera, David Weinberg and Arthur Kosowsky for very informative discussions regarding the observed matter power spectrum from Ly-$\\alpha $ and the CMB.",
"We are also thankful to Solèene Chabanier for sharing the matter power spectrum data with us.",
"FM is grateful to the Institutode Astrofisica de Canarias in Tenerife and to Purple Mountain Observatory in Nanjing, China for their hospitality while part of this research was carried out."
]
] | 2011.14053 |
[
[
"Deep Learning for Regularization Prediction in Diffeomorphic Image\n Registration"
],
[
"Abstract This paper presents a predictive model for estimating regularization parameters of diffeomorphic image registration.",
"We introduce a novel framework that automatically determines the parameters controlling the smoothness of diffeomorphic transformations.",
"Our method significantly reduces the effort of parameter tuning, which is time and labor-consuming.",
"To achieve the goal, we develop a predictive model based on deep convolutional neural networks (CNN) that learns the mapping between pairwise images and the regularization parameter of image registration.",
"In contrast to previous methods that estimate such parameters in a high-dimensional image space, our model is built in an efficient bandlimited space with much lower dimensions.",
"We demonstrate the effectiveness of our model on both 2D synthetic data and 3D real brain images.",
"Experimental results show that our model not only predicts appropriate regularization parameters for image registration, but also improving the network training in terms of time and memory efficiency."
],
[
"Introduction",
"Diffeomorphic image registration is a fundamental tool for various medical image analysis tasks, as it provides smooth and invertible smooth mapping (also known as a diffeomorphism) between pairwise images.",
"Examples include atlas-based image segmentation [4], [13], anatomical shape analysis based on geometric changes [35], [49], [18], and motion correction in spatial-temporal image sequences [10], [23], [42].",
"The nice properties of diffeomorphisms keep topological structures of objects intact in images.",
"Artifacts (i.e., tearing, folding, or crossing) that generate biologically meaningless images can be effectively avoided, especially when large deformation occurs.",
"The problem of diffeomorphic image registration is typically formulated as an optimization over transformation fields, such as a free-form deformation using B-splines [28], a LogDemons algorithm based on stationary velocity fields (SVF) [3], and a large diffeomorphic deformation metric mapping (LDDMM) method utilizing time-varying velocity fields [8].",
"To ensure the smoothness of transformation fields, a regularization term defined on the tangent space of diffeomorphisms (called velocity fields) is often introduced in registration models.",
"Having such a regularity with proper model parameters is critical to registration performance because they greatly affect the estimated transformations.",
"Either too large or small-valued regularity can not achieve satisfying registration results (as shown in Fig .REF ).",
"Models of handling the regularity parameter mainly include (i) direct optimizing a Bayesian model or treating it as a latent variable to integrate out via Expectation Maximization (EM) algorithm [2], [1], [48], [38], (ii) exhaustive search in the parameter space [20], [36], [29], and (iii) utilizing parameter continuation methods [15], [14], [25], [26].",
"Direct optimization approaches define a posterior of transformation fields that includes an image matching term as a likelihood and a regularization as a prior to support the smoothness of transformations [51], [1], [34].",
"Estimating regularization parameters of these models using direct optimization is not straightforward due to the complex structure of the posterior distribution.",
"Simpson et al.",
"infer the level of regularization in small deformation registration model by mean-field VB inference [21], which allows tractable approximation of full Bayesian inference in a hierarchical probabilistic model [32], [33].",
"However, these aforementioned algorithms are heavily dependent on initializations, and are prone to getting stuck in the local minima of high-dimensional and non-linear functions in the transformation space.",
"A stochastic approximative expectation maximization (SAEM) algorithm [1] was developed to marginalize over the posterior distribution of unknown parameters using a Markov Chain Monte Carlo (MCMC) sampling method.",
"Later, Zhang et al.",
"estimate the model parameters of regularization via a Monte Carlo Expectation Maximization (MCEM) algorithm for unbiased atlas building problem [48].",
"A recent model of Hierarchical Bayesian registration [38] further characterizes the regularization parameters as latent variables generated from Gamma distribution, and integrates them out by an MCEM method.",
"Figure: Left to right: examples of transformation fields overlaid with deformed images with under-regularized and over-regularized registration models.",
"A small regularization introduces crossing artifacts on the transformations vs. a large regularization discourages sufficient transformations between images.Despite the achievement of the aforementioned methods, estimating the regularization parameter in a high-dimensional and nonlinear space of 3D MRIs (i.e., dimension is typically $128^3$ or higher) inevitably leads to expensive computational cost through iterative optimizations.",
"To address this issue, we present a deep learning approach to fast predict registration parameters.",
"While there exist learning-based registration models for transformations [22], [7], [9], we are particularly interested in learning the relationship between pairwise images and optimal regularizations of transformations via regression.",
"In order to produce \"ground truth\" regularization parameters, we first introduce a low-dimensional Bayesian model of image registration to estimate the best regularity from the data itself.",
"Following a recent work of [40], we construct a posterior distribution of diffeomorphic transformations entirely in a bandlimited space with much lower dimensions.",
"This greatly reduces the computational cost of data generation in training.",
"The theoretical tools developed in this work are generic to various deformable registration models, e.g, stationary velocity fields that remain constant over time [3].",
"The model recommends optimal registration parameters for registration in real-time and has great potential in clinical applications (i.e., image-guided navigation system for brain shift compensation during surgery [24]).",
"To summarize, our main contributions are three folds: To the best of our knowledge, we are the first to present a predictive regularization estimation method for diffeomorphic image registration through deep learning.",
"We develop a low-dimensional Bayesian framework in a bandlimited Fourier space to speed up the training data generation.",
"Our model significantly speeds up the parameter estimation, while maintaining comparable registration results.",
"The paper is organized as follows.",
"In sec.",
", we lay out the basics of image registration optimization in the LDDMM framework.",
"In sec.",
"REF , we first develop a low-dimensional posterior distribution that is parametrized by bandlimited velocity fields.",
"We then estimate the regularization parameter by maximizing the posterior.",
"In sec.",
"REF , we design a deep convolutional neural network that takes an image pair as input and adaptively predicts the optimal smoothness level for registration.",
"In sec.",
", we validate our model on both 2D synthetic data and 3D brain MRI scans."
],
[
"Background: Fast LDDMM With Geodesic Shooting",
"In this section, we first briefly review a fast image registration algorithm FLASH in the setting of LDDMM with geodesic shooting [46], [47].",
"The core idea of the FLASH is to reparameterize diffeomorphic transformations effectively in its tangent space (also known as velocity fields), where the signals are smooth without developing high frequencies in the Fourier space.",
"This allows all computations of the original LDDMM with geodesic shooting [37], [45] to be carried out in the bandlimited space of velocity fields with much lower dimensions.",
"As a result, the FLASH algorithm significantly speeds up diffeomorphic image registration with little to no loss of accuracy."
],
[
"Geodesic Shooting in Fourier Spaces",
"Given time-dependent velocity field $\\tilde{v}_t \\in \\tilde{V}$ , the diffeomorphism $\\phi ^{-1}_t \\in \\widetilde{{\\rm Diff}}(\\Omega )$ in the finite-dimensional Fourier domain can be computed as $\\frac{d \\tilde{\\phi }^{-1}_t }{dt}&=-\\tilde{\\mathcal {D}} \\tilde{\\phi }^{-1}_t \\ast \\tilde{v}_t,$ where $\\tilde{\\mathcal {D}} \\tilde{\\phi }^{-1}_t$ is a tensor product $\\tilde{\\mathcal {D}} \\otimes \\tilde{\\phi }^{-1}_t$ , representing the Fourier frequencies of a Jacobian matrix $\\tilde{\\mathcal {D}}$ .",
"The $\\ast $ is a circular convolution with zero padding to avoid aliasing.",
"We truncate the output of the convolution in each dimension to a suitable finite set to avoid the domain growing to infinity.",
"Geodesic shooting algorithm states that the transformation can be uniquely determined by integrating a partial differential equation with a given initial velocity $v_0$ forward in time.",
"In contrast to the original LDDMM that optimizes over a collection of time-dependent velocity fields, geodesic shooting estimates an optimal initial velocity $v_0$ .",
"In this work, we adopt an efficient variant of geodesic shooting defined in Fourier spaces.",
"We first discretize the Jacobian and divergence operators using finite difference approximations (particularly the central difference scheme) and then compute their Fourier coefficients.",
"Detailed derivations can be found in the appendix section of FLASH [47].",
"The Fourier representation of the geodesic constraint that satisfies Euler-Poincaré differential (EPDiff) equation is [46] $\\frac{\\partial \\tilde{v}_t}{\\partial t} ={\\rm ad}^{\\dagger }_{\\tilde{v}_t} \\tilde{v}_t =-\\tilde{\\mathcal {K}}\\left[(\\tilde{\\mathcal {D}} \\tilde{v}_t)^T \\star \\tilde{\\mathcal {L}}(\\alpha )\\tilde{v}_t + \\tilde{\\nabla } \\cdot \\tilde{\\mathcal {L}}(\\alpha )\\tilde{v}_t \\otimes \\tilde{v}_t) \\right],$ where $\\tilde{\\mathcal {L}}$ is a symmetric, positive-definite differential operator that is a function of parameter $\\alpha $ (details are in Sec.",
"REF ).",
"Here $\\tilde{\\mathcal {K}}$ is the inverse operator of $\\tilde{\\mathcal {L}}$ , and $\\star $ is the truncated matrix-vector field auto-correlation The output signal maintains bandlimited after the auto-correlation operates on zero-padded input signal followed by truncating it back to the bandlimited space..",
"The ${\\rm ad}^{\\dagger }$ is an adjoint operator to the negative Jacobi–Lie bracket of vector fields, ${\\rm ad}_{\\tilde{v}} \\tilde{w} = -[\\tilde{v}, \\tilde{w}]= \\tilde{\\mathcal {D}}\\tilde{v} \\ast \\tilde{w} - \\tilde{\\mathcal {D}}\\tilde{w} \\ast \\tilde{v}$ .",
"The operator $\\tilde{\\nabla } \\cdot $ is the discrete divergence (computed by summing the Fourier coefficients of different directions over in $\\tilde{\\mathcal {D}}$ ) of a vector field."
],
[
"FLASH: Fast Image Registration",
"Consider a source image $S$ and a target image $T$ as square-integrable functions defined on a torus domain $\\Omega = \\mathbb {R}^d / \\mathbb {Z}^d$ ($S(x), T(x) : \\Omega \\rightarrow \\mathbb {R}$ ).",
"The problem of diffeomorphic image registration is to find the geodesic (shortest path) of diffeomorphic transformations $\\phi _t \\in {\\rm Diff}(\\Omega ): \\Omega \\rightarrow \\Omega , t \\in [0, 1]$ , such that the deformed image $S \\circ \\phi ^{-1}_1$ at time point $t=1$ is similar to $T$ .",
"The objective function can be formulated as a dissimilarity term plus a regularization that enforces the smoothness of transformations $E(\\tilde{v}_0) &= \\frac{\\gamma }{2}\\text{Dist} (S \\circ \\phi _1^{-1}, T) + \\frac{1}{2}(\\tilde{\\mathcal {L}}(\\alpha ) \\tilde{v}_0, \\tilde{v}_0), \\, s.t., \\text{Eq}.",
"(\\ref {eq:finalleftinvariantfft}) \\, \\& \\, \\text{Eq}.",
"(\\ref {eq:epdiffleft}).$ The $\\text{Dist}(\\cdot , \\cdot )$ is a distance function that measures the dissimilarity between images.",
"Commonly used distance metrics include sum-of-squared difference ($L_2$ norm) of image intensities [8], normalized cross-correlation (NCC) [6], and mutual information (MI) [41].",
"The $\\phi _1^{-1}$ denotes the inverse of deformation that warps the source image $S$ in the spatial domain when $t = 1$ .",
"Here $\\gamma $ is a weight parameter balancing between the distance function and regularity term.",
"The distance term stays in the full spatial domain, while the regularity term is computed in bandlimited space."
],
[
"Our Model: Deep Learning for Regularity Estimation in Image Registration",
"In this section, we present a supervised learning model based on CNN to predict the regularity of image registration for a given image pair.",
"Analogous to [44], we run optimization-based image registration to obtain training data.",
"We introduce a low-dimensional Bayesian model of image registration to produce appropriate regularization parameters for training."
],
[
"Low-dimensional Bayesian Model of Registration",
"In contrast to previous approaches, our proposed model is parameterized in a bandlimited velocity space $\\tilde{V}$ , with parameter $\\alpha $ enforcing the smoothness of transformations.",
"Assuming an independent and identically distributed (i.i.d.)",
"Gaussian noise on image intensities, we obtain the likelihood $p(T \\, | \\, S\\circ \\phi _1^{-1}, \\sigma ^2) = \\frac{1}{(\\sqrt{2 \\pi } \\sigma ^2 )^{M}} \\exp { \\left(-\\frac{1}{2\\sigma ^2}|| S \\circ \\phi _1^{-1} -T ||_2^2 \\right)},$ where $ \\sigma ^2 $ is the noise variance and $M$ is the number of image voxels.",
"The deformation $\\phi _1^{-1}$ corresponds to $\\tilde{\\phi }_1^{-1}$ in Fourier space via the Fourier transform $\\mathcal {F}(\\phi _1^{-1})=\\tilde{\\mathcal {\\phi }}_1^{-1}$ , or its inverse $\\phi _1^{-1}=\\mathcal {F}^{-1}(\\tilde{\\mathcal {\\phi }}_1^{-1})$ .",
"The likelihood is defined by the residual error between a target image and a deformed source at time point $t=1$ .",
"We assume the definition of this distribution is after the fact that the transformation field is observed through geodesic shooting; hence is not dependent on the regularization parameters.",
"Analogous to [39], we define a prior on the initial velocity field $\\tilde{v}_0$ as a complex multivariate Gaussian distribution, i.e., $p({\\tilde{v}}_0 | \\alpha ) = \\frac{1}{(2 \\pi )^{\\frac{M}{2}} | {\\tilde{\\mathcal {L}}}^{-1}(\\alpha ) |^{\\frac{1}{2}}} \\exp { \\left(-\\frac{1}{2}({\\tilde{\\mathcal {L}}(\\alpha )} {\\tilde{v}}_0, {\\tilde{v}}_0) \\right)},$ where $|\\cdot |$ is matrix determinant.",
"The Fourier coefficients of $\\tilde{\\mathcal {L}}$ is, i.e., $\\tilde{\\mathcal {L}} = \\left( \\alpha \\tilde{\\mathcal {A}} + 1\\right)^3$ , $\\tilde{\\mathcal {A}}(\\xi _1 , \\ldots , \\xi _d) = -2 \\sum _{j = 1}^d \\left(\\cos (2\\pi \\xi _j) - 1 \\right)$ .",
"Here $\\tilde{\\mathcal {A}}$ denotes a negative discrete Fourier Laplacian operator with a $d$ -dimensional frequency vector $(\\xi _1 , \\ldots , \\xi _d)$ , where $d$ is the dimension of the bandlimited Fourier space.",
"Combining the likelihood in Eq.",
"(REF ) and prior in Eq.",
"(REF ) together, we obtain the negative log posterior distribution on the deformation parameter parameterized by $\\tilde{v}_0$ as $-\\ln \\, p(\\tilde{v}_0 \\, | \\, S, T, \\sigma ^2, \\alpha ) = & \\frac{1}{2}(\\tilde{\\mathcal {L}} \\tilde{v}_0, \\tilde{v}_0) +\\frac{\\Vert S \\circ \\phi _1^{-1} - T \\Vert _2^2}{2\\sigma ^2} - \\frac{1}{2}\\ln |\\tilde{\\mathcal {L}}| +2M \\ln \\sigma + M\\ln (2\\pi ).$ Next, we optimize Eq.",
"(REF ) over the regularization parameter $\\alpha $ and the registration parameter $\\tilde{v}_0$ by maximum a posterior (MAP) estimation using gradient descent algorithm.",
"Other optimization schemes, such as BFGS [27], or the Gauss-Newton method [5] can also be applied.",
"Gradient of $\\alpha $ .",
"To simplify the notation, first we define $f(\\tilde{v}_0) \\triangleq -\\ln \\, p(\\tilde{v}_0 \\, | \\, S, T, \\sigma ^2, \\alpha )$ .",
"Since the discrete Laplacian operator $\\tilde{\\mathcal {L}}$ is a diagonal matrix in Fourier space, its determinant can be computed as $\\displaystyle \\prod _{j=1}^d (\\alpha \\tilde{\\mathcal {A}}_j + 1 )^3$ .",
"Therefore, the log determinant of $\\tilde{\\mathcal {L}}$ operator is $\\ln |\\tilde{\\mathcal {L}}| = 3 \\sum _{j=1}^d (\\alpha \\tilde{\\mathcal {A}}_j + 1 ).",
"$ We then derive the gradient term $\\nabla _{\\alpha }f(\\tilde{v}_0)$ as $\\nabla _{\\alpha }f(\\tilde{v}_0) = -\\frac{3}{2}[ \\sum _{j=1}^d \\frac{\\tilde{\\mathcal {A}}_j}{\\alpha \\tilde{\\mathcal {A}}_j + 1} - \\langle (\\alpha \\tilde{\\mathcal {A}} +1)^{5}\\tilde{\\mathcal {A}} \\tilde{v}_0, \\tilde{v}_0 \\rangle ].$ Gradient of $\\tilde{v}_0$ .",
"We compute the gradient with respect to $\\tilde{v}_0$ by using a forward-backward sweep approach developed in [50].",
"Steps for obtaining the gradient $\\nabla _{\\tilde{v}_0}f(\\tilde{v}_0)$ are as follows: Forward integrating the geodesic shooting equation Eq.",
"(REF ) to compute $\\tilde{v}_1$ , Compute the gradient of the energy function Eq.",
"(REF ) with respect to $\\tilde{v}_1$ at $t = 1$ , $\\nabla _{\\tilde{v}_1}f(\\tilde{v}_0) = \\tilde{\\mathcal {L}}^{-1} (\\alpha ) \\left( \\frac{1}{\\sigma ^2}(S \\circ \\phi _1^{-1} - T) \\cdot \\nabla (S \\circ \\phi _1^{-1}) \\right).$ Bring the gradient $\\nabla _{\\tilde{v}_1}f(\\tilde{v}_0)$ back to $t=0$ by integrating adjoint Jacobi fields backward in time [50], $\\frac{d{\\hat{v}}}{dt} = -{\\rm ad}^{\\dagger }_{\\tilde{v}}{\\hat{h}}, \\quad \\frac{d{\\hat{h}}}{dt} = -\\hat{v} -{\\rm ad}_{\\tilde{v}} \\hat{h} + {\\rm ad}^{\\dagger }_{\\hat{h}} \\tilde{v},$ where $\\hat{v} \\in V$ are introduced adjoint variables with an initial condition $\\hat{h} = 0, \\hat{v} = \\nabla _{\\tilde{v}_1}f(\\tilde{v}_0)$ at $t=1$ .",
"A summary of the optimization is in Alg.",
"REF .",
"It is worthy to mention that our low-dimensional Bayesian framework developed in bandlimited space dramatically speeds up the computational time of generating training regularity parameters by approximately ten times comparing with high-dimensional frameworks.",
"[!htb] textnormal InputInput OutputOutput optimal smoothness level $\\alpha ^{opt}$ , and registration solution $\\tilde{v}_0$ Low-dimensional MAP Estimation for $\\alpha $ $i=1$ to $r$ 1.",
"Compute gradient $\\nabla _{\\alpha }f(\\tilde{v}_0)$ by Eq.",
"(REF ); Update $\\alpha $ when its gradient is greater than zero; 2.",
"$\\Vert \\nabla _{\\alpha }f(\\tilde{v}_0) \\Vert ^2 > 1e-6$ $ \\alpha ^{opt} \\leftarrow \\alpha ^{opt}- \\tau \\nabla _{\\alpha }f(\\tilde{v}_0);$ break; 3.",
"Forward integrating the geodesic evolution equation in Eq.",
"(REF ) with initial velocity $\\tilde{v}_0$ ; 4.",
"Compute gradient $\\nabla _{\\tilde{v}_1}f(\\tilde{v}_0) $ by Eq.",
"(REF ) then backward integrating adjoint equations Eq.",
"(REF ) to generate $\\nabla _{\\tilde{v}_0}f(\\tilde{v}_0) $ at $t=0$ ; Update $\\tilde{v}_0$ when its gradient is greater than zero; 5.",
"$\\Vert \\nabla _{\\tilde{v}_0}f(\\tilde{v}_0) \\Vert ^2 > 1e-6$ $ \\tilde{v}_0 \\leftarrow \\tilde{v}_0- \\epsilon \\nabla _{\\tilde{v}_0 }f(\\tilde{v}_0);$ break; Compute algorithm stop rate 6.",
"Compute total energy Eq.",
"(REF ) for $i-$ th iteration as $\\rm {Obj}_i$ ; 7.",
"$\\frac{\\rm {Obj}_i - \\rm {Obj}_{i-1}}{\\rm {Obj}_i} < u$ $ q = q + 1 $ ; continue; Convergence check 8.",
"$ q \\ge q_{min}$ break ; continue; End MAP of low-dimensional Bayesian registration model for training data generation."
],
[
"Network Architecture",
"Now we are ready to introduce our network architecture by using the estimated $\\alpha $ and given image pairs as input training data.",
"Fig.",
"REF shows an overview flowchart of our proposed learning model.",
"With the optimal registration regularization parameter $\\alpha ^{opt}$ obtained from an image pair (as described in Sec.",
"REF ), a two-stream CNN-based regression network takes source images, target images as training data to produce a predictive regularization parameter.",
"We optimize the network $\\mathcal {H} (S, T; W)$ with the followed objective function, $E_\\text{loss} = \\text{Err} [\\mathcal {H} (S, T; W), \\alpha ^{opt}] + \\text{Reg}(W),$ where $\\text{Reg}(W)$ denotes the regularization on convolution kernel weights $W$ .",
"$\\text{Err}[\\cdot ,\\cdot ]$ denotes the data fitting term between the ground truth and the network output.",
"In our model, we use $L_2$ norm for both terms.",
"Other network architectures, e.g.",
"3D Residual Networks (3D-ResNet) [16] and Very Deep Convolutional Networks (3D-VGGNet) [31], [43] can be easily applied as well.",
"Figure: Illustration of our proposed network.",
"From left to right: training data includes pairwise images, a CNN-based neural network, and a loss computed between the network output and the optimal parameter α opt \\alpha ^{opt}, which is estimated from Alg.",
".In our network, we input the 3D source and target images into separate channels that include four convolutional blocks.",
"Each 3D convolutional block is composed of a 3D convolutional layer, a batch normalization (BN) layer with activation functions (PReLU or ReLU), and a 3D max-pooling layer.",
"Specifically, we apply $5\\times 5 \\times 5$ convolutional kernel and $2\\times 2 \\times 2$ max-pooling layer to encode a batch (size as $B$ ) of source and target images ($128^3$ ) to feature maps ($16^3 $ ).",
"After extracting the deep features from source and target channels, we combine them into a fusion channel, which includes three convolutional blocks, a fully connected layer, and an average pooling layer to produce the network output."
],
[
"Experiment",
"To demonstrate the effectiveness of the proposed low-dimensional Bayesian registration model, we validate it through three sets of experiments.",
"For 2D synthetic data registration, we deform a binary source image with velocity fields sampled from the prior distribution in Eq.",
"(REF ) with known regularization parameters to simulate target images.",
"We show three convergence graphs of our MAP estimation and compare them with the ground truth parameters.",
"Similarly, we synthesize 900 image pairs using regularization parameters at different scales respectively, i.e., $\\alpha = \\lbrace 0.1, 1.0, 10.0\\rbrace $ , to test the performance of our predictive model.",
"We then use the predicted parameter $\\alpha $ to run registration model and show the error maps between target and deformed images.",
"For 3D brain MRI registration, we show results on both MAP and our network prediction.",
"We first show the numerical difference between the MAP estimation and our prediction (i.e.",
"predicted regularization parameter), and then report the mean error of deformed images between both methods across all datasets.",
"We visualize the transformation grids and report the value of regularization parameters for both methods.",
"To further investigate the accuracy of parameters generated by our model, we perform registration-based segmentation and examine the resulting segmentation accuracy over nine brain structures, including cortex, putamen, cerebellum, caudate, gyrus, brain stem, precuneus, cuneus, and hippocampus.",
"We evaluate a volume-overlapping similarity measurement, also known as S$ø$ rensen$-$ Dice coefficient [11], between the propagated segmentation and the manual segmentation.",
"The statistics of dice evaluation over 150 registration pairs are reported.",
"We last compare the computational efficiency on both time and memory consumption of the proposed method with a baseline model that performs Bayesian estimation of regularization parameter in the full spatial domain [48].",
"To generate the training data of initial velocities, we run the proposed low-dimensional Bayesian registration algorithms until convergence.",
"We use the Euler integrator in geodesic shooting and set the of integration steps as 10.",
"We set algorithm stop rate $u$ as $1e-6$ and minimal convergence iteration $q_{min}$ as 30.",
"We use optimal truncated dimension for $\\tilde{v}_0$ as 16 and $\\sigma = 0.03$ according to [50].",
"For the network setting, We initialize the convolution kernel weights using the He normal initializer [17] and use the Adam optimizer with a learning rate of $5e-4$ until convergence.",
"We set 16 and $1.0e-4$ as batch size and weight decay.",
"The maximum epoch for 2D and 3D network training is 1000."
],
[
"Data",
"We run experiments on both 2D synthetic dataset and 3D real brain MRI scans.",
"2D synthetic data.",
"We generate synthetic bull-eye images with the size of $100 \\times 100$ (as shown in Fig.",
"REF ).",
"We manipulate the width $a$ and height $b$ of two ellipses by using equation $\\frac{(x-50)^2}{a^2} + \\frac{(y-50)^2}{b^2} = 1$ .",
"3D brain MRIs.",
"We include 1500 public T1-weighted brain MRI scans from Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset [19], Open Access Series of Imaging Studies (OASIS) [12], and LONI Probabilistic Brain Atlas Individual Subject Data (LPBA40) [30], among which 260 subjects have manual delineated segmentation labels.",
"All 3D data were carefully pre-processed as $128\\times 128\\times 128$ , $1.25mm^{3}$ isotropic voxels, and underwent skull-stripped, intensity normalized, bias field corrected, and pre-aligned with affine transformation.",
"For both 2D and 3D datasets, we split the images by using $70\\%$ as training images, $15\\%$ as validation images, and $15\\%$ as testing images such that no subjects are shared across the training, validation, and testing stage.",
"We evaluate the hyperparameters of models and generate preliminary experiments on the validation dataset.",
"The testing set is only used for computing the final results."
],
[
"Results",
"Fig.",
"REF displays our MAP estimation of registration results including appropriate regularity parameters on 2D synthetic images.",
"The middle panel of Fig.",
"REF reports the convergence of $\\alpha $ estimation vs. ground truth.",
"It indicates that our low-dimensional Bayesian model provides trustworthy regularization parameters that are fairly close to ground truth for network training.",
"The bottom panel of Fig.",
"REF shows the convergence graph of the total energy for our MAP approach.",
"Figure: Top panel: source image, velocity fields generated from prior distribution and transformation fields (with known regularization parameter α=3,6,11\\alpha = 3, 6, 11), and target images produced by deformed source images; Middle panel: convergence graphs of estimated α\\alpha by MAP for training data; Bottom panel: convergence graphs of total energy Eq. ().Fig.",
"REF further investigates the consistency of our network prediction.",
"The left panel shows estimates of regularization parameter at multiple scales, i.e., $\\alpha =0.1, 1.0, 10.0$ , over 900 2D synthetic image pairs respectively.",
"The right panel shows the mean error of image differences between deformed source images by transformations with predicted $\\alpha $ and target images.",
"While there are small variations on estimated regularization parameters, the registration results are very close (with averaged error at the level of $10^{-5}$ ).",
"Figure: Left: our prediction of regularization parameters over 900 image pairs synthesized with different ground truth parameters α=0.1,1.0,10.0\\alpha = 0.1, 1.0, 10.0; Right: error maps of image differences between deformed and target images.Fig.",
"REF shows examples of 2D pairwise image registration with regularization estimated by MAP and our predictive deep learning model.",
"We obtain the regularization parameter $\\alpha = 11.34$ (MAP) vs. $\\alpha = 13.20$ (network prediction), and $\\alpha = 5.44$ (MAP) vs. $\\alpha = 6.70$ (network prediction).",
"The error map of deformed images indicates that both estimations obtain fairly close registration results.",
"Figure: Left to right: two examples of 2D synthetic source, target, deformed images by MAP and predictive deep learning method, and error maps of deformed image intensity differences.Fig.",
"REF displays deformed 3D brain images overlaid with transformation grids for both methods.",
"The parameter estimated from our model produces a comparable registration result.",
"From our observation, a critical pattern between the optimal $\\alpha ^{opt}$ and its associate image pairs is that the value of $\\alpha $ is relatively smaller when large deformation occurs.",
"This is because the image matching term (encoded in the likelihood) requires a higher weight to deform the source image.",
"Figure: Left to right: source, target, deformed images overlaid with the transformation grids generated by both methods.",
"The optimal and predicted value of α\\alpha for two showcases are (8.90,9.20)(8.90, 9.20) and (3.44,2.60)(3.44, 2.60).Fig.",
"REF investigates the consistency of our network prediction over three different datasets of 3D brain MRI.",
"The left panel shows the absolute value of numerical differences between predicted regularization parameters and MAP estimations.",
"The right panel shows the voxel-wise mean error of image differences between deformed images by transformations with predicted $\\alpha $ and deformed images by MAP.",
"While slight numerical difference on estimated regularization parameters exists, the 3D deformed images are fairly close (with averaged voxel-wise error at the level of $10^{-6}$ ).",
"Figure: Left: statistics of the difference between our prediction and MAP estimation over 300 image pairs from three data sets, ADNI, LPBA40, and OASIS; Right: error maps of image differences between deformed image by our prediction and MAP estimation.Fig.",
"REF visualizes three views of the deformed brain images (overlay with propagated segmentation label) that are registered by MAP and our prediction.",
"Our method produces a registration solution, which is highly close to the one estimated by MAP.",
"The propagated segmentation label fairly aligns with the target delineation for each anatomical structure.",
"While we show different views of 2D slices of brains, all computations are carried out fully in 3D.",
"Figure: Axial, coronal and sagittal view of 3D segmentation labels with nine anatomical structures overlaid with source, target, deformed images by our low-dimensional MAP (α=8.91\\alpha = 8.91) and our network prediction (α=6.80\\alpha = 6.80).Fig.",
"REF reports the volume overlapping of nine anatomical structures for both methods, including Cor(cortex), Puta (putamen), Cere (cerebellum), Caud (caudate), gyrus, Stem (brain stem), Precun (precuneus), Cun (cuneus), and Hippo (hippocampus).",
"Our method produces comparable dice scores comparing with MAP estimations.",
"This indicates that the segmentation-based registration by using our estimation achieves comparable registration performance with little to no loss of accuracy.",
"Figure: Dice scores of propagated manual segmentations for both methods for 150 registration pairs.",
"Evaluations are performed over nine anatomical brain structures, Cor (cortex), Puta (putamen), Cere (cerebellum), Caud (caudate), gyrus, Stem (brain stem), Precun (precuneus), Cun (cuneus) and Hippo (hippocampus).Table.",
"REF quantitatively reports the averaged time and memory consumption of MAP estimation in full spatial image domain and our method.",
"The proposed predictive model provides appropriate regularization parameters approximately 1000 times faster than the conventional optimization-based registration method with a much lower memory footprint.",
"Table: Time and memory consumption of MAP estimation of regularization in full-dimensional image space vs. our proposed low-dimensional Bayesian model, as well as network prediction."
],
[
"Conclusion",
"In this paper, we proposed a deep learning-based approach to model the relationship between the regularization of image registration and the input image data.",
"We first developed a low-dimensional Bayesian model that defines image registration entirely in a bandlimited space.",
"We then learned the mapping between regularization parameters and spatial images through a CNN-based neural network.",
"To the best of our knowledge, we are the first to predict the optimal regularization parameter of diffeomorphic image registration by deep learning approaches.",
"In contrast to existing methods, our developed model substantially improves the efficiency and the robustness of image registration.",
"Our work has great potential in a variety of clinical applications, e.g.",
"image-guided navigation system for neurosurgery in real-time.",
"Potential future work may include: i) extending the current model to further consider adversarial examples, i.e., image outliers with significant differences; and ii) developing an unsupervised learning of registration parameter estimation to eliminate the need of training data generation (ground truth labels)."
]
] | 2011.14229 |
[
[
"Quantum speed limit time for topological qubit influenced by fermionic\n and bosonic environment"
],
[
"Abstract Quantum theory sets a limit on the minimum time required to transform from an initial state to a target state.",
"It is known as quantum speed limit time.",
"quantum speed limit time can be used to determine the rate of quantum evolution for closed and open quantum systems.",
"Given that in the real world we are dealing with open quantum systems, the study of quantum speed limit time for such systems has particular importance.",
"In this work we consider the topological qubit realized by two Majorana modes.",
"We consider the case in which the topological qubit is influenced by fermionic and bosonic environment.",
"Fermionic and bosonic environments are assumed to have Ohmic-like spectral density.",
"The quantum speed limit time is investigated for various environment with different Ohmic parameter.",
"It is observed that for super-Ohmic environment with increasing Ohmic parameter the quantum speed limit time gradually reaches to a constant value and thus the speed of evolution reaches to a uniform value.",
"The effects of external magnetic field on the evolution rate are also studied.",
"It is observed that with increasing magnitude of magnetic field, the quantum speed limit time decreases"
],
[
"Introduction",
"The minimum time required for the transformation of a quantum system from an initial state to a target state is known as quantum speed limit (QSL) time.",
"It can be said that the QSL time stems from time-energy uncertainty principle.",
"The maximum speed of quantum evolution can be obtained using QSL time.",
"QSL time is used in many topics of quantum information theory such as quantum communication [1], investigation of exact bounds in quantum metrology [2], computational bounds of physical systems [3] and quantum optimal control algorithms [4].",
"For closed quantum systems whose evolution is described using unitary operations, the QSL time is obtained using distance measures such as Bures angle and relative purity [5], [6], [7], [8], [9], [10], [11], [12].",
"Among the most important QSL time bound for closed systems, one can mention two bounds, one is Mandelstam-Tamm (MT) bound [11] and other is Margolus-Levitin (ML) bound [12].",
"Given that isolating a quantum system from its surroundings is difficult and almost impossible, and any real quantum system interacts inevitably with its surroundings, the study of open quantum systems is one of the fascinating topics in quantum information theory [13], [14], [15].",
"Therefore, due to the importance of open quantum systems and their role in quantum information theory, the study of QSL time for these systems has been considered in many recent works [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41].",
"In general, Mandelstam-Tamm bound and Margolus-Levitin bound are used to describe QSL time in closed and open quantum systems.",
"The generalizations of these two bounds for open quantum systems are given in Refs.",
"[7], [8].",
"In Ref.",
"[24], Deffner et al.",
"present a comprehensive and unified bound for non-Markovian dynamics that includes both MT and ML bounds .",
"In Ref.",
"[28], Zhang et al.",
"provide the QSL time bound for arbitrary initial states .",
"They have shown that the QSL time is depend on quantum coherence of initial state.",
"Based on the definition they provide, QSL time is the minimum time required for the evolution of an open quantum system from an initial state at time $\\tau $ to target state at time $\\tau +\\tau _D$ , where $\\tau _D$ is driving time.",
"In this work we will consider the QSL time bound which has introduced by Zhang et al.[28].",
"It has been observed that topological quantum computing is a promising design for the realization of quantum computers with stable qubits [42].",
"According to recent studies, there exist different and new types of topological ordered states that are physically achievable, such as topological insulators and superconductors [43], [44], [45].",
"For these systems, some of the excitations are topologically protected, provided that some symmetries, such as time inversion, are maintained.",
"In other words, the local perturbations that maintain these symmetries cannot disentangle the topological excitations.",
"The most interesting of these topological excitations are Majorana modes localized on topological defects, which follow the non-Abelian anyonic statistics [46], [47], [48].",
"The Kitaev 1D spineless p-wave superconductor chain model is the most common model for realizing such Majorana modes.",
"Each on-site fermion can be decomposed into two Majorana modes.",
"By properly adjusting the model, Majorana modes can be dangling at the end of the chain without pairing with other nearby Majorana modes to form common fermions.",
"So, these two separate Majorana modes can creat a topological qubit.",
"There exist two meanings to the word topological here : One meaning is that it is composed of Majorana modes that are topological excitations, another implication is that the topological qubit itself is non-local, meaning that the two majorana states are very separate and therefore cannot be combined into a common fermion.",
"From quantum information insight, the topological qubit is EPR-like, because it encode quantum state non-locally.",
"Both characteristics explain its resistance to local disturbances.",
"Given that topological excitations are robust against local perturbations, The question that may arise is whether topological qubits are also robust against decoherence, when they are considered as an open quantum system coupling to the non-topological environment.",
"The open system setup is more logical and realistic when performing quantum computations.",
"Since quantum information is carried by physical excitations, stability against decoherence indicates stability against local perturbations, but the opposite is not true.",
"Even if local excitations are stable to local perturbations, however the quantum information carried by the topological qubit may still penetrate into the environment.",
"However, since topological qubit is non-local, its interaction with the environment is quite different from that of conventional fermions, and quantum quantum decoherence behaviors are expected to be unusual.",
"This motivates us, in this work, to examine QSL time for decoherence of topological qubit.",
"In this paper the QSL time for the dynamics of a topological qubit realized by two Majorana modes coupled to a fermionic and bosonic Ohmic-like reservoir is discussed in detail."
],
[
"Decoherence of topological qubits",
"A topological qubit composed of two Majorana modes of the one-dimensional Kitaev’s chain that are spatially separated.",
"These Majorana modes are located at the two ends of a quantum wire and are denoted by $\\gamma _1$ and $\\gamma _2$ , and the following relations are established for them $\\gamma _{a}^{\\dagger }=\\gamma _{a}, \\quad \\left\\lbrace \\gamma _{a}, \\gamma _{b}\\right\\rbrace =2 \\delta _{a b},$ where $a, b \\in \\lbrace 1,2\\rbrace $ .",
"The Majorana modes are influenced by their surroundings in an incoherent way, which causes the decoherence of the topological qubit.",
"The Hamiltonian describing the intended general system is defined as follows $\\hat{H}=\\hat{H}_S+\\hat{H}_E+\\hat{H}_I,$ where $\\hat{H}_S$ represents the Hamiltonian of topological qubit system, $\\hat{H}_E$ describes the Hamiltonian of the environment and $\\hat{H}_I$ is the interaction Hamiltonian describes the influence of the environment on topological qubit which reads $\\hat{H}_I=G_1 \\gamma _1 \\hat{\\mathcal {Q}}_1 + G_2 \\gamma _2 \\hat{\\mathcal {Q}}_2,$ where $G_{1(2)}$ describes the real coupling constant and $\\hat{Q}_{1)(2)}$ is the composite operator consist of the electron creation operator $a^{\\dag }$ and annihilation operator $a$ .",
"According to hermeticity condition of interaction Hamiltonian i.e.",
"$\\hat{H}_I^{\\dag }=H_I$ we have $\\hat{\\mathcal {Q}}_a^{\\dag }=-\\hat{\\mathcal {Q}}_a.$ When the system is affected by a fermionic environment, the Majorana modes are located at two ends of a quantum wire which is placed over a s-wave superconductor.",
"Majorana modes are affected by a magnetic field $B$ whose direction is along the quantum wire.",
"Each of the Majorana modes is paired with a metallic nanowire through a tunnel junction with tunneling strength $B_i$ that is adjusted by an external gate voltage.",
"The schematic of this type of interaction is drawn in Fig.",
"(REF ).",
"Figure: Schematic representation of topological qubitrealized by two Majorana modes γ 1 \\gamma _1 and γ 2 \\gamma _2, interacting with a fermionic environment.When the system is affected by a bosonic environment, the Majorana modes are placed at two ends of a quantum ring with an empty space between them.",
"In this case the some environmental bosonic operator interact with the two Majorana modes locally.",
"the frequency dependence in the bosonic environment is provided by a magnetic flux $\\Phi $ passing through the quantum ring.",
"The schematic of this type of interaction is drawn in Fig.",
"(REF ).",
"In this work we consider Ohmic-like environmental spectral density $J(\\omega ) \\propto \\omega ^s$ , for both fermionic and bosonic environment.",
"When $s<1$ , $s=1$ and $s>1$ we have sub-Ohmic, Ohmic and super-Ohmic environment respectively.",
"Before interaction, the Majorana modes form a topological qubit with states $\\vert 0 \\rangle $ and $\\vert 1 \\rangle $ , which are related as follows $\\frac{1}{2}\\left(\\gamma _{1}-i \\gamma _{2}\\right)|0\\rangle =|1\\rangle , \\quad \\frac{1}{2}\\left(\\gamma _{1}+i \\gamma _{2}\\right)|1\\rangle =|0\\rangle $ Given that $\\gamma _a$ 's must satisfy Eq.",
"(REF ), they can be selected as $\\gamma _{1}=\\sigma _{1}, \\quad \\gamma _{2}=\\sigma _{2}, \\quad i \\gamma _{1} \\gamma _{2}=\\sigma _{3},$ where $\\sigma _i$ 's are Pauli matrices.",
"Here it is assumed that the state of the whole system $S+E$ is product i.e.",
"$\\rho _S(0) \\otimes \\rho _E$ and $\\rho _{S}(0)=\\sum _{i, j=0}^{1} \\rho _{i j}|i\\rangle \\langle j|$ is the initial state of the topological qubit.",
"For the case where the topological qubit is affected by a fermionic environment, the state of the topological qubit at time $t$ is obtained as $\\rho _{S}^{F}(t)=\\frac{1}{2}\\left(\\begin{array}{cc}1+\\left(2 \\rho _{00}-1\\right) \\alpha ^{2}(t) & 2 \\rho _{01} \\alpha (t) \\\\2 \\rho _{10} \\alpha (t) & 1+\\left(2 \\rho _{11}-1\\right) \\alpha ^{2}(t)\\end{array}\\right),$ and for the case where the topological qubit is influenced by a bosonic environment, the state of the topological qubit at time $t$ is obtained as $\\rho _{S}^{B}(t)=\\left(\\begin{array}{cc}\\rho _{00} & \\rho _{01} \\alpha (t) \\\\\\rho _{10} \\alpha (t) & \\rho _{11}\\end{array}\\right),$ where $\\alpha (t)=e^{-2 B^{2}\\left|\\beta _{F, B}\\right| \\mathcal {I}_{s}(t)},$ and decay parameter $\\mathcal {I}_{s}(t)$ is given by $\\mathcal {I}_{s}(t)=\\left\\lbrace \\begin{array}{ll}2 \\Gamma _{0}^{s-1} \\Gamma \\left(\\frac{s-1}{2}\\right)\\left(1-{ }_{1} F_{1}\\left(\\frac{s-1}{2} ; \\frac{1}{2} ; \\frac{-\\Gamma _{0}^{2} t^{2}}{4}\\right)\\right) & s \\ne 1 \\\\\\frac{\\Gamma _{0}^{2} t^{2}}{2}{ }_{2} F_{2}\\left(\\lbrace 1,1\\rbrace ;\\left\\lbrace \\frac{3}{2}, 2\\right\\rbrace ; \\frac{-\\Gamma _{0}^{2} t^{2}}{4}\\right) & \\quad s=1\\end{array}\\right.$ where $\\Gamma _0$ describes the cutoff frequency of the environment, $\\Gamma (z)$ is the Gamma function and ${ }_{i} F_{j}$ is the generalized hypergeometric function.",
"$\\beta _F$ and $\\beta _B$ are time-independent coefficients in Fermion and boson environments, respectively and are defined as $\\beta _{F}=\\frac{-4 \\pi }{\\Gamma \\left(\\frac{s+1}{0}\\right)}\\left(\\Gamma _{0}\\right)^{-(s+1)}$ and $\\beta _{B}=\\left\\lbrace \\begin{array}{ll}-\\frac{N_{s s}^{2} \\Gamma (3-\\Delta ) \\epsilon ^{2(\\Delta -4)}}{4 \\pi ^{2} \\Gamma (\\Delta -2) 2^{2} \\Delta -5} \\sin \\pi \\Delta & 2<\\Delta \\notin \\mathbf {N} \\\\-\\frac{N_{s c}^{2} e^{2(\\Delta -4)}}{4 \\pi (\\Delta -3) !",
"22^{2 \\Delta -5}} & 2 \\le \\Delta \\in \\mathbf {N}\\end{array}\\right.$ where $N_{sc}$ represents the number of degrees of freedom of the dual conformal field theory, $\\epsilon $ is the $UV$ cutoff of the length, $\\Delta =(s+4)/2$ shows the conformal dimension and $N$ stands for the set of natural numbers.",
"Figure: Schematic representation of topological qubitrealized by two Majorana modes γ 1 \\gamma _1 and γ 2 \\gamma _2, interacting with a bosonic environment."
],
[
"Quantum speed limit time",
"Quantum mechanics sets a bound on the evolution speed of a quantum process for a closed or open quantum systems.",
"In general, the minimum time required to transform from an initial state to a target state is known as QSL time.",
"Mandelstam and Tamm have introduced the QSL time bound known as (MT) bound for close quantum system as [11] $\\tau \\ge \\tau _{Q S L}=\\frac{\\pi \\hbar }{2 \\Delta \\mathcal {E}},$ where $\\Delta \\mathcal {E} =\\sqrt{ \\langle \\hat{H}^2 \\rangle - \\langle \\hat{H}\\rangle ^2} $ is the variance of energy of initial state and $\\hat{H}$ is the time-independent Hamiltonian describing the evolution of the closed quantum system.",
"Another bound for closed quantum systems, have introduced by Margolus and Levitin [12].",
"It is known as (ML) bound, which is defined as follows $\\tau \\ge \\tau _{Q S L}=\\frac{\\pi \\hbar }{2 \\mathcal {E}},$ where $\\mathcal {E}= \\langle \\hat{H} \\rangle $ .",
"By combining the two ML and MT bounds for close quantum system, one can obtain an unified bound for QSL time as follows $\\tau \\ge \\tau _{Q S L}=\\max \\left\\lbrace \\frac{\\pi \\hbar }{2 \\Delta E}, \\frac{\\pi \\hbar }{2 E}\\right\\rbrace .$ In the real world, the interaction of a system with its environment is inevitable, so in practice the study of open quantum systems is of particular importance.",
"The evolution of an open quantum system is described using a time-dependent master equation as $\\dot{\\rho }_t=\\mathcal {L}_t \\rho _t,$ where $\\rho _t$ is the state of the open quantum system at time $t$ and $\\mathcal {L}_t$ is the positive generator [15].The main goal here is to find the minimum time required to evolve from an initial state $\\rho _\\tau $ to a target state $\\rho _{\\tau +\\tau _D}$ for open quantum system, where $\\tau $ is initial time and $\\tau _D$ is driving time.",
"This minimal time is called QSL time.",
"In order to quantify the bound of QSL time one should use an appropriate distance measure.",
"In Refs.",
"[23], [28], the authors use relative purity to quantify the bound for QSL time.",
"A notable feature of the bound defined by them is that their bound can also be used for mixed initial states.",
"The relative purity between the initial state $\\rho _\\tau $ and the target state $\\rho _{\\tau +\\tau _D}$ is defined as follows $f\\left(\\tau +\\tau _{D}\\right)=\\frac{\\operatorname{tr}\\left(\\rho _{\\tau } \\rho _{\\tau +\\tau _{D}}\\right)}{\\operatorname{tr}\\left(\\rho _{\\tau }^{2}\\right)}$ By following the method given in Ref.",
"[28], the ML bound of QSL-time can be obtain as $\\tau \\ge \\frac{\\left|f\\left(\\tau +\\tau _{D}\\right)-1\\right| t r\\left(\\rho _{\\tau }^{2}\\right)}{\\sum _{i=1}^{n} \\kappa _{i} \\varrho _{i}}$ where $\\kappa $ and $\\varrho $ are the singular value of $\\mathcal {\\rho _t}$ and $\\rho _\\tau $ respectively and $\\overline{\\square }=\\frac{1}{\\tau _{D}} \\int _{\\tau }^{\\tau +\\tau _{D}} \\square d t$ .",
"By following a similar method, the MT bound of QSL-time for open quantum system can be obtain as $\\tau \\ge \\frac{\\left|f\\left(\\tau +\\tau _{D}\\right)-1\\right| \\operatorname{tr}\\left(\\rho _{\\tau }^{2}\\right)}{\\sqrt{\\sum _{i=1}^{n} \\kappa _{i}^{2}}}$ By combining ML and MT bounds, an unified bound can be achieved as follows $\\begin{aligned}\\tau _{(Q S L)}=& \\max \\left\\lbrace \\frac{1}{\\sum _{i=1}^{n} \\kappa _{i} \\rho _{i}}, \\frac{1}{\\sqrt{\\sum _{i=1}^{n} \\kappa _{i}^{2}}}\\right\\rbrace \\\\& \\times \\left|f\\left(\\tau +\\tau _{D}\\right)-1\\right| t r\\left(\\rho _{\\tau }^{2}\\right).\\end{aligned}$"
],
[
"Results",
"In this section we want to find the QSL time for the topological qubit when they interact with the bosonic or fermionic environment.",
"Let us consider the initial mixed state for topological qubit as $\\rho _{0}=\\frac{1}{2}\\left(\\begin{array}{cc}1+v_{z} & v_{x}-i v_{y} \\\\v_{x}+i v_{y} & 1-v_{z}\\end{array}\\right).$ When topological qubit interacts with fermionic environment its time evolution reads $\\rho _S^F(t)=\\frac{1}{2}\\left(\\begin{array}{cc}1+v_{z}^{\\prime } & v_{x}^{\\prime }-i v_{y}^{\\prime } \\\\v_{x}^{\\prime }+i v_{y}^{\\prime } & 1-v_{z}^{\\prime }\\end{array}\\right).$ Now we can find the singular value of $\\rho _\\tau $ and $\\mathcal {L}_t(\\rho _t)$ .",
"the singular value of $\\rho _\\tau $ are $\\begin{array}{l}\\varrho _{1}=\\frac{1}{2}\\left(1-\\sqrt{v_{x}^{\\prime 2}+v_{y}^{\\prime 2}+v_{z}^{\\prime 2}}\\right) \\\\\\varrho _{2}=\\frac{1}{2}\\left(1+\\sqrt{v_{x}^{\\prime 2}+v_{y}^{\\prime 2}+v_{z}^{\\prime 2}}\\right)\\end{array}$ where $v_{x}^{\\prime }=\\alpha (\\tau ) v_x$ , $v_{y}^{\\prime }=\\alpha (\\tau ) v_y$ and $v_{z}^{\\prime }=\\alpha (\\tau )^{2} v_{z}$ .",
"The singular value $\\kappa _i$ of $\\mathcal {L}_t(\\rho _t)$ can be written as $\\begin{array}{l}\\kappa _{1}=\\kappa _{2}=\\frac{1}{2} \\vert \\dot{\\alpha }(t)\\sqrt{v_x^2+v_y^2+4 \\alpha (t)^2 v_z^2} \\vert ,\\end{array}$ Therefore, it can be concluded that $\\varrho _1 \\kappa _1 + \\varrho _2 \\kappa _2$ is always less than $\\sqrt{\\kappa _1^2+\\kappa _2^2}$ and so the ML bound on QSL time is tighter than MT bound for open quantum systems.",
"For the bosonic environment we follow our calculations in the same way as before .",
"We consider the initial state in Eq.",
"(REF ).",
"For the case in which the topological qubit is influenced by bosonic environment its time evolution reads $\\rho _S^B(t)=\\frac{1}{2}\\left(\\begin{array}{cc}1+v_{z}^{\\prime } & v_{x}^{\\prime }-i v_{y}^{\\prime } \\\\v_{x}^{\\prime }+i v_{y}^{\\prime } & 1-v_{z}^{\\prime }\\end{array}\\right).$ The singular value of $\\rho _\\tau $ and $\\mathcal {L}_t(\\rho _t)$ can be obtain easily like fermionic environment.",
"The singular value of $\\rho _\\tau $ are $\\begin{array}{l}\\varrho _{1}=\\frac{1}{2}\\left(1-\\sqrt{v_{x}^{\\prime 2}+v_{y}^{\\prime 2}+v_{z}^{\\prime 2}}\\right) \\\\\\varrho _{2}=\\frac{1}{2}\\left(1+\\sqrt{v_{x}^{\\prime 2}+v_{y}^{\\prime 2}+v_{z}^{\\prime 2}}\\right)\\end{array}$ where $v_{x}^{\\prime }=\\alpha (\\tau ) v_x$ , $v_{y}^{\\prime }=\\alpha (\\tau ) v_y$ and $v_{z}^{\\prime }= v_{z}$ .",
"The singular value $\\kappa _i$ of $\\mathcal {L}_t(\\rho _t)$ can be written as $\\begin{array}{l}\\kappa _{1}=\\kappa _{2}=\\frac{1}{2} \\vert \\dot{\\alpha }(t)\\sqrt{v_x^2+v_y^2} \\vert ,\\end{array}$ In this work we consider the maximally coherent initial state with $v_x=v_y=1/\\sqrt{2}$ and $v_z=0$ .",
"So the QSL time for topological qubit inside both fermionic and bosonic environment can be obtained as $\\tau _{QSL}=\\frac{\\vert \\alpha (\\tau )^2-\\alpha (\\tau )\\alpha (\\tau +\\tau _D)\\vert }{\\frac{1}{\\tau _D}\\int _\\tau ^{\\tau +\\tau _D}\\vert \\dot{\\alpha }(t)\\vert }.$ Although we have reached a similar relation for two different environments, however due to the difference in $\\alpha (t)$ coefficient for both bosonic and fermionic environments, the results will definitely be different.",
"Figure: (Color online) QSL time versus Ohmic parameter ss for different bosonic and fermionic environment, B=0.4, τ D =1\\tau _D=1 and τ=1\\tau =1.Figure: (Color online) QSL time versus initial time parameter τ\\tau for different bosonic and fermionic environment when τ D =1\\tau _D=1 (a)s=0.1, (b)s=1, (c)s=1.5 and (d)s=2.5.Figure: (Color online) QSL time versus magnetic field parameter BB for different bosonic and fermionic environment when τ D =1\\tau _D=1 (a)s=0.1, (b)s=1, (c)s=1.5 and (d)s=2.5In Fig.REF , QSL time is plotted as a function of Ohmic parameter for both bosonic and fermionic environment.",
"As can be seen for bosonic environment the QSL time reaches to zero for small value of $s$ and increases again with increasing $s$ .",
"The situation is slightly different for the fermion environment.",
"For this environment, the QSL time starts from zero for small values of $s$ and increases steadily with increasing $s$ .",
"It is important to note that in any case the QSL time is smaller than the driving time $\\tau _D$ otherwise the miscalculations are done.",
"Also, as can be seen from the diagram, for the interval $s<0.2$ and $s \\in [0.2,1.68]$ , the QSL time in the fermionic environment is greater than the bosonic environment, and in other intervals the QSL time in the bosonic environment is higher than QSL time in the fermionic environment.",
"In Fig.REF , the QSL time is plotted in terms of initial time $\\tau $ for different value of Ohmic parameter for both fermionic and bosonic environment.",
"Fig.REF (a) shows the QSL time as a function of initial time $\\tau $ for sub-Ohmic environment.",
"From Fig.REF (a) one can see for sub-Ohmic bosonic and fermionic environment the QSL time starts to decrease at the beginning of the evolution until it reaches to zero.",
"It is also observed that the QSL time for bosonic environment is greater than fermionic environment since the value of $s$ is smaller than $0.2$ .",
"In Fig.4(b), QSL time is plotted for Ohmic bosonic and fermionic environment.",
"For Ohmic environment the behavior of QSL time is similar with sub-Ohmic environment.",
"It is also observed that the QSL time for fermionic environment is greater than QSL time for bodsonic environment.",
"In Fig.REF (c) and Fig.REF (d), the QSL time is plotted for both super-Ohmic bosonic and fermionic environment.",
"As can be seen by increasing $s$ the QSL time increases for both fermionic and bosonic environment.",
"As can be seen from Fig.REF (d) for super-Ohmic environment with larger value of $s$ due to the occurrence of coherence trapping [50], the QSL time would be gradually trapped to a fixed value, and so leads to a uniform evolution speed for the open system.",
"In Fig.REF , the QSL time is plotted as a function of magnetic parameter $B$ for both bosonic and fermionic environment.",
"As can be seen from Fig.REF the QSL time decreases with increasing magnetic parameter for sub-Ohmic, Ohmic and super-Ohmic bosonic and fermionic environment."
],
[
"Conclusion",
"In this work, we have studied the QSL time for topological qubit influenced by fermionic and bosonic environment.",
"We assume that the environments have an Ohmic-like spectral density.",
"We have shown that for Ohmic and sub-Ohmic environment the QSL time reduces with the starting point in time, in other words, it can be said that the open system experiences a speeded-up dynamics evolution process.",
"It is observed that for super-Ohmic environment with increasing Ohmic parameter the quantum speed limit time gradually reaches to a constant value and thus the speed of evolution reaches to a uniform value.",
"We have also studied the effect of magnetic field on QSL time.",
"It has shown that the QSL time decreases with increasing the value of magnetic field for both bosonic and fermionic environment with different Ohmic parameter."
]
] | 2011.14199 |
[
[
"On compact 4th order finite-difference schemes for the wave equation"
],
[
"Abstract We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\\geq 1$.",
"Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\\geq 2$.",
"The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schr\\\"{o}dinger ones.",
"The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for $n\\geq 2$.",
"For $n=1$ and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme.",
"We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation.",
"Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP.",
"The main schemes are generalized for non-uniform rectangular meshes.",
"We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well."
],
[
"Introduction",
"Compact higher-order finite-difference schemes for PDEs is a popular subject and a vast literature is devoted to them.",
"The case of such type schemes for the wave equation have recently attracted a lot of interest, in particular, see [2], [4], [8], [12], where much more related references can be found.",
"We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$ -dimensional wave equation with constant coefficients, $n\\geqslant 1$ .",
"Their construction on uniform meshes is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation related to the polylinear finite element method (FEM), together with further necessary improvements of the arising scheme for $n\\geqslant 2$ .",
"This alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schrödinger equations (TDSE).",
"The constructed schemes are implicit and three-point in each spatial direction and time.",
"For $n\\geqslant 2$ , there is a scheme with a splitting operator among them.",
"Notice that we use implicit approximations for the second initial condition in the spirit of the approximations for the equation.",
"Curiously, for $n=1$ and the mesh on characteristics of the equation, the 4th order scheme becomes explicit and very close to an exact scheme on a four-point stencil.",
"We present a conditional stability theorem covering the cases of stability in strong (standard) and weak energy norms with respect to both initial functions and free term in the equation.",
"Its corollary rigorously ensures the 4th order error bound in the case of smooth solutions to the IBVP.",
"Note that stability is unconditional for similar compact schemes on uniform meshes for other type PDEs, for example, see [3], [11].",
"Our approach is applied in a unified manner for any $n\\geqslant 1$ (not separately for $n=1$ , 2 or 3 as in many papers), the uniform rectangular (not only square) mesh is taken, the stability results are of standard kind in the theory of finite-difference schemes and proved by the energy techniques (not only by getting bounds for harmonics of the numerical solution as in most papers).",
"In particular, the last point allows us to prove rigorously the 4th order error estimate in the strong energy norm for smooth solutions.",
"Moreover, enlarging of most schemes to the case of the wave equation with the variable coefficient $\\rho (x)$ in front of $\\partial _t^2u$ is simple, and there exists some connection to [2], [12].",
"Also the main schemes are rather easily generalized for non-uniform rectangular meshes in space and time; we apply averaging technique to both aims.",
"Concerning compact schemes on non-uniform meshes for other (1D in space) equations, in particular, see [14], [5], [15], [17].",
"In our 1D numerical experiments, we first concentrate on demonstrating the sensitive dependence of the error orders in the mesh $L^2$ , uniform and strong energy norms on the weak smoothness order of the both initial functions and the weak dominating mixed smoothness order of the free term.",
"The cases of the delta-shaped, discontinuious or with discontinuos derivatives data are covered.",
"The higher-order practical error behavior is shown compared to standard 2nd approximation order schemes [19], [16] thus confirming the essential advantages of 4th order schemes over them in the non-smooth case as well.",
"Second, we present numerical results in the case of non-uniform spatial meshes with various node distribution functions (for the smooth data).",
"The paper is organized as follows.",
"Auxiliary Section contains results on stability of general symmetric three-level method with a weight for hyperbolic equations in the strong and weak energy norms that we need to apply.",
"The main Section is devoted to construction and analysis of the compact 4th order finite-difference schemes.",
"In Section , the main compact schemes are generalized to the case of non-uniform rectangular meshes.",
"The results of these sections have been received by A. Zlotnik.",
"Section contains results of numerical experiments have been accomplished by O. Kireeva."
],
[
"General symmetric three-level method for second order hyperbolic equations and its stability theorem",
"Let $H_h$ be a family of Euclidean spaces endowed with an inner product $(\\cdot ,\\cdot )_h$ and the corresponding norm $\\Vert \\cdot \\Vert _h$ , where $h$ is the parameter (related to a spatial discretization).",
"Let linear operators $B_h$ and $A_h$ act in $H_h$ and have the properties $B_h=B_h^*>0$ and $A_h=A_h^*>0$ .",
"Define the norms $\\Vert w\\Vert _{B_h}=(B_hw,w)_h^{1/2}$ and $\\Vert w\\Vert _{A_h}=(A_hw,w)_h^{1/2}$ in $H_h$ generated by them.",
"We assume that they are related by the following inequality $\\Vert w\\Vert _{A_h}\\leqslant \\alpha _h\\Vert w\\Vert _{B_h}\\ \\ \\ \\forall w\\in H_h \\ \\ \\Leftrightarrow \\ \\ A_h\\leqslant \\alpha _h^2B_h.$ For methods of numerical solving 2nd order elliptic equations, usually $\\alpha _h=c_0/h_{\\min }$ , where $h_{\\min }$ is a minimal size of the spatial discretization.",
"We introduce the uniform mesh $\\overline{\\omega }_{h_t}=\\lbrace t_m=mh_t\\rbrace _{m=0}^M$ on a segment $[0,T]$ , with the step $h_t=T/M>0$ and $M\\geqslant 2$ .",
"Let $\\omega _{h_t}=\\lbrace t_m\\rbrace _{m=1}^{M-1}$ .",
"We introduce the mesh averages and difference operators $\\bar{s}_ty=\\frac{\\check{y}+y}{2},\\,\\ s_ty=\\frac{y+\\hat{y}}{2},\\,\\ \\bar{\\delta }_ty=\\frac{y-\\check{y}}{h_t},\\,\\ \\delta _ty=\\frac{\\hat{y}-y}{h_t},\\,\\ \\mathring{\\delta }_ty=\\frac{\\hat{y}-\\check{y}}{2h_t}$ and $\\Lambda _ty=\\delta _t\\bar{\\delta }_ty=\\frac{\\hat{y}-2y+\\check{y}}{h_t^2}$ with $y^m=y(t_m)$ , $\\check{y}^{m}=y^{m-1}$ and $\\hat{y}^{m}=y^{m+1}$ , as well as the summation operator with the variable upper limit $I_{h_t}^my=h_t\\sum _{l=1}^m y^l$ for $1\\leqslant m\\leqslant M$ and $I_{h_t}^0y=0$ .",
"We consider a general symmetric three-level in $t$ method with a weight $\\sigma $ : $\\big (B_h+\\sigma h_t^2A_h\\big )\\Lambda _tv+A_hv=f\\ \\ \\text{in}\\ \\ H_h\\ \\ \\text{on}\\ \\ \\omega _{h_t},\\\\[1mm]\\big (B_h+\\sigma h_t^2A_h\\big )\\delta _tv^0+{\\textstyle \\frac{1}{2}}h_tA_hv^0=u_1+{\\textstyle \\frac{1}{2}}h_tf^0\\ \\ \\text{in}\\ \\ H_h$ where $v$ : $\\overline{\\omega }_{h_t}\\rightarrow H_h$ is the sought function and the functions $v^0,u_1\\in H_h$ and $f$ : $\\lbrace t_m\\rbrace _{m=0}^{M-1}\\rightarrow H_h$ are given; we omit their dependence on $h$ for brevity.",
"Note that the parameter $\\sigma $ can depend on $\\mathbf {h}:=(h,h_t)$ .",
"Recall that linear algebraic systems with one and the same operator $B_h+\\sigma h_t^2A_h$ has to be solved at time levels $t_m$ to find the solution $v^{m+1}$ , $0\\leqslant m\\leqslant M-1$ .",
"Note that () can be rewritten in the form closer to (REF ): $\\frac{\\big (B_h+\\sigma h_t^2A_h\\big )\\delta _tv^0-u_1}{0.5h_t}+A_hv^0=f^0$ .",
"Let the following conditions related to $\\sigma $ hold: either $\\sigma \\geqslant \\frac{1}{4}$ and $\\varepsilon _0=1$ , or $\\sigma <\\tfrac{1}{4},\\ \\ (\\tfrac{1}{4}-\\sigma )h_t^2\\alpha _h^2\\leqslant 1-\\varepsilon _0^2\\ \\ \\text{for some}\\ \\ 0<\\varepsilon _0<1.$ Then one can introduce the following $\\sigma $ - and $h_t$ -dependent norm in $H_h$ and bound it from below: $\\varepsilon _0\\Vert w\\Vert _{B_h}\\leqslant \\Vert w\\Vert _{0,\\mathbf {h}}:=\\big [\\Vert w\\Vert _{B_h}^2+(\\sigma -\\tfrac{1}{4})h_t^2\\Vert w\\Vert _{A_h}^2\\big ]^{1/2}\\ \\ \\forall w\\in H_h.$ Obviously, for $\\sigma \\leqslant \\tfrac{1}{4}$ , one also has $\\Vert w\\Vert _{0,\\mathbf {h}}\\leqslant \\Vert w\\Vert _{B_h}$ , and then the norms $\\Vert \\cdot \\Vert _{0,\\mathbf {h}}$ and $\\Vert \\cdot \\Vert _{B_h}$ are equivalent uniformly in $\\mathbf {h}$ .",
"We present the stability theorem for method (REF )-() with respect to the initial data $v^0$ and $u_1$ and the free term $f$ in the strong (standard) and weak energy mesh norms.",
"Define the norm $\\Vert y\\Vert _{L_{h_t}^1(H_h)}=\\tfrac{1}{4} h_t\\Vert y^0\\Vert _h+I_{h_t}^{M-1}\\Vert y\\Vert _h$ for $y$ : $\\lbrace t_m\\rbrace _{m=0}^{M-1}\\rightarrow H_h$ .",
"Theorem 2.1 For the solution to method (REF )-(), the following bounds hold: (1) in the strong energy norm $\\max _{1\\leqslant m\\leqslant M}\\big [\\Vert \\bar{\\delta }_tv^m\\Vert _{B_h}^2+(\\sigma -\\tfrac{1}{4})h_t^2\\Vert \\bar{\\delta }_tv^m\\Vert _{A_h}^2+\\Vert \\bar{s}_tv^m\\Vert _{A_h}^2\\big ]^{1/2}\\nonumber \\\\[1mm]\\leqslant \\big (\\Vert v^0\\Vert _{A_h}^2+\\varepsilon _0^{-2}\\Vert B_h^{-1/2}u_1\\Vert _h^2\\big )^{1/2}+2\\varepsilon _0^{-1}\\Vert B_h^{-1/2}f\\Vert _{L_{h_t}^1(H_h)};$ one can replace the $f$ -term with $2I_{h_t}^{M-1}\\Vert A_h^{-1/2}\\bar{\\delta }_tf\\Vert _h+3\\max \\limits _{0\\leqslant m\\leqslant M-1}\\Vert A_h^{-1/2}f^m\\Vert _h$ ; (2) in the weak energy norm $\\max _{0\\leqslant m\\leqslant M}\\max \\Big \\lbrace \\big [\\Vert v^m\\Vert _{B_h}^2+(\\sigma -\\tfrac{1}{4})h_t^2\\Vert v^m\\Vert _{A_h}^2\\big ]^{1/2},\\,\\Vert I_{h_t}^m\\bar{s}_tv\\Vert _{A_h}\\Big \\rbrace \\nonumber \\\\[1mm]\\leqslant \\big [\\Vert v^0\\Vert _{B_h}^2+(\\sigma -\\tfrac{1}{4})h_t^2\\Vert v^0\\Vert _{A_h}^2\\big ]^{1/2}+2\\Vert A_h^{-1/2}u_1\\Vert _h+2\\Vert A_h^{-1/2}f\\Vert _{L_{h_t}^1(H_h)}.$ For $f=\\delta _tg$ , one can replace $2\\Vert A_h^{-1/2}f\\Vert _{L_{h_t}^1(H_h)}$ with $\\tfrac{2}{\\varepsilon _0}I_{h_t}^{M}\\Vert B_{h}^{-1/2}\\big (g-s_tg^0\\big )\\Vert _h$ .",
"Similar bounds have recently been proved in [20] for the method $\\big (\\tau B_h+\\sigma h_t^2A_h\\big )\\Lambda _tv+B_{1h}\\mathring{\\delta }_tv+A_hv=f\\ \\ \\text{in}\\ \\ H_h\\ \\ \\text{on}\\ \\ \\omega _{h_t},\\\\[1mm]\\big (\\tau B_h+{\\textstyle \\frac{1}{2}}h_t B_{1h}+\\sigma h_t^2A_h\\big )\\delta _tv^0+{\\textstyle \\frac{1}{2}}h_tA_hv^0=u_1+{\\textstyle \\frac{1}{2}}h_tf^0\\ \\ \\text{in}\\ \\ H_h$ of a more general form, with the parameter $\\tau >0$ and an operator $B_{1h}=B_{1h}^*>0$ acting in $H_h$ .",
"In these bounds, one can take $\\tau =1$ and easily see from their proofs that the bounds mainly remain valid for $B_{1h}=B_{1h}^*\\geqslant 0$ , in particular, $B_{1h}=0$ (the case considered here), up to the norm of $f$ standing in (REF ) and the norm of $g-s_tg^0$ mentioned in Item 2.",
"To verify the validity of the bounds precisely with the norms of $f$ and $\\tilde{g}:=g-s_tg^0$ indicated in this theorem, it suffices to modify bounds for the following summands with $f$ in the strong energy equality in [20] ${\\textstyle \\frac{1}{2}}h_t(f^0,\\bar{\\delta }_tv^1)_h+2I_{h_t}^{m-1}(f,\\mathring{\\delta }_tv)_h\\leqslant {\\textstyle \\frac{1}{2}}h_t\\Vert B_h^{-1/2}f^0\\Vert _h\\Vert \\bar{\\delta }_tv^1\\Vert _{B_h}\\\\[1mm]+2I_{h_t}^{M-1}\\Vert B_h^{-1/2}f\\Vert _h\\max _{1\\leqslant m\\leqslant M}\\Vert \\bar{\\delta }_tv^m\\Vert _{B_h}\\leqslant \\tfrac{2}{\\varepsilon _0}\\Vert B_h^{-1/2}f\\Vert _{L_{h_t}^1(H_h)}\\max _{1\\leqslant m\\leqslant M}\\Vert \\bar{\\delta }_tv^m\\Vert _{0,\\mathbf {h}}$ and, setting $\\check{I}_{h_t}^mf=I_{h_t}^{m-1}f$ , in the weak energy equality in [20] $2I_{h_t}^m\\big ({\\textstyle \\frac{1}{2}}h_tf^0+\\check{I}_{h_t}f,\\bar{s}_tv\\big )_h=2I_{h_t}^m\\big (\\tilde{g},\\bar{s}_tv\\big )_h\\leqslant \\tfrac{2}{\\varepsilon _0}I_{h_t}^M\\Vert B_{h}^{-1/2}\\tilde{g}\\Vert _h\\max _{1\\leqslant m\\leqslant M}\\Vert \\bar{s}_tv^m\\Vert _{0,\\mathbf {h}},$ for $1\\leqslant m\\leqslant M$ , and the relations $\\mathring{\\delta }_t={\\textstyle \\frac{1}{2}}(\\delta _t+\\bar{\\delta }_t)$ and (REF ) have been applied.",
"Clearly in fact the norm $\\Vert \\cdot \\Vert _{0,\\mathbf {h}}$ stands on the left in (REF ) and on both sides in (REF ).",
"Bounds of type (REF ) with a stronger norm of $f$ can be found in [11].",
"Below we also refer to the following stability result.",
"Remark 2.1 Under assumptions (REF ) with $\\varepsilon _0=0$ , instead of bound (REF ) the following one holds $\\max _{1\\leqslant m\\leqslant M}\\big [\\Vert \\bar{\\delta }_tv^m\\Vert _{B_h}^2+(\\sigma -\\tfrac{1}{4})h_t^2\\Vert \\bar{\\delta }_tv^m\\Vert _{A_h}^2+\\Vert \\bar{s}_tv^m\\Vert _{A_h}^2\\big ]^{1/2} \\leqslant \\big [\\Vert v^0\\Vert _{A_h}^2\\nonumber \\\\[1mm]+\\Vert (B_h+\\sigma h_t^2A_h)^{-1/2}u_1\\Vert _h^2\\big ]^{1/2}+2I_{h_t}^{M-1}\\Vert A_h^{-1/2}\\bar{\\delta }_tf\\Vert _h+3\\max \\limits _{0\\leqslant m\\leqslant M-1}\\Vert A_h^{-1/2}f^m\\Vert _h,$ whereas bound (REF ) remains valid (its proof does not change for $\\varepsilon _0\\geqslant 0$ ).",
"To be convinced of the latter bound, it is necessary to transform and bound differently the terms with $v^0$ and $u_1$ in the case $f=0$ in the strong energy equality in [20].",
"Namely, using the formula $\\bar{s}_tv^1=v^0+{\\textstyle \\frac{1}{2}}h_t\\bar{\\delta }_tv^1$ and equation () with $f^0=0$ , we can set $C_h:=(B_h+\\sigma h_t^2A_h)^{-1}$ and obtain $(A_hv^0,\\bar{s}_tv^1)_h+(u_1,\\bar{\\delta }_tv^1)_h=\\Vert v^0\\Vert _{A_h}^2+\\big ({\\textstyle \\frac{1}{2}}h_tA_hv^0+u_1,\\bar{\\delta }_tv^1\\big )_h=\\Vert v^0\\Vert _{A_h}^2\\nonumber \\\\[1mm]+\\big (C_h\\big ({\\textstyle \\frac{1}{2}}h_tA_hv^0+u_1\\big ),-{\\textstyle \\frac{1}{2}}h_tA_hv^0+u_1\\big )_h=\\Vert v^0\\Vert _{A_h}^2+\\Vert u_1\\Vert _{C_h}^2-\\Vert {\\textstyle \\frac{1}{2}}h_tA_hv^0\\Vert _{C_h}^2$ since $C_h=C_h^*>0$ .",
"This implies the first bound of this Remark.",
"Notice that $B_h+\\sigma h_t^2A_h\\geqslant \\varepsilon _0B_h+\\tfrac{1}{4} h_t^2A_h$ under the assumptions either $\\sigma \\geqslant \\frac{1}{4}$ and $\\varepsilon _0=1$ , or (REF ) with $0\\leqslant \\varepsilon _0<1$ and, as a corollary, $C_h\\leqslant \\varepsilon _0^{-1}B_h^{-1}$ (for $\\varepsilon _0\\ne 0$ ) and $C_h\\leqslant 4h_t^{-2}A_h^{-1}$ .",
"But, for $\\varepsilon _0=0$ , the quantity $\\Vert w\\Vert _{0,\\mathbf {h}}$ could be (in general) only a semi-norm in $H_h$ , and its lower bound by $\\Vert w\\Vert _{B_h}$ uniformly in $\\mathbf {h}$ is not valid any more.",
"It is well-known that each of bounds (REF )-(REF ) implies existence and uniqueness of the solution to method (REF )-() for any given $v^0,u_1\\in H_h$ and $f$ : $\\lbrace t_m\\rbrace _{m=0}^{M-1}\\rightarrow H_h$ .",
"The same concerns finite-difference schemes below."
],
[
"Construction and properties of compact finite-difference schemes of the 4th order of approximation",
"We consider the following IBVP with the nonhomogeneous Dirichlet boundary condition for the slightly generalized wave equation $\\partial _t^2u-a_i^2\\partial _i^2u =f(x,t)\\ \\ \\text{in}\\ \\ Q_T=\\Omega \\times (0,T);\\\\[1mm]u|_{\\Gamma _T}=g(x,t);\\ \\ u|_{t=0}=u_0(x),\\ \\ \\partial _tu|_{t=0}=u_1(x),\\ \\ x\\in \\Omega .$ Here $a_1>0,\\ldots ,a_n>0$ are constants, $x=(x_1,\\ldots ,x_n)$ , $\\Omega =(0,X_1)\\times \\ldots \\times (0,X_n)$ , $n\\geqslant 1$ , $\\partial \\Omega $ is the boundary of $\\Omega $ and $\\Gamma _T=\\partial \\Omega \\times (0,T)$ is the lateral surface of $Q_T$ .",
"Hereafter the summation from 1 to $n$ over the repeated indices $i,j$ (and only over them) is assumed.",
"Below $\\delta ^{(ij)}$ is the Kronecker symbol.",
"Define the uniform rectangular mesh $\\bar{\\omega }_h=\\lbrace x_{\\mathbf {k}}=(k_1h_1,\\ldots ,k_nh_n);\\, 0\\leqslant k_1\\leqslant N_1,\\ldots ,0\\leqslant k_n\\leqslant N_n\\rbrace $ in $\\bar{\\Omega }$ with the steps $h_1=X_1/N_1,\\ldots ,h_n=X_n/N_n$ , $h=(h_1,\\ldots ,h_n)$ and $\\mathbf {k}=(k_1,\\ldots ,k_n)$ .",
"Let $\\omega _h=\\lbrace x_{\\mathbf {k}};\\, 1\\leqslant k_1\\leqslant N_1-1,\\ldots ,1\\leqslant k_n\\leqslant N_n-1\\rbrace $ and $\\partial \\omega _h=\\bar{\\omega }_h\\backslash \\omega _h$ be the internal part and boundary of $\\bar{\\omega }_h$ .",
"Define the meshes $\\omega _{\\mathbf {h}}:=\\omega _h\\times \\omega _{h_t}$ in $Q_T$ and $\\partial \\omega _{\\mathbf {h}}=\\partial \\omega _h\\times \\lbrace t_m\\rbrace _{m=1}^M$ on $\\bar{\\Gamma }_T$ .",
"We introduce the well-known difference operators $ (\\Lambda _lw)_{\\mathbf {k}}=\\tfrac{1}{h_l^2}(w_{\\mathbf {k}+\\mathbf {e}_l}-2w_{\\mathbf {k}}+w_{\\mathbf {k}-\\mathbf {e}_l})$ , $l=1,\\ldots ,n$ , on $\\omega _{h}$ , where $w_{\\mathbf {k}}=w(x_{\\mathbf {k}})$ and $\\mathbf {e}_1,\\ldots ,\\mathbf {e}_n$ is the standard coordinate basis in $\\mathbb {R}^n$ .",
"Let below $H_h$ be the space of functions defined on $\\bar{\\omega }_h$ , equal 0 on $\\partial \\omega _h$ and endowed with the inner product $ (v,w)_h=h_1\\ldots h_n\\sum \\nolimits _{x_{\\mathbf {k}}\\in \\omega _h}v_{\\mathbf {k}}w_{\\mathbf {k}}$ and the norm $\\Vert w\\Vert _h=(w,w)_h^{1/2}$ .",
"Lemma 3.1 For the sufficiently smooth in $\\bar{Q}_T$ solution $u$ to equation (REF ), the following formula holds $\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )\\Lambda _tu-a_j^2s_{N\\hat{j}}\\Lambda _ju-f_N=O(|\\mathbf {h}|^4)\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},$ where $s_N:=I+{\\textstyle \\frac{1}{12}}h_i^2\\Lambda _i,\\,\\ s_{N\\hat{j}}:=I+(1-\\delta ^{(ij)}){\\textstyle \\frac{1}{12}}h_i^2\\Lambda _i,\\,\\ f_N:=f+{\\textstyle \\frac{1}{12}}h_t^2\\Lambda _tf+{\\textstyle \\frac{1}{12}}h_i^2\\Lambda _if,$ and $I$ is the identity operator.",
"Note that $s_{N\\hat{j}}=I$ for $n=1$ .",
"We give two different proofs.",
"1.",
"The first one follows to the classical Numerov approach.",
"We take the simplest explicit three-level discretization of equation (REF ) having the form $\\Lambda _tv-a_i^2\\Lambda _iv=f\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}}$ (the particular case of equation (REF ) for $B_h=I$ , $A_h=-a_i^2\\Lambda _iv$ and $\\sigma =0$ ) and, under the assumption of sufficient smoothness of $u$ , select the leading term of its approximation error $\\psi _e:=\\Lambda _tu-a_i^2\\Lambda _iu-f$ : $\\psi _e=\\Lambda _tu-\\partial _t^2u-a_i^2(\\Lambda _iu-\\partial _i^2u)={\\textstyle \\frac{1}{12}}h_t^2\\partial _t^4u-{\\textstyle \\frac{1}{12}}h_i^2a_i^2\\partial _i^4u+O(|\\mathbf {h}|^4).$ We express the derivatives $\\partial _t^4u$ and $\\partial _k^4u$ in terms of mixed derivatives by differentiating equation (REF ): $\\partial _t^4u=a_i^2\\partial _i^2\\partial _t^2u+\\partial _t^2f,\\ \\ a_k^2\\partial _k^4u=\\partial _k^2\\partial _t^2u-(1-\\delta ^{(kj)})a_j^2\\partial _k^2\\partial _j^2u-\\partial _k^2f.$ Then formula (REF ) takes the form $\\psi _e=\\tfrac{h_t^2}{12}a_i^2\\partial _i^2\\partial _t^2u-\\tfrac{h_i^2}{12}\\partial _i^2\\partial _t^2u+\\tfrac{h_i^2}{12}(1-\\delta ^{(ij)})a_j^2\\partial _i^2\\partial _j^2u+\\tfrac{h_t^2}{12}\\partial _t^2f+\\tfrac{h_i^2}{12}\\partial _i^2f+O(|\\mathbf {h}|^4).$ Here all the 2nd order derivatives can be replaced by the corresponding symmetric three-point difference discretizations preserving the order of the remainder: $\\psi _e=\\tfrac{h_t^2}{12}a_i^2\\Lambda _i\\Lambda _tu-\\tfrac{h_i^2}{12}\\Lambda _i\\Lambda _tu+\\tfrac{h_i^2}{12}(1-\\delta ^{(ij)})a_j^2\\Lambda _i\\Lambda _ju+\\tfrac{h_t^2}{12}\\Lambda _tf+\\tfrac{h_i^2}{12}\\Lambda _if+O(|\\mathbf {h}|^4).$ Recalling the definition of $\\psi _e$ in (REF ), we can rewrite the last formula as (REF ).",
"2.",
"The second proof is based on averaging of equation (REF ) related to the polylinear finite elements.",
"We define the well-known average in the variable $x_k$ related to the linear finite elements $(q_kw)(x_k)=\\tfrac{1}{h_k}\\int _{-h_k}^{h_k}w(x_k+\\xi )\\big (1-\\tfrac{|\\xi |}{h_k}\\big )\\,d\\xi .$ For a function $w(x_k)$ smooth on $[0,X_k]$ , the following formulas hold $q_k\\partial _k^2w=\\Lambda _kw,\\ \\ \\\\[1mm]q_kw=w+{\\textstyle \\frac{1}{12}}h_k^2\\partial _k^2w+q_k\\rho _{k4}(\\partial _k^4w)=w+{\\textstyle \\frac{1}{12}}h_k^2\\Lambda _kw+\\tilde{\\rho }_{k4}(\\partial _k^4w),\\nonumber \\\\[1mm]\\hspace{-8.0pt} |q_k\\rho _{ks}(\\partial _k^sw)|\\leqslant c_sh_k^s\\Vert \\partial _k^sw\\Vert _{C(I_{kl})},\\, s=2,4,\\ |\\tilde{\\rho }_{k4}(\\partial _k^4w)|\\leqslant \\tilde{c}_4h_k^4\\Vert \\partial _k^4w\\Vert _{C(I_{kl})}$ and $q_kw=w+q_k\\rho _{k2}(\\partial _k^2w)$ at the nodes $x_k=x_{kl}:=lh_k$ , $1\\leqslant l\\leqslant N_k-1$ , with $I_{kl}:=[x_{k(l-1)},x_{k(l+1)}]$ .",
"The first formula is checked by integrating by parts and other formulas hold owing to the Taylor formula at $x_{kl}$ with the residual in the integral form $\\rho _{ks}(w)(x_k)=\\tfrac{1}{(s-1)!",
"}\\int _{x_{kl}}^{x_k}w(\\xi )(x_k-\\xi )^{s-1}\\,d\\xi ,$ for $s=2,4$ , together with $\\tfrac{1}{h_k}\\int _{-h_k}^{h_k}{\\textstyle \\frac{1}{2}}\\xi ^2\\big (1-\\tfrac{|\\xi |}{h_k}\\big )\\,d\\xi ={\\textstyle \\frac{1}{12}}h_k^2$ .",
"The respective formulas hold for the averaging operator $q_t$ in the variable $t=x_{n+1}$ as well (since one can set $X_{n+1}=T$ and $h_{n+1}=h_t$ ).",
"We apply the operator $\\bar{q}q_t$ with $\\bar{q}:=q_1\\ldots q_n$ to equation (REF ) at the nodes of $\\omega _{\\mathbf {h}}$ and get $\\bar{q}\\Lambda _tu-a_i^2 \\bar{q}_{\\hat{i}}q_t\\Lambda _iu=\\bar{q}q_tf\\ \\ \\text{with}\\ \\ \\bar{q}_{\\hat{i}}:=\\prod _{1\\leqslant k\\leqslant n,\\, k\\ne i}q_k.$ The multiple application of the above formulas for the averages leads to $\\Lambda _tu+{\\textstyle \\frac{1}{12}}h_i^2\\Lambda _i\\Lambda _tu-a_i^2\\big [\\Lambda _i^2u+(1-\\delta ^{(ij)}){\\textstyle \\frac{1}{12}}h_j^2\\Lambda _j\\Lambda _iu+{\\textstyle \\frac{1}{12}}h_t^2\\Lambda _i\\Lambda _tu\\big ]\\\\[1mm]=f+{\\textstyle \\frac{1}{12}}h_i^2\\Lambda _if+{\\textstyle \\frac{1}{12}}h_t^2\\Lambda _tf+O(|\\mathbf {h}|^4),$ and thus formula (REF ) is derived once again.",
"Remark 3.2 For the first order in time parabolic equation or TDSE, one should apply the simpler averaging $q_ty^m=\\frac{1}{h_t}\\int _{t_{m-1}}^{t_m}y(t)\\,dt$ in time to derive two-level higher-order compact schemes.",
"Formula (REF ) means that the discretization of equation (REF ) of the form $\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )\\Lambda _tv-a_i^2s_{N\\hat{i}}\\Lambda _iv=f_N\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}}$ has the approximation error of the order $O(|\\mathbf {h}|^4)$ .",
"Notice that the coefficients of formulas $y+{\\textstyle \\frac{1}{12}}h_t^2\\Lambda _ty={\\textstyle \\frac{1}{12}}(\\hat{y}+10f+\\check{y}),\\,\\ {\\textstyle \\frac{1}{12}}h_i^2\\Lambda _iw_{\\mathbf {k}}={\\textstyle \\frac{1}{12}}\\delta ^{(ii)}(w_{\\mathbf {k}-\\mathbf {e}_i}+w_{\\mathbf {k}+\\mathbf {e}_i})-\\tfrac{n}{6}w_{\\mathbf {k}}$ respectively on $\\omega _{h_t}$ and $\\omega _h$ are independent of $\\mathbf {h}$ .",
"For discretization (REF ), we consider the corresponding equation at $t_0=0$ $\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )\\delta _tv^0-{\\textstyle \\frac{1}{2}}h_ta_i^2s_{N\\hat{i}}\\Lambda _iv^0=u_{1N}+{\\textstyle \\frac{1}{2}}h_tf_N^0\\ \\ \\text{on}\\ \\ \\omega _h,$ cp.",
"(REF )-(), and find out for which $u_{1N}$ and $f_N^0$ its approximation error also has the order $O(|\\mathbf {h}|^4)$ .",
"Let $0<\\bar{h}_t\\leqslant T$ and $h_t\\leqslant \\bar{h}_t$ .",
"Lemma 3.2 For the sufficiently smooth in $\\bar{Q}_{\\bar{h}_t}$ solution $u$ to equation (REF ) satisfying the initial conditions from (), under the choice $u_{1N}=\\big (s_N+{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )u_1,\\,\\ \\\\[1mm]f_N^0=f_{dh_t}^{(0)}+{\\textstyle \\frac{1}{12}}h_i^2\\Lambda _if^0,\\ \\ f_{dh_t}^{(0)}=f_{d}^{(0)}+O(h_t^3)$ on $\\omega _h$ , where $f_{d}^{(0)}:=f_0+\\tfrac{1}{3} h_t(\\partial _tf)_0+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^2f)_0$ with $y_0:=y|_{t=0}$ , the approximation error of equation (REF ) satisfies the following formula $\\psi _e^0:=\\big (s_N-\\tfrac{h_t^2}{12}a_i^2\\Lambda _i\\big )(\\delta _tu)^0-\\tfrac{h_t}{2}a_i^2s_{N\\hat{i}}\\Lambda _iu_0-u_{1N}-\\tfrac{h_t}{2}f_N^0=O(|\\mathbf {h}|^4).$ Notice that $f_{d}^{(0)}$ is not the term $f_0+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^2f)_0$ of type approximated above.",
"Let $0\\leqslant t\\leqslant \\bar{h}_t$ .",
"Once again we give two proofs.",
"1.",
"Using Taylor's formula in $t$ and grouping separately terms with the time derivatives of odd and even orders, we obtain $\\psi _e^0=\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )(\\partial _tu)_0+\\tfrac{1}{6} h_t^2(\\partial _t^3u)_0+{\\textstyle \\frac{1}{2}}h_t\\big [\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )(\\partial _t^2u)_0\\\\[1mm]+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^4u)_0-a_i^2s_{N\\hat{i}}\\Lambda _iu_0\\big ]-u_{1N}-{\\textstyle \\frac{1}{2}}h_tf_N^0+O(|\\mathbf {h}|^4).$ In virtue of equation (REF ) we have $\\partial _t^3u=a_i^2\\partial _i^2\\partial _tu+\\partial _tf=a_i^2\\Lambda _i^2\\partial _tu+\\partial _tf+O(|h|^2).$ Moreover, $(\\partial _tu)_0=u_1$ , therefore we find $\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )(\\partial _tu)_0+\\tfrac{1}{6} h_t^2(\\partial _t^3u)_0\\nonumber \\\\[1mm]=\\big (s_N+{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )u_1+\\tfrac{1}{6} h_t^2(\\partial _tf)_0+O(|\\mathbf {h}|^4).$ Next, the first formula (REF ) implies $\\partial _t^4u=a_i^2\\Lambda _i\\partial _t^2u+\\partial _t^2f+O(|h|^2)$ and thus $\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )(\\partial _t^2u)_0+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^4u)_0=s_N(\\partial _t^2u)_0+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^2f)_0+O(|\\mathbf {h}|^4).$ Using (REF ) for $t=0$ and the formula $s_N=s_{N\\hat{k}}+{\\textstyle \\frac{1}{12}}h_k^2\\Lambda _k$ , we also have $s_N(\\partial _t^2u)_0-a_i^2s_{N\\hat{i}}\\Lambda _iu_0=s_N(a_i^2\\partial _i^2u_0+f_0)-a_i^2s_{N\\hat{i}}\\Lambda _iu_0\\\\[1mm]=a_i^2s_{N\\hat{i}}(\\partial _i^2u_0-\\Lambda _iu_0)+{\\textstyle \\frac{1}{12}}h_i^2a_i^2\\Lambda _i\\partial _i^2u_0+s_Nf_0\\\\[1mm]=a_i^2s_{N\\hat{i}}\\big (-{\\textstyle \\frac{1}{12}}h_i^2\\partial _i^4u_0\\big )+{\\textstyle \\frac{1}{12}}h_i^2a_i^2\\Lambda _i\\partial _i^2u_0+s_Nf_0+O(|h|^4)\\\\[1mm]={\\textstyle \\frac{1}{12}}h_i^2a_i^2(\\Lambda _i\\partial _i^2u_0-\\partial _i^4u_0)+s_Nf_0+O(|h|^4)=s_Nf_0+O(|h|^4).$ Therefore we have proved the formula $\\big (s_N-{\\textstyle \\frac{1}{12}}h_t^2a_i^2\\Lambda _i\\big )(\\partial _t^2u)_0+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^4u)_0-a_i^2s_{N\\hat{i}}\\Lambda _iu_0=s_Nf_0+{\\textstyle \\frac{1}{12}}h_t^2(\\partial _t^2f)_0+O(|h|^4).$ This formula and (REF ) under choice (REF )-() lead to formula (REF ).",
"2.",
"Again the second proof is based on averaging of equation (REF ).",
"We define the related one-sided average in $t$ over $(0,h_t)$ $q_ty^0=\\tfrac{2}{h_t}\\int _0^{h_t}y(t)\\big (1-\\tfrac{t}{h_t}\\big )\\,dt$ and apply $\\tfrac{h_t}{2}q_t(\\cdot )^0$ to (REF ).",
"Since $\\tfrac{h_t}{2}(q_t\\partial _tu)^0=(\\delta _tu)^0-(\\partial _tu)_0$ , we get $\\bar{q}(\\delta _tu)^0-\\tfrac{h_t}{2}a_i^2\\bar{q}_{\\hat{i}}\\Lambda _iq_tu^0=\\bar{q}u_1+\\tfrac{h_t}{2}\\bar{q}q_tf^0.$ Using Taylor's formula at $t=0$ and calculating the arising integrals, we find $\\tfrac{h_t}{2}q_tf^0=\\tfrac{h_t}{2}f_0+\\tfrac{h_t^2}{6}(\\partial _tf)_0+\\tfrac{h_t^3}{24}(\\partial _t^2f)_0+O(h_t^4)=\\tfrac{h_t}{2}f_d^{(0)}+O(h_t^4).$ Here we omit the integral representations for $O(h_t^4)$ -terms for brevity.",
"As in the proof of Lemma REF and owing to the last expansion, we have $\\bar{q}(\\delta _tu)^0=s_N(\\delta _tu)^0+O(|h|^4)$ and $\\bar{q}u_1=s_Nu_1+O(|h|^4),\\ \\ \\tfrac{h_t}{2}q_t\\bar{q}f^0=\\tfrac{h_t}{2}f_d^{(0)}+\\tfrac{1}{12}h_i^2\\Lambda _if_0+O(|\\mathbf {h}|^4).$ Also owing to Taylor's formula in $t$ at $t=0$ we can write down $u(\\cdot ,t)=u_0+tu_1+\\tfrac{t^2}{h_t}((\\delta _tu)^0-u_1)+O(t^3).$ Thus similarly first to (REF ) and second to (REF ) we obtain $\\tfrac{h_t}{2}a_i^2\\bar{q}_{\\hat{i}}\\Lambda _iq_tu^0=\\tfrac{h_t}{2}a_i^2\\bar{q}_{\\hat{i}}\\Lambda _iu_0+\\tfrac{h_t^2}{6}a_i^2\\bar{q}_{\\hat{i}}\\Lambda _iu_1+\\tfrac{h_t^2}{12}a_i^2\\bar{q}_{\\hat{i}}\\Lambda _i((\\delta _tu)^0-u_1)+O(h_t^4)\\\\[1mm]=\\tfrac{h_t}{2}a_i^2s_{N\\hat{i}}\\Lambda _iu_0+\\tfrac{h_t^2}{12}a_i^2\\Lambda _iu_1+\\tfrac{h_t^2}{12}a_i^2s_{N\\hat{i}}\\Lambda _i(\\delta _tu)^0+O(|\\mathbf {h}|^4).$ Inserting all the derived formulas into (REF ), we again obtain the desired result.",
"Remark 3.3 If $f$ is sufficiently smooth in $t$ in $\\bar{Q}_{\\bar{h}_t}$ , then the property $f_{dh_t}^{(0)}=f_{d}^{(0)}+O(h_t^3)$ (see ()) holds for the following three- and two-level approximations $f_{dh_t}^{(0)}=\\tfrac{7}{12}f^0+{\\textstyle \\frac{1}{2}}f^1-{\\textstyle \\frac{1}{12}}f^2,\\ \\ f_{dh_t}^{(0)}=\\tfrac{1}{3}f^0+\\tfrac{2}{3}f^{1/2}\\ \\ \\text{with}\\ \\ f^{1/2}:=f|_{t=h_t/2}.$ One can easily check this using the Taylor formula in $t$ at $t=0$ .",
"If $f$ is sufficiently smooth in $t$ in $\\bar{\\Omega }\\times [-\\bar{h}_t,\\bar{h}_t]$ , then clearly the same property holds for the one more three-level approximation $f_{dh_t}^{(0)}=f^0+\\tfrac{1}{3} h_t\\mathring{\\delta }_tf^0+{\\textstyle \\frac{1}{12}}h_t^2\\Lambda _tf^0=-{\\textstyle \\frac{1}{12}}f^{-1}+\\tfrac{5}{6}f^0+\\tfrac{1}{4} f^1\\ \\text{with}\\ f^{-1}:=f|_{t=-h_t}.$ Remark 3.4 Below we consider the case of non-smooth $f$ .",
"Namely the above second proofs of Lemmas REF -REF clarify that then $f_N^m$ should be replaced with $\\bar{q}q_tf^m$ , $0\\leqslant m\\leqslant M-1$ , according to (REF ) and (REF ) and identically to the polylinear FEM with the weight [19], or with some its suitable approximation.",
"In the simplest case $n=1$ , equations (REF )-(REF ) supplemented with the boundary condition take the following form $\\big [I+\\tfrac{1}{12}(h_1^2-a_1^2h_t^2)\\Lambda _1\\big ]\\Lambda _tv-a_1^2\\Lambda _1v=f_N,\\\\[1mm]v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\ \\big [I+\\tfrac{1}{12}(h_1^2-a_1^2h_t^2)\\Lambda _1\\big ]\\delta _tv^0-{\\textstyle \\frac{1}{2}}h_ta_1^2\\Lambda _1v^0=u_{1N}+{\\textstyle \\frac{1}{2}}h_tf_N^0,$ where equations are valid respectively on $\\omega _{\\mathbf {h}}$ and $\\omega _h$ .",
"Hereafter we assume that the function $v^0$ is given on $\\bar{\\omega }_h$ and take the general nonhomogeneous Dirichlet boundary condition.",
"This scheme can be interpreted as the particular case of scheme (REF )-() with the operators $B_h=I$ and $A_h=-a_1^2\\Lambda _1$ and the weight $\\sigma =\\sigma (\\mathbf {h})=\\tfrac{1}{12}\\big (1-\\tfrac{h_1^2}{a_1^2h_t^2}\\big )$ (a similar choice of $\\sigma $ was used in [11] in the 1D parabolic case) or the bilinear finite element method [19] with $B_h=I+\\tfrac{1}{6}h_1^2\\Lambda _1$ , $A_h=-a_1^2\\Lambda _1$ and $\\sigma =\\sigma (\\mathbf {h})=\\tfrac{1}{12}\\big (1+\\tfrac{h_1^2}{a_1^2h_t^2}\\big )$ (though the right-hand sides of the equations are not the same; but see also Remark REF ).",
"But for $n\\geqslant 2$ the above constructed equations (REF )-(REF ) are not of type (REF )-().",
"Therefore we replace them with the following one $\\big (s_N+{\\textstyle \\frac{1}{12}}h_t^2A_N\\big )\\Lambda _tv+A_Nv=f_N\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},\\\\[1mm]v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\ (s_N+{\\textstyle \\frac{1}{12}}h_t^2A_N)\\delta _tv^0+{\\textstyle \\frac{1}{2}}h_tA_Nv^0=u_{1N}+{\\textstyle \\frac{1}{2}}h_tf_N^0\\ \\ \\text{on}\\ \\ \\omega _h,$ where $A_N:=-a_i^2s_{N\\hat{i}}\\Lambda _i$ , that corresponds to the case $B_h=s_N$ , $A_h=A_N$ and $\\sigma ={\\textstyle \\frac{1}{12}}$ .",
"Since $A_N+a_i^2\\Lambda _i=a_i^2(I-s_{N\\hat{i}})\\Lambda _i$ , we have $h_t^2(A_N+a_i^2\\Lambda _i)\\Lambda _tu=O(|\\mathbf {h}|^4)$ and $h_t^2(A_N+a_i^2\\Lambda _i)(\\delta _tu)^0=O(|\\mathbf {h}|^4)$ for a function $u$ sufficiently smooth in $\\bar{Q}_T$ , and thus the approximation errors of the both equations of this scheme are also of the order $O(|\\mathbf {h}|^4)$ .",
"But the latter scheme fails for $n\\geqslant 3$ similarly to [3] in the case of the TDSE.",
"The point is that $s_N$ should approximate $I$ adequately, but for the minimal and maximal eigenvalues of $s_N<I$ as the operator in $H_h$ we have $\\lambda _{\\min }(s_N)=1-\\tfrac{1}{3}\\sin ^2\\tfrac{\\pi (N_i-1)}{2N_i}<\\lambda _{\\max }(s_N)<1.$ Therefore $\\lambda _{\\min }(s_N)>1-\\tfrac{n}{3}$ and $\\lambda _{\\min }(s_N)=1-\\tfrac{n}{3}+O\\big (\\delta ^{(ii)}\\frac{1}{N_i^2}\\big )$ that is suitable for $n=1,2$ , but $s_N$ becomes almost singular for $n=3$ and even $\\lambda _{\\min }(s_N)<0$ (i.e., $s_N$ is not positive definite any more) for $n\\geqslant 4$ , for small $|h|$ .",
"Thus for $n=3$ it is of sense to replace the last scheme with the scheme $\\big (\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2A_N\\big )\\Lambda _tv+A_Nv=f_N\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},\\\\[1mm]v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\ (\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2A_N)\\delta _tv^0+{\\textstyle \\frac{1}{2}}h_tA_Nv^0=u_{1N}+{\\textstyle \\frac{1}{2}}h_tf_N^0\\ \\ \\text{on}\\ \\ \\omega _h.$ Moreover, for any $n\\geqslant 1$ we can use the following scheme $\\big (\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2\\bar{A}_N\\big )\\Lambda _tv+\\bar{A}_Nv=f_N\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},\\\\[1mm]v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\ \\big (\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2\\bar{A}_N\\big )\\delta _tv^0+{\\textstyle \\frac{1}{2}}h_t\\bar{A}_Nv^0=u_{1N}+{\\textstyle \\frac{1}{2}}h_tf_N^0\\ \\ \\text{on}\\ \\ \\omega _h$ (cp.",
"[3] in the case of the TDSE); for $n=1$ it coincides with (REF )-().",
"Here the operators $\\bar{s}_N:=\\prod _{k=1}^ns_{kN},\\,\\ \\bar{s}_{N\\hat{l}}:=\\prod _{1\\leqslant k\\leqslant n,\\,k\\ne l}s_{kN},\\,\\ s_{kN}:=I+{\\textstyle \\frac{1}{12}}h_k^2\\Lambda _k,\\,\\ \\bar{A}_N:=-a_i^2\\bar{s}_{N\\hat{i}}\\Lambda _i$ are used, with $\\bar{s}_{N\\hat{l}}=I$ for $n=1$ .",
"The operator $\\bar{s}_N$ is the splitting version of $s_N$ , and $\\bar{s}_{N\\hat{l}}$ is the $(n-1)$ -dimensional case of $\\bar{s}_N$ .",
"Clearly $\\bar{A}_N=A_N$ for $n=1,2$ .",
"Herewith for the minimal and maximal eigenvalues of $\\bar{s}_N<I$ as the operator in $H_h$ we have $(\\tfrac{2}{3})^n<\\lambda _{\\min }(\\bar{s}_N)=\\prod _{k=1}^n 1-\\tfrac{1}{3}\\sin ^2\\tfrac{\\pi (N_k-1)}{2N_k}<\\lambda _{\\max }(\\bar{s}_N)<1.$ Moreover, the following relation between $\\bar{s}_N$ and $s_N$ holds $\\bar{s}_N=s_N+\\sum _{k=2}^n\\sum _{1\\leqslant i_1<\\ldots <i_k\\leqslant n}{\\textstyle \\frac{1}{12}}h_{i_1}^2\\Lambda _{i_1}\\ldots {\\textstyle \\frac{1}{12}}h_{i_k}^2\\Lambda _{i_k}.$ In virtue of the last formula we have $(\\bar{s}_N-s_N)\\Lambda _tu=O(|h|^4)$ and $(\\bar{s}_N-s_N)(\\delta _tu)^0=O(|h|^4)$ for a function $u$ sufficiently smooth in $\\bar{Q}_T$ , thus the approximation errors of the both equations of scheme (REF )-() still have the order $O(|\\mathbf {h}|^4)$ as for the previous scheme (REF )-().",
"Since $\\bar{A}_N-A_N=-a_i^2(\\bar{s}_{N\\hat{i}}-s_{N\\hat{i}})\\Lambda _i$ , in virtue of (REF ) we have $(\\bar{A}_N-A_N)y=O(|h|^4)$ for $y=\\Lambda _tu, u, (\\delta _tu)^0$ and a function $u$ sufficiently smooth in $\\bar{Q}_T$ , and thus the approximation errors of the both equations of scheme (REF )-() also have the order $O(|\\mathbf {h}|^4)$ as for the previous scheme (REF )-().",
"Finally, we recommend to apply scheme (REF )-(REF ) only in the case $n=1$ when it takes the form (REF )-().",
"Instead, for $n=2$ and 3, respectively schemes (REF )-() and (REF )-() can be applied.",
"Scheme (REF )-() is more universal and can be applied for any $n\\geqslant 1$ ; for $n=1$ , it coincides with (REF )-() but for $n=2$ and 3 its operators are more complicated than in (REF )-() and (REF )-() and thus it can be more spatially dissipative in practice.",
"Remark 3.5 Importantly, for example, scheme (REF )-() could be derived immediately like in the second proofs of Lemmas REF -REF by applying more direct though more complicated approximations of the averages in (REF ) and (REF ): $\\bar{q}\\Lambda _tu-a_i^2\\bar{q}_{\\hat{i}}q_t\\Lambda _iu=\\bar{s}_N\\Lambda _tu-a_i^2\\bar{s}_{N\\hat{i}}(I+\\tfrac{h_t^2}{12}\\Lambda _t)\\Lambda _iu+O(|\\mathbf {h}|^4)\\\\[1mm]=(\\bar{s}_N+\\tfrac{h_t^2}{12}\\bar{A}_N)\\Lambda _tu+\\bar{A}_Nu+O(|\\mathbf {h}|^4),\\\\[1mm]\\bar{q}(\\delta _tu)^0-\\tfrac{h_t}{2}a_i^2\\bar{q}_{\\hat{i}}\\Lambda _iq_tu^0=\\bar{s}_N(\\delta _tu)^0-a_i^2\\bar{s}_{\\hat{i}}\\Lambda _i\\big (\\tfrac{h_t}{2}u_0+\\tfrac{h_t^2}{12}u_1+\\tfrac{h_t^2}{12}(\\delta _tu)^0\\big )\\\\[1mm]+O(|\\mathbf {h}|^4)=\\big (\\bar{s}_N+\\tfrac{h_t^2}{12}\\bar{A}_N\\big )(\\delta _tu)^0+\\tfrac{h_t}{2}\\bar{A}_Nu_0-\\tfrac{h_t^2}{12}a_i^2\\Lambda _iu_1+O(|\\mathbf {h}|^4).$ For $n=1$ , implementation of scheme (REF )-() is simple and at each time level $\\lbrace t_m\\rbrace _{m=1}^M$ comes down to solving systems of linear algebraic equations with the same tridiagonal matrix.",
"For $n\\geqslant 2$ , all the constructed schemes can be effectively implemented by means of solving the systems of linear algebraic equations with the same matrix arising at each time level using FFT with respect to sines in all (or $n-1$ ) spatial directions (after excluding the given values $\\hat{v}|_{\\partial \\omega _h}=\\hat{g}$ in the equations at the nodes closest to $\\partial \\omega _h$ ).",
"The matrices are non-singular (more exactly, symmetric and positive definite after the mentioned excluding) that is definitely guaranteed under the hypotheses of Theorem REF below.",
"Note that the FFT-based algorithms have been very effective in practice in the recent study [21].",
"Remark 3.6 It is not difficult to extend the constructed schemes to the case of more general equation $\\rho \\partial _t^2u-a_i^2\\partial _i^2u=f$ with $\\rho =\\rho (x)>0$ sufficiently smooth in $\\bar{\\Omega }$ .",
"Namely, applying the alternative technique, one should simply replace the terms $s_N\\Lambda _tu$ , $s_N(\\delta _tu)^0$ and $s_Nu_1$ with $s_N(\\rho \\Lambda _tu)$ , $s_N(\\rho (\\delta _tu)^0)$ and $s_N(\\rho u_1)$ in (REF ), (REF ) and (REF ) keeping the same approximation orders.",
"Consequently the terms $s_N\\Lambda _tv$ , $s_N\\delta _tv^0$ , $\\bar{s}_N\\Lambda _tv$ and $\\bar{s}_N\\delta _tv^0$ are generalized as $s_N(\\rho \\Lambda _tv)$ , $s_N(\\rho \\delta _tv^0)$ , $\\bar{s}_N(\\rho \\Lambda _tv)$ and $\\bar{s}_N(\\rho \\delta _tv^0)$ in (REF )-(REF ), (REF )-(), (REF )-() and (REF )-() keeping the same approximation order $O(|\\mathbf {h}|^4)$ .",
"Also the following expansions in $\\Lambda _k$ for the arising operators at the upper level hold, for $n=2$ and 3, respectively $s_N(\\rho w)+{\\textstyle \\frac{1}{12}}h_t^2A_Nw=\\rho w+{\\textstyle \\frac{1}{12}}\\big [h_i^2\\Lambda _i(\\rho w)-a_i^2h_t^2\\Lambda _iw\\big ]\\\\[1mm]-({\\textstyle \\frac{1}{12}})^2h_t^2\\big (a_1^2h_2^2+a_2^2h_1^2\\big )\\Lambda _1\\Lambda _2w,\\\\[1mm]\\bar{s}_N(\\rho w)+{\\textstyle \\frac{1}{12}}h_t^2\\bar{A}_Nw=\\rho w+{\\textstyle \\frac{1}{12}}\\big [h_i^2\\Lambda _i(\\rho w) -a_i^2h_t^2\\Lambda _iw\\big ]\\\\[1mm]+({\\textstyle \\frac{1}{12}})^2\\sum _{1\\leqslant k<l\\leqslant 3}\\big [h_k^2h_l^2\\Lambda _k\\Lambda _l(\\rho w)-h_t^2(a_k^2h_l^2+a_lh_k^2)\\Lambda _k\\Lambda _lw\\big ]\\\\[1mm]+({\\textstyle \\frac{1}{12}})^3\\big [h_1^2 h_2^2h_3^2\\Lambda _1\\Lambda _2\\Lambda _3(\\rho w)-h_t^2(a_1^2h_2^2h_3^2+a_2^2h_1^2h_3^2+a_3^2h_1^2h_2^2)\\Lambda _1\\Lambda _2\\Lambda _3w\\big ].$ For $a_i$ and $h_i$ independent on $i$ , the formulas are simplified, and there, on the left, the operators differ only up to factors from ones appearing in the related formulas (21)-(22) in [2] and (11) in [12].",
"Moreover, one can show that in this case generalized equations (REF ) for $n=2$ and (REF ) for $n=3$ are equivalent to respective methods from [2], [12] up to approximations of $f$ .",
"But the stability and implementation issues in the generalized case are more complicated and are beyond the scope of this paper.",
"For $n\\geqslant 2$ , we also write down the scheme $\\bar{B}_N\\Lambda _tv+\\bar{A}_Nv=f_N\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},\\\\[1mm]v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\ \\bar{B}_N\\delta _tv^0+{\\textstyle \\frac{1}{2}}h_t\\bar{A}_Nv^0=u_{1N}+{\\textstyle \\frac{1}{2}}h_tf_N^0\\ \\ \\text{on}\\ \\ \\omega _h$ with the following splitting operator at the upper time level $\\hspace{-6.0pt} \\bar{B}_N:=B_{1N}\\ldots B_{nN},\\ B_{kN}:=s_{kN}-{\\textstyle \\frac{1}{12}}h_t^2a_k^2\\Lambda _k=I+{\\textstyle \\frac{1}{12}}(h_k^2-h_t^2a_k^2\\big )\\Lambda _k.$ Splitting of such type is well-known and widely used, in particular, see [11], [19], and the implementation of this scheme is most simple and comes down to sequential solving of systems with tridiagonal matrices in all $n$ spatial directions which are definitely non-singular under the hypotheses of Theorem REF below.",
"The following relation between $\\bar{B}_N$ and $\\bar{s}_N$ holds $\\bar{B}_N=\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2\\bar{A}_N+R$ with the “residual” operator $R:=\\sum _{k=2}^n\\big ({\\textstyle \\frac{1}{12}}h_t^2\\big )^k \\sum _{1\\leqslant i_1<\\ldots <i_k\\leqslant n}a_{i_1}^2\\ldots a_{i_k}^2\\Big (\\prod _{1\\leqslant l\\leqslant n,\\,l\\ne i_1,\\ldots i_k}s_{lN}\\Big )(-\\Lambda _{i_1})\\ldots (-\\Lambda _{i_k}).$ Clearly $R$ as the operator in $H_h$ satisfies $R=R^*>0$ .",
"In particular, one has $R=({\\textstyle \\frac{1}{12}}h_t^2\\big )^2a_1^2a_2^2\\Lambda _1\\Lambda _2\\ \\ \\text{for}\\ \\ n=2,\\\\[1mm]R=({\\textstyle \\frac{1}{12}}h_t^2\\big )^2\\big (a_1^2a_2^2s_{3N}\\Lambda _1\\Lambda _2+a_1^2a_3^2s_{2N}\\Lambda _1\\Lambda _3+a_2^2a_3^2s_{1N}\\Lambda _2\\Lambda _3\\big )\\\\[1mm]-({\\textstyle \\frac{1}{12}}h_t^2\\big )^3a_1^2a_2^2a_3^2\\Lambda _1\\Lambda _2\\Lambda _3\\ \\ \\text{for}\\ \\ n=3.$ Since $R\\Lambda _tu=O(h_t^4)$ and $R(\\delta _tu)^0=O(h_t^4)$ for a function $u$ sufficiently smooth in $\\bar{Q}_T$ , scheme (REF )-() has the approximation error $O(|\\mathbf {h}|^4)$ as scheme (REF )-().",
"Note that some other known methods of splitting are able to deteriorate this order of approximation.",
"Now we study the operator inequality in (REF ) for the above arisen operators.",
"Lemma 3.3 For the pairs of operators $(B_h,A_h)=(s_N,A_N)$ for $n=2$ , $(B_h,A_h)=(\\bar{s}_N,A_N)$ for $n=3$ , $(\\bar{s}_N,\\bar{A}_N)$ for $n\\geqslant 1$ and $(\\bar{s}_N+R,\\bar{A}_N)$ for $n\\geqslant 2$ , the following inequality holds $A_h\\leqslant \\alpha _h^2B_h\\ \\ \\text{in}\\ \\ H_h\\ \\ \\text{with}\\ \\ \\alpha _h^2<6C_0\\tfrac{a_i^2}{h_i^2},$ where $C_0=\\tfrac{4}{3}$ in the first case of $(B_h,A_h)$ or $C_0=1$ in other cases.",
"Let $1\\leqslant k\\leqslant n$ and $\\lbrace \\lambda _l^{(k)}:=\\tfrac{4}{h_k^2}\\sin ^2\\tfrac{\\pi lh_k}{2X_k}\\rbrace _{l=1}^{N_k-1}$ be the collection of eigenvalues of the operator $-\\Lambda _k$ in $H_h$ , with the maximal of them $\\lambda _{\\max }^{(k)}=\\tfrac{4}{h_k^2}\\sin ^2\\tfrac{\\pi (N_k-1)}{2N_k}<\\tfrac{4}{h_k^2}$ .",
"The inequality $-\\Lambda _k\\leqslant \\alpha _{1h}^2s_{kN}$ in $H_h$ is equivalent to the following inequality between the eigenvalues of these operators $\\lambda _l^{(k)}\\leqslant \\alpha _{1h}^2\\big (1-{\\textstyle \\frac{1}{12}}h_k^2\\lambda _l^{(k)}\\big ),\\ \\ 1\\leqslant l\\leqslant N_k-1.$ Consequently the sharp constant is $\\alpha _{1h}^2=\\max _{1\\leqslant l\\leqslant N_k-1}\\frac{\\lambda _l^{(k)}}{1-{\\textstyle \\frac{1}{12}}h_k^2\\lambda _l^{(k)}}=\\frac{\\lambda _{\\max }^{(l)}}{1-{\\textstyle \\frac{1}{12}}h_k^2\\lambda _{\\max }^{(k)}}<\\tfrac{3}{2}\\lambda _{\\max }^{(k)}<6\\tfrac{1}{h_k^2}.$ Herewith $\\alpha _{1h}^2=6\\tfrac{1}{h_k^2}\\big (1+O\\big (\\tfrac{1}{N_k^2}\\big )\\big )$ , thus the last bound is asymptotically sharp.",
"Similarly for $n=2$ the inequality $A_N\\leqslant \\alpha _h^2s_N$ in $H_h$ holds with $\\alpha _h^2=\\max _{1\\leqslant k\\leqslant N_1-1,\\,1\\leqslant l\\leqslant N_2-1}\\frac{\\big (1-{\\textstyle \\frac{1}{12}}h_1^2\\lambda _k^{(1)}\\big )a_2^2\\lambda _l^{(2)}+\\big (1-{\\textstyle \\frac{1}{12}}h_2^2\\lambda _l^{(2)}\\big )a_1^2\\lambda _k^{(1)}}{1-{\\textstyle \\frac{1}{12}}h_1^2\\lambda _k^{(1)}-{\\textstyle \\frac{1}{12}}h_2^2\\lambda _l^{(2)}}.$ It is not difficult to check that the function under the $\\max $ sign has the positive partial derivatives with respect to arguments $\\lambda _k^{(1)}$ and $\\lambda _l^{(2)}$ on the natural intervals of their values and thus $\\alpha _h^2=\\frac{\\big (1-{\\textstyle \\frac{1}{12}}h_1^2\\lambda _{\\max }^{(1)}\\big )a_2^2\\lambda _{\\max }^{(2)}+\\big (1-{\\textstyle \\frac{1}{12}}h_2^2\\lambda _{\\max }^{(2)}\\big )a_1^2\\lambda _{\\max }^{(1)}}{1-{\\textstyle \\frac{1}{12}}h_1^2\\lambda _{\\max }^{(1)}-{\\textstyle \\frac{1}{12}}h_2^2\\lambda _{\\max }^{(2)}}<2\\big (a_1^2\\lambda _{\\max }^{(1)}+a_2^2\\lambda _{\\max }^{(2)}\\big ).$ This implies (REF ) in the first case.",
"The last bound is asymptotically sharp too.",
"Next, in virtue of the inequalities $s_{N\\hat{i}}<\\bar{s}_{N\\hat{i}}$ for $n=3$ (see formula (REF ) for $n=2$ ) and $-\\Lambda _k<\\tfrac{3}{2}\\lambda _{\\max }^{(k)}s_{kN}$ in $H_h$ , the following inequalities in $H_h$ hold: $A_N=-a_i^2s_{N\\hat{i}}\\Lambda _i<a_i^2\\bar{s}_{N\\hat{i}}\\big (\\tfrac{3}{2} \\lambda _{\\max }^{(i)}s_{iN}\\big )=\\tfrac{3}{2}\\big (a_i^2\\lambda _{\\max }^{(i)}\\big )\\bar{s}_N\\ \\ \\text{for}\\ \\ n=3,\\\\[1mm]\\bar{A}_N=-a_i^2\\bar{s}_{N\\hat{i}}\\Lambda _i<a_i^2\\bar{s}_{N\\hat{i}}\\big (\\tfrac{3}{2}\\lambda _{\\max }^{(i)}s_{iN}\\big )=\\tfrac{3}{2}\\big (a_i^2\\lambda _{\\max }^{(i)}\\big )\\bar{s}_N\\leqslant \\tfrac{3}{2}\\big (a_i^2\\lambda _{\\max }^{(i)}\\big )(\\bar{s}_N+R)$ for $n\\geqslant 2$ .",
"Therefore inequality (REF ) has been proved in all the cases.",
"Now we state a result on conditional stability in two norms for the constructed schemes.",
"Theorem 3.1 Let $g=0$ in () and $0<\\varepsilon _0<1$ .",
"Let us consider schemes (REF )-(), (REF )-(), (REF )-() and (REF )-() under the condition $C_0h_t^2\\tfrac{a_i^2}{h_i^2}\\leqslant 1-\\varepsilon _0^2$ with the pairs of operators respectively $(B_h,A_h)=(s_N,A_N)$ for $n=2$ , $(B_h,A_h)=(\\bar{s}_N,A_N)$ for $n=3$ , $(\\bar{s}_N,\\bar{A}_N)$ for $n\\geqslant 1$ (for $n=1$ , this covers also scheme (REF )-()) and $(\\bar{s}_N+R,\\bar{A}_N)$ for $n\\geqslant 2$ .",
"Here $C_0$ is the same as in Lemma REF .",
"Then the solutions to all the listed schemes satisfy the following bounds $\\max _{1\\leqslant m\\leqslant M}\\big [\\varepsilon _0^2\\Vert \\bar{\\delta }_tv^m\\Vert _{B_h}^2+\\Vert \\bar{s}_tv^m\\Vert _{A_h}^2\\big ]^{1/2}\\nonumber \\\\[1mm]\\leqslant \\big (\\Vert v^0\\Vert _{A_h}^2+\\varepsilon _0^{-2}\\Vert B_h^{-1/2}u_{1N}\\Vert _h^2\\big )^{1/2}+2\\varepsilon _0^{-1}\\Vert B_h^{-1/2}f_N\\Vert _{L_{h_t}^1(H_h)};$ the $f_N$ -term can be taken as $2I_{h_t}^{M-1}\\Vert A_h^{-1/2}\\bar{\\delta }_tf_N\\Vert _h+3\\max \\limits _{0\\leqslant m\\leqslant M-1}\\Vert A_h^{-1/2}f^m\\Vert _h$ as well, and $\\max _{0\\leqslant m\\leqslant M}\\max \\big \\lbrace \\varepsilon _0\\Vert v^m\\Vert _{B_h},\\,\\Vert I_{h_t}^m\\bar{s}_tv\\Vert _{A_h}\\big \\rbrace \\\\[1mm]\\leqslant \\Vert v^0\\Vert _{B_h}+2\\Vert A_h^{-1/2}u_{1N}\\Vert _h+2\\Vert A_h^{-1/2}f_N\\Vert _{L_{h_t}^1(H_h)};$ for $f_N=\\delta _tg$ , the $f_N$ -term can be replaced with $2\\varepsilon _0^{-1}I_{h_t}^{M}\\Vert B_{h}^{-1/2}\\big (g-s_tg^0\\big )\\Vert _h$ .",
"Importantly, the both bounds hold for any free terms $u_{1N}\\in H_h$ and $f_N$ : $\\lbrace t_m\\rbrace _{m=0}^{M-1}\\rightarrow H_h$ (not only for those defined in Lemmas REF -REF ).",
"The theorem follows immediately from the above general stability Theorem REF applying assumption (REF ) for $\\sigma =1/12$ , in virtue of inequality (REF ) and Lemma REF .",
"Corollary 3.1 For the sufficiently smooth in $\\bar{Q}_T$ solution $u$ to the IBVP (REF )-(), $v^0=u_0$ on $\\omega _h$ and under the hypotheses of Theorem REF excluding $g=0$ , for all the schemes listed in it, the following 4th order error bound in the strong energy norm holds $\\max _{1\\leqslant m\\leqslant M}\\big [\\varepsilon _0^2\\Vert \\bar{\\delta }_t(u-v)^m\\Vert _{B_h}^2+\\Vert \\bar{s}_t(u-v)^m\\Vert _{A_h}^2\\big ]^{1/2}=O(|h|^4).$ The proof is standard (for example, see [11]) and follows from the stability bound (REF ) applied to the error $r:=u-v$ (herewith $r|_{\\partial \\omega _{\\mathbf {h}}}=0$ , $r^0=0$ ).",
"The approximation errors play the role of $f_N^m$ , $1\\leqslant m\\leqslant M-1$ , and $u_{1N}$ in the equations of the schemes, and the above checked conclusion that they have the order $O(|\\mathbf {h}|^4)$ for all the listed schemes is essential, as well as $h_t=O(|h|)$ in Theorem REF .",
"Notice that, in the very particular case $\\tfrac{h_1}{a_1}=\\ldots \\tfrac{h_n}{a_n}=h_t$ , schemes (REF )-() and (REF )-() become explicit (since then $\\bar{B}_N=I$ , see (REF )) and, moreover, the latter one differs from the simplest explicit scheme only by the above derived approximations of the free terms in its equations.",
"Herewith, for scheme (REF )-(), condition (REF ) is valid with $C_0=1$ and only $\\varepsilon _0=0$ (actually, with some $0<\\varepsilon _0=\\varepsilon _0(h)<1$ as one can check).",
"But, for scheme (REF )-() and $n\\geqslant 2$ , the condition even with $\\varepsilon _0=0$ fails; more careful analysis of inequality (REF ) for this scheme still allows to improve the bound for $\\alpha _h^2$ but not the drawn conclusion itself.",
"According to Remark REF , for scheme (REF )-(), even in this particular case some stability bounds still hold.",
"The bounds contain terms of the following type $\\Vert w\\Vert _{0,\\mathbf {h}}^2=\\big ((I+\\tfrac{h_i^2}{4}\\Lambda _1)w,w\\big )_h\\geqslant \\cos ^2\\tfrac{\\pi (N_1-1)}{2N_1}\\Vert w\\Vert _h^2\\ \\ \\forall w\\in H_h.$ Thus $\\Vert w\\Vert _{0,\\mathbf {h}}$ remains a norm in $H_h$ but clearly is no longer bounded from below by $\\Vert w\\Vert _h$ uniformly in $h$ (since the constant in the last inequality is sharp and has the order $O\\big (\\frac{1}{N_1^2}\\big )$ ).",
"The explicit scheme for $n=1$ is very specific.",
"Its equations are rewritten using a 4-point stencil simply as $v_k^{m+1}=v_{k-1}^m+v_{k+1}^m-v_k^{m-1}+h_t^2f_{Nk}^m\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},\\ \\ 1\\leqslant m\\leqslant M-1,\\\\[1mm]v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\ v_k^1={\\textstyle \\frac{1}{2}}(v_{k-1}^0+v_{k+1}^0)+h_tu_{1Nk}+{\\textstyle \\frac{1}{2}}h_t^2f_{Nk}^0\\ \\ \\text{on}\\ \\ \\omega _h.$ For clarity, let us pass to the related Cauchy problem with any $k\\in \\mathbb {Z}$ , $x_k=kh$ , $h=h_1$ , $a=a_1$ and the omitted boundary condition.",
"Then the following explicit formula holds $v_k^m={\\textstyle \\frac{1}{2}}(v_{k-m}^0+v_{k+m}^0)+\\sum \\nolimits _{l\\in I_k^{m}}h_tu_{1Nl}+{\\textstyle \\frac{1}{2}}h_t^2f_{Nl}^0+h_t^2\\sum _{p=1}^{m-1}\\sum \\nolimits _{l\\in I_k^{m-p}}f_{Nl}^p,$ where $k\\in \\mathbb {Z}$ , $1\\leqslant m\\leqslant M$ and $I_k^{m-p}$ is the set of indices $\\lbrace k-(m-p-1),k-(m-p+1),\\ldots ,k+(m-p-1)\\rbrace $ .",
"It can be verified most simply by induction with respect to $m$ .",
"Notice that all the mesh nodes lie on the characteristics $x-x_k=\\pm a_1t$ of the equation.",
"Of course, the stability of the scheme can be directly proved applying this formula.",
"Let us take $v^0=u_0$ and reset $u_{1Nk}=\\frac{1}{2h}\\int _{x_{k-1}}^{x_{k+1}}u_1(x)\\,dx$ and $f_{Nk}^0=\\frac{1}{hh_t}\\int _{T_k^1}f(x,t)\\,dxdt,\\ \\ f_{Nk}^m=\\frac{1}{2hh_t}\\int _{R_k^m}f(x,t)\\,dxdt;$ here $T_k^m$ and $R_k^m$ are the triangle and rhomb with the vertices $\\lbrace (x_{k\\pm m},0)$ , $(x_k,t_m)\\rbrace $ and $\\lbrace (x_{k\\pm 1},t_m),(x_k,t_{m\\pm 1})\\rbrace $ .",
"Then the above formula for $v_k^m$ takes the form $v_k^m={\\textstyle \\frac{1}{2}}\\big (u_0(x_{k-m})+u_0(x_{k+m})\\big )+\\tfrac{1}{2a_1}\\int _{x_{k-m}}^{x_{k+m}}u_1(x)\\,dx+\\tfrac{1}{2a_1}\\int _{T_k^m}f(x,t)\\,dxdt,$ thus at the mesh nodes it reproduces the classical d'Alembert formula for the solution $u$ to the Cauchy problem for the 1D wave equation, where the approximate and exact solution coincide: $v_k^m\\equiv u(x_k,t_m)$ for any $k\\in \\mathbb {Z}$ and $0\\leqslant m\\leqslant M$ .",
"Concerning exact schemes, see also [7]."
],
[
"The case of non-uniform rectangular meshes",
"This section is devoted to a generalization to the case of non-uniform rectangular meshes.",
"Let $1\\leqslant k\\leqslant n$ .",
"Define the general non-uniform meshes $0=x_{k0}<x_{k1}<\\ldots <x_{kN_k}=X_k$ in $x_k$ with the steps $h_{kl}=x_{kl}-x_{k(l-1)}$ and $\\overline{\\omega }_{h_t}$ with the nodes $0=t_0<t_1<\\ldots <t_M=T$ and steps $h_{tm}=t_m-t_{m-1}$ .",
"Let $\\omega _{hk}=\\lbrace x_{kl}\\rbrace _{l=1}^{N_k-1}$ .",
"We set $h_{k+,l}=h_{k(l+1)},\\ \\ h_{*k}={\\textstyle \\frac{1}{2}}(h_k+h_{k+}),\\ \\ h_{t+,m}=h_{t(m+1)},\\ \\ h_{*t}={\\textstyle \\frac{1}{2}}(h_t+h_{t+})$ as well as $h_{k\\max }=\\max _{1\\leqslant l\\leqslant N_k} h_{kl}$ and $h_{t\\max }=\\max _{1\\leqslant m\\leqslant M} h_{tm}$ .",
"Define the difference operators $\\delta _kw_l=\\tfrac{1}{h_{k+,l}}(w_{l+1}-w_l),\\ \\ \\bar{\\delta }_kw_l=\\tfrac{1}{h_{kl}}(w_l-w_{l-1}),\\ \\ \\Lambda _kw=\\tfrac{1}{h_{*k}}(\\delta _kw-\\bar{\\delta }_kw),\\\\[1mm]\\delta _ty^m=\\tfrac{1}{h_{t+,m}}(y^{m+1}-y^m),\\ \\ \\bar{\\delta }_ty^m=\\tfrac{1}{h_{tm}}(y^m-y^{m-1}),\\ \\ \\Lambda _ty=\\tfrac{1}{h_{*t}}(\\delta _ty-\\bar{\\delta }_ty),$ where $w_l=w(x_{kl})$ and $y^m=y(t_m)$ .",
"The last four operators generalize those defined above so that their notation is the same.",
"We extend the above technique based on averaging equation (REF ) and generalize the above average in $x_k$ : $q_kw(x_{kl})=\\frac{1}{h_{*k,l}}\\int _{x_{k(l-1)}}^{x_{k(l+1)}}w(x_k)e_{kl}(x_k)\\,dx_k,\\\\[1mm]e_{kl}(x_k)=\\tfrac{x_k-x_{k(l-1)}}{h_{kl}}\\,\\ \\text{on}\\ [x_{k(l-1)},x_{kl}],\\ e_{kl}(x_k)=\\tfrac{x_{k(l+1)}-x_k}{h_{k+,l}}\\,\\ \\text{on}\\ [x_{kl},x_{k(l+1)}].$ For a function $w(x_k)$ smooth on $[0,X_k]$ , formula (REF ) remains valid and $q_kw=w+q_k\\rho _{k1}(\\partial _kw),\\nonumber \\\\[1mm]q_kw=w+\\tfrac{1}{3}(h_{k+}-h_k)\\partial _kw+{\\textstyle \\frac{1}{12}}\\big [(h_{k+})^2-h_{k+}h_k+h_k^2\\big ]\\partial _k^2w+q_k\\rho _{k3}(\\partial _k^3w)$ on $\\omega _{hk}$ , and the first bound () remains valid for $s=1,3$ with $h_k$ replaced with $h_{*k}$ , see also (REF ), that follows from Taylor's formula after calculating the arising integrals over $[x_{k(l-1)},x_{k(l+1)}]$ .",
"Due to Taylor's formula we also have $\\partial _kw={\\textstyle \\frac{1}{2}}(\\bar{\\delta }_kw+\\delta _kw)-\\tfrac{1}{4}(h_{k+}-h_k)\\partial _k^2w+\\rho _{k}^{(1)}(\\partial _k^3w),\\partial _k^2w=\\Lambda _kw+\\rho _{k3}^{(2)}(\\partial _k^3w),\\\\[1mm]|\\rho _{k}^{(s)}(\\partial _k^3w)|\\leqslant c^{(s)}h_{*k}^{3-2(s-1)}\\Vert \\partial _k^3w\\Vert _{C(I_{kl})},\\ \\ s=1,2,$ thus the second expansion for $q_kw$ implies that $q_kw=s_{kN}w+\\tilde{\\rho }_{k3}(\\partial _k^3w),\\ \\ |\\tilde{\\rho }_{k3}(\\partial _k^3w)|\\leqslant \\tilde{c}_3h_{*k}^3\\Vert \\partial _k^3w\\Vert _{C(I_{kl})},\\\\[1mm]s_{kN}:=I+\\tfrac{1}{3}(h_{k+}-h_k)\\big [{\\textstyle \\frac{1}{2}}(\\bar{\\delta }_k+\\delta _k)-\\tfrac{1}{4}(h_{k+}-h_k)\\Lambda _k\\big ]\\nonumber \\\\[1mm]+{\\textstyle \\frac{1}{12}}\\big [(h_{k+})^2-h_{k+}h_k+h_k^2\\big ]\\Lambda _k=I+\\tfrac{1}{6}(h_{k+}-h_k)(\\bar{\\delta }_k+\\delta _k)+{\\textstyle \\frac{1}{12}}h_kh_{k+}\\Lambda _k,$ i.e., $s_{kN}=I+{\\textstyle \\frac{1}{12}}(h_{k+}\\beta _k\\delta _k-h_k\\alpha _k\\bar{\\delta }_k)$ or, in the averaging form, $s_{kN}w_l={\\textstyle \\frac{1}{12}}(\\alpha _{kl}w_{l-1}+10\\gamma _{kl}w_l+\\beta _{kl}w_{l+1}),\\\\\\alpha _k=2-\\tfrac{h_{k+}^2}{h_kh_{*k}},\\ \\beta _k=2-\\tfrac{h_k^2}{h_{k+}h_{*k}},\\ \\gamma _k=1+\\tfrac{(h_{k+}-h_k)^2}{5h_k h_{k+}},\\ \\alpha _k+10\\gamma _k+\\beta _k=12;$ all the presented formulas are valid on $\\omega _{hk}$ .",
"The operator $s_{kN}$ generalizes one defined above.",
"Its another derivation was originally given in [5], see also [14], [10].",
"Recall that the natural property $\\alpha _{kl}\\geqslant 0$ and $\\beta _{kl}\\geqslant 0$ (not imposed below) is equivalent to the rather restrictive condition on the ratio of the adjacent mesh steps $0.618\\approx \\tfrac{2}{\\sqrt{5}+1}\\leqslant \\tfrac{h_{k(l+1)}}{h_{kl}}\\leqslant \\tfrac{\\sqrt{5}+1}{2}\\approx 1.618.$ On $\\omega _{h_t}$ , the average $q_tw=q_{n+1}w$ is defined similarly, and thus $q_tw=s_{tN}w+\\tilde{\\rho }_{t3}(\\partial _t^3w),\\ \\ |\\tilde{\\rho }_{t3}(\\partial _t^3w)|\\leqslant \\tilde{c}_3h_{*t}^3\\Vert \\partial _t^3w\\Vert _{C[t_{m-1},t_{m+1}]}$ with $s_{tN}=I+{\\textstyle \\frac{1}{12}}(h_{t+}\\beta _t\\delta _t-h_t\\alpha _t\\bar{\\delta }_t)$ or, in the averaging form, $s_{tN}y={\\textstyle \\frac{1}{12}}(\\alpha _t\\check{y}+10\\gamma _ty+\\beta _t\\hat{y}),\\,\\alpha _t=2-\\tfrac{h_{t+}^2}{h_th_{*t}},\\ \\beta _t=2-\\tfrac{h_t^2}{h_{t+}h_{*t}},\\,\\gamma _t=1+\\tfrac{(h_{t+}-h_t)^2}{5h_t h_{t+}}.$ Let $\\omega _{h}=\\omega _{h1}\\times \\ldots \\times \\omega _{hn}$ .",
"Formula (REF ) for $u$ remains valid and implies $\\bar{s}_N\\Lambda _tu-a_i^2\\bar{s}_{N\\hat{i}}s_{tN}\\Lambda _iu=\\bar{q}q_tf+O(\\mathbf {h}_{\\max }^3)\\ \\ \\text{on}\\ \\ \\omega _{\\mathbf {h}},$ where $\\mathbf {h}_{\\max }=\\max \\lbrace h_{1\\max },\\ldots ,h_{n\\max },h_{t\\max }\\rbrace $ .",
"Formula (REF ) for $u$ remains valid as well.",
"It involves only two first time levels thus easily covers the case of the non-uniform mesh in $t$ and implies now $\\big (\\bar{s}_N+\\tfrac{h_{t1}^2}{12}a_i^2\\bar{s}_{N\\hat{i}}\\Lambda _i\\big )(\\delta _tu)^0+\\tfrac{h_{t1}}{2}a_i^2\\bar{s}_{N\\hat{i}}\\Lambda _iu_0=\\bar{q}u_1+\\tfrac{h_{t1}^2}{12}a_i^2\\bar{s}_{N\\hat{i}}\\Lambda _iu_1+\\bar{q}q_tf^0+O(\\mathbf {h}_{\\max }^3)$ on $\\omega _h$ , where $q_ty^0$ is given by formula (REF ) with $h_{t1}$ in the role of $h_t$ .",
"Owing to the above formulas, see also Remark REF , the last two formulas with $u$ lead us to the generalized scheme (REF )-(): $\\tfrac{1}{h_{*t}}\\big [\\big (\\bar{s}_N+\\tfrac{h_{*t}h_{t+}}{12}\\beta _t\\bar{A}_N\\big )\\delta _tv-\\big (\\bar{s}_N+\\tfrac{h_{*t}h_t}{12}\\alpha _t\\bar{A}_N\\big )\\bar{\\delta }_tv\\big ]+\\bar{A}_Nv=\\bar{s}_Ns_{tN}f,\\\\\\hspace{-8.0pt} v|_{\\partial \\omega _{\\mathbf {h}}}=g,\\ \\big (\\bar{s}_N+\\tfrac{h_{t1}^2}{12}\\bar{A}_N\\big )(\\delta _tv)^0+\\tfrac{h_{t1}}{2}\\bar{A}_Nv_0=(\\bar{s}_N-\\tfrac{h_{t1}^2}{12}\\bar{A}_N)u_1+\\tfrac{h_{t1}}{2}f_N^0$ with $f_N^0=\\bar{s}_Nf_0+\\tfrac{h_{t1}}{3}(\\delta _tf)^0$ , where equations are valid respectively on $\\omega _{\\mathbf {h}}$ and $\\omega _h$ and have the approximation errors of the order $O(\\mathbf {h}_{\\max }^3)$ .",
"For the uniform mesh in $t$ , the left-hand side of (REF ) takes the previous form whereas the term $\\bar{s}_Ns_{tN}f$ can be simplified keeping the same order of the approximation error: $(\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2\\bar{A}_N)\\Lambda _tv+\\bar{A}_Nv=(\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_t^2\\Lambda _t)f.$ The splitting version of equation (REF ) can be got by replacing the operators in front of $\\delta _tv$ and $\\bar{\\delta }_tv$ by the operators of the form $\\bar{B}_N=(s_1+{\\textstyle \\frac{1}{12}}h_{*t}\\tilde{h}_t\\sigma _ta_1^2\\Lambda _1)\\ldots (s_n+{\\textstyle \\frac{1}{12}}h_{*t}\\tilde{h}_t\\sigma _ta_n^2\\Lambda _n),$ where respectively $\\tilde{h}_t=h_{t+}$ and $\\sigma _t=\\beta _t$ , or $\\tilde{h}_t=h_t$ and $\\sigma _t=\\alpha _t$ .",
"Since $\\bar{B}_N=\\bar{s}_N+{\\textstyle \\frac{1}{12}}h_{*t}\\tilde{h}_t\\sigma _ta_i^2\\bar{s}_{N\\hat{i}}\\Lambda _i+R,$ where the operator $R$ satisfies formula (REF ) with $h_t^2$ replaced with $h_{*t}\\tilde{h}_t\\sigma _t$ , this replacement conserves the approximation error of the order $O(\\mathbf {h}_{\\max }^3)$ .",
"The splitting version of equation () is got simply by replacing $\\bar{s}_N+\\tfrac{h_{t1}^2}{12}\\bar{A}_N$ with the above operator (REF ) with $h_{t1}$ in the role of $h_t$ .",
"One can check also that the approximation errors still has the 4th order $O(\\mathbf {h}_{\\max }^4)$ for smoothly varying non-uniform meshes, cp.",
"[15], provided that, for example, $f_N^0=\\bar{s}_Nf^0-f^0+f_{dh_t}^{(0)}$ .",
"Here we do not touch the stability study in the case of the non-uniform mesh (even only in space) but this is noticeably more cumbersome like in [15] (since the operator $s_{kN}$ is not self-adjoint any more) and, moreover, imposes stronger conditions on $h_t$ , see also [17], [18]."
],
[
"Numerical experiments",
".1.",
"In the IBVP (REF )-() in the 1D case, we now take $\\Omega :=(-X/2,X/2)$ and rewrite the boundary condition as $u|_{x=-X/2}=g_0(t)$ and $u|_{x=X/2}=g_1(t)$ , $t\\in (0,T)$ .",
"We intend to analyze the practical error orders $\\gamma _{pr}$ of $r=u-v$ in three uniform in time mesh norms $\\max _{0\\leqslant m\\leqslant M}\\Vert r^m\\Vert _h,\\ \\ \\max _{0\\leqslant m\\leqslant M,\\,0\\leqslant k\\leqslant N}|r_k^m|,\\ \\ \\max _{1\\leqslant m\\leqslant M}\\max \\big \\lbrace \\Vert \\bar{\\delta }_tr^m\\Vert _h,\\,\\Vert \\bar{\\delta }_1r^m\\Vert _{\\tilde{h}}\\big \\rbrace ,$ which below are denoted respectively as $L_h^2$ , $C_h$ and $\\mathcal {E}_h$ (the 2nd and 3rd norms are the uniform and strong energy-type ones).",
"Here $\\Vert w\\Vert _{\\tilde{h}}=\\big (h\\sum _{k=1}^{N}w_k^2\\big )^{1/2}$ and $N=N_1$ .",
"The respective expected theoretical error orders $\\gamma _{th}$ are $\\min \\big \\lbrace \\tfrac{4}{5}\\alpha ,4\\big \\rbrace ,\\,\\ \\alpha \\geqslant 0;\\ \\ \\tfrac{4}{5}(\\alpha -\\tfrac{1}{2}),\\,\\ \\tfrac{1}{2}<\\alpha \\leqslant \\tfrac{11}{2};\\ \\ \\tfrac{4}{5}(\\alpha -1),\\,\\ 1\\leqslant \\alpha \\leqslant 6$ (in the spirit of [1]), where $\\alpha $ is the parameter defining the weak smoothness of the data, see details below (concerning the first order, for $\\alpha \\leqslant 1$ , it should refer to the continuous $L^2$ norm rather than the mesh one but that we will ignore).",
"The proof of the first order in the case $u_1=f=0$ see in [6].",
"For comparison, recall that for the 2nd approximation order methods the corresponding theoretical error orders $\\gamma _{th}^{(2)}$ are $\\min \\big \\lbrace \\tfrac{2}{3}\\alpha ,2\\big \\rbrace , \\alpha \\geqslant 0;\\ \\min \\big \\lbrace \\tfrac{2}{3}(\\alpha -\\tfrac{1}{2}),2\\big \\rbrace , \\alpha >\\tfrac{1}{2};\\ \\min \\big \\lbrace \\tfrac{2}{3}(\\alpha -1),2\\big \\rbrace , \\alpha \\geqslant 1,$ according to [19]; recall that the middle error order is derived from two other ones.",
"These orders also have recently been confirmed practically in [16].",
"Let $P_0(x)=(\\operatorname{sgn}x+1)/2$ be the Heaviside-type function, $P_1(x)=1-2|x|$ , $P_k(x)=(\\operatorname{sgn}x)(2x)^k$ ($k\\geqslant 2$ ) and $Q_l(t)= P_0(t-t_*)(t-t_*)^l$ ($l\\geqslant 0$ and $0<t_*<T$ ) be piecewise-polynomial functions.",
"For uniformity, we also set $P_{-1}(x)=\\delta (x)$ and $Q_{-1}(t)=\\delta (t-t_*)$ as the Dirac delta-functions concentrated at $x=0$ and $t=t_*$ .",
"We put $X=T=1$ .",
"We consider six typical Examples $E_{\\alpha }$ , $\\alpha =1/2,3/2,\\ldots ,11/2$ , of non-smooth data supplementing the study in [16].",
"The initial functions $u_0=P_{[\\alpha ]}$ and $u_1=c_1P_{[\\alpha ]-1}$ are piecewise-polynomial functions of the degree $[\\alpha ]$ and $[\\alpha ]-1$ respectively, with a unique singularity point $x=0$ , excluding the case $[\\alpha ]=0$ for $u_1$ , where $u_1(x)=c_1\\delta (x)$ .",
"Thus $u_0$ belongs to the Nikolskii space $H_2^\\alpha (\\Omega )$ [9] (and to the Sobolev-Slobodetskii space $W_2^{\\alpha -\\varepsilon }(\\Omega )$ , $0<\\varepsilon <1/2$ ), and $u_1\\in H_2^{\\alpha -1}(\\Omega )$ (for $\\alpha >1$ ).",
"The free term $f(x,t)=c_2P_{-1}(x)Q_{-1}(t)=c_2\\delta (x,t-t_*)$ is concentrated at $(x,t)=(0,t_*)$ for $\\alpha =1/2$ , or has the form $f(x,t)=f_1(x)f_2(t)=c_2P_0(x)Q_{-1}(t)$ for $\\alpha =3/2$ , or the form of two such type summands $f(x,t)=c_2P_0(x)Q_{[\\alpha ]-2}(t)+c_3P_1(x)Q_{[\\alpha ]-3}(t)$ for $\\alpha \\geqslant 5/2$ .",
"The term $f_1$ is piecewise-constant (the case $\\alpha _1=1/2$ ) for $\\alpha \\geqslant 3/2$ or also piecewise-linear (the case $\\alpha _1=3/2$ ) for $\\alpha \\geqslant 5/2$ , with a unique singularity point $x=0$ .",
"Respectively the term $f_2(t)=\\delta (t-t_*)$ for $\\alpha =3/2$ or $f_2(t)=Q_{\\alpha _2}(t)$ is a piecewise-polynomial function of the degree $\\alpha _2=\\alpha -2-\\alpha _1=[\\alpha ]-2-[\\alpha _1]$ for $\\alpha \\geqslant 5/2$ , with a unique singularity point $t=t_*$ .",
"Recall that, for $\\alpha _2>0$ , such $f_2$ belongs to the Sobolev-Nikolskii space $WH_1^{\\alpha _2}(0,T)$ (for example, see [19]), though not to the less broad Sobolev space $W_1^{\\alpha _2}(0,T)$ .",
"Thus $f$ itself or its both summands has the so called weak dominated mixed smoothness of the order $\\alpha _1$ in $x$ and $\\alpha _2$ in $t$ , with $\\alpha _1+\\alpha _2=\\alpha -1$ .",
"Recall that this property is much broader than the standard weak smoothness of the order $\\alpha -1$ in both $x$ and $t$ in $L^2(Q)$ ; in particular, the case of $f$ discontinuous in $x$ is covered for any considered $\\alpha $ .",
"Here $(c_1,c_2)=(0.4,0.4),(1.9,1.1)$ and $(c_1,c_2,c_3)=(0.58,2.1,2.3),(2.8,6.8$ , $7.3)$ , $(3.7,13,31)$ , $(4.6,24,51)$ for $\\alpha =\\frac{1}{2},\\frac{3}{2},\\ldots ,\\frac{11}{2}$ respectively.",
"We use these multipliers to make the contributions to $r(\\cdot ,T)$ due to $u_0$ , $u_1$ and $f$ of the similar magnitude and thus significant.",
"We also take smooth $g_0$ and $g_1$ (not affecting $\\gamma _{pr}$ ) to simplify the explicit forms of $u$ (which we omit here) based on the d'Alembert formula.",
"Namely, we set $g_0=0$ and $g_1(t)=(c_1t)^k$ for $\\alpha =\\frac{1}{2},\\frac{3}{2}$ ; $g_0=(-1)^{k}(-g_1^{(0k)}+c_1g_1^{(1k)})$ and $g_1=g_1^{(0k)}+c_1g_1^{(1k)}$ for $\\alpha \\geqslant \\frac{5}{2}$ , where $k=[\\alpha ]$ and $g_1^{(0k)}(t)={\\textstyle \\frac{1}{2}}\\big [(1-2at)^k+(1+2at)^k\\big ],\\ \\ k\\geqslant 2,\\\\g_1^{(1k)}(t)=0,\\ \\ k=2,\\ \\ g_1^{(1k)}(t)=\\tfrac{1}{4ak}\\big [(1+2at)^k-(1-2at)^k\\big ],\\ \\ k\\geqslant 3.$ The properties of $u$ in Example $E_{\\alpha }$ have been described in [16] or are similar.",
"Recall that, for example, $u$ is piecewise-constant and discontinuous on $\\bar{Q}$ for $\\alpha =\\frac{1}{2}$ , or $u$ is piecewise-linear with discontinuous piecewise-constant derivatives on $\\bar{Q}$ for $\\alpha =\\frac{3}{2}$ , etc.",
"The straight singularity lines are characteristics and $t=t_*$ .",
"Notice that $u$ is not the classical solution for any $\\alpha $ but is strong one for $\\alpha \\geqslant \\frac{7}{2}$ and one of several weak solutions for $\\alpha \\leqslant \\frac{5}{2}$ , see details in [16] (but note that, for $f(x,t)=P_1(x)\\delta (t-t_*)$ , $\\alpha =\\frac{5}{2}$ , the jump of $\\partial _tu$ across $t=t_*$ was not taken into account there).",
"We set $v^0=u_0$ ; $u_{1N}=q_1u_1$ for $\\alpha \\leqslant \\frac{3}{2}$ , or as in (REF ) for $\\alpha \\geqslant \\frac{5}{2}$ , and $f_N^m=(q_1f_1)q_tf_2^m$ on $\\omega _{\\mathbf {h}}$ , see Remark REF .",
"For $x_k\\in \\omega _h$ and even $N$ , we have $q_1\\delta (\\cdot )_k=\\tfrac{1}{h}$ for $x_k=0$ , or $q_1\\delta (\\cdot )_k=0$ otherwise, and $(q_1P_0)_k=P_0(x_k)$ .",
"Also, if $t_*=t_{m_*}\\in \\omega _{h_t}$ and $1\\leqslant k\\leqslant 5$ , then $(q_tQ_k)^m=(s_{tN}Q_k)^m$ for $t_m\\in \\omega _{h_t}$ , $m\\ne m_*$ , or $(q_tQ_k)^{m_*}=\\tfrac{\\tau ^k}{(k+1)(k+2)}$ .",
"We choose $a=\\frac{1}{\\sqrt{5}}$ , $t_*=\\frac{T}{2}$ and $\\tau =h$ (so the mesh is not adjusted to the characteristics).",
"To identify error orders more reliably, we compute the errors for respectively $N=200,400$ , $\\ldots ,N_{\\max }$ , where $N_{\\max }=3200,2000,800$ respectively for $\\frac{1}{2}\\leqslant \\alpha \\leqslant \\frac{5}{2},\\alpha =\\frac{7}{2},\\frac{9}{2}$ ; also $N=200,300,\\ldots ,N_{\\max }$ with $N_{\\max }=600$ for $\\alpha =\\frac{11}{2}$ ($N_{\\max }$ is lesser for $\\alpha \\geqslant \\frac{7}{2}$ to avoid an impact of the round-off errors on ${\\gamma _{pr}}$ ).",
"We plot graphs of $\\log _{10}\\Vert r\\Vert $ versus $\\log _{10}N$ , where $\\Vert r\\Vert $ is each of the three norms (REF ), and seek the almost linear dependence between them by the least square method.",
"Thus we calculate the dependence $\\Vert r\\Vert \\approx c_0h^{\\gamma _{pr}}=c_0(\\frac{X}{N})^{\\gamma _{pr}}$ .",
"For $\\alpha =\\frac{3}{2},\\frac{5}{2},\\frac{7}{2},\\frac{9}{2}$ and the extended set $N=200,400,\\ldots ,3200$ , we present $\\mathcal {E}$ , $C_h$ , $L_h^2$ -norms of the error denoted respectively by $\\triangle , \\Box , \\diamondsuit $ on Figs.",
"REF -REF .",
"Notice the abrupt decrease of the error range as $\\alpha $ grows.",
"We also observe the slight oscillation of the data for $\\alpha =\\frac{3}{2}$ that is an exception (they also present for $\\alpha =\\frac{1}{2}$ ); instead, the linear behavior is typical for other $\\alpha $ and the values $200\\leqslant N\\leqslant N_{\\max }$ .",
"The slight growth of $L_h^2$ -norm for $\\alpha =\\frac{7}{2}$ and much more significant growth of all the norms for $\\alpha =\\frac{9}{2}$ as $N$ increases reflect the impact of the round-off errors; the value of $N$ when the error begins to increase depends on the norm.",
"For $\\alpha =\\frac{11}{2}$ the situation is even more strong (not presented).",
"Figure: Examples E 3/2 E_{3/2} (left) and E 5/2 E_{5/2} (right): ℰ,C h ,L h 2 \\mathcal {E},C_h,L_h^2-norms of the errordenoted respectively by▵,□,♦\\triangle , \\Box , \\diamondsuit , for N=200,400,...,3200N=200,400,\\ldots ,3200Figure: Examples E 7/2 E_{7/2} (left) and E 9/2 E_{9/2} (right): ℰ,C h ,L h 2 \\mathcal {E},C_h,L_h^2-norms of the errordenoted respectively by▵,□,♦\\triangle , \\Box , \\diamondsuit , for N=200,400,...,3200N=200,400,\\ldots ,3200The computed $c_0$ and $\\gamma _{pr}$ together with the respective theoretical orders $\\gamma _{th}$ and $\\gamma _{th}^{(2)}$ , see (REF )-(REF ), and the error norms $\\Vert r_{N}\\Vert $ and $\\Vert r_{N}^{(2)}\\Vert $ for $N=200,N_{\\max }$ are collected in Table REF .",
"For more visibility, here we include the error norms $\\Vert r_{N}^{(2)}\\Vert $ for the standard second order scheme like (REF )-() but with the multiplier $-\\sigma a_1^2h_t^2$ substituted for $\\tfrac{1}{12}(h_1^2-a_1^2h_t^2)$ , with the weight $\\sigma =\\tfrac{1}{2}$ , the same $v^0$ and $f_N$ as well as $u_{1N}=q_1u_1$ for $\\alpha \\leqslant \\frac{3}{2}$ , or $u_{1N}=u_1$ for $\\alpha \\geqslant \\frac{5}{2}$ .",
"Table: Numerical results for the uniform meshThe main observation is the nice agreement between $\\gamma _{pr}$ and $\\gamma _{th}$ for all three norms in all Examples $E_{\\alpha }$ , thus the sensitive dependence of $\\gamma _{pr}$ on the data smoothness order $\\alpha $ becomes quite clear.",
"This agreement is mainly better for the first and second norms (REF ) (similarly to [16]).",
"Notice that $\\gamma _{th}^{(2)}/\\gamma _{pr}$ and the error $\\Vert r_{200}\\Vert $ in each norm decrease rapidly as $\\alpha $ grows.",
"Clearly the errors $\\Vert r_{N}\\Vert $ are much smaller than $\\Vert r_{N}^{(2)}\\Vert $ for $N=200$ and $N_{\\max }$ especially as $\\alpha $ grows.",
"This demonstrates the essential advantages of the 4th approximation order scheme over the 2nd order one in the important case of non-smooth data as well.",
"This is essential, in particular, in some optimal control problems [13].",
"We also remind the explicit scheme (REF )-().",
"For the same $X$ and $a$ but $h_t=h/a$ and $T=Mh_t>1$ , for example, the $C_h$ -norm of the error equals $0.311E$$-14$ even for $N=20$ and $M=10$ already in Example $E_{3/2}$ ; thus clearly it is caused purely by the round-off errors.",
".2.",
"Also we analyze numerically scheme (REF ) and () (with $f_N^0=s_Nf^0+\\frac{2}{3}(f^{\\tau /2}-f^0)$ ) on non-uniform spatial meshes such that $x_k=\\varphi (\\frac{k}{N})-\\frac{X}{2}$ , $0\\leqslant k\\leqslant N$ , and $h_k=x_k-x_{k-1}$ .",
"Here $\\varphi \\in C[0,1]$ is a given increasing node distribution function with the range $\\varphi ([0,1])=[0,X]$ .",
"We take again $X=T=1$ and $a=\\frac{1}{\\sqrt{5}}$ but consider only the smooth (analytic) exact solution $u$ for the data $u_0(x)=\\sin (2\\pi (x+0.5)),\\ u_1(x)=4\\sin (3\\pi (x+0.5)),\\ f(x,t)=e^{x+0.5-t},\\\\[1mm]g_0(t)=\\tfrac{1}{2a}\\big (\\tfrac{1}{a+1}e^{at}+\\tfrac{1}{a-1}e^{-at}-\\tfrac{2a}{a^2-1}e^{-t}\\big )\\ (a\\ne 1),\\ \\ g_1(t)=eg_0(t).$ We base on the practical stability condition $h_t^2\\frac{a^2}{h_{\\min }^2}\\leqslant {\\textstyle \\frac{1}{2}}$ with $h_{\\min }=\\min _{1\\leqslant k\\leqslant N}h_k$ (cp.",
"(REF ) for $C_0=1$ and $\\varepsilon _0^2={\\textstyle \\frac{1}{2}}$ ), thus we set $M=M_0:=\\big \\lfloor \\frac{\\sqrt{2}aT}{h_{\\min }}\\big \\rfloor $ , where $\\lfloor b\\rfloor $ is the maximal integer less or equal $b$ .",
"It turns out to be accurate in practice, see below.",
"We take $N=50,100,\\ldots ,1000$ .",
"In Table REF , the error behavior in the $C_h$ norm is represented for several functions $\\varphi _l$ , $0\\leqslant l\\leqslant 6$ .",
"Clearly $\\varphi _0(t)=t$ sets the uniform mesh and is included for comparison only.",
"Notice that $\\varphi _3^{\\prime }(0)=0$ whereas $\\varphi _l^{\\prime }(+0)=+\\infty $ , $l=4,5,6$ ; both cases are more complicated than the standard one $0<\\underline{c}\\leqslant \\varphi _l^{\\prime }(\\xi )\\leqslant \\bar{c}$ on $[0,1]$ , $l=1,2$ , in the existing theory [15].",
"The error orders $\\gamma _{pr}$ are close to 4 for $0\\leqslant l\\leqslant 3$ but decrease down to $2.411$ as in $\\varphi _l(\\xi )=\\xi ^{a_l}$ the power $a_l=\\frac{3}{4},\\frac{5}{8},\\frac{1}{2}$ diminishes, $l=4,5,6$ .",
"Thus the approximation orders 3 or 4, see Section , are not always the practical error orders as well.",
"For $l=2,6$ , the values of $c_0$ and $\\gamma _{pr}$ are marked by $^*$ meaning that the results are yet too rough for $N=50,100,150$ and thus ignored in their computation.",
"For any $l$ , the graphs of $\\log _{10}\\Vert r_N\\Vert _{C_h}$ versus $\\log _{10}N$ are very close to straight lines (omitted for brevity).",
"The mesh data $\\frac{h_{\\max }}{h_{\\min }},\\rho _{\\min }:=\\min _{1\\leqslant k\\leqslant N-1}\\frac{h_{k+1}}{h_k},\\rho _{\\max }:=\\max _{1\\leqslant k\\leqslant N-1}\\frac{h_{k+1}}{h_k}$ and $\\frac{M}{N}$ , all for $N=800$ only, are also included into the table.",
"Note that condition (REF ) is violated for $l=3,5,6$ , but this does not essentially affect the results.",
"For $l=1$ , $\\rho _{\\min }=\\rho _{\\max }$ since the steps $h_k$ form a geometric progression.",
"Also $\\varphi _l$ is strictly convex (or concave) on $[0,1]$ for $l=1,3$ (or $l=2,4,5,6$ ), accordingly $h_k=\\varphi _l^{\\prime }(\\xi _k)$ , where $\\xi _k\\in (\\frac{k-1}{N},\\frac{k}{N})$ , increases and $\\rho _{\\min }>1$ (or decreases and $\\rho _{\\max }<1$ ) as $k$ grows.",
"The ratios $\\frac{M}{N}$ are not high except $l=1,3$ .",
"Taking smaller $M$ by replacing $\\sqrt{2}$ with $\\frac{1}{\\sqrt{2}}$ in the above formula, for $l=0,1$ (the cases of the uniform and non-uniform meshes), leads us to highly unstable computations for $N\\geqslant 100$ : the $C_h$ -norm of numerical solutions grows exponentially.",
"Table: Numerical results for non-uniform spatial meshes, with ϕ 2 (ξ)=ln(60ξ+1) ln61\\varphi _2(\\xi )=\\frac{\\ln (60\\xi +1)}{\\ln 61}Acknowledgements The publication was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2019–2020 (grant no.",
"19-01-021) and by the Russian Academic Excellence Project “5-100” as well as by the Russian Foundation for the Basic Research, grant no.",
"19-01-00262."
]
] | 2011.14104 |
[
[
"Efficient Attention Network: Accelerate Attention by Searching Where to\n Plug"
],
[
"Abstract Recently, many plug-and-play self-attention modules are proposed to enhance the model generalization by exploiting the internal information of deep convolutional neural networks (CNNs).",
"Previous works lay an emphasis on the design of attention module for specific functionality, e.g., light-weighted or task-oriented attention.",
"However, they ignore the importance of where to plug in the attention module since they connect the modules individually with each block of the entire CNN backbone for granted, leading to incremental computational cost and number of parameters with the growth of network depth.",
"Thus, we propose a framework called Efficient Attention Network (EAN) to improve the efficiency for the existing attention modules.",
"In EAN, we leverage the sharing mechanism (Huang et al.",
"2020) to share the attention module within the backbone and search where to connect the shared attention module via reinforcement learning.",
"Finally, we obtain the attention network with sparse connections between the backbone and modules, while (1) maintaining accuracy (2) reducing extra parameter increment and (3) accelerating inference.",
"Extensive experiments on widely-used benchmarks and popular attention networks show the effectiveness of EAN.",
"Furthermore, we empirically illustrate that our EAN has the capacity of transferring to other tasks and capturing the informative features.",
"The code is available at https://github.com/gbup-group/EAN-efficient-attention-network."
],
[
"Introduction",
"Recently, many plug-and-play and straightforward self-attention modules that utilize the interior information of a network to enhance instance specificity [21] are proposed to boost the generalization of deep convolutional neural networks (CNNs) [13], [31], [18], [15], [5], [30].",
"Figure: Comparison of network structures between (a) ResNet, (b) Org-full attention network, (c) Share-full attention network, and (d) our EAN network.",
"The detailed introduction of different networks is shown in Section .The self-attention module is usually plugged into every block of a residual network (ResNet) [11] (see Fig.",
"REF (a) for the structure of a ResNet and Fig.",
"REF (b) for a network with attention modules).",
"In general, the implementation of the attention module can be divided into three steps [15]: (1) Extraction: the plug-in module extracts internal features of a network by computing their statistics, like mean, variance, or higher-order moments [13], [17]; (2) Processing: the module leverages the extracted features to adaptively generate a mask that measures the importance of the feature maps via a fully connected layer [13], convolution layer [31], or feature-wise linear transformation [21], [18] etc.",
"; (3) Recalibration: the mask is used to calibrate the feature maps of the network by element-wise multiplication or addition [13], [5].",
"The operations and trainable modules in the implementation of self-attention inevitably require extra computational cost and parameters, resulting in slow inference and cumbersome network [2].",
"This limits self-attention usage on applications that need a real-time response or small memory consumption, such as robotics, self-driving car, and mobile device.",
"Previous works mainly focus on improving the capacity of self-attention module [31], [15] or reducing the parameters [21], [18], [15].",
"However, these methods do not balance the inference time and parameters$/$ performance.",
"For example, Huang et al.",
"[15] propose sharing mechanism to share an attention module with the same set of parameters to different blocks in the same stages as Fig.",
"REF (c).",
"Though the shared module significantly reduces the trainable parameters, the computational cost remains the same.",
"Besides, the lightweight attention design, such as feature-wise linear transformation used in SGE [18], reduces the parameters of individual modules and achieves good performance, but it still increases inference time as in Fig.",
"REF .",
"Naturally, a question would be asked: Can we modify the existing self-attention networks such that the modified networks can (1) achieve faster inference speed, (2) contain fewer parameters, and (3) still maintain comparable accuracy at the same time?",
"The major obstacle of accelerating the attention network is the connections between modules and the network.",
"The previous implementation of attention modules follows conventional practice where the attention modules are individually plugged into every block throughout a CNN shown in Fig.",
"REF (b) [13], [31], [18], [15], [5], and hence the computational cost increases with the growing number of blocks.",
"However, how many modules should be used and where to plug the module are rarely discussed.",
"To improve the efficiency of the attention modules in CNNs, in this paper, a simple idea is proposed to reduce the number of interactions between blocks and attention modules instead of plugging the attention modules into each block.",
"Meanwhile, to reduce the parameter cost, we adopt the sharing mechanism [15], and achieve the outcome in Fig.",
"REF (d).",
"Comparing to Fig.",
"REF (b) and (c), our advantages are two-folded, 1 smaller parameter increment 2 less computational cost increment because of fewer connections between backbone and attention module.",
"However, to achieve satisfactory performance, the dense connections [14] and the adequate number of trainable parameters [28] in networks are two critical factors in general.",
"Thus, to balance efficiency and satisfactory performance, we propose a novel reinforcement-learning-based method to search for the optimal connection scheme.",
"Compared with the popular architecture search methods [29], [24], [7], our proposed method can better achieve all the goals where we can obtain attention network with sparse connections between the backbone and modules while (1) accelerating inference (2) reducing extra parameter increment and (3) maintaining accuracy.",
"Our method is called Efficient Attention Network (EAN), which shares the attention module within the backbone and searches where to connect the shared attention module.",
"Our Contribution.",
"1.",
"We propose an effective connection searching method to improve the efficiency of the various attention network while maintaining the original accuracy, reducing the extra parameters increment, and accelerating the inference.",
"2.",
"Our results show that disconnecting some modules from the backbone can be harmless or even improve the model performance.",
"Such empirical finding differs from the intuition behind the conventional approach which applies attention modules to each block.",
"3.",
"Through our empirical experiments, we illustrate that the attention network searched by our method has the capacity of transferring to other tasks and capturing the informative features, which means our EAN networks have potential to apply to wide fields."
],
[
"Related Works",
"Neural Architecture Search (NAS).",
"Designing a satisfactory neural architecture automatically, also known as neural architecture search, is of significant interest for academics and industrial AI research.",
"Such a problem may always be formulated as searching for the optimal combination of different network granularities.",
"The early NAS works require expensive computational cost for scratch-training a massive number of architecture candidates [34], [35].",
"To alleviate the searching cost, the recent advances of one-shot approaches for NAS bring up the concept of supernet based on the weight-sharing heuristic.",
"Supernet serves as the search space embodiment of the candidate architectures, and it is trained by optimizing different sub-networks from the sampling paths, e.g., SPOS [10], GreedyNAS [32] and FairNAS [6].",
"The most conceptually related work [19] aims to propose a lightweight non-local (LightNL) block [30] and searches for the optimal configuration to incorporate the non-local block into mobile neural networks.",
"Although the inserted location of the LightNL is also considered in their NAS objective, the construction of the LightNL blocks is also jointly optimized in their objective.",
"As both the inserted location and the construction of LightNL are integrated completely after the searching, it is hard to differentiate the net contribution of their proposed inserted location of LightNL blocks.",
"However, in our work, we tailor to identify the importance of where to plug in the attention module, and we do not concentrate on only one design of the existing attention modules, compared to Li et al.",
"[19] that only specializes on non-local block [30].",
"To sum up, the difference in the research target and more general consideration differentiate our work with Li et al. [19].",
"Self-Attention Mechanism.",
"The self-attention mechanism is widely used in CNNs for computer vision [13], [30], [15], [5], [18], [21].",
"The self-attention module is modularized as a network component and inserted into different layers of the network to emphasize informative features and their importance according to the internal information.",
"Many works focus on the design of the attention module for specific functionality.",
"Squeeze-and-Excitation (SE) module [13] leverages global average pooling to extract the channel-wise statistics and learns the non-mutually-exclusive relationship between channels.",
"Spatial Group-wise Enhance (SGE) module [18] learns to recalibrate features by saliency factors learned from different groups of the feature maps.",
"Dense-Implicit-Attention (DIA) module [15] captures the layer-wise feature interrelation with a recurrent neural network (RNN)."
],
[
"Preliminaries",
"In this section, we briefly review ResNet [11].",
"Then, we formulate two types of attention networks: Org-full network (Fig.",
"REF (b)), and Share-full network (Fig.",
"REF (c)).",
"The structure of ResNet is shown in Fig.",
"REF (a).",
"In general, the ResNet architecture has several stages, and each stage, whose feature maps have the same size, is a collection of consecutive blocks.",
"Suppose a ResNet has $m$ blocks.",
"Let $x_\\ell $ be the input of the $\\ell ^\\text{th}$ block and $f_\\ell (\\cdot )$ be the residual mapping, and then the output $x_{\\ell +1}$ of the $\\ell ^\\text{th}$ block is defined as $x_{\\ell +1} = x_\\ell + f_\\ell (x_\\ell ).$"
],
[
"Org-full Attention Network",
"We describe an attention network as an Org-full network (Fig.",
"REF (b)) if the attention module is individually defined for each block.",
"Note that the term “full” refers to a scenario that all blocks in a network connect to the attention modules, while “Org” is short for “Original”.",
"Many popular attention modules adopt this way to connect the ResNet backbone [13], [18], [31].",
"We denote the attention module in the $\\ell ^\\text{th}$ block as $M(\\cdot ; W_\\ell )$ , where $W_\\ell $ are the parameters.",
"Then the attention will be formulated as $M(f_\\ell (x_\\ell );W_\\ell )$ which consists of the extraction and processing operations introduced in Section .",
"In the recalibration step, the attention is applied to the residual output $f_\\ell (x_\\ell )$ , i.e., $x_{\\ell +1} = x_\\ell + M(f_\\ell (x_\\ell );W_\\ell )\\odot f_\\ell (x_\\ell ),$ where $\\ell =1,\\cdots , m$ and $\\odot $ is the element-wise multiplication.",
"Eqn.",
"REF indicates that the computational cost and number of parameters grow with the increasing number of blocks $m$ ."
],
[
"Share-full Attention Network",
"We denote an attention network as a Share-full network (Fig.",
"REF (c)) if the blocks within one stage are connected to the same attention module defined for the stage.",
"The idea of Share-full network is first proposed in Huang et al. [15].",
"We denote attention module defined in the stage $k$ as $M(\\cdot ;W_{k})$ .",
"If the $\\ell ^\\text{th}$ block belongs to the $k_\\ell $ stage, then the attention is modeled as $M(f_\\ell (x_\\ell );W_{k_\\ell })$ .",
"The building block becomes $x_{\\ell +1} = x_\\ell + M(f_\\ell (x_\\ell );W_{k_\\ell })\\odot f_\\ell (x_\\ell ),$ where $\\ell =1,\\cdots , m$ .",
"Distinct from the Org-full attention network, the number of extra parameters of the Share-full network depends on the number of stages, instead of the number of blocks $m$ .",
"Typically, a ResNet has 3$\\sim $ 4 stages but has tens of blocks, which indicates a Share-full network can significantly reduce the extra parameters."
],
[
"Proposed Method",
"In this section, we systematically introduce the proposed Efficient Attention Network (EAN) method, which consists of two parts: First, we pre-train a supernet as the search space, and the supernet has the same network structure as a Share-full network.",
"Second, we use a policy-gradient-based method to search for an optimal connection scheme from the supernet.",
"The basic workflow of our method is shown in Alg. .",
"[t] Searching optimal connection scheme Input: Training set $D_\\text{train}$ ; validation set $D_\\text{val}$ ; a Share-full network $\\Omega (\\mathbf {x}|\\mathbf {1})$ ; learning rate $\\eta $ ; pre-training step $K$ ; searching step $T$ ; time step $h$ to apply PPO.",
"Output: The trained controller $\\chi _\\theta (x_0)$ .",
"[1] Pre-train the supernet $t$ from 1 to $K$ $\\mathbf {a} \\sim [Bernoulli(0.5)]^m$ train $\\Omega (\\mathbf {x}|\\mathbf {a})$ with $D_\\text{train}$ Policy-gradient-based search $t$ from 1 to $T$ $\\mathbf {p}_\\theta \\leftarrow \\chi _\\theta (x_0)$ $\\mathbf {a} \\sim \\mathbf {p}_\\theta $ $g_\\text{spa} \\leftarrow $ Eqn.",
"REF $g_\\text{val} \\leftarrow \\Omega (D_\\text{val}|\\mathbf {a})$ , $g_\\text{rnd}$ $\\leftarrow $ $\\left\\Vert \\sigma _1(\\mathbf {a})-\\sigma _2(\\mathbf {a};\\phi )\\right\\Vert _2^2$ calculate the reward $G(\\mathbf {a})$ by Eqn.",
"REF update $\\theta $ by Eqn.",
"REF update $\\phi $ by minimizing $\\left\\Vert \\sigma _1(\\mathbf {a})-\\sigma _2(\\mathbf {a};\\phi )\\right\\Vert _2^2$ put $\\left(\\mathbf {p}_\\theta ,\\mathbf {a},G(\\mathbf {a})\\right)$ into replay buffer Update $\\theta $ from buffer t $\\ge $ h sample $\\left(\\mathbf {p}_\\theta ,\\mathbf {a},G(\\mathbf {a})\\right)$ from replay buffer update $\\theta $ by Eqn.",
"REF $\\chi _\\theta (x_0)$"
],
[
"Problem Description",
"We consider a supernet $\\Omega (\\mathbf {x}|\\mathbf {a})$ with $m$ blocks and input $\\mathbf {x}$ , and $\\Omega (\\mathbf {x}|\\mathbf {a})$ has the same network structure as a Share-full network.",
"A sequence $\\mathbf {a} = (a_1,a_2,\\cdots ,a_m)$ denotes an attention connection scheme, where $a_i =1$ when the $i^\\text{th}$ block is connected to the shared attention module, otherwise it is equal to 0.",
"A sub-network specified by a scheme $\\mathbf {a}$ can be formulated as follows: $\\begin{split}x_{\\ell +1} = x_\\ell + \\Big ( a_\\ell \\cdot M(f_\\ell (x_\\ell );W_{k_\\ell })&+(1-a_\\ell )\\mathbf {1} \\Big ) \\odot f_\\ell (x_\\ell ),\\end{split}$ where $\\mathbf {1}$ denotes an all-one vector and $\\ell $ is from 1 to $m$ .",
"In particular, $\\Omega (\\mathbf {x}|\\mathbf {a})$ becomes a Share-full network if $\\mathbf {a}$ is all-one vector, or a vanilla ResNet while $\\mathbf {a}$ is a zero vector.",
"Our goals are: (1) to find a connection scheme $\\mathbf {a}$ , which is sparse enough for less computation cost, from $2^m$ possibilities; (2) to ensure that the network $\\Omega (\\mathbf {x}|\\mathbf {a})$ possesses good generalization."
],
[
"Pre-training the Supernet",
"To determine the optimal architecture from the pool of candidates, it is costly to evaluate all their individual performance after training.",
"In many related works on NAS, candidates' validation accuracy from a supernet serve as a satisfactory performance proxy [10], [32], [6].",
"Similarly, to obtain the optimal connection scheme for the attention module, we propose to train the supernet as the search space following the idea of co-adaption [22].",
"We consider the validation performance of the sampled sub-networks as the proxy for their stand-aloneTrain the sub-networks from scratch performance.",
"Specifically, given a dataset, we split all training samples into the training set $D_\\text{train}$ and the validation set $D_\\text{val}$ .",
"To train the supernet, we activate or deactivate the attention module in each block of it randomly during optimization.",
"We first initialize a supernet $\\Omega (\\mathbf {x}|\\mathbf {a}^{(0)})$ , where $\\mathbf {a}^{(0)} = (1,\\cdots ,1)$ .",
"At iteration $t$ , we randomly draw a connection scheme $\\mathbf {a}^{(t)}=(a^t_1, \\cdots , a^t_m)$ , where $a^t_i$ is sampled from a Bernoulli distribution $B(0.5)$ .",
"Since every attention module connects or disconnects to the block, without any prior knowledge, it is reasonable to choose probability $0.5$ for randomly drawing a connection scheme.",
"Then, we train sub-network $\\Omega (\\mathbf {x}|\\mathbf {a}^{(t)})$ with the scheme $\\mathbf {a}^{(t)}$ from $\\Omega (\\mathbf {x}|\\mathbf {a}^{(0)})$ on $D_\\text{train}$ via weight-sharing.",
"Table: Comparison of relative inference time increment (see Eqn.",
"), number of parameters, and test accuracy between various attention models on CIFAR100 and ImageNet 2012.",
"“Org” stands for ResNet164 backbone in CIFAR100 and ResNet50 backbone in ImageNet.",
"EAN networks have faster inference speed among the networks with the same type of attention module compared with the same type Share-full attention network.Figure: The illustration of our policy-gradient-based method to search an optimal scheme."
],
[
"Training Controller with Policy Gradient",
"In this part, we introduce the step to search for the optimal connection scheme.",
"Concretely, we use a controller to generate connection schemes and update the controller by policy gradient, as illustrated in Fig.",
"REF .",
"We use a fully connected network as controller $\\chi _\\theta (x_0)$ to produce the connection schemes, where $\\theta $ are the learnable parameters, and $x_0$ is a constant vector $\\mathbf {0}$ .",
"The output of $\\chi _\\theta (x_0)$ is $\\mathbf {p_\\theta }$ , where $\\mathbf {p_\\theta } = (p_\\theta ^1,p_\\theta ^2,...,p_\\theta ^m)$ and $p_\\theta ^i$ represents the probability of connecting the attention to the $i^\\text{th}$ block.",
"A realization of $\\mathbf {a}$ is sampled from the controller output, i.e., $\\mathbf {a} \\sim \\mathbf {p}_\\theta $ .",
"The probability associated with the scheme $\\mathbf {a}$ is $\\mathbf {\\hat{p}_\\theta } = (\\hat{p}_\\theta ^1,\\hat{p}_\\theta ^2,...,\\hat{p}_\\theta ^m)$ , where $\\hat{p}_\\theta ^i = (1-a_i)(1-p_\\theta ^i)+a_ip_\\theta ^i$ .",
"We denote $G(\\mathbf {a})$ as a reward for $\\mathbf {a}$ .",
"The parameter set $\\theta $ can be updated via policy gradient with learning rate $\\eta $ , i.e., $\\begin{split}R_\\theta &= G(\\mathbf {a})\\cdot \\sum _{i=1}^m\\log \\hat{p}_\\theta ^i,\\\\\\theta &= \\theta + \\eta \\cdot \\nabla {R}_\\theta .\\end{split}$ In this way, the controller tends to output the probability that results in a large reward $G$ .",
"Therefore, designing a reasonable $G$ can help us search for a good structure.",
"Sparsity Reward.",
"One of our goals is to accelerate the inference of the attention network.",
"To achieve, we complement a sparsity reward $g_\\text{spa}$ to encourage the controller to generate the schemes with fewer connections between attention modules and backbone.",
"We define $g_\\text{spa}$ by $g_\\text{spa} = 1 - \\frac{\\left\\Vert \\mathbf {a}\\right\\Vert _0}{m},$ where $\\left\\Vert \\cdot \\right\\Vert _0$ is a zero norm that counts the number of non-zero entities, and $m$ is the number of blocks.",
"Validation Reward.",
"The other goal is to find the schemes with which the networks can maintain the original accuracy.",
"Hence, we use the validation accuracy of the sub-network $\\Omega (\\mathbf {x}|\\mathbf {a})$ sampled from the supernet as a reward, which depicts the performance of its structure.",
"The accuracy of $\\Omega (\\mathbf {x}|\\mathbf {a})$ on $D_\\text{val}$ is denoted as $g_\\text{val}$ .",
"In fact, it is popular to use validation accuracy of a candidate network as a reward signal in NAS [24], [34], [10], [35], [32].",
"Furthermore, it has been empirically proven that the validation performance of the sub-networks sampled from a supernet can be positively correlated to their stand-alone performance [1].",
"We evaluate the correlation between the validation accuracy of subnetworks sampled from a supernet and their stand-alone performance on CIFAR100 with ResNet and SE module over 42 samples and obtain the Pearson coefficient is 0.71.",
"Curiosity Bonus.",
"To encourage the controller to explore more potentially useful connection schemes, we add the Random Network Distillation (RND) curiosity bonus [3] in our reward.",
"Two extra networks with input $\\mathbf {a}$ are involved in the RND process, including a target network $\\sigma _1(\\cdot )$ and a predictor network $\\sigma _2(\\cdot ;\\phi )$ , where $\\phi $ is the parameter set.",
"The parameters of $\\sigma _1(\\cdot )$ are randomly initialized and fixed after initialization, while $\\sigma _2(\\cdot ;\\phi )$ is trained with the connection schemes collected by the controller.",
"The basic idea of RND is to minimize the difference between the outputs of these two networks, which is denoted by term $\\sigma _\\phi (\\cdot ) = \\left\\Vert \\sigma _1(\\cdot )-\\sigma _2(\\cdot ;\\phi )\\right\\Vert _2^2$ , over the seen connection schemes.",
"If the controller generates a new scheme $\\mathbf {a}$ , $\\sigma _\\phi (\\mathbf {a})$ is expected to be larger because the predictor $\\sigma _2(\\cdot ;\\phi )$ never trains on scheme $\\mathbf {a}$ .",
"Then, we denote the term $\\left\\Vert \\sigma _1(\\mathbf {a})-\\sigma _2(\\mathbf {a};\\phi )\\right\\Vert _2^2$ as $g_\\text{rnd}$ , which is used as curiosity bonus to reward the controller for exploring a new scheme.",
"Besides, in Fig.",
"REF , we empirically show that RND bonus mitigates the fast convergence of early training iterations, leading to exploration for more schemes.",
"To sum up, our reward $G(\\mathbf {a})$ becomes $G(\\mathbf {a}) = \\lambda _1\\cdot g_\\text{spa} + \\lambda _2 \\cdot g_\\text{val}+ \\lambda _3 \\cdot g_\\text{rnd},$ where $\\lambda _1, \\lambda _2, \\lambda _3$ are the coefficient for each bonus.",
"Data Reuse.",
"To improve the utilization efficiency of sampled connection schemes and speed up the training of the controller, we incorporate Proximal Policy Optimization (PPO) [26] in our method.",
"As shown in Alg.",
", after the update of parameter $\\theta $ and $\\phi $ , we put the tuple $(\\mathbf {p}_\\theta ,~\\mathbf {a},~G(\\mathbf {a}))$ into a buffer.",
"At the later step, we retrieve some used connection scheme and update $\\theta $ as follows: $\\begin{split}\\kappa &=\\mathbb {E}_{\\mathbf {a}\\sim \\mathbf {p}_{\\theta _{old}}}\\left[G(\\mathbf {a})\\sum _{i=1}^m\\frac{\\hat{p}^i_\\theta }{\\hat{p}^i_{\\theta _{old}}}\\nabla _\\theta \\log \\hat{p}_\\theta ^i\\right],\\\\\\theta &=\\theta +\\eta \\cdot \\kappa ,\\end{split}$ where $\\eta $ is learning rate and the $\\theta _{old}$ denotes the $\\theta $ sampled from buffer."
],
[
"Datasets and Settings",
"On CIFAR100 [16] and ImageNet 2012 [25] datasets, we conduct experiments on ResNet [11] backbone with different attention modules, including Squeeze-Excitation (SE) [13], Spatial Group-wise Enhance (SGE) [18] and Dense-Implicit-Attention (DIA) [15] modules.",
"In our supplementary, we describe these modules as well as the training settings of controller and networks.",
"Since the networks with attention modules have extra computational cost from the vanilla backbone inevitably, we formulate the relative inference time increment to represent the relative speed of different attention networks, i.e., $\\frac{I_t(w.~Attention) - I_t(wo.~Attention)}{I_t(wo.~Attention)} \\times 100\\%,$ where $I_t(\\cdot )$ denotes the inference time of the network and the notation $w/wo.~Attention$ represents the network with$/$ without the attention module.",
"The inference time is measured by forwarding the data of batch size 50 for 1000 times on a server with Intel(R) Xeon(R) Gold 5122 CPU @ 3.60GHz and 1 Tesla V100 GPU.",
"Figure: Comparison of the validation accuracy distribution between EAN and Random schemes for SE module.",
"The validation accuracy is obtained by training from scratch the model on CIFAR100 with ResNet164 backbone.Figure: Comparison of the convergence speed between ENAS and EAN.",
"The controller tends to generate a deterministic scheme when 𝐩 ¯\\bar{\\mathbf {p}} is close to 1.CIFAR100.",
"CIFAR100 consists of 50k training images and 10k test images of size 32 by 32.",
"In our implementation, we choose 10k images from the training images as a validation set (100 images for each class, 100 classes in total), and the remainder images as a sub-training set.",
"Regarding the experimental settings of ResNet164 [11] backbone with different attention modules, the supernet is trained for 150 epochs, and the search step $T$ is set to be 1000.",
"ImageNet 2012.",
"ImageNet 2012 comprises 1.28 million training images.",
"We split 100k images (100 from each class and 1000 classes in total) as the validation set and the remainder as the sub-training set.",
"The testing set includes 50k images.",
"Besides, the random cropping of size 224 by 224 is used in ImageNet experiments.",
"Regarding the experimental settings of ResNet50 [11] backbone with different attention modules, the supernet is trained for 40 epochs, and the search step $T$ is set to be 300.",
"Table: The connection schemes searched by ENAS or EAN.",
"The experiment is conducted on CIFAR100 with SE module and ResNet164 backbone.Table: Transfer the optimal architecture searched by EAN from image classification to crowd counting task."
],
[
"Results",
"The concrete connection schemes found by EAN are presented in our supplementary.",
"Table REF shows the test accuracy, the number of parameters, and relative inference time increment on CIFAR100 and ImageNet 2012.",
"Fig.",
"REF visualizes the ImageNet results from Table REF .",
"Since EAN and Share-full network use sharing mechanism [15] for the attention module, over the vanilla ResNet, they both have fewer parameters increment than the Org-full network.",
"Note that EAN networks have faster inference speed among the networks with the same type of attention module compared with the same type Share-full attention network.",
"Furthermore, Share-full networks have higher accuracy than Org-full networks, but in most cases, the accuracy of EAN networks surpass that of Share-full networks.",
"It implies that disconnecting the interaction between the attention and backbone in the appropriate location can maintain or even improve the performance of attention models."
],
[
"Performance distribution of random schemes and EAN schemes",
"In this part, we demonstrate that our EAN method can find effective connection schemes.",
"For the comparison, we draw 180 random connection schemes and obtain 40 connection schemes by EAN search, both under SE module with CIFAR100.",
"Fig.",
"REF displays the distribution of their stand-alone performance.",
"The validation accuracy of random schemes ranges from 71 to $75.6$ while ours EAN accuracy clusters on the right side of random scheme distribution (i.e., on the interval $[74, 76]$ ), which implies the EAN can readily find effective connection schemes.",
"Specifically, EAN accuracy (average: 75.10) is greater than random schemes (average: 74.29) with P-value $4\\times 10^{-7}<0.05$ under t-test.",
"Besides, the standard derivation of EAN (std.",
": 0.45) is much smaller than random sampling (std.",
": 0.81)."
],
[
"Comparison with other searching methods",
"In this part, we compare our EAN search method with heuristic selection policy (HSP), Genetic Algorithm (GA) [29], ENAS [24] and DARTS [7].",
"HSP is a heuristic policy that makes connection every $N$ layers.",
"For example, when $N=2$ , the schemes can be $10101\\cdots $ or $01010\\cdots $ .",
"Table REF displays the experiments conducted on CIFAR100 with ResNet 164 and SE-module for different searching methods.",
"From Table REF , our EAN outperforms the heuristic method such as HSP or GA.",
"Different from DARTS that searches schemes by minimizing the validation loss, the reinforcement-learning-based method (e.g., ENAS and our method) can directly consider the validation accuracy as a reward although the accuracy or sparsity constraint is not differentiable.",
"However, ENAS tends to converge to some periodic-alike schemes, which indicates that ENAS does not learn effective scheme from the reward.",
"Table: Comparison of the testing accuracy and relative inference time increment of the searched network for different methods.From our empirical results, the controller of ENAS tends to converge to some periodic-alike schemes at a fast speed.",
"In this case, it will conduct much less exploration of the potential efficient structures.",
"The majority of the schemes searched by ENAS are “111...111” (Share-full network) or “000...000” (Vanilla network), which shows that it can not get the balance between the performance and inference time.",
"The list of schemes searched by ENAS is presented in our supplementary.",
"In Table REF , the minority of the periodic-alike schemes searched by ENAS are shown, e.g., “001” in ENAS (a).",
"Such schemes may come from the input mode of ENAS, i.e., for a connection scheme $\\mathbf {a} = (a_1,a_2,...,a_m)$ , the value of component $a_l$ depends on $a_{l-1},a_{l-2},...,a_1$ .",
"Such strong sequential correlations let the sequential information dominate in the RNN controller instead of the policy rewards.",
"Compared with the periodic-alike connection schemes from ENAS, the schemes from EAN demonstrate better performance.",
"Besides, our experiment indicates that ENAS explores a much smaller number of candidate schemes.",
"We quantify the convergence of the controller using $\\bar{\\mathbf {p}}=\\frac{1}{m}\\sum _{i=1}^m\\hat{p}_\\theta ^i$ , which is the mean of the probability $\\hat{\\mathbf {p}}$ associated with the scheme.",
"When $\\bar{\\mathbf {p}}$ is close to 1, the controller tends to generate a deterministic scheme.",
"Fig.",
"REF shows the curve of $\\bar{\\mathbf {p}}$ with the growth of searching iterations, where $\\bar{\\mathbf {p}}$ of ENAS shows the significant tendency for convergence in 20 iterations and converges very fast within 100 iterations.",
"Generally speaking, methods in NAS [24], [34] require hundreds or thousands of iterations for convergence."
],
[
"Transferring Connection Schemes",
"To further investigate the generalization of EAN, we conduct experiments on transferring the optimal architecture from image classification to crowd counting task [33], [4], [20], [12], semantic segmentation [9].",
"Crowd counting.",
"Crowd counting aims to estimate the density map and predict the total number of people for a given image, whose efficiency is also crucial for many real-world applications, e.g., video surveillance and crowd analysis.",
"However, most state-of-the-art works still rely on the heavy pre-trained backbone networks [23] for obtaining satisfactory performance on such dense regression problems.",
"The experiments show that the network obtained by EAN on ImageNet serves as an efficient backbone network and can extract the representative features for crowd counting.",
"The networks pre-trained on the ImageNet dataset serve as the backbone of crowd counting models.",
"We evaluate the transferring performance on the commonly-used Shanghai Tech dataset [33], which includes two parts.",
"Shanghai Tech part A (SHHA) has 482 images with 241,677 people counting, and Shanghai Tech part B (SHHB) contains 716 images with 88,488 people counting.",
"Following the previous works, SHHA and SHHB are split into train/validation/test set with 270/30/182 and 360/40/316 images, respectively.",
"The performance on the test set is reported using the standard Mean Square Error (MSE) and Mean Absolute Error (MAE), as shown in Table REF .",
"Our EAN can outperform the baseline (Org-full and Share-full) while reducing the inference time increment by over 40% compared with the baseline.",
"Figure: Grad-CAM visualization of different attention models.",
"The red region indicates an essential place for a network to obtain a target score (𝐏\\mathbf {P}) while the blue region is the opposite.Semantic segmentation.",
"We verify the transferability of the attention networks obtained by EAN on semantic segmentation task in Pascal VOC 2012 [8] dataset.",
"Table REF shows the performance comparison of the backbone with different attention modules, e.g., DIA and SE.",
"Again, our results indicate that the EAN-network can maintain the performance of the Share-full network and significantly reduce the time increment compared with the Share-full network, which shows EAN has the capacity of transferring to semantic segmentation.",
"Table: Performance and relative inference time increment comparison on Pascal VOC 2012 val set."
],
[
"Capturing Discriminative Features",
"To study the ability of EAN in capturing and exploiting features of a given target, we apply Grad-CAM [27] to compare the regions where different models localize on with respect to their target prediction.",
"Grad-CAM is a technique to generate the heatmap highlighting network attention by the gradient related to the given target.",
"Fig.",
"REF shows the visualization results and the softmax scores for the target with vanilla ResNet50, Share-full-SE, and EAN-SE on the validation set of ImageNet 2012.",
"The red region indicates an essential place for a network to obtain a target score while the blue region is the opposite.",
"The results show that EAN-SE can extract similar features as Share-full-SE, and in some cases, EAN can even capture much more details of the target associating with higher confidence for its prediction.",
"This implies that the searched attention connection scheme may have a more vital ability to emphasize the more discriminative features for each class than the two baselines (Vanilla ResNet and Share-full-SE).",
"Therefore it is reasonable to bring additional improvement on the final classification performance with EAN in that the discrimination is crucial for the classification task, which is also validated from ImageNet test results in Table REF ."
],
[
"Conclusion",
"To improve the efficiency of using the attention module in a network, we propose an effective EAN method to search for an optimal connection scheme to plug the modules.",
"Our numerical results show that the attention network searched by our method can preserve the original accuracy while reducing the extra parameters and accelerating the inference.",
"We empirically illustrate that our attention networks have the capacity of transferring to other tasks and capturing the informative features."
]
] | 2011.14058 |
[
[
"Geometric parametrization of $SO(D+1)$ phase space of all dimensional\n loop quantum gravity"
],
[
"Abstract To clarify the geometric information encoded in the $SO(D+1)$ spin-network states for the higher dimensional loop quantum gravity, we generalize the twisted-geometry parametrization of the $SU(2)$ phase space for $(1+3)$ dimensional loop quantum gravity to that of the $SO(D+1)$ phase space for the all-dimensional case.",
"The Poisson structure in terms of the twisted geometric variables suggests a new gauge reduction procedure, with respect to the discretized Gaussian and simplicity constraints governing the kinematics of the theory.",
"Endowed with the geometric meaning via the parametrization, our reduction procedure serves to identify proper gauge freedom associated with the anomalous discretized simplicity constraints and subsequently leads to the desired classical state space of the (twisted) discrete ADM data."
],
[
"Introduction",
"Loop quantum gravity (LQG) [1][2][3][4] as a candidate theory of quantum gravity provides a possibility of deriving general relativity (GR) from the foundation of plank-scale quantum geometry.",
"Thus the theory, in a broader context, provides a concrete platform for exploring the relation between the continuum classical GR variables and the discretized geometric quantum data, such as those of the twistor theory and Regge calculus [5][6].",
"On the other hand, it has been realized that the correspondence between the field-geometric and the quantum data is far beyond the issue of merely taking the continuum limits.",
"This is due to the fact that canonical GR is governed by a constraint system, and the correspondence may be fully revealed only for the physical degrees of freedom— with all the constraints properly imposed in the quantum theory.",
"From the opposite direction of this view, the concrete goal of recovering the familiar ADM data from LQG may provide useful instructions in tackling the abstract problems of quantum constraint reductions in the theory.",
"A series of illuminating analysis in this direction has been carried out in the case of the $SU(2)$ formulation of $(1+3)$ -dimensional loop quantum gravity.",
"Based on the Ashtekar formulation of canonical GR using the $SU(2)$ desitized triad and connection conjugate variables, LQG in this formulation has a kinematic Hilbert space spanned by the spin-network states, each of which is given by a network of the connection holonomies, with each edge of the graph of the network colored by a specific $SU(2)$ representation, and each of the vertices colored by an intertwiner specifying a coupling among the neighboring $SU(2)$ representations.",
"Under the well-defined flux-holonomy geometric operators, the $SU(2)$ representations indicate the quanta of the triad-fluxes as the area elements dual to the graph's edges, while the intertwiners indicate the intersection angles amongst these triad-fluxes at the vertices.",
"This discretized distribution of the 2-dimensional spatial area elements with the intersection angles leads to a specific notion of quantum geometry that is the foundation of LQG.",
"The classical constraints– the scalar, vector and $SU(2)$ Gauss constraints—can be represented via the flux-holonomy operators for the quantum theory.",
"It has been shown that the imposition of the quantum Gauss constraints on the coherent spin-network states gives rise to a proper semi-classical symplectic reduction, in the holonomy-flux phase of the discretized Ashtekar formulation on the given graph.",
"Remarkably, in the reduced state space, the coherent spin-network states satisfying the quantum Gauss constraints not only describe the intrinsic spatial geometry built from the polytope-cells dual to the network [5][7][8], but also carry precisely the right data to specify the extrinsic curvature of the hypersurface made of the polytopes.",
"[6][9].",
"Through this first stage of the semi-classical gauge reduction, a notion of kinematic ADM data may thus appear in the discrete form of Regge geometry, upon which the further reductions with the momentum and scalar constraints should to be carried out.",
"The quantum vector and scalar constraints take much more complicated forms in the flux-holonomy operators, and unlike the quantum Gauss constraint, their anomalous algebra is no longer of first class.",
"With the quantum anomaly hindering the standard Dirac procedure mirroring the classical gauge reduction, the treatment of these loop-quantized ADM constraints remains a crucial challenge for LQG tackled by many ongoing projects.",
"As we introduced above, loop quantum gravity is first constructed as a quantum theory of GR in four dimensional spacetime.",
"Nevertheless, with the various classical and quantum gravity theories in higher-dimensional spacetimes (i.e., Kaluza-Klein theory, super string theories) showing remarkable potentials in unifying gravity and other fundamental interactions, it has been recognized that the framework of arbitrary dimensional loop quantum gravity may serve as a novel approach toward the higher-dimensional ideas of unification, upon the background-independent and non-perturbative construction of the discretized quantum geometry.",
"Pioneered by Bodendorfer, Thiemann and Thurn [10][11][12], the loop quantization approach for general relativity in all dimensions has been developed.",
"In the context of the higher dimensional loop quantum gravity, the challenge of loop quantum anomaly already exists at the kinematic level before the accounts of the quantum ADM constraints; though, here it is in a simpler form for us to develop concrete insights and solutions to the problem.",
"In detail, the all dimensional LQG is based on the universal Ashtekar formulation of $(1+D)$ dimensional general relativity in the form of the $SO(D+1)$ Yang-Mills theory, with the kinematic phase space coordinatized by the canonical pairs $(A_{aIJ},\\pi ^{bKL})$ , consisting of the spatial $SO(D+1)$ connection fields $A_{aIJ}$ and the vector fields $\\pi ^{bKL}$ .",
"In this formulation, the theory is governed by the first class system of the $SO(D+1)$ Gauss constraints, the $(D+1)$ -dimensional ADM constraints and the additional constraints called the simplicity constraints.",
"Taking the form $S^{ab}_{IJKL}:=\\pi ^{a[IJ}\\pi ^{|b|KL]}$ , the simplicity constraints generate extra gauge symmetries in the $SO(D+1)$ Yang-Mills phase space.",
"It is known that the phase space correctly reduces to the familiar ADM phase space after the symplectic reductions with respected to the Gauss and simplicity constraints.",
"Similar to the case of the $SU(2)$ formulation, the loop quantization of the $SO(D+1)$ formulation leads to the spin-network states of the $SO(D+1)$ holonomies carrying the quanta of the flux operators representing the flux of $\\pi ^{bKL}$ over a $(D-1)$ surface.",
"Following the previous experience, one may attempt to look for the all-dimensional Regge ADM data encoded in the $SO(D+1)$ spin-network states, through a gauge reduction procedure with respect to both the quantum $SO(D+1)$ Gaussian constraints and the quantum simplicity constraints.",
"This is where the challenge arises— the standard quantum simplicity constraints in LQG carry serious quantum anomaly.",
"As a result of the loop quantization, the abelian algebra of the classical simplicity constraints becomes the deformed algebra of the quantum simplicity constraints that is not even close [12].",
"As an important consequence, the transformations generated by these anomalous quantum simplicity constraints can happen between states supposed to be physically distinct in terms of the semiclassical limits.",
"Strong impositions of the quantum simplicity constraints thus lead to over-constrained physical states unable to reproduce the semi classical degrees of freedom.",
"In a closer look, the quantum simplicity constraints in LQG consist of two types of local constraints due to the network discretization— the edge-simplicity constraints and the vertex-simplicity constraints.",
"Importantly, the algebra anomaly happens only amongst the vertex-simplicity constraints, while the edge-simplicity constraints remain anomaly free in the sense of having a weakly abelian algebra.",
"Previously, we have proposed a new method [13] of weakly imposing the anomalous vertex-simplicity constraints for the vanishing expectation values and minimal quantum fluctuations, upon a special class of states in the space of the $SO(D+1)$ -invariant spin-network states satisfying the quantum Gauss constraints.",
"With their edges labeled by only the simple representations and their vertices by specific coherent states of intertwiners, this class of states strongly satisfy the quantum edge-simplicity constraints and are sharply peaked for the flux operators.",
"We found that, among this class of states, each weak solution of the vertex-simplicity constraints describes a set of quantum $D$ -dimensional polytopes dual to the vertices of its graph.",
"Also, in large quantum-number limits, these weak solutions indeed recover all the degrees of freedom in the classical $D$ -dimensional polytopes, which may be assembled to describe all possible states of quantum spatial geometry.",
"Concerning the proper gauge-reduction procedure, this remarkable result suggests that, in the space of strong solutions to the first class system of the quantum Gauss constraints and edge-simplicity constraints, the vertex-simplicity constraints should serve as additional constraints—unrelated to the quantum gauge symmetries—and select the special gauge-invariant states capable of giving the desired quantum discrete spatial intrinsic geometry.",
"Toward the full ADM data of a hypersurface, our remaining task is to show how the polytopes at the various vertices are correlated, so that both of the intrinsic and extrinsic geometry can be recovered.",
"Clearly, this task would be completing the other half gauge-reduction procedure following our strategy above: identifying the proper gauge orbits associated with the quantum simplicity constraints and finding the geometrical interpretation to the invariant degrees of freedom.",
"It is known that the classical simplicity constraints transform only the pure-gauge components of the $SO(D+1)$ Ashtekar connection $A_{aIJ}$ , while leaving the vector fields $\\pi ^{bKL}$ invariant.",
"We will demonstrate that this picture could emerge in the LQG flux-holonomy phase space associated with a graph, following the other half of our reduction procedure at the classical and discrete level under a satisfying geometric interpretation, in spite of the anomaly of quantum simplicity constraint already appearing in this classical and discrete formulation.",
"In the crucial step for establishing such an interpretation, we will generalize the existing twisted-geometry parametrization for the $SU(2)$ flux-holonomy phase space, into that for the $SO(D+1)$ setting.",
"These new geometric coordinates for the phase space, along with their well-formulated expressions for the symplectic structure, enable the full analysis of the gauge reductions in the language of the twisted geometry.",
"Our result shows that, the discretized classical Gaussian, edge-simplicity and vertex-simplicity constraints catching the anomaly of quantum vertex simplicity constraint define a constraint surface in the discrete phase space of all dimensional LQG, and the kinematic physical degrees of freedom parameterized by the generalized hypersurface twisted-geometry are given by the gauge orbits in the constraint surface generated by the first class system of discrete Gaussian and edge-simplicity constraints.",
"In particular, we find the orbits of the edge-simplicity constraints to be along the angle variables of the twisted geometry, which indeed represent the smeared form of the pure-gauge components of the Ashtekar connection in the continuous theory.",
"Finally, the complete ADM data of a Regge hypersurface can be identified as the degrees of freedom of the reduced generalized twisted geometry space, under an additional condition called the shape matching condition.",
"In our brief review of the classical Ashtekar formulation of all dimensional GR in Section 2, we will also introduce the flux-holonomy phase space for the discretized formulation with the anomalous vertex simplicity constraints.",
"In Section 3 and Section 4 we will introduce the twisted-geometry parametrization for the $SO(D+1)$ phase space, and analyze the Poisson structures among the new geometric parametrization variables and the discretized simplicity constraints.",
"Finally in Section 5 we will combine the obtained gauge transformations with the geometric interpretations, and formalize the gauge reduction procedure that leads to the desired ADM data.",
"We will then conclude with the outlook for the possible next steps of the future research."
],
[
"Phase space of all dimensional loop quantum gravity and simplicity constraint",
"The classical Ashtekar formulation of general relativity with arbitrary spacetime dimensionality of $(D+1)$ has been developed by Bodendofer, Thiemann and Thurn in [10].",
"The continuum connection phase space of the theory is coordinatized by a $so(D+1)$ valued canonical pair $(A_{aIJ}, \\pi ^{bKL})$ with the non-trivial Poisson brackets $\\lbrace A_{aIJ}(x), \\pi ^{bKL}(y)\\rbrace =2\\kappa \\beta \\delta _a^b\\delta _{[I}^K\\delta _{J]}^L\\delta ^{(D)}(x-y),$ where $\\beta $ is the Barbero-Immirzi parameter and $\\kappa $ is the gravitational constant.",
"It is known that this phase space correctly reduces to the familiar ADM phase space after the standard sympletic reduction procedure with respect to the first-class constraint system of the Gauss constraints $\\mathcal {G}^{IJ}\\approx 0$ and simplicity constraints $S^{ab}_{IJKL}:=\\pi ^{a[IJ}\\pi ^{|b|KL]}\\approx 0$ .",
"Specifically, the spatial metric $q_{ab}$ is given by $q_{ab}=e_{aI}e_b^I$ , where $e_a^I$ is a D-bein field parametrizing the simplicity constraint solutions in the form $\\pi ^{aIJ}=2\\sqrt{q}{\\mathcal {N}}^{[I}e^{|a|J]}$ together with a chosen field ${\\mathcal {N}}^I$ satisfying ${\\mathcal {N}}^I{\\mathcal {N}}_I=1$ and ${\\mathcal {N}}^Ie_{aI}=0$ .",
"The densitized extrinsic curvature is given by $\\tilde{K}_a^{\\ b}=K_{aIJ}\\pi ^{bIJ}$ where $K_{aIJ}$ is the component of $A_{aIJ}$ under the splitting $A_{aIJ}\\equiv \\Gamma _{aIJ}(e)+\\beta K_{aIJ}$ on simplicity constraint surface, where $\\Gamma _{aIJ}(e)$ is the unique torsionless spin connection compatible with the D-bein $e_{aI}$ .",
"Let us look into the simplicity constraints from the perspectives of the corresponding reductions.",
"First, the solutions $\\pi ^{aIJ}=2\\sqrt{q}{\\mathcal {N}}^{[I}e^{|a|J]}$ to the quadratic simplicity constraints introduced above defines the constraint surface of the simplicity constraints.",
"It is easy to check that the infinitesimal gauge transformations induced by simplicity constraints are given by $\\delta K_{c}^{PQ}=\\lbrace \\int _{\\sigma }d^Dxf_{ab}^{IJKL}\\pi ^a_{[IJ}\\pi ^b_{KL]}(x), K_{c}^{PQ}(y)\\rbrace =4\\kappa \\beta f_{cb}^{[PQKL]}\\pi ^b_{KL}(y).$ On the simplicity constraint surface we have $\\pi ^{aIJ}=2\\sqrt{q}{\\mathcal {N}}^{[I}e^{|a|J]}$ and thus $\\delta K_{c}^{IJ}{\\mathcal {N}}_I=0$ .",
"Therefore, introducing the decomposition of $K_{aIJ}$ as $K_{aIJ}\\equiv 2{\\mathcal {N}}_{[I}K_{|a|J]}+\\bar{K}_{aIJ},$ where $\\bar{K}_{aIJ}:=\\bar{\\eta }_I^K\\bar{\\eta }_J^LK_{aKL}$ with $\\bar{\\eta }^I_J=\\delta ^I_J-{\\mathcal {N}}^I {\\mathcal {N}}_J$ and $\\bar{K}_{aIJ}{\\mathcal {N}}^I=0$ , we immediately see that the longitudinal components $\\bar{K}_{aIJ}$ parametrize the gauge redundancy, while the transverse components $2{\\mathcal {N}}_{[I}K_{|a|J]}$ are gauge invariant based on the transformations given in (REF ).",
"From the expressions for the ADM variables $\\tilde{\\tilde{q}}^{ab}=\\frac{1}{2}\\pi ^{aIJ}\\pi ^b_{IJ}$ and $\\tilde{K}_a^{\\ b}=K_{aIJ}\\pi ^{bIJ}$ , it is easy to see that these variables are indeed invariant under the gauge transformations by the simplicity constraints.",
"Through the sympletic gauge-reduction procedure, the simplicity constraints thus eliminate the two sets of degrees of freedom– setting $\\bar{\\pi }^{aIJ}:=\\pi ^{aIJ}-2\\sqrt{q}n^{[I}e^{|a|J]}\\approx 0$ by the restriction to the constraint surface and removing the pure-gauge components $\\bar{K}_{aIJ}:=\\bar{\\eta }_I^K\\bar{\\eta }_J^LK_{aKL}$ .",
"The foundation leading to the quantum geometry of loop quantum gravity is the use of the spatially smeared variables– the D-bein fluxes over surfaces and connection holonomies over paths– for the conjugate pairs of elementary variables.",
"The quantization of the flux-holonomy algebra leads to the space of spin-network states mentioned above, spanned by the basis states of holonomy networks each labeled by a graph with the representation and intertwiner colorings.",
"We will focus on the holonomies and fluxes based on one specific graph for the following.",
"The edges of the given graph naturally provide the set of paths for a fixed set of holonomies, and the cell decomposition dual to the graph provides the set of (D-1)-faces specifying a fixed set of fluxes.",
"In this setting, the holonomy over one of the edges is naturally conjugating to the flux over the face traversed by the edge, and the pairs associated with the given graph satisfy the smeared version of the algebra (REF ) and form a new phase space.",
"More precisely, given the graph $\\gamma $ embedded in the spatial manifold, we consider a new algebra given by replacing $(A_{aIJ},\\pi ^{bKL})$ with the pairs $(g_e, X_e)\\in SO(D+1)\\times so(D+1)$ over all edges $e$ of $\\gamma $ .",
"These pairs of variables represent the discretized version of the connection and its conjugate momentum $\\pi ^{aIJ}$ , respectively via the holonomies $g_e=\\mathcal {P}\\exp \\int _e A$ with $\\mathcal {P}$ denoting the path-ordered product, and fluxes $X_e=\\int _{e^\\star }(g\\pi g^{-1})^an_ad{}^{D-1}\\!S$ with $e^\\star $ being the dual (D-1)-dimensional face to the edge $e$ , with the normal $n_a$ and infinitesimal coordinate area element $d{}^{D-1}\\!S$ and $g$ is the parallel transport from one fixed vertex to the point of integration along a path adapted to the graph.",
"Since $SO(D+1)\\times so(D+1)\\cong T^\\ast SO(D+1)$ , this new discrete phase space called the phase space of $SO(D+1)$ loop quantum gravity on a fixed graph, is a direct product of $SO(D+1)$ cotangent bundles.",
"Finally, the complete phase space of the theory is given by taking the union over the phase spaces of all possible graphs.",
"Just like the $SU(2)$ case, the new variables $(g_e, X_e)$ of the phase space of $SO(D+1)$ loop quantum gravity can be seen as a discretized version of the continuum phase space.",
"A series of studies following the original works by Freidel and Speziale show that the mentioned phase space of $SU(2)$ loop quantum gravity carries the notion of what is called the twisted geometry [5][6], and this space can undergo a symplectic reduction with respect to the discretized Gauss constraints (associated with the quantum Gauss constraint operators), giving rise to a reduced phase space containing the discretized ADM data of a polyhedral Regge hypersurface.",
"Based on such a foundation, our first goal is providing a generalization to the above approach for the $SO(D+1)$ formulation.",
"This includes the generalization of the twisted-geometry parametrization for the $SO(D+1)$ phase space, which should provide a clear correspondence between the original variables $(g_e, X_e)$ and the hypersurface geometry data.",
"Our second goal is addressing the proper treatment of the (discretized) Gaussian and simplicity constraints, following the geometric meaning of the phase space under the new parametrization.",
"We will use the standard forms of the (discretized) Gaussian and simplicity constraints in agreement with the quantum constraints.",
"With $X_{-e}=-g_e^{-1}X_eg_e\\equiv \\tilde{X}_e$ , the (discretized) Gauss constraints $G_v^{IJ}\\approx 0$ for each vertex $v\\in \\gamma $ of the graph take the form $G_v^{IJ}=\\sum _{e|s(e)=v}X_e^{IJ}+\\sum _{e|t(e)=v}\\tilde{X}_e^{IJ}\\approx 0,$ where $s(e)$ and $t(e)$ respectively denote the source and target vertices of the oriented edge $e$ .",
"The (discretized) simplicity constraints consist of the edge-simplicity constraints $S^{IJKL}_e\\approx 0$ and vertex-simplicity constraints $S^{IJKL}_{v,e,e^{\\prime }}\\approx 0$ taking the forms $S_e^{IJKL}\\equiv X^{[IJ}_e X^{KL]}_e\\approx 0, \\ \\forall e\\in \\gamma ,\\quad S_{v,e,e^{\\prime }}^{IJKL}\\equiv X^{[IJ}_e X^{KL]}_{e^{\\prime }}\\approx 0,\\ \\forall e,e^{\\prime }\\in \\gamma , s(e)=s(e^{\\prime })=v.$ As we mentioned in the introduction, since the commutative set of conjugate momentum varaibles $\\lbrace \\pi ^{bKL}\\rbrace $ becomes non-commutative set of flux variables $\\lbrace X^{KL}_e\\rbrace $ after the smearing, these discrete version of simplicity constraints become non-commutative and thus anomalous."
],
[
"Bivector parametrization of intrinsic geometry",
"In our previous work [13], we have explicitly constructed specific flux-coherent states based on a chosen graph $\\gamma $ , which are sharply peaked in every pair of flux variables associated with the source and target points of each of the edges, while having the coloring of the edges restricted to the $SO(D+1)$ simple representations.",
"Such restriction to the irreducible representations has been shown to strongly solve the quantum edge-simplicity constraints.",
"Subsequently, this implies that the flux expectation values associated to the source and target points of an edge must respectively take the form $N_e V_e$ and $N_e \\tilde{V}_e$ , with $V_e$ and $\\tilde{V}_e$ given by normalized bivectors in $\\mathbb {R}^{D+1}$ and the shared norm $N_e$ is the Casimir value labeling the simple representation.",
"We have shown that, further, the quantum Gauss and quantum vertex-simplicity constraints can be weakly imposed upon our flux-coherent states by restricting and correlating the values of $(V_e, \\tilde{V}_e, N_e)$ over all the edges , and the resulted states describe the familiar discrete geometry of a set of D-polytopes dual to the graph $\\gamma $ , with the corresponding faces dual to the same edge having the same area.",
"In more details, with the fixed oriented graph $\\gamma $ , each of our flux-coherent states has a $SO(D+1)$ simple representation $N_e\\in \\mathbb {R}$ to each of the edges, assigning the area of the corresponding set of dual (D-1)-dimensional surfaces; for each vertex $v\\in \\gamma $ connected to $n_v$ number of edges, the state is also peaked at the $n_v$ number of unit bi-vectors $V^{IJ}_e(v)$ assigning the directions of the $n_v$ number of the (D-1)-surfaces.",
"These parameters describe a direct product space $P^{\\textrm {aux}}_\\gamma \\equiv \\times _e \\mathbb {R}_e\\times _v P_v, \\quad P_v\\equiv \\times _{e:v=b(e)\\ \\textrm {or}\\ v=t(e)} Q^e_{D-1},$ where $Q_{D-1}:=SO(D+1)/(SO(D-1)\\times SO(2))$ is the space of unit bi-vectors $V^{IJ}_e(v)$ .",
"Note that by our assignment, each edges is labelled by $N_e$ and two unit bi-vectors.",
"Calling $s(e)$ the source vertex and $t(e)$ the target vertex of an edge $e$ , we denote the two bivectors as $V_e\\equiv V^{IJ}_e=V^{IJ}_e(s(e))$ and $\\tilde{V}_e\\equiv \\tilde{V}^{IJ}_e=V^{IJ}_e(t(e))$ .",
"We may use this notation to factorize the space as $P^{\\textrm {aux}}_\\gamma =\\times _e P^{\\textrm {aux}}_e, \\quad P^{\\textrm {aux}}_e= Q^e_{D-1}\\times Q^e_{D-1}\\times \\mathbb {R}_e,$ and the variables associated to each edge of the graph are thus a triple $(V_e,\\tilde{V}_e,N_e)$ .",
"For our flux-coherent states [13], the weak imposition of the quantum Gauss and vertex-simplicity constraints amounts to imposing the corresponding constraints in the space $P^{\\textrm {aux}}_\\gamma $ .",
"The weak imposition of the quantum Gauss constraints at a vertex $v$ implies $C_{\\vec{N}_v}\\equiv \\sum _{e:v=b(e)}N_eV^{IJ}_e+\\sum _{e: v=t(e)}N_e\\tilde{V}^{IJ}_e=0,$ and that of the vertex-simplicity constraints at a vertex $v$ (weakly) implies $S^{IJKL}_{v}\\equiv V^{[IJ}_{e_\\imath }(v)V^{KL]}_{e_\\jmath }(v)=0,\\quad \\forall e_\\imath , e_\\jmath :v=e_\\imath \\cap e_\\jmath .$ The variables $V^{IJ}_e(v)$ for the vertex $v$ satisfying both conditions defining the common constraint surface $\\mathcal {P}^{\\textrm {s.}}_{\\vec{N}_v}=\\lbrace (V_{e_1}^{IJ}(v),...,V_{e_{n_v}}^{IJ}(v))\\in P_v| C_{\\vec{N}_v}=0,\\ S^{IJKL}_{v}=0 \\rbrace $ must take the form of $V_{e}^{IJ}(v)=\\mathcal {N}^{[I}(v) V_{e}^{J]}(v)$ , where the vectors $V_{e}^{J}(v)\\in \\mathbb {R}^{D+1}$ for each $v$ lie in the subspace $\\mathbb {R}^{D}\\subset \\mathbb {R}^{D+1}$ orthogonal to an unit vector $\\mathcal {N}^{I}(v)\\in \\mathbb {R}^{D+1}$ , and they satisfy the familiar $D$ -dimensional Minkowski closure conditions applied through the closure constraints $ C_{\\vec{N}_v}=0$ .",
"Therefore these solutions define the space of flat D-polytopes embedded in the flat space $\\mathbb {R}^{D}\\subset \\mathbb {R}^{D+1}$ , and each of the $n_v$ -valent vertex $v$ can be thought of as dual to a flat convex D-polytope whose $n_v$ number of (D-1)-faces' areas and normal vectors are given by repectively $\\lbrace N_e| b(e)\\ \\text{or}\\ t(e)=v\\rbrace $ and $\\lbrace V^{J}_e,\\tilde{V}^{J}_{e^{\\prime }}| b(e)=v,\\ \\text{and}\\ t(e^{\\prime })=v\\rbrace $ satisfying the closure conditions.",
"Since the shape of a D-polytope is invariant under the rotations, it is useful to introduce the space $\\mathfrak {P}^{\\textrm {s.}}_{\\vec{N}_v}$ of shapes of the D-polytopes, i.e.",
"the space of closed normals modulo the vertex-wise $SO(D+1)$ rotations as [13] $\\mathfrak {P}^{\\textrm {s.}}_{\\vec{N}_v}\\equiv \\mathcal {P}^{\\textrm {s.}}_{\\vec{N}_v}/SO(D+1).$ Therefore, we see that the bivector variables $( N_e{V}_e, N_e\\tilde{V}_e)$ carried by the flux-coherent states, when taking the onshell values of the Gauss and vertex-simplicity constraints, may give a notion of discretized spatial geometry as an assembly of the locally flat D-polytopes dual to the vertices, with the identical areas for the pairs of corresponding faces amongst neighboring polytopes.",
"We want to complete the description of such geometry and extend it to the extrinsic part, so that a notion of hypersurface ADM data could be identified for the relevant region of the LQG phase space.",
"Since the $D$ -polytope geometry arises only after imposing the edge-simplicity constraints, in this context the relevant region of the LQG phase space is expected to be the edge-simplicity constraint surface.",
"This surface, denoted as $\\times _e T_s^\\ast SO(D+1)_e$ , is obtained from the discrete phase space $\\times _e T^\\ast SO(D+1)_e$ by restricting the $X_e$ to be of the bivector form of $N_e V^{IJ}_e $ .",
"As shown in [14][15], the space $Q_{D-1}$ is a $2(D-1)$ -dimensional phase space with the invariant Kahler form $\\Omega _{N^2/2}$ , and the $SO(D+1)$ orbits in $P_v$ are generated precisely by the closure constraints; therefore we may construct the $SO(D+1)$ -reduced phase space given by $\\mathfrak {P}_{\\vec{N}_v}=\\lbrace (V_{e_1}^{IJ}(v),...,V_{e_{n_v}}^{IJ}(v))\\in P_v| C_{\\vec{N}_v}=0 \\rbrace /SO(D+1).$ The Poisson structure on this $n_v(\\frac{D(D+1)}{2}-1-\\frac{(D-2)(D-1)}{2})-D(D+1)=2n_v(D-1)-D(D+1)$ -dimensional space is obtained from $\\Omega _{N^2/2}$ defined in $Q_{D-1}$ , via the standard symplectic reduction.",
"However, subject to the additional vertex-simplicity constraints the space $\\mathfrak {P}^{\\textrm {s.}}_{\\vec{N}_v}$ describing the shapes of the D-polytopes is not a phase space because the imposition of vertex-simplicity constraints clearly does not give a symplectic reduction [13].",
"Just as mentioned, we will demonstrate that a true reduction can be carried out in the discrete phase space extended from $P^{\\textrm {aux}}_\\gamma $ , which includes the information about the extrinsic curvature and contains the gauge degrees of freedom for the discretized simplicity constraints.",
"From now on, we denote the symplectic reduction by double quotient $\\mathfrak {P}_{\\vec{N}_v}=P_v//C_{\\vec{N}_v}$ .",
"Considering the space $P_{\\gamma }^{\\textrm {aux}}$ defined for the whole graph $\\gamma $ , we can accordingly apply the symplectic reduction by $\\mathcal {C}_\\gamma =\\lbrace C_{\\vec{N}_v}|v\\in \\gamma \\rbrace $ and impose both the closure conditions and vertex-simplicity constraints on all the vertices.",
"The result is $\\mathcal {K}_\\gamma \\equiv P_{\\gamma }^{\\textrm {aux}}//\\mathcal {C}=\\times _e\\mathbb {R}_e\\times _v \\mathfrak {P}_{\\vec{N}_v}, \\quad \\mathcal {K}_\\gamma ^{\\textrm {s.}}\\equiv \\mathcal {K}_\\gamma |_{S_v^{IJKL}=0, \\forall v\\in \\gamma }=\\times _e\\mathbb {R}_e\\times _v \\mathfrak {P}^{\\text{s.}}_{\\vec{N}_v}.$"
],
[
"Full parametrization",
"According to our discussion above, the space $(g_e, X_e)$ is parametrized by $ (g_e, N_e V_e)$ in the constraint surface $\\times _e T_s^\\ast SO(D+1)_e$ .",
"However, this is not the end of the story.",
"For describing the hypersurface geometry, it is also important to express $(g_e, V_e, \\tilde{V}_e, N_e)$ in terms of the variables clearly describe the extrinsic geometry distinctly from the intrinsic geometry, so that a concrete notion of hypersurface can emerge after proper impositions of the constraints.",
"A method for this goal has been studied in the $SU(2)$ formulation, and in the following we will explicitly work out the generalization of such method for our $SO(D+1)$ case.",
"In both cases the key lies in extracting the an angle variables from the values $(g_e, V_e, \\tilde{V}_e, N_e)$ , that capture the extrinsic curvature of the hypersurface.",
"To extract the extrinsic data, we first identify the intrinsic geometric data completely.",
"Recall the emerging D-polytopes dual to the vertices, that for any two D-polytope next to each across an edge, the neighboring pair of (D-1)-faces associated to an edge always have the same area.",
"As observed in the $SU(2)$ case, when the neighboring pair of faces are subject to an additional “shape matching\" condition that they have the same shape in addition to the same area, each of these special values of $( N_e{V}_e, N_e\\tilde{V}_e)$ assigns one discretized intrinsic-geometry of a D-dimensional hypersurface, given by simply gluing the neighboring identical faces of the D-polytopes.",
"Such geometry is just the spatial Regge geometry, with the local geometry within each D-polytope flat and the curvature of the hypersurface captured in the parallel transports amongst the constituent D-polytopes.",
"The area-matching D-polytopes dual to a graph $\\gamma $ without the shape matching conditions define a more general notion of geometry, which is called twisted geometry in the existing literature.",
"Now let us look at the construction for the twisted geometry associated to $\\gamma $ in all-dimensional case.",
"Note that two neighboring D-polytopes prescribed by the flux variables must be rotated by a specific $SO(D+1)$ element, for their identified pair of faces to aligned in the normal directions.",
"In this manner, the flux data can specify one $SO(D+1)$ element to every edge $e$ as the necessary operation to align and glue the pair of faces dual to $e$ , and this element should rotate the inward (area-weighted) normal $-N_e\\tilde{V}_e$ of the (D-1)-face for the the target D-polytope, into the outward normal $N_e{V}_e$ of the corresponding (D-1)-face for the source D-polytope.",
"Under the generalized Regge geometry interpretation [16], these transformations across the edges carry the meaning of the Levi-Civita holonomies.",
"We thus define the $SO(D+1)$ valued Levi-Civita holonomy $h^{\\Gamma }_{e}$ for every edge $e$ as a function of the bivector variables associated to the neighborhood of $e$ .",
"Note that, by construction we have ${V}_e= -h^{\\Gamma }_{e}\\circ \\tilde{V}_e:=-h^{\\Gamma }_{e}\\tilde{V}_e(h^{\\Gamma }_{e})^{-1}$ .",
"We now adopt a decomposition of the holonomy as a $SO(D+1)$ element in the following way.",
"First, we choose once for all a fixed generator $\\tau _0 \\in so(D+1)$ as a reference bivector $\\tau ^{IJ}_0\\equiv (\\frac{\\partial }{\\partial x_1})^{[I}(\\frac{\\partial }{\\partial x_2})^{J]}$ .",
"Then for each edge $e\\subset \\gamma $ , we specify a special pair of differentiable $SO(D+1)$ -valued functions of the bi-vector variables called the Hopf sections, denoted as $u_e( {V}_{e})$ and $\\tilde{u}_e( \\tilde{V}_{e})$ ; the Hopf sections for each edge $e$ are defined by the conditions $V_e=u_e \\tau _0 u^{-1}_e\\,\\,,\\,\\, \\tilde{V}_e=-\\tilde{u}_e \\tau _0 \\tilde{u}^{-1}_e\\,\\,\\text{and }\\,\\, u_e( -{V}_{e})=u_e( {V}_{e})e^{2\\pi \\tau _{13}}, \\tilde{u}_e( -{\\tilde{V}}_{e})=\\tilde{u}_e( {\\tilde{V}}_{e})e^{2\\pi \\tau _{13}}$ with $\\tau _{13}:= (\\frac{\\partial }{\\partial x_1})^{[I}(\\frac{\\partial }{\\partial x_3})^{J]}$ and $e^{2\\pi \\tau _{13}}\\tau _{0}e^{-2\\pi \\tau _{13}}=-\\tau _{0}$ .",
"Observe that the choice for the Hopf sections is clearly non-unique, and from now on our parametrization will be given under one fixed choice of $\\lbrace u_e,\\tilde{u}_e\\rbrace $ for every edge $e$ , under which the Levi-Civita holonomy $h^{\\Gamma }_{e}$ can be expressed in the form $h^{\\Gamma }_{e}( {V}_{e^{\\prime }},\\tilde{V}_{e^{\\prime }})\\equiv u_e\\,\\, (e^{\\bar{\\eta }_{e}^{\\mu }\\, \\bar{\\tau }_\\mu }\\, e^{\\eta _{e}\\,\\tau _o})\\,\\,\\tilde{u}_e^{-1},$ where the $ e^{\\bar{\\eta }^\\mu \\bar{\\tau }_\\mu }$ takes value in the subgroup $SO(D-1)\\subset SO(D+1)$ preserving both $\\frac{\\partial }{\\partial x_1}$ and $\\frac{\\partial }{\\partial x_2}$ .",
"Note that the bivector functions $\\eta _{e}$ and $\\bar{\\eta }_{e}^{\\mu }$ are well-defined via the given $h^{\\Gamma }_{e}$ and the chosen Hopf sections.",
"Accordingly, the holonomy $g_e$ assigned to edge $e$ can also be decomposed as $&&g_e(V_{e^{\\prime }},\\tilde{V}_{e^{\\prime }},\\xi _e,\\bar{\\xi }_e^\\mu )\\equiv u_e\\,\\, (e^{\\bar{\\xi }_e^\\mu \\bar{\\tau }_\\mu }\\, e^{\\xi _e\\tau _o})\\,\\,\\tilde{u}_e^{-1}.$ Observe that while the $\\bar{\\eta }_{e}^{\\mu }$ and $\\eta _{e}$ are already fixed by the given $h^{\\Gamma }_{e}( {V}_{e^{\\prime }},\\tilde{V}_{e^{\\prime }})$ and the Hopf sections, the free variables $\\bar{\\xi }_e^\\mu $ and $\\xi _e$ , which we will call the angle variables, parametrize the additional degrees of freedom in $g_e$ .",
"Moreover, we can factor out $h^{\\Gamma }_{e}$ from $g_e$ through the expressions $g_e= h^{\\Gamma }_{e}\\,\\, \\left(e^{-\\bar{\\eta }_e^\\mu \\tilde{u}_{e}\\!\\bar{\\tau }_\\mu \\!\\tilde{u}^{-1}_{e}}\\,e^{\\bar{\\xi }_e^\\mu \\tilde{u}_{e}\\!\\bar{\\tau }_\\mu \\!\\tilde{u}^{-1}_{e}}\\, e^{-(\\xi _e- \\eta _e) \\tilde{V}_e}\\right) =\\left(e^{\\bar{\\xi }_e^\\mu {u}_{e}\\!\\bar{\\tau }_\\mu \\!",
"{u}^{-1}_{e}}\\,e^{ -\\bar{\\eta }_e^\\mu {u}_{e}\\!\\bar{\\tau }_\\mu \\!",
"{u}^{-1}_{e}}\\, e^{(\\xi _e- \\eta _e) {V}_e}\\right)\\,\\,h^{\\Gamma }_{e},$ where the ${u}_{e}\\bar{\\tau }_\\mu {u}^{-1}_{e}$ or $\\tilde{u}_{e}\\bar{\\tau }_\\mu \\tilde{u}^{-1}_{e}$ takes values from the subgroups $SO(D-1)\\subset SO(D+1)$ respectively preserving the bivector ${V}_e$ or $\\tilde{V}_e$ .",
"Having introduced the parametrization defined by (REF )-(REF ) for all points in the phase space, we now focus on the points parametrized by the special angle-bivector values with the flux data $( N_e{V}_e, N_e\\tilde{V}_e)$ describing a Regge intrinsic geometry, so that we can identify the extrinsic curvature data for these states through the angle variables in the following manner.",
"The above decomposition with the angle-bivector variables suggests a splitting of the the Ashtekar connection as $A_a=\\Gamma _a+\\beta K_a$ on a given graph.",
"For that, consider the integral of $A_a=\\Gamma _a+\\beta K_a\\in so(D+1)$ along an infinitesimal edge direction $\\ell ^a_e$ leading to $A_e\\equiv A_a\\ell ^a_e$ , $\\Gamma _e\\equiv \\Gamma _a\\ell ^a_e$ and $K_e\\equiv K_a\\ell ^a_e$ .",
"Clearly, we have the following obvious correspondence of $g_e= e^{A_e} \\,\\,\\,\\text{and}\\,\\,\\,h^{\\Gamma }_{e}= e^{\\Gamma _e}.$ The remaining factor should account for the $K_e$ .",
"Here we adopt the Regge interpretation that the descritized extrinsic curvature $K$ , just like the intrinsic curvature, is distributed only at the faces of the polytope-decomposition dual to $\\gamma $ .",
"According to the above discussion, the value of $K_e$ may thus be expressed in either the local gauge for the source polytope or that for the target polytope, respectively as $\\left(e^{\\bar{\\xi }_e^\\mu {u}_{e}\\!\\bar{\\tau }_\\mu \\!",
"{u}^{-1}_{e}}\\,e^{ -\\bar{\\eta }_e^\\mu {u}_{e}\\!\\bar{\\tau }_\\mu \\!",
"{u}^{-1}_{e}}\\, e^{(\\xi _e- \\eta _e) {V}_e}\\right)=e^{\\beta K_e} \\,\\,\\,\\text{or}\\,\\, \\,\\left(e^{-\\bar{\\eta }_e^\\mu \\tilde{u}_{e}\\!\\bar{\\tau }_\\mu \\!\\tilde{u}^{-1}_{e}}\\,e^{\\bar{\\xi }_e^\\mu \\tilde{u}_{e}\\!\\bar{\\tau }_\\mu \\!\\tilde{u}^{-1}_{e}}\\, e^{-(\\xi _e- \\eta _e) \\tilde{V}_e}\\right)= e^{\\beta K_e}\\,.\\nonumber \\\\$ A well-known feature of a Regge hypersurface is that the the extrinsic curvature distribution at a specific face of a constituent polytope must be a vector 1-form distribution parallel to the normal of the face.",
"This knowledge then suggests the further correspondence of $\\frac{1}{\\beta }(\\xi _e- \\eta _e) {V}_e= K^{\\perp }_e \\,\\,\\,\\text{or}\\,\\, \\,\\frac{1}{\\beta }(\\xi _e- \\eta _e) \\tilde{V}_e= -K^{\\perp }_e,$ when expressed in the source frame or target frame.",
"Finally, this leaves the remaining degrees of freedom to account for $K^{/\\!/}_e$ via $\\frac{1}{\\beta }\\ln (e^{\\bar{\\xi }_e^\\mu {u}_{e}\\!\\bar{\\tau }_\\mu \\!",
"{u}^{-1}_{e}}\\,e^{ -\\bar{\\eta }_e^\\mu {u}_{e}\\!\\bar{\\tau }_\\mu \\!",
"{u}^{-1}_{e}})= K^{/\\!/}_e \\,\\,\\,\\text{or}\\,\\, \\,\\frac{1}{\\beta }\\ln (e^{-\\bar{\\eta }_e^\\mu \\tilde{u}_{e}\\!\\bar{\\tau }_\\mu \\!\\tilde{u}^{-1}_{e}}\\,e^{\\bar{\\xi }_e^\\mu \\tilde{u}_{e}\\!\\bar{\\tau }_\\mu \\!\\tilde{u}^{-1}_{e}})= K^{/\\!/}_e.$ In general, the data in $(N_e, V_e, \\tilde{V}_e)$ contains information about both intrinsic and extrinsic geometry.",
"Out of these $4D-3$ degrees of freedom of $(N_e, V_e, \\tilde{V}_e)$ , only $2D-1$ of them would be interpretable as intrinsic-geometry property of the D-dimensional slice, while the other $2D-2$ of them carry information about the extrinsic geometry.",
"The extra angle $\\xi _e$ is the missing ingredient necessary in order to reconstruct the $(2D-1)_{\\textrm {th}}$ component of $K^{\\perp }_e$ .",
"As we will demonstrate in more details, the angles $\\bar{u}_e$ containing the information about the components $K^{/\\!/}_e$ of $K_e$ are purely redundant variables, in views of both the Regge hypersurface geometry and gauge reduction involving the discretized simplicity constraints.",
"The set of angle-bivector variables $(N_e, V_e, \\tilde{V}_e,\\xi _e,\\bar{\\xi }_e^\\mu )$ gives the generalization of twisted geometry parametrization for $SO(D+1)$ phase space.",
"We will now carry out an analysis of the canonical correspondence between these variables and the LQG phase space, before coming back to provide more support on the discrete hypersurface interpretation and drawing insights on the proper treatment of the gauge reduction with the anomalous discretized simplicity constraints."
],
[
"Sympletic structure of $SO(D+1)$ LQG phase space",
"Recall that the phase space of $SO(D+1)$ loop quantum gravity associated with each edge of a given graph can be given by the group tangent space $TSO(D+1)$ .",
"Since this space is bundle-isomorphic to $T^*SO(D+1)$ , as a phase space it enjoys the natural symplectic structure of the $SO(D+1)$ cotangent bundle.",
"Explicitly, the bundle isomorphism $TSO(D+1) \\rightarrow T^*SO(D+1) $ is given by the trivialization $( so(D+1), SO(D+1)) \\rightarrow TSO(D+1)$ using a basis of right-invariant $so(D+1)$ vector fields, followed by the identification $so(D+1) \\rightarrow so^*(D+1)$ that leads to the trivialization of the cotangent bundle $(so^*(D+1), SO(D+1)) \\rightarrow T^*SO(D+1)$ .",
"A right-invariant vector field $\\hat{X}$ associated to the Lie algebra element $X\\in \\mathfrak {g}$ , acts on a function on the group manifold via the right derivative $\\nabla _X^R$ as $\\nabla _X^Rf(g)\\equiv \\frac{d}{dt}f(e^{-tX}g)|_{t=0};$ under the adjoint transformation $X\\mapsto -gXg^{-1}$ , we obtain the corresponding left derivative $\\nabla _X^Lf(g)\\equiv \\frac{d}{dt}f(ge^{tX})|_{t=0}=-\\nabla ^R_{gXg^{-1}}f(g).$ It is straight forward to show that the map from the right invariant vector fields $\\hat{X}$ to the corresponding elements $X$ of the algebra is provided by the algebra-valued, right-invariant 1-form $dgg^{-1}$ satisfying $i_{\\hat{X}}(dgg^{-1})=(\\mathcal {L}_{\\hat{X}}g)g^{-1}=-X,$ where $i$ denotes the interior product, and $\\mathcal {L}_{\\hat{Y}}\\equiv i_{\\hat{Y}}d+di_{\\hat{Y}}$ denotes the Lie derivative.",
"It is clear from the above, that a basis for $\\mathfrak {g}$ is then associated to a set of right-invariant vector fields, which serves as a global tangent-space basis providing the trivialization as $(so(D+1), SO(D+1)) \\rightarrow TSO(D+1)$ .",
"Moreover, there is also a (local) coordinate system $G_{IJ}: SO(D+1) \\rightarrow so(D+1)$ for the base manifold $SO(D+1)$ , such that for any element $g$ , we have $dG_{IJ}|_g\\equiv (dgg^{-1})_{IJ}$ .",
"Using such a local coordinate system, the trivialization thus locally takes the form of $(X_{IJ}, G_{IJ}) \\rightarrow TSO(D+1)$ .",
"Next, to describe the cotangent fiber bundle, we note that for every $X\\in \\mathfrak {g}$ there is a corresponding element $h_X$ in the dual algebra $\\mathfrak {g}^\\ast $ , which as a linear function of $Y\\in \\mathfrak {g}$ is defined by $h_X(Y)\\equiv \\textrm {Tr}(XY ) \\equiv -2\\textrm {tr}(X_{IJ}\\tau ^{IJ}Y_{KL}\\tau ^{KL})=2X_{IJ}Y_{KL}\\delta ^{K[I}\\delta ^{J]L}=2X^{KL}Y_{KL},$ with the duality map given by the non-degenerate operator $\\frac{1}{2}\\textrm {Tr}(\\tau ^{IJ}\\tau ^{KL})=\\delta ^{K[I}\\delta ^{J]L}.$ Using this ad-invariant pairing we have identified $so(D+1)$ to $so(D+1)^\\ast $ and specified the trivial cotangent bundle structure $(so^*(D+1), SO(D+1)) \\rightarrow T^*SO(D+1)$ .",
"Thereby, the above (local) coordinate system describes the trivialization of the cotangent bundle in the explicit form $(X^{IJ}, G_{IJ}) \\rightarrow T^*SO(D+1)$ .",
"Recognizing that by construction $(X^{IJ}, G_{IJ})$ (locally) forms the dual coordinate pair of the cotangent bundle, we can now simply read off the natural symplectic potential for $TSO(D+1)$ as $\\Theta \\equiv X^{IJ}dG_{IJ} =\\frac{1}{2}\\textrm {Tr}(Xdgg^{-1}).$ The symplectic 2-form then follows as $\\Omega \\equiv -d\\Theta =- \\frac{1}{2}d\\textrm {Tr}(Xdgg^{-1})=\\frac{1}{4}\\textrm {Tr}(d\\tilde{X}\\wedge g^{-1}dg-dX\\wedge dgg^{-1})$ where we have introduced $\\tilde{X}\\equiv -g^{-1}Xg$ .",
"Among the interesting phase space functions in $TSO(D+1)$ (or equivalently in $T^*SO(D+1)$ ), we will specifically study the ones of the form $f\\equiv f(g)$ and $h_Y\\equiv h_Y(X)$ .",
"From the symplectic 2-form we can compute the following important Poisson brackets among them: $\\lbrace h_Y,h_Z\\rbrace =2h_{[Y,Z]},\\quad \\lbrace h_Y,f(g)\\rbrace =2\\nabla ^R_Yf(g),\\quad \\lbrace f(g),h(g)\\rbrace =0.$ Proof.",
"Let us identify $so(D+1)$ with $\\mathbb {R}^{\\frac{D(D+1)}{2}}$ via $X^i=\\textrm {Tr}(\\tau ^iX)=h_{\\tau ^i}(X)$ , where $i\\in \\lbrace 1,...,\\frac{D(D+1)}{2}\\rbrace $ and $\\tau ^i$ is an element of the orthogonal basis of $so(D+1)$ .",
"Consider the following vector field on $T^\\ast \\!SO(D+1)$ , $\\hat{Y}\\equiv \\nabla ^R_Y+[X,Y]^i\\frac{\\partial }{\\partial X^i}.$ This vector field is such that $i_{\\hat{Y}}\\Theta &=&- \\frac{1}{2}\\textrm {Tr}(XY),\\\\\\nonumber \\mathcal {L}_{\\hat{Y}}\\Theta &=& \\frac{1}{2}\\textrm {Tr}([X,Y]dgg^{-1})- \\frac{1}{2}\\textrm {Tr}(X[Y,dgg^{-1}])=0.$ Therefore we have $i_{\\hat{Y}}\\Omega =di_{\\hat{Y}}\\Theta -\\mathcal {L}_{\\hat{Y}}\\Theta =- \\frac{1}{2}d\\textrm {Tr}(XY),$ which implies that $\\hat{Y}$ is the Hamiltonian vector field of $ \\frac{1}{2}h_Y(X)$ and $\\lbrace \\frac{1}{2}h_Y, \\frac{1}{2}h_Z\\rbrace =\\Omega (\\hat{Y},\\hat{Z})=- \\frac{1}{2}i_{\\hat{Z}}dh_Y= \\frac{1}{2}h_{[Y,Z]}.$ Next, the Hamiltonian vector field of a function $f(g)$ on the group is $\\hat{f}=- 2\\nabla ^R_{X^i}f\\frac{\\partial }{\\partial X^i},$ since $i_{\\hat{f}}\\Omega = \\nabla ^R_{X^i}f\\textrm {Tr}(\\tau ^idgg^{-1})\\equiv -df.$ It is then easy to see that any two functions of the forms $f(g)$ and $h(g)$ would have a vanishing Poisson bracket as given by $\\Omega _{T^\\ast \\!G}(\\hat{X}_f,\\hat{X}_h)=0$ .",
"Finally, we have $\\lbrace \\frac{1}{2}h_Y,f\\rbrace =i_{\\hat{Y}}df=-i_{\\hat{X}_f}dh_Y=\\nabla ^R_Yf.$ $\\square $ We see from the brackets (REF ) that the Poisson action of $h_Y(X)$ generates left derivatives.",
"Similarly, the right derivative $\\lbrace \\tilde{h}_Y,f(g)\\rbrace =2\\nabla ^L_Yf(g)$ is generated by the action of $\\tilde{h}_Y(X)\\equiv \\textrm {Tr}(Y\\tilde{X})$ with $\\tilde{X}=-g^{-1}Xg$ .",
"Finally, the two Hamiltonians commute as given by $\\lbrace h_Y,\\tilde{h}_Z\\rbrace =0$ .",
"Using the obtained Poisson brackets, one may evaluate the algebra amongst the discretized Gauss constraints, edge-simplicity constraints and vertex-simplicity constraints defined in (REF ) and (REF ).",
"It turns out that $G_v\\approx 0$ and $S_e\\approx 0$ form a first class constraint system, with the algebra $\\lbrace S_e, S_e\\rbrace \\propto S_e\\,,\\,\\, \\lbrace S_e, S_v\\rbrace \\propto S_e,\\,\\,\\lbrace G_v, G_v\\rbrace \\propto G_v,\\,\\,\\lbrace G_v, S_e\\rbrace \\propto S_e,\\,\\,\\lbrace G_v, S_v\\rbrace \\propto S_v, \\quad b(e)=v,$ where the brackets within $G_v\\approx 0$ is just the $so(D+1)$ algebra, and the ones within $S_e\\approx 0$ weakly vanish.",
"The algebra involving the vertex-simplicity constraint are the problematic ones, with the open anomalous brackets $\\lbrace S_{v,e,e^{\\prime }},S_{v,e,e^{\\prime \\prime }}\\rbrace \\propto \\emph {anomaly term}$ where the terms $ \\emph {anomaly term}$ are not proportional to any of the existing constraints in the phase space."
],
[
" Symplectomorphism between edge-simplicity constraint surface and angle-bivector space",
"Having discussed the symplectic structure of the $T^*SO(D+1)$ phase space, we recall the angle-bivector parametrization for the edge-simplicity constraint surface this space using the twisted-geometry variables $(V,\\tilde{V},\\xi , N,\\bar{\\xi }^\\mu )\\in P:=Q_{D-1}\\times Q_{D-1}\\times T^*S\\times SO(D-1)$ , where $e^{\\bar{\\xi }^\\mu \\bar{\\tau }_\\mu }:=\\bar{u}$ , and $\\bar{\\tau }_\\mu \\in so(D-1)$ , $\\mu \\in \\lbrace 1,...,\\frac{(D-1)(D-2)}{2}\\rbrace $ .",
"To capture the intrinsic curvature, we have specified one pair of the $so(D+1)$ valued Hopf sections– $u(V)$ and $\\tilde{u}( \\tilde{V})$ – for each edge.",
"With the specified $u(V)$ and $\\tilde{u}( \\tilde{V})$ , the parametrization associated with each edge is given by the map $(V,\\tilde{V},\\xi ,N,\\bar{\\xi }^\\mu )\\mapsto (X,g):&& X=N\\,V=N\\,u(V)\\tau _ou(V)^{-1}\\\\\\nonumber &&g=u(V)\\,e^{\\bar{\\xi }^\\mu \\bar{\\tau }_\\mu }e^{\\xi \\tau _o}\\,\\tilde{u}(\\tilde{V})^{-1}$ which implies that $\\tilde{X}\\equiv -g^{-1}Xg=N\\tilde{V}$ .",
"We first note that the map is a two-to-one double covering of the image that takes the bi-vector form $X=Nu\\tau _ou^{-1}$ solving the edge-simplicity constraint $X^{[IJ}X^{KL]}=0$ .",
"Let us denote this bi-vector subset as $so(D+1)_s$ , and denote the image as $T_{s}^\\ast \\!SO(D+1)\\equiv T^\\ast \\!SO(D+1)|_{X^{[IJ}X^{KL]}=0}$ that is the edge-simplicity constraint surface in the phase space.",
"Clearly, under the map introduced above from $P$ to $T_{s}^\\ast \\!SO(D+1)$ , the two points $(V,\\tilde{V},\\xi , N,\\bar{\\xi }^\\mu )$ and $(-V,-\\tilde{V},-\\xi ,-N,\\dot{\\xi }^\\mu )$ related with $e^{\\dot{\\xi }^\\mu \\bar{\\tau }_{\\mu }}=e^{-2\\pi \\tau _{13}}e^{\\bar{\\xi }^\\mu \\bar{\\tau }_{\\mu }}e^{2\\pi \\tau _{13}}$ and $\\tau _{13}=\\delta _1^{[I}\\delta _3^{J]}$ are mapped to the same point $(X, g)\\in T_{s}^\\ast \\!SO(D+1) $ .",
"A bijection map can thus be established in the region $|X|\\ne 0$ by selecting either branch among the two signs, leading to the corresponding one of the two inverse maps from the region with $|X|\\ne 0$ given by: $N=|X|,\\quad V=\\frac{X}{|X|}, \\quad \\tilde{V}=-\\frac{g^{-1}Xg}{|X|},\\quad \\xi =\\textrm {Tr}(\\tau _o\\ln (u^{-1}g\\tilde{u})),\\quad \\bar{\\xi }^\\mu =\\textrm {Tr}(\\bar{\\tau }^\\mu \\ln (e^{-\\xi \\tau _o}u^{-1}g\\tilde{u}))$ or $N=-|X|,\\quad V=-\\frac{X}{|X|}, \\quad \\tilde{V}=\\frac{g^{-1}Xg}{|X|},\\quad \\xi =-\\textrm {Tr}(\\tau _o\\ln (u^{-1}g\\tilde{u})),\\quad \\dot{\\xi }^\\mu =\\textrm {Tr}(\\bar{\\tau }^\\mu \\ln (e^{\\xi \\tau _o}e^{-2\\pi \\tau _{13}}u^{-1}g\\tilde{u}e^{2\\pi \\tau _{13}})).$ Thus, we have an isomorphism between the two sets $P^\\ast /\\mathbb {Z}^2\\rightarrow T_{s}^\\ast \\!SO(D+1)\\!\\setminus \\!\\lbrace |X|=0\\rbrace ,$ where $P^\\ast \\equiv P|_{N\\ne 0}$ denotes the region with ${N\\ne 0}$ , and the identifying $\\mathbb {Z}^2$ operation is defined by $(V,\\tilde{V},\\xi , N,\\bar{\\xi }^\\mu ) \\rightarrow (-V,-\\tilde{V},-\\xi ,-N,\\dot{\\xi }^\\mu )$ in the region $N\\ne 0$ .",
"Since $P^\\ast \\equiv P|_{N\\ne 0}$ provides a double-covering coordinate system for $ T_{s}^\\ast \\!SO(D+1)\\!\\setminus \\!\\lbrace |X|=0\\rbrace $ , we may use the bivector-angle variables to express the induced presymplectic structure of $T_{s}^\\ast \\!SO(D+1)\\!\\setminus \\!\\lbrace |X|=0\\rbrace $ inherited from the phase space $T^*\\!SO(D+1)$ .",
"First, the induced presymplectic potential can be expressed as $\\Theta _{T_s^\\ast \\!SO(D+1)} |_{|X|>0}&=& \\frac{1}{2}\\textrm {Tr}(Xdgg^{-1})|_{T_{s}^\\ast \\!SO(D+1); |X|>0}\\\\\\nonumber &=& \\frac{1}{2}N\\textrm {Tr}(u\\tau _{o}u^{-1} (duu^{-1}+u(d\\xi \\tau _o+d\\bar{\\xi }^\\mu \\bar{\\tau }_\\mu )u^{-1}-ue^{\\bar{\\xi }^\\mu \\bar{\\tau }_\\mu }e^{\\xi \\tau _o} \\tilde{u}^{-1}d\\tilde{u}\\tilde{u}^{-1}\\tilde{u}e^{-\\bar{\\xi }^\\mu \\bar{\\tau }_\\mu }e^{-\\xi \\tau _o} u^{-1})) \\\\\\nonumber &=& \\frac{1}{2}N\\textrm {Tr}(Vduu^{-1})+ \\frac{1}{2}Nd\\xi - \\frac{1}{2}N\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1}).$ From the point of view of the space $P$ , we may extend this potential in the limit $N\\rightarrow 0$ and simply define $\\Theta _{P}\\equiv \\frac{1}{2}N\\textrm {Tr}(Vduu^{-1})+ \\frac{1}{2}Nd\\xi - \\frac{1}{2}N\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1})$ as the presymplectic potential in $P$ .",
"This potential gives the presympletic form $\\Omega _P$ as $\\Omega _P=-d\\Theta _P &=& \\frac{1}{2}N\\textrm {Tr}(Vduu^{-1}\\wedge duu^{-1})-\\frac{1}{2}N\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1}\\wedge d\\tilde{u}\\tilde{u}^{-1}) \\\\\\nonumber && -\\frac{1}{2}dN\\wedge (d\\xi +\\textrm {Tr}(Vduu^{-1})-\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1})).$ It is clear that the $N=0$ region of the above presymplectic structure is degenerate, as expected due to the degeneracy in the parametrization itself in the ${N= 0}$ region of $T_{s}^\\ast \\!SO(D+1)$ .",
"More importantly, as we shall demonstrate in the next section, the induced presymplectic structure for $P^\\ast $ coincides with the natural symplectic structures of the two constituent spaces — the $Q_{D-1}$ and $T^*S^1$ , while leaving the third component $SO(D-1)$ completely degenerate.",
"Therefore, this $SO(D-1)$ component faithfully parametrizes the symplectic degeneracies of $T_{s}^\\ast \\!SO(D+1)\\!\\setminus \\!\\lbrace |X|=0\\rbrace $ as a presymplectic manifold.",
"Since the edge-simplicity constraints form a first class system with the discretized Gauss constraints, we expect the $SO(D-1)$ degenerate degrees of freedom to be generated by the first-class constraints.",
"Indeed, we note that the induced symplectic form on $T_s^\\ast \\!SO(D+1)\\!\\setminus \\!\\lbrace |X|=0\\rbrace $ given by $\\Omega _{T^\\ast \\!SO(D+1)}\\equiv -d\\Theta _{T^\\ast \\!SO(D+1)}$ is different from $\\Omega _{P^\\ast }:=-d\\Theta _{P^\\ast }$ obtained from the induced symplectic potential $\\Theta _{P^\\ast }$ , since the Hamiltonian vector fields of any function on $P^\\ast $ given by the two symplectic forms always differ by a transformation induced by the edge-simplicity constraints.",
"More explicitly, we can evaluate the transformations induced by the edge-simplicity constraints in the LQG discrete phase space, and obtain $\\lbrace S_e^{IJKL}, X_e\\rbrace |_{S_e=0}=0\\,\\,\\text{and}\\,\\,\\,\\lbrace S_e^{IJKL}, g_e\\rbrace |_{S_e=0}\\propto X_e^{[IJ}(\\tau ^{KL]}g_e)|_{S_e=0}\\propto V_e^{[IJ}(\\tau ^{KL]}g_e)|_{V_e=u_e \\tau _0 u_e^{-1}}.$ Now it is easy to see that the edge-simplicity constraint transforms the holonomy $g_e$ by the left action of an $SO(D-1)$ element preserving the two vectors forming $V_e$ .",
"Then, via the parametrization (REF ) of $g_e$ , we conclude that the edge-simplicity constraints generate the transformation of the $SO(D-1)$ angles, which are precisely the degenerate component with respected to the presymplectic form $\\Omega _{P^\\ast }$ .",
"Lastly, let us view the above transformations induced by $S_e^{IJKL}$ under the discrete Regge geometry interpretation proposed in section 3.2.",
"Since the edge-simplicity constraints commute (on-shell) with the flux variables, it is clear that the transformations act trivially on the intrinsic geometry as desired.",
"Moreover, the above shows that the transformations change only the $SO(D-1)$ angles $\\bar{\\xi }_e^\\mu $ among the twisted-geometry variables, then according to our interpretation (REF ) the transformations act only upon $K^{/\\!/}_e$ , which are indeed the pure gauge components in the original Ashtekar formulation.",
"To go further and study the gauge reductions in the new geometric point of view, we need to compute the Poisson brackets between the twisted-geometry variables using the presymplectic form $\\Omega _P$ .",
"In order to do that, in the following section we will study the Hopf sections $u(V)$ and $\\tilde{u}(\\tilde{V})$ in the perspectives of their contributions to the Hamiltonian fields on $P$ defined by $\\Omega _P$ ."
],
[
"Hopf map and Geometric action on the Hopf section",
"The Hopf map is defined as a special projection map $\\pi : SO(D+1)\\mapsto Q_{D-1}$ with $Q_{D-1}:=SO(D+1)/(SO(2)\\times SO(D-1))$ , such that every element in $Q_{D-1}$ comes from the maximal subgroup of $SO(D+1)$ that fixed $\\tau _{o}$ .",
"The maximal subgroup takes the form $SO(2)\\times SO(D-1)$ , and in the definition representation of $SO(D+1)$ the Hopf map reads $\\pi : \\quad SO(D+1) &\\rightarrow & Q_{D-1} \\\\\\nonumber g &\\rightarrow & V(g)=g\\tau _og^{-1}.$ Note that the vector $V(g)$ is invariant under $g\\mapsto g^{\\alpha ,\\beta ^\\mu }=ge^{\\alpha \\tau _o+\\beta ^\\mu \\bar{\\tau }_\\mu }$ , thus it is a function of $2D-2$ variables only.",
"This result shows that $SO(D+1)$ can be seen as a bundle (we would call it the Hopf bundle) over $Q_{D-1}$ with a $SO(2)\\times SO(D-1)$ fiber.",
"On this bundle we can introduce the Hopf sections, each as an inverse map to the above projection $u:\\quad Q_{D-1} &\\rightarrow & SO(D+1)\\\\\\nonumber V&\\mapsto & u(V),$ such that $\\pi (u(V))=V$ .",
"This section assigns a specific $SO(D+1)$ element $u$ to each member of the $Q_{D-1}$ , and it is easy to see that any given section $u$ is related to all other sections via $u^{\\alpha ,\\alpha ^{\\prime \\mu }}\\equiv ue^{\\alpha \\tau _o+\\alpha ^{\\prime \\mu }\\bar{\\tau }_\\mu }$ ; therefore the free angles $\\lbrace \\alpha ,\\alpha ^{\\prime \\mu }\\rbrace $ parametrize the set of all possible Hopf sections.",
"Let us identify $so(D+1)$ with $\\mathbb {R}^{\\frac{D(D+1)}{2}}$ via the representation $X=X^{IJ}$ .",
"Then, an element $V\\in Q_{D-1}$ is identified with a unit bi-vector in $\\mathbb {R}^{\\frac{D(D+1)}{2}}$ , and we have a natural action of rotations by the group $SO(D+1)$ in this space.",
"Since this action is given via the co-adjoint representation, we can further associate each algebra element $X\\in so(D+1)$ to a vector field $\\hat{X}$ on $Q_{D-1}$ , which acts on a function of $Q_{D-1}$ as $\\mathcal {L}_{\\hat{X}}f(V):=\\frac{d}{dt}f(e^{-tX}Ve^{tX})|_{t=0}.$ Specifically in the case of linear functions we have $\\mathcal {L}_{\\hat{X}}V=-[X,V].$ Next, we observe that the $SO(D+1)$ action on $Q_{D-1}$ as a symplectic manifold is Hamiltonian; by explicit calculation one can verify that $\\hat{X}$ is a Hamiltonian vector field associated to the function $ \\frac{1}{2}h_X(V)\\equiv NV^{IJ}X_{IJ}$ on $Q_{D-1}$ , and the action above can be obtained from the Poisson bracket between $V$ and $ \\frac{1}{2}h_X$ , which results to $\\lbrace \\frac{1}{2}h_X,V\\rbrace =N\\Omega _{Q}(\\hat{X},\\hat{V})=-[X,V]=\\mathcal {L}_{\\hat{X}}V.$ We are especially interested in the action of the algebra on the Hopf section.",
"Let us first note that $\\mathcal {L}_{\\hat{X}}V(u)=(\\mathcal {L}_{\\hat{X}}u)\\tau _ou^{-1} +u\\tau _o(\\mathcal {L}_{\\hat{X}}u^{-1})=[(\\mathcal {L}_{\\hat{X}}u)u^{-1}, V(u)].$ Comparing this with (REF ), we deduce that $(\\mathcal {L}_{\\hat{X}}u)u^{-1}=-X+V(u)F_X(V)+\\sum _{\\mu }\\bar{V}_\\mu (u)L^\\mu _X(V),$ where $\\bar{V}_\\mu (u)\\equiv u\\bar{\\tau }_\\mu u^{-1}$ , $F_X(V)$ and $L^\\mu _X(V)$ are functions of $V\\in Q_{D-1}$ , with both $V(u)F_X(V)$ and $\\bar{V}_\\mu (u)L^\\mu _X(V)$ commuting with the element $V(u)$ for all $\\mu $ .",
"Lemma.",
"Define $\\bar{V}^\\mu _{IJ}=u(V)\\bar{\\tau }_{IJ}^\\mu u(V)^{-1}$ , the solution function $L^{IJ}\\equiv L: Q_{D-1}\\mapsto so(D+1)$ of the equations $\\textrm {Tr}(Lduu^{-1})=0, \\quad L^{IJ}V_{IJ}=1,\\quad L^{IJ}\\bar{V}^\\mu _{IJ}=0,\\ \\forall \\mu ,$ appears in the Lie derivative of the Hopf map section $u(V)$ as, $L_X=2F_X$ and it satisfies the key coherence identity $\\mathcal {L}_{\\hat{X}}L_Y-\\mathcal {L}_{\\hat{Y}}L_X=L_{[X,Y]}.$ Finally, the general solution to this identity satisfying the conditions $L^{IJ}V_{IJ}= 1, L^{IJ}\\bar{V}^\\mu _{IJ}=0$ is given by $L^{\\prime }=L+d\\alpha $ where $\\alpha $ is a function on $Q_{D-1}$ .",
"Proof.",
"Takeing the interior product of an arbitrary vector field $\\hat{X}$ with the defining expression $\\textrm {Tr}(Lduu^{-1})=0$ and recalling that by definition of Lie derivative $(\\mathcal {L}_{\\hat{X}}u)u^{-1}=i_{\\hat{X}}(duu^{-1})$ , we have $0=i_{\\hat{X}}\\textrm {Tr}(Lduu^{-1})=\\textrm {Tr}(L(\\mathcal {L}_{\\hat{X}}u)u^{-1}) =-\\textrm {Tr}(LX)+F_X\\textrm {Tr}(LV)=-L_X+2F_X,$ where we used $L^{IJ}V_{IJ}=1, L^{IJ}\\bar{V}^\\mu _{IJ}=0,$ and (REF ).",
"Hence we proved $2F_X=L_X$ .",
"To prove (REF ) we first observe that $\\mathcal {L}_{\\hat{X}}(duu^{-1})&=&i_{\\hat{X}}(duu^{-1}\\wedge duu^{-1})+d[(\\mathcal {L}_{\\hat{X}}u)u^{-1}]\\\\\\nonumber &=&[-X+\\frac{1}{2}VL_X+\\sum _{\\mu }\\bar{V}_\\mu L^\\mu _X,duu^{-1}]+d(-X+\\frac{1}{2}VL_X+\\sum _{\\mu }\\bar{V}_\\mu L^\\mu _X)\\\\\\nonumber &=&\\frac{1}{2}VdL_X+\\bar{V}_\\mu dL^\\mu _X-[X,duu^{-1}],$ where we used the definition of Lie derivative in the first equality, (REF ) in the second and $dV=[duu^{-1},V]$ with $d\\bar{V}^\\mu =[duu^{-1},\\bar{V}^\\mu ]$ in the third.",
"The above then leads to $0=\\mathcal {L}_{\\hat{X}}\\textrm {Tr}(Lduu^{-1})=\\textrm {Tr}((\\mathcal {L}_{\\hat{X}}L-[L,X])duu^{-1}) +dL_X$ with the help of the equalities $L^{IJ}V_{IJ}=1$ and $L^{IJ}\\bar{V}^\\mu _{IJ}=0$ .",
"Finally, by taking the interior product of the last equation with $\\hat{Y}$ we get $\\mathcal {L}_{\\hat{Y}}L_X&=& \\textrm {Tr}((\\mathcal {L}_{\\hat{X}}L-[L,X] )(Y-\\frac{1}{2}VL_Y-\\sum _{\\mu }\\bar{V}_\\mu L^\\mu _Y))\\\\\\nonumber &=&\\mathcal {L}_{\\hat{X}}L_Y-L_{[X,Y]}-\\frac{1}{2}L_Y(\\textrm {Tr}((\\mathcal {L}_{\\hat{X}}L)V) -\\textrm {Tr}(L[X,V]))-\\sum _{\\mu } L^\\mu _Y(\\textrm {Tr}(\\mathcal {L}_{\\hat{X}}L\\bar{V}_\\mu ) -\\textrm {Tr}(L[X,\\bar{V}_\\mu ]))\\\\\\nonumber &=&\\mathcal {L}_{\\hat{X}}L_Y-L_{[X,Y]}-\\frac{1}{2}L_Y\\mathcal {L}_{\\hat{X}}(\\textrm {Tr}(LV) )-\\sum _{\\mu } L^\\mu _Y\\mathcal {L}_{\\hat{X}}\\textrm {Tr}(L\\bar{V}_\\mu )$ and since the last two terms vanish, we obtain the coherence identity (REF ).",
"Suppose we have another solution $L^{\\prime }$ to the coherence identity and also the conditions $L^{IJ}V_{IJ}=1$ and $L^{IJ}\\bar{V}^\\mu _{IJ}=0$ .",
"Using the 1-form $\\beta \\equiv -\\textrm {Tr}(L^{\\prime }duu^{-1})$ we see can that its contraction with $\\hat{X}$ $\\beta _X\\equiv i_{\\hat{X}}\\beta =-\\textrm {Tr}(L^{\\prime }(\\mathcal {L}_{\\hat{X}}u)u^{-1})=L^{\\prime }_X-L_X$ is the difference between the two solutions and thus also a solution to the coherence identity.",
"This, together with the definition of the differential $i_{\\hat{X}}i_{\\hat{Y}}d\\beta =\\mathcal {L}_{\\hat{Y}}\\beta _X -\\mathcal {L}_{\\hat{X}}\\beta _Y+\\beta _{[X,Y]}$ , implies that $d\\beta =0$ , which means that there exist a function $\\alpha $ locally such that $\\beta =d\\alpha $ at least, and thus $L^{\\prime }_X=L_X+\\mathcal {L}_{\\hat{X}}\\alpha $ .",
"This proves the gauge freedom (REF ).",
"$\\square $ Finally, let us recall that the freedom in choosing the Hopf section lies in the two function parameters $\\alpha (V)$ and $\\alpha ^{\\prime \\mu }(V)$ in the expression $u^{\\prime }(V)\\equiv u(V)e^{\\alpha (V)\\tau _o+\\alpha ^{\\prime \\mu }(V)\\tau _\\mu }$ for all possible choices of the sections.",
"Applying the equation (REF ) to this $u^{\\prime }$ , we immediately get $L^{\\prime }_X= L_X+ i_{\\hat{X}}d\\alpha $ .",
"Referring to (REF ), we see now that the set of functions $L$ satisfying the above three key conditions is exactly the set of the function coefficients for the component of $(du)u^{-1}$ in the $V$ direction, given under all possible choices of the Hopf section $u$ .",
"Applying these conditions in the presympletic form $\\Omega _P$ , we will now identify the Hamiltonian fields in $P$ and compute the Poisson brackets."
],
[
"Computation of Hamiltonian vector fields in pre-symplectic manifold $P$",
"Recall that we have obtained the pre-symplectic potential $\\Theta _{P}:= \\frac{1}{2}N\\textrm {Tr}(Vduu^{-1})+ \\frac{1}{2}Nd\\xi - \\frac{1}{2}N\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1})$ induced from the edge-simplicity constraint surface in the $SO(D+1)$ phase space.",
"The potential defines a presympletic form $\\Omega _P$ as $\\Omega _P=-d\\Theta _P &=& \\frac{1}{2}N\\textrm {Tr}(Vduu^{-1}\\wedge duu^{-1})-\\frac{1}{2}N\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1}\\wedge d\\tilde{u}\\tilde{u}^{-1}) \\\\\\nonumber && -\\frac{1}{2}dN\\wedge (d\\xi +\\textrm {Tr}(Vduu^{-1})-\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1})).$ To compute the associated Poisson brackets, we first need to compute the Hamiltonian vector fields on $P$ .",
"Let us denote the Hamiltonian vector field for the function $f$ as $\\chi _f$ , where $f\\in \\lbrace N, \\xi , h_X\\equiv NV_X, \\tilde{h}_X\\equiv N\\tilde{V}_X\\rbrace $ .",
"Using the definition and $i_{\\chi _f}\\Omega _P=-df$ , in the $N\\ne 0$ region the vector fields could be checked to be given by $\\chi _{h_X} &=& 2\\hat{X}-L_X(V)\\partial _\\xi ,\\quad \\chi _{\\tilde{h}_X} = - 2 \\hat{\\tilde{X}}-L_X(\\tilde{V})\\partial _\\xi , \\\\\\nonumber \\chi _N&=& -2\\partial _\\xi ,\\quad \\quad \\quad \\quad \\quad \\chi _\\xi =2\\partial _N+\\frac{4}{N}\\widehat{[L,V]}+\\frac{4}{N}\\widehat{[L,\\tilde{V}]}.$ Here $\\hat{X}$ and $\\widehat{[L,V]}$ are the vector fields generating the adjoint action on $Q_{D-1}$ labelled by $V$ , associated respectively to the algebra elements $X$ and $[L(V),V]$ .",
"Similarly, $\\hat{\\tilde{X}}$ and $\\widehat{[L,\\tilde{V}]}$ are the vector fields generating the adjoint action on $Q_{D-1}$ labelled by $\\tilde{V}$ , associated respectively to the algebra elements $X$ and $[L(\\tilde{V}),\\tilde{V}]$ .",
"Proof.",
"To check the first equation of (REF ), we first note that for a constant $X$ we have $i_{\\hat{X}}\\Omega _P=-\\frac{1}{2}\\textrm {Tr}(d(NV)X)+\\frac{1}{4}L_X(V)dN.$ Since we have $i_{\\partial _\\xi }\\Omega _P=\\frac{1}{2}dN$ , the first equation of (REF ) follows immediately.",
"The computation for $\\chi _{\\tilde{h}_X}$ is similar with an opposite sign due to the reversal of the orientation.",
"To check for $\\chi _\\xi $ , we first evaluate $i_{\\partial _N}\\Omega _P=-\\frac{1}{2}d\\xi -\\frac{1}{2}\\textrm {Tr}(Vduu^{-1})+\\frac{1}{2}\\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1}),$ and then we have $i_{\\widehat{[L,V]}}\\Omega &=& -\\frac{1}{2}N \\textrm {Tr}([V,[L,V]]duu^{-1}) -\\frac{1}{2}dN\\textrm {Tr}((V-L)[L,V])\\\\\\nonumber &=& - \\frac{1}{2}N \\textrm {Tr}(\\frac{1}{4}(L-\\textrm {Tr}(LV)V)duu^{-1}) -\\frac{1}{2}dN\\textrm {Tr}((V-L)[L,V])= \\frac{1}{4}N \\textrm {Tr}(Vduu^{-1}),$ where we decomposed $L$ as $L=(L-\\textrm {Tr}(LV)V)+\\textrm {Tr}(LV)V$ and used the definitional properties of $L$ .",
"A similar calculation shows that $i_{\\widehat{[L,\\tilde{V}]}}\\Omega &=& -\\frac{1}{4}N \\textrm {Tr}(\\tilde{V}d\\tilde{u}\\tilde{u}^{-1}),$ and thus the last equation of (REF ) follows.",
"$\\square $ Let us now address the degeneracy of $\\Omega _P$ resulting to the non-uniqueness of the Hamiltonian vector fields.",
"While $\\Omega _P$ is trivally closed as coming from a local symplectic potential, it has degeneracies in the directions tangent to the $SO(D-1)$ fiber and also in the boundary region with $N=0$ .",
"There are mainly two ways to reduce the manifold $P$ to obtain a sympletic manifold.",
"The first way is to simply consider a new space $P^\\ast :=P|_{N\\ne 0}$ and then reduce it respected to the $SO(D-1)$ fiber, then the result would be a $(4D-2)$ -dimensional sympletic manifold denoted by $\\check{P}^\\ast $ .",
"The second way is to reduce the pre-symplectic manifold by the kernel of $\\Omega _P$ , i.e.",
"to consider the quotient manifold $\\bar{\\check{P}}\\equiv P/\\textrm {Ker}(\\Omega _P)$ ; the result would be a symplectic manifold with non-degenerate 2-form given by the quotient projection of $\\Omega _P$ .",
"In obtaining the space $\\bar{\\check{P}}$ , we have introduced the equivalence classes under the equivalence relation $p\\sim p^{\\prime }$ whenever $p^{\\prime }=e^{\\hat{D}}p$ , with $\\hat{D}\\in \\textrm {Ker}(\\Omega _P)$ and $p, p^{\\prime }\\in P$ .",
"The operation is thus determined by the vector fields in the kernel of $\\Omega _P$ .",
"Since it is obvious that all tangent vector fields $\\hat{T}_{SO(D-1)}$ of the fiber $SO(D-1)$ belong to $\\textrm {Ker}(\\Omega _P)$ , we may first construct $\\check{P}=P/\\hat{T}_{SO(D-1)}=Q_{D-1}\\times T^\\ast \\!S^1\\times Q_{D-1}$ .",
"Then, to remove the remaining kernel in the region with $N=0$ , we look for the vector fields preserving the region while having the interior products with $\\Omega _P$ proportional to $N$ .",
"The set of such vector fields turn out to be given by $\\hat{D}_X\\equiv \\chi _{h_X}-\\chi _{\\tilde{h}_Y},$ where $Y=-g^{-1}Xg$ with $g=ue^{\\xi \\tau _0}e^{\\bar{\\xi }^\\mu \\bar{\\tau }_\\mu }\\tilde{u}^{-1}$ being a group element rotating $V$ to $\\tilde{V}=-g^{-1}Vg$ .",
"Indeed, using the fact that $V_X=\\tilde{V}_Y$ , the interior product of the field with the symplectic 2-form is $i_{\\hat{D}_X}\\Omega _P=-d(NV_X-N\\tilde{V}_Y)-N\\textrm {Tr}(\\tilde{V}dY) =-N\\textrm {Tr}([V,X]dgg^{-1}),$ which vanishes at $N=0$ .",
"Next, to find the equivalence class generated by the vector fields $\\hat{D}_X $ , we note that the actions of the fields should rotate jointly the vectors $V$ and $\\tilde{V}$ , that is we have $\\hat{D}_X(V)=-[X,V]$ , $\\hat{D}_X(\\tilde{V})=-g^{-1}[X,V]g$ .",
"Further, the actions preserves the group element $g$ , as demonstrated by the fact that $\\hat{D}_X(g)=-Xg-gY=0.$ Therefore, given $p\\equiv (V,\\tilde{V},0,\\xi )$ and $p^{\\prime }\\equiv (V^{\\prime },\\tilde{V}^{\\prime },0,\\xi ^{\\prime })$ , we have $ p^{\\prime }\\sim p$ if and only if the two are related by a joint rotation in $V$ and $\\tilde{V}$ and a $g$ -preserving translations in $\\xi $ .",
"The two copies of $Q_{D-1}$ at the ends of each edge are thus identified under this equivalence relation, and after the quotient we are left with a manifold $SO(D+1)/SO(D-1)$ parametrized by only $V$ and $\\xi $ .",
"Let us observe that the two quotient operations with respected to $\\hat{T}_{SO(D-1)}$ and $\\hat{D}_X$ commute, since $\\hat{D}_X$ doesn't change $e^{\\xi ^\\mu \\bar{\\tau }_\\mu }\\in SO(D-1)$ which is the degrees of freedom reduced by $\\hat{T}_{SO(D-1)}$ .",
"This fact can be illustrated as ${}@C=2.5cm{P[d]^{\\hat{T}_{SO(D-1)}} [r]^{\\hat{D}_X} & \\bar{P}[d]^{\\hat{T}_{SO(D-1)}}\\\\\\check{P} [r]^{\\hat{D}_X}&\\bar{\\check{P}} }$ where $\\bar{P}$ span $P^\\ast $ for $N\\ne 0$ and $SO(D+1)$ for $N=0$ ; Similarly, $\\bar{\\check{P}}$ span $\\check{P}^\\ast $ for $N\\ne 0$ and $SO(D+1)/SO(D-1)$ for $N=0$ .",
"Finally, let us point out that the symplectic potential is invariant under the $\\mathbb {Z}_2$ transformation $(V,\\tilde{V},N,\\xi ,\\bar{\\xi }^{\\mu })\\rightarrow (-V,-\\tilde{V},-N,-\\xi ,\\dot{\\xi }^\\mu ).$ This can be seen via the transformations of the Hopf sections in the form of $u\\rightarrow ue^{2\\pi \\tau _{13}}$ and $\\tilde{u}\\rightarrow \\tilde{u}e^{2\\pi \\tau _{13}}$ , with $\\tau _{13}=\\delta _1^{[I}\\delta _3^{J]}$ .",
"Clearly these transformations leave $\\Theta _P$ invariant since $d(ue^{2\\pi \\tau _{13}})(ue^{2\\pi \\tau _{13}})^{-1}=duu^{-1}$ .",
"Hence (REF ) is a canonical transformation, and both $\\check{P}^\\ast /\\mathbb {Z}_2$ and $\\bar{\\check{P}}/\\mathbb {Z}_2$ are again symplectic manifolds."
],
[
"Consistency with natural Poisson structures of constituent spaces",
"We have seen that the manifold $P=Q_{D-1}\\times Q_{D-1}\\times T^\\ast \\!S^1\\times SO(D-1)$ , viewed essentially as the edge-simplicity constraint surface of the LQG phase space, is equipped with the induced pre-symplectic potential $\\Theta _{P}$ .",
"On the other hand, the space is also a product space of the components $Q_{D-1}$ and $T^\\ast \\!S^1$ each having a natural phase space structure.",
"Therefore, the product space $P$ is also endowed with a class of natural Poisson structures given by the consistent gluing of the constituent spaces' symplectic structures.",
"As it turns out, the Poisson structure given by $\\Theta _{P}$ indeed belongs to such a class.",
"The natural phase space structure of the constituent spaces $Q_{D-1}$ and $T^\\ast \\!S^1$ are well-known, and they are given by: The cotangent bundle $T^\\ast S^1$ with the symplectic 2-form $\\Omega _{T^\\ast S^1}:=\\frac{1}{2}dN\\wedge d\\xi $ , giving the Poisson bracket $\\lbrace \\xi ,N\\rbrace =2$ .",
"The manifold $Q_{D-1}$ with the natural invariant Kahler metric and the corresponding Kahler form $\\Omega _Q$ , which is induce from the the standard Hermitian metric on $\\mathbb {C}^{D+1}$ and re-scaled into the form $\\pm \\Omega _{N^2/2}:=\\pm N\\Omega _Q$ .",
"The sympletic form $\\pm N\\Omega _Q$ gives the Poisson brackets $\\lbrace NV^{IJ}, NV^{KL}\\rbrace =\\pm \\frac{N}{2}(\\delta ^{IL}V^{JK}+\\delta ^{ JK}V^{IL }-\\delta ^{IK}V^{JL}-\\delta ^{JL}V^{IK})$ , where it becomes clear that $N=\\sqrt{2NV^{IJ}NV_{IJ}}$ is a Casimir quantity satisfying $\\lbrace N,NV_{IJ}\\rbrace =0$ .",
"Using the pre-symplectic potential $\\Theta _{P}$ , one could compute the Poisson brackets and obtain $&&\\lbrace \\xi ,N\\rbrace =2,\\nonumber \\\\&&\\lbrace NV^{IJ}, NV^{KL}\\rbrace =\\frac{N}{2}(\\delta ^{IL}V^{JK}+\\delta ^{ JK}V^{IL }-\\delta ^{IK}V^{JL}-\\delta ^{JL}V^{IK}),\\nonumber \\\\&&\\lbrace N\\tilde{V}^{IJ}, N\\tilde{V}^{KL}\\rbrace =-\\frac{N}{2}(\\delta ^{IL}\\tilde{V}^{JK}+\\delta ^{ JK}\\tilde{V}^{IL }-\\delta ^{IK}\\tilde{V}^{JL}-\\delta ^{JL}\\tilde{V}^{IK}),\\nonumber \\\\&&\\lbrace V^{IJ},N\\rbrace = \\lbrace \\tilde{V}^{IJ},N\\rbrace =0,$ $\\lbrace V^{IJ},\\tilde{V}^{KL}\\rbrace =0,$ and $\\lbrace \\xi ^\\mu ,\\ \\cdot \\ \\rbrace =0, \\quad \\forall \\ \\cdot \\,.$ From the above, the $\\Theta _{P}$ indeed endows the source and target $Q_{D-1}$ spaces respectively with the symplectic forms $N\\Omega _Q$ and $-N\\Omega _Q$ .",
"Also, from Eq.",
"(REF ) the two spaces truly Poisson commute.",
"As for the space $T^*S^1$ , the induced symplectic form is also identical with $\\Omega _{T^\\ast S^1}$ .",
"Lastly, the vanishing brackets in Eq.",
"(REF ) indicate the degeneracy in $\\Theta _{P}$ in the $SO(D-1)$ directions.",
"Separately in the sympletic manifolds $T^\\ast S^1$ and $Q_{D-1}$ , the Hamiltonian vector fields of the functions $\\lbrace \\,h_X, \\,\\tilde{h}_X,\\, N,\\, \\xi \\,\\rbrace $ generating the above brackets can be obtained respectively according to $\\Omega _{T^\\ast S^1}$ and $\\Omega _Q$ .",
"In comparison, the Hamiltonian vector fields in $P$ of the same functions according to $\\Omega _P$ clearly differ by the terms depending on the $L$ as given in (REF ).",
"As expected, these difference terms are generated by $\\Omega _P$ via its mixing components between $T^\\ast S^1$ and $Q_{D-1}$ , which in turn is a result of $N$ becoming a phase space degree of freedom in $P$ .",
"The Poisson brackets given by $\\Theta _{P}$ between $\\xi $ and $V$ , or the ones between $\\xi $ and $\\tilde{V}$ , turn out to be non-trivial.",
"The results of the brackets are given by a function $L: Q_{D-1}\\rightarrow so(D+1)$ in the form $\\lbrace \\xi ,NV^{IJ}\\rbrace \\equiv L^{IJ}(V), \\quad \\lbrace \\xi ,N\\tilde{V}^{IJ}\\rbrace \\equiv L^{IJ}(\\tilde{V}).$ Remarkably, the equations (REF ) taken as the definition equations for the function $L$ , together with the brackets (REF ), already constrained the set of possible $L^{IJ}$ to be exactly the set of results of the brackets $\\lbrace \\xi ,NV^{IJ}\\rbrace $ and $ \\lbrace \\xi ,N\\tilde{V}^{IJ}\\rbrace $ given by the potential $\\Theta _{P}$ corresponding to our choice of the Hopf sections.",
"This result can be verified by the fact that, the function $L$ defined by Eqs.",
"(REF ) is constrained by three conditions given by the above Poisson brackets (REF ), and these three conditions are exactly the definition of $L$ in Lemma in section 4.3, which can be illustrated as follows.",
"The first of the conditions comes from the equation $NV_{IJ}L^{IJ}=NV_{IJ}\\lbrace \\xi ,NV^{IJ}\\rbrace =\\frac{1}{4}\\lbrace \\xi ,N^2\\rbrace =\\frac{1}{2}N\\lbrace \\xi ,N\\rbrace =N,$ which gives the normalization condition $L^{IJ}(V)V_{IJ}=1$ .",
"The second condition comes from that $NV^{IJ}L^{KL}\\epsilon _{IJKL\\bar{M}}=NV^{IJ}\\lbrace \\xi ,NV^{KL}\\rbrace \\epsilon _{IJKL\\bar{M}} =\\frac{1}{2}\\lbrace \\xi ,NV^{IJ}NV^{KL}\\rbrace \\epsilon _{IJKL\\bar{M}}=0,$ where we use the fact that $V$ as bi-vector satisfies $V^{IJ}V^{KL}\\epsilon _{IJKL\\bar{M}}=0$ , with $\\bar{M}$ being a $(D-3)-$ tuple asymmetry index.",
"This result implies the orthogonality condition $L^{IJ}(V)\\bar{V}^\\mu _{IJ}(V)=0, \\forall \\mu $ , where $\\bar{V}^\\mu _{IJ}(V)\\tau ^{IJ}\\in so(D+1)$ denotes the basis members that commutes with $V_{IJ}\\tau ^{IJ}\\in so(D+1)$ .",
"Finally, the third constraint just comes from the Jacobi identity $\\lbrace \\xi ,\\lbrace NV^{IJ},NV^{KL}\\rbrace \\rbrace +\\lbrace NV^{IJ},\\lbrace NV^{KL},\\xi \\rbrace \\rbrace +\\lbrace NV^{KL},\\lbrace \\xi ,NV^{IJ}\\rbrace \\rbrace \\equiv 0,$ from which we get the following coherence identity, $\\lbrace NV^{IJ},L^{KL}(V)\\rbrace - \\lbrace NV^{KL},L^{ IJ}(V)\\rbrace \\equiv \\frac{1}{2}(\\delta ^{IL}L^{JK}(V)+\\delta ^{ JK}L^{IL }(V)-\\delta ^{IK}L^{JL}(V)-\\delta ^{JL}L^{IK}(V)).$ Similarly, the we have the conditions $L^{IJ}(\\tilde{V})\\tilde{V}_{IJ}=1$ , $L^{IJ}(\\tilde{V})\\bar{V}^\\mu _{IJ}(\\tilde{V})=0, \\forall \\mu $ and $\\lbrace N\\tilde{V}^{IJ},L^{KL}(\\tilde{V})\\rbrace - \\lbrace N\\tilde{V}^{KL},L^{ IJ}(\\tilde{V})\\rbrace \\equiv -\\frac{1}{2}(\\delta ^{IL}L^{JK}(\\tilde{V})+\\delta ^{ JK}L^{IL }(\\tilde{V})-\\delta ^{IK}L^{JL}(\\tilde{V})-\\delta ^{JL}L^{IK}(\\tilde{V})).$ The Hamiltonian action (REF ) can be used to write the coherence identity (REF ) and (REF ) as an identity involving Lie derivatives: contracting (REF ) and (REF ) with $X^{IJ}$ and $Y^{KL}$ , we get $\\mathcal {L}_{\\hat{X}}L_Y-\\mathcal {L}_{\\hat{Y}}L_X=L_{[X,Y]},$ where $L_X\\equiv \\textrm {Tr}(LX)$ is the component of $L$ along the algebra element $X$ .",
"Now it is easy to see these three conditions makes the Lemma above applicable and we can verify the result given in the beginning of this paragraph."
],
[
"Scheme of discretized Gauss constraints and simplicity constraints reduction procedure",
"So far we have discussed the phase space structure mainly associated with a single edge of the graph $\\gamma $ , for studying the edge-simplicity constraint surface.",
"To carry on the constraint reduction including the Gauss constraint $G_v^{IJ}\\approx 0$ and vertex-simplicity constraint $S_v^{IJKL}\\approx 0$ , we should now switch to the discrete phase space corresponding to the full graph $\\gamma $ .",
"Clearly, this phase space is just given by the direct product $\\mathcal {P}_\\gamma \\equiv \\times _e T^\\ast SO(D+1)_e$ , with any two flux-holonomy variables associated with distinct edges Poisson commuting with each other.",
"Then, by solving the edge-simplicity constraint equations on all of the edges of $\\gamma $ , the above study can be applied to the result constraint surface $\\mathcal {P}^{\\text{s}}_\\gamma \\equiv \\times _e T_{\\text{s}}^\\ast SO(D+1)_e$ in a direct manner.",
"Recall that, the set $\\lbrace G_v^{IJ}\\approx 0, S_e^{IJKL}\\approx 0\\rbrace $ of the discretized Gauss and edge-simplicity constraints form a first class constraint system in $\\mathcal {P}_\\gamma $ , with the algebra given in (REF ).",
"Therefore, we may perform a standard sympletic reduction with respect to this constraint system.",
"Then, we may treat the vertex-simplicity constraint $S_v^{IJKL}\\approx 0$ as additional conditions, selecting from the reduced phase space the correct physical degrees of freedom.",
"Now we proceed with the reductions upon $\\mathcal {P}_\\gamma $ through the following steps.",
"Symplectic reduction with respected to edge-simplicity constraint $ S_e^{IJKL}\\approx 0$ From our previous analysis, the edge-simplicity constraint surface $\\mathcal {P}^{\\text{s}}_\\gamma $ in $\\mathcal {P}_\\gamma $ would be given by $\\times _e T_{\\text{s}}^\\ast SO(D+1)_e$ , which is related to the fullangle-bivector space $P_\\gamma $ defined as $P_\\gamma \\equiv \\times _e P_e,\\quad P_e={Q}^e_{D-1}\\times Q^e_{D-1}\\times T^\\ast S_e^1\\times SO(D-1)_e,$ where each $D(D+1)-\\frac{(D-1)(D-2)}{2}$ dimensional space $P_e$ is described by the coordinates $(N_e,V_e,\\tilde{V}_e,\\xi _e,\\bar{u}_e)$ .",
"Following our analysis above, we conclude that $P_\\gamma |_{N_e\\ne 0}$ provides a double-covering coordinatization for $\\mathcal {P}^{\\text{s}}_\\gamma |_{X_e\\ne 0}$ , and the symplectomorphism (up to some gauge transformation) $\\mathcal {P}^{\\text{s}}_\\gamma \\cong \\bar{P}_\\gamma /\\mathbb {Z}_2,$ with $ \\bar{P}_\\gamma :=\\times _{e\\in \\gamma }\\bar{P}_e$ and $\\bar{P}_e$ as defined in section 4.4.",
"Moreover, the the gauge orbits generated by edge-simplicity constraints in $\\mathcal {P}_\\gamma $ correspond to the degrees of freedom of $\\bar{u}_e$ .",
"Therefore, the resulted reduced phase space $\\mathcal {P}^{\\text{S}}_\\gamma $ with respected to edge-simplicity constraint can be characterized by the symplectomorphism $\\mathcal {P}^{\\text{S}}_\\gamma \\cong \\bar{\\check{P}}_\\gamma /\\mathbb {Z}_2,$ with $\\bar{\\check{P}}_\\gamma :=\\times _{e\\in \\gamma }\\bar{\\check{P}}_e$ , $\\check{P}_\\gamma :=\\times _{e\\in \\gamma }\\check{P}_e$ and $\\check{P}_e:=Q^e_{D_1}\\times Q^e_{D-1}\\times T^*S_e^1$ and $\\bar{\\check{P}}_e$ as defined in section 4.4.",
"In particular, the reduced angle-bivector variables $(N_e,V_e,\\tilde{V}_e,\\xi _e,)$ provide a double-covering coordinatization for the reduced phase space $\\mathcal {P}^{S}_\\gamma $ in the ${X_e\\ne 0}$ region.",
"Symplectic reduction with respected to discretized Gauss constraints $G_v^{IJ}\\approx 0$ Recall that the discretized Gauss constraints acting upon $\\mathcal {P}_\\gamma $ take the form $G_v=\\sum _{e|s(e)=v}X_e+\\sum _{e|t(e)=v}\\tilde{X}_e\\approx 0$ , and it is straight forward to see that the constraints they induce in $\\mathcal {P}^{\\text{S}}_\\gamma $ are just the closure constraints $C_v:=\\sum _{e|s(e)=v}N_eV_e+\\sum _{e|t(e)=v}N_e\\tilde{V}_e\\approx 0$ we mentioned in the beginning.",
"The symplectic reduction inside $\\mathcal {P}^{\\text{S}}_\\gamma $ can be perform using the closure constraint with the results given by the analysis described section $3.1$ .",
"Utilizing the solutions (REF ), the obtained reduced phase space $\\mathcal {P}^{\\text{S,G}}_\\gamma $ is characterized by the symplectomorphism $\\mathcal {P}^{\\text{S,G}}_\\gamma \\cong \\bar{\\check{H}}_\\gamma /\\mathbb {Z}_2,$ where we define $\\bar{\\check{H}}_\\gamma :=\\bar{\\check{P}}_\\gamma /\\!/SO(D+1)^{V(\\gamma )},\\quad \\check{H}_\\gamma :=\\check{P}_\\gamma /\\!/SO(D+1)^{V(\\gamma )}=\\left(\\times _e T^\\ast S_e^1\\right)\\times \\left(\\times _v \\mathfrak {P}_{\\vec{N}_v}\\right)$ with $V(\\gamma )$ being the number of the vertices in $\\gamma $ .",
"Observe that the double quotient operation in (REF ) is “non-local\" in terms of the original phase space variables, due to the fact that the variables across the two connected vertices for each edge-subspace are correlated by the condition $X_{-e}=-g^{-1}_e X_eg_e$ .",
"This technical difficulty is removed by the parametrization (REF ), through which the bi-vectors $V_e$ and $\\tilde{V}_e$ are assigned independently, with the relation $X_{-e}=-g^{-1}_e X_eg_e$ implicitly ensured by the definition of the angle variables.",
"The imposition of the closure constraints and the quotient by $SO(D+1)^{V_\\gamma }$ can then be taken at each of the vertices separately.",
"Further, the reduced space carries the $T^\\ast \\!S^1$ degrees of freedom at every edge in the following manner.",
"The $SO(D+1)^{V_\\gamma }$ gauge orbits in this context are generated by the closure constraints acting on the remaining connection variables in $\\mathcal {P}^{S}_\\gamma $ — $\\xi _e$ ; according to (REF ) the actions are given by $\\lbrace \\xi _e, C_{s(e)}^{IJ}\\rbrace =L^{IJ}(V_e),\\quad \\lbrace \\xi _e, C_{t(e)}^{IJ}\\rbrace =L^{IJ}(\\tilde{V}_e).$ Since $g_e$ and $h^\\Gamma _e$ transform identically as an $SO(D+1)$ holonomy over $e$ and notice their decomposition (REF ), referring to (REF ) we infer that $\\xi _e$ and $\\eta _e$ behave the same under the transformations by the closure constraint: $\\lbrace \\eta _e, C_{s(e)}^{IJ}\\rbrace = \\lbrace \\xi _e, C_{s(e)}^{IJ}\\rbrace =L^{IJ}(V_e),\\quad \\lbrace \\eta _e, C_{t(e)}^{IJ}\\rbrace =\\lbrace \\xi _e, C_{t(e)}^{IJ}\\rbrace =L^{IJ}(\\tilde{V}_e).$ This implies that the extrinsic curvature 1-form $K^{\\perp }_e$ identified in (REF ) is indeed $SO(D+1)$ invariant.",
"Hence, assuming that the graph $\\gamma $ is such that $\\eta _e$ can be given globally without ambiguities to ensure that the Levi-Civita holonomy $h^{\\Gamma }_e$ is properly defined to capture the intrinsic curvature by Eq.",
"(REF ), we may use the gauge invariant $\\xi ^o_e:=\\xi _e-\\eta _e$ in place of the $\\xi _e$ in $(N_e,\\xi _e)\\in T^\\ast \\!",
"S^1$ and obtain the description of the $SO(D+1)$ -invariant degrees of freedom in $T^\\ast \\!S^1$ under the coordinates $(N_e,\\xi ^o_e)$ .",
"Imposing vertex-simplicity constraints $S_v^{IJKL}\\approx 0$ As mentioned, here we treat the vertex-simplicity constraint $S_v^{IJKL}\\approx 0$ as second-class constraints in selecting the physical states of the discrete geometries from which the ADM data can be recovered.",
"In the space $\\mathcal {P}^{\\text{S,G}}_\\gamma $ and $\\check{H}_\\gamma $ , the vertex-simplicity constraints take the form $S_v^{IJKL}\\equiv V^{[IJ}_{e_\\imath }V^{KL]}_{e_\\jmath }\\approx 0, $ ( $\\forall e_\\imath , e_\\jmath : b(e_\\imath )= b(e_\\jmath )=v$ ).",
"Denoting the subspace satisfying $S_v^{IJKL}=0$ as $\\check{H}^{\\textrm {s.}}_\\gamma \\subset \\check{H}_\\gamma $ , we refer again to the results in Sec.",
"$3.1$ and find that the subspace is characterized by $\\check{H}^{\\textrm {s.}}_\\gamma =\\left(\\times _e T^\\ast S_e^1\\right)\\times \\left(\\times _v \\mathfrak {P}_{\\vec{N}_v}^{\\textrm {s.}}\\right).$ To finalize our procedure, we divide $\\check{H}^{\\textrm {s.}}_\\gamma $ by the kernel of the reduced symplectic 2-form $\\Omega _\\gamma \\equiv \\times _e\\Omega _{P_e}/\\!/SO(D+1)^{V_\\gamma }$ , to remove the artificial degeneracy resulted from the parametrization singularity described in section $4.4$ .",
"Then we arrive the final space $\\bar{\\check{H}}^{\\textrm {s.}}_\\gamma :=\\check{H}^{\\textrm {s.}}_\\gamma /\\text{Ker}(\\Omega _\\gamma )$ , which is isomorphic by construction to the vertex-simplicity constraint surface in $\\mathcal {P}^{\\text{S,G}}_\\gamma $ as $\\bar{\\check{H}}^{\\textrm {s.}}_\\gamma /\\mathbb {Z}^2\\cong \\mathcal {P}^{\\text{S,G}}_\\gamma |_{S_v^{IJKL}=0}.$ after moduling the identifying operation $\\mathbb {Z}^2$ .",
"Based upon this reduction procedure, we claim that the kinematic physical degrees of freedom of the theory on a given graph $\\gamma $ are captured by the collection of solutions of verticex-simplicity constraints in the phase space $\\bar{\\check{H}}_\\gamma /\\mathbb {Z}^2$ .",
"Let us supply the above reduction procedure in the discrete LQG phase space with a classical picture under proper continuous limits for the Regge sector discussed in Section 3.2, with the interpretation of the variables $(V_e,\\tilde{V}_e,N_e,\\xi _e,\\xi ^\\mu _e)$ as the extrinsic and intrinsic geometrical data.",
"Specifically, recall that the $N_e$ has the meaning of the area of the (D-1)-face dual to $e$ , and $\\xi ^o_e$ represents the norm of the extrinsic curvature 1-form integrated along $e$ are clear.",
"A concrete translation between $(g_e, X_e)$ and the hypersuface ADM data $(\\pi ^{aIJ}(x), K_{aIJ}(x))$ can be thus established using the straight forward conditions $g_e\\simeq \\mathbb {I}+A_e$ and $X_e\\simeq \\pi _e$ , with which we have $\\frac{1}{2\\beta }\\textrm {Tr}(X_edg_eg_e^{-1})\\simeq \\frac{1}{\\beta }\\pi ^e_{IJ}dA_e^{IJ}.$ In the limit of infinitely short edges one may then read off the sympletic form $\\Omega =-d\\Theta =\\frac{1}{\\beta }dA\\wedge d\\pi $ , and the familiar brackets of loop quantum gravity follow, $\\lbrace A_{e}^{IJ}(x),\\pi ^{e^{\\prime }}_{KL}(y)\\rbrace =2\\beta \\delta ^{[I}_{K}\\delta ^{J]}_L\\delta _e^{e^{\\prime }}\\delta ^{(D)}(x-y).$ Also, recalling the splitting $A_{a}^{IJ}=\\Gamma _a^{IJ}(\\pi )+\\beta K_a^{IJ}$ with $\\Gamma _a^{IJ}(\\pi )$ being a function of $\\pi ^{bKL}$ satisfying $\\Gamma _a^{IJ}(\\pi )=\\Gamma _a^{IJ}(e)$ on simplicity constraint surface, one recovers the Poisson bracket $\\lbrace K_{e}^{IJ}(x),\\pi ^{e^{\\prime }}_{KL}(y)\\rbrace =2\\delta ^{[I}_{K}\\delta ^{J]}_L\\delta _e^{e^{\\prime }}\\delta ^{(D)}(x-y)$ .",
"The same continuous limit also reveals the classical counterpart to our simplicity constraint reduction in the discrete phase space.",
"Through the correspondence $g_e=u_ee^{\\xi ^e\\tau _o}e^{\\bar{\\xi }_e^\\mu \\bar{\\tau }_\\mu }\\tilde{u}_e^{-1}$ and $u_ee^{\\bar{\\eta }_e^\\mu \\bar{\\tau }_\\mu }e^{\\eta ^e\\tau _o}\\tilde{u}_e^{-1}\\simeq \\mathbb {I}+\\Gamma _e$ in the continuous limits being taken, we have $K_e \\simeq \\frac{1}{\\beta }u_e (\\xi ^o_e\\tau _o+\\check{\\xi }_e^\\mu \\bar{\\tau }_\\mu )u_e^{-1},$ with the notation $e^{\\check{\\xi }_e^\\mu \\bar{\\tau }_\\mu }\\equiv e^{-\\bar{\\eta }_e^\\mu \\bar{\\tau }_\\mu }e^{\\bar{\\xi }_e^\\mu \\bar{\\tau }_\\mu }$ .",
"Recalling our correspondence $K^{\\perp }_e:=\\frac{1}{\\beta }u_e (\\xi ^o_e\\tau _o)u_e^{-1}$ , $K^{/\\!/}_e:=\\frac{1}{\\beta }u_e (\\check{\\xi }_e^\\mu \\bar{\\tau }_\\mu )u_e^{-1}$ , we can clearly see that despite of the anomaly in the vertex-simplicity constraints, our reduction procedure correctly remove the component $K^{/\\!/}_e$ , while preserving the component $K^{\\perp }_e$ that contributes to the extrinsic curvature as expressed in the same form as in the classical Ashtekar formulation: $\\text{tr}(K_e\\pi ^{e^{\\prime }})=\\frac{1}{\\beta } \\text{tr}( u_e (\\xi _e^o\\tau _o+\\check{\\xi }_e^\\mu \\bar{\\tau }_\\mu )u_e^{-1}\\pi ^{e^{\\prime }}) =\\frac{1}{\\beta }\\text{tr}( u_e (\\xi _e^o\\tau _o)u_e^{-1}\\pi ^{e^{\\prime }})=\\text{tr}(K^{\\perp }_e\\pi ^{e^{\\prime }}), \\ \\ b(e)=b(e^{\\prime }).$ Indeed, as a generator of the group preserving $\\mathcal {N}_{b(e)}=\\mathcal {N}_{b(e^{\\prime })}$ , the component $K^{/\\!/}_e$ has no projection on the bivector $\\pi ^{e^{\\prime }}\\simeq X^{e^{\\prime }}=N_{e^{\\prime }}V_{e^{\\prime }}=N_{e^{\\prime }}\\mathcal {N}^{[I}_{b(e^{\\prime })}V^{J]}_{e^{\\prime }}$ and thus provides no contribution to the extrinsic curvature as it showed in above Eq.",
"(REF ).",
"This procedure is thus consistent to the symplectic reduction with respect to simplicity constraint in connection phase space in which it act as a well first class constraint, where $K^{/\\!/}_e$ play the same role as the component $\\bar{K}_{aIJ}$ in connection phase space, and $\\text{tr}(K_e\\pi ^{e^{\\prime }})$ is proportion to the densitized extrinsic curvature $\\tilde{K}_a^{\\ b}$ along the graph in continuum limit.",
"This result means that we can choose the $SO(D-1)$ fibers as the “gauge orbit” of simplicity constraint in the discrete phase space, because the redundant degrees of freedom $K^{/\\!/}_e$ are precisely those transformed along the $SO(D-1)$ fibers, which is same as how $\\bar{K}_{aIJ}$ acts along the true gauge orbits of simplicity constraint in continuum connection phase space.",
"Now based on the above discussions, we have demonstrated that On both edge and vertex-simplicity constraints surface, the degrees of freedoms in $\\bar{\\xi }^\\mu _e$ (or equivalent, the $SO(D-1)$ fiber) of the discrete LQG phase space represent “gauge” degrees of freedom playing the same role of the components $K^{/\\!/}$ eliminated in the symplectic reduction with respect to the simplicity constraints in the original Ashtekar formulation of all dimensional LQG in the continuous phase space.",
"Remakably in this sense, under the correspondence between the generalized twisted geometry variables and smeared Ashtekar variables in the Regge sector, the continuous limit of our reduction procedure indeed recovers the symplectic reduction in the Ashtekar formulation with respect to the original Gauss and simplicity constraints."
],
[
"Conclusion and outlook",
"To better explore the spacetime geometry information encoded in the higher-dimensional spin-network states, we proposed a new kinematic gauge-reduction procedure for the $SO(D+1)$ LQG at the classical and discrete level.",
"The reduction takes place with respect to the anomalous kinematic constraint system consisting of the discrete simplicity constraints $ \\lbrace S_e^{IJKL}\\approx 0,\\,S_v^{IJKL}\\approx 0 \\rbrace $ and the discrete $SO(D+1)$ Gauss constraints $G_v^{IJ}\\approx 0$ , defined in the $SO(D+1)$ LQG phase space associated with a given graph for the spin network states.",
"Motivated by our previous work on the weak solutions of the quantum vertex-simplicity constraints given by the coherent intertwiners, we generalized the twisted-geometry parametrization of the $SU(2)$ LQG phase space, into the angle-bivector parametrization of the constraint surface of edge-simplicity constraints $S_e^{IJKL}\\approx 0$ in the $SO(D+1)$ LQG phase space.",
"Further, when restricted to the common constraint surface of the full kinematic constraints $ \\lbrace G_v^{IJ}\\approx 0,\\,S_e^{IJKL}\\approx 0,\\,S_v^{IJKL}\\approx 0 \\rbrace $ , the new parametrization endows the angle-bivector variables with the meaning of the constrained smeared formulations of Ashtekar variables.",
"In particular, the $SO(D-1)$ angle variables are identified with the smeared Ashtekar connection components that are pure-gauge corresponding to simplicity constraint in the original classical and continuous theory.",
"Through studying the properties of the Hopf sections in $SO(D+1)$ Hopf fibre bundle, we obtained the Poisson algebra among the angle-bivector variables, and subsequently the action of the constraint system on the twisted-geometry variables.",
"Then, the full symplectic reduction with respect to the first-class sub-system of the discrete constraints $\\lbrace G_v^{IJ}\\approx 0,\\, S_e^{IJKL}\\approx 0\\rbrace $ is performed and results to the gauge-invariant reduced phase space $\\mathcal {P}^{\\text{S,G}}_\\gamma $ .",
"Crucially as we discovered, when again restricted to the common constraint surface of the full system of the discrete constraints, the first-class subsystem generates the orbits that recover the gauge orbits generated by the original continuous simplicity and Gauss constraints in the continuous limits.",
"In particular, the edge-simplicity constraints generate precisely the transformations in the $SO(D-1)$ angle variables.",
"Finally, we demonstrated that when the remaining anomalous vertex-simplicity constraints are imposed as additional constraints upon the gauge-invariant reduced phase space, the selected state space $ \\bar{\\check{H}}^{\\textrm {s.}}_\\gamma \\subset \\mathcal {P}^{\\text{S,G}}_\\gamma $ truly describes the discrete ADM data in the form of Regge hypersurface geometry, up to the shape matching condition.",
"We are thus led to the new point of view, in which the (quantum) discrete Gauss and edge-simplicity constraints are the true generators of the kinematic gauge symmetry for all-dimensional LQG , while the anomalous vertex-simplicity constraints only act as addition state-selection laws.",
"From a different view, this new gauge reduction has supplemented the missing pieces in our previous prescription of the weak solutions to the quantum vertex-simplicity constraint, where the weak solutions are interpreted as semi-classical polytopes used to assemble the spatial geometry.",
"Indeed, the absent description of the extrinsic curvature components in the Ashtekar connection are captured by the angle variables parametrizing the holonomies, left out from the previous prescriptions based on only the bivectors labelling coherent intertwiners.",
"Remarkably, our gauge orbit reductions leave the only angle variable $ \\xi ^o_e $ giving precisely the (smeared) Regge extrinsic curvature, for the states on the vertex-simplicity constraint surface in the reduced phase space.",
"Our results point to two interesting future research directions based upon the generalized twisted-geometry parametrization of the $SO(D+1)$ LQG phase space.",
"Firstly, we have assumed the existence of the coherent spin-network states sharply peaked in the angle-bivector variables— this was based on the known explicit construction of the $SU(2)$ coherent spin-network states sharply peaked in the twisted-geometry variables [9].",
"As indicated in this earlier study, these states are highly valuable for not only do they recover the boundary semi-classical states for the spinfoam models (a covariant version of LQG), but also serve as a special type of the Thiemann-Hall's complexifier coherent states in canonical LQG [17][18][19].",
"Through these connections, the clear intrinsic and extrinsic geometry interpretations via the twisted-geometry variables have illuminated many important perspectives of the classical limits for both canonical and covariant LQG in $(1+3)$ dimension.",
"In the same manner, we expect our angle-bivector parametrization to offer valuable insights to the covariant and canonical LQG in higher dimensions, for which the geometric meaning of the coherent states has been even more elusive.",
"In fact, it is known that the currently prevailing Thiemann-Hall's $SO(D+1)$ coherent states are too complicated for explicit computations.",
"It is our hope that the angle-bivector coherent states, which could be constructed and studied based on the recent works [20], may serve as the alternative coherent states with the much simpler Gaussian distribution formulation and clear geometric meanings, for clarifying the semi-classical behavior of the $SO(D+1)$ LQG.",
"The second direction is toward the physical evolutions in canonical $SO(D+1)$ LQG.",
"This may be pursuit either in the context of the Dirac theory with the Hamiltonian operators (arbitrary combinations of the quantum scalar and vector constraints) to be solved as additional quantum constraints and the local observables to be constructed, or in the context of a classically deparametrized theory with one physical Hamiltonian operator giving the evolution in a specified notion of time.",
"In both cases, the new crucial challenge here is to deal with the algebra involving the quantum Hamiltonian operators.",
"As mentioned, although the full system of Gauss, simplicity and Hamiltonian constraints are of first class in the continuous classical theory, it inevitably becomes anomalous under the loop quantization [11] [21]— especially with the typical closed loop holonomy representation for the curvature factors in the Hamiltonian operators.",
"Our new insights in the quantum orbits for the simplicity constraints may provide an approach to the problem that is closely guided by the physical and geometric picture.",
"For instance in the context of the deparametrized theory, our results suggest quantizing the discretized physical Hamiltonian associated to each graph that is gauge invariant with respect to just the quantum Gaussian constraints and edge-simplicity constraints.",
"This would lead to the dynamics preserving the gauge symmetry, with which one could then study the weak stability of the vertex-simplicity constraints under the dynamics.",
"Indeed, the program in this manner would be guided by the ultimate goal for the quantum evolutions of the Regge ADM data."
],
[
"Acknowledgments",
"This work is supported by the National Natural Science Foundation of China (NSFC) with Grants No.",
"11775082, No.",
"11875006 and No.",
"11961131013."
]
] | 2011.14152 |
[
[
"Symbolic dynamics and transfer operators for Weyl chamber flows: a class\n of examples"
],
[
"Abstract We provide special cross sections for the Weyl chamber flow on a sample class of Riemannian locally symmetric spaces of higher rank, namely the direct product spaces of Schottky surfaces.",
"We further present multi-parameter transfer operator families for the discrete dynamical systems on Furstenberg boundary that are related to these cross sections."
],
[
"Introduction",
"Discretizations of flows on various types of spaces and symbolic dynamics for them are useful for many purposes.",
"The applications that motivate this article are transfer operator approaches to Laplace eigenfunctions, resonances and dynamical zeta functions as, e.g., in [22], [7], [11], [12], [21], [15], [4], [16], [13], [14], [18], [17], [1], [6], [20], [3].",
"At the current state of art, these are limited to hyperbolic spaces, and mostly even to hyperbolic surfaces.",
"The necessary discretizations of the geodesic flow on these spaces are typically constructed by means of cross sections.",
"For the dynamical approaches to Laplace eigenfunctions referred to above it was discovered to be crucial to deviate from the classical notion of cross sections.",
"For these applications a relaxed notion is used where we require from a cross section that it detects all periodic geodesics, but it need not have “time unbounded” intersections with all geodesics, in particular not with those that eventually stay in an end of the considered space.",
"In contrast, the classical notion would require that a cross section has infinitely many intersections with all geodesics in both of their “time directions”, past and future.",
"Also transfer operator approaches to dynamical zeta functions such as Ruelle and Selberg zeta functions were seen to benefit from this more flexible notion of cross section.",
"In this article we discuss a notion of cross section for the Weyl chamber flow on Riemannian locally symmetric spaces that is modelled in analogy to this relaxed notion of cross sections for one rank spaces.",
"We construct such cross sections for a sample class of spaces.",
"We further provide related discrete dynamical systems on the Furstenberg boundary of the Riemannian globally symmetric spaces covering these spaces and we present associated multi-parameter transfer operator families.",
"This work may be seen as a first step of an attempt towards transfer operator approaches for higher rank spaces.",
"However, cross sections for Weyl chamber flows are sought for also for other purposes.",
"See, e.g., [8].",
"Our considerations here might also be of use for such questions.",
"In a nutshell, we define a cross section for the Weyl chamber flow to be a subset of the Weyl chamber bundle (or the set of Weyl chambers) that is intersected by all compact oriented flats and for which each intersection with any oriented flat is discrete in the regular directions of the flow.",
"The first requirement is motivated by the fact that compact oriented flats are the Weyl-chamber-flow equivalent of periodic geodesics for the geodesic flow.",
"For spaces of rank one, the compact oriented flats are precisely the periodic geodesics.",
"The second requirement is motivated by the idea that intersections should be only momentarily.",
"The restriction of this requirement to regular directions results from the fact that in singular directions there are some space dimensions of the considered flat in which no motion happens.",
"We allow non-discreteness in the space directions without motion.",
"The sample spaces of higher rank that we consider here are the direct products of Schottky surfaces.",
"These spaces have a rather simple structure, but nevertheless their study in regard to our goals is very instructive.",
"For Schottky surfaces, cross sections for the geodesic flow, induced discrete dynamical systems as well as one-parameter transfer operator families are well-known.",
"We will take advantage of this knowledge.",
"As cross section for the geodesic flow on a Schottky surface one typically takes the set of unit tangent vectors that are based on the boundary of a standard fundamental domain for the considered surface and that are directed to the interior of the fundamental domain.",
"We call such a cross section standard for the moment and refer to Section for details and precise definitions.",
"Our main results regarding these spaces are essentially as follows.",
"Theorem (Coarse statement) Let $r\\in \\mathbb {N}$ .",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ let $\\mathbb {Y}_j$ be a Schottky surface and $\\widehat{C}_j$ a standard cross section for the geodesic flow on $\\mathbb {Y}_j$ .",
"Let $\\mathbb {Y}\\mathbb {Y}_1\\times \\cdots \\times \\mathbb {Y}_r$ be the Riemannian locally symmetric space of rank $r$ which is the direct product of the Schottky surfaces $\\mathbb {Y}_j$ , $j\\in \\lbrace 1,\\ldots , r\\rbrace $ .",
"The direct product $\\widehat{C} \\widehat{C}_1 \\times \\cdots \\times \\widehat{C}_r$ provides a cross section for the Weyl chamber flow on $\\mathbb {Y}$ .",
"The first return map of $\\widehat{C}$ is (semi-)conjugate to a discrete dynamical system $F$ on the Furstenberg boundary of the Riemannian globally symmetric space $\\mathbb {X}$ covering $\\mathbb {Y}$ .",
"The map $F$ is piecewise given by the action of certain elements of the fundamental group of $\\mathbb {Y}$ .",
"The multi-parameter transfer operator family associated to $F$ can be provided in explicit form.",
"The refined version of this theorem features explicit expressions and formulas at each level.",
"It appears as Theorems REF , REF and REF and as Section .",
"A few remarks are in order.",
"The choice of $\\widehat{C}$ as direct product of the cross sections $\\widehat{C}_j$ for the Schottky surfaces may seen trivial at first glimpse.",
"However, it shows important properties of cross sections for Weyl chamber flows of spaces of higher rank.",
"For each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , the base point set of $\\widehat{C}_j$ is the full boundary of the fundamental domain for $\\mathbb {Y}_j$ .",
"The base point set of $\\widehat{C}$ is only a small part of the boundary of a fundamental domain for $\\mathbb {Y}$ .",
"Thus, the cross section $\\widehat{C}$ is a rather sparse subset of the Weyl chamber bundle of $\\mathbb {Y}$ , and it does not arise from a Koebe–Morse method (as the $\\widehat{C}_j$ do).",
"The first return map relies on the notion of “next” intersections.",
"Since the flow is multi-dimensional as soon as the rank $r$ is larger than 1, the concept of next intersections needs to be discussed with care.",
"It is introduced in Section REF .",
"The multi-parameter transfer operator family associated to the map $F$ is not the direct product of the one-parameter transfer operator families for $\\mathbb {Y}_j$ , $j\\in \\lbrace 1,\\ldots , r\\rbrace $ .",
"This family is discussed in detail in Section .",
"This article is structured as follows.",
"In Section we first survey the necessary background on Riemannian globally and locally symmetric spaces.",
"In Section REF we then introduce the notions of cross sections, next intersections, first return map and induced discrete dynamical systems on Furstenberg boundary.",
"Section – are devoted to the study of our sample class of spaces.",
"In Section we briefly present the well-known cross sections and related objects for Schottky surfaces.",
"In Section we provide and discuss a cross section for the Weyl chamber flow.",
"In Section we introduce the discrete dynamical system on Furstenberg boundary that is induced by a well-chosen set of representatives for the cross section.",
"In the final Section we present the multi-parameter transfer operator family associated to this discrete dynamical system."
],
[
"Acknowledgement",
"This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project no.",
"264148330."
],
[
"Geometry and dynamics of Riemannian locally symmetric\nspaces",
"In this section we introduce the elements from the geometry and dynamics of Riemannian globally and locally symmetric spaces that we will use further on.",
"Since there are many excellent treatises of this subject, we will keep our presentation to the bare minimum necessary for our purposes and will omit any proofs.",
"For comprehensive expositions, we refer in particular to [10], [5], [9].",
"We further present the notion of cross section for the Weyl chamber flow, discuss the concept of next intersections and the existence of the first return map.",
"We also provide the notion of a set of representatives for the cross section and the notion of an associated discrete dynamical system on Furstenberg boundary."
],
[
"Globally symmetric spaces",
"Let $\\mathbb {X}$ be a Riemannian symmetric space of noncompact type.",
"A flat of $\\mathbb {X}$ is any totally geodesic, flat submanifold of $\\mathbb {X}$ of maximal dimension among all such submanifolds.",
"The common dimension of all flats is the rang of $\\mathbb {X}$ .",
"Let $F$ be a flat of $\\mathbb {X}$ , pick a point $x\\in F$ and consider the union of all flats that are distinct from $F$ but contain the point $x$ .",
"This union intersects $F$ in a finite union of hyperplanes of $F$ .",
"The connected components of the (open) complement of this union of hyperplanes in $F$ are the Weyl chambers in $F$ with base point $x$ .",
"We let $\\operatorname{C\\!",
"}\\mathbb {X}\\lbrace \\text{Weyl chamber in~$F$ with base point~$x$}\\mid \\text{$F$ flat, $x\\in F$} \\rbrace $ denote the set of all Weyl chambers of $\\mathbb {X}$ , i.e., the family of all Weyl chambers of all flats with all base points.",
"We note that each Weyl chamber $c\\in \\operatorname{C\\!",
"}\\mathbb {X}$ is contained in a unique flat $F$ and has a unique base point $x$ .",
"For the introduction of further objects, in particular the Weyl chamber flow, we take advantage of some elements of the structure theory of $\\mathbb {X}$ .",
"We fix an arbitrary point $x_0$ in $\\mathbb {X}$ , an origin or reference point, let $G$ be the identity component of the group of Riemannian isometries of $\\mathbb {X}$ , and let $K\\operatorname{Stab}_{G}(x_0)$ denote the stabilizer group of $x_0$ in $G$ .",
"Then $G$ is a real semisimple Lie group with finite center, and $K$ is a maximal compact subgroup of $G$ .",
"We may identify the symmetric space $\\mathbb {X}$ with the homogeneous space $G/K$ via the isomorphism $G/K \\rightarrow \\mathbb {X}\\,,\\quad gK \\mapsto g(x_0)\\,.$ We fix a flat $F_0$ through $x_0$ , a reference flat.",
"Then there exists a unique maximal abelian subgroup $A$ of $G$ such that $F_0 = A(x_0)$ .",
"Indeed, $F_0$ may be identified with $A$ via the isomorphism $A \\rightarrow F_0\\,,\\quad a \\mapsto a(x_0)\\,.$ We now fix a Weyl chamber $c_0$ in $F_0$ with base point $x_0$ , a reference Weyl chamber, and let $M Z_K(A)$ denote the centralizer of $A$ in $K$ .",
"Then we may interpret the set $\\operatorname{C\\!",
"}\\mathbb {X}$ as the homogeneous space $G/M$ via the isomorphism $G/M \\rightarrow \\operatorname{C\\!",
"}\\mathbb {X}\\,,\\quad gM \\mapsto g(c_0)\\,.$ Let $\\pi _B\\colon \\operatorname{C\\!",
"}\\mathbb {X}\\rightarrow \\mathbb {X}\\,,\\quad c\\mapsto x_c\\,,$ denote the map from a Weyl chamber $c$ to its base point $x_c$ .",
"Under the isomorphisms from (REF ) and (REF ), the map $\\pi _B$ becomes the quotient map $G/M \\rightarrow G/K\\,,\\quad gM \\mapsto gK\\,,$ which also turns $\\operatorname{C\\!",
"}\\mathbb {X}$ into a bundle over $\\mathbb {X}$ .",
"We further let $M^{\\prime }N_K(A)$ denote the normalizer of $A$ in $K$ , and let $WM^{\\prime }/M$ denote the Weyl group.",
"With respect to the isomorphism in (REF ), the action of $G$ on $\\operatorname{C\\!",
"}\\mathbb {X}$ becomes $G\\times G/M \\rightarrow G/M\\,,\\quad (h,gM) \\mapsto hgM\\,,$ and the action of the Weyl group $W$ on $\\operatorname{C\\!",
"}\\mathbb {X}$ becomes $W\\times G/M \\rightarrow G/M\\,,\\quad (m^{\\prime }M, gM) \\mapsto gm^{\\prime }M\\,.$ We immediately see that the set of all Weyl chambers in $F_0$ with base point $x_0$ are the (finitely many and pairwise distinct) elements of the orbit of the reference Weyl chamber $c_0$ under the Weyl group $W$ .",
"Further, the set of all flats of $\\mathbb {X}$ consists of all $G$ -translates of the reference flat $F_0$ .",
"The Weyl chamber flow on $\\mathbb {X}$ is the action of $A$ on $\\operatorname{C\\!",
"}\\mathbb {X}$ which under the isomorphism in (REF ) becomes $A\\times G/M \\rightarrow G/M\\,,\\quad (a,gM)\\mapsto gaM\\,.$ The isomorphism in (REF ) implies that the Lie group $A$ is isomorphic to $\\mathbb {R}^r$ , where $r$ is the rang of $\\mathbb {X}$ .",
"Thus, the Weyl chamber flow is a flow on $\\operatorname{C\\!",
"}\\mathbb {X}$ with $r$ “time” dimensions.",
"The (unoriented) flow “lines” of the Weyl chamber flow on $\\mathbb {X}$ are the flats.",
"More precisely, for each each Weyl chamber $gM$ , the base point set of its $A$ -orbit is $\\pi _B(gAM) = gAK\\,,$ which is the unique flat that contains $gM$ .",
"For Riemannian symmetric spaces of rang one, the Weyl chamber flow coincides with the geodesic flow.",
"In what follows we introduce a rather coarse notion of orientation of flats (following [10]), which provides an appropriate notion of directions for the Weyl chamber flow and will be crucial for our notion of cross sections.",
"We further introduce the Furstenberg boundary, which provides the appropriate geometry at infinity for our codings in Section .",
"To that end we note that the choice of the reference Weyl chamber $c_0$ distinguishes an open subset, $A^+$ , of $A$ by means of the isomorphism in (REF ).",
"The subset $A^+$ is often called the open positive Weyl chamber in $A$ , subject to the choice of $c_0$ .",
"We let $\\mathfrak {g}$ and $\\mathfrak {a}$ denote the Lie algebra of $G$ and $A$ , respectively, and let $\\Lambda $ be the set of roots of $(\\mathfrak {g},\\mathfrak {a})$ .",
"We let $\\mathfrak {a}^+$ denote the subset of $\\mathfrak {a}$ that corresponds to $A^+$ under the exponential map.",
"Then $\\Lambda ^+ \\lbrace \\lambda \\in \\Lambda \\mid \\forall \\, H\\in \\mathfrak {a}^+\\colon \\lambda (H) > 0\\rbrace $ is a choice of positive roots of $(\\mathfrak {g},\\mathfrak {a})$ .",
"We set $\\mathfrak {n} \\bigoplus _{\\lambda \\in \\Lambda ^+}\\mathfrak {g}_\\lambda \\,,$ where $\\mathfrak {g}_\\lambda \\lbrace X\\in \\mathfrak {g} \\mid \\forall \\, H\\in \\mathfrak {a}\\colon \\operatorname{ad}_H(X) = \\lambda (H)X \\rbrace $ is the root space of $\\lambda $ , and $\\operatorname{ad}$ is the adjoint representation of $\\mathfrak {g}$ .",
"We let $N \\exp \\mathfrak {n}$ be the (unipotent) subgroup associated to $\\mathfrak {n}$ .",
"We note that $N$ completes the pair $(K,A)$ to an Iwasawa decomposition of $G$ .",
"Thus, the map $N\\times A \\times K \\rightarrow G\\,,\\quad (n,a,k) \\mapsto nak\\,,$ is an isomorphism of Lie groups.",
"We let $P NAM$ be the associated minimal parabolic subgroup of $G$ .",
"The Furstenberg boundary of $\\mathbb {X}$ is the homogeneous space $G/P \\,.$ We let $\\nu \\colon G/M \\rightarrow G/P\\,,\\quad gM \\mapsto gP\\,,$ denote the canonical projection from the Weyl chamber bundle to the Furstenberg boundary.",
"In order to define a notion of orientation of flats, we call two Weyl chambers $gM$ and $hM$ asymptotic if they project to the same point in the Furstenberg boundary, thus if $\\nu (gM) = \\nu (hM)\\,.$ This property induces an equivalence relation on the set of all Weyl chambers.",
"The combined map $\\alpha (\\pi _B,\\nu )\\colon G/M \\rightarrow G/K \\times G/P$ is an isomorphism.",
"Therefore, the Furstenberg boundary $G/P$ can be interpreted as the set of equivalence classes of asymptotic Weyl chambers.",
"An orientation or direction of a flat $F$ is an equivalence class of asymptotic Weyl chambers that has a representative in $F$ .",
"An oriented flat is a flat endowed with a distinguished orientation.",
"The set of oriented flats can be identified with the homogeneous space $G/AM$ via the isomorphism $G/AM \\rightarrow \\lbrace \\text{oriented flats}\\rbrace \\,,\\quad gAM \\mapsto \\bigl (g(F_0),gP\\bigr )\\,.$ The map $\\nu $ in (REF ) splits into the two canonical maps $\\nu _1\\colon G/M\\rightarrow G/AM\\qquad \\text{and}\\qquad \\nu _2\\colon G/AM\\rightarrow G/P\\,,$ where $\\nu _1$ maps a Weyl chamber $gM$ to the flat that contains it and endows this flat with the equivalence class of $gM$ as orientation, and $\\nu _2$ projects an oriented flat to its orientation, identified with the point in the Furstenberg boundary $G/P$ .",
"The actions of $G$ and $W$ on $G/M$ from (REF ) and (REF ) descend to actions on $G/AM$ and $G/P$ , turning $\\nu _1$ and $\\nu _2$ into $G$ -equivariant as well as $W$ -equivariant maps.",
"In particular, we have $W\\times G/AM \\rightarrow G/MA\\,,\\quad (m^{\\prime }M, gMA) \\mapsto gm^{\\prime }MA\\,,$ and $W\\times G/P\\rightarrow G/P\\,,\\quad (m^{\\prime }M, gP) \\mapsto gm^{\\prime }P\\,.$ For any flat $gF_0$ , the possible orientations are therefore characterized by the finitely many points $gm^{\\prime }P$ , where $m^{\\prime }$ runs through a representative set of $W$ in $M^{\\prime }$ ."
],
[
"Locally symmetric spaces",
"We continue to use the notation from the previous section and now apply the identifications discussed there without mentioning the isomorphisms.",
"In particular, we allow ourselves to write $\\mathbb {X}= G/K$ , and analogously for other objects.",
"We let $\\Gamma $ be a discrete subgroup of $G$ .",
"The quotient space $\\mathbb {Y}\\Gamma \\backslash \\mathbb {X}= \\Gamma \\backslash G/K$ is a locally symmetric space or more precisely, if $\\Gamma $ has torsion, an orbifold.",
"We let $\\pi ^\\Gamma \\colon \\mathbb {X}\\rightarrow \\mathbb {Y}\\,,\\quad gK \\mapsto \\Gamma gK\\,,$ denote the canonical quotient map.",
"Most of the objects defined for $\\mathbb {X}$ in the previous section descend to analogous objects for $\\mathbb {Y}$ , via $\\pi ^\\Gamma $ .",
"The rank, $r$ , of $\\mathbb {Y}$ is the rank of $\\mathbb {X}$ .",
"The $\\pi ^\\Gamma $ -images of the flats of $\\mathbb {X}$ are called the flats of $\\mathbb {Y}$ .",
"We remark that flats of $\\mathbb {Y}$ are not necessarily isometric to $\\mathbb {R}^r$ , much in contrast to flats of $\\mathbb {X}$ .",
"In particular, a flat of $\\mathbb {Y}$ might be compact (as a subset of $\\mathbb {Y}$ ).",
"For locally symmetric spaces of rang one, compact flats coincide with periodic geodesics, more precisely, with the subsets of $\\mathbb {Y}$ traced out by periodic geodesics.",
"The set of Weyl chambers of $\\mathbb {Y}$ is $\\operatorname{C\\!",
"}\\mathbb {Y}= \\Gamma \\backslash G/M\\,,$ the Weyl chamber flow on $\\mathbb {Y}$ is $A\\times \\Gamma \\backslash G/M\\rightarrow \\Gamma \\backslash G/M\\,,\\quad (a,\\Gamma gM)\\mapsto \\Gamma gaM\\,,$ and the set of oriented flats of $\\mathbb {Y}$ is $\\Gamma \\backslash G/AM\\,.$ Since the projection maps $\\pi _B$ and $\\nu _1$ in (REF ) and (REF ) are $\\Gamma $ -equivariant, they induce the analogous maps on $\\mathbb {Y}$ : $\\pi _B^\\Gamma &\\colon \\operatorname{C\\!",
"}\\mathbb {Y}\\rightarrow \\mathbb {Y}\\,,\\quad \\Gamma gM\\mapsto \\Gamma gK\\,,\\multicolumn{2}{l}{\\text{and}}\\\\\\nu _1^\\Gamma &\\colon \\Gamma \\backslash G/M \\rightarrow \\Gamma \\backslash G/AM\\,,\\quad \\Gamma gM \\mapsto \\Gamma gAM\\,.$"
],
[
"Cross sections and induced discrete dynamical\nsystems",
"We resume the notation from the previous two sections.",
"We let $A^\\textnormal {reg}W(A^+)\\,,$ denote the set of regular elements in $A$ .",
"For any subset $S\\subseteq G/M$ , we say that an oriented flat $gAM$ intersects $S$ in $hM$ if $\\nu _1(hM) = gAM\\,.$ We say that the intersection is discrete if there exists a neighborhood $U$ of the identity element $\\operatorname{id}$ in $A$ such that for all $a\\in U\\cap A^\\textnormal {reg}$ we have $haM \\notin S\\,.$ We note that in this case, the Weyl chamber $hM$ is contained in the oriented flat $gAM$ and determines its orientation.",
"Analogously, for any subset $\\widehat{S} \\subseteq \\Gamma \\backslash G/M$ we say that an oriented flat $\\Gamma gAM$ of $\\mathbb {Y}$ intersects $\\widehat{S}$ in the Weyl chamber $\\Gamma hM$ of $\\mathbb {Y}$ if $\\nu _1^\\Gamma (\\Gamma hM) = \\Gamma gAM\\,,$ and we call the intersection discrete if there exists a neighborhood $U$ of $\\operatorname{id}$ in $A$ such that for all $a\\in U\\cap A^\\textnormal {reg}$ we have $\\Gamma haM\\notin \\widehat{S}\\,.$ With these preparations we can now propose the following notion of cross section.",
"Definition 2.1 We call a subset $\\widehat{C}$ of $\\Gamma \\backslash G/M$ a cross section for the Weyl chamber flow on $\\mathbb {Y}$ if every compact oriented flat of $\\mathbb {Y}$ intersects $\\widehat{C}$ , and each intersection of any flat of $\\mathbb {Y}$ with $\\widehat{C}$ is discrete.",
"We emphasize the following aspects of this definition.",
"We do not request that every oriented flat shall intersect $\\widehat{C}$ .",
"Definition REF is motivated by presumed properties necessary for transfer-operator-based investigations of the spectral theory of $\\mathbb {Y}$ .",
"For rank one spaces, our previous investigations showed that for such applications, we only needed to request that all periodic geodesics intersect a cross section for the geodesic flow.",
"By density properties of these geodesics and a certain smoothness of the cross sections, it automatically meant that all geodesics that returned infinitely often to the compact core of the considered space intersect the cross section.",
"For several results, it was crucial that those geodesics that eventually stay in the ends of the space, do not need to intersect the cross section at all or, when travelling along these geodesics, eventually stop intersecting it.",
"For this reason, also for higher rank spaces, we only request that at least all compact oriented flats are detected by the cross section.",
"In our examples in the following sections, we will see that all flats that are “returning” intersect the cross sections constructed there, but that flats “vanishing to infinity” eventually will not intersect anymore.",
"We require discreteness of intersections only in the regular directions of the Weyl chamber flow, thus, for the action of $A^\\textnormal {reg}$ .",
"The application of $a\\in A^\\textnormal {reg}$ on a Weyl chamber $\\Gamma gM$ causes motion in each space dimension of the flat that contains $\\Gamma gM$ .",
"In stark contrast, for $a\\in A\\setminus A^\\textnormal {reg}$ , there is no motion in some space dimensions.",
"We allow non-discrete intersections in these dimensions.",
"Since a set $\\widehat{C}$ as in Definition REF is not a cross section in the classical sense, one might want to call it a “cross section for the returning parts of the Weyl chamber flow in the regular time directions.” We now turn to the definition of a first return map for a cross section of the Weyl chamber flow, where we aim to preserve the idea that this map should be given as follows.",
"We pick a Weyl chamber $\\Gamma gM$ in $\\widehat{C}$ and consider the oriented flat $F = \\Gamma gAM$ that contains $\\Gamma gM$ .",
"We move along $F$ in the direction given by $gM$ , starting at $\\Gamma gM$ and ask for the “next” intersection of $F$ with $\\widehat{C}$ , say $\\Gamma g^{\\prime }M$ .",
"The first return map should map $\\Gamma gM$ to $\\Gamma g^{\\prime }M$ .",
"Moving in the direction $\\nu (gM)$ is the same as restricting the flow to the positive Weyl chamber $A^+$ (that deduces with the choice of the reference Weyl chamber $c_0$ , see Section REF ).",
"However, if the rank $r$ of $\\mathbb {Y}$ is larger than 1, then $A^+$ has $r$ time parameters and hence there does not need to be a well-defined “first” next intersection.",
"We overcome this issue with Definition REF below, for which we start with a brief preparation.",
"The positive Weyl chamber $A^+$ can be parametrized by an open cone in $(\\mathbb {R}_{>0})^r$ , that is, by a convex subset $\\tau ^+$ of $(\\mathbb {R}_{>0})^r$ such that for each $t=(t_1,\\ldots , t_r)\\in \\tau ^+$ the whole open ray $\\mathbb {R}_{>0}\\cdot t = \\lbrace (ct_1,\\ldots ,ct_r) \\mid c>0\\rbrace $ is contained in $\\tau ^+$ .",
"We fix such a parametrization $\\tau ^+ \\rightarrow A^+\\,,\\quad t\\mapsto a_t\\,.$ Definition 2.2 Let $\\widehat{C}\\subseteq \\Gamma \\backslash G/M$ be a cross section for the Weyl chamber flow on $\\mathbb {Y}$ .",
"We say that $\\Gamma gM \\in \\widehat{C}$ has a future intersection with $\\widehat{C}$ if $\\Gamma gA^+M\\cap \\widehat{C} \\ne \\emptyset \\,.$ In this case, let $T\\lbrace t = (t_1,\\ldots , t_r)\\in \\tau ^+ \\mid \\Gamma ga_tM \\in \\widehat{C}\\rbrace $ be the set of time vectors of the future intersections.",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , let $\\operatorname{pr}_j \\colon \\mathbb {R}^r \\rightarrow \\mathbb {R}\\,, (t_1,\\ldots , t_r) \\mapsto t_j\\,,$ be the projection on the $j$ -th component, and set $T_j &\\operatorname{pr}_j(T)\\\\& \\ = \\lbrace t_j \\mid \\exists \\, t_1,\\ldots , t_{j-1},t_{j+1},\\ldots , t_r\\colon (t_1,\\ldots , t_r) \\in T\\rbrace \\,.$ We say that $\\Gamma gM$ has a next intersection with $\\widehat{C}$ if for all $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , $t_{0,j}\\min T_j$ exists, $t_0 \\bigl (t_{0,1},\\ldots , t_{0,r}\\bigr ) \\in \\tau ^+\\,$ and $\\Gamma g a_{t_0} M \\in \\widehat{C}\\,.$ In this case, we call $t_0$ the first return time vector of $\\Gamma gM$ .",
"Let $\\widehat{C}_1$ be the subset of $\\widehat{C}$ for which the first return vector exists.",
"The first return map is the map $\\widehat{R}\\colon \\widehat{C}_1 \\rightarrow \\widehat{C}\\,,\\quad \\widehat{v}=\\Gamma g M \\mapsto \\Gamma ga_{t(\\widehat{v})} M\\,,$ where $t(\\widehat{v})$ is the first return time vector of $\\widehat{v}$ .",
"For the applications that motivate this article we need to be able to (semi-)conjugate the first return map to a function on the Furstenberg boundary of $\\mathbb {X}$ .",
"In what follows we present the necessary structures.",
"Let $\\widehat{C}\\subseteq \\Gamma \\backslash G/M$ be a cross section for the Weyl chamber flow of $\\mathbb {Y}$ .",
"To determine the subset of $\\widehat{C}$ on which the first return map $\\widehat{R}$ becomes a self-map, we define iteratively for $n\\in \\mathbb {N}$ , $n\\ge 2$ , the sets $\\widehat{C}_2 R\\bigl (\\widehat{C}_1\\bigr ) \\cap \\widehat{C}_1\\,,\\quad \\widehat{C}_3R\\bigl (\\widehat{C}_2\\bigr )\\cap \\widehat{C}_1\\,,\\quad \\ldots \\,,$ hence $\\widehat{C}_{n} R\\bigl (\\widehat{C}_{n-1}\\bigr ) \\cap \\widehat{C}_{1}\\qquad \\text{for$n\\in \\mathbb {N}$, $n\\ge 2$}\\,,$ where $\\widehat{C}_1$ was defined in Definition REF .",
"Then $\\widehat{C}_\\mathnormal {st}\\bigcap _{n\\in \\mathbb {N}} \\widehat{C}_n$ is the subset of $\\widehat{C}$ that consists of all those Weyl chambers in $\\widehat{C}$ that yield an infinite sequence of successive next intersections with $\\widehat{C}$ .",
"For each element in $\\widehat{C}_\\mathnormal {st}$ , the next intersection is obviously also contained in $\\widehat{C}_\\mathnormal {st}$ .",
"Thus, if $\\widehat{C}_\\mathnormal {st}$ is nonempty, then the first return map $\\widehat{R}$ restricts to a self-map of $\\widehat{C}_\\mathnormal {st}$ : $\\widehat{R}\\colon \\widehat{C}_\\mathnormal {st}\\rightarrow \\widehat{C}_\\mathnormal {st}\\,.$ The set $\\widehat{C}_\\mathnormal {st}$ may constitute a cross section on its own, in which case we call it the strong cross section contained in $\\widehat{C}$ .",
"(The subscript $\\mathnormal {st}$ refers to “strong” and is used here for the same motivation as in [19].)",
"We say that $C\\subseteq G/M$ is a set of representatives for $\\widehat{C}$ if the quotient map $\\pi ^\\Gamma \\colon G/M \\rightarrow \\Gamma \\backslash G/M\\,,\\quad gM \\mapsto \\Gamma gM\\,,$ restricts to a bijection between $C$ and $\\widehat{C}$ .",
"(We use $\\pi ^\\Gamma $ to denote both the map in (REF ) and the map in (REF ).",
"This double use is motivated by the joint property of these maps to project to $\\Gamma $ -equivalence classes.",
"The context will always clarify which instance of $\\pi ^\\Gamma $ is used.)",
"If $C$ is any set of representatives for $\\widehat{C}$ and $C_1$ is the subset of $C$ that corresponds to $\\widehat{C}_1$ , then the first return map $\\widehat{R}$ induces a map $R\\colon C_1\\rightarrow C$ which makes the diagram ${C_1 [r]^{R} [d]_{\\pi ^\\Gamma } & C [d]^{\\pi ^\\Gamma }\\\\\\widehat{C}_1 [r]^{\\widehat{R}} & \\widehat{C}}$ commutative.",
"If $C_\\mathnormal {st}$ denotes the subset of $C$ that corresponds to $\\widehat{C}_\\mathnormal {st}$ , then $R$ restricts to a self-map of $C_\\mathnormal {st}$ .",
"We recall the map $\\nu \\colon G/M\\rightarrow G/P$ from (REF ) that projects the Weyl chambers of $\\mathbb {X}$ to the points in the Furstenberg boundary that are identified with their equivalence class of asymptotic Weyl chambers (and hence, in a certain sense, the direction of the Weyl chamber).",
"For a well-chosen pair $(\\widehat{C}, C)$ one may find a (unique) map $F\\colon \\nu (C_1) \\rightarrow \\nu (C)$ such that the diagram ${C_1 [r]^R [d]_{\\nu } & C [d]^{\\nu }\\\\\\nu (C_1) [r]^{F} & \\nu (C)}$ commutes.",
"In this case and if $C_\\mathnormal {st}\\ne \\emptyset $ , $F$ restricts to a self-map of $\\nu (C_\\mathnormal {st})$ .",
"If $\\widehat{C}_\\mathnormal {st}$ is a cross section, we call $F\\colon \\nu (C_\\mathnormal {st}) \\rightarrow \\nu (C_\\mathnormal {st})$ the discrete dynamical system on the Furstenberg boundary induced by $(\\widehat{C}, C)$ .",
"Typically, the map $F$ is piecewise given by the action of certain elements from $\\Gamma $ on subsets of $G/P$ .",
"The orbits of Weyl chambers $\\Gamma gM$ under the first return map $\\widehat{R}$ relate to orbits of $F$ and hence to sequences of the acting elements from $\\Gamma $ .",
"These sequences are often called coding sequences for the oriented flats or Weyl chambers, and the shift along coding sequences provides a symbolic dynamics for the Weyl chamber flow.",
"We remark that the set $C$ completely determines the cross section $\\widehat{C}$ .",
"Therefore, for constructions of cross section we may start by finding a “nice” set $C$ and define a cross section as $\\pi ^\\Gamma (C)$ .",
"We also note that the knowledge of $C$ is sufficient to determine the discrete dynamical system on the Furstenberg boundary.",
"Thus, also this is induced by $C$ alone.",
"In Sections and we will present for a certain class of locally symmetric spaces cross sections for their Weyl chamber flows as well as codings and associated discrete dynamical systems"
],
[
"Schottky surfaces",
"The locally symmetric spaces for which we will demonstrate the existence of cross sections for the Weyl chamber flow and induced discrete dynamical systems in the sense of Section REF are product spaces of Schottky surfaces.",
"Schottky surfaces are certain hyperbolic surfaces, hence locally symmetric spaces of rank one.",
"Therefore the Weyl chamber flow on Schottky surfaces coincides with the geodesic flow, and is a well-studied object.",
"The classical Koebe–Morse method gives rise to cross sections and codings for the geodesic flow on Schottky surfaces, which have already been used for many different purposes.",
"Also we will take advantage of these results for our constructions in Sections and .",
"In this section, we briefly present these classical results, with an emphasis on the dynamical aspects.",
"We refer to [2] for details and proofs.",
"The Riemannian symmetric space we consider in this section is the hyperbolic plane.",
"We will use throughout the upper half-plane model $2 \\lbrace z\\in \\operatorname{Im}z > 0\\rbrace \\,,\\quad ds^2_z\\frac{dz\\,d\\overline{z}}{(\\operatorname{Im}z)^2}\\,,$ where $\\operatorname{Im}z$ denotes the imaginary part of $z\\in .",
"We identify the identitycomponent~$ G$ of the group of Riemannian isometries of~$ with the Lie group $\\operatorname{PSL}_2(\\mathbb {R}) = \\operatorname{SL}_2(\\mathbb {R})/\\lbrace \\pm I\\rbrace $ , where $I$ denotes the identity matrix in $\\operatorname{SL}_2(\\mathbb {R})$ .",
"We denote an element of $G=\\operatorname{PSL}_2(\\mathbb {R})$ by $\\begin{bmatrix} a&b\\\\c&d\\end{bmatrix}$ if it is represented by the matrix $\\left({\\begin{matrix} a&b \\\\ c&d\\end{matrix}}\\right)\\in \\operatorname{SL}_2(\\mathbb {R})$ .",
"With respect to this identification, $G$ acts on $ by fractional lineartransformations.",
"Thus,$$g(z) = \\frac{az+b}{cz+d}$$for all $ g=[ ab cd ]G$, $ z.",
"As origin of $ wepick~$ i$, as reference flat the imaginary axis $ iR>0$ and as reference Weylchamber the upper half of the reference flat, thus $ i(1,)$.",
"Thestabilizer group of~$ i$ is$$K = \\operatorname{PSO}(2) = \\operatorname{SO}(2)/\\lbrace \\pm I\\rbrace \\,,$$the maximal abelian subgroup~$ A$ of~$ G$ is$$A = \\left\\lbrace a_t \\begin{bmatrix} e^{t/2}&0\\\\0&e^{-t/2}\\end{bmatrix} \\ \\left|\\ t\\in \\mathbb {R}\\vphantom{\\begin{bmatrix} e^{t/2}&0\\\\0&e^{-t/2}\\end{bmatrix}}\\right.\\right\\rbrace \\,,$$and the positive Weyl chamber in~$ A$ is$$A^+ = \\lbrace a_t \\mid t>0\\rbrace \\,.$$The centralizer of~$ A$ in~$ K$ is the trivial group~$ M = { id}$, andthe normalizer of~$ A$ in~$ K$ is$$M^{\\prime } = \\left\\lbrace \\operatorname{id}, \\begin{bmatrix} 0&1\\\\-1&0\\end{bmatrix} \\right\\rbrace \\,.$$$ We may identify the reference Weyl chamber with the unit tangent vector at $i$ that is tangent to $i[1,\\infty )$ (i.e., the vector at $i$ that points upwards).",
"The unipotent subgroup is $N = \\left\\lbrace \\begin{bmatrix} 1&x\\\\0&1\\end{bmatrix} \\ \\left|\\ x\\in \\mathbb {R}\\vphantom{\\begin{bmatrix} 1&x\\\\0&1\\end{bmatrix}} \\right.\\right\\rbrace \\,.$ Hence, the associated minimal parabolic subgroup $P=NAM$ is the subgroup of $G$ that is represented by upper triangular matrices in $\\operatorname{SL}_2(\\mathbb {R})$ .",
"The Furstenberg boundary $G/P$ coincides with the geodesic boundary of $, and the action of $ G$ on $ G/K$ extends continuouslyto~$ G/P$.",
"We may identify the Furstenberg boundary~$ G/P$ with $ P1R= R{}$ by means of the isomorphism$$G/P \\rightarrow P^1_\\mathbb {R}\\,,\\quad gP \\mapsto g(\\infty )\\,,$$where$$g(\\infty ) = \\frac{a}{c}$$for $ g=[ ab cd ]G$, using the convention $ a/0 = $.$ A Schottky surface is a hyperbolic surface of infinite area without cusps and conical singularities.",
"Each Schottky surface (and only those) arises from the following construction.",
"We pick $q\\in \\mathbb {N}$ and fix $2q$ closed Euclidean disks in $ that are centered in~$ R$ and that are pairwise disjoint.",
"We fixa pairing of these disks, that we shall indicate by indices with oppositesigns.",
"Let$$\\mathcal {D}_1, \\mathcal {D}_{-1},\\,\\ldots \\,, \\mathcal {D}_q, \\mathcal {D}_{-q}$$be the chosen disks.",
"For $ k{1,..., q}$ we pick an element~$ gkG$that maps the exterior of the disk~$ Dk$ to the interior of thedisk~$ D-k$.",
"The subgroup~$$ of~$ G$ generated by theelements~$ g1,..., gq$,$$\\Gamma = \\langle g_1,\\ldots , g_q\\rangle \\,,$$is a \\emph {Schottky group}, and the hyperbolic surface $ Y is a Schottky surface.",
"The complement of the union of the disks in $,$$\\mathcal {F} \\bigcup _{k=1}^q \\bigl ( \\mathcal {D}_{k} \\cup \\mathcal {D}_{-k}\\bigr )\\,,$$is an open fundamental domain for~$ Y$, all of whose sides in~$ are given by geodesics (namely the part of the boundary of the disks that is in $).",
"Its side-pairings are given by the elements~$ g1,..., gk$.$ In order to provide a cross section for the geodesic flow on $\\mathbb {Y}$ , we set $\\mathcal {I} \\lbrace \\pm 1,\\ldots , \\pm q\\rbrace $ and, for any $k\\in \\mathcal {I}$ , let $s_k$ be the boundary of $\\mathcal {D}_k$ in $, and $ Ik$ denote the part of theFurstenberg boundary~$ G/P = R{}$ that is exterior to~$ Dk$.",
"Werefer to~$ Ik$ as \\emph {forward interval}.",
"We recall the map~$$from~(\\ref {eq:combmap}) and set$$C_k \\alpha ^{-1}(s_k\\times I_k)\\,.$$The set~$ Ck$ may be identified with the set of unit tangent vectors of~$ that are based on $s_k$ and “point into” the fundamental domain $\\mathcal {F}$ .",
"We set $C \\bigcup _{k\\in \\mathcal {I}} C_k\\qquad \\text{and}\\qquad \\widehat{C}\\pi ^\\Gamma (C)\\,.$ The following statement is immediately implied from the definitions of $\\widehat{C}$ and $C$ and the property of $\\mathcal {F}$ to be geodesically convex.",
"It is also an immediate consequence of the Koebe–Morse method, which discusses instead the neighboring $\\Gamma $ -translates of $\\mathcal {F}$ .",
"Proposition 3.1 The set $\\widehat{C}$ is a cross section for the geodesic flow on the Schottky surface $\\Gamma \\backslash , and $ C$ is a set of representatives for~$ C$.$ For the presentation of the induced discrete dynamical system, we restrict the consideration to the strict cross section contained to $\\widehat{C}$ .",
"To that end we denote by $L$ the limit set of $\\Gamma $ , that is the set of limit points of the $\\Gamma $ -orbit $\\Gamma (z)$ in the Furstenberg boundary $P^1_\\mathbb {R}$ , where $z$ is any point of $.",
"We let\\begin{equation}C_{\\mathnormal {st}} \\lbrace c\\in C \\mid \\nu (c) \\in L \\rbrace \\end{equation}be the subset of Weyl chambers (or unit tangent vectors) in~$ C$ that projectto the limit set~$ L$, and we set\\begin{equation}\\widehat{C}_\\mathnormal {st}\\pi ^\\Gamma (C_\\mathnormal {st})\\,.\\end{equation}(For convenience we allow ourselves here this slight abuse of notation: we havenot yet shown that $ Cst$ coincides with the set definedin~(\\ref {eq:crossstrong}).",
"This will be done inProposition~\\ref {prop:cross_schottky}.",
")For $ kI$ we set$$I_{\\mathnormal {st},k}^c L \\cap (P^1_\\mathbb {R}\\setminus I_k)\\,,$$which is the part of the limit set~$ L$ contained in the interior of thedisk~$ Dk$.",
"We call~$ Ist,kc$ a \\emph {strong coding set}, awording whose meaning will become clear in Section~\\ref {sec:coding}.",
"Then $ L$is the disjoint union of these sets:$$L = \\bigcup _{k\\in \\mathcal {I}} I_{\\mathnormal {st},k}^c\\,.$$We define a self-map $ FLL$ by$$F\\vert _{I_{\\mathnormal {st},k}^c} \\colon I_{\\mathnormal {st},k}^c \\rightarrow L\\,,\\quad x\\mapsto g_{k}(x)\\,,$$for $ kI$.$ As above in Proposition REF , the following statements follow immediately from the definitions of $\\widehat{C}_\\mathnormal {st}$ , $C_\\mathnormal {st}$ and $F$ as well as the properties of limit sets of Schottky groups.",
"For the convenience of the reader, we provide a sketch of the proof.",
"Proposition 3.2 The set $\\widehat{C}_\\mathnormal {st}$ is a cross section for the geodesic flow on the Schottky surface $\\Gamma \\backslash .",
"It is intersected by every geodesicon~$ that is contained in a compact subset of $\\Gamma \\backslash , which may depend\\footnote {Due to the specialstructure of Schottky surfaces, we may choose a uniform compact set, namely thecompact core of~\\Gamma \\backslash .}",
"on the considered geodesic.",
"It is thestrong cross section contained in~$ C$.",
"The set~$ Cst$ is a set ofrepresentatives for~$ Cst$, and the map $ F$ is the discrete dynamicalsystem on the Furstenberg boundary that is induced by~$ Cst$.$ We recall that oriented flats on $\\Gamma \\backslash are precisely thegeodesics on~$ .",
"Let $\\widehat{\\gamma }$ be such a geodesic and suppose that $\\gamma $ is one of its representative geodesics on $.",
"If$$ is directed towards a point in the limit set~$ L$, then while movingtowards this points, $$ intersects infinitely many $$-translatesof the fundamental domain~$ F$.",
"In turn, $$ stays ``far away^{\\prime \\prime } fromthe ends of~$ and intersects $\\widehat{C}$ in an unbounded set of times.",
"However, if $\\gamma $ is directed towards a point not in $L$ , then eventually $\\gamma $ will stay in a single $\\Gamma $ -translate of $\\mathcal {F}$ .",
"In turn, $\\widehat{\\gamma }$ will travel to an end of $\\Gamma \\backslash , and willintersect~$ C$ only finitely many times in this direction.",
"From thisdichotomy, one can easily deduce that $ Cst$ as definedin~(\\ref {eq:crossstrong_schottky}) coincides with the set definedin~(\\ref {eq:crossstrong}).",
"We remark that this dichotomy takes advantage ofproperties of limit sets that are rather specific to Schottky groups.$ To show that $\\widehat{C}_\\mathnormal {st}$ is indeed a cross section, we first note that the compact oriented flats on $\\Gamma \\backslash are precisely the periodicgeodesics of~$ .",
"Let $\\widehat{\\gamma }=\\Gamma gAM$ be such a periodic geodesic.",
"By Proposition REF , $\\gamma $ intersects $\\widehat{C}$ at least once, say in $\\widehat{v}$ .",
"Since all intersections of $\\widehat{\\gamma }$ and $\\widehat{C}$ are discrete by Proposition REF , and the group $A$ is one-dimensional, every future intersection with $\\widehat{C}$ is a next intersection.",
"Since $\\widehat{\\gamma }$ is periodic, the intersection in $\\widehat{v}$ will repeatedly be among the future intersections, showing that these exist unboundedly.",
"Thus, $\\widehat{v}$ is in $\\widehat{C}_\\mathnormal {st}$ and $\\widehat{\\gamma }$ intersects $\\widehat{C}_\\mathnormal {st}$ .",
"Obviously, $C_\\mathnormal {st}$ is a set of representatives of $\\widehat{C}_\\mathnormal {st}$ .",
"It remains to indicate why $F$ is the induced discrete dynamical system.",
"Let $\\widehat{v} = \\Gamma g M \\in \\widehat{C}_\\mathnormal {st}$ .",
"The geodesic $\\widehat{\\gamma }$ determined by $\\widehat{v}$ is $\\Gamma g AM$ .",
"Without loss of generality, we may assume that $g\\in G$ is chosen such that $vgM$ is the unique representative of $\\widehat{v}$ in $C_\\mathnormal {st}$ .",
"Then the geodesic $\\gamma $ on $ determined by~$ v$ is~$ gAM$, or,from a more dynamical point of view, the trajectory$$t\\mapsto ga_tM\\,.$$Let$$b_0 \\lim _{t\\rightarrow \\infty } ga_tM = g(\\infty )$$be the point in the Furstenberg boundary of~$ to which $\\gamma $ is oriented (or projects).",
"Due to the relation between $C$ and $\\mathcal {F}$ , the time-minimal intersection between $\\gamma ^+\\lbrace ga_tM \\mid t > 0\\rbrace $ and the $\\Gamma $ -translates of $C$ is located at the (unique) boundary component of $\\mathcal {F}$ through which $\\gamma ^+$ passes.",
"The structure of $\\mathcal {F}$ implies that this intersection is in $g_{-k}(C_{-k})$ for some $k\\in \\mathcal {I}$ if and only if $b_0\\in I_{\\mathnormal {st},-k}^c$ .",
"Suppose that the intersection is in $g_{-k_0}(C_{-k_0})$ at time $t_0>0$ , with $k_0\\in \\mathcal {I}$ .",
"Then the next intersection of $\\widehat{v}$ with $\\widehat{C}_\\mathnormal {st}$ is in $\\Gamma g a_{t_0} M$ , which corresponds via $\\bigl (\\pi ^\\Gamma \\vert _C\\bigr )^{-1}$ to the element $g_{-k_0}^{-1}\\bigl ( ga_{t_0}M \\bigr ) = g_{k_0}\\bigl ( ga_{t_0}M\\bigr )$ of $C_\\mathnormal {st}$ .",
"In turn, $F(b_0) & = F\\bigl ( \\nu (gM) \\bigr ) = \\nu \\bigl ( R(gM) \\bigr ) = \\nu \\bigl (g_{k_0}(ga_{t_0}M) \\bigr ) = g_{k_0}\\nu \\bigl ( ga_{t_0}M \\bigr ) = g_{k_0}(b_0)\\,.$ This shows that $F$ is indeed the induced discrete dynamical system."
],
[
"Cross sections for the Weyl chamber flow on\nproduct spaces",
"Let $r\\in \\mathbb {N}$ .",
"We recall that $2 = denotes the hyperbolic plane andconsider the Riemannian symmetric space$$\\mathbb {X}(2)^r = \\cdots \\times of rank~$ r$, given by the direct product of $ r$ copies of~$ .",
"We identify the identity component of the group of Riemannian isometries of $\\mathbb {X}$ with $G \\operatorname{PSL}_2(\\mathbb {R})^r\\,.$ Then the action of $G$ on $\\mathbb {X}$ is $g(z) = \\bigl ( g_1(z_1),\\ldots , g_r(z_r) \\bigr )$ for all $g=(g_1,\\ldots , g_r)\\in G$ and $z=(z_1,\\ldots , z_r)\\in \\mathbb {X}$ .",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ we choose a (Fuchsian) Schottky group $\\Gamma _j$ in $\\operatorname{PSL}_2(\\mathbb {R})$ and set $\\Gamma \\Gamma _1\\times \\cdots \\times \\Gamma _r\\,.$ In this section we construct a cross section for the Weyl chamber flow on the locally symmetric space $\\mathbb {Y}\\Gamma \\backslash \\mathbb {X}\\,.$ We start with some preparatory considerations.",
"Since $\\mathbb {X}$ as well as $\\mathbb {Y}= \\Gamma _1\\backslash \\cdots \\times \\Gamma _r\\backslash enjoy clear product structures, several of the necessary objects are the directproducts of the analogous objects of the single factors.",
"As originof~$ X$ we choose$$x_0 (i,\\ldots , i) \\in \\mathbb {X}\\,.$$The flats and Weyl chambers of~$ X$ are the direct products of theflats and Weyl chambers of~$ .",
"Therefore we choose $F_0(i\\mathbb {R}_{>0})^r$ as reference flat of $\\mathbb {X}$ , and $c_0(i(1,\\infty ))^r$ as reference Weyl chamber of $\\mathbb {X}$ , which are the direct products of our chosen reference flat and reference Weyl chamber of $.$ In Section we discussed the groups and maps associated to our choices of reference objects of $.",
"In what follows we will use for thesegroups and maps the notation from Section~\\ref {sec:schottky} but with theadditional subscript ``1^{\\prime \\prime }.",
"Thus, $ Gu = PSL2(R)$, $ Ku= PSO(2)$, etc.The subscript-free notation is preserved for the objects relatedto~$ X$ and~$ Y$.",
"The stabilizer group of~$ x0$ in~$ G$ is$$K = \\operatorname{Stab}_G(x_0) = K_u^r = \\operatorname{PSO}(2)^r\\,,$$the maximal abelian subgroup of~$ G$ determined by~$ F0$ is~$ A = Aur$, thepositive Weyl chamber in~$ A$ determined by~$ c0$ is $ A+ = (Au+)r$.",
"Thecentralizer group and normalizer group of~$ A$ in~$ K$ are $ M = Mur$ and$ M'=(M'u)r$, respectively, and the Weyl group is $ W = Wur$.",
"Theunipotent subgroup is $ N = Nur$, and the minimal parabolic subgroup is $ P = Pur$.",
"Thus, the Furstenberg boundary of~$ X$ is$$G/P = G_u/P_u \\times \\cdots \\times G_u/P_u\\,,$$the $ r$-times direct product of the Furstenberg boundary of~$ , which we identify with $\\bigl ( P^1_\\mathbb {R}\\bigr )^r$ via the isomorphism $gP = (g_1P_u,\\ldots , g_rP_u) \\mapsto \\bigl (g_1(\\infty ),\\ldots , g_r(\\infty )\\bigr )\\,.$ For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ we fix a fundamental domain $\\mathcal {F}_j$ for the Fuchsian Schottky group $\\Gamma _j$ in $ of the form as inSection~\\ref {sec:schottky}, arising from the choice of $ qj$ Euclidean disks.We set $ Ij = {1,..., qj}$, let\\begin{equation}g_{j,k}\\,,\\ s_{j,k}\\,,\\ I_{j,k}\\qquad \\text{for $k\\in \\mathcal {I}_j$}\\end{equation}denote the side-pairing elements in~$ j$, the geodesic sides of~$ Fj$, and the forward intervals, respectively.",
"We recall the map~$$from~(\\ref {eq:combmap}) and set$$\\mathcal {J} _{j=1}^r \\mathcal {I}_j\\,.$$For each $ m = (m1,..., mr)J$ we set$$Q_m _{j=1}^r \\bigl ( s_{j,m_j} \\times I_{j,m_j}\\bigr )\\,,\\qquad C_m \\alpha ^{-1}( Q_m )\\,,$$and$$C \\bigcup _{m\\in \\mathcal {J}} C_m\\qquad \\text{and}\\qquad \\widehat{C} \\pi ^\\Gamma (C)\\,,$$where $$ is the map in~(\\ref {eq:projWC}).$ Theorem 4.1 The set $\\widehat{C}$ is a cross section for the Weyl chamber flow on $\\mathbb {Y}$ , and $C$ is a set of representatives for $\\widehat{C}$ .",
"Preparatory for the proof we briefly discuss the product structure of $\\widehat{C}$ and $C$ .",
"To that end, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ and $m_j \\in \\mathcal {I}_j$ , we set $C_{j,m_j} \\alpha _u^{-1}\\bigl ( s_{j,m_j} \\times I_{j,m_j} \\bigr )\\,,$ where $\\alpha _u$ denotes the map in (REF ) for $G_u$ .",
"Further, we set $C_j \\bigcup _{m_j\\in \\mathcal {I}_j} C_{j, m_j}\\qquad \\text{and}\\qquad \\widehat{C}_j \\pi _u^\\Gamma \\bigl ( C_j \\bigr )\\,.$ Then, for any $m = (m_1,\\ldots , m_r)\\in \\mathcal {J}$ , we have $C_m = _{j=1}^r C_{j,m_j}\\,,$ and further $\\widehat{C} = _{j=1}^r \\widehat{C}_j\\,.$ For each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , the set $\\widehat{C}_j$ is a cross section for the Weyl chamber flow (geodesic flow) on the Schottky surface $\\Gamma _j\\backslash with $ Cj$ as set of representatives byProposition~\\ref {prop:cross_schottky}.$ Let $\\Gamma g AM$ be an oriented compact flat of $\\mathbb {Y}$ .",
"With $g=(g_1,\\ldots , g_r)$ we have $\\Gamma g AM = (\\Gamma _1 g_1 A_uM_u,\\cdots ,\\Gamma _r g_r A_uM_u)\\,.$ Thus, for each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , $\\gamma _j \\Gamma _j g_j A_uM_u$ is an oriented flat (oriented geodesic) on $\\Gamma _j\\backslash that iscontained in a compact subset of~$ j.",
"It is even a periodic geodesic.",
"Thus, $\\gamma _j$ intersects $\\widehat{C}_j$ by Proposition REF , say in $\\Gamma _jg_ja_jM_u$ .",
"Then $\\Gamma g AM$ intersects $\\widehat{C}$ in $\\Gamma g (a_1,\\ldots , a_r)M$ .",
"This establishes REF for $\\widehat{C}$ .",
"In order to show REF let $\\Gamma gM\\in \\widehat{C}$ and suppose that $g=(g_1,\\ldots , g_r)$ .",
"Then $\\Gamma g M = (\\Gamma _1 g_1 M_u,\\ldots , \\Gamma _r g_r M_u) \\in \\widehat{C}_1 \\times \\cdots \\times \\widehat{C}_r\\,.$ Thus, for each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , the Weyl chamber (unit tangent vector) $\\Gamma _j g_j M_u$ is in the cross section $\\widehat{C}_j$ for the geodesic flow on $\\Gamma _j\\backslash .",
"Hence we find $ j>0$ such that for all$ t(-j,j)$, $ t=0$,$$\\Gamma _j g_j a_t M_u \\notin \\widehat{C}_j\\,,$$where$$a_t = \\begin{bmatrix} e^{t/2}&0\\\\0&e^{-t/2}\\end{bmatrix} \\in G_u = \\operatorname{PSL}_2(\\mathbb {R})\\,.$$For$$U (-\\varepsilon _1,\\varepsilon _1)\\times \\cdots \\times (-\\varepsilon _r,\\varepsilon _r)$$we have that each element $ aUAreg$ is of the form$$a = (a_{t_1},\\ldots , a_{t_r})$$with $ tj(-j, j)$, $ tj=0$, for $ j{1,..., r}$, andhence$$\\Gamma ga M = \\bigl ( \\Gamma _1g_1a_{t_1}M_u, \\ldots , \\Gamma _rg_ra_{t_r}M_u\\bigr )\\notin \\widehat{C}\\,.$$This establishes~(\\ref {cs2}) for~$ C$ and finishes the proof that~$ C$ isa cross section for the Weyl chamber flow on~$ Y$.",
"Finally for each$ j{1,..., r}$, the set $ Cj$ is a set of representatives for~$ Cj$.",
"This property is stable under direct products, and hence $ C$ is indeeda set of representatives for~$ C$.$ We briefly discuss an aspect of the structure and thickness of $\\widehat{C}$ .",
"For each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ the set of base points of the cross section $\\widehat{C}_j$ is the full boundary of the fundamental domain $\\mathcal {F}_j$ .",
"A fundamental domain for $\\mathbb {Y}= \\Gamma \\backslash \\mathbb {X}$ is given by $\\mathcal {F} _{j=1}^r \\mathcal {F}_j\\,.$ The set of base points of $\\widehat{C}$ , however, is only a rather sparse subset of the boundary of $\\mathcal {F}$ , getting sparser if the rank becomes larger.",
"This shows that the cross section $\\widehat{C}$ is not implied by a Koebe–Morse method, but is a genuinely related to the Weyl chamber bundle.",
"In a way similar to the construction of $\\widehat{C}$ we can find the strong cross section contained in $\\widehat{C}$ .",
"To that end, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , we let $L_j$ denote the limit set of $\\Gamma _j$ (in the rank one situation), set $C_{\\mathnormal {st}, j} \\lbrace c\\in C_j \\mid \\nu _1(c) \\in L_j\\rbrace $ as well as $\\widehat{C}_{\\mathnormal {st}, j} \\pi _u^\\Gamma \\bigl ( C_{\\mathnormal {st}, j}\\bigr )\\,.$ As stated in Proposition REF , $\\widehat{C}_{\\mathnormal {st}, j}$ is the strong cross section contained in $\\widehat{C}_j$ for the geodesic flow on $\\Gamma _j\\backslash .",
"We further define$$C_\\mathnormal {st}_{j=1}^k C_{\\mathnormal {st},j}\\qquad \\text{and}\\qquad \\widehat{C}_\\mathnormal {st}_{j=1}^r \\widehat{C}_{\\mathnormal {st},j}\\,.$$Then $ Cst= (Cst)$.$ Theorem 4.2 The set $\\widehat{C}_\\mathnormal {st}$ is the strong cross section contained in $\\widehat{C}$ , and $C_\\mathnormal {st}$ is a set of representatives.",
"These statements can be proven analogously to those in Theorem REF , using Proposition REF instead of Proposition REF .",
"Codings and discrete dynamical systems on Furstenberg boundary In this section we present the discrete dynamical system on the Furstenberg boundary that is induced by the set of representatives $C_\\mathnormal {st}$ for the strong cross section $\\widehat{C}_\\mathnormal {st}$ from Section .",
"We resume the notation from Section .",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , we denote the strong coding sets determined by the choice of the fundamental domain $\\mathcal {F}_j$ for $\\Gamma _j$ by $I_{\\mathnormal {st},j,k}^c\\qquad \\text{for $k\\in \\mathcal {I}_j$}.$ For $m\\in \\mathcal {J} = _{j=1}^k \\mathcal {I}_j$ , $m=(m_1,\\ldots , m_r)$ , we set $I_{\\mathnormal {st}, m}^c _{j=1}^r I_{\\mathnormal {st}, j, m_j}^c$ and $D \\bigcup _{m\\in \\mathcal {J}} I_{\\mathnormal {st},m}^c\\,.$ We define the map $F\\colon D\\rightarrow D$ as follows: for $m\\in \\mathcal {J}$ , $m=(m_1,\\ldots , m_r)$ , we set $g_m \\bigl (g_{1,m_1},\\ldots , g_{r,m_r}\\bigr )\\,,$ where $g_{j,m_j}$ are the side-pairing elements of $\\mathcal {F}_j$ (see ()).",
"Restricted to the subset $I_{\\mathnormal {st},m}^c$ of $D$ , the map $F$ is $F\\vert _{I_{\\mathnormal {st},m}^c}\\colon I_{\\mathnormal {st},m}^c\\rightarrow D\\,,\\quad x\\mapsto g_m(x)\\,.$ Further, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ let $F_j\\colon L_j\\rightarrow L_j$ denote the discrete dynamical system induced on the Furstenberg boundary $P^1_\\mathbb {R}$ of $ induced by the cross section~$ Cst,j$ and its set of representatives~$ Cst,j$ for the geodesic flowon~$ j. Theorem 5.1 The map $F$ is the discrete dynamical system on Furstenberg boundary that is induced by $C_{\\mathnormal {st}}$ .",
"For any $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we have $F\\vert _{I_{\\mathnormal {st}, m}^c} = \\left( F_1\\vert _{I_{\\mathnormal {st},1,m_1}^c},\\ldots ,F_r\\vert _{I_{\\mathnormal {st},r,m_r}^c} \\right)\\,.$ The statement in (REF ) follows immediately from the definition of the map $F$ .",
"To establish (REF ), we recall that the $r$ -dimensional Weyl chamber flow on $\\mathbb {Y}$ is the direct product of the 1-dimensional geodesic flows on the factors $\\Gamma _j\\backslash , $ j{1,...,r}$,of~$ Y$.",
"We recall further that the cross section~$ Cst$ and itsset of representatives~$ Cst$ are direct products of the cross sections andsets of representatives for these factors.",
"Therefore, this product structuredescends to the induced discrete dynamical systems on Furstenberg boundary,which yields~(\\ref {mapi}) due to the identity in~(\\ref {mapii}).$ Transfer operators In this final section we propose a definition of a transfer operator family for the multi-dimensional discrete dynamical system $F$ in Section .",
"We continue to use the notation from Sections and and start by presenting the well-known definition of transfer operator families for one-dimensional flows, specialized to our setup.",
"Let $j\\in \\lbrace 1,\\ldots , r\\rbrace $ and recall from (REF ) and Section the discrete dynamical system $F_j\\colon L_j\\rightarrow L_j$ induced by the set of representatives $C_j$ of the cross section $\\widehat{C}_j$ for the geodesic flow on the Schottky surface $\\Gamma _j\\backslash .",
"TheRuelle-type \\emph {transfer operator}~$ Lj,s$ with parameter~$ s associated to $F_j$ is (at least initially) an operator on the space of functions $\\psi \\colon L_j\\rightarrow , given by\\begin{equation}\\mathcal {L}_{j,s}\\psi (x) \\sum _{y\\in F_j^{-1}(x)} | F^{\\prime }_j(y)|^{-s} \\psi (y)\\qquad (x\\in L_j)\\,.\\end{equation}Here, the derivative of~$ F'j$ at~$ y$ is understood as follows.",
"The point~$ y$is contained in~$ Ist,j,kc$ for a unique~$ kIj$.",
"Then~$ Fj$ actson a small neighborhood of~$ y$ in~$ Lj$ by the fractional lineartransformation~$ gj,k$, which extends to an analytic map in a smallneighborhood of~$ y$ in~$ R$.",
"We use the derivative of this extended mapfor~$ F'j(y)$.$ The function space which one uses as domain for the transfer operator $\\mathcal {L}_{j,s}$ depends on its further applications.",
"One may choose spaces of functions with larger domain or with some regularity properties.",
"We will refrain here from these discussions and will use the space of functions on the limit sets as place holder.",
"We shall now provide another presentation of $\\mathcal {L}_{j,s}$ that takes advantage of the explicit description of $F_j$ .",
"To that end we note that $F_j$ restricts to the bijections $I_{\\mathnormal {st},j,k}^c \\rightarrow L_j\\setminus I_{\\mathnormal {st},j,-k}^c\\,,\\quad x\\mapsto g_{j,k}(x)\\,,$ for each $k\\in \\mathcal {I}_j$ .",
"Thus, for each $k\\in \\mathcal {I}_j$ , each $x\\in I_{\\mathnormal {st},j,k}^c$ has the ($|\\mathcal {I}_j|-1$ ) preimages $\\lbrace g_{j,\\ell }^{-1}(x) \\mid \\ell \\in \\mathcal {I}_j, \\ell \\ne -k\\rbrace \\,.$ For any function $\\psi \\colon L_j\\rightarrow , we set$$\\psi _{k} \\psi \\cdot 1_{I_{\\mathnormal {st},j,k}^c}\\qquad \\text{for $k\\in \\mathcal {I}_j$}\\,,$$where $ 1A$ denotes the characteristic function of the set~$ A$.",
"Then$$\\psi = \\sum _{k\\in \\mathcal {I}_j} \\psi _k\\,.$$Further, for $ hj$, $ s, any subset $A\\subseteq \\mathbb {R}$ and any function $\\varphi \\colon A\\rightarrow we set\\begin{equation}\\tau _s(h^{-1})\\varphi (x) \\bigl (h^{\\prime }(x)\\bigr )^s \\varphi \\bigl ( h(x)\\bigr )\\qquad (x\\in A)\\,,\\end{equation}whenever it is well-defined (as it will be in all our applications).",
"For each$ kIj$ we have then$$\\bigl (\\mathcal {L}_{j,s}\\psi \\bigr )_k = \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}}\\tau _s\\bigl (g_{j,\\ell }\\bigr )\\psi _\\ell $$or, in a more compact form,$$\\mathcal {L}_{j,s} = \\sum _{k\\in \\mathcal {I}_j} 1_{I_{\\mathnormal {st},j,k}^c} \\cdot \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}} \\tau _s\\bigl (g_{j,\\ell }\\bigr )\\,.$$$ For the transfer operator family associated to the multi-dimensional map $F$ we propose a definition analogous to those in () but allowing a multi-dimensional parameter $s\\in r$ .",
"(We recall that the rank of the considered Riemannian locally symmetric space $\\mathbb {Y}$ is $r$ .)",
"We first consider the parameter-free transfer operator $\\mathcal {L}f(x) \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-1} f(y)\\,,$ acting on functions $f\\colon D\\rightarrow , where the derivative $ F'$ is understoodanalogously to above and $ |F'(y)|$ is the absolute value of the determinant ofthe linear map $ F'(y)$.",
"For any $ yD$ we find a unique element $ mJ$,$ m=(m1,..., mr)$, such that $ yIst,mc$.",
"Letting$$y = (y_1,\\ldots , y_r)\\qquad \\text{and}\\qquad g_m = \\bigl ( g_{1,m_1},\\ldots ,g_{r,m_r} \\bigr )$$we then have $ F=gm$ is a small neighborhood of~$ y$ in~$ D$.",
"Thus,$$F(y) = g_m(y) = \\bigl ( g_{1,m_1}(y_1),\\ldots , g_{r,m_r}(y_r) \\bigr )$$and the Jacobi matrix of~$ F$ at~$ y$ is the diagonal matrix\\begin{equation}J_F(y) =\\begin{pmatrix}g_{1,m_1}^{\\prime }(y_1)\\\\& \\ddots \\\\& & g_{r,m_r}^{\\prime }(y_r)\\end{pmatrix}\\,.\\end{equation}Thus,\\begin{equation}|F^{\\prime }(y)|^{-1} = |\\det J_F(y)|^{-1} = \\prod _{j=1}^r\\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-1}\\,.\\end{equation}Motivated by the diagonal structure of the Jacobi matrix in~(\\ref {eq:JM}), wepropose to endow each non-zero entry separately with a weight.",
"Thus, the\\emph {parametrized transfer operator}~$ Ls$ with $ s=(s1,..., sr)r$ is defined as\\begin{equation}\\mathcal {L}_sf(x) = \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-s} f(y)\\,,\\end{equation}where$$|F^{\\prime }(y)|^{-s} \\prod _{j=1}^r \\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-s_j}$$in the notation from~(\\ref {eq:det_unweight}).",
"This use of the parameter alsoreflects well the independence of the $ r$ dimensions of the oriented flatsof~$ Y$.$ The analogy between the transfer operators in () and () goes further.",
"For $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we set $B(m) \\lbrace n=(n_1,\\ldots , n_r)\\in \\mathcal {J} \\mid \\exists \\, j\\in \\lbrace 1,\\ldots ,r\\rbrace \\colon n_j = -m_j\\rbrace \\,.$ The map $F\\colon D\\rightarrow D$ restricts to the bijections $I_{\\mathnormal {st},m}^c \\rightarrow D\\setminus \\!\\!\\bigcup _{n\\in B(m)}I_{\\mathnormal {st},n}^c\\,,\\quad x\\mapsto g_m(x)\\,.$ In analogy to (), we define for $h=(h_1,\\ldots , h_n)\\in \\Gamma $ , $s=(s_1,\\ldots , s_r)\\in r$ , any subset $A\\subseteq \\mathbb {R}^r$ and any function $\\varphi \\colon A\\rightarrow ,{\\begin{@align}{1}{-1}\\omega _s(h^{-1})\\varphi (x) & = \\bigl | h^{\\prime }(y) \\bigr |^{-s} \\varphi \\bigl (h(y)\\bigr )\\\\& = \\bigl |h^{\\prime }_1(y_1)\\bigr |^{-s_1} \\cdots \\bigl |h^{\\prime }_r(y_r)\\bigr |^{-s_r}\\varphi \\bigl (h(y)\\bigr )\\,.",
"\\nonumber \\end{@align}}Then\\begin{equation}\\mathcal {L}_s = \\sum _{m\\in \\mathcal {J}} 1_{I_{\\mathnormal {st},m}^c} \\cdot \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\n\\notin B(m)\\end{array}} \\omega _s(g_n)\\,,\\end{equation}or, if we set$$f_m f\\cdot 1_{I_{\\mathnormal {st},m}^c} \\qquad (m\\in \\mathcal {J})$$for any function $ fD, then $(\\mathcal {L}_sf)_m = \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\ n\\notin B(m)\\end{array}} \\omega _s(g_n)f_n\\,.$ We remark that this multi-parameter transfer operator for the Weyl chamber flow on $\\mathbb {Y}$ is not the direct sum of the one-parameter transfer operators for the geodesic flow on the Schottky surfaces from which $\\mathbb {Y}$ is build.",
"This is consistent with the fact that the spectral theory of $\\mathbb {Y}$ is not just the “direct product” of the spectral theories of its factors.",
"We leave any further investigation into this transfer operator family for future work.",
"Let $r\\in \\mathbb {N}$ .",
"We recall that $2 = denotes the hyperbolic plane andconsider the Riemannian symmetric space$$\\mathbb {X}(2)^r = \\cdots \\times of rank~$ r$, given by the direct product of $ r$ copies of~$ .",
"We identify the identity component of the group of Riemannian isometries of $\\mathbb {X}$ with $G \\operatorname{PSL}_2(\\mathbb {R})^r\\,.$ Then the action of $G$ on $\\mathbb {X}$ is $g(z) = \\bigl ( g_1(z_1),\\ldots , g_r(z_r) \\bigr )$ for all $g=(g_1,\\ldots , g_r)\\in G$ and $z=(z_1,\\ldots , z_r)\\in \\mathbb {X}$ .",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ we choose a (Fuchsian) Schottky group $\\Gamma _j$ in $\\operatorname{PSL}_2(\\mathbb {R})$ and set $\\Gamma \\Gamma _1\\times \\cdots \\times \\Gamma _r\\,.$ In this section we construct a cross section for the Weyl chamber flow on the locally symmetric space $\\mathbb {Y}\\Gamma \\backslash \\mathbb {X}\\,.$ We start with some preparatory considerations.",
"Since $\\mathbb {X}$ as well as $\\mathbb {Y}= \\Gamma _1\\backslash \\cdots \\times \\Gamma _r\\backslash enjoy clear product structures, several of the necessary objects are the directproducts of the analogous objects of the single factors.",
"As originof~$ X$ we choose$$x_0 (i,\\ldots , i) \\in \\mathbb {X}\\,.$$The flats and Weyl chambers of~$ X$ are the direct products of theflats and Weyl chambers of~$ .",
"Therefore we choose $F_0(i\\mathbb {R}_{>0})^r$ as reference flat of $\\mathbb {X}$ , and $c_0(i(1,\\infty ))^r$ as reference Weyl chamber of $\\mathbb {X}$ , which are the direct products of our chosen reference flat and reference Weyl chamber of $.$ In Section we discussed the groups and maps associated to our choices of reference objects of $.",
"In what follows we will use for thesegroups and maps the notation from Section~\\ref {sec:schottky} but with theadditional subscript ``1^{\\prime \\prime }.",
"Thus, $ Gu = PSL2(R)$, $ Ku= PSO(2)$, etc.The subscript-free notation is preserved for the objects relatedto~$ X$ and~$ Y$.",
"The stabilizer group of~$ x0$ in~$ G$ is$$K = \\operatorname{Stab}_G(x_0) = K_u^r = \\operatorname{PSO}(2)^r\\,,$$the maximal abelian subgroup of~$ G$ determined by~$ F0$ is~$ A = Aur$, thepositive Weyl chamber in~$ A$ determined by~$ c0$ is $ A+ = (Au+)r$.",
"Thecentralizer group and normalizer group of~$ A$ in~$ K$ are $ M = Mur$ and$ M'=(M'u)r$, respectively, and the Weyl group is $ W = Wur$.",
"Theunipotent subgroup is $ N = Nur$, and the minimal parabolic subgroup is $ P = Pur$.",
"Thus, the Furstenberg boundary of~$ X$ is$$G/P = G_u/P_u \\times \\cdots \\times G_u/P_u\\,,$$the $ r$-times direct product of the Furstenberg boundary of~$ , which we identify with $\\bigl ( P^1_\\mathbb {R}\\bigr )^r$ via the isomorphism $gP = (g_1P_u,\\ldots , g_rP_u) \\mapsto \\bigl (g_1(\\infty ),\\ldots , g_r(\\infty )\\bigr )\\,.$ For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ we fix a fundamental domain $\\mathcal {F}_j$ for the Fuchsian Schottky group $\\Gamma _j$ in $ of the form as inSection~\\ref {sec:schottky}, arising from the choice of $ qj$ Euclidean disks.We set $ Ij = {1,..., qj}$, let\\begin{equation}g_{j,k}\\,,\\ s_{j,k}\\,,\\ I_{j,k}\\qquad \\text{for $k\\in \\mathcal {I}_j$}\\end{equation}denote the side-pairing elements in~$ j$, the geodesic sides of~$ Fj$, and the forward intervals, respectively.",
"We recall the map~$$from~(\\ref {eq:combmap}) and set$$\\mathcal {J} _{j=1}^r \\mathcal {I}_j\\,.$$For each $ m = (m1,..., mr)J$ we set$$Q_m _{j=1}^r \\bigl ( s_{j,m_j} \\times I_{j,m_j}\\bigr )\\,,\\qquad C_m \\alpha ^{-1}( Q_m )\\,,$$and$$C \\bigcup _{m\\in \\mathcal {J}} C_m\\qquad \\text{and}\\qquad \\widehat{C} \\pi ^\\Gamma (C)\\,,$$where $$ is the map in~(\\ref {eq:projWC}).$ Theorem 4.1 The set $\\widehat{C}$ is a cross section for the Weyl chamber flow on $\\mathbb {Y}$ , and $C$ is a set of representatives for $\\widehat{C}$ .",
"Preparatory for the proof we briefly discuss the product structure of $\\widehat{C}$ and $C$ .",
"To that end, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ and $m_j \\in \\mathcal {I}_j$ , we set $C_{j,m_j} \\alpha _u^{-1}\\bigl ( s_{j,m_j} \\times I_{j,m_j} \\bigr )\\,,$ where $\\alpha _u$ denotes the map in (REF ) for $G_u$ .",
"Further, we set $C_j \\bigcup _{m_j\\in \\mathcal {I}_j} C_{j, m_j}\\qquad \\text{and}\\qquad \\widehat{C}_j \\pi _u^\\Gamma \\bigl ( C_j \\bigr )\\,.$ Then, for any $m = (m_1,\\ldots , m_r)\\in \\mathcal {J}$ , we have $C_m = _{j=1}^r C_{j,m_j}\\,,$ and further $\\widehat{C} = _{j=1}^r \\widehat{C}_j\\,.$ For each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , the set $\\widehat{C}_j$ is a cross section for the Weyl chamber flow (geodesic flow) on the Schottky surface $\\Gamma _j\\backslash with $ Cj$ as set of representatives byProposition~\\ref {prop:cross_schottky}.$ Let $\\Gamma g AM$ be an oriented compact flat of $\\mathbb {Y}$ .",
"With $g=(g_1,\\ldots , g_r)$ we have $\\Gamma g AM = (\\Gamma _1 g_1 A_uM_u,\\cdots ,\\Gamma _r g_r A_uM_u)\\,.$ Thus, for each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , $\\gamma _j \\Gamma _j g_j A_uM_u$ is an oriented flat (oriented geodesic) on $\\Gamma _j\\backslash that iscontained in a compact subset of~$ j.",
"It is even a periodic geodesic.",
"Thus, $\\gamma _j$ intersects $\\widehat{C}_j$ by Proposition REF , say in $\\Gamma _jg_ja_jM_u$ .",
"Then $\\Gamma g AM$ intersects $\\widehat{C}$ in $\\Gamma g (a_1,\\ldots , a_r)M$ .",
"This establishes REF for $\\widehat{C}$ .",
"In order to show REF let $\\Gamma gM\\in \\widehat{C}$ and suppose that $g=(g_1,\\ldots , g_r)$ .",
"Then $\\Gamma g M = (\\Gamma _1 g_1 M_u,\\ldots , \\Gamma _r g_r M_u) \\in \\widehat{C}_1 \\times \\cdots \\times \\widehat{C}_r\\,.$ Thus, for each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , the Weyl chamber (unit tangent vector) $\\Gamma _j g_j M_u$ is in the cross section $\\widehat{C}_j$ for the geodesic flow on $\\Gamma _j\\backslash .",
"Hence we find $ j>0$ such that for all$ t(-j,j)$, $ t=0$,$$\\Gamma _j g_j a_t M_u \\notin \\widehat{C}_j\\,,$$where$$a_t = \\begin{bmatrix} e^{t/2}&0\\\\0&e^{-t/2}\\end{bmatrix} \\in G_u = \\operatorname{PSL}_2(\\mathbb {R})\\,.$$For$$U (-\\varepsilon _1,\\varepsilon _1)\\times \\cdots \\times (-\\varepsilon _r,\\varepsilon _r)$$we have that each element $ aUAreg$ is of the form$$a = (a_{t_1},\\ldots , a_{t_r})$$with $ tj(-j, j)$, $ tj=0$, for $ j{1,..., r}$, andhence$$\\Gamma ga M = \\bigl ( \\Gamma _1g_1a_{t_1}M_u, \\ldots , \\Gamma _rg_ra_{t_r}M_u\\bigr )\\notin \\widehat{C}\\,.$$This establishes~(\\ref {cs2}) for~$ C$ and finishes the proof that~$ C$ isa cross section for the Weyl chamber flow on~$ Y$.",
"Finally for each$ j{1,..., r}$, the set $ Cj$ is a set of representatives for~$ Cj$.",
"This property is stable under direct products, and hence $ C$ is indeeda set of representatives for~$ C$.$ We briefly discuss an aspect of the structure and thickness of $\\widehat{C}$ .",
"For each $j\\in \\lbrace 1,\\ldots , r\\rbrace $ the set of base points of the cross section $\\widehat{C}_j$ is the full boundary of the fundamental domain $\\mathcal {F}_j$ .",
"A fundamental domain for $\\mathbb {Y}= \\Gamma \\backslash \\mathbb {X}$ is given by $\\mathcal {F} _{j=1}^r \\mathcal {F}_j\\,.$ The set of base points of $\\widehat{C}$ , however, is only a rather sparse subset of the boundary of $\\mathcal {F}$ , getting sparser if the rank becomes larger.",
"This shows that the cross section $\\widehat{C}$ is not implied by a Koebe–Morse method, but is a genuinely related to the Weyl chamber bundle.",
"In a way similar to the construction of $\\widehat{C}$ we can find the strong cross section contained in $\\widehat{C}$ .",
"To that end, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , we let $L_j$ denote the limit set of $\\Gamma _j$ (in the rank one situation), set $C_{\\mathnormal {st}, j} \\lbrace c\\in C_j \\mid \\nu _1(c) \\in L_j\\rbrace $ as well as $\\widehat{C}_{\\mathnormal {st}, j} \\pi _u^\\Gamma \\bigl ( C_{\\mathnormal {st}, j}\\bigr )\\,.$ As stated in Proposition REF , $\\widehat{C}_{\\mathnormal {st}, j}$ is the strong cross section contained in $\\widehat{C}_j$ for the geodesic flow on $\\Gamma _j\\backslash .",
"We further define$$C_\\mathnormal {st}_{j=1}^k C_{\\mathnormal {st},j}\\qquad \\text{and}\\qquad \\widehat{C}_\\mathnormal {st}_{j=1}^r \\widehat{C}_{\\mathnormal {st},j}\\,.$$Then $ Cst= (Cst)$.$ Theorem 4.2 The set $\\widehat{C}_\\mathnormal {st}$ is the strong cross section contained in $\\widehat{C}$ , and $C_\\mathnormal {st}$ is a set of representatives.",
"These statements can be proven analogously to those in Theorem REF , using Proposition REF instead of Proposition REF .",
"Codings and discrete dynamical systems on Furstenberg boundary In this section we present the discrete dynamical system on the Furstenberg boundary that is induced by the set of representatives $C_\\mathnormal {st}$ for the strong cross section $\\widehat{C}_\\mathnormal {st}$ from Section .",
"We resume the notation from Section .",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , we denote the strong coding sets determined by the choice of the fundamental domain $\\mathcal {F}_j$ for $\\Gamma _j$ by $I_{\\mathnormal {st},j,k}^c\\qquad \\text{for $k\\in \\mathcal {I}_j$}.$ For $m\\in \\mathcal {J} = _{j=1}^k \\mathcal {I}_j$ , $m=(m_1,\\ldots , m_r)$ , we set $I_{\\mathnormal {st}, m}^c _{j=1}^r I_{\\mathnormal {st}, j, m_j}^c$ and $D \\bigcup _{m\\in \\mathcal {J}} I_{\\mathnormal {st},m}^c\\,.$ We define the map $F\\colon D\\rightarrow D$ as follows: for $m\\in \\mathcal {J}$ , $m=(m_1,\\ldots , m_r)$ , we set $g_m \\bigl (g_{1,m_1},\\ldots , g_{r,m_r}\\bigr )\\,,$ where $g_{j,m_j}$ are the side-pairing elements of $\\mathcal {F}_j$ (see ()).",
"Restricted to the subset $I_{\\mathnormal {st},m}^c$ of $D$ , the map $F$ is $F\\vert _{I_{\\mathnormal {st},m}^c}\\colon I_{\\mathnormal {st},m}^c\\rightarrow D\\,,\\quad x\\mapsto g_m(x)\\,.$ Further, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ let $F_j\\colon L_j\\rightarrow L_j$ denote the discrete dynamical system induced on the Furstenberg boundary $P^1_\\mathbb {R}$ of $ induced by the cross section~$ Cst,j$ and its set of representatives~$ Cst,j$ for the geodesic flowon~$ j. Theorem 5.1 The map $F$ is the discrete dynamical system on Furstenberg boundary that is induced by $C_{\\mathnormal {st}}$ .",
"For any $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we have $F\\vert _{I_{\\mathnormal {st}, m}^c} = \\left( F_1\\vert _{I_{\\mathnormal {st},1,m_1}^c},\\ldots ,F_r\\vert _{I_{\\mathnormal {st},r,m_r}^c} \\right)\\,.$ The statement in (REF ) follows immediately from the definition of the map $F$ .",
"To establish (REF ), we recall that the $r$ -dimensional Weyl chamber flow on $\\mathbb {Y}$ is the direct product of the 1-dimensional geodesic flows on the factors $\\Gamma _j\\backslash , $ j{1,...,r}$,of~$ Y$.",
"We recall further that the cross section~$ Cst$ and itsset of representatives~$ Cst$ are direct products of the cross sections andsets of representatives for these factors.",
"Therefore, this product structuredescends to the induced discrete dynamical systems on Furstenberg boundary,which yields~(\\ref {mapi}) due to the identity in~(\\ref {mapii}).$ Transfer operators In this final section we propose a definition of a transfer operator family for the multi-dimensional discrete dynamical system $F$ in Section .",
"We continue to use the notation from Sections and and start by presenting the well-known definition of transfer operator families for one-dimensional flows, specialized to our setup.",
"Let $j\\in \\lbrace 1,\\ldots , r\\rbrace $ and recall from (REF ) and Section the discrete dynamical system $F_j\\colon L_j\\rightarrow L_j$ induced by the set of representatives $C_j$ of the cross section $\\widehat{C}_j$ for the geodesic flow on the Schottky surface $\\Gamma _j\\backslash .",
"TheRuelle-type \\emph {transfer operator}~$ Lj,s$ with parameter~$ s associated to $F_j$ is (at least initially) an operator on the space of functions $\\psi \\colon L_j\\rightarrow , given by\\begin{equation}\\mathcal {L}_{j,s}\\psi (x) \\sum _{y\\in F_j^{-1}(x)} | F^{\\prime }_j(y)|^{-s} \\psi (y)\\qquad (x\\in L_j)\\,.\\end{equation}Here, the derivative of~$ F'j$ at~$ y$ is understood as follows.",
"The point~$ y$is contained in~$ Ist,j,kc$ for a unique~$ kIj$.",
"Then~$ Fj$ actson a small neighborhood of~$ y$ in~$ Lj$ by the fractional lineartransformation~$ gj,k$, which extends to an analytic map in a smallneighborhood of~$ y$ in~$ R$.",
"We use the derivative of this extended mapfor~$ F'j(y)$.$ The function space which one uses as domain for the transfer operator $\\mathcal {L}_{j,s}$ depends on its further applications.",
"One may choose spaces of functions with larger domain or with some regularity properties.",
"We will refrain here from these discussions and will use the space of functions on the limit sets as place holder.",
"We shall now provide another presentation of $\\mathcal {L}_{j,s}$ that takes advantage of the explicit description of $F_j$ .",
"To that end we note that $F_j$ restricts to the bijections $I_{\\mathnormal {st},j,k}^c \\rightarrow L_j\\setminus I_{\\mathnormal {st},j,-k}^c\\,,\\quad x\\mapsto g_{j,k}(x)\\,,$ for each $k\\in \\mathcal {I}_j$ .",
"Thus, for each $k\\in \\mathcal {I}_j$ , each $x\\in I_{\\mathnormal {st},j,k}^c$ has the ($|\\mathcal {I}_j|-1$ ) preimages $\\lbrace g_{j,\\ell }^{-1}(x) \\mid \\ell \\in \\mathcal {I}_j, \\ell \\ne -k\\rbrace \\,.$ For any function $\\psi \\colon L_j\\rightarrow , we set$$\\psi _{k} \\psi \\cdot 1_{I_{\\mathnormal {st},j,k}^c}\\qquad \\text{for $k\\in \\mathcal {I}_j$}\\,,$$where $ 1A$ denotes the characteristic function of the set~$ A$.",
"Then$$\\psi = \\sum _{k\\in \\mathcal {I}_j} \\psi _k\\,.$$Further, for $ hj$, $ s, any subset $A\\subseteq \\mathbb {R}$ and any function $\\varphi \\colon A\\rightarrow we set\\begin{equation}\\tau _s(h^{-1})\\varphi (x) \\bigl (h^{\\prime }(x)\\bigr )^s \\varphi \\bigl ( h(x)\\bigr )\\qquad (x\\in A)\\,,\\end{equation}whenever it is well-defined (as it will be in all our applications).",
"For each$ kIj$ we have then$$\\bigl (\\mathcal {L}_{j,s}\\psi \\bigr )_k = \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}}\\tau _s\\bigl (g_{j,\\ell }\\bigr )\\psi _\\ell $$or, in a more compact form,$$\\mathcal {L}_{j,s} = \\sum _{k\\in \\mathcal {I}_j} 1_{I_{\\mathnormal {st},j,k}^c} \\cdot \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}} \\tau _s\\bigl (g_{j,\\ell }\\bigr )\\,.$$$ For the transfer operator family associated to the multi-dimensional map $F$ we propose a definition analogous to those in () but allowing a multi-dimensional parameter $s\\in r$ .",
"(We recall that the rank of the considered Riemannian locally symmetric space $\\mathbb {Y}$ is $r$ .)",
"We first consider the parameter-free transfer operator $\\mathcal {L}f(x) \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-1} f(y)\\,,$ acting on functions $f\\colon D\\rightarrow , where the derivative $ F'$ is understoodanalogously to above and $ |F'(y)|$ is the absolute value of the determinant ofthe linear map $ F'(y)$.",
"For any $ yD$ we find a unique element $ mJ$,$ m=(m1,..., mr)$, such that $ yIst,mc$.",
"Letting$$y = (y_1,\\ldots , y_r)\\qquad \\text{and}\\qquad g_m = \\bigl ( g_{1,m_1},\\ldots ,g_{r,m_r} \\bigr )$$we then have $ F=gm$ is a small neighborhood of~$ y$ in~$ D$.",
"Thus,$$F(y) = g_m(y) = \\bigl ( g_{1,m_1}(y_1),\\ldots , g_{r,m_r}(y_r) \\bigr )$$and the Jacobi matrix of~$ F$ at~$ y$ is the diagonal matrix\\begin{equation}J_F(y) =\\begin{pmatrix}g_{1,m_1}^{\\prime }(y_1)\\\\& \\ddots \\\\& & g_{r,m_r}^{\\prime }(y_r)\\end{pmatrix}\\,.\\end{equation}Thus,\\begin{equation}|F^{\\prime }(y)|^{-1} = |\\det J_F(y)|^{-1} = \\prod _{j=1}^r\\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-1}\\,.\\end{equation}Motivated by the diagonal structure of the Jacobi matrix in~(\\ref {eq:JM}), wepropose to endow each non-zero entry separately with a weight.",
"Thus, the\\emph {parametrized transfer operator}~$ Ls$ with $ s=(s1,..., sr)r$ is defined as\\begin{equation}\\mathcal {L}_sf(x) = \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-s} f(y)\\,,\\end{equation}where$$|F^{\\prime }(y)|^{-s} \\prod _{j=1}^r \\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-s_j}$$in the notation from~(\\ref {eq:det_unweight}).",
"This use of the parameter alsoreflects well the independence of the $ r$ dimensions of the oriented flatsof~$ Y$.$ The analogy between the transfer operators in () and () goes further.",
"For $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we set $B(m) \\lbrace n=(n_1,\\ldots , n_r)\\in \\mathcal {J} \\mid \\exists \\, j\\in \\lbrace 1,\\ldots ,r\\rbrace \\colon n_j = -m_j\\rbrace \\,.$ The map $F\\colon D\\rightarrow D$ restricts to the bijections $I_{\\mathnormal {st},m}^c \\rightarrow D\\setminus \\!\\!\\bigcup _{n\\in B(m)}I_{\\mathnormal {st},n}^c\\,,\\quad x\\mapsto g_m(x)\\,.$ In analogy to (), we define for $h=(h_1,\\ldots , h_n)\\in \\Gamma $ , $s=(s_1,\\ldots , s_r)\\in r$ , any subset $A\\subseteq \\mathbb {R}^r$ and any function $\\varphi \\colon A\\rightarrow ,{\\begin{@align}{1}{-1}\\omega _s(h^{-1})\\varphi (x) & = \\bigl | h^{\\prime }(y) \\bigr |^{-s} \\varphi \\bigl (h(y)\\bigr )\\\\& = \\bigl |h^{\\prime }_1(y_1)\\bigr |^{-s_1} \\cdots \\bigl |h^{\\prime }_r(y_r)\\bigr |^{-s_r}\\varphi \\bigl (h(y)\\bigr )\\,.",
"\\nonumber \\end{@align}}Then\\begin{equation}\\mathcal {L}_s = \\sum _{m\\in \\mathcal {J}} 1_{I_{\\mathnormal {st},m}^c} \\cdot \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\n\\notin B(m)\\end{array}} \\omega _s(g_n)\\,,\\end{equation}or, if we set$$f_m f\\cdot 1_{I_{\\mathnormal {st},m}^c} \\qquad (m\\in \\mathcal {J})$$for any function $ fD, then $(\\mathcal {L}_sf)_m = \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\ n\\notin B(m)\\end{array}} \\omega _s(g_n)f_n\\,.$ We remark that this multi-parameter transfer operator for the Weyl chamber flow on $\\mathbb {Y}$ is not the direct sum of the one-parameter transfer operators for the geodesic flow on the Schottky surfaces from which $\\mathbb {Y}$ is build.",
"This is consistent with the fact that the spectral theory of $\\mathbb {Y}$ is not just the “direct product” of the spectral theories of its factors.",
"We leave any further investigation into this transfer operator family for future work."
],
[
"Codings and discrete dynamical systems on Furstenberg\nboundary",
"In this section we present the discrete dynamical system on the Furstenberg boundary that is induced by the set of representatives $C_\\mathnormal {st}$ for the strong cross section $\\widehat{C}_\\mathnormal {st}$ from Section .",
"We resume the notation from Section .",
"For $j\\in \\lbrace 1,\\ldots , r\\rbrace $ , we denote the strong coding sets determined by the choice of the fundamental domain $\\mathcal {F}_j$ for $\\Gamma _j$ by $I_{\\mathnormal {st},j,k}^c\\qquad \\text{for $k\\in \\mathcal {I}_j$}.$ For $m\\in \\mathcal {J} = _{j=1}^k \\mathcal {I}_j$ , $m=(m_1,\\ldots , m_r)$ , we set $I_{\\mathnormal {st}, m}^c _{j=1}^r I_{\\mathnormal {st}, j, m_j}^c$ and $D \\bigcup _{m\\in \\mathcal {J}} I_{\\mathnormal {st},m}^c\\,.$ We define the map $F\\colon D\\rightarrow D$ as follows: for $m\\in \\mathcal {J}$ , $m=(m_1,\\ldots , m_r)$ , we set $g_m \\bigl (g_{1,m_1},\\ldots , g_{r,m_r}\\bigr )\\,,$ where $g_{j,m_j}$ are the side-pairing elements of $\\mathcal {F}_j$ (see ()).",
"Restricted to the subset $I_{\\mathnormal {st},m}^c$ of $D$ , the map $F$ is $F\\vert _{I_{\\mathnormal {st},m}^c}\\colon I_{\\mathnormal {st},m}^c\\rightarrow D\\,,\\quad x\\mapsto g_m(x)\\,.$ Further, for $j\\in \\lbrace 1,\\ldots , r\\rbrace $ let $F_j\\colon L_j\\rightarrow L_j$ denote the discrete dynamical system induced on the Furstenberg boundary $P^1_\\mathbb {R}$ of $ induced by the cross section~$ Cst,j$ and its set of representatives~$ Cst,j$ for the geodesic flowon~$ j. Theorem 5.1 The map $F$ is the discrete dynamical system on Furstenberg boundary that is induced by $C_{\\mathnormal {st}}$ .",
"For any $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we have $F\\vert _{I_{\\mathnormal {st}, m}^c} = \\left( F_1\\vert _{I_{\\mathnormal {st},1,m_1}^c},\\ldots ,F_r\\vert _{I_{\\mathnormal {st},r,m_r}^c} \\right)\\,.$ The statement in (REF ) follows immediately from the definition of the map $F$ .",
"To establish (REF ), we recall that the $r$ -dimensional Weyl chamber flow on $\\mathbb {Y}$ is the direct product of the 1-dimensional geodesic flows on the factors $\\Gamma _j\\backslash , $ j{1,...,r}$,of~$ Y$.",
"We recall further that the cross section~$ Cst$ and itsset of representatives~$ Cst$ are direct products of the cross sections andsets of representatives for these factors.",
"Therefore, this product structuredescends to the induced discrete dynamical systems on Furstenberg boundary,which yields~(\\ref {mapi}) due to the identity in~(\\ref {mapii}).$ Transfer operators In this final section we propose a definition of a transfer operator family for the multi-dimensional discrete dynamical system $F$ in Section .",
"We continue to use the notation from Sections and and start by presenting the well-known definition of transfer operator families for one-dimensional flows, specialized to our setup.",
"Let $j\\in \\lbrace 1,\\ldots , r\\rbrace $ and recall from (REF ) and Section the discrete dynamical system $F_j\\colon L_j\\rightarrow L_j$ induced by the set of representatives $C_j$ of the cross section $\\widehat{C}_j$ for the geodesic flow on the Schottky surface $\\Gamma _j\\backslash .",
"TheRuelle-type \\emph {transfer operator}~$ Lj,s$ with parameter~$ s associated to $F_j$ is (at least initially) an operator on the space of functions $\\psi \\colon L_j\\rightarrow , given by\\begin{equation}\\mathcal {L}_{j,s}\\psi (x) \\sum _{y\\in F_j^{-1}(x)} | F^{\\prime }_j(y)|^{-s} \\psi (y)\\qquad (x\\in L_j)\\,.\\end{equation}Here, the derivative of~$ F'j$ at~$ y$ is understood as follows.",
"The point~$ y$is contained in~$ Ist,j,kc$ for a unique~$ kIj$.",
"Then~$ Fj$ actson a small neighborhood of~$ y$ in~$ Lj$ by the fractional lineartransformation~$ gj,k$, which extends to an analytic map in a smallneighborhood of~$ y$ in~$ R$.",
"We use the derivative of this extended mapfor~$ F'j(y)$.$ The function space which one uses as domain for the transfer operator $\\mathcal {L}_{j,s}$ depends on its further applications.",
"One may choose spaces of functions with larger domain or with some regularity properties.",
"We will refrain here from these discussions and will use the space of functions on the limit sets as place holder.",
"We shall now provide another presentation of $\\mathcal {L}_{j,s}$ that takes advantage of the explicit description of $F_j$ .",
"To that end we note that $F_j$ restricts to the bijections $I_{\\mathnormal {st},j,k}^c \\rightarrow L_j\\setminus I_{\\mathnormal {st},j,-k}^c\\,,\\quad x\\mapsto g_{j,k}(x)\\,,$ for each $k\\in \\mathcal {I}_j$ .",
"Thus, for each $k\\in \\mathcal {I}_j$ , each $x\\in I_{\\mathnormal {st},j,k}^c$ has the ($|\\mathcal {I}_j|-1$ ) preimages $\\lbrace g_{j,\\ell }^{-1}(x) \\mid \\ell \\in \\mathcal {I}_j, \\ell \\ne -k\\rbrace \\,.$ For any function $\\psi \\colon L_j\\rightarrow , we set$$\\psi _{k} \\psi \\cdot 1_{I_{\\mathnormal {st},j,k}^c}\\qquad \\text{for $k\\in \\mathcal {I}_j$}\\,,$$where $ 1A$ denotes the characteristic function of the set~$ A$.",
"Then$$\\psi = \\sum _{k\\in \\mathcal {I}_j} \\psi _k\\,.$$Further, for $ hj$, $ s, any subset $A\\subseteq \\mathbb {R}$ and any function $\\varphi \\colon A\\rightarrow we set\\begin{equation}\\tau _s(h^{-1})\\varphi (x) \\bigl (h^{\\prime }(x)\\bigr )^s \\varphi \\bigl ( h(x)\\bigr )\\qquad (x\\in A)\\,,\\end{equation}whenever it is well-defined (as it will be in all our applications).",
"For each$ kIj$ we have then$$\\bigl (\\mathcal {L}_{j,s}\\psi \\bigr )_k = \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}}\\tau _s\\bigl (g_{j,\\ell }\\bigr )\\psi _\\ell $$or, in a more compact form,$$\\mathcal {L}_{j,s} = \\sum _{k\\in \\mathcal {I}_j} 1_{I_{\\mathnormal {st},j,k}^c} \\cdot \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}} \\tau _s\\bigl (g_{j,\\ell }\\bigr )\\,.$$$ For the transfer operator family associated to the multi-dimensional map $F$ we propose a definition analogous to those in () but allowing a multi-dimensional parameter $s\\in r$ .",
"(We recall that the rank of the considered Riemannian locally symmetric space $\\mathbb {Y}$ is $r$ .)",
"We first consider the parameter-free transfer operator $\\mathcal {L}f(x) \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-1} f(y)\\,,$ acting on functions $f\\colon D\\rightarrow , where the derivative $ F'$ is understoodanalogously to above and $ |F'(y)|$ is the absolute value of the determinant ofthe linear map $ F'(y)$.",
"For any $ yD$ we find a unique element $ mJ$,$ m=(m1,..., mr)$, such that $ yIst,mc$.",
"Letting$$y = (y_1,\\ldots , y_r)\\qquad \\text{and}\\qquad g_m = \\bigl ( g_{1,m_1},\\ldots ,g_{r,m_r} \\bigr )$$we then have $ F=gm$ is a small neighborhood of~$ y$ in~$ D$.",
"Thus,$$F(y) = g_m(y) = \\bigl ( g_{1,m_1}(y_1),\\ldots , g_{r,m_r}(y_r) \\bigr )$$and the Jacobi matrix of~$ F$ at~$ y$ is the diagonal matrix\\begin{equation}J_F(y) =\\begin{pmatrix}g_{1,m_1}^{\\prime }(y_1)\\\\& \\ddots \\\\& & g_{r,m_r}^{\\prime }(y_r)\\end{pmatrix}\\,.\\end{equation}Thus,\\begin{equation}|F^{\\prime }(y)|^{-1} = |\\det J_F(y)|^{-1} = \\prod _{j=1}^r\\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-1}\\,.\\end{equation}Motivated by the diagonal structure of the Jacobi matrix in~(\\ref {eq:JM}), wepropose to endow each non-zero entry separately with a weight.",
"Thus, the\\emph {parametrized transfer operator}~$ Ls$ with $ s=(s1,..., sr)r$ is defined as\\begin{equation}\\mathcal {L}_sf(x) = \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-s} f(y)\\,,\\end{equation}where$$|F^{\\prime }(y)|^{-s} \\prod _{j=1}^r \\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-s_j}$$in the notation from~(\\ref {eq:det_unweight}).",
"This use of the parameter alsoreflects well the independence of the $ r$ dimensions of the oriented flatsof~$ Y$.$ The analogy between the transfer operators in () and () goes further.",
"For $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we set $B(m) \\lbrace n=(n_1,\\ldots , n_r)\\in \\mathcal {J} \\mid \\exists \\, j\\in \\lbrace 1,\\ldots ,r\\rbrace \\colon n_j = -m_j\\rbrace \\,.$ The map $F\\colon D\\rightarrow D$ restricts to the bijections $I_{\\mathnormal {st},m}^c \\rightarrow D\\setminus \\!\\!\\bigcup _{n\\in B(m)}I_{\\mathnormal {st},n}^c\\,,\\quad x\\mapsto g_m(x)\\,.$ In analogy to (), we define for $h=(h_1,\\ldots , h_n)\\in \\Gamma $ , $s=(s_1,\\ldots , s_r)\\in r$ , any subset $A\\subseteq \\mathbb {R}^r$ and any function $\\varphi \\colon A\\rightarrow ,{\\begin{@align}{1}{-1}\\omega _s(h^{-1})\\varphi (x) & = \\bigl | h^{\\prime }(y) \\bigr |^{-s} \\varphi \\bigl (h(y)\\bigr )\\\\& = \\bigl |h^{\\prime }_1(y_1)\\bigr |^{-s_1} \\cdots \\bigl |h^{\\prime }_r(y_r)\\bigr |^{-s_r}\\varphi \\bigl (h(y)\\bigr )\\,.",
"\\nonumber \\end{@align}}Then\\begin{equation}\\mathcal {L}_s = \\sum _{m\\in \\mathcal {J}} 1_{I_{\\mathnormal {st},m}^c} \\cdot \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\n\\notin B(m)\\end{array}} \\omega _s(g_n)\\,,\\end{equation}or, if we set$$f_m f\\cdot 1_{I_{\\mathnormal {st},m}^c} \\qquad (m\\in \\mathcal {J})$$for any function $ fD, then $(\\mathcal {L}_sf)_m = \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\ n\\notin B(m)\\end{array}} \\omega _s(g_n)f_n\\,.$ We remark that this multi-parameter transfer operator for the Weyl chamber flow on $\\mathbb {Y}$ is not the direct sum of the one-parameter transfer operators for the geodesic flow on the Schottky surfaces from which $\\mathbb {Y}$ is build.",
"This is consistent with the fact that the spectral theory of $\\mathbb {Y}$ is not just the “direct product” of the spectral theories of its factors.",
"We leave any further investigation into this transfer operator family for future work."
],
[
"Transfer operators",
"In this final section we propose a definition of a transfer operator family for the multi-dimensional discrete dynamical system $F$ in Section .",
"We continue to use the notation from Sections and and start by presenting the well-known definition of transfer operator families for one-dimensional flows, specialized to our setup.",
"Let $j\\in \\lbrace 1,\\ldots , r\\rbrace $ and recall from (REF ) and Section the discrete dynamical system $F_j\\colon L_j\\rightarrow L_j$ induced by the set of representatives $C_j$ of the cross section $\\widehat{C}_j$ for the geodesic flow on the Schottky surface $\\Gamma _j\\backslash .",
"TheRuelle-type \\emph {transfer operator}~$ Lj,s$ with parameter~$ s associated to $F_j$ is (at least initially) an operator on the space of functions $\\psi \\colon L_j\\rightarrow , given by\\begin{equation}\\mathcal {L}_{j,s}\\psi (x) \\sum _{y\\in F_j^{-1}(x)} | F^{\\prime }_j(y)|^{-s} \\psi (y)\\qquad (x\\in L_j)\\,.\\end{equation}Here, the derivative of~$ F'j$ at~$ y$ is understood as follows.",
"The point~$ y$is contained in~$ Ist,j,kc$ for a unique~$ kIj$.",
"Then~$ Fj$ actson a small neighborhood of~$ y$ in~$ Lj$ by the fractional lineartransformation~$ gj,k$, which extends to an analytic map in a smallneighborhood of~$ y$ in~$ R$.",
"We use the derivative of this extended mapfor~$ F'j(y)$.$ The function space which one uses as domain for the transfer operator $\\mathcal {L}_{j,s}$ depends on its further applications.",
"One may choose spaces of functions with larger domain or with some regularity properties.",
"We will refrain here from these discussions and will use the space of functions on the limit sets as place holder.",
"We shall now provide another presentation of $\\mathcal {L}_{j,s}$ that takes advantage of the explicit description of $F_j$ .",
"To that end we note that $F_j$ restricts to the bijections $I_{\\mathnormal {st},j,k}^c \\rightarrow L_j\\setminus I_{\\mathnormal {st},j,-k}^c\\,,\\quad x\\mapsto g_{j,k}(x)\\,,$ for each $k\\in \\mathcal {I}_j$ .",
"Thus, for each $k\\in \\mathcal {I}_j$ , each $x\\in I_{\\mathnormal {st},j,k}^c$ has the ($|\\mathcal {I}_j|-1$ ) preimages $\\lbrace g_{j,\\ell }^{-1}(x) \\mid \\ell \\in \\mathcal {I}_j, \\ell \\ne -k\\rbrace \\,.$ For any function $\\psi \\colon L_j\\rightarrow , we set$$\\psi _{k} \\psi \\cdot 1_{I_{\\mathnormal {st},j,k}^c}\\qquad \\text{for $k\\in \\mathcal {I}_j$}\\,,$$where $ 1A$ denotes the characteristic function of the set~$ A$.",
"Then$$\\psi = \\sum _{k\\in \\mathcal {I}_j} \\psi _k\\,.$$Further, for $ hj$, $ s, any subset $A\\subseteq \\mathbb {R}$ and any function $\\varphi \\colon A\\rightarrow we set\\begin{equation}\\tau _s(h^{-1})\\varphi (x) \\bigl (h^{\\prime }(x)\\bigr )^s \\varphi \\bigl ( h(x)\\bigr )\\qquad (x\\in A)\\,,\\end{equation}whenever it is well-defined (as it will be in all our applications).",
"For each$ kIj$ we have then$$\\bigl (\\mathcal {L}_{j,s}\\psi \\bigr )_k = \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}}\\tau _s\\bigl (g_{j,\\ell }\\bigr )\\psi _\\ell $$or, in a more compact form,$$\\mathcal {L}_{j,s} = \\sum _{k\\in \\mathcal {I}_j} 1_{I_{\\mathnormal {st},j,k}^c} \\cdot \\sum _{\\begin{array}{c}\\ell \\in \\mathcal {I}_j \\\\ \\ell \\ne -k\\end{array}} \\tau _s\\bigl (g_{j,\\ell }\\bigr )\\,.$$$ For the transfer operator family associated to the multi-dimensional map $F$ we propose a definition analogous to those in () but allowing a multi-dimensional parameter $s\\in r$ .",
"(We recall that the rank of the considered Riemannian locally symmetric space $\\mathbb {Y}$ is $r$ .)",
"We first consider the parameter-free transfer operator $\\mathcal {L}f(x) \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-1} f(y)\\,,$ acting on functions $f\\colon D\\rightarrow , where the derivative $ F'$ is understoodanalogously to above and $ |F'(y)|$ is the absolute value of the determinant ofthe linear map $ F'(y)$.",
"For any $ yD$ we find a unique element $ mJ$,$ m=(m1,..., mr)$, such that $ yIst,mc$.",
"Letting$$y = (y_1,\\ldots , y_r)\\qquad \\text{and}\\qquad g_m = \\bigl ( g_{1,m_1},\\ldots ,g_{r,m_r} \\bigr )$$we then have $ F=gm$ is a small neighborhood of~$ y$ in~$ D$.",
"Thus,$$F(y) = g_m(y) = \\bigl ( g_{1,m_1}(y_1),\\ldots , g_{r,m_r}(y_r) \\bigr )$$and the Jacobi matrix of~$ F$ at~$ y$ is the diagonal matrix\\begin{equation}J_F(y) =\\begin{pmatrix}g_{1,m_1}^{\\prime }(y_1)\\\\& \\ddots \\\\& & g_{r,m_r}^{\\prime }(y_r)\\end{pmatrix}\\,.\\end{equation}Thus,\\begin{equation}|F^{\\prime }(y)|^{-1} = |\\det J_F(y)|^{-1} = \\prod _{j=1}^r\\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-1}\\,.\\end{equation}Motivated by the diagonal structure of the Jacobi matrix in~(\\ref {eq:JM}), wepropose to endow each non-zero entry separately with a weight.",
"Thus, the\\emph {parametrized transfer operator}~$ Ls$ with $ s=(s1,..., sr)r$ is defined as\\begin{equation}\\mathcal {L}_sf(x) = \\sum _{y\\in F^{-1}(x)} |F^{\\prime }(y)|^{-s} f(y)\\,,\\end{equation}where$$|F^{\\prime }(y)|^{-s} \\prod _{j=1}^r \\bigl (g_{j,m_j}^{\\prime }(y_j)\\bigr )^{-s_j}$$in the notation from~(\\ref {eq:det_unweight}).",
"This use of the parameter alsoreflects well the independence of the $ r$ dimensions of the oriented flatsof~$ Y$.$ The analogy between the transfer operators in () and () goes further.",
"For $m=(m_1,\\ldots , m_r)\\in \\mathcal {J}$ we set $B(m) \\lbrace n=(n_1,\\ldots , n_r)\\in \\mathcal {J} \\mid \\exists \\, j\\in \\lbrace 1,\\ldots ,r\\rbrace \\colon n_j = -m_j\\rbrace \\,.$ The map $F\\colon D\\rightarrow D$ restricts to the bijections $I_{\\mathnormal {st},m}^c \\rightarrow D\\setminus \\!\\!\\bigcup _{n\\in B(m)}I_{\\mathnormal {st},n}^c\\,,\\quad x\\mapsto g_m(x)\\,.$ In analogy to (), we define for $h=(h_1,\\ldots , h_n)\\in \\Gamma $ , $s=(s_1,\\ldots , s_r)\\in r$ , any subset $A\\subseteq \\mathbb {R}^r$ and any function $\\varphi \\colon A\\rightarrow ,{\\begin{@align}{1}{-1}\\omega _s(h^{-1})\\varphi (x) & = \\bigl | h^{\\prime }(y) \\bigr |^{-s} \\varphi \\bigl (h(y)\\bigr )\\\\& = \\bigl |h^{\\prime }_1(y_1)\\bigr |^{-s_1} \\cdots \\bigl |h^{\\prime }_r(y_r)\\bigr |^{-s_r}\\varphi \\bigl (h(y)\\bigr )\\,.",
"\\nonumber \\end{@align}}Then\\begin{equation}\\mathcal {L}_s = \\sum _{m\\in \\mathcal {J}} 1_{I_{\\mathnormal {st},m}^c} \\cdot \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\n\\notin B(m)\\end{array}} \\omega _s(g_n)\\,,\\end{equation}or, if we set$$f_m f\\cdot 1_{I_{\\mathnormal {st},m}^c} \\qquad (m\\in \\mathcal {J})$$for any function $ fD, then $(\\mathcal {L}_sf)_m = \\sum _{\\begin{array}{c}n\\in \\mathcal {J}\\\\ n\\notin B(m)\\end{array}} \\omega _s(g_n)f_n\\,.$ We remark that this multi-parameter transfer operator for the Weyl chamber flow on $\\mathbb {Y}$ is not the direct sum of the one-parameter transfer operators for the geodesic flow on the Schottky surfaces from which $\\mathbb {Y}$ is build.",
"This is consistent with the fact that the spectral theory of $\\mathbb {Y}$ is not just the “direct product” of the spectral theories of its factors.",
"We leave any further investigation into this transfer operator family for future work."
]
] | 2011.14098 |
[
[
"Transverse single-spin asymmetries of midrapidity $\\pi^0$ and $\\eta$\n mesons in polarized $p$$+$$p$ collisions at $\\sqrt{s}=200$ GeV"
],
[
"Abstract We present a measurement of the transverse single-spin asymmetry for $\\pi^0$ and $\\eta$ mesons in $p^\\uparrow$$+$$p$ collisions in the pseudorapidity range $|\\eta|<0.35$ and at a center-of-mass energy of 200 GeV with the PHENIX detector at the Relativistic Heavy Ion Collider.",
"In comparison with previous measurements in this kinematic region, these results have a factor of 3 smaller uncertainties.",
"As hadrons, $\\pi^0$ and $\\eta$ mesons are sensitive to both initial- and final-state nonperturbative effects for a mix of parton flavors.",
"Comparisons of the differences in their transverse single-spin asymmetries have the potential to disentangle the possible effects of strangeness, isospin, or mass.",
"These results can constrain the twist-3 trigluon collinear correlation function as well as the gluon Sivers function."
],
[
"Introduction",
"Spin-momentum correlations in hadronic collisions have attracted increasing experimental and theoretical interest in the past two decades.",
"In particular, transverse single-spin asymmetries (TSSAs) have been one of the primary means to probe transverse partonic dynamics in the nucleon.",
"In the context of proton-proton collisions, one transversely-polarized proton collides with another unpolarized proton and the TSSA measures the asymmetry in yields of particles that travel to the left versus the right of the polarized-proton-going direction.",
"Large azimuthal asymmetries of up to $\\approx \\,$ 40% have been observed from transversely polarized $ p^\\uparrow + p $ collisions in light meson production at large Feynman-x ($ x_F = 2 p_L/\\sqrt{s}$ ), from center of mass energies of $ \\sqrt{s} = 4.9 ~{\\rm GeV}$ up to $500~~{\\rm GeV}$ [1], [2], [3], [4], [5], [6].",
"Next-to-leading order perturbative quantum chromodynamics (QCD) calculations that only include spin-momentum correlations from parton scattering predict small asymmetries on the order of $m_q/Q$ [7], where $m_q$ is the bare quark mass and $Q$ is the hard scale, indicating that significant nonperturbative effects must dominate the large measured asymmetries.",
"Two different approaches have been proposed to describe the large asymmetries observed in hadronic interactions.",
"In the first approach, nonperturbative parton distribution functions (PDFs) and fragmentation functions (FFs) are explicitly dependent on transverse momentum in the transverse-momentum-dependent (TMD) framework.",
"These functions depend on a soft ($k_T$ ) and hard ($Q$ ) momentum scale such that $\\Lambda _{QCD}\\lesssim k_T\\ll Q$ .",
"One possible origin of the large TSSAs is the Sivers TMD PDF [8], which correlates the nucleon transverse spin with the parton transverse momentum, $k_T$ .",
"Another possible origin of the TSSA is the Collins TMD FF [9], which correlates the transverse polarization of a fragmenting quark to the angular distribution of hadrons.",
"The second approach to describe the large asymmetries relies on collinear higher-twist effects with multiparton correlations.",
"In the twist-3 approach, interference arises between scattering amplitudes with one and two collinear partons, which leads to a nonzero TSSA.",
"This approach applies to observables in which only one sufficiently hard momentum scale is measured, such that $Q \\gg \\Lambda _{QCD}$ [10].",
"To keep the multiparton correlation functions process independent, the initial- and final-state interactions between the struck parton and the proton remnants are included in the hard perturbative part of the twist-3 collinear factorization [11].",
"Collinear twist-3 correlation functions are split into two types: the quark-gluon-quark functions (qgq) and the trigluon functions (ggg).",
"In the context of initial-state effects, the qgq functions describe the interference from scattering off of one quark versus scattering off of a gluon and a quark of the same flavor, while the ggg functions capture the interference between scattering off of one gluon versus scattering off of two.",
"The twist-3 approach is well suited to describe observed inclusive forward hadron asymmetries because the observed hadron $p_T$ can be used as a proxy for the hard scale, and unlike the TMD approach, these correlation functions do not explicitly depend on a soft-scale transverse momentum.",
"However, the twist-3 approach has been related to $k_T$ moments of TMD PDFs and TMD FFs and has been shown to be equivalent to the TMD approach in the overlapping kinematic regime [12].",
"Because the Sivers function is PT-odd, in order to be nonzero it must include a soft-gluon exchange with the proton remnant, which can occur before and/or after the hard partonic scattering depending on the process [13].",
"Significant nonzero asymmetries due to the Sivers TMD PDF have been measured in semi-inclusive deep-inelastic lepton-nucleon scattering (SIDIS) [14], [15], where the soft-gluon exchange can only happen in the final state.In hadronic interactions where at least one final-state hadron is measured, both initial- and final-state interactions can play competing roles in the measured asymmetries; here TMD-factorization breaking has been predicted due to soft gluon exchanges that are possible in both the initial and final states simultaneously [16].",
"Additional leading-power spin asymmetries have been predicted in hadronic collisions due to this breakdown, without which these asymmetries would be subleading [17], but further work is needed to connect TMD-factorization breaking to experimentally measured asymmetries.",
"Note that inclusive hadron TSSA measurements in hadronic collisions appear to plateau at $p_T$ up to 5 GeV/c [5], [3] and have been measured to be nonzero at up to $p_T\\,\\approx 7$ GeV/c [18].",
"Recent studies in the twist-3 framework have successfully described the $p_T$ dependence of these forward asymmetries by including twist-3 effects in hadronization [19].",
"The twist-3 perturbative prediction is that the asymmetry should eventually decrease as the hard scale $p_T$ continues increasing [17].",
"Since the inception of the collinear twist-3 and TMD factorization pictures, there has been theoretical evidence that they could combine to form a unified picture of TSSAs in hard processes.",
"This concept was recently tested with the first simultaneous global analysis of TSSAs in SIDIS, Drell-Yan, $e^+ e^-$ annihilation, and proton-proton collisions [10].",
"This study used quark TMD PDFs and FFs to describe the asymmetries in processes that are sensitive to the soft-scale momentum, i.e.",
"SIDIS, Drell-Yan, and $e^+ e^-$ annihilation.",
"These TMD functions were also used to calculate collinear twist-3 qgq correlation functions which were applied to inclusive forward pion asymmetry measurements from the Relativistic Heavy Ion Collider (RHIC).",
"This simultaneous description of TSSAs across multiple collision species indicates that all TSSAs have a common origin that is related to multiparton correlations.",
"Additional questions about the origin of the TSSAs in hadronic interactions remain.",
"Forward jet measurements indicate that the TSSA is significantly smaller than neutral pion asymmetries at similar $x_F$ and $\\sqrt{s}$ [20].",
"Nonzero kaon and antiproton asymmetries observed at forward rapidities show that the measured asymmetries cannot be due only to proton valence quark contributions as naively predicted in a valence-like model, where the Sivers effect from sea-quarks and/or gluons is ignored, and that the fragmentation of quarks into hadrons in which they are not valence quarks could play a role in the observed nonzero asymmetries [4], [21].",
"Eta meson measurements, sensitive to potential effects from strange quark contributions, isospin, and/or hadron mass show forward asymmetries similar in magnitude to neutral pions [22].",
"At midrapidity at RHIC, nonzero TSSAs have been measured for charged pion pair production [23], [24].",
"Even four decades after the initial discovery of large TSSAs in hadronic interactions [1], there remain many unresolved questions about their origin.",
"Therefore it is crucial to continue extending measurements to try to better understand the nonperturbative dynamics which are responsible for the TSSAs in hadronic collisions.",
"In this paper we report a measurement of the TSSA of $\\mbox{$\\pi ^0$} $ and $\\eta $ mesons in $p^\\uparrow $$+$$p$ collisions at $\\sqrt{s}$ = 200 GeV in the midrapidity region $|\\eta |<$ 0.35.",
"The data was taken during the 2015 RHIC run and a total integrated luminosity of approximately 60 pb$^{-1}$ was collected.",
"This measurement extends previous measurements from RHIC to higher $p_T$ and reduces the statistical uncertainties by a factor of three in the overlapping $p_T$ region."
],
[
"Analysis",
"The asymmetries are measured with transversely polarized proton beams where the average polarization of the clockwise beam was $0.58\\pm 0.02$ and that of the counter-clockwise beam was $0.60 \\pm 0.02$ [25].",
"The direction of the beam polarization was found to be consistent with the vertical within statistical uncertainties.",
"The polarization direction of each beam independently changes bunch to bunch which reduces systematic uncertainties associated with variations in detector performance with time.",
"The relative luminosity is the ratio of the integrated luminosity for bunches that were polarized in opposite directions.",
"It is determined by the number of times each crossing fires a minimum-bias (MB) trigger and is measured to better than $10^{-4}$ .",
"The relative luminosity values for both the beams were limited to the range of 0.91 to 1.09 for all beam fills used in these measurements.",
"The bunch-to-bunch changes in polarization direction also allow for polarization-averaged measurements and, for a single-spin asymmetry analysis, provide two ways to measure the TSSA with the same data set.",
"This is done by sorting the particle yields for the polarization directions of one beam at a time, effectively averaging over the polarization of the other beam.",
"RHIC uses eight different spin patterns for sequential fills which are carefully chosen to minimize potential effects from nonzero average polarization of either beam.",
"The statistically independent asymmetries measured from the two beams are used to verify the analysis and are averaged together for the final result.",
"The data analysis procedure is similar to our previous measurements [5].",
"Neutral pion and eta mesons are reconstructed via their two-photon decays by using the midrapidity electromagnetic calorimeter (EMCal).",
"The EMCal is located in two central arms, each covering $\\Delta \\phi =\\pi /2$ in azimuth and $|\\eta |<0.35$ in pseudorapidity, centered at $\\phi =\\pi /16$ and $15\\pi /16$ .",
"The EMCal comprises two different types of calorimeters: six sectors of sampling lead-scintillator calorimeters and two sectors of Čerenkov lead-glass calorimeters [26].",
"The two calorimeter systems have different granularity ($\\delta \\phi \\times \\delta \\eta =0.011\\times 0.011$ in the lead scintillator and $0.008\\times 0.008$ in the lead glass) and also different responses to charged hadrons, which provides important systematic cross checks for these measurements.",
"A tracking system includes a drift chamber to measure track momentum and pad chamber stations to measure the charged particle hit position [27].",
"The measurement of the track positions in front of the calorimeter is used to veto charged particles from the photon sample.",
"The beam-beam counters (BBC) are arrays of quartz Čerenkov radiators that surround the beam pipe and are placed $\\pm 144$ cm away from the nominal collision point.",
"The BBC covers full azimuth and $3.0<|\\eta |<3.9$ in pseudorapidity.",
"They measure the z-vertex position; a vertex cut of $\\pm 30$ cm around the nominal collision point is used for this analysis.",
"The MB trigger requires at least one charged particle to be measured in both sides of the BBC.",
"This analysis is based on the data sample selected with the EMCal-based high-energy-photon trigger with energy threshold of 1.5 GeV, which is taken in coincidence with the MB trigger.",
"Photons are identified as clusters in the EMCal and are required to pass a shower profile cut which suppresses clusters from hadrons.",
"High-$p_T$ trigger photons are paired with another photon in the same event that is also on the same side of the detector.",
"A charged track veto cut eliminates clusters that geometrically match with a measured charged track, reducing background from electrons.",
"The contribution of EMCal detector noise is reduced by a minimum energy cut of 0.5 GeV and a time-of-flight cut of $|TOF|< 5$ ns.",
"The timing of the cluster is measured by the EMCal and the time zero reference of the event is provided by the BBC.",
"Each photon pair is required to pass an energy asymmetry cut: $\\alpha =|E_1-E_2|/(E_1+E_2)<0.8$ .",
"The $\\pi ^{0}$ yields comprise photon pairs with invariant mass in the signal region $\\pm 25~{\\rm MeV/c^2}$ from the $\\pi ^{0}$ mass peak and $\\eta $ meson yields are measured in the range $\\pm 70~{\\rm MeV}/c^2$ around the $\\eta $ mass peak.",
"The transverse single-spin asymmetries are determined with the “relative luminosity\" formula $A_N = \\frac{1}{P \\, \\left<\\cos (\\phi )\\right>} \\frac{{N^{\\uparrow }}-\\mathcal {R}{N^{\\downarrow }} }{{N^{\\uparrow }}+\\mathcal {R}{N^{\\downarrow }}},$ which compares the yield of particles for when the beam was polarized up versus down.",
"Here $P$ is the beam polarization, $N$ refers to the meson yield, the arrows refer to the up ($\\uparrow $ ) or down ($\\downarrow $ ) directions of beam polarization, and $\\mathcal {R}=\\mathcal {L}^\\uparrow /\\mathcal {L}^\\downarrow $ is the relative luminosity.",
"The acceptance factor, $\\left<\\cos (\\phi )\\right>$ , accounts for the detector azimuthal coverage, where $\\phi =0$ points $90^\\circ $ from the (vertical) spin axis.",
"This correction is calculated as a function of photon pair $p_T$ because the diphoton azimuthal acceptance depends heavily on the decay angle and ranges from 0.95 at low $p_T$ to 0.89 at high $p_T$.",
"The asymmetry is calculated separately for the two detector arms and then the average weighed by the statistical error is taken for the final result.",
"As written, Eq.",
"(REF ) is for the arm to the left of the direction of travel of the beam that is being taken as polarized.",
"An overall minus sign is needed for the asymmetry of particle yields in the arm to the right of the polarized-beam-going direction.",
"An alternative method of calculating the asymmetry is the “square root” formula $A_N = \\frac{1}{P \\, \\left<\\cos (\\phi )\\right>}\\frac{\\sqrt{N_L^{\\uparrow }N_R^{\\downarrow }}-\\sqrt{N_L^{\\downarrow }N_R^{\\uparrow }}}{\\sqrt{N_L^{\\uparrow }N_R^{\\downarrow }}+\\sqrt{N_L^{\\downarrow }N_R^{\\uparrow }}},$ which is used as a cross check.",
"This formula combines data from the two arms (left and right) and both beam polarization directions (up and down).",
"The subscripts in Eq.",
"(REF ) refer to the yields to the left ($L$ ) and right ($R$ ) side of the polarized-beam-going direction.",
"Figure: Example invariant mass distributions around the (a) π 0 \\pi ^{0} and (b) η\\eta peak for photon pairs with 4<p T <5 GeV /c{4<p_T<5 ~{\\rm GeV}/c} in one ofthe detector arms.",
"The minus-45 ∘ 45^{\\circ } hatched [blue] region at thecenter of each plot corresponds to the invariant mass region underthe peak which is used to calculate A N A_N in Eq.",
"() and theplus-45 ∘ 45^{\\circ } [red] side band regionscorrespond to the photon pairs that are used to calculateA N BG A_N^{\\rm BG}.",
"The bold solid [green] curves correspond to the fitto the combinatorial background, which is used to calculatethe background fraction.The measured asymmetries are also corrected for background $A_N^{\\rm Sig} = \\frac{A_N - r \\cdot A_N^{\\rm BG}}{1-r},$ where $r$ is the fractional contribution of photon pairs from combinatorial background within the invariant mass peak.",
"The background fraction is calculated from fits to the invariant mass spectra where a Gaussian is used to describe the invariant mass peak, and a third order polynomial is used to describe the combinatorial background, as shown in the green curves in Fig.",
"REF .",
"Using this method, the contribution of combinatorial background under the $\\pi ^{0}$ peak is determined to vary from 10% in the lowest $p_T$ bin to 6% in the highest.",
"Under the $\\eta $ meson invariant mass peak, the background fraction varies from 71% to 47% in the lowest to highest $p_T$ bins.",
"In Eq.",
"(REF ), the background asymmetry, $A_N^{\\rm BG}$ , is evaluated with photon pairs in side band regions located on either side of the signal peak, as represented in the red regions in Fig.",
"REF .",
"For the $\\pi ^{0}$ analysis these side band regions are $47<M_{\\gamma \\gamma }<97~{\\rm MeV}/c^2$ and $177<M_{\\gamma \\gamma }<227~{\\rm MeV}/c^2$ , and for the $\\eta $ meson analysis these regions are $300<M_{\\gamma \\gamma }<400~{\\rm MeV}/c^2$ and $700<M_{\\gamma \\gamma }<800~{\\rm MeV}/c^2$ .",
"These background regions match the ranges that were used in previous results [5] and are chosen to approximate the behavior of the combinatorial background under the invariant mass peak.",
"They are selected to be close to the peak while still far enough away to contain negligible contributions from signal photon pairs.",
"The background asymmetries are consistent with zero across all $p_T$ bins.",
"The background asymmetries between the low-mass and high-mass regions are also consistent with zero and with each other.",
"Table: The measured A N A_N of η\\eta mesons in pp++pp collisions at s\\sqrt{s} =200 GeV as a function of p T p_T.",
"An additional scale uncertainty of 3.4% due to the polarization uncertainty is not shown.",
"The total σ syst \\sigma _{\\rm syst} in the lowest p T p_T bin includes an additional systematic uncertainty of 6.20×10 -4 6.20\\times 10^{-4} from bunch shuffling.Tables REF and REF show the asymmetries with statistical and systematic uncertainties.",
"The total systematic uncertainty is the sum of the three sources of systematic uncertainty added in quadrature.",
"The systematic uncertainty on the asymmetry due to the background fraction in Eq.",
"(REF ) is determined by varying the fit ranges when computing $r$ and calculating how much the background-corrected asymmetry changes.",
"While the asymmetries calculated with the “relative luminosity\" [Eq.",
"(REF )] and the “square root\" [Eq.",
"(REF )] formulas were found to be statistically consistent, their difference was assigned as a conservative systematic uncertainty due to possible variations in detector performance and beam conditions.",
"This dominates the total systematic uncertainty for most $p_T$ bins.",
"Bunch shuffling is a technique used to investigate potential sources of systematic uncertainty that could cause the measured asymmetry results to vary from their true values beyond statistical fluctuations.",
"Bunch shuffling involves randomizing the assigned bunch-by-bunch polarization directions of the beam such that the physical asymmetry disappears, thereby isolating the statistical variations present in the data.",
"All asymmetry values have bunch shuffling results consistent with statistical variations except for the lowest $p_T$ bin where there is 7% and 6% more variation beyond what is expected from statistical fluctuations in the $\\pi ^{0}$ and $\\eta $ meson analyses, respectively.",
"These values are used to assign additional systematic uncertainties to the lowest $p_T$ bin of the $\\pi ^{0}$ and $\\eta $ meson asymmetries and dominate the total systematic uncertainty for those bins.",
"Additional cross checks included examining the asymmetries in the two arms separately using Eq.",
"(REF ) and measuring the asymmetry as an explicit function of $\\phi $ .",
"All checks were statistically consistent with the main asymmetry results."
],
[
"Results and discussion",
"Figure REF shows the $A_N$ of neutral pions at midrapidity in $p^\\uparrow $$+$$p$ collisions at $\\sqrt{s}$ =200 GeV, where the bands represent the systematic uncertainty and the bars represent the statistical uncertainty.",
"The comparison to previous results [5] demonstrates the improvement in statistical precision.",
"The inset in Fig.",
"REF shows a zoomed-in comparison at small $p_T$.",
"The new measurement is consistent with our previous measurement and improves the precision on average by a factor of 3.",
"The new measurement of $A_N$ of neutral pions is consistent with zero in the entire $p_T$ range.",
"Figure: Transverse single-spin asymmetry of eta mesons measured at|η|<0.35|\\eta |<0.35 in p ↑ p^\\uparrow ++pp collisions at s=200 GeV \\sqrt{s} = 200 ~{\\rm GeV}.",
"An additionalscale uncertainty of 3.4% due to the polarization uncertainty is notshown.The measurement of $A_N$ of $\\eta $ mesons in $p^\\uparrow $$+$$p$ collisions at $\\sqrt{s}$ = 200 GeV is shown in Fig.",
"REF .",
"This measurement is also compared to the previous result, similarly to Fig.",
"REF .",
"The new measurement is consistent with the previous result and with zero across the entire $p_T$ range.",
"In principle, comparisons of $\\pi ^{0}$ and $\\eta $ meson TSSAs may indicate additional effects from strange quarks, isospin differences, or hadron mass.",
"At forward rapidity, existing measurements [22], [28] do not yet clearly resolve whether the $\\eta $ meson asymmetry is larger than the $\\pi ^{0}$ asymmetry as predicted in some models [29].",
"At midrapidity, there is a larger contribution from gluon dynamics and, as shown in Fig.",
"REF , both asymmetries are consistent with zero and therefore show no evidence for differences due to strangeness, isospin, or mass.",
"Figure: Comparison of the π 0 \\pi ^{0} and η\\eta meson asymmetriesmeasured at |η|<0.35|\\eta |<0.35 in p ↑ p^\\uparrow ++pp collisions at s=200 GeV \\sqrt{s}=200~{\\rm GeV}.An additional scale uncertainty of 3.4% due to the polarizationuncertainty is not shown.Figure: This π 0 \\pi ^{0} asymmetry result plotted with theory calculations forthe asymmetry in both the collinear twist-3 andTMD theoretical frameworks.",
"See text for details.Figure REF shows this $\\pi ^{0}$ TSSA result plotted with theoretical predictions.",
"The qgq curve shows the predicted contribution from collinear twist-3 qgq functions from both the polarized proton and the process of hadronization.",
"This curve was calculated with fits that were published in Ref.",
"[10] and has been reevaluated in the rapidity range of PHENIX.",
"Midrapidity $\\pi ^{0}$ production includes a large fractional contribution from gluons in the proton, so a complete collinear twist-3 description of the midrapidity $\\pi ^{0}$ TSSA would also need to include the contribution from the trigluon correlation function.",
"Given the small expected contribution from the qgq correlation function, this measurement can constrain future calculations of the ggg correlation function, such as those in Ref. [31].",
"The other theory curves in Fig.",
"REF show predictions for the midrapidity $\\pi ^{0}$ TSSA generated by the Sivers TMD PDF.",
"These curves include contributions from both the quark and gluon Sivers functions and have been evaluated for $x_F = 0$ , which approximates the measured kinematics.",
"These calculations use the generalized parton model (GPM) which takes the first $k_T$ moment of the Sivers function (e.g.",
"$\\int {k_T\\cdot q(k_T)}$ ) and does not include next-to-leading-order interactions with the proton fragments.",
"The “GPM” curve uses the parameters stated in Eq.",
"(32) of Ref. [30].",
"The color-gauge-invariant generalized parton model (CGI-GPM) expands on the GPM by including initial- and final-state interactions through the one-gluon exchange approximation.",
"This model has been shown to reproduce the predicted sign change for the quark Sivers function in SIDIS and Drell-Yan.",
"The CGI-GPM curves plotted in Fig.",
"REF show two different scenarios for this model, the specifics of which can be found in Eq.",
"(34) of Ref. [30].",
"The values that are used for the Scenario 1 curve are chosen to maximize the open heavy flavor TSSA generated by the gluon Sivers function while still keeping this asymmetry within the statistical error bars of the published result in Ref.",
"[32] and simultaneously describing the previously published midrapidity $\\pi ^{0}$ TSSA from Ref. [5].",
"The values used in the Scenario 2 curve are similarly calculated, except that they minimize the open heavy flavor TSSA within the range of the published statistical error bars.",
"As shown in the zoomed-in inset of Fig.",
"REF , this $\\pi ^{0}$ TSSA result has the statistical precision at low $p_T$ needed to distinguish between the GPM and CGI-GPM frameworks, preferring CGI-GPM Scenario 2.",
"Measurements of TSSAs in $p$$+$$p$ collisions are essential to understanding the underlying nonperturbative processes which generate them.",
"In particular, further measurements are necessary to clarify certain questions in the interpretations of the TSSAs.",
"For example, the small forward jet asymmetries measured in Ref.",
"[20] have been interpreted as a cancellation of up and down quark asymmetries, implying that the comparatively forward large neutral pion asymmetries include significant contributions from spin-momentum correlations in hadronization [19].",
"Additionally, the $p_T$ dependence of these forward rapidity measurements remains to be clearly understood; measurements of nonzero asymmetries out to even higher $p_T$ would help confirm that these twist-3 observables eventually fall off with increasing hard scale.",
"While the midrapidity measurements here are all consistent with zero, they still provide the highest available statistical precision and $p_T$ reach available at the PHENIX experiment.",
"While forward rapidity light hadron TSSAs are dominated by valence quark spin-momentum correlations in the polarized proton, these midrapidity TSSA measurements are sensitive to both quark and gluon dynamics at leading order.",
"Thus these data also provide further constraints to gluon spin-momentum correlations in transversely polarized protons [31], [33]."
],
[
"Summary",
"The measurements presented here were motivated by the outstanding questions regarding the physical origin of TSSAs.",
"The TSSAs of $\\pi ^{0}$ and $\\eta $ mesons were measured at midrapidity in $p$$+$$p$ collisions at $\\mbox{$\\sqrt{s}$} =200$ GeVby the PHENIX experiment.",
"The measured $\\pi ^{0}$ ($\\eta $ ) meson asymmetry is consistent with zero in the presented $p_T$ range, up to precision of $3\\times 10^{-4}$ $(2\\times 10^{-3})$ in the lowest $p_T$ bins.",
"Both measurements have a significant reduction in uncertainty from previous measurements at midrapidity at RHIC.",
"These data extend previous constraints to any presence of gluon spin-momentum correlations in transversely polarized protons.",
"We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.",
"We also thank D. Pitonyak, S. Yoshida, U.",
"D'Alesio, F. Murgia and C. Pisano for helpful discussions.",
"We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (People's Republic of China), Croatian Science Foundation and Ministry of Science and Education (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), J. Bolyai Research Scholarship, EFOP, the New National Excellence Program (ÚNKP), NKFIH, and OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), Basic Science Research and SRC(CENuM) Programs through NRF funded by the Ministry of Education and the Ministry of Science and ICT (Korea).",
"Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S.",
"Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, the US-Hungarian Fulbright Foundation, and the US-Israel Binational Science Foundation."
]
] | 2011.14170 |
[
[
"Counting centralizers and z-classes of some F-groups"
],
[
"Abstract A finite group $G$ is called an F-group if for every $x, y \\in G \\setminus Z(G)$, $C(x) \\leq C(y)$ implies that $C(x) = C(y)$.",
"On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their centralizers are conjugate in the group.",
"In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$-classes to be equal to the index of its center.",
"We also give a necessary and sufficient condition for the number of $z$-classes of a finite F-group to attain its maximal number (which extends an earlier result).",
"Among other results, we have computed the number of element centralizers and $z$-classes of some finite groups and extend some previous results."
],
[
"Introduction",
"In 1953, Ito [21] introduced the notion of the class of F-groups, consisting of finite groups $G$ in which for every $x, y \\in G \\setminus Z(G)$ , $C(x) \\le C(y)$ implies that $C(x)=C(y)$ , where $C(x)$ and $C(y)$ are centralizers of $x$ and $y$ respectively.",
"Since then the influence of the element centralizers on the structure of groups has been studied extensively.",
"An interesting subclass of F-groups is the class of I-groups, consisting of groups in which all centralizers of non-central elements are of same order.",
"Ito in [21] proved that I-groups are nilpotent and direct product of an abelian group and a group of prime power order.",
"Later on in 2002, Ishikawa [19] proved that I-groups are of class at most 3.",
"In 1971, Rebmann [33] investigated and classified F-groups.",
"In 2007, R. S. Kulkarni [26], [27] introduced the notion of $z$ -classes in a group.",
"Two elements of a group are said to be $z$ -equivalent or in the same $z$ -class if their centralizers are conjugate in the group.",
"$z$ -equivalence is an equivalence relation which is weaker that conjugacy relation.",
"An infinite group generaly contains infinitely many conjugacy classes, but may have finitely many $z$ -classes.",
"In [27] the author observed the influence of the $z$ -classes in the groups of automorphisms of classical geometries and apart from other results he concludes that this finiteness of $z$ -classes can be related to the idea of finiteness of dynamical types of transformation to the geometry.",
"It may be mentioned here that apart from the geometric motivation, finding $z$ -classes of a group itself is of independent interest as a pure combinatoral problem.",
"More information on this and related concepts may be found in [14], [15], [16], [23], [29].",
"In a recent work, the authors in [28], investigated $z$ -classes in finite $p$ -groups.",
"Among other results, they proved that a non-abelian $p$ -group can have at most $\\frac{p^k-1}{p-1}+1$ number of $z$ -classes, where $\\mid \\frac{G}{Z(G)}\\mid =p^k$ and gave the necessary condition to attain the maximal number which is not sufficient.",
"Recently, the authors in [6], gave a necessary and sufficient condition for a finite $p$ -group of conjugate type $(n, 1)$ to attain this maximal number.",
"In this paper, apart from other results, we extend this result and give a necessary and sufficient condition for a finite $F$ -group to attain this maximal number.",
"For a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$ -classes to be equal to the index of its center.",
"As a consequence, we confirm a Conjecture in [4], namely, if $G$ is a finite group such that the number of centralizers is equal to the index of its center, then $G$ is an F-group.",
"Among other results, we have computed the number of element centralizers and $z$ -classes of some groups and improve some earlier results.",
"It may be mentioned here that characterization of groups in terms of the number of element centralizers have been considered by many researchers (see for example [37] for finite groups and [39] for infinite groups).",
"Throughout this paper, $G$ is a group with center $Z(G)$ , commutator subgroup $G^{\\prime }$ and the set of element centralizers $\\operatorname{Cent}(G)$ .",
"We write $Z(x)$ to denote the center of the proper centralizer $C(x)$ and `$z$ -class' to denote the set of $z$ -classes in $G$ ."
],
[
"Preliminaries",
"We begin with some Remarks which will be used in the sequel.",
"Remark 2.1 (See [36]) A collection $\\Pi $ of non-trivial subgroups of a group $G$ is called a partition if every non-trivial element of $G$ belongs to a unique subgroup in $\\Pi $ .",
"If $\\mid \\Pi \\mid =1$ , the partition is said to be trivial.",
"The subgroups in $\\Pi $ are called components of $\\Pi $ .",
"Following Miller, any abelian group having a non-trivial partition is an elementary abelian $p$ -group of order $\\ge p^2$ .",
"Let $S$ be a subgroup of $G$ .",
"A set $\\Pi = \\lbrace H_1, H_2, \\dots , H_n \\rbrace $ of subgroups of $G$ is said to be a strict $S$ -partition of $G$ if $S \\le H_i$ ($i=1, 2, \\dots , n$ ) and every element of $G\\setminus S$ belongs to one and only one subgroup $H_i$ ($i=1, 2, \\dots , n$ ).",
"For more information about partition see [34].",
"Given a group $G$ , let $\\mathcal {A}= \\lbrace C(x) \\mid x \\in G \\setminus Z(G) \\rbrace $ and $\\mathcal {B}= \\lbrace Z(x) \\mid x \\in G \\setminus Z(G) \\rbrace $ .",
"$\\mathcal {A}$ and $\\mathcal {B}$ are partially ordered sets with respect to inclusion and they have the same length.",
"The length of $\\mathcal {A}$ (and of $\\mathcal {B}$ ) is called the rank of $G$ .",
"A group $G$ has rank 1 if and only if $\\mathcal {B}$ is a strict $Z(G)$ partition.",
"Recall that a finite group $G$ is called an F-group if for every $x, y \\in G \\setminus Z(G)$ , $C(x) \\le C(y)$ implies that $C(x) = C(y)$ .",
"Following [12], a finite non-abelian group $G$ is an F-group if and only if $\\mathcal {B}$ is a strict $Z(G)$ partition.",
"Hence being a group of rank 1 is equivalent to being an F-group.",
"Remark 2.2 Given a group $G$ , two elements $x, y \\in G$ are said to be $z$ -equivalent or in the same $z$ -class if their centralizers are conjugate in $G$ , i.e., if $C(x)=gC(y)g^{-1}$ for some $g \\in G$ .",
"It is well known that “being $z$ -equivalent\" is an equivalenve relation on $G$ .",
"Following [27], the order of the $z$ -class of $x$ , if finite, is given by $\\mid z-class \\; of\\; x \\mid =\\mid G : N_G(C(x)) \\mid .",
"\\mid F_x^{\\prime } \\mid ,$ where $F_x^{\\prime }:= \\lbrace y \\in G \\mid C(x)=C(y)\\rbrace $ .",
"From this it is easy to see that the number of $z$ -classes in $G$ i.e., $\\mid z-class \\mid = \\mid \\operatorname{Cent}(G) \\mid $ if and only if $C(x) \\lhd G$ for all $x \\in G $ .",
"The following Theorems will be used to obtain some of our results.",
"For basic notions of isoclinism, see [17], [28].",
"Theorem 2.3 (p.135 [17]) Every group is isoclinic to a group whose center is contained in the commutator subgroup.",
"Theorem 2.4 (Theorem A [22], Lemma 3.2 [39]) Any two isoclinic groups have the same number of centralizers.",
"Theorem 2.5 (Theorem 11 [22], Theorem 3.3 [39]) The representatives of the families of isoclinic groups with $n$ -centralizers ($n \\ne 2, 3$ ) can be chosen to be finite group.",
"Theorem 2.6 (Lemma 4, p. 303 [11]) Let $G$ be a finite group with an abelian normal subgroup of prime index $p$ .",
"Then $\\mid G \\mid =p.\\mid Z(G)\\mid .",
"\\mid G^{\\prime } \\mid $ ."
],
[
"The main results",
"In this section, we prove the main results of the paper.",
"Let $D_8$ be the dihedral group of order 8.",
"It is easy to verify that $\\mid \\operatorname{Cent}(D_8)\\mid =\\mid \\frac{D_8}{Z(D_8)}\\mid $ .",
"The following result gives a necessary and sufficient condition for $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ , where $G$ is a finite group.",
"Recall that $Z(x)$ denotes the center of the proper centralizer $C(x)$ .",
"Proposition 3.1 Let $G$ be a finite group.",
"Then $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ iff $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ .",
"Let $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid =n$ and $C(x_i), 1\\le i \\le n-1$ be the proper centralizers of $G$ .",
"Suppose $x_iZ(G)=x_jZ(G)$ for some $1 \\le i, j \\le n-1$ ; $i \\ne j$ .",
"Then $x_iz_1=x_jz_2$ for some $z_1, z_2 \\in Z(G)$ and consequently, $C(x_i)=C(x_iz_1)=C(x_jz_2)=C(x_j)$ , which is a contradiction.",
"Hence $G=Z(G) \\sqcup x_1Z(G) \\sqcup x_2Z(G) \\sqcup \\dots \\sqcup x_{n-1}Z(G).$ Now, for some $i, 1 \\le i \\le n-1$ suppose $Z(x_i)=Z(G) \\sqcup x_iZ(G) \\sqcup X$ , where $X= \\sqcup x_jZ(G)$ for some $1 \\le j \\le n-1, i \\ne j$ .",
"Then $C(x_i)=C(x_l)$ for some $1 \\le l \\le n-1, i \\ne l$ , which is a contradiction.",
"Therefore $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ .",
"Conversely, suppose $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ .",
"Then $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ , noting that in the present scenario, for any $x \\in G \\setminus Z(G)$ , $Z(x)$ will contain exactly one right coset of $Z(G)$ other than $Z(G)$ .",
"The authors in [28] proved that if $G$ is a group in which $Z(G)$ has finite index, then the number of $z$ -classes is at most the index $[G:Z(G)]$ .",
"In the following result, for a finite group we give a necessary and sufficient condition for $\\mid z-class \\mid $ to attain its upper bound.",
"We also confirm a Conjecture in [4], namely if $G$ is a finite group such that $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ , then $G$ is an F-group.",
"Theorem 3.2 Let $G$ be a finite group.",
"(a) If $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ , then $G$ is an F-group.",
"([4]).",
"(b) If $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ , then $\\frac{G}{Z(G)}$ is elementary abelian 2-group.",
"([4]) (c) $\\mid z-class \\mid =\\mid \\frac{G}{Z(G)}\\mid $ iff $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ .",
"(d) If $\\mid z-class \\mid =\\mid \\frac{G}{Z(G)}\\mid $ , then $G$ is an F-group.",
"a) By Proposition REF , we have $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ and hence $Z(x) \\cap Z(y)=Z(G)$ for any $x, y \\in G \\setminus Z(G)$ with $C(x) \\ne C(y)$ .",
"Therefore in view of Remark REF , $G$ is an F-group.",
"b) By Proposition REF , we have $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ .",
"Therefore $\\Pi = \\lbrace \\frac{Z(x)}{Z(G)} \\mid x \\in G \\setminus Z(G)\\rbrace $ is a partition of $\\frac{G}{Z(G)}$ and hence the result follows.",
"c) Suppose $\\mid z-class \\mid =\\mid \\frac{G}{Z(G)}\\mid $ .",
"By [28], $\\frac{G}{Z(G)}$ is abelian and hence using Remark REF , $\\mid \\operatorname{Cent}(G) \\mid =\\mid z-class \\mid =\\mid \\frac{G}{Z(G)}\\mid $ .",
"Now, the result follows from Proposition REF .",
"Converesely, suppose $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ .",
"Then in view of Proposition REF , Corollary REF and Remark REF , $\\mid z-class \\mid =\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ .",
"d) From(c), we have $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ and hence $Z(x) \\cap Z(y)=Z(G)$ for any $x, y \\in G \\setminus Z(G)$ with $C(x) \\ne C(y)$ .",
"Therefore in view of Remark REF , $G$ is an F-group.",
"Recall that a $p$ -group ($p$ a prime) is said to be special if its center and commutator subgroup coincide and are elementary abelian.",
"Furthermore, a group $G$ is extraspecial if $G$ is a special $p$ -group and $\\mid G^{\\prime } \\mid =\\mid Z(G) \\mid =p$ .",
"We now give the following result concerning the upper and lower bounds of $\\mid \\operatorname{Cent}(G) \\mid $ .",
"For lower bounds of $\\mid \\operatorname{Cent}(G) \\mid $ one may also see [38].",
"Theorem 3.3 Let $G$ be a finite group and $p$ be the smallest prime divisor of its order.",
"Then (a) $p+2 \\le \\mid \\operatorname{Cent}(G) \\mid $ ; with equality if and only if $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ ; equivalently, if and only if $G$ is isoclinic to an extraspecial group of order $p^3$ .",
"(b) $\\mid \\operatorname{Cent}(G) \\mid \\le \\mid \\frac{G}{Z(G)}\\mid $ ; with equality if and only if $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ ; equivalently, $G$ is isoclinic to a special 2-group.",
"a) Using arguments similar to [7], we have $p+2 \\le \\mid \\operatorname{Cent}(G) \\mid $ with equality if and only if $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ .",
"Next, suppose $G$ is isoclinic to an extraspecial group of order $p^3$ .",
"Then by Theorem REF and [10], we have $p+2 = \\mid \\operatorname{Cent}(G) \\mid $ .",
"Conversely, Suppose $p+2 = \\mid \\operatorname{Cent}(G) \\mid $ .",
"Then we have $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ .",
"In the present scenario, by Theorem REF and Theorem REF , $G$ is isoclinic to a finite group $H$ of order $p^n$ with $Z(H) \\subseteq H^{\\prime }$ .",
"Moreover, since $\\frac{H}{Z(H)} \\cong C_p \\times C_p$ , therefore $Z(H) = H^{\\prime }$ and $H$ has an abelian subgroup of index $p$ .",
"Hence using Theorem REF , we have $p^n=p.\\mid Z(H) \\mid .\\mid H^{\\prime }\\mid $ and consequently, $H$ is an extraspecial group of order $p^3$ .",
"b) Since $C(xz_1)=C(xz_2)$ for all $xz_1, xz_2 \\in xZ(G)$ , therefore $\\mid \\operatorname{Cent}(G) \\mid \\le \\mid \\frac{G}{Z(G)}\\mid $ ; with equality if and only if $\\mid \\frac{Z(x)}{Z(G)}\\mid =2$ for all $x \\in G \\setminus Z(G)$ by Proposition REF .",
"For the last part, suppose $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ .",
"Then in view of Theorem REF , $\\frac{G}{Z(G)}$ is elementary abelian 2-group.",
"Therefore $G=A \\times H$ , where $A$ is an abelian group and $H$ is the sylow 2-subgroup of $G$ with $\\frac{H}{Z(H)}$ elementary abelian.",
"Consequently, $G$ is isoclinic to $H$ .",
"In the present scenario, by Theorem REF and Theorem REF , $H$ is isoclinic to a finite group $H_1$ of order $p^n$ with $Z(H_1) \\subseteq H_1^{\\prime }$ .",
"Since $\\frac{H_1}{Z(H_1)}$ is abelian, therefore $Z(H_1)=H_1^{\\prime }$ .",
"It now follows that $H_1^{\\prime }$ and $\\frac{H_1}{Z(H_1)}$ have same exponent (by [2]).",
"In the present scenario, we have $H_1=Z(H_1)$ is elementary abelian 2-group and hence $H_1$ is a special 2-group isoclinic to $G$ .",
"We now give our first counting formula for number of distinct centralizers: Proposition 3.4 Let $G$ be a finite F-group such that $\\mid \\frac{G}{Z(G)}\\mid =p^k$ .",
"If $\\mid \\frac{Z(x)}{Z(G)}\\mid =p^m$ for all $x \\in G \\setminus Z(G)$ , then $\\mid \\operatorname{Cent}(G) \\mid =\\frac{p^k-1}{p^m-1}+1$ .",
"Since $G$ is an F-group, therefore by Remark REF we have $Z(x) \\cap Z(y)=Z(G)$ for any $x, y \\in G \\setminus Z(G)$ with $C(x) \\ne C(y)$ .",
"Hence the result follows, noting that each $Z(x)$ contains exactly $(p^m-1)$ distinct right cosets of $Z(G)$ other than $Z(G)$ .",
"The following result generalizes [3].",
"Theorem 3.5 Let $G$ be a finite group and $p$ a prime.",
"Then $\\mid \\frac{Z(x)}{Z(G)}\\mid =p$ for all $x \\in G \\setminus Z(G)$ if and only if (a) $G$ is an F-group.",
"(b) $\\frac{G}{Z(G)}$ is of exponent $p$ .",
"(c) $\\mid \\operatorname{Cent}(G) \\mid =\\frac{p^k-1}{p-1}+1$ , where $\\mid \\frac{G}{Z(G)}\\mid =p^k$ .",
"Suppose $\\mid \\frac{Z(x)}{Z(G)}\\mid =p$ for all $x \\in G \\setminus Z(G)$ .",
"a) Clearly, $\\Pi =\\lbrace \\frac{Z(x)}{Z(G)} \\mid x \\in G \\setminus Z(G)\\rbrace $ is a partition of $\\frac{G}{Z(G)}$ .",
"Hence by Remark REF , $G$ is an F-group.",
"b) It is clear from the proof of (a) that every element of $\\frac{G}{Z(G)}$ is of order $\\le p$ .",
"c) Clearly, for any $x \\in G \\setminus Z(G)$ , $Z(x)$ contains exactly $(p-1)$ distinct right cosets of $Z(G)$ different from $Z(G)$ .",
"And hence the result follows.",
"Conversely, suppose $G$ is an F-group, $\\mid \\frac{G}{Z(G)}\\mid =p^k$ and $\\mid \\operatorname{Cent}(G) \\mid =\\frac{p^k-1}{p-1}+1 (=l)$ .",
"Let $C(x_i),1 \\le i \\le l-1 $ be the proper centralizers of $G$ .",
"Since $G$ is an F-group, therefore by Remark REF , $\\Pi = \\lbrace \\frac{Z(x_i)}{Z(G)} \\mid 1\\le i \\le l-1\\rbrace $ is a partition of $\\frac{G}{Z(G)}$ .",
"In the present scenario, we have $p^k=\\mid \\frac{Z(x_1)}{Z(G)} \\mid + \\mid \\frac{Z(x_2)}{Z(G)} \\mid + \\dots +\\mid \\frac{Z(x_{l-1})}{Z(G)} \\mid -\\frac{p^k-1}{p-1}+1$ .",
"Consequently, $\\mid \\frac{Z(x)}{Z(G)} \\mid =p$ for all $x \\in G \\setminus Z(G)$ .",
"As an immediate corollary, we have the following result ([3]).",
"Recall that a finite group $G$ is said to be a CA-group if centralizer of every noncentral element of $G$ is abelian.",
"Corollary 3.6 Let $G$ be a finite F-group such that $\\mid \\frac{G}{Z(G)}\\mid =p^4$ .",
"If $G$ is not a CA-group, then $\\mid \\operatorname{Cent}(G) \\mid =p^3+p^2+p+2$ .",
"It follows from Theorem REF , noting that in the present scenario, we have $\\mid \\frac{Z(x)}{Z(G)}\\mid =p$ for all $x \\in G \\setminus Z(G)$ .",
"Following Ito [21], a finite group $G$ is said to be of conjugate type $(n, 1)$ if every proper centralizer of $G$ is of index $n$ .",
"He proved that a group of conjugate type $(n, 1)$ is nilpotent and $n=p^a$ for some prime $p$ .",
"Moreover, he also proved that a group of conjugate type $(p^a, 1)$ is a direct product of a $p$ -group of the same type and an abelian group.",
"The author in [20] classified finite $p$ -groups of conjugate type $(p, 1)$ and $(p^2, 1)$ upto isoclinism.",
"In the following result, we calculate the number of element centralizers and $z$ -classes of a finite group of conjugate type $(p, 1)$ .",
"Given a group $G$ , $nacent(G)$ denotes the set of non-abelian centralizers of $G$ .",
"For more information about $nacent(G)$ see [5], [24].",
"Proposition 3.7 Let $G$ be a finite group such that $ \\mid \\frac{G}{C(x)} \\mid =p$ for all $x \\in G \\setminus Z(G)$ .",
"Then (a) $\\frac{G}{Z(G)}$ is elementary abelian $p$ -group of order $p^k$ for some $k$ .",
"(b) $\\mid \\operatorname{Cent}(G) \\mid =\\mid $ z$-class\\mid =\\frac{p^k-1}{p-1}+1$ .",
"(c) $\\mid nacent(G)\\mid = 1$ iff $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ .",
"(d) $\\mid nacent(G)\\mid = \\mid \\operatorname{Cent}(G) \\mid $ iff $\\frac{G}{Z(G)}$ is of order $> p^2$ .",
"a) In view of Ito [21], $G = A \\times P$ , where $A$ is an abelian group and $P$ is a $p$ -group of conjugate type $(p, 1)$ .",
"Therefore using Remark REF , $\\frac{G}{Z(G)}$ is elementary abelian $p$ -group of order $p^k$ for some $k$ .",
"b) The result follows from Theorem REF and Remark REF , noting that in the present scenario, by [32], we have $\\mid \\frac{Z(x)}{Z(G)}\\mid =p$ for all $x \\in G \\setminus Z(G)$ .",
"c) If $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ , then $G$ is a CA-group and hence $\\mid nacent(G)\\mid = 1$ .",
"Conversely, suppose $\\mid nacent(G)\\mid = 1$ .",
"Then view of [32], we have $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ .",
"d) In view of [32], we have $\\mid \\frac{Z(x)}{Z(G)}\\mid =p$ for all $x \\in G \\setminus Z(G)$ .",
"Hence the result follows.",
"We now compute the number of distinct centralizers and $z$ -classes of an extraspecial $p$ -group.",
"It is well known that for any prime $p$ and every positive integer $a$ , there exists, upto isomorphism, exactly two extraspecial groups of order $p^{2a+1}$ .",
"Proposition 3.8 Let $G$ be an extraspecial $p$ -group of order $p^{2a+1}$ for some prime $p$ .",
"Then $\\mid \\operatorname{Cent}(G) \\mid =\\mid z-class \\mid =\\frac{p^{2a}-1}{p-1}+1$ .",
"Using [31], we have $\\mid C(x) \\mid = p^{2a}$ for all $x \\in G \\setminus Z(G)$ .",
"Now the result follows from Proposition REF , noting that here we have $\\mid Z(G) \\mid =p$ .",
"Our next result concerns about finite groups of conjugate type $(p^2, 1)$ .",
"Proposition 3.9 Let $G$ be a finite group such that $ \\mid \\frac{G}{C(x)} \\mid =p^2$ for all $x \\in G \\setminus Z(G)$ .",
"Then one of the following assertions hold: (a) $\\frac{G}{Z(G)}$ is elementary abelian $p$ -group.",
"(b) $\\frac{G}{Z(G)}$ is non-abelian of order $p^3$ ($p$ odd) and of exponent $p$ ; and $\\mid \\operatorname{Cent}(G) \\mid =p^2+p+2$ .",
"If $\\frac{G}{Z(G)}$ is abelian, then by Remark REF , $\\frac{G}{Z(G)}$ is elementary abelian.",
"Next, suppose $\\frac{G}{Z(G)}$ is non-abelian.",
"In view of [32], $\\frac{Z(x)}{Z(G)}=p$ for all $x \\in G \\setminus Z(G)$ .",
"In the present scenario, if $p=2$ , then $\\mid \\operatorname{Cent}(G) \\mid =\\mid \\frac{G}{Z(G)}\\mid $ and hence using Corollary REF , $\\frac{G}{Z(G)}$ is elementary abelian, which is a contradiction.",
"Therefore $p$ is odd and consequently, using [20], $G$ is isoclinic to a group of order $p^5$ with center of order $p^2$ .",
"Hence $\\frac{G}{Z(G)}$ is non-abelian of order $p^3$ .",
"Since $G$ is an F-group, therefore by Remark REF , $\\Pi =\\lbrace \\frac{Z(x)}{Z(G)} \\mid x \\in G \\setminus Z(G)\\rbrace $ is a partition of $\\frac{G}{Z(G)}$ and hence $\\frac{G}{Z(G)}$ is of exponent $p$ .",
"Moreover, by Theorem REF we have $\\mid \\operatorname{Cent}(G) \\mid =p^2+p+2$ .",
"It may be mentioned here that for the group G:= Small group (64, 73) in [40], we have $\\frac{G}{Z(G)} \\cong C_2 \\times C_2 \\times C_2$ and $ \\mid \\frac{G}{C(x)} \\mid =4$ for all $x \\in G \\setminus Z(G)$ .",
"The following Proposition generalizes [3].",
"Proposition 3.10 Let $G$ be a finite F-group such that $\\mid \\frac{G}{Z(G)}\\mid =p^k$ .",
"If $\\mid \\frac{Z(x)}{Z(G)} \\mid \\le p^2$ for all $x \\in G \\setminus Z(G)$ , then $\\mid \\operatorname{Cent}(G) \\mid =p^{k-1}+p^{k-2}+ \\dots +p+2-vp$ , where $v$ is the number of centralizers for which $\\frac{Z(x)}{Z(G)} = p^2$ .",
"Since $G$ is a finite F-group, therefore by Remark REF , $Z(x) \\cap Z(y)=Z(G)$ for all $x, y \\in G \\setminus Z(G)$ with $C(x) \\ne C(y)$ .",
"Let $v$ be the number of centralizers for which $\\frac{Z(x)}{Z(G)} = p^2$ .",
"Then the centers of these $v$ number of centralizers will contain exactly $v(p^2-1)$ distinct right cosets of $Z(G)$ different from $Z(G)$ .",
"On the other-hand the center of each of the remaining proper centralizers will contain exactly $(p-1)$ distinct right cosets of $Z(G)$ other than $Z(G)$ .",
"Consequently, $\\mid \\operatorname{Cent}(G) \\mid = \\frac{(p^k-1)-v(p^2-1)}{p-1}+v+1=p^{k-1}+p^{k-2}+ \\dots +p+2-vp.$ As an immediate corollary we obtain the following result for finite groups of conjugate type $(p^2, 1)$ .",
"As we have already mentioned, Ishikawa [19] proved that I-groups are of class at most 3.",
"For finite groups of class 3 having conjugate type $(p^2, 1)$ , we obtained in Proposition REF that $\\mid \\operatorname{Cent}(G) \\mid =p^2+p+2$ .",
"Corollary 3.11 Let $G$ be a finite group such that $ \\mid \\frac{G}{C(x)} \\mid =p^2$ for all $x \\in G \\setminus Z(G)$ .",
"Then $\\mid \\operatorname{Cent}(G) \\mid =p^{k-1}+p^{k-2}+ \\dots +p+2-vp$ , where $\\mid \\frac{G}{Z(G)}\\mid =p^k$ and $v$ is the number of centralizers for which $\\frac{Z(x)}{Z(G)}=p^2$ .",
"It follows from Proposition REF , noting that in the present scenario, in view of [32], we have $\\mid \\frac{Z(x)}{Z(G)} \\mid \\le p^2$ for all $x \\in G \\setminus Z(G)$ .",
"We now prove the following result which improves [7].",
"A group $G$ is said to be $n$ -centralizer if $\\mid \\operatorname{Cent}(G) \\mid =n$ .",
"Proposition 3.12 Let $G$ be a finite $n(=p^2+2)$ -centralizer group.",
"Then $\\mid \\frac{G}{C(x)}\\mid =p^2$ for all $x \\in G \\setminus Z(G)$ iff $\\frac{G}{Z(G)} \\cong C_p \\times C_p \\times C_p \\times C_p$ and $G$ is an F-group.",
"Suppose $\\mid \\frac{G}{C(x)}\\mid =p^2$ for all $x \\in G \\setminus Z(G)$ .",
"Then $G$ is an F-group.",
"In view of Proposition REF , $\\frac{G}{Z(G)}$ is elementary abelian.",
"Let $X_i=C(x_i), 1\\le i \\le n-1$ where $x_i \\in G \\setminus Z(G)$ .",
"We have $G=\\overset{n-1}{\\underset{i=1}{\\cup }}X_i$ and $\\mid G \\mid =\\overset{n-1}{\\underset{i=1}{\\sum }} \\mid X_i \\mid $ .",
"Therefore by [13], we have $G=X_1X_2$ and $X_1 \\cap X_2=Z(G)$ .",
"Hence $\\frac{G}{Z(G)} \\cong C_p \\times C_p \\times C_p \\times C_p$ .",
"Conversely, suppose $\\frac{G}{Z(G)} \\cong C_p \\times C_p \\times C_p \\times C_p$ and $G$ is a finite F-group.",
"In view of Corollary REF , $G$ is a CA-group.",
"Now, suppose $\\mid \\frac{G}{C(x)}\\mid =p$ for some $x \\in G \\setminus Z(G)$ .",
"Then in view of Theorem REF and [9], $\\mid \\operatorname{Cent}(G) \\mid \\ne p^2+2$ .",
"Therefore $\\mid \\frac{Z(x)}{Z(G)} \\mid \\le p^2$ for all $x \\in G \\setminus Z(G)$ and consequently, using Corollary REF we have $\\mid \\frac{G}{C(x)}\\mid =p^2$ for all $x \\in G \\setminus Z(G)$ .",
"Remark 3.13 Let $G$ be a finite group and let $H$ be a subgroup of $G$ .",
"Then $(G, H)$ is called a Camina pair if $x$ is conjugate in $G$ to $xz$ for all $x \\in G \\setminus H$ .",
"A finite group $G$ is called a Camina group if $(G, G^{\\prime })$ is a Camina pair.",
"Recall that a $p$ -group $G$ is semi-extraspecial, if $G$ satisfies the property for every maximal subgroup $N$ of $Z(G)$ that $\\frac{G}{N}$ is an extraspecial group.",
"It is known that every semi-extraspecial group is a special group ([31]) and for such groups $\\mid C(x) \\mid $ is equal to the index $[G: G^{\\prime }]$ for all $x \\in G \\setminus G^{\\prime }$ ([31]).",
"Following [31], a group $G$ is a semi-extraspecial $p$ -group for some prime $p$ if and only if $G$ is a Camina group of nilpotent class 2.",
"A group $G$ is said to be ultraspecial if $G$ is semi-extraspecial and $\\mid G^{\\prime } \\mid = \\sqrt{\\mid G: G^{\\prime } \\mid }$ .",
"It is known that all of the ultraspecial groups of order $p^6$ are isoclinic ([31]).",
"As an immediate application to the above Proposition, we have the following result: Proposition 3.14 Let $G$ be a finite $n(=p^2+2)$ -centralizer group.",
"Then $\\mid \\frac{G}{C(x)}\\mid =p^2$ for all $x \\in G \\setminus Z(G)$ if and only if $G$ is isoclinic to an ultraspecial group of order $p^6$ .",
"In view of Proposition REF , $\\frac{G}{Z(G)}$ is abelian.",
"Therefore $G$ is nilpotent of class 2 and hence by [20], Proposition REF and Remark REF we have the result.",
"In [8], the author proved that if $G$ is a finite 6-centralizer group, then $\\frac{G}{Z(G)} \\cong D_8, A_4, C_2 \\times C_2 \\times C_2$ or $C_2 \\times C_2 \\times C_2 \\times C_2$ .",
"In this connection, we have the following result: Proposition 3.15 Let $G$ be a finite group such that $\\mid \\frac{G}{Z(G)} \\mid =16$ .",
"Then $G$ is 6-centralizer if and only if $G$ is isoclinic to an ultraspecial group of order 64.",
"Suppose $\\mid \\operatorname{Cent}(G) \\mid =6$ .",
"Using [1], $G$ is a CA-group.",
"Now, suppose $\\mid \\frac{G}{C(x)}\\mid =2$ for some $x \\in G \\setminus Z(G)$ .",
"Then in view of Theorem REF and [9], $\\mid \\operatorname{Cent}(G) \\mid \\ne 6$ .",
"Consequently, by Proposition REF , we have $\\mid \\frac{G}{C(x)}\\mid =4$ for all $x \\in G \\setminus Z(G)$ .",
"Therefore by Proposition REF , $G$ is isoclinic to an ultraspecial group of order 64.",
"Conversely, suppose $G$ is isoclinic to an ultraspecial group of order 64.",
"Then by Remark REF , $\\mid G^{\\prime } \\mid =4$ and $\\mid \\frac{G}{C(x)}\\mid =4$ for all $x \\in G \\setminus Z(G)$ .",
"Since we have $\\mid \\frac{G}{Z(G)} \\mid =16$ , therefore by Proposition REF , $\\mid \\operatorname{Cent}(G) \\mid = 6$ .",
"We now compute the number of centralizers of a finite group with maximal centralizers (maximal among the subgroups).",
"It may be mentioned here that Kosvintsev in 1973 [25] studied and characterised these groups.",
"He proved that in a finite nilpotent group every centralizer is maximal if and only if $G$ is of the conjugate type $(p, 1)$ .",
"Moreover, he proved that in a finite non-nilpotent group $G$ the centralizer of every non-central is a maximal subgroup if and only if $G=MZ(G)$ , where $M$ is a biprimary subgroup in $G$ that is a Miller-Moreno group.",
"Recently, in 2020 the authors [35] studied these groups.",
"It seems that the authors are unaware of the paper of Kosvintsev [25].",
"However they have given a characterization of such non-nilpotent group $G$ in terms of $\\frac{G}{Z(G)}$ by proving that if $G$ is a non-nilpotent group of such type, then $\\frac{G}{Z(G)}$ is either abelian or $\\frac{G}{Z(G)}=\\frac{P}{Z(G)} \\rtimes \\frac{Q}{Z(G)}$ is a minimal non-abelian group (Miller and Moreno analyzed minimal non-abelian groups in [30]) with $\\mid \\frac{P}{Z(G)} \\mid =p^a$ and $\\mid \\frac{Q}{Z(G)} \\mid =q$ , where $p$ and $q$ are primes.",
"In this connection, we notice that there is a minor error in the result.",
"In this case $\\frac{G}{Z(G)}$ cannot be abelian.",
"Proposition 3.16 Let $G$ be a finite group in which the centralizer of any non-central element of $G$ is maximal (maximal among all subgroups).",
"(a) If $G$ is nilpotent, then $\\mid \\frac{G}{Z(G)} \\mid =p^k$ for some $k \\in N $ and $\\mid \\operatorname{Cent}(G) \\mid =\\mid $ z$-class\\mid =\\frac{p^k-1}{p-1}+1$ .",
"Moreover, $\\mid nacent(G)\\mid = 1$ iff $\\frac{G}{Z(G)} \\cong C_p \\times C_p$ and $\\mid nacent(G)\\mid = \\mid \\operatorname{Cent}(G) \\mid $ iff $\\frac{G}{Z(G)}$ is of order $> p^2$ .",
"(b) If $G$ is non-nilpotent, then $\\mid \\operatorname{Cent}(G) \\mid = \\mid \\operatorname{Cent}(\\frac{G}{Z(G)}) \\mid =p^a+2$ , where $p^a$ is the order of the $p$ -Sylow subgroup of $\\frac{G}{Z(G)}$ .",
"Moreover, $G$ is a CA-group.",
"a) Since $G$ is nilpotent, therefore in view of [25], we have $\\mid \\frac{G}{C(x)}\\mid =p$ ($p$ a prime) for all $x \\in G \\setminus Z(G)$ .",
"Now, the result follows using Proposition REF .",
"b) In view of [35], we have $\\frac{G}{Z(G)}=\\frac{P}{Z(G)} \\rtimes \\frac{Q}{Z(G)}$ is a minimal non-abelian group with $\\mid \\frac{P}{Z(G)} \\mid =p^a$ and $\\mid \\frac{Q}{Z(G)} \\mid =q$ , where $p$ and $q$ are primes.",
"Moreover, by [18], $\\frac{G}{Z(G)}$ has trivial center.",
"In the present scenario, we have $\\frac{C(x)}{Z(G)}=C(xZ(G))$ for any $x \\in G \\setminus Z(G)$ .",
"Hence $\\mid \\operatorname{Cent}(G) \\mid = \\mid \\operatorname{Cent}(\\frac{G}{Z(G)}) \\mid =p^a+2$ and $G$ is a CA-group, noting that in the present scenario $G$ is an F-group.",
"The authors in [6], gave a necessary and sufficient condition for a finite $p$ -group of type $(n, 1)$ to attain the maximal number of $z$ -classes.",
"In the following Theorem we extend this result as follows: Theorem 3.17 Let $G$ be a finite F-group with $\\mid \\frac{G}{Z(G)}\\mid =p^k$ , where $p$ is a prime.",
"Then $G$ has $\\frac{p^k-1}{p-1}+1$ $z$ -classes if and only if (a) $\\frac{G}{Z(G)}$ is elementary abelian and (b) For all $x \\in G \\setminus Z(G)$ , $Z(x)= \\langle x, Z(G)\\rangle $ .",
"Let $G$ be a finite F-group such that $\\mid \\frac{G}{Z(G)}\\mid =p^k$ and $G$ has $\\frac{p^k-1}{p-1}+1$ $z$ -classes.",
"By [28], $G$ is isoclinic to a finite $p$ -group $H$ and by [28], $H$ has $\\frac{p^k-1}{p-1}+1$ $z$ -classes.",
"In the present scenario, by [28], $\\frac{H}{Z(H)}$ is elementary abelian and consequently, $\\frac{G}{Z(G)}$ is elementary abelian.",
"Hence $C(x) \\lhd G$ for all $x \\in G $ and therefore, by Remark REF , $\\mid \\operatorname{Cent}(G) \\mid =\\frac{p^k-1}{p-1}+1$ .",
"Now, the result follows from Theorem REF .",
"Conversely, suppose (a) and (b) holds.",
"Then $C(x) \\lhd G$ for all $x \\in G $ and consequently, by Remark REF and Theorem REF we have $\\mid z-class \\mid =\\frac{p^k-1}{p-1}+1$ .",
"We conclude the paper with the following result: Proposition 3.18 Let $G$ be an finite group such that $\\mid G^{\\prime } \\mid =p$ ($p$ a prime) and $G^{\\prime } \\subseteq Z(G)$ .",
"Then $G$ is isoclinic to an extraspecial $p$ -group of order $p^{2a+1}$ and $\\mid \\operatorname{Cent}(G) \\mid =\\mid z-class \\mid =\\frac{p^{2a}-1}{p-1}+1$ .",
"Since $G^{\\prime } \\subseteq Z(G)$ therefore $G=A \\times H$ , where $A$ is an abelian subgroup and $H$ is the sylow $p$ -subgroup of $G$ with $\\mid H^{\\prime } \\mid =p$ .",
"Consequently, $G$ is isoclinic to $H$ .",
"In the present scenario, by Theorem REF and Theorem REF , $H$ is isoclinic to a finite $p$ -group $H_1$ with $Z(H_1) \\subseteq H_1^{\\prime }$ .",
"Therefore we have $\\mid Z(H_1) \\mid =\\mid H_1^{\\prime } \\mid =p$ .",
"It now follows that $H_1^{\\prime }$ and $\\frac{H_1}{Z(H_1)}$ have same exponent (by [2]).",
"Thus $H_1$ is an extraspecial $p$ -group of order $p^{2a+1}$ and $G$ is isoclinic to $H_1$ .",
"Moreover, by Proposition REF and Theorem REF we have $\\mid \\operatorname{Cent}(G) \\mid =\\mid z-class \\mid =\\frac{p^{2a}-1}{p-1}+1$ .",
"Acknowledgment I would like to thank Prof. Mohammad Zarrin for his valuable suggestions and comments on the earlier draft of the paper."
],
[
"Acknowledgment",
"I would like to thank Prof. Mohammad Zarrin for his valuable suggestions and comments on the earlier draft of the paper."
]
] | 2011.14071 |
[
[
"On characterization of functions preserving metric-type conditions via\n triangular and polygonal structures"
],
[
"Abstract Following the train of thought from our previous paper we revisit the theorems of Pongsriiam and Termwuttipong by further developing their characterization of certain property-preserving functions using the so-called triangle triplets.",
"We develop more general analogues of disjoint sum lemmas for broader classes of metric-type spaces and we apply these to extend results of Bors\\`ik an Dobo\\v{s} as well as those obtained by Khemaratchatakumthorn and Pongsriiam.",
"As a byproduct we obtain methods of generating non-trivial and infinite strong $b$-metric spaces which are not metric."
],
[
"Introduction",
"The first appearances of metric transforms in the literature date back to 1935, where Wilson [36] has investigated functions preserving the triangle inequality and some other properties.",
"Most researchers in this field agree that the second important paper in this field was due to Sreenivasan, twelve years later [32].",
"Additionally, the well-known monograph of Kelley [21] presents some results from this theory as exercises.",
"The theory of metric preserving functions was developed by Borsík, Doboš and Piotrowski [4], [5], [12], [13], [14], [11].",
"See also lectures on this theory by Doboš [10] and an introductory article by Corazza [9].",
"Of course this list is by no means exhaustive, as many other mathematicians, including Pokorný [28] and Vallin [33], [34], [35], had contributed to this topic.",
"In 2013, Petruşel et al.",
"[27] have shown applications of metric transforms in metric fixed point theory.",
"Another research on this topic which emphasizes the fixed point theory, was simultaneously conducted by Kirk and Shahzad [25], well known for their broad contributions to the field of metric fixed point theory.",
"In their monograph [26], they highlighted the problem of very few non-trivial, natural examples of strong $b$ -metric spaces in the literature.",
"We would like to tackle this issue to some extent in this paper.",
"Also, another follow-up in this topic was made by Pongsriiam and Termwuttipong [29], who generalized the results of Kirk and Shahzad from their paper [25].",
"This group of Thai mathematicians, namely, Khemaratchatakumthorn, Pongsriiam, Samphavat and Termwuttipong [22], [23], [24], [29], [30], [31] have made a huge contribution to extension of theory of metric preserving functions.",
"The functions they have investigated were preserving some other (not necessarily metric) axioms of the distance-type functions.",
"This general name refers to all semimetrics satisfying a certain condition resembling the triangle inequality.",
"In particular, this group of mathematicians focused mainly on $b$ -metric spaces and ultrametric spaces.",
"It is worth noticing that the latter class of spaces was also investigated by other researchers, see [15], [16].",
"Although the concept of a $b$ -metric space (also called the quasimetric space) was introduced as early as 1937 by Frink [18], it seemed to be out of favour for quite a long time.",
"Recently, $b$ -metrics and many other generalizations of the concept of metric space attract many scientists and appear in an increasing number of fields.",
"For a survey on these generalizations and their applications, refer to a paper of Van An et al.",
"[1].",
"For review of recent progress in the fixed point theory in $b$ -metric spaces, see the survey article of Karapınar [20].",
"We also recommend a paper of Cobzaş [8] and a broad list of references therein.",
"The purpose of this paper is to extend the already known characterizations (based on triangle-like structures) of some classes of functions which preserve certain properties of semimetric spaces.",
"These extensions will include both strong $b$ -metric spaces and the ones satisfying condition known as relaxed polygonal inequality, introduced by Fagin et al.",
"[17].",
"As we approach this issue, we also obtain as a byproduct some equivalents of wide-known results on combining multiple metric spaces into a single one.",
"These supporting lemmas also partially answer the aforementioned issue raised by Kirk and Shahzad.",
"We then follow with a corresponding result for functions connected with semimetric spaces satisfying the relaxed polygonal inequality.",
"We discuss some of the obtained answers and provide some examples to illustrate both the notions used and theorems obtained."
],
[
"Preliminaries",
"We begin with the definition of a semimetric space, as this is one of the primary concepts we will need to use later on.",
"Definition 2.1 A semimetric space is a non-empty set $X$ equipped with a function $d:X\\times X\\rightarrow [0,+\\infty )$ , satisfying the following conditions: (S1) For all $x,y\\in X$ , $d(x,y)=0$ if and only if $x=y$ ; (S2) For all $x,y\\in X$ , $d(x,y)=d(y,x)$ .",
"Then, the function $d$ is called a semimetric.",
"The following definition introduces various properties which a semimetric space may possess.",
"Definition 2.2 Let $(X,d)$ be a semimetric space.",
"If a function $d$ satisfies (U) $d(x,z)\\leqslant \\max \\lbrace d(x,y),d(y,z) \\rbrace $ for any $x,y,z\\in X$ , then we call $(X,d)$ an ultrametric space; (M) $ d(x,z)\\leqslant d(x,y)+d(y,z)$ for all $x,y,z\\in X$ , then $(X,d)$ is a metric space; (S) $d(x,z)\\leqslant Kd(x,y) + d(y,z)$ for any $x,y,z\\in X$ , where $K\\geqslant 1$ is fixed, then $(X,d)$ is called a strong $b$-metric space; (P) $d(x_0, x_n) \\leqslant K \\cdot \\sum _{i=1}^n d(x_{i-1},x_i)$ for any $n\\in \\mathbb {N}$ and $x_0,x_1,\\dots , x_n\\in X$ where $K\\geqslant 1$ is fixed, then we say that the space $(X,d)$ satisfies the $K$-relaxed polygonal inequality ($K$ -rpi for short); (B) $d(x,z)\\leqslant K\\left(d(x,y) + d(y,z) \\right)$ for any $x,y,z\\in X$ , where $K\\geqslant 1$ is fixed, then $(X,d)$ is called a $b$ -metric space.",
"The coefficients $K$ , which appear in definitions (S), (P) and (B) will be called the relaxation constants.",
"Throughout the article, we will mainly focus on functions which, in a sense, preserve or transform the aforementioned inequalities (although sometimes at the cost of the topological structure).",
"Therefore, let us introduce an appropriate definition: Definition 2.3 Let ($A_1$ ), ($A_2$ ) be two properties of a semimetric space.",
"We say that $f:[0,+\\infty ) \\rightarrow [0,+\\infty )$ is ($A_1$ )-($A_2$ )-preserving, if for any semimetric space $(X,d)$ satisfying the property ($A_1$ ), $f\\circ d$ is a semimetric and the space $(X,f\\circ d)$ satisfies the condition ($A_2$ ).",
"For more convenient presentation of inclusions between given classes of functions, we denote by $P_{A_1, A_2}$ the class of all ($A_1$ )-($A_2$ )-preserving functions.",
"Following this train of thought, by $P_{A_1}$ we will denote the class of all ($A_1$ )-($A_1$ )-preserving functions.",
"Now, let us introduce some definitions connected with behaviour of property-preserving functions.",
"Definition 2.4 A function $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ is said to be amenable if $f^{-1}[\\lbrace 0\\rbrace ]=\\lbrace 0\\rbrace $ ; subadditive if $f(a+b)\\leqslant f(a)+f(b)$ for all $a,b\\in [0,+\\infty )$ ; quasi-subadditive if $f(a+b)\\leqslant K\\cdot (f(a)+f(b))$ for some $K\\geqslant 1$ and all $a,b\\in [0,+\\infty )$ ; concave if $f(ta+(1-t)b)\\leqslant tf(a)+(1-t)f(b)$ for every $t\\in [0,1]$ and $a,b\\in [0,+\\infty )$ ; tightly bounded if there exists $v>0$ such that $f(a)\\in [v,2v]$ whenever $a>0$ .",
"Below we quote a few results from papers of originators of this topic [4], [9], [10] composed into two lemmas, which will be used in the latter part of this paper.",
"Lemma 2.1 (Sufficient conditions for metric preservation) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ be an amenable function.",
"If any of the conditions below is satisfied: (i) $f$ is concave; (ii) $f$ is increasing and subadditive; (iii) $f$ is amenable and tightly bounded; then $f\\in P_M$ , i.e., $f$ is metric preserving.",
"Lemma 2.2 (Necessary conditions for metric preservation) Let $f\\in P_M$ .",
"Then $f$ is both amenable and subadditive.",
"For a broader survey of results in the topic of metric preserving functions, see the aforementioned lectures of Doboš [10] as well as a nice paper summarizing some recent results by Samphavat et al.",
"[31]."
],
[
"Construction tools for $b$ -metric and strong {{formula:6319686a-24e4-4a2b-9358-dd7ae8c7e378}} -metric spaces",
"It is well known that a disjoint union of two metric spaces, say $(X_1,d_1)$ and $(X_2,d_2)$ , is metrizable.",
"In case when we deal with spaces of finite diameter the case is easy.",
"Otherwise, we may replace both metrics by bounded metrics equivalent to them (in particular, this mapping $t\\mapsto \\frac{t}{t+1}$ is an example of (M)-preserving function).",
"$d_i^\\prime (x,y):= \\frac{d_i(x,y)}{1+d_i(x,y)} \\text{ for } x,y\\in X_i, i=1,2.$ The resulting spaces obviously are not isometric to the original ones in the general case.",
"Nevertheless, they retain the same topology.",
"In the case of more general distance-type functions, tampering with their values may alter the relaxation constants from their definitions, as the following example depicts.",
"Example 3.1 Let $X:=\\lbrace 1,2,3\\rbrace $ and $d$ be a semimetric satisfying: $d(1,2)=d(2,3) = 1 \\qquad d(1,3)=4.$ It can be easily seen that remaining values of $d$ stem from the axioms (S1) and (S2).",
"A simple calculation also shows that $d$ satisfies condition (B) for $K=2$ .",
"Now, as previously, let $d^\\prime (x,y):= \\frac{d(x,y)}{1+d(x,y)}$ for all $x,y\\in X$ .",
"Thus, $d^\\prime (1,2)=d^\\prime (2,3) = \\frac{1}{2} \\qquad d^\\prime (1,3)=\\frac{4}{5}.$ One can immediately notice that such modification of the distances of this simple $b$ -metric space resulted in transforming it into a metric space.",
"Remark 3.1 By changing the original values of $d(1,2)$ and $d(2,3)$ from 1 to $\\frac{1}{4}$ , $(X,d)$ would be a $b$ -metric space with relaxation constant $K=8$ .",
"In such case $d^\\prime (1,2)=d^\\prime (2,3) = \\frac{1}{5}$ .",
"This means that $(X,d^\\prime )$ would no longer be a metric space, but rather a $b$ -metric one with relaxation constant $K=2$ .",
"If we restrict ourselves to the case of finite diameter, then we are able to perform the operation of joining two spaces in such way, that the result satisfies the same type of inequality with the relaxation constant equal to the greater one from the initial spaces.",
"Let us now put this claim in the formal setting.",
"For a semimetric space $(X,d)$ and its subset $A\\subset X$ , by $\\operatorname{diam}_{d}(A)$ we will denote its diameter with respect to the semimetric $d$ , i.e.",
"$\\operatorname{diam}_{d}(A) := \\sup \\lbrace d(x,y)\\ : \\ x,y\\in A \\rbrace .$ We will skip the subscript in situations where it does not lead to any ambiguities.",
"Lemma 3.1 (Concatenation lemma for $b$ -metric and strong $b$ -metric spaces) Let $(X_1,d_1)$ , $(X_2,d_2)$ be a pair of disjoint, strong $b$ -metric spaces ($b$ -metric spaces respectively) with relaxation constants $K_1$ , $K_2$ .",
"Additionally, assume that both spaces have finite diameter, i.e., $r_1:=\\operatorname{diam}_{d_1}(X_1)<\\infty $ and $r_2:=\\operatorname{diam}_{d_2}(X_2)<\\infty $ and $X_1\\cup X_2$ have at least three elements.",
"Let $X:=X_1\\cup X_2$ .",
"Then, there exists an extension of $d_1,\\ d_2$ , namely $d:X\\times X \\rightarrow [0,+\\infty )$ which is a strong $b$ -metric space (respectively $b$ -metric space) with relaxation constant $K:=\\max \\lbrace K_1,K_2\\rbrace $ and $\\operatorname{diam}_d(X)=\\max \\lbrace r_1, r_2\\rbrace $ .",
"We begin by defining $d$ .",
"Put $d(x,y):={\\left\\lbrace \\begin{array}{ll}d_1(x,y), & x,y\\in X_1\\\\d_2(x,y), & x,y\\in X_2\\\\\\frac{\\max \\lbrace r_1,r_2\\rbrace }{1+K}, & \\text{everywhere else.}\\end{array}\\right.",
"}$ From (REF ) we may infer that $d_{|X_i\\times X_i} = d_i$ for $i=1,2$ , thus it is an extension of both $d_1$ and $d_2$ .",
"The definition is correct thanks to $X_1$ and $X_2$ being disjoint.",
"It is easily seen that the pair $(X,d)$ satisfies the conditions of a semimetric space, as having at least three elements guarantees that $\\max \\lbrace r_1,r_2\\rbrace >0$ .",
"Moreover, the fact, that the resulting space diameter equals the greater one of the diameters of composing spaces is a direct consequence of the definition of $d$ .",
"What is left to check is whether the strong $b$ -metric (or just $b$ -metric) condition holds.",
"Consider any three distinct points $x,y,z\\in X$ .",
"We have the following cases: $x,y,z\\in X_1$ or $x,y,z\\in X_2$ .",
"In such cases $d$ coincides with either $d_1$ or $d_2$ , so the desired inequality holds.",
"$x,y\\in X_1$ and $z \\in X_2$ .",
"We need to check three variants of a strong $b$ -metric inequality.",
"$\\begin{aligned}d(x,y) &=& d_1(x,y) \\leqslant r_1 &\\leqslant & \\frac{\\max \\lbrace r_1,r_2\\rbrace }{1+K} \\cdot (1+K) = Kd(x,z)+d(z,y),\\\\d(x,z) &=& \\frac{\\max \\lbrace r_1,r_2\\rbrace }{1+K} &\\leqslant & K\\cdot 0 + \\frac{\\max \\lbrace r_1,r_2\\rbrace }{1+K} \\leqslant K d_1(x,y)+d(y,z),\\\\d(y,z) &=& \\frac{\\max \\lbrace r_1,r_2\\rbrace }{1+K} &\\leqslant & K\\cdot 0 + \\frac{\\max \\lbrace r_1,r_2\\rbrace }{1+K} \\leqslant K d_1(y,x) + d(x,z).\\\\\\end{aligned}$ One can easily notice, that the $b$ -metric inequality follows from this reasoning as well.",
"$x\\in X_1$ and $y,z\\in X_2$ .",
"The desired inequalities in this case are obtained analogously as above.",
"Therefore, $(X,d)$ is a strong $b$ -metric space (or $b$ -metric space).",
"Remark 3.2 In the case where we are interested in concatenating $b$ -metric spaces, we may replace $1+K$ with $2K$ , which yields a shorter distance between these two parts of space.",
"Now we know that in a case when we have a finite family of semimetric spaces satisfying a given property, their disjoint sum also does satisfy such condition.",
"The problem appears when we want to concatenate an infinite family of metric spaces.",
"As we will see in the following results, sometimes it is desired to construct an unbounded space, but the lemma above does not offer a help in such case either.",
"Therefore, we need to introduce the following Lemma 3.2 (On the union of an increasing sequence of semimetric spaces) Let $ be a linearly ordered set and $ F:= {(Xt,dt) : t}$ be an increasing family of strong $ b$-metric ($ b$-metric respectively) with a fixed constant $ K1$ spaces i.e.",
"for every pair of indices $ st$ we have $ Xs Xt$ and $ ds dt$ (which means $ ds(x,y) = dt(x,y)$ for $ x,yXs$).",
"Then a pair $ (X,d)$ defined as$$X:=\\bigcup _{t\\in X_t \\qquad \\qquad d(x,y):=d_p(x,y), \\text{ where } p \\text{ is any index such that } x,y\\in X_pis strong b-metric (b-metric respectively) with the relaxation constant K\\geqslant 1.",
"}{\\begin{xmlelement*}{proof}Since \\mathcal {F} is increasing then d(x,y) is well defined for all x,y\\in X.We can then proceed to checking axioms of semimetric\\begin{itemize}\\item [(S1)] d(x,y)=0 implies that d_{t_0} (x,y)=0 for some t_0\\in .",
"Therefore x=y.",
"The reverse implication is obvious;\\item [(S2)] d(x,y)=d_t(x,y)=d_t(y,x)=d(y,x) for some t\\in .\\end{itemize}Lastly, we need to prove the (S) condition (or (B) condition) for (X,d).",
"Take any three distinct points x,y,z\\in X.",
"There exists t_0\\in such that x,y,z\\in X_{t_0}.",
"Since (X_{t_0},d_{t_0}) satisfies (S) (or (B), respectively), we haved(x,z) = d_{t_0} (x,z)\\leqslant K d_{t_0} (x,y) + d_{t_0}(y,z) = Kd(x,y)+d(y,z).Consequently, (X,d) satisfies (S) condition.",
"The proof of b-metric case goes analogously.\\end{xmlelement*}}$ Remark 3.3 It is worth noting that we do not employ this lemma in its full extent, as in the subsequent reasoning we will restrict to the cases where $\\mathbb {N}$ .",
"Notice that these two tools allow us to construct an increasing sequence of strong $b$ -metric spaces having some desired property and then gluing them into a single, larger space – perhaps of infinite diameter.",
"Although strong $b$ -metric spaces constructed in such a way can hardly be seen as natural, this approach partially refers to [26] of Kirk and Shahzad on providing more examples of strong $b$ -metric spaces."
],
[
"Characterizing certain property-preserving functions",
"In this section we introduce the main result of this paper, foreshadowed in the abstract.",
"However, to proceed with introducing the new result, we need to extend [22].",
"Definition 4.1 Let $K\\geqslant 1$ and $a,b,c \\in [0,+\\infty )$ be such that $a\\geqslant b \\geqslant c$ .",
"We will say that the $(a,b,c)$ forms a triangle triplet (respectively $K$ -triangle triplet or strong $K$ -triangle triplet) if the respective set of inequalities: (TT), ($K$ -TT) or (S$K$ -TT), holds: (TT) $a\\leqslant b+c$ ; ($K$ -TT) $a\\leqslant K\\left( b+c\\right)$ ; (S$K$ -TT) $a\\leqslant K c+b$ .",
"Remark 4.1 Usually, no assumptions on relations between $a,b$ and $c$ are made.",
"Adding these relations enables us to shorten the latter part of the definitions as well as shorten some parts of the proofs.",
"While trying not to confuse the reader, we will call a triplet $(a,b,c)$ a triangle triplet etc.",
"if rearrangement of those three numbers form the respective variant of the triangle triplet.",
"For example we will refer to $(2,4,5)$ as the triangle triplet due to the fact, that $(5,4,2)$ satisfies the discussed definition.",
"We shall now quote a well-known result from the theory of metric preserving functions, which we will try to replicate [4], [9], [10].",
"Lemma 4.1 (Triangle triplet characterization of $P_M$ ) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ be an amenable function.",
"Then $f$ is metric preserving iff for any triangle triplet $(a,b,c)$ , values $\\lbrace f(a),f(b),f(c)\\rbrace $ can be arranged into a triangle triplet.",
"Let us focus for a moment on the particular subclass of functions preserving properties (B) and (S), where the relaxation constant on the original space is fixed.",
"Lemma 4.2 (Triangle-like triplet characterization of property-preserving functions with fixed relaxation constant) Let $f:[0,+\\infty )\\longrightarrow [0,+\\infty )$ be a function such that for every (strong) $b$ -metric space $(X,d)$ with fixed relaxation constant $K\\geqslant 1$ , the space $(X,f\\circ d)$ is a (strong) $b$ -metric space.",
"Then, there exists $K^\\prime \\geqslant 1$ such that: a) for every (strong) $K$ -triangle triplet $(a,b,c)$ , the values $f(a)$ , $f(b)$ and $f(c)$ can be arranged into a (strong) $K^\\prime $ -triangle triplet.",
"b) the relaxation constant of the resulting space $(X,f\\circ d)$ is bounded by $K^\\prime $ .",
"Remark 4.2 This lemma actually gives us insight into six families of property-preserving functions, namely $P_{B}$ , $P_{B,S}$ , $P_{S,B}$ , $P_{S}$ , $P_{M,B}$ and $P_{M,S}$ (the last two obtained by fixing $K=1$ ).",
"Also, one can also immediately deduce b) from a).",
"Due to the fact, that each version of this lemma is proved by the same method, we will simply prove the version for particular subclass of $P_{B,S}$ .",
"Let $f$ be a function satisfying the assumptions of our lemma, i.e.",
"for every $b$ -metric space $(X,d)$ with relaxation constant $K\\geqslant 1$ we have that $(X,f\\circ d)$ is a strong $b$ -metric space.",
"Suppose the contrary, i.e., for every $K^\\prime \\geqslant 1$ (in particular for every $n\\in \\mathbb {N}$ ) there exists a $K$ -triangle triplet $\\lbrace a_n,b_n,c_n\\rbrace $ which is mapped to $\\left\\lbrace f(a_n),f(b_n),f(c_n)\\right\\rbrace $ , the latter one not forming a strong $n$ -triangle triplet.",
"Without the loss of generality, we may assume that $f(a_n)>n\\cdot f(b_n)+f(c_n)$ (this fact becomes clear in the further part of the proof).",
"Let $\\hat{X}_1 = X_1:=\\lbrace (1,1),(1,2),(1,3)\\rbrace $ and $d_1((1,1),(1,2))=a_1 \\qquad d_1((1,2),(1,3))=b_1, \\qquad d_1((1,1),(1,3))=c_1.",
"$ The rest of the values, again, stems from extending $d_1$ to the whole $X_1\\times X_1$ as a semimetric (i.e., (S1) and (S2) conditions imply the remaining function values), yielding $\\hat{d}_1$ .",
"Thus $(\\hat{X}_1,\\hat{d}_1)$ is not a strong $b$ -metric space with relaxation constant 1.",
"Nonetheless, it is still a strong $b$ -metric space if we replace 1 by some larger constant.",
"Now, we will proceed inductively.",
"Having defined $\\left(\\hat{X}_{n-1},\\hat{d}_{n-1}\\right)$ , which is a finite (thus bounded) $b$ -metric space with relaxation constant $K$ , we try to define $\\hat{X}_{n}$ .",
"Put $X_n:=\\lbrace (n,1),(n,2),(n,3)\\rbrace $ and $d_n:X_n\\times X_n \\rightarrow [0,+\\infty ),$ as a unique semimetric satisfying $\\hat{d_n}((n,1),(n,2)) = a_n, \\qquad \\hat{d_n}((n,2),(n,3)) = b_n, \\qquad \\hat{d_n}((n,1),(n,3)) = c_n.$ Again, $(X_n,d_n)$ is a bounded $b$ -metric space with relaxation constant $K$ .",
"We can now use Lemma REF to obtain larger $b$ -metric space $\\hat{X}_n:=X_{n}\\cup \\hat{X}_{n-1}$ , equipped with the extended $b$ -metric $\\hat{d}_{n}$ , which equals $d_n$ on $X_n\\times X_n$ and $\\hat{d}_{n-1}$ on $\\hat{X}_{n-1}\\times \\hat{X}_{n-1}$ .",
"Successively, we obtain the increasing family of $b$ -metric spaces $\\left\\lbrace \\left(\\hat{X}_n,\\hat{d}_n\\right) \\ : \\ n\\in \\mathbb {N} \\right\\rbrace $ .",
"Due to Lemma REF we obtain a $b$ -metric space $(X,D)$ defined as $X:=\\bigcup _{n\\in \\mathbb {N}} X_n, \\qquad D(x,y):=\\hat{d}_{\\min \\lbrace p \\; : \\; x,y\\in \\hat{X}_p\\rbrace }(x,y).$ What is left to prove is that $(X,f\\circ D)$ fails to be a strong $b$ -metric space.",
"Suppose that (S) inequality holds in $(X,D)$ for some $K^\\prime \\geqslant 1$ and let $n_0\\geqslant K^\\prime $ be a fixed natural number.",
"This way, the fact that $(X,D)$ satisfies the strong $b$ -metric inequality with constant $K^\\prime $ implies satisfying it for $n_0$ .",
"Considering $(n_0,1),(n_0,2),(n_0,3)\\in X$ yields $D\\left((n_0,1),(n_0,2)\\right) &=& d_{n_0}\\left( (n_0,1),(n_0,2)\\right) = a_{n_0}\\\\ &>& n_0 \\cdot b_{n_0}+c_{n_0} = n_0 \\cdot D\\left((n_0,2),(n_0,3)\\right) + D\\left((n_0,3),(n_0,1)\\right)\\\\&=& n_0 \\cdot d_{n_0}\\left( (n_0,2),(n_0,3)\\right) + d_{n_0}\\left( (n_0,3),(n_0,1)\\right).$ Of course $f(a_n)$ , $f(b_n)$ and $f(c_n)$ do not have to be aligned in the same order as $a_n$ , $b_n$ and $c_n$ , but the result stays the same as we can interchange the appropriate pairs of points in the reasoning above to obtain the proper counterexample.",
"This shows that $(X,f\\circ D)$ fails to satisfy strong $b$ -metric condition for $n_0$ , thus it fails for $K^\\prime $ as well.",
"Since $K^\\prime $ was arbitrary, $(X,f\\circ D)$ is not a strong $b$ -metric space despite $(X,D)$ being a $b$ -metric one.",
"This concludes the proof of both part a) and b), since it yields a contradiction with the assumption that $(X,f\\circ d)$ is a strong $b$ -metric space whenever $(X,d)$ is a $b$ -metric space with relaxation constant $K$ .",
"The reasoning for the remaining variants of this lemma is exactly the same – all that needs to be changed is the type of triangle triplets used in constructing the proper counterexample.",
"The said lemma allows us to draw somewhat surprising conclusion, that a single property-preserving mapping $f$ does not allow us to obtain arbitrarily large values of relaxation constant on resulting space whenever the relaxation constant of the initial space is bounded.",
"This seems particularly interesting for classes $P_{MS}$ and $P_{MB}$ .",
"We shall now proceed with first of the main results of this paper, which is an extension of analogous characterization already proven by other authors (see [4], [9], [22]).",
"Their results have been incorporated in this theorem in points (i) and (ii) – in particular, we will not provide the proofs for those, as they were well-described in the respective papers they were taken from.",
"Theorem 4.1 (Characterization of triangle-type inequality preserving functions via triangle-like triplets:) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ be an amenable function.",
"Then: $f\\in P_{MB}=P_{B}$ $\\iff $ there exists $K\\geqslant 1$ for every triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b),f(c)\\right)$ form a $K$ -triangle triplet.",
"$f\\in P_{M}$ $\\iff $ for any triangle triplet $(a,b,c)$ , the resulting values $\\left(f(a),f(b), f(c)\\right)$ form a triangle triplet as well.",
"$f\\in P_{BS}$ $\\iff $ for every $K_1\\geqslant 1$ there exists $K_2\\geqslant 1$ , such that for every $K_1$ -triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K_2$ -strong triangle triplet.",
"$f\\in P_{S}$ $\\iff $ for every $K_1\\geqslant 1$ there exists $K_2\\geqslant 1$ , such that for every $K_1$ -strong triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K_2$ -strong triangle triplet.",
"$f\\in P_{SB}$ $\\iff $ for every $K_1\\geqslant 1$ there exists $K_2\\geqslant 1$ , such that for every $K_1$ -strong triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K_2$ -triangle triplet.",
"$f\\in P_{MS}$ $\\iff $ there exists $K\\geqslant 1$ , such that for every triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K$ -strong triangle triplet.",
"$f\\in P_{SM}$ $\\iff $ for every $K$ -strong triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a triangle triplet.",
"Once again, we would like to remind that $f$ does not have to be monotone, thus the relation $f(a)\\geqslant f(b) \\geqslant f(c)$ does not have to hold (see Remark REF ).",
"The necessity in each case follows as a simple conclusion from Lemma REF .",
"In the last, seventh equivalence another short proof of necessity can be given, as considering a three-point space $X:=\\lbrace 1,2,3\\rbrace $ is enough.",
"Supposing that for some $K$ a strong $K$ -triangle triplet $\\lbrace a,b,c\\rbrace $ which is not mapped to a triangle one exists yields an instant contradiction if we define $d(1,2):=a, \\ d(2,3):=b, \\ d(3,1):=c$ and allow the assumptions (S1), (S2) to fill in the rest of values.",
"Applying $f$ to such defined strong $b$ -metric space maps it into a semimetric structure which is not a metric space.",
"The sufficient part of this last implication follows from the same method as described in point f. above.",
"A sufficiency part can be shown as follows.",
"Assume that $f$ satisfies the right-hand side of equivalence (3) and let $(X,d)$ be an arbitrary $b$ -metric space.",
"Let us denote the relaxation constant of this space by $K$ .",
"Consider three distinct points $x_1,x_2,x_3\\in X$ .",
"WLOG we can assume that $f\\circ d \\left( x_1,x_3\\right) \\geqslant f\\circ d \\left( x_1,x_2\\right) \\geqslant f\\circ d \\left( x_2,x_3\\right).$ Since $(X,d)$ is a $b$ -metric space, then values $d \\left( x_1,x_3\\right), d \\left( x_1,x_2\\right), d \\left( x_2,x_3\\right)$ can be arranged into a $K$ -triangle triplet.",
"From our assumption there exists $K^\\prime $ such that $\\lbrace f\\circ d \\left( x_1,x_3\\right),f\\circ d \\left( x_1,x_2\\right),f\\circ d \\left( x_2,x_3\\right) \\rbrace $ is a strong $K$ -triangle triplet.",
"From the definition of such triplet we obtain $f\\circ d \\left( x_1,x_3\\right) \\leqslant K^\\prime \\cdot f\\circ d \\left( x_2,x_3\\right)+ f\\circ d \\left( x_1,x_2\\right).$ Due to $K^\\prime $ being independent of the choice of $x_1,x_2,x_3$ (since it depends solely on $K$ ) the proof of equivalence (3) is finished.",
"The sufficiency proofs for points (4-7) are almost identical.",
"This characterization might allow us to construct other results describing the properties of metric-type property preserving functions.",
"Before moving to the next part of this paper, we point out two issues we are concerned about.",
"First one of them is the observation that for a fixed function $f$ , the right-hand side of each of the implications (3), (4) and (5) in fact define a mapping $g:[1,+\\infty )\\rightarrow [1,+\\infty )$ , which for every $K_1\\geqslant 1$ assigns the smallest possible $K_2\\geqslant 1$ such that $(X,f\\circ d)$ is $K_2$ -parametrized space, whenever $(X,d)$ is the space of appropriate type with relaxation constant $K_1$ .",
"We know that in all three cases $g(1) = 1$ and the function $g$ is obviously nondecreasing.",
"It is reasonable to think, that different functions $f_1$ , $f_2$ are described by the same mapping $g$ , but an unanswered question remains – what properties of $f$ can we infer based on $g$ and conversely?",
"Moreover, we would like to know what kinds of functions $g$ can be obtained in this way i.e., what attributes do we require from such functions to be obtainable from the discussed theorem.",
"Theorem REF also shifts the research on this functions from metric and topological setting to more real-function theoretical field, connected with functional inequalities.",
"We already know several of the inclusions between the listed above $P_{A_1, A_2}$ families.",
"Hopefully, this characterization will allow us to explore this topic further, allowing us to prove equalities or proper inclusions for some families of property-preserving functions.",
"Thus, we pose the following Open problem 1: Establish the relations between $P_{S,M}$ , $P_{S}$ , $P_{S,B}$ , $P_{B,S}$ and other classes of property preserving functions.",
"In particular, determine which of the inclusions are proper."
],
[
"Further characterizations for semimetrics satisfying relaxed polygonal inequalities",
"It turns out, that the reasoning that was applied to obtain the Theorem REF can be extended on functions satisfying (P) axiom.",
"However, we need to introduce additional definitions.",
"Definition 5.1 A finite tuple of non-negative real numbers $A:=(a_1,\\dots ,a_n)$ such that $a_1 \\geqslant a_2 \\geqslant \\dots \\geqslant a_n.$ is said to be a $K$-relaxed polygon if the following inequality holds: $a_1 \\leqslant K \\cdot \\sum _{i=2}^{n} a_i .$ Remark 5.1 Analogously to the Remark REF , we will refer to a tuple $(a_1,\\dots ,a_n)$ as $K$-relaxed polygon whenever its elements, sorted in nonincreasing order, satisfy Definition REF .",
"Example 5.1 For example, a tuple $A:=(120, 20, 10, 10, 10, 10)$ is an example of $K$ -relaxed polygon with $K=2$ .",
"Indeed $120 \\leqslant 2\\cdot \\left( 20 +10 +10 +10 +10\\right) = 120.$ Notice, that we do not require the largest number to be bounded by any subset sum of the remaining values, i.e.",
"for example we do not require 120 to be bound from above by $K\\cdot (10+10)$ .",
"One can observe, that such a tuple $A$ is a $K$ -relaxed polygon if and only if $\\left(1+K\\right)\\cdot a_1 \\; \\leqslant \\; K\\cdot \\sum _{i= 2}^{n} a_i.$ Then, we introduce the following notion of implementation.",
"Definition 5.2 Consider a $K$ -relaxed polygon $A:=(a_1,\\dots ,a_n), \\qquad a_1 \\geqslant \\dots \\geqslant a_n$ and a semimetric space $(X,d)$ satisfying $K$ -rpi.",
"If there exists a sequence of $n$ -points, $x_1,\\dots ,x_n\\in X$ such that (I1) $d(x_1,x_n):=a_1$ ; (I2) for each $2\\leqslant i \\leqslant n$ , $d(x_{i-1},x_{i}):=a_{i}$ ; then we say that $(X,d)$ implements a $K$ -relaxed polygon $A$ .",
"One may wonder if for any $K$ -relaxed polygon there exists a semimetric space $(X,d)$ which implements it and the answer to that question is, luckily, positive.",
"Lemma 5.1 (Implementation lemma:) Let $A:=(a_1,\\dots ,a_n)$ be a $K$ -relaxed polygon.",
"There exists an $n+1$ element set $X$ with a semimetric $d$ which satisfies $K$ -rpi and implements $A$ .",
"We will construct such space in two steps.",
"Let $X:=\\lbrace 1,\\dots , n\\rbrace $ and put $d(1,n):=a_1$ ; for each $1\\leqslant i \\leqslant n$ , $d({i-1},{i}):=a_{i}$ ; Thus, all of the requirements (I1)-(I2) are satisfied.",
"This definition leaves us with $n$ -gon with no diagonal distances defined.",
"Consider now $i,j\\in X$ such that they are not adjacent i.e.",
"$|i-j|\\notin \\lbrace 1,n\\rbrace $ .",
"For such pair (for the sake of simplicity let us assume $i<j$ ) one can define the distance as follows $d(i,j):=\\min \\left\\lbrace \\sum _{k=i}^{j-1} d(k,k+1), \\left(\\sum _{k=j}^{n-1} d(k,k+1) + d(1,n) + \\sum _{k=1}^{i-1}d(k,k+1) \\right) \\right\\rbrace ,$ where the sums in the latter part of the minimum might be empty (for example when $i=1$ and $j<n$ or $j=n$ and $i>1$ ).",
"Figure: Informally, the distance between two points (for example, red line between 2 and ii) can be described as the shorter of the paths over the perimeter.",
"Notice, that such distance does not always have to be the left-oriented path (here denoted by lime green).Notice, that for any two non-adjacent points $i,j$ thanks to the formulation of $d(i,j)$ the distance can be bound from above by the distance over the perimeter of the discussed polygon.",
"Indeed, for any sequence of point $i,x_1,x_2,\\dots ,x_k,j\\in X$ .",
"$K\\cdot \\left( d(i,x_1) + \\dots + d(x_k,j)\\right) &\\geqslant & K\\cdot \\min \\left\\lbrace \\sum _{k=i}^{j-1} d(k,k+1), \\left( d(1,n) + \\sum _{k=j}^{n-1} d(k,k+1) + \\sum _{k=1}^{i-1}d(k,k+1) \\right) \\right\\rbrace \\\\ &\\geqslant & d(i,j).$ Perhaps this part of the proof can be reworked to be nicer.",
"Now, consider any two adjacent points of the space, $i$ and $i+1$ (the case of $n$ and 1 is analogous).",
"Consider any point $x\\in X$ .",
"WLOG we can assume that $x>i$ .",
"Then we have three posibilities: $d(i,x) = \\sum _{k=i}^{x-1} d(k,k+1)$ , then obviously $d(i,x)\\geqslant d(i,i+1)$ ; $d(i,x) = d(1,n) + \\sum _{k=x}^{n-1} d(k,k+1) + \\sum _{k=1}^{i-1}d(k,k+1)$ .",
"If $d(x,i+1) =d(1,n) + \\sum _{k=x}^{n-1} d(k,k+1) + \\sum _{k=1}^{i}d(k,k+1)$ then $d(x,i+1)\\geqslant d(i,i+1)$ and the case is once again trivial.",
"$d(i,x) = d(1,n) + \\sum _{k=x}^{n-1} d(k,k+1) + \\sum _{k=1}^{i-1}d(k,k+1)$ and $d(x,i+1) = \\sum _{k=i+1}^{x-1} d(k,k+1)$ .",
"Then, using the fact that $A$ is a $K$ -relaxed polygon $d(i,i+1) \\leqslant K\\cdot \\left( d(1,n) - d(i,i+1) + \\sum _{i=1}^n d(k,k+1)\\right),$ which translates to $d(i,i+1) \\cdot (1+K) \\leqslant a_1 \\leqslant K\\cdot \\sum _{k=1}^n d(k,k+1) = K\\cdot \\sum _{k=1}^n a_k.$ When more points are considered, the reasoning remains the same (although it might need some more formal touch) – we either use the fact that length of one of the edges exceeds the distance $d(i,i+1)$ or the sum of the distances is greater or equal to the remaining part of the perimeter of the initial $K$ -relaxed polygon.",
"In any case, the property (P) from Definition 2. holds.",
"To obtain analogous result to Theorem REF we need lemmas which are analogous to Lemmas REF and REF .",
"We provide only the short sketches of proofs to the lemmas, since the proofs are similar to the respective results from the previous sections.",
"Lemma 5.2 (Concatenation lemma for semimetric spaces satisfying $K$ -rpi) Let $(X_1,d_1)$ , $(X_2,d_2)$ be a pair of disjoint, semimetric spaces satisfying property (P) with relaxation constants $K_1$ , $K_2$ .",
"Assume that $\\operatorname{diam}_{d_1}(X_1)=r_1<\\infty $ and $\\operatorname{diam}_{d_2}(X_2)=r_2<\\infty $ .",
"Let $X:=X_1\\cup X_2$ .",
"There exists an extension of $d_1,\\ d_2$ , namely $d:X\\times X \\rightarrow [0,+\\infty )$ which satisfies $K$ -rpi, where $K:=\\max \\lbrace K_1,K_2\\rbrace $ .",
"Additionally, $\\operatorname{diam}_d(X)=\\max \\lbrace r_1, r_2\\rbrace $ .",
"Let us begin with defining $d$ as $d(x,y):={\\left\\lbrace \\begin{array}{ll}d_1(x,y), & x,y\\in X_1\\\\d_2(x,y), & x,y\\in X_2\\\\\\frac{\\max \\lbrace r_1,r_2\\rbrace }{2K}, & \\text{everywhere else,}\\end{array}\\right.",
"}$ where $K:=\\max \\lbrace r_1,r_2\\rbrace $ .",
"One can immediately notice, that such space $(X,d)$ will have its diameter at most $r:=\\max \\lbrace r_1,r_2\\rbrace $ .",
"What is left to prove is that $d$ fulfills the property $K$ -rpi.",
"Let $n\\in \\mathbb {N}$ and $x_1, \\dots , x_n\\in X$ .",
"The cases where both $x_1$ and $x_n$ belong to the same $X_i$ are relatively obvious, since in the cases where all remaining $x_k$ , $1\\leqslant k <n$ belong to the same component, the respective relaxed polygonal property steps in.",
"At the same time, if there is at least one $x_{k_1}$ from the second component, then $d(x_1,x_n)\\leqslant r_i \\leqslant 2K \\cdot \\frac{r}{2K} \\leqslant K\\cdot \\sum _{i=1}^{n} d(x_{i-1},x_i).$ The last inequality follows from the fact, that the polygon consisting of points $x_1,\\dots ,x_n$ consists of at least two indices, $1\\leqslant i_0,i_1 < n$ , for which $x_{i_0}\\in X_i$ and $x_{i_0+1}\\notin X_i$ , as well as $x_{i_1}\\notin X_i$ and $x_{i_1+1}\\in X_i$ .",
"If $x_1, x_n$ belong to distinct components, then there exists $1\\leqslant k< n$ such that $x_k$ belongs to the same component as $x_1$ and $x_{k+1}$ does not.",
"Therefore $\\frac{r}{2K} = d(x_1,x_n) \\leqslant K\\cdot d(x_k,x_{k+1}) \\leqslant K\\cdot \\sum _{i=1}^n d(x_{i-1},x_i).$ This finishes the proof of the fact that $(X,d)$ satisfies $K$ -rpi.",
"Lemma 5.3 (Summation lemma for semimetric spaces satisfying $K$ -rpi) Let $ be a linearly ordered set and $ F:= {(Xt,dt) : t}$ be an increasing family of semimetric spaces satisfying $ K$-rpi with a fixed constant $ K1$.",
"Then a semimetric space $ (X,d)$ given by$$X:=\\bigcup _{t\\in X_t \\qquad \\qquad d(x,y):=d_p(x,y), \\text{ where } p \\text{ is any index s.t. }",
"x,y\\in X_psatisfies K-rpi as well.",
"}{\\begin{xmlelement*}{proof}Since (P) property considers only finite sequences of points (albeit of arbitrary length), then the proof of this lemma is almost exactly the same as of the Lemma \\ref {summationLemma}.\\end{xmlelement*}}$ Using these definitions and lemmas, one can formulate and prove the following theorem, analogous to the Theorem REF .",
"Theorem 5.1 (Characterization of (P)-preserving mappings) Consider an amenable function $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ .",
"Then $f$ is (P)-preserving iff for any $K_1\\geqslant 1$ there exists $K_2$ such that $f$ maps any $K_1$ -relaxed polygon to $K_2$ -relaxed polygon, i.e.",
"$\\forall _{K_1\\geqslant 1} \\ \\exists _{K_2 \\geqslant 1} \\ \\Bigg ( \\forall _{n\\in \\mathbb {N}} \\ \\forall _{a_1,\\dots ,a_n \\in [0,+\\infty )} \\ \\left( \\ \\left(1+K_1\\right)\\cdot \\max \\lbrace a_i \\ : \\ i\\leqslant n \\rbrace \\; \\leqslant \\; K_1\\cdot \\sum _{1\\leqslant i\\leqslant n}a_i \\ \\right)\\nonumber \\\\\\Rightarrow \\left( \\left(1+K_2\\right)\\cdot \\max \\lbrace f(a_i) \\ : \\ i\\leqslant n \\rbrace \\; \\leqslant \\; K_2\\cdot \\sum _{1\\leqslant i\\leqslant n}f(a_i) \\right) \\ \\Bigg ).$ We begin by showing the sufficiency of the proposed condition.",
"Assume that (REF ) holds.",
"Let $(X,d)$ be any semimetric space satisfying $K_1$ -relaxed polygonal inequality.",
"Consider a finite sequence of points $x_1,\\dots , x_n\\in X$ .",
"Since $d$ satisfies $K_1$ -rpi, then the tuple $A:=\\left( d\\left(x_1,x_n\\right), d\\left(x_1,x_2\\right),d\\left(x_2,x_3\\right), \\dots , d\\left(x_{n-1},x_{n}\\right) \\right)$ is (up to an order of elements) a $K_1$ -relaxed polygon.",
"Therefore, the values of $f$ at these respective tuple $A_f:=\\left(f\\circ d\\left(x_1,x_n\\right), f\\circ d\\left(x_1,x_2\\right),f\\circ d\\left(x_2,x_3\\right), \\dots , f\\circ d\\left(x_{n-1},x_{n}\\right) \\right)$ can be arranged into a $K_2$ -relaxed polygon.",
"This implies that $f\\circ d (x_1,x_n) &\\leqslant & \\max A_f \\leqslant K_2 \\cdot \\left( f\\circ d (x_1,x_n) +\\sum _{i=2}^n f\\circ d\\left(x_{i-1},x_{i}\\right)\\right) - K_2 \\max A_f\\\\&\\leqslant & K_2 \\cdot \\left( f\\circ d (x_1,x_n) +\\sum _{i=2}^n f\\circ d\\left(x_{i-1},x_{i}\\right)\\right) - K_2 f\\circ d (x_1,x_n)\\\\&=& K_2 \\cdot \\sum _{i=2}^n f\\circ d\\left(x_{i-1},x_{i}\\right),$ so once again we are able to observe that the particular order in which the elements appear in $A_f$ does not matter as long as the condition (P) holds.",
"Since the points $x_1,\\dots ,x_n$ were chosen arbitrarily, $(X,f\\circ d)$ satisfies $K_2$ -rpi.",
"The proof of the other implication goes analogously to the scheme from the proof of Theorem REF .",
"Fix function $f$ .",
"If we suppose the contrary, we obtain the existence of such $K_1\\geqslant 1$ that for every $K_2\\geqslant 1$ there exists some $K_1$ -relaxed polygon which is not mapped to a $K_2$ -relaxed polygon (albeit again it might be a $K_2^{\\prime }$ -relaxed polygon for some greater constant $K_2^{\\prime }$ ).",
"Consider a sequence of such $K_1$ polygons $A_n$ .",
"Due to Lemma REF , this implies the existence of spaces $(X_n,d_n)$ , all of which satisfy $K_1$ -rpi.",
"Therefore, using Lemma REF we obtain an increasing sequence of spaces with (P)-property with the same constant $K_1$ .",
"What remains is to sum them up using Lemma REF , obtaining a large space $(X,d)$ which satisfies $K_1$ -rpi as well.",
"The function $f$ fails to preserve property (P) due to the fact that for any natural number $n$ there exists a set of points $\\lbrace x_1,\\dots ,x_k\\rbrace $ (coming from the space $X_n$ ) in $X$ such that $A_n:=\\left(d(x_1,x_k),d(x_1,x_2),\\dots ,d(x_{k-1},x_k) \\right)$ is a $K$ -relaxed polygon.",
"Therefore $B:=\\left(f\\circ d(x_1, x_k), \\dots , f\\circ d(x_{k-1},x_k) \\right)$ does not form a $n$ -relaxed polygon and thus $(X,f\\circ d)$ fails to satisfy $n$ -rpi.",
"It is also possible to investigate some results concerning families $P_{P,S}, P_{P,B}, P_{P,M}$ as well as $P_{S,P}, P_{B,P}, P_{M,P}$ .",
"The problem of obtaining analogous results for such classes of spaces is left as another open problem for the Readers.",
"This part of the paper would be somewhat lacking without any examples of functions in $P_{P}$ .",
"But before we provide any such characterization, we suggest some easy to check sufficient conditions for a function to be (P)-preserving.",
"We will need the characterization of spaces satisfying $K$ -rpi, which is due to Fagin et al.",
"[17].",
"Theorem 5.2 (Characterization of $K$ -rpi spaces via metric bounds) Let $(X,d)$ be a semimetric space.",
"The following are equivalent: (i) $(X,d)$ satisfies (P) with relaxation constant $K\\geqslant 1$ ; (ii) there exists a metric $d^{\\prime }$ on $X$ such that for all $x,y\\in X$ $d^{\\prime }(x,y)\\leqslant d(x,y) \\leqslant K\\cdot d^{\\prime }(x,y)$ Having this characterization at our disposal we can proceed with the following result.",
"Lemma 5.4 (Sufficient conditions for (P)-preservation) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ .",
"If there exist $a,b\\in \\mathbb {R}$ such that $ax\\leqslant f(x) \\leqslant bx$ for all $x\\geqslant 0$ , then $f\\in P_P$ .",
"Assume that $f$ is bounded from both sides by linear functions $x\\mapsto ax$ and $x\\mapsto bx$ .",
"Let $(X,d)$ be any semimetric space satisfying condition (P).",
"From Theorem REF we obtain the existence of a metric $d^{\\prime }$ on $X$ such that the inequalities (REF ) hold.",
"Since linear functions belong to $P_M$ due to being subadditive, increasing and amenable (see Lemma REF ), their compositions with $d^{\\prime }$ are also metrics.",
"Thus we have $ad^{\\prime }(x,y)\\leqslant ad(x,y)\\leqslant f(d(x,y)) \\leqslant bd(x,y) = \\frac{b}{a}\\cdot ad^{\\prime }(x,y)$ Therefore $f\\circ d$ is bounded from below by the metric $ad^{\\prime }(x,y)$ and from above by $\\frac{b}{a} \\cdot ad^{\\prime }(x,y)$ .",
"Due to Theorem REF we have that $f(d(x,y))$ satisfies condition (P).",
"Due to $(X,d)$ being arbitrary, $f\\in P_P$ .",
"This allows us to construct this somewhat nontrivial example of a function from $P_P$ .",
"Example 5.2 Let $f(x):=5x-4\\lfloor x \\rfloor $ , where $\\lfloor x \\rfloor $ denotes the largest integer not exceeding $x$ .",
"Figure: A graph depicting sawtooth function ff.Obviously $x\\leqslant f(x) \\leqslant 5x$ , thus $f\\in P_P$ .",
"One can also notice that $f$ does not belong to $P_M$ , as it fails to be subadditive.",
"$f(\\frac{3}{2}) = \\frac{15}{2} >f(1)+f\\left(\\frac{1}{2}\\right) = 1+\\frac{5}{2} = \\frac{7}{2}.$ From Lemma REF we obtain that $f\\notin P_{M}$ .",
"Remark 5.2 As a conclusion we obtain that although $P_M$ is not disjoint with $P_P$ (as identity belongs to both of those classes) we have $P_M\\ne P_P$ ."
],
[
"Acknowledgments",
"As usual, I would like to express my utmost gratitude to both of my supervisors, that is Jacek Jachymski and Mateusz Krukowski for multitude of fruitful discussions which included priceless mathematical and writing tips.",
"Lastly, I would like to acknowledge Piotr Nowakowski and Mateusz Lichman for adding some interesting comments on the topic."
],
[
"Characterizing certain property-preserving functions",
"In this section we introduce the main result of this paper, foreshadowed in the abstract.",
"However, to proceed with introducing the new result, we need to extend [22].",
"Definition 4.1 Let $K\\geqslant 1$ and $a,b,c \\in [0,+\\infty )$ be such that $a\\geqslant b \\geqslant c$ .",
"We will say that the $(a,b,c)$ forms a triangle triplet (respectively $K$ -triangle triplet or strong $K$ -triangle triplet) if the respective set of inequalities: (TT), ($K$ -TT) or (S$K$ -TT), holds: (TT) $a\\leqslant b+c$ ; ($K$ -TT) $a\\leqslant K\\left( b+c\\right)$ ; (S$K$ -TT) $a\\leqslant K c+b$ .",
"Remark 4.1 Usually, no assumptions on relations between $a,b$ and $c$ are made.",
"Adding these relations enables us to shorten the latter part of the definitions as well as shorten some parts of the proofs.",
"While trying not to confuse the reader, we will call a triplet $(a,b,c)$ a triangle triplet etc.",
"if rearrangement of those three numbers form the respective variant of the triangle triplet.",
"For example we will refer to $(2,4,5)$ as the triangle triplet due to the fact, that $(5,4,2)$ satisfies the discussed definition.",
"We shall now quote a well-known result from the theory of metric preserving functions, which we will try to replicate [4], [9], [10].",
"Lemma 4.1 (Triangle triplet characterization of $P_M$ ) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ be an amenable function.",
"Then $f$ is metric preserving iff for any triangle triplet $(a,b,c)$ , values $\\lbrace f(a),f(b),f(c)\\rbrace $ can be arranged into a triangle triplet.",
"Let us focus for a moment on the particular subclass of functions preserving properties (B) and (S), where the relaxation constant on the original space is fixed.",
"Lemma 4.2 (Triangle-like triplet characterization of property-preserving functions with fixed relaxation constant) Let $f:[0,+\\infty )\\longrightarrow [0,+\\infty )$ be a function such that for every (strong) $b$ -metric space $(X,d)$ with fixed relaxation constant $K\\geqslant 1$ , the space $(X,f\\circ d)$ is a (strong) $b$ -metric space.",
"Then, there exists $K^\\prime \\geqslant 1$ such that: a) for every (strong) $K$ -triangle triplet $(a,b,c)$ , the values $f(a)$ , $f(b)$ and $f(c)$ can be arranged into a (strong) $K^\\prime $ -triangle triplet.",
"b) the relaxation constant of the resulting space $(X,f\\circ d)$ is bounded by $K^\\prime $ .",
"Remark 4.2 This lemma actually gives us insight into six families of property-preserving functions, namely $P_{B}$ , $P_{B,S}$ , $P_{S,B}$ , $P_{S}$ , $P_{M,B}$ and $P_{M,S}$ (the last two obtained by fixing $K=1$ ).",
"Also, one can also immediately deduce b) from a).",
"Due to the fact, that each version of this lemma is proved by the same method, we will simply prove the version for particular subclass of $P_{B,S}$ .",
"Let $f$ be a function satisfying the assumptions of our lemma, i.e.",
"for every $b$ -metric space $(X,d)$ with relaxation constant $K\\geqslant 1$ we have that $(X,f\\circ d)$ is a strong $b$ -metric space.",
"Suppose the contrary, i.e., for every $K^\\prime \\geqslant 1$ (in particular for every $n\\in \\mathbb {N}$ ) there exists a $K$ -triangle triplet $\\lbrace a_n,b_n,c_n\\rbrace $ which is mapped to $\\left\\lbrace f(a_n),f(b_n),f(c_n)\\right\\rbrace $ , the latter one not forming a strong $n$ -triangle triplet.",
"Without the loss of generality, we may assume that $f(a_n)>n\\cdot f(b_n)+f(c_n)$ (this fact becomes clear in the further part of the proof).",
"Let $\\hat{X}_1 = X_1:=\\lbrace (1,1),(1,2),(1,3)\\rbrace $ and $d_1((1,1),(1,2))=a_1 \\qquad d_1((1,2),(1,3))=b_1, \\qquad d_1((1,1),(1,3))=c_1.",
"$ The rest of the values, again, stems from extending $d_1$ to the whole $X_1\\times X_1$ as a semimetric (i.e., (S1) and (S2) conditions imply the remaining function values), yielding $\\hat{d}_1$ .",
"Thus $(\\hat{X}_1,\\hat{d}_1)$ is not a strong $b$ -metric space with relaxation constant 1.",
"Nonetheless, it is still a strong $b$ -metric space if we replace 1 by some larger constant.",
"Now, we will proceed inductively.",
"Having defined $\\left(\\hat{X}_{n-1},\\hat{d}_{n-1}\\right)$ , which is a finite (thus bounded) $b$ -metric space with relaxation constant $K$ , we try to define $\\hat{X}_{n}$ .",
"Put $X_n:=\\lbrace (n,1),(n,2),(n,3)\\rbrace $ and $d_n:X_n\\times X_n \\rightarrow [0,+\\infty ),$ as a unique semimetric satisfying $\\hat{d_n}((n,1),(n,2)) = a_n, \\qquad \\hat{d_n}((n,2),(n,3)) = b_n, \\qquad \\hat{d_n}((n,1),(n,3)) = c_n.$ Again, $(X_n,d_n)$ is a bounded $b$ -metric space with relaxation constant $K$ .",
"We can now use Lemma REF to obtain larger $b$ -metric space $\\hat{X}_n:=X_{n}\\cup \\hat{X}_{n-1}$ , equipped with the extended $b$ -metric $\\hat{d}_{n}$ , which equals $d_n$ on $X_n\\times X_n$ and $\\hat{d}_{n-1}$ on $\\hat{X}_{n-1}\\times \\hat{X}_{n-1}$ .",
"Successively, we obtain the increasing family of $b$ -metric spaces $\\left\\lbrace \\left(\\hat{X}_n,\\hat{d}_n\\right) \\ : \\ n\\in \\mathbb {N} \\right\\rbrace $ .",
"Due to Lemma REF we obtain a $b$ -metric space $(X,D)$ defined as $X:=\\bigcup _{n\\in \\mathbb {N}} X_n, \\qquad D(x,y):=\\hat{d}_{\\min \\lbrace p \\; : \\; x,y\\in \\hat{X}_p\\rbrace }(x,y).$ What is left to prove is that $(X,f\\circ D)$ fails to be a strong $b$ -metric space.",
"Suppose that (S) inequality holds in $(X,D)$ for some $K^\\prime \\geqslant 1$ and let $n_0\\geqslant K^\\prime $ be a fixed natural number.",
"This way, the fact that $(X,D)$ satisfies the strong $b$ -metric inequality with constant $K^\\prime $ implies satisfying it for $n_0$ .",
"Considering $(n_0,1),(n_0,2),(n_0,3)\\in X$ yields $D\\left((n_0,1),(n_0,2)\\right) &=& d_{n_0}\\left( (n_0,1),(n_0,2)\\right) = a_{n_0}\\\\ &>& n_0 \\cdot b_{n_0}+c_{n_0} = n_0 \\cdot D\\left((n_0,2),(n_0,3)\\right) + D\\left((n_0,3),(n_0,1)\\right)\\\\&=& n_0 \\cdot d_{n_0}\\left( (n_0,2),(n_0,3)\\right) + d_{n_0}\\left( (n_0,3),(n_0,1)\\right).$ Of course $f(a_n)$ , $f(b_n)$ and $f(c_n)$ do not have to be aligned in the same order as $a_n$ , $b_n$ and $c_n$ , but the result stays the same as we can interchange the appropriate pairs of points in the reasoning above to obtain the proper counterexample.",
"This shows that $(X,f\\circ D)$ fails to satisfy strong $b$ -metric condition for $n_0$ , thus it fails for $K^\\prime $ as well.",
"Since $K^\\prime $ was arbitrary, $(X,f\\circ D)$ is not a strong $b$ -metric space despite $(X,D)$ being a $b$ -metric one.",
"This concludes the proof of both part a) and b), since it yields a contradiction with the assumption that $(X,f\\circ d)$ is a strong $b$ -metric space whenever $(X,d)$ is a $b$ -metric space with relaxation constant $K$ .",
"The reasoning for the remaining variants of this lemma is exactly the same – all that needs to be changed is the type of triangle triplets used in constructing the proper counterexample.",
"The said lemma allows us to draw somewhat surprising conclusion, that a single property-preserving mapping $f$ does not allow us to obtain arbitrarily large values of relaxation constant on resulting space whenever the relaxation constant of the initial space is bounded.",
"This seems particularly interesting for classes $P_{MS}$ and $P_{MB}$ .",
"We shall now proceed with first of the main results of this paper, which is an extension of analogous characterization already proven by other authors (see [4], [9], [22]).",
"Their results have been incorporated in this theorem in points (i) and (ii) – in particular, we will not provide the proofs for those, as they were well-described in the respective papers they were taken from.",
"Theorem 4.1 (Characterization of triangle-type inequality preserving functions via triangle-like triplets:) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ be an amenable function.",
"Then: $f\\in P_{MB}=P_{B}$ $\\iff $ there exists $K\\geqslant 1$ for every triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b),f(c)\\right)$ form a $K$ -triangle triplet.",
"$f\\in P_{M}$ $\\iff $ for any triangle triplet $(a,b,c)$ , the resulting values $\\left(f(a),f(b), f(c)\\right)$ form a triangle triplet as well.",
"$f\\in P_{BS}$ $\\iff $ for every $K_1\\geqslant 1$ there exists $K_2\\geqslant 1$ , such that for every $K_1$ -triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K_2$ -strong triangle triplet.",
"$f\\in P_{S}$ $\\iff $ for every $K_1\\geqslant 1$ there exists $K_2\\geqslant 1$ , such that for every $K_1$ -strong triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K_2$ -strong triangle triplet.",
"$f\\in P_{SB}$ $\\iff $ for every $K_1\\geqslant 1$ there exists $K_2\\geqslant 1$ , such that for every $K_1$ -strong triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K_2$ -triangle triplet.",
"$f\\in P_{MS}$ $\\iff $ there exists $K\\geqslant 1$ , such that for every triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a $K$ -strong triangle triplet.",
"$f\\in P_{SM}$ $\\iff $ for every $K$ -strong triangle triplet $(a,b,c)$ , the values $\\left(f(a),f(b), f(c)\\right)$ form a triangle triplet.",
"Once again, we would like to remind that $f$ does not have to be monotone, thus the relation $f(a)\\geqslant f(b) \\geqslant f(c)$ does not have to hold (see Remark REF ).",
"The necessity in each case follows as a simple conclusion from Lemma REF .",
"In the last, seventh equivalence another short proof of necessity can be given, as considering a three-point space $X:=\\lbrace 1,2,3\\rbrace $ is enough.",
"Supposing that for some $K$ a strong $K$ -triangle triplet $\\lbrace a,b,c\\rbrace $ which is not mapped to a triangle one exists yields an instant contradiction if we define $d(1,2):=a, \\ d(2,3):=b, \\ d(3,1):=c$ and allow the assumptions (S1), (S2) to fill in the rest of values.",
"Applying $f$ to such defined strong $b$ -metric space maps it into a semimetric structure which is not a metric space.",
"The sufficient part of this last implication follows from the same method as described in point f. above.",
"A sufficiency part can be shown as follows.",
"Assume that $f$ satisfies the right-hand side of equivalence (3) and let $(X,d)$ be an arbitrary $b$ -metric space.",
"Let us denote the relaxation constant of this space by $K$ .",
"Consider three distinct points $x_1,x_2,x_3\\in X$ .",
"WLOG we can assume that $f\\circ d \\left( x_1,x_3\\right) \\geqslant f\\circ d \\left( x_1,x_2\\right) \\geqslant f\\circ d \\left( x_2,x_3\\right).$ Since $(X,d)$ is a $b$ -metric space, then values $d \\left( x_1,x_3\\right), d \\left( x_1,x_2\\right), d \\left( x_2,x_3\\right)$ can be arranged into a $K$ -triangle triplet.",
"From our assumption there exists $K^\\prime $ such that $\\lbrace f\\circ d \\left( x_1,x_3\\right),f\\circ d \\left( x_1,x_2\\right),f\\circ d \\left( x_2,x_3\\right) \\rbrace $ is a strong $K$ -triangle triplet.",
"From the definition of such triplet we obtain $f\\circ d \\left( x_1,x_3\\right) \\leqslant K^\\prime \\cdot f\\circ d \\left( x_2,x_3\\right)+ f\\circ d \\left( x_1,x_2\\right).$ Due to $K^\\prime $ being independent of the choice of $x_1,x_2,x_3$ (since it depends solely on $K$ ) the proof of equivalence (3) is finished.",
"The sufficiency proofs for points (4-7) are almost identical.",
"This characterization might allow us to construct other results describing the properties of metric-type property preserving functions.",
"Before moving to the next part of this paper, we point out two issues we are concerned about.",
"First one of them is the observation that for a fixed function $f$ , the right-hand side of each of the implications (3), (4) and (5) in fact define a mapping $g:[1,+\\infty )\\rightarrow [1,+\\infty )$ , which for every $K_1\\geqslant 1$ assigns the smallest possible $K_2\\geqslant 1$ such that $(X,f\\circ d)$ is $K_2$ -parametrized space, whenever $(X,d)$ is the space of appropriate type with relaxation constant $K_1$ .",
"We know that in all three cases $g(1) = 1$ and the function $g$ is obviously nondecreasing.",
"It is reasonable to think, that different functions $f_1$ , $f_2$ are described by the same mapping $g$ , but an unanswered question remains – what properties of $f$ can we infer based on $g$ and conversely?",
"Moreover, we would like to know what kinds of functions $g$ can be obtained in this way i.e., what attributes do we require from such functions to be obtainable from the discussed theorem.",
"Theorem REF also shifts the research on this functions from metric and topological setting to more real-function theoretical field, connected with functional inequalities.",
"We already know several of the inclusions between the listed above $P_{A_1, A_2}$ families.",
"Hopefully, this characterization will allow us to explore this topic further, allowing us to prove equalities or proper inclusions for some families of property-preserving functions.",
"Thus, we pose the following Open problem 1: Establish the relations between $P_{S,M}$ , $P_{S}$ , $P_{S,B}$ , $P_{B,S}$ and other classes of property preserving functions.",
"In particular, determine which of the inclusions are proper."
],
[
"Further characterizations for semimetrics satisfying relaxed polygonal inequalities",
"It turns out, that the reasoning that was applied to obtain the Theorem REF can be extended on functions satisfying (P) axiom.",
"However, we need to introduce additional definitions.",
"Definition 5.1 A finite tuple of non-negative real numbers $A:=(a_1,\\dots ,a_n)$ such that $a_1 \\geqslant a_2 \\geqslant \\dots \\geqslant a_n.$ is said to be a $K$-relaxed polygon if the following inequality holds: $a_1 \\leqslant K \\cdot \\sum _{i=2}^{n} a_i .$ Remark 5.1 Analogously to the Remark REF , we will refer to a tuple $(a_1,\\dots ,a_n)$ as $K$-relaxed polygon whenever its elements, sorted in nonincreasing order, satisfy Definition REF .",
"Example 5.1 For example, a tuple $A:=(120, 20, 10, 10, 10, 10)$ is an example of $K$ -relaxed polygon with $K=2$ .",
"Indeed $120 \\leqslant 2\\cdot \\left( 20 +10 +10 +10 +10\\right) = 120.$ Notice, that we do not require the largest number to be bounded by any subset sum of the remaining values, i.e.",
"for example we do not require 120 to be bound from above by $K\\cdot (10+10)$ .",
"One can observe, that such a tuple $A$ is a $K$ -relaxed polygon if and only if $\\left(1+K\\right)\\cdot a_1 \\; \\leqslant \\; K\\cdot \\sum _{i= 2}^{n} a_i.$ Then, we introduce the following notion of implementation.",
"Definition 5.2 Consider a $K$ -relaxed polygon $A:=(a_1,\\dots ,a_n), \\qquad a_1 \\geqslant \\dots \\geqslant a_n$ and a semimetric space $(X,d)$ satisfying $K$ -rpi.",
"If there exists a sequence of $n$ -points, $x_1,\\dots ,x_n\\in X$ such that (I1) $d(x_1,x_n):=a_1$ ; (I2) for each $2\\leqslant i \\leqslant n$ , $d(x_{i-1},x_{i}):=a_{i}$ ; then we say that $(X,d)$ implements a $K$ -relaxed polygon $A$ .",
"One may wonder if for any $K$ -relaxed polygon there exists a semimetric space $(X,d)$ which implements it and the answer to that question is, luckily, positive.",
"Lemma 5.1 (Implementation lemma:) Let $A:=(a_1,\\dots ,a_n)$ be a $K$ -relaxed polygon.",
"There exists an $n+1$ element set $X$ with a semimetric $d$ which satisfies $K$ -rpi and implements $A$ .",
"We will construct such space in two steps.",
"Let $X:=\\lbrace 1,\\dots , n\\rbrace $ and put $d(1,n):=a_1$ ; for each $1\\leqslant i \\leqslant n$ , $d({i-1},{i}):=a_{i}$ ; Thus, all of the requirements (I1)-(I2) are satisfied.",
"This definition leaves us with $n$ -gon with no diagonal distances defined.",
"Consider now $i,j\\in X$ such that they are not adjacent i.e.",
"$|i-j|\\notin \\lbrace 1,n\\rbrace $ .",
"For such pair (for the sake of simplicity let us assume $i<j$ ) one can define the distance as follows $d(i,j):=\\min \\left\\lbrace \\sum _{k=i}^{j-1} d(k,k+1), \\left(\\sum _{k=j}^{n-1} d(k,k+1) + d(1,n) + \\sum _{k=1}^{i-1}d(k,k+1) \\right) \\right\\rbrace ,$ where the sums in the latter part of the minimum might be empty (for example when $i=1$ and $j<n$ or $j=n$ and $i>1$ ).",
"Figure: Informally, the distance between two points (for example, red line between 2 and ii) can be described as the shorter of the paths over the perimeter.",
"Notice, that such distance does not always have to be the left-oriented path (here denoted by lime green).Notice, that for any two non-adjacent points $i,j$ thanks to the formulation of $d(i,j)$ the distance can be bound from above by the distance over the perimeter of the discussed polygon.",
"Indeed, for any sequence of point $i,x_1,x_2,\\dots ,x_k,j\\in X$ .",
"$K\\cdot \\left( d(i,x_1) + \\dots + d(x_k,j)\\right) &\\geqslant & K\\cdot \\min \\left\\lbrace \\sum _{k=i}^{j-1} d(k,k+1), \\left( d(1,n) + \\sum _{k=j}^{n-1} d(k,k+1) + \\sum _{k=1}^{i-1}d(k,k+1) \\right) \\right\\rbrace \\\\ &\\geqslant & d(i,j).$ Perhaps this part of the proof can be reworked to be nicer.",
"Now, consider any two adjacent points of the space, $i$ and $i+1$ (the case of $n$ and 1 is analogous).",
"Consider any point $x\\in X$ .",
"WLOG we can assume that $x>i$ .",
"Then we have three posibilities: $d(i,x) = \\sum _{k=i}^{x-1} d(k,k+1)$ , then obviously $d(i,x)\\geqslant d(i,i+1)$ ; $d(i,x) = d(1,n) + \\sum _{k=x}^{n-1} d(k,k+1) + \\sum _{k=1}^{i-1}d(k,k+1)$ .",
"If $d(x,i+1) =d(1,n) + \\sum _{k=x}^{n-1} d(k,k+1) + \\sum _{k=1}^{i}d(k,k+1)$ then $d(x,i+1)\\geqslant d(i,i+1)$ and the case is once again trivial.",
"$d(i,x) = d(1,n) + \\sum _{k=x}^{n-1} d(k,k+1) + \\sum _{k=1}^{i-1}d(k,k+1)$ and $d(x,i+1) = \\sum _{k=i+1}^{x-1} d(k,k+1)$ .",
"Then, using the fact that $A$ is a $K$ -relaxed polygon $d(i,i+1) \\leqslant K\\cdot \\left( d(1,n) - d(i,i+1) + \\sum _{i=1}^n d(k,k+1)\\right),$ which translates to $d(i,i+1) \\cdot (1+K) \\leqslant a_1 \\leqslant K\\cdot \\sum _{k=1}^n d(k,k+1) = K\\cdot \\sum _{k=1}^n a_k.$ When more points are considered, the reasoning remains the same (although it might need some more formal touch) – we either use the fact that length of one of the edges exceeds the distance $d(i,i+1)$ or the sum of the distances is greater or equal to the remaining part of the perimeter of the initial $K$ -relaxed polygon.",
"In any case, the property (P) from Definition 2. holds.",
"To obtain analogous result to Theorem REF we need lemmas which are analogous to Lemmas REF and REF .",
"We provide only the short sketches of proofs to the lemmas, since the proofs are similar to the respective results from the previous sections.",
"Lemma 5.2 (Concatenation lemma for semimetric spaces satisfying $K$ -rpi) Let $(X_1,d_1)$ , $(X_2,d_2)$ be a pair of disjoint, semimetric spaces satisfying property (P) with relaxation constants $K_1$ , $K_2$ .",
"Assume that $\\operatorname{diam}_{d_1}(X_1)=r_1<\\infty $ and $\\operatorname{diam}_{d_2}(X_2)=r_2<\\infty $ .",
"Let $X:=X_1\\cup X_2$ .",
"There exists an extension of $d_1,\\ d_2$ , namely $d:X\\times X \\rightarrow [0,+\\infty )$ which satisfies $K$ -rpi, where $K:=\\max \\lbrace K_1,K_2\\rbrace $ .",
"Additionally, $\\operatorname{diam}_d(X)=\\max \\lbrace r_1, r_2\\rbrace $ .",
"Let us begin with defining $d$ as $d(x,y):={\\left\\lbrace \\begin{array}{ll}d_1(x,y), & x,y\\in X_1\\\\d_2(x,y), & x,y\\in X_2\\\\\\frac{\\max \\lbrace r_1,r_2\\rbrace }{2K}, & \\text{everywhere else,}\\end{array}\\right.",
"}$ where $K:=\\max \\lbrace r_1,r_2\\rbrace $ .",
"One can immediately notice, that such space $(X,d)$ will have its diameter at most $r:=\\max \\lbrace r_1,r_2\\rbrace $ .",
"What is left to prove is that $d$ fulfills the property $K$ -rpi.",
"Let $n\\in \\mathbb {N}$ and $x_1, \\dots , x_n\\in X$ .",
"The cases where both $x_1$ and $x_n$ belong to the same $X_i$ are relatively obvious, since in the cases where all remaining $x_k$ , $1\\leqslant k <n$ belong to the same component, the respective relaxed polygonal property steps in.",
"At the same time, if there is at least one $x_{k_1}$ from the second component, then $d(x_1,x_n)\\leqslant r_i \\leqslant 2K \\cdot \\frac{r}{2K} \\leqslant K\\cdot \\sum _{i=1}^{n} d(x_{i-1},x_i).$ The last inequality follows from the fact, that the polygon consisting of points $x_1,\\dots ,x_n$ consists of at least two indices, $1\\leqslant i_0,i_1 < n$ , for which $x_{i_0}\\in X_i$ and $x_{i_0+1}\\notin X_i$ , as well as $x_{i_1}\\notin X_i$ and $x_{i_1+1}\\in X_i$ .",
"If $x_1, x_n$ belong to distinct components, then there exists $1\\leqslant k< n$ such that $x_k$ belongs to the same component as $x_1$ and $x_{k+1}$ does not.",
"Therefore $\\frac{r}{2K} = d(x_1,x_n) \\leqslant K\\cdot d(x_k,x_{k+1}) \\leqslant K\\cdot \\sum _{i=1}^n d(x_{i-1},x_i).$ This finishes the proof of the fact that $(X,d)$ satisfies $K$ -rpi.",
"Lemma 5.3 (Summation lemma for semimetric spaces satisfying $K$ -rpi) Let $ be a linearly ordered set and $ F:= {(Xt,dt) : t}$ be an increasing family of semimetric spaces satisfying $ K$-rpi with a fixed constant $ K1$.",
"Then a semimetric space $ (X,d)$ given by$$X:=\\bigcup _{t\\in X_t \\qquad \\qquad d(x,y):=d_p(x,y), \\text{ where } p \\text{ is any index s.t. }",
"x,y\\in X_psatisfies K-rpi as well.",
"}{\\begin{xmlelement*}{proof}Since (P) property considers only finite sequences of points (albeit of arbitrary length), then the proof of this lemma is almost exactly the same as of the Lemma \\ref {summationLemma}.\\end{xmlelement*}}$ Using these definitions and lemmas, one can formulate and prove the following theorem, analogous to the Theorem REF .",
"Theorem 5.1 (Characterization of (P)-preserving mappings) Consider an amenable function $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ .",
"Then $f$ is (P)-preserving iff for any $K_1\\geqslant 1$ there exists $K_2$ such that $f$ maps any $K_1$ -relaxed polygon to $K_2$ -relaxed polygon, i.e.",
"$\\forall _{K_1\\geqslant 1} \\ \\exists _{K_2 \\geqslant 1} \\ \\Bigg ( \\forall _{n\\in \\mathbb {N}} \\ \\forall _{a_1,\\dots ,a_n \\in [0,+\\infty )} \\ \\left( \\ \\left(1+K_1\\right)\\cdot \\max \\lbrace a_i \\ : \\ i\\leqslant n \\rbrace \\; \\leqslant \\; K_1\\cdot \\sum _{1\\leqslant i\\leqslant n}a_i \\ \\right)\\nonumber \\\\\\Rightarrow \\left( \\left(1+K_2\\right)\\cdot \\max \\lbrace f(a_i) \\ : \\ i\\leqslant n \\rbrace \\; \\leqslant \\; K_2\\cdot \\sum _{1\\leqslant i\\leqslant n}f(a_i) \\right) \\ \\Bigg ).$ We begin by showing the sufficiency of the proposed condition.",
"Assume that (REF ) holds.",
"Let $(X,d)$ be any semimetric space satisfying $K_1$ -relaxed polygonal inequality.",
"Consider a finite sequence of points $x_1,\\dots , x_n\\in X$ .",
"Since $d$ satisfies $K_1$ -rpi, then the tuple $A:=\\left( d\\left(x_1,x_n\\right), d\\left(x_1,x_2\\right),d\\left(x_2,x_3\\right), \\dots , d\\left(x_{n-1},x_{n}\\right) \\right)$ is (up to an order of elements) a $K_1$ -relaxed polygon.",
"Therefore, the values of $f$ at these respective tuple $A_f:=\\left(f\\circ d\\left(x_1,x_n\\right), f\\circ d\\left(x_1,x_2\\right),f\\circ d\\left(x_2,x_3\\right), \\dots , f\\circ d\\left(x_{n-1},x_{n}\\right) \\right)$ can be arranged into a $K_2$ -relaxed polygon.",
"This implies that $f\\circ d (x_1,x_n) &\\leqslant & \\max A_f \\leqslant K_2 \\cdot \\left( f\\circ d (x_1,x_n) +\\sum _{i=2}^n f\\circ d\\left(x_{i-1},x_{i}\\right)\\right) - K_2 \\max A_f\\\\&\\leqslant & K_2 \\cdot \\left( f\\circ d (x_1,x_n) +\\sum _{i=2}^n f\\circ d\\left(x_{i-1},x_{i}\\right)\\right) - K_2 f\\circ d (x_1,x_n)\\\\&=& K_2 \\cdot \\sum _{i=2}^n f\\circ d\\left(x_{i-1},x_{i}\\right),$ so once again we are able to observe that the particular order in which the elements appear in $A_f$ does not matter as long as the condition (P) holds.",
"Since the points $x_1,\\dots ,x_n$ were chosen arbitrarily, $(X,f\\circ d)$ satisfies $K_2$ -rpi.",
"The proof of the other implication goes analogously to the scheme from the proof of Theorem REF .",
"Fix function $f$ .",
"If we suppose the contrary, we obtain the existence of such $K_1\\geqslant 1$ that for every $K_2\\geqslant 1$ there exists some $K_1$ -relaxed polygon which is not mapped to a $K_2$ -relaxed polygon (albeit again it might be a $K_2^{\\prime }$ -relaxed polygon for some greater constant $K_2^{\\prime }$ ).",
"Consider a sequence of such $K_1$ polygons $A_n$ .",
"Due to Lemma REF , this implies the existence of spaces $(X_n,d_n)$ , all of which satisfy $K_1$ -rpi.",
"Therefore, using Lemma REF we obtain an increasing sequence of spaces with (P)-property with the same constant $K_1$ .",
"What remains is to sum them up using Lemma REF , obtaining a large space $(X,d)$ which satisfies $K_1$ -rpi as well.",
"The function $f$ fails to preserve property (P) due to the fact that for any natural number $n$ there exists a set of points $\\lbrace x_1,\\dots ,x_k\\rbrace $ (coming from the space $X_n$ ) in $X$ such that $A_n:=\\left(d(x_1,x_k),d(x_1,x_2),\\dots ,d(x_{k-1},x_k) \\right)$ is a $K$ -relaxed polygon.",
"Therefore $B:=\\left(f\\circ d(x_1, x_k), \\dots , f\\circ d(x_{k-1},x_k) \\right)$ does not form a $n$ -relaxed polygon and thus $(X,f\\circ d)$ fails to satisfy $n$ -rpi.",
"It is also possible to investigate some results concerning families $P_{P,S}, P_{P,B}, P_{P,M}$ as well as $P_{S,P}, P_{B,P}, P_{M,P}$ .",
"The problem of obtaining analogous results for such classes of spaces is left as another open problem for the Readers.",
"This part of the paper would be somewhat lacking without any examples of functions in $P_{P}$ .",
"But before we provide any such characterization, we suggest some easy to check sufficient conditions for a function to be (P)-preserving.",
"We will need the characterization of spaces satisfying $K$ -rpi, which is due to Fagin et al.",
"[17].",
"Theorem 5.2 (Characterization of $K$ -rpi spaces via metric bounds) Let $(X,d)$ be a semimetric space.",
"The following are equivalent: (i) $(X,d)$ satisfies (P) with relaxation constant $K\\geqslant 1$ ; (ii) there exists a metric $d^{\\prime }$ on $X$ such that for all $x,y\\in X$ $d^{\\prime }(x,y)\\leqslant d(x,y) \\leqslant K\\cdot d^{\\prime }(x,y)$ Having this characterization at our disposal we can proceed with the following result.",
"Lemma 5.4 (Sufficient conditions for (P)-preservation) Let $f:[0,+\\infty )\\rightarrow [0,+\\infty )$ .",
"If there exist $a,b\\in \\mathbb {R}$ such that $ax\\leqslant f(x) \\leqslant bx$ for all $x\\geqslant 0$ , then $f\\in P_P$ .",
"Assume that $f$ is bounded from both sides by linear functions $x\\mapsto ax$ and $x\\mapsto bx$ .",
"Let $(X,d)$ be any semimetric space satisfying condition (P).",
"From Theorem REF we obtain the existence of a metric $d^{\\prime }$ on $X$ such that the inequalities (REF ) hold.",
"Since linear functions belong to $P_M$ due to being subadditive, increasing and amenable (see Lemma REF ), their compositions with $d^{\\prime }$ are also metrics.",
"Thus we have $ad^{\\prime }(x,y)\\leqslant ad(x,y)\\leqslant f(d(x,y)) \\leqslant bd(x,y) = \\frac{b}{a}\\cdot ad^{\\prime }(x,y)$ Therefore $f\\circ d$ is bounded from below by the metric $ad^{\\prime }(x,y)$ and from above by $\\frac{b}{a} \\cdot ad^{\\prime }(x,y)$ .",
"Due to Theorem REF we have that $f(d(x,y))$ satisfies condition (P).",
"Due to $(X,d)$ being arbitrary, $f\\in P_P$ .",
"This allows us to construct this somewhat nontrivial example of a function from $P_P$ .",
"Example 5.2 Let $f(x):=5x-4\\lfloor x \\rfloor $ , where $\\lfloor x \\rfloor $ denotes the largest integer not exceeding $x$ .",
"Figure: A graph depicting sawtooth function ff.Obviously $x\\leqslant f(x) \\leqslant 5x$ , thus $f\\in P_P$ .",
"One can also notice that $f$ does not belong to $P_M$ , as it fails to be subadditive.",
"$f(\\frac{3}{2}) = \\frac{15}{2} >f(1)+f\\left(\\frac{1}{2}\\right) = 1+\\frac{5}{2} = \\frac{7}{2}.$ From Lemma REF we obtain that $f\\notin P_{M}$ .",
"Remark 5.2 As a conclusion we obtain that although $P_M$ is not disjoint with $P_P$ (as identity belongs to both of those classes) we have $P_M\\ne P_P$ ."
],
[
"Acknowledgments",
"As usual, I would like to express my utmost gratitude to both of my supervisors, that is Jacek Jachymski and Mateusz Krukowski for multitude of fruitful discussions which included priceless mathematical and writing tips.",
"Lastly, I would like to acknowledge Piotr Nowakowski and Mateusz Lichman for adding some interesting comments on the topic.",
"As usual, I would like to express my utmost gratitude to both of my supervisors, that is Jacek Jachymski and Mateusz Krukowski for multitude of fruitful discussions which included priceless mathematical and writing tips.",
"Lastly, I would like to acknowledge Piotr Nowakowski and Mateusz Lichman for adding some interesting comments on the topic."
]
] | 2011.14110 |
[
[
"Analysing the Epoch of Reionization with three-point correlation\n functions and machine learning techniques"
],
[
"Abstract Three-point and high-order clustering statistics of the high-redshift 21cm signal contain valuable information about the Epoch of Reionization.",
"We present 3PCF-Fast, an optimised code for estimating the three-point correlation function of 3D pixelised data such as the outputs from numerical and semi-numerical simulations.",
"After testing 3PCF-Fast on data with known analytic three-point correlation function, we use machine learning techniques to recover the mean bubble size and global ionisation fraction from correlations in the outputs of the publicly available 21cmFAST code.",
"We assume that foregrounds have been perfectly removed and negligible instrumental noise.",
"Using ionisation fraction data, our best MLP model recovers the mean bubble size with a median prediction error of around 10%, or from the 21cm differential brightness temperature with median prediction error of around 14%.",
"A further two MLP models recover the global ionisation fraction with median prediction errors of around 4% (using ionisation fraction data) or around 16% (using brightness temperature).",
"Our results indicate that clustering in both the ionisation fraction field and the brightness temperature field encode useful information about the progress of the Epoch of Reionization in a complementary way to other summary statistics.",
"Using clustering would be particularly useful in regimes where high signal-to-noise ratio prevents direct measurement of bubble size statistics.",
"We compare the quality of MLP models using the power spectrum, and find that using the three-point correlation function outperforms the power spectrum at predicting both global ionisation fraction and mean bubble size."
],
[
"Introduction",
"A few hundred million years after the Big Bang, the first stars and galaxies began to form [7].",
"The radiation emitted from these luminous structures interacted with the surrounding neutral hydrogen and caused it to become ionized.",
"These initially isolated ionized bubbles grew over time.",
"Around 1 billion years after the Big Bang the Universe became fully-ionized, see for instance [5] and [26].",
"The phase shift from a fully neutral to a fully ionized Universe occurred during the so-called Epoch of Reionization (EoR).",
"Many particulars about this process remain unconstrained by current data, predominantly because there are precious few sources of observable radiation during this time.",
"Another way to observe the process of reionization would be to distinguish regions of ionized hydrogen in the neutral background.",
"The most promising probe for this is the 21cm hyperfine transition of hydrogen, which is only observed in neutral hydrogen.",
"Measurements of the 21cm signal on the sky thus provide a map of which parts of the Universe were neutral.",
"By observing this signal at different redshifts, these maps can be extended into three-dimensional maps of the neutral hydrogen.",
"The size and clustering properties of the ionized hydrogen bubbles change throughout the EoR.",
"The 21cm signal is much weaker than other foreground sources at the same frequencies.",
"These strong foregrounds make it difficult to extract the actual 21cm signal.",
"Past and ongoing purpose-built experiments such as the Murchison Widefield Arrayhttp://www.mwatelescope.org/telescope (MWA, [73]), the Low Frequency Arrayhttp://www.lofar.org/ (LOFAR, [58]), and the Precision Array for Probing the Epoch of Reionizationhttp://eor.berkeley.edu/ (PAPER, [2]) have begun to place upper limits on the overall intensity of the signal.",
"The Experiment to Detect the Global EoR Signaturehttps://www.haystack.mit.edu/ast/arrays/Edges/ (EDGES) last year claimed a first detection of the 21cm signal.",
"This 21cm absorption profile was observed at redshifts between $15 < z < 20$ with an amplitude of $500 \\text{mK}$ , published in [6].",
"This exciting result has generated much attention in the past year, as the amplitude is significantly more negative than that anticipated by standard reionization models.",
"The strongly negative amplitude is difficult to explain without considering additional cooling mechanisms or a higher background radiation than that of the Cosmic Microwave Background.",
"Several recent publications have considered possible modifications that could explain the discrepancy, for instance: considering dark matter interactions ([13], [4], [56]); the properties of dark matter ([80], [14], [79], [41]); axionic dark matter ([54], [71], [39]); the effects of radio-wave background ([12]); and considerations of mirror neutrinos ([3]).",
"Other attempts have been made to explain the amplitude in terms of the foreground analysis method (e.g.",
"[72]).",
"However, until other 21cm observations confirm this detection it is still sensible to continue work with the standard fiducial models that exclude such exotic physics.",
"Upcoming experiments such as the Hydrogen Epoch of Reionization Arrayhttp://reionization.org/ (HERA, [11]) and the Square Kilometre Arrayhttps://www.skatelescope.org/ (SKA, [50]) will be able to provide more detailed measurements and should allow us to understand the processes of reionization in detail and confirm the scenarios proposed to explain the EDGES detection.",
"The most detailed theoretical modelling of the 21cm signal currently makes use of simulations.",
"Numerical and semi-numerical simulations encapsulate many aspects of the complex non-linear reioinization processes.",
"Common numerical simulations include C2-RAY [49], which models the ionising photons emission processes and traces these rays from source to absorption; GRIZZLY ([17]), which uses one-dimensional radiative transfer simulations to model the radiation profiles around different source types, and then stamps these profiles onto source locations; and many codes which use adaptive refinement [38] to model both large scales and small scales in a single simulation (see for example the Cosmic Reionization on Computers program [21], and LICORICE [66]).",
"Such simulations can provide theoretical predictions for a range of possible reionization scenarios, by specifying different values for simulation input parameters.",
"Comparisons between 21cm data and theory often make use of fast approximate semi-numerical simulations such as 21cmFAST [53] and SimFast21 [64].",
"By running a large number of simulations for a range of different reionization scenarios, we can determine which scenarios give rise to the best match between simulated and observed data.",
"Two techniques can make this process more efficient.",
"First, sampling methods such as Markov chain Monte Carlo (MCMC) ([23], [27], [61], [24], [25]) reduce the total number of simulations that are needed in order to hone-in on the best regions of parameter-space.",
"Second, the simulated and observed data can be compressed before comparing them by using summary statistics.",
"These summary statistics reduce the total size of the data while retaining much of the useful information, and are more robust to modelling and sample variance errors.",
"Common summary statistics are the power spectrum and its higher-order equivalent the bispectrum ([69], [74], [46], [19], [75], and [31]).",
"Both statistics contain information about the clustering properties of ionized hydrogen bubbles.",
"In this paper we use machine learning techniques to investigate using the three-point correlation function (3PCF) as another summary statistic for 21cm data.",
"In particular, we determine whether the 3PCF can inform us about the mean size of ionized bubbles ($R_{\\mathrm {bubble}}$ ) and about the global ionization fraction $\\langle x_{\\mathrm {HII}}\\rangle $ .",
"These statistics provide information about the progress of the EoR and encode useful information about different physical scenarios.",
"They also provide a means to reduce the effect of thermal noise, since they are statistical quantities that are averaged over the entire map.",
"Although the 3PCF should be less affected by noise than full 21cm maps, the power spectrum should be even less affected.",
"We compare the relative performances of using either the power spectrum or the 3PCF with our methodology.",
"This indicates whether the 3PCF likely encodes any extra information about bubble size statistics than does the power spectrum.",
"As well as recent work using the bispectrum, some research has focussed on using the 3PCF as a tool for investigating the EoR.",
"[22] use a derived statistic from the 3PCF to concentrate on phase information.",
"[28] investigate whether the 3PCF of 21cm data can be modelled using a local bias model.",
"Their resulting model makes predictions with around $20 \\%$ accuracy for large ionized regions at early times, but breaks down for other scenarios.",
"Machine learning has already been suggested and used for a number of different applications with 21cm data: to emulate power spectrum outputs quickly from 21cmFAST ([33], [65], [35]), to derive reionization parameters directly from the 21cm power spectrum ([68]), and to derive reionization parameters from 21cm images ([18]).",
"[33] also present a mapping between 21cmFAST and SimFast21 providing a proof of concept for mapping between simulations that predict different EoR histories.",
"In this paper, we run a large representative sample of semi-numerical simulations using 21cmFAST.",
"For each simulation, we calculate the 3PCF of the resulting 21cm maps.",
"We also measure the characteristic reionization features: the global ionization fraction, and the size distribution of the ionized bubbles.",
"We then use machine learning techniques to determine the relationships between the 3PCF measurements and the characteristic reionization features.",
"The rest of the paper is split in to the following sections.",
"Section describes the mathematical concept behind the 3PCF, and a description of the code implementation.",
"We also test the code on data with known analytical 3PCF.",
"In Section we describe current physical models of the reionization process.",
"We include a description of the 21cmFAST code and a summary of the range of reionization scenarios considered in this paper.",
"Section gives an overview of the machine learning techniques we use, including the search strategy that we use to find the best possible model.",
"We also summarise the methods used to analyse the performance of the resulting models.",
"In the remaining sections we use our data to learn about the characteristic reionization features: the mean bubble size in Section , and the global ionization fraction in Section .",
"We end the paper in Section with our conclusions.",
"For Cosmological parameters, we use $\\Omega _{\\mathrm {M}} = 0.3153$ , $\\Omega _{\\mathrm {b}} = 0.0493$ , $\\Omega _{\\mathrm {\\Lambda }} = 0.6847$ , $H_0 = 67.36 \\text{km s}^{-1}\\text{Mpc}^{-1}$ , $n_{\\mathrm {s}} = 0.9649$ , $\\sigma _8 = 0.8111$ , the latest results using the default Plik likelihood from [60]."
],
[
"Three point correlation calculation",
"The three-point correlation function $\\xi ^{(3)}(\\mathbf {r_1}, \\mathbf {r_2}, \\mathbf {r_3})$ is defined as the ensemble average over triplets of points in real space, $\\xi ^{(3)} \\left( \\mathbf {r_1}, \\mathbf {r_2}, \\mathbf {r_3} \\right) = \\left\\langle \\delta (\\mathbf {r_1}) \\delta (\\mathbf {r_2}) \\delta (\\mathbf {r_3}) \\right\\rangle $ The angular brackets denote an ensemble average over a large region of space (or over a large number of universe realisations) to mitigate the effect of statistical fluctuations.",
"If the signal is translationally invariant then the ensemble average can be replaced by a spatial average.",
"If the signal is also rotationally invariant then the 3PCF depends only on the lengths ($r_1 = |\\mathbf {r_1}$ ) of the real-space vectors and not on their directions.",
"These invariance assumptions are broken in real observations, in part due to the light cone effect (for example [10]) and redshift space distortions (for example [45]).",
"Note that the three vectors $\\mathbf {r_1}, \\mathbf {r_2}, \\mathbf {r_3}$ connect three points in real space and so have a vector sum of $\\mathbf {0}$ , i.e.",
"they form a closed triangle.",
"In practice, $\\xi ^{(3)}$ measurements are actually made over configurations of triangles, by which we mean over sets of unique triangle side-lengths in order to beat down statistical noise.",
"The three-point correlation function (3PCF) for a single triangle configuration is an average over all triangles with those side lengths $r_1, r_2, r_3$ , namely $\\xi ^{(3)} \\left( r_1, r_2, r_3 \\right) = \\Big \\langle \\delta (\\mathbf {r_1}) \\delta (\\mathbf {r_2}) \\delta (\\mathbf {r_3}) \\Big \\rangle _ { (|\\mathbf {r_1}|, |\\mathbf {r_2}|, |\\mathbf {r_3}|)\\ =\\ (r_1, r_2, r_3) }.$"
],
[
"Code Implementation",
"Calculating the 3PCF involves placing differently-sized triangles into the data field.",
"The product of the data values at each of the three triangle vertices is summed over a large number of similarly-sized triangles, and an estimate of the 3PCF is built up.",
"The final output of the algorithm is the 3PCF estimates $\\xi ^{(3)}(\\mathbf {r_i})$ at a discrete set of radius values $\\mathbf {r_i} = (r_1, r_2, r_3)$ , corresponding to a discrete set of radius bins.",
"The 3PCF estimate for each radius bin is calculated by using a set of many triangles with similar (but not identical) side lengths.",
"In this section we first describe how to find these sets of triangles, by matching triangles whose side-lengths lie within a given binned range of radii $R_\\mathrm {min} \\le r < R_\\mathrm {max}$ .",
"We also provide pseudocode for our C++ 3PCF-Fast algorithm (publicly available on GitHubhttps://github.com/wdjennings/3PCF-Fast).",
"Finally we discuss how we use the output statistics from the 3PCF-Fast code to estimate 3PCF values.",
"Our algorithm is similar in nature to other high-order codes (see for instance [16]), although we subsample both the triangle configurations and the number of lattice points and measure the level of approximation needed for robust estimates of the 3PCF.",
"Efficiently finding sets of similarly-sized triangles is a key preparation stage of the algorithm.",
"The data in this section are represented as a pixelised set of scalar values in three-dimensions.",
"For each radius bin we find all the triangles whose edge lengths $r_1, r_2, r_3$ lie within a fixed range of side-lengths $R_\\mathrm {min} \\le r_i < R_\\mathrm {max}$ .",
"There are a finite number of such triangles because the three vertices are constrained to lie on the centres of pixels in the data.",
"To find explicit matching triangles we place the first vertex at the origin.",
"We then find all possible second vertices ($\\mathbf {r_2}$ ) which lie within the spherical shell $R_\\mathrm {min} \\le | \\mathbf {r_2} | < R_\\mathrm {max}$ of the origin.",
"From each of the matching second vertex points, we find the third vertex points ($\\mathbf {r_3}$ ) which are a valid distance both from $\\mathbf {r_2}$ and from the origin.",
"This last step is effectively finding pixels which lie in the overlap of two spherical shells.",
"Figure REF shows an example in two dimensions: with the first triangle vertex at the origin, the dark purple annulus indicates the allowed region for the second vertex between $R_\\mathrm {min}$ and $R_\\mathrm {max}$ .",
"The orange region then shows the allowed region of third vertices from one of the possible second vertices.",
"The final matching triangles (of which there are two) are outlined in black in the figure.",
"To prevent repeated calculations we use a python script to search for these matching triangles and store the resulting pairs of vectors $(\\mathbf {r_2},\\mathbf {r_3})$ in a binary file.",
"This binary file can be loaded by the main C++ algorithm many times.",
"we refer to these binary files as vertices files.",
"Measurements of the 3PCF are calculated by looping over possible lattice points $\\mathbf {r_1}$ and summing the contributions for all triangle configurations $(\\mathbf {r_2}, \\mathbf {r_3})$ at that pixel.",
"Both the number of triangle configurations and the number of lattice points are sampled to give faster calculations, and we investigate different levels of sampling in Section REF .",
"Figure: Triangles matching the radius bin condition 2.5pixels≤r<3.5pixels2.5\\ \\text{pixels} \\le r < 3.5\\ \\text{pixels}.",
"The two regions shown are the radius conditions around the first and second points.",
"The allowed third point(s) then lie in the overlap of these annuli.The three-point correlation function of a data field is usually calculated in comparison to a random field without clustering.",
"The correlation function then quantifies the extent to which the data field is more clustered than the random unclustered field.",
"The purpose of the random field is to create a comparison for the data field.",
"Using a uniform field can thus be seen as a method for counting the number of triangle configuration occurrences.",
"The outputs from our three-point code are the auto- and cross-correlation statistics between the data (D) and random (R) fields.",
"For the 3PCF these statistics are the data-data-data statistic (DDD), data-data-random (DDR), data-random-random (DRR) and random-random-random (RRR).",
"DDD is the auto-correlation found by multiplying the data field at all three vertices.",
"DDR is the cross-correlation found by multiplying the data at two vertices and the random field at the final vertex; and so on.",
"These statistics will later be combined to give an estimate of the 3PCF.",
"For a scalar data field, the random field should be uniform with mean equal to the data mean.",
"Instead, it is practically simpler and mathematically identical to normalise the data field to have a mean of unity, so that the random field averaged within in each pixel is also everywhere unity.",
"This allows our code to skip the correlation calculations for the random field, since the value of RRR is equal to the known integer count of triangles.",
"Algorithm REF shows the pseudocode for our algorithm, taking as inputs a data field $D[\\mathbf {r}]$ and a binned vertices file, and outputting the three-point correlation statistics (DDD, DDR, DRR, and RRR) for each radius bin.",
"[H] Three-point correlation algorithm [1] 3PCF$D[\\mathbf {r}], R_\\mathrm {min}, R_\\mathrm {max}$ DDD, DDR, DRR, RRR $\\leftarrow 0$ Initialise to zero Load $(\\mathbf {r_2},\\mathbf {r_3})$ using ($R_\\mathrm {min}$ , $R_\\mathrm {max}$ ) vertices file all $\\mathbf {r_1}$ over all data pixels each $\\mathbf {r_2}$ ,$\\mathbf {r_3}$ pair over matching triangles DDD $\\mathrel {+}\\mathrel {=} D[\\mathbf {r_1}] \\times D[\\mathbf {r_1}+\\mathbf {r_2}] \\times D[\\mathbf {r_1}+\\mathbf {r_3}]$ DDR $\\mathrel {+}\\mathrel {=} D[\\mathbf {r_1}] \\times D[\\mathbf {r_1}+\\mathbf {r_2}]$ DRR $\\mathrel {+}\\mathrel {=} D[\\mathbf {r_1}]$ RRR $\\mathrel {+}\\mathrel {=} 1$ return DDD, DDR, DRR, RRR Algorithm REF outputs the correlation statistics (DDD, DDR, DRR, RRR).",
"An estimate of the 3PCF is found by combining these statistics.",
"The simplest such estimator is given by ratios of the data- and random-field auto-correlations, $\\xi ^{(3)} = \\frac{DDD - RRR}{RRR}.$ Another estimator, $\\xi ^{(3)} = \\frac{DDD - 3DDR + 3DRR - RRR}{RRR},$ from [40] generally leads to less biased results, because it takes account of cross-correlations between the data and random fields which the simple estimator ignores.",
"The number of triangles found by this matching algorithm can quickly exceed hundreds of thousands for side lengths larger than around ten pixels.",
"Even running the matching algorithm itself for such side lengths can take several days and, more significantly, using such an exhaustive set of triangles in the correlation algorithm would require years of CPU time.",
"An accurate measurement of the 3PCF can be obtained efficiently by subsampling a small number of triangles from all valid matches.",
"In this section we sample 5000 occurrences of triangle configurations from the total available number of valid matches.",
"In Section REF we discuss the effect of sampling and why a value of 5000 was chosen."
],
[
"Testing ",
"We test our code by generating three-dimensional realisations for a distribution with a known 3PCF.",
"We compare the measured 3PCF from our code to the theoretical form, to get an indication of the regimes in which the code has good accuracy and precision.",
"Our testing distribution consists of three-dimensional realisations made up of a set of points in a box.",
"First, a large set of points are uniformly placed on the surfaces of many identically-sized spheres.",
"The data are then saved to a data file by overlaying a three-dimensional pixelised grid and counting the number of points in each grid: zero for no points, one for a single point, and so on.",
"For all realisations in this section, the data are represented as a box with side length 100 arbitrary units pixelised into $512^3$ pixels.",
"The amplitude and shape of the theoretical 3PCF for these realisations depend on the sphere radius $R$ and the number density of spheres $n_\\mathrm {s}$ .",
"We describe a scenario as a particular pair of these two parameters.",
"We also use the number of spheres $N_\\mathrm {s} = n_\\mathrm {s} \\times 10^6$ , since all realisations in this section have a fixed box size of 100 arbitrary units in each of the three dimensions.",
"The equilateral 3PCF of points-on-sphere realisations has a closed analytic form [43].",
"For a scenario with parameters $n_\\mathrm {s}$ and $R$ , the 3PCF for equilateral triangles as a function of the triangle side length $r$ is given by $\\xi ^{(3)}(r; R, n_\\mathrm {s}) =\\begin{dcases}\\frac{1}{16 \\pi ^3 R^3 n_\\mathrm {s}^2 r^2 \\sqrt{3 R^2 - r^2}} & \\text{ if } r < R \\sqrt{3}, \\\\0 & \\text{ otherwise}\\\\\\end{dcases}$ Generating a realisation for a scenario involves choosing where to put the spheres and then placing points on the surfaces of those spheres.",
"A uniformly random set of $N_\\mathrm {S}$ points are chosen to be the centres of the spheres.",
"Points are then placed randomly onto the surface of each sphere.",
"Ensuring that the points are indeed uniformly distributed across the spheres surface is most easily done using the method from [55]: sample three random variables $x,y,z$ from the normal distribution $\\mathcal {N}(0,1)$ and divide by the Euclidean norm of these three coordinates.",
"The distribution of the normalised vector $\\mathbf {r} = \\frac{R}{\\sqrt{x^2 + y^2 + z^2}}\\begin{bmatrix}x \\\\ y \\\\ z\\end{bmatrix},$ is then uniform across the surface of a sphere with radius $R$ .",
"After storing the locations of all points on all spheres, the final pixelised realisation of the scalar field is generated by rounding the point coordinates to the nearest integer.",
"Figure REF shows a slice through an example realisation of the testing distribution.",
"All data in this testing section have pixel size of around $0.2$ arbitrary physical units.",
"Figure: Slice through an example realisation of points-on-spheres data.",
"This scenario uses spheres with R=10R=10 and N s =200N_\\mathrm {s}=200.",
"Each sphere appears as a circular annulus as it has been horizontally sliced for this figure.",
"Some annuli appear thicker than others because slicing a thick spherical shell near its pole gives a wider region when viewed from above."
],
[
"Test results",
"We test our code by generating points-on-spheres realisations for many $R$ and $N_\\mathrm {s}$ scenarios.",
"We compare the outputs of our code to that of the true theoretical 3PCF using Equation (REF ).",
"Figure REF shows the theoretical and measured equilateral 3PCFs for seven scenarios with a range of $R$ values and fixed $n_\\mathrm {s} = 5 \\times 10^{-5}$ , using the Landy-Szalay estimator in Equation (REF ).",
"We plot the dimensionless 3PCF defined as $r^3 \\xi ^{(3)}$ (see for example [28]).",
"The theoretical 3PCF is shown in each case as the dashed line.",
"The measured 3PCF estimates are subject to sample variance, meaning that the output from the code depends on the randomly-seeded initial conditions.",
"We use five realisations with different random seeds to determine whether the theoretical 3PCF lies inside the spread of the five measured code outputs.",
"The shaded regions in Figure REF show the standard deviation of the measured 3PCF across these five realisations.",
"Figure REF similarly shows the theoretical and measured 3PCFs for scenarios with fixed $R=5$ and various $N_\\mathrm {s}$ values.",
"Figure: Equilateral 3PCFs for points-on-spheres scenarios with varying sphere radius RR.",
"The dashed lines show the theoretical 3PCF for each scenario whose parameters are shown in the legend.",
"The shaded regions show the standard deviation of the measured 3PCF across five realisations, using our code with the Landy-Szalay estimator.",
"The theory lines lie within the shaded regions for most triangle side lengths, except at radius values near the upper valid limits.",
"Vertical dotted lines show the theory asymptotes at R3R\\sqrt{3} for each line.Figure: Equilateral 3PCFs for points-on-spheres data with varying n s n_\\mathrm {s} between 1×10 -5 1\\times 10^{-5} and 7×10 -5 7\\times 10^{-5}, using the LS estimator.",
"The dashed theory lines again lie within the measured shaded regions for most triangle side lengths.",
"Vertical dotted line shows the theory asymptote at R3=53R\\sqrt{3} = 5\\sqrt{3}.The measured and theoretical 3PCFs match closely across most of the triangle side lengths.",
"The theoretical 3PCF in Equation (REF ) has a vertical asymptote at the maximum allowed radius $R \\sqrt{3}$ .",
"This can be seen in Figure REF as a slight upturn near the right-hand sides of each dashed line.",
"Our code slightly over-predicts the theory in each case near the maximum valid radius.",
"This is due to the binning of triangles: each binned output is calculated using equilateral triangles with a range of side lengths as described in Section REF .",
"Averaging the 3PCF over these differently-sized triangles (some of which are larger than the valid maximum radius) causes a discrepancy between measured and theoretical 3PCF."
],
[
"Optimisation",
"A number of steps were taken to optimise and improve the code.",
"First, we added multi-threading to make better use of available computational resources.",
"Second, we allow for subsampling of triangle configurations and jackknifing to allow for calculation of errors.",
"The number of triangles found by the matching algorithm in Section REF can quickly exceed hundreds of thousands for side lengths larger than around ten pixels.",
"An accurate measurement of the 3PCF can be obtained more efficiently by subsampling a small number of triangles from all valid matches.",
"We test how this subsampling affects the robustness of the final 3PCF estimate.",
"The 3PCF is calculated on $x_{\\mathrm {HII}}(\\mathbf {r})$ data from five randomly-seeded 21cmFAST realisations using input the canonical input parameters $T_{\\mathrm {vir}}= 10^{4} \\text{K}$ , $\\zeta _{\\mathrm {ion}}= 30.0$ , $R_{\\mathrm {max}}= 15.0 \\text{Mpc}$ and $E_0 = 200\\ \\text{eV}$ .",
"The variance in our code's output over the five realisations is plotted in Figure REF , as a function of the number of triangles used in the 3PCF algorithm.",
"The variance is large when only a few triangles are used but decreases with a larger number of triangles.",
"For more than around 2000 triangles, the scatter plateaus indicating that adding more triangles is unlikely to result in improve final 3PCF estimates.",
"The remaining variance across the five runs is likely due to sample variance.",
"We use a conservative value of 5000 triangles in all the 3PCF estimates from hereafter.",
"Figure: Effect of subsampling triangles in the 3PCF algorithm.",
"When few triangles are used, the outputs from the code show a larger variance than when more triangles are used.",
"For more than around 2000 triangles the variance plateaus indicating that adding more triangles provides minimal extra information.The 21cm differential brightness temperature $\\delta T_{\\mathrm {b}}$ is defined as the difference between the measured 21cm brightness temperature and the uniform background CMB brightness temperature.",
"By removing the background CMB temperature, the value of $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ then specifies the extent of 21cm emission ($\\delta T_{\\mathrm {b}}> 0$ ) or absorption ($\\delta T_{\\mathrm {b}}< 0$ ).",
"The actual observable for radio interferometers is $\\delta T_{\\mathrm {b}}- \\langle \\delta T_{\\mathrm {b}}\\rangle $ , where $\\langle \\delta T_{\\mathrm {b}}\\rangle $ is the global reionization signal averaged across the whole sky.",
"[15] gives an approximate relationship for the 21cm brightness temperature $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ as $\\delta T_{\\mathrm {b}}(\\mathbf {r}) = 27& x_{\\mathrm {HI}}(\\mathbf {r})\\ \\big [ 1 + \\delta (\\mathbf {r}) \\big ]\\left(\\frac{\\Omega _{\\mathrm {b}} h^2}{0.023}\\right)\\left(\\frac{0.15}{\\Omega _{\\mathrm {M}} h^2}\\right)^{1/2} \\\\& \\left(1 - \\frac{T_{\\mathrm {\\gamma }}}{T_{\\mathrm {spin}}}\\right)\\left(\\frac{1+z}{10}\\right)^{1/2}\\left(\\frac{H(z)}{H(z) + \\delta _r v_r (\\mathbf {r})}\\right)\\ \\text{mK}\\,.$ This approximation includes the effects of neutral hydrogen fraction $x_{\\mathrm {HI}}(\\mathbf {r})$ ; total matter density contrast $\\delta (\\mathbf {r})$ ; cosmological parameters for the densities of baryonic matter $\\Omega _{\\mathrm {b}}$ and total matter $\\Omega _{\\mathrm {M}}$ ; the CMB temperature $T_{\\mathrm {\\gamma }}$ ; the spin temperature $T_{\\mathrm {S}}$ which quantifies the relative populations of electrons in the higher and lower energy states of the 21cm transition; the Hubble parameter $H(z)$ ; and $\\delta _r v_r (\\mathbf {r})$ , the radial velocity gradient.",
"The spin temperature can be written [15] as a sum of three parts, $T_{\\mathrm {spin}}^{-1} = \\frac{T_{\\gamma } + x_\\alpha T_\\alpha ^{-1} + x_\\mathrm {c} T_\\mathrm {K}^{-1}}{1 + x_\\alpha + x_\\mathrm {c} }$ with $T_{\\gamma }$ the background CMB temperature, $T_\\mathrm {K}$ the kinetic gas temperature, $T_\\alpha $ the Lyman-alpha colour temperature (closely linked to the gas temperature for all redshifts of interest), and the coupling coefficients for collisions ($x_\\mathrm {c}$ ) and the Wouthysen-Field coefficient ($x_\\alpha $ ) from [78].",
"In particular, the kinetic gas temperature and the Lyman-alpha background radiation have a strong effect on the global and local evolution of the spin temperature.",
"These two features change throughout the EoR as different physical processes interact with the growth of structure in the Universe."
],
[
"We use the publicly available semi-numerical code 21cmFAST to generate our data.",
"We briefly describe the algorithm in this subsection.",
"The simulation begins by seeding an initial linear density field onto a three-dimensional grid at very high redshift.",
"This linear density field is evolved using first-order perturbation theory (see [82]) to approximate gravitational collapse, giving an approximate gravitationally-evolved density field $\\delta (\\mathbf {r})$ .",
"The simulation then finds the highest density regions where the matter will collapse to form luminous structures and thus contribute ionizing photons towards the reionization process.",
"The extent of collapse is calculated directly from the non-linear density field following the model of spherical collapseIn order to match the more accurate ellipsoidal collapse model [67], 21cmFAST afterwards normalises the spherical collapse fractions so that their average value matches that expected from ellipsoidal collapse.",
"[62].",
"If the mean enclosed density in a region exceeds a theoretical critical value then the region is assumed to collapse.",
"The collapse fraction $f_{\\mathrm {coll}}(\\mathbf {r}, R)$ on decreasing scales $R$ is then found from the contributions of both resolved and unresolved halos.",
"The default 21cmFAST implementation has a minimum halo mass $M_{\\mathrm {min}}$ for a halo to host star-forming galaxies that evolves with redshift, corresponding to a minimum virial temperature $T_{\\mathrm {vir}}$ for ionising photons.",
"The ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ is found by determining whether the collapsed matter in a region generates enough ionizing photons to ionize the enclosed hydrogen atoms.",
"An ionizing efficiency parameter $\\zeta _{\\mathrm {ion}}$ specifies how many ionizing photons are sourced per unit of collapsed matter.",
"If $f_{\\mathrm {coll}}(\\mathbf {r}, R) \\ge \\zeta _{\\mathrm {ion}}^{-1}$ for any particular region, then the central pixel is painted as fully ionized using the method in [81].",
"this differs from the default method of another common semi-numerical simulation SimFast21, which instead paints the full spherical region as ionized if there are enough photons using the method in [52].",
"See [30] for a discussion of these two methods.",
"Fluctuations in the spin temperature are calculated by considering the kinetic gas temperature and the Lyman-$\\alpha $ background temperature.",
"The kinetic gas temperature $T_\\mathrm {K}$ is determined by considering the balance between a number of important heating and cooling mechanisms including X-ray emissions, Hubble expansion, adiabatic heating and cooling, and gas particle density changes due to ionization events.",
"The dominant heating effect in 21cmFAST is from X-rays.",
"The rate of emitted X-ray photons is assumed to be proportional to the growth rate of collapsed matter in the dark matter halos.",
"Photons are emitted with a range of wavelengths, the luminosities for which are assumed to follow a power-law relationship $L(\\nu ) \\propto (\\nu / \\nu _0)^{-\\alpha }$ .",
"The parameter $\\alpha $ controls the slope of this spectral energy density function, and the parameter $\\nu _0$ controls the minimum frequency of X-rays which can escape into the Inter-Galactic Medium (IGM).",
"This minimum frequency can also be written in terms of a minimum energy value, $E_0 = h \\nu _0$ , using the Planck constant.",
"See [53] for a full derivation of the calculations and assumptions that 21cmFAST makes for the spin temperature fluctuations.",
"The final step is to use Equation (REF ) and calculate the 21cm brightness temperature field $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ using the non-linear density field $\\delta (\\mathbf {r})$ , the neutral fraction field $x_{\\mathrm {HI}}(\\mathbf {r}) = 1 - x_{\\mathrm {HII}}(\\mathbf {r})$ and the spin temperature fluctuation field $T_{\\mathrm {spin}}(\\mathbf {r})$ .",
"In this paper we consider different reionization scenarios by changing three of these simulation parameters: The ionization efficiency $\\zeta _{\\mathrm {ion}}$ , specifying how many ionising photons are sourced per unit of collapsed matter; The $E_0$ parameter which controls the minimum energy (or frequency) of X-ray photons which are able to escape into the IGM; The minimum virial temperature $T_{\\mathrm {vir}}$ which specifies a lower mass limit $M_{\\mathrm {min}}$ of collapsed matter which produces ionizing photons and X-rays.",
"Fixing the other simulation parameters involves setting the efficiency of X-rays to a constant value.",
"We use $\\zeta _{\\mathrm {X}} = 10^{-57} M_{\\odot }^{-1}$ to match the assumption in [53], equivalent to approximately a single X-ray photon for each stellar baryon as motivated by observations of low-redshift galaxies.",
"The uncertain intergalactic-medium X-ray properties are then parametrised by $E_0$ ."
],
[
"Training and testing data details",
"We run 1000 21cmFAST simulations in total for our data.",
"Each simulation generates three-dimensional realisations of the $\\delta T_{\\mathrm {b}}$ field in a cube of size $250\\text{Mpc}$ resolved into $256^3$ pixels (smoothed from density fields resolved into $768^3$ pixels).",
"Each simulation uses a different random seed for the initial conditions.",
"The resulting redshifts from this algorithm are between $z=5$ and $z=26.6$ (see [53] for a description of the iterative algorithm that generates these steps).",
"These redshifts are 5.0, 5.6, 6.3, 7.0, 7.78, 8.7, 9.6, 10.7, 11.9, 13.2, 14.6, 16.1, and 17.8.",
"We ignore simulated results for higher redshifts, because the mean ionization fraction is extremely small and the mean bubble size is generally smaller than the resolution of our simulations.",
"For each simulation, we calculate the statistics of interest: the 3PCF using 3PCF-Fast described in Section ; the bubble size distribution, described in this section; and the global ionization fraction, found by trivially averaging the ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ for each redshift.",
"In order to sample a range of different reionization scenarios, we use a Latin Hypercube [47] approachUsing the implementation from [1].",
"This method efficiently samples the input space with far fewer simulations than a naive exhaustive grid-search would require.",
"The following ranges and scales of simulation parameters are used: $T_{\\mathrm {vir}}$ in the logarithmic range $[10^4, 2 \\times 10^5]\\ \\text{K}$ $\\zeta _{\\mathrm {ion}}$ in the linear range $[5, 100]$ $E_0$ in the linear range $[100, 1500]\\ \\text{eV}$ These ranges were chosen to match those by the simulation authors (for example [23]).",
"The lower $T_{\\mathrm {vir}}$ limit comes from a minimum temperature for the cooling of atomic hydrogen accreting onto halos.",
"The upper limit arises from observations of high-redshift Lyman break galaxies [23].",
"The $\\zeta _{\\mathrm {ion}}$ upper and lower limits correspond roughly to escape fractions of 5% to 100% for ionizing photons for standard values of the other controlling factors in [23] such as the number of ionising photons produced per stellar baryon.",
"The range for $E_0$ was chosen in a similar way to [57], motivated by hydrodynamic simulations [9] and considering the energy that would allow an X-ray photon to travel a distance of roughly one Hubble length when travelling through a medium with $\\langle x_{\\mathrm {HII}}(z) \\rangle = 0.5$ ."
],
[
"Three-point correlation-function measurements",
"We use the code described in Section to calculate the three-point correlation function.",
"We calculate $\\xi ^{(3)}$ of both the ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ and of the 21cm differential brightness temperature field $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ .",
"In the code we use 28 equilateral triangle bin configurations with side lengths spaced in bins between $5\\ \\text{Mpc}$ and $109\\ \\text{Mpc}$ .",
"These bins are spaced linearly for radii less than $20 \\text{Mpc}$ , with logarithmically spaced bins for higher radiiThe radius values for these 28 bins are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 29, 35, 42, 51, 62, 75, 91, and 109 Mpc.. Increasing the number of r-vector configurations beyond these equilateral triangles would almost certainly improve our ability to predict the mean bubble size or ionization fraction history.",
"Further work would be needed however to investigate what size and shape of triangle configurations encode the most information about the topology of the EoR."
],
[
"Mean-free-path measurements for $x_{\\mathrm {HII}}(\\mathbf {r})$",
"In order to measure the mean bubble size we use our own implementation of the mean-free path method described in [52].",
"The mean-free path method simulates the emission of photons from random locations within the transparent regions.",
"The distance travelled by each photon before it hits a phase change (from ionized to neutral) is measured and the resulting number of rays in a range of radius bins is calculated as $dP/dR$ .",
"We use $10^5$ simulated photons in our measurements, and the resulting distances are rounded to the nearest pixel size ($L/N = 250.0 \\text{Mpc}/ 256 = 0.98 \\text{Mpc}$ ).",
"Figure REF shows the resulting distributions for $R dP/dR$ from a simulation with canonical parameter values $T_{\\mathrm {vir}}= 10^4 \\text{K}$ , $\\zeta _{\\mathrm {ion}}= 30$ and $E_0 = 200\\ \\text{eV}$ .",
"We use the mean of these mean-free path distributions (hereafter written $R_{\\mathrm {bubble}}$ ) as a statistic to trace the mean bubble size.",
"Figure: Example mean free path measurements of RdP/dRR dP/dR using ionization fraction field data x HII (𝐫)x_{\\mathrm {HII}}(\\mathbf {r}).",
"Each line shows RdP/dRR dP/dR for a single redshift taken from a simulation with T vir =10 4 KT_{\\mathrm {vir}}= 10^4 \\text{K}, ζ ion =30.0\\zeta _{\\mathrm {ion}}= 30.0 and E 0 =200evE_0 = 200 \\text{ev}."
],
[
"Machine learning techniques",
"The three-point correlation function of 21cm data likely encodes much information about the underlying reionization processes.",
"We can uncover the relationships between $\\xi ^{(3)}$ and these physical processes by looking at how the 3PCF changes over a range of physical scenarios.",
"We use machine learning techniques to learn these relationships.",
"Our models extract relationships between physical processes and $\\xi ^{(3)}$ measurements by using simulated data.",
"Our trained models can then use unseen measurements of 3PCF data to make predictions about the physical status of reionization in the data.",
"In particular, we train models that predict the global ionization fraction and the mean bubble size from $\\xi ^{(3)}$ .",
"Training these models is effectively a form of high-dimensional curve fitting: learning a best-fit functional form $f(\\mathbf {x})$ that maps from a set of input values $\\mathbf {x} = \\xi ^{(3)}$ to a set of output values ($R_{\\mathrm {bubble}}$ or $\\langle x_{\\mathrm {HII}}(z) \\rangle $ ).",
"After training, our models can make predictions for new unseen data.",
"For instance, the mean bubble size model can take measurements of $\\xi ^{(3)}$ and predict the mean bubble size.",
"In this section we describe the machine learning techniques we use along with a theoretical description of how they are trained.",
"All models are trained on the same architecture, each on a single node using 16 Xeon E5-2650 cores and 128GB RAM.",
"700 of our simulations are used for training and validation, and 300 simulations are held back for testing."
],
[
"Artificial neural networks",
"Artificial neural networks (ANNs) are a common regression technique for learning a complex non-linear relationship between two sets of variables the `inputs' and `outputs'.",
"An ANN represents the relationship in functional form $y_i = f(\\mathbf {x_i})$ by manipulating the inputs $\\mathbf {x_i}$ through a series of weighted summations and function evaluations.",
"For ANNs this series of repeated operations occurs in a series of distinct layers.",
"The values in the first layer $\\mathbf {h^{(0)}}$ are the input variables $\\mathbf {x_i}$ .",
"The values from one layer $h^{(l-1)}_j$ affect the values in the following layer $h^{(l)}_j$ according to $\\begin{array}{l}\\mathbf {h^{(l)}} = h^{(l)}_j = \\phi _\\theta \\left(\\ {\\sum \\limits _{i=1}^{N_i}} W^{(l)}_{ij} h^{(l-1)}_{j} \\right)\\,.\\end{array}$ The values in each layer are thus a sum over the values in the previous layer, weighted using a set of trainable values $W^{(l)}_{ij}$ .",
"The summations into each neuron are passed through an activation function $\\phi _\\theta (x)$ , which determines the resulting output values that are passed on to the next layer of neurons.",
"At the end of this process, the final layer contains the network's fitted evaluations of the function, $y_i = f(\\mathbf {x_i})$ .",
"Training these models involves choosing the set of weights $W^{(l)}_{ij}$ which most closely mimic the function's behaviour.",
"The `closeness' with which the model mimics the relationship in the training data is quantified using an objective function, $\\text{Objective} = \\frac{1}{2 N}\\sum \\limits _{n=1}^{N} \\big [ f(\\mathbf {x_n}) - y_n \\big ]^2 - \\frac{\\alpha }{2} \\sum \\limits _{i,j,l} \\left(W^{(l)}_{ij}\\right)^2$ so that training is then done by finding the values $W^{(l)}_{ij}$ which minimize this objective function for some training data $(\\mathbf {x_n}, y_n)$ .",
"The regularization parameter $\\alpha $ in this equation allows finer control over the complexity of the model.",
"A high value of $\\alpha $ encourages the training towards simpler models, with more of the weight values $W^{(l)}_{ij}$ being near zero.",
"Three common activation functions $\\phi _\\theta (x)$ are the hyperbolic tangent function $\\phi _\\theta (x) = \\tanh (x)$ ; the logistic function $\\phi _\\theta (x) = 1/(1+\\exp (-x))$ the rectified linear unit ('relu`) function $\\phi _\\theta (x) = \\text{max}(0, x)$ .",
"All three activation functions are used during our hyperparameter search method.",
"The weights are initialised randomly and are then updated iteratively in order to improve the objective function.",
"Each iteration is known as a single `epoch'.",
"In each epoch, the weights are updated using the current gradient of the objective.",
"By using this gradient, the weights are moved towards a value that should cause the objective function to improve.",
"Our ANNs use the backpropagation algorithm [77], a common technique for efficiently calculating the gradient of the objective function (see [63] for a more detailed description of this algorithm).",
"The coarseness with which the weights are updated is controlled by a parameter known as the learning rate.",
"A high learning rate means that the weights are changed with a large magnitude at each step.",
"The learning rate can be set to a constant value for all epochs, but it can also adapt to the current speed of the learning.",
"An adaptive learning rate is usually set to decrease if the objective function plateaus (i.e.",
"begins to fall slowly between epochs).",
"It is common to set an upper limit for the number of epochs allowed.",
"We discuss this and other choices made in our models in Section REF later.",
"Multilayer perceptrons (MLPs) are a subclass of artificial neural networks, with the restrictions that they contain at least one hidden layer and have a non-linear activation function.",
"Figure REF shows a typical MLP's layer structure, with lines representing the weighted connections between values.",
"Circles represent the neurons which hold the values $h^{(l)}_j$ and pass the weighted inputs through the activation function.",
"Our MLPs are implemented using scikit-learn from [59].",
"We use the `adam' optimization method [36] which terminates either when the maximum number of eoochs has been reached, or when the objective function falls below a tolerance of $10^{-10}$ for at least two consecutive iterations.",
"Figure: Visualization of a multilayer perceptron with two hidden layers.",
"Lines are weighted connections from left to right.",
"Circles are neurons which hold the values and pass them to the following layer.We measure the goodness of fit between predicted output values $y^*(k,z)$ and measured output values $y(k,z)$ using a mean squared error function, $\\text{MSE} \\big [ y, y^* \\big ] = \\frac{1}{N} \\sum \\limits _{i}^{N} \\left( \\frac{y_i - y_i^*}{y_i} \\right)^2\\ $ also making use of the root mean-squared error $\\text{RMSE} = \\sqrt{\\text{MSE}}$ and the percentage mean-squared error $= 100.0 \\times \\text{RMSE}$ .",
"A percentage RMSE of 100 indicates that the predicted and measured output values are wrong by an average factor of 2."
],
[
"Hyperparameter search",
"The weighted-connection values $W_{\\mathrm {ij}}$ of a multilayer perceptron are updated during the training process in order to find the best match between the input and outputs in the training data.",
"Several aspects of models must also be fixed before even starting to train the model.",
"We refer to these values as hyperparameters.",
"The hyperparameters can have a strong effect on the final accuracy of predictions but it is rarely obvious what hyperparameter values will result in the most accurate model.",
"We use a random search method with cross-validation to find the best hyperparmeters for each of our model applications.",
"This process is described here.",
"In order to determine the best hyperparameter values, we train and compare a large number of models with a range of initial hyperparameter values.",
"Each model is trained using a set of randomised hyperparameters and the model with highest prediction accuracy is selected as the best model.",
"Two of the most important hyperparameters are the number of hidden layers and the sizes of these layers, collectively known as the network architecture.",
"The architecture affects the model's ability to represent complex functions: a network with fewer and smaller layers is only be able to model simple relationships, whereas a larger network with more layers (or larger layers) will be able to represent more complex relationships.",
"Using a model that is too small will result in poor prediction accuracy.",
"Using a model that is too large will result in overfitting.",
"There are no prescribed rules for deciding what range of architectures to consider, but a common technique is to use one's knowledge both about the complexity and the dimensionality of the function that is being modelled.",
"When using the 3PCF measurements as the inputs, there are around 30 input dimensions to the model.",
"We use networks with between one and three hidden layers, with layer sizes randomly chosen uniformly in the range $[0,500]$ .",
"This range of layer sizes was chosen as being a similar order of magnitude to the input dimensionality while also remaining computationally feasible.",
"The full set of parameters which were randomly varied for each model in the hyperparmaeter search are: Number of hidden layers uniformly in the linear range $[1, 3]$ Size of each layer uniformly in the linear range $[0, 500]$ Training batch size uniformly in the linear range $[30, 500]$ Number of training epochs uniformly in the range $[50, 500]$ Initial learning rate uniformly in the log range $[10^{-4}, 10^{-2}]$ Learning rate either constant and adaptive with equal chance Activation from `relu', `tanh', or `logistic' with equal chance Regularization parameter $\\alpha $ from equation Equation (REF ) uniformly in the log range $[10^{-4}, 10^{-2}]$ These ranges match those suggested by the scikit-learn website [59].",
"We use fixed default values for the `adam' parameters $\\textsc {\\small {beta\\_1}} = 0.9$ , $\\textsc {\\small {beta\\_2}}=0.999$ , $\\textsc {\\small {epsilon}} = 1e-08$ and $\\textsc {\\small {tol}} = 0.0001$ .",
"For all models the weight values are initialised using the Xavier initialiation strategy [20].",
"This method sets the weights in the $i$ th layer by sampling uniform values in the range $[-U_i, U_i]$ .",
"The normalising value $U_i = \\sqrt{6} / \\sqrt{n_i + n_{i+1}}$ is different for each layer, using values for the total number of input weight connections ($n_i$ , also known as `fan in') and output weight connections ($n_{i+1}$ , also known as `fan out').",
"Note that there are seven hyper-parameters being varied in this random search.",
"Given that we choose only 1000 different random sets from the above ranges, it is unlikely that we have identified the optimal model."
],
[
"Cross-validation",
"By trying a range of different hyperparameter values as described we can usually find a model with better prediction accuracy.",
"However this process is sensitive to overfitting.",
"In order to determine which model has the highest accuracy while reducing the chance of overfitting, we use five-fold cross-validation approach.",
"Five models are trained with the same fixed hyperparameters, where each model is provided with data from only four of the five folds.",
"In each case the fifth excluded fold is used to calculate the prediction performance, using Equation (REF ).",
"The performances are thus measured on unseen data, so that the `best' model with highest performance is one which performs well on the unseen validation data.",
"The overall accuracy score is taken as the the mean of the validation scores.",
"This cross-validation approach is used to compare the performance of every combination of hyperparameter values.",
"After finding the best hyperparameter values, the model is trained for a final time using all the training data.",
"Standard practice for machine learning tasks is to retain a final segment of the data to check the final performance of the best model.",
"If the model performs well on this testing data, then we are more confident that it makes good predictions for completely unseen data."
],
[
"Input and output scaling",
"Data scaling can be used to improve the efficiency of artificial neural networks during training, and also to improve the quality of the final predictions.",
"The weight values in our neural networks are initialised at small values as described in Section REF .",
"In general, different input features into a model have different scales and magnitudes.",
"Ideally all inputs into the network would have similar orders of magnitude and simple distribution such as normal or uniform.",
"This can easily be achieved by separately normalising or standardising each input feature.",
"Normalising an input feature forces all values to lie in the range $[0, 1]$ by linearly scaling the minimum and maximum feature values.",
".",
"Standardising an input feature scales the feature to have a mean of zero and a standard deviation of 1.0.",
"Scaling the model output value(s) also has a beneficial effect on the final prediction accuracy.",
"Our neural networks use the scikit-learn objective function in Equation (REF ) to quantify the goodness of fit during training.",
"Scaling the output values using normalisation or standardisation can help mitigate the relative importance of output values with different magnitudes.",
"The input features to our models are the 3PCF measurements $\\xi ^{(3)}(r)$ for a range of different triangle sizes $r$ .",
"These 3PCF values span a wide range of magnitudes.",
"We use the MinMaxScaler method from scikit-learn to normalise separately each 3PCF bin.",
"We also compare the effect of scaling the 3PCF values by four different powers of the binned radius values: the raw 3PCF $\\xi ^{(3)}$ ; the dimensionless 3PCF $r^3 \\xi ^{(3)}(r)$ used for more natural visualisations (see for instance [28]); and two other powers of the radius for completeness: $r \\xi ^{(3)}(r)$ and $r^2 \\xi ^{(3)}(r)$ .",
"The output features to our models are either the bubble sizes $R_{\\mathrm {bubble}}$ or the global ionization fraction $\\langle x_{\\mathrm {HII}}(z) \\rangle $ .",
"We scale the $R_{\\mathrm {bubble}}$ function using the $\\text{sinh}^{-1}$ function as described by [44]."
],
[
"Learning typical bubble sizes from the 3PCF",
"The progress of the Epoch of Reionization can be tracked by measuring the mean size of ionized regions.",
"Ionized regions are initially small and isolated around the earliest ionising sources.",
"The regions continually grow throughout the EoR, and the precise details of this continued growth depends on the physical interactions between ionising sources and the surrounding neutral regions.",
"The sources themselves are seeded from the clustered non-linear density field and so show significant clustering [29], but the details of reionization also affect the clustering of the resulting ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ and 21cm brightness temperature field $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ .",
"Throughout the EoR, the mean bubble size $R_{\\mathrm {bubble}}$ will likely boost the 3PCF at characteristic triangle sizes.",
"Thus, the 3PCF contains information about the physics of reionization [48].",
"Similarly, higher-order clustering statistics contain information about the physical reionization parameters (see for instance [70]) which affect the morphology of the $x_{\\mathrm {HII}}(\\mathbf {r})$ and $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ fields.",
"In this section, we train models to predict the mean bubble size $R_{\\mathrm {bubble}}$ using the 3PCF from simulated data.",
"First, we use 3PCF measurements of the ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ to train our models.",
"The resulting model is a useful means of determining whether $\\xi ^{(3)}$ does indeed contain information about the mean bubble size.",
"In practice, however, the ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ is difficult to disentangle from the actual results of interferometer experiments.",
"In the second half of this section we train models to predict the mean bubble size using simulated $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ data, which would be directly available from interferometer observations.",
"As well as the data cleaning steps in Sections REF and REF , we also exclude data with global ionization fraction outside the range $0.01 \\le x_{\\mathrm {HII}}\\le 0.95$ ."
],
[
"Results training on $x_{\\mathrm {HII}}(\\mathbf {r})$ data",
"In this subsection we train a model to learn how the mean bubble size $R_{\\mathrm {bubble}}$ is related to the 3PCF of ionization fraction data $x_{\\mathrm {HII}}(\\mathbf {r})$ .",
"Our training and testing data are from the range of simulated reionization scenarios described in Section REF , and we use the multilayer perceptron model described in Section .",
"Figure REF shows the measured $x_{\\mathrm {HII}}(\\mathbf {r})$ 3PCF for a range of redshifts, showing the true mean bubble size as vertical lines.",
"This figure is for a scenario with canonical parameter values $T_{\\mathrm {vir}}= 10^4 \\text{K}$ , $\\zeta _{\\mathrm {ion}}= 30$ and $E_0 = 200\\ \\text{eV}$ .",
"The amplitude of the dimensionless 3PCF seen in Figure REF reaches a peak at intermediate scales.",
"Either side of the peak the amplitude decreases, although at larger scales above 20Mpc the amplitude near the start and end of the EoR ($x_{\\mathrm {HII}}= 0.82$ and $x_{\\mathrm {HII}}= 0.10$ ) show a second rise in the amplitude.",
"Figure: Example measurements of r 3 ξ (3) r^3 \\xi ^{(3)} for ionization fraction field data x HII (𝐫)x_{\\mathrm {HII}}(\\mathbf {r}).",
"Each line shows the measured statistic for a single redshift, all taken from a simulation with ζ ion =30.0\\zeta _{\\mathrm {ion}}= 30.0, T vir =10 4 KT_{\\mathrm {vir}}= 10^4 \\text{K} and E 0 =200eVE_0 = 200\\ \\text{eV}.",
"The redshifts and corresponding global ionization fraction are shown for each line in the legend.Before running a full hyperparameter search, we first compare four options for different input-scaling types.",
"We train one model for each of the four possible input scaling types, namely $\\xi ^{(3)}$ , $r \\xi ^{(3)}$ , $r^2 \\xi ^{(3)}$ and $r^3 \\xi ^{(3)}$ .",
"The MLP models in this section all have the same architecture, namely two hidden layers both containing 100 nodes.",
"The following values are used for the other hyperparameters: a training batch size of 200; 200 maximum epochs; a constant learning rate of $10^{-3}$ ; the `relu' activation function; and fixed regularization parameter $\\alpha = 10^{-3}$ .",
"These hyperparameters were chosen as the midpoints of the allowed random-search ranges or, for categorical choices, as the default parameters suggested by the code authors [59].",
"Table REF shows the resulting overall RMSE values for models using each of the four different scaling types.",
"Our results indicate that scaling the 3PCF by $r^2$ or $r^3$ generates more accurate models than scaling by $r$ or not scaling at all.",
"Using $\\xi ^{(3)}$ or $r \\xi ^{(3)}$ as inputs makes it harder for our MLP models to uncover a relationship between the 3PCF and the mean bubble size.",
"We use $r^2 \\xi ^{(3)}(r)$ as inputs to our models hereafter, as these have the best overall RMSE value.",
"Table: RMSE performance on unseen testing data using the four different input scaling types in Section .",
"The model using r 2 ξ (3) r^2 \\xi ^{(3)} inputs has the best performance, with the two lowest powers of rr having the worst performance.",
"These RMSE values are only for a single cross-validated model with the fixed hyperparameters given in Section , but this indicates that the relationship between r 2 ξ (3) r^2 \\xi ^{(3)} and the mean bubble size is easier to learn than the other inputs.We now find the best MLP model to predict the mean bubble sizes from the 3PCF of ionization fraction field data $x_{\\mathrm {HII}}(\\mathbf {r})$ .",
"We use the full hyper-parameter search method described in Section REF , comparing 1000 randomly chosen models and selecting the one with best cross-validated performance.",
"The resulting best MLP model uses three hidden layers with sizes [148, 142, 93]; training batch size of 296; a maximum of 563 epochs (of which the model used all epochs before terminating); adaptive learning rate starting at $4.7 \\times 10^{-3}$ ; the `relu' activation function; and L2 regularization parameter $3.9 \\times 10^{-4}$ .",
"Figure REF shows the accuracy of the best MLP model's predictions for unseen testing data.",
"We plot all predicted $R_{\\mathrm {bubble}}$ values as a function of the true values.",
"Marker colours are used to indicate the value of $\\langle x_{\\mathrm {HII}}(z) \\rangle $ for each measurement.",
"A model with perfect predictions would lie exactly on the dotted black diagonal line.",
"Deviations from this diagonal represents less accurate predictions.",
"Figure REF shows the distribution of errors predicted by this model.",
"The median prediction error from these distributions is a good measure of model performance.",
"The model in this subsection has a median prediction error of $10.1\\%$ .",
"Figure: Predicted bubble size vs true bubble size for the best model in Section .",
"These predictions are made on unseen testing data, using only the 3PCF of ionization fraction field data as inputs to the model.",
"The predicted values and true values generally lie along the diagonal for values of R bubble <70MpcR_{\\mathrm {bubble}}< 70 \\text{Mpc}.",
"Larger mean bubble sizes are harder to model and show much lager scatter away from the diagonal, as discussed in the text.The accuracy of the model depends strongly on the magnitude of the true bubble size.",
"The model struggles to make accurate predictions for mean bubble sizes that are larger than $70\\ \\text{Mpc}$ : predictions for $R_{\\mathrm {bubble}}< 70 \\text{Mpc}$ lie close to the diagonal, but predictions for $R_{\\mathrm {bubble}}> 70 \\text{Mpc}$ show much larger scatter.",
"This can be understood in terms of the relationship between the 3PCF and the mean bubble size.",
"Near the end of the EoR, the widespread overlap of ionized bubbles gives rise to a larger average mean free path of ionising photons, but also blurs the definition of a mean bubble size.",
"Many bubbles have merged, and thus the `mean' bubble size is a less clear feature.",
"The model's ability to learn the mean bubble size from 3PCF measurements reflects this."
],
[
"Results training on $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ data",
"The situation is more complicated when using measurements of the 21cm differential brightness temperature field $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ instead of the ionization fraction field $x_{\\mathrm {HII}}(\\mathbf {r})$ .",
"The relationship between $\\delta T_{\\mathrm {b}}$ and the ionization fraction $x_{\\mathrm {HII}}$ given in Equation (REF ) is assumed to be linear, but the other terms in this equation also impact the morphology of the 21cm brightness temperature field.",
"Most notably, local spin temperature fluctuations $T_{\\mathrm {spin}}(\\mathbf {r})$ and local density fluctuations $\\delta (\\mathbf {r})$ can both change the local values of $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ .",
"Fluctuations in these values confuse the otherwise simple relationship between the 3PCF and the mean bubble size.",
"Figure REF shows the measured $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ 3PCF from a simulation with parameters $\\zeta _{\\mathrm {ion}}= 30.0$ , $T_{\\mathrm {vir}}= 10^4 \\text{K}$ and $E_0 = 200\\ \\text{eV}$ .",
"The true mean bubble sizes (calculated as the mean of the mean-free path distributions) are shown as vertical lines.",
"The 3PCF of the brightness temperature data has a more complex evolution over the EoR than the 3PCF of ionization fraction data shown in Figure REF .",
"In general, the amplitude of the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ 3PCF decreases until around $\\langle x_{\\mathrm {HII}}(z) \\rangle = 0.25$ , before increasing to a maximum near the end of the EoR.",
"The complex evolution of other features is less obvious and justifies the need for machine learning models here.",
"Figure: Example measurements of r 3 ξ (3) r^3 \\xi ^{(3)} for 21cm differential brightness temperature field data δT b (𝐫)\\delta T_{\\mathrm {b}}(\\mathbf {r}), using the same simulation as Figure .",
"These data have also been processed using the radius bins in Section .Using the same method as for the ionization fraction field model, we train a model to predict the mean bubble sizes using the 3PCF of simulated $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ data.",
"The resulting best model uses three hidden layers with sizes [158, 188, 187]; training batch size of 169; a maximum of 864 epochs; adaptive learning rate starting at $1.3 \\times 10^{-3}$ ; the `relu' activation function; and L2 regularlization parameter $4.3 \\times 10^{-3}$ .",
"Figure: Predicted bubble size vs true bubble size for unseen testing data, using the best model in Section .",
"This model uses the 3PCF of δT b (𝐫)\\delta T_{\\mathrm {b}}(\\mathbf {r}) data to predict the mean bubble size.",
"The predicted values and true values generally lie along the main diagonal for middling values of R bubble R_{\\mathrm {bubble}} between 25 and 100 Mpc.",
"The model can accurately predict the mean bubble size in these scenarios.",
"Deviations from the diagonal line at larger and smaller bubble sizes are worse for the reasons discussed in the text.This $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ MLP model has a median prediction error of $13.4 \\%$ .",
"This performance is slightly worse than using $x_{\\mathrm {HII}}(\\mathbf {r})$ data, indicating that the extra complexities of including local spin temperature fluctuations and local density field fluctuations do indeed contaminate the relationship between the mean bubble size and the data-field correlations.",
"The model cannot distinguish between correlations of ionized regions and correlations of low density contrast regions (`under-dense' regions), because both of these scenarios give rise to lower values for $\\delta T_{\\mathrm {b}}$ .",
"Similarly, regions with low local values for the spin temperature $T_{\\mathrm {spin}}$ can mimic ionized regions.",
"We plot the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ model's predicted mean bubble sizes for unseen testing data in Figure REF , as a function of the true mean bubble size.",
"Two features are worth nothing in comparison to the previous $x_{\\mathrm {HII}}(\\mathbf {r})$ MLP model.",
"First, although the average performance of the $\\delta T_{\\mathrm {b}}$ model is worse, the performance at larger bubble sizes is better.",
"Whereas the $x_{\\mathrm {HII}}(\\mathbf {r})$ model's predictions showed a large scatter around the diagonal for $R_{\\mathrm {bubble}}> 70 \\text{Mpc}$ , the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ model's predictions show a more consistent relationship with the mean bubble size: all predictions with $R_{\\mathrm {bubble}}> 25 \\text{Mpc}$ are made with a roughly consistent accuracy.",
"In particular for larger bubble sizes, the 3PCF of brightness temperature data appears to encode more information about bubble sizes than does the ionization fraction field.",
"A likely reason for this is the effect of neutral regions.",
"Whereas neutral regions in the ionization fraction field have a uniform value of $x_{\\mathrm {HI}}= 1.0$ , these regions can have different values in the brightness temperature field owing to the other terms in Equation (REF ).",
"This relationship could encode information in the brightness temperature field correlations that does not exist in the ionization fraction field, thus allowing our MLP model to learn the mean bubble size more easily.",
"The second interesting feature is the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ model's poorer performance at low bubble sizes, seen as the large solid cluster of markers in the bottom left of Figure REF .",
"It is not immediately obvious why this occurs.",
"Including spin temperature fluctuations certainly causes a more complex relationship between the ionization fraction field and the 3PCF of the brightness temperature field.",
"It is possible that this effect is worse at earlier times, when the mean bubble sizes are generally smaller.",
"Figure REF shows the histograms of prediction errors for both final best MLP models: one using $x_{\\mathrm {HII}}(\\mathbf {r})$ data, and one using $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ data.",
"Ideally, all predictions would be near zero percentage RMSE.",
"The distribution of errors for these two models does not depend strongly on which data are used ($x_{\\mathrm {HII}}(\\mathbf {r})$ or $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ ) although, as mentioned above, each model has different prediction accuracies for different mean bubble size regimes.",
"Figure: Histogram of prediction errors for mean bubble size models.",
"Visibly, the overall distribution of errors does not depend strongly on which data are used.",
"However, it can be seen from Figures and that the prediction accuracies of these two MLP models depend strongly on the mean bubble size: the δT b (𝐫)\\delta T_{\\mathrm {b}}(\\mathbf {r}) model makes better predictions at higher bubble sizes, and the x HII (𝐫)x_{\\mathrm {HII}}(\\mathbf {r}) model makes better predictions at lower bubble sizes."
],
[
"Effect of modelling weights",
"Training an MLP model involves finding the optimal `weight' parameters.",
"These weights are usually initialised to random values as discussed in Section REF .",
"Different initial weight values will result in different final weight values at the end of training.",
"Thus the performance of an MLP model depends on the choice of initial weight values.",
"It is interesting to determine the impact that the choice of initial weight values has on model performance.",
"The black line in Figure REF shows the distribution of median prediction errors for a set of 500 models, each of which has different randomised initial weights but identical hyperparameters to our best model in Section REF .",
"The lighter orange line shows the distribution of median prediction errors for all 1000 MLP models in the full hyperparameter search.",
"For our best model, weight initialisation clearly has a strong effect on the model performance: the median prediction error can vary between $10 \\%$ and $30\\%$ .",
"Although it is possible that our best model is particularly susceptible to different weight initialisations, it is likely that other MLP models would also have a similar magnitude of spread in performances.",
"This likely puts an upper limit on the possible performance from any MLP model, even if a deeper hyperparameter search were performed.",
"Note that the best model's RMSE value of roughly $13\\%$ lies close to the best performance found by varying random initial weights.",
"Our best model has almost certainly benefited somewhat from a `lucky' weight initialisation, in the sense that retraining the MLP model with different initial weights would likely lead to a worse RMSE performance.",
"Our model is a good one – it has an acceptable RMSE performance on unseen testing data – but a broader investigation into the full hyperparameter space could potentially lead to a higher accuracy model.",
"Figure: Histogram showing the spread of RMSE model performance, either for different weight initialisations of the MLP models or from all 1000 models in the hyperparameter.",
"For the `weight initialisations' line, all models have the best hyper-parameters as determined in Section ."
],
[
"Comparison to Power Spectrum",
"The 21cm line is subject to many sources of noise.",
"In particular, thermal noise in the raw observed data affects our ability to make inferences from 21cm maps.",
"In order to reduce the effect of noise, statistical quantities such as clustering statistics can be used.",
"These metrics are less affected by noise since they are calculated as averages across the entire map.",
"The 3PCF is a higher-order clustering statistic and so should reduce the effect of noise.",
"However, it is also interesting to check whether using a lower-order statistic such as the power spectrum provides equally good results.",
"In this section, we use the full hyper-parameter search method described in Section REF to find an MLP model that predicts the mean bubble sizes from the power spectrum of the differential brightness temperature field $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ .",
"As in the previous sections, we compare 1000 randomly chosen models and select the one with best cross-validated performance.",
"The resulting best MLP model uses three hidden layers with sizes [191, 110, 76]; training batch size of 194; a maximum of 596 epochs (of which the model used all epochs before terminating); constant learning rate starting at $4.8 \\times 10^{-3}$ ; the `relu' activation function; and L2 regularization parameter $4.5 \\times 10^{-4}$ .",
"Figure: Predicted bubble size vs true bubble size for unseen testing data, using the best model in Section .",
"This model uses the power spectrum of δT b (𝐫)\\delta T_{\\mathrm {b}}(\\mathbf {r}) data to predict the mean bubble size.",
"The predicted values and true values generally lie along the main diagonal, although in comparison to the equivalent 3PCF model in Figure this model makes significantly worse predictions.Figure REF shows the accuracy of the best $P(k)$ model's predictions for unseen testing data.",
"This model has a median prediction error of $18.3\\%$ , somewhat worse than the equivalent model in Section REF which uses 3PCF measurements instead of power spectrum measurements as inputs to the MLP model.",
"The information encoded in the power spectrum appears to be less strongly related to the mean bubble size than the information encoded in the 3PCF.",
"It is worth nothing that if noise was added to the underlying simulated $\\delta T_{\\mathrm {b}}$ maps then the performances of the 3PCF models would likely be impacted.",
"More investigation would be needed to determine whether this impact would be greater for the 3PCF model than for the power spectrum models."
],
[
"Learning the global ionization fraction from the 3PCF",
"In the previous section we investigate using 3PCF measurements to predict the mean bubble size.",
"The mean bubble size is a useful metric for tracking the growth of ionising regions, but the global ionization fraction $\\langle x_{\\mathrm {HII}}(z) \\rangle $ is a more direct measurement for the overall progress of the Epoch of Reionization.",
"The redshift history of $\\langle x_{\\mathrm {HII}}(z) \\rangle $ can be strongly affected by the reionization parameters: different ionising efficiency $\\zeta _{\\mathrm {ion}}$ scenarios have a different abundance of ionising photons, which affects the EoR duration; different $T_{\\mathrm {vir}}$ scenarios have different halo mass function distributions, leading to more or fewer ionising sources and also affect the EoR duration.",
"In this section, we train a model to predict the value of $\\langle x_{\\mathrm {HII}}(z) \\rangle $ from 3PCF measurements.",
"Our models learn the relationship between the 3PCF and global ionization fraction by using the same simulated data in Section .",
"Measurements of the 3PCF and bubble size distribution use the methods described in Sections REF and REF respectively.",
"The data are cleaned using the same ionization fraction filters, namely $0.01 \\le \\langle x_{\\mathrm {HII}}(z) \\rangle \\le 0.95$ ."
],
[
"Results training on $x_{\\mathrm {HII}}(\\mathbf {r})$ data",
"We use the same search strategy as in the previous section.",
"The best model uses three hidden layers with sizes [192, 150, 50]; training batch size of 261; a maximum of 365 epochs (of which the model used all epochs before terminating); adaptive learning rate starting at $2.00 \\times 10^{-3}$ ; the r̀elu' activation function; and L2 regularization parameter $3.72 \\times 10^{-4}$ .",
"The model has an extremely good median prediction error of $3.6 \\%$ .",
"Figure REF indicates the performance from this model, showing the predicted values of $\\langle x_{\\mathrm {HII}}(z) \\rangle $ as a function of the true $\\langle x_{\\mathrm {HII}}(z) \\rangle $ values in the testing data.",
"Marker colours show the mean bubble size.",
"All markers lie close to the perfect-model diagonal in Figure REF , confirming that this model makes extremely accurate predictions.",
"As in the previous section, the model accuracy is higher for $\\langle x_{\\mathrm {HII}}(z) \\rangle < 0.6$ than for $\\langle x_{\\mathrm {HII}}(z) \\rangle > 0.6$ .",
"Ionisation fraction 3PCF measurements have a very strong relationship with the global ionization fraction.",
"Ionisation fraction field data contain a range of bubble sizes.",
"The 3PCF measures clustering on a range of scales and this information is apparently strong enough to provide immediate and accurate predictions for the mean ionization fraction.",
"The predictions begin to worsen near the end of the EoR for $\\langle x_{\\mathrm {HII}}(z) \\rangle > 0.6$ , when overlap causes a more complex bubble size distribution.",
"However, the predictions are still visibly good and still have a low RMSE value.",
"Figure: Predicted global ionization fraction vs true global ionization fraction for unseen testing data, using the ionization fraction 3PCF as inputs.",
"The predicted and true values lie very closely along the diagonal, particularly for values 〈x HII (z)〉<0.6\\langle x_{\\mathrm {HII}}(z) \\rangle < 0.6.",
"Predictions for 〈x HII (z)〉>0.6\\langle x_{\\mathrm {HII}}(z) \\rangle > 0.6 are slightly worse as discussed in the text."
],
[
"Results training on $\\delta T_{\\mathrm {b}}$ data",
"In this subsection we train a model to predict the global ionization fraction $\\langle x_{\\mathrm {HII}}(z) \\rangle $ from $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ 3PCF data.",
"We use the same search strategy as the previous subsections.",
"The best model uses three hidden layers with sizes [168, 174, 70]; training batch size of 361; a maximum of 506 epochs (of which the model used all epochs before terminating); adaptive learning rate starting at $4.44 \\times 10^{-3}$ ; the r̀elu' activation function; and L2 regularization parameter $3.65 \\times 10^{-3}$ .",
"As seen in Figure REF , predicting the global ionization fraction using $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ 3PCF data gives less accurate results than using $x_{\\mathrm {HII}}(\\mathbf {r})$ data.",
"The $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ model's median prediction error is $16.0 \\%$ , much worse than the error of $3.6 \\%$ for the $x_{\\mathrm {HII}}(\\mathbf {r})$ model.",
"Figure REF gives the final prediction histograms for the two global ionization fraction models, using either ionization fraction data $x_{\\mathrm {HII}}(\\mathbf {r})$ or brightness temperature field data $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ .",
"Predictions of the global ionization fraction depend strongly on which data are used: the prediction errors for the model using $x_{\\mathrm {HII}}(\\mathbf {r})$ data are much lower than those for the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ model.",
"Figure: Histogram of prediction errors for predicting the global ionization fraction.",
"Each line shows the histogram of errors for a single model.",
"The model using x HII (𝐫)x_{\\mathrm {HII}}(\\mathbf {r}) 3PCF data has a much more accurate median prediction error (3.6%3.6 \\%) than the model using δT b (𝐫)\\delta T_{\\mathrm {b}}(\\mathbf {r}) data (16.0%)16.0 \\%).The model predictions shown in Figure REF deviate more widely from the perfect diagonal than do the predictions in Figure REF .",
"Interestingly, this model's accuracy increases for the later stages of the EoR with $\\langle x_{\\mathrm {HII}}(z) \\rangle > 0.6$ , as opposed to decreasing as did the accuracy of the model using $x_{\\mathrm {HII}}(\\mathbf {r})$ 3PCF data.",
"This can be understood by considering the impact of density and spin temperature fluctuations.",
"Local fluctuations have a more significant impact on the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ field at early times than at later times.",
"Thus, the morphology of the $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ field is more closely linked to that of the $x_{\\mathrm {HII}}(\\mathbf {r})$ field at later times.",
"Figure: Predicted global ionization fraction vs true global ionization fraction for unseen testing data, using δT b (𝐫)\\delta T_{\\mathrm {b}}(\\mathbf {r}) as inputs.",
"The predictions generally lie along the diagonal, but with larger scatter than using x HII (𝐫)x_{\\mathrm {HII}}(\\mathbf {r}) as model inputs."
],
[
"Conclusions",
"The three-point correlation function (3PCF) of the 21cm signal contains valuable information about the morphology and history of the Epoch of Reionization.",
"We present an optimised code for estimating the three-point correlation function of 3D pixelised data, such as the outputs from semi-numerical simulations.",
"The code includes jackknifing for error estimates and user-changeable parameters for choosing a level of approximate sampling.",
"We test the code on a testing distribution with known analytic 3PCF, finding that the estimates from our code match the true 3PCF closely.",
"After testing, we use our code to calculate the 3PCF for a range of simulated reionization scenarios using 21cmFAST.",
"Throughout we assume an idealised case where instrumental noise is negligible, and 21cm foregrounds have been perfectly removed.",
"We use machine learning techniques and train models to recover both the typical bubble size and the global ionization fraction from measured 3PCF outputs of semi-numerical simulations.",
"We first train models to recover the typical bubble size, from the 3PCF of either ionization fraction data or 21cm differential brightness temperature data.",
"The two models are both able to determine the general trend of increasing typical bubble size and have similar overall accuracy.",
"The model using $x_{\\mathrm {HII}}(\\mathbf {r})$ 3PCF data has better performance at small bubble sizes ($1 < R_{\\mathrm {bubble}}< 70 \\text{Mpc}$ , whereas the model using $\\delta T_{\\mathrm {b}}(\\mathbf {r})$ has better performance for larger bubble sizes ($R_{\\mathrm {bubble}}> 25 \\text{Mpc}$ ).",
"Both features can be understood in terms of how the data field morphologies evolve over the EoR.",
"We compare the performances of predict the typical bubble size using either the 3PCF or the power spectrum.",
"We find that using the 3PCF instead of the power spectrum leads a noticeable improvement in the final MLP model's prediction accuracy, with median prediction accuracies of around $10\\%$ and $14\\%$ respectively.",
"We then train a model to recover the global ionization fraction from ionization fraction 3PCF data.",
"The resulting model has extremely accurate predictions and shows the three-point clustering of $x_{\\mathrm {HII}}(\\mathbf {r})$ data is strongly related to the evolution of the global ionization fraction.",
"Our model is able to uncover this relationship with median prediction accuracy of $4\\%$ , although the predictions are slightly less accurate for the later stages of the EoR with $\\langle x_{\\mathrm {HII}}(z) \\rangle > 0.6$ .",
"Unfortunately this model would practically not be useful in EoR analysis because the ionization fraction field is difficult to probe directly.",
"Instead, observations are made in terms of the differential brightness temperature.",
"We train a fourth and final model to predict the global ionization fraction from the 3PCF of the differential brightness temperature field.",
"This MLP model has a median prediction accuracy of $16\\%$ .",
"The resulting model makes accurate predictions for the late stages of the EoR ($\\langle x_{\\mathrm {HII}}(z) \\rangle > 0.6$ ), but struggles with the early stages.",
"As with all machine learning projects, our models to predict the typical bubble size and global ionization fraction could likely be improved by gathering more data from a wider range of reionizaion scenarios.",
"This would allow the models to learn more general connections between the 3PCF measurements and characteristic reionization features.",
"Providing other brightness temperature field summary statistics could also improve our models, for instance the distribution of pixel brightnesses [32] or the size distribution of bright regions [34].",
"We also note that our models assume a constant value for the X-ray efficiency.",
"Ideally this constraint should be lifted and the X-ray efficiency allowed to vary as with the other simulation parameters.",
"Further studies will be necessary to evaluate the effectiveness of such an approach in the presence of instrumental effects and noise, as well as foreground residuals.",
"The techniques in this paper are tested on simulated data.",
"We have assumed that instrumental noise is negligible at our scales and lower redshifts of operation, as is expected during the EoR upcoming experiments such as the SKA ([37]).",
"Instrumental smoothing will predominantly affect smaller-scale features on the same scale as the instrument's point spread function, and the effect on larger-scale features would be minimal.",
"Whilst as noted by [75] the bispectrum of Gaussian noise is zero, there will be noise and possibly bias on both the 3PCF and the bispectrum due to sample variance, instrumental systematic effects, ionispheric effects, finite number of baselines, restricted field-of-view, and radio frequency interference.",
"All of these will need to be considered in future studies.",
"Furthermore we have assumed a best-case scenario where 21cm foregrounds have been perfectly removed.",
"This assumption is not uncommon in recent literature (see for example [68], [18], [33]) but remains the subject of much discussion.",
"Several studies ([42], [51], [8]) have claimed that foreground removal can be effective for the power spectrum.",
"[76] show that foregrounds could be a problem for recovering the 21cm bispectrum.",
"More work would be needed to understand the impact of foreground residuals on the 3PCF signal.",
"There are several other possible avenues of future work to build on these results.",
"First, using similar machine learning techniques to predict the full bubble size distribution $dP/dR$ from 3PCF data.",
"The full bubble size distribution provides a more detailed description of the morphology than the typical bubble size alone.",
"Secondly, using a larger selection of triangle configurations (both sizes and shapes) would likely provide more information and make it easier to recover the bubble size statistics.",
"Thirdly, training models to map from 3PCF measurements directly to parameters in a similar way to [68].",
"Such inference models can only make estimates of the `best' parameters and do not provide uncertainty regions in the same way as MCMC analysis.",
"Instead, training emulators to forward-model the 3PCF outputs directly from the simulation input parameters would effectively remove the need for further simulations.",
"Finally, investigating the effect of realistic experiment conditions would indicate whether the 3PCF of future 21cm measurements could be used to extract physically-meaningful bubble size statistics.",
"This work presents the first attempt to predict fundamental properties of the Epoch of Reionization using the three-point correlation function and machine learning techniques.",
"We provide a publicly available code 3PCF-Fast to help the community perform similar analyses in the future."
],
[
"Acknowledgements",
"WDJ was supported by the Science and Technology Facilities Council (ST/M503873/1) and from the European Community through the DEDALE grant (contract no.",
"665044) within the H2020 Framework Program of the European Commission.",
"CAW's research is supported by a UK Research and Innovation Future Leaders Fellowship, grant number MR/S016066/1.",
"However, the research presented in this paper was carried out with financial support from the European Research Council under ERC grant number 638743-FIRSTDAWN (held by Jonathan Pritchard).",
"FBA acknowledges support from the DEDALE grant, from the UK Science and Technology Research Council (STFC) grant ST/M001334/1, and from STFC grant ST/P003532/1.",
"The data underlying this article will be shared on reasonable request to the corresponding author."
]
] | 2011.14157 |
[
[
"Risk assessment for airborne disease transmission by poly-pathogen\n aerosols"
],
[
"Abstract In the case of airborne diseases, pathogen copies are transmitted by droplets of respiratory tract fluid that are exhaled by the infectious and, after partial or full drying, inhaled as aerosols by the susceptible.",
"The risk of infection in indoor environments is typically modelled using the Wells-Riley model or a Wells-Riley-like formulation, usually assuming the pathogen dose follows a Poisson distribution (mono-pathogen assumption).",
"Aerosols that hold more than one pathogen copy, i.e.",
"poly-pathogen aerosols, break this assumption even if the aerosol dose itself follows a Poisson distribution.",
"For the largest aerosols where the number of pathogen in each aerosol can sometimes be several hundred or several thousand, the effect is non-negligible, especially in diseases where the risk of infection per pathogen is high.",
"Here we report on a generalization of the Wells-Riley model and dose-response models for poly-pathogen aerosols by separately modeling each number of pathogen copies per aerosol, while the aerosol dose itself follows a Poisson distribution.",
"This results in a model for computational risk assessment suitable for mono-/poly-pathogen aerosols.",
"We show that the mono-pathogen assumption significantly overestimates the risk of infection for high pathogen concentrations in the respiratory tract fluid.",
"The model also includes the aerosol removal due to filtering by the individuals which becomes significant for poorly ventilated environments with a high density of individuals, and systematically includes the effects of facemasks in the infectious aerosol source and sink terms and dose calculations."
],
[
"S5 Appendix. Numerical Considerations",
"If $\\alpha $ , $\\beta _k$ , $\\gamma $ , and $w$ are all constant with respect to time; the model has both an explicit and recursive solution for the concentration density and dose.",
"Otherwise, it may not be possible to get a closed form analytical solution from the general solution and one would need to solve the model numerically.",
"Of course, even with an analytical solution, one can solve it numerically.",
"There are a number of numerical pitfalls with both the analytical and numerical solutions, which arise as $M_c$ becomes large.",
"For a numerical solution, the time step of integration must be small compared to the smallest time scale in the model.",
"The smallest time scale could be the time scale on which $\\alpha $ , $\\beta _k$ , $\\gamma $ , and/or $w$ change over time; but for large enough $M_c$ it will always be time scale of the total sink for $k = M_c$ aerosols.",
"The total sink for $n_k$ is $-\\alpha (d_0,t) - k \\gamma (t)$ , which scales linearly in $M_c$ for $n_{M_c}$ when $M_c$ is large enough.",
"The time step $\\delta t$ must be $\\delta t < (\\alpha + M_c \\gamma )^{-1}$ .",
"Thus the total number of timesteps $N_t$ scales as $N_t \\sim M_c$ for large $M_c$ .",
"Since the total number of terms scales linearly in $M_c$ as long as $\\partial n_k / \\partial t$ is not calculated in vector-matrix form or $\\mathbf {A}$ is stored in a sparse format, the total computational effort scales as $\\mathcal {O}\\left(M_c^2\\right)$ .",
"Also note, the model is stiff when $M_c \\gamma (t) \\gg \\alpha (d_0, t) + \\gamma (t)$ since the total sink timescale for $k = M_c$ is much shorter than the total sink timescale for $k = 1$ .",
"For the analytical solution for coefficients constant in time; there are different difficulties for the explicit solution and the recursive solution.",
"The number of terms scales as $\\mathcal {O}\\left(M_c^3\\right)$ in the explicit solution and $\\mathcal {O}\\left(M_c^2\\right)$ in the recursive solution, since there are $M_c$ equations and they have double and single sums respectively, with each sum scaling as $M_c$ .",
"The analytical solutions have additional difficulties — avoiding numerical overflow and maintaining accuracy.",
"The problem is the binomial coefficients $\\binom{i}{k}$ and $\\binom{i - k}{p}$ where $i \\rightarrow M_c$ and $p \\in [0, i - k]$ .",
"They can be calculated naively by computing each factorial and then doing the multiplication and division ($k!$ will be the largest); or carefully with cancellation handled explicitly or using logarithms of the gamma function, which overflows later.",
"And then even if all the binomial coefficients don't overflow, their products and sums can still overflow (the explicit version is worse than the recursive version in this regard due to the products of binomial coefficients).",
"To see this, we will find the upper bound for $V_k\\left(\\vec{y},x\\right)$ in the recursive solution.",
"Consider its formula $V_k\\left(\\vec{y},x\\right) = \\sum _{i=k}^{M_c} \\binom{i}{k} y_i (1 - x)^{i - k} \\quad .$ In the model, all the elements of $\\vec{y}$ , which is always either $\\vec{\\beta }$ or $\\vec{n}_0$ , are non-negative.",
"Since $x \\in [0, 1]$ , $1 - x \\in [0, 1]$ and is therefore also non-negative.",
"This means that all elements in the sum in $V_k$ are non-negative, meaning $V_k \\ge 0$ and thus we only need to find the upper bound to determine the risk of overflow (the lower bound, zero, is not a worry).",
"Now, $y_i \\le \\max {\\left(\\vec{y}\\right)}$ for all $i$ where $\\max {\\left(\\vec{y}\\right)}$ shall denote the maximum element of $\\vec{y}$ .",
"Since $x \\in [0, 1]$ , $0 \\le (1 - x)^{i - k} \\le 1$ .",
"Then we just need to get an upper bound for the binomial coefficients.",
"The binomial coefficient $\\binom{j}{m} = k!",
"/ m!",
"(j - m)!$ is at its greatest value for fixed $j$ when $m = \\lfloor j / 2 \\rfloor $ or $m = \\lceil j / 2 \\rceil $ where $\\lfloor \\cdot \\rfloor $ and $\\lceil \\cdot \\rceil $ denote the floor and ceil operators respectively.",
"This value is $\\binom{j}{\\lfloor j / 2 \\rfloor } = \\binom{j}{\\lceil j / 2 \\rceil } = \\frac{j!",
"}{{\\lfloor j / 2 \\rfloor }!",
"{\\lceil j / 2 \\rceil }!}",
"\\quad .$ This will be maximized when $i = j = M_c$ .",
"Then, the upper bound for $V_k$ is $V_k\\left(\\vec{y},x\\right) \\le \\sum _{i=k}^{M_c} \\frac{M_c!",
"}{{\\lfloor M_c / 2 \\rfloor }!",
"{\\lceil M_c / 2 \\rceil }!}",
"\\max {\\left(\\vec{y}\\right)} = \\frac{(M_c + 1 - k) M_c!",
"\\max {\\left(\\vec{y}\\right)}}{{\\lfloor M_c / 2 \\rfloor }!",
"{\\lceil M_c / 2 \\rceil }!}",
"\\quad .$ For any $M_c$ , the largest upper bound for any of the $V_k$ is found for $k = 1$ , so the upper bound we need to worry about is $V_k\\left(\\vec{y},x\\right) \\le \\frac{M_c M_c!",
"\\max {\\left(\\vec{y}\\right)}}{{\\lfloor M_c / 2 \\rfloor }!",
"{\\lceil M_c / 2 \\rceil }!}",
"\\quad .$ Now, the factorial function has the bounds [1] $\\sqrt{2 \\pi } m^{m + \\frac{1}{2}} e^{-m} \\exp {\\left[\\frac{1}{12 m + 1}\\right]} < m!",
"< \\sqrt{2 \\pi } m^{m + \\frac{1}{2}} e^{-m} \\exp {\\left[\\frac{1}{12 m}\\right]} \\quad .$ We will still have an upper bound for $V_k$ if we use the factorial lower bounds in place of the factorials in the denominator of Eq (REF ) and the factorial upper bound for the factorial in the numerator.",
"This leads to $V_k\\left(\\vec{y},x\\right) < \\frac{\\max {\\left(\\vec{y}\\right)}M_c^{M_c + \\frac{3}{2}} \\exp {\\left[\\frac{1}{12 M_c} + {\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor } + {\\left\\lceil \\frac{M_c}{2} \\right\\rceil }\\right]}}{{\\sqrt{2 \\pi } {\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor }^{{\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor } + \\frac{1}{2}} {\\left\\lceil \\frac{M_c}{2} \\right\\rceil }^{{\\left\\lceil \\frac{M_c}{2} \\right\\rceil } + \\frac{1}{2}} \\exp {\\left[M_c + \\frac{1}{12 {\\lfloor \\frac{M_c}{2} \\rfloor } + 1} + \\frac{1}{12 {\\lceil \\frac{M_c}{2} \\rceil } + 1}\\right]}}} \\quad .$ If we take the $\\log _2$ , we get the value of the base-2 exponent.",
"Taking the $\\log _2$ of both sides, $\\log _2{V_k\\left(\\vec{y},x\\right)} <\\log _2{\\left[\\max {\\left(\\vec{y}\\right)}\\right]}+ \\left(M_c + \\frac{3}{2}\\right) \\log _2{(M_c)} \\\\- \\left({\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor } + \\frac{1}{2}\\right) \\log _2{{\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor }}- \\left({\\left\\lceil \\frac{M_c}{2} \\right\\rceil } + \\frac{1}{2}\\right) \\log _2{{\\left\\lceil \\frac{M_c}{2} \\right\\rceil }} \\\\+ \\left(\\log _2{e}\\right) \\left(\\frac{1}{12 M_c} - \\frac{1}{12 {\\lfloor \\frac{M_c}{2} \\rfloor } + 1} - \\frac{1}{12 {\\lceil \\frac{M_c}{2} \\rceil } + 1} \\right) \\\\- \\frac{1}{2} - \\frac{1}{2} \\log _2{\\pi } \\quad ,$ where we have used the fact that ${\\lfloor M_c / 2 \\rfloor } + {\\lceil M_c / 2 \\rceil } = M_c$ .",
"To represent all the $V_k$ for a particular $M_c$ without overflowing, a floating point format's maximum supported base-2 exponent, $emax$ (using the same notation as the IEEE 754 standard [2]), must be at least this value ($emax \\ge \\log _2(V_k)$ ).",
"But, even if the value $\\log _2(V_k)$ might not overflow with this minimum value, calculating the binomial coefficient when $\\max {\\left(\\vec{y}\\right)} < 1$ could overflow before the multiplication drops the magnitude.",
"So for the minimum $emax$ , we must replace the $\\max {\\left(\\vec{y}\\right)}$ with $\\max {\\left[1,\\max {\\left(\\vec{y}\\right)}\\right]}$ and we get $emax\\left(\\vec{y}\\right) \\ge \\Bigg \\lceil \\log _2{\\left[\\max {\\left[1,\\max {\\left(\\vec{y}\\right)}\\right]}\\right]}+ \\left(M_c + \\frac{3}{2}\\right) \\log _2{(M_c)} \\\\- \\left({\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor } + \\frac{1}{2}\\right) \\log _2{{\\left\\lfloor \\frac{M_c}{2} \\right\\rfloor }}- \\left({\\left\\lceil \\frac{M_c}{2} \\right\\rceil } + \\frac{1}{2}\\right) \\log _2{{\\left\\lceil \\frac{M_c}{2} \\right\\rceil }} \\\\+ \\left(\\log _2{e}\\right) \\left(\\frac{1}{12 M_c} - \\frac{1}{12 {\\lfloor \\frac{M_c}{2} \\rfloor } + 1} - \\frac{1}{12 {\\lceil \\frac{M_c}{2} \\rceil } + 1} \\right) \\\\- \\frac{1}{2} - \\frac{1}{2} \\log _2{\\pi }\\Bigg \\rceil \\quad .$ To see how much of an overestimate this is for $emax$ , let's compare it to $\\log _2{\\max {\\left(V_k\\right)}}$ for a few simple cases.",
"First, lets compare it to $V_k\\left(\\vec{1},0\\right)$ since $\\vec{y} = \\vec{1}$ has all elements equal to the maximum element, $x = 0$ maximizes $(1 - x)^{i - k}$ , and all terms in the sum are integers.",
"In Fig REF (left panel), $\\log _2{\\max {\\left(V_k\\right)}}$ is compared to $emax$ from Eq (REF ) and the $emax$ values for the four smallest IEEE-754-219 binary floating point formats [2].",
"We calculated $\\max {\\left(V_k\\right)}$ using variable sized integers before being converting to multi-precision floating point numbers for taking the $\\log _2$ with the largest supported exponent with the GNU Multiple Precision Floating-point Reliable Library (MPFR, see https://www.mpfr.org) and the GNU Multiple Precision Arithmetic Library (GMP, see https://gmplib.org) in Python using the gmpy2 package (https://pypi.org/project/gmpy2).",
"The minimum $emax$ from Eq (REF ) is only barely larger in a logarithmic sense than $\\log _2{\\max {\\left(V_k\\right)}}$ except for small $M_c$ , so it isn't an excessive overestimate of the required $emax$ .",
"Figure: Comparison of Maximum Vk Values to Required emax Overestimate And Floating Point Format Limits.",
"log 2 maxV k \\log _2{\\max {\\left(V_k\\right)}} is compared to the overestimate of the required emax(y →)emax(\\vec{y}) from Eq () as well as the maximum exponents supported by the four smallest IEEE-754-2019 binary floating point formats .",
"Note that the x87 FPU 80-bit floating point format has the same maximum as binary128 .",
"(Left) For y →=1 →\\vec{y} = \\vec{1} and x=0x = 0 as a function of M c M_c.",
"(Right) For y →=ψ →\\vec{y} = \\vec{\\psi } from Eq () and x=0x = 0 as a function of the average multiplicity k j =π 6d 0 3 ρ p,j \\left<k\\right>_j = \\frac{\\pi }{6} d_0^3 \\rho _{p,j}.Second, we will compare it to typical $\\vec{y}$ used in the model.",
"We will base $\\vec{y}$ on $\\vec{\\beta }_{I,k}$ for single infectious person but remove all of the environment parameters and person specific parameters except for the expected average multiplicity $\\left<k\\right>_j = \\frac{\\pi }{6} d_0^3 \\rho _{p,j}$ .",
"We shall use the vector $\\psi _k = \\frac{V}{\\rho _j \\lambda _{I,j} S_{I,m,out,j}} \\beta _{I,k} = P_P\\left(\\left<k\\right>_j, k\\right) \\quad .$ For a range of $\\left<k\\right>_j$ , $M_c$ was determined with the single infectious person production heuristic $M_{c,I,j}$ for thresholds $T$ of $10^{-1}$ and $10^{-9}$ .",
"Then the $V_k\\left(\\vec{\\psi },0\\right)$ were calculated in binary floating point with a 256 bit mantissa, $emax = 32768$ , and minimum exponent $emin = -32767$ with gmpy2, MPFR, and GMP as before.",
"The binomial coefficients were calculated exactly before conversion to floating point.",
"To improve the accuracy, the sum in Eq (REF ) was done as a sorting sum where all the terms were sorted in a list, the two smallest terms removed, those terms added together and inserted back into list; and this process repeated till only a single term (the total sum) remained.",
"The right panel of Fig REF compares $\\log _2{\\max {\\left(V_k\\right)}}$ against the $emax$ calculated from Eq (REF ) and the maximum exponents of some of the smallest IEEE 754 binary floating point formats.",
"We can see that Eq (REF ) overestimates the required $emax$ by quite a bit in a logarithmic sense for $\\left<k\\right>_j < 10$ , but not by much in a logarithmic sense for larger $\\left<k\\right>_j$ .",
"For small $\\left<k\\right>_j$ , it suggests an overkill $emax$ but binary16 (half precision) and binary32 (single precision) are typically the smallest floating point formats with hardware support on most computers and the bulk of the computational effort is spent on diameter bins with the largest $M_c$ , so there isn't much reason to use a smaller format for small $\\left<k\\right>_j$ .",
"binary64 (double precision) is sufficient for $\\vec{\\psi }$ in this case up to $\\left<k\\right>_j = 1000$ .",
"The exact values of other terms in $\\vec{\\beta }_I$ would determine if binary64 would be safe for that expected multiplicity with $\\vec{\\beta }_I$ .",
"The extra $emax$ requirement would additively increase by $\\log _2{\\left[\\max {\\left(1, \\frac{1}{V} \\rho _j \\lambda _{I,j} S_{I,m,out,j}\\right)}\\right]}$ .",
"binary128 and x87 FPU 80-bit floating point numbers which have the same $emax$ [3] would have some headroom for the magnitude of the other coefficients in $\\vec{\\beta }_I$ even for $\\left<k\\right>_j = 10^4$ .",
"To investigate the required floating point precision to calculate $n_k$ , we will look at the required precision in bits required to calculate all $U_k\\left(d_0,\\vec{y},x\\right)$ for a given $M_c$ to within a specified relative tolerance $\\delta $ of the exact values.",
"We chose $\\alpha & = & 1 \\quad , \\\\\\gamma & = & 3 \\quad , \\\\\\vec{y} & = & \\vec{1} \\quad , \\\\x & \\in & \\left\\lbrace 0, \\frac{5}{13}, 1\\right\\rbrace \\quad ,$ which makes $U_k\\left(d_0,\\vec{y},x\\right)$ a rational number for any $M_c$ .",
"For several $M_c$ , the $U_k\\left(d_0,\\vec{y},x\\right)$ were calculated exactly using variable sized rational arithmetic.",
"Then, the smallest floating point precision was found for which the maximum relative error in the $U_k\\left(d_0,\\vec{y},x\\right)$ calculated in floating point in that precision (note, the binomial coefficents were calculated first as variable sized integers and then converted to floating point) is less than $\\delta $ using the explicit and recursive formulas for $U_k\\left(d_0,\\vec{y},x\\right)$ .",
"The calculations were done using gmpy2, MPFR, and GMP as before.",
"The required precisions for $\\delta $ of 10%, 10 PPM (Parts per Million), and 1 PPB (Parts per Billion) are shown in Fig REF and compared to the precisions of various standard floating point formats.",
"Figure: Required Floating Point Precision to Calculate Uk.",
"The required floating point precision in bits to get the largest relative error in the U k d 0 ,y →,xU_k\\left(d_0,\\vec{y},x\\right) to be less than the tolerance δ\\delta for three values of xx using the (Left) explicit formula and (Right) recursive formula.",
"Horizontal lines show the precisions provided by the four smallest IEEE-754-2019 binary floating point formats and the x87 FPU 80-bit floating point format .",
"Both panels share the same legend, which is in the left panel.Using the explicit formula, even quadruple precision (binary128) is insufficient by $Mc = 80$ for this $\\vec{y}$ even for a tolerance of 10%.",
"But using the recursive formula, double precision (binary64) is good enough for a tolerance of 1 PPB even for the largest $M_c = 5000$ that was checked.",
"From this and the recursive solution's number of terms scaling as $\\mathcal {O}\\left(M_c^2\\right)$ instead of $\\mathcal {O}\\left(M_c^3\\right)$ , the recursive solution is much more amenable to calculations than the explicit solution.",
"Considering both Fig REF and REF , double precision (binary64) seems to be suitable, depending on the largest values in the $\\vec{y}$ , for moderate $M_c$ of a few hundred.",
"To go to $M_c$ of a few thousand, quadruple precision (binary128) or x87 FPU 80-bit floating point numbers are required.",
"Above this, either multi-precision floating point with higher exponents must be used or the model must be solved numerically.",
"The number of terms to compute scales quadratically in $M_c$ for both the numerical solution and recursive analytical solution, and cubically for the explicit analytical solution; and the analytical solutions have the additional problems of avoiding numerical overflow as well as the computational effort to calculate some terms increasing with $M_c$ .",
"For any particular fixed size number format, there is an $M_c$ above which the analytical result will overflow and one must solve the model numerically instead."
]
] | 2011.14118 |
[
[
"Insulated primes"
],
[
"Abstract The degree of insulation of a prime $p$ is defined as the largest interval around it in which no other prime exists.",
"Based on this, the $n$-th prime $p_{n}$ is said to be insulated if and only if its degree of insulation is higher than its neighboring primes.",
"Consequently, a new special sequence emerges given as 7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, and so on.",
"This paper presents several properties and intriguing relations concerning degree of insulation and insulated primes.",
"Finally, the reader is left with a captivating open problem."
],
[
"Introduction",
"Since centuries, prime numbers have both puzzled and fascinated researchers.",
"Several interesting subsets of primes have been identified and analyzed in literature [3].",
"For example, primes that are present in a specific form such as Proth prime, Mersenne prime, and Fermat prime [7].",
"Sequences of primes are also defined if they obey certain property, for instance, Chen prime [1], Lucky prime [4], Wilson prime [2], Ramanujan prime [11], etc.",
"which have captured huge attention due to associated unsolved problems.",
"Similarly, this paper defines a new special subset of primes denoted by $\\mathbb {I}$ , called the insulated primes, which is defined based on conditions related to the gaps between consecutive primes.",
"Sequences such as twin primes, cousin primes, sexy primes, etc.",
"are also based on the gaps between consecutive primes.",
"Other sequence associated with gaps is the sequence of least prime of distance $n$ from nearest prime $(n = 1, ~ \\text{even})$ called the lonely/isolated primes which is given as 2, 5, 23, 53, 409, 293, 211, 1847, 3137, 2179, 3967, 23719, 16033, 40387, 44417, and so on [9].",
"Another interesting sequence is prime islands which is the least prime whose adjacent primes are exactly $2n$ apart given as 3, 5, 7, 29, 23, 53, 89, 223, 113, 331, 631, 211, 1381, 1129, 1637, 4759, 2579, 3433, and so on [10].",
"In order to pursue the said objective, we begin by defining degree of insulation and then proceed to the insulated primes.",
"Definition 1.1 Degree of insulation $D(p)$ of a prime $p$ is defined as the maximum value of the set $X=\\lbrace m:\\pi (p-m)=\\pi (p+m)-1, m\\in \\mathbb {N} \\rbrace $ , where $\\pi (n)$ is the prime counting function.",
"Definition 1.2 The $n$ -th prime $p_{n}$ is said to be insulated if and only if $D(p_{n-1})<D(p_{n})$ and $D(p_{n+1})<D(p_{n})$ .",
"The degree of insulation essentially gives the largest bracket around $p$ such that $p$ is the only prime in that spread, thus insulating $p$ from neighboring primes.",
"That is why, the set $\\mathbb {I}$ is named insulated primes.",
"Let us begin with an illustration using the prime triplet $(19, 23, 29)$ .",
"Consider number 23 which is solved as $\\pi (23-1) &\\overset{\\mathrm {?",
"}}{=}\\pi (23+1)-1 \\Rightarrow 8\\overset{\\mathrm {?",
"}}{=}9-1 \\Rightarrow 8 = 8 \\\\\\pi (23-2) &\\overset{\\mathrm {?",
"}}{=}\\pi (23+2)-1 \\Rightarrow 8\\overset{\\mathrm {?",
"}}{=}9-1 \\Rightarrow 8 = 8 \\\\\\pi (23-3) &\\overset{\\mathrm {?",
"}}{=}\\pi (23+3)-1 \\Rightarrow 8\\overset{\\mathrm {?",
"}}{=}9-1 \\Rightarrow 8 = 8 \\\\\\pi (23-4) &\\overset{\\mathrm {?",
"}}{=}\\pi (23+4)-1 \\Rightarrow 8\\overset{\\mathrm {?",
"}}{=}9-1 \\Rightarrow 8 = 8 \\\\\\pi (23-5) &\\overset{\\mathrm {?",
"}}{=}\\pi (23+5)-1 \\Rightarrow 7\\overset{\\mathrm {?",
"}}{=}9-1 \\Rightarrow 7 \\ne 8 ,$ so $D(23)=4$ .",
"Now, for 19 and 29, we have $\\pi (19-2) \\overset{\\mathrm {?",
"}}{=}\\pi (19+2)-1 \\Rightarrow 7\\overset{\\mathrm {?",
"}}{=}8-1 \\Rightarrow 7 = 7$ and $\\pi (29-1) \\overset{\\mathrm {?",
"}}{=}\\pi (29+1)-1 \\Rightarrow 9\\overset{\\mathrm {?",
"}}{=}10-1 \\Rightarrow 9 = 9 ,$ so $D(19)=2$ and $D(29)=1$ , hence 23 is an insulated prime.",
"As the determination of insulated primes requires the calculation of $D(p)$ , so the above procedure is coded in python and the resulting line plot of the values of $D(p)$ vs primes is shown in Figure REF .",
"Figure: Plot of D(p)D(p) for primes less than 1000The sequence of insulated primes can be understood as the set of local maximas in the $D(p)$ plot.",
"Thus, the sequence $\\mathbb {I}$ is 7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, 337, 359, 389, 409, 421, 449, 479, 491, 509, 547, 577, 607, 631, 653, 691, 709, 751, 761, 797, 811, 823, 839, 863, 887, 919, 953, 983, and so on.",
"Figure REF is the plot of $n$ -th insulated prime $i_n$ vs $n$ .",
"Figure: Plot of i n i_n versus nn for primes less than 1000Note that this sequence is neither present on the Online Encyclopedia of Integer Sequences (OEIS) nor its explored earlier, hence, this paper aims to investigate it."
],
[
"Analysis of degree of insulation",
"Certain patterns can be observed from the graphs shown earlier, but these need to be rigorously proven, which is done below along with other critical results.",
"Theorem 2.1 Primes just adjacent to an insulated prime can never be insulated.",
"Let $p_n$ be insulated prime, then $D(p_{n-1})<D(p_{n})$ and $D(p_{n+1})<D(p_{n})$ by definition.",
"Now, for $p_{n-1}$ to be insulated prime, the conditions $D(p_{n-2})<D(p_{n-1})$ and $D(p_{n})<D(p_{n-1})$ must hold.",
"Clearly, the latter condition is contradictory to condition of $p_n$ , therefore, $p_{n-1}$ is not an insulated prime.",
"For $p_{n+1}$ to be an insulated prime, the conditions $D(p_{n})<D(p_{n+1})$ and $D(p_{n+2})<D(p_{n+1})$ must hold.",
"But the prior condition is not feasible, therefore, $p_{n+1}$ is also not an insulated prime.",
"Hence, proved.",
"Theorem 2.2 If $\\alpha \\notin X$ then $(\\alpha +r)\\notin X$ for all $r\\ge 0$ .",
"For some $r\\ge 0$ , we have $\\pi (p+\\alpha +r)\\ge \\pi (p+\\alpha )$ and $\\pi (p-\\alpha )\\ge \\pi (p-\\alpha -r)$ because $\\pi (n)$ is an increasing function.",
"This implies $\\pi (p+\\alpha +r)-\\pi (p-\\alpha -r)\\ge \\pi (p+\\alpha )-\\pi (p-\\alpha )$ .",
"As $X$ contains all the possible candidates for being $D(p)$ and $\\alpha \\notin X$ , then $\\pi (p+\\alpha )-\\pi (p-\\alpha )> 1$ because there exists at least one prime $p$ such that $p-\\alpha \\le p \\le p+\\alpha $ .",
"Using this gives $\\pi (p+\\alpha +r)-\\pi (p-\\alpha -r)\\ge \\pi (p+\\alpha )-\\pi (p-\\alpha ) > 1$ which implies $\\pi (p+\\alpha +r)-\\pi (p-\\alpha -r)\\ne 1$ .",
"Thus, $\\alpha +r$ is also not a possible candidate for $D(p)$ .",
"Corollary 2.3 For prime $p$ , if $\\alpha \\notin X$ then $D(p)<\\alpha $ .",
"As $(\\alpha +r) \\notin X$ that is $\\alpha +r$ is not a candidate for being $D(p)$ for all $r\\ge 0$ .",
"Now, if non-zero $D(p)$ exists then $D(p)$ must be less than $\\alpha $ .",
"Theorem 2.4 For a prime $p$ , every $m\\in X$ obeys $ \\log (p+m)\\log (p-m) < m\\log \\left((p-m)^c (p+m)\\right) + p\\log \\left(\\frac{(p-m)^c}{p+m}\\right)$ provided $p\\ge 17$ , where $c=30\\frac{\\ln {113}}{113}$ .",
"From [8], we have $\\frac{x}{\\log {x}} < \\pi (x) < \\frac{cx}{\\log {x}}$ which gives $\\pi (p+m) - \\pi (p-m) < \\frac{c(p+m)}{\\log (p+m)} - \\frac{p-m}{\\log (p-m)} .$ For $m$ to belong to set $X$ , the left hand side of the inequality must be one.",
"On solving the expression and rearranging the terms, we get the desired result.",
"Theorem 2.5 For any $N > N_0$ , there exists an $n\\in [N+1,2N+1]$ such that $D(p_n) \\approx g_n$ where $g_n = p_{n+1} - p_n$ .",
"By definition of degree of insulation, we also have $D(p_n) &\\approx \\min (p_{n+1}-p_n, p_n-p_{n-1}) \\\\&= (p_n - p_{n-1}) \\times \\min \\left(1,~\\frac{p_{n+1}-p_n}{p_n - p_{n-1}}\\right)$ The ratio of consecutive prime gaps appears in the above expression.",
"Using Theorem 3.8 in [6], for $n\\in [N+1,2N+1]$ , we have $\\frac{p_{n+1}-p_n}{p_n - p_{n-1}} > \\frac{c_0 \\log {N} \\log _2{N} \\log _4{N}}{(\\log _3{N})^2}$ where $c_0$ is a constant and $\\log _{t}{N}$ is the t-fold iterated logarithmic function.",
"As $N$ is sufficiently large, so it turns out the right side of the inequality is greater than one.",
"Thus, $D(p_n) \\approx p_n - p_{n-1} = g_n$ .",
"For a particular prime $p$ , the natural procedure for finding $D(p)$ is inspired from the definition itself, which was illustrated in the previous section for number 23.",
"The process began from $m=1$ and kept on checking until the relation is violated.",
"So, it either requires to compute prime counting function or to determine the surrounding primes of the given prime.",
"For extremely large primes, the second approach is highly difficult.",
"However, the prior approach can be improved in view of the above theorems to reduce the effort of checking at all the values, as they allow to use bracketing techniques for faster numerical computation of $D(p)$ .",
"Additionally, reader may refer to Appendix where a faster algorithm is presented for large $p$ .",
"In addition to the above results, which confirm the connection of $D(p)$ with gaps between primes, the definition of $D(p)$ also allows us to directly show the following theorem.",
"Theorem 2.6 For a prime $p$ , we have $p-(D(p)+1)$ or $p+(D(p)+1)$ is prime if $D(p)$ is odd, and $p-D(p)$ or $p+D(p)$ is prime if $D(p)$ is even.",
"Corollary 2.7 If $D(p)=1$ then either $p-2$ or $p+2$ is a prime.",
"Therefore, counting the number of primes with $D(p)=1$ gives the count of twin-prime pairs.",
"Let $f_k$ be the probability of finding a prime $p<1000$ with $D(p)=k$ , then Figure REF is a line plot of $f_k$ with $k$ for primes upto 1000.",
"Figure: Plot of f k f_k for 1≤k≤111\\le k\\le 11 with primes p<1000p<1000The quantity $f_k$ is simply the fraction of primes with $D(p)=k$ over total primes below 1000.",
"The graph depicts that probability falls with increasing $k$ .",
"The probability of prime with $D(p)=1$ is highest.",
"Note that $D(p)=1$ corresponds to twin primes.",
"For primes upto $10^5$ , the pattern is slightly deviated as evident from Figure REF .",
"It is expected to behave nicely for extremely set of primes.",
"Figure: Plot of f k f_k for 1≤k≤301\\le k\\le 30 with primes p<10 5 p<10^5If the above trend follows, that is, the number of twin primes are larger than number of primes with other gaps, then its convenient to show the infinitude of number of twin-primes.",
"There is huge literature in this direction as well, but this problem lies out of the scope of this paper."
],
[
"Formula for insulated primes",
"In addition to the results listed above, one must also be curious about an approximate/asymptotic formula for the insulated primes.",
"Figure REF indicated an almost linear growth, which was further strengthened by evaluating larger insulated primes.",
"Figure: Comparison of i n i_n plots for primes less than 10 5 10^5Using MATLAB command cftool for the Curve fitting toolbox, a variety of curves (with different settings) were tested.",
"It was observed that the equation $y = 19.18 n^{1.093}$ is an extremely good fit as indicated in Figure REF .",
"Figure: Comparison of i n i_n plots for primes less than 10 6 10^6The equation performs even better when tested for primes upto one million as shown in Figure REF .",
"It has an R-square value of unity which is extremely good.",
"This allows to conclude that $i_n \\sim 19.18 n^{1.093}$ is heuristically an accurate fit.",
"However, this result needs to be rigorously proven, but this analysis is sufficient to convince that insulated primes are definitely well-behaved in comparison to primes or other prime subsets."
],
[
"Conclusion and Future scope",
"This paper defines and explores the properties of a new sequence called the insulated primes.",
"The sequence is 7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, and so on.",
"The insulated primes are defined using the concept of degree of insulation, which is also shown to have deep connection with gaps between consecutive primes.",
"A detailed mathematical analysis is conducted which unravels several characteristics.",
"Based on this analysis, a faster algorithm for evaluation of $D(p)$ was suggested as well as a quick discussion on Twin-Prime conjecture was also conducted.",
"Finally, heuristics show that $n$ -th insulated prime $i_n$ obeys a power law.",
"A strong form of this result is the conjecture that $i_n$ is varies almost linearly for $n\\rightarrow \\infty $ .",
"The authors encourage inquisitive readers to ponder upon the ideas presented in the paper.",
"An extension of the insulated primes is the highly insulated primes.",
"The set of highly insulated primes $\\mathbb {I}_H \\subset \\mathbb {I} \\subset \\mathbb {P}$ can be defined by treating insulation as an operation on the set $\\mathbb {I}$ .",
"This is done by applying the definition of insulation on $i_n$ , that is, $i_n$ is highly insulated prime if $D(i_{n-1})<D(i_n)$ and $D(i_{n+1})<D(i_n)$ .",
"This is demonstrated by investigating the triplet of insulated primes $(i_{n-1},i_n,i_{n+1})=(13,23,37)$ .",
"As $D(13)=2$ , $D(23)=4$ and $D(37)=3$ , so 23 is also an highly insulated prime.",
"Similarly, other highly insulated primes are 23, 53, 89, 211, 293, and so on.",
"The interested readers may investigate this sequence."
],
[
"Modified algorithm for $D(p)$ when {{formula:9c768377-5a37-49a7-b2d8-2ead3c7fefac}} is large",
"For large $X$ and $y\\in (0,X]$ , the prime number theorem shows that on average for $x\\in [X,2X]$ , we have $\\pi (x+y) - \\pi (x) \\approx \\frac{y}{\\log {x}} .$ Substitute $x+y=p+m$ and $x=p-m$ then $\\pi (p+m) - \\pi (p-m) \\approx \\frac{2m}{\\log (p-m)}$ for $2m\\in (0,X]$ and $p\\in [X+m,2X+m]$ .",
"For $D(p)$ , the left hand side must be unity, therefore we get $2m \\approx \\log (p-m)$ .",
"Let $\\operatorname{W}$ (.)",
"is Lambert's W function then $m$ can be explicitly written as $m = p - 0.5 \\operatorname{W}(2e^{2p})$ .",
"For large primes $p$ , evaluation of $e^{2p}$ is an tough task, so using the asymptotic relation for Lambert's W function, we get $m \\sim p - 0.5 (\\log {2e^{2p}} - \\log {\\log {2e^{2p}}})$ which simplifies to give $m \\sim \\frac{1}{2} \\log \\left(\\frac{2p}{\\log {2}} + 1\\right)$ .",
"The above step provides a better way to determine $m$ rather than aimlessly beginning from 1 and linearly searching as it was illustrated for prime 23.",
"However, the derivation of the above step imposes certain conditions which can be relaxed by not treating the obtained $m$ as the final answer.",
"This is done by considering $m$ as just a better candidate to be $D(p)$ which is then substituted in $\\pi (p+m)-\\pi (p-m)-1$ to cross-check.",
"If its value is zero, then assign $m_0=m$ and $m_1=p$ , else set $m_0=1$ and $m_1=m$ .",
"Now, apply bisection method to search $D(p)$ in the interval $[m_0,m_1]$ .",
"This procedure immensely reduces the number of times one needs to evaluate $\\pi (x)$ .",
"An additional note here is that using sharper results on gaps between primes, one can choose a better $m_1$ than just assigning to $p$ in the first case."
]
] | 2011.14210 |
[
[
"On Ramsey-minimal infinite graphs"
],
[
"Abstract For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \\to (G,H)$, i.e.",
"every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$.",
"We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$.",
"This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves.",
"We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph.",
"We use these to prove, for example, that if $G=S_\\infty$ is an infinite star and $H=nK_2$, $n \\ge 1$ is a matching, then the pair $(S_\\infty,nK_2)$ admits no Ramsey-minimal graphs."
],
[
"Introduction",
"Let $F$ , $G$ and $H$ be possibly infinite, simple graphs with no isolated vertices.",
"We follow some notation in [11].",
"We say that $F$ arrows $(G,H)$ or that $F \\rightarrow (G,H)$ if for every red-blue coloring of the edges of $F$ , there exists either a red $G$ or a blue $H$ contained in $F$ .",
"In this case, we say that $F$ is an $(G,H)$ -arrowing graph.",
"A red-blue coloring of $F$ is called $(G,H)$ -good if $F$ does not contain a red $G$ or a blue $H$ with respect to the coloring.",
"An alternate definition for $F \\rightarrow (G,H)$ would then be that the graph $F$ admits no $(G,H)$ -good coloring.",
"A $(G,H)$ -arrowing graph $F$ is said to be $(G,H)$ -minimal if there is no proper subgraph $F^{\\prime } \\subset F$ such that $F^{\\prime } \\rightarrow (G,H)$ .",
"In other words, $F$ is $(G,H)$ -minimal if it arrows $(G,H)$ and $F-e \\lnot \\rightarrow (G,H)$ for every $e \\in E(F)$ .",
"The collection of all $(G,H)$ -minimal graphs is denoted as $\\mathcal {R}(G,H)$ , and it satisfies the symmetric property $\\mathcal {R}(G,H)=\\mathcal {R}(H,G)$ .",
"The problem involving $(G,H)$ -minimal graphs is classically done for finite $G$ and $H$ , as introduced in [5].",
"One of the major problems that arose was determining whether $\\mathcal {R}(G,H)$ is finite or infinite.",
"Following the studies done by Beardon [1] on magic labelings, Cáceres et al.",
"[6] on metric dimensions, and Stein [13] on extremal graph theory, we attempt to extend this finite problem to an infinite one.",
"To our knowledge, this is the first serious attempt to do so.",
"It appears that some properties which are expected to be true for finite graphs do not hold in the scope of infinite graphs.",
"For finite graphs $G$ and $H$ , it is known that $\\mathcal {R}(G,H)$ is nonempty.",
"This is because we can obtain a $(G,H)$ -minimal graph from an arbitrary $(G,H)$ -arrowing graph by iteratively deleting enough edges.",
"However, if one of $G$ or $H$ is infinite, then $\\mathcal {R}(G,H)$ might be empty.",
"As we shall see in Example , the ray $P_\\infty $ and $K_2$ as a pair do not admit any minimal graph.",
"If we consider the double ray $P_{2\\infty }$ instead of $P_\\infty $ , we have that $\\mathcal {R}(P_{2\\infty },K_2)=\\lbrace P_{2\\infty }\\rbrace $ , and thus a minimal graph exists.",
"An intriguing but difficult problem in general would be to classify which pairs $(G,H)$ induce an empty (resp., nonempty) $\\mathcal {R}(G,H)$ .",
"For which pairs of graphs $(G,H)$ is $\\mathcal {R}(G,H)$ empty?",
"The study of Ramsey-minimal properties of infinite graphs is naturally related to graphs which are isomorphic to some proper subgraph of themselves.",
"We will call such graphs self-embeddable.",
"Note that if $F$ is a self-embeddable graph, then we can pick an $F^{\\prime } \\subset F$ isomorphic to $F$ .",
"Thus, if $F \\rightarrow (G,H)$ is self-embeddable, then it is not $(G,H)$ -minimal since we can choose a proper subgraph $F^{\\prime } \\rightarrow (G,H)$ .",
"A $(G,H)$ -arrowing graph $F$ that is self-embeddable cannot be minimal.",
"The notion of a self-embeddable graph differs from that of a self-contained graph [12], which is one isomorphic to a proper induced subgraph of itself.",
"While self-contained graphs have applications in other problems, such as the tree alternative conjecture [3], we do not require the proper subgraph to be induced in our case, hence the differing vocabulary.",
"The general outline of this paper is as follows.",
"In Section , we present the common notation and conventions that we use for this paper.",
"We then give some compactness results for Ramsey-minimal graphs in Section .",
"In Section , we obtain some general progress on Problem .",
"In Section , we turn our attention to the case where $G$ is an infinite graph and $H$ is a matching $nK_2$ .",
"For an example of previous work on Ramsey-minimal finite graphs involving matchings, see Burr et al.",
"[4]."
],
[
"Preliminaries",
"In this paper, we exclusively work with simple graphs $G=(V(G),E(G))$ with no isolated vertices (i.e.",
"every vertex of $G$ is adjacent to another vertex).",
"Our graphs are taken to be countable (including finite), with the exception of the graphs of Section which may be uncountable.",
"Let $\\mathbb {N}=\\lbrace 1,2,\\ldots \\rbrace $ be the set of natural numbers.",
"For $n \\in \\mathbb {N}$ , $nG$ denotes the graph consisting of $n$ disjoint copies of $G$ .",
"We say that $H$ is a subgraph of $G$ (or simply $H \\subseteq G$ ) if $V(H) \\subseteq V(G)$ and $E(H) \\subseteq E(G)$ .",
"A subgraph $H$ of $G$ is proper, and written as $H \\subset G$ , if $E(H)$ is a proper subset of $E(G)$ .",
"Also, we say that $G$ (properly) contains $H$ , or that $H$ (properly) embeds into $G$ , if there is a (proper) subgraph $H^{\\prime }$ of $G$ such that $H^{\\prime } \\cong H$ .",
"Figure: A kk-ray P k∞ P_{k\\infty }.The ray $P_\\infty $ is an infinite graph of the form $(\\lbrace x_0,x_1,\\ldots \\rbrace ,\\lbrace x_0x_1,x_1x_2,\\ldots \\rbrace )$ , where $x_0$ is its endpoint.",
"The double ray $P_{2\\infty }$ , on the other hand, is of the form $(\\lbrace x_n: n \\in \\mathbb {Z}\\rbrace ,\\lbrace x_nx_{n+1}:n \\in \\mathbb {Z}\\rbrace )$ .",
"In general, a $k$ -ray $P_{k\\infty }$ (shown in Figure REF ), $k \\ge 1$ , is formed by identifying the endpoints of $k$ distinct rays.",
"A family of graphs of particular interest is the family of comb graphs.",
"Let $\\ell \\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ be a function.",
"The comb $C^\\ell $ is a graph obtained from a base ray $P_\\infty $ (called the spine) by attaching, for every $n$ , a path $P_{\\ell (n)}$ of order $\\ell (n)$ by one of its endpoints to the vertex $x_n$ of $P_\\infty $ .",
"Other infinite graphs of interest include the (countably) infinite complete graph $K_\\infty $ and the (countably) infinite star $S_\\infty =K_{1,\\infty }$ .",
"We use some terminologies of embeddings from [2], [9], [10].",
"Recall that a graph homomorphism $G \\rightarrow H$ is a map $\\varphi \\colon V(G) \\rightarrow V(H)$ such that if $vw \\in E(G)$ , then $\\varphi (v)\\varphi (w) \\in E(H)$ .",
"If $\\varphi $ is injective, then the homomorphism is called an embedding $G \\hookrightarrow H$ .",
"An embedding $G \\hookrightarrow G$ is said to be a self-embedding of $G$ .",
"A self-embedding is nontrivial if its image, seen as a graph with vertex set $\\varphi (V(G))$ and edge set $\\lbrace \\varphi (v)\\varphi (w):vw \\in E(G)\\rbrace $ , is a proper subgraph of $G$ .",
"A graph $G$ is said to be self-embeddable if it has a nontrivial self-embedding.",
"In other words, a self-embeddable graph is a graph that properly embeds into itself.",
"We say that $G$ is strongly self-embeddable if it admits an embedding into $G-v$ for every $v \\in V(G)$ .",
"A strongly self-embeddable graph is clearly self-embeddable, but the converse does not hold in general, as shown in the following example.",
"The infinite star $S_\\infty $ is self-embeddable but not strongly so (since $S_\\infty $ does not embed into the null graph $S_\\infty -c$ , where $c$ is its center vertex).",
"Another example can be found in the graph $G$ of Figure REF .",
"It is self-embeddable, with a right translation as its nontrivial self-embedding.",
"However, $G$ does not embed into the disconnected graph $G-v$ , where $v$ is the vertex indicated in the figure.",
"Figure: A self-embeddable graph GG that is not strongly self-embeddable.",
"The red subgraph illustrates the image of a nontrivial self-embedding of GG in the form of a right translation by 1.By induction, we easily obtain the following stronger properties for strongly self-embeddable graphs.",
"Let $G$ be strongly self-embeddable.",
"Then: $G-V$ contains $G$ for every finite $V \\subset V(G)$ ; $G-E$ contains $G$ for every finite $E \\subset E(G)$ .",
"(i) Suppose that $V=\\lbrace v_1,\\ldots ,v_n\\rbrace $ and that $H=G-\\lbrace v_1,\\ldots ,v_{n-1}\\rbrace $ contains $G$ .",
"In other words, there is a subgraph $G^{\\prime } \\subseteq H$ isomorphic to $G$ .",
"Since $G^{\\prime }$ and $G$ are isomorphic, $G^{\\prime }$ is strongly self-embeddable.",
"Thus, $G^{\\prime }-v_n$ must contain $G^{\\prime } \\cong G$ .",
"Since $G^{\\prime }-v_n \\subseteq H-v_n = G-V$ , we can conclude that $G-v$ contains $G$ .",
"(ii) Suppose that $e=uv$ .",
"Since $G-v$ contains a copy of $G$ by hypothesis, we have that $G-e \\supseteq G-v$ contains $G$ as well.",
"The statement for any $G-E$ then follows by induction."
],
[
"Compactness results",
"The aim of this section is, for infinite (possibly uncountable) graphs $F$ , $G$ , $H$ , to express what $F \\rightarrow (G,H)$ means in terms of their finite subgraphs $\\hat{F}$ , $\\hat{G}$ , $\\hat{H}$ .",
"Theorem confirms that all $(G,H)$ -minimal graphs are finite if $G$ and $H$ are finite.",
"This result assures us that we only need to deal with the case where one of $G$ and $H$ is infinite, since taking both as finite graphs would produce a completely finite problem.",
"We prove the theorem by topological means using Tychonoff's theorem.",
"Recall that, for a family of topological spaces $S_i$ , $i \\in I$ , the product topology on $\\prod S_i$ is generated by basic open sets of the form $\\prod U_i$ , where each $U_i$ is open in $S_i$ , and $U_i=S_i$ except for finitely many values of $i$ .",
"Tychonoff's theorem states that whenever each $S_i$ is compact, $\\prod S_i$ is also compact.",
"Let $F$ be a graph, and $G$ and $H$ be finite graphs.",
"If $F \\rightarrow (G,H)$ , then there is a finite $\\hat{F} \\subseteq F$ such that $\\hat{F} \\rightarrow (G,H)$ .",
"Let $X$ be the product of $|E(F)|$ -many copies of the discrete space $S = \\lbrace \\text{red},\\text{blue} \\rbrace $ , equipped with the product topology.",
"We can identify $X$ with the set of all functions $E(F) \\rightarrow S$ , i.e.",
"the set of all red-blue colorings of $F$ 's edges.",
"By Tychonoff's theorem, $X$ is compact.",
"By assumption, $F \\rightarrow (G,H)$ , so for every coloring $c \\in X$ , we can pick a finite set of edges $D_c$ forming either a red $G$ or a blue $H$ .",
"Then, each $c|_{D_c}$ determines a basic open set $O_c:=\\lbrace d\\colon E(F) \\rightarrow S: d|_{D_c}=c|_{D_c}\\rbrace .$ Since $c \\in O_c$ for all $c \\in X$ , the collection $\\lbrace O_c: c \\in X \\rbrace $ covers $X$ .",
"By compactness, there is a finite sequence $c_1, \\ldots , c_n \\in X$ so that $O_{c_1} \\cup \\cdots \\cup O_{c_n} = X$ .",
"Let $\\hat{F}$ be the subgraph of $F$ induced by $D_{c_1} \\cup \\cdots \\cup D_{c_n}$ .",
"We claim $\\hat{F} \\rightarrow (G,H)$ .",
"Pick a coloring $\\hat{d}\\colon E(\\hat{F}) \\rightarrow S$ , and extend it arbitrarily to a coloring $d\\colon E(F) \\rightarrow S$ .",
"Then, there is an $i \\le n$ such that $d \\in O_{c_i}$ , so $D_{c_i} \\subseteq E(\\hat{F})$ either forms a red $G$ or a blue $H$ under $\\hat{d}$ , as required.",
"Now, we want to be able to characterize embeddability of a graph $G$ into another graph $F$ in terms of embeddability of finite subgraphs $\\hat{G} \\subseteq G$ into $F$ .",
"We might want to prove something such as: $G$ embeds into $F$ if and only if every finite subgraph $\\hat{G} \\subseteq G$ embeds into $F$ .",
"Figure: Every finite subgraph G ^⊆G\\hat{G} \\subseteq G embeds into FF, but GG itself does not.However, this statement is not true; a counterexample is shown in Figure REF .",
"The problem is that the embeddings $\\hat{G} \\hookrightarrow F$ are incompatible in some sense—larger finite subgraphs $\\hat{G} \\subseteq G$ must be embedded further down $F$ , and so there is no way to “stitch together” these embeddings to get an embedding $G \\hookrightarrow F$ .",
"To ensure compatibility between partial embeddings, we instead work with the following notion of a pointed graph.",
"We also need to ensure that our graphs are locally finite: that is, $\\deg (v)<\\infty $ for each vertex $v$ .",
"A pointed graph is a triple $G = (V,E,*)$ , where $(V,E)$ is a graph, and $* \\in V$ is a specified vertex of $G$ , called the basepoint of $G$ .",
"A pointed subgraph $H$ is a subgraph of $G$ such that $*_H=*_G$ .",
"A pointed homomorphism is a graph homomorphism mapping basepoints to basepoints—we call it a pointed embedding if it is injective.",
"A pointed graph is locally finite, connected, etc.",
"if the underlying graph is.",
"Using pointed graphs, we can obtain a version of our desired result.",
"We first provide the following form of Kőnig's infinity lemma to pave way for the compactness argument used in the proof of Proposition .",
"[[7]] Let $V_0, V_1, \\ldots $ be an infinite sequence of disjoint nonempty finite sets, and let $K$ be a graph on their union.",
"Assume that every vertex in $V_{n+1}$ has a neighbor in $V_n$ .",
"Then, $K$ contains a ray $v_0v_1\\ldots $ such that $v_n \\in V_n$ for all $n$ .",
"Let $F$ and $G$ be locally finite, pointed graphs, and suppose $G$ is connected.",
"If every connected, finite, pointed subgraph $\\hat{G}$ of $G$ admits a pointed embedding into $F$ , then $G$ admits a pointed embedding into $F$ .",
"Since $G$ is locally finite and connected, it must be countable (see Exercise 1, Chapter 8 of [7]).",
"The lemma clearly holds if $G$ is finite, so assume that $G$ is countably infinite.",
"Enumerate $V(G)$ as follows: let $v_0 = *$ , let $v_1, \\ldots , v_k$ be the neighbors of $*$ , let $v_{k+1}, \\ldots , v_\\ell $ be the vertices of distance 2 to $*$ , etc.",
"This indeed enumerates $G$ by the assumption that $G$ is locally finite and connected.",
"Then, the induced subgraphs $\\hat{G}_n:=G[v_0, \\ldots , v_n]$ are all connected, finite, pointed subgraphs of $G$ .",
"For each $n$ , let $V_n$ be the set of pointed embeddings $\\hat{G}_n \\hookrightarrow F$ .",
"Each $V_n$ is nonempty by assumption.",
"Inductively, we show each $V_n$ is finite.",
"We see that $V_0$ is finite, since there is a unique embedding $\\hat{G}_0 \\hookrightarrow F$ mapping $*_G$ to $*_F$ .",
"Now assume $V_n$ is finite, and pick $f \\in V_n$ .",
"Since $\\hat{G}_{n+1}$ is connected, $v_{n+1}$ is adjacent to some $v_j$ for $j \\le n$ .",
"Since $F$ is locally finite, $f(v_j)$ has finite degree, so there are only finitely many ways to define $f(v_{n+1})$ and extend $f$ to an embedding in $V_{n+1}$ .",
"Since there are also only finitely many ways to choose $f \\in V_n$ , it follows that $V_{n+1}$ is finite, and so all the $V_n$ are finite by induction.",
"Similarly to before, let $K$ be the graph on $\\bigcup _{n=0}^\\infty V_n$ , where we insert all edges between $f \\in V_{n+1}$ and $f|_{\\hat{G}_n} \\in V_n$ .",
"By Lemma , $K$ contains an infinite ray $f_0f_1\\ldots $ such that $f_n \\in V_n$ for all $n$ .",
"Define $f\\colon V(G) \\rightarrow V(F)$ by $f(v_n) = f_n(v_n)$ .",
"We claim $f$ is a pointed embedding $G \\hookrightarrow F$ .",
"$f$ is a graph homomorphism: suppose $v_nv_m \\in E(G)$ , where $n \\le m$ .",
"We have $f(v_n) = f_n(v_n) = f_m(v_n)$ and $f(v_m) = f_m(v_m)$ .",
"Since $f_m$ is a graph homomorphism, $f(v_n)f(v_m)=f_m(v_n)f_m(v_m) \\in E(F)$ .",
"$f$ is injective: suppose $f(v_n) = f(v_m)$ for $n \\le m$ .",
"Then, $f_m(v_n) = f_n(v_n) = f_m(v_m) \\Rightarrow v_n = v_m$ since $f_m$ is injective.",
"$f$ is pointed: $f(*_G) = f_0(*_G) = *_F$ since $f_0$ is pointed.",
"Interestingly, Proposition actually generalizes Kőnig's lemma, in its more standard, graph-theoretical form (as found in [7]).",
"[Kőnig's lemma] Every locally finite, connected, infinite graph $F$ contains a ray.",
"Pick an arbitrary basepoint $* \\in F$ .",
"For every $n$ , we claim that $P_n$ (with $*_{P_n}$ chosen as an endpoint) admits a pointed embedding into $F$ .",
"If $P_n$ does not admit a pointed embedding into $F$ , then there is no vertex $v \\in V(F)$ such that $d(*_F,v) \\ge n$ .",
"Since $F$ is connected and locally finite, we can enumerate $V(F)$ as follows: let $v_0 = *$ , let $v_1, \\ldots , v_k$ be the neighbors of $*$ , let $v_{k+1}, \\ldots , v_\\ell $ be the vertices of distance 2 to $*$ , etc.",
"By stage $n$ , we will have enumerated all of $F$ , hence $F$ is finite; contradiction.",
"The result now follows from Proposition , with $G = P_\\infty $ and $*_{P_\\infty }$ as its endpoint.",
"For pointed graphs $F$ , $G$ , and a non-pointed graph $H$ , we write $F \\xrightarrow{}(G,H)$ if for every red-blue coloring of the edges of $F$ , there exists either a red $G$ as a pointed subgraph of $F$ or a blue $H$ in the underlying graph of $F$ .",
"This definition gives a stronger condition for $F$ than $F \\rightarrow (G,H)$ , and, unlike regular arrowing, $F \\xrightarrow{}(G,H)$ and $F \\xrightarrow{}(H,G)$ are not necessarily equivalent.",
"We also note that $F\\xrightarrow{}(G,H)$ only if $H \\subseteq K_{1,\\deg (*_F)}$ .",
"Let $F$ , $G$ be locally finite, pointed graphs and let $H$ be a graph.",
"Suppose $G$ is connected and for every connected, finite, pointed $\\hat{G} \\subseteq G$ , we have $F \\xrightarrow{}(\\hat{G},H)$ .",
"Then, $F \\xrightarrow{}(G,H)$ .",
"Take an arbitrary red-blue coloring of $F$ such that $F$ does not contain a blue $H$ .",
"Denote $F^{\\prime }$ as the pointed subgraph of $F$ induced by all the red edges.",
"By assumption, all connected, finite, pointed $\\hat{G} \\subseteq G$ admits a pointed embedding into $F^{\\prime }$ .",
"By Proposition , $G$ admits a pointed embedding into $F^{\\prime }$ , so a red $G$ exists as a pointed subgraph of $F$ .",
"Proposition is precisely Theorem for the case $H=K_2$ .",
"Thus, Theorem generalizes both Proposition and Kőnig's lemma."
],
[
"General progress on Problem ",
"Throughout this section and the next, we fix a pair of (potentially infinite) graphs $G$ and $H$ .",
"We provide a sufficient condition under which $\\mathcal {R}(G,H)$ is empty by first finding some suitable family of graphs $\\mathcal {F}$ for the pair $(G,H)$ .",
"Suppose that $\\mathcal {F}$ is a (possibly infinite) collection of graphs such that: $F \\rightarrow (G,H)$ for every $F \\in \\mathcal {F}$ ; Every $(G,H)$ -arrowing graph $\\Gamma $ contains some graph $F \\in \\mathcal {F}$ .",
"We have the following: $\\mathcal {R}(G,H) \\subseteq \\lbrace F \\in \\mathcal {F}: \\text{$F$ is not self-embeddable}\\rbrace $ ; If every $F \\in \\mathcal {F}$ is self-embeddable, then $\\mathcal {R}(G,H)$ is empty.",
"(i) Fix a $(G,H)$ -minimal graph $\\Gamma $ .",
"By condition (2), there is an $F \\in \\mathcal {F}$ such that $F$ is contained in $\\Gamma $ .",
"Since $F \\rightarrow (G,H)$ by condition (1), we must have $F=\\Gamma $ as $\\Gamma $ is $(G,H)$ -minimal.",
"Therefore, $\\Gamma \\in \\mathcal {F}$ .",
"By Observation , $\\Gamma $ is not self-embeddable, and we are done.",
"(ii) follows directly from (i).",
"Conditions (1) and (2) are not sufficient to ensure that $\\mathcal {R}(G,H)$ and $\\lbrace F \\in \\mathcal {F}: \\text{$F$ is not self-embeddable}\\rbrace $ coincide.",
"For example, take $G=P_\\infty $ , $H=K_2$ and $\\mathcal {F}=\\lbrace P_\\infty ,P_{2\\infty }\\rbrace $ .",
"While conditions (1) and (2) hold, $\\mathcal {R}(P_\\infty ,K_2)$ can be shown to be empty while $P_{2\\infty }$ is not self-embeddable.",
"Hence, $\\mathcal {R}(G,H) \\subset \\lbrace F \\in \\mathcal {F}: \\text{$F$ is not self-embeddable}\\rbrace $ .",
"That said, we can create an extra condition to make both sets equal.",
"Let $\\mathcal {F}$ be a collection of graphs such that conditions (1) and (2) of Theorem hold.",
"Suppose that we also have the following condition: $F_1$ and $F_2$ do not contain each other for every different $F_1,F_2 \\in \\mathcal {F}$ .",
"We have the following: $\\mathcal {R}(G,H)=\\lbrace F \\in \\mathcal {F}: \\text{$F$ is not self-embeddable}\\rbrace $ ; $\\mathcal {R}(G,H)$ is empty if and only if every $F \\in \\mathcal {F}$ is self-embeddable.",
"(i) We prove that $\\lbrace F \\in \\mathcal {F}: \\text{$F$ is not self-embeddable}\\rbrace \\subseteq \\mathcal {R}(G,H)$ .",
"Suppose $F \\in \\mathcal {F}$ is not self-embeddable.",
"Since $F \\rightarrow (G,H)$ by condition (1), it remains to show that no proper $F^{\\prime } \\subset F$ arrows $(G,H)$ .",
"If we assume the contrary, then we have an $F^{\\prime \\prime } \\in \\mathcal {F}$ contained in $F^{\\prime }$ by condition (2).",
"This implies that $F$ contains $F^{\\prime \\prime }$ and contradicts condition (3).",
"(ii) follows easily from (i).",
"The preceding theorems have a few applications.",
"For example, in order to prove Theorem of the next section, we will need to use Theorem (ii).",
"Also, we can consider the special case where $\\mathcal {F}$ is chosen as $\\lbrace G\\rbrace $ .",
"This yields the following results: We have the following: $\\mathcal {R}(G,K_2)$ is empty if and only if $G$ is self-embeddable.",
"If it is nonempty, then $\\mathcal {R}(G,K_2)=\\lbrace G\\rbrace $ ; If $H \\ne K_2$ , then $G \\rightarrow (G,H)$ implies that $\\mathcal {R}(G,H)$ is empty.",
"(i) Take $\\mathcal {F}=\\lbrace G\\rbrace $ .",
"Conditions (1)–(3) of Theorems and all hold when $H=K_2$ .",
"The statement directly follows from Theorem .",
"(ii) Again, take $\\mathcal {F}=\\lbrace G\\rbrace $ .",
"Conditions (1) and (2) of Theorem are both satisfied.",
"Now we just need to show that $G$ is self-embeddable.",
"Color an arbitrary edge of $G$ blue and the rest of the edges red.",
"Since $H \\ne K_2$ , there is no blue $H$ in $G$ .",
"So by the fact that $G \\rightarrow (G,H)$ , there is a red copy of $G$ in $G$ .",
"This proves that $G$ properly contains itself, and thus self-embeddable.",
"The following examples demonstrate some direct applications of Theorem for some pairs of graphs: $\\mathcal {R}(P_{k\\infty },K_2)$ is empty if and only if $k=1$ .",
"When $k>1$ , we have $\\mathcal {R}(P_{k\\infty },K_2)=\\lbrace P_{k\\infty }\\rbrace $ .",
"These observations are obtained directly from Theorem (i).",
"By Theorem (ii), $\\mathcal {R}(S_\\infty ,K_{1,n})$ is empty for all $2 \\le n \\le \\infty $ since $S_\\infty \\rightarrow (S_\\infty ,K_{1,n})$ .",
"$\\mathcal {R}(K_\\infty ,H)$ is empty for all graphs $H$ .",
"This follows from Theorem since $K_\\infty $ is self-embeddable and $K_\\infty \\rightarrow (K_\\infty ,H)$ for all $H$ by the infinite Ramsey theorem."
],
[
"Some results involving matchings",
"We saw in Theorem (i) an answer to Problem whenever $H=K_2$ .",
"Now, let us consider the more general case where $H=nK_2$ .",
"It becomes apparent that the characteristics of $\\mathcal {R}(G,nK_2)$ , $n \\ge 2$ , are still related to whether $G$ is (strongly) self-embeddable.",
"We have the following: If $G$ is connected and not self-embeddable, then for all $n \\ge 2$ , we have $nG \\in \\mathcal {R}(G,nK_2)$ so that $\\mathcal {R}(G,nK_2)$ is nonempty; If $G$ is strongly self-embeddable, then $\\mathcal {R}(G,nK_2)$ is empty for all $n \\ge 2$ .",
"(i) Fix a red-blue coloring of $nG$ which does not create a blue $nK_2$ .",
"It follows that there must be a component of $nG$ isomorphic to $G$ which is colored all red.",
"Hence, $nG \\rightarrow (G,nK_2)$ .",
"Now let $e \\in E(nG)$ be an arbitrary edge located in some component $G^{\\prime }$ of $nG$ .",
"We show that $nG-e \\lnot \\rightarrow (G,nK_2)$ by constructing a $(G,nK_2)$ -good coloring of $nG-e$ as follows: for every component of $nG$ other than $G^{\\prime }$ , color one of its edges blue; color the rest of the edges red.",
"Since $G$ is connected, there are exactly $n-1$ components in $nG$ other than $G^{\\prime }$ , so this coloring only manages to produce a blue $(n-1)K_2$ .",
"Also, since $G$ is not self-embeddable, there cannot be a red $G$ in any of the components of $nG$ .",
"By the connectivity of $G$ , there cannot be a red $G$ in all of $nG$ either.",
"Therefore, this coloring is indeed $(G,nK_2)$ -good.",
"(ii) By appealing to Theorem (ii), it suffices to prove that $G \\rightarrow (G,nK_2)$ .",
"Fix a red-blue coloring of $G$ which does not create a blue $nK_2$ .",
"We claim that this coloring creates a red $G$ .",
"We construct a set of vertices $V \\subset V(G)$ using the following algorithm: initialize $V=\\emptyset $ while $G-V$ contains a blue edge do choose a blue edge $e=uv$ in $G-V$ $V \\leftarrow V \\cup \\lbrace u,v\\rbrace $ output $V$ This algorithm must terminate after at most $n-1$ while loop iterations since the edge chosen at each iteration must be independent from the edges chosen at previous iterations.",
"It is then clear that the output $V$ is finite and that $G-V$ only contains red edges.",
"By Proposition (i), we have a copy of $G$ in $G-V$ .",
"It follows that there exists a red copy of $G$ in $G$ , and we are done.",
"We note that Theorem (i) can fail to hold if $G$ is disconnected.",
"For example, given $G=2P_{2\\infty }$ , it can be shown that $(n+1)P_{2\\infty } \\rightarrow (G,nK_2)$ .",
"This implies that $(2n)P_{2\\infty }$ is not minimal for $n \\ge 2$ , so $\\mathcal {R}(2P_{2\\infty },nK_2)$ cannot be shown to be nonempty using the previous line of reasoning.",
"Observe that $P_\\infty $ is strongly self-embeddable, while $P_{k\\infty }$ is connected and not self-embeddable for $k>1$ .",
"By Theorem , we have for every $n \\ge 2$ , $\\mathcal {R}(P_{k\\infty },nK_2)$ is empty if and only if $k=1$ .",
"We note that the converse of Theorem (ii) does not necessarily hold.",
"There indeed exists a graph $G=S_\\infty $ not strongly self-embeddable such that $\\mathcal {R}(G,nK_2)$ is empty for all $n$ .",
"For all $n \\ge 1$ , $\\mathcal {R}(S_\\infty ,nK_2)$ is empty.",
"We prove Theorem by first defining a collection of graphs $\\mathcal {F}_n$ such that conditions (1) and (2) of Theorem hold.",
"For every $n \\ge 1$ , define $\\mathcal {F}_n$ to be the collection of all graphs $F$ satisfying the following two conditions: $F$ contains exactly $n$ vertices of infinite degree forming the set $X=\\lbrace x_1,\\ldots ,x_n\\rbrace $ ; If $uv \\in E(F)$ , then at least one of $u$ , $v$ is an element of $X$ .",
"Then, $\\mathcal {F}_n$ satisfies conditions (1) and (2) of Theorem , with $G=S_\\infty $ and $H=nK_2$ .",
"To prove condition (1), we show that $F \\rightarrow (S_\\infty ,nK_2)$ for all $F \\in \\mathcal {F}_n$ by induction on $n$ .",
"The base case $n=1$ follows from the fact that $\\mathcal {F}_1=\\lbrace S_\\infty \\rbrace $ and $S_\\infty \\rightarrow (S_\\infty ,K_2)$ .",
"Now assume that every graph in $\\mathcal {F}_n$ arrows $(S_\\infty ,nK_2)$ , and let $F \\in \\mathcal {F}_{n+1}$ be arbitrary.",
"Suppose that $c$ is a red-blue coloring of $F$ which produces no red $S_\\infty $ .",
"Pick an arbitrary vertex $u$ adjacent to $x_{n+1} \\in X$ such that $u \\notin X$ and the edge $x_{n+1}u$ is colored blue.",
"This can be done since otherwise, $x_{n+1}$ is incident to infinitely many red edges.",
"Observe that $F^{\\prime }:=F-\\lbrace x_{n+1},u\\rbrace $ is an element of $\\mathcal {F}_n$ , so it contains a blue $nK_2$ with respect to the coloring $c|_{F^{\\prime }}$ .",
"Since this blue $nK_2$ and the blue $x_{n+1}u$ form an independent set of edges, there exists a blue $(n+1)K_2$ in $F$ with respect to $c$ .",
"This proves that $F \\rightarrow (S_\\infty ,(n+1)K_2)$ and completes the induction.",
"Now we prove condition (2).",
"Suppose that $\\Gamma $ arrows $(S_\\infty ,nK_2)$ .",
"We claim that $\\Gamma $ contains at least $n$ vertices of infinite degree.",
"Assume that $Y$ , where $|Y|<n$ , is the set of vertices of $\\Gamma $ having an infinite degree.",
"By coloring all edges incident to a vertex in $Y$ blue and the rest of the edges red, we obtain a $(S_\\infty ,nK_2)$ -good coloring of $\\Gamma $ ; contradiction.",
"Now, we can take arbitrary vertices $x_1,\\ldots ,x_n$ of $\\Gamma $ of infinite degree.",
"The subgraph of $\\Gamma $ induced by all edges that are incident to at least one of $x_1,\\ldots ,x_n$ is an element of $\\mathcal {F}_n$ , therefore condition (2) holds.",
"[Proof of Theorem ] Let $n \\ge 1$ .",
"Take $\\mathcal {F}_n$ as defined in Lemma .",
"Invoking Theorem (ii), we need to show that every $F \\in \\mathcal {F}_n$ is self-embeddable.",
"Let $F \\in \\mathcal {F}_n$ be arbitrary.",
"We aim to define a nontrivial self-embedding $\\varphi $ of $F$ .",
"Denote $U$ as the complement of $X$ (i.e.",
"$U:=V(F)\\setminus X$ ), and for all $1 \\le i \\le n$ , let $\\rho _i\\colon U \\rightarrow \\lbrace 0,1\\rbrace $ be such that $\\rho _i(u)=1$ iff $u$ is adjacent to $x_i$ .",
"Define $\\rho (u)=(\\rho _1(u),\\ldots ,\\rho _n(u))$ for every $u \\in U$ .",
"Since the image of $\\rho $ is finite, then by the infinite pigeonhole principle, there exists an $x \\in \\lbrace 0,1\\rbrace ^n$ such that $\\rho ^{-1}(x)$ is an infinite set, say $\\rho ^{-1}(x)=\\lbrace u_1,u_2,\\ldots \\rbrace $ .",
"Define $\\varphi \\colon V(F) \\rightarrow V(F)$ as $\\varphi (v)={\\left\\lbrace \\begin{array}{ll}u_{i+1}, & v=u_i,\\ i \\ge 1, \\\\ v, & \\text{otherwise}.\\end{array}\\right.",
"}$ It is clear that $\\varphi $ is injective, and it is nontrivial since $u_1 \\notin \\varphi (V(F))$ .",
"It remains to show that $\\varphi $ is a graph homomorphism.",
"Suppose that $uv \\in E(F)$ .",
"By condition (b), we can assume without loss of generality that $v \\in X$ , say $v=x_j$ for some $1 \\le j \\le n$ .",
"Since $v \\notin U$ , we must have $v \\notin \\rho ^{-1}(x)$ , so $\\varphi (v)=v$ .",
"If $u \\notin \\rho ^{-1}(x)$ , then $\\varphi (u)\\varphi (v) = uv \\in E(F)$ , and we are done.",
"So assume that $u=u_i$ for some $i \\ge 1$ .",
"Since $uv=u_ix_j$ is an edge, we have that $\\rho _j(u_i)=1$ .",
"Thus, we also have $\\rho _j(u_{i+1})=1$ since $\\rho (u_{i+1})=x=\\rho (u_i)$ .",
"It follows that $\\varphi (u)\\varphi (v)=\\varphi (u_i)\\varphi (x_j)=u_{i+1}x_j \\in E(F).$ The proof that $\\varphi $ is a nontrivial self-embedding is then complete.",
"Now, let $C^\\ell $ be a comb with ray $x_0x_1\\ldots $ as its spine and a path $P_\\ell (n)$ of order $\\ell (n)$ attached to every $x_n$ .",
"Suppose that the value of $s_{C^\\ell }:=\\min _{\\ell (n)>1} n$ , which is equal to the smallest natural number $n$ such that $\\deg (x_n)=3$ , exists (that is, $C^\\ell $ is not a ray).",
"We can assume that our combs satisfy $s_{C^\\ell } \\ge \\ell (s_{C^\\ell })-1$ without loss of generality.",
"Indeed, if $s=s_{C^\\ell }$ is such that $s<\\ell (s)-1$ , then we can define a function $\\ell ^*(n)={\\left\\lbrace \\begin{array}{ll}1, & 1 \\le n < \\ell (s)-1 \\\\ s+1, & n=\\ell (s)-1, \\\\ \\ell (n-\\ell (s)+s+1), & n \\ge \\ell (s),\\end{array}\\right.",
"}$ so that $C^{\\ell ^*}\\cong C^\\ell $ and $s^*=s_{C^{\\ell ^*}}$ satisfies $s^* \\ge \\ell ^*(s^*)-1$ .",
"Here, $C^{\\ell ^*}$ is basically the comb obtained from $C^\\ell $ by exchanging the positions of $x_0\\ldots x_s$ and $P_{\\ell (s)}$ in the comb.",
"The following theorem gives equivalent formulations to the statement that $\\mathcal {R}(C^\\ell ,nK_2)$ is empty for all $n \\ge 2$ .",
"In particular, the theorem gives an answer for Problem whenever $G$ is a comb and $H$ is a matching.",
"Let $C^\\ell $ be a comb which is not a ray with $x_0x_1\\ldots $ as its spine.",
"Suppose that the value of $s=\\min _{\\ell (n)>1} n$ is at least $\\ell (s)-1$ .",
"The following statements are equivalent: $\\mathcal {R}(C^\\ell ,nK_2)$ is empty for all $n \\ge 2$ ; $\\mathcal {R}(C^\\ell ,nK_2)$ is empty for some $n \\ge 2$ ; $C^\\ell $ is self-embeddable; $C^\\ell $ is strongly self-embeddable; There exists a $p \\ge 1$ such that $\\ell (n) \\le \\ell (n+p)$ for all $n \\in \\mathbb {N}$ .",
"Implication $(1 \\rightarrow 2)$ is trivial, while implications $(2 \\rightarrow 3)$ and $(4 \\rightarrow 1)$ are both consequences of Theorem .",
"So we just need to prove $(5 \\rightarrow 4)$ and $(3 \\rightarrow 5)$ .",
"$5 \\rightarrow 4$ : Let $v \\in V(C^\\ell )$ .",
"If $v=x_0$ , then by positively translating $C^\\ell $ by $p$ (so that $x_n \\mapsto x_{n+p}$ and $P_{\\ell (n)}$ maps into $P_{\\ell (n+p)}$ ), we obtain an embedding $C^\\ell \\hookrightarrow C^\\ell -v$ .",
"Otherwise, there exists a $k \\in \\mathbb {N}$ such that $v$ is located in the path $P_{\\ell (k)}$ attached to $x_k$ .",
"In this case, positively translate $C^\\ell $ by $ap$ , where $a$ is chosen such that $ap>k$ , to obtain the desired embedding into $C^\\ell -v$ .",
"$3 \\rightarrow 5$ : Let $\\varphi \\colon V(C^\\ell ) \\rightarrow V(C^\\ell )$ be a nontrivial self-embedding given by the assumption that $C^\\ell $ is self-embeddable.",
"We cannot have $\\varphi (x_0)=x_0$ , since that would mean $\\varphi $ is the identity map, hence not nontrivial.",
"Thus, $\\varphi (x_0)$ is located in the path $P_{\\ell (k)}$ , which is attached to $x_k$ , for some $k \\in \\mathbb {N}$ .",
"Suppose that $P_{\\ell (k)}:=x_ky_1\\ldots y_{\\ell (k)-1}$ .",
"If $\\varphi (x_0)=x_k$ , then $\\varphi $ must be a positive translation by $k$ .",
"Hence, statement 5 holds by taking $p=k$ since $\\varphi $ maps each $P_{\\ell (n)}$ into $P_{\\ell (n+k)}$ .",
"So assume that $\\varphi (x_0)=y_j$ for some $1 \\le j \\le \\ell (k)-1$ .",
"This has an implication that $\\ell (k)>1$ , and thus $s \\le k$ since $s=\\min _{\\ell (n)>1} n$ .",
"Observe that we cannot have $j>s$ since that would mean $\\varphi (x_s)=y_{j-s}$ has degree 2, which is less than $\\deg (x_s)=3$ ; contradiction.",
"In summary, we have the inequality $j \\le s \\le k$ .",
"We claim that $j<k$ .",
"Assume for the sake of contradiction that $j=s=k$ .",
"Since we have established that $j \\le \\ell (k)-1$ and $s \\ge \\ell (s)-1=\\ell (k)-1$ , we then have $\\ell (k)-1=j=s=k$ .",
"It follows that $\\varphi $ must be the map $\\varphi (v)={\\left\\lbrace \\begin{array}{ll}y_{j-n}, & v=x_n,\\ 0 \\le n \\le s-1, \\\\ x_{s-m}, & v=y_m,\\ 1 \\le m \\le \\ell (k)-1, \\\\ v, & \\text{otherwise},\\end{array}\\right.",
"}$ which is not nontrivial; contradiction.",
"Now we can define $p=k-j \\ge 1$ .",
"We prove that $\\ell (n) \\le \\ell (n+p)$ for all $n \\ge s$ (the case where $n<s$ is trivial since then $\\ell (n)=1$ ).",
"Case 1.",
"$j=s$ .",
"We have previously established the inequalities $j \\le \\ell (k)-1$ and $s \\ge \\ell (s)-1$ .",
"We thus have $\\ell (s) \\le s+1 = j+1 \\le \\ell (k) = \\ell (s+p).$ In addition, $\\varphi $ necessarily maps each $P_{\\ell (n)}$ , $n>s$ , into $P_{\\ell (n+(k-s))}$ .",
"Hence, we also have $\\ell (n) \\le \\ell (n+p)$ for all $n>s$ .",
"Case 2.",
"$j<s$ .",
"We can see that $\\varphi $ necessarily maps each $P_{\\ell (n)}$ , $n \\ge s$ , into $P_{\\ell (n+(k-j))}$ .",
"It follows that $\\ell (n) \\le \\ell (n+p)$ for all $n \\ge s$ .",
"In both cases, we see that statement 5 holds for the chosen $p=k-j$ .",
"This completes the proof.",
"Figure: Two combs C ℓ C^\\ell such that (a) ℓ(n)=n\\ell (n)=n, and (b) ℓ(1)=3\\ell (1)=3 and ℓ(n)=2\\ell (n)=2 for n>1n>1.",
"For each of the two combs, the red subgraph illustrates the graph image of its nontrivial self-embedding.If $\\ell (n)=n$ , then $C^\\ell $ satisfies $s \\ge \\ell (s)-1$ (with $s=2$ ) as well as statement 5 of Theorem for any choice of $p \\ge 1$ .",
"As such, $C^\\ell $ is strongly self-embeddable via a positive translation.",
"Figure REF (a) shows a positive translation of $C^\\ell $ by 1.",
"In addition, we have that $\\mathcal {R}(C^\\ell ,nK_2)$ is empty for all $n \\ge 2$ by Theorem .",
"Suppose that $\\ell (1)=3$ and $\\ell (n)=2$ for $n>1$ .",
"We have $s<\\ell (s)-1$ (with $s=1$ ).",
"So we define, using formula (REF ) preceding Theorem , a function $\\ell ^*$ such that $\\ell ^*(n)={\\left\\lbrace \\begin{array}{ll}1, & n=1 \\\\ 2, & n>1.\\end{array}\\right.",
"}$ The comb $C^{\\ell ^*}$ is illustrated in Figure REF (b) as a red subgraph of $C^\\ell $ .",
"We see that $\\mathcal {R}(C^{\\ell ^*},nK_2)$ is always empty since $C^{\\ell ^*}$ satisfies statement 5 of Theorem .",
"By the fact that $C^\\ell \\cong C^{\\ell ^*}$ , we have that $\\mathcal {R}(C^{\\ell ^*},nK_2)$ is always empty as well.",
"If $\\ell (1)=2$ and $\\ell (n)=1$ for $n>1$ , then $s \\ge \\ell (s)-1$ (with $s=1$ ).",
"However, statement 5 does not hold since $\\ell (1)>\\ell (1+p)$ for all $p \\ge 1$ .",
"This implies that $C^\\ell $ is not self-embeddable and $\\mathcal {R}(C^\\ell ,nK_2)$ is nonempty for all $n \\ge 2$ .",
"From Theorem (i), we can infer that $nC^\\ell \\in \\mathcal {R}(C^\\ell ,nK_2)$ in this case."
],
[
"Concluding remarks",
"Problem , in its full generality, is quite a challenging problem to attack.",
"For this reason, we chose to devote a significant part of this study to the particular case where $H$ is a matching.",
"Even then, we were not able to completely answer Problem .",
"While Theorem managed to get us closer, we still have the following problem involving $\\mathcal {R}(G,nK_2)$ .",
"Is there a sufficient and necessary condition for $G$ under which $\\mathcal {R}(G,nK_2)$ is empty for all $n \\ge 2$ ?",
"Further studies can also be done on other specific cases of the pair $(G,H)$ .",
"Of course, another avenue of research would be to consider multi-color Ramsey-minimal infinite graphs, as done in [8] for finite graphs.",
"While the compactness results of Section are interesting in their own right, our only application of them was to eliminate consideration of the case where both graphs $G$ , $H$ are finite (Theorem ).",
"We believe that these results, and other compactness results, could prove extremely useful in future studies of Ramsey-type problems for infinite graphs.",
"Apart from Section 3, we have only considered countable graphs in this paper.",
"It would be interesting and worthwhile to study $\\mathcal {R}(G,H)$ when at least one of $G$ , $H$ is uncountable.",
"Presumably, this problem is significantly harder, and one would have to consider set-theoretic concerns."
],
[
"Acknowledgements",
"We would like to thank Fawwaz Fakhrurrozi Hadiputra for addressing Theorem in the beginning of our study, which inspired us to formulate Theorems and .",
"Also, we would like to thank the anonymous referees for their valuable feedback and comments on this paper."
]
] | 2011.14074 |
[
[
"HexGen and Hex2Spline: Polycube-based Hexahedral Mesh Generation and\n Spline Modeling for Isogeometric Analysis Applications in LS-DYNA"
],
[
"Abstract In this paper, we present two software packages, HexGen and Hex2Spline, that seamlessly integrate geometry design with isogeometric analysis (IGA) in LS-DYNA.",
"Given a boundary representation of a solid model, HexGen creates a hexahedral mesh by utilizing a semi-automatic polycube-based mesh generation method.",
"Hex2Spline takes the output hexahedral mesh from HexGen as the input control mesh and constructs volumetric truncated hierarchical splines.",
"Through B\\'{e}zier extraction, Hex2Spline transfers spline information to LS-DYNA and performs IGA therein.",
"We explain the underlying algorithms in each software package and use a rod model to explain how to run the software.",
"We also apply our software to several other complex models to test its robustness.",
"Our goal is to provide a robust volumetric modeling tool and thus expand the boundary of IGA to volume-based industrial applications."
],
[
"Introduction",
"Isogeometric analysis (IGA) [18] is a computational technique that integrates computer aided design (CAD) with simulation methods such as finite element analysis (FEA).",
"It adopts the idea of design-through-analysis and enables direct analysis of the designed geometry.",
"IGA has many advantages over traditional FEA such as exact, smooth geometric representation and superior numerical performance.",
"Many software packages have been developed for IGA and there are mainly two directions.",
"The first direction is to incorporate IGA with commercial finite element software.",
"For example, the user subroutine UEL in Abaqus is used to define IGA elements and perform IGA in Abaqus [20], [21].",
"The second direction is to develop open software packages.",
"GeoPDEs [7] and igatools [27] work on NURBS (Non-Uniform Rational B-Spline) patches and provide a general framework to implement IGA methods.",
"PetIGA [6] is another framework for IGA based on PETSc [3].",
"These software packages help boost the use of IGA in engineering applications.",
"However, these packages are analysis-oriented.",
"Currently, there is no available toolkit from the geometric modeling side, especially for volume parameterization.",
"Therefore, the motivation of our work is to develop a geometric modeling tool to bridge the gap between geometric design and IGA analysis.",
"There are two major challenges in volume parameterization, control mesh generation and volumetric spline construction.",
"A control mesh is generally an unstructured hexahedral (hex) mesh.",
"Various strategies have been proposed in the literature [42] for unstructured hex mesh generation, such as grid-based or octree-based [32], [33], medial surface [29], [28], plastering [4], [36], whisker weaving [10] and vector field-based methods [26].",
"These methods have created hex meshes for certain geometries, but are not robust and reliable for arbitrary geometries.",
"The polycube-based method [37], [14] is another appealing approach for all-hex meshing.",
"A smooth harmonic field [39] was used to generate polycubes for arbitrary genus geometries.",
"Boolean operations [23] were introduced to cope with arbitrary genus geometries.",
"In [24], polycube structure was generated based on the skeleton branches of a geometric model.",
"For these methods, the hex mesh quality is directly affected by the polycube structure and mapping distortion.",
"Computing the polycube structure with a low-distortion mapping remains an open problem for arbitrary geometries.",
"It is essential to improve the mesh quality for analysis by using methods such as pillowing, smoothing and optimization [31], [43], [44], [30].",
"Pillowing is a sheet insertion technique that eliminates the situations where two neighboring hex elements share more than one face.",
"Smoothing and optimization are used to further improve mesh quality by relocating vertices.",
"In our software, we implement all the above mentioned methods for quality improvement.",
"The second ingredient in volume parameterization is volumetric spline construction.",
"Several algorithms have been developed.",
"The initial development of IGA was based on NURBS.",
"Since it adopts a global tensor-product structure, it does not support local refinement.",
"T-splines were initially developed to support local refinement for surfaces [35], [34].",
"For solid models, the rational T-spline basis functions were used to convert unstructured hex meshes to solid T-splines [40].",
"Boolean operations [23] and skeletons [24] are used to create hex meshes, which are later converted to T-meshes.",
"However, local refinement using T-splines requires extensive mesh manipulation to satisfy desired properties such as linear independence.",
"Hierarchical spline is an alternative to T-spline to avoid this issue.",
"Several techniques were then developed based on hierarchical B-splines (HB-splines) [11], [38], such as truncated hierarchical B-splines (THB-splines) [12], [13].",
"In this paper, we integrate our semi-automatic polycube-based mesh generation with the volumetric truncated hierarchical spline construction (TH-spline3D) [41] to perform IGA on volumetric models in LS-DYNA.",
"The developed software packages feature: 1) semi-automatic polycube-based all-hex mesh generation from a CAD model; 2) TH-spline3D construction on hex meshes; and 3) Bézier extraction for LS-DYNA.",
"We first overview the entire pipeline and explain the algorithm behind each module of the pipeline.",
"We then provide various examples to explain how to run the software package.",
"The main objective of the software package is to make our pipeline accessible to industrial and academic communities who are interested in real-world engineering applications.",
"Our software favors versatility over efficiency.",
"We will use a concrete example to go through all the steps in running the software.",
"In particular, when user intervention is needed, we will explain details of the involved manual work.",
"The paper is outlined as follows.",
"In Section 2, we overview the pipeline.",
"In Section 3, we present the HexGen software package that conducts semi-automatic polycube-based all-hex mesh generation from a CAD file.",
"In Section 4, we talk about Hex2Spline that constructs TH-spline3D on hex meshes and performs Bézier extraction for IGA in LS-DYNA.",
"Finally, in Section 5, we demonstrate several complex models using our software package."
],
[
"Pipeline design",
"Our pipeline incorporates two software packages to bridge the gap between the input CAD model with IGA in LS-DYNA, as shown in Fig.",
"REF .",
"We first use the HexGen software package to build an all-hex mesh for the CAD model.",
"With a high quality all-hex mesh generated, we then use the Hex2Spline software package to construct TH-spline3D and extract Bézier information for LS-DYNA.",
"As shown in Fig.",
"REF , we first generate a triangle mesh from the CAD model by using the free software LS-PrePost, which is a pre and post-processor for LS-DYNA.",
"Then we use centroidal Voronoi tessellation (CVT) segmentation [17] to create a polycube structure [37], which is used to generate all-hex meshes via parametric mapping [9] and octree subdivision [44].",
"The quality of the all-hex mesh is evaluated to ensure that the resulting volumetric spline model can be used in IGA.",
"In case that a poor quality hex mesh is generated, the program has several quality improvement functions, including pillowing [43], smoothing, and optimization [30].",
"Each quality improvement function can be run independently and one can use these functions to improve the mesh quality.",
"Once a good quality hex mesh is obtained, one can run the Hex2Spline program to build volumetric splines.",
"In particular, TH-spline3D is built on the unstructured hex mesh and it also supports local refinement.",
"The Hex2Spline can output the Bézier extraction information of TH-spline3D in a format that can be imported into LS-DYNA to perform IGA.",
"Currently, our software only has a command-line interface (CLI).",
"Users need to specify necessary options via the command line to run the software.",
"In Sections and , we will explain the algorithms implemented in each software as well as how to run the software in detail."
],
[
"HexGen: Polycube-based hex mesh generation",
"Surface segmentation, polycube construction, parametric mapping, and octree subdivision are used together in the HexGen software package to construct an all-hex mesh from the boundary representation given by the input CAD model.",
"Given an triangle mesh generated from the CAD model, we first use surface segmentation to divide the mesh into several surface patches that satisfy the polycube structure constraints, which will be discussed in Section REF .",
"Then, the corner vertices, edges and face information of each surface patch are extracted from the surface segmentation result to construct a polycube structure.",
"Each component of the polycube structure is topologically equivalent to a cube.",
"Finally, we generate the all-hex mesh through parametric mapping and octree subdivision.",
"Quality improvement techniques can be used to further improve the mesh quality.",
"In this section, we introduce the main algorithm for each module of the HexGen software package, namely surface segmentation, polycube construction, parametric mapping and octree subdivision, and quality improvement.",
"We use a rod model (see Fig.",
"REF ) to explain how to run CLI for each module.",
"We also discuss the user intervention that is involved in the semi-automatic polycube-based hex mesh generation."
],
[
"Surface segmentation",
"The surface segmentation in the pipeline framework is implemented based on CVT segmentation [17].",
"CVT segmentation is used to classify vertices into different groups by minimizing an energy function.",
"Each group is called a Voronoi region $\\lbrace {V_j}\\rbrace $ and it has a corresponding center called a generator $\\lbrace {g_j}\\rbrace $ .",
"The Voronoi region and its corresponding generator are updated iteratively in the minimization process.",
"In [17], each element of the surface triangle mesh is assigned to one of the six Voronoi regions $\\lbrace {V_j}\\rbrace _{j = 1}^6$ based on the normal vector ${\\varkappa _{_{\\mathcal {T}(i)}}}$ of the surface, where ${{\\mathcal {T}(i)}}$ is the $i^{th}$ element of the surface triangle mesh ${{\\mathcal {T}}}$ .",
"The initial generators of the Voronoi regions are the three principal normal vectors and their opposite normals vectors ($ \\pm X$ , $ \\pm Y$ , $ \\pm Z$ ).",
"Two energy functions and their corresponding distance functions are used together in [17].",
"The classical energy function and its corresponding distance function provide initial Voronoi regions and generators.",
"Then the harmonic boundary-enhanced (HBE) energy function and its corresponding distance function are applied to eliminate non-monotone boundaries.",
"The detailed definitions of energy functions and their corresponding distance functions are described in [17].",
"Here, we summarize the surface segmentation process inalg:1.",
"[H] Surface Segmentation Algorithm [1] Manifold triangular surface mesh $\\mathcal {T}$ , weighting factor $\\omega $ Manifold triangular surface mesh including segmentation information Calculate the unit normal ${\\varkappa _{_{\\mathcal {T}(i)}}}$ of the triangle mesh Use six principal axes ($ \\pm X$ , $ \\pm Y$ , $ \\pm Z$ ) as the initial generators $\\lbrace {g_j}\\rbrace _{j = 1}^6$ classical energy not converge Associate ${\\varkappa _{_{\\mathcal {T}(i)}}}$ with $\\lbrace {g_j}\\rbrace _{j = 1}^6$ by using classical distance functions each unit normal ${\\varkappa _{_{\\mathcal {T}(i)}}}$ in group $\\lbrace {V_j}\\rbrace $ Update generators $ {g_j }$ based on classical energy function Use CVT results as the input for HBE CVT HBE energy not converge Associate ${\\varkappa _{_{\\mathcal {T}(i)}}}$ with $\\lbrace {g_j}\\rbrace _{j = 1}^6$ by using the HBE distance functions controlled by weighting factor $\\omega $ each unit normal ${\\varkappa _{_{_{\\mathcal {T}(i)}}}}$ in group $\\lbrace {V_j}\\rbrace $ Update generators ${g_j}$ based on HBE energy function Through the above pseudocode inalg:1, we describe two energy minimization processes, which are combined together to yield a monotone segmentation.",
"When we use the HBE distance function to define Voronoi regions, we use a weighting factor $\\omega $ to control the balance between the classical distance and the boundary-enhanced term (see Eq.",
"4 in [17]).",
"Based onalg:1, we implement and organize the code into a CLI program (Segmentation.exe), which can segment a given triangle mesh into 6 Voronoi regions.",
"Users can give options through the command line to run Segmentation.exe.",
"Taking the rod model as an example, we first generate a triangle mesh from its CAD model by using LS-PrePost.",
"Then we segment the triangle mesh by running the following command: Segmentation.exe -i rod_tri.k -o rod_initial_write.k -m rod_manual.txt -l 0.1 There are four options used in the command: -i: Surface triangle mesh of the input geometry (rod_tri.k); -o: Output segmentation result (rod_initial_write.k); -m: Input file with user intervention (rod_manual.txt); and -l: Weighting factor $\\omega $ used in HBE distance function.",
"The input and output are .k files, which can be read by LS-PrePost.",
"We refer readers to [25], which explains the file format.",
"We use -l to control the balance between the distance and the boundary-enhanced term.",
"The weighting factor $\\omega $ can be assigned any arbitrary positive value; however, to obtain the best segmentation behavior, $\\omega $ must take small value.",
"We find that when $\\omega =0.1$ , the segmentation result of the rod model has fewer zig-zags and outliers.",
"Users need to do a trial and error to obtain a good weighting factor.",
"Note that zig-zags and outliers may still exist regardless of the choice of $\\omega $ .",
"To fix this issue, user intervention is needed to prepare a file that stores the correct segmentation result for such elements.",
"Segmentation.exe can read this file through option -m to improve the segmentation result.",
"The snippets of the input text file for the rod model are given in Appendix A1.",
"Once we get the initial segmentation result, we need to further segment each Voronoi region into several patches to satisfy the topological constraints for polycube construction (see Fig.",
"REF (d)).",
"The following three conditions should be satisfied during the further segmentation: 1) two patches with opposite orientations (e.g., +X and -X) cannot share a boundary; 2) each corner vertex must be shared by more than two patches; and 3) each patch must have four boundaries.",
"Note that we define the corner vertex as a vertex locating at the corner of the cubic region in the model.",
"The further segmentation is done manually by using the patch ID reassigning function in LS-PrePost.",
"The detailed operation is shown in Appendix A2.",
"In addition to the issue with zig-zags and outliers, the algorithm has several limitations.",
"For example, alg:1 cannot guarantee a good quality polycube structure, which will affect the quality of hex mesh.",
"Elements with small or negative scaled Jacobian in a hex mesh may appear.",
"Some adjustments on the polycube structure and quality improvement are needed as a follow-up step."
],
[
"Polycube construction",
"In this section, we discuss the detailed algorithm of polycube construction using the segmented triangle mesh.",
"A polycube consisting of multiple cubes is topologically equivalent to the original geometry.",
"Several automatic polycube construction algorithms have been proposed in the literature [16], [22], [17], but it is challenging to generalize these methods to general CAD models.",
"To achieve versatility for real industrial applications, we develop a semi-automatic polycube construction software based on the segmented surface.",
"However, for some complex geometries, it may slow down the process because of the potentially heavy user intervention.",
"The key information we need for a polycube is its corners and the connectivity relationship among them.",
"For the surface of polycube, we can automatically get the corners and build their connectivity based on the segmentation result by using alg:2.",
"However, it is usually difficult to obtain inner vertices and their connectivity as we only have a surface input without any information about the interior volume.",
"Indeed, this is also the place that involves user intervention, where we use LS-PrePost to manually build the interior connectivity.",
"The detailed operation is shown in Appendix A3.",
"As the auxiliary information for this user intervention, alg:2 will output corners and connectivity of the segmented surface patches into .k file.",
"Finally, the generated polycube structure is the cubic regions splitting the volumetric domain of the geometry.",
"[H] Polycube Boundary Surface Construction Algorithm [1] Manifold triangular surface mesh including segmentation information The boundary surface of the polycube structure each vertex $v_i$ Get the number of patches $n$ which the vertex $v_i$ surrounding $\\mathbf {n}\\ge 3$ Mark the vertex as a corner $v_i^c$ Output file including corner coordinates each patches $\\lbrace {S_j}\\rbrace $ Find its four corners $v_i^c$ which define a quad $\\mathcal {Q}_j$ Extract edge information from $\\mathcal {Q}_j$ Output files of connectivity relationship including edges and faces We implement and organize the code into a CLI program (PolyCube.exe) based on alg:2.",
"For the rod model, we run the following command to extract the corners and their connectivity for the boundary surface of its polycube: PolyCube.exe -i rod_initial_read.k -o rod_polycube_structure.k -c 1 There are four options used in the command: -i: Surface triangle mesh with the segmentation information (rod_initial_read.k); -o: Polycube surface connectivity (rod_polycube_structure.k) for polycube construction in LS-PrePost; and -c: Control indicator if some additional file needs to be output -c 0: No output; and -c 1: Output corner points, edges, faces of polycube structure.",
"The output .k file contains the corners and their connectivity for the boundary surface of the polycube (see Fig.",
"REF (a)).",
"Users need to import it into LS-PrePost and manually create interior corners and corresponding connectivity (see Fig.",
"REF (b)) to build a polycube structure (see Fig.",
"REF (f)).",
"We also provide option -c to output the corners, edges, and faces of the polycube structure if users intend to use other software to build a polycube structure.",
"Users can find their file format in Appendix A4.",
"Figure: The polycube construction and the parametric mappingprocess.",
"(a) The boundary surface of the polycube generatedby alg:2; (b) The interior corners and connectivitymanually created in LS-PrePost to generate polycube structure.",
"Wesplit the polycube into multiple cubes and use each individual cubeas the parametric domain.",
"S 0 S_0, P 0 P_0 and U 0 U_0 are used forparametric mapping.",
"I 1 I_1 and U 1 U_1 are used for linearinterpolation to create interior vertices of the mesh."
],
[
"Parametric mapping and octree subdivision",
"After the polycube is constructed, we need to build a bijective mapping between the input triangle mesh and the boundary surface of the polycube structure.",
"In our software, we implement the same idea as in [24] to use a unit cube as the parametric domain for polycube structure.",
"As a result, we can construct a generalized polycube structure (see Fig.",
"REF (b)) that can align with the given geometry better and generate a high quality hex mesh.",
"Through the pseudocode in alg:3, we describe how the segmented surface mesh, the polycube structure and the unit cube are combined to create a (volume) parametric mapping and octree subdivision.",
"Let $\\lbrace S_i\\rbrace _{i=1}^N$ be the segmented surface patches coming from the segmentation result (see Fig.",
"REF (a)).",
"Each segmented surface patch corresponds to one boundary surface of the polycube $P_i$ $(1\\le i \\le N)$ (see Fig.",
"REF (b)), where $N$ is the number of the boundary surface.",
"There are also interior surfaces, denoted by $I_j$ $(1\\le j \\le M)$ , where $M$ is the number of the interior surface.",
"The union of $\\lbrace P_i\\rbrace _{i=1}^N$ and $\\lbrace I_j\\rbrace _{j=1}^M$ is the set of surfaces of the polycube structure.",
"For the parametric domain, let $\\lbrace U_k\\rbrace _{k=1}^6$ denote the six surface patches of the unit cube (see Fig.",
"REF (b)).",
"Each cubic region in the polycube structure represents one volumetric region of the geometry and has a unit cube as its parametric domain.",
"Fig.",
"REF (b) shows the example of one cubic region and its corresponding volume domain of the geometry marked in the dashed rectangle.",
"Therefore, for each cube in the polycube structure, we can find its boundary surface $P_i$ and map the segmented surface patch $S_i$ to its corresponding parametric surface $U_k$ of the unit cube.",
"To map $S_i$ to $U_k$ , we first map its corresponding boundary edges of $S_i$ to the boundary edges of $U_k$ .",
"Then we get the parameterization of $S_i$ by using the cotangent Laplace operator to compute the harmonic function [44], [8].",
"Note that for an interior surface $I_j$ of the polycube structure, we skip the parametric mapping step.",
"An all-hex mesh can be obtained from this surface parameterization combined with the octree subdivision.",
"We generate the hex element for each cubic region in the following process.",
"To obtain vertex coordinates on the segmented patch $S_i$ , we first subdivide the unit cube (see Fig.",
"REF (b)) recursively to get their parametric coordinates.",
"The physical coordinates can be obtained by using the parametric mapping, which has a one-to-one correspondence between the parametric domain $U_k$ and the physical domain $S_i$ .",
"To obtain the vertices on the interior surface of the cubic region, we skip the parametric mapping step and directly use the linear interpolation to calculate the physical coordinates.",
"Fig.",
"REF shows the example of the rod model.",
"A composition of mappings among $S_0$ , $P_0$ and $U_0$ is done to build parametric mapping and obtain vertex coordinates on the surface $S_0$ .",
"$I_1$ and $U_1$ are combined for linear interpolation to obtain the vertices on the interior surface of the cubic region.",
"Finally, the vertices inside the cubic region are calculated by linear interpolation.",
"The entire all-hex mesh is built by going through all the cubic regions.",
"Based on alg:3, we implement and organize the code into a CLI program ( ParametricMapping.exe) that can generate an all-hex mesh by combining parametric mapping with the octree subdivision.",
"Here, we run the following command to generate an all-hex mesh for the rod model: ParametricMapping.exe -i rod_indexPatch_read.k -p rod_polycube_structure.k -o rod_hex.vtk -s 2 There are three options used in the command: -i: Surface triangle mesh of the input geometry with segmentation information (rod_indexPatch_read.k); -o: Unstructured hex mesh (rod_hex.vtk); -p: Polycube structure (rod_polycube_structure.k); and -s: Octree level.",
"We use -i to set the segmentation file generated in Section REF and use -p to set the polycube structure created in Section REF .",
"Option -s is used to set the level of recursive subdivision to be applied.",
"There is no subdivision if we set -s to be 0.",
"In the rod model, we set -s to be 2 to create a level-2 all-hex mesh.",
"The output all-hex mesh is stored in the VTK format (see Fig.",
"REF (g)) and it can be visualized in Paraview [1].",
"[H] Parametric Mapping Algorithm [1] Segmented triangle mesh $\\mathcal {T}=\\lbrace S_i\\rbrace _{i=1}^N$ , polycube structure All-hex mesh Find boundary surfaces $\\lbrace P_i\\rbrace _{i=1}^N$ and interior surfaces $\\lbrace I_j\\rbrace _{j=1}^M$ in the polycube structure each cube in the polycube structure Create a unit cube $\\lbrace U_k\\rbrace _{k=1}^6$ as the parametric domain each surface in the cube it is a boundary surface $P_i$ Get the surface parameterization $f: S_i \\rightarrow U_k \\subset \\mathbb {R}^2$ each cube in the polycube structure Subdivide the unit cube $\\lbrace U_k\\rbrace _{k=1}^6$ recursively to get parametric coordinates $v_{_{para}}$ each surface in the cube it is a boundary surface $P_i$ Obtain physical coordinates using $f^{-1}(v_{_{para}})$ it is an interior surface $I_j$ Obtain physical coordinates using linear interpolation Obtain interior vertices in the cubic region using linear interpolation Combine hex elements from each cubic region"
],
[
"Quality improvement",
"If the quality of the hex mesh is not satisfactory, quality improvement needs to be applied to the hex mesh.",
"We integrate three quality improvement techniques in the software package, namely pillowing, smoothing and optimization.",
"Users can improve mesh quality through the command line options before building volumetric splines.",
"We first use pillowing to insert one layer around the boundary [44].",
"By using the pillowing technique, we ensure that each element has at most one face on the boundary, which can help improve the mesh quality around the boundary.",
"After pillowing, smoothing and optimization [44] are used to further improve mesh quality.",
"For smoothing, different relocation methods are applied to three types of vertices: vertices on sharp edges on the boundary, vertices on the boundary surface, and interior vertices.",
"For each sharp-edge vertex, we first detect its two neighboring vertices on the curve, and then calculate their middle point.",
"For each vertex on the boundary surface, we calculate the area center of its neighboring boundary quadrilaterals (quads).",
"For each interior vertex, we calculate the weighted volume center of its neighboring hex elements as the new position.",
"We relocate a vertex in an iterative way.",
"Each time the vertex moves only a small step towards the new position and this movement is done only if the new location results in an improved local Jacobian.",
"If there are still poor quality elements after smoothing, we run the optimization whose objective function is the Jacobian.",
"Each vertex is then moved toward an optimal position that maximizes the worst Jacobian.",
"We present alg:4 for quality improvement.",
"Here, we show how to improve mesh quality for the rod model.",
"We first run the following command to perform pillowing on the rod model: Quality.exe -i rod_hex.vtk -Q -m 1 -n 1 -o rod_hex_pillow.vtk There are four options used in the command: -i: Unstructured hex mesh (rod_hex.vtk); -o: The hex mesh after quality improvement (rod_hex_pillow.vtk); -m: Improvement method.",
"Pillowing when -m 1; and -n: Number of pillowing layer.",
"Option -n allows users to specify the number of layers to be inserted.",
"With -n 1, we insert one layer around the boundary, which is enough to ensure each element have at most one face on the boundary.",
"The result is shown in Fig.",
"REF (a).",
"After pillowing, we can use the following command to smooth the mesh: Quality.exe -i rod_hex_pillow.vtk -Q -m 2 -p 0.001 -n 50 -s 2 -o rod_hex_pillow_lap.vtk There are seven options used in the command: -i: The input unstructured hex mesh in the vtk format (rod_hex_pillow.vtk); -o: The output hex mesh after quality improvement (rod_hex_pillow_lap.vtk); -m: Improvement method.",
"Smoothing when -m 2; and -s: Sharp feature preservation -s 0: No sharp features are preserved; -s 1: Detect sharp features automatically, and set tolerance -t; and -s 2: Manually select sharp feature points and store the indices in the \"sharp.txt\" file.",
"-t: Tolerance for automatically detecting sharp features; -p: Step size for smoothing; and -n: Number of steps for smoothing.",
"By using the above command, we relocate a vertex only if the new location will improve the local scaled Jacobian.",
"Option -s is used to preserve sharp features.",
"Here, sharp feature detection is only based on the mesh normal information.",
"Therefore, it is not robust for complex geometries and manual work is needed to adjust sharp features.",
"When we use automatic detection with option -s 1, we need to set a tolerance -t. There is no typical number for this option.",
"We need to do a trial and error to get the optimal value.",
"For the rod model, we set it to be 0.8.",
"However, some sharp features may not be detected regardless of the tolerance.",
"User intervention is needed with the option -s 2 if automatic detection is not satisfactory.",
"Through this command line option, Quality.exe will read an input file that includes the user-defined sharp features.",
"The snippets of the related file is shown in Appendix A5.",
"Then we run the optimization step by using the command: Quality.exe -i rod_hex_pillow_smooth.vtk -Q -m 3 -p 0.001 -n 15 -o rod_hex_pillow_smooth_opt.vtk There are five options used in the command: -i: Unstructured hex mesh (rod_hex_pillow_smooth.vtk); -o: The hex mesh after quality improvement (rod_hex_pillow_smooth_opt.vtk); -t: Tolerance related to sharp feature preservation; -m: Improvement method.",
"Optimization when -m 3; -p: Step size for optimization; and -n: Number of steps for optimization.",
"The quality improvement result for the rod model is shown in Fig.",
"REF with a boundary layer created using pillowing, followed up by smoothing and optimization.",
"Figure: Mesh quality improvement.",
"(a) The original mesh before quality improvement; (b) The mesh after pillowing, smoothing and optimization.",
"Part of the mesh is removed to show the interior.",
"The elements on the original cross section are labeled in green color while the inserted boundary layer is labeled in red color.Quality Improvement Algorithm [1] Hex mesh, step number $n$ Hex mesh with improved quality Insert an outer layer to the input mesh iteration < $n$ each vertex $v_i$ $v_i$ is on a sharp edge improving the local Jacobian Relocate $v_i$ a small step towards the middle of its neighboring vertices $v_i$ is on the boundary surface improving the local Jacobian Relocate $v_i$ a small step towards the the area center of its neighboring boundary quads $v_i$ is an interior vertex improving the local Jacobian Relocate $v_i$ a small step towards the weighted volume center of its neighboring elements iteration < $n$ each negative Jacobian element improving the local Jacobian Relocate $v_i$ to maximize the Jacobian Relocate vertices where Jacobian is minimum to maximize the worst Jacobian With the generated hex mesh as the input control mesh, now we present the Hex2Spline software package to build TH-spline3D.",
"TH-spline3D can define spline functions on arbitrarily unstructured hex meshes.",
"It further supports local refinement for adaptive IGA.",
"Hex2Spline can output the Bézier information of constructed volumetric splines, which can be easily used in LS-DYNA or any other existing IGA frameworks.",
"In the following, we introduce the main algorithm for each component of the Hex2Spline software package, including blending functions on an unstructured hex mesh, TH-spline3D with local refinement, and Bézier extraction."
],
[
"Blending functions on an unstructured hex mesh",
"In this section, we describe how to build blending functions on an all-hex mesh.",
"Hex2Spline supports arbitrarily unstructured all-hex mesh.",
"In the following, we denote $\\Omega _e$ as an hex element indexed by $e$ .",
"There are three types of elements in the hex mesh: boundary elements, interior regular elements and interior irregular elements.",
"The element is defined as a boundary element if it contains a boundary vertex; otherwise, it is an interior element.",
"For an interior element, if it contains an extraordinary edgeAn extraordinary edge is an interior edge shared by other than four hexahedral elements., we call it an irregular element; otherwise, it is a regular element.",
"In the following, we discuss the main algorithm for building blending functions on boundary elements and interior irregular elements.",
"A regular interior element is a special case of an irregular interior element whose blending functions are merely tricubic B-splines.",
"Note that the following construction only applies to tricubic splines with uniform knot intervals (i.e., the same knot interval for every edge).",
"alg:bzmat1 shows the pseudocode to obtain the blending functions defined on an interior irregular element.",
"They are obtained through the Bézier extraction matrix $\\mathbf {M}$ .",
"$\\mathbf {M}$ can be obtained by computing the 64 Bézier control points $\\mathbf {Q}_e$ from a local spline control mesh $\\mathcal {N}$ that consists of $\\Omega _e$ and its one-ring neighborhood.",
"Each of these Bézier points is obtained by a convex combination of the vertices in the local control mesh, and we have $\\mathbf {Q}_e=\\mathbf {M}\\mathbf {P}$ , where $\\mathbf {P}$ is the vector of vertices in the local control mesh.",
"Then the blending functions on $\\Omega _e$ are defined by using the transpose of $\\mathbf {M}$ , that is, $\\mathbf {B}_e=\\mathbf {M}^T\\mathbf {b}$ , where $\\begin{aligned}\\mathbf {b}=&[N_{0}(u)N_{0}(v)N_{0}(w),\\ldots ,N_{0}(u)N_{1}(v)N_{0}(w),N_{1}(u)N_{1}(v)N_{0}(w),\\\\&\\ldots ,N_{0}(u)N_{0}(v)N_{1}(w),N_{1}(u)N_{0}(v)N_{1}(w),\\ldots ,N_{3}(u)N_{3}(v)N_{3}(w)]^T,\\end{aligned}$ is the vector of 64 tricubic Bernstein polynomials.",
"Each univariate cubic Bernstein polynomial is given as $N_k(t)=\\dbinom{3}{k}(1-t)^{3-k}t^k$ ($k=0,\\ldots ,3$ ).",
"Readers can refer to [41] for the coefficients of $\\mathbf {M}$ .",
"Note that we only need $\\mathbf {M}$ to define $\\mathbf {B}_e$ , and we do not actually compute Bézier points.",
"[H] Blending Functions Algorithm (Interior) [1] An interior element $\\Omega _e$ and its local control mesh $\\mathcal {N}$ The blending functions $\\mathbf {B}_e$ each Bézier point $Q_{e,i}$ of $\\Omega _e$ , where $i=0,\\ldots ,63$ $Q_{e,i}$ is body point Compute its coordinates based on $\\mathbf {P}$ $Q_{e,i}$ is face, edge, corner point Compute its coordinates by averaging the nearest body points Ensemble matrix $\\mathbf {M}$ such that $\\mathbf {Q}_e=\\mathbf {M}\\mathbf {P}$ $\\mathbf {B}_e=\\mathbf {M}^T\\mathbf {b}$ alg:bzmat2 shows the pseudocode to define blending functions on a boundary element.",
"The body Bézier points and the Bézier points on the interior corners, edges or faces are obtained the same way as in alg:bzmat1, while the Bézier points on the boundary is defined using only the boundary quadrilateral mesh.",
"The 16 Bézier points on a boundary face can be obtained by convex combinations of the vertices on the local quad control mesh.",
"The detailed computation method is explained in [41].",
"We finally get all the Bézier points as $\\mathbf {Q}_e=\\mathbf {M}\\mathbf {P}$ .",
"The blending functions are then defined by $\\mathbf {B}_e=\\mathbf {M}^T\\mathbf {b}$ .",
"Note that alg:bzmat2 can also be used to preserve sharp features in the mesh by adjusting the Bézier extraction matrix $\\mathbf {M}$ .",
"Reader can refer to [41] for details.",
"[H] Blending Functions Algorithm (Boundary) [1] A boundary element $\\Omega _e$ and its local control mesh $\\mathcal {N}$ ; sharp feature information The blending functions $\\mathbf {B}_e$ each Bézier point $Q_{e,i}$ of $\\Omega _e$ , where $i=0,\\ldots ,63$ $Q_{e,i}$ corresponds to a sharp corner $P_k$ $Q_{e,i}=P_k$ $Q_{e,i}$ is on a sharp edge Compute its coordinates as a convex combination of the two end points of the sharp edge $Q_{e,i}$ is on the boundary $Q_{e,i}$ is a face point Compute its coordinates based on $P_k$ $Q_{e,i}$ is an edge or a corner point Compute its coordinates by averaging the nearest boundary face points $Q_{e,i}$ is a body point or on an interior surface Calculate $Q_{e,i}$ the same as alg:bzmat1 Ensemble matrix $\\mathbf {M}$ such that $\\mathbf {Q}_e=\\mathbf {M}\\mathbf {P}$ $\\mathbf {B}_e=\\mathbf {M}^T\\mathbf {b}$"
],
[
"TH-spline3D for local refinement",
"The next step is to introduce local refinement to achieve computational efficiency and accuracy.",
"TH-spline3D employs a hierarchical structure and uses the truncation mechanism to perform local refinement.",
"Global refinement is also supported since it is a special case of local refinement, and it is done by Catmull-Clark subdivision for solids [2], [5].",
"alg:THSpline3D1 shows the pseudocode to construct TH-spline3D based on the blending functions developed in Section REF .",
"Locally refined meshes as well as spline functions between different levels are related through Catmull-Clark subdivision for solids.",
"The program allows users to specify a list of target elements to be locally refined.",
"This is enabled by reading a series of user-defined files to the program, each of which contains indices of target elements at a certain level.",
"The input mesh is treated as the level-0 mesh by default, one needs to provide a file named \"lev_rfid.txt\" to refine certain level-0 elements.",
"As a result, a level-1 mesh is generated that can be used to define multi-level local refinement.",
"One can check the hierarchical control meshes and add more elements in the \"lev_rfid.txt\" file to further refine the mesh.",
"The remaining procedure to construct TH-spline3D is automatic and can be divided into three steps: i) refine target elements by Catmull-Clark subdivision for solids, ii) select certain blending functions from hierarchical meshes, and iii) truncate some blending functions on the coarse mesh.",
"Then with the help of alg:bzmat1 and alg:bzmat2, we can construct TH-spline3D on hierarchical control meshes.",
"Readers can refer to [41] for details.",
"[H] TH-spline3D Algorithm [1] Initial control points $\\mathbf {P}^0$ and their associated blending functions $\\mathcal {B}^0$ , element $\\Omega _e$ and its local control mesh $\\mathcal {N}$ The truncated blending functions $\\mathcal {B}_{\\text{TH-spline3D}}$ local refinement is needed Generate mesh $\\mathcal {M}^{\\ell +1}$ from mesh $\\mathcal {M}^\\ell $ based on Catmull-Clark subdivision [2], [5] Ensemble matrix $\\mathbf {C}$ so that $\\mathbf {P}^{\\ell +1}=\\mathbf {C}\\mathbf {P}^{\\ell }$ Compute the Bézier control points on hierarchical structure $\\mathbf {Q}^{\\ell +1}=\\mathbf {M}\\mathbf {\\mathbf {P}^{\\ell +1}}$ Construct the hierarchical B-splines on the hierarchical control meshes by using alg:bzmat1 or alg:bzmat2 Select the blending function to be active Truncate chosen blending functions Collect all the blending functions ($\\mathcal {B}_{\\text{TH-spline3D}}^{\\ell +1}$ ) up to Level $\\ell +1$"
],
[
"Bézier extraction for LS-DYNA",
"After blending functions are defined based on Bernstein polynomials, the Bézier information of the constructed volumetric splines can be written in the BEXT file for LS-DYNA.",
"The program will also output files for the visualization of Bézier mesh in Paraview [1].",
"The BEXT file contains all the control points and the Bézier extraction matrix $\\mathbf {M}^{T}$ for each Bézier element.",
"We reduce the file size by using both sparse and dense formats to write $\\mathbf {M}^{T}$ .",
"The matrix is output row by row.",
"In a sparse format, only non-zeros of a row are output, where an index is paired with each non-zero coefficient to indicate its column location in the matrix.",
"On the other hand, an entire row is output in the dense format without additional column indices.",
"The choice of the two formats depend on the number of non-zeros in a row.",
"The sparse format is favored when the row only has a few non-zeros.",
"The snippets of the BEXT format file is shown in Appendix A6."
],
[
"Applying Hex2Spline to the rod model",
"Based on the above algorithms, we implement and organize the code into a CLI program (Hex2Spline.exe) that can construct TH-spline3D on an unstructured hex mesh and extract Bézier information for analysis.",
"During the spline construction, users can specify if refinement is needed.",
"In the end, Hex2Spline generates the BEXT file for LS-DYNA.",
"The program also supports sharp feature preservation.",
"All the available options for this program are explained as follows: -i: Unstructured hex mesh (rod_hex.vtk); -o: The BEXT file for LS-DYNA (rod_hex_BEXT.txt); -S: Spline construction mode; -s: Sharp feature preservation; -s 0: No sharp features need to be preserved; -s 1: Detect sharp feature automatically, and set tolerance -t; and -s 2: Manually select sharp feature points and store them in \"sharp.txt\".",
"-g: Set the level of global refinement; -l: Enable local refinement; and -t: Tolerance related to sharp feature preservation.",
"Here, we apply local refinement to create a hierarchical mesh of the rod model (see Fig.",
"REF (g)) by using the command: Hex2Spline.exe -i rod_hex.vtk -S -s 2 -l -o rod_hex.BEXT In the command, we use the option -l to switch on the local refinement mode and construct TH-spline3D with local refinement.",
"Unlike global refinement, users need to prepare a \"lev_rfid.txt\" file to specify indices of target elements.",
"Fig.",
"REF (b) shows the spline construction with one level local refinement.",
"Users can perform further refinements level by level.",
"For example, users can edit the \"lev_rfid.txt\" file to include more elements and use the same command to perform two levels of local refinement and the result is shown in Fig.",
"REF (c): Hex2Spline.exe -i rod_hex.vtk -S -s 2 -l 2 Figure: The visualization of the output Bézier mesh with: (a)global refinement; (b) one level local refinement; and (c) two levels oflocal refinement.Users can also use the following command to perform spline construction with one level global refinement and the result is shown in Fig.",
"REF (a): Hex2Spline.exe -i rod_hex.vtk -o rod_hex -S -s 2 -g 1 Here we use the option -g to switch on global refinement mode and set the argument to 1 to construct spline with one level global refinement."
],
[
"Applications using HexGen and Hex2Spline",
"The algorithms discussed in Sections and are implemented in C++.",
"The Eigen library [15] and Intel MKL [19] are used for matrix and vector operations and numerical linear algebra.",
"We also take advantage of openMP to support multi-threading computation.",
"We use a compiler-independent building system (CMake) and a version-control system (Git) to support software development.",
"We have compiled the source code into two software packages, Hex2Gen software package: Segmentation module (Segmentation.exe); Polycube construction module (Polycube.exe); All-hex mesh generation module (ParametricMapping.exe); and Quality improvement module (Quality.exe).",
"Hex2Spline software package: Volumetric spline construction module (Hex2Spline.exe).",
"The software is open-source and can be found in the following Github link (https://github.com/yu-yuxuan/HexGen_Hex2Spline).",
"We have applied the software packages to several models and generated all-hex meshes with good quality.",
"For each model, we show the HBECVT based segmentation result, further segmentation result, corresponding polycube structure, and the all-hex mesh.",
"These models include: two types of mount and hepta models (Fig.",
"REF ); engine mount and lower arm from Honda Co. along with rockerarm (Fig.",
"REF ); ant, bust, and fertility models (Fig.",
"REF ); and the joint model from Honda Co. (Fig.",
"REF ).",
"Table REF shows the statistics of all testing models.",
"We use the scaled Jacobian to evaluate the quality of all-hex meshes.",
"From Table REF , we can observe that the obtained all-hex meshes have good quality (minimal Jacobian $>$ 0.1).",
"Figs.",
"REF -REF (a) show HBECVT segmentation results of testing models, we can observe that the initial segmentation results generated by the HBECVT do not satisfy the topological constraints for polycube construction.",
"We need to further segment each Voronoi region into several patches.",
"The generated polycubes (Figs.",
"REF -REF (b)) align with the given geometry better, which in turn induces less mesh distortion and yields a mesh of better quality.",
"After generating all-hex meshes (Figs.",
"REF -REF (c)), we tested all the models for IGA by using TH-spline3D.",
"Bézier elements are extracted for the IGA analysis (Figs.",
"REF -REF (d)).",
"For each testing model in Figs.",
"REF -REF , we use LS-DYNA to perform eigenvalue analysis and show the first mode result.",
"For the testing model in Fig.",
"REF , we show the result of solving a Poisson problem in LS-DYNA.",
"From the results we can observe that our algorithm yields valid TH-spline3D for IGA applications in LS-DYNA.",
"Table: Statistics of all the tested models.Figure: Results of two types of mount and hepta models.",
"(a) Surfacetriangle meshes and segmentation results; (b) Polycube structures;(c) All-hex control meshes; (d) Volumetric splines with IGA results of eigenvalue analysis in LS-DYNA.Figure: Results of engine mount, lower arm, and rockerarm model.",
"(a) Surfacetriangle meshes and segmentation results; (b) Polycube structures;(c) All-hex control meshes; (d) Volumetric splines with IGA results of eigenvalue analysis in LS-DYNA.Figure: Results of ant, bust, and fertility model.",
"(a) Surfacetriangle meshes and segmentation results; (b) Polycube structures;(c) All-hex control meshes; (d) Volumetric splines with IGA results of eigenvalue analysis in LS-DYNA.Figure: Results of joint model.",
"(a) Surfacetriangle meshes and segmentation results; (b) Polycube structures;(c) All-hex control meshes; (d) Volumetric splines with IGA results of solving Poisson equation in LS-DYNA."
],
[
"Conclusion and future work",
"In this paper, we present two software packages (HexGen and Hex2Spline) for IGA applications in LS-DYNA.",
"The main goal of HexGen and Hex2Spline is to make our pipeline accessible to industrial and academic communities who are interested in real-world engineering applications.",
"The all-hex mesh generation program (HexGen) can generate all-hex meshes.",
"It consists of four executable files, namely segmentation module (Segmentation.exe), polycube construction module (Polycube.exe), all-hex mesh generation module (ParametricMapping.exe) and quality improvement module (Quality.exe).",
"The volumetric spline construction program (Hex2Spline.exe) is developed based on the spline construction method in [41].",
"Users can generate a volumetric spline model given any unstructured hex mesh and output a BEXT file to perform IGA in LS-DYNA.",
"Both programs are compiled in executable files and can be easily run in the Command Prompt (cmd) in platform.",
"The rod model is used to explain how to use these two programs in detail.",
"We also tested our software package using several other models.",
"In conclusion, we integrate our hex mesh generation and volumetric spline construction techniques and develop a software platform to create IGA models for LS-DYNA.",
"Our software also has limitations that we will address in our future work.",
"First, the hex mesh generation module is semi-automatic and needs user intervention to create polycube structure.",
"We will improve the underneath algorithm and make polycube construction more automatic.",
"In addition, our software cannot guarantee to generate good quality hex mesh for complex geometry.",
"Therefore, in the future we will expand our software package to use hex-dominant meshing methods to create hybrid meshes for IGA applications."
],
[
"Acknowledgment",
"Y. Yu, A. Li, J. Liu and Y. Zhang were supported in part by Honda funds.",
"X. Wei is partially supported by the ERC AdG project CHANGE n. 694515, as well as the Swiss National Science Foundation project HOGAEMS n.200021_188589.",
"We also acknowledge the open source scientific library Eigen and its developers.",
"The authors would like to thank Kenji Takada for providing the CAD geometries of engine mount, lower arm and joint.",
"The authors would also like to thank Attila P. Nagy and David J. Benson for various fruitful discussions about the commercial software LS-DYNA."
],
[
"Input text file to correct segmentation result from Segmentation.exe",
"In this section, we describe the data format of the input text file used in Segmentation.exe to correct the segmentation result.",
"One can prepare this file to move elements on the wrong patch to the desired patch.",
"In this text file, each row has two values to define this modification of one element (see List ).",
"The first value indicates the element index in the triangular mesh and the second value is the desired patch index.",
"Segmentation.exe can read this file through option -m to improve the segmentation result.",
"355 1 356 1 361 1 362 1 365 1 366 1 369 1 370 1 495 6 496 6 499 6 500 6"
],
[
"Further segmentation in LS-PrePost",
"In this section, we introduce how to perform further segmentation in LS-PrePost and obtain an admissible segmentation for polycube construction.",
"It mainly involves reassigning elements to different patches and there are four steps to achieve this (see Fig.",
"REF ): 1) click move/copy tab; 2) select elements; 3) reassign the patch ID; and 4) click the Apply button to finish.",
"Figure: The detailed operations for further segmentation."
],
[
"Building the interior connectivity of polycube structure in LS-PrePost",
"In this section, we use the rod model to introduce how to build the interior connectivity of polycube structure in LS-PrePost.",
"There are four steps to create one cubic region ( see Fig.",
"REF ): 1) click EleEdt tab; 2) select Elem Type as Hexa; 3) select eight nodes to define cubic regions, you can also check the selection on the float box; and 4) click the Accept button to finish.",
"By repeating the same operation, we generate the polycube structure with multiple cubic regions to split the volumetric domain of the geometry.",
"Figure: The detailed operations to build the interior connectivity of polycube structure."
],
[
"Output text file from Polycube.exe",
"In CLI program (PolyCube.exe), we output a .k file which contains the corners and their connectivity for the boundary surface of the polycube.",
"It can be directly opened by LS-PrePost.",
"If one intends to use other software to build a polycube structure, we also provide option to output the corners, edges, and faces of the polycube in three separate text files (see Lists -).",
"In the corner file, each row depicts the associated vertex ($v_i$ ) information.",
"The first value indicates the index of the vertex in the triangles mesh, the last three values are its x, y, z coordinates ($x_i, y_i, z_i$ ).",
"In the edge file, each row uses the indices of two corners to define the edge between them.",
"These indices should agree with the corner file.",
"The face file stores the information of boundary faces on polycube structure.",
"Each row contains four vertex indices in counter-clockwise order to define the connectivity of one face.",
"21 4.06622 0.0052 2.31336 143 16.4588 0.7317 2.29912 153 4.0543 3.3803 2.31406 371 16.4604 2.67402 4.12666 391 13.5206 0.65268 2.07416 71,446 436,446 436,442 371,442 1464,1639 1601,1639 71,446,436,442 1464,1639,1601,1503 1439,1664,1639,1464 1744,1917,1877,1784 1246,1784,1877,153"
],
[
"Sharp feature file for Quality.exe and Hex2Spline.exe",
"This section describes how to manually define sharp features for Quality.exe and Hex2Spline.exe.",
"One can prepare an input file including the sharp feature information with the help of Paraview.",
"There are four steps (see Fig.",
"REF ): 1) select the points along the sharp feature; 2) use Extract selection function to extract the indices of these nodes; 3) check the selected points information under \"Properties\"; and 4) copy the index to \"sharp.txt\" file and use it as the input sharp feature file for Quality.exe and Hex2Spline.exe.",
"Figure: The detailed operations to create user-defined sharp features."
],
[
"Input BEXT file for LS-DYNA",
"In this section, we describe the data format of BEXT file for IGA in LS-DYNA.",
"The BEXT file consists of two parts to store the spline information: 1) control point (Fig.",
"REF A); and 2) Bézier element including the indices of control points supported by this Bézier element (Fig.",
"REF B) and the Bézier extraction matrix (Fig.",
"REF C).",
"The Bézier extraction matrix is output row by row and the format of each row depends on the number of non-zero values in this row.",
"If this row has less than 20 non-zeros, a sparse format is used to store their column indices and values; otherwise, a dense format is used to store all values in this row.",
"To distinguish between two formats, a sparse row begins with s while a dense row begins with d. Figure: Snippets of the BEXT file."
]
] | 2011.14213 |
[
[
"New Galois Hulls of GRS Codes and Application to EAQECCs"
],
[
"Abstract Galois hulls of linear codes have important applications in quantum coding theory.",
"In this paper, we construct some new classes of (extended) generalized Reed-Solomon (GRS) codes with Galois hulls of arbitrary dimensions.",
"We also propose a general method on constructing GRS codes with Galois hulls of arbitrary dimensions from special Euclidean orthogonal GRS codes.",
"Finally, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and MDS EAQECCs by utilizing the above results."
],
[
"Introduction",
" Let $q=p^h$ , where $p$ is an odd prime.",
"Denote by $\\mathbb {F}_{q}$ the finite field with $q$ elements.",
"An $[n,k,d]_{q}$ code $\\mathcal {C}$ is a linear code over $\\mathbb {F}_{q}$ with length $n$ , dimension $k$ and minimum distance $d$ .",
"The Singleton bound states that $k\\le n-d+1$ .",
"If the parameters of $\\mathcal {C}$ reach the Singleton bound (i.e., $k=n-d+1$ ), then $\\mathcal {C}$ is called a maximum distance separable (MDS) code.",
"Due to the optimal properties, MDS codes play an important role in coding theory and related fields, see [REF , REF ].",
"Let $\\mathcal {C}^\\bot $ be the dual code of $\\mathcal {C}$ .",
"The hull of $\\mathcal {C}$ is defined by $Hull(\\mathcal {C})=\\mathcal {C}\\bigcap \\mathcal {C}^\\bot $ .",
"Two special cases are extremely interesting for researchers.",
"One is $Hull(\\mathcal {C})=\\lbrace \\mathbf {0}\\rbrace $ .",
"In this case, $\\mathcal {C}$ is called a linear complementary dual (LCD) code.",
"In [REF ], Carlet et al.",
"constructed LCD codes by utilizing cyclic codes, Reed-Solomon codes and generalized residue codes, together with direct sum, puncturing, shortening, extension, $(u|u+v)$ construction and suitable automorphism action.",
"In [REF ] and [REF ], Carlet et al.",
"showed that any linear code over $\\mathbb {F}_{q}$ $(q>3)$ is equivalent to a Euclidean LCD code and any linear code over $\\mathbb {F}_{q^{2}}$ $(q>2)$ is equivalent to a Hermitian LCD code.",
"The other case is $Hull(\\mathcal {C})=\\mathcal {C}$ (resp.",
"$\\mathcal {C}^\\bot $ ), in which $\\mathcal {C}$ is called a self-orthogonal (resp.",
"dual containing) code.",
"In particular, the code $\\mathcal {C}$ satisfying $\\mathcal {C}=\\mathcal {C}^\\bot $ is called a self-dual code.",
"The construction of MDS Euclidean self-dual code is a popular issue in recent years.",
"In [REF ], Grassl and Gulliver showed that the problem has been completely solved over finite fields of characteristic 2.",
"In [REF ] and [REF ], Jin, Xing and Yan constructed some classes of new MDS self-dual codes through (extended) GRS codes.",
"In 1995, the first quantum error-correcting code was constructed.",
"Shortly after that, Shor et al.",
"established a connection between quantum codes and classical error-correcting codes satisfying certain self-orthogonal or dual containing property in [REF ].",
"In [REF ], Brun et al.",
"introduced entanglement-assisted quantum error-correcting code (EAQECC), which did not require the dual-containing property for standard quantum error-correcting codes.",
"Thus we could construct EAQECCs via classical linear codes without self-orthogonality.",
"However, the determination of the number of shared pairs was not an easy thing.",
"In [REF ], Guenda et al.",
"showed that the number of shared pairs was related to the hull of classical linear code.",
"So the results on hulls of classical linear codes can be applied in the constructions of EAQECCs and some new families of EAQECCs were discovered through the hulls of classical linear codes.",
"Some people have studied the hulls of classical linear codes and constructed EAQECCs and MDS EAQECCs.",
"In [REF ] and [REF ], several infinite families of MDS codes with Euclidean hulls of arbitrary dimensions were presented.",
"Then they were applied to construct some families of MDS EAQECCs.",
"Also utilizing (extended) GRS codes, in [REF ], several new MDS codes with Euclidean or Hermitian hulls of arbitrary dimensions were proposed and these MDS codes were also applied in the constructions of MDS EAQECCs.",
"Recently, in [REF ], Cao gave several new families of MDS codes with Galois hulls of arbitrary dimensions and constructed nine new families of MDS EAQECCs.",
"In [REF ], Qian et al.",
"gave a general construction on MDS codes with Galois hulls of arbitrary dimensions.",
"Based on the above results, we construct some new MDS codes with Galois hulls of arbitrary dimensions.",
"Furthermore, we propose a mechanism on how to find MDS codes with Galois hulls of arbitrary dimensions from GRS codes with special Euclidean orthogonal property.",
"All the known results on the constructions of Galois hulls of MDS codes are listed in Table 1.",
"This paper is organized as follows.",
"In Section 2, we will introduce some basic knowledge and useful results on GRS codes and Galois hulls.",
"In Section 3, we construct some new (extended) GRS codes with Galois hulls of arbitrary dimensions.",
"In Section 4, we propose a mechanism for the constructions of some special cases.",
"In Section 5, we will present our main results on the constructions of new EAQECCs and MDS EAQECCs.",
"In Section 6, we will make a conclusion.",
"|c|c|c|c| Known constructions on Galois hulls of MDS codes $q$ $n$ $k$ Reference $q=p^h$ $n \\mid q-1$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h$ $n|(p^e-1)$ $1\\le k\\le \\lfloor \\frac{n}{2}\\rfloor $ [REF ] $q=p^h$ is odd, $2e\\mid h$ $n\\le p^e$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h$ is odd, $2e\\mid h$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h$ is odd, $2e\\mid h$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h$ is odd, $2e\\mid h$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h$ is odd, $2e\\mid h$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h$ is odd $2e\\mid h$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h$ is odd, $2e\\mid h$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q = p^h $ is even $n\\le q$ , $\\frac{m}{\\gcd (e,m)}$ and $m>1$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h>3$ $n\\le r$ , $r=p^m$ with $m\\mid h$ and $p^e+1\\mid \\frac{q-1}{r-1}$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q = p^h>3 $ $n\\mid q$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h>3$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h>3$ $(n-1)|(q -1)$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h>3$ $n=2n^{\\prime }$ and $n^{\\prime }|q$ $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^h>3$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ [REF ] $q=p^{h}>3$ $n\\mid (q-1)$ $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h$ $n=2n^{\\prime }$ and $n^{\\prime }|(q-1)$ $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ] $q=p^h>3$ Table: NO_CAPTION $1\\le k\\le \\lfloor \\frac{p^e+n}{p^e+1}\\rfloor $ [REF ]"
],
[
"Preliminaries",
" In this section, we introduce some basic notations and useful results on (extended) GRS codes and Galois hulls.",
"Readers are referred to [REF , Chapter 10] for more details on (extended) GRS codes.",
"Let $\\mathbb {F}_{q}$ be a finite field with $q$ elements.",
"Denote by $\\mathbb {F}_q^*=\\mathbb {F}_q\\backslash \\lbrace 0\\rbrace $ .",
"In this paper, we always assume $q=p^h$ , where $p$ is an odd prime.",
"For $1\\le n\\le q$ , choose two vectors $\\mathbf {v}=(v_{1},v_{2},\\ldots ,v_{n})\\in (\\mathbb {F}_{q}^{*})^{n}$ and $\\mathbf {a}=(a_{1},a_{2},\\ldots ,a_{n})\\in \\mathbb {F}_{q}^{n}$ , where $a_{i}(1\\le i\\le n)$ are distinct.",
"For an integer $k$ with $1\\le k\\le n$ , the GRS code of length $n$ associated with $\\mathbf {v}$ and $\\mathbf {a}$ is defined as follows: $\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v})=\\lbrace (v_{1}f(a_{1}),\\ldots ,v_{n}f(a_{n})):f(x)\\in \\mathbb {F}_{q}[x], \\mathrm {deg}(f(x))\\le k-1\\rbrace .$ The code $\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v})$ is a $q$ -ary $[n,k]$ MDS code and its dual is also MDS [REF , Chapter 11].",
"The extended GRS code associated with $\\mathbf {v}$ and $\\mathbf {a}$ is defined by: $\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v},\\infty )=\\lbrace (v_{1}f(a_{1}),\\ldots ,v_{n}f(a_{n}),f_{k-1}):f(x)\\in \\mathbb {F}_{q}[x],\\mathrm {deg}(f(x))\\le k-1\\rbrace ,$ where $f_{k-1}$ is the coefficient of $x^{k-1}$ in $f(x)$ .",
"The code $\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v},\\infty )$ is a $q$ -ary $[n+1,k]$ MDS code and its dual is also MDS [REF , Chapter 11].",
"For $1\\le i\\le n$ , we define $u_i:=\\prod _{1\\le j\\le n,j\\ne i}(a_{i}-a_{j})^{-1},$ which will be used frequently in this paper.",
"For any two vectors $\\mathbf {a}=(a_{1},a_{2},\\ldots ,a_{n})$ and $\\mathbf {b}=(b_{1},b_{2},\\ldots ,b_{n})$ in $\\mathbb {F}_{q}^{n}$ , we define their Euclidean inner product as: $\\langle \\mathbf {a},\\mathbf {b}\\rangle =\\sum \\limits _{i=1}^{n}a_{i}b_{i}.$ The Euclidean dual code of $\\mathcal {C}$ is defined as: $\\mathcal {C}^\\perp =\\left\\lbrace \\mathbf {a} \\in \\mathbb {F}_{q}^{n}:\\langle \\mathbf {a},\\mathbf {b}\\rangle =0 \\text{ for any }\\mathbf {b} \\in \\mathcal {C} \\right\\rbrace .$ The Euclidean hull of $\\mathcal {C}$ is defined by $Hull(\\mathcal {C})=\\mathcal {C}\\cap \\mathcal {C}^\\perp $ .",
"Similarly, if $h$ is even, for any two vectors $\\mathbf {a}=(a_{1},a_{2},\\ldots ,a_{n})$ and $\\mathbf {b}=(b_{1},b_{2},\\ldots ,b_{n})$ in $\\mathbb {F}_{q}^{n}$ , the Hermitian inner product is defined as: $\\langle \\mathbf {a},\\mathbf {b}\\rangle _{\\frac{h}{2}}=\\sum \\limits _{i=1}^{n}a_{i}b_{i}^{p^{\\frac{h}{2}}}.$ The Hermitian dual code of $\\mathcal {C}$ is defined as: $\\mathcal {C}^{\\perp _{\\frac{h}{2}}}=\\left\\lbrace \\mathbf {a} \\in \\mathbb {F}_{q}^{n}:\\langle \\mathbf {a},\\mathbf {b}\\rangle _{\\frac{h}{2}}=0 \\text{ for any }\\mathbf {b} \\in \\mathcal {C} \\right\\rbrace .$ The Hermitian hull of $\\mathcal {C}$ is defined by $Hull_{\\frac{h}{2}}(\\mathcal {C})=\\;\\mathcal {C}\\cap \\mathcal {C}^{\\perp _{\\frac{h}{2}}}.$ In [REF ], Fan et al.",
"gave the definition of $e$ -Galois inner product with $0\\le e\\le h-1$ .",
"It is a generalization of Euclidean inner product and Hermitian inner product.",
"For any two vectors $\\mathbf {a}=(a_{1},a_{2},\\ldots ,a_{n})$ and $\\mathbf {b}=(b_{1},b_{2},\\ldots ,b_{n})$ in $\\mathbb {F}_{q}^{n}$ , the $e$ -Galois inner product is defined as: $\\langle \\mathbf {a},\\mathbf {b}\\rangle _e=\\sum \\limits _{i=1}^{n}a_{i}b_{i}^{p^e}.$ The $e$ -Galois dual code of $\\mathcal {C}$ is defined as: $\\mathcal {C}^{\\perp _e}=\\left\\lbrace \\mathbf {a} \\in \\mathbb {F}_{q}^{n}:\\langle \\mathbf {a},\\mathbf {b}\\rangle _e=0 \\text{ for any }\\mathbf {b} \\in \\mathcal {C} \\right\\rbrace .$ The $e$ -Galois hull of $\\mathcal {C}$ is defined by $Hull_{e}(\\mathcal {C})=\\;\\mathcal {C}\\cap \\mathcal {C}^{\\perp _{e}}.$ In the following, we list some useful results, which will be used in our constructions.",
"Lemma 2.1 ([REF , Lemmas 2 and 3]) Let $\\mathcal {C}$ be a linear code and $\\mathbf {c}$ be a codeword of $\\mathcal {C}$ .",
"(i).",
"For $\\mathcal {C}=\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v})$ , the codeword $\\mathbf {c}=(v_{1}f(a_{1}),\\ldots ,v_{n}f(a_{n}))\\in \\mathcal {C}\\bigcap \\mathcal {C}^{\\bot _e}$ if and only if there exists a polynomial $g(x)\\in \\mathbb {F}_{q}[x]$ with $\\mathrm {deg}(g(x))\\le n-k-1$ , such that $(v_{1}^{p^e+1}f^{p^e}(a_{1}),v_{2}^{p^e+1}f^{p^e}(a_{2}),\\ldots ,v_{n}^{p^e+1}f^{p^e}(a_{n}))=(u_{1}g(a_{1}),u_{2}g(a_{2}),\\ldots ,u_{n}g(a_{n}))$ .",
"(ii).",
"For $\\mathcal {C}=\\mathbf {GRS}_{k}(\\mathbf {a}, \\mathbf {v}, \\infty )$ , the codeword $\\mathbf {c}=(v_{1}f(a_{1}),\\ldots ,v_{n}f(a_{n}), f_{k-1})\\in \\mathcal {C}\\bigcap \\mathcal {C}^{\\bot _e}$ if and only if there exists a polynomial $g(x)\\in \\mathbb {F}_{q}[x]$ with $\\mathrm {deg}(g(x))\\le n-k$ , such that $(v_{1}^{p^e+1}f^{p^e}(a_{1}),v_{2}^{p^e+1}f^{p^e}(a_{2}),\\ldots ,v_{n}^{p^e+1}f^{p^e}(a_{n}),f_{k-1}^{p^e})=(u_{1}g(a_{1}),u_{2}g(a_{2}),\\ldots ,u_{n}g(a_{n}), -g_{n-k})$ .",
"Lemma 2.2 ([REF , Corollaries 2.1 and 2.2]) Let $\\mathcal {C}(m)$ be an (extended) GRS code, where $m$ represents dimension.",
"(i).",
"Assume that $\\mathcal {C}(m)=\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v})$ and $1\\le m\\le \\lfloor \\frac{n}{2}\\rfloor $ .",
"Then $\\mathcal {C}(m)^\\bot =\\mathcal {C}(n-m)$ if and only if there exists $\\lambda \\in \\mathbb {F}_{q}^{*}$ such that $\\lambda u_{i}=v_{i}^{2}$ for all $i=1,2,\\ldots ,n$ .",
"In particular, when $n$ is even and $m=\\frac{n}{2}$ , $\\mathcal {C}(\\frac{n}{2})$ is MDS self-dual (see [REF , Corollary 2.4]).",
"(ii).",
"Assume that $\\mathcal {C}(m)=\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v},\\infty )$ and $1\\le m\\le \\lfloor \\frac{n+1}{2}\\rfloor $ .",
"Then $\\mathcal {C}(m)^{\\bot }=\\mathcal {C}(n+1-m)$ if and only if $-u_{i}=v_{i}^{2}$ for all $i=1,2 ,\\ldots ,n$ .",
"In particular, when $n$ is odd and $m=\\frac{n+1}{2}$ , $\\mathcal {C}(\\frac{n+1}{2})$ is MDS self-dual (see Lemma 2.2 of [REF ]).",
"Denote by $E:=\\lbrace x^{p^e+1}|x\\in \\mathbb {F}_q^*\\rbrace $ .",
"In fact, $E$ is a multiplicative subgroup of $\\mathbb {F}_q^*$ (see [REF ]).",
"We have the following result.",
"Lemma 2.3 ([REF , Lemma 4]) Let $q=p^h$ and $0\\le e\\le h-1$ .",
"Then $\\mathbb {F}_{p^e}^*\\subseteq E$ if and only if $2e\\mid h$ .",
"At the end of this section, we define two polynomials $\\Psi _B(x)$ and $\\Delta _B(x)$ over $\\mathbb {F}_q$ as $\\Psi _B(x)=\\prod \\limits _{b\\in B}(x-b)$ and $\\Delta _B(x)=\\Psi _B^{\\prime }(x)$ for any $B\\subseteq \\mathbb {F}_q$ .",
"It is easy to see that $\\Delta _B(b)=\\Psi _B^{\\prime }(b)=\\prod \\limits _{b^{\\prime }\\in B,b^{\\prime }\\ne b}(b-b^{\\prime })$ , for any $b\\in B$ ."
],
[
"Some constructions of MDS codes with $e$ -Galois hulls of arbitrary dimensions",
" In this section, we will present several new MDS codes with $e$ -Galois hulls of arbitrary dimensions via (extended) GRS codes.",
"Theorem 1 Let $q=p^{em}$ with $p$ odd prime.",
"Assume that $t\\mid (p^{e}-1)$ , $m$ is even, $r\\le m-1$ and $n=tp^{er}$ .",
"Then for any $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , there exists an $[n+1,k]_q$ MDS code $\\mathcal {C}$ with $l$ -dimensional $e$ -Galois hull.",
"Let $V$ be an $r$ -dimensional $\\mathbb {F}_{p^e}$ -vector subspace in $\\mathbb {F}_{q}$ with $V\\cap \\mathbb {F}_{p^e}=0$ .",
"Choose $\\omega \\in \\mathbb {F}_{p^e}^*$ , which is a $t$ -th primitive root of unity.",
"Let $V_j=\\omega ^{j}+V(j=0,1,\\ldots ,t-1)$ and $\\bigcup \\limits _{j=0}^{t-1}V_{j}=\\lbrace a_1,\\ldots ,a_n\\rbrace $ .",
"For any $a_i\\in V_{j_0}$ , $\\begin{aligned}u_i&=\\Delta _{V_{j_0}}(a_i)\\prod _{j=0,j\\ne j_0}^{t-1}\\Psi _{V_j}(a_i)&\\\\&=\\left(\\prod _{0\\ne \\gamma \\in V}\\gamma \\right)\\cdot \\left(\\prod _{j=0,j\\ne j_0}^{t-1}\\prod _{\\gamma \\in V}(\\omega ^{j_0}-\\omega ^{j}+\\gamma )\\right)&\\\\&=\\left(\\prod _{0\\ne \\gamma \\in V}\\gamma \\right)\\cdot \\left(\\prod _{\\gamma \\in V}\\omega ^{j_0(t-1)}\\prod _{d=1}^{t-1}\\left(1+\\omega ^{-j_0}\\gamma -\\omega ^{d}\\right)\\right)&\\\\&=\\omega ^{-j_0p^{er}}\\cdot \\left(\\prod _{0\\ne \\gamma \\in V}\\gamma \\right)\\cdot \\left(\\prod _{\\gamma \\in V}\\prod _{d=1}^{t-1}(1+\\gamma -\\omega ^{d})\\right)&\\end{aligned}$ where $d=j-j_0$ and the last equality follows from that $\\prod \\limits _{\\gamma \\in V}\\omega ^{j_0(t-1)}=\\omega ^{-j_0p^{er}}$ and $\\omega ^{-j_0}\\gamma $ runs through $V$ when $\\gamma $ runs through $V$ .",
"Let $b=\\left(\\prod \\limits _{0\\ne \\gamma \\in V}\\gamma \\right)\\cdot \\left(\\prod \\limits _{\\gamma \\in V}\\prod \\limits _{d=1}^{t-1}(1+\\gamma -\\omega ^{d})\\right)$ , which is independent of $i$ .",
"It follows that $u_i=\\omega ^{-j_0p^{er}}\\cdot b$ , for any $1\\le i\\le n$ .",
"Choose $\\lambda =b^{-1}\\in \\mathbb {F}_q^*$ .",
"Since $\\omega \\in \\mathbb {F}_{p^e}^*$ and Lemma REF , then $\\lambda u_i\\in \\mathbb {F}_{p^e}^*\\subseteq E$ with $1\\le i\\le n$ .",
"Let $v_{i}^{p^e+1}=\\lambda u_i(1\\le i\\le n)$ .",
"Choose $\\mathbf {a}=(a_1,\\ldots ,a_{n})$ and $\\mathbf {v}=(\\alpha v_{1},\\alpha v_{2},\\ldots ,\\alpha v_{s},v_{s+1},\\ldots ,v_{n})$ , where $\\alpha \\in \\mathbb {F}_{q}^*$ and $\\alpha ^{p^e+1}\\ne 1$ .",
"Denote by $\\mathcal {C}:=\\mathbf {GRS}_k(\\mathbf {a},\\mathbf {v},\\infty )$ and $\\xi =\\alpha ^{p^e+1}$ .",
"For any $\\mathbf {c}=(\\alpha v_{1}f(a_{1}),\\ldots ,\\alpha v_{s}f(a_{s}),v_{s+1}f(a_{s+1}), \\ldots ,v_{n}f(a_{n}),f_{k-1})\\in Hull_e(\\mathcal {C})$ with $\\deg (f(x))\\le k-1$ , by Lemma REF (ii), there exists a polynomial $g(x)\\in \\mathbb {F}_{q}[x]$ with $\\deg (g(x))\\le n-k$ such that $\\begin{aligned}&(\\xi v_{1}^{p^e+1}(a_1)f^{p^e}(a_{1}),\\cdots ,\\xi v_{s}^{p^e+1}(a_s)f^{p^e}(a_{s}),v_{s+1}^{p^e+1}(a_{s+1})f^{p^e}(a_{s+1}),\\cdots ,v_{n}^{p^e+1}(a_{n})f^{p^e}(a_{n}),f^{p^e}_{k-1})&\\\\=&(u_{1}g(a_{1}),\\cdots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\cdots ,u_{n}g(a_{n}),-g_{n-k}).&\\end{aligned}$ From $(\\ref {xi})$ and $v_i^{p^e+1}=\\lambda u_i$ ($1\\le i\\le n$ ), we derive $\\begin{aligned}&(\\xi \\lambda u_{1}f^{p^e}(a_{1}),\\cdots ,\\xi \\lambda u_{s}f^{p^e}(a_s),\\lambda u_{s+1}f^{p^e}(a_{s+1}),\\cdots , \\lambda u_{n}f^{p^e}(a_{n}),f^{p^e}_{k-1})&\\\\=&(u_{1}g(a_{1}),\\cdots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\cdots ,u_{n}g(a_{n}),-g_{n-k}).&\\end{aligned}$ When $s+1\\le i\\le n$ , we get $\\lambda f^{p^e}(a_i)=g(a_i)$ .",
"Note that $\\deg (f^{p^e}(x))\\le p^e(k-1)\\le n-k-1$ from $k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $\\deg (g(x))\\le n-k$ .",
"It implies that $\\lambda f^{p^e}(x)=g(x)$ from $n-s\\ge n-k+1$ .",
"Assume that $f_{k-1}\\ne 0$ .",
"By $\\lambda f^{p^e}(x)=g(x)$ , we have $\\deg (f^{p^e}(x))=\\deg (g(x))$ , that is, $p^e(k-1)=n-k$ , which yields a contradiction to $k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ .",
"Hence, $f_{k-1}=0$ , which implies that $\\deg (f(x))\\le k-2$ .",
"Comparing the first $s$ coordinates on both sides of (REF ), we have $\\xi \\lambda u_if^{p^e}(a_i)= u_ig(a_i)= \\lambda u_if^{p^e}(a_i)$ for $i=1,\\ldots ,s$ .",
"Hence $f^{p^e}(a_i)=0$ , i.e., $f(a_i)=0$ for $i=1,\\ldots ,s$ with $\\xi \\ne 1$ .",
"Then $f(x)=c(x)\\prod _{i=1}^{s}(x-a_{i}),$ for some $c(x)\\in \\mathbb {F}_{q}[x]$ with $\\deg (c(x))\\le k-2-s$ .",
"It follows that $\\dim (Hull_e(\\mathcal {C}))\\le k-1-s$ .",
"Conversely, put $f(x)=c(x)\\prod \\limits _{i=1}^{s}(x-a_{i})$ , where $c(x)\\in \\mathbb {F}_{q}[x]$ and $\\deg (c(x))\\le k-2-s$ , which yields $f_{k-1}=0$ .",
"Assume that $g(x)=\\lambda f^{p^e}(x)$ , then $\\deg (g(x))\\le n-k-1$ , which yields $g_{n-k}=0$ .",
"Therefore, $\\begin{aligned}&(\\xi \\lambda u_{1}f^{p^e}(a_{1}),\\ldots ,\\xi \\lambda u_{s}f^{p^e}(a_{s}),\\lambda u_{s+1}f^{p^e}(a_{s+1}),\\ldots ,\\lambda u_{n}f^{p^e}(a_{n}),0)&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n}),0).&\\end{aligned}$ According to Lemma REF (ii), $(\\alpha v_{1}f(a_{1}),\\ldots ,\\alpha v_{s}f(a_{s}), v_{s+1}f(a_{s+1}),\\ldots , v_{n}f(a_{n}),0)\\in Hull_e(\\mathcal {C})$ .",
"Thus $\\dim (Hull_e(\\mathcal {C}))\\ge k-1-s$ .",
"Consequently, $\\dim (Hull_e(\\mathcal {C}))=k-1-s=l$ .",
"The proof has been completed.",
"Let $1\\le t\\le p^{e}$ .",
"We fix an $\\mathbb {F}_p$ -linear subspace $B\\subseteq \\mathbb {F}_{p^e}$ , such that $|B|\\ge t$ .",
"Set $b_1=0,b_2,\\ldots ,b_t$ be $t$ distinct elements of $B$ .",
"Put $Tr$ represents trace function from $\\mathbb {F}_q$ to $\\mathbb {F}_{p^e}$ , with $q=p^h$ and $e\\mid h$ .",
"Define $T_i:=\\lbrace x\\in \\mathbb {F}_q: Tr(x)=b_i\\rbrace $ , where $1\\le i\\le t$ .",
"Then $|T_i|=p^{h-e}$ and $T_i\\bigcap T_j=\\emptyset $ , for any $1\\le i\\ne j\\le t$ .",
"The following result can be obtained easily.",
"Lemma 3.1 The symbols are the same as above.",
"Then $\\Delta _{T_i}(x)=1$ .",
"By the definition of $T_i$ and $Tr$ , it is easy to get $\\Psi _{T_i}(x)=\\prod \\limits _{\\alpha \\in T_i}(x-\\alpha )=Tr(x)-b_i=x+x^{p^e}+\\cdots +x^{p^{h-e}}-b_i.$ Take the derivative of both sides of (REF ), $\\Delta _{T_i}(x)=\\Psi _{T_i}^{\\prime }(x)=1.$ Remark 3.1 The case $e=\\frac{h}{2}$ has been shown in [REF ].",
"Now, we give the second construction.",
"Theorem 2 Let $q=p^h$ , $2e\\mid h$ and $n=tp^{h-e}$ with $1\\le t\\le p^{e}$ .",
"For any $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , there exists an $[n+1,k]_q$ MDS code $\\mathcal {C}$ with $l$ -dimensional $e$ -Galois hull.",
"Since $T_i$ and $T_j$ are pairwise disjoint for any $1\\le i\\ne j\\le t$ , it follows that $|\\bigcup \\limits _{i=1}^t T_i|=tp^{h-e}=n$ .",
"Denote by $A=\\bigcup \\limits _{i=1}^t T_i=\\lbrace a_1,\\ldots ,a_n\\rbrace $ .",
"For any $a_i\\in T_{j_0}$ , by Lemma REF , we have $\\begin{aligned}u_i=\\Delta _{A}(a_i)=\\Delta _{T_{j_0}}(a_i)\\cdot \\left(\\prod _{j\\ne j_0,j=1}^t\\Psi _{T_j}(a_i)\\right)=\\prod _{j\\ne j_0,j=1}^t(Tr(a_i)-b_j).\\end{aligned}$ From the definition of $Tr$ , we know $Tr(a_i)\\in \\mathbb {F}_{p^e}$ ($1\\le i\\le n$ ).",
"Since $b_j\\in B\\subseteq \\mathbb {F}_{p^e}$ ($1\\le j\\le t$ ), one has $u_i\\in \\mathbb {F}_{p^e}$ .",
"By the definition of $u_i$ , it is easy to see that $u_i\\ne 0$ with $1\\le i\\le n$ .",
"Therefore, for any $1\\le i\\le n$ , it takes $u_i\\in \\mathbb {F}_{p^e}^*\\subseteq E$ by Lemma REF .",
"Let $v_{i}^{p^e+1}= u_i$ ($1\\le i\\le n$ ) and $s:=k-1-l$ .",
"Choose $\\mathbf {a}=(a_1,\\ldots ,a_{n})$ and $\\mathbf {v}=(\\alpha v_{1},\\alpha v_{2},\\ldots ,\\alpha v_{s},v_{s+1},\\ldots ,v_{n})$ , where $\\alpha \\in \\mathbb {F}_{q}^*$ and $\\alpha ^{p^e+1}\\ne 1$ .",
"Consider the $e$ -Galois hull of the $[n+1,k]_q$ MDS code $\\mathcal {C}:=\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v},\\infty )$ .",
"Similarly as the proof of Theorem REF , we can obtain the result."
],
[
"Constructions via MDS codes with special Euclidean orthogonal property",
" When $\\frac{h}{e}$ is odd, we can make the constructions from MDS codes satisfying Euclidean orthogonal properties.",
"Theorem 3 Let $q=p^h$ , where $p$ is an odd prime.",
"Assume $1\\le m\\le \\lfloor \\frac{n}{2}\\rfloor $ and $\\frac{h}{e}$ is odd.",
"Suppose $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v})^\\perp =\\mathbf {GRS}_{n-m}(\\mathbf {a},\\mathbf {v}).$ For any $0\\le l\\le k \\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ , there exists a $q$ -ary $[n,k]$ MDS code $\\mathcal {C}$ with $\\dim (Hull_e(\\mathcal {C}))=l$ .",
"From $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v})^\\perp =\\mathbf {GRS}_{n-m}(\\mathbf {a},\\mathbf {v})$ and Lemma REF (i), $v_i^{2}=\\lambda u_i\\ne 0(1\\le i\\le n),$ where $\\lambda \\in \\mathbb {F}_{q}^*$ .",
"Since $\\frac{h}{e}$ and $p$ are odd, then $\\gcd (p^e+1,p^h-1)=2$ .",
"So there exist two integers $\\mu $ and $\\nu $ , such that $\\mu (p^e+1)+\\nu (p^h-1)=2$ .",
"Substituting into (REF ), we have $v_i^{\\mu (p^e+1)}=\\lambda u_i\\ne 0(1\\le i\\le n)$ .",
"Set $v^{\\prime }_i=v_i^\\mu (1\\le i\\le n)$ .",
"Then ${v^{\\prime }}_i^{p^e+1}=\\lambda u_i\\ne 0(1\\le i\\le n).$ Denote by $s:=k-l$ , $\\mathbf {a}=(a_{1},a_{2},\\ldots ,a_{n})$ and $\\mathbf {v}^{^{\\prime }}=(\\alpha v^{\\prime }_{1},\\alpha v^{\\prime }_{2},\\ldots ,\\alpha v^{\\prime }_{s},v^{\\prime }_{s+1},\\ldots ,v^{\\prime }_{n})$ , where $\\alpha \\in \\mathbb {F}_{q}^*$ and $\\alpha ^{p^e+1}\\ne 1$ .",
"Consider the $e$ -Galois hull of the $[n,k]_q$ MDS code $\\mathcal {C}:=\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v}^{\\prime })$ .",
"Then for any $\\mathbf {c}=(\\alpha v^{\\prime }_{1}f(a_{1}),\\ldots ,\\alpha v^{\\prime }_{s}f(a_{s}),v^{\\prime }_{s+1}f(a_{s+1}),\\ldots ,v^{\\prime }_{n}f(a_{n}))\\in Hull_e(\\mathcal {C})$ with $\\deg (f(x))\\le k-1$ , according to Lemma REF (i), there exists a polynomial $g(x)\\in \\mathbb {F}_{q}[x]$ with $\\deg (g(x))\\le n-k-1$ such that $\\begin{aligned}&(\\alpha ^{p^e+1} {v^{\\prime }}_{1}^{p^e+1}f^{p^e}(a_{1}),\\ldots ,\\alpha ^{p^e+1}{v^{\\prime }}_{s}^{p^e+1}f^{p^e}(a_{s}),{v^{\\prime }}_{s+1}^{p^e+1}f^{p^e}(a_{s+1}),\\ldots ,{v^{\\prime }}_{n}^{p^e+1}f^{p^e}(a_{n}))&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n})).&\\end{aligned}$ Set $\\xi =\\alpha ^{p^e+1}$ .",
"Then $\\begin{aligned}&(\\xi \\lambda u_{1}f^{p^e}(a_{1}),\\ldots ,\\xi \\lambda u_{s}f^{p^e}(a_{s}),\\lambda u_{s+1}f^{p^e}(a_{s+1}),\\ldots ,\\lambda u_{n}f^{p^e}(a_{n}))&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n})).&\\end{aligned}$ Considering the last $n-s$ coordinates of (REF ), we get $\\lambda f^{p^e}(a_i)=g(a_i)$ ($s+1\\le i\\le n$ ).",
"Hence the number of distinct roots of $\\lambda f^{p^e}(x)-g(x)$ is at least $n-s\\ge n-k$ .",
"Since $k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ , we have $\\deg (f^{p^e}(x))\\le p^e(k-1)\\le n-k-1$ , which derives that $\\deg (\\lambda f^{p^e}(x)-g(x))\\le n-k-1$ together with $\\deg (g(x))\\le n-k-1$ .",
"Hence $\\lambda f^{p^e}(x)=g(x)$ .",
"Comparing the first $s$ coordinates of (REF ), $\\xi \\lambda u_if^{p^e}(a_i)= u_ig(a_i)=\\lambda u_if^{p^e}(a_i)$ for $i=1,\\ldots ,s$ .",
"Hence $f(a_i)=0$ with $\\xi \\ne 1$ and $\\lambda u_i\\ne 0$ $(i=1,\\ldots ,s)$ .",
"Then $f(x)$ can be expressed as $f(x)=c(x)\\prod _{i=1}^{s}(x-a_{i}),$ for some $c(x)\\in \\mathbb {F}_{q}[x]$ with $\\deg (c(x))\\le k-1-s$ .",
"Therefore, $\\dim (Hull_e(\\mathcal {C}))\\le k-s$ .",
"Conversely, put $f(x)=c(x)\\prod \\limits _{i=1}^{s}(x-a_{i})$ , where $c(x)\\in \\mathbb {F}_{q}[x]$ and $\\deg (c(x))\\le k-1-s$ .",
"Assume that $g(x)=\\lambda f^{p^e}(x)$ , which yields $\\deg (g(x))\\le n-k-1$ .",
"Then $\\begin{aligned}&(\\xi \\lambda u_{1}f^{p^e}(a_{1}),\\ldots ,\\xi \\lambda u_{s}f^{p^e}(a_{s}),\\lambda u_{s+1}f^{p^e}(a_{s+1}),\\ldots ,\\lambda u_{n}f^{p^e}(a_{n}))&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n})).&\\end{aligned}$ By Lemma REF (i), $(\\alpha v^{\\prime }_{1}f(a_{1}),\\ldots ,\\alpha v^{\\prime }_{s}f(a_{s}), v^{\\prime }_{s+1}f(a_{s+1}),\\ldots , v^{\\prime }_{n}f(a_{n}))\\in Hull_e(\\mathcal {C})$ .",
"Therefore, $\\dim (Hull_e(\\mathcal {C}))\\ge k-s$ .",
"Hence $\\dim (Hull_e(\\mathcal {C}))= k-s=l$ .",
"Example 4.1 In Theorem 1 of [REF ], we have $v_i^2=\\lambda u_i$ , where $\\lambda =g^{\\frac{\\sqrt{q}+1}{2}}$ and $g$ is a primitive element of $\\mathbb {F}_q$ .",
"By Lemma REF , $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v})^\\perp =\\mathbf {GRS}_{n-m}(\\mathbf {a},\\mathbf {v})$ is satisfied.",
"Therefore, for any $0\\le l\\le k \\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ , there exists a $q$ -ary $[n,k]$ MDS code $\\mathcal {C}$ with $\\dim (Hull_e(\\mathcal {C}))=l$ by Theorem REF .",
"Afterwards, we apply extended GRS codes to construct MDS codes with $e$ -Galois hulls of arbitrary dimensions.",
"Theorem 4 Let $q=p^h$ ($p$ is an odd prime), $\\frac{h}{e}$ is odd and $1\\le m\\le \\lfloor \\frac{n+1}{2}\\rfloor $ .",
"Suppose $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v},\\infty )^{\\bot }=\\mathbf {GRS}_{n+1-m}(\\mathbf {a},\\mathbf {v},\\infty ).$ For any $1\\le k \\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , there exists a $q$ -ary $[n+1,k]$ MDS code $\\mathcal {C}$ with $\\dim (Hull_e(\\mathcal {C}))=l$ .",
"(i).",
"Since $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v},\\infty )^{\\bot }=\\mathbf {GRS}_{n+1-m}(\\mathbf {a},\\mathbf {v},\\infty )$ and by Lemma REF (ii), we obtain $v_{i}^{2}=-u_{i}\\ne 0(1\\le i \\le n).$ It follows that $\\gcd (p^e+1,p^h-1)=2$ , since $\\frac{h}{e}$ and $p$ are odd.",
"Similar to Theorem REF , it takes $\\mu (p^e+1)+\\nu (p^l-1)=2$ ($\\mu $ and $\\nu $ are two integers) and $v_i^{\\mu (p^e+1)}=-u_i\\ne 0(1\\le i\\le n)$ .",
"Set $v^{\\prime }_i=v_i^\\mu (1\\le i\\le n)$ .",
"Then ${v^{\\prime }}_i^{p^e+1}= -u_i\\ne 0(1\\le i\\le n).$ Denote by $s:=k-l-1$ , $\\mathbf {a}=(a_{1},a_{2},\\ldots ,a_{n})$ and $\\mathbf {v}^{^{\\prime }}=(\\alpha v^{\\prime }_{1},\\alpha v^{\\prime }_{2},\\ldots ,\\alpha v^{\\prime }_{s},v^{\\prime }_{s+1},\\ldots ,v^{\\prime }_{n})$ , where $\\alpha \\in \\mathbb {F}_{q}^*$ and $\\alpha ^{p^e+1}\\ne 1$ .",
"Consider the $e$ -Galois hull of the $[n,k]_q$ MDS code $\\mathcal {C}:=\\mathbf {GRS}_{k}(\\mathbf {a},\\mathbf {v}^{\\prime },\\infty )$ .",
"Then for any $\\mathbf {c}=(\\alpha v^{\\prime }_{1}f(a_{1}),\\ldots ,\\alpha v^{\\prime }_{s}f(a_{s}),v^{\\prime }_{s+1}f(a_{s+1}),\\ldots ,v^{\\prime }_{n}f(a_{n}),f_{k-1})\\in Hull_e(\\mathcal {C})$ with $\\deg (f(x))\\le k-1$ , according to Lemma REF (ii), there exists a polynomial $g(x)\\in \\mathbb {F}_{q}[x]$ with $\\deg (g(x))\\le n-k$ such that $\\begin{aligned}&(\\alpha ^{p^e+1} {v^{\\prime }}_{1}^{p^e+1}f^{p^e}(a_{1}),\\ldots ,\\alpha ^{p^e+1}{v^{\\prime }}_{s}^{p^e+1}f^{p^e}(a_{s}),{v^{\\prime }}_{s+1}^{p^e+1}f^{p^e}(a_{s+1}),\\ldots ,{v^{\\prime }}_{n}^{p^e+1}f^{p^e}(a_{n}),f_{k-1}^{p^e})&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n}),-g_{n-k}).&\\end{aligned}$ Set $\\xi =\\alpha ^{p^e+1}$ .",
"It yields $\\begin{aligned}&(-\\xi u_{1}f^{p^e}(a_{1}),\\ldots ,-\\xi u_{s}f^{p^e}(a_{s}),-u_{s+1}f^{p^e}(a_{s+1}),\\ldots ,-u_{n}f^{p^e}(a_{n}),f_{k-1}^{p^e})&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n}),-g_{n-k}).&\\end{aligned}$ When $s+1 \\le i \\le n$ , we get $-f^{p^e}(a_i)=g(a_i)$ .",
"Therefore, the number of distinct roots of $f^{p^e}(x)+g(x)$ is at least $n-s\\ge n-k+1$ .",
"We know $\\deg (f^{p^e}(x))\\le p^e(k-1)\\le n-k-1$ from $k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ , and $\\deg (g(x))\\le n-k$ .",
"Thus it derives that $\\deg (f^{p^e}(x)+g(x))\\le n-k$ .",
"So $-f^{p^e}(x)=g(x)$ .",
"From the first $s$ coordinates of (REF ), we have $-\\xi u_if^{p^e}(a_i)=u_ig(a_i)= -u_if^{p^e}(a_i)$ with $i=1,\\ldots ,s$ , which derives $f(a_i)=0$ with $\\xi \\ne 1$ and $u_i\\ne 0$ $(i=1,\\ldots ,s)$ .",
"Then $f(x)$ can be expressed as $f(x)=c(x)\\prod _{i=1}^{s}(x-a_{i}),$ for some $c(x)\\in \\mathbb {F}_{q}[x]$ with $\\deg (c(x))\\le k-2-s$ .",
"It follows that $\\dim (Hull_e(\\mathcal {C}))\\le k-1-s$ .",
"Conversely, put $f(x)=c(x)\\prod \\limits _{i=1}^{s}(x-a_{i})$ , where $c(x)\\in \\mathbb {F}_{q}[x]$ and $\\deg (c(x))\\le k-2-s$ , which yields $f_{k-1}=0$ .",
"Assume that $g(x)=-f^{p^e}(x)$ .",
"Then $\\deg (g(x))\\le n-k-1$ , which yields $g_{n-k}=0$ .",
"It takes $\\begin{aligned}&(-\\xi u_{1}f^{p^e}(a_{1}),\\ldots ,-\\xi u_{s}f^{p^e}(a_{s}),-u_{s+1}f^{p^e}(a_{s+1}),\\ldots ,-u_{n}f^{p^e}(a_{n}),0)&\\\\=&(u_{1}g(a_{1}),\\ldots ,u_{s}g(a_{s}),u_{s+1}g(a_{s+1}),\\ldots ,u_{n}g(a_{n}),0).&\\end{aligned}$ According to Lemma REF (ii), $(\\alpha v^{\\prime }_{1}f(a_{1}),\\ldots ,\\alpha v^{\\prime }_{s}f(a_{s}), v^{\\prime }_{s+1}f(a_{s+1}),\\ldots , v^{\\prime }_{n}f(a_{n}),0)\\in Hull_e(\\mathcal {C})$ .",
"Therefore, $\\dim (Hull_e(\\mathcal {C}))\\ge k-1-s$ .",
"As a result, $\\dim (Hull_e(\\mathcal {C}))= k-1-s=l$ .",
"Example 4.2 In Theorem 1(ii) of [REF ], we have $v_i^2=-u_i$ .",
"By Lemma REF , $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v},\\infty )^{\\perp }=\\mathbf {GRS}_{n+1-m}(\\mathbf {a},\\mathbf {v},\\infty )$ is satisfied.",
"Therefore, for any $1\\le k \\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , there exists a $q$ -ary $[n+1,k]$ MDS code $\\mathcal {C}$ with $\\dim (Hull_e(\\mathcal {C}))=l$ .",
"Remark 4.1 These two theorems build relationships between $e$ -Galois orthogonal property of GRS codes and Euclidean orthogonal property of GRS codes.",
"Therefore, when $\\frac{h}{e}$ is odd, we can construct GRS codes with $e$ -Galois hulls of arbitrary dimensions via GRS codes satisfying Euclidean orthogonal property.",
"In fact, if an (extended) GRS code is a Euclidean self-orthogonal code (including Euclidean self-dual code), the conditions of Theorems REF and REF can be satisfied."
],
[
"Applications to EAQECCs and MDS EAQECCs",
" In this section, we apply the results in Sections 3 and 4 to construct several families of EAQECCs and MDS EAQECCs, which are more general than previous works.",
"More details on EAQECCs are referred to [REF ].",
"An $[[n,k,d;c]]_q$ EAQECC $\\mathcal {C}$ means that under the assist of $c$ pairs of maximally entangled Bell states, the quantum code $\\mathcal {C}$ can encode $k$ information qubits into $n$ channel qubits and $d$ represents the minimum distance.",
"Similar to classical linear codes, EAQECC also satisfies the quantum Singleton bound, which is given in the following lemma.",
"Lemma 5.1 ([REF , REF , REF ]) Assume that $d\\le \\frac{n+2}{2}$ .",
"Then $[[n,k,d;c]]_{q}$ EAQECC satisfies $n+c-k\\ge 2(d-1)$ , where $0\\le c\\le n-1$ .",
"$\\square $ Remark 5.1 When $d\\le \\frac{n+2}{2}$ , an EAQECC attaining the quantum Singleton bound, that is $n+c-k=2(d-1)$ , is called an MDS EAQECC.",
"For a matrix $M=(m_{ij})$ over $\\mathbb {F}_q$ , define $M^{(p^{h-e})}=(m_{ij}^{p^{h-e}})$ and put $M^{T_e}=\\left(M^{(p^{h-e})}\\right)^T$ .",
"In [REF ] and [REF ], the authors proposed methods for constructing EAQECCs by utilizing classical linear codes with $e$ -Galois inner products over finite fields in the following.",
"Lemma 5.2 ([REF , Corollary 3.2] and [REF , Corollary 5.2]) Let $H$ be a parity check matrix of an $[n,k,d]$ linear code $\\mathcal {C}$ over $\\mathbb {F}_{q}$ .",
"Then there exists an $[[n,2k-n+c,d;c]]_q$ EAQECC, where $c=rank(HH^{T_e})$ is the required number of maximally entangled states.",
"$\\square $ In [REF ], Guenda et al.",
"showed the relationship between the value of $rank(HH^{T_e})$ and the hull dimension of linear code with parity check matrix $H$ , that is $rank(HH^{T_e})=n-k-\\dim (Hull_e(\\mathcal {C})).$ As a direct consequence of Lemmas REF and REF , one has the following result, which has been shown in [REF ].",
"Lemma 5.3 ([REF ]) Let $\\mathcal {C}$ be an $[n,k,d]$ linear code over $\\mathbb {F}_q$ and its $e$ -Galois dual $\\mathcal {C}^{\\perp _e}$ has parameters $[n,n-k,d^{\\perp _e}]$ .",
"Then there exists $[[n,k-\\dim (Hull_e(\\mathcal {C})),d;n-k-\\dim (Hull_e(\\mathcal {C}))]]_q$ EAQECC and $[[n,n-k-\\dim (Hull_{m-e}(\\mathcal {C})),d^\\perp ;k-\\dim (Hull_{m-e}(\\mathcal {C}))]]_q$ EAQECC.",
"Let $\\mathcal {C}$ be an $[n,k,n-k+1]$ -MDS code over $\\mathbb {F}_q$ .",
"Its $e$ -Galois dual code $\\mathcal {C}^{\\bot _e}$ is also an MDS code with $[n,n-k,k+1]$ .",
"Denote by $l=\\dim (Hull_e(\\mathcal {C}))$ and $l^{\\prime }=\\dim (Hull_{m-e}(\\mathcal {C}))$ .",
"Then we can obtain the following result by Lemma REF .",
"Corollary 5.1 ([REF ]) Assume that $\\mathcal {C}$ is an $[n,k]$ -MDS code over $\\mathbb {F}_q$ .",
"If $k\\le \\lfloor \\frac{n}{2}\\rfloor $ , then there exists an $[[n,k-l,n-k+1;n-k-l]]_q$ EAQECC and an $[[n,n-k-l,k+1;k-l]]_q$ MDS EAQECC.",
"$\\square $ From Corollary REF and all the theorems in Sections 3 and 4, we have the following results directly.",
"Theorem 5 Let $q=p^{em}$ with $p$ odd prime.",
"Assume that $t\\mid (p^{e}-1)$ , $m$ is even, $r\\le m-1$ and $n=tp^{er}$ .",
"(i).",
"For any $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , then there exists an $\\big [[n+1,k-l,n-k+2;n+1-k-l]\\big ]_{q}$ EAQECC over $\\mathbb {F}_q$ .",
"(ii).",
"For any $1\\le k\\le \\lfloor \\frac{p^{h-e}+n-1}{p^{h-e}+1}\\rfloor $ and $0\\le l^{\\prime }\\le k-1$ , then there exists an $\\big [[n+1,n+1-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ MDS EAQECC over $\\mathbb {F}_q$ .",
"Example 5.1 Choose $(p,m,r,e,t)=(5,4,3,3,31)$ .",
"It is easy to see that $t=31\\mid 5^3-1=p^e-1$ .",
"By Theorem REF , there exist EAQECCs with parameters $\\big [[60546876,k-l,60546877-k;60546876-k-l]\\big ]_{q}$ , where $1\\le k\\le 480531$ and $0\\le l\\le k-1$ and MDS EAQECCs with parameters $\\big [[60546876,60546876-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ , where $1\\le k\\le 31$ and $0\\le l^{\\prime }\\le k-1$ .",
"The two classes of EAQECCs are new in the sense that their parameters can not be covered by previous results.",
"Theorem 6 Let $q=p^h$ , $2e\\mid h$ and $n=tp^{h-e}$ with $1\\le t\\le p^{e}$ .",
"(i).",
"For any $1\\le k\\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , there exists an $\\big [[n+1,k-l,n-k+2;n+1-k-l]\\big ]_{q}$ EAQECC over $\\mathbb {F}_q$ .",
"(ii).",
"For any $1\\le k\\le \\lfloor \\frac{p^{h-e}+n-1}{p^{h-e}+1}\\rfloor $ and $0\\le l^{\\prime }\\le k-1$ , there exists an $\\big [[n+1,n+1-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ MDS EAQECC over $\\mathbb {F}_q$ .",
"Example 5.2 Choose $(p,h,e,t)=(3,16,4,73)$ .",
"Then $p^e=3^4=81>73=t$ .",
"By Theorem REF , there exist EAQECCs with parameters $\\big [[38795194,k-l,38795195-k;38795194-k-l]\\big ]_{q}$ , where $1\\le k\\le 473113$ and $0\\le l\\le k-1$ and MDS EAQECCs with parameters $\\big [[38795194,38795194-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ , where $1\\le k\\le 73$ and $0\\le l^{\\prime }\\le k-1$ .",
"These two classes of EAQECCs also have new parameters which have not been reported previously.",
"Theorem 7 Let $q=p^h$ .",
"Assume $1\\le m\\le \\lfloor \\frac{n}{2}\\rfloor $ and $\\frac{h}{e}$ is odd.",
"Suppose $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v})^\\perp =\\mathbf {GRS}_{n-m}(\\mathbf {a},\\mathbf {v}).$ (i).",
"For any $1\\le k \\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k$ , there exists an $\\big [[n,k-l,n-k+1;n-k-l]\\big ]_{q}$ EAQECC over $\\mathbb {F}_q$ .",
"(ii).",
"For any $1\\le k \\le \\lfloor \\frac{p^{h-e}+n-1}{p^{h-e}+1}\\rfloor $ and $0\\le l^{\\prime }\\le k$ , there exists an $\\big [[n,n-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ MDS EAQECC over $\\mathbb {F}_q$ .",
"Example 5.3 Similarly as Example REF , from Theorem 1 of [REF ], we know there exists a $q$ -ary $[n,\\frac{n}{2}]$ MDS Euclidean self-dual code, where $n=s(\\sqrt{q}-1)+t(\\sqrt{q}+1)$ , $\\sqrt{q}$ and $s$ satisfy “$\\sqrt{q}\\equiv 1 \\pmod {4}$ and $s$ is even\" or “$\\sqrt{q}\\equiv 3 \\pmod {4}$ and $s$ is odd\".",
"By Remark REF and Theorem REF , there exists an $\\big [[n,n-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ MDS EAQECC or $\\big [[n,k-l,n-k+1;n-k-l]\\big ]_{q}$ EAQECC over $\\mathbb {F}_q$ .",
"The length of the MDS EAQECC or EAQECC is $n=s(\\sqrt{q}-1)+t(\\sqrt{q}+1)$ , which is more flexible than the previous results.",
"Theorem 8 Let $q=p^h$ , $\\frac{h}{e}$ is odd and $1\\le m\\le \\lfloor \\frac{n+1}{2}\\rfloor $ .",
"Suppose $\\mathbf {GRS}_{m}(\\mathbf {a},\\mathbf {v},\\infty )^{\\perp }=\\mathbf {GRS}_{n+1-m}(\\mathbf {a},\\mathbf {v},\\infty ).$ (i).",
"For any $1\\le k \\le \\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ and $0\\le l\\le k-1$ , there exists an $\\big [[n+1,k-l,n-k+2;n+1-k-l]\\big ]_{q}$ EAQECC over $\\mathbb {F}_q$ .",
"(ii).",
"For any $1\\le k \\le \\lfloor \\frac{p^{h-e}+n-1}{p^{h-e}+1}\\rfloor $ and $0\\le l^{\\prime }\\le k-1$ , there exists an $\\big [[n+1,n+1-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ MDS EAQECC over $\\mathbb {F}_q$ .",
"Example 5.4 Similarly as Example REF , from Theorem 1 of [REF ], there exists a $q$ -ary MDS Euclidean self-dual code of length $n+1$ .",
"The parameters $n,n^{\\prime },n_1,n_2,t$ satisfy the following conditions: (i).",
"$n=tn^{\\prime }$ is odd; (ii).",
"$n^{\\prime }=n_1n_2$ and $n^{\\prime }\\mid (q-1)$ ; (iii).",
"$n_1=\\gcd (n^{\\prime },\\sqrt{q}+1)$ ; (iv).",
"$n_2=\\frac{n^{\\prime }}{\\gcd (n^{\\prime },\\sqrt{q}+1)}$ ; (v).",
"$1\\le t\\le \\frac{\\sqrt{q}-1}{n_2}$ .",
"So by Remark REF and Theorem REF , there exists an $\\big [[n+1,n+1-k-l^{\\prime },k+1;k-l^{\\prime }]\\big ]_{q}$ MDS EAQECC or an $\\big [[n+1,k-l,n-k+2;n+1-k-l]\\big ]_{q}$ EAQECC over $\\mathbb {F}_q$ .",
"The length of the MDS EAQECC or EAQECC is $n+1=tn^{\\prime }+1$ , which is more flexible than the previous results.",
"Remark 5.2 Theorems REF and REF propose a mechanism for the constructions of EAQECCs and MDS EAQECCs via MDS codes with Euclidean orthogonal property.",
"In fact, all Euclidean self-orthogonal (extended) GRS codes can be used to construct EAQECCs and MDS EAQECCs."
],
[
"Conclusion",
" Inspired by [REF , REF ], we propose some new MDS codes with Galois hulls of arbitrary dimensions and several new families of EAQECCs and MDS EAQECCs.",
"In our constructions, the lengths of codes($n$ or $n+1$ ) are flexible.",
"However, the dimension $k$ is roughly upper bounded by $\\lfloor \\frac{p^e+n-1}{p^e+1}\\rfloor $ or $\\lfloor \\frac{p^{h-e}+n-1}{p^{h-e}+1}\\rfloor $ .",
"How to increase the upper bound is not an easy task, which is left as an open problem.",
"When $\\frac{h}{e}$ is odd, we can associate Galois hulls of GRS codes with GRS codes satisfying Euclidean orthogonal property.",
"Precisely, if there exists an (extended) GRS code satisfying one of the Euclidean orthogonal properties of Theorems REF and REF , then we can construct an $[n,k]_q$ (extended) GRS code $\\mathcal {C}$ with $\\dim (Hull_e(\\mathcal {C}))=l$ , where $0\\le l\\le k-1$ .",
"For the case $\\frac{h}{e}$ is even, some constructions are given.",
"However, how to propose a mechanism for the constructions of MDS codes with Galois hulls of arbitrary dimensions is also an open problem."
],
[
"Acknowledgements",
"The authors thank anonymous reviewers, the editor and the associate editor for their suggestions and comments to improve the readability of this paper.",
"This research is supported by National Natural Science Foundation of China under Grant 11471008 and the Fundamental Research Funds for the Central Universities of CCNU under grant CCNU20TD002."
]
] | 2011.14061 |
[
[
"Deterministic Certification to Adversarial Attacks via Bernstein\n Polynomial Approximation"
],
[
"Abstract Randomized smoothing has established state-of-the-art provable robustness against $\\ell_2$ norm adversarial attacks with high probability.",
"However, the introduced Gaussian data augmentation causes a severe decrease in natural accuracy.",
"We come up with a question, \"Is it possible to construct a smoothed classifier without randomization while maintaining natural accuracy?\".",
"We find the answer is definitely yes.",
"We study how to transform any classifier into a certified robust classifier based on a popular and elegant mathematical tool, Bernstein polynomial.",
"Our method provides a deterministic algorithm for decision boundary smoothing.",
"We also introduce a distinctive approach of norm-independent certified robustness via numerical solutions of nonlinear systems of equations.",
"Theoretical analyses and experimental results indicate that our method is promising for classifier smoothing and robustness certification."
],
[
"Introduction",
"Neural Network models achieve an enormous breakthrough in image classification these years but are vulnerable to imperceptible perturbations known as adversarial attacks, as evidenced by [17] [45].",
"Since then, many researchers began to develop their adversarial attacks such as DeepFool [36], Jacobian based Saliency Map Attack (JSMA) [39], CW attack [7], etc.",
"On the other hand, researchers also attempted to build defense mechanisms that are invincible to adversarial attacks.",
"Please see [58] for the thorough surveys of adversarial attacks and defense approaches.",
"However, we still have a long way to go in that making an indestructible machine learning model is still an open problem in the AI community.",
"Empirical defense strategies can be categorized into several groups.",
"The first one is adversarial detection [13] [30].",
"However, [6] bypassed ten detection methods and showed that it is extremely challenging to detect adversarial attacks.",
"The second one is denoising [28] [56] [57].",
"The kind of methods is computationally inefficient (some works trained on hundreds of GPUs) but useful in practice.",
"The third one is called adversarial training, which is the most effective but suffers from attack dependency [47] [31].",
"In addition to the aforementioned methods, there are still various defense strategies proposed.",
"[2] showed that most of the defense methods are ineffective when encountering sophisticated designed attacks due to lack of a provable defense guarantee.",
"Figure: A 2D example to visualize our result.",
"Left:The classifier's decision boundary from the neural network is very “sharp\".",
"Pointsnear the sharp region are vulnerable to adversarial attacks.",
"Right: We can seethat the boundary of the classifier from our method is smoothed, and thecertification (“safe zone\") is mostly accurate.To avoid this arms race, a series of researches [43] on provable defenses or certified robustness have emerged with theoretical guarantees.",
"Specifically, for any point $x$ , there is a set containing the provably robust $x$ , implying that every point in this set gives the same prediction.",
"Assume that adversarial examples are caused due to the irregularity of the decision boundary.",
"Randomized smoothing [8] introduced Gaussian random noise for smoothing the base classifier.",
"Nonetheless, many challenges remain.",
"Firstly, Gaussian data augmentation causes a severe decrease in natural accuracy.",
"Secondly, the prediction and certification require numerous samples during inference, which is a time-consuming process.",
"Thirdly, there is still a chance, even slightly, that the certification is inaccurate.",
"In many applications, like self-driving vehicles, we do not want to take this kind of risk.",
"In this paper, we propose a new method to “smooth\" the decision boundary of the classifier.",
"Our method can also provide a “safe zone\" that predicts the same class as the input deterministically.",
"We use a 2D example to demonstrate our method in Figure REF .",
"Note that the regions near the decision boundary are more comfortable to certify by our approach, while the areas far from the decision boundary are less accurate.",
"It is due to the initial guess of the solver of nonlinear systems of equations.",
"To overcome this problem, we set the error tolerance of the solution to an acceptable threshold [49].",
"Furthermore, we care more about data near the decision boundary than those far away from it since we can find adversarial examples with our bare eyes if the attack is strong enough.",
"Our contributions are summarized as follows: To our knowledge, we are the first to introduce Bernstein polynomial into the adversarial example community for which we devise a deterministic algorithm to smooth the base classifier.",
"Our method, compared with [8] and [42], can maintain higher natural accuracy while achieving comparable robustness (certified accuracy) in CIFAR-10Other results obtained from different datasets such as MNIST, SVHN, and CIFAR-100 would be discussed in the Appendix [21].",
"We can certify a smoothed classifier with arbitrary norms (norm independent).",
"Our method can be used in different aspects of machine learning beyond certified robustness, such as over-fitting alleviation."
],
[
"Related Works",
"We roughly categorize adversarial defense researches into empirical defenses and certified defenses.",
"Empirical defenses seem to be robust to existing adversarial perturbations but lack of formal robustness guarantee.",
"In this section, we will study previous empirical defenses and certified defenses with the focus on randomized smoothing."
],
[
"Empirical defenses",
"In practice, the best empirical defense is Adversarial Training (AT) [17] [23] [31].",
"AT is operated by first generating the adversarial examples by projected gradient descent and then augmenting them into the training set.",
"Although the classifier yielded based on AT is robust to most gradient-based attacks, it is still difficult to tell whether this classifier is undoubtedly robust.",
"Moreover, adaptive attacks break most empirical defenses.",
"A pioneering study [2] showed that most of the methods are ineffective by using sophisticated designed attacks.",
"To stop this arms race between attackers and defenders, some researchers try to focus on building a verification mechanism with a robustness guarantee."
],
[
"Certified defenses",
"Certified defenses can be divided into exact (a.k.a “complete\") methods and conservative (a.k.a “incomplete\") methods.",
"Exact methods are usually based on Satisfiability Modulo Theories solvers [19] [20] or mixed-integer linear programming [46], but they are computationally inefficient.",
"Conservative methods are guaranteed to find adversarial examples if they exist but might mistakenly judge a safe data point as an adversarial one (false positive) [54] [55] [43] [52] [51] [12] [40] [41] [9][15] [34] [44] [53] [18] [60]."
],
[
"Introducing randomness into neural network classifiers has been used as a heuristic defense method against adversarial perturbation.",
"Some works [29] [5] introduced randomness without provable guarantees.",
"[25] first used inequalities from differential privacy to prove robustness guarantees of $\\ell _2$ and $\\ell _1$ norm with Gaussian and Laplacian noises, respectively.",
"Later on, [27] used information theory to prove a stronger $\\ell _2$ robustness guarantee for Gaussian noise.",
"All these robustness guarantees are still loose.",
"[8] provided a tight robustness guarantee for randomized smoothing.",
"[42] integrated randomized smoothing with adversarial training to improve the performance of smoothed classifiers.",
"[59] [14] both considered natural and robust errors in their loss function.",
"The authors tried to maximize the certified radius during training.",
"These methods have great success using randomized smoothing but some researchers [16] [22] [4] raised concerns about randomized smoothing.",
"[22] [4] claimed that extending the smoothing technique to defend against other attacks can be challenging, especially in the high-dimensional regime.",
"The largest $\\ell _p$ -radius that can be certified decreases as $O(1/d^{\\frac{1}{2} - \\frac{1}{p}})$ with dimension $d$ for $p > 2$ .",
"This means that the radius decreases due to the curse of dimensionality."
],
[
"Problem Setup",
"In this section, we will set up our problem mathematically to make this paper self-contained.",
"The notations frequently used in this paper are described in Table REF in the Appendix.",
"Here, we consider the conventional classification problem.",
"Suppose that $\\mathcal {X} \\subseteq \\mathbb {R}^m$ is the set of data and $\\mathcal {Y} \\subseteq \\mathbb {R}^K$ is the set of labels.",
"We can train a (base) classifier $f : \\mathcal {X} \\rightarrow \\mathcal {Y}$ .",
"If for each $x$ one can find a $\\Vert \\delta \\Vert _p \\le \\epsilon $ such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x+\\delta ) \\ne \\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x)$ , where $\\Vert \\cdot \\Vert _p$ denotes $p$ -norm for $p > 1$ and $[K] = \\lbrace 1,2,\\cdots ,K\\rbrace $ .",
"We call $x_{adv} = x + \\delta $ adversarial example.",
"The following are our goals: To find a transformation $\\tau : C(\\mathcal {X};\\mathbb {R}^K) \\rightarrow C^\\infty (\\mathcal {X};\\mathbb {R}^K)$ , with the purpose of translating a classifier into a smoothed classifier, denoted as $\\tilde{f} = \\tau (f)$ .",
"To find a minimal $R$ with respect to $p$ -norm, where $p>1$ , such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}\\tilde{f_i}(x+R) \\ne \\operatornamewithlimits{arg\\,max}_{i\\in [K]}\\tilde{f_i}(x)$ .",
"We call $R$ the certified radius of $x$ .",
"Figure: Architecture Design.",
"Top: A typical image classifier specialized in spectral normalization, adding to each layer of a feature extractor.",
"We compress the layer xx to a small dimension.",
"Bottom: We adopt Bernstein polynomial to smooth the classifier for prediction and certification."
],
[
"Proposed Method",
"In this paper, we concern about “What is a smooth classifier?”.",
"We can think it as a function that can be differentiated infinitely times, and the function we differentiate is still continuous.",
"The simplest example we can think of is polynomials.",
"Hence, we try to approximate a neural network classifier via Bernstein polynomial.",
"Unlike convolution with a Gaussian distribution or other distributions, the exact computation of Bernstein polynomial is more natural to apply.",
"Our method contains three parts, as illustrated in Figure REF .",
"The first part is the spectral normalization.",
"This mechanism is usually introduced to stably train the discriminator in Generative Adversarial Networks (GANs).",
"Spectral normalization is adopted here because it can make the feature extractor's output be bounded due to the change of input.",
"The second part is the Bernstein polynomial approximation, which is the most important one.",
"Bernstein polynomial takes a feature vector and the classifier into consideration (See Equation (REF )).",
"We can also consider this form as uniform sampling in $[0,1]^d$ and place these samples into the base classifier to obtain the coefficients.",
"With these coefficients, we manage to use a linear combination of Bernstein basis to construct a smoother version of the base classifier.",
"To overcome the computational complexity of high dimensional Bernstein polynomials, we compress $x$ to have small (say 3$\\sim $ 6) dimensions.",
"The last part is the process of calculating the certified radius via root-finding.",
"We construct nonlinear systems of equations to characterize the points on the decision boundary and solve them by numerical root-finding techniques.",
"After applying this process, we obtain a certified radius for each input data."
],
[
"Spectral Normalization (SN)",
"According to [35], spectral normalization controls the Lipschitz constant of neural networks that were originally used in stabilizing the training of the discriminator in GANs.",
"The formulation is stated as: $W_{SN}^{(i)} = \\frac{W_i}{\\sigma (W_i)},\\sigma (W_i) = \\max _{x:x\\ne 0}\\frac{\\Vert W_ix\\Vert _2}{\\Vert x\\Vert _2},$ which is equivalent to being divided by the greatest singular value of $W_i$ , the weights of $i$ -th layer.",
"Suppose that our feature extractor $G$ (see Figure 2) is the composition function composed of Lipschitz continuous functions $g_i(x^{(i)};W_{SN}^{(i)}):= ReLU(W_{SN}^{(i)}x^{(i)} + b^{(i)})$ , where $x^{(i)}$ is the output of $g_{i-1}$ and $b^{(i)}$ is the bias for $i=1,2,\\cdots ,M$ , $i.e.$ , $G(x^{(1)}) := g_M(g_{M-1}\\cdots g_1(x^{(1)};W_{SN}^{(1)});W_{SN}^{(2)})\\cdots ;W_{SN}^{(M)}),$ where $x^{(1)} = I$ is the input data.",
"For each $g_i$ we have $\\Vert g_i(x^{(i)}+\\delta ;W_{SN}^{(i)}) - g_i(x^{(i)};W_{SN}^{(i)})\\Vert _2 \\le \\Vert W_{SN}^{(i)}\\Vert _2\\Vert \\delta \\Vert _2.$ Given two input data $I$ and $I^{\\prime }$ , and by Equation (REF ), we have $\\Vert W_{SN}^{(i)}\\Vert _2 = 1$ for $i = 1,2,\\cdots ,M$ .",
"We can derive that the Lipschitz constant of $G$ is one by Equation (REF ) and (REF ) as: If $G(I) \\ne G(I^{\\prime })$ , there exists an $R>0$ such that $R \\le \\Vert G(I) - G(I^{\\prime })\\Vert _2 \\le \\Vert I-I^{\\prime }\\Vert _2.$ We can further observe from Equation (REF ) that the upper bound of the certified radius is tighter in feature domain than that in the image domain."
],
[
"Smoothing the Classifier via Bernstein Polynomial",
"Weierstrass approximation theorem asserts that a continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.",
"All of the formal definitions are described as follows [10].",
"Theorem 1 (Weierstrass Approximation Theorem) Let $f \\in C([a,b])$ .",
"Given $\\epsilon >0$ , there exists a polynomial $p(x)$ such that $|f(x) - p(x)| \\le \\epsilon , \\hspace{8.5pt} \\forall x\\in [a,b].$ Bernstein proved the Weierstrass Approximation theorem in a probability approach [3].",
"The rationale behind Bernstein polynomial is to compute the expected value $\\mathbb {E}[f(\\eta /n)]$ , where $\\eta \\sim Bin(n,x)$ and $Bin(n,x)$ is the binomial distribution.",
"Note that the parameter $n$ , relating to how it can be used to smooth the classifier, will be elaborated in detail later.",
"The following Definition REF and Theorem REF are from [3].",
"Definition 1 (Bernstein Polynomial) Let $f(x)$ be a function defined on $[0,1]$ .",
"The $n$ -th Bernstein Polynomial is defined by $B_n(f;x) = \\sum \\limits _{k=0}^nf\\left(\\frac{k}{n}\\right)\\binom{n}{k}x^k(1-x)^{n-k}.$ Note that $B_n(f;0) = f(0), \\hspace{8.5pt} B_n(f;1) = f(1).",
"$ Theorem 2 (Bernstein) Let $f(x)$ be bounded on $[0,1]$ .",
"Then $\\lim _{n \\rightarrow \\infty }B_n(f;x) = f(x)$ at any point $x\\in [0,1]$ at which $f$ is continuous.",
"If $f \\in C([0,1])$ , the limit holds uniformly in $[0,1]$ .",
"Back to Theorem REF , we can take it as a corollary of Theorem REF .",
"Since the neural network is a multi-dimensional function, we have to generalize the one dimensional Bernstein polynomial (Definition REF ) into a multi-dimensional version.",
"The following is one kind of generalization which considers independent random variables expectation [48].",
"Definition 2 ($\\mathbf {d}$ -dimensional Bernstein Polynomial) Let $f: [0,1]^d \\rightarrow \\mathbb {R}$ .",
"We can define $d$ -dimensional Bernstein Polynomial by $\\begin{split}&B_{n_1,\\cdots ,n_d}(f;x_1,\\cdots ,x_d)\\\\&= \\sum \\limits _{\\begin{array}{c}0\\le k_j \\le n_j \\\\ j \\in \\lbrace 1,\\cdots ,d\\rbrace \\end{array}}f\\left(\\frac{k_1}{n_1},\\cdots ,\\frac{k_d}{n_d}\\right)\\prod \\limits _{j=1}^d \\binom{n_j}{k_j}{x_j}^{k_j}(1-x_j)^{n_j - k_j}.\\end{split}$ For simplicity, we abbreviate the notation by setting $\\frac{k}{n} := \\left(\\frac{k_1}{n_1},\\cdots ,\\frac{k_d}{n_d}\\right)$ and $b_j := \\binom{n_j}{k_j}{x_j}^{k_j}(1-x_j)^{n_j - k_j}$ , and have $B_n(f;x) = \\sum \\limits _{\\begin{array}{c}0\\le k_j \\le n_j \\\\ j \\in \\lbrace 1,\\cdots ,d\\rbrace \\end{array}} f\\left(\\frac{k}{n}\\right)\\prod \\limits _{j=1}^d b_j.$"
],
[
"How to Smooth the Classifier via Bernstein Polynomial?",
"As we can see from Theorem 2, as $n$ becomes larger, the Bernstein polynomial will gradually approximate the original function.",
"Thus, the rationale behind our Bernstein polynomial-based smoothed classifier is that we can choose a proper $n$ to smooth and approximate the base classifier.",
"Obviously, there is a trade-off between smoothness (robustness) and accuracy (natural accuracy).",
"We use an example to visualize Theorem REF in Figure REF .",
"Figure: Decision boundaries of the neural network variants under different nns.",
"As we can see, the smoothed decision boundary obtained with a larger nn gradually approximates the original neural network.By Definition REF , we can transform our classifier $f:[0,1]^d\\rightarrow \\mathbb {R}^K$ , $f = [f_1,f_2,\\cdots , f_K]$ , into a smoothed classifier as $\\tilde{f}(x) = [B_n(f_1;x),\\cdots ,B_n(f_K;x)],$ with a proper choice of $n$ .",
"Without any prior knowledge of the pre-trained neural network, we set $n_1 = n_2 = \\cdots = n_d = n$ , which denotes the uniform sampling in all directions.",
"We can see that the complexity of computing $d$ -dimensional Bernstein Polynomial is $O(n^d)$ , which is too high in the image domain even if we take $n=2$ .",
"To overcome this problem, we propose to exploit dimension reduction in the feature layer.",
"As we will see later in our empirical results, less than $1\\%$ natural accuracy will be sacrificed in MNIST [24] and CIFAR-10 due to dimensionality reduction in the feature space."
],
[
"Certified Radius via Root-Finding",
"To find the certified radius of the input data, different from [8], we are not trying to find a closed form solution.",
"Instead, we propose to use numerical root-finding techniques to find an adversarial example in the feature domain.",
"We abbreviate the notation by setting $\\beta _i = B_n(f_{i};x)$ and $S(\\beta )_i = \\exp (\\beta _i)/\\sum _{j = 1}^{K}\\exp (\\beta _j)$ , where $K$ is the number of classes.",
"Note that $S(\\beta )$ is the traditional softmax function which represents the probability vector.",
"Also, recall that $d \\le K$ is the dimension of a feature vector $x_0$ .",
"In the worst case analysis, the runner-up class is the easiest target for attackers.",
"Hence, we assume that the closest point to a feature vector $x_0$ on the decision boundary between the top two predictions of the smoothed classifier $\\tilde{f}(x) = \\beta = [\\beta _1,\\cdots ,\\beta _K]$ follows $\\beta _{\\rho (1)} = \\beta _{\\rho (2)}$ , where $\\rho $ is the mapping from ranks to predictions.",
"Therefore, the nonlinear systems of equations for characterizing the points on the decision boundary of smoothed classifier $\\tilde{f}(x)$ are described as: ${\\left\\lbrace \\begin{array}{ll}\\phi _0 := \\beta _{\\rho (1)} - \\beta _{\\rho (2)} = 0\\\\\\phi _1 := S(\\beta )_{\\rho (1)} - 0.5 = 0\\\\\\phi _2 := S(\\beta )_{\\rho (2)} - 0.5 = 0\\\\\\phi _3 := S(\\beta )_{\\rho (3)} = 0\\\\\\hspace{34.0pt}\\vdots \\\\\\phi _d := S(\\beta )_{\\rho (d)} = 0\\\\\\end{array}\\right.",
"}$ where $\\phi _0$ can be seen as the decision boundary between the top two predictions, and $\\phi _1$ and $\\phi _2$ are the probabilities of the top two predictions, respectively.",
"The remaining equations indicate that the probabilities other than $\\rho (1)$ and $\\rho (2)$ are all zeros.",
"The only thing we need to do is to start from the initial guess (feature vector $x_0$ ) and find the adversary nearest to it.",
"To solve Equation (REF ), we transform it into a minimization problem.",
"We consider the function $\\Phi : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ , $\\Phi = (\\phi _0, \\phi _1, \\cdots , \\phi _d)$ , and minimize $\\frac{1}{2}\\Vert \\Phi (x)\\Vert _2^2$ is equivalently to finding $x^* = \\operatornamewithlimits{arg\\,min}_{x}\\lbrace F(x)\\rbrace ,$ where $F(x)=\\frac{1}{2}\\sum \\limits _{i=0}^{d}(\\phi _i(x))^2=\\frac{1}{2}\\Vert \\Phi (x)\\Vert _2^2.$ Suppose that $x^*$ is the optimal solution to Equation (REF ), that is, $\\Phi (x^*) = 0$ .",
"According Equation (REF ), we have $F(x^*) = 0$ and $F(x) > 0$ if $\\Phi (x) \\ne 0$ .",
"The details of how to solve Equation (REF ) are described in the Appendix.",
"Let $\\emph {sol}$ denote the solution of Equation (REF ), we can easily compute $R = \\Vert x_0 - \\emph {sol}\\Vert _p$ for $p>1$ , which is known as the certified radius.",
"Remark that this radius is computed in the feature domain (see Equation (REF )).",
"On the other hand, we prove that our method satisfies the following proposition specialized in norm independence.",
"Proposition 1 Let $\\tilde{f}: [0,1]^d \\rightarrow \\mathbb {R}^K$ be a smoothed function and $A=\\lbrace \\tilde{y}\\in \\mathbb {R}^K : \\tilde{y}_i\\ne \\tilde{y}_j, \\forall i \\ne j\\rbrace $ denote the area without the smoothed decision boundary.",
"For any $x \\in [0,1]^d$ and $\\tilde{f}(x) \\in A$ , there exists an $R>0$ such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}\\tilde{f_i}(x+\\hat{\\delta })= \\operatornamewithlimits{arg\\,max}_{i\\in [K]}\\tilde{f_i}(x)$ whenever $\\Vert \\hat{\\delta }\\Vert _p < R$ for $p>1$ .",
"The proof can be found in the Appendix."
],
[
"Inference procedure",
"Suppose that we have a pre-trained model for the purpose of either prediction or certification.",
"The prediction is a simple substitution of a smoothed classifier for the base classifier.",
"Hence, we mainly illustrate our smoothing and certification procedures in Algorithm REF and REF , respectively.",
"A classifier $f$ , a feature $x_0$ , and a parameter $n$ $\\beta (= \\tilde{f}(x_0))$ , the output of the smoothed classifier Initialize an empty array $\\beta = [ \\hspace{4.25pt}]$ i=1,2,$\\cdots $ , K Compute Equation (REF ) by $f_i, x_0$ and $n$ to get $\\beta _i$ Append $\\beta _i$ to $\\beta $ $\\beta $ Smooth($f$ , $x_0$ , $n$ ) An input data $I$ , a feature extractor $G$ , a classifier $f$ , and a parameter $n$ Certified Radius $R$ $x_0 = G(I)$ $\\beta $ = Smooth($f$ , $x_0$ , $n$ ) Setup Equation (REF ) by $\\beta $ and the dimension of $x_0$ Let $x_0$ be the initial guess Transform Equation (REF ) into Equation (REF ) Solve Equation (REF ) by a least-squares solver Project $\\emph {sol}$ onto $[0,1]^d$ $R = \\Vert x_0 - \\emph {sol}\\Vert _p$ $R$ Certification($I$ , $G$ , $f$ , $n$ )"
],
[
"Deterministic algorithm",
"Note that our deterministic algorithm has two-fold meanings, $i.e.$ , deterministic smoothing and deterministic certification.",
"By the definition of the deterministic algorithm, given a particular input, it always produces the same output.",
"Hence, Algorithm REF and REF are both deterministic.",
"There are some advantages and disadvantages to the robust classification task.",
"The most significant advantage is that our algorithm does not depend on any random value; it always produces a stable output (robustness).",
"The disadvantage is that for attackers, attacking the smoothed classifier is no more difficult than attacking the non-smoothed classifier since it is end-to-end differentiable.",
"Thus, we do not claim that our method is a defense mechanism but a certification/verification method."
],
[
"Experiments",
"In this section, we aim to compare our method with state of the art certification methods, randomized smoothing [8] [42], on CIFAR-10.",
"We also compare the empirical upper bound using PGD attack [31] with our certified accuracy at all radii.",
"Other results obtained from different datasets such as MNIST, SVHN [37], and CIFAR-100 [21] would be discussed in the Appendix.",
"In addition, we perform the ablation study."
],
[
"Important Metrics",
"We are interested in three significant metrics, natural accuracy, robust accuracy, and certified accuracy.",
"Natural accuracy is the number of correctly identified clean testing examples divided by the number of total testing examples.",
"Robust accuracy is the number of correctly identified adversarial testing examples divided by the number of total testing examples.",
"Certified accuracy at radius $R$ is defined as the fraction of testing examples correctly predicted by the classifier and are provably robust within an $\\ell _2$ ball of radius $R$ .",
"For brevity, we define the certified accuracies at all radii as “certified curve\".",
"Figure: Comparing our adversarially-trained model vs. and with σ=0.12\\sigma =0.12 and σ=0.25\\sigma =0.25 on CIFAR-10.",
"Our result exhibits the highest natural accuracy under n=1n = 1 and d=5d = 5.",
"For fair comparison with their methods, we chose the NN models with the top two natural accuracy (the certified accuracy at radius equals to zero)."
],
[
"Experimental Setup",
"The architecture of our method is illustrated in Figure REF .",
"We trained the model by adding spectral normalization to every layer of feature extractor $G$ and reducing the feature vector's dimension to a small number.",
"According to [42], they employ PGD adversarial training to improve the certified curve empirically.",
"Hence, we aslo incorporated it into our model architecture at the expense of decreasing the natural accuracy slightly.",
"As for inference, we applied the reduced feature $d$ to find the exact distance to the decision boundary by using root-finding algorithms from scipy.optimize package [49].",
"Details of the algorithm we used will be further discussed in the Appendix.",
"We employed a 110-layer residual network as our base model for a reasonable comparison, which was also adopted in [8] and [42].",
"Furthermore, there are two primary hyperparameters: the parameter $n$ and the dimension $d$ of the feature vector, affecting our smoothed model."
],
[
"Evaluation Results",
"We certified a subset of 500 examples from the CIFAR-10 test set, as done in [8] and [42], that was released on the Github pages.",
"Our method took 6.48 seconds to certify each example on an NVIDIA RTX 2080 Ti with $n=1$ , while 15 seconds are required in [8] under the same setting.",
"We primarily compare with [8] and [42] as they were shown to outperform all other provable $\\ell _2$ -defenses thoroughly.",
"Recall that our method is norm-independent, as described in Proposition REF , we did the experiment in $\\ell _2$ norm to compare with their results.",
"As our work claims to maintain high natural accuracy, we utilized $\\sigma =0.12$ and $\\sigma =0.25$ from [8] and [42], which lead to the top two natural accuracy of their results.",
"For [42], we took one of their results: PGD adversarial training with two steps and $\\epsilon =64/255$ for comparison since it has the highest natural accuracy among all of their settings.",
"Note that the results are directly taken from their GitHub pages.",
"Although we added additional components such as spectral normalization (SN) to the base model for training, the natural accuracy only dropped from 93.7% to 93.5%, which nearly carries no harm than adding Gaussian noise for data augmentation.",
"Then, by applying PGD adversarial training with 20 steps and $\\Vert \\epsilon \\Vert _2=0.1$ to the base model with SN, the accuracy dropped from 93.5% to 87.6%.",
"We call the adversarial trained model “baseline\" in Table REF .",
"We applied our method with $n = 1$ to the “baseline\" model and got 87.4% accuracy, which is the highest accuracy among the methods used for comparison.",
"As shown in Figure REF , we have the following observations for comparison results: Our method obtains a higher natural accuracy at $R = 0$ .",
"Our method exhibits higher certified accuracies at lower radii ($R < 0.5$ ) but still obtains non-zero accuracies at larger radii ($R > 1.0$ ).",
"To check the rationality of our theoretical results, the empirical upper bounds of the certified curve obtained from practical attacks (PGD attack) were used for comparison.",
"Although there are some uncertainties in root finding, the default of the sum of squares error and the approximate solution error are set to $1.49\\times 10^{-8}$ which is an acceptable error.",
"Therefore, we can observe from Figure REF that our certification results approximate the empirical upper bounds.",
"The errors caused by spectral normalization and other possible reasons are discussed in the Appendix.",
"Figure: The empirical upper bounds vs. our certified curve on CIFAR-10.",
"The upper bounds are obtained by attacking the smoothed model with ∥ϵ∥ 2 ∈\\Vert \\epsilon \\Vert _2\\in {0,0.1,0.2,0.3,0.5,1,1.2,1.4,1.5} via the PGD attack with 20 steps.",
"Our certified curve is relatively conservative, and appears to be beneath the empirical upper bounds.We can observe from Algorithm REF , where $K$ is the number of the classes of a dataset, $n$ and $d$ are the parameters in Equation (REF ), the complexity is $O(Kn^d)$ .",
"As $K$ increases, the algorithm may cost an unacceptable time.",
"Therefore, due to the high computational complexity, ImageNet [11] was not considered in our results.",
"The scalability of our work would be an interesting issue for future study."
],
[
"Ablation study",
"Our method involves some issues, including dimension reduction, different model architectures, and hyperparameters, that may affect the performance.",
"To explore the effectiveness of each part, we conduct ablation studies.",
"Table: Empirical result comparison between the base classifier and the smoothed classifier.",
"The perturbation of the attacks were set to ∥ϵ∥ ∞ =2/255\\Vert \\epsilon \\Vert _\\infty =2/255 in PGD with 20 steps.",
"Acc denotes natural accuracy.",
"Note that CIFAR-10 refers to a subset of 500 examples from the test set.First, the robust accuracy should be proportional to the certified accuracy theoretically.",
"We conducted an experiment to find $n$ that results in the best robust accuracy.",
"From Table REF , we observe that as $n = 1$ and $n = 7$ , the natural accuracy and the robust accuracy under PGD are the highest among all $n$ s. Since our method with $n = 1$ spends the lowest computation time, we used $n = 1$ in all the other experiments.",
"Remark that our method is a certification instead of a defense method.",
"Thus, as we chose $n = 1 \\sim 7$ , natural accuracy and robust accuracy are slightly lower than the baseline.",
"However, we discover an intriguing phenomenon in other datasets and models in that we may have better natural accuracy and robust accuracy than the baseline with a proper choice of $n$ (see Table REF in the Appendix).",
"Figure: Left: Comparison between our method with and without adversarial training.",
"Right: Certified curve under different dimensions in our non-adversarially trained smoothed classifier.",
"Note that the dimension represents the number of equations we solve to find the root.Second, on the left side of Figure REF , we show that adversarial training is still a useful heuristic strategy that improves the certified curve while sacrificing only limited natural accuracy at radius = 0.",
"In addition, the dimension of the feature vector influences the certified curve a lot.",
"We can observe from the right side of Figure REF that the feature dimensions of length five and six lead to the best certified curve.",
"However, the larger dimension, the longer the computation time.",
"This is why the feature dimension is set to be five in our experiments.",
"Since our feature domain is constrained in the $[0,1]^d$ , there is an upper bound of the certified radius, which is the dimension's square root if we choose $\\ell _2$ norm.",
"Finally, we find that most black-box certification methods only exploit one model in their experiments, which is considered insufficient in some cases.",
"Different model architectures may differ a lot in the certified curve.",
"Hence, we employ several models, including ResNet, WideResNet, VGG, and DenseNet, to support our claim.",
"(See Figure REF in the Appendix.)"
],
[
"Remedy Over-fitting of Regression Model",
"In a regression problem, one often has a large amount of features.",
"To prevent over-fitting to the training dataset, feature engineering provides a sophisticated design.",
"Most of them are based on statistical tools to select very few representative data.",
"However, it might also not be able to prevent over-fitting and perform worse with respect to unseen data.",
"Our method provides a post-processing step to smooth the original regression model and preserve its shape.",
"Figure REF shows an example to verify our claim.",
"Figure: Regression Model.",
"In both graphs, orange dots are noisy training data and green curve denotes the model prediction.",
"Left: The original over-fitting regression model.",
"Right: After applying our method, the shape is smoothed without over-fitting to outliers."
],
[
"Conclusion and Future Work",
"We are first to achieve deterministic certification to adversarial attacks via Bernstein polynomial.",
"Moreover, we demonstrate through 2D example visualization, theoretical analyses, and extensive experimentation that our method is a promising research direction for classifier smoothing and robustness certification.",
"Besides smoothing the decision boundary, our method can also be applied to other tasks like over-fitting alleviation.",
"Since the idea “smoothing\" is a general regularization technique, it may be beneficial in various machine learning domains."
],
[
"Appendix of Deterministic Certification to Adversarial Attacks via Bernstein Polynomial Approximation",
"The appendix is organized as follows.",
"Firstly, we provide a notation table.",
"Secondly, proofs mentioned in the main paper are provided.",
"Thirdly, we offer more experimental results using MNIST, SVHN, and CIFAR-100.",
"Fourthly, we discuss how to solve non-linear systems of equations in detail.",
"Finally, we discuss the certification errors resulted from spectral normalization with skip connections, the singularity of Jacobian matrices, and the conservative constant."
],
[
"Proof of Proposition ",
"Proposition 1.",
"(also stated in the main manuscript) Let $\\tilde{f}: [0,1]^d \\rightarrow \\mathbb {R}^K$ be a smoothed function and let $A=\\lbrace \\tilde{y}\\in \\mathbb {R}^K : \\tilde{y}_i\\ne \\tilde{y}_j, \\forall i \\ne j\\rbrace $ denote the area without the smoothed decision boundary.",
"For any $x \\in [0,1]^d$ and $\\tilde{f}(x) \\in A$ , there exists an $R>0$ such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}\\tilde{f_i}(x+\\hat{\\delta })= \\operatornamewithlimits{arg\\,max}_{i\\in [K]}\\tilde{f_i}(x)$ whenever $\\Vert \\hat{\\delta }\\Vert _p < R$ for $p>1$ .",
"We can prove a more general case of Proposition REF due to the smooth condition is unnecessary as follows: Lemma 1 Let $f:[0,1]^d \\rightarrow \\mathbb {R}^K$ be a continuous function, let $A = \\lbrace y \\in \\mathbb {R}^K : y_i \\ne y_j \\forall i \\ne j \\rbrace \\subset \\mathbb {R}^K$ , and let $X = \\lbrace x \\in [0,1]^d : f(x) \\in A\\rbrace \\subset [0,1]^d$ .",
"For any $x \\in X$ , there exists an $R>0$ such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x+\\delta )= \\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x)$ whenever $\\Vert \\delta \\Vert _p < R$ for $p>1$ .",
"The proof contains two cases.",
"The first case states that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x+\\delta )$ is a set and the second case states that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x+\\delta ) \\ne \\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x)$ .",
"Case 1 : Suppose that there exists a $\\Vert \\delta \\Vert _p < R$ , and let $x+\\delta = x^{\\prime }$ such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x^{\\prime })$ is a set containing multiple elements.",
"Then $f(x^{\\prime }) \\notin A \\Rightarrow x^{\\prime } \\notin X$ .",
"Case 2 : Suppose that there exists a $\\Vert \\delta \\Vert _p < R$ , and let $x+\\delta = x^{\\prime }$ such that $\\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x^{\\prime }) \\ne \\operatornamewithlimits{arg\\,max}_{i\\in [K]}f_i(x)$ .",
"Define the distance between $f(x)$ and $f(x^{\\prime })$ by $d(f(x),f(x^{\\prime })) = r$ , there is an open ball $\\mathbb {B}(f(x),r)$ .",
"By the definition of continuous functions in topological spaces, $i.e.$ , for each open set in the codomain, its preimage is an open set in that domain.",
"Thus, $f^{-1}(\\mathbb {B}(f(x),r)) = \\mathbb {B}(x,\\delta )$ should be an open subset of $X$ .",
"However, there exists some $\\delta _1 < \\delta $ such that $x+\\delta _1 \\notin X$ .",
"This indicates that $f$ is not continuous on $X$ .",
"This is a contradiction.",
"By the results of Case 1 and Case 2, the lemma follows.",
"Hence, our method can also be applied to neural networks directly.",
"However, due to the massive amounts of deep neural network parameters, the result may be inaccurate.",
"Besides, as $n$ grows larger or the feature vector's dimension extends, the certification process will be intractable.",
"Thus, we only solve the problem with small $n$ and small dimensions of the feature vector."
],
[
"Additional Experiments",
"The datasets we use are summarized in Table REF .",
"Among them, MNIST and CIFAR-10 have been popularly used in this area.",
"Since we do not have the certified results of other methods in other datasets, we only conduct experimental comparison between the base classifier and the smoothed classifier to explore the effect of trials empirically.",
"Table: The evaluation datasets are sorted based on descending order of image size."
],
[
"For CIFAR-10",
"As mentioned before, different model architecture may differ a lot in the certified curve.",
"In Figure REF , we can see that the certified curve varies a lot.",
"For certified radius at 0, the accuracy differs from 89% (Wide ResNet) to 81% (VGG19).",
"As for radius at 0.2, the accuracy also varies in a wide range.",
"Although the tendency of certified curves look similar, their gaps may differ by more than 10%, which is a significant difference.",
"Figure: Employing different adversarially trained models to show the certified curve is actually model-dependent."
],
[
"For MNIST",
"We include MNIST [24] to demonstrate some of our claims in a simple manner.",
"As we mentioned in Table REF of the main paper, if we take $n=5$ , the natural accuracy 98.51% and robust accuracy 59.16% in our method are both larger than the baseline with the natural accuracy 98.50% and robust accuracy 52.25%.",
"As we can also see in Figure REF that there is a trade-off between the number of $n$ and certified accuracy.",
"Table: (MNIST) Empirical result comparison between the base classifier and the smoothed classifier.",
"Note that ∥ϵ∥ ∞ =0.1\\Vert \\epsilon \\Vert _\\infty =0.1 was set for attacks in FGSM and PGD (20 steps).",
"Acc denotes natural accuracy.Figure: Trade-off between the number of nn and certified accuracy (MNIST is taken as an example here.",
"We see similar phenomena in other datasets.)",
"Smaller/larger nns exhibit larger/smaller certified radii but suffer from lower/higher natural accuracies.Table: (SVHN) Empirical result comparison between the base classifier and the smoothed classifier.",
"Note that ∥ϵ∥ ∞ =0.01\\Vert \\epsilon \\Vert _\\infty =0.01 was set for attacks in FGSM and PGD with 20 steps.",
"Acc denotes natural accuracy.Table: (CIFAR-100) Empirical result comparison between the base classifier and the smoothed classifier.",
"Note that ∥ϵ∥ ∞ =2/255\\Vert \\epsilon \\Vert _\\infty =2/255 was set for attacks in FGSM and PGD with 20 steps.",
"Acc denotes natural accuracy.Figure: Comparison with different Cs.",
"As C is a conservative parameter, the larger the C is, the closer it is towards the empirical upper bounds."
],
[
"Non-Linear Least Squares Problems and Root-Finding",
"In Algorithm REF of the main paper, we adopt a least-square solver.",
"[32] is a great resource for solving nonlinear least squares problem so we suggest readers to find more details in this book.",
"To make this paper self-contained, we describe some important methods from [32] below.",
"First we consider a function $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}^m$ with $m \\ge n$ .",
"Minimizing $\\Vert f(x)\\Vert $ is equivalent to finding $x^* &= \\operatornamewithlimits{arg\\,min}_{x}\\lbrace F(x)\\rbrace .\\\\F(x) &= \\frac{1}{2}\\sum \\limits _{i=1}^{m}(f_i(x))^2 = \\frac{1}{2}\\Vert f(x)\\Vert ^2 = \\frac{1}{2}f(x)^Tf(x).$ Let $f \\in C^2(\\mathbb {R}^n;\\mathbb {R}^m)$ , and consider Taylor Expansion $f(x+h) = f(x) + J(x)h + O(\\Vert h\\Vert ^2),$ where $J(x) \\in \\mathbb {R}^{m \\times n}$ is the Jacobian matrix, we have $(J(x))_{ij} = \\frac{\\partial f_i}{\\partial x_j}(x)$ .",
"Since $F:\\mathbb {R}^n \\rightarrow \\mathbb {R}$ , we can denote the first derivative and second derivative of $F$ , respectively, in Equations (12) (13) and Equations (14) (15) as: $\\frac{\\partial F}{\\partial x_j} &= \\sum \\limits _{i=1}^mf_i(x)\\frac{\\partial f_i(x)}{\\partial x_j}.\\\\F^{\\prime }(x) &= J(x)^Tf(x).",
"\\\\\\frac{\\partial ^2F}{\\partial x_j \\partial x_k} &= \\sum \\limits _{i=1}^m \\frac{\\partial f_i(x)}{\\partial x_j}\\frac{\\partial f_i(x)}{\\partial x_k} + f_i(x)\\frac{\\partial f_i(x)}{\\partial x_j \\partial x_k}.\\\\F^{\\prime \\prime }(x) &= J(x)^TJ(x) + \\sum \\limits _{i=1}^m f_i(x)f_i^{\\prime \\prime }(x).$"
],
[
"Gradient Descent (Newton-Raphson's method)",
"Let $f(x+h) \\approx l(h) := f(x) + J(x)h$ , which is the first order Taylor expansion.",
"Suppose that we find an $h$ such that $f(x+h) = 0$ .",
"We have the following iteration form: $J(x_k)h &= -f(x_k)\\\\x_{k+1} &= x_k + h.$"
],
[
"Gauss-Newton method",
"First, we also consider $f(x+h) \\approx l(h) := f(x) + J(x)h$ and $F(x+h) \\approx L(h)$ , where $L(h) &:= \\frac{1}{2}l(h)^Tl(h), \\hspace{21.25pt}(with \\ f=f(x) \\ and \\ J=J(x)) \\\\&= \\frac{1}{2} [(f^T+h^TJ^T)(f+Jh)] \\\\&= \\frac{1}{2} [f^Tf + f^TJh + h^TJ^Tf + h^TJ^TJh]\\\\&= \\frac{1}{2}f^Tf + h^TJ^Tf + \\frac{1}{2}h^TJ^TJh\\\\&= F(x) + h^TJ^Tf + \\frac{1}{2}h^TJ^TJh.$ We can further compute the first and second derivative of $L(h)$ , respectively, as $L^{\\prime }(h) &= J^Tf + J^TJh \\\\L^{\\prime \\prime }(h) &= J^TJ.",
"$ From Equations (REF ) and (REF ), we have $L^{\\prime }(0) = F^{\\prime }(x)$ .",
"If $J$ has full rank, $L^{\\prime \\prime }(h)$ is positive definite.",
"We have the following iteration form: $J^T(x_k)J(x_k)h_{gn} &= -J^T(x_k)f(x_k) \\\\x_{k+1} &= x_k + \\alpha h_{gn}.$ Note that $h_{gn} = \\operatornamewithlimits{arg\\,min}\\limits _h\\lbrace L(h)\\rbrace $ is called Gauss-Newton step and it is a descent direction since $h_{gn}^TF^{\\prime }(x) = h_{gn}^T(J^Tf) = -h_{gn}^TJ^TJh_{gn} < 0.",
"$ Provided that $\\lbrace x | F(x) \\le F(x_0)\\rbrace $ is bounded and the Jacobian $J(x)$ has full rank in all steps, Gauss-Newton method has a convergence guarantee.",
"To overcome the constraint that the Jacobian should have full rank in all steps, the Levenberg–Marquardt method adds a damping parameter $\\mu $ to modify Equation (REF )."
],
[
"The Levenberg–Marquardt (LM) method",
"[26] and [33] defined the step $h_{lm}$ by modifying Equation (REF ) as: $(J^T(x_k)J(x_k) + \\mu I)h_{lm} &= -J^T(x_k)f(x_k).$ The above modification has lots of benefits; we list some of them as follows: $\\forall \\mu > 0$ , $(J^TJ + \\mu I)$ is positive definite which can be proved by the definition of positive definite.",
"This ensures $h_{lm}$ is a descent direction; the proof is similar to Equation (REF ).",
"For large values of $\\mu $ , we get $h_{lm} \\approx -\\frac{1}{\\mu }J^Tf = -\\frac{1}{\\mu }F^{\\prime }(x)$ .",
"This is good when $x_k$ is far from the solution.",
"If $\\mu $ is very small, then $h_{lm} \\approx h_{gn}$ .",
"This is also a good direction if x is close to the solution $x^*$ .",
"We employ Levenberg–Marquardt method (“LM\") [26] [33] for $\\ell _2$ norm certification.",
"The LM method, which is consumed with the advantages of both the Gauss-Newton method and Gradient Descent, is commonly adopted to solve Equation (REF )."
],
[
"Trust region method",
"Recall from Equation (REF ), we can also add a conservative constant $\\xi $ which is similar to the idea of soft margin in the support vector machine.",
"We abbreviate the notation by setting $\\beta _i = B_n(f_{i};x)$ and $S(\\beta )_i = \\exp (\\beta _i)/\\sum _{j = 1}^{K}\\exp (\\beta _j)$ , where $K$ is the number of classes.",
"Note that $S(\\beta )$ is the traditional softmax function which represents the probability vector.",
"Also, recall that $d \\le K$ is the dimension of a feature vector $x_0$ .",
"In the worst case analysis, the runner-up class is the easiest target for attackers.",
"Hence, we assume that the closest point to a feature vector $x_0$ on the decision boundary between the top two predictions of the smoothed classifier $\\tilde{f}(x) = \\beta = [\\beta _1,\\cdots ,\\beta _K]$ follows $\\beta _{\\rho (1)} = \\beta _{\\rho (2)}$ , where $\\rho $ is the mapping from ranks to predictions.",
"Therefore, the nonlinear systems of equations for characterizing the points on the decision boundary of $\\tilde{f}(x)$ are described as: ${\\left\\lbrace \\begin{array}{ll}\\phi _0 := \\beta _{\\rho (1)} - \\beta _{\\rho (2)} - \\xi = 0\\\\\\phi _1 := S(\\beta )_{\\rho (1)} - \\frac{1}{1+\\exp (-\\xi )} = 0\\\\\\phi _2 := S(\\beta )_{\\rho (2)} - \\frac{1}{1+\\exp (\\xi )} = 0\\\\\\phi _3 := S(\\beta )_{\\rho (3)} = 0\\\\\\hspace{34.0pt}\\vdots \\\\\\phi _d := S(\\beta )_{\\rho (d)} = 0\\\\\\end{array}\\right.",
"}$ where $\\xi = \\frac{\\beta _{\\rho (1)} - \\beta _{\\rho (2)}}{C}$ is a conservative constant and C is a parameter (see Figure REF ).",
"In addition, $\\phi _0$ can be seen as the decision boundary between the top two predictions if $C \\rightarrow \\infty $ .",
"By $\\phi _0$ , we know that $\\beta _{\\rho (2)} = \\beta _{\\rho (1)} - \\xi $ .",
"We can omit $\\beta _{\\rho (3)},\\cdots ,\\beta _{\\rho (K)}$ since they are too small.",
"By employing softmax function, we have $S(\\beta )_{\\rho (1)} = \\frac{\\exp (\\beta _{\\rho (1)})}{\\exp (\\beta _{\\rho (1)}) + \\exp (\\beta _{\\rho (2)})} = \\frac{\\exp (\\beta _{\\rho (1)})}{\\exp (\\beta _{\\rho (1)}) + \\exp (\\beta _{\\rho (1)} - \\xi )} = \\frac{1}{1+\\exp (-\\xi )}$ .",
"Also, $S(\\beta )_{\\rho (2)}$ can be computed in the same way as $S(\\beta )_{\\rho (1)}$ by setting $\\beta _{\\rho (1)} = \\beta _{\\rho (2)} + \\xi $ .",
"We can observe from $S(\\beta )_{\\rho (1)}$ and $S(\\beta )_{\\rho (2)}$ that the probability of top two predictions has a controllable gap $\\frac{1}{1+\\exp (-\\xi )} - \\frac{1}{1+\\exp (\\xi )}$ .",
"The remaining equations indicate that the probabilities other than $\\rho (1)$ and $\\rho (2)$ are all zeros.",
"Equation (REF ) reveals the decision boundary of the smoothed classifier if $\\xi $ is zero, otherwise it represents a margin slightly away from the decision boundary.",
"The only thing we need to do is to start from the initial guess (feature vector $x_0$ ) and find the adversary nearest to it.",
"We can replace Equation (REF ) with Equation (REF ) and solve Equation (REF ) by means of the trust region method [38].",
"First we consider the model function $m_k(s_k)$ by first order Taylor expansion of $\\Phi (x_k + s_k)$ , that is, $F(x_k + s_k) \\approx m_k(s_k) = \\frac{1}{2}\\Vert \\Phi (x_k) + J_ks_k\\Vert _2^2$ .",
"To be precise, $m_k(s_k)$ can be written as: $m_k(s_k) &= F(x_k) + s_k^TJ_k^T\\Phi (x_k) + \\frac{1}{2}s_k^TB_ks_k\\\\\\mathcal {T}_k &= \\lbrace x \\in \\mathbb {R}^n : \\Vert x - x_k\\Vert _p \\le \\Delta _k \\rbrace ,$ where $J_k = \\nabla \\Phi (x_k)$ , $B_k = J_k^TJ_k$ is an approximation of $\\nabla ^2 \\Phi (x_k)$ , $\\mathcal {T}_k$ is called the trust region, and $\\Delta _k$ is the radius of the trust region.",
"Besides, we need to define a step ratio $r_k$ by $r_k &= \\frac{F(x_k) - F(x_k+s_k)}{m_k(0) - m_k(s_k)} \\\\&= \\frac{\\Vert \\Phi (x_k)\\Vert _2^2 - \\Vert \\Phi (x_k+s_k)\\Vert _2^2}{\\Vert \\Phi (x_k)\\Vert _2^2 - \\Vert \\Phi (x_k) + J_ks_k\\Vert _2^2},$ where we call the numerator actual reduction and the denominator predicted reduction.",
"$r_k$ is used to determine the acceptance or not of the step $s_k$ and update (expand or contract) $\\Delta _k$ .",
"Note that the step $s_k$ is obtained by minimizing the model $m_k$ over the region, including 0.",
"Hence, the predicted reduction is always nonnegative.",
"If $r_k$ is negative, then $F(x_k+s_k)$ is greater than $F(x_k)$ and $s_k$ must be rejected.",
"On the other hand, if $r_k$ is large, it is safe to expand the trust region.",
"The trust region method [38] is depicted in Algorithm REF .",
"For each iteration, the step $s$ is computed by solving an approximate solution of the following subproblem, $\\min _{s\\in \\mathbb {R}^n}m_k(s), \\hspace{5.0pt} \\text{subject to} \\hspace{2.0pt} \\Vert s\\Vert _p &\\le \\Delta _k,$ where $\\Delta _k$ is the radius of the trust region.",
"$\\hat{\\Delta } > 0$ , $\\Delta _0 \\in (0, \\hat{\\Delta })$ and $\\eta \\in [0,\\frac{1}{4})$ Solution $x_k$ $k = 0,1,2\\cdots $ Get $s_k$ by approximately solving Equation (REF ) Evaluate $r_k$ from Equation (REF ) $r_k < \\frac{1}{4}$ $\\Delta _{k+1} = \\frac{1}{4}\\Delta _k$ $r_k > \\frac{3}{4}$ and $\\Vert s_k\\Vert = \\Delta _k$ $\\Delta _{k+1} = \\min (2\\Delta _k,\\hat{\\Delta })$ $\\Delta _{k+1} = \\Delta _k$ $r_k > \\eta $ $x_{k+1} = x_k + s_k$ $x_{k+1} = x_k$ Trust region method [38] Due to different $\\ell _p$ norm, we should change our approach to solve Equation (REF ).",
"As mentioned in the main paper, we choose LM to solve Equation (REF ) since the trust region is based on $\\ell _2$ norm.",
"If we solve Equation (REF ) in $\\ell _\\infty $ norm, we will adopt the dogbox method [50], which is a modification of the traditional dogleg trust region method [38].",
"As for other norms, we can either use the traditional dogleg method with different trust regions or solve the minimization problem by other approaches mentioned in scipy.optimize package.",
"In the 2D example (see Figure REF ), we consider a simpler version of Equation (REF ), which is $\\phi _0$ , and F(x) becomes $\\frac{1}{2}\\phi _0(x)^2$ .",
"Recall that our goal is to find a point on the decision boundary which is the closest point to the feature $x_0$ .",
"Thus, we define the objective function as $\\hat{F}(x) = \\frac{1}{2}\\phi _0(x)^2 + \\Vert x_0 - x\\Vert ^2_2$ , where $\\Vert x_0 - x\\Vert ^2_2$ is a regularization term, and adopt the trust region method [38] to minimize $\\hat{F}(x)$ .",
"Minimizing $\\hat{F}(x)$ is a generalized optimization problem with arbitrary dimensions of $x$ .",
"Thus, $-\\hat{F}(x)$ can be added to the loss function as a regularization term to fine-tune the model.",
"The certification procedure (Algorithm REF ) is time-consuming as $n$ becomes large.",
"To solve this problem, we can adopt the neural network approach.",
"As [1] stated, the convex optimization problem is unfolded into a neural network architecture.",
"Hence, we can use the fashion of neural networks to accelerate the certification procedure, which is an interesting future work."
],
[
"The uncertainty of root-finding",
"Unlike most model architectures contain skip connections, in spectral normalization, we do not consider skip connections.",
"If the feature extractor $G$ contains a skip connection, the Lipschitz constant should be computed by $\\Vert G(x) + x - G(x^{\\prime }) - x^{\\prime }\\Vert _2 &< \\Vert G(x) - G(x^{\\prime })\\Vert _2 + \\Vert x-x^{\\prime }\\Vert _2\\\\&< 2\\Vert x-x^{\\prime }\\Vert _2.$ If more skip connections are considered, the Lipschitz constant will grow in the power of two.",
"This is a reason why spectral normalization will cause the error.",
"In the training procedure, there might be other ways to control the Lipschitz constant.",
"It is an interesting future work to discuss how to control the Lipschitz constant with skip connections.",
"Consider Equation (REF ), we can see that the Jacobian is singular, which is a bad condition for solving minimization problem.",
"In this case, the solution might be inaccurate.",
"However, in practice, we have an acceptable result (see Figure REF ) due to the modification of LM (see Equation REF ).",
"A better description of the decision boundary is needed to make the Jacobian non-singular which is also an interesting topic to discuss in the future.",
"The other uncertainty is due to the conservative parameter $\\xi = \\frac{\\beta _{\\rho (1)} - \\beta _{\\rho (2)}}{C}$ in Equation (REF ).",
"We can observe from Figure REF that as $C$ gradually grows, the certified curve will approximate the empirical upper bounds.",
"However, in some scenarios, we can only choose small $C$ to make the certified result conservative enough."
]
] | 2011.14085 |
[
[
"Theoretical Accuracy Analysis of RSS-Based Range Estimation for Visible\n Light Communication"
],
[
"Abstract In this paper, an improved channel model of visible light communication (VLC) for ranging in presented.",
"For indoor channel model of VLC, distance is estimated based on received signal strength.",
"In this model, received shot noise as a distance-dependent parameter is considered in range estimation accuracy.",
"Moreover, based on this model, the Cramer-Rao lower bound is computed as the theoretical limits on the performance and accuracy of any unbiased estimator.",
"In this way, the effects of horizontal and vertical distances are investigated.",
"In addition, the transmitted power effect on RSN and accordingly on CRLB is demonstrated."
],
[
"Introduction",
"Recently, visible light communication (VLC) has gradually become a research hotspot in indoor environments because of its advantages such as illumination and a high capacity for data transmission [1].",
"Additionally, the optical communication systems are safe where radio frequency (RF) is hazardous or even forbidden (same as underground mines, airplanes, submarines, intrinsically safe environments like petrochemical industries and so on).",
"Also, the radio wave is more vulnerable to multipath effects than visible light.",
"Furthermore, the light-emitting diode (LED) based localization is a promising approach as it can provide highly accurate positioning, inexpensively.",
"Within the last years, increasing research in visible light positioning (VLP) can be observed which can make VLC as a more practical technology.",
"The VLP systems use the photodiode (PD), extracting features from the received signals such as received signal strength (RSS), angle of arrival (AOA), time of arrival (TOA), and time difference of arrival (TDOA) [2], [3], [4], [5].",
"By using the least-squares method, the accuracy of indoor visible light communication localization system based on RSS in non-line-of-sight environments is analyzed in [6].",
"In [7], the authors have investigated the simultaneous position and orientation estimation problem using RSS.",
"Cramer-Rao lower bound (CRLB) as a fundamental limit on RSS based range estimation is investigated in [8].",
"In [9], the CRLB of the position error is determined using RSS measurements.",
"It is demonstrated in [10] that RSS-based distance estimation in VLP is biased.",
"In [11], a low-complexity TDOA-based VLP system using a novel and practical localization scheme based on cross-correlation has been proposed and experimentally demonstrated.",
"Performance improvements over AOA based positioning are illustrated by simulation in which combines AOA and RSS information to enhance positioning accuracy in an asynchronous VLP system [12].",
"Moreover, [13] applies the phase difference of arrival (PDOA) as the optical wireless positioning technique.",
"Generally, four noise sources are considered in VLP, which are thermal noise, shot noise of background radiation, shot noise of dark current and shot noise of received signal (RSN).",
"The total effect of these four noises is integrated as a total noise.",
"It is assumed that the total noise has Gaussian probability distribution function (p.d.f.)",
"and the variance of this noise equals to the sum of the variance of four noise sources. [14].",
"Note that the first three noise sources are constant respect to the received signal power transmitted by LED and received in the photodetector.",
"However, opposite to other noise sources, RSN depends on received signal power and range of receiver from the transmitter.",
"Dependence of RSN variance to the distance affects the total noise variance considerably which can increase $\\sqrt{\\text{CRLB}}$ in RSS.",
"Whereas it is shown in other fields such as channel capacity [15], [16] that the effect of input-dependent Gaussian noise (RSN) can not be ignored, this effect has not been considered so far in VLP to the best of our knowledge.",
"In this letter, CRLB is calculated theoretically by taking into account received signal power-dependent RSN.",
"In this way, the effects of horizontal and vertical distances are investigated.",
"In addition, the transmitted power effect on RSN and accordingly on CRLB is demonstrated.",
"The rest of the paper is organized as follows: in Section , we describe the system model and present the basic assumptions.",
"In Section , we derive the exact closed-form expression for the CRLB.",
"Simulations and related discussions are given in Section .",
"Finally, conclusions are drawn in Section ."
],
[
"System Model and Basic Assumptions",
"We suppose a VLC system with the corresponding transmitter and receiver.",
"Let $\\phi $ be the angle of irradiance respect to the normal axis of transmitter surface and $\\varphi $ be the angle of incidence respect to the normal axis of desk plane.",
"Angles $\\phi $ and $\\varphi $ are associated with the locations of the LED source and the receiver.",
"In this case, the channel LOS gain is given by $\\small {\\eta _{\\text{LOS}} = \\left\\lbrace \\begin{array}{rl}\\frac{(m+1)S}{2\\pi d^2}\\,\\text{cos}^m\\phi \\,T_{\\text{s}}(\\varphi )\\,g(\\varphi )\\,\\text{cos}\\varphi ,\\quad 0\\le \\varphi \\le \\varphi _\\text{con}\\\\0,\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\varphi >\\varphi _\\text{con}\\end{array}, \\right.",
"}$ where $m$ is the order of Lambertian emission, $S$ is the active area of the detector, and $d$ denotes the distance between transmitter and the receiver surface.",
"$\\varphi _\\text{con}$ is the field of view (FOV) of the receiver.",
"$T_{\\text{s}}(\\varphi )$ and $g(\\varphi )$ are the gain of the optical filter and the gain of the optical concentrator, respectively.",
"For sake of simplicity, the parameters are assumed such that $T_{\\text{s}}(\\varphi )=g(\\varphi )=1$ [17].",
"In general, the multipath transmission in VLC is not challenging like RF.",
"Typically, the power of the reflected components will be much lower than LOS path [18].",
"The channel gain is almost flat and the received power is given by the channel DC gain on the directed path and diffuse links [19].",
"The channel gain is given by $H(0)=\\eta _{\\text{LOS}}+\\eta _{\\text{diff}},$ where $\\eta _{\\text{diff}}$ expresses the channel diffuse gain.",
"Thus the received optical power can be summarized by $x=R_\\text{p}\\, H(0)\\, P_\\text{t}+w=P_0+P_{\\text{diff}}+w,$ where $P_0=R_\\text{p}\\, \\eta _{\\text{LOS}} \\, P_\\text{t}$ and $P_{\\text{diff}}=R_\\text{p}\\, \\eta _{\\text{diff}} \\, P_\\text{t}$ .",
"$\\omega $ is additive white Gaussian noise (AWGN) and $R_\\text{p}$ denotes the detector responsivity.",
"$P_\\text{t}$ denotes the transmitted optical power.",
"In practice, eye safety limitations impose a fundamental constraint on the $P_\\text{t}$ .",
"Figure REF illustrates the geometry of an indoor VLC system with one LED lamp located on the ceiling.",
"The receiver has a sensor which estimates the distance between itself and the transmitter based on the measured RSS.",
"Generally, in the VLC systems, the receiver is pointed straight up at the ceiling and the transmitter is pointed straight down from the ceiling and it is placed within the receiver-field of view.",
"As a result, $\\text{cos}\\phi = \\text{cos}\\varphi = \\frac{h}{d}$ and the LOS power can be expressed as $P_0=R_\\text{p} P_\\text{t}\\frac{(m+1)S}{2\\pi }\\frac{h^{m+1}}{d^{m+3}},$ where vertical distance is represented by $h$ .",
"Based on diffuse links considered in [20], the influence to CRLB of multipath reflections is ignorable when diffuse channel gain has already been measured in a specific scenario and the modulation speed is much slower than the channel cut-off frequency.",
"Thus, $\\eta _{\\text{diff}}$ can be treated as a constant.",
"In the visible light system, the noise model could be separated into several sources as follows [21]: 1) Thermal Noise $\\sigma _{\\text{TH}}$ : The thermal (Johnson) noise is given as $\\sigma _{\\text{T H}}=\\sqrt{\\frac{8\\pi \\kappa T_{\\text{e}}}{G}\\eta SI_\\text{2}B^2+\\frac{16\\pi ^2\\kappa T_{\\text{e}}\\Gamma }{g_\\text{m}}\\eta ^2S^2I_\\text{3}B^3}$ , where $k$ denotes the Boltzmann constant, $T_{\\text{e}}$ represents the absolute temperature, $G$ is the open-loop voltage gain, $\\eta $ is the fixed capacitance of photodetector per unit area, $\\Gamma $ is the channel noise factor, $B$ is the equivalent noise bandwidth, $g_\\text{m}$ is the transconductance, and the noise bandwidth factors are $I_\\text{2}= 0.562$ and $I_\\text{3} = 0.0868$ .",
"2) Shot Noise From Background Radiation $\\sigma _{\\text{BG}}$ : The current noise from background radiation is $\\sigma _{\\text{BG}}=\\sqrt{2qR_\\text{p}p_{\\text{BS}}S\\breve{\\lambda }B}$ , where $q$ is the electron charge, $p_{\\text{BS}}$ is the background spectral irradiance, $\\breve{\\lambda }$ is the bandwidth of the optical filter.",
"3) Shot Noise From Dark Current $\\sigma _{\\text{DC}}$ : The noise from the dark current of a photodiode is $\\sigma _{\\text{DC}}=\\sqrt{2qI_{\\text{DC}}B}$ , where $I_{\\text{DC}}$ refers to the photodiode dark current.",
"4) RSN $\\sigma _{\\text{SS}}$ : The shot noise can be expressed as $\\sigma _{\\text{SS}}=\\sqrt{2q(P_0+P_{\\text{diff}})B}$ .",
"Clearly, RSN depends on LOS power.",
"Nevertheless, RSN is supposed to be constant in previous VLP works to the best of our knowledge.",
"Thus the variance of Gaussian noise detected by the receiver can be summarized by $\\sigma _0^2=\\sigma _{\\text{T H}}^2+\\sigma _{\\text{BG}}^2+\\sigma _{\\text{DC}}^2+\\sigma _{\\text{SS}}^2=\\sigma _{\\text{T}}^2+\\sigma _{\\text{SS}}^2.$ One can find a detailed discussion about noise sources and an in-depth analysis of their influence on the signal-to-noise ratio (SNR) in [22]."
],
[
"Cramer-Rao Lower Bound",
"In this part, theoretical limits on ranging accuracy are investigated by deriving the CRLB for the provided model.",
"Note that RSS can estimate only one parameter, so it is assumed that vertical distance (height) is known which is a common assumption [20].",
"First, we write the probability density function (PDF) of the received optical power using Equation (REF ) as $p(x; d)=\\frac{1}{\\sigma _0\\sqrt{2\\pi }}\\,\\exp \\left\\lbrace -\\frac{1}{2\\sigma _0^2}(x-P_0-P_{\\text{diff}})^2\\right\\rbrace ,$ By using the natural logarithm, we have $\\ln p(x; d)=-\\ln \\sigma _0\\sqrt{2\\pi }-\\frac{1}{2\\sigma _0^2}(x-P_0-P_{\\text{diff}})^2.$ Differentiating the log-likelihood function, we have $\\frac{\\partial \\ln p(x; d)}{\\partial d}=-\\frac{\\frac{\\partial \\sigma _0}{\\partial d}}{\\sigma _0}+{\\frac{\\frac{\\partial \\sigma _0}{\\partial d}}{\\sigma _0^3}}(x-P_0-P_{\\text{diff}})^2+\\frac{1}{\\sigma _0^2}(x-P_0-P_{\\text{diff}})\\frac{\\partial P_0}{\\partial d},$ concerning the relationship between the RSN and the received optical power, $\\frac{\\partial \\sigma _0}{\\partial d}$ is equal to $\\frac{\\partial \\sigma _0}{\\partial d}=qB\\big (\\sigma _{\\text{T}}^2+2q(P_0+P_{\\text{diff}})B\\big )^{-\\frac{1}{2}}\\frac{\\partial P_0}{\\partial d}=\\frac{qB}{\\sigma _0}\\frac{\\partial P_0}{\\partial d}.$ Alternatively, by using Equation (REF ), we can rewrite Equation (REF ) as $\\frac{\\partial \\ln p(x; d)}{\\partial d}=-\\frac{qB}{\\sigma _0^2}\\frac{\\partial P_0}{\\partial d}+\\frac{qB}{\\sigma _0^4}(x-P_0-P_{\\text{diff}})^2\\frac{\\partial P_0}{\\partial d}+\\frac{1}{\\sigma _0^2}(x-P_0-P_{\\text{diff}})\\frac{\\partial P_0}{\\partial d}.$ By using Equation (REF ), the second derivative would be $&\\frac{\\partial ^2\\ln p(x; d)}{{\\partial d}^2}=\\frac{2(qB)^2}{\\sigma _0^4}(\\frac{\\partial P_0}{\\partial d})^2-\\frac{qB}{\\sigma _0^2}\\frac{\\partial ^2P_0}{{\\partial d}^2}-4\\frac{(qB)^2}{\\sigma _0^6}(x-P_0-P_{\\text{diff}})^2(\\frac{\\partial P_0}{\\partial d})^2\\nonumber \\\\&-\\frac{2qB}{\\sigma _0^4}(x-P_0-P_{\\text{diff}})(\\frac{\\partial P_0}{\\partial d})^2+\\frac{qB}{\\sigma _0^4}(x-P_0-P_{\\text{diff}})^2\\frac{\\partial ^2P_0}{{\\partial d}^2}\\nonumber \\\\&-2\\frac{qB}{\\sigma _0^4}(x-P_0-P_{\\text{diff}})(\\frac{\\partial P_0}{\\partial d})^2-\\frac{1}{\\sigma _0^2}(\\frac{\\partial P_0}{\\partial d})^2+\\frac{1}{\\sigma _0^2}(x-P_0-P_{\\text{diff}})\\frac{\\partial ^2P_0}{{\\partial d}^2}.$ Upon taking the negative expected value we have $-\\mathbb {E}\\bigg [\\frac{\\partial ^2\\ln p(x; d)}{{\\partial d}^2}\\bigg ]=(\\frac{\\partial P_0}{\\partial d})^2[\\frac{2(qB)^2}{\\sigma _0^4}+\\frac{1}{\\sigma _0^2}].$ Then, the variance of any unbiased estimator must satisfy $\\sqrt{\\text{var}(\\widehat{d_0})}\\ge \\sqrt{\\frac{1}{\\sigma _0^2+2(qB)^2}}\\,\\frac{\\sigma _0^2}{|\\frac{\\partial P_0}{\\partial d}|},$ where $|\\frac{\\partial P_0}{\\partial d}|$ refers to absolute value of $\\frac{\\partial P_0}{\\partial d}$ which is given by $\\frac{\\partial P_0}{\\partial d}=-\\frac{R_\\text{p}P_\\text{t}S}{2\\pi }(m+1)(m+3)\\frac{h^{m+1}}{d^{m+4}}.$"
],
[
"Numerical Results and Discussion",
"In this section, the simulation results are provided to evaluate the performance of the RSS method.",
"The CRLB results (by taking account of the impact of the channel DC gain, as well as the RSN,) are simulated and studied in this section.",
"The simulations parameters are listed in Table REF .",
"Table: Some typical parameter valuesBy considering the RSN, the Gaussian noise variance consists of a constant parameter ($\\sigma _{\\text{T}}^2$ ) and a variable one ($\\sigma _{\\text{SS}}^2$ ).",
"According to the parameters of Table REF , we can calculate $\\sigma _{\\text{TH}}^2=1.12\\,10^{-13}$ , $\\sigma _{\\text{BG}}^2=2.38\\,10^{-14}$ , and $\\sigma _{\\text{DC}}^2=6.40\\,10^{-22}$ which in total is $\\sigma _{\\text{T}}^2=1.36\\,10^{-13}$ .",
"The value of $m$ can be expressed as $m = -\\frac{\\text{ln}2}{\\text{ln}(\\theta _\\frac{1}{2})}$ , where $\\theta _\\frac{1}{2}$ is defined as the view angle when irradiance is half of the value at $0^\\circ $ .",
"When the value of $m$ increases, it means the value of $\\theta _\\frac{1}{2}$ decreases.",
"The signal is more concentrated and the SNR is higher.",
"Therefore, the accuracy is higher.",
"By considering this fact, CRLB is plotted in [20] versus source optical power with several numbers of $m$ .",
"The effect of the variable parameter is shown in Fig.",
"REF which is a contour plot under the three-dimensional shaded surface where $m=50$ , and $P_t=1 \\text{W}$ .",
"Obviously, the effect of the RSN is mainly visible just below the LED, while further away (horizontally) from the LED, the effect is diminishable.",
"Although, because of benefits provided by higher SNRs, it is more likely to utilize the receiver near to the transmitter.",
"If so, RSN ($\\sigma _{\\text{SS}}^2$ ) is not negligible and has a noticeable effect on the analysis of indoor VLC system.",
"In Fig.",
"REF , the effect of distance over $\\sqrt{\\text{CRLB}}$ is shown where it is assumed $m=1$ , $P_t=1 \\text{W}$ .",
"$\\ell $ refers to the horizontal distance between the transmitter and receiver and $\\ell =\\sqrt{d^2-h^2}$ .",
"$\\ell $ varies from 1 to 2 meters, and $h$ varies from 1 to 3 meters.",
"If the receiver is placed at a larger distance, $\\sqrt{\\text{CRLB}}$ will increase.",
"More importantly, this figure represents that under the new model, the $\\sqrt{\\text{CRLB}}$ is higher than what was previously supposed.",
"This considerable difference originates from the nature of VLC noise model, illustrating that RSN can't be ignored and has a significant effect on the theoretical accuracy analysis of VLP.",
"As mentioned before, the other noise components are assumed to be more dominant in the literature.",
"We emphasize that without solving the CRLB in detail, the relation between RSN and CRLB could not be found.",
"Figure: The influence on the variance of Gaussian noise of RSN for a typical room environment.Figure: CRLB\\sqrt{\\text{CRLB}} for proposed accurate formula by Equation () against inaccurate one (equation (16) of ) as a common formula in literature.In Fig.",
"REF , for several values of $P_t$ , CRLB is plotted where it is assumed $m=1$ , $\\ell $ varies from 1 to 2 meters, and $h$ varies from 1 to 3 meters.",
"Obviously, by increasing the transmitted optical power, the CRLB decreases toward lower values.",
"More significantly, this figure shows that the relation between the transmitted optical power and the CRLB is so complicated.",
"$\\sqrt{\\text{var}(\\widehat{d_0})}$ is proportional to $\\frac{1}{P_t}$ for lower values of received optical power, whereas for higher values of received optical power, $\\sqrt{\\text{var}(\\widehat{d_0})}$ is proportional to $\\frac{1}{\\sqrt{P_t}}$ .",
"In other words, CRLB will decrease for higher values of received optical power.",
"Although increasing the $P_t$ is one way to rise the received optical power, eye safety regulations will limit the amount of transmitted optical power.",
"However, received optical power could be risen by some other ways such as increasing $R_p$ or $S$ , reducing $d$ , and utilizing tilting technique.",
"Figure: CRLB\\sqrt{\\text{CRLB}} for proposed accurate formula by Equation () with several transmitted optical powers.Figure: The proportion of accurate formula to inaccurate formula with several transmitted optical powers.The proportion of accurate formula of $\\sqrt{\\text{CRLB}}$ to inaccurate one is shown in Fig.",
"REF .",
"The figures present the mean of $\\sqrt{\\text{CRLB}}$ with regard to horizontal distance where $m=1$ , $\\ell $ varies from 1 to 2 meters, and $h$ varies from 1 to 3 meters.",
"It can be clearly seen from Fig.",
"REF that the figures grow by increasing $P_t$ .",
"However, the rate of growth declines for higher values of transmitted optical power.",
"This is because the impact of $P_t$ on CRLB will decrease sharply for higher values of received optical power.",
"In addition, Fig.",
"REF illustrates that the relation between the transmitted optical power and the CRLB is complicated which should be discussed in the future.",
"Figure: CRLB\\sqrt{\\text{CRLB}} against the order of Lambertian emission with several values of horizontal distance and vertical distance.In Fig.",
"REF , the value of $\\sqrt{\\text{CRLB}}$ is evaluated by varying the horizontal distance and vertical distance.",
"The radiation model of LEDs is not perfect Lambertian distribution in practical applications and $m$ might be different in different LED manufacturers.",
"We vary the value of $m$ from 1 to 80.",
"Fig.",
"REF indicates that by increasing horizontal distance for the given height, $\\sqrt{\\text{CRLB}}$ rises where $P_t=1 \\text{W}$ .",
"Moreover, this figure shows that $\\sqrt{\\text{CRLB}}$ is a convex function, i.e.",
"$\\sqrt{\\text{CRLB}}$ has a minimum value where $m=m_\\text{opt}$ .",
"It is proved in [23] that $m_\\text{opt}\\approx -(2+\\frac{1}{\\text{lncos}\\phi })+\\sqrt{1+(\\frac{1}{\\text{lncos}\\phi })^2}$ .",
"In the future works, we would calculate $m_\\text{opt}$ more accurately."
],
[
"Conclusion",
"For an indoor positioning system based on visible light, we have presented an analysis of the CRLB of RSS-based ranging.",
"In this work, RSN is considered in modelling and showed that can not be negligible.",
"By taking into account RSN and its relation to distance parameter, a more accurate noise model and closed-form expression for CRLB are achieved.",
"A detailed discussion about the parameters that determine the CRLB and their dependence on the geometry of the system is also presented.",
"In addition, we have investigated the effect of Lambertian emission order on the $\\sqrt{\\text{CRLB}}$ and showed that $\\sqrt{\\text{CRLB}}$ is a convex function."
]
] | 2011.14080 |
[
[
"Quantum body in uniform magnetic fields"
],
[
"Abstract In this article it will be presented the first attempt made in order to perform gauge invariant calculations of eigenstates of a quantum body in its condensed phase, the latter reacting to an external uniform magnetic field.",
"The target is achieved introducing a new unitary translation operator transforming eigenstates into a new set of eigenstates having different total linear momentum.",
"This new quantum representation solves the problem of calculating the magnetic response of quantum eigenstates of finite or either infinite periodic systems to uniform magnetic fields, where equivalence between the customarily used representation and the new representation has been made."
],
[
"Introduction",
"Since last decades many attempts has been made in order calculating the response of matter compounds to external magnetic fields[2], [1] where also problems have been faced, in computational physics[3],of calculating the orbital magnetisation of a quantum body[2], [1].",
"There has been the case of periodic boundary conditions applied to the system at null external fields in order reach a first clear understanding of the magnetism at the quantum level, but still on debate, here differently it is put on relevance the use can be made of a transformation of the eigenstates via a new set of unitary operators translating the linear momentum of the set of eigenstates, the latter interacting with a magnetic field.",
"The appearance of this unitary transformation allows calculating directly the Hamiltonian of the system passing from a position dependent kinetic energy operator to an independent one[9], [8], [1] called boundary free, showing the representability of the Hamiltonian being still in a gauge invariant form where measuring of the Hamiltonian operator can be brought forward in the linear momentum space.",
"There changing of its values can be directly measured by experimental means making the theory here presented extremely useful when calculating eigenstates dependent on the wave vector ${\\bf {k}}$ .",
"It is chosen eigenstates of matter being in a Bloch form[4], [5], [6], [1], [2] or either the new representation introduced in[7] making it worth calculating Hamiltonian eigenstates for an estimation of the quantum magnetization $\\bf {M}$ of material compounds at the quantum level, wherethe latter are dependent on external magnetic fields.",
"In the first section will be dervived the unitary transformation of eigenstates of the quantal system, with the aim to introduce for the first time this new representation and find in the second section a valuable expression of an energy functional, then allowing to calculate a quantum weighted magnetization.",
"Conclusions are reported at the end of this article."
],
[
"Unitary Transformation of Quantum Eigenstates",
"In this section it will be derived an algorithm in order to model the quantum Hamiltonian of a quantum sample of matter being it in a finite or infinite shape, still considering a boundary free[1], [2] formalism when is the case of the avoidance of the position operator by the kinetic energy part of the Hamiltonian.",
"Differently by Zak models[8] here it is considered the set of quantum eigenstates of matter system being in a Bloch form not recurring to a representation where the position coordinates have a lack of a direct interpretation still considering anyhow eigenstates of matter being dependent on the wave vector ${\\bf {k}}$ .",
"It is worth considering ${\\bf {k}}$ vectors of a quantum eigenstate of matter representing still the linear momentum of the system[8], [4], [9] where only the Bloch gauge is employed, i.e.",
"where the form of the eigenstates is still represented as a phase factor multiplied by an eigen amplitude spanning the Hilbert space of the system: $\\Psi _{n,{\\bf {k}}}=e^{i{\\bf {k}}\\cdot \\bf {r}}u_{n,{\\bf {k}}}$ The unclear idea of a quasi momentum ${\\bf {k}}$ required by Bloch and Zak[4], [8] during their first study of the transport of electrons in crystals, where only periodicity is asked, makes difficult recognizing the behaviour of electrons should have on a finite sample where still eigen energies are amenable having dispersion relations dependent by the wave vector ${\\bf {k}}$ of the state.",
"Either in an amorphous system appears the ${\\bf {k}}$ dependence of eigenstates as so that the latter should not be confined to a prescribed unitary cell of a reciprocal space[6] considering the sampling of the ${\\bf {k}}$ vectors always due to a scattering process may affect the linear momentum of the system itself.",
"Here considering the special case of a uniform magnetic field acting on a quantum body the wave vector should be scattered by a linear momentum, as it is well known[9], [10], proportional to the potential vector ${\\bf {A}}={\\frac{e}{c}}\\bf {B} \\times \\bf {r}$ of the system generating then on it a total linear momentum $\\bar{P}$ : $\\bar{P}=\\langle \\Psi {|\\bf {p}} - {\\frac{e}{c }}{\\bf {A}}|\\Psi \\rangle $ it making a position dependent kinetic energy term in the Hamiltonian.",
"The need of having a consistent unitary theory in order describe amorphous systems as well as periodic systems such as crystals makes then mandatory need of having an Hamiltonian operator represented by a boundary free operator, i.e.",
"not dependent on the shape of the matter compound under study.",
"The target is reached by use of a particular Unitary transformation $U$ of eigenstates $\\Psi _{n,{\\bf {k}}}$ that brings eigenstates at wave vector ${\\bf {k}}$ to eigenstates of scattered linear momentum ${{\\bf {k}}}-\\frac{e}{c}\\bf {A}$ .",
"The operator $U$ reads: $U=e^{-i{\\frac{e}{c}(\\bf {B} \\times \\bf {i\\nabla _{{\\bf {k}}}})\\cdot \\bf {r}}}$ Latter operator can be recasted as follows: $U=e^{i{\\frac{e}{c}(\\bf {B} \\times \\bf {r})\\cdot {i\\nabla _{{\\bf {k}}}}}}$ and by definition of a translation operator[9] it acts shifting the eigenstates from ${\\bf {k}}$ to ${{\\bf {k}}}-\\frac{e}{c}\\bf {B \\times r}$ : $U\\Psi _{n,{\\bf {k}}}=e^{i{\\frac{e}{c}(\\bf {B} \\times \\bf {r})\\cdot {i\\nabla _{{\\bf {k}}}}}}\\Psi _{n,{\\bf {k}}}=\\Psi _{n,{\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}}$ making then possible evaluating at the knowledge of eigenstates $\\Psi _{n,{\\bf {k}}}$ the whole set of eigenstates $\\Psi _{n,{\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}} $ of the quantum body interacting with a uniform magnetic field $\\bf {B}$ .",
"The Hamiltonian operator $H$ acting on $\\Psi _{n,{\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}}$ becomes: $H\\Psi _{n,{\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}}={[\\frac{\\bf {p}^2}{2m}+V(\\bf {r})]}\\Psi _{n,{\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}}$ where operator $H$ is formally transformed to the Hamiltonian operator $\\tilde{H}=U^+HU$ ,the latter acting on eigenstates $\\Psi _{n,{\\bf {k}}}$ , giving back the following eigenvalue equation: $\\tilde{H}\\Psi _{n,{\\bf {k}}}={[\\frac{({\\bf {p}}-\\frac{e}{c}\\bf {B}\\times i\\nabla _{{\\bf {k}}})^2}{2m}+V(\\bf {r})]}\\Psi _{n,{\\bf {k}}}$ In order to show the equivalence between eq.",
"(REF ) and the general Hamiltonian eigenvalue equation customarily employed in condensed matter physics, with the aim of calculating eigenstates of matter interacting with a uniform magnetic field: $H\\Psi _{n,{\\bf {k}}}={[\\frac{({\\bf {p}}+\\frac{e}{c}\\bf {B}\\times \\bf {r})^2}{2m}+V(\\bf {r})]}\\Psi _{n,{\\bf {k}}}$ it is worth to notice, being the shift of ${\\bf {k}}$ to ${\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}$ , the eigenvalue equation for the quantum chiral eigenstate $\\Psi $ is reached as: $H\\Psi _{n,{\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}}={[\\frac{({\\bf {p}}+\\frac{e}{c}\\bf {B}\\times \\bf {r})^2}{2m}+V(\\bf {r})]}\\Psi _{n,{\\bf {k}}}$ by a direct shift of the reference system of wave vector ${\\bf {k}}$ to ${\\bf {k}}-\\frac{e}{c}\\bf {B \\times r}$ , then showing that a macroscopic energy observable is still definable and still invariant by the shifting of the quantum linear momentum in $\\bf {k}$ space.",
"The latter enable us reaching a quantification of the amount of the gauge invariant energy of a magnetic systems, solving then the targeted problem of calculating matter response to external uniform magnetic fields.",
"In the next section it will be reported a derivation of the quantum energy functional should be employed in order calculate eigenstates of matter in a boundary free form while also deriving the expression of an Orbital Magnetisation.",
"Conclusions are left to the last part of the article."
],
[
"Energy functional of a body in uniform magnetic fields",
"In this section it will be introduced an energy functional, we shall perform a functional derivative, in order to obtain a set of quantum eigenvalue differential equations, that can be directly implemented in first principle codes[3] with the aim of modeling the ab initio response of quantum matter to uniform magnetic fields.",
"It is clear by the definition of the Hamiltonian operator reported in eq.",
"(REF ) that the energy observable can be written in terms of powers of $\\frac{1}{c}$ as in the following: $&&\\langle \\Psi _{n,{\\bf {k}}}|\\tilde{H}|\\Psi _{n,{\\bf {k}}}\\rangle ={\\langle \\Psi _{n,{\\bf {k}}}|[\\frac{(\\bf {p}-\\frac{e}{c}\\bf {B}\\times i\\nabla _{{\\bf {k}}})^2}{2m}+V(\\bf {r})]}|\\Psi _{n,{\\bf {k}}}\\rangle \\\\\\nonumber $ and the Hamiltonian operator becomes, $&&{\\tilde{H}=[\\frac{{\\bf {p}^2}}{2m}-\\frac{e{\\bf {p}}}{mc}\\cdot {(\\bf {B}\\times i\\nabla _{{\\bf {k}} })}+ \\frac{e^2}{2mc^2}{(\\bf {B}\\times i\\nabla _{{\\bf {k}}})^2}+V(\\bf {r})]}$ allowing to evaluate the energy functional derivatives with respect to eigenstates of the body, where are considered variations of the total energy with respect to the quantum eigenfunctions $\\Psi ^*_{n,{\\bf {k}}}$ : $\\frac{\\delta [ \\langle \\Psi | \\tilde{H} |\\Psi \\rangle -E\\langle \\Psi | \\Psi \\rangle ]}{\\delta \\Psi ^*}=0 \\\\\\nonumber $ A linear energy term can be evaluated,t due to the magnetic field coupling with a quantum vector field $\\bf {M}$ , the latter amenable of being recognized the quantum orbital magnetisation of matter[2], [1] where we expand the Hamiltonian operator only to first powers of $\\frac{1}{c}$ reaching the following eigenvalue equation: $\\tilde{H}|\\Psi _{n,{\\bf {k}}}\\rangle ={[\\frac{\\bf {p}^2}{2m}-\\frac{e{\\bf {p}}}{mc}\\cdot {(\\bf {B}\\times i\\nabla _{{\\bf {k}} })}+V(\\bf {r})]}|\\Psi _{n,{\\bf {k}}}\\rangle $ Demonstration brought forward until now determines an expectation energy functional linear, in terms of the uniform magnetic field, then write the energy as a scalar product of the magnetic field and of a new field, we may recognize, related to the orbital magnetization $\\bf {M}$ of the body found in the case of null external uniform magnetic field[2], [1] where it is the case of a crystal.",
"In fact energy functional can be written as: ${\\langle \\Psi _{n,{\\bf {k}}}|\\tilde{H}|\\Psi _{n,{\\bf {k}}}\\rangle =\\langle \\Psi _{n,{\\bf {k}}}|\\frac{{\\bf {p}^2}}{2m}|\\Psi _{n,{\\bf {k}}}\\rangle + E_{M,n} +\\langle \\Psi _{n,{\\bf {k}}}|V(\\bf {r})}|\\Psi _{n,{\\bf {k}}}\\rangle \\\\\\nonumber $ where the amount of energy due to the magnetization of the body is then: $&&E_{M,n}= \\bf {B}\\cdot \\bf {M_n}\\\\\\nonumber &&{\\bf {M_n}}= \\frac{e}{mc}\\langle \\Psi _{n,{\\bf {k}}}|{\\bf {p}\\times i\\nabla _{{\\bf {k}} }}|\\Psi _{n,{\\bf {k}}}\\rangle $ The evaluation of the energy term $E_{M,n}$ allows also having at hands a linear response of matter to magnetic fields and still define an orbital magnetization density either in the general case of finite amorphous systems as an invariant trace of the vector $M_n$ or either its weigthed average value as further reported : $&&{\\bf {M}_{T}}=\\frac{1}{V}\\sum _n{\\bf {M_n}}\\\\\\nonumber &&{\\bf {M}}=\\frac{1}{V}\\sum _n f_n{\\bf {M_n}}$ being $f_n$ the occupation numbers of the eigenstate $n$ of the quantum body and $V$ its volume.",
"The formula stays invariant either in the thermodynamic limit for infinite crystals or for the more appealing finite periodic systems routinely encountered in computational physics, then allowing to have a unifided theory making possible calculate magnetic properties of quantum matter or either compute them at the first principles level in an abinitio framework[3].",
"Put on forward these main results it will be reported conclusionsof the article in the next section."
],
[
"Conclusions",
"In this article it has been introduced a new quantum representation by the use of a gauge transformation of quantum eigenstates of matter in order to solve the problem of calculating the magnetic response of quantum eigenstates of finite or either infinite periodic systems to uniform magnetic fields.",
"Equivalence between the customarily used representation and the new representation has been made, it solving the problem of calculating quantum Hamiltonian operators in a boundary free context, i.e.",
"having a form of the kinetic energy operator independent of the position operator, it leaving to the freedom of calculating the magnetic response of matter in crystals as well as finite amorphous samples.",
"This algorithm may appear then useful in a first principles context where an ab initio modelling of matter is made in order to simulate numerically the magnetic response of a quantum body to uniform magnetic fields[3]."
]
] | 2011.14082 |
[
[
"Curvature Regularization to Prevent Distortion in Graph Embedding"
],
[
"Abstract Recent research on graph embedding has achieved success in various applications.",
"Most graph embedding methods preserve the proximity in a graph into a manifold in an embedding space.",
"We argue an important but neglected problem about this proximity-preserving strategy: Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models.",
"To address the problem, we propose curvature regularization, to enforce flatness for embedding manifolds, thereby preventing the distortion.",
"We present a novel angle-based sectional curvature, termed ABS curvature, and accordingly three kinds of curvature regularization to induce flat embedding manifolds during graph embedding.",
"We integrate curvature regularization into five popular proximity-preserving embedding methods, and empirical results in two applications show significant improvements on a wide range of open graph datasets."
],
[
"Introduction",
" Recent research on graph embedding has achieved considerable success to represent graph data in various applications, such as node classification [1], [2], link prediction [3], [4], community discovery [5], [6], and recommendation [7], [8].",
"Graph embedding aims to encode the topology patterns in a graph into distributed vector representations, which can be readily fed into machine learning models for downstream applications.",
"Thus, graph embedding bridges the gap between non-Euclidean graph structures and machine learning models operating in an Euclidean space.",
"Most graph embedding methods aim to preserve the proximity in a graph into an embedding space.",
"For instance, the objective of node embedding is that proximal nodes in a graph should have similar representations in the embedding space, wherein proximal nodes are usually defined as co-occurrence nodes in a random walk path [9] or neighboring nodes some hops away [10].",
"Such proximity-preserving embedding methods are efficient and flexible, and the commonly employed models include matrix factorization, deep neural networks, and edge reconstruction models [11].",
"As graph embedding places nodes on an embedding manifold (a metric space containing a set of points and a geodesic distance function) in an ambient embedding Euclidean space, we argue an important but neglected problem about the proximity-preserving graph embedding.",
"Graph topology patterns, while preserved well into an embedding manifold by preserving proximity, may distort in the ambient embedding Euclidean space, and hence to detect them becomes difficult for machine learning models that operate in an Euclidean space.",
"Motivating case study.",
"We present a case study to illustrate and analyze such pattern distortion, as shown in Figure REF .",
"For a toy graph in panel A, we seek its 2-D node embedding by preserving proximity, as shown in panel B1.",
"Meanwhile, an oracle node embedding is estimated through isometric embedding as a comparison in panel B2.",
"The adopted embedding methods are Isomap [12] and its variant (see Appendix for technical details).",
"We have three observations.",
"Figure: A case study on pattern distortion in graph embedding.",
"Observation 1: Graph topology-patterns are preserved in an embedding manifold via proximity-preserving graph embedding.",
"For both B1 and B2 embeddings, the connected nodes in the graph tend to be close in the embedding space since the proximity in the graph is preserved.",
"From the perspective of geometry, proximity-preserving graph embedding is a mapping from a graph to its isometric homeomorphic manifold [13].",
"In other words, for any two nodes, their shortest-path distance in the graph is consistent with their geodesic distanceThe geodesic distance between two nodes in an embedding manifold is defined as the length of the shortest polygonal curve between the two nodes (points).",
"A shortest polygonal curve is shown as the orange curve in Figure REF B1.",
"A formal definition is in problem formulation section.",
"in the embedding manifold.",
"As shown in panel C, the two embeddings in B1 and B2 both preserve well the consistency between the two kinds of distance, so they both preserve graph proximity– and thus also the implied topology patterns– in their embedding manifolds via proximity-preserving embedding.",
"E.g., the homophily in the graph (i.e., similar nodes are more likely to connect than dissimilar ones [14]) is presented as spatial clustering (the green boundary) of similar nodes, in term of node labels, in the embeddings in both B1 and B2.",
"Observation 2: However, such patterns inevitably distort in the ambient embedding space.",
"For the embedding in panel B1, nodes far apart in embedding manifold, as measured by geodesic distance (the orange curve with arrows), may have a short straight-line distance (grey lines with arrows) in the ambient Euclidean space of the embedding manifold.",
"As the embedding manifold may be folded, twisted, and curved in the ambient Euclidean space, such distance divergence is inevitable for embeddings learned by preserving proximity.",
"As shown in panel D, the distance divergence is much larger in a curved embedding manifold (panel B1) than in a flat embedding manifold (panel B2).",
"The divergence between graph and Euclidean distances reflects that geometry patterns in the embedding manifold distort in its ambient Euclidean space.",
"Thus, we define pattern distortion as the distance divergence between an embedding manifold and its ambient Euclidean space.",
"Mathematically, we say the manifold has distortion $\\rho $ in the ambient Euclidean space, $\\rho = \\frac{1}{n(n-1)} \\sum _{i \\ne j} \\frac{d_\\mathcal {M}(\\mathbf {x}_i,\\mathbf {x}_j)}{d_\\mathcal {E}(\\mathbf {x}_i,\\mathbf {x}_j)},$ where $n$ is the number of nodes, vector $\\mathbf {x}_i$ and $\\mathbf {x}_j$ are the representations of node $v_i$ and $v_j$ in the embedding space, and $d_\\mathcal {M}(\\cdot ,\\cdot )$ and $d_\\mathcal {E}(\\cdot ,\\cdot )$ denote the geodesic distance function in the embedding manifold and the Euclidean distance function in the ambient Euclidean space, respectively.",
"The minimum of the distortion is 1.0, which indicates a flat manifold where geodesic distance is equal to Euclidean distance, e.g., a straight line.",
"Observation 3: Such distortion hinders machine learning applications operating in an embedding space.",
"The embedding in B2 exhibits more discriminative spatial distribution than B1, which is desirable for machine learning applications, e.g., node classification, because of a large margin.",
"Obviously, the difficulties to detect patterns from the two embeddings are very different, although both of them similarly preserve the graph topology patterns in their embedding manifolds (Observation 1).",
"Since the geometry patterns preserved in an embedding manifold may distort in its ambient Euclidean space (Observation 2), which most of popular machine learning models [15] operate in and hence “see” only the Euclidean patterns, detecting those patterns becomes difficult or even impossible.",
"Now, we can conclude that a good embedding should not only preserve topology patterns in a graph (Observation 1), but also eliminate the distortion of its embedding manifold in the ambient Euclidean space (Observation 2), so as to present Euclidean patterns that can be detected effectively by machine learning models (Observation 3).",
"This case study, although leveraging specific embedding methods (Isomap and its variant), is representative of proximity-preserving graph embedding methods, because such distortion is inevitable due to their common strategy of (only) proximity preservation.",
"We note that many popular graph embedding methods (e.g., matrix factorization and deep learning-based methods) use dot product to measure the similarity between representations of nodes in an embedding space, which is different from Euclidean distance used in this case study.",
"However, the distortion measure $\\rho $ defined in Eq.REF also apply to reflect the pattern distortion for those embedding methods because dot product is highly related to Euclidean distance [16].",
"It's important to note that the pattern distortion not only exists in Euclidean embedding, but also exists in non-Euclidean embedding [17].",
"Existing proximity-preserving embedding methods mostly focus on how to preserve the topology patterns effectively and efficiently, but rarely notice the problem to eliminate the distortion, which motivates us to study this neglected but important problem.",
"In the literature, a straightforward strategy to prevent the distortion is isometric embedding, e.g., Isomap, the method used to obtain the oracle embedding in Figure REF B2.",
"However, isometric embedding involve scalability problem because of its very high time complexity, which limit their applications in practice.",
"Our solution.",
"In this paper, we address the distortion problem in proximity-preserving graph embedding through an intuitive idea: Since the distortion is caused by curved embedding manifolds, if we can enforce the embedding manifolds to be flat, the distortion will be prevented.",
"Fortunately, curvature, a concept from differential geometry [18], aims to measure how far a manifold (e.g., a curve or a surface) is from being flat.",
"If we have one curvature that can apply to embedding manifold, we can restrict the curvature during proximity-preserving embedding, so as to induce a flat embedding manifold which has a low distortion $\\rho $ in ambient Euclidean space.",
"The challenge to implement the idea is lacking a existing curvature that can measure embedding manifold in graph embedding setting.",
"In the literature, most of curvatures, e.g., Gaussian curvature, are defined by the gradient of manifold function which cannot get in graph embedding setting where we only know the node/edge representations, i.e., a set of points, in embedding space.",
"In this paper, we fill this void by proposing a novel angle based sectional curvature, termed ABS curvature, to measure the curvature of the embedding manifold.",
"Accordingly, we present three kinds of curvature regularization that can be readily incorporated into existing proximity-preserving node embedding methods so as to induce flat embedding manifolds with a low distortion.",
"Finally, we empirically validate the algorithm by incorporating the regularization term into five popular node embedding methods for performing node classification and link prediction tasks on eight open datasets of graphs.",
"In summary, the contribution of this paper is four-fold: 1) We raise and formulate an important but neglected problem in proximity-preserving graph embedding, the geometry patterns in embedding manifold distort in the ambient Euclidean space; 2) We provide a solution to prevent the distortion, to restrict the curvature of embedding manifold during embedding; 3) We propose a novel curvature, ABS curvature, for embedding manifold, present three kinds of curvature regularization, and develop a curvature regularization optimizing algorithm; 4) We validate our method by incorporating the regularization term into five popular node embedding methods on eight open datasets of graphs."
],
[
"Problem Formulation",
" For convenience of presentation, this paper only considers the setting of proximity-preserving node embedding.",
"The developed method can be readily extended to other graph embedding tasks, e.g., edge embedding.",
"We firstly describe proximity-preserving node embedding from a geometry perspective.",
"Let $\\mathcal {G}=(V,E)$ be a graph, where $V$ and $E$ are node set and edge set, respectively.",
"Each edge $e_{i,j} \\in E$ connects two nodes $v_{i},v_{j} \\in V$ .",
"Let $f: v_i \\rightarrow \\mathbf {x}_i$ be an embedding function mapping a node $v_i$ in graph to a representation vector $\\mathbf {x}_i$ of $d$ dimension, $d \\ll |V|$ .",
"The representation $\\mathbf {x}_i \\in X$ can be considered as a point in an embedding (Riemannian) manifold $\\mathcal {M}$ , and $X$ is a set of points.",
"The geodesic polygonal curve in the manifold, like straight line in the Euclidean space, is denoted by $P_{i,j}$ and defined by the ordered nodes along the shortest paths in the graph.",
"The length of geodesic polygonal curve is geodesic distance $d_{\\mathcal {M}}$ .",
"The manifold is embedded in an ambient Euclidean space, and around any point $\\mathbf {x}_i$ , the manifold locally resembles a Euclidean tangent space $\\mathcal {T}_{\\mathbf {x}_i}\\mathcal {M}$ .",
"Definition 1.",
"(Geodesic Polygonal Curve) For two points $\\mathbf {x}_i$ and $\\mathbf {x}_j$ in embedding manifold $\\mathcal {M}$ of graph $\\mathcal {G}$ , the geodesic polygonal curve $P_{i,j}$ between them is a polygonal curve specified by a sequence of edges $e_{q^{^{\\prime }},q^{^{\\prime \\prime }}},(q^{^{\\prime }},q^{^{\\prime \\prime }}) \\in \\Gamma _{i,j}$ , where $\\Gamma _{i,j}$ is an ordered set and contains the indexs of edges along the shortest path between node $i$ and $j$ in graph $\\mathcal {G}$ .",
"Definition 2.",
"(Geodesic Distance) Geodesic distance in embedding manifold $\\mathcal {M}$ is the length of a geodesic polygonal curve.",
"Specifically, for two points $\\mathbf {x}_i$ and $\\mathbf {x}_j$ in $\\mathcal {M}$ , $d_{\\mathcal {M}}(\\mathbf {x}_i,\\mathbf {x}_j) = \\sum _{(q^{^{\\prime }},q^{^{\\prime \\prime }}) \\in ~ \\Gamma _{i,j}} d_{\\mathcal {E}}(\\mathbf {x}_{q^{^{\\prime }}},\\mathbf {x}_{q^{^{\\prime \\prime }}}),$ where $d_{\\mathcal {E}}(\\mathbf {x}_{q^{^{\\prime }}},\\mathbf {x}_{q^{^{\\prime \\prime }}})$ is the local Euclidean distance between the two consecutive points along $P_{i,j}$ , the geodesic polygonal curve.",
"For any node $v_i$ , proximity-preserving graph embedding requires its neighbor node $v_{j}$ has a similar representation $\\mathbf {x}_j$ with $\\mathbf {x}_i$ in local Euclidean tangent space $\\mathcal {T}_{\\mathbf {x}_i}\\mathcal {M}$ .",
"Specifically, the proximal nodes are commonly defined as co-occurrence nodes in a random walk path [9] or neighbor nodes one hop or $r$ hops away [10].",
"And the similarity in local tangent space is usually measured by Euclidean distance [19], [20] or dot product [5], [21].",
"The problem we study in this paper is how to minimize the distortion $\\rho $ defined by Eq.REF during above proximity-preserving node embedding."
],
[
"Preliminaries",
" Curvature.",
"Curvature is a concept from differential geometry, and it measures how far a manifold (e.g., a curve or a surface) is from being flat.",
"Taking curve as an example, we can intuitively consider curvature as the difference from a curve to its tangent at a point.",
"The curvature of a straight line is zero.",
"Mathematically, the curvature of a curve is defined as the rate of change of direction of tangent at a point that moves on the curve at a constant speed.",
"For 2-dimensional surface, normal curvature is usually used to measure how a surface differs from its flat tangent plane around one point.",
"Consider planes that are perpendicular to the tangent plane at a point on the surface, i.e., the planes contain the normal vector of the surface at that point.",
"Those planes are called normal plane.",
"The intersection of a normal plane with the surface forms a normal section (i.e., a curve) whose curvature is able to profile the curvature of the surface along the curve.",
"The curvatures of the normal sections formed by all normal planes are called normal curvatures of the surface at the point, as illustrated in Figure REF A.",
"The maximum and minimum normal curvatures are called principal curvatures; the product of the two principal curvatures is the Gaussian curvature.",
"For $n$ -dimensional manifold embedded in a ambient Euclidean space, sectional curvature is one way to define its curvature, which is an extension of normal curvature to high dimensional space.",
"Like surfaces have tangent planes, a $n$ -dimensional manifold at a point also has a Euclidean tangent space of $n$ -dimension.",
"For a plane in the tangent space, there is a surface in the manifold, the surface has the plane as its tangent plane.",
"That is, the surface consists of geodesic curves in the manifold emanating from the point through all the directions in the tangent plane.",
"The curvatures of these geodesic curves reflect how the manifold differs from its tangent space along the directions.",
"The Gaussian curvature of the surface is called the sectional curvature of the manifold for the tangent plane, and the sectional curvatures for all tangent planes can completely describe the curvature of the manifold around the tangent point.",
"Figure: Panel A is an illustration of normal curvature of 2-dimensional surface.",
"The intersection of a normal plane with the surface forms a normal section (red curve) whose curvature is used to profile the curvature of the surface along the curve.",
"Panel B is an illustration of angle based curvature of polygonal curve.",
"The angle based curvature at point 𝐱 i \\mathbf {x}_i is defined as the turning angle, κ i =θ i \\kappa _i = \\theta _i.",
"Red arrows denote the sign of the angles.Discrete curvature.",
"All above curvatures are designed for continuous differentiable object.",
"Now we describe angle based curvature, which aims to measure the curvature of discrete polygonal curve, e.g., the aforementioned geodesic polygonal curve in embedding manifold.",
"Recall that curve curvature is modeled as the rate of change of direction of tangent at a point that moves on the curve.",
"Extending this concept to polygonal curve, as shown in Figure REF B, the direction of tangent (dash arrow) of polygonal curve (black line) at point $\\mathbf {x}_i$ changes by a signed angle $\\theta _i$ , i.e., the turning angle at point $\\mathbf {x}_i$ .",
"In other words, the curvature of polygonal curve at a point can be measure by the turning angle, and hence the turning angle is called the angle based curvature.",
"Mathematically, the angle based curvature at the point $\\mathbf {x}_i$ is defined as $\\kappa _i=\\theta _i$ ."
],
[
"Methods",
" To address the distortion problem, our basic idea is to restrict the curvature of embedding manifold during graph embedding, thereby inducing a flat embedding manifold with a low distortion $\\rho $ , Eq.REF .",
"In this section, we start by proposing a novel angle based sectional curvature for embedding manifold, and then accordingly present three kinds of curvature regularization that can be readily incorporated into existing proximity-preserving node embedding methods."
],
[
"Angle-based Sectional Curvature",
" In the literature, there is no off-the-shelf curvature which can apply to embedding manifold.",
"We proposed a novel angle based sectional curvature, termed ABS curvature, to fill the void.",
"ABS curvature measures how an embedding manifold differ from its tangent space via the curvature of geodesic polygonal curve in embedding manifold.",
"We design ABS curvature from two basic assumptions.",
"The first is similar to the aforementioned sectional curvature, i.e., the curvature of geodesic curve can be used to reflect how the manifold differs from its tangent space along the direction of the curve.",
"However, it's difficult to calculate the curvature of geodesic curve in graph embedding setting because we cannot obtain the functions of geodesic curves in embedding manifold.",
"Thus, the second assumption we made is geodesic polygonal curve formed by points can approximate the geodesic curve in embedding manifold.",
"We give a formal definition of ABS curvature.",
"Definition 3.",
"(Angle-based Sectional Curvature) For any geodesic polygonal curve $P_{i,j}$ that passes through a point $\\mathbf {x}_q$ in embedding manifold $\\mathcal {M}$ , the angle-based sectional curvature at $\\mathbf {x}_q$ along $P_{i,j}$ , $K_{q}(P_{i,j})$ , is the angle-based curvature of $P_{i,j}$ at $\\mathbf {x}_q$ , $K_{q}(P_{i,j}) = \\kappa _q,$ where $\\kappa _q$ is defined on the geodesic polygonal curve $P_{i,j}$ .",
"The ABS curvature $K_{q}(P_{i,j})$ is defined on a geodesic polygonal curve $P_{i,j}$ (i.e., a section of embedding manifold), and is an extrinsic curvature.",
"We then present the notations to denote the curvature of the entire embedding manifold.",
"Curvature vector $K_{q}$ is formed by all curvatures $K_{q}(P_{i,j})$ defined on every geodesic polygonal curve passing through the point $\\mathbf {x}_q$ , which describes the curvature of $\\mathcal {M}$ at $\\mathbf {x}_q$ .",
"The curvature vectors $K_{q}$ at every point in $\\mathcal {M}$ form a curvature vector field $\\mathcal {K}$ , which provides a completed description for the curvature of embedding manifold $\\mathcal {M}$ .",
"We then provide a connection between the proposed ABS curvature and the distortion to support our basic idea theoretically.",
"Theorem 1 states that the distortion $\\rho $ can be prevented by decreasing the absolute value of ABS curvature of embedding manifold under given conditions.",
"Those conditions limit the theorem in the scope that ABS curvature can effects independently, since curvature and torsion can both influence the distortion in high-dimensional space [13].",
"To meet the conditions during graph embedding, we design specific optimization strategy in the following algorithm.",
"Lemma 1.",
"In a 2-dimensional embedding manifold $\\mathcal {M}$ , the distortion $\\rho $ is an increasing function of $|K_{q}(P_{i,j})|$ , the absolute value of ABS curvature at any $\\mathbf {x}_q$ along $P_{i,j}$ , if the absolute value of summation of curvatures along any part of $P_{i,j}$ is less than $\\frac{\\pi }{2}$ , $|\\sum _p K_{p}(P^s_{i,j})| < \\frac{\\pi }{2}$ .",
"$P^s_{i,j}$ is a part of $P_{i,j}$ and $\\mathbf {x}_p$ is a point along $P^s_{i,j}$ .",
"Please see the Appendix for proof.",
"Theorem 1.",
"In a n-dimensional embedding manifold $\\mathcal {M}$ , the distortion $\\rho $ is an increasing function of $|K_{q}(P_{i,j})|$ , the absolute value of ABS curvature at any $\\mathbf {x}_q$ along $P_{i,j}$ , if the absolute value of summation of curvatures along any part of $P^{^{\\prime }}_{i,j}$ is less than $\\frac{\\pi }{2}$ , $|\\sum _p K_{p}(P^{^{\\prime }s}_{i,j})| < \\frac{\\pi }{2}$ .",
"$P^{^{\\prime }}_{i,j}$ is the projection of $P_{i,j}$ in the 2-dimensional subspace of embedding manifold $\\mathcal {M}$ , $P^{^{\\prime }s}_{i,j}$ is a part of $P^{^{\\prime }}_{i,j}$ and $\\mathbf {x}_p$ is a point along $P^{^{\\prime }s}_{i,j}$ .",
"Please see the Appendix for proof."
],
[
"Curvature Regularization",
" In this section, we firstly present curvature regularization, a form of regularization based on the proposed ABS curvature.",
"we then design two efficient variants of curvature regularization to solve the scalability of the curvature regularization.",
"Finally, we develop an optimizing algorithm to optimize the curvature regularization during proximity-preserving graph embedding, so as to induce flat embedding manifolds.",
"From Theorem 1, we expect to decrease the absolute value of ABS curvature of embedding manifold.",
"As it's difficult to optimize the absolute value of a angle directly, we design the curvature regularization as the cosine of the ABS curvature because the cosine value of a angle increases with the absolute value of the angle when the angle is in $[-\\pi ,\\pi ]$ and the cosine function is convex and easy to calculate through vector product.",
"The curvature regularization is given by $\\Omega _c(X) = \\sum \\nolimits _{K_q(P_{i,j}) \\in \\mathcal {K}} \\cos (K_q(P_{i,j})).$ To obtain the ABS curvature vector filed $\\mathcal {K}$ involve the problem of scalability because it need to calculate the shortest path between every node pair in a graph.",
"We adopt a sampling strategy to solve the problem, and the sampled curvature regularization is given by $\\Omega _{s}(X) = \\sum \\nolimits _{v_i, v_j \\in S} \\cos (K_q(P_{i,j})),$ where $S$ is a set of sampled nodes.",
"The sampled curvature regularization only restricts the ABS curvature along geodesic polygonal curve $P_{i,j}$ between sampled nodes, $v_i, v_j \\in S$ .",
"A comprehensive and unbiased sampling is crucial for this regularization.",
"In real-world applications, random walk-based embedding methods are usually leveraged to deal with very large graphs in which it is also very computationally difficult to obtain sampled shortest paths.",
"We design a approximated curvature regularization for random walk-based embedding methods specifically.",
"$\\Omega _{a}(X) = \\sum \\nolimits _{r \\in R} \\cos (K_q(r)),$ where $r$ is a acyclic path generated by random walk in embedding method, and $R$ is a set of random-walk paths.",
"In this regularization, we replace shortest paths by random-walk paths because random-walk paths may include lots of shortest paths in graph.",
"Based on curvature regularization, we propose an algorithm to optimize the curvature regularization during proximity-preserving graph embedding, as shown in Algorithm 1.",
"There are two phases in the algorithm.",
"In first phase, we minimize the embedding loss term $\\mathcal {L}(\\mathcal {G})$ and the curvature regularization term $\\Omega (X)$ separately, so as to obtain an embedding manifold with low enough ABS curvature, thereby making most of geodesic polygonal curves meet the conditions in the theorem 1.",
"In second phase, we minimize the two terms jointly to get a flat embedding manifold which preserves well the proximity in graph and has a low distortion $\\rho $ .",
"Weight $\\lambda $ is a trade-off hyperparameter.",
"Curvature regularization optimizing algorithm.",
"[1] input: graph $\\mathcal {G}$ output: node representations $X$ preprocessing: get paths in $\\mathcal {G}$ Shortest paths ($\\Omega _{c}$ and $\\Omega _{s}$ ) or random-walk paths ($\\Omega _{a}$ ) $t$ iterations First phase not converged minimize embedding loss term $\\mathcal {L}(\\mathcal {G})$ not converged minimize curvature regularization term $\\Omega (X)$ Specific term is $\\Omega _{c}$ , $\\Omega _{s}$ , or $\\Omega _{a}$ not converged Second phase minimize the two terms jointly $\\mathcal {L}(\\mathcal {G}) + \\lambda \\Omega (\\mathbf {X})$ $\\lambda $ is a trade-off hyperparameter"
],
[
"Experiments",
" We comprehensively validate our method on both node classification (NC) and link prediction (LP) tasks, on eight open graph datasets.",
"We integrate curvature regularization into five popular proximity-preserving node embedding methods, and then compare performance of the embedding methods with the curvature regularization against the original embedding methods, and empirically demonstrate significant improvements."
],
[
"Experimental Setup",
"Datasets.",
"We evaluate the proposed method on eight open graph datasets described below (more details are available in the Appendix).",
"In all experiments, we do not use node attribute information, and only use node representations learned by node embedding methods.",
"Empirical results show significant improvements on a wide range of open graph datasets.",
"(1) Citation networks.",
"Cora, Citeseer and Pubmed are citation network benchmark datasets [22], [23], where nodes are papers and edges are citation links.",
"Node labels are the academic topic of papers.",
"(2) WebKB.",
"WebKB contains a subset of web pages collected from computer science departments.",
"Specifically, we use the subgraphs from four universities, i.e., Cornell, Texas, Washington and Wisconsin.",
"In these networks, nodes are web pages and edges are hyperlinks between pages.",
"The web pages are classified into the five categories, student, project, course, staff, and faculty.",
"(3) Polblogs.",
"Political blog network [24] consists of blogs about US politics and the links between them.",
"Blogs are divided into two communities based on their political labels (liberal and conservative).",
"Proximity-preserving node embedding methods.",
"We integrate curvature regularization into the following five popular node embedding methods to learn node representations.",
"Matrix Factorization (MF) represents graph property with a matrix, and it can factorize this matrix to obtain low-dimensional vector representations of nodes [25].",
"Laplacian Eigenmaps (LE) learns node representations by factorizing graph laplacian eigenmaps [19].",
"DeepWalk learns node representations from random walk paths by leveraging Skip-Gram neural networks [9].",
"Node2vec is an extension of DeepWalk which adopts biased random walk and Skip-Gram neural networks to learn node representations [21].",
"Structural Deep Network Embedding (SDNE) learns node representations that preserve the first-order and second-order proximity with a deep autoencoder network [10].",
"Parameter search.",
"For all node embedding models, we perform a random sampling hyper-parameter search on validation set of each dataset to get competitor models (See detailed hyper-parameter setting in the Appendix).",
"The hyper-parameters searched over include the dimension of node representation as well as hyper-parameters specific to each model.",
"We then integrate the curvature regularization term into those competitors, and only adjust the number of iteration $t$ and the weight $\\lambda $ in algorithm."
],
[
"Experimental Results",
"Node classification.",
"With the node representations learned by each embedding method, we split randomly $60\\%$ of the nodes in a graph as the training set and the remaining $40\\%$ of nodes in a graph as the test set.",
"Then we use a one-vs-rest logistic regression classifier to predict the labels.",
"Accuracy is adopted as the evaluation metric in node classification task.",
"We repeat the above process 10 times and average these results.",
"Final results are summarized in Table REF .",
"The reported numbers denote the mean classification accuracy in percent.",
"We use suffix '-c', '-s', or '-a' to respectively denote $\\Omega _c$ , $\\Omega _s$ , or $\\Omega _a$ , the regularization term used.",
"In general, curvature regularization improves the performance of those node embedding methods.",
"The best performing method is highlighted in bold.",
"Table: Mean Classification Accuracy (Percent)Link prediction.",
"In the link prediction task, we randomly remove $40\\%$ of the links in the graph and then learn node representations from the subgraph induced from the remaining links.",
"In order to predict the unobserved links, we need to construct link representations.",
"Specifically, here we use link representations computed from the Hadamard product of the node representations of the nodes connected by a link.",
"We adopt mean average precision to evaluate the performance, which is used to summarise precision-recall curves [26].",
"Table: Mean Average Precision (Percent)Analysis of convergence and distortion reduction.",
"We conduct comparison experiments to analyze the convergence of curvature loss and the deduction of distortion (Eq.",
"REF ) during optimizing on the Cora dataset, as shown in the Appendix."
],
[
"Discussions",
" In this section, we first discuss what are \"good\" curvatures and \"bad\" curvatures in graph embedding, then review the negative sampling strategy from the perspective of distortion prevention, and finally connect the curvature regularization to the betweenness centrality of graph."
],
[
"\"Good\" Curvatures and \"Bad\" Curvatures",
" Curving an embedding space is indeed required for graph embedding, where a useful curvature is of benefit to preserving graph topology.",
"For instance, the circles in Figure REF B2 indicate useful curvatures that form good node distributions, where the connected nodes are close and disconnected nodes are far apart.",
"Hyperbolic space is also an example, whose useful negative curvature is suitable for embedding tree-structured graphs [17].",
"However, useless or harmful curvatures are also inevitable in proximity-preserving embedding because the existing strategies have no restriction on the \"non-local\" curves of an embedding space, such as the global \"swirling\" curve in Figure REF B1.",
"Such harmful curvatures bring undesired distortions because unsupervised graph embedding aims to faithfully preserve graph topology patterns into an embedding space.",
"In other words, any changes/distortions in patterns is not desired.",
"We stress that the proposed algorithm differentiates useful from harmful curvatures because the former will contribute to the reconstruction term in the objective function while the latter will not and thus be eliminated by the curvature regularization during optimizing."
],
[
"Reviewing Negative Sampling from the Perspective of Distortion Prevention",
" Negative sampling is an approximation strategy to alleviate the computational problem of Skip-Gram neural network [27] and it has been widely used in graph embedding methods [9].",
"In node embedding methods, the objective of negative sampling is to enlarge the distance between two disconnected nodes in Euclidean embedding space.",
"From distortion perspective, negative sampling is one way to decrease the distortion of embedding manifold.",
"Suppose node $v_j$ is a negative sample of $v_i$ and the geodesic distance between them $d_\\mathcal {M}(\\mathbf {x}_i,\\mathbf {x}_j)$ in embedding manifold is fixed, if the Euclidean distance between them $d_\\mathcal {E}(\\mathbf {x}_i,\\mathbf {x}_j)$ in ambient Euclidean space is enlarged by negative sampling, the term $\\frac{d_\\mathcal {M}(\\mathbf {x}_i,\\mathbf {x}_j)}{d_\\mathcal {E}(\\mathbf {x}_i,\\mathbf {x}_j)}$ in distortion $\\rho $ , Eq.REF , will decrease.",
"However, negative sampling strategy may destroy preserving proximity in graph.",
"If the sampled node is a proximal node to the target node, to enlarge the Euclidean distance between them will increase inevitably their geodesic distance and influence proximity preservation.",
"Besides, negative sampling cannot guarantee the decrease of the distortion $\\rho $ because to decrease the terms in $\\rho $ corresponding to the sampled nodes may increase the other terms in $\\rho $ .",
"Therefore, good negative samplers are very crucial for negative sampling strategy [28]."
],
[
"A Connection to Betweenness Centrality of Graph",
" In graph theory, betweenness centrality is a measure of centrality in a graph, which is defined as the number of shortest paths that pass through a node.",
"The proposed curvature regularization is based on shortest paths and constrains more on nodes with high betweenness than others– which we think makes sense to prevent distortion: A node with high betweenness in a graph is corresponding to a “bottleneck” region in an embedding manifold, i.e., a small region where lots of geodesic curves pass through.",
"Taking Figure REF B2 as an example, the nodes in the middle have high betweenness in graph.",
"If we curve or fold the embedding space around these nodes, the Euclidean distances of most node pairs will be changed, which will bring large distortions in terms of Eq.",
"REF , i.e., the divergence between the geodesic and Euclidean metric.",
"To contrast, if we curve the embedding space around the nodes with low betweenness, it brings limited distortion.",
"Thus, the curvature around high-betweenness nodes should be constrained more due to their high influence on distortion."
],
[
"Conclusion",
" For the first time, we raised and formulated the pattern distortion problem in proximity-preserving graph embedding, which is important but neglected in existing works.",
"As the distortion is caused by the curved embedding manifold, our basic idea to address the problem is to leverage curvature regularization to enforce flatness for embedding manifolds, thereby preventing the distortion.",
"We proposed a novel angle-based sectional curvature, termed ABS curvature, for embedding manifold, and accordingly presented a curvature regularization to induce flat embedding manifolds during graph embedding.",
"We further designed two efficient variants of curvature regularization to solve the scalability of curvature regularization.",
"We developed an optimizing algorithm to induce an embedding manifold with low distortion during graph embedding.",
"Finally, we evaluate the proposed algorithm by integrating curvature regularization into five popular proximity-preserving embedding methods.",
"Comparisons on the NC and LP tasks show significant improvements on eight open graph datasets.",
"As future work, we will attempt to apply the curvature regularization to more applications of graph embedding, such as graph alignment [29], community detection [30], structural comprehension [31], [32], and epidemic dynamic prediction [33], [34]."
],
[
"Broader Impact",
"This work raised the distortion problem in graph embedding for the first time.",
"As the problem is important but neglected in the existing works, it will attract many researchers to design new approaches to address it.",
"In this work, three kinds of curvature regularization and an optimizing algorithm have been proposed to prevent the pattern distortion during graph embedding.",
"Obviously, the results of the work will have an immediate impact on improving the performance of various proximity-preserving graph embedding methods.",
"This work will also benefit graph analysis applications in the real world, such as social network analysis, recommendation system, and knowledge graph mining.",
"The proposed models and algorithm advance the development of graph representation models which may bring negative societal consequences including privacy leak and fairness issues.",
"For example, sensitive personal information, e.g., political orientation, occupation, and disease, may encode implicitly in user connections in social network.",
"Those privacy can be learned effectively by graph representation models, and then may be leaked illegally by someone with bad intentions.",
"Meanwhile, the links in social network also encode the information about population subgroups, such as gender and ethnic group.",
"If such information is extracted by graph representation models and fed into downstream machine learning models, the trained model may lead to unfair predictions.",
"For example, if one company scarcely employees women, models trained on this data would prefer man."
],
[
"Acknowledgments",
" We are very grateful to the anonymous reviewers for their constructive comments.",
"This work was supported by the National Natural Science Foundation of China under grant 61876069; Jilin Province Key Scientific and Technological Research and Development Project under Grant Nos.",
"20180201067GX and 20180201044GX; Jilin Province Natural Science Foundation under Grant No.",
"20200201036JC; National Science Foundation IIS 16-19302 and IIS 16-33755; Zhejiang University ZJU Research 083650; Futurewei Technologies HF2017060011 and 094013; UIUC OVCR CCIL Planning Grant 434S34; UIUC CSBS Small Grant 434C8U; and IBM-Illinois Center for Cognitive Computing Systems Research (C3SR); and China Scholarships Council under scholarship 201806170202.",
"Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the funding agencies."
]
] | 2011.14211 |
[
[
"Group superschemes"
],
[
"Abstract We develop a general theory of algebraic group superschemes, which are not necessarily affine.",
"Our key result is a category equivalence between those group superschemes and Harish-Chandra pairs, which generalizes the result known for affine algebraic group superschemes.",
"Then we present the applications, including the Barsotti-Chevalley Theorem in the super context, and an explicit construction of the quotient superscheme $\\mathbb{G}/\\mathbb{H}$ of an algebraic group superscheme $\\mathbb{G}$ by a group super-subscheme $\\mathbb{H}$."
],
[
"Introduction",
"The purpose of this paper is to generalize a description of algebraic supergroups, which uses Harish-Chandra pairs, to the category of locally algebraic group superschemes.",
"The first results of this kind were developed in [10], where it has been shown, using Hopf superalgebra technique, that any algebraic supergroup is the product of its largest even super-subgroup and a purely-odd super-subscheme.",
"An explicit construction of such a decomposition was first proposed in the article [5], and then significantly revised and generalized in the papers [11], [12].",
"The main result of these papers can be formulated as a fundamental equivalence between the category of (affine) algebraic supergroups and specific Harish-Chandra pairs.",
"In the broadest sense, a Harish-Chandra pair is the couple $(\\mathsf {G}, \\mathsf {V})$ , where $\\mathsf {G}$ is a group scheme, and $\\mathsf {V}$ is a finite-dimensional $\\mathsf {G}$ -module, equipped with a bilinear symmetric $\\mathsf {G}$ -equivariant map from $\\mathsf {V}\\times \\mathsf {V}$ to the Lie algebra $\\mathsf {g}$ of $\\mathsf {G}$ (see Section 11.1 for more details).",
"We call a Harish-Chandra pair affine, algebraic, or locally algebraic, provided $\\mathsf {G}$ is an affine, algebraic, or locally algebraic group scheme, respectively.",
"To every affine algebraic group superscheme $\\mathbb {G}$ , we can associate the affine algebraic Harish-Chandra pair $(\\mathbb {G}_{ev}, \\mathfrak {g}_1)$ , where $\\mathbb {G}_{ev}$ is the largest purely-even group super-subscheme of $\\mathbb {G}$ and $\\mathfrak {g}$ is the Lie superalgebra of $\\mathbb {G}$ .",
"The action of $\\mathbb {G}_{ev}$ on $\\mathfrak {g}_1$ is induced by the adjoint action of $\\mathbb {G}$ on $\\mathfrak {g}$ .",
"The corresponding bilinear map is the restriction of Lie super-bracket to $\\mathfrak {g}_1$ .",
"The functor $\\mathbb {G}\\mapsto (\\mathbb {G}_{ev}, \\mathfrak {g}_1)$ is called the Harish-Chandra functor.",
"There is a functor that, given an affine algebraic pair $(\\mathsf {G}, \\mathsf {V})$ , constructs an algebraic supergroup as a superscheme product $\\mathsf {G}\\times {\\bf E}$ , where ${\\bf E}$ is a purely-odd superscheme isomorphic to $\\mathrm {SSp}(\\Lambda (\\mathsf {V}))$ .",
"Moreover, this functor is a quasi-inverse of the Harish-Chandra functor.",
"The Harish-Chandra functor can be extended to the larger category of locally algebraic group superschemes.",
"We prove that this functor has a quasi-inverse as well.",
"It means that every locally algebraic group superscheme $\\mathbb {G}$ is isomorphic to a superscheme product $\\mathbb {G}_{ev}\\times {\\bf E}$ as above, where $\\mathbb {G}_{ev}$ is regarded as a group scheme.",
"Although the proof mostly follows the papers [11], [12], we introduce a new object that plays a crucial role for non-affine group superschemes.",
"We show that every group superscheme $\\mathbb {G}$ contains a normal group subfunctor $\\mathbb {S}$ , which can be regarded as a formal supergroup (see Remark 3.6, [7]).",
"We call $\\mathbb {S}$ a formal neighborhood of the identity.",
"The superscheme $\\bf E$ is contained in $\\mathbb {S}$ .",
"In particular, $\\mathbb {G}=\\mathbb {G}_{ev}\\mathbb {S}$ , i.e., $\\mathbb {S}$ is sufficiently large as a group subfunctor.",
"Moreover, $\\mathbb {S}$ is \"quasi-affine\" in the sense that a complete Hopf superalgebra $\\widehat{\\mathcal {O}_e}$ can represent it (see Section 8).",
"Note that $\\mathbb {S}$ is \"negligible\" from the topological point of view.",
"That is, for every open super-subscheme $\\mathbb {U}$ of $\\mathbb {G}$ , we have $\\mathbb {U}\\mathbb {S}\\subseteq \\mathbb {U}$ .",
"The reason for introducing $\\mathbb {S}$ is that $\\bf E$ is not a group subfunctor, but every product $xy$ of elements of $\\bf E$ can be uniquely expressed as $f z$ , where $f\\in \\mathbb {S}_{ev}$ and $z\\in {\\bf E}$ .",
"In our proof, we also exploit a Hopf superalgebra pairing between the Hopf superalgebra $\\widehat{\\mathcal {O}_e}$ and the hyperalgebra $\\mathrm {hyp}(\\mathbb {G})$ of $\\mathbb {G}$ .",
"For example, each $\\mathbb {S}(R)$ can be identified with the subgroup of $(\\mathrm {hyp}(\\mathbb {G})\\otimes R)^{\\times }$ consisting of all group-like elements.",
"Finally, to show that $\\mathbb {G}$ coincides with $\\mathbb {G}_{ev}{\\bf E}$ , we use the endofunctor $X\\mapsto \\mathsf {gr}(X)=\\mathsf {gr}_{\\mathcal {I}_X}(X)$ of the category of geometric superschemes, more precisely, its functor-of-points counterpart.",
"The functor $\\mathsf {gr}$ is interesting on its own.",
"Indeed, it preserves immersions and induces an endofunctor of the category of group superschemes.",
"Moreover, a morphism $f : X\\rightarrow Y$ of superschemes (of locally finite type) is an isomorphism if and only if $\\mathsf {gr}(f)$ is.",
"The group superscheme $\\mathsf {gr}(\\mathbb {G})$ is a semi-direct product of $\\mathsf {gr}(\\mathbb {G})_{ev}\\simeq \\mathbb {G}_{ev}$ and a normal purely-odd group super-subscheme $\\mathbb {G}_{odd}$ .",
"In this case, $\\mathbb {G}_{odd}=\\bf E$ .",
"Then Proposition REF implies that the graded companion of the natural embedding $\\mathbb {G}_{ev}{\\bf E}\\rightarrow \\mathbb {G}$ is an isomorphism, hence $\\mathbb {G}_{ev}{\\bf E}=\\mathbb {G}$ .",
"The remainder of the paper is devoted to applications of the fundamental equivalence.",
"First, we prove that a slightly modified (super)version of the well-known Barsotti-Chevalley theorem occurs in the category of algebraic group superschemes.",
"More precisely, a connected algebraic group superscheme $\\mathbb {G}$ has normal group super-subschemes $\\mathbb {G}_1\\le \\mathbb {G}_2$ such that $\\mathbb {G}_1$ is affine, $\\mathbb {G}_2/\\mathbb {G}_1$ is an abelian group variety, and $\\mathbb {G}/\\mathbb {G}_2$ is again affine.",
"Further, we describe abelian supervarieties and anti-affine algebraic group superschemes.",
"Contrary to the purely-even case, we show that the class of pseudoabelian group superschemes is extensive.",
"Regardless of whether the ground field is perfect or not, we construct pseudoabelian group superschemes, which are neither abelian supervarieties nor even solvable group functors (compare with [9], chapter 8).",
"In the last section, we prove that for any algebraic group superscheme $\\mathbb {G}$ and its group super-subscheme $\\mathbb {H}$ , the sheaf quotient $\\mathbb {G}/\\mathbb {H}$ is a superscheme of finite type.",
"Using Remark 9.11, [14], and Theorem 1.1 from [2], we reduce the general case to the case when $\\mathbb {G}_{ev}=\\mathsf {M}\\times \\mathsf {N}$ , where $\\mathsf {M}$ is an affine group subscheme, $\\mathsf {N}$ is an abelian variety, and $\\mathbb {H}_{ev}\\le \\mathsf {M}$ .",
"This immediately implies that $\\mathbb {H}$ is affine.",
"Note also that $\\mathsf {N}$ is a central group subscheme of $\\mathbb {G}_{ev}$ .",
"Under these conditions, $\\mathbb {G}$ has an open covering by (finitely many) affine $\\mathbb {H}$ -saturated super-subschemes $\\mathsf {U}{\\bf E}$ , where $\\mathsf {U}=\\mathsf {U}_{aff}\\times \\mathsf {U}_{ab}$ , $\\mathsf {U}_{aff}$ and $\\mathsf {U}_{ab}$ form open affine coverings of $\\mathsf {M}$ and $\\mathsf {N}$ , respectively, and $\\mathsf {U}/\\mathsf {H}\\simeq \\mathsf {U}_{aff}/\\mathsf {H}\\times \\mathsf {U}_{ab}$ is an affine superscheme.",
"Modifying the proof of the main theorem from [13], we show that $\\mathbb {U}$ is isomorphic to a homogeneous fiber quotient $\\mathbb {X}\\times ^{\\mathsf {H}}\\mathbb {H}$ , where $\\mathbb {X}$ is an affine superscheme on which $\\mathsf {H}$ acts on the right, and such that $(\\mathbb {X}\\times ^{\\mathsf {H}}\\mathbb {H})/\\mathbb {H}$ is affine.",
"The morphism $\\mathbb {X}\\times ^{\\mathsf {H}}\\mathbb {H}\\rightarrow \\mathbb {U}$ is constructed in a more straightforward way than in [13].",
"To prove that it is an isomorphism, we use the above functor $\\mathsf {gr}$ .",
"Each $\\mathbb {U}/\\mathbb {H}$ is affine, and the standard arguments imply that they form an open covering of $\\mathbb {G}/\\mathbb {H}$ .",
"The article is organized as follows.",
"In the first section, we have collected all necessary notations, definitions, and results on superschemes.",
"Throughout the article, we distinguish between superschemes as functors and geometric superschemes, although the categories of superschemes and geometric superschemes are equivalent to each other by the Comparison Theorem.",
"For example, to introduce the notions of separated and proper morphisms in the second section, the language of geometric superschemes seems to be more natural and convenient.",
"However, the group superschemes are introduced in two incarnations, as group functors and as group objects in the category of geometric superschemes.",
"In the fourth section, we prove that every group superscheme has the largest affine quotient, again using the language of geometric superschemes.",
"In the fifth section, for every $K$ -functor $\\mathbb {X}$ , we define its tangent functor at a $K$ -point and investigate its properties in the case when $\\mathbb {X}$ is a superscheme of locally finite type.",
"In the sixth section, the Lie superalgebra of a group superscheme is introduced as functor and a superalgebra.",
"In the seventh section, we develop a fragment of the differential calculus on locally algebraic group superschemes.",
"In the eighth section we introduce the normal group subfunctor $\\mathbb {S}$ and discuss how it relates to the Hopf superalgebra $\\mathrm {hyp}(\\mathbb {G})$ .",
"The ninth section is devoted to studying the functor $\\mathsf {gr}$ .",
"We prove that it commutes with direct products, thus it induces an endofunctor of the category of geometric group superschemes.",
"Moreover, a superscheme $\\mathsf {gr}(X)$ is always \"split\" in the sense that there are two superscheme morphisms $i_X : X_{ev}\\rightarrow \\mathsf {gr}(X)$ and $q_X : \\mathsf {gr}(X)\\rightarrow X_{ev}$ such that $q_X i_X=\\mathrm {id}_{X_{ev}}$ , and $i_X$ induces an isomorphism onto $\\mathsf {gr}(X)_{ev}$ .",
"In particular, if $\\mathbb {G}$ is a locally algebraic group superscheme, then $\\mathsf {gr}(\\mathbb {G})\\simeq \\mathbb {G}_{ev}\\ltimes \\mathbb {G}_{odd}$ , where $\\mathbb {G}_{odd}=\\ker {\\bf q}_G$ is a purely-odd group superscheme.",
"In the tenth section, we show that $\\mathbb {G}_{odd}\\simeq \\mathrm {SSp}(\\Lambda (\\mathfrak {g}_1^*))$ and for any morphism $\\mathbb {G}\\rightarrow \\mathbb {H}$ of group superschemes, the induced morphism $\\mathbb {G}_{odd}\\rightarrow \\mathbb {H}_{odd}$ is uniquely defined by the corresponding linear map $\\mathfrak {g}_1\\rightarrow \\mathfrak {h}_1$ .",
"In the eleventh section, using the techniques developed previously, we prove that the category of locally algebraic group superschemes is equivalent to locally algebraic Harish-Chandra pairs.",
"The content of the twelfth and thirteenth sections has already been discussed in detail above."
],
[
"Superschemes as $K$ -functors",
"For the content of this section, we refer to [8], [14], [18].",
"Throughout this article, $K$ is the ground field of odd or zero characteristic.",
"Let $\\mathsf {SAlg}_K$ denote the category of super-commutative $K$ -superalgebras.",
"If $A\\in \\mathsf {SAlg}_K$ , then let $I_A$ and $\\overline{A}$ denote the super-ideal $AA_1$ and the factor-algebra $A/I_A$ , respectively.",
"A $K$ -functor $\\mathbb {X}$ is a functor from the category $\\mathsf {SAlg}_K$ to the category of sets $\\mathsf {Sets}$ .",
"The category of $K$ -functors is denoted by $\\mathcal {F}$ .",
"A $K$ -functor $\\mathbb {X}$ is called an affine superscheme if it is representable by a superalgebra $A$ , that is, $\\mathbb {X}(B)=\\mathrm {Hom}_{\\mathsf {SAlg}_K}(A, B) \\text{ for } B\\in \\mathsf {SAlg}_K.$ In this case, $\\mathbb {X}$ is denoted by $\\mathrm {SSp}(A)$ .",
"A closed affine super-subscheme of $\\mathbb {X}=\\mathrm {SSp}(A)$ is defined as $\\mathbb {V}(I)(B)=\\lbrace \\phi \\in \\mathbb {X}(B)\\mid \\phi (I)=0\\rbrace \\text{ for } B\\in \\mathsf {SAlg}_K,$ where $I$ is a super-ideal of $A$ .",
"It is easy to see that $\\mathbb {V}(I)\\simeq \\mathrm {SSp}(A/I)$ .",
"Similarly, a super-ideal $I$ of $A$ defines an open $K$ -subfunctor of $\\mathbb {X}=\\mathrm {SSp}(A)$ as $\\mathbb {D}(I)=\\lbrace \\phi \\in \\mathbb {X}(B)\\mid \\phi (I)B=B\\rbrace \\text{ for } B\\in \\mathsf {SAlg}_K .$ In general, a $K$ -subfunctor $\\mathbb {Y}$ of a $K$ -functor $\\mathbb {X}$ is called closed (respectively, open) if for any morphism of functors ${\\bf f} : \\mathrm {SSp}(A)\\rightarrow \\mathbb {X}$ the inverse image ${\\bf f}^{-1}(\\mathbb {Y})$ is closed (respectively, open) in $\\mathrm {SSp}(A)$ .",
"A collection of open $K$ -subfunctors $\\lbrace \\mathbb {X}_i\\rbrace _{i\\in I}$ is said to be an open covering of a $K$ -functor $\\mathbb {X}$ , provided $\\mathbb {X}(F)=\\cup _{i\\in I}\\mathbb {X}_i(F)$ for any field extension $K\\subseteq F$ .",
"A $K$ -functor $\\mathbb {X}$ is called local if for any $K$ -functor $ \\mathbb {Y}$ and any open covering $\\lbrace \\mathbb {Y}_i\\rbrace _{i\\in I}$ of $\\mathbb {Y}$ the diagram $\\mathrm {Mor}_{\\mathcal {F}}(\\mathbb {Y} , \\mathbb {X})\\rightarrow \\prod _{i\\in I}\\mathrm {Mor}_{\\mathcal {F}}(\\mathbb {Y}_i , \\mathbb {X})\\rightrightarrows \\prod _{i, j\\in I}\\mathrm {Mor}_{\\mathcal {F}}(\\mathbb {Y}_i\\cap \\mathbb {Y}_j , \\mathbb {X})$ is exact.",
"A local $K$ -functor $\\mathbb {X}$ is called a superscheme whenever $\\mathbb {X}$ has an open covering by affine super-subschemes.",
"Superschemes form a full subcategory of $\\mathcal {F}$ , denoted by $\\mathcal {SF}$ .",
"Let $\\mathbb {X}$ be a $K$ -functor.",
"If $\\mathbb {Y}$ is a subfunctor of $\\mathbb {X}$ , such that for any $A\\in \\mathsf {SAlg}_K$ , the set $\\mathbb {Y}(A)$ contains exactly one element, then $\\mathbb {Y}$ is called a one-point subfunctor of $\\mathbb {X}$ .",
"Lemma 1.1 Assume that $\\mathbb {X}$ is a superscheme.",
"Then every one-point subfunctor of $\\mathbb {X}$ is closed.",
"Let $\\mathbb {Y}$ be a one-point subfunctor of $\\mathbb {X}$ .",
"For any superalgebra $A$ , we have $\\mathbb {Y}(A)=\\lbrace y_A\\rbrace $ .",
"Choose a covering of $\\mathbb {X}$ by open affine super-subschemes $\\mathbb {X}_i\\simeq \\mathrm {SSp}(A_i)$ for $i\\in I$ .",
"A verbatim superization of [8], I.1.7(6), implies that $\\mathbb {X}_i$ meets $\\mathbb {Y}$ if and only if $y_K$ belongs to $\\mathbb {X}_i(K)$ .",
"In the latter case, $y_K$ corresponds to a superalgebra morphism $\\phi : A_i\\rightarrow K$ and $\\mathbb {Y}\\cap \\mathbb {X}_i=\\mathbb {V}(\\ker \\phi )$ .",
"Lemma 9.1, [14], concludes the proof."
],
[
"Geometric superschemes",
"Recall that a geometric superspace $X$ consists of a topological space $X^e$ and a sheaf of super-commutative superalgebras $\\mathcal {O}_X$ such that all stalks $\\mathcal {O}_{X, x}$ for $x\\in X^e$ are local superalgebras.",
"A morphism of superspaces $f : X\\rightarrow Y$ is a pair $( f^e, f^*)$ , where $f^e : X^e\\rightarrow Y^e$ is a morphism of topological spaces and $f^* : \\mathcal {O}_Y\\rightarrow f^e_{*}\\mathcal {O}_X$ is a morphism of sheaves such that $f^*_x : \\mathcal {O}_{Y , f (x)}\\rightarrow \\mathcal {O}_{X,x}$ is a local morphism for any $x\\in X^e$ .",
"Let $\\mathcal {V}$ denote the category of geometric superspaces.",
"To simplify the notations, we use $\\mathcal {O}(X)$ instead of $\\mathcal {O}_X(X^e)$ .",
"Let $X$ be a geometric superspace.",
"If $U$ is an open subset of $X^e$ , then $(U, \\mathcal {O}_X|_U)$ is again a geometric superspace, called an open super-subspace of $X$ .",
"In what follows $(U, \\mathcal {O}_X|_U)$ is denoted by $U$ .",
"Let $R$ be a superalgebra.",
"An affine superscheme $\\mathrm {SSpec}(R)$ can be defined as follows.",
"The underlying topological space of $\\mathrm {SSpec}(R)$ coincides with the prime spectrum of $R$ , endowed with the Zariski topology.",
"For any open subset $U\\subseteq (\\mathrm {SSpec}(R))^e$ , the super-ring $\\mathcal {O}_{\\mathrm {SSpec}(R)}(U)$ consists of all locally constant functions $h : U\\rightarrow \\sqcup _{\\mathfrak {p}\\in U} R_{\\mathfrak {p}}$ such that $h(\\mathfrak {p})\\in R_{\\mathfrak {p}} , \\mathfrak {p}\\in U$ .",
"A superspace $X$ is called a (geometric) superscheme if there is an open covering $X^e =\\cup _{i\\in I} U_i$ , such that each open super-subspace $U_i$ is isomorphic to an affine superscheme $\\mathrm {SSpec}(R_i)$ .",
"Superschemes form a full subcategory of $\\mathcal {V}$ , denoted by $\\mathcal {SV}$ .",
"If $X$ is a superscheme, then every open super-subspaces of $X$ is a superscheme, called an open super-subscheme.",
"A superscheme $Z$ is a closed super-subscheme of $X$ if there is a closed embedding $\\iota : Z^e\\rightarrow X^e$ such that the sheaf $\\iota _*\\mathcal {O}_Z$ is an epimorphic image of the sheaf $\\mathcal {O}_X$ .",
"Example 1.2 The sheafification of the presheaf $U\\mapsto I_{\\mathcal {O}_X(U)}=\\mathcal {O}_X(U) (\\mathcal {O}_X )(U)_1$ is a sheaf of $\\mathcal {O}_X$ -super-ideals, which is denoted by $\\mathcal {I}_X$ .",
"Then $(X^e, \\mathcal {O}_X/\\mathcal {I}_X)$ is the largest purely-even closed super-subscheme of $X$ , denoted by $X_{ev}$ .",
"Regarded as a geometric scheme, $X_{ev}$ is denoted by $X_{res}$ .",
"Moreover, every morphism $f : X\\rightarrow Y$ induces the morphism $f_{ev} : X_{ev}\\rightarrow Y_{ev}$ so that $X\\mapsto X_{ev}$ is an endofunctor of $\\mathcal {SV}$ ."
],
[
"Comparison Theorem",
"(see [14], Theorem 5.14, and [4], I, §1, 4.4) Theorem 1.3 The functor $X\\mapsto \\mathbb {X}$ , where $\\mathbb {X}(A)=\\mathrm {Mor}_{\\mathcal {SV}}(\\mathrm {SSpec}(A), X)$ for $A\\in \\mathsf {SAlg}_K$ , defines an equivalence $\\mathcal {SV}\\simeq \\mathcal {SF}$ ."
],
[
"Immersions",
"A morphism $f : X\\rightarrow Y$ in the category $\\mathcal {SV}$ is called an immersion if there is an open subset $U$ in $Y^e$ such that $f^e$ induces a homeomorphism from $X^e$ onto the closed subset $f^e(X^e)\\subseteq U$ and the induced morphism of sheaves $f^* : \\mathcal {O}|_U\\rightarrow f_* \\mathcal {O}_X$ is surjective.",
"The latter is equivalent to the condition that for every $x\\in X^e$ , the induced superalgebra morphism $\\mathcal {O}_{Y, f^e(x)}\\rightarrow \\mathcal {O}_{X, x}$ is surjective.",
"If $f^e(X^e)=U$ , then $f$ is called an open immersion.",
"Finally, if $U=Y^e$ , then $f$ is called a closed immersion.",
"The proofs of the following lemmas are standard, and we leave them for the reader.",
"Lemma 1.4 A morphism $f : X\\rightarrow Y$ in the category $\\mathcal {SV}$ is closed, open or immersion if and only if for any open covering $\\lbrace V_i\\rbrace _{i\\in I}$ of $Y$ , each morphism $f|_{f^{-1}(V_i)} : f^{-1}(V_i)\\rightarrow V_i$ is closed, open or immersion, correspondingly.",
"Lemma 1.5 Let $f : X\\rightarrow Y$ be a morphism in the category $\\mathcal {SV}$ such that $f^e$ is injective.",
"Then $f$ is closed, or an immersion, respectively if and only if for any finite open covering $\\lbrace U_j\\rbrace _{j\\in J}$ of $X$ , each morphism $f|_{U_j} : U_j\\rightarrow Y$ is closed, or an immersion, respectively.",
"Finally, $f$ is an open immersion if and only if for any open covering $\\lbrace U_j\\rbrace _{j\\in J}$ of $X$ , each morphism $f|_{U_j} : U_j\\rightarrow Y$ is an open immersion.",
"Proposition 1.6 (cf.",
"[9], I.l) Let $f : Z\\rightarrow X$ be a morphism of geometric superschemes.",
"Then $f$ is a closed immersion if and only if $\\bf f$ is an isomorphism of $\\mathbb {Z}$ onto a closed super-subscheme of $\\mathbb {X}$ .",
"Let $\\lbrace U_j\\rbrace _{j\\in J}$ be an open covering of $X$ .",
"Then $\\lbrace Z_j=i^{-1}(U_j)\\rbrace _{j\\in J}$ is an open covering of $Z$ .",
"By Lemma 5.2(2,3), [14], $\\lbrace \\mathbb {U}_j\\rbrace _{j\\in J}$ is an open covering of $\\mathbb {X}$ .",
"Moreover, each $\\mathbb {U}_j$ is affine whenever $U_j$ is.",
"Observe that, for every superalgebra $A$ , and every open subset $U\\subseteq X^e$ , the subset $\\mathbb {U}(A)\\subseteq \\mathbb {X}(A)$ consists of all superscheme morphism $h : \\mathrm {SSpec}(A)\\rightarrow X$ such that $h^e((\\mathrm {SSpec}(A))^e)\\subseteq U$ .",
"Thus ${\\bf f}^{-1}(\\mathbb {U}_j)=\\mathbb {Z}_j$ for any $j\\in J$ .",
"Assume now that $f$ is a closed immersion.",
"Then $\\bf f$ is an embedding of $K$ -functors.",
"The above remarks, combined with Lemma 9.1 (1), [14], reduce the general case to $X=\\mathrm {SSpec}(A)$ for $A\\in \\mathsf {SAlg}_K$ .",
"There is a super-ideal $I$ of $A$ such that $f$ can be identified with the closed immersion $\\mathrm {SSpec}(A/I)\\rightarrow \\mathrm {SSpec}(A)$ .",
"Lemma 4.1, [14], implies that $\\bf f$ is the canonical isomorphism of $\\mathrm {SSp}(A/I)$ onto the closed super-subscheme $\\mathbb {V}(I)$ of $\\mathrm {SSp}(A)$ .",
"For the reverse statement, we use Lemma 9.1 (1), [14] again, and the general case can be reduced to $X=\\mathrm {SSpec}(A)$ .",
"There are a super-ideal $I$ of $A$ and an isomorphism ${\\bf h} : \\mathbb {Z}\\rightarrow \\mathrm {SSp}(A/I)$ such that ${\\bf f}=\\mathrm {SSp}(\\pi ){\\bf h}$ , where $\\pi $ is the canonical epimorphism $A\\rightarrow A/I$ .",
"By Comparison Theorem, there is an isomorphism $h : Z\\rightarrow \\mathrm {SSpec}(A/I)$ such that $f=\\mathrm {SSpec}(\\pi )h$ .",
"In other words, for every open affine covering $\\lbrace U_j\\rbrace _{j\\in J}$ of $X$ , each morphism $f|_{Z_j} : Z_j\\rightarrow U_j$ is a closed immersion.",
"Therefore, $f$ is a closed immersion, proving the proposition.",
"Example 1.7 The closed immersion $X_{ev}\\rightarrow X$ corresponds to the embedding $\\mathbb {X}_{ev}\\rightarrow \\mathbb {X}$ , where the subfunctor $\\mathbb {X}_{ev}$ is defined as $\\mathbb {X}_{ev}(A)=\\mathbb {X}(\\iota )(\\mathbb {X}(A_0))\\simeq \\mathbb {X}(A_0) \\text{ for } A\\in \\mathsf {SAlg}_K ,$ and $\\iota : A_0\\rightarrow A$ is the canonical algebra monomorphism.",
"In particular, $\\mathbb {X}_{ev}$ is a closed super-subscheme of $\\mathbb {X}$ (see also [14], Proposition 9.2).",
"As above, $\\mathbb {X}\\rightarrow \\mathbb {X}_{ev}$ is an endofunctor of the category $\\mathcal {SF}$ .",
"Note that if $\\mathbb {X}$ is not a superscheme, then the above map $\\mathbb {X}(\\iota )$ is no longer injective.",
"For example, consider the functor $A\\mapsto \\overline{A}$ for $A\\in \\mathsf {SAlg}_K$ .",
"Let $f : X\\rightarrow Y$ be an immersion.",
"Assume that $f$ factors through an open super-subscheme $U$ of $Y$ .",
"Let $\\mathcal {J}$ denote $\\ker (f_*\\mathcal {O}_U\\rightarrow \\mathcal {O}_X)$ .",
"For any non-negative integer $n$ , one can define the $n$ -th neighborhood of $f$ as a closed super-subscheme of $U$ , given by the super-ideal sheaf $\\mathcal {J}^{n+1}$ , denoted by $Y^n_f$ .",
"Lemma 1.8 The definition of $Y^n_f$ does not depend on $U$ .",
"If $f$ factors through another open super-subscheme $V$ of $Y$ , then $f$ factors through $U\\cap V$ .",
"Without loss a generality, one can assume that $V\\subseteq U$ .",
"Then for every $x\\in X^e\\subseteq V^e$ there is a natural isomorphism $\\mathcal {O}_{Y^n_f, x}\\simeq \\mathcal {O}_{U, x}/\\mathcal {J}^{n+1}_x\\simeq \\mathcal {O}_{V, x}/(\\mathcal {J}|_V)_x^{n+1}$ .",
"For example, let $B$ be a local superalgebra with nilpotent maximal super-ideal $\\mathfrak {m}$ such that $B/\\mathfrak {m}=K$ .",
"Every superscheme morphism $\\mathrm {SSpec}(B)\\rightarrow X$ is uniquely defined by a $K$ -point $x\\in X^e$ and by a local morphism of superalgebras $\\mathcal {O}_x\\rightarrow B$ .",
"In particular, such morphism factors through any open neighborhood of $x$ , and it is a closed immersion if and only if $\\mathcal {O}_x\\rightarrow B$ is surjective.",
"In particular, we have a canonical closed immersion $\\mathrm {SSpec}(\\mathcal {O}_x/\\mathfrak {m}_x^{n+1})\\rightarrow X$ , which is just the $n$ -th neighborhood of the closed immersion $i_x : \\mathrm {SSpec}(\\mathcal {O}_x/\\mathfrak {m}_x)\\rightarrow X$ ."
],
[
"Morphisms of (locally) finite type",
"Recall that a morphism $f : X\\rightarrow Y$ of geometric superschemes is said to be of locally finite type if there is an open covering of $Y$ by affine super-subschemes $V_i\\simeq \\mathrm {SSpec}(B_i)$ such that for every $i$ , the open super-subscheme $f^{-1}(V_i)$ is covered by open super-subschemes $U_{ij}\\simeq \\mathrm {SSpec}(A_{ij})$ , where each $A_{ij}$ is a finitely generated $B_i$ -superalgebra.",
"If the above covering of each $f^{-1}(V_i)$ is finite, then $f$ is said to be a morphism of finite type (cf.",
"[6], II.3).",
"Lemma 1.9 Let $\\phi : B\\rightarrow A$ be a superalgebra morphism.",
"Then $A$ , regarded as a $B$ -superalgebra via $\\phi $ , is finitely generated if and only if there are $b_1, \\ldots , b_s\\in B_0$ such that $\\sum _{1\\le i\\le s}B_0 b_i=B_0$ and $A_{\\phi (b_i)}$ is a finitely generated $B_{b_i}$ -superalgebra for each $1\\le i\\le s$ .",
"There is a super-subalgebra $C$ of $A$ , finitely generated over $B$ , such that for every $i$ , there is $A_{\\phi (b_i)}=C_{\\phi (b_i)}$ .",
"Lemma 1.2, [18], concludes the proof.",
"Combining Lemma REF with Lemma 3.5, [14], one can easily derive the following characterization of morphisms of finite type (cf.",
"[6], Exercise II.3.3(b)).",
"Lemma 1.10 A morphism $f : X\\rightarrow Y$ in $\\mathcal {SV}$ is of locally finite (respectively, finite) type if and only if for every open super-subscheme $V\\simeq \\mathrm {SSpec}(B)$ of $Y$ , the open super-subscheme $U=f^{-1}(V)$ has a (finite) open covering by super-subschemes $U_i\\simeq \\mathrm {SSpec}(A_i)$ such that each $A_i$ is a finitely generated $B$ -superalgebra.",
"A superscheme $X$ is said to be of (locally) finite type if the canonical morphism $X\\rightarrow \\mathrm {SSpec}(K)$ is of (locally) finite type."
],
[
"Separated and proper morphisms of superschemes",
"Let $\\mathcal {P}$ be a property of a class of geometric superschemes or a class of morphisms in $\\mathcal {SV}$ .",
"We say that this property is even reducible, provided $X$ satisfies $\\mathcal {P}$ if and only if $X_{ev}$ does (respectively, $f$ satisfies $\\mathcal {P}$ if and only if $f_{ev}$ does).",
"For example, the property of a Noetherian (geometric) superscheme to be affine is even reducible (see [17], Theorem 3.1).",
"The equivalence $\\mathcal {SV}\\simeq \\mathcal {SF}$ naturally translates this definition to the category $\\mathcal {SF}$ .",
"That is, $\\mathcal {P}$ is even reducible in $\\mathcal {SV}$ if and only if it is even reducible in $\\mathcal {SF}$ .",
"Since the categories $\\mathcal {SV}$ and $\\mathcal {SF}$ are equivalent and $\\mathcal {SF}$ has fibered products, $\\mathcal {SV}$ has as well.",
"Let $p_X$ and $p_Y$ denote the canonical projection morphisms $X\\times _S Y\\rightarrow X$ and $X\\times _S Y\\rightarrow Y$ , respectively.",
"The fibered product $X\\times _{SSpec \\ K} Y$ , denoted by $X\\times Y$ , is a direct product of superschemes $X$ and $Y$ in $\\mathcal {SV}$ .",
"The following lemma will be used later.",
"Lemma 2.1 Let $X$ and $Y$ be superschemes over a superscheme $S$ .",
"Let $\\lbrace U_i\\rbrace _{i\\in I}$ be an open covering of $X$ .",
"Then the superschemes $U_i\\times _S Y$ form an open covering of $X\\times _S Y$ .",
"By Theorem 5.14 and remark after Proposition 5.12, [14], one can work in the category $\\mathcal {SF}$ .",
"The easy superization of both I.1.7(3) and I.1.7(4) from [8] implies the statement.",
"A morphism $f : X\\rightarrow Y$ in $\\mathcal {SV}$ is called separated if the diagonal morphism $\\delta _f : X\\rightarrow X\\times _Y X$ is a closed immersion.",
"We also say that $X$ is separated over $Y$ .",
"In particular, a superscheme $X$ is called separated, provided $X$ is separated over $SSpec \\ K$ (cf.",
"[6], II, §4).",
"For example, any morphism of affine superschemes is separated (see Proposition II.4.1, [6]).",
"In particular, any affine superscheme is separated.",
"The following lemma superizes Corollary II.4.2, [6].",
"Lemma 2.2 A morphism $f : X\\rightarrow Y$ is separated if and only $\\delta _f^e(X^e)$ is a closed subset of $(X\\times _Y X)^e$ .",
"The proof of Corollary II.4.2, [6] can be copied verbatim, provided we prove the following.",
"If $V$ is an affine super-subscheme of $Y$ and $U$ is an affine super-subscheme of $X$ such that $f(U)\\subseteq V$ , then the natural morphism $U\\times _V U\\rightarrow X\\times _Y X$ is an open immersion.",
"By the remark after Proposition 5.12, [14], all we need is to check the analogous statement in the category $\\mathcal {SF}$ which is evident (see [8], I.1.7(3)).",
"Lemma 2.3 For any morphisms $f : X\\rightarrow S$ and $g : Y\\rightarrow S$ in $\\mathcal {SV}$ , the fibred product $X_{ev}\\times _{S_{ev}} Y_{ev}$ is isomorphic to $(X\\times _S Y)_{ev}$ .",
"For every $L\\in \\mathcal {SV}$ , let $L_0$ denote a purely-even superscheme $(L^e, (\\mathcal {O}_L)_0)$ .",
"The morphisms $L\\rightarrow Z_{ev}$ are in one-to-one correspondence with the morphisms $L_0\\rightarrow Z$ , which factor through $L_0\\rightarrow Z_{ev}$ .",
"Therefore, by the universality of a fiber product, the morphisms $L\\rightarrow X_{ev}\\times _{S_{ev}} Y_{ev}$ are in one-to-one correspondence with the morphisms $L_0\\rightarrow X\\times _S Y$ hence with the morphisms $L\\rightarrow (X\\times _S Y)_{ev}$ , proving the lemma.",
"Proposition 2.4 A morphism $f : X\\rightarrow Y$ is separated if and only if $f_{ev}$ is separated, if and only if $f_{res}$ is separated (the latter is regarded as a morphism of geometric schemes).",
"Lemma REF implies that $\\delta _f|_{X_{ev}}$ can be identified with $\\delta _{f|_{X_{ev}}}$ .",
"The last equivalence is now apparent.",
"Corollary 2.5 The property of a superscheme morphism to be separated is even reducible.",
"In particular, all standard properties of separated morphisms of schemes, formulated in Corollary II.4.6, [6], are valid for superschemes.",
"A morphism of geometric superschemes $f : X\\rightarrow Y$ is called closed if $f^e$ takes closed subsets of $X^e$ to closed subsets of $Y^e$ .",
"It is called universally closed if for any morphism $Y^{\\prime }\\rightarrow Y$ the projection $X\\times _Y Y^{\\prime }\\rightarrow Y^{\\prime }$ is closed.",
"Obviously, the first property is even reducible.",
"Hence, by Lemma REF , the second one is, too.",
"A morphism of geometric superschemes $f : X\\rightarrow Y$ is called proper if it is separated, universally closed, and of finite type.",
"Lemma 2.6 If $X$ is Noetherian, then $f : X\\rightarrow Y$ is of finite type if and only if $f_{ev}$ is of finite type, if and only if $f_{res}$ is of finite type.",
"By Lemma REF , it is enough to consider the case $X=SSpec \\ A$ and $Y=SSpec \\ B$ , where $A$ is a Noetherian superalgebra, and $\\overline{A}$ is finitely generated over $\\overline{B}$ .",
"Since $A_1^n/A_1^{n+1}$ is a finitely generated $\\overline{A}$ -module for every $n\\ge 1$ and $A_1^N =0$ for sufficiently large $N$ , $A$ is finitely generated over $B_0$ , hence over $B$ .",
"Corollary 2.7 In the full subcategory consisting of Noetherian superschemes, Corollary II.4.8, [6], can be superized verbatim.",
"A Noetherian superscheme $X$ is called complete if it is separated and proper over $SSpec \\ K$ , i.e., the morphism $X\\rightarrow SSpec \\ K$ is separated and proper simultaneously.",
"By the above, this property is even reducible.",
"Recall that a superalgebra $A$ is called reduced, provided the algebra $\\overline{A}$ is.",
"A geometric superscheme $X$ is called reduced if $X_{res}$ is a reduced scheme (cf.",
"[16], 2.3).",
"This property is local, i.e., $X$ is reduced if and only if the superalgebra $\\mathcal {O}_x$ is reduced for every $x\\in X^e$ .",
"In particular, an affine superscheme $SSpec \\ A$ is reduced if and only if the superalgebra $A$ is reduced.",
"Thus a superscheme $X$ is reduced if and only if every open affine super-subscheme of $X$ is reduced.",
"Following [9], we call a geometric superscheme $X$ (algebraic) supervariety if $X$ is of finite type, separated and geometrically reduced.",
"The latter means that the superscheme $X_{\\overline{K}}=X\\times SSpec \\overline{K}$ is reduced.",
"By the above, a supervariety is even reducible.",
"A superalgebra $A$ is called Grassman-like if $\\overline{A}=K$ or, equivalently, if odd elements generate $A$ .",
"Lemma 2.8 A connected supervariety $SSpec \\ A$ is complete if and only if $A$ is a Grassman-like superalgebra.",
"Use even reducibility of this property and [9], A.114(f).",
"Let $f : X\\rightarrow Y$ be a morphism in $\\mathcal {SV}$ and let $X$ be a Noetherian superscheme.",
"Then there is a closed super-subscheme $Z$ of $Y$ such that $f^e(X^e)\\subseteq Z^e$ , the morphism $f$ factors through the closed immersion $Z\\rightarrow Y$ , and for any closed super-subscheme $Z^{\\prime }$ of $Y$ such that $f$ factors through the closed immersion $Z^{\\prime }\\rightarrow Y$ , $Z\\rightarrow Y$ factors through $Z^{\\prime }\\rightarrow Y$ .",
"Since $X$ is a Noetherian superscheme, the third remark after Proposition 3.1 in [17] implies that $f_*\\mathcal {O}_X$ is a coherent sheaf of $\\mathcal {O}_Y$ -supermodules.",
"Then, by Corollary 3.2, [17] and Proposition 2.5 from [16], the superideal sheaf $\\ker (\\mathcal {O}_Y\\rightarrow f_*\\mathcal {O}_X)$ is a coherent sheaf of $\\mathcal {O}_Y$ -supermodules, which defines the superscheme $Z$ , called a superscheme-theoretic image of $f$ and denoted by $f(X)$ .",
"Observe also that if $X$ is (geometrically) reduced, then $f(X)$ is also (geometrically) reduced.",
"Proposition 2.9 The following statements hold: If $Z$ is a closed super-subscheme of a complete Noetherian superscheme $X$ , then $Z$ is complete.",
"Let $f : X\\rightarrow Y$ be a morphism of algebraic supervarieties.",
"If $X$ is complete, then $f^e(X^e)$ is closed, and $f(X)$ is complete.",
"If $X$ is a complete connected supervariety, then $\\mathcal {O}_X(X^e)$ is a Grassman-like superalgebra.",
"The first statement follows by superized Corollary II.4.8(a, b), [6].",
"It is easy to see that $f_{res}(X_{res})$ coincides with $f(X)_{res}$ .",
"Then the second statement follows by [9], A.114(d).",
"Let $A$ denote $\\mathcal {O}(X)$ .",
"The induced morphism $f : X\\rightarrow SSpec \\ A$ factors through its scheme-theoretic image $f(X)$ .",
"Since $f(X)$ is closed, $f(X)=C(I)$ for a superideal $I$ of $A$ .",
"On the other hand, the composition of superalgebra morphisms $A\\rightarrow A/I\\rightarrow \\mathcal {O}(X)$ is an identity map.",
"Thus $I=0$ , hence $f(X)=SSpec \\ A$ is a complete connected supervariety.",
"Lemma REF concludes the proof."
],
[
"Group superschemes",
"A group object in the category $\\mathcal {SF}$ is called a group superscheme.",
"Similarly, a group object in the category $\\mathcal {SV}$ is called a geometric group superscheme.",
"These objects form subcategories in $\\mathcal {SF}$ and $\\mathcal {SV}$ , respectively, with morphisms preserving their group structures.",
"They are denoted by $\\mathcal {SFG}$ and $\\mathcal {SVG}$ , correspondingly.",
"Theorem REF implies that $\\mathcal {SFG}\\simeq \\mathcal {SVG}$ .",
"A geometric group superscheme is called locally algebraic if it is of locally finite type as a superscheme.",
"Symmetrically, a group superscheme $\\mathbb {G}$ is called locally algebraic if its geometric counterpart $G$ is locally algebraic.",
"Locally algebraic group superschemes form a full subcategory of $\\mathcal {SFG}$ , denoted by $\\mathcal {SFG}_{la}$ .",
"The category $\\mathcal {SFG}_{la}$ is equivalent to the full subcategory $\\mathcal {SVG}_{la}$ of $\\mathcal {SVG}$ , consisting of all locally algebraic geometric group superschemes.",
"A geometric group superscheme is called algebraic if it is of finite type as a superscheme.",
"As above, one can define algebraic group superschemes as two equivalent categories $\\mathcal {SVG}_a$ and $\\mathcal {SFG}_a$ .",
"Let $\\mathbb {E}$ denote the trivial group superscheme.",
"That is, for every $A\\in \\mathsf {SAlg}_K$ , the group $\\mathbb {E}(A)=\\lbrace e_A\\rbrace $ is trivial.",
"Its geometric counterpart is isomorphic to $e\\simeq \\mathrm {SSpec}(K)$ with the trivial group structure.",
"The unique point of $(\\mathrm {SSpec}(K))^e$ is also denoted by $e$ .",
"A sequence $e\\rightarrow H\\rightarrow G\\stackrel{f}{\\rightarrow } R\\rightarrow e$ in the category $\\mathcal {SVG}$ is called exact if the corresponding sequence $\\mathbb {E}\\rightarrow \\mathbb {H}\\rightarrow \\mathbb {G}\\stackrel{\\bf f}{\\rightarrow } \\mathbb {R}\\rightarrow \\mathbb {E}$ in the category $\\mathcal {SFG}$ is exact, that is, for any superalgebra $A$ we have $\\mathbb {H}(A)=\\ker {\\bf f}(A)$ , and the sheafification of the naive quotient functor $A\\rightarrow (\\mathbb {G}/\\mathbb {H})_{(n)}(A)=\\mathbb {G}(A)/\\mathbb {H}(A)\\subseteq \\mathbb {R}(A),$ with respect to the Grothendieck topology of fppf coverings, coincides with $\\mathbb {R}$ .",
"The latter is equivalent to the following condition.",
"For any superalgebra $A$ and any $g\\in \\mathbb {R}(A)$ , there is a finitely presented $A$ -superalgebra $A^{\\prime }$ , which is a faithfully flat $A$ -supermodule such that $\\mathbb {R}(\\iota _A)(g)\\in {\\bf f}(A^{\\prime })(\\mathbb {G}(A^{\\prime }))$ , where $\\iota _A : A\\rightarrow A^{\\prime }$ is the corresponding monomorphism of superalgebras (cf.",
"[14], page 144, and Proposition 5.15 therein).",
"A geometric group superscheme $G$ (or the corresponding group superscheme $\\mathbb {G}$ ) is called an abelian supervariety if $G$ is a complete supervariety.",
"Since both properties are even reducible, we obtain that $G$ is an abelian supervariety if and only if $G_{ev}$ is an abelian variety (cf.",
"[9], Definition 10.13).",
"Let $\\mathbb {R}$ be a group superscheme that acts on a superscheme $\\mathbb {X}$ on the right, and on a superscheme $\\mathbb {Y}$ on the left.",
"Then $\\mathbb {R}$ acts on $\\mathbb {X}\\times \\mathbb {Y}$ as $(x, y)\\cdot r=(x\\cdot r, r^{-1}\\cdot y)$ for $x\\in \\mathbb {X}(B), y\\in \\mathbb {Y}(B), r\\in \\mathbb {R}(B)$ and $B\\in \\mathsf {SAlg}_K$ .",
"The sheafification of the naive quotient $B\\rightarrow (\\mathbb {X}(B)\\times \\mathbb {Y}(B))/\\mathbb {R}(B)$ is called the homogeneous fiber quotient, and it is denoted by $\\mathbb {X}\\times ^{\\mathbb {R}}\\mathbb {Y}$ .",
"This construction is functorial in both $\\mathbb {X}$ and $\\mathbb {Y}$ .",
"Assume that all the previous superschemes are affine, say $\\mathbb {X}\\simeq \\mathrm {SSp}(\\mathbb {A}), \\mathbb {Y}\\simeq \\mathrm {SSp}(\\mathbb {B})$ , and $\\mathbb {R}\\simeq \\mathrm {SSp}(\\mathbb {D})$ .",
"In particular, $\\mathbb {A}$ and $\\mathbb {B}$ are right and left $\\mathbb {D}$ -coideal superalgebras (cf.",
"[18]).",
"Lemma 3.1 If $\\mathbb {X}\\times ^{\\mathbb {R}}\\mathbb {Y}$ is an affine superscheme, it is canonically isomorphic to $\\mathrm {SSp}(\\mathbb {A}\\square _{\\mathbb {D}}\\mathbb {B})$ .",
"Moreover, if there are $\\mathbb {R}$ -equivariant morphisms $\\mathbb {X}^{\\prime }\\rightarrow \\mathbb {X}$ and $\\mathbb {Y}^{\\prime }\\rightarrow \\mathbb {Y}$ , where $\\mathbb {X}^{\\prime }\\simeq \\mathrm {SSp}(\\mathbb {A}^{\\prime })$ , $\\mathbb {Y}^{\\prime }\\simeq \\mathrm {SSp}(\\mathbb {B}^{\\prime })$ , and $\\mathbb {X}^{\\prime }\\times ^{\\mathbb {R}}\\mathbb {Y}^{\\prime }$ are affine, then the induced morphism $\\mathbb {X}^{\\prime }\\times ^{\\mathbb {R}}\\mathbb {Y}^{\\prime }\\rightarrow \\mathbb {X}\\times ^{\\mathbb {R}}\\mathbb {Y}$ is dual to the natural superalgebra morphism $\\mathbb {A}^{\\prime }\\square _{\\mathbb {D}}\\mathbb {B}^{\\prime }\\rightarrow \\mathbb {A}\\square _{\\mathbb {D}}\\mathbb {B}$ .",
"Use Proposition 4.1, [18]."
],
[
"The affinization of group superschemes",
"Let $X$ be a geometric superscheme, and $U$ be its open super-subscheme.",
"As it has been already observed, a (geometric) superscheme morphism $f : Z\\rightarrow X$ factors through $U$ if and only if $f^e(Z^e)\\subseteq U$ .",
"In particular, if $X\\times _Z Y$ is a fibred product in $\\mathcal {SV}$ , then for every open super-subschemes $U$ and $V$ of $X$ and $Y$ , respectively, the open super-subscheme $p_X^{-1}(U)\\cap p_Y^{-1}(V)$ of $X\\times _Z Y$ is naturally isomorphic to $U\\times _Z V$ .",
"Here $p_X : X\\times _Z Y\\rightarrow X$ and $p_Y : X\\times _Z Y\\rightarrow Y$ are the canonical projections.",
"Note that if $\\mathbb {X}(K)\\ne \\emptyset $ (respectively, $\\mathbb {Y}(K)\\ne \\emptyset $ ), then $p_X^*$ (respectively, $p^*_Y$ ) is injective.",
"Note that the sheaf morphisms $p_X^*$ and $p_Y^*$ induce a superalgebra morphism $\\mathcal {O}(X)\\otimes _{\\mathcal {O}(Z)}\\mathcal {O}(Y)\\rightarrow \\mathcal {O}(X\\times _Z Y)$ .",
"Let $G$ be a group superscheme, and let $m, \\iota $ and $\\epsilon $ denote the multiplication morphism $G\\times G\\rightarrow G$ , the inversion morphism $G\\rightarrow G$ , and the closed immersion $e\\rightarrow G$ , respectively.",
"We also denote by $p_1$ and $p_2$ the corresponding projections from $G\\times G$ to $G$ .",
"Lemma 4.1 Let $H\\simeq \\mathrm {SSpec}(B)$ be an affine geometric group superscheme.",
"Then every morphism $G\\rightarrow H$ in $\\mathcal {SVG}$ is uniquely defined by a superalgebra morphism $\\phi : B\\rightarrow \\mathcal {O}(G)$ , that satisfies the following conditions: The superalgebra morphism $B\\otimes B\\stackrel{\\phi \\otimes \\phi }{\\rightarrow }\\mathcal {O}(G)\\otimes \\mathcal {O}(G)\\stackrel{p_1^*\\otimes p_2^*}{\\rightarrow }\\mathcal {O}(G\\times G)$ makes the diagram $\\begin{array}{ccc}B\\otimes B & \\rightarrow & \\mathcal {O}(G\\times G) \\\\\\uparrow & & \\uparrow \\\\B & \\rightarrow & \\mathcal {O}(G)\\end{array},$ where the vertical arrows are $\\Delta _B$ and $m^*$ , respectively, to be commutative.",
"$\\epsilon ^* \\phi =\\epsilon _B$ .",
"Use Lemma 4.1, [14].",
"It is clear that there is the largest super-subalgebra $C$ of $\\mathcal {O}(G)$ , such that the superalgebra morphism $\\Delta _C : C\\rightarrow \\mathcal {O}(G)\\stackrel{m^*}{\\rightarrow } \\mathcal {O}(G\\times G)$ maps $C$ to $(p_1^*\\otimes p_2^*)(C\\otimes C)$ .",
"Proposition 4.2 $C$ is a Hopf superalgebra with the coproduct $\\Delta _C$ , the antipode $\\iota ^*|_C$ and the counit $\\epsilon ^*|_C$ .",
"Let $q_{12}$ and $q_3$ denote the canonical projections $(G\\times G)\\times G\\rightarrow G\\times G$ and $(G\\times G)\\times G\\rightarrow G$ , respectively.",
"Symmetrically, let $q_{1}$ and $q_{23}$ denote the canonical projections $G\\times (G\\times G)\\rightarrow G$ and $G\\times (G\\times G)\\rightarrow G\\times G$ , respectively.",
"The natural isomorphism $\\alpha : (G\\times G)\\times G\\rightarrow G\\times (G\\times G)$ is uniquely defined by the identities: $q_1\\alpha =p_1 p_{12}, \\ p_1 q_{23}\\alpha =p_2 q_{12} \\text{ and } \\ q_3=p_2q_{23}\\alpha .$ Moreover, the following identities are satisfied: $m q_{12}=p_1(m\\times \\mathrm {id}_G), \\ q_3=p_2(m\\times \\mathrm {id}_G), \\ mq_{23}=p_2(\\mathrm {id}_G\\times m),\\text{ and } q_1=p_1(\\mathrm {id}_G\\times m).$ Then there is a commutative diagram $\\begin{array}{ccccccc}C^{\\otimes 3} & \\rightarrow & \\mathcal {O}(G)^{\\otimes 3} & \\stackrel{p_1^*\\otimes p_2^*\\otimes \\mathrm {id}_{\\mathcal {O}(G)}}{\\rightarrow } & \\mathcal {O}(G\\times G)\\otimes \\mathcal {O}(G) & \\stackrel{q_{12}^*\\otimes q_3^*}{\\rightarrow } & \\mathcal {O}((G\\times G)\\times G) \\\\\\uparrow & & & & \\uparrow & & \\uparrow \\\\C^{\\otimes 2} & & & \\longrightarrow & \\mathcal {O}(G)^{\\otimes 2} & \\stackrel{p_1^*\\otimes p_2^*}{\\rightarrow } &\\mathcal {O}(G\\times G) \\\\\\uparrow & & & & & & \\uparrow \\\\C & & & & \\longrightarrow & & \\mathcal {O}(G)\\end{array},$ where the upper vertical arrows are $\\Delta _C\\otimes \\mathrm {id}_C, m^*\\otimes \\mathrm {id}_{\\mathcal {O}(G)}$ , and $(m\\times \\mathrm {id}_G)^*$ , respecitvely.",
"The lower vertical arrows are $\\Delta _C$ and $m^*$ , respectively.",
"Symmetrically, there is a commutative diagram $\\begin{array}{ccccccc}\\mathcal {O}(G\\times (G\\times G)) & \\stackrel{q_1^*\\otimes q_{23}^*}{\\leftarrow } & \\mathcal {O}(G)\\otimes \\mathcal {O}(G\\times G) & \\stackrel{\\mathrm {id}_{\\mathcal {O}(G)}\\otimes p_1^*\\otimes p_2^*}{\\leftarrow } & \\mathcal {O}(G)^{\\otimes 3} & \\leftarrow & C^{\\otimes 3} \\\\\\uparrow & & \\uparrow & & & & \\uparrow \\\\\\mathcal {O}(G\\times G) & \\stackrel{p_1^*\\otimes p_2^*}{\\leftarrow } & \\mathcal {O}(G)^{\\otimes 2} & \\longleftarrow & & &C^{\\otimes 2} \\\\\\uparrow & & & & & & \\uparrow \\\\\\mathcal {O}(G) & & \\longleftarrow & & & & C\\end{array},$ where the upper vertical arrows are $(\\mathrm {id}_G\\times m)^*, \\mathrm {id}_{\\mathcal {O}(G)}\\otimes m^*$ , and $\\mathrm {id}_C\\otimes \\Delta _C$ , respectively.",
"Using the isomorphism $\\alpha $ and the axiom of associativity, we obtain that $\\Delta _C$ is a coproduct.",
"Therefore, $C$ is a superbialgebra.",
"We leave it for the reader to verify the identity $\\Delta _C\\iota ^*=t_{C, C}(\\iota ^*\\otimes \\iota ^*)\\Delta _C,$ where $t_{C, C}$ is a braiding $c_1\\otimes c_2\\mapsto (-1)^{|c_1||c_2|}c_2\\otimes c_1$ for $c_1, c_2\\in C$ .",
"Therefore, $\\Delta _C$ maps $\\iota ^*(C)$ to $\\iota ^*(C)^{\\otimes 2}$ , hence $\\iota ^*(C)\\subseteq C$ .",
"The proposition is proven.",
"Corollary 4.3 The geometric group superscheme $\\mathrm {SSpec}(C)$ is the largest affine quotient of $G$ .",
"This means that every homomorphism from $G$ to an affine geometric group superscheme uniquely factors through $G\\rightarrow \\mathrm {SSpec}(C)$ .",
"Similarly, $\\mathrm {SSp}(C)$ is the largest affine quotient of $\\mathbb {G}$ .",
"We denote the superschemes $\\mathrm {SSpec}(C)$ and $\\mathrm {SSp}(C)$ by $G^{aff}$ and $\\mathbb {G}^{aff}$ , respectively.",
"In general, $C$ is a proper super-subalgebra of $\\mathcal {O}(G)$ .",
"But if $G$ is algebraic, then $C=\\mathcal {O}(G)$ .",
"Superizing Lemma 1.8 from [4], I, §2, one can show that if $X$ and $Y$ are geometric superschemes of finite type, then $p^*_X\\otimes p_Y^*$ induces an isomorphism $\\mathcal {O}(X)\\otimes \\mathcal {O}(Y)\\simeq \\mathcal {O}(X\\times Y)$ , which is functorial in both $X$ and $Y$ ."
],
[
"The tangent functor at specific points",
"The content of this section extends [4], [18].",
"Let $\\mathbb {X}$ be a $K$ -functor.",
"For every point $x\\in \\mathbb {X}(K)$ , one can define a $K$ -functor $\\mathrm {T}_x(\\mathbb {X})$ , called a tangent $K$ -functor at point $x$ , as follows.",
"For each superalgebra $R$ , set $x_R=\\mathbb {X}(\\iota ^R_K)(x)$ , where $\\iota _K^R : K\\rightarrow R$ is the canonical embedding.",
"Observe that $x=x_K$ .",
"Let $R[\\epsilon _0, \\epsilon _1]$ denote the superalgebra of dual super-numbers, that is $R[\\epsilon _0, \\epsilon _1]=R[x|y]/<x^2,y^2, xy>, \\text{ where } |x|=0, |y|=1,$ and $\\epsilon _0$ and $\\epsilon _1$ denote the residue classes of $x$ and $y$ , respectively.",
"We have two $R$ -superalgebra morphisms $p_R : R[\\epsilon _0, \\epsilon _1]\\rightarrow R$ and $i_R : R\\rightarrow R[\\epsilon _0, \\epsilon _1]$ such that $p_R(r)=r$ , $p_R(\\epsilon _i)=0$ for $i=0, 1$ , and $i_R(r)=r$ .",
"Since, $p_r i_R=\\mathrm {id}_R$ , the map $\\mathbb {X}(p_R)$ is surjective.",
"Set $\\mathrm {T}_x(\\mathbb {X})(R)=\\mathbb {X}(p_R)^{-1}(x_R)$ .",
"It is clear that $\\mathrm {T}_x(\\mathbb {X})$ is a $K$ -functor.",
"Assume that $\\mathbb {X}$ is a superscheme.",
"Let $\\mathbb {U}\\simeq \\mathrm {SSp}(A)$ be an open affine super-subscheme of $\\mathbb {X}$ .",
"As it has been already observed, $x_R$ belongs to $\\mathbb {U}(R)$ if and only if $x$ belongs to $\\mathbb {U}(K)$ .",
"Besides, $x$ can be identified with a superalgebra morphism $\\epsilon _A : A\\rightarrow K$ and $x_R$ can be identifed with $\\iota ^R_K\\epsilon _A$ .",
"Let $\\mathfrak {m}_A$ denote $\\ker \\epsilon _A$ .",
"Recall that there is a superscheme morphism ${\\bf i}_x : \\mathrm {SSp}(\\mathcal {O}_x)\\rightarrow \\mathbb {X}$ (corresponding to the closed immersion $i_x : \\mathrm {SSpec}(\\mathcal {O}_x)\\rightarrow X$ ) that factors through every open neighborhood of $x$ .",
"Lemma 5.1 For every open super-subscheme $\\mathbb {U}$ of $\\mathbb {X}$ and every $R\\in \\mathsf {SAlg}_K$ , there is $\\mathbb {X}(p_R)^{-1}(\\mathbb {U}(R))\\subseteq \\mathbb {U}(R[\\epsilon _0, \\epsilon _1])$ .",
"In particular, $\\mathrm {T}_x(\\mathbb {X})(R)=\\mathrm {T}_x(\\mathbb {U})(R)$ , provided $x\\in \\mathbb {U}(K)$ .",
"Let $U$ be an open super-subscheme of $X$ that corresponds to $\\mathbb {U}$ .",
"Then the statement of the lemma is equivalent to the following.",
"If $f : \\mathrm {SSpec}(R[\\epsilon _0, \\epsilon _1])\\rightarrow X$ satisfies $(f\\mathrm {SSpec}(p_R))^e(\\mathrm {SSpec}(R)^e)\\subseteq U^e$ , then $f^e((\\mathrm {SSpec}(R[\\epsilon _0, \\epsilon _1]))^e)\\subseteq U^e$ .",
"Note that any prime super-ideal of $R[\\epsilon _0, \\epsilon _1]$ has a form $\\mathfrak {p}+\\epsilon _0 R +\\epsilon _1 R$ , where $\\mathfrak {p}\\in (\\mathrm {SSpec}(R))^e$ .",
"Since $(\\mathrm {SSpec}(p_R))^e(\\mathfrak {p})=\\mathfrak {p}+\\epsilon _0 R +\\epsilon _1 R$ , the statement follows.",
"Let $V$ be a superspace.",
"Denote by $\\mathbb {V}_a$ the $K$ -functor given by $\\mathbb {V}_a(R)=V\\otimes R$ for $R\\in \\mathsf {SAlg}_K$ .",
"From now on, assume that $\\mathbb {X}$ is of locally finite type, i.e., its geometric counterpart $X$ is.",
"Lemma REF implies that the superspace $\\mathfrak {m}_x/\\mathfrak {m}_x^2$ is finite-dimensional.",
"Lemma 5.2 The functor $\\mathrm {T}_x(\\mathbb {X})$ is isomorphic to $\\mathbb {V}_a$ , where $V=(\\mathfrak {m}_x/\\mathfrak {m}_x^2)^*$ .",
"By Lemma REF , we have a commutative diagram $\\begin{array}{ccccc}\\mathrm {T}_x(\\mathbb {X})(R) & \\rightarrow & \\mathbb {X}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {X}(R) \\\\\\parallel & & \\uparrow & & \\uparrow \\\\\\mathrm {T}_x(\\mathbb {U})(R) & \\rightarrow & \\mathbb {U}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {U}(R) \\\\\\parallel & & \\uparrow & & \\uparrow \\\\\\mathrm {T}_x(\\mathrm {SSp}(\\mathcal {O}_x))(R) & \\rightarrow & \\mathrm {SSp}(\\mathcal {O}_x)(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathrm {SSp}(\\mathcal {O}_x)(R),\\end{array}$ where $\\mathbb {U}$ is an open affine super-subscheme of $\\mathbb {X}$ with $x\\in \\mathbb {X}(K)$ .",
"The lower middle and rightmost vertical arrows are ${\\bf i}_x(R[\\epsilon _0, \\epsilon _1])$ and ${\\bf i}_x(R)$ , respectively.",
"In particular, if $\\mathbb {U}\\simeq \\mathrm {SSp}(A)$ , then $(\\mathfrak {m}_A/\\mathfrak {m}_A^2)^*\\otimes R\\simeq \\mathrm {T}_x(\\mathbb {U})(R)$ via $(\\phi \\otimes r)(a)=\\epsilon _A(a)+(-1)^{|r||a|}\\epsilon _{k}\\phi (\\bar{a})r,$ where $r\\in R, \\phi \\in (\\mathfrak {m}_A/\\mathfrak {m}_A^2)^*, a\\in A, \\bar{a}=a-\\epsilon _A(a)$ and $k\\equiv |r|+|\\phi | \\pmod {2}$ .",
"Similarly, we have $(\\mathfrak {m}_x/\\mathfrak {m}_x^2)^*\\otimes R\\simeq \\mathrm {T}_x(\\mathrm {SSp}(\\mathcal {O}_x))(R)$ , and both isomorphisms are functorial in $R$ .",
"Since $\\mathfrak {m}_A/\\mathfrak {m}_A^2\\simeq \\mathfrak {m}_x/\\mathfrak {m}_x^2$ , the lemma follows.",
"To an homogeneous element $r\\in R$ , we associate an endomorphism $\\hat{r}$ of the superalgebra $R[\\epsilon _0, \\epsilon _1]$ defined by $\\hat{r}(a+\\epsilon _0 b+\\epsilon _1 c)=a+(-1)^{|a||r|}(\\epsilon _{|r|}br+\\epsilon _{1+|r|}cr).$ In fact, we have $\\hat{r}((a+\\epsilon _0 b+\\epsilon _1 c)(a^{\\prime }+\\epsilon _0 b^{\\prime }+\\epsilon _1 c^{\\prime }))=$ $\\hat{r}(aa^{\\prime }+(ba^{\\prime }+ab^{\\prime })\\epsilon _0+((-1)^{|a|}ac^{\\prime }+ca^{\\prime })\\epsilon _1)=$ $aa^{\\prime } +(-1)^{|r|(|a|+|a^{\\prime }|)}[\\epsilon _{|r|}(ba^{\\prime }+ab^{\\prime })r+\\epsilon _{1+|r|}((-1)^{|a|}ac^{\\prime }+ca^{\\prime })r] =$ $(a+(-1)^{|r||a|}(\\epsilon _{|r|}br+\\epsilon _{1+|r|}cr))(a^{\\prime }+(-1)^{|r||a^{\\prime }|}(\\epsilon _{|r|}b^{\\prime }r+\\epsilon _{1+|r|}c^{\\prime }r))=$ $\\hat{r}(a+b\\epsilon _0+c\\epsilon _1)\\hat{r}(a^{\\prime }+b^{\\prime }\\epsilon _0+c^{\\prime }\\epsilon _1).$ Moreover, if $r^{\\prime }$ is another homogeneous element, then $\\widehat{r r^{\\prime }}=\\widehat{r^{\\prime }}\\widehat{r}$ .",
"This proves the following lemma.",
"Lemma 5.3 An endomorphism $\\hat{r}$ satisfies $p_R\\hat{r}=p_R$ and $\\hat{r}i_R=i_R$ .",
"Therefore, $\\mathbb {X}(\\hat{r})$ maps $\\mathrm {T}_x(\\mathbb {X})(R)$ to itself.",
"Moreover, if we identify $\\mathrm {T}_x(\\mathbb {X})(R)$ with $(\\mathfrak {m}_x/\\mathfrak {m}_x^2)^*\\otimes R$ , then $\\mathbb {X}(\\hat{r})$ coincides with the map $\\phi \\otimes r^{\\prime }\\mapsto \\phi \\otimes r^{\\prime }r$ .",
"In particular, $\\mathrm {T}_x(\\mathbb {X})(R)$ has a natural structure of the right $R$ -supermodule.",
"Let ${\\bf f} : \\mathbb {X}\\rightarrow \\mathbb {Y}$ be a morphism of $K$ -functors.",
"Choose points $x\\in \\mathbb {X}(K)$ and $\\mathbb {Y}(K)$ such that ${\\bf f}(K)(x)=y$ .",
"For every superalgebra $R$ , the map ${\\bf f}(R)$ takes $x_R$ to $y_R$ .",
"Therefore, there is a commutative diagram $\\begin{array}{ccccc}\\mathrm {T}_x(\\mathbb {X})(R) & \\rightarrow & \\mathbb {X}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {X}(R) \\\\\\downarrow & & \\downarrow & & \\downarrow \\\\\\mathrm {T}_y(\\mathbb {Y})(R) & \\rightarrow & \\mathbb {Y}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {Y}(R)\\end{array}.$ The induced map $\\mathrm {T}_x(\\mathbb {X})(R)\\rightarrow \\mathrm {T}_y(\\mathbb {Y})(R)$ is functorial in $R$ .",
"That is, $\\bf f$ induces a $K$ -functor morphism $\\mathrm {d}_x{\\bf f} : \\mathrm {T}_x(\\mathbb {X})\\rightarrow \\mathrm {T}_y(\\mathbb {Y})$ , called the differential of $\\bf f$ at the point $x$ .",
"Let $f : X\\rightarrow Y$ be a morphism of locally algebraic geometric superschemes.",
"If $f^e(x)=y$ , then $f^*_x$ induces morphism of superspaces $d_x f : (\\mathfrak {m}_x/\\mathfrak {m}_x^2)^*\\rightarrow (\\mathfrak {n}_y/\\mathfrak {n}_y^2)^*$ .",
"Lemma 5.4 If we identify $\\mathrm {T}_x(\\mathbb {X})$ and $\\mathrm {T}_y(\\mathbb {Y})$ with $\\mathbb {V}_a$ and $\\mathbb {W}_a$ , respectively, where $V=(\\mathfrak {m}_x/\\mathfrak {m}_x^2)^*$ and $W=(\\mathfrak {n}_y/\\mathfrak {n}_y^2)^*$ , then $\\mathrm {d}_x{\\bf f}$ is naturally induced by $d_x f$ .",
"Replace $\\mathbb {X}$ and $\\mathbb {Y}$ by open affine neighborhoods $\\mathbb {U}$ and $\\mathbb {V}$ of $x$ and $y$ , respectively, such that $\\mathbb {U}\\subseteq {\\bf f}^{-1}(\\mathbb {V})$ , and use the diagram from Lemma REF ."
],
[
"Lie superalgebra of a group superscheme",
"Let $\\mathbb {G}$ be a locally algebraic group superscheme, $e$ be the identity element of $\\mathbb {G}(K)$ , and $\\mathrm {Lie}(\\mathbb {G})$ be the tangent functor $\\mathrm {T}_e(\\mathbb {G})$ .",
"Then $\\mathrm {Lie}(\\mathbb {G})$ is called the Lie superalgebra functor of $\\mathbb {G}$ .",
"Arguing as in Lemma REF , we obtain the exact sequence $\\begin{array}{ccccccccc}1 & \\rightarrow & \\mathrm {Lie}(\\mathbb {G})(R) & \\rightarrow & \\mathbb {G}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {G}(R) & \\rightarrow & 1\\\\& & \\parallel & & \\uparrow & & \\uparrow & & \\\\0 & \\rightarrow & R\\otimes (\\mathfrak {m}_e/\\mathfrak {m}^2_e)^* & \\rightarrow & \\mathrm {SSp}(\\mathcal {O}_e)(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow &\\mathrm {SSp}(\\mathcal {O}_e)(R) & & .\\end{array}$ Denote the superspace $(\\mathfrak {m}_e/\\mathfrak {m}^2_e)^*$ by $\\mathfrak {g}$ and call it the Lie superalgebra of $\\mathbb {G}$ .",
"Since $\\mathbb {G}(p_R)$ and $\\mathbb {G}(i_R)$ are group homomorphisms, there is an exact split sequence of groups: $1\\rightarrow \\mathrm {Lie}(\\mathbb {G})(R)\\rightarrow \\mathbb {G}(R[\\epsilon _0, \\epsilon _1])\\stackrel{\\mathbb {G}(p_R)}{\\rightarrow }\\mathbb {G}(R)\\rightarrow 1.$ In other words, $\\mathbb {G}(R[\\epsilon _0, \\epsilon _1])\\simeq \\mathbb {G}(R)\\ltimes \\mathrm {Lie}(\\mathbb {G})(R)$ , where $\\mathbb {G}(R)$ is identified with the subgroup $\\mathbb {G}(i_R)(\\mathbb {G}(R))\\le \\mathbb {G}(R[\\epsilon _0, \\epsilon _1])$ .",
"The group $\\mathbb {G}(R)$ acts on $\\mathrm {Lie}(\\mathbb {G})(R)$ as $\\mathrm {Ad}(g)(z)=\\mathbb {G}(i_R)(g)z \\mathbb {G}(i_R)(g)^{-1} \\text{ for } g\\in \\mathbb {G}(R) \\text{ and } z\\in \\mathrm {Lie}(\\mathbb {G})(R).$ This action is called adjoint.",
"Lemma 6.1 The adjoint action of $\\mathbb {G}(R)$ on $\\mathrm {Lie}(\\mathbb {G})(R)$ is $R$ -linear and functorial in $R$ .",
"Moreover, $\\mathbb {G}(R)$ acts on $\\mathrm {Lie}(\\mathbb {G})(R)$ by parity preserving operators.",
"The first statement can be easily derived from $\\hat{r}i_R=i_R$ .",
"The second is obvious.",
"To prove the last statement, we define an endomorphism $\\iota $ of superalgebra $R[\\epsilon _0, \\epsilon _1]$ by $a+\\epsilon _0 b+\\epsilon _1 c\\mapsto a+\\epsilon _0 b-\\epsilon _1 \\text{ for } c, a, b, c\\in R.$ It satisfies $p_R\\iota =p_R, \\iota i_R=i_R$ and commutes with each $\\hat{r}$ .",
"Thus $\\mathbb {G}(\\iota )$ induces an automorphism of $R$ -supermodule $\\mathrm {Lie}(\\mathbb {G})(R)$ that commutes with the adjoint action of $\\mathbb {G}(R)$ .",
"On the other hand, if we identify $\\mathrm {Lie}(\\mathbb {G})(R)$ with $(\\mathfrak {m}_e/\\mathfrak {m}^2_e)^*\\otimes R$ , then one immediately sees that $\\mathbb {G}(\\iota )$ sends a homogeneous element $x\\in \\mathrm {Lie}(\\mathbb {G})(R)$ to $(-1)^{|x|}x$ ."
],
[
"A fragment of differential calculus",
"In this section, we superize a fragment of differential calculus from [4], II, §4, for locally algebraic group superschemes (see also [3], [18]).",
"Let $V$ be a finite-dimensional superspace.",
"Recall that the affine (algebraic) group superscheme $\\mathrm {GL}(V)$ is defined as $\\mathrm {GL}(V)(R)=\\mathrm {End}_R(V\\otimes R)^{\\times }_0 \\text{ for } R\\in \\mathsf {SAlg}_K.$ In other words, each $\\mathrm {GL}(V)(R)$ consists of all graded (parity-preserving) $R$ -linear automorphisms of $R$ -supermodule $V\\otimes R$ .",
"The group superscheme $\\mathrm {GL}(V)$ is called the general linear supergroup.",
"Let $\\mathbb {G}$ be a group superscheme.",
"Then a (finite-dimensional) superspace $V$ is called a (left) $\\mathbb {G}$ -supermodule, provided there is a group superscheme morphism ${\\bf f} :\\mathbb {G}\\rightarrow \\mathrm {GL}(V)$ .",
"From now on, all $\\mathbb {G}$ -supermodules are assumed to be finite-dimensional unless stated otherwise.",
"Note that $\\mathbb {G}$ -supermodules form an abelian category (with graded morphisms).",
"For example, $V$ is a $\\mathrm {GL}(V)$ -supermodule.",
"If $\\mathbb {G}$ is a locally algebraic group superscheme, then Lemma REF implies that the Lie superalgebra $\\mathfrak {g}$ of $\\mathbb {G}$ is a $\\mathbb {G}$ -supermodule via $\\mathrm {Ad} : \\mathbb {G}\\rightarrow \\mathrm {GL}(\\mathfrak {g})$ .",
"Corollary REF immediately implies the following lemma.",
"Lemma 7.1 Let $\\mathbb {G}$ be a group superscheme.",
"Then the category of $\\mathbb {G}$ -supermodules is naturally isomorphic to the category of $\\mathbb {G}^{aff}$ -supermodules.",
"It is also isomorphic to the category of (finite-dimensional) right $\\mathcal {O}(G^{aff})$ -supercomodules (cf.",
"[18]).",
"Lemma 7.2 If $\\mathbb {E} \\rightarrow \\mathbb {H}\\stackrel{\\bf i}{\\rightarrow }\\mathbb {G}\\stackrel{\\bf p}{\\rightarrow }\\mathbb {R}\\rightarrow \\mathbb {E}$ is an exact sequence of locally algebraic group superschemes, then we have an exact sequence of $K$ -functors $\\mathbb {E}\\rightarrow \\mathrm {Lie}(\\mathbb {H})\\stackrel{\\mathrm {d}_e\\bf i}{\\rightarrow } \\mathrm {Lie}(\\mathbb {G})\\stackrel{\\mathrm {d}_e\\bf p}{\\rightarrow } \\mathrm {Lie}(\\mathbb {R}),$ that is, the sequence of $A$ -supermodules $0\\rightarrow \\mathrm {Lie}(\\mathbb {H})(A)\\stackrel{\\mathrm {d}_e{\\bf i}(A)}{\\rightarrow } \\mathrm {Lie}(\\mathbb {G})(A)\\stackrel{\\mathrm {d}_e{\\bf p}(A)}{\\rightarrow } \\mathrm {Lie}(\\mathbb {R})(A)$ is exact for every $A\\in \\mathsf {SAlg}_K$ .",
"The standard diagram chasing implies the first statement.",
"The second statement follows by Lemma REF .",
"From now on, all group superschemes are locally algebraic.",
"Following [4], [18], we denote the image of $x\\otimes r\\in \\mathfrak {g}\\otimes r$ in $\\mathbb {G}(R[\\epsilon _0, \\epsilon _1])$ by $e^{\\epsilon _{|x|+|r|} x\\otimes r}$ .",
"If ${\\bf f} : \\mathbb {G}\\rightarrow \\mathbb {H}$ is a morphism of group superschemes, then ${\\bf f}(R[\\epsilon _0, \\epsilon _1])(e^{\\epsilon _{|x|+|r|} x\\otimes r})=e^{\\epsilon _{|x|+|r|}\\mathrm {d}_e {\\bf f}(R)(x\\otimes r)}=e^{\\epsilon _{|x|+|r|}\\mathrm {d}_e(K)(x)\\otimes r}.$ In particular, if $V$ is a $\\mathbb {G}$ -supermodule with respect to a homomorphism ${\\bf f} :\\mathbb {G}\\rightarrow \\mathrm {GL}(V)$ , then we have ${\\bf f}(R[\\epsilon _0, \\epsilon _1])(e^{\\epsilon _{|x|+|r|} x\\otimes r})=\\mathrm {id}_V +\\epsilon _{|x|+|r|}\\mathrm {d}_e {\\bf f}(K)(x)\\otimes r.$ If we denote $({\\bf f}(R[\\epsilon _0, \\epsilon _1])(e^{\\epsilon _{|x|+|r|} x\\otimes r}))(v\\otimes 1) \\ \\mbox{and} \\ (\\mathrm {d}_e{\\bf f} (R)(x\\otimes r))(v\\otimes 1)$ by $e^{\\epsilon _{|x|+|r|} x\\otimes r}\\cdot (v\\otimes 1) \\ \\mbox{and} \\ (x\\otimes r)\\cdot (v\\otimes 1),$ respectively, where $v\\in V$ , then the above formula can be recorded as $e^{\\epsilon _{|x|+|r|} x\\otimes r}\\cdot (v\\otimes 1)=v\\otimes 1+(-1)^{|r||v|}\\epsilon _{|x|} (x\\cdot v)\\otimes r.$ Consider the category of pairs $(\\mathbb {G}, V)$ , where $\\mathbb {G}$ is a group superscheme, and $V$ is a $\\mathbb {G}$ -supermodule.",
"The morphisms in this category are couples $({\\bf f}, h)$ , where ${\\bf f} : \\mathbb {G}\\rightarrow \\mathbb {H}$ is a group superscheme morphism and $h : V\\rightarrow W$ is a linear map of superspaces, such that for every $R\\in \\mathsf {SAlg}_K$ the diagram $\\begin{array}{ccc}\\mathbb {G}(R)\\times \\mathbb {V}_a(R) & \\rightarrow & \\mathbb {V}_a(R) \\\\\\downarrow & & \\downarrow \\\\\\mathbb {H}\\times \\mathbb {W}_a(R) & \\rightarrow & \\mathbb {W}_a(R)\\end{array}$ is commutative.",
"Here the horizontal maps correspond to the actions of $\\mathbb {G}(R)$ and $\\mathbb {H}(R)$ on $\\mathbb {V}_a(R)$ and $\\mathbb {W}_a(R)$ , respectively, the right vertical map is ${\\bf h}_a(R)=h\\otimes \\mathrm {id}_R$ , and the left vertical map is ${\\bf f}(R)\\times {\\bf h}_a(R)$ .",
"Lemma 7.3 If $({\\bf f}, h)$ is a morphism of pairs $(\\mathbb {G}, V)\\rightarrow (\\mathbb {H}, W)$ , then for every $x\\in \\mathfrak {g}, r\\in R$ , and $v\\in V$ there is ${\\bf h}_a(R)((x\\otimes r)\\cdot (v\\otimes 1))=(-1)^{|r||v|}(\\mathrm {d}_e{\\bf f}(K)(x)\\cdot h(v))\\otimes r.$ We have ${\\bf h}_a(R)(e^{\\epsilon _{|x|+|r|}x\\otimes r}\\cdot v)=h(v)\\otimes 1+\\epsilon _{|x|+|r|}{\\bf h}_a(R)((x\\otimes r)\\cdot (v\\otimes 1))=$ $e^{\\epsilon _{|x|+|r|}\\mathrm {d}_e{\\bf f}(K)(x)\\otimes r}\\cdot (h(v)\\otimes 1)=h(v)\\otimes 1+(-1)^{|r||v|}\\epsilon _{|x|+|r|}(\\mathrm {d}_e{\\bf f}(K)(x)\\cdot h(v))\\otimes r,$ proving the formula.",
"Let $\\mathrm {ad}$ denote $\\mathrm {d}_e\\mathrm {Ad} : \\mathrm {Lie}(\\mathbb {G})\\rightarrow \\mathrm {Lie}(\\mathrm {GL}(\\mathfrak {g}))=\\mathfrak {gl}(\\mathfrak {g})_a$ .",
"For every $x, y\\in \\mathfrak {g}=\\mathrm {Lie}(\\mathbb {G})(K)$ and $r, r^{\\prime }\\in R$ we set $[x\\otimes r, y\\otimes r^{\\prime }]=(\\mathrm {ad}(R)(x\\otimes r))(y\\otimes r^{\\prime })$ .",
"Example 7.4 The Lie superalgebra of $\\mathrm {GL}(V)$ is canonically isomorphic to $\\mathfrak {gl}(V)=\\mathrm {End}_K(V)$ , regarded as a superspace with $\\mathfrak {gl}(V)_i=\\lbrace \\phi \\mid \\phi (V_j)\\subseteq V_{i+j\\pmod {2}}\\rbrace $ for $i=0, 1$ .",
"Moreover, if $X, Y\\in \\mathfrak {gl}(V)$ and $r, r^{\\prime }\\in R$ , then $ [X\\otimes r, Y\\otimes r^{\\prime }]=(-1)^{|r||Y|}[X, Y]\\otimes rr^{\\prime }=(-1)^{|r||Y|}(XY-(-1)^{|X||Y|}YX)\\otimes rr^{\\prime }.$ Indeed, we have $(e^{\\epsilon _{|X|+|r|}X\\otimes r})\\cdot (Y\\otimes r^{\\prime })=(\\mathrm {id}_V\\otimes 1 +\\epsilon _{|X|+|r|}X\\otimes r)(Y\\otimes r^{\\prime })(\\mathrm {id}_V\\otimes 1 -\\epsilon _{|X|+|r|}X\\otimes r)=$ $Y\\otimes r^{\\prime }+\\epsilon _{|X|+|r|}((-1)^{|r||Y|}XY\\otimes rr^{\\prime }-(-1)^{(|X|+|r|)(|Y|+|r^{\\prime }|)+|r^{\\prime }||X|}YX\\otimes r^{\\prime }r)=$ $Y\\otimes r^{\\prime } +(-1)^{|r||Y|}\\epsilon _{|X|+|r|}(XY-(-1)^{|X||Y|}YX)\\otimes rr^{\\prime },$ proving the formula.",
"Lemma 7.5 Let ${\\bf f} : \\mathbb {G}\\rightarrow \\mathbb {H}$ be a morphism of locally algebraic group superschemes.",
"Then for every $x, y\\in \\mathfrak {g}$ and $r, r^{\\prime }\\in R$ there is $\\mathrm {d}_e{\\bf f}(R)([x\\otimes r, y\\otimes r^{\\prime }])=(-1)^{|r||y|}[\\mathrm {d}_e{\\bf f}(K)(x), \\mathrm {d}_e{\\bf f}(K)(y)]\\otimes rr^{\\prime }.$ In particular, $\\mathfrak {g}\\otimes R$ and $\\mathfrak {h}\\otimes R$ are Lie superalgebras for the operation $[ \\ , \\ ]$ and $\\mathrm {d}_e{\\bf f}$ is a morphism of Lie superalgebra functors.",
"We have $\\mathrm {d}_e{\\bf f}(R)\\mathrm {Ad}(R)(g)=\\mathrm {Ad}(R)({\\bf f}(R)(g))\\mathrm {d}_e{\\bf f}(R)$ for every superalgebra $R$ and $g\\in \\mathbb {G}(R)$ .",
"That is, $({\\bf f}, \\mathrm {d}_e{\\bf f}(K))$ is a morphism of pairs $(\\mathbb {G}, \\mathfrak {g})\\rightarrow (\\mathbb {H}, \\mathfrak {h})$ , where $\\mathfrak {g}$ and $\\mathfrak {h}$ are regarded as $\\mathbb {G}$ -supermodule and $\\mathbb {H}$ -supermodule with respect to the adjoint actions.",
"Lemma REF implies the first statement.",
"To prove the second statement, one needs to show that the operation $[ \\ , \\ ]$ on $\\mathfrak {g}$ satisfies the identities $(B2)$ , $(B3)$ , and $(B4)$ from [11] (recall that $char K\\ne 2$ ).",
"Applying the first statement to $\\mathrm {Ad} : \\mathbb {G}\\rightarrow \\mathrm {GL}(\\mathfrak {g})$ , one obtains $[[x, y], z]=(XY-(-1)^{|x||y|}YX)(z),$ where $X=\\mathrm {ad}(K)(x), \\ Y=\\mathrm {ad}(K)(y)$ .",
"Therefore, $(B4)$ (or the super Jacobi identity) $[[x, y], z]=[x, [y, z]]-(-1)^{|x||y|} [y, [x, z]]$ follows.",
"Next, let $R$ denote $K[\\epsilon ^{\\prime }_0, \\epsilon ^{\\prime }_1]$ .",
"Then the group commutator $[e^{\\epsilon _{|x|}x}, e^{\\epsilon ^{\\prime }_{|y|}y}]=e^{\\epsilon _{|x|}x}e^{\\epsilon ^{\\prime }_{|y|}y}e^{-\\epsilon _{|x|}x}e^{-\\epsilon ^{\\prime }_{|y|}y},$ calculated in $\\mathbb {G}(R[\\epsilon _0, \\epsilon _1])$ , is equal to $e^{\\mathrm {Ad}(R[\\epsilon _0, \\epsilon _1])(e^{\\epsilon _{|x|}x})((-1)^{|y|}y\\otimes \\epsilon ^{\\prime }_{|y|})}e^{-\\epsilon ^{\\prime }_{|y|}y}=e^{\\epsilon ^{\\prime }_{|y|}y+(-1)^{|x||y|}\\epsilon _{|x|}\\epsilon ^{\\prime }_{|y|}(\\mathrm {ad}(K))(x))(y)}e^{-\\epsilon ^{\\prime }_{|y|}y}=$ $e^{(-1)^{|x||y|}\\epsilon _{|x|}\\epsilon ^{\\prime }_{|y|}[x, y]}.$ The group identity $[g, h]=[h, g]^{-1}$ implies $\\epsilon _{|x|}\\epsilon ^{\\prime }_{|y|}[x, y]=-\\epsilon ^{\\prime }_{|y|}\\epsilon _{|x|}[y, x]$ , hence $(B3)$ (or the super skew-symmetry) $[x, y]=-(-1)^{|x||y|}[y, x]$ follows.",
"The identity $(B2)$ states $[[x, x], x]=0$ for every $x\\in \\mathfrak {g}_1$ .",
"If $char K\\ne 3$ , then $(B2)$ follows from $(B4)$ .",
"If $char(K)=3$ , then $(B4)$ does not imply $(B2)$ .",
"However, in Lemma REF , we prove the identity $(B2)$ in general.",
"This finishes the proof.",
"Lemma 7.6 The adjoint action of $\\mathbb {G}$ on $\\mathrm {Lie}(\\mathbb {G})$ commutes with the super-bracket functor.",
"For every $g\\in \\mathbb {G}(R)$ we have $\\mathrm {Ad}(R)(g)e^{(-1)^{|x||y|}\\epsilon _{|x|}\\epsilon ^{\\prime }_{|y|}[x, y]}=e^{(-1)^{|x||y|}\\epsilon _{|x|}\\epsilon ^{\\prime }_{|y|}\\mathrm {Ad}(R)(g)[x, y]}=$ $\\mathrm {Ad}(R)(g)[e^{\\epsilon _{|x|}x}, e^{\\epsilon ^{\\prime }_{|y|}y}]=[\\mathrm {Ad}(R)(g)e^{\\epsilon _{|x|}x}, \\mathrm {Ad}(R)(g)e^{\\epsilon ^{\\prime }_{|y|}y}]=$ $[e^{\\epsilon _{|x|}\\mathrm {Ad}(R)(g)x} , e^{\\epsilon ^{\\prime }_{|y|}\\mathrm {Ad}(R)(g)y}]=e^{(-1)^{|x||y|}\\epsilon _{|x|}\\epsilon ^{\\prime }_{|y|}[\\mathrm {Ad}(R)(g)x, \\mathrm {Ad}(R)(g)y]},$ proving our statement."
],
[
"The formal neighborhood of the identity",
"Let $G$ be a geometric group superscheme.",
"Let $\\epsilon $ denote the identity morphism $e\\rightarrow G$ .",
"We have a commutative diagram $\\begin{array}{ccccc}& & G & & \\\\& & \\uparrow & & \\\\& & G\\times G & & \\\\& \\swarrow p_1 & \\uparrow & p_2 \\searrow & \\\\G & \\stackrel{\\epsilon }{\\leftarrow } & e & \\stackrel{\\epsilon }{\\rightarrow } & G\\end{array},$ where the closed immersion $e\\rightarrow G\\times G$ is induced by the universality of the direct product.",
"For every affine open neighborhood $U$ of $e$ , the immersion $e\\rightarrow G\\times G$ factors through $U\\times U\\subseteq G\\times G$ .",
"If we denote the image of the point $e$ in $(G\\times G)^e$ by $e\\times e$ , then $\\mathcal {O}_{e\\times e}$ is naturally isomorphic to $(\\mathcal {O}_e\\otimes \\mathcal {O}_e)_{\\mathfrak {n}}$ , where $\\mathfrak {n}={\\mathcal {O}_e\\otimes \\mathfrak {m}_e+\\mathfrak {m}_e\\otimes \\mathcal {O}_e}$ .",
"Moreover, $m^*_{e\\times e}$ sends $\\mathcal {O}_e$ to $(\\mathcal {O}_e\\otimes \\mathcal {O}_e)_{\\mathfrak {n}}$ , and $\\mathfrak {m}_e$ to $\\mathfrak {n}_{\\mathfrak {n}}$ .",
"For any non-negative integer $n$ , let $S_n\\simeq \\mathrm {SSpec}(\\mathcal {O}_e/\\mathfrak {m}_e^{n+1})$ denote the $n$ th neighborhood of $\\epsilon $ .",
"Lemma 8.1 For any non-negative integers $k$ and $t$ , there are commutative diagrams $\\begin{array}{ccc}G\\times G & \\stackrel{m}{\\rightarrow } & G \\\\\\uparrow & & \\uparrow \\\\S_k\\times S_t & \\rightarrow & S_{k+t}\\end{array}$ and $\\begin{array}{ccc}G & \\stackrel{\\iota }{\\rightarrow } & G \\\\\\uparrow & & \\uparrow \\\\S_k & \\rightarrow & S_k\\end{array}.$ The local superalgebra $\\mathcal {O}_e/\\mathfrak {m}_e^{k+1}\\otimes \\mathcal {O}_e/\\mathfrak {m}_e^{t+1}$ is canonically isomorphic to $(\\mathcal {O}_e\\otimes \\mathcal {O}_e)_{\\mathfrak {n}}/({\\mathcal {O}_e\\otimes \\mathfrak {m}^{k+1}_e+\\mathfrak {m}^{t+1}_e\\otimes \\mathcal {O}_e})_{\\mathfrak {n}}$ .",
"Moreover, $m^*_{e\\times e}$ induces a local superalgebra morphism $\\mathcal {O}_e/\\mathfrak {m}_e^{k+t+1}\\rightarrow (\\mathcal {O}_e\\otimes \\mathcal {O}_e)_{\\mathfrak {n}}/(\\mathfrak {m}^{k+1}_e\\otimes \\mathcal {O}_e+\\mathcal {O}_e\\otimes \\mathfrak {m}^{t+1})_{\\mathfrak {n}}$ that makes the diagram $\\begin{array}{ccc}\\mathcal {O}_{e\\times e} & \\stackrel{m^*_{e\\times e}}{\\leftarrow } & \\mathcal {O}_e \\\\\\downarrow & & \\downarrow \\\\\\mathcal {O}_e/\\mathfrak {m}_e^{k+1}\\otimes \\mathcal {O}_e/\\mathfrak {m}_e^{t+1} & \\leftarrow & \\mathcal {O}_e/\\mathfrak {m}_e^{k+t+1}\\end{array}$ commutative.",
"The proof of the second statement is analogous.",
"Translating to the category $\\mathcal {SFG}$ , one sees that the group superscheme $\\mathbb {G}$ contains an ascending chain of closed super-subschemes $\\mathbb {S}_0\\subseteq \\mathbb {S}_1\\subseteq \\ldots $ , such that each $\\mathbb {S}_n$ is isomorphic to $\\mathrm {SSp}(\\mathcal {O}_e/\\mathfrak {m}_e^{n+1})$ .",
"Moreover, the terms satisfy $\\mathbb {S}_n^{-1}\\subseteq \\mathbb {S}_n$ and $\\mathbb {S}_n\\mathbb {S}_m\\subseteq \\mathbb {S}_{m+n}$ for any positive integers $m$ and $n$ .",
"Therefore, $\\mathbb {S}=\\cup _{n\\ge 0}\\mathbb {S}_n$ is a group subfunctor of $\\mathbb {G}$ such that for every $R\\in \\mathsf {SAlg}_K$ , the group $\\mathbb {S}(R)$ consists of all superalgebra morphisms $\\mathcal {O}_e\\rightarrow R$ vanishing on some power of $\\mathfrak {m}_e$ .",
"We call $\\mathbb {S}$ the formal neighborhood of the identity in $\\mathbb {G}$ .",
"The group functor $\\mathbb {S}$ is \"quasi-affine\" in the following sense.",
"The complete local superalgebra $\\widehat{\\mathcal {O}_e}=\\varprojlim _n \\mathcal {O}_e/\\mathfrak {m}_e^{n+1}$ is a complete Hopf superalgebra for the comultiplication $\\Delta : \\widehat{\\mathcal {O}_e}\\rightarrow \\widehat{\\mathcal {O}_e}\\widehat{\\otimes } \\widehat{\\mathcal {O}_e}$ and the antipode $S : \\widehat{\\mathcal {O}_e}\\rightarrow \\widehat{\\mathcal {O}_e}$ , induced by $m^*_{e\\times e}$ and $\\iota ^*_e$ , respectively (see [7], Definition 3.3).",
"Moreover, there is $\\Delta (\\widehat{\\mathfrak {m}_e})\\subseteq \\widehat{\\mathcal {O}_e}\\widehat{\\otimes } \\widehat{\\mathfrak {m}_e}+\\widehat{\\mathfrak {m}_e}\\widehat{\\otimes } \\widehat{\\mathcal {O}_e} \\ \\mbox{and} \\ S(\\widehat{\\mathfrak {m}_e})\\subseteq \\widehat{\\mathfrak {m}_e},$ that is, $\\widehat{\\mathfrak {m}_e}$ is a (closed) Hopf superideal of $\\widehat{\\mathcal {O}_e}$ .",
"For every $R\\in \\mathsf {SAlg}_K$ , the group $\\mathbb {S}(R)$ consists of all continuous superalgebra morphisms $\\widehat{\\mathcal {O}_e}\\rightarrow R$ , where $R$ is regarded as a discrete superalgebra.",
"The group operations of $\\mathbb {S}(R)$ are defined by $(gh)(a)=(g\\otimes h)(\\Delta (a))$ and $g^{-1}(a)=g(S(a))$ for $g, h\\in \\mathbb {S}(R)$ and $a\\in \\widehat{\\mathcal {O}_e}$ .",
"We use the \"Sweedler notation\" and write $\\Delta (a)= a_{(0)}\\otimes a_{(1)}$ , by omitting the summation symbol.",
"The sum on the left is convergent in the $\\mathfrak {n}$ -adic topology.",
"Then $(gh)(a)=g(a_{(0)}) h(a_{(1)})$ , where the sum on the right contains only finitely many non-zero summands.",
"Assume again that $\\mathbb {G}$ is locally algebraic.",
"For a non-negative integer $n$ , let $\\mathrm {hyp}_n(\\mathbb {G})$ denote the superspace $(\\widehat{\\mathcal {O}_e}/\\widehat{\\mathfrak {m}_e}^{n+1})^*$ .",
"Set $\\mathrm {hyp}(\\mathbb {G})=\\cup _{n\\ge 0}\\mathrm {hyp}_n(\\mathbb {G})\\subseteq (\\widehat{\\mathcal {O}_e})^*$ .",
"The superspace $\\mathrm {hyp}(\\mathbb {G})$ has a natural Hopf superalgebra structure with respect to the product $(\\phi \\psi )(a)= (-1)^{|\\psi ||a_{(0)}|}\\phi (a_{(0)})\\psi (a_{(1)}),$ the coproduct $\\Delta ^*(\\phi )=\\phi _{(0)}\\otimes \\phi _{(1)}$ , uniquely defined by the identity $\\phi (ab)= (-1)^{|\\phi _{(1)}||a|}\\phi _{(0)}(a)\\phi _{(1)}(b) \\text{ for } \\phi , \\psi \\in \\mathrm {hyp}(\\mathbb {G})\\text{ and } a, b\\in \\widehat{\\mathcal {O}_e},$ and the antipode $S^*$ , such that $(S^*(\\phi ))(a)=\\phi (S(a))$ .",
"Furthermore, for every $R\\in \\mathsf {SAlg}_K$ , $\\widehat{\\mathcal {O}_e}\\otimes R=\\varprojlim _n(\\mathcal {O}_e\\otimes R)/(\\mathfrak {m}_e\\otimes R)^{n+1}$ is a complete Hopf $R$ -superalgebra with respect to the comultiplication $\\Delta \\otimes \\mathrm {id}_R$ and the antipode $S\\otimes \\mathrm {id}_R$ , and $\\mathrm {hyp}(\\mathbb {G})\\otimes R$ has the unique structure of Hopf $R$ -superalgebra for every $R\\in \\mathsf {SAlg}_K$ such that the pairing $(\\mathrm {hyp}(\\mathbb {G}))\\otimes R)\\times (\\widehat{\\mathcal {O}_e}\\otimes R)\\rightarrow R$ given by $<\\phi \\otimes r, a\\otimes r^{\\prime }>=(-1)^{|r||a|}\\phi (a)rr^{\\prime }$ , is a Hopf pairing in the sense of [11].",
"The following lemma is folklore.",
"Lemma 8.2 For every non-negative integers $k$ and $t$ , there is: $\\mathrm {hyp}_k(\\mathbb {G})\\mathrm {hyp}_t(\\mathbb {G})\\subseteq \\mathrm {hyp}_{k+t}(\\mathbb {G})$ , $\\Delta ^*(\\mathrm {hyp}_k(\\mathbb {G}))\\subseteq \\oplus _{0\\le s\\le k}\\mathrm {hyp}_s(\\mathbb {G})\\otimes \\mathrm {hyp}_{k-s}(\\mathbb {G})$ , $S^*(\\mathrm {hyp}_k(\\mathbb {G}))\\subseteq \\mathrm {hyp}_k(\\mathbb {G})$ .",
"For a Hopf superalgebra $H$ , let $\\mathsf {Gpl}(H)$ denote the subgroup of $H^{\\times }$ consisting of all group-like elements of $H$ .",
"Lemma 8.3 The group functor $\\mathbb {S}$ is canonically identified with the group functor $R\\mapsto \\mathsf {Gpl}(\\mathrm {hyp}(\\mathbb {G})\\otimes R)$ .",
"Additionally, we have $\\mathfrak {s}\\simeq \\mathrm {hyp}_1(\\mathbb {G})^+=\\lbrace \\phi \\in \\mathrm {hyp}_1(\\mathbb {G})\\mid \\phi (1)=0\\rbrace $ and the Lie super-bracket on $\\mathfrak {s}$ is defined by $[\\phi , \\psi ]=\\phi \\psi -(-1)^{|\\phi ||\\psi |}\\psi \\phi $ .",
"In particular, this operation satisfies the identity $(B2)$ .",
"We identify the superspace $\\mathrm {hyp}(\\mathbb {G})\\otimes R$ with $\\varinjlim _n \\mathrm {Hom}_K(\\widehat{\\mathcal {O}_e}/\\mathfrak {m}_e^{n+1}, R)$ via the above pairing.",
"Then the first statement follows.",
"Furthermore, the previous dual super-numbers technique shows that the Lie superalgebra $\\mathfrak {s}$ of $\\mathbb {S}$ can be identified with the super-subspace $\\mathsf {P}$ of $\\mathrm {hyp}(\\mathbb {G})^+$ consisting of all primitive elements.",
"Also, $\\dim \\mathfrak {s}=\\dim \\mathfrak {g}$ (see Lemma REF below) and $\\mathrm {hyp}_1(\\mathbb {G})^+\\subseteq \\mathsf {P}$ .",
"Since $\\mathbb {S}$ is \"represented\" by the complete Hopf superalgebra $\\widehat{\\mathcal {O}_e}$ , to prove the last statement, one can mimic the calculation from Lemma 3.4, [18].",
"Finally, $[[\\phi , \\phi ], \\phi ]=[2\\phi ^2, \\phi ]=0$ for every $\\phi \\in \\mathfrak {s}_1$ .",
"Lemma 8.4 Let $\\mathbb {G}$ be a locally algebraic group superscheme.",
"Then the operation $[ \\ , \\ ]$ on $\\mathfrak {g}$ satisfies the identity $(B2)$ .",
"Since each immersion $\\mathbb {S}_n\\rightarrow \\mathbb {G}$ factors through any open affine neighborhood $\\mathbb {U}$ of $e=e_K\\in \\mathbb {G}(K)$ , we have the following commutative diagram $\\begin{array}{ccccccccc}1 & \\rightarrow & \\mathrm {Lie}(\\mathbb {G})(R) & \\rightarrow & \\mathbb {G}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {G}(R) & \\rightarrow & 1\\\\& & \\parallel & & \\uparrow & & \\uparrow & & \\\\& & R\\otimes (\\mathfrak {m}_e/\\mathfrak {m}^2_e)^* & \\rightarrow & \\mathbb {S}_n(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow &\\mathbb {S}_n(R) & & \\\\& & \\parallel & & \\uparrow & & \\uparrow & & \\\\& & R\\otimes (\\mathfrak {m}_e/\\mathfrak {m}^2_e)^* & \\rightarrow & \\mathbb {S}_k(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow &\\mathbb {S}_k(R) & &\\end{array},$ where $k\\le n$ .",
"Using Lemma REF , we obtain a commutative diagram of groups $\\begin{array}{ccccccccc}1 & \\rightarrow & \\mathrm {Lie}(\\mathbb {G})(R) & \\rightarrow & \\mathbb {G}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow & \\mathbb {G}(R) & \\rightarrow & 1\\\\& & \\parallel & & \\uparrow & & \\uparrow & & \\\\1 & \\rightarrow & R\\otimes (\\mathfrak {m}_e/\\mathfrak {m}^2_e)^* & \\rightarrow & \\mathbb {S}(R[\\epsilon _0, \\epsilon _1]) & \\rightarrow &\\mathbb {S}(R) & \\rightarrow & 1\\end{array}.$ Thus $\\mathfrak {g}=\\mathfrak {s}$ and using the same trick with group commutators as in Lemma REF , one can derive that the Lie super-bracket on $\\mathfrak {g}$ coincides with the Lie super-bracket on $\\mathfrak {s}$ .",
"Lemma REF concludes the proof.",
"Lemma 8.5 Each $\\mathbb {S}_n$ is invariant with respect to the conjugation action of $\\mathbb {G}$ on itself.",
"Therefore, $\\mathbb {S}$ is a normal group subfunctor of $\\mathbb {G}$ .",
"We use the equivalent language of geometric superschemes.",
"The conjugation action is defined by a superscheme morphism $ c : G\\times G\\rightarrow G$ .",
"Let $U\\simeq \\mathrm {SSpec}(A)$ be an affine neighborhood of $e$ .",
"For every affine open super-subscheme $V\\simeq \\mathrm {SSpec}(B)$ , there is a commutative diagram $\\begin{array}{ccc}V\\times U & \\stackrel{c|_{V\\times U}}{\\rightarrow } & G \\\\\\uparrow & & \\uparrow \\\\V\\times e & \\stackrel{c|_{V\\times e}}{\\rightarrow } & e\\end{array}.$ Then $c^{-1}|_{V\\times U}(U)=(V\\times U)\\cap c^{-1}(U)$ can be covered by open affine supersubschemes $\\mathrm {SSpec}((B\\otimes A)_{g})$ .",
"There is a dual commutative diagram $\\begin{array}{ccc}(B\\otimes A)_g & \\stackrel{c^*}{\\leftarrow } & A \\\\\\downarrow & & \\downarrow \\\\B_{\\overline{g}} & \\leftarrow & K\\end{array},$ where the vertical arrows are $\\mathrm {id}_B\\otimes \\epsilon _A$ and $\\epsilon _A$ , respectively, and $\\overline{g}=(\\mathrm {id}_B\\otimes \\epsilon _A)(g)$ .",
"Since localization is a faithful functor, we obtain $c^*(\\mathfrak {m}_A)\\subseteq (B\\otimes \\mathfrak {m}_A)_g$ .",
"Since $c^*(\\mathfrak {m}^{n+1}_A)\\subseteq (B\\otimes \\mathfrak {m}_A)^{n+1}_g=(B\\otimes \\mathfrak {m}^{n+1}_A)_g$ , $c^*$ induces a superalgebra morphism $(B\\otimes A/\\mathfrak {m}_A^{n+1})_g \\leftarrow A/\\mathfrak {m}^{n+1}_A$ .",
"Considering all $g$ and $V$ , one sees that $c$ sends $G\\times S_n$ to $S_n$ , which proves the lemma.",
"Lemma 8.6 Let $X$ be a geometric superscheme.",
"For every finite-dimensional superalgebra $A$ , we have $\\mathcal {O}(X\\times \\mathrm {SSpec}(A))\\simeq \\mathcal {O}(X)\\otimes A$ .",
"Let $Y$ denote $\\mathrm {SSpec}(A)$ .",
"Choose a covering of $X$ by affine open super-subschemes $U_i$ .",
"Then $V_i=U_i\\times Y$ is an open covering of $X\\times Y$ by affine super-subschemes.",
"If the elements $a_1, \\ldots , a_k$ form a basis of $A$ , then each element $f\\in \\mathcal {O}(V_i)\\simeq \\mathcal {O}(U_i)\\otimes A$ can be uniquely expressed as $\\sum _{1\\le j\\le k} f_j\\otimes a_j$ .",
"Moreover, if for some $f^{\\prime }\\in \\mathcal {O}(V_s)$ we have $f|_{V_i\\cap V_s}=f^{\\prime }|_{V_i\\cap V_s}$ , then $f_j|_{W}=f^{\\prime }_j|_W$ for any open affine super-subscheme $W$ of $V_i\\cap V_s$ and $1\\le j\\le k$ , that is, $f_j|_{V_i\\cap V_s}=f^{\\prime }_j|_{V_i\\cap V_s}$ for every $1\\le j\\le k$ .",
"Thus, our lemma follows.",
"Lemma 8.7 The conjugation action of $\\mathbb {G}$ on $\\mathbb {S}$ factors through the action of $\\mathbb {G}^{aff}$ on $\\mathbb {S}$ .",
"Let $c_n$ denote $c|_{G\\times S_n}$ .",
"By Lemma REF , $c_n^*$ maps $\\mathcal {O}(S_n)$ to $\\mathcal {O}(G)\\otimes \\mathcal {O}(S_n)$ , that is, in Sweedler's notation, $c_n^*(a)=a_{(0)}\\otimes a_{(1)}$ for $a\\in \\mathcal {O}(S_n)$ .",
"The superspace generated by the coefficients $a_{(0)}$ , where $a$ ranges over $\\mathcal {O}(S_n)$ , is called the coefficient superspace of $\\mathcal {O}(S_n)$ , and it is denoted by $\\mathrm {cf}(\\mathcal {O}(S_n))$ .",
"The fact that $c_n$ defines an action implies the commutativity of the diagram $\\begin{array}{cccccc}\\mathcal {O}(G)\\otimes \\mathcal {O}(S_n) & \\stackrel{m^*\\otimes \\mathrm {id}_{\\mathcal {O}(S_n)}}{\\rightarrow } & \\mathcal {O}(G\\times G)\\otimes \\mathcal {O}(S_n) & \\stackrel{\\beta ^*}{\\leftarrow } & \\mathcal {O}(G\\times (G\\times S_n)) \\\\\\uparrow & & & & \\uparrow \\\\\\mathcal {O}(S_n) & & \\stackrel{c_n^*}{\\longrightarrow } & & \\mathcal {O}(G)\\otimes \\mathcal {O}(S_n)\\end{array},$ where the vertical arrows are $c_n^*$ and $(\\mathrm {id}_G\\times c_n)^*$ , respectively, and $\\beta $ is the canonical isomorphism $(G\\times G)\\times S_n\\rightarrow G\\times (G\\times S_n)$ .",
"Arguing as in Proposition REF , one sees that $(\\mathrm {id}_G\\times c_n)^*$ factors through $\\mathrm {id}_{\\mathcal {O}(G)}\\otimes c_n^*$ .",
"Moreover, the composition of the morphism $\\mathcal {O}(G)^{\\otimes 2}\\otimes \\mathcal {O}(S_n)\\simeq \\mathcal {O}(G)\\otimes \\mathcal {O}(G\\times S_n)\\rightarrow \\mathcal {O}(G\\times (G\\times S_n))$ with $\\beta ^*$ can be identified with $p_1^*\\otimes p_2^*\\otimes \\mathrm {id}_{\\mathcal {O}(S_n)}$ .",
"Therefore, $\\mathrm {cf}(\\mathcal {O}(S_n))\\subseteq \\mathcal {O}(G^{aff})$ , and $\\mathcal {O}(S_n)=\\mathcal {O}_e/\\mathfrak {m}_e^{n+1}$ is a left $\\mathcal {O}(G^{aff})$ -coideal superalgebra (cf.",
"[18]).",
"Furthermore, for any $t\\ge n$ , there is a commutative diagram $\\begin{array}{ccc}\\mathcal {O}_e/\\mathfrak {m}_e^{t+1} & \\stackrel{c^*_t}{\\rightarrow } & \\mathcal {O}(G^{aff})\\otimes \\mathcal {O}_e/\\mathfrak {m}_e^{t+1} \\\\\\downarrow & & \\downarrow \\\\\\mathcal {O}_e/\\mathfrak {m}_e^{n+1} & \\stackrel{c^*_n}{\\rightarrow } & \\mathcal {O}(G^{aff})\\otimes \\mathcal {O}_e/\\mathfrak {m}_e^{n+1},\\end{array}$ hence $c^*=\\varprojlim c^*_n$ defines a structure of left $\\mathcal {O}(G^{aff})$ -coideal superalgebra on $\\widehat{\\mathcal {O}_e}$ .",
"In other words, the conjugation action of $\\mathbb {G}$ on $\\mathbb {S}$ is defined by $(g\\cdot h)(a)=(\\overline{g}\\cdot h)(a)=\\overline{g}(a_{(0)})h(a_{(1)}),$ where $g\\in \\mathbb {G}(R), h\\in \\mathbb {S}(R), a\\in \\widehat{\\mathcal {O}_e}, c^*(a)=a_{(0)}\\otimes a_{(1)}, R\\in \\mathsf {SAlg}_K$ , and $\\overline{g}$ is the image of $g$ under the group homomorphism $\\mathbb {G}(R)\\rightarrow \\mathbb {G}^{aff}(R)$ .",
"Since $\\widehat{\\mathcal {O}_e}$ is a left $\\mathcal {O}(G^{aff})$ -coideal superalgebra, $\\mathbb {G}(R)$ acts on $\\widehat{\\mathcal {O}_e}\\otimes R$ on the right by $R$ -superalgebra automorphisms $(a\\otimes r)\\cdot g=\\overline{g}(a_{(0)}) a_{(1)}\\otimes r \\text{ for } g\\in \\mathbb {G}(R) \\text{ and } r\\in R,$ and this action is functorial in $R$ .",
"We also have $<g\\cdot h, x>=<h, x\\cdot g> \\text{ for } x\\in \\widehat{\\mathcal {O}_e}\\otimes R, $ where $h\\in \\mathbb {S}(R)$ is regarded as a group-like element of $\\mathrm {hyp}(\\mathbb {G})\\otimes R$ .",
"Lemma 8.8 The group superscheme $\\mathbb {G}$ acts on $\\widehat{\\mathcal {O}_e}$ by Hopf superalgebra automorphisms.",
"Moreover, if we define its action on $\\mathrm {hyp}(\\mathbb {G})$ by the above formula $<g\\cdot \\phi , a>=<\\phi , a\\cdot g> \\text{ for } g\\in \\mathbb {G}(R), \\phi \\in \\mathrm {hyp}(\\mathbb {G})\\otimes R, \\text{ and }a\\in \\widehat{\\mathcal {O}_e}\\otimes R,$ then this action also preserves the Hopf superalgebra structure of $\\mathrm {hyp}(\\mathbb {G})\\otimes R$ and is functorial in $R$ .",
"Since $< \\ , \\ >$ is a Hopf duality, the second statement is obvious.",
"We have a commutative diagram $\\begin{array}{ccc}\\mathbb {G}^{aff}\\times \\mathbb {S} & \\stackrel{{\\bf c}}{\\rightarrow } & \\mathbb {S} \\\\& & \\uparrow \\\\\\uparrow & & \\mathbb {S}\\times \\mathbb {S} \\\\& & \\uparrow \\\\\\mathbb {G}^{aff}\\times \\mathbb {S}\\times \\mathbb {S} & \\rightarrow & (\\mathbb {G}^{aff}\\times \\mathbb {S})\\times (\\mathbb {G}^{aff}\\times \\mathbb {S}),\\end{array}$ where the lower horizontal arrow is defined as $(g, s, s^{\\prime })\\mapsto (g, s, g, s^{\\prime })$ , the left vertical arrow is $\\mathrm {id}_{\\mathbb {G}^{aff}}\\times {\\bf m}$ , and the right vertical arrows are ${\\bf c}\\times {\\bf c}$ and $\\bf m$ , read from the bottom to the top, respectively.",
"The commutativity of the corresponding diagram of superalgebras $\\begin{array}{ccc}\\mathcal {O}(G^{aff})\\otimes \\widehat{\\mathcal {O}_e} & \\stackrel{c^*}{\\leftarrow } & \\widehat{\\mathcal {O}_e} \\\\& & \\downarrow \\\\\\downarrow & & \\widehat{\\mathcal {O}_e}\\otimes \\widehat{\\mathcal {O}_e}\\\\& & \\downarrow \\\\\\mathcal {O}(G^{aff})\\otimes \\widehat{\\mathcal {O}_e}\\otimes \\widehat{\\mathcal {O}_e} & \\leftarrow & (\\mathcal {O}(G^{aff})\\otimes \\widehat{\\mathcal {O}_e})\\otimes (\\mathcal {O}(G^{aff})\\otimes \\widehat{\\mathcal {O}_e}),\\end{array}$ is equivalent to the identity $(\\star ) \\ a_{(0)}\\otimes (a^{\\prime }_{(1)})^{\\prime }_{(0)}\\otimes (a^{\\prime }_{(1)})^{\\prime }_{(1)}= (-1)^{|(a^{\\prime }_{(1)})_{(0)}||(a^{\\prime }_{(0)})_{(1)}|}(a^{\\prime }_{(0)})_{(0)}(a^{\\prime }_{(1)})_{(0)}\\otimes (a^{\\prime }_{(0)})_{(1)}\\otimes (a^{\\prime }_{(1)})_{(1)},$ for any $a\\in \\widehat{\\mathcal {O}_e}$ , where $\\Delta (a)=a^{\\prime }_{(0)}\\otimes a^{\\prime }_{(1)}, \\ c^*(a)=a_{(0)}\\otimes a_{(1)}, \\ \\Delta (a_{(1)})=(a_{(1)})^{\\prime }_{(0)}\\otimes (a_{(1)})^{\\prime }_{(1)}, $ $c^*(a^{\\prime }_{(0)})=(a^{\\prime }_{(0)})_{(0)}\\otimes (a^{\\prime }_{(0)})_{(1)}, \\ c^*(a^{\\prime }_{(1)})= (a^{\\prime }_{(1)})_{(0)}\\otimes (a^{\\prime }_{(1)})_{(1)} .$ Applying $g\\otimes \\mathrm {id}_{\\widehat{\\mathcal {O}_e}}^{\\otimes 2}$ to both parts of $(\\star )$ , we obtain $\\Delta (a\\cdot g)= a^{\\prime }_{(0)}\\cdot g\\otimes a_{(1)}^{\\prime }\\cdot g$ .",
"Corollary 8.9 The above action of $\\mathbb {G}$ on $\\mathfrak {s}\\simeq \\mathrm {hyp}(\\mathbb {G})_1^+$ coincides with $\\mathrm {Ad}$ .",
"Lemma 8.10 If $\\mathbb {U}$ is an open super-subscheme of $\\mathbb {G}$ , then $\\mathbb {S}\\mathbb {U}\\subseteq \\mathbb {U}$ , and $\\mathbb {U}\\mathbb {S}\\subseteq \\mathbb {U}$ .",
"All one needs to show is that for any $n\\ge 0$ , the morphism $U\\times S_n\\stackrel{m}{\\rightarrow } G$ factors through $U$ .",
"That is, $m^e$ maps $(U\\times S_n)^e$ to $U^e$ .",
"Without losing generality, one can assume that $U$ is affine, say $U\\simeq \\mathrm {SSpec}(A)$ .",
"Then every prime super-ideal of $A\\otimes \\mathcal {O}_e/\\mathfrak {m}_e^{n+1}$ has a form $\\mathfrak {p}\\otimes \\mathcal {O}_e/\\mathfrak {m}_e^{n+1}+A\\otimes \\mathfrak {m}_e/\\mathfrak {m}_e^{n+1}$ , where $\\mathfrak {p}\\in (\\mathrm {SSpec}(A))^e$ .",
"Thus $(U\\times S_n)^e=(U\\times S_0)^e=(U\\times e)^e$ and our statement follows by one of the group axioms."
],
[
"Filtered superalgebras and their graded companions",
"A (downward) filtered superalgebra is a couple $(A, \\mathsf {I}$ ), where $A$ is a superalgebra, and $\\mathsf {I}$ is a super-ideal filtration $A=I_0\\supseteq I_1\\supseteq I_2\\supseteq \\ldots $ of $A$ such that $I_k I_l\\subseteq I_{k+l}$ for every $k, l\\ge 0$ .",
"We associate with any filtered superalgebra $(A, \\mathsf {I})$ a graded superalgebra $\\mathsf {gr}_{\\mathsf {I}}(A)=\\oplus _{k\\ge 0} I_k/I_{k+1}$ .",
"If there is a super-ideal $I$ such that $I_k=I^k$ for each $k\\ge 0$ , then this filtration is called $I$ -adic, and the corresponding graded superalgebra is denoted by $\\mathsf {gr}_I(A)$ .",
"If $(B, \\mathsf {J})$ is another filtered superalgebra, then the tensor product $A\\otimes B$ is a filtered superalgebra with respect to the filtration $T_k=I_0\\otimes J_k+I_1\\otimes J_{k-1}+\\ldots +I_k\\otimes J_0 \\text{ for } k\\ge 0.$ We call the filtration $\\mathsf {T}$ a tensor product of filtrations $\\mathsf {I}$ and $\\mathsf {J}$ .",
"A morphism of filtered superalgebras from $(A, \\mathsf {I})$ to $(B, \\mathsf {J})$ , is a superalgebra morphism $\\phi : A\\rightarrow B$ such that $\\phi (I_k)\\subseteq J_k$ for every $k\\ge 0$ .",
"Any such morphism induces a superalgebra morphism $\\mathsf {gr}_{\\mathsf {I}}(A)\\rightarrow \\mathsf {gr}_{\\mathsf {J}}(B)$ .",
"If $(A, \\mathsf {I})$ is a filtered superalgebra, then for every $n\\ge 0$ the superalgebra $A/I_{n+1}$ has a finite filtration $A/I_{n+1}=I_0/I_{n+1}\\supseteq I_1/I_{n+1}\\supseteq \\ldots \\supseteq I_n/I_{n+1}\\supseteq 0,$ which is denoted by $\\mathsf {I}_{\\le n}$ .",
"Proposition 9.1 There is a natural isomorphism $\\mathsf {gr}_{\\mathsf {I}}(A)\\otimes \\mathsf {gr}_{\\mathsf {J}}(B)\\simeq \\mathsf {gr}_{\\mathsf {T}}(A\\otimes B),$ induced by the canonical embeddings $A\\rightarrow A\\otimes B$ and $B\\rightarrow A\\otimes B$ , which is functorial in both $A$ and $B$ .",
"Both $A\\rightarrow A\\otimes B$ and $B\\rightarrow A\\otimes B$ are morphisms of filtered superalgebras.",
"Therefore, they induce the canonical morphisms of graded superalgebras $\\mathsf {gr}_{\\mathsf {I}}(A)\\rightarrow \\mathsf {gr}_{\\mathsf {T}}(A\\otimes B) \\text{ and } \\mathsf {gr}_{\\mathsf {J}}(B)\\rightarrow \\mathsf {gr}_{\\mathsf {T}}(A\\otimes B)$ given by $a+I_{k+1}\\mapsto a\\otimes 1+T_{k+1} \\text{ and } b+J_{l+1}\\mapsto 1\\otimes b+T_{l+1}.$ Thus, there is a superalgebra morphism $\\mathsf {gr}_{\\mathsf {I}}(A)\\otimes \\mathsf {gr}_{\\mathsf {J}}(B)\\rightarrow \\mathsf {gr}_{\\mathsf {T}}(A\\otimes B)$ which takes $(a+I_{k+1})\\otimes (b+J_{l+1})$ to $a\\otimes b +T_{k+l+1}$ for every $a\\in I_k, b\\in J_l$ .",
"Moreover, if we define a graded superalgebra structure on $\\mathsf {gr}_{\\mathsf {I}}(A)\\otimes \\mathsf {gr}_{\\mathsf {J}}(B)$ as $(\\mathsf {gr}_{\\mathsf {I}}(A)\\otimes \\mathsf {gr}_{\\mathsf {J}}(B))(n)=\\oplus _{0\\le k\\le n} I_k/I_{k+1}\\otimes J_{n-k}/J_{n-k+1} \\text{ for } n\\ge 0,$ then the above map is graded superalgebras' morphism.",
"For every $k\\ge 0$ , choose elements $a_{i, k}\\in I_k$ and $b_{j, k}\\in J_k$ , where $i$ and $j$ run over index sets $S_k$ and $L_k$ , respectively, such that they form bases of $I_k$ modulo $I_{k+1}$ , and $J_k$ modulo $J_{k+1}$ , respectively.",
"Then the homogeneous component $(\\mathsf {gr}_{\\mathsf {I}}(A)\\otimes \\mathsf {gr}_{\\mathsf {J}}(B))_n$ has a basis consisting of all elements $(a_{i, k}+I_{k+1})\\otimes (b_{j, n-k}+J_{n-k+1}) \\text{ for } 0\\le k\\le n, i\\in S_k \\text{ and } j\\in L_{n-k}.$ Moreover, the elements $a_{i, k}\\otimes b_{j, n-k} \\text{ for } 0\\le k\\le n, i\\in S_k \\text{ and }j\\in L_{n-k}$ generate $T_n$ modulo $T_{n+1}$ .",
"Since our morphism induces a one-to-one correspondence between the basis of $(\\mathsf {gr}_{\\mathsf {I}}(A)\\otimes \\mathsf {gr}_{\\mathsf {J}}(B))_n$ and the generating set of $T_n/T_{n+1}$ , all we need is to show that the elements $a_{i, k}\\otimes b_{j, n-k}$ are linearly independent over $T_{n+1}$ .",
"Consider the filtered superalgebras $(A/I_{n+1}, \\mathsf {I}_{\\le n})$ and $(B/J_{n+1}, \\mathsf {J}_{\\le n})$ .",
"The elements $\\overline{a_{i, k}}=a_{i, k}+I_{n+1}$ and $\\overline{b_{j, l}}=b_{j, l}+I_{n+1}$ , where $0\\le k, l\\le n, i\\in S_k,$ and $j\\in L_l$ , form bases of vector (super)spaces $A/I_{n+1}$ and $B/J_{n+1}$ , respectively.",
"The superalgebra $A/I_{n+1}\\otimes B/J_{n+1}\\simeq (A\\otimes B)/(I_{n+1}\\otimes B +A\\otimes J_{n+1})$ has a basis $\\overline{a_{i, k}}\\otimes \\overline{b_{j, l}} \\text{ for } 0\\le k, l\\le n, i\\in S_k \\text{ and } j\\in L_l.$ Let $\\mathsf {T}^{\\prime }$ denote the tensor product of filtrations $\\mathsf {I}_{\\le n}$ and $\\mathsf {J}_{\\le n}$ .",
"Then $T^{\\prime }_{2n+1}=0$ and for every $0\\le s\\le 2n$ the superspace $T^{\\prime }_s/T^{\\prime }_{s+1}$ is spanned by the part of the above basis consisting of the elements $\\overline{a_{i, k}}\\otimes \\overline{b_{j, s-k}}$ , where $k, s-k\\le n$ .",
"In particular, the latter elements are linearly independent over $T^{\\prime }_{s+1}$ .",
"If we apply this remark to $s=n$ , and observe that $T^{\\prime }_n/T^{\\prime }_{n+1}$ is naturally isomorphic to $T_n/T_{n+1}$ , the proof concludes."
],
[
"Graded companions of geometric superschemes",
"Recall that if $\\mathcal {F}$ is a presheaf of abelian groups on a topological space $X$ , then its sheafification is denoted by $\\mathcal {F}^+$ .",
"If the index set $I$ is infinite, then a direct sum of sheaves $\\mathcal {F}=\\oplus _{i\\in I}\\mathcal {F}_i$ is a presheaf but not necessary a sheaf (see [6], Exercise II.1.10).",
"The direct product $\\overline{\\mathcal {F}}=\\prod _{I\\in I}\\mathcal {F}_i$ is a sheaf on $X$ .",
"Let $\\widehat{\\mathcal {F}}$ denote a sub-presheaf of $\\overline{\\mathcal {F}}$ such that for every open subset $U\\subseteq X$ a section $f\\in \\overline{\\mathcal {F}}(U)$ belongs to $\\widehat{\\mathcal {F}}(U)$ if and only if there is an open covering $\\lbrace U_j\\rbrace _{j\\in J}$ of $U$ with $f|_{U_j}\\in \\mathcal {F}(U_j)$ for each $j\\in J$ .",
"The following lemma can be easily derived from Proposition-Definition II.1.2, [6].",
"Lemma 9.2 $\\widehat{\\mathcal {F}}$ is a sheaf isomorphic to $\\mathcal {F}^+$ .",
"Let $X$ be a geometric superscheme.",
"A quasi-coherent super-ideal sheaf $\\mathcal {J}$ on $X$ is called locally nilpotent if for every $x\\in X^e$ the stalk $\\mathcal {J}_x$ is a locally nilpotent super-ideal of $\\mathcal {O}_{X, x}$ .",
"Let $\\mathsf {gr}_{\\mathcal {J}}(\\mathcal {O}_X)$ denote the sheaf of superalgebras $(\\oplus _{n\\ge 0}\\mathcal {J}^n/\\mathcal {J}^{n+1})^+ .$ Proposition 9.3 The following statements hold: A geometric superspace $\\mathsf {gr}_{\\mathcal {J}}(X)=(X^e, \\mathsf {gr}_{\\mathcal {J}}(\\mathcal {O}_X))$ is a superscheme; $X\\rightarrow \\mathsf {gr}_{\\mathcal {I}_X}(X)$ is an endofunctor of the category $\\mathcal {SV}$ that takes immersions to immersions; A morphism $f : X\\rightarrow Y$ of superschemes of locally finite type is an isomorphism if and only if $\\mathsf {gr}(f) : \\mathsf {gr}_{\\mathcal {I}_X}(X)\\rightarrow \\mathsf {gr}_{\\mathcal {I}_Y}(Y)$ is.",
"Without loss of generality, one can assume that $X$ is affine, say $X=\\mathrm {SSpec}(A)$ .",
"Then $\\mathcal {J}=\\widetilde{J}$ , where $J$ is a super-ideal of $A$ such that $J_{\\mathfrak {p}}$ is locally nilpotent for every $\\mathfrak {p}\\in (\\mathrm {SSpec}(A))^e$ .",
"Therefore, $J$ is locally nilpotent.",
"Since $(\\mathrm {SSpec}(\\mathsf {gr}_J(A)))^e=(\\mathrm {SSpec}(A/J))^e=(\\mathrm {SSpec}(A))^e$ , Proposition 2.1 (3), [17] (see also [6], Proposition II.5.2 (c)) implies $\\mathsf {gr}_{\\mathcal {J}}(\\mathcal {O}_X)\\simeq (\\oplus _{n\\ge 0}\\widetilde{J^n/J^{n+1}})^+\\simeq \\mathcal {O}_{\\mathrm {SSpec}(\\mathsf {gr}_J(A))},$ hence $\\mathsf {gr}_{\\mathcal {J}}(X)\\simeq \\mathrm {SSpec}(\\mathsf {gr}_J(A))$ .",
"If $f : X\\rightarrow Y$ is a morphism in $\\mathcal {SV}$ , then $f^*(\\mathcal {I}_Y)\\subseteq f_*^e\\mathcal {I}_X$ , and $f^*$ induces the required morphism of sheaves $\\mathsf {gr}_{\\mathcal {I}_Y}(\\mathcal {O}_Y)\\rightarrow f_*^e\\mathsf {gr}_{\\mathcal {I}_X}(\\mathcal {O}_X)$ .",
"Since the functor $A\\rightarrow \\mathsf {gr}_{I_A}(A)$ commutes with localizations, for every $x\\in X^e$ there is $\\mathcal {O}_{\\mathsf {gr}_{\\mathcal {I}_X}(X), x}\\simeq \\mathsf {gr}_{I_{\\mathcal {O}_x}}(\\mathcal {O}_x),$ and this isomorphism is functorial in $X$ .",
"Thus, the second statement follows.",
"Similarly, if $\\mathsf {gr}(f)$ is an isomorphism, then $f^e$ is a homeomorphism of topological spaces and for every $x\\in X^e$ the local morphism $f^*_e$ induces the isomorphism $\\mathsf {gr}_{I_{\\mathcal {O}_{Y, y}}}(\\mathcal {O}_{Y, y})\\simeq \\mathsf {gr}_{I_{\\mathcal {O}_{X, x}}}(\\mathcal {O}_{X, x}),$ where $y=f^e(x)$ .",
"By Proposition 1.10, [16], the local superalgebras $\\mathcal {O}_{X, x}$ , and $\\mathcal {O}_{Y, y}$ are Hausdorff spaces with respect to their $I_{\\mathcal {O}_{X, x}}$ -adic, and $I_{\\mathcal {O}_{Y, y}}$ -adic topologies, respectively.",
"Hence $\\mathcal {O}_{Y, y}\\simeq \\mathcal {O}_{X, x}$ and the proposition is proven.",
"Proposition 9.4 For every geometric superschemes $X$ and $Y$ , there is an isomorphism $\\mathsf {gr}_{\\mathcal {I}_{X\\times Y}}(X\\times Y)\\simeq \\mathsf {gr}_{\\mathcal {I}_X}(X)\\times \\mathsf {gr}_{\\mathcal {I}_Y}(Y).$ Moreover, for every morphisms $X\\rightarrow Z$ and $Y\\rightarrow T$ in $\\mathcal {SV}$ , the diagram $\\begin{array}{ccc}\\mathsf {gr}_{\\mathcal {I}_{X\\times Y}}(X\\times Y) & \\simeq & \\mathsf {gr}_{\\mathcal {I}_X}(X)\\times \\mathsf {gr}_{\\mathcal {I}_Y}(Y)\\\\\\downarrow & & \\downarrow \\\\\\mathsf {gr}_{\\mathcal {I}_{Z\\times T}}(Z\\times T) & \\simeq & \\mathsf {gr}_{\\mathcal {I}_Z}(Z)\\times \\mathsf {gr}_{\\mathcal {I}_T}(T)\\end{array}$ is commutative.",
"Let $U\\simeq \\mathrm {SSpec}(A)$ and $V\\simeq \\mathrm {SSpec}(B)$ be open super-subschemes of $X$ and $Y$ , respectively.",
"Let $p_X$ and $p_Y$ denote the canonical projections $X\\times Y\\rightarrow X$ and $X\\times Y\\rightarrow Y$ respectively.",
"As we have already observed, $p_X^{-1}(U)\\cap p_Y^{-1}(V)\\simeq U\\times V\\simeq \\mathrm {SSpec}(A\\otimes B)$ is an open super-subscheme of $X\\times Y$ .",
"Proposition REF and Proposition REF imply that there is the unique open immersion $\\phi _{U V} : \\mathsf {gr}_{\\mathcal {I}_{U\\times V}}(U\\times V)\\rightarrow \\mathsf {gr}_{\\mathcal {I}_{X}}(X)\\times \\mathsf {gr}_{\\mathcal {I}_{Y}}(Y)$ such that $p_{\\mathsf {gr}_{\\mathcal {I}_{X}}(X)}\\phi _{U V}=\\mathsf {gr}(p_{U}) \\text{ and } \\ p_{\\mathsf {gr}_{\\mathcal {I}_{Y}}(Y)}\\phi _{U V}=\\mathsf {gr}(p_{V}).$ Besides, $\\phi _{U V}$ induces an isomorphism onto the open affine super-subscheme $\\mathsf {gr}_{\\mathcal {I}_{U}}(U)\\times \\mathsf {gr}_{\\mathcal {I}_{V}}(V)$ of $\\mathsf {gr}_{\\mathcal {I}_{X}}(X)\\times \\mathsf {gr}_{\\mathcal {I}_{Y}}(Y)$ .",
"Furthermore, for every open affine super-subschemes $U^{\\prime }\\subseteq U$ and $V^{\\prime }\\subseteq V$ , the composition of the natural open immersion $\\mathsf {gr}_{\\mathcal {I}_{U^{\\prime }\\times V^{\\prime }}}(U^{\\prime }\\times V^{\\prime })\\rightarrow \\mathsf {gr}_{\\mathcal {I}_{U\\times V}}(U\\times V)$ and $\\phi _{U, V}$ coincides with $\\phi _{U^{\\prime } V^{\\prime }}$ .",
"Using these remarks, one can construct a collection of open immersions $\\phi _{U V} : \\mathsf {gr}_{\\mathcal {I}_{U\\times V}}(U\\times V)\\rightarrow \\mathsf {gr}_{\\mathcal {I}_{X}}(X)\\times \\mathsf {gr}_{\\mathcal {I}_{Y}}(Y)$ compatible with each other, where $U$ and $V$ run over open affine coverings of $X$ and $Y$ , respectively.",
"Thus the first statement follows.",
"The same reduction to affine open coverings can be used to prove the second statement.",
"We leave the details for the reader.",
"From now on, the graded companion $\\mathsf {gr}_{\\mathcal {I}_X}(X)$ of a geometric superscheme $X$ is denoted just by $\\mathsf {gr}(X)$ .",
"Similarly, $\\mathsf {gr}_{I_A}(A)$ is denoted by $\\mathsf {gr}(A)$ .",
"Proposition 9.5 $G\\mapsto \\mathsf {gr}(G)$ is an endofunctor of the category $\\mathcal {SVG}$ .",
"Moreover, if $G$ is (locally) algebraic, then $\\mathsf {gr}(G)$ is.",
"Let $m, \\iota $ , and $\\epsilon $ denote the multiplication, the inverse, and the identity morphisms of $G$ , respectively.",
"Composition of $\\mathsf {gr}(m)$ with the above isomorphism $\\mathsf {gr}(G\\times G)\\simeq \\mathsf {gr}(G)\\times \\mathsf {gr}(G)$ defines the multiplication morphism $m^{\\prime } : \\mathsf {gr}(G)\\times \\mathsf {gr}(G)\\rightarrow \\mathsf {gr}(G)$ .",
"Moreover, if $f : G\\rightarrow H$ is a morphism of (geometric) group superschemes, then $\\mathsf {gr}(f)$ is a morphism of (geometric) groupoid superschemes.",
"Similarly, $\\mathsf {gr}(\\iota )$ and $\\mathsf {gr}(\\epsilon )$ are the inverse and the unit morphisms of $\\mathsf {gr}(G)$ , respectively.",
"Using Proposition REF , one obtains the following commutative diagram $\\begin{array}{ccccc}\\mathsf {gr}(G\\times G\\times G) & \\simeq & \\mathsf {gr}(G)\\times \\mathsf {gr}(G\\times G) & \\simeq & \\mathsf {gr}(G)\\times \\mathsf {gr}(G)\\times \\mathsf {gr}(G) \\\\\\downarrow & & \\downarrow & & \\downarrow \\\\\\mathsf {gr}(G\\times G) & \\simeq & \\mathsf {gr}(G)\\times \\mathsf {gr}(G) & = & \\mathsf {gr}(G)\\times \\mathsf {gr}(G) \\\\\\downarrow & & \\downarrow & & \\downarrow \\\\\\mathsf {gr}(G) & = & \\mathsf {gr}(G) & = & \\mathsf {gr}(G) ,\\end{array}$ where the top vertical arrows are $\\mathsf {gr}(\\mathrm {id}_G\\times m)$ , $\\mathrm {id}_{\\mathsf {gr}(G)}\\times \\mathsf {gr}(m)$ and $\\mathrm {id}_{\\mathsf {gr}(G)}\\times m^{\\prime }$ , respectively, and the bottom vertical arrows are $\\mathsf {gr}(m)$ and twice $m^{\\prime }$ , respectively.",
"Constructing the symmetric diagram for $G\\times G\\times G\\stackrel{m\\times \\mathrm {id}_G}{\\rightarrow } G\\times G\\stackrel{m}{\\rightarrow } G,$ one can derive that $m^{\\prime }$ satisfies the axiom of associativity.",
"We leave for the reader the routine checking of all remaining group axioms.",
"Lemma 9.6 Let $X$ be a geometric superscheme.",
"There are natural morphisms $i_X : X_{ev}\\rightarrow \\mathsf {gr}(X)$ and $q_X : \\mathsf {gr}(X)\\rightarrow X_{ev}$ , both functorial in $X$ , such that $q_X i_X=\\mathrm {id}_{X_{ev}}$ .",
"Moreover, $i_X$ is a closed immersion that maps $X_{ev}$ isomorphically onto $\\mathsf {gr}(X)_{ev}$ .",
"Since the underlying topological spaces of all the appearing superschemes are the same, all we need is to define morphisms of their superalgebra sheaves.",
"We have two morphisms of superalgebra presheaves $\\mathcal {O}_{X_{ev}}=\\mathcal {O}_{X}/\\mathcal {I}_X\\rightarrow \\oplus _{n\\ge 0}\\mathcal {I}_X^n/\\mathcal {I}_X^{n+1}$ and $\\oplus _{n\\ge 0}\\mathcal {I}_X^n/\\mathcal {I}_X^{n+1}\\rightarrow \\mathcal {O}_{X}/\\mathcal {I}_X=\\mathcal {O}_{X_{ev}},$ where the first morphism is the isomorphism onto the 0th component of $\\oplus _{n\\ge 0}\\mathcal {I}_X^n/\\mathcal {I}_X^{n+1}$ and the second morphism is the projection onto its 0th component.",
"These morphisms uniquely extend to morphisms of superalgebra sheaves $ q^*_X : \\mathcal {O}_{X_{ev}}\\rightarrow (\\oplus _{n\\ge 0}\\mathcal {I}_X^n/\\mathcal {I}_X^{n+1})^+=\\mathcal {O}_{\\mathsf {gr}(X)}$ and $i_X^* : \\mathcal {O}_{\\mathsf {gr}(X)}=(\\oplus _{n\\ge 0}\\mathcal {I}_X^n/\\mathcal {I}_X^{n+1})^+\\rightarrow \\mathcal {O}_{X_{ev}}.$ Since the composition of morphisms of presheaves is an identity map, we have $i_X^* q_X^*=\\mathrm {id}_{\\mathcal {O}_{X_{ev}}}$ .",
"The functoriality follows.",
"Finally, since the last statement is local, one can assume that $X$ is affine, say $X=\\mathrm {SSpec}(A)$ .",
"Then $i_X$ is induced by the projection $\\mathsf {gr}{A}\\mapsto \\overline{A}$ (and $q_X$ is induced by the embedding $\\overline{A}\\mapsto \\mathsf {gr}(A)$ ).",
"The lemma is proven.",
"Lemma 9.7 For every geometric superschemes $X$ and $Y$ , there are the following commutative diagrams $\\begin{array}{ccc}(X\\times Y)_{ev} & \\simeq & X_{ev}\\times Y_{ev} \\\\\\downarrow & & \\downarrow \\\\\\mathsf {gr}(X\\times Y) & \\simeq & \\mathsf {gr}(X)\\times \\mathsf {gr}(Y)\\end{array} \\ \\mbox{and} \\ \\begin{array}{ccc}\\mathsf {gr}(X\\times Y) & \\simeq & \\mathsf {gr}(X)\\times \\mathsf {gr}(Y) \\\\\\downarrow & & \\downarrow \\\\(X\\times Y)_{ev} & \\simeq & X_{ev}\\times Y_{ev}\\end{array},$ where the vertical arrows in the first diagram are $i_{X\\times Y}$ and $i_X\\times i_Y$ , while the vertical arrows in the second diagram are $q_{X\\times Y}$ and $q_X\\times q_Y$ .",
"Arguing as in Proposition REF , one can reduce the general case to $X=\\mathrm {SSpec}(A)$ and $ Y=\\mathrm {SSpec}(B)$ .",
"Then the commutativity of the above diagrams is equivalent to the commutativity of the diagrams $\\begin{array}{ccc}\\overline{A\\otimes B} & \\simeq & \\overline{A}\\otimes \\overline{B} \\\\\\uparrow & & \\uparrow \\\\\\mathsf {gr}(A\\otimes B) & \\simeq & \\mathsf {gr}(A)\\otimes \\mathsf {gr}(B)\\end{array} \\ \\mbox{and} \\ \\begin{array}{ccc}\\mathsf {gr}(A\\otimes B) & \\simeq & \\mathsf {gr}(A)\\otimes \\mathsf {gr}(B) \\\\\\uparrow & & \\uparrow \\\\\\overline{A\\otimes B} & \\simeq & \\overline{A}\\otimes \\overline{B}\\end{array},$ where the vertical arrows are the corresponding projections and embeddings.",
"Corollary 9.8 If $G$ is a group superscheme, then $i_G$ and $q_G$ are morphisms of group superschemes.",
"For example, let us consider $i_G$ .",
"We have two commutative diagrams $\\begin{array}{ccc}(G\\times G)_{ev} & \\simeq & G_{ev}\\times G_{ev} \\\\\\downarrow & & \\downarrow \\\\\\mathsf {gr}(G\\times G) & \\simeq & \\mathsf {gr}(G)\\times \\mathsf {gr}(G)\\end{array} \\ \\mbox{and} \\ \\begin{array}{ccc}(G\\times G)_{ev} & \\stackrel{m_{ev}}{\\rightarrow } & G_{ev} \\\\\\downarrow & & \\downarrow \\\\\\mathsf {gr}(G\\times G) & \\stackrel{\\mathsf {gr}(m)}{\\rightarrow }& \\mathsf {gr}(G)\\end{array},$ where $m : G\\times G\\rightarrow G$ is the multiplication morphism.",
"Since the multiplication morphisms of $G_{ev}$ and $\\mathsf {gr}(G)$ are factored through the isomorphisms (horizontal arrows) of the first diagram, our statement follows.",
"The case of $q_G$ is similar."
],
[
"An interpretation of the functor of points and some useful consequences",
"Due to the category equivalence $\\mathcal {SVG}\\simeq \\mathcal {SFG}$ , the results of the previous section can be reformulated as follows.",
"There is an endofunctor of the category $\\mathcal {SFG}$ that assigns to each group superscheme $\\mathbb {G}$ a group superscheme $\\mathsf {gr}(\\mathbb {G})$ such that there are natural morphisms (in $\\mathcal {SFG}$ ) ${\\bf i}_{\\mathbb {G}} : \\mathbb {G}_{ev}\\rightarrow \\mathsf {gr}(\\mathbb {G}), \\ {\\bf q}_{\\mathbb {G}} : \\mathsf {gr}(\\mathbb {G})\\rightarrow \\mathbb {G}_{ev},$ where ${\\bf i}_{\\mathbb {G}}$ is an isomorphism onto $\\mathsf {gr}(\\mathbb {G})_{ev}$ and ${\\bf q}_{\\mathbb {G}}{\\bf i}_{\\mathbb {G}}=\\mathrm {\\bf id}_{\\mathbb {G}_{ev}}$ .",
"Let $\\mathbb {G}_{odd}$ denote $\\ker {\\bf q}_{\\mathbb {G}}$ .",
"Lemma REF implies that $\\mathbb {G}_{odd}$ is a closed normal group super-subscheme of $\\mathsf {gr}(\\mathbb {G})$ .",
"Moreover, we have $\\mathsf {gr}(\\mathbb {G})=\\mathsf {gr}(\\mathbb {G})_{ev}\\ltimes \\mathbb {G}_{odd}\\simeq \\mathbb {G}_{ev}\\ltimes \\mathbb {G}_{odd}.$ If ${\\bf f} : \\mathbb {G}\\rightarrow \\mathbb {H}$ is a morphism of group superschemes, then Lemma REF implies $\\mathsf {gr}({\\bf f})(\\mathbb {G}_{odd})\\le \\mathbb {H}_{odd}$ .",
"Thus, $\\mathbb {G}\\rightarrow \\mathbb {G}_{odd}$ is an endofunctor of the category $\\mathcal {SFG}$ .",
"Proposition 10.1 Let $\\mathbb {G}$ be a locally algebraic group superscheme.",
"The group superscheme $\\mathbb {G}_{odd}$ is affine, and represented by a local graded Hopf superalgebra $\\Lambda (\\mathfrak {g}^*_1)$ , where the elements of $\\mathfrak {g}^*_1$ are assumed to be primitive.",
"Moreover, if ${\\bf f} : \\mathbb {G}\\rightarrow \\mathbb {H}$ is a morphism of locally algebraic group superschemes, then $\\mathbb {G}_{odd}\\rightarrow \\mathbb {H}_{odd}$ is induced by the (dual) linear map $(\\mathrm {d}_e{\\bf f})^* : \\mathfrak {h}_1^*\\rightarrow \\mathfrak {g}^*_1$ .",
"We have the exact sequence $e\\rightarrow G_{odd}\\rightarrow \\mathsf {gr}(G)\\stackrel{q_G}{\\rightarrow } G_{ev}\\rightarrow e,$ that is the geometric counterpart of the exact sequence $\\mathbb {E}\\rightarrow \\mathbb {G}_{odd}\\rightarrow \\mathsf {gr}(\\mathbb {G})\\stackrel{{\\bf q}_{\\mathbb {G}}}{\\rightarrow }\\mathbb {G}_{ev}\\rightarrow \\mathbb {E}.$ Since for every field extension $K\\subseteq F$ and every group superscheme $\\mathbb {H}$ there is $\\mathbb {H}(F)=\\mathbb {H}_{ev}(F)$ , the underlying topological space of $G_{odd}$ consists of the unit element $e$ only (see Lemma 5.5, [14]).",
"Therefore, $G_{odd}$ is affine, and represented by a local Hopf superalgebra $A$ with the nilpotent maximal superideal $A^+$ .",
"Since ${\\bf q}_{\\mathbb {G}}{\\bf i}_{\\mathbb {G}}={\\bf id}_{\\mathbb {G}_{ev}}$ , Lemma REF implies that the Lie superalgebra of $\\mathbb {G}_{odd}$ is purely-odd.",
"Hence, $A$ is generated by odd elements.",
"In particular, $I_A=A^+$ .",
"Furthermore, $\\mathsf {gr}(G)$ is naturally isomorphic to $G_{ev}\\times G_{odd}$ so that the projection onto $G_{ev}$ is identified with $q_{G}$ .",
"Thus $\\mathcal {O}_{\\mathsf {gr}(G), e}\\simeq \\mathcal {O}_{G_{ev}, e}\\otimes \\mathcal {O}_{G_{odd}, e}=\\mathcal {O}_{G_{ev}, e}\\otimes A$ and the maximal superideal $\\mathfrak {n}$ of $\\mathcal {O}_{\\mathsf {gr}(G), e}$ is identified with $\\overline{\\mathfrak {m}_{e}}\\otimes A +\\mathcal {O}_{G_{ev}, e}\\otimes A^+$ , where $\\overline{\\mathfrak {m}_e}=\\mathfrak {m}_e/I_{\\mathcal {O}_e}$ .",
"Since the superideal $A^+$ is nilpotent, the $\\mathfrak {n}$ -adic topology on $\\mathcal {O}_{\\mathsf {gr}(G), e}$ coincides with its $\\overline{\\mathfrak {m}_e}$ -adic topology.",
"Besides, the grading of $\\mathcal {O}_{\\mathsf {gr}(G), e}$ is given by $\\mathcal {O}_{\\mathsf {gr}(G), e}(k)\\simeq \\mathcal {O}_{G_{ev}, e}\\otimes (A^+)^k/(A^+)^{k+1} \\text{ for } k\\ge 0.$ In particular, there is an isomorphism of complete graded superalgebras $\\widehat{\\mathcal {O}_{\\mathsf {gr}(G), e}}\\simeq \\widehat{\\mathcal {O}_{G_{ev}, e}}\\otimes A$ , such that $\\widehat{\\mathcal {O}_{\\mathsf {gr}(G), e}}(k)\\simeq \\widehat{\\mathcal {O}_{G_{ev}, e}}\\otimes (A^+)^k/(A^+)^{k+1} \\text{ for } k\\ge 0.$ Moreover, the epimorphism of complete Hopf superalgebras $\\widehat{\\mathcal {O}_{\\mathsf {gr}(G), e}}\\rightarrow A$ , induced by the closed immersion $G_{odd}\\rightarrow \\mathsf {gr}(G)$ , is identified with $\\widehat{\\mathcal {O}_{\\mathsf {gr}(G), e}}\\rightarrow \\widehat{\\mathcal {O}_{\\mathsf {gr}(G), e}}/\\widehat{\\mathcal {O}_{\\mathsf {gr}(G), e}}\\widehat{\\mathcal {O}_{G_{ev}, e}}^+.$ Thus $A\\simeq \\mathsf {gr}(A)$ is a graded Hopf superalgebra, generated by component $A(1)=A^+/(A^+)^2$ , whose elements are primitive.",
"Let $J$ denote the kernel of the natural epimorphism $\\Lambda (A(1))\\rightarrow A$ of graded Hopf superalgebras.",
"Choose a basis $a_1, \\ldots , a_t$ of $A(1)$ .",
"The induction on $t$ shows that for every $1\\le i\\le t$ , the induced epimorphism $\\Lambda (A(1))/\\Lambda (A(1))a_i\\rightarrow A/Aa_i$ is an isomorphism.",
"Hence $J\\subseteq \\cap _{1\\le i\\le t}\\Lambda (A(1))a_i=Ka_1\\ldots a_t$ .",
"Since $Ka_1\\ldots a_t$ is not Hopf superideal, $J=0$ .",
"Finally, $A(1)$ is the odd component of $\\mathfrak {n}/\\mathfrak {n}^2\\simeq \\mathfrak {m}_e/\\mathfrak {m}_e^2$ .",
"That is, $A(1)\\simeq \\mathfrak {g}_1^*$ , and Lemma REF concludes the proof.",
"Lemma 10.2 The induced morphism $\\mathsf {gr}(S_k)\\rightarrow \\mathsf {gr}(G)$ coincides with the $k$ th neighborhood of the closed embedding $e\\rightarrow \\mathsf {gr}(G)$ .",
"It is enough to show that the kernel of the induced epimorphism $\\mathsf {gr}(\\mathcal {O}_e)\\rightarrow \\mathsf {gr}(\\mathcal {O}_e/\\mathfrak {m}_e^{k+1})$ coincides with $\\mathfrak {n}^{k+1}$ , where $\\mathfrak {n}$ is the maximal superideal of $\\mathsf {gr}(\\mathcal {O}_e)\\simeq \\mathcal {O}_{\\mathsf {gr}(G), e}$ .",
"Let $V$ denote the superspace $I_{\\mathcal {O}_e}/I_{\\mathcal {O}_e}^2$ .",
"As it was already shown in the previous lemma, the graded superalgebra $\\mathsf {gr}(\\mathcal {O}_e)$ is naturally isomorphic to $\\overline{\\mathcal {O}_e}\\otimes \\Lambda (V)$ .",
"Besides, $\\mathfrak {n}$ is identified with the graded superideal $\\overline{\\mathfrak {m}_e}\\otimes 1 + \\overline{\\mathcal {O}_e}\\otimes \\Lambda (V)^+$ , where $\\overline{\\mathfrak {m}_e}=\\mathfrak {m}_e/I_{\\mathcal {O}_e}$ .",
"For every $k\\ge 0$ , there is $\\mathfrak {n}^{k+1}=\\sum _{k+1-t\\le l\\le k+1}\\overline{\\mathfrak {m}_e}^l\\otimes (\\oplus _{k+1-l\\le s\\le t}\\Lambda ^s(V))=\\oplus _{0\\le i\\le t}\\overline{\\mathfrak {m}_e}^{k+1-i}\\otimes \\Lambda ^i(V),$ where $t=\\dim V$ .",
"In particular, the graded superalgebra $\\mathsf {gr}(\\mathcal {O}_e)/\\mathfrak {n}^{k+1}$ is isomorphic to $\\oplus _{0\\le i\\le t}\\overline{\\mathcal {O}_e}/\\overline{\\mathfrak {m}_e}^{k+1-i}\\otimes \\Lambda ^i(V).$ On the other hand, the isomorphism $\\mathsf {gr}(\\mathcal {O}_{e})\\simeq \\overline{\\mathcal {O}_e}\\otimes \\Lambda (V)$ implies that each element $f\\in \\mathcal {O}_e$ has the \"canonical\" form $f=\\sum _{0\\le s\\le t, 1\\le i_1<\\ldots < i_s\\le t} f_{i_1, \\ldots , i_s}v_{i_1}\\ldots v_{i_s}, $ where the elements $v_1, \\ldots v_t\\in (\\mathcal {O}_e)_1$ form a basis of $V$ , and the coefficients $f_{i_1, \\ldots , i_s}\\in (\\mathcal {O}_e)_0$ are uniquely defined modulo $I_{\\mathcal {O}_e}$ , or equivalently, modulo $(\\mathcal {O}_e)_1^2$ .",
"In particular, $f$ belongs to $\\mathfrak {m}_e$ if and only if its \"free\" coefficient (corresponding to $s=0$ ) does.",
"Moreover, $f$ belongs to $\\mathfrak {m}_e^{k+1}$ if and only if each $f_{i_1, \\ldots , i_s}$ belongs to $\\mathfrak {m}_e^{k+1-s}$ modulo $I_{\\mathcal {O}_e}$ .",
"Therefore, every element $\\overline{f}$ of superalgebra $\\mathcal {O}_e/\\mathfrak {m}_e^{k+1}$ has a form $\\sum _{0\\le s\\le t, 1\\le i_1<\\ldots < i_s\\le t} \\overline{f}_{i_1, \\ldots , i_s}\\overline{v_{i_1}}\\ldots \\overline{v_{i_s}}, \\text{ where } \\overline{f}_{i_1, \\ldots , i_s}\\in (\\mathcal {O}_e/\\mathfrak {m}_e^{k+1})_0,$ and each $f_{i_1, \\ldots , i_s}$ is uniquely defined modulo $\\mathfrak {m}_e^{k+1-s}+I_{\\mathcal {O}_e}$ .",
"The lemma follows.",
"Corollary 10.3 The formal neighborhood of the identity in $\\mathsf {gr}(\\mathbb {G})$ coincides with $\\cup _{k\\ge 0}\\mathsf {gr}(\\mathbb {S}_k)$ .",
"If we denote the formal neighborhood of the identity in $\\mathsf {gr}(\\mathbb {G})$ by $\\mathsf {gr}(\\mathbb {S})$ , then the above corollary states that $\\mathsf {gr}(\\mathbb {S})$ is \"represented\" by complete Hopf superalgebra $\\mathsf {gr}(\\widehat{\\mathcal {O}_e})\\simeq \\widehat{\\mathsf {gr}(\\mathcal {O}_e)}$ , whose comultiplication and antipode are $\\mathsf {gr}(\\Delta )$ and $\\mathsf {gr}(S)$ , respectively.",
"Moreover, arguing as in Lemma REF , one sees that each element $f\\in \\widehat{\\mathcal {O}_e}$ has a \"canonical\" form $\\sum _{0\\le s\\le t, 1\\le i_1<\\ldots < i_s\\le t} f_{i_1, \\ldots , i_s}v_{i_1}\\ldots v_{i_s}, \\text{ where } f_{i_1, \\ldots , i_s}\\in (\\widehat{\\mathcal {O}_e})_0,$ and the coefficients $f_{i_1, \\ldots , i_s}$ are uniquely defined modulo $(\\widehat{\\mathcal {O}_e})_1^2$ .",
"It is clear that $\\mathbb {S}_{ev}=\\mathbb {S}\\cap \\mathbb {G}_{ev}$ is the formal neighborhood of the identity in $\\mathbb {G}_{ev}$ .",
"Moreover, $\\mathrm {hyp}(\\mathbb {G}_{ev})$ can be naturally identified with the purely-even subalgebra of $\\mathrm {hyp}(\\mathbb {G})$ , consisting of all $\\phi \\in \\mathrm {hyp}(\\mathbb {G})$ such that $\\phi (I_{\\widehat{\\mathcal {O}_e}})=0$ .",
"More generally, let $\\mathrm {hyp}^{(k)}(\\mathbb {G})$ denote the super-subspace $\\lbrace \\phi \\in \\mathrm {hyp}(\\mathbb {G})\\mid \\phi (I^{k+1}_{\\widehat{\\mathcal {O}_e}})=0\\rbrace $ .",
"In particular, $\\mathrm {hyp}(\\mathbb {G}_{ev})=\\mathrm {hyp}^{(0)}(\\mathbb {G})$ and $\\mathrm {hyp}(\\mathbb {G})=\\mathrm {hyp}^{(t)}(\\mathbb {G})$ .",
"The following lemma is similar to Lemma REF .",
"Lemma 10.4 For every nonnegative integers $0\\le k, l\\le t$ there is: $\\mathrm {hyp}^{(k)}(\\mathbb {G})\\mathrm {hyp}^{(t)}(\\mathbb {G})\\subseteq \\mathrm {hyp}^{(k+t)}(\\mathbb {G})$ , $\\Delta ^*(\\mathrm {hyp}^{(k)}(\\mathbb {G}))\\subseteq \\sum _{0\\le s\\le k} \\mathrm {hyp}^{(s)}(\\mathbb {G})\\otimes \\mathrm {hyp}^{(k-s)}(\\mathbb {G})$ , $S^*(\\mathrm {hyp}^{(k)}(\\mathbb {G}))\\subseteq \\mathrm {hyp}^{(k)}(\\mathbb {G})$ .",
"Lemma REF immediately implies that $\\mathsf {gr}(\\mathrm {hyp}(\\mathbb {G}))=\\oplus _{0\\le k\\le t}\\mathrm {hyp}^{(k)}(\\mathbb {G})/\\mathrm {hyp}^{(k-1)}(\\mathbb {G})$ has a natural structure of (finitely) graded Hopf superalgebra.",
"Proposition 10.5 There is a canonical isomorphism $\\mathsf {gr}(\\mathrm {hyp}(\\mathbb {G}))\\simeq \\mathrm {hyp}(\\mathsf {gr}(\\mathbb {G}))$ of graded Hopf superalgebras.",
"First of all, the superspace $\\mathrm {hyp}(\\mathsf {gr}(\\mathbb {G}))$ is isomorphic to $\\oplus _{0\\le k\\le t}(I_{\\widehat{\\mathcal {O}_e}}^k/I_{\\widehat{\\mathcal {O}_e}}^{k+1})^{\\underline{*}}$ , where $(I_{\\widehat{\\mathcal {O}_e}}^k/I_{\\widehat{\\mathcal {O}_e}}^{k+1})^{\\underline{*}}$ consists of all continuous linear maps $I_{\\widehat{\\mathcal {O}_e}}^k/I_{\\widehat{\\mathcal {O}_e}}^{k+1}\\rightarrow K$ , and $K$ is regarded as a discrete vector space.",
"In fact, arguing as in Lemma REF , one sees that the $\\widehat{\\mathfrak {n}}$ -adic topology of $\\mathsf {gr}(\\widehat{\\mathcal {O}_e})$ (respectively, the factor-topology of $I_{\\widehat{\\mathcal {O}_e}}^k/I_{\\widehat{\\mathcal {O}_e}}^{k+1}$ ) is equivalent to its $\\widehat{\\overline{\\mathfrak {m}_e}}$ -adic topology.",
"Thus, $\\mathsf {gr}(\\widehat{\\mathcal {O}_e})$ is isomorphic to $\\oplus _{0\\le k\\le t}I_{\\widehat{\\mathcal {O}_e}}^k/I_{\\widehat{\\mathcal {O}_e}}^{k+1}$ as a (linear) topological superspace.",
"Next, we have a natural superspace morphism $\\mathrm {hyp}^{(k)}(\\mathbb {G})/\\mathrm {hyp}^{(k-1)}(\\mathbb {G})\\rightarrow (I_{\\widehat{\\mathcal {O}_e}}^{k-1}/I_{\\widehat{\\mathcal {O}_e}}^k)^{\\underline{*}},$ induced by the Hopf pairing $< \\ , \\ >$ , which is obviously injective.",
"On the other hand, every $\\phi \\in (I_{\\widehat{\\mathcal {O}_e}}^{k-1}/I_{\\widehat{\\mathcal {O}_e}}^k)^{\\underline{*}}$ is some $\\psi \\in \\mathsf {gr}(\\widehat{\\mathcal {O}_e})^{\\underline{*}}$ restricted on $I_{\\widehat{\\mathcal {O}_e}}^{k-1}/I_{\\widehat{\\mathcal {O}_e}}^k$ .",
"Since $\\psi $ is continuous, there is $l\\ge 0$ such that $\\psi (\\widehat{\\mathfrak {n}}^{l+1})=0$ .",
"Choose a basis of $\\mathsf {gr}(\\widehat{\\mathcal {O}_e})$ consisting of elements $\\overline{f} v_{i_1}\\ldots v_{i_s}$ , where $1\\le i_1<\\ldots < i_s\\le t, 0\\le s\\le t$ , and $f\\in (\\widehat{\\mathcal {O}_1})_0$ .",
"We also assume that for every $k\\ge 0$ , the elements $\\overline{f}$ , that belong to $\\widehat{\\overline{\\mathfrak {m}_e}}^k\\setminus \\widehat{\\overline{\\mathfrak {m}_e}}^{k+1}$ , form a basis of $\\widehat{\\overline{\\mathfrak {m}_e}}^k /\\widehat{\\overline{\\mathfrak {m}_e}}^{k+1}$ .",
"Then the elements $f v_{i_1}\\ldots v_{i_s}$ form a basis of $\\widehat{\\mathcal {O}_e}$ .",
"Moreover, a superideal $\\widehat{\\mathfrak {m}_e}^{k+1}$ has a basis consisting of all $f v_{i_1}\\ldots v_{i_s}$ such that $f\\in \\widehat{\\mathfrak {m}_e}^{k+1-s}+ I_{\\widehat{\\mathcal {O}_e}}$ (or equivalently, $\\overline{f}\\in \\widehat{\\overline{\\mathfrak {m}_e}}^{k+1-s}$ ).",
"Similarly, a superideal $I_{\\widehat{\\mathcal {O}_e}}^k$ has a basis consisting of all $f v_{i_1}\\ldots v_{i_s}$ with $s\\ge k$ .",
"Set $\\psi ^{\\prime }(f v_{i_1}\\ldots v_{i_s})=\\psi (\\overline{f} v_{i_1}\\ldots v_{i_s})$ for each basic element $f v_{i_1}\\ldots v_{i_s}$ .",
"Then $\\psi ^{\\prime }$ belongs to $\\mathrm {hyp}^{(k)}(\\mathbb {G})$ and its image in $(I_{\\widehat{\\mathcal {O}_e}}^{k-1}/I_{\\widehat{\\mathcal {O}_e}}^k)^{\\underline{*}}$ coincides with $\\phi $ .",
"In other words, we have a natural graded pairing $\\mathsf {gr}(\\mathrm {hyp}(\\mathbb {G}))\\times \\mathsf {gr}_{I_{\\widehat{\\mathcal {O}_e}}}(\\widehat{\\mathcal {O}_e})\\rightarrow K$ that induces an isomorphism $\\mathsf {gr}(\\mathrm {hyp}(\\mathbb {G}))\\simeq \\mathrm {hyp}(\\mathsf {gr}(\\mathbb {G}))$ of superspaces.",
"It is a Hopf pairing by the definition of Hopf superalgebra structure on $\\mathsf {gr}(\\widehat{\\mathcal {O}_e})$ .",
"The proposition is proven.",
"Let $\\gamma _1, \\ldots , \\gamma _t$ be a basis of $\\mathfrak {g}_1\\simeq V^*$ , dual to the basis $v_1, \\ldots , v_t$ of $V$ .",
"Proposition 10.6 Every element $\\phi \\in \\mathrm {hyp}_k(\\mathbb {G})$ can be uniquely expressed as $\\phi =\\sum _{0\\le s\\le t, 1\\le i_1<\\ldots < i_s\\le t}\\phi _{i_1, \\ldots , i_s}\\gamma _{i_1}\\ldots \\gamma _{i_s}, \\text{ where }\\phi _{i_1, \\ldots , i_s}\\in \\mathrm {hyp}_{k-s}(\\mathbb {G}_{ev}).$ It is sufficient to prove this statement for a (locally algebraic) group superscheme $\\mathbb {G}\\simeq \\mathsf {gr}(\\mathbb {G})$ .",
"In this case, we have $\\widehat{\\mathcal {O}_e}\\simeq A\\otimes \\Lambda (V)$ , where $A\\simeq \\overline{\\widehat{\\mathcal {O}_e}}$ is a closed Hopf subalgebra of $\\widehat{\\mathcal {O}_e}$ .",
"Since $\\widehat{\\mathcal {O}_e}$ is also a graded Hopf superalgebra with $\\widehat{\\mathcal {O}_e}(1)=A\\otimes V$ , then each $v\\in V$ is skew primitive, i.e.",
"$\\Delta (v)=v\\otimes a+ b\\otimes v$ , where $a$ and $b$ are group-like elements from $A$ .",
"Further, the Hopf pairing $\\mathrm {hyp}(\\mathbb {G})\\times \\widehat{\\mathcal {O}_e}\\rightarrow K$ is graded.",
"Therefore, it is enough to show that $<\\phi \\gamma _{i_1}\\ldots \\gamma _{i_s}, f v_{j_1}\\ldots v_{j_l}>\\ne 0$ implies $s=l, i_1=j_1, \\ldots , i_s=j_s$ .",
"Moreover, in this case $<\\phi \\gamma _{i_1}\\ldots \\gamma _{i_s}, f v_{i_1}\\ldots v_{i_s}>=<\\phi , fb_{i_1}\\ldots b_{i_s}>.$ We use an induction on $s$ .",
"If $s=0$ , then the statement is obvious.",
"Let $s\\ge 1$ .",
"We have $\\Delta (f v_{j_1}\\ldots v_{j_l})=( f_{(0)}\\otimes f_{(1)})\\prod _{1\\le p\\le l}(v_{j_p}\\otimes a_{j_p}+ b_{j_p}\\otimes v_{j_p})=$ $\\sum _{K\\subseteq \\lbrace j_1, \\ldots , j_l\\rbrace } \\pm f_{(0)}b_{\\overline{K}}v_K\\otimes f_{(1)} a_{K}v_{\\overline{K}},$ where $\\Delta (f)=f_{(0)}\\otimes f_{(1)}$ and $v_K$ (respectively, $a_K$ or $b_K$ ) denotes a product of all $v_{j_p}$ (respectively, $a_{j_p}$ or $b_{j_p}$ ) for $j_p\\in K$ (in an arbitrary order), and $\\overline{K}=\\lbrace j_1, \\ldots , j_l\\rbrace \\setminus K$ is the complement of $K$ .",
"Then $<(\\phi \\gamma _{i_1}\\ldots \\gamma _{i_{s-1}})\\gamma _{i_s}, f v_{j_1}\\ldots v_{j_l}>=$ $\\sum _{K\\subseteq \\lbrace j_1, \\ldots , j_l\\rbrace }\\pm <\\phi \\gamma _{i_1}\\ldots \\gamma _{i_{s-1}}, f_{(0)}b_{\\overline{K}}v_K><\\gamma _{i_s}, f_{(1)} a_{K}v_{\\overline{K}}>$ contains nonzero summands for $K=\\lbrace i_1, \\ldots , i_{s-1}\\rbrace $ only.",
"Moreover, $\\overline{K}$ should be a singleton, say $\\overline{K}=\\lbrace j_p\\rbrace $ .",
"Thus $s=l$ and $<\\gamma _{i_s}, f_{(1)} a_{K}v_{j_p}>=\\epsilon _A(f_{(1)})<\\gamma _{i_s}, v_{j_p}>\\ne 0$ implies $i_s=j_p$ .",
"Besides, $<(\\phi \\gamma _{i_1}\\ldots \\gamma _{i_{s-1}})\\gamma _{i_s}, f v_{j_1}\\ldots v_{j_l}>$ is equal to $<\\phi \\gamma _{i_1}\\ldots \\gamma _{i_{s-1}}, f_{(0)} b_{i_s} v_{i_1}\\ldots v_{i_{s-1}}>\\epsilon _A(f_{(1)})=<\\phi , fb_{i_1}\\ldots b_{i_s}>.$ The proposition is proven."
],
[
"The category of Harish-Chandra pairs",
"In this subsection, we follow [11].",
"A pair $(\\mathsf {G}, \\mathsf {V})$ , where $\\mathsf {G}$ is an algebraic group scheme, and $\\mathsf {V}$ is a $\\mathsf {G}$ -module, is a Harish-Chandra pair if the following conditions hold: There is a functor $\\mathsf {V}_a\\times \\mathsf {V}_a\\rightarrow \\mathsf {g}_a$ , denoted by $[ \\ , \\ ]$ , where $\\mathsf {g}$ is the Lie algebra of $\\mathsf {G}$ , such that for every $R\\in \\mathsf {Alg}_K$ , the map $\\mathsf {V}_a(R)\\times \\mathsf {V}_a(R)\\rightarrow \\mathsf {g}_a(R)$ is $R$ -bilinear and symmetric.",
"The functor $\\mathsf {V}_a\\times \\mathsf {V}_a\\rightarrow \\mathsf {g}_a$ is $\\mathsf {G}$ -equivariant for the diagonal action of $\\mathsf {G}$ on $\\mathsf {V}_a\\times \\mathsf {V}_a$ and the adjoint action of $\\mathsf {G}$ on $\\mathsf {g}_a$ .",
"The induced action of $\\mathsf {g}$ on $\\mathsf {V}$ satisfies $[v, v]\\cdot v=0$ for $v\\in \\mathsf {V}$ .",
"The morphism of Harish-Chandra pairs $(\\mathsf {G}, \\mathsf {V})\\rightarrow (\\mathsf {H}, \\mathsf {W})$ is a morphism of pairs $(\\mathsf {f}, u)$ as in Section 7, such that the diagram $\\begin{array}{ccc}\\mathsf {V}_a\\times \\mathsf {V}_a & \\rightarrow & \\mathsf {g}_a \\\\\\downarrow & & \\downarrow \\\\\\mathsf {W}_a\\times \\mathsf {W}_a & \\rightarrow & \\mathsf {h}_a\\end{array}$ is commutative, where the first vertical map is $\\mathsf {u}_a\\times \\mathsf {u}_a$ and the second vertical map is $\\mathrm {d}_e\\mathsf {f}$ .",
"The category of Harish-Chandra pairs is denoted by $\\mathsf {HCP}$ .",
"To every algebraic group superscheme $\\mathbb {G}$ , we associate a Harish-Chandra pair $(\\mathsf {G}, \\mathsf {V})$ with $\\mathsf {G}=\\mathbb {G}_{ev}$ (regarded as a group scheme), and $\\mathsf {V}=\\mathfrak {g}_1$ .",
"Besides, the action of $\\mathsf {G}$ on $\\mathsf {V}$ is induced by the adjoint action of $\\mathbb {G}$ , and $[ \\ , \\ ]$ is the restriction of Lie super-bracket of $\\mathfrak {g}$ on its odd component.",
"Lemma 11.1 The correspondence $\\Phi : \\mathbb {G}\\mapsto (\\mathsf {G}, \\mathsf {V})$ is a functor from $\\mathcal {SFG}_{la}$ to $\\mathsf {HCP}$ .",
"We call $\\Phi $ the Harish-Chandra functor.",
"Recall that each group $\\mathbb {S}(R)$ is identified with $\\mathsf {Gpl}(\\mathrm {hyp}(\\mathbb {G})\\otimes R)$ .",
"For every $v\\in \\mathsf {V}=\\mathfrak {g}_1, x\\in \\mathsf {g}=\\mathfrak {g}_0$ and $a\\in R_1, b\\in R_0$ such that $b^2=0$ , define the group-like elements $f(b , x)=\\epsilon _e^*\\otimes 1+x\\otimes b, e(a, v)=\\epsilon _e^* \\otimes 1+v\\otimes a .$ These elements are images of the group-like elements $e^{\\epsilon _0(x\\otimes 1)}$ and $e^{-\\epsilon _1(v\\otimes 1)}$ from $\\mathbb {S}(R[\\epsilon _0, \\epsilon _1])$ under a homomorphism of groups, induced by $\\epsilon _0\\mapsto b, \\epsilon _1\\mapsto a$ .",
"Using the identity involving group commutators from Lemma REF , we obtain the first three relations in the list: $[e(a, v), e(a^{\\prime }, v^{\\prime })]=f(-aa^{\\prime }, [v, v^{\\prime }])$ , $[f(b, x), e(a, v)]=e(ba, [x, v])$ , $[f(b, x), f(b^{\\prime }, x^{\\prime })]=f(bb^{\\prime }, [x, x^{\\prime }])$ , $e(a, v)e(a^{\\prime }, v)=f(-aa^{\\prime }, \\frac{1}{2}[v, v])e(a+a^{\\prime }, v)$ .",
"A direct computation can derive the fourth relation.",
"The relations $(1), (2), (3)$ are the relations $(\\mathrm {i}), (\\mathrm {iii}), (\\mathrm {iv})$ from Lemma 4.2 in [11], and the relation $(4)$ is $(\\mathrm {ii})$ therein.",
"Also, for every $g\\in \\mathsf {G}(R)$ we have $e(a, v)^g=e(a, \\mathrm {Ad}(g)v)) \\text{ and } f(b, x)^g=f(b, \\mathrm {Ad}(g)x))$ .",
"Define the group subfunctors $\\bf \\Sigma $ and $\\mathsf {F}$ of $\\mathbb {S}$ and $\\mathsf {S}=\\mathbb {S}_{ev}$ respectively, such that $\\bf \\Sigma (R)$ is generated by all the elements $f(b, x)$ and $e(a, v)$ , but $\\mathsf {F}(R)$ is generated only by $f(b, x)$ , where $a, b\\in R$ and $R\\in \\mathsf {SAlg}_K$ .",
"Lemma 11.2 The following statements hold: $\\mathsf {F}(R)={\\bf \\Sigma }(R)\\cap \\mathsf {S}(R)$ .",
"If $v_1, \\ldots , v_t$ is a basis of $\\mathsf {V}$ , then each element of ${\\bf \\Sigma }(R)$ is uniquely expressed in the form $f e(a_1, v_1)\\ldots e(a_t, v_t),$ where $f\\in \\mathsf {F}(R)$ and $a_i\\in R_1$ for $1\\le i\\le t$ .",
"Both group subfunctors $\\mathsf {F}$ and $\\bf \\Sigma $ are invariant for the conjugation action of $\\mathsf {G}$ .",
"To prove the first two statements, use Proposition REF , and copy the proof of Proposition 4.3 from [11].",
"The third statement follows from the relation $(5)$ above.",
"Remark 11.3 The symmetric version of Lemma REF (2), where $f$ appears on the right-hand side, is also valid.",
"Now we can define a group subfunctor $\\mathbb {G}^{\\prime }$ of $\\mathbb {G}$ , such that $\\mathbb {G}^{\\prime }(R)=\\mathsf {G}(R){\\bf \\Sigma }(R)$ for $R\\in \\mathsf {SAlg}_K$ .",
"Corollary 11.4 Each element of $\\mathbb {G}^{\\prime }(R)$ is uniquely expressed in the form $ge(a_1, v_1)\\ldots e(a_t, v_t),$ where $g\\in \\mathsf {G}(R)$ , and $a_i\\in R_1$ for $1\\le i\\le t$ .",
"Let $\\bf E$ denote a subfunctor of $\\bf \\Sigma $ such that ${\\bf E}(R)$ consists of all products $e(a_1, v_1)\\ldots e(a_t, v_t), \\text{ where } a_i\\in R_1 \\text{ for } 1\\le i\\le t.$ It is clear that ${\\bf E}\\simeq \\mathrm {SSp}(\\Lambda (\\mathsf {V}))$ is a purely-odd affine superscheme.",
"Then $\\mathbb {G}^{\\prime }$ is isomorphic to $\\mathsf {G}\\times {\\bf E}$ as a superscheme.",
"In particular, $\\mathbb {G}^{\\prime }$ is a (locally algebraic) group superscheme.",
"Theorem 11.5 We have $\\mathbb {G}=\\mathbb {G}^{\\prime }$ .",
"By Proposition REF , one has to prove that the induced morphism $\\mathsf {gr}(\\mathbb {G}^{\\prime })\\rightarrow \\mathsf {gr}(\\mathbb {G})$ of group superschemes is an isomorphism.",
"Since $\\mathbb {G}_{ev}=\\mathbb {G}^{\\prime }_{ev}$ , Lemma REF implies that $\\mathsf {gr}(\\mathbb {G}^{\\prime })_{ev}$ is mapped isomorphically onto $\\mathsf {gr}(\\mathbb {G}^{\\prime })_{ev}$ .",
"By Proposition REF , it is sufficient to prove that the embedding $\\mathbb {G}^{\\prime }\\rightarrow \\mathbb {G}$ induces an isomorphism of odd components of their Lie superalgebras.",
"But as it has been already observed, in both $\\mathbb {G}$ and $\\mathbb {G}^{\\prime }$ , the elements $e^{\\epsilon _1 (v_i\\otimes 1)}$ coincide with $e(-\\epsilon _1, v_i)$ .",
"The theorem is proven.",
"Corollary 11.6 Let $\\mathsf {S}$ denote $\\mathbb {S}_{ev}$ .",
"Then $\\mathbb {S}=\\mathsf {S}{\\bf E}={\\bf E}\\mathsf {S}$ .",
"If $s\\in \\mathbb {S}(R)$ is expressed as $gx$ , where $g\\in \\mathsf {G}(R)$ and $x\\in {\\bf E}(R)$ , then $g=sx^{-1}\\in \\mathbb {G}_{ev}(R)\\cap \\mathbb {S}=\\mathbb {S}_{ev}(R)$ , since ${\\bf E}\\subseteq \\mathbb {S}$ .",
"It is clear that ${\\bf E}\\mathsf {S}\\subseteq \\mathbb {S}$ .",
"Moreover, for any element $sx$ , where $s\\in \\mathsf {S}(R)$ and $x\\in {\\bf E}(R),$ we have $sx=x^{s}s=x^{\\prime } (f s)$ for $f\\in \\mathsf {F}(R)$ and $x^{\\prime }\\in {\\bf E}(R)$ .",
"Then $x^{s}=x^{\\prime } f$ by the symmetric version of Lemma REF (2)."
],
[
"A quasi-inverse of the Harish-Chandra functor",
"Let $(\\mathsf {G}, \\mathsf {V})$ be a Harish-Chandra pair.",
"Note that $\\mathsf {g}\\oplus \\mathsf {V}$ has a natural Lie superalgebra structure, induced by the bilinear map $\\mathsf {V}\\times \\mathsf {V}\\rightarrow \\mathsf {g}$ , and by the differential of the action of $\\mathsf {G}$ on $\\mathsf {V}$ .",
"Let $\\mathfrak {g}$ denote this Lie superalgebra.",
"Fix a basis $x_1, \\ldots , x_l$ of $\\mathfrak {g}_0=\\mathsf {g}$ .",
"Following [11], one can define a group functor $\\bf \\Sigma ^{\\prime }$ that associates with each superalgebra $R$ a group generated by symbols $e^{\\prime }(a, v), f^{\\prime }(b, x)$ for $a\\in R_1, b\\in R_0$ such that $b^2=0, v\\in \\mathsf {V}$ and $x\\in \\mathsf {g}$ , subject to the relations $(1)-(4)$ .",
"Note that if either one of the parameters $a, b, v, x$ is zero, then the corresponding generator is the unit.",
"Let $\\mathsf {F}^{\\prime }$ be a group subfunctor of $\\bf \\Sigma ^{\\prime }$ , such that $\\mathsf {F}^{\\prime }(R)$ is generated only by the symbols $f^{\\prime }(b, x)$ .",
"Since the proof of Lemma 4.4, [11] uses only the above relations, one can conclude again that any element of ${\\bf \\Sigma }^{\\prime }(R)$ has a form $f^{\\prime }e(a_1, v_1)\\ldots e^{\\prime }(a_t, v_t)$ , where $f^{\\prime }\\in \\mathsf {F}^{\\prime }(R)$ .",
"On the other hand, there is a natural morphism of the group functor ${\\bf \\Sigma }^{\\prime }$ to the group functor $R\\mapsto \\mathsf {Gpl}(\\mathsf {U}(\\mathfrak {g})\\otimes R)$ that sends $e^{\\prime }(a, v)$ to $1\\otimes 1+v\\otimes a$ and $f^{\\prime }(b, x)$ to $1\\otimes 1+x\\otimes b$ , respectively.",
"Then Proposition 4.3, [11] implies that the above form is unique.",
"Further, the action of $\\mathsf {G}$ on $\\mathsf {V}$ induces the action on the generators by $g\\cdot e^{\\prime }(a, v)=e^{\\prime }(r_1a, v_1)\\ldots e^{\\prime }(r_k a, v_t) \\text{ and } g\\cdot f^{\\prime }(b, x)=f^{\\prime }(s_1b, x_1)\\ldots f^{\\prime }(s_l b, x_l),$ where $g\\cdot v=\\sum _{1\\le i\\le t} v_i\\otimes r_i$ and $\\mathrm {Ad}(g)(x)=\\sum _{1\\le j\\le l} x_j\\otimes s_j$ .",
"Lemma 11.7 The above action preserves the relations $(1)-(4)$ .",
"Hence, it defines a homomorphism $\\alpha (R) : \\mathsf {G}(R)\\rightarrow \\mathrm {Aut}({\\bf \\Sigma }^{\\prime }(R))$ .",
"Consider the relation $(1)$ .",
"Suppose that $g\\cdot v^{\\prime }=\\sum _{1\\le i\\le t} v_i\\otimes r^{\\prime }_i$ .",
"Then for every $x, y, z\\in \\lbrace e^{\\prime }(r_i a, v_i), e^{\\prime }(r^{\\prime }_k a^{\\prime }, v_k)\\mid 1\\le i, k\\le t\\rbrace $ we have $[[x, y], z]=1$ .",
"Thus, the well-known commutator identity $[xy, z]=[x, z]^y [y, z]$ implies $[xy, z]=[x, z][y, z]$ .",
"Therefore, $g\\cdot [e^{\\prime }(a, v), e^{\\prime }(a^{\\prime }, v^{\\prime })]=[g\\cdot e^{\\prime }(a, v), g\\cdot e^{\\prime }(a^{\\prime }, v^{\\prime })]=\\prod _{1\\le i, k\\le t}[e^{\\prime }(r_i a, v_i), e^{\\prime }(r^{\\prime }_k a^{\\prime }, v_k)]=$ $\\prod _{1\\le i, k\\le t}f^{\\prime }(-r_iar^{\\prime }_k a^{\\prime }, [v_i, v_k])=f^{\\prime }(-aa^{\\prime }, g\\cdot [v, v^{\\prime }])=g\\cdot f^{\\prime }(-aa^{\\prime }, [v, v^{\\prime }]).$ The identities (3) and (4) can be derived similarly.",
"The identity $(2)$ follows from $g\\cdot [x, v]=[\\mathrm {Ad}(g)(x), g\\cdot v]$ , which is obtained by using the identity at the beginning of the proof of Lemma REF .",
"The lemma is proven.",
"The map $f^{\\prime }(b, x)\\mapsto f(b, x)$ induces a morphism of group functors $i : \\mathsf {F}^{\\prime }\\rightarrow \\mathsf {F}$ .",
"In fact, it is the composition of the above morphism from $\\mathsf {F}^{\\prime }$ to the group functor $R\\mapsto \\mathsf {Gpl}(\\mathsf {U}(\\mathsf {g})\\otimes R_0)$ and the natural morphism from the latter to $\\mathsf {F}$ , induced by the Hopf algebra morphism $\\mathsf {U}(\\mathsf {g})\\rightarrow \\mathrm {hyp}(\\mathbb {G}_{ev})$ .",
"It is clear that $({\\bf \\Sigma }^{\\prime }(R), \\mathsf {F}^{\\prime }(R), \\mathsf {G}(R), i(R), \\alpha (R))$ is a quintuple in the sense of [11], with the only difference that $\\alpha (R)$ is a group homomorphism.",
"We define a group ${\\bf \\Gamma }(R)$ as a factor-group of $\\mathsf {G}(R)\\ltimes {\\bf \\Sigma }^{\\prime }(R)$ by a normal subgroup ${\\bf \\Xi }(R)=\\lbrace (i(R)(f), f^{-1})\\mid f\\in \\mathsf {F}^{\\prime }(R)\\rbrace $ .",
"Remark 11.8 ${\\bf \\Gamma }(R)$ is isomorphic to the factor-group of the amalgamated free product $\\mathsf {G}(R)*_{\\mathsf {F}^{\\prime }(R)} {\\bf \\Sigma }^{\\prime }(R)$ modulo the relations $s^g=(\\alpha (R)(g))(s)$ for $g\\in \\mathsf {G}(R)$ and $s\\in {\\bf \\Sigma }^{\\prime }(R)$ .",
"Lemma 2.1 from [11] implies that every element of ${\\bf \\Gamma }(R)$ can be uniquely expressed as $g e^{\\prime }(a_1, v_1)\\ldots e^{\\prime }(a_t, v_t)$ , where $g\\in \\mathsf {G}(R)$ and $a_i\\in R_1$ for $1\\le i\\le t$ .",
"The following lemma is now apparent.",
"Lemma 11.9 $R\\mapsto {\\bf \\Gamma }(R)$ is a group functor, hence a locally algebraic group superscheme.",
"Moreover, $\\Psi : (\\mathsf {G}, \\mathsf {V})\\rightarrow {\\bf \\Gamma }$ is a functor from $\\mathsf {HCP}$ to $\\mathcal {SFG}_{la}$ .",
"Theorem 11.10 The functors $\\Phi $ and $\\Psi $ are quasi-inverse to each other.",
"Since the group structure of ${\\bf \\Gamma }(R)$ is defined by the relations similar to $(1)-(5)$ , Theorem REF implies $\\Psi \\circ \\Phi \\simeq \\mathrm {id}_{\\mathcal {SFG}_{la}}$ .",
"Further, the odd component of the Lie superalgebra of $\\bf \\Gamma $ can be identified with the (commutative) subgroup of ${\\bf \\Gamma }(K[\\epsilon _0, \\epsilon _1])$ consisting of all elements $e^{\\prime }(c_1\\epsilon _1, v_1)\\ldots e^{\\prime }(c_t\\epsilon _1, v_t)$ , where $c_i \\in K$ for $1\\le i\\le t$ , which can be identified with $\\mathsf {V}$ .",
"Moreover, the adjoint action of $\\mathsf {G}$ is identified with the original action of $\\mathsf {G}$ .",
"Thus $\\Phi \\circ \\Psi \\simeq \\mathrm {id}_{\\mathsf {HCP}}$ .",
"The theorem is proven.",
"In what follows, let $\\mathsf {E}$ denote $\\mathbb {E}_{ev}$ .",
"Let $\\mathbb {G}$ be a locally algebraic group superscheme.",
"If $\\Phi (\\mathbb {G})\\simeq (\\mathsf {G}, \\mathsf {V})$ or $\\Psi ((\\mathsf {G}, \\mathsf {V}))\\simeq \\mathbb {G}$ , then we say that $\\mathbb {G}$ is represented by $(\\mathsf {G}, \\mathsf {V})$ .",
"A sequence $(\\mathsf {R}, \\mathsf {W})\\rightarrow (\\mathsf {G}, \\mathsf {V})\\rightarrow (\\mathsf {H}, \\mathsf {U})$ in $\\mathsf {HCP}$ is called exact whenever the following conditions hold: The sequences $0\\rightarrow \\mathsf {W}\\rightarrow \\mathsf {V}\\rightarrow \\mathsf {U}\\rightarrow 0$ and $\\mathsf {E}\\rightarrow \\mathsf {R}\\rightarrow \\mathsf {G}\\rightarrow \\mathsf {H}\\rightarrow \\mathsf {E}$ are exact in the categories of superspaces and group schemes, respectively.",
"$\\mathsf {W}$ is a $\\mathsf {G}$ -submodule of $\\mathsf {V}$ .",
"$\\mathsf {R}$ acts trivially on $\\mathsf {V}/\\mathsf {W}$ .",
"$[\\mathsf {V}, \\mathsf {W}]\\subseteq \\mathrm {Lie}(\\mathsf {R})$ .",
"Theorem 11.11 A sequence of group superschemes $\\mathbb {R}\\rightarrow \\mathbb {G}\\rightarrow \\mathbb {H}$ is exact if and only if the sequence $\\Phi (\\mathbb {R})\\rightarrow \\Phi (\\mathbb {G})\\rightarrow \\Phi (\\mathbb {H})$ is exact.",
"Suppose that $\\mathbb {R}, \\mathbb {G}$ and $\\mathbb {H}$ are represented by the pairs $(\\mathsf {R}, \\mathsf {W}), (\\mathsf {G}, \\mathsf {V})$ , and $(\\mathsf {H}, \\mathsf {U})$ , respectively.",
"Without loss of a generality, one can replace $\\mathbb {R}\\rightarrow \\mathbb {G}\\rightarrow \\mathbb {H}$ by the sequence $\\Psi ((\\mathsf {R}, \\mathsf {W}))\\rightarrow \\Psi ((\\mathsf {G}, \\mathsf {V}))\\rightarrow \\Psi ((\\mathsf {H}, \\mathsf {U}))$ .",
"Then $\\mathbb {R}$ coincides with $\\ker (\\mathbb {G}\\rightarrow \\mathbb {H})$ if and only if $\\mathsf {R}=\\ker (\\mathsf {G}\\rightarrow \\mathsf {H}), \\mathsf {W}=\\ker (\\mathsf {V}\\rightarrow \\mathsf {U})$ and $\\mathsf {W}$ satisfies the conditions $(2a)-(2c)$ .",
"For example, since $e^{\\prime }(a, v)he^{\\prime }(a, v)^{-1}\\in \\mathbb {R}(A)$ for every $h\\in \\mathbb {R}(A), a\\in A_1$ and $v\\in \\mathsf {V}$ , then $e^{\\prime }(a, v)he^{\\prime }(a, v)^{-1}h^{-1}=e^{\\prime }(a, v)e^{\\prime }(a, -h\\cdot v)=e^{\\prime }(a, v-h\\cdot v)\\in \\mathbb {R}(A)$ implies $v-h\\cdot v\\in \\mathsf {W}$ , hence $(2b)$ .",
"The converse statement is obvious.",
"Suppose that $\\mathbb {G}\\rightarrow \\mathbb {H}$ is surjective in the Grothendieck topology.",
"Consider a couple $h\\in \\mathsf {H}(A_0)=\\mathbb {H}(A_0)$ and $e^{\\prime }(a, u)\\in \\mathbb {H}(A)$ .",
"There is an fppf covering $A^{\\prime }$ of $A$ , such that $e^{\\prime }(\\iota (a), u)=e^{\\prime }(a^{\\prime }, \\overline{v}),$ where $v\\in \\mathsf {V}$ , and $\\overline{v}$ is the image of $v$ in $\\mathsf {U}$ .",
"That is, $\\mathsf {V}\\rightarrow \\mathsf {U}$ is a surjective map.",
"Finally, let $\\iota : A_0\\rightarrow B$ be an fppf covering of $A_0$ , such that $\\mathbb {H}(\\iota )(h)$ belongs to the image of $\\mathbb {G}_{ev}(B)=\\mathsf {G}(B_0)$ .",
"Then $B_0$ is an fppf covering of $A_0$ .",
"Conversely, if $h\\in \\mathsf {H}(A_0)$ belongs to the image of $\\mathsf {G}$ up to an fppf covering $B$ of $A_0$ (in the category of algebras), then $h$ belongs to the image of $\\mathbb {G}$ up to an fppf covering $B\\otimes _{A_0} A$ of $A$ .",
"The theorem is proven."
],
[
"Radicals",
"Let $\\mathbb {G}$ be an algebraic group superscheme, represented by a Harish-Chandra pair $(\\mathsf {G}, \\mathsf {V})$ , where $\\mathsf {V}=\\mathrm {Lie}(\\mathbb {G})_1$ , and $\\mathsf {G}=\\mathbb {G}_{ev}$ is regarded as an algebraic group scheme.",
"Let $\\mathsf {R}$ be a normal group subscheme of $\\mathsf {G}$ .",
"The largest normal group super-subscheme $\\mathbb {H}$ of $\\mathbb {G}$ such that $\\mathsf {H}=\\mathbb {H}_{ev}\\le \\mathsf {R}$ , is called the $\\mathsf {R}$ -radical of $\\mathbb {G}$ .",
"If the $\\mathsf {R}$ -radical of $\\mathbb {G}$ is trivial, then $\\mathbb {G}$ is called $\\mathsf {R}$ -semisimple.",
"Let $\\mathsf {W}$ be a $\\mathsf {G}$ -submodule of $\\mathsf {V}$ such that $[\\mathsf {W}, \\mathsf {V}]\\subseteq \\mathrm {Lie}(\\mathsf {R})$ and $[\\mathsf {W}, \\mathsf {V}, \\mathsf {V}]\\subseteq \\mathsf {W}$ .",
"Such a submodule is called $\\mathsf {R}$ -subordinated.",
"The sum of two $\\mathsf {R}$ -subordinated submodules is again $\\mathsf {R}$ -subordinated.",
"Thus, there is the largest $\\mathsf {R}$ -subordinated submodule, denoted by $\\mathsf {W}_{\\mathsf {R}}$ .",
"Set $\\mathsf {H}_{\\mathsf {R}}=\\ker (\\mathsf {R}\\rightarrow \\mathrm {GL}(\\mathsf {V}/\\mathsf {W}_{\\mathsf {R}}))$ .",
"Since $\\mathsf {W}_{\\mathsf {R}}$ is a $\\mathsf {G}$ -submodule of $\\mathsf {V}$ , we have $\\mathsf {H}_{\\mathsf {R}}\\unlhd \\mathsf {G}$ .",
"Lemma 12.1 The Harish-Chandra sub-pair $(\\mathsf {H}_{\\mathsf {R}}, \\mathsf {W}_{\\mathsf {R}})$ represents the $\\mathsf {R}$ -radical of $\\mathbb {G}$ .",
"Let $(\\mathsf {H}, \\mathsf {W})$ be a Harish-Chandra sub-pair of $(\\mathsf {G}, \\mathsf {V})$ that represents a normal group super-subscheme of $\\mathbb {G}$ with $\\mathsf {H}\\le \\mathsf {R}$ .",
"Since $\\mathsf {H}$ acts trivially on $\\mathsf {V}/\\mathsf {W}$ , we have $ [\\mathsf {W}, \\mathsf {V}, \\mathsf {V}]\\subseteq [\\mathrm {Lie}(\\mathsf {H}), \\mathsf {V}]\\subseteq \\mathsf {W},$ that is, $\\mathsf {W}$ is $\\mathsf {R}$ -subordinated.",
"Thus $\\mathsf {W}\\subseteq \\mathsf {W}_{\\mathsf {R}}$ and $\\mathsf {H}\\le \\mathsf {H}_{\\mathsf {R}}$ .",
"To complete the proof, it remains to show that $[\\mathsf {W}_{\\mathsf {R}}, \\mathsf {V}]\\subseteq \\mathrm {Lie}(\\mathsf {H_R})$ .",
"Recall that $\\mathrm {Lie}(\\mathsf {H}_{\\mathsf {R}})=\\lbrace x\\in \\mathrm {Lie}(\\mathsf {R})\\mid [x, \\mathsf {V}]\\subseteq \\mathsf {W}_{\\mathsf {R}}\\rbrace ,$ hence $[\\mathsf {W}_{\\mathsf {R}}, \\mathsf {V}]\\subseteq \\mathrm {Lie}(\\mathsf {H_R})$ if and only if $[\\mathsf {W}_{\\mathsf {R}}, \\mathsf {V}, \\mathsf {V}]\\subseteq \\mathsf {W}_{\\mathsf {R}}$ .",
"The lemma is proven.",
"Remark 12.2 The submodule $\\mathsf {W}_{\\mathsf {R}}$ can be also defined as $ \\mathsf {W}_{\\mathsf {R}}=\\lbrace w\\in \\mathsf {V}\\mid \\mbox{for any odd positive integer} \\ n, \\mbox{ there is } \\ [w, \\underbrace{\\mathsf {V}, \\ldots , \\mathsf {V}}_{n-\\mbox{times}}]\\subseteq \\mathrm {Lie}(\\mathsf {R})\\rbrace .$ Corollary 12.3 Every algebraic group superscheme $\\mathbb {G}$ contains the largest connected normal affine group super-subscheme, denoted by $\\mathbb {G}_{aff}$ .",
"The group super-subschemes $\\mathbb {G}_{aff}$ can be characterized by the property that $\\mathbb {G}/\\mathbb {G}_{aff}$ does not contain non-trivial normal connected affine group super-subschemes.",
"Let $\\mathsf {G}_{aff}$ denote the largest connected normal affine group subscheme of $\\mathsf {G}$ (cf.",
"[9], Proposition 8.1).",
"Then the sub-pair $(\\mathsf {H}^0_{\\mathsf {G}_{Aff}}, \\mathsf {W}_{\\mathsf {G}_{Aff}})$ represents $\\mathbb {G}_{aff}$ , i.e., $\\mathbb {G}_{aff}$ is nothing else but the connected component of the $\\mathsf {G}_{aff}$ -radical of $\\mathbb {G}$ .",
"The second statement is now apparent.",
"Let $\\mathbb {G}$ be an algebraic group superscheme, represented by a Harish-Chandra pair $(\\mathsf {G}, \\mathsf {V})$ .",
"For every group subscheme $\\mathsf {S}$ of $\\mathsf {G}$ , let $\\mathsf {V}_{\\mathsf {S}}$ denote the smallest $\\mathsf {G}$ -submodule of $\\mathsf {V}$ such that $\\mathsf {S}\\le \\ker (\\mathsf {G}\\rightarrow \\mathrm {GL}(\\mathsf {V}/\\mathsf {V}_{\\mathsf {S}}))$ .",
"Lemma 12.4 Let $\\mathsf {R}$ be a normal group subscheme of $\\mathsf {G}=\\mathbb {G}_{ev}$ .",
"Then $\\mathbb {G}$ is $\\mathsf {R}$ -semisimple if and only if the following conditions hold: For any normal group subscheme $\\mathsf {S}$ of $\\mathbb {G}$ such that $\\mathsf {S}\\le \\mathsf {R}$ , there is $[\\mathsf {V}, \\mathsf {V}_{\\mathsf {S}}]\\lnot \\subseteq \\mathrm {Lie}(\\mathsf {S})$ ; There is no non-zero $\\mathsf {G}$ -submodule $\\mathsf {W}$ of $\\mathsf {V}$ such that $[\\mathsf {V}, \\mathsf {W}]=0$ .",
"A connected algebraic group superscheme $\\mathbb {G}$ is called pseudoabelian if $\\mathbb {G}_{aff}=\\mathbb {E}$ .",
"Let $\\mathbb {G}$ be a pseudoabelian group superscheme and $\\mathbb {G}\\ne \\mathbb {E}$ .",
"Then $\\mathbb {G}_{ev}=\\mathsf {G}$ is not affine.",
"Recall that $\\mathsf {G}^{aff}$ denotes the largest affine quotient of $\\mathsf {G}$ .",
"Set $\\mathsf {N}=\\ker (\\mathsf {G}\\rightarrow \\mathsf {G}^{aff})$ .",
"Then $\\mathsf {N}$ is a nontrivial anti-affine algebraic group, hence smooth and connected normal group subscheme of positive dimension (cf.",
"[9], Corollary 8.14 and Proposition 8.37).",
"Since the natural morphism $\\mathsf {G}\\rightarrow \\mathrm {GL}(\\mathsf {V})$ factors through $\\mathsf {G}\\rightarrow \\mathsf {G}^{aff}$ , $\\mathsf {N}$ acts trivially on $\\mathsf {V}$ .",
"Besides, by Corollary 8.13, [9], $\\mathsf {N}$ is central in $\\mathsf {G}$ .",
"Lemma 12.5 The group scheme $\\mathsf {N}$ is an abelian group variety such that $\\mathsf {G}=\\mathsf {G}_{aff}\\mathsf {N}$ .",
"The sub-pair $(\\mathsf {N}_{aff}, 0)$ represents a connected normal affine group super-subscheme of $\\mathbb {G}$ .",
"Thus, $\\mathsf {N}_{aff}=\\mathsf {E}$ , and by Theorem 8.28, [9], $\\mathsf {N}$ is an abelian group variety.",
"The group scheme $\\mathsf {G}/\\mathsf {G}_{aff}\\mathsf {N}$ is a quotient of both $\\mathsf {G}/\\mathsf {G}_{aff}$ and $\\mathsf {G}/\\mathsf {N}$ .",
"Recall that $\\mathsf {G}/\\mathsf {G}_{aff}$ is an abelian variety by the same Theorem 8.28, [9].",
"Therefore, $\\mathsf {G}/\\mathsf {G}_{aff}\\mathsf {N}$ is a complete and affine connected scheme simultaneously, hence trivial (cf.",
"[9], A.75(g)).",
"The following example shows that the class of pseudoabelian group superschemes is extensive.",
"Example 12.6 For every connected algebraic group scheme $\\mathsf {G}$ such that $\\mathsf {N}=\\ker (\\mathsf {G}\\rightarrow \\mathsf {G}^{aff})$ is an abelian variety, and $\\mathsf {G}=\\mathsf {G}_{aff}\\mathsf {N}$ , there exists a pseudoabelian group superscheme $\\mathbb {G}$ with $\\mathbb {G}_{ev}=\\mathsf {G}$ .",
"Let $\\mathsf {W}$ be a faithful $\\mathsf {G}_{aff}/(\\mathsf {G}_{aff}\\cap \\mathsf {N})$ -module and $\\mathsf {V}=\\mathsf {W}\\oplus \\mathsf {W}^*$ .",
"Then $\\mathsf {V}$ has the natural structure of $\\mathsf {G}$ -module for the diagonal action of $\\mathsf {G}_{aff}$ and the trivial action of $\\mathsf {N}$ .",
"Furthermore, we have a bilinear symmetric $\\mathsf {G}$ -equivariant map $\\mathsf {V}\\times \\mathsf {V}\\rightarrow \\mathrm {Lie}(\\mathsf {G})$ defined by the rule $[v, w]=[\\phi , \\psi ]=0, [\\phi , v]=\\phi (v)x \\text{ for } v, w\\in \\mathsf {W} \\text{ and } \\phi , \\psi \\in \\mathsf {W}^*,$ where $x\\in \\mathrm {Lie}(\\mathsf {N})\\setminus 0$ .",
"Then $(\\mathsf {G}, \\mathsf {V})$ is a Harish-Chandra pair representing a group superscheme $\\mathbb {G}$ .",
"By Lemma REF , the group superscheme $\\mathbb {G}$ is pseudoabelian.",
"The following theorem is a super-version of well-known Barsotti-Chevalley theorem (cf.",
"[9], Theorem 8.27).",
"Theorem 12.7 Let $\\mathbb {G}$ be a connected algebraic group superscheme.",
"Then there are normal group super-subschemes $\\mathbb {G}_1$ and $\\mathbb {G}_2$ of $\\mathbb {G}$ such that $\\mathbb {G}_1$ is affine and connected, $\\mathbb {G}_2/\\mathbb {G}_1$ is a (purely-even) abelian group variety, and $\\mathbb {G}/\\mathbb {G}_2$ is affine.",
"Set $\\mathbb {G}_1=\\mathbb {G}_{aff}$ .",
"Then $\\mathbb {H}=\\mathbb {G}/\\mathbb {G}_1$ is a pseudoabelian group superscheme represented by a Harish-Chandra pair $(\\mathsf {H}, \\mathsf {V})$ .",
"As above, let $\\mathsf {N}$ denote the kernel $\\ker (\\mathsf {H}\\rightarrow \\mathsf {H}^{aff})$ .",
"Then $(\\mathsf {N}, 0)$ represents a normal (purely-even) group subvariety $\\mathbb {N}$ of $\\mathbb {H}$ .",
"Moreover, Lemma REF implies that $\\mathbb {H}/\\mathbb {N}$ is an affine group superscheme.",
"We define $\\mathbb {G}_2$ as $\\pi ^{-1}(\\mathbb {N})$ , where $\\pi : \\mathbb {G}\\rightarrow \\mathbb {G}/\\mathbb {G}_1$ is the quotient morphism.",
"Corollary 12.8 Let $\\mathbb {G}$ be an abelian group supervariety.",
"Then $\\mathbb {G}_{ev}\\unlhd \\mathbb {G}$ , and $\\mathbb {G}_{ev}$ is an abelian group supervariety; $\\mathbb {G}_{aff}\\cap \\mathbb {G}_{ev}=\\mathbb {E}$ , hence $\\mathbb {G}_{aff}$ is a (purely-odd) affine unipotent group superscheme; $\\mathbb {G}/(\\mathbb {G}_{aff}\\times \\mathbb {G}_{ev})$ is a (purely-odd) affine unipotent group superscheme.",
"We have seen that $\\mathbb {G}$ is an abelian group supervariety if and only if $\\mathsf {G}=\\mathbb {G}_{res}$ is an abelian group variety.",
"Thus $\\mathsf {G}$ acts trivially on $\\mathsf {V}=\\mathrm {Lie}(\\mathbb {G})_1$ and $(\\mathbb {G}_1)_{ev}$ is trivial, which implies the first and second statements.",
"The third statement is clear.",
"Remark 12.9 If $\\mathbb {G}$ is an abelian group supervariety, then the group super-subscheme $\\mathbb {G}_{aff}$ is represented by the pair $(\\mathsf {E}, \\mathsf {W})$ , where $\\mathsf {W}=\\lbrace w\\in \\mathsf {V}\\mid [w, \\mathsf {V}]=0\\rbrace $ ."
],
[
"Anti-affine group superschemes",
"An algebraic group superscheme $\\mathbb {G}$ is called anti-affine whenever $\\mathcal {O}(G)$ is a Grassman-like superalgebra.",
"Set $\\Phi (\\mathbb {G})=(\\mathsf {G}, \\mathsf {V})$ .",
"Theorem REF implies that $\\mathbb {G}$ is anti-affine if and only if the algebraic group scheme $\\mathsf {G}$ is (cf.",
"[9], page 39).",
"If $\\mathsf {N}=\\ker (\\mathsf {G}\\rightarrow \\mathsf {G}^{aff})$ , then $(\\mathsf {N}, 0)$ represents a central anti-affine group super-subscheme $\\mathbb {N}$ of $\\mathbb {G}$ such that $\\mathbb {G}/\\mathbb {N}\\simeq \\mathbb {G}^{aff}$ .",
"But an anti-affine group super-subscheme of $\\mathbb {G}$ is not necessarily central in $\\mathbb {G}$ , contrary to the purely-even case (cf.",
"[9], Corollary 8.13).",
"For example, if $\\mathbb {G}$ is a psedoabelian group superscheme from Example REF , then its group super-subscheme, represented by the pair $(\\mathsf {N}, \\mathsf {V})$ , is anti-affine but not central.",
"Similarly, an anti-affine group superscheme is no longer commutative, but it is always nilpotent.",
"If $\\mathbb {G}$ is anti-affine, then $\\mathbb {G}_{ev}$ is central, and $\\mathbb {G}/\\mathbb {G}_{ev}$ is a purely-odd unipotent group superscheme."
],
[
"A reduction",
"Let $\\mathbb {G}$ be an algebraic group superscheme, and $\\mathbb {H}$ its group super-subscheme.",
"They are represented by Harish-Chandra pairs $(\\mathsf {G}, \\mathsf {V})$ and $(\\mathsf {H}, \\mathsf {W})$ , respectively.",
"Recall that if $\\mathbb {H}_1$ and $\\mathbb {H}_2$ are group super-subschemes of $\\mathbb {G}$ such that $\\mathbb {H}_1\\le \\mathbb {H}_2$ , $\\mathbb {G}/\\mathbb {H}_2$ is a superscheme, and $\\mathbb {H}_2/\\mathbb {H}_1$ is an affine superscheme, then $\\mathbb {G}/\\mathbb {H}_1$ is a superscheme as well (see Remark 9.11, [14]).",
"As above, let $\\mathsf {N}$ denote $\\ker (\\mathsf {G}\\rightarrow \\mathsf {G}^{aff})$ .",
"Proposition 13.1 The following statements are equivalent: $\\mathbb {G}/\\mathbb {H}$ is a superscheme for arbitrary $\\mathbb {H}$ .",
"$\\mathbb {G}/\\mathbb {H}$ is a superscheme, provided $\\mathsf {G}=\\mathsf {M}\\times \\mathsf {N}$ , where $\\mathsf {M}$ is an affine group subscheme of $\\mathsf {G}$ such that $\\mathsf {H}\\le \\mathsf {M}$ .",
"All we need is to show that $(2)$ implies $(1)$ .",
"Let us start with two elementary observations.",
"First, suppose $\\mathsf {S}$ is a group sub-scheme of $\\mathsf {N}$ .",
"In that case, the pair $(\\mathsf {S}, 0)$ represents a central group super-subscheme of $\\mathbb {G}$ .",
"Besides, the group superscheme $\\mathbb {G}/\\mathbb {S}$ is represented by the pair $(\\mathsf {G}/\\mathsf {S}, \\mathsf {V})$ such that $\\ker (\\mathsf {G}/\\mathsf {S}\\rightarrow (\\mathsf {G}/\\mathsf {S})^{aff})=\\mathsf {N}/\\mathsf {S}$ , i.e., $(\\mathsf {G}/\\mathsf {S})^{aff}$ is naturally isomorphic to $\\mathsf {G}^{aff}$ .",
"Second, if $\\mathsf {R}$ is an affine normal group subscheme of $\\mathsf {G}$ , then $(\\mathsf {G}/\\mathsf {R})_{aff}=\\mathsf {G}_{aff}\\mathsf {R}/\\mathsf {R}$ .",
"Every normal affine connected group subscheme of $\\mathsf {G}/\\mathsf {R}$ has the form $\\mathsf {L}/\\mathsf {R}$ , where $\\mathsf {L}$ is a normal affine group subscheme of $\\mathbb {G}$ (use Proposition 8.1, [9]).",
"By Proposition 1.52, [9], $\\mathsf {L}^0$ is normal, hence $\\mathsf {L}^0\\le \\mathsf {G}_{aff}$ .",
"Since $(\\mathsf {L}/\\mathsf {R})/(\\mathsf {L}^0\\mathsf {R}/\\mathsf {R})\\simeq \\mathsf {L}/\\mathsf {L}^0\\mathsf {R}$ is etale and $\\mathsf {L}^0\\mathsf {R}/\\mathsf {R}$ is connected, we have $\\mathsf {L}/\\mathsf {R}=\\mathsf {L}^0\\mathsf {R}/\\mathsf {R}\\le \\mathsf {G}_{aff}\\mathsf {R}/\\mathsf {R}$ .",
"By the first observation, $(\\mathsf {H}\\cap \\mathsf {N}, 0)$ represents a central group super-subscheme $\\mathbb {L}$ of $\\mathbb {G}$ such that $\\mathbb {L}\\le \\mathbb {H}$ .",
"Since $\\mathbb {G}/\\mathbb {H}\\simeq (\\mathbb {G}/\\mathbb {L})/(\\mathbb {H}/\\mathbb {L})$ , without loss of generality, one can assume that $\\mathsf {H}\\cap \\mathsf {N}=\\mathsf {E}$ .",
"In particular, $\\mathsf {H}$ is a group subscheme of $\\mathsf {G}^{aff}$ , hence affine.",
"Let $\\mathsf {R}$ denote the group scheme $\\mathsf {G}_{aff}\\mathsf {H}\\cap \\mathsf {N}$ .",
"The second observation implies that $(\\mathsf {G}/\\mathsf {R})_{aff}=\\mathsf {G}_{aff}\\mathsf {R}/\\mathsf {R}$ and $(\\mathsf {G}_{aff}\\mathsf {H})/\\mathsf {R}\\cap \\mathsf {N}/\\mathsf {R}=\\mathsf {E}$ .",
"Since $\\mathbb {H}\\mathbb {R}/\\mathbb {H}\\simeq \\mathbb {R}$ is affine, $\\mathbb {G}/\\mathbb {H}$ is a superscheme provided $\\mathbb {G}/\\mathbb {H}\\mathbb {R}\\simeq (\\mathbb {G}/\\mathbb {R})/(\\mathbb {H}\\mathbb {R}/\\mathbb {R})$ is.",
"Therefore, one can assume that $\\mathsf {G}_{aff}\\mathsf {H}\\cap \\mathsf {N}=\\mathsf {E}$ .",
"Arguing as in Lemma REF , one can show that $\\mathsf {N}$ is an abelian variety and $\\mathsf {G}^0=\\mathsf {G}_{aff}\\mathsf {N}$ .",
"In particular, $\\mathsf {N}$ can be identified with $(\\mathsf {G}/\\mathsf {G}_{aff}\\mathsf {H})^0$ .",
"By Theorem 1.1, [2], there is a finite, hence affine, group subscheme $\\mathsf {F}$ of $\\mathsf {G}/\\mathsf {G}_{aff}\\mathsf {H}$ such that $\\mathsf {F}\\mathsf {N}=\\mathsf {G}/\\mathsf {G}_{aff}\\mathsf {H}$ .",
"The inverse image of $\\mathsf {F}$ in $\\mathsf {G}$ is an affine group subscheme $\\mathsf {M}$ such that $\\mathsf {G}=\\mathsf {M}\\mathsf {N}$ .",
"Repeating the above arguments with $\\mathsf {R}=\\mathsf {M}\\cap \\mathsf {N}$ , one can reduce the general case to $\\mathsf {M}\\cap \\mathsf {N}=\\mathsf {E}$ .",
"The proposition is proven."
],
[
"Sheaf quotient",
"Let $\\mathbb {G}$ be an algebraic group superscheme, represented by a Harish-Chandra pair $(\\mathsf {G}, \\mathsf {V})$ .",
"Let $\\mathbb {H}$ be its closed group super-subscheme, represented by a sub-pair $(\\mathsf {H}, \\mathsf {W})$ .",
"In this section, we prove the following theorem.",
"Theorem 13.2 A sheaf quotient $\\mathbb {G}/\\mathbb {H}$ is a superscheme of finite type.",
"The proof will be given in the series of lemmas.",
"Without loss of generality, one can assume that $\\mathbb {G}$ and $\\mathbb {H}$ satisfy the condition $(2)$ in Proposition REF .",
"Since $\\mathsf {X}=\\mathsf {G}/\\mathsf {H}$ is a scheme of finite type (see Theorem 7.35 and Remark 7.36 from [9]), any open affine subscheme $\\mathsf {U}^{\\prime }$ of $\\mathsf {G}/\\mathsf {H}$ has a form $\\mathsf {U}/\\mathsf {H}$ , where $\\mathsf {U}$ is an open affine $\\mathsf {H}$ -saturated subscheme of $\\mathsf {G}$ (cf.",
"[8], I.5.7.1(1)).",
"Set $\\mathrm {Sp}(A)=\\mathsf {U}$ and $\\mathrm {Sp}(D)=\\mathsf {H}$ .",
"Then $A^{\\prime }=\\mathcal {O}(\\mathsf {U}^{\\prime })\\simeq A^D$ .",
"Recall that the superschemes $\\mathbb {G}$ and $\\mathbb {H}$ can be represented as $\\mathsf {G}{\\bf E}$ and $\\mathsf {H}{\\bf E}^{\\prime }$ , respectively.",
"Lemma 13.3 The subfunctor $\\mathbb {U}=\\mathsf {U}{\\bf E}={\\bf E}\\mathsf {U}$ is an open affine $\\mathbb {H}$ -saturated super-subscheme of $\\mathbb {G}$ .",
"It is clear that $\\mathsf {U}{\\bf E}$ is open in $\\mathbb {G}$ .",
"Further, we have $\\mathbb {U}=\\mathsf {U}{\\bf E}=\\mathsf {U}\\mathsf {S}{\\bf E}=\\mathsf {U}\\mathbb {S}$ .",
"Since $\\mathbb {S}$ is a normal group subfunctor, we also have $\\mathbb {U}=\\mathbb {S}\\mathsf {U}={\\bf E}\\mathsf {S}\\mathsf {U}={\\bf E}\\mathsf {U}$ .",
"Finally, for every $hx\\in \\mathbb {H}$ , where $h\\in \\mathsf {H}$ and $x\\in {\\bf E}^{\\prime }$ , there is $\\mathbb {U}(hx)=\\mathsf {U}\\mathbb {S}(hx)=(\\mathsf {U}h) (\\mathbb {S}^{h^{-1}} x)\\subseteq \\mathsf {U}\\mathbb {S}=\\mathbb {U}.$ The lemma is proven.",
"Set $\\mathrm {SSp}(\\mathbb {A})=\\mathbb {U}$ and $\\mathrm {SSp}(\\mathbb {D})=\\mathbb {H}$ .",
"Let $\\tau : \\mathbb {A}\\rightarrow \\mathbb {A}\\otimes \\mathbb {D}$ be the corresponding $\\mathbb {D}$ -coideal superalgebra map.",
"Recall that $\\mathsf {G}=\\mathsf {M}\\times \\mathsf {N}$ , where $\\mathsf {H}\\le \\mathsf {M}$ and $\\mathsf {M}$ is affine.",
"Thus $\\mathsf {U}$ can be chosen so that $\\mathsf {U}=\\mathsf {U}_{aff}\\times \\mathsf {U}_{ab}$ , where $\\mathsf {U}_{aff}$ is an open affine subscheme of $\\mathsf {M}$ and $\\mathsf {U}_{ab}$ is an open affine subscheme of $\\mathsf {N}$ .",
"Moreover, the algebra $A$ is isomorphic to $A_{aff}\\otimes A_{ab}$ as a $D$ -comodule, where $A_{aff}\\simeq K[\\mathsf {U}_{aff}], A_{ab}\\simeq K[\\mathsf {U}_{ab}]$ , and $A_{ab}$ is a trivial $D$ -comodule.",
"Following notations from [13], one can define a morphism of $\\mathbb {D}$ -coideal superalgebras $\\omega _{\\theta } : \\mathbb {A}\\stackrel{\\tau ^{\\prime }}{\\rightarrow } (\\wedge (\\mathsf {V^*})\\otimes A)\\square _D\\mathbb {D}\\stackrel{\\wedge (\\theta ^{\\prime })\\square _D\\mathrm {id}_{\\mathbb {D}}}{\\rightarrow } (A\\otimes \\wedge (\\mathsf {Z}))\\square _{D}\\mathbb {D},$ where $\\mathsf {Z}=\\ker (\\mathsf {V^*}\\rightarrow \\mathsf {W^*})$ .",
"The first morphism makes sense since $\\tau $ factors through the natural embedding $\\mathbb {A}\\square _D\\mathbb {D}\\subseteq \\mathbb {A}\\otimes \\mathbb {D}$ .",
"The proofs of Lemma 4.1, Proposition 4.2, and Lemma 4.4 in [13] can be repeated verbatim.",
"Only the definition of $\\theta ^{\\prime }$ and the proof of Lemma 4.6, [13] needs a commentary.",
"The structure of the right $\\mathsf {G}$ -module of $\\mathsf {V}^*$ is completely defined by the structure of $\\mathsf {M}$ -module because $\\mathsf {N}$ acts trivially on $\\mathsf {V}^*$ .",
"Therefore, one can define the map $\\kappa =\\kappa _A : A\\otimes \\mathsf {V}^*\\rightarrow \\mathsf {V}^*\\otimes A$ by $a_1\\otimes a_2\\otimes v^*\\mapsto v^*_{(0)}\\otimes a_1 j(v^*_{(1)})\\otimes a_2,$ where $a_1\\in A_{aff}, a_2\\in A_{ab}$ , $v^*\\mapsto v_{(0)}\\otimes v^*_{(1)}$ is the corresponding comodule map $\\mathsf {V}^*\\rightarrow \\mathsf {V}^*\\otimes K[\\mathsf {M}]$ , and $j : K[\\mathsf {M}]\\rightarrow A_{aff}$ is dual to the open immersion $\\mathsf {U}_{aff}\\rightarrow \\mathsf {M}$ .",
"We will show that $\\omega _{\\theta }$ is an isomorphism.",
"Remark 13.4 The definition of $\\omega _{\\theta }$ is consistent with a base change, and once needed, $K$ can be replaced by a suitable field extension $L\\supseteq K$ .",
"It has been proven that $\\mathsf {gr}(\\mathbb {G})\\simeq \\mathsf {G}\\ltimes \\mathbb {G}_{odd}$ and $\\mathsf {gr}(\\mathbb {H})\\simeq \\mathsf {H}\\ltimes \\mathbb {H}_{odd}$ , where $\\mathbb {G}_{odd}\\simeq \\mathrm {SSp}(\\Lambda (\\mathsf {V}^*))$ , $\\mathbb {H}_{odd}\\simeq \\mathrm {SSp}(\\Lambda (\\mathsf {W}^*))$ and $\\mathsf {V}^*$ and $\\mathsf {W}^*$ consist of primitive elements.",
"Using Proposition REF and arguing as in Proposition REF , one sees that $\\mathsf {gr}(\\mathbb {U})\\simeq \\mathrm {SSp}(\\mathsf {gr}(\\mathbb {A}))$ is an open $\\mathsf {gr}(\\mathbb {H})$ -saturated super-subscheme of $\\mathsf {gr}(\\mathbb {G})$ .",
"Proposition 13.5 The following statements hold: $\\mathsf {gr}(\\mathbb {U})\\simeq \\mathsf {U}\\mathbb {G}_{odd}=\\mathbb {G}_{odd}\\mathsf {U}$ .",
"As the right and left $D$ -coideal superalgebras, $\\mathsf {gr}(\\mathbb {A})$ and $\\mathsf {gr}(\\mathbb {D})$ are naturally isomorphic to $\\mathbb {A}$ and $\\mathbb {D}$ , respectively.",
"$\\mathsf {gr}(\\tau ^{\\prime })=\\mathsf {gr}(\\tau )^{\\prime }$ .",
"The geometric counterpart of $\\mathsf {U}\\mathbb {G}_{odd}$ is the open super-subscheme $q_G^{-1}(U_{ev})\\simeq U_{ev}\\times G_{odd}$ , where $U$ is the geometric counterpart of $\\mathbb {U}$ in $G$ .",
"Since $\\mathsf {gr}(U)^e=U^e=U_{ev}^e=(q_G^{-1}(U_{ev}))^e$ , the condition $(1)$ follows.",
"Further, the right $\\mathsf {gr}(\\mathbb {D})$ -coideal superalgebra structure on $\\mathsf {gr}(\\mathbb {A})$ is defined by a superalgebra morphism $\\mathsf {gr}(\\mathbb {A})\\stackrel{\\mathsf {gr}(\\tau )}{\\rightarrow } \\mathsf {gr}(\\mathbb {A}\\otimes \\mathbb {D})\\simeq \\mathsf {gr}(\\mathbb {A})\\otimes \\mathsf {gr}(\\mathbb {D})$ .",
"More precisely, for an element $a\\in I_{\\mathbb {A}}^n$ , let $\\tau (a)=\\sum _{0\\le k\\le n} a_{(0), k}\\otimes a_{(1), n-k}$ be adapted to the $I_{\\mathbb {A}\\otimes \\mathbb {D}}$ -adic filtration.",
"That is, $a_{(0), k}\\in I^k_{\\mathbb {A}}$ and $a_{(1), n-k}\\in I^{n-k}_{\\mathbb {D}}$ for every $0\\le k\\le n$ .",
"Then $\\mathsf {gr}(\\tau )$ is defined by $a+I_{\\mathbb {A}}^{n+1}\\mapsto \\sum _{0\\le k\\le n} (a_{(0), k}+I_{\\mathbb {A}}^{k+1})\\otimes (a_{(1), n-k}+I_{\\mathbb {D}}^{n-k+1}).$ It follows that the right $D$ -coideal superalgebra structure of $\\mathsf {gr}(\\mathbb {A})$ is defined as $a+I_{\\mathbb {A}}^{n+1}\\mapsto (a_{(0), n}+I_{\\mathbb {A}}^{n+1})\\otimes (a_{(1), 0}+I_{\\mathbb {D}}),$ where $a\\mapsto a_{(0), n}\\otimes (a_{(1), 0}+I_{\\mathbb {D}})$ is the right $D$ -coideal superalgebra map of $\\mathbb {A}$ .",
"On the other hand, if we identify $\\mathbb {A}$ with $\\Lambda (\\mathsf {V}^*)\\otimes A$ , then the latter map is $x\\otimes a\\mapsto x\\otimes a_{(0)}\\otimes a_{(1)}$ , where $x\\in \\Lambda (\\mathsf {V}^*), a\\in A$ and $a\\mapsto a_{(0)}\\otimes a_{(1)}$ is the $D$ -coideal algebra map of $A$ .",
"Moreover, we have $I_{\\mathbb {A}}^k=(\\oplus _{s\\ge k}\\Lambda ^s(\\mathsf {V}^*))\\otimes A$ for $0\\le k\\le \\dim \\mathsf {V}$ .",
"Then $\\mathbb {A}$ and $\\mathsf {gr}(\\mathbb {A})$ are isomorphic as $D$ -coideal superalgebras.",
"The case of $\\mathbb {D}$ is similar.",
"Furthermore, $\\mathbb {A}\\square _D\\mathbb {D}$ is isomorphic to $\\Lambda (\\mathsf {V}^*)\\otimes B\\otimes \\Lambda (\\mathsf {W}^*)$ , where $B=\\lbrace a_{(0)}\\otimes a_{(1)}\\mid a\\in A\\rbrace \\subseteq A\\otimes D .$ In particular, $I^k_{\\mathbb {A}\\square _D\\mathbb {D}}=\\mathbb {A}\\square _D\\mathbb {D}\\cap I^k_{\\mathbb {A}\\otimes \\mathbb {D}}$ for $0\\le k\\le \\dim \\mathsf {V}+\\dim \\mathsf {W}$ , and we obtain a natural isomorphism $\\mathsf {gr}(\\mathbb {A}\\square _D\\mathbb {D})\\simeq \\mathsf {gr}(\\mathbb {A})\\square _D\\mathsf {gr}(\\mathbb {D})$ that makes the diagram $\\begin{array}{ccccc}\\mathsf {gr}(\\mathbb {A}) & \\stackrel{\\mathsf {gr}(\\tau ^{\\prime })}{\\rightarrow } & \\mathsf {gr}(\\mathbb {A}\\square _D\\mathbb {D}) & \\rightarrow & \\mathsf {gr}(\\mathbb {A}\\otimes \\mathbb {D}) \\\\\\parallel & & \\downarrow & & \\downarrow \\\\\\mathsf {gr}(\\mathbb {A}) & \\stackrel{\\mathsf {gr}(\\tau )^{\\prime }}{\\rightarrow } & \\mathsf {gr}(\\mathbb {A})\\square _D\\mathsf {gr}(\\mathbb {D}) & \\rightarrow & \\mathsf {gr}(\\mathbb {A})\\otimes \\mathsf {gr}(\\mathbb {D})\\end{array}$ commutative.",
"The proposition is proven.",
"Lemma 13.6 The morphism $\\mathsf {gr}(\\wedge (\\theta ^{\\prime })\\square _D\\mathrm {id}_{\\mathbb {D}})$ is identified with $\\wedge (\\theta ^{\\prime })\\square _D\\mathrm {id}_{\\mathsf {gr}(\\mathbb {D})}$ .",
"Recall that the morphism $\\wedge (\\theta ^{\\prime })$ is defined as $v^*_1\\wedge \\ldots \\wedge v^*_k\\otimes a\\mapsto \\sum _{i_1, \\ldots , i_k} a_{i_1}\\ldots a_{i_k}a\\otimes z_{i_1}\\wedge \\ldots \\wedge z_{i_k},$ where $\\theta ^{\\prime }(v^*_s)=\\sum _{i_s} a_{i_s}\\otimes z_{i_s}$ for $1\\le s\\le k$ .",
"Moreover, the morphism $\\wedge (\\theta ^{\\prime })\\square \\mathrm {id}$ is defined as $v^*_1\\wedge \\ldots \\wedge v^*_k\\otimes a_{(0)}\\otimes a_{(1)}\\otimes \\wedge w^*_1\\wedge \\ldots \\wedge w^*_l\\mapsto $ $\\sum _{i_1, \\ldots , i_k} a_{i_1}\\ldots a_{i_k}a_{(0)}\\otimes z_{i_1}\\wedge \\ldots \\wedge z_{i_k}\\otimes a_{(1)}\\otimes \\wedge w^*_1\\wedge \\ldots \\wedge w^*_l,$ and is compatible with the corresponding filtrations of the superalgebras $(\\wedge (\\mathsf {V}^*)\\otimes A)\\square _D\\mathbb {D}$ and $(A\\otimes \\wedge (\\mathsf {Z}))\\square _{D}\\mathbb {D}$ .",
"The following lemma is now obvious.",
"Lemma 13.7 The morphism $\\mathsf {gr}(\\omega _{\\theta })$ can be naturally identified with $\\mathsf {gr}(\\mathbb {A})\\stackrel{\\mathsf {gr}(\\tau )^{\\prime }}{\\rightarrow } (\\wedge (\\mathsf {V}^*)\\otimes A)\\square _D\\mathsf {gr}(\\mathbb {D})\\stackrel{\\wedge (\\theta ^{\\prime })\\square _D\\mathrm {id}_{\\mathsf {gr}(\\mathbb {D})}}{\\rightarrow } (A\\otimes \\wedge (\\mathsf {Z}))\\square _{D}\\mathsf {gr}(\\mathbb {D}).$ Let $\\theta $ be a retract of the inclusion $A\\otimes \\mathsf {Z}\\rightarrow A\\otimes \\mathsf {V}^*$ from Lemma 4.4 , [13].",
"Let $\\mathsf {V}^*=\\mathsf {Z}\\oplus \\mathsf {T}$ , and elements $t_j$ form a basis of $\\mathsf {T}$ for $1\\le j\\le s=\\dim \\mathsf {W}^*$ .",
"The free $A$ -module $\\ker \\theta $ has a basis consisting of the elements $t^{\\prime }_j$ such that $t^{\\prime }_j = t_j\\mathsf {}\\pmod {A\\otimes \\mathsf {Z}}$ .",
"Choose a closed point $x$ in $\\mathsf {U}$ .",
"By Remark REF , one can assume that $x\\in \\mathsf {U}(K)$ .",
"That is, $x$ is an algebra morphism $A\\rightarrow K$ .",
"The natural embedding $\\mathbb {H}_{odd}\\rightarrow \\mathsf {U}\\times \\mathbb {G}_{odd}\\simeq \\mathsf {gr}(\\mathbb {U})$ , defined as $h\\mapsto (x, h)$ , is dual to the superalgebra morphism induced by $x$ and $\\mathsf {V}^*\\rightarrow \\mathsf {W}^*$ .",
"The natural (right-hand side) action of group $\\mathbb {H}_{odd}$ on $\\mathsf {gr}(\\mathbb {U})$ is defined by a coideal superalgebra map $z\\mapsto z\\otimes 1 \\text{ and } t^{\\prime }_j\\mapsto t^{\\prime }_j\\otimes 1+1\\otimes \\overline{t_j} \\text{ for } z\\in \\mathsf {Z},$ where $\\overline{t_j}$ is the image of $t_j$ in $\\mathsf {W}^*$ .",
"Finally, the $A$ -superalgebra morphism $\\wedge (\\theta ): A\\otimes \\Lambda (\\mathsf {V}^*)\\rightarrow A\\otimes \\Lambda (\\mathsf {Z})$ induces the embedding ${\\bf s} :\\mathsf {U}\\times \\mathrm {SSp}(\\Lambda (\\mathsf {Z}))\\rightarrow \\mathsf {gr}(\\mathbb {U})$ .",
"Lemma 13.8 There is a natural isomorphism $(\\mathsf {U}\\times \\mathrm {SSp}(\\Lambda (\\mathsf {Z})))\\times \\mathbb {H}_{odd}\\simeq \\mathsf {gr}(\\mathbb {U}),$ induced by the above two embeddings and the multiplication map.",
"Choose a basis $z_1, \\ldots , z_{t-s}$ of $\\mathsf {Z}$ .",
"We use notations from Proposition REF .",
"Any couple $(g, h)\\in (\\mathsf {U}\\times \\mathrm {SSp}(\\Lambda (\\mathsf {Z})))\\times \\mathbb {H}_{odd}$ is taken to a superalgebra morphism $(g, h)(a z_I t^{\\prime }_J)=\\sum _{S\\subseteq J} g(\\wedge (\\theta )(az^I t^{\\prime S}))h(\\overline{t}^{J\\setminus S})=g(az_I)h(\\overline{t}_J).$ The proof concludes by the observation that $A\\otimes \\Lambda (\\mathsf {V}^*)\\simeq (A\\otimes \\Lambda (\\mathsf {Z}))\\otimes \\Lambda (\\mathsf {T}^{\\prime })$ , where $\\mathsf {T}^{\\prime }$ is the $K$ -span of $t^{\\prime }_j$ for $1\\le j\\le s$ .",
"Lemma 13.9 The superalgebra morphism $\\mathsf {gr}(\\omega _{\\theta })$ is an isomorphism, and therefore $\\omega _{\\theta }$ is as well.",
"Consider the following sequence of superscheme morphisms $(\\star ) \\ \\mathsf {gr}(\\mathbb {U})\\stackrel{\\bf a}{\\leftarrow } \\mathsf {gr}(\\mathbb {U})\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})\\stackrel{\\bf b}{\\leftarrow } (\\mathsf {U}\\times \\mathbb {G}_{odd})\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})\\stackrel{\\bf c}{\\leftarrow } (\\mathsf {U}\\times \\mathrm {SSp}(\\Lambda (\\mathsf {Z})))\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})\\simeq $ $\\mathrm {SSp}((A\\otimes \\wedge (\\mathsf {Z}))\\square _{D}\\mathsf {gr}(\\mathbb {D})).$ The morphism $\\bf a$ is induced by $(u, h)\\mapsto uh$ for $u\\in \\mathsf {gr}(\\mathbb {U})$ and $h\\in \\mathsf {gr}(\\mathbb {H})$ .",
"Since $\\mathsf {gr}(\\mathbb {H})\\simeq \\mathsf {H}\\times \\mathbb {H}_{odd}$ , the superscheme $\\mathsf {gr}(\\mathbb {U})\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})$ is canonically isomorphic to $\\mathsf {gr}(\\mathbb {U})\\times \\mathbb {H}_{odd}$ , hence it is affine.",
"Lemma REF implies that $\\mathsf {gr}(\\mathbb {U})\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})\\simeq \\mathrm {SSp}(\\mathsf {gr}(\\mathbb {A})\\square _D\\mathsf {gr}(\\mathbb {D}))$ and the morphism $\\bf a$ is dual to $\\mathsf {gr}(\\tau )^{\\prime }$ .",
"Similarly, we have $(\\mathsf {U}\\times \\mathbb {G}_{odd})\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})\\simeq (\\mathsf {U}\\times \\mathbb {G}_{odd})\\times \\mathbb {H}_{odd}\\simeq \\mathrm {SSp}((\\Lambda (\\mathsf {V}^*)\\otimes A)\\square _D\\mathsf {gr}(\\mathbb {D})),$ and the isomorphism $\\bf b$ is dual to the superalgebra isomorphism induced by $\\kappa ^{-1}$ , $(\\mathsf {U}\\times \\mathrm {SSp}(\\Lambda (\\mathsf {Z})))\\times ^{\\mathsf {H}}\\mathsf {gr}(\\mathbb {H})\\simeq (\\mathbb {G}_{odd}\\times \\mathsf {U})\\times \\mathbb {H}_{odd}\\simeq \\mathrm {SSp}((A\\otimes \\Lambda (\\mathsf {Z}))\\square _D\\mathsf {gr}(\\mathbb {D}))$ and the morphism ${\\bf c}={\\bf s}\\times ^{\\mathsf {H}}\\mathrm {id}_{\\mathsf {gr}(\\mathbb {H})}$ is just $\\mathrm {SSp}(\\wedge (\\theta )\\square \\mathrm {id}_{\\mathsf {gr}(\\mathbb {D})})$ .",
"Thus the composition ${\\bf a}{\\bf b}{\\bf c}$ coincides with $\\mathrm {SSp}(\\mathsf {gr}(\\omega _{\\theta }))$ .",
"The above isomorphisms allow us to identify ${\\bf a}{\\bf b}{\\bf c}$ with the isomorphism from Lemma REF .",
"Thus, the lemma follows.",
"Lemma 13.10 The sheaf quotient $\\mathbb {U}/\\mathbb {H}$ is an affine superscheme.",
"By Theorem 3.1 (1), [13], we need to show that the superalgebra morphism $\\alpha : \\mathbb {A}\\otimes \\mathbb {A}\\rightarrow \\mathbb {A}\\otimes \\mathbb {D}$ given by $a\\otimes a^{\\prime }\\mapsto \\sum aa^{\\prime }_{(0)}\\otimes a^{\\prime }_{(1)}$ , is surjective.",
"Equivalently, the functor morphism $\\mathrm {SSp}(\\alpha ): \\mathbb {U}\\times \\mathbb {H}\\rightarrow \\mathbb {U}\\times \\mathbb {U}$ given by $(u, h)\\mapsto (u, uh),$ maps $\\mathbb {U}\\times \\mathbb {H}$ isomorphically onto a closed super-subscheme of $\\mathbb {U}\\times \\mathbb {U}$ .",
"Define a functor morphism ${\\bf p} : \\mathbb {G}\\times \\mathbb {G}\\rightarrow \\mathbb {G}$ by $(g, g^{\\prime })\\mapsto g^{-1}g^{\\prime }$ .",
"Then $\\mathrm {SSp}(\\alpha )(\\mathbb {U}\\times \\mathbb {H})=(\\mathbb {U}\\times \\mathbb {U})\\cap {\\bf p}^{-1}(\\mathbb {H})$ , hence it is closed in $\\mathbb {U}\\times \\mathbb {U}$ .",
"The lemma follows.",
"Let $\\mathsf {V}^{\\prime }$ be an open affine subscheme of $\\mathsf {U}^{\\prime }$ .",
"As above, $\\mathsf {V}^{\\prime }=\\mathsf {V}/\\mathsf {H}$ , where $\\mathsf {V}\\simeq \\mathrm {Sp}(B)$ is an affine subscheme of $\\mathsf {U}$ and $B^{\\prime }=\\mathcal {O}(\\mathsf {V}^{\\prime })\\simeq B^D$ .",
"Set $\\mathbb {V}=\\mathsf {V}{\\bf E}$ .",
"Then $\\mathbb {V}$ is an open affine $\\mathbb {H}$ -saturated super-subscheme of $\\mathbb {U}$ .",
"Lemma 13.11 $\\mathbb {V}/\\mathbb {H}$ is an open affine super-subscheme of $\\mathbb {U}/\\mathbb {H}$ .",
"The inclusion $\\mathsf {V}^{\\prime }\\subseteq \\mathsf {U}^{\\prime }$ is defined by a morphism $\\phi : A\\rightarrow B$ of $D$ -coideal algebras.",
"By Lemma 3.5, [14], there are elements $a_1, \\ldots , a_k\\in A^{\\prime }$ such that $\\sum _{1\\le i\\le k}B^{\\prime }\\phi (a_i)=B^{\\prime }$ , and for each $i$ , the induced morphism $A^{\\prime }_{a_i}\\rightarrow B^{\\prime }_{\\phi (a_i)}$ is an isomorphism.",
"Thus, $B=\\sum _{1\\le i\\le k}B\\phi (a_i)=B$ and $A_{a_i}\\simeq B_{\\phi (a_i)}$ for $1\\le i\\le k$ .",
"There is $\\mathbb {B}=\\mathcal {O}(\\mathbb {V})\\simeq (B\\otimes \\Lambda (\\mathsf {Z}))\\square _D\\mathbb {D}$ , and the morphism of $\\mathbb {D}$ -coideal superalgebras $\\mathbb {A}\\rightarrow \\mathbb {B}$ , dual to the inclusion $\\mathbb {V}\\subseteq \\mathbb {U}$ , is $(\\phi \\otimes \\mathrm {id}_{\\Lambda (\\mathsf {Z})})\\square _D\\mathrm {id}_{\\mathbb {D}}$ .",
"Moreover, by Proposition 4.2(1), [13], $\\mathbb {V}/\\mathbb {H}\\simeq \\mathrm {SSp}((B\\otimes \\Lambda (\\mathsf {Z}))^D)$ and $\\mathbb {U}/\\mathbb {H}\\simeq \\mathrm {SSp}((A\\otimes \\Lambda (\\mathsf {Z}))^D)$ .",
"Therefore, the inclusion $\\mathbb {V}/\\mathbb {H}\\subseteq \\mathbb {U}/\\mathbb {H}$ is dual to $\\phi \\otimes \\mathrm {id}_{\\Lambda (\\mathsf {Z})}$ , and the superalgebras $\\mathcal {O}(\\mathbb {U}/\\mathbb {H})$ and $\\mathcal {O}(\\mathbb {V}/\\mathbb {H})$ satisfy the conditions of Lemma 3.5, [14], for the elements $a_i\\otimes 1$ , where $1\\le i\\le k$ .",
"The lemma is proven.",
"Corollary 13.12 The statement of Lemma REF remains valid for an arbitrary open subscheme $\\mathsf {V}^{\\prime }$ of $\\mathsf {U}^{\\prime }$ .",
"Indeed, $\\mathsf {V}^{\\prime }$ is a finite union of open affine subschemes.",
"Fix a finite open covering of $\\mathsf {G}/\\mathsf {H}$ by affine subschemes $\\mathsf {U}_i^{\\prime }$ for $1\\le i\\le t$ .",
"Then there is a collection of affine superschemes $\\mathbb {U}_i^{\\prime }=\\mathbb {U}_i/\\mathbb {H}$ and quotient morphisms $\\mathbb {U}_i\\rightarrow \\mathbb {U}_i^{\\prime }$ such that for every pair of indices $1\\le i\\ne j\\le t$ we have $(\\mathbb {U}_i\\cap \\mathbb {U}_j)/\\mathbb {H}=\\mathbb {U}_i^{\\prime }\\cap \\mathbb {U}_j^{\\prime }$ .",
"Consider the collection of corresponding geometric superschemes $U_i^{\\prime }$ .",
"Since the underlying topological space of each $U^{\\prime }_i$ coincides with $(\\mathsf {U}_i^{\\prime })^e$ (see Proposition 4.2, [13]), one can construct a geometric superscheme $Z$ with $Z^e=\\mathsf {X}^e$ as follows.",
"For any open subset $W\\subseteq Z^e$ , we define $\\mathcal {O}_Z(W)=\\ker (\\prod _{1\\le i\\le t}\\mathcal {O}_{U^{\\prime }_i}(W\\cap (U_i^{\\prime })^e)\\rightrightarrows \\prod _{1\\le i\\ne j\\le t}\\mathcal {O}_{U_i^{\\prime }\\cap U_j^{\\prime }}(W\\cap (U^{\\prime }_i\\cap U^{\\prime }_j)^e)),$ where $U^{\\prime }_i\\cap U^{\\prime }_j$ denotes the geometric counterpart of $\\mathbb {U}_i^{\\prime }\\cap \\mathbb {U}_j^{\\prime }$ .",
"It is clear that $\\mathcal {O}_Z$ is a superalgebra sheaf such that $\\mathcal {O}_Z|_{(U^{\\prime }_i)^e}\\simeq \\mathcal {O}_{U^{\\prime }_i}$ for $1\\le i\\le t$ .",
"The following lemma concludes the proof of the main theorem.",
"Lemma 13.13 There is the unique morphism $\\mathbb {G}\\rightarrow \\mathbb {Z}$ , the restriction of which on each $\\mathbb {U}_i$ coincides with $\\mathbb {U}_i\\rightarrow \\mathbb {U}_i^{\\prime }\\subseteq \\mathbb {Z}$ .",
"In particular, $\\mathbb {Z}\\simeq \\mathbb {G}/\\mathbb {H}$ .",
"The open immersions $U^{\\prime }_i\\rightarrow Z$ induce the open embeddings $\\mathbb {U}^{\\prime }_i\\rightarrow \\mathbb {Z}$ .",
"Thus, $\\mathbb {U}^{\\prime }_i$ form an open covering of $\\mathbb {Z}$ .",
"Since $\\mathbb {Z}$ is a local functor, the collection of morphisms $\\mathbb {U}_i\\rightarrow \\mathbb {U}_i^{\\prime }$ uniquely extends to a morphism $\\mathbb {G}\\rightarrow \\mathbb {Z}$ , which is constant on $\\mathbb {H}$ -orbits.",
"Moreover, any morphism $\\mathbb {G}\\rightarrow \\mathbb {Y}$ , where $\\mathbb {Y}$ is a faisceau, is uniquely defined by its restrictions on $\\mathbb {U}_i$ for $1\\le i\\le t$ .",
"Since such morphisms $\\mathbb {U}_i\\rightarrow \\mathbb {Y}$ factor through the morphisms $\\mathbb {U}_i^{\\prime }\\rightarrow \\mathbb {Y}$ , $\\mathbb {G}\\rightarrow \\mathbb {Y}$ factors through the unique morphism $\\mathbb {Z}\\rightarrow \\mathbb {Y}$ .",
"Lemma is proven."
]
] | 2011.14207 |
[
[
"On the triple tensor product of nilpotent Lie algebras"
],
[
"Abstract In this paper, we give the explicit structure of $ \\otimes^{3} H $ and $ \\wedge^{3} H $ where $ H $ is a generalized Heisenberg Lie algebra of rank at most $ 2.",
"$ Moreover, for a non-abelian nilpotent Lie algebra $ L, $ we obtain an upper bound for the dimension of $ \\otimes^{3} L."
],
[
" Introduction",
"It is known from [8] that the dimension of tensor product of a Lie algebra $ L $ of dimension $ n $ when $ \\dim L^{2}=m $ is bounded by $ (n-m)(n-1)+2 $ and the structure of the tensor product of a Heisenberg Lie algebra is given.",
"Let $ H $ be a generalized Heisenberg Lie algebra of rank at most $ 2.",
"$ In this paper, we complete the explicit structure of the tensor square and the exterior square of $ H. $ Moreover, we give the triple tensor product and the triple exterior product of $H.",
"$ Finally, we show that dimension of the triple tensor of a Lie algebra $ L $ with derived subalgebra of dimension $ m $ is bounded by $ n(n-m)^{2}.",
"$"
],
[
" Preliminary and known results",
"Let $ \\mathbb {F} $ be a fixed field and $ [, ] $ is used to denote the Lie bracket.",
"For any two Lie algebras $ L $ and $ K, $ we say there exists an action $ L $ on $ K $ if an $ \\mathbb {F} $ -bilinear map $ L\\times K\\rightarrow K, $ $ (l, k)\\mapsto ~^{l}k $ satisfying $^{[l, l^{\\prime }]}k= ^{l}(^{l^{\\prime }}k)-^{l^{\\prime }}(^{l}k)\\qquad \\text{and}\\qquad ^{l}[k, k^{\\prime }]=[^{l}k, k^{\\prime }]+[k, ^{l}k^{\\prime }]$ for all $ l, l^{\\prime }\\in L $ and $ k, k^{\\prime } \\in K. $ The actions are compatible if $^{^{l}k}l^{\\prime }=[l^{\\prime }, ^{k}l] \\qquad \\text{and} \\qquad ^{^{k}l}k^{\\prime }=[k^{\\prime }, ^{l}k]$ for all $ k, k^{\\prime } \\in K, $ $ l,l^{\\prime }\\in L. $ Clearly, a Lie algebra $ L $ acts on itself by $ ^{l^{\\prime }}l=[l^{\\prime },l] $ for all $ l, l^{\\prime } \\in L. $ Morover, for all Lie algebras $ Q, $ a bilinear function $ \\varphi : L\\times K\\rightarrow Q $ is called a Lie pairing if for all $ l, l^{\\prime }\\in L $ and $ k, k^{\\prime }\\in K, $ we have $& \\varphi ([l, l^{\\prime }], k)=\\varphi (l, ^{l^{\\prime }}k)-\\varphi (l^{\\prime }, ^{l}k),\\cr & \\varphi (l, [k, k^{\\prime }])=\\varphi (^{k^{\\prime }}l,k)-\\varphi (^{k}l, k^{\\prime }),\\cr & \\varphi (^{k}l, ^{l^{\\prime }}k^{\\prime })=-[\\varphi (l,k), \\varphi (l^{\\prime },k^{\\prime })].$ The non-abelian tensor product $ L\\otimes K $ is the Lie algebra generated by symbols $ l\\otimes k $ for all $ l\\in L $ and $ k\\in K $ with the following defining relations $&c(l\\otimes k)=cl\\otimes k=l\\otimes ck, \\cr & (l+l^{\\prime })\\otimes k=l\\otimes k+l^{\\prime }\\otimes k,\\cr & l\\otimes (k+k^{\\prime })=l\\otimes k+l\\otimes k^{\\prime },\\cr & [l, l^{\\prime }]\\otimes k=l\\otimes ^{l^{\\prime }}k-l^{\\prime }\\otimes ~ ^{l}k,\\cr & l\\otimes [k, k^{\\prime }]= ^{k^{\\prime }}l\\otimes k-^{k}l\\otimes k^{\\prime },\\cr & [(l\\otimes k), (l\\otimes k^{\\prime })]=-^{k}l\\otimes ~^{l^{\\prime }}k^{\\prime }$ for all $ c\\in \\mathbb {F}, $ $ l, l^{\\prime }\\in L $ and $ k, k^{\\prime } \\in K. $ If $ L=K $ and all actions are Lie multiplication, then $ L\\otimes L $ is called the non-abelian tensor square of $ L. $ The next two propositions are used in the rest of paper.",
"Proposition 2.1 [3] There are actions of both $ L $ and $ K $ on $L\\otimes K $ given by $&^{l^{\\prime }}(l\\otimes k)=[l, l^{\\prime }]\\otimes k+l\\otimes (^{l^{\\prime }}k),\\cr & ^{k^{\\prime }}(l\\otimes k)=(^{k^{\\prime }}l)\\otimes k+l\\otimes [k^{\\prime }, k]$ for all $ l, l^{\\prime } \\in L $ and $ k, k^{\\prime } \\in K. $ Recall that a Lie algebra $ H $ is called a generalized Heisenberg of rank $ n $ if $ H^{2}=Z(H) $ and $ \\dim H^{2}=n.",
"$ If $ n=1, $ then $ H $ is a Heisenberg Lie algebra that is more well-known.",
"Such Lie algebras are odd dimension and $ H\\cong H(m)= \\langle x_{i}, y_{i}, z\\mid [x_{i}, y_{i}]=z, 1\\le i \\le m \\rangle .$ Proposition 2.2 [8] Let $ H(m) $ be a Heisenberg Lie algebra.",
"Then $H(m)\\otimes H(m)\\cong H(m)/H(m)^{2}\\otimes H(m)/H(m)^{2}$ for all $ m $ such that $ m\\ge 2.",
"$ In the case $ m=1, $ $ H(1)\\otimes H(1) $ is an abelian Lie algebra of dimension $ 6.",
"$ Let $ L\\square L $ be the submodule of $ L\\otimes L $ generated by the elements $ l\\otimes l. $ Then the exterior square $ L\\wedge L $ of $ L $ is the quotient $ L\\wedge L\\cong L\\otimes L/ L\\square L. $ For all $ l\\otimes l^{\\prime } \\in L\\otimes L,$ we denote the coset $ l\\otimes l^{\\prime }+L\\square L $ by $ l\\wedge l^{\\prime }.",
"$ The Schur multiplier of a Lie algebra $ L $ is defined as $ \\mathcal {M}(L)\\cong R\\cap F^{2}/[R, F] $ where $ L\\cong F/R $ and $ F $ is a free Lie algebra.",
"It is known that the Lie algebra $ \\mathcal {M}(L) $ is abelian and independent of the choice of the free Lie algebra $ F$ (see [1], [2] for more information).",
"Lemma 2.3 [5] Let $ L $ be an $ n $ -dimensional abelian Lie algebra.",
"Then $ \\dim \\mathcal {M}(L)=\\frac{1}{2}n(n-1).",
"$ Lemma 2.4 [6] Let $ L $ be a nilpotent Lie algebra of class two.",
"Then $ L\\wedge L\\cong \\mathcal {M}(L)\\oplus L^{2}.",
"$ Let $ A(n) $ be used to denote an abelian Lie algebra of dimension $ n. $ Then we have Lemma 2.5 [11] (i).",
"$ A(n)\\wedge A(n)\\cong \\mathcal {M}(A(n)), $ (ii).",
"$ H(1)\\wedge H(1)\\cong A(3), $ (iii).",
"$ H(m)\\wedge H(m)\\cong A(2m^{2}-m) $ for all $ m\\ge 2.",
"$ The next theorem shows that the exterior product of a Lie algebra $ L $ is a direct summand of $ L\\otimes L $ when $ L/L^{2} $ is of finite dimension.",
"Theorem 2.6 [9] Let $ L/L^{2} $ be a finite dimensional Lie algebra.",
"Then $L\\otimes L\\cong L\\wedge L\\oplus L\\square L.$ In the following, the notations $ L_{6,22}(\\varepsilon ), $ $L_{5,8}, $ $L_{6, 7}^{(2)}(\\eta )$ and $ L_{1} $ are taken from [4] and [10].",
"Proposition 2.7 [6] The Schur multiplier of Lie algebras $ L_{6,22}(\\varepsilon ), $ $ L_{5,8}, $ $L_{6, 7}^{(2)}(\\eta ),$ and $ L_{1} $ are abelian Lie algebras of dimension $ 8, 6, 8 $ and $ 9, $ respectively.",
"A Lie algebra $ L $ is called capable provided that $ L\\cong H/Z(H) $ for a Lie algebra $ H. $ Proposition 2.8 [10] Let $ H $ be a non-capable generalized Heisenberg Lie algebra of rank 2 such that $ \\dim H=n.",
"$ Then $\\dim \\mathcal {M}(H)=\\frac{1}{2}(n-3)(n-2)-2$ or $\\dim \\mathcal {M}(H)=\\frac{1}{2}(n-1)(n-4)+1.$"
],
[
"main results",
"We know from Proposition REF , $ L $ acts on $ L\\otimes L. $ On the other hand, the tensor product $ L\\otimes L $ acts on $ L $ by $ ^{t}l=~~ ^{\\lambda (t)}l$ for all $ t\\in L\\otimes L $ and $ l\\in L $ such that $ \\lambda : L\\otimes L\\rightarrow L $ is a homomorphism given by $ a\\otimes b\\mapsto [a, b].$ These actions are compatible and we can construct the triple tensor product $ \\otimes ^{3}L=(L\\otimes L)\\otimes L. $ In this section, we give the explicit structure of $ L\\otimes L=\\otimes ^{2} L $ and $ L\\wedge L=\\wedge ^{2} L$ when $ L $ is a generalized Heisenberg of rank $ 2.",
"$ Moreover, we obtain $ \\otimes ^{3}L $ and $ \\wedge ^{3}L $ when $ L $ is a generalized Heisenberg of rank at most 2.",
"Also, for a non-abelian nilpotent Lie algebra $ L, $ we give an upper bound for the triple tensor product of $ L. $ The following lemmas are useful instruments in the next.",
"Lemma 3.1 Let $ L $ be a Lie algebra of nilpotency class two.",
"Then (i).",
"$ L\\otimes L $ acts trivially on $ L. $ (ii).",
"$ (L\\otimes L)\\otimes L $ is an abelian Lie algebra.",
"(i).",
"We know that $ L $ is a Lie algebra of nilpotency class two, so $ \\gamma _{3}(L)=0.",
"$ By considering action of $ L\\otimes L $ on $ L, $ we have $ ^{\\lambda (a\\otimes b)}c=[[a, b], c]=0 $ for all $ a\\otimes b\\in L\\otimes L $ and $ c\\in L. $ Therefore $ L\\otimes L $ acts trivially on $ L. $ (ii).",
"By using the relations of the non-abelian tensor on $ (L\\otimes L)\\otimes L $ and the part (i), we have $[(a\\otimes b)\\otimes c, (a^{\\prime }\\otimes b^{\\prime })\\otimes c^{\\prime }]&= -(^{c}(a\\otimes b)) \\otimes ^{a^{\\prime }\\otimes b^{\\prime }}c\\cr &=-^{c}(a\\otimes b)\\otimes 0=0$ Therefore $ [(a\\otimes b)\\otimes c, (a^{\\prime }\\otimes b^{\\prime })\\otimes c^{\\prime }]=0 $ for all $ (a\\otimes b)\\otimes c, (a^{\\prime }\\otimes b^{\\prime })\\otimes c^{\\prime } \\in (L\\otimes L)\\otimes L, $ and so $ (L\\otimes L)\\otimes L $ is an abelian Lie algebra.",
"Let $ L $ be a nilpotent Lie algebra of class $ k $ , $ i_{L}: L\\rightarrow L $ be the identity homomorphism and $ \\gamma _{k}(L) $ be the $ k $ -th term of the lower central series of $ L $ and $ \\varphi : \\gamma _{k}(L) \\rightarrow L$ be a natural homomorphism.",
"Define homomorphisms $ \\overline{\\varphi }=(\\varphi \\otimes i_{L})\\otimes i_{L} : (\\gamma _{k}(L)\\otimes L)\\otimes L \\rightarrow \\otimes ^{3}L $ and $ \\gamma : (L\\otimes L) \\otimes \\gamma _{k}(L) \\rightarrow \\otimes ^{3} L $ given by $ (a\\otimes b)\\otimes c\\mapsto (a\\otimes b) \\otimes c $ .",
"Then Lemma 3.2 Let $ L $ be a non-abelian nilpotent Lie algebra of class $ k, $ then $ \\mathrm {Im} \\gamma \\subseteq \\mathrm {Im} \\overline{\\varphi }.",
"$ By using the relation $ k\\otimes [l, l^{\\prime }]=[l^{\\prime }, k] \\otimes l- [l, k] \\otimes l^{\\prime }$ on $ L\\otimes L, $ we can see that $ (a\\otimes b) \\otimes [x_{1}, \\dots , x_{k-1}, x_{k}]\\in (\\gamma _{k}(L)\\otimes L)\\otimes L$ for all $ a, b, x_{1}, \\dots , x_{k-1}, x_{k} \\in L.$ Therefore the result follows.",
"The next result plays a key role in proving the next theorem.",
"Proposition 3.3 If $ L $ is a nilpotent Lie algebra of class $ k, $ then $(\\gamma _{k}(L)\\otimes L)\\otimes L \\overset{(\\varphi \\otimes i_{L})\\otimes i_{L}}{\\longrightarrow }\\otimes ^{3}L \\longrightarrow \\otimes ^{3} L/\\gamma _{k}(L)\\rightarrow 0,$ is exact.",
"By using Lemma REF and [13], the result follows.",
"In the following proposition, we are going to determine the structure of $ \\otimes ^{3} H(m) $ and $ \\wedge ^{3} H(m) $ for all $ m\\ge 1.",
"$ Proposition 3.4 Let $ H(m) $ be a Heisenberg Lie algebra.",
"Then $\\otimes ^{3}H(m)\\cong {\\left\\lbrace \\begin{array}{ll}A(12) &\\text{if }m=1,\\\\A(2^{3}m^{3}) &\\text{if } m\\ge 2\\\\\\end{array}\\right.",
"}$ and $\\wedge ^{3}H(m)\\cong {\\left\\lbrace \\begin{array}{ll}A(2) &\\text{if }m=1,\\\\A(4m^{3}-4m^{2}-m) &\\text{if } m\\ge 2.\\\\\\end{array}\\right.",
"}$ We claim that $( \\varphi (L^{2})\\otimes L)\\otimes L \\cong (\\varphi (L^{2})\\otimes L^{ab})\\otimes L^{ab}.$ It is clear that $ \\varphi (L^{2})$ and $ L $ act trivially on each other.",
"Hence $( \\varphi (L^{2})\\otimes L)\\otimes L \\cong (\\varphi (L^{2})\\otimes L^{ab})\\otimes L.$ Also, $ \\varphi (L^{2})\\otimes L^{ab} $ and $ L $ act trivially on each other.",
"Thus $( \\varphi (L^{2})\\otimes L)\\otimes L \\cong (\\varphi (L^{2})\\otimes L^{ab})\\otimes L^{ab}.$ By using the following exact sequence $L^{2}\\otimes L\\overset{\\varphi \\otimes i_{L}}{\\longrightarrow }L\\otimes L\\longrightarrow L/L^{2}\\otimes L/L^{2}\\longrightarrow 0,$ we have $\\dim L\\otimes L=\\dim L^{ab}\\otimes L^{ab}+\\dim \\mathrm {Im} \\varphi \\otimes i_{L} =\\dim L^{ab}\\otimes L^{ab}+\\dim \\varphi (L^{2})\\otimes L.$ Let $ L\\cong H(1).",
"$ Since $ H(1)\\otimes H(1)\\cong A(6) $ and $ H(1)^{ab}\\otimes H(1)^{ab}\\cong A(4), $ we have $ \\dim \\varphi (L^{2})\\otimes L=2 $ by using (REF ).",
"Since $( \\varphi (L^{2})\\otimes L)\\otimes L \\cong (\\varphi (L^{2})\\otimes L^{ab})\\otimes L^{ab}$ , $\\dim (\\varphi (L^{2})\\otimes L)\\otimes L = (\\dim \\varphi (L^{2})\\otimes L^{ab})\\dim L^{ab}.$ Now, by using (REF ), Proposition REF and Lemma REF (ii), we have $\\dim \\otimes ^{3} L=\\dim \\otimes ^{3} L^{ab}+ \\dim (\\varphi (L^{2})\\otimes L^{ab})\\otimes L^{ab}=12$ and so $ \\otimes ^{3} L\\cong A(12).",
"$ Similarly, if $ L\\cong H(m) $ for all $ m\\ge 2, $ then $ \\otimes ^{3}L\\cong A(2^{3}m^{3}).",
"$ Let $ L\\cong H(1), $ then $ \\dim \\varphi (L^{2})\\wedge L= \\dim \\wedge ^{2} L-\\dim \\wedge ^{2} L^{ab}= 2$ by using the following exact sequence $L^{2}\\wedge L\\overset{\\varphi \\wedge i_{L}}{\\longrightarrow }L\\wedge L\\longrightarrow L/L^{2}\\wedge L/L^{2}\\longrightarrow 0$ and Lemma REF .",
"We can easily see $ (\\varphi (L^{2})\\wedge L) \\wedge L\\cong (\\varphi (L^{2})\\wedge L^{ab}) \\wedge L^{ab} $ and since $ \\varphi (L^{2})\\wedge L\\cong A(2), $ we have $ (\\varphi (L^{2})\\wedge L) \\wedge L\\cong A(2)\\wedge A(2).",
"$ Thus $ (\\varphi (L^{2})\\wedge L) \\wedge L\\cong A(1) $ by using Lemma REF .",
"On the other hand, $ \\wedge ^{3} L/L^{2}\\cong (A(2)\\wedge A(2))\\wedge A(2), $ hence $ \\wedge ^{3} L/L^{2}\\cong A(1) $ by using Lemma REF and [7].",
"Consider the following exact sequence $(L^{2} \\wedge L)\\wedge L \\overset{(\\varphi \\wedge i_{L})\\wedge i_{L}}{\\longrightarrow }\\wedge ^{3}L \\longrightarrow \\wedge ^{3} L/L^{2}\\rightarrow 0.$ Therefore $\\dim \\wedge ^{3}L &=\\dim \\wedge ^{3} L/L^{2}+\\dim \\mathrm {Im} (\\varphi \\wedge i_{L})\\cr &=\\dim \\dim \\wedge ^{3} L/L^{2}+\\dim (\\varphi (L^{2})\\wedge L) \\wedge L\\cr &=2.$ Since $ \\otimes ^{3} L $ is an abelian Lie algebra by using Lemma REF (ii), we have $ \\wedge ^{3} L\\cong A(2).",
"$ Similarly, if $ L\\cong H(m) $ for all $ m\\ge 2, $ then $ \\wedge ^{3}L\\cong A(4m^{3}-4m^{2}-m).",
"$ The following theorem determines all $ n $ -dimensional capable generalized Heisenberg Lie algebras of rank $ 2.",
"$ Theorem 3.5 [10] Let $ H $ be a generalized Heisenberg Lie algebra such that $ \\dim H=n $ and $ \\dim H^{2}=2.",
"$ Then $ H $ is capable if and only if $ n=5, 6 $ or 7 and $ H\\cong L_{5,8}=\\langle x_1,\\ldots , x_5\\big {|}[x_1, x_2] = x_4, [x_1, x_3] = x_5\\rangle , $ $ H\\cong L_{6,22}(\\varepsilon )=\\langle x_1,\\ldots , x_6\\big {|}[x_1, x_2] = x_5= [x_3, x_4], [x_1, x_3] = x_6,[x_2, x_4] = \\epsilon x_6 \\rangle $ where $ \\varepsilon \\in \\mathbb {F}/(\\stackrel{*}{\\sim }), $ $ L_{6, 7}^{(2)}(\\eta )=\\langle x_1,\\ldots , x_6 \\mid [x_{1}, x_{2}]=x_{5}, [x_{1}, x_{3}]=x_{6}, [x_{2}, x_{4}]=\\eta x_{6}, [x_{3}, x_{4}]=x_{5}+x_{6}$ where $ \\eta \\in \\lbrace 0, \\omega \\rbrace $ or $ H\\cong L_{1}=\\langle x_1,\\ldots , x_7\\big {|}[x_1, x_2] = x_6=[x_3, x_4],[x_1, x_5] = x_7= [x_2, x_3]\\rangle .",
"$ The next propositions determine the structure of $ \\otimes ^{2} H, $ $ \\wedge ^{2}H, $ $ \\otimes ^{3} H $ and $ \\wedge ^{3}H $ when $ H $ is a generalized Heisenberg Lie algebra of rank $ 2.",
"$ Proposition 3.6 Let $ H$ be an $ n $ -dimensional generalized Heisenberg Lie algebra of rank $ 2.",
"$ (i).",
"If $ H$ is non-capable, then $\\otimes ^{2}H \\cong {\\left\\lbrace \\begin{array}{ll}A((n-2)^{2}) &\\text{if }Z^{\\wedge }(H)=H^{2},\\\\A((n-2)^{2}+2) &\\text{otherwise} \\\\\\end{array}\\right.",
"}$ and $\\wedge ^{2}H \\cong {\\left\\lbrace \\begin{array}{ll}A(\\frac{1}{2}(n-3)(n-2)) &\\text{if }Z^{\\wedge }(H)=H^{2},\\\\A(\\frac{1}{2}(n-1)(n-4)+3) &\\text{otherwise.}",
"\\\\\\end{array}\\right.",
"}$ (ii).",
"If $ H $ is capable, then $\\otimes ^{2}H\\cong {\\left\\lbrace \\begin{array}{ll}A(14) &\\text{if }H\\cong L_{5,8},\\\\A(20) &\\text{if } H\\cong L_{6,22}(\\varepsilon )~~ \\text{or}~~L_{6, 7}^{(2)}(\\eta ),\\\\A(26) &\\text{if } H\\cong L_{1}\\\\\\end{array}\\right.",
"}$ and $\\wedge ^{2}H\\cong {\\left\\lbrace \\begin{array}{ll}A(8) &\\text{if }H\\cong L_{5,8},\\\\A(10) &\\text{if } H\\cong L_{6,22}(\\varepsilon )~~ \\text{or}~~L_{6, 7}^{(2)}(\\eta ),\\\\A(11) &\\text{if } H\\cong L_{1}.\\\\\\end{array}\\right.",
"}$ (i).",
"First, we obtain the exterior product of $ H. $ By invoking Lemma REF and Proposition REF , we have $ \\wedge ^{2}H=\\mathcal {M}(H)\\oplus H^{2}\\cong A(\\frac{1}{2}(n-3)(n-2))$ when $ Z^{\\wedge }(H)=H^{2}.",
"$ By a similar way, we can see that $\\wedge ^{2}H= A(\\frac{1}{2}(n-1)(n-4)+3) $ when $ Z^{\\wedge }(H)\\ne H^{2}.",
"$ Since $ H\\square H\\cong H^{ab}\\square H^{ab}\\cong A(\\frac{1}{2}(n-m)(n-m+1))$ by [9] and using Theorem REF , we have $ \\otimes ^{2}H=H\\square H\\oplus H\\wedge H \\cong A(n^{2}-2n+3) $ when $ Z^{\\wedge }(H)=H^{2}.",
"$ By a similar way, we can see that $ \\otimes ^{2}H \\cong A(n^{2}-2n+5) $ when $ Z^{\\wedge }(H)\\ne H^{2}.",
"$ (ii).",
"The proof is similar to the part (i).",
"Proposition 3.7 Let $ H$ be $ n $ -dimensional generalized Heisenberg Lie algebra of rank $ 2.",
"$ (i).",
"Let $H $ is non-capable.",
"Then $\\otimes ^{3}H \\cong {\\left\\lbrace \\begin{array}{ll}A((n-2)^{3}) &\\text{if }Z^{\\wedge }(H)=H^{2},\\\\A((n^{2}-4n+6)(n-2)) &\\text{otherwise.}",
"\\\\\\end{array}\\right.",
"}$ and $\\wedge ^{3}H \\cong {\\left\\lbrace \\begin{array}{ll}A(\\frac{1}{2}(n-2)(n^{2}-6n+7)) &\\text{if }Z^{\\wedge }(H)=H^{2},\\\\A(\\frac{1}{2}n(n^{2}-8n+23)-14)&\\text{otherwise.}",
"\\\\\\end{array}\\right.",
"}$ (ii).",
"If $H $ is capable, then $\\otimes ^{3}H\\cong {\\left\\lbrace \\begin{array}{ll}A(42) &\\text{if }H\\cong L_{5,8},\\\\A(80) &\\text{if } H\\cong L_{6,22}(\\varepsilon )~~ \\text{or}~~L_{6, 7}^{(2)}(\\eta ),\\\\A(130) &\\text{if } H\\cong L_{1}\\\\\\end{array}\\right.",
"}$ and $\\wedge ^{3}H\\cong {\\left\\lbrace \\begin{array}{ll}A(12) &\\text{if }H\\cong L_{5,8},\\\\A(20) &\\text{if } H\\cong L_{6,22}(\\varepsilon )~~ \\text{or}~~L_{6, 7}^{(2)}(\\eta ),\\\\A(39) &\\text{if } H\\cong L_{1}.\\\\\\end{array}\\right.",
"}$ The proof is obtained by a similar way used in the proof of Proposition REF .",
"Theorem 3.8 Let $ L $ be an $ n $ -dimensional non-abelian nilpotent Lie algebra with derived subalgebra of dimension $ m. $ Then $\\dim \\otimes ^{3} L \\le n(n-m)^{2}.$ In particular, for $ m=1 $ the equality holds if and only if $ L\\cong H(1).",
"$ Let $ L $ be an $ n $ -dimesional nilpotent Lie algebra of nilpotency class $ k. $ We proceed by induction on $ n. $ Since $ L $ is non-abelian, $ n\\ge 3.",
"$ If $ n=3, $ then $ L\\cong H(1) $ by the classification of all nilpotent Lie algebras is given in [4].",
"By using Proposition REF , we have $ \\otimes ^{3} H(1)\\cong A(12).",
"$ Therefore the result holds.",
"Since $ \\gamma _{k}(L) $ is central, $ \\gamma _{k}(L) $ and $ L $ act trivially on each other and so we have $ \\gamma _{k}(L)\\otimes L\\cong \\gamma _{k}(L)\\otimes L^{ab}$ by using [3].",
"By the same reason and using [3], we have $ (\\gamma _{k}(L)\\otimes L)\\otimes L\\cong (\\gamma _{k}(L)\\otimes L^{ab})\\otimes L^{ab}.",
"$ Consider two cases.",
"Let $ L/\\gamma _{k}(L) $ is abelian.",
"Then $ L $ is a nilpotant Lie algebra of class two and $ \\gamma _{k}(L)=L^{2}.",
"$ Thus $ \\dim \\otimes ^{3} L/\\gamma _{k}(L)=\\dim \\otimes ^{3} L/L^{2}=(n-m)^{3}.",
"$ By using Proposition REF , we have $\\dim \\otimes ^{3}L & \\le \\dim \\otimes ^{3}(L/\\gamma _{k}(L))+\\dim ((\\gamma _{k}(L)\\otimes L)\\otimes L)\\cr & =(n-m)^{3}+m(n-m)^{2}=n(n-m)^{2}.$ If $ L/\\gamma _{k}(L) $ is non-abelian, then $ \\dim L/\\gamma _{k}(L)=n-\\dim \\gamma _{k}(L) $ and $ \\dim (L/\\gamma _{k}(L))^{2}=m-\\dim \\gamma _{k}(L) , $ hence $\\dim \\otimes ^{3}(L/\\gamma _{k}(L)) \\le (n-\\dim \\gamma _{k}(L))(n-m)^{2}$ by using the induction hypothesis.",
"Now, Proposition REF and (REF ) imply that $\\dim \\otimes ^{3}L&\\le \\dim \\otimes ^{3}(L/\\gamma _{k}(L))+\\dim ((\\gamma _{k}(L)\\otimes L)\\otimes L)\\cr &\\le (n-\\dim \\gamma _{k}(L))(n-m)^{2}+\\dim \\gamma _{k}(L)(n-m)^{2} \\cr & = n(n-m)^{2}.$ Let $ m=1.",
"$ Then $ L\\cong H(k) \\oplus A(n-2k-1)$ for all $ k\\ge 1 $ by using [12].",
"By using Proposition REF the equality holds if and only if $ L\\cong H(1).",
"$"
]
] | 2011.14100 |
[
[
"Unsupervised Constrained Community Detection via Self-Expressive Graph\n Neural Network"
],
[
"Abstract Graph neural networks (GNNs) are able to achieve promising performance on multiple graph downstream tasks such as node classification and link prediction.",
"Comparatively lesser work has been done to design GNNs which can operate directly for community detection on graphs.",
"Traditionally, GNNs are trained on a semi-supervised or self-supervised loss function and then clustering algorithms are applied to detect communities.",
"However, such decoupled approaches are inherently sub-optimal.",
"Designing an unsupervised loss function to train a GNN and extract communities in an integrated manner is a fundamental challenge.",
"To tackle this problem, we combine the principle of self-expressiveness with the framework of self-supervised graph neural network for unsupervised community detection for the first time in literature.",
"Our solution is trained in an end-to-end fashion and achieves state-of-the-art community detection performance on multiple publicly available datasets."
],
[
"Introduction",
"Graphs or networks are ubiquitous in our daily life.",
"Graph representation learning [23], [14] is the task of mapping different components of a graph (such as nodes, edges or the entire graph) to a vector space to facilitate downstream graph mining tasks.",
"Among various types of graph representation techniques, graph neural networks (GNNs) [30] have received significant attention as they are able to apply neural networks directly on the graph structure.",
"Most of the GNNs can be represented in the form of a message passing network, where each node updates its vector representation by aggregating messages from neighboring nodes with its own [12], [13].",
"GNNs are traditionally trained in a semi-supervised way [19] on a node classification loss when a subset of node labels are available.",
"More recently, unsupervised and self-supervised graph neural networks have been proposed where a reconstruction loss [18], [3] or noise contrastive loss [27], [36] is used to train the networks.",
"Community detection is one of the most important tasks for network analysis and has been studied for decades in classical network analysis community [11], [31].",
"However, compared to other tasks such as node classification [19], [27] and link prediction [18], [34], community detection has not been explored much in the framework of graph neural networks.",
"Being inherently unsupervised in nature, it is challenging to train GNNs for community detection directly.",
"Traditionally, methods have been proposed where a graph representation learning algorithm is trained on a generic unsupervised loss and then a clustering algorithm is applied as a post-processing step to discover communities [23], [2].",
"Such approaches are sub-optimal in nature as the node representation learning module and the clustering algorithm work independently.",
"More recently, there have been efforts to train graph neural networks directly for community detection in graphs [6], [32] (Section ).",
"In contrast to a fully unsupervised approach, a graph neural architecture is proposed in [8] for a supervised version of community detection .",
"In classical machine learning, constraint clustering has been shown to be very efficient where must-link or no-link constraints are given as input [28].",
"But, obtaining direct ground truth community labels or such pair-wise constraints is expensive for real-world networks.",
"In this paper, we aim to derive such constraints in an unsupervised way, by using the principle of self-expressiveness of data [16].",
"This allows to express each data point by a linear combination of other data points which potentially lie in the same subspace.",
"The principle of self-expressiveness has been successfully applied in computer vision and image processing for object detection and segregation [33], [20].",
"However, inherent computational demand to build pair-wise similarity matrix and subsequent use of spectral clustering makes it infeasible to directly apply the principle of self-expressiveness and subspace clustering in domains like graphs where number of nodes can be very large [10], [16], [17].",
"We have taken a different approach in this paper to address these computational challenges.",
"Our solution uses a self-supervised GNN and generate node communities from the embeddings obtained.",
"To guide the generated communities, we use the principle of self-expressiveness on randomly sampled batches of nodes to generate a set of soft must-link and no-link constraints.",
"Following are the novel contributions made in this paper: We propose a novel community detection algorithm, referred as SEComm (Self-Expressive Community detection in graph).",
"To the best of our knowledge, we are the first in literature to combine the principle of self-expressiveness with the framework of self-supervised graph neural network for unsupervised community detection.",
"Our solution is able to use both link structure and the node attributes of a graph to detect node communities.",
"To address the computational issues, our solution uses the principle of self-expressiveness to generate a set of soft must-link or no-link constraints on a subset of nodes divided into batches.",
"In contrast to existing literature on self-expressiveness (which typically applies spectral clustering as a post-processing step), our solution is trained in an end-to-end fashion.",
"To show the merit of the proposed algorithm, we conduct experiments with multiple publicly available graph datasets and compare the results with a diverse set of algorithms.",
"SEComm is able to improve the state-of-the-art performance of unsupervised community detection with a significant margin in almost all the real-world datasets we used.",
"Model ablation study and sensitivity analysis give further insights of the algorithm.",
"Source code of SEComm is available at https://github.com/viz27/SEComm."
],
[
"Related Work",
"We have presented the related work into three categories.",
"Graph Neural Networks: Graph neural networks have gained significant attention in last few years with their success on a diverse set of applications [30], [9].",
"Typically, GNNs are trained on node-classification, link prediction and graph reconstruction losses [19], [13].",
"Recently, self-supervised learning has been able to achieve performance close to supervised learning for multiple downstream tasks [4], [15].",
"Extending the concept of information maximization, DGI [27] and GRACE [36] have been proposed where information between different graph entities (graph-level to node-level, corrupted versions of a graph etc.)",
"are maximized.",
"However, none of the GNNs above handles community detection in their respective objectives.",
"Principle of Self-Expressiveness: The concept of self expressiveness was proposed to cluster data drawn from multiple low dimensional linear or affine subspaces embedded in a high dimensional space [10].",
"Given enough samples, each data point in a union of subspaces can always be written as a linear or affine combination of all other points [10], [16].",
"Subspace clustering exploits this to build a similarity matrix, from which the segmentation of the data can be easily obtained using spectral clustering [21], [16].",
"Recently, a deep learning based subspace clustering method has been proposed where an encoder is used to map data to some embedding space before building the pair-wise similarity matrix and applying spectral clustering [17].",
"However, inherent computational demand to build pair-wise similarity matrix and subsequent use of spectral clustering makes it infeasible to directly apply the principle of self-expressiveness in domains like graphs where the number of nodes can be very large.",
"Community Detection with GNNs: As explained in Section , a disjoint approach of applying clustering on node embeddings obtained by some representation algorithm is inherently sub-optimal in nature [3].",
"In [35], the authors have used an adaptive graph convolution method that performs high-order graph convolution to obtain smooth node embeddings that capture global cluster structure.",
"The node embeddings obtained are subsequently used to detect communities using spectral clustering.",
"In [25], a probabilistic generative model is proposed to learn community membership and node representation collaboratively.",
"More recently, researchers have tried to propose GNN algorithms that can operate directly for community detection in a graph [6].",
"In [32], authors propose to derive node community membership in the hidden layer of an encoder and introduced a community-centric dual decoder to reconstruct network structures and node attributes in an unsupervised fashion.",
"Our work is towards this direction of obtaining node communities directly in the framework of graph neural networks."
],
[
"Problem Formulation",
"Let us denote an input graph by $G=(V,E,X)$ , where $V = \\lbrace 1,2,\\cdots ,N\\rbrace $ is the set of nodes and $E \\subseteq V \\times V$ is the set of edges.",
"We assume that each node has some attribute values present in a vector $x_i \\in \\mathbb {R}^F$ and $X = [x_1 \\; x_2 \\; \\cdots \\; x_N]^T \\in \\mathbb {R}^{N \\times F}$ is the node attribute matrix of the graph.",
"The goal of our work is to learn a function $f: V \\mapsto [K]$ , where $[K] = \\lbrace 1,2,\\cdots ,K\\rbrace $ is the set of community (or cluster) indices, to map each node to a community by exploiting the link structure and node attributes of the graph.",
"We want to achieve this without having any ground truth community information of a node.",
"Intuitively, nodes which are closely connected in the graph or have similar attributes, should be members of the same community.",
"Important notations in the paper are summarized in Table REF .",
"Table: Different notations used in the paperFigure: Training phases of SEComm"
],
[
"Our Solution: SEComm",
"There are multiple steps in our proposed solution SEComm as shown in Figure REF .",
"We discuss each of them."
],
[
"Self-supervised Node Embedding",
"The first step of SEComm is to learn node representation in an unsupervised way.",
"Self-supervised learning [15] has been used recently for obtaining both node embeddings [27], [36] and graph-level embeddings [26].",
"Potentially, any self-supervised differentiable approach to obtain node representation can be integrated with our solution.",
"In the following, we have adopted the principle of mutual information maximization between two corrupted versions of the given graph, motivated from [27], [36], which is then used to formulate the final objective of SEComm in Section REF .",
"Given the input graph $G=(V,E,X)$ , two graph views $G_1$ and $G_2$ are generated from it by employing a corruption function.",
"The corruption function randomly removes a small portion of edges from the input graph and also randomly masks a fraction of dimensions with zeros in node features.",
"The vertex sets of $G_1$ and $G_2$ remain the same.",
"These views are used for contrastive learning at both graph topology and node feature levels.",
"We use a GCN encoder to generate node embeddings for both $G_1$ and $G_2$ .",
"For a graph $G$ , GCN encoder derives node representations as follows: $Z = f(X, A) = \\text{ReLU}(\\hat{A} \\; \\text{ReLU}(\\hat{A}XW^{(0)}) \\; W^{(1)})$ where each row of $Z \\in \\mathbb {R}^{|V| \\times F^{\\prime }}$ contains the corresponding node representation.",
"$A$ is the adjacency matrix of the graph $G$ .",
"We compute $\\tilde{A} = A + I$ , where $I \\in \\mathbb {R}^{|V| \\times |V|}$ is the identity matrix and the degree diagonal matrix $\\tilde{D_{ii}}$ with $\\tilde{D_{ii}} = \\sum \\limits _{j \\in V} \\tilde{A}_{ij}$ , $\\forall i \\in V$ .",
"We set $\\hat{A} = \\tilde{D}^{-\\frac{1}{2}} \\tilde{A} \\tilde{D}^{-\\frac{1}{2}}$ .",
"$W^{(0)}$ and $W^{(1)}$ are the trainable parameter matrices of GCN.",
"Let us use $Z_1$ and $Z_2$ to denote the node embedding matrices for the two views $G_1$ and $G_2$ obtained from the GCN encoder (parameters shared).",
"Next, the following noise contrastive objective (via a discriminator) is used.",
"For any node $i \\in V$ , let us denote the corresponding nodes in $G_1$ and $G_2$ as $G_1(i)$ and $G_2(i)$ respectively.",
"For each $i \\in V$ , the pair $(G_1(i), G_2(i))$ is considered as a positive example.",
"Negative examples are sampled from both the views for each node $i \\in V$ .",
"More formally, we randomly select a set of nodes $V_{- i} = \\lbrace j \\in V \\;|\\; j \\ne i\\rbrace $ such that $|V_{- i}| = N_{-}$ (number of negative samples), $\\forall i \\in V$ .",
"Both $(G_1(i), G_1(j))$ and $(G_1(i), G_2(j))$ are considered as negative examples.",
"Then, the following objective function is minimized: $\\begin{aligned}&\\underset{W_{SS}}{\\text{min}} \\; \\mathcal {L}_{SS} \\; = \\; &&\\sum \\limits _{i \\in V} \\Bigg [ - \\frac{cos(Z_{1i},Z_{2i})}{\\tau } \\\\&&& + \\log \\Big ( \\sum \\limits _{j \\in V_{- i}} e^{\\frac{cos(Z_{1i},Z_{1j})}{\\tau }} + e^{\\frac{cos(Z_{1i},Z_{2j})}{\\tau }} \\Big ) \\Bigg ]\\end{aligned}$ where $Z_{1i}$ and $Z_{2i}$ denotes the $i$ th row of $Z_1$ and $Z_2$ respectively, $cos()$ is the cosine similarity between the two embeddings and $\\tau $ is a temperature parameter.",
"This essentially maximizes the agreement between the embeddings of $i$ th node in two views.",
"Both link structure and node attributes of the graph are considered because of the use of GCN encoder in Equation REF .",
"Node embedding matrix $Z$ of the original input graph $G$ can be obtained by Equation REF , where the purpose of $G_1$ and $G_2$ were just to enable the training of the self-supervised loss in Equation REF ."
],
[
"Learning Node Similarities through Self-Expressiveness",
"Vector node representations obtained in Section REF are quite generic.",
"There is no guarantee that similarities between the nodes captured through such embeddings are suitable to discover communities in the graph.",
"In contrast to node classification or other supervised tasks, lack of any ground truth in training for community detection makes the problem highly non-trivial.",
"To tackle this, we use the principle of self-expressiveness [10], [17] which aims to approximate a data point by a sparse linear sum of a subset of other points, which stays in the same subspace.",
"This is more prevalent when the graph has large number of nodes and embedding dimension is also reasonably high [10].",
"Based on the contribution of a point to reconstruct some other point, it is possible to learn a pair-wise similarity matrix using this principle.",
"Such a pair-wise similarity can guide the generation of communities from the node embeddings obtained by the self-supervised layer.",
"However, computation of pair-wise similarity matrix for a graph can be too expensive as it needs $O(N^2)$ storage and processing.",
"Hence, we propose a batch-wise learning procedure, as discussed later.",
"Given the node embedding matrix $Z \\in \\mathbb {R}^{N \\times F^{\\prime }}$ obtained from Section REF , we want to derive a node similarity matrix $S \\in \\mathbb {R}^{N \\times N}$ using the principle of self-expressiveness.",
"For each node $i \\in V$ , we first try to express $Z_i$ ($i$ th row of $Z$ ) by a linear sum of few other node embeddings $Z_j$ , $j \\ne i$ .",
"So, $Z_i = \\sum \\limits _{j \\in V} q_{ij} Z_j$ , where $q_{ij}$ is the $(i,j)$ th element of a coefficient matrix $Q \\in \\mathbb {R}^{N \\times N}$ and we enforce $Q_{ii}=0$ to avoid the trivial solution of Q being assigned to a identity matrix.",
"We need to learn this coefficient matrix $Q$ which will be used to generate similarity matrix $S$ .",
"It can be shown [16] under the assumption of subspace independence that, by minimizing certain norms of $Q$ , it is possible to have a block-diagonal structure (up to a permutation) of $Q$ .",
"In that case, each block in $Q$ would contain nodes which belong to the same subspace.",
"This can be posed as the following optimization problem.",
"$\\begin{aligned}\\underset{Q}{\\text{min}} \\; ||Q||_p \\;\\; \\text{such that,} \\; Z=QZ; \\; \\text{diag}(Q)=0\\end{aligned}$ where $||Q||_p$ is $p$ th matrix norm of $Q$ and diag$(Q)$ denotes the diagonal entries of $Q$ .",
"Based on the choice in some existing literature [21], [16], we use square Frobenius matrix norm for our implementation.",
"However, exact reconstruction of $Z$ using the this principle may not be possible.",
"So, we relax the hard constraint $Z=QZ$ with square Frobenius norm of $(Z - QZ)$ (soft constraint).",
"This gives us the following objective function.",
"$\\begin{aligned}\\underset{Q}{\\text{min}} \\; \\mathcal {L}_{SE} \\; = \\; || Z - QZ ||_F^2 + \\lambda _1 ||Q||_F^2 \\;\\; \\text{such that,} \\; \\text{diag}(Q)=0\\end{aligned}$ where $\\lambda _1$ is a weight parameter of this optimization.",
"In principle, while a pairwise similarity matrix $S$ can be constructed trivially as $Q$ +$Q^T$ , many heuristics have been proposed to improve the clustering performance of $S$ (when using methods such as spectral clustering directly on $S$ ).",
"We follow the heuristics proposed in [16] to construct the node similarity matrix $S$ as: $Q^{\\prime } = \\frac{1}{2}(Q + Q^T)$ Compute the $r$ rank SVD of $Q^{\\prime }$ , ie.",
"$Q^{\\prime } = U \\Sigma V^T$ , where $r = dK+1$ , $K$ is the number of communities and $d$ is the maximal intrinsic dimension of subspaces which is set to 4 in all our experiments.",
"Compute $L = U \\Sigma ^\\frac{1}{2}$ and normalize each row of $L$ to have unit norm.",
"Set negative values in $L$ to zero to obtain $L^{\\prime }$ .",
"Construct similarity matrix $S$ as $S = (L^{\\prime } + L^{\\prime T})/||L||_\\infty $ , so that $s_{ij} \\in [0,1]$ .",
"As mentioned before, an inherent difficulty to compute the pair-wise similarity matrix is the computation and storage of $N \\times N$ dimensional matrix $S$ .",
"So, instead of computing this matrix for all pairs of nodes, we use batch-wise learning.",
"We sample batches of randomly selected nodes with batch size $M$ , where $M \\le N$ .",
"We train the loss in Equation REF for each batch.",
"The required computation in each batch is $O(M^3)$ (for solving Equation REF and the subsequent use of SVD decomposition) which is much lesser than $O(N^3)$ for a significantly smaller $M$ .",
"However, the problem with this approach is that one would not get complete similarity matrix for the graph.",
"It only computes $s_{ij}$ if nodes $i$ and $j$ belong to a same batch.",
"Let us denote $\\mathfrak {S}$ to be the set of node pairs for which the similarity is computed in the batch-wise learning.",
"Clearly, $\\mathfrak {S} \\subsetneq V \\times V$ and $|\\mathfrak {S}| << N^2$ (when $M<N)$ .",
"This makes it difficult to use with spectral clustering, as most of the subspace clustering algorithms do [17].",
"But as explained next, our overall solution does not need all the node-pair similarities.",
"Rather, it filters the existing similarities computed with the batch-wise solution using a simple trick explained next."
],
[
"Constrained Node Community Detection",
"Instead of applying expensive spectral clustering on the complete matrix $S$ as a post processing step to find node clusters, we use a neural network based solution which is significantly more scalable.",
"We use a fully-connected multilayer perceptron (MLP) with the set of trainable parameters $W_{MLP}$ to map each node embedding to its corresponding soft community membership vector as follows.",
"$C_i = \\text{Softmax}(\\text{MLP}(Z_i)) \\in \\mathbb {R}^K$ where the MLP maps each $Z_i \\in \\mathbb {R}^{F^{\\prime }}$ to a $K$ dimensional vector, $K$ is the number of communities.",
"We assume to know $K$ beforehand.",
"The softmax layer converts the $K$ dimensional vector to a probability distribution such that $c_{ik}$ ($k$ th element of $C_i$ ) denotes the probability that $i$ th node belongs to $k$ th community, $\\forall k = [K]$ .",
"Equation REF ensures that nodes having similar embeddings will be mapped to similar positions in the $(K-1)$ dimensional probability simplex.",
"However, relying completely on embeddings to detect communities is not desirable since the embeddings are generated with generic objectives.",
"So, they may not be optimal to generate node communities.",
"Hence, we use the information learned in node-pair similarities in Section REF to guide both the detection of node communities by training the parameters of MLP in Equation REF and updating node embeddings.",
"Let us form the community membership matrix $C = [C_1,\\cdots ,C_N]^T$ $\\in \\mathbb {R}^{N \\times K}$ .",
"If the complete node similarity matrix $S$ is available, one may try to minimize the following objective.",
"$\\begin{aligned}\\underset{W_{SS},\\; W_{MLP}}{\\text{min}} || C C^T - S ||_F^2\\end{aligned}$ There are multiple drawbacks present in the objective function above.",
"First, it needs us to compute the complete node similarity matrix $S$ in Section REF which prevents the batch-learning mechanism explained before.",
"Next, the computation involved is $O(N^2)$ in Equation REF .",
"Further, there is another issue if one wants to use all pair-wise node similarities in $S$ to guide the community detection.",
"Due to noise in the dataset, many of the pair-wise similarities may not reflect the actual similarities between the nodes.",
"The similarity values which are around 0.5 neither express a strong similarity nor a strong dissimilarity between a node pair.",
"So they are less informative compared to the similarity values which are close to 0 or 1.",
"But they can still influence the parameters of the neural network because of Equation REF .",
"Hence, instead of considering all the pair-wise similarity values, we only consider the ones in $\\mathfrak {S}$ computed over the batches as discussed in Section REF .",
"Further, we have observed experimentally in Section REF that for a larger dataset, it is okay even if some nodes are not part of any of the batches selected randomly.",
"Thus, the number of batches can be significantly smaller than $\\frac{N}{M}$ for a larger dataset.",
"We also introduce two thresholds $\\theta _{low}$ and $\\theta _{high}$ to use only those node-pair similarities which are extreme in their values, thus more informative in nature.",
"We set $0 < \\theta _{low} \\le \\theta _{high} < 1$ .",
"We also set $\\theta _{high} = 1 - \\theta _{low}$ , as this choice works well in the experiments and also reduces the number of hyperparameters.",
"Let us introduce the set $\\mathfrak {S}_{ext} \\subseteq \\mathfrak {S}$ as follows.",
"$\\mathfrak {S}_{ext} = \\Big \\lbrace (i,j) \\in \\mathfrak {S} \\;|\\; S_{ij} \\le \\theta _{low} \\;\\text{or}\\; S_{ij} \\ge \\theta _{high} \\Big \\rbrace $ Here, a node pair $(i,j)$ in $\\mathfrak {S}_{ext}$ should be roughly constrained to be in the same cluster when $S_{ij}$ value is very high or in different clusters when $S_{ij}$ is very low.",
"Thus, we derive a set of soft version of must-link and no-link constraints in an unsupervised way to guide the formation of communities.",
"With these, we formulate the following optimization to detect the communities: $\\begin{aligned}\\underset{W_{SS},\\; W_{MLP}}{\\text{min}} \\sum \\limits _{(i,j) \\in \\mathfrak {S}_{ext}} \\Big ( C_i^T C_j - S_{ij} \\Big )^2\\end{aligned}$ By considering only the node pairs in $\\mathfrak {S}_{ext}$ , we are able to ignore the pairs which are neither too similar nor too dissimilar, to contribute to the learning of community memberships.",
"As $C_i$ is a probability distribution over all the $K$ communities for a node $i$ , we want to avoid trivial community formations where each node is assigned to all the communities with roughly uniform probabilities, or all the nodes are assigned to a single community [5].",
"So, we update the main objective in Equation REF as: $\\begin{aligned}&\\underset{W_{SS},\\; W_{MLP}}{\\text{min}} \\; \\mathcal {L}_{Com} = &\\sum \\limits _{(i,j) \\in \\mathfrak {S}_{ext}} \\Big ( C_i^T C_j - S_{ij} \\Big )^2 \\\\&&+ \\lambda _2 \\biggl \\vert \\biggl \\vert \\frac{C^T C}{||C^T C||_F} - \\frac{I_K}{\\sqrt{K}} \\biggl \\vert \\biggl \\vert _F^2\\end{aligned}$ The second component in the equation above ensures that communities are close to orthogonal and they are balanced in size.",
"Please note that due to the use of neural network to generate community membership for each node in Equation REF , the optimization in Equation REF is not a discrete optimization.",
"Rather, we solve it with respect to the parameters $W_{SS}$ of the self-supervised layer (Eq.",
"REF ) and $W_{MLP}$ of the MLP (Eq.",
"REF ).",
"The total loss to train the node embeddings and community detection can be written as a weighted sum of self-supervised loss and community detection loss, which is shown below: $\\begin{aligned}\\underset{W_{SS},\\; W_{MLP}}{\\text{min}} \\; \\mathcal {L}_{total} = \\alpha \\mathcal {L}_{SS} + \\mathcal {L}_{Com}\\end{aligned}$ where $\\alpha $ is a weight factor of the optimization.",
"The node-pair similarity values are obtained by solving the batch-learning technique in Section REF .",
"The overall algorithm SEComm proceeds in an iterative way by solving the self-expressive layer for each batch and then updating the parameters of the neural network by minimizing Equation REF .",
"The pseudo code of SEComm is presented in REF .",
"SEComm - Self-Expressive Community Detection [1] Input: The graph $G=(V,E,X)$ , $|V|=N$ , $K$ : Number of communities in the graph, $M$ : Batch size for the self-expressive layer, $P$ : Number of batches used for training self expressive layer, Thresholds $\\theta _{low}$ and $\\theta _{high}$ .",
"Output: Community membership vector $C_i \\in \\mathbb {R}^{K}$ for each node $i \\in V$ .",
"Initialize the parameters of the self-supervised GNN and clustering MLP (in Eq.",
"REF ).",
"pre-training step: Obtain node embeddings $Z \\in \\mathbb {R}^{N \\times F^{\\prime }}$ by training the self-supervised GNN.",
"Initialize $\\mathfrak {S}$ as empty.",
"$batch \\in \\lbrace 1,2,\\cdots ,P\\rbrace $ Sample a batch of $M$ nodes from $V$ Learn the pair-wise node similarity matrix $S_m$ for the selected nodes by optimizing Eq.",
"REF .",
"Add all the node-pairs from the batch to $\\mathfrak {S}$ .",
"Construct $\\mathfrak {S}_{ext}$ according to Eq.",
"REF $iter \\in \\lbrace 1,2,\\cdots ,T\\rbrace $ Generate node embedding matrix $Z \\in \\mathbb {R}^{N \\times F^{\\prime }}$ using the self-supervised GNN.",
"Generate cluster membership vector $C_i \\in \\mathbb {R}^{K}$ for each node $i \\in V$ .",
"Update the parameters of the GNN and clustering MLP by optimizing Eq.",
"REF"
],
[
"Training and Analysis of SEComm",
"We use ADAM with default parameterization to solve the optimization formulations in Equations REF and REF .",
"For the self-expressive loss in Equation REF , we train until the loss saturates.",
"For the total loss in Equation REF , we particularly focus on the saturation of the regularization $\\biggl \\vert \\biggl \\vert \\frac{C^T C}{||C^T C||_F} - \\frac{I_K}{\\sqrt{K}} \\biggl \\vert \\biggl \\vert _F^2$ .",
"Experimentally, using the convergence of this component explicitly as a stopping criteria for SEComm gives slightly better result for all the datasets, than checking the total convergence.",
"But as explained in Section REF , different components of the loss function have similar contributions.",
"Hence they saturate almost in the same time for most of the cases.",
"This can also be observed in Section REF .",
"Time Complexity: Time complexity of the self-supervised GNN in Section REF is $O(|E| + NFF^{\\prime }N_-)$ , where $N_-$ is the number of negative samples used.",
"The self-expressive layer takes another $O(P M^3)$ time, where $P$ is the number of batches sampled, and $M$ is the size of each batch.",
"Finally, generating community membership takes $O(NK^2)$ time because of solving the loss in Equation REF .",
"Thus, the overall run time of each iteration of SEComm is linearly dependent on the number of nodes and number of edges in the graph."
],
[
"Experimental Evaluation",
"This section presents the details of the experiments that we conducted and the analysis of the results.",
"Table: Summary of the datasets used"
],
[
"Datasets Used",
"To show the merit of SEComm, we conduct experiments on 5 publicly available graph datasets [19], [35].",
"Different statistics of the datasets are summarized in Table REF .",
"Cora, Citeseer and Pubmed are citation datasets where nodes correspond to papers and are connected by an edge if one cites the other.",
"Wiki is a collection of webpages where nodes are webpages and are connected if one links to other.",
"Physics is a co-authorship network where nodes are authors, that are connected by an edge if they have co-authored a paper [24].",
"Each of these datasets have attribute vector associated with each node.",
"They also have ground truth community membership of each node, which we use to evaluate the performance of our proposed and baseline algorithms.",
"Table: Performance of Community Detection by SEComm and other baseline algorithmsTable: Community Detection on Physics DatasetFigure: Loss vs Accuracy progression during training"
],
[
"Baseline Algorithms",
"We use a diverse set of baselines to compare the performance of SEComm.",
"We divide them into the following categories.",
"Using only Node Features: As each node is associated with some attribute vectors, we use k-means and spectral clustering (Spectral-f) algorithms directly on the node attributes to cluster nodes into different communities.",
"Naturally, these approaches ignore the graph structure completely.",
"Using only Graph Structure: We also use spectral clustering (Spectral-g) on the graph structure.",
"Here we consider adjacency matrix of a graph as the similarity matrix between the nodes.",
"We also use popular unsupervised node embedding techniques DeepWalk [23], which is a random walk based technique and DNGR [7], which is an auto-encoder based technique.",
"Using both Node Features and Graph Structure: We use a set of unsupervised graph neural network based techniques.",
"GNN based approaches are naturally able to use both link structure and node attribute of the graph.",
"They are: graph autoencoder (GAE) and graph variational autoencoder (VGAE) [18], marginalized graph autoencoder (MGAE) [29], adversarially regularized graph autoencoder (ARGE) and variational graph autoencoder (ARVGE) [22].",
"These methods typically learn the node embeddings and use clustering on the embeddings as a post processing step.",
"Finally, we use two recently proposed community detection methods - AGC [35], which uses high-order graph convolution to get node embeddings and detect communities via spectral clustering on the embeddings and GUCD [32], which uses an auto-encoder based framework to obtain direct community assignments for every node."
],
[
"Experimental Setup",
"Our proposed algorithm SEComm generates community membership of each node in a graph in the framework of graph neural networks.",
"As each node has a single ground truth community membership in all the datasets that we use, we consider the index of the maximum value of $C_i \\in \\mathbb {R}^K$ (from Equation REF ) as the community index of the node generated by SEComm.",
"There are multiple hyperparameters present in SEcomm.",
"For weight factors in optimization such as $\\lambda _1$ , $\\lambda _2$ and $\\alpha $ , we check the contribution of different components in a loss function at the beginning of the algorithm, and set these parameters to values such that effective contributions of those components become roughly the same.",
"This ensures that the optimization pays similar importance to different components of SEComm.",
"For the temperature parameter $\\tau $ , we use the same values used in the literature [36].",
"For threshold parameters $\\theta _{low}$ ($0 < \\theta _{low} \\le 0.5$ ), we set it to 0.5 for relatively smaller datasets as we do not want to discard any information for them.",
"For Pubmed, we set it to 0.05 as considering more node-pair similarity values adds noise and also increases runtime of SEComm.",
"However on Physics, the training convergence is not smooth when we set $\\theta _{low}$ to a very small number.",
"So, we set it to $0.3$ on this dataset.",
"As mentioned in Section REF , we set $\\theta _{high}=1-\\theta _{low}$ .",
"We have also conducted sensitivity analysis of SEComm with respect to some of these hyperparameters in Section REF ."
],
[
"Results of Community Detection",
"Tables REF and REF show the performance of community detection by different baseline algorithms and SEComm.",
"We use three popularly used metrics to evaluate the quality of community detection.",
"They are clustering accuracy (Acc), normalized mutual information (NMI), and macro F1-score (F1) [1], [35].",
"We use ground truth community information of the nodes only to calculate these quality metrics.",
"While reporting the performance of baseline algorithms for the first four datasets in Table REF , we have collected the best results from the available literature [35], [32] which adopted the same experimental setup.",
"We mark some entry as `NA' if the result of that algorithm for a dataset is not publicly available.",
"For Physics dataset, the baseline results are not available in the literature.",
"So, we have run and reported results only for better-performing and diverse subset of baselines in Table REF with adequate hyperparameter tuning.",
"Additionally, we have also reported the runtime on Physics dataset for these algorithms to give more insight about scalability.",
"We run SEComm 10 times on each dataset and report the average performance.",
"Tables REF and REF show that SEComm is able to achieve state-of-the-art (SOTA) performance for all the datasets, and for all the metrics, except on Citeseer-F1% and Physics-NMI% scores, where SEComm is next to AGC.",
"In terms of performance improvement by clustering accuracy, SEComm is able to improve SOTA by 10.1% on Cora, 4.2% on Citeseer, 6.7% on Pubmed, 5.9% on Wiki and 3.4% on Physics.",
"We also check the standard deviation of the performance of SEComm over 10 runs in each dataset.",
"Standard deviation lies in the range of 0.5% - 1% on all the datasets, which shows the robustness of SEComm.",
"Among the baselines, AGC mostly performs better than others.",
"But, AGC is computationally much expensive because of their adaptive strategy for determining k in k-order graph convolution.",
"As shown in Table REF , runtime of AGC on Physics dataset is ~16 times more than that of SEComm.",
"As expected, the algorithms which use both graph structure and node attributes perform better than the ones which use only one of those.",
"The consistent performance of SEComm on all the datasets shows the importance of integrating the objective of community detection directly into the framework of self-supervised graph neural network (the loss from these components propagates to each other through backpropagation).",
"Further, use of the principle of self-expressiveness regularizes the communities formed in SEComm to achieve better performance.",
"Run-time of SEComm and its various components on all the datasets is shown in Table REF .",
"Usefulness of the individual components of SEComm are presented in Section REF .",
"Table: Runtime of SEComm on various datasets"
],
[
"Loss and Accuracy of SEComm",
"Typically for an unsupervised algorithm, the loss that it minimizes and the metric that is used to evaluate the performance of the algorithm are not necessarily the same.",
"So, it is important to see if reducing the loss over the epochs actually increases the performance of the algorithm with respect to the quality metric.",
"For SEComm, we plot the loss in Equation REF and the clustering accuracy that it achieves over different epochs of the algorithm for the datasets Cora and Wiki in Figure REF .",
"One can see that with the decreasing loss, overall clustering accuracy improves with some minor fluctuations.",
"Thus, the unsupervised loss that SEComm minimizes essentially helps to improve the performance of clustering.",
"This is also another reason of consistent performance (improved metric scores with less standard deviation) of SEComm on multiple datasets.",
"Figure: Performance of SEComm with varying embedding dimensionFigure: Performance of SEComm with varying threshold θ low \\theta _{low}Figure: Performance of SEComm with varying number of batches sampled and batch size respectivelyTable: Model Ablation Study of SEComm"
],
[
"Sensitivity to Hyperparameters",
"In this section, we show the sensitivity of SEComm to different hyperparameters.",
"We keep all other hyperparameters fixed while changing only the hyperparameter of interest.",
"We vary the embedding dimension $F^{\\prime }$ from 64 to 512 for Cora and Wiki and show the performance of community detection in Figure REF .",
"We can see some fluctuation in the performance for Cora.",
"As other hyperparameters are tuned keeping $F^{\\prime }=256$ for Cora, there are some sudden lows around it.",
"For Wiki, the performance was low at $F^{\\prime }=64$ as that is not sufficient enough to preserve all the information about the graph in the embeddings.",
"It increases at $F^{\\prime }=128$ .",
"There is a gradual decrease of performance beyond that as the embeddings start holding noisy information when dimension increases more.",
"We check the performance on Cora, Wiki and Pubmed in Figure REF with varying $\\theta _{low}$ (we set $\\theta _{high} = 1 - \\theta _{low}$ ).",
"As we decrease $\\theta _{low}$ , we are filtering out more pair-wise similarities, especially the ones which lies in the mid zone of the range $[0,1]$ .",
"Filtering out such similarity values might lead to less amount of data to regularize the communities learned by SEComm in Equation REF for a smaller dataset.",
"Thus, the performance on Cora is affected when $\\theta _{low}$ is very low in Figure REF .",
"But on a larger dataset, filtering out such less informative similarity values (as explained in Section REF ) can lead to removing noise and help improving the performance.",
"Thus in Pubmed, better performance is observed around $\\theta _{low} = 0.05$ (which implies $\\theta _{high} = 0.95$ ).",
"Below which the amount data becomes too less to train the algorithm properly, and above which it was adding noise.",
"As discussed in Section REF , we do not need to train the self-expressive layer (in Eq.",
"REF ) on the complete dataset.",
"In Figure REF , we vary the number of batches sampled, where each batch contains 2000 nodes.",
"We can see that the performance improved initially and then saturates when number of batches is 5 or more.",
"Thus, optimal performance on Pubmed for community detection can be achieved by using only ~50% (or more) of data points to train the self-expressive layer.",
"In Figure REF , we change the batch size, keeping number of batches as 6.",
"More is the batch size, more computation resource and time needed.",
"We observe that SEComm is able to achieve reasonably good performance when the batch size is 2000 or more."
],
[
"Model Ablation Study",
"In this section, we show the usefulness of different components of SEComm.",
"In particular, we check the community detection performance in the following scenarios.",
"SEComm-GNN We run k-means on the node embeddings produced by the self-supervised GNN used in SEComm, without running the other modules of SEComm.",
"SEComm-Spectral We run spectral clustering on the complete similarity matrix $S$ to find node clusters.",
"However, $S$ can be computed only for smaller graphs and hence this experiment cannot be performed on larger datasets like Pubmed and Physics.",
"SEComm-Embeddings We run k-means on the node embeddings generated after the complete training of SEComm (including the self-expressive and clustering modules) We compare the results of the above with the community detection output of the complete model of SEComm in Table REF .",
"Again we use three metrics clustering accuracy, NMI and F1 score to evaluate the quality of community detection.",
"Interestingly, we do not see any clear winners between the three model variants.",
"But it is clear from the reported performance numbers that the complete model of SEComm outperforms its variants (except Citeseer-F1%)."
],
[
"Discussion and Conclusion",
"In this work, we have proposed a novel graph neural network that can directly be used for node community detection in a graph.",
"We use the principle of self-expressiveness to derive a set of soft node-pair constraints to regularize the formation of the communities.",
"To the best of our understanding, this is the first work to integrate a self-expressive layer into a self-supervised GNN.",
"Our approach is highly scalable, without compromising the performance of community detection.",
"SEComm is able to achieve state-of-the-art performance on all the datasets that we used for community detection.",
"Due to the use of graph neural network to directly generate community memberships of nodes, SEComm can work in an inductive setup.",
"It would be interesting to analyze the performance of SEComm on newly added nodes or even new graphs without retraining.",
"We want to thank Prof. M. Narasimha Murty from CSA, IISc for his feedback on this work."
]
] | 2011.14078 |
[
[
"Softening of the Euler buckling criterion under discretisation of\n compliance"
],
[
"Abstract Euler solved the problem of the collapse of tall thin columns under unexpectedly small loads in 1744.",
"The analogous problem of the collapse of circular elastic rings or tubes under external pressure was mathematically intractable and only fully solved recently.",
"In the context of carbon nanotubes, an additional phenomenon was found experimentally and in atomistic simulations but not explained: the collapse pressure of smaller diameter tubes deviates below the continuum mechanics solution [Torres-Dias et al., Carbon 123, 145 (2017)].",
"Here, this deviation is shown to occur in discretized straight columns and it is fully explained in terms of the phonon dispersion curve.",
"This reveals an unexpected link between the static mechanical properties of discrete systems and their dynamics described through dispersion curves."
],
[
"Softening of the Euler buckling criterion under discretisation of compliance D.J.",
"Carter School of Physics and Astronomy, Queen Mary University of London, London E1 4NS D.J.",
"Dunstan [][email protected] School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, United Kingdom W. Just School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom O.F.",
"Bandtlow School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom A. San Miguel Université de Lyon, F-69000 Lyon, France and Institut Lumière Matière, CNRS, UMR 5306, Université Lyon 1, F-69622 Villeurbanne, France Euler solved the problem of the collapse of tall thin columns under unexpectedly small loads in 1744.",
"The analogous problem of the collapse of circular elastic rings or tubes under external pressure was mathematically intractable and only fully solved recently.",
"In the context of carbon nanotubes, an additional phenomenon was found experimentally and in atomistic simulations but not explained: the collapse pressure of smaller diameter tubes deviates below the continuum mechanics solution [Torres-Dias et al., Carbon 123, 145 (2017)].",
"Here, this deviation is shown to occur in discretized straight columns and it is fully explained in terms of the phonon dispersion curve.",
"This reveals an unexpected link between the static mechanical properties of discrete systems and their dynamics described through dispersion curves.",
"The tendency of tall thin columns to collapse under unexpectedly small loads was already known to the ancient Greeks - it is suggested that their use of entasis (Fig.REF a) was to strengthen columns [1], and the Romans based their structures on short fat columns, as in the Pont du Gard (Fig.REF b).",
"The phenomenon was explained by Euler in a classic work [2]; his explanation is now usually expressed in terms of the elastic energy required for a lateral deflection of the column compared with the work done by the advance of the load for the same deflection.",
"Figure: (a) Entasis in Greek columns (exaggerated).",
"(b) Schematic of the Pont du Gard.",
"(c) A continuous pillar, buckling (dashed).",
"(d) A discretized pillar, buckling (dashed).",
"(e) The collapse of a continuous elastic ring.",
"(f) A discretised elastic ring (a polygon).",
"(g) A continuous tube.",
"(h) An example of a discretised tube (a nanotube).The collapse of circular tubes under external pressure follows the same simple physics, but is mathematically intractable.",
"Although it was of great importance in early steam engine boilers, and later in oil wells, engineers had to rely on empirical testing [3], [4], as only partial theoretical solutions were available [5].",
"The full solution for a continuous elastic ring was given only in 2011 [6], [7].",
"Most recently, it was observed that small-diameter atomistic rings or tubes collapse at pressures lower than the continuum solution.",
"If the continuum solution is normalised against the bending stiffness of the tube wall, $D$ , and the diameter, $d$ , of the tube, the collapse pressure, $P_{\\text{C}}=\\frac{3D}{d^{3}}$ , can be expressed as $P_{\\text{C}}^N= P_C \\frac{d^3}{D}= 3$ .",
"In a recent study of the collapse pressure of single-wall carbon nanotubes [8], a surprising result was reported.",
"Experimental measurements and simulations (molecular dynamics modelling and density-functional modelling) were in agreement in finding $P_{\\text{C}}^N = 3(1 - \\frac{\\beta ^{2}}{d^{2}})$ , with a value of $\\beta $ about 0.4nm (this is therefore the diameter of the smallest stable nanotubes).",
"Both the analytic form of this expression and the value of $\\beta $ were wholly unexplained.",
"Nanotubes are not continuous elastic rings, as they have a radius of only a few atoms.",
"To find the collapse pressure of discretised elastic rings, Sun et al.",
"[9] modelled them as polygons of area $A$ with rigid sides joined by angular springs at the vertices (spring constant $k$ ), under an external pressure, $P$ .",
"The total energy is written as a function of the angles of the hinges, $\\sum \\frac{1}{2}k\\theta _i^2 +PA$ , and minimised.",
"Results were similar, but still unexplained.",
"Here we set out to find an explanation by investigating the simpler problem of an atomistic or discretised column.",
"The same phenomenon occurs; the collapse force is reduced for small numbers of segments.",
"Neither numerical nor analytical analysis reveal any explanation.",
"Recasting the problem in terms of the Euler buckling of an infinitely long column, constrained to buckle at finite wavelengths, does provide an explanation.",
"This approach reveals a link between the static mechanical properties of discrete systems and phonon dispersion curves.",
"In continuum mechanics, a column of length $L$ with unconstrained ends and a bending stiffness, $D$ (defined by $E = \\frac{1}{2}DR^{-2}$ where $R$ is the radius of curvature and $E$ is the stored elastic energy per unit length) has a buckling force of $F_{\\text{C}} = \\pi ^{2}\\frac{D}{L^{2}}$ .",
"This force is to be applied along the column axis.",
"The normalised collapse force, $F_{\\text{CN}}=F_{\\text{C}}\\frac{L^{2}}{D}$ , is thus $\\pi ^{2} \\approx 9.870$ .",
"Concentrating the bending compliance, $S = \\frac{1}{D}$ , at a number, $N$ , of points (atoms or hinges with angular springs) does indeed give a reduction in the normalised collapse force for small $N$ .",
"We used the numerical method of Sun et al.",
"[9] for polygons, energy minimised, but for straight columns under an endload (Fig.REF ).",
"Note that we have divided the column into $N$ equal intervals and then moved all the compliance in each interval to the centre of that interval.",
"That gives $N+1$ rigid links or rods, joined by the compliant hinges, and the two end rods are half the length of the others.",
"The calculations confirm that the collapse force is reduced at small $N$ for the discretised columns, as for the polygons.",
"However, the calculations give no hint of the reason for this behaviour.",
"Figure: A pillar along the xx-axis is discretised to have compliance only at NN angular hinges of stiffness c=1 a 0 Sc=\\frac{1}{a_{0}S}.",
"The lower part of the graph shows a chain of rigid rods with N+1N+1 angles shown by ν\\nu .",
"Buckling occurs causing yy-axis displacement y ν y_\\nu .Indeed, for small $N$ , it is not difficult to find analytic expressions for the collapse force, e.g.",
"for $N=1$ , the normalised collapse force is 4 and for $N=6$ it is 9.646.",
"One might think, by considering the case of $N=1$ , that the compliance has been moved from where, in a continuous column, much of it is wasted (regions of smaller curvature), to the point of maximum curvature.",
"That explanation is, however valid only for $N=1$ .",
"For larger $N$ , it is easy to show that as much compliance has been moved to where it is less useful as to where it is more useful.",
"An alternative to the energy calculation is to calculate the lateral oscillation frequency of the column as a function of the longitudinal compression or tension.",
"Compression softens the oscillatory mode.",
"The collapse force is found by setting the frequency (found via eigenvalues for a matrix of equations of motion) to zero and solving for the force.",
"These results of course agree with the energy calculation, but again do not give any hint as to the physical reason for the phenomenon.",
"The explanation is found by removing the restriction of the column length, $L$ .",
"Consider an infinite chain of point-mass hinges of mass $\\rho a_{0}$ , connecting light rods of length $a_{0}$ , with angular springs in the hinges of compliance $Sa_{0}$ and under a tension $T$ .",
"The equation of motion for the $n^{\\text{th}}$ hinge is readily set up, as Eq.",
"(REF ) where the $y$ -coordinate is perpendicular to the chain.",
"Eq.",
"(REF ) – (REF ) show its development using Bloch's theorem [10] to obtain the phonon dispersion curve $\\omega (k)$ .",
"yn = Ta02(yn-1-2yn+yn+1)-1a04S(yn-2-4yn-1+6yn-4yn+1+yn+2) ynei a0 k nun(x)= un(x) ei a0 k n (Ta02 (e-i a0 k+ei a0 k-2)-1a04 S(e-i 2 a0 k-4 e-i a0 k-4 ei a0 k+e-i 2 a0 k-6)) yn ==-2 = -4 (2 + a02 S T - 2 (a0 k)) (a0 k2]2)a04 S (k) = 212a0k2+a02ST-2a0ka04S This dispersion relation, Eq.",
"(REF ), is interesting because it has solutions for negative tension, i.e.",
"compression.",
"As the problem is set up, these solutions are unphysical, because an infinite or indefinitely long system will collapse even under an indefinitely small compression.",
"However, we can extract interesting special cases which are physically realisable.",
"First, we are not interested in running waves, but in standing waves.",
"We can set the wavelength $\\lambda $ of a standing wave by imposing constraints $y=0$ every half-wavelength along the chain, to constrain the positions of the nodes.",
"This also prevents buckling of the chain under compression at wavelengths longer than $\\lambda $ , and under compressive forces less than required for buckling at this wavelength.",
"We can do this for the special cases where the half-wavelength $\\lambda /2 = \\pi /k$ is an integer multiple of the length $a_0$ , i.e.",
"$k = \\pi /(N a_0)$ , and even more specifically we set the nodes halfway between two hinges (previously our unconstrained endpoints).",
"In this way the section between two adjacent nodes replicates the model of Fig.REF , and of course it has the frequency given by Eq.",
"(REF ).",
"The compressive force at which Eq.",
"(REF ) gives zero frequency is, precisely, the Euler buckling force for the finite, half-wavelength system.",
"Substituting $L = \\frac{1}{2}\\lambda = \\frac{\\pi }{k}$ into Eq.",
"(REF ), we have an expression for the frequency as a function of $L$ , with $a_{0}$ still as a variable.",
"We may now express $a_{0}$ in terms of $N$ , the number of atoms and $L$ , the length: $a_{0} = \\frac{L}{N}$ , but there is no need yet to make $N$ an integer.",
"Keeping $N$ as a real number, we solve for the collapse tension $T_{\\text{C}}$ or compressive force $F_{\\text{C}} = -T_{\\text{C}}$ by setting the frequency equal to zero, and we obtain $F_{\\text{C}}=-T_{\\text{C}}=\\frac{2 N ^{2}}{L^{2}S}\\left(1-\\cos \\frac{\\pi }{N}\\right).$ Normalising, $F_{C}^N=2 N ^{2}\\left(1-\\cos \\frac{\\pi }{N}\\right).$ Expanding the cosine term as $1-\\frac{1}{2}\\frac{\\pi ^2}{N^2} +$ .",
".",
".",
"gives $F_{C}^N=\\pi ^{2}\\left(1-\\frac{\\pi ^{2}}{12N^{2}}+O(N^{-4})\\right).$ Dropping the higher-order terms, this can be written for columns as $F_{C}^N =\\pi ^2 \\left(1-\\frac{\\beta ^2}{N^{2}}\\right)$ with $\\beta ^2=\\pi ^2/12$ .",
"For comparison with polygons and nanotubes, noting that $P_C$ corresponds to a tangential force in the circumscribed circle of $F_C = P_CR$ , and that the length of the circle $(2 \\pi R)$ corresponds to two buckling wavelengths rather than the half-wavelength for columns, the equivalent expressions are $F_{C}^{N} =\\frac{3}{4}\\pi ^2 \\left(1-\\frac{\\beta ^2}{N^{2}}\\right)$ Each problem has its own value of $\\beta ^2$ .",
"Eq.",
"(REF ) is plotted in Fig.REF .",
"For comparison with the numerical and analytic solutions, we may pick out the values of Eq.",
"(REF ) where $N$ is an integer: And of course they agree exactly.",
"However, Eq.",
"(REF ) provides both the explanation of the functional form of the dependence of the collapse force on $N$ and the explanation of the value of the parameter $\\beta $ .",
"The functional form is very close to $N^{-2}$ (Eq.",
"(REF )) until $N$ is small enough that the higher terms in the cosine expansion become important, which is only at $N=1$ (Fig.REF ).",
"The value of $\\beta ^2$ in Eq.",
"(REF ) is simply given by $\\beta ^2 = \\frac{\\pi ^{2}}{12} = 0.823$ for large $N$ , and rather less if we fit Eq.",
"(REF ) to the plotted data for $N = 2$ and above.",
"This fit, in the inset of Fig.REF , and the still more linear behaviour of the polygon data in Fig.REF , explains why Torres-Dias et al.",
"found the $N^{-2}$ behaviour rather than the trigonometric form of Eq.",
"(REF ) from their studies of carbon nanotubes.",
"Figure: The normalised collapse force F C N F_C^N of Eq.",
"() is plotted against N -2 N^{-2} where NN is the number of hinges per half-wavelength of the buckling mode of an infinite chain (solid black line), or, equivalently, the number of hinges in a column of length LL.",
"Results from calculations for integer NN are plotted (solid circles), for N=1-4,6,∞N = 1-4, 6,\\infty .",
"The dashed line in the inset is the least-squares fit of Eq.5 to the data for n=2-4,6,∞n = 2-4, 6,\\infty , with the fitted value β 2 =0.755\\beta ^2 = 0.755.",
"The crosses are the numerical data for the collapse of mm-gons (m=4nm = 4n) plotted for m=4-8,10,12,∞m = 4-8, 10,12,\\infty as described in the text, and the dotted line is a least-squares fit of Eq.6 to these data, with β 2 =0.53\\beta ^2=0.53.An early approximation for the problem of the elastic ring was to equate the tangential force at collapse to the collapse force of a column of length one-quarter of the circumference.",
"Levy [5] appreciated that this is not exact, and in fact gives a normalised collapse pressure of 4 instead of the correct value of 3.",
"This is why the numerical calculations for the collapse pressure of polygons [9] rescaled for the same wavelength of collapse as the column, and converted from pressure to circumferential force, extrapolate to $\\frac{3}{4} \\pi ^{2}$ at $N\\rightarrow \\infty $ as seen in Fig.REF .",
"We see also that the inverse-square dependence on $N$ is even better obeyed by the polygons than by the linear column, fully justifying the conclusion of Torres-Dias et al.",
"that the nanotube experiments and theory fitted this dependence.",
"Clearly, the next step will be to solve the polygon problem in such a way as to explain this behaviour, before continuing to the more complicated nanotube problem.",
"We set out to explain the reduction of the collapse pressure of atomistic rings or tubes such as carbon nanotubes at small diameters.",
"We have shown that is the consequence of small numbers of discrete bodies in an Eulerian buckling wavelength.",
"There remains two practical issues for future work.",
"The first is the extension of these calculations to discretised circular rings (polygons), where we do not expect the value of $\\beta ^2$ to be the same as for straight columns.",
"The second is the scaling factor between the value of $\\beta ^2$ for simple polygonal rings and carbon nanotubes.",
"This can be investigated by standard techniques, such as the study of structures intermediate in complexity between carbon nanotubes and linear discretised columns.",
"More fundamentally, attributing the reduction of the collapse pressure to the properties of the phonon dispersion curve might be considered to be a complete explanation of the phenomenon under study, or it might be considered to beg the question, why does the phonon dispersion curve behave in this way?",
"We are not aware of a good answer to this question.",
"Very generally, the phase velocity of a wave, given by the square root of stiffness over inertia (mass density), decreases as the wavelength is reduced toward the lattice constant.",
"Viewed macroscopically, should this be interpreted as a decrease in the effective stiffness?",
"Whatever that explanation may be, it would then fully explain why the Euler buckling force decreases when $a_{0}$ of a discretised pillar approaches $L$ .",
"There is a strong tendency to consider that phenomena arising on the atomic scale (such as phonon dispersion curves) are properties of the very small, and fully explained by a mathematical derivation invoking e.g.",
"Bloch's theorem, without considering if the same phenomena would arise at the macroscopic scale.",
"Yet here the reduction in collapse pressure is the same if $a_{0}$ is $1{}\\text{ or }3{}$ , for the same $N$ .",
"Thus the static properties of structures at all scales are here linked to the softening of acoustic phonon modes at wavelengths approaching the atomic scale.",
"Acknowledgement: DJD is grateful to the University of Lyons 1 (Labex iMust) for support for the initiation of this work."
]
] | 2011.14120 |
[
[
"Bayesian noise wave calibration for 21-cm global experiments"
],
[
"Abstract Detection of millikelvin-level signals from the 'Cosmic Dawn' requires an unprecedented level of sensitivity and systematic calibration.",
"We report the theory behind a novel calibration algorithm developed from the formalism introduced by the EDGES collaboration for use in 21-cm experiments.",
"Improvements over previous approaches are provided through the incorporation of a Bayesian framework and machine learning techniques such as the use of Bayesian evidence to determine the level of frequency variation of calibration parameters that is supported by the data, the consideration of correlation between calibration parameters when determining their values and the use of a conjugate-prior based approach that results in a fast algorithm for application in the field.",
"In self-consistency tests using empirical data models of varying complexity, our methodology is used to calibrate a 50 $\\Omega$ ambient-temperature load.",
"The RMS error between the calibration solution and the measured temperature of the load is 8 mK, well within the 1$\\sigma$ noise level.",
"Whilst the methods described here are more applicable to global 21-cm experiments, they can easily be adapted and applied to other applications, including telescopes such as HERA and the SKA."
],
[
"Introduction",
"For nearly a century, scientists have been using radio-frequency instruments to advance the study of astronomy and complement information from the visual regime of the electromagnetic spectrum [18].",
"As we begin to take measurements of the early universe, these instruments must continue to evolve to support observations.",
"Unexplored cosmic information from the Epoch of Reionisation and Cosmic Dawn redshifted into the radio spectrum could provide constraints on fundamental physics such as primordial black holes, galaxy formation, and universal curvature as discussed in [11].",
"A unique probe of phenomena from the early cosmos is the hydrogen that inundates the intergalactic medium (IGM).",
"Heating and cooling of the IGM associated with hydrogen's absorption and emission of 21-cm photons produce a dynamic brightness temperature relative to the cosmic microwave background temperature, tracing the evolution of surrounding structure during the Cosmic Dawn.",
"The brightness temperature of this 21-cm photon signal can be described by $\\begin{aligned}T_{21}(z) \\approx & \\ 0.023 \\mathrm {K} \\ \\times \\\\& x_{{H}{i}}(z) \\left[ \\left(\\frac{0.15}{\\Omega _{\\mathrm {m}}} \\right)\\left(\\frac{1+z}{10}\\right) \\right]^{\\frac{1}{2}} \\left(\\frac{\\Omega _{\\mathrm {b}}h}{0.02}\\right)\\left[1-\\frac{T_{\\mathrm {R}}(z)}{T_{\\mathrm {S}}(z)}\\right],\\end{aligned}$ which is heavily dependent on environmental factors of the early universe such as $x_{{H}{i}}$ , the fraction of neutral hydrogen, $\\Omega _{\\mathrm {m}}$ and $\\Omega _{\\mathrm {b}}$ , the matter and baryon densities with respect to the universal critical density for a flat universe and Hubble's constant.",
"Here, the $0.023$ is a constant from atomic-line physics.",
"$T_{\\mathrm {R}}$ is the background radiation temperature and $T_{\\mathrm {S}}$ is known as the `21-cm spin temperature', which is related to the kinetic temperature of neutral hydrogen gas in the IGM [23], [10].",
"This cosmic hydrogen signature measurable in the spectral sky has been redshifted to wavelengths under 200 MHz through the expansion of the universe as discussed in [18].",
"There has been a recent surge in the field of 21-cm cosmology following the reported detection of an absorption feature consistent with a Cosmic Dawn signature.",
"This was reported by the Experiment to Detect the Global EoR Signature (EDGES) in early 2018 from measurements of a sky-averaged radio spectrum [6].",
"The signal, centred at 78 MHz with a width corresponding to a period between 180 million and 270 million years after the Big Bang, matches the theoretical position in frequency, but its depth of $\\sim 0.5$ K is a factor of two greater than the largest predictions from theoretical models [7].",
"This discrepancy would suggest that the temperature difference between the IGM and the cosmic microwave background was much larger than previously thought and would require new physics to explain, such as dark matter-baryon interactions [4] or excess radio backgrounds [9].",
"Another possible explanation for this discrepancy is that the measured signal is not cosmological but of systematic origin.",
"This may be the case in EDGES due to some of the methodology used, such as a potentially unphysical foreground removal method and calibration of the receiver in a separate environment from the data acquisition [13], [19].",
"In this paper, we present a novel calibration algorithm that improves on the work of the EDGES team [20] through the utilisation of a Bayesian framework to promote efficient use of the data to remove systematics.",
"Using conjugate priors and machine learning techniques, our pipeline can be applied in the field with the collection of data with additional capabilities for optimising individual noise wave parameters and incorporating correlations between them.",
"This paper is organised as follows.",
"In theory we review the methodology behind calibration using noise waves as well as present a Bayesian framework that provides greater flexibility in radiometer calibration.",
"mockdata describes the process of using mock data sets modelled after empirical measurements of reflection coefficients with the incorporation of a realistic noise model to evaluate our pipeline."
],
[
"Methods",
"In this section, we detail the methodology behind radiometer calibration using noise wave parameters.",
"An overview of global signal measurement are outlined in measSig.",
"edgesCalibration summarises the basic procedure with some mathematical improvements while chap:bayes describes our Bayesian framework and its associated advantages."
],
[
"Measuring the global signal",
"The noise necessitating calibration emerges during measurement-taking.",
"In an averaged or global experiment, the sky temperature $(\\Omega , \\nu , t)$ is a function of the direction $\\Omega $ , frequency $\\nu $ and time $t$ .",
"This can be broken down into two primary components: the global 21-cm signal $T_{21}$ and astrophysical foregrounds $f̰$ $(\\Omega , \\nu , t) = T_{21}(\\nu ) + f̰(\\Omega , \\nu , t).$ The antenna measures the sky signal convolved with the normalised antenna directivity $B$ .",
"The process of measurement introduces the random noise term $N_{\\mathrm {data}}$ .",
"$D(\\nu , t) = \\int (\\Omega , \\nu , t) B(\\Omega , \\nu )\\mathrm {d}\\Omega + N_{\\mathrm {data}}.$ Our 21-cm signature can thus be represented as $T_{21} \\approx D(\\nu , t) - \\int f̰(\\Omega , \\nu , t)B(\\Omega , \\nu )\\mathrm {d}\\Omega - N_{\\mathrm {data}}.$ Here, the integral is assessed through foreground and beam modelling techniques such as those discussed in [2] while modelling of $N_{\\mathrm {data}}$ from the statistical properties of $D(\\nu , t)$ is accomplished by a calibration algorithm as articulated in this paper and outlined in fig:nsfig.",
"Having a fully Bayesian framework when modelling the beam, the sky and the systematics has major advantages for global 21-cm experiments such as REACH [1], as it provides the greatest flexibility in being able to model all effects and jointly fit for them.",
"Figure: Diagram showing the evolution of the 21-cm signal hampered by astrophysical foregrounds, convolvution with the antenna beam and the emergence of measurement noise before calibration to retrieve the antenna temperature."
],
[
"Calibration methodology",
"The standard calibration strategy follows the method introduced by Dicke to characterise systematic features in radio frequency instruments [8] and is widely used in experiments such as EDGES [15] and LOFAR [5] to evaluate the spectral index of the sky's diffuse radio background [20].",
"This technique involves measurements of two internal reference standards; a load and a noise source, in addition to a series of external calibration sources attached to the receiver input in lieu of the antenna.",
"These include an ambient-temperature ‘cold’ load, a ‘hot’ load heated to $\\sim 400$ K, an open-ended cable and a shorted cable.",
"A block diagram showing this arrangement is presented in f:dickeswitchpic.",
"When calibrating the receiver, reflection coefficients are taken of the calibration source connected to the receiver input ($\\Gamma _{\\mathrm {cal}}$ ) and of the receiver itself ($\\Gamma _{\\mathrm {rec}}$ ) as well as power spectral densities (PSDs) of the input ($P_{\\mathrm {cal}}$ ), the internal reference load ($P_{\\mathrm {L}}$ ) and the internal reference noise source ($P_{\\mathrm {NS}}$ ) [15].",
"These measurements are used to calculate a preliminary `uncalibrated' antenna temperature $^*$ Figure: Diagram of a typical calibration setup.",
"For characterisation of the receiver, a switch cycles between a calibrator connected to the input and the two internal references.$^* = \\left(\\frac{P_{\\mathrm {cal}}-P_{\\mathrm {L}}}{P_{\\mathrm {NS}}-P_{\\mathrm {L}}}\\right) + L̰,$ where $L̰$ and $$ are assumptions for the noise temperature of the internal reference load and excess noise temperature of the internal noise source above ambient, respectively.",
"This initial calculation is used to calibrate out any time-dependent system gain that emerges from a series of filters, amplifiers and cables, as well as the analogue-to-digital converter within the experimental apparatus [15].",
"Each PSD measurement can be expressed in terms of specific response contributions as detailed in [6] $\\begin{aligned}P_{\\mathrm {cal}} = g_{\\mathrm {sys}} \\Bigg [ &\\left(1-|\\Gamma _{\\mathrm {cal}}|^2\\right)\\left|\\frac{\\sqrt{1 - |\\Gamma _{\\mathrm {rec}}|^2}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}}\\right|^2 \\\\+ & |\\Gamma _{\\mathrm {cal}}|^2\\left|\\frac{\\sqrt{1 - |\\Gamma _{\\mathrm {rec}}|^2}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}}\\right|^2 \\\\+ & \\operatorname{Re}\\left(\\Gamma _{\\mathrm {cal}}\\frac{\\sqrt{1 - |\\Gamma _{\\mathrm {rec}}|^2}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}}\\right) \\\\+ & \\operatorname{Im}\\left(\\Gamma _{\\mathrm {cal}}\\frac{\\sqrt{1 - |\\Gamma _{\\mathrm {rec}}|^2}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}}\\right)+ T_0 \\Bigg ].\\end{aligned}$ Here, $g_{\\mathrm {sys}}$ is the system gain referenced to the receiver input and $$ is our calibrated input temperature.",
"$$ , $$ , and $$ are the ‘noise wave parameters’ introduced by [14] to calibrate the instrument.",
"$$ represents the portion of noise reflected by the antenna that is uncorrelated with the output noise of the low noise amplifier (LNA).",
"$$ and $$ are the portions of reflected noise correlated with noise from the LNA [15], [20].",
"In the EDGES experiment, these calibration quantities are modelled using seven-term polynomials in frequency.",
"The PSDs for the internal reference load and noise source can similarly be expressed as in eqn:pant.",
"However, since the reflection coefficients of the internal references are typically less than 0.005, they are taken to be zero in order to simplify the equations $P_{\\mathrm {L}} = g_{\\mathrm {sys}}^*[L̰\\left(1-|\\Gamma _{\\mathrm {rec}}|^2\\right)+T_{0}^*],$ $P_{\\mathrm {NS}} = g_{\\mathrm {sys}}^*[\\left(L̰+\\right)\\left(1-|\\Gamma _{\\mathrm {rec}}|^2\\right)+T_{0}^*].$ As shown in f:dickeswitchpic, the internal references may be on a separate reference plane than the receiver input, resulting in a system gain $g_{\\mathrm {sys}}^*$ and a noise offset $T_{0}^*$ different from those defined in eqn:pant.",
"This effect is taken into account by two additional scale and offset parameters, $C_1$ and $C_2$ , introduced by EDGES [15].",
"Since $C_1$ and $C_2$ also correct for first-order assumptions in the noise temperatures of the internal reference load and noise source, we have chosen to absorb these terms into $L̰$ and $$ .",
"This adjustment allows all calibration parameters, $$ , $$ , $$ , and an ‘effective’ $$ and $L̰$ , to be solved for in units of kelvin, facilitating a joint solution of parameters.",
"Expanding eqn:tantstar using eqn:pant,eqn:pl,eqn:pns yields a linear identity providing a relationship between the uncalibrated input temperature and a final calibrated temperature of any device connected to the receiver input $\\begin{aligned}\\left( \\frac{P_{\\mathrm {cal}} - P_{\\mathrm {L}}}{P_{\\mathrm {NS}} - P_{\\mathrm {L}}} \\right) + L̰&= \\left[ \\frac{1-|\\Gamma _{\\mathrm {cal}}|^2}{|1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}|^2} \\right] \\\\& + \\left[ \\frac{|\\Gamma _{\\mathrm {cal}}|^2}{|1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}|^2} \\right] \\\\& + \\left[ \\frac{\\operatorname{Re}\\left(\\frac{\\Gamma _{\\mathrm {cal}}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}}\\right)}{\\sqrt{1-|\\Gamma _{\\mathrm {rec}}|^2}} \\right] \\\\& + \\left[ \\frac{\\operatorname{Im}\\left(\\frac{\\Gamma _{\\mathrm {cal}}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}}\\right)}{\\sqrt{1-|\\Gamma _{\\mathrm {rec}}|^2}} \\right], \\\\\\end{aligned}$ where all parameters are frequency-dependent.",
"This is not explicitly shown for simplicity of notation.",
"For estimation of the noise wave parameters, $$ , $\\Gamma _{\\mathrm {cal}}$ and $\\Gamma _{\\mathrm {rec}}$ are measured along with the PSDs while $g_{\\mathrm {sys}}$ and $$ are calibrated out.",
"The cold and hot loads exhibit the main temperature references needed for $L̰$ and $$ .",
"The cables facilitate the derivation of the noise wave parameters describing spectral ripples from the noise properties of the receiver by acting as antennas looking at an isotropic sky with temperatures equal to the cables' physical temperatures [20]."
],
[
"Bayesian calibration framework",
"One possible source of systematics in the calibration methodology used by EDGES comes from measuring the response of the four external calibrators along with the receiver reflection coefficient in a laboratory away from where the instrument is actually deployed [6].",
"This process, especially with regards to how calibration parameters change, can be non-trivial.",
"Furthermore, the fixed polynomial order used by EDGES for all noise wave parameters may underfit or overfit individual parameters and thus `fit out' data useful for determining systematics or potentially even the 21-cm signal itself if a joint fit is performed.",
"In response to these issues, we have developed a calibration pipeline that improves on the strategies presented in edgesCalibration.",
"We introduce a novel Bayesian methodology using conjugate priors for a dynamic application of our algorithm to be run with data collection regardless of system complexity.",
"Also included are model selection methods using machine learning techniques for the optimisation of individual noise wave parameters to combat overfitting and underfitting, the results of which converge with that of a least-squares approach when wide priors are adopted.",
"Our pipeline easily incorporates many more calibrators than the standard four shown in f:dickeswitchpic to increase constraints on noise wave parameters while identifying possible correlations between them.",
"A schematic of the improved calibration method is shown in flowchart.",
"Figure: Outline of the Bayesian calibration algorithm.",
"Blue blocks represent data to be taken, red blocks represent calculations and green blocks represent calculation outputs.In order to simplify our calibration approach, we first define the following terms $X_{\\mathrm {unc}} = -\\frac{|\\Gamma _{\\mathrm {cal}}|^2}{ 1-|\\Gamma _{\\mathrm {cal}}|^2},$ $X_{\\mathrm {L}} = \\frac{|1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}|^2}{1-|\\Gamma _{\\mathrm {cal}}|^2},$ $X_{\\mathrm {cos}} = -\\operatorname{Re}\\left(\\frac{\\Gamma _{\\mathrm {cal}}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}} \\times \\frac{X_{\\mathrm {L}}}{\\sqrt{1-|\\Gamma _{\\mathrm {rec}}|^2}}\\right),$ $X_{\\mathrm {sin}} = -\\operatorname{Im}\\left(\\frac{\\Gamma _{\\mathrm {cal}}}{1-\\Gamma _{\\mathrm {cal}}\\Gamma _{\\mathrm {rec}}} \\times \\frac{X_{\\mathrm {L}}}{\\sqrt{1-|\\Gamma _{\\mathrm {rec}}|^2}}\\right),$ $X_{\\mathrm {NS}} = \\left( \\frac{P_{\\mathrm {cal}}-P_{\\mathrm {L}}}{P_{\\mathrm {NS}}-P_{\\mathrm {L}}} \\right) X_{\\mathrm {L}},$ which represent initial calibration measurements on $D$ in the frequency domain for the characterisation of $N_{\\mathrm {data}}$ from bayestsource via our noise wave parameters.",
"It is expected that calibration-related deviations of $D$ in the time domain are sufficiently curtailed through practical strategies such as temperature control of the receiver environment.",
"Incorporating these into eqn:caleqn, with some rearrangement, then gives the equation $X_{\\mathrm {unc}} + X_{\\mathrm {cos}} + X_{\\mathrm {sin}} + X_{\\mathrm {NS}} + X_{\\mathrm {L}}L̰ = ,$ at each frequency.",
"Here, there are no squared or higher-order terms, allowing us to take advantage of the linear form by grouping the data and noise wave parameters into separate matrices X Xunc Xcos Xsin XNS XL , Tunc Tcos Tsin TNS TL.",
"In these equations, all of our data; the reflection coefficient measurements and power spectral densities, are grouped in an ${X}$ vector which forms a matrix where one of the axes is frequency.",
"The calibration parameters as frequency-dependent polynomials of varying degree are collected into a ${{\\Theta }}$ vector which serves as our model describing $N_{\\mathrm {data}}$ .",
"Applying these definitions condenses the calibration equation into ${T}_{\\mathrm {cal}}= {X}{{\\Theta }}+\\sigma ,$ where ${T}_{\\mathrm {cal}}$ is a vector over frequency and $\\sigma $ is a noise vector representing our error.",
"Since EDGES assumes that each power spectral density measurement is frequency independent, we have assumed that $\\sigma $ is a multivariate normal distribution.",
"This assumption is implicit in the EDGES analysis in which they use a least-squares minimisation approach for solving model parameters.",
"For calibration of the receiver, we are concerned with the construction of predictive models of the noise wave parameters, ${\\Theta }$ , in the context of some dataset, ${T}$ .",
"We can use ${\\Theta }$ to calculate the probability of observing the data given a specific set of noise wave parameters: $\\begin{aligned}p\\big ({T} \\:\\big \\vert \\: {\\Theta }, \\sigma ^2\\big ) &= \\\\ & \\frac{1}{2\\pi \\sigma ^2}^{N/2}\\exp { \\Bigg \\lbrace -\\frac{1}{2\\sigma ^2}\\left({T}-{X}{\\Theta }\\right)^{\\top }\\left({T} -{X}{\\Theta }\\right) \\Bigg \\rbrace },\\end{aligned}$ where, $N$ is the number of measurements.",
"This distribution on the data is the likelihood.",
"For the purposes of calibration, ${T}$ may be ${T}_{\\mathrm {cal}}$ measurements or alternatively, ${T}_{\\mathrm {sky}}$ for prediction of a sky signal.",
"Our model must also specify a prior distribution, quantifying our initial assumptions on the values and spread of our noise wave parameters which we specify as a multivariate normal inverse gamma distribution: $\\begin{aligned}p\\left({\\Theta }, \\sigma ^2\\right) \\propto & \\left(\\frac{1}{\\sigma ^2}\\right)^{a+1+\\left(d/2\\right)} \\times \\\\ &\\exp \\left[ -\\frac{1}{\\sigma ^2}\\lbrace b+\\frac{1}{2}\\left({\\Theta }-{\\mu }_{{\\Theta }}\\right)^{\\top }{V}_{{\\Theta }}^{-1}\\left({\\Theta }-{\\mu }_{{\\Theta }}\\right)\\rbrace \\right],\\end{aligned}$ which is proportional up to an integration constant.",
"Here, $a$ and $b$ , which are greater than zero, along with ${V}_{{\\Theta }}$ and ${\\mu }_{{\\Theta }}$ represent our prior knowledge on the noise wave parameters.",
"$d$ is the length of our vector ${\\Theta }$ .",
"likelihood is determined by a set of values for our model ${\\Theta }$ .",
"We can marginalise out the dependence on ${\\Theta }$ and our noise term by integrating over the prior distribution by both ${\\Theta }$ and $\\sigma ^2$ at once.",
"Following the steps in [3] $\\begin{aligned} p\\left({T}_{\\mathrm {cal}}\\right) &= \\int p\\left({T}_{\\mathrm {cal}}\\:\\big \\vert \\: {\\Theta }, \\sigma ^2\\right) p\\left({\\Theta }, \\sigma ^2\\right) \\mathrm {d}{\\Theta } \\mathrm {d}\\sigma ^2\\\\&= \\frac{b^a\\Gamma \\left(a^*\\right)\\sqrt{|{V}^*|}}{{b^*}^{a^*}\\Gamma \\left(a\\right)\\sqrt{|{V}_{{\\Theta }}|}}(2\\pi )^{-N/2}, \\\\\\end{aligned}$ where $\\begin{aligned}a^* &= a + \\frac{N}{2}, \\\\b^* &= b + \\frac{1}{2}[{\\mu }_{{\\Theta }}^{\\top }{V}_{{\\Theta }}^{-1}{\\mu }_{{\\Theta }} + {T}_{\\mathrm {cal}}^{\\top }{T}_{\\mathrm {cal}}- {\\mu }^{*\\top }{V}^{*-1}{\\mu }^*], \\\\{\\mu }^* &= \\left({V}_{{\\Theta }}^{-1} + {X}^{\\top }{X}\\right)^{-1}\\left({V}_{{\\Theta }}^{-1}{\\mu }_{{\\Theta }} + {X}^{\\top }{T}_{\\mathrm {cal}}\\right), \\\\{V}^* &= \\left({V}_{{\\Theta }}^{-1} + {X}^{\\top }{X}\\right)^{-1}, \\\\\\end{aligned}$ and $\\Gamma \\left(x\\right)$ represents the Gamma function, not to be confused with the notation for our reflection coefficients.",
"eqn:ev is the evidence, which gives the probability of observing the data ${T}_{\\mathrm {cal}}$ given our model.It is in fact better to use the equivalent more numerically stable expression $b^*=b + {q}^{\\top } {q} + {q}^{\\top } {X} {V}_{{\\Theta }} {X}^{\\top } {q}$ , where ${q}= {T}_{\\mathrm {cal}}-{X}{\\mu }^*$ to avoid cancellation of large terms.",
"With the prior distribution specified, we use Bayes' equation to invert the conditioning of the likelihood and find the posterior using the likelihood, prior and evidence: $p\\left({\\Theta }, \\sigma ^2 \\:\\big \\vert \\: {T}_{\\mathrm {cal}}\\right) = \\frac{p\\left({T}_{\\mathrm {cal}}\\:\\big \\vert \\: {\\Theta }, \\sigma ^2\\right)p\\left({\\Theta }, \\sigma ^2\\right)}{p\\left({T}_{\\mathrm {cal}}\\right)}.$ Similarly from [3], this can be written as $\\begin{aligned} p\\Bigl ({\\Theta },\\sigma ^2 \\:\\big \\vert \\: & {T}_{\\mathrm {cal}}\\Bigl ) \\propto \\left(\\frac{1}{\\sigma ^2}\\right)^{a^* + \\frac{d}{2} + 1} \\times \\\\& \\exp { \\Bigg \\lbrace -\\frac{1}{\\sigma ^2} \\Bigg [ b^* + \\frac{1}{2}\\left({\\Theta } - {\\mu }^*\\right)^{\\top }{V}^{*-1}\\left({\\Theta } - {\\mu }^*\\right) \\Bigg ] \\Bigg \\rbrace }.\\end{aligned}$ The posterior distribution represents the uncertainty of our parameters after analysis, reflecting the increase in information [16].",
"We highlight the difference between the `likelihood-only' least-squares approach versus the Bayesian approach with the former being a special case of the latter with very wide priors demonstrable when ${V}_{{\\Theta }} \\rightarrow \\infty \\Rightarrow {V}_{{\\Theta }}^{-1} \\rightarrow 0$ , and ${\\mu }^*$ becomes ${\\Theta }$ .",
"The transition from `non-starred' variables to `starred' variables represents our `Bayesian update' of the prior to the posterior noise wave parameters in light of the calibration data ${T}_{\\mathrm {cal}}$ .",
"As we can see, the posterior distribution is in the same probability distribution family as eqn:prior, making our prior a conjugate prior on the likelihood distribution.",
"The use of conjugate priors gives a closed-form solution for the posterior distribution through updates of the prior hyperparameters via the likelihood function [3], [17].",
"The resulting numerical computation is many orders of magnitude faster than MCMC methods relying on full numerical sampling and permits an in-place calculation in the same environment as the data acquisition.",
"This becomes particularly useful for the speed of the algorithm as frequency dependence is introduced in which the computations would not be manageable without conjugate gradients.",
"To allow for a smooth frequency dependency, we promote each of our noise wave parameters in eqn:theta to a vector of polynomial coefficients $ḭ = \\begin{pmatrix}ḭ^{[0]}, & ḭ^{[1]}, & ḭ^{[2]}, & ..., & ḭ^{[n]}\\end{pmatrix},$ where $i$ is our noise wave parameter label; $i \\in \\lbrace \\mathrm {unc, \\ cos, \\ sin , \\ NS, \\ L}\\rbrace $ , modelled using $n+1$ polynomial coefficients.",
"Likewise ${X}_{i} = \\begin{pmatrix}{X}_{i}, & {X}_{i}\\left(\\frac{\\nu }{\\nu _0}\\right), & {X}_{i}{\\left(\\frac{\\nu }{\\nu _0}\\right)}^2, & ..., & {X}_{i}{\\left(\\frac{\\nu }{\\nu _0}\\right)}^{n}\\end{pmatrix},$ where $\\nu $ is a vector of input frequencies which are raised to powers up to $n$ .",
"For a vector of $n$ 's attributed to our calibration parameters, under this notation multiplication in eqn:linearmodel is element-wise and eqn:ev is effectively $p\\left({T}_{\\mathrm {cal}}|{n}\\right)$ .",
"Assuming a uniform prior on ${n}$ , inverting Bayes' theorem gives $p\\left({n}|{T}_{\\mathrm {cal}}\\right)$ for use in model comparison in which the relative probabilities of models can be evaluated in light of the data and priors.",
"Occam’s razor advises whether the extra complexity of a model is needed to describe the data [21], permitting optimisation of the polynomial orders for individual noise wave parameters as detailed in chap:opt.",
"By taking a random sampling of the resulting posterior, we characterise the noise wave parameters as multivariate distributions depicted in contour plots which exhibit a peak value accompanied by $1\\sigma $ and $2\\sigma $ variance as well as correlation between parameters inferred from a covariance matrix.",
"Following characterisation of the receiver, we next apply the ${T}_{\\mathrm {cal}}$ from our calibration to a set of raw antenna data $\\hat{{X}}$ for prediction of our sky signal, ${T}_{\\mathrm {sky}}$ , from bayestsource.",
"The predictions for the data follow from the posterior predictive distribution $p\\left({T}_{\\mathrm {sky}} \\:\\big \\vert \\: {T}_{\\mathrm {cal}} \\right) = \\int p\\left( {T}_{\\mathrm {sky}} \\:\\big \\vert \\: {\\Theta },\\sigma ^2 \\right) p \\left( {\\Theta },\\sigma ^2 \\:\\big \\vert \\: {T}_{\\mathrm {cal}} \\right) \\mathrm {d}{\\Theta }\\mathrm {d}\\sigma ^2.$ The first probability in the integral is the likelihood for our antenna measurement ${T}_{\\mathrm {sky}}$ and the second is our posterior from eqn:post.",
"Following the steps in [3], this can be shown to be a multivariate Student's t-distribution written as: $\\begin{aligned}p\\Big ( & {T}_{\\mathrm {sky}} \\:\\big \\vert \\: {T}_{\\mathrm {cal}} \\Big ) = \\frac{\\Gamma \\left( a^* + \\frac{d}{2} \\right)}{\\Gamma \\left( a^* \\right)\\pi ^{\\frac{d}{2}}|2b^*\\left( I + \\hat{{X}}{V}^*\\hat{{X}}^{\\top }\\right)|^{\\frac{1}{2}}}\\\\ & \\times \\left[ 1 + \\frac{\\left( {T}_{\\mathrm {sky}} - \\hat{{X}}{\\mu }^* \\right)^{\\top } \\left( I + \\hat{{X}}{V}^*\\hat{{X}}^{\\top } \\right)^{-1} \\left( {T}_{\\mathrm {sky}} - \\hat{{X}}{\\mu }^* \\right)}{2b^*} \\right]^{-\\left( a^* + \\frac{d}{2} \\right)},\\end{aligned}$ where $I$ is the $N \\times N$ identity matrix and $a^*$ , $b^*$ , ${\\mu }^*$ and ${V}^*$ are defined in starred.",
"This new distribution on ${T}_{\\mathrm {sky}}$ corresponds to a set of points with error bars and represents the calibrated sky temperature as the output of the receiver.",
"To verify the performance of our pipeline and highlight features of the algorithm, we evaluate the results of self-consistency checks using empirical models of data based on measurements taken in the laboratory.",
"To make this data as realistic as possible, we used actual measurements of the reflection coefficients of many types of calibrators (see tab:calibrators) to generated power spectral densities using eqn:pant,eqn:pl,eqn:pns given a set of realistic model noise wave parameters along with some assumptions about the noise, which are described in chap:solution.",
"The impedance of the calibrators which were measured with a vector network analyser (VNA) and used in our pipeline are shown on a Smith chart in f:smith Table: Table of calibrators used in the creation of our empirical data models for analysis.",
"Calibrators are added in pairs in the order below when increasing the number of calibration sources used by our algorithm.Figure: Smith chart (Argand diagram) showing the measured complex impedance of the calibrators used in the Bayesian pipeline across a range of frequencies.We start by demonstrating the importance of correlation between noise wave parameters when determining their values to provide a better calibration solution for the reduction of systematic features in the data such as reflections (chap:correlation).",
"We then show the increased constraints on these noise wave parameters attributed to the inclusion of more calibrators than the standard number of four (chap:multCal).",
"Following this, we illustrate the effectiveness of model selection for the optimisation of individual noise wave parameters to prevent the loss of information resulting from overfitting or underfitting of the data (chap:opt).",
"Finally, these features are incorporated into a calibration solution applied to a $50 \\ \\Omega $ load (chap:solution)."
],
[
"Correlation between noise wave parameters",
"In this section, we show the first major feature of our Bayesian pipeline; the consideration of correlation between noise wave parameters when deriving their values.",
"This is best demonstrated when noise is introduced in an idealised way as to retain a form matching the Gaussian form of our mathematical model.",
"To do this, empirical models of power spectral densities are calculated from eqn:pant,eqn:pl,eqn:pns using measurements of $\\Gamma _{\\mathrm {rec}}$ , $\\Gamma _{\\mathrm {cal}}$ and $$ for the cold and hot loads, as well as a set of realistic noise wave parameters.",
"Gaussian noise of one unit variation is then added to the $$ measurements after the calculation to conserve its Gaussian form.",
"This data is submitted to our algorithm and the resulting posterior distributions for coefficients of the polynomial noise wave parameters are compared to the initial values.",
"Such posterior distributions can be seen in f:goodplot showing the results of models using only the cold load (grey posterior), only the hot load (red posterior) and using both loads in tandem (blue posterior).",
"For these calculations we chose a set of model noise wave parameters as constants across the frequency band; = 250 K = 190 K = 90 K = 1200 K L̰ = 298 K In f:goodplot, a strong correlation between the $L̰$ and $$ is evident as the hot-load posterior is highly skewed as expected from eqn:xl,eqn:xns.",
"The resulting intersection of posteriors from the individual loads facilitate the derivation of noise wave parameters as the dual-load posterior is found within the region of posterior overlap crossing with the values of the model shown in the inset of f:goodplot.",
"Retrieval of the noise wave parameter values using correlations between them found in the data demonstrate the relevance of this information which is not taken into account in previous calibration techniques.",
"Figure: Plot showing the joint posteriors of L̰L̰ and for models using the cold load, the hot load, and both loads concurrently shown as the grey, red and blue posteriors respectively.",
"The black cross hairs mark the noise wave parameter values used to generate data submitted to the pipeline.",
"A zoom-in of the posterior intersection is provided to illustrate the constraint of noise wave parameter values attributed to the correlation between parameters."
],
[
"Constraints with additional calibrators",
"A nice feature of our pipeline is the ability to include as many calibrators as required to constrain the calibration parameters.",
"For analysis, six more calibrators are introduced in pairs following the order presented in tab:calibrators.",
"We include data generated from measurements of multiple resistors terminating a high quality 25 m cable made by GORE®.",
"Data for these calibrators is once again generated using fixed terms and Gaussian noise of one unit variation added to $$ as discussed above.",
"f:linearall shows the results of models using four, six, and eight calibrators.",
"Figure: Posterior results of our pipeline using data from four, six and eight calibrators shown in grey, red and blue, respectively.",
"Cross hairs mark the values of noise wave parameters used to generate the data.",
"These values fall within 1σ1\\sigma of the posterior mean values.",
"We can see that the constraint on noise wave parameter values increases with the number of calibrators used in our pipeline which is encouraging.As shown, the inclusion of more calibrators increases the constraint on the resulting noise wave parameters.",
"However, we note that after the inclusion of four calibrators, the relative additional constraint decreases with each additional calibrator and thus the use of more than eight calibrators would be unnecessary.",
"The values of noise wave parameters used to generate the data as indicated by the cross hairs in f:linearall all fall within $1\\sigma $ of our pipeline's resulting posterior averages for models using all eight calibrators."
],
[
"Optimisation of individual noise wave parameters",
"The final highlight of our Bayesian pipeline is a the use of machine learning techniques to optimise individual noise wave parameters.",
"This is advantageous as a blanket set of order-seven polynomials applied to all noise wave parameters, such as done in the EDGES experiment, may underfit or overfit individual parameters and misidentify systematics or information about the signal being measured.",
"The optimisation procedure compares the evidences (eqn:ev) of different models to determine the vector of noise wave parameter polynomial coefficients ${n}$ that best describes the data as briefly mentioned at the end of chap:bayes.",
"Since the model favoured by the data will have the highest evidence, we use a steepest descent procedure to compare models in `${n}$ -space' and determine the direction of the gradient in `evidence-space'.",
"After multiple iterations, this brings us to the model with the maximal evidence.",
"Since ${n}$ consists of five numbers corresponding to the number of polynomial coefficients for each of the five noise wave parameters, models are generated by individually increasing each index of ${n}$ by 1.",
"We expect the evidence to follow an `Occam's cliff,' in which the evidence sharply increases preceding the optimal ${n}$ with a slow fall off following the maximum.",
"To demonstrate this, data is generated using measurements from all eight calibrators of tab:calibrators and noise wave parameters as second-order polynomials = x2 -3x + 250 K = 2x2 + 190 K = 3x2 + 8x + 90 K = 4x2 + 5x + 1200 K L̰ = 5x2 + 10x + 298 K where $x$ is our normalised frequency.",
"Gaussian noise of one unit variation is applied to the calibrator input temperatures as before.",
"The evidences of various models are plotted in f:evidence in which an Occam's cliff can be seen peaking at polynomial order two.",
"As expected from the plot, the steepest descent algorithm finds that noise wave parameters modelled as second-order polynomials best describe the data.",
"Figure: Evidence of multiple models are plotted which display the Occam's cliff.",
"Data is generated using noise wave parameters as order-2 polynomials.",
"We see that for the model with the highest evidence, that is, the model favoured by the data, the number of polynomial coefficients matches that of the model noise wave parameters."
],
[
"Application with realistic noise",
"To demonstrate the robustness of our pipeline, we conducted self-consistency checks using empirically modelled data with a more complicated noise model.",
"This data was generated using reflection coefficients of eight calibrators and the receiver measured in the laboratory.",
"These reflection coefficients were then smoothed using a cubic smoothing spline [22] in order to maintain their approximate shape over frequency.",
"The same second-order noise wave parameters detailed in chap:opt are used with the reflection coefficients to generate our model power spectral densities.",
"Following this, we added of order 1% Gaussian noise independently to the smoothed $\\Gamma _{\\mathrm {rec}}$ and $\\Gamma _{\\mathrm {cal}}$ as well as $P_{\\mathrm {cal}}$ to more accurately represent the instrument noise from measurement equipment such as vector network analysers.",
"No noise was added to the calibrator input temperatures.",
"This results in a model that does not match the Gaussian form of our mathematical model as in the previous sections and thus does not demonstrate the features of our pipeline as explicitly, but is more representative of data set expected from measurements in the field.",
"Data for the receiver and the cold load generated using this noise model are shown in f:calQualities.",
"Figure: Power spectral densities and reflection coefficients for the receiver and the cold load generated under our realistic noise model.Using data generated for all eight calibrators with our realistic noise model, the calibration algorithm selects optimal polynomial orders matching those of the model noise wave parameters whose values fall within within $1\\sigma $ of the posterior peak values as shown in f:fgxSamples.",
"For these higher order tests, we use fgivenx plots which condense noise wave parameter posteriors into samples that can be compared to the model parameter values instead of comparing each individual coefficient [12].",
"When this calibration model is used to calibrate an ambient-temperature $50 \\ \\Omega $ load, the RMS error between the calibrated temperature and the measured temperature is 8 mK, well within the $1\\sigma $ noise level (bottom right panel of f:fgxSamples).",
"This level of accuracy is comparable to the 26 mK noise floor estimated for the EDGES pipeline in 2016 [15].",
"By individually adjusting each component of noise arising in our realistic noise model, we may determine what kind of noise our calibration algorithm is most sensitive to, as well as calculate the maximum amount of noise permissible for a specified level of systematic feature reduction.",
"These topics are intended to be explored in a future work.",
"Figure: Results from 1000 samples using data generated with our more realistic noise model (shown in black).",
"The second-order noise wave parameters shown in red are used to generate the data inputted to our pipeline.",
"The polynomial order and values of the noise wave parameters that best suit the data according to our algorithm match that of the empirical model.",
"This solution is applied to an ambient-temperature load, shown in the bottom right panel as our predictive y ^\\hat{y} from predictive, and calibrates it to within 1σ1\\sigma of ambient temperature.Here we presented the development of a calibration methodology based on the procedure used by EDGES but with key improvements to characterise reflections arising at connections within the receiver.",
"Our pipeline utilises the Dicke switching technique and a Bayesian framework in order to individually optimise calibration parameters while identifying correlations between them using a dynamic algorithm to be applied in the same environment as the data acquisition.",
"In a comprehensive investigation, we have evaluated our algorithm's interpretation of empirical models of data which have been generated from known noise wave parameters and a realistic noise model.",
"The solution, applied to an ambient-temperature $50 \\ \\Omega $ load, produces a calibrated temperature with an RMS residual temperature of 8 mK.",
"Future work for the pipeline regards application of real calibrator data, optimisation of noise wave parameter coefficients through marginalisation techniques and incorporation into an end-to-end simulation based on an entire experimental apparatus to better understand error tolerances.",
"The flexibility of the algorithm attributed to our novel approach allows its application to any experiment relying on similar forms of calibration such as REACH [1], were we intend to use this for in-the-field and on-the-fly radiometer calibration."
],
[
"Acknowledgements",
"ILVR would like to thank S. M. Masur for her helpful comments.",
"WJH was supported by a Gonville & Caius Research Fellowship, STFC grant number ST/T001054/1 and a Royal Society University Research Fellowship.",
"We would like to thank the The Cambridge-Africa ALBORADA Research Fund for their support.",
"We would also like to thank the Kavli Foundation for their support of the REACH experiment.",
"The data underlying this article will be shared on reasonable request to the corresponding author."
]
] | 2011.14052 |
[
[
"AdaBins: Depth Estimation using Adaptive Bins"
],
[
"Abstract We address the problem of estimating a high quality dense depth map from a single RGB input image.",
"We start out with a baseline encoder-decoder convolutional neural network architecture and pose the question of how the global processing of information can help improve overall depth estimation.",
"To this end, we propose a transformer-based architecture block that divides the depth range into bins whose center value is estimated adaptively per image.",
"The final depth values are estimated as linear combinations of the bin centers.",
"We call our new building block AdaBins.",
"Our results show a decisive improvement over the state-of-the-art on several popular depth datasets across all metrics.",
"We also validate the effectiveness of the proposed block with an ablation study and provide the code and corresponding pre-trained weights of the new state-of-the-art model."
],
[
"Introduction",
"This paper tackles the problem of estimating a high quality dense depth map from a single RGB input image.",
"This is a classical problem in computer vision that is essential for many applications [27], [31], [17], [7].",
"In this work, we propose a new architecture building block, called AdaBins that leads to a new state-of-the-art architecture for depth estimation on the two most popular indoor and outdoor datasets, NYU [37] and KITTI [14].",
"The motivation for our work is the conjecture that current architectures do not perform enough global analysis of the output values.",
"A drawback of convolutional layers is that they only process global information once the tensors reach a very low spatial resolution at or near the bottleneck.",
"However, we believe that global processing is a lot more powerful when done at high resolution.",
"Our general idea is to perform a global statistical analysis of the output of a traditional encoder-decoder architecture and to refine the output with a learned post-processing building block that operates at the highest resolution.",
"As a particular realization of this idea, we propose to analyze and modify the distribution of the depth values.",
"Depth distribution corresponding to different RGB inputs can vary to a large extent (see Fig.",
"REF ).",
"Some images have most of the objects located over a very small range of depth values.",
"Closeup images of furniture will, for example, contain pixels most of which are close to the camera while other images may have depth values distributed over a much broader range, e.g.",
"a corridor, where depth values range from a small value to the maximum depth supported by the network.",
"Along with the ill-posed nature of the problem, such a variation in depth distribution makes depth regression in an end-to-end manner an even more difficult task.",
"Recent works have proposed to exploit assumptions about indoor environments such as planarity constraints [26], [22] to guide the network, which may or may not hold for a real-world environment, especially for outdoors scenes.",
"Instead of imposing such assumptions, we investigate an approach where the network learns to adaptively focus on regions of the depth range which are more probable to occur in the scene of the input image.",
"Our main contributions are the following: We propose an architecture building block that performs global processing of the scene's information.",
"We propose to divide the predicted depth range into bins where the bin widths change per image.",
"The final depth estimation is a linear combination of the bin center values.",
"We show a decisive improvement for supervised single image depth estimation across all metrics for the two most popular datasets, NYU [37] and KITTI [14].",
"We analyze our findings and investigate different modifications on the proposed AdaBins block and study their effect on the accuracy of the depth estimation."
],
[
"Related Work",
"The problem of 3D scene reconstruction from RGB images is an ill-posed problem.",
"Issues such as lack of scene coverage, scale ambiguities, translucent or reflective materials all contribute to ambiguous cases where geometry cannot be derived from appearance.",
"Recently, methods that rely on convolutional neural networks (CNNs) are able to produce reasonable depth maps from a single RGB input image at real-time speeds.",
"Monocular depth estimation has been considered by many CNN methods as a regression of a dense depth map from a single RGB image [8], [25], [45], [16], [46], [11], [19], [1], [26], [22].",
"As the two most important competitors, we consider BTS [26] and DAV [22].",
"BTS uses local planar guidance layers to guide the features to full resolution instead of standard upsampling layers during the decoding phase.",
"DAV uses a standard encoder-decoder scheme and proposes to exploit co-planarity of objects in the scene via attention at the bottleneck.",
"Our results section compares to these (and many other) methods.",
"Encoder-decoder networks have made significant contributions in many vision related problems such as image segmentation [35], optical flow estimation [10], and image restoration [28].",
"In recent years, the use of such architectures have shown great success both in the supervised and the unsupervised setting of the depth estimation problem [15], [41], [21], [48], [1].",
"Such methods typically use one or more encoder-decoder networks as a sub part of their larger network.",
"In this paper we adapted the baseline encoder-decoder network architecture used by [1].",
"This allows us to more explicitly study the performance attribution of our proposed extension on the pipeline which is typically a difficult task.",
"Transformer networks are gaining greater attention as a viable building block outside of their traditional use in NLP tasks and into computer vision tasks [32], [43], [2], [6].",
"Following the success of recent trends that combine CNNs with Transformers [2], we propose to leverage a Transformer encoder as a building block for non-local processing on the output of a CNN.",
"Figure: Choices for bin widths.",
"Uniform and Log-uniform bins are pre-determined.",
"`Trained bins' vary from one dataset to another.",
"Adaptive bins vary for each input image."
],
[
"Methodology",
"In this section, we present the motivation for this work, provide details of the AdaBins architecture, and describe the corresponding loss functions used."
],
[
"Motivation",
"Our idea could be seen as a generalization of depth estimation via an ordinal regression network as proposed by Fu et al. [11].",
"Fu et al.",
"observed that a performance improvement could be achieved if the depth regression task is transformed into a classification task.",
"They proposed to divide the depth range into a fixed number of bins of predetermined width.",
"Our generalization solves multiple limitations of the initial approach.",
"First, we propose to compute adaptive bins that dynamically change depending on the features of the input scene.",
"Second, a classification approach leads to a discretization of depth values which results in poor visual quality with obvious sharp depth discontinuities.",
"This might still lead to good results with regard to the standard evaluation metrics, but it can present a challenge for downstream applications, e.g.",
"computational photography or 3D reconstruction.",
"Therefore, we propose to predict the final depth values as a linear combination of bin centers.",
"This allows us to combine the advantages of classification with the advantages of depth-map regression.",
"Finally, compared to other architectures, e.g.",
"DAV [22], we compute information globally at a high resolution and not primarily in the bottleneck part at a low resolution."
],
[
"AdaBins design",
"Here, we discuss four design choices of our proposed architecture that are most important for the obtained results.",
"First, we employ an adaptive binning strategy to discretize the depth interval $D = (d_{min}, d_{max})$ into $N$ bins.",
"This interval is fixed for a given dataset and is determined by dataset specification or manually set to a reasonable range.",
"To illustrate our idea of dividing a depth interval into bins, we would like to contrast our final solution with three other possible design choices we evaluated: Fixed bins with a uniform bin width: the depth interval $D$ is divided into $N$ bins of equal size.",
"Fixed bins with a log scale bin width: the depth interval $D$ is divided into bins of equal size in log scale.",
"Trained bin widths: the bin widths are adaptive and can be learned for a particular dataset.",
"While the bin widths are general, all images finally share the same bin subdivision of the depth interval $D$ .",
"AdaBins: the bin widths $\\textbf {b}$ are adaptively computed for each image.",
"We recommend the strategy of AdaBins as the best option and our ablation study validates this choice by showing the superiority of this design over its alternatives.",
"An illustration of the four design choices for bin widths can be seen in Fig.",
"REF .",
"Second, discretizing the depth interval $D$ into bins and assigning each pixel to a single bin leads to depth discretization artifacts.",
"We therefore predict the final depth as a linear combination of bin centers enabling the model to estimate smoothly varying depth values.",
"Third, several previous architectures propose performing global processing using attention blocks to process information after an encoder block in the architecture (e.g., image captioning [5], [18] or object detection [2]).",
"Also, the current state-of-the-art in depth estimation uses this strategy [22].",
"Such an architecture consists of three blocks ordered as such: encoder, attention, followed by a decoder.",
"We initially followed this approach but noticed that better results can be achieved when using attention at the spatially higher resolution tensors.",
"We therefore propose an architecture that also has these three blocks, but ordered as follows: encoder, decoder, and finally attention.",
"Fourth, we would like to build on the simplest possible architecture to isolate the effects of our newly proposed AdaBins concept.",
"We therefore build on a modern encoder-decoder [1] using EfficientNet B5 [40] as the backbone for the encoder.",
"In the next subsection, we provide a description of the entire architecture.",
"Figure: An overview of the mini-ViT block.",
"The input to the block is a multi-channel feature map of the input image.",
"The block includes a Transformer encoder that is applied on patch embeddings of the input for the purpose of learning to estimate bin widths bb and a set of convolutional kernels needed to compute our Range-Attention-Maps RR."
],
[
"Architecture description",
"Fig.",
"REF shows an overview of our proposed depth estimating architecture.",
"Our architecture consists of two major components: 1) an encoder-decoder block built on a pre-trained EfficientNet B5 [40] encoder and a standard feature upsampling decoder; 2) our proposed adaptive bin-width estimator block called AdaBins.",
"The first component is primarily based on the simple depth regression network of Alhashim and Wonka [1] with some modifications.",
"The two basic modifications are switching the encoder from DenseNet [20] to EfficientNet B5 and using a different appropriate loss function for the new architecture.",
"In addition, the output of the decoder is a tensor $\\textbf {x}_d \\in \\mathbb {R}^{h \\times w \\times C_d}$ , not a single channel image representing the final depth values.",
"We refer to this tensor as the “decoded features\".",
"The second component is a key contribution in this paper, the AdaBins module.",
"The input to the AdaBins module are decoded features of size $h \\times w \\times C_d$ and the output tensor is of size $h \\times w \\times 1$ .",
"Due to memory limitations of current GPU hardware, we use $h=H/2$ and $w=W/2$ to facilitate better learning with larger batch sizes.",
"The final depth map is computed by simply bilinearly upsampling to $H\\times W\\times 1$ .",
"The first block in the AdaBins module is called mini-ViT.",
"An overview of this block is shown in Fig.",
"REF .",
"It is a simplified version of a recently proposed technique of using transformers for image recognition [6] with minor modifications.",
"The details of mini-ViT are explained in the next paragraph.",
"There are two outputs of mini-ViT: 1) a vector $\\textbf {b}$ of bin-widths, which defines how the depth interval $D$ is to be divided for the input image, and 2) Range-Attention-Maps $\\mathcal {R}$ of size $h \\times w \\times C$ , that contain useful information for pixel-level depth computation.",
"Table: Mini-ViT architecture details."
],
[
"Mini-ViT.",
"Estimating sub-intervals within the depth range $D$ which are more probable to occur for a given image would require a combination of local structural information and global distributional information at the same time.",
"We propose to use global attention in order to calculate a bin-widths vector $\\textbf {b}$ for each input image.",
"Global attention is expensive both in terms of memory and computational complexity, especially at higher resolutions.",
"However, recent rapid advances in transformers provide some efficient alternatives.",
"We take inspiration from the Vision Transformer ViT [6] in designing our AdaBins module with transformers.",
"We also use a much smaller version of the transformer proposed as our dataset is smaller and refer to this transformer as mini-ViT or mViT in the following description."
],
[
"Bin-widths.",
"We first describe how the bin-widths vector $\\textbf {b}$ is obtained using mViT.",
"The input to the mViT block is a tensor of decoded features $\\mathbf {x_d}\\in ~\\mathbb {R}^{h \\times w \\times C_d}$ .",
"However, a transformer takes a sequence of fixed size vectors as input.",
"We first pass the decoded features through a convolutional block, named as Embedding Conv (see Fig REF ), with kernel size $p\\times p$ , stride $p$ and number of output channels $E$ .",
"Thus, the result of this convolution is a tensor of size $h/p~\\times ~w/p~\\times E$ (assuming both $h$ and $w$ are divisible by $p$ ).",
"The result is reshaped into a spatially flattened tensor $\\mathbf {x_p}~\\in ~\\mathbb {R}^{S\\times E}$ , where $S=\\frac{hw}{p^2}$ serves as the effective sequence length for the transformer.",
"We refer to this sequence of $E$ -dimensional vectors as patch embeddings.",
"Following common practice [2], [6], we add learned positional encodings to the patch embeddings before feeding them to the transformer.",
"Our transformer is a small transformer encoder (see Table.",
"REF for details) and outputs a sequence of output embeddings $\\mathbf {x_o}~\\in ~\\mathbb {R}^{S\\times E}$ .",
"We use an MLP head over the first output embedding (we also experimented with a version that has an additional special token as first input, but did not see an improvement).",
"The MLP head uses a ReLU activation and outputs an N-dimensional vector $\\textbf {b}^{\\prime }$ .",
"Finally, we normalize the vector $\\textbf {b}^{\\prime }$ such that it sums up to 1, to obtain the bin-widths vector $\\textbf {b}$ as follows: $b_i = \\frac{b^{\\prime }_i + \\epsilon }{\\sum _{j=1}^{N} (b^{\\prime }_j + \\epsilon )},$ where $\\epsilon =10^{-3}$ .",
"The small positive $\\epsilon $ ensures each bin-width is strictly positive.",
"The normalization introduces a competition among the bin-widths and conceptually forces the network to focus on sub-intervals within $D$ by predicting smaller bin-widths at interesting regions of $D$ .",
"In the next subsection, we describe how the Range-Attention-Maps $\\mathcal {R}$ are obtained from the decoded features and the transformer output embeddings."
],
[
"Range attention maps.",
"At this point, the decoded features represent a high-resolution and local pixel-level information while the transformer output embeddings effectively contain more global information.",
"As shown in Fig.",
"REF , output embeddings 2 through $C+1$ from the transformer are used as a set of $1\\times 1$ convolutional kernels and are convolved with the decoded features (following a $3\\times 3$ convolutional layer) to obtain the Range-Attention Maps $\\mathcal {R}$ .",
"This is equivalent to calculating the Dot-Product attention weights between pixel-wise features treated as `keys' and transformer output embeddings as `queries'.",
"This simple design of using output embeddings as convolutional kernels lets the network integrate adaptive global information from the transformer into the local information of the decoded features.",
"$\\mathcal {R}$ and $\\textbf {b}$ are used together to obtain the final depth map."
],
[
"Hybrid regression.",
"Range-Attention Maps $\\mathcal {R}$ are passed through a $1\\times 1$ convolutional layer to obtain $N$ -channels which is followed by a Softmax activation.",
"We interpret the $N$ Softmax scores $p_k$ , $k=1,...,N$ , at each pixel as probabilities over $N$ depth-bin-centers $c(\\textbf {b}) := \\lbrace c(b_1), c(b_2), ..., c(b_N)\\rbrace $ calculated from bin-widths vector $\\textbf {b}$ as follows: $c(b_i) = d_{min} + (d_{max} - d_{min})(b_i/2 + \\sum _{j=1}^{i-1} b_j)$ Finally, at each pixel, the final depth value $\\tilde{d}$ is calculated from the linear combination of Softmax scores at that pixel and the depth-bin-centers $c(\\textbf {b})$ as follows: $\\tilde{d} = \\sum _{k=1}^N c(b_k) p_k$ Compared to Fu et al.",
"[11] we do not predict the depth as the bin center of the most likely bin.",
"This enables us to predict smooth depth maps without the discretization artifacts as can bee seen in Fig.",
"REF .",
"Figure: Demonstration of artifacts introduced by the discretization of the depth interval.",
"Our hybrid regression results in smoother depth maps.Pixel-wise depth loss.",
"Inspired by [26], we use a scaled version of the Scale-Invariant loss (SI) introduced by Eigen [8]: $\\mathcal {L}_{pixel} = \\alpha \\sqrt{\\frac{1}{T}\\sum _i g_{i}^{2} - \\frac{\\lambda }{T^2}(\\sum _i g_i)^2}$ where $g_i = \\log \\tilde{d_i} - \\log d_i$ and the ground truth depth $d_i$ and $T$ denotes the number of pixels having valid ground truth values.",
"We use $\\lambda = 0.85$ and $\\alpha = 10$ for all our experiments.",
"Bin-center density loss.",
"This loss term encourages the distribution of bin centers to follow the distribution of depth values in the ground truth.",
"We would like to encourage the bin centers to be close to the actual ground truth depth values and the other way around.",
"We denote the set of bin centers as $c(\\textbf {b})$ and the set of all depth values in the ground truth image as $X$ and use the bi-directional Chamfer Loss [9] as a regularizer: $\\mathcal {L}_{bins} = chamfer(X, c(\\textbf {b})) + chamfer(c(\\textbf {b}), X)$ Finally, we define the total loss as: $\\mathcal {L}_{total} = \\mathcal {L}_{pixel} + \\beta \\mathcal {L}_{bins}$ We set $\\beta = 0.1$ for all our experiments.",
"We experimented with different loss functions including the RMSE loss, and the combined SSIM [42] plus $L_1$ loss suggested by [1].",
"However, we were able to achieve the best results with our proposed loss.",
"We offer a comparison of the different loss functions and their performance in our ablation study.",
"Table: Comparison of performances on the NYU-Depth-v2 dataset.",
"The reported numbers are from the corresponding original papers.",
"Best results are in bold, second best are underlined.Table: Comparison of performances on the KITTI dataset.",
"We compare our network against the state-of-the-art on this dataset.",
"The reported numbers are from the corresponding original papers.",
"Measurements are made for the depth range from 0m0m to 80m80m.",
"Best results are in bold, second best are underlined.Table: Comparison of performance with respect to the choice of loss function.We conducted an extensive set of experiments on the standard depth estimation from a single image datasets for both indoor and outdoor scenes.",
"In the following, we first briefly describe the datasets and the evaluation metrics, and then present quantitative comparisons to the state-of-the-art in supervised monocular depth estimation."
],
[
"NYU Depth v2",
"is a dataset that provides images and depth maps for different indoor scenes captured at a pixel resolution of $640\\times 480$ [37].",
"The dataset contains 120K training samples and 654 testing samples [8].",
"We train our network on a 50K subset.",
"The depth maps have an upper bound of 10 meters.",
"Our network outputs depth prediction having a resolution of $320\\times 240$ which we then upsample by $2\\times $ to match the ground truth resolution during both training and testing.",
"We evaluate on the pre-defined center cropping by Eigen et al.",
"[8].",
"At test time, we compute the final output by taking the average of an image's prediction and the prediction of its mirror image which is commonly used in previous work."
],
[
"KITTI",
"is a dataset that provides stereo images and corresponding 3D laser scans of outdoor scenes captured using equipment mounted on a moving vehicle [14].",
"The RGB images have a resolution of around $1241\\times 376$ while the corresponding depth maps are of very low density with lots of missing data.",
"We train our network on a subset of around 26K images, from the left view, corresponding to scenes not included in the 697 test set specified by [8].",
"The depth maps have an upper bound of 80 meters.",
"We train our network on a random crop of size $704\\times 352$ .",
"For evaluation, we use the crop as defined by Garg [13] and bilinearly upsample the prediction to match the ground truth resolution.",
"The final output is computed by taking the average of an image's prediction and the prediction of its mirror image."
],
[
"SUN RGB-D",
"is an indoor dataset consisting of around $10K$ images with high scene diversity collected with four different sensors [39], [44], [23].",
"We use this dataset only for cross-evaluating pre-trained models on the official test set of 5050 images.",
"We do not use it for training."
],
[
"Evaluation metrics.",
"We use the standard six metrics used in prior work [8] to compare our method against state-of-the-art.",
"These error metrics are defined as: average relative error (REL): $\\frac{1}{n}\\sum _p^n \\frac{\\vert y_p-\\hat{y}_p \\vert }{y}$ ; root mean squared error (RMS): $\\sqrt{\\frac{1}{n}\\sum _p^n (y_p-\\hat{y}_p)^2)}$ ; average ($\\log _{10}$ ) error: $\\frac{1}{n}\\sum _p^n \\vert \\log _{10}(y_p)-\\log _{10}(\\hat{y}_p) \\vert $ ; threshold accuracy ($\\delta _i$ ): $\\%$ of $y_p$ s.t.",
"$\\text{max}(\\frac{y_p}{\\hat{y}_p},\\frac{\\hat{y}_p}{y_p}) = \\delta < thr$ for $thr=1.25,1.25^2,1.25^3$ ; where $y_p$ is a pixel in depth image $y$ , $\\hat{y}_p$ is a pixel in the predicted depth image $\\hat{y}$ , and $n$ is the total number of pixels for each depth image.",
"Additionally for KITTI, we use the two standard metrics: Squared Relative Difference (Sq.",
"Rel): $\\frac{1}{n}\\sum _p^n \\frac{\\Vert y_p-\\hat{y}_p \\Vert ^2}{y}$ ; and RMSE log: $\\sqrt{\\frac{1}{n}\\sum _p^n \\Vert \\log y_p - \\log \\hat{y}_p\\Vert ^2}$ ."
],
[
"Implementation details",
"We implement the proposed network in PyTorch [33].",
"For training, we use the AdamW optimizer [30] with weight-decay $10^{-2}$ .",
"We use the 1-cycle policy [38] for the learning rate with $max\\_lr~=~3.5\\times ~10^{-4}$ , linear warm-up from $max\\_lr/25$ to $max\\_lr$ for the first $30\\%$ of iterations followed by cosine annealing to $max\\_lr/75$ .",
"Total number of epochs is set to 25 with batch size 16.",
"Training our model takes 20 min per epoch on a single node with four NVIDIA V100 32GB GPUs.",
"For all results presented we train for 25 epochs.",
"Our main model has about 78M parameters: 28M for the CNN encoder, 44M for the CNN decoder, and 5.8M for the new AdaBins module.",
"Figure: Effect of number of bins (N) on performance as measured by Absolute Relative Error metric.",
"we can observe interesting behaviour for lower values of N. As N increases, performance starts to saturate."
],
[
"Comparison to the state-of-the-art",
"We consider the following two methods to be our main competitors: BTS [26] and DAV [22].",
"For completeness, we also include selected previous related methods in the comparison tables.",
"For BTS and DAV we report the corresponding evaluation numbers from their papers.",
"For BTS we also verified these numbers by retraining their network using the authors code.",
"DAV did not have code available by the deadline, but the authors sent us the resulting depth images used in our figures.",
"In our tables we report the numbers given by the authors in their paper The authors of DAV clarified in an email that they compute the depth maps at 1/4th the resolution and then downsample the ground truth for evaluation.",
"However, we believe that all other methods, including ours, evaluate at the full resolution.. NYU-Depth-v2: See Table REF for the comparison of the performance on the official NYU-Depth-v2 test set.",
"While the state of the art performance on NYU has been saturated for quite some time, we were able to significantly outperform the state of the art in all metrics.",
"The large gap to the previous state of the art emphasises that our proposed architecture addition makes an important contribution to improving the results.",
"KITTI: Table REF lists the performance metrics on the KITTI dataset.",
"Our proposed architecture significantly outperforms previous state-of-the-art across all metrics.",
"In particular, our method improves the RMS score by about $13.5\\%$ and Squared Relative Difference by $22.4\\%$ over the previous state-of-the-art.",
"SUN RGB-D: To compare the generalisation performance, we perform a cross-dataset evaluation by training our network on the NYU-Depth-v2 dataset and evaluate it on the test set of the SUN RGB-D dataset without any fine-tuning.",
"For comparison, we also used the same strategy for competing methods for which pretrained models are available [26], [47], [4] and report results in Table.",
"REF .",
"Table: Results of models trained on the NYU-Depth-v2 dataset and tested on the SUN RGB-D dataset without fine-tuning.Table: Comparison of different design choices for bin-widths and regression.",
"AdaBins module results in a significant boost in performance.",
"Base: encoder-decoder with an EfficientNet B5 encoder.",
"R: standard regression.",
"HR: Hybrid Regression.",
"(Log)Uniform-Fix: Fixed (log) uniform bin-widths.",
"Train-Fix: Trained bin-widths but Fixed for each dataset."
],
[
"Ablation study",
"For our ablation study, we evaluate the influence of the following design choices on our results: AdaBins: We first evaluate the importance of our AdaBins module.",
"We remove the AdaBins block from the architecture and use the encoder-decoder to directly predict the depth map by setting $C_d=1$ .",
"We then use the loss given by Eq.",
"REF to train the network.",
"We call this design standard regression and compare it against variants of our AdaBins module.",
"Table.",
"REF shows that the architecture without AdaBins (Row 1) performs worse than all other variants (Rows 2-5).",
"Bin types: In this set of experiments we examine the performance of adaptive bins over other choices as stated in Sec.",
"REF .",
"Table.",
"REF lists results for all the discussed variants.",
"The Trained-but-Fixed variant performs worst among all choices and our final choice employing adaptive bins significantly improves the performance and outperforms all other variants.",
"Number of bins ($N$ ): To study the influence of the number of bins, we train our network for various values of $N$ and measure the performance in terms of Absolute Relative Error metric.",
"Results are plotted in Fig.",
"REF .",
"Interestingly, starting from $N=20$ , the error first increases with increasing $N$ and then decreases significantly.",
"As we keep increasing $N$ above 256, and with higher values the gain in performance starts to diminish.",
"We use $N=256$ for our final model.",
"Loss function: Table.",
"REF lists performance corresponding to the three choices of loss function.",
"Firstly, the $L_1$ /SSIM combination does not lead to the state-of-the-art performance in our case.",
"Secondly, we trained our network with and without the proposed Chamfer loss (Eq.",
"REF ).",
"Introducing the Chamfer loss clearly gives a boost to the performance.",
"For example, introducing the Chamfer loss reduces the Absolute Relative Error from $10.6\\%$ to $10.3\\%$ ."
],
[
"Conclusion",
"We introduced a new architecture block, called AdaBins for depth estimation from a single RGB image.",
"AdaBins leads to a decisive improvement in the state of the art for the two most popular datasets, NYU and KITTI.",
"In future work, we would like to investigate if global processing of information at a high resolution can also improve performance on other tasks, such as segmentation, normal estimation, and 3D reconstruction from multiple images."
],
[
"Geometric Consistency",
"We provide a qualitative evaluation of the geometric consistency of depth maps predicted by our model.",
"Surface normal maps provide a good way to visualize the orientation and texture details of surfaces present in the scene.",
"Fig REF shows the visualization of the normals extracted from the depth maps for our model and for DAV [22] and BTS [26].",
"Although the orientations predicted by DAV seems to be consistent, the texture details are almost completely lost.",
"BTS, on the other hand, preserves the texture but sometimes results in erroneous orientation details.",
"Our method exhibits detailed texture and consistent orientations without explicitly imposing geometric constraints, such as co-planarity, used by other methods [22], [26].",
"Figure: Visualization of surface normals extracted from predicted depth maps."
],
[
"Generlization Analysis",
"Here we qualitatively analyze the capability of our method to generalise to unseen data.",
"We use the models (AdaBins and BTS [26]) trained on NYU-Depth-v2 [37] but show predictions on SUN RGB-D [39] dataset in Fig REF .",
"Depth maps predicted by BTS have conspicuous artifacts whereas our method provides consistent results on the unseen data.",
"Figure: Qualitative comparison of generalization from NYU-Depth-v2 to SUN RGB-D dataset.Darker pixels are farther.",
"Missing ground truth values are shown in white."
],
[
"More Results on KITTI dataset",
"Fig REF shows a qualitative comparison of BTS [26] and our method on the KITTI dataset.",
"For better visualization, we have removed the sky regions from the visualized depth maps using segmentation masks predicted by a pretrained segmentation model[3].",
"We can observe that our method demonstrates superior performance particularly in predicting extents and edges of the on-road vehicles, sign-boards and thin poles.",
"Additionally, BTS tends to blend the farther away objects with background whereas our method preserves the structure with clear separation.",
"Figure: Qualitative comparison on KITTI dataset."
],
[
"MLP Head Details",
"We use a three-layer MLP on the first output embedding of the transformer in the mini-ViT module.",
"The architecture details with parameters are given in Table REF .",
"Table: Architecture details of MLP head.",
"FC: Fully Connected layer, E: Embedding dimension, N: Number of bins"
]
] | 2011.14141 |
[
[
"FaceGuard: A Self-Supervised Defense Against Adversarial Face Images"
],
[
"Abstract Prevailing defense mechanisms against adversarial face images tend to overfit to the adversarial perturbations in the training set and fail to generalize to unseen adversarial attacks.",
"We propose a new self-supervised adversarial defense framework, namely FaceGuard, that can automatically detect, localize, and purify a wide variety of adversarial faces without utilizing pre-computed adversarial training samples.",
"During training, FaceGuard automatically synthesizes challenging and diverse adversarial attacks, enabling a classifier to learn to distinguish them from real faces and a purifier attempts to remove the adversarial perturbations in the image space.",
"Experimental results on LFW dataset show that FaceGuard can achieve 99.81% detection accuracy on six unseen adversarial attack types.",
"In addition, the proposed method can enhance the face recognition performance of ArcFace from 34.27% TAR @ 0.1% FAR under no defense to 77.46% TAR @ 0.1% FAR."
],
[
"Introduction",
" With the advent of deep learning and availability of large datasets, Automated Face Recognition (AFR) systems have achieved impressive recognition rates .",
"The accuracy, usability, and touchless acquisition of state-of-the-art (SOTA) AFR systems have led to their ubiquitous adoption in a plethora of domains.",
"However, this has also inadvertently sparked a community of attackers that dedicate their time and effort to manipulate faces either physically , or digitally , in order to evade AFR systems .",
"AFR systems have been shown to be vulnerable to adversarial attacks resulting from perturbing an input probe , , , .",
"Even when the amount of perturbation is imperceptible to the human eye, such adversarial attacks can degrade the face recognition performance of SOTA AFR systems .",
"With the growing dissemination of “fake news\" and “deepfakes\" , research groups and social media platforms alike are pushing towards generalizable defense against continuously evolving adversarial attacks.https://bit.ly/2IkfSxk Figure: (Top Row) Adversarial faces synthesized via 6 adversarial attacks used in our study.",
"(Bottom Row) Corresponding adversarial perturbations (gray indicates no change from the input).",
"Notice the diversity in the perturbations.",
"ArcFace scores between adversarial image and the unaltered gallery image (not shown here) are given below each image.",
"A score above 0.36\\textbf {0.36} indicates that two faces are of the same subject.",
"Zoom in for details.A considerable amount of research has focused on synthesizing adversarial attacks , , , , , .",
"Obfuscation attempts (faces are perturbed such that they cannot be identified as the attacker) are more effective , computationally efficient to synthesize , , and widely adopted compared to impersonation attacks (perturbed faces can automatically match to a target subject).",
"Similar to prior defense efforts , , this paper focuses on defending against obfuscation attacks (see Fig.",
"REF ).",
"Given an input probe image, $\\mathbf {x}$ , an adversarial generator has two requirements under the obfuscation scenario: (1) synthesize an adversarial face image, $\\mathbf {x}_{adv} = \\mathbf {x}+ \\delta $ , such that SOTA AFR systems fail to match $\\mathbf {x}_{adv}$ and $\\mathbf {x}$ , and (2) limit the magnitude of perturbation $||\\delta ||_p$ such that $\\mathbf {x}_{adv}$ appears very similar to $\\mathbf {x}$ to humans.",
"A number of approaches have been proposed to defend against adversarial attacks.",
"Their major shortcoming is generalizability to unseen adversarial attacks.",
"Adversarial face perturbations may vary significantly (see Fig.",
"REF ).",
"For instance, gradient-based attacks, such as FGSM and PGD , perturb every pixel in the face image, whereas, AdvFaces and SemanticAdv perturb only the salient facial regions, e.g., eyes, nose, and mouth.",
"On the other hand, GFLM performs geometric warping to the face.",
"Since the exact type of adversarial perturbation may not be known a priori, a defense system trained on a subset of adversarial attack types may have degraded performance on other unseen attacks.",
"To the best of our knowledge, we take the first step towards a complete defense against adversarial faces by integrating an adversarial face generator, a detector, and a purifier into a unified framework, namely FaceGuard (see Fig.",
"REF ).",
"Robustness to unseen adversarial attacks is imparted via a stochastic generator that outputs diverse perturbations evading an AFR system, while a detector continuously learns to distinguish them from real faces.",
"Concurrently, a purifier removes the adversarial perturbations from the synthesized image.",
"This work makes the following contributions: topsep=0mm [noitemsep] A new self-supervised framework, namely FaceGuard, for defending against adversarial face images.",
"FaceGuard combines benefits of adversarial training, detection, and purification into a unified defense mechanism trained in an end-to-end manner.",
"With the proposed diversity loss, a generator is regularized to produce stochastic and challenging adversarial faces.",
"We show that the diversity in output perturbations is sufficient for improving FaceGuard's robustness to unseen attacks compared to utilizing pre-computed training samples from known attacks.",
"Synthesized adversarial faces aid the detector to learn a tight decision boundary around real faces.",
"FaceGuard's detector achieves SOTA detection accuracies of $99.81\\%$ , $98.73\\%$ , and $99.35\\%$ on 6 unseen attacks on LFW , Celeb-A , and FFHQ .",
"As the generator trains, a purifier concurrently removes perturbations from the synthesized adversarial faces.",
"With the proposed bonafide loss, the detector also guides purifier's training to ensure purified images are devoid of adversarial perturbations.",
"At 0.1% False Accept Rate, FaceGuard's purifier enhances the True Accept Rate of ArcFace from $34.27\\%$ under no defense to $77.46\\%$ ."
],
[
"Related Work",
" Table: Related work in adversarial defenses used as baselines in our study.",
"Unlike majority of prior work, FaceGuard is self-supervised where no pre-computed adversarial examples are required for training.Defense Strategies.",
"In literature, a common defense strategy, namely robustness is to re-train the classifier we wish to defend with adversarial examples , , , .",
"However, adversarial training has been shown to degrade classification accuracy on real (non-adversarial) images , .",
"In order to prevent degradation in AFR performance, a large number of adversarial defense mechanisms are deployed as a pre-processing step, namely detection, which involves training a binary classifier to distinguish between real and adversarial examples , , , , , , , , , , , , , , .",
"The attacks considered in these studies , , , were initially proposed in the object recognition domain and they often fail to detect the attacks in a feature-extraction network setting, as in face recognition.",
"Therefore, prevailing detectors against adversarial faces are demonstrated to be effective only in a highly constrained setting where the number of subjects is limited and fixed during training and testing , , .",
"Figure: FaceGuard employs a detector (𝒟\\mathcal {D}) to compute an adversarial score.",
"Scores below detection threshold (τ\\tau ) passes the input to AFR, and high value invokes a purifier and sends the purified face to the AFR system.Another pre-processing strategy, namely purification, involves automatically removing adversarial perturbations in the input image prior to passing them to a face matcher , , , .",
"However, without a dedicated adversarial detector, these defenses may end up “purifying\" a real face image, resulting in high false reject rates.",
"In Tab.",
"REF , we summarize a few studies on adversarial defenses that are used as baselines in our work.",
"Adversarial Attacks:.",
"Numerous adversarial attacks have been proposed in literature , , , , .",
"For example, Fast Gradient Sign Method (FGSM) generates an adversarial example by back-propagating through the target model .",
"Other approaches optimize adversarial perturbation by minimizing an objective function while satisfying certain constraints , , .",
"We modify the objective functions of these attacks in order to craft adversarial faces that evade AFR systems.",
"We evaluate FaceGuard on six unseen adversarial attacks that have high success rates in evading ArcFace : FGSM , PGD , DeepFool , AdvFaces , GFLM , and SemanticAdv (see Tab.",
"REF )."
],
[
"Limitations of State-of-the-Art Defenses",
"Robustness.",
"Adversarial training is regarded as one of the most effective defense method , , on small datasets including MNIST and CIFAR10.",
"Whether this technique can scale to large datasets and a variety of different attack types (perturbation sets) has not yet been shown.",
"Adversarial training is formulated as , : $\\min _{\\theta }~\\underset{(x,y)\\sim \\mathcal {P}_{data}}{\\mathbb {E}}\\left[\\max _{\\delta \\in \\Delta }\\ell \\left(f_{\\theta }\\left(x+\\delta \\right), y\\right)\\right],$ where $(x,y)\\sim \\mathcal {P}_{data}$ is the (image, label) joint distribution of data, $f_{\\theta }\\left(x\\right)$ is the network parameterized by $\\theta $ , and $\\ell \\left(f_{\\theta }\\left(x\\right), y\\right)$ is the loss function (usually cross-entropy).",
"Since the ground truth data distribution, $\\mathcal {P}_{data}$ , is not known in practice, it is later replaced by the empirical distribution.",
"Here, the network, $f_\\theta $ is made robust by training with an adversarial noise ($\\delta $ ) that maximally increases the classification loss.",
"In other words, adversarial training involves training with the strongest adversarial attack.",
"Figure: (a) Adversarial training degrades AFR performance of FaceNet matcher on real faces in LFW dataset compared to standard training.",
"(b) A binary classifier trained to distinguish between real faces and FGSM attacks fails to detect unseen attack type, namely PGD .Figure: Overview of training the proposed FaceGuard in a self-supervised manner.",
"An adversarial generator, 𝒢\\mathcal {G}, continuously learns to synthesize challenging and diverse perturbations that evade a face matcher.",
"At the same time, a detector, 𝒟\\mathcal {D}, learns to distinguish between the synthesized adversarial faces and real face images.",
"Perturbations residing in the synthesized adversarial faces are removed via a purifier, 𝒫ur\\mathcal {P}ur.The generalization of adversarial training has been in question , , , , .",
"It was shown that adversarial training can significantly reduce classification accuracy on real examples , .",
"In the context of face recognition, we illustrate this by training two face matchers on CASIA-WebFace: (i) FaceNet trained via the standard training process, and (ii) FaceNet by adversarial training (FGSMWith max perturbation hyperparameter as $\\epsilon = 8/256$ .",
").",
"We then compute face recognition performance across training iterations on a separate testing dataset, LFW .",
"Fig.",
"REF shows that adversarial training drops the accuracy from $99.13\\% \\xrightarrow{} 98.27\\%$ .",
"We gain the following insight: adversarial training may degrade AFR performance on real faces.",
"Detection.",
"Detection-based approaches employ a pre-processing step to “detect\" whether an input face is real or adversarial , , , .",
"A common approach is to utilize a binary classifier, $\\mathcal {D}$ , that maps a face image, $\\mathbf {x}\\in \\mathbb {R}^{H\\times W\\times C}$ to $\\lbrace 0, 1\\rbrace $ , where 0 indicates a real and 1 an adversarial face.",
"We train a binary classifier to distinguish between real and FGSM attack samples in CASIA-WebFace .",
"In Fig.",
"REF , we evaluate its detection accuracy on FGSM and PGD samples in LFW .",
"We find that prevailing detection-based defense schemes may overfit to the specific adversarial attacks utilized for training."
],
[
"FaceGuard",
" Our defense aims to achieve robustness without sacrificing AFR performance on real face images.",
"We posit that an adversarial defense trained alongside an adversarial generator in a self-supervised manner may improve robustness to unseen attacks.",
"The main intuitions behind our defense mechanism are as follows: [noitemsep,leftmargin=*] Since adversarial training may degrade AFR performance, we opt to obtain a robust adversarial detector and purifier to detect and purify adversarial attacks.",
"Given that prevailing detection-based methods tend to overfit to known adversarial perturbations (see Supp.",
"), a detector and purifier trained on diverse synthesized adversarial perturbations may be more robust to unseen attacks.",
"Sufficient diversity in synthesized perturbations can guide the detector to learn a tighter boundary around real faces.",
"In this case, the detector itself can serve as a powerful supervision for the purifier.",
"Lastly, pixels involved in the purification process may serve to indicate adversarial regions in the input face."
],
[
"Adversarial Generator",
" The generalizability of an adversarial detector and purifier relies on the quality of the synthesized adversarial face images output by FaceGuard's adversarial generator.",
"We propose an adversarial generator that continuously learns to synthesize challenging and diverse adversarial face images.",
"The generator, denoted as $\\mathcal {G}$ , takes an input real face image, $\\mathbf {x}\\in \\mathbb {R}^{H\\times W\\times C}$ , and outputs an adversarial perturbation $\\mathcal {G}(\\mathbf {x},\\mathbf {z})$ , where $\\mathbf {z}\\sim \\mathcal {N}(0, \\mathbf {I})$ is a random latent vector.",
"Inspired by prevailing adversarial attack generators , , , , , we treat the output perturbation $\\mathcal {G}(\\mathbf {x},\\mathbf {z})$ as an additive perturbation mask.",
"The final adversarial face image, $\\mathbf {x}_{adv}$ , is given by $\\mathbf {x}_{adv} = \\mathbf {x}+ \\mathcal {G}(\\mathbf {x},\\mathbf {z})$ .",
"In an effort to impart generalizability to the detector and purifier, we emphasize the following requirements of $\\mathcal {G}$ : [noitemsep] Adversarial: Perturbatation, $\\mathcal {G}(\\mathbf {x},\\mathbf {z})$ , needs to be adversarial such that an AFR system cannot identify the adversarial face image $\\mathbf {x}_{adv}$ as the same person as the input probe $\\mathbf {x}$ .",
"Visually Realistic: Perturbation $\\mathcal {G}(\\mathbf {x},\\mathbf {z})$ should also be minimal such that $\\mathbf {x}_{adv}$ appears as a legitimate face image of the subject in the input probe $\\mathbf {x}$ .",
"Stochastic: For an input $\\mathbf {x}$ , we require diverse adversarial perturbations, $\\mathcal {G}(\\mathbf {x},\\mathbf {z})$ , for different latents $\\mathbf {z}$ .",
"For satisfying all of the above requirements, we propose multiple loss functions to train the generator.",
"Obfuscation Loss To ensure $\\mathcal {G}(\\mathbf {x},\\mathbf {z})$ is indeed adversarial, we incorporate a white-box AFR system, $\\mathcal {F}$ , to supervise the generator.",
"Given an input face, $\\mathbf {x}$ , the generator aims to output an adversarial face, $\\mathbf {x}_{adv} = \\mathbf {x}+ \\mathcal {G}(\\mathbf {x},\\mathbf {z})$ such that the face representations, $\\mathcal {F}(\\mathbf {x})$ and $\\mathcal {F}(\\mathbf {x}_{adv})$ , do not match.",
"In other words, the goal is to minimize the cosine similarity between the two face representationsFor brevity, we denote $\\mathbb {E}_\\mathbf {x}\\equiv \\mathbb {E}_{\\mathbf {x}\\in \\mathcal {P}_{data}}$ .",
": $\\mathcal {L}_{obf} = \\mathbb {E}_{\\mathbf {x}}\\left[\\frac{\\mathcal {F}(\\mathbf {x}) \\cdot \\mathcal {F}(\\mathbf {x}_{adv})}{\\left|\\left|\\mathcal {F}(\\mathbf {x})\\right|\\right|\\left|\\left|\\mathcal {F}(\\mathbf {x}_{adv})\\right|\\right|}\\right].$ Perturbation Loss With the identity loss alone, the generator may output perturbations with large magnitudes which will (a) be trivial for the detector to reject and (b) violate the visual realism requirement of $x_{adv}$ .",
"Therefore, we restrict the perturbations to be within $[-\\epsilon , \\epsilon ]$ via a hinge loss: $\\mathcal {L}_{pt} = \\mathbb {E}_{\\mathbf {x}}\\left[\\max \\left(\\epsilon , \\left|\\left|\\mathcal {G}(\\mathbf {x},\\mathbf {z})\\right|\\right|_2\\right)\\right].$ Diversity Loss The above two losses jointly ensure that at each step, our generator learns to output challenging adversarial attacks.",
"However, these attacks are deterministic; for an input image, we will obtain the same adversarial image.",
"This may again lead to an inferior detector that overfits to a few deterministic perturbations seen during training.",
"Motivated by studies of preventing mode collapse in GANs , we propose maximizing a diversity loss to promote stochastic perturbations per training iteration, $i$ : $\\mathcal {L}_{div} = -\\frac{1}{N_{ite}}\\sum _{i=1}^{N_{ite}} \\frac{\\left|\\left|\\mathcal {G}(\\mathbf {x}, \\mathbf {z}_1)^{(i)} - \\mathcal {G}(\\mathbf {x}, \\mathbf {z}_2)^{(i)}\\right|\\right|_1}{\\left|\\left|\\mathbf {z}_1 - \\mathbf {z}_2\\right|\\right|_1},$ where $N_{ite}$ is the number of training iterations, $\\mathcal {G}(\\mathbf {x}, \\mathbf {z})^{(i)}$ is the perturbation output at iteration $i$ , and $(\\mathbf {z}_1,\\mathbf {z}_2)$ are two i.i.d.",
"samples from $\\mathbf {z}\\sim \\mathcal {N}(0,\\mathbf {I})$ .",
"The diversity loss ensures that for two random latent vectors, $\\mathbf {z}_1$ and $\\mathbf {z}_2$ , we will obtain two different perturbations $\\mathcal {G}(\\mathbf {x}, \\mathbf {z}_1)^{(i)}$ and $\\mathcal {G}(\\mathbf {x}, \\mathbf {z}_2)^{(i)}$ .",
"GAN Loss Akin to prior work on GANs , , we introduce a discriminator to encourage perceptual realism of the adversarial images.",
"The discriminator, $Dsc$ , aims to distinguish between probes, $\\mathbf {x}$ , and synthesized faces $\\mathbf {x}_{adv}$ via a GAN loss: $\\mathcal {L}_{GAN} = \\mathbb {E}_{\\mathbf {x}}\\left[{\\log Dsc(\\mathbf {x})}\\right] +\\mathbb {E}_{\\mathbf {x}}{\\left[\\log (1-Dsc(\\mathbf {x}_{adv}))\\right]}.$"
],
[
"Adversarial Detector",
"Similar to prevailing adversarial detectors, the proposed detector also learns a decision boundary between real and adversarial images , , , .",
"A key difference, however, is that instead of utilizing pre-computed adversarial images from known attacks (e.g.",
"FGSM and PGD) for training, the proposed detector learns to distinguish between real images and the synthesized set of diverse adversarial attacks output by the proposed adversarial generator in a self-supervised manner.",
"This leads to the following advantage: our proposed framework does not require a large collection of pre-computed adversarial face images for training.",
"We utilize a binary CNN for distinguishing between real input probes, $\\mathbf {x}$ , and synthesized adversarial samples, $\\mathbf {x}_{adv}$ .",
"The detector is trained with the Binary Cross-Entropy loss: $\\mathcal {L}_{BCE} = \\mathbb {E}_{\\mathbf {x}}\\left[\\text{log}\\mathcal {D}(\\mathbf {x})\\right] + \\mathbb {E}_{\\mathbf {x}}\\left[\\text{log}\\left(1 - \\mathcal {D}(\\mathbf {x}_{adv})\\right)\\right].$"
],
[
"Adversarial Purifier",
" The objective of the adversarial purifier is to recover the real face image $\\mathbf {x}$ given an adversarial face $\\mathbf {x}_{adv}$ .",
"We aim to automatically remove the adversarial perturbations by training a neural network $\\mathcal {P}ur$ , referred as an adversarial purifier.",
"The adversarial purification process can be viewed as an inverted procedure of adversarial image synthesis.",
"Contrary to the obfuscation loss in the adversarial generator, we require that the purified image, $\\mathbf {x}_{pur}$ , successfully matches to the subject in the input probe $\\mathbf {x}$ .",
"Note that this can be achieved via a feature recovery loss, which is the opposite to the obfuscation loss, i.e., $\\mathcal {L}_{fr}=-\\mathcal {L}_{obf}$ .",
"Note that an adversarial face image, $\\mathbf {x}_{adv} = \\mathbf {x}+ \\delta $ , is metrically close to the real image, $\\mathbf {x}$ , in the input space.",
"If we can estimate $\\delta $ , then we can retrieve the real face image.",
"Here, the perturbations can be predicted by a neural network, $\\mathcal {P}ur$ .",
"In other words, retrieving the purified image, $\\mathbf {x}_{pur}$ involves: (1) subtracting the perturbations from the adversarial image, $\\mathbf {x}_{pur} = \\mathbf {x}_{adv} - \\mathcal {P}ur(\\mathbf {x}_{adv})$ and (2) ensuring that the purification mask, $\\mathcal {P}ur(\\mathbf {x}_{adv})$ , is small so that we do not alter the content of the face image by a large magnitude.",
"Therefore, we propose a hybrid perceptual loss that (1) ensures $\\mathbf {x}_{pur}$ is as close as possible to the real image, $\\mathbf {x}$ via a $\\ell _{1}$ reconstruction loss and (2) a loss that minimizes the amount of alteration, $\\mathcal {P}ur(\\mathbf {x}_{adv})$ : $\\begin{split}\\mathcal {L}_{perc} = \\mathbb {E}_{\\mathbf {x}}\\left|\\left|\\mathbf {x}_{pur} - \\mathbf {x}\\right|\\right|_1 + \\left|\\left|\\mathcal {P}ur(\\mathbf {x}_{adv})\\right|\\right|_2.\\end{split}$ Finally, we also incorporate our detector to guide the training of our purifier.",
"Note that, due to the diversity in synthesized adversarial faces, the proposed detector learns a tight decision boundary around real faces.",
"This can serve as a strong self-supervisory signal to the purifier for ensuring that the purified images belong to the real face distribution.",
"Therefore, we also incorporate the detector as a discriminator for the purifier via the proposed bonafide loss: $\\mathcal {L}_{bf} = \\mathbb {E}_{\\mathbf {x}}\\left[\\text{log} \\mathcal {D}(\\mathbf {x}_{pur})\\right].$ Table: 2D t-SNE visualization of face representations extracted via ArcFace from 1,4561,456 (a) real, (b) AdvFaces , and (c) purified images belonging to 10 subjects in LFW .",
"Example AdvFaces pertaining to a subject moves farther from its identity cluster while the proposed purifier draws them back."
]
] | 2011.14218 |
[
[
"Rewrite to Reinforce: Rewriting the Binary to Apply Countermeasures\n against Fault Injection"
],
[
"Abstract Fault injection attacks can cause errors in software for malicious purposes.",
"Oftentimes, vulnerable points of a program are detected after its development.",
"It is therefore critical for the user of the program to be able to apply last-minute security assurance to the executable file without having access to the source code.",
"In this work, we explore two methodologies based on binary rewriting that aid in injecting countermeasures in the binary file.",
"The first approach injects countermeasures by reassembling the disassembly whereas the second approach leverages a full translation to a high-level IR and lowering that back to the target architecture."
],
[
"Introduction",
"Nowadays, the Fault Injection (FI) hardware attacks are becoming more prevalent.",
"Successful fault attacks lead to information leakage [1], [2] or privilege escalation.",
"While fault injection is targeted at the hardware (e.g., clock glitching), consequences of the resulting faults may affect the software running on a processor.",
"For example, ARM's secure boot can be affected by voltage glitching to enable an attacker to load controlled values in the program counter (PC) [3].",
"Furthermore, Vasselle et al.",
"[4] show how laser injected faults can bypass the secure bootloader on an android smartphone.",
"To defend against these attacks, an extensive amount of countermeasures have been proposed that can be categorized in three groups [5], namely those that can be applied to the software source code, those that are implemented within the compiler tool-chain, and those that are directly applied to the execution binary.",
"The first two categories require access to the source code of the program which may not be practical in some scenarios.",
"For instance, binary-level protection is useful for legacy binary code, or for third-party library code, or even for binary code for which the source code has been lost.",
"In this paper, we target the problem of applying FI countermeasures when we do not have access to the source code of the program.",
"Knowing that applying countermeasures directly to the binary file is not easy, we demonstrate how static binary rewriting approaches help with instrumenting the program with our countermeasures.",
"We apply and evaluate two static binary rewriting schemes; One reassembleable disassembly and the other complete translation.",
"Using the reassembleable disassembly method, we demonstrate how we can apply simple fixes to the binary file with low overhead.",
"Using the full-translation to LLVM-IR approach, we show how more complex countermeasures can be implemented exploiting the power of an intermediate representation (IR).",
"The rest of the paper is organized as the following: In Section , we discuss the related work.",
"In Section , we give a brief background on binary rewriting.",
"In Section , we introduce our countermeasure insertion methodologies.",
"In Section , we show the results of simple FI countermeasures implemented using our proposed approaches.",
"Finally, in Section , we conclude the paper."
],
[
"Related Work",
"While the bulk of fault countermeasures is based in detecting faults in redundant design, researchers have made many different proposals regarding the format and abstraction level of these redundancies.",
"Some of the related work start from the source code and add the countermeasure at that high level of abstraction.",
"For instance, Lalande et al.",
"[6] proposed a counter-based approach in which a counter is incremented and checked after a set number of instructions to detect jump attacks.",
"In other works, the countermeasures are added at compile time and require access to the source code.",
"Barry et al.",
"modified the LLVM back-end for insertion of countermeasures against instruction-skip fault attacks.",
"For this purpose, they duplicated instructions based on whether the operation is idempotent [7].",
"Our focus is on scenarios where the source code of the program is unavailable and we need to protect the executable file against fault attacks.",
"Given-Wilson et al.",
"[8] propose a methodology to detect vulnerabilities of program binaries to fault injection.",
"In their methodology, they annotate the source code with safety properties and detect vulnerabilities when these properties are shown to not be held using model-checking.",
"Bréjon et al.",
"[9] propose a framework consisting of symbolic execution, static analysis, and model-checking to find vulnerabilities in the binary file.",
"Both of these works focus on finding vulnerabilities but once the vulnerabilities are found, the source code should be accessible to add corresponding countermeasures.",
"De Keulenaer et al.",
"[10] use the link-time optimizer tool Diablo [11] and look for patterns of instructions in the assembly-level IR that are known to be vulnerable to fault injection and replace them with hardened code.",
"They show the approach of using the assembly-level IR results in a more compact hardened code, compared to the compiler-level IR, which is important for small embedded systems.",
"In another work, O'sullivan et al.",
"[12] propose a lifting and rewriting methodology based on the SecondWrite tool [13] for hardening a binary file against low-level software attacks such as buffer overflow attacks.",
"Fault injection attacks are not considered in their work.",
"In this project, we propose two different approaches to find the vulnerabilities in the binary code against fault injection attacks and add corresponding countermeasures.",
"As the first contribution of this paper, we propose a simulation-driven countermeasure insertion.",
"In this method, only the vulnerable parts of the binary file are patched and hence the overhead of the protected code is decreased compared to a full application of the countermeasure.",
"As a second contribution, this work for the first time shows how lifting the binary to an IR can help in adding countermeasures against fault attacks.",
"We use two different IR levels to this end and compare their resulting overhead."
],
[
"Binary Rewritering",
"In this section, we will go through the publicly available binary rewriting solutions and compare their approach with regard to structural recovery, data type extraction, and limitations on supported architectures."
],
[
"Definition",
"Binary rewriting is denoted as the process of modifying a compiled program in such a way that it remains executable and functional without having access to the source-code.",
"There are two types of binary modifiers, static and dynamic.",
"In the static approach, results of the modification will be stored on a persistent memory like disk for future execution.",
"However, dynamic rewriting is applied during the program execution.",
"In this paper we focus on static binary rewriting technique and compare the tools following this scheme.",
"Based on Figure REF , from a high-level view, in the first step ([baseline=(char.base)] shape=circle,draw,inner sep=1pt] (char) black1;), a binary rewriter receives a file in a binary stream format as input and passes it to the disassembler for decoding the instructions, and retrieving global variables and sections.",
"Decoded instructions help step [baseline=(char.base)] shape=circle,draw,inner sep=1pt] (char) black2; in building the control and data flow, recovering data types, and function boundaries to semantically enriching the context with metadata lost during the compilation.",
"Transformation step ([baseline=(char.base)] shape=circle,draw,inner sep=1pt] (char) black3;) modifies the target binary in a way that mutated output is a working executable."
],
[
"Static Binary Rewriting",
"There are three known rewriting schemes.",
"The oldest one is based on detouring at assembly level.",
"Detouring works by hooking out the underlying instruction.",
"There are two flavors of the detouring technique, patch-based instrumentation and replica-based instrumentation.",
"Patch-based instrumentation replaces the instruction with an unconditional branch to a new section containing instrumentation, replaced instruction, and a control flow transfer back to the patch point.",
"Detouring is a direct rewriting and is ISA dependant which makes the approach inconvenient.",
"This approach introduces a high performance degradation given the two control transfers at patch points.",
"Replica-based instrumentation method inserts jump instructions to a replicated code section containing both a copy of the original code and the instrumentation.",
"All memory references in this section are modified to maintain fewer control flow transfers between original and replicated section.",
"While the performance of this approach is better compared to the patch-based instrumentation, the size of the resulting binary is noticeably increased.",
"Reassembleable disassembly works by recovering relocatable assembly code, the instrumentation of which could be inlined and reassembled back to a working binary.",
"This approach first introduced by UROBOROS [14] and then expanded by improving on top of the idea in Ramblr [15].",
"This approach enhances the performance since inlined assembly avoids inserting control flow changing instructions at instrumentation points.",
"As a result, performance penalty caused by jump instructions are alleviated in this technique.",
"Full-translation approach works upon translating a low-level machine code to a high-level intermediate representation (IR) using a compiler-based front-end for architecture independent binary rewriting.",
"This process is called lifting the binary and assembling the IR back to a working executable is denoted as lowering.",
"The advantage of lifting the binary to a high-level IR are two fold.",
"First, relying on IR makes the rewriting framework ISA-agnostic; Second, working on a high-level IR provides the ability to apply program analysis techniques like Value Set Analysis (VSA) [16] and optimization passes like Simple Expression Tracker (SET) and Offset Shifted Register Analysis (OSRA) [17].",
"On the other hand, complete translation suffers from changing the structural integrity such as cache locality and Control Flow Graph (CFG).",
"While each approach has its own drawbacks and benefits, we focused our evaluation on two recent research: Datalog Disassembly (Ddisasm) [18] for reassembling the disassembly and Rev.ng [17] as the candidate for full-translation."
],
[
"Comparison of Binary Rewriters",
"In this section, we briefly describe the reasons behind choosing the above-mentioned two binary rewriters as our candidates.",
"During the linking phase, linker replaces the symbolic labels with concrete memory addresses which results in losing the relocation information.",
"Hence, to perform rewriting tasks, we need to recover the symbolic references from absolute addresses.",
"A process which is called symbolization.",
"Symbolization aims to distinguish whether an intermediate value belongs to a symbol or treat the value as a constant integer.",
"Comparing reassembling methods, Ramblr provided counter-examples in real-world binaries for which the UROBOROS symbol categorization fails.",
"UROBOROS scans the data section linearly and considers any machine word-sized buffer whose integer representation falls in a memory region as a memory reference.",
"This assumption with the compiler optimization introduces False-positive and False-negatives.",
"Ramblr improved the content classification by applying strong heuristics like localized VSA and Intra-function data dependence analysis.",
"To improve the binary rewriting results, Ramblr depends heavily on symbolic execution for accurate CFG recovery which slows down the rewriting process and brings up scalability issues.",
"Apart from Ramblr's heuristics, Ddisasm incorporated register value analysis as an alternative over traditional VSA.",
"In addition, they introduced Data Access Pattern (DAP) analysis which is a def-use analysis combined with the results of register value analysis for a refined register value inference at any given data access point.",
"Rev.ng relies on full binary translation by lifting the binary to TCG (the IR used in QEMU [19]) and para-lifting TCG to LLVM-IR to benefit from more advanced transformation and analysis passes for CFG and function boundary recovery.",
"While frameworks like angr [20] use lifting to apply more advanced binary analysis on top of the intermediate-level representation, they do not lower the resulting transformation back to the binary.",
"Moreover, Rev.ng heavily relies on code pointers for identifying function entry points and leverages VSA for a more precise value boundary tracking.",
"As the rewriter tools to harden the binary code against fault injection, in this work, we chose Ddisasm and Rev.ng to show the difference between two different rewriting schemes for this purpose."
],
[
"Countermeasure Insertion Methodology",
"For complex architectures, like x86-64, it is not straight forward to group the bits in the binary file to form full instructions.",
"Neither is it easy to group the instructions to form basic blocks at this level of abstraction.",
"Therefore, manipulating the binary file directly is not trivial.",
"We propose a procedure in which we use an open source disassembler and binary manipulation tools as well as binary lifters to make the binary hardening process more manageable."
],
[
"Rewriting the Binary",
"Using the disassemblers and working on the assembly code, compared to the binary file, can help in finding patterns of instructions and applying fixes locally.",
"However, at the assembly code level, the register allocation and memory usage are fixed.",
"Therefore, applying fixes at this level requires extra caution not to overwrite the allocated registers in use.",
"A favorable property of this level of hierarchy is that we know which part of the assembly code corresponds to which part of the machine code exactly.",
"We take advantage of this property in our proposed methodology and build an iterative process that, using simulation of fault effects, can locally apply countermeasures only to the parts that they are required.",
"While simple and small fixes can be applied at the assembly level, more complex fixes are not easily applicable.",
"In this case, a higher level of abstraction that enables modification of the code and different types of analysis is useful.",
"Since LLVM-IR is in Single Static Assignment (SSA) [21] format and supports different levels of hierarchy (namely module, function, basic-block, and instruction), we choose it as our high-level IR.",
"Support of different levels of hierarchy in the IR makes it easier to perform static analyses on the program.",
"Additionally, being a part of the LLVM tool-chain, has the advantage of being open-source, having a big number of active contributors, and a well-maintained documentation.",
"Despite the aforementioned advantages, however, lifting the binary to such a high level of abstraction will eliminate the mapping between the abstraction levels.",
"This results from the fact that the high level of abstraction lacks the low-level target-dependent information.",
"Consequently, applying targeted and local fixes to the binary files at this high level of abstraction is not readily available.",
"In the following subsections, we discuss the mentioned approaches to countermeasure injection."
],
[
"Faulter+Patcher Approach",
"Our first approach injects countermeasures at the abstraction level of assembly code and is thus able to patch the binary file in a targeted manner.",
"Fig.",
"REF shows the overall scheme.",
"In this approach, we have a fault simulation-driven, iterative method to mitigate fault injection vulnerabilities in the binary file.",
"The system consists of two main components: a faulter and a patcher.",
"The faulter is simulating faults under a certain fault model in a target binary and produces a list of vulnerabilities, meaning faults where unwanted behavior in the target binary is triggered.",
"The patcher uses the list produced by the faulter to patch the binary.",
"The patcher will patch each fault vulnerability as localized as possible, without affecting its surrounding code.",
"The patched binary is then again run through the faulter and patcher.",
"We repeat this process until no more faults are present or can be fixed.",
"In the following, the faulter and the patcher are discussed in detail."
],
[
"Faulter",
"For our purpose, fault injection vulnerabilities are vulnerabilities where an attacker, i.e.",
"an unauthorized user, is able to trigger a behavior in a target binary that should be reserved only for authorized users.",
"For example, consider a pincode checker that receives an input pin and checks if the inserted value is correct.",
"For a correct pin, the program will proceed to run some sensitive operations.",
"An attacker does not know the correct pincode, but may be able to skip an instruction in the target binary such that the program will conclude the inserted pin is correct and therefore run the sensitive operations.",
"These faults are labeled “successful faults”.",
"Faults that do not trigger the unwanted behavior or cause the program to crash are ignored.",
"We first choose a fault model that we want to protect our binary file against.",
"Regardless of the fault model, and the number of faults injected per run, the faulter takes a target binary and two inputs; a “good” input and a “bad” input.",
"For instance, in the pincheck example, the “good” input is the correct pincode and the “bad” input is any value other than the correct pincode.",
"First, the “good” and the “bad” inputs are executed to see the difference of execution traces between them.",
"When running the target binary with the “bad” input, we effectively can record a trace of all the instructions executed.",
"For each offset in that trace, taking the “single bit flip model” as an example, we run the target binary normally up to that offset in the trace, flip a bit in the instruction at the trace offset, and then resume execution.",
"The target binary either crashes, executes as an incorrect input, or behaves differently (as a correct input).",
"If it behaves as a correct input, the trace offset and the fault that caused it, in this case a bit offset into an instruction at the trace offset, is recorded.",
"We implemented this simple faulter in Python using the Qiling binary emulator package.",
"We fork each fault simulation to speed up the process.",
"Our faulter supports x86-64 Linux binaries only, but including support for other architectures supported by Unicorn should be straightforward."
],
[
"Patcher",
"The list of “successfult faults” coming out of the faulter is addressed locally in the patcher.",
"The patcher replaces the vulnerable patterns of instructions with known hardened patterns.",
"For example, consider a run of the faulter under the “instruction skip” fault model that identified that at timestamp 40, the skipping of a mov instruction is a successful fault.",
"A local countermeasure is to perform the mov twice, or add a compare instruction to verify the mov has been executed prior.",
"Note that these countermeasures cause duplicate reads, as redundancy is key to mitigate fault injection attacks.",
"We implemented a proof of concept patcher based on GrammaTech's Ddisasm tool and their Python binary manipulation libraries.",
"The Ddisasm tool performs pointer analysis on an executable and produces an IR in the form of GrammaTech Intermediate Representation for Binaries (GTIRB) [22] that can then be manipulated and recompiled into an executable by the Python GTIRB libraries."
],
[
"Rinse and repeat",
"After running the faulter and the patcher once, we end up with a patched binary.",
"Running the faulter on the patched binary may reveal that, since we added code and changed distances between instructions, we added new vulnerabilities.",
"These new vulnerabilities then can be addressed by running the patcher iteratively until a fixed point is reached.",
"In our second approach, we inject countermeasures at the abstraction level of compiler IR.",
"To be able to implement more complex countermeasures, working at the level of assembly code is cumbersome if not impractical.",
"For example, consider a countermeasure where extra registers are needed to hold some intermediate values.",
"If the assembly code had been generated with a high level of optimization and no register is available in the unprotected program, extra steps should be taken to spill some data to the memory to make a few registers available and load them back to the same registers after the countermeasure.",
"This requires knowledge of the state of the memory at different locations.",
"However, there is no guarantee that these steps are possible therefore implementing some countermeasures on certain programs might not be feasible.",
"To overcome this problem, we propose a process as shown in the upper half of Fig.",
"REF .",
"This process consists of three steps; First, we transform the program from binary to a compiler IR.",
"Second, we implement the countermeasure on the IR.",
"Last, we transform the protected IR back to the executable binary format, hence, achieve the goal of protecting the binary file.",
"These steps are elaborated in the rest if this section.",
"Figure: High-level overview of the Faulter+Patcher (lower half) and the Hybrid (upper half) approaches"
],
[
"Transforming to an intermediate representation",
"The goal of this step is to have a representation of the binary file which, while preserving the functionality, is easier to modify and supports a format in which different types of analysis can be performed.",
"As discussed earlier, we choose LLVM-IR.",
"There are several open-source tools that are able to lift the binary file of different architectures to the architecture-independent form of LLVM-IR.",
"In this work, we use Rev.ng.",
"As mentioned in Section , Rev.ng is a binary analysis framework based on QEMU and LLVM.",
"As part of this framework, it is possible to extract the LLVM-IR of a program from its binary.",
"The binary file can be for any of the x86, x86-64, ARM, MIPS, s390x, or AArch64 (WIP) architectures.",
"In this project, without loss of generality, our focus is on the x86-64 architecture."
],
[
"Implementing the countermeasure",
"After the lifting step is performed, we have the LLVM-IR representation of the algorithm.",
"Therefore the countermeasure can be implemented in the form of a combination of optimization and/or back-end passes depending on the desired protection.",
"If the protection algorithm does not have any dependencies specific to the target architecture, the entire countermeasure can be in the form of an optimization pass.",
"Otherwise, back-end passes would be required."
],
[
"Generating the protected executable",
"Finally, the protected IR needs to be compiled to an executable.",
"In LLVM, the llc tool is responsible for translating the LLVM-IR to an architecture-specific executable file.",
"Specific steps might need to be added in the form of back-end passes to make sure the implemented countermeasures are retained unchanged through this process.",
"Once the hardened binary file is generated, we use the same faulter system to detect remaining vulnerabilities."
],
[
"Choosing the Right Method",
"The targeted insertion of countermeasures in the Faulter+Patcher scheme makes the overhead of the applied assurance smaller than a holistic approach.",
"Furthermore, the mere act of lifting the binary to LLVM-IR and translating it back to binary in the Hybrid approach adds extra overhead to the program.",
"This stems from the internal functions of the binary rewriting tools, which in our case is Rev.ng.",
"On the other hand, applying countermeasures in the Hybrid approach is easily automated and is guaranteed to be feasible.",
"The hierarchy levels supported by LLVM-IR as well as its SSA format eases many static analyses such as finding idempotent pieces, finding and replacing all the uses of a variable, and many more.",
"The aforementioned trade-off between these methods, makes each method suitable for a different use case scenario.",
"In size-constrained applications, such as programs for small embedded systems, the Faulter+Patcher method is more favorable due to its smaller footprint.",
"In scenarios where the code size is not of critical concern, the Hybrid approach provides a simpler and guaranteed assurance for inserting complex countermeasures."
],
[
"Experimental Results",
"In this section, we show how our proposed approaches can apply countermeasures against a chosen fault model.",
"We first show the local protections that we add by our Faulter+Patcher approach.",
"We then demonstrate a holistic protection that can be used in our Hybrid approach.",
"Finally, we show the results of the inserted countermeasures in our case studies."
],
[
"Local Protections",
"In the Faulter+Patcher approach, we are able to insert protected code patterns locally.",
"The following is the description of these redundant computation-based protections.",
"To protect the mov operation against fault attacks, after executing the mov operation, the result of the two memory locations are compared and in case of an inconsistency, a faulthandler is called (Table REF )."
],
[
"Protecting the ",
"We can protect a cmp instruction against fault attacks by executing the comparison twice and comparing their resulting flags (Table REF ).",
"To this end, we use the pushfq instruction in x86-64 ISA which requires a valid stack pointer (rsp).",
"Due to Intel's red zone, we have to subtract 128 bytes from rsp to jump out of the red zone."
],
[
"Protecting the ",
"By hardening the conditional jump operations, we detect glitches that change the jump condition.",
"In the protected code shown in Table REF , we use the flags register and match this to the expected flag in the jump-target and the fall-through of a branch."
],
[
"Holistic Protection",
"[] cmp rs1, rs2 bne target2 ... target1: ... target2: ...",
"Figure: Assembly code and CFG of a simple branch instructionIn this section, we describe a conditional branch hardening method that we later use in our case studies for the Hybrid countermeasure implementation approach.",
"Imagine a simple program that receives a pin code and only if the pin code is correct, resumes the program to execute some operations.",
"In the assembly code (equally the executable file) of this program, there will be a comparison instruction to compare the inserted pin code and its expected value, as well as a conditional branch that, based on the result of the comparison, jumps to a successor basic block.",
"Figure REF shows the assembly code and the control flow graph (CFG) of this branch operation.",
"In the case of Control Flow Integrity (CFI), going from BB1 to either of BB2 or BB3 does not raise any issues since they both are valid paths in the CFG.",
"However, if an injected fault results in taking the wrong branch, it will be an unnoticed fault.",
"In this conditional branch hardening method, our goal is to protect against this outcome of FI.",
"In this method, we assign a unique ID to each basic block ($UID_{BB}$ ) at compile time.",
"We then use an algorithm, $h(UID_{src},UID_{dst},cmp\\_res)$ , which calculates a checksum at run-time based on the fixed $UID$ s assigned to the source and destination blocks at compile-time, i.e.",
"$UID_{src}$ and $UID_{dst}$ , and the dynamically-evaluated compare result, $cmp\\_res$ .",
"The calculated checksum will be stored in a register and checked in the destination basic blocks.",
"At the destination blocks, since the expected $cmp\\_res$ for the taken edge is known, the expected value of $h$ is known.",
"Therefore checking the evaluated value only requires reading the register value and comparing it with the expected value.",
"When the register does not contain the expected value, we jump to a fault-response basic block.",
"The simplicity level of the $h$ function can be decided based on the required security properties of the program.",
"As an example, we chose a simple option for $h$ and implemented the countermeasure as an optimization pass in the LLVM tool-chain.",
"In this example, the checksum is calculated as the XOR result of the UID of the taken destination block and that of the source block ($UID_{dst} \\oplus UID_{src}$ ).",
"The pseudo-code of the calculation procedure of this checksum in LLVM is shown in Algorithm REF where $cmp\\_res$ is the result of the comparison for the conditional branch, and $UID_{Tdst}$ , $UID_{Fdst}$ , and $UID_{src}$ are the UIDs assigned to the true destination (the destination taken when the comparison result is true), the false destination (the destination taken when the comparison result is false), and the source block respectively.",
"The $mask$ shown on line 4 will have the value of all ones if the comparison result is false and the value of all zeros if the comparison result is true.",
"The checksum will be located in one register and each of the destination blocks will evaluate whether the value of the checksum is correct.",
"Furthermore, we made this evaluation duplicated; Figure REF shows the CFG of this implementation protecting the conditional branch shown in Figure REF .",
"We run the comparison instruction once (C1), and calculate the checksum based on its result and keep it in a register (D1).",
"We perform this calculation another duplicated time and keep the result in a new register (D2).",
"We then perform the comparison again and run the branch based on the result of the second comparison (C2).",
"As shown by the orange boxes, the expected value of the checksum is different for the out-going edges of the source block (N1 vs. N2).",
"In the destination blocks, we check both copies of the checksum stored in registers against the expected values in a nested fashion.",
"The green blocks show the nested checksum validations and the blue blocks represent the fault-response.",
"As a simple example, the fault-response can be aborting the execution (in the flt_respx basic blocks).",
"Figure: CFG of the example conditional branch hardening[t] Simple example algorithm for $h$ [1] $cmp\\_res$ , $UID_{Tdst}$ , $UID_{Fdst}$ , $UID_{src}$ $checksum$ Generate unique checksums for edges : $const_{Tdst} \\leftarrow UID_{Tdst} \\oplus UID_{src}$ $const_{Fdst} \\leftarrow UID_{Fdst} \\oplus UID_{src}$ Calculate checksum of the branch : $cmp\\_ext \\leftarrow zero\\_extend(cmp\\_res)$ $mask \\leftarrow cmp\\_ext - 1$ $checksum \\leftarrow (\\lnot mask \\wedge const_{Tdst}) \\vee (mask \\wedge const_{Fdst})$ In this scenario, if the attacker tries to skip one of the comparison instructions or change it to compute the inverse output, the checksum validation process will be able to catch the injected error.",
"If the adversary intends to bypass this process, they would need to inject the exact same fault in both comparison results.",
"In Section REF , we show a simulated analysis of the effectiveness of this countermeasure.",
"The overall overhead of this countermeasure depends on the number of conditional branches that we want to protect and therefore is highly application-specific.",
"Table REF shows how many instructions are required to replace a simple conditional branch by this method.",
"We implement the conditional branch hardening as an optimization pass in the LLVM compiler tool-chain.",
"Therefore, its translation will differ for different target architectures based on how the instruction-lowering is done for that architecture in the LLVM back-end.",
"Table: Qualitative overhead of the conditional branch hardening"
],
[
"Case Studies",
"We choose two applications for the proof of concept of our proposed methods.",
"The first application is a simple pin-check program that receives an input password and checks the correctness of the inserted password.",
"The second application is a secure bootloader in which the hash of the content of a memory location is calculated and compared with an expected hash value.",
"For each case study, we use the faulter described in Section REF with the two fault models of “instruction skip” and “single bit flip” and verify that the code has vulnerabilities to these fault models.",
"All of these vulnerabilities were caused by the conditional jumps (mov, cmp, and jmp instructions related to a jump operation) in the program.",
"We then insert the code patterns described in Section REF using the Faulter+Patcher approach for each vulnerable point and reevaluate the hardened binary file iteratively until confirmed that no more vulnerabilities exist.",
"Furthermore, we apply the conditional branch hardening countermeasure from Section REF using our Hybrid approach and verify that the vulnerabilities have been mitigated.",
"In the case of the “instruction skip” fault model, we were able to resolve all the vulnerabilities using the mentioned countermeasures.",
"In the case of the “single bit flip” fault model we were able to reduce the number of vulnerable points by 50% using both methodologies.",
"Table REF shows the overhead caused by the inserted countermeasure in each approach.",
"The overhead caused by the Hybrid approach is 2 to 5 times bigger than that of the iterative method.",
"This is an expected outcome since the Hybrid approach applies the countermeasure to the entire program whereas the Faulter+Patcher approach only does so to the vulnerable points.",
"Furthermore, duplicating every instruction, which is the go-to protection scheme against fault injection, implies at least 300% overhead in code size (since for each instruction, it will add another copy of the instruction and a comparison procedure between their results).",
"Therefore, both of our methods perform better than a simple duplication scheme.",
"Table: Overhead of adding the protections"
],
[
"Conclusion and Future Work",
"In this work, we proposed two approaches for hardening the binary file against fault attacks based on binary rewriting.",
"In the first approach, we disassemble the binary file and insert countermeasures which enables us to insert local countermeasure and keep the structure of the original binary file.",
"We propose an iterative and simulation-driven framework that only inserts countermeasures to the vulnerable parts of the program.",
"In the second approach, we lift the binary file to LLVM-IR and insert countermeasures at a higher level of abstraction.",
"This enables us to implement more complex countermeasures and perform static analyses on the program.",
"As the future work, we intend to study the mapping between the lifted LLVM-IR of a binary code and the original binary file and enable an iterative countermeasure insertion for the Hybrid methodology."
]
] | 2011.14067 |
[
[
"A general family of MSRD codes and PMDS codes with smaller field sizes\n from extended Moore matrices"
],
[
"Abstract We construct six new explicit families of linear maximum sum-rank distance (MSRD) codes, each of which has the smallest field sizes among all known MSRD codes for some parameter regime.",
"Using them and a previous result of the author, we provide two new explicit families of linear partial MDS (PMDS) codes with smaller field sizes than previous PMDS codes for some parameter regimes.",
"Our approach is to characterize evaluation points that turn extended Moore matrices into the parity-check matrix of a linear MSRD code.",
"We then produce such sequences from codes with good Hamming-metric parameters.",
"The six new families of linear MSRD codes with smaller field sizes are obtained using MDS codes, Hamming codes, BCH codes and three Algebraic-Geometry codes.",
"The MSRD codes based on Hamming codes, of minimum sum-rank distance $ 3 $, meet a recent bound by Byrne et al."
],
[
"Introduction",
"Maximum distance separable (MDS) codes are “optimal” in the sense that their minimum Hamming distance [20] attains the Singleton bound [45].",
"Although this bound is a crude bound on the minimum Hamming distance of codes over small alphabets, MDS codes are optimal erasure codes from an information-theoretical perspective: Over large enough alphabets and for a fixed information rate and block length, any Hamming-metric erasure pattern (any subset of coordinates to be erased) that can be corrected by some code can be corrected by any MDS code.",
"For this reason, in erasure scenarios where alphabets need not be too small and where information rates and block lengths are fixed, MDS codes offer the best erasure correction capability.",
"One of such erasure scenarios is that of node repair in distributed storage, where MDS codes were traditionally a popular choice.",
"In this scenario, a node may need to be repaired due to its data being lost, but also due to being unresponsive or unavailable.",
"However, repairing a single node out of $ n $ nodes using an MDS code of rate $ k/n $ requires contacting $ k $ other nodes.",
"With the large amounts of data stored in nowadays' distributed storage systems, repairing a single node (the most frequent erasure pattern) results in a high latency due to reading and decoding large amounts of data.",
"Locally repairable codes (LRCs) [17], [24] may repair one node (or more generally, $ \\delta -1 $ nodes per local set) by contacting only a small number $ r $ (called locality) of other nodes.",
"Simultaneously, they are able to correct a large number of global erasures in catastrophic cases, in contrast with simply using a Cartesian product of MDS codes of dimension $ r $ and block length $ r+\\delta - 1 $ .",
"Although extensions of the Singleton bound are known for LRCs (see, e.g., [17] or [24]), they do not have the same information-theoretical meaning as the classical bound [45].",
"More concretely, an LRC whose (global) minimum Hamming distance attains such a Singleton bound is not necessarily capable of correcting all the erasure patterns correctable by some other LRC of the same parameters.",
"LRCs that may correct all the erasure patterns correctable by some other LRC with the same locality constraints, over a large enough alphabet, are called partial MDS (PMDS) codes or maximally recoverable (MR) LRCs.",
"They were introduced independently in [3], [16] and may correct any $ \\delta -1 $ erasures per local set, plus any extra $ h $ erasures elsewhere.",
"Here, $ \\delta $ is the distance of the local codes, and $ h $ is the co-dimension of the global code after puncturing it on any $ \\delta -1 $ coordinates per local set, hence called the number of global or heavy parities.",
"Several constructions of PMDS codes exist in the literature [3], [4], [5], [7], [9], [14], [16], [19], [22], [36], [37].",
"In Construction 1 in [36], it was shown that any maximum sum-rank distance (MSRD) code [31] may be easily and explicitly turned into a PMDS code [36].",
"Moreover, such Construction 1 enjoys further flexibility and dynamic properties such as being compatible with an arbitrary choice of local codes, locally replacing such local codes without changing the overall storage architecture or recoding all the stored data, and enabling any hierarchical structure of local codes with any number of levels and always being able to correct any information-theoretically correctable erasure pattern for the corresponding locality constraints (see [36]).",
"As another application of the flexibility enabled by MSRD codes, optimal LRCs with multiple disjoint repair sets were obtained based on MSRD codes in [8].",
"Apart from being used as PMDS codes for repair in distributed storage [36], MSRD codes have found applications in universal error correction and security in multishot network coding [38], [35], rate-diversity optimal space-time codes with multiple fading blocks and minimum delay [29], [44], and private information retrieval on PMDS-coded databases or where communication with servers is through a linearly coded network [32].",
"They may be applicable in a multishot or multilayer version of crisscross error and erasure correction, extending [41].",
"In this work, MSRD codes are considered as those codes whose minimum sum-rank distance [38] attains the Singleton bound given in [36].",
"By the same result, such MSRD codes may be defined as MDS codes that remain MDS after being multiplied by any invertible block-diagonal matrix of the appropriate sizes (see Definition REF ).",
"It is precisely because of this mathematical property that any MSRD code may be turned into a PMDS code as in Construction 1 in [36], enjoying all the properties described above (see [36]).",
"In scenarios where large alphabets are allowed, codes over small fields are preferable, as they enjoy lower computational complexity while being able to fit the larger alphabets (by encoding data by “chunks”).",
"In contrast with MDS codes, PMDS codes with linear field sizes in the code length do not exist for general parameters [18].",
"Hence the same holds for MSRD codes by Construction 1 in [36].",
"See Subsection REF for a detailed discussion on field sizes.",
"Any maximum rank distance (MRD) code [12], [13], [41] may be used as an MSRD or PDMS code.",
"However, the field size of any MRD code is exponential in the code length, rendering them impractical in most cases.",
"PMDS codes with sub-exponential field sizes were obtained in [3], [4], [5], [7], [14], [16], [19], [22], [36], [37].",
"The first and only known MSRD codes with sub-exponential field sizes are linearized Reed-Solomon codes [31] (obtained later independently in [11], [39]), which recover as particular cases (generalized) Reed-Solomon codes [40] and Gabidulin codes [12], [13], [41] whenever the sum-rank metric recovers the Hamming metric and the rank metric, respectively.",
"Recently, a few MSRD codes were found in [6] for minimum sum-rank distance 2 or block length minus 1, or for parameters with trivial matrix sizes in some components (see Section ).",
"In this work, we obtain a general family of MSRD codes (thus PMDS codes) whose field sizes are smaller than those obtained before for many parameter regimes.",
"See Section for a detailed summary and comparisons, and the Appendix for concrete tables.",
"Interestingly, for minimum sum-rank distance 3 (co-dimension $ h=2 $ ), we obtain MSRD codes whose parameters meet the bound recently given in [6] (see Subsections REF and REF ).",
"Remarkably, such codes are the first and only known MSRD codes with minimum sum-rank distance at least 3 and linear field sizes in the block length when the number columns per matrix is arbitrary but constant.",
"We obtain the general family of MSRD codes as follows.",
"We define extended Moore matrices (Definition REF ), which coincide with the matrices introduced in [31] but where evaluation points per conjugacy class need not be linearly independent over the base field.",
"We then characterize when extended Moore matrices are the parity-check matrix (or generator matrix) of an MSRD code (Theorem REF ).",
"The obtained sufficient and necessary conditions on the evaluation points constitute a generalization of the concept of (partial) spread in projective geometry.",
"Extending Segre's construction of spreads [42], we construct sequences of evaluation points satisfying the required conditions by using tensor products of a basis of a small finite-field extension with a sequence of $ t $ -wise independent [16] elements over a larger finite-field extension (Theorem REF ).",
"As $ t $ -wise independent sequences coincide with linear Hamming-metric codes by vectorizing finite-field extensions (Lemma REF ), what is left is to use families of Hamming-metric codes with small redundancy.",
"Our choices (which seem to be the best) of such Hamming-metric codes are: Trivial codes, yielding linearized Reed-Solomon codes (Subsection REF ); MDS codes (Subsection REF ); Hamming codes, equivalent to spreads as constructed by Segre (Subsection REF ); primitive BCH codes (Subsection REF ); and Algebraic-Geometry (AG) codes, including Hermitian AG codes (Subsection REF ); Suzuki AG codes (Subsection REF ); and García-Stichtenoth's second sequence of AG codes (Subsection REF ).",
"As mentioned above, our general family of MSRD codes recovers linearized Reed-Solomon codes when using a trivial code to construct the evaluation points.",
"Even though linearized Reed-Solomon codes recover as particular cases (generalized) Reed-Solomon codes [40] and Gabidulin codes [12], [13], [41], our general family of MSRD does not seem to have an analogue in the Hamming metric or the rank metric (see Remark REF ).",
"The remainder of the manuscript is organized as follows.",
"In Section , we collect some preliminaries on MDS, MSRD and PMDS codes, together with some considerations and known bounds on field sizes.",
"In Section , we characterize when a sequence of evaluation points turn an extended Moore matrix into the parity-check matrix of an MSRD code.",
"In Section , we construct such sequences via tensor products and a range of known Hamming-metric codes.",
"Finally, in Section , we provide a summary of the obtained explicit MSRD and PMDS codes and compare their parameters with codes from the literature.",
"The Appendix contains several tables with achievable field sizes in the binary case (characteristic 2)."
],
[
"Basic notation",
"We will denote $ \\mathbb {N} = \\lbrace 0,1,2, \\ldots \\rbrace $ and $ \\mathbb {Z}_+ = \\lbrace 1,2,3, \\ldots \\rbrace $ .",
"For positive integers $ m \\le n $ , we denote $ [n] = \\lbrace 1,2, \\ldots , n\\rbrace $ and $ [m,n] = \\lbrace m, m+1, \\ldots , n \\rbrace $ .",
"For a field $ \\mathbb {F} $ , we denote $ \\mathbb {F}^* = \\mathbb {F} \\setminus \\lbrace 0 \\rbrace $ and we use $ \\langle \\cdot \\rangle _\\mathbb {F} $ and $ \\dim _\\mathbb {F}(\\cdot ) $ to denote $ \\mathbb {F} $ -linear span and dimension over $ \\mathbb {F} $ , respectively.",
"We denote by $ \\mathbb {F}^{m \\times n} $ the set of $ m \\times n $ matrices with entries in $ \\mathbb {F} $ , and we denote $ \\mathbb {F}^n = \\mathbb {F}^{1 \\times n} $ .",
"The group of invertible matrices in $ \\mathbb {F}^{n \\times n} $ is denoted by $ {\\rm GL}_n(\\mathbb {F}) $ .",
"A code in $ \\mathbb {F}^n $ is any subset $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ , and we say that $ \\mathcal {C} $ is a linear code if it is an $ \\mathbb {F} $ -linear vector subspace of $ \\mathbb {F}^n $ .",
"For matrices $ A_1, A_2, \\ldots , $ $ A_g $ $ \\in \\mathbb {F}^{r \\times s} $ , for some positive integers $ g $ , $ r $ and $ s $ , we define the block-diagonal matrix $ {\\rm diag} (A_1, A_2, \\ldots , A_g) = \\left( \\begin{array}{cccc}A_1 & 0 & \\ldots & 0 \\\\0 & A_2 & \\ldots & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\ldots & A_g\\end{array} \\right) \\in \\mathbb {F}^{gr \\times gs} .",
"$ We will also denote by $ \\mathbf {c} \\cdot \\mathbf {d} \\in \\mathbb {F} $ the conventional inner product of $ \\mathbf {c}, \\mathbf {d} \\in \\mathbb {F}^n $ (i.e., $ \\mathbf {c} \\cdot \\mathbf {d} = \\mathbf {c} \\mathbf {d}^T $ ), and we denote the dual of a linear code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ by $ \\mathcal {C}^\\perp = \\lbrace \\mathbf {d} \\in \\mathbb {F}^n \\mid \\mathbf {c} \\cdot \\mathbf {d} = 0, \\textrm { for all } \\mathbf {c} \\in \\mathcal {C} \\rbrace \\subseteq \\mathbb {F}^n .",
"$ For a prime power $ q $ , we denote by $ \\mathbb {F}_q $ the finite field with $ q $ elements.",
"Throughout this manuscript, we will fix a prime power $ q $ and a finite-field extension $ \\mathbb {F}_q \\subseteq \\mathbb {F}_{q^m} $ , for some positive integer $ m $ .",
"The field $ \\mathbb {F}_q $ will be called the base field throughout the manuscript.",
"Our target codes will be linear codes $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^n $ , hence we will usually call $ \\mathbb {F}_{q^m} $ the field of linearity of $ \\mathcal {C} $ .",
"We will use the conventional big O, big Omega and big Theta notations for a positive real-valued function $ f $ , written $ \\mathcal {O}(f) $ , $ \\Omega (f) $ and $ \\Theta (f) $ , respectively.",
"However, we will try to provide exact bounds as much as possible, and we will leave the simplified asymptotic notation when comparisons become complicated."
],
[
"MDS codes",
"For a positive integer $ n $ and a field $ \\mathbb {F} $ , we define the Hamming weight [20] of a vector $ \\mathbf {c} = (c_1, c_2, \\ldots , $ $ c_n) $ $ \\in \\mathbb {F}^n $ by $ {\\rm wt}_H(\\mathbf {c}) = | \\lbrace i \\in [n] \\mid c_i \\ne 0 \\rbrace |.",
"$ We define the Hamming metric $ {\\rm d}_H : \\left( \\mathbb {F}^n \\right) ^2 \\longrightarrow \\mathbb {N} $ by $ {\\rm d}_H(\\mathbf {c}, \\mathbf {d}) = {\\rm wt}_H(\\mathbf {c} - \\mathbf {d}) $ , for all $ \\mathbf {c}, \\mathbf {d} \\in \\mathbb {F}^n $ .",
"For a (linear or non-linear) code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ , we define its minimum Hamming distance by $ {\\rm d}_H(\\mathcal {C}) = \\min \\left\\lbrace {\\rm d}_H(\\mathbf {c}, \\mathbf {d}) \\mid \\mathbf {c}, \\mathbf {d} \\in \\mathcal {C}, \\mathbf {c} \\ne \\mathbf {d} \\right\\rbrace .",
"$ We next revisit the Singleton bound [45].",
"Proposition 1 (Singleton bound [45]) For any (linear or non-linear) code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ , it holds that $| \\mathcal {C} | \\le |\\mathbb {F}|^{n - {\\rm d}_H(\\mathcal {C}) + 1}.$ We define maximum distance separable (MDS) codes as usual.",
"Definition 2 (MDS codes [45]) We say that a (linear or non-linear) code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ is maximum distance separable (MDS) if equality holds in (REF ).",
"Recall that, for a linear code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ of dimension $ k $ , we say that $ G \\in \\mathbb {F}^{k \\times n} $ and $ H \\in \\mathbb {F}^{h \\times n} $ , $ h = n-k $ , are a generator matrix and a parity-check matrix of $ \\mathcal {C} $ , respectively, if $ \\mathcal {C} = \\left\\lbrace \\mathbf {x} G \\in \\mathbb {F}^n \\mid \\mathbf {x} \\in \\mathbb {F}^k \\right\\rbrace = \\left\\lbrace \\mathbf {y} \\in \\mathbb {F}^n \\mid \\mathbf {y} H = \\mathbf {0} \\right\\rbrace , $ respectively.",
"As is well known, $ H $ and $ G $ form a generator matrix and a parity-check matrix, respectively, of the dual code $ \\mathcal {C}^\\perp \\subseteq \\mathbb {F}^n $ .",
"The following result can be found in [30] and [23].",
"This lemma will be crucial for our purposes (see Subsection REF ).",
"Lemma 3 Let $ t $ be a positive integer, let $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ be a linear code of dimension $ k $ , and let $ H \\in \\mathbb {F}^{h \\times n} $ be one of its parity-check matrices, where $ h = n-k $ .",
"It holds that $ {\\rm d}_H(\\mathcal {C}) \\ge t+1 $ if, and only if, any $ t $ columns of $ H $ are linearly independent.",
"In particular, $ \\mathcal {C} $ is MDS if, and only if, any $ h $ columns of $ H $ are linearly independent."
],
[
"MSRD codes",
"Fix positive integers $ m $ and $ r $ , and an ordered basis $ \\alpha = ( \\alpha _1, \\alpha _2, \\ldots , \\alpha _m ) \\in \\mathbb {F}_{q^m}^m $ of $ \\mathbb {F}_{q^m} $ over $ \\mathbb {F}_q $ .",
"We define the matrix representation map $ M_{\\alpha } : \\mathbb {F}_{q^m}^r \\longrightarrow \\mathbb {F}_q^{m \\times r} $ by $M_{\\alpha } \\left( \\sum _{i=1}^m \\alpha _i \\mathbf {c}_i \\right) = \\left( \\begin{array}{cccc}c_{1,1} & c_{1,2} & \\ldots & c_{1,r} \\\\c_{2,1} & c_{2,2} & \\ldots & c_{2,r} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\c_{m,1} & c_{m,2} & \\ldots & c_{m,r} \\\\\\end{array} \\right) \\in \\mathbb {F}_q^{m \\times r},$ where $ \\mathbf {c}_i = (c_{i,1}, c_{i,2}, \\ldots , c_{i,r}) \\in \\mathbb {F}_q^m $ , for $ i = 1,2, \\ldots , m $ .",
"In order to define sum-rank weights on vectors with components in $ \\mathbb {F}_{q^m} $ , we will subdivide them into subvectors as $ \\mathbf {c} = (\\mathbf {c}^{(1)}, $ $ \\mathbf {c}^{(2)}, $ $ \\ldots , $ $ \\mathbf {c}^{(g)}) \\in \\mathbb {F}_{q^m}^{g r} $ , where $ \\mathbf {c}^{(i)} \\in \\mathbb {F}_{q^m}^r $ , for $ i = 1,2, \\ldots , g $ , for a positive integer $ g $ .",
"Now using (REF ), we may consider $ \\mathbf {c} \\in \\mathbb {F}_{q^m}^{gr} $ as a list of $ g $ matrices of size $ m \\times r $ over $ \\mathbb {F}_q $ : $\\mathbf {c} = \\left( \\mathbf {c}^{(1)}, \\mathbf {c}^{(2)}, \\ldots , \\mathbf {c}^{(g)} \\right) \\equiv \\left( M_{\\alpha } \\left( \\mathbf {c}^{(1)} \\right) , M_{\\alpha } \\left( \\mathbf {c}^{(2)} \\right), \\ldots , M_{\\alpha } \\left( \\mathbf {c}^{(g)} \\right) \\right) \\in \\left( \\mathbb {F}_q^{m \\times r} \\right)^g .$ In this work, the matrix sizes at different positions all have size $ m \\times r $ .",
"See Remark REF below regarding different matrix sizes at different positions.",
"The sum-rank metric was defined in [38], under the name extended rank distance, as follows.",
"Definition 4 (Sum-rank metric [38]) Let $ g $ be a positive integer, and let $ \\mathbf {c} = (\\mathbf {c}^{(1)}, $ $ \\mathbf {c}^{(2)}, $ $ \\ldots , $ $ \\mathbf {c}^{(g)}) \\in \\mathbb {F}_{q^m}^{g r} $ , where $ \\mathbf {c}^{(i)} \\in \\mathbb {F}_{q^m}^r $ , for $ i = 1,2, \\ldots , g $ .",
"We define the sum-rank weight of $ \\mathbf {c} $ , for the length partition $ (g,r) $ over the base field $ \\mathbb {F}_q $ , by $ {\\rm wt}_{SR}(\\mathbf {c}) = \\sum _{i=1}^g {\\rm Rk} \\left( M_{\\alpha } ( \\mathbf {c}^{(i)} ) \\right) .",
"$ Finally, we define the sum-rank metric $ {\\rm d}_{SR} : \\left( \\mathbb {F}_{q^m}^{g r} \\right) ^2 \\longrightarrow \\mathbb {N} $ , for the length partition $ (g,r) $ over the base field $ \\mathbb {F}_q $ , by $ {\\rm d}_{SR}(\\mathbf {c}, \\mathbf {d}) = {\\rm wt}_{SR}(\\mathbf {c} - \\mathbf {d}) $ , for all $ \\mathbf {c}, \\mathbf {d} \\in \\mathbb {F}_{q^m}^{g r} $ .",
"For a code $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^{gr} $ (linear or non-linear), we define its minimum sum-rank distance (for the length partition $ (g,r) $ over the base field $ \\mathbb {F}_q $ ) by ${\\rm d}_{SR}(\\mathcal {C}) = \\min \\lbrace {\\rm d}_{SR}(\\mathbf {c}, \\mathbf {d}) \\mid \\mathbf {c}, \\mathbf {d} \\in \\mathcal {C}, \\mathbf {c} \\ne \\mathbf {d} \\rbrace .$ The number $ g $ will be called the number of matrix sets.",
"In view of Remark REF below, we will assume from now on that $ m \\ge r $ , hence $ r $ is the maximum possible rank of an $ m \\times r $ matrix.",
"For brevity in the notation, we will omit the length partition $ (g,r) $ and the base field $ \\mathbb {F}_q $ when they are understood from the context.",
"However, it is important to keep in mind that the definition of sum-rank metric in $ \\mathbb {F}_{q^m}^{gr} $ depends on $ (g,r) $ and $ \\mathbb {F}_q $ , since the ($ \\mathbb {F}_q $ -linear) vector space isomorphism $ \\mathbb {F}_{q^m}^{gr} \\cong \\left( \\mathbb {F}_q^{m \\times r} \\right)^g $ given by (REF ) depends on the triplet $ (g,r,q) $ .",
"Considering codes in $ \\mathbb {F}_{q^m}^{gr} $ instead of $ \\left( \\mathbb {F}_q^{m \\times r} \\right)^g $ will allow us to consider $ \\mathbb {F}_{q^m} $ -linear codes and to characterize MSRD codes in terms of MDS codes (Proposition REF ).",
"Observe that the Hamming metric [20] and the rank metric [12], [13], [41] are recovered from the sum-rank metric by setting $ r = 1 $ and $ g = 1 $ , respectively.",
"We have the following extension of the Singleton bound from the Hamming metric (Proposition REF ) to the sum-rank metric, that is, from the case $ r = 1 $ to the case $ r \\ge 1 $ .",
"This result was given in [36].",
"Proposition 5 (Singleton bound [36]) Let $ g $ be a positive integer, and let $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^{gr} $ be a (linear or non-linear) code.",
"It holds that $| \\mathcal {C} | \\le q^{m (gr - {\\rm d}_{SR}(\\mathcal {C}) + 1)}.$ Furthermore, equality holds in (REF ) if, and only if, $ \\mathcal {C} \\cdot {\\rm diag}(A_1, A_2, \\ldots , A_g) \\subseteq \\mathbb {F}_{q^m}^{gr} $ is MDS, for all $ A_1, A_2, \\ldots , A_g \\in {\\rm GL}_{r}(\\mathbb {F}_q) $ .",
"The main objects of study in this manuscript are maximum sum-rank distance (MSRD) codes, introduced in [31], which are a natural extension of MDS codes (Definition REF ).",
"Definition 6 (MSRD codes [31]) For a positive integer $ g $ , we say that a (linear or non-linear) code $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^{gr} $ is maximum sum-rank distance (MSRD), for the length partition $ (g,r) $ and the base field $ \\mathbb {F}_q $ , if equality holds in (REF ), or equivalently, if $ \\mathcal {C} \\cdot {\\rm diag}(A_1, A_2, \\ldots , A_g) \\subseteq \\mathbb {F}_{q^m}^{gr} $ is MDS, for all $ A_1, A_2, \\ldots , A_g \\in {\\rm GL}_{r}(\\mathbb {F}_q) $ , by Proposition REF .",
"Remark 7 It was shown in [36] that, if $ m < r $ , then there is no MSRD code in $ \\mathbb {F}_{q^m}^{gr} $ as in Definition REF with minimum sum-rank distance larger than 1.",
"For this reason, we will assume from now on that $ m \\ge r $ .",
"An alternative Singleton bound exists for the case $ m < r $ [36], but any MSRD code as in Definition REF achieves such a bound for the same number of matrix sets $ g $ by transposing the matrices in (REF ).",
"The resulting codes are however only $ \\mathbb {F}_q $ -linear.",
"Remark 8 Given a linear MSRD code in $ \\mathbb {F}_{q^m}^{gr} $ , one may obtain a linear MSRD code in $ \\mathbb {F}_{q^m}^{r_1 + r_2 + \\cdots + r_g} $ , with different numbers of columns $ r_1, r_2, \\ldots , r_g \\le r $ per matrix as in (REF ), by puncturing or shortening on some coordinates [33].",
"Singleton bounds and MSRD code constructions for the case of different numbers of both rows and columns at different positions in the matrices in (REF ) can be found in [6].",
"However, the codes constructed in [6] are only $ \\mathbb {F}_q $ -linear, have minimum sum-rank distance 2 or $ \\sum _{i=1}^g r_i - 1 $ , or require the number of rows and columns to be 1 at some positions.",
"The following result was proven in [33].",
"Lemma 9 ([33]) For a positive integer $ g $ , a linear code $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^{gr} $ is MSRD if, and only if, its dual $ \\mathcal {C}^\\perp \\subseteq \\mathbb {F}_{q^m}^{gr} $ is MSRD, in both cases for the length partition $ (g,r) $ and base field $ \\mathbb {F}_q $ .",
"What Lemma REF implies is that, in order to construct linear MSRD codes, we may utilize either generator matrices or parity-check matrices, and focus on either low-dimensional or high-dimensional linear codes.",
"In this work, we will construct high-dimensional linear MSRD codes by building their parity-check matrices with a small number of rows.",
"This is because, although not always, information rates are generally larger than $ 1/2 $ in real scenarios."
],
[
"PMDS codes",
"In this subsection, we briefly recall the definitions of locally repairable codes [17], [24] and PMDS codes [3], [16].",
"We conclude with Construction 1 from [36] that turns any MSRD code into a PMDS code, essentially showing that the former is a subfamily of the latter.",
"Definition 10 (Locally repairable codes [17], [24]) For positive integers $ g $ , $ r $ and $ \\delta $ , we say that a code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ is a locally repairable code (LRC) with $ (r,\\delta ) $ -localities if $ n = g(r+\\delta -1) $ and we may partition $ [n] = \\Gamma _1 \\cup \\Gamma _2 \\cup \\ldots \\cup \\Gamma _g $ , such that, if we denote $ \\nu = r + \\delta - 1 $ , then $ \\Gamma _i = [ (i-1)\\nu + 1, i \\nu ] $ , thus $ |\\Gamma _i| = \\nu $ , and $ {\\rm d}_H(\\mathcal {C}_{\\Gamma _i}) \\ge \\delta $ , where $ \\mathcal {C}_{\\Gamma _i} \\subseteq \\mathbb {F}^\\nu $ denotes the projection of $ \\mathcal {C} $ onto the coordinate in $ \\Gamma _i $ , for $ i = 1,2, \\ldots , g $ .",
"The set $ \\Gamma _i $ is called the $ i $ th local set and $ \\nu $ is the local-set size.",
"In many occasions, we only use the term locality for the number $ r $ , whereas $ \\delta $ is called the local distance.",
"Partial MDS (PMDS) codes, introduced in [3], [16], are those LRCs that may correct any erasure pattern that is information-theoretically correctable given the locality constraints in Definition REF .",
"Such patterns are exactly those with $ \\delta - 1 $ erasures per local set and an extra $ h = gr - k $ erasures anywhere else, where $ k = \\dim (\\mathcal {C}) $ .",
"This is equivalent to obtaining an MDS code after puncturing on any $ \\delta - 1 $ coordinates per local set.",
"We will follow this formulation.",
"Definition 11 (PMDS codes [3], [16]) We say that a linear code $ \\mathcal {C} \\subseteq \\mathbb {F}^n $ is a partial MDS (PMDS) code with $ (r,\\delta ) $ -localities if it is an LRC with $ (r,\\delta ) $ -localities and, for any $ \\Delta _i \\subseteq \\Gamma _i $ with $ | \\Delta _i | = r $ , for $ i = 1,2, \\ldots ,g $ , the restricted code $ \\mathcal {C}_\\Delta \\subseteq \\mathbb {F}^{g r} $ is MDS, where $ \\Delta = \\bigcup _{i=1}^g \\Delta _i $ .",
"The following construction is Construction 1 in [36].",
"Construction 1 ([36]) Fix positive integers $ g $ and $ r $ .",
"Choose a base field size $ q $ and an extension degree $ m \\ge r $ , and define the field of linearity of our target codes as $ \\mathbb {F} = \\mathbb {F}_{q^m} $ .",
"Next choose: Outer code: A linear code $ \\mathcal {C}_{out} \\subseteq \\mathbb {F}_{q^m}^{g r} $ that is MSRD for the length partition $ (g,r) $ over $ \\mathbb {F}_q $ .",
"Local codes: MDS codes $ \\mathcal {C}_{loc}^{(i)} \\subseteq \\mathbb {F}_q^{r + \\delta - 1} $ , linear over the base field $ \\mathbb {F}_q $ and of dimension $ r $ , for $ i = 1,2, \\ldots , g $ .",
"Global code: Let $ \\mathcal {C}_{glob} \\subseteq \\mathbb {F}_{q^m}^n $ , where $ n = g (r + \\delta - 1) $ , be given by $ \\mathcal {C}_{glob} = \\mathcal {C}_{out} \\cdot {\\rm diag}(A_1, A_2, \\ldots , A_g), $ where $ A_1, A_2, \\ldots , A_g \\in \\mathbb {F}_q^{r \\times (r + \\delta - 1)} $ are arbitrary generator matrices of $ \\mathcal {C}_{loc}^{(i)} $ , for $ i = 1,2, \\ldots , g $ .",
"The following result is [36].",
"Proposition 12 ([36]) The linear code $ \\mathcal {C}_{glob} \\subseteq \\mathbb {F}_{q^m}^n $ from Construction REF has dimension $ k = \\dim (\\mathcal {C}_{out}) = \\dim (\\mathcal {C}_{glob}) $ and is a PMDS code with $ (r,\\delta ) $ -localities.",
"In conclusion, any MSRD code as in Definition REF naturally gives a PMDS code via Construction REF .",
"By puncturing on any $ \\delta - 1 $ coordinates per local set, we recover a linear code that is sum-rank isometric to the original MSRD code $ \\mathcal {C}_{out} $ .",
"If the generator matrices $ A_1, A_2, \\ldots , A_g \\in \\mathbb {F}_q^{r \\times (r + \\delta - 1)} $ of the local codes are chosen to be systematic, and we puncture on the $ \\delta - 1 $ coordinates corresponding to parity symbols in each local set, then we exactly recover the MSRD code $ \\mathcal {C}_{out} $ .",
"Remark 13 If $ \\delta = 1 $ or $ \\delta = 2 $ , then there is no additional restriction on the base field $ \\mathbb {F}_q $ , as MDS codes with minimum Hamming distance 1 or 2 exist over any finite field.",
"However, if $ \\delta > 2 $ , then we need to assume that $ q \\ge r + \\delta - 1 $ in order to use known MDS codes over $ \\mathbb {F}_q $ , such as Reed-Solomon codes [40]."
],
[
"Field sizes in applications of MSRD codes",
"Before constructing MSRD codes (thus PMDS codes by Construction REF ), it is important to know what we want in an MSRD code.",
"Otherwise we are lost before starting.",
"The parameters of the ambient space are $ m $ , $ r $ (matrix sizes), $ g $ (number of matrix sets) and $ q $ (base field size).",
"However, the computational complexity of encoding and decoding with a linear (over $ \\mathbb {F}_{q^m} $ ) code in $ \\mathbb {F}_{q^m}^{gr} $ is strongly governed by the size of the field of linearity: $ q^m $ .",
"In some applications of MSRD codes, such as constructing PMDS codes (Subsection REF ) or universal error-correcting codes in multishot linear network coding [35], the base field $ \\mathbb {F}_q $ is an artifice only constrained to contain a given finite field $ \\mathbb {F}_{q_0} $ .",
"The field $ \\mathbb {F}_{q_0} $ is the field of linearity of the local codes in PMDS codes (Construction REF ) or the field of coefficients for linear network coding [35].",
"In these scenarios, $ q_0 $ is generally much smaller than the size of an erased unit, e.g., an erased storage node or a network packet in error (we are comparing $ q_0 = 2, 2^2, 2^3, \\ldots $ with MiB $ = 2^{20 \\cdot 8} $ , GiB $ = 2^{30 \\cdot 8} $ , TiB $ = 2^{40 \\cdot 8} $ , $ \\ldots $ ).",
"The final constraints on the pair $ (m,q) $ are that $ \\mathbb {F}_{q_0} \\subseteq \\mathbb {F}_q $ (i.e., $ q $ is a power of $ q_0 $ ) and the size of an erased unit, measured in number of bits, is a multiple of $ m \\log _2(q) $ (the erased unit is a vector with components in $ \\mathbb {F}_{q^m} $ ).",
"This means that, in such applications, we have almost full freedom on the pair $ (m,q) $ (to construct PMDS codes we only need $ q \\ge \\nu $ if $ \\delta > 2 $ , and there is no restriction on $ q $ if $ \\delta = 2 $ , by Remark REF ).",
"Thus the main focus is on obtaining a size $ | \\mathbb {F}_{q^m} | = q^m $ as small as possible in order to reduce the computational complexity of encoding and decoding, without worrying about the exact pair $ (m,q) $ .",
"For instance, to construct PMDS codes, $ (m_1, q_1) $ is better than $ (m_2, q_2) $ if, and only if, $ q_1^{m_1} < q_2^{m_2} $ , as long as both $ q_1 \\ge \\nu $ and $ q_2 \\ge \\nu $ , regardless of the relation between $ q_1 $ and $ q_2 $ , and between $ m_1 $ and $ m_2 $ .",
"However, in other applications, such as rate-diversity optimal multiblock space-time codes [29], [44] or multilayer/multishot versions of criss-cross error correction [41], we may not have such flexibility on the pair of parameters $ (m,q) $ .",
"In criss-cross error correction [41], errors occur along rows and columns of matrices in $ \\mathbb {F}_q^{m \\times r} $ , where in many cases, $ q = 2 $ .",
"Here, codewords need to fit such structures and we do not have any flexibility on the pair $ (m,q) $ .",
"Hence we just need to find an MSRD code with suitable parameters $ m $ , $ r $ , $ g $ and $ q $ .",
"When building rate-diversity optimal multiblock space-time codes, the base field size $ q $ corresponds to the constellation size, $ m $ corresponds to the time delay, $ r $ is the number of transmit antennas and $ g $ is the number of fading blocks (see [44]).",
"Therefore, in this case, small $ q $ may be desirable for implementation purposes.",
"As an example, space-time codes based on linearized Reed-Solomon codes [31] achieve minimum possible delay $ m = r $ and constellation size $ q \\approx g $ , while space-time codes based on cyclic division algebras [43] require constellation sizes that are exponential in $ g $ and which suffer from approximation errors in the neighbourhood of the complex-plane origin (see [44]).",
"If we fix $ q $ , then it is desirable to obtain linear MSRD codes with smallest possible value of $ m $ .",
"This is because of the next proposition, which is left to the reader to prove.",
"It means that if we find a linear MSRD code for a pair $ (q,m) $ , then we may easily obtain a linear MSRD code for the pair $ (q, mM) $ , for any positive integer $ M $ , being all other parameters equal.",
"Thus an MSRD code with a smaller value of $ m $ enables a wider range of attainable values of $ m $ .",
"Proposition 14 For positive integers $ m $ , $ r $ and $ g $ , and for a linear code $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^{gr} $ , define $ \\mathcal {C} \\otimes \\mathbb {F}_{q^{mM}} = \\left\\lbrace \\lambda \\mathbf {c} \\mid \\mathbf {c} \\in \\mathcal {C}, \\lambda \\in \\mathbb {F}_{q^{mM}} \\right\\rbrace \\subseteq \\mathbb {F}_{q^{mM}}^{gr} , $ for any positive integer $ M $ .",
"Then $ \\mathcal {C} \\otimes \\mathbb {F}_{q^{mM}} $ is $ \\mathbb {F}_{q^{mM}} $ -linear, $ \\dim _{\\mathbb {F}_{q^m}} \\left( \\mathcal {C} \\right) = \\dim _{\\mathbb {F}_{q^{mM}}} \\left( \\mathcal {C} \\otimes \\mathbb {F}_{q^{mM}} \\right), $ and the minimum sum-rank distances of $ \\mathcal {C} $ and $ \\mathcal {C} \\otimes \\mathbb {F}_{q^{mM}} $ are the same, in both cases for the length partition $ (g,r) $ over the field $ \\mathbb {F}_q $ .",
"In particular, $ \\mathcal {C} $ is MSRD if, and only if, so is $ \\mathcal {C} \\otimes \\mathbb {F}_{q^{mM}} $ .",
"A difficult research problem, still open in most cases, is to determine constraints in $ m $ , $ q $ and $ q^m $ for the existence of MSRD codes and PMDS codes.",
"This problem is a highly non-trivial extension of the well known MDS conjecture (not even the asymptotic order of possible MSRD or PMDS codes is known in general, whereas we know that MDS codes exist if, and only if, the code length is at most linear in the field size).",
"Recently, the following bounds were given in [6] for MSRD codes.",
"Proposition 15 ([6]) For positive integers $ m $ , $ r $ and $ g $ , let $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^{gr} $ be a (linear or non-linear) MSRD code.",
"If we set $ h = {\\rm d}_{SR}(\\mathcal {C}) - 1 \\ge 2 $ , then $g \\le \\left\\lfloor \\frac{h-2}{r} \\right\\rfloor + \\left\\lfloor (q-1) \\cdot \\frac{q^m}{q^r - 1} \\right\\rfloor + 1 , \\quad \\textrm {or} \\quad q^m \\ge \\frac{q^r - 1}{q - 1} \\cdot \\left( g - \\left\\lfloor \\frac{h-2}{r} \\right\\rfloor - 1 \\right) .$ For the case $ h = 2 $ and arbitrary $ m $ and $ r $ (recall that $ m \\ge r $ ), we have the tighter bound $g \\le \\left\\lfloor (q-1) \\cdot \\frac{q^m + 1}{q^r - 1} \\right\\rfloor , \\quad \\textrm {or} \\quad q^m \\ge \\frac{q^r - 1}{q - 1} \\cdot g - 1 ,$ and if, furthermore, $ r $ divides $ m $ and $ r \\ge 2 $ , then (REF ) implies the bound $g \\le (q-1) \\cdot \\frac{(q^m - 1)}{q^r - 1} , \\quad \\textrm {or} \\quad q^m \\ge \\frac{q^r - 1}{q - 1} \\cdot g + 1 .$ Finally, for the case $ m = r $ and arbitrary $ g $ , we have the bound $g \\le \\left\\lfloor \\frac{h-2}{r} \\right\\rfloor + q + 1 , \\quad \\textrm {or} \\quad q \\ge g - \\left\\lfloor \\frac{h-2}{r} \\right\\rfloor - 1 .$ Similarly in the PMDS literature, the following bounds were proven in [18].",
"Proposition 16 ([18]) Let $ \\mathcal {C} \\subseteq \\mathbb {F}_{q^m}^n $ be a PMDS code as in Definition REF , with $ (r,\\delta ) $ -localities, $ g $ local sets and $ h = gr - \\dim (\\mathcal {C}) $ .",
"If $ \\delta + 1 \\le h \\le g $ , then $q^m \\ge \\left\\lfloor \\frac{g}{h^2} \\right\\rfloor \\cdot {r + \\delta - 1 \\atopwithdelims ()\\delta } - 1 \\ge \\left\\lfloor \\frac{g}{h^2} \\right\\rfloor \\cdot \\left( \\frac{r+\\delta - 1}{\\delta } \\right)^\\delta - 1 ,$ and if $ h < \\delta + 1 $ and $ h \\le g $ , then $q^m \\ge \\left\\lfloor \\frac{g}{h^2} \\right\\rfloor \\cdot {r + h - 2 \\atopwithdelims ()h-1} - 1 \\ge \\left\\lfloor \\frac{g}{h^2} \\right\\rfloor \\cdot \\left( \\frac{r+ h - 2}{h-1} \\right)^{h-1} - 1 .$ Note that the lower bounds in Proposition REF , when setting $ \\delta = 1 $ , are smaller than those in Proposition REF .",
"This makes sense as MSRD codes can be seen as a subfamily of PMDS codes by setting $ \\delta = 1 $ (see Subsection REF )."
],
[
"Extended Moore matrices",
"This section contains the main method for constructing parity-check matrices of MSRD codes.",
"The section concludes with a definition of a general family of MSRD codes (Definition REF ).",
"Such codes exist and are explicit as long as certain sequence $ \\left( \\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r} \\right) \\in \\mathbb {F}_{q^m}^{\\mu r} $ is known.",
"Explicit constructions of such sequences will be deferred to Section ."
],
[
"The definitions",
"Throughout this subsection, we define the field automorphism $ \\begin{array}{rcl}\\sigma : \\mathbb {F}_{q^m} & \\longrightarrow & \\mathbb {F}_{q^m} \\\\a & \\mapsto & a^q.\\end{array} $ We now define the conjugacy relation.",
"The following definition is a particular case of [26], but already appeared in [25].",
"Definition 17 (Conjugacy [25], [26]) We say that $ a,b \\in \\mathbb {F}_{q^m} $ are conjugate in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ if there exists $ c \\in \\mathbb {F}_{q^m}^* $ such that $ b = \\sigma (c)c^{-1} a = c^{q-1} a $ .",
"Conjugacy is an equivalence relation, whose classes are called conjugacy classes.",
"It was shown in [28] that there are exactly $ q-1 $ non-zero conjugacy classes in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ , each of size $ (q^m-1) / (q-1) $ .",
"Furthermore, they are represented by consecutive powers of a primitive element of $ \\mathbb {F}_{q^m} $ , as observed in the paragraph after [36].",
"Lemma 18 ([28], [36]) Let $ \\gamma \\in \\mathbb {F}_{q^m}^* $ be a primitive element of $ \\mathbb {F}_{q^m} $ .",
"Then $ \\gamma ^0, \\gamma ^1, \\ldots , \\gamma ^{q-2} $ are pair-wise non-conjugate and represent the $ q-1 $ distinct non-zero conjugacy classes in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ .",
"Recently, it was shown in [34] that in some cases we may take the elements in $ \\mathbb {F}_q^* $ as the $ q-1 $ representatives of the conjugacy classes in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ .",
"Lemma 19 ([34]) The $ q-1 $ elements in $ \\mathbb {F}_q^* $ are pair-wise non-conjugate in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ if, and only if, $ q-1 $ and $ (q^m-1)/(q-1) $ are coprime.",
"We now turn to extended Moore matrices.",
"We start by defining truncated norms.",
"Again, the following definition is a particular case of [26], but already appeared in [25].",
"Definition 20 (Truncated norms [25], [26]) Fix $ a \\in \\mathbb {F}_{q^m} $ .",
"We define its $ i $ th truncated norm as $ N_i(a) = \\sigma ^{i-1}(a) \\cdots \\sigma (a)a = a^{\\frac{q^i-1}{q-1}} , $ for all $ i \\in \\mathbb {N} $ .",
"Note that if $ a \\in \\mathbb {F}_q $ , then $ N_i(a) = a^i $ , for all $ i \\in \\mathbb {N} $ .",
"Observe that the map $ N_i $ depends on $ \\sigma $ , but we do not write this dependency for simplicity in the notation.",
"We may now define extended Moore matrices.",
"Definition 21 (Extended Moore matrices) Let $ \\mathbf {a} = ( a_1, a_2, \\ldots , a_\\ell ) \\in (\\mathbb {F}_{q^m}^*)^\\ell $ be a vector of $ \\ell $ pair-wise non-conjugate elements in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ .",
"Let $ \\beta _i = ( \\beta _{i,1}, \\beta _{i,2}, \\ldots , \\beta _{i, \\eta _i} ) \\in \\mathbb {F}_{q^m}^{\\eta _i} $ be an arbitrary vector, for some positive integer $ \\eta _i $ , for $ i = 1,2, \\ldots , \\ell $ .",
"Define $ \\beta = (\\beta _1, \\beta _2, \\ldots , \\beta _\\ell ) \\in \\mathbb {F}_{q^m}^N $ , where $ N = \\eta _1 + \\eta _2 + \\cdots + \\eta _\\ell $ .",
"For $ h = 1,2, \\ldots , N $ , we define the extended Moore matrix $ M_h(\\mathbf {a}, \\beta ) \\in \\mathbb {F}_{q^m}^{h \\times N} $ by $M_h(\\mathbf {a}, \\beta ) = \\left( \\begin{array}{lll|c|lll}\\beta _{1,1} & \\ldots & \\beta _{1, \\eta _1} & \\ldots & \\beta _{\\ell ,1} & \\ldots & \\beta _{\\ell , \\eta _\\ell } \\\\\\beta _{1,1}^q a_1 & \\ldots & \\beta _{1, \\eta _1}^q a_1 & \\ldots & \\beta _{\\ell ,1}^q a_\\ell & \\ldots & \\beta _{\\ell , \\eta _\\ell }^q a_\\ell \\\\\\beta _{1,1}^{q^2} N_2(a_1) & \\ldots & \\beta _{1, \\eta _1}^{q^2} N_2(a_1) & \\ldots & \\beta _{\\ell ,1}^{q^2} N_2(a_\\ell ) & \\ldots & \\beta _{\\ell , \\eta _\\ell }^{q^2} N_2(a_\\ell ) \\\\\\vdots & \\ddots & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\\\beta _{1,1}^{q^{h-1}} N_{h-1}(a_1) & \\ldots & \\beta _{1, \\eta _1}^{q^{h-1}} N_{h-1}(a_1) & \\ldots & \\beta _{\\ell ,1}^{q^{h-1}} N_{h-1}(a_\\ell ) & \\ldots & \\beta _{\\ell , \\eta _\\ell }^{q^{h-1}} N_{h-1}(a_\\ell ) \\\\\\end{array} \\right).$ Such matrices extend the well known Moore matrices (see [27]) from one to several conjugacy classes.",
"They extend the matrices in [31] in the sense that the $ \\eta _i $ components of $ \\beta _i \\in \\mathbb {F}_{q^m}^{\\eta _i} $ over $ \\mathbb {F}_{q^m} $ need not be linearly independent over $ \\mathbb {F}_q $ , for $ i = 1,2, \\ldots , \\ell $ .",
"The main objective of this manuscript is to turn the matrix $ M_h(\\mathbf {a}, \\beta ) $ into the parity-check matrix of an MSRD code of length $ N $ and dimension $ N - h $ (Definition REF ).",
"For convenience, we define MDS matrices and MSRD matrices as follows.",
"Definition 22 (MDS matrices) If $ 1 \\le h \\le N $ , we say that a matrix $ M \\in \\mathbb {F}_{q^m}^{h \\times N} $ is MDS if any $ h $ (distinct) columns of $ M $ form an invertible $ h \\times h $ matrix.",
"Definition 23 (MSRD matrices) Assume that we partition $ N = gr $ , for positive integers $ g $ and $ r $ .",
"If $ 1 \\le h \\le N $ , we say that a matrix $ M \\in \\mathbb {F}_{q^m}^{h \\times N} $ is MSRD for the length partition $ (g, r) $ over the field $ \\mathbb {F}_q $ if $ M \\cdot {\\rm diag}(A_1, A_2, \\ldots , A_g) \\in \\mathbb {F}_{q^m}^{h \\times N} $ is an MDS matrix, for all matrices $ A_i \\in {\\rm GL}_r(\\mathbb {F}_q) $ , for $ i = 1,2, \\ldots , g $ .",
"By Lemma REF , an MDS matrix is nothing but the parity-check matrix of a linear MDS code.",
"By Proposition REF , an MSRD matrix is nothing but the parity-check matrix of a linear MSRD code.",
"Our strategy to characterize MSRD extended Moore matrices will follow two steps.",
"First, we characterize when extended Moore matrices are MDS.",
"Second, we characterize when an MDS extended Moore matrix remains an MDS matrix after multiplication on the right by an appropriate block-diagonal matrix."
],
[
"MDS extended Moore matrices",
"In this subsection, we characterise when an extended Moore matrix (Definition REF ) is an MDS matrix (Definition REF ).",
"We will need the concept of $ h $ -wise independence, introduced in [16].",
"Definition 24 ($ h $ -wise independence [16]) We say that a subset $ T \\subseteq \\mathbb {F}_{q^m} $ is $ h $ -wise independent over $ \\mathbb {F}_q $ if any subset of at most $ h $ (distinct) elements of $ T $ is linearly independent over $ \\mathbb {F}_q $ .",
"Analogously, for a positive integer $ \\eta $ , we say that a vector $ \\beta = (\\beta _1, \\beta _2, \\ldots , \\beta _\\eta ) \\in \\mathbb {F}_{q^m}^\\eta $ is $ h $ -wise independent if $ T = \\lbrace \\beta _1, \\beta _2, \\ldots , \\beta _\\eta \\rbrace $ has size $ \\eta $ and is $ h $ -wise independent.",
"Note that the size of $ T $ in Definition REF is not restricted.",
"For $ | T | \\le h $ , $ T $ is $ h $ -wise independent over $ \\mathbb {F}_q $ if, and only if, $ T $ is linearly independent over $ \\mathbb {F}_q $ .",
"We will also need the following four auxiliary lemmas.",
"The first of these is immediate from the $ \\mathbb {F}_q $ -linearity of $ \\sigma $ .",
"Lemma 25 Fix integers $ 1 \\le \\eta \\le h $ and an element $ a \\in \\mathbb {F}_{q^m}^* $ .",
"Assume that there exist $ \\lambda _1, \\lambda _2 , \\ldots , \\lambda _\\eta \\in \\mathbb {F}_q $ such that $ \\lambda _1 \\beta _1 + \\lambda _2 \\beta _2 + \\cdots + \\lambda _\\eta \\beta _\\eta = 0 $ , for elements $ \\beta _1, \\beta _2, \\ldots , \\beta _\\eta \\in \\mathbb {F}_{q^m} $ .",
"Then it holds that $ \\left( \\begin{array}{llll}\\beta _1 & \\beta _2 & \\ldots & \\beta _\\eta \\\\\\beta _1^q a & \\beta _2^q a & \\ldots & \\beta _\\eta ^q a \\\\\\beta _1^{q^2} N_2(a) & \\beta _2^{q^2} N_2(a) & \\ldots & \\beta _\\eta ^{q^2} N_2(a) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\beta _1^{q^{h-1}} N_{h-1}(a) & \\beta _2^{q^{h-1}} N_{h-1}(a) & \\ldots & \\beta _\\eta ^{q^{h-1}} N_{h-1}(a)\\\\\\end{array} \\right) \\left( \\begin{array}{c}\\lambda _{1} \\\\\\lambda _{2} \\\\\\vdots \\\\\\lambda _\\eta \\end{array} \\right) = \\mathbf {0} .",
"$ The next lemma follows immediately from the invertibility of Moore matrices [27] and Lemma REF .",
"Lemma 26 Fix integers $ 1 \\le \\eta \\le h $ and an element $ a \\in \\mathbb {F}_{q^m}^* $ .",
"The dimension of the $ \\mathbb {F}_q $ -linear subspace generated by the elements $ \\beta _1, \\beta _2, \\ldots , \\beta _\\eta \\in \\mathbb {F}_{q^m} $ equals the rank of the matrix $ \\left( \\begin{array}{llll}\\beta _1 & \\beta _2 & \\ldots & \\beta _\\eta \\\\\\beta _1^q a & \\beta _2^q a & \\ldots & \\beta _\\eta ^q a \\\\\\beta _1^{q^2} N_2(a) & \\beta _2^{q^2} N_2(a) & \\ldots & \\beta _\\eta ^{q^2} N_2(a) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\beta _1^{q^{h-1}} N_{h-1}(a) & \\beta _2^{q^{h-1}} N_{h-1}(a) & \\ldots & \\beta _\\eta ^{q^{h-1}} N_{h-1}(a)\\\\\\end{array} \\right) \\in \\mathbb {F}_{q^m}^{h \\times \\eta } .",
"$ The next lemma may be easily derived by simplifying telescopic products (see [31] for more general formulations).",
"Lemma 27 With notation and assumptions as in Definition REF , it holds that $ M_h(\\mathbf {a}, \\beta ) \\cdot {\\rm diag} \\left( \\beta _{1,1}^{-1}, \\ldots , \\beta _{1, \\eta _1}^{-1} | \\ldots | \\beta _{\\ell ,1}^{-1}, \\ldots , \\beta _{\\ell , \\eta _\\ell }^{-1} \\right) = $ $\\left( \\begin{array}{lll|c|lll}1 & \\ldots & 1 & \\ldots & 1 & \\ldots & 1 \\\\\\beta _{1,1}^{q-1} a_1 & \\ldots & \\beta _{1, \\eta _1}^{q-1} a_1 & \\ldots & \\beta _{\\ell ,1}^{q-1} a_\\ell & \\ldots & \\beta _{\\ell , \\eta _\\ell }^{q-1} a_\\ell \\\\N_2(\\beta _{1,1}^{q-1} a_1) & \\ldots & N_2(\\beta _{1, \\eta _1}^{q-1} a_1) & \\ldots & N_2(\\beta _{\\ell ,1}^{q-1} a_\\ell ) & \\ldots & N_2(\\beta _{\\ell , \\eta _\\ell }^{q-1} a_\\ell ) \\\\\\vdots & \\ddots & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\N_{h-1}(\\beta _{1,1}^{q-1} a_1) & \\ldots & N_{h-1}(\\beta _{1, \\eta _1}^{q-1} a_1) & \\ldots & N_{h-1}(\\beta _{\\ell ,1}^{q-1} a_\\ell ) & \\ldots & N_{h-1}(\\beta _{\\ell , \\eta _\\ell }^{q-1} a_\\ell ) \\\\\\end{array} \\right) .$ The next lemma is a particular case of [25].",
"Lemma 28 ([25]) Let the notation and assumptions be as in Definition REF , and further assume that $ h = N = \\eta _1 + \\eta _2 + \\cdots + \\eta _\\ell $ .",
"Then it holds that ${\\rm Rk} \\left( \\begin{array}{lll|c|lll}1 & \\ldots & 1 & \\ldots & 1 & \\ldots & 1 \\\\\\beta _{1,1}^{q-1} a_1 & \\ldots & \\beta _{1,\\eta _1}^{q-1} a_1 & \\ldots & \\beta _{\\ell ,1}^{q-1} a_\\ell & \\ldots & \\beta _{\\ell , \\eta _\\ell }^{q-1} a_\\ell \\\\N_2(\\beta _{1,1}^{q-1} a_1) & \\ldots & N_2(\\beta _{1, \\eta _1}^{q-1} a_1) & \\ldots & N_2(\\beta _{\\ell ,1}^{q-1} a_\\ell ) & \\ldots & N_2(\\beta _{\\ell , \\eta _\\ell }^{q-1} a_\\ell ) \\\\\\vdots & \\ddots & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\N_{h-1}(\\beta _{1,1}^{q-1} a_1) & \\ldots & N_{h-1}(\\beta _{1, \\eta _1}^{q-1} a_1) & \\ldots & N_{h-1}(\\beta _{\\ell ,1}^{q-1} a_\\ell ) & \\ldots & N_{h-1}(\\beta _{\\ell , \\eta _\\ell }^{q-1} a_\\ell ) \\\\\\end{array} \\right) =$ $ \\sum _{i=1}^\\ell {\\rm Rk} \\left( \\begin{array}{llll}1 & 1 & \\ldots & 1 \\\\\\beta _{i,1}^{q-1} a_i & \\beta _{i,2}^{q-1} a_i & \\ldots & \\beta _{i, \\eta _i}^{q-1} a_i \\\\N_2(\\beta _{i,1}^{q-1} a_i) & N_2(\\beta _{i,2}^{q-1} a_i) & \\ldots & N_2(\\beta _{i,\\eta _i}^{q-1} a_i) \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\N_{h-1}(\\beta _{i,1}^{q-1} a_i) & N_{h-1}(\\beta _{i,2}^{q-1} a_i) & \\ldots & N_{h-1}(\\beta _{i,\\eta _i}^{q-1} a_i)\\end{array} \\right) .",
"$ The main result of this subsection is the following theorem.",
"Theorem 1 Let the notation and assumptions be as in Definition REF .",
"For $ h = 1,2, \\ldots , N $ , the extended Moore matrix $ M_h(\\mathbf {a}, \\beta ) \\in \\mathbb {F}_{q^m}^{h \\times N} $ is MDS if, and only if, the vector $ \\beta _i = ( \\beta _{i,1}, \\beta _{i,2}, $ $ \\ldots , $ $ \\beta _{i,\\eta _i} ) \\in \\mathbb {F}_{q^m}^{\\eta _i} $ is $ h $ -wise independent over $ \\mathbb {F}_q $ , for all $ i = 1,2, \\ldots , \\ell $ .",
"First, assume that $ ( \\beta _{i,1}, \\beta _{i,2}, \\ldots , \\beta _{i,\\eta _i} ) $ is not $ h $ -wise independent over $ \\mathbb {F}_q $ , for some $ i =1,2, \\ldots , \\ell $ .",
"Then $ M_h(\\mathbf {a}, \\beta ) $ contains an $ h \\times h $ submatrix that is not invertible by Lemma REF .",
"Conversely, assume that $ ( \\beta _{i,1}, \\beta _{i,2}, $ $ \\ldots , $ $ \\beta _{i,\\eta _i} ) $ is $ h $ -wise independent over $ \\mathbb {F}_q $ , for $ i = 1,2, \\ldots , \\ell $ .",
"Take an arbitrary $ h \\times h $ submatrix $ M^\\prime \\in \\mathbb {F}_{q^m}^{h \\times h} $ of $ M_h(\\mathbf {a}, \\beta ) $ , and let $ 0 \\le \\eta ^\\prime _i \\le \\min \\lbrace h, \\eta _i \\rbrace $ be the number of columns from the $ i $ th block of $ \\eta _i $ columns in $ M_h(\\mathbf {a}, \\beta ) $ appearing in $ M^\\prime $ , for $ i = 1,2, \\ldots , \\ell $ .",
"Note that $ h = \\eta ^\\prime _1 + \\eta ^\\prime _2 + \\cdots + \\eta ^\\prime _\\ell $ .",
"Since $ (\\beta _{i,1}, \\beta _{i,2}, $ $ \\ldots , $ $ \\beta _{i,\\eta _i} ) $ is $ h $ -wise independent over $ \\mathbb {F}_q $ and $ \\eta ^\\prime _i \\le h $ , then the $ i $ th block of $ \\eta ^\\prime _i $ columns in $ M^\\prime $ forms an $ \\eta ^\\prime _i \\times h $ matrix of full rank $ \\eta ^\\prime _i $ by Lemma REF , for $ i = 1,2, \\ldots , \\ell $ .",
"Finally, by combining Lemmas REF and REF , we conclude that $ {\\rm Rk}(M^\\prime ) = \\eta ^\\prime _1 + \\eta ^\\prime _2 + \\cdots + \\eta ^\\prime _\\ell = h, $ and therefore $ M^\\prime \\in \\mathbb {F}_{q^m}^{h \\times h} $ is invertible.",
"Hence $ M_h(\\mathbf {a}, \\beta ) $ is MDS and we are done."
],
[
"MSRD extended Moore matrices",
"In this subsection, we characterise when an extended Moore matrix (Definition REF ) is an MSRD matrix (Definition REF ).",
"The first characterization is simply combining Proposition REF with Theorem REF and the $ \\mathbb {F}_q $ -linearity of the map $ \\sigma $ .",
"Proposition 29 Let the notation and assumptions be as in Definition REF .",
"Further assume that $ \\eta _1 = \\eta _2 = \\ldots = \\eta _\\ell = \\mu r $ , for positive integers $ \\mu $ and $ r $ .",
"Hence $ N = g r $ , for $ g = \\ell \\mu $ .",
"For $ h = 1,2, \\ldots , N $ , the extended Moore matrix $ M_h(\\mathbf {a}, \\beta ) \\in \\mathbb {F}_{q^m}^{h \\times N} $ is MSRD for the length partition $ (g,r) $ over $ \\mathbb {F}_q $ if, and only if, the vector $ ( \\beta _{i,1}^\\prime , \\beta _{i,2}^\\prime , $ $ \\ldots , $ $ \\beta _{i,\\mu r}^\\prime ) $ is $ h $ -wise independent over $ \\mathbb {F}_q $ , where $ (\\beta _{i,1}^\\prime , \\beta _{i,2}^\\prime , \\ldots , \\beta _{i, \\mu r}^\\prime ) = (\\beta _{i,1}, \\beta _{i,2}, \\ldots , \\beta _{i, \\mu r}) \\cdot {\\rm diag} (A_{i,1}, A_{i,2}, \\ldots , A_{i,\\mu }) \\in \\mathbb {F}_{q^m}^{\\mu r}, $ for all matrices $ A_{i,1}, A_{i,2}, \\ldots , A_{i,\\mu } \\in {\\rm GL}_r (\\mathbb {F}_q) $ , for all $ i = 1,2, \\ldots , \\ell $ .",
"The main result of this subsection is the following theorem.",
"Theorem 2 Let the notation and assumptions be as in Proposition REF .",
"Assume that $ \\beta _1 = \\beta _2 = \\ldots = \\beta _\\ell = (\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) \\in \\mathbb {F}_{q^m}^{\\mu r} $ , and define the $ \\mathbb {F}_q $ -linear subspace $\\mathcal {H}_i = \\left\\langle \\beta _{(i-1)r+1}, \\beta _{(i-1)r+2}, \\ldots , \\beta _{ir} \\right\\rangle _{\\mathbb {F}_q} \\subseteq \\mathbb {F}_{q^m},$ for $ i = 1,2, \\ldots , \\mu $ .",
"Then the extended Moore matrix $ M_h(\\mathbf {a}, \\beta ) \\in \\mathbb {F}_{q^m}^{h \\times N} $ is MSRD for the length partition $ (g,r) $ over $ \\mathbb {F}_q $ if, and only if, the following two conditions hold for all $ i = 1,2, \\ldots , \\mu $ : $ \\dim _{\\mathbb {F}_q}(\\mathcal {H}_i) = r $ , i.e., $ \\beta _{(i-1)r+1}, \\beta _{(i-1)r+2}, \\ldots , \\beta _{ir} $ are linearly independent over $ \\mathbb {F}_q $ , and $ \\mathcal {H}_i \\cap \\left( \\sum _{j \\in \\Gamma } \\mathcal {H}_j \\right) = \\lbrace 0 \\rbrace $ , for any set $ \\Gamma \\subseteq [\\mu ] $ , such that $ i \\notin \\Gamma $ and $ |\\Gamma | \\le \\min \\lbrace h,\\mu \\rbrace -1 $ .",
"We prove both implications separately.",
"$ \\Longleftarrow $ ): Take matrices $ A_1, A_2, \\ldots , A_\\mu \\in {\\rm GL}_r(\\mathbb {F}_q) $ .",
"Condition 1 implies that $ \\beta ^\\prime _{(i-1)r+1}, \\beta ^\\prime _{(i-1)r+2}, $ $ \\ldots , $ $ \\beta ^\\prime _{ir} $ $ \\in \\mathbb {F}_{q^m} $ are linearly independent over $ \\mathbb {F}_q $ , where $ (\\beta ^\\prime _{(i-1)r+1}, \\beta ^\\prime _{(i-1)r+2}, \\ldots , \\beta ^\\prime _{ir}) = (\\beta _{(i-1)r+1}, \\beta _{(i-1)r+2}, \\ldots , \\beta _{ir}) \\cdot A_i \\in \\mathbb {F}_{q^m}^r, $ for all $ i = 1,2, \\ldots , \\mu $ .",
"Next, fix an index $ i = 1,2, \\ldots , \\mu $ , and take a subset $ \\Gamma \\subseteq [\\mu ] $ , such that $ i \\notin \\Gamma $ and $ |\\Gamma | \\le \\min \\lbrace h,\\mu \\rbrace -1 $ .",
"Condition 2 and the $ \\mathbb {F}_q $ -linear independence of each set $ \\lbrace \\beta ^\\prime _{(j-1)r+1}, \\beta ^\\prime _{(j-1)r+2}, $ $ \\ldots , $ $ \\beta ^\\prime _{jr} \\rbrace $ imply that the set $\\bigcup _{j \\in \\Gamma \\cup \\lbrace i\\rbrace } \\left\\lbrace \\beta ^\\prime _{(j-1)r+1}, \\beta ^\\prime _{(j-1)r+2}, \\ldots , \\beta ^\\prime _{jr} \\right\\rbrace \\subseteq \\mathbb {F}_{q^m}$ is linearly independent over $ \\mathbb {F}_q $ .",
"Since every subset of size at most $ h $ of $ \\lbrace \\beta ^\\prime _1, \\beta ^\\prime _2, \\ldots , \\beta ^\\prime _{\\mu r} \\rbrace $ is contained in a set of the form (REF ), we deduce that the vector $ ( \\beta ^\\prime _1, \\beta ^\\prime _2, \\ldots , \\beta ^\\prime _{\\mu r} ) $ is $ h $ -wise linearly independent over $ \\mathbb {F}_q $ .",
"Hence the extended Moore matrix $ M_h(\\mathbf {a}, \\beta ) \\in \\mathbb {F}_{q^m}^{h \\times N} $ is MSRD by Proposition REF .",
"$ \\Longrightarrow $ ): Assume first that Condition 1 does not hold for some $ i = 1,2, \\ldots , \\mu $ .",
"Without loss of generality, we may assume that there exist $ \\lambda _1, \\lambda _2, \\ldots , \\lambda _{r-1} \\in \\mathbb {F}_q $ such that $ \\sum _{j=1}^{r-1} \\lambda _j \\beta _{(i-1)r + j} + \\beta _{ir} = 0.",
"$ Thus if we define the invertible matrix $ A_i = \\left( \\begin{array}{c|c}I_{r-1} & \\begin{array}{c}\\lambda _{1} \\\\\\vdots \\\\\\lambda _{r-1}\\end{array} \\\\\\hline \\begin{array}{ccc}0 & \\ldots & 0\\end{array} & 1\\end{array} \\right) \\in {\\rm GL}_r(\\mathbb {F}_q), $ where $ I_{r-1} \\in {\\rm GL}_{r-1}(\\mathbb {F}_q) $ denotes the $ (r-1) \\times (r-1) $ identity matrix, then it holds that $ (\\beta _{(i-1)r+1}, \\ldots , \\beta _{ir -1}, \\beta _{ir}) \\cdot A_i = (\\beta _{(i-1)r+1}, \\ldots , \\beta _{ir -1}, 0).",
"$ Clearly, $ (\\beta _{(i-1)r+1}, \\ldots , \\beta _{ir -1}, 0) \\in \\mathbb {F}_{q^m}^r $ is not $ h $ -wise independent, thus $ M_h(\\mathbf {a}, \\beta ) $ is not MSRD by Proposition REF .",
"Next, assume that Condition 2 does not hold for some $ i = 1,2, \\ldots , \\mu $ .",
"Then we may assume, without loss of generality, that there exists a subset $ \\Gamma \\subseteq [\\mu ] $ such that $ i \\in \\Gamma $ , $ |\\Gamma | \\le h $ , and there exist $ \\lambda _{j,u} \\in \\mathbb {F}_q $ , for $ u = 1,2, \\ldots , r $ , for $ j \\in \\Gamma $ , such that $ \\lambda _{j,r} = 1 $ , for $ j \\in \\Gamma $ , and $ \\sum _{j \\in \\Gamma } \\sum _{u=1}^r \\lambda _{j,u} \\beta _{(j-1)r + u} = 0.",
"$ Define, for each $ j \\in \\Gamma $ , the invertible matrix $ A_j = \\left( \\begin{array}{c|c}I_{r-1} & \\begin{array}{c}\\lambda _{j,1} \\\\\\vdots \\\\\\lambda _{j,r-1}\\end{array} \\\\\\hline \\begin{array}{ccc}0 & \\ldots & 0\\end{array} & 1\\end{array} \\right) \\in {\\rm GL}_r(\\mathbb {F}_q), $ and define, for convenience, $ A_j = I_r \\in {\\rm GL}_r(\\mathbb {F}_q) $ if $ j \\notin \\Gamma $ .",
"If we set $ (\\beta ^\\prime _1, \\beta ^\\prime _2, \\ldots , \\beta ^\\prime _{\\mu r}) = (\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) \\cdot {\\rm diag}(A_1, A_2, \\ldots , A_\\mu ), $ then it holds that $ \\sum _{j \\in \\Gamma } \\beta ^\\prime _{jr} = \\sum _{j \\in \\Gamma } \\sum _{u=1}^r \\lambda _{j,u} \\beta _{(j-1)r + u} = 0.",
"$ Since $ |\\Gamma | \\le h $ , then the vector $ (\\beta ^\\prime _1, \\beta ^\\prime _2, \\ldots , \\beta ^\\prime _{\\mu r}) $ is not $ h $ -wise independent over $ \\mathbb {F}_q $ , hence $ M_h(\\mathbf {a}, \\beta ) $ is not MSRD by Proposition REF .",
"Remark 30 Observe that, in the case $ 2 = h \\le \\mu $ , Conditions 1 and 2 in Theorem REF are equivalent to the set $ \\lbrace \\mathcal {H}_1, \\mathcal {H}_2 , \\ldots , \\mathcal {H}_\\mu \\rbrace $ being a partial spread of size $ \\mu $ of $ r $ -dimensional subspaces of $ \\mathbb {F}_{q^m} \\cong \\mathbb {F}_q^m $ .",
"In Subsection REF , we describe this case in more detail.",
"General sets of vector spaces satisfying Conditions 1 and 2 in Theorem REF constitute therefore a natural generalization of the concept of partial spread.",
"Remark 31 Observe that Conditions 1 and 2 in Theorem REF , combined, are equivalent to $ \\dim _{\\mathbb {F}_q} \\left( \\mathcal {H}_i \\right) = r $ , for $ i = 1,2, \\ldots , \\mu $ , and $ \\dim _{\\mathbb {F}_q} \\left( \\sum _{i \\in \\Gamma } \\mathcal {H}_i \\right) = r |\\Gamma | = r \\min \\lbrace h, \\mu \\rbrace $ , for any set $ \\Gamma \\subseteq [\\mu ] $ of size $ |\\Gamma | = \\min \\lbrace h, \\mu \\rbrace $ .",
"In particular, if Conditions 1 and 2 hold, then $m \\ge r \\min \\lbrace h,\\mu \\rbrace , \\quad \\textrm {or} \\quad |\\mathbb {F}_{q^m}| = q^m \\ge q^{r \\min \\lbrace h,\\mu \\rbrace } .$ Note that the only additional assumption in Theorem REF is that $ \\beta _1 = \\beta _2 = \\ldots = \\beta _\\ell $ .",
"For different vectors $ \\beta _i $ we still have the same requirements, including (REF ).",
"Hence the field size in (REF ) is necessary for extended Moore matrices as in Theorem REF to be MSRD.",
"However, observe that $ q^m = q^{r \\min \\lbrace h, \\mu \\rbrace } $ is much smaller than the smallest field size required by an MRD code [12], [13], [41] with base field size $ q $ , which would be $ q^m = q^{gr} = q^{\\ell \\mu r} $ (recall that $ g = \\ell \\mu $ ).",
"In subsections REF and REF , we will obtain MSRD extended Moore matrices with $ m = r \\min \\lbrace h, \\mu \\rbrace $ .",
"In that case, we may obtain $ m = r $ if, and only if, $ h = 1 $ or $ \\mu = 1 $ (as in Subsection REF ).",
"Recall from Remark REF that $ m \\ge r $ is necessary for MSRD codes to exist.",
"However, in later subsections, we will obtain field sizes $ q^m $ which may be larger than $ q^{r \\min \\lbrace h, \\mu \\rbrace } $ but smaller relative to the parameters $ g $ and $ r $ (and $ \\delta $ for PMDS codes).",
"In conclusion, we have the following general family of MSRD codes.",
"Definition 32 Let the notation and assumptions be as in Theorem REF .",
"That is, let $ \\mathbf {a} = ( a_1, a_2, \\ldots , a_\\ell ) \\in (\\mathbb {F}_{q^m}^*)^\\ell $ be a vector of $ \\ell $ pair-wise non-conjugate elements in $ \\mathbb {F}_{q^m} $ with respect to $ \\sigma $ .",
"Let $ \\beta _1 = \\beta _2 = \\ldots = \\beta _\\ell = (\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) \\in \\mathbb {F}_{q^m}^{\\mu r} $ satisfy Conditions 1 and 2 in Theorem REF .",
"Let $ N = g r $ , where $ g = \\ell \\mu $ .",
"For $ h = 1,2, \\ldots , N $ , we define the following $ k $ -dimensional linear MSRD code, where $ k = N-h $ , for the length partition $ (g,r) $ over the base field $ \\mathbb {F}_q $ : $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) = \\left\\lbrace \\mathbf {y} \\in \\mathbb {F}_{q^m}^N \\mid \\mathbf {y} M_h(\\mathbf {a}, \\beta ) = \\mathbf {0} \\right\\rbrace .",
"$ Remark 33 Theorem REF and the family of MSRD codes in Definition REF have no meaningful analogue in the Hamming-metric case ($ r = 1 $ ) or the rank-metric case ($ g = 1 $ ).",
"Setting $ r = 1 $ , extended Moore matrices require polynomial field sizes $ q^m = q^{\\min \\lbrace h, \\mu \\rbrace } $ in the code length $ g \\le q-1 $ (thus worse than classical Reed-Solomon codes [40]), unless $ h = 1 $ or $ \\mu = 1 $ .",
"In either of those cases, we may choose $ m = \\min \\lbrace h, \\mu \\rbrace = 1 $ and then an extended Moore matrix is simply a (rectangular) Vandermonde matrix with distinct evaluation points and possibly with column multipliers.",
"Hence the corresponding MDS codes in Definition REF are just generalized Reed-Solomon codes [40].",
"Setting $ g = 1 $ , extended Moore matrices become classical (rectangular) Moore matrices with possibly $ \\mathbb {F}_q $ -linearly dependent evaluation points.",
"However, by Lemma REF , such rectangular Moore matrices form parity-check matrices of MRD codes if, and only if, all evaluation points are $ \\mathbb {F}_q $ -linearly independent.",
"Hence the corresponding MRD codes in Definition REF are just Gabidulin codes [13], [41]."
],
[
"Explicit constructions of MSRD codes",
"What is missing in Definition REF is finding the sequence $ (\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) \\in \\mathbb {F}_{q^m}^{\\mu r} $ .",
"In this section, we provide a technique for constructing explicit sequences $ (\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) \\in \\mathbb {F}_{q^m}^{\\mu r} $ satisfying Conditions 1 and 2 in Theorem REF .",
"This method provides several explicit subfamilies of the codes in Definition REF , where the vector $ \\mathbf {a} \\in (\\mathbb {F}_{q^m}^*)^\\ell $ can be explicitly chosen as in Lemmas REF or REF ."
],
[
"The technique of tensor products",
"In this subsection, we explore the method of performing tensor products of sequences over $ \\mathbb {F}_{q^r} $ and $ \\mathbb {F}_{q^m} $ .",
"This technique is inspired by that used in [14].",
"However, the codes obtained in [14] and in this work are not equivalent (by inspecting the attained parameters).",
"For the remainder of this section, we will fix an ordered basis $ \\alpha = ( \\alpha _1, \\alpha _2, \\ldots , \\alpha _r ) \\in \\mathbb {F}_{q^r}^r $ of $ \\mathbb {F}_{q^r} $ over $ \\mathbb {F}_q $ .",
"We will also assume from now on that $ m = r \\rho $ (hence $ \\mathbb {F}_{q^r} \\subseteq \\mathbb {F}_{q^m} $ ), for some positive integer $ \\rho $ .",
"Choose a vector $\\gamma = (\\gamma _1, \\gamma _2, \\ldots , \\gamma _\\mu ) \\in \\mathbb {F}_{q^m}^\\mu .$ Define the tensor product of $ \\alpha $ with $ \\gamma $ as $(\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) = \\alpha \\otimes \\gamma = (\\alpha _1 \\gamma _1, \\ldots , \\alpha _r \\gamma _1 | \\ldots | \\alpha _1 \\gamma _\\mu , \\ldots , \\alpha _r \\gamma _\\mu ) \\in \\mathbb {F}_{q^m}^{\\mu r}.$ In other words, for $ i = 1,2, \\ldots , \\mu $ , we define $ (\\beta _{(i-1)r+1}, \\beta _{(i-1)r+2}, \\ldots , \\beta _{ir}) = \\alpha \\gamma _i = (\\alpha _1 \\gamma _i , \\alpha _2 \\gamma _i, \\ldots , \\alpha _r \\gamma _i) \\in \\mathbb {F}_{q^m}^r.",
"$ The main result of this section is the following theorem.",
"The proof is straightforward and is left to the reader.",
"Theorem 3 The vector $ (\\beta _1, \\beta _2, \\ldots , \\beta _{\\mu r}) \\in \\mathbb {F}_{q^m}^{\\mu r} $ in (REF ) satisfies Conditions 1 and 2 in Theorem REF , for $ i = 1,2, \\ldots , \\mu $ , if and only if, the vector $ \\gamma = (\\gamma _1, \\gamma _2, \\ldots , \\gamma _\\mu ) \\in \\mathbb {F}_{q^m}^\\mu = \\mathbb {F}_{q^{r \\rho }}^\\mu $ is $ t $ -wise independent over $ \\mathbb {F}_{q^r} $ , for $ t = \\min \\lbrace h,\\mu \\rbrace $ .",
"Remark 34 Observe that, if $ \\rho \\ge \\mu $ , then we may simply choose $ \\gamma = (\\gamma _1, \\gamma _2, \\ldots , \\gamma _\\mu ) \\in \\mathbb {F}_{q^m}^\\mu $ in Theorem REF such that $ \\gamma _1, \\gamma _2, \\ldots , \\gamma _\\mu $ are linearly independent over $ \\mathbb {F}_{q^r} $ .",
"However, in that case, $ m = r \\rho \\ge r \\mu $ , and we do not gain anything by considering $ \\rho > \\mu $ .",
"Hence, we may focus only on the case $ \\rho \\le \\mu $ .",
"In other words, we only need to focus on constructing a vector $ \\gamma \\in \\mathbb {F}_{q^m}^\\mu $ that is $ t $ -wise independent over $ \\mathbb {F}_{q^r} $ , and only in the case $ \\rho \\le \\mu $ , where we always assume that $ m = r \\rho $ .",
"The next subsection provides a coding-theoretic method to construct such vectors."
],
[
"Minimum Hamming distance means $ t $ -wise independence",
"In this subsection, we revisit the equivalence between the concept of $ t $ -wise independent set in $ \\mathbb {F}_{q^m} $ over $ \\mathbb {F}_{q^r} $ and that of $ \\mathbb {F}_{q^r} $ -linear code in $ \\mathbb {F}_{q^r}^\\mu $ with minimum Hamming distance larger than $ t $ .",
"This equivalence has been used previously in the PMDS literature in [16], [14] and throughout [19], among others, but it seems new in the context of MSRD codes.",
"In view of Remark REF , we will assume from now on that $ \\rho \\le \\mu $ .",
"The following result is immediate from combining Lemma REF , Definition REF and the $ \\mathbb {F}_{q^r} $ -linearity of the map $ M_{\\delta } $ given in (REF ).",
"Lemma 35 Let $ \\delta \\in \\mathbb {F}_{q^m}^\\rho $ be an ordered basis of $ \\mathbb {F}_{q^m} = \\mathbb {F}_{q^{r \\rho }} $ over $ \\mathbb {F}_{q^r} $ .",
"Consider the matrix representation map $ M_{\\delta } : \\mathbb {F}_{q^{r \\rho }}^\\mu \\longrightarrow \\mathbb {F}_{q^r}^{\\rho \\times \\mu } $ , as in (REF ), and define $H_{\\gamma } = M_{\\delta }(\\gamma ) \\in \\mathbb {F}_{q^r}^{\\rho \\times \\mu }.$ The vector $ \\gamma \\in \\mathbb {F}_{q^m}^\\mu $ is $ t $ -wise independent over $ \\mathbb {F}_{q^r} $ if, and only if, $ {\\rm d}_H(\\mathcal {C}_{\\gamma }) \\ge t+1 $ , for the $ \\mathbb {F}_{q^r} $ -linear code $\\mathcal {C}_{\\gamma } = \\left\\lbrace \\mathbf {y} \\in \\mathbb {F}_{q^r}^\\mu \\mid \\mathbf {y} H_{\\gamma } = \\mathbf {0} \\right\\rbrace \\subseteq \\mathbb {F}_{q^r}^\\mu .$ Here, we are considering $ {\\rm d}_H(\\mathcal {C}_{\\gamma }) = \\mu +1 $ if $ \\mathcal {C}_{\\gamma } = \\lbrace \\mathbf {0} \\rbrace $ , which is equivalent to $ t = \\rho = \\mu $ and $ H_{\\gamma } \\in {\\rm GL}_{\\mu }(\\mathbb {F}_{q^r}) $ , i.e., $ \\gamma \\in \\mathbb {F}_{q^{r\\mu }}^{\\mu } $ is an ordered basis of $ \\mathbb {F}_{q^{r \\mu }} $ over $ \\mathbb {F}_{q^r} $ .",
"In conclusion, to construct $ \\gamma \\in \\mathbb {F}_{q^r}^\\mu $ , we may choose a known Hamming-metric code $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ as in (REF ), with code length $ \\mu $ , dimension $ \\mu - \\rho $ and minimum Hamming distance at least $ t + 1 $ .",
"As in related works [14], [16], [19], the field size $ q^m = \\left( q^r \\right)^\\rho $ has as exponent $ \\rho $ the redundancy (i.e., codimension) of the code $ \\mathcal {C}_{\\gamma } $ .",
"Hence, the objective is to use known Hamming-metric codes $ \\mathcal {C}_{\\gamma } $ over the field $ \\mathbb {F}_{q^r} $ with minimum Hamming distance larger than $ t = \\min \\lbrace h, \\mu \\rbrace $ , with large code length $ \\mu $ and small redundancy $ \\rho $ (i.e., large dimension $ \\mu - \\rho $ )."
],
[
"Using trivial codes: Recovering linearized RS codes",
"As a first choice of $ \\mathcal {C}_{\\gamma } $ , we choose a trivial code $ \\mathcal {C}_{\\gamma } = \\lbrace \\mathbf {0} \\rbrace $ and recover duals of linearized Reed-Solomon codes [31].",
"As in Lemma REF , we define $ {\\rm d}_H(\\mathcal {C}_{\\gamma }) = \\mu +1 $ if $ \\mathcal {C}_{\\gamma } = \\lbrace \\mathbf {0} \\rbrace \\subseteq \\mathbb {F}_{q^{r \\mu }}^\\mu $ .",
"Theorem 4 Choose $ \\mu = \\rho = 1 $ , thus $ m = r \\rho = r $ , $ \\gamma = 1 \\in \\mathbb {F}_{q^r}^1 $ , $ \\mathcal {C}_{\\gamma } = \\lbrace 0 \\rbrace \\subseteq \\mathbb {F}_{q^r}^1 $ , hence $ (\\beta _1, \\beta _2, \\ldots , \\beta _r) = \\alpha = (\\alpha _1, \\alpha _2, \\ldots , \\alpha _r) \\in \\mathbb {F}_{q^r}^r.",
"$ Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^r}^{g r} $ in Definition REF is the dual of a linearized Reed-Solomon code [31], [11], also called linearized Goppa code in [10].",
"The redundancy $ h $ is arbitrary with $ 1 \\le h \\le gr-1 $ , and the number of matrix sets $ g $ may be arbitrary satisfying that $ 1 \\le g = \\ell \\le q-1 .",
"$ The base field is $ \\mathbb {F}_q $ , with $ q > g $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = q^r , \\quad \\textrm {that is,} \\quad m = r,$ Remark 36 There is an additive gap of $ \\lfloor (h-2)/r \\rfloor + 2 $ for the parameter $ g $ between the upper bound (REF ) and the value of $ g $ achievable by the MSRD codes in Theorem REF .",
"If $ h \\le r+1 $ , then (REF ) reads $ q \\ge g - 1 $ , and such a gap is reduced to 2.",
"Asymptotically, if $ h = \\mathcal {O}(r q) $ , then (REF ) implies that $ g = \\mathcal {O}(q) $ for the case $ m = r \\ge 2 $ .",
"In this case, the MSRD codes in Theorem REF attain such an asymptotic bound.",
"By [35], such duals are precisely linearized Reed-Solomon codes for $ \\mathbf {a} \\in (\\mathbb {F}_{q^r}^*)^\\ell $ chosen as in Lemma REF (see also [34] for other cases, and [10] for a general description).",
"Note that we have chosen to define the MSRD codes in Definition REF as those whose parity-check matrix is an MSRD extended Moore matrix.",
"The MSRD code with such a matrix as its generator matrix is therefore simply a linearized Reed-Solomon code, if $ \\mathcal {C}_{\\gamma } = \\lbrace 0 \\rbrace $ as in this subsection.",
"In [31], the vectors $ \\beta _i \\in \\mathbb {F}_{q^r}^r $ , for $ i = 1,2, \\ldots , \\ell $ , were allowed to be different, have lengths smaller than $ r $ , and a $ \\sigma $ -derivation was allowed to be used in its generator matrix.",
"However, such extensions do not provide further parameter regimes for finite fields (to obtain distinct lengths $ r_i \\le r $ per block, we may simply use puncturing, see Remark REF ).",
"Linearized Reed-Solomon codes were proposed as PMDS codes, via Construction REF , originally in [36].",
"As stated there, the field sizes attainable by such codes are $|\\mathbb {F}_{q^m}| = q^r = (g+1)^r,$ for $ g = q-1 $ local sets, and locality $ r $ (Definition REF ), assuming that $ q \\ge \\nu = r + \\delta - 1 $ if $ \\delta > 2 $ .",
"If we do not wish $ g+1 $ to be a prime power, but $ q $ is even, then we may guarantee the field size $|\\mathbb {F}_{q^m}| = q^r \\le \\left( 2 \\max \\lbrace \\nu , g \\rbrace \\right) ^r ,$ by choosing $ q $ to be the smallest power of 2 larger than $ \\max \\lbrace \\nu , g \\rbrace $ .",
"We obtain the same MSRD codes if we choose, more generally, the trivial code $ \\mathcal {C}_{\\gamma } = \\lbrace \\mathbf {0} \\rbrace \\subseteq \\mathbb {F}_{q^r}^\\mu $ , i.e., $ \\rho = \\mu \\ge 1 $ and $ H_{\\gamma } \\in {\\rm GL}_\\rho (\\mathbb {F}_{q^r}) $ .",
"In other words, if we choose $ \\gamma \\in \\mathbb {F}_{q^{r \\rho }}^\\rho $ to be an ordered basis of $ \\mathbb {F}_{q^{r \\rho }} $ over $ \\mathbb {F}_{q^r} $ .",
"As is well known, the tensor product $ (\\beta _1, \\beta _2, \\ldots , \\beta _{r \\rho }) = \\alpha \\otimes \\gamma = (\\alpha _1 \\gamma _1, \\ldots , \\alpha _r \\gamma _1 | \\ldots | \\alpha _1 \\gamma _\\rho , \\ldots , \\alpha _r \\gamma _\\rho ) \\in \\mathbb {F}_{q^{r \\rho }}^{r \\rho } $ is in turn an ordered basis of $ \\mathbb {F}_{q^{r \\rho }} $ over $ \\mathbb {F}_q $ .",
"This is the classical proof that, given finite-dimensional field extensions $ K_1 \\subseteq K_2 \\subseteq K_3 $ , then $ [K_3 : K_1] = [K_3 : K_2] \\cdot [K_2 : K_1] $ .",
"Thus, we obtain duals of linearized Reed-Solomon codes, as before, but with length $ \\ell (r \\rho ) $ , for the length partition $ (\\ell , r \\rho ) $ over the field $ \\mathbb {F}_q $ , which does not add anything, as we may choose $ r \\rho $ instead of $ r $ from the beginning."
],
[
"Using MDS codes",
"In this subsection, we explore the case where $ \\mathcal {C}_{\\gamma } $ is an MDS code.",
"Theorem 5 Choose any $ \\mu \\le q^r + 1 $ and $ \\rho = t = \\min \\lbrace h, \\mu \\rbrace $ , being $ h $ arbitrary with $ 1 \\le h \\le gr-1 $ .",
"Choose $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ in (REF ) as an MDS code of dimension $ \\mu - t $ , thus $ {\\rm d}_H(\\mathcal {C}_{\\gamma }) = t+1 $ .",
"For instance, $ \\mathcal {C}_{\\gamma } $ can be chosen as the projective extension [23] of a classical Reed-Solomon code [40].",
"Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ in Definition REF has $ g $ matrix sets satisfying that $ g = \\ell \\mu \\le (q-1) \\left( q^r + 1 \\right), $ where $ \\ell $ and $ \\mu $ may be arbitrary such that $ 1 \\le \\ell \\le q-1 \\quad \\textrm {and} \\quad 1 \\le \\mu \\le q^r + 1.",
"$ The base field is $ \\mathbb {F}_q $ , with $ q \\ge \\max \\left\\lbrace \\ell + 1, \\@root r \\of {\\mu - 1} \\right\\rbrace $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = q^{r \\min \\lbrace h, \\mu \\rbrace } , \\quad \\textrm {that is,} \\quad m = r \\min \\lbrace h, \\mu \\rbrace .$ Choosing $ \\ell = q-1 $ and $ \\mu = q^r + 1 $ , thus $ g = \\ell \\mu = (q-1) \\left( q^r + 1 \\right) $ , then the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = \\left( \\frac{g}{q-1} - 1 \\right) ^{\\min \\left\\lbrace h, \\frac{g}{q-1} \\right\\rbrace } .$ The parameters attainable by such codes are similar to those achieved by the codes in [19].",
"Although not proven nor remarked in [19], the codes in [19] are PMDS codes that indeed are built using MSRD codes via Construction REF .",
"However, for the codes in Theorem REF , $ g $ may be up to $ (q-1) \\left( q^r + 1 \\right) $ , while in [19], $ g $ may only be up to $ \\lceil q^r / r \\rceil $ , being all other parameters equal.",
"We now plug the MSRD codes from Theorem REF into Construction REF .",
"The following corollary holds by Proposition REF and Theorem REF .",
"Corollary 37 In Construction REF , choose $ \\mathcal {C}_{out} = \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ to be the MSRD code in Theorem REF .",
"Assume that $ q $ is even and such that $ \\mu \\le q^r + 1 $ and $ q > \\nu = r+\\delta - 1 $ .",
"Furthermore, choose $ q $ satisfying also that, either 1) $ \\mu = q^r + 1 $ , or 2) $ q $ the smallest power of 2 satisfying $ q > \\nu $ .",
"Set $ \\ell = q-1 $ and $ g = \\ell \\mu \\le (q-1) \\left( q^r + 1 \\right) $ .",
"Then $ \\mathcal {C}_{glob} \\subseteq \\mathbb {F}_{q^m}^{g \\nu } $ in Construction REF is a PMDS code with $ (r,\\delta ) $ -localities, and its field of linearity has size $|\\mathbb {F}_{q^m}| = q^{r \\min \\lbrace h, \\mu \\rbrace } \\le \\max \\left\\lbrace \\left( 2 \\nu \\right)^r , \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor - 1 \\right\\rbrace ^{\\min \\left\\lbrace h, \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor \\right\\rbrace } .$"
],
[
"Using Hamming codes (or spreads)",
"We now investigate the case $ h = 2 \\le \\mu $ (see Remark REF ).",
"As we show next, in this case we obtain the first and only known MSRD codes with arbitrary parameters except for $ h = 2 $ (minimum sum-rank distance 3) and with field sizes $ q^m $ that are linear in $ g $ .",
"In addition, such MSRD codes meet the bounds (REF ) with equality, hence $ g $ may not be larger relative to $ q^m $ , or $ q^m $ smaller relative to $ g $ , for $ h = 2 $ and arbitrary $ q $ and $ r $ .",
"As shown in Remark REF , when $ h = 2 \\le \\mu $ , Conditions 1 and 2 in Theorem REF hold if, and only if, the set $ \\lbrace \\mathcal {H}_1, \\mathcal {H}_2 , \\ldots , $ $ \\mathcal {H}_\\mu \\rbrace $ is a partial spread of size $ \\mu $ of $ r $ -dimensional $ \\mathbb {F}_q $ -linear subspaces of $ \\mathbb {F}_{q^m} \\cong \\mathbb {F}_q^m $ .",
"Recall that a partial spread is a set of $ r $ -dimensional $ \\mathbb {F}_q $ -linear subspaces $ \\mathcal {H}_1, \\mathcal {H}_2 , \\ldots , \\mathcal {H}_\\mu \\subseteq \\mathbb {F}_{q^m} $ such that $ \\mathcal {H}_i \\cap \\mathcal {H}_j = \\lbrace 0 \\rbrace $ if $ i \\ne j $ .",
"In the case $ m = r \\rho $ , which we are assuming in this section, there exist partial spreads of maximum possible size $ \\mu $ whose union form the total space $ \\mathbb {F}_{q^m} $ , and are therefore simply called spreads.",
"The first known construction of spreads when $ m = r \\rho $ was provided by Segre [42], and coincides exactly with our tensor-product technique (Subsection REF ) when choosing $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ as a $ (\\mu - \\rho ) $ -dimensional Hamming code [20] (see also [30] or [23]).",
"We thus obtain the following subfamily of MSRD codes from Definition REF .",
"Theorem 6 Consider $ 1 \\le \\rho < \\mu $ , and choose $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ in (REF ) as a $ (\\mu - \\rho ) $ -dimensional Hamming code.",
"In other words, choose the vector $ \\gamma = (\\gamma _1, \\gamma _2, \\ldots , \\gamma _\\mu ) \\in \\left( \\mathbb {F}_{q^{r \\rho }}^* \\right) ^\\mu $ in (REF ) such that its components form the projective space $ \\mathbb {P}_{\\mathbb {F}_{q^r}}(\\mathbb {F}_{q^{r \\rho }}) = \\lbrace [\\gamma _1], [\\gamma _2], \\ldots , [\\gamma _{\\mu }] \\rbrace , $ where $ [\\gamma ] = \\lbrace \\lambda \\gamma \\in \\mathbb {F}_{q^{r \\rho }}^* \\mid \\lambda \\in \\mathbb {F}_{q^r}^* \\rbrace $ , for $ \\gamma \\in \\mathbb {F}_{q^{r \\rho }}^* $ .",
"Then we have that $ \\mu = \\frac{q^{r \\rho } - 1}{q^r - 1} \\quad \\textrm {and} \\quad t = h = {\\rm d}_H \\left( \\mathcal {C}_{\\gamma } \\right) - 1 = 2.",
"$ Finally, set $ \\ell = q-1 $ and $ g = \\ell \\mu $ .",
"Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ in Definition REF satisfies that $ {\\rm d}_{SR}(\\mathcal {C}_k(\\mathbf {a}, \\beta )) = 3 $ (i.e., $ h = 2 $ ) and has $ g $ matrix sets, where $ g = (q-1) \\cdot \\frac{q^{r \\rho } - 1}{q^r - 1} .",
"$ The base field is $ \\mathbb {F}_q $ , being $ q $ an arbitrary prime power, and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = q^{r \\rho } = \\frac{q^r-1}{q - 1} \\cdot g + 1 .$ In particular, for $ r \\ge 2 $ , the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ meets the bounds (REF ) with equality.",
"Remark 38 Observe that, setting $ \\rho = 1 $ , the MSRD codes in Theorem REF coincide with those from Theorem REF for $ h = 2 $ .",
"Setting $ \\rho = 2 $ , the MSRD codes in Theorem REF coincide with those from Theorem REF for $ h = 2 $ .",
"Remark 39 Theorem REF above makes use of partial spreads when $ r $ divides $ m $ .",
"There exist constructions of partial spreads when $ r $ does not divide $ m $ .",
"Using results from [1], [2], one may prove that the maximum size $ \\mu $ of a partial spread of $ r $ -dimensional $ \\mathbb {F}_q $ -linear subspaces $ \\mathcal {H}_1, \\mathcal {H}_2 , \\ldots , \\mathcal {H}_\\mu \\subseteq \\mathbb {F}_{q^m} $ satisfies that $ \\frac{q^m - q^s}{q^r - 1} - q^s + 1 \\le \\mu \\le \\frac{q^m - q^s}{q^r - 1}, $ where $ s \\ge 0 $ is the remainder of the Euclidean division of $ m $ by $ r $ .",
"Choosing $ g = (q-1) \\mu $ , the corresponding MSRD codes in Definition REF satisfy that $ (q-1) \\cdot \\left( \\frac{q^m - q^s}{q^r - 1} - q^s + 1 \\right) \\le g \\le (q-1) \\cdot \\frac{q^m - q^s}{q^r - 1} .",
"$ We will not provide the corresponding construction of PMDS codes via Construction REF , as there exist linear PMDS codes for $ h = 2 $ with smaller field sizes [4]."
],
[
"Using BCH codes",
"In this subsection, we explore the case where $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ is a BCH code.",
"Assume in this subsection that $ q $ and $ \\mu $ are coprime.",
"Denote the order of $ q^r $ modulo $ \\mu $ by $s = {\\rm ord}_{\\mu } \\left( q^r \\right) = \\min \\left\\lbrace \\widetilde{s} \\in \\mathbb {Z}_+ \\mid \\mu \\textrm { divides } q^{r \\widetilde{s}} - 1 \\right\\rbrace .$ Consider the code $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ in (REF ) to be a BCH code, see [30] [23].",
"By the BCH bound [30] [23], we have that $ {\\rm d}_H \\left( \\mathcal {C}_{\\gamma } \\right) \\ge \\partial $ if the minimal generator polynomial of $ \\mathcal {C}_{\\gamma } $ vanishes in $ a^b, a^{b+1}, a^{b+2}, \\ldots , a^{b+\\partial - 2} \\in \\mathbb {F}_{q^{rs}}, $ for integers $ b \\ge 0 $ and $ 2 \\le \\partial \\le n $ , where $ a \\in \\mathbb {F}_{q^{rs}} $ is a primitive root of $ x^{\\mu } - 1 $ .",
"If we choose $ \\mathcal {C}_{\\gamma } $ to be the largest BCH code whose minimal generator polynomial has such roots, then by [23], we have that $ \\rho = \\mu - \\dim \\left( \\mathcal {C}_{\\gamma } \\right) = | C_b \\cup C_{b+1} \\cup C_{b+2} \\cup \\ldots \\cup C_{b+\\partial -2} |, $ where $ C_i \\subseteq \\lbrace 0,1,\\ldots , \\mu -1 \\rbrace $ is the $ i $ th $ q^{r} $ -cyclotomic coset modulo $ \\mu $ [30] [23], given by $ C_i = \\lbrace i, i q^{r}, i q^{2r}, i q^{3r}, \\ldots \\rbrace \\quad (\\textrm {mod } \\mu ), $ for $ i = 0,1, \\ldots , \\mu - 1 $ .",
"The integer $ \\partial $ is called the prescribed distance of the BCH code $ \\mathcal {C}_{\\gamma } $ , and the set $ C_b \\cup C_{b+1} \\cup C_{b+2} \\cup \\ldots \\cup C_{b+\\partial -2} $ is called the defining set of $ \\mathcal {C}_{\\gamma } $ .",
"By the discussion above, the following theorem holds.",
"Theorem 7 As above, assume that $ q $ and $ \\mu $ are coprime, and set $ s = {\\rm ord}_{\\mu } \\left( q^r \\right) $ , as in (REF ).",
"Take a positive integer $ b \\ge 0 $ , and choose the code $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ in (REF ) to be a BCH code, as above, with prescribed distance $ \\partial = t+1 $ and defining set $ C_b \\cup C_{b+1} \\cup C_{b+2} \\cup \\ldots \\cup C_{b+ t-1} $ , being $ h $ arbitrary with $ 1 \\le h \\le gr - 1 $ , and being $ t = \\min \\lbrace h, \\mu \\rbrace $ .",
"Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ in Definition REF has $ g = \\ell \\mu $ matrix sets, where $ \\ell $ may be arbitrary with $ 1 \\le \\ell \\le q-1 $ .",
"The base field is $ \\mathbb {F}_q $ , where $ q $ is coprime with $ \\mu $ and satisfies $ q > \\ell $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = q^{r \\rho } = \\left( q^r \\right) ^{| C_b \\cup C_{b+1} \\cup C_{b+2} \\cup \\ldots \\cup C_{b+ t-1} |} ,$ that is, $ m = r \\cdot | C_b \\cup C_{b+1} \\cup C_{b+2} \\cup \\ldots \\cup C_{b+ t-1} | $ .",
"We now upper bound the size of the defining set $ C_b \\cup C_{b+1} \\cup C_{b+2} \\cup \\ldots \\cup C_{b+ t-1} $ .",
"The following observations are trivial and well known: $C_0 = \\lbrace 0 \\rbrace , \\quad |C_i| \\le s, \\quad \\textrm {and} \\quad C_{i q^r} = C_i,$ for $ i = 0,1,2, \\ldots , \\mu - 1 $ .",
"Therefore, we take $ b = 0 $ , and then we have $ |C_0 | = 1 $ and we may remove from $ C_0 \\cup C_1 \\cup C_2 \\cup \\ldots \\cup C_{t-1} $ each cyclotomic coset $ C_i $ where $ i $ is a multiple of $ q^r $ .",
"Hence $| C_0 \\cup C_1 \\cup C_2 \\cup \\ldots \\cup C_{t-1} | \\le 1 + s \\cdot \\left\\lceil \\frac{q^r - 1}{q^r} \\cdot (t-1) \\right\\rceil .$ Therefore, we have proven the following enhancement of Theorem REF .",
"Theorem 8 Let the assumptions and notation be as in Theorem REF .",
"Assume further that $ b = 0 $ .",
"Then the base field is $ \\mathbb {F}_q $ , where $ q $ is coprime with $ \\mu $ and satisfies $ q > \\ell $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = q^{r \\rho } \\le \\left( q^r \\right) ^{1 + s \\left\\lceil \\frac{q^r - 1}{q^r} (\\min \\lbrace h , \\mu \\rbrace -1) \\right\\rceil } .$ We may further choose $ \\mathcal {C}_{\\gamma } $ to be a primitive BCH code, meaning that we choose $ \\mu = q^{rs} - 1 $ , where $ s $ may be chosen arbitrary (it then follows that $ s = {\\rm ord}_{\\mu }\\left( q^r \\right) $ ), and then choose $ \\ell = q-1 $ .",
"Then the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) $ has size $|\\mathbb {F}_{q^m}| = q^{r \\rho } \\le q^r \\cdot \\left( \\frac{g}{q-1} + 1 \\right) ^{ \\left\\lceil \\frac{q^r - 1}{q^r} (h -1) \\right\\rceil } .$ We now plug the MSRD codes from Theorem REF into Construction REF .",
"The following corollary holds by Proposition REF and Theorem REF .",
"Corollary 40 In Construction REF , choose $ \\mathcal {C}_{out} = \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ to be the MSRD code in Theorem REF .",
"We further assume that $ \\mu = q^{rs} - 1 $ , for an arbitrary positive integer $ s $ , and where $ q $ is the smallest power of 2 such that $ q > \\nu = r + \\delta - 1 $ .",
"Set $ \\ell = q-1 $ and $ g = \\ell \\mu = (q-1) \\left( q^{rs} - 1 \\right) $ .",
"Then $ \\mathcal {C}_{glob} \\subseteq \\mathbb {F}_{q^m}^{g \\nu } $ in Construction REF is a PMDS code with $ (r,\\delta ) $ -localities, and its field of linearity has size $|\\mathbb {F}_{q^m}| = q^{r \\rho } \\le \\left( 2 \\nu \\right)^r \\cdot \\left( \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor + 1 \\right) ^{ \\left\\lceil \\frac{q^r - 1}{q^r} (h -1) \\right\\rceil } .$ The field size (REF ) is not smaller than that achieved by linearized Reed-Solomon codes (REF ) in general if $ h > r $ .",
"However, if $ h \\le r $ , then $ h \\le q^r $ , and observe that $\\textrm {if } h \\le q^r, \\quad \\textrm {then} \\quad \\left\\lceil \\frac{q^r - 1}{q^r} (h -1) \\right\\rceil = h-1 .$ Hence, for the parameter regime in which the field sizes (REF ) improve (REF ), that is, for $ h \\le r $ , it holds that $|\\mathbb {F}_{q^m}| \\le \\left( 2 \\nu \\right)^r \\cdot \\left( \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor + 1 \\right) ^{ \\left\\lceil \\frac{q^r - 1}{q^r} (h -1) \\right\\rceil } = \\left( 2 \\nu \\right)^r \\cdot \\left( \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor + 1 \\right) ^{ h -1 } .$"
],
[
"Using Algebraic-Geometry (AG) codes",
"In this subsection, we explore the case where $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ is an Algebraic-Geometry code, or AG code for short.",
"AG codes have only been proposed to construct PMDS codes in [19], to the best of our knowledge.",
"However, it is not clear whether or not the PMDS codes in [19] are built from or may produce MSRD codes, as the obtained parameters in this work and in [19] are different.",
"Before starting, it is important to note that we do not obtain asymptotically smaller field sizes $ q^m $ than in previous subsections (neither does [19]).",
"Perhaps surprisingly, the main disadvantage of using AG codes here is that they are asymptotically good.",
"Assume that $ {\\rm d}_H(\\mathcal {C}_{\\gamma }) = \\Omega (\\mu ) $ , meaning that $ {\\rm d}_H(\\mathcal {C}_{\\gamma }) \\ge C \\mu $ , for some constant $ C > 0 $ .",
"Since $ \\rho > {\\rm d}_H(\\mathcal {C}_{\\gamma }) $ by the Singleton bound (Proposition REF ), then $ \\rho > C \\mu .",
"$ Hence, for the base field $ \\mathbb {F}_q $ , we may choose $ \\mu \\le g \\le (q-1) \\mu $ , and the obtained field sizes satisfy $ q^m = q^{r \\rho } > \\left( q^r \\right)^{C \\mu } \\ge \\left( q^r \\right)^{ \\frac{C g}{q-1} }.",
"$ Such field sizes are therefore exponential in the number of matrix sets $ g $ .",
"For PMDS codes, the resulting field sizes are asymptotically much larger than (REF ), since $ (2g)^r \\ll \\left( q^{ \\frac{C g}{q-1} } \\right)^r , \\quad \\textrm {due to} \\quad (2g)^{q-1} \\ll q^{ C g } , \\quad \\textrm {asymptotically}, $ since for such AG codes we will have that $ g \\ge \\mu \\ge q^r \\gg q $ if $ r \\ge 2 $ .",
"A similar fact was already observed in [36], when comparing the field size (REF ) (obtained in [36]) with those obtained in [19] using AG codes.",
"However, the codes in Subsection REF (linearized RS codes [31] and their duals [10]) require $ q > g $ .",
"Hence, we may not use such MSRD codes over a fixed base field $ \\mathbb {F}_q $ if we want $ q $ to be small and/or fixed and $ g $ be large and/or grow.",
"Even if this is not the case for building PMDS codes, it may be the case for other applications of MSRD codes where we do not have flexibility on $ q $ (see Subsection REF ).",
"In such cases, using AG codes will provide MSRD codes with smaller $ m $ , thus smaller $ q^m $ , than in previous subsections, relative to the other code parameters being $ q $ smaller than $ g $ .",
"For these reasons, we will only describe MSRD codes from now on, and we do not provide further PMDS codes.",
"In order to describe the obtained parameters of the MSRD codes in Definition REF , we need to briefly revisit AG codes.",
"For further details, the reader is referred to [46].",
"Consider an irreducible projective curve $ \\mathcal {X} $ over $ \\mathbb {F}_{q^r} $ (meaning irreducible over the algebraic closure of $ \\mathbb {F}_{q^r} $ ) with algebraic function field $ \\mathcal {F} $ , and let $ \\mathfrak {g} = \\mathfrak {g}(\\mathcal {X}) = \\mathfrak {g}(\\mathcal {F}) $ be its genus.",
"Points in $ \\mathcal {X} $ correspond to places in $ \\mathcal {F} $ (we may work indistinctly with $ \\mathcal {X} $ or $ \\mathcal {F} $ ) and we say that they are rational if they are rational over $ \\mathbb {F}_{q^r} $ (their coordinates lie in $ \\mathbb {F}_{q^r} $ ).",
"A divisor over $ \\mathcal {X} $ is a formal sum $ D = \\sum _{P \\in \\mathcal {X}} \\mu _P P $ , for integers $ \\mu _P \\in \\mathbb {Z} $ which are all zero except for a finite number.",
"The support of $ D $ is defined as $ \\lbrace P \\in \\mathcal {X} \\mid \\mu _P \\ne 0 \\rbrace $ , and $ D $ is called rational if all points in its support are rational.",
"We define the degree of the rational divisor $ D $ as $ \\deg (D) = \\sum _{P \\in \\mathcal {X}} \\mu _P \\in \\mathbb {Z} $ .",
"All divisors considered in this paper will be rational (over $ \\mathbb {F}_{q^r} $ ).",
"For divisors $ D = \\sum _{P \\in \\mathcal {X}} \\mu _P P $ and $ E = \\sum _{P \\in \\mathcal {X}} \\lambda _P P $ , we write $ D \\preceq E $ if $ \\mu _P \\le \\lambda _P $ , for all $ P \\in \\mathcal {X} $ .",
"For an algebraic function $ f \\in \\mathcal {F} $ , we define its divisor as $ (f) = \\sum _{P \\in \\mathcal {X}} \\nu _P(f) P $ , where $ \\nu _P $ is the valuation at the point $ P $ (see [46]).",
"Hence we may define the Riemann-Roch space (see [46]) of a divisor $ D $ as the vector space over $ \\mathbb {F}_{q^r} $ given by $\\mathcal {L}(D) = \\lbrace f \\in \\mathcal {F} \\mid (f) + D \\succeq 0 \\rbrace .$ Finally, fix rational divisors $ D = P_1 + P_2 + \\cdots + P_\\mu $ and $ G $ over $ \\mathcal {X} $ with disjoint supports and where the points $ P_1, P_2, \\ldots , P_\\mu $ are all distinct.",
"We define the corresponding Algebraic-Geometry code (see [46]), or AG code for short, as the linear code $\\mathcal {C}(D,G) = \\left\\lbrace \\left( f(P_1), f(P_2), \\ldots , f(P_\\mu ) \\right) \\mid f \\in \\mathcal {L}(G) \\right\\rbrace \\subseteq \\mathbb {F}_{q^r}^\\mu .$ For our purposes, the most important results are the following two well known lemmas on the parameters of AG codes.",
"The first is the well known Goppa bound [46].",
"Lemma 41 (Goppa bound [46]) If $ \\deg (G) < \\mu $ , then ${\\rm d}_H(\\mathcal {C}(D,G)) \\ge \\mu - \\dim (\\mathcal {C}(D,G)) - \\mathfrak {g} + 1.$ The following lemma is [46].",
"Lemma 42 ([46]) If $ 2 \\mathfrak {g} -2 < \\deg (G) < \\mu $ , then $ \\dim (\\mathcal {C}(D,G)) = \\deg (G) - \\mathfrak {g} + 1.",
"$ Hence we obtain the following theorem from Lemmas REF and REF .",
"Theorem 9 Assume that $ 2 \\mathfrak {g} -2 < \\deg (G) = \\mu - h - 1 $ , in particular $ t = \\min \\lbrace h,\\mu \\rbrace = h $ if $ \\mathfrak {g} > 0 $ .",
"Define $ \\rho = \\mu - \\deg (G) + \\mathfrak {g} - 1 $ , which thus satisfies $ \\mathfrak {g} - 1 < \\rho < \\mu - \\mathfrak {g} + 1 $ .",
"Choose the code $ \\mathcal {C}_{\\gamma } \\subseteq \\mathbb {F}_{q^r}^\\mu $ in (REF ) to be the AG code $ \\mathcal {C}_{\\gamma } = \\mathcal {C}(D,G) $ , as above.",
"Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ in Definition REF has $ g = \\ell \\mu $ matrix sets, where $ \\ell $ may be arbitrary with $ 1 \\le \\ell \\le q-1 $ .",
"The base field is $ \\mathbb {F}_q $ , where $ q > \\ell $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{gr} $ has size $|\\mathbb {F}_{q^m}| = q^{r \\rho } = \\left( q^r \\right)^{\\mu - \\deg (G) + \\mathfrak {g} -1} = \\left( q^r \\right)^{h + \\mathfrak {g}} .$ As usual in the AG-code literature, the name of the game is: 1) to be able to pick as many rational points $ P_1, P_2, \\ldots , P_\\mu $ as possible in $ \\mathcal {X} $ but outside the support of $ G $ , and 2) to have control over $ \\deg (G) $ .",
"This is achieved by using one-point AG codes.",
"We assume that $ \\mathcal {X} $ has at least $ \\mu + 1 $ distinct rational points (over $ \\mathbb {F}_{q^r} $ ), $ P_1, P_2, \\ldots , P_\\mu , Q $ , and we define $ G = (\\mu - h - 1) Q $ , which thus satisfies that $ \\deg (G) = \\mu - h - 1 $ .",
"The corresponding one-point AG code is $ \\mathcal {C}(D,G) $ , being $ D = P_1 + P_2 + \\cdots + P_\\mu $ and $ G $ as above.",
"We will explore different choices of $ \\mathcal {X} $ in the following subsections."
],
[
"Using Hermitian AG codes",
"We start by exploring Hermitian curves $ \\mathcal {X} $ (see [46]).",
"Throughout the subsection, we assume that there is a positive integer $ s $ such that $ q^r = p^{2s} , \\quad \\textrm {that is,} \\quad 2s = r \\log _p(q), $ where $ p $ is the prime number that divides $ q $ , which can be arbitrary, e.g., $ p = 2 $ .",
"The Hermitian curve is the projective plane curve with homogeneous equation $ x^{q^{ \\frac{r}{2} } + 1} - y^{q^{ \\frac{r}{2} }} z - y z^{q^{ \\frac{r}{2} }} = 0.",
"$ This curve is called a Hermitian curve and has $ q^{ \\frac{3r}{2} }+1 $ rational points (over $ \\mathbb {F}_{q^r} $ ), and genus $ \\mathfrak {g} = \\mathfrak {g} (\\mathcal {X}) = \\frac{q^{ \\frac{r}{2} } \\left( q^{ \\frac{r}{2} }-1 \\right) }{2} .",
"$ Therefore, we may choose $ \\mu = q^{3r/2} $ in Theorem REF , and we deduce the following consequence.",
"Corollary 43 Let the notation and assumptions be as in Theorem REF , but where $ \\mathcal {X} $ is the Hermitian curve above.",
"Further assume that $ G = (\\mu - h - 1)Q $ (recall that $ \\mu - h \\ge 2 \\mathfrak {g} $ ), for a rational point $ Q $ in $ \\mathcal {X} $ , different than $ P_1, P_2, \\ldots , P_\\mu $ that form the support of $ D $ .",
"Assume moreover that $ \\ell = q-1 \\quad \\textrm {and} \\quad \\mu = q^{ \\frac{3r}{2} } .",
"$ Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ in Definition REF has $ g = \\ell \\mu = (q-1) q^{ \\frac{3r}{2} } $ matrix sets.",
"The base field is $ \\mathbb {F}_q $ , where $ q = \\ell + 1 $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{gr} $ has size $|\\mathbb {F}_{q^m}| = \\left( q^r \\right)^{h + \\mathfrak {g}} = \\mu ^{ \\frac{2}{3} (h + \\mathfrak {g}) } = \\mu ^{ \\frac{1}{3} ( 2h + \\mu ^{2/3} - \\mu ^{1/3} ) } ,$ that is, $ m = r \\left( h + \\frac{1}{2} \\left( \\mu ^{ \\frac{2}{3} } - \\mu ^{ \\frac{1}{3} } \\right) \\right) $ , where $ \\mu = \\frac{g}{q-1} $ .",
"Observe that the field size (REF ) may be asymptotically smaller than the field sizes (REF ) and (REF ) if $ h < \\mu $ and $ \\mu ^{2/3} - \\mu ^{1/3} < \\varepsilon h $ for a sufficiently small $ \\varepsilon > 0 $ .",
"In other words, we may reduce the exponent $ h $ to roughly $ 2h / 3 $ for a small enough $ \\varepsilon > 0 $ .",
"In the next subsection, we reduce the exponent to roughly $ h/2 $ by making use of the so-called Suzuki curves."
],
[
"Using Suzuki AG codes",
"In this subsection, we explore Suzuki curves $ \\mathcal {X} $ (see [21]).",
"Throughout the subsection, we assume that there is a positive integer $ s $ such that $ r $ divides $ 2s + 1 $ (hence $ r $ is odd), and we consider the even field size $ q^r = 2^{2s + 1} , \\quad \\textrm {that is,} \\quad 2s+1 = r \\log _2(q).",
"$ The Suzuki curve is the projective plane curve with homogeneous equation $ x^{2^s} \\left( y^{q^r} + y x^{q^r - 1} \\right) = z^{2^s} \\left( z^{q^r} + z x^{q^r - 1} \\right) .",
"$ This curve has $ q^{2r} + 1 $ rational points over $ \\mathbb {F}_{q^r} $ by [21], and genus $ \\mathfrak {g} = \\mathfrak {g} (\\mathcal {X}) = 2^s \\left( q^r - 1 \\right) $ by [21].",
"Therefore, we may choose $ \\mu = q^{2r} $ in Theorem REF , hence $ \\mathfrak {g} = 2^s \\left( \\mu ^{ \\frac{1}{2} } - 1 \\right) \\le \\mu ^{ \\frac{1}{4} } \\left( \\mu ^{ \\frac{1}{2} } - 1 \\right) = \\mu ^{ \\frac{3}{4} } - \\mu ^{ \\frac{1}{4} } , $ and we deduce the following consequence.",
"Corollary 44 Let the notation and assumptions be as in Theorem REF , but where $ \\mathcal {X} $ is the Suzuki curve above.",
"Further assume that $ G = (\\mu - h - 1)Q $ (recall that $ \\mu - h \\ge 2 \\mathfrak {g} $ ), for a rational point $ Q $ in $ \\mathcal {X} $ , different than $ P_1, P_2, \\ldots , P_\\mu $ that form the support of $ D $ .",
"Assume moreover that $ \\ell = q-1 \\quad \\textrm {and} \\quad \\mu = q^{2r} .",
"$ Then the MSRD code $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{g r} $ in Definition REF has $ g = \\ell \\mu = (q-1) q^{2r} $ matrix sets.",
"The base field is $ \\mathbb {F}_q $ , where $ q = \\ell + 1 $ , and the field of linearity of $ \\mathcal {C}_k(\\mathbf {a}, \\beta ) \\subseteq \\mathbb {F}_{q^m}^{gr} $ has size $|\\mathbb {F}_{q^m}| = \\left( q^r \\right)^{h + \\mathfrak {g}} = \\mu ^{ \\frac{1}{2} (h + \\mathfrak {g}) } \\le \\mu ^{ \\frac{1}{2} \\left( h + \\mu ^{ \\frac{3}{4} } - \\mu ^{ \\frac{1}{4} } \\right) } ,$ that is, $ m = r \\left( h + 2^s \\left( \\mu ^{ \\frac{1}{2} } - 1 \\right) \\right) \\le r \\left( h + \\mu ^{ \\frac{3}{4} } - \\mu ^{ \\frac{1}{4} } \\right) $ , where $ \\mu = \\frac{g}{q-1} $ .",
"Observe now that the field size (REF ) may be asymptotically smaller than the field sizes (REF ), (REF ) and (REF ) if $ h < \\mu $ and $ \\mu ^{3/4} - \\mu ^{ 1/4 } < \\varepsilon h $ for a sufficiently small $ \\varepsilon > 0 $ .",
"In other words, we may reduce the exponent $ h $ to roughly $ h / 2 $ for a small enough $ \\varepsilon > 0 $ .",
"Finally, in the next subsection, we reduce the exponent to $ 4h/i $ , where $ i $ may grow unbounded, by making use of the second sequence of curves by García and Stichtenoth."
],
[
"Using García-Stichtenoth's AG codes",
"In this subsection, we explore the second sequence of curves $ (\\mathcal {X}_i)_{i=1}^\\infty $ given by García and Stichtenoth (see [15] or [46]).",
"As in Subsection REF , we assume throughout the subsection that there is a positive integer $ s $ such that $ q^r = p^{2s} , \\quad \\textrm {that is,} \\quad 2s = r \\log _p(q), $ where $ p $ is the prime number that divides $ q $ , which can be arbitrary, e.g., $ p = 2 $ .",
"In this case, rather than giving the implicit equations of the projective curve, it is more convenient to define recursively the associated sequence of algebraic function fields $ (\\mathcal {F}_i)_{i=1}^\\infty $ .",
"First we define $ \\mathcal {F}_1 = \\mathbb {F}_{q^r}(x_1) $ , where $ x_1 $ is transcendental over $ \\mathbb {F}_{q^r} $ , and then we define recursively $ \\mathcal {F}_{i+1} = \\mathcal {F}_i(x_{i+1}) $ , where $ x_{i+1} $ is algebraic over $ \\mathcal {F}_i $ satisfying the equation $ x_{i+1}^{q^{ \\frac{r}{2} }} + x_{i+1} = \\frac{x_i^{q^{ \\frac{r}{2} }}}{x_i^{q^{ \\frac{r}{2} } - 1} + 1 } , $ for all $ i \\in \\mathbb {Z}_+ $ .",
"The $ i $ th curve $ \\mathcal {X}_i $ has $ q^{ \\frac{ir}{2} } \\left( q^{ \\frac{r}{2} } -1 \\right) + 1 $ rational points, and its genus is given by $\\mathfrak {g} (\\mathcal {X}_i) = \\left\\lbrace \\begin{array}{ll}\\left( q^{\\frac{i r}{4}} - 1 \\right) ^2 & \\textrm {if } i \\textrm { is even}, \\\\\\left( q^{\\frac{(i+1) r}{4}} - 1 \\right) \\cdot \\left( q^{\\frac{(i-1)r}{4}} - 1 \\right) & \\textrm {if } i \\textrm { is odd}, \\\\\\end{array} \\right.$ by [15].",
"This means that the number of rational places divided by the genus, for the $ i $ th curve, converges to an optimal value, the so-called Ihara's constant, as $ i $ goes to infinity.",
"See [46] for more details on the asymptotic behaviour of AG codes.",
"To build MSRD codes as in Definition REF , we may choose $ \\mu _i = q^{ \\frac{ir}{2} } \\left( q^{ \\frac{r}{2} } -1 \\right) $ in Theorem REF , hence $ \\mathfrak {g}_i = \\mathfrak {g}(\\mathcal {X}_i) \\le q^{ \\frac{ir}{2} } = \\frac{\\mu _i}{ q^{ \\frac{r}{2} } - 1 } , $ for all $ i \\in \\mathbb {Z}_+ $ , and we deduce the following consequence.",
"Corollary 45 Let the notation and assumptions be as in Theorem REF , but where $ \\mathcal {X}_i $ is the $ i $ th curve in the García-Stichtenoth's sequence given above, for $ i \\in \\mathbb {Z}_+ $ .",
"We will also add an index $ i $ in every parameter that depends on the $ i $ th curve.",
"Assume that $ G_i = (\\mu _i - h_i - 1)Q_i $ (recall that $ \\mu _i - h_i \\ge 2 \\mathfrak {g}_i $ ), for a rational point $ Q_i $ in $ \\mathcal {X}_i $ , different than $ P_{i,1}, P_{i,2}, \\ldots , P_{i,\\mu _i} $ that form the support of $ D_i $ .",
"Assume moreover that $ \\mu _i = q^{ \\frac{ir}{2} } \\left( q^{ \\frac{r}{2} } -1 \\right) \\quad \\textrm {and} \\quad \\ell = q-1 .",
"$ Then the MSRD code $ \\mathcal {C}_{k_i}(\\mathbf {a}_i, \\beta _i) \\subseteq \\mathbb {F}_{q^{m_i}}^{g_i r} $ in Definition REF , for each $ i \\in \\mathbb {Z}_+ $ , has $ g_i = \\ell \\mu _i = (q-1) \\left( q^{ \\frac{r}{2} } -1 \\right) q^{ \\frac{ir}{2} } $ matrix sets.",
"The base field is $ \\mathbb {F}_q $ , where $ q = \\ell + 1 $ , constant in $ i $ , and the field of linearity of $ \\mathcal {C}_{k_i}(\\mathbf {a}_i, \\beta _i) \\subseteq \\mathbb {F}_{q^{m_i}}^{g_i r} $ has size $|\\mathbb {F}_{q^{m_i}}| = \\left( q^r \\right)^{h_i + \\mathfrak {g}_i} \\le \\left( q^r \\right)^{ h_i + q^{ \\frac{ir}{2} } } = \\left( \\frac{\\mu _i}{ q^{ \\frac{r}{2} } - 1 } \\right)^{ \\frac{2}{i} \\left( h_i + \\frac{\\mu _i}{ q^{ \\frac{r}{2} } - 1 } \\right) } .$ that is, $ m_i \\le r \\left( h_i + q^{ \\frac{ir}{2} } \\right) = r \\left( h_i + \\frac{\\mu _i}{ q^{ \\frac{r}{2} } - 1 } \\right) $ , where $ \\mu _i = \\frac{g_i}{q-1} $ .",
"Observe now that the field size (REF ) may be asymptotically smaller than the field sizes (REF ), (REF ), (REF ) and (REF ) in general if $ \\mu _i / (q^{r/2} - 1) < h_i < \\mu _i $ (a range where $ h_i = \\Theta (\\mu _i) = \\Theta (g_i) = \\Theta (g_i r) $ ).",
"In such cases, we may reduce the exponent $ h_i $ to $ 4h_i / i $ , where $ i $ may grow unbounded.",
"Hence the coefficient of $ h_i $ in the exponent may be as small as wanted."
],
[
"Summary of results and comparisons with the literature",
"In this final section, we will summarize the parameters of the MSRD codes and PMDS codes obtained throughout this work, and compare them to those from the literature.",
"The parameters of the MSRD codes obtained in Subsections REF , REF , REF , REF , REF , REF , and REF are summarized in Table REF .",
"The parameters of the PMDS codes obtained in Subsections REF , REF , and REF are summarized in Table REF .",
"Table: Table summarizing the code parameters of the linear MSRD codes obtained in this work throughout Subsections , , , , , , and .",
"They are 𝔽 q m \\mathbb {F}_{q^m} -linear codes in 𝔽 q m gr \\mathbb {F}_{q^m}^{gr} with code length N=gr N = gr , dimension k=gr-h k = gr - h , minimum sum-rank distance d=h+1 d = h+1 .",
"Their codewords can be seen as lists of g g matrices over 𝔽 q \\mathbb {F}_q of size m×r m \\times r , where m=rρ m = r \\rho , ρ∈ℤ + \\rho \\in \\mathbb {Z}_+ , and m=r m = r only in the first row.",
"The linear MSRD code in the first row was obtained in , and later independently in and .Table: Table summarizing the code parameters of the linear PMDS codes obtained in this work throughout Subsections , , and .",
"They are 𝔽 q m \\mathbb {F}_{q^m} -linear codes in 𝔽 q m gν \\mathbb {F}_{q^m}^{g \\nu } , where r r is the locality, δ \\delta is the local distance, g g is the number of local sets, h h is the number of global parities, ν=r+δ-1 \\nu = r+\\delta -1 is the local-set size and q q is a power of 2.",
"The field size of the local codes is a subfield of 𝔽 q \\mathbb {F}_q .",
"The linear PMDS code in the first row was obtained in ."
],
[
"Comparison with MSRD codes in the literature",
"We start by discussing MSRD codes.",
"First of all, smaller values of $ g $ and $ r $ in Table REF may be obtained in each row by puncturing or shortening the corresponding MSRD codes (see Remark REF ).",
"However, the comparison between $ q^m $ and $ g $ (or $ r $ ) would then be lost, as $ q $ and $ m $ remain unchanged after puncturing or shortening.",
"As discussed in the Introduction and after Remark REF , any MRD code [12], [13], [41] is an MSRD code, however, their fields of linearity have size $ q^m \\ge q^{gr} $ , thus exponential in the code length $ N = gr $ .",
"Equivalently, they require $ m \\ge gr $ for the matrix sizes in (REF ).",
"As the reader may check, all of the field sizes $ q^m $ in Table REF are sub-exponential in $ N = gr $ and much smaller than $ q^{gr} $ .",
"The first known construction of MSRD codes with sub-exponential field sizes is that of linearized Reed-Solomon codes and their duals, which moreover admit any value of $ q $ , $ r $ , $ h $ and $ g $ as long as $ g < q $ .",
"They were introduced in [31], and later independently in [11], [39].",
"As discussed in Subsection REF , this construction corresponds to the codes in the first row in Table REF .",
"Thus the comparison with the rest of the obtained MSRD codes can be directly inspected in that table.",
"We note that these are the only known MSRD codes satisfying $ m = r $ when $ g > 1 $ (thus yielding square matrices in (REF )).",
"However, they require $ q > g $ and in particular $ q = 2 $ may not be attained, which is not the case for the rest of the codes in Table REF .",
"The MSRD codes based on MDS, Hamming and primitive BCH codes admit a smaller value of $ q^m $ when $ h $ is small relative to $ r $ , although not necessarily only when $ h < r $ .",
"As discussed in Subsection REF , MSRD codes based on AG codes always require field sizes $ q^m $ much larger than linearized Reed-Solomon codes if $ q > g $ .",
"However, MSRD codes based on AG codes admit values $ q \\ll g $ , and in such cases, they admit smaller coefficients of $ h $ in the exponent of the field size $ q^m $ than the rest of MSRD codes.",
"Some constructions of MSRD codes were recently given in [6].",
"As explained in Remark REF , such codes are only $ \\mathbb {F}_q $ -linear, and have minimum sum-rank distance equal to 2 or $ \\sum _{i=1}^g r_i - 1 $ (total number of columns, across all matrices, minus 1), or require the number of rows or columns to be 1 at certain positions in the matrices in (REF ).",
"Finally, as we wrote in Subsection REF , the codes in the third row in Table REF , based on Hamming codes and valid for $ h = 2 $ (i.e., minimum sum-rank distance 3), achieve the maximum possible value of $ g $ with respect to the other parameters, in view of the bound (REF ).",
"Equivalently, they achieve the smallest possible value of $ m $ with respect to the other parameters.",
"As noted in Table REF , the attainable values of $ q $ and $ r $ for such construction are arbitrary.",
"Finally, this is the first and only known family of MSRD codes with field size $ q^m $ linear in $ g $ , hence linear in the code length $ N =gr $ if the number of columns per matrix, $ r $ , is upper bounded by a constant.",
"Since their minimum sum-rank distance is 3, they are either 2-erasure-correcting or 1-error-correcting."
],
[
"Comparison with PMDS codes in the literature",
"We now turn to discussing PMDS codes.",
"For small specific values of $ r $ , $ \\delta $ , $ g $ or $ h $ , there exist PMDS codes with smaller field sizes than in Table REF .",
"More concretely, PMDS codes with field sizes that are linear in the code length $ n = g \\nu $ were obtained in [3] for $ h = 1 $ and any $ \\delta $ , and for $ h > 1 $ and $ \\delta = 2 $ based on the irreducibility of certain polynomials, which are only known to cover some parameter values.",
"Finally, PMDS codes with smaller field sizes than in Table REF were obtained for $ h = 2 $ and $ g=2 $ in [4] and [22], respectively.",
"To the best of our knowledge, the PMDS codes available for general parameters with the smallest known field sizes are those in [14], [19], [36].",
"First of all, the PMDS codes in the first row in Table REF are exactly those obtained in [36] (more concretely, in [36]).",
"They have smaller field sizes than the codes in the second and third rows in general when $ \\nu > g $ or when $ r < \\min \\left\\lbrace \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor , h-1 \\right\\rbrace .",
"$ In [14], PMDS codes are obtained with field sizes $q^m = \\mathcal {O} \\left( \\max \\left\\lbrace (2 \\nu )^{\\delta + h} , g \\right\\rbrace ^h \\right) ,$ and later, in [19], PMDS codes are obtained with the following field sizes: $\\begin{split}q^m & = \\left( \\max \\left\\lbrace \\widetilde{\\mathcal {O}}(g), (2 \\nu )^{ \\left\\lfloor \\frac{h+1}{2} \\right\\rfloor } \\right\\rbrace \\right)^{\\min \\lbrace h, g \\rbrace } , \\\\q^m & = \\left( \\max \\left\\lbrace \\widetilde{\\mathcal {O}}(g), 2^r \\right\\rbrace \\right)^{\\min \\lbrace h, g \\rbrace } , \\\\q^m & = \\left( \\max \\left\\lbrace \\widetilde{\\mathcal {O}}(g), (2 \\nu )^{ h + \\delta - 1 } \\right\\rbrace \\right)^{\\min \\lbrace h, g \\rbrace } , \\\\q^m & = \\left( \\max \\left\\lbrace \\widetilde{\\mathcal {O}}(g), (2 \\nu )^\\nu \\right\\rbrace \\right)^{\\min \\lbrace h, g \\rbrace } ,\\end{split}$ where $ \\widetilde{\\mathcal {O}} $ corresponds to the big O notation but disregarding logarithmic multiplicative factors.",
"The PMDS codes in the second and third rows in Table REF , based on MDS codes and primitive BCH codes, have smaller field sizes than those in (REF ) and (REF ) in most cases when $ g $ is large relative to the other parameters.",
"For instance, if $ \\nu = \\mathcal {O}(1) $ (i.e., $ r = \\mathcal {O}(1) $ and $ \\delta = \\mathcal {O}(1) $ ), but $ g $ is unbounded, then the field sizes in the second and third rows of Table REF are $q^m = \\mathcal {O} \\left( \\left( \\frac{g}{\\nu } \\right)^{ \\min \\left\\lbrace h, \\left\\lfloor \\frac{g}{\\nu } \\right\\rfloor \\right\\rbrace } \\right) \\quad \\textrm {and} \\quad q^m = \\mathcal {O} \\left( \\left( \\frac{g}{\\nu } \\right)^{h-1} \\right),$ respectively, which are smaller than (REF ) and (REF ) in such a parameter regime.",
"In addition, if $ \\nu = \\mathcal {O}(1) $ and also $ h = \\mathcal {O}(1) $ , and $ g $ is the only unbounded parameter, then the best field size among all known PMDS codes is that in the third row in Table REF , which reads $q^m = \\mathcal {O} \\left( g^{h-1} \\right) = \\mathcal {O} \\left( n^{h-1} \\right),$ where $ n = g \\nu $ is the code length.",
"The existence of PMDS codes with field size (REF ) has been proven recently in [5] for $ r = 2 $ .",
"In contrast, the codes in the third row of Table REF can be explicitly constructed and admit any value of $ r $ .",
"Simultaneously and independently to this work, a construction of PMDS codes was obtained in [7] with field sizes $q^m = q^h = \\Theta \\left( n^{\\frac{h}{2}} \\right) = \\Theta \\left( n r^{h-2} \\right),$ for parameters satisfying $ g = \\Theta (q) $ , $ r = \\Theta (q) $ , and $ h \\le \\min \\lbrace g, \\delta +1 \\rbrace $ .",
"For this parameter regime, the field sizes (REF ) meet the lower bound from Proposition REF asymptotically, hence they have optimal field asymptotic order.",
"We remark that the parameter regime of our PMDS codes (second and third rows in Table REF ), is disjoint from the regime considered in [7], since in our case, we have that $ g = \\Omega \\left( q^r \\right) \\gg r $ , hence it may not happen at the same time that $ g = \\Theta (r) $ .",
"Finally, as discussed in Subsection REF and in [36], the PMDS codes obtained in [19] based on AG codes have larger field sizes than those in the first row of Table REF , which were already obtained in [36]."
],
[
"Appendix: Tables with even field sizes for MSRD codes",
"It is usual in the PMDS literature to consider and compare the obtained field sizes asymptotically, where some parameters grow unbounded (mostly the code length $ n = g \\nu $ , i.e., the number of local sets $ g $ and/or the local-set size $ \\nu $ ).",
"There are not many constructions of MSRD codes so far, so no trend exists when comparing their parameters.",
"However, in possible applications of MSRD and PMDS codes, it is reasonable to think that code parameters will rarely be large (even more so due to the required field sizes).",
"In this appendix, we provide several tables of attainable field sizes $ q^m $ , divisible by 2, among the linear MSRD codes obtained in this work.",
"First, we give a summary in Table REF , which is similar to Table REF , but where the field size $ q^m $ is not compared to $ g $ , but written as a function of $ q $ , $ r $ and $ h $ , excluding $ g $ .",
"The reason behind this is that typically the maximum attainable value of $ g $ is quite large for most of these codes, and in most cases we would puncture them in order to have a much smaller number of matrix sets $ g $ .",
"The motivation behind this is given in Table REF .",
"That table contains the case $ q = 2 $ , where other parameters vary.",
"Note that, in that case, linearized Reed-Solomon codes (Subsection REF ) require $ g = 1 $ , thus not being different than a Gabidulin code.",
"In Tables REF , REF and REF , we fix $ g $ and let other parameters vary.",
"In contrast, in Tables REF and REF , we fix the code length $ N = gr $ and let other parameters vary.",
"In these tables, bold numbers indicate field sizes that are the smallest among MSRD codes of the same parameters.",
"As linearized Reed-Solomon codes have the same field sizes for all $ h $ , a bold number in that row means that the field size is the smallest for the corresponding parameters for some $ h $ .",
"The field sizes attained by linear MSRD codes based on AG codes (Subsections REF , REF and REF ) are quite larger than those obtained by the other linear MSRD codes for small parameters.",
"In general, MSRD codes based on AG codes (as in Subsection REF ) are mostly of asymptotic interest.",
"For this reason, they are not included in Tables REF , REF , REF , REF , REF and REF .",
"Due to Remark REF , the linear MSRD codes based on Hamming codes (Subsection REF ) are only described for $ \\rho = 3 $ .",
"Finally, at the end of each table we consider the smallest field size attainable by an MRD code for the corresponding parameters.",
"As it can be seen, MRD codes always require significant larger field sizes than the MSRD codes from this work, for the same parameters.",
"Table: Table summarizing the code parameters of the linear MSRD codes obtained in this work.",
"In contrast with Table , field sizes are described in terms of q q , r r and h h , excluding g g .Table: Table for fixed q=2 q = 2 , while other parameters vary.Table: Table for fixed g=7 g = 7 , while other parameters vary.Table: Table for fixed g=15 g = 15 , while other parameters vary.Table: Table for fixed g=31 g = 31 , while other parameters vary.Table: Table for fixed N=gr=30 N = gr = 30 , while other parameters vary.Table: Table for fixed N=gr=62 N = gr = 62 , while other parameters vary."
]
] | 2011.14109 |
[
[
"Vacuum-Ultraviolet Absorption and Emission Spectroscopy of Gaseous,\n Liquid, and Supercritical Xenon"
],
[
"Abstract Bose-Einstein condensation, an effect long known for material particles as cold atomic gases, has in recent years also been observed for photons in microscopic optical cavitites.",
"Here, we report absorption and emission spectroscopic measurements on the lowest electronic transition ($5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$) of xenon, motivated by the search for a thermalization medium for photon Bose-Einstein condensation in the vacuum-ultraviolet spectral regime.",
"We have recorded pressure-broadened xenon spectra in the 135 nm to 190 nm wavelength regime at conditions near the critical point.",
"The explored pressure and temperature range includes high pressure gaseous xenon below the critical pressure and supercritical xenon at room temperature, as well as liquid xenon close to the boiling point near the critical pressure."
],
[
"Introduction",
"Bose-Einstein condensation has been experimentally observed with ultracold atomic gases, exciton polaritons, and more recently with photons, e.g.",
"in dye solution-filled optical microcavities [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].",
"In the latter system, a short mirror spacing in the wavelength regime provides a low-frequency cutoff, and visible spectral range photons confined in the resonator thermalize to the dye, which is at room temperature, by repeated absorption re-emission processes.",
"The dye molecules fulfill the thermodynamic Kennard-Stepanov relation between absorption and emission spectral profiles.",
"When photons in the cavity thermalize faster than they are lost through e.g.",
"mirror transmission, Bose-Einstein condensation to a macroscopically occupied ground state is observed [8], [11], [12].",
"Other than in the well-known laser, no inverted active medium is required and spontaneous emission is retrapped.",
"Thus, the usual argument of ultraviolet coherent sources showing disfavourable pump power scaling and thus making it hard to operate at high optical frequencies here does not apply.",
"In previous work, we have proposed photon condensation as a possible means to realize coherent optical sources in the UV spectral regime, and discussed the use of noble gases at high pressure as possible thermalization media with closed electronic transitions from the ground to the lowest electronically excited state lying in the vacuum ultraviolet spectral regime (VUV) [13].",
"Frequent collisions of the gas atoms can lead to a thermalization of quasimolecular states, which is a prerequisite for the Boltzmann-like Kennard-Stepanov scaling between absorption and re-emission in this gaseous system [14], [15], [16], [17].",
"A further important prerequisite for thermalization of UV photons by repeated absorption re-emission processes of course is a sufficient spectral overlap between absorption and emission spectral profiles, such that the emission can be reabsorbed by the gas atoms during the cavity lifetime.",
"In a binary collisional model (and when neglecting the effect of pressure shifts), the molecular xenon emission line has a Stokes shift of 25 with respect to the 147 absorption line.",
"The motivation of the present work is to obtain data of absorption and emission spectral profiles in dense xenon samples, which is a regime where spectral data on this system is limited.",
"In earlier works studying VUV absorption of xenon in the gaseous regime, broadband measurements up to 30 pressure at room temperature have been reported [13].",
"For monochromatic light at 173 wavelength at a pressure approaching the critical pressure of 58.4 complete absorption in a 13 long cell has been reported, indicating strong absorption far off-resonant from the $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$ transition centered at 147 wavelength [18].",
"We are not aware of other published measurements for supercritical xenon in absorption or emission spectroscopy, despite its wide use in chromatography, and as a non-linear medium for temporally compressing ultra-short laser pulses [19], [20].",
"Xenon in the liquid state is to date used in large scale scintillation detectors in the search for dark matter and neutrino research [21], [22].",
"For this use mainly two properties are of interest - namely the absorption length for the scintillation light emitted by xenon and the accurate wavelength of this emitted light.",
"For the absorption length in the literature two contradicting results are available.",
"Modern measurements find an attenuation length for the scintillation emission for liquid xenon at close to -110 temperature of up to 100, with the value being mainly determined by impurities in the xenon [22], [23], [24].",
"On the other hand, an early broadband absorption measurement of liquid xenon close to its boiling point, which at pressures of approximately 50 is a few degrees centigrade below room temperature, reported a complete absorption of light of wavelengths up to 226.5 through a cell of several millimeter length [25].",
"Since the two measurements were recorded at different temperatures, they do not directly contradict each other although such a large difference in absorption seems hard to justify theoretically.",
"Work studying the emission spectrum of xenon in the liquid phase has reported the emission to be centred around 1780.6 [26] or 174 [27], respectively, while the most recent report by Fujii et al.",
"reports a center wavelength of [parse-numbers=false]174.8 0.1 (stat) 0.1 (sys.)",
"[28].",
"Here we report measurements of the absorption and emission of the $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$ electronic transition of xenon.",
"While earlier work of our group has studied absorption spectra in the gaseous regime up to 30 pressure at room temperature, here we extend the investigated pressure range up to 130, thus well reaching into the supercritical regime.",
"We have moreover recorded data for samples cooled to the liquid phase.",
"Besides the described absorption spectral measurements, also emission spectral data for the described phases are reported.",
"In the following, chapter II describes the used experimental setup and chapter III contains absorption measurements of xenon samples.",
"In the subsequent chapter IV, emission measurements are reported, and finally chapter V gives conclusions."
],
[
"Experimental setup",
"A schematic representation of the experimental setup used to analyse vacuum-ultraviolet absorption and emission properties of xenon is shown in Fig.",
"REF .",
"A high pressure cell containing xenon is placed inside a vacuum chamber which is evacuated to 5e-5 to supress absorption of light of a wavelength of less than 193 by atmospheric oxygen and nitrogen molecules.",
"The absorption spectroscopy setup is similar as previously reported [13].",
"In brief, for absorption spectroscopy a 150 water-cooled deuterium lamp (Hamamatsu L1835) is used as a light source, permitting spectroscopic measurements down to a wavelength of 115.",
"Given that the deuterium lamp emission spectrum has large variations of the emission spectral density (near two orders of magnitudes over the spectral regime of interest), a double monochromator setup is used.",
"A home-built first monochromator utilizes a curved grating with 1200 groves/mm, with its blaze angle optimized for 200 wavelength radiation in a constant deviation monochromator scheme.",
"With a 1 wide exit slit the width of the produced spectral bandpass is 5.5.",
"The transmitted radiation passing the first monochromator is subsequently split up, with one part passing through the high-pressure cell containing the gas to be analyzed and the second one being sent around the cell to act as a reference path, allowing for normalization of the spectroscopic signal.",
"The radiation passing the cell enters a second monochromator, for which a commercial device (McPherson 234/302 200) is used.",
"This monochromator is equipped with a 2400 groves/mm grating.",
"For a slit width of 10 a wavelength resolution of 0.05 is reached.",
"The resulting wavelength-resolved signal is detected using a cooled open nose VUV sensitive CCD camera.",
"A flip mirror placed before the detection monochromator allows for path selection between the reference path and the path through the spectroscopic pressure cell.",
"The used cell is constructed to sustain gas pressures up to 200.",
"We use a metal cell with MgF$_2$ windows.",
"On the optical axis used for absorption measurements the cell is equipped with step windows, such that the optical path length in the cell can be tuned by using windows of a suitable step height.",
"In the presented measurements an effective cell length of 180 on the optical axis is used in the absorption measurements.",
"For spectroscopic measurements of the emission, a different excitation light source is used.",
"Here, a Nd:YAG laser (Quanta-Ray GCR-12S) is utilized, which produces 10 long pulses near 1064 wavelength with a repetition rate of 10 and a pulse energy of 250.",
"With the fourth harmonic of this laser near 266 wavelength and an optical pulse energy of 10, we can excite the pressure-broadened $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{p}$ xenon two-photon transition, which has a transition wavelength of 128, off-resonantly.",
"It is interesting to note that at higher xenon densities a second two-photon excitation channel near 133 transition wavelength arises, which in earlier work has been attributed to the formation of excitons [29].",
"This second channel can be excited almost resonantly with the used irradiation wavelength.",
"The pump light near 266 is focussed into the high pressure cell with a lens of 20 focal length along a direction perpedicular to the path utilized for absorption spectroscopy.",
"In this way, the same analysis optics and spectrometer as described above can be used to record the time integrated spectrally resolved fluorescence and phosphorescence signal of the xenon sample.",
"For the part of the measurements using gas pressures above a pressure of used xenon gas bottle pressure (near 56), a manual spindle press (Sitec 750.1030) is used to compress the xenon gas.",
"To allow for spectroscopic measurements of xenon in the liquid phase, a cooling of the cell is required.",
"For this, the xenon cell can be equipped with a cooling jacket, through which liquid nitrogen is fed at a variable flow rate.",
"The reachable cell temperature of $\\approx {-20}{}$ are clearly below the critical temperature of xenon (16.58), thus spectroscopic measurements in the liquid phase can well be carried out with the present setup."
],
[
"Xenon VUV absorption measurements",
"To begin with, absorption spectra of the xenon $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$ electronic transition have been recorded, i.e.",
"the transition from the ground state to the lowest-lying electronically excited state, in the wavelength range around 147.",
"From the ground and singly excited potential energy curves for two colliding xenon atoms, see Fig.",
"REF , it can be seen that for not to small internuclear distances (above 4), when the ground state energy curve still is mostly flat, the (singly) excited state energy curve already bends to lower energies for decreasing internuclear distance.",
"The described shape of the potential curves is the origin of the significantly stronger broadening of the red wing of the pressure-broadened absorption line in comparison to the blue wing in a regime beyond the impact limit at the used high gas pressures [31].",
"Fig.",
"REF shows pressure broadened xenon VUV absorption spectra observed for different values of the gas pressure.",
"Besides the $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$ transition (centered at 147 wavelength in the thin vapor regime) on the short wavelength side also the red wing of the next higher energetic xenon line, centered at 129.5 wavelength, is observed starting with the data recorded at 62 pressure.",
"For the data recorded at the highest used pressure (130) the two absorption line signals already overlap.",
"Near the line center, the $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$ experimental absorption data is saturated, with corresponding data being here omitted for clarity.",
"We observe a strong pressure broadening and shift to the red side of the resonance, the latter understood from the already above mentioned enhancement of the red wing originating from the form of the quasimolecular potential curves.",
"A more detailed inspection of the red wing of the absorption profile, for which also at higher pressures no overlap with other lines is present, shows that the observed line shape far of resonance here well matches an exponential decay.",
"This observation remains valid throughout the gaseous and supercritical phases of the xenon gas (above and below near 58.4 respectively) at the used room temperature conditions.",
"In order to compare the individual measurements, in the following the wavelength at which the red wing of the absorption profile reaches an absorption coefficient of 100 is evaluated and denoted as $\\lambda _{{100}{}}$ .",
"The corresponding quantity is relevant as to judge whether the enhanced red wing results in a reabsorption of the emission spectrum of the transition despite the large (25 in the absence of pressure broadening) Stokes shift.",
"For a gas-filled microcavity with e.g.",
"1.5 mirror spacing, fluorescent radiation of wavelength $\\lambda _{{100}{}}$ would be reabsorbed following 30 cavity round trips, which is in the order of magnitude of the realizable cavity finesse values with dielectic mirrors in this wavelength range.",
"Table REF summarizes the obtained results for room temperature conditions and the data points in Fig.",
"REF a display the corresponding variation with cell pressure, showing an increase of the wavelength shift from the unperturbed line center at which an absorption coefficient of 100 is reached.",
"Figure: Room temperature (T≈1.04T c T \\approx 1.04 T_c where T c =16.6T_c = {16.6}{} denotes the critical temperature of xenon) absorption spectra of xenon in the wavelength range of 137.5170 for pressures between 20;130.Saturated regions are omitted for clarity.The vertical dashed line gives the position of the line center of the 5p 6 →5p 5 6s5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s} transition in the absence of pressure broadening and shift.Table: Measured values for the wavelength at which the absorption coefficient of the spectrum of the lowest energetic xenon absorption line at room temperature reaches a value of 100 on the red and the blue wing of the resonance.To allow for a comparison of this data with theory, we have carried out numerical simulations.",
"These numerical simulations are based on a calculation of the Franck-Condon factors for the system [30].",
"For the simulation, it is assumed that the absorption on the wing of the profile is mainly due to the effect of two-body collisions, and the quasistatic approximation is used [32], [33], [34], [35].",
"The influence of multiparticle collisions, which can be relevant at the used high densities, should be mostly visible close to the resonance wavelength of the observed transition.",
"Thus higher order collisions are not considered in the numerical simulation and the whole profile can be derived from the potential energy curves in Fig.",
"REF .",
"Figure: (a) Experimentally determined wavelength λ 100 \\lambda _{{100}{}} at which the absorption coefficient on the red wing xenon 5p 6 →5p 5 6s5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s} resonance reaches a value of 100 (data points) along with theory (solid red line) versus the gas pressure at room temperature.",
"(b) Variation of λ 100 \\lambda _{{100}{}} on the cell temperature, for a constant pressure of 57 (P≈0.98P c P \\approx 0.98 P_c,where P c =58.4 ¯P_c = {58.4}{\\bar{}} denotes the critical pressure of xenon).The observed jump at the wavelength λ 100 \\lambda _{{100}{}} occurs at a temperature consistent with the liquid to gaseous phase transition temperature corresponding to 15.38 at the used pressure value.The red solid line in Fig.",
"REF a shows the result of our simulations for the expected value of $\\lambda _{{100}{}}$ versus pressure.",
"Our experimental results are in good agreement with the calculations, which supports the assumption that the far red wing of the absorption profile is governed by the effect of two-body collisions.",
"At the highest experimentally investigated pressure of 130 xenon exhibits a density of 1988.",
"A further increase in pressure up to 4194, the pressure at which xenon at room temperature would become solid [36], would lead to a further increase of the density by a factor 1.7, and, following our simulations, lead to an additional red shift of $\\lambda _{{100}{}}$ by only about 1.1.",
"We however point out that at such densities, which approach those of a solid, the influence of multiparticle collisions should be carefully evaluated, which may well result into more significant corrections to the line shape than in our presently experimentally accessible parameter regime.",
"In further measurements, we have recorded corresponding data for different cell temperatures, which at the used constant gas pressure of 57 allows both the gaseous and, upon cooling to below a temperature of 15.38 (corresponding to the expected value of this phase transition at this pressure value), the liquid phase to be analyzed.",
"Fig.",
"4b shows the variation of the wavelength $\\lambda _{{100}{}}$ versus the cell temperature (data points) along with the dependence expected from our simulation (red solid line).",
"Upon cooling to below the phase transition temperature, we observe a shift of $\\lambda _{{100}{}}$ to the red by about 1.1, with the experimental results being in good agreement with our theory predictions.",
"The corresponding increased width of the absorption line in the red wing is well understood by the density increase (here corresponding to a factor of about 1.7) upon entry into the liquid phase.",
"The results are in stark contrast to previous measurements by McLennan and Turnbull who under comparable conditions found a complete absorption of light up to a wavelength of 226.5 through a cell of several length [25].",
"It is, however, consistent with previous findings for liquid xenon at a temperature of -108, where an absorption length for light at 178 of more than 100 was observed [22]."
],
[
"Xenon emission measurements",
"We have next carried out spectroscopic measurements of the vacuum-ultraviolet emission of gaseous xenon.",
"At pressures above 100, upon excitation of the lowest energetic electronically excited of the xenon atom, the $5\\text{p}^5 6\\text{s}$ state, it is well known that Stokes-shifted radiation centered at near 172 wavelength is emitted, the so-called second excimer continuum [31], [37], [38], [39], [40].",
"This is well understood from the potential curves of the xenon dimer system (Fig.",
"REF ), with the singly excited state being bound thus forming an excimer system.",
"Accordingly, when exciting the $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{p}$ two-photon transition off-resonantly, the emission is found spectrally peaked around the second excimer continuum at 172 wavelength.",
"We attribute de-excitation from the $5\\text{p}^5 6\\text{p}$ state to the $5\\text{p}^5 6\\text{s}$ state in the used high-pressure regime to be dominated by collisional processes given the occurring multiple crossings of the quasimolecular electronically excited states curves [41].",
"In our experiment, we monitor the emission from the subsequently expected decay to the electronic ground state, which is in the vacuum-ultraviolet.",
"In Fig.",
"REF spectrally resolved emission measurements recorded at room temperature for three different xenon pressures are presented.",
"All three spectra are normalized to their respective maximum value for better comparability since the excitation efficiency of the used two-photon transition depends strongly on the xenon density.",
"For gaseous xenon at 40 pressure the peak of the emission spectrum is found at 172.20.1, which agrees within experimental uncertainties with previous measurements of the emission of gaseous xenon [42], [38], [40].",
"The two spectra shown by blue dashed lines, recorded at 62 and 180 pressure respectively, corresponding to pressure values above the critical pressure, are clearly shifted to longer wavelengths.",
"For the highest pressure data (with 180 pressure), the maximum of the emission is centered at 175.3 0.1 wavelength, corresponding to a red shift of 3.1 with respect to the 40 data.",
"The full width of the emission spectrum for the three shown spectra remains nearly constant at 13.30.2.",
"Given that the observed width stays nearly constant despite the large density difference in spectra, we conjecture that the observed red shift of the supercritical and liquid phase emission data with respect to the gas emission is not due to reabsorption effects.",
"Figure: Measured spectra of the xenon second excimer continuum for three different values of the xenon pressure at room temperature (T=1.04T c T = 1.04 T_c).Figure: a) Observed variation of the spectral maximum of the xenon second excimer continuum emission versus the xenon pressure recorded at room temperature (T≈1.04T c T \\approx 1.04 T_c), see b) for the dependence of the xenon density on pressure.c) Gives corresponding experimental data for the variation of the peak emission with temperature recorded at the constant pressure of 53 (P≈0.91P c P \\approx 0.91 P_c), see d) for the dependence of xenon density on temperature.Here the phase transition temperature is T≈11.8T \\approx {11.8}{}.Fig.",
"REF a shows the variation of the peak emission wavelength versus pressure.",
"The observed red shift coincides with the increase of the gas density upon pressure variation, see Fig.",
"REF b.",
"In the supercritical regime, a continuous variation of density with gas pressure occurs.",
"Given that the used room temperature conditions are only a few degrees above the critical temperature of xenon, the density increase at values near the critical pressure visible in Fig.",
"REF b is relatively rapid, which leads to the fast increase of the pressure shift in this regime observed in Fig.",
"REF a.",
"Fig.",
"REF c shows data for the variation of the peak emission wavelength on temperature, recorded at the constant value of the pressure above the liquid of 53; see Fig.",
"6d for the corresponding variation of density.",
"The density here exhibits a discontinuity upon cooling from the gaseous to the liquid phase, given that the used pressure is below the critical pressure (phase transition temperature: 11.8 at a pressure of 53).",
"The observed sudden variation of the peak of the emission visible in Fig.",
"REF c near this temperature is in good agreement with the expected discontinuous density change upon crossing the phase transition from liquid to gaseous.",
"In order to compare our results to earlier work, we average our available data both above and below the phase transition temperature, yielding results for the peak emission wavelength of the second eximer continuum of 174.9 0.4 and 172.0 1.0 wavelength in the liquid and gaseous phase of xenon respectively.",
"Other works report the emission of liquid xenon to be centred around 1780.6 [26] or 174 [27], while the here performed measurement most closely agrees with the recent work of Fujii et al.",
"with a reported center wavelength of [parse-numbers=false]174.8 0.1 (stat) 0.1 (syst.)",
"[28]."
],
[
"Conclusions",
"To conclude, absorption and emission spectroscopic measurements of the xenon $5\\text{p}^6 \\rightarrow 5\\text{p}^5 6\\text{s}$ electronic transition at densities close to the critical point are reported.",
"In the absorption measurements, in agreement with e.g.",
"the lower pressure regime data of [31], we observe an asymmetric lineshape with an enhanced red wing.",
"Data for the far red wing has been compared to numerical simulations based on two-body collisions also described in our work, which gives good agreement with our experimental results even at densities up to 2000.",
"In contrast to earlier measurements by Koehler et al.",
"for supercritical xenon [18], as well as by McLennan and Turnbull for liquid xenon [25] no significant broadening of the absorption profile extending above what can be expected from the binary potential energy curves was observed.",
"In our measurements, other than in earlier work, xenon VUV emission spectra have also been observed in the supercritical phase.",
"It was shown that the peak emission wavelength of supercritical xenon at room temperature exhibits a similar shift compared to the dilute gas emission as observed in liquid xenon.",
"As applies to the liquid phase case, also for the supercritical xenon case at high pressure, the observed shift of the peak emission wavelength of in both cases near 3 cannot be explained by the binary potential energy curves, based on which no change in the peak emission wavelength would be expected.",
"Future theory work here should analyse for the possible influence of multiparticle collisions.",
"Within the supercritical phase, the magnitude of the observed shift of the peak emission wavelength may be utilized to monitor whether the sample is more in a \"gas-like\" or a \"liquid-like\" state, for small and large spectral shifts of the emission respectively.",
"This can provide a spectroscopic means to determine the position of the so-called Widom-line in supercritical xenon [43], [44].",
"In the investigated pressure and temperature regimes, the pressure shifts reduce the Stokes shift between absorption and emission not yet by an amount to allow for sufficient reabsorption enabling the thermalization of photons by absorption and re-emission processes as in visible spectral range dye microcavity experiments [8], [45].",
"For the future, the influence of manybody collisional effects on the lineshapes should be investigated, which can be relevant for higher gas pressures than experimentally investigated in the present work.",
"A different possible approach is to investigate the influence of higher gas temperatures, as numerical simulations predict a sufficient spectral overlap at temperatures of 1000 and above.",
"The increased possible reabsorption of fluorescent radiation is understood from that ground state atoms at such elevated temperatures during diatomic collisions can climb up to smaller internuclear distances in the quasimolecular potential curves (Fig.",
"REF ) than at room temperature.",
"Metal-coated multilayer mirrors [46] can allow for the realization of suitable microcavities in this temperature regime, where the use of dielectic cavity mirrors is not feasible.",
"Once sufficient spectral overlap is established, it will be important to test for the Boltzmann-like Kennard-Stepanov scaling between absorption and re-emission profiles.",
"Other attractive candidates for photon gas thermalization in the vacuum-ultraviolet regime are mixtures of small molecules that are gaseous at room temperatures as the CO molecule with a light noble gas acting as a buffer gas, i.e.",
"which itself is transparent in the spectral regime of the molecular lines.",
"We acknowledge support of the Deutsche Forschungsgemeinschaft within SFB/TR 185 (277625399)."
]
] | 2011.14131 |
[
[
"Delegated RingCT: faster anonymous transactions"
],
[
"Abstract We present a modification to RingCT protocol with stealth addresses that makes it compatible with Delegated Proof of Stake based consensus mechanisms called Delegated RingCT.",
"Our scheme has two building blocks: a customised version of an Integrated Signature and Encryption scheme composed of a public key encryption scheme and two signature schemes (a digital signature and a linkable ring signature); and non-interactive zero knowledge proofs.",
"We give a description of the scheme, security proofs and a prototype implementation whose benchmarking is discussed.",
"Although Delegated RingCT doesn't have the same degree of anonymity as other RingCT constructions, we argue that the benefits that the compatibility with DPoS consensus mechanisms brings constitutes a reasonable trade-off for being able to develop an anonymous decentralised cryptocurrency that is faster and more scalable than existing ones."
],
[
"Introduction",
"Bitcoin appeared in 2008 [1] and is widely considered to be the first decentralised cryptocurrency.",
"Its ingenious design, that uses a blockchain as a distributed ledger to store the transactions that happen on the network and Nakamoto consensus [2] (which centres around the proof-of-work mechanism and the “longest-chain-win” rule) to reach a decentralised consensus about the state of that blockchain, was revolutionary at the time.",
"Even today Bitcoin is the most well known and most valuable cryptocurrency.",
"Since then, the industry has grown and the term cryptocurrency is not solely a synonym of currency anymore, but has extended to other use cases (e.g.",
"smart contracts).",
"Still, more than ten years later, we do not have a cryptocurrency that is widely used as a currency, as Bitcoin was supposed to be as the title of its original paper states: a peer-to-peer electronic cash system.",
"One can argue that this is due to external factors, such as government regulations, lack of knowledge or necessity by societies, ideological motives, etc.",
"But we can also argue that the intrinsic technical limitations of current cryptocurrencies, due to their design, have contributed to this situation.",
"These design flaws include the inability to scale, insufficient maximum throughput, slow confirmation times, ledger size or lack of anonymity.",
"In our opinion, the ideal cryptocurrency is decentralised, fast, scalable, anonymous, has a transparent monetary policy and is environmentally friendly.",
"Many cryptocurrencies have been created in the last few years that have tried to fulfil these goals but, so far, none of them has been able to reach them all.",
"Some, like Monero [3] and ZCash [4], solve the anonymity issue but still share the same other limitations of Bitcoin.",
"Other cryptocurrencies based on Delegated Proof of Stake (DPoS), like Tezos [5] (Liquid Proof of Stake) and Nano [6] (Open Representative Voting), improve on the maximum throughput and slow confirmation times but are only pseudo-anonymous, meaning that anonymity is only maintained as long as a node on the network is not associated to a \"real world\" identity.",
"These consensus mechanisms are typically faster than others that use hashrate power competition to select the node that proposes the new transactions, like Bitcoin, allowing for a greater throughput of transactions, and have a much lower carbon footprint.",
"The goal of this paper is to bring together some of the strengths of these designs and develop a protocol that can be used as a building block for a cryptocurrency with the properties mentioned above, specifically an anonymous decentralised cryptocurrency faster and more scalable than the current ones."
],
[
"Contributions",
"We present an extension to the base protocol of Monero, RingCT with stealth addresses, that makes it compatible with Delegated Proof of Stake, a family of consensus mechanisms where the weight of a node in the consensus for validating transactions is proportional to its delegated stake on the network, called Delegated RingCT.",
"We first present a generic version of Delegated RingCT constructed from two cryptographic primitives: a customised version of an Integrated Signatures and Encryption scheme (ISE) [7], which is composed of a public key encryption scheme (PKE) and two signature schemes, a digital signature (DS) [8] and a linkable ring signature (LRS) [9]; and non-interactive zero knowledge proofs (NIZK).",
"We, then, give a concrete efficient instantiation of Delegated RingCT and a prototype implementation whose benchmarking shows that the scheme can be used to build a faster and more scalable anonymous decentralised cryptocurrency.",
"Our scheme has some limitations and, despite being secure according to our security model, our definition is somewhat weaker than other RingCT constructions.",
"We argue that the benefits outweigh the cons, as we will discuss later."
],
[
"Overview and Intuition",
"For completeness, we give a brief summary of the RingCT protocol and the DPoS based consensus mechanisms.",
"We explain the reasoning behind our modifications to combine the two and construct Delegated RingCT.",
"On a basic level, a transaction has a sender, a receiver and a transferred amount.",
"To achieve anonymity of all three components, RingCT protocol uses the following: Linkable ring signature to obfuscate the real sender of a transaction within a ring of possible senders and the linkability to detect double spends, since each transaction must have a unique linkable tag (also called key image or serial number).",
"Confidential transactions to obfuscate the transaction amount, typically using an additive homomorphic commitment scheme like Pedersen commitments [10].",
"These are used to make a range proof, proving that the balance lies within a certain range, and a balance proof, proving that the total balance of the input accounts spent is equal to the total amount of the created output accounts.",
"Stealth addresses to obfuscate the receiver of a transaction.",
"Every node has a pair of long term keys (a long term public key and a long term secret key) and every transaction as a one-time pair of keys (a one-time public key and a one-time secret key).",
"The sender can derive a one-time public key and a public auxiliary information from the receiver's long term public key.",
"The receiver can recover the one-time secret key of the account created in the transaction using his long term secret key and the auxiliary information.",
"Delegated Proof of Stake is a family of consensus mechanisms that is based on two basic concepts: The weight that a node has on the consensus of the network is proportional to his stake (balance) on the network.",
"The stake of a node can be delegated to another node, transferring its weight on the consensus to that node.",
"With this in mind, we first need to introduce the concept of stake delegation in RingCT.",
"We follow the terms used in Nano [6], and call representative a node to which has been assigned some stake by another node.",
"This concept is distinct from the owner of an account, but they can be the same.",
"We do this by switching the representation of a coin in RingCT from a commitment of an additive homomorphic commitment scheme (e.g., Pedersen commitments [10]) to a ciphertext of an additive homomorphic public encryption scheme (e.g., exponential ElGamal [11]).",
"This allows the owner of an account, that initially has no representative, to encrypt its balance with a long term public key of a node, making him the representative of that account.",
"The representative can then prove by decryption that a certain amount of stake was delegated to him using his long term secret key.",
"A NIZK proof is required to prove that the encryption is well formed.",
"Since the PKE is additive homomorphic and the consensus algorithm only needs to know the total amount of delegated stake to a given node, the representative reveals the total amount of delegated balance to him without revealing the individual balances of the accounts delegated to him, otherwise there would be no obfuscation of balances.",
"A node can redelegate the stake of an account at anytime proving he knows the one-time secret key of that account (digital signature), by reencrypting the balance with a new long term public key and proving with a NIZK that both ciphertexts are equivalent, i.e., encrypt the same balance.",
"Transactions are modified from standard ring confidential transactions and, because of that, the ciphertext needs to be compatible with the range proof protocol and the balance proof.",
"The linkable ring signatures and the stealth addresses remain the same.",
"We construct a customised version of an Integrated Signature and Encryption scheme to express the fact that the same keys are used for the encryption/decryption, the digital signature and the linkable ring signature."
],
[
"Related Work",
"Since Monero and ZCash are the two most valuable anonymous cryptocurrencies by market capitalisation, research in this area is mostly based on the two different technologies they use.",
"The base protocol used by Monero was first described in [3].",
"Since then, it has evolved into a ring of confidential transactions, which combines linkable ring signatures [9], [12] with confidential transactions [13].",
"Reference [14] gives the first formal syntax of RingCT and improves it with a new version and [15], [16], [17], [18] improve on the size and the efficiency of the linkable ring signature component and [19] on the range proof.",
"Compatibility with smart-contracts was achieved in [20].",
"ZCash uses zero-knowledge Succinct Non-interactive AR-guments of Knowledge (zk-SNARKs) to construct a decentralised payment scheme [4], but requires a trusted setup.",
"Since then, other zero-knowledge proofs for arithmetic circuits were developed by improving on efficiency, decreasing the amount of \"trust\" required or increasing the scope of use [21], [22], [23], [24], [25].",
"Another relevant approach to anonymous cryptocurrencies is Zerocoin protocol [26], improved in [27]."
],
[
"Basic Notation",
"We use additive notation and define $\\mathbb {G}$ as a cyclic group of prime order $p$ in which the discrete logarithm problem is hard and $\\mathbb {F}$ as the scalar field of $\\mathbb {G}$ .",
"Let $\\mathcal {H}: \\lbrace 0, 1\\rbrace ^{*} \\rightarrow \\mathbb {F}$ be a cryptographic hash function and $G$ a generator of $\\mathbb {G}$ with unknown discrete logarithm relationship.",
"A function is negligible in the security parameter $\\lambda $ , written negl($\\lambda $ ), if it vanishes faster than the inverse of any polynomial in $\\lambda $ .",
"$\\mathcal {A}$ probabilistic polynomial time (PPT) algorithm is a randomised algorithm that runs in time poly($\\lambda $ ).",
"In a randomised algorithm $\\mathcal {A}$ , the input randomness $r \\in \\mathbb {F}$ is explicit and we write $z \\leftarrow A(x_1,..., x_n; r)$ .",
"We use $x \\overset{r}{\\leftarrow } X$ to denote sampling $x$ uniformly at random from $X$ .",
"For readability, we denote the set of elements $\\lbrace x_n\\rbrace _{n=0}^{N-1}$ by just $x_n$ when is clear in the context that we are referring to a set of elements instead of a single element."
],
[
"Integrated Signatures and Encryption Scheme",
"The concept of combining public key schemes was first introduced by [7].",
"Reference [28], inspired by [29], combines one signature scheme with one encryption scheme.",
"In this work we go a little further and use a combination of a public key encryption scheme with two signatures schemes, a standard digital signature and a linkable ring signature.",
"It is composed of the following polynomial time algorithms.",
"$pp \\leftarrow $ Setup($1^{\\lambda }$ ) on input a security parameter $1^{\\lambda }$ , output public parameters $pp$ .",
"KeyGen: this algorithm is divided in three steps to capture the concept of stealth addresses, in the following way.",
"- $(ltpk, ltsk) \\leftarrow $ LongTermKeyGen($pp$ ).",
"On input public parameters $pp$ , it randomly generates a keypair ($ltpk, ltsk$ ).",
"- $(pk$ , aux) $\\leftarrow $ OneTimePKGen($ltpk; r$ ).",
"On input a long term public key $ltpk$ , it outputs a random one-time public key $pk$ and the auxiliary information aux.",
"- $sk$ /$\\perp \\leftarrow $ OneTimeSKGen($pk$ , aux, $ltsk$ ).",
"On input a one-time public key $pk$ , an auxiliary information aux and a long term secret key $ltsk$ , it outputs the one-time secret key $sk$ if $ltsk$ is valid.",
"If not, it returns $\\perp $ .",
"$C \\leftarrow $ Encrypt($pk, m; r$ ): on input a public key $pk$ and a plaintext $m$ , it outputs a random ciphertext $C$ .",
"$m \\leftarrow $ Decrypt($sk, C$ ): on input a secret key $sk$ and a ciphertext $C$ , output a plaintext $m$ .",
"$\\sigma \\leftarrow $ Sign($sk, m$ ): on input $sk$ and a message $m$ , output a signature $\\sigma $ .",
"0/1 $\\leftarrow $ Verify($pk, m, \\sigma $ ): on input $pk$ , a message $m$ , and a signature $\\sigma $ , output “1” if the signature is valid and “0” if it is not.",
"$\\sigma \\textsubscript {ring} \\leftarrow $ Signring($sk, m, R$ ): Generates a linkable ring signature $\\sigma \\textsubscript {ring}$ on a message $m$ with respect to a ring $R$ of one-time public keys, provided that $sk$ is a one-time secret key corresponding to some $pk$ in the ring.",
"0/1 $\\leftarrow $ Verifyring($\\sigma \\textsubscript {ring}, m, R$ ): Verifies a signature $\\sigma $ on a message $m$ with respect to a ring of public keys $R$ .",
"Outputs \"0\" is the signature is rejected, and \"1\" if accepted.",
"0/1 $\\leftarrow $ Link($\\sigma \\textsubscript {ring}$ , $\\sigma ^{\\prime }\\textsubscript {ring}$ ): Determines if signatures the linkable ring signatures $\\sigma \\textsubscript {ring}$ and $\\sigma ^{\\prime }\\textsubscript {ring}$ were signed using the same private key.",
"Outputs \"0\" if the signatures were signed using different private keys and \"1\" if they were signed using the same private key.",
"Since each component being individually secure does not imply that the composition of all the components is also secure, we need to have a joint security model, i.e., a model that evaluates the security of each component in the presence of the others, which are simulated by oracles.",
"The only component that does not need to be simulated by an oracle in the public key setting is the PKE, since an adversary can easily do it.",
"Definition 1 (Joint Security for ISE) We say an ISE is jointly secure if: Its PKE component is IND-CPA secure (1-plaintext/2-recipient) in the presence of two signing oracles, one for the DS and the other for the LRS.",
"Its DS component is EUF-CMA secure in the presence of a signing oracle simulating the LRS component.",
"Its LRS component is secure, following the security model of [18], in the presence of a signing oracle simulating the DS component."
],
[
"Non-Interactive Zero-Knowledge Proof",
"A NIZK proof system in the CRS model consists of the following four PPT algorithms [30]: $pp \\leftarrow $ Setup($1^\\lambda $ ): on input $1^\\lambda $ , outputs public parameters $pp$ .",
"$crs \\leftarrow $ CRSGen($pp$ ): on input $pp$ , outputs a common reference string $crs$ .",
"$\\pi \\leftarrow $ Prove($crs, x, w$ ): on input a $crs$ and a statement-witness pair ($x, w$ ), outputs a proof $\\pi $ .",
"0/1 $\\leftarrow $ Verify($crs, x, \\pi $ ): on input $crs$ , a statement $x$ , and a proof $\\pi $ , outputs “0” if rejects and “1” if accepts."
],
[
"Data Structures",
"We begin by describing the data structures used by a Delegated RingCT system.",
"Blockchain.",
"A Delegated RingCT protocol operates on top of a blockchain $B$ , which is a publicly accessible and append-only database.",
"At any given time $t$ , all users have access to $B_t$ , which is a sequence of transactions.",
"If $t < t^{\\prime }$ , state of $B_t$ is anterior to the state of $B_{t^{\\prime }}$ .",
"Public parameters.",
"A trusted party generates the public parameters $pp$ , which are used by the protocol's algorithms.",
"These include the group in which the algorithms perform operations, generators of the group, cryptographic hash functions and parameters regarding transactions, namely: $V$ , which specifies the maximum possible number of coins that the protocol can handle.",
"Any balance and transfer must lie in the integer interval $V = [0, v\\textsubscript {max}]$ .",
"$N$ , the maximum size of the ring used in a delegated ring confidential transaction DRingCTx, i.e., the maximum number of input accounts.",
"$M$ , the maximum number of spend accounts in a DRingCTx, such that $M \\subset N$ .",
"$T$ , the maximum number of output accounts of a DRingCTx.",
"Keys.",
"There are two pairs of keys: long term keys, which are composed of a long term public key $ltpk$ , and a long term secret key $ltsk$ and are associated with a unique node on the network; and one-time keys, which are composed of a one-time public key $pk$ and a one-time secret key $sk$ and are associated with a unique account.",
"One-time keys are derived from long term keys and one node on the network can have multiple accounts.",
"Account.",
"Each account is associated with a one-time keypair $(pk, sk)$ and a coin $C$ , which is a ciphertext of an additive homomorphic PKE scheme and encrypts an amount/balance $a$ with randomness $k$ , also known as coin key.",
"The one-time public key $pk$ acts as a stealth address and can only receive one transaction.",
"The secret key $sk$ is kept privately and is used to spend the balance of the account once.",
"Delegated ring confidential transaction.",
"A delegated ring confidential transaction DRingCTx consists of a ring of input accounts, a linkable ring signature, output accounts with zero-knowledge proofs of encryption, range and balance, and an auxiliary information to help the owners of the destination addresses recover the one-time secret keys of the output accounts.",
"Typically, one of the output accounts belongs to the sender and its balance is the \"change\" of a transaction.",
"Change representative transaction.",
"A change representative transaction CRx consists of a digital signature based on the one-time secret key of the account and a zero-knowledge proof of equivalence between the old ciphertext $C$ and the new ciphertext $C^{\\prime }$ , assuring that the balance is the same."
],
[
"Algorithms",
"A Delegated RingCT scheme is a tuple of polynomial-time algorithms defined as below: $pp \\leftarrow $ Setup($1^\\lambda $ ): on input a security parameter $\\lambda $ , output public parameters $pp$ .",
"$(act$ , aux) $\\leftarrow $ CreateAccount$(a, ltpk)$ : on input an amount $a$ and a long term public key $ltpk$ , outputs an auxiliary information aux and an account $act = (pk, C)$ , composed of a one time public key $pk$ and a coin $C$ .",
"$a \\leftarrow $ RevealBalance$(ltsk, C)$ : on input a long time secret key $ltsk$ and a coin $C$ , outputs the balance $a$ in plaintext.",
"This algorithm is used by the representative of an account.",
"DRingCTx = ($act_n$ , $\\sigma \\textsubscript {ring}$ , aux, $act_t, \\pi \\textsubscript {range}, \\pi \\textsubscript {enc}, \\pi \\textsubscript {bal})$ /$\\perp \\leftarrow $ CreateDRingCTx$(act_n, ask_m, a_t$ , $ltpk_t, m)$ : on input a ring of $N$ input accounts, $M$ account secret keys corresponding to some of those accounts, $T$ output amounts, $T$ destination addresses and a transaction message $m$ , it outputs the input accounts $act_n$ , a linkable ring signature $\\sigma \\textsubscript {ring}$ , auxiliary information aux, $T$ output accounts, a range proof $\\pi \\textsubscript {range}$ , an encryption proof $\\pi \\textsubscript {enc}$ and a balance proof $\\pi \\textsubscript {bal}$ , if the accounts secret keys are valid.",
"It outputs $\\perp $ otherwise.",
"0/1/-1 $\\leftarrow $ VerifyDRingCTx(DRingCTx): on input a delegated ring confidential transaction DRingCTx, outputs “0” if the transaction is valid, “1” if it is invalid and \"-1\" if the transaction is linked to a previous valid transaction, i.e., if any of $act_n$ has been spent previously.",
"If DRingCTx is valid, it is recorded on the blockchain $B$ .",
"Otherwise, it is discarded.",
"CRx = $(\\sigma , act^{\\prime }, \\pi \\textsubscript {equal})/ \\perp \\leftarrow $CreateCRx$(ask, act, ltpk, m)$ : on input an account $act$ with the corresponding account secret key $ask$ , a long term public key $ltpk$ and a transaction message $m$ , it outputs a digital signature $\\sigma $ , a new account $act^{\\prime }$ with the same amount $a$ encrypted with the new $ltpk$ and a proof of equivalence $\\pi \\textsubscript {equal}$ , if the account secret keys is valid.",
"It outputs $\\perp $ otherwise.",
"0/1 $\\leftarrow $ VerifyCRx(CRx): on input a change representative transaction CRx and a digital signature $\\sigma $ , it outputs \"0\" if its invalid.",
"Otherwise, it outputs \"1\" and the CRx transaction is appended to the blockchain $B$ ."
],
[
"Correctness",
"Correctness of Delegated RingCT requires that: A valid delegated ring confidential transaction DRingCTx will always be accepted and recorded on the blockchain $B$ .",
"A valid change representative transaction CRx will always be accepted and recorded on the blockchain $B$ ."
],
[
"Security Model",
"We focus only on the transaction layer of a cryptocurrency, and assume that network-level and consensus-level attacks are out of scope.",
"Intuitively, a Delegated RingCT protocol should have the following properties.",
"Unforgeability.",
"This property captures the idea that only someone who knows the secret key of an account can spend it and change its representative.",
"Anonymity.",
"This property captures the idea that an outside party, other than the owner or the representative of the sender account, cannot know who is the real sender, who is the receiver or what is the amount of a DRingCTx transaction.",
"Linkability.",
"This property captures the idea that you can only spend once from an account, i.e., you cannot double spend.",
"Non-frameability.",
"This property captures the idea that a malicious party cannot construct a transaction that invalidates a valid transaction.",
"We formalise the above intuitions into a game-based security model between an adversary $\\mathcal {A}$ and a challenger $\\mathcal {CH}$ .",
"The capabilities of the adversary are modelled by the queries that he can make to oracles implemented by the challenger, which are described bellow.",
"$ltpk_i \\leftarrow $ KeyOracle($i$ ): on the $i^{th}$ query, the challenger $\\mathcal {CH}$ runs $(ltsk_i, ltpk_i) \\leftarrow $ ISE.LongTermKeyGen$(pp\\textsubscript {ise})$ and returns $ltpk_i$ to $\\mathcal {A}$ .",
"$act_i \\leftarrow $ AccountOracle($ltpk_i, a$ ): the challenger $\\mathcal {CH}$ runs $act_i \\leftarrow $ CreateAccount($a, ltpk_i$ ) if $ltpk_i$ was generated by a query to KeyOracle.",
"Returns the $act_i$ to $\\mathcal {A}$ .",
"$sk_i \\leftarrow $ CorruptOracle($act_i$ ): on input an account $act_i$ that corresponds to a query to AccountOracle, it runs $sk_i \\leftarrow $ ISE.KeyGen.OneTimeSK($pk_i, {\\fontfamily {cmss}\\selectfont {aux}}, ltsk_i$ ) and returns the associated account secret key $sk_i$ .",
"DRingCTx $\\leftarrow $ TransOracle($act_n$ , $act_m$ , $a_t$ , $ltpk_t, m$ ): on input a set of $N$ input accounts, $M$ spend accounts, $T$ amounts, $T$ destination addresses and a transaction message $m$ , $\\mathcal {CH}$ runs DRingCTx $\\leftarrow $ CreateDRingCTx($act_n$ , $ask_m$ , $a_t$ , $ltpk_t, m)$ and returns DRingCTx to $\\mathcal {A}$ .",
"CRx $\\leftarrow $ ChangeOracle($act_i, ltpk_i, m$ ): on input an account $act_i$ , a $ltpk_i$ and a transaction message $m$ , it runs CRx $\\leftarrow $ CreateCRx ($ask_i, act_i, ltpk_i, m$ ) and returns CRx to $\\mathcal {A}$ .",
"Definition 2 (Unforgeability) The probability of an adversary being able to forge a valid delegated ring confidential transaction DRingCTx or a valid change representative transaction CRx without knowing any secret key of the public keys of the ring, is negligible.",
"This property is captured by a game between a challenger $\\mathcal {CH}$ and a probabilistic polynomial-time adversary $\\mathcal {A}$ , where $\\mathcal {A}$ can query all oracles and outputs: DRingCTx = ($act_n$ , $\\sigma \\textsubscript {ring}$ , aux, $acts_t, \\pi \\textsubscript {range}, \\pi \\textsubscript {enc}, \\pi \\textsubscript {bal}$ ), such that all of the $N$ input accounts were generated by queries to AccountOracle and none was used as input to CorruptOracle or TransOracle.",
"CRx = ($\\sigma , act^{\\prime }_{i}, \\pi \\textsubscript {equal}$ ), such that ChangeOracle was not queried with ($act_i, \\cdot $ ) and $act_i$ was not corrupted by CorruptOracle.",
"$\\mathcal {A}$ wins if Pr[VerifyDRingCTx(DRingCTx) = 1] $\\ge $ negl($\\lambda $ ) or if Pr[VerifyCRx(CRx) = 1] = $\\ge $ negl($\\lambda $ ).",
"Definition 3 (Anonymity) As long as a ring of a transaction contains two uncorrupted input accounts, an adversary can do no better than guessing at determining the sender of a valid transaction.",
"This property is captured by the following game between a challenger $\\mathcal {CH}$ and a probabilistic polynomial-time adversary $\\mathcal {A}$ : $\\mathcal {A}$ has access to all oracles.",
"He chooses a ring of input accounts $act_n$ , where all the accounts are generated by AccountOracle, and two indices $i_0, i_1$ , such that $act_{i_0}$ and $act_{i_1}$ were not corrupted by the CorruptOracle.",
"The challenger $\\mathcal {CH}$ picks the sender of the transaction by selecting a uniformly random bit $b \\in \\lbrace 0,1\\rbrace $ and outputs a DRingCTx.",
"$\\mathcal {A}$ tries to guess which account is the real sender with $b^{\\prime }$ and wins if $|\\text{Pr}[b^{\\prime } = b]| > \\frac{1}{2}$ .",
"Definition 4 (Linkability) An adversary is unable to produce $k + 1$ non-linked valid transactions on a combined anonymity set of $k$ input accounts.",
"This property is captured in a game between a challenger $\\mathcal {CH}$ and a probabilistic polynomial-time adversary $\\mathcal {A}$ , where $\\mathcal {A}$ has access to all oracles and, for $i \\in [0, k-1]$ , produces a delegated ring confidential transaction DRingCTx$i$.",
"He then produces another DRingCTx(DRingCTx and sends them all to the challenger.",
"$\\mathcal {A}$ wins if the following checks: $|K|$ = $k$ , where $K \\equiv \\cup _{i=0}^{k-1} R_i$ .",
"Each $act_i \\in K$ .",
"Each $R_i \\subset K$ .",
"VerifyDRingCTx(DRingCTx$i$) = 1 for all $i$ .",
"VerifyDRingCTx(DRingCTx) = 1.",
"Definition 5 (Non-frameability/Non-slanderability) An adversary is unable to generate a valid transaction that links with another previous valid transaction that was generated honestly.",
"This property is captured by the following game between a challenger $\\mathcal {CH}$ and a probabilistic polynomial-time adversary $\\mathcal {A}$ : $\\mathcal {A}$ has access to all oracles pre and pos-challenge.",
"In the challenge stage he chooses an uncorrupted account $act_{i^{*}}$ that was generated by a query to the AccountOracle, and a ring $act_n$ such that $act_{i^{*}} \\in act_n$ , and sends them to the challenger.",
"$\\mathcal {CH}$ responds with a DRingCTx.",
"In the forge stage $\\mathcal {A}$ produces another DRingCTx' and wins if DRingCTx' links with DRingCTx with non-negligible probability."
],
[
"A Generic Construction",
"We present a generic construction of Delegated RingCT from ISE and NIZK, in the following way: Let ISE = (Setup, KeyGen, Sign, Signring, Verify, Verifyring, Encrypt, Decrypt) be an ISE scheme whose PKE is additively homomorphic and used to encrypt the balance of an account.",
"The LRS component is used to authenticate a DRingCTx and the DS component to authenticate a CRx.",
"Let NIZKcorrect = (Setup, CRSGen, Prove, Verify) be a NIZK proof system for $L\\textsubscript {correct}$ .",
"It is used to construct a valid DRingCTx, is composed of: $L\\textsubscript {enc} = \\lbrace (pk, C) \\, | \\, \\exists \\, a, k \\, s.t.",
"\\, C = \\text{ISE.",
"{\\fontfamily {cmss}\\selectfont {Encrypt}}}(pk, a; k) \\rbrace $ $L\\textsubscript {range} = \\left\\lbrace C \\, | \\, \\exists \\, a \\, s.t.",
"\\, a \\in V\\right\\rbrace $ $L\\textsubscript {bal} = \\lbrace ({C}_m, {C}_t) \\, | \\, \\exists \\, a_m, a_t \\, s.t.",
"\\, \\sum _{m=0}^{M-1}{a_m} = \\sum _{t=0}^{T-1}{a_t}\\rbrace $ Let NIZKequal = (Setup, CRSGen, Prove, Verify) be a NIZK proof system for $L\\textsubscript {equal}$ .",
"It is used to make a valid CRx: $L\\textsubscript {equal} = \\lbrace (pk_1, pk_2, c_1, c_2) \\, | \\, \\exists \\, a_1, a_2 \\, s.t.",
"\\, a_1 = a_2 \\rbrace $ A Delegated RingCT contruction is composed of the following algorithms.",
"$pp \\leftarrow $ Setup($1^{\\lambda }$ ): on input a security parameter $1^{\\lambda }$ , it runs $pp\\textsubscript {ise} \\leftarrow $ ISE.Setup($1^\\lambda $ ), $pp\\textsubscript {nizk} \\leftarrow $ NIZK.Setup($1^\\lambda $ ), $crs \\leftarrow $ NIZK.CRSGen($pp\\textsubscript {nizk})$ , outputs $pp = (pp\\textsubscript {ise}, pp\\textsubscript {nizk}, crs)$ .",
"$(act$ , aux) $\\leftarrow $ CreateAccount$(a, ltpk)$ .",
"On input a one-time public key $ltpk$ and an amount $a$ , it runs $(pk$ , aux) $\\leftarrow $ ISE.KeyGen.OneTimePKGen$(ltpk; r)$ and computes the coin $C \\leftarrow $ ISE.Encrypt$(pk, a; k)$ .",
"It outputs the account $act = (pk, C)$ and the auxiliary information aux.",
"$a \\leftarrow $ RevealBalance$(ltsk, C)$ .",
"On input a ciphertext $C$ and a long time secret key $ltsk$ , it runs: - $a \\leftarrow $ ISE.Decrypt$(sk, C)$ .",
"DRingCTx = $(\\sigma \\textsubscript {ring}$ , aux, $act_t, \\pi \\textsubscript {range}, \\pi \\textsubscript {enc}, \\pi \\textsubscript {bal})$ $\\leftarrow $ CreateDRingCTx$(act_n, ask_m, a_t$ , $ltpk_t, m)$ : on input a ring of $N$ accounts, $M$ spend accounts, $T$ amounts, $T$ long term public keys and a transaction message $m$ , it creates a ring confidential transaction via the following steps: run $(act_t$ , auxt) $\\leftarrow $ CreateAccount($a_t, ltpk_t$ ) to generate the output accounts.",
"run $\\pi $correct $\\leftarrow $ NIKZcorrect.Prove for all output accounts to generate a zero-knowledge proof $\\pi $correct for $L\\textsubscript {correct}$ .",
"compute $\\sigma \\textsubscript {ring} \\leftarrow $ ISE.Signring$(pk_n, sk_m)$ .",
"output the delegated ring confidential transaction DRingCTx.",
"$1/0/-1 \\leftarrow $ VerifyDRingCTx(DRingCTx, $act_n$ )): on input a DRingCTx and the corresponding ring of input accounts, it outputs \"1\" if both the following algorithms output \"1\".",
"It outputs \"-1\" if ISE.Link outputs \"1\" and \"0\" otherwise.",
"ISE.Verifyring($\\sigma \\textsubscript {ring}, m, pk_n$ ): on input a LRS $\\sigma \\textsubscript {ring}$ , a message $m$ and $N$ input public keys, it outputs \"1\" if the signature is valid and \"0\" if it is invalid.",
"NIZKcorrect.Verify($crs, x, \\pi $correct): on input $crs$ , a statement $x$ and a proof $\\pi $correct, it outputs \"1\" if the proof is valid and \"0\" if it is invalid.",
"CRx = $(\\sigma , \\pi \\textsubscript {equal})/ \\perp $ $\\leftarrow $ CreateCRx($act, sk, ltpk, m$ ).",
"On input an account $act$ , the corresponding account secret key $ask$ , the long term public key $ltpk$ as the new representative and a transaction message $m$ , it outputs: - $\\sigma \\leftarrow $ ISE.Sign($sk)$ , which outputs a digital signature proving the knowledge of $sk$ .",
"- $\\pi \\textsubscript {equal} \\leftarrow $ NIZKequal.Prove, which outputs a proof of equivalence of ciphertexts.",
"- it returns $(\\sigma , \\pi \\textsubscript {equal})$ .",
"If $ask$ is invalid, it returns $\\perp $ .",
"$1/0 \\leftarrow $ VerifyCRx(CRx) $= (\\sigma , \\pi \\textsubscript {equal})).$ - check if ISE.Verify($\\sigma )=1$ and NIZKequal.Verify ($\\pi \\textsubscript {equal}) = 1$ .",
"- if both the above tests pass, return 1.",
"- else, return 0.",
"Correctness of our generic Delegated RingCT construction follows from the correctness of ISE and the completeness of all the NIZK used and security is captured by the following theorem and lemmas.",
"As defined in our security model, our Delegated RingCT is secure if it satisfies four properties: unforgeability, anonymity, linkability and non-frameability.",
"In order to prove that the scheme satisfies each one of these properties, we assume the security of ISE and the zero-knowledge of NIZK.",
"We then simulate a game in which an adversary $\\mathcal {A}$ tries to break the property in question.",
"This game can be simulated by another adversary $\\mathcal {B}$ , which acts as the challenger in $\\mathcal {A}$ 's game.",
"$\\mathcal {B}$ himself wants to break the security of ISE in its own game, so he only needs to make sure that $\\mathcal {A}$ 's response to the challenge can be used as an answer to $\\mathcal {B}$ 's challenge.",
"If $\\mathcal {A}$ has a non-negligible advantage in his game, then $\\mathcal {B}$ will have it as well.",
"However, we assumed that the ISE is secure, so, we prove by contradiction that $\\mathcal {A}$ cannot have a non-negligible advantage in his game and that the property of Delegated RingCT he wants to break is satisfied.",
"Theorem 1 Assuming the security of ISE and NIZK, the above Delegated RingCT construction is secure.",
"We prove this theorem via the following four lemmas.",
"Lemma 1.1 Assuming the security of the ISE and the zero-knowledge property of NIZK, our Delegated RingCT construction satisfies unforgeability.",
"We proceed via a sequence of games.",
"Game 1.1.1 The real experiment.",
"$\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below.",
"Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow $ ISE.Setup($1^{\\lambda }$ ), $pp\\textsubscript {nizk} \\leftarrow $ NIZK.Setup($1^{\\lambda }$ ), $crs \\leftarrow $ NIZK.CRSGen($pp\\textsubscript {nizk}$ ) and sends $pp = (pp\\textsubscript {ise}, pp\\textsubscript {nizk}, crs)$ to $\\mathcal {A}$ .",
"Queries: Throughout the experiment, $\\mathcal {A}$ can make queries to all oracles.",
"The challenger $\\mathcal {CH}$ answers these queries as defined in the security model.",
"Forge: $\\mathcal {A}$ outputs a ring confidential transaction DRingCTx, such that all the input accounts are generated by the AccountOracle and uncorrupted ($\\mathcal {A}$ does not know any of the accounts secret keys) and were not used as a query to TransOracle; and a change representative transaction CRx for an uncorrupted account that was generated by AccountOracle.",
"$\\mathcal {A}$ wins if any of the transactions is valid with non-negligible probability.",
"Game 1.1.2 Same as Game REF , except that $\\mathcal {CH}$ uses a simulator to generate $\\pi \\textsubscript {correct}$ for the TransOracle queries and $\\pi \\textsubscript {equal}$ for the ChangeOracle queries without knowing any of the accounts secret keys.",
"The zero-knowledge proofs are indistinguishable from the real ones, by definition.",
"By a direct reduction to the zero-knowledge property of the underlying NIZK, we have: $| \\text{Pr}[S_1] - \\text{Pr}[S_0]| \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 1.1.1 Assuming the unforgeability of the ISE’s linkable ring signature component, $\\text{Pr}[S_1] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ for all PPT adversary $\\mathcal {A}$ .",
"Suppose there exists a PPT adversary $\\mathcal {A}$ that has non-negligible advantage in Game REF .",
"We can build an adversary $\\mathcal {B}$ that breaks the security of the the ISE with the same advantage.",
"Given the public parameters $pp\\textsubscript {ise}$ by its challenger, $\\mathcal {B}$ simulates Game REF as follows: Setup: $\\mathcal {B}$ runs $(crs, \\tau ) \\leftarrow \\mathcal {S}(pp$nizk) and sends $(crs, \\tau , pp\\textsubscript {ise})$ to $\\mathcal {A}$ , where $\\mathcal {S}$ is a simulator.",
"Queries: $\\mathcal {A}$ can query all oracles and $\\mathcal {B}$ answers them in the following way: $ltpk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {B}$ computes the $ltpk_i$ according to the $pk_i$ given by its challenger.",
"$act_i \\leftarrow $ AccountOracle($ltpk_i, a$ ): $\\mathcal {B}$ queries the GenOracle of its own game to get $pk_i$ , runs $C \\leftarrow $ ISE.Encrypt$(pk_{i}, a; k)$ and responds with $act_{i} = (pk_{i}, C)$ .",
"$ltsk_i \\leftarrow $ CorruptOracle($act_i$ ): $\\mathcal {B}$ queries the CorruptOracle of its own game with $pk_i$ to get $sk_i$ , and returns $(sk_i, a, k)$ to $\\mathcal {A}$ .",
"DRingCTx $\\leftarrow $ TransOracle($act_n, act_m, a_t, ltpk_t, m$ ): $\\mathcal {B}$ queries the SignringOracle of its own game with ($pk_m, m, pk_n$ ) to get $\\sigma \\textsubscript {ring}$ and runs the simulator to get $\\pi \\textsubscript {correct}$ .",
"It returns DRingCTx = ($act_n, \\sigma \\textsubscript {ring}$ , aux, $act_t, \\pi \\textsubscript {correct}$ ) to $\\mathcal {A}$ .",
"CRx $\\leftarrow $ ChangeOracle($act_i, ltpk_i, m$ ): $\\mathcal {B}$ queries the SignOracle of its own game with ($pk_i, m$ ) to get $\\sigma $ and runs the simulator to get $\\pi \\textsubscript {equal}$ .",
"It returns CRx = ($\\sigma , act^{\\prime }, \\pi \\textsubscript {equal}$ ) to $\\mathcal {A}$ .",
"Forge: $\\mathcal {A}$ submits a delegated ring confidential transaction DRingCTx and a change representative CRx.",
"$\\mathcal {B}$ forwards the linkable ring signature and the digital signature to its own challenger and breaks the ISE security with the same advantage as $\\mathcal {A}$ (if any of the two is valid).",
"$\\mathcal {B}$ ’s simulation of Game REF is perfect.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma 1.2 Assuming the anonymity property of ISE’s linkable ring signature component, the zero-knowledge property of NIZK, our Delegated RingCT construction satisfies anonymity.",
"We proceed via a sequence of games.",
"Game 1.2.1 The real experiment.",
"$\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below.",
"Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow $ ISE.Setup($1^{\\lambda }$ ), $pp\\textsubscript {nizk} \\leftarrow $ NIZK.Setup($1^{\\lambda }$ ), $crs \\leftarrow $ NIZK.CRSGen($pp\\textsubscript {nizk}$ ) and sends $pp = (pp\\textsubscript {ise}, pp\\textsubscript {nizk}, crs)$ to $\\mathcal {A}$ .",
"Pre-challenge queries: Throughout the experiment, $\\mathcal {A}$ can query all oracles.",
"$\\mathcal {B}$ answers them as defined in the security model.",
"Challenge: $\\mathcal {A}$ picks a ring of input accounts $act_n$ , chooses two indices $i_0$ , $i_1$ and sends it to $\\mathcal {CH}$ , such that $act_{i_0}$ and $act_{i_1}$ are uncorrupted.",
"$\\mathcal {CH}$ chooses one of the two accounts to be the sender by selecting a uniformly random bit $b \\in \\lbrace 0,1\\rbrace $ , runs CreateDRingCTx($act_n, act_{i_b}, a_t, ltpk_t$ ) and sends DRingCTx to $\\mathcal {A}$ .",
"Post-challenge queries: $\\mathcal {A}$ can query all oracles in the same way as in the pre-challenge stage, except querying the CorruptOracle with $act_{i_b}$ .",
"Guess: $\\mathcal {A}$ chooses a bit $b^{\\prime } \\in \\lbrace 0,1\\rbrace $ and wins if: $|$ Pr[$b^{\\prime } = b]| \\ge \\frac{1}{2}$ .",
"Game 1.2.2 Same as Game REF , except $\\mathcal {CH}$ makes a ring guess of random $N$ indexes out of $Q$ , the maximum number of queries that $\\mathcal {A}$ can make to the AccountOracle, and a guess of the two indices $i_0, i_1$ .",
"If $\\mathcal {A}$ picks a different ring for the DRingCTx or different spend accounts, $\\mathcal {CH}$ aborts.",
"Let $E$ be the event that $\\mathcal {CH}$ does not abort and $S_0$ the event that $\\mathcal {A}$ wins in Game REF .",
"The probability that $E$ occurs is $\\text{Pr}[E] \\ge \\frac{N!(Q-N)!",
"}{Q} \\cdot \\frac{1}{N^2}$ , where $N$ is the size of the ring of the DRingCTx.",
"Conditioned on $\\mathcal {CH}$ does not abort, $\\mathcal {A}$ ’s view in Game REF is identical to that in Game REF .",
"Therefore, we have: $\\text{Pr}[S_1] \\ge \\text{Pr}[S_0] \\cdot \\frac{N!(Q-N)!}{Q!}",
"\\cdot \\frac{1}{N^2}$ Game 1.2.3 Same as Game REF , except that $\\mathcal {CH}$ generates the necessary zero-knowledge proofs for the TransOracle queries without knowing the secret keys via running the simulator.",
"By a direct reduction to the zero-knowledge property of the underlying NIZK, we have: $| \\text{Pr}[S_2] - \\text{Pr}[S_1]| \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 1.2.1 Assuming the anonymity property of the ISE’s linkable ring signature component, $\\text{Pr}[S_2] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ for all PPT adversary $\\mathcal {A}$ .",
"Suppose there exists a PPT adversary $\\mathcal {A}$ has non-negligible advantage in Game REF , we can build an adversary $\\mathcal {B}$ that breaks the anonymity property of the ISE’s linkable ring signature component with the same advantage.",
"Given ($pp\\textsubscript {ise}, LRS_{i_b}$ ) by its challenger, $\\mathcal {B}$ simulates Game REF as follows: Setup: $\\mathcal {B}$ runs $(crs, \\tau ) \\leftarrow \\mathcal {S}(pp$nizk) and sends $(crs, \\tau , pp\\textsubscript {ise})$ to $\\mathcal {A}$ , where $\\mathcal {S}$ is a simulator.",
"$\\mathcal {B}$ randomly picks $N$ indices $\\in [Q]$ and two indices $j_0, j_1 \\in [N]$ .",
"Pre-challenge queries: Throughout the experiment, $\\mathcal {A}$ can query all oracles.",
"$\\mathcal {B}$ answers them as defined in the security model, except for: $ltpk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {B}$ computes the $ltpk_i$ according to the $pk_i$ given by its challenger.",
"$act_i \\leftarrow $ AccountOracle($ltpk_i, a$ ): $\\mathcal {B}$ queries the GenOracle of its own game to get $pk_i$ , runs $C \\leftarrow $ ISE.Encrypt$(pk_{i}, a; k)$ and responds with $act_{i} = (pk_{i}, C)$ .",
"$ltsk_i \\leftarrow $ CorruptOracle($act_i$ ): $\\mathcal {B}$ queries the CorruptOracle with $pk_i$ and returns $(sk_i, a, k)$ to $\\mathcal {A}$ .",
"DRingCTx $\\leftarrow $ TransOracle($act_n, act_m, a_t, ltpk_t, m$ ): $\\mathcal {B}$ queries the SignringOracle of its own game with ($pk_m, m, pk_n$ ) to get $\\sigma \\textsubscript {ring}$ and runs the simulator to get $\\pi \\textsubscript {correct}$ .",
"It returns DRingCTx = ($act_n, \\sigma \\textsubscript {ring}, {\\fontfamily {cmss}\\selectfont {aux}}, act_t, \\pi \\textsubscript {correct}$ ) to $\\mathcal {A}$ .",
"CRx $\\leftarrow $ ChangeOracle($act_i, ltpk_i, m$ ): $\\mathcal {B}$ queries the SignOracle of its own game with ($pk_i, m$ ) to get $\\sigma $ and runs the simulator to get $\\pi \\textsubscript {equal}$ .",
"It returns CRx = ($\\sigma , act^{\\prime }, \\pi \\textsubscript {equal}$ ) to $\\mathcal {A}$ .",
"Challenge: $\\mathcal {A}$ picks a ring of input accounts $\\lbrace act_n\\rbrace _{n=0}^{N-1}$ and two of those accounts.",
"If any of the ring accounts contains a different public key than the ones chosen by $\\mathcal {B}$ or any of the two is corrupted or have been used as spend account in queries to TransOracle, $\\mathcal {B}$ aborts.",
"Otherwise, $\\mathcal {B}$ uses the $LRS_{i_b}$ given by its challenger to construct the delegated ring confidential transaction DRingCTx and sends it to $\\mathcal {A}$ .",
"Post-challenge queries: $\\mathcal {A}$ can query all oracles in the same as in the pre-challenge stage, except querying the CorruptOracle with $act_{i_b}$ .",
"Guess: $\\mathcal {A}$ guesses which account is the real sender of DRingCTx with $b^{\\prime } \\in \\lbrace {0,1}\\rbrace $ and $\\mathcal {B}$ forwards the guess to its own challenger to try to guess which $pk$ is the real signer of the linkable ring signature.",
"$\\mathcal {B}$ wins with the same advantage as $\\mathcal {A}$ .",
"$\\mathcal {B}$ ’s simulation for Game REF is perfect.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma 1.3 Assuming the linkability property of ISE’s linkable ring signature component and the zero-knowledge property of NIZK, our Delegated RingCT construction satisfies linkability.",
"We prove via a sequence of games.",
"Game 1.3.1 The real experiment.",
"$\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below.",
"Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow $ ISE.Setup($1^{\\lambda }$ ), $pp\\textsubscript {nizk} \\leftarrow $ NIZK.Setup($1^{\\lambda }$ ), $crs \\leftarrow $ NIZK.CRSGen($pp\\textsubscript {nizk}$ ) and sends $pp = (pp\\textsubscript {ise}, pp\\textsubscript {nizk}, crs)$ to $\\mathcal {A}$ .",
"Queries: Throughout the experiment, $\\mathcal {A}$ can query all oracles and $\\mathcal {CH}$ answers as defined in the security model.",
"Forge: $\\mathcal {A}$ produces, for $i \\in [0, k-1]$ , a delegated ring confidential transaction DRingCTx$i$.",
"He then produces another DRingCTx and sends them all to the challenger.",
"$\\mathcal {A}$ wins if the following checks: $|K|$ = $k$ , where $K \\equiv \\cup _{i=0}^{k-1} R_i$ .",
"Each $act_i \\in K$ .",
"Each $R_i \\subset K$ .",
"VerifyDRingCTx(DRingCTx)$i$) = 1 for all $i$ .",
"VerifyDRingCTx(DRingCTx) = 1.",
"Game 1.3.2 Same as Game REF , except that $\\mathcal {CH}$ generates the necessary zero-knowledge proofs for the TransOracle queries without knowing the secret keys via running the simulator.",
"By a direct reduction to the zero-knowledge property of the underlying NIZK, we have: $| \\text{Pr}[S_1] - \\text{Pr}[S_0]| \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 1.3.1 Assuming the linkability property of the ISE’s linkable ring signature component, $\\text{Pr}[S_1] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ for all PPT adversary $\\mathcal {A}$ .",
"Suppose there exists a PPT adversary $\\mathcal {A}$ has non-negligible advantage in Game REF , we can build an adversary $\\mathcal {B}$ that breaks the linkability property of the ISE’s linkable ring signature component with the same advantage.",
"Given $pp\\textsubscript {ise}$ , $\\mathcal {B}$ simulates Game REF as follows: Setup: $\\mathcal {B}$ runs $(crs, \\tau ) \\leftarrow \\mathcal {S}(pp$nizk) and sends $(crs, \\tau , pp\\textsubscript {ise})$ to $\\mathcal {A}$ , where $\\mathcal {S}$ is a simulator.",
"Queries: Throughout the experiment, $\\mathcal {A}$ can query all oracles.",
"$\\mathcal {B}$ answers them in the same way as in the security model, except for: $ltpk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {B}$ computes the $ltpk_i$ according to the $pk_i$ given by its challenger.",
"$act_i \\leftarrow $ AccountOracle($ltpk_i, a$ ): $\\mathcal {B}$ queries the GenOracle of its own game to get $pk_i$ , runs $C \\leftarrow $ ISE.Encrypt$(pk_{i}, a; k)$ and responds with $act_{i} = (pk_{i}, C)$ .",
"$ltsk_i \\leftarrow $ CorruptOracle($act_i$ ): $\\mathcal {B}$ queries the CorruptOracle with $pk_i$ and returns $(sk_i, a, k)$ to $\\mathcal {A}$ .",
"DRingCTx $\\leftarrow $ TransOracle($act_n, act_m, a_t, ltpk_t, m$ ): $\\mathcal {B}$ queries the SignringOracle of its own game with ($pk_m, m, pk_n$ ) to get $\\sigma \\textsubscript {ring}$ and runs the simulator to get $\\pi \\textsubscript {correct}$ .",
"It returns DRingCTx = ($act_n, \\sigma \\textsubscript {ring}, {\\fontfamily {cmss}\\selectfont {aux}}, act_t, \\pi \\textsubscript {correct}$ ) to $\\mathcal {A}$ .",
"CRx $\\leftarrow $ ChangeOracle($act_i, ltpk_i, m$ ): $\\mathcal {B}$ queries the SignOracle of its own game with ($pk_i, m$ ) to get $\\sigma $ and runs the simulator to get $\\pi \\textsubscript {equal}$ .",
"It returns CRx = ($\\sigma , act^{\\prime }, \\pi \\textsubscript {equal}$ ) to $\\mathcal {A}$ .",
"Forge: $\\mathcal {A}$ submits $k+1$ ring confidential transaction DRingCTx.",
"$\\mathcal {B}$ forwards linkable ring signatures of the transactions to its own challenger and wins with the same advantage as $\\mathcal {A}$ .",
"$\\mathcal {B}$ ’s simulation for Game REF is perfect.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma 1.4 Assuming the non-frameability property of ISE’s linkable ring signature component and the adaptive zero-knowledge property of NIZK, our Delegated RingCT construction satisfies non-frameability.",
"We prove via a sequence of games.",
"Game 1.4.1 The real experiment.",
"$\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below.",
"Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow $ ISE.Setup($1^{\\lambda }$ ), $pp\\textsubscript {nizk} \\leftarrow $ NIZK.Setup($1^{\\lambda }$ ), $crs \\leftarrow $ NIZK.CRSGen($pp\\textsubscript {nizk}$ ) and sends $pp = (pp\\textsubscript {ise}, pp\\textsubscript {nizk}, crs)$ to $\\mathcal {A}$ .",
"Pre-challenge queries: Throughout the experiment, $\\mathcal {A}$ can query all oracles and $\\mathcal {CH}$ answers these queries as described in the security model.",
"Challenge: $\\mathcal {A}$ chooses a ring of accounts $act_n$ and an uncorrupted account $act_{i^{*}}$ , from the ring and generated by the AccountOracle, to be the spend account.",
"$\\mathcal {CH}$ runs DRingCTx $\\leftarrow $ CreateDRingCTx($act_n, act_{i^{*}}, a_t, ltpk_t$ ) and returns DRingCTx to $\\mathcal {A}$ .",
"Post-challenge queries: The same as the pre-challenge queries.",
"Forge: $\\mathcal {A}$ outputs a delegated ring confidential transaction DRingCTx that was not queried by the TransOracle and wins if the transaction is linked with the previous valid transaction produced by $\\mathcal {CH}$ in the challenge stage with non-negligible probability.",
"Game 1.4.2 Same as Game REF , except $\\mathcal {CH}$ makes a ring guess of random $N$ indexes out of $Q$ , the maximum number of queries that $\\mathcal {A}$ can make to the AccountOracle, and a guess of the index of the spend account.",
"If $\\mathcal {A}$ picks different indices, $\\mathcal {CH}$ aborts.",
"Let $E$ be the event that $\\mathcal {CH}$ does not abort and $S_0$ the event that $\\mathcal {A}$ wins in Game REF .",
"The probability that $E$ occurs is $\\text{Pr}[E] \\ge \\frac{N!(Q-N)!",
"}{Q} \\cdot \\frac{1}{N}$ , where $N$ is the size of the ring of the DRingCTx.",
"Conditioned on $\\mathcal {CH}$ does not abort, $\\mathcal {A}$ ’s view in Game REF is identical to that in Game REF .",
"Therefore, we have: $\\text{Pr}[S_1] \\ge \\text{Pr}[S_0] \\cdot \\frac{N!(Q-N)!}{Q!}",
"\\cdot \\frac{1}{N}$ Game 1.4.3 Same as Game REF , except that $\\mathcal {CH}$ generates the necessary zero-knowledge proofs for the TransOracle queries without knowing the secret keys via running the simulator.",
"By a direct reduction to the zero-knowledge property of the underlying NIZK, we have: $| \\text{Pr}[S_2] - \\text{Pr}[S_1]| \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 1.4.1 Assuming the non-frameability property of the ISE’s linkable ring signature component, $\\text{Pr}[S_2] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ for all PPT adversary $\\mathcal {A}$ .",
"Suppose there exists a PPT adversary $\\mathcal {A}$ has non-negligible advantage in Game REF , we can build an adversary $\\mathcal {B}$ that breaks the non-frameability property of the ISE’s linkable ring signature component with the same advantage.",
"Given the ($pp\\textsubscript {ise}, \\sigma \\textsubscript {ring}$ ) by its challenger, $\\mathcal {B}$ simulates Game REF as follows: Setup: $\\mathcal {B}$ runs $(crs, \\tau ) \\leftarrow \\mathcal {S}(pp$nizk) and sends $(crs, \\tau , pp\\textsubscript {ise})$ to $\\mathcal {A}$ , where $\\mathcal {S}$ is a simulator.",
"Pre-challenge queries: Throughout the experiment, $\\mathcal {A}$ can query all oracles.",
"$\\mathcal {B}$ answers them in the following way: $ltpk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {B}$ computes the $ltpk_i$ according to the $pk_i$ given by its challenger.",
"$act_i \\leftarrow $ AccountOracle($ltpk_i, a$ ): $\\mathcal {B}$ queries the GenOracle of its own game to get $pk_i$ , runs $C \\leftarrow $ ISE.Encrypt$(pk_{i}, a; k)$ and responds with $act_{i} = (pk_{i}, C)$ .",
"$ltsk_i \\leftarrow $ CorruptOracle($act_i$ ): $\\mathcal {B}$ queries the CorruptOracle with $pk_i$ and returns $(sk_i, a, k)$ to $\\mathcal {A}$ .",
"DRingCTx $\\leftarrow $ TransOracle($act_n, act_m, a_t, ltpk_t, m$ ): $\\mathcal {B}$ queries the SignringOracle of its own game with ($pk_m, m, pk_n$ ) to get $\\sigma \\textsubscript {ring}$ and runs the simulator to get $\\pi \\textsubscript {correct}$ .",
"It returns DRingCTx = ($act_n, \\sigma \\textsubscript {ring}, {\\fontfamily {cmss}\\selectfont {aux}}, act_t, \\pi \\textsubscript {correct}$ ) to $\\mathcal {A}$ .",
"CRx $\\leftarrow $ ChangeOracle($act_i, ltpk_i, m$ ): $\\mathcal {B}$ queries the SignOracle of its own game with ($pk_i, m$ ) to get $\\sigma $ and runs the simulator to get $\\pi \\textsubscript {equal}$ .",
"It returns CRx = ($\\sigma , act^{\\prime }, \\pi \\textsubscript {equal}$ ) to $\\mathcal {A}$ .",
"Challenge: $\\mathcal {A}$ chooses an uncorrupted account $act_{i^{*}}$ that was generated by a query to the AccountOracle, and a ring $act_n$ such that $act_{i^{*}} \\in act_n$ , and sends them to the challenger.",
"$\\mathcal {CH}$ responds with a DRingCTx constructed from the $\\sigma \\textsubscript {ring}$ of its own challenge and the simulated zero-knowledge proof $\\pi \\textsubscript {correct}$ .",
"Pre-challenge queries: The same as pre-challenge queries.",
"Forge: $\\mathcal {A}$ submits a ring confidential transaction DRingCTx' that was not produced by the TransOracle.",
"$\\mathcal {B}$ forwards the the linkable ring signature of the transaction to its own challenger and wins with the same advantage as $\\mathcal {A}$ .",
"$\\mathcal {B}$ ’s simulation of Game REF is perfect.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma REF , Lemma REF , Lemma REF and Lemma REF prove Theorem REF ."
],
[
"A Concrete Instantiation",
"We use the following instantiations for ISE and NIZK components of Delegated RingCT: Setup: On input security parameter $1^\\lambda $ , it outputs public parameters $pp$ .",
"The cyclic group $\\mathbb {G}$ used is the elliptic curve Curve25519 [31] and the corresponding scalar field $\\mathbb {F}$ is $Z_p$ , where $p = 2^{255} - 19$ .",
"The hash function used is SHA-3 and $v\\textsubscript {max} = 2^{32}$ .",
"ISE.Keygen: we use the instantiation of [15].",
"- LongTermPKGen: The user picks his long term secret key $ltsk = (x_1, x_2) \\in Z^2_p$ and computes his long term public key $ltpk = (x_1G, x_2G)$ .",
"- OneTimePKGen: On input a long term public key $ltpk = (x_1G, x_2G)$ , it picks a random $r \\in Z_p$ and computes a one-time public key $pk = x_1G \\cdot \\mathcal {H}(x_2rG)G$ .",
"It outputs $pk$ and the auxiliary information $R = rG$ .",
"- OneTimeSKGen: On input a one-time public key pk, an auxiliary information $R$ and a long term secret key $ltsk = (x_1, x_2)$ , it checks if $pk = x_1G \\cdot \\mathcal {H}(x_2R)G$ .",
"If it is correct, then it outputs the one-time secret key $sk = x_1 + \\mathcal {H}(x_2R)$ .",
"ISE.PKE: Twisted ElGamal [28].",
"ISE.DS: Schnorr [32].",
"ISE.LRS: Triptych [18].",
"NIZK for $L\\textsubscript {enc}$ : $\\pi \\textsubscript {enc}$ of [28].",
"NIZK for $L\\textsubscript {bal}$ : Balance proof from [15].",
"NIZK for $L\\textsubscript {range}$ : Bulletproofs [19].",
"NIZK for $L\\textsubscript {equal}$ : $\\pi \\textsubscript {equal}$ of [28]."
],
[
"Analysis",
"Correctness follows from the correctness of ISE components instantiations and the completeness of NIZK's intantiations.",
"Regarding security, we want to prove that each component of the ISE instantiation is secure in the presence of one or two signing oracles (the PKE oracle can be easily simulated by the adversary in the public setting).",
"First, we assume the standalone security of that component.",
"Then, we construct a challenger $\\mathcal {CH}$ that is able to simulate the signing oracle(s) without the knowledge of the secret keys and an adversary $\\mathcal {A}$ that can query the oracle(s) and tries to break the security of the ISE.",
"Then, we construct another adversary $\\mathcal {B}$ that acts as the challenger $\\mathcal {CH}$ in $\\mathcal {A}$ 's game and wants to break the standalone security of one of the components of the ISE in his own game.",
"$\\mathcal {B}$ can use the attempt of $\\mathcal {A}$ for his own attempt, and, so, he will have the same advantage as $A$ .",
"If $\\mathcal {A}$ has a non-negligible advantage in winning his game, $\\mathcal {B}$ will have too.",
"However, we assume that the component in question is secure, thus, we prove by contradiction, that the game $\\mathcal {A}$ is trying to break is also secure, which is indistinguishable from the real experiment.",
"Theorem 2 The obtained ISE scheme is jointly secure if the twisted ElGamal is IND-CPA secure (1-plaintext/2-recipient), the Schnorr signature is EUF-CMA secure and the Triptych LRS is secure.",
"We prove this theorem via the following three lemmas.",
"Lemma 2.1 The PKE component is IND-CPA secure in the presence of two signing oracles, one for the Schnorr signature and the other for the Triptych LRS.",
"We prove via a sequence of games.",
"Game 2.1.1 The real security experiment for ISE’s PKE component.",
"Challenger $\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below: Setup: $\\mathcal {CH}$ runs $pp \\leftarrow $ ISE.Setup and sends $pp$ to $\\mathcal {A}$ .",
"Queries: the adversary $\\mathcal {A}$ can make queries to the following oracles: $pk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {CH}$ runs $(ltpk_i, ltsk_i) \\leftarrow $ LongTermKeyGen($pp)$ and $(pk_i$ , aux$i$) $\\leftarrow $ OneTimePKGen($ltpk_i; r$ ) and returns $pk_i$ to $\\mathcal {A}$ .",
"$\\xi \\leftarrow $ HashOracle(data): $\\mathcal {CH}$ emulates a random oracle by using the lazy sampling technique.",
"He maintains an initially empty list $L\\textsubscript {hash}$ , an on a given query with some data, if there is an entry (data, $\\xi $ ) in the list, $\\mathcal {CH}$ returns $\\xi $ .",
"Else, $\\mathcal {CH}$ picks $\\xi \\overset{r}{\\leftarrow } Z_p$ and inserts (data, $\\xi $ ) in $L\\textsubscript {hash}$ , then returns $\\xi $ .",
"$\\sigma \\leftarrow $ SignOracle($pk_{i}, m$ ): on input a public key $pk_{i}$ and a message $m$ , $\\mathcal {CH}$ runs ISE.Sign($sk_i$ , $m$ ) $\\rightarrow \\sigma $ and returns $\\sigma $ to $\\mathcal {A}$ .",
"$\\sigma \\textsubscript {ring} \\leftarrow $ SignringOracle($pk_{i}, m, R$ ): on input a public key $pk_{i}$ , a message $m$ and a ring $R$ of public keys, $\\mathcal {CH}$ runs ISE.Signring($sk_{i}$ , $m$ , $R) \\leftarrow \\sigma \\textsubscript {ring}$ and returns $\\sigma \\textsubscript {ring}$ to $\\mathcal {A}$ .",
"Challenge: $\\mathcal {A}$ submits two public keys $pk_1$ and $pk_2$ generated by the KeyOracle and two messages $m_1$ and $m_2$ .",
"$\\mathcal {CH}$ picks a random bit $\\beta $ and randomness $r$ , computes $X_1 = pk_1^r, X_2 = pk_2^r, Y = g^rh^{m_{\\beta }}$ and sends $C = (X_1, X_2, Y)$ to $\\mathcal {A}$ .",
"Guess: $\\mathcal {A}$ outputs its guess $\\beta ^{\\prime }$ for $\\beta $ and wins if $\\beta ^{\\prime } = \\beta $ .",
"According to the definition of Game REF , we have: $\\text{{\\fontfamily {cmss}\\selectfont {Adv}}}\\textsubscript {\\mathcal {A}}(\\lambda ) = \\text{Pr}[S_0] - 1/2$ Game 2.1.2 Same as Game REF , but $\\mathcal {CH}$ picks two indices from $[Q]$ , being $Q$ the maximum queries $\\mathcal {A}$ can make to the KeyOracle.",
"If $\\mathcal {A}$ picks a public key with a different index, $\\mathcal {CH}$ aborts.",
"Let $E$ be the event that $\\mathcal {CH}$ does not abort and $S_1$ the event that $\\mathcal {A}$ wins in Game REF .",
"Conditioned on $\\mathcal {CH}$ does not abort, $\\mathcal {A}$ ’s view in Game REF is identical to that in Game REF .",
"Therefore, we have: $\\text{Pr}[S_1] \\ge \\text{Pr}[S_0] \\cdot (\\frac{2!(Q-2)!}{Q!",
"})$ Game 2.1.3 The same as Game REF except that $\\mathcal {CH}$ simulates the signing oracles by programming a random oracle $\\mathcal {H}$ , rather than using the real secret keys, in the following way.",
"Queries: the adversary $\\mathcal {A}$ can make queries to the following oracles: $pk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {CH}$ picks random elements $J_i = j_iG$ (such that $j_i$ is known), $\\xi _i$ , $\\lbrace P_j\\rbrace _{j=0}^{m-1}$ and $z_i$ .",
"He computes $pk_i = (z_iG + \\sum _{j=0}^{m-1}P_j\\xi ^j) \\cdot \\xi _i^{-m}$ and $U = j \\cdot pk_i$ .",
"Finally, he sends $pk_i$ to $\\mathcal {A}$ .",
"$\\xi \\leftarrow $ HashOracle(data): $\\mathcal {CH}$ emulates a random oracle by using the lazy sampling technique.",
"He maintains an initially empty list $L$hash, an on a given query with some data, if there is an entry (data, $\\xi $ ) in $L$hash, $\\mathcal {CH}$ returns $\\xi $ .",
"Else, $\\mathcal {CH}$ picks $\\xi \\overset{r}{\\leftarrow } Z_p$ and inserts (data, $\\xi $ ) in $L$hash, then returns $\\xi $ .",
"$\\sigma \\leftarrow $ SignOracle($pk_i, m$ ): on input a public key $pk_i$ and a message $m$ , $\\mathcal {CH}$ picks $z, \\xi \\overset{r}{\\leftarrow } Z_p$ , sets $A = zG - \\xi pk_{i})$ and returns $\\sigma = (A, z)$ to $\\mathcal {A}$ .",
"Else, $\\mathcal {CH}$ aborts to avoid possible inconsistency in programming.",
"$\\sigma \\textsubscript {ring} \\leftarrow $ SignringOracle($pk_i, m, R$ ): on input a public key $pk_i$ , a message $m$ and a ring $R$ of public keys, $\\mathcal {CH}$ runs returns $\\sigma \\textsubscript {ring} = (a_i, z_i, J_i)$ to $\\mathcal {A}$ , where $a_i$ is computed from the other elements.",
"Else, $\\mathcal {CH}$ aborts to avoid possible inconsistency in programming.",
"Denote the event that $\\mathcal {CH}$ aborts in Game REF by $E$ .",
"Conditioned on $E$ does not occur, $\\mathcal {A}$ ’s view in Game REF and Game REF are identical.",
"This follows from the fact that $\\mathcal {CH}$ perfectly mimics the hash oracle and signing oracle.",
"Let Qhash, Qsign and Qsignring be the maximum number of hash queries and signing queries that $\\mathcal {A}$ makes during security experiment.",
"By the union bound, we conclude that $\\text{Pr}[E] \\le (Q\\textsubscript {hash}Q\\textsubscript {sign}Q\\textsubscript {sign\\textsubscript {ring}})/p$ , which is negligible in $\\lambda $ .",
"In summary, we have: $|\\text{Pr}[S_2] - \\text{Pr}[S_1]| \\le \\text{Pr}[E] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 2.1.1 Assuming the IND-CPA security (1-plaintext/2-recipient) of twisted ElGamal PKE, $\\text{Pr}[S_1]$ is negligible in $\\lambda $ for any PPT adversary $\\mathcal {A}$ .",
"We prove this claim by showing that if there exists a PPT adversary $\\mathcal {A}$ has non-negligible advantage in Game REF , we can build a PPT adversary $\\mathcal {B}$ that breaks the IND-CPA security (single-message, two-recipient) of twisted ElGamal PKE with the same advantage, since $\\mathcal {CH}$ can simulate the signing oracles without using the secret keys.",
"$\\mathcal {B}$ forwards the guess of $\\mathcal {A}$ to its own challenger and wins with the same advantage.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma 2.2 The DS component is EUF-CMA secure in the presence of a signing oracle for the Triptych LRS.",
"We prove via a sequence of games.",
"Game 2.2.1 The real security experiment for ISE’s DS component.",
"The challenger $\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below: Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow \\text{\\fontfamily {cmss}\\selectfont {Setup}}(1^{\\lambda })$ and sends it to $\\mathcal {A}$ .",
"Queries: the adversary $\\mathcal {A}$ can make queries to the following oracles: $pk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {CH}$ runs $(ltpk_i, ltsk_i) \\leftarrow $ LongTermKeyGen($pp)$ and $(pk_i$ , aux$i$) $\\leftarrow $ OneTimePKGen($ltpk_i; r$ ) and returns $pk_i$ to $\\mathcal {A}$ .",
"$\\xi \\leftarrow $ HashOracle(data): $\\mathcal {CH}$ emulates a random oracle by using the lazy sampling technique.",
"He maintains an initially empty list $L$hash, and on a given query with some data, if there is an entry (data, $\\xi $ ) in $L$hash, $\\mathcal {CH}$ returns $\\xi $ .",
"Else, $\\mathcal {CH}$ picks $\\xi \\overset{r}{\\leftarrow } Z_p$ and inserts (data, $\\xi $ ) in $L$hash, then returns $\\xi $ .",
"$\\sigma \\textsubscript {ring} \\leftarrow $ SignringOracle($pk_i, m, R$ ): on input a public key $pk_i$ , a message $m$ and a ring $R$ of public keys, $\\mathcal {CH}$ runs ISE.Signring($sk_i$ , $m$ , $R) \\leftarrow \\sigma \\textsubscript {ring}$ and returns $\\sigma \\textsubscript {ring}$ to $\\mathcal {A}$ .",
"Forge: $\\mathcal {A}$ outputs a signature $\\sigma $ and wins if Verify($pk, m, \\sigma ) = 1$ .",
"Game 2.2.2 The same as Game REF except that $\\mathcal {CH}$ simulates signing oracle by programming a random oracle $\\mathcal {H}$ , rather than using the real secret keys, in the following way.",
"Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow \\text{\\fontfamily {cmss}\\selectfont {Setup}}(1^{\\lambda })$ and sends it to $\\mathcal {A}$ .",
"Queries: the adversary $\\mathcal {A}$ can make queries to the following oracles: $pk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {CH}$ picks random elements $J_i = j_iG$ (such that $j_i$ is known), $\\xi _i$ , $\\lbrace P_j\\rbrace _{j=0}^{m-1}$ and $z_i$ .",
"He computes $pk_i = (z_iG + \\sum _{j=0}^{m-1}P_j\\xi ^j) \\cdot \\xi _i^{-m}$ and $U = j \\cdot pk_i$ .",
"Finally, he sends $pk_i$ to $\\mathcal {A}$ .",
"$\\xi \\leftarrow $ HashOracle(data): $\\mathcal {CH}$ emulates a random oracle by using the lazy sampling technique.",
"He maintains an initially empty list $L$hash, an on a given query with some data, if there is an entry (data, $\\xi $ ) in $L$hash, $\\mathcal {CH}$ returns $\\xi $ .",
"Else, $\\mathcal {CH}$ picks $\\xi \\overset{r}{\\leftarrow } Z_p$ and inserts (data, $\\xi $ ) in $L$hash, then returns $\\xi $ .",
"$\\sigma \\textsubscript {ring} \\leftarrow $ SignringOracle($pk_i, m, R$ ): on input a public key $pk_i$ , a message $m$ and a ring $R$ of public keys, $\\mathcal {CH}$ runs returns $\\sigma \\textsubscript {ring} = (a_i, z_i, J_i)$ to $\\mathcal {A}$ , where $a_i$ is computed from the other elements.",
"Else, $\\mathcal {CH}$ aborts to avoid possible inconsistency in programming.",
"Denote the event that $\\mathcal {CH}$ aborts in Game REF by $E$ .",
"Conditioned on $E$ does not occur, $\\mathcal {A}$ ’s view in Game REF and Game REF are identical.",
"This follows from the fact that $\\mathcal {CH}$ perfectly mimics the hash oracle and signing oracle.",
"Let $Q$hash and $Q$signring ring be the maximum number of hash queries and signing queries that $\\mathcal {A}$ makes during security experiment.",
"By the union bound, we conclude that $\\text{Pr}[E] \\le ($ Q$\\textsubscript {hash}$ Q$\\textsubscript {sign\\textsubscript {ring}})/p$ , which is negligible in $\\lambda $ .",
"In summary, we have: $| \\text{Pr}[S_1] - \\text{Pr}[S_0]| \\le \\text{Pr}[E] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 2.2.1 Assuming the EUF-CMA security of Schnorr digital signature, $\\text{Pr}[S_1]$ is negligible in $\\lambda $ for any PPT adversary $\\mathcal {A}$ .",
"We prove this claim by showing that if there exists a PPT adversary $\\mathcal {A}$ has non-negligible advantage in Game REF , we can build a PPT adversary $\\mathcal {B}$ that breaks the the EUF-CMA security of Schnorr digital signature with the same advantage, since $\\mathcal {CH}$ can simulate the signing oracle for the LRS component without using the secret keys.",
"$\\mathcal {B}$ forwards the digital signature $\\sigma $ forged by $\\mathcal {A}$ to its challenger and wins with the same advantage.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma 2.3 The LRS component is secure in the presence of a signing oracle for the Schnorr signature.",
"We prove via a sequence of games.",
"Game 2.3.1 The real security experiment for ISE’s LRS component.",
"Challenger $\\mathcal {CH}$ interacts with $\\mathcal {A}$ as below: Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow \\text{\\fontfamily {cmss}\\selectfont {Setup}}(1^{\\lambda })$ and sends it to $\\mathcal {A}$ .",
"Queries: the adversary $\\mathcal {A}$ can make queries to the following oracles: $pk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {CH}$ runs $(ltpk_i, ltsk_i) \\leftarrow $ LongTermKeyGen($pp)$ and $(pk_i$ , aux$i$) $\\leftarrow $ OneTimePKGen($ltpk_i; r$ ) and returns $pk_i$ to $\\mathcal {A}$ .",
"$\\xi \\leftarrow $ HashOracle(data): $\\mathcal {CH}$ emulates a random oracle by using the lazy sampling technique.",
"He maintains an initially empty list $L$hash, an on a given query with some data, if there is an entry (data, $\\xi $ ) in $L$hash, $\\mathcal {CH}$ returns $\\xi $ .",
"Else, $\\mathcal {CH}$ picks $\\xi \\overset{r}{\\leftarrow } Z_p$ and inserts (data, $\\xi $ ) in $L$hash, then returns $\\xi $ .",
"$\\sigma \\leftarrow $ SignOracle($pk_i, m$ ): on input a public key $pk_i$ and a message $m$ , $\\mathcal {CH}$ runs ISE.Sign($sk_i$ , $m$ ) $\\rightarrow \\sigma $ and returns $\\sigma $ to $\\mathcal {A}$ .",
"Forge: $\\mathcal {A}$ outputs a signature $\\sigma \\textsubscript {ring}$ and wins if Verify($pk, m, \\sigma \\textsubscript {ring}) = 1$ .",
"Game 2.3.2 The same as Game REF except that $\\mathcal {CH}$ simulates signing oracle by programming a random oracle $\\mathcal {H}$ , rather than using the real secret keys, in the following way.",
"Setup: $\\mathcal {CH}$ runs $pp\\textsubscript {ise} \\leftarrow \\text{\\fontfamily {cmss}\\selectfont {Setup}}(1^{\\lambda })$ and sends it to $\\mathcal {A}$ .",
"Queries: the adversary $\\mathcal {A}$ can make queries to the following oracles: $pk_i \\leftarrow $ KeyOracle($i$ ): $\\mathcal {CH}$ runs $(ltpk_i, ltsk_i) \\leftarrow $ LongTermKeyGen($pp)$ and $(pk_i$ , aux$i$) $\\leftarrow $ OneTimePKGen($ltpk_i; r$ ) and returns $pk_i$ to $\\mathcal {A}$ .",
"$\\xi \\leftarrow $ HashOracle(data): $\\mathcal {CH}$ emulates a random oracle by using the lazy sampling technique.",
"He maintains an initially empty list $L$hash, an on a given query with some data, if there is an entry (data, $\\xi $ ) in $L$hash, $\\mathcal {CH}$ returns $\\xi $ .",
"Else, $\\mathcal {CH}$ picks $\\xi \\overset{r}{\\leftarrow } Z_p$ and inserts (data, $\\xi $ ) in $L$hash, then returns $\\xi $ .",
"$\\sigma \\leftarrow $ SignOracle($pk_i, m$ ): on input a public key $pk_i$ and a message $m$ , $\\mathcal {CH}$ picks $z, \\xi \\overset{r}{\\leftarrow } Z_p$ , sets $A = zG - \\xi pk_{i})$ and returns $\\sigma = (A, z)$ to $\\mathcal {A}$ .",
"Else, $\\mathcal {CH}$ aborts to avoid possible inconsistency in programming.",
"Denote the event that $\\mathcal {CH}$ aborts in Game REF by $E$ .",
"Conditioned on $E$ does not occur, $\\mathcal {A}$ ’s view in Game REF and Game REF are identical.",
"This follows from the fact that $\\mathcal {CH}$ perfectly mimics the hash oracle and signing oracle.",
"Let $Q$hash and $Q$sign be the maximum number of hash queries and signing queries that $\\mathcal {A}$ makes during security experiment.",
"By the union bound, we conclude that $\\text{Pr}[E] \\le (Q\\textsubscript {hash} Q\\textsubscript {sign})/p$ , which is negligible in $\\lambda $ .",
"In summary, we have: $| \\text{Pr}[S_1] - \\text{Pr}[S_0]| \\le \\text{Pr}[E] \\le \\text{{\\fontfamily {cmss}\\selectfont {negl}}}(\\lambda )$ We now argue that no PPT adversary has non-negligible advantage in Game REF .",
"Claim 2.3.1 Assuming the security of Triptych LRS, $\\text{Pr}[S_1]$ is negligible in $\\lambda $ for any PPT adversary $\\mathcal {A}$ .",
"We prove this claim by showing that if there exists a PPT adversary $\\mathcal {A}$ has non-negligible advantage in Game REF , we can build a PPT adversary $\\mathcal {B}$ that breaks the security of Triptych LRS with the same advantage, since $\\mathcal {CH}$ can simulate the signing oracle for the DS component without using the secret keys.",
"$\\mathcal {B}$ forwards the LRS forged by $\\mathcal {A}$ to its own challenger and wins with same advantage.",
"The claim immediately follows.",
"This proves Lemma REF .",
"Lemma REF , Lemma REF and Lemma REF prove Theorem REF ."
],
[
"Performance",
"We implemented DelegatedRingCT in Ubuntu 18.04, Intel Core i7-4790 3.60GHz, 16GB RAM.",
"We used the dalek cryptography repositoryhttps://github.com/dalek-cryptography, which has implementations for Curve 25519, Bulletproofs and ed25519 signatures in Rust, and where each element in $\\mathbb {G}$ and $Z_p$ are represented by 32 bytes.",
"The benchmarking results for the proving time, verification time and sizeWe used offsets, like Monero, for the input accounts and assumed a size of 3 bytes each.",
"of DRingCTx for a typical transaction of $M=2$ spend accounts and $T=2$ output accounts are in Figures 1, 2 and 3, respectively.",
"CRx has a size of 352 bytes, a verification time of 660 us and a proving time of $261.89$ us.",
"Figure: Running time of CreateDRingCTxFigure: Running time of VerifyDRingCTxFigure: Size of DRingCTx"
],
[
"Discussion and Future Work",
"On one hand, our Delegated RingCT can vastly improve on the verification times of a transaction and, consequently, on the scalability of a cryptocurrency that implements this protocol, as shown in the previous section.",
"Even though we have not implemented any distributed consensus mechanism, validation times are not expected to increase too much and will mostly depend on the type of ledger used and the blocks time interval (asynchronous vs synchronous) .",
"On the other hand, anonymity of Delegated RingCT is not as strong as in RingCT, since, if you delegate an account to a representative, a malicious actor can easily know the balance of that individual account by decrypting the ciphertext, so some amount of trust in the representative is required.",
"Besides that, we do not capture in our security model the possibility of the receiver changing the representative of the output accounts of a transaction.",
"If the representative is the same as of the sender account, he would be able to decrypt both accounts and see that balances match, revealing the real sender of the transaction.",
"However, even if the representative knows which account is the real sender, he will not know what other accounts belong to the same master address or what the master address is.",
"Moreover, delegation is optional and you can delegate to yourself, increasing your anonymity, or even not delegate at all (even though your stake will not contribute to the consensus).",
"Nevertheless, further investigation and formal security proofs are required regarding anonymity against its own representative.",
"Another limitation of our scheme is that the balance size cannot be too large if using a variation of additive ElGamal, because, after decryption, the discrete logarithm problem must be solved to reveal the balance.",
"We used 32 bits in our implementation, which seems reasonable for the total amount of coins of a cryptocurrency, but can be insufficient if the market capitalisation becomes too high.",
"However, if necessary, two concatenated ciphertexts can be used instead of just one, which come at a cost of greater size of transaction and higher verification times.",
"Still, this approach is more efficient than other additive homomorphic public key encryption schemes.",
"Future work will focus on choosing a Delegated Proof of Stake consensus mechanism, as well as the ledger structure, and investigate how and when the stake weights will be updated.",
"If it is too frequent it can hinder anonymity.",
"If too infrequent, it can create new attack vectors."
]
] | 2011.14159 |
[
[
"A Deductive Verification Framework For Higher Order Programs"
],
[
"Abstract In this report, we present the preliminary work developed for our research project for the APDC (\\'Area Pr\\'atica de Desenvolvimento Curricular) course.",
"The main goal of this project is to develop a framework, on top of the Why3 tool, for the verification of effectful higher-order programs.",
"We use defunctionalization as an intermediate transformation from higher-order OCaml implementations into first order ones.",
"The target for our translation is WhyML, the Why3's programming language.",
"We believe defunctionalization can be an interesting route for the automated verification of higher-order programs, since one can employ off-the-shelf automated program verifiers to prove the correctness of the generated first-order program.",
"This report also serves to introduce the reader to the subject of deductive program verification and some of the tools and concepts used to prove higher order effectful programs."
],
[
"Introduction",
"Producing error free code for a complex system is, regardless of the programmer’s skills, a herculean task.",
"Customarily, the only weapons we have to combat the plethora of bugs rotting away our code are tests and plenty of caution.",
"But caution can only get you so far and program testing, in Dijkstra's words, \"can be a very effective way to show the presence of bugs, but is hopelessly inadequate for showing their absence\".",
"Therefore, in order to make sure that our code does what we want, we need formal proofs of its correctness.",
"There is no one definition of correctness, however, considering it depends on how we want to specify our program.",
"We might simply want to make sure that there are no runtime errors involving type safety, arithmetic overflows or unhandled exceptions.",
"We might also want to prove that the program eventually terminates.",
"More generally, we might want to prove some relation to its input and/or the state with its output.",
"For each definition of correctness there are better ways of proving it.",
"For the class of programs we will analyse, that is, functional programs, the best alternative is deductive verification.",
"Dispatching proofs for imperative implementations can be challenging, as they heavily rely on modifying data structures after they have been created, which can be hard to model using first order logic.",
"Therefore, writing in a programming language that stays as close as possible to logical definitions, that is, a functional language, means we will have an easier time identifying the necessary assertions to create an adequate proof.",
"OCaml is a good candidate for writing effectful, verifiable code, not only because it is primarily a functional language, but also because it allows us to use imperative constructs, such as mutable records.",
"To write specifications for code with side effects, we need additional infrastructure on top of first order logic.",
"One of the most popular is Separation Logic , which allows us to connect physical memory to logical statements.",
"Although it is very expressive, its intrinsic technicalities prevent it from being widely adopted.",
"In this work, we will be using GOSPEL (Generic OCaml SPEcification Language) , which is built on top of Separation Logic, as a higher level alternative to define changes in the heap and, to prove the specification, we will use Why3 , a program verifier tailored towards automated verification.",
"But effectful code is only one half of this project; the other being higher-order functions.",
"As previously said, code with side effects can be difficult to map onto logical definitions, but this problem becomes exponentially more complex when these effects originate from first class functions, due to the way programs are proved.",
"In short, we annotate them with pre and postconditions and check that an implementation conforms to the provided specification.",
"This is normally done by employing some technique of verification conditions generation, such as Dijkstra Weakest Precondition Calculus .",
"However, if we don't know the behaviour of a part of our routine, and this unknown part might alter our program's state, then it can be hard to prove if a given program is correct.",
"Some work has been put into bringing closer automated verification and higher-order programs with effects, such as the tool Who .",
"Nonetheless, a framework of that caliber can be complex to implement and maintain.",
"Another option would be Iris , one of the most powerful tools available in the realm of program verification, but whose power comes at a cost.",
"The logic employed by it is, albeit very rich, extremely complex and almost inaccessible to anyone without years of training in the field.",
"In addition, there is no way to automate proofs with it, this being the main reason why Iris falls out of the scope of our project.",
"Most options available for handling this problem present their own hindrances, such as, for example, their complexity, cross-compatibility and their practical usefulness .",
"We have put forth a small contribution to solve these limitations by creating a Why3 plugin which makes use of defunctionalization , to handle the verification of such programs.",
"Defunctionalization works by translating a higher-order implementation into an equivalent first-order counterpart.",
"We believe this technique can open the door to automatic verification of higher-order code, since we can use an existing first-order verifier to prove the defunctionalized program.",
"Previous work studied the combination of defunctionalization and software verification. .",
"However, no prototype was developed and all examples were made entirely by hand.",
"In this project, we lay the groundwork for a fully-fledged verification framework for effectful higher-order programs, by exploring the technique of defunctionalization.",
"To the best of our knowledge, this is the first practical implementation of defunctionalization as a proof technique.",
"Our framework will accept as input a higher-order program, annotated with assertions written in GOSPEL, translate it into a first-order equivalent WhyML program so that we can use Why3 to prove it satisfies the assertions."
],
[
"Why3",
"Why3 is a verification framework that allows us to interface with multiple theorem provers and automated solvers using the same surface syntax and underlying logic , .",
"This means that, if one such solver is not able to discharge a verification condition, one can easily switch to another which might be able to complete our proof.",
"To understand how it works, we will present the defenition of an array list type written in Why3's dedicated programming language WhyML: type arrayList 'a = mutable arr : array 'a; mutable size : int This type definition features a field arr of type array 'a, the polymorphic type of arrays defined in Why3 standard library, and an integer size representing the number of elements in a value of type arrayList.",
"Although our type declaration has all we need to build a comprehensive set of functions for our arrayList, from a logical point of view we are missing crucial information.",
"In particular, that the size attribute cannot be greater than the length of the array.",
"To express this, we will add the following type invariant: invariant 0 <= size <= arr.length Next, we will write a function that replaces the element at a given index and returns the old element: let set (al: arrayList 'a) (i : int) (e : 'a) : 'a For simplicity’s sake, we will add a precondition requiring i to be within the bounds of our list: requires 0 <= i < al.size Had we not included such precondition, we would have to include in the body of set an if..then..else expression to avoid accessing an index out of bounds.",
"This a common practice of defensive programming.",
"Next, we will need to specify that the result of our function is the old value at index i and that the new value is e. To do this, we will will introduce two postconditions, where we use the old tag to refer to our array list before the function was called and result for our function's return value ensures result = old al.arr[i] ensures al.arr[i] = e Lastly, we also need to specify that every other element in the array, as well as its size, is unchanged.",
"ensures al.size = old al.size ensures forall j.",
"0 <= j < al.size / not j = i -> old al.arr[j] = al.arr[j] Now we simply write the body of the function, trivial as it is: = let ret = al.arr[i] in al.arr[i] <- e; ret The full annotated function can be found in annex .",
"When fed into Why3, the above set function generates six verification conditions.",
"These include safety properties (access index in array bounds, which is proved with our invariant) and functional ones (type invariant preservation and postcondition entailment).",
"All of these are automatically verified in less then a second by the Alt-Ergo SMT solver, one of the provers Why3 allows us to interface with."
],
[
"Defunctionalization",
"Before jumping into our framework's implementation lets first take a gander at a few quick (original) examples to understand how defunctionalization works, how it can be mechanized and how we can apply it to program verification."
],
[
"A simple example",
"As previously mentioned, defunctionalization is the practice of turning higher-order programs into equivalent first-order implementations.",
"To illustrate this method, we will convert a higher-order OCaml program that reverses a list.",
"The code will be written in CPS (Continuation Passing Style) , which, instead of letting the program return a value to the function’s caller, passes the return values as a parameter to a “continuation” function that is given as a first order value.",
"The implementation is as follows: ocaml let rec reverseauxcps l k = match l with | [] -> k [] | x::t -> reverseauxcps t (fun r -> x :: (k r)) let reverse l = reverseauxcps l (fun r -> r) There are two anonymous functions: the identity function (fun r -> r) and the function which builds the reverse of the list (fun r -> x::(k r)).",
"In order to begin our gentle descent towards a first order version of this program, we introduce a new algebraic data type to represent continuations: ocaml type 'a kont = Kid | Krev of 'a * 'a kont The constructors Kid and Krev match, respectively, the identity function and the one that builds the reverse.",
"Krev‘s two extra arguments represent its free variables, namely the appended value and the continuation function.",
"Next, we need to introduce an apply function which will replace all first order functions calls: ocaml let rec apply kont arg = match kont with | Kid -> let r = arg in r | Krev(x, k) -> let r = arg in x::(apply r k) The defunctionalization process terminates by replacing, in the original code, native function application with calls to apply and anonymous functions with constructors of the kont data type.",
"ocaml let rec reverseauxdefun l k = match l with | [] -> apply [] k | x::t -> reverseauxdefun t (Krev(x, k)) let reversedefun l = reverseauxdefun l Kid The full defunctionalized implementation can be found in annex and if the reader wishes to understand in more detail how this function would run, we have included an example of the program running in annex ."
],
[
"Proving a defunctionalized program",
"With defunctionalization's bases covered, let us introduce how it's used in a proof context.",
"For this, we consider a simpler function, namely one that computes a list's size, also written in CPS.",
"The WhyML implementation is as follows: let rec lengthcps (l : list 'a) (k : int -> 'b) : 'b = match l with | [] -> k 0 | :: t -> lengthcps t (fun l -> k (1 + l)) end let len (l : list 'a) : int = lengthcps l (fun x -> x) Now we annotate our functions with their respective postconditions.",
"The len function's specification is fairly straightforward: ensures length l = result where length is a purely logical function, issued from the Why3 standard library, which can only be used within specifications and shall never be mixed with executable code.",
"The CPS function's specification is a bit more intricate.",
"We could, naively, write something like ensures k (length l) = result However, this leaves us with the problem of how to interpret the use of functions in our logic, since these may have side effects (e.g., divergence) which can lead to logical inconsistencies.",
"Therefore, we introduce a layer of abstraction to allow us to use potentially effectful programs in our assertions: instead of thinking of functions as a computation of an output from a given input, such as f: t1 -> t2, we will think of them as a pair of predicates, namely their precondition (pre : t1 -> prop) and postcondition (post : t1 -> t2 -> prop).",
"This is strongly inspired by the work of Régis-Gianas and Pottier .",
"Under this setting, we would provide with lengthcps the following specification: ensures post k (length l) result and the continuation defined in lengthcps with ensures post k (l + 1) result Now we will proceed with our program's defunctionalization as we did before, first by defining a type to represent our first order functions: type kont = Kid | Klen kont The next step, that is, the apply function, is quite similiar to our previous example, which can be found in annex .",
"Finally, since we have no higher order functions, we must define our own post predicate, as follows: predicate post (k : kont) (length : int) (result : int) = match k with | Kid -> result = length | Klen k -> post k (length + 1) result end For the above first-order implementation and specification, all the verification conditions generated by Why3 are automatically discharged in no time.",
"As we can observe, the defunctionalized definition of predicate post is a simple matter of collecting the predicate's uses from the original higher-order implementation.",
"As for the specification of function apply, since this function simulates native application, we provide it with a postcondition stating that applying k to arg must be directly related to result.",
"As we will show, the translation of the post predicate and the generation of the specification of the apply are indeed amenable to automatic tool translation."
],
[
"The GOSPEL to Why3 Pipeline",
"In spite of the fact defunctionalization can be a bit of a difficult concept to grasp, once it is understood it becomes clear that we can divide the process into neatly defined steps and that these steps can very easily be mechanized.",
"Although we give a complete overview of how OCaml code is converted in our translation scheme, found in annex , we will explain in detail some of the trickier rules and exemplify with some case studies how our converter fares when confronted with decently complex programs.",
"The full implementation for our translation scheme, along with some test cases, can be found in our GitLab https://gitlab.com/releaselab/defunctionalization/-/tree/patch-1"
],
[
"Defunctionalization at Work",
"Explaining in detail every single one of our translation scheme's minutia would be a tad redundant, seeing as WhyML and OCaml are very similar, given they both derive from the ML dialect.",
"Indeed, most of our translations simply amount to lightly re-organizing the OCaml code to fit WhyML's syntax, picking up the free variables and figuring out the type of the expression.",
"Nevertheless, we will convert a handful of expressions to explain some of the peculiarities of how we approach defunctionalization before we get into the relatively more complex case studies.",
"First we will be looking at the translation of the following expression : x + y.",
"When converting an OCaml expression we will return, along with its WhyML translation, its set of free variables and its type.",
"Seeing as this expression belongs to the class $e1\\ o\\ e2$ , x being $e1$ , + being o and y being $e2$ , we will need $e1$ and $e2$ 's translation and free variables.",
"Seeing as $e1$ and $e2$ are simply two variables, we will translating both directly, meaning e1's translation will be x and $e2^{\\prime }s$ will be y and their set of free variables will be a singleton set each only containing a tuple with the variable's name and its type, which we obtain from a dictionary we refer to in our translation scheme as varType which binds the name of user defined variables to their types.",
"Finally we need only to combine the two translations with o's, which will simply be +, producing \"x + y\", and the two sets of free variables, giving us {(x, int), (y, int)}.",
"Note however that we do not define this expression's type attribute, seeing as it is only used when translating function applications and we know that an addition can only produce integers, meaning we will never apply the result of this expression as a function.",
"It is worth noting that in order to use the addition operator in WhyML we need to import the int.Int module.",
"To handle this, we import, along with int.Int, several other libraries so we can naturally use everything in the OCaml standard library, such as lists and arithmetic operations.",
"Next, we will bind the previous expression to a lambda and translate it with the fun arg* : $\\tau $ -> e1 construction fun (x : int) (z : int) : int -> x + y Important to note that z is not used and y is not defined in the scope of this expression.",
"First and foremost, we will be adding the function's arguments' type to the aforementioned varType dictionary.",
"After computing $e1$ 's free variables, we must remove $x$ and $z$ from the set.",
"Since $e1$ is our last translated addition (x + y), our new set of free variables is simply $\\lbrace (y, int)\\rbrace $ , since x is defined in the scope of this expression.",
"Additionally, we will defunctionalize this lambda using the buildApply function, which takes the lambda's free variables, type and arguments' name and returns its defunctionalized alias.",
"Given the previous defunctionalization examples, the reader might be expecting something like this type kont1 = K int let apply (k : kont1) (arg1 : int) (arg2 : int) : int = match k with K y -> let x = arg1 in let z = arg2 in x + y Although this would be an accurate translation, it is not fully defunctionalized, due to the fact OCaml technically does not support functions with more than one argument, which might seem unintuitive seeing as our original lambda has two.",
"When OCaml parses it, however, it produces a one argument function that returns another one argument function, such as: fun (x : int) -> (fun z : int -> x + y) This means we can partially apply arguments and, instead of throwing an error, OCaml will simply return a lambda where the applied arguments are now free variables.",
"Due to this detail, we defunctionalize our functions in the way OCaml defines them instead of how the user does.",
"Hence, we will first translate the innermost lambda into K1 y x with the generated defunctionalization being similar to: type kont1 = K1 int int predicate post1 (k : kont1) (arg :int) (result : int) = match k with K1 y x -> let z = arg in true let apply1 (k : kont1) (arg : int) : int = ensures post1 k arg result match k with | K1 y x -> let z = arg in y + x Seeing as we didn't annotate our original lambda with a postcondition, we fill in the post predicate with the default value true.",
"Before moving on to the outermost lambda, we will bind the type int -> int to the function apply1 the the predicate post1.",
"Now all that is left is to translate the outermost lambda, which is largely identical to the first one, except we now bind int -> (int -> int) to the function apply2 and the predicate post2 and, seeing as the body of this function will be our previously generated expression, we must append it a specification that describes its return value so we may use this function in future specifications.",
"We then conclude that our post predicate is as follows: type kont2 = K2 int predicate post2 (k : kont2) (arg : int) (result : kont1) = match k with K2 y -> let x = arg in result = K1 y x In conclusion, we create two apply functions, one that takes the first argument and returns a continuation and another which accepts that continuation along with the final argument returning the result of the original function.",
"Another detail is that, unlike in our previous examples, we defined two types (kont1 and kont2) with one constructor each instead of one type with two constructors.",
"This is due to the fact that we need different apply functions for each different function type since WhyML does not support generalized algebraic data types as arguments or return values.",
"We cannot put all our constructors in the same definition because that would mean, if we had more than one apply function, their pattern matching wouldn't be exhaustive and Why3 wouldn't be able to dispatch a proof.",
"Finally, we will set this expression's type as int -> (int -> int) and reset the changes made to varType at the start of the translation, that is, remove the binding of x to int and the binding of z to int.",
"Important to note that removing a pairing means that, if there was any other value associated with the key before adding the removed value, that becomes the new value, if not, the binding now becomes undefined.",
"To conclude, we will now convert the application of a function to a set of arguments: (k 3) 4.",
"First, we will translate the expression within brackets with the $e1\\ e2$ rule.",
"To do so, we will get the translation of $e1$ and $e2$ , both of which are directly obtained; their free variables (which are not relevant for this example), and their type, e2's is irrelevant, due to the fact it is a literal value and those are never used as functions, and e1's is obtained in the same manner as before: by checking the current mapping for the variable name k. For the sake of argument, we will assume k is bound to the previous lambda, whose type is int -> (int -> int).",
"With both sub expressions now converted, it is time to turn our attention to our original application, which will be converted as follows: apply2 k 3 and it's type will be int -> int, seeing as we have already applied k's first argument.",
"Next, we do the same thing for the next argument, the only notable difference is that our expression's type is now int -> int, which is bound to the integer 1 and thus we will use the apply1 function, giving us: apply1 (apply2 k 3) 4 As the one might have noticed, the previous expressions were defunctionalized even though they weren't used in a higher order context.",
"This is another quirk of our translation: any and all function declarations and applications, when converted, are also defunctionalized, regardless of whether or not they are used as parameters.",
"The main reason for this is that, when a function is declared we generally have no way of knowing if it is going to be used as a first order value.",
"Despite the fact that we are defunctionalizing more than what we strictly need, translating every function the same way makes the conversion process more uniform and easier to define and implement."
],
[
"Extending GOSPEL",
"Since we have covered the most relevant rules for translating and defunctionalizing OCaml code, let us take a look at GOSPEL.",
"We won't go over its syntax in much detail, seeing as its specifications are quite similar to Why3's, although there are some interesting divergences.",
"Namely, GOSPEL annotations are added as comments beginning with the symbol \"@\" at the tail end of our implementations or, for anonymous functions, between the fun keyword and the function's arguments like so ocaml fun [@gospel | ensures Q |] args... -> exp There are some other which are less relevant, for example, having to redefine the name of each argument and the name for the result and the absence of curly brackets, but for the most part we will use similar constructions as we did for Why3 specifications.",
"Due to this, our WhyML conversions will be almost identical to their GOSPEL counterparts, with one small difference: the post \"predicate\".",
"To understand how we have extended GOSPEL to allow usages of post, let us consider a small example.",
"ocaml (*@ r = f g x ensures post (g : int -> int) x r *) This will be the specification for a function f which takes a function g as a parameter and an integer x. we have omitted the implementation since we will be focusing on how we will deal with the post condition.",
"As previously explained, post is used to avoid directly using non-pure functions (that is, with side-effects) in our specifications.",
"Because post has to work for any function applied to however many arguments of any types, it cannot be defined through WhyML.",
"Instead, as previously shown, we create several posts, one for each apply function and whenever we encounter a post in our specifications, we will replace its application with the respective defunctionalized post.",
"That is why we put predicate in quotation marks; post is not really a predicate nor a function, but more of a meta-predicate: a predicate which, depending on the situation, can be any predicate you need.",
"The only restriction for using it is that the first argument, that is, the function, must have its type explicitly declared so we know which post to use.",
"Now that we understand the keyword, let us look at how we convert it.",
"For one argument functions it is almost direct.",
"For the aforementioned specification, we would find the post predicate for functions of type int -> int, and replace the call to post with post2 from our previous translation: ensures post2 g x r It gets interesting, however, when we apply post to a function that takes more than one argument.",
"Let us imagine that the g function was of type int -> (int -> int) and was applied to an extra argument y, such as, post g x y r. The translation of such post would be: ensures forall var0 : kont1.",
"post1 g x var0 -> post2 var0 y r In simpler terms, if we assume the return value for g when applied to x is var0, whose type would be kont1, then when var0 is applied to y, it will return r."
],
[
"Case studies",
"With the most important rules detailed and understood, we will examine how our tool handles relatively complex programs.",
"The following case studies, unlike the ones presented in Section , are not original and were presented by one of the project's supervisors in their precursor paper on defunctionalized proofs."
],
[
"Height of a tree",
"Firstly, we will look at at a function that computes the height of a binary tree with the following signature ocaml let heighttreecps (t: int tree): int = ... (*@ r = heighttree t ensures (height t) = r *) An interesting peculiarity of this program is that it only accepts a tree of integers, even though, intuitively, a height function should work for any type of tree.",
"However, we won't be dealing with any kind of polymorphism, for that would add certain complications when building our constructors .",
"Another thing worth mentioning is that OCaml does not have a binary tree type, although WhyML does.",
"This means that, even though the OCaml program we will present technically does not compile on its own unless the programmers defines a tree exactly as it's done in WhyML, it will result in a valid program after the conversion if we import the tree.Tree module.",
"Even so, one could wonder why we simply don't define the tree type in our code instead of going in this roundabout way.",
"The reason is that we are going to need a logical function from Why3 standard library, namely the height function, in our specification, which we can only apply in conjunction with WhyML's tree type.",
"This solution, flawed as it may be, will be sufficient to prove our program.",
"First let us define the recursive auxiliary function that performs the calculation ocaml let rec heighttree (t: int tree) (k: int -> int) : int = match t with | Empty -> k 0 | Node((lt : int tree), , (rt : int tree)) -> heighttree lt (fun [@gospel | ensures post (k : (integer -> integer)) (1 + max hl (height rt)) result |] (hl : int) : int -> heighttree rt (fun [@gospel | ensures post (k : (integer -> integer)) (1 + max hl hr) result |] (hr : int) : int -> k (1 + max hl hr))) (*@ r = heighttree t k ensures post (k : (integer -> integer)) (height t) r*) This one is a bit trickier than our previous examples, due to the nested continuations.",
"In short, we create two continuations, one for each tree branch, which will calculate their heights.",
"Finally, they will compare the two and return the highest value plus 1.",
"Then, all our main function has to do is run the recursive auxiliary function with the identity function.",
"ocaml let heighttreecps (t: int tree): int = heighttree t (fun [@gospel |ensures result = x|] (x : int) : int -> x) (*@ r = heighttree t ensures (height t) = r*) The fully annotated program and its conversion can be found in our GitLab and can be proved using the CVC4 prover, which dispatches the proof in less than a second."
],
[
"The final example we will present is an interpreter for a small language that consists entirely of integer literals and subtractions thereof: ocaml type exp = Const of integer | Sub of exp * exp To define how expressions are reduced, we will present two rules, first we assert that a $Sub$ node with two $Const$ can be reduced to a single $Const$ : $Sub (Const\\ v1) (Const\\ v2) \\xrightarrow{} Const (v1 - v2)$ Additionally, we will also need an inference rule that allows us to properly parse larger expressions: ${C[e] \\xrightarrow{} C[e^{\\prime }]}{e \\xrightarrow{} e^{\\prime }}$ C representing the following reduction context: $\\begin{array}{crl}C & ::= & \\square \\\\& | & C[\\mathtt {Sub} \\; \\square \\; \\mathtt {e}] \\\\& | & C[\\mathtt {Sub} \\; (\\mathtt {Const} \\: \\mathtt {v}) \\; \\square ]\\end{array}$ Simply put, $C[e]$ represents a context where the expression $e$ has completely replaced the symbol $\\square $ , meaning our inference rule reads: if an expression $e$ can be reduced to a simpler $e^{\\prime }$ , then any context $C[e]$ can be reduced to $C[e^{\\prime }]$ With our interpreter clearly described let us move on to how we represent our contexts.",
"Although the most natural translation would be of an algebraic data type with three constructors, one for each of our rules, we will represent them as first order functions that take an expression and return an expression, leading to much cleaner and compact code.",
"Next, we will implement a function that quite directly translates our inference rule by taking a context $C$ and an expression $e$ which returns a context and a new expression making sure that $C[e] = C^{\\prime }[e^{\\prime }]$ , $C^{\\prime }$ and $e^{\\prime }$ being the returned context and expression respectively.",
"ocaml let rec decomposeterm (e: exp) (c : exp -> exp) : ((exp -> exp) * exp) = match e with | Sub (Const (v1 : int), Const (v2 : int)) -> (c, e) | Sub (Const (v : int), (e : exp)) -> decomposeterm e (fun [@gospel | ensures post (c : (exp -> exp)) (Sub (Const v) x) result |] (x : exp) : exp -> c (Sub(Const v, x))) | Sub ((e1 : exp), (e2 : exp)) -> decomposeterm e1 (fun [@gospel | ensures post (c : (exp -> exp)) (Sub x e2) result |] (x : exp) : exp -> c (Sub(x, e2))) (*@ cres, eres = decomposeterm e c requires not (isvalue e) ensures isredex eres forall res.",
"post (c : (exp -> exp)) e res -> post (cres : (exp -> exp)) eres res*) The first thing one might notice is the fact we only match e with Subs and no Const.",
"When pattern matching is non-exhaustive, WhyML will add an extra branch of the following type | -> absurd and attempt to prove that it is never reached.",
"In this case, it will meet this obligation knowing that e is not a value, in other words, not a constant.",
"This knowledge comes from the isvalue predicate, which we do not define above.",
"We haven't added its implementation, nor the others we use, seeing as their quite simple.",
"Moreover, functions with preconditions are handled a bit differently: any function presenting these will be converted directly instead of defunctionalized, seeing as there are no interesting case studies of stateless continuations with preconditions.",
"Therefore we will assume that any function with these will never be used as a first order value and therefore has no need in being defucntionalized, thereby simplifying our proof obligations.",
"Now that we have a function that satisfies $C^{\\prime }[e^{\\prime }] = C[e]$ we need one for the condition $C^{\\prime }[e^{\\prime }] = e$ .",
"For this purpose, all we need is to apply our original function with a context so that $C[e] = e$ , in other words, the identity function.",
"ocaml let decompose (e: exp) : (exp -> exp) * exp = decomposeterm e (fun [@gospel | ensures result = x |] (x : exp) : exp -> x) (*@ cres, eres = decompose e requires not (isvalue e) ensures isredex eres post (cres : (exp -> exp)) eres e *) With decomposition out of the way, all that's left is to properly evaluate our expressions.",
"To do this, we will define a function which takes an expression and keeps decomposing and performing head reductions, that is, applying the first rule we presented that reduces a $Sub$ into a $Const$ , until it reaches a constant: ocaml let red (e : exp) : int = match e with | Const (v : int) -> v | -> match decompose e with ((c : exp -> exp), (r : exp)) -> let r : exp = headreduction r in red (c r) (*@ r = red e ensures r = eval e *) When converted, this program, whose translation also available in our GitLab, is once again successfully proved, this time by the Alt-ergo prover, in less then a second.",
"However to reach it, we need to supply it with the following lemma, which says that for any two expressions, if they evaluate to the same value and are applied to the same context, their evaluations will stay equal relative to one another.",
"lemma posteval: forall c: exp -> exp, arg1: exp, arg2: exp, r1: exp, r2 : exp.",
"eval arg1 = eval arg2 -> post0 c arg1 r1 -> post0 c arg2 r2 -> eval r1 = eval r2 Since this lemma is much broader than our target function's post condition, there could be some value in making it part of GOSPEL's (and, naturally, Why3's) standard library, along with other general axioms regarding higher order functions, so that the programmer has a fairly decent safety net when their provers are unable to reach the required proof obligations.",
"Of course we couldn't put it as is, given that it depends on the eval function, which is specific to our case study.",
"Nonetheless, we could turn it into a kind of meta-lemma where we take a logical function of type 'a -> 'a as an argument.",
"To use such a \"meta\"-lemma to prove our transitive property would be just a matter of instantiating it with the proper 'eval'-like function.",
"Due to the fact that GOSPEL does not yet support lemmas, it will have to be placed directly in our translation.",
"An interesting limitation for this lemma is that it could cease to work if we introduced more first order functions of type exp -> exp that are divorced from the contexts we have described.",
"To handle this, our tool would have to know which specific continuations the user is referring to and create a lemma tailored to them.",
"A hypothetical translation that solves this problem can be found in annex ."
],
[
"Limitations",
"Due to time constraints, we could only hope to produce a rough prototype that could be used less as a fully fledged tool and more of a proof of concept.",
"Naturally, what we created presents itself with the following non-exhaustive list of restrictions, along with those previously mentioned in Section : To prove an OCaml program, we first needed a way to convert the source code into a more digestible format, such as an abstract syntax tree.",
"Two choices emerged; the Typedtree module and the Parsetree module, both from OCaml's compiler-libs library; the main difference between the two being that the Typedtree has already inferred the types for all expressions and the Parsetree only has the type information that the user puts in the source code.",
"Seeing as, to build our defunctionalized constructors, we need to know the types of the free variables, our options were to simply use the Typedtree or, alternatively, the Parsetree, with the caveat that the user would have to explicitly add the types for all variables defined in 'let' bindings, match cases and function definitions.",
"We decided on using the later seeing as it would be the easiest option to convert programs, although it does pose a restriction on the type of programs we can convert.",
"Due to this choice, our program does not actually type-check the original code and leaves this to WhyML.",
"This means that, theoretically speaking, we could have a situation where a badly typed program is converted and the resulting WhyML program is correctly typed, leading to a nonsensical specification.",
"Although we believe this situation is impossible, we have no formal proof that this is the case.",
"In order to make the best of the time given, we decided, instead of translating it in its entirety, to focus our efforts onto a subset of the OCaml language.",
"Although we are missing some very useful syntactic sugar such as module declarations and optional arguments, the constructions allowed by our converter (e.g., match expressions, let..in, if..then..else, lambda expressions) give us plenty of breathing room to write rich, functional code.",
"Due to the fact our apply functions will be mutually recursive, proving termination, while not impossible ,would be quite difficult, therefore, we only prove partial correctness; in other words, we do not prove that these functions are terminating.",
"Defunctionalization can only be used knowing a priori which functions will be used as first order values.",
"Therefore, if a function takes, for example, a lambda of type int -> int, and no functions of this type are defined, then our converter throws an error.",
"Although we hope to eventually use defunctionalization to prove functions with side effects, our prototype can only be used to prove pure programs."
],
[
"Conclusions and future work",
"In this report we have proposed a translator that can convert GOSPEL annotated OCaml programs into WhyML in order to prove their specification, possibly involving higher order functions.",
"This represents not only the first work involving the verification of GOSPEL implementations, but also, as far as we know, the first usage of defunctionalization in the context of function specification.",
"While our tool covers an interesting subset of OCaml programs, we have only scratched the surface of what defunctionalization is capable of.",
"We will save for future work patching up some of the limitations presented in Section , namely dealing with effectful programs and polymorphism.",
"To tackle the former, we anticipate the need to by pass the state of the program as an extra parameter to our posts ; for the latter, a much more profound extension to Why3 is needed, namely extending Why3's type system with generalized algebraic data types ."
],
[
"Annotated set function",
"use array.Array use int.Int type arrayList 'a = mutable arr : array 'a; mutable size : int invariant 0 <= size <= arr.length let set (al : arrayList 'a) (i : int) (e : 'a) : 'a requires 0 <= i < al.size ensures al.arr[i] = e ensures result = old al.arr[i] ensures al.size = old al.size ensures forall j.",
"0 <= j < al.size / not j = i -> old al.arr[i] = result.arr[i] = let ret = al.arr[i] in al.arr[i] <- e; ret"
],
[
"Defunctionalized reverse function",
"ocaml type 'a kont = Kid | Krev of 'a * 'a kont let rec apply kont arg = match kont with | Kid -> let r = arg in r | Krev(x, k) -> let r = arg in x::(apply r k) let rec reverseauxdefun l k = match l with | [] -> apply [] k | x::t -> reverseauxdefun t (Krev(x, k)) let reversedefun l = reverseauxdefun l Kid"
],
[
"Reverse function stack trace",
"ocaml reversedefun [1;2;3] which then has the following stack trace ocaml reverseauxdefun [1;2;3] Kid reverseauxdefun [2;3] Krev(1, Kid) reverseauxdefun [3] Krev(2, Krev(1, Kid)) reverseauxdefun [] Krev(3, Krev(2, Krev(1, Kid))) apply [] Krev(3, Krev(2, Krev(1, Kid))) 3::(apply [] Krev(2, Krev(1, Kid)) 3::2::(apply [] Krev(1, Kid)) 3::2::1::(apply [] Kid) 3::2::1::[] Given this, we can see that this function returns the list 3::2::1::[], that is, [3;2;1], the reverse of [1;2;3].",
"Interesting to note that for each call to apply the first argument, namely the list, doesn't change.",
"With this in mind, we could have had the apply function take a void value (called unit in ocaml) as its first argument and had the Kid function simply return the empty list.",
"We chose to do it this way, despite this, to keep the reversecps function as close to CPS conventions as possible."
],
[
"Defunctionalized length",
"use int.Int use list.List use list.Length let rec function apply (k : kont) (arg : int) : int ensures post k arg result = match k with | Kid -> let x = arg in x | Klen kont -> let l = arg in apply kont (1 + l) end let rec function lengthdefun (l : list 'a) (k : kont) : int ensures post k (length l) result = match l with | Nil -> apply k 0 | Cons t -> lengthdefun t (Klen k) end let lengthfin (l : list 'a) ensures result = length l = lengthdefun l Kid"
],
[
"Translation Scheme",
"-20cm plus -1fill | c | c | l | Symbol Construction Translation 14eme () [l]e.trans := \"()\"; s [l]e.trans := s; e.type := varType[s].top(); e.freeVar := [(s, e.type)]; [] [l]e.trans := \"Nil\" n [l]e.trans := n (s)?",
"(e1 (,e2)*) [l]if is_constructor() then c.trans := format(\"%s %s..., s, e1.trans, ...) else c.trans := format(\"(%s, ...), e1.trans...) e.freeVar := union(c1.vars,...) e1 o e2 [l]e.trans := format(\"%s %s %s\", e1.trans, o.trans, e2.trans); e.freeVar := union(e1.freeVar, e2.freeVar) e1 :: e2 [l]e.trans := format(\"Cons (%s) (%s)\", e1.trans, e2.trans); e.freeVar = union(e1.freeVar, e2.freeVar); e1;e2 [l] e.trans := format(\"( %s) ; (%s)\", e1.trans, e2.trans); e.freeVar := union(e1.freeVar, e2.freeVar); e.type := e2.type var_def in e2 [l]e.trans := format(\"%s in %s\", var_def.trans, e2.trans); e.freeVar := union(var_def.freeVar, e2.freeVar.except(var_def.vname)); e.type := e2.type; varType[var_def.vname].pop();\" match e1 with (|c -> e2)+ [l] e.trans := format(\"match %s with |%s -> %s... end\", e1.trans, c.trans, e2.trans, ...); e.freeVar := union(e1, e2.except(c.vars),..); e.type := e2.type; if e1 then e2 else e3 [l] e.trans := format(\"if %s then %s else %s\", e1.trans, e2.trans, e3.trans); e.freeVar := union(e1.freeVar, e2.freeVar, e3.freeVar); e.type := e2.type fun arg+ : $\\tau $ -> e1 [l] for all arg do varType[arg.name].push(arg.type); e.freeVar := e1.freeVar.except(arg1,...); let name = buildApply(e.freeVar, e1.trans, arg1, ..., $\\tau $ .trans) in e.trans := format(%s(%s...,), name, e.freeVar[0]...); e.type := ($\\tau $ , arg1, ...); for all arg do varType[arg.name].pop(); e1 e2 [l]e.trans := format(\"%s %s %s\", getApply(e1.type), e1.trans, e2.trans e.freeVar := union(e1.freeVar, e2.freeVar) e.type := consumeArg(e1.type) 54emt type ('s1 (,'s)*)?",
"s2 = (| s3 (of s4(* s)*)?",
")+ [l]t.trans := fomat(\"type %s %s... = |\"%s %s... = %s...\", s2, s1,..., arg.trans) type ('s1 (,'s)*)?",
"s2 = arg(;arg)* [l]t.trans := fomat(\"type %s %s... = |\"%s %s... = %s...\", s2, s1,..., arg.trans) type ('s1 (,'s)*)?",
"s2 = $\\tau $ [l]t.trans := format(\"type %s %s... = %s, s2, s1,..., $\\tau $ .trans) 14emc n [l]c.trans := n (s)?",
"(c1 (,c2)*) [l]if is_constructor() then c.trans := format(\"%s %s..., s, c1.trans, ...) else c.trans := format(\"(%s, ...), c1.trans...) c.vars := union(c1.vars, c2.vars, ...) arg[l]l.trans := arg.trans; l.vars = {(arg.name)} [ ] [l]c.trans := \"Nil\" c1::c2 [l]l.trans := format(\"Cons %s %s\", c1.trans, c2.trans); c1.vars := union(c1.vars, c2.vars) 14emo +(.)?",
"[l]o.trans := \"+\" -(.)?",
"[l]o.trans := \"-\" *(.)?",
"[l]o.trans := \"*\" /(.)?",
"[l]o.trans := \"/\" && [l]o.trans := \"&&\" ||[l]o.trans := \"||\" =[l]o.trans := \"=\" 14em$\\tau $ s [l]$\\tau $ .trans := s s (-> $\\tau $ )+ $\\tau $ .trans := [l]String.format(\"%s -> %s, s, $\\tau $ .trans) () [l]$\\tau $ .trans :=\"()\" 14emarg () [l]arg.trans := \"()\" (s : $\\tau $ ) [l]arg.trans := format(\"s : %s\", $\\tau $ .trans); arg.type := $\\tau $ .trans; arg.var := s 14emvar_def let (rec)?",
"s arg* : $\\tau $ = e1 [l] varDef.vname := s; if isRec() then varType[s].push(($\\tau $ , arg1,...)); if isFun() then buildApply(s, e1.freeVar, arg1, ..., $\\tau $ .trans); var_def.trans := format(\"let %s = %s in \", s, e1.trans); if not isRec() then varType[s].push((arg1.type,..., $\\tau $ )); 14emtop_level t;; [l]top_level.trans := t.trans var_def;; [l]top_level.trans := var_def.trans e;; top_level.trans := [l]format(\"let %s = %s\", VAR_NAME, e.trans)"
],
[
"Defunctionalized lemma",
"let rec lemma posteval (k: kont0) (arg1 arg2 r1 r2: exp) requires eval arg1 = eval arg2 requires post0 k arg1 r1 post0 k arg2 r2 ensures eval r1 = eval r2 variant k = match k with | K2 -> () | K1 e c -> posteval c (Sub arg1 e) (Sub arg2 e) r1 r2 | K0 n c -> posteval c (Sub (Const n) arg1) (Sub (Const n) arg2) r1 r2 | -> true end"
]
] | 2011.14044 |
[
[
"Dynamics of non-Abelian strings in supersymmetric gauge theories"
],
[
"Abstract This thesis is devoted to studying strong coupling phenomena (and confinement in particular) in supersymmetric gauge theories.",
"The central object of investigation is the non-Abelian string that is responsible for the \"instead-of-confinement\" phase for monopoles in 4D ${\\mathcal N} = 2$ supersymmetric QCD with the U($N$) gauge group and $N_f$ flavors of quark hypermultiplets, $N \\leqslant N_f \\leqslant 2 N$.",
"Here it is shown that the non-Abelian strings and confined monopoles survive when we transition to the ${\\mathcal N} = 1$ supersymmetric QCD.",
"To this end we consider a mass term $\\mu$ for the adjoint matter, and in the limit of large $\\mu$ the bulk theory flows to ${\\mathcal N} = 1$.",
"We consider this transition both from the bulk point of view and from the world sheet theory, which is the two-dimensional $\\mathbb{CP}(N-1)$ model.",
"Survival of monopoles in the ${\\mathcal N} = 1$ supersymmetric QCD is important for the \"instead-of-confinement\" phase, and also for the Seiberg-Witten picture of confinement.",
"Apart from that, we also consider non-Abelian vortex strings in 4D $\\mathcal{N} = 2$ supersymmetric QCD with U$(N=2)$ gauge group and $N_f=4$ flavors of quark hypermultiplets.",
"It has been recently shown that these vortices behave as critical superstrings.",
"In particular, the lowest string state appears to be a massless BPS \"baryon.\"",
"Here we show the occurrence of this stringy baryon using a purely field-theoretic method.",
"Moreover, we explicitly demonstrate the \"instead-of-confinement\" phase, when the screened quarks and gauge bosons of weak coupling are replaced by the confined monopole-antimonopole pairs of strong coupling.",
"English version.",
"Full text (Russian and English) is available at the official web page https://go.spbu.ru/20a2711 ."
],
[
"[display] 1 8pt 1 1em 1 1em citnum"
]
] | 2011.14121 |
[
[
"Robotic grasp detection using a novel two-stage approach"
],
[
"Abstract Recently, deep learning has been successfully applied to robotic grasp detection.",
"Based on convolutional neural networks (CNNs), there have been lots of end-to-end detection approaches.",
"But end-to-end approaches have strict requirements for the dataset used for training the neural network models and it's hard to achieve in practical use.",
"Therefore, we proposed a two-stage approach using particle swarm optimizer (PSO) candidate estimator and CNN to detect the most likely grasp.",
"Our approach achieved an accuracy of 92.8% on the Cornell Grasp Dataset, which leaped into the front ranks of the existing approaches and is able to run at real-time speeds.",
"After a small change of the approach, we can predict multiple grasps per object in the meantime so that an object can be grasped in a variety of ways."
],
[
"Introduction",
"Robotic grasping is a core function for intelligent robot to perform a variety of autonomous manipulation tasks[6].",
"Humans can instinctively perceive the unstructured environment, find out the characteristics of the grasped object and grasp the object directly.",
"But the robot can’t do that and if we model the surrounding environment around the robot, it is not only time-consuming but also hard to model.",
"Recently, with the development of deep learning,many end-to-end robotic grasp detection approaches based on CNN are developed.",
"Most of these new approaches are one-stage approaches, which find a good grasp in one step using CNN and get a high accuracy on the datasets.",
"But most of these end-to-end approaches[1][5][10] get the results in one step using trained CNN models, so those approaches have very high request to accuracy of the model which relies heavily on the quality of datasets.",
"But in practical use, it’s very hard to make a high-quality dataset.",
"Hence, in order to garner better results in practice, we need to find a robotic grasp detection approach which has a lower request to accuracy of the model.",
"Sliding window detection is a robotic grasp detection which was often used in the past.",
"The approach uses the classifier to test the selected parts of the image one by one.",
"The part which has the highest score is considered to be the best grasp of the object[10].",
"Although this approach is a little time-consuming, it can take measures such as randomization, conditional constraints and so on to reduce the influence of model’s inaccuracy on the final result.",
"Based on sliding window detection and CNN, we propose a two-stage robotic grasp detection approach which can extract the best grasp from the object's RGB image using PSO candidate estimator and CNN.",
"First, we design a robotic grasp identification model based deep CNN to determine whether the input is a good grasp.",
"Then, we find good candidate grasps using candidate estimator.",
"This candidate estimator finds proper candidate grasps using PSO algorithm.",
"Using this approach, we lower the requirement of the model accuracy.",
"We evaluate our approach on the Cornell Grasp Detection Dataset (see Figure 1).",
"Our approach achieved an accuracy of 92.8%.",
"This paper is organized as follows: in section 2, we discuss the related work.",
"In section 3, we present our identification models.",
"In section 4, we describe our candidate estimator.",
"In section 5, we present our experiment and evaluation.",
"In section 6, We present our results and then compare it with other approaches.",
"In section 7, we present our experiment on real robots.",
"Finally, we provide our conclusions.",
"Figure: Cornell Grasp Detection Dataset contains a variety of grasp objects, every object has many graspable feature labels and ungraspable feature labels."
],
[
"Related work",
"In the past few decades, with the development of robots and artificial intelligence, a number of robotic grasp detection approaches have been produced.",
"Most of the early approaches depend on the accurate information about environment and object to find good grasps.",
"Based on accurate information, some approaches[2][7][8][9] are used to realize successful and stable grasps.",
"But the disadvantage of these approaches is that many robots usually don’t know beforehand the model of the grasp object, so it’s hard to build the complex 3D models of the objects.",
"Recent years, with the development of deep learning, many researchers propose to detect the grasp using deep learning.",
"Here are the known approaches.",
"Ian Lenz proposed a two-stage approach based on sliding window detection[6].",
"First, they find some candidate grasp rectangles.",
"Then they test all these candidate grasp rectangles using a CNN with simple structure to get a set of grasps with high probability.",
"Finally, they use a complex CNN to rank all the grasps which are screened out for the first stage and find the best grasp of the object.",
"Because the sliding window detection is too time-consuming, Zhichao Wang proposed a novel robotic grasp detection system[11].",
"In this system, they first reuse the result of the former grasping detection for feedback to find candidate grasp with higher probability.",
"Then they use a CNN to evaluate the candidate grasps and find the best grasp.",
"Joseph Redmon proposed a new CNN model by simplifying AlexNet network[10].",
"Using this model, they can directly get a regression to the grasp and the classification of the grasp object.",
"With the development of deep learning, Sulabh Kumra proposed a novel CNN model for robotic grasp detection[5].",
"This model consists of two 50-layer CNNs and a shallow CNN.",
"They merge the outputs of these two 50-layer CNNs and input them into the shallow CNN to predict the grasp configuration.",
"Among these approaches, Ian Lenz and Zhichao Wang’s approaches are two-stage, but the accuracy is not good (see Table 1).",
"Joseph Redmon and Sulabh Kumra’s approaches are end-to-end,which have high requirements for the accuracy of the dataset.",
"It’s hard to achieve in practical use.",
"So we decide to attempt a better two-stage approach."
],
[
"Identification model",
"We use CNN as the grasp identification model.",
"In order to enable the identification model to learn how to distinguish between grasping and non-grasping features, we first train the network as a binary classifier.",
"After the model training, the classification probability of the softmax layer is taken as the model output.",
"In order to reduce the computing cost and increase the computing speed, we used smaller networks and tried the following models."
],
[
"Simplified AlexNet{{cite:fd1db5ac44e0e3a1862ed914398c6c0950b07e23}} model",
"Our first model is based on a simplified version of AlexNet proposed by Krizhevsky.",
"AlexNet achieved high accuracy on target recognition tasks.",
"Our first model used the AlexNet architecture, but changed the configuration.",
"The model (see Figure 2) includes 3 convolutional layers,3 maximum pooling layers, and 2 fully connected layers.",
"Finally, softmax activation function is used to activate the output.",
"Figure: Simplified AlexNet model.We trained the model.",
"And the highest training accuracy of the model is 83.39%, and the highest validation accuracy is 82.12%, which is not ideal.",
"We use the method of feature map visualization to find the problems existing in the model.",
"We output the feature map behind each layer (see Figure 3).",
"From the input image on the far left of figure 3, we can see that the key features of the image should be located in the middle of the image.",
"But in the end the critical feature our model learned (the yellow part of the image) was to the left of the image.",
"Figure: The result of Simplified AlexNet model’s feature map visualization.This shows that our model only extracts part of the texture details, while the critical features of the image are not extracted.",
"In addition, due to the existence of pooling layer, the further back, the less image information will be retained."
],
[
"Original GraspingNet model",
"Based on the problems with Simplified Alexnet model, we decided to try using a more complex model to extract deep, critical features.",
"We increase the size and number of convolution kernels.",
"In the new model (see Figure 4), the size of the convolution kernel of the three convolutional layers is 3, 5 and 7, and the number is 32,64,128, respectively.",
"In addition, we remove the Maxpooling layer after each convolutional layer to retain more information.",
"And we add the Batch Normalization operation between the convolution and activation functions of the convolution layer to speed up convergence.",
"After the two fully connected layers, we use the softmax activation function to output.",
"Finally, we select Relu as the activation function of the convolution layer.",
"Figure: Original GraspingNet model.We trained the model and achieved 98.50% train accuracy and 85.50% validation accuracy.",
"It is obvious that the model has the phenomenon of overfitting.",
"And this modelis time-consuming, it takes four times as long as the first model.",
"In addition, during the process of feature map visualization (see Figure 5), we found that Original GraspingNet model was still not so ideal in extracting critical features.",
"As can be seen from Figure 5, the critical features extracted by the model are still to the left of the picture.",
"Figure: The result of Original GraspingNet model’s feature map visualization."
],
[
"Final GraspingNet model",
"We modified our identification model based on the former 2 models.",
"A CNN usually consists of convolution layer, pooling layer and full connection layer.",
"But in Final GraspingNet model, we also remove the pooling layer and add the Batch Normalization operation between the convolution and activation functions of the convolution layer.",
"Then, we remove the full connection layer with a large number of participants and use the GAP layer instead.",
"By doing so, on the one hand, it can prevent overfitting and make the classification more natural; on the other hand, it can also increase the speed of the model.",
"So our final identification model is an 8-layer CNN, which is mainly composed of Convolution layer, Global Average Pooling (GAP) layer and the output layer (see Figure 6).",
"Figure: The architecture of the identification model.",
"There are 8 convolution layers in the hidden layer.",
"And in each convolution layer, we do convolution operation, Batch Normalization operation and Relu operation."
],
[
"Convolution layer",
"In this model, we decide to replace the large convolution kernels of Original GraspingNet with 3 by 3 convolution kernels so that it can improve the ability of the model to extract critical features while increasing the speed of the model.",
"In each convolution layer, the model needs to perform convolution, Batch Normalization, and Relu activation function operations in turn, before being input to the next convolution layer.",
"In the first convolution layer, we use 32 3*3 convolutional kernels to calculate with stride 1, and ignore the pixels of the image edge by setting padding=valid.",
"After the convolution operation, we'll do the Batch Normalization and Relu activation function operations.",
"The operation of the first convolution layer is defined as $F_{c1}(.",
"),$ the input is defined as $I_{c1},$ and the output is defined as $R_{c1}.$ Then the operation of the first convolution layer is as follows: $&&R_{c1}=&F_{c1}(I_{c1})&$ In the second, third and fourth convolution layers, we use 64 3*3 convolution kernels with stride 2 for operation.",
"After the convolution operation, we'll do the Batch Normalization and Relu activation function operations.",
"This operation is defined as $F_{c2}(.",
"),$ and the output of the second, third and fourth layer is defined as $R_{c2},R_{c3},R_{c4},$ then the operation of the second, third and fourth convolution layer is as follows: $&&R_{c2}=&F_{c2}(R_{c1})& \\\\&&R_{c3}=&F_{c2}(R_{c2})& \\\\&&R_{c4}=&F_{c2}(R_{c3})&$ In the last four convolution layers, we use 128 3*3 convolution kernels to operate with stride 2.",
"This operation is defined as $F_{c3}(.",
"),$ and the output of the last four convolution layers is defined as $R_{c5},R_{c6},R_{c7},R_{c8},$ then the operation of the last four convolution layers is as follows: $&&R_{c5}=&F_{c3}(R_{c4})& \\\\&&R_{c6}=&F_{c3}(R_{c5})& \\\\&&R_{c7}=&F_{c3}(R_{c6})& \\\\&&R_{c8}=&F_{c3}(R_{c7})&$"
],
[
"GAP layer",
"The Global Average Pooling operation conducted by GAP layer is defined as $F_{GAP}(.",
"),$ and the output is defined as $R_{GAP},$ then, $&&R_{GAP}=&F_{GAP}(R_{c8})&$"
],
[
"Output layer",
"Before the final output of the model, we first use the convolution of 1*1 to perform ascending and reducing operations on the output of the GAP layer to increase the information combination across channels.",
"In addition, we will use dropout to further prevent overfitting of the model before ascending and reducing operations.",
"We define the dropout operation as $F_{D}(.",
"),$ the 1*1 convolution ascending dimension operation as $F_{A}(.",
")$ and the output of the dropout operation and ascending dimension operation as $R_{A},$ then, $&&\\quad R_{A}=&F_{D}(F_{A}(R_{GAP}))&$ We define 1*1 convolution reducing operation as $F_{R}(.",
")$ and the output of the dropout operation and ascending dimension operation as $R_{R},$ then, $&&\\quad \\, R_{R}=&F_{D}(F_{R}(R_{A}))&$ Finally, we activate it using the softmax function, which outputs the results of the model identification in probabilistic form.",
"We define the operation of the softmax function as $F_{S}(.",
")$ and the output of the output layer as $O,$ then, $&&\\quad \\ O=&F_{S}(R_{R})&$"
],
[
"Model training",
"The training process of the convolutional neural network model is to find the global optimal solution in the parameter space.",
"In this process, we may encounter many local optima, but we have to skip the local optima to find the global optima in the parameter space to train the best model.",
"Therefore, we used the stochastic gradient descent method to train the CNN model and set the learning rate in sections.",
"During training (see Figure 7), the learning rate decayed every 60 epochs.",
"In this way, when the model starts training, because the training speed is faster, some local optima are skipped.",
"Finally, when the model is trained near the optimal solution, the learning rate converges to the optimal solution of the model at a very low learning rate.",
"In addition, during the training, we also increased the value of momentum.",
"On the one hand, it is more helpful for the model to get rid of the local optimal solution, on the other hand, it can also increase the stability of model training to some extent.",
"Figure: Final GraspingNet training curve"
],
[
"Five-dimensional representation",
"There are mainly two kinds of grasp representation.",
"One is a seven-dimensional representation which is proposed by Jiang Yun[12].",
"The other is a five-dimensional representation which is proposed by Ian Lenz[6].",
"The five-dimensional representation is a simplification of the seven-dimensional representation.",
"In order for computational convenience, we use the five-dimensional representation.",
"In the five-dimensional representation, the grasp is a rectangle which is determined by location, size and direction as: $g = \\left\\lbrace x,y,\\theta ,h,w \\right\\rbrace $ $(x,y)$ is the location of a rectangle, $\\theta $ is the direction of the rectangle relative to the horizontal axis,h is the length of the gripper,w is the width of the gripper opening(see Figure 8).",
"Figure: five-dimensional grasp representation"
],
[
"Candidate estimator",
"In the sliding window detection, every image to be detected is a large and continuous state space.",
"If we use enumeration approach to search the grasp, we need to detect a great number of candidate grasps.",
"For example, if we detect an image whose size is 224 by 224, we may need to consider 50176(224*224=50176) locations.",
"Suppose that there are 70 different lengths of grippers, the width of the gripper opening ranges from 30 to 100 and that the angle of the gripper ranges from 0° degree to 180° and the angle changes in one degree, we should consider 632,217,600(50176*70*180 = 632,217,600) candidates.",
"If we do that, the cost of computing is very large.",
"In order to solve this problem, we design a candidate estimator based on PSO.",
"This estimator transfers the search of the grasp to an optimization problem.",
"The parameters of the optimization problem are the five-dimensional representation of every candidate.",
"The processed image is the state space of the optimization problem and the constraint of the problem is the edges of the image.",
"Our optimization goal is to find a candidate to optimize the score function of the grasp.",
"As for the score function, we use the grasp identification model in section 3.",
"As can be seen above, the grasp identification model outputs the probability of grasping.",
"Then, we use PSO to solve the problem."
],
[
"PSO algorithm",
"PSO is a kind of stochastic optimization technique based on population, which was proposed by Eberhart and Kennedy in 1995.",
"PSO algorithm mimics the clustering behavior of animals such as insects, herds, birds, and so on.",
"These populations search for food in a cooperative way.",
"Every particle of the population constantly changes the way it searches by learning from its own experience and that of other members.",
"During the search of the algorithm, each particle searches for the optimal solution separately in the search space, and marks it as the current individual extremum.",
"Then, the particle shares its individual extremum with other particles in the particle swarm and the particle swarm finds the optimal individual extremum as the current global optimal solution.",
"All particles of the particle swarm adjust their speed and position according to their current individual extremums and the current global optimal solution shared by the particle swarm.",
"Therefore, the algorithm can be used to deal with the optimization problem of multivariable functions with multiple local optima.",
"In this article, we use a particle $x_{i}$ to represent a candidate.",
"Every particle has a speed $v_{i}^{k}$ at iteration $k.$ The movement of the particle is determined by its current individual extremum $p_{best_{i}}$ and the current global optimal solution $g_{best}.$ The updated formula of particle velocity and position is as follows: $&&v_{i}^{k+1}=wv_{i}^{k}+c_{1}&r_{1}p_{best_{i}}+c_{2}r_{2}g_{best}&\\nonumber \\\\&& x_{i}^{k+1}=&x_{i}^{k}+v_{i}^{k+1}& \\nonumber $ $w$ is the inertia factor.",
"$c_{1}$ and $c_{2}$ are accelerating factors.",
"$r_{1}$ and $r_{2}$ are 2 random numbers, which are used for increasing the randomness of the search.",
"The is shown in Algortithm 1.",
"[h] PSO Algortithm [1] $g\\_fit=0,initial\\_time=0$ .",
"$(g\\_fit<init)and(initial\\_time<max\\_init)$ Randomly initialize particle swarm $x_{i}$ ,$v_{i}$ ; Use identification model to caculate score for $x_{i}$ ,$v_{i}$ ; Update $p\\_fit_{i},g\\_fit$ ; $initial\\_time=initial\\_time+1$ ; $(g\\_best<prob)and(iteration<max\\_iter)$ $v_{i}^{k+1}=wv_{i}^{k}+c_{1}r_{1}p_{best_{i}}+c_{2}r_{2}g_{best}$ ; $x_{i}^{k+1}=x_{i}^{k}+v_{i}^{k+1}$ ; Use identification model to caculate score for $x_{i}$ ,$v_{i}$ ; Update $p\\_fit_{i},g\\_fit$ ; Update $p\\_best_{i},g\\_best$ ; $iteration=iteration+1$ ; Output the best candidate grasp.",
"In Algorithm 1,$g\\_fit$ is the highest score of all particles and $p\\_fit$ is the highest score of every particle.",
"$initial\\_time$ is the iteration during initializing and $iteration$ is the number of searches.",
"$max\\_init$ is the maximum number of initializations and $max\\_init$ is the maximum number of searches.$init$ is the minimum threshold that $g\\_fit$ should meet during initialization and $prob$ is the minimum threshold that $g\\_fit$ should meet during searching.Through this candidate estimator, we can find a good grasp fast.",
"The iterative convergence process of particles is as follows (see Figure 9): the particles are randomly distributed inside the image during initialization.",
"And as the algorithm iterates, the particles gradually gather near the optimal location.",
"If the center of a particle is near the edges of the image during initialization or iteration so that some components of the particle exceed the boundaries of the image when it is segmented, the particle will be eliminated and we will add a new particle randomly.",
"The convergence condition is that the number of iterations exceeds the maximum number of iterations or the global optimal particle adaptation value reaches the threshold.",
"Figure: PSO iteration process.The first behavior of the image is the current global best position, and the second behavior is the current position of all particles.And if we make a small change to candidate estimator, the estimator can output the particles whose scores are above a certain threshold and rank high among all the particles at the same time so that we can multiple grasps."
],
[
"Improvements to PSO",
"In order to make the PSO algorithm converge faster, we add some constraints to the particles during initialization or iteration.",
"First, we hope that the particles are distributed as close to the object as possible.",
"So when initializing particles, we distribute the particles within the center of the image and there has to be a particle with a score greater than 0.7.",
"Second, we get the histogram of gray image corresponding to the RGB image before initializing the particles.",
"According to the histogram which we get, we can roughly estimate the size of the target object.",
"Then, we will initialize the particles differently according to the size of the target object.",
"We can make sure the particle size is proper and reduce the effect of unproper particle size on particle score so that we can accelerate the convergence of the algorithm.",
"Finally, we limit the size, aspect ratio and area of particles to a certain extent.",
"If a particle is out of the limit during initialization or iteration, we will get the particle size back to a reasonable range by multiplying it by a corresponding correction factor."
],
[
"The Cornell Grasp Detection Dataset",
"The dataset contains 885 images, 400 objects, 240 different categories and corresponding grasp labels.",
"This data set is specially designed for parallel gripper.",
"Each image contains multiple grasp labels.",
"The labels are comprehensive and varied in orientation, location and scale, but it can't contain all possible grasps.",
"And there are also errors in the labeling of partial positive samples and negative samples.",
"Nonetheless, they are excellent examples of grasps.",
"So we choose to analyze and evaluate based on the dataset."
],
[
"Image preprocessing",
"Before inputting the image into the identification model, we preprocess the image.",
"We clipped the original image to the center of the object, keeping 300 by 300 pixels.",
"Then in order to compare with the previous work, we scaled the image to 224 by 224 pixels.",
"As for the grasp identification model, we set the rectangular image horizontally along the long axis, fill the top and bottom ends with 0 pixels and scale the image to 24 by 24 pixels before inputting the feature image to the model.",
"Before training the model, we made necessary augmentation of the training set, including some operations such as translation, scaling, and rotation."
],
[
"The grasp identification model",
"Before the input of the feature image into the model, we will preprocess the image, scale the image to 24*24 and fill it.",
"So we can input the preprocessed feature image directly.",
"And our identification model outputs in the form of probability.",
"And the output is probability of the network softmax layer classification of the graspable features (see Figure 10).",
"Figure: The output of the identification.If we use direct classification, take the third example in Figure 10.",
"The model will classify it as graspable because the output is 0.617 which is greater than 0.5.",
"If the model only outputs in the form of probability, it will not impact on subsequent iterations.",
"So we don't directly evaluate models by how accurate they are, and it can be used as a reference for selecting identification model."
],
[
"Multigrasp detection approach",
"We have made a small modification to the original method.",
"Instead of only printing the highest-scoring grasps, we output several of the highest-scoring grasps at the same time.",
"These grasps converge to different positions of the target object, so we can have multiple ways to grab the same object."
],
[
"Grasp evaluation",
"If we evaluate on the Cornell Grasp Detection Dataset, there have been two different evaluation criteria.",
"The first approach is point evaluation, which evaluates whether the predicted grasp is a successful grasp through judging whether the distance between the center of the predicted fetch point and the center of the tag is below a certain threshold.",
"There has been a lot of discussion about this evaluation criteria.",
"The biggest problem with this evaluation criteria is that it doesn't take into account the size and angle of the grasp.",
"And under this evaluation criteria, previous work has rarely revealed thresholds for evaluating comparisons.",
"So it is difficult to compare results.",
"Therefore, we do not use such evaluation criteria.",
"The second approach is rectangle evaluation.",
"This approach considers the whole grasp rectangle.",
"In this evaluation criteria, satisfying the following two points is considered to be a correct grab point: The angle between the grasp and the graspable label should not exceed $30^{\\circ }.$ The intersection ratio of grasp and the graspable label should be no less than 20%.",
"In our work, we use the second approach to evaluate the model.",
"Although we use a more proper approach to evaluate, there are still some problems.",
"Although some predicted grasps are not evaluated to meet the above two points, they are actually graspable(see Figure 11).",
"Figure: The first line is the grasps labeled by the dataset, and the second line is the result of our detection.",
"Take the results of column 1 and column 4 from the left in the picture as an example.",
"Although they do not meet the requirements of rectangle evaluation, it can still be considered to be graspable."
],
[
"Results and comparisons",
"We tested on a computer configured with a single CPU (i9-9900k 3.6G), a single GPU (NVDIA 1080Ti), and 16 gigabytes of memory.",
"According to the evaluation criteria in part iii, we achieved good results in the Cornell dataset."
],
[
"Comparison of different approaches",
"We compared the accuracy of our method with the results of previous experiments on the Cornell dataset (as shown in Table 1).",
"Our approach achieved a success rate of 92.8%.",
"Among all known approaches, our success rate is only slightly lower than Guo et al.",
"Compared to the existing two-stage method, our method improved the success rate by 11 percentage points.",
"Table: We compare our approach with the previous work.In terms of speed, our algorithm needs 378ms to process an image.",
"Although our method is still much slower than the one-stage method of Kumra et al.",
"[5] and Redmon et al.",
"[10], it should be able to run in real time."
],
[
"The result of multigrasp",
"In the front, we mentioned that with a small change in our method, we can get multiple grab points at once, which can provide multiple solutions for grabbing an object.",
"In this regard, we also carried out relevant experiments.",
"The results are seen in Figure 12.",
"Figure: The result of the multigrasp approach.From the figure, we can see that the multigrasp method provides multiple correct grasps for each target object, and we also make statistics on the success rate and time consumption (see Table 2).",
"In the evaluation of multigrasp detection, as long as one predicted grasp candidate of an image meets the requirements, we consider that the detection of capture points of this image is successful.",
"It can be seen from the results that compared with the original method, the method of mutligrasp has a higher success rate.",
"The increase in average time was only 5ms.",
"Table: We compare our approach with the previous work.In the multigrasp approach, we output multiple grasps at different locations.",
"The fault tolerance of the identification model is further improved, because even if the model erroneously judges an ungraspable candidate to be the highest score of a target object, it is possible to find a correct grasp candidate from other grasps in the output."
],
[
"Summary",
"We show that deep learning model can better learn grasping features, and PSO optimization algorithm can well solve the problem of multi-local optimal value.",
"We used an uncomplicated network, greatly reducing computing costs and making the model easy to train and deploy.",
"We turn the detection problem into an optimization problem for processing, so that the detection result is not too dependent on label data, but has multiple possibilities in the whole state space.",
"We use PSO algorithm to deal with this kind of multi-variable complex function optimization problem with multi-local optimal advantages, which has low computational complexity, fast convergence speed and can find a better solution in considerable time.",
"In the problem of grasp detection, we combined deep network learning characteristics and PSO optimization algorithm to solve the problem quickly, and obtained very advanced results."
],
[
"The introduction of the experiment",
"After our approach worked well on the lab computer, we decided to transplant it to a real robot for testing.",
"The process is as follows (see Figure 13): first, we selected six objects that did not appear in the dataset.",
"Then, we use the watershed algorithm to process the original image taken in the robot camera to extract the target object, and place the extracted object in the same position on the light white background image.",
"Finally, we use our method to detect the capture point of the image obtained in the previous step, and draw the grasp on the original image.",
"Figure: The process of experiment on the robot."
],
[
"Result and discussion",
"We used our method to detect the grasping points on the robot and obtained the following results (see Figure 14): Figure: The result of the single grasp detection.While there are no major differences in kind between the objects we select and those in the Cornell dataset, there are significant differences in detail.",
"However, this did not have a great impact on our detection results.",
"Among the six selected objects, except the pink plastic blocks in the second row and the first column, the other predicted grasps were correct.",
"This shows that our detection approach can not only be used on the robot, but also achieve good results in the case that the unfamiliar object is not marked, as long as the object in the data set belongs to the same category as the unfamiliar object.",
"In practical use, we can select a few typical objects in the same kind of objects to mark accurately, so as to achieve a good effect in the detection of grasping points of most objects in this class, thus reducing the number of objects required by the data set, conducive to the application of our method in the actual detection."
],
[
"Conclusion",
"In this article, we present a robotic grasp detection approach based on deep learning and PSO algorithm.",
"We transfer the detection problem to an optimization problem.",
"We first use the CNN to learn graspable feature.",
"Then we take the identification model as the objective function of optimization, take the whole image as the state space, take the parameters of the rectangle identification points as variables and use the PSO algorithm to solve the optimization problem.",
"In terms of accuracy, our algorithm is at the top of the Cornell Grasp Detection Dataset.",
"Moreover, our approach is two-stage.",
"Compared to end-to-end approach, it has a lower request to accuracy of the model and the dataset, which contributes to our practical use.",
"In the future work, we will improve the speed and robustness of the algorithm so that it can achieve better results in real industrial deployment."
],
[
"My Appendix",
"All the codes used for experiment can be downloaded at https://github.com/KathylinLawes/Robotic-grasp-detection.",
"And the color should be used for all figures in print.",
"figs/1.jpg Zhe Chu is an undergraduate at northwestern polytechnical university.",
"He came to northwestern polytechnical university in 2017.",
"He is a member of the soccer robot base at northwestern polytechnical university.He is interested in robotics,machine learning and computer vision.",
"figs/16.jpg Mengkai Hu received the bachelor’s degree in electronics and information engineering from Northwestern Polytechnical University in 2017.",
"He is now pursuing the master's degree in School of Electronics Engineering and Computer Science, Peking University.",
"His research focus on signal processing.",
"figs/15.png Xiangyu Chen received the B.E.",
"degree and the M.E.",
"degree from Northwestern Polytechnical University, Xi’an, China, in 2017 and 2020 respectively.",
"Currently he is a research assistant in Shenzhen Institutes of Advanced Technology, Chinese Academy of Science.",
"His research focus on robotics, deep learning and computer vision."
]
] | 2011.14123 |
[
[
"Discrete Math with Programming: A Principled Approach"
],
[
"Abstract Discrete mathematics is the foundation of computer science.",
"It focuses on concepts and reasoning methods that are studied using math notations.",
"It has long been argued that discrete math is better taught with programming, which takes concepts and computing methods and turns them into executable programs.",
"What has been lacking is a principled approach that supports all central concepts of discrete math -- especially predicate logic -- and that directly and precisely connects math notations with executable programs.",
"This paper introduces such an approach.",
"It is based on the use of a powerful language that extends the Python programming language with proper logic quantification (\"for all\" and \"exists some\"), as well as declarative set comprehension (also known as set builder) and aggregation (e.g., sum and product).",
"Math and logical statements can be expressed precisely at a high level and be executed directly on a computer, encouraging declarative programming together with algorithmic programming.",
"We describe the approach, detailed examples, experience in using it, and the lessons learned."
],
[
"Introduction",
"Discrete mathematics is the foundation of computer science.",
"The central concepts in it—from logic and reasoning, to sets and functions, to sequences and recursion, to relations and graphs—are essential mental tools for modeling real-world objects and developing programming solutions, whether for basic problem solving or for advanced software development.",
"At the same time, the core discipline of computer science is program development.",
"It is taught in introductory programming sequences, branching at upper levels to projects in courses such as databases, networking, security, and especially compilers that have to deal with sophisticated discrete structures for representing computer programs themselves.",
"As a result, the two primary courses in computer science are typically discrete math and program development, the two largest-by-hours areas in [32], but with very different course activities: The former teaches concepts and reasoning methods, with the help of paper and pencil or text formatting tools for writing math notations and natural language.",
"The latter emphasizes transforming concepts and computing methods into executable programs, overcoming idiosyncratic issues of programming languages and systems.",
"Students can see some common concepts underlying both, e.g., sets and sequences in the former are realized as some collection types in the latter.",
"However, there are no direct, precise connections between the two kinds of activities—the same concepts are studied in completely different contexts, and computations involving these concepts are expressed completely differently in math notations than in most programming languages, especially Java, the current dominant language used in teaching program development.",
"Clearly, the instruction of discrete math and programming should be integrated to connect theory with practice, to let them reinforce each other, and to help students better understand and master both.",
"This in fact has been pursued, in a great deal of prior work and effort, as discussed in Section .",
"What is lacking is a principled approach for doing this, an approach that covers all central concepts in discrete math and promotes disciplined uses of these concepts in problem specifications and programming.",
"This paper presents such a principled approach.",
"The approach has four main features: It covers all central concepts in discrete math, especially including the fundamental, challenging topic of predicate logic.",
"It is based on a powerful language with precise syntax and semantics, directly connecting math notations with programming language notations.",
"It supports clear and precise specifications of problem statements using any combination of the language elements for all concepts.",
"It promotes declarative expression of complex computation problems but also supports easy expression of algorithmic steps.",
"The language we use, referred to as DA in this paper, extends the Python programming language.",
"It supports proper logic quantification, as well as declarative set comprehension (also known as set former and set builder) and aggregation (e.g., sum and product) over sets and sequences.",
"Declarative specifications of problem statements are directly executable in Python with the module for DA extensions, just as algorithmic steps are.",
"This approach was used in teaching Foundations of Computer Science at Stony Brook University in Spring 2020, entirely as extra-credit programming problems added to regular homework assignments.",
"Our results and analysis of using this approach support broader deployment, and we give suggestions for future adoption."
],
[
"Related work",
"Many methods for teaching discrete math have been studied since the early ages of computer science, e.g., [6], [24], especially those involving programming, e.g. [21].",
"Naturally, algorithms introduced in discrete math are frequently selected for implementation in a programming language.",
"For example, students could be required to implement an algorithm of their choosing in any programming language [20].",
"Even an entire follow-up lab course could be developed, e.g., for students who are already familiar with languages such as Java [30].",
"However, these programming components do not require writing declarative, logical specifications that are central to discrete math.",
"There have been many uses of various declarative languages, not only functional and logic languages, but also the SQL database language.",
"Examples include using FP, ML, and Prolog in a complementary fashion [11], using SML extensively to write recursive functions over lists for many set and logic operations [34], and using SQL to program with sets and relations and especially its EXISTS operator [25].",
"All of these efforts had to get around the lack of real universal and existential quantifiers.",
"Other approaches used dedicated logic and modeling systems.",
"For example, an automated system could take a list of facts, and generate a list of support facts to give students insight into how first-order logic works [23].",
"The Alloy modeling language [36] is excellent for writing specifications using sets, relations, and predicate logic and then finding models that satisfy a specification [33], although it suffers from issues with recursion.",
"The powerful SMT solver Z3 was also used, for solving puzzles [12].",
"These are farther away from introductory programming.",
"There are also many studies of using supporting tools, especially visual tools and more powerful proof tools.",
"For example, specialized programs were used for visualizing graphs and algorithms that do depth-first search, etc.",
"[2], for learning rules in solving visual logic puzzles [4], for proof editing with helpful checks [3], and for giving meta-level support [22].",
"There were also efforts that encourage students to make their own tools, e.g., a proof checker of a natural deduction system, a database management system, a propositional logic proof system, and a symbolic execution engine [15].",
"Additionally, dedicated books have been written on the topic of teaching discrete math with a programming language, e.g., C [1] and Python [31], [26].",
"However, these do not cover writing logical and declarative specifications for the central topic of predicate logic, instead opting for writing iterations or recursions for traditional written exercises on the subject, or avoiding the subject altogether."
],
[
"Approach",
"We first discuss all central topics that a principled approach must cover.",
"We then describe the use of a powerful and precise language, the support for clear and precise problem specifications, the fostering of declarative as well as algorithmic programming, and the use of programming as an enhancement.",
"Central topics in discrete math.",
"Discrete math is typically one of the first two courses in computer science, the other being introductory programming.",
"Despite many textbooks written and used for the subject, the central topics are well-known, as captured in commonly used textbooks, e.g., [8], [7], [27], [5], as well as [32].",
"Logic.",
"This includes propositional logic, the more general predicate logic that includes quantifiers, and proof methods.",
"Sets, functions, and relations.",
"This includes definitions, operations, and properties over sets, functions, and relations.",
"Sequences and recursion.",
"This includes definitions of sequences, summation and product forms, recursive formulas, and proofs by induction.",
"Additional topics are often included, but they are generally special, expanded cases of the core topics.",
"Typical such topics are: Graphs and trees.",
"A graph is essentially a set of vertices plus a binary relation on the vertices.",
"A tree is simply a graph where each vertex has one incoming edge and multiple outgoing edges forming no cycles.",
"Counting and probability.",
"Counting corresponds to the cardinality of sets of interest.",
"Probability is essentially the count of elements of interest divided by the count of all elements.",
"All topics above include aspects of reasoning and proofs, as well as computations and algorithms.",
"These aspects are often included in the expositions of the topics above, e.g., proofs with logic in [27] and both proofs and algorithms with many topics in [8], [7].",
"These aspects are sometimes also covered as separate topics, e.g., proofs in [32], [5] and algorithms in [27].",
"In the expositions of all these topics, examples and applications from number theory are often used [27], [7], and corresponding concepts such as Boolean algebra are often introduced [27], [7], [5].",
"Among all topics in discrete math, logic is typically viewed as the most fundamental topic—it is usually the first topic to study [7], [27], [5] and is also emphasized particularly as driving the entire subject [10].",
"Therefore, a principled approach for discrete math with programming must cover logic as well as all other topics.",
"Powerful and precise language.",
"To cover all central topics, one can see the variety of discrete values, operations on them, and properties about them that must be expressed.",
"To express all of them precisely and unambiguously, and to connect them with programming, a powerful language with precise syntax (i.e., forms) and semantics (i.e., meanings) is needed.",
"We describe the use of such a language that combines the advantages of the two kinds of languages used in introductory courses: Traditional mathematical notations used in discrete math.",
"These notations are high-level and concise.",
"However, they generally do not have formal semantics, and allow loose usage with no automated checking for syntax or semantics.",
"Programming languages, such as the dominant language Java, used in introductory programming.",
"These languages have precise syntax and semantics and are automatically checked for the syntax and executed following the semantics.",
"However, they are lower-level, tedious, and verbose.",
"The language we use, DA, extends Python.",
"Python is well-known for being significantly more concise and higher-level than languages like Java and C/C++, and is already widely used by scientists and high-schoolers alike and taught to non-CS and novice students, demonstrating its power and ease of use.",
"Python already supports sets and sequences, comprehensions over them, and generator expressions with operators all, any, sum, max, etc.",
"on top of commonly used loops and recursive functions for programming at a high level like pseudocode.",
"The DA extensions we use support the following main language constructs that are not in Python, but are essential for expressing all central concepts clearly and directly.",
"Proper universal and existential quantifications.",
"These capture the exact meaning of quantified statements (that use universal quantifier $\\forall $ and existential quantifier $\\exists $) in predicate logic.",
"Comprehensions over sets and sequences with logic/pattern variables.",
"These correspond to set builder notations for forming expressions over sets, relations, and sequences.",
"Aggregations over sets and sequences with logic/pattern variables.",
"These are similar to comprehensions but support summation, product, counting, maximum, and minimum.",
"The quantification forms are built on the best previous languages that support quantified expressions, SETL [13], [29], designed exactly as a set-theoretic programming language, and ABC [9], designed exactly to teach introductory programming.In fact, SETL is one of the earliest and most powerful programming languages.",
"ABC is a descendant of SETL, and Python is a descendant of ABC and C [35].",
"These are discussed in detail in Section .",
"Use of what we call logic variables, or pattern variables, in comprehension and aggregation, as well as quantification, was motivated by a history of informal use in writing declarative set expressions, e.g., [18], [19], [14], which led to its precise formalization with patterns by Liu et al [17], [16].",
"These are discussed in Section .",
"Together, the extended language DA supports simpler and clearer problem specification as well as expression of computations and algorithms.",
"In particular, logical statements about mathematical concepts can be directly executed for computation and checking, unlike math notations on paper that are disjoint from low-level programs executed on computers.",
"The only aspect not supported in DA is formal development of complete proofs, but such proof development is well-known to be challenging even for the best experts.",
"Support for easier writing of proofs remains a direction for future study [28].",
"Clear and precise problem specification.",
"With a powerful language for expressing logical statements, problem specifications can be written more easily and clearly.",
"Whether for computations or for proofs, precise problem specification is the most critical task.",
"The language we use supports such specification for any aspect that needs it.",
"Input specification.",
"This specifies all sets, functions, etc.",
"given, plus logical statements specifying additional relationships among the given structures.",
"Output specification.",
"This similarly specifies the sets, numbers, etc.",
"to be produced, plus logical statements specifying how the output is related to the input.",
"Auxiliary value specification.",
"This specifies auxiliary sets, functions, etc.",
"to use, plus logical statements relating them to input, output, and to each other.",
"The language constructs we use for quantification, comprehension, and aggregation can be arbitrarily combined in writing logical statements.",
"Section gives example specifications.",
"Declarative as well as algorithmic programming.",
"Dominant programming languages such as Java and C/C++ do not support quantification, comprehension, or aggregation, so they must be programmed using iteration or recursion.",
"For commonplace programming taught in courses and used in practice, the most important language constructs for expressing computations are iterations carried out in loops, where assignments and conditions are used to set and test values for starting and ending the loop and to update variables in between.",
"Languages like Python with DA extensions support quantification, comprehension, and aggregation as built-ins, which are compiled into loops automatically for execution.",
"When a specification specifies an output using these constructs, which can be executed automatically, the specification serves as a good example of higher-level, declarative programming.",
"Section gives examples.",
"Programming as enhancement.",
"When teaching discrete math with programming in the DA language, all discrete structures, operations on them, and properties about them can be expressed directly and precisely, and then executed directly.",
"The programming part is exactly to write these precisely and directly.",
"To ensure that students still learn and build their full skills at least as well as they learn discrete math without programming, we used three general methods: (1) assign the programming part several days later than the non-programming part, (2) give in-class exercises for doing the most important of these problems before the programming part was assigned, and (3) assign more or larger problems for the programming part so as to benefit more from automatic execution.",
"Method (1) allows students to do homework problems first in their head.",
"Method (2) forces students to do, or at least try, homework problems earlier.",
"Method (3) helps show that programming is an enhancement."
],
[
"Predicate logic with programming",
"The topic on logic starts with propositional logic.",
"It is about using propositions and logical operators.",
"Programming with these is straightforward.",
"Propositional logical statements can be directly and precisely written in commonly-used programming languages: use program variables for propositions, and use Boolean operators for conjunction, etc.",
"A small issue is that implication (e.g., p implies q) is not in common programming languages, but it can be easily defined or directly written using its equivalent form with negation and disjunction (i.e., not p or q).",
"Predicate logic extends propositional logic to include the use of predicates with arguments and quantifiers quantifying over values of these arguments.",
"It is significantly more sophisticated and cannot be expressed directly in most programming languages, not even Prolog.",
"Our approach for teaching predicate logic with programming is as follows: (1) introduce the language, emphasizing quantifications, (2) write specifications, especially through examples, and (3) execute the specifications, directly in Python with DA extensions.",
"Language.",
"Logical operators in Python are used, because they are simple and easy to read: and for conjunction ($\\,\\wedge \\,$ in math notation), or for disjunction ($\\,\\vee \\,$), and not for negation ($\\sim \\,$ or $\\lnot $).",
"Quantifications, for writing statements with universal and existential quantifiers ($\\,\\forall \\,$ and $\\,\\exists \\,$, respectively), are first discussed in class as usual.",
"Then, only two additional slides, shown in Figure REF , are discussed.",
"Figure: Slides introducing precise quantifications and relationships.The first slide gives an overview relating logic and practice of programming, including programming languages related to Python.",
"In particular, Python and C are widely-used languages in practice, and Python's roots ABC and SETL were meant for beginners and actually based on set theory, respectively.",
"It ends with the name of the language to be used, DistAlgo [17], [16], plus a tentative new name, Alda, for it, abbreviated for both as DA in this paper.",
"The second slide shows the precise language constructs for quantifications: the universal quantification means that each element x in set S has property P(x), and the existential quantification means that some element x in S has property P(x).",
"The first line with $\\,\\forall \\,$ and $\\,\\exists \\,$ is from the textbook [7], [8] used for the course.",
"The second and third lines show a few other math notations.",
"The first line with each and some is from ABC and ideal for reading.",
"The next line is the form in DA as implemented in Python.",
"The last two lines show the forms in Python and SETL.",
"All ABC, DA, and SETL forms match the math notations better than the Python form.",
"More critically, the constructs in ABC, DA, SETL, and informally in math notations—but not in Python—also give a witness: When the existential quantification is true, variable x is bound to a value in set S that makes P(x) true.",
"This powerful feature is important for expressing search using math and logic at a high level [13], [29].",
"We see that the second slide directly and precisely connects the many different math notations with the best, easy-to-read programming language constructs.",
"No more study of Python or DA was done in class, for three reasons.",
"(1) Time diverted from teaching all regularly taught materials should be minimized.",
"(2) The homework gave program files that contained examples.",
"(3) We were confident in the power and ease of Python and DA extensions from past teaching experience.",
"Specification.",
"We show the use of DA quantifications in specifying examples with different combinations.",
"Two main examples are used in the textbook: a college cafeteria with students choosing items at different stations, and Tarski's world as a grid of blocks of various colors and shapes.",
"We name them cafe and tarski, respectively, and use parts of cafe as examples.",
"For cafe, the textbook provides a figure with example students, food stations, items in those stations, and the items each student chose.",
"It then lists four statements in math notations and discusses their truth values in English.",
"The first example has: $\\,\\exists \\,$ an item $I$ such that $\\,\\forall \\,$ students $S$, $S$ chose $I$.",
"“There is an item that was chosen by every student.” We write the corresponding precise statement as an example in the program file given to students: some(I in items, has= each(S in students, has= chose(S,I))) The homework then asks that several statements written in math notations or English be written in DA.",
"For example, an exercise problem in the textbook was used in the homework, asking for the truth values of a list of statements.",
"The first statement is: $\\,\\forall \\,$ students $S$, $\\,\\exists \\,$ a dessert $D$ such that $S$ chose $D$ The expected answer is: each(S in students, has= some(D in desserts, has= chose(S,D))) In total, in the programming part on predicate logic, 5 statements for cafe and 2 for tarski were asked, all involving nested alternating quantifiers as the example above, and some also involving and, or, and not.",
"Execution.",
"The most practically attractive aspect in teaching discrete math with programming is that programs are executable to give the specified meaning.",
"We show execution of the statements written, using the cafe example.",
"To execute a statement for cafe, the sets of stations, items, and students with their choices for predicate chose must be given.",
"The given program file for cafe contains the following definitions, followed by example quantifications as discussed earlier, plus printing of the values of the quantifications.",
"pystyle # page 87 of textbook: Example 3.3.3.",
"# given knowledge from the first paragraph and shown in Figure 3.3.2: salads = {'green salad', 'fruit salad'} main_courses = {'spaghetti', 'fish'} desserts = {'pie', 'cake'} beverages = {'milk', 'soda', 'coffee'} stations = [salads, main_courses, desserts, beverages] choices = { 'Uta': {'green salad', 'spaghetti', 'pie', 'milk'}, 'Tim': {'fruit salad', 'fish', 'pie', 'cake', 'milk', 'coffee'}, 'Yuen': {'spaghetti', 'fish', 'pie', 'soda'} } students = choices.keys() # helper set and function, to capture English more easily: items = setof(item, sta in stations, item in sta) def chose(student, item): return item in choices[student] Later extra-credit programming assignments gradually include writing more or all of such sets and definitions by students.",
"Later assignments also tell students to use witnesses, e.g., print the value of I in the first precise statement earlier in this section.",
"The resulting programs as well as the given programs can be executed directly in Python with the module for DA extensions.",
"Students are given two files, cafe.da and tarski.da, and two commands to run, python -m pip install pyDistAlgo for installing DA, and python -m da cafe.da for running the cafe example.",
"Programming after thinking and preparing.",
"Programming should be an enhancement to traditional discrete math coursework, while minimizing the chores of working with a computer system.",
"The homework instructions try to accomplish this.",
"For the predicate logic part, there are two prerequisites to doing extra-credit programming.",
"First, for the cafe problem, students were given in-class exercises to do the exercise problem in the textbook that will be used in the programming part, before the programming part was posted.",
"This requires them to think about the problem more before programming.",
"Only 3 of the 5 statements for cafe were given for the written part of the homework.",
"Second, students should install Python and be able to run it on a command line.",
"This was not needed if students used machines in the computer lab."
],
[
"Other topics",
"The same approach was also used in teaching all other topics.",
"We highlight the power of DA extensions to facilitate this.",
"Language.",
"Topics other than predicate logic require simpler language constructs, and no dedicated slides are used—only one or two slides for each topic with some notes on the side.",
"Figure REF shows two examples.",
"Most such slides only have one or two lines of notes on them; these slides are the exceptions.",
"Figure: Slides on set properties and operations, and on programming withsequences.The first slide shows set operations and properties, all of which can be directly written in DA.",
"The notes added on the bottom-right use set intersection as an example, showing it is in given file sets.da, and giving the exact comprehension constructs for expressing it in DA, in Python (thus in DA too, as DA extends Python), and in an ideal notation.",
"The comprehension in DA means the set of x satisfying membership clauses x in A and x in B.",
"In general, any membership and Boolean conditions can be written; the semantics of DA automatically avoids the well-known Russell's paradox.",
"In particular, x is a logic variable, meaning that different occurrences of it in a comprehension automatically have the same value.",
"So either order of the two membership clauses has the same meaning, and the DA compiler can decide how to implement them efficiently.",
"It is critical to note that the comprehension in Python, with for followed by if, is different (and so are comprehensions in SETL): it means to iterate over elements in A and, for each element x, if it is also in B, put it in the resulting set.",
"Aggregations such as sum and product over sets and sequences can also be expressed easily in DA.",
"For example, given a set of sequence S, sumof(x, x in S) means the sum of all elements of S. The second slide shows pseudocode for programming with sequences.",
"The notes added on the right show corresponding constructs in Python, in green, and alternatives in DA, in blue.",
"In particular, for computing summation, a for-loop block can be programmed as a 1-line aggregation for sum, in both Python and DA.",
"In general, Python is both powerful and easy to use and is well-known to be close to pseudocode when used to program algorithms.",
"Comprehensions and aggregations in DA improve over those in Python by being completely declarative, exactly as in math.",
"Specification.",
"Using more powerful constructs in DA, specifications of problem statements and computations for other topics can also be written easily and precisely as for predicate logic, and be made executable in Python.",
"Dozens of extra-credit programming problems were given on expressing operations on and properties of sets, sequences, functions, and relations, e.g., writing operations on sequences using both aggregations and recursive definitions; writing definitions of 1-1 and onto functions and using them to check given functions; writing definitions of reflexivity, symmetry, and transitivity and using them to check given relations; writing Euclid algorithm using both iteration and recursion; writing recursion for Hanoi Tower; and expressing transitive closure with an existential quantification with witness in a while loop.",
"Execution.",
"All specifications in Python and DA can be executed directly as discussed in Section .",
"The main difference is that, for later topics, students are asked to write more parts or even all parts of the solution to a problem on their own.",
"Programming as enhancement.",
"Besides similar use of programming as for predicate logic, more computation problems are given, such as Hanoi Tower.",
"There was even an extra extra-credit programming part on solving the online exam scheduling problem that the course itself had, by writing quantifications, comprehensions, and aggregations in DA and feeding them to a solver."
],
[
"Results, analysis, and adoption",
"To give insight into whether extra-credit programming was truly beneficial, we mainly considered two metrics: student test performance, a quantitative metric; and student surveys, a qualitative metric.",
"Student background and course prerequisites.",
"Most students in the course were Computer Science (CS) majors or pre-majors.",
"Most students were in their first year at Stony Brook, but most students had some kind of programming experience before this course.",
"The only prerequisite for the course was basic calculus (Calculus A or Calculus I in a series of 3 or 2 courses on calculus).",
"To do the extra-credit programming, students only needed to be able to run Python (Python 3.7 was used) on a command line.",
"Specifically, 88 out of 115 students total in the class roster were listed as CS majors (33) or pre-majors (55).",
"22 were listed as freshman and 75 as sophomore; many in their first year were listed as sophomore because they passed a certain number of required credits from their first semester and/or their high school Advanced Placement courses.",
"To better understand the background of the students, a questionnaire was given out at the start of the course.",
"Out of 115 students enrolled, 108 students responded, including 53 of the 56 students who did one or more of the programming tasks.",
"Out of the 108 respondents, 98 (91%) had some programming background, from a course, job, and/or recreation.",
"Of 102 who indicated their class year, 86 (84%) indicated they were in their first year.",
"Of 106 who indicated their majors, 91 (86%) indicated they were CS majors or pre-majors.",
"Out of the 53 respondents who did one of more programming tasks, 49 (92%) had some programming background.",
"Of 51 who indicated their class year, 43 (84%) indicated they were in their first year.",
"Of 53 who indicated their majors, 44 (83%) indicated they were CS majors or pre-majors.",
"These percentages are the same or very close to those for the entire class.",
"Student test performance.",
"To see if extra-credit programming helped students learn better, we examine whether students who did more programming on certain topics performed better on exams covering those topics.",
"We consider three groups of programming assignments: the first two assignments before Midterm 1, the next three before Midterm 2, and all six before the Final, grouped by their relevancy to the exam.",
"In each group, students are categorized by how many they submitted for that group, and the test average from each category was taken.",
"Figure REF shows the results.",
"Figure: Exam performance vs. number of relevant programming assignmentsubmissions.",
"The number of students in each category is shown on thebottom of the bar.With a very slight exception (0.8%) in group 2, there is a clear positive correlation between the number of programming assignments submitted and exam performance.",
"Unfortunately, this is not enough to prove that the assignments by themselves improved student performance, due to self-selection bias—it is probable that students who would do better on exams would do more extra-credit work.",
"In the future, it may be worthwhile to introduce these assignments in a way that eliminates self-selection bias.",
"Student surveys.",
"For each homework, an online survey was created, containing a section asking for qualitative feedback on extra-credit programming.",
"For the first programming assignment, on predicate logic, of 115 students enrolled, 109 submitted the homework; 53 (48.7%) did not try the extra credit, 20 (18.3%) tried but had issues installing or running Python, and 36 (33.0%) completed all or part of the programming.",
"For the 53 who did not try, and the 20 who tried but failed, the optional nature of the extra credit, being $<$1% of the course grade, mostly likely did not motivate them enough.",
"Of the 36 who did programming successfully, 32 (88.9%) completed both cafe and tarski, and 4 (11.1%) completed one of them.",
"Overall, 21 of 36 (58.3%) indicated they enjoyed the programming, and 24 (66.7%) indicated they wanted more programming like this.",
"For comparison, the last programming assignment had 96 students respond; only 8 (8.4%) failed to install or run Python—much less than before—but it also reveals that these students did not try sooner.",
"44 (46.3%) did not try the extra credit, and 43 (45.3%) successfully did one or more programming problems—again an improvement.",
"Of the 43 who succeeded, 32 (74.4%) indicated they enjoyed the programming—another positive result.",
"Follow-up questionnaire.",
"A survey was conducted six months after the course ended, with two questions: (1) Did you feel that the programming tasks helped you learn concepts in the course?",
"(2) Did you feel that the programming tasks helped you connect to other CS courses?",
"The survey was sent to the 56 students who did one or more of the programming tasks.",
"Out of the 34 respondents, 30 (88.2%) replied Yes to the first question, and 24 (70.6%) replied Yes to the second.",
"It is worth noting that, among the 4 who replied No to the first question, 2 were near the top of the class, and 2 were near the bottom.",
"The large majority of students in the middle, who were more likely to benefit from these activities, felt that the extra credit programming was helpful.",
"12 students also wrote additional comments voluntarily.",
"Most of them explained how they liked the programming tasks.",
"Two mentioned that the syntax was confusing at first, but the work was still helpful.",
"Suggestions for adoption.",
"The feedback we have received, from both the surveys and in-person interactions with students, supports that students enjoy discrete math more with programming.",
"For students who succeeded installing and running Python, extra-credit programming was well received.",
"Clearly not all students come to a discrete math course equally capable of configuring the minutia of their programming environment.",
"As programming was optional, system configuration was only briefly covered in lectures.",
"For the future, we suggest making programming required to a degree, and providing in-class configuration sessions.",
"Once required, the programming part could be given more weight in the course grade, and could be tested in the exams as well.",
"Because it was extra-credit programming worth $<$1% of the course grade for each assignment, and because it came with entirely different and much longer problem descriptions, it was harder to initially motivate the average student.",
"However, this can be easily overcome by increasing its weight in the course grade.",
"Indeed, most students who did not do extra-credit programming indicated they would have liked to, and some asked about exercises after the course ended.",
"Already, we have had positive results with 56 students who successfully did some or all programming problems.",
"Some went beyond—running distributed algorithms using DA, asking deep questions about Python, and asking to do research projects.",
"We will make problem descriptions and program files for the programming assignments publicly available.",
"Programming solutions will be made available to instructors by requests."
],
[
"Conclusion",
"We have presented a principled approach for teaching discrete math with programming by using a powerful language that extends Python.",
"The approach and language cover all central topics, and allows novice users to understand the concepts precisely, write them rigorously in specifications, and use them directly in executions.",
"Our results and analysis of using the approach support broader deployment.",
"Exploiting Python also allows the approach to be built on to teach more advanced subjects later on, especially with the increasing growth of Python libraries."
]
] | 2011.14059 |
[
[
"Singularities in Euler flows: multivalued solutions, shock waves, and\n phase transitions"
],
[
"Abstract In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations.",
"We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions.",
"We use ideas from the geometrical theory of PDEs, in particular, symmetries and differential constraints to find solutions to the Euler system.",
"Solutions obtained are multivalued, have singularities of projection to the plane of independent variables.",
"We analyze the propagation of the shock wave front along with phase transitions."
],
[
"Introduction",
"Various types of critical phenomena, such as singularities, discontinuities, wave fronts and phase transitions, have always been of interest from both mathematical [1], [2], [3] and practical [4] viewpoints.",
"In the context of gases, discontinuous solutions to the Euler system, describing their motion, are usually treated as shock waves.",
"In the past decades such phenomena have widely been studied, see for instance, [5] for the case of Chaplygin gases, [6], [7], where the weak shocks are considered.",
"It is worth mentioning also [8], [9], where the influence of turbulence on shocks and detonations is emphasized.",
"This paper can be seen as a natural continuation of [10], where we have considered the case of ideal gas flows.",
"Here, we elaborate the case of more complicated and at the same time more interesting from the singularity theory viewpoint model, the van der Waals model, known as one of the most popular in the description of phase transitions.",
"So, singularities of shock wave type that can be viewed as in some sense singular solutions to the Euler system are analyzed together with singularities of purely thermodynamic nature, phase transitions.",
"Our approach to finding and investigating such phenomena is essentially based on the geometric theory of PDEs [11], [13], [12], [15], [14].",
"Namely, we find a class of multivalued solutions to the Euler system (see also [16]), and singularities of their projection to the plane of independent variables are exactly what drives the appearance of the shock wave [17].",
"Similar ideas were used in a series of works [18], [19], [20], where multivalued solutions to filtration equations were obtained along with analysis of shocks.",
"To find such solutions we use the idea of adding a differential constraint to the original PDE in such a way that the resulting overdetermined system of PDEs is compatible [21].",
"The same concepts were also used in [22], where a general solution to the Hunter-Saxton equation was found, in [23], where the two-dimensional Euler system was considered, and in [24], where this approach was applied to the Khokhlov-Zabolotskaya equation.",
"The paper is organized as follows.",
"Section is preliminary, we describe there the necessary concepts from thermodynamics.",
"In Section , we analyze a multivalued solution to Euler equations and its singularities, including shock waves and phase transitions.",
"In the last section, we discuss the results.",
"The essential computations for this paper were made with the DifferentialGeometry package [25] in Maple."
],
[
"Thermodynamics",
"In this section, we give necessary concepts from thermodynamics.",
"As we shall see, geometrical interpretation of thermodynamic states allows one to use Arnold's ideas from the theory of Legendrian and Lagrangian singularities [1], [2], [3], which are crucial in description of phase transitions.",
"The geometrical approach to thermodynamics was initiated already by Gibbs [26], and was further developed in, for example, [27], [28], and recently in [29].",
"For more detailed analysis we also refer to [30]."
],
[
"Legendrian and Lagrangian manifolds",
"Consider the contact space $\\left(\\mathbb {R}^{5},\\theta \\right)$ with coordinates $(s,e,\\rho ,p,T)$ standing for specific entropy, specific inner energy, density, pressure and temperature.",
"The contact structure $\\theta $ is given by $\\theta =T^{-1}de-ds-pT^{-1}\\rho ^{-2}d\\rho .$ Then, a thermodynamic state is a Legendrian manifold $\\widehat{L}\\subset \\left(\\mathbb {R}^{5},\\theta \\right)$ , i.e.",
"$\\theta \\left|_{\\widehat{L}}\\right.=0$ and $\\dim \\widehat{L}=2$ .",
"From the physical viewpoint this means that the first law of thermodynamics holds on $\\widehat{L}$ .",
"Due to (REF ), it is natural to choose $(e,\\rho )$ as coordinates on $\\widehat{L}$ .",
"Then, a two-dimensional manifold $\\widehat{L}\\subset \\left(\\mathbb {R}^{5},\\theta \\right)$ is given by $\\widehat{L}=\\left\\lbrace s=S(e,\\rho ),\\, T=\\frac{1}{S_{e}},\\, p=-\\rho ^{2}\\frac{S_{\\rho }}{S_{e}}\\right\\rbrace ,$ where the function $S(e,\\rho )$ specifies the dependence of the specific entropy on $e$ and $\\rho $ .",
"Note that determining a Legendrian manifold $\\widehat{L}$ by means of (REF ) requires the knowledge of $S(e,\\rho )$ , while in experiments one usually gets relations between pressure, density and temperature.",
"To this reason, we get rid of the specific entropy $s$ by means of projection $\\pi \\colon \\mathbb {R}^{5}\\rightarrow \\mathbb {R}^{4}$ , $\\pi (s,e,\\rho ,p,T)=(e,\\rho ,p,T)$ and consider an immersed Lagrangian manifold $\\pi \\left(\\widehat{L}\\right)=L\\subset \\left(\\mathbb {R}^{4},\\Omega \\right)$ in a symplectic space $\\left(\\mathbb {R}^{4},\\Omega \\right)$ , where the structure symplectic form $\\Omega $ is $\\Omega =d\\theta =d(T^{-1})\\wedge de-d(pT^{-1}\\rho ^{-2})\\wedge d\\rho .$ Then, one can treat thermodynamic state manifolds as Lagrangian manifolds $L\\subset \\left(\\mathbb {R}^{4},\\Omega \\right)$ , i.e.",
"$\\Omega \\left|_{L}\\right.=0$ .",
"In coordinates $(T,\\rho )$ , a thermodynamic Lagrangian manifold $L$ is given by two functions $L=\\left\\lbrace p=P(T,\\rho ),\\,e=E(T,\\rho )\\right\\rbrace .$ Since $\\Omega \\left|_{L}\\right.=0$ , the functions $P(T,\\rho )$ and $E(T,\\rho )$ are not arbitrary, but are related by $\\left[p-P(T,\\rho ),e-E(T,\\rho )\\right]\\left|_{L}\\right.=0,$ where $[f,g]$ is a Poisson bracket of functions $f$ and $g$ on $\\left(\\mathbb {R}^{4},\\Omega \\right)$ uniquely defined by the relation $[f,g]\\,\\Omega \\wedge \\Omega =df\\wedge dg\\wedge \\Omega .$ Equation (REF ) forces the following relation between $P(T,\\rho )$ and $E(T,\\rho )$ : $(-\\rho ^{-2}T^{-1}P)_{T}=(T^{-2}E)_{\\rho }$ , and therefore the following theorem is valid: Theorem 1 The Lagrangian manifold $L$ is given by means of the Massieu-Planck potential $\\phi (\\rho ,T)$ $p=-\\rho ^{2}T\\phi _{\\rho },\\quad e=T^{2}\\phi _{T}.$ Remark 2 Having given the Lagrangian manifold $L$ by means of (REF ), one can find out the entropy function $S(e,\\rho )$ solving the overdetermined system $T=\\frac{1}{S_{e}},\\, p=-\\rho ^{2}\\frac{S_{\\rho }}{S_{e}}$ with compatibility condition (REF )."
],
[
"Riemannian structures, singularities, phase transitions",
"There is one more important structure arising, as it was shown in [29], from measurement approach to thermodynamics.",
"Indeed, if one considers equilibrium thermodynamics as a theory of measurement of random vectors, whose components are inner energy and volume $v=\\rho ^{-1}$ , one drives to the universal quadratic form on $(\\mathbb {R}^{4},\\Omega )$ of signature $(2,2)$ : $\\kappa =d(T^{-1})\\cdot de-\\rho ^{-2}d(pT^{-1})\\cdot d\\rho ,$ where $\\cdot $ is the symmetric product, and areas on $L$ , where the restriction $\\kappa |_{L}$ of $\\kappa $ to $L$ is negative, are those where the variance of a random vector $(e,v=\\rho ^{-1})$ is positive [29], [31].",
"Using (REF ), we get $\\kappa |_{L}=-(2T^{-1}\\phi _{T}+\\phi _{TT})dT\\cdot dT+(2\\rho ^{-1}\\phi _{\\rho }+\\phi _{\\rho \\rho })d\\rho \\cdot d\\rho ,$ and taking into account (REF ), we conclude that the condition of positive variance is satisfied at points on $L$ , where $e_{T}>0,\\quad p_{\\rho }>0,$ which is known as the condition of the thermodynamic stability.",
"Let us now explore singularities of Lagrangian manifolds.",
"We will be interested in the singularities of their projection to the plane of intensive variables $(p,T)$ , i.e.",
"points where the form $dp\\wedge dT$ degenerates.",
"We will assume that extensive variables $(e,\\rho )$ may serve as global coordinates on $L$ , i.e.",
"the form $de\\wedge d\\rho $ is non-degenerated everywhere.",
"The set where $dp\\wedge dT=0$ coincides with that where $2\\rho ^{-1}\\phi _{\\rho }+\\phi _{\\rho \\rho }=0$ , or, equivalently, where the from $\\kappa |_{L}$ degenerates.",
"A manifold $L$ turns out to be divided into submanifolds $L_{i}$ , where both $(e,\\rho )$ and $(p,T)$ may serve as coordinates, or, equivalently, the form (REF ) is non-degenerated.",
"Such $L_{i}$ are called phases.",
"Additionally, those of $L_{i}$ , where (REF ) is negative, are called applicable phases.",
"Thus we end up with the observation that singularities of projection of thermodynamic Lagrangian manifolds are related with the theory of phase transitions.",
"Indeed, by a phase transition of the first order we mean a jump from one applicable state to another, governed by the conservation of intensive variables $p$ , $T$ , and specific Gibbs potential $\\gamma =e-Ts+p/\\rho ,$ which in terms of the Massieu-Planck potential is expressed as $\\gamma =-T(\\phi +\\rho \\phi _{\\rho })$ [30].",
"Consequently, to find the points of phase transition, one needs to solve the system $p=-\\rho _{1}^{2}T\\phi _{\\rho }(T,\\rho _{1}),\\quad p=-\\rho _{2}^{2}T\\phi _{\\rho }(T,\\rho _{2}),\\quad \\phi (T,\\rho _{1})+\\rho _{1}\\phi _{\\rho }(T,\\rho _{1})=\\phi (T,\\rho _{2})+\\rho _{2}\\phi _{\\rho }(T,\\rho _{2}),$ where $p$ and $T$ are the pressure and the temperature of the phase transition, and $\\rho _{1}$ and $\\rho _{2}$ are the densities of gas and liquid phases.",
"Example 3 (Ideal gas) The simplest example of a gas is an ideal gas model.",
"In this case the Legendrian manifold is given by $\\widehat{L}=\\left\\lbrace p=R\\rho T,\\, e=\\frac{n}{2}RT,\\, s=R\\ln \\left(\\frac{T^{n/2}}{\\rho }\\right)\\right\\rbrace ,$ where $R$ is the universal gas constant, and $n$ is the degree of freedom.",
"The differential quadratic form $\\kappa |_{L}$ is $\\kappa |_{L}=-\\frac{Rn}{2}\\frac{dT^{2}}{T^{2}}-R\\rho ^{-2}d\\rho ^{2}.$ It is negative definite on the entire $\\widehat{L}$ , and there are no phase transitions, nor are there singularities of projection of $\\widehat{L}$ to the $p-T$ plane.",
"Example 4 (van der Waals gas) To define the Legendrian manifold for van der Waals gases we will use reduced state equations: $\\widehat{L}=\\left\\lbrace p=\\frac{8T\\rho }{3-\\rho }-3\\rho ^{2},\\, e=\\frac{4nT}{3}-3\\rho ,\\,s=\\ln \\left(T^{4n/3}(3\\rho ^{-1}-1)^{8/3}\\right)\\right\\rbrace .$ The differential quadratic form $\\kappa |_{L}$ is $\\kappa |_{L}=-\\frac{4n}{3T^{2}}dT^{2}+\\frac{6(\\rho ^{3}-6\\rho ^{2}-4T+9\\rho )}{\\rho ^{2}T(\\rho -3)^{2}}d\\rho ^{2}.$ In this case it changes its sign, the manifold $\\widehat{L}$ has a singularity of cusp type.",
"The singular set of $\\widehat{L}$ , called also caustic, and the curve of phase transition are shown in Figure 1.",
"Figure: Singularities of the van der Waals Legendrian manifold.",
"Caustic (black line) and phase transition curve (red line) in coordinates (p,T)(p,T) lam1; the curve of phase transition in (p,ρ,T)(p,\\rho ,T) lam2.",
"Gas phase on the right of the curve, liquid phase on the left of the curve, wet steam inside the curve."
],
[
"Euler equations",
"In this paper we will study non-stationary, one-dimensional flows of gases, described by the following system of differential equations: Conservation of momentum $\\rho (u_{t}+uu_{x})=-p_{x},$ Conservation of mass $\\rho _{t}+(\\rho u)_{x}=0,$ Conservation of entropy along the flow $s_{t}+us_{x}=0.$ Here $u(t,x)$ is the flow velocity, $\\rho (t,x)$ is the density of the medium, and $s(t,x)$ is the specific entropy.",
"System (REF )-(REF ) is incomplete.",
"It becomes complete once extended by equations of thermodynamic state (REF ).",
"We will be interested in homentropic flows, i.e.",
"those with $s(t,x)=s_{0}$ .",
"On the one hand, this assumption satisfies (REF ) identically, on the other hand, it allows us to express all the thermodynamic variables in terms of $\\rho $ .",
"Indeed, the entropy $s$ has the following expression in terms of the Massieu-Planck potential $\\phi (T,\\rho )$ : $s=\\phi +T\\phi _{T}$ [30].",
"Putting $s=s_{0}$ , we get an equation $s_{0}=\\phi +T\\phi _{T}$ , which determines $T(\\rho )$ uniquely, since its derivative with respect to $T$ is positive due to the negativity of $\\kappa |_{L}$ .",
"Substituting $T(\\rho )$ into (REF ), one gets $p=p(\\rho )$ .",
"Thus we end up with the following two-component system of PDEs: $u_{t}+uu_{x}+A(\\rho )\\rho _{x}=0,\\quad \\rho _{t}+(\\rho u)_{x}=0,$ where $A(\\rho )=p^{\\prime }(\\rho )/\\rho $ .",
"We do not specify the function $A(\\rho )$ yet, we shall do this while solving (REF )."
],
[
"Finding solutions",
"To find solutions to system (REF ), we use the idea of adding a differential constraint to (REF ), compatible with the original system.",
"It is worth mentioning that a solution is an integral manifold of the Cartan distribution on (REF ) (see [11], [13], [12] for details).",
"This geometrical interpretation of a solution to a PDE allows to find ones in the form of manifolds, which, in general, may not be globally given by functions.",
"This approach gives rise to investigation of singularities in a purely geometrical manner, which is shown in this paper.",
"In general, finding differential constraints is not a trivial problem.",
"But having found ones, the problem of finding solutions is reduced to the integration of a completely integrable Cartan distribution of the resulting compatible overdetermined system.",
"In case the Cartan distribution has a solvable transversal symmetry algebra, whose dimension equals the codimension of the Cartan distribution, we are able to get explicit solutions in quadratures by applying the Lie-Bianchi theorem (for details we refer to [11], [13], [12]).",
"We will look for a differential constraint compatible with (REF ) in the form of a quasilinear equation $u_{x}-\\rho _{x}\\left(\\alpha (\\rho )u+\\beta (\\rho )\\right)=0,$ where functions $\\alpha (\\rho )$ and $\\beta (\\rho )$ are to be determined.",
"We will denote system (REF )-(REF ) by $\\mathcal {E}$ .",
"Theorem 5 System (REF )-(REF ) is compatible if $\\alpha (\\rho )=-\\frac{1}{\\rho (C_{3}\\rho -1)},\\quad \\beta (\\rho )=\\frac{C_{2}}{\\rho (C_{3}\\rho -1)},\\quad A(\\rho )=C_{1}+\\frac{C_{5}}{\\rho ^{3}}\\left(C_{3}+\\frac{C_{7}}{\\rho }\\right)^{C_{6}},$ where $C_{i}$ are constants.",
"The proof of Theorem REF is more technical rather than conceptual.",
"First of all, we lift system (REF )-(REF ) to the space of 3-jets $J^{3}(\\mathbb {R}^{2})$ by applying total derivatives $D_{t}&=&\\partial _{t}+u_{t}\\partial _{u}+\\rho _{t}\\partial _{\\rho }+u_{tt}\\partial _{u_{t}}+\\rho _{tt}\\partial _{\\rho _{t}}+\\ldots ,\\\\D_{x}&=&\\partial _{x}+u_{x}\\partial _{u}+\\rho _{x}\\partial _{\\rho }+u_{xx}\\partial _{u_{x}}+\\rho _{xx}\\partial _{\\rho _{x}}+\\ldots .$ to equations of $\\mathcal {E}$ the required number of times consequently.",
"The resulting system $\\mathcal {E}_{3}\\subset J^{3}(\\mathbb {R}^{2})$ , consisting of equations only of the third order, contains 9 equations for 8 variables of purely third order, i.e.",
"$u_{ttt}$ , $u_{xxx}$ , $u_{txx}$ and so on.",
"Eliminating them from $\\mathcal {E}_{3}$ , we get 7 relations (6 obtained by lifting $\\mathcal {E}$ to $J^{2}(\\mathbb {R}^{2}$ ) plus one remained from eliminations of third-order variables).",
"Again, we eliminate all the variables of the second order and we get four relations of the first order.",
"Eliminating $u_{x}$ , $u_{t}$ and $\\rho _{t}$ we end up with the expression of the form $\\rho _{x}^{3}G(\\rho ,u)=0$ , where $G(\\rho ,u)$ is a polynomial in $u$ , whose coefficients are ODEs on $\\alpha (\\rho )$ , $\\beta (\\rho )$ and $A(\\rho )$ , solving which we get (REF ).",
"It is worth saying that these computations are algebraic and are well suited for computer algebra systems.",
"Remark 6 Using (REF ) and (REF ), one can show that the function $A(\\rho )=p^{\\prime }(\\rho )/\\rho $ given in (REF ) corresponds to that of ideal gas in case of $C_{1}=C_{3}=0,\\quad C_{5}=R\\left(1+\\frac{2}{n}\\right)\\exp \\left(\\frac{2s_{0}}{Rn}\\right),\\quad C_{6}=-2-\\frac{2}{n},\\quad C_{7}=1.$ van der Waals gas in case of $C_{1}=-6,\\quad C_{3}=-1,\\quad C_{5}=24\\left(1+\\frac{2}{n}\\right)\\exp \\left(\\frac{3s_{0}}{4n}\\right),\\quad C_{6}=-2-\\frac{2}{n},\\quad C_{7}=3.$ The case of ideal gases was thoroughly investigated in [10].",
"Here, we are interested in the case of van der Waals gases.",
"Summarizing, we have a compatible overdetermined system of PDEs $\\mathcal {E}=\\left\\lbrace F_{1}=u_{t}+uu_{x}+A(\\rho )\\rho _{x}=0,\\, F_{2}=\\rho _{t}+(\\rho u)_{x}=0,\\, F_{3}=u_{x}-\\rho _{x}\\left(\\alpha (\\rho )u+\\beta (\\rho )\\right)=0\\right\\rbrace \\subset J^{1}(\\mathbb {R}^{2}),$ where functions $\\alpha (\\rho )$ , $\\beta (\\rho )$ and $A(\\rho )$ are specified in (REF ).",
"This system is a smooth manifold $\\mathcal {E}$ in the space of 1-jets $J^{1}(\\mathbb {R}^{2})$ of functions on $\\mathbb {R}^{2}$ .",
"Since $\\dim J^{1}(\\mathbb {R}^{2})=8$ , and $\\mathcal {E}$ consists of 3 relations on $J^{1}(\\mathbb {R}^{2})$ , $\\dim \\mathcal {E}=5$ .",
"The dimension of the Cartan distribution $\\mathcal {C}_{\\mathcal {E}}$ on $\\mathcal {E}$ equals 2, therefore $\\mathrm {codim}\\,\\mathcal {C}_{\\mathcal {E}}=3$ .",
"Let us choose $(t,x,u,\\rho ,\\rho _{x})$ as internal coordinates on $\\mathcal {E}$ .",
"Then the Cartan distribution $\\mathcal {C}_{\\mathcal {E}}$ is generated by differential 1-forms $\\omega _{1}&=&du-u_{x}dx-u_{t}dt,\\\\\\omega _{2}&=&d\\rho -\\rho _{x}dx-\\rho _{t}dt,\\\\\\omega _{3}&=&d\\rho _{x}-\\rho _{xx}dx-\\rho _{xt}dt,$ where $\\rho _{xx}$ , $\\rho _{xt}$ , $u_{t}$ , $u_{x}$ , $\\rho _{t}$ are expressed due to $\\mathcal {E}$ and its prolongation $\\mathcal {E}_{2}=\\left\\lbrace D_{t}(F_{1})=0,\\,D_{t}(F_{2})=0,\\,D_{t}(F_{3})=0,\\,D_{x}(F_{1})=0,\\,D_{x}(F_{2})=0,\\,D_{x}(F_{3})=0\\right\\rbrace $ : $\\rho _{xx}=\\frac{\\rho _{x}^{2}\\left(\\rho (C_{3}\\rho -1)^{3}A^{\\prime }+(C_{3}\\rho -1)^{2}A+3C_{3}(C_{2}-u)^{2}\\right)}{(C_{3}\\rho -1)\\left((C_{2}-u)^{2}-A\\rho (C_{3}\\rho -1)^{2}\\right)},\\quad \\rho _{t}=\\frac{\\rho _{x}(C_{3}\\rho u+C_{2}-2u)}{1-C_{3}\\rho },$ $u_{x}=\\frac{\\rho _{x}(C_{2}-u)}{\\rho (C_{3}\\rho -1)},\\quad u_{t}=-\\frac{\\rho _{x}\\left(A\\rho (C_{3}\\rho -1)+u(C_{2}-u)\\right)}{\\rho (C_{3}\\rho -1)},$ $\\begin{split}\\rho _{xt}=&\\frac{\\rho _{x}^{2}}{\\rho (C_{3}\\rho -1)^{2}\\left(A\\rho (C_{3}\\rho -1)^{2}-(C_{2}-u)^{2}\\right)}\\left(\\rho ^{2}(C_{3}\\rho -1)^{3}(C_{3}\\rho u+C_{2}-2u)A^{\\prime }\\right.+{}\\\\&+\\left.\\rho A(C_{3}\\rho -1)^{2}(C_{3}\\rho u+3C_{2}-4u)+(C_{2}-u)^{2}(3C_{3}^{2}\\rho ^{2}u+3C_{3}\\rho (C_{2}-2u)-2C_{2}+2u)\\right),\\end{split}$ where $A(\\rho )$ is given by (REF ).",
"We shall look for integrals of the distribution (REF )-(REF ), which give us an (implicit) solution to (REF )-(REF ).",
"Theorem 7 The distribution (REF )-(REF ) is a completely integrable distribution with a 3-dimensional Lie algebra $\\mathfrak {g}$ of transversal infinitesimal symmetries generated by vector fields $X_{1}=t\\partial _{t}+x\\partial _{x}-\\rho _{x}\\partial _{\\rho _{x}},\\quad X_{2}=\\partial _{t},\\quad X_{3}=\\partial _{x}$ with brackets $[X_{1},X_{3}]=-X_{3}$ , $[X_{1},X_{2}]=-X_{2}$ , $[X_{2},X_{3}]=0$ .",
"The Lie algebra $\\mathfrak {g}$ is solvable, and its sequence of derived algebras is $\\mathfrak {g}=\\langle X_{1}, X_{2}, X_{3}\\rangle \\supset \\langle X_{2}, X_{3}\\rangle \\supset 0.$ Thus the Lie-Bianchi theorem [11], [13], [12] can be applied to integrate (REF )-(REF ).",
"Let us choose another basis $\\langle \\varkappa _{1},\\varkappa _{2},\\varkappa _{3}\\rangle $ in $\\mathcal {C}_{\\mathcal {E}}$ by the following way: $\\begin{pmatrix}\\varkappa _{1}\\\\\\varkappa _{2}\\\\\\varkappa _{3}\\end{pmatrix}=\\begin{pmatrix}\\omega _{1}(X_{1}) & \\omega _{1}(X_{2}) & \\omega _{1}(X_{3})\\\\\\omega _{2}(X_{1}) & \\omega _{2}(X_{2}) & \\omega _{2}(X_{3})\\\\\\omega _{3}(X_{1}) & \\omega _{3}(X_{2}) & \\omega _{3}(X_{3})\\end{pmatrix}^{-1}\\begin{pmatrix}\\omega _{1}\\\\\\omega _{2}\\\\\\omega _{3}\\end{pmatrix}.$ Due to the structure of the symmetry Lie algebra $\\mathfrak {g}$ , the form $\\varkappa _{1}$ is closed [11], [12], and therefore locally exact, i.e.",
"$\\varkappa _{1}=dQ_{1}$ , where $Q_{1}\\in C^{\\infty }(J^{1})$ , while restrictions $\\varkappa _{2}|_{M_{1}}$ and $\\varkappa _{3}|_{M_{1}}$ to the manifold $M_{1}=\\left\\lbrace Q_{1}=\\mathrm {const}\\right\\rbrace $ are closed and locally exact too.",
"Integrating the differential 1-form $\\varkappa _{1}$ we observe that variables $u$ , $\\rho $ , $t$ , $x$ can be chosen as local coordinates on $M_{1}$ and $M_{1}=\\left\\lbrace \\rho _{x}=\\frac{\\alpha _{1}\\rho ^{2}(C_{3}\\rho -1)}{\\rho A(C_{3}\\rho -1)^{2}-(C_{2}-u)^{2}}\\right\\rbrace ,$ where $\\alpha _{1}$ is a constant.",
"Integrating restrictions $\\varkappa _{2}|_{M_{1}}$ and $\\varkappa _{3}|_{M_{1}}$ , we get two more relations that give us a solution to (REF )-(REF ) implicitly: $t+\\alpha _{2}+\\frac{C_{2}-u}{\\alpha _{1}\\rho }+\\frac{C_{3}u}{\\alpha _{1}}=0,$ and $\\begin{split}0&=x+\\alpha _{3}+\\frac{1}{\\alpha _{1}}\\left(C_{1}\\ln \\rho -C_{1}C_{3}\\rho +\\frac{C_{3}u^{2}}{2}+\\frac{u(C_{2}-u)}{\\rho }-C_{5}\\left(C_{3}+\\frac{C_{7}}{\\rho }\\right)^{C_{6}+1}\\cdot \\right.",
"{}\\\\&\\cdot \\left.\\frac{2\\rho ^{2}C_{3}^{2}-C_{7}^{2}(C_{6}+1)(C_{3}\\rho (C_{6}+3)-C_{6}-2)+C_{3}C_{7}\\rho (C_{3}\\rho (C_{6}+3)-2C_{6}-2)}{(C_{6}+1)(C_{6}+2)(C_{6}+3)C_{7}^{3}\\rho ^{2}}\\right),\\end{split}$ where we have already substituted $A(\\rho )$ from (REF ), and $\\alpha _{2}$ , $\\alpha _{3}$ are constants.",
"The graph of a multivalued solution for the density is shown in Figure 2.",
"We used substitution (REF ), where $C_{5}=240$ , $n=3$ , together with $C_{2}=1$ , $\\alpha _{1}=1$ , $\\alpha _{2}=2$ , $\\alpha _{3}=1$ .",
"Figure: Graph of the density in case of n=3n=3 for time moments t=0t=0, t=30t=30"
],
[
"Caustics and shock waves",
"We can see that solution given by (REF )-(REF ) is, in general, multivalued.",
"To figure out where the two-dimensional manifold $N$ given by (REF )-(REF ) has singularities of projection to the plane of independent variables, one needs to find zeroes of the two-form $dt\\wedge dx$ .",
"Condition $(dt\\wedge dx)|_{N}=0$ gives us a curve in the plane $\\mathbb {R}^{2}(t,x)$ called caustic.",
"Choosing $\\rho $ as a coordinate on the caustic, we get its equations in a parametric form: $\\begin{split}x(\\rho )&=-\\frac{1}{2\\alpha _{1}}\\left(2C_{1}\\ln \\rho +C_{1}(C_{3}^{3}\\rho ^{3}-4\\rho ^{2}C_{3}^{2}+3C_{3}\\rho -2)+C_{3}C_{2}^{3}+2\\alpha _{1}\\alpha _{3}\\right)\\pm {}\\\\&\\pm \\frac{C_{2}(C_{3}\\rho -1)^{2}}{\\alpha _{1}\\rho ^{2}}\\sqrt{C_{1}\\rho ^{3}+C_{5}\\left(C_{3}+\\frac{C_{7}}{\\rho }\\right)^{C_{6}}}-\\frac{C_{5}\\left(C_{3}+\\frac{C_{7}}{\\rho }\\right)^{C_{6}}}{2(C_{6}+2)(C_{6}+3)C_{7}^3\\alpha _{1}(C_{6}+1)\\rho ^{3}}\\cdot {}\\\\&\\cdot \\left(C_{3}^{3}(-4+C_{7}^{3}(C_{6}^3+6C_{6}^2+11C_{6}+6)+(-2C_{6}-6)C_{7})\\rho ^3\\right.-{}\\\\&-\\left.2C_{7}((2(C_{6}^3+6C_{6}^2+11C_{6}+6))C_{7}^2+(-C_{6}^2-3C_{6})C_{7}-C_{6})C_{3}^2\\rho ^2+\\right.",
"{}\\\\&+\\left.C_{7}^2(C_{6}+1)((C_{6}+3)(5C_{6}+12)C_{7}-2C_{6})C_{3}\\rho -2C_{7}^3(C_{6}+4)(C_{6}+2)(C_{6}+1)\\right),\\end{split}$ $t(\\rho )=-\\alpha _{2}-\\frac{C_{2}C_{3}}{\\alpha _{1}}\\pm \\frac{(C_{3}\\rho -1)^{2}}{\\alpha _{1}\\rho ^{2}}\\sqrt{C_{1}\\rho ^{3}+C_{5}\\left(C_{3}+\\frac{C_{7}}{\\rho }\\right)^{C_{6}}}.$ To construct a discontinuous solution from the multivalued one given by (REF )-(REF ), we use the mass conservation law.",
"Let us write down equation (REF ) with the velocity $u$ found from (REF ) in terms $t$ and $\\rho $ : $\\rho _{t}+\\left(\\rho \\frac{\\alpha _{1}\\rho (t+\\alpha _{2})+C_{2}}{1-C_{3}\\rho }\\right)_{x}=0,$ and therefore the conservation law has the form $\\Theta =\\rho dx-\\rho \\frac{\\alpha _{1}\\rho (t+\\alpha _{2})+C_{2}}{1-C_{3}\\rho }dt.$ Its restriction $\\Theta |_{N}$ to the manifold $N$ given by (REF )-(REF ) is a closed form, locally $\\Theta |_{N}=dH$ , and the potential $H(\\rho ,t)$ equals $\\begin{split}H(\\rho ,t)&=\\frac{\\rho }{2\\alpha _{1}(C_{3}\\rho -1)^{2}}\\left(C_{1}C_{3}^{3}\\rho ^{3}-4C_{1}C_{3}^{2}\\rho ^{2}+\\rho \\left( C_{2}^2C_{3}^2+(2C_{2}(t+\\alpha _{2})\\alpha _{1}+5C_{1})C_{3}+\\alpha _{1}^{2}(t+\\alpha _{2})^{2}\\right)-2C_{1}\\right)-{}\\\\&-\\frac{C_{5}\\left(C_{3}+\\frac{C_{7}}{\\rho }\\right)^{C_{6}}}{(C_{6}+2)\\alpha _{1}C_{7}^{2}(C_{6}+1)\\rho ^2}(C_{3}\\rho +C_{7})(C_{3}(1+(C_{6}+2)C_{7})\\rho -(C_{6}+1)C_{7}).\\end{split}$ The discontinuity line, or a shock wave front is found from the system of equations $H(\\rho _{1},t)=H(\\rho _{2},t),\\quad x(\\rho _{1},t)=x(\\rho _{2},t),$ where $x(\\rho ,t)$ is obtained from (REF )-(REF ) by eliminating $u$ .",
"Caustics along with the shock wave front are shown in Figure 3.",
"Note that the picture is similar to that in case of phase transitions.",
"Figure: Caustic (black) and shock wave front (red) for n=3n=3.The final result here is the expression for the time interval, within which the solution (REF )-(REF ) is smooth.",
"Theorem 8 Solution given by (REF )-(REF ) is smooth and unique in the time interval $t\\in [0,t^{*})$ , where $t^{*}=\\frac{1}{\\alpha _{1}}\\left(-C_{2}C_{3}-\\alpha _{1}\\alpha _{2}+(C_{3}-3)^{2}\\sqrt{\\frac{C_{1}}{27}+C_{5}(C_{3}+3C_{7})^{C_{6}}}\\right),$ and in case of (REF ), where $C_{5}=240$ , $n=3$ , together with $C_{2}=1$ , $\\alpha _{1}=1$ , $\\alpha _{2}=2$ , $\\alpha _{3}=1$ approximately $t^{*}=12.53$ ."
],
[
"Phase transitions",
"Having a solution, one can remove the phase transition curve from the space of thermodynamic variables to $\\mathbb {R}^{2}(t,x)$ .",
"Indeed, on the one hand, we have all the thermodynamic parameters as functions of $(t,x)$ .",
"On the other hand, we have conditions on phase transitions (REF ) in the space of thermodynamic variables.",
"In combination they give us a curve of phase transitions in $(t,x)$ plane.",
"Phase transitions together with the shock wave are presented in Figure 4.",
"Figure: Phase transition curve (dash line) and shock wave front (red line)."
],
[
"Discussion",
"In the present work we analyzed critical phenomena in gas flows of purely thermodynamic nature, which are phase transitions, and shock waves arising from singularities of solutions to the Euler system.",
"To obtain such solutions we used a differential constraint compatible with the original system.",
"In this work it was found in a purely computational way, and it seems interesting how to get it in a more regular way.",
"One of possible ways to find such constraints is using differential invariants.",
"Then, constraints can be found constructively by solving quotient PDEs, which was successfully realized in [22].",
"We hope to make use of this method in the future research.",
"The analysis of phase transitions showed that sometimes shock waves can be accompanied with phase transitions, which is shown in Figure 4, since the phase transition curve intersects the shock wave front, and on the one side of the discontinuity curve we observe a pure gas phase, while on the other side we can see a wet steam."
],
[
"Acknowledgements",
"This work was partially supported by the Russian Foundation for Basic Research (project 18-29-10013) and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (project 19-7-1-13-3)."
]
] | 2011.14175 |
[
[
"Approximate Midpoint Policy Iteration for Linear Quadratic Control"
],
[
"Abstract We present a midpoint policy iteration algorithm to solve linear quadratic optimal control problems in both model-based and model-free settings.",
"The algorithm is a variation of Newton's method, and we show that in the model-based setting it achieves cubic convergence, which is superior to standard policy iteration and policy gradient algorithms that achieve quadratic and linear convergence, respectively.",
"We also demonstrate that the algorithm can be approximately implemented without knowledge of the dynamics model by using least-squares estimates of the state-action value function from trajectory data, from which policy improvements can be obtained.",
"With sufficient trajectory data, the policy iterates converge cubically to approximately optimal policies, and this occurs with the same available sample budget as the approximate standard policy iteration.",
"Numerical experiments demonstrate effectiveness of the proposed algorithms."
],
[
"Introduction",
"With the recent confluence of reinforcement learning and data-driven optimal control, there is renewed interest in fully understanding convergence, sample complexity, and robustness in both “model-based” and “model-free” algorithms.",
"Linear quadratic problems in continuous spaces provide benchmarks where strong theoretical statements can be made.",
"In practice, it is often difficult or impossible to develop a model of a system from first-principles.",
"In this case, one may use so-called “model-based” system identification methods which attempt to estimate a model of the dynamics from observed sample data, then solve the Riccati equation using the identified system matrices.",
"An approximately optimal control policy is then computed assuming certainty-equivalence , , or using robust control approaches to explicitly account for model uncertainty , , , .",
"The analyses in these recent works has focused on providing finite-sample performance/suboptimality guarantees.",
"As an alternative, so-called “model-free” methods may also be used, which do not attempt to learn a model of the dynamics.",
"The category of policy optimization methods which directly attempt to optimize the control policy, including policy gradient, has received significant attention recently for standard LQR , , multiplicative-noise LQR , Markov jump LQR , and LQ games related to $\\mathcal {H}_\\infty $ robust control , .",
"Between the fully model-based system identification approaches and the fully model-free policy optimization approaches lies another category of methods, which we denote as value function approximation methods.",
"These methods attempt to estimate value functions then compute policies which are optimal with respect to these value functions.",
"This class of methods includes approximate dynamic programming, exemplified by approximate value iteration, which estimates state-value functions, and approximate policy iteration, which estimates state-action value functions.",
"In particular, for LQR problems, approximate policy iteration was studied by , and by , under the guise of a quasi-Newton method.",
"For LQ games, approximate policy iteration was studied by under the guise of Q-learning, and by , .",
"Note that approximate policy iteration is sometimes called quasi-Newton or Q-learning.",
"In stochastic optimal control, the functional Bellman equation gives a sufficient and necessary condition for optimality of a control policy ().",
"It has been long-known, but perhaps underappreciated, that application of Newton's method to find the root of the functional Bellman equation in stochastic optimal control is equivalent to the dynamic programming algorithm of policy iteration (, ).",
"In this most general setting, conditions for and rates of convergence are available (, ), but may be difficult or impossible to verify in practice.",
"Furthermore, even representing the value functions and policies and executing the policy iteration updates may be intractable.",
"This motivates the basic approximation of such problems by linear dynamics and quadratic costs over finite-dimensional, infinite-cardinality state and action spaces.",
"In linear-quadratic problems, the Bellman equation becomes a matrix algebraic Riccati equation, and application of the Newton method to the Riccati equation yields the well-known Kleinman-Hewer algorithm.",
"introduced this for continuous-time systems, and studied it for discrete-time systems.",
"The Newton method has many variations devised to improve the convergence rate and information efficiency, including higher-order methods (such as Halley (), super-Halley (), and Chebyshev ()), and multi-point methods (), which compute derivatives at multiple points and of which the midpoint method is the simplest member.",
"Some of these have been applied to solving Riccati equations by , , , , , but without consideration of the situation when the dynamics are not perfectly known.",
"Our main contributions are: We present a midpoint policy iteration algorithm to solve linear quadratic optimal control problems when the dynamics are both known (Algorithm ) and unknown (Algorithm REF ).",
"We demonstrate that the method converges, and does so at a faster cubic rate than standard policy iteration or policy gradient, which converge at quadratic and linear rates, respectively.",
"We show that approximate midpoint policy iteration converges faster in the model-free setting even with the same available sample budget as the approximate standard policy iteration.",
"We present numerical experiments that illustrate and demonstrate the effectiveness of the algorithms and provide an open-source implementation to facilitate their wider use."
],
[
"Preliminaries",
"The infinite-horizon average-cost time-invariant linear quadratic regulator (LQR) problem is 2 minimize T 1T Ex0, wt t=0T xt ut Qxx Qxu Qux Quu xt ut , subject to xt+1 = A xt + B ut + wt, where $x_t \\in \\mathbb {R}^n$ is the system state, $u_t \\in \\mathbb {R}^m$ is the control input, and $w_t$ is i.i.d.",
"process noise with zero mean and covariance matrix $W$ .",
"The state-to-state system matrix $A \\in \\mathbb {R}^{n \\times n}$ and input-to-state system matrix $B \\in \\mathbb {R}^{n \\times m}$ may or may not be known; we present algorithms for both settings.",
"The optimization is over the space $\\Pi $ of (measurable) history dependent feedback policies $\\pi = \\lbrace \\pi _t \\rbrace _{t=0}^{\\infty }$ with $u_t = \\pi _t(x_{0:t}, u_{0:t-1})$ .",
"The penalty weight matrix Q = Qxx Qxu Qux Quu Sn+m has blocks $Q_{xx}$ , $Q_{uu}$ , $Q_{xu}=Q_{ux}^\\intercal $ which quadratically penalize deviations of the state, input, and product of state and input from the origin, respectively.",
"We assume the pair $(A, B)$ is stabilizable, the pair $(A, Q_{xx}^{1/2})$ is detectable, and the penalty matrices satisfy the definiteness condition $Q \\succ 0$ , in order to ensure feasibility of the problem (see ).",
"An LQR problem is fully described by the tuple of problem data $(A, B, Q)$ , which are fixed after problem definition.",
"In general, operators denoted by uppercase calligraphic letters depend on the problem data $(A, B, Q)$ , but we will not explicitly notate this for brevity; dependence on other parameters will be denoted explicitly by functional arguments.",
"We index over time in the evolution of a dynamical system with the letter $t$ , and index over iterations of an algorithm with the letter $k$ .",
"Dynamic programming can be used to show that the optimal policy that solves () is linear state-feedback ut = Kxt , where the gain matrix $K=\\mathcal {K}(P)$ is expressed through the linear-fractional operator $\\mathcal {K}$ K(P) -( Quu + BP B )-1(Qux + BP A) , and $P$ is the optimal value matrix found by solving the algebraic Riccati equation (ARE) R(P) = 0, where $\\mathcal {R}$ is the quadratic-fractional Riccati operator R(P) -P + Qxx + AP A - (Qxu + AP B) ( Quu + BP B )-1(Qux + BP A) .",
"The optimal gain and value matrix operators can be expressed more compactly as K(P) = -Huu-1(P) Hux(P) , R(P) = -P + Hxx(P) - Hxu(P) Huu-1(P) Hux(P) where $\\mathcal {H}$ is the state-action value matrix operator H(P) = Hxx(P) Hxu(P) Hux(P) Huu(P) Q + A B P A B .",
"The discrete-time Lyapunov equation with matrix $F$ and symmetric matrix $S$ is X = FX F + S, whose solution we denote by $X = \\texttt {DLYAP}(F, S)$ , which is unique if $F$ is Schur stable."
],
[
"Derivatives of the Riccati operator",
"The first total derivative In infinite dimensions, the first total derivative is called the Fréchet derivative, and the first directional derivative is called the Gateaux derivative.",
"As we are only considering finite-dimensional systems, we do not need the full generality of these objects.",
"of the Riccati operator evaluated at point $P \\in \\mathbb {S}^{n}$ is denoted as $\\mathcal {R}^\\prime (P) \\in \\mathbb {S}^{n} \\times \\mathbb {S}^{n}$ .",
"With a slight abuse of notation, the first directional derivative of the Riccati operator evaluated at point $P$ in direction $X$ is denoted as $\\mathcal {R}^\\prime (P, X) \\in \\mathbb {S}^{n}$ .",
"Computation of the first directional derivative is straightforward and follows e.g.",
"the derivation given by .",
"The general limit definition of this derivative is R(P, X) 0 R(P + X) - R(P) = .",
"dR(P + X)d |=0 Notice that since $\\mathcal {R}: \\mathbb {S}^{n} \\rightarrow \\mathbb {S}^{n}$ it follows that $\\mathcal {R}^\\prime (\\cdot , \\cdot ): \\mathbb {S}^{n} \\times \\mathbb {S}^{n} \\rightarrow \\mathbb {S}^{n}$ .",
"In evaluating the first directional derivative, it will be useful note that H(P + X) = Q + A B (P + X) A B .",
"The first derivative is then R(P, X) = .",
"dR(P + X)d |=0 = .",
"dd [ -(P + X) + Hxx(P + X) - Hxu(P + X) Huu-1(P + X) Hux(P + X) ] |=0 At this point it will be useful to evaluate the following expressions: .",
"d H(P + X)d |=0 = .",
"dd [ Q + A B (P + X) A B ] |=0 = A B X A B , and .",
"d Huu-1(P + X)d |=0 = -Huu-1(P) .",
"d Huu(P + X)d |=0 Huu-1(P) = -( Quu + BP B )-1 ( BX B ) ( Quu + BP B )-1 , where we used the rule for a derivative of a matrix inverse e.g.",
"as in .",
"Continuing with the first derivative, R(P, X) = .",
"dd [ -(P + X) + Hxx(P + X) - Hxu(P + X) Huu-1(P + X) Hux(P + X) ] |=0 = - .",
"dd [ P + X ] |=0 + .",
"dd [Hxx(P + X) ] |=0 - .",
"dd [ Hxu(P + X) ] |=0 Huu-1(P) Hux(P) - Hxu(P) .",
"dd [ Huu-1(P + X) ] |=0 Hux(P) - Hxu(P) Huu-1(P) .",
"dd [ Hux(P + X) ] |=0 = -X + A X A - A X B(Quu + B P B)-1 (Qux + B P A) - (Qxu + A P B) ( Quu + B P B)-1 B X A +A P B(Quu+B P B)-1 B X B(Quu + B P B)-1 B P A where we used the product rule for matrix derivatives."
],
[
"Identities",
"Considering two symmetric matrices $P, X$ and the related gains K = K(P) = -( Quu + BP B )-1 (Qux + BP A), L = K(X) = -( Quu + BX B )-1 (Qux + BX A), we have Qux + Quu K = - BP (A+BK), KQux + KQuu K = - (BK)P (A+BK) .",
"Thus the Riccati operator $\\mathcal {R}(P)$ can be rewritten as R(P) = - P + Qxx + Qxu K + AP (A + BK) = - P + Qxx + Qxu K - (BK)P (A + BK) + (A + BK)P (A + BK) = - P + I K Qxx Qxu Qux Quu I K + (A + BK)P (A + BK) = -P + I K ( Q + A B P A B ) I K , and the derivative $\\mathcal {R}^\\prime (X, P)$ can be written using (REF ) as R(X, P) = -P + (A+BL)P (A+BL).",
"and we have the identity R(P) - R(X, P) = I K Q I K + (A + BK)P (A + BK) - (A+BL)P (A+BL) .",
"In the case of $X=P$ , identity (REF ) specializes to R(P) - R(P, P) = I K Q I K ."
],
[
"Generic Newton methods",
"First we consider finding a solution to the equation $f(x)=0$ where $f: \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ , whose total derivative at a point $x$ is $f^\\prime (x) \\in \\mathbb {R}^{n \\times n}$ .",
"The methods under consideration can be understood and derived as the numerical integration of the following Newton-Leibniz integral from the second fundamental theorem of calculus: 0 = f(x) = f(xk)+xkx f(t) dt."
],
[
"Newton method",
"The Newton method, due originally in heavily modified form to , and originally in the general differential form to (see the historical notes of , ), begins with an initial guess $x_0$ then proceeds with iterations xk+1 = xk - f(xk)-1 f(xk), until convergence.",
"Intuitively, the Newton method forms a linear approximation $f(x_k) + f^\\prime (x_k)(x-x_k)$ to the function $f$ at $x_k$ , and assigns the point where the linear approximation crosses 0 as the next iterate.",
"The Newton method can be derived from () by using left rectangular integration.",
"The Newton update can be rearranged into the Newton equation f(xk) (xk+1 - xk) = - f(xk), where the left-hand side is recognized as the directional derivative of $f$ evaluated at point $x_k$ in direction $x_{k+1} - x_k$ .",
"This rearrangement implies that the Newton method does not require explicit evaluation of the entire total derivative $f^\\prime (x_k)$ so long as a suitable direction $x_{k+1} - x_k$ can be found which solves the Newton equation.",
"This will become important in the LQR setting as we use this fact to avoid notating and computing large order-4 tensors.",
"This technique uses derivative information at a single point and is known to achieve quadratic convergence in a neighborhood of the root ().",
"In the setting of both continuous- and discrete-time LQR, this algorithm is known to achieve quadratic convergence globally, as shown by , , ."
],
[
"Mid-point Newton method",
"The midpoint Newton method, due originally to , begins with an initial guess $x_0$ then proceeds with iterations xNk+1 = xk - f(xk)-1 f(xk), xMk = 12 (xk+ xNk+1), xk+1 = xk - f( xMk )-1 f(xk), until convergence.",
"The midpoint Newton method can be derived from () by using midpoint rectangular integration.",
"Intuitively, much like the Newton method, the midpoint Newton method forms a linear approximation $f(x_k) + f^\\prime (x^M_k)(x-x_k)$ to the function $f$ at $x_k$ , and assigns the point where the linear approximation crosses 0 as the next iterate.",
"The distinction is that the slope of the linear approximation is not evaluated at $x_k$ as in the Newton method, but rather at the midpoint $x^M_{k} = \\frac{1}{2} (x_k + x^N_{k+1})$ where $x^N_{k+1}$ is the Newton iterate.",
"The updates can be rearranged into the Newton equations f(xk) (xNk+1 - xk) = - f(xk), f(xMk) (xk+1 - xk) = - f(xk), where the left-hand side of the first equation is recognized as the directional derivative of $f$ evaluated at point $x_k$ in direction $x^N_{k+1} - x_k$ ; the second equation is of the same form.",
"This rearrangement implies that the midpoint Newton method does not require explicit evaluation of the entire total derivative $f^\\prime (x_k)$ so long as a suitable direction $x_{k+1} - x_k$ can be found which solves the Newton equation.",
"This will become important in the LQR setting as we use this fact to avoid notating and computing large order-4 tensors.",
"Each iteration in this technique uses derivative information at two points, $x_k$ and $x^M_{k}$ .",
"This method has been shown to achieve cubic convergence in a neighborhood of the root by , , ."
],
[
"Exact midpoint policy iteration",
"We now consider application of the midpoint Newton method to the Riccati equation ().",
"Although () could be brought to the vector form $f(x)=0$ by vectorization with $x=\\operatorname{svec}(P)$ and $f(x) = \\operatorname{svec}(\\mathcal {R}(\\operatorname{smat}(x)))$ , it will be simpler to leave the equations in matrix form, which is possible due to the special form of the Newton-type updates, which only involve directional derivatives (and not total derivatives).",
"Applying the midpoint Newton update to () yields PNk+1 = Pk - R(Pk)-1 (R(Pk)), PMk = 12 ( PNk+1 + Pk ), Pk+1 = Pk - R(PMk)-1 (R(Pk)), The updates can be rearranged into the Newton equations R(Pk, PNk+1-Pk) = -R(Pk) R(PMk, Pk+1-Pk) = -R(Pk) and further by linearity of $\\mathcal {R}^\\prime (\\cdot , X)$ in $X$ to R(Pk, PNk+1) = R(Pk, Pk) - R(Pk), R(PMk, Pk+1) = R(PMk, Pk) - R(Pk).",
"Recalling the expression for $\\mathcal {R}^\\prime $ in (REF ) for the left-hand sides and applying the identities in (REF ) and (REF ) to the right-hand sides, these become the Lyapunov equations PNk+1 = (A+BKk)PNk+1 (A+BKk) + Q + KkR Kk + (A+BKk)PNk+1 (A+BKk), Pk+1 = (A+BLk)PNk+1 (A+BLk) + Q + KkR Kk + (A+BKk)Pk (A+BKk) - (A+BLk)Pk (A+BLk), or more compactly, PNk+1 = DLYAP(FN, SN), Pk+1 = DLYAP(FM, SM), where r@ c@ l@ r@ c@ l@ FN = A+BKk, SN = I Kk Q I Kk FM = A+BLk, SM = I Kk Q I Kk + (A+BKk)Pk (A+BKk) - (A+BLk)Pk (A+BLk) where Kk = K(Pk), Lk = K(Mk), Mk = 12 (Pk + PNk+1) .",
"These updates are collected in the full midpoint policy iteration in Algorithm .",
"Exact midpoint policy iteration (MPI) [1] System matrices $A,B$ , penalty matrix $Q$ , initial stabilizing gain $K_0$ , tolerance $\\varepsilon $ Initialize: $k = 0$ , $P_{-1} = \\infty I_n$ , and $P_0 = \\texttt {DLYAP}(F, S)$ where $F = A+BK_0$ and $S = \\begin{bmatrix} I & K_0^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & K_0^\\intercal \\end{bmatrix}^\\intercal .$ $\\Vert P_{k} - P_{k-1}\\Vert > \\varepsilon $ Compute $K_k = \\mathcal {K}(P_k).$ Compute $F^N = A+BK_k$ , and $S^N = \\begin{bmatrix} I & K_k^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & K_k^\\intercal \\end{bmatrix}^\\intercal .$ Solve $P^N_{k+1} = \\texttt {DLYAP}(F^N, S^N)$ Compute $M_k = \\frac{1}{2} (P_k + P^N_{k+1})$ and $L_k = \\mathcal {K}(M_k).$ Compute $F^M = A+BL_k$ , and $S^M = \\begin{bmatrix} I & K_k^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & K_k^\\intercal \\end{bmatrix}^\\intercal + (A+BK_k)^\\intercal P_{k} (A+BK_k) - (A+BL_k)^\\intercal P_{k} (A+BL_k).$ Solve $P_{k+1} = \\texttt {DLYAP}(F^M, S^M).$ $k \\leftarrow k+1$ $P_k$ , $K_k = \\mathcal {K}(P_k)$ Consider Exact Midpoint Policy Iteration in Algorithm .",
"For any feasible problem instance, there exists a neighborhood around the optimal gain $K^*$ from which any initial gain $K_0$ yields cubic convergence, i.e.",
"$\\Vert K_{k+1} - K^* \\Vert \\le \\mathcal {O} \\left(\\Vert K_{k} - K^* \\Vert ^3 \\right)$ where $\\Vert \\cdot \\Vert $ is any matrix norm.",
"By the assumptions on $(A, B, Q)$ , the closed-loop matrix under the optimal gain satisfies $\\rho (A+BK^*) < 1$ .",
"Since the spectral radius of the closed-loop matrix $A+BK$ is continuous with respect to (each entry of) the gain $K$ (see ), it follows that there exists a radius $\\varepsilon _0 > 0$ and ball $\\mathcal {B}_0 = \\lbrace K : \\ \\Vert K - K^*\\Vert < \\varepsilon _0 \\rbrace $ around the optimal gain $K^*$ within which any gain $K$ is stabilizing.",
"For use later, define the following quantities and operators in terms of the system data $(A, B, Q)$ and gain $K$ .",
"Let $P$ be the solution to P = FP F + S where $F = A + BK$ and $S = \\begin{bmatrix} I & {K}^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & {K}^\\intercal \\end{bmatrix}^\\intercal $ .",
"Let $K^N = \\mathcal {K}(P)$ .",
"Let $P^N$ be the solution to PN = FNP FN + SN where $F^N = A+BK^N$ and $S^N = \\begin{bmatrix} I & {K^N}^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & {K^N}^\\intercal \\end{bmatrix}^\\intercal $ .",
"Let $P^M = \\frac{1}{2}(P + P^N)$ .",
"Let $K^M = \\mathcal {K}(P^M)$ and $F^M = A+BK^M$ .",
"Define the operator Y(K) = PN - FMPN FM + R(P).",
"Now, at the optimal gain $K = K^* = \\mathcal {R}(P^*)$ , we have that $K^N = K^M = K^*$ and $P^N = P^M = P^*$ where $P^*$ solves the Riccati equation $\\mathcal {R}(P^*) = 0$ .",
"Therefore, Y(K*) = P* - (A+BK*)P* (A+BK*) + 0 = I K* Q I K*0 By inspection of all the preceding relevant quantities, $\\mathcal {Y}(K)$ is continuous with respect to $K$ , and therefore there exists a radius $\\varepsilon _1 > 0$ and ball $\\mathcal {B}_1 = \\lbrace K : \\ \\Vert K - K^*\\Vert < \\varepsilon _1 \\rbrace $ around the optimal gain $K^*$ within which any gain $K$ satisfies $\\mathcal {Y}(K) \\succ 0$ .",
"Define $\\varepsilon = \\min (\\varepsilon _0, \\varepsilon _1)$ and likewise $\\mathcal {B} = \\lbrace K : \\ \\Vert K - K^*\\Vert < \\varepsilon \\rbrace $ .",
"Consider an arbitrary matrix $P$ computed from an arbitrary gain $K$ within $\\mathcal {B}$ as the solution to the Lyapunov equation P = (A+BK)P (A+BK) + I K Q I K, which is well defined since $K$ is stabilizing and $Q \\succ 0$ .",
"Define the set P = {P : P solves () with K B }.",
"Theorem 2 of requires that the inverse $\\mathcal {R}(P)^{-1}$ exist everywhere in $\\mathcal {B}$ ; however the proof of Theorem 2 of only uses this assumption in order to ensure that solutions to the Newton equations () and () exist and are unique.",
"Therefore, it suffices to prove just that solutions to the Newton equations () and () exist and are unique for any $P$ in $\\mathcal {P}$ .",
"Write the first Newton equation () as R(P, PN - P) = -R(P).",
"Using the expression (REF ), this can be rewritten as P - PN = (A+BKN)(P - PN)(A+BKN) -R(P).",
"where $K^N = \\mathcal {K}(P)$ .",
"By e.g.",
"the matrix $-\\mathcal {R}(P) \\succ 0$ (this is related to convergence of value iteration).",
"Also, because $K^N = \\mathcal {K}(P)$ we may apply the Wonham-like identity developed in P = FNP FN + S where $F^N = A+BK^N$ and S = I KN Q I KN+ (K - KN)(Quu + BP B) (K - KN) 0 , which shows that the gain $K^N$ is stabilizing i.e.",
"$A+BK^N$ is Schur stable.",
"Therefore the solution to () is unique and well defined (and positive definite).",
"Similarly, write the second Newton equation () as R(PM, P+ - P) = -R(P) where $P^M = \\frac{1}{2}(P + P^N)$ and $P^N$ solves the first Newton equation (), equivalently PN = FNPN FN + SN where $S^N = \\begin{bmatrix} I & {K^N}^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & {K^N}^\\intercal \\end{bmatrix}^\\intercal $ .",
"Using the expression (REF ), the second Newton equation () can be further rewritten as P - P+ = (A+BKM)(P - P+)(A+BKM) -R(P).",
"where $K^M = \\mathcal {K}(P^M)$ .",
"Again, by e.g.",
"the matrix $-\\mathcal {R}(P) \\succ 0$ .",
"By construction $P^M = \\frac{1}{2}(P + P^N)$ , so combining the expressions P = FNP FN + S, PN = FNPN FN + SN we obtain PM = 12(P + PN) = 12( FNP FN + S + FNPN FN + SN ) = FN( 12 (P+PN) ) FN + 12(S + SN) = FNPM FN + 12(S + SN) and using $K^M = \\mathcal {K}(P^M)$ and the Wonham-like identity of again we obtain PM = FMPM FM + SM where $F^M = A + B K^M$ and SM = 12(S + SN) + (KN - KM)(Quu + BPM B) (KN - KM) 0, which shows that the gain $K^M$ is stabilizing i.e.",
"$A + BK^M$ is Schur stable.",
"Therefore the solution to () is unique and well defined (and positive definite).",
"Since $P$ in $\\mathcal {P}$ was arbitrary, we have proved the assertion that solutions to the Newton equations () and () exist and are unique for any $P$ in $\\mathcal {P}$ .",
"Furthermore, $P^+$ equivalently solves the Lyapunov equation P+ = FMP+ FM + S+ where S+ = PN - FMPN FM + R(P) 0 where the positive definiteness of $S^+$ follows by the restriction of $K$ to $\\mathcal {B}_1$ , and therefore $P^+$ proves stability of $K^M$ .",
"Theorem 2 of also requires that $\\mathcal {R}$ be sufficiently smooth with bounded derivatives up to third order in $\\mathcal {B}$ .",
"Any $P \\in \\mathcal {P}$ is positive definite and bounded above since $P$ solves () and $K \\in \\mathcal {K}$ is stable.",
"Also, by assumption we have $Q_{uu} \\succ 0$ .",
"Therefore, the term $Q_{uu} + B^\\intercal P B \\succeq Q_{uu} \\succ 0$ so its inverse is well defined and bounded above.",
"By examination of () and (REF ) it is evident that $\\mathcal {R}$ and $\\mathcal {R}^\\prime $ are analytic functions with upper bounds by the preceding arguments.",
"Higher-order derivatives of $\\mathcal {R}$ follow similar Lyapunov equations as $\\mathcal {R}^\\prime $ and are thus also upper bounded on $\\mathcal {P}$ .",
"The assumptions of Theorem 2 of are satisfied, and thus we conclude that the iterates converge $\\Vert P_{k+1} - P^*\\Vert < c \\Vert P_{k} - P^*\\Vert $ for some $c \\in [0, 1)$ and do so at a cubic rate $\\Vert P_{k+1} - P^* \\Vert \\le \\mathcal {O} \\left(\\Vert P_{k} - P^* \\Vert ^3 \\right)$ .",
"Consider $K_{k} = K$ so $P_{k+1} = P^+$ , and $L_k = K^M$ .",
"Since $K_{k+1} = \\mathcal {K}(P_{k+1})$ and $P_{k+1}$ proves stability of a gain matrix $K^M$ , this implies Schur stability of every $A+BK_k$ (using the Wonham-like identity of ).",
"Likewise, since the sequence of $P_k$ approach the limit $P^*$ and $K_k = \\mathcal {K}(P_k)$ , the sequence of $K_k$ approach the limit $K^*$ .",
"Since $\\Vert K_k - K^*\\Vert = \\Vert \\mathcal {K}(P_k) - \\mathcal {K}(P^*) \\Vert = \\mathcal {O}(\\Vert P_{k} - P^*\\Vert )$ , we conclude the same cubic convergence result holds for $K_k$ ."
],
[
"Approximate midpoint policy iteration",
"In the model-free setting we do not have access to the dynamics matrices $(A,B)$ , so we cannot execute the updates in Algorithm .",
"However, the gain $K=\\mathcal {K}(P)$ can be computed solely from the state-action value matrix $H=\\mathcal {H}(P)$ as $K=-H_{uu}^{-1}H_{ux}$ .",
"Thus, if we can obtain accurate estimates of $H$ , we can use the estimate of $H$ to compute $K$ and we need not perform any other updates that depend explicitly on $(A,B)$ .",
"We begin by summarizing an existing method in the literature for estimating state-action value functions from observed state-and-input trajectories."
],
[
"State-action value estimation",
"First, we connect the matrix $H$ with the (relative) state-action value ($\\mathcal {Q}$ ) function, which determines the (relative) cost of starting in state $x = x_0$ , taking action $u = u_0$ , then following the policy $u_t=K x_t$ thereafter: Tr(PW) + QK(x,u) = x u Qxx Qxu Qux Quu x u + wE [ (Ax+Bu+w)P (Ax+Bu+w) ] = x u Hxx Hxu Hux Huu x u + Tr(PW) or simply QK(x,u) = x u Hxx Hxu Hux Huu x u where $H=\\mathcal {H}(P)$ and $P$ is the solution to DLYAP(A+BK, I K Qxx Qxu Qux Quu I K ) From this expression it is clear that a state-input trajectory, or “rollout,” $\\mathcal {D}= \\lbrace x_t, u_t \\rbrace _{t=0}^{\\ell }$ must satisfy this cost relationship, which can be used to estimate $H$ .",
"In particular, least-squares temporal difference learning for $\\mathcal {Q}$ -functions (LSTDQ) was originally introduced by and analyzed by , , and is known to be a consistent and unbiased estimator of $H$ .",
"Following the development of , the LSTDQ estimator is summarized in Algorithm REF .",
"LSTDQ: Least-squares temporal difference learning for $Q$ -functions [1] Rollout $\\mathcal {D}= \\lbrace x_t, u_t \\rbrace _{t=0}^{\\ell }$ , gain matrix $K^{\\text{eval}}$ , penalty matrix $Q$ , noise covariance $W$ .",
"Compute augmented rollout $\\lbrace z_t, v_t, c_t \\rbrace _{t=0}^{\\ell }$ where $z_t = \\begin{bmatrix} x \\\\ u \\end{bmatrix}, \\ v_t = \\begin{bmatrix} x \\\\ K^{\\text{eval}} x \\end{bmatrix}, \\ c_t = z_t^\\intercal Q z_t.$ Use feature map $\\phi (z) = \\operatorname{svec}\\left(z z^\\intercal \\right)$ and noise quantity $\\psi = \\operatorname{svec}\\left( \\begin{bmatrix} I \\\\ K^{\\text{eval}} \\end{bmatrix} W \\begin{bmatrix} I \\\\ K^{\\text{eval}} \\end{bmatrix}^\\intercal \\right)$ and compute the parameter estimate $ \\hat{\\Theta } = \\Big ( \\sum _{t=1}^\\ell \\phi (z_t) (\\phi (z_t) - \\phi (v_{t+1}) + \\psi )^\\intercal \\Big )^\\dag \\sum _{t=1}^\\ell \\phi (z_t) c_t.",
"$ $\\hat{H} = \\operatorname{smat}(\\hat{\\Theta })$ .",
"We collect rollouts to feed into Algorithm REF via Algorithm REF , i.e.",
"by initializing the state with $x_0$ drawn from the given initial state distribution $\\mathcal {X}_0$ , then generating control inputs according to $u_t = K^{\\text{play}} x_t + u^{\\text{explore}}_t$ where $K^{\\text{play}}$ is a stabilizing gain matrix, and $u^{\\text{explore}}_t$ is an exploration noise drawn from a distribution $\\mathcal {U}_t$ , assumed Gaussian in this work, to ensure persistence of excitation.",
"$\\texttt {ROLLOUT}$ : Rollout collection [1] Gain $K^{\\text{play}}$ , rollout length $\\ell $ , initial state distribution $\\mathcal {X}_0$ , exploration distributions $\\lbrace \\mathcal {U}_t \\rbrace _{t=0}^\\ell $ .",
"Initialize state $x_0 \\sim \\mathcal {X}_0$ $t=0,1,2,\\ldots ,\\ell $ Sample exploratory control input $u^{\\text{explore}}_t \\sim \\mathcal {U}_t$ and disturbance $w_t \\sim W$ Generate control input $u_t = K^{\\text{play}} x_t + u^{\\text{explore}}_t$ Record state $x_t$ and input $u_t$ Update state according to $x_{t+1} = A x_t + B u_t + w_t$ $\\mathcal {D}= \\lbrace x_t, u_t \\rbrace _{t=0}^{\\ell }$ .",
"Note that LSTDQ is an off-policy method, and thus the gain $K^{\\text{play}}$ used to generate the data in Algorithm REF and the gain $K^{\\text{eval}}$ whose state-action value matrix is estimated in Algorithm REF need not be identical.",
"We will use this fact in the next section to give an off-policy, offline (OFF) and on-policy, online (ON) version of our algorithm.",
"Likewise, the penalty matrix $Q$ used in Algorithm REF need not be the same as the one in the original problem statement, which is critical to developing the model-free midpoint update in the next section."
],
[
"Derivation of approximate midpoint policy iteration",
"We have shown that estimates of the state-action value matrix $H$ can be obtained by $\\texttt {LSTDQ}$ using either off-policy or on-policy data.",
"In the following development, (OFF) denotes a variant where a single off-policy rollout $\\mathcal {D}$ is collected offline before running the system, and (ON) denotes a variant where new on-policy rollouts are collected at each iteration.",
"Also, an overhat symbol “ $\\hat{}$ ” denotes an estimated quantity while the absence of one denotes an exact quantity.",
"In approximate policy iteration, we can simply form the estimate $\\hat{H}_k$ using $\\texttt {LSTDQ}$ (see ).",
"For approximate midpoint policy iteration, the form of $\\hat{H}_k$ is more complicated and requires multiple steps.",
"To derive approximate midpoint policy iteration, we will re-order some of the steps in the loop of Algorithm .",
"Specifically, move the gain calculation in step 3 to the end after step 9.",
"We will also replace explicit computation of the value function matrices with estimation of state-action value matrices, i.e.",
"subsume the pairs of steps 4, 5 and 8,9 into single steps, and work with $H$ instead of $P$ .",
"Thus, at the beginning of each iteration we have in hand an estimated state-action value matrix $\\hat{H}_k$ and gain matrix $\\hat{K}_k$ satisfying $\\hat{K}_k = -\\hat{H}_{uu,k}^{-1} \\hat{H}_{ux,k}$ .",
"First we translate steps 4, 5, 6, and 7 to a model-free version.",
"Working backwards starting with step 7, in order to estimate $L_k$ , it suffices to estimate $\\mathcal {H}(M_k)$ since $L_k = -\\mathcal {H}(M_k)_{uu}^{-1} \\mathcal {H}(M_k)_{ux}$ .",
"In order to find $\\mathcal {H}(M_k)$ , notice that the operator $\\mathcal {H}(X)$ is linear in $X$ , so H(Mk) = H( 12 (Pk + PNk+1) ) = 12 ( H(Pk) + H(PNk+1) ) .",
"Therefore we can estimate $\\mathcal {H}(M_k)$ by estimating $\\mathcal {H}(P_k)$ and $\\mathcal {H}(P^N_{k+1})$ separately and taking their midpoint.",
"Since the estimate $\\hat{H}_k$ of $\\mathcal {H}(P_k)$ is known from the prior iteration, what remains is to find an estimate $\\hat{H}^N_{k+1}$ of $\\mathcal {H}(P^N_{k+1})$ by collecting DN = ROLLOUT(Kk, , X0, { Ut }t=0) ON, or using DN = DOFF, and estimating $\\hat{H}^N_{k+1} = \\texttt {LSTDQ}(\\mathcal {D}^N, \\hat{K}_k, Q)$ .",
"Then we form the estimated gain $\\hat{L}_k = -{\\hat{H}^M_{uu,k}}{}^{-1} \\hat{H}^M_{ux,k}$ where $\\hat{H}^M_k = \\frac{1}{2} (\\hat{H}_k + \\hat{H}^N_{k+1})$ .",
"Now we translate steps 8, 9, and 2 to a model-free version.",
"Working backwards, starting with step 2, in order to estimate $K_{k+1}$ , it suffices to find an estimate $\\hat{H}_{k+1}$ of matrix $\\mathcal {H}(P_{k+1})$ since $K_{k+1} = -\\mathcal {H}(P_{k+1})_{uu}^{-1} \\mathcal {H}(P_{k+1})_{ux}$ .",
"From steps 8 and 9, we want to estimate Hk+1 = H(Pk+1) = Q + A B Pk+1 A B , where Pk+1 = DLYAP( FM, SM ), FM = A+BLk, SM = I Kk ( Q + A B Pk A B ) I Kk - I Lk A B Pk A B I Lk .",
"Comparing the two arguments to $\\texttt {DLYAP}(\\cdot , \\cdot )$ in (REF ) and (REF ), we desire both A+BK = A+BLk, I K QM I K = SM.",
"Clearly it suffices to take $K=L_k$ in (REF ).",
"Notice that, critically, all quantities in $S^M$ on the right-hand side of (REF ) have been estimated already, i.e.",
"$\\hat{K}_k$ , $\\hat{L}_k$ , $\\hat{H}_k$ have been calculated already and Q + A B Pk A B = Hk, A B Pk A B = Hk - Q.",
"Substituting $K=L_k$ in (REF ) and comparing coefficients, it suffices to estimate $Q^M$ by QM = I Kk Hk I Kk 0 0 0 - (Hk - Q) .",
"At this point, establish the rollout $\\mathcal {D}^M$ either by collecting DM = ROLLOUT(Lk, , X0, { Ut }t=0), ON or using DM = D. OFF Then the matrix $\\hat{H}^O_{k+1} = \\texttt {LSTDQ}(\\mathcal {D}^M, \\hat{L}_k, \\hat{Q}^M)$ estimates HOk+1 = QM + A B Pk+1 A B .",
"However, we need Hk+1 = Q + A B Pk+1 A B , which is easily found by offsetting $H^O_{k+1}$ as Hk+1 = HOk+1 + (Q - QM), and thus Hk+1 = HOk+1 + (Q - QM) estimates $H_{k+1}$ .",
"One further consideration to address is the initial estimate $\\hat{H}_0$ ; since we do not have a prior iterate to use, we simply collect $\\mathcal {D}= \\texttt {ROLLOUT}(\\hat{K}_0, \\ell , \\mathcal {X}_0, \\lbrace \\mathcal {U}_t \\rbrace _{t=0}^\\ell )$ and estimate $\\hat{H}_0 = \\texttt {LSTDQ}(\\mathcal {D}, \\hat{K}_0, Q)$ i.e.",
"the first iteration will be a standard approximate policy iteration/Newton step.",
"Importantly, the initial gain $\\hat{K}_0$ must stabilize the system so that the value functions are finite-valued.",
"Also, although a convergence criterion such as $\\Vert \\hat{H}_{k} - \\hat{H}_{k-1}\\Vert > \\varepsilon $ could be used, it is more straightforward to use a fixed number of iterations $N$ so that the influence of stochastic errors in $\\hat{H}_k$ does not lead to premature termination of the program.",
"Likewise, a schedule of increasing rollout lengths $\\ell $ could be used for the (ON) variant to achieve increasing accuracy, but finding a meaningful schedule which properly matches the fast convergence rate of the algorithm requires more extensive analysis.",
"The full set of updates are compiled in Algorithm REF .",
"Approximate midpoint policy iteration (AMPI) [1] Penalty $Q$ , gain $\\hat{K}_0$ , number of iterations $N$ , rollout length $\\ell $ , distributions $\\mathcal {X}_0$ , $\\lbrace \\mathcal {U}_t \\rbrace _{t=0}^\\ell $ .",
"Initialize: $\\hat{H}_{-1} = \\infty I_{n+m}$ and $k = 0$ Collect $\\mathcal {D}= \\texttt {ROLLOUT}(\\hat{K}_0, \\ell , \\mathcal {X}_0, \\lbrace \\mathcal {U}_t \\rbrace _{t=0}^\\ell )$ Estimate value matrix $\\hat{H}_0 = \\texttt {LSTDQ}(\\mathcal {D}, K_0, Q)$ .",
"$k < N$ Set $\\mathcal {D}^N =\\mathcal {D}$ (OFF), or collect $\\mathcal {D}^N = \\texttt {ROLLOUT}(\\hat{K}_k, \\ell , \\mathcal {X}_0, \\lbrace \\mathcal {U}_t \\rbrace _{t=0}^\\ell )$ (ON) Estimate value matrix $\\hat{H}^N_{k+1} = \\texttt {LSTDQ}(\\mathcal {D}^N, \\hat{K}_k, Q)$ .",
"Form the midpoint value estimate $\\hat{H}^M_k = \\frac{1}{2} (\\hat{H}_k + \\hat{H}^N_{k+1}).$ Compute the midpoint gain $\\hat{L}_k = -{\\hat{H}^M_{uu,k}}{}^{-1} \\hat{H}^M_{ux,k}.$ Set $\\mathcal {D}^M =\\mathcal {D}$ (OFF), or collect $\\mathcal {D}^M = \\texttt {ROLLOUT}(\\hat{L}_k, \\ell , \\mathcal {X}_0, \\lbrace \\mathcal {U}_t \\rbrace _{t=0}^\\ell )$ (ON) Estimate $\\hat{H}^O_{k+1} = \\texttt {LSTDQ}(\\mathcal {D}^M, \\hat{L}_k, \\hat{Q}^M)$ where $Q^M=\\begin{bmatrix}\\begin{bmatrix}I \\\\ \\hat{K}_k\\end{bmatrix}^\\intercal \\hat{H}_k\\begin{bmatrix}I \\\\ \\hat{K}_k\\end{bmatrix}& 0 \\\\0 & 0\\end{bmatrix}-(\\hat{H}_k -Q).$ Compute the estimated value matrix $\\hat{H}_{k+1} = \\hat{H}^O_{k+1} + ( Q - \\hat{Q}^M )$ .",
"Compute the gain $\\hat{K}_{k+1} = -\\hat{H}_{uu,k+1}^{-1} \\hat{H}_{ux,k+1}.$ $k \\leftarrow k+1$ $\\hat{H}_k$ , $\\hat{K}_k$ Consider Approximate Midpoint Policy Iteration in Algorithm REF .",
"As the rollout length $\\ell $ grows to infinity, the state-action value matrix estimate $\\hat{H}_k$ converges to the exact value.",
"Thus, in the infinite data limit, for any feasible problem instance, there exists a neighborhood around the optimal gain $K^*$ from which any initial gain $\\hat{K}_0$ converges cubically to $K^*$ .",
"The claim follows by Proposition and the fact that $\\texttt {LSTDQ}$ is a consistent estimator , , i.e.",
"as $\\ell \\rightarrow \\infty $ the estimates $\\hat{H}$ used in Algorithm REF approach the true values $H$ indirectly used in Algorithm ."
],
[
"Numerical experiments",
"In this section we compare the empirical performance of proposed midpoint policy iteration (MPI) with standard policy iteration (PI), as well as their approximate versions (AMPI) and (API).",
"In all experiments, regardless of whether the exact or approximate algorithm is used, we evaluated the value matrix $P_k$ associated to the policy gains $K_k$ at each iteration $k$ on the true system, i.e.",
"the solution to $P_k = \\texttt {DLYAP}\\left(A+BK_k, \\begin{bmatrix} I & K_k^\\intercal \\end{bmatrix} Q \\begin{bmatrix} I & K_k^\\intercal \\end{bmatrix}^\\intercal \\right)$ .",
"We then normalized the deviation $\\Vert P_k - P^*\\Vert $ , where $P^*$ solves the Riccati equation (), by the quantity $\\operatorname{Tr}(P^*)$ .",
"This gives a meaningful metric to compare different suboptimal gains.",
"We also elected to use the off-policy version (OFF) of AMPI and API in order to achieve a more direct and fair comparison between the midpoint and standard methods; each is given access to precisely the same sample data and initial policy, so differences in convergence are entirely due to the algorithms.",
"Python code which implements the proposed algorithms and reproduces the experimental results is available at https://github.com/TSummersLab/midpoint-policy-iteration."
],
[
"Representative example",
"Here we consider one of the simplest tasks in the control discipline: regulating an inertial mass using a force input.",
"The stochastic continuous-time dynamics of the second-order system are dx = Ac x dt + Bc u dt + dw where Ac = 0 1 0 0 , Bc = 0 1 , with mass $\\mu > 0$ , state $x \\in \\mathbb {R}^2$ where the first state is the position and the second state is the velocity, force input $u \\in \\mathbb {R}$ , and $dw \\in \\mathbb {R}^2$ is a Wiener process with covariance $W_c \\succeq 0$ .",
"Forward-Euler discretization of the continuous-time dynamics with sampling time $\\Delta t$ yields the discrete-time dynamics xt+1 = A xt + B ut + wt where A = 1 t 0 1 , B = 0 t , with $w_t \\sim \\mathcal {N}(0, W)$ with $W = \\Delta t \\cdot W_c $ .",
"We used $\\mu = 1$ , $\\Delta t = 0.01$ , $W_c = 0.01 I_2$ , $Q = I_3$ .",
"The initial gain was chosen by perturbing the optimal gain $K^*$ in a random direction such that the initial relative error $\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) = 10$ ; in particular the initial gain was $K_0 = \\begin{bmatrix} -0.035 & -2.087 \\end{bmatrix}$ .",
"For the approximate algorithms, we used the hyperparameters $\\ell = 300$ , $\\mathcal {X}_0 = \\mathcal {N}(0, I_2)$ , $\\mathcal {U}_t = \\mathcal {N}(0, I_2)$ for $t=0,1,\\ldots , \\ell $ .",
"The results of applying midpoint policy iteration and the standard policy iteration are plotted in Figure .",
"Clearly MPI and AMPI converge more quickly to the (approximate) optimal policy than PI and API, with MPI converging to machine precision in 7 iterations vs 9 iterations for PI, and AMPI converging to noise precision in 6 iterations vs 8 iterations for API.",
"Figure: Relative value error ∥P k -P * ∥/Tr(P * )\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) vs iteration count kk using PI and MPI on the inertial mass control problem."
],
[
"Randomized examples",
"Next we apply the exact and approximate PI algorithms on 10000 unique problem instances in a Monte Carlo-style approach, where problem data was generated randomly with $n=4$ , $m=2$ , entries of $A$ drawn from $\\mathcal {N}(0,1)$ and $A$ scaled so $\\rho (A) \\sim \\text{Unif}([0, 2])$ , entries of $B$ drawn from $\\text{Unif}([0, 1])$ , and $Q = U \\Lambda U^\\intercal \\succ 0$ with $\\Lambda $ diagonal with entries drawn from $\\text{Unif}([0, 1])$ and $U$ orthogonal by taking the QR-factorization of a square matrix with entries drawn from $\\mathcal {N}(0,1)$ , where we denote the uniform distribution on the interval $[a,b]$ by $\\text{Unif}([a, b])$ and the multivariate Gaussian distribution with mean $\\mu $ and variance $\\Sigma $ by $\\mathcal {N}(\\mu , \\Sigma )$ .",
"We used a small process noise covariance of $W = 10^{-6} I_4$ to avoid unstable iterates due to excessive data-based approximation error of $H$ , over all problem instances.",
"All initial gains $K_0$ were chosen by perturbing the optimal gain $K^*$ in a random direction such that the initial relative error $\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) = 10$ .",
"For the approximate algorithms, we used the hyperparameters $\\ell = 100$ , $\\mathcal {X}_0 = \\mathcal {N}(0, I_2)$ , $\\mathcal {U}_t = \\mathcal {N}(0, I_2)$ for $t=0,1,\\ldots , \\ell $ .",
"In Figures , , , we plot the relative value error $\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*)$ over iterations.",
"In Figures , each scatter point represents a unique Monte Carlo sample, i.e.",
"a unique problem instance, initial gain, and rollout.",
"Figures and show the same data, but using a hexagonal bin 2D histogram to visualize the empirical probability density of outcomes.",
"Each plot shows the empirical distribution of errors at the iteration count $k$ labeled in the subplot titles above each plot.",
"The x-axis is the spectral radius of $A$ which characterizes open-loop stability.",
"Figures and show the results of the exact algorithms i.e.",
"Algorithm , while Figures and show the results of the approximate algorithms i.e.",
"Algorithm REF .",
"In sub-Figures (b), (b), (b), (b), scatter points lying below 1.0 on the y-axis indicate that the midpoint method achieves lower error than the standard method on the same problem instance.",
"From Figure (a), it is clear that MPI achieves extremely fast convergence to the optimal gain, with the relative error being less than $10^{-13}$ , essentially machine precision, on almost all problem instances after just 5 iterations.",
"From Figure (b), we see that MPI achieves significantly lower error than PI on iteration counts $2,3,4,5$ for almost all problem instances.",
"The relative differences in error on iteration counts $6,7$ are due to machine precision error and are negligible for the purposes of comparison i.e.",
"after 6 iterations both algorithms have effectively converged to the same solution.",
"We observe very similar results using the approximate algorithms.",
"From Figure (a), it is clear that AMPI achieves extremely fast convergence to a good approximation of the optimal gain, with the relative error being less than $10^{-6}$ on almost all problem instances after just 4 iterations.",
"From Figure (b), we see that AMPI achieves significantly lower error than API on iteration counts $2,3,4,5$ for almost all problem instances; recall that Algorithm REF takes a standard PI step on the first iteration, explaining the identical performance on $k=1$ .",
"Figure: (a) Relative value error ∥P k -P * ∥/Tr(P * )\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) using MPI and (b) ratio of relative error using MPI divided by that using PI.Figure: (a) Relative value error ∥P k -P * ∥/Tr(P * )\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) using AMPI and (b) ratio of relative error using AMPI divided by that using API.Figure: (a) Relative value error ∥P k -P * ∥/Tr(P * )\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) using MPI and (b) ratio of relative error using MPI divided by that using PI.Figure: (a) Relative value error ∥P k -P * ∥/Tr(P * )\\Vert P_k - P^*\\Vert /\\operatorname{Tr}(P^*) using AMPI and (b) ratio of relative error using AMPI divided by that using API."
],
[
"Conclusions and future work",
"Empirically, we found that regardless of the stabilizing initial policy chosen, convergence to the optimum always occurred when using the exact midpoint method.",
"Likewise, we also found that approximate midpoint and standard PI converge to the same approximately optimal policy, and hence value matrix $P$ , after enough iterations when evaluated on the same fixed off-policy rollout data $\\mathcal {D}$ .",
"We conjecture that such robust, finite-data convergence properties can be proven rigorously, which we leave to future work.",
"This algorithm is perhaps most useful in the regime of practical problems where it is relatively expensive to collect data and relatively cheap to perform the linear algebraic computations required to execute the updates.",
"Both the exact and approximate midpoint PI incur a computation cost double that of their standard PI counterparts.",
"Theoretically, the faster cubic convergence rate of MPI over the quadratic convergence rate of PI should dominate this order constant (2$\\times $ ) cost with sufficiently many iterations.",
"However, unfortunately, due to finite machine precision, the total number of useful iterations that increase the precision of the optimal policy is limited, and the per-iteration cost largely counteracts the faster over-iteration convergence of MPI.",
"This phenomenon becomes even more apparent in the model-free case where the “noise floor” is even higher.",
"However, this disadvantage may be reduced by employing iterative Lyapunov equation solvers in Algorithm or iterative (recursive) least-squares solvers in Algorithm REF and warm-starting the midpoint equation with the Newton solution.",
"Furthermore, the benefit of the faster convergence of the midpoint PI may become more important in LQR problems with large state- and input-dimensions, and in extensions to nonlinear systems, where the order constants in Propositions and REF are smaller.",
"The current methodology is certainty-equivalent in the sense that we treat the estimated value functions as correct.",
"Future work will explore ways to estimate and account for uncertainty in the value function estimate explicitly to minimize regret risk in the initial transient stage of learning when the amount of information is low and uncertainty is high."
]
] | 2011.14212 |
[
[
"Is Support Set Diversity Necessary for Meta-Learning?"
],
[
"Abstract Meta-learning is a popular framework for learning with limited data in which an algorithm is produced by training over multiple few-shot learning tasks.",
"For classification problems, these tasks are typically constructed by sampling a small number of support and query examples from a subset of the classes.",
"While conventional wisdom is that task diversity should improve the performance of meta-learning, in this work we find evidence to the contrary: we propose a modification to traditional meta-learning approaches in which we keep the support sets fixed across tasks, thus reducing task diversity.",
"Surprisingly, we find that not only does this modification not result in adverse effects, it almost always improves the performance for a variety of datasets and meta-learning methods.",
"We also provide several initial analyses to understand this phenomenon.",
"Our work serves to: (i) more closely investigate the effect of support set construction for the problem of meta-learning, and (ii) suggest a simple, general, and competitive baseline for few-shot learning."
],
[
"Introduction",
"The ability to learn from limited experience is a defining aspect of human intelligence.",
"The domain of few-shot learning aims to evaluate this criterion within machine learning [10].",
"For few-shot learning problems, meta-learning [30] techniques have attracted increasing attention.",
"Meta-learning methods typically construct few-shot classification tasks in an episodic manner by sampling support ($S$ ) and query ($Q$ ) examples from a fixed number of classes.",
"The objective is then to deliver an algorithm that can perform well on the query points of a task by training on only a few support samples.",
"Conventional wisdom is that the performance of meta-learning methods will improve as we train on more diverse tasks.",
"Therefore, during meta-training, we typically allow a task to be constructed from any possible pair of support and query sets.",
"In this work, we question this notion, specifically investigating the effect that support set diversity has on meta-learning.",
"Surprisingly, we find that reducing the total number of unique support sets not only does not result in adverse effects—in most cases, it in fact yields significant performance improvements.",
"The main contributions of this work are as follows: (1) We propose a simple modification to the meta-learning objective that applies to a broad set of methods, in which we aim to optimize a biased objective by restricting the number of support sets used to construct tasks.",
"(2) On multiple datasets and model architectures we empirically demonstrate the performance improvement of the learned algorithm when trained with our proposed objective.",
"(3) Finally, we explore a framework to understand this surprising phenomenon and attempt to provide some insights into the observed gains.",
"Our work delivers a simple, competitive baseline for few-shot learning and, more generally, investigates the role that task construction plays in meta-learning."
],
[
"Background & Related Work",
"Meta-learning formulations typically rely on episodic training, wherein an algorithm is learned to adapt to a task, given its support set, so as to minimize the loss incurred on the query set.",
"Meta-learning methods differ in terms of the algorithms they learn: Gradient-based meta-learning methods [11], [12], [23], [27], [18] consider learning gradient-based algorithms, which are typically parameterized by an initialization (possibly together with some pre-conditioning matrix).",
"Recently, last-layer meta-learning methods [18], [22], [3], [26] have gained popularity.",
"These methods don't learn an optimal model parameter initialization for the task distribution but instead learn a feature extractor for the covariates in the support and query sets.",
"The extracted support set features along with their labels are used to learn the parameters of the last layer, e.g., in a non-parametric fashion [26] or by finding the unique solution for convex problems [18], [2].",
"Several recent works [6], [7], [21] on few-shot learning have proposed methods that don't directly optimize the original meta-learning objective.",
"Instead they chose to minimize a transfer-learning based loss where a deep network is trained in a supervised manner on all the $(x, y)$ pairs present in the dataset.",
"In contrast to episodic training, they show that a model trained without any notion of a task can be used to construct an algorithm whose performance matches or improves upon meta-learning methods across many benchmarks.",
"Similar to these works, we propose to optimize an alternate objective for meta-learning; however, our work differs by retaining the episodic structure from the meta-learning framework but restricting the support set (and by consequence task) diversity.",
"In addition to using other objectives for few-shot learning, recent work aims to improve meta-learning by explicitly looking at the task structure and their relationships.",
"Among these, [32] propose to handle the lack of mutual exclusiveness among different tasks through an information-theoretic regularized objective.",
"[8] propose a minimax objective to make the learned algorithm task-robust, which improves the performance of the algorithm on the most difficult task in the face of a task pool with diverse difficulty.",
"[20] propose to augment the set of possible tasks by augmenting the pre-defined set of classes that generate the tasks with varying degrees of rotated inputs as new classes.",
"In addition, several popular meta-learning methods [26], [18], in order to improve the meta-test performance, change the number of ways or shots of the sampled meta-training tasks, thus increasing the complexity and diversity of the tasks.",
"In contrast to these works, we look at the structure and diversity of tasks specifically through the lens of support set diversity, and show that, surprisingly, reducing diversity (by fixing support set) not only maintains—but in many cases significantly improves—the performance of meta-learning.",
"Our results indicate that this simple modification to meta-learning is an effective method worthy of future study."
],
[
"Meta-Learning with a Fixed Support Pool",
"We begin with some notation that we will use to re-formulate the meta-learning objective.",
"This will allow us to introduce the concept of a support pool in a manner amenable with current meta-learning methods.",
"Notation.",
"In a standard offline meta supervised learning problem, we are given a dataset $\\subseteq \\mathcal {X} \\times [N]$ with examples from $[N] 1, \\ldots , N$ different classes.",
"A task $(C, S, Q)$ is constructed by first choosing a subset of classes $C \\subseteq [N]$ , and conditioning on $C$ , a support set $S$ and a query set $Q$ are sampled i.i.d.",
"from $^{C} \\lbrace (x, y) \\in : y \\in C\\rbrace $ .",
"The goal is to learn the parameters of an algorithm $w \\in ^d$ so that when given a support set $S$ , the algorithm will output a model that achieves a low loss value on the query set.",
"We denote this loss value as a function of the algorithm parameter, support, and query set: $\\ell : ^d \\times {S} \\times {Q} \\rightarrow .$ To identify the optimal parameter of the algorithm $w$ , we optimize the objective in [A] below: $\\begin{split}\\textrm {[A]:}\\, \\min _w _C _{(S, Q) \\; \\mid \\; C} \\ell (w, S, Q)\\quad \\displaystyle \\equiv \\quad \\textrm {[B]:}\\, \\min _w _{S_p \\sim ()} _{C} _{Q \\;\\mid \\; C} \\ell (w; S_p^C, Q)\\end{split}$ Reformulating the meta-learning objective.",
"We now give another view of this objective, which will motivate our proposed modified objective.",
"Instead of sampling from the set of classes $C$ when constructing a task, we consider first sampling uniformly from $$ , a collection of support pools.",
"A support pool $S_p \\in $ is a smaller subset of the entire dataset $D$ that contains some examples from every class in $D$ , ${y: (x, y) \\in S_p}=[N]$ .",
"With $S_p$ , we sample a subset of classes $C \\subseteq [N]$ as before and take all the examples from $S_p$ that are of the classes in $C$ .",
"This set of examples is now our support set which we denote by $S_p^C = {(x, y) \\in S_p: y \\in C}$ .",
"With the classes $C$ determined, we can then sample the query set $Q$ of examples from class $C$ the same way as described previously.",
"In this view, the marginal distribution of the support query set pair $(S_p^C, Q)$ is exactly the same as that of $(S, Q)$ discussed before.",
"As a result, we can rewrite our meta-learning objective as in (REF )[B].",
"This reformulation prompts us to first check how big the collection of support pools $$ typically is.",
"In the most unconstrained form, $$ can contain all possible subsets of a dataset where we have at least one sample from every class $(\\forall c \\in [N])$ .",
"However, most meta-learning methods only train on episodic tasks where the support $S$ is of a specific configuration, i.e., the support of each task is comprised of exactly $n$ ways (classes) with each class having $k$ shots (examples).",
"For instance, when training on miniImagenet [31] 5-way 5-shot ($n=5, k=5$ ) tasks, all tasks that are sampled consist of exactly 5 shots from each of the 5 classes.",
"Hence, it also makes sense to limit $$ to only support pools that obey a certain structure.",
"In this work, we consider $$ to consist only of subsets of $$ that have exactly $k$ samples from each class i.e.",
"$= \\lbrace S_p: {\\footnotesize \\sum }_{(x, y)\\in S_p} I(y = c)=k, \\;\\forall c \\in [N] \\rbrace $ .",
"The meta-training set of miniImagenet [24] has 64 classes with 600 examples each, resulting in $|| = \\binom{600}{5}^{64} \\approx 3\\times 10^{755}$ support pools.",
"This makes us wonder: Is this astronomically large number of support pools necessary or possibly redundant to achieve good meta-test performance?",
"Can we use a much smaller number of support pools instead?",
"Fixing the support pool.",
"Motivated from the questions above, we take the most extreme reduction approach: before training begins, we randomly construct one support pool $S_{p,0}$ and let $= {S_{p, 0}}$ (just this single support pool) and solve for the corresponding optimization problem: $\\min _w _{C} _{Q \\;\\mid \\; C} \\ell (w; S_{p,0}^C, Q).$ Note that the expectation over $S_p$ is removed because there is only a single possible realization of the random support pool.",
"In terms of the total number of different possible support sets that the algorithm being learned can see, we have also achieved a reduction factor of $\\binom{600}{5}^{5}$ times for the aforementioned miniImagenet example.",
"In the rest of the paper, we first show the performance obtained by optimizing this reduced objective (REF ) for some of the best-performing meta-learning methods on multiple meta-learning problems and/or with different model architectures.",
"We then make an initial attempt at understanding the effectiveness of using this fixed support pool objective."
],
[
"Experiments",
"We now aim to understand the differences in generalization performance between the algorithm learned by optimizing (REF ), i.e.",
"our proposed objective (which we denote ), and the algorithm obtained by optimizing the original meta-learning objective (REF ) (which we denote ).",
"For every experiment, we randomly sample $k$ samples (shots) from each class in the dataset to form the support pool $S_{p,0}$ , which is then fixed throughout training.",
"Here, different runs of on the exact same problem and meta-learning method can use different (but fixed) support pools.",
"Later in this section, we will discuss how this randomness affects meta-test performance over multiple runs.",
"For an unbiased estimation and a fair comparison, when we evaluate a learned algorithm on the meta-validation or meta-test set, we compute the loss and accuracy using the objective given in (REF ) (i.e., an average over all possible support pools), just as we would for the learned algorithm.",
"To fully evaluate the performance differences, we compare and along three axes: Meta-learning methods: We focus our initial experiments on protoypical-networks (PN) [26] which uses a non-parametric last layer solver, as Protonets is a highly competitive approach, and also has the benefit of training faster and with less memory than competitors such as SVM [18], Ridge Regression (RR) [3].",
"However, we also experiment with SVM and RR-based solvers to confirm that the trends observed with Protonets generalize to these methods.",
"As initialization-based methods are harder to optimize for highly over-parameterized models, we defer such evaluations to future work.",
"Datasets: Our study mainly uses two of the most widely-used benchmarks in few-shot learning: (i) miniImagenet () [31], which consists of 100 classes of $84 \\times 84$ images, split into 64 train, 16 validation and 20 test, and (ii) CIFAR-FS () [3] with an identical split but with $32 \\times 32$ images.",
"Additionally for the SVM method we elect to compare on the more difficult FC-100 () dataset, as the SVM-based solver has achieved significantly superior performance on this benchmark [18].",
"Architectures: We conduct our main experiments using Resnet-12, as many meta-learning methods [2], [22] have relied on this architecture to achieve good performance.",
"Additionally, we experiment with another wider but shallower over-parameterized architecture, WideResNet-16-10 [33] (also used by [25]), and Conv64, which was used by some initial works in meta-learning [11], [26].",
"Figure: Performance gains observed upon reducing support set diversity by fixing a single support pool.",
"In (a) we compare and across datasets, algorithms and train/test configurations (XwYsZq X-ways, Y-shots and Z-query), in (b) we evaluate the architecture's role in a comparison between and on mini-5w5s tasks and in (c) we show how the meta-test performance varies across runs for both and ."
],
[
"Main Results.",
"Figure REF (a) consolidates the results of our experiments on different datasets with a myriad of training task configurations.",
"Surprisingly, contrary to our intuition about task diversity, we can clearly see that optimizing the fix-ml objective performs similarly or in many cases better with respect to its ml counterpart.",
"In all the cases except for the dataset, we evaluate the validation and test accuracies on 2000 randomly drawn tasks consisting of novel classes and for each of them we observe a $95\\%$ confidence interval of $\\pm (0.32-0.36) \\%$ around their respective means.",
"For , since the performance improvements are lower, we evaluate on 10000 tasks instead and observe a $95\\%$ interval of $\\pm (0.13-0.14) \\%$ (around the reported mean) which confirms the significance of the improvement.",
"Specifically, we see that on all of the and 5w1s and 5w5s evaluations, Protonets trained with achieve better performance.",
"In addition, for each of these evaluations, we consider training with different number of ways and shots as proposed in [26], [18].",
"Our results also confirm that is better for a range of meta-training specifics.",
"Besides, for other meta-learning methods like SVM and RR, fixing the support pool can also improve performance.",
"For all the experiments in Figure REF (a) we use the Resnet-12 backbone.",
"In order to further validate our findings, we show mini-5w5s meta test results on two other architectures in Figure REF (b).",
"WRN-16-10 has roughly 50% more parameters than Resnet-12 and we see that continues to perform better than in this over-parameterized regime.",
"On the other hand, it seems that fixing the support pool for shallower backbones like Conv64 hinders the meta-test performance.",
"For this we hypothesize that is effective for over-parameterized models and algorithms but less so for shallower models with fewer parameters.",
"Further analysis on how model architecture and over-parameterization affects is a future research direction.",
"Recall that earlier in this section, we suspected that fixing the support pool randomly prior to the start of training may induce higher variance in the final meta-test performance over different runs.",
"From the observations in Figure REF (c), we confirm that this is not true.",
"We compare and on -5w5s and -5w1s over five different runs each, where for every run of a different random support pool was sampled and then fixed.",
"We can clearly see that the performance standard deviation is not only not worse but actually better than .",
"Finally, in Figure REF (a) we see that Protonets trained using objective achieves competitive results on -5w5s when compared with other other meta-learning methods and with a simple transfer learning method [6] devoid of any self-supervised learning [21] tricks or a transductive setting [14]."
],
[
"Implementation.",
"We follow the learning rate schedule and data augmentation scheme of [18] and use a task batch size of 4-8 when training with lower number of ways (e.g., 5) due to observed improvements in convergence and generalization.",
"Additionally, we experiment with varying number of query points per class (in a single task).",
"We observe that for some cases in Figure REF (a) this further improves the advantage of over ."
],
[
"Discussion",
"To understand the improved meta-test performance of the -learned algorithms, we first analyze their performance on the meta-training set.",
"Evaluating the -learned algorithm's performance averaged over all possible support pools (equivalent to the standard training loss) is important because, by the generalization theory using uniform convergence bounds (e.g.",
"VC dimension, Rademacher complexity [1]), the test loss tracks training loss more closely with more training samples.",
"In the case of meta-learning, the “samples” are tasks in the form of $(S,Q)$ pairs.",
"Therefore, we see that an algorithm's -training loss which considers all possible $(S,Q)$ pairs, should track the algorithm's loss on the meta-test set more reliably than the training loss which uses only a single support pool (inducing much fewer $(S,Q)$ pairs).",
"Figure: In (a) we compare learned Protonets with other popular methods on -5w5s.",
"On -5w5s (b) and -5w1s (c) we evaluate the -learned algorithm after each training epoch using three criterions; blue: 's loss on the specific support pool (S p,0 S_{p,0}) used in its objective (); red: the meta-learning objective in ()[B] and green: the objective in () but on 10 randomly sampled support pools different from S p,0 S_{p,0}.For MetaOptNet-SVM we report the performance observed by [18] without label smoothing.",
"Naively, one might think that because is optimized on a single fixed support pool, the training loss on any other support pool would be much worse.",
"To verify this idea, we re-evaluate the -learned algorithms 1) over 10 other fixed support pools by replacing $S_{p,0}$ in (REF ) with $S_p \\ne S_{p, 0}$ ; 2) on the original meta-learning objective (REF )[B] by averaging over all possible support pools $S_p$ (because of the large number of possible support pools, we estimate this average using a very large number of sampled tasks).",
"We conduct this evaluation for Protonet on -5w5s and -5w1s tasks, for all the trained algorithm snapshots over the entire 60 training epochs.",
"Here, the random class set $C$ in (REF )[B] is set to always contain exactly 5 classes ($|C| = 5$ ) in order to match the meta-test scenario.",
"The results are shown in Figure REF (b),(c).",
"As we see in both plots, despite optimizing for one fixed support pool (blue curve), the training losses of other support pools (green curves) and the training loss (red curve) are all decreasing consistently in the same trend as the blue curve throughout the entire optimization trajectory.",
"To understand this consistent decrease, we note that the gradient of objective is a biased gradient of the original objective.",
"When the gradient bias is not too large, stochastic gradient descent using a biased gradient can still lead to a reduction in objective loss and convergence [4], which is what we observe for the objective when using biased stochastic gradients.",
"However, this does not give us a complete picture for meta-training.",
"Even though training loss is considerably reduced for the -learned algorithm, we need to compare this value against -training loss of the corresponding -learned algorithm to understand the meta-test performance difference.",
"In Figure REF (c), we compare the end-of-training solutions with the corresponding solutions using the training objective (REF )[B] on 4 different dataset/task configurations.",
"We see that still consistently achieves higher -training losses than their counterparts.",
"With a relatively lower test loss but a much higher training loss, the solution must have a smaller generalization gap than that of the solution.",
"Since there isn't a single best way to understand neural networks' generalization gap, we look at some of the recently proposed metrics: 1) Sharpness of the training loss landscape: A popular hypothesis (dating back to [13]) is that the flatness of the training loss landscape is correlated with (and possibly leads to) a smaller generalization gap [17], [5].",
"Several sharpness metrics based on this hypothesis were called into question by [9], showing that a functionally equivalent neural network with differently scaled parameters could have arbitrarily different sharpness according to these metrics.",
"However, it was argued in [16] that even though sharp minima with similar test performance as flat minima do exist, stochastic gradient descent will not converge to them.",
"While further understanding sharpness's relationship with generalization gap is still an ongoing research topic, we still consider two sharpness analyses here.",
"1a) Sharpness visualization using 1d interpolation plots was first proposed by [17] and also used by [15] to compare the flatness of two different solutions' training loss landscape when explaining the two solutions' generalization gap differences.",
"For two models with the same neural architecture but different weights, training and test loss are evaluated at model parameters that are different linear interpolations of the two models' weights.",
"In our case, we interpolate between the parameters of end-of-training -learned algorithm and -learned algorithm from Protonet for miniImagenet 5w5s.",
"This result is shown in Figure REF (a),(b).",
"In REF (a), we see that is located at a relatively “sharper” minima than despite having a smaller generalization gap.",
"This is inconsistent with observations in [17], [15] possibly because in our case, the loss landscape between meta-train and meta-test are not necessarily shifted versions of each other (confirmed also in Figure REF (b)).",
"Instead, we see that the trends of these two graphs align well with each other.",
"Therefore, we cannot solely rely on 1d visualization-based sharpness analysis to explain the generalization gap.",
"Figure: In (a),(b) (two different runs) we plot the train and test loss (averaging over all support sets) for network weights obtained by linearly interpolating between the (w fml w_{fml} at 0.0.)",
"and (w ml w_{ml} at 1.1.)",
"Protonet solutions for -5w5s tasks.",
"We evaluate ()[B] with w=(1-α)w fml +αw ml ,α∈[-0.2,1.2]w = (1-\\alpha )w_{fml} + \\alpha w_{ml},\\,\\, \\alpha \\in [-0.2, 1.2].",
"In (c) we compare the train losses for and learned Protonets at the end of training.",
"In (d) we show the correlation between the generalization gap and () ()\\frac{()}{()} on 5w5s tasks for , (Protonet) and (SVM).1b) In addition to visualizations, quantitative measures of the training loss landscape sharpness has also been used.",
"Specifically, one notion of the loss landscape sharpness is in the curvature of the loss function.",
"This is typically measured by the maximum eigenvalue [19] or the trace $(\\textrm {tr}(\\mathbf {H}))$ of the Hessian matrix ($\\mathbf {H}$ ) [16] for the parameters at the end of training .",
"However, directly computing $\\mathbf {H}$ accurately is both computationally expensive and memory-wise “impossible\" for over-parameterized models like Resnet-12.",
"Thus, $\\mathbf {H}$ is typically approximated with the Fisher information matrix ($\\mathbf {F}$ ) at the same parameter [29].",
"$\\textrm {tr}(\\mathbf {F})$ can be easily estimated without constructing the entire matrix $\\mathbf {F}$ .",
"However, $\\mathbf {F}$ is only equal to $\\mathbf {H}$ when the parameters are the true parameters of the training set (in practice, achieve training loss extremely close to 0).",
"On the other hand, this approximation is very crude when the training loss is much larger than 0 in the case of (Figure REF (c)).",
"Hence, $\\textrm {tr}()$ for 's parameters cannot be accurately estimated using $(F)$ .",
"As a result, for our analysis, this approach is not readily applicable.",
"2) Another quantitative measure which approximates the Takeuchi Information Criterion (TIC) [28] was empirically shown to correlate more with the generalization gap as compared to measures of the loss landscape's flatness [29].",
"Specifically, [29] ([29]) proposes the quantity $\\frac{()}{()}$, where $$ is the uncentered covariance matrix of the gradient with respect to the training data distribution.",
"To evaluate how well this measure tracks the generalization gap difference between and , we evaluate this approximate TIC value for three pairs of and -learned algorithms in Figure REF (d).",
"We see that for all three datasets, has both higher $\\frac{()}{()}$ and larger generalization gap than its counterpart, indicating some possible correlation between these two.",
"Thus it would be interesting to understand what intuitive property of the solutions is captured by this quantity and why optimizing the objective leads to solutions with lower $\\frac{()}{()}$."
],
[
"Conclusion",
" In this paper, we have proposed a simple yet effective modification of the meta-learning objective.",
"We begin by re-formulating the traditional meta-learning objective, which motivates our approach of reducing the support set diversity.",
"Then, we experimentally show that for a variety of meta-learning methods, datasets, and model architectures, the algorithms learned using our objective result in superior generalization performance relative to the original objective.",
"We analyze this improvement by understanding the optimization loss and the generalization gap separately.",
"While the optimization loss trade-off is easier to reason about using our framework, the generalization gap is still not fully understood based on several initial analyses, which we hope would pave the way for future work in this area."
]
] | 2011.14048 |
[
[
"Truly shift-invariant convolutional neural networks"
],
[
"Abstract Thanks to the use of convolution and pooling layers, convolutional neural networks were for a long time thought to be shift-invariant.",
"However, recent works have shown that the output of a CNN can change significantly with small shifts in input: a problem caused by the presence of downsampling (stride) layers.",
"The existing solutions rely either on data augmentation or on anti-aliasing, both of which have limitations and neither of which enables perfect shift invariance.",
"Additionally, the gains obtained from these methods do not extend to image patterns not seen during training.",
"To address these challenges, we propose adaptive polyphase sampling (APS), a simple sub-sampling scheme that allows convolutional neural networks to achieve 100% consistency in classification performance under shifts, without any loss in accuracy.",
"With APS, the networks exhibit perfect consistency to shifts even before training, making it the first approach that makes convolutional neural networks truly shift-invariant."
],
[
"Non-linear activation functions and shift invariance",
"We saw in Section of the paper that anti-aliasing a signal before downsampling restores sum-shift-invariance.",
"In particular, consider a 1-D signal $x_0(n)$ and its 1-pixel shift $x_1(n) = x_0(n-1)$ .",
"Anti-aliasing the two signals (with an ideal low pass filter) followed by downsampling with stride 2 results in $y_0^a(n)$ and $y_1^a(n)$ with DTFTs Y0a() = X0(/2)2, Y1a() = X0(/2)e-j/22, that satisfy $Y_0^{a}(0) = Y_1^{a}(0)$ .",
"Azulay and Weiss pointed out in that the sum-shift invariance obtained via anti-aliasing is lost due to the action of non-linear activation functions like ReLU in convolutional neural networks.",
"They postulated that this happens through the generation of high-frequency content after applying ReLU.",
"We elaborate on this phenomenon here and also show that high frequencies alone do not provide a full picture.",
"Let $g(\\cdot )$ be a generic pointwise non-linear activation function applied to the outputs of anti-aliased downsampling.",
"Owing to the pointwise nature of $g$ , the stride operation and the non-linearity can be interchanged, making the network block in Fig.",
"REF (a) equivalent to the one in Fig.",
"REF (b).",
"Notice in Fig.",
"REF (b) that despite anti-aliasing $x_0$ with an ideal low pass filter LPF, $g$ generates additional high frequencies which can result in aliasing on downsampling.",
"One can not simply use another low pass filter to get rid of these newly generated aliased components.",
"For example, a new low pass filter block added after $g$ in Fig.",
"REF (a) can be interchanged with the stride operation to result in a dilated filter which is not low pass any more (Fig.",
"REF (b)).",
"While high frequencies generated by non-linear activations can lead to invariance loss for various choices of $g$ , we show in Section REF that this might not always be necessary.",
"For example, polynomial activations, despite generating aliased components, do not impact sum-shift-invariance.",
"Therefore, in addition to its high frequency generation ability, we also take a closer look at how the ReLU non-linearity affects sum-shift invariance in terms of its thresholding behavior in Section REF .",
"In Theorem 1 from Section in the paper, we stated that for any integer $m>1$ , non-linear activation functions of the form $g(y) = y^m$ do not impact sum-shift-invariance.",
"We provide the proof below.",
"Figure: Pointwise non-linearity gg can be interchanged with the stride operation.",
"Despite anti-aliasing x 0 x_0 with LPF block, gg generates high frequencies which can lead to additional aliasing during downsampling.Figure: Additional low pass filtering after gg in (a) does not eliminate the impact of aliasing.",
"This is because, as shown in (b), interchanging the final LPF block with stride operation, results in a dilated version of the filter which is not low-pass any more.Let the DTFTs of $z_0 = g(y_0^a)$ and $z_1 = g(y_1^a)$ be $Z_0()$ and $Z_1()$ .",
"Then by definition of the DTFT, Z0(0) = nz0(n) , and Z1(0) = nz1(n).",
"Since $z_0 = (y_0^a)^m$ , and $z_1 = (y_1^a)^m$ , we have Z0() = ( Y0a() Y0a() ...Y0a()m times ), Z1() = ( Y1a() Y1a() ...Y1a()m times ), where $\\circledast $ represents circular convolution.",
"For $i \\in \\lbrace 0,1\\rbrace $ , we can write Zi() = (12)m-1 Yia(1) Yia(- i=1m-1 i)d, where $\\bar{\\alpha } = (\\alpha _1, \\alpha _2,\\ldots \\alpha _{m-1})$ .",
"From (), we have $Y_1^{a}() = Y_0^{a}()e^{-j\\omega /2}.$ Using (REF ) and (REF ), we can write $Z_1() = Z_0() e^{-\\frac{j}{2}} $ , which when combined with (REF ) gives $\\sum _{n\\in }z_0(n) = \\sum _{n\\in }z_1(n).$ Using linearity of Fourier transform, the result in Theorem 1 can be extended to arbitrary polynomial activation functions of the form $g(y) = \\sum _{i=0}^m a_i y^i$ with $m>1$ ."
],
[
"ReLU spoils sum-shift-invariance",
"We now consider the ReLU non-linear activation function, $h(y) = \\mathrm {relu}(y)$ , which clips all negative values of signal $y$ to zero.",
"Unlike the case with polynomials in Section REF , deriving a closed form expression for the DTFTs of $h(y_0^a)$ and $h(y_1^a)$ , for arbitrary $x_0$ and $x_1$ is non-trivial.",
"We therefore analyze a simpler case where $x_0$ is assumed to be a cosine signal, and illustrate how sum-shift invariance is lost due to ReLUs.",
"Let $x_0$ be an $N$ length 1-D cosine and $x_1 = x_0(n-1)$ be its 1-pixel shift.",
"We define the two signals as x0 = (2n N), and x1 = (2(n-1) N) n{0, 1, ...N-1}.",
"For any $N>4$ , $x_0$ satisfies the Nyquist criterion and is anti-aliased by default.",
"For $N^{\\prime } = N/2$ and $n\\in \\lbrace 0, 1, \\ldots N^{\\prime }-1\\rbrace $ , the downsampled outputs $y_0^a$ and $y_1^a$ are then given by y0a(n) = x0(2n) = (2n N'), y1a(n) = x1(2n) = (2(n-1/2) N').",
"Note that $y_0^a$ and $y_1^a$ are structurally similar signals, and can be interpreted as half-pixel shifted versions of each other.",
"The action of $h$ on $y_i^a$ can be regarded as multiplication by a window which is zero for any $ n$ where $y_i^a(n)<0$ .",
"We construct sets $\\lbrace S_i^+\\rbrace _{i=0}^1$ containing $n$ where $y_i^a(n)>0$ .",
"For simplicity in constructing the sets, we assume $N^{\\prime }>6$ and divisible by 4 (similar conclusions from below can be reached without these simplifying assumptions as well).",
"Then we have S0+ = {n: n, n[0, N'4-1] [3N'4+1 , N'-1] }, S1+ = {n: n, n[0, N'4] [3N'4+1 , N'-1 ] }.",
"Notice that the supports $S_0^+$ and $S_1^+$ differ by 1 pixel near $n=\\frac{N^{\\prime }}{4}$ .",
"This is because despite being structurally similar, $y_0^a$ and $y_1^a$ have slightly different zero crossings, which results in some differences in the support of thresholded outputs.",
"We can now compute the sums $\\sum h(y_0^a)$ and $\\sum h(y_1^a)$ .",
"n h(y0a)(n) = n S0+(2n N') = ( n S0+ ej2n N' ) = (2/N)(2/N).",
"Similarly, $\\sum _{n \\in } h(y_1^a)(n)$ is given by n h(y1a)(n) = ( n S1+ ej2(n-1/2)N' ) = (e-jN'n S1+ ej2n N' ).",
"We can rewrite (REF ) in terms of (REF ), and get n h(y1a)(n) = (e-jN'n S0+ ej2n N' + e-jN'ej2n N'|n=N'/4 ) =(2N)n h(y0a)(n) + (2N).",
"(REF ) illustrates the loss in sum-shift-invariance caused by ReLU.",
"Notice that the differences in $\\sum h(y_0^a)$ and $\\sum h(y_1^a)$ arise due to minor differences in the signal content in $y_0^a$ and $y_1^a$ , which are amplified by ReLU.",
"The term $sin(2\\pi /N)$ arises due to a 1-pixel difference in the supports of $h(y_0^a)$ and $h(y_1^a)$ , whereas the cosine term is associated with $e^{-j/2}$ from (), again depicting the impact of small differences in $y_0^a$ and $y_1^a$ .",
"Figure: Illustration of baseline ResNet architectures used in our experiments.",
"(a) ResNet-20, (b) ResNet-18 used in CIFAR-10 classification.",
"(c) Baseline ResNet-18 used in the ImageNet classification experiments.We trained ResNet models with APS, anti-aliasing and baseline conventional downsampling approaches on CIFAR-10 and ImageNet datasets, and compared their achieved classification consistency and accuracy.",
"For CIFAR-10 experiments, four variants of the architecture were used: ResNet-20, 56, 18 and 50.",
"ResNet 20 and 56 were originally introduced in for CIFAR-10 classification and are smaller models with number of channels: $\\lbrace 16, 32, 64\\rbrace $ in different layers, and use stride 2 twice, which results in a resolution of $8\\times 8$ in the final convolutional feature maps.",
"On the other hand, ResNet-18 and 50 contain $\\lbrace 64, 128, 256, 512\\rbrace $ number of channels, and downsample three times with a stride 2, resulting in final feature map resolution of $4\\times 4$ .",
"Similar to the experiments with CIFAR-10 in , we use a convolution with stride 1 and kernel size of $3\\times 3$ in the first convolutional layer.",
"In all architectures, global average pooling layers are used at the end of the convolutional part of the networks.",
"Fig.",
"REF (a)-(b) illustrate the baseline architectures of ResNet-20 and 18 used in our experiments.",
"The original training set of the CIFAR-10 dataset was split into training and validation subsets of size 45k and 5k.",
"All models were trained with batch size of 256 for 250 epochs using stochastic gradient descent (SGD) with momentum $0.9$ and weight decay $5\\mathrm {e}{-4}$ .",
"The initial learning rate was chosen to be 0.1 and was decayed by a factor of 0.1 every 100 epochs.",
"Training was performed on a single NVIDIA V-100 GPU.",
"All the models were randomly initialized with a fixed seed before training.",
"The models with the highest validation accuracy were used for evaluation on the test set.",
"For ImageNet classification, we used standard ResNet-18 model as baseline whose architecture is illustrated in Fig.",
"REF (c).",
"In all experiments, input image size of $224\\times 224$ was used.",
"The models were trained with batch size of 256 for 90 epochs using SGD with momentum 0.9 and weight decay of $1\\mathrm {e}{-4}$ .",
"An initial learning rate of $0.1$ was chosen which was decayed by a factor of $0.1$ every 30 epochs.",
"The models were trained in parallel on four NVIDIA V-100 GPUs.",
"We report results for models with the highest validation accuracy.",
"We were able to show significant improvements in consistency and accuracy with APS over baseline and anti-aliased downsampling without substantial hyper-parameter tuning.",
"Further improvements in the results with better hyper-parameter search are therefore possible."
],
[
"Embedding APS in ResNet architecture",
"We replace the baseline stride layers in the ResNet architectures with APS modules.",
"To ensure shift invariance, a consistent choice of polyphase components in the main and residual branch stride layer is needed.",
"APS uses a permutation invariant criterion (like $\\mathrm {argmax}$ ) to choose the component to be sampled in the main branch.",
"The index of the chosen component is passed to the residual branch where the polyphase component with the same index is sampled.",
"An illustration is provided in Fig.",
"REF .",
"Figure: Residual connection block with (a) baseline stride, (b) APS layer."
],
[
"Impact of polyphase component selection method on classification accuracy",
"In the paper, we saw that APS achieves perfect shift invariance by selecting the polyphase component with the highest $l_2$ norm, i.e.",
"$= y_{i_1j_1},\\\\\\text{where }\\text{ } i_1, j_1 = _{i,j} \\lbrace \\Vert y_{ij} \\Vert _2 \\rbrace _{i,j=0}^1.$ This can also be achieved, however, with other choices of shift invariant criteria.",
"Here, we study the impact of different such criteria on the accuracy obtained on CIFAR-10 classification.",
"In particular, we explore maximization of $l_p$ norms with $p=1$ and $\\infty $ in addition to $p=2$ .",
"We also consider minimization of $l_1$ and $l_2$ norms.",
"We run the experiments on ResNet-18 architecture with 9 different initial random seeds and report the mean and standard deviation of achieved accuracy on the test set.",
"Table REF shows that choosing the polyphase component with the largest $l_\\infty $ norm provides the highest classification accuracy which is then followed by choosing the one with the highest $l_2$ norm and $l_1$ norm.",
"Additionally, the accuracy obtained when choosing polyphase component with minimum $l_2$ norm is somewhat lower than the case which chooses maximum $l_2$ norm.",
"We believe this could be due to the polyphase components with higher energy containing more discriminative features.",
"Note that for all cases in Table REF , the achieved classification accuracy is $\\sim 2\\%$ higher than that of baseline ResNet-18 (reported in the paper).",
"This is because in each case, APS enables stronger generalization via perfect shift invariance prior.",
"Table: Impact of polyphase component selection method used by APS on CIFAR-10 classification accuracy.Table: Impact of APS on classification consistency and accuracy (evaluated on unshifted images) obtained using models trained with random shifts in data augmentation.",
"Models trained without data augmentation are also shown for reference."
],
[
"Experiments with data augmentation",
"We saw in Section of the paper that APS results in $100\\%$ classification consistency and more than $2\\%$ improvement in accuracy on CIFAR-10 dataset for models trained without any random shifts (data augmentation).",
"Here, we assess how baseline sampling compares with APS when the models are trained on CIFAR-10 dataset with data augmentation (labelled as DA).",
"The results are reported in Table REF .",
"We observe that while data augmentation does improve classification consistency for baseline models, it is still lower than APS which yields perfect shift invariance.",
"Classification accuracy, on the other hand, for both the baseline and APS is comparable (within the limits of training noise) when the models are trained with random shifts.",
"This is not surprising because data augmentation is known to improve classification accuracy on images with patterns similar to the ones seen in training set.",
"Note that, as reported in Section of the paper, accuracy of networks with APS is more robust to image corruptions, and the models continue to yield $100\\%$ classification consistency on all image distributions."
],
[
"Downsampling circularly shifted images with odd dimensions",
"With circular shift, pixels that exit from one end of a signal roll back in from the other, thereby preventing any information loss.",
"While this makes circular shifts convenient for evaluating the impact of downsampling on shift invariance over finite length signals, they can lead to additional artifacts at the boundaries when sampling odd-sized signals.",
"For example, as illustrated in Fig.",
"REF , while the polyphase components $y_1$ and $\\tilde{y}_0$ are identical, $y_0$ and $\\tilde{y}_1$ do not contain the same pixels near the boundaries.",
"This is because downsampling an odd-sized signal with stride-2 breaks the periodicity associated with circular shifts, resulting in minor differences in the sets of polyphase components near the boundaries.",
"We investigate the impact of these artifacts by training ResNet-18 models with different downsampling modules on CIFAR-10 dataset with images center-cropped to size $30\\times 30$ .",
"These images result in odd-sized feature maps inside the networks which generate boundary artifacts after downsampling.",
"The models were then evaluated on $30\\times 30$ center-cropped CIFAR-10 test set.",
"Results in Table REF show that despite the presence of artifacts, both the classification consistency and accuracy on unshifted images is greater for models that use APS.",
"Figure: Boundary artifacts associated with circular shifts.",
"Downsampling an odd length signal and its circular shift can result in minor differences in polyphase components near their boundaries.Table: Classification consistency and accuracy obtained with different variants of ResNet-18 when evaluated on CIFAR-10 test set with images cropped to size 30×3030\\times 30.",
"The models were trained without seeing random shifts during training.",
"Despite the presence of boundary effects caused by circular shifts on odd-sized feature maps, we observe higher consistency and accuracy with models containing APS."
],
[
"Timing analysis",
"APS computes the norms of polyphase components for downsampling consistently to shifts.",
"This leads to a modest increase in the time required to perform a forward pass in comparison with baseline network.",
"For example, a forward pass on a $224\\times 224$ image with a circular padded baseline ResNet-18 takes $8.15\\pm 0.47$ ms on a single V-100 GPU.",
"In comparison, ResNet-18 with APS layers takes $11.88\\pm 0.06$ ms."
]
] | 2011.14214 |
[
[
"Polarization dynamics in a nuclear spin gyroscope"
],
[
"Abstract Self-compensated comagnetometers, employing overlapping samples of spin-polarized alkali and noble gases (for example K-$^3$He) are promising sensors for exotic beyond-the-standard-model fields and high-precision metrology such as rotation sensing.",
"When the comagnetometer operates in the so-called self-compensated regime, the effective field, originating from contact interactions between the alkali valence electrons and the noble-gas nuclei, is compensated with an applied magnetic field.",
"When the comagnetometer begins operation in a given magnetic field, spin-exchange optical pumping establishes equilibrium between the alkali electron-spin polarization and the nuclear-spin polarization.",
"Subsequently, when the magnetic field is tuned to the compensation point, the spin polarization is brought out of the equilibrium conditions.",
"This causes a practical issue for long measurement times.",
"We report on a novel method for closed-loop control of the compensation field.",
"This method allows optimization of the operating parameters, especially magnetic field gradients, in spite of the inherently slow (hours to days) dynamics of the system.",
"With the optimization, higher stable nuclear polarization, longer relaxation times and stronger electron-nuclear coupling are achieved which is useful for nuclear-spin-based quantum memory, spin amplifiers and gyroscopes.",
"The optimized sensor demonstrates a sensitivity comparable to the best previous comagnetometer but with four times lower noble gas density.",
"This paves the way for applications in both fundamental and applied science."
],
[
"Introduction",
"Over the past 20 years, vapor-cell-based atomic sensors [1] have received growing attention due to the possibilities of miniaturization and low power consumption [2], features, which compare favorably to cryogenically cooled sensors such as superconducting interferometers (SQUIDs).",
"In a gas containing alkali atoms and polarized noble gas atoms, the electrons of the alkalis feel an effective magnetic field resulting from contact interactions between the electrons and the noble-gas nuclei.",
"An applied external magnetic field can be tuned to cancel this effective field, then such a comagnetometer operates at the so-called compensation point.",
"Under these conditions, the polarized noble-gas nuclei can adiabatically follow a slowly changing magnetic field.",
"This renders the device insensitive to transverse magnetic fields that vary slower than the response time of the system, typically, several hundreds of milliseconds [3].",
"Another advantage of operating near the compensation field is that the alkali vapor may be brought into the spin-exchange-relaxation-free (SERF) regime [4], if the frequency of the alkali-spin Larmor precession is much lower than the rate of spin-exchange collisions.",
"Operation in the SERF regime significantly improves the sensitivity of the comagnetometer [5] to nonmagnetic perturbations.",
"While designed to be insensitive to magnetic fields, self-compensated comagnetometers are exquisitely sensitive to nonmagnetic interactions.",
"First demonstrated in the early 2000s [3], these devices were extensively improved in follow-up studies.",
"It was shown that the effects of light shifts and radiation trapping can be minimized if the probed alkali species is polarized by spin-exchange optical pumping (hybrid pumping) [6].",
"Recently, the response of the system to a low-frequency field modulation was explored [7], which can be used for in-situ characterization of the comagnetometer at the compensation point.",
"Noble-gas comagnetometers have proven to be sensitive gyroscopes [8], [9], [10], [11], [12] and powerful tools for exotic physics searches [13], [14], [15], [16], [12].",
"Due to the long coherence time of noble-gas spins and the ability to optically manipulate alkali-atom spins, comagnetometers have found applications in quantum memory assisted by spin exchange [17], [18], [19].",
"Although extremely sensitive, self-compensated comagnetometers are often difficult to operate for a long period of time (days) because of the need of frequent field zeroing, alignment of laser beams, etc.",
"One way to improve the stability of the system is to use a high noble-gas pressure (for example 12 amg was used in Ref.",
"[13], where $1\\,\\text{amg}=2.69\\times 10^{19}\\,$ cm$^{-3}$ ).",
"Although this approach is applicable for individual experiments, it poses challenges of producing and handling multiple gas-containing vapor cells on larger scales and thus acts as a bottleneck for wide usability in applications.",
"Here, we report on a novel method for following, in real time, the build-up of nuclear magnetization through spin-exchange optical pumping.",
"This enables reaching stationary working conditions, where the effective magnetic field experienced by alkali atoms is approximately zero.",
"A ${^3}$ He-$^{39}$ K-$^{87}$ Rb comagnetometer is successfully operated at equilibrium nuclear polarization above 100 nT with a cell pressure below 3 amg In our work, the nuclear polarization in the self-compensated regime was above 3%.",
"For comparison, the nuclear polarization in Ref.",
"[13] was comparable however with four times the noble gas density.",
"without frequent field zeroing.",
"This is enabled by precise compensation of the field gradients inside the magnetic shield.",
"In Sec.",
", we review the effect of magnetic field gradients on the comagnetometer, showing that for a given cell, equilibrium nuclear magnetization can only be achieved at high pressure or low field gradients.",
"In Sec.",
", we present the experimental setup and show the measured effect of field gradients on the transverse and longitudinal relaxation rates and the evolution of nuclear magnetization over time.",
"Polarization dynamics is further discussed in Sec.",
"and the conclusions are drawn in Sec.",
"."
],
[
"Comagnetometer theory",
"We are interested in understanding how field gradients affect the equilibrium nuclear magnetization.",
"Consider an ensemble of polarized electronic ($^{39}$ K) and nuclear ($^3$ He) spins enclosed in a perfectly spherical cell.",
"The dynamics of the longitudinal component (along the light propagation direction, $z$ ) of the nuclear polarization follows [3] $ \\frac{d}{dt}P^n(t)={R^{ne}_\\text{se}}(P^e-P^n)-\\frac{1}{T_1^n}P^n\\,,$ where $P^n(t)$ is the time-dependent fractional $^3$ He polarization, for which the steady-state value is determined as $ P^n(t\\rightarrow \\infty )=\\frac{R^{ne}_\\text{se}}{1/T_1^n+R^{ne}_\\text{se}}P^e\\,.$ Here $P^e$ is the electronic spin polarization (in typical high sensitivity experiments, $P^e \\approx 50\\%$ [21]), $R^{ne}_\\text{se}=\\mathcal {N}_\\text{e}\\sigma ^{ne}_{\\text{se}}v$ is the spin-exchange rate for a noble gas atom, $\\sigma ^{ne}_{\\text{se}}$ is the spin-exchange cross-section, $v$ is the characteristic relative velocity in alkali-noble-gas collisions, $\\mathcal {N}_\\text{e}$ is the concentration of alkali atoms, and $T_1^n$ is the longitudinal relaxation rate of nuclear spins.",
"The main contribution to the relaxation time $T_1^n$ in many experiments comes from magnetic field inhomogeneities throughout the cell.",
"In this case and for sufficiently high pressures [22], [23], $ \\frac{1}{T^n_1}=D_{n}\\frac{|\\nabla B_\\perp |^2}{B_z^2},$ where $B_z$ is the leading field seen by the nuclear spins and the transverse field gradients are characterized by $|\\nabla B_\\perp |^2 = |\\nabla B_x|^2 + |\\nabla B_y|^2$ .",
"There are various contributions to the field gradient, including the residual gradients from the magnetic shields, the field coils, as well as gradients due to the polarized spins (in our case dominated by nuclear spins).",
"Note that the gradient due to polarized spins only occurs when the cell is not spherical.",
"This gradient is proportional to $P^n \\mathcal {N}_n$ and cannot be fully compensated by first-order gradient coils.",
"Here we describe this effect using a phenomenological factor $\\alpha $ , accounting for the asphericity of the cell, making the overall gradient $ |\\nabla B_\\perp |^2=|\\nabla B_{\\perp ,0}|^2 + |\\alpha P^n \\mathcal {N}_n|^2\\,,$ with $|\\nabla B_{\\perp ,0}|^2 $ representing the first-order gradient that can be zeroed by gradient coils.",
"The noble gas self-diffusion coefficient $D_{n}$ is a function of the temperature $\\Theta $ and the noble gas density $\\mathcal {N}_{n}$ .",
"It can be approximated by $D_{n}=D_0~\\frac{\\sqrt{\\Theta /273\\,\\textrm {K}}}{\\mathcal {N}_{n}/1\\,\\textrm {amg}}\\,.$ For $D_0=1.65\\,\\text{cm}^2/\\text{s}$ in the case of $^3$ He [24], $\\Theta =458\\,\\text{K}$ ($185^\\circ $ C), and $\\mathcal {N}_{n} \\approx 3\\,\\text{amg}$ , the self-diffusion coefficient is roughly $0.71\\,\\text{cm}^2/\\text{s}$ ."
],
[
"Compensation point",
"In the comagnetometer, the principal mechanism via which nuclear and electronic spins couple to each other is spin-exchange collisions.",
"Under particular conditions it turns out that this coupling can result in a suppression of sensitivity to slowly changing transverse magnetic fields.",
"This suppression is maximal when the applied longitudinal magnetic field is tuned to the so-called compensation point $B_c$ , where the coupling of nuclear and electronic spins is the strongest [3], $B_c=- B_n - B_e = -\\lambda M_n P^n - \\lambda M_e P^e.$ Here $\\lambda =2\\kappa _0\\mu _0/3$ In CGS units: $\\lambda =8\\pi \\kappa _0/3$ .",
"with $\\mu _0$ being the vacuum permeability and $\\kappa _0$ being the spin-exchange enhancement factor due to the overlap of the alkali electron wave function and the nucleus of the noble gas [26]; $M=\\mu \\,\\mathcal {N}$ corresponds to a fully polarized sample with atoms of magnetic moment $\\mu $ and density $\\mathcal {N}$ .",
"As an aside, note that the transverse noble-gas nuclear spin damping rate, hence the comagnetometer bandwidth, is maximal if the applied longitudinal field matches [27]: $ B_d= - B_n + B_e Q\\frac{\\gamma _n}{\\gamma _e},$ where $Q$ is the slowing down factor.",
"This point is often referred to as the fast-damping field, which is better suited for high-bandwidth rotation measurements.",
"Under our experimental conditions, electronic spin magnetization $ B_e $ is small ($\\sim 2$ nT) compared to nuclear spin magnetization $ B_n $ ($\\sim 100$ nT) [8], leading to $ B_c \\approx B_d \\approx - B_n= -\\lambda M_n P^n.$ At this point, alkali spins are to first-order insensitive to transverse magnetic fields but are still sensitive to non-magnetic interactions [8].",
"Figure: Absolute value of the steady state compensation field as a function of leading field B z B_z at high gradient (1 nT/cm, red solid line) and low gradient (0.5 nT/cm, green solid line).",
"When the system is brought to such a field, it will generally self-adjust over time.",
"However, the intersections of the green line with the dashed black line (|B c |=B z |B_c|=B_z), marked with stars, correspond to equilibrium compensation points.",
"Here, we make a distinction between the lower stationary compensation point (orange star) and the upper one (green star).",
"These two points correspond to the roots of Eq.",
"(); however, the system is stable near the upper point and unstable near the lower one.",
"In the simulation, the asphericity factor is α𝒩 n =25\\alpha \\mathcal {N}_n = 25 nT/cm and the spin-exchange polarization rate is R se ne =4.5×10 -6 R^{ne}_\\text{se}=4.5\\times 10^{-6} s -1 ^{-1}."
],
[
"Equilibrium condition",
"Suppose the comagnetometer has reached steady-state conditions for a given $B_z$ , which is not the compensation point.",
"Generally, if the device is brought to the compensation point by setting $B_z=B_c$ , we will find that the system is no longer in equilibrium: the nuclear spin polarization will change over time.",
"This can be seen, for instance, from the combination of Eqs.",
"(REF ) and (REF ), which shows that the steady-state polarization depends on $B_z$ (if the transverse gradient $\\nabla B_\\perp $ is not proportional to $B_z$ ).",
"Consequently, $B_c$ changes because $B_c \\approx -B_n = -\\lambda M_n P^n$ , and the system is no longer at the compensation point.",
"If we wish to keep the system at the compensation point over an extended measurement time, it is possible to experimentally track it and lock the leading field such that $B_z=B_c$ [28], [29].",
"In many cases, in closed-loop operation, however, the nuclear polarization will gradually decrease to zero, degrading the sensitivity of the comagnetometer.",
"It turns out, there exist special values of $B_z$ where the system is in stable equilibrium at the compensation point.",
"Here, the polarization does not change with time as the equilibrium polarization corresponds to the compensation point.",
"From Eqs.",
"(REF ), (REF ) and (REF ), one finds an equation for the leading field corresponding to the stationary system $B_z^2+ \\lambda M_n P^e B_z+ \\frac{D_n}{ R_{\\text{se}}^{ne}}|\\nabla B_\\perp |^2=0\\,.$ A solution for $B_z$ exists only if $\\left(\\lambda M_n P^e \\right)^2-4\\frac{D_n}{R_{\\text{se}}^{ne}}|\\nabla B_\\perp |^2\\geqslant 0\\,,$ that is $\\sqrt{\\frac{R_{\\text{se}}^{ne}}{4D_n}} \\lambda M_n P^e \\geqslant |\\nabla B_\\perp |\\,.$ This illustrates why high noble-gas pressures are desirable.",
"Indeed, noting that the diffusion coefficient $D_n$ is inversely proportional to $\\mathcal {N}_n$ , the magnitude of tolerable gradients scales as $\\mathcal {N}_n^{3/2}$ .",
"Because $P^e$ is limited to unity, and $\\lambda $ is a constant, the only ways to ensure the comagnetometer operates at the equilibrium compensation point is to use high nuclear spin density and/or more accurately zero the gradients.",
"In Fig.",
"REF , one sees that, for a constant gradient $|\\nabla B_\\perp |$ , the nuclear polarization, and hence the compensation field $B_c$ , grows with external magnetic field $B_z$ .",
"In this figure, stationary compensation points can be found by looking for the intersection of the $B_c$ curve with the diagonal (black dashed) line.",
"In the case of $|\\nabla B_\\perp |=0.5$ nT/cm (green solid line), these stationary points are indicated with orange and green stars.",
"However, when $|\\nabla B_\\perp |=1$ nT/cm (red solid line), no stationary compensation point exists.",
"In practice, part of the gradient may be related to the coils generating the leading field, therefore the field gradient may change with the $B_z$ field.",
"This effect is not included in the results presented in Fig.",
"REF ."
],
[
"Experimental setup",
"The experimental setup is sketched in Fig.",
"REF .",
"We use a 20-mm-diameter spherical cell filled with 3 amg of $^3$ He and 50 torr of N$_2$ .",
"The cell has a sidearm loaded with a drop of alkali-metal mixture, 1% $^{87}$ Rb and 99% K molar fractions.",
"The vapor cell is placed in a Twînleaf MS-1LF magnetic shield.",
"To minimize the effect of cell asphericity [30], the comagnetometer was mounted up-right ($z$ -axis along gravity) so that the cell sidearm is plugged by the liquid droplet of alkali metals.",
"Copper wires (not shown in Fig.",
"REF ) are looped around the cylindrical layers for degaussing of the shield and its content.",
"The working temperature of $185^\\circ $ C is achieved with AC resistive heating.",
"At this temperature, the ratio of Rb to K concentrations in the vapor is 3:97, which was measured by fitting the transmission spectrum of Rb D$_2$ and K D$_1$ lines.",
"Figure: Sketch of the experimental setup.",
"LIA – lock-in amplifier, PD – photodetector, L – lens, CS – low-noise current source, P – polarizer, M – mirror, BE – beam expander, λ/4\\lambda /4 – quarter-wave plate, λ/2\\lambda /2 – half-wave plate, MS1-LF – magnetic shield.",
"The inset shows the directions and polarizations of the laser beams, magnetic field modulations and the directions of the generated electronic and nuclear magnetization.",
"The B x B_x modulation is only used for closed-loop control of the compensation point and 3 ^3He magnetization measurement, see Sec.",
".Rubidium atoms are optically pumped by 70 mW of circularly-polarized light from a Toptica TA Pro laser in resonance with the Rb D$_1$ line.",
"For uniform pumping over the cell, the beam is expanded to 20-mm diameter.",
"Potassium (and helium) spins are pumped by spin-exchange collisions with Rb.",
"The comagnetometer readout is realized by monitoring the polarization rotation of a linearly polarized $\\approx 16\\,\\text{mW}/\\text{cm}^2$ (average intensity of 7 mW and beam diameter of 7.5 mm) probe beam (Toptica DL Pro) detuned about 0.5 nm toward shorter wavelength from the K D$_1$ line.",
"Because K atoms are pumped by spin-exchange optical pumping, the SERF magnetometer is much less sensitive to light shifts of the pump beam [6].",
"Both pump and probe beams are guided to the setup with optical fibers.",
"No active stabilization of the lasers is performed apart from that of temperature and current of the diode lasers.",
"To perform low-noise detection of the response to perturbations along the $y$ -axis, the $B_z$ field is modulated with a sine wave (800 Hz, 35 nT peak-to-peak) [31], [9] and the comagnetometer signal is analyzed with a lock-in amplifier.",
"Details on the parametric modulation and chosen parameters are given in Appendix REF .",
"In typical SERF magnetometers, the lock-in detection is achieved by modulating the incident or output probe-beam polarization with photoelastic or Faraday modulators.",
"Using magnetic field modulation instead is helpful to improve the compactness as well as lowering the cost but the SERF resonance is then slightly broadened because of the modulation field, see Appendix REF .",
"After zeroing the magnetic field in the cell and optimizing the pump and probe beam frequencies and powers, the width of the SERF resonance was reduced to 8 nT, leading to a sensitivity to $B_y$ field better than $10\\,\\text{fT}/\\sqrt{\\text{Hz}}$ at 20 Hz, limited by photon shot noise.",
"This characterization was done prior to the gradient optimization and operating the system in the self-compensated regime."
],
[
"Gradient optimization",
"From Eq.",
"(REF ), one sees that the key to maximizing the nuclear polarization is reduction of transverse gradients $\\begin{split}|\\nabla B_\\perp |^2={\\left(\\partial _xB_x\\right)}^2&+{\\left(\\partial _yB_x\\right)}^2+{\\left(\\partial _zB_x\\right)}^2 \\\\&+{\\left(\\partial _xB_y\\right)}^2+{\\left(\\partial _yB_y\\right)}^2\\\\&+{\\left(\\partial _zB_y\\right)}^2\\,.\\end{split}$ The Twînleaf MS-1LF magnetic shield provides 5 independent first-order gradient coils in a built-in flexible printed circuit board, allowing the shimming of the field gradient tensor.",
"To zero transverse gradients, we use a zeroing procedure with two steps.",
"The first step, as we will explain below (Sec.",
"REF ), is to use $T^n_2$ as an indicator to zero the gradients of the longitudinal field $B_z$ .",
"Then, because ${\\nabla } \\times {B} = 0$ , the two $T^n_1$ -relevant components appearing in Eq.",
"(REF ), $\\partial _xB_z=\\partial _zB_x$ and $\\partial _yB_z=\\partial _zB_y$ , are also nulled.",
"The second step is to zero the other remaining independent components using the change of nuclear polarization as an indicator, see Sec.",
"REF ."
],
[
"Optimization at low nuclear polarization",
"At first, the gradients of the longitudinal field $B_z$ are optimized away from the compensation point, at low nuclear polarization.",
"The transverse decay rate of $^3$ He spins is known to be a function of the diffusion across $z$ -field gradients [32] $\\frac{1}{T^n_2}=\\frac{8a^4\\gamma _n^2|\\nabla B_z|^2}{175 D_{n}}\\,,$ where $a$ is the cell radius and $\\gamma _n$ is the nuclear gyromagnetic ratio.",
"To accurately zero $|\\nabla B_z|$ , the following procedure is used: (i) Helium is polarized for 5 min with $B_z=100$ nT.",
"(ii) The $B_y$ field is incremented by 5.5 nT to adiabatically tip the helium spins away from the $z$ -axis.",
"(iii) After turning the $B_y$ field off, the spins are left to precess.",
"The precession is measured for 5 s by monitoring probe beam before the pump beam is turned off (the weak probe beam is left on).",
"Then the spins precess in the dark for 5 min before the pump beam is turned on to measure the precession signal again.",
"The two precession amplitudes are compared to estimate $T^n_2$ .",
"(iv) Nuclear polarization is destroyed by applying a $\\partial _zB_z$ gradient of the order of 30 nT/cm along the $z$ -axis for 5 s. This is long enough to depolarize He spins, for details, see Appendix REF .",
"The last step is important to make sure all measurements are realized at the same nuclear polarization.",
"The procedure is then repeated iteratively for different values of $\\partial _iB_z$ gradients, in the sequence $i=z,x,y$ .",
"After each sequential step the respective gradient is set to its optimum value.",
"The results are depicted in Fig.",
"REF ; the nuclear spin transverse relaxation time is seen to increase from about 200 s (no applied field gradients) to 15 h after a first round of optimization.",
"Figure: Optimization of the transverse relaxation time of He nuclear spins at low polarization in a leading field of 100 nT.",
"(a) Samples of 3 ^3He free decay (left and right panels) separated by a dark time (pump laser off) of 37660 s (about 10.510.5 hours).",
"Simultaneous fitting of the two data sets leads to T 2 n =19080(40)T^n_2 = 19080(40) s (5.3 hours).",
"(b) Evolution of transverse relaxation time as a function of the B z B_z magnetic field gradients.",
"The order of optimization goes as (i)(i) ∂ z B z \\partial _zB_z, (ii)(ii) ∂ x B z \\partial _xB_z, (iii)(iii) ∂ y B z \\partial _yB_z.",
"After a first round of gradient optimization, T 2 n T^n_2 increases from about 200 s to 15 hours as suggested by the fits (solid lines), see Eq.",
"().Let us note that this method is time-consuming: not only because of the need to destroy $^3$ He polarization between measurements, but also because spins have to precess in the dark for some time for a precise measurement of $T^n_2$ .",
"This becomes increasingly problematic in the course of the optimization procedure as the transverse relaxation time increases.",
"Each point in Fig.",
"REF (b) necessitates about 10 min acquisition time.",
"Figure: Gradient-zeroing method at the compensation point.",
"(a) Change of ∂ y B z \\partial _yB_z gradient value as a function of time.",
"(b) Effect of applied ∂ y B z \\partial _yB_z gradient on the compensation field as a function of time."
],
[
"Optimization at the compensation point",
"Once at the compensation point, we modulate the $B_x$ field with a 40-Hz sine wave of about 0.1 nT amplitude The modulation frequency should be higher than the Larmor frequency of $^3$ He, while not being too far away to obtain a reasonable sensitivity..",
"The response of the comagnetometer with respect to $B_z$ exhibits a dispersive resonance centered at $-B_n$ [7], which in our case is $\\approx B_c$ , see Eq.",
"(REF ).",
"This resonance can be used for closed-loop control of the compensation point based on direct reference to the nuclear polarization which is an alternative to previous approaches locking to electronic resonance [28], [29].",
"Our closed-loop control of the compensation field allows the comagnetometer to always be operated at optimum sensitivity, and the mechanisms affecting $T^n_1$ or $T^n_2$ can be studied in real time.",
"Figure: Gradient optimization at the compensation point.",
"The top panel shows the derivative of the time evolution of the compensation point as a function of the applied B z B_z gradients.",
"The results for the other two independent gradient channels are shown on the bottom panel.",
"The data are fit with a parabola, see Eq.",
"(), for which the center corresponds to the best compensated gradient.",
"For discussions on the slope of parabolas, see the text and Tab.",
".As coils intrinsically generate field gradients, it is important to be able to compensate for them at the working point.",
"There are five independent first-order gradients, however, the optimization discussed in Sec.",
"REF only involves gradients of the longitudinal field.",
"To maximise $T^n_1$ , we use the following routine: we start by locking the field to the compensation point.",
"We then vary one of the currents through the gradient coils over time: values are changed every 25 s following a saw-tooth pattern, see Fig.",
"REF (a).",
"This way, the optimum gradient is sampled from different directions at different times, ensuring that effects of drifts are minimized.",
"As a result of changing the gradient, the equilibrium nuclear polarization is changed, and so is the compensation point, see Fig.",
"REF (b), which is observed in real time with the closed-loop control.",
"Then, each 25 s data set corresponding to a field gradient value is differentiated with respect to time in order to obtain the rate of change (in pT/s) of the compensation field as a function of the gradient value.",
"The results are depicted for each of the five independent gradient channels in Fig.",
"REF .",
"Fitting the slope as a function of applied gradient with a parabola (the derivation is given in Appendix ), the center, which corresponds to the best compensated gradient, is extracted.",
"If the new center differs from the previous known value, we observe a buildup of the compensation field after moving to the new value, which indicates that a higher equilibrium field can be reached.",
"This method has to be applied after large changes of the stationary compensation field (typically above 10 nT), since with different leading fields, the gradients change as well.",
"Note that this gradient optimization procedure works also when not at equilibrium nuclear polarization.",
"Indeed, polarization buildup or decay linear in time does not affect the centers of the parabolas shown in Fig.",
"REF .",
"If the polarization changes in a nonlinear fashion, this is no longer the case.",
"Thus, it is important to start with close-to-optimum values for the gradients, which can be obtained using the $T^n_2$ method as discussed in Sec.",
"REF .",
"Table: Calculated and measured slope ratios of the parabolas for each independent magnetic gradient channel.",
"The contribution of applied gradients to the slope is calculated using Eq.",
"(), and compared to that of the ∂ x B z \\partial _xB_z channel.Note that the magnetic field gradient generated by a gradient coil affects at least two gradient components.",
"The actual gradient generated in our setup and the expected slope of parabolas in Fig.",
"REF are summarized in Table REF .",
"Our results show that, as predicted, the $\\partial _xB_y$ gradient is of greatest concern, while the $\\partial _zB_z$ has about four times less impact.",
"However, a sizable deviation between expected and measured ratios is also observed.",
"We attribute this error to not well-known calibration factors and coefficients of the gradient coils.",
"Indeed, estimates are provided by the gradient coils manufacturer, obtained by simulating the generated field in the absence of the magnetic shield.",
"This suggest our method for gradient optimization could also be used for in-situ calibration of gradient coils which we plan to further investigate.",
"Figure: Dynamics of spin-exchange optical pumping of 3 ^{3}He nuclear spins in a leading field of 150 nT.",
"The circles are obtained when there was no applied gradients.",
"The squares are obtained when the gradients were exaggerated by changing the sign of the optimized applied gradients (triangles).",
"Each data set was fit (solid lines) assuming an exponential approach of the nuclear field to the equilibrium value over time, see Eq.",
"()."
],
[
"Polarization dynamics",
"In Fig.",
"REF , we use the experimental method presented in Ref.",
"[7] to show how gradients affect the dynamics of spin-exchange optical pumping of $^3$ He nuclear spins for a leading field of 150 nT and an electronic polarization about 50%.",
"The electronic polarization is chosen to optimize the sensitivity of the comagnetometer, while the nuclear polarization is determined by the requirement of operation at a stable compensation point, see Eq.",
"(REF ).",
"When the gradients are uncompensated the steady-state nuclear field reaches 115 nT.",
"When the gradients are exaggerated (reversal of all optimized gradients) the steady state value drops to 70 nT.",
"No stationary compensation point could be found for either of these configurations.",
"When the gradients are fully optimized, the nuclear field reaches above $B_{n}\\approx 140$ nT (corresponding to a nuclear polarization of $\\approx 3\\%$ ) Higher nuclear spin polarization can be achieved, for example, $30\\%$ in Ref. [18].",
"However, this was done at 100% electronic polarization and at a higher leading field.",
"Therefore, the $30\\%$ polarization is not at the stable compensation point.",
"The polarization in our experiment is similar to that in Ref. [13].",
"In this regime a stable compensation point exists.",
"It was found at 131 nT, determined by the conditions described by Eq.",
"(REF ).",
"Note this is slightly below the maximum polarization of 140 nT achieved with a leading field of 150 nT displayed in Fig.",
"REF .",
"At this compensation field, a noise level corresponding to a sensitivity to nuclear spin-dependent energy shifts of $\\approx 3 \\times 10^{-22}\\,\\rm {eV}/\\sqrt{\\rm {Hz}}$ was achieved in the range 0.1 to 1 Hz.",
"In terms of gyroscopic sensitivity, this corresponds to about $0.5\\, \\mu \\text{rad~s}^{-1}/\\sqrt{\\text{Hz}}$ ($\\approx 80\\,\\rm {nHz}/\\sqrt{\\rm {Hz}}$ ), which is $2.5\\,\\text{fT}/\\sqrt{\\text{Hz}}$ in terms of pseudo-magnetic field sensitivity.",
"To estimate the sensitivity of the system we measured the spectrum of the output signal and calibrated it with a routine based on magnetic field modulation adapted from Ref. [35].",
"Once the system's parameters are determined, different strategies can be employed to reach the stable stationary compensation point starting from an unpolarized system.",
"The nuclear polarization dynamics for different strategies are illustrated by numerical simulations displayed in Fig.",
"REF .",
"Considered strategies include: $(i)$ the leading field is set to the upper stable compensation field value and the optical pumping rate is adjusted to have maximum sensitivity of the comagnetometer, corresponding to $P_e \\approx 0.5$ ; $(ii)$ the leading field is set to a value much larger than the upper stable compensation field (in our conditions 400 nT, practically limited by, for example, the coil current source) together with a higher optical pumping rate, such that $P_e \\approx 1$ , until the upper stationary compensation field is reached.",
"Then the leading field can be locked and the optical pumping rate adjusted such that the system has the highest sensitivity [the conditions of strategy $(i)$ ]; $(iii)$ the parameters are set to strategy $(ii)$ until the polarization passes the lower stationary compensation point, after which the parameters are set to strategy $(i)$ .",
"In this case, the system's nuclear polarization field will continue to grow until it reaches the upper stationary compensation field.",
"The benefit of the latter strategy is that the comagnetometer is operational in the fastest time, for the given simulation parameters, within 35 mins.",
"This could be beneficial, for example, for gyroscopic applications on board vehicles.",
"The changing nuclear polarization while employing strategy $(iii)$ affects the signal-to-noise ratio of the gyroscope, but the device remains operational in the process.",
"Our simulations show, however, that with this strategy, it takes about 50 hours for the system to reach steady state nuclear polarization.",
"Strategy $(ii)$ is a good alternative to have the system quickly operational at the upper stationary compensation point, i.e.",
"at steady state nuclear polarization.",
"Figure: Calculated dynamics of three nuclear polarization strategies.",
"(a) Applied magnetic field as a function of time.",
"(b) Dynamics of the nuclear polarization.",
"Calculations are made with parameters similar to that in Fig.",
"and |∇B ⊥ |=0.5|\\nabla B_\\perp |=0.5\\,nT.",
"The gray shaded area shows nuclear field in between the two stationary compensation points, located at B n =40B_n=40\\,nT and B n =150B_n=150\\,nT, see also Fig. .",
"When the nuclear field is in this area, the closed-loop control of the compensation point can be turned-on and leave the nuclear field move to the highest compensation point over time.",
"Strategy (i)(i) consists in applying a zz-field equal to the known upper stationary compensation field, B n =150B_n=150\\,nT.",
"Strategies (ii)(ii) and (iii)(iii) consist in applying, at t=0t=0, a larger field than in (i)(i) (here, 400 nT) and a higher pumping rate (such that P e =1P_e=1) during the initial stage.",
"Closed-loop control of the compensation field can be turned-on either when reaching the upper stationary compensation field [strategy (ii)(ii)] or right after passing over the lower one [B n =40B_n=40\\,nT, strategy (iii)(iii)]."
],
[
"Conclusion",
"Motivated by applications in NMR-based rotation sensing, searches for ultralight bosonic dark matter and experiments measuring exotic spin-dependent interactions, we constructed a dual-species (K-$^3$ He) comagnetometer with hybrid optical pumping (via Rb atoms) operating in self-compensating regime.",
"Building on the body of earlier work by several groups, we developed a method for closed-loop control of the compensation point allowing practical optimization of the operating parameters in spite of the inherently slow (hours to days) dynamics of the system.",
"When the comagnetometer operates at the compensation point, it is generally insensitive to magnetic fields.",
"However, we showed that the system can still be optimized in terms of magnetic gradients and fields without changing the operation mode.",
"The presented gradient optimization method facilitates achieving much higher longitudinal and transverse relaxation times as is otherwise possible, improving the polarization level and stability.",
"At the compensation point, this also increases the coupling between the nuclear and electronic spins and improves stability of the compensation point.",
"With these, our method has potential to boost the performance of comagnetometer nuclear-spin-ensemble-based quantum memory [18], [17], [19], amplifiers [36], [37], [38], [39] and gyroscopes [28], [10], [40].",
"Finally, the device shows a sensitivity comparable to the best previous comagnetometers [13], however, at a several times smaller density of $^3$ He.",
"The helium number density of 3 amg used in this work is lower than in most previous studies.",
"Since lower-pressure cells are easier to manufacture and safer to operate, the optimization technique allows for wide usability of vapor-cell-based nuclear spin sensors in various application areas."
],
[
"SERF magnetometer",
"We measured the polarization of K via Faraday rotation of a linearly polarized probe beam.",
"The rotation magnitude is in proportional to the component of K polarization $P^e_y$ according to [21] $ \\theta =\\frac{1}{2}n_e r_e c D f_{D_1} \\frac{\\nu -\\nu _{D_1}}{(\\nu -\\nu _{D_1})^2+\\left[\\Gamma ^{D_1}_\\text{tot}/(2\\pi )\\right]^2} P^e_y\\,,$ where $n_e = 7.5 \\times 10^{13}$ cm$^{-3}$ is the number density of K, $r_e = 2.8\\times 10^{-15}$ m is the classical electron radius, $c=3\\times 10^{8}$ m/s is the speed of light, $D=2$ cm is the cell diameter, $f_{\\text{D}_1}=1/3$ is the oscillator strength for the D$_1$ transition at frequency $\\nu _{\\text{D}_1}=389.236$ THz, $\\nu =389.539$ THz is the laser frequency, and $\\Gamma ^{\\text{D}_1}_\\text{tot}/(2\\pi )=48$ GHz is the half-width at half-maximum (HWHM) of the transition.",
"With these values, the maximum rotation angle, corresponding to $P^e_y=1$ , is 0.7 rad.",
"An optical polarimeter was used to read out the rotation angle as shown in Fig.",
"REF , and the decrossing angle between two polarizers was set to $\\pi /4$ .",
"Thus, the signal $I \\propto \\sin ^2(\\theta -\\pi /4) \\approx 0.5-\\theta $ is linearly dependent on the rotation angle.",
"To suppress low-frequency drifts, we applied a modulation field $ B_z^m \\sin ( \\omega _m t)$ along the pumping axis (see, for example, Ref. [41]).",
"Consequently, $P^e_y$ is a sum of harmonics.",
"Under the assumption that the field component $B_z$ is small, the approximate solution to the first harmonic is [31], [9] $ \\begin{split}P^e_y(\\omega _m)=&-\\frac{2 P^e_z R_\\text{tot} \\gamma _e Q}{ Q^2R_\\text{tot}^2+ (\\gamma _e B_y)^2} J_0\\left(\\frac{\\gamma _e B_z^m}{Q \\omega _m}\\right)\\\\&\\times J_1\\left(\\frac{\\gamma _e B_z^m}{Q \\omega _m}\\right) B_y \\sin (\\omega _mt)\\,,\\end{split}$ where $J_i(\\cdot )$ are Bessel functions of the first kind of order $i$ , $Q$ is the slowing-down factor, and $R_\\text{tot}$ is the total relaxation rate of K. For a modulation frequency of 800 Hz and $Q=5.2$ (assuming $P^e= 50\\%$ ), the factor $ J_0(\\frac{\\gamma _e B_m}{Q \\omega _m}) J_1(\\frac{\\gamma _e B_z^m}{Q \\omega _m})$ is maximized for $B_z^m=161$ nT.",
"However when $B_z^m$ is large, the modulation field causes rf broadening of magnetic resonance transitions, and the relaxation rate is given by [13] $ R_\\text{rf}=\\frac{5}{36}\\frac{(\\gamma _e B_z^m)^2}{Q^2 R_\\text{se}}\\,,$ where $R_\\text{se}=n_e \\sigma _{ee} v = 79000$ s$^{-1}$ is the alkali-alkali spin-exchange rate with the collision cross section $\\sigma _{ee} =1.5\\times 10^{-14}$ cm$^2$ .",
"For $B_z^m=161$ nT, the rate is $R_\\text{rf}=52$ s$^{-1}$ , being 5 times larger than the spin destruction ratio of K on He.",
"We experimentally found that the sensitivity of the K magnetometer peaks at $B_z^m=35$ nT, which balances the beneficial effect of larger $B_z^m$ on the signal amplitude with its deleterious effect on the linewidth due to the RF broadening.",
"The signal was fed into a lock-in amplifier and demodulated at the first-harmonic to retrieve the rotation angle.",
"To suppress the alkali polarization gradient, we employ a hybrid pumping technique where an optically thin sample of Rb atoms is optically pumped and the K atoms are polarized via spin-exchange collisions with the Rb atoms.",
"Uniform optical pumping can be achieved with a high ratio of the receiver to donor densities.",
"In general, both K and Rb can be the spin donor.",
"Because of the high He pressure in our cell, the spin destruction collisions between alkali atoms and He atoms are the dominant source of relaxation.",
"Given that the spin-destruction cross section of K on He is around 18 times smaller than that of Rb on He [26], [42], we chose K as the receiver in the hope of reaching a better magnetometer sensitivity.",
"Note that the modulation of $B_x$ or $B_y$ fields also results in a usable magnetic resonance.",
"However, in that case the projection of the electronic spins on the $z$ -axis is then modulated periodically too, leading to a lower average electronic spin polarization along the $z$ -axis and therefore a lower equilibrium nuclear polarization, in turn affecting the stability of the comagnetometer [Eq.",
"(REF )]."
],
[
"Depolarization of nuclear spins with applied gradients",
"In Sec.",
"REF , we proposed a method to optimize the gradients at low nuclear polarization.",
"To make sure all the measurements are performed at the same nuclear polarization, the step $(iv)$ consists of destroying nuclear polarization by applying a 30 nT/cm $\\partial _zB_z$ gradient for 5 s. It should be stressed that by implementing this step, one needs to make sure that nuclear polarization is completely destroyed.",
"One parameter to consider in this context is the diffusion rate between slices in which the rotation angle of the spins changes by $\\pi $ .",
"The thickness of such a slice is $\\delta z = \\frac{1}{2\\gamma _n \\nabla B_z \\delta t}\\,,$ where $\\gamma _n=3.24\\times 10^{-2}\\,$ Hz/nT is the gyromagnetic ratio of $^3$ He nuclear spins, $\\nabla B_z$ is the applied $z$ -gradient and $\\delta t$ is the time it is applied.",
"With $\\nabla B_z=30\\,$ nT/cm and $\\delta t = 5\\,$ s, we find $\\delta z=0.11\\,$ cm.",
"Noting that, for our experimental conditions, $D_{^3\\text{He}}\\approx 0.71\\,$ cm$^2$ /s, $^3$ He spins diffuse through multiple slices during step $(iv)$ , which leads to depolarization.",
"Note that, during step $(ii)$ , the nuclear magnetization was rotated in the $y-z$ plane.",
"Besides, applying a $\\partial _zB_z$ gradient automatically generates a $\\partial _yB_y$ gradient, contributing together to the depolarization of longitudinal and transverse nuclear spins [Eqs.",
"(REF ) and (REF )]."
],
[
"Time derivative of polarization in gradient optimization process",
"The solution of Eq.",
"(REF ) gives the time evolution of the nuclear polarization of the following form $ \\begin{split}P^n(t)=P^n_s+&[P^n(0)-P^n_s]\\\\&\\times \\left\\lbrace 1-\\exp \\left[-(R_{se}^{ne}+1/T^n_1)t\\right]\\right\\rbrace \\,,\\end{split}$ where $P^n(0)$ is an initial value of the polarization and $P^n_s$ is the steady state polarization defined in Eq.",
"(REF ) by $P^n(t\\rightarrow \\infty )$ .",
"Therefore, the time derivative of the nuclear polarization is $ \\begin{split}\\frac{d}{dt} P^n(t) = \\left[R^{ne}_{\\text{se}}+1/T^n_1\\right] [P^n_s - P^n(0)] \\\\ \\times \\exp [-(R_{\\text{se}}^{ne}+1/T^n_1)t]\\,.\\end{split}$ In our experiment, the measurement period for a given gradient value lasts for 25 s, which is much shorter than $T_1^n$ , therefore we can approximate $\\exp (-t/T_1^n)\\approx 1$ .",
"In addition, $\\exp (-R_{se}^{ne}t) \\approx 1$ since $R_{\\text{se}}^{ne} \\lesssim 1/T^n_1$ .",
"Moreover, throughout the whole zeroing procedure the initial polarization for each measurement varies by less than 5%, therefore for simplicity we assume it to be constant over all measurements within the zeroing procedure.",
"This leads to the following equation for time derivative of the polarization in each measurement $ \\begin{split}\\frac{d }{dt}P^n(t) &= (R_{\\text{se}}^{ne}+1/T^n_1)(P^n_s-P^n(0))\\,,\\end{split}$ then using Eqs.",
"(REF ) and (REF ), one can get an expression of polarization time derivative as a function of the gradients of transversal fields $|\\nabla B_\\perp |$ (see Fig.",
"REF ) $\\frac{d }{dt}P^n(t) = R_{\\text{se}}^{ne}[P^e-P^n(0)]-\\frac{D_{n}}{B_z^2}P^n(0) |\\nabla B_\\perp (t)|^2\\,,$ where time dependence of the gradient is due to the change of its value during the zeroing routine."
],
[
"Acknowledgements",
"We thank Wei Ji for fruitful discussions.",
"This work was supported by the German Federal Ministry of Education and Research (BMBF) within the Quantentechnologien program (FKZ 13N15064).",
"The work of D.F.J.K.",
"was supported by U.S. National Science Foundation (NSF) Grant No.",
"PHY-2110388.",
"SP and MP acknowledge support by the National Science Centre, Poland within the OPUS program (2020/39/B/ST2/01524)."
]
] | 2210.07687 |
[
[
"Addressing energy density functionals in the language of path-integrals\n II: Comparative study of functional renormalization group techniques applied\n to the (0+0)-D $O(N)$-symmetric $\\varphi^{4}$-theory"
],
[
"Abstract The present paper is the second of a series of publications that aim at investigating relevant directions to turn the nuclear energy density functional (EDF) method as an effective field theory (EFT).",
"The EDF approach has known numerous successes in nuclear theory over the past decades and is currently the only microscopic technique that can be applied to all atomic nuclei.",
"However, the phenomenological character of the EDF method also comes with important limitations, such as the lack of an explicit connection with quantum chromodynamics (QCD).",
"As was argued in the first paper of this series, reformulating the EDF framework as an EFT would enable us to overcome these limitations.",
"In particular, path-integral (PI) techniques are suited to achieve such a purpose as they allow to design numerous non-perturbative approximations and can take Lagrangians possibly derived from EFTs of QCD as inputs.",
"In our previous paper, we have illustrated such technical features for diagrammatic PI techniques in a study of the (0+0)-D $O(N)$-symmetric $\\varphi^{4}$-theory.",
"In the present work, we consider another class of PI techniques, i.e.",
"functional renormalization group (FRG) approaches, that we apply on the same toy model.",
"Despite our explicit interest for the nuclear many-body problem, the presented study is also directed towards FRG practitioners from various fields: technical details are provided for FRG techniques based on 1-particle-irreducible (1PI), 2-particle-irreducible (2PI) and 2-particle-point-irreducible (2PPI) effective actions (EAs), coined respectively as 1PI-, 2PI- and 2PPI-FRGs, and the treatment of the $O(N)$ symmetry is also addressed thoroughly.",
"Connections between these various FRG methods are identified as well."
],
[
"Context: Energy density functional method and path-integral formalism",
"The energy density functional (EDF) method is arguably one of the most exploited microscopic approach in theoretical nuclear physics.",
"This has actually been the case for the past several decades.",
"As a matter of fact, nuclear physicists have estimated ground state (gs) observables and excited state properties throughout the whole nuclear chart thanks to EDF calculations [1], [3].",
"In the first paper of this series, coined as paper I, we have outlined the main features of the EDF method: i) The EDF approach relies on functionals of the nucleonic density that depend on free parameters fitted on nuclear data and the analytical forms of these functionals are postulated consistently with the symmetries of the two-body nucleon interaction; ii) The recipe underlying EDF calculations is essentially split into two steps: the single-reference implementation used as a first step, which is basically a mean-field approach obtained through a variational treatment of the aforementioned functional, and the multi-reference scheme, where all symmetries spontaneously broken at the single-reference level are restored, owing to the fact that mesoscopic systems such as nuclei can not exhibit spontaneous symmetry breakings (SSBs).",
"Despite the presence of density functionals in the EDF formalism according to point i), the EDF approach should not be confused with density functional theory (DFT), as we have argued in more detail in paper I.",
"Actually, the difference between EDF and DFT is notably due to point ii) and the presence of SSB in the EDF framework since DFT, and more precisely the Hohenberg-Kohn (HK) theorems [4] and the Kohn-Sham scheme [5], [6], are only formulated in a symmetry-conserving framework [7].",
"The phenomenological nature of the functionals (point i)) and the spontaneous breakdown and subsequent restoration of symmetries (point ii)) both allow for describing collective properties across the whole nuclear chart at low cost within the EDF framework.",
"Naturally, this also comes with a price as this phenomenological character induces problematic limitations of the EDF approach as well, such as the lack of possibility to systematically improve the obtained results or the absence of explicit connection with quantum chromodynamics (QCD).",
"In paper I, we explained technically that all of these limitations could be overcome by turning the EDF approach into an effective field theory (EFT) [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].",
"Although we pointed out other parallel directions aiming at designing more reliable approaches also based on density functionals (like the density-matrix expansion [21], [22]), we argued in addition that a framework of choice to achieve the reformulation of the EDF method as an EFT is the path-integral (PI) formulation of quantum field theory (QFT).",
"This is due to the fact that PI techniques take Lagrangians (or classical actions), potentially derived from EFTs of QCD, as inputs.",
"Most PI techniques are also themselves systematically improvable, as was illustrated in paper I.",
"For the nuclear models we aim at constructing, this implies that both the nuclear interaction (i.e.",
"the Lagrangian resulting from an EFT) and the many-body treatment (i.e.",
"the PI technique) rely on an expansion and can therefore be the subject of systematically improvable approximations.",
"Furthermore, there are several ways to introduce collective degrees of freedom (dofs) within the PI framework whereas such dofs also play a pivotal role for the EDF method, either at the level of the functional through densities or via order parameters introduced at the multi-reference level.",
"On the side of the PI formalism, collective dofs can be introduced by means of a Hubbard-Stratonovich transformation (HST) and/or using higher-order effective actions (EAs), i.e.",
"$n$ -particle-(point-)irreducible ($n$ P(P)I) EAs with $n>1$ .",
"More precisely, 2PI EAs are functionals of propagators whereas their local versions, i.e.",
"2PPI EAs, are by definition functionals of the diagonal parts of these propagators, which can be identified as densities."
],
[
"Recall from paper I",
"Let us then recall what was done in the first part of our study within paper I.",
"In the latter, we have investigated several PI techniques: the loop expansion (LE) which notably implements perturbation theory in the PI formalism, optimized perturbation theory (OPT) and self-consistent perturbation theory (SCPT).",
"Some of these methods were also combined with various resummation techniques when relevant.",
"The playground chosen for these applications was the (0+0)-D $O(N)$ -symmetric $\\varphi ^4$ -theory, also called (0+0)-D $O(N)$ model, which is specified by the classical action: $S\\Big (\\vec{\\widetilde{\\varphi }}\\Big )=\\frac{m^{2}}{2}\\vec{\\widetilde{\\varphi }}^{2}+\\frac{\\lambda }{4!",
"}\\left(\\vec{\\widetilde{\\varphi }}^{2}\\right)^{2} \\;,$ and relies e.g.",
"on the generating functionalWe have set $\\hbar =1$ in Eq.",
"(REF ) and follow this convention throughout this entire paper.",
": $Z\\Big (\\vec{J},K\\Big ) = e^{W\\big (\\vec{J},K\\big )} = \\int _{\\mathbb {R}^N} d^N\\vec{\\widetilde{\\varphi }} \\ e^{-S_{JK}\\big (\\vec{\\widetilde{\\varphi }}\\big )} \\;,$ withWe slightly change our notations as compared to paper I, labeling color (supercolor) indices by $a_{1}$ , $a_{2}$ , $a_{3}$ , ... ($b_{1}$ , $b_{2}$ , $b_{3}$ , ...) instead of $a$ , $b$ , $c$ , ... ($\\alpha $ , $\\beta $ , $\\gamma $ , ...) for convenience.",
"Recall that color (supercolor) indices run by definition from 1 to $N$ ($N+1$ ) and label e.g.",
"the components of $\\vec{\\widetilde{\\varphi }}$ ($\\widetilde{\\Psi }$ ).",
"$S_{JK}\\Big (\\vec{\\widetilde{\\varphi }}\\Big )\\equiv S\\Big (\\vec{\\widetilde{\\varphi }}\\Big )-\\sum _{a=1}^{N} J_{a}\\widetilde{\\varphi }_{a} -\\frac{1}{2} \\sum _{a_{1},a_{2}=1}^{N} \\widetilde{\\varphi }_{a_{1}} K_{a_{1} a_{2}} \\widetilde{\\varphi }_{a_{2}} \\;,$ $\\vec{J}$ and $K$ being the sources and the integration in Eq.",
"(REF ) runs over the $N$ components of the field $\\vec{\\widetilde{\\varphi }}$ , i.e.",
": $\\vec{\\widetilde{\\varphi }} \\equiv \\begin{pmatrix}\\widetilde{\\varphi }_{1} \\\\\\vdots \\\\\\widetilde{\\varphi }_{N}\\end{pmatrix} \\;.$ In paper I, we have investigated the effects of a HST on the performances of the aforementioned PI techniques by considering three mathematically equivalent representations of the (0+0)-D $O(N)$ model.",
"Firstly, there is the original representation based on classical action (REF ), involving only the original dofs (REF ).",
"Then, the mixed representation can be obtained by applying a HST to $Z\\big (\\vec{J}=\\vec{0},K=0\\big )$ , thus yielding: $Z_{\\mathrm {mix}}\\big (\\mathcal {J},\\mathcal {K}\\big ) = e^{W_{\\mathrm {mix}}(\\mathcal {J},\\mathcal {K})} = \\int _{\\mathbb {R}^{N+1}} d^N \\vec{\\widetilde{\\varphi }} d\\widetilde{\\sigma } \\ e^{-S_{\\mathrm {mix},\\mathcal {J}\\mathcal {K}}\\big (\\vec{\\widetilde{\\varphi }},\\widetilde{\\sigma }\\big )}\\;,$ with $\\begin{split}S_{\\text{mix},\\mathcal {J}\\mathcal {K}}\\Big (\\widetilde{\\Psi }\\Big ) \\equiv & \\ S_\\text{mix}\\Big (\\widetilde{\\Psi }\\Big ) - \\sum _{b=1}^{N+1} \\mathcal {J}_{b} \\widetilde{\\Psi }_{b} \\\\& - \\frac{1}{2} \\sum _{b_{1},b_{2}=1}^{N+1} \\widetilde{\\Psi }_{b_{1}} \\mathcal {K}_{b_{1}b_{2}} \\widetilde{\\Psi }_{b_{2}} \\;,\\end{split}$ $S_\\text{mix}\\Big (\\widetilde{\\Psi }\\Big ) = \\frac{m^{2}}{2}\\vec{\\widetilde{\\varphi }}^{2} + \\frac{1}{2} \\widetilde{\\sigma }^{2} + i \\sqrt{\\frac{\\lambda }{12}} \\widetilde{\\sigma } \\vec{\\widetilde{\\varphi }}^{2} \\;,$ and $\\widetilde{\\Psi } \\equiv \\begin{pmatrix}\\vec{\\widetilde{\\varphi }} \\\\\\widetilde{\\sigma }\\end{pmatrix} \\;,$ $\\mathcal {J} \\equiv \\begin{pmatrix}\\vec{J} \\\\j\\end{pmatrix} \\;,$ $\\mathcal {K} \\equiv \\begin{pmatrix}K & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & k\\end{pmatrix} \\;.$ The theory now involves also a collective dof under the form of the fluctuating auxiliary field $\\widetilde{\\sigma }$ .",
"The original dofs can also be integrated out within the mixed representation by means of Gaussian integration, which gives us the collective representation of the system, involving the collective dof $\\widetilde{\\sigma }$ only.",
"A generating functional of this collective representation is: $Z_{\\mathrm {col}}(\\mathcal {J}) = e^{W_{\\mathrm {col}}(\\mathcal {J})} = \\int _{\\mathbb {R}} d\\widetilde{\\sigma } \\ e^{-S_{\\mathrm {col},\\mathcal {J}}(\\widetilde{\\sigma })}\\;,$ with $S_{\\mathrm {col},\\mathcal {J}}(\\widetilde{\\sigma }) = S_{\\mathrm {col}}(\\widetilde{\\sigma }) - j \\widetilde{\\sigma } - \\frac{1}{2} \\sum _{a_{1},a_{2}=1}^{N} J_{a_{1}} G_{\\widetilde{\\sigma },a_{1}a_{2}} J_{a_{2}} \\;,$ $S_{\\text{col}}(\\widetilde{\\sigma }) = \\frac{1}{2} \\widetilde{\\sigma }^2 -\\frac{1}{2} \\mathrm {Tr}_{a}\\left[\\ln (2\\pi G_{\\widetilde{\\sigma }})\\right] \\;,$ and $G^{-1}_{\\widetilde{\\sigma },a_{1}a_{2}} = \\left(m^2 + i\\sqrt{\\frac{\\lambda }{3}}\\widetilde{\\sigma }\\right)\\delta _{a_{1}a_{2}} \\;,$ and $\\mathrm {Tr}_{a}$ denotes the trace over color indicesWe refer to paper I for the expressions of the generating functionals (REF ), (REF ) and (REF ) in arbitrary dimensions..",
"The introduction of a collective dof via HST turned out to be particularly fruitful in our study of paper I.",
"In particular, the LE in the collective representation and SCPT based on a 2PI EA in the mixed representation both yielded much better performances than their counterparts implemented within the original representation.",
"Among the different families of approaches tested in paper I, only SCPT is implemented in the EA framework.",
"We have thus also investigated the effect of introducing collective dofs via 2PI EAs within SCPT.",
"More precisely, SCPT has the peculiarity to rely on optimized expansions of EAs, where the optimization is carried out by extremizing the EA under consideration.",
"The variational equations thus obtained are actually self-consistent in finite dimensions which, combined with the fact that some EAs are functionals of densities, enable us to say that SCPT is the PI technique closest in spirit to the EDF method (although it should be stressed that SCPT overcomes the aforementioned limitations of the EDF approach, enabling for instance to systematically improve Hartree-Fock(-Bogoliubov) results, as was illustrated in paper I)."
],
[
"Focus on paper II",
"In the present paper, i.e.",
"paper II, we investigate other means to treat EAs by exploiting FRG approaches, which rely instead on (integro-)differential equations.",
"The (0+0)-D $O(N)$ -symmetric $\\varphi ^{4}$ -theory will remain our playground and our study will therefore benefit once again from various features of this model.",
"Firstly, its simplicity (and more specifically its zero-dimensional nature) allows for pushing our approaches to quite high truncation orders, to identify more readily connections between the different tested techniques and provides us with exact (analytical) solutions for benchmarking.",
"Secondly, despite its simplicity, the classical action of this toy model possesses a symmetry, i.e.",
"the $O(N)$ symmetry which is continuous for $N>1$ : this can also be seen as an advantage considering the pivotal role of symmetries within the EDF method.",
"We stress that the exact solution of this zero-dimensional toy model always conserves the $O(N)$ symmetryThis feature of the exact solution of the (0+0)-D $O(N)$ model can be traced back to Mermin-Wagner theorem [23], [24], [25], as was shown in Ref. [26].",
"(explicitly and spontaneously), which constitutes an interesting link (illustrated in paper I) with mesoscopic systems such as nuclei that can not spontaneously break symmetries either.",
"However, we point out that the analysis of the possible occurrence of spontaneous breakdown of the $O(N)$ symmetry will be much more concise here as compared to that of paper I, notably because most of the tested FRG implementations (all those based on 2PI or 2PPI EAs) can not break the $O(N)$ symmetry by construction.",
"Regarding the tested FRG methods actually, we will also follow the steps of paper I by exploiting the aforementioned techniques to introduce collective (bosonic) dofs in the theory and examining their impacts on the performances of those methods.",
"Besides, we also believe that this approach could shed some light on the foundations and the origins of the successes of the EDF approach (e.g.",
"the following questions still remain open: why is the EDF method already so efficient at the single-reference level?",
"What is the role of the density-dependences of the functionals in such performances?).",
"When possible, we will notably discuss the effect of a HST by considering the mixed and/or collective representation(s) of the studied model.",
"Higher-order EAs will be exploited as well since we will implement FRG techniques formulated from 1PI, 2PI and 2PPI EAs, respectively referred to as 1PI-, 2PI- and 2PPI-FRGs.",
"Although higher-order EAs occupy a rather small place in the FRG community as most FRG applications are based on 1PI EAs, this is certainly not the case in nuclear physics where a particular interest has been manifested for 2PPI EAs.",
"Although PI techniques remain clearly marginal in the landscape of nuclear theory, several works exploring the 2PPI EA framework are to be mentioned like the contributions of Furnstahl and coworkers (already discussed in paper I) who pointed out the possibility and advantages to turn the EDF framework into an EFT in several of their papers [27], [28], [29], [30], [31], [32].",
"A particular reason to consider this direction is the connection between DFT and 2PPI EAs, as it has been shown that the latter are density functionals in the sense of DFT [33], even in the sense of Kohn-Sham DFT [34].",
"The approach exploited in this way by Furnstahl and collaborators is thus coined as ab initio DFT for nuclei [30] but it should be stressed that it treats 2PPI EAs with SCPT, which has also been discussed within our (0+0)-D $O(N)$ model in paper IIn paper I, we have more precisely implemented SCPT based on 2PI EAs for the (0+0)-D $O(N)$ model and found that the optimal solutions of the corresponding gap equations always fully conserve the $O(N)$ symmetry.",
"Furthermore, in the absence of SSB, the 2PI and 2PPI EAs of our zero-dimensional toy model coincide, which is why we have also tested SCPT based on 2PPI EAs in this comparative study.. We will investigate in the present paper the possibility to design such an ab initio DFT using 2PPI-FRG instead of SCPT, thus following the direction set notably by Kemler and Braun [35], [36], [37] as well as Yokota and collaborators [38], [39], [40], [41], [42], [43] in the framework of different models.",
"However, the present study differentiates itself from the latter works by comparing 2PPI-FRG techniques with other FRG approaches, for various truncation orders.",
"This will notably enable us to identify connections between 2PPI-FRG implementations and the 2PI-FRG formalism.",
"In what follows, sections , and contain our analysis based on 1PI-, 2PI- and 2PPI-FRG approaches, respectively.",
"As in paper I, the results obtained for the gs energy and density of the (0+0)-D $O(N)$ model from the tested PI techniques will be benchmarked against the corresponding exact solutions, $E^{\\mathrm {exact}}_{\\mathrm {gs}}$ and $\\rho ^{\\mathrm {exact}}_{\\mathrm {gs}}$ , whose analytical representations are already given in paper I.",
"Furthermore, once again following the steps of paper I, those numerical applications will be performed at $N=1$ or 2 (mostly at $N=2$ for the sake of conciseness) with $\\mathcal {R}e(\\lambda )\\ge 0$ , investigating the two following phases: the phase or regime with $m^2>0$ (coined as unbroken-symmetry phase) and that with $m^2<0$ (coined as broken-symmetry phaseIn this so-called broken-symmetry phase, the $O(N)$ symmetry is broken only spontaneously and only at the classical level.).",
"Finally, section contains our conclusion, which is twofold.",
"We first conclude on the FRG study of the present paper, comparing the most performing tested 1PI-, 2PI- and 2PPI-FRG approaches.",
"Then, we conclude on our whole comparative study made of papers I and II, comparing this time the most performing FRG method of paper II with the best LE, OPT and SCPT implementations identified in paper I.",
"In particular, since 2PI- and 2PPI-FRG approaches are significantly less present in the literature than their 1PI counterparts, we will keep the discussion general throughout most of sections and , presenting notably the corresponding formalisms in arbitrary dimensions.",
"In any case, technical details will be provided for all tested FRG techniques, in the main part of the text as well as in appendices: Appendix : an introduction to the bosonic index formalism used in the 2PI EA framework.",
"Appendix : a discussion on two specific implementations of the 2PI-FRG, coined as C-flow [44] and CU-flow [45], illustrating notably, with numerical results for the (0+0)-D $O(N)$ -symmetric $\\varphi ^4$ -theory, that the C-flow is not suited to construct reliable approximation schemes.",
"Appendix : an explanation on how to simplify the flow equations within the 1PI- and 2PI-FRGs using the conservation of the $O(N)$ symmetry.",
"Appendix : a presentation of additional flow equations (with complementary derivations when considered relevant) that were used to obtain the numerical results presented in this paper.",
"This study has been carried out as part of a PhD project and we refer to the corresponding PhD thesis [46] for readers interested in further technical details."
],
[
"Generalities",
"As already mentioned, the most widespread FRG approach is the 1PI-FRG, that was proposed by Wetterich in Refs.",
"[47], [48], [49], [50] alongside with others [51], [52], [53], [54], [55], [56].",
"It remains an active area in numerous fields of physics, as e.g.",
"in QCD [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], quantum gravity [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], condensed matter physics [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90] and even in out-of-equilibrium physics [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], [101].",
"A few applications in the context of nuclear physics have been performed as well [102], [103].",
"Various toy model studies based on the 1PI-FRG are to be noted too, like those of the quantum anharmonic oscillator, i.e.",
"the (0+1)-D $\\varphi ^4$ -theory [104], [105].",
"In particular, the 1PI-FRG has already been used to treat the (0+0)-D $O(N)$ model [106], [26], [107], [108], notably in the framework of a recent project designing novel implementations of the 1PI-FRG in connection with fluid dynamics [109], [26], [107], [108].",
"We do not aim at presenting here all FRG implementations based on 1PI EAs but rather refer to reviews [110], [111], [112] or textbooks [113], [114] for complete overviews on this topic.",
"We discuss instead the main ingredients of the 1PI-FRG approach that we will use throughout this entire study, which is still presently the most widespread FRG implementation for fermionic systems.",
"To define those main ingredients, we consider a local source, that we denote as $J_{\\alpha }$ , and a Schwinger functional $W[J]$ , that we express as: $Z[J]=e^{W[J]}= \\int \\mathcal {D}\\widetilde{\\psi } \\ e^{-S\\big [\\widetilde{\\psi }\\big ] +\\int _{\\alpha } J_{\\alpha }\\widetilde{\\psi }_{\\alpha }}\\;,$ where $\\widetilde{\\psi }$ is either a bosonic or a Grassmann field.",
"The index $\\alpha $ encompasses all types of internal indices $i$ as well as coordinates in Euclidean $D$ -dimensional spacetime, such that the corresponding integration reads: $\\int _{\\alpha } \\equiv \\sum _{i} \\int ^{1/T}_{0} d\\tau \\int d^{D-1} r \\;,$ with $T$ the temperature.",
"The 1PI EA $\\Gamma ^{(\\mathrm {1PI})}[\\psi ]$ is related to $W[J]$ via the Legendre transform: $\\Gamma ^{(\\mathrm {1PI})}[\\psi ] = - W[J] + \\int _{\\alpha } J_{\\alpha } \\psi _{\\alpha } \\;,$ with $\\psi _{\\alpha } = \\frac{\\delta W[J]}{\\delta J_{\\alpha }} \\;.$ Within the FRG framework, one introduces a scale-dependence into the generating functionals of interests via a cutoff function.",
"For the Schwinger functional and the 1PI EA introduced in Eqs.",
"(REF ) and (REF ), this can be done as follows: $Z_{k}[J]=e^{W_{k}[J]}= \\int \\mathcal {D}\\widetilde{\\psi } \\ e^{-S\\big [\\widetilde{\\psi }\\big ] - \\Delta S_{k}\\big [\\widetilde{\\psi }\\big ] +\\int _{\\alpha } J_{\\alpha }\\widetilde{\\psi }_{\\alpha }}\\;,$ $\\Gamma _{k}^{(\\mathrm {1PI})}[\\psi ] = - W_{k}[J] + \\int _{\\alpha } J_{\\alpha } \\psi _{\\alpha } - \\Delta S_{k}[\\psi ] \\;,$ where $\\psi _{\\alpha } \\equiv \\psi _{k,\\alpha }[J] =\\frac{\\delta W_{k}[J]}{\\delta J_{\\alpha }}\\;,$ and $\\Delta S_{k}$ is defined in terms of the cutoff function $R_{k}$ as: $\\Delta S_{k}\\Big [\\widetilde{\\psi }\\Big ]=\\frac{1}{2}\\int _{\\alpha _{1},\\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{1}}R_{k,\\alpha _{1} \\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{2}}\\;.$ By then differentiating Eq.",
"(REF ) with respect to the scale $k$ , one can derive the following exact differential equation for the effective average action or flowing EA $\\Gamma _{k}^{(\\mathrm {1PI})}[\\psi ]$ : $\\begin{split}\\dot{\\Gamma }^{(\\mathrm {1PI})}_{k}[\\psi ] \\equiv & \\ \\partial _{k} \\Gamma ^{(\\mathrm {1PI})}_{k}[\\psi ] \\\\= & \\ \\frac{1}{2}\\mathrm {STr}\\left[\\dot{R}_{k}\\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{k}[\\psi ]+R_{k}\\right)^{-1}\\right] \\;,\\end{split}$ which is the celebrated Wetterich equation [48].",
"The functional matrix $\\Gamma _{k}^{(\\mathrm {1PI})(2)}[\\psi ]\\equiv \\frac{\\delta ^{2}\\Gamma ^{(\\mathrm {1PI})}_{k}[\\psi ]}{\\delta \\psi \\delta \\psi }$ is the Hessian of $\\Gamma ^{(\\mathrm {1PI})}_{k}$ and $\\mathrm {STr}$ denotes the supertrace with respect to $\\alpha $ -indices.",
"In practice, the Wetterich equation can not be solved directly (except for some toy models, such as the $O(N)$ model considered in this study [106]) but must rather be turned into a truncated set of coupled differential equations using a given expansion scheme.",
"In the present study, the Wetterich equation is always treated within one expansion scheme, namely the vertex expansion [55], [115].",
"The latter consists in expanding all sides of the Wetterich equation (REF ) in powers of the field $\\psi $ .",
"In particular, such an expansion for the left-hand side of Eq.",
"(REF ) directly follows from the following Taylor expansion of $\\Gamma ^{(\\mathrm {1PI})}_{k}$ : $\\begin{split}\\Gamma ^{(\\mathrm {1PI})}_{k}[\\psi ] = & \\ \\overline{\\Gamma }^{(\\mathrm {1PI})}_{k} \\\\& + \\sum _{n=2}^{\\infty }\\frac{1}{n!",
"}\\int _{\\alpha _{1},\\cdots ,\\alpha _{n}} \\overline{\\Gamma }_{k,\\alpha _{1} \\cdots \\alpha _{n}}^{(\\mathrm {1PI})(n)} \\\\& \\times \\left(\\psi -\\overline{\\psi }_{k}\\right)_{\\alpha _{1}} \\cdots \\left(\\psi -\\overline{\\psi }_{k}\\right)_{\\alpha _{n}} \\;,\\end{split}$ where $\\overline{\\psi }_{k,\\alpha }=\\left.\\frac{\\delta W_{k}[J]}{\\delta J_{\\alpha }}\\right|_{J=0}$ , $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k}\\equiv \\Gamma ^{(\\mathrm {1PI})}_{k}\\big [\\psi =\\overline{\\psi }_{k}\\big ]$ , $\\overline{\\Gamma }_{k,\\alpha _{1} \\cdots \\alpha _{n}}^{(\\mathrm {1PI})(n)}\\equiv \\left.\\frac{\\delta ^{n}\\Gamma _{k}^{(\\mathrm {1PI})}[\\psi ]}{\\delta \\psi _{\\alpha _{1}}\\cdots \\delta \\psi _{\\alpha _{n}}}\\right|_{\\psi =\\overline{\\psi }_{k}}$ and $\\overline{\\psi }_{k}$ must extremize the flowing EA, i.e.",
": $\\left.\\frac{\\delta \\Gamma ^{(\\mathrm {1PI})}_{k}[\\psi ]}{\\delta \\psi _{\\alpha }}\\right|_{\\psi =\\overline{\\psi }_{k}} = 0 \\quad \\forall \\alpha , k\\;.$ The final step of the vertex expansion amounts to identifying the terms of identical powers in the field in the expanded Wetterich equation to infer an infinite tower or hierarchy of coupled integro-differential equations for the 1PI vertices $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(n)}$ .",
"This tower is of course always truncated in practice: in current treatments of fermionic systems for instance [116], [117], [118], [119], [120], [121], one typically uses the so-called level-2 truncation in which ($\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k}$ ,) $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(2)}$ and $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(4)}$ are the only flowing vertices.",
"Although this approach based on the vertex expansion can be seen as the current canonical 1PI-FRG implementation to tackle fermionic systems, it is not suited to describe strongly-coupled fermions since the expansion (REF ) can only be performed around the trivial configuration $\\overline{\\psi }_{k,\\alpha }=0$ $\\forall \\alpha $ if $\\psi $ is a Grassmann field.",
"This implies that, in this situation, the FRG reduces basically to a LE and is therefore perturbative in essence.",
"One can however directly remedy this by e.g.",
"bosonizing the system with a HST and then perform the vertex expansion around a non-trivial configuration for the Hubbard-Stratonovich field, the resulting approach being henceforth non-perturbative.",
"This highlights the importance of our 1PI-FRG investigations within the mixed and collective representations of the (0+0)-D $O(N)$ model: indeed, even if the latter model is originally purely bosonic, we will often discuss the relevance of the tested methods for (realistic) fermionic problems and analyze the possibility to extend the conclusions of our comparative study towards such problems.",
"Note also that there are other very performing expansion schemes of the Wetterich equation for bosonic systems, and especially to treat $O(N)$ models.",
"In that respect, one can quote in particular the derivative expansion (DE) [122], [123], [124], [125], [126], [127] and the Blaizot-Méndez-Galain-Wschebor (BMW) approximationSee Refs.",
"[128], [112] for pedagogical presentations of the DE and the BMW approximation, and Ref.",
"[112] for e.g.",
"a comparison of the performances of those various expansion schemes in the determination of critical exponents.",
"[129], [130], [131], [132], [133], [134], [135], [136], [137].",
"However, it should be noted that, in (0+0)-D, the DE amounts to rewriting exactly the Wetterich equation, i.e.",
"the DE is no longer an approximation, as discussed in Ref. [106].",
"Furthermore, as the main motivation for using the BMW approximation is to better grasp the momentum dependence of 1PI vertices (which is absent in (0+0)-D), we will neither investigate the DE nor the BMW approximation in the present toy model study and content ourselves with the vertex expansion.",
"Finally, we want to address a key aspect of FRG approaches, i.e.",
"the starting point of the flow.",
"These approaches indeed all rely on coupled differential equations that require initial conditions to be solved numerically.",
"To discuss this, it is crucial to understand that the original formulation of the 1PI-FRG based on the Wetterich equation implements the Wilsonian momentum-shell integration, which allows for an efficient description of critical phenomena [49], [138], [139], [140], [141], [142], [143], [144], [112], and the flow parameter $k$ introduced via $R_{k}$ in Eq.",
"(REF ) thus corresponds to a momentum scale.",
"More precisely, the differential equations underlying this FRG approach are solved by evolving the flow parameter $k$ from a chosen ultraviolet cutoff $k_{\\mathrm {i}}=\\Lambda $ down to $k_{\\mathrm {f}}=0$ .",
"Although we will not discuss in detail the different possible relevant choices for the analytical expressions of $R_{k}$ in finite dimensionsWidespread choices for $R_{k}$ are the exponential regulator or the Litim regulator [145], [146].",
"See Refs.",
"[147], [110], [112] for more exhaustive discussions on this topic., we will just point out that this cutoff function must satisfy in the present case: [left=]align Rk=ki,12 = 1,2.",
"Rk=kf,12 = 0 1,2.",
"Condition (REF ) directly follows from the fact that we must recover the theory under consideration, specified by Eq.",
"(REF ), at the end of the flow.",
"Furthermore, it can also be shown that the divergence of the cutoff function imposed by Eq.",
"(REF ) at $k=k_{\\mathrm {i}}$ imposes that the starting point of the flow coincides with the classical theory, as a consequence of the modification of the Legendre transform in Eq.",
"(REF ).",
"This translates into: [left=]align (1PI)k=ki[] = S[=].",
"(1PI)k=kf[] = (1PI)[].",
"Therefore, the 1PI-FRG procedure originally formulated by Wetterich starts from the classical theory, incorporates progressively quantum correlations on top of it throughout the flow by solving integro-differential equations, so as to reach the corresponding quantum theory.",
"It should be noted however that several drawbacks of this 1PI-FRG formulation, based on both the Wetterich equation and on the Wilsonian momentum-shell integration, have been identified along the years [148], such as violations of Ward identities at finite values of $R_{k}$ [149], [150].",
"Hence, other flow schemes, still treating the Wetterich but using different initial conditions and other flow parameters (such as the temperature), became popular notably in the condensed matter theory community: we can mention for example the temperature flow [151], the interaction flow [152] and the $\\Omega $ -flow [153], [154] in that respect.",
"Once again, the framework chosen for the present study is certainly not relevant to compare all these flow schemesThe $\\Omega $ - and temperature flows have already been compared in the framework of a (2+1)-D Hubbard model [155]., owing to the vanishing of momentum and frequency dependences in (0+0)-D notably.",
"Moreover, the aforementioned drawbacks do not apply to the (0+0)-D model studied here either and we will therefore only consider the FRG approach described so far, which uses the classical theory as starting point, in our 1PI-FRG analysis.",
"Regardless of the chosen flow scheme, one only reaches an approximated version of the quantum theory specified by $\\Gamma ^{(\\mathrm {1PI})}[\\psi ]$ at the end of the flow in practice, i.e.",
"equality (REF ) is only approximatively fulfilled in actual calculations owing to the truncation of the infinite hierarchy of equations resulting from the treatment of the Wetterich equation.",
"One might actually say that this is where lies the power of this approach since such a truncation implies that approximations within the FRG scheme do not rely on the smallness of coupling constants and can therefore provide us with the possibility to design various non-perturbative approaches.",
"This will be thoroughly illustrated in the following sections.",
"We can however already appreciate such a potential for our aim of reformulating the EDF method, considering the strongly-coupled nature of the nuclear many-body problem."
],
[
"Original 1PI functional renormalization group",
"We now turn to our 1PI-FRG applications.",
"As opposed to the other sections dedicated to the 2PI- and 2PPI-FRGs, we mainly consider the formalism in (0+0)-D for our 1PI-FRG analysis, notably since the corresponding developments at finite dimensions are already exhaustively discussed in the available literature [113], [114], [110], [111], [112].",
"We thus consider the (0+0)-D limit from now on until the end of section .",
"In this framework, the Wetterich equation for a 1PI EA $\\Gamma _{k}^{(\\mathrm {1PI})}\\big (\\vec{\\phi }\\big )$ reads: $\\begin{split}& \\dot{\\Gamma }^{(\\mathrm {1PI})}_{k}\\Big (\\vec{\\phi }\\Big ) = \\frac{1}{2}\\mathrm {STr}\\left[\\dot{R}_{k}\\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{k}\\Big (\\vec{\\phi }\\Big )+R_{k}\\right)^{-1}\\right] \\\\& = \\frac{1}{2}\\sum _{a_{1},a_{2}=1}^{N} \\dot{R}_{k,a_{1}a_{2}} \\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{k}\\Big (\\vec{\\phi }\\Big )+R_{k}\\right)_{a_{2}a_{1}}^{-1} \\;,\\end{split}$ with $\\Gamma ^{(\\mathrm {1PI})(2)}_{k,a_{1}a_{2}}\\Big (\\vec{\\phi }\\Big ) \\equiv \\frac{\\partial ^{2}\\Gamma ^{(\\mathrm {1PI})}_{k}\\big (\\vec{\\phi }\\big )}{\\partial \\phi _{a_{1}}\\partial \\phi _{a_{2}}} \\;.$ As can be deduced from the general formalism presented in section REF , the vertex expansion procedure applied to Eq.",
"(REF ) starts from the Taylor expansion of $\\Gamma ^{(\\mathrm {1PI})}_{k}\\big (\\vec{\\phi }\\big )$ around its flowing extremum at $\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k}$ , which reduces in (0+0)-D to: $\\begin{split}\\Gamma ^{(\\mathrm {1PI})}_{k}\\Big (\\vec{\\phi }\\Big ) = & \\ \\overline{\\Gamma }^{(\\mathrm {1PI})}_{k} \\\\& + \\sum _{n=2}^{\\infty } \\frac{1}{n!}",
"\\sum _{a_{1},\\cdots ,a_{n}=1}^{N} \\overline{\\Gamma }_{k,a_{1} \\cdots a_{n}}^{(\\mathrm {1PI})(n)} \\\\& \\times \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{1}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{n}} \\;.\\end{split}$ We have used in Eq.",
"(REF ) the definitions: $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k} \\equiv \\Gamma ^{(\\mathrm {1PI})}_{k}\\Big (\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k}\\Big ) \\quad \\forall k \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k,a_{1} \\cdots a_{n}} \\equiv \\left.",
"\\frac{\\partial ^{n} \\Gamma ^{(\\mathrm {1PI})}_{k}\\big (\\vec{\\phi }\\big )}{\\partial \\phi _{a_{1}} \\cdots \\partial \\phi _{a_{n}}} \\right|_{\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k}} \\quad \\forall a_{1},\\cdots ,a_{n},k \\;,$ and $\\overline{\\Gamma }_{k,a}^{(\\mathrm {1PI})(1)} = 0 \\quad \\forall a, k \\;,$ by construction.",
"We will then distinguish two situations to pursue the vertex expansion further.",
"On the one hand, in the unbroken-symmetry regime (i.e.",
"in the phase with $m^{2}>0$ ) where $\\vec{\\overline{\\phi }}_{k}=\\vec{0}$ $\\forall k$ , the infinite set of differential equations resulting from the vertex expansion includesWe have performed the replacement $\\overline{G}_{k}\\rightarrow \\overline{G}_{k}-\\overline{G}^{(0)}_{k}$ in the output of the vertex expansion procedure so as to obtain Eq.",
"(REF ) (the same remark applies to Eq.",
"(REF ) given afterwards).",
"Physical observables are not affected by this shift and the necessity of it can be seen by the fact that no quantum corrections must be added to $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k}$ throughout the flow if $\\lambda =0$ , i.e.",
"the relation $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k=k_{\\mathrm {f}}}=\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k=k_{\\mathrm {i}}}$ must hold in the free case, which is indeed satisfied thanks to this operation.",
": $\\dot{\\overline{\\Gamma }}_{k} = \\frac{N}{2} \\dot{R}_{k} \\left( \\overline{G}_{k} - \\overline{G}^{(0)}_{k} \\right) \\;,$ $\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2)}_{k} = -\\frac{N+2}{6} \\dot{R}_{k} \\overline{G}^{2}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\;,$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(4)}_{k} = & \\ \\frac{N+8}{3}\\dot{R}_{k} \\overline{G}^{3}_{k}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\right)^2 \\\\& - \\frac{N+4}{10}\\dot{R}_{k} \\overline{G}^{2}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(6)}_{k}\\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(6)}_{k} = & - \\frac{5N+130}{3} \\dot{R}_{k} \\overline{G}^{4}_{k} \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\right)^{3} \\\\& + \\left(N+14\\right) \\dot{R}_{k} \\overline{G}^{3}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(6)}_{k} \\\\& - \\frac{N+6}{14}\\dot{R}_{k} \\overline{G}^{2}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(8)}_{k}\\;,\\end{split}$ where, as shown later by Eqs.",
"(REF ) and (REF ), $\\overline{G}_{k}$ and $\\overline{G}^{(0)}_{k}$ are respectively the diagonal parts of the propagators $\\overline{G}_{k}$ and $\\overline{G}^{(0)}_{k}$ defined by: $\\overline{G}^{ \\ -1}_{k,a_{1}a_{2}} \\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{k,a_{1}a_{2}} + R_{k,a_{1}a_{2}} \\;,$ $\\left(\\overline{G}^{(0)}_{k}\\right)_{a_{1}a_{2}}^{-1} \\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{k=k_{\\mathrm {i}},a_{1}a_{2}} + R_{k,a_{1}a_{2}} \\;,$ with $R_{k,a_{1}a_{2}} = R_{k} \\ \\delta _{a_{1}a_{2}} \\;.$ In Eqs.",
"(REF ) to (REF ), we have also used the following relations resulting from the $O(N)$ symmetry: $\\overline{\\Gamma }_{k,a_{1}a_{2}}^{(\\mathrm {1PI})(2)}=\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(2)} \\ \\delta _{a_{1}a_{2}} \\quad \\forall a_{1},a_{2}\\;,$ $\\begin{split}\\overline{\\Gamma }_{k,a_{1}a_{2}a_{3}a_{4}}^{(\\mathrm {1PI})(4)}= & \\ \\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(4)}\\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}+\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}} \\right.",
"\\\\& \\left.",
"+\\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\right) \\quad \\forall a_{1},a_{2},a_{3},a_{4}\\;,\\end{split}$ $\\begin{split}\\overline{\\Gamma }_{k,a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}}^{(\\mathrm {1PI})(6)} = & \\ \\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(6)} \\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}\\delta _{a_{5}a_{6}}\\right.",
"\\\\& +\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{5}}\\delta _{a_{4}a_{6}}+\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{6}}\\delta _{a_{4}a_{5}} \\\\& +\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}}\\delta _{a_{5}a_{6}} + \\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{5}}\\delta _{a_{4}a_{6}} \\\\& + \\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{6}}\\delta _{a_{4}a_{5}} + \\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\delta _{a_{5}a_{6}} \\\\& + \\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{5}}\\delta _{a_{3}a_{6}} + \\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{6}}\\delta _{a_{3}a_{5}} \\\\& + \\delta _{a_{1}a_{5}}\\delta _{a_{2}a_{3}}\\delta _{a_{4}a_{6}} + \\delta _{a_{1}a_{5}}\\delta _{a_{2}a_{4}}\\delta _{a_{3}a_{6}} \\\\& + \\delta _{a_{1}a_{5}}\\delta _{a_{2}a_{6}}\\delta _{a_{3}a_{4}} + \\delta _{a_{1}a_{6}}\\delta _{a_{2}a_{3}}\\delta _{a_{4}a_{5}} \\\\& \\left.",
"+ \\delta _{a_{1}a_{6}}\\delta _{a_{2}a_{4}}\\delta _{a_{3}a_{5}} + \\delta _{a_{1}a_{6}}\\delta _{a_{2}a_{5}}\\delta _{a_{3}a_{4}} \\right) \\\\& \\quad \\forall a_{1},\\cdots ,a_{6}\\;,\\end{split}$ $\\overline{\\Gamma }_{k,a_{1}\\cdots a_{n}}^{(\\mathrm {1PI})(n)} = 0 \\quad \\forall a_{1},\\cdots ,a_{n},~\\forall n ~ \\mathrm {odd}\\;.$ As a consequence of Eq.",
"(REF ), the propagators $\\overline{G}_{k}$ and $\\overline{G}^{(0)}_{k}$ satisfy: $\\overline{G}_{k,a_{1}a_{2}} = \\overline{G}_{k} \\ \\delta _{a_{1}a_{2}} = \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{k} + R_{k}\\right)^{-1} \\delta _{a_{1}a_{2}} \\;,$ $\\overline{G}^{(0)}_{k,a_{1}a_{2}} = \\overline{G}^{(0)}_{k} \\ \\delta _{a_{1}a_{2}} = \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{k=k_{\\mathrm {i}}} + R_{k}\\right)^{-1} \\delta _{a_{1}a_{2}} \\;.$ Note that the differential equations (REF ) to (REF ) are already given in Ref.",
"[106] which indeed already treats the unbroken-symmetry phase of the (0+0)-D $O(N)$ model in its original representation with the 1PI-FRG considered here, i.e.",
"that based on the vertex expansion of the Wetterich equation.",
"This 1PI-FRG implementation is however not exploited to tackle the broken-symmetry phase of our toy model in the study of Ref.",
"[106]Although Ref.",
"[106] discusses the performances of several approximated PI techniques (perturbation theory, $1/N$ -expansion, 1PI-FRG based on the vertex expansion of the Wetterich equation) only in the framework of the unbroken-symmetry phase of the (0+0)-D $O(N)$ model, it also reports a calculation of the effective potential by an exact resolution of the Wetterich equation in the broken-symmetry phase.",
"It shows in this way that the 1PI-FRG flow restores the $O(N)$ symmetry broken by its classical starting point, which is consistent with Fig.",
"1 of paper I together with the Mermin-Wagner theorem..",
"In this phase (i.e.",
"in the phase with $m^{2}<0$ ), the infinite tower resulting from the vertex expansion contains the following differential equations at $N=1$ : $\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{k} = \\frac{1}{2} \\dot{R}_{k} \\left( \\overline{G}_{k} - \\overline{G}^{(0)}_{k} \\right)\\;,$ $\\dot{\\overline{\\phi }}_{k}=\\frac{1}{2\\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{k}}\\dot{R}_{k}\\overline{G}_{k}^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k} \\;,$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2)}_{k} = & \\ \\dot{\\overline{\\phi }}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k} + \\dot{R}_{k}\\overline{G}_{k}^{3}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k}\\right)^{2} \\\\& -\\frac{1}{2}\\dot{R}_{k}\\overline{G}_{k}^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(3)}_{k} = & \\ \\dot{\\overline{\\phi }}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}-3\\dot{R}_{k}\\overline{G}_{k}^{4}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k}\\right)^{3} \\\\& + 3\\dot{R}_{k}\\overline{G}_{k}^{3}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k} \\\\& -\\frac{1}{2}\\dot{R}_{k}\\overline{G}_{k}^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(5)}_{k}\\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(4)}_{k} = & \\ \\dot{\\overline{\\phi }}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(5)}_{k}+12\\dot{R}_{k}\\overline{G}_{k}^{5}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k}\\right)^{4} \\\\& -18\\dot{R}_{k}\\overline{G}_{k}^{4}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k}\\right)^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k} \\\\& + 4\\dot{R}_{k}\\overline{G}^{3}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(5)}_{k} \\\\& + 3\\dot{R}_{k}\\overline{G}^{3}_{k}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\right)^{2}-\\frac{1}{2}\\dot{R}_{k}\\overline{G}^{2}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(6)}_{k}\\;,\\end{split}$ with $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k,1 \\cdots 1} \\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k}$ .",
"Moreover, the propagators $\\overline{G}_{k}\\equiv \\overline{G}_{k,11}$ and $\\overline{G}^{(0)}_{k}\\equiv \\overline{G}^{(0)}_{k,11}$ are still given by Eqs.",
"(REF ) and (REF ), respectively.",
"At the present stage, we still have not specified the model under consideration: all we know is that we are dealing with a 1PI EA depending on a single field $\\vec{\\phi }$ which is a vector in color space and lives in a (0+0)-D spacetime.",
"In the framework of FRG approaches, the model (i.e.",
"the classical action under consideration) is often only specified via the initial conditions used to solve the underlying set of differential equations.",
"The initial conditions used to solve the above two sets of differential equations for our (0+0)-D $O(N)$ model can be obtained by assuming that the $O(N)$ symmetry can only be spontaneously broken in the direction set by $a=N$ in color space (i.e.",
"by assuming that $\\vec{\\phi }^2=\\phi _{N}^2$ ), thus following the convention already used in paper I.",
"In this case, they are given byThe logarithm term in Eq.",
"(REF ) was added to shift the calculated gs energy $E_{\\mathrm {gs}}$ so that the latter coincides with the corresponding exact solution given in paper I for $\\lambda =0$ and $m^{2}>0$ .",
": $\\overline{\\phi }_{k=k_{\\mathrm {i}},a} = \\overline{\\varphi }_{\\mathrm {cl},a} = \\left\\lbrace \\begin{array}{lll}\\displaystyle {0 \\quad \\forall a, ~ \\forall m^2 > 0\\;,} \\\\\\\\\\displaystyle {\\pm \\sqrt{-\\frac{6 m^2}{\\lambda }} \\ \\delta _{a N} \\quad \\forall a, ~ \\forall m^2 < 0 ~ \\mathrm {and} ~ \\lambda \\ne 0\\;,}\\end{array}\\right.$ $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k=k_{\\mathrm {i}}} = \\left\\lbrace \\begin{array}{lll}\\displaystyle {- \\frac{N}{2}\\ln \\bigg (\\frac{2\\pi }{m^{2}}\\bigg ) + S\\Big (\\vec{\\widetilde{\\varphi }}=\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}\\Big ) = - \\frac{N}{2}\\ln \\bigg (\\frac{2\\pi }{m^{2}}\\bigg ) \\quad \\forall m^2 > 0\\;,} \\\\\\\\\\displaystyle {S\\Big (\\vec{\\widetilde{\\varphi }}=\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}\\Big ) = \\frac{m^{2}}{2} \\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}^{2} + \\frac{\\lambda }{4!",
"}\\left(\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}^{2}\\right)^{2} \\quad \\forall m^2 < 0\\;,}\\end{array}\\right.$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k=k_{\\mathrm {i}},a_{1} \\cdots a_{n}} = \\left.\\frac{\\partial ^{n}S\\big (\\vec{\\widetilde{\\varphi }}\\big )}{\\partial \\widetilde{\\varphi }_{a_{1}}\\cdots \\partial \\widetilde{\\varphi }_{a_{n}}}\\right|_{\\vec{\\widetilde{\\varphi }}=\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}} \\quad \\forall a_{1}, \\cdots , a_{n} \\;.$ From Eq.",
"(REF ), we then directly deduce the initial conditions for the symmetric part $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(n)}$ of the 1PI vertices of even order $n$ introduced via Eqs.",
"(REF ) to (REF ) for $m^2>0$ : $\\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{k=k_{\\mathrm {i}}} = m^2 \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k=k_{\\mathrm {i}}} = \\lambda \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k=k_{\\mathrm {i}}} = 0 \\quad \\forall n \\ge 6 \\;.$ The truncation of the infinite tower of differential equations containing either Eqs.",
"(REF ) to (REF ) (for all $N$ and $m^{2}>0$ ) or Eqs.",
"(REF ) to (REF ) (for $N=1$ and $m^{2}<0$ ) is implemented by the condition: $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k} = \\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k=k_{\\mathrm {i}}} \\quad \\forall k,~ \\forall n > N_{\\mathrm {max}} \\;,$ where $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(n)}$ corresponds to: i) the symmetric part of the 1PI vertices of (even) order $n$ (as defined via Eqs.",
"(REF ) to (REF ) up to $n=6$ ) for all $N$ and $m^{2}>0$ ; ii) the 1PI vertices themselves for $N=1$ and $m^{2}<0$ according to the definition $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(n)}\\equiv \\overline{\\Gamma }_{k,1 \\cdots 1}^{(\\mathrm {1PI})(n)}$ .",
"We finally infer the gs energy from $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{k}$ at the end of the flow using the equality: $E^\\text{1PI-FRG;orig}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {1PI})}_{k=k_{\\mathrm {f}}} \\;.$ Furthermore, the gs density $\\rho _{\\mathrm {gs}}$ is determined at $m^{2}>0$ from the relation: $\\begin{split}\\rho ^\\text{1PI-FRG;orig}_{\\mathrm {gs}} = & \\ \\frac{1}{N} \\sum _{a=1}^{N} \\left.",
"\\frac{\\partial ^{2} W_{k=k_{\\mathrm {f}}}\\big (\\vec{J}\\big )}{\\partial J_{a}^{2}} \\right|_{\\vec{J}=\\vec{0}} \\\\= & \\ \\frac{1}{N} \\sum _{a=1}^{N} \\left(\\overline{\\Gamma }_{k=k_{\\mathrm {f}}}^{(\\mathrm {1PI})(2)}\\right)_{aa}^{-1} \\\\= & \\ \\left( \\overline{\\Gamma }_{k=k_{\\mathrm {f}}}^{(\\mathrm {1PI})(2)} \\right)^{-1} \\;,\\end{split}$ which results from Eq.",
"(REF ).",
"We will actually neither calculate $E_{\\mathrm {gs}}$ nor $\\rho _{\\mathrm {gs}}$ in the regime with $m^{2}<0$ , as explained below.",
"Finally, the chosen cutoff function for both $m^2<0$ and $m^2>0$ is: $R_{k,a_{1}a_{2}} = R_{k} \\ \\delta _{a_{1}a_{2}} = \\left(k^{-1} - 1\\right) \\delta _{a_{1}a_{2}} \\quad \\forall a_{1}, a_{2}\\;,$ which indeed satisfies the boundary conditions (REF ) (i.e.",
"we have indeed $R_{k=k_{\\mathrm {i}}} = \\infty $ and $R_{k=k_{\\mathrm {f}}} = 0$ ) as the flow parameter $k$ runs from $k_{\\mathrm {i}}=0$ to $k_{\\mathrm {f}}=1$ in the present case.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} (upper panel) or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} (lower panel) and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Regarding the regime with $m^{2}>0$ , we solve the differential equations (REF ) to (REF ) up to $N_{\\mathrm {max}}=6$ , with the initial conditions (REF ) to (REF ) and the cutoff function (REF ).",
"Due to the symmetry constraint (REF ), this enables us to determine the first two non-trivial orders of the 1PI-FRG, the first one being set by the truncation $N_{\\mathrm {max}}=4$ and the second one by $N_{\\mathrm {max}}=6$ .",
"The results thus obtained are displayed in Fig.",
"REF for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ with $N=2$ .",
"While the first non-trivial order results all lie within a few percents away from the corresponding exact solution over the whole range of tested values for the coupling constant (i.e.",
"$\\lambda /4!",
"\\in [0,10]$ as for most figures of paper I), the second one's are hardly distinguishable from their exact solutions.",
"The performances of this FRG approach are also barely affected as the coupling constant $\\lambda /4!$ evolves, hence the non-perturbative character of this approach.",
"However, the resolution of the system given by Eqs.",
"(REF ) to (REF ) with $m^{2}<0$ , using the initial conditions (REF ) to (REF ) and the cutoff function (REF ), is prevented, the system being too stiff, at least for the numerical tools used in this studyIn the present study, all sets of differential equations underpinning FRG approaches are solved numerically using the $\\mathtt {NDSolve}$ function of $\\mathtt {Mathematica~12.1}$ ..",
"Note that we have checked that this problem still occurs at $N=2$ and for $N_{\\mathrm {max}}$ up to 6 [46].",
"Hence, we only display original 1PI-FRG results for the unbroken-symmetry phase.",
"In conclusion, we can directly see from Fig.",
"REF that, at first and especially at second non-trivial orders (i.e.",
"at $N_{\\mathrm {max}}=4$ and 6), the 1PI-FRG based on a vertex expansion of the Wetterich equation does manage to describe reliably the strongly-coupled regime of our bosonic toy model in its original representation.",
"We have however discussed earlier that fermionic systems, and therefore nuclear systems in principleOne should stress however that nuclear Lagrangians are not necessarily based on fermionic dofs: one can for instance simply exploit $\\alpha $ -particles as dofs for the description of $^{16}O$ .",
"The choice of relevant dofs in nuclear EFTs is still a very active debate [156]., can not be treated at strong couplings via this 1PI-FRG implementation.",
"We have also pointed out that the use of HSTs within the fermionic 1PI-FRG framework can drastically change the situation and give us access to non-perturbative physics, as a result of vertex expansions around non-trivial minima for Hubbard-Stratonovich fields.",
"With this in mind, we thus address in what follows the mixed and collective representations of the (0+0)-D $O(N)$ model.",
"We will notably check if the stiffness of the equation systems to solve within the 1PI-FRG is sufficiently reduced in these situations to tackle the regime with $m^{2}<0$ ."
],
[
"Mixed 1PI functional renormalization group",
"We have stressed before that the 1PI-FRG implementation based on a vertex expansion of the Wetterich equation has already been applied to the (0+0)-D $O(N)$ model in its unbroken-symmetry phase and in its original representation.",
"This is to our knowledge not the case for the mixed or the collective representation, regardless of the values of $m^2$ and $\\lambda $ .",
"Let us thus now consider the 1PI-FRG in the mixed representation, i.e.",
"the mixed 1PI-FRG.",
"The Wetterich equation for our zero-dimensional $O(N)$ model in this representation reads: $\\begin{split}& \\dot{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big )+\\mathcal {R}_{k}\\right)^{-1}\\right] \\\\& = \\frac{1}{2}\\sum _{b_{1},b_{2}=1}^{N+1} \\dot{\\mathcal {R}}_{k,b_{1}b_{2}}\\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big )+\\mathcal {R}_{k}\\right)_{b_{2}b_{1}}^{-1} \\;,\\end{split}$ where the Hessian of the flowing EA is now given by: $\\begin{split}\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = \\ & \\begin{pmatrix}\\frac{\\partial ^{2}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}}{\\partial \\vec{\\phi }\\partial \\vec{\\phi }} & \\frac{\\partial ^{2}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}}{\\partial \\vec{\\phi }\\partial \\eta } \\\\\\frac{\\partial ^{2}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}}{\\partial \\eta \\partial \\vec{\\phi }} & \\frac{\\partial ^{2}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}}{\\partial \\eta \\partial \\eta }\\end{pmatrix} \\\\\\equiv \\ & \\begin{pmatrix}\\Gamma ^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k} & \\Gamma ^{(\\mathrm {1PI})(1\\phi ,1\\eta )}_{\\mathrm {mix},k} \\\\\\Gamma ^{(\\mathrm {1PI})(1\\phi ,1\\eta )}_{\\mathrm {mix},k} & \\Gamma ^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k}\\end{pmatrix} \\;,\\end{split}$ and the cutoff function $\\mathcal {R}_{k}$ exhibits the following matrix structure in extended color space: $\\mathcal {R}_{k} = \\begin{pmatrix}R^{(\\phi )}_{k} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & R^{(\\eta )}_{k}\\end{pmatrix} = \\begin{pmatrix}R_{k} \\mathbb {I}_{N} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & R_{k}\\end{pmatrix} = R_{k} \\mathbb {I}_{N+1} \\;,$ with $\\mathbb {I}_{D}$ the $D$ -dimensional identity matrix.",
"We then apply the vertex expansion procedure to the Wetterich equation in the form (REF ), starting from the Taylor series: $\\begin{split}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = & \\ \\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k} \\\\& + \\sum _{n=2}^{\\infty } \\frac{1}{n!}",
"\\sum _{m=0}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\\\& \\times \\sum _{a_{1},\\cdots ,a_{m}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m)\\eta )} \\\\& \\times \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{1}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{m}} \\\\& \\times \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ with $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k} \\equiv \\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k},\\eta =\\overline{\\eta }_{k}\\Big ) \\quad \\forall k \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k,a_{1} \\cdots a_{n}} \\equiv \\left.\\frac{\\partial ^{n+m}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\big (\\vec{\\phi },\\eta \\big )}{\\partial \\phi _{a_{1}}\\cdots \\partial \\phi _{a_{n}}\\partial \\eta ^{m}}\\right|_{\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k} \\atop \\eta =\\overline{\\eta }_{k}} \\quad \\forall a_{1},\\cdots ,a_{n}, k \\;,$ andThe relations $\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi )}_{\\mathrm {mix},k}\\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,0\\eta )}_{\\mathrm {mix},k}$ and $\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\eta )}_{\\mathrm {mix},k}\\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(0\\phi ,n\\eta )}_{\\mathrm {mix},k}$ are assumed for all $n$ in Eq.",
"(REF ) as well as in subsequent equations.",
"$\\overline{\\Gamma }^{(\\mathrm {1PI})(1\\phi )}_{\\mathrm {mix},k,a} = \\overline{\\Gamma }^{(\\mathrm {1PI})(1\\eta )}_{\\mathrm {mix},k}=0 \\quad \\forall a, k \\;,$ since the flowing EA is now extremal at $\\begin{pmatrix} \\vec{\\phi } & \\eta \\end{pmatrix} = \\begin{pmatrix} \\vec{\\overline{\\phi }}_{k} & \\overline{\\eta }_{k} \\end{pmatrix}$ .",
"Furthermore, as this EA now depends on several fields, it will be convenient to use the following definition: $\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {mix},k}+\\mathcal {R}_{k}=\\mathcal {P}_{k}+\\mathcal {F}_{k}\\;,$ where the fluctuation matrix $\\mathcal {F}_{k}$ contains all the field dependence, i.e.",
": $\\mathcal {F}_{k} = \\begin{pmatrix}\\Delta \\Gamma _{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\phi )} & \\Delta \\Gamma _{\\mathrm {mix},k}^{(\\mathrm {1PI})(1\\phi ,1\\eta )} \\\\\\Delta \\Gamma _{\\mathrm {mix},k}^{(\\mathrm {1PI})(1\\phi ,1\\eta )} & \\Delta \\Gamma _{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\eta )}\\end{pmatrix}\\;,$ with $\\Delta \\Gamma _{\\mathrm {mix},k}^{(\\mathrm {1PI})\\mathrm {X}}=\\Gamma ^{(\\mathrm {1PI})\\mathrm {X}}_{\\mathrm {mix},k}-\\overline{\\Gamma }^{(\\mathrm {1PI})\\mathrm {X}}_{\\mathrm {mix},k} \\;,$ which imposes that $\\mathcal {P}_{k}$ satisfies: $\\mathcal {P}_{k} = \\begin{pmatrix}R^{(\\phi )}_{k}+\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k} & \\overline{\\Gamma }^{(\\mathrm {1PI})(1\\phi ,1\\eta )}_{\\mathrm {mix},k} \\\\\\overline{\\Gamma }^{(\\mathrm {1PI})(1\\phi ,1\\eta )}_{\\mathrm {mix},k} & R^{(\\eta )}_{k}+\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k}\\end{pmatrix}\\;.$ For $m^2>0$ , we will use the counterparts of Eqs.",
"(REF ) to (REF ) introducing the symmetric parts of 1PI vertices: $\\overline{\\Gamma }_{\\mathrm {mix},k,a_{1}a_{2}}^{(\\mathrm {1PI})(2\\phi ,n\\eta )}=\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\phi ,n\\eta )} \\ \\delta _{a_{1}a_{2}} \\quad \\forall a_{1},a_{2},n\\;,$ $\\begin{split}\\overline{\\Gamma }_{\\mathrm {mix},k,a_{1}a_{2}a_{3}a_{4}}^{(\\mathrm {1PI})(4\\phi ,n\\eta )}= & \\ \\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(4\\phi ,n\\eta )}\\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}+\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}} \\right.",
"\\\\& \\left.",
"+\\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\right) \\quad \\forall a_{1},a_{2},a_{3},a_{4},n\\;,\\end{split}$ $\\overline{\\Gamma }_{\\mathrm {mix},k,a_{1}\\cdots a_{n}}^{(\\mathrm {1PI})(n\\phi ,m\\eta )} = 0 \\quad \\forall a_{1},\\cdots ,a_{n},m, ~ \\forall n ~ \\mathrm {odd}\\;.$ The next step of the vertex expansion consists in carrying out matrix products between $\\mathcal {P}_{k}^{-1}$ and $\\mathcal {F}_{k}$ and the rest of the recipe towards the set of coupled differential equations to solve is detailed in appendix REF .",
"Although we only give our derived expressions for these flow equations in the latter appendix, we would like to point out here, for the needs of our discussion, that the underlying derivations require to define the following (inverse) propagators: $\\begin{split}& \\left(\\overline{G}^{(\\phi )}_{k}\\right)^{-1}_{a_{1}a_{2}} = \\overline{\\Gamma }^{(2\\phi )}_{\\mathrm {mix},k,a_{1}a_{2}} + R^{(\\phi )}_{k,a_{1}a_{2}} \\\\& = \\left(\\overline{G}^{(\\phi )}_{k}\\right)^{-1} \\delta _{a_{1}a_{2}} = \\left( \\overline{\\Gamma }^{(2\\phi )}_{\\mathrm {mix},k} + R_{k} \\right) \\delta _{a_{1}a_{2}} \\quad \\forall a_{1}, a_{2} \\;,\\end{split}$ $\\left(\\overline{G}^{(\\eta )}_{k}\\right)^{-1} = \\overline{\\Gamma }^{(2\\eta )}_{\\mathrm {mix},k} + R_{k} \\;,$ which are the mixed counterparts of Eq.",
"(REF ) or (REF ) and the last line of Eq.",
"(REF ) is only valid in the unbroken-symmetry phase.",
"Hence, as a first level of approximation which is basically mean-field theory (MFT), we can set all bosonic entries of $\\mathcal {P}_{k}^{-1}$ equal to zero [157].",
"For the unbroken-symmetry regime of the toy model under consideration, this amounts to neglecting the bottom-right component of $\\mathcal {P}_{k}^{-1}$ , i.e.",
"this amounts to setting $\\mathcal {P}_{k,N+1 \\hspace{1.13791pt} N+1}^{-1}=\\overline{G}^{(\\eta )}_{k}=0$ .",
"The sets of differential equations to solve in the framework of MFT can therefore be directly inferred from those resulting from the standard vertex expansion procedure and given in appendix REF .",
"Whether we restrict ourselves to MFT or not, the initial conditions used to solve the differential equations within the mixed 1PI-FRG are directly inferred for all $N$ from the classical action $S_{\\mathrm {mix}}$ expressed in Eq.",
"(REF ), which gives usThe remark of footnote REF also applies to the logarithm term in Eq.",
"(REF ).",
": $\\begin{pmatrix}\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}} \\\\\\overline{\\eta }_{k=k_{\\mathrm {i}}}\\end{pmatrix} = \\begin{pmatrix}\\vec{\\overline{\\varphi }}_{\\mathrm {cl}} \\\\\\overline{\\sigma }_{\\mathrm {cl}}\\end{pmatrix} = {\\left\\lbrace \\begin{array}{ll} \\displaystyle {\\begin{pmatrix}\\vec{0} \\\\0\\end{pmatrix} \\quad \\forall m^2 > 0\\;,} \\\\\\\\\\displaystyle {\\begin{pmatrix}0 \\\\\\vdots \\\\0 \\\\\\pm \\sqrt{\\frac{-6m^2}{\\lambda }} \\\\im^2\\sqrt{\\frac{3}{\\lambda }}\\end{pmatrix} \\quad \\forall m^2< 0 ~ \\text{and} ~ \\lambda \\ne 0 \\;,} \\end{array}\\right.}",
"\\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k=k_{\\mathrm {i}}} = \\left\\lbrace \\begin{array}{lll}\\displaystyle {- \\frac{N}{2}\\ln \\bigg (\\frac{2\\pi }{m^{2}}\\bigg ) + S_{\\mathrm {mix}}\\Big (\\vec{\\widetilde{\\varphi }}=\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}},\\widetilde{\\sigma }=\\overline{\\eta }_{k=k_{\\mathrm {i}}}\\Big ) = - \\frac{N}{2}\\ln \\bigg (\\frac{2\\pi }{m^{2}}\\bigg ) \\quad \\forall m^2 > 0\\;,} \\\\\\\\\\displaystyle {S_{\\mathrm {mix}}\\Big (\\vec{\\widetilde{\\varphi }}=\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}},\\widetilde{\\sigma }=\\overline{\\eta }_{k=k_{\\mathrm {i}}}\\Big ) = \\frac{m^{2}}{2}\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}^{2} + \\frac{1}{2} \\overline{\\eta }_{k=k_{\\mathrm {i}}}^{2} + i \\sqrt{\\frac{\\lambda }{12}} \\overline{\\eta }_{k=k_{\\mathrm {i}}} \\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}}^{2} \\quad \\forall m^2 < 0\\;,}\\end{array}\\right.$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}},a_{1} \\cdots a_{n}} = \\left.\\frac{\\partial ^{n+m}S_{\\mathrm {mix}}\\big (\\vec{\\widetilde{\\varphi }},\\widetilde{\\sigma }\\big )}{\\partial \\widetilde{\\varphi }_{a_{1}}\\cdots \\partial \\widetilde{\\varphi }_{a_{n}}\\partial \\widetilde{\\sigma }^{m}}\\right|_{\\vec{\\widetilde{\\varphi }}=\\vec{\\overline{\\phi }}_{k=k_{\\mathrm {i}}} \\atop \\widetilde{\\sigma }=\\overline{\\eta }_{k=k_{\\mathrm {i}}}} \\quad \\forall a_{1}, \\cdots , a_{n} \\;,$ where the latter relation translates in the unbroken-symmetry regime to: $\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} = m^2 \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} = 1 \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} = i\\sqrt{\\frac{\\lambda }{3}} \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} = 0 \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(m\\phi ,n\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} = 0 \\quad \\forall \\ m+n \\ge 4 \\;.$ The truncation of the infinite set of differential equations resulting from the vertex expansion is now imposed by: $\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k} = \\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} \\quad \\forall k,~ \\forall \\ n + m > N_{\\mathrm {max}} \\;.$ Finally, the gs energy is deduced in the present case from: $E^\\text{1PI-FRG;mix}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k=k_{\\mathrm {f}}} \\;,$ whereas the gs density is determined from: $\\rho ^\\text{1PI-FRG;mix}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k=k_{\\mathrm {f}}}\\right)_{aa}^{-1} = \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k=k_{\\mathrm {f}}}\\right)^{-1} \\;,$ where $\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k}$ is introduced in the right-hand side using Eq.",
"(REF ) (at $n=0$ ) and assuming that $m^{2}>0$ .",
"For all our mixed 1PI-FRG calculations, we have also exploited the cutoff function $R_k$ of Eq.",
"(REF ) for both the original and auxiliary field sectors.",
"Let us first concentrate our discussion on the regime with $m^{2}>0$ .",
"Without the MFT approximation, our mixed 1PI-FRG results exhibit a distinct convergence towards the exact solution, as can be seen in Fig.",
"REF for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ with $N=2$ .",
"For example, at $N_{\\mathrm {max}}=4$ , the mixed 1PI-FRG outperforms the original 1PI-FRG in all these situations, although one can also point out that $N_{\\mathrm {max}}=4$ corresponds to the third non-trivial order of the mixed 1PI-FRG but only to the first one of the original 1PI-FRG (the first three non-trivial orders of the mixed 1PI-FRG approach correspond to $N_{\\mathrm {max}}=2,3$ and 4).",
"The superiority of the mixed 1PI-FRG as compared to the original one for a given $N_{\\mathrm {max}}$ can notably be attributed to the 1-point correlation function of the auxiliary field taking non-trivial values, as illustrated by Fig.",
"REF .",
"This echoes very clearly our comparison between SCPT based on the original and mixed 2PI EAs in paper I where the 1-point correlation function of the auxiliary field was also put forward to explain the difference between the bare vertex approximation (BVA) and the original Hartree-Fock result.",
"Figure: Imaginary part of the 1-point correlation function of the auxiliary field in the framework of the mixed representation at N=2N=2 and m 2 =+1m^{2}=+1 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} (upper panel) or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} (lower panel) and the corresponding LOAF approximation results E gs LOAF E_{\\mathrm {gs}}^{\\mathrm {LOAF}} or ρ gs LOAF \\rho _{\\mathrm {gs}}^{\\mathrm {LOAF}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} (upper panel) or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} (lower panel) and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).We have also implemented the MFT by setting the propagator $\\overline{G}^{(\\eta )}_{k}$ equal to zero for all $k$ .",
"This amounts to setting the mass of the bosonic field $\\widetilde{\\sigma }$ (or equivalently the associated cutoff function $R^{(\\eta )}_{k}$ introduced in Eq.",
"(REF )) to infinity.",
"This completely freezes the fluctuations of this field.",
"In other words, the MFT can not capture radiative corrections associated with the auxiliary field.",
"This notably excludes all contributions beyond the leading order of the collective LE.",
"Therefore, the MFT can not, by construction, outperform the leading order of the collective LE, which coincides with the auxiliary field LE (LOAF) approximation [158], [159], [160], [161], [162], [163], [164], [165], [166], [167] for the studied model.",
"This means that the MFT should tend to the LOAF approximation as the truncation order $N_{\\mathrm {max}}$ increases, which is illustrated by Fig.",
"REF .",
"Finally, although Fig.",
"REF shows that the approximation underlying MFT induces a significant loss in the accuracy of mixed 1PI-FRG results, they also illustrate that its efficiency is not affected in the strongly-coupled regime: the MFT can therefore be considered as a first level of non-perturbative approximations.",
"Regarding the phase with $m^{2}<0$ , we encounter the same limitation as in the original representation: the set of differential equations resulting from the vertex expansion procedure applied to Eq.",
"(REF ) is too stiff to be solved from $k_{\\mathrm {i}}=0$ to $k_{\\mathrm {f}}=1$ with the chosen numerical toolsREF .",
"We will therefore turn to the collective representation as our last attempt to describe the broken-symmetry phase with the 1PI-FRG."
],
[
"Collective 1PI functional renormalization group",
"For the collective representation of the (0+0)-D $O(N)$ model, the Wetterich equation reduces to: $\\dot{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {col},k}\\Big (\\vec{\\phi }\\Big ) = \\frac{1}{2} \\dot{R}_{k} \\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {col},k}(\\eta )+R_{k}\\right)^{-1} \\;,$ with $\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {col},k}(\\eta ) \\equiv \\frac{\\partial ^{2}\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {col},k}(\\eta )}{\\partial \\eta ^{2}} \\;.$ Hence, we are basically in the same situation as when treating the broken-symmetry phase of the original representation with $N=1$ .",
"The output of the vertex expansion procedure applied to Eq.",
"(REF ) can therefore be directly deduced from the set of differential equations containing notably Eqs.",
"(REF ) to (REF ), which yields: $\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{\\mathrm {col},k} = \\frac{1}{2} \\dot{R}_{k} \\left( \\overline{D}_{k} - \\overline{D}^{(0)}_{k} \\right)\\;,$ $\\dot{\\overline{\\eta }}_{k}=\\frac{1}{2\\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{\\mathrm {col},k}}\\dot{R}_{k}\\overline{D}_{k}^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k} \\;,$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2)}_{\\mathrm {col},k} = & \\ \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k} + \\dot{R}_{k}\\overline{D}_{k}^{3}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k}\\right)^{2} \\\\& -\\frac{1}{2}\\dot{R}_{k}\\overline{D}_{k}^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{\\mathrm {col},k} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k} = & \\ \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{\\mathrm {col},k}-3\\dot{R}_{k}\\overline{D}_{k}^{4}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k}\\right)^{3} \\\\& + 3\\dot{R}_{k}\\overline{D}_{k}^{3}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{\\mathrm {col},k} \\\\& -\\frac{1}{2}\\dot{R}_{k}\\overline{D}_{k}^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(5)}_{\\mathrm {col},k} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(4)}_{\\mathrm {col},k} = & \\ \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(5)}_{\\mathrm {col},k}+12\\dot{R}_{k}\\overline{D}_{k}^{5}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k}\\right)^{4} \\\\& -18\\dot{R}_{k}\\overline{D}_{k}^{4}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k}\\right)^{2}\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k} \\\\& + 4\\dot{R}_{k}\\overline{D}^{3}_{k}\\overline{\\Gamma }^{(\\mathrm {1PI})(3)}_{\\mathrm {col},k}\\overline{\\Gamma }^{(\\mathrm {1PI})(5)}_{\\mathrm {col},k} \\\\& + 3\\dot{R}_{k}\\overline{D}^{3}_{k}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{\\mathrm {col},k}\\right)^{2}-\\frac{1}{2}\\dot{R}_{k}\\overline{D}^{2}_{\\mathrm {col},k}\\overline{\\Gamma }^{(\\mathrm {1PI})(6)}_{\\mathrm {col},k} \\;,\\end{split}$ with $\\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {col},k} \\equiv \\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {col},k}(\\eta =\\overline{\\eta }_{k}) \\quad \\forall k \\;,$ $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{\\mathrm {col},k} \\equiv \\left.\\frac{\\partial ^{n} \\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {col},k}(\\eta )}{\\partial \\eta ^{n}} \\right|_{\\eta =\\overline{\\eta }_{k}} \\quad \\forall k \\;,$ and $\\overline{D}^{ \\ -1}_{k} \\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{\\mathrm {col},k} + R_{k} \\;,$ $\\left(\\overline{D}^{(0)}_{k}\\right)^{-1} \\equiv \\overline{\\Gamma }^{(\\mathrm {1PI})(2)}_{\\mathrm {col},k=k_{\\mathrm {i}}} + R_{k} \\;.$ The corresponding initial conditions are: $\\overline{\\eta }_{k=k_{\\mathrm {i}}} = \\overline{\\sigma }_{\\mathrm {cl}} = i\\left(\\frac{\\sqrt{3}m^{2}-\\sqrt{3m^{4}+2N\\lambda }}{2\\sqrt{\\lambda }}\\right) \\;,$ $\\begin{split}& \\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {col},k=k_{\\mathrm {i}}} = S_{\\mathrm {col}}\\big (\\widetilde{\\sigma }=\\overline{\\eta }_{k=k_{\\mathrm {i}}}\\big ) \\\\& = \\frac{1}{2}\\left(\\overline{\\eta }_{k=k_{\\mathrm {i}}}\\right)^{2} - \\frac{N}{2} \\ln \\hspace{-1.42271pt}\\left(\\frac{2\\pi }{m^{2}+i\\sqrt{\\frac{\\lambda }{3}}\\overline{\\eta }_{k=k_{\\mathrm {i}}}}\\right) \\;,\\end{split}$ $\\begin{split}& \\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{\\mathrm {col},k=k_{\\mathrm {i}}} = \\left.\\frac{\\partial ^{n}S_{\\mathrm {col}}\\big (\\widetilde{\\sigma }\\big )}{\\partial \\widetilde{\\sigma }^{n}}\\right|_{\\widetilde{\\sigma }=\\overline{\\eta }_{k=k_{\\mathrm {i}}}} \\\\& = \\delta _{n 2} + (-1)^{n+1} \\frac{N}{2} \\left(n-1\\right)!",
"\\left(\\frac{i\\sqrt{\\frac{\\lambda }{3}}}{m^{2}+i\\sqrt{\\frac{\\lambda }{3}}\\overline{\\eta }_{k=k_{\\mathrm {i}}}}\\right)^{n} \\\\& \\quad \\forall n \\ge 2 \\;.\\end{split}$ In addition, the infinite tower of differential equations including Eqs.",
"(REF ) to (REF ) is truncated by imposing: $\\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{\\mathrm {col},k} = \\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{\\mathrm {col},k=k_{\\mathrm {i}}} \\quad \\forall k,~ \\forall n > N_{\\mathrm {max}} \\;.$ Furthermore, the gs energy is deduced from: $E^\\text{1PI-FRG;col}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {col},k=k_{\\mathrm {f}}} \\;,$ whereas the gs density is estimated with the relation: $\\rho ^\\text{1PI-FRG;col}_{\\mathrm {gs}} = \\frac{i}{N} \\sqrt{\\frac{12}{\\lambda }} \\overline{\\eta }_{k=k_{\\mathrm {f}}} \\;.$ Finally, we still use the cutoff function $R_k$ given by Eq.",
"(REF ).",
"Note that all analytical results given since Eq.",
"(REF ) are valid for both the unbroken- and broken-symmetry regimes.",
"This follows from the fact that the $O(N)$ symmetry does not constrain the auxiliary field as it does for the original field via Eqs.",
"(REF ) to (REF ).",
"Hence, as opposed to the original situation, there is no additional difficulty in treating the phase with $m^2<0$ instead of $m^2>0$ in the framework of the collective representation.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} (upper panel) or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} (lower panel) and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =±1m^{2}=\\pm 1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).From this implementation of the collective 1PI-FRG, we are able to calculate $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ for all signs of $m^{2}$ , which yields notably our first 1PI-FRG results for $m^{2}<0$ .",
"These results, shown in Fig.",
"REF at $N=2$ , are however disappointing in the sense that, for $m^{2}>0$ , they are all outperformed by the original and mixed 1PI-FRG approaches at a given $N_{\\mathrm {max}}$ .",
"Actually, for both $m^{2}<0$ and $m^{2}>0$ , Fig.",
"REF shows for $E_{\\mathrm {gs}}$ at $N=2$ that the collective 1PI-FRG must be pushed beyond $N_{\\mathrm {max}}=4$ to yield an accuracy below $10\\%$ , whereas this is already achieved by the mixed 1PI-FRG at $N_{\\mathrm {max}}=2$ for $m^{2}>0$ .",
"The connection between the collective 1PI-FRG and MFT is also clear.",
"The starting point of the collective 1PI-FRG procedure coincides with the collective classical action, i.e.",
"with the LOAF approximation towards which the MFT tends.",
"Hence, the collective 1PI-FRG incorporates quantum corrections (which correspond to the bosonic fluctuations neglected by the MFT) on top of the LOAF approximation throughout the flow: it is therefore by construction more efficient than the MFT version of the mixed 1PI-FRG."
],
[
"Generalities",
"We have argued in the introduction (especially in section REF ) that higher-order EAs such as 2PI and 2PPI EAs play an important role in our aim of reformulating the nuclear EDF approach with PI techniques.",
"We thus continue our FRG study by considering the 2PI EA framework here before treating the 2PPI EA one in the next section.",
"Formulations of FRG approaches for 2PI EAs have started since the early 2000s [168], [169], [44], [170], [147].",
"Some of these approaches discuss the implementation of symmetries in an exhaustive manner [168], [147], as e.g.",
"the conservation of $U(1)$ Ward identities for the description of superconductivity.",
"The detailed discussion of Ref.",
"[147] also outlines the recipe to construct flow equations for any $n$ PI EA (i.e.",
"for $n$ PI EAs with $n \\ge 1$ ) by interpreting cutoff functions as shifts for the sources.",
"Moreover, in Ref.",
"[169], the ideas of the work of Alexandre, Polonyi and Sailer [171], [172] deriving a generalization of the Callan-Symanzik equation [173], [174], [175] via the 1PI EA are exploited to determine a flow equation for the 2PI EA.",
"The resulting approach, called internal space (IS) renormalization group (RG), has been compared with other RG methods, including the standard Callan-Symanzik RG, the Wegner-Houghton RG [176], the local potential approximation (LPA) [177], [178] of the DE within the 1PI-FRG, in the framework of a comparative study on a (0+1)-D $O(N)$ -symmetric $\\varphi ^4$ -theory [179].",
"We will rather focus in this study on the more recent 2PI-FRG formalism put forward by Dupuis in Refs.",
"[44], [45], and more specifically on its different versions called C-flow [44], U-flow [45] and CU-flow [45] that we will define further below.",
"The U-flow and CU-flow can be both formulated via a modification of the Legendre transform defining the 2PI EA, in the same way as for the 1PI-FRG with the extra term $\\Delta S_{k}[\\phi ]$ in Eq.",
"(REF ).",
"In any case, the aim remains to obtain a starting point as convenient as possible for the flow: the presence of $\\Delta S_{k}[\\phi ]$ in Eq.",
"(REF ) enables us to start the 1PI-FRG procedure at the classical theory whereas the 2PI-FRG flow can begin at the result of SCPT in this way.",
"In fact, Wetterich also developed a 2PI-FRG approach based on a modified Legendre transform as well [170].",
"However, as opposed to this work, Dupuis' 2PI-FRG is based on flow equations for the Luttinger-Ward (LW) functional and not for the 2PI EA itself, which significantly improves its convergenceThis improvement will be thoroughly illustrated via the numerical applications presented in section REF on the 2PPI-FRG..",
"Moreover, both Wetterich and Dupuis ignore the field dependence of the 2PI EA in their 2PI-FRG formulations, i.e.",
"they consider $\\Gamma ^{(\\mathrm {2PI})}[\\phi =0,G]$ , which is also referred to as bosonic EA [170].",
"Some appealing features of Dupuis' 2PI-FRGIn what follows, 2PI-FRG refers implicitly to the 2PI-FRG à la Dupuis developed in Refs.",
"[44], [45].",
"are listed below, stressing some advantages of this 2PI-FRG as compared to its 1PI counterpart: Contrary to a 1PI EA depending only on Grassmann fields, expansions of $\\Gamma ^{(\\mathrm {2PI})}[\\phi =0,G]$ around non-trivial minima are possible since the propagator $G$ is a bosonic variable.",
"As a result, the 2PI-FRG is well suited to grasp non-perturbative physics in fermionic systems, even without HSTs.",
"When the 2PI-FRG flow is designed to take results of self-consistent PT as inputs, the 2PI-FRG offers the possibility to start the flow in a broken-symmetry phase, which enables us to avoid phase transitions (and the associated problematic divergences encountered in the 1PI-FRG) during the flow.",
"Besides its convenient starting points, the 2PI-FRG is designed itself to avoid the undesirable divergences from which the 1PI-FRG suffers.",
"This is simply because the quantities calculated during the flow are different: one calculates the 1PI vertices (i.e.",
"derivatives of the 1PI EA) during the 1PI-FRG flow and the 2PI vertices (i.e.",
"derivatives of the LW functional) during the 2PI-FRG flow.",
"A concrete example is given in Ref.",
"[44] for an application of the C-flow implementation of the 2PI-FRG to the BCS theory.",
"However, the most performing implementations of the 2PI-FRG, i.e.",
"the U-flow and the CU-flow, both require to solve the Bethe-Salpeter equation at each step of the flow, as will explained technically later in this section.",
"This increases significantly the numerical cost of these FRG approaches, although we will argue throughout this study that such a cost is worth paying in various situations (when treating strongly-coupled fermions notably) and some ideas can actually be put forward to lower this cost (see section REF ).",
"Nonetheless, this is probably due to this important numerical weight that very few applications of the 2PI-FRG have been carried out so far.",
"Among these, we can mention the work of Rentrop, Jakobs and Meden on the (0+1)-D $\\varphi ^4$ -theory [180] and on the (0+1)-D Anderson impurity modelThe work of Refs.",
"[180], [181] echoes that of Ref.",
"[104] which treats the same models with the 1PI-FRG. [181].",
"Later on, the 2PI-FRG has been designed by Katanin so as to take the 2PI vertices calculated from another non-perturbative method, i.e.",
"dynamical mean-field theory (DMFT) [182], [183], as inputs [184], thus developing a 2PI counterpart for the so-called DMF$^2$ RG already developed in the 1PI framework [185], [186], [187], [120].",
"Most importantly, the tower of differential equations resulting from this approach are tractable enough to perform a quantitative description of a (2+1)-D extendedThe adjective “extended” for the Hubbard model indicates that the latter involves non-local Coulomb interaction, i.e.",
"Coulomb interaction between fermions on different lattice sites.",
"Hubbard model [184].",
"This is to our knowledge the only 2PI-FRG application to a model with finite space dimensions, which illustrates that this promising 2PI approach has barely been beyond the stage of toy model applications.",
"Before presenting the 2PI-FRG formalism, we first define some notations and review the symmetry properties of the main objects involved in the 2PI EA framework.",
"All 2PI-FRG implementations of this method investigated in the present study are based on the generating functional: $Z[K]=e^{W[K]}=\\int \\mathcal {D}\\widetilde{\\psi } \\ e^{-S\\big [\\widetilde{\\psi }\\big ] + \\frac{1}{2}\\int _{\\alpha ,\\alpha ^{\\prime }}\\widetilde{\\psi }_{\\alpha }K_{\\alpha \\alpha ^{\\prime }}\\widetilde{\\psi }_{\\alpha ^{\\prime }}} \\;,$ where $\\widetilde{\\psi }$ is either a real bosonic field ($\\zeta =+1$ ) or a real Grassmann field ($\\zeta =-1$ ).",
"The index $\\alpha \\equiv (i,x)$ used in Eq.",
"(REF ) combines this time all types of internal indices $i$ with $x\\equiv (r, \\tau , c)$ including the space coordinate $r$ , the imaginary time $\\tau $ and a charge index $c$ if necessary.",
"The latter index is defined as followsTherefore, with this notation based on a charge index, $\\widetilde{\\psi }_{\\alpha }$ is always mathematically treated as a real field with an extra index, instead of a complex one.",
": $\\widetilde{\\psi }_{\\alpha }=\\widetilde{\\psi }_{i,x}=\\left\\lbrace \\begin{array}{lll}\\displaystyle {\\widetilde{\\psi }_{i}(r,\\tau ) \\quad \\mathrm {for}~c=-\\;.}",
"\\\\\\\\\\displaystyle {\\widetilde{\\psi }^{\\dagger }_{i}(r,\\tau ) \\quad \\mathrm {for}~c=+\\;.",
"}\\end{array}\\right.$ The shorthand notation for integration exploited in Eq.",
"(REF ) is set by: $\\int _{\\alpha } \\equiv \\sum _{i,c} \\int ^{1/T}_{0} d\\tau \\int d^{D-1}r\\;,$ assuming that the studied system lives in a $D$ -dimensional spacetime.",
"It will be also most convenient to group $\\alpha $ -indices by pairs via a bosonic index: $\\gamma \\equiv (\\alpha ,\\alpha ^{\\prime }) \\;,$ and the main features of the 2PI EA formalism based on such indices are presented in appendix .",
"For instance, the connected correlation functions can be expressed in terms of such indices: $W^{(n)}_{\\gamma _{1} \\cdots \\gamma _{n}}[K] \\equiv \\frac{\\delta ^{n}W[K]}{\\delta K_{\\gamma _{1}} \\cdots \\delta K_{\\gamma _{n}}} = \\frac{\\delta ^{n}W[K]}{\\delta K_{\\alpha _{1}\\alpha ^{\\prime }_{1}} \\cdots \\delta K_{\\alpha _{n}\\alpha ^{\\prime }_{n}}} \\;,$ which defines the connected propagator: $G_{\\gamma } = W^{(1)}_{\\gamma }[K] \\;,$ for $n=1$ .",
"As the components of the source $K$ satisfy $K_{\\alpha \\alpha ^{\\prime }} = \\zeta K_{\\alpha ^{\\prime }\\alpha }$ , the correlation functions of Eq.",
"(REF ) possess the symmetry properties: [left=]align W(n)1(i,i')n[K]=W(n)1(i',i)n[K] , W(n)1n[K]=W(n)P(1)P(n)[K] , with $P$ denoting an arbitrary element of the permutation group of order $n$ , and especially: $G_{\\alpha \\alpha ^{\\prime }}=\\zeta G_{\\alpha ^{\\prime }\\alpha } \\;,$ at $n=1$ .",
"With this bosonic index notation, the Legendre transform defining the 2PI EA under consideration reads: $\\Gamma ^{(\\mathrm {2PI})}[G] = -W[K] + \\frac{1}{2} \\int _{\\gamma } K_{\\gamma } G_{\\gamma } \\;.$ where an integration over bosonic indices amounts to integrating (or just summing) over all its constituent indices, i.e.",
": $\\int _{\\gamma } \\equiv \\int _{\\alpha ,\\alpha ^{\\prime }} \\;.$ In what follows, we will use a DeWitt-like notation for the integration over bosonic indices.",
"For $n$ arbitrary bosonic matrices $M_{m}$ (with $m=1,\\dots ,n$ ), it takes the following form: $\\begin{split}& M_{1,\\gamma _{1}\\hat{\\gamma }_{1}} \\cdots M_{n,\\hat{\\gamma }_{n-1}\\gamma _{2}} \\\\& = \\frac{1}{2^{n-1}}\\int _{\\gamma _{3},\\cdots ,\\gamma _{n+1}} M_{1,\\gamma _{1}\\gamma _{3}} \\cdots M_{n,\\gamma _{n+1}\\gamma _{2}} \\;,\\end{split}$ where the hatted indices are all dummy and the non-hatted ones are all free by convention.",
"In the 2PI EA framework, it is also natural to consider the LW functional $\\Phi [G]$ , which corresponds to the interaction part of $\\Gamma ^{(\\mathrm {2PI})}[G]$ , i.e.",
": $\\Phi [G]\\equiv \\Gamma ^{(\\mathrm {2PI})}[G]-\\Gamma ^{(\\mathrm {2PI})}_{0}[G] \\;,$ where the non-interacting part of the 2PI EA can be expressed from Gaussian integration, thus yieldingSince we have now two types of indices ($\\alpha $ - and bosonic indices), we stress that $\\mathrm {STr}$ still denotes the trace with respect to $\\alpha $ -indices in the whole section , as in the rest of this paper.",
": $\\Gamma _{0}^{(\\mathrm {2PI})}[G] = -\\frac{\\zeta }{2} \\mathrm {STr} \\left[ \\mathrm {ln}(G) \\right] + \\frac{\\zeta }{2} \\mathrm {STr} \\left[GC^{-1}-\\mathbb {I}\\right]\\;,$ with $\\mathbb {I}$ denoting the identity with respect to $\\alpha $ -indices (i.e.",
"$\\mathbb {I}_{\\alpha _{1}\\alpha _{2}}=\\delta _{\\alpha _{1}\\alpha _{2}}=\\delta _{i_{1}i_{2}}\\delta _{c_{1}c_{2}}\\delta ^{(D-1)}(r_{1}-r_{2})\\delta ^{(1)}(\\tau _{1}-\\tau _{2})$ ) and $C$ being the free propagator, i.e.",
": $C_{\\alpha \\alpha ^{\\prime }}^{-1} = \\left.",
"\\frac{\\delta ^{2} S\\big [\\widetilde{\\psi }\\big ]}{\\delta \\widetilde{\\psi }_{\\alpha }\\delta \\widetilde{\\psi }_{\\alpha ^{\\prime }}} \\right|_{\\widetilde{\\psi }=0} \\;.$ Recall that the LW functional is the sum of 2PI diagrams, with propagator lines corresponding to the full propagator $G$ .",
"The so-called 2PI vertices correspond to its derivatives, i.e.",
"$\\Phi _{\\gamma _{1}\\cdots \\gamma _{n}}^{(n)}[G]\\equiv \\frac{\\delta ^{n}\\Phi [G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}}$ .",
"Another important relation of the 2PI formalism is the Dyson equation that follows from Eq.",
"(REF ), i.e.",
": $\\frac{\\delta \\Gamma ^{(\\mathrm {2PI})}[G]}{\\delta G_{\\gamma }} = K_{\\gamma } =- G^{-1}_{\\gamma } + C^{-1}_{\\gamma } - \\Sigma _{\\gamma }[G] \\;,$ where the self-energy $\\Sigma $ is directly connected to the LW functional: $\\Sigma _{\\gamma }[G]\\equiv -\\frac{\\delta \\Phi [G]}{\\delta G_{\\gamma }} \\;.$ All equations introduced since the beginning of section REF underlie the 2PI EA framework, but are not inherent to the 2PI-FRG formalism.",
"All 2PI-FRG approaches treated in this study can be applied to any system whose classical action can be put in the form: $\\begin{split}S\\Big [\\widetilde{\\psi }\\Big ] = & \\ \\frac{1}{2}\\int _{\\alpha _{1},\\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{1}}C^{-1}_{\\alpha _{1}\\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{2}} \\\\& +\\frac{1}{4!",
"}\\int _{\\alpha _{1},\\alpha _{2},\\alpha _{3},\\alpha _{4}}U_{\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{4}}\\widetilde{\\psi }_{\\alpha _{1}}\\widetilde{\\psi }_{\\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{3}}\\widetilde{\\psi }_{\\alpha _{4}} \\;,\\end{split}$ with the two-body interaction $U$ satisfying the symmetry property: $U_{\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{4}}=\\zeta ^{N(P)}U_{\\alpha _{P(1)}\\alpha _{P(2)}\\alpha _{P(3)}\\alpha _{P(4)}} \\;,$ $N(P)$ being the number of inversions in the permutation $P$ .",
"Hence, in the present 2PI-FRG study, we are only treating systems with two-body interactions at most.",
"However, it is straightforward to generalize the formalism presented in the whole section to interactions which are three-body or more after including in $S\\big [\\widetilde{\\psi }\\big ]$ terms being sextic in $\\widetilde{\\psi }_{\\alpha }$ or more.",
"This is in an important remark to prepare the ground for applications to nuclear system as it is acknowledged that a treatment of three-body interactions is necessary for many-body techniques grounded on EFTs of QCD and aiming at a quantitative description of atomic nuclei.",
"Regarding the 2PI-FRG considered here, three main implementations can be defined from action (REF ) depending on the way cutoff functions $R_{\\mathfrak {s}}$ are introduced: either to dress the free propagator via e.g.",
"$C^{-1} \\rightarrow C_{\\mathfrak {s}}^{-1} = C^{-1} + R_{\\mathfrak {s}}$ , or to dress the two-body interaction via e.g.",
"$U \\rightarrow U_{\\mathfrak {s}} = U + R_{\\mathfrak {s}}$ , or both.",
"These three options define respectively the C-flow, the U-flow and the CU-flow of the 2PI-FRG.",
"We stress that, in principle, all these three versions can implement the Wilsonian momentum-shell integration, but this is not always the case depending on the choice of cutoff function(s).",
"The flow parameter can thus be dimensionless and we will therefore denote it by $\\mathfrak {s}$ instead of $k$ in what follows.",
"After performing the substitution(s) $C^{-1} \\rightarrow C_{\\mathfrak {s}}^{-1}$ and/or $U \\rightarrow U_{\\mathfrak {s}}$ , all relevant functionals become dependent on $\\mathfrak {s}$ , including the 2PI EA defined by Eq.",
"(REF ) that becomes in this way: $\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G] = -W_{\\mathfrak {s}}[K] + \\frac{1}{2} \\int _{\\gamma } K_{\\gamma } G_{\\gamma } \\;,$ with $G_{\\gamma } \\equiv G_{\\mathfrak {s},\\gamma }[K] = W^{(1)}_{\\mathfrak {s},\\gamma }[K]\\;.$ In particular, for any approach based on the EA (REF ) including the entire 2PI-FRG framework, the 1-point correlation function $\\phi =\\big \\langle \\widetilde{\\psi }\\big \\rangle $ is imposed to vanish and spontaneous breakdowns of the $O(N)$ symmetry can therefore not be exhibited.",
"An exact flow equation can be derived for $\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ , i.e.",
": $\\begin{split}& \\dot{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})}[G] \\equiv \\partial _{\\mathfrak {s}} \\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G] \\\\& = \\underbrace{\\dot{C}_{\\mathfrak {s},\\hat{\\gamma }}^{-1} G_{\\hat{\\gamma }}}_{C^{-1}\\rightarrow C_{\\mathfrak {s}}^{-1}} + \\underbrace{\\frac{1}{6} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} \\left(W_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{1}}^{(2)}[K]+\\frac{1}{2}\\Pi _{\\hat{\\gamma }_{2}\\hat{\\gamma }_{1}}[G]\\right)}_{U\\rightarrow U_{\\mathfrak {s}}} \\;,\\end{split}$ which is the counterpart of the Wetterich equation for the 2PI-FRG.",
"The underbraces indicate which terms are generated by the flow dependence of either $C_{\\mathfrak {s}}$ or $U_{\\mathfrak {s}}$ , which means that the term induced by $U\\rightarrow U_{\\mathfrak {s}}$ ($C^{-1}\\rightarrow C^{-1}_{\\mathfrak {s}}$ ) is to be discarded for the $C$ -flow ($U$ -flow).",
"Eq.",
"(REF ) also involves the pair propagator $\\Pi [G]$ that can be expressed as: $\\Pi _{\\gamma _{1} \\gamma _{2}}[G] \\equiv \\frac{\\delta ^{2}W_{0}[K(G)]}{\\delta K_{\\gamma _{1}} \\delta K_{\\gamma _{2}}} = G_{\\alpha _{1} \\alpha ^{\\prime }_{2}} G_{\\alpha ^{\\prime }_{1} \\alpha _{2}} + \\zeta G_{\\alpha _{1} \\alpha _{2}} G_{\\alpha ^{\\prime }_{1} \\alpha ^{\\prime }_{2}} \\;,$ where $W_{0}[K]$ is by definition $W[K]$ evaluated at vanishing $U$ , i.e.",
"$W_{0}[K]$ is the non-interacting version of the Schwinger functional defined by Eq.",
"(REF ).",
"As in the fermionic 1PI-FRG, the conventional treatment of the exact flow equation (REF ) is the vertex expansion, which now consists in Taylor expanding both sides of Eq.",
"(REF ) with respect to $G$ .",
"In particular, the corresponding expansion for $\\dot{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ is directly deduced from: $\\begin{split}\\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}[G] = & \\ \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}} \\\\& + \\sum _{n=2}^{\\infty }\\frac{1}{n!",
"}\\int _{\\gamma _{1},\\cdots ,\\gamma _{n}} \\overline{\\Gamma }_{\\mathfrak {s},\\gamma _{1} \\cdots \\gamma _{n}}^{(\\mathrm {2PI})(n)} \\\\& \\times \\left(G-\\overline{G}_{\\mathfrak {s}}\\right)_{\\gamma _{1}} \\cdots \\left(G-\\overline{G}_{\\mathfrak {s}}\\right)_{\\gamma _{n}} \\;,\\end{split}$ with $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}}\\equiv \\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ , $\\overline{\\Gamma }_{\\mathfrak {s},\\gamma _{1} \\cdots \\gamma _{n}}^{(\\mathrm {2PI})(n)}\\equiv \\left.\\frac{\\delta ^{n}\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]}{\\delta G_{\\gamma _{1}}\\cdots \\delta G_{\\gamma _{n}}}\\right|_{G=\\overline{G}_{\\mathfrak {s}}}$ and $\\overline{G}_{\\mathfrak {s}}$ must satisfy: $\\left.\\frac{\\delta \\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]}{\\delta G_{\\gamma }}\\right|_{G=\\overline{G}_{\\mathfrak {s}}} = 0 \\quad \\forall \\gamma , \\mathfrak {s} \\;.$ An infinite tower of coupled differential equations can then be derived for the 2PI vertices $\\overline{\\Phi }^{(n)}_{\\mathfrak {s}}\\equiv \\Phi ^{(n)}_{\\mathfrak {s}}[G=\\overline{G}_{\\mathfrak {s}}]$ by comparing the two expanded sides of Eq.",
"(REF ) and equalizing the terms with identical powers of $G-\\overline{G}_{\\mathfrak {s}}$ .",
"The 2PI vertices are introduced in those equations after replacing the derivatives $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})(n)}$ with $\\overline{\\Phi }^{(n)}_{\\mathfrak {s}}$ using splitting (REF ).",
"In what follows, we will deduce these coupled flow equations for 2PI vertices by directly using the functional chain rule, which is equivalent to exploiting the vertex expansion outlined here.",
"This equivalence is actually illustrated in section by comparing numerical results obtained from 2PI- and 2PPI-FRGsSee Fig.",
"REF more specifically.. All versions of the 2PI-FRG can be applied to the (0+0)-D $O(N)$ model in its original representation.",
"Indeed, after setting: $\\widetilde{\\psi }_{\\alpha } = \\widetilde{\\varphi }_{a} \\;,$ $\\int _{\\gamma } = \\sum _{a,a^{\\prime }=1}^{N} \\;,$ $\\gamma \\equiv (\\alpha ,\\alpha ^{\\prime }) = (a,a^{\\prime }) \\;,$ one can see that the generic classical action (REF ) reduces to $S\\big (\\vec{\\widetilde{\\varphi }}\\big )$ expressed by Eq.",
"(REF ), i.e.",
"we obtain in this way: $\\begin{split}S\\Big [\\widetilde{\\psi }\\Big ] = & \\ S\\Big (\\vec{\\widetilde{\\varphi }}\\Big ) \\\\= & \\ \\frac{1}{2}\\sum _{a_{1},a_{2}=1}^{N}\\widetilde{\\varphi }_{a_{1}}C^{-1}_{a_{1}a_{2}}\\widetilde{\\varphi }_{a_{2}} \\\\& + \\frac{1}{4!",
"}\\sum _{a_{1},a_{2},a_{3},a_{4}=1}^{N} U_{a_{1}a_{2}a_{3}a_{4}}\\widetilde{\\varphi }_{a_{1}}\\widetilde{\\varphi }_{a_{2}}\\widetilde{\\varphi }_{a_{3}}\\widetilde{\\varphi }_{a_{4}} \\;,\\end{split}$ with the free inverse propagator: $C^{-1}_{a_{1}a_{2}} = m^{2} \\delta _{a_{1}a_{2}} \\;,$ and the two-body interaction: $U_{a_{1}a_{2}a_{3}a_{4}} = \\frac{\\lambda }{3}\\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}+\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}}+\\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\right) \\;,$ which is consistent with Eq.",
"(REF ) at $\\zeta =+1$ .",
"However, some implementations of the 2PI-FRG (the U-flow and the CU-flow) are not suited to treat the mixed and collective representations of our toy model, because the interaction part of the classical action can no longer be put in the form $\\int _{\\alpha _{1},\\alpha _{2},\\alpha _{3},\\alpha _{4}}U_{\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{4}} \\widetilde{\\psi }_{\\alpha _{1}}\\widetilde{\\psi }_{\\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{3}}\\widetilde{\\psi }_{\\alpha _{4}}$ in those situations.",
"We will discuss the mixed case in more details in section REF .",
"In the rest of section , we will further develop the 2PI-FRG formalism in a generic $D$ -dimensional framework instead of specifying to the (0+0)-D $O(N)$ model, as opposed to the 1PI-FRG section.",
"The equations for the studied toy model can be directly inferred from the generic equations thus derived by imposing equalities (REF ), (REF ), (REF ), (REF ) and (REF ) with $\\zeta =+1$ .",
"Before moving to section on the 2PPI-FRG, we will actually only treat the U-flow implementation of the 2PI-FRG, in its standard version, i.e.",
"the plain U-flow (pU-flow), and its improved version, i.e.",
"the modified U-flow (mU-flow).",
"The underpinning reason is that we want to illustrate that the 2PI-FRG provides us with excellent correlated starting points given by SCPT based on 2PI EAs, which is particularly appealing in our aim to treat strongly correlated systems.",
"We will discuss the price to pay (in terms of numerical cost) for such powerful starting points and all of this can be addressed in the U-flow (and more specifically in the mU-flow) framework.",
"In addition, there already exists studies of 2PI-FRG methods comparing numerical results of C-flow, U-flow and CU-flow approaches [180], [181], with Ref.",
"[181] notably discussing how to construct $\\Phi $ -derivable approximations from some of these 2PI-FRG implementations.",
"From this, we can expect the mU-flow to be the most performing version of the 2PI-FRG (in general and for the studied $O(N)$ model in particular), alongside with the CU-flow which yields similar performances.",
"Moreover, the C-flow is not suited to design reliable approximation schemes.",
"The last remarks on the CU-flow and the C-flow are thoroughly illustrated in appendix .",
"Finally, we stress that the present work is to our knowledge the first to apply the 2PI-FRG developed in Refs.",
"[44], [45] to an $O(N)$ model with $N>1$ (see appendix REF in particular) and we will also exploit the simplicity of our zero-dimensional toy model to reach higher truncation orders than those investigated in previous studies [180], [181], allowing notably the 2PI three-particle vertex $\\overline{\\Phi }_{\\mathfrak {s}}^{(3)}$ to flow."
],
[
"Plain U-flow",
"The U-flow scheme, developed in Refs.",
"[45], [180], is based on the substitution $U \\rightarrow U_{\\mathfrak {s}} = R_{\\mathfrak {s}} U$ or equivalently $U \\rightarrow U_{\\mathfrak {s}} = U + R_{\\mathfrak {s}}$ .",
"The U-flow implements in principle the Wilsonian momentum-shell integration, with $\\mathfrak {s}$ being connected to the momentum scale.",
"This is notably the case with the cutoff function $R_{\\mathfrak {s}}$ put forward in section IV.C.",
"of Ref.",
"[45] which plays the role of an IR regulator for (low-energy) collective fluctuations, thus preventing problematic divergences during the flow.",
"However, a perfectly valid choice for $R_{\\mathfrak {s}}$ , used for all our 2PI-FRG applications in (0+0)-D with the U-flow, could be set by $U_{\\mathfrak {s}} = R_{\\mathfrak {s}} U = \\mathfrak {s} U$ , with $\\mathfrak {s}$ a dimensionless parameter running from $\\mathfrak {s}_{\\mathrm {i}}=0$ to $\\mathfrak {s}_{\\mathrm {f}}=1$ .",
"This follows the philosophy of the 2PPI-FRG discussed in section and the resulting 2PI-FRG implementation does not carry out the momentum-shell integration à la Wilson.",
"However, we will explain later that such a choice for $R_{\\mathfrak {s}}$ can enable us to start the flow in broken-symmetry phases, which could also be advantageous in the description of critical physics.",
"The boundary conditions for $U_{\\mathfrak {s}}$ are: [left=]align Us=si,12 = 0 1, 2 .",
"Us=sf = U .",
"Hence, the starting point of the flow corresponds to the free theory, at least for the standard implementation of the U-flow called the pU-flow.",
"We will see in section REF that, after a suitable transformation of the LW functional, condition (REF ) implies that the 2PI-FRG flow starts from SCPT results, which defines the mU-flow.",
"Regarding the infinite hierarchy of differential equations to solve within the pU-flow, one can start by differentiating the Dyson equation (REF ) with respect to $\\mathfrak {s}$ at vanishing source $K$ : $\\dot{\\overline{G}}_{\\mathfrak {s},\\alpha _{1}\\alpha ^{\\prime }_{1}}=\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\overline{G}_{\\mathfrak {s},\\alpha _{1} \\alpha _{2}} \\dot{\\overline{\\Sigma }}_{\\mathfrak {s},\\alpha _{2} \\alpha ^{\\prime }_{2}} \\overline{G}_{\\mathfrak {s},\\alpha ^{\\prime }_{2} \\alpha ^{\\prime }_{1}} \\;,$ where we have used $\\dot{C}_{\\gamma }^{-1}=\\dot{C}_{\\mathfrak {s},\\gamma }^{-1}=0$ $\\forall \\gamma $ , which holds by definition of the U-flow.",
"Moreover, a differential equation expressing $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}}$ is directly found from the master equation (REF ): $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{1}{6} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1} \\hat{\\gamma }_{2}} \\left(\\overline{W}_{\\mathfrak {s}}^{(2)} + \\frac{1}{2}\\overline{\\Pi }_{\\mathfrak {s}}\\right)_{\\hat{\\gamma }_{2}\\hat{\\gamma }_{1}} \\;,$ Following the derivations of Refs.",
"[45], [180], the corresponding flow equations for the 2PI vertices can be directly inferred from the chain rule based on bosonic indices (see appendix ): [left=]align s, = -s,1(1) - Gs, s,(2) = s, - Gs, s,(2) , s,1n(n) = s,1n(n) + Gs, s,1n(n+1) n 2 , with $\\overline{\\dot{\\Phi }}_{\\mathfrak {s}}^{(n)}\\equiv \\dot{\\Phi }_{\\mathfrak {s}}^{(n)}[G=\\overline{G}_{\\mathfrak {s}}]$ .",
"The derivatives $\\overline{\\dot{\\Phi }}_{\\mathfrak {s}}^{(n)}$ are then expressed by differentiating the flow equation for the 2PI EA (REF ), combined with the definition of the LW functional given by Eqs.",
"(REF ) and (REF ).",
"Up to $n=3$ , this leads to: $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s},\\gamma } = & -\\frac{1}{3}\\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}}_{\\gamma \\hat{\\gamma }_{1}} \\left[ 2 \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}} \\dot{U}_{\\mathfrak {s}} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}} + \\dot{U}_{\\mathfrak {s}} \\right]_{\\hat{\\alpha }_{1} \\hat{\\alpha }_{2} \\hat{\\alpha }^{\\prime }_{2} \\hat{\\alpha }^{\\prime }_{1}} \\overline{G}_{\\mathfrak {s},\\hat{\\gamma }_{2}} \\\\& +\\frac{1}{6} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}}_{\\gamma \\hat{\\gamma }_{1}} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{(2)} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }_{1} \\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}^{(3)} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{2}}^{(2)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}}^{(2)} = & \\ \\frac{1}{3} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} \\Bigg [\\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}^{(3)}\\right)\\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{6} \\hat{\\gamma }_{7}}^{(2)} \\\\& \\times \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7} \\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8} \\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9} \\hat{\\gamma }_{10}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{2} \\hat{\\gamma }_{7} \\hat{\\gamma }_{10}}^{(3)}\\right) \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{10} \\hat{\\gamma }_{1}}^{(2)} \\\\& - \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{6}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{6} \\hat{\\gamma }_{7}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7} \\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{8} \\hat{\\gamma }_{1}}^{(2)} \\\\& + \\frac{1}{2} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta ^{2} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}} \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}^{(4)}\\right) \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{6} \\hat{\\gamma }_{1}}^{(2)} + \\frac{1}{4} \\frac{\\delta ^{2}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{1}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}} \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}}\\Bigg ] + \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }\\gamma _{1}\\gamma _{2}}^{(3)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}\\gamma _{3}} = & \\ \\frac{1}{3} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} \\Bigg [ 3 \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}}\\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}\\right)\\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}} \\bigg (\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} \\\\& - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{2}\\hat{\\gamma }_{7}\\hat{\\gamma }_{10}}\\bigg ) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{11}} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{11}\\hat{\\gamma }_{12}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{12}\\hat{\\gamma }_{13}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{13}\\hat{\\gamma }_{14}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{3}\\hat{\\gamma }_{11}\\hat{\\gamma }_{14}}\\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{14}\\hat{\\gamma }_{1}} \\\\& + \\Bigg ( \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}\\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}} \\\\& \\times \\left( \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}}\\frac{\\delta ^{2} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{2}\\gamma _{3}\\hat{\\gamma }_{7}\\hat{\\gamma }_{10}}^{(4)} \\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{1}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\Bigg ) \\\\& -2 \\Bigg ( \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}\\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}} \\\\& \\times \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{11}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{11}\\hat{\\gamma }_{12}}^{\\mathrm {inv}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{12}\\hat{\\gamma }_{1}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\Bigg ) \\\\& + 3 \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{1}} \\\\& - \\Bigg ( \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} \\frac{\\delta ^{2}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}} \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{1}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\Bigg ) \\\\& -\\frac{1}{2} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\overline{\\Phi }^{(5)}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}\\gamma _{3}\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{1}} \\Bigg ] + \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }^{(4)}_{\\mathfrak {s},\\hat{\\gamma }\\gamma _{1}\\gamma _{2}\\gamma _{3}}\\;,\\end{split}$ where the following shorthand notations were used: $\\frac{\\delta ^{n}\\overline{\\Pi }_{\\mathfrak {s}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}} \\cdots \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{n}}} \\equiv \\left.\\frac{\\delta ^{n}\\Pi [G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}}\\right|_{G=\\overline{G}_{\\mathfrak {s}}} \\;,$ and $\\begin{split}\\mathcal {F}_{\\gamma _{1}\\gamma _{2}\\gamma _{3}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\equiv & \\ \\mathcal {F}_{\\gamma _{1}\\gamma _{2}\\gamma _{3}} + \\mathcal {F}_{\\gamma _{2}\\gamma _{1}\\gamma _{3}} \\\\& + \\mathcal {F}_{\\gamma _{3}\\gamma _{1}\\gamma _{2}}\\;,\\end{split}$ valid for any functional $\\mathcal {F}$ .",
"Eqs.",
"(REF ) to (REF ) also involve the inverse pair propagator $\\Pi ^{\\mathrm {inv}}$ (and more specifically $\\overline{\\Pi }_{\\mathfrak {s}}^{\\mathrm {inv}}\\equiv \\Pi ^{\\mathrm {inv}}[G=\\overline{G}_{\\mathfrak {s}}]$ ), which satisfies: $\\Pi ^{\\mathrm {inv}}_{\\gamma _{1} \\gamma _{2}}[G] = G^{-1}_{\\alpha _{1} \\alpha ^{\\prime }_{2}} G^{-1}_{\\alpha ^{\\prime }_{1} \\alpha _{2}} + \\zeta G^{-1}_{\\alpha _{1} \\alpha _{2}} G^{-1}_{\\alpha ^{\\prime }_{1} \\alpha ^{\\prime }_{2}} \\;,$ as “$\\mathrm {inv}$ ” labels inverses with respect to bosonic indicesInverses with respect to $\\alpha $ -indices are on the other hand always indicated by a “$-1$ ” as exponent in our notations.",
"by definition, i.e.",
": $\\mathcal {I}_{\\gamma _{1}\\gamma _{2}} = M_{\\gamma _{1}\\hat{\\gamma }} M^{\\mathrm {inv}}_{\\hat{\\gamma }\\gamma _{2}} \\;,$ with $M$ an arbitrary bosonic matrix and $\\mathcal {I}$ the bosonic identity matrix defined via: $\\mathcal {I}_{\\gamma _{1}\\gamma _{2}} \\equiv \\frac{\\delta G_{\\gamma _{1}}}{\\delta G_{\\gamma _{2}}} = \\delta _{\\alpha _{1}\\alpha _{2}} \\delta _{\\alpha ^{\\prime }_{1}\\alpha ^{\\prime }_{2}} + \\zeta \\delta _{\\alpha _{1}\\alpha ^{\\prime }_{2}} \\delta _{\\alpha ^{\\prime }_{1}\\alpha _{2}} \\;.$ Hence, Eqs.",
"(REF ) and (REF ) together with the hierarchy made of Eqs.",
"(REF ) to (REF ) constitute the set of differential equations to solve within the pU-flowIn the framework of the (0+0)-D $O(N)$ model, these equations can be further simplified using the $O(N)$ symmetry, which is discussed in detail in appendix REF .. To solve these equations, the derivative $\\overline{W}_{\\mathfrak {s}}^{(2)}$ must be determined from the flowing 2PI vertex $\\overline{\\Phi }^{(2)}_{\\mathfrak {s}}$ at each value taken by $\\mathfrak {s}$ throughout the flow.",
"This can be achieved by determining $\\overline{W}_{\\mathfrak {s}}^{(2)}[K]$ self-consistently from the relation: $\\overline{W}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}}^{(2)}[K] = \\overline{\\Pi }_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}} - \\overline{\\Pi }_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{1}} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}}^{(2)} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\gamma _{2}}^{(2)} \\;,$ or by inverting a bosonic matrix according to: $\\overline{W}_{\\mathfrak {s}}^{(2)} = \\left(\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})(2)}\\right)^{\\mathrm {inv}} = \\left(\\overline{\\Pi }_{\\mathfrak {s}}^{\\mathrm {inv}} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}} \\;.$ Eqs.",
"(REF ) and (REF ), which can be derived from each other, are two equivalent versions of the Bethe-Salpeter equation.",
"This is why the latter equation must be solved at each step of the flow in the framework of the U-flow implementation of the 2PI-FRG.",
"It remains in general a challenging task for finite dimensional models but some recent progress in the treatment of the Bethe-Salpeter equation should also be pointed out [188], [189], especially since they have led to the successful 2PI-FRG application to the (2+1)-D Hubbard model mentioned earlier [184].",
"One can of course apply drastic approximations to circumvent the need for repeating the resolution of this equation at each step of the flow but this leads in principle to significant losses in the quality of the obtained results.",
"One can mention for example the truncated U-flow (tU-flow) which is a version of the pU-flow approximating Eq.",
"(REF ) as: $\\overline{W}_{\\mathfrak {s}}^{(2)}=\\overline{\\Pi }_{\\mathfrak {s}}+\\mathcal {O}\\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right) \\simeq \\overline{\\Pi }_{\\mathfrak {s}} \\;.$ We will not investigate the tU-flow further in this study since it has already been illustrated that it is a perturbative approach in essence [180] and we are mainly interested in methods suited for quantitative descriptions of strongly-coupled systems.",
"Before going any further, we also point out that the resolution of the Bethe-Salpeter equation in the form of Eq.",
"(REF ) and in the framework of the (0+0)-D $O(N)$ model is discussed in detail in appendix REF .",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).At the starting point of the pU-flow, the flowing objects are simply set equal to their non-interacting versions.",
"In particular, the LW functional and all its derivatives vanish in the framework of the free theory by construction.",
"This yields: $\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}=C \\;,$ $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = \\Gamma ^{(\\mathrm {2PI})}_{0}\\big [G=\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}\\big ] = -\\frac{\\zeta }{2} \\mathrm {STr} \\left[ \\mathrm {ln}(C) \\right] \\;,$ $\\overline{\\Sigma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma }=0 \\quad \\forall \\gamma \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma _{1},\\cdots ,\\gamma _{n}}^{(n)}=0 \\quad \\forall \\gamma _{1}, \\cdots , \\gamma _{n} , ~ \\forall n \\ge 2 \\;.$ Moreover, the truncation of the infinite hierarchy underpinning the pU-flow is set by choosing an integer $N_{\\mathrm {max}}$ such that: $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}=\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(n)} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;.$ Finally, for our numerical applications to the (0+0)-D $O(N)$ model, the gs energy and density are directly deduced from $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})}$ and $\\overline{G}_{\\mathfrak {s}}$ at the end of the flow, i.e.",
": $E^{\\text{2PI-FRG;pU-flow}}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\;,$ and $\\rho ^{\\text{2PI-FRG;pU-flow}}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},aa} \\;.$ Our pU-flow results for the (0+0)-D $O(N)$ model at $N=2$ are reported in Fig.",
"REF .",
"Only the unbroken-symmetry regime is shown in this figure since the starting point of the pU-flow, namely the free theory, is ill-defined in the phase with $m^{2}<0$ as the generating functional (REF ) diverges at $\\lambda =0$ , $m^{2}<0$ and at vanishing sources $\\vec{J}$ and $K$ .",
"Fig.",
"REF clearly illustrates the convergence of the pU-flow results towards the exact solutions for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ as the truncation order $N_{\\mathrm {max}}$ increases from $N_{\\mathrm {max}}=1$ to $N_{\\mathrm {max}}=3$ , thus reaching an accuracy of about $2\\%$ or less for $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ at $N_{\\mathrm {max}}=3$ and $\\lambda /4!=10$ .",
"However, the truncation order $N_{\\mathrm {max}}=3$ requires to calculate the 2PI three-particle vertex $\\overline{\\Phi }^{(3)}_{\\mathfrak {s}}$ throughout the flow, which is very involved for realistic theories.",
"We will thus examine in what follows how we can reach such an accuracy at lowest truncation orders by using correlated starting points for the 2PI-FRG flow."
],
[
"Modified U-flow",
"As was already mentioned, the mU-flow is an implementation of the U-flow version of the 2PI-FRG based on correlated starting points in the form of SCPT results.",
"Its formulation relies on a diagrammatic expression of the LW functional, which reads for the theory based on the classical action (REF ): $\\begin{split}& \\Phi [G] = \\Phi _{\\mathrm {SCPT}}[U,G] \\\\& \\equiv \\frac{1}{8} \\ {\\begin{array}{c}\\begin{fmffile}{Diagrams/2PIFRGphi_HartreeFock}\\begin{fmfgraph}(25,20){i}{o}{decor.shape=circle,decor.filled=full,decor.size=0.2cm}{v1}{phantom,tension=10}{i,i1}{phantom,tension=10}{o,o1}{plain,left,tension=0.5,foreground=(1,,0,,0)}{i1,v1,i1}{plain,right,tension=0.5,foreground=(1,,0,,0)}{o1,v1,o1}\\end{fmfgraph}\\end{fmffile}\\end{array}} - \\frac{1}{48} \\hspace{7.11317pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/2PIFRGphi_RPA}\\begin{fmfgraph}(20,20){vUp}{vDown}{decor.shape=circle,decor.filled=full,decor.size=0.2cm}{vUp}{decor.shape=circle,decor.filled=full,decor.size=0.2cm}{vDown}{plain,left,foreground=(1,,0,,0)}{vUp,vDown}{plain,right,foreground=(1,,0,,0)}{vUp,vDown}{plain,left=0.4,foreground=(1,,0,,0)}{vUp,vDown}{plain,right=0.4,foreground=(1,,0,,0)}{vUp,vDown}\\end{fmfgraph}\\end{fmffile}\\end{array}} \\hspace{5.69046pt} + \\mathcal {O}\\big (U^{3}\\big ) \\\\& = \\frac{1}{8}\\int _{\\gamma _{1},\\gamma _{2}}U_{\\gamma _{1}\\gamma _{2}}G_{\\gamma _{1}}G_{\\gamma _{2}} \\\\& \\hspace{8.5359pt} - \\frac{1}{48} \\int _{\\gamma _{1},\\gamma _{2},\\gamma _{3},\\gamma _{4}} U_{\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{4}} U_{\\alpha ^{\\prime }_{1}\\alpha ^{\\prime }_{2}\\alpha ^{\\prime }_{3}\\alpha ^{\\prime }_{4}} G_{\\gamma _{1}}G_{\\gamma _{2}}G_{\\gamma _{3}}G_{\\gamma _{4}} \\\\& \\hspace{8.5359pt} + \\mathcal {O}\\big (U^{3}\\big ) \\;.\\end{split}$ We have introduced in this way the functional $\\Phi _{\\mathrm {SCPT}}[U,G]$ , which is identical to the LW functional, although we stress its dependence with respect to the interaction $U$ (usually left implicit) and the subscript “SCPT” indicates that we consider its expression (REF ) usually taken as input for the variational procedure underlying SCPT.",
"From this functional, one can define the transformation of the LW functional on which the mU-flow is based, i.e.",
": $\\Phi _{\\mathfrak {s}}[G] \\equiv \\Phi _{\\mathfrak {s}}[G] + \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G] -\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U_{\\mathfrak {s}},G] \\;,$ where $\\Phi _{\\mathfrak {s}}[G]$ will be referred to as the modified LW functional and the functional $\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G]$ only contains the terms of order $\\mathcal {O}\\big (U^{N_{\\mathrm {SCPT}}}\\big )$ or less by definition.",
"According to definitions (REF ) and (REF ), we can also define a modified 2PI EA $\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ as: $\\begin{split}\\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}[G] \\equiv & \\ \\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}[G] + \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G] \\\\& -\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U_{\\mathfrak {s}},G] \\;.\\end{split}$ In the same way, the modified 2PI vertices $\\Phi ^{(n)}_{\\mathfrak {s},\\gamma _{1}\\cdots \\gamma _{n}}[G] \\equiv \\frac{\\delta ^{n} \\Phi _{\\mathfrak {s}}[G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}}$ are related to the derivatives $\\Phi ^{(n)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}},\\gamma _{1}\\cdots \\gamma _{n}}[U,G] \\equiv \\frac{\\delta ^{n} \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}}$ via: $\\begin{split}\\Phi ^{(n)}_{\\mathfrak {s}}[G] \\equiv & \\ \\Phi ^{(n)}_{\\mathfrak {s}}[G] + \\Phi ^{(n)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G] \\\\& -\\Phi ^{(n)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U_{\\mathfrak {s}},G] \\quad \\forall n \\in \\mathbb {N}^{*} \\;,\\end{split}$ which, at $n=1$ , gives us the self-energy: $\\Sigma _{\\mathfrak {s},\\gamma }[G] \\equiv - \\frac{\\delta \\Phi _{\\mathfrak {s}}[G]}{\\delta G_{\\gamma }} \\;.$ Furthermore, we introduce the configuration $\\overline{G}_{\\mathfrak {s}}$ of the propagator $G$ which extremizes the bold 2PI EA of Eq.",
"(REF ) according to: $\\left.",
"\\frac{\\delta \\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]}{\\delta G_{\\gamma }}\\right|_{G=\\overline{G}_{\\mathfrak {s}}} = 0 \\quad \\forall \\gamma ,\\mathfrak {s} \\;.$ Similarly to Eq.",
"(REF ) at vanishing source $K$ and at $G=\\overline{G}_{\\mathfrak {s}}$ , definition (REF ) can also be rewritten in the form of a Dyson equation, i.e.",
": $\\overline{G}^{-1}_{\\mathfrak {s}} = C^{-1}-\\Sigma _{\\mathfrak {s}}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ] \\;,$ as follows from Eq.",
"(REF ).",
"Hence, we have just assigned bold counterparts to all flowing objects of the 2PI-FRG framework.",
"Let us then consider the boundary conditions (REF ) for $U_{\\mathfrak {s}}$ which still hold within the mU-flow.",
"From Eq.",
"(REF ), one can directly deduce that $\\Phi _{\\mathfrak {s}}[G]$ and all its derivatives $\\Phi ^{(n)}_{\\mathfrak {s}}[G]$ vanish at the starting of the flow.",
"According to definitions (REF ) and (REF ), it also implies that, at $\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}$ , the modified LW functional and its derivatives coincide with their perturbative expression inferred from Eq.",
"(REF ), i.e.",
": $\\Phi _{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}[G] = \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G] \\;,$ $\\Phi ^{(n)}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}[G] = \\Phi ^{(n)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G] \\quad \\forall n \\in \\mathbb {N}^{*} \\;,$ whereas, according to Eq.",
"(REF ), they reduce to the original counterparts at $\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}$ , namely: $\\Phi _{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}[G] = \\Phi _{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}[G] \\;,$ $\\Phi ^{(n)}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}[G] = \\Phi ^{(n)}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}[G] \\quad \\forall n \\in \\mathbb {N}^{*} \\;.$ Still regarding the initial conditions of the mU-flow, inserting Eq.",
"(REF ) at $n=1$ into the Dyson equation (REF ) at $\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}$ yields the relation: $\\overline{G}^{-1}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = C^{-1} + \\Phi ^{(1)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big [U,G=\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}\\big ] \\;,$ which is nothing other than the gap equation underlying SCPT based on $\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G]$ , with $\\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}$ being a solution for the propagator $G$ .",
"The initial conditions of the mU-flow are thus fully determined from SCPT solutions.",
"For a given value of $N_{\\mathrm {SCPT}}$ , this translates into: $\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = \\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}} \\;,$ $\\begin{split}\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = \\ & \\Gamma ^{(\\mathrm {2PI})}_{0}\\big [G=\\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big ] \\\\& + \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big [U,G=\\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big ] \\;,\\end{split}$ $\\overline{\\Sigma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = -\\Phi ^{(1)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big [U,G=\\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big ] \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(n)}= \\Phi ^{(n)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big [U,G=\\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big ] \\quad \\forall n \\ge 2 \\;,$ and the truncation of the infinite tower of flow equations is set in the same spirit as in the pU-flow by: $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}=\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(n)} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;.$ In summary, the quantities $\\overline{G}_{\\mathfrak {s}}$ , $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}}\\equiv \\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ , $\\overline{\\Sigma }_{\\mathfrak {s}}\\equiv \\Sigma _{\\mathfrak {s}}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}\\equiv \\Phi _{\\mathfrak {s}}^{(n)}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ (with $n\\ge 2$ ) are the flowing objects in the framework of the mU-flow, in contrast with $\\overline{G}_{\\mathfrak {s}}$ , $\\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ , $\\Sigma _{\\mathfrak {s}}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ and $\\Phi _{\\mathfrak {s}}^{(n)}\\big [G=\\overline{G}_{\\mathfrak {s}}\\big ]$ (with $n\\ge 2$ ) for the pU-flow.",
"To clarify, in this section REF and as opposed to the notations used in section REF , the upper bars always label functionals evaluated at $G=\\overline{G}_{\\mathfrak {s}}$ (with $\\overline{G}_{\\mathfrak {s}}$ defined by Eq.",
"(REF )) instead of $G=\\overline{G}_{\\mathfrak {s}}$ (with $\\overline{G}_{\\mathfrak {s}}$ defined by Eq.",
"(REF )).",
"Furthermore, according to our previous explanation based on Eqs.",
"(REF ) and (REF ), physical quantities are recovered at the end of the flow in the present case in the same manner as in the pU-flow.",
"For the gs energy and density of the (0+0)-D $O(N)$ model, this gives us the following relations from their pU-flow counterparts (i.e.",
"Eqs.",
"(REF ) and (REF )): $E^{\\text{2PI-FRG;mU-flow}}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\;,$ and $\\rho ^{\\text{2PI-FRG;mU-flow}}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},aa} \\;.$ Before going further with discussing our numerical results obtained from the mU-flow, we address three additional points related to the starting points of this approach: Starting the flow at the Hartree-Fock result: According to our notations, the Hartree-Fock result for SCPT based on the 2PI EA is obtained at $N_{\\mathrm {SCPT}}=1$ , in which case the LW functional is approximated as follows: $\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}[U,G] = \\frac{1}{2} U_{\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}}G_{\\hat{\\gamma }_{1}}G_{\\hat{\\gamma }_{2}} \\;,$ as follows from Eq.",
"(REF ).",
"Plugging expression (REF ) into definition (REF ) of the modified 2PI EA leads to: $\\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}[G] = \\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}[G] + \\frac{1}{2}\\left(U-U_{\\mathfrak {s}}\\right)_{\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}}G_{\\hat{\\gamma }_{1}}G_{\\hat{\\gamma }_{2}} \\;.$ Choosing a multiplicative cutoff function $U_{\\mathfrak {s}}=U+R_{\\mathfrak {s}}$ for the sake of our comparison, the rightmost term of Eq.",
"(REF ) becomes: $\\frac{1}{2}\\left(U-U_{\\mathfrak {s}}\\right)_{\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}}G_{\\hat{\\gamma }_{1}}G_{\\hat{\\gamma }_{2}} = -\\frac{1}{2} G_{\\hat{\\gamma }_{1}} R_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} G_{\\hat{\\gamma }_{2}}\\;.$ We can thus see that the rightmost term of Eq.",
"(REF ) has a very similar expression to that of $-\\Delta S_{k}\\big [\\widetilde{\\psi }\\big ]$ (given by Eq.",
"(REF )) that modifies the Legendre transform defining the 1PI EA (in Eq.",
"(REF )) within the FRG based on the Wetterich equation.",
"Hence, both in the 1PI-FRG based on the Wetterich equation and in the mU-flow version of the 2PI-FRG, the Legendre transform defining the underlying flowing EA is modified to obtain a convenient starting point for the flow: the classical theory in the former case and the Hartree-Fock theory (for $N_{\\mathrm {SCPT}}=1$ ) or higher orders of SCPT (for $N_{\\mathrm {SCPT}}>1$ ) in the latter one.",
"Before moving to the next point, we also refer to appendix REF for the expressions of flow equations underpinning the mU-flow at $N_{\\mathrm {SCPT}}=1$ at arbitrary dimensions as well as for the specific case of the (0+0)-D $O(N)$ model.",
"Starting the flow in broken-symmetry phases: Let us consider the situation where the choice for the flow-dependent interaction $U_{\\mathfrak {s}}$ is such that the Wilsonian momentum-shell integration is not implemented through the U-flow.",
"At first sight, this could be taken as a severe limitation of this approach for the purpose of describing critical phenomena.",
"However, this issue is remarkably circumvented in the framework of the mU-flow which can be designed to start in a broken-symmetry phase if necessary (or more generally in the phase that we aim to describe) so as to avoid undesirable phase transitions during the flow, depending on the solution $\\overline{G}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}$ chosen for the initial conditions.",
"This remark only applies if the different solutions of the Dyson equation (REF ) give us access to each phase that we seek to describe, which works in principle for the $U(1)$ symmetry notably, and therefore for the description of superfluid systems, as the relevant order parameters can be identified as components of the propagator $G$ in such a situationHowever, the mU-flow does not provide us with a similar freedom to tackle an $O(N)$ symmetry as neither $\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ nor $\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ is capable of spontaneously breaking such a symmetry, as was already explained right below Eq.",
"(REF )..",
"Starting the flow with SCPT based on mixed 2PI EAs: In paper I, SCPT based on a mixed 2PI EA clearly stood out among all tested PI techniques.",
"We have illustrated in particular, still in paper I, that SCPT based on a mixed 2PI EA manages to grasp further correlations, especially in the strongly-coupled regime of our toy model, as compared to its counterpart formulated without HST (i.e.",
"in the original representation).",
"This enhancement of performance can be attributed to the role of the 1-point correlation function of the Hubbard-Stratonovich field efficiently used as variational parameter in this implementation of SCPT.",
"For that reason, it is clearly appealing to design a mU-flow exploiting SCPT based on mixed 2PI EAs as starting point for the flow.",
"However, for the $O(N)$ model treated in this study and for models based on a two-body interaction in general, the mixed 2PI EA will be defined from a classical action with a Yukawa interaction, which causes problems in the derivations of the (m)U-flow equations.",
"Such problems actually already arise in the derivation of the master equation for the mixed 2PI EA and can be simply illustrated for the studied toy model based on the mixed generating functional (REF ).",
"In the derivation of the master equation for the original 2PI EA $\\Gamma ^{(\\mathrm {2PI})}_{\\mathfrak {s}}$ for the U-flow, one needs to introduce the bosonic matrix $W_{\\mathfrak {s}}^{(2)}$ that must be connected to the 2PI vertices to solve the resulting coupled flow equations.",
"As explained earlier, this connection is achieved by solving the Bethe-Salpeter equation which amounts to inverting a bosonic matrix according to definition (REF ).",
"For our (0+0)-D $O(N)$ model, the latter translates into: $\\mathcal {I}_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} = \\frac{1}{2} \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} M_{(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})} M^{\\mathrm {inv}}_{(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{2})} \\;.$ By following the same procedure in the mixed representation of this model, we would have to invert instead matrices with components of the form $\\mathcal {M}_{b_{1}(b_{2},b^{\\prime }_{2})}$ (labeled by a supercolor index $b$ and a bosonic index $(b,b^{\\prime })$ ), as a result of the Yukawa interaction $\\mathcal {Y}_{b_{1}(b_{2},b^{\\prime }_{2})}$ introduced at the expense of the two-body one $U_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}\\equiv U_{a_{1}a^{\\prime }_{1}a_{2}a^{\\prime }_{2}}$ .",
"Unfortunately, it is not straightforward to find inverses for such matrices.",
"For instance, the relations: $\\mathcal {I}_{(b_{1},b^{\\prime }_{1})(b_{2},b^{\\prime }_{2})} = \\sum _{b_{3}=1}^{N+1} \\mathcal {M}_{(b_{1},b^{\\prime }_{1})b_{3}} \\mathcal {M}^{\\mathrm {inv}}_{b_{3}(b_{2},b^{\\prime }_{2})} \\;,$ and $\\delta _{b_{1}b^{\\prime }_{2}} = \\frac{1}{2}\\sum _{b_{3},b^{\\prime }_{3}=1}^{N+1} \\mathcal {M}_{b_{1}(b_{3},b^{\\prime }_{3})} \\mathcal {M}^{\\mathrm {inv}}_{(b_{3},b^{\\prime }_{3})b_{2}} \\;,$ do not provide the right number of conditions to fix the components of $\\mathcal {M}^{\\mathrm {inv}}$ in an unambiguous manner.",
"Hence, extending the mU-flow implementation of the 2PI-FRG in the framework of the mixed representation of $O(N)$ models (in (0+0)-D as well as in finite dimensions), and more generally to theories based on Yukawa interactions, still requires a consequent work on the side of the formalism, that we postpone to subsequent projects.",
"With Figs.",
"REF , REFThe caption of Fig.",
"4.19 in Ref.",
"[46], which shows the same results as Fig.",
"REF , should indicate “$m^2=\\pm 1$ ” instead of “$m^2=+ 1$ ”.",
"and REF , we then discuss our mU-flow results exploiting SCPT based on the original 2PI EA as starting point.",
"We can already appreciate from Fig.",
"REF the gain in accuracy achieved by the mU-flow with its simplest correlated starting point (i.e.",
"the Hartree-Fock result) as compared to the pU-flow starting from the free theory.",
"For instance, the mU-flow results for $E_{\\mathrm {gs}}$ in Fig.",
"REF are already less than $2\\%$ away from the exact solution at $N_{\\mathrm {max}}=1$ over the whole tested range for the coupling constant (i.e.",
"$\\lambda /4!\\in [0,10]$ ) whereas such a performance is barely reached for $\\lambda /4!>1$ (i.e.",
"beyond the weakly-coupled regime) at $N_{\\mathrm {max}}<3$ for the pU-flow.",
"Actually, Fig.",
"REF shows that, over most of the range defined by $\\lambda /4!\\in [0,10]$ and at $m^{2}=1$ , the pU-flow at $N_{\\mathrm {max}}=2$ (where the two-particle vertex $\\Phi ^{(2)}_{\\mathfrak {s}}[G=\\overline{G}_{\\mathfrak {s}}]$ is flowing) is outperformed by the mU-flow at $N_{\\mathrm {max}}=1$ (where the two-particle vertex $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ is not flowing).",
"Put differently, we can achieve, even in strongly-coupled regimes, fairly satisfactory quantitative performances (accuracy below $2\\%$ for $E_{\\mathrm {gs}}$ and below $1\\%$ for $\\rho _{\\mathrm {gs}}$ in Fig.",
"REF ) with the mU-flow at $N_{\\mathrm {max}}=1$ , i.e.",
"by discarding the flow of $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ .",
"These statements have very profound consequences as the 2PI two-particle vertex $\\Phi ^{(2)}_{\\mathfrak {s}}[G=\\overline{G}_{\\mathfrak {s}}]$ (or its modified version $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ equivalently) is already significantly demanding to calculate for realistic theories, just like the 1PI two-particle vertex $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\equiv \\Gamma ^{(\\mathrm {1PI})(4)}_{k}\\big [\\psi =\\overline{\\psi }_{k}\\big ]$ in the 1PI-FRG framework.",
"We have thus illustrated in this way the power and usefulness of FRG implementations based on correlated starting points.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =±1m^{2}=\\pm 1 and N=1N=1 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =±1m^{2}=\\pm 1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).",
"The SCPT result labeled “SCPT 2PI EA 𝒪(U)\\mathcal {O}(U)” is identical to that corresponding to the legend “SCPT original 2PI EA 𝒪ℏ 2 \\mathcal {O}\\big (\\hbar ^{2}\\big )” in the figures of paper I.We stress nonetheless that the above remarks based on the results of Fig.",
"REF concern the mU-flow at $N_{\\mathrm {SCPT}}=1$ .",
"It is then quite natural to test the mU-flow for more involved starting points, i.e.",
"for $N_{\\mathrm {SCPT}}>1$ , which has never been done so far to our knowledge.",
"We exploit the simplicity of our toy model at $N=1$ to achieve this (see appendix REF for the expressions of the underlying flow equations), which will be sufficient to make our point.",
"Hence, Fig.",
"REF shows mU-flow results up to $N_{\\mathrm {max}}=3$ with $N_{\\mathrm {SCPT}}=1,2$ and 3 for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ in the unbroken- and broken-symmetry regimes of our toy model.",
"We point out first of all the appearance of the same stiffness issues as before with the used numerical toolsREF for the mU-flow at $N_{\\mathrm {SCPT}}=2$ and 3 with $N_{\\mathrm {max}}=2$ , hence explaining the absence of the corresponding curves in Fig.",
"REF .",
"This does not prevent us from noticing in this figure that, besides a few exceptions, the mU-flow at $N_{\\mathrm {SCPT}}=1$ is more performing than at $N_{\\mathrm {SCPT}}=2$ or 3 for a given truncation order $N_{\\mathrm {max}}$ , for $E_{\\mathrm {gs}}$ as well as for $\\rho _{\\mathrm {gs}}$ .",
"Although the starting point contains more and more information about the system to describe as $N_{\\mathrm {SCPT}}$ increases, we recall that it is the bare SCPT results (i.e.",
"without resummation) that the mU-flow procedure takes as inputs.",
"Such results take the form of diverging asymptotic series for the original 2PI EA $\\Gamma ^{(\\mathrm {2PI})}(G)$ (as was shown in paper I notably) and the corresponding estimates for $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ all worsen as the truncation order $N_{\\mathrm {SCPT}}$ increases, except for particularly small values of the coupling constant $\\lambda $ .",
"It was indeed only after applying a resummation procedure that we managed to turn SCPT into a systematically improvable technique in paper I.",
"In the light of the latter comments, we can conclude that, unlike resummation procedures, the mU-flow approach is not suited to extract the full information from the asymptotic series representing the 2PI EA (and the corresponding 2PI vertices) taken as input(s).",
"The mU-flow is thus most efficient at $N_{\\mathrm {SCPT}}=1$ , which is why we have also tested this approach in the unbroken- and broken-symmetry regimes for $N=2$ , as reported by Fig.",
"REF .",
"Our qualitative conclusions on the latter figure are particularly positive and thus echo those inferred from the mU-flow results of Fig.",
"REF : for $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ and for both signs of $m^{2}$ , we see that the mU-flow procedure at $N_{\\mathrm {max}}=1$ clearly improves the Hartree-Fock curve representing its starting point, and this mU-flow result is itself improved by increasing the truncation order $N_{\\mathrm {max}}$ until the curve corresponding to $N_{\\mathrm {max}}=3$ becomes barely distinguishable from the exact solution.",
"To conclude on section , the mU-flow with $N_{\\mathrm {SCPT}}=1$ clearly stands out from our 2PI-FRG study.",
"We have thus illustrated that FRG approaches designed by dressing interaction parts of classical actions with cutoff functions can exploit very efficiently correlated starting points.",
"However, we have also shown why such an introduction of cutoff functions leads to complications of the flow equations, translating into the appearance of the Bethe-Salpeter in the U-flow formalism, but this has to be contrasted with the possibility to achieve satisfactory performances by ignoring the flow of complicated objects such as 2PI two-particle vertices."
],
[
"Generalities",
"We now consider FRG methods based on density functionals in the form of 2PPI EAs.",
"The explicit present of the density as dof brings us one step closer to the nuclear EDF formalism, although we recall that we still rely on completely different types of equations (differential equations for FRG and self-consistent equations for EDF).",
"Since 2PPI EAs can be viewed as simplified versions of 2PI EAs, the 2PPI-FRG approaches designed so far rely on flow equations that are less demanding to treat numerically, as compared to those presented in the latter section for the 2PI-FRG.",
"The question is then whether one can implement a 2PPI-FRG approach as performing as the mU-flow version of the 2PI-FRG, but at a lower numerical cost.",
"This will be addressed in this section where we will once again present FRG formalisms in a general setting before focusing on the (0+0)-D $O(N)$ model for our numerical applications.",
"Before doing so, we make a state of play of the 2PPI-FRG approaches and corresponding applications achieved hitherto.",
"The 2PPI-FRG was first developed in the early 2000s by Polonyi and Sailer [190] and discussed in the context of quantum electrodynamics (QED).",
"Links between this approach and DFT were also emphasized in Ref.",
"[190], as well as in Ref.",
"[191] which puts forward the 2PPI-FRG as a means for calculating properties of nuclear systems in a systematic manner.",
"The first numerical application of this method came out almost a decade later with the work of Kemler and Braun [35], who took as theoretical laboratories the (0+0)-D $\\varphi ^4$ -theory in its unbroken-symmetry regime (i.e.",
"the studied toy model with $N=1$ and $m^2>0$ ) and to the (0+1)-D $\\varphi ^4$ -theory, still in the phase without SSB.",
"Corrections of the application to the latter toy model were pointed out subsequently by Rentrop and collaborators in Ref. [180].",
"A few years later, an extension of the 2PPI-FRG formalism, coined as Kohn-Sham FRG (KS-FRG) due to its connection with the Kohn-Sham scheme of DFT, was developed by Liang, Niu and Hatsuda [192].",
"To our knowledge, the KS-FRG has only been applied to the toy model considered in the present study with $N=1$ and in its unbroken-symmetry regime [192].",
"The 2PPI-FRG practitioners have also managed to treat a (1+1)-D model [36], [37], [38], [39], [40], called the Alexandrou-Negele nuclei [193].",
"Such a model reproduces some basic properties of the nuclear force (short-range repulsive and long-range attractive).",
"As any other toy model, the Alexandrou-Negele nuclei have been used to benchmark different theoretical approaches (see Ref.",
"[194] for the similarity renormalization group (SRG)) but they have also been exploited to describe real physical systems such as ultracold fermionic atoms interacting via a dipolar interaction [195].",
"For that reason in addition to the technical difficulties related to the inclusion of a space dimension, the work of Kemler and Braun presented in Ref.",
"[36] can be considered as a pioneering work for the 2PPI-FRG community.",
"More specifically, it presents results obtained for the gs energies (in comparison with Monte Carlo results [193]), intrinsic densities and density correlation functions.",
"Other subsequent applications were carried out for an infinite number of particles by Yokota and collaborators in order to study spinless nuclear matter with this model: this led to the determination of the nuclear saturation curve and other gs properties on the one hand [38] and to the calculation of spectral functions for the study of excited states on the other hand [39].",
"Then, applications to higher-dimensional systems were performed recently, still by Yokota and collaborators, on (2+1)-D [196], [40] and (3+1)-D [42] homogeneous electron gases [42], thus achieving the first two-dimensional and three-dimensional applications of the 2PPI-FRG.",
"Note also the work of Ref.",
"[43] which designs a 2PPI-FRG approach to describe classical liquids.",
"Finally, the 2PPI-FRG formalism has also been generalized to treat superfluid systems [41], thus marking a significant step towards the description of systems with competing instabilities.",
"The resulting approach can actually be considered as a DFT for systems with pairing correlations and echoes the work of Furnstahl and collaborators with SCPT based on 2PPI EAs [29].",
"As a next step, we outline the basic ingredients of the 2PPI EA formalism.",
"The generating functional underlying this FRG approach is given byAlthough the notations for the generating functionals $Z[K]$ and $W[K]$ are identical to those of Eq.",
"(REF ) related to the 2PI-FRG, we stress that $Z[K]$ and $W[K]$ are always defined by Eq.",
"(REF ) in all our discussions related to the 2PPI-FRG.",
": $Z[K] = e^{W[K]} = \\int \\mathcal {D}\\widetilde{\\psi }^{\\dagger }\\mathcal {D}\\widetilde{\\psi } \\ e^{-S\\big [\\widetilde{\\psi }^{\\dagger },\\widetilde{\\psi }\\big ] + \\int _{\\alpha } K_{\\alpha }\\widetilde{\\psi }_{\\alpha }^{\\dagger }\\widetilde{\\psi }_{\\alpha }} \\;,$ where we consider here a complex field $\\widetilde{\\psi }$ which is either bosonic or Grassmannian.",
"The different configurations of $\\widetilde{\\psi }$ are now specified by an $\\alpha $ -index which is essentially the same as that used in section in our presentation of the 2PI-FRG, with the exception of the charge index $c$ .",
"Namely, we now have $\\alpha \\equiv (i,x)$ with $x\\equiv (r, \\tau )$ and the following shorthand notation holds throughout the entire section : $\\int _{\\alpha } \\equiv \\sum _{i} \\int ^{1/T}_{0} d\\tau \\int d^{D-1}r \\;.$ The connected correlation functions deduced from Eq.",
"(REF ) are: $W_{\\alpha _{1}\\cdots \\alpha _{n}}^{(n)}[K] \\equiv \\frac{\\delta ^{n} W[K]}{\\delta K_{\\alpha _{1}} \\cdots \\delta K_{\\alpha _{n}}} \\;,$ which yields the density: $\\rho _{\\alpha } = W_{\\alpha }^{(1)}[K] \\;,$ at $n=1$ .",
"As opposed to the 2PI-FRG, the source $K$ is now a bosonic object regardless of the nature of $\\widetilde{\\psi }$ (i.e.",
"Grassmannian or not), which implies notably the following symmetry property for the correlation functions of Eq.",
"(REF ): $W^{(n)}_{\\alpha _{1}\\cdots \\alpha _{n}}[K]=W^{(n)}_{\\alpha _{P(1)}\\cdots \\alpha _{P(n)}}[K] \\;,$ to be compared with Eqs.",
"(REF ) for the 2PI-FRG formalism.",
"Finally, the following 2PPI EA is defined from the Schwinger functional $W[K]$ introduced in Eq.",
"(REF ): $\\begin{split}\\Gamma ^{(\\mathrm {2PPI})}[\\rho ] = -W[K] + \\int _{\\alpha } K_{\\alpha } \\rho _{\\alpha } \\;.\\end{split}$ All implementations of the 2PPI-FRG treated in this paper are applicable to any system whose classical action can be written as: $\\begin{split}S\\Big [\\widetilde{\\psi }^{\\dagger },\\widetilde{\\psi }\\Big ] = & \\ \\int _{\\alpha } \\widetilde{\\psi }_{\\alpha }^{\\dagger }\\left(\\partial _{\\tau } + \\hat{O}_{\\mathrm {kin}} + V_{\\alpha } - \\mu \\right)\\widetilde{\\psi }_{\\alpha } \\\\& + \\frac{1}{2}\\int _{\\alpha _{1},\\alpha _{2}} \\widetilde{\\psi }_{\\alpha _{1}}^{\\dagger }\\widetilde{\\psi }_{\\alpha _{2}}^{\\dagger } U_{\\alpha _{1}\\alpha _{2}} \\widetilde{\\psi }_{\\alpha _{2}}\\widetilde{\\psi }_{\\alpha _{1}} \\;,\\end{split}$ where $U_{\\alpha _{1}\\alpha _{2}} = U_{\\alpha _{2}\\alpha _{1}} \\;,$ $\\hat{\\mathcal {O}}_{\\mathrm {kin}}$ corresponds to the kinetic operatorFor instance, $\\hat{\\mathcal {O}}_{\\mathrm {kin}}=-\\frac{\\nabla ^{2}}{2m}$ for any non-relativistic system of mass $m$ .",
"and we place ourselves in the grand canonical ensemble here by using a chemical potential $\\mu $ to monitor the particle numberAlternatively, one can also simply impose a given particle number at the initial conditions of the 2PPI-FRG procedure since the particle number is conserved during the flow, as shown in Ref. [36]..",
"In the framework of the 2PPI-FRG, the flow parameter $\\mathfrak {s}$ is always dimensionless and the resulting approach thus does not implement the Wilsonian momentum-shell integrationTo be more specific, none of the 2PPI-FRG implementations exploited so far, i.e.",
"the approach of Polonyi and Sailer [190] and the KS-FRG of Liang and collaborators [192], implements the Wilsonian momentum-shell integration, although this is in principle possible via suitable choices of cutoff functions, as for the 2PI-FRG..",
"It is also typically the one-body potential $V$ and the two-body interaction $U$ which are rendered flow-dependent by introducing cutoff functions through the substitutions $U \\rightarrow U_{\\mathfrak {s}}$ and $V \\rightarrow V_{\\mathfrak {s}}$ applied to Eq.",
"(REF ).",
"After doing so, definition (REF ) becomes: $\\begin{split}\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PPI})}[\\rho ] = -W_{\\mathfrak {s}}[K] + \\int _{\\alpha } K_{\\alpha } \\rho _{\\alpha } \\;,\\end{split}$ where $\\rho _{\\alpha } \\equiv \\rho _{\\mathfrak {s},\\alpha }[K] = W^{(1)}_{\\mathfrak {s},\\alpha }[K] \\;.$ Moreover, although the classical action (REF ) only contains a two-body interaction, the 2PPI-FRG approaches discussed in this study can be straightforwardly extended to treat three-body and even higher-body interactions by including higher powers of $\\widetilde{\\psi }^{\\dagger }\\widetilde{\\psi }$ into Eq.",
"(REF ), similarly to Eq.",
"(REF ) for the 2PI-FRG.",
"We can also draw a parallel with our discussions of section on the 2PI-FRG by noticing that setting: $\\widetilde{\\psi }_{\\alpha } = \\widetilde{\\psi }^{\\dagger }_{\\alpha } = \\widetilde{\\varphi }_{a} \\;,$ $\\int _{\\alpha } = \\sum _{a=1}^{N} \\;,$ and imposing all kinetic terms to vanish enable us to get back the classical action $S\\big (\\vec{\\widetilde{\\varphi }}\\big )$ of the (0+0)-D $O(N)$ model as follows: $\\begin{split}S\\Big [\\widetilde{\\psi }^{\\dagger },\\widetilde{\\psi }\\Big ] = \\ & S\\Big (\\vec{\\widetilde{\\varphi }}\\Big ) \\\\= \\ & \\sum _{a=1}^{N} \\widetilde{\\varphi }_{a} V_{a} \\widetilde{\\varphi }_{a} + \\frac{1}{2} \\sum _{a_{1},a_{2}=1}^{N} \\widetilde{\\varphi }_{a_{1}} \\widetilde{\\varphi }_{a_{2}} U_{a_{1}a_{2}} \\widetilde{\\varphi }_{a_{2}} \\widetilde{\\varphi }_{a_{1}} \\;,\\end{split}$ with the one-body potential: $V_{a} = \\frac{m^{2}}{2} \\quad \\forall a \\;,$ and the two-body interaction: $U_{a_{1}a_{2}} = \\frac{\\lambda }{12} \\quad \\forall a_{1}, a_{2} \\;.$ Therefore, as for the 2PI-FRG, the 2PPI-FRG formalism can be directly applied to the studied toy model in its original representation, as opposed to its mixed and collective counterparts.",
"The 2PPI-FRG flow equations to treat for the (0+0)-D $O(N)$ model can thus be straightforwardly inferred from the generic ones given in this section after exploiting Eqs.",
"(REF ), (REF ) and (REF ).",
"We finally stress that, to our knowledge, the present study is the first presenting an application of the 2PPI-FRG to an $O(N)$ model with $N>1$ ."
],
[
"Standard 2PPI functional renormalization group",
"We start by discussing the standard version of the 2PPI-FRG, as proposed by Polonyi and Sailer [190] and then exploited e.g.",
"by Kemler and Braun [35], [36].",
"After performing the substitutions $V \\rightarrow V_{\\mathfrak {s}}$ and $U \\rightarrow U_{\\mathfrak {s}}$ into the classical action (REF ), we can show that the corresponding flow-dependent 2PPI EA $\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PPI})}[\\rho ]$ , defined by Eq.",
"(REF ), satisfies the master equation: $\\begin{split}\\dot{\\Gamma }^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ] = & \\ \\int _{\\alpha } \\dot{V}_{\\mathfrak {s},\\alpha } \\rho _{\\alpha } + \\frac{1}{2} \\mathrm {STr}\\left[\\dot{U}_{\\mathfrak {s}} \\left(\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PPI})(2)}[\\rho ]\\right)^{-1} \\right] \\\\& + \\frac{1}{2} \\int _{\\alpha _{1},\\alpha _{2}} \\dot{U}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\rho _{\\alpha _{1}} \\rho _{\\alpha _{2}} \\;,\\end{split}$ where the 2PPI vertices satisfy: $\\Gamma _{\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}}^{(\\mathrm {2PPI})(n)}[\\rho ] \\equiv \\frac{\\delta ^{n}\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PPI})}[\\rho ]}{\\delta \\rho _{\\alpha _{1}} \\cdots \\delta \\rho _{\\alpha _{n}}} \\;.$ Result (REF ) is thus the counterpart of the Wetterich equation for the standard 2PPI-FRG.",
"The flow equation (REF ) via a vertex expansion procedure based on the following expression of $\\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ]$ : $\\begin{split}\\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ] = & \\ \\overline{\\Gamma }^{(\\mathrm {2PPI})}_{\\mathfrak {s}} \\\\& + \\sum _{n=2}^{\\infty } \\frac{1}{n!}",
"\\int _{\\alpha _{1},\\cdots ,\\alpha _{n}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}}^{(\\mathrm {2PPI})(n)} \\\\& \\times \\left(\\rho -\\overline{\\rho }_{\\mathfrak {s}}\\right)_{\\alpha _{1}} \\cdots \\left(\\rho -\\overline{\\rho }_{\\mathfrak {s}}\\right)_{\\alpha _{n}} \\;,\\end{split}$ where $\\overline{\\Gamma }^{(\\mathrm {2PPI})}_{\\mathfrak {s}} \\equiv \\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho =\\overline{\\rho }_{\\mathfrak {s}}]$ , $\\overline{\\Gamma }^{(\\mathrm {2PPI})(n)}_{\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}} \\equiv \\left.\\frac{\\delta ^{n}\\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PPI})}[\\rho ]}{\\delta \\rho _{\\alpha _{1}}\\cdots \\delta \\rho _{\\alpha _{n}}}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}}$ and the flowing density $\\overline{\\rho }_{\\mathfrak {s}}$ extremizes the flowing 2PPI EA: $\\left.",
"\\frac{\\delta \\Gamma _{\\mathfrak {s}}^{(\\mathrm {2PPI})}[\\rho ]}{\\delta \\rho _{\\alpha }}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}} = 0 \\quad \\forall \\alpha ,\\mathfrak {s} \\;,$ similarly to Eq.",
"(REF ) for the 1PI-FRG.",
"Expanding both sides of Eq.",
"(REF ) around the configuration $\\rho =\\overline{\\rho }_{\\mathfrak {s}}$ and then identifying the terms involving identical powers of $\\rho -\\overline{\\rho }_{\\mathfrak {s}}$ gives us the infinite set of differential equations underlying the standard 2PPI-FRG.",
"The first equations of this hierarchy areIn principle, the chemical potential $\\mu $ also depends on the flow parameter $\\mathfrak {s}$ in the present situation, which translates into an extra flow equation in the infinite hierarchies underpinning 2PPI-FRG approaches.",
"However, we discard this flow equation here as it is not relevant for our (0+0)-D applications.",
": $\\dot{\\overline{\\Gamma }}_{\\mathfrak {s}}^{(\\mathrm {2PPI})} = \\int _{\\alpha } \\dot{V}_{\\mathfrak {s},\\alpha } \\overline{\\rho }_{\\mathfrak {s},\\alpha } + \\frac{1}{2} \\int _{\\alpha _{1},\\alpha _{2}} \\dot{U}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\left( \\overline{G}_{\\mathfrak {s},\\alpha _{2}\\alpha _{1}} + \\overline{\\rho }_{\\mathfrak {s},\\alpha _{2}} \\overline{\\rho }_{\\mathfrak {s},\\alpha _{1}} \\right) \\;,$ $\\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{1}} = \\int _{\\alpha _{2}} \\overline{G}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\left(- \\dot{V}_{\\mathfrak {s},\\alpha _{2}} + \\frac{1}{2}\\int _{\\alpha _{3},\\cdots ,\\alpha _{6}} \\dot{U}_{\\mathfrak {s},\\alpha _{3}\\alpha _{4}}\\overline{G}_{\\mathfrak {s},\\alpha _{3}\\alpha _{5}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{2}\\alpha _{5}\\alpha _{6}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{6}\\alpha _{4}} - \\int _{\\alpha _{3}} \\dot{U}_{\\mathfrak {s},\\alpha _{2}\\alpha _{3}}\\overline{\\rho }_{\\mathfrak {s},\\alpha _{3}} \\right) \\;,$ $\\begin{split}\\dot{\\overline{\\Gamma }}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}}^{(\\mathrm {2PPI})(2)} = & \\int _{\\alpha _{3}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{3}} \\overline{\\Gamma }^{(\\mathrm {2PPI})(3)}_{\\mathfrak {s},\\alpha _{3}\\alpha _{1}\\alpha _{2}} + \\dot{U}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} + \\int _{\\alpha _{3},\\cdots ,\\alpha _{8}} \\dot{U}_{\\mathfrak {s},\\alpha _{3}\\alpha _{4}} \\overline{G}_{\\mathfrak {s},\\alpha _{3}\\alpha _{5}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{1}\\alpha _{5}\\alpha _{6}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{6}\\alpha _{7}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{2}\\alpha _{7}\\alpha _{8}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{8}\\alpha _{4}} \\\\& - \\frac{1}{2}\\int _{\\alpha _{3},\\cdots ,\\alpha _{6}} \\dot{U}_{\\mathfrak {s},\\alpha _{3}\\alpha _{4}}\\overline{G}_{\\mathfrak {s},\\alpha _{3}\\alpha _{5}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}\\alpha _{5}\\alpha _{6}}^{(\\mathrm {2PPI})(4)} \\overline{G}_{\\mathfrak {s},\\alpha _{6}\\alpha _{4}} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}\\alpha _{3}}^{(\\mathrm {2PPI})(3)} = & \\ \\int _{\\alpha _{4}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{4}} \\overline{\\Gamma }^{(\\mathrm {2PPI})(4)}_{\\mathfrak {s},\\alpha _{4}\\alpha _{1}\\alpha _{2}\\alpha _{3}} \\\\& - \\Bigg ( \\int _{\\alpha _{4},\\cdots ,\\alpha _{11}} \\dot{U}_{\\mathfrak {s},\\alpha _{4}\\alpha _{5}} \\overline{G}_{\\mathfrak {s},\\alpha _{4}\\alpha _{6}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{1}\\alpha _{6}\\alpha _{7}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{7}\\alpha _{8}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{2}\\alpha _{8}\\alpha _{9}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{9}\\alpha _{10}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{3}\\alpha _{10}\\alpha _{11}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{11}\\alpha _{5}} \\\\& + (\\alpha _{2},\\alpha _{1},\\alpha _{3}) + (\\alpha _{1},\\alpha _{3},\\alpha _{2}) \\Bigg ) + \\Bigg (\\int _{\\alpha _{4},\\cdots ,\\alpha _{9}} \\dot{U}_{\\mathfrak {s},\\alpha _{4}\\alpha _{5}} \\overline{G}_{\\mathfrak {s},\\alpha _{4}\\alpha _{6}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}\\alpha _{6}\\alpha _{7}}^{(\\mathrm {2PPI})(4)} \\overline{G}_{\\mathfrak {s},\\alpha _{7}\\alpha _{8}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{3}\\alpha _{8}\\alpha _{9}}^{(\\mathrm {2PPI})(3)} \\overline{G}_{\\mathfrak {s},\\alpha _{9}\\alpha _{5}} \\\\& + (\\alpha _{1},\\alpha _{3},\\alpha _{2}) + (\\alpha _{2},\\alpha _{3},\\alpha _{1}) \\Bigg ) - \\frac{1}{2}\\int _{\\alpha _{4},\\cdots ,\\alpha _{7}} \\dot{U}_{\\mathfrak {s},\\alpha _{4}\\alpha _{5}}\\overline{G}_{\\mathfrak {s},\\alpha _{4}\\alpha _{6}} \\overline{\\Gamma }_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{6}\\alpha _{7}}^{(\\mathrm {2PPI})(5)} \\overline{G}_{\\mathfrak {s},\\alpha _{7}\\alpha _{5}} \\;,\\end{split}$ where we have used the definition: $\\overline{G}^{-1}_{\\mathfrak {s}} \\equiv \\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})(2)} \\;,$ together with the notation set by Eq.",
"(REF ) for $\\alpha $ -indices.",
"Regarding then the choice of cutoff functions, $U_{\\mathfrak {s}}$ must satisfy the same boundary conditions as those encountered in the U-flow implementation of the 2PI-FRG, i.e.",
": [left=]align Us=si,1 2 = 0 1, 2 .",
"Us=sf = U .",
"For the analytical form of the one-body potential $V_{\\mathfrak {s}}$ , all aforementioned state-of-the-art implementations of the 2PPI-FRG simply use $V_{\\mathfrak {s}}=V$ $\\forall \\mathfrak {s}$ , for which the starting point of the flow coincide with the free theoryRef.",
"[191] also puts forward the alternative $V_{\\mathfrak {s}}=(1-\\mathfrak {s})V_{\\mathrm {KS}}$ $\\forall \\mathfrak {s}$ (with $\\mathfrak {s}_{\\mathfrak {i}}=0$ and $\\mathfrak {s}_{\\mathfrak {f}}=1$ ), for which the starting point of the flow corresponds to the Kohn-Sham system specified by the Kohn-Sham potential $V_{\\mathrm {KS}}$ .",
"However, this option has never been exploited to treat any model in the framework of published studies to our knowledge and we defer the investigation of this approach to future work.",
"(according to Eq.",
"(REF )), as for the pU-flow version of the 2PI-FRG.",
"We will therefore also follow this path to obtain all numerical results presented in this section , using also the same cutoff function for $U_{\\mathfrak {s}}$ as for our 2PI-FRG study of section (i.e.",
"$U_{\\mathfrak {s}}=\\mathfrak {s}U$ with $\\mathfrak {s}$ still running from $\\mathfrak {s}_{\\mathrm {i}}=0$ to $\\mathfrak {s}_{\\mathrm {f}}=1$ ).",
"However, the initial conditions for the 2PPI-FRG are not as easily determined from the free theory as it was for the pU-flow implementation of the 2PI-FRG.",
"The underlying reason is that the Legendre transforms defining 2PPI EAs can not be done explicitlySee Refs.",
"[197], [46] for technical explanations on this point., which implies that we can not express $\\Gamma ^{(\\mathrm {2PPI})}[\\rho ]$ , or its non-interacting version $\\Gamma _{0}^{(\\mathrm {2PPI})}[\\rho ]$ , explicitly in terms of $\\rho $ at finite dimensions.",
"As a result, one must express the 2PPI vertices: $\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\alpha _{1}\\cdots \\alpha _{n}}^{(\\mathrm {2PPI})(n)} = \\left.\\frac{\\delta ^{n}\\Gamma _{0}^{(\\mathrm {2PPI})}[\\rho ]}{\\delta \\rho _{\\alpha _{1}}\\cdots \\delta \\rho _{\\alpha _{n}}}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}} \\;,$ in terms of derivatives of the non-interacting version $W_{0}[K]$ of the Schwinger functional expressed by Eq.",
"(REF ) (with Eq.",
"(REF ) as classical action), i.e.",
": $\\begin{split}W_{0}[K] = \\zeta \\mathrm {STr}[\\ln (G_{K})] \\;,\\end{split}$ where $G^{-1}_{K,\\alpha _{1}\\alpha _{2}} = \\left(\\partial _{\\tau _{1}} + \\hat{O}_{\\mathrm {kin}} + V_{\\alpha _{1}} - \\mu - K_{\\alpha _{1}}\\right) \\delta _{\\alpha _{1}\\alpha _{2}}\\;.$ and $\\zeta =\\pm 1$ still distinguishes the situations where $\\widetilde{\\psi }$ is a bosonic or a Grassmann field, as in section .",
"The connection between the 2PPI vertices $\\overline{\\Gamma }^{(\\mathrm {2PPI})(n)}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}$ and the derivatives of $W_{0}[K]$ can be achieved from the chain rule: $\\frac{\\delta }{\\delta \\rho _{\\alpha _{1}}} = \\int _{\\alpha _{2}} \\frac{\\delta K_{\\alpha _{2}}}{\\delta \\rho _{\\alpha _{1}}} \\frac{\\delta }{\\delta K_{\\alpha _{2}}} = \\int _{\\alpha _{2}} \\left(W^{(2)}_{\\mathfrak {s}}[K]\\right)_{\\alpha _{1}\\alpha _{2}}^{-1} \\frac{\\delta }{\\delta K_{\\alpha _{2}}} \\;,$ which follows from Eq.",
"(REF ), and we have used $W^{(n)}_{\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}}[K] \\equiv \\frac{\\delta ^{n} W_{\\mathfrak {s}}[K]}{\\delta K_{\\alpha _{1}} \\cdots \\delta K_{\\alpha _{n}}}$ .",
"From this, we can indeed infer the relationsNote that Eq.",
"(red4.452) of Ref.",
"[46] is erroneous and should be replaced by Eq.",
"(REF ).",
": $\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\alpha _{1}\\alpha _{2}}^{(\\mathrm {2PPI})(2)} = \\left(W^{(2)}_{0}[K=0]\\right)_{\\alpha _{1}\\alpha _{2}}^{-1} \\;,$ $\\begin{split}\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\alpha _{1} \\cdots \\alpha _{n}}^{(\\mathrm {2PPI})(n)} = & \\ \\int _{\\alpha _{2n-2}} \\left(W^{(2)}_{0}[K]\\right)_{\\alpha _{n}\\alpha _{2n-2}}^{-1} \\frac{\\delta }{\\delta K_{\\alpha _{2n-2}}} \\cdots \\\\& \\times \\int _{\\alpha _{n+1}} \\left(W^{(2)}_{0}[K]\\right)_{\\alpha _{3}\\alpha _{n+1}}^{-1} \\\\& \\times \\frac{\\delta }{\\delta K_{\\alpha _{n+1}}} \\left(W^{(2)}_{0}[K]\\right)_{\\alpha _{1}\\alpha _{2}}^{-1} \\bigg |_{K=0} \\quad \\forall n\\ge 3 \\;,\\end{split}$ whereas the initial condition for the flowing density is simply given by: $\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\alpha } = W_{0,\\alpha }^{(1)}[K=0] \\;.$ Some 2PPI-FRG studies like that of Ref.",
"[36] exploit the master equation expressing $\\dot{W}_{\\mathfrak {s}}[K]$ (instead of Eq.",
"(REF ) expressing $\\dot{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})}[\\rho ]$ ) and treat both sides of this equation with an expansion around $K_{\\alpha }=0$ $\\forall \\alpha $ (instead of the vertex expansion around $\\rho =\\overline{\\rho }_{\\mathfrak {s}}$ ).",
"In this way, such a lengthy determination of initial conditions can be avoided since the flowing objects are $W^{(n)}_{\\mathfrak {s}}[K=0]$ and the 2PPI vertices can be recovered at the end of the flow if necessary through relations such as Eqs.",
"(REF ) and (REF ).",
"Fortunately, in (0+0)-D, the 2PPI EA $\\Gamma ^{(\\mathrm {2PPI})}(\\rho )$ (defined by Eq.",
"(REF )) coincides with the 2PI one $\\Gamma ^{(\\mathrm {2PI})}(G)$ (defined by Eq.",
"(REF )) for which we have an explicit expression in terms of $G$ in the non-interacting case, as given by Eq.",
"(REF ) in arbitrary dimensions.",
"This enables us to determine the initial conditions for the 2PPI vertices of the (0+0)-D $O(N)$ model as follows: $\\begin{split}\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},a_{1}\\cdots a_{n}}^{(\\mathrm {2PPI})(n)} = & -\\frac{1}{2} \\left.\\frac{\\partial ^{n}}{\\partial \\rho _{a_{1}}\\cdots \\partial \\rho _{a_{n}}} \\sum _{a_{n+1}={1}}^{N} \\ln (\\rho _{a_{n+1}}) \\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}} \\\\& \\quad \\forall n\\ge 2 \\;,\\end{split}$ and the corresponding flowing density satisfies: $\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},a} = \\frac{1}{m^{2}} \\quad \\forall a \\;.$ Then, to truncate the infinite hierarchy of equations underpinning the standard 2PPI-FRG, one can design the three following conditions: For the standard U-flow (sU-flow): $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})(n)} = 0 \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;.$ For the plain U-flow (pU-flow): $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})(n)} = \\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PPI})(n)} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;.$ For the improved U-flow (iU-flow): $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})(n)} = \\left.\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PPI})(n)}\\right|_{\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}\\rightarrow \\overline{\\rho }_{\\mathfrak {s}}} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;.$ Although the standard 2PPI-FRG has already been applied to the (0+0)-D model considered here (only in its unbroken-symmetry regime with $N=1$ ) [35], it should be stressed at the present stage that only the simplest truncation scheme, i.e.",
"the sU-flow, was exploited in this case.",
"The sU-flow truncation is indeed clearly the most drastic whereas the iU-flow one is a priori the most refined.",
"The condition underpinning the sU-flow enables us to content ourselves with the determination of the initial conditions $\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PPI})(n)}$ up to $n=N_{\\mathrm {max}}$ .",
"However, the pU-flow requires us to pursue this procedure up to $n=N_{\\mathrm {max}}+2$ since the differential equations (resulting from the vertex expansion) expressing $\\dot{\\overline{\\Gamma }}_{\\mathfrak {s}}^{(\\mathrm {2PPI})(n)}$ (with $2\\le n \\le N_{\\mathrm {max}}$ ) depend on other 2PPI vertices $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})(m)}$ of order up to $m=n+2$ .",
"Finally, the iU-flow was first introduced in Ref.",
"[35] under the name “RG improvement” or “RGi”The substitution $\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}\\rightarrow \\overline{\\rho }_{\\mathfrak {s}}$ can not be directly carried out in the framework of finite-dimensional theories for which it is not possible to express $\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PPI})(n)}$ explicitly in terms of $\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}$ (despite this, Ref.",
"[35] exploited an alternative definition of the iU-flow allowing for an application to a (0+1)-D $\\varphi ^{4}$ -theory).",
"This is why the iU-flow has not been fully worked out for finite-dimensional theories up to now..",
"Finally, the gs energy and density of our (0+0)-D $O(N)$ model are obtained at the end of the standard 2PI-FRG flow from the relations: $E^\\text{s2PPI-FRG}_{\\mathrm {gs}} = \\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}^{(\\mathrm {2PPI})} \\;,$ $\\rho ^\\text{s2PPI-FRG}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},a} \\;.$ Our numerical results for the standard 2PPI-FRG up to $N_{\\mathrm {max}}=3$ and at $N=2$ are reported in Fig.",
"REF .",
"The sU-flow can not be implemented at $N_{\\mathrm {max}}=1$ since condition (REF ) implies that $\\overline{G}^{-1}_{\\mathfrak {s},a_{1}a_{2}}=\\overline{\\Gamma }_{\\mathfrak {s},a_{1}a_{2}}^{(\\mathrm {2PPI})(2)}=0$ $\\forall a_{1},a_{2},\\mathfrak {s}$ at this truncation order, which renders the associated flow equations (REF ) and (REF ) ill-defined.",
"Moreover, in accordance with a previous remark on the pU-flow implementation of the 2PI-FRG, Fig.",
"REF only displays results in the unbroken-symmetry regime as the free theory, used as starting point for the flow, is ill-defined at $m^{2}<0$ .",
"From this figure and for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ , it can clearly be seen that, for a given truncation order $N_{\\mathrm {max}}$ , the sU-flow is outperformed by the pU-flow which is itself clearly less performing than the iU-flow, as expected.",
"As a next step, we will examine what are the connections between these various versions of the standard 2PPI-FRG and the KS-FRG.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0)."
],
[
"Kohn-Sham functional renormalization group",
"The KS-FRG was put forward as a “novel optimization theory of FRG with faster convergence” [192], as compared to other 2PPI-FRG approaches (especially the sU-flow and pU-flow of the standard 2PPI-FRG).",
"It is based on the splitting of the total 2PPI EA $\\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ] \\equiv \\Gamma ^{(\\mathrm {2PPI})}[\\rho ;V_{\\mathfrak {s}},U_{\\mathfrak {s}}]$ (introduced in Eq.",
"(REF )) into the mean-field part $\\Gamma _{\\mathrm {KS},\\mathfrak {s}}[\\rho ] \\equiv \\Gamma ^{(\\mathrm {2PPI})}[\\rho ;V_{\\mathfrak {s}}=V_{\\mathrm {KS},\\mathfrak {s}},U_{\\mathfrak {s}}=0]$ (specified by a flow-dependent Kohn-Sham potential $V_{\\mathrm {KS},\\mathfrak {s}}$ ) and the correlation part $\\gamma _{\\mathfrak {s}}[\\rho ]$ , i.e.",
": $\\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ] = \\Gamma _{\\mathrm {KS},\\mathfrak {s}}[\\rho ] + \\gamma _{\\mathfrak {s}}[\\rho ] \\;.$ Plugging this decomposition into the exact flow equation for $\\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ]$ , expressed by Eq.",
"(REF ), gives us: $\\begin{split}\\dot{\\gamma }_{\\mathfrak {s}}[\\rho ] = & \\ \\int _{\\alpha _{1}} \\rho _{\\alpha _{1}} \\left(\\dot{V}_{\\mathfrak {s},\\alpha _{1}} + \\int _{\\alpha _{2}} \\overline{\\Gamma }^{(2)}_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{2}} \\right) \\\\& + \\frac{1}{2} \\mathrm {STr}\\left[\\dot{U}_{\\mathfrak {s}} \\left( \\Gamma ^{(2)}_{\\mathrm {KS},\\mathfrak {s}}[\\rho ] + \\gamma ^{(2)}_{\\mathfrak {s}}[\\rho ] \\right)^{-1} \\right] \\\\& + \\frac{1}{2} \\int _{\\alpha _{1},\\alpha _{2}} \\dot{U}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\rho _{\\alpha _{1}} \\rho _{\\alpha _{2}} \\;,\\end{split}$ after using the following chain rule: $\\begin{split}\\dot{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}[\\rho ] = & \\ \\int _{\\alpha _{1},\\alpha _{2}} \\frac{\\delta \\Gamma _{\\mathrm {KS},\\mathfrak {s}}[\\rho ]}{\\delta V_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}}} \\frac{\\delta V_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}}}{\\delta \\overline{\\rho }_{\\mathfrak {s},\\alpha _{2}}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{2}} \\\\= & - \\int _{\\alpha _{1},\\alpha _{2}} \\rho _{\\alpha _{1}} \\overline{\\Gamma }^{(2)}_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{2}} \\;,\\end{split}$ which assumes that $\\frac{\\delta ^{n} \\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}}{\\delta \\overline{\\rho }_{\\mathfrak {s},\\alpha _{1}}\\cdots \\delta \\overline{\\rho }_{\\mathfrak {s},\\alpha _{n}}} = \\left.\\frac{\\delta ^{n} \\Gamma _{\\mathrm {KS},\\mathfrak {s}}[\\rho ]}{\\delta \\rho _{\\alpha _{1}}\\cdots \\delta \\rho _{\\alpha _{n}}}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}} \\equiv \\overline{\\Gamma }^{(n)}_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}}$ $\\forall \\alpha _{1},\\cdots ,\\alpha _{n},\\mathfrak {s}$ .",
"Eq.",
"(REF ) is the master equation of the KS-FRG.",
"It is also turned into an infinite hierarchy of coupled differential equations through a vertex expansion procedure, which relies in this situation on the expansion of the correlation part $\\gamma _{\\mathfrak {s}}[\\rho ]$ : $\\begin{split}\\gamma _{\\mathfrak {s}}[\\rho ] = & \\ \\overline{\\gamma }_{\\mathfrak {s}} \\\\& + \\sum _{n=2}^{\\infty } \\int _{\\alpha _{1},\\cdots ,\\alpha _{n}} \\overline{\\gamma }^{(n)}_{\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}} \\\\& \\times \\left(\\rho -\\overline{\\rho }_{\\mathfrak {s}}\\right)_{\\alpha _{1}} \\cdots \\left(\\rho -\\overline{\\rho }_{\\mathfrak {s}}\\right)_{\\alpha _{n}} \\;,\\end{split}$ with $\\overline{\\gamma }_{\\mathfrak {s}} \\equiv \\gamma _{\\mathfrak {s}}[\\rho =\\overline{\\rho }_{\\mathfrak {s}}]$ , $\\overline{\\gamma }^{(n)}_{\\mathfrak {s},\\alpha _{1}\\cdots \\alpha _{n}} \\equiv \\left.\\frac{\\delta ^{n}\\gamma _{\\mathfrak {s}}[\\rho ]}{\\delta \\rho _{\\alpha _{1}}\\cdots \\delta \\rho _{\\alpha _{n}}}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}}$ and $\\overline{\\rho }_{\\mathfrak {s}}$ now verifies the equality: $\\left.",
"\\frac{\\delta \\gamma _{\\mathfrak {s}}[\\rho ]}{\\delta \\rho _{\\alpha }}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}} = 0 \\quad \\forall \\alpha ,\\mathfrak {s} \\;.$ The derivatives $\\overline{\\Gamma }^{(n)}_{\\mathrm {KS},\\mathfrak {s}}$ involved in the coupled differential equations, obtained from this vertex expansion and given in appendix REF , are specified from the flow-dependent Kohn-Sham potential $V_{\\mathrm {KS},\\mathfrak {s}}$ which is itself determined by solving, at each step of the flow, the Kohn-Sham equation [5], [6] in the formFrom Eqs.",
"(REF ) and (REF ) together with splitting (REF ), one can see that the flowing density $\\overline{\\rho }_{\\mathfrak {s}}$ of the KS-FRG still fulfills condition (REF ) exploited within the standard 2PPI-FRG.",
": $\\left.",
"\\frac{\\delta \\Gamma _{\\mathrm {KS},\\mathfrak {s}}[\\rho ]}{\\delta \\rho _{\\alpha }}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}} = 0 \\quad \\forall \\alpha ,\\mathfrak {s} \\;.$ Therefore, the enhanced convergence of the KS-FRG, as compared to some standard 2PPI-FRG approaches, comes at the price of solving the Kohn-Sham equation at each step of the flow.",
"Although this extra cost is negligible for the studied (0+0)-D toy model (see appendix REF ), one should stress that it might substantially burden the numerical resolution of the KS-FRG flow equations for more realistic models.",
"The KS-FRG is designed to use the same starting points as the standard 2PPI-FRG.",
"The free theoryREF will thus still be exploited to initialize the flow for our KS-FRG applications.",
"In any case, the initial conditions for the flowing objects $\\overline{\\gamma }_{\\mathfrak {s}}$ and $\\overline{\\gamma }^{(n)}_{\\mathfrak {s}}$ are trivial and read: $\\overline{\\gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = 0 \\;,$ $\\overline{\\gamma }^{(n)}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\alpha _{1}, \\cdots , \\alpha _{n}} = 0 \\quad \\forall \\alpha _{1}, \\cdots , \\alpha _{n}, ~ \\forall n \\ge 2 \\;,$ whereas the truncation of the infinite tower of differential equations generated by the vertex expansion of Eq.",
"(REF ) is simply implemented by: $\\overline{\\gamma }_{\\mathfrak {s}}^{(n)} = \\overline{\\gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(n)} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;,$ for a given truncation order $N_{\\mathrm {max}}$ .",
"The gs energy and density of the (0+0)-D $O(N)$ model are estimated from the flowing objects of the KS-FRG via the relations (see appendix REF for the determination of $\\Gamma _{\\mathrm {KS},\\mathfrak {s}}(\\rho )$ ): $\\begin{split}E^\\text{KS-FRG}_{\\mathrm {gs}} = & \\ \\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}^{(\\mathrm {2PPI})} = \\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} + \\overline{\\gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\\\= & - \\frac{1}{2} \\sum _{a=1}^{N} \\ln \\big (2\\pi \\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},a}\\big ) + \\overline{\\gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\;,\\end{split}$ $\\rho ^\\text{KS-FRG}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},a} \\;.$ Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Finally, we analyze numerical results for the KS-FRG at $N=2$ with Figs.",
"REF and REF .",
"The KS-FRG was introduced to improve the convergence of the standard 2PPI-FRG in its most basic implementations, i.e.",
"with the sU-flow or pU-flow as truncation schemes.",
"This improvement is illustrated by Fig.",
"REF , which clearly shows that, at the same truncation order $N_{\\mathrm {max}}$ , the KS-FRG outperforms the pU-flow version of the standard 2PPI-FRG (which itself outperforms its sU-flow counterpart according to our previous discussion on Fig.",
"REF ) over the whole range of tested values for the coupling constant (i.e.",
"for $\\lambda /4!",
"\\in [0,10]$ ) for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ up to $N_{\\mathrm {max}}=3$ .",
"We also point out interesting connections between different FRG approaches tested so far: Fig.",
"REF illustrates that the pU-flow version of the 2PI-FRG, the iU-flow version of the 2PPI-FRG and the KS-FRG lead to identical results for the studied toy model.",
"The equivalence between the pU-flow of the 2PI-FRG on the one hand and the latter two 2PPI-FRG approaches on the other hand would no longer be valid at finite dimensions since the 2PI EA $\\Gamma ^{(\\mathrm {2PI})}(G)$ and the 2PPI EA $\\Gamma ^{(\\mathrm {2PPI})}(\\rho )$ only coincide in (0+0)-D. However, the connection between the iU-flow and the KS-FRG remains unaffected as dimension increases since both of these approaches are based on the 2PPI EA $\\Gamma ^{(\\mathrm {2PPI})}(\\rho )$ .",
"The latter remark enables us to characterize the KS-FRG as a more easily implementable version of the iU-flow, notably since it is in general not straightforward (if not impossible) to generalize the substitution $\\overline{\\rho }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}\\rightarrow \\overline{\\rho }_{\\mathfrak {s}}$ defining the iU-flow in Eq.",
"(REF ) to finite-dimensional theoriesREF .",
"Furthermore, the KS-FRG completely bypasses the cumbersome determination of the initial conditions $\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PPI})(n)}$ that must be performed up to $n=N_{\\mathrm {max}}+2$ within the iU-flow but this should be put in contrast with the evaluation of the derivatives $\\overline{\\Gamma }^{(n)}_{\\mathrm {KS},\\mathfrak {s}}$ through the KS-FRG flow, requiring notably the resolution of the Kohn-Sham equation at each step of the flow.",
"To conclude this section on the 2PPI-FRG, we get back to the equivalence between the most performing 2PPI-FRG approaches tested in this study and the pU-flow version of the 2PI-FRG illustrated with Fig.",
"REF .",
"Although this only holds in (0+0)-D, it certainly also highlights the relevance of a mU-flow implementation of the 2PPI-FRG even for certain finite-dimensional theories, considering the excellent performances of the 2PI-FRG mU-flow reported in section .",
"Such an implementation of the mU-flow in the 2PPI EA framework would indeed be significantly less demanding to implement numerically (notably since it would not rely on the Bethe-Salpeter equation, as opposed to its 2PI counterpart).",
"However, on the side of the formalism, the initial conditions can not be as readily determined as in the 2PI case, still because the 2PPI EA $\\Gamma ^{(\\mathrm {2PPI})}_{\\mathfrak {s}}[\\rho ]$ (and the 2PPI vertices) can not be explicitly expressed in terms of $\\rho $ in general.",
"Although such a feature might be handled with the inversion method (developed by Fukuda and collaborators [33], [198]) to successfully construct a mU-flow for finite-dimensional systems within the 2PPI EA framework, it clearly complexifies the underlying formalism as compared to the 2PI case.",
"This illustrates that the 2PPI-FRG framework provides us with numerically less demanding alternatives to 2PI-FRG approaches, but the underlying formalism is also less flexible, notably to exploit correlated starting points."
],
[
"Conclusion on the functional renormalization group study",
"In this comparative study, we have investigated three FRG schemes, the 1PI-FRG, the 2PI-FRG and the 2PPI-FRG, formulated respectively from 1PI, 2PI and 2PPI EAs.",
"The results of sections , and were presented mostly for a single value of $N$ (i.e.",
"at $N=2$ ) notably since, as opposed to the original LE with the divergences produced by the Goldstone modes at $N\\ge 2$ (see paper I), the FRG formalisms considered here do not discriminate specific values of $N$ .",
"One can actually check in Ref.",
"[46] that the performances of the tested FRG methods are hardly modified at $N=1$ instead of 2.",
"As the KS-FRG stands out as the most efficient 2PPI-FRG approach tested in the present study, we compare the three first non-trivial orders of this method in Fig.",
"REF , still at $N=2$ , with those of the most performing 1PI-FRG and 2PI-FRG techniques tested in previous sections, i.e.",
"the mixed 1PI-FRG and the mU-flow version of the 2PI-FRG at $N_{\\mathrm {SCPT}}=1$ .",
"This comparison is done for both the unbroken- and broken-symmetry regimes of the (0+0)-D $O(N)$ model, except for the mixed 1PI-FRG which is limited to the phase with $m^{2}>0$ owing to the stiffness issues reported in section .",
"Since the pU-flow version of the 2PI-FRG is equivalent to the KS-FRG for the toy model under consideration, Fig.",
"REF basically compares the mU-flow and pU-flow implementations of the 2PI-FRG.",
"As was pointed out in section , the better performances of the mU-flow can be attributed to the quality of its starting point: the Hartree-Fock theory for the mU-flow and the free theory (at squared mass $|m^{2}|$ ) for the pU-flow or the KS-FRG.",
"Furthermore, one can see in Fig.",
"REF (and for the regime with $m^{2}>0$ ) that the mU-flow of the 2PI-FRG clearly outperforms the mixed 1PI-FRG whereas the KS-FRG yields a similar or a slightly better accuracy than the latter.",
"In addition to the absence of mixed 1PI-FRG results in the broken-symmetry phase, it can be said that our best 1PI-FRG results are rather disappointing as compared to the 2PI-FRG and 2PPI-FRG ones.",
"However, it can not be concluded from this study that the 1PI-FRG is less efficient or performing than the 2PI- and 2PPI-FRGs on a general setting, especially considering that the full-fledged FRG machinery can not be completely exploited in the chosen (0+0)-D framework.",
"We will actually come back to this last point later in this concluding section.",
"That said, we do expect the excellent performances of the mU-flow version of the 2PI-FRG to hold in the framework of more realistic (fermionic) systems, notably because they are designed to tackle both particle-hole and particle-particle channels on an equal footing.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =±1m^{2}=\\pm 1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).",
"For the sake of comparison with the 2PI-FRG mU-flow results, the KS-FRG was extended to the broken symmetry phase by exploiting the flow dependent one-body potential V 𝔰 =(2𝔰-1)V=(2𝔰-1)m 2 2V_{\\mathfrak {s}}=(2\\mathfrak {s}-1)V=(2\\mathfrak {s}-1)\\frac{m^{2}}{2} ∀𝔰\\forall \\mathfrak {s} for m 2 <0m^{2}<0, alongside with the usual two-body interaction U 𝔰 =𝔰UU_{\\mathfrak {s}}=\\mathfrak {s}U ∀𝔰\\forall \\mathfrak {s}.",
"In this way, the starting point of the KS-FRG corresponds to the free theory with m 2 \\left|m^{2}\\right| as squared mass.Besides the numerical results, some striking features of the formalisms underlying the tested FRG methods should also be highlighted.",
"All FRG implementations exploited in this study, whether they are formulated in the 1PI, the 2PI or the 2PPI framework, are based on a vertex expansion of the exact flow equation for the corresponding EA (although this may not appear clearly in the 2PI-FRG formalism).",
"For fermionic systems, we are somehow limited to the vertex expansion in the 1PI framework since functionals of Grassmann variables can only be exploited via their Taylor expansions, unless of course one studies an auxiliary bosonic system by making use of e.g.",
"a HST.",
"This highlights an advantage of the 2PI- and 2PPI-FRGs over their 1PI counterpart: the 2PI and 2PPI EAs are functionals of propagators or densities, which are not Grassmann variables even for fermionic systems.",
"This implies that, even for fermionic systems (and without HST), one can directly exploit vertex expansions around non-trivial minima and thus describe non-perturbative phenomena.",
"However, to exploit 2PI EAs with FRG in a reliable and performing manner, the limitations of the C-flow scheme within the 2P(P)I-FRG (see appendix REF ) forces us to introduce cutoff functions in the interaction parts of the classical action taken as input.",
"This results in consequent complications of the underlying flow equations, notably with the resolution of the Bethe-Salpeter equation required at each step of the flow within the U-flow and CU-flow schemes of the 2PI-FRG.",
"In that respect, the 1PI-FRG based on the Wetterich equation, which does not suffer from such complications, is certainly appealing, with or without HST.",
"There are however some developments, currently available in the literature, that could help reducing significantly the complexity of U-flow and CU-flow equations, as will be discussed further below.",
"Another remark should be done concerning a drawback that one might assign to the 2PI and 2PPI frameworks.",
"One should have in mind that the 1PI vertices correspond to the physical interactions in the studied quantum system and are therefore, in many contexts, the objects of interest that we want to calculate in the end (see Refs.",
"[44], [45] for the relations which allow for determining 1PI vertices from the 2PI ones).",
"In condensed matter physics for instance, physical susceptibilities are usually extracted from the two-particle vertex $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}$ at the end of the flow (see Refs.",
"[199], [117], [200], [201] for examples in the context of the 1PI-FRG).",
"However, in nuclear physics and more specifically in nuclear structure, one rather focuses e.g.",
"on gs energies or densities, or maybe on the propagators of the theory and these objects are all directly accessible in the 2PI scheme (i.e.",
"their determination does not necessitate an intermediate calculation of 1PI vertices), as we have illustrated throughout the present study.",
"Hence, we have just concluded on the results presented in sections to as well as on the formalisms of the FRG techniques that led to these results.",
"Before going further with a general conclusion on papers I and II, we believe it is also important to conclude on what could not be discussed in this paper.",
"In particular, the 1PI-FRG is very widespread and has been subject to numerous theoretical developments, even very recently.",
"Although we have exploited its conventional implementation for fermionic systems (based on a vertex expansion of the Wetterich equation), there are several state-of-the-art extensions or refinements of this approach that were not discussed in this study, either because their implementation in the framework of our zero-dimensional toy model was not relevant or simply for the sake of conciseness.",
"We have already mentioned alternatives to the vertex expansion, i.e.",
"the DE and the BMW approximation, whose relevance could not be illustrated here owing to the (0+0)-D character of the model under consideration.",
"The following features of the 1PI-FRG are all relevant in the context of the vertex expansion, especially to treat (finite-dimensional) fermionic systems: Flowing bosonization: A very fruitful FRG implementation is the so-called scale-dependent or flowing bosonization that was developed by combining the 1PI-FRG approach with HSTs in the following manners: either by exploiting a HST that is scale-dependent itself [202], [203], or rather by simply letting the expectation value of the Hubbard-Stratonovich field depend on the momentum scaleThis formulation based on scale-dependent expectation values has been generalized [147] such that only the fluctuating composite operators introduced via HSTs (and not their expectation values) are scale-dependent (see Ref.",
"[67] for a concrete example of application in the context of QCD).",
"A further generalization aiming at treating explicit symmetry breaking is developed in Ref. [69].",
"[204], [205], [206].",
"The flowing bosonization enable us for instance to monitor the Fierz ambiguityFermionic bilinears can be reordered by using the so-called Fierz transformations which should not change physical observables.",
"In the framework of some many-body techniques, this nonetheless affects physical results, which is known as the Fierz ambiguity.",
"in the framework of the fermionic 1PI-FRG [157]: even if one integrates out (by means of a HST) the fermionic bilinears that produce such an ambiguity at the starting point of the flow, these bilinears would be automatically regenerated throughout the flow, hence the need for flowing bosonization to integrate out these problematic interaction terms at each step of the flow.",
"In the same spirit, one can also integrate out diverging couplings at each step of the flow in the context of phase transitions, which also renders flowing bosonization extremely relevant to tackle critical physics.",
"DMF$^2$ RG: We have already mentioned several times that the 1PI-FRG based on a vertex expansion of the Wetterich equation is not suited to described strongly-coupled fermions and that such a limitation can be overcome with a HST.",
"Another option to tackle strongly-coupled fermionic systems is provided by a recent extension of the fermionic 1PI-FRG, developed notably for the Hubbard model and coined as DMF$^2$ RG [185], [186], [187], [120].",
"In the latter approach, the flow starts at the non-perturbative results of DMFT instead of the classical theory and the FRG procedure then grasps non-local correlations on top of the local ones captured by DMFT.",
"Truncated unity (TU) FRG: The TUFRG was introduced to capture more efficiently the momentum dependences of the 1PI vertices in the context of the fermionic 1PI-FRG [153], [116].",
"The underlying idea is to introduce truncated completeness relations, involving sums over form factors, in the FRG equations, hence the name “TU”.",
"In this way, some momentum dependences of the vertices can be represented by a set of coordinates associated to a few form factors instead of relying on the discretization of the full momentum space (or the Brillouin zone).",
"Moreover, another version of the TUFRG, called TU$^2$ FRG [119], was put forward recently to unify the original TU formulation in both real and momentum spaces and exhibits a much better numerical scaling for a certain class of models breaking translational symmetry.",
"Multiloop FRG: For our original 1PI-FRG applications in the unbroken-symmetry phase, we have identified the first two non-trivial orders as those associated to the truncation orders $N_{\\mathrm {max}}=4$ and $N_{\\mathrm {max}}=6$ , respectively.",
"The truncation at $N_{\\mathrm {max}}=4$ , known as the level-2 truncation, already relies on the calculation of the two-particle vertex $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(4)}$ throughout the flow, which is already quite involved, whereas the next truncation order (i.e.",
"that at $N_{\\mathrm {max}}=6$ ) includes as flowing object the three-particle vertex $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(6)}$ , whose direct calculation is currently out of reach for most finite dimensional models.",
"Before discussing very briefly some state-of-the-art treatments of $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}$ within the FRG framework, we would like to point out another recently developed expansion scheme of the Wetterich equation, coined as multiloop FRG [207], [208], [209], [199], [200], [201], which consists in taking into account the effects of the three-particle vertex $\\overline{\\Gamma }^{(\\mathrm {1PI})(6)}$ in a systematic manner on top of the level-2 truncation of the vertex expansion.",
"Single-boson-exchange (SBE) decomposition: A conventional treatment of the two-particle vertex $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}$ is based on the parquet approximation combined with a decomposition of its 2-particle-reducible (2PR) part in terms of high-frequency asymptotics [210].",
"There is also the SBE decomposition of $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}$ [211], which rather retains the $U$ -reducible (i.e.",
"1-vertex-reducible) diagrams contributing to $\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}$ instead of the 2PR ones.",
"In the 1PI-FRG framework, the implementation of the SBE decomposition has recently shown some significant computational advantages as compared to its conventional counterpart based on high-frequency asymptotics [121].",
"For instance, regarding the most recent developments of these various techniques, a merging of the SBE formalism with the the DMF$^2$ RG has already been achieved successfully in the framework of the (2+1)-D Hubbard model [120] and the multiloop FRG formalism based on the SBE decomposition has already been published [212].",
"Some of the above features of the 1PI-FRG could be relevant to improve the 2PI- and 2PPI-FRG approaches as well.",
"As mentioned in section REF , the DMF$^2$ RG has also been exploited within the 2PI-FRG framework [184].",
"An advantage of this DMF$^2$ RG formulation over its counterpart based on the 1PI framework and the Wetterich equation is the possibility to properly treat models with non-local two-body interactions.",
"Furthermore, the idea of TUFRG would also be very relevant in the 2PI and 2PPI frameworks, as was already pointed out by Katanin in the 2PI case [213].",
"Notably, this would drastically simplify the resolution of the Bethe-Salpeter equation within the U-flow and CU-flow versions of the 2PI-FRG, and therefore render the 2PI-FRG scheme more amenable to realistic physical systems."
],
[
"Conclusion on the whole comparative study of path-integral techniques",
"To conclude on the study of papers I and II, we finally consider Figs.",
"REF and REF , where respectively the first and third non-trivial orders of the most performing tested PI techniques (combined with resummation theory when relevant) are compared, namely the LE in the collective representation (i.e.",
"the collective LE), OPT based on the principle of minimal sensitivity (PMS), SCPT for the mixed 2PI EA and the mU-flow version of the 2PI-FRG with Hartree-Fock starting point (i.e.",
"with $N_{\\mathrm {SCPT}}=1$ ).",
"Before comparing their numerical results, we point out that, to our knowledge, the study of paper I is the first to push the collective LE and SCPT based on the mixed 2PI EA up to their third non-trivial orders (with or without combination with resummation theory) whereas paper II discusses first applications of 2PI-FRG and 2PPI-FRG methods to an $O(N)$ model with $N>1$ , pushing notably the 2PI-FRG mU-flow up to truncation orders not investigated so far.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =±1m^{2}=\\pm 1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =±1m^{2}=\\pm 1 and N=2N=2 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).",
"The legend 𝒫ℬ[U/V]\\mathcal {PB}[U/V] or 𝒫ℬ X [U/V]\\mathcal {PB}_{X}[U/V] indicates that the Padé-Borel resummation was exploited with a [U/V][U/V] Padé approximant, whereas XX in “𝒫ℬ X [U/V]\\mathcal {PB}_{X}[U/V]” specifies that the Padé approximant is calculated from the series representing XX (see paper I for more details on the implementation of Padé-Borel resummation in the framework of the LE or SCPT).The estimates of both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ determined from those four PI techniques depicted in Figs.",
"REF and REF are all fairly close, in both the unbroken- and broken-symmetry regimes of the (0+0)-D $O(N)$ model.",
"That said, one should stress that all results presented in those figures are obtained at $N=2$ and, as opposed to the other three approaches, the performances of the collective LE are expected to vary substancially as $N$ varies, and more specifically to improve as $N$ increasesThis can already be observed in paper I by comparing results at $N=1$ and 2. owing to its connection with the $1/N$ -expansion identified in paper I.",
"One might also notice from the results of Figs.",
"REF and REF that OPT based on the PMS is slightly less performing than the collective LE, SCPT for the mixed 2PI EA and the mU-flow version of the 2PI-FRG.",
"However, this should be put in contrast with a certain ease of implementation underlying the OPT approach.",
"Indeed, among all performing PI techniques represented in Figs.",
"REF and REF , OPT is probably the only method whose numerical implementation is not expected to get significantly more complicated in the framework of finite-dimensional theories.",
"In particular, the equations underlying the tested optimization procedures in OPT (like the PMS) should remain polynomial in the framework of finite-dimensional $O(N)$ models while the gap equations encountered in SCPT and the flow equations underpinning FRG implementations would respectively become self-consistent and integro-differential.",
"We have nonetheless pointed out in section REF some directions relevant to reduce the cost of the numerical implementation of FRG approaches for finite-dimensional models.",
"The same can be done for other PI techniques: see e.g.",
"recent applications of the TU approximation [153], [116] to the resolution of parquet equations in Refs.",
"[214], [215].",
"On the side of the formalism, all four performing PI methods discussed here were implemented by introducing various bosonic dofs in the problem: i) the Hubbard-Stratonovich field for the collective LE and SCPT based on the mixed 2PI EA; ii) the classical auxiliary field for OPT; iii) the propagator(s) for the 2PI-FRG and SCPT based on the mixed 2PI EA.",
"This illustrates the richness of the PI framework as well as its relevance to reformulate the nuclear EDF framework, whose efficiency also relies on bosonic dofs in the form of the nucleonic density.",
"We also recall that PI techniques are actually suited to turn the EDF method into a more reliable approach owing to their ability of treating classical actions (derived e.g.",
"from an EFT of QCD) as inputs, thus potentially providing an explicit connection with the dynamics of quarks and gluons which can not be accessed via current EDF calculations.",
"Furthermore, we have also highlighted several links between PI techniques and DFT, which is also of special interest for those aiming at improving the EDF framework (or at finding more reliable alternatives) [27], [191], [28], [29], [30], [31], [36], [37], [32].",
"In particular, all PI approaches relying on 2PPI EAs, such as the 2PPI-FRG treated in section , can be viewed as an implementation of DFT [33].",
"One can also mention the self-consistent condition (SCC) of OPT, which was tested in paper I and is in spirit quite close to the Kohn-Sham scheme.",
"The above remarks, alongside with the whole studies of papers I and II, emphasize the power and richness of the PI formalism and its ability to describe strongly-coupled quantum systems (whether nuclear or not) with its various methods, sometimes clearly connected to celebrated approaches of quantum many-body theory.",
"Further connections with the nuclear EDF method might still be identified: for example, one could investigate if the mU-flow implementation of the 2PI-FRG has the power to describe collective phenomena by restoring the symmetries broken down by the SCPT results used as its starting point, which is precisely what is achieved within the multi-reference scheme of the EDF framework."
],
[
"Acknowledgments",
"The authors thank Thomas Duguet, Vittorio Somà, Nicolas Dupuis and Fabian Rennecke for stimulating discussions.",
"K.F.",
"also acknowledges fruitful exchanges with colleagues at the Institut für Theoretische Physik in Tübingen (Aiman Al-Eryani, Sarah Heinzelmann and Sabine Andergassen) and particularly thanks Sabine Andergassen for her patience while writing the manuscript.",
"The data generated for this study can be made available on reasonable request."
],
[
"Introduction to the bosonic index formalism",
"The formalism related to bosonic indices involves additional subtleties as compared to that based solely on the fermionic ones, denoted by $\\alpha $ in our notations.",
"This stems from the symmetry properties (expressed by Eqs.",
"(REF ) and (REF )) of objects like $G_{\\alpha \\alpha ^{\\prime }}$ or $K_{\\alpha \\alpha ^{\\prime }}$ .",
"In particular, this bosonic index formalism relies on an identity matrix $\\mathcal {I}$ constructed so as to exhibit such symmetries, i.e.",
": $\\begin{split}\\mathcal {I}_{\\gamma _{1}\\gamma _{2}}=\\mathcal {I}_{\\gamma _{2}\\gamma _{1}}\\equiv \\ & \\frac{\\delta G_{\\gamma _{1}}}{\\delta G_{\\gamma _{2}}} = \\frac{\\delta K_{\\gamma _{1}}}{\\delta K_{\\gamma _{2}}} \\\\= & \\ \\delta _{\\alpha _{1}\\alpha _{2}}\\delta _{\\alpha ^{\\prime }_{1}\\alpha ^{\\prime }_{2}}+\\zeta \\delta _{\\alpha _{1}\\alpha ^{\\prime }_{2}}\\delta _{\\alpha ^{\\prime }_{1}\\alpha _{2}}\\;.\\end{split}$ An expression for the functional derivative $\\frac{\\delta W[K]}{\\delta K_{\\gamma }}$ in terms of the propagator $G_{\\gamma }$ directly follows from definition (REF ) and from the generating functional expressed by Eq.",
"(REF ): $\\begin{split}\\frac{\\delta W[K]}{\\delta K_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} = & \\ \\frac{1}{Z[K]}\\int \\mathcal {D}\\widetilde{\\psi } \\left[\\frac{1}{2}\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}} \\widetilde{\\psi }_{\\alpha _{2}}\\frac{\\delta K_{\\alpha _{2}\\alpha ^{\\prime }_{2}}}{\\delta K_{\\alpha _{1}\\alpha ^{\\prime }_{1}}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{2}}\\right] \\\\& \\times e^{-S\\big [\\widetilde{\\psi }\\big ] + \\frac{1}{2}\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha _{2}}K_{\\alpha _{2}\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{2}}} \\\\= & \\ \\frac{1}{Z[K]}\\int \\mathcal {D}\\widetilde{\\psi } \\Bigg [\\frac{1}{2} \\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}} \\widetilde{\\psi }_{\\alpha _{2}}\\big (\\delta _{\\alpha _{2}\\alpha _{1}}\\delta _{\\alpha ^{\\prime }_{2}\\alpha ^{\\prime }_{1}} \\\\& +\\zeta \\delta _{\\alpha _{2}\\alpha ^{\\prime }_{1}}\\delta _{\\alpha ^{\\prime }_{2}\\alpha _{1}}\\big )\\widetilde{\\psi }_{\\alpha ^{\\prime }_{2}}\\Bigg ] \\\\& \\times e^{-S\\big [\\widetilde{\\psi }\\big ] + \\frac{1}{2}\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha _{2}}K_{\\alpha _{2}\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{2}}} \\\\= & \\ \\frac{1}{Z[K]}\\int \\mathcal {D}\\widetilde{\\psi } \\ \\frac{1}{2} \\Big (\\widetilde{\\psi }_{\\alpha _{1}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{1}}+\\zeta \\underbrace{\\widetilde{\\psi }_{\\alpha ^{\\prime }_{1}}\\widetilde{\\psi }_{\\alpha _{1}}}_{\\zeta \\widetilde{\\psi }_{\\alpha _{1}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{1}}}\\Big ) \\\\& \\times e^{-S\\big [\\widetilde{\\psi }\\big ] + \\frac{1}{2}\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha _{2}} K_{\\alpha _{2}\\alpha ^{\\prime }_{2}} \\widetilde{\\psi }_{\\alpha ^{\\prime }_{2}}} \\\\= & \\ \\frac{1}{Z[K]}\\int \\mathcal {D}\\widetilde{\\psi } \\ \\widetilde{\\psi }_{\\alpha _{1}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{1}} \\\\& \\times e^{-S\\big [\\widetilde{\\psi }\\big ] + \\frac{1}{2}\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha _{2}}K_{\\alpha _{2}\\alpha ^{\\prime }_{2}}\\widetilde{\\psi }_{\\alpha ^{\\prime }_{2}}} \\\\= & \\ G_{\\alpha _{1}\\alpha ^{\\prime }_{1}} \\;,\\end{split}$ or, in terms of bosonic indices, $\\frac{\\delta W[K]}{\\delta K_{\\gamma }} = G_{\\gamma }\\;.$ Another important point is the matrix multiplication with respect to bosonic indices.",
"For two bosonic matrices $M$ and $N$ , this gives us: $\\begin{split}\\left(MN\\right)_{\\gamma _{1}\\gamma _{2}} = & \\ \\frac{1}{2}\\int _{\\gamma _{3}} M_{\\gamma _{1}\\gamma _{3}} N_{\\gamma _{3}\\gamma _{2}} \\\\= & \\ \\frac{1}{2} \\int _{\\alpha _{3},\\alpha ^{\\prime }_{3}} M_{\\gamma _{1}(\\alpha _{3},\\alpha ^{\\prime }_{3})} N_{(\\alpha _{3},\\alpha ^{\\prime }_{3})\\gamma _{2}} \\\\= & \\ M_{\\gamma _{1}\\hat{\\gamma }} N_{\\hat{\\gamma }\\gamma _{2}} \\;,\\end{split}$ which is generalized to $n$ bosonic matrices by Eq.",
"(REF ).",
"Hence, it involves an extra $1/2$ factor as compared to the standard matrix multiplication with respect to $\\alpha $ -indices, which is convenient considering the symmetry properties discussed right above Eq.",
"(REF ).",
"For instance, the bosonic identity matrix involving two terms in its definition (REF ) so as to satisfy those symmetries, such a $1/2$ factor is usually canceled out as follows: $\\begin{split}\\frac{\\delta }{\\delta G_{\\gamma _{1}}} G_{\\hat{\\gamma }} M_{\\hat{\\gamma }\\gamma _{2}} = & \\ \\frac{1}{2}\\int _{\\gamma _{3}} \\underbrace{\\frac{\\delta G_{\\gamma _{3}}}{\\delta G_{\\gamma _{1}}}}_{\\mathcal {I}_{\\gamma _{1}\\gamma _{3}}} M_{\\gamma _{3}\\gamma _{2}} \\\\= & \\ \\frac{1}{2}\\int _{\\gamma _{3}} \\left( \\delta _{\\alpha _{1}\\alpha _{3}}\\delta _{\\alpha ^{\\prime }_{1}\\alpha ^{\\prime }_{3}} + \\zeta \\delta _{\\alpha _{1}\\alpha ^{\\prime }_{3}}\\delta _{\\alpha ^{\\prime }_{1}\\alpha _{3}} \\right) M_{\\gamma _{3}\\gamma _{2}} \\\\= & \\ \\frac{1}{2} \\big ( M_{\\gamma _{1}\\gamma _{2}} + \\zeta \\underbrace{M_{(\\alpha ^{\\prime }_{1},\\alpha _{1})\\gamma _{2}}}_{\\zeta M_{(\\alpha _{1},\\alpha ^{\\prime }_{1})\\gamma _{2}}} \\big ) \\\\= & \\ M_{\\gamma _{1} \\gamma _{2}} \\;,\\end{split}$ with $M$ being an arbitrary bosonic matrix possessing also the symmetry properties of Eq.",
"(REF ).",
"Note in addition that a specific realizations of Eq.",
"(REF ) are the chain rule: $\\frac{\\delta W[K]}{\\delta G_{\\gamma _{1}}}=\\frac{1}{2}\\int _{\\gamma _{2}} \\frac{\\delta K_{\\gamma _{2}}}{\\delta G_{\\gamma _{1}}} \\frac{\\delta W[K]}{\\delta K_{\\gamma _{2}}}\\;.$ and the definition of an inverse $M^{\\mathrm {inv}}$ of a bosonic matrix $M$ : $\\mathcal {I}_{\\gamma _{1}\\gamma _{2}} = \\left(M M^{\\mathrm {inv}}\\right)_{\\gamma _{1}\\gamma _{2}} = \\frac{1}{2}\\int _{\\gamma _{3}} M_{\\gamma _{1}\\gamma _{3}} M^{\\mathrm {inv}}_{\\gamma _{3}\\gamma _{2}}\\;,$ which is already given by Eq.",
"(REF ).",
"We can also evaluate the derivative $\\frac{\\delta \\Gamma ^{(\\mathrm {2PI})}[G]}{\\delta G_{\\gamma }}$ with the help of the chain rule expressed by Eq.",
"(REF ).",
"This is achieved by differentiating both sides of Eq.",
"(REF ) with respect to $G$ , thus yielding: $\\begin{split}\\frac{\\delta \\Gamma ^{(\\mathrm {2PI})}[G]}{\\delta G_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} = & -\\frac{\\delta W[K]}{\\delta G_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} + \\frac{1}{2}\\int _{\\gamma _{2}} \\frac{\\delta K_{\\gamma _{2}}}{\\delta G_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} G_{\\gamma _{2}} \\\\& + \\frac{1}{2}\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}} K_{\\alpha _{2}\\alpha ^{\\prime }_{2}} \\frac{\\delta G_{\\alpha _{2}\\alpha ^{\\prime }_{2}}}{\\delta G_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} \\\\= & -\\frac{1}{2}\\int _{\\gamma _{2}} \\frac{\\delta K_{\\gamma _{2}}}{\\delta G_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} \\underbrace{\\frac{\\delta W[K]}{\\delta K_{\\gamma _{2}}}}_{G_{\\gamma _{2}}} + \\frac{1}{2}\\int _{\\gamma _{2}} \\frac{\\delta K_{\\gamma _{2}}}{\\delta G_{\\alpha _{1}\\alpha ^{\\prime }_{1}}} G_{\\gamma _{2}} \\\\& + \\frac{1}{2}\\big (K_{\\alpha _{1}\\alpha ^{\\prime }_{1}} + \\zeta \\underbrace{K_{\\alpha ^{\\prime }_{1}\\alpha _{1}}}_{\\zeta K_{\\alpha _{1}\\alpha ^{\\prime }_{1}}}\\big ) \\\\= & \\ K_{\\alpha _{1}\\alpha ^{\\prime }_{1}} \\;,\\end{split}$ and, in terms of bosonic indices, $\\frac{\\delta \\Gamma ^{(\\mathrm {2PI})}[G]}{\\delta G_{\\gamma }} = K_{\\gamma }\\;,$ which is consistent with the leftmost equality in result (REF )."
],
[
"C-flow",
"The underlying idea of the C-flow version of the 2PI-FRG [44] remains to implement the momentum-shell integration à la Wilson.",
"By definition, the C-flow consists in considering a flow-dependent free propagator $C_{\\mathfrak {s}}$ .",
"This amounts to inserting a cutoff function $R_{\\mathfrak {s}}$ in the classical action (REF ) via the substitution $C^{-1} \\rightarrow C^{-1}_{\\mathfrak {s}} = R_{\\mathfrak {s}} C^{-1}$ or equivalently $C^{-1} \\rightarrow C^{-1}_{\\mathfrak {s}} = C^{-1} + R_{\\mathfrak {s}}$ , which is exactly what is done for the 1PI-FRG based on the Wetterich equation by introducing the term $\\Delta S_{k}$ in Eq.",
"(REF ).",
"The C-flow is therefore close in spirit to Wetterich's approach.",
"Owing to such a connection, we can deduce the required values for $C_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}$ and $C_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}$ from the boundary conditions for $R_{k}$ set by Eqs.",
"(REF ): [left=]align Cs=si, = 0 , Cs=sf = C , with $\\mathfrak {s}_{\\mathrm {i}}$ and $\\mathfrak {s}_{\\mathrm {f}}$ still denoting the values of the flow parameter at the beginning and at the end of the flow, respectively.",
"Moreover, the LW functional $\\Phi [G]$ , defined as the interacting part of the 2PI EA $\\Gamma ^{(\\mathrm {2PI})}[G]$ through Eq.",
"(REF ), does not depend on the free propagator $C$ , and therefore not on $\\mathfrak {s}$ for the C-flow.",
"Consequently, it is an invariant of the flow in this case, which translates into: $\\dot{\\Phi }_{\\mathfrak {s}}[G]=0 \\quad \\forall \\mathfrak {s} \\;.$ In particular, Eq.",
"(REF ) implies that all components of $\\overline{\\dot{\\Sigma }}_{\\mathfrak {s}}$ and $\\overline{\\dot{\\Phi }}_{\\mathfrak {s}}^{(n)}$ (with $n \\ge 2$ ) vanish, which enables us to simplify the chain rules (REF ) for the C-flow: [left=]align s, = - Gs, s,(2) , s,1n(n) = Gs, s,1n(n+1) n 2 , The flow dependence of $\\overline{G}_{\\mathfrak {s}}$ , involved in both Eqs.",
"(REF ) to (REF ), follows from that of $C_{\\mathfrak {s}}$ according to: $\\dot{\\overline{G}}_{\\mathfrak {s},\\alpha _{1} \\alpha ^{\\prime }_{1}}=-\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\overline{G}_{\\mathfrak {s}, \\alpha _{1} \\alpha _{2}}\\left(\\dot{C}_{\\mathfrak {s}}^{-1}-\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}\\right)_{\\alpha _{2} \\alpha ^{\\prime }_{2}} \\overline{G}_{\\mathfrak {s}, \\alpha ^{\\prime }_{2} \\alpha ^{\\prime }_{1}} \\;.$ Result (REF ) can be derived by differentiating the Dyson equation (REF ) with respect to $\\mathfrak {s}$ at vanishing source $K$ , similarly to Eq.",
"(REF ) for the U-flow.",
"A flow equation for $\\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})}$ can also be directly inferred from the master equation (REF ) expressing $\\dot{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ , which gives us: $\\dot{\\overline{\\Gamma }}_{\\mathfrak {s}}^{(\\mathrm {2PI})} = \\dot{C}_{\\mathfrak {s},\\hat{\\gamma }}^{-1} \\overline{G}_{\\mathfrak {s},\\hat{\\gamma }} \\;,$ for the C-flow.",
"However, Eq.",
"(REF ) is actually not tractable.",
"To see this, one can consider the Dyson equation for $\\overline{G}_{\\mathfrak {s}}$ , i.e.",
": $\\overline{G}_{\\mathfrak {s}} = \\left(C_{\\mathfrak {s}}^{-1} - \\overline{\\Sigma }_{\\mathfrak {s}}\\right)^{-1} \\;,$ to notice that $\\overline{G}_{\\mathfrak {s}}$ must satisfy $\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma }=0$ $\\forall \\gamma $ , as a result of condition (REF ).",
"According to Eq.",
"(REF ), this induces a divergence of $\\frac{\\zeta }{2} \\mathrm {STr}\\left[ \\mathrm {ln}(\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}) \\right]$ in $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{0,\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}$ , and in $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}$ as a result.",
"Therefore, we will calculate during the flow the following quantity: $\\begin{split}\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}} \\equiv & \\ \\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})} - \\Gamma _{0,\\mathfrak {s}}^{(\\mathrm {2PI})}[G = C_{\\mathfrak {s}}] \\\\= & \\ \\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})} + \\frac{\\zeta }{2} \\mathrm {STr} \\left[ \\mathrm {ln}(C_{\\mathfrak {s}}) \\right] \\;,\\end{split}$ instead of $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}}$ .",
"The extra term $\\Gamma _{0,\\mathfrak {s}}[G = C_{\\mathfrak {s}}]$ eliminates the aforementioned divergence problem, thanks to condition (REF ).",
"In conclusion, the tower of differential equations underlying the C-flow version of the 2PI-FRG is thus given by: $\\dot{\\overline{G}}_{\\mathfrak {s},\\alpha _{1} \\alpha ^{\\prime }_{1}}=-\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\overline{G}_{\\mathfrak {s}, \\alpha _{1} \\alpha _{2}}\\left(\\dot{C}_{\\mathfrak {s}}^{-1}-\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}\\right)_{\\alpha _{2} \\alpha ^{\\prime }_{2}} \\overline{G}_{\\mathfrak {s}, \\alpha ^{\\prime }_{2} \\alpha ^{\\prime }_{1}} \\;,$ $\\Delta \\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\dot{C}_{\\mathfrak {s},\\hat{\\gamma }}^{-1} \\left(\\overline{G}_{\\mathfrak {s}}-C_{\\mathfrak {s}}\\right)_{\\hat{\\gamma }} \\;,$ $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s},\\gamma } = - \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }\\gamma }^{(2)} \\;,$ $\\dot{\\overline{\\Phi }}_{\\mathfrak {s},\\gamma _{1}\\cdots \\gamma _{n}}^{(n)} = \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }\\gamma _{1}\\cdots \\gamma _{n}}^{(n+1)} \\quad \\forall n \\ge 2 \\;.$ The quantity $\\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}^{(\\mathrm {2PI})}$ is readily obtained at the end of the flow of $\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}}$ through definition (REF ) in the form $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}=\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}-\\frac{\\zeta }{2} \\mathrm {STr} \\left[ \\mathrm {ln}(C) \\right]$ .",
"We then discuss the initial conditions for the C-flow.",
"It was explained above that the following relation follows from both Dyson equation (in the form of Eq.",
"(REF )) and the initial condition for $C_{\\mathfrak {s}}$ (given by (REF )): $\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma }=0 \\quad \\forall \\gamma \\;.$ After combining Eq.",
"(REF ) with the diagrammatic expression of $\\Phi [G]$ given by result (REF ), one can find the initial conditions for all 2PI vertices in the framework of the C-flow according to: $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma _{1}\\cdots \\gamma _{n}}^{(n)} = \\left.",
"\\frac{\\delta ^{n} \\Phi _{\\mathrm {SCPT}}[U,G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}} \\right|_{G=\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}} \\;,$ which translates for specific values of $n$ into: $\\overline{\\Sigma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma } = 0 \\quad \\forall \\gamma \\;,$ $\\overline{\\Phi }_{\\mathfrak {s} = \\mathfrak {s}_{\\mathrm {i}},\\gamma _{1}\\gamma _{2}}^{(2)} = U_{\\gamma _{1}\\gamma _{2}} \\;,$ $\\begin{split}\\overline{\\Phi }_{\\mathfrak {s} = \\mathfrak {s}_{\\mathrm {i}},\\gamma _{1}\\gamma _{2}\\gamma _{3}\\gamma _{4}}^{(4)} = & -\\frac{1}{2} \\big [\\big (\\big \\lbrace \\big [ U_{\\alpha _{1} \\alpha _{2} \\alpha _{3} \\alpha _{4}} U_{\\alpha ^{\\prime }_{1} \\alpha ^{\\prime }_{2} \\alpha ^{\\prime }_{3} \\alpha ^{\\prime }_{4}} \\\\& + \\zeta \\left(\\alpha _{1} \\leftrightarrow \\alpha ^{\\prime }_{1}\\right) \\big ] + \\zeta \\left(\\alpha _{2} \\leftrightarrow \\alpha ^{\\prime }_{2}\\right) \\big \\rbrace \\\\& + \\zeta \\left(\\alpha _{3} \\leftrightarrow \\alpha ^{\\prime }_{3}\\right) \\big ) + \\zeta \\left(\\alpha _{4} \\leftrightarrow \\alpha ^{\\prime }_{4} \\right) \\big ] \\;,\\end{split}$ $\\overline{\\Phi }_{\\mathfrak {s} = \\mathfrak {s}_{\\mathrm {i}},\\gamma _{1} \\cdots \\gamma _{n}}^{(n)}=0 \\quad \\forall \\gamma _{1}, \\cdots , \\gamma _{n} , ~ \\forall n ~ \\mathrm {odd} \\;.$ Even though Eqs.",
"(REF ) to (REF ) are derived from a truncated result in practice, these equations are exact at $\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}$ because of condition (REF ).",
"It remains to determine the initial condition for $\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}}$ .",
"Combining definition (REF ) with Eqs.",
"(REF ) and (REF ) at $\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}$ gives us: $\\begin{split}\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = & \\ \\overline{\\Gamma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PI})} - \\Gamma _{0,\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(\\mathrm {2PI})}[G=C_{\\mathfrak {s}}] \\\\= & -\\frac{\\zeta }{2} \\mathrm {STr} \\left[ \\mathrm {ln}\\big (\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}\\big ) \\right] + \\frac{\\zeta }{2} \\mathrm {STr}\\left[\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}C_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{-1}-\\mathbb {I}\\right] \\\\& + \\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} + \\frac{\\zeta }{2} \\mathrm {STr} \\left[ \\mathrm {ln}(C_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}) \\right] \\;,\\end{split}$ which, according to the latter results, simply reduces to: $\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}} = 0 \\;.$ As a next step, we address the truncation of the infinite hierarchy of flow equations for the 2PI vertices within the C-flow.",
"Two truncation schemes have been designed for this version of the 2PI-FRG [44].",
"We discuss both of them in what follows: Truncated C-flow (tC-flow): The tC-flow is a specific implementation of the C-flow in which the infinite tower of differential equations given by Eqs.",
"(REF ) to (REF ) is rendered finite according to the condition: $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}=\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(n)} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;.$ In particular, it is shown in Ref.",
"[180] that the tC-flow scheme with truncation order $N_{\\mathrm {max}}=2 N_{\\mathrm {SCPT}}-1$ or $2 N_{\\mathrm {SCPT}}$ (with $N_{\\mathrm {SCPT}} \\in \\mathbb {N}^{*}$ ) is equivalent to SCPT carried out up to order $\\mathcal {O}\\big (U^{N_{\\mathrm {SCPT}}}\\big )$ , e.g.",
"the tC-flow with $N_{\\mathrm {max}}=1$ or 2 reproduces Hartree-Fock results.",
"Modified C-flow (mC-flow): The mC-flow is an alternative to the tC-flow and relies on the following condition: $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}=\\left.\\overline{\\Phi }_{\\mathrm {SCPT},N_{\\mathrm {SCPT}},\\mathfrak {s}}^{(n)}\\right|_{U \\rightarrow \\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s}}^{(2)}} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;,$ with $\\begin{split}\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s},\\alpha _{1} \\alpha _{2} \\alpha _{3} \\alpha _{4}}^{(2)} = & \\ \\frac{1}{3}\\Big (\\overline{\\Phi }_{\\mathfrak {s},\\alpha _{1} \\alpha _{2} \\alpha _{3} \\alpha _{4}}^{(2)}+\\overline{\\Phi }_{\\mathfrak {s},\\alpha _{2} \\alpha _{3} \\alpha _{1} \\alpha _{4}}^{(2)} \\\\& +\\overline{\\Phi }_{\\mathfrak {s},\\alpha _{3} \\alpha _{1} \\alpha _{2} \\alpha _{4}}^{(2)}\\Big ) \\;,\\end{split}$ instead of Eq.",
"(REF ).",
"The truncation condition (REF ) is an ansatz based on the perturbative expression of the LW functional $\\Phi _{\\mathrm {SCPT}}[U,G]$ given by Eq.",
"(REF ), and involves its truncated version $\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G]$ already defined between Eqs.",
"(REF ) and (REF ).",
"From this functional, we introduce the following quantities: $\\overline{\\Phi }_{\\mathrm {SCPT},N_{\\mathrm {SCPT}},\\mathfrak {s}}\\equiv \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}\\big [U,G=\\overline{G}_{\\mathfrak {s}}\\big ] \\;,$ and $\\overline{\\Phi }^{(n)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}},\\mathfrak {s},\\gamma _{1}\\cdots \\gamma _{n}}\\equiv \\left.",
"\\frac{\\delta ^{n} \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}}[U,G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}} \\right|_{G=\\overline{G}_{\\mathfrak {s}}} \\;.$ Hence, we have for example at $N_{\\mathrm {SCPT}}=1$ : $\\begin{split}& \\left.\\overline{\\Phi }_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1,\\mathfrak {s}}\\right|_{U\\rightarrow \\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s}}^{(2)}} \\\\& \\equiv \\frac{1}{8}\\int _{\\gamma _{1},\\gamma _{2}}\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s},\\gamma _{1} \\gamma _{2}}^{(2)}\\overline{G}_{\\mathfrak {s},\\gamma _{1}}\\overline{G}_{\\mathfrak {s},\\gamma _{2}} \\;.\\end{split}$ Furthermore, the motivation for replacing $U$ by $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ is set by the initial condition for $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ , i.e.",
"Eq.",
"(REF ).",
"Generalizing the latter relation to all $\\mathfrak {s}$ indeed suggests to substitute $U$ by $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ in Eq.",
"(REF ) but the problem is that $U$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ have different symmetry properties.",
"Indeed, condition (REF ), i.e.",
": $U_{\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{4}}=\\zeta ^{N(P)}U_{\\alpha _{P(1)}\\alpha _{P(2)}\\alpha _{P(3)}\\alpha _{P(4)}} \\;,$ imposes an invariance of $U$ (up to a sign) under $4!=24$ transformations as opposed to Eq.",
"(REF ), i.e.",
": $\\begin{split}\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(\\alpha _{1},\\alpha ^{\\prime }_{1})(\\alpha _{2},\\alpha ^{\\prime }_{2})} = & \\ \\zeta \\overline{\\Phi }^{(2)}_{\\mathfrak {s},(\\alpha ^{\\prime }_{1},\\alpha _{1})(\\alpha _{2},\\alpha ^{\\prime }_{2})} \\\\= & \\ \\zeta \\overline{\\Phi }^{(2)}_{\\mathfrak {s},(\\alpha _{1},\\alpha ^{\\prime }_{1})(\\alpha ^{\\prime }_{2},\\alpha _{2})} \\\\= & \\ \\overline{\\Phi }^{(2)}_{\\mathfrak {s},(\\alpha ^{\\prime }_{1},\\alpha _{1})(\\alpha ^{\\prime }_{2},\\alpha _{2})} \\;,\\end{split}$ which relates to 4 transformations only.",
"This brings us to the relevance of $\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s}}^{(2)}$ which is constructed so as to possess the same symmetry properties as $U$ [180]: $\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s},\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{4}}^{(2)}=\\zeta ^{N(P)}\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s},\\alpha _{P(1)}\\alpha _{P(2)}\\alpha _{P(3)}\\alpha _{P(4)}}^{(2)} \\;,$ hence the substitution $U \\rightarrow \\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s}}^{(2)}$ in Eq.",
"(REF ).",
"Finally, we note that, according to expression (REF ) of the LW functional, a choice $N_{\\mathrm {SCPT}}\\le N_{\\mathrm {max}}/2$ induces that $\\overline{\\Phi }_{\\mathrm {SCPT},N_{\\mathrm {SCPT}},\\mathfrak {s}}^{(N_{\\mathrm {max}}+1)}[U,G]$ vanishes for all $\\mathfrak {s}$ , which implies that the mC-flow reduces to the tC-flow in this case.",
"Let us then focus on our numerical applications of the C-flow to the (0+0)-D $O(N)$ model.",
"We will more particularly focus on the situation where $N=1$ , which will enable us to investigate more readily high truncation orders (up to $N_{\\mathrm {max}}=10$ to be specific), thus illustrating clearly why the C-flow implementation of the 2PI-FRG is not suited to construct reliable approximation schemes.",
"For the the (0+0)-D $O(N)$ model at $N=1$ , the flow equations (REF ) to (REF ) reduce to: $\\dot{\\overline{G}}_{\\mathfrak {s}} =- \\overline{G}_{\\mathfrak {s}}^{2} \\left(\\dot{C}_{\\mathfrak {s}}^{-1}-\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}\\right) \\;,$ $\\Delta \\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{1}{2} \\dot{C}_{\\mathfrak {s}}^{-1} \\left(\\overline{G}_{\\mathfrak {s}}-C_{\\mathfrak {s}}\\right) \\;,$ $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = - \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\;,$ $\\dot{\\overline{\\Phi }}_{\\mathfrak {s}}^{(n)} = \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(n+1)} \\quad \\forall n \\ge 2 \\;,$ where we have introduced the shorthand notations $\\overline{G}_{\\mathfrak {s}}\\equiv \\overline{G}_{\\mathfrak {s},11}$ , $\\overline{\\Sigma }_{\\mathfrak {s}}\\equiv \\overline{\\Sigma }_{\\mathfrak {s},11}$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}\\equiv \\overline{\\Phi }_{\\mathfrak {s},(1,1)\\cdots (1,1)}^{(n)}$ $\\forall n$ .",
"To implement the tC-flow up to $N_{\\mathrm {max}}=10$ , we will need to consider to consider the perturbative expression of the LW functional up to order 5 in $\\lambda $ , which reads according to Ref.",
"[216]: $\\begin{split}\\Phi _{\\mathrm {SCPT}}(U,G) = & \\ \\frac{1}{8} \\lambda G^{2} - \\frac{1}{48} \\lambda ^{2} G^{4} + \\frac{1}{48} \\lambda ^{3} G^{6} - \\frac{5}{128} \\lambda ^{4} G^{8} \\\\& + \\frac{101}{960} \\lambda ^{5} G^{10} + \\mathcal {O}\\big (\\lambda ^{6}\\big ) \\;,\\end{split}$ still at $N=1$ .",
"From this, we infer the following initial conditions for the 2PI vertices: $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(2)} = \\lambda \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(4)} = - 8 \\lambda ^{2} \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(6)} = 960 \\lambda ^{3} \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(8)} = - 403200 \\lambda ^{4} \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(10)} = 390942720 \\lambda ^{5} \\;,$ where we have taken into account that the identity matrix of the bosonic index formalism (given by Eq.",
"(REF )) reduces to 2 (and not 1) in the (0+0)-D limit at $N=1$ , i.e.",
"Eq.",
"(REF ) becomes in the (0+0)-D limit: $\\mathcal {I}_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\equiv \\frac{\\partial G_{a_{1}a^{\\prime }_{1}}}{\\partial G_{a_{2}a^{\\prime }_{2}}} = \\delta _{a_{1}a_{2}}\\delta _{a^{\\prime }_{1}a^{\\prime }_{2}} + \\delta _{a_{1}a^{\\prime }_{2}}\\delta _{a^{\\prime }_{1}a_{2}} \\;,$ which yields at $N=1$ : $\\mathcal {I} \\equiv \\frac{\\partial G}{\\partial G} = 2 \\;,$ as opposed to standard derivation rulesThe 2PI-FRG flow equations of the (0+0)-D $O(N)$ model can either be derived via standard derivation rules based on the identity $\\mathcal {I} \\equiv \\frac{\\partial G}{\\partial G} = 1$ or by taking the (0+0)-D limit of their more general versions (written in terms of bosonic indices) using Eq.",
"(REF ).",
"We always follow the latter procedure in this study but solving the equations thus obtained in both situations leads in principle to identical results (see appendix F.6 of Ref.",
"[46] for a more detailed discussion on this equivalence)..",
"The truncation conditions underpinning the mC-flow can also be derived from expression (REF ) alongside with Eq.",
"(REF ) as derivation rule.",
"We have notably investigated the following combinations for $(N_{\\mathrm {max}},N_{\\mathrm {SCPT}})$Recall that, as discussed right below Eq.",
"(REF ), it is pointless to investigate the mC-flow with $N_{\\mathrm {SCPT}} \\le N_{\\mathrm {max}}/2$ as it reduces to the tC-flow in this situation.According to the definition of $\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s}}^{(2)}$ given by Eq.",
"(REF ), we have $\\overline{\\Phi }_{\\mathrm {sym},\\mathfrak {s}}^{(2)}=\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ for the studied toy model at $N=1$ .",
": At $N_{\\mathrm {max}}=2$ : At $N_{\\mathrm {SCPT}}=2$ : $\\begin{split}\\overline{\\Phi }^{(3)}_{\\mathfrak {s}} = & \\ \\left.",
"\\overline{\\Phi }^{(3)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=2,\\mathfrak {s}} \\right|_{\\lambda \\rightarrow \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}} \\\\= & - 4 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{2} \\overline{G}_{\\mathfrak {s}} \\;.\\end{split}$ At $N_{\\mathrm {SCPT}}=3$ : $\\begin{split}\\overline{\\Phi }^{(3)}_{\\mathfrak {s}} = & \\ \\left.",
"\\overline{\\Phi }^{(3)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=3,\\mathfrak {s}} \\right|_{\\lambda \\rightarrow \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}} \\\\= & - 4 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{2} \\overline{G}_{\\mathfrak {s}} + 20 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{3} \\overline{G}_{\\mathfrak {s}}^{3} \\;.\\end{split}$ At $N_{\\mathrm {max}}=3$ : At $N_{\\mathrm {SCPT}}=2$ : $\\begin{split}\\overline{\\Phi }^{(4)}_{\\mathfrak {s}} = & \\ \\left.",
"\\overline{\\Phi }^{(4)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=2,\\mathfrak {s}} \\right|_{\\lambda \\rightarrow \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}} \\\\= & - 8 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{2} \\;.\\end{split}$ At $N_{\\mathrm {SCPT}}=3$ : $\\begin{split}\\overline{\\Phi }^{(4)}_{\\mathfrak {s}} = & \\ \\left.",
"\\overline{\\Phi }^{(4)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=3,\\mathfrak {s}} \\right|_{\\lambda \\rightarrow \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}} \\\\= & - 8 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{2} + 120 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{3} \\overline{G}_{\\mathfrak {s}}^{2} \\;.\\end{split}$ At $N_{\\mathrm {max}}=4$ : At $N_{\\mathrm {SCPT}}=3$ : $\\begin{split}\\overline{\\Phi }^{(5)}_{\\mathfrak {s}} = & \\ \\left.",
"\\overline{\\Phi }^{(5)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=3,\\mathfrak {s}} \\right|_{\\lambda \\rightarrow \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}} \\\\= & \\ 480 \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{3} \\overline{G}_{\\mathfrak {s}} \\;.\\end{split}$ Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=1N=1 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).",
"The SCPT result labeled “SCPT 2PI EA 𝒪(U n )\\mathcal {O}(U^{n})” is identical to that corresponding to the legend “SCPT original 2PI EA 𝒪ℏ n+1 \\mathcal {O}\\big (\\hbar ^{n+1}\\big )” in the figures of paper I.Moreover, for all our C-flow applications, we use the same cutoff function as in our 1PI-FRG study, namely at $N=1$ : $C_{\\mathfrak {s}}^{-1} = C^{-1} + R_{\\mathfrak {s}} = m^{2} + R_{\\mathfrak {s}} \\;,$ with $R_{\\mathfrak {s}} = \\mathfrak {s}^{-1} - 1 \\;,$ which satisfies the required boundary conditions set by Eqs.",
"(REF ) as the flow parameter $\\mathfrak {s}$ still runs from $\\mathfrak {s}_{\\mathrm {i}}=0$ to $\\mathfrak {s}_{\\mathrm {f}}=1$ during the flow.",
"Finally, the gs energy and density of our toy model are determined within the C-flow also from Eqs.",
"(REF ) and (REF ) (that were previously exploited for the pU-flow), i.e.",
": $E^{\\text{2PI-FRG;C-flow}}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\underset{N=1}{=} \\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} -\\frac{1}{2} \\mathrm {ln}\\bigg (\\frac{2\\pi }{m^{2}}\\bigg ) \\;,$ $\\rho ^{\\text{2PI-FRG;C-flow}}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},aa} \\underset{N=1}{=} \\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\;.$ Figure: Gs density ρ gs \\rho _{\\mathrm {gs}} calculated at m 2 =+1m^{2}=+1 and N=1N=1 (ℛe(λ)≥0\\mathcal {R}e(\\lambda ) \\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0), and compared with the corresponding exact solution (black dots).",
"The SCPT result labeled “SCPT 2PI EA 𝒪(U n )\\mathcal {O}(U^{n})” is identical to that corresponding to the legend “SCPT original 2PI EA 𝒪ℏ n+1 \\mathcal {O}\\big (\\hbar ^{n+1}\\big )” in the figures of paper I.The tC-flow and mC-flow results obtained in this way for the (0+0)-D $O(N)$ model at $N=1$ are reported in Fig.",
"REF .",
"In this figure, we can see that, for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ , the tC-flow curves are further and further away from the exact solution (except for $\\lambda /4!\\ll 1$ ) as the truncation order $N_{\\mathrm {max}}$ increases.",
"This is certainly an odd feature for an FRG approach but it is consistent with the equivalence between the tC-flow and SCPT mentioned earlier and highlighted in Ref.",
"[180]: such a worsening is thus a manifestation of the asymptotic character of the series underlying SCPT, which is at the heart of paper I.",
"We can also note that there are no tC-flow results for the truncation orders $N_{\\mathrm {max}}=3 ~ \\mathrm {or} ~ 4$ and $N_{\\mathrm {max}}=7 ~ \\mathrm {or} ~ 8$ in Fig.",
"REF as we face the same stiffness issues with the used numerical toolsREF as those encountered in section for our 1PI-FRG applications in the broken-symmetry phase.",
"However, all those tC-flow calculations (with or without stiffness issues) have been performed in the unbroken-symmetry phase, which suggests that the origin of these stiffness problems for the 1PI-FRG flow equations (only occurring in the broken-symmetry phase) on the one hand and for the 2PI-FRG tC-flow on the other hand are different.",
"Despite such limitations, it is rather fruitful to further exploit the equivalence between the tC-flow implementation of the 2PI-FRG and SCPT.",
"To that end, we recall that, as we have done in paper I, SCPT based on $\\Gamma ^{(\\mathrm {2PI})}(G)$ is carried out by solving the gap equations extremizing $\\Gamma ^{(\\mathrm {2PI})}(G)$ with respect to $G$ and then picking up the physical solutionThe physical solution $\\overline{G}$ leading to our SCPT results shown in Figs.",
"REF and REF is defined as the solution of the gap equation (for the propagator $G$ ) yielding the calculated complex gs density $\\rho ^{\\mathrm {calc}}_{\\mathrm {gs},\\mathrm {comp}}$ that is closest to the corresponding exact solution $\\rho _{\\mathrm {gs}}^{\\mathrm {exact}}$ , i.e.",
"that gives us the smallest norm $\\big \\vert \\rho ^{\\mathrm {calc}}_{\\mathrm {gs},\\mathrm {comp}}-\\rho _{\\mathrm {gs}}^{\\mathrm {exact}} \\big \\vert $ .",
"The term “norm” should be understood here as the norm of a complex number as $\\rho ^{\\mathrm {calc}}_{\\mathrm {gs},\\mathrm {comp}}$ might have a non-zero imaginary part.",
"The results displayed in Figs.",
"REF and REF are then defined as $\\rho _{\\mathrm {gs}}^{\\mathrm {calc}} = \\mathcal {R}e\\big (\\rho _{\\mathrm {gs},\\mathrm {comp}}^{\\mathrm {calc}}\\big )$ .",
"$\\overline{G}$ .",
"Remarkably, the initial conditions of the C-flow are such that the tC-flow results coincide with those of our physical solutions for $N_{\\mathrm {max}}=1 ~ \\mathrm {or} ~ 2$ , $N_{\\mathrm {max}}=5 ~ \\mathrm {or} ~ 6$ and $N_{\\mathrm {max}}=9 ~ \\mathrm {or} ~ 10$ according to Fig.",
"REF .",
"Nevertheless, there is a change of physical solutions in the perturbative regime (i.e.",
"for $\\lambda /4!\\ll 1$ ) for SCPT applied up to order $\\mathcal {O}(\\lambda ^{2})=\\mathcal {O}(U^{2})$ and $\\mathcal {O}(\\lambda ^{4})=\\mathcal {O}(U^{4})$ , as can be seen in Fig.",
"REF .",
"These correspond respectively to the tC-flow approach with truncation orders $N_{\\mathrm {max}}=3 ~ \\mathrm {or} ~ 4$ and $N_{\\mathrm {max}}=7 ~ \\mathrm {or} ~ 8$ , which are precisely the $N_{\\mathrm {max}}$ values where the stiffness problem arises.",
"This illustrates that the tC-flow is not suited to fully reproduce SCPT when there is a change of physical solutions involved in the latter framework for the chosen truncation of the 2PI EA.",
"This limitation can be attributed to the fact that the initial conditions for the tC-flow are fixed once and for all (i.e.",
"regardless of the values of coupling constants) from the perturbative expression (REF ) of the LW functional, which does not allow for reproducing the change of solutions observed in Fig.",
"REF .",
"This also implies that the stiffness problem arising in our tC-flow calculations is inherent to the C-flow formalism and not to the numerical tools exploited in this studyREF .",
"Regarding the mC-flow, we can see in Fig.",
"REF that the corresponding ansatz underlying the truncation manages to cure the aforementioned stiffness problem for $N_{\\mathrm {max}}= 3 ~ \\mathrm {or} ~ 4$ , but not for all choices of $N_{\\mathrm {SCPT}}$ : the combination $(N_{\\mathrm {max}},N_{\\mathrm {SCPT}})=(3,2)$ set by Eq.",
"(REF ) still suffers from it for instance.",
"Besides, it also introduces this issue at the truncation order $N_{\\mathrm {max}}=2$ which is not affected in the framework of the tC-flow: this problem arises e.g.",
"at the truncation $(N_{\\mathrm {max}},N_{\\mathrm {SCPT}})=(2,3)$ set by Eq.",
"(REF ), hence its absence from Fig.",
"REF .",
"Furthermore, we can also see that mC-flow results might deteriorate as $N_{\\mathrm {max}}$ and/or $N_{\\mathrm {SCPT}}$ increase(s), as can be noticed by comparing the curves associated with $(N_{\\mathrm {max}},N_{\\mathrm {SCPT}})=(2,2)$ and $(N_{\\mathrm {max}},N_{\\mathrm {SCPT}})=(3,3)$ in Fig.",
"REF .",
"This is an important drawback as it shows that we loose accuracy while incorporating explicitly more information in our truncation.",
"The ansatz underlying the mC-flow truncation is therefore not reliable.",
"We have only discussed applications of the C-flow version of the 2PI-FRG to the regime of the studied toy model with $m^{2}>0$ so far.",
"The corresponding formalism is actually not suited to treat the regime with $m^{2}<0$ : if one sets $m^{2}$ equal to a negative value to solve the equation system made of Eqs.",
"(REF ) to (REF ), the results thus obtained would be unphysical (with e.g.",
"a negative estimate of the gs density $\\rho _{\\mathrm {gs}}$ ).",
"A more relevant way the tackle the unbroken-symmetry regime with a C-flow approach would be to add a linear source $J$ in the generating functional (REF ), i.e.",
"to add a source term $\\int _{\\alpha } J_{\\alpha } \\widetilde{\\psi }_{\\alpha }$ alongside with $\\frac{1}{2}\\int _{\\alpha ,\\alpha ^{\\prime }} \\widetilde{\\psi }_{\\alpha } K_{\\alpha \\alpha ^{\\prime }} \\widetilde{\\psi }_{\\alpha ^{\\prime }}$ in definition (REF ).",
"However, besides complicating the C-flow formalism presented in this appendix, we do not expect such an extension to lift the limitations of the C-flow approach identified from Fig.",
"REF .",
"Hence, although we have only discussed here numerical applications of the C-flow to the unbroken-symmetry regime of our toy model at $N=1$See Ref.",
"[46] for additional results from the 2PI-FRG C-flow at $N=1$ or 2 in the original or mixed representation of the (0+0)-D $O(N)$ model., we have highlighted here important limitations of this 2PI-FRG implementation, which are inherent to this approach rather than to the model under consideration.",
"More precisely, neither the tC-flow nor the mC-flow are systematically improvable in a reliable fashion: i) the tC-flow worsens with increasing truncation orders and does not enable us to go reliably beyond the Hartree-Fock result of SCPT since it just reproduces SCPT without resummation; ii) the ansatz underlying the mC-flow does not seem reliable either as the corresponding results might also deteriorate as $N_{\\mathrm {max}}$ and/or $N_{\\mathrm {SCPT}}$ increase(s).",
"As an outlook of the present study, one could further investigate the properties of the series (REF ) representing the 2PI EA in the 2PI-FRG framework to see whether a combination of the tC-flow with resummation theory might be relevant to overcome limitation i).",
"We thus conclude that, unless one considers particularly weakly-coupled regimes where SCPT without resummation (and therefore the tC-flow) can improve with increasing truncation orders, the 2PI-FRG C-flow exploited here is not suited to design well-controlled systematically improvable approximation schemes."
],
[
"CU-flow",
"The CU-flow version of the 2PI-FRG [45] essentially amounts to combining the C-flow and the U-flow, and more specifically the tC-flow and the pU-flow, together.",
"A noticeable fact is that, as the C-flow and (if the corresponding cutoff function is chosen accordingly) the U-flow, such an approach also carries out the Wilsonian momentum-shell integration.",
"Hence, the CU-flow relies on the transformation $C^{-1} \\rightarrow C^{-1}_{\\mathfrak {s}}=C^{-1} + R^{(C)}_{\\mathfrak {s}}$ (or equivalently $C^{-1} \\rightarrow C^{-1}_{\\mathfrak {s}}=R^{(C)}_{\\mathfrak {s}} C^{-1}$ ) combined with $U \\rightarrow U_{\\mathfrak {s}} = U + R^{(U)}_{\\mathfrak {s}}$ (or equivalently $U \\rightarrow U_{\\mathfrak {s}} = R^{(U)}_{\\mathfrak {s}} U$ ), with: [left=]align Cs=si, = 0 .",
"Cs=sf = C .",
"Us=si,12 = 0 1, 2 .",
"Us=sf = U .",
"As can be seen from Eqs.",
"(REF ) and (REF ), the boundary conditions for $C_{\\mathfrak {s}}$ and $U_{\\mathfrak {s}}$ are therefore exactly identical to those of the C-flow and the U-flow, respectively.",
"Due to the initial condition (REF ), the starting point of the CU-flow suffers from the same divergence problem as that of the C-flow, which is why we will consider the functional $\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}}$ (defined by Eq.",
"(REF )) in the present framework as well.",
"Furthermore, we can already expect from the master equation (REF ) expressing $\\dot{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PI})}[G]$ that the differential equations underlying the CU-flow contain contributions from both the C-flow and the U-flow, besides the flow equation expressing $\\dot{\\overline{G}}_{\\mathfrak {s}}$ which is given by Eq.",
"(REF ) as in the C-flow.",
"Actually, since the LW functional is an invariant of the C-flow (according to Eq.",
"(REF )), the CU-flow equations expressing $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}$ and $\\dot{\\overline{\\Phi }}^{(n)}_{\\mathfrak {s}}$ (with $n \\ge 2$ ) coincide in principle with those of the pU-flow, and notably with Eq.",
"(REF ) for $\\dot{\\overline{\\Phi }}^{(2)}_{\\mathfrak {s}}$ and Eq.",
"(REF ) for $\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s}}$ .",
"There is however a subtlety that implies that Eq.",
"(REF ) (which expresses $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}$ for the pU-flow) is not valid in the present situationWe refer to appendix F.4.3 of Ref.",
"[46] for a technical explanation on this point.. With the help of the chain rule of Eq.",
"(REF ), we can rewrite this equation in a form exploitable for the CU-flow, which gives us: $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s},\\gamma } = & -\\frac{1}{3} \\bigg [ 2 \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}} \\dot{U}_{\\mathfrak {s}} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}} \\\\& + \\dot{U}_{\\mathfrak {s}} \\bigg ]_{\\alpha \\hat{\\alpha } \\hat{\\alpha }^{\\prime } \\alpha ^{\\prime }} \\overline{G}_{\\mathfrak {s},\\hat{\\gamma }} \\\\& +\\frac{1}{6} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1} \\hat{\\gamma }_{2}} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\overline{\\Phi }_{\\mathfrak {s},\\gamma \\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{(3)} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{1}}^{(2)} - \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }^{(2)}_{\\mathfrak {s},\\hat{\\gamma }\\gamma } \\;.\\end{split}$ We then discuss the initial conditions for the CU-flow.",
"On the one hand, as for the C-flow, imposing $C_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma }=0$ $\\forall \\gamma $ (i.e.",
"Eq.",
"(REF )) induces that: $\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma }=0 \\quad \\forall \\gamma \\;,$ $\\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}=0 \\;.$ On the other hand, as for the pU-flow, the condition $U_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma _{1}\\gamma _{2}}=0$ $\\forall \\gamma _{1},\\gamma _{2}$ (i.e.",
"Eq.",
"(REF )) implies that: $\\overline{\\Sigma }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma }=0 \\quad \\forall \\gamma \\;,$ $\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}},\\gamma _{1} \\cdots \\gamma _{n}}^{(n)}=0 \\quad \\forall \\gamma _{1},\\cdots , \\gamma _{n}, ~ \\forall n \\ge 2 \\;.$ Moreover, the infinite tower of differential equations for the CU-flow is closed by enforcing the same truncation condition as in the tC-flow and the pU-flow, i.e.",
": $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}=\\overline{\\Phi }_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {i}}}^{(n)} \\quad \\forall \\mathfrak {s}, ~ \\forall n > N_{\\mathrm {max}} \\;,$ which is why we mentioned that the CU-flow is essentially a merger of the latter two approaches.",
"Figure: Difference between the calculated gs energy E gs calc E_{\\mathrm {gs}}^{\\mathrm {calc}} or density ρ gs calc \\rho _{\\mathrm {gs}}^{\\mathrm {calc}} and the corresponding exact solution E gs exact E_{\\mathrm {gs}}^{\\mathrm {exact}} or ρ gs exact \\rho _{\\mathrm {gs}}^{\\mathrm {exact}} at m 2 =+1m^{2}=+1 and N=1N=1 (ℛe(λ)≥0\\mathcal {R}e(\\lambda )\\ge 0 and ℐm(λ)=0\\mathcal {I}m(\\lambda )=0).",
"The mC-flow curve represents the best C-flow result up to N max =3N_{\\mathrm {max}}=3 whereas the green, orange and purple mU-flow curves correspond to the best U-flow results obtained at N max =1,2 and 3N_{\\mathrm {max}}=1,2~\\mathrm {and}~3, respectively.As for the C-flow, we will only apply the CU-flow to the (0+0)-D $O(N)$ model at $N=1$ , which we will consider sufficient to make our point, showing notably that, in the unbroken-symmetry regime, the CU-flow version of the 2PI-FRG can achieve similar performances to those of the mU-flow at $N_{\\mathrm {SCPT}}=1$ .",
"We will thus exploit the following CU-flow equations to treat our toy model at $N=1$ : $\\dot{\\overline{G}}_{\\mathfrak {s}} = -\\overline{G}_{\\mathfrak {s}}^{2} \\left( \\dot{C}_{\\mathfrak {s}}^{-1} - \\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} \\right) \\;,$ $\\begin{split}\\Delta \\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = & \\ \\frac{1}{2} \\dot{C}_{\\mathfrak {s}}^{-1} \\left( \\overline{G}_{\\mathfrak {s}} - C_{\\mathfrak {s}} \\right) \\\\& + \\frac{\\dot{U}_{\\mathfrak {s}}}{24} \\left( 4\\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-1} + \\overline{G}_{\\mathfrak {s}}^{2} \\right) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = & \\ \\frac{\\dot{U}_{\\mathfrak {s}}}{3} \\overline{G}_{\\mathfrak {s}} \\left( 2 + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-2} \\left( \\frac{1}{2} \\overline{G}_{\\mathfrak {s}}^{3} \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\right) \\\\& - \\frac{\\dot{U}_{\\mathfrak {s}}}{6} \\overline{G}_{\\mathfrak {s}} - \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s}}^{(2)} = & \\ \\frac{\\dot{U}_{\\mathfrak {s}}}{6} \\bigg ( 2\\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-3} \\left( 8 \\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right)^{2} \\\\& - 64 \\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-2} \\overline{G}_{\\mathfrak {s}}^{-4} \\\\& + \\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-2} \\left(16\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\right) + 2 \\bigg ) \\\\& + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s}}^{(3)} = & \\ \\frac{\\dot{U}_{\\mathfrak {s}}}{6} \\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-2} \\bigg ( 6 \\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-2} \\\\& \\times \\left( 8 \\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right)^{3} + 6 \\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-1} \\\\& \\times \\left( 8 \\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right) \\left( 16 \\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\right) \\\\& - 384 \\overline{G}_{\\mathfrak {s}}^{-4} \\bigg ( \\left( 2 \\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-1}\\left(8\\overline{G}_{\\mathfrak {s}}^{-3}-\\overline{\\Phi }_{\\mathfrak {s}}^{(3)}\\right) \\\\& - \\overline{G}_{\\mathfrak {s}}^{-1} \\bigg ) - \\overline{\\Phi }_{\\mathfrak {s}}^{(5)} \\bigg ) + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\;,\\end{split}$ where we used once again the shorthand notations $\\overline{G}_{\\mathfrak {s}}\\equiv \\overline{G}_{\\mathfrak {s},11}$ , $\\overline{\\Sigma }_{\\mathfrak {s}}\\equiv \\overline{\\Sigma }_{\\mathfrak {s},11}$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}\\equiv \\overline{\\Phi }_{\\mathfrak {s},(1,1)\\cdots (1,1)}^{(n)}$ $\\forall n$ .",
"We will also use the same cutoff functions as in our previous C-flow and U-flow applications, i.e.",
": $C_{\\mathfrak {s}}^{-1} = C^{-1} + R^{(C)}_{\\mathfrak {s}} = m^{2} + \\mathfrak {s}^{-1} - 1 \\;,$ $U_{\\mathfrak {s}} = R^{(U)}_{\\mathfrak {s}} U = \\mathfrak {s}\\lambda \\;,$ still at $N=1$ , and with $\\mathfrak {s}$ also running from $\\mathfrak {s}_{\\mathrm {i}}=0$ to $\\mathfrak {s}_{\\mathrm {f}}=1$ .",
"The gs energy and density of the (0+0)-D $O(N)$ model will be extracted from the flowing objects of the CU-flow exactly as within the C-flow, namely: $E^{\\text{2PI-FRG;CU-flow}}_{\\mathrm {gs}} = \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\underset{N=1}{=} \\Delta \\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} -\\frac{1}{2} \\mathrm {ln}\\bigg (\\frac{2\\pi }{m^{2}}\\bigg ) \\;,$ $\\rho ^{\\text{2PI-FRG;CU-flow}}_{\\mathrm {gs}} = \\frac{1}{N} \\sum _{a=1}^{N} \\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}},aa} \\underset{N=1}{=} \\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}} \\;.$ In Fig.",
"REF which shows the results thus obtained, the CU-flow estimates for $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ clearly outperform at $N_{\\mathrm {max}}=2$ the mC-flow curve which can be considered as our best C-flow result, as can be seen from Fig.",
"REF .",
"Furthermore, Fig.",
"REF shows that the CU-flow results are comparable to the best mU-flow ones (i.e.",
"to the mU-flow results at $N_{\\mathrm {SCPT}}=1$ ) up to $N_{\\mathrm {max}}=3$ for both $E_{\\mathrm {gs}}$ and $\\rho _{\\mathrm {gs}}$ .",
"Similarly to the previous analysis of our C-flow results in section REF , we also expect the latter remarks to be general and notably that the CU-flow could also efficiently tackle $O(N)$ models at $N>1$See Ref.",
"[46] for additional results from the 2PI-FRG CU-flow in the framework of the (0+0)-D $O(N)$ model at $N=2$ ..",
"Note however that, for the same reasons as the C-flow, the CU-flow is not suited to treat the unbroken-symmetry regime of the studied toy model."
],
[
"1PI formalism",
"In this appendix, we will explain how the $O(N)$ symmetry is exploited to simplify the equations underlying the vertex expansion of the Wetterich equation within the unbroken-symmetry phase of the (0+0)-D $O(N)$ model.",
"To that end, we start by considering the corresponding Wetterich equation, given by Eq.",
"(REF ) that can be put in the form: $\\dot{\\Gamma }^{(\\mathrm {1PI})}_{k}\\Big (\\vec{\\phi }\\Big ) = \\frac{1}{2}\\sum _{a_{1},a_{2}=1}^{N} \\dot{R}_{k,a_{1}a_{2}} G_{k,a_{2}a_{1}}\\Big (\\vec{\\phi }\\Big ) \\;.$ where the propagator $G_{k}\\big (\\vec{\\phi }\\big )$ is defined by: $G^{-1}_{k,a_{1}a_{2}}\\Big (\\vec{\\phi }\\Big ) \\equiv \\Gamma ^{(\\mathrm {1PI})(2)}_{k,a_{1}a_{2}}\\Big (\\vec{\\phi }\\Big )+R_{k,a_{1}a_{2}} \\;.$ We then briefly present the main ingredients of the vertex expansion of Eq.",
"(REF ) in the unbroken-symmetry phase.",
"In the latter phase, we have $\\vec{\\overline{\\phi }}_{k}=\\vec{0}$ $\\forall k$ and all 1PI vertices of odd order, i.e.",
"$\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(n)}$ with $n$ odd, also vanish according to Eq.",
"(REF ).",
"According to this, we can simplify the Taylor expansion (REF ) of $\\Gamma _{k}^{(\\mathrm {1PI})}\\big (\\vec{\\phi }\\big )$ as follows: $\\begin{split}\\Gamma ^{(\\mathrm {1PI})}_{k}\\Big (\\vec{\\phi }\\Big ) = & \\ \\overline{\\Gamma }^{(\\mathrm {1PI})}_{k} \\\\& + \\sum _{\\underset{\\lbrace \\text{n even}\\rbrace }{n=2}}^{\\infty }\\frac{1}{n!",
"}\\sum _{a_{1},\\cdots ,a_{n}=1}^{N} \\overline{\\Gamma }_{k,a_{1} \\cdots a_{n}}^{(\\mathrm {1PI})(n)} \\phi _{a_{1}} \\cdots \\phi _{a_{n}} \\;,\\end{split}$ where the curly braces indicate below discrete sums indicate a constraint that must be satisfied by all terms of the sum in question and the last line of Eq.",
"(REF ) thus only involves terms with even values of $n$ .",
"From Eq.",
"(REF ), we can infer an expansion of the left-hand side of the Wetterich equation (REF ) in powers of the fields, i.e.",
": $\\begin{split}\\dot{\\Gamma }^{(\\mathrm {1PI})}_{k}\\Big (\\vec{\\phi }\\Big ) = & \\ \\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{k} \\\\& + \\sum _{\\underset{\\lbrace \\text{n even}\\rbrace }{n=2}}^{\\infty }\\frac{1}{n!",
"}\\sum _{a_{1},\\cdots ,a_{n}=1}^{N} \\dot{\\overline{\\Gamma }}_{k,a_{1} \\cdots a_{n}}^{(\\mathrm {1PI})(n)} \\phi _{a_{1}} \\cdots \\phi _{a_{n}} \\;.\\end{split}$ Focusing then on the right-hand side of the Wetterich equation (REF ), one can first expand the propagator $G_{k}\\big (\\vec{\\phi }\\big )$ around $\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k}=0$ $\\forall k$ , still using Eq.",
"(REF ) to simplify the relations thus derived.",
"This leads to: $\\begin{split}& G_{k,a_{1}a_{2}}\\Big (\\vec{\\phi }\\Big ) \\\\& = \\hspace{14.22636pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag1}\\begin{fmfgraph*}(20,20){i0,i1,i2,i3}{o0,o1,o2,o3}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label=a_{1}}{v1}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label=a_{2}}{v2}{phantom}{i1,v1}{phantom}{i2,v1}{plain,tension=0.6}{v1,v2}{phantom}{v2,o1}{phantom}{v2,o2}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{11.38092pt} - \\frac{1}{2} \\hspace{19.91684pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag2}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{plain}{v1,vtopLeft}{plain}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\& \\hspace{8.5359pt} + \\frac{1}{4} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag3}\\begin{fmfgraph*}(30,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{iUp}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{oUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=50}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-50}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-50}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=50}{vDownR}{plain,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{plain,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=0.7}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=4.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=4.0}{vDownR,vDown}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v2,vDownR}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{7.11317pt} - \\frac{1}{24} \\hspace{19.91684pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag4}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{decor.shape=cross,decor.size=3.5thick,decor.angle=-15}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=15}{vDownR}{plain}{v1,vtopLeft}{plain}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v1,vDownR}{phantom,tension=3.0}{vDownL,vDown}{phantom,tension=1.0}{vDownL,vbottomLeft}{phantom,tension=3.0}{vDownR,vDown}{phantom,tension=1.0}{vDownR,vbottomRight}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} - \\frac{1}{8} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag5}\\begin{fmfgraph*}(30,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{iUp}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{oUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v3}{decor.shape=cross,decor.size=3.5thick,decor.angle=50}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-50}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-50}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=50}{vDownR}{decor.shape=cross,decor.size=3.5thick}{vUp}{decor.shape=cross,decor.size=3.5thick}{vDown}{plain,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{plain,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=1.0}{v1,v3}{plain,tension=1.0}{v3,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v2,vDownR}{dashes,tension=0}{v3,vUp}{dashes,tension=0}{v3,vDown}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{7.11317pt} + \\frac{1}{48} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag6}\\begin{fmfgraph*}(25,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{iUp}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{oUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{vDownR}{decor.shape=cross,decor.size=3.5thick}{iMid}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{vUpL}{plain,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{plain,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=0.5}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v2,vDownR}{dashes,tension=0}{iMid,v1}{phantom,tension=1.0}{vUpL,iUp}{phantom,tension=2.0}{vUpL,vUp}{dashes,tension=0}{v1,vUpL}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} + \\frac{1}{48} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag7}\\begin{fmfgraph*}(25,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{iUp}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{oUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{vDownR}{decor.shape=cross,decor.size=3.5thick}{oMid}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{vUpR}{plain,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{plain,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=0.5}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v2,vDownR}{dashes,tension=0}{oMid,v2}{phantom,tension=1.0}{vUpR,oUp}{phantom,tension=2.0}{vUpR,vUp}{dashes,tension=0}{v2,vUpR}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{9.95863pt} - \\frac{1}{720} \\hspace{19.91684pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_Diag8}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{vDown}{i1}{o1}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=8,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{decor.shape=cross,decor.size=3.5thick,decor.angle=-15}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=15}{vDownR}{decor.shape=cross,decor.size=3.5thick}{o1}{decor.shape=cross,decor.size=3.5thick}{i1}{plain}{v1,vtopLeft}{plain}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v1,vDownR}{phantom,tension=3.0}{vDownL,vDown}{phantom,tension=1.0}{vDownL,vbottomLeft}{phantom,tension=3.0}{vDownR,vDown}{phantom,tension=1.0}{vDownR,vbottomRight}{dashes,tension=0}{v1,o1}{dashes,tension=0}{v1,i1}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} + \\mathcal {O}\\bigg (\\left|\\vec{\\phi }\\right|^8\\bigg ) \\;,\\end{split}$ where we have used the diagrammatic rules: ${\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_FeynRuleCross}\\begin{fmfgraph*}(20,20){i0,i1,i2,i3}{o0,o1,o2,o3}{label=a,decor.shape=cross,decor.size=3.5thick}{v2}{phantom}{i1,v1}{phantom}{i2,v1}{dashes,tension=0.6}{v1,v2}{phantom}{v2,o1}{phantom}{v2,o2}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\quad &\\rightarrow \\quad \\phi _{a} \\;, \\\\{\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_FeynRule1PIvertex}\\begin{fmfgraph*}(8,6){iDown,i1,iUp}{oDown,o1,oUp}{vUpL,vUp,vUpR}{vDownL,vDown,vDownR}{label=a_{1},label.angle=180,label.dist=0.06cm}{i1}{label=a_{n},label.angle=135,label.dist=0.1cm}{iUp}{label=a_{2},label.angle=-135,label.dist=0.06cm}{iDown}{label=.,label.dist=0cm}{o1}{label=.,label.angle=-135,label.dist=-0.03cm}{oUp}{label=.,label.angle=135,label.dist=-0.03cm}{oDown}{label=.,label.angle=-90,label.dist=0.07cm}{vDown}{label=.,label.angle=90,label.dist=0.07cm}{vUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=n,label.dist=0}{v1}{plain,tension=0.5}{i1,v1}{plain,tension=0.5}{iDown,v1}{plain,tension=0.5}{iUp,v1}{phantom,tension=0.5}{vDown,v1}{phantom,tension=0.5}{v1,o1}{phantom,tension=0.5}{v1,oUp}{phantom,tension=0.5}{v1,oDown}{phantom,tension=0.5}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\quad &\\rightarrow \\quad \\overline{\\Gamma }_{k,a_{1} \\cdots a_{n}}^{(\\mathrm {1PI})(n)} \\equiv \\left.\\frac{\\partial ^{n}\\Gamma _{k}^{(\\mathrm {1PI})}\\big (\\vec{\\phi }\\big )}{\\partial \\phi _{a_{1}}\\cdots \\partial \\phi _{a_{n}}}\\right|_{\\vec{\\phi }=\\vec{0}} \\;, \\\\{\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_Gexpansion_FeynRuleG}\\begin{fmfgraph*}(20,20){i0,i1,i2,i3}{o0,o1,o2,o3}{label=a_{1}}{v1}{label=a_{2}}{v2}{phantom}{i1,v1}{phantom}{i2,v1}{plain,tension=0.6}{v1,v2}{phantom}{v2,o1}{phantom}{v2,o2}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\quad &\\rightarrow \\quad \\overline{G}_{k,a_{1}a_{2}} \\equiv G_{k,a_{1}a_{2}}\\Big (\\vec{\\phi }=\\vec{\\overline{\\phi }}_{k}\\Big )\\;,$ and the empty dots just indicate external points.",
"As a next step, we insert this diagrammatic expression into the right-hand side of Eq.",
"(REF ).",
"As a result of sums over color indices in the latter relation, this basically amounts to joining the external points in Eq.",
"(REF ) as follows: ${\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_ConsequenceSTrOperator_Diag1}\\begin{fmfgraph*}(20,20){i0,i1,i2,i3}{o0,o1,o2,o3}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label=a_{1}}{v1}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label=a_{2}}{v2}{decor.shape=circle,decor.filled=shaded,decor.size=8.0thick}{v3}{phantom}{i1,v1}{phantom}{i2,v1}{plain,tension=0.6}{v1,v3}{plain,tension=0.6}{v3,v2}{phantom}{v2,o1}{phantom}{v2,o2}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{22.76228pt} \\xrightarrow{} \\hspace{-2.84544pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_ConsequenceSTrOperator_Diag2}\\begin{fmfgraph*}(20,20){i0,i1,i2,i3}{o0,o1,o2,o3}{vUp}{decor.shape=circle,decor.filled=shaded,decor.size=8.0thick}{v3}{decor.shape=cross,decor.size=6.0thick}{vUp}{phantom}{i1,v1}{phantom}{i2,v1}{phantom,tension=0.6}{v1,v3}{phantom,tension=0.6}{v3,v2}{phantom}{v2,o1}{phantom}{v2,o2}{plain,left,tension=0}{v3,vUp}{plain,right,tension=0}{v3,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{-5.69046pt} \\;,$ where the cross indicates an insertion of $\\dot{R}_{k}$ according to the ruleNote that rule (redE.24) and Eqs.",
"(redE.65) to (redE.67) and (redE.75) to (redE.78) of Ref.",
"[46] are erroneous and should be replaced by rule (REF ) and Eqs.",
"(REF ) to (REF ) and (REF ) to (REF ), respectively.",
"We also point out that the last lines of Eqs.",
"(red4.55), (red4.56) and (red4.57) of Ref.",
"[46] contain errors as well and should respectively read: “$=\\left(m^{2}+\\frac{\\lambda }{6}\\overline{\\phi }^{2}_{k=k_{\\mathrm {i}},N}\\right)\\delta _{a_{1}a_{2}}+\\frac{\\lambda }{3}\\overline{\\phi }^{2}_{k=k_{\\mathrm {i}},N}\\delta _{a_{1}N}\\delta _{a_{2}N} \\quad \\forall a_{1},a_{2}$ ”, “$=\\frac{\\lambda }{3} \\overline{\\phi }_{k=k_{\\mathrm {i}},N} \\big (\\delta _{a_{1}N}\\delta _{a_{2}a_{3}}+\\delta _{a_{2}N}\\delta _{a_{1}a_{3}}+\\delta _{a_{3}N}\\delta _{a_{1}a_{2}}\\big ) \\quad \\forall a_{1}, a_{2}, a_{3}$ ” and “$=\\frac{\\lambda }{3} \\big ( \\delta _{a_{1}a_{2}} \\delta _{a_{3}a_{4}} + \\delta _{a_{1}a_{3}} \\delta _{a_{2}a_{4}} + \\delta _{a_{1}a_{4}} \\delta _{a_{2}a_{3}} \\big ) \\quad \\forall a_{1}, a_{2}, a_{3}, a_{4}$ ”.",
": ${\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_FeynRuleRkdot}\\begin{fmfgraph*}(8,4){i1}{o1}{decor.shape=cross,decor.size=6.0thick,label=a_{1},label.angle=0,label.dist=16}{v1}{label=a_{2},label.angle=180,label.dist=16}{v2}{plain}{i1,v1}{plain}{v1,o1}{plain}{i1,v2}{plain}{v2,o1}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\quad \\quad \\rightarrow \\quad \\dot{R}_{k,a_{1}a_{2}} = \\dot{R}_{k} \\delta _{a_{1}a_{2}}\\;,$ and the shaded blobs encompass any combinations of 1PI vertices and field insertions (REF ) that can be found in Eq.",
"(REF ).",
"Hence, the right-hand side of Eq.",
"(REF ) becomes in this way: $\\begin{split}& \\frac{1}{2}\\sum _{a_{1},a_{2}=1}^{N} \\dot{R}_{k,a_{1}a_{2}} G_{k,a_{2}a_{1}}\\Big (\\vec{\\phi }\\Big ) \\\\& = \\frac{1}{2} \\hspace{5.69046pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag1}\\begin{fmfgraph*}(10,10){vUp}{vDown}{decor.shape=cross,decor.size=6.0thick}{vUp}{plain,left}{vUp,vDown}{plain,right}{vUp,vDown}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{11.38092pt} - \\frac{1}{4} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag2}\\begin{fmfgraph*}(15,15){vUp}{vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=cross,decor.size=6.0thick}{vUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{phantom}{v1,vtopLeft}{phantom}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{plain,left,tension=0}{v1,vUp}{plain,right,tension=0}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\& \\hspace{8.5359pt} + \\frac{1}{8} \\hspace{2.84544pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag3}\\begin{fmfgraph*}(26.25,11.25){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=45}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-45}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-45}{iUp}{decor.shape=cross,decor.size=3.5thick,decor.angle=45}{oUp}{decor.shape=cross,decor.size=6.0thick}{vCross}{dashes,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{dashes,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,right,tension=0.7}{v1,v2}{plain,left,tension=0.7}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=4.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=4.0}{vDownR,vDown}{phantom,tension=8.0}{vCross,vUp}{phantom,tension=1.0}{vCross,vDown}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{2.84544pt} - \\frac{1}{48} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag4}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{vDown}{vUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{decor.shape=cross,decor.size=3.5thick,decor.angle=-15}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=15}{vDownR}{decor.shape=cross,decor.size=6.0thick}{vUp}{phantom}{v1,vtopLeft}{phantom}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v1,vDownR}{phantom,tension=3.0}{vDownL,vDown}{phantom,tension=1.0}{vDownL,vbottomLeft}{phantom,tension=3.0}{vDownR,vDown}{phantom,tension=1.0}{vDownR,vbottomRight}{plain,left,tension=0}{v1,vUp}{plain,right,tension=0}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} - \\frac{1}{16} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag5}\\begin{fmfgraph*}(30,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick}{iUp}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick}{oUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v3}{decor.shape=cross,decor.size=3.5thick,decor.angle=58}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-58}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=56}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=-56}{vDownR}{decor.shape=cross,decor.size=3.5thick,decor.angle=-58}{iUp}{decor.shape=cross,decor.size=3.5thick,decor.angle=58}{oUp}{decor.shape=cross,decor.size=6.0thick}{vUp}{dashes,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{dashes,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=1.0}{v1,v3}{plain,tension=1.0}{v3,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v3,vDownL}{dashes,tension=0}{v3,vDownR}{phantom,tension=1.0}{v3,vUp}{phantom,tension=1.0}{v3,vDown}{plain,left=0.35,tension=0}{v1,vUp}{plain,right=0.35,tension=0}{v2,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{2.84544pt} + \\frac{1}{48} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag6}\\begin{fmfgraph*}(25,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{vDownR}{decor.shape=cross,decor.size=3.5thick}{iMid}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{iUp}{decor.shape=cross,decor.size=6.0thick}{vUp}{dashes,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{phantom,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=0.5}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v2,vDownR}{dashes,tension=0}{iMid,v1}{phantom,tension=1.0}{vUpL,iUp}{phantom,tension=2.0}{vUpL,vUp}{plain,left=0.45,tension=0}{v1,vUp}{plain,right=0.45,tension=0}{v2,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} - \\frac{1}{1440} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag7}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{vDown}{vUp}{i1}{o1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=8,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{decor.shape=cross,decor.size=3.5thick,decor.angle=-15}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=15}{vDownR}{decor.shape=cross,decor.size=3.5thick}{o1}{decor.shape=cross,decor.size=3.5thick}{i1}{decor.shape=cross,decor.size=6.0thick}{vUp}{phantom}{v1,vtopLeft}{phantom}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v1,vDownR}{phantom,tension=3.0}{vDownL,vDown}{phantom,tension=1.0}{vDownL,vbottomLeft}{phantom,tension=3.0}{vDownR,vDown}{phantom,tension=1.0}{vDownR,vbottomRight}{dashes,tension=0}{v1,o1}{dashes,tension=0}{v1,i1}{plain,left,tension=0}{v1,vUp}{plain,right,tension=0}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} + \\mathcal {O}\\bigg (\\left|\\vec{\\phi }\\right|^8\\bigg ) \\;.\\end{split}$ According to Eq.",
"(REF ), we can equate the right-hand sides of Eqs.",
"(REF ) and (REF ) and then identify the terms with identical powers of the field, thus yielding: $\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{k} = \\frac{1}{2} \\hspace{5.69046pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag1}\\begin{fmfgraph*}(10,10){vUp}{vDown}{decor.shape=cross,decor.size=6.0thick}{vUp}{plain,left}{vUp,vDown}{plain,right}{vUp,vDown}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{5.69046pt} \\;,$ $\\sum _{a_{1},a_{2}=1}^{N} \\dot{\\overline{\\Gamma }}_{k,a_{1} a_{2}}^{(\\mathrm {1PI})(2)} \\phi _{a_{1}} \\phi _{a_{2}} = - \\frac{1}{2} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag2}\\begin{fmfgraph*}(15,15){vUp}{vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=cross,decor.size=6.0thick}{vUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{phantom}{v1,vtopLeft}{phantom}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{plain,left,tension=0}{v1,vUp}{plain,right,tension=0}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{5.69046pt} \\;,$ $\\begin{split}& \\sum _{a_{1},a_{2},a_{3},a_{4}=1}^{N} \\dot{\\overline{\\Gamma }}_{k,a_{1} a_{2} a_{3} a_{4}}^{(\\mathrm {1PI})(4)} \\phi _{a_{1}} \\phi _{a_{2}} \\phi _{a_{3}} \\phi _{a_{4}} \\\\\\\\& = 3 \\hspace{2.84544pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag3}\\begin{fmfgraph*}(26.25,11.25){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=45}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-45}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-45}{iUp}{decor.shape=cross,decor.size=3.5thick,decor.angle=45}{oUp}{decor.shape=cross,decor.size=6.0thick}{vCross}{dashes,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{dashes,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,right,tension=0.7}{v1,v2}{plain,left,tension=0.7}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=4.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=4.0}{vDownR,vDown}{phantom,tension=8.0}{vCross,vUp}{phantom,tension=1.0}{vCross,vDown}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{2.84544pt} - \\frac{1}{2} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag4}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{vDown}{vUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{decor.shape=cross,decor.size=3.5thick,decor.angle=-15}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=15}{vDownR}{decor.shape=cross,decor.size=6.0thick}{vUp}{phantom}{v1,vtopLeft}{phantom}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v1,vDownR}{phantom,tension=3.0}{vDownL,vDown}{phantom,tension=1.0}{vDownL,vbottomLeft}{phantom,tension=3.0}{vDownR,vDown}{phantom,tension=1.0}{vDownR,vbottomRight}{plain,left,tension=0}{v1,vUp}{plain,right,tension=0}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{5.69046pt} \\;,\\end{split}$ $\\begin{split}& \\sum _{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}=1}^{N} \\dot{\\overline{\\Gamma }}_{k,a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}^{(\\mathrm {1PI})(6)} \\phi _{a_{1}} \\phi _{a_{2}} \\phi _{a_{3}} \\phi _{a_{4}} \\phi _{a_{5}} \\phi _{a_{6}} \\\\\\\\& = - 45 {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag5}\\begin{fmfgraph*}(30,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick}{iUp}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick}{oUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v3}{decor.shape=cross,decor.size=3.5thick,decor.angle=58}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-58}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=56}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=-56}{vDownR}{decor.shape=cross,decor.size=3.5thick,decor.angle=-58}{iUp}{decor.shape=cross,decor.size=3.5thick,decor.angle=58}{oUp}{decor.shape=cross,decor.size=6.0thick}{vUp}{dashes,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{dashes,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=1.0}{v1,v3}{plain,tension=1.0}{v3,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v3,vDownL}{dashes,tension=0}{v3,vDownR}{phantom,tension=1.0}{v3,vUp}{phantom,tension=1.0}{v3,vDown}{plain,left=0.35,tension=0}{v1,vUp}{plain,right=0.35,tension=0}{v2,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} + 15 \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag6}\\begin{fmfgraph*}(25,12.75){iDown,iMid,iUp}{oDown,oMid,oUp}{vUp}{vDown}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=6,label.dist=0}{v1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{iDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{oDown}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=60}{vDownR}{decor.shape=cross,decor.size=3.5thick}{iMid}{decor.shape=cross,decor.size=3.5thick,decor.angle=-60}{iUp}{decor.shape=cross,decor.size=6.0thick}{vUp}{dashes,tension=1.0}{v1,iUp}{dashes,tension=1.0}{v1,iDown}{phantom,tension=1.0}{v2,oUp}{dashes,tension=1.0}{v2,oDown}{plain,tension=0.5}{v1,v2}{phantom,tension=1.0}{vDownL,iDown}{phantom,tension=2.0}{vDownL,vDown}{phantom,tension=1.0}{vDownR,oDown}{phantom,tension=2.0}{vDownR,vDown}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v2,vDownR}{dashes,tension=0}{iMid,v1}{phantom,tension=1.0}{vUpL,iUp}{phantom,tension=2.0}{vUpL,vUp}{plain,left=0.45,tension=0}{v1,vUp}{plain,right=0.45,tension=0}{v2,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\\\\\& \\hspace{8.5359pt} - \\frac{1}{2} \\hspace{11.38092pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_RHSexpansion_Diag7}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{vDown}{vUp}{i1}{o1}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=8,label.dist=0}{v1}{decor.shape=cross,decor.size=3.5thick,decor.angle=40}{vbottomLeft}{decor.shape=cross,decor.size=3.5thick,decor.angle=-40}{vbottomRight}{decor.shape=cross,decor.size=3.5thick,decor.angle=-15}{vDownL}{decor.shape=cross,decor.size=3.5thick,decor.angle=15}{vDownR}{decor.shape=cross,decor.size=3.5thick}{o1}{decor.shape=cross,decor.size=3.5thick}{i1}{decor.shape=cross,decor.size=6.0thick}{vUp}{phantom}{v1,vtopLeft}{phantom}{v1,vtopRight}{dashes}{v1,vbottomLeft}{dashes}{v1,vbottomRight}{dashes,tension=0}{v1,vDownL}{dashes,tension=0}{v1,vDownR}{phantom,tension=3.0}{vDownL,vDown}{phantom,tension=1.0}{vDownL,vbottomLeft}{phantom,tension=3.0}{vDownR,vDown}{phantom,tension=1.0}{vDownR,vbottomRight}{dashes,tension=0}{v1,o1}{dashes,tension=0}{v1,i1}{plain,left,tension=0}{v1,vUp}{plain,right,tension=0}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{11.38092pt} \\;.\\end{split}$ We are now arriving at the core of our derivations since we are about to explain how to simplify the sums over color indices involved in Eqs.",
"(REF ) to (REF ) by using the definitions of symmetric parts of the cutoff function (given by Eq.",
"(REF )), of the 1PI vertices (given by Eqs.",
"(REF ) to (REF )) and of the propagator (given by Eq.",
"(REF )).",
"We will explain in this way the appearance of the $N$ -dependent factors in the right-hand sides of the resulting flow equations, expressed notably by Eqs.",
"(REF ) to (REF ).",
"As a first step, we can directly evaluate the sums over color indices in Eq.",
"(REF ) (using Eqs.",
"(REF ) and (REF )) to obtain: $\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{k} = \\frac{N}{2} \\overline{G}_{k} \\;,$ since the sums over color indices yield a factor $N$ for any closed propagator loop in the present situation.",
"In order to evaluate such sums in (REF ) to (REF ), we must open up the 1PI vertices in favor of their symmetric part.",
"For $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(4)}$ , this amounts to writing: $\\begin{split}\\hspace{14.22636pt}{\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamma4_Diag1}\\begin{fmfgraph*}(15,15){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vbottomLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{3}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{4}}{vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v1}{plain}{v1,vtopLeft}{plain}{v1,vtopRight}{plain}{v1,vbottomLeft}{plain}{v1,vbottomRight}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{14.22636pt} = & \\ \\frac{1}{3} \\left(\\rule {0cm}{1.7cm}\\right.",
"\\hspace{14.22636pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamma4_Diag2}\\begin{fmfgraph*}(20,25){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vbottomLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{3}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{4}}{vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{plain}{v1,vtopLeft}{plain}{v1,vbottomLeft}{dots,tension=1.0}{v1,v2}{dots,tension=1.0}{v2,v3}{plain}{v3,vtopRight}{plain}{v3,vbottomRight}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\& + \\hspace{14.22636pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamma4_Diag3}\\begin{fmfgraph*}(20,25){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vbottomLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{3}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{4}}{vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{phantom}{v1,vtopLeft}{plain,tension=0}{v1,vtopRight}{plain}{v1,vbottomLeft}{dots,tension=1.0}{v1,v2}{dots,tension=1.0}{v2,v3}{phantom}{v3,vtopRight}{plain,tension=0}{v3,vtopLeft}{plain}{v3,vbottomRight}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\& + \\hspace{14.22636pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamma4_Diag4}\\begin{fmfgraph*}(20,25){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vbottomLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{3}}{vtopRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{4}}{vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{phantom}{v1,vtopLeft}{plain}{v1,vbottomLeft}{plain,tension=0}{v1,vbottomRight}{dots,tension=1.0}{v1,v2}{dots,tension=1.0}{v2,v3}{plain}{v3,vtopRight}{phantom}{v3,vbottomRight}{plain,tension=0}{v3,vtopLeft}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{14.22636pt} \\left.\\rule {0cm}{1.7cm}\\right) \\;,\\end{split}$ with the following rule: ${\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamman_FeynRule}\\begin{fmfgraph*}(15,10){iDown,i1,iUp}{oDown,o1,oUp}{vUpL,vUp,vUpR}{vDownL,vDown,vDownR}{label=.,label.dist=-0.4cm}{o1}{label=.,label.angle=-135,label.dist=0.325cm}{oUp}{label=.,label.angle=135,label.dist=0.325cm}{oDown}{label=.,label.angle=-90,label.dist=-0.14cm}{vDown}{label=.,label.angle=90,label.dist=-0.14cm}{vUp}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=n,label.dist=0}{v1}{dots,tension=0.5}{i1,v1}{dots,tension=0.5}{iDown,v1}{dots,tension=0.5}{iUp,v1}{phantom,tension=0.5}{vDown,v1}{phantom,tension=0.5}{v1,o1}{phantom,tension=0.5}{v1,oUp}{phantom,tension=0.5}{v1,oDown}{phantom,tension=0.5}{v1,vUp}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{2.84544pt} \\rightarrow \\quad \\overline{\\Gamma }^{(\\mathrm {1PI})(n)}_{k} \\;,$ where there are $n/2$ dotted lines that leave the empty circle containing the integer $n$ (which is always even here).",
"Rule (REF ) is to be distinguished from that numbered () since the right-hand side of the latter (former) only involves the full (symmetric part of the) 1PI vertices.",
"After inserting Eqs.",
"(REF ) and (REF ) into result (REF ), we obtain: $\\begin{split}\\dot{\\overline{\\Gamma }}_{k}^{(\\mathrm {1PI})(2)} \\delta _{a_{1} a_{2}} = & - \\frac{1}{6} \\left(\\rule {0cm}{1.7cm}\\right.",
"\\hspace{14.22636pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamma4_Diag5}\\begin{fmfgraph*}(25,25){vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{o1}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vtopLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vbottomLeft}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=6.0thick}{o1}{plain}{v1,vtopLeft}{plain}{v1,vbottomLeft}{dots,tension=1.0}{v1,v2}{dots,tension=1.0}{v2,v3}{phantom,tension=0.5}{v3,vtopRight}{phantom,tension=0.5}{v3,vbottomRight}{plain,left,tension=0}{v3,o1}{plain,right,tension=0}{v3,o1}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\\\& + 2 \\hspace{14.22636pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/pure1PIFRG_OpenGamma4_Diag6}\\begin{fmfgraph*}(20,25){vUp}{vDown}{vtopLeft,vtopRight}{vbottomLeft,vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{1}}{vbottomLeft}{decor.shape=circle,decor.filled=empty,decor.size=1.5thick,label.dist=0.15cm,label=a_{2}}{vbottomRight}{decor.shape=circle,decor.filled=empty,decor.size=0.5cm,label=4,label.dist=0}{v2}{decor.shape=cross,decor.size=6.0thick}{vUpbis}{phantom}{v1,vtopLeft}{plain}{v1,vbottomLeft}{dots,tension=1.0}{v1,v2}{dots,tension=1.0}{v2,v3}{phantom}{v3,vtopRight}{plain}{v3,vbottomRight}{phantom,tension=4.5}{vUp,vUpbis}{phantom,tension=1.0}{vDown,vUpbis}{plain,left=0.45,tension=0}{v1,vUpbis}{plain,right=0.45,tension=0}{v3,vUpbis}\\end{fmfgraph*}\\end{fmffile}\\end{array}} \\hspace{14.22636pt} \\left.\\rule {0cm}{1.7cm}\\right) \\\\\\\\= & -\\frac{N+2}{6} \\dot{R}_{k} \\overline{G}^{2}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k} \\delta _{a_{1}a_{2}} \\;,\\end{split}$ which, at $a_{1}=a_{2}$ , corresponds to our final expression for $\\dot{\\overline{\\Gamma }}_{k}^{(\\mathrm {1PI})(2)}$ : $\\dot{\\overline{\\Gamma }}_{k}^{(\\mathrm {1PI})(2)} = -\\frac{N+2}{6} \\dot{R}_{k} \\overline{G}^{2}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k} \\;.$ We have thus derived the flow equation (REF ).",
"To derive the homologous flow equations for the vertices of higher orders, we must open up vertices of order $n>4$ .",
"For Eq.",
"(REF ) that expresses $\\dot{\\overline{\\Gamma }}_{k}^{(\\mathrm {1PI})(4)}$ in terms of $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(4)}$ and $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(6)}$ , we also need to open up $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(6)}$ using this time the 15 permutations involved in Eq.",
"(REF ).",
"This results in: $\\begin{split}& \\frac{1}{3}\\dot{\\overline{\\Gamma }}_{k}^{(\\mathrm {1PI})(4)}\\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}+\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}}+\\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\right) \\\\& = \\left(\\frac{N+8}{9}\\dot{R}_{k} \\overline{G}^{3}_{k}\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4)}_{k}\\right)^2 - \\frac{N+4}{30}\\dot{R}_{k} \\overline{G}^{2}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(6)}_{k}\\right) \\\\& \\hspace{8.5359pt} \\times \\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}+\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}}+\\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\right) \\;,\\end{split}$ which is equivalent to Eq.",
"(REF ) after equating the free indices.",
"To then derive the flow equation (REF ) expressing $\\dot{\\overline{\\Gamma }}_{k}^{(\\mathrm {1PI})(6)}$ , the 1PI vertex $\\overline{\\Gamma }_{k}^{(\\mathrm {1PI})(8)}$ must also be opened up and so on."
],
[
"2PI formalism",
"As explained right below Eq.",
"(REF ), the $O(N)$ symmetry can not be spontaneously broken down in the 2PI-FRG framework.",
"For our toy model study, this means that no SSB can occur, whether we work in the unbroken- or broken-symmetry regime of the (0+0)-D $O(N)$ model.",
"To explain clearly how such a conservation of the $O(N)$ symmetry impacts the 2PI-FRG formalism, it will be convenient to disentangle the resulting simplifications from those related to symmetry properties inherent to the 2PI EA framework.",
"We have thus essentially two symmetry arguments that allow us to simplify the flow equations underlying all 2PI-FRG (C-flow, U-flow and CU-flow) approaches tested in this study: Symmetry argument inherent to the 2PI EA framework: The symmetry properties of the correlation functions $W^{(n)}$ given by Eqs.",
"(REF ) are also exhibited by the propagator $\\overline{G}_{\\mathfrak {s}}$ , the self-energy $\\overline{\\Sigma }_{\\mathfrak {s}}$ and all other 2PI vertices $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}$ (with $n \\ge 2$ ).",
"For the 2PI two-particle vertex $\\overline{\\Phi }^{(2)}_{\\mathfrak {s}}$ of the (0+0)-D $O(N)$ model at $N=2$ for instance, we have a priori $2^{4}=16$ components $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}$ to consider for the flow.",
"However, since $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}=\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a^{\\prime }_{1},a_{1})(a_{2},a^{\\prime }_{2})}=\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a^{\\prime }_{2},a_{2})}=\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a^{\\prime }_{1},a_{1})(a^{\\prime }_{2},a_{2})}$ according to Eq.",
"(REF ) and $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}=\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{2},a^{\\prime }_{2})(a_{1},a^{\\prime }_{1})}$ according to Eq.",
"(REF ), this set reduces to 6 flowing components, which are for instance: $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,1)(1,1)}$ , $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,1)(1,2)}$ , $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,1)(2,2)}$ , $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,2)(1,2)}$ , $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,2)(2,2)}$ and $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(2,2)(2,2)}$ .",
"Symmetry argument inherent to the $O(N)$ symmetry: Since the 2PI-FRG formalism was developed in a framework that can not exhibit any spontaneous breakdown of the $O(N)$ symmetry, all matrices reduce to scalars in color space throughout the entire flow.",
"For the (0+0)-D $O(N)$ model, this translates intoIn particular, the cutoff function $R_{\\mathfrak {s}}$ defining the propagator $C_{\\mathfrak {s}}$ (via e.g.",
"the relation $C^{-1}_{\\mathfrak {s}} = C^{-1} + R_{\\mathfrak {s}}$ ) within the C-flow or the CU-flow must therefore be chosen such that condition (REF ) is fulfilled.",
": $C^{(-1)}_{\\mathfrak {s},a a^{\\prime }} = C^{(-1)}_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }} \\quad \\forall \\mathfrak {s} \\;,$ $\\overline{G}_{\\mathfrak {s},a a^{\\prime }} = \\overline{G}_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }} \\quad \\forall \\mathfrak {s} \\;,$ $\\overline{\\Sigma }_{\\mathfrak {s},a a^{\\prime }} = \\overline{\\Sigma }_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }} \\quad \\forall \\mathfrak {s} \\;.$ After rewriting for example the C-flow equations (REF ) to (REF ) for the (0+0)-D $O(N)$ model, i.e.",
": $\\dot{\\overline{G}}_{\\mathfrak {s},a_{1} a^{\\prime }_{1}}=-\\sum _{a_{2},a^{\\prime }_{2}=1}^{N} \\overline{G}_{\\mathfrak {s}, a_{1} a_{2}}\\left(\\dot{C}_{\\mathfrak {s}}^{-1}-\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}\\right)_{a_{2} a^{\\prime }_{2}} \\overline{G}_{\\mathfrak {s}, a^{\\prime }_{2} a^{\\prime }_{1}} \\;,$ $\\Delta \\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{1}{2} \\sum _{a,a^{\\prime }=1}^{N} \\dot{C}_{\\mathfrak {s},a a^{\\prime }}^{-1} \\left(\\overline{G}_{\\mathfrak {s}}-C_{\\mathfrak {s}}\\right)_{a a^{\\prime }} \\;,$ $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s},a_{1} a^{\\prime }_{1}} = - \\frac{1}{2} \\sum _{a_{2},a^{\\prime }_{2} = 1}^{N} \\dot{\\overline{G}}_{\\mathfrak {s},a_{2} a^{\\prime }_{2}} \\overline{\\Phi }_{\\mathfrak {s},(a_{2}, a^{\\prime }_{2})(a_{1}, a^{\\prime }_{1})}^{(2)} \\;,$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s},(a_{1}, a^{\\prime }_{1})\\cdots (a_{n}, a^{\\prime }_{n})}^{(n)} = & \\ \\frac{1}{2} \\sum _{a_{n+1},a^{\\prime }_{n+1} = 1}^{N} \\dot{\\overline{G}}_{\\mathfrak {s},a_{n+1} a^{\\prime }_{n+1}} \\\\& \\times \\overline{\\Phi }_{\\mathfrak {s},(a_{n+1}, a^{\\prime }_{n+1})(a_{1}, a^{\\prime }_{1})\\cdots (a_{n}, a^{\\prime }_{n})}^{(n+1)} \\\\& \\quad \\forall n \\ge 2 \\;,\\end{split}$ using Eqs.",
"(REF ) to (REF ), we can see that the components of the 2PI vertices $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}$ which have at least one bosonic index with distinct color indices (i.e.",
"at least one index $\\gamma =(a,a^{\\prime })$ with $a \\ne a^{\\prime }$ ) are somehow cut out of the flow.",
"In particular, this translates into the fact that they do not affect $\\overline{\\Gamma }^{(\\mathrm {2PI})}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}$ and $\\overline{G}_{\\mathfrak {s}=\\mathfrak {s}_{\\mathrm {f}}}$ which are the quantities of interest for us in the present study since the gs energy and density are directly determined from them (as can be seen e.g.",
"from Eqs.",
"(REF ) and (REF )).",
"Getting back to our example on $\\overline{\\Phi }^{(2)}_{\\mathfrak {s}}$ , the number of corresponding components of interest for the flow is thus further reduced from 6 to 3: $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,1)(1,1)}$ , $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,1)(2,2)}$ and $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(2,2)(2,2)}$ .",
"Finally, since the color space is isotropic in the absence of SSBFor $\\overline{\\Phi }_{\\mathfrak {s}}^{(3)}$ , this implies $\\overline{\\Phi }^{(3)}_{\\mathfrak {s},(2,2)(2,2)(2,2)}=\\overline{\\Phi }^{(3)}_{\\mathfrak {s},(1,1)(1,1)(1,1)}$ $\\forall \\mathfrak {s}$ , but also $\\overline{\\Phi }^{(3)}_{\\mathfrak {s},(1,1)(2,2)(2,2)}=\\overline{\\Phi }^{(3)}_{\\mathfrak {s},(1,1)(1,1)(2,2)}$ $\\forall \\mathfrak {s}$ ., we have $\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(2,2)(2,2)}=\\overline{\\Phi }^{(2)}_{\\mathfrak {s},(1,1)(1,1)}$ $\\forall \\mathfrak {s}$ , thus ending up with only 2 relevant components of $\\overline{\\Phi }^{(2)}_{\\mathfrak {s}}$ for the flow.",
"Therefore, with all the symmetry considerations outlined here, the coupled equations (REF ) to (REF ) with $n=2$ reduce at $N=2$ to the simpler set: $\\dot{\\overline{G}}_{\\mathfrak {s}} =- \\overline{G}_{\\mathfrak {s}}^{2} \\left(\\dot{C}_{\\mathfrak {s}}^{-1}-\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}\\right) \\;,$ $\\Delta \\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\dot{C}_{\\mathfrak {s}}^{-1} \\left(\\overline{G}_{\\mathfrak {s}}-C_{\\mathfrak {s}}\\right) \\;,$ $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = - \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\left(\\overline{\\Phi }_{\\mathfrak {s},(1,1)(1,1)}^{(2)} + \\overline{\\Phi }_{\\mathfrak {s},(1,1)(2,2)}^{(2)}\\right) \\;,$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s},(1,1)(1,1)}^{(2)} = & \\ \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\Big (\\overline{\\Phi }_{\\mathfrak {s},(1,1)(1,1)(1,1)}^{(3)} \\\\& + \\overline{\\Phi }_{\\mathfrak {s},(1,1)(1,1)(2,2)}^{(3)}\\Big ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s},(1,1)(2,2)}^{(2)} = & \\ \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\Big (\\overline{\\Phi }_{\\mathfrak {s},(1,1)(1,1)(2,2)}^{(3)} \\\\& + \\overline{\\Phi }_{\\mathfrak {s},(1,1)(2,2)(2,2)}^{(3)}\\Big ) \\;,\\end{split}$ with $\\dot{C}^{-1}_{\\mathfrak {s},a a^{\\prime }} = \\dot{C}^{-1}_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }}$ $\\forall \\mathfrak {s}$ , $\\dot{\\overline{G}}_{\\mathfrak {s},a a^{\\prime }} = \\dot{\\overline{G}}_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }}$ $\\forall \\mathfrak {s}$ and $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s},a a^{\\prime }} = \\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }}$ $\\forall \\mathfrak {s}$ as a result of Eqs.",
"(REF ) to (REF )."
],
[
"Mixed 1PI functional renormalization group",
"We then outline the main steps of the vertex expansion procedure treating the Wetterich equation for (0+0)-D $O(N)$ model in the framework of the mixed representation.",
"The Wetterich equation to consider is already given by Eq.",
"(REF ) in the form: $\\dot{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big )+\\mathcal {R}_{k}\\right)^{-1}\\right] \\;,$ with $\\mathcal {R}_{k} = \\begin{pmatrix}R^{(\\phi )}_{k} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & R^{(\\eta )}_{k}\\end{pmatrix} = \\begin{pmatrix}R_{k} \\mathbb {I}_{N} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & R_{k}\\end{pmatrix} = R_{k} \\mathbb {I}_{N+1} \\;.$ As a first step, we expand the mixed 1PI EA around an extremum ($\\overline{\\Gamma }_{\\mathrm {mix},k,a}^{(\\mathrm {1PI})(1\\phi )}=\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(1\\eta )}=0$ $\\forall a,k$ ) via Eq.",
"(REF ) recalled below: $\\Gamma ^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = \\overline{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k} + \\sum ^{\\infty }_{n=2} \\frac{1}{n!}",
"\\sum _{m=0}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\sum _{a_{1},\\cdots ,a_{m}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m)\\eta )} \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{1}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{m}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,$ from which we deduce the following expression of the left-hand side of Eq.",
"(REF ): $\\begin{split}\\dot{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = & \\ \\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{\\mathrm {mix},k} - \\sum _{a_{1}=1}^{N} \\left( \\sum _{a_{2}=1}^{N} \\dot{\\overline{\\phi }}_{k,a_{2}} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{2} a_{1}}^{(\\mathrm {1PI})(2\\phi )} + \\dot{\\overline{\\eta }}_{k} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1}}^{(\\mathrm {1PI})(1\\phi ,1\\eta )} \\right) \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{1}} \\\\& - \\left( \\sum _{a_{1}=1}^{N} \\dot{\\overline{\\phi }}_{k,a_{1}} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1}}^{(\\mathrm {1PI})(1\\phi ,1\\eta )} + \\dot{\\overline{\\eta }}_{k} \\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\eta )} \\right) \\left(\\eta -\\overline{\\eta }_{k}\\right) \\\\& + \\sum ^{\\infty }_{n=2} \\frac{1}{n!}",
"\\sum _{m=0}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\sum _{a_{1},\\cdots ,a_{m}=1}^{N} \\Bigg (\\dot{\\overline{\\Gamma }}_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m)\\eta )} - \\sum _{a_{m+1}=1}^{N} \\dot{\\overline{\\phi }}_{k,a_{m+1}} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{m+1} a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})((m+1)\\phi ,(n-m)\\eta )} \\\\& - \\dot{\\overline{\\eta }}_{k} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m+1)\\eta )} \\Bigg ) \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{1}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{m}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;.\\end{split}$ Then, the right-hand side of Eq.",
"(REF ) is expanded by exploiting the matrices $\\mathcal {P}_{k}$ and $\\mathcal {F}_{k}$ already defined by Eqs.",
"(REF ) to (REF ).",
"Each component of $\\mathcal {F}_{k}$ can also be expanded in the same way as in Eq.",
"(REF ), thus leading toREF : $\\begin{split}& \\mathcal {F}_{k,a_{1}a_{2}} \\equiv \\Gamma ^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k,a_{1}a_{2}}-\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k,a_{1}a_{2}} \\\\& = \\sum ^{\\infty }_{n=1} \\frac{1}{n!}",
"\\sum _{m=0}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\sum _{a_{3},\\cdots ,a_{m+2}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m+2}}^{(\\mathrm {1PI})((m+2)\\phi ,(n-m)\\eta )} \\\\& \\hspace{8.5359pt} \\times \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{3}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{m+2}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ $\\begin{split}& \\mathcal {F}_{k,a_{1} \\hspace{1.13791pt} N+1} = \\mathcal {F}_{k,N+1 \\hspace{1.13791pt} a_{1}} \\equiv \\Gamma ^{(\\mathrm {1PI})(1\\phi ,1\\eta )}_{\\mathrm {mix},k,a_{1}}-\\overline{\\Gamma }^{(\\mathrm {1PI})(1\\phi ,1\\eta )}_{\\mathrm {mix},k,a_{1}} \\\\& = \\sum ^{\\infty }_{n=1} \\frac{1}{n!}",
"\\sum _{m=0}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\sum _{a_{2},\\cdots ,a_{m+1}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m+1}}^{(\\mathrm {1PI})((m+1)\\phi ,(n-m+1)\\eta )} \\\\& \\hspace{8.5359pt} \\times \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{2}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{m+1}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ $\\begin{split}& \\mathcal {F}_{k,N+1 \\hspace{1.13791pt} N+1} \\equiv \\Gamma ^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k}-\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k} \\\\& = \\sum ^{\\infty }_{n=1} \\frac{1}{n!}",
"\\sum _{m=0}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\sum _{a_{1},\\cdots ,a_{m}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m+2)\\eta )} \\\\& \\hspace{8.5359pt} \\times \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{1}} \\cdots \\Big (\\vec{\\phi }-\\vec{\\overline{\\phi }}_{k}\\Big )_{a_{m}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;.\\end{split}$ The expansion of the right-hand side of Eq.",
"(REF ) is then carried out by matrix multiplications between $\\mathcal {P}_{k}^{-1}$ and $\\mathcal {F}_{k}$ according to: $\\begin{split}& \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\left(\\Gamma ^{(\\mathrm {1PI})(2)}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big )+\\mathcal {R}_{k}\\right)^{-1}\\right] \\\\& = \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\left(\\mathcal {P}_{k}+\\mathcal {F}_{k}\\right)^{-1}\\right] \\\\& = \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\mathcal {P}_{k}^{-1}\\left(\\mathbb {I}_{N+1}+\\mathcal {P}_{k}^{-1}\\mathcal {F}_{k}\\right)^{-1}\\right] \\\\& = \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\mathcal {P}_{k}^{-1}\\left(\\mathbb {I}_{N+1}+\\sum _{n=1}^{\\infty }(-1)^{n}\\left(\\mathcal {P}_{k}^{-1}\\mathcal {F}_{k}\\right)^{n}\\right)\\right] \\\\& = \\frac{1}{2}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\mathcal {P}_{k}^{-1}\\right] \\\\& \\hspace{8.5359pt} + \\frac{1}{2}\\sum _{n=1}^{\\infty }(-1)^{n}\\mathcal {ST}r\\left[\\dot{\\mathcal {R}}_{k}\\mathcal {P}_{k}^{-1}\\left(\\mathcal {P}_{k}^{-1}\\mathcal {F}_{k}\\right)^{n}\\right] \\;.\\end{split}$ In the unbroken-symmetry phase, the $O(N)$ symmetry can be used to simplify the products between $\\mathcal {P}_{k}^{-1}$ and $\\mathcal {F}_{k}$ , especially since it enables us to replace the 1PI vertices in favor of their symmetric parts (according to Eq.",
"(REF ) in particular) and imposes that the 1PI vertices $\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k}$ with $n$ odd all vanish (according to Eq.",
"(REF )).",
"Those simplifications do not apply in the framework of the broken-symmetry regime and the corresponding equations are thus particularly cumbersome.",
"Hence, we will focus on the unbroken-symmetry phase from now on until the end of appendix REF , although the general recipe of the vertex expansion outlined here is essentially the same for all phases.",
"Using the aforementioned simplifications, definition (REF ) of $\\mathcal {P}_{k}$ reduces to: $\\mathcal {P}_{k} = \\begin{pmatrix}\\left(R_{k} + \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k}\\right) \\mathbb {I}_{N} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & R_{k} + \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k}\\end{pmatrix} \\;,$ and the inversion of $\\mathcal {P}_{k}$ thus becomes trivial such that $\\mathcal {P}_{k}^{-1}$ is now a diagonal matrix, i.e.",
": $\\begin{split}\\mathcal {P}_{k}^{-1} = & \\ \\begin{pmatrix}\\left(R_{k} + \\overline{\\Gamma }^{(2\\phi )}_{\\mathrm {mix},k}\\right)^{-1} \\mathbb {I}_{N} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & \\left(R_{k} + \\overline{\\Gamma }^{(2\\eta )}_{\\mathrm {mix},k}\\right)^{-1}\\end{pmatrix} \\\\\\equiv & \\ \\begin{pmatrix}\\overline{G}^{(\\phi )}_{k} \\mathbb {I}_{N} & \\vec{0} \\\\\\vec{0}^{\\mathrm {T}} & \\overline{G}^{(\\eta )}_{k}\\end{pmatrix} \\;.\\end{split}$ Moreover, as $\\vec{\\overline{\\phi }}_{k}=\\vec{0}$ $\\forall k$ in the unbroken-symmetry regime, the expressions of the components of $\\mathcal {F}_{k}$ given by Eqs.",
"(REF ) to (REF ) can be simplified asREF : $\\begin{split}\\mathcal {F}_{k,a_{1}a_{2}} = & \\ \\sum ^{\\infty }_{n=1} \\frac{1}{n!}",
"\\sum _{\\underset{\\lbrace {red}{\\text{m even}} \\rbrace }{m=0}}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\\\& \\times \\sum _{a_{3},\\cdots ,a_{m+2}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m+2}}^{(\\mathrm {1PI})((m+2)\\phi ,(n-m)\\eta )} \\\\& \\times \\phi _{a_{3}} \\cdots \\phi _{a_{m+2}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ $\\begin{split}\\mathcal {F}_{k,a_{1} \\hspace{1.13791pt} N+1} = & \\ \\mathcal {F}_{k,N+1 \\hspace{1.13791pt} a_{1}} \\\\= & \\ \\sum ^{\\infty }_{n=1} \\frac{1}{n!}",
"\\sum _{\\underset{\\lbrace {red}{\\text{m odd}} \\rbrace }{m=0}}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\\\& \\times \\sum _{a_{2},\\cdots ,a_{m+1}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m+1}}^{(\\mathrm {1PI})((m+1)\\phi ,(n-m+1)\\eta )} \\\\& \\times \\phi _{a_{2}} \\cdots \\phi _{a_{m+1}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ $\\begin{split}\\mathcal {F}_{k,N+1 \\hspace{1.13791pt} N+1} = & \\ \\sum ^{\\infty }_{n=1} \\frac{1}{n!}",
"\\sum _{\\underset{\\lbrace {red}{\\text{m even}} \\rbrace }{m=0}}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\\\& \\times \\sum _{a_{1},\\cdots ,a_{m}=1}^{N} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m+2)\\eta )} \\\\ & \\times \\phi _{a_{1}} \\cdots \\phi _{a_{m}} \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ on the one hand and, on the other hand, the left-hand side of Eq.",
"(REF ), given by Eq.",
"(REF ), reduces toREF : $\\begin{split}\\dot{\\Gamma }^{(\\mathrm {1PI})}_{\\mathrm {mix},k}\\Big (\\vec{\\phi },\\eta \\Big ) = & \\ \\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{\\mathrm {mix},k} - \\dot{\\overline{\\eta }}_{k} \\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\eta )} \\left(\\eta -\\overline{\\eta }_{k}\\right) \\\\& + \\sum ^{\\infty }_{n=2} \\frac{1}{n!}",
"\\sum _{\\underset{\\lbrace \\text{{red}{m even}} \\rbrace }{m=0}}^{n} \\begin{pmatrix}n \\\\m\\end{pmatrix} \\\\& \\times \\sum _{a_{1},\\cdots ,a_{m}=1}^{N} \\bigg (\\dot{\\overline{\\Gamma }}_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m)\\eta )} \\\\& - \\dot{\\overline{\\eta }}_{k} \\overline{\\Gamma }_{\\mathrm {mix},k,a_{1} \\cdots a_{m}}^{(\\mathrm {1PI})(m\\phi ,(n-m+1)\\eta )} \\bigg ) \\phi _{a_{1}} \\cdots \\phi _{a_{m}} \\\\& \\times \\left(\\eta -\\overline{\\eta }_{k}\\right)^{n-m} \\;,\\end{split}$ where the restrictions “$\\text{{red}{m even}}$ ” and “$\\text{{red}{m odd}}$ ” for the sums are a direct consequence of Eq.",
"(REF ).",
"Finally, we carry out the matrix products $(\\mathcal {P}_{k}^{-1}\\mathcal {F}_{k})^{n}$ involved in Eq.",
"(REF ) (where the expansion is performed up to a finite truncation order $n=N_{\\mathrm {max}}$ in practice) with $\\mathcal {P}_{k}^{-1}$ given by Eq.",
"(REF ) and $\\mathcal {F}_{k}$ specified by Eqs.",
"(REF ) to (REF ).",
"In this way, we have expanded the RHS of the Wetterich equation given by Eq.",
"(REF ).",
"By identifying the terms of the relation thus derived with those of Eq.",
"(REF ) involving the same powers of the fields $\\vec{\\phi }$ and $\\eta -\\overline{\\eta }_{k}$ , we obtain the following tower of differential equations for $N=2$ : $\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})}_{\\mathrm {mix},k} = \\ \\left(\\overline{G}^{(\\phi )}_{k}-\\overline{G}^{(\\phi )(0)}_{k}\\right) + \\frac{1}{2}\\left(\\overline{G}^{(\\eta )}_{k}-\\overline{G}^{(\\eta )(0)}_{k}\\right) \\;,$ $\\dot{\\overline{\\eta }}_{k} = \\frac{\\dot{R}_{k}}{2\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k}} \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 + 2 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\phi )}_{k}\\right)^2\\right) \\;,$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2\\phi )}_{\\mathrm {mix},k} = & \\ \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\phi ,1\\eta )} -\\frac{1}{6} \\dot{R}_{k} \\bigg (3 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 + 4 \\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(4\\phi )} \\left(\\overline{G}^{(\\phi )}_{k}\\right)^2 \\\\& - 6 \\left(\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\phi ,1\\eta )}\\right)^2 \\overline{G}^{(\\eta )}_{k} \\overline{G}^{(\\phi )}_{k} \\left(\\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right)\\bigg ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2\\eta )}_{\\mathrm {mix},k} = & \\ \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(3\\eta )} + \\dot{R}_{k} \\bigg (-\\frac{1}{2} \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 + \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k}\\right)^2 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^3 - \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\phi )}_{k}\\right)^2 \\\\& + 2 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\left(\\overline{G}^{(\\phi )}_{k}\\right)^3\\bigg ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k} = \\ & \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(2\\phi ,2\\eta )} + \\dot{R}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 \\bigg ( \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\eta )}_{k} - \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\phi )}_{k} \\left(2 \\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right)\\bigg ) \\\\& + \\frac{1}{3} \\dot{R}_{k} \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\phi )}_{k} \\bigg (6 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\eta )}_{k} \\left( \\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right) + \\overline{G}^{(\\phi )}_{k} \\Big (4 \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\phi )}_{\\mathrm {mix},k} \\overline{G}^{(\\phi )}_{k} \\\\& - 3 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\eta )}_{k} \\left(\\overline{G}^{(\\eta )}_{k} + 2 \\overline{G}^{(\\phi )}_{k}\\right)\\Big )\\bigg ) \\\\& + \\mathcal {F}_{1}^{(N=2)}\\Big (\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k}; \\ n+m \\le 5\\Big ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k} = & \\ \\dot{\\overline{\\eta }}_{k}\\overline{\\Gamma }_{\\mathrm {mix},k}^{(\\mathrm {1PI})(4\\eta )} \\\\& + \\dot{R}_{k} \\bigg (3 \\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k} \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^3 - 3 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k}\\right)^3 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^4 + 6 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\phi )}_{k}\\right)^3 \\Big (\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\\\& - \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\phi )}_{k}\\Big )\\bigg ) \\\\& + \\mathcal {F}_{2}^{(N=2)}\\Big (\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k}; \\ n+m \\le 5\\Big ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(4\\phi )}_{\\mathrm {mix},k} = & \\ \\frac{1}{3} \\dot{R}_{k} \\bigg (9 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k}\\right)^2 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^3 - 18 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 \\overline{G}^{(\\phi )}_{k} \\left(2 \\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right) \\\\& + 2 \\left(\\overline{G}^{(\\phi )}_{k}\\right)^2 \\Big (5 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4\\phi )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\phi )}_{k} + 18 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^4 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 \\left(\\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right) \\\\& - 9 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\phi )}_{\\mathrm {mix},k} \\overline{G}^{(\\eta )}_{k} \\left(\\overline{G}^{(\\eta )}_{k} + 2 \\overline{G}^{(\\phi )}_{k}\\right)\\Big )\\bigg ) \\\\& + \\mathcal {F}_{3}^{(N=2)}\\Big (\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k}; \\ n+m \\le 6\\Big ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(4\\eta )}_{\\mathrm {mix},k} = & \\ 3 \\dot{R}_{k} \\bigg (\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(4\\eta )}_{\\mathrm {mix},k}\\right)^2 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^3 - 6 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^4 + 4 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k}\\right)^4 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^5 \\\\& + 2 \\left(\\overline{G}^{(\\phi )}_{k}\\right)^3 \\Big (\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k}\\right)^2 - 6 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\phi )}_{k} + 4 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^4 \\left(\\overline{G}^{(\\phi )}_{k}\\right)^2\\Big )\\bigg ) \\\\& + \\mathcal {F}_{4}^{(N=2)}\\Big (\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k}; \\ n+m \\le 6\\Big ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} = & \\ \\frac{1}{3} \\dot{R}_{k} \\bigg (12 \\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k} \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 \\overline{G}^{(\\phi )}_{k} \\bigg (\\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\phi )}_{k} \\left(\\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right) \\\\& - \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\left(2 \\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right)\\bigg ) + 3 \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\eta )}_{\\mathrm {mix},k} \\left(\\overline{G}^{(\\eta )}_{k}\\right)^2 \\bigg (\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\eta )}_{k} \\\\& - \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\phi )}_{k} \\left(2 \\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right)\\bigg ) + 3 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(3\\eta )}_{\\mathrm {mix},k}\\right)^2 \\left(\\overline{G}^{(\\eta )}_{k}\\right)^3 \\bigg (-3 \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\eta )}_{k} \\\\& + 2 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\phi )}_{k} \\left(3 \\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right)\\bigg ) + \\overline{G}^{(\\phi )}_{k} \\Big (6 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\eta )}_{k} \\left(\\overline{G}^{(\\eta )}_{k} + \\overline{G}^{(\\phi )}_{k}\\right) \\\\& + \\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,2\\eta )}_{\\mathrm {mix},k} \\overline{G}^{(\\phi )}_{k} \\Big (4 \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\phi )}_{\\mathrm {mix},k} \\overline{G}^{(\\phi )}_{k} - 15 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\eta )}_{k} \\left( \\overline{G}^{(\\eta )}_{k} + 2 \\overline{G}^{(\\phi )}_{k}\\right)\\Big ) \\\\& + 6 \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\left(\\overline{G}^{(\\phi )}_{k}\\right)^2 \\Big (-2 \\overline{\\Gamma }^{(\\mathrm {1PI})(4\\phi )}_{\\mathrm {mix},k} \\overline{G}^{(\\phi )}_{k} + \\left(\\overline{\\Gamma }^{(\\mathrm {1PI})(2\\phi ,1\\eta )}_{\\mathrm {mix},k}\\right)^2 \\overline{G}^{(\\eta )}_{k} \\left( \\overline{G}^{(\\eta )}_{k} + 3 \\overline{G}^{(\\phi )}_{k}\\right)\\Big )\\Big )\\bigg ) \\\\& + \\mathcal {F}_{5}^{(N=2)}\\Big (\\overline{\\Gamma }^{(\\mathrm {1PI})(n\\phi ,m\\eta )}_{\\mathrm {mix},k}; \\ n+m \\le 6\\Big ) \\;,\\end{split}$ where the propagators $\\overline{G}^{(\\phi )(0)}_{k}$ and $\\overline{G}^{(\\eta )(0)}_{k}$ are defined byNote that $\\overline{G}^{(\\phi )(0)}_{k}$ and $\\overline{G}^{(\\eta )(0)}_{k}$ are introduced in the flow equations for the same reasons as $\\overline{G}^{(0)}_{k}$ in the framework of the original 1PI-FRG.",
"We thus refer to footnote REF for more details on this point.",
": $\\begin{split}\\left(\\overline{G}^{(\\phi )(0)}_{k}\\right)^{-1}_{a_{1}a_{2}} = & \\ \\left(\\overline{G}^{(\\phi )(0)}_{k}\\right)^{-1} \\delta _{a_{1}a_{2}} \\\\= & \\ \\left( \\overline{\\Gamma }^{(2\\phi )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} + R_{k} \\right) \\delta _{a_{1}a_{2}} \\quad \\forall a_{1}, a_{2} \\;,\\end{split}$ $\\left(\\overline{G}^{(\\eta )(0)}_{k}\\right)^{-1} = \\overline{\\Gamma }^{(2\\eta )}_{\\mathrm {mix},k=k_{\\mathrm {i}}} + R_{k} \\;.$ The functions $\\mathcal {F}_{n}^{(N)}$ , involved in Eqs.",
"(REF ) to (REF ), depend on 1PI vertices of order 5 (and 6 for $n \\ge 3$ ) so that they do not contribute to the above equations for $N_{\\mathrm {max}} \\le 4$ , according to the initial conditions set by Eq.",
"(REF ).",
"In this appendix, we present additional pU-flow equations for the (0+0)-D $O(N)$ model, showing notably what the resolution of the Bethe-Salpeter equation amounts to in this framework.",
"Owing to the full conservation of the $O(N)$ symmetry within the 2PI-FRG scheme, the propagator $\\overline{G}_{\\mathfrak {s}}$ and the self-energy $\\overline{\\Sigma }_{\\mathfrak {s}}$ satisfy respectively Eqs.",
"(REF ) and (REF ) in the present situation.",
"This implies that the corresponding pair propagator, defined previously by Eq.",
"(REF ), reduces to: $\\begin{split}\\overline{\\Pi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1}) (a_{2},a^{\\prime }_{2})} = & \\ \\overline{G}_{\\mathfrak {s},a_{1} a^{\\prime }_{2}} \\overline{G}_{\\mathfrak {s},a^{\\prime }_{1} a_{2}} + \\overline{G}_{\\mathfrak {s},a_{1} a_{2}} \\overline{G}_{\\mathfrak {s},a^{\\prime }_{1} a^{\\prime }_{2}} \\\\= & \\ \\overline{G}_{\\mathfrak {s}}^{2} \\left(\\delta _{a_{1} a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1} a_{2}} + \\delta _{a_{1} a_{2}} \\delta _{a^{\\prime }_{1} a^{\\prime }_{2}}\\right) \\;,\\end{split}$ and, according to Eqs.",
"(REF ) and (REF ) expressing respectively the bosonic identity matrix and the derivative $\\overline{W}_{\\mathfrak {s}}^{(2)}$ , this also leads to: $\\begin{split}& \\overline{W}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}^{(2)} = \\left(\\overline{\\Pi }_{\\mathfrak {s}}^{\\mathrm {inv}} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\\\& = \\frac{1}{2} \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\overline{\\Pi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\\\& = \\overline{G}_{\\mathfrak {s}}^{2} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\;,\\end{split}$ with $\\begin{split}& \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\\\& = \\mathcal {I}_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} + \\frac{1}{2} \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\overline{\\Pi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})} \\overline{\\Phi }_{\\mathfrak {s},(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{2})}^{(2)} \\\\& = \\delta _{a_{1}a_{2}} \\delta _{a^{\\prime }_{1}a^{\\prime }_{2}} + \\delta _{a_{1}a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1}a_{2}} + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\;.\\end{split}$ Furthermore, as mentioned in section REF , we have exploited the following analytical expression of $U_{\\mathfrak {s}}$ for all our U-flow calculations: $\\begin{split}U_{\\mathfrak {s},a_{1}a_{2}a_{3}a_{4}} = & \\ R_{\\mathfrak {s}} U_{a_{1}a_{2}a_{3}a_{4}} \\\\= & \\ \\mathfrak {s} U_{a_{1}a_{2}a_{3}a_{4}} \\\\= & \\ \\frac{\\mathfrak {s}\\lambda }{3}\\left(\\delta _{a_{1}a_{2}}\\delta _{a_{3}a_{4}}+\\delta _{a_{1}a_{3}}\\delta _{a_{2}a_{4}}+\\delta _{a_{1}a_{4}}\\delta _{a_{2}a_{3}}\\right) \\;,\\end{split}$ where the last line was obtained using expression (REF ) of the two-body interaction $U$ .",
"The Kronecker deltas introduced via Eqs.",
"(REF ) and (REF ) as well as Eqs.",
"(REF ) and (REF ) can be exploited to evaluate sums over color induces involved in the generic pU-flow equations (REF ), (REF ) and (REF ) to (REF ).",
"For the flow equations expressing $\\dot{\\overline{\\Gamma }}_{\\mathfrak {s}}^{(\\mathrm {2PI})}$ and $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}}$ (i.e.",
"Eqs.",
"(REF ) and (REF )) for instance, this gives us: $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{\\lambda }{72} \\overline{G}_{\\mathfrak {s}}^{2} \\left(\\sum _{a_{1},a_{2}=1}^{N}\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a_{1})(a_{2},a_{2})}^{\\mathrm {inv}} + 2\\sum _{a_{1},a^{\\prime }_{1}=1}^{N}\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a^{\\prime }_{1})(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} + N\\left(2+N\\right) \\right) \\;,$ $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = & - \\frac{\\lambda }{72} \\overline{G}_{\\mathfrak {s}} \\sum _{a_{1},a^{\\prime }_{1},a_{2}=1}^{N} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} \\\\& \\times \\Bigg ( \\sum _{a_{3},a_{4}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a_{2})(a_{3},a_{3})}^{\\mathrm {inv}} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{4},a_{4})(a_{2},a^{\\prime }_{1})}^{\\mathrm {inv}} \\\\& + 2 \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a_{2})(a_{3},a^{\\prime }_{3})}^{\\mathrm {inv}} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{1})}^{\\mathrm {inv}} \\Bigg ) \\\\& - \\frac{\\lambda }{36} \\overline{G}_{\\mathfrak {s}} \\left(N+2\\right) \\sum _{a_{1}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a_{1})}^{\\mathrm {inv}} \\\\& + \\frac{\\lambda }{576} \\overline{G}_{\\mathfrak {s}}^{4} \\sum _{a_{1},a^{\\prime }_{1},a_{2},a_{3},a_{4},a^{\\prime }_{4},a_{5},a^{\\prime }_{5}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{2},a_{2})(a_{4},a^{\\prime }_{4})}^{\\mathrm {inv}} \\\\& \\times \\overline{\\Phi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{4},a^{\\prime }_{4})(a_{5},a^{\\prime }_{5})}^{(3)} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{5},a^{\\prime }_{5})(a_{3},a_{3})}^{\\mathrm {inv}} \\\\& + \\frac{\\lambda }{288} \\overline{G}_{\\mathfrak {s}}^{4} \\sum _{a_{1},a^{\\prime }_{1},a_{2},a^{\\prime }_{2},a_{3},a^{\\prime }_{3},a_{4},a^{\\prime }_{4}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{2},a^{\\prime }_{2})(a_{3},a^{\\prime }_{3})}^{\\mathrm {inv}} \\\\& \\times \\overline{\\Phi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})(a_{4},a^{\\prime }_{4})}^{(3)} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{4},a^{\\prime }_{4})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\;,\\end{split}$ where the color indices set equal to 1 in the right-hand side of Eq.",
"(REF ) just result from our convention $\\overline{\\Sigma }_{\\mathfrak {s}} \\equiv \\overline{\\Sigma }_{\\mathfrak {s},11}$ (which is arbitrary in the sense that $\\overline{\\Sigma }_{\\mathfrak {s},11} = \\overline{\\Sigma }_{\\mathfrak {s},a a}$ $\\forall a$ according to Eq.",
"(REF )).",
"Rewriting the pU-flow equations explicitly for our toy model as in Eqs.",
"(REF ) and (REF ) can become quickly cumbersome, especially for the flow equations (REF ) and (REF ) expressing respectively $\\dot{\\overline{\\Phi }}^{(2)}_{\\mathfrak {s}}$ and $\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s}}$ .",
"One could also just replace $U_{\\mathfrak {s}}$ , $\\overline{\\Pi }_{\\mathfrak {s}}$ , $\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma }}$ and $\\frac{\\delta ^{2}\\overline{\\Pi }_{\\mathfrak {s}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}}$ by their expressions in terms of $\\overline{G}_{\\mathfrak {s}}$ and Kronecker deltas in the generic pU-flow equations given in section REF , and then solve numerically the (quite lengthy) differential equations thus obtained.",
"Another possibility is to exploit the (0+0)-D character of the studied model with the equality $\\Gamma ^{(\\mathrm {2PI})}(G)=\\Gamma ^{(\\mathrm {2PPI})}(\\rho )$ .",
"More specifically, this amounts to rederiving the flow equations underlying a 2PI-FRG approach in the 2PPI-FRG framework, yielding in this way less cumbersome flow equations (as compared to Eqs.",
"(REF ), (REF ) and their counterparts expressing $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}$ with $n\\ge 2$ ).",
"This is achieved by treating the exact flow equation for $\\Gamma ^{(\\mathrm {2PPI})}(\\rho )$ (i.e.",
"the (0+0)-D version of Eq.",
"(REF )) with a vertex expansion: for the 2PI-FRG pU-flow, such a procedure is equivalent to implementing the KS-FRG according to our findings of section See also appendix F.8.2 of Ref.",
"[46] for an application of this idea for the 2PI-FRG mU-flow, where we develop in this way a mU-flow version of the 2PPI-FRG only valid in the (0+0)-D framework.. We can also see e.g.",
"from Eqs.",
"(REF ) and (REF ) that, for the (0+0)-D $O(N)$ model, the resolution of the Bethe-Salpeter equation within the pU-flow simply amounts to inverting the bosonic matrix $\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ , i.e.",
"to solve: $\\begin{split}\\mathcal {I}_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} = & \\ \\frac{1}{2} \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})} \\\\& \\times \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\;,\\end{split}$ which is nothing other than a set of $N^4$ coupled algebraic equations.",
"We will discuss in more detail the resolution of such a system of equations at $N=2$ in the upcoming section on the mU-flow."
],
[
"Modified U-flow",
"As a next step, we address further the mU-flow equations.",
"As opposed to the pU-flow (with Eqs.",
"(REF ), (REF ) and (REF ) to (REF )), generic differential equations underpinning the mU-flow were not given in section .",
"We remedy this below for the mU-flow with Hartree-Fock starting point, i.e.",
"with $N_{\\mathrm {SCPT}}=1$ : $\\dot{\\overline{G}}_{\\mathfrak {s},\\alpha _{1}\\alpha ^{\\prime }_{1}}=\\int _{\\alpha _{2},\\alpha ^{\\prime }_{2}}\\overline{G}_{\\mathfrak {s},\\alpha _{1} \\alpha _{2}} \\dot{\\overline{\\Sigma }}_{\\mathfrak {s},\\alpha _{2} \\alpha ^{\\prime }_{2}} \\overline{G}_{\\mathfrak {s},\\alpha ^{\\prime }_{2} \\alpha ^{\\prime }_{1}} \\;,$ $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{1}{6} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1} \\hat{\\gamma }_{2}} \\left(\\overline{W}_{\\mathfrak {s}}^{(2)} - \\overline{\\Pi }_{\\mathfrak {s}}\\right)_{\\hat{\\gamma }_{2}\\hat{\\gamma }_{1}} \\;,$ $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s},\\gamma } = & - \\frac{1}{3} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}}_{\\gamma \\hat{\\gamma }_{1}} \\\\& \\times \\Bigg ( \\left[ 2 \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - U + U_{\\mathfrak {s}} \\right)\\right)^{\\mathrm {inv}} \\dot{U}_{\\mathfrak {s}} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\left(\\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - U + U_{\\mathfrak {s}} \\right)\\right)^{\\mathrm {inv}} + \\dot{U}_{\\mathfrak {s}} \\right]_{\\hat{\\alpha }_{1} \\hat{\\alpha }_{2} \\hat{\\alpha }^{\\prime }_{2} \\hat{\\alpha }^{\\prime }_{1}} \\overline{G}_{\\mathfrak {s},\\hat{\\gamma }_{2}} \\\\& - \\frac{1}{2} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{(2)} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }_{1} \\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}^{(3)} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{2}}^{(2)} - 3 \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} \\overline{G}_{\\mathfrak {s},\\hat{\\gamma }_{2}} \\Bigg ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}}^{(2)} = & \\ \\frac{1}{3} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} \\Bigg [\\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}^{(3)}\\right)\\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{6} \\hat{\\gamma }_{7}}^{(2)} \\\\& \\times \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7} \\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8} \\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9} \\hat{\\gamma }_{10}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{2} \\hat{\\gamma }_{7} \\hat{\\gamma }_{10}}^{(3)}\\right) \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{10} \\hat{\\gamma }_{1}}^{(2)} \\\\& - \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{6}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{6} \\hat{\\gamma }_{7}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7} \\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{8} \\hat{\\gamma }_{1}}^{(2)} \\\\& + \\frac{1}{2} \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{2} \\hat{\\gamma }_{3}}^{(2)} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3} \\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta ^{2} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4} \\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}} \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5} \\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}^{(4)}\\right) \\overline{W}_{\\mathfrak {s},\\hat{\\gamma }_{6} \\hat{\\gamma }_{1}}^{(2)} + \\frac{1}{4} \\frac{\\delta ^{2}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{1}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}} \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}}\\Bigg ] - \\dot{U}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}} \\\\& + \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }_{\\mathfrak {s},\\hat{\\gamma }\\gamma _{1}\\gamma _{2}}^{(3)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}\\gamma _{3}} = & \\ \\frac{1}{3} \\dot{U}_{\\mathfrak {s},\\hat{\\gamma }_{1}\\hat{\\gamma }_{2}} \\Bigg [ 3 \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}}\\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}\\right)\\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}} \\bigg (\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} \\\\& - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{2}\\hat{\\gamma }_{7}\\hat{\\gamma }_{10}}\\bigg ) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{11}} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{11}\\hat{\\gamma }_{12}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{12}\\hat{\\gamma }_{13}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{13}\\hat{\\gamma }_{14}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{3}\\hat{\\gamma }_{11}\\hat{\\gamma }_{14}}\\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{14}\\hat{\\gamma }_{1}} \\\\& + \\Bigg ( \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}\\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}} \\\\& \\times \\left( \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}}\\frac{\\delta ^{2} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} - \\overline{\\Phi }_{\\mathfrak {s},\\gamma _{2}\\gamma _{3}\\hat{\\gamma }_{7}\\hat{\\gamma }_{10}}^{(4)} \\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{1}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\Bigg ) \\\\& -2 \\Bigg ( \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\left(\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}}\\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} - \\overline{\\Phi }^{(3)}_{\\mathfrak {s},\\gamma _{1}\\hat{\\gamma }_{3}\\hat{\\gamma }_{6}}\\right) \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}} \\\\& \\times \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{11}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{11}\\hat{\\gamma }_{12}}^{\\mathrm {inv}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{12}\\hat{\\gamma }_{1}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\Bigg ) \\\\& + 3 \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{9}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{9}\\hat{\\gamma }_{10}}^{\\mathrm {inv}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{10}\\hat{\\gamma }_{1}} \\\\& - \\Bigg ( \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}}^{\\mathrm {inv}} \\frac{\\delta \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{5}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{5}\\hat{\\gamma }_{6}}^{\\mathrm {inv}} \\frac{\\delta ^{2}\\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{6}\\hat{\\gamma }_{7}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{2}} \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{3}}} \\overline{\\Pi }_{\\mathfrak {s},\\hat{\\gamma }_{7}\\hat{\\gamma }_{8}}^{\\mathrm {inv}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{8}\\hat{\\gamma }_{1}} + (\\gamma _{2},\\gamma _{1},\\gamma _{3}) + (\\gamma _{3},\\gamma _{1},\\gamma _{2}) \\Bigg ) \\\\& -\\frac{1}{2} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{2}\\hat{\\gamma }_{3}} \\overline{\\Phi }^{(5)}_{\\mathfrak {s},\\gamma _{1}\\gamma _{2}\\gamma _{3}\\hat{\\gamma }_{3}\\hat{\\gamma }_{4}} \\overline{W}^{(2)}_{\\mathfrak {s},\\hat{\\gamma }_{4}\\hat{\\gamma }_{1}} \\Bigg ] + \\dot{\\overline{G}}_{\\mathfrak {s},\\hat{\\gamma }} \\overline{\\Phi }^{(4)}_{\\mathfrak {s},\\hat{\\gamma }\\gamma _{1}\\gamma _{2}\\gamma _{3}}\\;,\\end{split}$ where we have used again notation (REF ) and exploited a slight modification of definition (REF ), i.e.",
": $\\frac{\\delta ^{n}\\overline{\\Pi }_{\\mathfrak {s}}}{\\delta \\overline{G}_{\\mathfrak {s},\\gamma _{1}} \\cdots \\delta \\overline{G}_{\\mathfrak {s},\\gamma _{n}}} \\equiv \\left.\\frac{\\delta ^{n}\\Pi [G]}{\\delta G_{\\gamma _{1}} \\cdots \\delta G_{\\gamma _{n}}}\\right|_{G=\\overline{G}_{\\mathfrak {s}}} \\;.$ In short, Eq.",
"(REF ) is obtained by differentiating the Dyson equation in the form of Eq.",
"(REF ) whereas the Eqs.",
"(REF ) to (REF ) are derived in the same way as their pU-flow counterparts (i.e.",
"Eqs.",
"(REF ) and (REF ) to (REF )) but using in addition the following substitutions in order to replace the original flowing quantities of the pU-flow by their bold counterpartsWe refer to appendix F.4.2 of Ref.",
"[46] for more details on these derivations.",
": [left=]align (2PI)s[G] (2PI)s[G] + 12Us,12G1G2 , s,[G] s,[G] - Us,G , (2)s,12[G] (2)s,12[G] + Us,12 , (n)s,1n[G] (n)s,1n[G] n 3 , where $\\Delta U_{\\mathfrak {s}} \\equiv U_{\\mathfrak {s}} - U$ .",
"Note that these substitutions can all be inferred from definition (REF ) at $N_{\\mathrm {SCPT}}=1$ .",
"Moreover, we stress that, according to transformation (REF ) (combined with Eq.",
"(REF )), the Bethe-Salpeter equation takes the form: $\\overline{W}^{(2)}_{\\mathfrak {s}} = \\left(\\overline{\\Pi }_{\\mathfrak {s}}^{\\mathrm {inv}} + \\overline{\\Phi }^{(2)}_{\\mathfrak {s}} + \\Delta U_{\\mathfrak {s}} \\right)^{\\mathrm {inv}} \\;.$ in the framework of the mU-flow at $N_{\\mathrm {SCPT}}=1$ .",
"We then continue with a discussion similar to that of section REF .",
"To that end, we still focus on the mU-flow with $N_{\\mathrm {SCPT}}=1$ and give the corresponding flow equations in the framework of the (0+0)-D $O(N)$ model for all $N$ .",
"The underlying idea here is once again to illustrate how the conservation of the $O(N)$ symmetry is used to simplify generic flow equations (given e.g.",
"by Eqs.",
"(REF ) to (REF ) in the studied case) and explain how the Bethe-Salpeter equation is solved throughout the flow.",
"For the toy model under consideration, the bold quantities underlying the mU-flow at $N_{\\mathrm {SCPT}}=1$ rely on the following modified LW functional: $\\begin{split}\\Phi _{\\mathfrak {s}}(G) \\equiv & \\ \\Phi _{\\mathfrak {s}}(G) + \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}(U,G) \\\\& - \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}(U_{\\mathfrak {s}},G) \\\\= & \\ \\Phi _{\\mathfrak {s}}(G) + \\lambda \\left(1-\\mathfrak {s}\\right) \\Bigg ( \\frac{1}{24}\\left(\\sum _{a_{1}=1}^{N} G_{a_{1}a_{1}} \\right)^{2} \\\\& + \\frac{1}{12} \\sum _{a_{1},a_{2}=1}^{N} G_{a_{1}a_{2}}^{2} \\Bigg ) \\;,\\end{split}$ with $\\begin{split}\\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}(U,G) = & \\ \\frac{1}{24} \\hspace{2.27626pt} {\\begin{array}{c}\\begin{fmffile}{Diagrams/2PIEA_Hartree}\\begin{fmfgraph}(30,20){i}{o}{phantom,tension=10}{i,i1}{phantom,tension=10}{o,o1}{plain,left,tension=0.5,foreground=(1,,0,,0)}{i1,v1,i1}{plain,right,tension=0.5,foreground=(1,,0,,0)}{o1,v2,o1}{zigzag,foreground=(0,,0,,1)}{v1,v2}\\end{fmfgraph}\\end{fmffile}\\end{array}} \\\\& +\\frac{1}{12}{\\begin{array}{c}\\begin{fmffile}{Diagrams/2PIEA_Fock}\\begin{fmfgraph}(15,15){i}{o}{phantom,tension=11}{i,v1}{phantom,tension=11}{v2,o}{plain,left,tension=0.4,foreground=(1,,0,,0)}{v1,v2,v1}{zigzag,foreground=(0,,0,,1)}{v1,v2}\\end{fmfgraph}\\end{fmffile}\\end{array}} \\\\= & \\ \\frac{\\lambda }{24}\\left(\\sum _{a_{1}=1}^{N} G_{a_{1}a_{1}} \\right)^{2} \\\\& + \\frac{\\lambda }{12} \\sum _{a_{1},a_{2}=1}^{N} G_{a_{1}a_{2}}^{2} \\;.\\end{split}$ Moreover, the conservation of the $O(N)$ symmetry implies: $\\overline{G}_{\\mathfrak {s},a a^{\\prime }} = \\overline{G}_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }} \\quad \\forall \\mathfrak {s} \\;,$ $\\overline{\\Sigma }_{\\mathfrak {s},a a^{\\prime }} = \\overline{\\Sigma }_{\\mathfrak {s}} \\ \\delta _{a a^{\\prime }} \\quad \\forall \\mathfrak {s} \\;,$ $\\overline{\\Pi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1}) (a_{2},a^{\\prime }_{2})} = \\overline{G}_{\\mathfrak {s}}^{2} \\left(\\delta _{a_{1} a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1} a_{2}} + \\delta _{a_{1} a_{2}} \\delta _{a^{\\prime }_{1} a^{\\prime }_{2}}\\right) \\;,$ and the expression of $\\overline{W}_{\\mathfrak {s}}^{(2)}$ to consider now can be directly inferred from Eq.",
"(REF ) (alongside with Eqs.",
"(REF ), (REF ) and (REF )): $\\begin{split}& \\overline{W}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}^{(2)} \\\\& = \\Big (\\overline{\\Pi }_{\\mathfrak {s}}^{\\mathrm {inv}} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} + \\Phi ^{(2)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}\\big (U_{\\mathfrak {s}},G=\\overline{G}_{\\mathfrak {s}}\\big ) \\\\& \\hspace{8.5359pt} - \\Phi ^{(2)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}\\big (U,G=\\overline{G}_{\\mathfrak {s}}\\big ) \\Big )_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\\\& = \\frac{1}{2} \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\overline{\\Pi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\\\& = \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\;,\\end{split}$ with $\\begin{split}& \\overline{\\Upsilon }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\\\& \\equiv \\Big (\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}} \\Big ( \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} + \\Phi ^{(2)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}\\big (U_{\\mathfrak {s}},G=\\overline{G}_{\\mathfrak {s}}\\big ) \\\\& \\hspace{8.5359pt} - \\Phi ^{(2)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}\\big (U,G=\\overline{G}_{\\mathfrak {s}}\\big ) \\Big )\\Big )_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\\\& = \\delta _{a_{1}a_{2}} \\delta _{a^{\\prime }_{1}a^{\\prime }_{2}} + \\delta _{a_{1}a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1}a_{2}} + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\\\& \\hspace{8.5359pt} - \\frac{\\lambda }{3}(1-\\mathfrak {s}) \\overline{G}^{2}_{\\mathfrak {s}} \\left( \\delta _{a_{1} a^{\\prime }_{1}} \\delta _{a_{2} a^{\\prime }_{2}} + \\delta _{a_{1} a_{2}} \\delta _{a^{\\prime }_{1} a^{\\prime }_{2}} + \\delta _{a_{1} a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1} a_{2}} \\right) \\;,\\end{split}$ where the last equality simply follows from definition (REF ) of the bosonic identity matrix and: $\\begin{split}\\left(\\Phi ^{(2)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}\\big (U_{\\mathfrak {s}},G=\\overline{G}_{\\mathfrak {s}}\\big ) - \\Phi ^{(2)}_{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=1}\\big (U,G=\\overline{G}_{\\mathfrak {s}}\\big )\\right)_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} = & -\\frac{\\lambda }{3}(1-\\mathfrak {s}) \\left( \\delta _{a_{1} a^{\\prime }_{1}} \\delta _{a_{2} a^{\\prime }_{2}} + \\delta _{a_{1} a_{2}} \\delta _{a^{\\prime }_{1} a^{\\prime }_{2}} \\right.",
"\\\\& \\left.",
"+ \\delta _{a_{1} a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1} a_{2}} \\right) \\;,\\end{split}$ as a result of Eq.",
"(REF ).",
"We also stress once again that the upper bars label functionals evaluated at $G=\\overline{G}_{\\mathfrak {s}}$ instead of $G=\\overline{G}_{\\mathfrak {s}}$ in the framework of the mU-flow.",
"With relations (REF ) to (REF ), we can rewrite for example the mU-flow equations (REF ) and (REF ) for the (0+0)-D $O(N)$ model and obtain: $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{\\lambda }{72} \\overline{G}_{\\mathfrak {s}}^{2} \\left( \\sum _{a_{1},a_{2}=1}^{N} \\overline{\\Upsilon }^{\\mathrm {inv}}_{\\mathfrak {s},(a_{1},a_{1})(a_{2},a_{2})} + 2 \\sum _{a_{1},a^{\\prime }_{1}=1}^{N} \\overline{\\Upsilon }^{\\mathrm {inv}}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{1},a^{\\prime }_{1})} - 2 N \\left( N + 2 \\right) \\right) \\;,$ $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = & -\\frac{\\lambda }{72} \\overline{G}_{\\mathfrak {s}} \\sum _{a_{1},a^{\\prime }_{1},a_{2}=1}^{N} \\left(\\mathcal {I} + \\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} \\\\& \\times \\Bigg ( \\sum _{a_{3},a_{4}=1}^{N} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{1},a_{2})(a_{3},a_{3})}^{\\mathrm {inv}} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{4},a_{4})(a_{2},a^{\\prime }_{1})}^{\\mathrm {inv}} + 2 \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{1},a_{2})(a_{3},a^{\\prime }_{3})}^{\\mathrm {inv}} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{1})}^{\\mathrm {inv}} \\Bigg ) \\\\& + \\frac{\\lambda }{18} \\overline{G}_{\\mathfrak {s}} \\left(N+2\\right) \\sum _{a_{1}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a_{1})}^{\\mathrm {inv}} \\\\& + \\frac{\\lambda }{576} \\overline{G}_{\\mathfrak {s}}^{4} \\sum _{a_{1},a^{\\prime }_{1},a_{2},a_{3},a_{4},a^{\\prime }_{4},a_{5},a^{\\prime }_{5}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{2},a_{2})(a_{4},a^{\\prime }_{4})}^{\\mathrm {inv}} \\overline{\\Phi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{4},a^{\\prime }_{4})(a_{5},a^{\\prime }_{5})}^{(3)} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{5},a^{\\prime }_{5})(a_{3},a_{3})}^{\\mathrm {inv}} \\\\& + \\frac{\\lambda }{288} \\overline{G}_{\\mathfrak {s}}^{4} \\sum _{a_{1},a^{\\prime }_{1},a_{2},a^{\\prime }_{2},a_{3},a^{\\prime }_{3},a_{4},a^{\\prime }_{4}=1}^{N} \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(a_{1},a^{\\prime }_{1})}^{\\mathrm {inv}} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{2},a^{\\prime }_{2})(a_{3},a^{\\prime }_{3})}^{\\mathrm {inv}} \\overline{\\Phi }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})(a_{4},a^{\\prime }_{4})}^{(3)} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{4},a^{\\prime }_{4})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\;,\\end{split}$ where the color indices equal to 1 in the right-hand side of Eq.",
"(REF ) follow from the convention $\\overline{\\Sigma }_{\\mathfrak {s}} \\equiv \\overline{\\Sigma }_{\\mathfrak {s},11} = \\overline{\\Sigma }_{\\mathfrak {s},a a}$ $\\forall a$ , similarly to Eq.",
"(REF ).",
"We stress also that our previous comments, put forward right below Eq.",
"(REF ) and concerning the derivation of the pU-flow equations for our (0+0)-D toy model, also apply in the mU-flow framework.",
"In particular, one can exploit the 2PPI-FRG framework in (0+0)-D so as to implement the 2PI-FRG mU-flow by solving flow equations simpler than Eqs.",
"(REF ), (REF ) and their counterparts expressing $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}$ with $n\\ge 2$REF .",
"Let us then focus more deeply on the quantities involved in Eqs.",
"(REF ) and (REF ), which will enable us to further clarify the Bethe-Salpeter equation is solved within our (0+0)-D applications of the mU-flow at $N_{\\mathrm {SCPT}}=1$ .",
"The bosonic matrix $\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ satisfies the equality: $\\begin{split}\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} = & \\ \\delta _{a_{1}a_{2}} \\delta _{a^{\\prime }_{1}a^{\\prime }_{2}} + \\delta _{a_{1}a^{\\prime }_{2}} \\delta _{a^{\\prime }_{1}a_{2}} \\\\& + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }^{(2)}_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} \\;,\\end{split}$ which can be obtained by following derivation (REF ) with $\\overline{G}_{\\mathfrak {s}}$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ respectively replaced by $\\overline{G}_{\\mathfrak {s}}$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ .",
"As can be seen from Eqs.",
"(REF ) and (REF ), solving the Bethe-Salpeter equation now translates into inverting the bosonic matrices $\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ and $\\overline{\\Upsilon }_{\\mathfrak {s}}$ .",
"Actually, by comparing Eqs.",
"(REF ) and (REF ), we can notice that $\\overline{\\Upsilon }_{\\mathfrak {s}}$ reduces to $\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}$ by setting $\\mathfrak {s}=1$ in the very last line of expression (REF ).",
"Hence, the components of $\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}}$ can be directly deduced from those of $\\overline{\\Upsilon }_{\\mathfrak {s}}^{\\mathrm {inv}}$ as well and we can content ourselves to invert $\\overline{\\Upsilon }_{\\mathfrak {s}}$ , i.e.",
"to solve: $\\mathcal {I}_{(a_{1},a^{\\prime }_{1})(a_{2},a^{\\prime }_{2})} = \\frac{1}{2} \\sum _{a_{3},a^{\\prime }_{3}=1}^{N} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{1},a^{\\prime }_{1})(a_{3},a^{\\prime }_{3})} \\overline{\\Upsilon }_{\\mathfrak {s},(a_{3},a^{\\prime }_{3})(a_{2},a^{\\prime }_{2})}^{\\mathrm {inv}} \\;.$ At $N=1$ , the resolution of Eq.",
"(REF ) simply yields: $\\overline{\\Upsilon }_{\\mathfrak {s}}^{\\mathrm {inv}} = \\frac{4}{\\overline{\\Upsilon }_{\\mathfrak {s}}} \\;,$ with $\\overline{\\Upsilon }_{\\mathfrak {s}} \\equiv \\overline{\\Upsilon }_{\\mathfrak {s},(1,1)(1,1)}$ whereas, at $N=2$ , $\\overline{\\Upsilon }_{\\mathfrak {s}}^{\\mathrm {inv}}$ is determined by solving $2^{4}=16$ coupled algebraic equations deduced from Eq.",
"(REF ) combined with Eq.",
"(REF ).",
"Among the 16 components of $\\overline{\\Upsilon }_{\\mathfrak {s}}^{\\mathrm {inv}}$ at $N=2$ , only 8 differ from zero and are given by: $\\begin{split}\\overline{\\Upsilon }_{\\mathfrak {s},(1,1)(1,1)}^{\\mathrm {inv}} = & \\ \\overline{\\Upsilon }_{\\mathfrak {s},(2,2)(2,2)}^{\\mathrm {inv}} \\\\= & \\ \\frac{9\\left(2+\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{2}\\right)}{9+9\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{2}+2\\mathfrak {s}^{2}\\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{4}} \\;,\\end{split}$ $\\begin{split}\\overline{\\Upsilon }_{\\mathfrak {s},(1,1)(2,2)}^{\\mathrm {inv}} = & \\ \\overline{\\Upsilon }_{\\mathfrak {s},(2,2)(1,1)}^{\\mathrm {inv}} \\\\= & -\\frac{3\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{2}}{9+9\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{2}+2\\mathfrak {s}^{2}\\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{4}} \\;,\\end{split}$ $\\begin{split}\\overline{\\Upsilon }_{\\mathfrak {s},(1,2)(1,2)}^{\\mathrm {inv}} = & \\ \\overline{\\Upsilon }_{\\mathfrak {s},(1,2)(2,1)}^{\\mathrm {inv}} = \\overline{\\Upsilon }_{\\mathfrak {s},(2,1)(1,2)}^{\\mathrm {inv}} = \\overline{\\Upsilon }_{\\mathfrak {s},(2,1)(2,1)}^{\\mathrm {inv}} \\\\= & \\ \\frac{3}{3+\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{2}} \\;.\\end{split}$ Following our previous explanation between Eq.",
"(REF ) and (REF ), we directly infer from this the 8 non-vanishing components of $\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}}$ (and therefore of $\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)^{\\mathrm {inv}}$ involved in the pU-flow equations such as Eq.",
"(REF )): $\\begin{split}\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(1,1)}^{\\mathrm {inv}} = & \\ \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(2,2)(2,2)}^{\\mathrm {inv}} \\\\= & \\ \\frac{9\\left(2+\\lambda \\overline{G}_{\\mathfrak {s}}^{2}\\right)}{9+9\\lambda \\overline{G}_{\\mathfrak {s}}^{2}+2\\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{4}} \\;,\\end{split}$ $\\begin{split}\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,1)(2,2)}^{\\mathrm {inv}} = & \\ \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(2,2)(1,1)}^{\\mathrm {inv}} \\\\= & -\\frac{3\\lambda \\overline{G}_{\\mathfrak {s}}^{2}}{9+9\\lambda \\overline{G}_{\\mathfrak {s}}^{2}+2\\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{4}} \\;,\\end{split}$ $\\begin{split}\\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,2)(1,2)}^{\\mathrm {inv}} = & \\ \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(1,2)(2,1)}^{\\mathrm {inv}} \\\\= & \\ \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(2,1)(1,2)}^{\\mathrm {inv}} \\\\= & \\ \\left(\\mathcal {I}+\\overline{\\Pi }_{\\mathfrak {s}}\\overline{\\Phi }_{\\mathfrak {s}}^{(2)}\\right)_{(2,1)(2,1)}^{\\mathrm {inv}} \\\\= & \\ \\frac{3}{3+\\lambda \\overline{G}_{\\mathfrak {s}}^{2}} \\;.\\end{split}$ We have also treated the mU-flow with $N_{\\mathrm {SCPT}}$ up to 3 at $N=1$ .",
"To that end, we exploited the following expression of the modified LW functional: $\\begin{split}\\Phi _{\\mathfrak {s}}(G) = & \\ \\Phi _{\\mathfrak {s}}(G) + \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=3}(U,G) \\\\& - \\Phi _{\\mathrm {SCPT},N_{\\mathrm {SCPT}}=3}(U_{\\mathfrak {s}},G) \\\\= & \\ \\Phi _{\\mathfrak {s}}(G) + \\frac{1}{8} \\lambda G^{2} \\left(1-\\mathfrak {s}\\right) - \\frac{1}{48} \\lambda ^{2} G^{4} \\left(1-\\mathfrak {s}^{2}\\right) \\\\& + \\frac{1}{48} \\lambda ^{3} G^{6} \\left(1-\\mathfrak {s}^{3}\\right) \\;,\\end{split}$ which can be deduced from Eq.",
"(REF ) and from our choice (REF ) for the cutoff function, i.e.",
"$U_{\\mathfrak {s}}=\\mathfrak {s}\\lambda $ at $N=1$ .",
"With Eq.",
"(REF ), we derive the differential equations underlying the mU-flow up to $N_{\\mathrm {SCPT}}=3$ for the (0+0)-D $O(N)$ model at $N=1$ , namely: At $N_{\\mathrm {SCPT}}=1$ : $\\dot{\\overline{G}}_{\\mathfrak {s}} = \\overline{G}_{\\mathfrak {s}}^{2} \\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} \\;,$ $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{\\lambda }{24} \\left( 4\\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-1} + \\overline{G}_{\\mathfrak {s}}^{2} \\right) - \\frac{1}{8} \\lambda \\overline{G}_{\\mathfrak {s}}^{2} \\;,$ $\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = -\\frac{\\lambda }{3} \\left( 2 + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-1} \\bigg ( \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-2} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right) - 2\\overline{G}_{\\mathfrak {s}} \\bigg ) \\;,$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(2)}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{6} \\bigg ( 2 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-3} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right)^{2} -64 \\overline{G}_{\\mathfrak {s}}^{-4} \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-2} \\\\& + \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-2} \\left( 16\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\right) + 2 \\bigg ) - \\lambda + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{6} \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-2} \\bigg ( 6 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-2} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right)^{3} \\\\& + 6 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) \\right)^{-1} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\right) \\left( -48\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\right) + 384 \\overline{G}_{\\mathfrak {s}}^{-5} - \\overline{\\Phi }_{\\mathfrak {s}}^{(5)} \\bigg ) + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\;,\\end{split}$ At $N_{\\mathrm {SCPT}}=2$ : $\\dot{\\overline{G}}_{\\mathfrak {s}} = \\overline{G}_{\\mathfrak {s}}^{2} \\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} \\;,$ $\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = \\frac{\\lambda }{24} \\left( 4\\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-1} + \\overline{G}_{\\mathfrak {s}}^{2} \\right) - \\frac{1}{8} \\lambda \\overline{G}_{\\mathfrak {s}}^{2} + \\frac{1}{24} \\mathfrak {s} \\lambda ^{2} \\overline{G}_{\\mathfrak {s}}^{4} \\;,$ $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = & -\\frac{\\lambda }{3} \\left( 2 + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-1} \\bigg ( \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-2} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) \\right) \\\\& - 2\\overline{G}_{\\mathfrak {s}} + 2\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{3} \\bigg ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(2)}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{6} \\bigg ( 2 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-3} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) \\right)^{2} \\\\& -64 \\overline{G}_{\\mathfrak {s}}^{-4} \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-2} \\\\& + \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-2} \\left( 16\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} - 8\\lambda ^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right) + 2 \\bigg ) \\\\& - \\lambda + 2 \\mathfrak {s} \\lambda ^{2} \\overline{G}_{\\mathfrak {s}}^{2} + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{6} \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-2} \\\\& \\times \\bigg ( 6 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-2} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) \\right)^{3} \\\\& + 6 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right)^{-1} \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) \\right) \\\\& \\times \\left( -48\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} - 8\\lambda ^{2}\\left(1-\\mathfrak {s}^{2}\\right) \\right) + 384 \\overline{G}_{\\mathfrak {s}}^{-5} - \\overline{\\Phi }_{\\mathfrak {s}}^{(5)} \\bigg ) + 8 \\mathfrak {s} \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\;,\\end{split}$ At $N_{\\mathrm {SCPT}}=3$ : $\\dot{\\overline{G}}_{\\mathfrak {s}} = \\overline{G}_{\\mathfrak {s}}^{2} \\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} \\;,$ $\\begin{split}\\dot{\\overline{\\Gamma }}^{(\\mathrm {2PI})}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{24} \\left( 4\\left( 2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-1} + \\overline{G}_{\\mathfrak {s}}^{2} \\right) - \\frac{1}{8} \\lambda \\overline{G}_{\\mathfrak {s}}^{2} + \\frac{1}{24} \\mathfrak {s} \\lambda ^{2} \\overline{G}_{\\mathfrak {s}}^{4} \\\\& - \\frac{1}{16} \\mathfrak {s}^{2} \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{6} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Sigma }}_{\\mathfrak {s}} = & -\\frac{\\lambda }{3} \\left( 2 + \\overline{G}_{\\mathfrak {s}}^{2} \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} \\right)^{-1} \\bigg ( \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-2} \\\\& \\times \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) + 20 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{3} \\left(1-\\mathfrak {s}^{3}\\right) \\right) - 2\\overline{G}_{\\mathfrak {s}} + 2\\mathfrak {s}\\lambda \\overline{G}_{\\mathfrak {s}}^{3} - \\frac{9}{2}\\mathfrak {s}^{2}\\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{5} \\bigg ) \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(2)}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{6} \\bigg ( 2 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-3} \\\\& \\times \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) + 20 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{3} \\left(1-\\mathfrak {s}^{3}\\right) \\right)^{2} \\\\& -64 \\overline{G}_{\\mathfrak {s}}^{-4} \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-2} \\\\& + \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-2} \\\\& \\times \\left( 16\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} - 8\\lambda ^{2}\\left(1-\\mathfrak {s}^{2}\\right) + 120 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{2} \\left(1-\\mathfrak {s}^{3}\\right) \\right) + 2 \\bigg ) - \\lambda + 2 \\mathfrak {s} \\lambda ^{2} \\overline{G}_{\\mathfrak {s}}^{2} - \\frac{15}{2} \\mathfrak {s}^{2} \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{4} + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\Phi }}^{(3)}_{\\mathfrak {s}} = & \\ \\frac{\\lambda }{6} \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-2} \\\\& \\times \\bigg ( 6 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-2} \\\\& \\times \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) + 20 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{3} \\left(1-\\mathfrak {s}^{3}\\right) \\right)^{3} \\\\& + 6 \\left(2\\overline{G}_{\\mathfrak {s}}^{-2} + \\overline{\\Phi }_{\\mathfrak {s}}^{(2)} - \\lambda \\left(1-\\mathfrak {s}\\right) + \\lambda ^{2}\\overline{G}_{\\mathfrak {s}}^{2}\\left(1-\\mathfrak {s}^{2}\\right) - \\frac{5}{2}\\lambda ^{3}\\overline{G}_{\\mathfrak {s}}^{4}\\left(1-\\mathfrak {s}^{3}\\right) \\right)^{-1} \\\\& \\times \\left(8\\overline{G}_{\\mathfrak {s}}^{-3} - \\overline{\\Phi }_{\\mathfrak {s}}^{(3)} - 4 \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{2}\\right) + 20 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{3} \\left(1-\\mathfrak {s}^{3}\\right) \\right) \\\\& \\times \\left( -48\\overline{G}_{\\mathfrak {s}}^{-4} - \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} - 8\\lambda ^{2}\\left(1-\\mathfrak {s}^{2}\\right) + 120 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{2} \\left(1-\\mathfrak {s}^{3}\\right) \\right) + 384 \\overline{G}_{\\mathfrak {s}}^{-5} - \\overline{\\Phi }_{\\mathfrak {s}}^{(5)} + 480 \\lambda ^{3} \\overline{G}_{\\mathfrak {s}} \\left(1-\\mathfrak {s}^{3}\\right) \\bigg ) \\\\& + 8 \\mathfrak {s} \\lambda ^{2} \\overline{G}_{\\mathfrak {s}} - 60 \\mathfrak {s}^{2} \\lambda ^{3} \\overline{G}_{\\mathfrak {s}}^{3} + \\frac{1}{2} \\dot{\\overline{G}}_{\\mathfrak {s}} \\overline{\\Phi }_{\\mathfrak {s}}^{(4)} \\;,\\end{split}$ where we have used our usual shorthand notations at $N=1$ , i.e.",
"$\\overline{G}_{\\mathfrak {s}}\\equiv \\overline{G}_{\\mathfrak {s},11}$ , $\\overline{\\Sigma }_{\\mathfrak {s}}\\equiv \\overline{\\Sigma }_{\\mathfrak {s},11}$ and $\\overline{\\Phi }_{\\mathfrak {s}}^{(n)}\\equiv \\overline{\\Phi }_{\\mathfrak {s},(1,1)\\cdots (1,1)}^{(n)}$ $\\forall n$ ."
],
[
"Kohn-Sham functional renormalization group",
"The generic differential equations that we solved to implement the KS-FRG were not given in section that contains our 2PPI-FRG analysis.",
"These flow equations, which are the counterparts of Eqs.",
"(REF ) to (REF ) for the KS-FRG, belong to an infinite hierarchy of coupled differential equations for the vertices $\\overline{\\gamma }^{(n)}_{\\mathfrak {s}}$ and are obtained from a vertex expansion of the master equation (REF ), as was explained right below Eq.",
"(REF ).",
"The first equations of this hierarchy readREF : $\\dot{\\overline{\\gamma }}_{\\mathfrak {s}} = \\int _{\\alpha } \\dot{V}_{\\mathfrak {s},\\alpha } \\overline{\\rho }_{\\mathfrak {s},\\alpha } + \\int _{\\alpha _{1},\\alpha _{2}} \\overline{\\rho }_{\\mathfrak {s},\\alpha _{1}} \\overline{\\Gamma }^{(2)}_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{2}} + \\frac{1}{2} \\int _{\\alpha _{1},\\alpha _{2}} \\dot{U}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\left( \\overline{G}_{\\mathfrak {s},\\alpha _{2}\\alpha _{1}} + \\overline{\\rho }_{\\mathfrak {s},\\alpha _{2}} \\overline{\\rho }_{\\mathfrak {s},\\alpha _{1}} \\right) \\;,$ $\\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{1}} = \\int _{\\alpha _{2}} \\overline{G}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\left(- \\dot{V}_{\\mathfrak {s},\\alpha _{2}} + \\frac{1}{2}\\int _{\\alpha _{3},\\cdots ,\\alpha _{6}} \\dot{U}_{\\mathfrak {s},\\alpha _{3}\\alpha _{4}}\\overline{G}_{\\mathfrak {s},\\alpha _{3}\\alpha _{5}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{2}\\alpha _{5}\\alpha _{6}} \\overline{G}_{\\mathfrak {s},\\alpha _{6}\\alpha _{4}} - \\int _{\\alpha _{3}} \\dot{U}_{\\mathfrak {s},\\alpha _{2}\\alpha _{3}}\\overline{\\rho }_{\\mathfrak {s},\\alpha _{3}} \\right) \\;,$ $\\begin{split}\\dot{\\overline{\\gamma }}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}}^{(2)} = & \\int _{\\alpha _{3}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{3}} \\overline{\\gamma }^{(3)}_{\\mathfrak {s},\\alpha _{3}\\alpha _{1}\\alpha _{2}} + \\dot{U}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}} \\\\& + \\int _{\\alpha _{3},\\cdots ,\\alpha _{8}} \\dot{U}_{\\mathfrak {s},\\alpha _{3}\\alpha _{4}} \\overline{G}_{\\mathfrak {s},\\alpha _{3}\\alpha _{5}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{1}\\alpha _{5}\\alpha _{6}} \\overline{G}_{\\mathfrak {s},\\alpha _{6}\\alpha _{7}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{2}\\alpha _{7}\\alpha _{8}} \\overline{G}_{\\mathfrak {s},\\alpha _{8}\\alpha _{4}} \\\\& - \\frac{1}{2}\\int _{\\alpha _{3},\\cdots ,\\alpha _{6}} \\dot{U}_{\\mathfrak {s},\\alpha _{3}\\alpha _{4}}\\overline{G}_{\\mathfrak {s},\\alpha _{3}\\alpha _{5}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(4)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(4)}\\right)_{\\alpha _{1}\\alpha _{2}\\alpha _{5}\\alpha _{6}} \\overline{G}_{\\mathfrak {s},\\alpha _{6}\\alpha _{4}} \\;,\\end{split}$ $\\begin{split}\\dot{\\overline{\\gamma }}_{\\mathfrak {s},\\alpha _{1}\\alpha _{2}\\alpha _{3}}^{(3)} = & \\ \\int _{\\alpha _{4}} \\dot{\\overline{\\rho }}_{\\mathfrak {s},\\alpha _{4}} \\overline{\\gamma }^{(4)}_{\\mathfrak {s},\\alpha _{4}\\alpha _{1}\\alpha _{2}\\alpha _{3}} \\\\& - \\Bigg ( \\int _{\\alpha _{4},\\cdots ,\\alpha _{11}} \\dot{U}_{\\mathfrak {s},\\alpha _{4}\\alpha _{5}} \\overline{G}_{\\mathfrak {s},\\alpha _{4}\\alpha _{6}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{1}\\alpha _{6}\\alpha _{7}} \\overline{G}_{\\mathfrak {s},\\alpha _{7}\\alpha _{8}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{2}\\alpha _{8}\\alpha _{9}} \\overline{G}_{\\mathfrak {s},\\alpha _{9}\\alpha _{10}} \\\\& \\times \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{3}\\alpha _{10}\\alpha _{11}} \\overline{G}_{\\mathfrak {s},\\alpha _{11}\\alpha _{5}} + (\\alpha _{2},\\alpha _{1},\\alpha _{3}) + (\\alpha _{1},\\alpha _{3},\\alpha _{2}) \\Bigg ) \\\\& + \\Bigg (\\int _{\\alpha _{4},\\cdots ,\\alpha _{9}} \\dot{U}_{\\mathfrak {s},\\alpha _{4}\\alpha _{5}} \\overline{G}_{\\mathfrak {s},\\alpha _{4}\\alpha _{6}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(4)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(4)}\\right)_{\\alpha _{1}\\alpha _{2}\\alpha _{6}\\alpha _{7}} \\overline{G}_{\\mathfrak {s},\\alpha _{7}\\alpha _{8}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(3)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(3)}\\right)_{\\alpha _{3}\\alpha _{8}\\alpha _{9}} \\overline{G}_{\\mathfrak {s},\\alpha _{9}\\alpha _{5}} \\\\& + (\\alpha _{1},\\alpha _{3},\\alpha _{2}) + (\\alpha _{2},\\alpha _{3},\\alpha _{1}) \\Bigg ) - \\frac{1}{2}\\int _{\\alpha _{4},\\cdots ,\\alpha _{7}} \\dot{U}_{\\mathfrak {s},\\alpha _{4}\\alpha _{5}}\\overline{G}_{\\mathfrak {s},\\alpha _{4}\\alpha _{6}} \\left(\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(5)}+\\overline{\\gamma }_{\\mathfrak {s}}^{(5)}\\right)_{\\alpha _{1}\\alpha _{2}\\alpha _{3}\\alpha _{6}\\alpha _{7}} \\overline{G}_{\\mathfrak {s},\\alpha _{7}\\alpha _{5}} \\;,\\end{split}$ where the propagator $\\overline{G}_{\\mathfrak {s}}$ satisfies: $\\overline{G}^{-1}_{\\mathfrak {s}} \\equiv \\overline{\\Gamma }_{\\mathfrak {s}}^{(\\mathrm {2PPI})(2)} = \\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}}^{(2)} + \\overline{\\gamma }_{\\mathfrak {s}}^{(2)} \\;.$ Finally, the differential equations that we have solved to obtain the KS-FRG results presented in section for the (0+0)-D $O(N)$ model can be directly inferred from Eqs.",
"(REF ) to (REF ) combined with Eqs.",
"(REF ), (REF ) and (REF ).",
"Another important point to address is the resolution of the Kohn-Sham equation and the determination of the derivatives $\\overline{\\Gamma }^{(n)}_{\\mathrm {KS}}$ that should be carried out alongside the treatment of these differential equations to implement the KS-FRG.",
"For the studied toy model, we can once again benefit from the (0+0)-D framework and exploit the connection between 2PI and 2PPI EAs to deduce an explicit expression of $\\Gamma _{\\mathrm {KS}}(\\rho )$ in terms of $\\rho $ .",
"In particular, expression (REF ) of the free 2PI EA $\\Gamma _{0}^{(\\mathrm {2PI})}$ reduces for the (0+0)-D $O(N)$ model to: $\\Gamma _{0}^{(\\mathrm {2PI})}(G) = -\\frac{1}{2}\\mathrm {Tr}_{a}\\big [\\ln (2\\pi G)\\big ] + \\frac{1}{2}\\mathrm {Tr}_{a}\\big (C^{-1}G\\big ) -\\frac{N}{2} \\;.$ After comparing the classical actions (REF ) and (REF ) at vanishing two-body interaction $U$ , one deduces that the mean-field part $\\Gamma _{\\mathrm {KS},\\mathfrak {s}}(\\rho )$ of our toy model can also be expressed after performing the substitutions $G_{a_{1}a_{2}}\\rightarrow \\rho _{a_{1}}\\delta _{a_{1}a_{2}}$ and $C^{-1}_{a_{1}a_{2}} \\rightarrow V_{\\mathrm {KS},\\mathfrak {s},a_{1}}\\delta _{a_{1}a_{2}}$ in Eq.",
"(REF ).",
"This yields: $\\Gamma _{\\mathrm {KS},\\mathfrak {s}}(\\rho ) = - \\frac{1}{2} \\sum _{a=1}^{N} \\ln (2\\pi \\rho _{a}) + \\frac{1}{2} \\sum _{a=1}^{N} V_{\\mathrm {KS},\\mathfrak {s},a} \\rho _{a} - \\frac{N}{2} \\;.$ As a result, the derivatives $\\overline{\\Gamma }^{(n)}_{\\mathrm {KS},\\mathfrak {s}}$ can be directly obtained by differentiating the latter equation, i.e.",
": $\\begin{split}\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s},a_{1}\\cdots a_{n}}^{(\\mathrm {2PPI})(n)} = & -\\frac{1}{2} \\left.\\frac{\\partial ^{n}}{\\partial \\rho _{a_{1}}\\cdots \\partial \\rho _{a_{n}}} \\sum _{a_{n+1}={1}}^{N} \\ln (\\rho _{a_{n+1}}) \\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}} \\\\& \\quad \\forall n\\ge 2 \\;.\\end{split}$ However, it is in principle not possible to find an explicit expression for $\\Gamma _{\\mathrm {KS},\\mathfrak {s}}[\\rho ]$ in terms of $\\rho $ (such as that of Eq.",
"(REF )) at finite dimensions for the same reason that we can not determine such an analytical expression for $\\Gamma ^{(\\mathrm {2PPI})}[\\rho ]$ or its non-interacting version $\\Gamma _{0}^{(\\mathrm {2PPI})}[\\rho ]$ , namely that Legendre transforms underlying 2PPI EAs can not be carried out explicitlyREF .",
"Therefore, for the implementation of the KS-FRG at finite dimensions, we rely on the same lengthy procedure to determine the derivatives $\\overline{\\Gamma }^{(n)}_{\\mathrm {KS},\\mathfrak {s}}$ as that required to determine the initial conditions of the standard 2PPI-FRG (see Eqs.",
"(REF ) and (REF )).",
"In this situation, the derivatives $\\overline{\\Gamma }^{(n)}_{\\mathrm {KS},\\mathfrak {s}}$ are thus determined from the relations: $\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1}\\alpha _{2}}^{(2)} = \\left(W^{(2)}_{\\mathrm {KS},\\mathfrak {s}}[K=0]\\right)_{\\alpha _{1}\\alpha _{2}}^{-1} \\;,$ $\\begin{split}\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s},\\alpha _{1} \\cdots \\alpha _{n}}^{(n)} = & \\ \\int _{\\alpha _{2n-2}} \\left(W^{(2)}_{\\mathrm {KS},\\mathfrak {s}}[K]\\right)_{\\alpha _{n}\\alpha _{2n-2}}^{-1} \\frac{\\delta }{\\delta K_{\\alpha _{2n-2}}} \\cdots \\\\& \\times \\int _{\\alpha _{n+1}} \\left(W^{(2)}_{\\mathrm {KS},\\mathfrak {s}}[K]\\right)_{\\alpha _{3}\\alpha _{n+1}}^{-1} \\\\& \\times \\frac{\\delta }{\\delta K_{\\alpha _{n+1}}} \\left(W^{(2)}_{\\mathrm {KS},\\mathfrak {s}}[K]\\right)_{\\alpha _{1}\\alpha _{2}}^{-1} \\bigg |_{K=0} \\\\& \\quad \\forall n\\ge 3 \\;,\\end{split}$ where $\\begin{split}& Z_{\\mathrm {KS},\\mathfrak {s}}[K]=e^{W_{\\mathrm {KS},\\mathfrak {s}}[K]} \\\\& =\\int \\mathcal {D}\\widetilde{\\psi }^{\\dagger }\\mathcal {D}\\widetilde{\\psi } \\ e^{-\\int _{\\alpha } \\widetilde{\\psi }_{\\alpha }^{\\dagger }\\left(\\partial _{\\tau } + \\hat{O}_{\\mathrm {kin}} + V_{\\mathrm {KS},\\mathfrak {s},\\alpha } - \\mu \\right)\\widetilde{\\psi }_{\\alpha }+\\int _{\\alpha } K_{\\alpha } \\widetilde{\\psi }_{\\alpha }^{\\dagger }\\widetilde{\\psi }_{\\alpha }} \\;.\\end{split}$ Finally, we explain how the Kohn-Sham equation is solved at each step of the KS-FRG flow in the framework of the (0+0)-D $O(N)$ model.",
"This amounts to determining the Kohn-Sham potential $V_{\\mathrm {KS},\\mathfrak {s}}$ that satisfies the equality: $\\left.",
"\\frac{\\partial \\Gamma _{\\mathrm {KS},\\mathfrak {s}}(\\rho )}{\\partial \\rho _{a}}\\right|_{\\rho =\\overline{\\rho }_{\\mathfrak {s}}} = 0 \\quad \\forall a,\\mathfrak {s} \\;,$ with $\\Gamma _{\\mathrm {KS},\\mathfrak {s}}(\\rho )$ given by Eq.",
"(REF ).",
"The solution of this procedure is simply: $V_{\\mathrm {KS},\\mathfrak {s},a} = \\overline{\\rho }_{\\mathfrak {s},a}^{-1} \\quad \\forall a, \\mathfrak {s} \\;.$ After plugging result (REF ) into Eq.",
"(REF ) at vanishing source $K$ , one directly obtains: $\\begin{split}\\overline{\\Gamma }_{\\mathrm {KS},\\mathfrak {s}} = - \\frac{1}{2} \\sum _{a=1}^{N} \\ln \\big (2\\pi \\overline{\\rho }_{\\mathfrak {s},a}\\big ) \\;,\\end{split}$ which is consistent with relation (REF )."
]
] | 2210.07748 |
[
[
"Young Star-Forming Complexes in the Ring of the S0 galaxy NGC 4324"
],
[
"Abstract We present the results of our study of starforming regions in the lenticular galaxy NGC 4324.",
"During a complex analysis of multiwavelength observational data -- the narrow-band emission-line images obtained with the 2.5-m telescope at the Caucasus Mountain Observatory of the Sternberg Astronomical Institute of the Moscow State University and the archival images in the broad bands of the SDSS, GALEX and WISE surveys -- we have detected young starforming complexes (clumps) located in the inner ring of the lenticular galaxy NGC 4324, and we have established a regular pattern of their distribution along the ring, which, nevertheless, changes with time (with age of starforming regions).",
"We suggest several possible evolutionary paths of the lenticular galaxy NGC 4324, of which the accretion of gas-rich satellites or giant clouds (the so-called minor merging) is the most probable one."
],
[
"INTRODUCTION",
"Lenticular galaxies, by the definition of this morphological type, are generally believed to be disk galaxies without star formation.",
"A deficit of gas in these early-type galaxies is again traditionally mentioned as being responsible for the absence of star formation in the disks of lenticular galaxies.",
"However, deeper surveys in radio lines have recently shown that, in fact, quite often there is a cold gas in lenticular galaxies, both neutral hydrogen (Sage and Welch 2006; Serra et al.",
"2012) and molecular gas (Welch and Sage 2003; Welch et al.",
"2010), that can serve as a fuel for star formation.",
"At the same time, the star formation being observed in some gas-rich lenticular galaxies is usually confined to ring structures (Pogge and Eskridge 1993; Salim et al.",
"2012) and, probably, may have a different trigger and slightly different physics than does the star formation in the arms of spiral galaxies.",
"There is statistical evidence that the star formation in rings is much more efficient than the star formation in spiral arms (Kormendy and Kennicutt 2004).",
"Figure: The image of NGC 4324 in combined colors taken from the SDSS survey, DR9 (Ahn et al.",
"2012)The nearby early-type galaxy NGC 4324 being investigated here is remarkable for its bright blue ring (Fig.",
"1) embedded in a large-scale stellar disk typical for lenticular galaxies, having a reddish color and without distinct structural features, except the ring.",
"The blue color of the ring points to current or recent star formation within it.",
"The galaxy NGC 4324 was included into the sample of the ATLAS-3D project (Cappellari et al.",
"2011) and was investigated by means of panoramic spectroscopy.",
"There are also photometric surveys that included NGC 4324.",
"In the ARRAKIS atlas (Comerón et al.",
"2014), where the ring structures noticeable in the 3.6-$\\mu $ m and 4.5-$\\mu $ m bands are collected, it is classified as a galaxy with an inner ring, (L)SA(r)0$+$ .",
"A bar is suspected when inspecting the image left after the subtraction of the galaxy model constructed from the parameters derived during the decomposition of the image from the S4G survey (Sheth et al.",
"2010) given in ARRAKIS, although only the ring is pointed out among the distinguished features in the catalogue.",
"However, the presence of a bar in this galaxy is also noted in the HyperLEDA database.",
"According to this database, the galaxy under study is a member of the NGC 4303 group (Garcia 1993).",
"Basic characteristics of the galaxy being investigated are given in Table 1.",
"Table: The main characteristics of NGC 4324"
]
] | 2210.07786 |
[
[
"Statistical significances and projections for proton decay experiments"
],
[
"Abstract We study the statistical significances for exclusion and discovery of proton decay at current and future neutrino detectors.",
"Various counterintuitive flaws associated with frequentist and modified frequentist statistical measures of significance for multi-channel counting experiments are discussed in a general context and illustrated with examples.",
"We argue in favor of conservative Bayesian-motivated statistical measures, and as an application we employ these measures to obtain the current lower limits on proton partial lifetime at various confidence levels, based on Super-Kamiokande's data, generalizing the 90\\% CL published limits.",
"Finally, we present projections for exclusion and discovery reaches for proton partial lifetimes in $p \\rightarrow \\overline \\nu K^+$ and $p \\rightarrow e^+ \\pi^0$ decay channels at Hyper-Kamiokande, DUNE, JUNO, and THEIA."
],
[
"Introduction",
"In order to account for the observed matter-antimatter asymmetry in our universe, baryon number must be violated as required by the Sakharov conditions [1].",
"Although baryon number is a global symmetry of the (renormalizable) Standard Model (SM) Lagrangian, it is violated by non-perturbative electroweak sphaleron effects that are heavily suppressed at temperatures much lower than the electroweak scale [2], [3].",
"The sphaleron effects, however, together with the CP-violation in the electroweak sector are not sufficient to explain the observed baryon asymmetry, and therefore provide a key motivation for theories beyond the SM with additional $B$ -violation.",
"Grand unified theories (GUTs), with or without supersymmetry, are well-motivated and generically predict baryon number violation, and therefore can lead to proton decay [4], [5], [6], [7], [8], [10], [9], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43].",
"After integrating out the heavy fields, the non-renormalizable operators built out of the SM fields that allow proton decay are of dimension-six or higher, with the suppression scale of order the GUT breaking scale.",
"In this paper, we consider proton decay in the $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay channels that are typically predicted to be the leading modes in supersymmetric [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43] and non-supersymmetric [5], [6], [7], [8], [10], [9], [11], [12], [13], [14], [15], [16], [17] GUTs, respectively.",
"At present, the strongest constraints on these proton partial lifetimes are from the Super-Kamioka neutrino detection experiment (Super-Kamiokande), where the most stringent published 90% CL lower limits are $5.9 \\times 10^{33}$ years for the $p \\rightarrow \\overline{\\nu } K^+$ mode [44] and $2.4 \\times 10^{34}$ years for the $p \\rightarrow e^+ \\pi ^0$ mode [45].",
"We will make projections for the exclusion and discovery reaches for these proton modes decays at future neutrino detectors at Deep Underground Neutrino Experiment (DUNE) [46], Jiangmen Underground Neutrino Observatory (JUNO) [47], Hyper-Kamiokande (the successor to Super-Kamiokande, and an order of magnitude larger) [48], and THEIA (a novel detector concept with water based liquid scintillator, 10% liquid scintillator and 90% water, that can detect and distinguish between Cerenkov and the scintillation light) [49].",
"In order to project the exclusion and discovery reaches, it is necessary to make choices regarding the statistical tools to be employed.",
"Indeed, the results for such projections are only meaningful in the context of those choices.",
"Here, we are interested in counting experiments with multiple independent channels with different signal rates and backgrounds, with uncertainties.",
"Our statistical analysis choices are guided by several requirements.",
"We aim for statistical measures that avoid reporting an exclusion or discovery when the experiment is actually not sensitive to the physics signal hypothesis under investigation.",
"As we will discuss, pure frequentist statistics can suffer from this problem.",
"We choose statistical measures such that the presence of a non-informative channel (one with a much higher background and/or a much lower signal rate than other channels) does not unduly affect the exclusion or discovery conclusion.",
"We avoid statistical measures that contain the subtle flaw that they could counterintuitively imply a greater sensitivity for an experiment if it increases its background.",
"Regarding this last point, in a previous paper [50], we have discussed the fact that the median expected significance for discovery or exclusion has just such a counterintuitive flaw in the context of frequentist $p$ -values for a single-channel counting experiment.",
"We proposed a solution to that problem.",
"As we will see below, this type of problem also occurs in the case of multi-channel counting experiments, and can be avoided using Bayesian-motivated statistical measures.",
"For these reasons, section of this paper is devoted to a rather extensive discussion of the statistical issues associated with multi-channel counting experiments with background and nuisance parameter uncertainties, in which we highlight some of the problems that can occur and explain our choices of statistical tools in a general context.",
"In Section we apply these statistical measures to discuss the present exclusions from Super-Kamiokande, and we project exclusion and discovery prospects for proton decay at DUNE, JUNO, Hyper-Kamiokande, and THEIA, for the proton decay modes $p \\rightarrow e^+ \\pi ^0$ and $p \\rightarrow \\overline{\\nu }K^+$ .",
"Section summarizes our findings for exclusion and discovery prospects for run-times of 10 and 20 years.",
"In this paper we are concerned with new physics signals and backgrounds, which are both assumed to occur as random discrete events governed by Poisson statistics, possibly in multiple independent channels.",
"In general, given data resulting from an experiment, the significance of a possible exclusion or discovery can be given in terms of a $p$ -value, defined as the probability of obtaining a result of equal or greater incompatibility with a null hypothesis $H_0$ .",
"In high-energy physics, the $p$ -value is often conventionally reported as a significance, defined by $Z &\\equiv & \\sqrt{2} \\ {\\rm erfc}^{-1}\\left(2 p\\right),$ which in the special case of a Gaussian distribution would coincide with the number of standard deviations.",
"The assumption for discovery is that the null hypothesis is a background-only hypothesis $H_0 = H_{b}$ , while for exclusion the null hypothesis is a signal plus background model $H_0 = H_{s+b}$ .",
"Consider a test-statistic $Q$ defined in such a way that larger $Q$ is more signal-like and smaller $Q$ is more background-like.",
"In a single-channel counting experiment, for example, $Q$ is simply the number of observed events.",
"Then, for an experimental outcome $Q_{\\rm obs}$ , one has the $p$ -value for discovery: $p_{\\rm disc} &=& P(Q \\ge Q_{\\rm obs} | H_{b}),$ and the $p$ -value for exclusion: $p_{\\rm excl} &=& P(Q \\le Q_{\\rm obs} | H_{s+b}).$ In a frequentist approach, the $p$ -value for a given data outcome is often used to provide a quantitative measure of the credence we give to $H_0$ .",
"However, the $p$ -value cannot be directly interpreted as the probability that the null hypothesis is true, given the data.",
"Nevertheless, small $p$ -values are considered a measure of evidence against $H_0$ in frequentist statistics.",
"In particle physics, two popular standards for exclusion are to require that $p_{\\rm excl} < 0.10$ or $0.05$ , commonly referred to as 90% or 95% exclusion.",
"For rejection of the background-only hypothesis in favor of some new model, a higher standard is almost always required, with either $Z_{\\rm disc} > 3$ $(p_{\\rm disc} < 0.001350)$ for “evidence\", or $Z_{\\rm disc}>5$ $(p_{\\rm disc} < 2.867 \\times 10^{-7})$ for “discovery\".",
"In high energy physics experiments in the 21st century, starting with the Higgs boson searches at the LEP $e^-e^+$ collider and for all kinds of searches for new phenomena at the Large Hadron Collider (LHC), it has become very common to use a modified frequentist statistical measure for exclusion, called the CL$_s$ method.",
"This is a more conservative approach to assigning exclusion significances than $p_{\\rm excl}$ .",
"The idea of CL$_s$ [51], [52], [53], [54] is to divide the usual $p$ -value for exclusion by the $p$ -value that would be obtained with the signal assumed absent: ${\\rm CL}_s (Q_{\\rm obs}) &=&\\frac{P(Q \\le Q_{\\rm obs} | H_{s+b})}{P(Q \\le Q_{\\rm obs} | H_{b})}.$ A specific motivation for using CL$_s$ rather than $p_{\\rm excl}$ is to avoid reporting an exclusion in cases for which the experiment is actually not sensitive to the purported signal hypothesis, but the observed data has a small $p$ -value anyway.",
"This can occur, for example, in a counting experiment if the observed number of events is significantly smaller than the background estimate, as we will discuss in detail shortly.",
"Note that, by design, CL$_s$ is not a $p$ -value or even a probability, but rather a ratio of probabilities.",
"Nevertheless, the exclusion is reported using CL$_s$ in place of the exclusion $p$ -value, so that one reports 95% (or 90%) exclusion if ${\\rm CL}_s < 0.05$ (or $0.1$ ).",
"Because the denominator is always less than 1, the modified frequentist measure CL$_s$ is always more conservative in reporting exclusions than the frequentist $p$ value, in the sense that using it reduces the false exclusion rate compared to using $p_{\\rm excl}$ .",
"In particle physics literature, ${\\rm CL}_s$ was introduced in ref.",
"[51] and detailed (along with its advantages, reviewed and illustrated below) in refs.",
"[52], [53], [54].",
"It is also useful to have a counterpart to the $p_{\\rm disc}$ statistic that similarly guards against claiming discovery in situations where the experiment is not sensitive to the signal model.",
"In ref.",
"[55], an approach to discovery significance was proposed using the Bayes factor [56], [57], [58] of the null hypothesis $H_0 = H_{b}$ to the alternative hypothesis $H_1 = H_{s+b}$ .",
"For an experiment investigating a putative signal with strength $s$ , the Bayes factor $B_{01}$ is (using the probabilities in place of the likelihoods, to which they are proportional): $B_{0 1} &=& \\frac{P(Q_{\\rm obs} | H_b)}{\\displaystyle \\displaystyle {0.85}{\\int _{{1.2}{0}}^{{1.2}{\\infty }}} ds^\\prime \\, \\pi (s^\\prime )\\, P(Q_{\\rm obs} |H_{s^\\prime + b} )} ,$ where $\\pi (s^{\\prime })$ is a Bayesian prior probability distribution for the signal strength.",
"As mentioned in [55], this expression is only meaningful in the case of a prior that is proper, i.e.",
"$\\int _0^\\infty ds^\\prime \\, \\pi (s^\\prime ) = 1$ , since otherwise the arbitrary normalization of an improper prior would make the Bayes factor $B_{01}$ also arbitrary.",
"This precludes the use of a flat prior, for example.",
"For a single-channel counting experiment with background mean $b$ , that reference argues in favor of the proper prior $\\pi (s^\\prime ) = {b}/{(s^\\prime + b)^2}$ , referred to as the objective signal prior.",
"However, we find it counterintuitive to use a prior for the signal that depends on the background.",
"Instead, we choose simply $\\pi (s^{\\prime }) = \\delta (s^{\\prime } - s)$ , expressing certainty in the prediction of the signal model.",
"If the signal model prediction is not perfectly well known, it is straightforward to generalize this with an appropriate $\\pi (s^{\\prime })$ .",
"We therefore define the simple likelihood ratio statistic for the confidence level in the discovery, ${\\rm CL}_{\\rm disc}(Q_{\\rm obs}) &=& \\frac{P(Q_{\\rm obs}|H_b)}{P(Q_{\\rm obs}|H_{s+b})}.$ While various scales have been proposed (see e.g.",
"Jeffreys' in [58] and Kass and Raftery's in [57]) to interpret the Bayes factor as a measure of evidence in favor of or against a null hypothesis, we propose to use ${\\rm CL}_{\\rm disc}$ in place of $p$ in eq.",
"(REF ) to obtain a discovery significance $Z$ , in exactly the same way that a frequentist $p_{\\rm disc}$ would be used.",
"As we will illustrate below, our choice gives results that are always more conservative than the significances obtained from $p_{\\rm disc}$ .",
"This is very similar to the way the modified frequentist measure ${\\rm CL}_s$ is now commonly used in place of $p$ in eq.",
"(REF ) to report an exclusion significance that is always more conservative than that of the standard frequentist method, even though ${\\rm CL}_s$ , like ${\\rm CL}_{\\rm disc}$ , is not a probability."
],
[
"Single-channel counting experiments",
"To illustrate the statistical methods discussed above let us consider the special case of a simple experiment that counts the number of events $n$ , with signal and background modeled as independent Poisson processes with means $s$ and $b$ respectively.",
"For a mean $\\mu $ , the Poisson probability to observe $n$ events is $P(n | \\mu ) &=& \\frac{\\mu ^n e^{-\\mu }}{n!",
"}.$ Therefore, in the idealized case of perfectly known background, the $p$ -value for discovery is the probability that data generated under hypothesis $H_0 = H_{b}$ is equally or more signal-like than the actual observed number of events $n$ : $p_{\\rm disc} (n, b) = \\sum _{k=n}^\\infty P(k|b) = {\\gamma (n, b)}/{\\Gamma (n)}.$ The $p$ -value for exclusion is the probability that data generated under hypothesis $H_0 = H_{s+b}$ is equally or more background-like than the actual observed number of events $n$ : $p_{\\rm excl} (n, b, s) = \\sum _{k=0}^n P(k |s + b) = {\\Gamma (n + 1, s + b)}/{\\Gamma (n + 1)}.$ In these equations, $\\gamma (z, x)$ and $\\Gamma (z, x)$ are the lower and upper incomplete gamma functions, respectively, defined by $\\gamma (z,x) \\:=\\: \\int _0^x dt \\: t^{z-1} e^{-t},\\qquad \\Gamma (z,x) \\:=\\: \\int _x^\\infty dt \\: t^{z-1} e^{-t},$ so that $\\Gamma (z) = \\gamma (z,x) + \\Gamma (z,x)$ is the ordinary gamma function.",
"The CL$_s$ statistic for exclusion in this case is ${\\rm CL}_s (n,b,s) &=&\\frac{p_{\\rm excl}(n,b,s)}{p_{\\rm excl}(n,b,0)}\\:=\\:\\frac{\\Gamma (n+1, s+b)}{\\Gamma (n+1, b)}.$ This is larger than $p_{\\rm excl}(n,b,s)$ by a factor $\\Gamma (n+1)/\\Gamma (n+1,b)$ .",
"Figure REF illustrates the idea of the ${\\rm CL}_s$ method [51], [52], [53], [54].",
"In the figure, $p_{\\rm excl}(n,b,s)$ (the shaded area under the blue histograms) is divided by $p_{\\rm excl}(n,b,0)$ (the shaded area under the red histograms) to give ${\\rm CL}_s$ .",
"The first panel shows the case $b=2.2$ , $s=8.4$ , and $n=5$ .",
"In situations like this, where the $H_b$ and $H_{s+b}$ hypothesis distributions do not have much overlap, $p_{\\rm excl}$ and CL$_{\\rm s}$ evaluate to very similar results due to the denominator of the CL$_{\\rm s}$ definition being close to 1.",
"For this particular case, one finds $p_{\\rm excl} = 0.0475$ and CL$_s = 0.0487$ , and by either criterion one would report a better than 95% exclusion.",
"Figure: Illustration of the idea of the CL s {\\rm CL}_s statistic for exclusion as an improvement over p excl p_{\\rm excl}.The Poisson distributions P(k|μ)P(k|\\mu ) are generated under the hypotheses that signal and background are both present μ=s+b\\mu = s + b (blue histograms) and that the signal is absent μ=b\\mu = b (red histograms).",
"For the observed number of events nn, p excl (n,b,s)p_{\\rm excl}(n,b,s) [from eq.",
"()] is shown bythe shaded area part under the blue histogram, and p excl (n,b,0)p_{\\rm excl}(n,b,0) is the shaded area partunder the red histogram, while CL s _s [from eq.",
"()] is their ratio.In the first plot, the Poisson means of the signal and background are taken to be s=8.4s=8.4 and b=2.2b=2.2, respectively, while in the second plot they are s=2.2s=2.2 and b=8.4b=8.4.",
"In both plots,the observed number of events is n=5n=5.",
"In the first plot, there is little overlap between thedistributions from the H b H_b and H s+b H_{s+b} hypotheses, and p excl =0.0475p_{\\rm excl} = 0.0475 and CL s =0.0487_s = 0.0487, so one would report better than 95% exclusion using either criterion.",
"In the second plot, the overlap is much larger.",
"Although p excl =0.0475p_{\\rm excl} = 0.0475 is the same (since s+bs+b and nn did not change), one finds CL s =0.3022_s = 0.3022, and one refrains from reporting an exclusion of the hypothesis H s+b H_{s+b}.The second panel of Figure REF illustrates the case $b = 8.4$ , $s=2.2$ , and $n=5$ , so that the overlap between the distributions for $H_b$ and $H_{s+b}$ is much larger.",
"In cases like this with a larger overlap (i.e.",
"the signal regions get polluted by the background) statistical conclusions based on $p_{\\rm excl}$ alone can be too aggressive.",
"Since we engineered this example to have the same $b+s$ and $n$ as for the first panel, we get the same$\\endcsname $The general fact that $p_{\\rm excl}(n,b,s)$ depends only on the sum $s+b$ , and not on $s$ or $b$ separately, is a clear reason to reject it as a measure of confidence in the presence of the signal model, because it says that any exclusion for signal $s$ and background $b$ would imply an equally strong exclusion for the case that the signal is $s=0$ if the background $b$ were increased by the numerical value of $s$ .",
"$p_{\\rm excl} = 0.0475$ , which taken at face value would again give a better than 95% exclusion.",
"However, proponents of the CL$_s$ criteria point out that here it must be recognized that for $b = 8.4$ , the outcome $n\\le 5$ would have been a low-probability occurrence no matter what$\\endcsname $Here we are taking it as a requirement that $s\\ge 0$ , although in some situations quantum interference with the background could allow for $s<0$ .",
"See, for example, the case of a digluon resonance at the LHC [59].",
"the signal mean $s$ was.",
"Thus, the frequentist $p_{\\rm excl}$ is really telling us more about the observed data than making a useful statement about the signal hypothesis.",
"One finds that CL$_s = 0.3022$ , and using this one would, sensibly and conservatively, refrain from excluding the signal hypothesis.",
"In fact, no matter the outcome for $n$ , the experiment with $b=8.4$ simply lacks the statistical ability to exclude the $s=2.2$ signal model at 90% confidence, according to the ${\\rm CL}_s$ statistic.",
"This can be seen by computing it for the least signal-like outcome, $n=0$ , which gives CL$_s = 0.1108$ .",
"One possible practical interpretation of the very small $p_{\\rm excl}$ in such cases with $n$ significantly less than $b$ might be that the background estimate could be wrong for reasons unknown, while another is that the background simply fluctuated low from its true mean.",
"In any case, the intuitive interpretation of the ${\\rm CL}_s$ statistic is that the quoted significance for exclusion should be reduced from the usual frequentist value, due to the large overlap between the signal+background region and the background-only region.",
"Indeed, if the number of events is sufficiently small, one finds that the usual frequentist $p$ -value would correspond to an exclusion even in cases that defy sensible practical interpretation.",
"Considering the case $n=0$ more generally, one finds $p_{\\rm excl}(n\\!=\\!0,s,b) = e^{-(s+b)}$ , which becomes arbitrarily small for any fixed $s$ , if $b$ is sufficiently large.",
"One could use this to make an absurd claim of exclusion for a model that predicted $s = 10^{-500}$ or even $s=0$ exactly, simply by observing a smaller than expected number of events, if the background is large enough!",
"In contrast, usage of the statistic ${\\rm CL}_s(n=0,b,s) = e^{-s}$ conforms to the intuitively reasonable idea that, as an absolute prerequisite for excluding a signal hypothesis, the expected signal strength must not be too small.",
"Specifically, only models that predict $s>-\\ln (0.05) \\approx 2.996$ can be excluded at 95% confidence according to the ${\\rm CL}_s$ measure, for any $b$ and for any possible experimental outcome $n$ .",
"Similarly, 90% exclusion by the ${\\rm CL}_s$ method requires $s > -\\ln (0.1) \\approx 2.303$ .",
"The dependence of the exclusion significance on $b$ is shown for fixed $s = 4$ and $n=0,1,2,3$ in Figure REF .",
"For very small $b$ , the two statistics are nearly equal, $p_{\\rm excl} \\simeq {\\rm CL}_s$ .",
"For any fixed $n$ , in the limit of large $b$ one has CL$_s = e^{-s}$ , while $p_{\\rm excl}$ becomes absurdly small in comparison, which would imply an absurdly large $Z_{\\rm excl}$ .",
"Figure: Comparison of significances ZZ obtained using eq.",
"() from p excl p_{\\rm excl} [dashed lines, from eq.",
"()] and CL s _s [solid lines, from eq.",
"()], for fixed s=4s=4 as a function of varying bb,for n=0,1,2,n=0,1,2, and 3.",
"For very small bb, the two statistics are nearly equal, p excl ≃ CL s p_{\\rm excl} \\simeq {\\rm CL}_s.",
"In the limit of large bb one has CL s =e -s _s = e^{-s}, independent of nn,while p excl p_{\\rm excl} becomes absurdly small in comparison.Non-observation of a significant excess above background expectations can be used to constrain new physics.",
"In particular, for a single-channel counting experiment, the minimum signal needed to claim an exclusion at a given confidence level $1-\\alpha $ , equivalent to significance $Z = \\sqrt{2}\\, {\\rm erfc}^{-1}(2 \\alpha )$ , for a perfectly known background mean $b$ , is obtained [60], [61] by solving for $s$ in either $\\alpha &=& \\frac{\\Gamma (n + 1, s + b)}{\\Gamma (n + 1)} \\qquad (\\text{$p_{\\rm excl}$ method})$ in the standard frequentist approach, or $\\alpha &=& \\frac{\\Gamma (n + 1, s + b)}{\\Gamma (n + 1, b)} \\qquad (\\text{CL$_s$ method})$ in the modified frequentist approach.",
"Figure REF shows the 90% CL ($\\alpha = 0.1$ , left panel) and 95% CL ($\\alpha = 0.05$ , right panel) upper limits on signal as functions of the background mean, for a fixed number of observed events $n = 0, 1, 2$ , using the $p_{\\rm excl}$ (red lines) and CL$_s$ (blue lines) criteria.",
"Also shown in the figure are the 90% CL and 95% CL upper limits on $s$ that are obtained using the Feldman-Cousins (FC) method based on an ordering principle introduced in ref. [62].",
"The upper limits obtained by the FC method for a fixed $n$ do not always decrease with increasing $b$ ; instead they have a sawtooth pattern, as can be seen from the dotted lines in the figure.",
"This behavior is because of the discreteness of Poisson distributions.",
"The solid black lines in Figure REF show the results obtained by the FC method after requiring them to be non-increasing as a function of background mean.",
"Figure: The 90% CL (left panel) and 95% CL (right panel) upper limits on signal asfunctions of the background mean bb, for a fixed number of observed events n=0,1,2n = 0, 1, 2,using the CL s {\\rm CL}_s technique [blue lines, from eq.",
"()], standard frequentist pp-value approach [red lines, from eq.",
"()],and Feldman-Cousins method [solid black lines, from ref.",
"].The dotted black lines show the results obtained by the Feldman-Cousins method before requiring them to benon-increasing as a function of background mean.It is clear from the figure that the upper limits on $s$ obtained using the standard frequentist $p_{\\rm excl}$ approach are the least conservative, and can even go negative in the case where the number of observed events $n$ is small compared to the expected background mean.",
"For a fixed $n$ , despite the upper limits given by the ${\\rm CL}_s$ and FC methods being very different from each other, we note that they are both almost flat at very small backgrounds and then decrease slowly (or stay constant) as a function of background, always remaining positive.",
"For small $b$ the FC upper limits are more conservative, and for large $b$ , the ${\\rm CL}_s$ upper limits are more conservative.",
"The other striking difference between these two upper limits is that, for $n=0$ , the FC upper limits decrease with $b$ , but the ${\\rm CL}_s$ upper limits are independent of $b$ .",
"In particular, at a chosen confidence level $1 - \\alpha $ , for $n=0$ the ${\\rm CL}_s$ upper limit on $s$ is $-\\ln (\\alpha )$ .",
"The same result also holds for any $n$ in the limit that the background is extremely large.",
"At 90% (95%) CL, the upper limit given by ${\\rm CL}_s$ for $n=0$ , or for any $n$ as $b \\rightarrow \\infty $ , is around 2.303 (2.996).",
"On the other hand, the upper limit given by the FC method decreases as a function of $b$ and approaches a constant value at large $b$ .",
"For example, for $n=0$ , the 90% (95%) CL upper limit given by the FC method, after requiring it to be non-increasing as a function of $b$ , is approximately 0.8 (1.34) at large $b$ .",
"It is important for the following that the result for CL$_s(n,b,s)$ in the case of a single Poisson channel in eq.",
"(REF ) can also be obtained [63] as a Bayesian credible interval, using a flat prior for the signal and likelihoods ${\\cal L}(s|n,b) \\propto P(n|s+b)$ : ${\\rm CL}_{\\rm excl}(n, b, s) &=&\\frac{\\displaystyle {0.85}{\\int _{{1.4}{s}}^{{1.2}{\\infty }}} ds^{\\prime }\\, {\\cal L}(s^{\\prime }|n,b)}{\\displaystyle {0.85}{\\int _{{1.2}{0}}^{{1.2}{\\infty }}} ds^{\\prime }\\, {\\cal L}(s^{\\prime }|n,b)}\\:=\\:\\frac{\\displaystyle {0.85}{\\int _{{1.4}{s}}^{{1.2}{\\infty }}}ds^\\prime \\, e^{-(s^\\prime + b)} \\, (s^\\prime + b)^n}{\\displaystyle {0.85}{\\int _{{1.2}{0}}^{{1.2}{\\infty }}}ds^\\prime \\, e^{-(s^\\prime + b)} \\, (s^\\prime + b)^n}.$ Performing the integrations, ${\\rm CL}_{\\rm excl}(n,b,s)$ as defined by eq.",
"(REF ) is precisely equal to ${\\rm CL}_s(n,b,s)$ as defined by eq.",
"(REF ).If the signal mean is instead allowed to be negative with $s + b \\ge 0$ (see previous footnote), then ${\\rm CL}_{\\rm excl}(n, b, s)$ can be defined as ${\\rm CL}_{\\rm excl}(n, b, s)=\\frac{\\displaystyle {0.85}{\\int _{{1.4}{s}}^{{1.2}{\\infty }}}ds^\\prime \\, e^{-(s^\\prime + b)} \\, (s^\\prime + b)^n}{\\displaystyle {0.85}{\\int _{{1.2}{-b}}^{{1.2}{\\infty }}}ds^\\prime \\, e^{-(s^\\prime + b)} \\, (s^\\prime + b)^n}$ .",
"After performing the integrations, ${\\rm CL}_{\\rm excl}(n,b,s)$ is now precisely equal to $p_{\\rm excl} (n,b,s)$ as defined in eq.",
"(REF ).",
"However, despite the numerical equivalence, the interpretation is quite different, since the ratio of frequentist $p$ -values is not directly a Bayesian confidence interval.",
"Moreover, the equivalence between CL$_s$ and ${\\rm CL}_{\\rm excl}$ is only approximate in more complicated generalizations.",
"Looking ahead to the case of experiments which collect counts in multiple independent channels governed by Poisson statistics, and which may have nuisance parameters including uncertainties in the backgrounds, we will argue for a generalization based straightforwardly on the Bayesian version ${\\rm CL}_{\\rm excl}$ as given in eq.",
"(REF ) rather than CL$_s$ given in eq.",
"(REF ) or its specialization eq.",
"(REF ).",
"For a single-channel counting experiment, the discovery confidence level statistic defined in eq.",
"(REF ) becomes ${\\rm CL}_{\\rm disc}(n,b,s) &=&\\frac{P(n|b)}{P(n|b+s)} \\:=\\:\\frac{e^s}{(1 + s/b)^n},$ which can be used in place of $p$ in eq.",
"(REF ) to obtain a discovery significance.",
"(If the result is greater than 1, then clearly no discovery claim should be contemplated.)",
"Note that unlike $p_{\\rm disc}(n,b)$ , the result for ${\\rm CL}_{\\rm disc}(n,b,s)$ depends on the strength of the signal whose discovery is under investigation.",
"It is always more conservative than $p_{\\rm disc}(n,b)$ in claiming discovery, just as CL$_{\\rm s}$ is more conservative than $p_{\\rm excl}$ in claiming exclusion.",
"For example, in the extreme case $s=0$ , one has ${\\rm CL}_{\\rm disc}(n,b,s=0) = 1$ for any $b$ and $n$ , so one would never claim discovery using that criteria.",
"In contrast, the frequentist statistic $p_{\\rm disc}(n,b)$ can be arbitrarily small, implying an arbitrarily large discovery significance $Z$ , even in situations where the physics provides absolutely no possible source for a signal.For example, imagine a search for a new fundamental particle of mass 1 TeV, conducted by dropping a bag of hammers from the top of a tall building, with a somewhat noisy detector surrounding the impact point on the sidewalk.",
"For this experiment, theoretical modeling confidently predicts $s=0$ , so one should reasonably refrain from announcing discovery even if one estimated $b = 0.01$ and observed $n=3$ .",
"As we will see below, ${\\rm CL}_{\\rm disc}$ also generalizes more straightforwardly to cases that have multiple independent channels governed by Poisson statistics, and which may have nuisance parameters including uncertainties in the backgrounds.",
"Figure REF compares the discovery significance obtained from $p_{\\rm disc}$ and ${\\rm CL}_{\\rm disc}$ as a function of $s$ for fixed $n$ , with different curves for different values of $b$ .",
"Note that the discovery significance obtained from ${\\rm CL}_{\\rm disc}$ , which is always more conservative than that of $p_{\\rm disc}$ , is maximized at $s = n - b$ .",
"Figure: Comparison of discovery significances obtained using eq.",
"() from p disc p_{\\rm disc}[dashed lines, from eq.",
"()]and CL disc {\\rm CL}_{\\rm disc} [solid lines, from eq.",
"()]as a function of ss for n=3n=3 (left panel) and n=10n=10 (right panel), for various choices of bb.Given the number of observed events $n$ and an expected background mean, the standard $p$ -value for discovery $p_{\\rm disc}$ does not depend on the signal.",
"So, for a perfectly known background mean $b$ , we can compute the number of events needed for discovery at a significance $Z$ by solving for $n$ from [see eqs.",
"(REF ) and (REF )] $\\frac{1}{2} {\\rm erfc}\\left(\\frac{Z}{\\sqrt{2}}\\right) &=& \\frac{\\gamma (n, b)}{\\Gamma (n)}\\qquad (p_{\\rm disc}\\:\\text{method}).$ On the other hand, ${\\rm CL}_{\\rm disc}$ depends also on the signal, in which case the number of events needed for discovery for a known background $b$ and signal mean $s$ at a given significance $Z$ can be obtained by solving for $n$ from [see eqs.",
"(REF ) and (REF )] $\\frac{1}{2} {\\rm erfc}\\left(\\frac{Z}{\\sqrt{2}}\\right) &=& \\frac{e^s}{(1 + s/b)^{n}}\\qquad ({\\rm CL}_{\\rm disc}\\: \\text{method}).$ Figure REF shows the observed number of events required for $Z=3$ evidence (left panel) and $Z=5$ discovery (right panel) given by the $p_{\\rm disc}$ approach (solid black lines), and the ${\\rm CL}_{\\rm disc}$ approach for two choices of the signal mean $s=2$ (dashed red lines) and 10 (dashed blue lines) as functions of $b$ .",
"It is clear from the figure that, for a given background mean, the observed number of events needed for discovery given by the ${\\rm CL}_{\\rm disc}$ approach are at least as large as the result given by the $p_{\\rm disc}$ criterion, and often much larger when the background is not very small.",
"Figure: The observed number of events nn needed for Z=3Z=3 evidence (left panel) andZ=5Z=5 discovery (right panel) as functions of the expected background mean bb.The solid black lines show the result obtained from eq.",
"() using the standard frequentist approach based on p disc p_{\\rm disc}, which is independent of the signal mean ss.",
"The dashed red and blue lines show the results obtained from eq.",
"() using CL disc {\\rm CL}_{\\rm disc} for the cases of signal mean s=2s =2 and 10, respectively.We now turn to the question of projecting expectations for exclusion and discovery at ongoing and future experiments.",
"In simulations or assessments of a proposed experiment, one considers the statistics of pseudo-data generated under an alternative hypothesis $H_1$ .",
"For assessments of prospects for exclusion the alternative hypothesis is that the signal source is absent, $H_1 = H_b$ , while for discovery, the pseudo-data is generated assuming that both signal and background are present, $H_1 = H_{s+b}$ .",
"A common way to project an expected result is to set the number of events $n$ equal to the median expected value under the hypothesis $H_1$ .",
"However, due to the discrete nature of Poisson statistics events, the median expected outcome has the striking flaw that it can predict smaller significances if an experiment takes more data or reduces its background.",
"This counterintuitive feature of the median expected significance was pointed out and studied in detail in refs.",
"[64], [65], and in [50] where it was referred to as the “sawtooth problem\".",
"It occurs for the median expected CL$_s$ and ${\\rm CL}_{\\rm disc}$ as well.",
"The sawtooth behavior of the median expected CL$_s$ and ${\\rm CL}_{\\rm disc}$ as a function of the background mean $b$ , for various values of signal mean $s$ , is evident from Figure REF .",
"For comparison, Figure REF also show the significances obtained from the exact Asimov expected CL$_s$ and ${\\rm CL}_{\\rm disc}$ (dashed lines), detailed below, that are smooth and sensible.",
"Figure: Median (solid lines) and exact Asimov (dashed lines) expected significances obtainedusing eq.",
"()from CL s = CL excl {\\rm CL}_s = {\\rm CL}_{\\rm excl} [from eqs.",
"() and ()] for exclusion (left panel) and CL disc {\\rm CL}_{\\rm disc} [from eqs.",
"() and ()] for discovery (right panel),as a function of the background mean bb for various values of signal mean ss,for a single-channel Poisson counting experiment.",
"Due to the discrete nature of Poisson statistics,the median expected significances suffer from a sawtooth behavior.On the other hand, the exact Asimov expected significances behave sensibly as they decrease monotonically with bb.Therefore, in ref.",
"[50], we proposed instead to use an exact Asimov approach for projecting sensitivities of planned experiments, where the observed number of events $n$ is replaced by its mean expected value $\\langle n_{\\rm excl} \\rangle = b$ for exclusion and $\\langle n_{\\rm disc} \\rangle = s + b$ for discovery.",
"From eqs.",
"(REF ) and (REF ) we thus obtain for the expected exclusion in the case of a single-channel counting experiment with signal and background means $s$ and $b$ : $p_{\\rm excl}^A &=& \\frac{\\Gamma (b+1, s+b)}{\\Gamma (b+1)},\\\\{\\rm CL}_s^A \\,=\\, {\\rm CL}_{\\rm excl}^A&=& \\frac{\\Gamma (b+1, s+b)}{\\Gamma (b+1, b)},$ Similarly, for the expected discovery significance, we obtain from eqs.",
"(REF ) and (REF ): $p_{\\rm disc}^A &=& \\frac{\\gamma (s+b, b)}{\\Gamma (s+b)},\\\\{\\rm CL}_{\\rm disc}^A&=& \\frac{e^s}{(1 + s/b)^{s+b}}.$ Figure REF compares the exact Asimov expected significances obtained from frequentist (dashed lines) and modified frequentist ${\\rm CL}_s$ /Bayesian ${\\rm CL}_{\\rm disc}$ (solid lines) confidence levels, for both exclusion (left panel) and discovery (right panel) cases.",
"Figure: The exact Asimov expected significances obtained from frequentist pp-values (dashed lines) and modified frequentist CL s {\\rm CL}_s/Bayesian CL disc {\\rm CL}_{\\rm disc} confidence levels (solid lines), converted to significances ZZ using eq.",
"(),for a single-channel Poisson counting experiment.",
"Results are presented as functions of the background mean bb for various values of signal mean ss.",
"The term “exact Asimov\" means that we set the number of events equal to the mean expected according to the hypothesis H 1 H_1, so n=bn=b for exclusion and n=s+bn=s+b for discovery.The left panel compares p excl A p_{\\rm excl}^A to CL s A {\\rm CL}_s^A for exclusion, from eqs.",
"() and ().The right panel compares p disc A p_{\\rm disc}^A to CL disc A {\\rm CL}_{\\rm disc}^A for discovery, from eqs.",
"() and ().This illustrates the more general fact that CL$_s$ and ${\\rm CL}_{\\rm disc}$ are more conservative than $p_{\\rm excl}$ and $p_{\\rm disc}$ , respectively.",
"In order to project expected exclusions based on the $p_{\\rm excl}$ or CL$_s$ approaches, we set eq.",
"(REF ) or () equal to the desired $\\alpha = 0.10$ or $0.05$ , and then solve for $s$ .",
"We also consider projections based on the FC method, in two different ways.",
"One is the Feldman-Cousins experimental sensitivity, advocated within ref.",
"[62], that is defined as the arithmetic mean of the upper limits obtained by the FC method at a chosen confidence level$\\endcsname $These upper limits on signal are defined in ref.",
"[62], and are shown as a function of background $b$ for $n = 0, 1, 2$ with solid black lines in Figure REF .",
"$s^{\\rm UL}_{\\rm FC} (n, b)$ in a large number of pseudo-experiments with data generated under background-only hypothesis:An implementation of the Feldman-Cousins method to evaluate the upper limits and the experimental sensitivity, advocated within ref.",
"[62], is made available with the Zstats v2.0 package [67].",
"$\\text{FC sensitivity} &=& \\sum _{n=0}^{\\infty } P(n | b) \\, s^{\\rm UL}_{\\rm FC} (n, b).$ The other way is to simply compute the upper limit on signal given by the FC method with the observed number of events taken to be the nearest integer to the expected background mean $n = {\\rm round}(b)$$\\endcsname $When rounding half-integral values of $b=0.5, 1.5, 2.5, 3.5, \\ldots $ , we follow the IEEE 754 standard of taking the nearest even integer such that $\\text{round}(b) = 0, 2, 2, 4, \\ldots $ .",
".",
"We consider the latter for future reference, as it was alluded to in ref.",
"[66] while projecting exclusion sensitivity for proton decay in $p \\rightarrow \\overline{\\nu } K^+$ channel at DUNE.",
"Figure: The expected 90% CL (left panels) and 95% CL (right panels) upper limit on signalas a function of the background mean, using the exact Asimov modified frequentist CL s {\\rm CL}_s [blue lines, from eq.",
"()] and standard frequentistpp-value [red lines, from eq.",
"()],the Feldman-Cousins experimental sensitivity[green lines, from ref.",
"and eq.",
"()],and the Feldman-Cousins method from ref.",
"with n= round (b)n=\\textrm {round}(b) [black lines].The top and bottom panels show the same information but with logarithmic and linear scales,respectively, for bb.In Figure REF , we compare the expected 90% CL (left panels) and 95% CL (right panels) upper limits on the signal mean $s$ , obtained using the exact Asimov ${\\rm CL}_s$ (blue lines) and $p_{\\rm excl}$ (red lines), FC experimental sensitivity$\\endcsname $In evaluating the FC sensitivity, we used the upper limits obtained by the Feldman-Cousins method for a fixed $n$ before requiring them to be non-increasing as a function of background mean.",
"This does not make much difference as the FC upper limit differs from its non-increasing (with $b$ ) version only when the number of observed events are few compared to the expected background mean $b$ , for which the probability of occurrence is small and will rapidly fall off for even smaller $n$ .",
"(green lines), and FC upper limit with $n={\\rm round}(b)$ (black lines).",
"We note the following from the figure.",
"First, unlike the case with the observed upper limits (i.e.",
"fixed $n$ ), the $p_{\\rm excl}$ method gives sensible positive expected upper limits with the exact Asimov approach for all $b$ , but still is less conservative than the CL$_s$ and FC sensitivity results.",
"Second, the upper limit given by the FC method with $n={\\rm round}(b)$ suffers from a sawtooth problem and is therefore counter-intuitive and flawed as a method of comparing experimental prospects for different scenarios, as it implies that an experiment could become more sensitive if it had larger background.",
"Finally, the FC sensitivity and the upper limits given by exact Asimov ${\\rm CL}_s$ are both sensible as they increase monotonically with $b$ , and are also comparable at small backgrounds.",
"At large backgrounds, however, the FC sensitivity is slightly more conservative.",
"We also note that ${\\rm CL}_s$ upper limits are much easier to evaluate than the FC upper limits.",
"We now turn to the issue of prospects for discovery, using the exact Asimov criterion.",
"The signal mean needed for an expected discovery at a significance $Z$ is given by the solution for $s$ in setting eq.",
"(REF ) for $p_{\\rm disc}$ , or () for ${\\rm CL}_{\\rm disc}$ , equal to $\\frac{1}{2} {\\rm erfc}\\left(\\frac{Z}{\\sqrt{2}}\\right)$ for the desired $Z$ .",
"Figure REF compares the signals $s$ needed for an expected $Z=3$ evidence or $Z=5$ discovery, as a function of background mean $b$ , based on $p_{\\rm disc}^A$ and ${\\rm CL}_{\\rm disc}^A$ .",
"We note that as expected the results from ${\\rm CL}_{\\rm disc}^A$ are more conservative than those obtained from $p_{\\rm disc}^A$ .",
"Figure: The signal needed for an expected Z=3Z=3 evidence (lower curves) or Z=5Z=5 discovery (higher curves), as a function of background mean bb,using the exact Asimov p disc p_{\\rm disc} [red lines, from eq.",
"()]and CL disc {\\rm CL}_{\\rm disc} [blue lines, from eq.",
"()].For very small $b$ , note that for $Z=3$ the $s$ needed in Figure REF is actually less than 1.",
"Here, it is important to note that the discovery statistics $p_{\\rm disc}$ and ${\\rm CL}_{\\rm disc}$ are not well-defined in the strict background-free limit $b \\rightarrow 0$ .",
"Specifically, $p_{\\rm disc} (n, 0) &=&{\\left\\lbrace \\begin{array}{ll}0 \\textrm { if } n \\ne 0\\\\[-5pt]1 \\textrm { if } n = 0,\\end{array}\\right.",
"}\\\\{\\rm CL}_{\\rm disc} (n, 0, s) &=&{\\left\\lbrace \\begin{array}{ll}0 \\textrm { if } n \\ne 0, s \\ne 0\\\\[-5pt]1 \\textrm { otherwise.}\\end{array}\\right.",
"}$ Since $\\langle n_{\\rm disc} \\rangle = s$ for $b=0$ , the above implies that the exact Asimov expected discovery significances are both infinite, $Z(p^{\\rm A}_{\\rm disc}) = Z({\\rm CL}^A_{\\rm disc}) = \\infty $ , for any non-zero $s$ (however small).",
"However, as a practical matter, it is clearly unreasonable to suggest an expectation of a discovery if the mean expected number of signal events is much less than 1.",
"Therefore, in order to be conservative, in cases with an extremely small background we can impose an additional requirement that $P(n \\ge 1)$ should be greater than some fixed value in order to claim an expected discovery.",
"Figure REF shows the probability of observing at least one event, $P(n \\ge 1) = \\sum _{n = 1}^\\infty P(n | s) = 1 - e^{-s},$ as a function of signal mean $s$ .",
"For example, if we require $P(n \\ge 1) > (50\\%, 63.2\\%, 95\\%)$ then the signal mean has to be $s > (0.693,\\, 1.0,\\, 2.996)$ respectively.",
"Requiring $s > \\ln 2 \\approx 0.693$ guarantees the median number of events is at least 1, and $s > 1$ guarantees the expected mean number of events $\\langle n_{\\rm disc} \\rangle > 1$ .",
"Figure: Probability of observing at least one event obtained from eq.",
"(),as a function of the signal mean ss, in the case with no background b=0b=0."
],
[
"Exclusion for multi-channel counting experiments",
"Consider a counting experiment with $N$ independent channels.",
"For each channel $i=1,\\ldots ,N$ , the background and possible signal are assumed to be governed by Poisson distributions with means $b_i$ and $s_i$ .",
"For future convenience, define $s &=& \\sum _{i=1}^N s_i,\\\\r_i &=& s_i/s,$ so that $s$ is the total mean expected signal in all channels, and the $r_i$ are the expected fractions of the total signal events for each channel.",
"Given an observation $\\lbrace n_i\\rbrace $ , the $p$ -value for exclusion isIn the following we use $\\vec{n}$ as the argument of a function to denote the dependence on the full set $\\lbrace n_i\\rbrace $ .",
"This applies similarly for $\\vec{b}$ and $\\vec{s}$ to represent the dependences on $\\lbrace b_i \\rbrace $ and $\\lbrace s_i\\rbrace $ .",
"$p_{\\rm excl}(\\vec{n}, \\vec{b}, \\vec{s})&=&\\sum _{\\lbrace k_i\\rbrace }\\,\\prod _{i=1}^N\\,P (k_i|s_i+b_i),$ where the sums over non-negative integer numbers of events $\\lbrace k_i\\rbrace $ are restricted according to the condition that $Q(\\vec{k}) \\le Q(\\vec{n}).$ where $Q$ is an appropriately chosen test-statistic with the property that larger $Q$ is more signal-like.",
"We can also compute: $p_{\\rm excl}(\\vec{n}, \\vec{b}, 0)&=&\\sum _{\\lbrace k_i\\rbrace }\\,\\prod _{i=1}^N\\,P (k_i|b_i),$ with the same restrictions on $k_i$ as in eq.",
"(REF ).",
"Then we have ${\\rm CL}_s(\\vec{n}, \\vec{b}, \\vec{s}) &=& \\frac{p_{\\rm excl}(\\vec{n}, \\vec{b}, \\vec{s})}{p_{\\rm excl}(\\vec{n}, \\vec{b}, 0)} ,$ which is interpreted as the confidence level in the hypothesis that the signal is present.",
"For the single channel case, the obvious choice for $Q$ is the observed number of events, but in the multi-channel case one can consider different choices for $Q$ .",
"A simple and good choiceThere are other choices, including the profile likelihood ratio, but these are more complicated and end up giving very similar (and often identical) results.",
"of test-statistic $Q$ is the likelihood ratio, $q(\\vec{n}, \\vec{b}, \\vec{s}) &=& \\prod _{i=1}^N \\frac{P(n_i|s_i+b_i)}{P(n_i|b_i)},$ which simplifies to $q &=& \\prod _{i=1}^N e^{-s_i} \\left(1 + \\frac{s_i}{b_i}\\right)^{n_i}.$ It is more convenient to use instead $Q = \\ln (q) = -s + \\sum _{i=1}^N n_i \\ln (1 + s_i/b_i),$ which gives exactly the same results for $p_{\\rm excl}$ and ${\\rm CL}_s$ as $Q=q$ , since $\\ln (q)$ increases monotonically with $q$ .",
"The contribution $-s$ is an irrelevant constant (independent of the data $\\lbrace n_i\\rbrace $ ), so the use of $Q = \\ln (q)$ amounts to taking the sum of the individual $n_i$ 's, but weighting each of the channels by the factor $w_i = \\ln (1 + s_i/b_i).$ This means that, using eq.",
"(REF ) in eq.",
"(REF ), the restriction on the $\\lbrace k_i\\rbrace $ appearing in the sums in eqs.",
"(REF ) and (REF ) becomes: $\\sum _{i=1}^N (n_i - k_i) \\ln (1 + s_i/b_i) &\\ge & 0.$ In contrast, the Bayesian way is to define, as a generalization of eq.",
"(REF ): ${\\rm CL}_{\\rm excl}(\\vec{n},\\vec{b},\\vec{s}) &=&\\frac{\\displaystyle {0.85}{\\int _{{1.4}{s}}^{{1.2}{\\infty }}} ds^{\\prime } \\,{0.85}{\\prod _{{1.2}{i=1}}^{{1.2}{\\hspace{1.9pt}N}}}\\, P(n_i | r_i s^{\\prime }+ b_i)}{\\displaystyle {0.85}{\\int _{{1.2}{0}}^{{1.2}{\\infty }}} ds^{\\prime } \\,{0.85}{\\prod _{{1.2}{i=1}}^{{1.2}{\\hspace{1.9pt}N}}}\\, P(n_i | r_i s^{\\prime }+ b_i)}.$ Unlike in the special case of a single channel, ${\\rm CL}_{\\rm excl}(\\vec{n},\\vec{b},\\vec{s})$ defined in this way is not exactly equal to ${\\rm CL}_s(\\vec{n},\\vec{b},\\vec{s})$ defined by eq.",
"(REF ).",
"Therefore, we will now study some simple test cases to illustrate the differences.",
"First, let us consider what happens when there are two channels, one of which (the “bad\", or non-informative channel) has a much lower signal and higher background than the other (the “good channel\").",
"As a specific numerical case, suppose: $&b_1 = 2,\\qquad &s_1 = 7,\\quad \\qquad \\: n_1 = 2,\\qquad \\qquad \\:\\:\\mbox{(good channel)},\\\\&b_2 = 10,\\qquad &s_2 = 0.01,\\qquad n_2 = \\text{varying},\\qquad \\mbox{(bad channel)}.$ In this case, because the bad channel 2 has a tiny expected signal $s_2$ and a large background $b_2$ , one intuitively expects it to provide essentially no information about the correctness of the signal hypothesis, no matter what $n_2$ is observed.",
"Considering only the good channel 1, we obtain $p_{\\rm excl} &=& 0.006232, \\qquad Z_{\\rm excl} = 2.4987\\qquad \\mbox{(channel 1 alone)},\\\\{\\rm CL}_s \\:=\\: {\\rm CL}_{\\rm excl}&=& 0.009210,\\qquad Z_{\\rm excl} =2.3571,\\qquad \\mbox{(channel 1 alone)}.$ However, combining both channels using the formulas (REF ), (REF ), and (REF ) above, we have the results shown in the left panel of Figure REF .",
"Figure: Comparison of exclusion significances ZZ in the case of a counting experiment with a good channel and a bad channel.",
"The solid lines are the modified frequentist CL s {\\rm CL}_s [solid red line, from eqs.",
"()-() and ()] and CL excl {\\rm CL}_{\\rm excl} [solid blue line, from eq. ()].",
"In this example, CL excl {\\rm CL}_{\\rm excl} is visually indistinguishable from the result obtained from channel 1 only, conforming with the fact that channel 2 contains essentially no information about the signal.",
"Also shown are the results for p excl p_{\\rm excl} obtained from considering channel 1 only [dashed blue line, from eq.",
"()] and from both channels [dashed red line, from eqs.",
"() and ()].Counterintuitively, adding another channel with a larger background and almost no expected signal has increased our confidence in the exclusion as measured by either the frequentist $p_{\\rm excl}$ or the modified frequentist ${\\rm CL}_s$ measures, when $n_2$ is small.",
"In contrast, ${\\rm CL}_{\\rm excl}$ behaves as intuitively expected; the result obtained including both channels is numerically almost independent of $n_2$ and almost identical to the result obtained only from channel 1.",
"To understand the origin of this counterintuitive effect for $p_{\\rm excl}$ and CL$_{s}$ , let us consider which integers $k_1, k_2$ contribute to the sums in eqs.",
"(REF ) and (REF ).",
"In general, $k_1=0$ and 1 each contribute for a very large range of $k_2$ , so that very nearly we have a factor $\\sum _{k_2 = 0}^\\infty P(k_2|s_2 + b_2) \\approx 1$ for channel 2 in eq.",
"(REF ).",
"However, for $k_1 = n_1 = 2$ , we only get a factor of $\\sum _{k_2 = 0}^{n_2} P(k_2|s_2 + b_2) < 1$ contributing to the $p$ -values.",
"The problem boils down to this fact: for the contributions with $k_1 = n_1$ , only a subset of the $k_2$ values contribute, even though any result for $k_2$ should give us essentially no information about the presence of the (tiny) signal.",
"This explains why the counterintuitive problem disappears for reasonably large $n_2$ , where we see from the left panel of Figure REF that ${\\rm CL}_{\\rm excl}\\approx {\\rm CL}_s$ and $p_{\\rm excl}$ agree with their counterparts from channel 1 only.",
"To show another facet of this disturbing effect, in the right panel of Figure REF we use the same data except that $n_2=0$ is fixed and $b_2$ is varying.",
"Again, we see that despite channel 2 containing essentially no information about the signal, the modified frequentist ${\\rm CL}_s$ including both channels depends on $b_2$ , while ${\\rm CL}_{\\rm excl}$ is almost exactly flat, conforming to intuitive expectation.",
"Another study case is shown in the first panel of Figure REF , with: $&&n_1 = 1,\\qquad b_1 = 1,\\qquad s_1 = 4,\\\\&&n_2 = 1,\\qquad b_2 = 1,\\qquad s_2 = {\\rm varying}.$ The variation of exclusion significances as a function of $s_2$ is shown in the first panel of Figure REF .",
"For $s_2 = s_1 = 4$ exactly, the results satisfy ${\\rm CL}_s = {\\rm CL}_{\\rm excl}$ and agree precisely with the result that would be obtained for a single combined channel with $n=2$ , $b=2$ , $s=8$ .",
"However, the $Z$ value for CL$_s$ has a small discontinuity at exactly $s_2 = 4$ , such that for all other values of $s_2$ , ${\\rm CL}_s$ has a higher exclusion significance $Z$ than ${\\rm CL}_{\\rm excl}$ .",
"Numerically: ${\\rm CL}_s &=& 0.004093,\\quad (Z = 2.644),\\qquad \\mbox{(for $s_2 = 4$)}\\\\{\\rm CL}_s &=& 0.003616,\\quad (Z = 2.686),\\qquad \\mbox{(for $s_2 = 4 \\pm \\epsilon $)},$ for $\\epsilon $ arbitrarily small but non-zero.",
"Figure: Comparison of exclusion significances ZZ obtained from CL s {\\rm CL}_s [red line, from eqs.",
"()-() and ()]and CL excl {\\rm CL}_{\\rm excl} [blue line, from eq.",
"()]and p excl p_{\\rm excl} [green line, from eqs.",
"() and ()],for the test cases of eqs.",
"()-() [left panel] and ()-() [right panel].The results for p excl p_{\\rm excl} and CL s _s exhibit discontinuities as s 2 s_2 is varied, due to abrupt changes in which outcomes k 1 k_1 and k 2 k_2 are summed over.",
"TheBayesian version CL excl {\\rm CL}_{\\rm excl} does not have such discontinuities.This discontinuity can be traced to the fact that for $s_2=4$ exactly, the weights satisfy $w_1 = w_2$ exactly for the two channels, which affects which integers are summed over due to eq.",
"(REF ).",
"There are also discontinuities in CL$_s$ at $s_2=\\sqrt{5}-1 \\approx 1.23607$ , where $w_1 = 2 w_2$ , and at $s_2 = 5^{1/3} - 1 \\approx 0.709976$ , where $w_1 = 3 w_2$ , etc.",
"For another case study, consider: $&&n_1 = 2,\\qquad b_1 = 2.4,\\qquad s_1 = 8.5,\\\\&&n_2 = 4,\\qquad b_2 = 2.3,\\qquad s_2 = {\\rm varying}.$ The results are depicted in the second panel of Figure REF , and show more pronounced discontinuities in both frequentist $p_{\\rm excl}$ and the modified frequentist ${\\rm CL}_s$ .",
"In contrast, the Bayesian result ${\\rm CL}_{\\rm excl}$ is smooth as we vary $s_2$ , and gives more conservative exclusion significances.",
"Let us now consider the question of projecting expected exclusion significances for future experiments.",
"In the multi-channel case, one can define Asimov results for $p_{\\rm excl}$ and ${\\rm CL}_s$ by replacing each $n_i$ in eqs.",
"(REF ) and (REF ) by the mean expected result $b_i$ in the restriction eq.",
"(REF ).",
"However, in the multi-channel case, the resulting sets of $\\lbrace k_i\\rbrace $ that contribute to the sums will depend discontinuously on the $\\lbrace s_i\\rbrace $ and $\\lbrace b_i\\rbrace $ , leading to the same sort of sawtooth problems that occurs in the median expected significance.",
"In particular, an increase in the backgrounds often leads, counterintuitively, to a larger expected significance.",
"(This problem did not occur in the single-channel case, because the sum $\\sum _{k=0}^n$ was evaluated in closed form in terms of incomplete $\\Gamma $ functions, after which the argument $n$ could be interpreted as a continuous real number rather than an integer.)",
"In contrast, if one uses ${\\rm CL}_{\\rm excl}(\\vec{n},\\vec{b},\\vec{s})$ , then the exact Asimov method is perfectly straightforward and continuous, since it does not involve sums over integers subject to restrictions.",
"Thus one can simply replace $n_i$ by $b_i$ in eq.",
"(REF ) to obtain the exact Asimov result.",
"The Asimov results for $p_{\\rm excl}$ , ${\\rm CL}_s$ , and ${\\rm CL}_{\\rm excl}$ are compared in Figure REF for two test cases, showing the sawtooth behavior of the first two and the smooth, monotonic (and more conservative) behavior of the latter.",
"Figure: Comparison of the Asimov expected exclusion significances ZZ obtained fromp excl p_{\\rm excl} [red lines, from eq.",
"()], CL s {\\rm CL}_s [blue lines, from eq.",
"()],and CL excl {\\rm CL}_{\\rm excl} [black lines, from eq.",
"()],for two test cases with two independent channels, as labeled.",
"The Asimov results are obtained by setting n i =b i n_i = b_i for each channel.Due to the non-continuous effect of the restriction of eq.",
"(), the Asimov p excl p_{\\rm excl} and CL s {\\rm CL}_s have a counterintuitive non-monotonic behavior as the first channel background mean b 1 b_1 is varied, while the Asimov CL excl {\\rm CL}_{\\rm excl} is monotonic in the expected way.In view of the preceding discussion, we propose ${\\rm CL}_{\\rm excl}$ in eq.",
"(REF ) as the preferred statistic for exclusion for multi-channel counting experiments.",
"Unlike $p_{\\rm excl}$ and CL$_s$ (with which it coincides in the single-channel case), it does not suffer from the problem of being affected significantly by the presence of a bad channel, and does not have discontinuities when signal and background means are changed infinitesimally.",
"The exact Asimov result is straightforward to obtain and behaves continuously and monotonically in the expected way with respect to changes in the background.",
"Furthermore, the introduction of background uncertainties and probability distributions for nuisance parameters is more straightforward, avoiding discontinuities in the integrand, as we will see below."
],
[
"Discovery for multi-channel counting experiments",
"For the discovery case, the frequentist $p$ -value is defined by $p_{\\rm disc}(\\vec{n}, \\vec{b}, \\vec{s}) &=& \\sum _{\\lbrace k_i\\rbrace } \\prod _{i=1}^N P(k_i| b_i).$ The sum over $\\lbrace k_i\\rbrace $ is restricted by the condition that the test-statistic $\\ln (q)$ defined by eq.",
"(REF ) is not smaller for $\\lbrace k_i\\rbrace $ than for the observed data $\\lbrace n_i\\rbrace $ , so: $\\sum _{i=1}^N (n_i - k_i) \\ln (1 + s_i/b_i) &\\le & 0.$ Unlike the single-channel special case, $p_{\\rm disc}$ depends on the signal strengths $s_i$ when there is more than one channel because of this restriction.",
"Note that the inequality has the opposite sense compared to the exclusion case, eq.",
"(REF ).",
"A more conservative, and simpler, alternative to $p_{\\rm disc}(\\vec{n}, \\vec{b}, \\vec{s})$ is the generalization of eq.",
"(REF ), ${\\rm CL}_{\\rm disc}(\\vec{n}, \\vec{b}, \\vec{s})&=&\\prod _{i=1}^N\\frac{ P(n_i|b_i)}{P(n_i| s_i + b_i)}.$ In order to compare these criteria for discovery, we first consider a case with one good channel and one bad channel, starting from the following values: $&b_1 = 2,\\qquad &s_1 = 9.5,\\qquad \\:\\: n_1 = 10,\\qquad \\mbox{(good channel)},\\\\&b_2 = 10,\\qquad &s_2 = 0.01,\\qquad n_2 = 10,\\qquad \\mbox{(bad channel)}.$ In Figure REF , we show the results for the discovery significance $Z$ obtained from $p_{\\rm disc}$ and ${\\rm CL}_{\\rm disc}$ , considering variations in both $n_2$ and $b_2$ as the other quantities are held fixed, and compare to the same results using only channel 1.",
"As in the exclusion case, we note that $p_{\\rm disc}$ is affected in a non-trivial way by the presence of the bad channel, contrary to intuitive expectations.",
"Figure: Comparison of discovery significance ZZ in the case of a counting experiment with a good channel and a bad channel.",
"The solid lines are obtained fromp disc p_{\\rm disc} [red lines, from eqs.",
"() and ()] and CL disc {\\rm CL}_{\\rm disc} [blue lines, from eq. ()].",
"The dashed lines are obtained in the same way, but considering only the data from channel 1.",
"In this example, CL disc {\\rm CL}_{\\rm disc} is more resistant to the effects of the non-informative channel, except in the case that b 2 b_2 is very small.The step function discontinuities in p disc p_{\\rm disc} in the right panelare not numerical artifacts, butoccur at values of b 2 b_2 such that the ratio of weights w 1 /w 2 =ln(1+s 1 /b 1 )/ln(1+s 2 /b 2 )w_1/w_2 = \\ln (1+s_1/b_1)/\\ln (1 + s_2/b_2) is rational.The step function discontinuities in $p_{\\rm disc}$ are not a numerical artifact, but occur at values of $b_2$ such that the ratio of weights $w_1/w_2 = \\ln (1+s_1/b_1)/\\ln (1 + s_2/b_2)$ is a rational number, so that the integer number of terms appearing in the $\\sum _{\\lbrace k_i\\rbrace }$ in eq.",
"(REF ) changes discontinuously.",
"In contrast, ${\\rm CL}_{\\rm disc}$ is seen to be much less affected by the presence of the bad channel.",
"The reason for this is that for any channel $i$ with very small $s_i$ , the numerator and denominator factors for that channel will cancel in the limit $s_i/b_i \\rightarrow 0$ in eq.",
"(REF ).",
"The exception (in the right panel of Figure REF ) occurs in the case that $b_2$ is also small, in which case $n_2=10$ is a surprising outcome for both the background-only and background+signal hypotheses.",
"Further comparisons between the significances obtained from $p_{\\rm disc}$ and ${\\rm CL}_{\\rm disc}$ for two test cases are shown in Figure REF .",
"Figure: Comparison of significances ZZ for discovery, obtained using p disc p_{\\rm disc} [red lines, from eqs.",
"() and ()] and CL disc {\\rm CL}_{\\rm disc} [blue lines, from eq.",
"()], for two 2-channel test cases with data as labeled.The results obtained from $p_{\\rm disc}$ have numerous discontinuities, which are small numerically but have the disturbing property of being non-monotonic as the background $b_2$ is varied.",
"The results from ${\\rm CL}_{\\rm disc}$ are reliably more conservative, as we have already noted, and do not suffer discontinuities because there is no restricted sum over integers in its definition.",
"For the purpose of projecting discovery prospects in future experiments, one can again define the Asimov values of $p_{\\rm disc}$ and ${\\rm CL}_{\\rm disc}$ by replacing $n_i$ with $b_i + s_i$ in eqs.",
"(REF ) and (REF ) respectively.",
"These are compared for two test cases in Figure REF .",
"In the case of $p_{\\rm disc}$ , the constraint put on the sum by eq.",
"(REF ) leads to a non-monotonic sawtooth behavior, although much less pronounced than in the exclusion case in Figure REF .",
"Figure: Comparison of the Asimov expected discovery significances ZZ obtained fromp disc p_{\\rm disc} [red lines, from eqs.",
"() and ()] and CL disc {\\rm CL}_{\\rm disc} [blue lines, from eq.",
"()],for two test cases with two independent channels, as labeled.",
"The Asimov results are obtained by setting n i =b i +s i n_i = b_i + s_i for each channel.Due to the non-continuous effect of the restriction of eq.",
"(), the Asimov p disc p_{\\rm disc} has a counterintuitive non-monotonic behavior as the first channel background mean b 1 b_1 is varied, while the Asimov CL disc {\\rm CL}_{\\rm disc} is monotonic in the expected way, and more conservative.For the reasons just discussed, and because of the ease of generalization to the case of background uncertainties as discussed in the next section, we propose to use ${\\rm CL}_{\\rm disc}$ as the figure of merit for the significance of a possible discovery, and for projecting the discovery reach of future experiments."
],
[
"Background uncertainty and other nuisance parameters",
"In the real world, the background level is never perfectly known.",
"Furthermore, the background and signal may depend on other nuisance parameter(s), to be called $\\nu $ below.",
"These can be dealt with in a Bayesian approach by assuming probability densities $f(b)$ and $g(\\nu )$ , subject to the normalization conditions $\\int _0^\\infty db\\, f(b) = 1$ and $\\int d\\nu \\, g(\\nu )= 1$ .",
"For example, following [50], we can model the background uncertainty in terms of an on-off problem [68], [69], [70], [71], [72], [73], where $m$ is the number of Poisson events in a signal-off (background-only) region, and the ratio of background means in the signal-off and signal-on regions is called $\\tau $ .",
"In terms of $m$ and $\\tau $ , the point estimate for the background and its variance are $\\hat{b} = m/\\tau , \\qquad \\Delta _{b} = \\sqrt{m}/\\tau ,$ or equivalently $\\tau = \\hat{b}/\\Delta _b^2,\\qquad m = \\hat{b}^2/\\Delta _b^2,$ so that the probability density of $b$ is $f(b) &=& f(b \\hspace{1.0pt}|\\hspace{1.0pt} \\hat{b}, \\Delta _b) \\:=\\: \\tau ^{m+1} b^m e^{-\\tau b}/m!,$ the posterior probability distribution for $b$ obtained by using Bayes' theorem with Poisson likelihood for background in the signal-off region $P(m | \\tau b)$ and flat prior for $b$ .",
"Note that this probability distribution can be used as a model even in situations where the estimates of the background and its uncertainty come partly or completely from theory rather than some signal-off region data.",
"In the case of eq.",
"(REF ), the probability for observing $n$ events in the signal-on region is obtained by averaging over $b$ [74], [71], [75], [72], [73] to obtain $\\Delta P (n,\\hat{b}, \\Delta _b, s) &=& \\int _0^\\infty db \\, f(b | \\hat{b}, \\Delta _b) P(n | s + b),$ We can then extend the definitions of frequentist $p$ -values and to the uncertain background case by simply replacing the Poisson probability $P(n | s + b)$ with $\\Delta P(n, \\hat{b}, \\Delta _b, s)$ [51]: $p_{\\rm excl}(n,\\hat{b}, \\Delta _b,s) &=& \\sum _{k=0}^n \\Delta P(k, \\hat{b}, \\Delta _b, s).\\\\p_{\\rm disc}(n, \\hat{b}, \\Delta _b) &=& \\sum _{k=n}^\\infty \\Delta P(k, \\hat{b}, \\Delta _b, 0),$ Explicit formulas for $\\Delta P(n, \\hat{b}, \\Delta _b, s)$ , $p_{\\rm excl}(n,\\hat{b}, \\Delta _b,s)$ , and $p_{\\rm disc}(n,\\hat{b}, \\Delta _b)$ can be found in eqs.",
"(12)-(15) of ref. [50].",
"Besides these, we note the simple formula: $p_{\\rm excl}(n,\\hat{b}, \\Delta _b,0) &=& \\frac{B(1/(1 + \\Delta _b^2/\\hat{b}), m+1, n+1)}{B(m+1, n+1)}.$ Similarly, the confidence levels discussed in the previous sections can be obtained in the uncertain background case as ${\\rm CL}_s(n,\\hat{b}, \\Delta _b,s) &=& \\frac{p_{\\rm excl}(n, \\hat{b}, \\Delta _b, s)}{p_{\\rm excl}(n, \\hat{b}, \\Delta _b, 0)},\\\\{\\rm CL}_{\\rm excl}(n,\\hat{b}, \\Delta _b,s) = \\frac{\\displaystyle {0.85}{\\int _{{1.4}{s}}^{{1.2}{\\infty }}} ds^{\\prime }\\, \\Delta P(n, \\hat{b}, \\Delta _b, s^{\\prime })}{\\displaystyle {0.85}{\\int _{{1.2}{0}}^{{1.2}{\\infty }}} ds^{\\prime }\\, \\Delta P(n, \\hat{b}, \\Delta _b, s^{\\prime })}&=& \\frac{p_{\\rm excl}(n, \\hat{b}, \\Delta _b, s)}{p_{\\rm excl}(n, \\hat{b}, \\Delta _b, 0)},\\\\{\\rm CL}_{\\rm disc}(n, \\hat{b}, \\Delta _b,s) &=& \\frac{\\Delta P(n,\\hat{b}, \\Delta _b,0)}{\\Delta P(n,\\hat{b}, \\Delta _b,s)}.$ Note that we retain the property ${\\rm CL}_{\\rm excl}= {\\rm CL}_s$ in the single-channel case with non-zero background uncertainty.",
"The exact Asimov expectations for $p_{\\rm excl}$ , ${\\rm CL}_s = {\\rm CL}_{\\rm excl}$ and $p_{\\rm disc}$ , ${\\rm CL}_{\\rm disc}$ in the uncertain background case are obtained by replacing $n$ in the preceding equations by its expected mean in each case: $\\langle n_{\\rm excl} \\rangle &=& \\sum _{n=0}^\\infty n\\, \\Delta P(n,\\hat{b}, \\Delta _b,0)\\:=\\:\\hat{b} + \\Delta ^2_{b}/\\hat{b},\\\\\\langle n_{\\rm disc} \\rangle &=& \\sum _{n=0}^\\infty n\\, \\Delta P(n,\\hat{b}, \\Delta _b,s)\\:=\\:s + \\hat{b} + \\Delta ^2_{b}/\\hat{b},$ for exclusion and discovery, respectively.",
"More generally, for any probability distributions $f(\\vec{b})$ and $g(\\nu )$ for the background and other nuisance parameters, one can marginalize (integrate) over $b_i$ and $\\nu $ .",
"In the case of exclusion, eq.",
"(REF ) generalizes to $p_{\\rm excl} &=&\\int d \\nu \\, g(\\nu )\\int d\\vec{b}\\, f(\\vec{b})\\:\\sum _{\\lbrace k_i\\rbrace }\\,\\prod _{i=1}^N\\,P (k_i|s_i+b_i),$ and similarly for eq.",
"(REF ), which then gives CL$_s$ .",
"However, note that the sum $\\sum _{\\lbrace k_i\\rbrace }$ is subject to the restriction eq.",
"(REF ), so that even the numbers of terms in the sum depends in a discontinuous way on $\\nu $ and $b_i$ as we integrate over them in the multichannel case.",
"Ref.",
"[76] contains a discussion of various ways to account for the uncertainties in the background and nuisance parameters in the frequentist methods.",
"As argued above, we prefer instead to generalize eq.",
"(REF ), resulting in: ${\\rm CL}_{\\rm excl}&=&\\frac{1}{D} \\int d \\nu \\, g(\\nu )\\: \\int d\\vec{b}\\, f(\\vec{b})\\,\\int _{s}^\\infty ds^{\\prime } \\,\\prod _{i=1}^N P(n_i | r_i s^{\\prime } + b_i).$ Here we have used a short-hand notation to be used several times below, such that the normalization factor $D$ is equal to the expression that follows it with $s=0$ .",
"Similarly, in the case of discovery in the presence of background uncertainties and nuisance parameters, we can generalize eq.",
"(REF ) to obtain $p_{\\rm disc} &=&\\int d \\nu \\, g(\\nu )\\int d\\vec{b}\\, f(\\vec{b})\\:\\sum _{\\lbrace k_i\\rbrace }\\,\\prod _{i=1}^N\\,P (k_i|b_i),$ this time subject to the constraint eq.",
"(REF ) on the terms in the sum.",
"However, as argued above, we prefer to use the more conservative ${\\rm CL}_{\\rm disc}&=&\\frac{\\displaystyle \\int d \\nu \\, g(\\nu ) \\int d\\vec{b}\\, f(\\vec{b}) \\: \\prod _{i=1}^N P(n_i|b_i)}{\\displaystyle \\int d \\nu \\, g(\\nu ) \\int d\\vec{b}\\, f(\\vec{b}) \\: \\prod _{i=1}^N P(n_i| s_i + b_i)}.$ To obtain the Asimov results, one can substitute in the mean expected values for $n_i$ , namely $\\langle n_{i, \\rm excl} \\rangle &=&\\int d\\nu \\, g(\\nu ) \\int _0^\\infty d\\vec{b}\\, f(\\vec{b})\\, \\sum _{n_i=0}^\\infty n_i P(n_i|b_i),\\\\\\langle n_{i, \\rm disc} \\rangle &=&\\int d\\nu \\, g(\\nu ) \\int _0^\\infty d\\vec{b}\\, f(\\vec{b})\\, \\sum _{n_i=0}^\\infty n_i P(n_i|s_i+b_i).$"
],
[
"Application to proton decay",
"In this section, we will first consider the application of the Bayesian statistic ${\\rm CL}_{\\rm excl}$ to estimate the current lower limits on proton partial lifetimes in $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ modes, based on Super-Kamiokande's data, at various confidence levels generalizing the 90% CL published limits.",
"We will then consider the prospects for exclusion or discovery of these proton decay modes for several planned future neutrino experiments: DUNE [46], JUNO [47], Hyper-Kamiokande [48], and THEIA [49].",
"We do this by applying the Bayesian approach of using ${\\rm CL}_{\\rm excl}$ and ${\\rm CL}_{\\rm disc}$ with the exact Asimov criterion of replacing the observed counts by their respective expected means.",
"As discussed above, the Bayesian approaches ${\\rm CL}_{\\rm excl}$ for exclusion and ${\\rm CL}_{\\rm disc}$ for discovery are ideal methods to obtain these limits and projections, as they: 1) guard against claiming exclusion (or discovery) when an experiment is actually not sensitive to the signal model, and therefore are more conservative than the frequentist $p_{\\rm excl}$ and $p_{\\rm disc}$ ; 2) are well-behaved in multi-channel counting experiments in the sense that, unlike the (modified) frequentist approach, ${\\rm CL}_{\\rm excl}$ and ${\\rm CL}_{\\rm disc}$ are not overly affected by the presence of non-informative channels and do not have any discontinuities as the signal and background means are varied; 3) are easily able to include uncertainties in the backgrounds and the signal selection efficiencies, especially for multi-channel counting experiments.",
"The estimates for the backgrounds and the signal selection efficiencies in a specific proton decay mode have been obtained by the DUNE, JUNO, and THEIA collaborations by modeling the experiments as single-channel counting experiments, whereas Hyper-Kamiokande searches for proton decay are modeled as multi-channel counting experiments based on the signal regions and search strategies used at Super-Kamiokande.",
"Before we present our results, we first review the methods we employ to obtain the limits/projections for proton partial lifetimes at single-channel and multi-channel counting experiments, based on the methods elucidated in section .",
"The number of decays in a specific decay channel at an experiment with $N_0$ initial number of protons for a runtime of $\\Delta t$ is given by $\\Delta N = N_0 \\Gamma \\Delta t,$ where the proton partial width $\\Gamma $ is extremely small.",
"(More generally, $\\Delta N = N_0 (1 - e^{-\\Gamma \\Delta t})$ .)",
"Therefore the signal can be computed as $s = \\epsilon ( \\Delta N ) = \\epsilon N_0 \\Gamma \\Delta t,$ where $0 \\le \\epsilon \\le 1$ is the signal selection efficiency.",
"In terms of the number of protons per kiloton of detector material $N_p$ and the exposure $\\lambda $ ($=$ runtime $\\times $ number of kilotons of detector material) of the experiment in units of kiloton-years, we can reexpress eq.",
"(REF ) as $s = \\Gamma N_p \\epsilon \\lambda .$ The present exclusion limit at confidence level $1-\\alpha $ for the proton partial lifetime is then provided by [77] $\\tau _p = 1/\\Gamma = N_p \\epsilon \\lambda /s,$ where $s$ is the number of signal events that gives ${\\rm CL}_{\\rm excl}$ equal to $\\alpha $ .",
"For a future experiment, the exclusion reach for the proton partial lifetime at confidence level $1-\\alpha $ is given by the same formula eq.",
"(REF ), where $s$ is now the signal that makes the exact Asimov ${\\rm CL}^A_{\\rm excl}$ equal to $\\alpha $ .",
"The discovery reach for a given significance $Z$ is likewise obtained from eq.",
"(REF ) using the $s$ that provides for ${\\rm CL}^A_{\\rm disc} = \\frac{1}{2}{\\rm erfc}(Z/\\sqrt{2})$ .",
"Eq.",
"(REF ) holds for an experiment with a single search channel with known background $b$ and signal selection efficiency $\\epsilon $ .",
"For the more general case of an experiment with one or more independent search channels with possibly uncertain backgrounds and signal efficiencies, we employ a Bayesian approach to obtain the limit/reach for proton partial lifetime, as discussed above.",
"First, for the exclusion case, given the number of observed events $n_i$ in each search channel labeled $i$ , the upper limit on proton partial width at a confidence level $1-\\alpha $ is obtained by solving for $\\Gamma $ in (see eq.",
"(REF ), and ref.",
"[78]): $\\alpha &=&\\frac{1}{D}\\int _\\Gamma ^\\infty d\\Gamma ^\\prime \\, \\prod _{i=1}^N \\, \\int _0^1d\\epsilon _{i} \\,g(\\epsilon _i)\\,\\int _0^\\infty db_i \\,f(b_i)\\,P(n_i | s_i^{\\prime } + b_i).$ Here, $D$ is a normalization factor, defined to equal the expression that follows it evaluated at $\\Gamma =0$ , and in each search channel labeled by $i$ , the signal rate is $s_i^{\\prime } = N_p \\epsilon _{i} \\lambda _i \\Gamma ^\\prime ,$ and $g(\\epsilon _i)$ and $f(b_i)$ are the probability distributions for the signal efficiency $\\epsilon _i$ and the background $b_i$ .",
"These distributions can take different forms to parameterize our lack of perfect knowledge of the efficiency and background, such that $\\int _0^1 d\\epsilon _{i} \\,g(\\epsilon _i)= 1$ and $\\int _0^\\infty db_i \\,f(b_i)= 1$ .",
"For example, the probability distribution of true signal selection efficiency $\\epsilon _{i}$ might be taken to be a truncated Gaussian distribution with central value $\\hat{\\epsilon }_i$ and variance $\\Delta _{\\epsilon _i}$ , as in the Super-Kamiokande search analyses in refs.",
"[44], [45]: $g(\\epsilon _{i} | \\hat{\\epsilon }_{i}, \\Delta _{\\epsilon _{i}}) &=& \\sqrt{\\frac{2}{\\pi }}\\frac{\\exp \\left[{- \\frac{(\\epsilon _{i} - \\hat{\\epsilon }_{i})^2}{2 \\Delta ^2_{\\epsilon _{i}}}}\\right]}{\\Delta _{\\epsilon _{i}}\\left({\\rm erf} \\left(\\frac{1 - \\hat{\\epsilon }_{i}}{\\sqrt{2} \\Delta _{\\epsilon _{i}}} \\right)+{\\rm erf} \\left(\\frac{\\hat{\\epsilon }_{i}}{\\sqrt{2} \\Delta _{\\epsilon _{i}}} \\right)\\right)}.$ The probability distribution of true background $b_i$ in the $i^{\\rm th}$ search channel $f(b_i | \\hat{b}_i, \\Delta _{b_i})$ can be taken to be given by eq.",
"(REF ) as in the on-off problem, in terms of quantities $m_i$ and $\\tau _i$ , related to the central value $\\hat{b}_i$ and variance $\\Delta _{b_i}$ by eq.",
"(REF ).",
"Eq.",
"(REF ) assumes that the search channels are independent.",
"If the background and the signal selection efficiencies are perfectly known, i.e.",
"$f(b_i | \\hat{b}_i, \\Delta _{b_i}) = \\delta (b_i - \\hat{b}_i)$ and $g(\\epsilon _i | \\hat{\\epsilon }_i, \\Delta _{\\epsilon _i}) = \\delta (\\epsilon _i - \\hat{\\epsilon }_i)$ , then we get $\\alpha &=&\\frac{1}{D} \\int _{\\Gamma }^\\infty \\,d\\Gamma ^{\\prime } \\, \\prod _i P(n_i| s_i^{\\prime } + b_i),$ with $s_i^{\\prime } = N_p \\epsilon _i \\lambda _i \\Gamma ^{\\prime }$ after identifying $\\hat{b}_i = b_i$ and $\\hat{\\epsilon }_i = \\epsilon _i$ .",
"This corresponds to eq.",
"(REF ).",
"Specializing further to a single search channel (dropping the subscript $i$ ), this reduces to eq.",
"(REF ) with $s^\\prime = N_p \\epsilon \\lambda \\Gamma ^\\prime $ .",
"For projecting the exclusion reach for partial lifetime at future experiments, we make use of the exact Asimov method by replacing the number of events $n_i$ in each search channel by their respective expected means, $\\langle b_i \\rangle &=& \\int _0^\\infty db_i\\,f(b_i)\\, b_i ,$ for example $\\langle b_i \\rangle = (m_i + 1)/\\tau _i =\\hat{b}_i + \\Delta _{b_i}^2/b_i$ if the on-off problem treatment is used for the background.",
"The expected confidence level $1-\\alpha $ upper limit on partial width $\\Gamma $ is then solved from eq.",
"(REF ) with $n_i$ replaced by $\\langle b_i \\rangle $ : $\\alpha &=&\\frac{1}{D}\\int _\\Gamma ^\\infty d\\Gamma ^\\prime \\, \\prod _i \\, \\int _0^1d\\epsilon _{i} \\,g(\\epsilon _i)\\,\\int _0^\\infty db_i \\,f(b_i)\\,P(\\langle b_i \\rangle | s_i^{\\prime } + b_i).$ Equation (REF ) gives the Asimov expected lower limit on the partial lifetime via $\\tau _p = 1/\\Gamma $ .",
"For the expected discovery reach for proton partial widths at future experiments, we use a method based on the exact Asimov evaluation of the statistic ${\\rm CL}_{\\rm disc}$ .",
"In particular, we solve for $\\Gamma $ from (see eq.",
"(REF )) $\\frac{1}{2} {\\rm erfc}\\left(\\frac{Z}{\\sqrt{2}}\\right)=\\frac{\\displaystyle \\prod _i \\, \\int _0^\\infty db_i \\,f(b_i)\\, P(\\langle n_i \\rangle | b_i)}{\\displaystyle \\prod _i \\, \\int _0^1 d\\epsilon _i \\,g(\\epsilon _i)\\, \\int _0^\\infty db_i \\,f(b_i)\\, P(\\langle n_i \\rangle | s_i + b_i)},$ where $s_i = N_p \\lambda _i \\epsilon _i \\Gamma $ and $\\langle n_i \\rangle = \\langle s_i \\rangle + \\langle b_i \\rangle $ , with $\\langle b_i \\rangle $ as given in eq.",
"(REF ), and $\\langle s_i \\rangle &=&\\Gamma N_p \\lambda _i \\, \\int _0^1 d\\epsilon _{i} \\,g(\\epsilon _i)\\, \\epsilon _{i}.$ This gives the expected discovery reach for partial lifetime using $\\tau _p = 1/\\Gamma $ corresponding to a chosen significance $Z$ .",
"Table: Super-Kamiokande's data for p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ andp→e + π 0 p \\rightarrow e^+ \\pi ^0 decay modes, taken from refs.",
"and, respectively.",
"In each decay mode, the exposures λ i \\lambda _i inkton-years, total backgrounds b ^ i ±Δ b i \\hat{b}_i \\pm \\Delta _{b_i}, signal efficienciesϵ ^ i ±Δ ϵ i \\hat{\\epsilon }_i \\pm \\Delta _{\\epsilon _i}, and the observed number of counts n i n_i are listed.〈s i 90CL 〉\\langle s_i^{90\\text{CL}} \\rangle are the expected signal events, defined in eq.",
"(),for proton partial lifetime set equal to its 90% CL lower limit.The last column gives a brief description of each of the channels referringto the detector period (SK I-IV) and the name of the search method used in refs.",
", .Based on Super-Kamiokande's data, taken from refs.",
"[44], [45], which we quote for completeness in Table REF , we now compute the upper limit on proton partial widths in the $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay modes that are excluded at various confidence levels (e.g.",
"95%, 90%, 68%, 50% CL) using eq.",
"(REF ), which can then be translated into corresponding lower limits on the proton partial lifetime.",
"Super-Kamiokande uses a water Cerenkov detector with a fiducial mass of 22.5 ktons, and the analysis for $p \\rightarrow e^+ \\pi ^0$ in ref.",
"[45] also includes data from an enlarged fiducial mass of 27.2 ktons.",
"While Super-Kamiokande can probe for proton decay in both $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay modes, it is less sensitive to the former decay mode, because the $K^+$ is produced below its Cerenkov threshold in water and is only identified from its decay constituents.",
"Figure REF shows our own$\\endcsname $Besides using the probability distribution for true background as in the on-off problem (eq.",
"(REF )), we have considered various other distributions such as a Gaussian, and a convolution of Gaussian and Poisson (only for search channels with extremely low backgrounds) as done in refs.",
"[44], [45], but there was no noticeable change in our results.",
"In Super-Kamiokande's analysis for $p \\rightarrow \\overline{\\nu } K^+$ decay mode in ref.",
"[44], the search channels with large backgrounds that are referred to as “$p_\\mu $ spectrum\" in Table REF were further divided into sub-channels, but due to insufficient data made available, we are not able to include that subdivision.",
"computed estimates of the current$\\endcsname $In this paper, we only consider the limits from data published in journal articles.",
"In the case of $p\\rightarrow \\overline{\\nu }K^+$ , there is more data from the continuation of run SK-IV, which was not used for the published limit in ref. [44].",
"It is therefore quite possible that a future limit, based on data already taken, will be stronger.",
"confidence levels for the exclusion of proton decay at Super-Kamiokande in $p \\rightarrow \\overline{\\nu } K^+$ (left panel) and $p \\rightarrow e^+ \\pi ^0$ (right panel) channels as a function of proton partial lifetime in the respective decay channels.",
"This generalizes the results presented by the Super-Kamiokande collaboration, which gave results only for 90% CL exclusions.",
"From the data in Table REF , we estimated the current lower limits on proton partial lifetimes to be $\\tau _p/{\\rm Br}(p \\rightarrow \\overline{\\nu } K^+)&>&{\\left\\lbrace \\begin{array}{ll}5.1 \\times 10^{33} \\text{ years at 95\\% CL},\\\\[-7pt]6.6 \\times 10^{33} \\text{ years at 90\\% CL},\\\\[-7pt]1.3\\times 10^{34} \\text{ years at 68\\% CL},\\\\[-7pt]2.2 \\times 10^{34} \\text{ years at 50\\% CL},\\end{array}\\right.",
"}$ and $\\tau _p/{\\rm Br}(p \\rightarrow e^+ \\pi ^0)&>&{\\left\\lbrace \\begin{array}{ll}1.9 \\times 10^{34} \\text{ years at 95\\% CL},\\\\[-7pt]2.4 \\times 10^{34} \\text{ years at 90\\% CL},\\\\[-7pt]4.9 \\times 10^{34} \\text{ years at 68\\% CL},\\\\[-7pt]8.1 \\times 10^{34} \\text{ years at 50\\% CL}.\\end{array}\\right.",
"}$ In comparison, the published $90\\%$ CL exclusion limit on proton partial lifetimes from the Super-Kamiokande collaboration are $\\tau _p/{\\rm Br}(p \\rightarrow \\overline{\\nu } K^+)&>& 5.9 \\times 10^{33} \\text{ years at 90\\% CL (SuperK 2014 \\cite {Super-Kamiokande:2014otb})},\\\\\\tau _p/{\\rm Br}(p \\rightarrow e^+ \\pi ^0)&>& 2.4 \\times 10^{34} \\text{ years at 90\\% CL (SuperK 2020 \\cite {Super-Kamiokande:2020wjk})}.$ shown as the shaded red regions in Figure REF .",
"We see that in the case of $p \\rightarrow \\overline{\\nu }K^+$ , our estimate for the 90% CL limit is somewhat stronger than the journal published limit in ref.",
"[44], but we note that it agrees perfectly with a later unpublished limit by Super-Kamiokande reported in a conference paper [79], based on the same data.",
"In the case of $p \\rightarrow e^+ \\pi ^0$ , our estimate for the 90% CL limit agrees perfectly with the Super-Kamiokande published limit in ref. [45].",
"Figure: The current confidence level for the exclusion of proton decay inp→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ (left panel) andp→e + π 0 p \\rightarrow e^+ \\pi ^0 (right panel) channels, as a function of the respectiveproton partial lifetimes.Our confidence level estimates (solid black lines) are obtained usingeq.",
"()based on Super-Kamiokande's data through2014 (left panel) and2020 (right panel),summarized in Table .The red shaded regions correspond to Super-Kamiokande's published exclusionson proton partial lifetimes at 90%90\\% CL, from and .We now discuss projections for exclusion and discovery of proton decay at possible future neutrino detectors DUNE, JUNO, Hyper-Kamiokande, and THEIA.",
"As both DUNE and JUNO can track and reconstruct charged kaons with high efficiency, they will be primarily searching for proton decay in $p \\rightarrow \\overline{\\nu } K^+$ decay mode.",
"For these searches, DUNE uses its far detector with a total of 40 kiloton (kton) fiducial mass of liquid argon [46], and JUNO uses its central detector with a 20 kton fiducial mass of a liquid scintillator [47].",
"On the other hand, Hyper-Kamiokande [48] uses a water Cerenkov detector with 186 ktons of fiducial mass and is sensitive to both $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay modes among others.",
"As was the case with Super-Kamiokande, Hyper-Kamiokande will be more sensitive to the $p \\rightarrow e^+ \\pi ^0$ mode, compared to the $p \\rightarrow \\overline{\\nu } K^+$ mode, due to much better reconstruction of the Cerenkov rings of the positron and the electromagnetic showers emanating from $\\pi ^0 \\rightarrow \\gamma \\gamma $ .",
"THEIA is a new detector concept with water-based liquid scintillator (10% liquid scintillator and 90% water) that will be able to detect and distinguish between the Cerenkov and the scintillation light [49].",
"Here, we project sensitivities for both THEIA-25 and THEIA-100 with fiducial masses 17 and 80 ktons, respectively, that were considered in ref. [49].",
"Due to the ability to detect scintillation signals from charged particles such as $K^+$ produced below its Cerenkov threshold, and Cerenkov signals, the THEIA detector aims to have enhanced sensitivity to the $p \\rightarrow \\overline{\\nu } K^+$ mode [49] while also being able to probe the $p \\rightarrow e^+ \\pi ^0$ mode [80].",
"The numbers of protons per kiloton of detector material are $N_p ={\\left\\lbrace \\begin{array}{ll}2.71 \\times 10^{32} \\quad \\text{(DUNE)},\\\\[-7pt]3.38 \\times 10^{32} \\quad \\text{(JUNO)},\\\\[-7pt]3.34 \\times 10^{32} \\quad \\text{(Hyper-Kamiokande)},\\\\[-7pt]3.35 \\times 10^{32} \\quad \\text{(THEIA)}.\\end{array}\\right.",
"}$ For the purposes of projecting sensitivities for THEIA and JUNO, we took the liquid scintillator in both detectors to have $6.75 \\times 10^{33}$ protons per 20 kilotons based on ref. [47].",
"Figure REF shows the runtimes at DUNE that are required for an expected 90% CL exclusion (first panel) and $Z=3$ evidence (second panel), in the $p \\rightarrow \\overline{\\nu } K^+$ decay mode, as a function of the background rate per megaton-year of exposure.",
"Figure: The required runtimes at DUNE (with 40 kiloton fiducial mass of liquid argon) for an expected 90%90\\% CL exclusion (top panel) and Z=3Z=3 evidence (bottom panel) as a function of background rate per megaton-year of exposure, for various proton partial lifetimes in thep→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ channel, as labeled.",
"The runtimes are computed using eq.",
"() where the signal needed for 90% CL exclusion (Z=3Z=3 evidence) is obtained from setting eq.",
"() (eq.",
"()) equal to 0.1 (0.00135).",
"We also require s≥1s \\ge 1 in the bottom panel, which yields the horizontal dashed lines for very small bb in the lower left corner.",
"The solid lines (and dashed lines in the bottom panel) assume the signal selection efficiency ϵ\\epsilon to be 40%, and the shaded bands encompassing them correspond to varying ϵ\\epsilon by ±10%\\pm 10\\%.The colored lines and bands correspond to various choices of proton partial lifetimes.",
"For the purposes of illustration, we chose a signal selection efficiency $\\epsilon = 40 \\pm 10 \\%$ that is plausible, based on various signal selection efficiencies that are considered in refs.",
"[66], [82], [81], [83], [84], [85].",
"The solid lines in the figure assume $\\epsilon = 40 \\%$ , and the shaded bands surrounding them vary $\\epsilon $ by $\\pm 10 \\%$ .",
"The required runtimes $\\Delta t$ in the figure are obtained using eq.",
"(REF ), which gives $\\Delta t &=& \\frac{s \\tau _p}{N_p N_{\\rm kton} \\epsilon },$ where $N_{\\rm kton}$ is the number of kilotons of detector material, and $s$ is the upper limit on signal for 90% CL exclusion obtained from setting ${\\rm CL}_{\\rm excl}^A$ (as in eq.",
"()) equal to 0.1, or the signal needed for $Z=3$ evidence obtained from setting ${\\rm CL}_{\\rm disc}^A$ (as in eq.",
"()) equal to 0.00135.",
"As discussed at the end of Section REF , the zero background limit for the discovery case is not well defined, in a sense that at $b=0$ , any non-zero signal, albeit arbitrarily small, would yield an infinite significance.",
"Therefore, to be conservative, we require that the mean expected number of signal events $s$ is at least 1 in order to have an expected discovery.",
"The dashed lines for very small $b$ /Mton-year in the lower left corner of the bottom panel (for discovery case) of Figure REF correspond to this additional requirement that $s\\ge 1$ .",
"It is clear from the figure that if the estimated background per megaton-year of exposure at DUNE increases, the required runtime increases more steeply for discovery than for exclusion.",
"In Figure REF , we show the expected 90% CL exclusion reach (first panel) and the expected $Z=3$ evidence reach (second panel) for proton partial lifetime in $p \\rightarrow \\overline{\\nu } K^+$ decay channel at DUNE as a function of the runtime in years.",
"Figure: Proton partial lifetimes in p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ channel that are expected to be excluded at 90%90\\% CL [top panel, from eqs.",
"() and ()] or discovered at Z=3Z=3 significance [bottom panel, from eqs.",
"(), () and ()] at 40kton DUNE, as a function of runtime for various background rates per megaton-year of exposure, as labeled.",
"The signal selection efficiency ϵ\\epsilon is taken to be 40% (solid lines) ±\\pm 10%10\\% (shaded bands).",
"The long dashed black line in each panel shows the idealized optimistic case of no background and ϵ=46%\\epsilon = 46\\% , with the expected mean number of events required to be s=1s=1in the second panel.",
"Our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit onproton partial lifetime, based on Super-Kamiokande's data from2014 , are shown ashorizontal dashed lines.The three colored lines/bands correspond to various assumed background rates per megaton-year of exposure taken from refs.",
"[46], [66], [82], [81], [83], [84], [85].",
"The signal selection efficiency is again taken to be $\\epsilon = 40\\%$ (solid colored lines) $\\pm $ $10 \\%$ (shaded bands).",
"The signals computed from setting eq.",
"() equal to 0.1, and eq.",
"() equal to $0.00135$ , are plugged into eq.",
"(REF ) to obtain the expected 90% CL exclusion, and $Z=3$ evidence, reaches for proton partial lifetime, respectively.",
"The black dashed curves correspond to a very optimistic scenario with $b=0$ and $\\epsilon = 46\\%$ [66], and using the requirement $s=1$ in the discovery case (bottom panel).",
"Also shown in Figure REF and other figures below are horizontal lines at our previously mentioned estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit based on Super-Kamiokande's data from 2014 [44].",
"The usual standard for discovery in particle physics is a significance of $Z=5$ .",
"Therefore, we show in Figure REF the expected reach for $Z=5$ in the $p \\rightarrow \\overline{\\nu }K^+$ channel at 40 kton DUNE, as a function of the runtime.",
"We note that even after 25 years, the discovery reach in this channel with nominal background rates remains below the value of $\\tau _p(p \\rightarrow \\overline{\\nu }K^+)$ that we estimate to be excluded at 50% CL by the Super-Kamiokande data already published in 2014.",
"Of course, a 50% CL exclusion is far from definitive, but this indicates the challenge being faced.",
"This could change if the background can be reduced to near 0, as indicated by the dashed line, while maintaining a high efficiency for the signal.",
"Figure: Proton partial lifetimes in the p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ channelthat are expected to be discovered with a significance Z=5Z=5 at 40kton DUNE, as a functionof the runtime, for various background rates per megaton-year of exposure, as labeled.",
"The results are obtained fromeqs.",
"(), () and().The horizontal dashed lines shown are our estimates of the current 95%, 90%, 68%, and 50% CLexclusion limit on proton partial lifetime,based on Super-Kamiokande's data from 2014 .Figure: Proton partial lifetimes in the p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ channelthat give different probabilities of observing at least one eventfrom eq.",
"(), which in turn correspond to different values of theexpected signal, as labeled.The signal selection efficiency ϵ\\epsilon is taken to be 46% based on .The horizontal dashed lines shown are our estimates of the current 95%, 90%, 68%, and 50% CLexclusion limit on proton partial lifetime,based on Super-Kamiokande's data from 2014 .As noted above in Section REF , if the mean expected number of signal events is $s=1$ , and one makes the optimistic assumption that the background is completely negligible ($b=0$ ), then the probability of obtaining at least one event is about 63.2%.",
"Figure REF shows the value of $\\tau $ /Br$(p \\rightarrow \\overline{\\nu }K^+)$ , as a function of the runtime, that would give various other probabilities of obtaining at least one event, again with the very optimistic assumption of absolutely no background $b=0$ and $\\epsilon = 46\\%$ [66].",
"Each of these choices for $P(n\\ge 1)$ is equivalent to a requirement on the signal $s$ , as labeled in the figure.",
"Ref.",
"[83] also provided a preliminary estimate for the background and signal efficiency for proton decay search in $p \\rightarrow e^+ \\pi ^0$ mode at DUNE.",
"Although DUNE is most sensitive to $p \\rightarrow \\overline{\\nu } K^+$ mode, for completeness, we will also show our expected reach estimates for proton partial lifetime in $p \\rightarrow e^+ \\pi ^0$ mode at DUNE after 10 years and 20 years of runtime in our summary plots in Figures REF and REF in the Outlook section below.",
"We now turn to projections for JUNO with 20 ktons of a liquid scintillator.",
"We again obtain the upper limit on the signal using eq.",
"() for exclusion reach, and the signal needed for discovery using eq.",
"() for discovery reach, then applying eq.",
"(REF ).",
"Figure REF shows the proton lifetime in $p \\rightarrow \\overline{\\nu } K^+$ decay channel that is expected to be excluded at 90% or 95% CL (top panel) or discovered at $Z=3$ or $Z=5$ significance (bottom panel) at JUNO, as a function of the runtime.",
"Figure: Proton partial lifetime in the p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ channel that is expected to be excluded at 90%90\\% or 95% CL [top panel, from eqs.",
"() and ()] or discovered at Z=3Z=3 or Z=5Z=5 significance [bottom panel, from eqs.",
"(), () and ()] at 20kton JUNO, as a function of runtime, for two different estimated, combinations of background rates per year and signal selection efficiencies (b/ year ,ϵ)(b/{\\rm year}, \\epsilon ), as labeled.Our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit on proton partial lifetime, based on Super-Kamiokande's data from 2014 , are shown as horizontal dashed lines.The two curves correspond to two different estimates of the background accumulated per megaton-year of exposure and the signal selection efficiency as labeled, takenActually, the signal efficiencies taken from refs.",
"[47], [86] do not already account for the branching ratio of about 84.5% of the $K^+$ decays that is included in JUNO's analysis.",
"Instead of multiplying this branching fraction with the right-hand side of eq.",
"(REF ) as done in ref.",
"[47], we redefined the signal efficiencies to account for this branching fraction.",
"from ref.",
"[47] ($b$ /Mton-year = 2.5, $\\epsilon = 55\\%$ ; upper curve) and ref.",
"[86] ($b$ /Mton-year = 1.5, $\\epsilon = 26\\%$ ; lower curve).",
"For comparison, our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit on proton partial lifetime, based on Super-Kamiokande's data from 2014 [44], are shown as horizontal dashed lines.",
"For projected exclusion sensitivities, both DUNE [66] and JUNO [47] experiments made use of the Feldman-Cousins (FC) method [62] to obtain the upper limit on the signal assuming a fixed number of observed events, e.g., $n=0$ .",
"This approach can be problematic for projections because the FC upper limits at a fixed $n$ decrease with $b$ (as can be seen from Figure REF ), and for projections it can imply that the expected sensitivity of the experiment gets better if the background increases.",
"Also considered in ref.",
"[66] is the usage of the FC method with $n=b$ .",
"For integer values of $b$ , the FC upper limit with $n=b$ sensibly increases as the background increases.",
"But for non-integer $b$ , $n$ is still an integer, and the FC upper limit with $n = {\\rm round}(b)$ does not always increase with $b$ , as shown above in Figure REF .",
"As a result, the projected sensitivity does not always decrease with $b$ .",
"This is why we chose to use the ${\\rm CL}_{\\rm excl}$ (= ${\\rm CL}_s$ for single-channel counting experiments) upper limit with the exact Asimov approximation given by eq.",
"() for DUNE and JUNO.",
"While the FC sensitivity of ref.",
"[62] from eq.",
"(REF ) also gives sensible projections for exclusion, we note that it is computationally more intense to evaluate (and gives only slightly more conservative results) than the exact Asimov expected ${\\rm CL}_{\\rm excl}$ upper limits.",
"We next turn to projections for Hyper-Kamiokande.",
"Figures REF and REF show our estimates for the proton partial lifetimes in $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay channels, respectively, that are expected to be excluded at 90% or 95% CL (top panels) or discovered at $Z=3$ or $Z=5$ significance (bottom panels), as a function of runtime at Hyper-Kamiokande.",
"Figure: Proton partial lifetime in the p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ channel thatis expected to be excluded at 90%90\\% or 95% CL [top panel; from eq.",
"()] or discovered at Z=3Z=3 or Z=5Z=5 significance [bottom panel; from eq.",
"()] at Hyper-Kamiokande with 186 kilotons of water, as a function of runtime, with the uncertainties in background and signal selection efficiency listed in Table , taken from ref. .",
"Our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit on proton partial lifetime, based on Super-Kamiokande's data from 2014 , are shown as horizontal dashed lines.Figure: Proton partial lifetime in the p→e + π 0 p \\rightarrow e^+ \\pi ^0 channel thatis expected to be excluded at 90%90\\% or 95% CL [top panel; from eq.",
"()] or discovered at Z=3Z=3 or Z=5Z=5 significance [bottom panel; from eq.",
"()] at Hyper-Kamiokande with 186 kilotons of water, as a function of runtime, with the uncertainties in background and signal selection efficiency listed in Table , taken from ref.",
".Our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit on proton partial lifetime, based on Super-Kamiokande's data from 2020 , are shown ashorizontal dashed lines.In order to obtain the exclusion and discovery reaches for $\\tau _p$ , the upper limit on partial width and the partial width needed for discovery are solved from eqs.",
"(REF ) and (REF ), respectively.",
"These equations are used to combine the independent search channels in each decay mode, based on the background means and the signal selection efficiencies, along with their uncertainties, given in ref.",
"[48] and summarized in our Table REF .",
"Table: Estimated backgrounds b ^ i ±Δ b i \\hat{b}_i \\pm \\Delta _{b_i} per megaton-year of exposureand signal efficiencies ϵ ^ i ±Δ ϵ i \\hat{\\epsilon }_i \\pm \\Delta _{\\epsilon _i} at Hyper-Kamiokande,taken from ref.",
",for p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ andp→e + π 0 p \\rightarrow e^+ \\pi ^0 decay modes.The last column gives a brief description of each of the channels referringto the name of the search methodused in ref.",
".Exposure in each channel for a 186kton Hyper-Kamiokande is given by λ i =0.186Mton×runtimeinyears\\lambda _i = 0.186 \\text{ Mton} \\times \\text{runtime in years}.Figures REF and REF also show our previously discussed estimates of the current exclusion limits at 95%, 90%, 68%, 50% CL in $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay modes based on the data from refs.",
"[44] and [45], respectively.",
"Finally, we turn to projections for THEIA.",
"In Figures REF and REF , we show the expected reaches, as a function of runtime, for proton partial lifetime in $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ decay modes, respectively, for 90% or 95% CL exclusion (top panels) and discovery at $Z=3$ or $Z=5$ significance (bottom panels).",
"Figure: Expected 90% or 95% CL exclusion reaches [top panel; from eqs.",
"() and ()] and Z=3Z=3 or Z=5Z=5 discovery reaches [bottom panel; from eqs.",
"(), () and ()] for proton partial lifetime in p→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ with THEIA-25 (red lines) and THEIA-100 (blue lines) with 17 and 80 ktons of water based liquid scintillator, respectively, as a function of runtime.",
"The estimates for the background (per megaton-year of exposure) and the signal efficiencies are taken from ref. .",
"Our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit on proton partial lifetime, based on Super-Kamiokande's data from 2014 , are shown as horizontal dashed lines.Figure: Expected 90% or 95% CL exclusion reaches [top panel; from eqs.",
"() and ()] and Z=3Z=3 or Z=5Z=5 discovery reaches [bottom panel; from eqs.",
"(), () and ()] for proton partial lifetime in p→e + π 0 p \\rightarrow e^+ \\pi ^0 with THEIA-25 (red lines) and THEIA-100 (blue lines) with 17 and 80 ktons of water based liquid scintillator, respectively, as a function of runtime.",
"The estimates for the background (per megaton-year of exposure) and the signal efficiencies are taken from ref. .",
"Our estimates of the current 95%, 90%, 68%, and 50% CL exclusion limit on proton partial lifetime, based on Super-Kamiokande's data from 2020 , are shown ashorizontal dashed lines.The lower (red) lines show the results for THEIA-25 with 17 ktons of fiducial mass of water based liquid scintillator, while the upper (blue) lines are for THEIA-100 with 80 ktons fiducial mass.",
"The expected reach for proton partial lifetime is computed using eq.",
"(REF ), where the expected signal for 90% CL exclusion ($Z=3$ evidence) is obtained from setting eq.",
"() (eq.",
"()) to 0.1 (0.00135).",
"The estimates for the background rate per megaton-year of exposure and the signal selection efficiency for the decays modes $p \\rightarrow \\overline{\\nu } K^+$ and $p \\rightarrow e^+ \\pi ^0$ are taken from refs.",
"[49] and [80], respectively.",
"As before, we also show our estimates for the current lower limits at various confidence levels based on the data from Super-Kamiokande [44], [45]."
],
[
"Outlook",
"We summarize our projections for future proton decay searches in the final states $e^+ \\pi ^0$ and $ \\overline{\\nu }K^+$ at DUNE, JUNO, and Hyper-Kamiokande in Figure REF for exclusion (assuming the signal is indeed absent), and in Figure REF for discovery (assuming the signal is actually present).",
"And in Figure REF we summarize our projections at THEIA for 95% CL exclusion and $Z=5$ discovery for various fiducial masses $N_{\\rm kton} = (10, 25, 50, 100)$ kton.",
"In each case, we show results for 10 years and 20 years of runtime.",
"The assumed backgrounds and signal efficiencies for DUNE$\\endcsname $For projections at DUNE in $p \\rightarrow \\overline{\\nu } K^+$ channel, we are using the optimistic choices based on ref. [81].",
"More pessimistic choices from refs.",
"[82], [83], [84], [85] will of course lead to lower reach estimates.",
", JUNO, and THEIA in each proton decay mode are labeled in the plots, while the corresponding information for the multi-channel Hyper-Kamiokande searches was given in Table REF above, quoted from ref. [48].",
"The vertical dashed lines correspond to our estimate of the current 90% CL (Fig.",
"REF , top panel of Fig.",
"REF , and bottom panel of Fig.",
"REF ) or 95% CL (bottom panel of Fig.",
"REF and top panel of Fig.",
"REF ) lower limit on proton partial lifetime in the respective decay channels, based on the published Super-Kamiokande data [44], [45].",
"Figure: Expected exclusion reaches at 90% CL (top panel) and 95% CL (bottom panel) for proton partial lifetime inp→e + π 0 p \\rightarrow e^+ \\pi ^0 (blue bars)andp→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ (red bars)decay channels at JUNO, DUNE, and Hyper-Kamiokande after 10 years (darker shading) and 20 years (lighter shading) of runtime.",
"The assumed backgrounds and signal efficiencies for JUNO and DUNE are labeled in the plots, and for Hyper-Kamiokande, the corresponding information is given in Table , quoted from ref. .",
"These results are based on preliminary estimates of the backgrounds and signal efficiencies, which are likely to change as the experiments progress, and therefore should be viewed with some caution as comparisons.The vertical dashed lines are our estimates of the current 90% CL (top panel) and 95% CL (bottom panel)lower limits based onSuper-Kamiokande's data from 2014 and 2020.Figure: Expected reaches for Z=3Z=3 evidence (top panel) and Z=5Z=5 discovery (bottom panel) for proton partial lifetime inp→e + π 0 p \\rightarrow e^+ \\pi ^0 (blue bars)andp→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ (red bars)decay channels, at JUNO, DUNE, and Hyper-Kamiokande after 10 years (darker shading) and 20 years (lighter shading) of runtime.",
"The assumed backgrounds and signal efficiencies for JUNO and DUNE are labeled in the plots, and for Hyper-Kamiokande, the corresponding information is given in Table , quoted from ref. .",
"These results are based on preliminary estimates of the backgrounds and signal efficiencies, which are likely to change as the experiments progress, and therefore should be viewed with some caution as comparisons.",
"The vertical dashed lines are our estimates of the current 90% CL lower limits based onSuper-Kamiokande's data from 2014 and 2020.Figure: Expected reaches at THEIA for 95% CL exclusion (top panel) andZ=5Z=5 discovery (bottom panel) for proton partial lifetime inp→e + π 0 p \\rightarrow e^+ \\pi ^0 (blue bars)andp→ν ¯K + p \\rightarrow \\overline{\\nu } K^+ (red bars)decay channels with various fiducial masses, as labeled, after 10 years (darker shading) and 20 years (lighter shading) of runtime.",
"The assumed background rates and signal efficiencies for THEIA are labeled in the plots.",
"These results are based on preliminary estimates of the backgrounds and signal efficiencies, which are likely to change as the experiment progresses.The vertical dashed lines are our estimates of the current 95% CL (top panel) and 90% CL (bottom panel)lower limits based onSuper-Kamiokande's data from 2014 and 2020.As noted above, our projections here are based on the exact Asimov evaluation of the Bayesian statistics ${\\rm CL}_{\\rm excl}$ and ${\\rm CL}_{\\rm disc}$ .",
"Our results are somewhat more conservative than previous projections appearing in refs.",
"[48] and the Snowmass report [80], which we have generalized to include 90% CL exclusion and $Z=3$ evidence reach estimates as a function of runtime (for various estimates of backgrounds and signal efficiencies, notably for DUNE and JUNO) as well as estimates for 95% CL exclusion and $Z=5$ discovery.",
"In the cases of single-channel searches for DUNE, JUNO, and THEIA, we have also investigated the use of the exact Asimov frequentist $p$ -value measures $p_{\\rm excl}$ and $p_{\\rm disc}$ .",
"These results are not shown in the figures; we find that they are only slightly less conservative than the estimates shown.",
"The two panels of Figure REF show the projected exclusion reaches at 90% and 95% confidence level, while the two panels of Figure REF give the projected reaches for $Z=3$ evidence and $Z=5$ discovery at DUNE, JUNO, and Hyper-Kamiokande.",
"And the top (bottom) panel of Figure REF shows the projected 95% CL exclusion ($Z=5$ discovery) reaches at THEIA with various fiducial masses of the detector material.",
"As expected, for each planned experiment the reaches for exclusion are substantially higher than the corresponding reaches for a possible discovery.",
"We note that the prospects for a definitive $Z=5$ discovery are particularly modest after one takes into account the limits already obtained by Super-Kamiokande.",
"The results shown in Figures REF , REF , and REF are preliminary estimates, as the presently available background and signal efficiency estimates vary significantly in their reliability, and more robust estimates will become available only when the experiments are closer to collecting data.",
"For the same reason, the results should be viewed with some caution as a direct comparison of the different experiments, which are at very different stages of planning and development.",
"Proton decay experiments prior to Super-Kamiokande have ruled out the the simplest variations of minimal $SU(5)$ GUT [5], and Super-Kamiokande has seemingly ruled out the minimal supersymmetric $SU(5)$ GUT [18], [19], [20], [21] with sfermion masses less than around the TeV scale.",
"However there are many other well-motivated GUT models that predict proton partial lifetimes well beyond the current lower limits (see summary tables in refs.",
"[80], [77] and references therein).",
"For example, non-supersymmetric GUTs such as some minimally extended $SU(5)$ models [7], [8] and minimal $SO(10)$ model [14] predict $p \\rightarrow e^+ \\pi ^0$ to be the dominant decay mode with partial lifetimes of order $10^{32} - 10^{36}$ years and $\\lesssim 5 \\times 10^{35}$ years, respectively.",
"Supersymmetric $SU(5)$ GUTs predict the proton partial lifetime for the leading mode $p \\rightarrow \\overline{\\nu } K^+$ to be $3 \\times 10^{34} - 2 \\times 10^{35}$ years in minimal supergravity framework (MSUGRA) and $3 \\times 10^{34} - 10^{36}$ years in supergravity models with non-universal gaugino masses (NUSUGRA), as discussed in ref.",
"[26] in the light of the observed Higgs mass.",
"Ref [22] revisited the minimal supersymmetric $SU(5)$ GUT and obtained $\\tau _p/{\\rm Br}(p \\rightarrow \\overline{\\nu } K^+)\\lesssim (2 - 6) \\times 10^{34}$ years assuming universality of the soft supersymmetry breaking parameters at the GUT scale with sfermion masses less than around $\\mathcal {O}(10)$ TeV.",
"There are also supersymmetric GUTs such as the split $SU(5)$ supersymmetry [31] and flipped $SU(5)$ supersymmetric GUTs [28], [29], [30], where the dominant decay mode can be $p \\rightarrow e^+ \\pi ^0$ with lifetimes of order $10^{35} - 10^{37}$ years.",
"From our estimates of the reaches summarized in Figures REF , REF , and REF , we can see that DUNE, JUNO, Hyper-Kamiokande, and THEIA can probe a significant fraction of the parameter space of various presently viable supersymmetric and non-supersymmetric GUTs and could eventually lead the way to a more complete theory.",
"The existing code repository Zstats [67] is updated with various statistical measures of significance for counting experiments with multiple independent search channels as investigated in this paper.",
"The updates include the significances based on our proposed Bayesian-motivated measures ${\\rm CL}_{\\rm disc}$ and ${\\rm CL}_{\\rm excl}$ , and their application to study the statistical significances for proton decay at current and future neutrino detectors.",
"To demonstrate the usage of the code, the repository also contains some code snippets in a Python notebook that generate the data in each of the figures in this paper.",
"Acknowledgments: We thank Jennifer Raaf and Edward Kearns for helpful discussions.",
"P.N.B.",
"thanks Tracy Slatyer for helpful discussions on Bayes factors and Zirui Wang for useful discussions on statistical combinations using RooStats package [87].",
"P.N.B.",
"and J.D.W.",
"thank Jianming Qian for useful discussions.",
"P.N.B.",
"and J.D.W.",
"also thank John Thiels for his help and support in using the Great Lakes cluster at University of Michigan.",
"This research was supported in part through computational resources and services provided by Advanced Research Computing (ARC), a division of Information and Technology Services (ITS) at the University of Michigan, Ann Arbor.",
"This research was also supported in part through computational resources and services provided by NICADD compute cluster at Northern Illinois University.",
"This work is supported in part by the National Science Foundation under grant number 2013340, and by the Department of Energy under grant number DE-SC0007859."
]
] | 2210.07735 |
[
[
"Understanding Multi-link Operation in Wi-Fi 7: Performance, Anomalies,\n and Solutions"
],
[
"Abstract Will Wi-Fi 7, conceived to support extremely high throughput, also deliver consistently low delay?",
"The best hope seems to lie in allowing next-generation devices to access multiple channels via multi-link operation (MLO).",
"In this paper, we aim to advance the understanding of MLO, placing the spotlight on its packet delay performance.",
"We show that MLO devices can take advantage of multiple contention-free links to significantly reduce their transmission time, but also that they can occasionally starve one another and surprisingly incur a higher delay than that of a well planned legacy single link operation.",
"We next examine and explain this anomaly, also putting forth practical workarounds to circumvent it.",
"We conclude by pointing to other disruptive features that, if successfully paired with MLO, can usher in exciting and unprecedented opportunities for Wi-Fi 8."
],
[
"Introduction",
"As we head out of a pandemic that made connectivity truly dependable, our appetite for mobile data is stronger than ever.",
"Myriad engineers behind the development of Wi-Fi, the technology carrying nearly two thirds of all wireless data, relentlessly feed this hunger by crafting ever more clever amendments, defining new Wi-Fi generations one after another.",
"At the time of writing, Wi-Fi 6 and 6E are a commercial reality, the making of Wi-Fi 7 is nearing completion, and the definition of Wi-Fi 8 starts catalyzing the interest of tech giants and avid researchers alike [1], [2], [3].",
"Yet before debating or fantasizing about what Wi-Fi 8 should be, what will Wi-Fi 7 deliver?",
"The IEEE 802.11be amendment, expected to be at the heart of Wi-Fi 7, will remain loyal to its legacy—and to its very name: EHT, short for `Extremely High Throughput'—by augmenting data rates through various upgrades ranging from wider bandwidths (up to 320 MHz) to higher modulation orders (up to 4096-QAM) [4], [5], [6].",
"But besides features boosting the nominal throughput, many experts point to multi-link operation (MLO) as the true paradigm shift Wi-Fi 7 will bring to the table.",
"MLO will allow Wi-Fi devices to concurrently operate on multiple channels through a single connection, aiming to support applications demanding not only higher capacity but also lower delay [7], [8].",
"Unlike merely multiplying the peak throughput gains provided by scaling up bandwidth and spectral efficiency, quantifying the advantages brought about by MLO in realistic scenarios is no straightforward endeavor.",
"And while several works have recently made valuable attempts at studying how MLO performs in terms of throughput and delay [9], [10], [11], [12], [13], [14], a deep and widespread understanding of the latter remains little more than wishful thinking.",
"Indeed, the exact benefits on a device employing MLO for delay-sensitive applications and the effects on coexisting basic service sets (BSSs) hinge on the specific MLO implementation, with several being defined in 802.11be to trade off complexity and flexibility.",
"Furthermore, as we will show in later sections, these benefits—or the lack thereof—highly depend on the traffic load, the surrounding environment, and the channel allocation strategy adopted.",
"In this paper, we shed light on the delay performance of STR EMLMR (standing for `Simultaneous Transmit and Receive Enhanced Multi-link Multi-radio'), arguably the most flexible MLO mode, under varying traffic demand, congestion, and channel allocation strategies.",
"We explain and quantify its main virtues with respect to legacy single-link (SL) as well as its caveats, also putting forth possible solutions to the latter.",
"Our main takeaways can be summarized as follows: In scenarios devoid of contention, STR EMLMR exploits additional available links to perform multiple transmissions in parallel, proportionally reducing the channel access delay.",
"In the presence of high load and contention, STR EMLMR devices frequently access multiple links thereby blocking contending neighbors, occasionally causing larger delays than those experienced with a static SL channel assignment.",
"For consistent worst-case delay reduction, STR EMLMR may require more channels than contending BSSs and/or performing a clever channel assignment that entirely circumvents delay anomalies caused by sporadic BSS starvation.",
"Compared to existing work, the novelty and contribution of the present paper is at least threefold: We introduce the reader to MLO, review its different implementations, and illustrate, for the first time, the intricate interactions it triggers between contending BSSs.",
"We demonstrate how such interplay may turn out being benign or unfavorable, depending on the traffic load and channel allocation strategies.",
"We identify, quantify, and explain, through novel results, the delay anomalies that may surprisingly arise when employing STR EMLMR in the presence of high load and contention.",
"We also propose multiple solutions to circumvent such anomalies and we evaluate and compare their effectiveness.",
"We provide a concise yet complete picture of the virtues and caveats of MLO by putting our findings into an even broader context.",
"We also invoke follow-up research on other disruptive features that, if successfully paired with MLO, can usher in exciting and unprecedented opportunities for Wi-Fi 8.",
"Figure: Example of STR EMLMR operations and packet interactions without (left) and with (right) contention.",
"Grey, orange, and blue slots denote occupied channels, ongoing backoffs, and successful transmissions, respectively.",
"Consecutive blue slots indicate aggregated packets.",
"For illustration purposes, all transmissions are downlink and the corresponding ACKs are omitted.",
"In the example, for AP 1, packet #3 experiences a lower delay than it would under SL operations.",
"For AP 2 instead, packet #1 undergoes a higher delay than it would with SL."
],
[
"A Primer on Multi-link Operation",
"In addition to legacy SL (Single-link) channel access—which follows the 802.11 Distributed Coordination Function (DCF) over one channel—Wi-Fi 7 will allow MLO through single association, with channel contention and access performed independently for each link.",
"The 802.11be amendment defines different MLO implementation flavors, with the main ones summarized as follows [7]."
],
[
"Enhanced Multi-link Single-radio (EMLSR)",
"EMLSR enables a single-radio multi-link device (MLD) to listen to two or more links simultaneously, e.g., by splitting its multiple antennas, performing clear channel assessment and receiving a limited type of control frames.",
"EMLSR supports opportunistic spectrum access at a reduced cost, as it requires a single fully functional 802.11be radio plus several other low-capability radios able only to decode 802.11 control frame preambles.",
"Upon reception of an initial control frame on one of the links, the EMLSR MLD can switch to the latter and operate using all antennas.",
"For a MLD implementing EMLMR, all radios are 802.11be-compliant and allow operating on multiple links concurrently.",
"EMLMR is further classified into two modes: Non-simultaneous Transmit and Receive (NSTR) EMLMR, where no simultaneous transmission and reception is allowed over a pair of links in order to prevent self interference at the MLD.",
"The latter entails ensuring near alignment in the end time of physical layer protocol data unit that are simultaneously transmitted, so as to avoid that subsequent incoming responses on one link, e.g., ACKs, overlap with the remaining transmission on another link.",
"Simultaneous Transmit and Receive (STR) EMLMR, where the above rule does not apply.",
"In order to avoid uplink-to-downlink intra-device interference, operating STR EMLMR requires sufficient frequency separation between the channels used by different links and/or sophisticated self-interference cancellation capabilities.",
"For instance, STR EMLMR with four links, each on an 80 MHz channel, could be implemented by using two channels each in the 5 GHz and 6 GHz bands, with a minimum channel separation of 160 MHz, and equipping MLDs with suitable radio-frequency filters.",
"A remark is in order about the `E' in EMLSR and EMLMR, standing for `enhanced'.",
"Indeed, non-enhanced versions of both have also been defined [7], their main features summarized as follows: MLSR, where unlike EMLSR, clear channel assessment and control frame reception (and of course, data transmission/reception) can only be performed on one channel at a time, thereby limiting opportunistic link selection.",
"MLMR, which compared to EMLMR only lacks extra capabilities to dynamically reconfigure spatial multiplexing over multiple links.",
"This difference is immaterial for the case studies of the present paper.",
"In the remainder of this article, we place the spotlight on STR EMLMR since it is the MLO operation mode that grants the highest degree of flexibility and requires the least amount of signaling, thus being the most likely to be adopted in first-wave Wi-Fi 7 commercial products.",
"Unlike previous work devoted to the achievable throughput of STR EMLMR, we focus on its delay performance as we deem it crucial to support ever more proliferating real-time applications."
],
[
"A Close-Up of STR EMLMR",
"As shown in Fig.",
"REF , exemplifying STR EMLMR in action over two links, it turns out that this mode of operation can affect the packet delay in multiple ways, depending on the particular scenario at hand.",
"In the following, we provide two examples that illustrate how STR EMLMR can respectively reduce and increase the delay with respect to legacy SL operations.",
"For the latter (not shown), we assume an orthogonal channel assignment as a benchmark, with AP 1 and AP 2 operating on link 1 and link 2 only, respectively."
],
[
"Delay reduction through STR EMLMR",
"Let us begin by focusing on the left hand side of Fig.",
"REF , where AP 2 is inactive and AP 1 can take advantage of two available links by routing traffic to either as needed.",
"In the example, packets #1 and #2 are aggregated and promptly transmitted over link 1.",
"As for packet #3, which arrives during an ongoing transmission, a new backoff is started on link 2, followed by a transmission.",
"Packet #3 thus enjoys a significant delay reduction compared to a legacy SL scenario, as in the latter it would have needed to wait for the ongoing transmissions on link 1 to be completed.",
"The right hand side of Fig.",
"REF illustrates a scenario where AP 1 and AP 2, both implementing STR EMLMR, contend for channel access.",
"In this example, AP 1 aggregates packets #4 and #5 upon backoff expiration and transmits them over link 1.",
"Meanwhile, more traffic arrives, namely packets #6 and #7 at AP 1 and packets #1 and #2 and AP 2.",
"Since link 1 is occupied by AP 1, both AP 1 and AP 2 undergo contention for link 2, with the backoff for AP 1 expiring first.",
"AP 1 thus aggregates and transmits packets #6 and #7 on link 2, thereby occupying both links concurrently.",
"It is only after the transmission of packets #4 and #5 by AP 1 is completed that AP 2 can eventually aggregate and transmit all its queued packets on link 1.",
"In the example, these packets experience a much higher delay than they would have under legacy SL operations.",
"Indeed, with SL and a static channel allocation (e.g., AP 1 on link 1 and AP 2 on link 2), AP 1 would have not been able to occupy both links simultaneously, and therefore would have not temporarily forced AP 2 into starvation.",
"We identify this phenomenon as an anomaly of MLO, and will devote Section IV to its better understanding.",
"As it can be seen through the above two examples, STR EMLMR is capable of taking advantage of multiple links to reduce the channel access time with respect to SL, but also to occasionally starve neighboring BSSs thereby increasing their delay.",
"In the sequel, we will confirm and quantify these two phenomena through targeted simulation campaigns."
],
[
"STR EMLMR in Contention-free Scenarios",
"We begin by considering a single, isolated BSS with one MLD station (STA) associated to an MLD access point (AP), and evaluate the delay performance of STR EMLMR in such a contention-free scenario.",
"Without loss of generality, we focus on downlink traffic and assume Poisson arrivals with constant packet size of 12000 bytes, 80 MHz channels, two spatial streams, and a modulation and coding scheme of 256-QAM 3/4 [7].",
"Packet aggregation is employed with the number of aggregated packets decided at the start of a transmission, up to a maximum of 1024.",
"A buffer size of 4096 packets is employed, ensuring sufficient room for the maximum allowed number of aggregated packets.",
"For this scenario, we study the effect of the traffic load on the packet delay under three schemes, namely: (i) SL, taken as the baseline, (ii) STR EMLMR:2, where the isolated AP can use two links at any time, and (iii) STR EMLMR:4, with four links simultaneously available."
],
[
"Packet delay",
"Fig.",
"REF shows the delay statistics with shaded curves ranging from 50%-tile to 99%-tile (i.e., median to 1%-worst) using SL and STR EMLMR with two or four links.",
"Intuitively, as more links are available and can be accessed dynamically, a certain delay requirement can be met for proportionally higher values of the traffic load, i.e., while supporting a proportionally higher throughput.",
"For instance, given a median delay of 1 ms, SL, STR EMLMR:2, and STR EMLMR:4 can roughly support up to half, one, and two Gbps, respectively.",
"Similarly, given a certain traffic load, availing of extra links decreases the delay, albeit with diminishing returns.",
"For instance at 0.5 Gbps, the three schemes incur 99%-tile delays of about 1.6, 0.7, and 0.5 ms.",
"Nonetheless, depending on the traffic load, accessing multiple links may be the only way to prevent the delay from growing unbounded.",
"E.g., a load of 1 Gbps exceeds the capacity of a single channel, thus SL incurs unbounded delay, whereas STR EMLMR:2 and STR EMLMR:4 keep the delay below 1.7 and 0.7 ms, respectively, 99% of the time.",
"Fig.",
"REF also displays the 99%-tile for the corresponding number of packets aggregated under each of the three schemes (dashed lines).",
"Even at moderate loads, owing to its inability of using multiple links, SL experiences a higher buffer congestion and is forced to aggregate a much larger number of packets per transmission than STR EMLMR.",
"The latter can instead parallelize access on multiple links, reducing the buffer congestion and thus the number of aggregated packets for each transmission.",
"In scenarios devoid of contention, STR EMLMR can exploits extra links—even across different frequency bands, something SL is not capable of—to operate on a wider bandwidth, and can therefore meet a certain delay requirement while supporting higher traffic loads (i.e., throughput) than SL.",
"While the above results are somewhat expected, they are in stark contrast to the delay anomaly experienced by MLDs in crowded scenarios, quantified in the next section.",
"Figure: Three modes of operation considered for a crowded scenario: (a) SL, (b) STR EMLMR:2, and (c) STR EMLMR:4.",
"Colors and numbers refer to different channels, letters denote BSSs, and dashed arrows indicate contention between BSSs."
],
[
"STR EMLMR in Crowded Scenarios",
"We now investigate when the delay reduction provided by STR EMLMR is maintained in the presence of contention, and when instead the delay is increased due to the starvation phenomenon, i.e., the anomaly, outlined in Section II-B.",
"To this end, we turn our attention to a more crowded enterprise scenario with 4 BSSs as depicted in Fig.",
"REF .",
"Each BSS comprises one AP and one associated STA, all BSSs are in the coverage range of each other, and the whole system has a limited amount of resources, namely four orthogonal 80 MHz channels.",
"For this challenging scenario, we consider three possible modes of operation, each making a different use of the four available channels: SL, with a single channel exclusively assigned to each BSS, as illustrated in Fig.",
"REF , and no contention.",
"Again, we take this mode as the baseline to assess STR EMLMR.",
"STR EMLMR:2, as shown in Fig.",
"REF , where each BSS employs two channels and shares both with one more contending BSS.",
"STR EMLMR:4, as shown in Fig.",
"REF , where all four BSSs employ and contend for all four channels.",
"Note that the above three arrangements assume statically assigning channels to BSSs according to a specific reuse scheme, and thus embody a hypothetical enterprise use case.",
"In this section, we assume the same values of total traffic load as in Section III, but this time evenly spread among all BSSs, i.e., one quarter each.",
"The scenarios in Section III (Fig.",
"REF ) vs.",
"Section IV (Fig.",
"REF ) can thus be regarded as an asymmetric vs. symmetric distribution of the same total load between contending BSSs."
],
[
"Packet delay",
"Fig.",
"REF shows the mean, 95%-tile, and 99%-tile delay using SL, STR EMLMR:2, and STR EMLMR:4 vs. the total traffic load.",
"For a relatively low load of 0.1 Gbps, the delay is decreased by adding multiple links since there is negligible contention and STR EMLMR can quickly find and exploit extra transmission opportunities, as previously shown in Section III.",
"However, once the load reaches higher values such as 1 Gbps and above, STR EMLMR worsens the delay compared to SL, and four links incur a higher delay than two.",
"These results stem from the anomaly illustrated on the right hand side of Fig.",
"REF , and can be further explained by the interplay between multi-link contention and packet aggregation, detailed as follows.",
"In Fig.",
"REF we dig deeper into the delay anomaly by observing how STR EMLMR devices occupy the available links depending on their traffic load.",
"The bars show, through different color opacity, the probability that an active BSS (i.e., with packets to transmit) will use a certain number of links concurrently.",
"For a high traffic load of 2.5 Gbps, SL is limited to transmit on one link only, whereas STR EMLMR:2 employs a second interface 26% of the time, and STR EMLMR:4 uses two or more interfaces 34% of the time.",
"A remarkable consequence (not shown for brevity) is that, with STR EMLMR:4, each contending BSS finds all four links occupied 24% of the time.",
"These events cause a deferral of the backoff countdown and prevent access to any wireless channel.",
"In other words, while SL mode allows—or better said, forces—each BSS to operate on its own dedicated link 100% of the time (Fig.",
"REF ), whenever STR EMLMR BSSs use multiple links opportunistically they inevitably prevent at least another BSS from accessing at least one of its allocated channels (Figs.",
"REF and REF ).",
"Due to a higher contention, which results in longer backoff times, whenever a STR EMLMR device does succeed in accessing the channel, it must occasionally aggregate a larger number of queued packets as exemplified on the right hand side of Fig.",
"REF .",
"While not shown, we observed that for a load of 2.5 Gbps, switching from SL to STR EMLMR:4 decreases the median number of aggregated packets from 138 to 91, but it also increases its 99%-tile value from 207 to 317.",
"The latter corresponds to occasional intervals of long channel occupancy and undesirable delay anomalies.",
"In the presence of high load and contention, STR EMLMR devices frequently access multiple links, thereby occasionally blocking contending neighbors for long periods of time and causing larger delays than those experienced by legacy SL under a static orthogonal channel allocation.",
"Figure: Crowded enterprise scenario: (a) delay and (b) number of links concurrently used by each active BSS vs. traffic load."
],
[
"Overcoming the Delay Anomaly",
"How to side-step the delay anomaly occasionally experienced by STR EMLMR in crowded environments?",
"In this section, we explore multiple practical options based on clever and/or extra channel assignment and compare their performance for the same enterprise scenario introduced in Section IV: EMLSR:2, detailed in Section II-A, with each MLD availing of two channels as in Fig.",
"REF but only equipped with one radio and thus only able to use one link at a time.",
"This setup still requires a total of four channels.",
"STR EMLMR:1+1, with each MLD using two links: one on a channel exclusively reserved (thus undergoing no contention) plus one on a channel shared with all other BSSs.",
"This hybrid arrangement requires a total of five channels as opposed to the four required in Fig.",
"REF .",
"STR EMLMR:5, with an overprovisioning of five links per MLD, each operating on a different channel, with all channels accessible by all four BSSs.",
"Like the previous one, this setup requires a total of five channels, but it additionally requires five radio interfaces per MLD.",
"Fig.",
"REF displays the delay experienced by the three above approaches when compared to SL (Fig.",
"REF ) and STR EMLMR:2 (Fig.",
"REF ).",
"We note how forcing each MLD to transmit on one link at a time with EMLSR:2 (purple) keeps the delay below or equal to that of SL (blue) across all values of load considered, while not increasing the total number of channels required.",
"At a load of 0.1 Gbps, all approaches experience low delays, with STR EMLMR:5 (green) achieving the lowest.",
"Indeed, the low contention arising in this regime makes it likely for a MLD to encounter multiple links available, and juggling up to five running backoffs further reduces the delay.",
"Interestingly, as the load grows to 2.5 Gbps, equipping each MLD with just two radios and operating STR EMLMR:1+1 (light blue) outperforms STR EMLMR:5, despite the latter employing as many as five radios per MLD.",
"Indeed, delay reduction is owed not only to a larger number of channels and an increased system throughput, but also to circumventing the delay anomaly by guaranteeing one contention-free channel per BSS.",
"Overall, to consistently outperform SL, STR EMLMR may thus require a total number of channels larger than the number of contending BSSs, and therefore equipping MLDs with additional radios, self-interference cancellation capabilities, and ensuring a sufficient inter-channel spacing.",
"For instance, operating both in the 5 GHz and 6 GHz bands could allow accessing five 80 MHz channels with a spacing of 160 MHz."
],
[
"Takeaway",
"For consistent worst-case delay reduction, one may resort to STR EMLMR with more channels than contending BSSs and/or to performing a clever channel assignment that entirely circumvents delay anomalies caused by sporadic traffic starvation.",
"Figure: Crowded enterprise scenario: delay vs. traffic load for difference schemes.",
"For each color, high/medium/low opacity respectively denotes the 50/95/99%-tile delay."
],
[
"Recap and Concluding Remarks",
"Our study confirmed that in scenarios devoid of contention, STR EMLMR exploits extra links to transmit opportunistically, supporting significantly higher traffic loads (and therefore throughput) than SL while meeting strict delay requirements.",
"Conversely, we discovered that in the presence of high load and contention, STR EMLMR devices frequently access multiple links, thereby blocking contending BSSs and occasionally causing larger delays than those experienced with a legacy SL operation with orthogonal channel assignment."
],
[
"STR vs. NSTR EMLMR",
"Though we focused on STR EMLMR for brevity, NSTR EMLMR too may incur delay anomalies.",
"Indeed, its required alignment of simultaneous transmissions comes at the expense of spectrum reuse efficiency, ultimately creating higher contention and further increasing the chances that a certain BSS is prevented from accessing any channel.",
"While delay anomalies may arise under high traffic load across all contending APs, their likelihood and relevance are reduced when the traffic is unevenly distributed across contenders.",
"Let us take the scenario studied in Section III as an extreme example, with all traffic handled by a single active BSS.",
"In such cases, STR EMLMR can efficiently map asymmetric traffic loads to all available links, drastically reducing the delay with respect to SL operations.",
"We observed that delay anomalies can be circumvented by employing EMLSR, which allows MLDs to opportunistically select a link among several but forces them to transmit on only one at a time.",
"Although EMLSR makes for a lower complexity mode than EMLMR to reduce the delay at low traffic loads, using one link at a time prevents MLDs from achieving higher throughputs than under SL operation.",
"While not shown in Fig.",
"REF , a traffic load beyond 0.5 Gbps would eventually exceed the channel capacity of EMLSR, just as it does with SL.",
"In this regime, availing of multiple radios would be the only approach to scale up the throughput so as to guarantee bounded delays.",
"We compared (i) a static channel assignment approach (SL, Fig.",
"REF ), (ii) an entirely dynamic approach (STR EMLMR, Figs.",
"REF and REF ), and (iii) a hybrid approach that cleverly reserves a certain channel for each BSS while leaving one more for contention (STR EMLMR:1+1).",
"We found the latter to be most effective at reducing worst-case delays, even more so—and more economical—than equipping MLDs with more radio interfaces (STR EMLMR:5).",
"Indeed, while STR EMLMR:1+1 guarantees at least one contention-free link for each BSS, STR EMLMR:5 merely spreads contention out over all available links, hence reducing the likelihood of a delay anomaly, but not necessarily overcoming it."
],
[
"Research Directions: Beyond Wi-Fi 7 and MLO",
"The highly scenario-dependent performance of MLO, paired with the inherent uncertainty of the unlicensed spectrum, call for a learning-based optimization [15].",
"Suitably trained models could foresee the most appropriate MLO configuration for a set of overlapping BSSs, given input features such as the traffic load and its corresponding QoS requirements, preventing undesirable phenomena like the delay anomaly described in this paper.",
"Under the right implementation and design, and possibly evolving towards learning-based operations, MLO is bound to keep its promises of throughput augmentation and delay reduction by making the most efficient use of all available radio resources.",
"Nonetheless, creating new transmission opportunities out of nowhere is not what MLO was meant for—a shortcoming that will eventually turn into a bottleneck as we, and our machines, demand more data delivered on time.",
"With Wi-Fi 7 defined and MLO up and running, beyond-802.11be technologies are expected to conquer new frequency bands and/or boost the spatial reuse of the old ones through advanced multi-antenna AP coordination."
],
[
"mmWave operations",
"Expanding Wi-Fi operations into the 60 GHz band is an alluring prospect as it could increase the available spectrum by nearly one order of magnitude.",
"And while 802.11 already defines modes of operation in this band (i.e., 802.11ad and 802.11ay), making mmWave Wi-Fi a greater commercial success may entail coming up with a more compatible design between sub- and above-8GHz operations, thereby reducing initial investment costs [3].",
"This paradigm is embodied by potential features like coordinated beamforming, aiming at packing APs with even more antennas not only to spatially multiplex their associated STAs, but also to suppress the interference generated/received to/from neighboring non-associated STAs.",
"By exploiting multiple spatial degrees of freedom to place radiation nulls, coordinated beamforming could make neighboring BSSs invisible to each other, making inter-BSS contention a thing of the past and creating fertile ground for MLO to exploit [2].",
"As we move towards Wi-Fi 8 with the formation of an 802.11 Task Group on Ultra High Reliability (UHR), above-8 GHz operations and AP coordination should not be added as a bolt-on, but rather conceived atop Wi-Fi 7 and embracing this MLO-native technology.",
"On the one hand, a marriage between MLO and mmWave or AP coordination could be offering unprecedented and exciting challenges to our research community.",
"But on the other, it creates the ultimate opportunity to meet the supreme goal of making unlicensed wireless the new wired.",
"Biographies Marc Carrascosa-Zamacois is a Ph.D. candidate at Univ.",
"Pompeu Fabra in Barcelona.",
"His research interests are wireless networks, with a focus on performance optimization and latency minimization.",
"Giovanni Geraci is an Assistant Professor at Univ.",
"Pompeu Fabra in Barcelona and the Head of Telecommunications.",
"He serves as an IEEE ComSoc Distinguished Lecturer, holds a dozen patents on wireless technologies, and received the IEEE ComSoc Outstanding Young Researcher Award for Europe, Middle East, and Africa.",
"Lorenzo Galati-Giordano is a Senior Research Engineer at Nokia Bell Labs in Stuttgart, focusing on unlicensed spectrum technologies.",
"He has 15+ years of academic and industrial experience and has co-authored tens of commercial patents, publications, and IEEE 802.11 standard contributions.",
"Anders Jonsson is a Full Professor at Univ.",
"Pompeu Fabra in Barcelona and the Head of the Artificial Intelligence and Machine Learning research group.",
"His research interests involve sequential decision problems, automated planning, and reinforcement learning.",
"Boris Bellalta is a Full Professor at Univ.",
"Pompeu Fabra in Barcelona and the Head of the Wireless Networking research group.",
"His research interests are in wireless networks, adaptive systems, machine learning and eXtended Reality.",
"Marc Carrascosa-Zamacois is a Ph.D. candidate at Univ.",
"Pompeu Fabra in Barcelona.",
"His research interests are wireless networks, with a focus on performance optimization and latency minimization.",
"Giovanni Geraci is an Assistant Professor at Univ.",
"Pompeu Fabra in Barcelona and the Head of Telecommunications.",
"He serves as an IEEE ComSoc Distinguished Lecturer, holds a dozen patents on wireless technologies, and received the IEEE ComSoc Outstanding Young Researcher Award for Europe, Middle East, and Africa.",
"Lorenzo Galati-Giordano is a Senior Research Engineer at Nokia Bell Labs in Stuttgart, focusing on unlicensed spectrum technologies.",
"He has 15+ years of academic and industrial experience and has co-authored tens of commercial patents, publications, and IEEE 802.11 standard contributions.",
"Anders Jonsson is a Full Professor at Univ.",
"Pompeu Fabra in Barcelona and the Head of the Artificial Intelligence and Machine Learning research group.",
"His research interests involve sequential decision problems, automated planning, and reinforcement learning.",
"Boris Bellalta is a Full Professor at Univ.",
"Pompeu Fabra in Barcelona and the Head of the Wireless Networking research group.",
"His research interests are in wireless networks, adaptive systems, machine learning and eXtended Reality."
]
] | 2210.07695 |
[
[
"A computational study on the energy efficiency of species production by\n pulsed streamer discharges in air"
],
[
"Abstract We study the energy efficiency of species production by streamer discharges with a single voltage pulse in atmospheric dry air, using a 2D axisymmetric fluid model.",
"Sixty different positive streamers are simulated by varying the electrode length, the pulse duration and the applied voltage.",
"Between these cases, the streamer radius and velocity vary by about an order of magnitude, but the variation in the maximal electric field is significantly smaller, about 30%.",
"We find that G-values for the production of N(4S), O(3P), NO and N2O, which have relatively high activation energies, vary by about 30% to 55%.",
"This variation is mainly caused by two factors: differences in the fraction of energy deposited in the streamer head region, and differences in the maximal electric field at the streamer head.",
"When accounting for both factors, our computed G-values are in good agreement with an analytic estimate proposed by Naidis (2012 Plasma Sources Sci.",
"Technol.",
"21 042001).",
"We also simulate negative streamers and find that their production of N(4S), O(3P) and NO is less energy efficient.",
"The results suggest that energy efficiency can be increased by reducing Joule heating in the streamer channel and by increasing the maximal electric field at the streamer head, for example by using short voltage pulses with a high applied voltage."
],
[
"Introduction",
"Streamer discharges are fast-moving ionization fronts with self-organized field enhancement at their heads [1].",
"In such discharges, chemically active species are produced by collisions of energetic electrons with gas molecules [2].",
"Due to their highly non-equilibrium nature [3], streamer discharges can efficiently produce species with a high activation energy without significant gas heating.",
"Streamer discharges are used for many plasma chemical applications such as air purification or ozone production [4], [5], [6], [7], removal of nitrogen oxides [8], [9], [10], liquid treatment [11], [12], surface modification [13], [14], plasma medicine [15], [16] and plasma assisted combustion [17], [18].",
"In this paper, we computationally study how streamer properties affect the energy efficiency with which chemically active species are produced.",
"Simulations are performed of both positive and negative streamers in atmospheric-pressure dry air, using a 2D axisymmetric fluid model.",
"An advantage of simulations is that they contain information on all species densities and fields, and that discharge parameters such as the voltage waveform can easily be modified.",
"However, it can be challenging to construct a suitable set of chemical reactions for given conditions and time scales of interest.",
"In this paper, we construct a set of 303 chemical reactions primarily based on the chemical reactions from [19], [20], [21], [22], [23], as shown in .",
"Past work.",
"Below, we briefly discuss some related computational and experimental work.",
"Several authors have numerically studied the energy efficiency of species production in pulsed streamer discharges.",
"This energy efficiency is often reported using $G$ -values, in units of atoms or molecules produced per 100 eV of input energy.",
"In simulations of positive streamers in air-methane [24], $G$ -values for O and N radicals were found of about 0.7 and 1.5–1.7, respectively.",
"It was observed that these $G$ -values were relatively insensitive to the applied voltage and the streamer length.",
"In [25], the authors simulated the production of N$_2(v$ = 1), O($^3$ P) and N($^4$ S) in primary and secondary streamers in atmospheric-pressure dry air.",
"When the applied voltage was increased, $G$ -values for O($^3$ P) increased and $G$ -values for N$_2(v$ = 1) decreased, since these species have different activation energies.",
"In [26], $G$ -values for the production of oxygen and nitrogen atoms in positive streamers in atmospheric air were computed, in sphere-plane gaps.",
"It was observed that the calculated $G$ -values weakly depended on the applied voltage and discharge conditions, which were found to be about 3–4 and 0.3–0.4 for oxygen and nitrogen atoms, respectively.",
"An analytical estimate was made for the $G$ -values of the production of chemically active species with high activation energies, which was further worked out in [27], as discussed in more detail in section REF .",
"We also give a few examples of relevant experimental work on the energy efficiency of NO-removal, which depends on the production of oxygen and nitrogen radicals.",
"In [28], the greatest removal efficiency was found when pulsed positive corona discharges were generated with short high-voltage pulses.",
"These pulses were shorter than the time required for primary streamers to bridge the discharge gap.",
"In [29], the authors used pulsed discharges to remove NO in a mixture of N$_2$ , O$_2$ and H$_2$ O.",
"They found the removal energy efficiency increased for shorter pulse widths.",
"The influence of shorter pulse duration on NO-removal energy efficiency was also confirmed by [30], in which a high energy efficiency (0.43 mol/kWh) for NO-removal by nanosecond pulsed discharges was found.",
"Relevant experimental work on the energy efficiency of O$_3$ production can be found in section REF ."
],
[
"Simulation model",
"We use a 2D axisymmetric drift-diffusion-reaction type fluid model with the local field approximation to simulate streamers in 80% $\\mathrm {N}_2$ and 20% $\\mathrm {O}_2$ , at 300 K and 1 bar.",
"Pulsed streamer discharges and their afterglows are simulated up to $t = 500$ ns using the open-source Afivo-streamer code [31]."
],
[
"Model equations",
"The temporal evolution of the electron density $n_{\\mathrm {e}}$ is given by $\\partial _t n_{\\mathrm {e}} = \\nabla \\cdot (\\mu _{\\mathrm {e}} {\\mathrm {E}} n_{\\mathrm {e}} + D_{\\mathrm {e}} \\nabla n_{\\mathrm {e}}) + S_{\\mathrm {e}} + S_{\\mathrm {ph}}\\,,$ where $\\mu _{\\mathrm {e}}$ is the electron mobility, ${\\mathrm {E}}$ the electric field, $D_{\\mathrm {e}}$ the electron diffusion coefficient.",
"Furthermore, $S_{\\mathrm {ph}}$ is the source term for non-local photoionization and $S_{\\mathrm {e}}$ is the sum of electron source terms from the reactions listed in table REF .",
"For photoionization, we use Zheleznyak's model [32] and the so-called Helmholtz approximation [33], [34], with the same parameters as in [35].",
"The temporal evolution of each ion species $n_{\\mathrm {i}}$ ($i = 1, 2, \\dots , n$ , listed in table REF ) is given by $\\partial _t n_{\\mathrm {i}} = - \\nabla \\cdot (\\pm \\mu _{\\mathrm {i}} {\\mathrm {E}} n_{\\mathrm {i}}) + S_{\\mathrm {i}}\\,,$ where the $\\pm $ accounts for the ion charge, $\\mu _{\\mathrm {i}}$ the ion mobility, and $S_{\\mathrm {i}}$ is the sum of ion source terms from the reactions listed in table REF .",
"For simplicity, we use a constant ion mobility $\\mu _{\\mathrm {i}}=2.2 \\times 10^{-4}$ m$^2$ V$^{-1}$ s$^{-1}$ [36] for all ion species, as in [37], [38].",
"For O$_2^+$ , the photoionization source term $S_{\\mathrm {ph}}$ is included in $S_{\\mathrm {i}}$ .",
"The electric field ${\\mathrm {E}}$ is calculated as ${\\mathrm {E}} = - \\nabla \\phi $ .",
"The electric potential $\\phi $ is obtained by solving Poisson's equation $\\nabla ^2 \\phi = - \\rho /\\varepsilon _0\\,,$ where $\\rho $ is the space charge density and $\\varepsilon _0$ is the vacuum permittivity.",
"Equation (REF ) is solved using the geometric multigrid method included in the Afivo library [39], [40]."
],
[
"Chemical reactions and input data",
"We have constructed a set of 303 chemical reactions, based on the reactions from [23] with additional reactions primarily from [19], [20], [21], [22].",
"A list of all 58 considered species is given in table REF .",
"The complete reaction list is given in , which also contains a list of the considered excited states of N$_2$ and O$_2$ .",
"The transport coefficients $\\mu _{\\mathrm {e}}$ and $D_{\\mathrm {e}}$ , and reaction rate coefficients for reactions R1–R31 in table REF are functions of the reduced electric field $E/N$ , where $E$ is the electric field and $N$ is the gas number density.",
"These coefficients were computed with BOLSIG$+$ , a two-term electron Boltzmann equation solver [41], using the temporal growth model.",
"Electron-neutral scattering cross sections for $\\mathrm {N}_2$ and $\\mathrm {O}_2$ were obtained from the Phelps database [42], [43]."
],
[
"Computational domain and initial condition",
"The 2D axisymmetric computational domain used in the simulations is illustrated in figure REF .",
"The domain measures 40 mm in the $r$ and $z$ directions, and it contains a plate-plate geometry with a needle protrusion at the upper plate.",
"To generate non-branching streamers with widely varying properties (such as radius and velocity), we use three different electrodes.",
"These electrodes are rod-shaped with semi-spherical tips, with lengths $L_\\mathrm {rod}$ of 2 mm, 4 mm and 6 mm, and radii of 0.2 mm, 0.4 mm and 0.6 mm, respectively.",
"Except for a small region near the rod electrode, the axial electric field is approximately uniform and equal to the average electric field between two plate electrodes, which is here defined as the background electric field $E_\\mathrm {bg}$ .",
"Figure: Schematic view of the 2D axisymmetric computational domain, measuring 40 mm ×\\times 40 mm.Right: the initial electric field without a streamer and the rod electrode geometry for L rod =L_\\mathrm {rod}= 4 mm.Boundary conditions for the electric potential φ\\phi are also indicated.Left: the axial electric field.E bg E_\\mathrm {bg} is the average electric field between two plate electrodes.For the electric potential, a Dirichlet boundary condition is applied on the upper plate and rod electrode (corresponding to the applied voltage).",
"The lower plate is grounded, and a homogeneous Neumann boundary condition is applied on the outer axial boundary.",
"For all plasma species densities, homogeneous Neumann boundary conditions are applied on all domain boundaries, including the rod electrode.",
"Secondary electron emission from electrodes due to ions and photons is not included.",
"As an initial condition, homogeneous background ionization with a density of $10^{11}\\,\\mathrm {m}^{-3}$ for both electrons and N$_2^+$ is included.",
"All other ion densities are initially zero.",
"For computational efficiency, the Afivo-streamer code includes adaptive mesh refinement.",
"We use the same refinement criteria for the grid spacing $\\Delta x$ as [35], which lead to a minimal grid spacing of $\\Delta x_\\mathrm {min}=$ 1.22 $\\mu $ m."
],
[
"Voltage waveform",
"A single voltage pulse with a rise time of 1 ns is used, during which it increases linearly.",
"The applied voltage is then constant until the streamer has reached a desired streamer length $L_\\mathrm {s}$ , after which the voltage is turned off with 1 ns fall time.",
"Here the streamer length $L_\\mathrm {s}$ is defined as the distance between the rod electrode tip and the streamer position $z_{\\mathrm {head}}$ at which the electric field has a maximum.",
"We use such a length-dependent voltage waveform so that we can study the effect of $L_\\mathrm {s}$ on energy efficiencies (due to more or less Joule heating in the channel, see section REF ).",
"Furthermore, it allows us to compare results at the same streamer lengths.",
"All simulations are performed until $t = 500$ ns."
],
[
"Streamer dynamics",
"We first present an example of a positive streamer in dry air.",
"A background field $E_\\mathrm {bg}$ of 14 kV/cm (half of the breakdown field of 28 kV/cm in air) was used, and the voltage was turned off after 20 ns.",
"Furthermore, a rod electrode length $L_\\mathrm {rod}=4$ mm and a desired streamer length $L_\\mathrm {s}=18$ mm were used.",
"Figure REF shows the time evolution of the electric field and electron density profiles for this example, together with its axial profiles.",
"Figure: Time evolution of (a) the electric field EE, (b) the electron density n e n_\\mathrm {e}, (c) the on-axis electric field EE, and (d) the line conductivity σ * \\sigma ^* for a positive streamer in E bg =14E_\\mathrm {bg}=14 kV/cm with L rod =4L_\\mathrm {rod}=4 mm and L s =18L_\\mathrm {s}=18 mm in air.Note that profiles of EE after 30 ns are shown on a different scale; for n e n_\\mathrm {e} the same scale is used at all times.Profiles of n e n_\\mathrm {e} are shown on a logarithmic scale.All panels in (a) and (b) are zoomed in into the region where 0 ⩽r⩽\\leqslant r \\leqslant 3 mm and 15 ⩽z⩽\\leqslant z \\leqslant 40 mm.The solid and dashed lines in panels (c) and (d) correspond to the streamer before and after the voltage is turned off, respectively.The streamer initiates from the rod electrode tip, after which its velocity and radius increase almost linearly with time, from about $0.7\\times 10^6$ m/s to $1.4\\times 10^6$ m/s and from about 0.5 mm to 0.9 mm, respectively.",
"The streamer radius is here defined as the electrodynamic radius at which the radial component of the electric field has a maximum.",
"The line conductivity $\\sigma ^*$ at the streamer head also increases as the streamer grows, due to its increasing radius, from about $3\\times 10^{-7}$ S$\\cdot $ m to $3\\times 10^{-6}$ S$\\cdot $ m. Here $\\sigma ^*$ is computed as $\\sigma ^*(z) = 2 \\pi e \\int _0^{10\\,\\mathrm {mm}} r n_\\mathrm {e} \\mu _\\mathrm {e} \\, \\mathrm {d} r\\,,$ where $e$ is the elementary charge.",
"In contrast, the electron density at the streamer head decreases slightly from about $6\\times 10^{19}$ m$^{-3}$ to $5\\times 10^{19}$ m$^{-3}$ .",
"Both the maximal electric field at the streamer head and the lowest internal on-axis electric field inside the streamer channel stay approximately constant during the voltage pulse; they are about 120 kV/cm and 6 kV/cm, respectively.",
"After 20 ns the voltage is turned off and the streamer stops, but the simulation continues up to 500 ns.",
"The streamer channel gradually loses its conductivity, with a 3–5 orders of magnitude decrease in the electron density and line conductivity.",
"The channel radius increases slightly from about 0.9 mm to 1.2 mm due to ion motion.",
"At 500 ns, the maximal electric field is about 9 kV/cm, and the lowest internal field is about 0.1 kV/cm.",
"The total deposited energy $Q_\\mathrm {total}$ at 500 ns is about 80 $\\mu $ J, which is here computed as $Q_\\mathrm {total} = \\int _{0}^{T_\\mathrm {total}}\\int _{\\Omega } j \\cdot E \\,\\mathrm {d} \\Omega \\mathrm {d} t\\,,$ where $j = e n_\\mathrm {e} \\mu _\\mathrm {e} E$ is the electron conduction current density, $\\Omega $ the computational domain, and $T_\\mathrm {total}=500$ ns."
],
[
"Plasma chemistry",
"We now look into the plasma chemistry of the streamer case mentioned above.",
"We consider 58 chemically active species and 303 chemical reactions, as shown in table REF .",
"A list of gross and net productions of all 58 species at $t=500$ ns is given in table REF , in units of the number of molecules/atoms produced in some particular state.",
"Here gross production means the total time and space integrated production of a specific species, without taking into account loss processes.",
"For net production, loss processes are taken into account.",
"Table: The gross and net productions of all 58 species for the streamer corresponding to figure at t=500t=500 ns.",
"Production below 10 3 10^3 has been replaced by ∼0\\sim 0.The time evolution of the gross and net production of 17 species is shown in figure REF , namely $e$ (electrons), N$_2(J)$ , N$_2(v)$ , O$_2(J)$ , O$_2(v)$ , O$_2$ (a), N($^4$ S), O($^3$ P), O$_2^+$ , O$_4^+$ , N$_4^+$ , O$_2^-$ , O$_4^-$ , O$_3$ , NO, NO$_2$ and N$_2$ O.",
"Note that the total vibrationally excited state N$_2(v)$ is the sum of each vibrationally excited state from N$_2(v_1)$ to N$_2(v_8)$ , and the total vibrationally excited state O$_2(v)$ is the sum of each vibrationally excited state from O$_2(v_1)$ to O$_2(v_4)$ .",
"Figure: Time evolution of the gross and net productions of 17 species for the streamer corresponding to figure .The vertical dotted black lines correspond to the moment when the voltage is turned off, namely t=20t=20 ns.Figure REF shows that the gross and net production are the same for N$_2(J)$ , N$_2(v)$ , O$_2(J)$ and O$_2(v)$ , because their loss reactions were not included.",
"For O$_2$ (a), N($^4$ S), O($^3$ P), O$_3$ , NO, NO$_2$ and N$_2$ O, gross and net production are almost equal due to relatively slow loss processes.",
"The gross and net production of O($^3$ P) differ because O($^3$ P) is gradually converted to O$_3$ and NO$_2$ , by the reactions $\\rm O(^3P) + O_2 + M \\rightarrow O_3 + M$ and $\\rm O(^3P) + NO \\rightarrow NO_2$ .",
"For charged species, net production is much lower than gross production since these species are rapidly converted to other species by attachment, detachment, ion conversion and recombination.",
"For most of the 17 species there is essentially no more production after 500 ns, with the exceptions being O$_3$ , NO$_2$ , O$_2^-$ and O$_4^-$ , which are produced relatively slowly."
],
[
"Parameter study of species production and energy efficiency",
"In this section, we simulate 60 positive streamers by varying the rod electrode length $L_\\mathrm {rod}$ from 2 mm to 6 mm, by varying the desired streamer length $L_\\mathrm {s}$ from 10 mm to 26 mm, and by varying the background field.",
"In figures REF and REF , we compare the total net production and energy efficiencies for nine neutral species: N$_2(J)$ , N$_2(v)$ , O$_2(J)$ , O$_2(v)$ , O$_2$ (a), N($^4$ S), O($^3$ P), NO and N$_2$ O.",
"To compare energy efficiencies, we use so-called $G$ -values, which give the net number of atoms or molecules produced per 100 eV of deposited energy, at $t=500$ ns.",
"We find the highest $G$ -values for N$_2(v)$ , about $10^2$ , and the lowest for N$_2$ O, about $10^{-2}$ .",
"Figure: The total net production at 500 ns of nine species versus the background field E bg E_\\mathrm {bg}.Results are shown for 60 positive streamers in different background fields with different electrode lengths (L rod =2,4,6L_\\mathrm {rod}=2, 4, 6 mm) and different streamer lengths (L s =10,18,26L_\\mathrm {s}=10, 18, 26 mm).Figure: GG-values for the production of nine species versus the background field E bg E_\\mathrm {bg} for the 60 streamers corresponding to figure .Figure REF shows that the total net production of the nine species increases with the background field and with the streamer length, as could be expected.",
"A longer rod electrode leads to a wider and faster streamer and therefore also to more production.",
"Figure REF shows that the $G$ -values for N$_2(v)$ , O$_2$ (a), N($^4$ S), O($^3$ P), NO and N$_2$ O vary by at most 30% to 55%.",
"Larger variation of about 70% to 130% can be observed for N$_2(J)$ , O$_2(J)$ and O$_2(v)$ .",
"The dependence of the $G$ -value on the background field for nine species can be grouped into three categories.",
"For N($^4$ S), O($^3$ P), NO and N$_2$ O, $G$ -values first slightly decrease and then increase with the background field, whereas for N$_2(J)$ , N$_2(v)$ , O$_2(J)$ and O$_2(v)$ $G$ -values monotonically decrease with the background field.",
"For O$_2$ (a), $G$ -values first decrease and then slightly increase with the background field.",
"These different dependencies can be explained by considering the activation energies for the reactions producing these species.",
"The key reactions producing N($^4$ S), O($^3$ P), NO and N$_2$ O have high activation energies ranging from 6 eV to 13 eV, so they are primarily produced in the high electric field near the streamer head, as illustrated in figure REF for N($^4$ S).",
"Specifically, N($^4$ S) and NO are mostly produced by an electron dissociation reaction, namely $\\rm e + N_2 \\rightarrow e + N(^4S) + N(^2D)$ , and the N$_2$ O production depends on electronic excitation reactions producing N$_2$ (A$_1$ ), N$_2$ (A$_2$ ) and N$_2$ (B).",
"O($^3$ P) production is determined by electron dissociation reactions of nitrogen and oxygen molecules, as well as electronic excitation reactions of nitrogen, e.g., N$_2$ (B).",
"Figure: The instantaneous production of N 2 (J)_2(J), O 2 _2(a) and N( 4 ^4S) at t=19t=19 ns for a positive streamer in E bg =14E_\\mathrm {bg}=14 kV/cm with L rod =4L_\\mathrm {rod}=4 mm and L s =18L_\\mathrm {s}=18 mm.N 2 (J)_2(J) is mostly produced in the channel, N( 4 ^4S) mostly in the head and O 2 _2(a) is produced in both.Profiles are shown using arbitrary units.Figure REF (a) shows the relation between the average maximal electric field $\\overline{E}_\\mathrm {max}$ and the background field $E_\\mathrm {bg}$ for the 60 streamer cases.",
"Here $\\overline{E}_\\mathrm {max}$ is an average over the time when the voltage is at its peak.",
"Note that $\\overline{E}_\\mathrm {max}$ first decreases and then increases with the background field, similar to the $G$ -values for N($^4$ S), O($^3$ P), NO and N$_2$ O.",
"This nonlinear $\\overline{E}_\\mathrm {max}$ dependence on the background field has also been observed in [44].",
"Figure REF (b) shows that there is an approximately linear relation between $\\overline{E}_\\mathrm {max}$ and these $G$ -values, taking N($^4$ S) as an example.",
"Figure: (a) The average maximal electric field E ¯ max \\overline{E}_\\mathrm {max} versus the background field E bg E_\\mathrm {bg} for the 60 streamers corresponding to figure .",
"(b) The simulated and estimated GG-values for N( 4 ^4S) versus E ¯ max \\overline{E}_\\mathrm {max} corresponding to panel (a).The symbols correspond to the 60 simulated cases, by taking an average over the time when the voltage is at its peak for E ¯ max \\overline{E}_\\mathrm {max}, by taking simulated GG-values at t=500t=500 ns and by taking estimated GG-values from equation () corrected with the factor Q head /Q total Q_\\mathrm {head}/Q_\\mathrm {total} from figure (d).In contrast, the electron-molecule reactions producing N$_2(J)$ , N$_2(v)$ , O$_2(J)$ and O$_2(v)$ have lower activation energies, below about 2 eV.",
"These reactions therefore primarily take place inside the streamer channel, where the electron density is high, as illustrated in figure REF for N$_2(J)$ .",
"Specifically, N$_2(J)$ and O$_2(J)$ are produced by rotational excitations of nitrogen and oxygen molecules, and N$_2(v)$ and O$_2(v)$ by vibrational excitations.",
"Figure REF shows that the production of N$_2(J)$ , N$_2(v)$ , O$_2(J)$ and O$_2(v)$ increases for higher background fields, but that the production of N($^4$ S), O($^3$ P), NO and N$_2$ O increases even more rapidly, so that $G$ -values for N$_2(J)$ , N$_2(v)$ , O$_2(J)$ and O$_2(v)$ decrease.",
"The production of O$_2$ (a) falls somewhat between these two regimes, because it can be produced by reactions with different activation energies.",
"There is a direct electronic excitation reaction generating O$_2$ (a) with an activation energy of 0.977 eV, but O$_2$ (a) is also produced indirectly from N$_2$ (A$_1$ ), N$_2$ (A$_2$ ) and N$_2$ (B), which correspond to higher activation energies, ranging from 6.17 eV to 8.16 eV.",
"Due to these different mechanisms, O$_2$ (a) is both produced inside the streamer channel and in the high electric field near the streamer head, as shown in figure REF ."
],
[
"Comparison with Naidis' estimate",
"In [27], Naidis proposed that “the maximum $G$ -values are governed by only one parameter – the reduced electric field in the streamer head $Y_\\mathrm {h}$ , and that their dependence on $Y_\\mathrm {h}$ in the typical range of propagating streamers is not strong, when both generation of active species and energy input occur mainly in the streamer head region.” The paper contains an analytic approximation for $G$ -values, that we for clarity rewrite as: $G_j = \\frac{2 n_j}{e E_\\mathrm {max}^2} \\int _{E_\\mathrm {ch}}^{E_\\mathrm {max}} \\frac{K_j}{\\mu _\\mathrm {e} E} \\, \\mathrm {d} E \\,,$ where $E_\\mathrm {max}$ is the maximal electric field at the streamer head, $E_\\mathrm {ch}$ the electric field in the streamer channel, $n_j$ the number density of molecules producing species $j$ , $K_j$ the field-dependent rate coefficient of electron-molecule reaction producing species $j$ , and $E$ is the electric field.",
"The integral is not sensitive to the particular value used for $E_\\mathrm {ch}$ , since it is meant to be used for species mainly produced in high electric fields, so we for simplicity use $E_\\mathrm {ch} = 0$ below.",
"Several approximations are made to derive equation (REF ): the streamer is assumed to be uniformly translating, the streamer head is locally assumed to be flat, so that one-dimensional integration can be performed, and Joule heating in the streamer channel is not taken into account.",
"As discussed in [27], it can be important to take Joule heating in the streamer channel into account.",
"We obtain this correction by measuring the deposited power in the streamer channel and in the streamer head region, as shown in figure REF .",
"The streamer head region was defined as the region where $E \\geqslant 28$ kV/cm.",
"There is still some Joule heating in the streamer channel after the voltage is turned off, which corresponds to currents that screen the electric field generated by the remaining space charge.",
"The fraction of power deposited in the channel increases with time.",
"However, even for short primary streamers, on time scales well below 10 ns, Joule heating in the streamer channel can still be important.",
"Figure: Time evolution of total deposited power P total P_\\mathrm {total}, power in the streamer head region P head P_\\mathrm {head} and in the streamer channel P channel P_\\mathrm {channel} for three positive streamers in E bg E_\\mathrm {bg} of (a) 10 kV/cm, (b) 18 kV/cm and (c) 26 kV/cm with L rod =4L_\\mathrm {rod}=4 mm and L s =18L_\\mathrm {s}=18 mm.The vertical dotted black lines correspond to the moment when the applied voltage has dropped to zero (after a 1 ns fall time).The streamer head region is defined as the region where E⩾28E \\geqslant 28 kV/cm.",
"(d) The fraction of energy deposited in the streamer head region Q head /Q total Q_\\mathrm {head}/Q_\\mathrm {total} versus the background field E bg E_\\mathrm {bg} for the 60 streamers corresponding to figure .A correction factor $Q_\\mathrm {head}/Q_\\mathrm {total}$ , i.e., the fraction of total energy deposited in the streamer head region, is computed by integrating the deposited power over time.",
"As shown in figure REF (d), this factor lies between 0.35 and 0.5 for the 60 cases considered in this paper.",
"Note that $Q_\\mathrm {head}/Q_\\mathrm {total}$ first decreases and then increases with the background field, similar to $\\overline{E}_\\mathrm {max}$ shown in figure REF (a).",
"In figure REF (b), we compare equation (REF ) corrected with the factors $Q_\\mathrm {head}/Q_\\mathrm {total}$ from figure REF (d) against simulation results, for the production of N($^4$ S).",
"For this comparison, we use $E_\\mathrm {max} = \\overline{E}_\\mathrm {max}$ and $E_\\mathrm {ch} = 0$ in equation (REF ).",
"The agreement is then surprisingly good, with deviations of up to about 25%.",
"We remark that variation of equation (REF ) is about 20% in the considered maximal field range, which is smaller than the variation in the correction factor from figure REF (d), which is about 35%."
],
[
"Comparison with negative streamers",
"We now compare the species production and $G$ -values for N($^4$ S), O($^3$ P) and NO between positive and negative streamers in different background fields.",
"A rod electrode length $L_\\mathrm {rod}=6$ mm and a desired streamer length $L_\\mathrm {s}=18$ mm were used.",
"Positive and negative streamers were obtained by changing the applied voltage polarity, while keeping all other simulation conditions the same.",
"In the background fields considered here, negative streamers have lower maximal electric fields than positive ones, as shown in figure REF and also obtained previously [45], [46].",
"Figure REF shows that total net production and $G$ -values for N($^4$ S), O($^3$ P) and NO are also lower for negative streamers, with the $G$ -values for N($^4$ S) and NO being up to 60% lower compared to positive streamers.",
"Figure: Comparison of positive and negative streamers.",
"(a) Time-averaged maximal electric field E ¯ max \\overline{E}_\\mathrm {max}, (b) velocity vv, (c) radius RR and (d) fraction of energy deposited in the streamer head region Q head /Q total Q_\\mathrm {head}/Q_\\mathrm {total}, all versus the background field E bg E_\\mathrm {bg}.The velocity and radius are given at the moment each streamer reached L s =18L_\\mathrm {s}=18 mm.",
"(e) The simulated and estimated GG-values for N( 4 ^4S) versus E ¯ max \\overline{E}_\\mathrm {max} corresponding to panel (a).",
"The estimated GG-values from equation () were corrected with the factor Q head /Q total Q_\\mathrm {head}/Q_\\mathrm {total} from panel (d).Figure: The total net production and GG-values for N( 4 ^4S), O( 3 ^3P) and NO versus the background field E bg E_\\mathrm {bg} for both positive and negative streamers with L rod =6L_\\mathrm {rod}=6 mm and L s =18L_\\mathrm {s}=18 mm.Furthermore, we again compare our results with Naidis' estimate, as discussed in section REF .",
"Figure REF (d) shows that the correction factor $Q_\\mathrm {head}/Q_\\mathrm {total}$ is smaller for negative streamers, indicating that more energy is deposited in the negative streamer channel.",
"As shown in figure REF (e), the simulated $G$ -values for N($^4$ S) agree well with estimated $G$ -values for both positive and negative streamers.",
"For both polarities the same approximately linear relationship between $G$ -values and $\\overline{E}_\\mathrm {max}$ can be observed.",
"We remark that in [47], the authors experimentally measured the energy efficiency of oxygen radical production in primary and secondary streamers for both voltage polarities.",
"In contrast to our results, they found a higher energy efficiency for negative streamers, but they noted that this efficiency could not accurately be measured for negative streamers due to a larger mismatch of the source impedance."
],
[
"Comparison with experimental data for O$_3$",
"O$_3$ is produced relatively slowly, on time scales significantly longer than the 500 ns considered here, as shown in figure REF .",
"In experiments, O($^3$ P) production is usually inferred from the measured O$_3$ production since most (e.g.",
"90%) of O($^3$ P) is converted to O$_3$ by $\\rm O(^3P) + O_2 + M \\rightarrow O_3 + M$ .",
"We can therefore estimate $G$ -values for O$_3$ based on the O($^3$ P) production, which gives $G$ -values of about 6–8, see figure REF (e).",
"Below, we list several experimentally obtained $G$ -values for O$_3$ production.",
"In [48], the authors found $G$ -values for O$_3$ production in O$_2$ (20%)/N$_2$ of about 4–6, using a 1 Hz pulse repetition rate in a point-plane gap.",
"Furthermore, it was observed that the energy efficiency of O$_3$ production was enhanced by reducing the pulse width, and that production was more efficient in primary streamers than in secondary streamers.",
"In [49], $G$ -values for O$_3$ in air of about 2.5–3.4 were found, in a large scale wire-plate reactor with a relative humidity ranging from 35% to 42% and pulse repetition rates of 10–400 Hz.",
"It was observed that short pulses increased energy efficiencies of O$_3$ production.",
"In [50], a higher $G$ -value of about 13 was measured for O$_3$ in dry air at a pulse repetition rate of 30 Hz in a wire-to-cylinder electrode reactor, using a short voltage pulse of 5 ns.",
"In [47], the authors proposed that primary streamers were more efficient than secondary streamers for O$_3$ production, and $G$ -values of about 10–16 were found for primary positive streamers in air in a wire-plate geometry with pulse repetition rates of 50–100 Hz, and relative humidity of 13–18%.",
"There are many differences between these experiments and our simulations, for example in the pulse repetition rate (which can lead to gas heating), pulse width, humidity, streamer branching and electrode geometry.",
"Our simulated $G$ -values agree reasonably well with some of the above experimental data.",
"The somewhat higher experimental $G$ -values of 10–16 may be a result of gas heating, which can increase reduced electric fields, making O$_3$ production more efficient."
],
[
"Conclusions and outlook",
"We have studied how the plasma chemistry of streamer discharges in dry air varies with streamer properties.",
"Simulations were performed at 300 K and 1 bar in a 40 mm plane-plane gap with a needle protrusion, using a 2D axisymmetric fluid model.",
"Sixty different positive streamers were obtained by varying the electrode length, the applied voltage and the duration of the voltage pulse.",
"The obtained streamers had diameters ranging from about 0.5 mm to 3.5 mm, velocities ranging from about $0.3\\times 10^6$ m/s to $5.7\\times 10^6$ m/s and maximal electric fields ranging from about 120 kV/cm to 150 kV/cm.",
"For species with a relatively high activation energy, such as N($^4$ S), O($^3$ P), NO and N$_2$ O, $G$ -values were found to vary by about 30% to 55% between the 60 positive cases.",
"The most important factors behind this variation were differences in the fraction of energy deposited in the streamer head region and variation in the maximal electric field at the streamer head.",
"When accounting for both factors, good agreement was obtained with an analytic estimate for $G$ -values proposed by Naidis [27].",
"For comparison, eight negative streamers were also simulated.",
"They were found to be less energy efficient for the production of N($^4$ S), O($^3$ P) and NO, mainly due to a lower maximal electric field and because more energy was deposited in the streamer channel.",
"Our results suggest two main ways to increase the energy efficiency with which species with a high activation energy are produced.",
"First, Joule heating losses in the streamer channel can be reduced, and second, the maximal reduced electric field at the streamer head can be increased.",
"Short voltage pulses with a high applied voltage can contribute to both factors.",
"Outlook.",
"In future computational work, it would be interesting to study the effect of repetitive pulses, as well as gas heating and humidity effects, and to directly compare with experimental data.",
"Another question is whether electric fields at streamer heads can be increased by reducing photoionization (for example through the addition of CO$_2$ [51]), as this leads to more frequent branching and narrower channels.",
"Finally, it would be interesting to study whether “minimal streamers” in low background fields can achieve a high energy efficiency for species production, due to their high maximal electric field [44].",
"B.G.",
"was funded by the China Scholarship Council (CSC) (Grant No. 201906280436).",
"We thank Dr. Atsushi Komuro from the University of Tokyo for sharing chemical reactions.",
"We acknowledge fruitful discussions with Prof. Dr. Ute Ebert, Dr. Bastiaan Braams and Mr. Hemaditya Malla at CWI."
],
[
"Data availability statement",
"The data that support the findings of this study are openly available at the following URL/DOI: https://doi.org/10.5281/zenodo.7093075."
],
[
"List of chemical reactions",
"Table REF shows the excited states of N$_2$ and O$_2$ included in the model, as well as their activation energies.",
"Furthermore, we give a complete list of 303 chemical reactions in table REF , with their reaction rate coefficients and references.",
"This reaction set contains different processes: rotational excitation, vibrational excitation, electron excitation, electron dissociation, electron impact ionization, electron attachment, electron detachment, negative ion conversion, positive ion conversion, electron-ion recombination, ion-ion recombination and neutral species conversion.",
"Note that reactions R37 and R226 are actually three-body reactions.",
"These reactions are included as two-body processes because the intermediate states have long lifetimes [19].",
"Table: Excited states of N 2 _2 and O 2 _2 with activation energies and the corresponding effective states used in the model.",
"The table is partially based on the one in .Table: List of reactions included in the model, with reaction rate coefficients and references.The symbol M denotes a neutral molecule (either N 2 \\mathrm {N_2} or O 2 \\mathrm {O_2}).Reaction rate coefficients are in units of cm 3 s -1 \\mathrm {cm^3\\,s^{-1}} and cm 6 s -1 \\mathrm {cm^6\\,s^{-1}} for two-body and three-body reactions, respectively.Reaction rate coefficients for reactions R1–R31 are functions of the reduced electric field E/NE/N, and they are computed with BOLSIG+ .TT(K) and T e T_e(K) are gas and electron temperatures, respectively.T e ( eV )T_e(\\mathrm {eV}) is also a measure of electron temperature but in a unit of eV, by multiplying T e (K)T_e(\\mathrm {K}) with the Boltzmann constant.Table: (Continued from previous page)Table: (Continued from previous page)Table: (Continued from previous page)Table: (Continued from previous page)Table: (Continued from previous page)"
]
] | 2210.07782 |
[
[
"Stellar feedback impact on the ionized gas kinematics in the dwarf\n galaxy Sextans A"
],
[
"Abstract Feedback from massive stars shapes the ISM and affects the evolution of galaxies, but its mechanisms acting at the small scales ($\\sim 10$ pc) are still not well constrained observationally, especially in the low-metallicity environments.",
"We present the analysis of the ionized gas (focusing on its kinematics, which were never studied before), and its connection to the massive stars in the nearby ($D \\sim 1.4$ Mpc) star-forming very metal-poor ($Z\\sim 0.07 Z_\\odot$) galaxy Sextans A.",
"The analysis is based on the observations with a scanning Fabry-Perot interferometer, long-slit spectroscopy and imaging in emission lines with narrow-band tunable filters.",
"We found 10 expanding superbubbles of ionized gas with ages of 1-3 Myr.",
"We argue that 3 of them are probable supernovae remnants, while the pre-supernovae feedback is an important source of energy for blowing-out the remaining superbubbles.",
"The two brightest sites of star formation exhibit signs of outflowing ionized gas, which is traced by its ionized and atomic gas kinematics and (in one case) by its emission line flux ratios.",
"Overall, the ionized gas kinematics in Sextans A is highly affected by the feedback from several generations of massive stars and inconsistent with the mere solid-body rotation observed in atomic hydrogen."
],
[
"Introduction",
"Massive stars play a crucial role in the evolution of the interstellar medium (ISM) and the galaxies as a whole.",
"Through the ionizing radiation, stellar winds and supernovae explosions are the main forces of stellar feedback , which are able to carve out low-density bubbles and shells in the ISM.",
"The cumulative contribution of energy and momentum input from multiple stars in OB associations and young star clusters lead to the formation of large superbubbles having sizes up to several hundred pc , .",
"The evolution of superbubbles can lead to the redistribution of star formation activity and chemical abundance in a galaxy .",
"The role of stellar feedback is especially prominent in dwarf irregular (dIrr) galaxies, where the larger structures can emerge due to their thick gas-rich discs and a lack of spiral density waves.",
"Nearby dIrr galaxies are often considered as ideal laboratories for studying the effects of massive stellar feedback on the surrounding ISM.",
"The resolved observational studies of the ISM and star formation regulated by stellar feedback were performed for several well-known nearby galaxies , , , , , , .",
"However, the number of such analyses for very low-metallicity galaxies ($Z<0.1Z_\\odot $ ) is still very limited , , .",
"At the same time, theoretical models and observations of stars suggest that the efficiency of the different feedback agents should vary with the metallicity.",
"For example, both the power of stellar winds and the rate of SNe type II decline in the low-metallicity regime.",
"demonstrated that the relative contribution of several feedback forces differ in superbubbles observed in the Small Magellanic Cloud (SMC) from that in the Large Magellanic Cloud (LMC), and they are also different from that in the Milky Way .",
"conclude that in the LMC the contributions from massive stars and SNe are almost equal while in the SMC SNe explosions are the dominant sources of the mechanical feedback.",
"It is important to check how the stellar feedback impacts the ISM at the small scales in very low-metallicity environments.",
"Table: Data Availability"
]
] | 2210.07726 |
[
[
"Multi-Task Learning based Video Anomaly Detection with Attention"
],
[
"Abstract Multi-task learning based video anomaly detection methods combine multiple proxy tasks in different branches to detect video anomalies in different situations.",
"Most existing methods either do not combine complementary tasks to effectively cover all motion patterns, or the class of the objects is not explicitly considered.",
"To address the aforementioned shortcomings, we propose a novel multi-task learning based method that combines complementary proxy tasks to better consider the motion and appearance features.",
"We combine the semantic segmentation and future frame prediction tasks in a single branch to learn the object class and consistent motion patterns, and to detect respective anomalies simultaneously.",
"In the second branch, we added several attention mechanisms to detect motion anomalies with attention to object parts, the direction of motion, and the distance of the objects from the camera.",
"Our qualitative results show that the proposed method considers the object class effectively and learns motion with attention to the aforementioned important factors which results in a precise motion modeling and a better motion anomaly detection.",
"Additionally, quantitative results show the superiority of our method compared with state-of-the-art methods."
],
[
"INTRODUCTION",
"With the growth of surveillance cameras, automatic analysis of video contents is called for.",
"Generally, the aim of this analysis is to detect anomalous events (i.e., unfamiliar or unexpected events in a given context [10]) in the video which may demand instant action.",
"Due to the rarity and diversity of anomalous events, adequate training anomaly samples are typically not available for supervised training.",
"Hence, researchers of the field dedicated more interest to semi-supervised approaches, in which normals are learned via a proxy task (i.e., a task that indirectly helps us to achieve our target goal) and detect the anomalies by finding the deviations from normality.",
"For example, reconstruction of current frames or prediction of masked frames are popular proxy tasks in video anomaly detection (VAD), in which the trained models on normals show a worse reconstruction or prediction result for anomalies, and the error of the estimation determines the anomaly score.",
"Researchers have employed different proxy tasks in multiple branches to consider different modalities (mostly appearance and motion) in their approaches.",
"Different proxy tasks are meant to be complementary to each other and consequently are combined towards a higher performance.",
"For example, Nguyen and Meunier [19] proposed a two-stream network in which one stream models the appearance features and detects appearance-based anomalies while the other models motion patterns and looks for motion anomalies.",
"Multiple similar strategies have been proposed and each work proposes a different combination of proxy tasks with different anomaly score fusion strategies [10], [19], [5], [13], [9].",
"Recently, researchers (e.g[10]) proposed to add more proxy tasks (in multiple branches) to cover more spatio-temporal patterns.",
"However, although recent methods attain better results, they still do not consider motion patterns thoroughly or do not explicitly analyze the class of the object for anomaly detection.",
"To address the aforementioned shortcomings, inspired by [10], we propose an improved multi-task learning based VAD method.",
"Different from [10] and following the success of the semantic segmentation task in considering the object class in VAD [5], we deploy the semantic segmentation proxy task, in the appearance branch.",
"Additionally, contrarily to [10], which performs anomaly detection at the object level, we propose a holistic VAD method, to not lose location information.",
"Baradaran and Bergevin [5] proposed to use optical flow magnitude (OFM) estimation for each object to detect motion anomalies.",
"However, the downside of their approach is that they neglect the motion direction information in motion estimation to overcome the confusion problem of the network in estimating optical flow (OF) from a single frame, as was proposed in [19].",
"Hence, they do not effectively detect the anomalies which are due to sudden direction changes (such as in fighting, jumping, etc).",
"Besides, to make a correspondence between each object and its motion magnitude (i.e., pixel-based object displacement through frames) some important factors, such as motion direction, object part, and distance of the object from the camera were not taken into account.",
"We address these issues by leveraging two different attention mechanisms.",
"We take advantage of a spatial and channel attention network and apply it to feature maps of the mid-layers in the encoder, which helps the network consider object parts (hands, legs, etc) for motion magnitude estimation.",
"Moreover, we designed a new attention network that helps estimate the motion magnitude for each object with attention to its distance from the camera and the direction of its motion (details in section 3).",
"Finally, we add future frame prediction as another proxy task, to find sudden motion changes.",
"However, to reduce the network size, we combine the semantic segmentation and future frame prediction tasks in a single task to be performed by a single network.",
"In summary, our contributions are multifold: A novel multi-task learning based video anomaly detection method which combines three different proxy tasks to more generally consider appearance and motion features for anomaly detection.",
"A combination of semantic segmentation and future frame prediction tasks in a single novel task to find both appearance and motion anomalies.",
"A novel attention network to estimate precise motion magnitude for an object with attention to its motion direction and its distance from the camera.",
"We also employ a spatial and channel attention mechanism in the backbone of the motion estimation branch to boost meaningful features and generate estimations specific to different object parts."
],
[
"PREVIOUS WORK",
"Researchers have formulated anomaly detection as frame(s) reconstruction [12], [7], [14], [1], [20], [11] or as prediction tasks [16], [3], [6], [24], [26], assuming that the unsupervised network (e.gA UNet) trained on normals generates a higher reconstruction/prediction error for anomalies.",
"However, all previously mentioned methods consider low-level features (color, intensity, etc) for anomaly detection and do not explicitly consider the class of objects for their evaluation.",
"Object-centric-based VAD methods [13], [23] although detect and crop objects out of frame (by a pre-trained object detector) for a better concentration on objects, they also consider low-level features in training and inference, since they reconstruct or predict cropped objects for VAD.",
"To address the mentioned challenge, inspired by [15], Baradaran and Bergevin [5] proposed a teacher-student-based VAD that uses semantic segmentation as the proxy task and hence is able to explicitly consider the class of the objects for VAD.",
"Baradaran and Bergevin [4] report that single-branch approaches (such as [12], [7]) do not effectively cover all motion cases and are usually dominated by appearance features.",
"Hence, to tackle the shortcomings of one stream methods, researchers [19], [5], [13], [9] have proposed two-stream VAD methods, to detect motion and appearance effectively in separate branches.",
"They mostly tackle the motion anomaly detection problem by reconstructing motion features (e.goptical flow features) [17].",
"One of the most noticeable related works is proposed by Neygun and Meunier [19] which formulates motion learning as a translation from the input frame to its corresponding optical flow map, trying to consider the correspondence between objects and their motion for the motion anomaly detection.",
"Baradaran and Bergevin [5] proposed to translate the input frame to its optical flow magnitude (considering only the magnitude of motion), as they reported that the network can be confused while predicting the complete optical flow from a single frame.",
"Although their approach addresses the issue of confusion and effectively learns correspondence between each object and its motion magnitude and detects related anomalies, their method neglects the direction information.",
"Hence their method may fail in the detection of anomalies which are due to sudden direction changes.",
"Moreover, they may fail in precise motion magnitude prediction since the perceived motion in frames is also a function of the distance from the camera (i.e.",
"the same motion magnitude looks smaller at distance) and also the direction of motion (objects moving parallel to the camera looks faster than the same motion away from the camera), and these factors have not been taken to account in their method.",
"Inspired by the success of multi-task learning based methods in considering different aspects essential for anomaly detection, we build on recent methods ([10], [5]) and we add future frame prediction for sudden direction change detection and attentive layers which estimate motion with attention to object part, motion direction, and distance of the object."
],
[
"METHOD",
"We propose a multi-task learning based video anomaly detection method that leverages three self-supervised proxy tasks in two separate branches in order to model normal patterns and consequently detect anomalies.",
"The pipeline of the proposed method is illustrated in Fig.",
"1(a) and described in detail in the following.",
"Figure: (a) The Framework of our proposed method (training stage).",
"Mask-RCNN and OF extractors (as teachers respectively for the appearance-motion and the motion branches) provide the required pseudo-GT for training their students.",
"During the inference stage, the output of each student network (estimation) is compared to its related pseudo-GT (expectation) to calculate the respective anomaly score.",
"(b) The motion branch in details.",
"Figures are best viewed in color."
],
[
"Multi-task learning",
"Inspired by [10] we propose a multi-task learning based VAD method, which leverages three proxy tasks in two branches for video anomaly detection.",
"The first branch (named as appearance-motion branch) combines two different tasks (semantic segmentation and future frame prediction) to model appearance and motion simultaneously.",
"The second branch (i.e., the motion branch) is in charge of learning the correspondence between each normal object and its normal motion magnitude, with attention to its distance from the camera, motion direction, and its body parts.",
"In this way, all three tasks are complementary to each other, each trying to find anomalies for which the other tasks may be sub-optimal."
],
[
"The appearance-motion branch",
"Experiments in [5] show that leveraging semantic segmentation as a proxy task helps to effectively find the appearance anomalies considering the class of objects.",
"Moreover, our experiments show that future frame prediction is a suitable task to detect sudden motion changes (e.gdirection changes, acceleration).",
"A network trained to predict the future frame of two consecutive normal frames fails to predict the precise location of the objects while having sudden motion changes (such as in fighting, jumping, etc).",
"To leverage the abilities of both tasks, the first branch of our method combines two different tasks of semantic segmentation and future frame prediction in a new single task and aims to predict the semantic segmentation map of the future frame by observing two consecutive frames.",
"In this way, not only does it learn the class of the normal objects in the frame during the training, but also it learns the normal evolution between two normal frames.",
"This branch follows a teacher-student strategy for anomaly detection.",
"During training, a student (resnet34-UNet in our method) gets two consecutive frames and learns to generate the semantic segmentation map of the future frame (i.e., the next frame), assuming that at inference time, the prediction error would be higher for anomalies.",
"The pseudo Ground-Truth (GT) for training the student network is generated by Mask-RCNN which is trained on a large dataset (MSCOCO in our experiments)."
],
[
"The motion branch",
"Although the first branch partially considers motion anomalies, it does not cover all motion cases.",
"Hence, in the second branch, we employ the idea proposed in [5] which is to learn a normal motion magnitude for each object by translating an input frame to its optical flow magnitude map.",
"This branch also follows a teacher-student strategy where the student (a vgg16-UNet here) learns to translate an input frame to its optical flow magnitude map, generated by a pre-trained optical flow extractor (considering its past frame) as a pseudo-GT.",
"In this way, the student network learns the correspondence between each object and its normal motion magnitude during training, assuming that it will make an imprecise motion estimation for objects moving faster than their normal motions.",
"However, this method has some challenges as follows: 1) some objects (such as humans) do not have a constant motion magnitude in all their parts.",
"For example, hands and legs usually have a larger motion magnitude compared to the chest and head.",
"This factor has not been taken into account in [5].",
"2) The motion magnitude perceived in frames (pixel displacement of objects) is a function of some variables such as motion direction and object distance from the camera.",
"Objects moving parallel to the camera show a larger motion (i.e, generates a larger optical flow magnitude) compared to objects moving away/towards from/to the camera.",
"Hence estimation of motion magnitude without attention to these factors would result in an imprecise prediction.",
"In our motion branch, we build on the method proposed in [5] by employing two different attention mechanisms to address the mentioned shortcomings.",
"We employ a spatial and channel attention network in the main network, to dedicate more attention to special body parts (such as legs and hands) and we also design a new attention network to help the network make predictions with attention to supplementary information (such as motion direction and relative distance information).",
"The details of the motion branch are illustrated in Fig.",
"1(b) and the details of the attention mechanisms are described next."
],
[
"Spatial and channel attention",
"By visualizing the generated feature maps of the basic vgg16-UNet in the motion branch, we observed that the encoder of the vgg16-UNet (trained semi-supervisedly) generates different levels of features through different layers and the mid-layers generate some feature maps that are activated for different object parts.",
"Hence, we applied the spatial and channel attention (SCSE) mechanism as proposed in [2] on the feature maps of mid-layers to help the network dedicate more attention to different body parts."
],
[
"Attention to distance and direction",
"To provide attention to distance and direction features, we designed another attention network, as illustrated in Fig.",
"2.",
"The network uses the direction and distance information as inputs and generates an attention map to apply to attentive layers.",
"Figure: The proposed attention network.",
"The generated attention map is applied to feature maps to provide attention to supplementary information (i.e., depth and direction).",
"Best viewed in color.In the motion branch, the teacher network extracts the optical flow of two consecutive frames (${I_{t-1}}$ , ${I_{t}}$ ) and provides the magnitude of the optical flow as the pseudo-GT to be utilized by the student to map the input frame (${I_{t}}$ ) to its OFM.",
"The generated OF features by the teacher (denoted as OF in (1)) encompasses the direction information in addition to magnitude information (1).",
"In (1) Mag stands for the magnitude of the optical flow and Ang shows the angle of the motion relative to the horizontal axis in the frame.",
"The Ang features can be supplied to our attention network as direction features.",
"However, we calculate the Cosine and Sine of Ang to normalize it and to generate two new different features: motion parallel to the camera (denoted as X in (2) and Fig.",
"2) and the motion towards/away to/from the camera (denoted as Y in (3) and Fig.",
"2).",
"$Mag, Ang = OF(I_{t-1},I_{t})\\\\X = |Cos(Ang)|\\\\Y = |Sin(Ang)|$ Since we do not have the actual information about the object's distance from the camera, we extract the depth maps of the frames to represent the relative distance information of the objects to the camera.",
"We use the MiDaS [22], [21] to estimate the relative depth maps of input frames.",
"MiDaS is pre-trained on 10 different datasets using multi-objective optimization which ensures high-quality depth map generation for a wide range of inputs.",
"We use the hybrid version of the method to keep the equilibrium between precision and execution time.",
"Fig.",
"2 shows how the extracted informations are combined and processed inside the attention network to generate the attention map."
],
[
"Inference",
"At the inference stage, we provide both normal and abnormal frames to each branch and we compare the estimation of each branch with their related expectations (pseudo-GT generated by teachers of each branch) to calculate the anomaly map of that frame.",
"We calculate the sum of activations in each frame as the anomaly score of that frame.",
"For temporal denoising, Savitzky–Golay filter [8] is applied on the anomaly scores.",
"Finally, we flag a frame as an anomaly if the anomaly score of any branch is larger than a predefined threshold.",
"As the networks of each branch have just been trained with normal frames, we assume to observe a considerable difference between estimations and expectations of each branch if it contains any anomalies specific to that branch.",
"These anomaly maps are supposed to contain activations at the position of the anomalies in the frame."
],
[
"EXPERIMENTS AND RESULTS",
"We evaluated the performance of our proposed method and the effectiveness of each contribution on the ShangahiTech [16] and UCSD-Ped2 [18] datasets.",
"The details of the experiments, qualitative and quantitative results, and also a comparison with state-of-the-art approaches appear next."
],
[
"Datasets",
"ShanghaiTech (ST) and UCSD-ped2 datasets are two of the benchmark datasets popularly used to evaluate semi-supervised VAD methods.",
"They provide only normal training frames in their training subsets and both normal and abnormal frames in the test subsets, along with frame-based and pixel-based annotations.",
"The definition of normality and anomaly is similar in both datasets.",
"People walking on the sidewalk (could be carrying bags or backpacks) are considered normal, however, the presence of some previously unseen objects (such as bikes, bicycles, cars, etc) or some previously unseen motion patterns (such as running, chasing, fighting, riding, etc) are considered as anomaly patterns.",
"Compared to UCSD-ped2, shanghaiTech is a more complicated dataset, since it has multiple different scenes (13 scenes) and much more anomalies.",
"On the other hand, low resolution and gray scale frames make the Ped2 dataset more challenging and prone to failures for the segmentation task."
],
[
"Evaluation metric",
"Following the state-of-the-art (SOTA) methods in the field, we provide our quantitative evaluation by measuring the frame-level AUC (Area Under Curve).",
"This curve is plotted by registering multiple True Positive Rate (TPR) and False Positive Rate (FPR) of the method by changing the anomaly score threshold from min to max.",
"The higher AUC indicates a better performance."
],
[
"Implementation details",
"Input frames in our experiments are resized to 256*256 for each branch.",
"We initialize encoders of both student branches (res34-UNet and vgg16-Unet) with parameters of the networks trained on Imagenet.",
"The learning rates of both branches are initialized to 0.001 however and are halved every 10 epochs.",
"We trained networks of both branches with the patch-based loss [5] (dividing input frame into 16 patches) by the Adam optimizer.",
"We employed Mask-RCNN (pre-trained on MSCOCO) as the teacher of the appearance branch and the Farneback algorithm from the OpenCV library as a teacher for pseudo-GT optical flow feature extraction from two consecutive frames $(\\Delta t=1)$ .",
"Finally, to discard the background and bring more attention to the foreground objects we mask the extracted optical flow maps with the extracted semantic segmentation maps."
],
[
"Qualitative analysis",
"To qualitatively analyze the effectiveness of the proposed method we visualize the anomaly maps for multiple normal and abnormal frames.",
"Figures 3, and 4 exhibits some qualitative results respectively for the appearance-motion and the motion branches.",
"Figure 3 contains multiple samples of anomalous frames (top row) and corresponding anomaly maps (bottom row), generated by the appearance-motion branch.",
"As can be seen in this figure, anomaly maps contain higher activations at anomalous objects (bicycle and bike in columns 1 and 2) or even at normal objects with sudden abnormal motions (legs and hands of a man while fighting in column 3 and for running or chasing persons in columns 4 and 5).",
"Figure: Qualitative results for the appearance-motion branch.Similar results can be observed in figure 4 for the motion branch.",
"As can be seen in this figure, our method generated larger activations for anamalous motions.",
"Figure: Qualitative results for the motion branch.",
"top: Input frames.",
"bottom: Anomaly map.Figure: Attention in motion estimation.",
"(a)Input frame.",
"(b)Anomaly map without attention.",
"(c)Anomaly map with attention.Figure 5 presents an anomalous frame (5a) and the generated anomaly map with and without applying the attention mechanisims.",
"As can be noticed, the generated anomaly map contains weaker activations at normals (inside the bounding boxes) in presence of attention compared to without attention.",
"It shows that precise motions have been estimated for normals.",
"For example, for the man close to the camera (5b) the basic network does not take the distance information into account and estimates a smaller motion compared to its GT which leads to a bigger difference between estimation and GT (and hence a larger activation in the anomaly map).",
"Additionally, we observe a larger activation at legs (i.e., a False Positive) in figure 5(b) however, as attention has been applied in 5(c) all mentioned failures have been addressed in our new attention-based method."
],
[
"Comparison with the SOTA methods and ablation study",
"Table 1 compares the performance of our proposed method with state-of-the-art semi-supervised VAD methods on two benchmark datasets (ST and Ped2).",
"The comparison is based on the frame-level AUC.",
"As can be seen, our method provides superior results compared to similar SOTA holistic semi-supervised VAD methods.",
"Tables 2, 3, and 4 confirm the effectiveness of each proposed contribution in increasing the total accuracy.",
"Table: Comparison of performance (AUC) with SOTA methods.",
"Best-performing method is denoted in boldface.Table: Contribution of each proxy task (Seg: semantic segmentatin, Pred: prediction, OFM: optical flow magnitude) in increasing the performance.Table: Contribution of each attention module (SCSE: spatial and channel wise attention , Att: attention to supplementary information) in increasing the performance of the motion branch.Table: Position of attention map.",
"This table shows how the position of attention map integration affects the performance of the motion branch.As can be noticed in table 2, all proxy tasks are complementary to each other and can detect more anomalies in an ensemble way.",
"Most importantly, we observe that by adding the future frame prediction task to our method, it detects more motion anomalies compared to just using the OFM.",
"Table 4 indicates that adding attention to any position in the network (encoder, decoder, skip connection, or the final layer) boosts the performance.",
"However, a higher performance has been observed for the encoder, decoder and final layer positions, compared to skip-connection.",
"In this paper we proposed a multi-task learning based video anomaly detection method with attentive layers, which benefits from three complementary proxy tasks, namely semantic segmentation, future frame prediction, and optical flow magnitude estimation, to detect anomalies in the video.",
"Our experiments show that adding each task results in higher performance in terms of AUC.",
"Importantly, experimental results confirm that our proposed idea of using attention to both direction of object motion and distance of the object from the camera introduces a new and meaningful way to engage context information in video anomaly detection and results in a precise motion estimation.",
"Our qualitative results show the explainability of estimations and detections and also the effectiveness of each contribution.",
"Quantitative results on the ShanghaiTech and Ped2 datasets, on the other hand, show the superior performance of our method, compared to SOTA methods."
]
] | 2210.07697 |
[
[
"Probing itinerant carrier dynamics at the diamond surface using single\n nitrogen vacancy centers"
],
[
"Abstract Color centers in diamond are widely explored for applications in quantum sensing, computing, and networking.",
"Their optical, spin, and charge properties have been extensively studied, while their interactions with itinerant carriers are relatively unexplored.",
"Here we show that NV centers situated within 10 nm of the diamond surface can be converted to the neutral charge state via hole capture.",
"By measuring the hole capture rate, we extract the capture cross section, which is suppressed by proximity to the diamond surface.",
"The distance dependence is consistent with a carrier diffusion model, indicating that the itinerant carrier lifetime can be long, even at the diamond surface.",
"Measuring dynamics of near-surface NV centers offers a new tool for characterizing the diamond surface and investigating charge transport in diamond devices."
],
[
"A. NV",
"To determine the NV- population of NVP ([NVP]-), the pulse sequence in Fig.",
"2a (main text) is used.",
"After initializing NVP with a 532 nm laser pulse, another 532 nm laser pulse is used to pump NVS.",
"During the readout period of NVP with the orange laser pulse, we see an exponential decay in the fluorescence (Fig.",
"REF inset).",
"The decay time is determined from an exponential fit of the fluorescence, which defines the length of an integration window.",
"A histogram of photon counts during this window in many experiment iterations is plotted (Fig.",
"REF ).",
"The histogram can be fitted with a double Poisson function.",
"The area under the Poisson fit of higher (lower) average photon counts gives us the NV- (NV0) population probability of NVP.",
"Figure: Determination of [NVP]- from double Poisson fit.",
"Blue (red) diagram corresponds to laser on NVS being ON (OFF).Using the pulse sequence shown in Fig.",
"2e (main text), the carrier generation rate of NVS is determined.",
"NVS is at first initialized into mostly NV0 charge state with an orange laser, and then a green laser is applied for readout.",
"During the readout, we see an exponential rise in the fluorescence, which can be fitted with an exponential fit (red solid line in Fig.",
"REF ).",
"The exponential fit provides us with a rise time, from which we calculate $\\gamma _{total}$ .",
"We have $\\gamma _{total} = \\gamma _{i} + \\gamma _{r}$ , where $\\gamma _{i}$ , $\\gamma _{r}$ , and $\\gamma _{total}$ are the ionization rate, recombination rate, and total rate for a particular excitation wavelength and power, respectively.",
"We calculate $\\gamma _r$ from the equation: $\\gamma _r = [NV]\\textsuperscript {-} \\gamma _{total}/100$ , where [NV]- is the steady state NV- population of NVS under green excitation.",
"For different powers of green excitation, we measure [NV]- for several NV centers, which falls in the range of [55%, 70%].",
"We use this range to calculate $\\gamma _r$ , and subsequently $\\gamma _i$ .",
"We then calculate the hole (or, electron) carrier generation rate, $\\gamma _{c} = \\gamma _{i}\\gamma _{r}/\\gamma _{tot}$ for a particular green readout power.",
"Figure: Calculation of carrier generation rate of NVS.",
"Red line is an exponentially rising fit of the fluorescence.The dark lifetime of NVP could be related to laser leakage, thermal activation, and/or tunneling of the NV center electron to the surrounding trap states (e.g.",
"vacancy complexes, surface traps).",
"To quantitatively understand from our obtained data points whether we can observe the dark lifetime of NVP in our experiment, we plot $\\beta $ versus $d$ , where $\\beta = (\\gamma _h/\\gamma _c)2\\pi d^2$ (Fig.",
"REF ).",
"From the plot, we see no gradual rise in the calculated values for longer $d$ .",
"If there was a gradual increase of $\\beta $ with $d$ , the increase would be the result of the dark lifetime of NVP, as at longer distances, the dark lifetime of NVP would play the dominant role in its ionization.",
"We also do not observe any decay of $\\beta $ with $d$ , which again supports our conclusion of very long $L_h$ .",
"Figure: Diagnosis of dark ionization rate of NVP.",
"β\\beta for 14 different NV pairs are plotted against dd to understand the influence of NVP dark lifetime in our experiment.To verify long diffusion length from reduced $\\chi ^2$ , we plot $\\gamma _h$ versus $\\gamma _ce^{-d/L_{h}}/(2\\pi d^2)$ after assuming an arbitrary value for $L_h$ (Fig.",
"REF (a)).",
"We then fit the calculated values with a linear fit (red solid line).",
"From the fit we extract reduced $\\chi ^2$ .",
"We plot the reduced $\\chi ^2$ for the corresponding $L_h$ (Fig.",
"REF (b)).",
"The reduced $\\chi ^2$ reduces until it becomes saturated with increasing $L_h$ .",
"The reduced $\\chi ^2$ saturates for $L_h$ of $\\sim $ 50 $\\mu $ m, which indicates that the diffusion length is much longer than the inter-NV separations we consider here.",
"Figure: (a) Plot used to calculate reduced χ 2 \\chi ^2 for different fit with L h L_h.",
"In this figure, we assumed L h L_h to be 50 μ\\mu m. Red solid line is a linear fit.",
"(b) Reduced χ 2 \\chi ^2 calculation from fits assuming different L h L_h.I.",
"Fitted values for $\\gamma _h$ in Fig.",
"2(c) of main text: | >X | >X | >X | NVS excitation power ($\\mu $ W) $\\gamma _h$ (kHz) 138 0.0043$\\pm $ 0.0001 168 0.0079$\\pm $ 0.0003 216 0.0123$\\pm $ 0.0007 II.",
"Fitted values for $\\gamma _h$ in Fig.",
"3(b) of main text: | >X | >X | >X | $d$ ($\\mu $ m) $\\gamma _h$ (kHz) 1.71 0.0049$\\pm $ 0.0005 1.29 0.0067$\\pm $ 0.0001 0.7 0.0175$\\pm $ 0.0020"
]
] | 2210.07820 |
[
[
"On Benefits and Challenges of Conditional Interframe Video Coding in\n Light of Information Theory"
],
[
"Abstract The rise of variational autoencoders for image and video compression has opened the door to many elaborate coding techniques.",
"One example here is the possibility of conditional interframe coding.",
"Here, instead of transmitting the residual between the original frame and the predicted frame (often obtained by motion compensation), the current frame is transmitted under the condition of knowing the prediction signal.",
"In practice, conditional coding can be straightforwardly implemented using a conditional autoencoder, which has also shown good results in recent works.",
"In this paper, we provide an information theoretical analysis of conditional coding for inter frames and show in which cases gains compared to traditional residual coding can be expected.",
"We also show the effect of information bottlenecks which can occur in practical video coders in the prediction signal path due to the network structure, as a consequence of the data-processing theorem or due to quantization.",
"We demonstrate that conditional coding has theoretical benefits over residual coding but that there are cases in which the benefits are quickly canceled by small information bottlenecks of the prediction signal."
],
[
"Introduction",
"The invention of autoencoders for image compression [1] has shaped the research in this area decisively for the last few years.",
"Similar to existing image and video compression standards like JPEG [2], JPEG2000 [3], HEVC [4], VVC [5], AV1 [6], or many more, this method relies on a transformation of the image into a sparse or low-dimensional domain.",
"Different to the traditional methods, transforms in learning-based compression are non-linear and data-driven.",
"Typically, these transforms are implemented as convolutional neural networks which are trained as a constrained autoencoder.",
"Learning-based approaches have also been proposed for video compression.",
"The general structure of the approaches has largely been taken over from known hybrid video coding solutions, such as VVC or AV1.",
"At first, motion is estimated, then the motion field is transmitted, typically in a lossy way.",
"Afterwards motion compensation is performed.",
"This yields a prediction frame which is used to reduce the temporal redundancy between frames.",
"The task at hand is now to transmit the remaining information to obtain the reconstructed frame.",
"In most published learning-based approaches [7], [8], [9], the strategy is taken over from traditional video compression, i.e., the difference between the original frame and the prediction frame, the so-called residual signal, is computed and transmitted.",
"At the decoder, the reconstructed residual is added to the prediction signal to obtain the reconstructed frame.",
"This strategy has been proven efficient in both traditional video coding and also in deep-learning-based video coding.",
"However, the latter case opens the door to a different and more general possibility: Conditional coding.",
"In conditional coding, we do not compress the residual, but rather the frame itself under the condition of knowing a prediction.",
"Using neural networks, conditional coders can be implemented using a conditional autoencoder.",
"This structure extends the autoencoder by adding an additional signal, the condition, to both encoder and decoder.",
"The network therefore learns by itself how to exploit the redundancy between the signals.",
"Conditional coding exploits a relationship from information theory: The entropy of the difference is greater or equal to the conditional entropy of the signal $x$ given its prediction $x_\\mathrm {p}$ : $H(x- x_\\mathrm {p}) \\ge H(x|x_\\mathrm {p})$ This inequality suggests that in theory conditional coding is always at least as good as conventional residual coding.",
"In this theoretical work, we examine the possibility of conditional coding for video compression in light of information theory.",
"The equation above suggests conditional coding is always superior to residual coding.",
"In the first part of this paper, we go one step further and precisely derive the theoretically possible gain of conditional coding over residual coding.",
"We furthermore take possible information bottlenecks into account, which may occur in practical implementations during the processing of the prediction signal, and we give bounds how such bottlenecks influence the coding performance.",
"In the second part of the paper we validate our theoretical results in simulations.",
"We gain more insights into when conditional coding is beneficial and where its limits are in practical scenarios."
],
[
"Related Work",
"All modern video coders make use of inter prediction to reduce the temporal redundancy between frames.",
"Block-based coders such as HEVC [4], VVC [5], VP9 [10] or AV1 [6] estimate motion on a block-level, yielding a prediction signal per block.",
"If no suitable block is available in the reference frame, the coders have the possibility to locally switch back to intra prediction using previously decoded content.",
"After prediction, a residual is computed by subtracting the prediction signal from the original.",
"The resulting residual is then transformed using a handcrafted frequency transform such as the discrete cosine transform, a discrete sine transform or a combination thereof.",
"Furthermore, AV1 and VVC allow dynamic switching between different transforms.",
"In 2019, the deep video compression framework (DVC) [7] was published.",
"This work introduced the first end-to-end trained video coder consisting of motion estimation, motion transmission, and residual compression.",
"Additionally, DVC contained a network to enhance the prediction frame.",
"For motion estimation, a SpyNet [11] architecture is used.",
"Both motion and residual compression use a standard autoencoder, similar to [1].",
"In subsequent work, the feature-space video coder (FVC) [9] shifts motion estimation and compensation into the feature space.",
"Here, the framework also uses residual coding to transmit the residual features.",
"DVC and FVC still follow the basic paradigm of residual coding, taken from traditional inter coding approaches.",
"In [12] and subsequently in [13], [14], Ladune et al.",
"proposed CodecNet, a conditional coding approach, together with auxiliary networks named ModeNet or MOFNet, which enable a skip mode.",
"Skip modes are commonly found in traditional codecs and describe copying of the prediction signal directly into the reconstructed frame without residual transmission.",
"As shown in Fig.",
"REF , the conditional coding approach used in CodecNet consists of two encoders.",
"One conditional branch encoder which generates a latent representation $y_\\mathrm {p}$ of the prediction signal $x_\\mathrm {p}$ and a main encoder, which has both the prediction signal and the original signal $x$ as input.",
"The main encoder compresses the original signal under the condition of knowing the prediction signal.",
"The resulting latent representation $y$ is coded and transmitted over the channel.",
"The decoder uses the compressed latent representation of the original and the latent representation of the prediction signal to reconstruct the original frame.",
"Note that the latent representation of the prediction signal is not transmitted since it can be constructed at the decoder.",
"The decoder therefore reconstructs the frame under the condition of knowing a latent representation of the prediction signal.",
"In previous work, we proposed the extended generalized difference coder (xGDC) [15], which is another instance of a conditional coder.",
"Here, we used trainable networks to replace difference and sum in traditional residual coding.",
"This generalized residual signal is then compressed jointly with the traditional linear residual.",
"Additionally the encoder signals which part of the image is to be reconstructed with the conditional coder and which part with the residual coder.",
"Figure: Left: Conventional residual coder.",
"Right: Schematic network structure of conditional autoencoder as proposed as CodecNet in and .",
"Dashed lines denote transmission over a channel."
],
[
"Ideal Conditional Coding",
"In order to get a better understanding of residual coding in video compression, we want to consider the different scenarios in the light of information theory.",
"The Shannon entropy serves as a lower bound of the bitrate needed to compress a signal.",
"In the following, we assume ideal lossless coders which reach entropy.",
"Let $H(x)$ denote the entropy of the distribution from which $x$ is drawn.",
"So, if $x$ symbolizes a natural image, $H(x)$ is the entropy of natural images.",
"In residual coding, we compress the residual $r=\\!x\\!-\\!x_\\mathrm {p}$ and in conditional coding, we compress the original frame $x$ under the condition of knowing $x_\\mathrm {p}$ .",
"We therefore have to compare the entropy of residual frames $H(r) = H(x- x_\\mathrm {p})$ and the conditional entropy $H(x|x_\\mathrm {p})$ .",
"We start by using Bayes law, to derive $H(x,x_\\mathrm {p}|r) + H(r) = H(x,x_\\mathrm {p},r) = H(x,x_\\mathrm {p}) + \\underbrace{H(r|x,x_\\mathrm {p})}_{=0}.$ We can easily see that the conditional entropy $H(r|x,x_\\mathrm {p})$ must be zero, since the residual is completely determined from knowing $x$ and $x_\\mathrm {p}$ .",
"We can therefore continue by summarizing and rearranging: $\\begin{split}H(r) &= H(x,x_\\mathrm {p}) - H(x,x_\\mathrm {p}|r) \\\\&=H(x|x_\\mathrm {p}) + H(x_\\mathrm {p}) - \\underbrace{H(x|x_\\mathrm {p},r)}_{=0} - H(x_\\mathrm {p}|r) \\\\&=H(x|x_\\mathrm {p}) + H(x_\\mathrm {p}) - H(x_\\mathrm {p}|r) = H(x|x_\\mathrm {p}) + I(x_\\mathrm {p},r).\\end{split}$ In the second line, we can see that $H(x|x_\\mathrm {p},r)$ must be zero because the original frame $x$ can be reconstructed from the prediction signal $x_\\mathrm {p}$ and the residual $r$ .",
"$H(x_\\mathrm {p}) - H(x_\\mathrm {p}|r)$ is the mutual information $I(x_\\mathrm {p},r)$ and hence we can write: $H(x-x_\\mathrm {p}) = H(x|x_\\mathrm {p}) + I(x_\\mathrm {p};r)$ Since the mutual information is non-negative, this implies $H(x-x_\\mathrm {p}) \\ge H(x|x_\\mathrm {p}),$ with equality if and only if $r$ and $x_\\mathrm {p}$ have no mutual information.",
"From this inequality we obtain general insights about the efficiency of conditional coding compared to residual coding.",
"We not only see that conditional coding is (in theory) at least as good as residual coding but we can also quantify how large the difference is.",
"The larger the mutual information between the residual and the prediction frame, the larger is the gain of conditional coding."
],
[
"Information Bottlenecks",
"When interpreting the results, we need to take into account possible bottlenecks between the prediction signal and the output.",
"In this context, a bottleneck describes a loss of information content in the prediction signal before the reconstruction of the frame.",
"It is easy to see that no bottleneck occurs in residual coding.",
"Here that prediction signal is directly used to reconstruct the original frame.",
"In conditional coding, however, the prediction signal has to be processed, so there are multiple possibilities where bottlenecks may occur in the processing chain.",
"One example where such a bottleneck appears is the conditional autoencoder proposed in [12] and [13].",
"The schematic structure of this coder is given in Fig.",
"REF .",
"There, an additional encoder is used to obtain a latent representation $y_\\mathrm {p}$ of the prediction signal $x_\\mathrm {p}$ , which is then used together with the transmitted latent representation $y$ to reconstruct the frame.",
"Hence, the decoder only sees a latent representation of the prediction signal, which can not contain the full information of the prediction signal.",
"Other ways for bottlenecks to appear in the processing chain come from the data processing theorem [16].",
"This theorem states that every processing on a signal will either decrease the entropy of that signal or keep it the same.",
"Therefore every processing step has the potential to reduce the information content in a non-reversible way.",
"Since (finite-size) convolutions are in general non-reversible (by finite-size convolutions), each convolutional layer processing the prediction signal potentially reduces the information content.",
"Additionally, it is easy to see that ReLU non-linearities also are prone to reducing entropy of the prediction.",
"Another bottleneck arises in practical scenarios when we consider quantized operations.",
"In order to obtain speed and bit-exact reproducibility, which are requirements of real-world coders, all calculations have to be performed in fixed-point arithmetic.",
"Ballé et al.",
"showed in [17] that it is possible to integerize networks, however this comes at a cost of reduced precision.",
"Preserving the information of the prediction signal poses another challenge to integer networks.",
"We can model any bottleneck with a general function $\\tilde{x}_\\mathrm {p}= f\\left(x_\\mathrm {p}\\right)$ , where $\\tilde{x}_\\mathrm {p}$ is a degraded version of $x_\\mathrm {p}$ .",
"It is clear that $H(x_\\mathrm {p}) \\ge H(\\tilde{x}_\\mathrm {p})$ and $H(x|x_\\mathrm {p}) \\le H(x|\\tilde{x}_\\mathrm {p})$ hold.",
"Furthermore, it can be shown that $H(x|x_\\mathrm {p}) = H(x|\\tilde{x}_\\mathrm {p}) - I(x;x_\\mathrm {p}|\\tilde{x}_\\mathrm {p})$ Plugging this result into (REF ), we obtain $H(x-x_\\mathrm {p}) = H(x|\\tilde{x}_\\mathrm {p}) - I(x;x_\\mathrm {p}|\\tilde{x}_\\mathrm {p}) + I(x_\\mathrm {p};r).$ Since $I(x;x_\\mathrm {p}|\\tilde{x}_\\mathrm {p})$ is non-negative, $H(x\\!-\\!x_\\mathrm {p})\\ge H(x|\\tilde{x}_\\mathrm {p})$ does not necessarily hold, depending on how much information is lost during the bottleneck $f$ ."
],
[
"Analysis",
"In the following section we analyze the behavior of the entropy in a simplified inter coding scenario.",
"Since there are no accurate probability models for entire pictures available, computing the precise entropy of an image is not possible."
],
[
"Setup",
"We consider the following scenario: We only consider one pixel of an image near an object boundary.",
"Let the value of this pixel be $x$ and let $x$ be uniformly distributed between 0 and 255.",
"It is easy to see that without further information $H(x) = 8\\,\\mathrm {bit}$ .",
"Furthermore, let $w$ be another uncorrelated pixel value from the reference frame which lies at the other side of the object boundary.",
"We model our prediction signal $x_\\mathrm {p}$ for $x$ as follows: with a probability of $(1-p)$ , $x$ is predicted correctly, i.e.",
"$x_\\mathrm {p}=x$ and with probability $p$ , the deviation in motion vector causes a prediction from $w$ , i.e.",
"$x_\\mathrm {p}=w$ .",
"The overall conditional probability mass function of the predictor is therefore: $f_{X_\\mathrm {p}|X}(x_\\mathrm {p}|x) = p\\cdot \\delta [p-w] + (1-p)\\cdot \\delta [p-x],$ with the discrete Dirac impulse $\\delta (x)$ .",
"To model the degradation by a bottleneck occurring during conditional coding we uniformly quantize our prediction signal to 7 bit or 6 bit.",
"Note that in an actual coder, the bottleneck might not occur in the pixel domain but in some latent representation.",
"However, quantization is able to reduce the information content of the effective prediction signal, and has therefore the same effect as a bottleneck.",
"We denote the degraded prediction signal as $\\tilde{x}_\\mathrm {p}$ .",
"During our experiments, we compare the entropies $H(x\\!-\\!x_\\mathrm {p})$, representing traditional residual coding, $H(x|x_\\mathrm {p})$ representing idealized conditional coding not taking into account bottlenecks, and $H(x|\\tilde{x}_\\mathrm {p})$ , which is a more realistic approximation of conditional coding.",
"Our goal is to show that idealized conditional coding never performs worse than residual coding and in most cases clearly outperforms the traditional method.",
"Second, we want to show in which cases potential bottlenecks have a large effect, introducing challenges for conditional coding."
],
[
"Results",
"We first examine the influence of the prediction error probability $p$ .",
"In Fig.",
"REF the blue and orange curve show $H(r)$ and $H(x|x_\\mathrm {p})$ , respectively, when we vary the probability $p$ of a wrong reference pixel.",
"We can see that $H(x|x_\\mathrm {p}) = H(r)$ only for $p= 0$ and $p= 1$ .",
"For all other cases, $H(r)$ is strictly greater than $H(x|x_\\mathrm {p})$ .",
"Interestingly, the curve of $H(x|x_\\mathrm {p})$ , almost looks like a linear function in $p$ .",
"When we look at the closed form expression $H(x|x_\\mathrm {p}) = \\frac{Np+1}{N+1}\\log _2\\left(Np+1\\right)-\\frac{Np}{N+1}\\log _2p,$ with $N = 2^8-1=255$ in this case, one can prove that it converges toward a linear function for $N\\rightarrow \\infty $ .",
"On the other hand, we see that $H(r)$ is a concave function with the same beginning and end point as $H(x|x_\\mathrm {p})$ .",
"Figure: Entropy of difference coding and conditional coding.",
"We show conditional coding with no bottleneck and with 7 and 6 bit bottleneck.When we add a bottleneck in the conditional coder, we need to transmit additional information.",
"In general, we see that we need more additional information when we have a tighter bottleneck.With increasing error probability, $H(x|\\tilde{x}_\\mathrm {p})$ gets closer to $H(r)$ until eventually the conditional coder performs better again.",
"From this we can already gain some intuitions, which we will confirm later with more experiments.",
"Bottlenecks in conditional coding increase the necessary rate.",
"This was expected, since the information content of the prediction signal decreases.",
"When the quality of the prediction decreases (in this case due to motion inaccuracy) and the rate increases, this effect is getting smaller and the loss due to the bottleneck decreases.",
"We see that the curves with bottleneck approach the ideal conditional coder.",
"Then the transmitted signal itself contains more information and does not rely on the prediction signal that much, therefore inaccuracies in the prediction signal are compensated better.",
"We also observe that the conditional coding gain depends on the prediction error probability $p$ .",
"A confusion with a non-correlated pixel due to motion inaccuracy is only one of the possible errors which can occur during inter prediction.",
"Even if the pixel was predicted from the correct reference pixel, present uncorrelated noise in both frames can lead to prediction inaccuracies.",
"Similarly, in highly correlated areas of an image, displacement errors can be modeled as additive noise.",
"For further reading on this topic see [18].",
"We model the error with additive Gaussian noise with zero mean and standard deviation $\\sigma _\\mathrm {p}$ .",
"In Fig.",
"REF , we examine how conditional coding behaves when we vary $\\sigma _\\mathrm {p}$ .",
"Figure: Entropies of conditional and residual coding when we vary σ p \\sigma _\\mathrm {p}.",
"We show the results for multiple values of pp.Here, we observe a key difference from the first experiment.",
"As expected, both $H(r)$ and $H(x|x_\\mathrm {p})$ increase with increasing prediction error $\\sigma _\\mathrm {p}$ .",
"However, the gap between the curves is constant over $\\sigma _\\mathrm {p}$ .",
"Recall that in the previous experiment, the gain depended on $p$ and thereby on the prediction quality.",
"This can be explained when we look at $p=0$ .",
"Here, we observe no conditional coding gain.",
"Recall that the conditional coding gain is equal to the mutual information between $r$ and $x_\\mathrm {p}$ , $I(r; x_\\mathrm {p})$ .",
"We see that since $r$ in this case is uncorrelated noise, this mutual information must be zero.",
"Therefore, this results fits our theoretical results from the previous section.",
"We also show the effects of bottlenecks in this scenario.",
"In this case we observe the case for a 7 bit bottleneck.",
"We see that especially for small $\\sigma _\\mathrm {p}$ , the presence of a bottleneck increases the entropy greatly.",
"However as $\\sigma _\\mathrm {p}$ increases, the curves more and more align.",
"This result again shows that bottlenecks are most relevant in areas with very good predictors.",
"Figure: Example to demonstrate the mutual information of the prediction signal x p x_\\mathrm {p} and the residual signal rr (Here displayed with increased amplitude and shifted such that gray represents zero).",
"The prediction in this example was computed using the optical flow from SpyNet .Finally, we want to demonstrate that $I(r;x_\\mathrm {p})$ is greater than zero for real images.",
"Since computing the precise mutual information for images is not possible, we show examples of a prediction signal and the corresponding residual, in Fig.",
"REF .",
"It is obvious that we can see structures of the prediction signal in the residual, especially around object boundaries, where small motion vector errors have a larger effect.",
"Since thereby obviously contains information, $I(r;x_\\mathrm {p})$ is strictly greater zero, which can be exploited by conditional coding."
],
[
"Benefits and Challenges",
"In the previous subsection, we clearly showed the potential of conditional coding to perform better than residual coding.",
"A downside of conditional coding is the need to process the data further, with can yield inadvertent information bottlenecks, which we also consider in our experiments.",
"From our computations and simulations, we arrive at the following benefits: Under ideal conditions conditional coding never performs worse than residual coding.",
"Even with moderate bottlenecks, conditional coding performs better when the prediction signal has small errors.",
"On the other hand, we also identified challenges when designing conditional interframe coders: Information bottlenecks between the prediction and the reconstruction signal must be avoided.",
"For low prediction quality, the gain of the ideal conditional coder is possible also with bottlenecks, for high quality, residual coding may be better.",
"Content adaptive coding is therefore desirable.",
"When we look at conditional inter coders that were published before, we see confirmation for our theoretical deliberations.",
"Ladune et al.",
"[13] showed that conditional coding performs better than residual coding.",
"Additionally, their method employs an additional network called ModeNet, which bypasses the conditional coder and copies the prediction signal directly into the reconstructed signal.",
"This yields a behavior similar to skip modes in traditional coders.",
"In our previous work [15], we proposed xGDC, which contains a switch between conditional coding and residual coding, thereby also bypassing the bottleneck, also beating a comparable residual coder.",
"In both approaches, the conditional coder is bypassed for areas with very good prediction signals.",
"This validates our theoretical assessment which calls for a strategy to avoid bottlenecks."
],
[
"Conclusion",
"In this paper we examined the theoretical properties of conditional inter coding compared to residual coding.",
"Conditional coding solves the general task of transmitting a frame when a prediction signal is known.",
"Residual compression, is one special case of conditional coding, therefore general conditional coding is always as least as good as residual coding.",
"In this work, we computed the gain of conditional coding compared to residual coding to be the mutual information of the residual and the prediction signal, $I(x_\\mathrm {p}, r)$ .",
"We showed this relationship in multiple experiments.",
"We additionally showed that conditional coding can create information bottlenecks between the prediction and the reconstructed signal, which can impair the quality of conditional coding, particularly for very good prediction quality and low rate.",
"We showed theoretical results how large the losses due to bottlenecks can be and showed in our simulations when this effect is particularly large.",
"We finally outlined the benefits and challenges of conditional coding compared to residual coding and showed that our theoretical deliberations explain results found in published conditional coding approaches.",
"In this paper we showed the large potential of conditional coding.",
"While residual coding, a low complexity approximation of true conditional coding was dominant for a long time, using end-to-end trained neural networks conditional is now possible.",
"This paper provides theoretical foundations for this coding approach and hopefully encourages further research into the efficient methods how conditional coding concepts can be applied in practice."
]
] | 2210.07737 |
[
[
"Competing Laughlin state and Wigner crystal in bilayer graphene"
],
[
"Abstract We study the fractional quantum Hall effect in the central Landau level of bilayer graphene.",
"By tuning the external applied magnetic field and the electric bias between the two layers one can access a regime where there is a degeneracy between Landau levels with orbital characters corresponding to N=0 and N=1 Galilean Landau levels.",
"While the Laughlin state is generically the ground state for filling $\\nu=1/3$ we find that it can be destroyed and replaced by a Wigner crystal at the same filling factor by tuning the bias and applied field.",
"This competition does not take place at $\\nu=2/3$ where the incompressible ground state remains stable.",
"The possibility of electrically inducing the Wigner crystal state opens a new range of studies of this state of matter."
],
[
"Introduction",
"Graphene-based samples have revealed an abundance of correlated phases in the quantum Hall regime [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].",
"Odd and even-denominator fractional quantum Hall states are observed in monolayer graphene as well as in bilayer graphene, fractional Chern insulators have been realized in samples involving graphene-hexagonal boron nitride structures.",
"Also of interest is the field-induced excitonic condensate in double-bilayer samples.",
"It has been pointed out that bilayer graphene (BLG) systems have a convenient parameter which can be tuned experimentally : the interlayer electric bias in addition to electronic density and external applied magnetic field.",
"The central Landau level of BLG has a nearly eightfold degeneracy due to the combination of several quantum numbers : the ordinary spin, the valley degree of freedom and also an orbital degeneracy.",
"The pattern of ordering in these levels is then very rich and complex.",
"It has been shown that there are tunable phase transitions in fractional quantum Hall states [17].",
"The electric bias directly controls the splitting between orbital levels and the Coulomb interaction between electrons is also impacted by the value of the external applied magnetic field as well as the bias.",
"Detailed investigations of the integer quantum Hall states have been performed on these systems [18] and have shown that an appropriate tight-binding model can capture the level ordering.",
"Recent advances have led to the observation of many fractional states and transitions between them.",
"This means that we have at our disposal a physical system where we can tune parameters affecting the fractional quantum Hall physics [19], [20], [21], [22], [23], [24], [25], [26], [27].",
"In the case of two-dimensional electron systems in GaAs it is known that there is a competition between incompressible electron liquids and crystal of electrons, the so-called Wigner crystal.",
"For filling factor $\\nu =1/3$ of the lowest Landau level, the ground state of the electronic system with Coulomb interactions is an incompressible liquid whose properties are well described by the Laughlin wavefunction [28] and it is only for small filling factors that the ground state is a crystal state [29].",
"Deciding the precise boundary between these phases has proven a difficult issue [30].",
"The crystal state appears as insulating state with a diverging longitudinal resistance when decreasing the temperature.",
"There are experimental evidences for reentrance of the Wigner crystal when one decreases the filling factor.",
"Studies of the crystal state are difficult due to the large values of the magnetic field needed to destroy the fractional quantum Hall liquids.",
"The crystal state is not the only competitor with the liquid states.",
"In higher Landau levels it is known that the electronic system may also form the so-called stripe or bubble phases.",
"As the Wigner crystal such states break translation symmetry and it is believed that they are in a distinct state of matter without topological order.",
"Their experimental signature is an insulating behavior with additional anisotropic properties.",
"We note that in two-dimensional GaAs electron or hole systems [31], [32], [33], [34], [35] there is a rich competition between several many-body ground states and that the Wigner crystal can be stabilized at or close to filling 1/3 by tuning gate potentials.",
"Graphene systems offer yet another arena for the study of such competing phases, in particular the bilayer graphene due to its tunability.",
"It is also known that mixing with higher Landau levels can bias the competition towards the Wigner crystal state.",
"Tuning the BLG system to obtain degeneracy of Landau levels with N=0 and N=1 character can be viewed as an extreme example of Landau level mixing albeit with absence of levels with $N>1$ .",
"So it is plausible that the competition between the Laughlin state and the Wigner crystal can be tuned.",
"In this paper we investigate the incompressible quantum Hall states that occur for filling factor $\\nu =1/3$ and $\\nu =2/3$ when the system is fully valley as well as spin polarized in the bilayer graphene system.",
"The interesting physics now emerges from the crossing of levels with orbital character N=0 and N=1.",
"This should happen in the central octet of Landau levels for fillings close to $\\nu =-3$ according to the present knowledge of level orderings [24].",
"Electrons form then an effective two-component system with a tunable anisotropic interaction given by the projection of the Coulomb potential into this two-component subspace.",
"We use exact diagonalization technique in the torus geometry which is well suited to study the competition between incompressible quantum Hall phases and the Wigner crystal [36], [37], [38], [39].",
"For the filling factor $\\nu =1/3$ we find that the Laughlin-like ground state is stable in a wide region of the phase diagram of the BLG system.",
"However there is a region enclosing the degeneracy point where it is no longer the ground state.",
"We observe the appearance of a set of quasi-degenerate ground states that form a lattice in reciprocal space as expected for spontaneous breakdown of translational symmetry.",
"The crystalline correlations are also revealed by computing an appropriate pair correlation function.",
"The crystal state is fully polarized in the $N=0$ orbitals since the exchange energy is larger than in the $N=1$ case.",
"It appears close to the boundary of a polarization transition.",
"The transition to the Laughlin state is either second order or weakly first order since the the ground state manifold includes a zero-momentum state that smooth deforms into the Laughlin state.",
"In the case of the fraction $\\nu =2/3$ we find no such crystal state but only a smooth crossover between the fully spin polarized state which is the particle-hole transform of the $\\nu =1/3$ state and the singlet state.",
"In section we describe the consequences of the band structure of bilayer graphene onto the Landau levels and discuss the effective Hamiltonian we use.",
"Section discusses our findings about the Laughlin state and the Wigner crystal.",
"Section is devoted to the fraction $\\nu =2/3$ where incompressible liquids are always stable.",
"Finally section presents our conclusions."
],
[
"quantum Hall effect in bilayer graphene",
"Bilayer graphene under a magnetic field has a peculiar Landau level structure.",
"Notably the zeroth level has an approximate eightfold degeneracy.",
"These eight levels have valley twofold indices $\\tau =K,K^\\prime $ , spin $\\sigma =\\uparrow ,\\downarrow $ and also an orbital index $N=0,1$ which is unique to the bilayer system.",
"Coulomb interactions are isotropic in spin and valley space to a very good approximation but there is no good symmetry between the two orbital states.",
"We denote the wavefunctions [40], [41] of these states as $\\psi _{N\\sigma \\tau }$ with $N=0,1$ .",
"The $\\psi _{1\\sigma \\tau }$ wavefunctions have weight concentrated onto the $n=0$ and $n=1$ cyclotron Galilean orbitals that we denote $\\phi _{0,1}$ : $\\psi _{1K\\uparrow }=(\\sqrt{1-\\gamma }\\phi _1,0,\\gamma \\phi _0,0)$ and $\\psi _{1K^\\prime \\uparrow }=(0,\\sqrt{1-\\gamma }\\phi _1,0,\\gamma \\phi _0)$ where the four components denote amplitudes on the sites $A,B^\\prime ,A^\\prime ,B$ where the sites are defined in ref.(McCann2012).",
"In what follows $N$ stands for BLG orbitals while $n$ refers to Galilean orbitals deriving from electrons with a parabolic dispersion relation.",
"Similarly we have $\\psi _{0K\\uparrow }=(\\phi _0,0,0,0)$ and $\\psi _{0K^\\prime \\uparrow }=(0,\\phi _0,0,0)$ .",
"The $\\gamma $ parameter can be adjusted by the external magnetic field and has an almost linear variation from $\\gamma =0.01$ for $B=2T$ to $\\gamma =0.35$ at $B=45T$ according to Ref.(Zibrov2017).",
"This parameter also varies with the electric bias $u$ between the layers.",
"In this work we concentrate on the case of spin and valley polarized configurations but we consider possible degeneracy of $N=0$ and $N=1$ states.",
"We thus project the Coulomb interaction onto this two-component subspace.",
"So we make the approximation of neglecting the mixing to higher Landau levels, necessary to reduce the Fock space dimension to a manageable size.",
"The second-quantized Hamiltonian can now be written as : $\\mathcal {H}=\\frac{1}{2A}\\sum _{q\\lbrace N_i\\rbrace }{\\tilde{V}}(q) F_{N_1 N_2}(q)F_{N_3 N_4}(-q)\\colon \\rho ^\\dag _{N_1N_2}(q) \\rho _{N_3N_4}(q)\\colon \\quad +\\quad \\Delta _{10}{\\hat{N}}_0$ The operators $\\rho _{nm}(q)$ are projected density operators with the following form factors : $F_{00}=\\textrm {e}^{-q^2/4},\\quad F_{11}=((1-\\gamma )L_1(q^2)+\\gamma L_0(q^2))\\textrm {e}^{-q^2/4},\\quad F_{10}=F_{01}^{*}=\\sqrt{1-\\gamma }\\,\\,\\frac{q_y-iq_x}{\\sqrt{2}}\\textrm {e}^{-q^2/4}$ Here we use the Coulomb potential ${\\tilde{V}}(q)=2\\pi e^2/\\epsilon |q|$ , $A$ is the area of the system.",
"$L_0$ and $L_1$ are Legendre polynomials.",
"In BLG systems the screening effects may modify the Coulomb potential.",
"We have used the modification proposed in Ref.",
"(PapicAbanin) and checked that our results do not depend sensitively upon screening effects as was observed previously [43].",
"So the results presented in this paper are for the bare Coulomb potential.",
"We measure all lengths in units of the magnetic length $\\ell =\\sqrt{\\hbar /eB}$ and energies are measured via the Coulomb scale $e^2/(\\epsilon \\ell )$ .",
"We explore the FQHE phase diagram as a function of the $\\gamma $ parameter which has a range $[0,1]$ and also as a function of the level splitting $\\Delta _{10}$ .",
"While strictly speaking the range of this parameter is unbounded all interesting variations occur if we vary it in the the interval $\\Delta _{10}=+1\\dots -1$ in units of the Coulomb energy scale.",
"In bilayer graphene the splitting $\\Delta _{10}$ is controlled mostly by the interlayer bias $u$ .",
"The precise relation has been explored in some details [18].",
"At zero bias $u=0$ there is nevertheless a nonzero splitting of $N=0$ and $N=1$ states due to the Lamb shift-like effect of the Fermi sea of all filled levels below the central octet.",
"Achieving degeneracy $\\Delta _{10}=0$ requires tuning $u$ to negative values in the conventions of ref.(Zibrov2017).",
"In Fig.",
"(REF ) we have plotted three lines giving the values of $\\gamma $ and $\\Delta _{10}$ when we vary the magnetic field at fixed interlayer bias $u=+80,0,-80$ meV.",
"Several limiting cases can be reached as a function of these parameters.",
"When $\\gamma =1$ the Coulomb interaction is identical to that of the lowest Landau level of nonrelativistic electrons irrespective of the orbital index since then we have $F_{00}=F_{11}$ and there are no Coulomb couplings between the orbitals $F_{10}=F_{01}=0$ .",
"So there is complete $SU(2)$ symmetry in the $n=0,1$ space and the problem is formally equivalent to a Galilean lowest Landau level system with zero Zeeman energy if $\\Delta _{10}=0$ .",
"Varying $\\Delta _{10}$ at $\\gamma =1$ is exactly like applying a Zeeman energy on the $SU(2)$ symmetric Coulomb interacting lowest Landau level.",
"So we recover the physics of the FQHE in the lowest Landau level with two components.",
"Many fractions are known to be present in this limiting case and their properties are mostly understood [44] at least for $\\nu =1/3$ and $\\nu =2/3$ most prominent cases.",
"If now we set $\\gamma $ to zero we have two orbitals with the form factors of the lowest $n=0$ and second $n=1$ Landau levels whose degeneracy is controlled again by $\\Delta _{10}$ .",
"For $\\Delta _{10}$ large and negative we are back to a single polarized lowest Landau level while for $\\Delta _{10}$ large and positive the physics is that of a single polarized second Landau level.",
"If $\\Delta _{10}=0$ we have maximal mixing between the two degenerate $n=0$ and $n=1$ orbitals, a situation unique to bilayer graphene.",
"This is a situation which is reminiscent of strong Landau level mixing.",
"Indeed when the cyclotron energy is not very large with respect to the Coulomb energy scale virtual transitions towards empty levels will modify the interactions between electrons residing in the LLL.",
"This effect in general destabilizes the FQHE states [48] and favors the appearance of the Wigner crystal state as is the case for example in two-dimensional hole systems [35] So the two-component system at $\\gamma =0$ can possibly share some of the physics of Landau level mixing.",
"When $\\Delta _{10}$ is very large and negative all electrons stay in the $N=0$ orbitals and the presence of the $N=1$ orbitals is manifest only through virtual interactions.",
"This effect vanishes for very large $\\Delta _{10}$ which means that the $\\gamma $ parameter does no longer play any role.",
"One observes then the physics of fully polarized electrons with the standard bare Coulomb interactions in the lowest Galilean Landau level.",
"For $\\Delta _{10}$ is very large and positive all electrons are now in the $N=1$ levels with a $\\gamma $ -dependent interaction as given by $F_{11}$ .",
"This is a fully polarized situation which interpolates between the lowest Landau level interaction for $\\gamma =1$ and the $n=1$ Landau level interaction for $\\gamma =0$ .",
"This last case is also known to harbor quantum Hall liquids but with the added complication of competing phases.",
"We note that for $\\gamma =1/2$ the interaction is exactly the one which is relevant for the $N=1$ Landau level of monolayer graphene.",
"To make progress we perform exact diagonalizations of the many-body problem defined by Eq.",
"(REF ) in a $L_x\\times L_y$ rectangular unit cell with periodic bounary conditions.",
"The total number of flux quanta through this system is quantized to an integer value $N_\\phi =L_xL_y/2\\pi $ .",
"This unit cell has an adjustable parameter called the aspect ratio $AR=L_x/L_y$ .",
"The periodic boundary conditions are imposed on this unit cell and we construct the many-body translation operators that are used to classify the eigenstates [36], [45].",
"The corresponding eigenvalues are $K_x=2\\pi s/L_x$ , $K_y=2\\pi t/L_y$ where $s$ and $t$ are integers varying in the interval $0,..,N$ where $N$ is the greatest common denominator of $N_e$ and $N_\\phi $ .",
"The origin of the quantum numbers $s,t$ is not always zero.",
"When $pq(N_e-1)=2k+1$ for $\\nu =p/q$ ($p,q$ coprime) the zero momentum state is located at $s_0=t_0=N/2$ , otherwise it is located at $s=t=0$ .",
"This shift is of no special physical significance.",
"Previous studies [43], [46], [47] for fillings $\\nu =1$ and $\\nu =1/2$ have revealed a rich physics."
],
[
"The fate of the Laughlin state $\\nu =1/3$",
"We now focus onto the most prominent fraction $\\nu =1/3$ and study its stability in the $\\Delta _{10}-\\gamma $ phase diagram.",
"With the limiting values for $\\gamma $ and $\\Delta _{10}$ the phase diagram can be drawn in a rectangle : see Fig.",
"(REF ).",
"Figure: The phase diagram at filling factor ν=1/3\\nu =1/3in the γ=0\\gamma =0 and Δ 10 \\Delta _{10} plane.",
"In the lower part corresponding to Δ 10 \\Delta _{10} negativeone recover the lowest Galilean Landau level physics of the Laughlin-like ν=1/3\\nu =1/3 incompressible state.In the upper part it is the second Landau level physics which is recovered with presumablyan incompressible state in the same universality class as the Laughlin state.",
"The right boundaryof the diagram with γ=1\\gamma =1 is a spinful system of electrons with the lowest Landau level form factor.The non-trivial region is the neighborhood of the γ=0\\gamma =0 and Δ 10 =0\\Delta _{10}=0 with full mixingof the two orbitally distinct Landau levels of BLG.",
"In the red region we find that the Laughlin stateis replaced by a Wigner crystal of electrons with a triangular structure.",
"Its extension goes up toγ≈0.7\\gamma \\approx 0.7 and it is located in a narrow range of orbital splitting of at mostΔ 10 ∈[-0.08,+0.06]\\Delta _{10}\\in [-0.08,+0.06] for γ=0\\gamma =0.",
"In devices based on BLGsamples , , typical values ofγ,Δ 10 \\gamma , \\Delta _{10} are given by the three curves with interlayer bias u=-80,0,+80meVu=-80,0,+80meVfor magnetic field varying between B=10TB=10T on the left-hand side(vertical blue dotted line at γ=0.06\\gamma =0.06) and B=45TB=45T on the right (vertical blue line at γ=0.25\\gamma =0.25).The Wigner crystal transition can be reached for realistic field strengths with negative enough bias.Along the $\\gamma =1$ line which is the vertical rightmost boundary the two orbitals have exactly the same form factor and there is no mixing between them since we have $F_{10}=F_{01}=0$ .",
"The external field $\\Delta _{10}$ is exactly analogous to a Zeeman field and the Coulomb interaction has the $SU(2)$ symmetry between the two components.",
"At filling factor $\\nu =1/3$ we should thus find the incompressible Coulomb FQHE state well approximated by the Laughlin wavefunction [28].",
"This FQHE state is fully polarized for $\\Delta _{10}$ large, be it positive or negative.",
"For the degeneracy point $\\Delta _{10}=0$ we have an unbroken $SU(2)$ symmetry and it is known that we have then an instance of quantum Hall ferromagnetism : the Laughlin state is a multiplet with total spin $S_{tot}=N_e/2$ where “spin” refers to the two components $N=0/1$ .",
"Adding a nonzero Zeeman-like energy will lift completely the degeneracy of the ground state multiplet favoring the $S^z_{tot}=+N_e/2$ or $-N_e/2$ according to the sign of $\\Delta _{10}$ .",
"Along the $\\Delta _{10}=-1$ line the system becomes one-component and the Coulomb interaction is governed by the form factor $F_{00}$ which is independent of $\\gamma $ .",
"So we recover the lowest Landau physics of the $\\nu =1/3$ Laughlin-like state.",
"If we now explore the $\\Delta _{10}=+1$ boundary the system is also one-component but interactions are ruled now by the form factor $F_{11}$ which is $\\gamma $ dependent.",
"For $\\gamma =1$ the interaction is exactly that of the Galilean lowest Landau level while for $\\gamma =0$ it is the interaction of the second Landau level.",
"In between we observe that the value $\\gamma =1/2$ gives the form factor appropriate to the $N=1$ Landau level of monolayer graphene.",
"We are not able to shed any new light on the nature of the $\\nu =1/3$ state in the second Landau level.",
"It is known that finite size effects become important and that a finite width of the electron gas is necessary to stabilize the Laughlin state.",
"Here we take the simple view that the Laughlin state is still a valid description of $\\nu =1/3$ even in this case.",
"As a consequence the set of boundaries (large negative $\\Delta _{10}$ , any $\\gamma $ ), (any $\\Delta _{10}$ , $\\gamma =1$ ), (large positive $\\Delta _{10}$ , any $\\gamma $ ) all share the same Laughlin-like physics differing only by polarization.",
"So the most interesting region is the small $\\gamma $ small $\\Delta _{10}$ region.",
"We expect that varying $\\Delta _{10}$ will induce a polarization transition between the two components.",
"So to locate phase boundaries we use two indicators : the polarization of the ground state and the quantum fidelity."
],
[
"Polarization transition",
"It is natural to define a polarization of the two-component system of any quantum state $\\Psi $ by the following quantity : $\\mathcal {P}=\\langle \\Psi |{\\hat{N}}_0|\\Psi \\rangle /N_e,$ that will track the orbital content of the state under consideration.",
"With the normalization we have chosen $\\mathcal {P}=1$ corresponds to all electrons residing in the $N=0$ orbital.",
"We infer that for any state $\\mathcal {P}$ will vary from +1 in the lower part of the phase diagram for $\\Delta _{10}$ negative to zero in the upper part for $\\Delta _{10}$ positive.",
"Exactly on the $\\gamma =1$ line the behavior of $\\mathcal {P}$ can be inferred from the nature of the spin multiplet of the ferromagnetic Laughlin state.",
"The curve is a simple staircase with two values according to the discussion about the quantum Hall ferromagnetism above.",
"For $\\gamma <1$ the curve is non-trivial.",
"We find that the polarization curve is now rounded and has a sharp decrease when $\\Delta _{10}$ goes up in all parts of the phase diagram.",
"However the inflection point of the curve is no longer pinned at $\\Delta _{10}=0$ but is at positive nonzero values for small enough $\\gamma $ .",
"We display in Fig.",
"(REF ) the case of $\\gamma =0$ which is the leftmost boundary of the phase diagram.",
"We find that the sharp transition happens for $\\Delta _{10}\\approx 0.06$ almost independently of the size of the system.",
"When $\\gamma $ is nonzero the sharp transition happens for decreasing values of $\\Delta _{10}$ .",
"A map of the polarization is given in Fig.",
"(REF ).",
"This is expected since the exchange energy is stronger in the lowest Landau level with respect to the second Landau level.",
"So even when $\\Delta _{10}=0$ it is more favorable energetically to polarize the system in the lowest Landau level.",
"One needs a finite amount of offset energy $\\Delta _{10}$ to overcome the exchange energy gain.",
"Figure: The polarization of the zero-momentum ground stateat ν=1/3\\nu =1/3 for γ=0\\gamma =0 as a function of Δ 10 \\Delta _{10}.",
"The curve is very weakly dependent uponthe aspect ratio of the unit cell.",
"For large negative values all electrons prefer to occupythe N=0 orbital while for large positive Δ 10 \\Delta _{10} they occupy the N=1 orbitals.We note that the transition between these two regimes happens for a non-trivial value of the baresplitting Δ 10 ≈0.06\\Delta _{10}\\approx 0.06 insensitive to the system sizes we have studied.Sizes are N e =6,7,8N_e=6,7,8 electronsIn the complete phase diagram we find a phase boundary given in Fig.",
"(REF ) where we see the polarization boundary evolving smoothly.",
"Figure: Polarization of the zero-momentum ground state for N=6 electrons as a function of the γ\\gamma parameterand the splitting Δ 10 \\Delta _{10} between the two levels.",
"In the SU(2)SU(2) symmetric case for γ=1\\gamma =1 thetransition happens right at Δ 10 =0\\Delta _{10}=0 while it is shifted in the upper half plane when γ<1\\gamma <1"
],
[
"Using fidelity to locate phase boundaries",
"To locate phase boundaries one can also compute overlaps of different ground states obtained by changing a parameter of the Hamiltonian.",
"An infinitesimal version of the overlap calculation is the so-called quantum fidelity : $\\mathcal {F}(\\delta )=|\\langle \\Psi (\\delta +\\epsilon )|\\Psi (\\delta )\\rangle |,$ where $\\delta $ is the parameter we choose to vary and $\\epsilon $ an infinitesimal increment.",
"Phase transitions are expected to appear as a strong dip away from unity and the bigger the size the stronger the dip.",
"We have computed the fidelity across the phase diagram and it is sensitive to the polarization transition what we observed above.",
"A sample behavior at fixed gamma as a function of $\\Delta _{10}$ is given in Fig.",
"(REF ).",
"The characteristic dip grows with the size of the system as expected for a phase transition.",
"With only these two indicators there is no sign of other phase boundaries.",
"Figure: The fidelityat ν=1/3\\nu =1/3 for γ=0\\gamma =0 as a function of Δ 10 \\Delta _{10}.",
"We track the zero-momentum ground state.There is evidence for a transition between two regimes happensfor a non-trivial value of the bare splitting Δ 10 ≈0.06\\Delta _{10}\\approx 0.06 weakly sensitive to the system sizeswe have studied.",
"This is numerically equal or very close to the value of maximal slope of thepolarization curve"
],
[
"The Wigner crystal",
"Beyond the simple two-phase picture given by the polarization and fidelity transition of the section above we nevertheless find that the physics changes drastically in the region of small $\\gamma $ and small $\\Delta _{10}$ .",
"Indeed we observe the appearance in the spectrum of a set of nearly-degenerate low-energy states well separated from higher-energy states by a gap.",
"We call these states the ground state manifold.",
"One has to fine tune the aspect ratio of the periodic unit cell used in the diagonalization [36], [37], [38] to obtain the degeneracy.",
"This is a typical occurrence of broken symmetry in a finite system.",
"This phenomenon has been observed in the lowest Galilean Landau level for small filling factors and is an evidence of a Wigner crystal ground state as found in refs.(HRY1999,HRY2000,YHR2001).",
"An example is displayed in the right panel of Fig.",
"(REF ) where the energy eigenvalues are plotted as a function of the momentum.",
"The set of almost degenerate states has an energy splitting of the order of at most $\\approx 10^{-4}$ in Coulomb scale units as opposed to excitation energies of $O( 10^{-2})$ .",
"If plotted in the Brillouin zone the momenta of the quasidegenerate states display a regular lattice which is the reciprocal lattice of a crystal structure as seen in the right panel of Fig.",
"(REF ).",
"In most cases we have studied the lattice is almost triangular.",
"Also the number of states in the quasidegenerate manifold is equal to the number of electrons, ruling out bubble or charge density wave states [36], [37], [38].",
"Figure: Energy spectrum for 6 electrons with γ=0\\gamma =0 and exact degeneracy Δ 10 =0\\Delta _{10}=0.The energies are plotted as a function of the momentum KK in a rectangular cellof aspect ratio 0.44.",
"The color indicates the polarization of the low-lying eigenstates.There is a set of quasidegenerate states clearly separated from the reminder of the spectrum.They are all essentially fully polarized in the N=0 orbital subspace.Figure: Excitation energies above the ground state as a function of Δ 10 \\Delta _{10}for γ=0\\gamma =0 for a system of N e =6N_e=6 particles at ν=1/3\\nu =1/3 in a rectangular unit cell of aspect ratioL x /L y =0.44L_x/L_y=0.44.",
"When there is orbital degeneracy Δ 10 =0\\Delta _{10}=0 we observe a set of quasidegenerate statespointing to broken translational symmetry.",
"An increase of the pseudo-Zeeman field lifts the degeneracies in a smooth crossover starting around Δ 10 ≈-0.02\\Delta _{10}\\approx -0.02This manifold of states is sensitive to the aspect ratio $L_x/L_y$ of the unit cell in which we diagonalize the Hamiltonian.",
"There is a optimal value which is size dependent as is the case of the low-filling factor Wigner crystal : $L_x/L_y=0.44$ for $N_e=6$ , $0.37$ for $N_e=7$ , $0.7$ for $N_e=8$ , $0.3$ for $N_e=9$ .",
"From observation of spectra like the ones presented in Fig.",
"(REF ) it is difficult to pinpoint a phase boundary separating the Wigner crystal from the Laughlin state.",
"We just observe a crossover with no striking discontinuity or (obvious) non-analyticity.",
"An example of the crossover is given in Fig.",
"(REF ).",
"For positive splitting $\\Delta _{10}$ the crystal does not survive the polarization transition and the degeneracy is destroyed.",
"If we increase $\\gamma $ we also find that the crystal signature disappears close to $\\gamma \\approx 0.7$ .",
"To characterize the crystalline order we also compute the pair correlation function : $g_{\\alpha \\beta }(R)=\\frac{1}{\\rho N_e}\\sum _{i\\ne j}\\delta (r_i-r_j-R)(|\\alpha \\rangle \\langle \\alpha |)_i(|\\beta \\rangle \\langle \\beta |)_j$ where $\\alpha ,\\beta =0,1$ is the orbital index and electron coordinates are $r_i,r_j$ , $\\rho $ is the density.",
"In this definition the pair correlation is periodic $g_{\\alpha \\beta }(x+L_x,y)=g_{\\alpha \\beta }(x,y)$ , $g_{\\alpha \\beta }(x,y+L_y)=g_{\\alpha \\beta }(x,y)$ .",
"The almost full polarization of the crystal states means that only $g_{00}$ is sizeable.",
"Indeed we find that other components $g_{11}$ ,$g_{10}$ ,$g_{01}$ track the pattern seen in $g_{00}$ .",
"In the region of degeneracies we observe directly a crystal structure in real space from $g_{00}$ as displayed in Fig.",
"(REF ).",
"A liquid state has instead a single crater-like feature around the position of the reference particle and a smooth background beyond a first ring of overdensity.",
"Here what we observe is definitely a state modulated in real space and the lattice of electrons has the reciprocal lattice we observe in the manifold of degenerate states.",
"We take this as evidence for a crystal state.",
"The number of overdensities plus the central electron is equal to the total number of electrons, again consistent with a Wigner crystal.",
"The case of $N_e=8$ is special : the lattice formed is square as can be seen in the central panel of Fig.",
"(REF ) and the momenta in the Brillouin zone also form a square lattice.",
"While we cannot ultimately conclude on the precise nature of the lattice, this is indicative of a close competition between triangular and square lattices.",
"To characterize the crystal strength we can measure the contrast around the first ring of overdensity in the pair correlation function which is prominent both in solid and liquid phases.",
"In the crystal state we find a contrast $(g_{max}-g_{min})/g_{min}\\approx 0.44$ when $\\gamma =0$ and $\\Delta _{10}=0$ for $N_e=9$ .",
"If we consider the liquid Laughlin state the contrast is only $0.07$ for a system of the same size.",
"The Coulomb ground state at $\\nu =1/3$ in a single polarized Landau level, well described by the Laughlin state, has even less contrast $\\approx 0.04$ .",
"We use this quantity to pin down the boundary of the crystal phase in the region with negative $\\Delta _{10}$ .",
"We decide quite arbitrarily that the state is liquid when the contrast is less than $0.1$ .",
"This gives a rough estimate of the stability region, leading to the red island in Fig.",
"(REF ).",
"As a comparison we display in Fig.",
"(REF ) the pair correlation for one-component spin-polarized electrons in the lowest Galilean Landau level for $N_e=8$ electrons at filling factor $\\nu =1/7$ where we expect a Wigner crystal [36].",
"Indeed the contrast is now close to 2 along the ring circling the reference electron at $r=0$ .",
"We attribute this difference to the fact that the unit cell is now larger in real space for the fraction $\\nu =1/7$ : this case has $N_\\phi =56$ leading to sizes $L_x=11.9$ and $L_y=29.6$ while our largest system for $N_e=9$ has $L_x=7.1$ and $L_y=23.8$ so finite size effects are presumably stronger for the bilayer case.",
"Figure: The pair correlation function g 00 (x,y)g_{00}(x,y) for N e =7N_e=7 (left),8 (center),9 (right) electronsat ν=1/3\\nu =1/3 for Δ 10 =0\\Delta _{10}=0 and γ=0\\gamma =0.",
"The electron located at the origingives a deep correlation hole in addition to the Pauli zero.",
"The aspect ratio is chosen soas to favor the appearance of the crystal structure.",
"This value of L x /L y L_x/L_y is size dependent.For N e =7,9N_e=7,9 (left and right) there is evidence for a triangular crystal state while it is a square latticethat appears for N e =8N_e=8 (center).",
"We note that revealing the square case requires a different aspect ratiothan for the triangular case.Figure: The pair correlation function g(x,y)g(x,y) for N=8N=8 electronsin the lowest Galilean Landau levelat filling factor ν=1/7\\nu =1/7.",
"The aspect ratio is taken tobe L x /L y =0.4L_x/L_y=0.4 to favor the appearance of the crystal structure as was observed for smaller system sizesin refs.(HRY1999,HRY2000,YHR2001).",
"There are eight overdensities which is consistent with a Wignercrystal with one electron per site and a triangular structure.",
"The density contrast is much higher for thissystem size and aspect ratio than in the case of the bilayer system.Finally we note that the Wigner state appears to be almost fully polarized : see the color bar in Fig.",
"(REF ) as a representative example.",
"An interesting question is whether this state is totally polarized $\\mathcal {P}=1$ or almost fully polarized $\\mathcal {P}\\lesssim 1$ .",
"From our numerical measurements we cannot discriminate between these two possibilities.",
"With the accessible sizes finite size scaling is not conclusive.",
"Partial polarization would be an interesting new phenomenon.",
"Non-trivial modifications of existing trial wavefunctions would be required to explain such a state."
],
[
"Behavior of the $\\nu =2/3$ state",
"We now turn to the study of the FQHE state at filling factor $\\nu =2/3$ .",
"Due to the two-component nature of the BLG states this is not simply the particle-hole conjugate of the $\\nu =1/3$ case which happens indeed at $\\nu =2-1/3=5/3$ .",
"We start by discussing the physics for $\\gamma =1$ as a function of the external field $\\Delta _{10}$ .",
"In this limiting case we have full $SU(2)$ symmetry in the orbital space only broken by the Zeeman-like field $\\Delta _{10}$ .",
"For two-component FQHE states at $\\nu =2/3$ we know that there are two incompressible candidate states that compete.",
"The first candidate is the particle-hole conjugate of the $\\nu =1/3$ state where the particle-hole symmetry does not involve the spin degrees of freedom.",
"It can be viewed as a composite fermion state with negative effective flux and two filled effective Landau levels [44].",
"This state is fully polarized and in the absence of Zeeman energy it gives rise to an exactly degenerate ferromagnetic spin multiplet.",
"The other candidate state is the $\\nu =2/3$ global spin singlet state which is known to be the ground state in the absence of Zeeman energy i.e.",
"it is definitely lower than the fully polarized 2/3 state.",
"For $\\gamma =1$ there is a sharp transition between these two ground states as a function of $\\Delta _{10}$ by translation of the known spin physics.",
"If we now consider the realistic case of $\\gamma <1$ , the $SU(2)$ symmetry is explicitly broken and nothing precludes a continuous transformation between these two states.",
"This is indeed what we observe in our diagonalizations.",
"We find no transitions in the interior region of the phase diagram.",
"At $\\gamma =1$ and varying $\\Delta _{10}$ the polarization $\\mathcal {P}$ has two sharp transitions from +1 to 1/2 and then to 0 which correspond to the expected spin transitions of $\\nu =2/3$ , the plateau at $\\mathcal {P}=1/2$ corresponding to the singlet state.",
"For $\\gamma <1$ these steps are rounded and no transitions remain.",
"We conclude that there is no Wigner crystal state at $\\nu =2/3$ ."
],
[
"Conclusions",
"We have studied by exact diagonalization techniques in the torus geometry the fate of the quantum Hall states at $\\nu =1/3$ and $\\nu =2/3$ in the bilayer graphene system when there is almost coincidence of Landau levels with N=0 and N=1 orbital character.",
"The detailed quantitative studies of ref.",
"(Hunt) have shown that this level coincidence can happen for $\\nu _{BLG}=-3+1/3,-3+2/3$ notably.",
"Previous studies of this coincidence [47] have shown a complex competition of phases for the half-filled case.",
"By tuning the applied magnetic field and the electric bias between the layers we find that it is possible to destabilize the Laughlin incompressible state for filling $\\nu =1/3$ and create a Wigner crystal of electrons.",
"This crystal is stabilized when the N=0 and N=1 are in almost coincidence creating a situation akin to extreme Landau level mixing [48] with the difference that there are no extra levels beyond N=1.",
"The crystal structure is revealed by the quasidegeneracies in the many-body spectrum as well as by the pair correlation function.",
"The crystal is favored by fine-tuning the rectangular unit cell we use in exact diagonalizations.",
"The crystal structure seen in real space from the measurement of the pair correlation has the same reciprocal lattice as observed in the magnetic Brillouin zone for the quasidegenerate states.",
"There is a polarisation transition that coincides with the boundary of the crystal phase for positive values of $\\Delta _{10}$ at least when $\\gamma $ is small enough.",
"For negative values of the level splitting we use the contrast of the ring of overdensity of the pair correlation to locate the other phase boundary.",
"The resulting map of the crystal state is given in our Fig.",
"(REF ).",
"The simplest way to observe the transition is to measure the longitudinal resistance that should have a sharp peak when there is level coincidence.",
"Thermodynamic measurements like the chemical potential can be used [16].",
"The interlayer bias should be large enough while there is no real restriction of the value of the magnetic field which controls the $\\gamma $ parameter.",
"The phase transition betwen the crystal state and the Laughlin state involves only a smooth lifting of the degeneracy of the ground state multiplet and the zero-momentum state is always a member of the degenerate states, deforming continuously into the Laughlin state.",
"This is compatible with a weakly first order or second order transition.",
"We acknowledge discussions with A. Assouline and thank M. Shayegan for useful correspondence.",
"We thank DRF and GENCI-CCRT for computer time allocation on the Topaze cluster."
]
] | 2210.07787 |
[
[
"Parameter-Free Average Attention Improves Convolutional Neural Network\n Performance (Almost) Free of Charge"
],
[
"Abstract Visual perception is driven by the focus on relevant aspects in the surrounding world.",
"To transfer this observation to the digital information processing of computers, attention mechanisms have been introduced to highlight salient image regions.",
"Here, we introduce a parameter-free attention mechanism called PfAAM, that is a simple yet effective module.",
"It can be plugged into various convolutional neural network architectures with a little computational overhead and without affecting model size.",
"PfAAM was tested on multiple architectures for classification and segmentic segmentation leading to improved model performance for all tested cases.",
"This demonstrates its wide applicability as a general easy-to-use module for computer vision tasks.",
"The implementation of PfAAM can be found on https://github.com/nkoerb/pfaam."
],
[
"Introduction",
"Convolutional neural networks have demonstrated an impressive ability to solve a broad range of computer vision tasks [1], [2], [3], [4].",
"Typically, a convolutional neural network is built modular and the local receptive field is increasing step-wise with network depth.",
"According to that architecture, the network captures hierarchical patterns based on input image representations within the network.",
"Increasing the representational power of neural networks is of ongoing research interest to emphasize the most important features for a given task.",
"Previous work has shown improvements based on adaptations regarding the inner connectivity [5], [6] or in utilizing an attention mechanism that globally highlight relevant features [7], [8], [9].",
"However, existing attention mechanisms rely on trainable parameters, only regard spatial or channel-wise attention, or introduce additional tunable hyperparameters.",
"Here, we introduce Parameter-free Average Attention Module (PfAAM) which improves performance solely by basic mathematical operations and is based on averaging input feature maps.",
"PfAAM can be introduced into network architectures of arbitrary form and do not add trainable parameters or non-trainable hyperparameters and therefore do not change the overall size or complexity of the network.",
"Furthermore, we show that the network performance of different architectures is enhanced by using PfAAM for both classification and semantic segmentation tasks.",
"While most previous work has focused on hand-crafted modules with additional parameters, we present PfAAM as a lightweight plug-and-play module that is compatible with most neural network architectures, enhancing their performance and can be used for various computer vision tasks."
],
[
"Related Work",
"In this section, we briefly review related model architectures and attention mechanisms."
],
[
"Network Architecture.",
"With the release of AlexNet [1] in 2012 for ImageNet LSVRC-2012 competition [10], outperforming all other submissions by a large margin, convolutional neural networks became state-of-the-art for computer vision tasks, having only recently been challenged by vision transformer models [11], which, however, require expensive pre-training on huge amounts of data.",
"Several improvements to the neural network architecture have been proposed to improve their discriminative abilities.",
"The application of deeper architectures [12], [13], [14], wider networks [15], increased connectivity [5], grouped convolutions [16], depthwise convolution [16] or reduced computational requirements [17], [18] have lead to a plethora of potential architectures.",
"Attempts have been made to automatically search for the best network architecture reducing the manual design of neural network architectures [19], [20].",
"However, building blocks that boost network performance at low computational costs and can be incorporated in any network architecture without the need for manual adjustment or hyperparameter search remain of high value and motivated the design of PfAAM."
],
[
"Attention Modules.",
"Human perception is highly selective and filters information based on its relevance for decision making.",
"According to this, so called attention mechnisms have been proposed for computer vision models.",
"See [21] for a comprehensive overview of attention mechanisms.",
"Generally, attention mechanisms can be split into methods related to channel attention [7], [9] focusing on 'what is important' in the image and spatial attention [22] highlighting 'where is the important information' in the image or a combination of both [8], [23].",
"However, most of these attention modules are implemented by adding learnable model parameters [7], [9], [22], [8] during training increasing computational cost and model size, relate only on spatial or channel attention, or depend on tunable hyperparameters [23].",
"As an extension to existing modules, PfAAM captures channel and spatial attention without adding parameters or hyperparameters and is simple by design, promoting a self-reinforcing effect by averaging activations."
],
[
"Parameter-free Average Attention Module",
"The general structure and computation of PfAAM is shown in figure REF .",
"Consider a feature map $F \\in \\mathbb {R} ^{H \\times W \\times C} $ as intermediate input, the PfAAM separates the input in a spatial attention part $A_{sp} \\in \\mathbb {R} ^{H \\times W \\times 1}$ by averaging the input along its channels and a channel attention part $A_{ch} \\in \\mathbb {R} ^{1 \\times 1 \\times C}$ by calculating the average of each channel.",
"The resulting attention maps are then expanded along their reduced dimensions and recombined to depict the attention to the most important parts of the feature input map.",
"The final recombined attention map uses a sigmoidal gating mechanism to enhance the representational power of the input.",
"The overall process can be summarized as follows: $F^{\\prime } = \\sigma ( A_{sp} \\otimes A_{ch} ) \\otimes F,$ with $\\otimes $ denoting the element-wise multiplication, $\\sigma $ the sigmoid function $\\sigma (x) = \\frac{1}{1+e^{-x}}$ and $F^{\\prime }$ the output of PfAAM.",
"By element-wise multiplication of $A_{sp}$ and $A_{ch}$ the values are broadcasted (copied) along the axes by which they were reduced during averaging to regain their input sizes.",
"In contrast to attention modules that emphasize features by learned parameters PfAAM is parameter-free and is solely highlighting features via averaging spatially and along channels."
],
[
"Spatial Attention Component",
"To emphasize the spatial attention in a feature map we produce a spatial attention map.",
"This is performed by averaging the spatial features along their channels.",
"As a result, the attention is focused on parts in the feature map where a feature is detected.",
"The average of each spatial element $x_{H \\times W} \\in \\mathbb {R} ^{C}$ can be calculated as follows: $A_{sp}(x_{H \\times W}) = \\frac{1}{C}\\sum \\limits _{i=1}^C x_{H \\times W}(i).$ By averaging along the channels, the dimension is reduced and produces a spatial map where each element represents the average across channels.",
"As a consequence the spatial areas with high activations are emphasized while areas with low activations are suppressed, thus highlighting positions with detected features."
],
[
"Channel Attention Component",
"In accordance to spatial attention the channel attention is calculated by averaging along the spatial dimensions of the feature map.",
"Formally, for each channel $y_C \\in \\mathbb {R} ^{H \\times W}$ the average along its spatial dimensions can be calculated as: $A_{ch}(y_C) = \\frac{1}{H \\times W}\\sum \\limits _{i=1}^H \\sum \\limits _{j=1}^W y_C(i,j).$ By averaging along the spatial dimensions, channels are emphasized in which a feature is detected and reducing the influence of channels with low activations for their corresponding features."
],
[
"Model Integration",
"The three-dimensional input to PfAAM is processed into a matrix with the same dimensions, which can be used as an element-wise multiplier to amplify the activations within the input.",
"Because of its simplicity, the PfAAM block can be easily integrated into different network architectures and positions, allowing it to be used as a general building block for convolutional neural networks.",
"In the following section, the position of PfAAM within the residual blocks and different pooling operations were analyzed, showing that averaging is slightly preferred over maximization.",
"Finally, PfAAM was successfully tested in different network architectures for classification and segmentation, showing an increase in performance."
],
[
"Experiments",
"In this section we tested the optimal PfAAM setup and network integration in an ablation study and performed experiments for classification and semantic segmentation with different network architectures."
],
[
"Ablation Study",
"To maximize the effect of the PfAAM block we tested different implementation options.",
"First, we tested averaging versus maximizing as channel and spatial pooling operations within PfAAM to analyze their effect on the overall performance.",
"Max pooling enhance the effect of individual strong activation whereas averaging increases areas with overall strong activation.",
"The performance of a baseline ResNet-164 [24] with PfAAM blocks added to each residual block of the network were compared based on classification error.",
"Furthermore, we tested the influence of an additional Batch Normalization [25] before each PfAAM.",
"The accuracy of the network was tested using the CIFAR-10 dataset [26], that consists of 50k training and 10k test images with a size of 32 x 32 RGB pixels belonging to 10 different classes.",
"See table REF for classification errors of the PfAAM implementations.",
"In total, there is no large difference in the resulting classification error and all PfAAM implementations improve the performance compared to the baseline model, from which we concluded that each of the implementations performs reasonably well.",
"Averaging without Batch Normalization showed the best performance overall, which is why we continued further experiments with this configuration, unless stated otherwise.",
"Table: Comparison of different PfAAM implementations CIFAR-10 using averaging or maximizing and an additional Batch Normalization (BN).",
"The lowest classification error is shown in bold."
],
[
"Experiments",
"To analyze the effect of PfAAM on neural network performance we used baseline architectures for classification and semantic segmentation and compared the performance of the regular architecture to the same architecture but with additional PfAAM blocks incorporated."
],
[
"Image Classification",
"To investigate the effect of PfAAM in a classification task, we conducted experiments using CIFAR-10 and CIFAR-100 as benchmark.",
"Both data sets have the same size, but are divided into 10 and 100 classes, respectively.",
"As model architectures Residual Networks [24] and Wide Residual Networks [15] with varying depth and width were used to cover basic architectures from shallow to deep and thin to wide.",
"Results in table REF show an reduction of the classification error for all tested architectures with integrated PfAAM.",
"For deeper architectures the effect of PfAAM is larger showing a reduction of the error rate of over 12% for ResNet-110 and ResNet-164 on CIFAR-10, thus ResNet-110+PfAAM almost matches the performance of the regular ResNet-164 which has 40% more trainable parameters.",
"For wider but shallower architectures with fewer residual blocks, the effect of PfAAM is smaller (1.4% reduction for WRN-16-8 on CIFAR-10), suggesting that the effect scales with the number of PfAAM units per network.",
"Since PfAAM does not introduce additional learnable parameters, it generally increases the performance of the network in image classification by improving the utilization of the existing parameters.",
"Table: Classification error (%) on CIFAR-10 and CIFAR-100.",
"The lowest error per model architecture and data set is shown in bold."
],
[
"Semantic Segmentation",
"To test PfAAM for semantic segmentation, we used the PASCAL VOC 2012 segmentation dataset [27] consisting of 1464 training and 1449 validation images of 20 categories and an additional background class.",
"Following previous work [28], [29], [2], we used the extended dataset with annotations from [30] resulting in 10582 training images.",
"We trained a U-Net [31] and a Feature Pyramid Network (FPN) [32] on the training images and compared the results to the same architectures with added PfAAM.",
"Each model used a ResNet-50 [24] as encoder-backbone, which was pre-trained on the ImageNet dataset [10].",
"The results in table REF show the mean intersection over union (mIoU) on the validation images.",
"Both models show increased performance when trained with PfAAM increasing the mIoU by 7.7% for U-Net and 5.3% for FPN, respectively.",
"The averaged validation mIoU for U-Net with and without PfAAM during training are depicted in figure REF showing a clear improvement for the PfAAM-model.",
"Similar to classification, the introduction of the PfAAM in the model architecture improves the performance, underlining its general applicability as neural network building block enhancing model performance.",
"Table: Segmentation results (mIoU, %) on PASCAL VOC 2012 validation set.",
"Best results per model architecture are shown in bold.Figure: Validation mIoU during training for a regular U-Net and the same architecture extended by PfAAM, shown is mean±stdmean \\pm std."
],
[
"Implementation Details",
"For CIFAR training we followed the established standard training procedure used by the original publications [24], [15].",
"Each 32x32 image or its its horizontally mirrored version was padded by 4 pixels and randomly cropped back to 32x32 pixel.",
"The neural networks were trained for 200 epochs by optimizing the cross-entropy loss using SGD (stochastic gradient decent) with a momentum of 0.9, a weight decay of 0.0005 a mini-batch size of 128 and an initial learning rate of 0.1.",
"The learning rate was step-wise decreased after 60 epochs, 120 epochs and 160 by a factor of 0.2.",
"For semantic segmentation using the PASCAL VOC dataset, the training images were randomly horizontally flipped and scaled by a factor of 0.5 to 2 for each axis, from which random 224x224 patches were cut and fed into the neural network.",
"Optimization was performed using SGD with a momentum of 0.9 and a constant learning rate of 0.0001 for 200 epochs, optimizing the cross-entropy loss function excluding pixels labeled as void.",
"Unless stated otherwise, all results are reported as the median over 5 runs."
],
[
"Conclusion",
"In this work, we present a novel attention mechanism PfAAM based on highlighting areas of high activation.",
"When PfAAM is used in different network architectures for classification and semantic segmentation, the performance increases for all tested architectures, while the network size remained unchanged and the computational cost is low.",
"Even though PfAAM does not add additional trainable parameters to the network and does not rely on other theoretical considerations, its positive effect is surprisingly robust, suggesting that it leads to a self-focusing effect on relevant features.",
"In summary, PfAAM provides a simple novel building block that might be considered for future neural network design in computer vision tasks."
]
] | 2210.07828 |
[
[
"Observational Constraints on the $f(\\phi,T)$ gravity theory"
],
[
"Abstract We investigate inflation in modified gravity framework by introducing a direct coupling term between a scalar field $\\phi$ and the trace of the energy momentum tensor $T$ as $f(\\phi,T) = 2 \\phi( \\kappa^{1/2} \\alpha T + \\kappa^{5/2} \\beta T^2) $ to the Einstein-Hilbert action.",
"We consider a class of inflaton potentials (i) $V_0 \\phi^p e^{-\\lambda\\phi}$, (ii) $V_0\\frac{ \\lambda \\phi^p}{1+\\lambda\\phi^p}$ and investigate the sensitivity of the modified gravity parameters $\\alpha$ and $\\beta$ on the inflaton dynamics.",
"We derive the potential slow-roll parameters, scalar spectral index $n_s$, and tensor-to-scalar ratio $r$ in the above $f(\\phi,T)$ gravity theory and analyze the following three choices of modified gravity parameters~(i) Case I:~ $\\alpha \\neq 0, ~\\beta=0$ i.e.",
"neglecting higher order terms, (ii) Case II:~ $\\alpha=0$, $\\beta \\neq 0$~ and do the analysis for $T^2$ term, (iii) Case III:~ $\\alpha \\neq 0$ and $\\beta \\neq 0$ i.e.",
"keeping all terms.",
"For a range of potential parameters, we obtain constraints on $\\alpha$ and $\\beta$ in each of the above three cases using the WMAP and the PLANCK data."
],
[
"Introduction",
"Several research has been conducted over the past few decades to describe the evolution of the Universe, both on a theoretical and observational level.",
"Recent observational results from the redshift of type Ia supernova[1], Cosmic Microwave Background (CMB) [2], [3] anisotropy from Planck[4], Wilkinson Microwave Anisotropy Probe (WMAP)[5], Baryon Acoustic Oscillations (BAO)[6], Large Scale Structures[7], altogether point to an expanding Universe which seems to be in harmony with the standard cosmological theory, i.e.",
"the famous $\\Lambda $ CDM[8], [9], [10], [11], [12] model within the General Relativity framework.",
"But, the evidence for isotropic and homogenous Universe appears to be detrimental to the conventional Cosmological model due to the horizon, flatness [13], fine tuning[8], [14], coincidence[8], [15] problems, etc.",
"However, cosmic inflation, a period of exponential expansion, in the early Universe offers quite a plausible solution to these problems [16], [2], [17], [18], [19], [20], [21].",
"This theoretical framework was developed by Guth [16], Linde[17], Starobinsky [18], Albrecht and Steinhardt [19] around forty years ago.",
"The most straightforward method to study inflation is to consider a scalar field $\\phi $ called Inflaton, which under the influence of a particular potential $V(\\phi )$ along with the slow-roll approximation (where the kinetic terms are neglected with respect to the potential term) is used to examine the inflationary expansion of the universe [22], [23].",
"A host of inflaton potential has been extensively studied [24], [25] along with various cosmological parameters using density perturbation and power-spectrum and has been verified by the CMBR anisotropy measurement [26], [4].",
"Though the Einstein's general relativity is accepted as the most suited model of gravity, it has few limitations in it.",
"It does not explain the requirement of dark matter and dark energy, to fit with the cosmological data which has been regarded as one of the primary drivers behind research into alternate theories of Einstein's gravity.",
"In this approach, the Einstein-Hilbert action is modified by adding some polynomial function of Ricci scalar $R$ (i.e.",
"$f(R)$ gravity)[27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], or some function of the Ricci scalar $R$ and/or the trace of the energy-momentum tensor $T$ ($f(R,T)$ gravity) [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52] or some Gauss-Bonnet function ($f(G)$ gravity) [53], [54], [55], [56] etc.",
"The first work of Inflation in modified gravity was done in [48] using a quadratic potential.",
"The same type of analysis has been done in the literature using several potentials like power-law and natural and hill-top potentials in modified gravity.",
"Starobinsky-type potential can also give compatible results with observational data in $f(R,T)$ gravity[49].",
"In this paper, we consider a class of modified gravity theory considering a specific form of $f(\\phi ,T)$ , discuss the slow-roll inflation in the context of this theory and obtain the limits on modified gravity parameters by comparing estimated values of CMB parameters with WMAP and PLANCK.",
"We have proposed a modified gravity model of the type $f(\\phi ,T) = 2 \\phi ( \\kappa ^{1/2} \\alpha T + \\kappa ^{5/2} \\beta T^2) $ which is an extension to the normal Einstein gravity or we can say that it is an add on to the usual $f(R,T)=R+2\\alpha \\kappa T$ gravity where we have promoted $\\alpha $ to a field by introducing $\\phi $ coupled with $T$ and $T^2$ terms, identified the $\\phi $ as an inflaton.",
"A comprehensive study with the inflaton $\\phi $ couples to $T$ to all order does not exist in the literature and the present work is the first step towards that direction.",
"Although a work with $\\phi $ coupled with $T$ exists in the literature [57], [58], but it was limited to the linear order, and we have extended that study by considering a higher order term $ \\sim \\phi ~T^2$ in $f(\\phi ,T)$ and considers a host of potentials, different from those chosen by Zhang et.",
"al [57].",
"Here, we have considered three different cases (i) Case I: $\\alpha \\ne 0$ and $\\beta =0$ which leads to $f(\\phi ,T) = 2 \\alpha \\sqrt{\\kappa } \\phi T $ , (ii) Case II: $\\alpha = 0$ and $\\beta \\ne 0$ leading to $f(\\phi ,T) = 2 \\alpha \\kappa ^{5/2} \\phi T^2 $ , and finally, (iii) Case III: $\\alpha \\ne 0$ and $\\beta \\ne 0$ where both the $T$ and $T^2$ terms are present in $f(\\phi ,T)$ .",
"In each of the above three cases we study the slow-roll inflation and investigate what kind of constraints follow on the modified gravity parameters $\\alpha , \\beta $ from the WMAP and Planck data in the parameter space of the inflaton potential.",
"Note that, in the limit, $\\alpha \\rightarrow 0$ and $\\beta \\rightarrow 0$ , this reduces to normal Einstein gravity where the inflaton $\\phi $ decays out i.e.",
"its number density falls to zero.",
"The paper is organized as follows: In section 2, we obtain the Einstein Field equations in the modified $f(\\phi , T)$ gravity and derive the slow-roll parameters in this modified gravity theory.",
"In section 3, we discuss the inflationary scenario for two different potentials, and derive the cosmological parameters such as scalar spectral index $n_s$ , tensor to scalar ratio $r$ , tensor spectral index $n_T$ , respectively.",
"These cosmological parameters have been subject to constraints in the parameter space of inflaton potential $(\\lambda ,p)$ within the context of modified $f(\\phi , T)$ gravity.",
"In section 4, we analyze our results and compare those with the PLANCK 2018[4] and the WMAP[5] data."
],
[
"Field equations in $f(\\phi ,T)$ gravity:",
"The action for the modified gravity $f(\\phi , T)$ model with the scalar field $\\phi $ coupled with the energy-momentum($T$ ) tensor can be written as, $\\mathcal {S}= \\frac{1}{2}\\int d^4x \\sqrt{-g} ~ F(R,T, \\phi ) +\\int d^4x\\sqrt{-g}~\\mathcal {L}_m (\\phi , \\partial _\\mu \\phi )$ where $F(R,T, \\phi ) = \\frac{1}{\\kappa } R + f(\\phi , T)$ , $R$ is the trace of the Ricci tensor $R_{\\mu \\nu }$ , and $T$ is the trace of energy-momentum tensor $T_{\\mu \\nu }$ of the matter present in the Universe.",
"The modified gravity term $f(\\phi , T)$ is a simple polynomial function of $\\phi $ and $T$ and we propose the following form $f(\\phi ,T) = 2 \\phi ( \\kappa ^{1/2} \\alpha T + \\kappa ^{5/2} \\beta T^2).$ $g$ is the determinant of the metric tensor $g_{\\mu \\nu }$ and G is the Newtonian constant of Gravitation.",
"We use the natural units, $c = \\hbar = 1$ , set $\\kappa =8\\pi G(=1/M^2_{Pl})=1$ and choose the metric signature $(+,-,-,-)$ .",
"If the Universe during the inflation era is dominated by a single inflaton field $\\phi (t)$ , which contributes to the matter Lagrangian $\\mathcal {L}_m$ given by, $\\mathcal {L}_m(\\phi , \\partial _\\mu \\phi ) = \\frac{1}{2}g^{\\mu \\nu }\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi -V(\\phi ) = \\frac{1}{2} \\dot{\\phi }^2 - V(\\phi )$ where in the last equality we have assumed the inflaton field is spatially homogeneous and it depends on $t$ only.",
"Varying the action Eq.",
"(REF ) with respect to gravity $g^{\\mu \\nu }$ , we get the modified Einstein equation as, $F_R(R,T, \\phi ) R_{\\mu \\nu }-\\frac{1}{2}F(R,T, \\phi ) g_{\\mu \\nu }+\\bigl (g_{\\mu \\nu }\\Box -\\nabla _{\\mu }\\nabla _{\\nu } \\bigr )F_R(R,T, \\phi )=T_{\\mu \\nu }-F_T(R,T, \\phi )T_{\\mu \\nu }-F_R(R,T, \\phi )\\Theta _{\\mu \\nu }$ Here $F_R(R,T,\\phi )=\\frac{\\partial F(R,T,\\phi )}{\\partial R}$ , $F_T(R,T,\\phi )=\\frac{\\partial F(R,T,\\phi )}{\\partial T}$ .",
"The energy-momentum tensor for the inflaton $\\phi (t)$ field is, $T_{\\mu \\nu }=-g_{\\mu \\nu }\\mathcal {L}_m+2 \\frac{\\delta \\mathcal {L}_m}{\\delta g^{\\mu \\nu }} = \\partial _\\mu \\phi \\partial _\\nu \\phi - g_{\\mu \\nu } \\Big ( \\frac{1}{2} \\dot{\\phi }^2 - V(\\phi )\\Big ),$ the trace of the energy-momentum tensor $T (= g^{\\mu \\nu } T_{\\mu \\nu }) = - \\dot{\\phi }^2 + 4 V(\\phi )$ and $\\Theta _{\\mu \\nu }=g^{\\alpha \\beta }\\frac{\\partial T_{\\alpha \\beta }}{\\partial g^{\\mu \\nu }}$ .",
"Taking the inflaton field is spatially homogeneous, it takes the form of a perfect fluid, with $\\mathcal {L}_m (= \\frac{1}{2} \\dot{\\phi }^2 - V(\\phi )) =-p $ which yields $\\Theta _{\\mu \\nu }=g^{\\alpha \\beta }\\frac{\\partial T_{\\alpha \\beta }}{\\partial g^{\\mu \\nu }}= -2T_{\\mu \\nu }-pg_{\\mu \\nu } = - 2 \\partial _\\mu \\phi \\partial _\\nu \\phi - 3 p g_{\\mu \\nu }$ The functional form of $f(\\phi ,T)$ considered here as $f(\\phi ,T) = 2 \\phi ( \\alpha T + \\beta T^2)$ (setting $\\kappa = 1$ ).",
"Accordingly, the Einstein equation takes the form as follows $R_{\\mu \\nu }-\\frac{1}{2}R g_{\\mu \\nu }=T_{\\mu \\nu }^{eff}$ where $T_{\\mu \\nu }^{eff}=T_{\\mu \\nu }+2\\phi \\Bigl \\lbrace T_{\\mu \\nu }\\bigl (\\alpha +2\\beta T \\bigr )+p g_{\\mu \\nu }\\bigl (\\alpha +2\\beta T \\bigr )+\\frac{1}{2}g_{\\mu \\nu } \\bigl (\\alpha T+\\beta T^2 \\bigr )\\Bigr \\rbrace $ As mentioned earlier, assuming that the Universe is filled up with a single and homogeneous inflaton field, the effective energy-momentum tensor of the inflaton field will take a diagonal form and we can define the effective energy density $\\rho _{eff}$ and pressure $p_{eff}$ as, $\\rho _{eff} = T^{eff}_{00}= \\frac{1}{2}\\dot{\\phi }^2+V+2\\phi \\dot{\\phi }^2 \\bigl (\\alpha -2\\beta \\dot{\\phi }^2+8\\beta V \\bigr )+\\phi \\bigl (-\\dot{\\phi }^2+4V \\bigr ) \\bigl (\\alpha -\\beta \\dot{\\phi }^2+4\\beta V \\bigr )$ $g_{ij} ~p_{eff} = T^{eff}_{ij} = \\left[\\frac{1}{2}\\dot{\\phi }^2-V-\\phi \\bigl (4V-\\dot{\\phi }^2 \\bigr ) \\bigl (\\alpha -\\beta \\dot{\\phi }^2+4\\beta V \\bigr ) \\right] g_{ij}$ The trace of energy-momentum tensor will take the form $T=\\rho _{eff}-3p_{eff}=-\\dot{\\phi }^2+4V+4\\phi \\bigl (-\\dot{\\phi }^2+4V \\bigr ) \\bigl (\\alpha -\\beta \\dot{\\phi }^2+4\\beta V \\bigr )+2\\phi \\dot{\\phi }^2 \\bigl (\\alpha -2\\beta \\dot{\\phi }^2+8\\beta V \\bigr )$ The line element for the Friedman-Lemaitre-Robertson-Walker (FLRW) metric in spherical coordinates has the following form, $ds^2 = dt^2-a^2(t)\\left[\\frac{dr^2}{1-k r^2} + r^2 d\\theta ^2 + r^2 \\sin ^2 \\theta d\\phi ^2 \\right]$ where $k=0$ is for flat universe.",
"For the background FLRW metric, the Friedman equations with effective density and pressure become $3H^2=\\rho _{eff}, ~-2\\dot{H}-3H^2=p_{eff}$ The continuity equation for the effective energy density and pressure will be, $\\begin{split}& \\ddot{\\phi }+3H\\dot{\\phi }\\biggl \\lbrace 1+2\\phi \\Bigl \\lbrace \\alpha +2\\beta \\bigl (-\\dot{\\phi }^2+4V \\bigr )\\Bigr \\rbrace \\biggr \\rbrace +V_{,\\phi }+2\\dot{\\phi }^2\\Bigl \\lbrace \\alpha +2\\beta \\bigl (-\\dot{\\phi }^2+4V \\bigr )\\Bigr \\rbrace +4\\phi \\ddot{\\phi }\\Bigl \\lbrace \\alpha +2\\beta (-\\dot{\\phi }^2+\\\\& 4V)\\Bigr \\rbrace +4\\beta \\phi \\dot{\\phi }^2\\bigl (-2\\ddot{\\phi }+4V_{,\\phi }\\bigr )+\\bigl (-\\dot{\\phi }^2+4V\\bigr )\\Bigl \\lbrace \\alpha +\\beta \\bigl (-\\dot{\\phi }^2+4V \\bigr )\\Bigr \\rbrace +\\alpha \\phi \\bigl (-2\\ddot{\\phi }+4V_{,\\phi }\\bigr )+2\\beta \\phi \\bigl (-\\dot{\\phi }^2+\\\\& 4V \\bigr )\\bigl (-2\\ddot{\\phi }+4V_{,\\phi } \\bigr )=0\\end{split}$"
],
[
"Slow-roll parameters and CMB constraints:",
"We assume that the universe is filled with spatially homogeneous scalar field which is minimally coupled with the trace of the energy-momentum tensor.",
"When the potential energy term prevails over the kinetic energy term i.e.",
"$V(\\phi ) >>\\frac{1}{2}\\dot{\\phi }^2 $ , a condition known as slow-roll condition, we enter the inflationary phase.",
"To study the inflation, we define the slow-roll parameters.",
"The Hubble slow-roll parameters are defined as $\\epsilon _H=-\\frac{\\dot{H}}{H^2}, ~~\\eta _H=-\\frac{1}{H}\\frac{\\ddot{\\phi }}{\\dot{\\phi }}$ The slow-roll approximation reads, $\\dot{\\phi }^2<< V, ~~ \\ddot{\\phi }<<3H\\dot{\\phi }, ~~ \\dot{\\phi }^2<<H \\dot{\\phi }$ Applying these conditions into Eq.",
"(REF ) along with Eq.",
"(REF ) and Eq.",
"(REF ), we get, $3H^2 = V \\bigl (1+4\\alpha \\phi +16\\beta \\phi V \\bigr ),$ $3H\\dot{\\phi } \\Bigl \\lbrace 1+2\\phi \\bigl (\\alpha +8\\beta V \\bigr )\\Bigr \\rbrace +\\frac{dV}{d\\phi }+4\\alpha \\phi V_{,\\phi }+32\\phi \\beta V V_{,\\phi }+4V \\bigl (\\alpha +4\\beta V \\bigr )=0$ In $f(\\phi ,T)$ gravity, we can find $\\dot{H}$ from Eq.",
"(REF ) as, $\\dot{H}=-\\frac{1}{2}\\bigl (\\rho _{eff}+p_{eff} \\bigr )=-\\frac{1}{2}\\dot{\\phi }^2 \\biggl [1+2\\phi \\Bigl \\lbrace \\alpha +2\\beta \\bigl (-\\dot{\\phi }^2+4V \\bigr ) \\Bigr \\rbrace \\biggr ]$ From Eq.",
"(REF ) and Eq.",
"(REF ) we can define the first potential slow-roll parameter as, $\\epsilon _v = \\frac{3\\dot{\\phi }^2}{2V} \\biggl \\lbrace \\frac{1+2\\alpha \\phi +16\\beta \\phi V}{1+4\\alpha \\phi +16\\beta \\phi V}\\biggr \\rbrace = \\frac{\\Bigl \\lbrace V_{,\\phi }\\bigl (1+4\\alpha \\phi \\bigr )+4V \\bigl (\\alpha +4\\beta V \\bigr )+32\\phi \\beta V V_{,\\phi }\\Bigr \\rbrace ^2}{2V^2 \\Bigl \\lbrace 1+2\\phi \\bigl (\\alpha +8\\beta V \\bigr )\\Bigr \\rbrace \\Bigl \\lbrace (1+4\\phi \\bigl (\\alpha +4\\beta V \\bigr )\\Bigr \\rbrace ^2}$ and taking the derivative of Eq.",
"(REF ), we can get the second potential slow-roll parameter as, $\\begin{split}\\eta _v& = \\eta _H+\\epsilon _H =\\frac{1}{V \\Bigl \\lbrace 1+4\\phi (\\alpha +4\\beta V)\\Bigr \\rbrace \\Bigl \\lbrace 1+2\\phi \\bigl (\\alpha +8\\beta V \\bigr )\\Bigr \\rbrace }\\biggl [V_{,\\phi \\phi } \\bigl (1+4\\alpha \\phi \\bigr )+4V_{,\\phi } \\bigl (2\\alpha + 4 \\beta V\\bigr ) +\\\\&\\quad 16\\beta V V_{,\\phi }+ 32\\beta \\bigl (V V_{,\\phi }+\\phi V_{,\\phi }^2+\\phi V V_{,\\phi \\phi } \\bigr )+\\frac{\\Bigl \\lbrace V_{,\\phi } \\bigl (1+4\\alpha \\phi \\bigr )+4V \\bigl (\\alpha +4\\beta V \\bigr )+32\\phi \\beta V V_{,\\phi } \\Bigr \\rbrace }{1+2\\phi \\bigl (\\alpha +8\\beta \\phi \\bigr )} \\times \\\\&\\quad \\bigl (2\\alpha +16\\beta V+16\\beta \\phi V_{,\\phi } \\bigr )\\biggr ]\\end{split}$ We see that in the limit $\\alpha \\rightarrow 0$ and $\\beta \\rightarrow 0$ , the expression for the potential slow-roll parameters leads to normal Einstein gravity.",
"The scalar spectral index and tensor-to-scalar ratio can be expressed in terms of potential slow-roll parameters as follows, $n_s=1-6\\epsilon _v+2\\eta _v, ~~ r=16\\epsilon _v$ Finally, the e-fold number $(N)$ , can be defined as the amount of inflation needed to produce an isotropic and homogeneous Universe, $N=\\int _{\\phi _{in}}^{\\phi _{final}}\\frac{H}{\\dot{\\phi }}d\\phi =\\int _{\\phi _{final}}^{\\phi _{in}} \\frac{V \\Bigl \\lbrace 1+4\\phi \\bigl (\\alpha +4\\beta V \\bigr )\\Bigr \\rbrace \\Bigl \\lbrace 1+2\\phi \\bigl (\\alpha +8\\beta V \\bigr ) \\Bigr \\rbrace }{V_{,\\phi } \\bigl (1+4\\alpha \\phi \\bigr )+4V \\bigl (\\alpha +4\\beta V \\bigr )+32\\phi \\beta VV_{,\\phi }}d\\phi $"
],
[
"Analysis of slow-roll inflation for different potentials:",
"We consider the following three cases: (i) $\\alpha \\ne 0$ , and $\\beta =0$ , (ii) $\\alpha = 0$ , and $\\beta \\ne 0$ and (iii) $\\alpha \\ne 0$ , and $\\beta \\ne 0$ .",
"In each case, we study the slow-roll inflationary cosmology with two inflaton potentials $ V=V_0 \\phi ^p e^{-\\lambda \\phi }$ and $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ .",
"We calculate the slow roll-parameters, e-fold, scalar spectral index and and tensor-to-scalar ratio and obtain the range of $\\alpha $ and $\\beta $ using those quantities in the parameter space of the two potentials, which are consistent with WMAP and Planck data."
],
[
"Case I: For $\\alpha \\ne 0$ , and {{formula:8ae9679e-fc28-4789-97e1-926383db519d}} :",
"We start with $\\beta = 0$ where $\\phi $ couples linearly with $T$ and $f(\\phi ,T)$ takes the simplest form $f(\\phi , T)= 2\\sqrt{\\kappa }~ \\alpha \\phi T$ .",
"We investigate the slow-roll inflationary expansion for two different inflaton potentials."
],
[
"Inflaton potential $V=V_0 \\phi ^p e^{-\\lambda \\phi }$",
"The first inflaton potential, which is a combination of the power law and exponential term, is of the following form $V=V_0 \\phi ^p e^{-\\lambda \\phi }$ where $V_0$ is a constant, p (power index) and $\\lambda $ are the potential parameters.",
"We find that for $\\lambda =0$ reduces to “chaotic potential $(\\phi ^p)$ \".",
"On the other hand, $p=0$ leads to exponential inflationary potentials with positive curvature discussed in [59].",
"Under the slow-roll approximation, the slow-roll parameters in terms of potential parameters takes the form: $\\epsilon _v = \\frac{1}{2(1+2 \\alpha \\phi )} \\left[ \\frac{4 \\alpha }{1+4 \\alpha \\phi } + \\frac{p-\\lambda \\phi }{\\phi } \\right]^2$ $\\begin{split}\\eta _v&= \\frac{1}{\\phi ^2 (1+2 \\alpha \\phi )^2 (1+4\\alpha \\phi )} \\times \\biggl [ p^2 \\bigl (1+2 \\alpha \\phi \\bigr ) \\bigl (1+ 4\\alpha \\phi \\bigr ) - p \\bigl \\lbrace 1+2 \\lambda \\phi \\bigl (1+2 \\alpha \\phi \\bigr ) \\bigl (1+ 4\\alpha \\phi \\bigr )\\bigl \\rbrace + \\phi ^2 \\Bigl \\lbrace \\lambda ^2 + \\\\&\\qquad 6 \\alpha \\lambda \\bigl (-1+\\lambda \\phi \\bigr ) + 8 \\alpha ^2 \\bigl \\lbrace -1+\\lambda \\phi \\bigl (-1+\\lambda \\phi \\bigr )\\bigl \\rbrace \\Bigr \\rbrace \\biggr ]\\end{split}$ The scalar spectral index($n_s$ ) and tensor-to-scalar ratio($r$ ) can be expressed in terms of potential parameters, modified gravity parameter(s), inflaton field $\\phi $ as follows: $\\begin{split}n_s&= 1 - \\frac{3}{1+2 \\alpha \\phi } \\biggl [ \\lambda - \\frac{p}{\\phi } -\\frac{4 \\alpha }{1+4 \\alpha \\phi } \\biggr ]^2 + \\frac{2}{\\phi ^2 (1+2 \\alpha \\phi )^2 (1+4\\alpha \\phi )} \\biggl [ p^2 \\bigl (1+2 \\alpha \\phi \\bigr ) \\bigl (1+ 4\\alpha \\phi \\bigr ) - p \\bigl \\lbrace 1+\\\\&\\qquad 2 \\lambda \\phi \\bigl (1+2 \\alpha \\phi \\bigr ) \\bigl (1+ 4\\alpha \\phi \\bigr )\\bigl \\rbrace + \\phi ^2 \\Bigl \\lbrace \\lambda ^2 + 6 \\alpha \\lambda \\bigl (-1+\\lambda \\phi \\bigr ) + 8 \\alpha ^2 \\bigl \\lbrace -1+\\lambda \\phi \\bigl (-1+\\lambda \\phi \\bigr )\\bigl \\rbrace \\Bigr \\rbrace \\biggr ]\\end{split}$ and $r = \\frac{8}{1+2 \\alpha \\phi } \\left[ \\frac{4 \\alpha }{1+4 \\alpha \\phi } + \\frac{p-\\lambda \\phi }{\\phi } \\right]^2$"
],
[
"Inflaton potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$",
"Next, we consider the fractional potential for inflationary expansion, $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ where $V_0$ is a constant, p and $\\lambda $ are the potential parameters.",
"We have taken $p = 2$ , 4 for this potential in this anaysis.",
"For $p=2$ , this takes the form of fractional potential which was first studied by Eshagli et al.[60].",
"For $p=4$ , this potential takes the form of Higgs inflation potential which was first studied by Maity[61].",
"Both of these potentials have been studied in the minimal scenario in normal Einstein gravity and non-minimal coupled gravity[25].",
"We have incorporated the modified gravity terms in the Lagrangian to see the effect of coupling between T and $\\phi $ .",
"Under the slow-roll approximation, the slow-roll parameters have the form: $\\epsilon _v = \\frac{1}{2 \\bigl (1+2 \\alpha \\phi \\bigr )} \\left[ \\frac{4 \\alpha }{1+4 \\alpha \\phi } + \\frac{p}{\\phi +\\lambda \\phi ^{1+p}} \\right]^2$ $\\eta _v = -\\frac{p + p\\lambda \\phi ^p +p^2 \\bigl ( 1+6 \\alpha \\phi + 8 \\alpha ^2 \\phi ^2 \\bigr ) \\bigl ( -1 + \\lambda \\phi ^p \\bigr ) +8 \\alpha ^2 \\phi ^2 \\bigl ( 1 + \\lambda \\phi ^p \\bigr )^2}{\\phi ^2 \\bigl (1+2 \\alpha \\phi \\bigr )^2 \\bigl (1+4 \\alpha \\phi \\bigr ) \\bigl (1+\\lambda \\phi ^p \\bigr )^2}$ The scalar spectral index and tensor-to-scalar ratio can be expressed in terms of potential parameters as follows: $\\begin{split}n_s&= 1 - 2 \\times \\biggl [\\frac{p + p\\lambda \\phi ^p +p^2 \\bigl ( 1+6 \\alpha \\phi + 8 \\alpha ^2 \\phi ^2 \\bigr ) \\bigl ( -1 + \\lambda \\phi ^p \\bigr ) +8 \\alpha ^2 \\phi ^2 \\bigl ( 1 + \\lambda \\phi ^p \\bigr )^2}{\\phi ^2 \\bigl (1+2 \\alpha \\phi \\bigr )^2 \\bigl (1+4 \\alpha \\phi \\bigr ) \\bigl (1+\\lambda \\phi ^p \\bigr )^2} \\biggr ] - \\frac{3}{1+2 \\alpha \\phi } \\times \\\\& \\qquad \\biggl [ \\frac{4 \\alpha }{1+4 \\alpha \\phi }+ \\frac{p}{\\phi +\\lambda \\phi ^{1+p}} \\biggr ]^2\\end{split}$ and $r = \\frac{8}{2(1+2 \\alpha \\phi )} \\left[ \\frac{4 \\alpha }{1+4 \\alpha \\phi } + \\frac{p}{\\phi +\\lambda \\phi ^{1+p}} \\right]^2$ We can determine the dependency of modified gravity parameter $\\alpha $ in the context where the scalar field $\\phi $ couples with the trace of energy-momentum tensor $T$ .",
"For the potential $V=V_0\\phi ^p e^{\\lambda \\phi }$ , $\\alpha $ lies between $[-0.00928,-0.00079],[-0.00912,-0.00625],[-0.00763,-0.00389]$ for $p=2$ and $\\lambda =0.01,0.05,0.1$ respectively which brings it to a better agreement with observational PLANCK 2018 data for the spectral index parameter $n_s$ and the tensor-to-scalar ratio $r$ along with the e-fold number($N$ ) lying in the range $40<N<70$ .",
"Similarly, we find the range of $\\alpha $ as $[-0.00711,-0.00669], [-0.00743,-0.00671]$ and $[-0.00736,-0.00588]$ for $p=4,\\lambda =0.01,0.05,0.1$ respectively, which gives $n_s$ within $2\\sigma $ limit of PLANCK 2018 data.",
"Similarly, for the potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ , the range of $\\alpha $ is given by $ [0.24900, 0.54260]$ , $[0.21520,0.72800]$ for $p=2,\\lambda =1,2$ and $[0.26870,0.84410], [0.34070,0.97690]$ for $p=4,\\lambda =1,2$ respectively.",
"In Table .",
"(REF ) and Table.",
"(REF ), we have tabulated the values of cosmological parameters for a particular value of $\\alpha $ (chosen from the given range) and $\\phi $ for both of these two potentials.",
"Table: For V=V 0 φ p e -λφ V=V_0\\phi ^{p} e^{-\\lambda \\phi }, the e-fold number NN and the spectral index parameters n s n_s and rr calculated for a fixed value of φ\\phi and α\\alpha are presented.Table: For V=V 0 λφ p 1+λφ p V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}, the e-fold number NN and the spectral index parameters n s n_s and rr, calculated for a fixed value of φ\\phi and α\\alpha are presented.We can observe that all the values nicely match the data given by Planck 2018.",
"Finally, in Fig.",
"REF we have plotted the results of two potentials for $N=40$ and 60.",
"The blue and red shaded region corresponds to WMAP data up to $95\\%$ and $68\\%$ C.L whereas grey, green and purple shaded regions corresponds to PLANCK, PLANCK+BK15, PLANCK+BK15+BAO respectively.",
"Figure: (Color online) Constraints on n s n_s and rr from CMB measurements of different potential.",
"Shaded regions are allowed by WMAP measuremnts, PLANCK alone, PLANCK+BK15, PLANCK+BK15+BAO upto 68%68\\% and 95%95\\% Confidence Level.From left side of Fig.",
"(REF ), we can infer that for $p=2$ values agree with PLANCK data for $N=60$ whereas for $p=4$ the values match with WMAP data.",
"All the values for $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ with $p=2,4$ and $\\lambda =1,2$ lies within PLANCK data region as shown in right side of Fig.",
"(REF )."
],
[
"Case II: For $\\alpha =0$ and {{formula:da7ab013-fc6c-4dde-8704-9466a83cd796}} :",
"Next we are to investigate the impact of higher order term in the modified gravity parameter $f(\\phi ,T)$ i.e.",
"$F(R,\\phi ,T)=R+2\\beta \\phi T^2$ on several cosmological parameters and their subsequent analysis."
],
[
"Inflaton Potential $V=V_0\\phi ^p e^{-\\lambda \\phi }$",
"As in the previous section, we derive the slow-roll parameters for this potential as: $\\epsilon _v = \\frac{e^{\\lambda \\phi } \\Bigl \\lbrace e^{\\lambda \\phi } \\bigl (-p + \\lambda \\phi \\bigr ) + 16 \\beta \\phi ^{1+p} \\bigl (-1-2p+2\\lambda \\phi \\bigr ) V_0 \\Bigr \\rbrace ^2 }{ 2 \\phi ^2 \\bigl (e^{\\lambda \\phi } +16~\\beta \\phi ^{1+p} ~V_0 \\bigr )^3 }$ and $\\begin{split}\\eta _v&= \\frac{1}{\\phi ^2 \\bigl (e^{\\lambda \\phi } +16~\\beta \\phi ^{1+p}~ V_0 \\bigr )^3} e^{\\lambda \\phi } \\biggl [ e^{2 \\lambda \\phi } \\Bigl \\lbrace p^2 + \\lambda ^2 \\phi ^2 -p \\bigl (1+2 \\lambda \\phi \\bigr )\\Bigr \\rbrace + 16 e^{\\lambda \\phi } \\beta \\phi ^{1+p} \\Bigl \\lbrace 6 p^2 +p \\bigl (2-12 \\lambda \\phi \\bigr ) \\\\& \\quad + \\lambda \\phi \\bigl (-5+6\\lambda \\phi \\bigr ) \\Bigr \\rbrace V_0 + 256 \\beta ^2 \\phi ^{2+2p} \\Bigl \\lbrace 1+6p^2 -7\\lambda \\phi +6 \\lambda ^2 \\phi ^2 +p\\bigl ( 5-12\\lambda \\phi \\bigr ) \\Bigr \\rbrace V_0^2 \\biggr ]\\end{split}$ The spectral index $n_s$ and tensor to scalar ratio $r$ can be derived using Eq.",
"(REF ) as, $\\begin{split}n_s&= \\frac{1}{\\phi ^2 \\bigl (e^{\\lambda \\phi } +16~\\beta \\phi ^{1+p}~ V_0 \\bigr )^3} \\biggl [ -e^{ 3\\lambda \\phi } \\Bigl \\lbrace p^2 + \\bigl ( -1+\\lambda ^2 \\bigr )\\phi ^2 +2 p \\bigl ( 1 - \\lambda \\phi \\bigr ) \\Bigr \\rbrace + 16 e^{2\\lambda \\phi }\\beta \\phi ^{1+p} \\Bigl \\lbrace -2p +\\phi \\times \\\\&\\quad \\bigl ( -4\\lambda +3 \\phi \\bigr ) \\Bigr \\rbrace V_0 + 256 ~e^{\\lambda \\phi }\\beta ^2 \\phi ^{2+2p} \\bigl ( -1-2p-2\\lambda \\phi +3 \\phi ^2 \\bigr ) V_0^2 + 4096 \\beta ^3 \\phi ^{5+3p} V_0^3 \\biggr ]\\end{split}$ and $r = \\frac{8~e^{\\lambda \\phi } \\Bigl \\lbrace e^{\\lambda \\phi } \\bigl (-p + \\lambda \\phi \\bigr ) + 16 \\beta \\phi ^{1+p} \\bigl (-1-2p+2\\lambda \\phi \\bigr ) V_0 \\Bigr \\rbrace ^2 }{ \\phi ^2 \\bigl (e^{\\lambda \\phi } +16~\\beta \\phi ^{1+p} ~V_0 \\bigr )^3 }$"
],
[
"Inflaton Potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$",
"We follow the same procedure for the potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ as before.",
"We calculate the slow-roll parameters as follows, $\\begin{split}\\epsilon _V = \\frac{\\Bigl \\lbrace p+p\\lambda \\phi ^p \\bigl (1+32V_0\\beta \\phi \\bigr )+16V_0\\beta \\phi ^{1+p}\\lambda \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2}{2\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )\\Bigr \\rbrace ^3}\\end{split}$ and $\\begin{split}\\eta _V&=\\frac{1}{\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr ) \\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr ) \\Bigr \\rbrace ^3}\\biggl [256V_0^2\\beta ^2\\lambda ^2\\phi ^{2+2p}(1+\\lambda \\phi ^p)^2+p(1+\\lambda \\phi ^p)\\Bigl \\lbrace -1+\\\\&\\quad 2\\lambda \\phi ^p \\bigl (-1+16V_0\\beta \\phi \\bigr )+\\lambda ^2\\phi ^{2p}\\bigl (-1+32V_0\\beta \\phi +1280V_0^2\\beta ^2\\phi ^2 \\bigr )\\Bigr \\rbrace +p^2\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+96V_0\\beta \\phi \\bigr )-\\\\&\\quad \\lambda ^3\\phi ^{3p}\\bigl (1+48V_0\\beta \\phi +512V_0^2\\beta ^2\\phi ^2 \\bigr )+\\lambda ^2\\phi ^{2p}\\bigl (-1+48V_0\\beta \\phi +1536V_0^2\\beta ^2\\phi ^2 \\bigr ) \\Bigr \\rbrace \\biggr ]\\end{split}$ Table: For V=V 0 φ p e -λφ V=V_0\\phi ^{p}e^{-\\lambda \\phi }, the e-fold number NN and the spectral index parameters n s n_s, rr and n T n_T, calculated for a fixed value of φ\\phi and β\\beta are presented.The scalar spectral index and tensor to scalar ratio are obtained as, $\\begin{split}n_s&= \\frac{1}{\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )\\Bigr \\rbrace ^3}\\Biggl [-2p \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+2\\lambda \\phi ^p \\bigl (1+8\\beta \\phi V_0 \\bigr )+\\lambda ^2\\phi ^{2p} \\bigl (1+16V_0\\beta \\phi +\\\\&\\quad 256V_0^2\\beta ^2\\phi ^2 \\bigr )\\Bigr \\rbrace -p^2 \\biggl \\lbrace 1+4\\lambda \\phi ^p+\\lambda ^2\\phi ^{2p}\\bigl (5+96V_0\\beta \\phi \\bigr )+2\\lambda ^3\\phi ^{3p}\\Bigl \\lbrace 3+96V_0\\beta \\phi +256V_0^2\\beta ^2 \\bigl (-1+3\\phi ^2 \\bigr )\\Bigr \\rbrace +\\\\&\\quad \\lambda ^3\\phi ^{3p}\\Bigl \\lbrace 1+48V_0\\beta \\phi +4096V_0^3\\beta ^3\\phi ^3+256V_0^2\\beta ^2 \\bigl (-1+3\\phi ^2 \\bigr )\\Bigr \\rbrace \\biggr \\rbrace \\Biggr ]\\end{split}$ and $r=\\frac{ 8 \\Bigl \\lbrace p+p\\lambda \\phi ^p \\bigl (1+32V_0\\beta \\phi \\bigr )+16V_0\\beta \\phi ^{1+p}\\lambda \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2}{2\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )\\Bigr \\rbrace ^3}$ We are to see now how the incorporation of the higher order terms in $f(\\phi ,T)$ affects the cosmological parameters and hence the constraints on $\\beta $ in the potential parameter space.",
"We find the range of $\\beta $ for $p=2,\\lambda =0,0.01,0.1$ are $[1\\times 10^{-6}, 4\\times 10^{-6}],[5.8\\times 10^{-7},6.4\\times 10^{-6}]$ and $[1.5\\times 10^{-5},6.9\\times 10^{-5}]$ whereas for $p=4,\\lambda =0,0.01,0.1$ the values of $\\beta $ lie within $[7.9\\times 10^{-10},6.6\\times 10^{-9}],[1.5\\times 10^{-9},8.5\\times 10^{-9}],[7\\times 10^{-9},1\\times 10^{-7}]$ which are compatible with the observational data.",
"We have shown our results in Table.",
"(REF ).",
"Similarly, we have analyzed our another potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ and we have displayed the range of $\\beta $ in table.",
"(REF ) Table: For V=V 0 λφ p 1+λφ p V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}, the e-fold number NN and the spectral index parameters n s n_s, rr and n T n_T, calculated for a fixed value of φ\\phi and β\\beta are presented.Figure: (Color online) Constraints on n s n_s and rr from CMB measurements of different potential.",
"Shaded regions are allowed by WMAP measuremnts, PLANCK alone, PLANCK+BK15, PLANCK+BK15+BAO upto 68%68\\% and 95%95\\% Confidence Level.In both Tables, the different cosmological parameter values are tabulated for particular value of $\\phi $ and $\\beta $ (chosen from the range as shown).",
"We clearly see that the inclusion of higher-order term produces large value of tensor-scalar ratio $r$ for $V_0\\phi ^p e^{\\lambda \\phi }$ and smaller values of $r$ for $V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ .",
"Finally, in Fig.",
"2 we have plotted the results of two potentials for N = 40 and 60.",
"From Fig.",
"(REF ) we can also realize that the potential $V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ produces larger $r$ and hence is a better inflaton potential than $\\phi ^p e^{\\lambda \\phi }$ in modified gravity with $F(R,\\phi ,T)=R+2\\beta \\phi T^2$ ."
],
[
"Case III: Both $\\alpha \\ne 0$ and {{formula:f8bb6275-f3eb-4de0-ad98-705fdfbef051}} :",
"Finally, we consider the most general form i.e.",
"$f(\\phi ,T)=2 \\phi ( \\kappa ^{1/2} \\alpha T + \\kappa ^{5/2} \\beta T^2)$ .",
"We are to investigate how the modifications in slow-roll parameters and spectral index parameters $n_s$ and $r$ results from this consideration."
],
[
"Inflaton Potential $V=V_0\\phi ^p e^{-\\lambda \\phi }$",
"After including both $T$ and $T^2$ terms, the slow-roll parameters become, $\\begin{split}\\epsilon _V = \\frac{e^{\\lambda \\phi } \\biggl \\lbrace e^{\\lambda \\phi } \\Bigl \\lbrace \\phi \\bigl (4 \\alpha \\lambda \\phi -4 \\alpha +\\lambda \\bigr )-p \\bigl (1 +4 \\alpha \\phi \\bigr ) \\Bigr \\rbrace +16 \\beta V_0 \\phi ^{p+1} \\bigl (2 \\lambda \\phi -2 p-1 \\bigr ) \\biggr \\rbrace ^2}{2 \\Bigl \\lbrace \\bigl ( 1 + 2 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+1}\\Bigr \\rbrace \\Bigl \\lbrace \\phi \\bigl ( 1 +4 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+2}\\Bigr \\rbrace ^2}\\end{split}$ and $\\begin{split}\\eta _V& = \\frac{1}{\\Bigl \\lbrace \\bigl ( 1+ 4 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+1}\\Bigr \\rbrace \\Bigl \\lbrace \\phi \\bigl ( 1+ 2 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+2} \\Bigr \\rbrace ^2} \\times e^{\\lambda \\phi } \\Biggl [ e^{2 \\lambda \\phi } \\biggl [ p^2 \\bigl ( 1+6 \\alpha \\phi + \\\\& \\quad 8 \\alpha ^2 \\phi ^2 \\bigr ) - p \\Bigl \\lbrace 1+ 2\\lambda \\phi +16\\alpha ^2 \\phi ^2 \\bigl ( -1+\\lambda \\phi \\bigr ) +4 \\alpha \\phi \\bigl ( -1+ 3 \\lambda \\phi \\bigr ) \\Bigr \\rbrace + \\phi ^2 \\Bigl \\lbrace \\lambda ^2 + 2\\alpha \\lambda \\bigl ( -5 +3 \\lambda \\phi \\bigr ) + \\\\& \\quad 8 \\alpha ^2 \\bigl ( 1- 3 \\lambda \\phi + \\lambda ^2 \\phi ^2 \\bigr ) \\Bigr \\rbrace \\biggr ] + 16 e^{\\lambda \\phi } \\beta \\phi ^{p+1} \\biggl \\lbrace 2p^2 \\bigl ( 3+ 8 \\alpha \\phi \\bigr )+ p \\Bigl \\lbrace 2 - 12 \\lambda \\phi -4 \\alpha \\phi \\bigl ( -5 + 8\\lambda \\phi \\bigr ) \\Bigr \\rbrace + \\phi \\\\& \\quad \\Bigl \\lbrace \\lambda \\bigl ( -5 + 6\\lambda \\phi \\bigr ) + 2 \\alpha \\bigl ( 3 -14 \\lambda \\phi +8 \\lambda ^2 \\phi ^2 \\bigr ) \\Bigr \\rbrace \\biggr \\rbrace V_0 +256 \\beta ^2 \\phi ^{2+2p} \\biggl \\lbrace 1+6p^2 -7 \\lambda \\phi +6 \\lambda ^2 \\phi ^2 + p \\bigl ( 5-\\\\& \\quad 12 \\lambda \\phi \\bigr ) \\biggr \\rbrace V_0^2\\Biggr ]\\\\\\end{split}$ The CMBR spectral index parameters $n_s$ and $r$ will be modified accordingly too, $\\begin{split}n_s&=1 - \\frac{3 e^{\\lambda \\phi } \\biggl \\lbrace e^{\\lambda \\phi } \\Bigl \\lbrace \\phi \\bigl (4 \\alpha \\lambda \\phi -4 \\alpha +\\lambda \\bigr )-p \\bigl (1 +4 \\alpha \\phi \\bigr ) \\Bigr \\rbrace +16 \\beta V_0 \\phi ^{p+1} \\bigl (2 \\lambda \\phi -2 p-1 \\bigr ) \\biggr \\rbrace ^2}{ \\Bigl \\lbrace \\bigl ( 1 + 2 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+1}\\Bigr \\rbrace \\Bigl \\lbrace \\phi \\bigl ( 1 +4 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+2}\\Bigr \\rbrace ^2} + \\\\& \\quad \\frac{2}{\\Bigl \\lbrace \\bigl ( 1+ 4 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+1}\\Bigr \\rbrace \\Bigl \\lbrace \\phi \\bigl ( 1+ 2 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+2} \\Bigr \\rbrace ^2} \\times e^{\\lambda \\phi } \\Biggl [ e^{2 \\lambda \\phi } \\biggl [ p^2 \\bigl ( 1+6 \\alpha \\phi + \\\\& \\quad 8 \\alpha ^2 \\phi ^2 \\bigr ) - p \\Bigl \\lbrace 1+ 2\\lambda \\phi +16\\alpha ^2 \\phi ^2 \\bigl ( -1+\\lambda \\phi \\bigr ) +4 \\alpha \\phi \\bigl ( -1+ 3 \\lambda \\phi \\bigr ) \\Bigr \\rbrace + \\phi ^2 \\Bigl \\lbrace \\lambda ^2 + 2\\alpha \\lambda \\bigl ( -5 +3 \\lambda \\phi \\bigr ) + \\\\& \\quad 8 \\alpha ^2 \\bigl ( 1- 3 \\lambda \\phi + \\lambda ^2 \\phi ^2 \\bigr ) \\Bigr \\rbrace \\biggr ] + 16 e^{\\lambda \\phi } \\beta \\phi ^{p+1} \\biggl \\lbrace 2p^2 \\bigl ( 3+ 8 \\alpha \\phi \\bigr )+ p \\Bigl \\lbrace 2 - 12 \\lambda \\phi -4 \\alpha \\phi \\bigl ( -5 + 8\\lambda \\phi \\bigr ) \\Bigr \\rbrace + \\phi \\\\& \\quad \\Bigl \\lbrace \\lambda \\bigl ( -5 + 6\\lambda \\phi \\bigr ) + 2 \\alpha \\bigl ( 3 -14 \\lambda \\phi +8 \\lambda ^2 \\phi ^2 \\bigr ) \\Bigr \\rbrace \\biggr \\rbrace V_0 +256 \\beta ^2 \\phi ^{2+2p} \\biggl \\lbrace 1+6p^2 -7 \\lambda \\phi +6 \\lambda ^2 \\phi ^2 + p \\bigl ( 5-\\\\& \\quad 12 \\lambda \\phi \\bigr ) \\biggr \\rbrace V_0^2\\Biggr ]\\\\\\end{split}$ and $\\begin{split}r =\\frac{8 e^{\\lambda \\phi } \\biggl \\lbrace e^{\\lambda \\phi } \\Bigl \\lbrace \\phi \\bigl (4 \\alpha \\lambda \\phi -4 \\alpha +\\lambda \\bigr )-p \\bigl (1 +4 \\alpha \\phi \\bigr ) \\Bigr \\rbrace +16 \\beta V_0 \\phi ^{p+1} \\bigl (2 \\lambda \\phi -2 p-1 \\bigr ) \\biggr \\rbrace ^2}{ \\Bigl \\lbrace \\bigl ( 1 + 2 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+1}\\Bigr \\rbrace \\Bigl \\lbrace \\phi \\bigl ( 1 +4 \\alpha \\phi \\bigr ) e^{\\lambda \\phi }+16 \\beta V_0 \\phi ^{p+2}\\Bigr \\rbrace ^2}\\end{split}$ Here, we have considered a fixed value of $\\beta = 10^{-6}$ and found the range of $\\alpha $ for each potential.",
"The range of $\\alpha $ for $p=2,\\lambda =0,0.01,0.1$ are found to be $[1\\times 10^{-3}, 0.025]$ , $[5.9\\times 10^{-3},0.01709]$ and $[2.3\\times 10^{-3},0.0252]$ whereas for $p=4,\\lambda =0,0.01,0.1$ the values of $\\alpha $ are found to lie within $[0.15,0.32]$ , $[0.325,0.429]$ , $[0.7876,0.9469]$ .",
"These ranges are obtained from the requirement of compatibility of spectral index parameters with the observational data and they are shown in Table (REF ).",
"Table: For V=V 0 φ p e -λφ V=V_0\\phi ^{p}e^{-\\lambda \\phi }, the e-fold number NN and the spectral index parameters n s n_s, rr and n T n_T, calculated for a fixed value of φ\\phi ,α\\alpha and β\\beta are presented."
],
[
"Inflaton Potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$",
"With the $T$ and $T^2$ terms in $f(\\phi ,T)$ , the slow-roll parameters are derived as $\\begin{split}\\epsilon _V = \\frac{\\Bigl \\lbrace p+p\\lambda \\phi ^p \\bigl (1+32V_0\\beta \\phi \\bigr )+4p\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )+4\\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\bigl (\\alpha +\\alpha \\lambda \\phi ^p+4V_0\\beta \\lambda \\phi ^p \\bigr ) \\Bigr \\rbrace ^2}{2\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace \\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+4\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2}\\end{split}$ and $\\begin{split}\\eta _V&= \\frac{1}{\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p(1+16V_0\\beta \\phi )+2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2 \\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+4\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr ) \\Bigr \\rbrace }\\times \\\\&\\quad \\Biggl [8\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 32V_0\\beta ^2\\lambda ^2\\phi ^{2p}+12V_0\\alpha \\beta \\phi ^p \\bigl (1+\\lambda \\phi ^p \\bigr )+\\bigl (\\alpha +\\alpha \\lambda \\phi ^p \\bigr )^2\\Bigr \\rbrace +p^2 \\biggl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+96V_0\\beta \\phi \\bigr )-\\\\&\\quad \\lambda ^3\\phi ^{3p}\\bigl (1+48V_0\\beta \\phi +512V_0\\beta ^2\\phi ^2 \\bigr )+\\lambda ^2\\phi ^{2p}\\bigl (-1+48V_0\\beta \\phi +1536 V_0^2\\beta ^2\\phi ^2 \\bigr )-8\\alpha ^2\\phi ^2 \\bigl (-1+\\lambda \\phi ^p \\bigr )\\times \\\\&\\quad \\bigl (1+\\lambda \\phi ^p \\bigr )^2-2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr ) \\Bigl \\lbrace -3-128 V_0\\beta \\lambda \\phi ^{1+p}+\\lambda ^2\\phi ^{2p} \\bigl (3+64V_0\\beta \\phi \\bigr )\\Bigr \\rbrace \\biggr \\rbrace +p \\bigl (1+\\lambda \\phi ^p \\bigr )\\biggl \\lbrace -1+\\\\&2\\lambda \\phi ^p \\bigl (-1+16V_0\\beta \\phi \\bigr )+\\lambda ^2\\phi ^{2p}\\bigl (-1+32V_0\\beta \\phi +1280V_0^2\\beta ^2\\phi ^2 \\bigr )+16\\alpha ^2\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )^2+4\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\times \\\\&\\quad \\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+80V_0\\beta \\phi \\bigr ) \\Bigr \\rbrace \\biggr \\rbrace \\Biggr ]\\end{split}$ The scalar spectral index $n_s$ and tensor-to-scalar ration $r$ are calculated as $\\begin{split}n_s&= \\frac{1}{\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr ) +2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2 \\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+4\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\rbrace }\\\\&\\quad \\Biggl [ 1-\\frac{3\\Bigl \\lbrace p+p\\lambda \\phi ^p \\bigl (1+32V_0\\beta \\phi \\bigr )+4p\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )+4\\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\bigl (\\alpha +\\alpha \\lambda \\phi ^p+4V_0\\beta \\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2}{\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace \\Bigr \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+4\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\rbrace ^2}+\\\\&\\quad 2\\biggl \\lbrace 8\\phi ^2(1+\\lambda \\phi ^p)^2 \\Bigl \\lbrace 32V_0^2\\beta ^2\\lambda ^2\\phi ^{2p}+12V_0\\alpha \\beta \\lambda \\phi ^p \\bigl (1+\\lambda \\phi ^p \\bigr )+\\bigl (\\alpha +\\alpha \\lambda \\phi ^p \\bigr )^2\\Bigr \\rbrace +p^2 \\Bigl \\lbrace 1+\\lambda \\phi ^p(1+96V_0\\beta \\phi )-\\\\&\\quad \\lambda ^3\\phi ^{3p}\\bigl (1+48V_0\\beta \\phi +512V_0^2\\beta ^2\\phi ^2 \\bigr )+\\lambda ^2\\phi ^{2p}\\bigl (-1+48V_0\\beta \\phi +1536V_0^2\\beta ^2\\phi ^2 \\bigr )-8\\alpha ^2\\phi ^2 \\bigl (-1+\\lambda \\phi ^p \\bigr )\\times \\\\&\\quad \\bigl (1+\\lambda \\phi ^p \\bigr )^2-2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr ) \\Bigl (-3-128V_0\\beta \\lambda \\phi ^{1+p}+\\lambda ^2\\phi ^{2p}\\bigl (3+64V_0\\beta \\phi \\bigr )\\Bigr )\\Bigr \\rbrace +p\\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace -1+\\\\&\\quad 2\\lambda \\phi ^p \\bigl (-1+16V_0\\beta \\phi \\bigr )+\\lambda ^2\\phi ^{2p}\\bigl (-1+32V_0\\beta \\phi +1280V_0^2\\beta ^2\\phi ^2 \\bigr )+16\\alpha ^2\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace \\biggr \\rbrace \\Biggr ]\\end{split}$ and $\\begin{split}r&=\\frac{8\\Bigl \\lbrace p+p\\lambda \\phi ^p \\bigl (1+32V_0\\beta \\phi \\bigr )+4p\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )+4\\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\bigl (\\alpha +\\alpha \\lambda \\phi ^p+4V_0\\beta \\lambda \\phi ^p \\bigr ) \\Bigr \\rbrace ^2}{\\phi ^2 \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+2\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace \\Bigl \\lbrace 1+\\lambda \\phi ^p \\bigl (1+16V_0\\beta \\phi \\bigr )+4\\alpha \\phi \\bigl (1+\\lambda \\phi ^p \\bigr )\\Bigr \\rbrace ^2}\\end{split}$ In Table (REF ), for the potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ , we have shown the range of $\\alpha $ where the different cosmological parameter values are tabulated for particular value of $\\phi $ and $\\alpha $ (chosen from the range(shown)).",
"The range of $\\alpha $ for $p=2,\\lambda =1,2$ are $[0.002,0.02]$ and $[0.0001,0.0018]$ whereas for $p=4,\\lambda =1,2$ $\\alpha $ lies within $[10^{-5},0.001]$ , $[10^{-5},0.002]$ .",
"We see the effect of both $T$ and $T^2$ terms in $f(\\phi ,T)$ - for the same choices of $\\alpha $ and $\\beta $ , a large value of $r$ is obtained for the potential $V_0 \\phi ^p e^{\\lambda \\phi }$ , whereas a smaller value of $r$ for $V_0 \\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ is obtained.",
"Table: For V=V 0 λφ p 1+λφ p V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}, the e-fold number NN and the spectral index parameters n s n_s, rr and n T n_T, calculated for a fixed value of φ\\phi ,α\\alpha and β\\beta are presented.Finally, in Fig.",
"(REF ), we have plotted the results of two potentials for $N = 40$ and 60.",
"From the Fig.",
"(REF ), we also see that in case where both $\\alpha \\ne 0,~\\beta \\ne 0$ , the potential $V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ is better compatible in the $n_s - r$ than the potential $ V_0 \\phi ^p e^{\\lambda \\phi }$ for $f(\\phi ,T)=2 \\phi (\\alpha T + \\beta T^2)$ .",
"Figure: (Color online) Constraints on n s n_s and rr from CMB measurements of different potential.",
"Shaded regions are allowed by WMAP measuremnts, PLANCK alone, PLANCK+BK15, PLANCK+BK15+BAO upto 68%68\\% and 95%95\\% Confidence Level."
],
[
"Conclusion",
"In this manuscript, we have proposed an extension of the modified gravity as $f(\\phi ,T) = 2 \\phi ( \\kappa ^{1/2} \\alpha T + \\kappa ^{5/2} \\beta T^2) $ where the inflaton $\\phi $ couples linearly and quadratically to the trace of energy-momentum tensor $T$ of the inflaton matter.",
"Such an extension seems to be interesting as it offers an alternative approach to deal with cosmological problems, including dark energy and dark matter.",
"We have investigated the paradigm of inflationary expansion with a specific form of the modified gravity parameter(as above) $f(\\phi ,T)$ for two distinct inflaton potentials (a)$V=V_0\\phi ^p e^{-\\lambda \\phi }$ and (b)$V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ .",
"We have considered three different cases - Case I: $\\alpha =0, \\beta \\ne 0$ , Case II: $\\alpha \\ne 0, \\beta =0$ and Case III: $\\alpha \\ne 0, \\beta \\ne 0$ .",
"We have derived the slow-roll parameters in each of these two potentials and computed the CMBR parameters i.e.",
"the scalar spectral index $n_s$ , the tensor to scalar ratio $r$ in order to study the inflation.",
"With $\\alpha \\ne 0$ and $\\beta = 0$ (Case I) in $f(\\phi ,T)$ , we found that the values of $n_s$ and $r$ obtained are in good agreement with the Planck 2018 data.",
"However, when the $\\alpha $ term is turned off, $\\beta $ is not (Case II), we obtain quite higher values of the scalar-to-tensor ratio while the $n_s$ values are still in good agreement with the Planck 2018 data (Table REF ).",
"In the case where $\\alpha \\ne 0$ and $\\beta \\ne 0$ , the $r$ values, obtained with the potential $V = V_0 \\phi ^p e^{-\\lambda \\phi }$ , are found to be still higher(by an order) than the one obtained with the potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ .",
"Note that if the coupling term is set to zero i.e.",
"$\\alpha = \\beta = 0$ , we recover the Einstein's gravity.",
"We also found that both our potentials agree quite well with experimental data in their respective parameter space for the simplest form of the modified gravity parameter $R+f(\\phi ,T)=R+2\\alpha \\phi T$ .",
"It is also to be noted that $\\beta $ , the coefficient of $T^2$ , is very small; hence its contribution to different cosmological parameters is negligible.",
"Also, the range of $\\alpha $ becomes smaller when we add higher order terms of $T$ .",
"We find that the inflationary dynamics and the $n_s-r$ values are very sensitive to the coupling parameters $\\alpha $ and $\\beta $ in the modified $(\\phi - T)$ gravity theory.",
"Finally, we conclude that the inflaton potential $V=V_0\\frac{\\lambda \\phi ^p}{1+\\lambda \\phi ^p}$ fits best for all three cases and all the cosmological parameters lie within $3\\sigma $ range of PLANCK 2018 data even for higher order terms."
],
[
"Acknowledgement",
"Ashmita would like to thank BITS Pilani K K Birla Goa campus for the fellowship support.",
"PS would like to thank Department of Science and Technology, Government of India for INSPIRE fellowship.",
"We thank Kinjal Banerjee and Rudranil Basu for the useful discussions and insightful comments related to this work."
]
] | 2210.07788 |
[
[
"Online Learning of Caching and Recommendation Policies in a Multi-BS\n Cellular Networks"
],
[
"Abstract Mobile edge computing is a key technology for the future wireless networks and hence, the efficiency of a cache-placement algorithm is important to seek the cache content which satisfies the maximum user demands.",
"Since recommendations personalizes an individual's choices, it is responsible for a significant percentage of user requests, and hence recommendation can be utilized to maximize the overall cache hit rate.",
"Hence, in this work, joint optimization of both recommendation and caching is proposed.",
"The influence of recommendation on the popularity of a file is modelled using a conditional probability distribution.",
"To this end, the concept of probability matrix is introduced and a Bayesian based model, specifically Dirichlet distribution is used to predict and estimate the content request probability and hence the average cache hit is derived.",
"Joint recommendation and caching algorithm is presented to maximize the average cache hits.",
"Subsequently, theoretical guarantees are provided on the performance of the algorithm.",
"Also, a heterogeneous network consisting of M small base stations and one macro base station is also presented.",
"Finally, simulation results confirm the efficiency of the proposed algorithms in terms of average cache hit rate, delay and throughput."
],
[
"Introduction",
"Due to the fast development of communication based applications, it is expected that there will be $5.3$ billion total Internet users (66 percent of global population) by 2023, up from $3.9$ billion (51 percent of global population) in 2018 [2].",
"In such a scenario, content delivery networks (CDNs) emerged as one of the promising technologies.",
"To further enhance the user's quality of experience, mobile edge computing (MEC) has been proposed which stores the popular contents close to the edge devices beforehand, and hence reduces the delay and alleviates the backhaul congestion [3], [4], [5].",
"While, on the other hand rapidly growing file sizes, reducing cache sizes (compared to the traditional content delivery networks), and unpredictable user demands make the task of caching algorithms even more difficult.",
"For example, the total data generated by Google per day is in the order of PBs, while installing 1TB in every small cell in the heterogeneous network will only shift less than 1 % of the data for even one content provider.",
"To overcome these issues, it has been observed that user demands are increasingly driven by recommendation based systems.",
"Recommendation based on an individual's preference have become an integral part of e-commerce, entertainment and other applications.",
"The success of recommender systems in Netflix and Youtube shows that 80% of hours streamed at Netflix and 30% of the overall videos viewed owes to recommender systems [6], [7].",
"With recommendation the user's request can be nudged towards locally cached contents, and hence resulting in lower access cost and latency.",
"The content placement in the wireless network largely depends on the user behaviour and file popularity.",
"The recent success of integration of artificial intelligence in the wireless communications has further led to better understanding of user behaviors and the characteristics of the network [8].",
"Especially the edge networks can now predict the content popularity profile hence increasing the average cache hit.",
"The high accuracy in prediction by the neural networks has resulted in many of the content popularity prediction models, such as, collaborative filtering with recurrent neural networks [9], the stack auto encoder [10], deep neural networks [11] and others.",
"However, the local content popularity profile need not match the global prediction by the central server.",
"Many of the recent works have proposed the edge caching strategies by learning the user preferences and content popularity [12].",
"Context awareness helps in classifying the environment, hence enabling the intelligent decisions at the edge to select the appropriate contents, for instance, Chen et al.",
"[13] presented the edge cooperative strategy based on neural collaborative filtering.",
"Jiang et al.",
"[14] used the offline user preferences data and statistical traffic patterns and proposed an online content popularity tracking algorithm.",
"However, the offline data will not be available always.",
"These works assume that users have identical preferences and no correlation amongst the data, which may not be the case in a practical situation.",
"Recommendation and caching can be individually approximated, however the joint optimization is NP hard without an optimal decomposition [15].",
"Hence, in this paper, two estimation procedures namely Point estimation and Bayesian estimation have been proposed for optimization.",
"Further, considering a real-time system, the estimation has been applied to a more practical heterogeneous network.",
"A typical heterogeneous network consists of a macro base station (MBS) and various low power nodes.",
"Similarly, in this work we have considered a heterogeneous system for the estimation model.",
"The main contributions of the paper are summarized as follows: For the first time two estimation procedures, Point and Bayesian estimation are provided.",
"A probabilistic model using Bayesian inference based on Dirichlet distribution is proposed.",
"Specifically, the influence of recommendation on the popularity profile is modelled using a conditional probability distribution.",
"A high probability guarantee on the estimated caching and recommendation strategies is provided.",
"Irrespective of the estimation method, it is shown that with a probability of $1-\\delta $ the proposed caching and recommendation strategy is $\\epsilon $ close to the optimal solution.",
"A high probability bound on the regret for Bayesian estimation method is provided.",
"To compare and contrast the obtained regret bound, we also derive a regret bound on the genie aided scenario using the Point estimation method.",
"First, we consider the Point estimation case and provide lower bound on the waiting time that is required to achieve an error $\\epsilon $ optimal solution with high probability.",
"Assuming a genie aided scenario, we prove a regret bound of $\\mathcal {O}(T^{2/3}\\sqrt{\\log T})$ .",
"Using the Martingale difference technique, we prove a high probability bound on the regret achieved by the Bayesian estimation method.",
"In particular, we show that $\\mathcal {O}(\\sqrt{T})$ regret is achievable, which is better than the genie aided scenario.",
"The proposed Point estimation and Bayesian estimation are further extended to a heterogeneous network consisting of $M$ SBSs with a central MBS.",
"The MBS computes an estimate of the probability transition matrix (PTM) and gives an update to each of the SBSs.",
"For computing guarantees, first $M$ is taken as two for the sake of simplicity and a lower bound on the waiting time is provided.",
"The same is generalized to a heterogeneous network consisting of $M$ SBSs, the estimation of the probability matrix has been derived and useful insights are drawn.",
"Numerical results are presented and it is shown that the proposed algorithm outperforms the existing least recently/frequently used (LRFU), least frequently used (LFU) and least recently used (LRU) in terms of average cache hit.",
"Notation: Bold uppercase letter denotes matrices.",
"$\\mathbb {E}(\\cdot )$ denotes the statistical expectation operator.",
"$f(\\cdot )$ represents the probability density function (PDF).",
"Superscript $(\\cdot )^{T}$ represents transposition.",
"$||\\cdot ||_F$ , $||\\cdot ||_{op}$ and $vec$ indicates the Frobenius norm, operator norm and vector respectively.",
"$I_d$ represents the $d \\times d$ identity matrix.",
"Further, $Dirch(\\alpha _1,\\alpha _2,\\ldots ,\\alpha _K)$ is the Dirichlet distribution with parameters $\\alpha _1,\\alpha _2,\\ldots ,\\alpha _K.$"
],
[
"System Model and Problem Statement",
"The system model consists of a wireless distributed content storage network with $M$ SBSs serving multiple users and one central MBS, as shown in Fig.",
"REF .",
"Each SBS can store up to $F$ contents/files of equal sizes from a catalog of contents denoted by $\\mathcal {C}:=\\lbrace 1,2,\\ldots ,F\\rbrace $ .",
"The requests are assumed to be independent and identically (iid) distributed across time.A more general model of non-stationary requests can be handled based on the insights provided in the later part of our paper.",
"As we know, recommending a file influences the users request process, and hence recommendation can provide “side information\" about the future requests.",
"Therefore, we consider the problem of jointly optimizing recommendation and caching policies in a cellular network.",
"We model the influence of recommendation on the request via a conditional probability distribution denoted $p_{ij,k}$ , which represents the probability that a user requested a file $i$ to the SBS $k$ given the content $j$ was recommended [16].",
"We assume that the time is slotted, and the PTM matrix for the $k$ -th SBS denoted by $(\\mathbf {P}_k)_{ij}:=p_{ij,k}$ , $i,j = 1,2,\\ldots ,F$ is assumed to be fixed across time slots.",
"For the sake of simplicity, it is assumed that at least one file is requested in every slot by each user $N$ in the network.This can be ensured if the slot duration is chosen to be large enough.",
"Let us use $u_{i}$ and $v_{j}$ to represent the probabilities with which a file $i$ is cached and file $j$ is recommended at any SBS, respectively.",
"This induces a set of caching and recommendation strategies denoted by $ \\mathcal {C}_{c,r}:= \\lbrace (\\mathbf {u},\\mathbf {v}) \\in [0,1]^{2 \\times F}: \\mathbf {u}^T \\mathbf {1} \\le c, \\mathbf {v}^T \\mathbf {1} \\le r\\rbrace ,$ where $r$ and $c$ are recommendation and cache constraints, respectively.",
"Figure: Distributed caching in a cellular network.In the sequel, the strategy is defined by the pair $(\\mathbf {u},\\mathbf {v})$ .",
"For a given strategy $(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}$ , the average cache hit at the SBS $k$ is given by $\\mathbf {u}^T \\textbf {P}_k\\mathbf {v}$ .",
"If the matrix $\\textbf {P}_k$ is known apriori at the SBS $k$ , the optimal strategy can be found by solving $\\max _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\textbf {P}_k \\mathbf {v}$ .",
"However, the matrix $\\textbf {P}_k$ is unknown, and therefore it needs to be estimated from the demands.",
"Let the variable $d_{k,i}^{(t)}$ denotes the demand at the SBS $k$ , and is defined as the total number of requests in the time slot $t$ for the file $i$ .",
"Since the demands arrive in a sequential manner, the PTMs need to be estimated and updated in an online fashion.",
"The performance of such algorithms is measured in terms of regret.",
"As apposed to adversarial setting of online learning, here we have assumed that there is an underlying distribution from which the requests are generated, namely the PTM.",
"Accordingly, the following provides the definition of the regret, which depends on the PTM: Definition 1 (Regret) The regret at the SBS $k$ after $T$ time slots with respect to any sequence of strategy $(\\mathbf {u}_{k,t},\\mathbf {v}_{k,t})$ , $t=1,2,\\ldots ,T$ is defined as $ \\texttt {Reg}_{k,T} &: =& T \\mathbf {u}_{k,*}^T \\mathbf {P}_k \\mathbf {v}_{k,*} - \\sum _{t = 1}^{T} \\mathbf {u}_t^T \\mathbf {P}_k \\mathbf {v_t},$ where $(\\mathbf {u}_{k,*}, \\mathbf {v}_{k,*}) := \\arg \\max _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T P_k \\mathbf {v}$ is the optimal strategy at the SBS $k$ .",
"The goal of the paper is to come up with a strategy at each SBS that results in a minimum regret.",
"Any caching and recommendation algorithm, either directly or indirectly estimates the PTM $\\mathbf {P}_k$ .",
"Therefore, the above goal translates to finding better estimates of the PTMs.",
"In this paper, we consider two approaches to finding the estimates of the PTMs in an online fashion, namely (i) Point estimation and (ii) Bayesian estimation methods.",
"Further, when there are multiple SBSs, any given SBS can potentially improve its estimate of PTM by fusing the estimates of the other SBSs.",
"The question of how to fuse the estimates that results in a good regret is another question to which we will shed some lights.",
"In the following section, we provide caching and recommendation algorithms for single SBS scenario, and provide theoretical guarantees for them."
],
[
"Joint Caching and Recommendation for Single SBS Scenario",
"In this section, we assume single SBS, and therefore, $M = 1$ as shown in Fig.",
"REF .",
"As mentioned above, using the demands obtained at the SBS, an estimate of the PTM matrix is computed using either Point estimation or Bayesian estimation method.",
"Given an estimate $\\hat{\\mathbf {P}}^{(t)}_k$ , the caching and recommendation strategies will be found by solving the following problemFor theoretical analysis, we assume that the problem can be solved exactly.",
"Figure: Distributed caching in a cellular network assuming single SBS.",
"$ (\\hat{\\mathbf {u}}_{o,t}^*, \\hat{\\mathbf {v}}_{o,t}^*) = \\arg \\max _{(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\hat{\\mathbf {P}}^{(t)}_k \\mathbf {v}.$ Now, we present the following two estimation procedures used in this paper.",
"Point estimation: Given any SBS $k$ , in this method, the demands until $t$ time slots is used to compute an estimate of the matrix $\\mathbf {P}_k$ .",
"During the first $t$ time slots, recommendation and caching are done with probabilities $q$ and $p$ , respectively.",
"Let $v_{jk}^{t} = 1$ if file $j$ was recommended in slot $t-1$ , and zero otherwise.",
"The recommendation and caching constraints in (REF ) are satisfied by choosing $q:=r/F$ and $p:= c/F$ .",
"We can see that as the value of $t$ increases, the estimate becomes better, and hence results in better performance.",
"The estimate of the $ij$ -th entry for SBS $k $ of the $\\mathbf {P}_k$ matrix is given by $ \\hat{p}_{ij,k}^{(t)} := \\frac{\\sum _{s=0}^{t-1} d_{ik}^{(s)} v_{jk}^{s-1} }{N \\sum _{s=0}^{t-1} v_{jk}^{s-1}}.$ The above is a naive estimate of the probabilities by using a simple counting of events.",
"The corresponding estimate of the matrix $\\mathbf {P}_k$ be denoted by $\\hat{\\mathbf {P}_k}^{(t)}$ .",
"Since $\\mathbb {E}\\lbrace \\hat{p}_{ij,k}^{(t+1)} | \\sum _{s=0}^{t-1} v_{jk}^s > 0\\rbrace = p_{ij,k}$ , the point estimator is an unbiased estimator.",
"After every time slot $t$ , the recommendation and caching probabilities are selected by solving the optimization problem in (REF ) with $\\hat{\\mathbf {P}}_k^{(t)}$ obtained in (REF ).",
"A procedure to find a strategy is given in Algorithm 1.",
"Bayesian estimation: In this method, for a given time slot, the rows of the matrix $\\mathbf {P}_k^{(t)}$ is sampled using a prior distribution, which is updated based on the past demands.",
"This may tradeoff the exploration versus exploitation while solving for the optimal recommendation and caching strategies.",
"Here, Dirichlet distribution is chosen as a prior.",
"The Dirichlet pdf is a multivariate generalization of the Beta distribution, and is given by $ f(x_1, \\ldots ,x_M, \\alpha _1, \\ldots , \\alpha _M) = \\frac{\\Gamma (\\sum _{j = 1}^{M}\\alpha _j)}{\\prod _{j = 1}^{M}\\Gamma (\\alpha _j)} \\prod _{j = 1}^{M}x_j^{\\alpha _j - 1},$ $\\alpha _j \\ge 0$ $\\forall $ $j$ .",
"The Dirichlet distribution is used as a conjugate pair in bayesian analysis and the shape of the distribution is determined by the parameter $\\alpha _j$ .",
"If $\\alpha _j = 1$ for all $j$ , then it leads to a uniform distribution.",
"The higher the value of $\\alpha _j$ , the greater the probability of occurence of $x_j$ .",
"The notation $(x_1,x_2,\\ldots ,x_M) \\sim \\texttt {Dirch}(\\alpha _1,\\alpha _2,\\ldots ,\\alpha _M)$ indicates that $(x_1,x_2,\\ldots ,x_M)$ is sampled from a Dirichlet distribution in (REF ).",
"An estimate in the beginning of the time slot $t$ of the $i$ -th row of the matrix $\\hat{\\mathbf {P}}_k^{(t)}$ is given by $ (\\hat{\\mathbf {P}_k}^{(t)})_i \\sim \\texttt {Dirch}\\left(\\sum _{s=1}^{t-1} d_{1k}^{(s)} v_{jk}^{s-1}, \\sum _{i=1}^{t-1}d_{2k}^{(s)} v_{jk}^{s-1}, \\sum _{s=1}^{t-1}d_{Fk}^{(s)} v_{jk}^{s-1}\\right),$ where $v_{jk}^{s-1}$ is as defined earlier with $v_{jk}^{0}$ sampled from $\\lbrace 0,1\\rbrace $ with probability $q:=r/F$ .",
"[h] Caching and recommendation algorithm (one SBS case) [1] Point estimation/Bayesian estimation $\\hat{\\mathbf {u}}_{b,0}^* \\stackrel{i.i.d.",
"}{\\sim } \\lbrace 0,1\\rbrace $ from $p = c/F$ , and $\\hat{\\mathbf {v}}_{b,0}^* \\stackrel{i.i.d.",
"}{\\sim } \\lbrace 0,1\\rbrace $ from $q = r/F$ .",
"Recommend and cache according to $\\hat{\\mathbf {v}}_{b,0}^*$ and $\\hat{\\mathbf {u}}_{b,0}^*$ .",
"$t=0,1,\\ldots ,T$ Observe demands $d^{(t)}_{ik}$ in slot $t$ .",
"Compute $\\hat{\\mathbf {P}}_{k}^{(t)}$ from (REF ) for point estimation Compute $\\hat{\\mathbf {P}}_k^{(t)}$ from (REF ) for Bayesian estimation Solve (REF ) Use $(\\hat{\\mathbf {v}}_{b,t}^*, \\hat{\\mathbf {u}}_{b,t}^*)$ to recommend and cache.",
"After every time slot $t$ , the recommendation and caching probabilities are selected by solving the optimization problem in (REF ) with $\\hat{\\mathbf {P}}_k^{(t)}$ obtained in (REF ).",
"A procedure to find a strategy is given in Algorithm 1.",
"In the following subsection, we provide theoretical guarantees of the above algorithm."
],
[
"Theoretical Guarantees",
"In this section, we provide a high probability bound on the regret for both Point estimation and Bayesian estimation.",
"For the Point estimation case, we start by providing a lower bound on the waiting time which is $\\epsilon $ close to the optimal caching strategies.",
"The result will be of the following form: With a probability of at least $1- \\delta $ , the following holds provided $t \\ge \\texttt {constant}$ $\\mathbf {u}_t^T \\mathbf {P}_k \\mathbf {v}_t \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\mathbf {P}_k \\mathbf {v} - \\epsilon ,$ where $(\\mathbf {u}_t, \\mathbf {v}_t)$ is the caching strategy obtained by using any algorithm.",
"The constant $\\epsilon $ depends on various parameters, as explained next.",
"Towards stating theoretical guarantees, the following definition is useful.",
"Definition 2 (Covering number) A set $\\mathcal {N}_{\\epsilon } := \\lbrace (\\mathbf {x}_1,\\mathbf {y}_1),(\\mathbf {x}_2,\\mathbf {y}_2),\\ldots ,(\\mathbf {x}_{\\mathcal {N}_{\\epsilon }},\\mathbf {y}_{\\mathcal {N}_{\\epsilon }})\\rbrace $ is said to be an $\\epsilon $ -cover of $\\mathcal {C}_{c,r}$ if for any $(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}$ , there exists $(\\mathbf {x}_j,\\mathbf {y}_j) \\in \\mathcal {N}_{\\epsilon }$ for some $j$ such that $\\Vert \\mathbf {u} - \\mathbf {x}_j\\Vert \\le \\frac{\\epsilon }{8}$ and $\\Vert \\mathbf {v} - \\mathbf {y}_j\\Vert \\le \\frac{\\epsilon }{8}$ .",
"The following theorem provides a bound that is useful to provide the final result.",
"Theorem 3.1 For a given estimate of the PTM denoted $\\hat{\\mathbf {P}}_k^{(t)}$ using Point estimation or Bayesian estimation, the following holds good $&&\\Pr \\left\\lbrace \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\mathbf {P}_k \\mathbf {v} - \\mathbf {u}_t \\mathbf {P}_k\\mathbf {v}_t \\ge \\epsilon \\right\\rbrace \\le |\\mathcal {N}_{\\epsilon }| \\Pr \\left\\lbrace \\Vert \\widehat{\\Delta P}^{(t)}\\Vert _F \\ge \\frac{\\epsilon }{4 \\kappa r c}\\right\\rbrace , $ where $(\\mathbf {u}_t, \\mathbf {v}_t)$ is the output of the Algorithm REF at time $t$ , and $\\widehat{\\Delta P}^{(t)}:= \\mathbf {P}_k - \\hat{\\mathbf {P}}_k^{(t)}$ .",
"Further, $\\kappa > 0$ is some constant.",
"Proof: See Appendix .",
"Using the above result, in the following, we provide our first main result on the performance of the Point estimation scheme.",
"Theorem 3.2 Using (REF ) for caching and recommendation in slot $t$ , for any $\\epsilon > 0$ , with a probability of at least $1-\\delta $ , $\\delta > 0$ , $(\\mathbf {u}_{o,t}^*)^T \\mathbf {P}_k \\mathbf {v}_{o,t}^* \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\mathbf {P}_k \\mathbf {v} - \\epsilon $ provided $ t \\ge \\frac{1}{q \\left(1 - \\exp \\lbrace -\\frac{N\\epsilon ^2}{8 \\kappa ^2 F^2 c^2 r^2}\\rbrace \\right)} \\log \\frac{2 \\mathcal {N}_\\epsilon F^2}{\\delta }.$ Proof: See Appendix .",
"As we know, the regret achieved by the Point estimation method is $\\mathcal {O}(T)$ as it incurs non-zero constant average error for all the slots $t$ satisfying (REF ).",
"In this method, the estimation of PTM is done using the samples obtained from the first $t$ slots, and the caching strategy is decided based on this estimate.",
"However, an improvement over this is to continuously update the estimates, and the caching/recommendation strategies.",
"Instead of analyzing the regret for this, we assume that at any time slot $t$ , a genie provides an estimate of the PTM as in (REF ) to compute the caching/recommendation strategies, and provide the corresponding approximate regret bound.",
"In particular, in Appendix , we show that a regret of $\\mathcal {O}(T^{2/3}\\sqrt{\\log T})$ can be achieved through the genie aided point estimation method.",
"As opposed to point estimation method, here (genie aided) the caching/recommendation decisions can be made using an improved estimate of the PTM in every time slot leading to a better regret.",
"We use this as a benchmark to compare the regret obtained from the Bayesian estimation method.",
"In the next section, we extend the result to two SBS scenario, and use the insights to extend it further in the later part of the paper to any number of SBSs."
],
[
"Bayesian Estimation: Single SBS Scenario",
"Note that unlike the analysis for point estimation, in this case, the strategies are correlated across time.",
"This makes the analysis non-trivial.",
"The approach we take is to convert a sequence of random variables (function of caching and recommendation across time) into a Martingale difference.",
"This enables us to use the Azuma's inequality, which can be used to provide high probability result on the regret.",
"In the following, we provide the result.",
"Theorem 3.3 For the Bayesian estimation in Algorithm REF , for any $\\epsilon > 0$ , with a probability of at least $1-\\delta $ , $\\delta > 0$ , the following bound on the regret holds $\\texttt {Reg}_T \\le {2rc\\max _{ij} p_{ij} |\\mathcal {N}_\\epsilon |}\\sum _t\\exp \\bigg \\lbrace -\\frac{8\\psi _t^2}{cr\\left|\\mathcal {N}_\\epsilon \\right|^2 \\bar{\\sigma }_t^2(t)} \\bigg \\rbrace + {2\\sum _t\\psi _t} + \\sqrt{128r^2c^2T\\log (1/\\delta )},$ where $\\alpha _{ij}^{(t)} = \\sum _{q=1}^{t-1}d_{i}^{(q)}v_{j}^{(q-1)}$ , $\\bar{\\sigma }_t^2:= \\left[\\sum _{j=1}^F \\frac{1}{\\left(\\sum _i \\alpha _{ij}^{(t)} +1\\right)^2}\\right]$ , and $\\psi _t$ is any non-negative number.",
"Proof: See Appendix .",
"Remark: Note that the above result is an algorithm dependent bound as it depends on the recommendation strategy, which is determined by the algorithm.",
"In order to provide more insights into the result, we will make certain assumption about the demands.",
"In particular, if the demands $d^{(q)}_i > 0$ almost surely for all $i$ and $q$ , then, $\\alpha _{ij}^{(t)}$ will be 0 when $v_j^{(q-1)} = 0$ for all $q \\le t-1$ or $\\mathcal {O}(t)$ in case of $\\sum _{q=1}^t v^{(q-1)}_j = \\mathcal {O}(\\sqrt{t})$ .",
"Note that this depends on the algorithm output, which we presume that at least files of higher probability transition values with recommendation will be sampled multiple times in a time frame.",
"This leads to $\\bar{\\sigma }_t^2 = \\mathcal {O}(\\frac{1}{t^2})$ .",
"Thus, assuming $\\Psi _t = \\mathcal {O}(\\sqrt{t})$ , the summation in the first term of the regret is $\\mathcal {O}(1)$ .",
"Overall, this results in $\\mathcal {O}(\\sqrt{T})$ regret.",
"Recall that an approximate regret of $\\mathcal {O}(T^{2/3} \\sqrt{\\log T})$ is shown for the genie aided case while the Bayesian estimation method achieves a regret of the order $\\sqrt{T}$ .",
"The genie aided regret is worse by an order of $T^{1/6}$ , which is partly due to the fact that the genie estimation of PTM is less accurate than the point estimation method.",
"In the next section, we extend our results to two SBS scenario."
],
[
"Proposed Caching and Recommendation Strategies With Multiple SBSs",
"In this section, we present caching and recommendation algorithms when there are multiple SBSs.",
"In particular, we provide insights on how to use the neighboring SBSs estimates to further improve the overall caching and recommendation performance of the network.",
"First, we present the results for two SBS scenario, and similar analysis will be used to extend the results to multiple SBSs."
],
[
"Two Small Base Station Scenario",
"In this subsection, we consider a two SBSs scenario, as shown in Fig.",
"REF .",
"As described in Section , $\\textbf {P}_1$ and $\\textbf {P}_2$ represents PTM for SBS-1 and SBS-2, respectively.",
"The central MBS sends the global update of the recommendation and caching decisions to each SBS.",
"Assume that the request across SBSs are independent.",
"Let each SBS use one of the estimation methods in Algorithm REF .",
"Let $\\hat{\\textbf {P}}_1^{(t)}$ and $ \\hat{\\textbf {P}}_2^{(t)}$ be the corresponding estimates (either point or Bayesian estimate) of $\\textbf {P}_1$ and $\\textbf {P}_2$ , respectively.",
"The two SBSs convey their respective PTM to the central MBS.",
"The central MBS computes an estimate $\\hat{\\textbf {Q}}_k^{(t)}$ , $k=1,2$ for SBS 1 and SBS 2 as a linear combination of the two estimates as given below Figure: Heterogeneous network with distributed caching consisting of two SBS.$ \\hat{\\textbf {Q}}_k^{(t)} = \\lambda _k \\hat{\\textbf {P}}_1^{(t)} + (1 - \\lambda _k) \\hat{\\textbf {P}}_2^{(t)},$ where $\\lambda _k \\in [0,1]$ , $k=1,2$ strikes a balance between the two estimates.",
"The above estimate is used to compute the respective caching and recommendation strategies for the two SBSs and will be communicated to the respective SBSs.",
"The above results in a better estimate, for example, when $\\textbf {P}_1 = \\textbf {P}_2$ or when the two matrices are close to each other.",
"The corresponding algorithm is shown below.",
"First, we prove the following guarantee for the Point estimation method.",
"Theorem 4.1 For Algorithm REF with point estimation, for any SBS $k$ and for any $\\epsilon > 0,$ with a probability of at least $1-\\delta $ , $\\delta > 0$ , the regret $\\texttt {Reg}_{k,T} < \\epsilon $ , i.e., $\\Pr \\bigg \\lbrace (\\mathbf {u}_{k,t}^*)^T \\textbf {P}_k \\mathbf {v}_{k,t}^* \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\hat{\\mathbf {Q}}^{(t)}_k \\mathbf {v} - \\epsilon _k \\bigg \\rbrace > 1- \\delta $ provided $ t \\ge \\max \\left\\lbrace \\tau \\left(\\frac{\\epsilon _k}{\\lambda _k}, \\frac{\\delta }{2}\\right), \\tau \\left(\\frac{\\epsilon _k}{(1-\\lambda _k)}, \\frac{\\delta }{2}\\right)\\right\\rbrace ,$ where $\\tau (\\epsilon , \\delta ) := \\frac{1}{q \\left(1 - \\exp \\lbrace -\\frac{N\\epsilon ^2}{8 \\kappa ^2 F^2 c^2 r^2}\\rbrace \\right)} \\log \\frac{2 |\\mathcal {N}_\\epsilon | F^2}{\\delta }.$ Further, $\\epsilon _k: = \\epsilon /2 - (1 - \\lambda _k) \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_2 - \\textbf {P}_1) \\mathbf {v}\\right|$ Proof: See Appendix .",
"[h] Caching and recommendation algorithm (two SBS case) [1] Point estimation/Bayesian estimation $\\hat{\\mathbf {u}}_{b,0}^* \\stackrel{i.i.d.",
"}{\\sim } \\lbrace 0,1\\rbrace $ from $p = c/F$ , and $\\hat{\\mathbf {v}}_{b,0}^* \\stackrel{i.i.d.",
"}{\\sim } \\lbrace 0,1\\rbrace $ from $q = r/F$ .",
"Recommend and cache according to $\\hat{\\mathbf {v}}_{b,0}^*$ and $\\hat{\\mathbf {u}}_{b,0}^*$ .",
"$t=0,1,\\ldots ,T$ Observe demands $d^{(t)}_{ijk}$ in slot $t$ , $k=1,2$ .",
"point estimation Compute $\\hat{\\mathbf {P}}_{k}^{(t)}$ from (REF ) for point estimation Compute $\\hat{\\mathbf {P}}_k^{(t)}$ from (REF ) for Bayesian estimation Choose $\\lambda _i$ , and find $\\hat{\\mathbf {Q}}^{(t)}_i$ from (REF ), and solve $ (\\hat{\\mathbf {u}}_{k,t}^*, \\hat{\\mathbf {v}}_{k,t}^*) = \\arg \\max _{(\\mathbf {u}, \\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\hat{\\mathbf {Q}}^{(t)}_k \\mathbf {v}$ Use $(\\hat{\\mathbf {v}}_{k,t}^*, \\hat{\\mathbf {u}}_{k,t}^*)$ to recommend and cache.",
"As in the single SBS case, to benchmark the performance of Bayesian estimation method, we consider a genie aided scenario, and in Appendix , we show that it achieves an approximate regret of $\\texttt {Reg}_{k,T} \\lessapprox \\max \\Bigg \\lbrace \\Theta \\lambda _k , \\Theta (1-\\lambda _k) \\Bigg \\rbrace T^{2/3} + 2T(1 - \\lambda _k)\\mathcal {V}_{12}, \\nonumber $ where $\\mathcal {V}_{12}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_2 - \\textbf {P}_1) \\mathbf {v}\\right|$ and $\\Theta = \\@root 3 \\of {\\frac{8 \\kappa ^2 F^2 c^2 r^2(\\log 4F^2T^2 + F)}{q N}} $ .",
"The above clearly shows the trade-off between the two terms.",
"The first term scales as $T^{2/3}$ while the second term scales with $T$ linearly.",
"This can be balanced by using $\\lambda _k = 1 - \\frac{1}{\\sqrt{T}}$ , which results in $\\mathcal {O}(\\sqrt{T})$ scaling of regret.",
"Note that the choice $\\lambda _k = 1 - \\frac{1}{\\sqrt{T}}$ reveals that as time progresses, i.e., as the BS $k$ collects more samples, the weights allocated to the neighboring BS should go down to zero, as expected.",
"Furthermore, by appropriately choosing $\\lambda _k$ as above, the regret obtained is of the order $T^{2/3}$ .",
"Next, we present the guarantees for Algorithm REF .",
"Theorem 4.2 For Algorithm REF with Bayesian estimation, for $ \\epsilon > 2 \\max _{k=1,2} (1 - \\lambda _k) T \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_1 - \\textbf {P}_2) \\mathbf {v}\\right|,$ with probability of at least $1-\\delta $ , $\\delta > 0$ , for any BS $k \\in \\lbrace 1, 2\\rbrace $ , the regret can be bounded as $\\texttt {Reg}_{k,T} \\le \\max \\left\\lbrace R_k\\left(\\frac{2\\epsilon _k}{\\lambda _k}, \\frac{\\delta }{2}\\right), R_k\\left(\\frac{2\\epsilon _k}{(1-\\lambda _k)}, \\frac{\\delta }{2}\\right)\\right\\rbrace .$ In the above, $ R_k(\\epsilon ,\\delta ) := 2rc\\max _{ijk} p_{ijk} |\\mathcal {N}_\\epsilon |\\sum _t\\exp \\bigg \\lbrace -\\frac{8\\psi _t^2}{cr\\left|\\mathcal {N_\\epsilon }\\right|^2 \\bar{\\sigma }^2_k(t)} \\bigg \\rbrace + 2\\sum _t\\psi _t + \\sqrt{128r^2c^2T\\log (1/\\delta )},$ $\\alpha _{ijk}^{(t)} = \\sum _{q=1}^{t-1}d_{ik}^{(q)}v_{jk}^{(q-1)}$ , $\\epsilon _k: = \\epsilon /2 - (1 - \\lambda _k) T \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_1 - \\textbf {P}_2) \\mathbf {v}\\right|$ , and $\\bar{\\sigma }^2_k(t):= \\left[\\sum _{j=1}^F \\frac{1}{\\left(\\sum _i \\alpha _{ijk}^{(t)} +1\\right)^2}\\right]$ .",
"Proof: See Appendix .",
"Remark: The result shows the tradeoff exhibited by $\\lambda _k$ .",
"In particular, larger $\\lambda _k$ makes the first regret inside the $\\max $ term in (REF ) larger, and smaller $\\lambda _k$ ensures that the second term inside the $\\max $ above dominates.",
"It is essential to balance the two depending on the number of samples received.",
"We further elaborate this in the simulation results.",
"Further, the above result is an algorithm dependent bound as the bound depends on the recommendation strategy, which is determined by the algorithm.",
"Following the single SBS analysis, we make similar assumptions in two SBS scenario.",
"In particular, if the demands $d^{(q)}_{ik} > 0$ almost surely for all $i$ and $q$ , then, $\\alpha _{ijk}^{(t)}$ will be 0 when $v_{jk}^{(q-1)} = 0$ for all $q \\le t-1$ or $\\mathcal {O}(t)$ in case of $\\sum _{q=1}^t v^{(q-1)}_{jk} = \\mathcal {O}(\\sqrt{t})$ .",
"Note that this depends on the algorithm output, which we presume that at least files of higher probability transition values with recommendation will be sampled multiple times in a time frame.",
"This leads to $\\bar{\\sigma }_k^2(t) = \\mathcal {O}(\\frac{1}{t^2})$ .",
"Thus, assuming $\\Psi _t = \\mathcal {O}(\\sqrt{t})$ , the summation in the first term of the regret is $\\mathcal {O}(1)$ .",
"Overall, this results in $\\mathcal {O}(\\sqrt{T})$ regret.",
"Furthermore, the regret obtained by the genie aided method is of the order $T^{2/3}$ , which is higher than the one achieved by the Bayesian estimation method.",
"This along with the experimental results establishes the superiority of the proposed Bayesian estimation method."
],
[
"Multiple Small Base Station Scenario",
"In this section, we consider a heterogeneous network with $M$ SBSs connected to a central MBS.",
"The requests at each SBS are assumed to be i.i.d.",
"with PTM $\\textbf {P}_1, \\textbf {P}_2, \\ldots , \\textbf {P}_M$ as described in Section .",
"Similar to the two SBS model, each SBS computes an estimate of the PTM as follows $ \\hat{\\textbf {Q}}_k^{(t)} = \\lambda ^{(k)}_1 \\hat{\\textbf {P}}_1^{(t)} + \\lambda _2^{(k)} \\hat{\\textbf {P}}_2^{(t)} + \\ldots + \\lambda ^{(k)}_M \\hat{\\textbf {P}}_M^{(t)},$ where $\\lambda ^{(k)}_1, \\lambda ^{(k)}_2, \\ldots , \\lambda ^{(k)}_M$ , $k = 1,2,\\ldots ,M$ are coefficients to be determined later.",
"Further, it satisfies $\\sum _{j=1}^M \\lambda _j^{(k)}$ .",
"The following theorem is a generalization of two BS model which provides a guarantee on the minimum time required to achieve a certail level of accuracy with high probability.",
"Theorem 4.3 Using (REF ) for any $\\epsilon > M^2 \\max _{k} \\lbrace (1 - \\lambda _1^{(k)}) \\mathcal {D}_{1} - \\lambda _2^{(k)} \\mathcal {D}_2 - \\ldots - \\lambda _M^{(k)} \\mathcal {D}_M\\rbrace ,$ for point estimation, with a probability of at least $1-\\delta $ , $\\delta > 0$ , for any BS $k$ , the regret ($\\texttt {Reg}_{k, T}$ ) is less than $\\epsilon $ $\\bigg \\lbrace $ i.e.",
"$(\\mathbf {u}_{o,t}^*)^T \\textbf {P}_1 \\mathbf {v}_{o,t}^* \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\textbf {P}_1 \\mathbf {v} - \\epsilon \\bigg \\rbrace $ provided $t \\ge \\max \\left\\lbrace \\tau \\left(\\frac{\\epsilon _1}{\\lambda _1^{(k)}}, \\frac{\\delta }{M}\\right), \\tau \\left(\\frac{\\epsilon _2}{\\lambda _2^{(k)}}, \\frac{\\delta }{M}\\right), \\ldots , \\tau \\left(\\frac{\\epsilon _M}{\\lambda _M^{(k)}}, \\frac{\\delta }{M}\\right) \\right\\rbrace , $ $ \\tau (\\epsilon , \\delta ) := \\frac{1}{q \\left(1 - \\exp \\lbrace -\\frac{N\\epsilon ^2}{8 \\kappa ^2 F^2 c^2 r^2}\\rbrace \\right)} \\log \\frac{2 \\mathcal {N}_\\epsilon F^2}{\\delta }, $ where, $\\epsilon _{k} := \\epsilon /M^2 - (1 - \\lambda _1^{(k)}) \\mathcal {D}_{1} + \\lambda _2^{(k)} \\mathcal {D}_2 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {D}_M$ , and $\\mathcal {D}_{k}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T P_k \\mathbf {v}\\right|$ $\\forall $ $k$ $= {1,2,\\ldots , M}$ .",
"Proof: See Appendix .",
"The regret for the genie aided case after appropriate choice for $\\lambda _k$ turns out to be of the order of $\\sqrt{T}$ .",
"The genie aided regret analysis is relegated to Appendix REF .",
"Next we present the regret bound for the Bayesian estimation method.",
"Theorem 4.4 Using (REF ) for any $\\epsilon > M^2 \\max _{k} \\lbrace (1 - \\lambda _1^{(k)}) \\mathcal {I}_{1} - \\lambda _2^{(k)} \\mathcal {I}_2 -, \\ldots , - \\lambda _M^{(k)} \\mathcal {I}_M\\rbrace ,$ with probability of at least $1-\\delta $ , $\\delta > 0$ , for any BS $k$ , $k \\in \\lbrace 1,2, \\ldots , M\\rbrace $ $\\texttt {Reg}_{k,T}$ , the regret of the Bayesian estimation method satisfies the following bound $\\texttt {Reg}_{k,T} \\le \\max \\left\\lbrace R_k\\left(\\frac{\\epsilon _1}{\\lambda _1^{(k)}}, \\frac{\\delta }{M}\\right), R_k\\left(\\frac{\\epsilon _2}{\\lambda _2^{(k)}}, \\frac{\\delta }{M}\\right), \\ldots , R_k\\left(\\frac{\\epsilon _M}{\\lambda _M^{(k)}}, \\frac{\\delta }{M}\\right) \\right\\rbrace .$ In the above $R_k(\\epsilon , \\delta ) := 2rc\\max _{ijk} p_{ijk} |\\mathcal {N}_\\epsilon |\\sum _t\\exp \\bigg \\lbrace -\\frac{8\\psi _t^2}{cr\\left|\\mathcal {N_\\epsilon }\\right|^2 \\bar{\\sigma }^2_k(t)} \\bigg \\rbrace + 2\\sum _t\\psi _t + \\sqrt{128r^2c^2T\\log (1/\\delta )}, \\nonumber $ $\\alpha _{ijk}^{(t)} = \\sum _{q=1}^{t-1}d_{ik}^{(q)}v_{jk}^{(q-1)}$ , $\\epsilon _{k} := \\epsilon /M^2 - (1 - \\lambda _1^{(k)}) \\mathcal {I}_{1} + \\lambda _2^{(k)} \\mathcal {I}_2 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {I}_M, \\\\ \\mathcal {I}_{k}: = \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T P_k \\mathbf {v}\\right|$ $\\forall $ $k$ $= {1,2,\\ldots , M}$ and $\\bar{\\sigma }^2_k(t):= \\left[\\sum _{j=1}^F \\frac{1}{\\left(\\sum _i \\alpha _{ijk}^{(t)} +1\\right)^2}\\right]$ .",
"Proof: See Appendix .",
"Remark: Note that the above result is an algorithm dependent bound as it depends on the recommendation strategy, which is determined by the algorithm.",
"Following the single SBS analysis, we make similar assumptions in multiple SBS scenario.",
"In particular, if the demands $d^{(q)}_{ik} > 0$ almost surely for all $i$ and $q$ , then, $\\alpha _{ijk}^{(t)}$ will be 0 when $v_{jk}^{(q-1)} = 0$ for all $q \\le t-1$ or $\\mathcal {O}(t)$ in case of $\\sum _{q=1}^t v^{(q-1)}_{jk} = \\mathcal {O}(\\sqrt{t})$ .",
"Note that this depends on the algorithm output, which we presume that at least files of higher probability transition values with recommendation will be sampled multiple times in a time frame.",
"This leads to $\\bar{\\sigma }_k^2(t) = \\mathcal {O}(\\frac{1}{t^2})$ .",
"Thus, assuming $\\Psi _t = \\mathcal {O}(\\sqrt{t})$ , the summation in the first term of the regret is $\\mathcal {O}(1)$ .",
"Overall, this results in $\\mathcal {O}(\\sqrt{T})$ regret.",
"As in the case of two SBS, the regret obtained is better than the genie aided scenario.",
"In the next section, we present experimental results that corroborates some of our theoretical observations."
],
[
"Simulation Results",
"In this section, simulation results are presented to highlight performance of the proposed caching and recommendation model.",
"The simulation setup consists of one SBS model, two-SBS model and a heterogeneous model with multiple users.",
"We assume a time-slotted system in the simulation setup.",
"For the heterogeneous model, the simulation consists of two scenarios as follows: Fixed Link Scenario: In this, the links between SBS and users are uniformly and independently distributed in $\\lbrace 0, 1\\rbrace $ with probability $1/2$ .",
"SINR Based Scenario: In this, the SBS and users are assumed to be distributed uniformly in a geographical area of radius 500m.",
"It is assumed that a SBS and user can communicate only if the corresponding SINR is greater than a threshold.",
"This SINR takes into account the fading channel, the path loss, power used, and the distance between the user and the SBS.",
"The minimum rate at which a file can be transferred from the SBS to a user is given by the threshold, and hence the reciprocal of the rate indicates the delay.",
"In the simulation, we have used $\\tau := \\frac{1}{\\log (1+\\texttt {SINR})}$ as a measure of the delay between a user and a SBS.",
"However, when the requested file is absent, a backhaul fetching delay of $\\alpha \\times \\tau $ is counted in addition to the down link delay of $\\tau $ , i.e., the overall delay when the file is absent is $(\\alpha + 1) \\tau $ , with $\\alpha = 10$ .",
"Also, if the threshold is $R$ , then at least $R$ bits can be sent in a time duration of at most $1/\\log (1 + \\texttt {SINR})$ seconds, and hence the throughput is roughly $R\\log (1 + \\texttt {SINR})$ bits/second.",
"Fig.",
"REF shows the plot for a heterogeneous system the metric used for comparison is throughput.",
"In Fig.",
"REF , the number of SBSs, users, the total number of files and threshold value for SINR are 5, 30, 100,and 12dB, respectively.",
"The throughput for the proposed algorithm with recommendation is 225 bits/s for a cache size of 24, while LRFU, LRU and LFU algorithm has a throughput of 100 bits/s, 85 bits/s and 70 bits/s respectively for the same cache size.",
"Thus from Fig.",
"REF we can see that the proposed algorithm has higher throughput as compared to the existing algorithms.",
"Fig.",
"REF corresponds to the SINR scenario for a two SBS model.",
"Fig.",
"REF shows the average delay versus cache size plot for (a) cache placement algorithm with recommendation (b) cache placement algorithm without recommendation, (c) LRFU algorithm, (d) LRU algorithm and (e) LFU algorithm.",
"In Fig.",
"REF , SBSs, the number of users, the total number of files and threshold value for SINR are 2, 25, 100 and 12dB, respectively.",
"From Fig.",
"REF we can observe that the delay of both the proposed algorithms is less as compared to the other benchmark algorithms, since pre-fetching files according to the estimated methods results in lower fetching costs from the backhaul and hence less delay.",
"Figure: Cache hit v/s λ\\lambda for 2 SBS and 𝐏 1 ≠𝐏 2 \\textbf {P}_1 \\ne \\textbf {P}_2.Fig.",
"REF shows the plot for two BS model.",
"The value of $\\lambda _1$ is varied between 0.1 and 1.",
"From the Fig.",
"REF , we can observe that as the value of $\\lambda _1$ approaches 0.5, the average cache hit increases, this is because for $\\lambda _1 = 0.5$ and $P_1 = P_2$ , the $Q$ popularity profile matrix of BS will have maximum similarity to the individual SBS popularity profile matrix and hence the cache hit will be maximum for $\\lambda _1 = 0.5$ and it will gradually decrease as we further increase the value of $\\lambda _1$ .",
"Fig.",
"REF shows the plot for average cache hit versus $\\lambda $ for 2 SBS when $\\textbf {P}_1 \\ne \\textbf {P}_2$ .",
"From the Fig.",
"REF , we can observe that for larger $T$ , the optimal lambda value is close to 1.",
"Also, for smaller value of $T$ , depending on the value of $\\Theta $ , the optimal value of $\\lambda $ is less than 1.",
"Thus, the simulation results prove that the recommendation helps in increasing the average cache hit when compared to the algorithm without recommendation and it also performs better than the existing popular LRFU, LRU and LFU algorithms."
],
[
"Conclusion",
"The deployment of MEC in the current wireless heterogeneous networks is an important application for the smooth transition to the distributed cloud based platform.",
"In this paper, a model that captures the caching decisions along with recommendation has been introduced.",
"Implications of recommendation on user requests has been studied and it has been observed that the recommendation does influence the demands from the users.",
"Bayesian and point estimation methods are used to determine the user request pattern.",
"An algorithm is then proposed to jointly optimize caching and recommendation.",
"A multi-tier heterogeneous model consisting of MBS and SBSs is also presented and an upper bound on the estimation accuracy of popularity profile is provided.",
"Finally, simulation results and theoretical proofs support the superior performance of the proposed method over the existing algorithms."
],
[
"Proof of Theorem ",
"From [17], it follows that $ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* - \\mathbf {u}^T \\mathbf {P}_k \\mathbf {v} \\le 2 \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|.$ Let $\\mathbf {x}^{*}$ and $\\mathbf {y}^*$ be solutions to $\\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|$ .",
"Since $\\mathbf {x}^{*}$ and $\\mathbf {y}^*$ belong to $\\mathcal {C}_{c,r}$ , for some $i=1,2,\\ldots , \\mathcal {N}_\\epsilon $ , there exist $\\mathbf {x}_i$ and $\\mathbf {y}_i$ in $\\mathcal {A}_{\\epsilon }$ such that $\\Vert \\mathbf {x}^{*} - \\mathbf {x}_i\\Vert _2 \\le \\epsilon /8$ , and $\\Vert \\mathbf {y}^{*} - \\mathbf {y}_i\\Vert _2 \\le \\epsilon /8$ .",
"Further, by adding and subtracting $\\mathbf {x}_i \\widehat{\\Delta P}^{(t)} \\mathbf {y}_i$ and $\\mathbf {x}_i \\widehat{\\Delta P}^{(t)} \\mathbf {y}_i$ , we get $ \\left|({\\mathbf {u}^*})^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}^*\\right| \\le \\left|\\mathbf {x}_i \\widehat{\\Delta P}^{(t)} \\mathbf {y}_i\\right| + \\frac{\\epsilon \\Vert \\widehat{\\Delta P}^{(t)}\\Vert _{op}}{4}.$ From (REF ) and (REF ) $\\Pr \\left\\lbrace \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\mathbf {P}_k \\mathbf {v} - \\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* \\ge \\epsilon \\right\\rbrace \\le \\Pr \\left\\lbrace \\bigcup _{i=1}^{\\mathcal {N}_{\\epsilon }} \\mathcal {B}_i\\ge g_\\epsilon \\right\\rbrace \\le \\sum _{i=1}^{\\mathcal {N}_{\\epsilon }} \\Pr \\left\\lbrace \\mathcal {B}_i \\ge \\frac{\\epsilon }{4}\\right\\rbrace , \\nonumber $ where $g_\\epsilon := \\frac{\\epsilon }{2} - \\epsilon \\Vert \\widehat{\\Delta P}^{(t)}\\Vert _{op}/4 \\le \\frac{\\epsilon }{4}$ , using $\\Vert \\widehat{\\Delta P}^{(t)}\\Vert _{op} \\le 1$ , and $\\mathcal {B}_i := \\left|\\mathbf {x}_i \\widehat{\\Delta P}^{(t)} \\mathbf {y}_i\\right|$ .",
"Using the fact that $\\left|\\mathbf {x}_i \\widehat{\\Delta P}^{(t)} \\mathbf {y}_i\\right| \\le \\kappa \\max _{j} \\Vert \\mathbf {x}_j\\Vert _1 \\max _{i} \\Vert \\mathbf {y}_i\\Vert _1 \\Vert \\widehat{\\Delta P}^{(t)}\\Vert _F \\le \\kappa rc \\Vert \\widehat{\\Delta P}^{(t)}\\Vert _F$ , the above can be further bounded to get $\\mathcal {N}_{\\epsilon } \\Pr \\left\\lbrace \\Vert \\widehat{\\Delta P}^{(t)}\\Vert _F \\ge \\frac{\\epsilon }{4 \\kappa r c}\\right\\rbrace $ .",
"This completes the proof."
],
[
"Proof of Theorem ",
"Consider the following $\\Pr \\left\\lbrace \\Vert \\widehat{\\Delta P}^{(t)}\\Vert ^2_F \\ge \\gamma \\right\\rbrace \\le \\Pr \\left\\lbrace \\max _{k,l}(p_{kl} - \\hat{p}^{(t)}_{kl})^2 \\ge \\frac{\\gamma }{F^2} \\right\\rbrace \\le F^2 \\Pr \\left\\lbrace (p_{kl} - \\hat{p}^{(t)}_{kl})^2 \\ge \\frac{\\gamma }{F^2} \\right\\rbrace , $ where $\\gamma := \\frac{\\epsilon }{4 \\kappa r c}$ , and the second inequality above follows from the union bound.",
"Conditioning on $V_{ls} := \\sum _{s=1}^{t} v_l^{(s-1)} = m$ , there are $Nm$ i.i.d.",
"samples available to estimate $p_{kl}$ .",
"Using Hoeffdings inequality $ \\nonumber \\mathbb {E}\\Pr \\left\\lbrace (p_{kl} - \\hat{p}^{(t)}_{kl})^2 \\ge \\frac{\\gamma }{F^2} \\left|\\right.",
"V_{ls} = m\\right\\rbrace &\\hspace{-8.5359pt}\\le \\hspace{-11.38092pt} & \\hspace{8.5359pt}2 \\mathbb {E} \\exp \\left\\lbrace -\\frac{2Nm \\gamma }{F^2} \\right\\rbrace .$ Since $V_{ls}$ is a binomial random variable with parameter $q$ , the above average with respect to $V_{ls}$ becomes $2 \\left(1- q \\left( 1 - \\exp \\lbrace -2N \\gamma /F^2\\rbrace \\right) \\right)^t.$ The following bound on the left hand side of (REF ) can be obtained using the above in (REF ), and substituting it in (REF ) $2 \\mathcal {N}_{\\epsilon } F^2 \\left(1- q \\left( 1 - \\exp \\lbrace -2N \\gamma /F^2\\rbrace \\right) \\right)^t.$ An upper bound on the above can be obtained by using $1 - x \\le e^{-x}$ .",
"Using the resulting bound, $\\Pr \\left\\lbrace \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\mathbf {P}_k \\mathbf {v} - \\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* \\ge \\epsilon \\right\\rbrace < \\delta $ provided $t$ satisfies the bound in the theorem."
],
[
"Genie Aided Regret Analysis: Heuristics for Two SBSs Case",
"Consider the instantaneous regret given by $\\texttt {Reg}_k(t) := (\\mathbf {u}_{k,t}^*)^T \\mathbf {P}_k \\mathbf {v}_{k,t}^* \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}_{k,t}^T \\mathbf {P}_k \\mathbf {v}_{k,t}$ at time $t$ .",
"Using the union bound, we can write $\\Pr \\left\\lbrace \\frac{1}{T} \\sum _{t = 1}^{T}\\texttt {Reg}_k(t) \\ge \\frac{1}{T}\\sum _{t=1}^{T}\\epsilon _t\\right\\rbrace \\le \\sum _{t=1}^T \\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\ge \\epsilon _t\\right\\rbrace , $ where $\\epsilon _t > 0$ .",
"From Theorem REF , it follows that for any $\\epsilon _t > 0$ , we have $\\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\le \\epsilon _t\\right\\rbrace \\ge 1- \\delta $ provided (REF ).",
"By choosing $\\delta = \\frac{1}{T^2}$ , the approximation $e^{-x} \\approx 1-x$ for small $x$ , and $|\\mathcal {N_\\epsilon }| \\lessapprox 1/\\epsilon ^F$ in Theorem REF , we get $\\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\le \\epsilon _t\\right\\rbrace \\ge 1- \\frac{1}{T^2}$ provided $t \\gtrapprox \\frac{8 \\kappa ^2 F^2 c^2 r^2}{q N\\epsilon _t^3}\\log \\frac{2F^2T^2}{\\epsilon _t^F}, $ where $\\gtrapprox $ is used to denote “approximately greater than or equal to\".",
"Assuming $\\epsilon _t <1 $ and using $\\log x \\approx x$ for small $x$ , we have $\\log \\frac{2F^2T^2}{\\epsilon _t^F} = \\log 2F^2T^2 + F \\log \\frac{1}{\\epsilon _t} \\le \\frac{(\\log 2F^2T^2 + F)}{\\epsilon _t}$ .The case of $\\epsilon _t >1$ can be handled in a similar fashion, and hence ignored.",
"Now, we can use (REF ) to write $\\epsilon _t$ in terms of $t$ to get $\\epsilon _t \\gtrapprox \\@root 3 \\of {\\frac{8\\kappa ^2F^2c^2r^2 (\\log 2F^2T^2 + F)}{qNt}}.$ In other words, with a probability of at least $1- \\frac{1}{T^2}$ , $\\texttt {Reg}_k(t) \\le \\@root 3 \\of {\\frac{8\\kappa ^2F^2c^2r^2 (\\log 2F^2T^2 + F)}{qNt}}$ .",
"Using this result in (REF ), we get the following result.",
"With a probability of at least $1-\\frac{1}{T}$ , $\\texttt {Reg}_{k,T} \\lessapprox \\@root 3 \\of {\\frac{8\\kappa ^2F^2c^2r^2 (\\log 2F^2T^2 + F)}{qN}}\\sum _{t=1}^T \\frac{1}{t^{1/3}} = \\mathcal {O}(T^{2/3} \\sqrt{\\log T}).$ Thus, the above shows that the regret achieved grows sub-linearly with time, and hence (genie aided) achieves a zero asymptotic average regret."
],
[
"Regret Analysis for Two SBS: Heuristics",
"The analysis here is very similar to the analysis of single BS case.",
"We repeat some of the analysis for the sake of clarity and completeness.",
"Let the instantaneous regret at the BS $k$ at time $t$ is given by $\\texttt {Reg}_k(t) := (\\mathbf {u}_{k,t}^*)^T \\mathbf {P}_k \\mathbf {v}_{k,t}^* \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}_{k,t}^T \\mathbf {P}_k \\mathbf {v}_{k,t}$ at time $t$ .",
"Using the union bound, we can write $\\Pr \\left\\lbrace \\frac{1}{T} \\sum _{t = 1}^{T}\\texttt {Reg}_k(t) \\ge \\frac{1}{T}\\sum _{t=1}^{T}\\epsilon _{k,t}\\right\\rbrace \\le \\sum _{t=1}^T \\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\ge \\epsilon _{k,T}\\right\\rbrace , $ where $\\epsilon _{k,t} > 0$ and $\\epsilon _{k,T} = \\frac{1}{T}\\sum _{t=1}^T \\epsilon _{k,t}$ .",
"From Theorem REF , it follows that for any $\\epsilon _{k,t} > 0$ , we have $\\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\le \\epsilon _{k,t}\\right\\rbrace \\ge 1- \\delta $ provided (REF ) is satisfied.",
"By choosing $\\delta = \\frac{1}{T^2}$ , assuming $\\epsilon _{k,t} <1 $ , using the approximations $e^{-x} \\approx 1-x$ for small $x$ , and $|\\mathcal {N_\\epsilon }| \\lessapprox 1/\\epsilon ^F$ , we have $\\log \\frac{2F^2T^2}{\\epsilon _{k,t}^F} \\le \\frac{(\\log 2F^2T^2 + F)}{\\epsilon _{k,t}}$ .",
"Using this in Theorem REF , we get $\\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\le \\epsilon _{k,t}\\right\\rbrace \\ge 1- \\frac{1}{T^2}$ provided $\\tau \\bigg (\\frac{\\epsilon _{k,t}}{\\lambda _k}, \\frac{\\delta }{2}\\bigg ) \\gtrapprox \\frac{8 \\kappa ^2 F^2 c^2 r^2 \\lambda _k^3(\\log 4F^2T^2 + F)}{q N\\epsilon _{k,t}^3}, $ where $\\gtrapprox $ is used to denote “approximately greater than or equal to\".",
"Using the above in (REF ), we get $t \\gtrapprox \\max \\left\\lbrace \\frac{8 \\kappa ^2 F^2 c^2 r^2 \\lambda _k^3 (\\log 4F^2T^2 + F)}{q N\\epsilon _{k,t}^3}, \\frac{8 \\kappa ^2 F^2 c^2 r^2 (1-\\lambda _k)^3(\\log 4F^2T^2 + F)}{q N\\epsilon _{k,t}^3}\\right\\rbrace .\\nonumber $ By rearranging and summing over $t$ , the error can be written as follows $\\epsilon _{k,t} \\lessapprox \\frac{1}{\\@root 3 \\of {t}}\\max \\Bigg \\lbrace \\Theta \\lambda _k, \\Theta (1- \\lambda _k) \\Bigg \\rbrace , \\nonumber $ where $\\Theta = \\@root 3 \\of {\\frac{8 \\kappa ^2 F^2 c^2 r^2(\\log 4F^2T^2 + F)}{q N}} $ .",
"Using this in the place of $\\epsilon _{k,t}$ in the above theorem, and summing over $t$ , we get with a probability of at least $1-\\frac{1}{T}$ , the following holds for BS $k$ $\\texttt {Reg}_{k,T} \\lessapprox \\max \\Bigg \\lbrace \\Theta \\lambda _k , \\Theta (1-\\lambda _k) \\Bigg \\rbrace \\frac{1}{\\@root 3 \\of {T}}\\sum _{t =1}^T 1 + 2T(1 - \\lambda _k)\\mathcal {V}_{12}, \\nonumber $ where $\\mathcal {V}_{12}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_2 - \\textbf {P}_1) \\mathbf {v}\\right|$ and $\\Theta = \\@root 3 \\of {\\frac{8 \\kappa ^2 F^2 c^2 r^2(\\log 4F^2T^2 + F)}{q N}} $ .",
"This completes the approximate analysis."
],
[
"Regret Analysis for Multiple SBS: Heuristics",
"The analysis here is again very similar to the analysis of single BS case.",
"From Theorem REF , it follows that for any $\\epsilon _{k,t} > 0$ , we have $\\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\le \\epsilon _{k,t}\\right\\rbrace \\ge 1- \\delta $ , where $\\epsilon _{k,t} > 0$ and $\\epsilon _k = \\frac{1}{T}\\sum _{t=1}^T \\epsilon _{k,t}$ .",
"By choosing $\\delta = \\frac{1}{T^2}$ , assuming $\\epsilon _{k,t} <1 $ , using the approximation $e^{-x} \\approx 1-x$ for small $x$ , and $|\\mathcal {N_\\epsilon }| \\lessapprox 1/\\epsilon ^F$ , we have $\\log \\frac{2F^2T^2}{\\epsilon _{k,t}^F} \\le \\frac{(\\log 2F^2T^2 + F)}{\\epsilon _{k,t}}$ .",
"Using this in Theorem REF , we get $\\Pr \\left\\lbrace \\texttt {Reg}_k(t) \\le \\epsilon _{k,t}\\right\\rbrace \\ge 1- \\frac{1}{T^2}$ provided $\\tau \\bigg (\\frac{\\epsilon _1}{\\lambda _1^{(k)}}, \\frac{\\delta }{M}\\bigg ) \\gtrapprox \\frac{8 \\kappa ^2 F^2 c^2 r^2 \\lambda _k^3(\\log 4F^2T^2 + F)}{q N\\epsilon _{k,t}^3}, $ where $\\gtrapprox $ is used to denote “approximately greater than or equal to\".",
"Using the above in (REF ), we get $t \\gtrapprox \\frac{\\Theta ^3}{\\epsilon _{k,t}^3} \\max \\left\\lbrace (\\lambda _1^{(k)})^3, (\\lambda _2^{(k)})^3, \\ldots , (\\lambda _M^{(k)})^3 \\right\\rbrace , \\nonumber $ where, $\\Theta = \\@root 3 \\of {\\frac{8 \\kappa ^2 F^2 c^2 r^2(\\log F^2T^2 + F)}{q N} }$ .",
"From the above, it is clear that the waiting time $t$ scales as the square of $F, c$ and $r$ , and is inversely proportional to the error $\\epsilon _{k,t}^3$ .",
"By rearranging and summing over $t$ , the error can be written as $\\epsilon _{k,t} \\lessapprox \\frac{1}{\\@root 3 \\of {t}}\\max \\Bigg \\lbrace \\Theta \\lambda _1^{(k)}, \\ldots , \\Theta \\lambda _M^{(k)} \\Bigg \\rbrace $ .",
"Using this in the place of $\\epsilon _{k,t}$ in the above theorem, and summing over $t$ , we get with a probability of at least $1-\\frac{1}{T}$ the following result on the regret for BS $k$ holds $\\texttt {Reg}_{k,T} &\\lessapprox & \\frac{\\Theta M^2}{\\@root 3 \\of {T}}\\max \\Bigg \\lbrace \\lambda _1^{(k)} , \\ldots , \\lambda _M^{(k)} \\Bigg \\rbrace \\sum _{t =1}^T 1 + M^2T\\big \\lbrace (1 - \\lambda _1^{(k)}) \\mathcal {D}_{1} - \\lambda _2^{(k)} \\mathcal {D}_2 -, \\ldots , -\\lambda _M^{(k)} \\mathcal {D}_M\\big \\rbrace \\nonumber $ Remark: Note that the value of regret depends on the values of $\\lambda _1^{(k)}, \\ldots , \\lambda _M^{(k)}$ and the term $\\mathcal {D}_k$ .",
"The first term scales as $T^{2/3}$ while the second term scales with $T$ linearly.",
"This can be balanced by using $\\lambda _1 = 1 - \\frac{1}{\\sqrt{T}}$ and $\\lambda _k = \\frac{1}{(M-1)\\sqrt{T}}$ , which results in $\\mathcal {O}(\\sqrt{T})$ scaling of regret.",
"Similar to the single SBS case, the choice $\\lambda _k = \\frac{1}{(M-1)\\sqrt{T}}$ reveals that as time progresses, i.e., as the BS $k$ collects more samples, the weights allocated to the neighboring BS should go down to zero, as expected.",
"Otherwise, one can optimize the above regret with respect to $\\lambda _k$ 's, and find the optimal choice."
],
[
"Proof of Theorem ",
"Similar to the proof of Theorem REF , from [17], it follows that at time $t$ , the performance gap of the proposed algorithm with respect to the optimal is given by $\\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* - \\mathbf {u}_t^T \\mathbf {P}_k \\mathbf {v}_t &\\le & 2 \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|.",
"\\nonumber $ Summing the above over all $t$ , we get $T\\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* - \\sum _t\\mathbf {u}^T_t \\mathbf {P}_k \\mathbf {v}_t &\\le & 2 \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|.",
"\\nonumber $ For a given $\\epsilon $ , the above implies that $ \\Pr \\bigg \\lbrace T\\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* - \\sum _t\\mathbf {u}^T_t \\mathbf {P}_k \\mathbf {v}_t \\ge \\frac{\\epsilon }{2} \\bigg \\rbrace &\\le & \\Pr \\bigg \\lbrace \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\ge \\frac{\\epsilon }{2} \\bigg \\rbrace \\nonumber \\\\&=& \\Pr \\bigg \\lbrace \\sum _t Y_t \\ge \\frac{\\epsilon }{2} - \\sum _t \\mathbb {E}\\left[ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}}\\hspace{-8.5359pt} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|\\right] \\hspace{-5.69046pt}\\bigg \\rbrace ,$ where $Y_t = \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u} \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| -\\mathbb {E}\\bigg [ \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }}\\left|\\mathbf {u} \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ]$ is a martingale difference, i.e., $\\mathbb {E}\\lbrace Y_t\\rbrace = 0$ , and $\\mathcal {N}_\\epsilon $ is the covering set of $\\mathcal {C}_{c,r}$ as in Definition I.",
"By Azuma's inequality, we have $ \\Pr \\bigg \\lbrace \\sum _t Y_t > \\frac{\\epsilon }{2} \\bigg \\rbrace \\le \\exp \\bigg \\lbrace \\frac{-\\epsilon ^2}{2 (4rc)^2}\\bigg \\rbrace .$ The above follows due to the fact that $\\left|Y_t\\right| \\le 4rc$ , which is explained below: $Y_t &\\le & \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| + \\mathbb {E} \\bigg [ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ]\\nonumber \\\\&\\le & \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T {P} \\mathbf {v}\\right| + \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{P}^{(t)} \\mathbf {v}\\right| + \\mathbb {E} \\bigg [ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T P \\mathbf {v}\\right| \\bigg ] + \\mathbb {E} \\bigg [ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{P}^{(t)} \\mathbf {v}\\right| \\bigg ] \\nonumber \\\\&\\le & 4rc,$ which follows from $\\left|Y_t\\right| \\le 4rc$ and $\\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T {P} \\mathbf {v}\\right| \\le rc$ .",
"Thus it follows from (REF ), $\\Pr \\bigg \\lbrace \\sum _t Y_t > \\frac{\\epsilon }{2} \\bigg \\rbrace \\le \\delta $ if $\\epsilon \\ge 32r^2c^2T \\log (1/\\delta )$ .",
"Using this definition of $\\left|Y_t\\right|$ , it follows that with a probability of at most $\\delta $ , we have $\\sum _t\\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u} \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|\\ge 2 \\sum _t \\mathbb {E} \\bigg [ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ] + \\sqrt{128r^2c^2T\\log (1/\\delta )}.",
"\\nonumber $ Choosing $\\epsilon = 32r^2c^2T \\log (1/\\delta )$ in (REF ), the following bound for regret is satisfied with a probability of at least $1 - \\delta $ : $ \\mathbf {u}^* \\mathbf {P}_k \\mathbf {v}^* - \\frac{1}{T}\\sum _{t = 1}^{T} \\mathbf {u}^T_t \\mathbf {P}_k \\mathbf {v}_t < \\frac{2}{T} \\sum _t \\mathbb {E} \\bigg [ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ] + \\sqrt{\\frac{128r^2c^2 \\log (1/\\delta )}{T}}.$ Now, it remains to bound the first term on the right hand side above.",
"For a given $\\psi _t > 0$ (to be chosen later), using total expectation rule, we get $\\sum _t \\mathbb {E} \\bigg [ \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ] &\\le & \\sum _t \\mathbb {E} \\bigg [ \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|,\\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\psi _t \\bigg ] \\nonumber \\\\&&\\times \\Pr \\bigg \\lbrace \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\psi _t \\bigg \\rbrace + \\sum _t \\psi _t \\nonumber \\\\&\\le & \\sum _t rc \\max _{kl} p_{kl} \\Pr \\bigg \\lbrace \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)}\\mathbf {v}\\right| > \\psi _t \\bigg \\rbrace + \\sum _t\\psi _t, \\nonumber $ where the first inequality above follows by using the bound $\\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\le \\psi _t$ .",
"Since $\\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right|$ , we get $ \\sum _t\\mathbb {E} \\bigg [ \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ] \\le rc |\\mathcal {N}_\\epsilon | \\max _{kl} p_{kl} \\sum _t \\Pr \\bigg \\lbrace \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\psi _t \\bigg \\rbrace + \\sum _t\\psi _t.$ Now, consider $ \\sum _t\\Pr \\bigg \\lbrace \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\psi _t \\bigg \\rbrace &\\stackrel{(a)}{\\le }& \\sum _t \\Pr \\bigg \\lbrace \\sum _{i,j=1}^{\\left|\\mathcal {N_\\epsilon }\\right|} \\mathbf {u}_i^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}_j > \\psi _{t}\\bigg \\rbrace \\nonumber \\\\&\\le & \\sum _t \\Pr \\bigg \\lbrace \\sum _{i, j=1}^{\\left|\\mathcal {N_\\epsilon }\\right|}\\sum _{l=1}^{\\left|\\mathcal {N_\\epsilon }\\right|} u_{ij}\\widehat{\\Delta P}^{(t)T}_{j} v_l> \\psi _t\\bigg \\rbrace \\nonumber \\\\&\\le & \\sum _t \\Pr \\bigg \\lbrace e^{s\\sum _{i,j=1}^{\\left|\\mathcal {N_\\epsilon }\\right|} u_{ij}^T X_j} > e^{s\\psi _t}\\bigg \\rbrace \\nonumber \\\\& \\stackrel{(b)}{\\le }& \\sum _t e^{-s\\psi _t} \\mathbb {E}[e^{s\\sum _{i,j=1}^{\\left|\\mathcal {N_\\epsilon }\\right|} u_{ij}^T X_j}]$ where $X_j := \\sum _{l=1}^{\\left|\\mathcal {N_\\epsilon }\\right|}v_l \\widehat{\\Delta P}^{(t)}_{lj}$ .",
"In the above, $(a)$ follows from the covering argument, and $(b)$ follows from the Chernoff bound.",
"From [18], using the optimal proxy variance, we get the following bound $\\sum _t \\Pr \\bigg \\lbrace \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\psi _t \\bigg \\rbrace &\\le & \\sum _t \\exp \\left\\lbrace -s\\psi _t+ s^2\\sum _{i,j,l=1}^{\\left|\\mathcal {N_\\epsilon }\\right|} \\frac{\\Vert u_{il}\\Vert ^2 \\Vert v_j\\Vert ^2\\sigma _j^2}{2}\\right\\rbrace ,$ where an upper bound on $\\sigma _l$ (see [18]) is given by $\\sigma _j \\le \\frac{1}{4(\\sum _i \\alpha _{ij}^{(t)} + 1)}$ and $\\alpha _{ij}^{(t)} = \\sum _{q=1}^{t-1}d_i^{(q)}v_j^{(q-1)}$ .",
"Optimizing the exponent in (REF ), the optimal $s^* = \\frac{\\psi _t}{\\sum _{i,j,l=1}^{\\left|\\mathcal {N_\\epsilon }\\right|} \\Vert u_{il}\\Vert ^2 \\Vert v\\Vert _F^2\\sigma _j^2}$ .",
"Further, $\\Vert u_{il}\\Vert ^2 \\le \\Vert u_{il}\\Vert \\le c$ and $\\Vert v\\Vert _F^2 \\le r |\\mathcal {N}_{\\epsilon }|^2$ .",
"Using these bounds, and the bound on $\\sigma _l$ above, (REF ) can be written as follows $\\sum _t\\Pr \\bigg \\lbrace \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\psi _t \\bigg \\rbrace &\\le & \\sum _t\\exp \\bigg \\lbrace -\\frac{8\\psi _t^2}{cr\\left|\\mathcal {N_\\epsilon }\\right|^2 \\bar{\\sigma }^2(t)} \\bigg \\rbrace ,$ where $\\bar{\\sigma }^2(t):= \\left[\\sum _{j} \\frac{1}{\\left(\\sum _i \\alpha _{ij}^{(t)} +1\\right)^2}\\right]$ .",
"Substituting the above in (REF ) results in $\\sum _t\\mathbb {E} \\bigg [ \\sup _{(u,v) \\in \\mathcal {N}_{\\epsilon }} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\bigg ] \\le rc \\max _{ij} p_{ij} |N_\\epsilon |\\sum _t\\exp \\bigg \\lbrace -\\frac{8\\psi _t^2}{cr\\left|\\mathcal {N_\\epsilon }\\right|^2 \\bar{\\sigma }^2(t)} \\bigg \\rbrace + \\sum _t\\psi _t.$ Thus, using the above, the regret can be written as $\\texttt {Reg}_T \\le {2rc\\max _{ij} p_{ij} |\\mathcal {N}_\\epsilon |}\\sum _t\\exp \\bigg \\lbrace -\\frac{8\\psi _t^2}{cr\\left|\\mathcal {N}_\\epsilon \\right|^2 \\bar{\\sigma }^2(t)} \\bigg \\rbrace + {2\\sum _t\\psi _t} + \\sqrt{128r^2c^2T\\log (1/\\delta )}.$"
],
[
"Proof of Theorem ",
"The analysis is done only for the first SBS as the analysis for the second SBS is similar.",
"As in (REF ), since $\\Pr \\bigg \\lbrace (\\mathbf {u}_{k,t}^*)^T \\textbf {P}_k \\mathbf {v}_{k,t}^* \\ge \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\mathbf {u}^T \\hat{\\mathbf {Q}}^{(t)}_k \\mathbf {v} - \\epsilon \\bigg \\rbrace \\le \\Pr \\left\\lbrace 2\\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| > \\epsilon \\right\\rbrace $ , it is sufficient to consider the following $ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| &=& \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_1 - \\hat{\\mathbf {Q}}^{(t)}_1 ) \\mathbf {v} \\right|\\nonumber \\\\&=& \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\lambda _1 \\mathbf {u}^T \\widehat{\\Delta P}_1^{(t)} \\mathbf {v} + (1 - \\lambda _1)\\mathbf {u}^T \\widehat{\\Delta P}_2^{(t)} \\mathbf {v} + (1 - \\lambda _1)\\mathbf {u}^T (\\textbf {P}_1 - \\textbf {P}_2) \\mathbf {v}\\right|\\nonumber \\\\&\\le & \\lambda _1 \\mathcal {V}_1 + (1-\\lambda _1) \\mathcal {V}_2 + (1 - \\lambda _1)\\mathcal {V}_{12},$ where $\\mathcal {V}_1 := \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P_1}^{(t)} \\mathbf {v}\\right|$ , $\\mathcal {V}_2 : = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P_2}^{(t)} \\mathbf {v}\\right|$ , and $\\mathcal {V}_{12}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_2 - \\textbf {P}_1) \\mathbf {v}\\right|$ .",
"Here, $\\widehat{\\Delta P}^{(t)} := \\textbf {P}_1 - \\hat{Q}_1^{(t)}$ , $\\widehat{\\Delta P_k}^{(t)} := \\textbf {P}_k - \\hat{\\textbf {P}}_k^{(t)}$ , $k=1,2$ .",
"Using the union bound, we get the following $\\Pr \\lbrace \\lambda _1 \\mathcal {V}_1 + (1-\\lambda _1) \\mathcal {V}_2 > \\epsilon _{1} \\rbrace \\hspace{-8.5359pt} &\\le & \\Pr \\left\\lbrace \\mathcal {V}_1 > \\frac{\\epsilon _{1}}{\\lambda _1 }\\right\\rbrace + \\Pr \\left\\lbrace \\mathcal {V}_2 > \\frac{\\epsilon _{1}}{(1-\\lambda _1)}\\right\\rbrace ,$ where $\\epsilon _{1} := \\epsilon /2 - (1 - \\lambda _1) \\mathcal {V}_{12}$ .Using results from Theorem REF to each of the above term with $\\epsilon $ replaced by $\\epsilon _{1}/\\lambda _1$ and $\\frac{\\epsilon _{1}}{(1-\\lambda _1)}$ with $\\delta $ replaced by $\\delta /2$ proves the theorem."
],
[
"Proof of Theorem ",
"Similar to the proof provided of Theorem 4.1, the analysis is done only for the first SBS using Bayesian estimate.",
"$ T\\mathbf {u}^* \\textbf {P}_1 \\mathbf {v}^* - \\sum _t\\mathbf {u}^T_t \\hat{\\mathbf {Q}}^{(t)}_k \\mathbf {v}_t &\\le & \\lambda _1 \\mathcal {U}_1 + (1-\\lambda _1) \\mathcal {U}_2 + (1 - \\lambda _1)\\mathcal {U}_{12}, \\nonumber $ where $\\mathcal {U}_1 := \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P_1}^{(t)} \\mathbf {v}\\right|$ , $\\mathcal {U}_2 : = \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P_2}^{(t)} \\mathbf {v}\\right|$ , and $\\mathcal {U}_{12}: = \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_1 - \\textbf {P}_2) \\mathbf {v}\\right|$ .",
"Here, $\\widehat{\\Delta P_k}^{(t)} := \\textbf {P}_k - \\hat{\\textbf {P}}_k^{(t)}$ , $k=1,2$ .",
"Consider the following $\\Pr \\lbrace T\\mathbf {u}^* \\textbf {P}_1 \\mathbf {v}^* - \\sum _t\\mathbf {u}^T_t \\hat{\\mathbf {Q}}^{(t)}_1 \\mathbf {v}_t \\ge \\epsilon \\rbrace &\\le & \\Pr \\lbrace \\lambda _1 \\mathcal {U}_1 + (1-\\lambda _1) \\mathcal {U}_2 + (1 - \\lambda _1)\\mathcal {U}_{12} \\ge \\epsilon \\rbrace \\nonumber \\\\&\\le & \\Pr \\left\\lbrace \\mathcal {U}_1 > \\frac{\\epsilon _1}{\\lambda _1 }\\right\\rbrace + \\Pr \\left\\lbrace \\mathcal {U}_2 > \\frac{\\epsilon _1}{(1-\\lambda _1)}\\right\\rbrace , \\nonumber $ where $\\epsilon _1 := \\epsilon /2 - (1 - \\lambda _1) \\mathcal {U}_{12}$ .",
"Using results from Theorem REF to each of the above term with $\\epsilon $ replaced by $2\\epsilon _1/\\lambda _1$ and $\\frac{2\\epsilon _1}{(1-\\lambda _1)}$ and $\\delta $ replaced by $\\delta /2$ , we get the desired result."
],
[
"Proof of Theorem ",
"The proof for multiple SBSs is a generalization of two SBSs and the analysis is done only for the first SBS, the rest of the SBSs are similar.",
"$ \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P}^{(t)} \\mathbf {v}\\right| \\hspace{-8.5359pt}&=& \\hspace{-11.38092pt} \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T (\\textbf {P}_1 - \\lambda ^{(k)}_1 \\hat{\\textbf {P}}_1^{(t)} - \\lambda _2^{(k)} \\hat{\\textbf {P}}_2^{(t)} + \\ldots - \\lambda ^{(k)}_M \\hat{\\textbf {P}}_M^{(t)}) \\mathbf {v}\\right| \\nonumber \\\\&=& \\hspace{-11.38092pt} \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} | \\lambda ^{(k)}_1\\mathbf {u}^T (\\textbf {P}_1 - \\hat{\\textbf {P}}_1^{(t)}) \\mathbf {v} + \\lambda _2^{(k)}\\mathbf {u}^T (\\textbf {P}_2 - \\hat{\\textbf {P}}_2^{(t)}) \\mathbf {v} + \\ldots + \\lambda _M^{(k)}\\mathbf {u}^T (\\textbf {P}_M -\\hat{\\textbf {P}}_M^{(t)}) \\mathbf {v} \\nonumber \\\\&& \\hspace{-11.38092pt} + (1 - \\lambda _1^{(k)})\\lbrace \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\textbf {P}_1 \\mathbf {v}\\right|\\rbrace \\ldots , - \\lambda _M^{(k)} \\lbrace \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T P_M \\mathbf {v}\\right|\\rbrace \\nonumber \\\\&\\le & \\hspace{-8.5359pt} \\lambda _1^{(k)} \\mathcal {V}_1 + \\lambda _2^{(k)} \\mathcal {V}_2 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {V}_M + (1 - \\lambda _1^{(k)})\\mathcal {D}_1 - \\lambda _2^{(k)} \\mathcal {D}_M -, \\ldots , - \\lambda _M^{(k)} \\mathcal {D}_M|,\\nonumber $ where $\\mathcal {V}_l := \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P_l}^{(t)} \\mathbf {v}\\right|$ , $l=1,2,\\ldots ,M$ , $\\mathcal {D}_{1}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\textbf {P}_1 \\mathbf {v}\\right|$ , $\\mathcal {D}_{2}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\textbf {P}_2 \\mathbf {v}\\right|$ , $\\ldots $ , $\\mathcal {D}_{M}: = \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T P_M \\mathbf {v}\\right|$ .",
"Here, $\\widehat{\\Delta P_k}^{(t)} := \\textbf {P}_1 - \\hat{\\textbf {Q}}_1^{(t)}$ , $\\widehat{\\Delta P_k}^{(t)} := \\textbf {P}_k - \\hat{\\textbf {P}}_k^{(t)}$ , $k=1,2, \\ldots , M$ .",
"Thus we can write the following: $\\Pr \\lbrace \\lambda _1^{(k)} \\mathcal {V}_1 + \\lambda _2^{(k)} \\mathcal {V}_2 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {V}_M > \\epsilon ^{^{\\prime }} \\rbrace &\\le & \\sum _{l=1}^M \\Pr \\left\\lbrace \\mathcal {V}_l > \\frac{\\epsilon _l}{\\lambda _l^{(k)} }\\right\\rbrace ,$ where $\\epsilon ^{^{\\prime }} := \\epsilon /M - (1 - \\lambda _1^{(k)}) \\mathcal {D}_{1} + \\lambda _2^{(k)} \\mathcal {D}_2 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {D}_M$ , and $\\epsilon _{k} := \\epsilon /M^2 - (1 - \\lambda _1^{(k)}) \\mathcal {D}_{1} + \\lambda _2^{(k)} \\mathcal {D}_2 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {D}_M, \\forall $ $k$ $= {1,2,\\ldots , M}$ .",
"Using results from Theorem REF to each of the above term with $\\epsilon $ replaced by $\\epsilon ^{^{\\prime }}$ and $\\epsilon _k$ and $\\delta $ replaced by $\\delta /M$ proves the theorem."
],
[
"Proof of Theorem ",
"The proof for multiple SBSs is a generalization of two SBSs and the analysis is done only for the first SBS for Bayesian estimate, the rest of the SBSs are similar.",
"First consider the following $ T\\mathbf {u}^* \\textbf {P}_1 \\mathbf {v}^* - \\sum _t\\mathbf {u}^T_t \\textbf {P}_1 \\mathbf {v}_t &\\le & \\sum _{j=1}^M \\lambda _j^{(k)} \\mathcal {W}_j + (1 - \\lambda _1^{(k)})\\mathcal {I}_1 - \\sum _{j=2}^M \\lambda _j^{(k)} \\mathcal {I}_j,$ where $\\mathcal {W}_j := \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\widehat{\\Delta P_j}^{(t)} \\mathbf {v}\\right|$ , $\\mathcal {I}_{j}: = \\sum _t \\sup _{(\\mathbf {u},\\mathbf {v}) \\in \\mathcal {C}_{c,r}} \\left|\\mathbf {u}^T \\textbf {P}_j \\mathbf {v}\\right|$ , $j=1,2,\\ldots ,M$ .",
"Here, $\\widehat{\\Delta P_k}^{(t)} := \\textbf {P}_k - \\hat{\\textbf {P}}_k^{(t)}$ , $k=1,2, \\ldots , M$ .",
"Thus we can write the following: $\\Pr \\lbrace \\lambda _1^{(k)} \\mathcal {W}_1 +, \\ldots , + \\lambda _M^{(k)} \\mathcal {W}_M \\ge \\epsilon ^{^{\\prime }} \\rbrace \\le \\sum _{l=1}^M \\Pr \\left\\lbrace \\mathcal {W}_l > \\frac{\\epsilon _l}{\\lambda _l^{(k)} }\\right\\rbrace ,$ where $\\epsilon ^{^{\\prime }} := \\epsilon /M - (1 - \\lambda _1^{(k)}) \\mathcal {I}_{1} + \\sum _{l=2}^M \\lambda _l^{(k)} \\mathcal {I}_k$ , and $\\epsilon _{k} := \\epsilon /M^2 - (1 - \\lambda _1^{(k)}) \\mathcal {I}_{1} + \\sum _{l=2}^M \\lambda _l^{(k)} \\mathcal {I}_k$ , for all $k = \\lbrace 1,2,\\ldots , M\\rbrace $ .",
"Using results from Theorem REF to each of the above term with $\\epsilon $ replaced by $\\epsilon ^{^{\\prime }}$ and $\\epsilon _k$ and $\\delta $ replaced by $\\delta /M$ , we get the regret bound described in the theorem."
]
] | 2210.07747 |
[
[
"Mutual-visibility in strong products of graphs via total\n mutual-visibility"
],
[
"Abstract Let $G$ be a graph and $X\\subseteq V(G)$.",
"Then $X$ is a mutual-visibility set if each pair of vertices from $X$ is connected by a geodesic with no internal vertex in $X$.",
"The mutual-visibility number $\\mu(G)$ of $G$ is the cardinality of a largest mutual-visibility set.",
"In this paper, the mutual-visibility number of strong product graphs is investigated.",
"As a tool for this, total mutual-visibility sets are introduced.",
"Along the way, basic properties of such sets are presented.",
"The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension.",
"Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given."
],
[
"Introduction",
"Let $G = (V(G), E(G))$ be a connected and undirected graph, and $X\\subseteq V(G)$ a subset of the vertices of $G$ .",
"If $x,y\\in V(G)$ , then we say that $x$ and $y$ are $X$ -visible, if there exists a shortest $x,y$ -path whose internal vertices are all not in $X$ .",
"$X$ is a mutual-visibility set if its vertices are pairwise $X$ -visible.",
"The cardinality of a largest mutual-visibility set is the mutual-visibility number of $G$ , and it is denoted by $\\mu (G)$ .",
"Each largest mutual-visibility set is also called $\\mu $ -set of $G$ .",
"These concepts were introduced by Di Stefano in [5].",
"They were in particular motivated by the significance that mutual-visibility properties play within problems that arise in mobile entity models.",
"Some of the numerous works that deal with such models are [1], [2], [4], [11].",
"Mutual-visibility sets in graphs are in a way dual to general position sets in graphs, the latter concepts being widely investigated in the last years [7], [8], [9], [10], [12].",
"Among other results, it was proved in [5] that the decision problem concerning the mutual-visibility number is NP-complete and the invariant was determined for several classes of graphs including block graphs, grids, and cographs.",
"The research was continued in [3] emphasizing on Cartesian products and graphs $G$ with $\\mu (G) = 3$ .",
"Interestingly, determining the mutual-visibility number of the Cartesian product of two complete graphs turns out to be equivalent to a case of the celebrated Zarankiewicz's problem which is a long-standing open combinatorial problem.",
"Continuing the investigation of the mutual-visibility in graph products, we investigate in this paper strong products.",
"In the next section we introduce the necessary concepts and recall some known results.",
"Then, in Section , we introduce total mutual-visibility sets which turned out to be useful for the investigation of mutual-visibility sets in strong products, and give some basic properties of total mutual-visibility sets.",
"In the subsequent section we first bound from below the (total) mutual-visibility number of strong products.",
"Then we determine the mutual-visibility number for the strong grids of arbitrary dimension which shows the tightness of the lower bound.",
"In addition, we find families of strong product graphs for which the bound is not tight and complete the section with another lower bound.",
"In Section we focus on strong prisms where we give a couple of tight bounds for the mutual-visibility number.",
"We conclude our exposition with several open problems and directions for further investigation."
],
[
"Preliminaries",
"Since two vertices from different components of a graph are not mutually visible, all graphs in the paper are connected unless stated otherwise.",
"For a natural number $n$ , we set $[n] = \\lbrace 1,\\ldots , n\\rbrace $ .",
"Given a graph $G = (V(G), E(G))$ , its order will be denoted by $n(G)$ .",
"The distance function $d_G$ on a graph $G$ is the usual shortest-path distance.",
"The subgraph $G^{\\prime }$ is convex, if for every two vertices of $G^{\\prime }$ , every shortest path in $G$ between them lies completely in $G^{\\prime }$ .",
"The convex hull of $V^{\\prime }\\subseteq V(G)$ , denoted as $\\mathit {hull}(V^{\\prime })$ , is defined as the smallest convex subgraph containing $V^{\\prime }$ .",
"The degree, $\\deg _G(x)$ , of a vertex $x$ is the number of its neighbors.",
"If $X\\subseteq V(G)$ , then $\\overline{X}$ denotes the complement of $X$ , that is the set containing all vertices of $G$ not in $X$ .",
"Moreover, $G[X]$ denotes the subgraph of $G$ induced by $X$ , that is the maximal subgraph of $G$ with vertex set $X$ .",
"The subgraph of $G$ induced by $\\overline{X}$ is denoted by $G-X$ , and by $G-v$ when $X=\\lbrace v\\rbrace $ .",
"Two vertices $u$ and $v$ are false twins if $uv\\notin E(G)$ and $N_G(u) = N_G(v)$ , where $N_G(x)$ is the open neighborhood of $x$ , and are true twins if $uv\\in E(G)$ and $N_G[u] = N_G[v]$ , where $N_G[x]$ is the closed neighborhood of $x$ .",
"Vertices are twins if they are true or false twins.",
"Adding a new vertex to a graph $G$ that is a true/false twin of an existing vertex of $G$ is an operation called splitting.",
"Another one-vertex extending operation is that of attaching a pendant vertex, that is a vertex connected by a single edge to an existing vertex of the graph.",
"A graph is a block graph if every block (i.e., a maximal 2-connected component) is a clique.",
"Block graphs can be generated by using true twins and pendant vertices.",
"Notice that the connected block graphs are exactly the graphs in which there is a unique induced path connecting every pair of vertices.",
"A graph is called cograph whenever it is obtained by a sequence of splittings starting from $K_1$ .",
"From this generative definition it follows a useful structural property.",
"Let $G$ be a cograph, and let $v_1$ be the starting vertex for a sequence of splitting operations that build $G$ .",
"If $G$ is connected, the first operation must be a true twin of $v_1$ (that produces $v_2$ adjacent to $v_1$ ).",
"Let $V_1 = \\lbrace v_1\\rbrace $ and $V_2 = \\lbrace v_2\\rbrace $ .",
"Now, for each further vertex $v$ which must be added to build $G$ , if $v$ is a twin of a vertex in $V_1$ ($V_2$ , respectively), then add it to $V_1$ (to $V_2$ , respectively).",
"We obtain that $V(G)$ can be partitioned into $V_1$ and $V_2$ where $v^{\\prime }v^{\\prime \\prime }\\in E(G)$ for each $v^{\\prime }\\in V_1$ and $v^{\\prime \\prime }\\in V_2$ .",
"Cographs include complete split graphs and complete $k$ -partite graphs.",
"A graph is a complete split graph if it can be partitioned into an independent set and a clique such that every vertex in the independent set is adjacent to every vertex in the clique.",
"A $k$ -partite graph (alias $k$ -chromatic graph) is a graph whose vertices are (or can be) partitioned into $k$ different independent sets; hence, a complete $k$ -partite graph is a $k$ -partite graph in which there is an edge between every pair of vertices from different independent sets.",
"The strong product $G\\boxtimes H$ of graphs $G$ and $H$ has vertex set $V(G\\boxtimes H) = V(G)\\times V(H)$ , with vertices $(g,h)$ and $(g^{\\prime },h^{\\prime })$ being adjacent in $G\\boxtimes H$ if either $gg^{\\prime }\\in E(G)$ and $h=h^{\\prime }$ , or $g=g^{\\prime }$ and $hh^{\\prime }\\in E(H)$ , or $gg^{\\prime }\\in E(G)$ and $hh^{\\prime }\\in E(H)$ , see [6].",
"A $G$ -layer through a vertex $(g,h)$ is the subgraph of $G\\boxtimes H$ induced by the vertices $\\lbrace (g^{\\prime },h):\\ g^{\\prime }\\in V(G)\\rbrace $ .",
"Analogously $H$ -layers are defined.",
"Finally, we recall the following result which is implicitly used throughout the paper.",
"Proposition 2.1 [6] If $(g, h)$ and $(g^{\\prime }, h^{\\prime })$ are vertices of a strong product $G\\boxtimes H$ , then $ d_{G\\boxtimes H} ((g, h), (g^{\\prime }, h^{\\prime })) = \\max \\lbrace d_G(g, g^{\\prime }), d_H (h, h^{\\prime })\\rbrace .$"
],
[
"Total mutual-visibility",
"The following definition introduces a variation of mutual-visibility.",
"It will be useful to provide bounds on the mutual-visibility number of strong product graphs, although we consider that the concept might be also of independent interest.",
"If $G$ is a graph and $X\\subseteq V(G)$ , then $X$ is a total mutual-visibility set of $G$ if every pair of vertices $x$ and $y$ of $G$ is $X$ -visible.",
"The term “total\" comes from observing that if $X$ is a total mutual-visibility set of $G$ , then for every pair $x,y\\in V(G)$ there exists a shortest $x,y$ -path whose internal vertices are all not in $X$ .",
"The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number of $G$ and is denoted by $\\mu _{\\rm t}(G)$ .",
"Notice that there could be graphs $G$ which do not contain total mutual-visibility sets, for such situations we set $\\mu _{\\rm t}(G) = 0$ .",
"For the sake of brevity, we say that $X$ is a $\\mu _{\\rm t}(G)$ -set (or $\\mu _{\\rm t}$ -set if we are not interested in the graph) if it is a total mutual-visibility set such that $|X| = \\mu _{\\rm t}(G)$ .",
"Clearly, every total mutual-visibility set is a mutual-visibility set, hence we have the following inequality $0\\le \\mu _{\\rm t}(G) \\le \\mu (G).$ In the following we show that such bounds can actually be achieved by the total mutual-visibility number.",
"Concerning the lower bound of (REF ), it can be easily checked that $\\mu _{\\rm t}(C_n) = 0$ for $n\\ge 5$ .",
"The variety of graphs with this property appears to be large as the next result confirms.",
"Proposition 3.1 Let $G$ be a graph.",
"If $V(G) = \\bigcup _{i=1}^k V_i$ , where $G[V_i]$ is a convex subgraph of $G$ and $\\mu _{\\rm t}(G[V_i]) = 0$ for each $i\\in [k]$ , then $\\mu _{\\rm t}(G) = 0$ .",
"Proof.",
"Suppose on the contrary that $G$ contains a total mutual-visibility set $X$ with $|X| \\ge 1$ .",
"Select an arbitrary vertex $x\\in X$ .",
"Then there exists an $i\\in [k]$ such that $x\\in V_i$ .",
"Hence clearly, $|X\\cap G[V_i]| \\ge 1$ .",
"However, since $G[V_i]$ is convex, we get that $X\\cap G[V_i]$ is a total mutual-visibility set of $G[V_i]$ , a contradiction to the assumption $\\mu _{\\rm t}(G[V_i]) = 0$ .",
"$\\Box $ In what follow we show that there also exist graphs $G$ with $\\mu _{\\rm t}(G) = 0$ such that they belong to well known graph classes and they are not covered by the Proposition REF .",
"To this end, recall that a cactus graph is a graph whose blocks are cycles and/or complete graphs $K_2$ .",
"Fig.",
"REF shows four examples of cactus graphs.",
"Proposition 3.2 Let $G$ be a cactus graph.",
"Then $\\mu _{\\rm t}(G)= 0$ if and only if $G$ has minimum degree 2 and for each cycle $C$ in $G$ with $n(C)\\le 4$ all the vertices in $C$ have degree at least 3.",
"Proof.",
"($\\Leftarrow $ ) Assume that $G$ does not contain pendant vertices and that for each cycle $C$ of $G$ , either $n(C)\\le 4$ and each vertex in $C$ has degree at least 3, or $n(C)\\ge 5$ .",
"Suppose now $\\mu _{\\rm t}(G)>0$ and consider any total mutual-visibility set $X$ of $G$ with $|X|\\ge 1$ and let $v\\in X$ .",
"If $v$ does not belong to any cycle of $G$ , since there are no pendant vertices, then $v$ must have at least two neighbors and such neighbors are not $X$ -visible, which is not possible.",
"Thus, we may consider $v$ belongs to a cycle $C$ .",
"If $n(C)\\ge 5$ , then the two neighbors of $v$ belonging to $C$ are not $X$ -visible.",
"If $n(C)\\le 4$ and each vertex in $C$ has degree at least 3, then again there must exist a pair of neighbors of $v$ which are non $X$ -visible, a contradiction again.",
"Hence $\\mu _{\\rm t}(G)=0$ must hold.",
"($\\Rightarrow $ ) It can be readily observed that each pendant vertex of $G$ forms a total mutual-visibility set of $G$ .",
"Thus, $G$ has minimum degree 2, since $\\mu _{\\rm t}(G)= 0$ .",
"Moreover, if $C$ is a cycle in $G$ such that $n(C)\\le 4$ and there exists $v\\in V(C)$ with $\\deg (v)=2$ , then the set $\\lbrace v\\rbrace $ is a total mutual-visibility set of $G$ , which is not possible.",
"Therefore, the second claim follows.",
"$\\Box $ Figure: Some cactus graphs.",
"The first two on the left do not fulfill the conditions of Proposition , and hence their total mutual-visibility number is greater than zero.As an application of this lemma, consider Fig.",
"REF .",
"From the left, the first two cactus graphs have total mutual-visibility number greater than zero since they both do not fulfill the conditions of the above lemma.",
"On the contrary, the other two graphs have total mutual-visibility number equal to zero.",
"Moreover, notice that among the cactus graph it is possible to find infinitely many graphs $G$ with $\\mu _{\\rm t}(G) = 0$ which are not covered by Proposition REF .",
"For instance, if $G$ is a cactus graph with minimum degree at least 2, girth at least 5, and contains at least one path of length at least 2 whose edges lie in no cycle, then $\\mu _{\\rm t}(G) = 0$ but $G$ might not admit a proper convex cover as in Proposition REF .",
"The rightmost graph in Fig.",
"REF is an example.",
"Concerning the upper bound in (REF ), we introduce the following definition.",
"A graph $G$ is a $(\\mu , \\mu _{\\rm t})$ -graph if $\\mu (G) = \\mu _{\\rm t}(G)$ .",
"Proposition 3.3 Block graphs (and hence trees and complete graphs) and graphs containing a universal vertex are all $(\\mu , \\mu _{\\rm t})$ -graphs.",
"Proof.",
"If $G$ is a complete graph, then $\\mu (G) = \\mu _{\\rm t}(G) = n(G)$ .",
"If $G$ is not complete and has a universal vertex, then it can be easily observed that $\\mu (G) = \\mu _{\\rm t}(G) = n(G)-1$ .",
"Assume that $G$ is a block graph.",
"From [5] we know that if $G$ is a block graph and $X$ the set of its cut-vertices, then $V(G)\\setminus X$ is a $\\mu $ -set of $G$ and hence $\\mu (G) = |V(G)\\setminus X|$ .",
"We now show that $V(G)\\setminus X$ is also a $\\mu _{\\rm t}$ -set of $G$ .",
"To this end, let us first observe that (1) each vertex in $V(G)\\setminus X$ is adjacent to a vertex in $X$ and that (2) $G[X]$ is a convex subgraph of $G$ .",
"Hence, every $x,y\\in V(G)$ are $(V(G)\\setminus X)$ -visible regardless their membership to $V(G)\\setminus X$ .",
"This proves that $V(G)\\setminus X$ is also a $\\mu _{\\rm t}$ -set of $G$ .",
"$\\Box $ In the following we characterize those cographs which are $(\\mu , \\mu _{\\rm t})$ -graphs.",
"To this aim we first recall a result from [5].",
"Lemma 3.4 [5] Given a graph $G$ , then $\\mu (G)\\ge n(G)-1$ if and only if there exists a vertex $v$ in $G$ adjacent to each vertex $u$ in $G-v$ such that $\\deg _{G-v} (u) < n(G) - 2$ .",
"In what follows, any vertex $v$ of $G$ fulfilling the condition in the above lemma will be called enabling.",
"Proposition 3.5 A cograph $G$ is a $(\\mu , \\mu _{\\rm t})$ -graph if and only if it has a universal vertex or no enabling vertices.",
"Proof.",
"($\\Leftarrow $ ) If $G$ has a universal vertex, then clearly $\\mu _{\\rm t}(G)=\\mu (G)$ .",
"If $G$ has no enabling vertices, then $\\mu (G)< n(G)-1$ by Lemma REF .",
"Since $\\mu (G)\\ge n(G)-2$ by [5], we get $\\mu (G) = n(G)-2$ .",
"According to the structural property of cographs recalled in Section , the vertices of $G$ can be partitioned into two sets $V_1$ and $V_2$ such that each vertex in $V_1$ is adjacent to each vertex of $V_2$ .",
"If $v_1$ ($v_2$ , respectively) is an arbitrary vertex in $V_1$ ($V_2$ , respectively), then it can be easily observed that $X=V(G) \\setminus \\lbrace v_1,v_2\\rbrace $ is a total mutual-visibility set.",
"Hence, $\\mu _{\\rm t}(G)=\\mu (G)= n(G)-2$ .",
"($\\Rightarrow $ ) We show that $G$ is not a $(\\mu , \\mu _{\\rm t})$ -graph by assuming that $G$ has an enabling vertex $v$ but no universal vertices.",
"In this case $V(G)$ can be partitioned in three sets: $A=\\lbrace v\\rbrace $ , $B$ the set of neighbors of $v$ , and $C$ that contains all the remaining vertices.",
"Notice that $C$ must be not empty otherwise $v$ would be a universal vertex, against the hypothesis.",
"By definition of enabling vertex, $B$ contains all the vertices $u\\in G$ such that $\\deg _{G-v}(v) < n(G)-2$ .",
"This implies that for each $u\\in C$ we have $\\deg _{G-v}(u) \\ge n(G)-2$ .",
"As a consequence, we have that (1) $G[C]$ is a clique, and (2) $bc \\in E(G)$ for each $b\\in B$ and $c\\in C$ .",
"Then $B\\cup C$ is a mutual-visibility set and hence $\\mu (G)\\ge n(G)-1$ .",
"As $G$ has no universal vertices, $\\mu (G)=n(G)-1$ .",
"We now show that $\\mu _{\\rm t}(G)$ cannot be equal to $n(G)-1$ .",
"In fact, let $u\\in V(G)$ and assume that $X= V(G)\\setminus \\lbrace u\\rbrace $ is a $\\mu _{\\rm t}$ -set.",
"Clearly, $u\\ne v$ because $v$ is not $X$ -visible with vertices in $C$ .",
"Vertex $u$ cannot be in $B$ since it is not a universal vertex and so there is a vertex $w \\in B$ such that $uw\\notin E(G)$ .",
"But then $u$ and $w$ are not $X$ -visible.",
"Finally, $u$ cannot be in $C$ , because in this case $u$ and $v$ are not $X$ -visible.",
"$\\Box $ The following result is a straightforward consequence of the characterization provided by Proposition REF .",
"Corollary 3.6 Complete split graphs and complete $k$ -partite graphs ($k \\ge 2$ ) with at least three vertices in each partition are $(\\mu , \\mu _{\\rm t})$ -graphs.",
"Observe that since $\\mu (C_n) = 3$ and $\\mu _{\\rm t}(C_n) = 2$ for $n\\le 4$ , the inequality $\\mu _{\\rm t}(G) \\le \\mu (G)$ can be strict.",
"Moreover, even if the equality is attained, it can happen that some $\\mu $ -sets are $\\mu _{\\rm t}$ -sets but some are not.",
"For an example consider the graph from Fig.",
"REF .",
"Figure: A graph GG with two μ\\mu -sets (represented by red vertices).",
"On the right-hand side a μ\\mu -set which is also a μ t \\mu _{\\rm t}-set is shown, while on the left-hand side the μ\\mu -set is not a μ t \\mu _{\\rm t}-set (the pair of vertices not in the μ\\mu -set are not visible)."
],
[
"Mutual-visibility in strong products",
"In this section we show how the total mutual-visibility of factor graphs can be used to provide lower bounds for the mutual-visibility number of their strong products.",
"Theorem 4.1 If $S_G$ and $S_H$ are total mutual-visibility sets of graphs $G$ and $H$ , respectively, where $|S_G| < n(G)$ and $|S_H| < n(H)$ , then $\\mu _{\\rm t}(G\\boxtimes H) \\ge |S_G| n(H) + |S_H| n(G) - |S_G| \\cdot |S_H|\\,.$ In particular, if $S_G$ and $S_H$ are $\\mu _{\\rm t}$ -sets, and $G$ and $H$ are non-complete graphs, then $\\mu _{\\rm t}(G\\boxtimes H) \\ge \\mu _{\\rm t}(G) n(H) + \\mu _{\\rm t}(H) n(G) - \\mu _{\\rm t}(G) \\mu _{\\rm t}(H)\\,.$ Proof.",
"Let $S = (V(G)\\times V(H)) \\setminus (\\overline{S_G} \\times \\overline{S_H})$ ; see Fig.",
"REF for an example of the construction of $S$ .",
"Figure: A representation of G⊠P 3 G\\boxtimes P_3, where GG is the graph in the right side of Fig. .",
"The represented μ t \\mu _{\\rm t}-set is that defined by Theorem .In the following we prove that $S$ is a total mutual-visibility set of $G\\boxtimes H$ .",
"Let $(g,h)$ and $(g^{\\prime },h^{\\prime })$ be arbitrary but distinct vertices from $V(G\\boxtimes H)$ .",
"Consider first the case in which $g\\ne g^{\\prime }$ and $h\\ne h^{\\prime }$ .",
"Regardless the membership of $g,g^{\\prime }$ to $S_G$ , since $S_G$ is a total mutual-visibility set of $G$ there exists a shortest $g,g^{\\prime }$ -path $P_G$ in $G$ such that no internal vertex of $P_G$ (if any) is in $S_G$ .",
"Let the consecutive vertices of $P_G$ be $g=g_0, g_1, \\ldots , g_k=g^{\\prime }$ , with $k\\ge 1$ since $g\\ne g^{\\prime }$ .",
"Similarly there is a shortest $h,h^{\\prime }$ -path $P_H$ in $H$ such that no internal vertex of $P_H$ (if any) is in $S_H$ .",
"Let the consecutive vertices of $P_H$ be $h=h_0, h_1, \\ldots , h_\\ell =h^{\\prime }$ , with $\\ell \\ge 1$ since $h\\ne h^{\\prime }$ .",
"Assume without loss of generality that $\\ell \\le k$ .",
"Then the vertices $(g,h) = (g_0,h_0), (g_1,h_1), \\ldots , (g_\\ell , h_\\ell ), (g_{\\ell +1}, h_\\ell ), \\ldots , (g_k, h_\\ell ) = (g^{\\prime },h^{\\prime })$ induce a shortest $(g,h),(g^{\\prime },h^{\\prime })$ -path $Q$ in $G\\boxtimes H$ .",
"Clearly, no internal vertex of $Q$ is in $S$ .",
"Consider now the remaining case in which $g=g^{\\prime }$ or $h=h^{\\prime }$ (but not both).",
"By the commutativity of the strong product we may without loss of generality assume $h= h^{\\prime }$ (and hence $g\\ne g^{\\prime }$ ).",
"Let $g = g_0, g_1, \\ldots , g_k = g^{\\prime }$ be the shortest $g,g^{\\prime }$ -path in $G$ , with $k\\ge 1$ .",
"If $k=1$ , then $(g,h)(g^{\\prime },h^{\\prime })\\in E(G\\boxtimes H)$ and there is nothing to prove.",
"Assume now $k\\ge 2$ .",
"Since $2\\le |S_H| < n(H)$ , there exists a vertex $z\\notin S_H$ such that $hz\\in E(H)$ .",
"Consider the path $Q^{\\prime }$ induced by the sequence of vertices $(g,h) = (g_0,h), (g_1,z), (g_2, z), \\ldots , (g_{k-1}, z), (g_k,h^{\\prime }) = (g^{\\prime },h^{\\prime })\\,.$ The length of $Q^{\\prime }$ is $k$ , hence $Q^{\\prime }$ is a shortest $(g,h),(g^{\\prime },h^{\\prime })$ -path.",
"Moreover, as $z\\notin S_H$ we get that each internal vertex of $Q$ does not belong to $S$ .",
"Consequently, the set $S$ is a total mutual-visibility set.",
"Since $|S| & = n(G)n(H) - (n(G) - |S_G|)(n(H) - |S_H|) \\\\& = |S_G| n(H) + |S_H| n(G) - |S_G|\\cdot |S_H|$ we are done with the first inequality.",
"When $S_G$ and $S_H$ are $\\mu _{\\rm t}$ -sets and $G$ and $H$ are non-complete graphs, the second inequality follows directly from the first one.",
"$\\Box $ Of course, when both $G$ and $H$ are $(\\mu , \\mu _{\\rm t})$ -graphs, the lower bound expressed by Theorem REF can be reformulated as follows: $\\mu (G\\boxtimes H) \\ge \\mu _{\\rm t}(G\\boxtimes H) \\ge \\mu (G)n(H) + \\mu (H)n(G) - \\mu (G) \\mu (H)\\,.$ Theorem REF extends to an arbitrary number of factors as follows.",
"Corollary 4.2 Let $H_k = G_1\\boxtimes G_2\\boxtimes \\cdots \\boxtimes G_k$ , $k\\ge 2$ .",
"If $G_i$ is a non-complete graph for each $1\\le i\\le k$ , then $\\mu _{\\rm t}(H_k) \\ge \\prod _{i=1}^k n(G_i) - \\prod _{i=1}^k (n(G_i) - \\mu _{\\rm t}(G_i)).$ Proof.",
"For each $1\\le i\\le k$ , let $X_i$ be a $\\mu _{\\rm t}(G_i)$ -set.",
"Let $S_k = (V(G_1)\\times \\cdots \\times V(G_k)) \\setminus (\\overline{X_1} \\times \\cdots \\times \\overline{X_k}).$ We prove that $S_k$ is a total mutual-visibility set of $H_k$ and proceed by induction on $k$ .",
"By Theorem REF we get that the assertion holds for $H_2$ .",
"Let us assume it is true for $H_{k}$ , $k\\ge 2$ , and consider $H_{k+1} = H_{k} \\boxtimes G_k$ .",
"Let us assume that $S_{k}$ is a total mutual-visibility set of $H_{k}$ .",
"By the proof of Theorem REF , $X_{k+1}$ is a total mutual-visibility set of $H_{k+1}$ .",
"Thus $\\begin{array}{rcl}\\mu _{\\rm t}(H_{k+1}) & \\ge & n(H_k) n(G_{k+1}) - (n(H_k) - \\mu _{\\rm t}(H_k))(n(G_{k+1}) - \\mu _{\\rm t}(G_{k+1})) \\\\& \\ge & n(H_k) n(G_{k+1}) - \\\\& & \\left( n(H_k) - \\left( \\prod _{i=1}^{k} n(G_i) - \\prod _{i=1}^{k} (n(G_i) - \\mu _{\\rm t}(G_i))\\right)\\right)(n(G_{k+1}) - \\mu _{\\rm t}(G_{k+1})) \\\\& = & n(H_{k+1}) - \\left(\\prod _{i=1}^k (n(G_i) - \\mu _{\\rm t}(G_i))\\right)(n(G_{k+1}) - \\mu _{\\rm t}(G_{k+1})) \\\\& = & n(H_{k+1}) - \\prod _{i=1}^{k+1} (n(G_i) - \\mu _{\\rm t}(G_i))\\end{array}$ and we are done.",
"$\\Box $ The following result (cf.",
"Theorem REF ) shows that there are $(\\mu , \\mu _{\\rm t})$ -graphs for which the lower bound provided by (REF ) coincides with the mutual-visibility number of the strong product.",
"Notice that it concerns the strong product of paths with at least three vertices, whereas Theorem REF (cf.",
"Section where strong prisms are considered) will provide the exact value of $\\mu (P_2\\boxtimes G)$ for every block graph $G$ (and hence also $\\mu (P_2\\boxtimes P_n)$ with $n\\ge 2$ ).",
"We first recall the following result that uses convex hulls to provide an upper bound to $\\mu (G)$ .",
"Lemma 4.3 [5] Given a graph $G$ , let $V_1,\\ldots ,V_k$ be subsets of $V(G)$ such that $\\bigcup _{i=1}^k V_i = V(G)$ .",
"Then, $\\mu (G) \\le \\sum _{i=1}^k \\mu ( \\mathit {hull}(V_i) )$ .",
"Theorem 4.4 If $H_k = P_{n_1}\\boxtimes \\cdots \\boxtimes P_{n_k}$ , where $k\\ge 2$ and $n(P_{n_i})\\ge 3$ for $i\\in [k]$ , then $\\mu (H_k) = \\prod _{i=1}^k n(P_{n_i}) - \\prod _{i=1}^k (n(P_{n_i}) - 2).$ Proof.",
"Let $X_i\\subseteq V(P_{n_i})$ be the (total) mutual-visibility set of $P_{n_i}$ formed by the end-vertices of the path.",
"According to the proof of Corollary REF , we get that $S_k = (V(P_{n_1})\\times \\cdots \\times V(P_{n_k})) \\setminus (\\overline{X_i} \\times \\cdots \\times \\overline{X_i})$ is a total mutual-visibility set of $H_k$ .",
"By the same corollary we also get the following lower bound: $\\mu (H_k) \\ge \\mu _{\\rm t}(H_k)\\ge \\prod _{i=1}^k n(P_{n_i}) - \\prod _{i=1}^k (n(P_{n_i}) - \\mu _{\\rm t}(P_{n_i}))= \\prod _{i=1}^k n(P_{n_i}) - \\prod _{i=1}^k (n(P_{n_i}) - 2) .$ Let the tuple $(\\ell _1,\\ldots ,\\ell _k)$ denote the generic vertex of $H_k$ , where $\\ell _i\\in [n(P_{n_i})]$ , $i\\in [k]$ .",
"We define the following two subsets of $V(H_k)$ : $V_\\mathrm {Int}= \\lbrace (\\ell _1,\\ldots ,\\ell _k):\\ \\forall ~ i\\in [k], \\ell _i\\ne 1 \\mbox{ and } \\ell _i\\ne n(P_{n_i}) \\rbrace $ ; $V_\\mathrm {Ext}= \\lbrace (\\ell _1,\\ldots ,\\ell _k):\\ \\exists ~ i\\in [k], \\ell _i= 1 \\mbox{ or } \\ell _i = n(P_{n_i}) \\rbrace $ .",
"From these definitions, it can be easily observed that $V_\\mathrm {Int}$ and $V_\\mathrm {Ext}$ form a partition of $V(H_k)$ .",
"Moreover, according to this notation we get the following characterization of the total mutual-visibility set $S_k$ defined in (REF ): $S_k=V_\\mathrm {Ext}.$ To prove the upper bound for $\\mu (H_k)$ , we use Lemma REF .",
"To this end, we determine the (minimum) number of induced and convex diagonals which cover all the vertices of $H_k$ .",
"A diagonal is either degenerated or non-degenerated: non-degenerated diagonals are paths of $H_k$ formed by at least two vertices and having the form $((\\ell _1,\\ldots ,\\ell _k), (\\ell _1+1,\\ldots ,\\ell _k+1), (\\ell _1+2,\\ldots ,\\ell _k+2), \\ldots )$ , whereas each degenerated diagonal consists of a single vertex.",
"These two kinds of diagonals are formally defined as follows (see.",
"Fig.",
"REF for two examples): $(i)$ Each vertex in $I= \\lbrace (\\ell _1,\\ldots ,\\ell _k):\\ \\exists ~ i\\in [k], \\ell _i=1 \\mbox{ and } \\forall j\\in [k], \\ell _j\\ne n(P_{n_j}) \\rbrace $ belongs to non-degenerated diagonals.",
"In particular, each vertex in $I$ is the initial vertex (i.e., one of its end-vertices) of such kind of diagonals.",
"$(ii)$ If $(\\ell _1,\\ldots ,\\ell _k)$ belongs to a non-degenerated diagonal $d$ , then also its neighbor $(\\ell _1+1,\\ldots ,\\ell _k+1)$ (if it exists in $H_k$ ) belongs to $d$ .",
"This property allows to define non-degenerated diagonals, along all their maximal length, till some terminating vertex having at least one coordinate $\\ell _i$ such that $\\ell _i = n(P_{n_i})$ .",
"We denote by $T$ all the terminating vertices of non-degenerated diagonals.",
"$(iii)$ The set $D= \\lbrace (\\ell _1,\\ldots ,\\ell _k) ~:~ \\exists ~ i,j\\in [k], \\ell _i=1 \\textit { and } \\ell _j = n(P_{n_j}) \\rbrace $ contains all vertices forming degenerated diagonals.",
"Figure: Visualization of diagonals as defined in the proof of Theorem .",
"(Left) In this strong product H 2 =P 5 ⊠P 6 H_2 = P_5\\boxtimes P_6, the thicker and bolder lines represent non-degenerated diagonals.",
"(Right) A representation of H 3 =P 5 ⊠P 6 ⊠P 6 H_3 = P_5\\boxtimes P_6\\boxtimes P_6 as an “opaque rectangular cuboid” where the position of the vertex with coordinates (1,1,1)(1,1,1) is shown.",
"Black vertices represents the elements of set II, that is the starting points of non-degenerate diagonals; white vertices represents the elements of set DD, that is vertices forming degenerated diagonal.",
"All such diagonals cover the whole graph H 3 H_3.Notice that the non-degenerated diagonals are pairwise vertex disjoint.",
"The requested covering of $H_k$ is given by all the maximal non-degenerated diagonals along with all the degenerated diagonals.",
"Now, let $X\\subseteq V(H_k)$ be the set containing the end-vertices of each non-degenerated diagonal and all the vertices forming degenerated diagonals; formally, $X=I \\cup T \\cup D$ .",
"According to Lemma REF we know that $\\mu (H_k)\\le |X|$ .",
"By Eq.",
"REF , we complete the proof by showing that $V_\\mathrm {Ext}= X$ .",
"Let $v=(\\ell _1,,\\ldots ,\\ell _k)\\in V_\\mathrm {Ext}$ .",
"By definition of $V_\\mathrm {Ext}$ , there exists a coordinate $\\ell _i$ of $v$ for which $\\ell _i = 1$ or $\\ell _i= n(P_{n_i})$ .",
"If both $\\ell _i = 1$ and $\\ell _i= n(P_{n_i})$ hold, then property $(iii)$ in the definition of diagonals holds.",
"This means that $v\\in D$ and hence $v\\in X$ .",
"If $\\ell _i = 1$ and $\\ell _i\\ne n(P_{n_i})$ hold, then property $(i)$ in the definition of diagonals holds.",
"This means that $v\\in I$ and hence $v\\in X$ .",
"If $\\ell _i \\ne 1$ and $\\ell _i= n(P_{n_i})$ hold for each $i$ , then consider the smallest coordinate $\\ell _j$ of $v$ .",
"According to property $(ii)$ , the vertex $v^{\\prime }= (\\ell _1-(\\ell _j-1),\\ldots ,\\ell _k-(\\ell _j-1))$ lies in the set $I$ from which a non-degenerated diagonal starts.",
"This implies that $v\\in T$ and hence $v\\in X$ .",
"So, in all cases we have $v\\in X$ .",
"Let $v=(\\ell _1,\\ldots ,\\ell _k)\\in X$ .",
"If $v\\in I\\cup D$ , then $v\\in V_\\mathrm {Ext}$ trivially holds.",
"Assume now $v\\in T$ , that is $v$ is the end-vertex of a non-degenerated diagonal $d$ starting at some vertex $v^{\\prime }=(\\ell _1^{\\prime },\\ldots ,\\ell _k^{\\prime })$ for which $(i)$ holds, and made maximal by iteratively applying property $(ii)$ .",
"According to $(ii)$ , an end-vertex of $d$ must be in $\\lbrace n(P_{n_1}), \\ldots , n(P_{n_k})\\rbrace $ , and hence $v\\in V_\\mathrm {Ext}$ .",
"This proves that $V_\\mathrm {Ext}= X$ holds.",
"$\\Box $ It seems worth pointing out that the result of Theorem REF for two- and three-dimensional strong grids reads as follows: $\\mu (P_{n_1}\\boxtimes P_{n_2}) & = 2n_1 + 2n_2 - 4\\,,\\\\\\mu (P_{n_1}\\boxtimes P_{n_2}\\boxtimes P_{n_3}) & = 2(n_1n_2 + n_1n_3 + n_2n_3) - 4(n_1 + n_2 + n_3) + 8\\,.$ To conclude the analysis, notice there are examples of graphs for which the bound of Theorem REF is not sharp.",
"An example of this situation is given in Fig.",
"REF .",
"Figure: The graph T⊠P 5 T\\boxtimes P_5 and its mutual-visibility set of cardinality 36.Let $T$ be the tree obtained from $K_{1,3}$ by subdividing each of its edges three times.",
"Then Theorem REF implies $\\mu (T\\boxtimes P_5)\\ge \\mu (P_5)n(T) + \\mu (T)n(P_5) - \\mu (P_5)\\mu (T) = 35$ , but in Fig.",
"REF we can see a mutual-visibility set of cardinality 36 found by computer search.",
"This example also shows that even when both factors of a strong product are $(\\mu , \\mu _{\\rm t})$ -graphs, their strong product does not achieve the equality in the bound of Theorem REF .",
"Note that this particular example can be generalized to an infinite family of graphs where the difference between the mutual-visibility number and the bound of the theorem becomes arbitrarily large.",
"This situation also suggests that generalizing Theorem REF (when $k=2$ ) to the strong product of two arbitrary trees might be a challenging problem.",
"Corollary 4.5 If $G_1, \\ldots , G_k$ are non-complete graphs, each containing a universal vertex, then $\\mu _{\\rm t}(G_1\\boxtimes \\cdots \\boxtimes G_k) = \\prod _{i=1}^k n(G_i) - 1\\,.$ Proof.",
"Since each $G_i$ is a $(\\mu , \\mu _{\\rm t})$ -graph, by Corollary REF we get $\\mu _{\\rm t}(G_1\\boxtimes \\cdots \\boxtimes G_k) \\ge \\prod _{i=1}^k n(G_i) - 1$ .",
"The claim follows by simply observing that $G_1\\boxtimes \\cdots \\boxtimes G_k$ is not a clique and hence $\\mu (G_1\\boxtimes \\cdots \\boxtimes G_k) < \\prod _{i=1}^k n(G_i)$ .",
"$\\Box $ We conclude the section with another lower bound on $\\mu (G\\boxtimes H)$ in terms of the mutual-visibility number of the factors.",
"Theorem 4.6 If $G$ and $H$ are graphs, then $\\mu (G\\boxtimes H) \\ge \\mu (G)\\mu (H)\\,.$ Proof.",
"Let $S_G$ be a $\\mu $ -set of $G$ and $S_H$ be a $\\mu $ -set of $H$ .",
"Then we claim that $S = S_G \\times S_H$ is a mutual-visibility set of $G\\boxtimes H$ .",
"Let $(g,h)$ and $(g^{\\prime },h^{\\prime })$ be arbitrary two vertices from $S$ .",
"Since $g,g^{\\prime }\\in S_G$ , there exists a shortest $g,g^{\\prime }$ -path $P_G$ in $G$ such that no internal vertex of $P_G$ is in $S_G$ .",
"Let the consecutive vertices of $P_G$ be $g=g_0, g_1, \\ldots , g_k=g^{\\prime }$ .",
"Similarly there is a shortest $h,h^{\\prime }$ -path $P_H$ in $H$ such that no internal vertex of $P_H$ is in $S_H$ .",
"Let the consecutive vertices of $P_H$ be $h=h_0, h_1, \\ldots , h_\\ell =h^{\\prime }$ .",
"Note that it is possible that $k=0$ or $\\ell = 0$ (but not both).",
"Assume without loss of generality that $\\ell \\le k$ .",
"Then the vertices $(g,h) = (g_0,h_0), (g_1,h_1), \\ldots , (g_\\ell , h_\\ell ), (g_{\\ell +1}, h_\\ell )), \\ldots , (g_k, h_\\ell )) = (g^{\\prime },h^{\\prime })$ induce a shortest $(g,h),(g^{\\prime },h^{\\prime })$ -path $Q$ in $G\\boxtimes H$ .",
"Clearly, no internal vertex of $Q$ is in $S$ , hence we conclude that $S$ is a mutual-visibility set.",
"$\\Box $"
],
[
"Mutual-visibility in strong prisms",
"In this section we study the mutual-visibility number of strong prisms, that is graphs in the form $G\\boxtimes P_2$ .",
"We begin with the following general lower bound.",
"Theorem 5.1 If $G$ is a graph, then $\\mu (G\\boxtimes P_2) \\ge \\max \\lbrace n(G), 2\\mu (G)\\rbrace $ .",
"Proof.",
"Assume that $V(P_2)=\\lbrace p,q\\rbrace $ and that $X$ is a $\\mu $ -set of $G$ .",
"We prove the statement by showing that both $S_1 = V(G) \\times \\lbrace p\\rbrace $ and $S_2 = X\\times P_2$ are mutual-visibility sets in $G\\boxtimes P_2$ .",
"Let $(g,p)$ and $(g^{\\prime },p)$ , with $q\\ne g^{\\prime }$ , be arbitrary two distinct vertices from $S_1$ .",
"Consider a shortest $g,g^{\\prime }$ -path $P_G$ in $G$ .",
"Let the consecutive vertices of $P_G$ be $g=g_0, g_1, \\ldots , g_k=g^{\\prime }$ .",
"Since $g\\ne g^{\\prime }$ we get $k\\ge 1$ .",
"If $k=1$ , then $(g,p)$ and $(g^{\\prime },p^{\\prime })$ are connected and there is nothing to prove.",
"If $k\\ge 2$ , then the vertices $(g,p) = (g_0,p), (g_1,q), \\ldots , (g_{k-1},q), (g_k, p) = (g^{\\prime }, p) $ induce a shortest $(g,p),(g^{\\prime },p)$ -path $Q$ in $G\\boxtimes P_2$ .",
"Clearly, no internal vertex of $Q$ is in $S_1$ , hence we conclude that $S_1$ is a mutual-visibility set of $G\\boxtimes P_2$ .",
"Concerning $S_2$ , let $(g,r)$ and $(g^{\\prime },r)$ , with $g\\ne g^{\\prime }$ , be two distinct vertices from $S_2$ .",
"We may without loss of generality assume that $r = p$ .",
"Since $g,g^{\\prime }\\in X$ , then there exists a shortest $(g,g^{\\prime })$ -path $P$ in $G$ such that the internal vertices of $P$ are not in $X$ .",
"Hence, $P\\times P_2$ is a shortest $(g,p),(g^{\\prime },p)$ -path in $G\\boxtimes P_2$ whose internal vertices are not in $S_2$ .",
"Let $(g,p)$ and $(g^{\\prime },q)$ be two vertices from $S$ located on different layers of $G\\boxtimes P_2$ .",
"If $g=g^{\\prime }$ these two vertices are adjacent and noting must be proved.",
"When $g\\ne g^{\\prime }$ , consider again the shortest $(g,p),(g^{\\prime },p)$ -path $P$ as before.",
"Now, from this path remove the last vertex and replace it with $(g^{\\prime },q)$ .",
"The resulting path $P^{\\prime }$ is a shortest $(g,p),(g^{\\prime },q)$ -path in $G\\boxtimes P_2$ whose internal vertices are not in $S_2$ .",
"Hence, $S_2$ is a mutual-visibility set of $G\\boxtimes P_2$ .",
"$\\Box $ Theorem REF can be improved for $(\\mu , \\mu _{\\rm t})$ -graphs as we next show.",
"Theorem 5.2 If $G$ is a $(\\mu , \\mu _{\\rm t})$ -graph, then $\\mu (G\\boxtimes P_2) \\ge \\mu (G) + n(G)$ .",
"Proof.",
"If $G$ is a complete graph, $G\\boxtimes P_2$ is also complete, then $\\mu (G\\boxtimes P_2) = 2 n(G) = \\mu (G) + n(G)$ .",
"Otherwise, consider a $\\mu _{\\rm t}$ -set (which is also a $\\mu $ -set) of $G$ and a total mutual-visibility set $X$ for $P_2$ composed by only one vertex.",
"Then we can apply the first inequality of Theorem REF and (REF ) as follows: $\\begin{array}{rcl}\\mu (G\\boxtimes P_2) & \\ge & \\mu (G)n(P_2) + |X|n(G) - \\mu (G)|X| \\\\& = & \\mu (G)\\cdot 2 + n(G) - \\mu (G) \\\\& = & \\mu (G) + n(G).\\end{array}$ $\\Box $ Next we show that the lower bound of Theorem REF is attained by block graphs.",
"To do so, we need the following lemma.",
"Lemma 5.3 Let $x$ be a cut vertex of a graph $G$ .",
"Then there exists a $\\mu $ -set of $G\\boxtimes K_2$ which contains at most one copy of $x$ in the two $G$ -layers.",
"Proof.",
"Let $S$ be a $\\mu $ -set of $G\\boxtimes K_2$ and suppose that $x^{\\prime }, x^{\\prime \\prime }\\in S$ , where $x^{\\prime }$ and $x^{\\prime \\prime }$ are the copies of $x$ in the $G$ -layers.",
"Let $H$ and $H^{\\prime }$ be two components of $(G\\boxtimes K_2) \\setminus \\lbrace x^{\\prime },x^{\\prime \\prime }\\rbrace $ .",
"Then $S\\cap V(H) = \\emptyset $ or $S\\cap V(H^{\\prime }) = \\emptyset $ , say $S\\cap V(H) = \\emptyset $ , for otherwise $S$ is not even a general position set.",
"Now the set $S^{\\prime } = (S \\cup \\lbrace z\\rbrace )\\setminus \\lbrace x^{\\prime }\\rbrace $ , where $z$ is an arbitrary vertex of $H$ , is also a gp-set of $G\\boxtimes K_2$ .",
"$\\Box $ Theorem 5.4 If $G$ is a block graph, then $\\mu (G\\boxtimes P_2) = n(G) + \\mu (G)$ .",
"Proof.",
"Let $X$ be the set of the cut vertices of $G$ .",
"By Lemma REF there exists a $\\mu $ -set $S$ with at most one copy of each vertex in $X$ .",
"We show that $S$ includes one copy of any vertex $v$ of $G$ if $v$ is a cut vertex and two copies of $v$ otherwise.",
"This proves the statement.",
"Consider two vertices $u,v$ of the same copy $G^{\\prime }$ of $G$ and the shortest $u,v$ -path in $G$ .",
"If $u$ and $v$ belong to the same block then they are adjacent since a block is a clique by definition.",
"Otherwise, consider the shortest $u,v$ -path in $G^{\\prime }$ : it is unique and passes only through cut vertices of $G^{\\prime }$ .",
"Since for each vertex in $X$ only one copy is in $S$ , there exists a shortest $u,v$ -path in $G\\boxtimes P_2$ without internal vertices in $S$ .",
"Then $u$ and $v$ are $X$ -visible.",
"Assume that $u$ and $v$ do not belong to the same copy of $G$ .",
"If they belong to two copies of the same block, then they are adjacent.",
"Otherwise, as above, there exists a shortest $u,v$ -path in $G\\boxtimes P_2$ passing through copies of cut vertices of $G$ and without internal vertices in $S$ .",
"$\\Box $ We conclude the paper by demonstrating sharpness of the bound of Theorem REF .",
"Theorem 5.5 If $n\\ge 3$ , then $\\mu (C_n\\boxtimes P_2) ={\\left\\lbrace \\begin{array}{ll}6; & n\\le 6,\\\\n; & \\mbox{otherwise}.",
"\\\\\\end{array}\\right.",
"}$ Proof.",
"Recall from [5] that $\\mu (C_n)= 3$ , $n\\ge 3$ .",
"Hence, Theorem REF implies that $\\mu (C_n\\boxtimes P_2) \\ge 6$ when $n\\le 6$ and $\\mu (C_n\\boxtimes P_2) \\ge n$ for $n\\ge 6$ .",
"It can be checked that $\\mu (C_n\\boxtimes P_2) \\le 6$ when $n\\le 6$ .",
"Hence $\\mu (C_n\\boxtimes P_2) = 6$ in these cases.",
"Assume in the rest that $n > 6$ which means that $\\mu (C_n\\boxtimes P_2) \\ge n$ .",
"Let $S$ be a $\\mu $ -set of $C_n\\boxtimes K_2$ .",
"We need to show that $|S| \\le n$ .",
"Let $v_0,v_1,\\ldots , v_{n-1}$ and $v^{\\prime }_0,v^{\\prime }_1,\\ldots , v^{\\prime }_{n-1}$ be the vertices of the two $C_n$ -layers.",
"Then a pair $v_i, v^{\\prime }_i$ is called a separating pair.",
"$S$ cannot contain three separating pairs since $|S| \\ge n\\ge 7$ for otherwise a vertex from $S$ which is not in three fixed separating pairs cannot be in visibility with all the vertices in the separating pairs.",
"Hence $|S| \\le n+2$ .",
"If $|S| \\in \\lbrace n+1, n+2\\rbrace $ , then consider one separating pair $v_j, v^{\\prime }_j$ .",
"Then there exists a vertex in $\\lbrace v_{j-1}, v^{\\prime }_{j-1}\\rbrace \\cap S$ and a vertex in $\\lbrace v_{j+1}, v^{\\prime }_{j+1}\\rbrace \\cap S$ which are not $S$ -visible.",
"We conclude that $|S| \\le n$ .",
"$\\Box $"
],
[
"Concluding remarks and future work",
"This work suggests some further research directions.",
"We have shown that block graphs and certain cographs are all $(\\mu , \\mu _{\\rm t})$ -graphs.",
"Notice that cographs can be generated by using true and false twins, and that block graphs can be generated by using true twins and pendant vertices.",
"A superclass of both cographs and block graphs is that formed by distance-hereditary graphs.",
"In fact, these graphs can be generated by using true twins, false twins, and pendant vertices.",
"It would be interesting to characterize all the distance-hereditary graphs that are $(\\mu , \\mu _{\\rm t})$ -graphs.",
"We left open the general question about characterizing the larger class $\\mathcal {G}$ of graphs formed by $(\\mu , \\mu _{\\rm t})$ -graphs.",
"In addition, another characterization that would be of interest concerns finding all graphs $G$ for which $\\mu _{\\rm t}(G)=0$ .",
"Concerning specific results, in view of Theorem REF (when we consider $k=2$ ), it would be interesting to study $\\mu (T\\boxtimes T^{\\prime })$ for any two trees $T$ and $T^{\\prime }$ .",
"Also, Theorem REF provides the lower bound $\\mu (G\\boxtimes P_2) \\ge \\mu (G) + n(G)$ for each $(\\mu , \\mu _{\\rm t})$ -graph $G$ , whereas Theorem REF states that the equality is attained in the case of block graphs.",
"We wonder if this equality holds for each $(\\mu , \\mu _{\\rm t})$ -graph.",
"Finally, another interesting point is studying other possible variations of the general concept of mutual-visibility sets and their relationships, as well as relationships with the concept of general position sets."
],
[
"Acknowledgments",
"S. Cicerone and G. Di Stefano were partially supported by the European project “Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies” (GEO-SAFE), contract no.",
"H2020-691161.",
"S. Klavžar was partially supported by the Slovenian Research Agency (ARRS) under the grants P1-0297, J1-2452, and N1-0285.",
"I. G. Yero has been partially supported by the Spanish Ministry of Science and Innovation through the grant PID2019-105824GB-I00.",
"Moreover, this investigation was partially developed while I. G. Yero was visiting the University of Ljubljana supported by “Ministerio de Educación, Cultura y Deporte”, Spain, under the “José Castillejo” program for young researchers (reference number: CAS21/00100)."
]
] | 2210.07835 |
[
[
"Performance boost of a collective qutrit refrigerator"
],
[
"Abstract A single qutrit with transitions selectively driven by weakly-coupled reservoirs can implement one of the world's smallest refrigerators.",
"We analyze the performance of $N$ such fridges that are collectively coupled to the reservoirs.",
"We observe a quantum boost, manifest in a quadratic scaling of the steady-state cooling current with $N$.",
"As $N$ grows further, the scaling reduces to linear, since the transitions responsible for the quantum boost become energetically unfavorable.",
"Fine-tuned inter-qutrit interactions may be used to maintain the quantum boost for all $N$ and also for not-perfectly collective scenarios."
],
[
"Model",
"Our generic starting point is a total Hamiltonian of the form $\\hat{H} = \\hat{H}_S + \\sum _\\nu \\hat{S}^\\nu \\otimes \\hat{B}^\\nu + \\sum _\\nu \\hat{H}_B^\\nu $ that is composed of a system part $\\hat{H}_S$ , different reservoirs $\\hat{H}_B^\\nu $ and the corresponding interactions written as products of bath operators $\\hat{B}^\\nu $ that couple to different system operators $\\hat{S}^\\nu $ .",
"In this paper, we will consider three reservoirs $\\nu \\in \\lbrace c,h,w\\rbrace $ that we label cold, hot, and work (hottest) reservoirs, respectively.",
"These reservoirs are modeled by standard harmonic oscillator baths $\\hat{H}_B^\\nu = \\sum _k \\omega _{k\\nu } \\hat{b}_{k\\nu }^\\dagger \\hat{b}_{k\\nu }$ throughout.",
"An introductory example for the system could be a single three-level model (qutrit) with ground state $\\left|0\\right>$ , first excited state $\\left|1\\right>$ , and most excited state $\\left|2\\right>$ , for which we could write $\\hat{H}_S = \\Delta \\left|2\\right>\\left<2\\right| + \\delta \\left|1\\right>\\left<1\\right|$ , where $\\Delta > \\delta > 0$ are the excitation energies and we have gauged the ground state energy to zero.",
"We consider the case where the coupling between system and reservoir drives the individual system transitions exclusively, e.g.",
"for the single qutrit via $\\hat{S}^c = [\\left|0\\right>\\left<1\\right|+\\left|1\\right>\\left<0\\right|]$ , $\\hat{S}^h = [\\left|0\\right>\\left<2\\right|+\\left|2\\right>\\left<0\\right|]$ , and $\\hat{S}^w = [\\left|1\\right>\\left<2\\right|+\\left|2\\right>\\left<1\\right|]$ , compare Fig.",
"REF bottom left panel.",
"In the appropriate regime (reservoirs held in thermal states with inverse temperatures obeying $\\beta _c > \\beta _h > \\beta _w$ and the cooling condition $\\beta _h \\Delta > \\beta _w (\\Delta -\\delta ) + \\beta _c \\delta $ ), the single-qutrit model implements a quantum absorption refrigerator (QAR): A device that implements stochastic cooling by on average absorbing heat from the coldest ($c$ ) and hottest ($w$ ) reservoirs and dumping the waste heat into the intermediate temperature ($h$ ) reservoir.",
"We provide more details on the working principles of this configuration and also for the case where the work and cold reservoirs are exchanged (as in Fig.",
"REF top left panel) in App. .",
"In this paper, we will instead as system consider $N$ identical qutrits $\\hat{H}_S &= \\Delta \\hat{N}_\\Delta + \\delta \\hat{N}_\\delta \\,,$ where $\\hat{N}_\\Delta = \\sum _{i=1}^N \\left(\\left|2\\right>\\left<2\\right|\\right)_i$ and $\\hat{N}_\\delta = \\sum _{i=1}^N \\left(\\left|1\\right>\\left<1\\right|\\right)_i$ count the total number of large and small excitations present in the system, and the introductory example is reproduced for $N=1$ .",
"The reservoirs are assumed to drive individual transitions as before, but the coupling to the $i$ th qutrit may in principle depend on its position.",
"Thus, we assume as system coupling operators the multi-qutrit operators $\\hat{S}^c &= \\sum _i \\left[h_i^c \\left(\\left|1\\right>\\left<0\\right|\\right)_i + {\\rm h.c.}\\right] \\equiv \\hat{S}^c_+ + \\hat{S}^c_-\\,,\\nonumber \\\\\\hat{S}^h &= \\sum _i \\left[h_i^h\\left(\\left|2\\right>\\left<0\\right|\\right)_i + {\\rm h.c.}\\right] \\equiv \\hat{S}^h_+ + \\hat{S}^h_-\\,,\\nonumber \\\\\\hat{S}^w &= \\sum _i \\left[h_i^w \\left(\\left|2\\right>\\left<1\\right|\\right)_i + {\\rm h.c.}\\right] \\equiv \\hat{S}^w_+ + \\hat{S}^w_-\\,,$ where $\\hat{S}^\\nu _- = (\\hat{S}^\\nu _+)^\\dagger $ .",
"The dependence of the coupling on the qutrit is encoded in the coefficients $h_i^\\nu \\in \\mathbb {C}$ , and furtheron we will denote the limit $h_{i\\nu }\\rightarrow 1$ as the collective limit, for which we write $\\hat{J}^\\nu = \\lim \\limits _{h_i^\\nu \\rightarrow 1} \\hat{S}^\\nu $ and $\\hat{J}^\\nu _\\pm = \\lim \\limits _{h_i^\\nu \\rightarrow 1} \\hat{S}^\\nu _\\pm $ .",
"Analogous to permutationally invariant many-qubit systems, where one can generalize single-qubit Pauli matrices – the generators of $su(2)$ – to large spin operators (see e.g.",
"[44] for potential applications), we can also define the collective generalizations of the generators of $su(3)$ .",
"We denote them by lower indices $\\hat{J}_\\alpha = \\frac{1}{2}\\sum _{i=1}^N \\hat{\\lambda }_i^\\alpha \\,,$ where $\\hat{\\lambda }_i^\\alpha $ denotes the Gell-Mann matrix $\\hat{\\lambda }^\\alpha $ (with $1 \\le \\alpha \\le 8$ ) acting on the $i$ th qutrit (with $1\\le i \\le N$ ).",
"In the collective limit, we can express the system coupling operators by the large qutrit operators, specifically we have $\\hat{S}_\\pm ^c \\rightarrow \\hat{J}_{\\pm }^c = \\hat{J}_6\\pm \\mathrm {i}\\hat{J}_7$ , $\\hat{S}_\\pm ^h \\rightarrow \\hat{J}_{\\pm }^h = \\hat{J}_4 \\pm \\mathrm {i}\\hat{J}_5$ , and $\\hat{S}_\\pm ^w \\rightarrow \\hat{J}_{\\pm }^w = \\hat{J}_1\\pm \\mathrm {i}\\hat{J}_2$ , see App. .",
"To represent the problem efficiently, it is advantageous to use common eigenstates of $\\hat{J}_3$ , $\\hat{J}_8$ and the collective Casimir operators of $su(3)$ like $\\hat{C}_2 = \\sum _{\\alpha =1}^8 \\hat{J}_\\alpha ^2$ .",
"We provide some example states in App. .",
"The question we address here is whether one can observe superradiant performance boosts [45] analogous to results for interacting qubits [35], [43] also for this system of non-interacting qutrits."
],
[
"Methods",
"We aim at the perturbative treatment of the system-reservoir interaction (i.e., the $\\hat{B}^\\nu $ operators) and a description of the system by master equations.",
"Depending on the microscopic implementation of (REF ), a non-perturbative treatment may require modifications to the global Hamiltonianian [46].",
"As in (REF ) all qutrits are identical and do not interact, the transformation into the interaction picture is straightforward as shown in Eq.",
"(REF ).",
"This facilitates the derivation of these master equations.",
"First, we consider the Redfield-II master equation [47] (Lamb-shift omitted, see App.",
"REF for details) $\\dot{\\rho }&= -\\mathrm {i}\\left[\\hat{H}_S, \\rho (t)\\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\frac{\\gamma _\\nu (-\\Omega _\\nu )}{2}\\left(\\left[\\hat{S}^\\nu _+ \\rho , \\hat{S}^\\nu \\right]+\\left[\\hat{S}^\\nu , \\rho \\hat{S}^\\nu _-\\right]\\right)\\nonumber \\\\&\\qquad +\\sum _\\nu \\frac{\\gamma _\\nu (+\\Omega _\\nu )}{2}\\left(\\left[\\hat{S}^\\nu , \\rho \\hat{S}^\\nu _+\\right]+\\left[\\hat{S}^\\nu _-\\rho , \\hat{S}^\\nu \\right]\\right)\\,,$ where $\\gamma _\\nu (\\omega )=\\Gamma _\\nu (\\omega )[1+n_\\nu (\\omega )]\\ge 0$ is the product of the reservoir spectral coupling density $\\Gamma _\\nu (\\omega )$ and the Bose distribution $n_\\nu (\\omega )=[e^{\\beta _\\nu \\omega }-1]^{-1}$ , evaluated at the system excitation frequencies $\\Omega _c=\\delta >0$ , $\\Omega _h=\\Delta >0$ , and $\\Omega _w=\\Delta -\\delta >0$ .",
"Since we analytically continue the spectral coupling density as an odd function $\\Gamma _\\nu (-\\omega )=-\\Gamma _\\nu (+\\omega )$ , we can express $\\gamma _\\nu (-\\Omega _\\nu )=\\Gamma _\\nu (\\Omega _\\nu ) n_\\nu (\\Omega _\\nu )\\ge 0$ .",
"The generator (REF ) need in general not preserve the positivity of the density matrix [2] and need not be thermodynamically consistent [48].",
"Nevertheless, it has for selective systems been shown to approach the true quantum dynamics in the appropriate regimes very well [49], [50], such that we use it as benchmark approach here.",
"Second, we consider the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) master equation [51], [52] (Lamb-shift neglected, see App.",
"REF for details) $\\dot{\\rho } &= -\\mathrm {i}\\left[\\hat{H}_S, \\rho (t)\\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\gamma _\\nu (+\\Omega _\\nu )\\left[\\hat{S}^\\nu _- \\rho \\hat{S}^\\nu _+ - \\frac{1}{2}\\left\\lbrace \\hat{S}^\\nu _+ \\hat{S}^\\nu _-, \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\gamma _\\nu (-\\Omega _\\nu )\\left[\\hat{S}^\\nu _+ \\rho \\hat{S}^\\nu _- - \\frac{1}{2}\\left\\lbrace \\hat{S}^\\nu _- \\hat{S}^\\nu _+, \\rho \\right\\rbrace \\right]\\,,$ which unconditionally preserves all density matrix properties and is thermodynamically consistent also for non-equilibrium reservoirs [53].",
"Third, we consider for the collective limit and fully symmetric initial conditions Pauli rate equations of the form (see App.",
"REF for details) $\\dot{P}_{Mm} &= \\sum _{M^{\\prime }m^{\\prime }} R_{Mm,M^{\\prime }m^{\\prime }} P_{M^{\\prime }m^{\\prime }}\\nonumber \\\\&\\qquad - \\sum _{M^{\\prime }m^{\\prime }} R_{M^{\\prime }m^{\\prime },Mm} P_{Mm}\\,.$ The quantities $P_{Mm}\\equiv \\left<M;m\\right| \\rho \\left|M;m\\right>$ are occupation probabilities of particular energy eigenstates $\\left|M;m\\right>$ of the system Hamiltonian (REF ) with $M\\ge 0$ large and $m\\ge 0$ small excitations with $0\\le m+M \\le N$ .",
"Specifically, these are completely symmetric under any permutation of the qutrits (see App.",
"for examples).",
"To obtain this set of permutationally completely symmetric states, we may apply combinations of collective raising operators $\\hat{J}^\\nu _+$ to the vacuum state ${\\left|\\Psi _{\\rm rep}^{\\rm vac}\\right> = \\left|0;0\\right>=\\left|0\\right>\\otimes \\ldots \\otimes \\left|0\\right>}\\,,$ which e.g.",
"generates the state with $N$ small and no large excitations $\\left|0;N\\right>=\\left|1\\ldots 1\\right>$ and the state with $N$ large and no small excitations $\\left|N;0\\right> = \\left|2\\ldots 2\\right>$ , and many others in between, see Fig.",
"REF right panel.",
"Correspondingly, the transition rate from $\\left|M^{\\prime };m^{\\prime }\\right>$ to $\\left|M;m\\right>$ is given by $R_{Mm,M^{\\prime }m^{\\prime }}=\\sum _\\nu \\Big [\\gamma _\\nu (+\\Omega _\\nu ) {\\left| \\left<M;m\\right| \\hat{J}^\\nu _-\\left|M^{\\prime };m^{\\prime }\\right> \\right|}^2+\\gamma _\\nu (-\\Omega _\\nu ) {\\left| \\left<M;m\\right| \\hat{J}^\\nu _+\\left|M^{\\prime };m^{\\prime }\\right> \\right|}^2\\Big ] \\ge 0$ .",
"As this approach is just a special case of LGKS equations obtained using a secular approximation (exactly valid for non-degenerate $\\hat{H}_S$ , here applied to isolated subspaces), it also obeys its favorable properties, manifest e.g.",
"in the fact that the rates for every reservoir respect local detailed balance [54].",
"To obtain the coefficients in Eq.",
"(REF ), we require the action of the collective ladder operators in the symmetric subspace, compare Fig.",
"REF right panel.",
"They can be evaluated by representing the symmetric subspace with two bosonic modes by means of a generalized Holstein-Primakoff transform, see App.",
", which yields $\\hat{J}^h_+ \\left|M;m\\right> &= \\sqrt{(N-M-m)(M+1)} \\left|M+1;m\\right>\\,,\\nonumber \\\\\\hat{J}^c_+ \\left|M;m\\right> &= \\sqrt{(N-M-m)(m+1)} \\left|M;m+1\\right>\\,,\\nonumber \\\\\\hat{J}^w_+ \\left|M;m\\right> &= \\sqrt{(M+1) m} \\left|M+1;m-1\\right>\\,,\\nonumber \\\\\\hat{J}^h_- \\left|M;m\\right> &= \\sqrt{(N-M-m+1)M} \\left|M-1;m\\right>\\,,\\nonumber \\\\\\hat{J}^c_- \\left|M;m\\right> &= \\sqrt{(N-M-m+1)m} \\left|M;m-1\\right>\\,,\\nonumber \\\\\\hat{J}^w_- \\left|M;m\\right> &= \\sqrt{M(m+1)} \\left|M-1;m+1\\right>\\,.$ Keeping $m=0$ and $M=j_z+N/2$ for $\\hat{J}^h_\\pm $ , or keeping $M=0$ and $m=j_z+N/2$ for $\\hat{J}^c_\\pm $ , or keeping $M+m=N$ and $M-m=2 j_z$ for $\\hat{J}^w_\\pm $ , we see that the transitions along the red or blue or green triangle facets in Fig.",
"REF right panel, respectively, precisely reproduce the usual $su(2)$ Clebsch-Gordan coefficients of the Dicke states with $j=N/2$ .",
"Accordingly, our model also includes the Dicke superradiant relaxation of two-level systems [9], [55], [10] if we couple only to one reservoir.",
"These Clebsch-Gordan coefficients are largest when $j_z=0$ (i.e., for one green/red/blue bath, they become maximal in the middle of the respective green/red/blue triangle facet in Fig.",
"REF right panel).",
"Accordingly, triangles where all coefficients are large have $m\\approx M \\approx N/3$ .",
"In Fig.",
"REF right panel, such a central cycle is marked (shaded), and our main findings are based on the properties of these most productive cycles.",
"Finally, we also use a coarse-grained rate equation [56] valid for infinite temperatures of the work reservoir $n_w(\\Delta -\\delta )\\rightarrow \\infty $ (see App.",
"REF for details) $\\dot{Q}_n &= \\sum _{n^{\\prime }} R_{nn^{\\prime }}^{\\rm cg} Q_{n^{\\prime }} - \\sum _{n^{\\prime }} R_{n^{\\prime }n}^{\\rm cg} Q_n\\,,\\nonumber \\\\Q_n &= \\sum _{M,m} P_{Mm} \\delta _{M+m,n}\\,,$ where the mesostate probabilities $Q_n$ are occupations of states with the total number of $n$ (small and/or large) excitations and $\\delta _{M+m,n}$ denotes a Kronecker symbol.",
"This reduction is possible because in this limit, the transitions along the green lines in Fig.",
"REF right panel become dominant, such that all populations connected by green transitions become identical.",
"The non-vanishing coarse-grained transition rates then become $R_{n,n+1}^{\\rm cg}&=[\\Gamma _c(1+n_c)+\\Gamma _h(1+n_h)] \\frac{(n+1)(N-n)}{2}\\,,\\nonumber \\\\R_{n,n-1}^{\\rm cg}&=[\\Gamma _c n_c + \\Gamma _h n_h] \\frac{(n+1)(N+1-n)}{2}$ with $\\Gamma _\\nu \\equiv \\Gamma _\\nu (+\\Omega _\\nu )$ and $n_\\nu \\equiv n_\\nu (+\\Omega _\\nu )$ and $0 \\le n \\le N-1$ in the first and $1 \\le n \\le N$ in the second line, respectively.",
"The coarse-grained rate equation provides a tremendous reduction of complexity by mapping our system for $n_w\\rightarrow \\infty $ to a tri-diagonal rate equation, and we provide analytic solutions for the cooling current in App.",
"REF .",
"We are dominantly aiming at the steady-state solutions to Eqns.",
"(REF ), (REF ), (REF ), and (REF ) in non-equilibrium scenarios.",
"While it is straightforward to evaluate the energy currents $I_{E,S}^\\nu (t)$ entering the system from balances of the system energy $\\frac{d}{dt} \\left< \\hat{H}_S \\right> = \\sum _\\nu I_{E,S}^\\nu (t)$ , we introduce also microscopically derived counting fields in App.",
", from which we demonstrate how to obtain the energy currents leaving the reservoirs $I_E^\\nu = - \\frac{d}{dt} \\left< \\hat{H}_B^\\nu \\right>$ and their fluctuations $S_{I_E^\\nu } = \\frac{d}{dt} \\left[\\left< (\\hat{H}_B^\\nu )^2 \\right>-\\left< \\hat{H}_B^\\nu \\right>^2\\right]$ in App.",
"."
],
[
"Results",
"In the collective and steady-state limits of the fully symmetric case (where the stationary solutions of LGKS (REF ) and Pauli (REF ) equations are identical), we can establish (see App.",
"), that the stationary currents are tightly coupled $\\bar{I}_E^w = \\frac{\\Delta -\\delta }{\\delta } \\bar{I}_E^c\\,,\\qquad \\bar{I}_E^h = -\\frac{\\Delta }{\\delta } \\bar{I}_E^c\\,,$ and that the $N$ -qutrit QAR is subject to the same cooling condition as a single qutrit QAR (see App.",
"REF ) $\\beta _h \\Delta &> \\beta _w (\\Delta -\\delta ) + \\beta _c \\delta \\,.$ Together with the inherent assumption $\\beta _c > \\beta _h > \\beta _w$ , this defines an operational cooling regime.",
"Due to the tight-coupling, we only quantify the cooling current below.",
"Our main result however is that, within appropriate regimes, the collective features of our model can support a quadratic scaling of the stationary cooling current with the number of qutrits $N$ , see Fig.",
"REF .",
"There, we approach the problem with methods of different complexity like the full Redfield treatment, the LGKS treatment, the rate equation treatment, and the coarse-grained rate equation treatment.",
"Figure: Double-logarithmic plot of the dimensionless stationary energy current entering the system from the cold reservoir (cooling current) versus the number of qutrits NN using different approaches and initial states (symbols, not all orange circles and blue triangles are shown).",
"The dashed curves represent the analytical expressions for n w →∞n_w\\rightarrow \\infty from Eq.",
"() in the limiting cases N≪n ¯N\\ll \\bar{n} and N≫n ¯N\\gg \\bar{n}.",
"Parameters: Γ c =Γ h =Γ w =0.1δ,Δ=10δ\\Gamma _c=\\Gamma _h=\\Gamma _w=0.1\\delta ,\\Delta =10\\delta , n c =10n_c=10, n h =1n_h=1 and n w =100n_w=100.",
"Red pentagons correspond to averages over 100 realizations with random-phase couplings like Eq.",
"() with h i ν =e iϕ i ν h_i^\\nu = e^{\\mathrm {i}\\varphi _i^\\nu } and randomly distributed ϕ i ν ∈[-0.1,+0.1]\\varphi _i^\\nu \\in [-0.1,+0.1].We see that all methods coincide for the case $N=1$ (bottom left), where a simple rate equation treatment is sufficient (see App. ).",
"For larger but still small $N$ , all methods applicable for a fully collective coupling predict a faster-than-linear growth of the current with the number of qutrits $N$ , manifesting the quantum boost of the working fluid.",
"This is seen both in LGKS (orange and green filled symbols) and Redfield (orange and green hollow symbols) approaches.",
"The perfect agreement of LGKS and Redfield approaches (solid versus hollow symbols) demonstrates that in the considered weak-coupling regime, the effect is not just a consequence of the secular approximation.",
"Furthermore, for collective couplings we also see that the cooling current in the fully symmetric subspace (orange symbols, generated with ladder operators from state (REF )) is larger than than that originating from a subspace with second largest Casimir operator eigenvalue (green symbols), which we constructed by acting with collective ladder operators $\\hat{J}^\\nu _\\pm $ on the representative state $\\left|\\Psi _{\\rm rep}\\right> &=\\frac{1}{\\sqrt{N}} \\Big [e^{\\mathrm {i}2\\pi 0/N} \\left|0 \\ldots 01\\right> + e^{\\mathrm {i}2\\pi 1/N} \\left|0 \\ldots 010\\right>\\nonumber \\\\&\\qquad + \\ldots + e^{\\mathrm {i}2\\pi (N-1)/N} \\left|10 \\ldots 0\\right>\\Big ]\\,,$ until no new orthogonal vectors were found (analogous to the discussion in App. ).",
"As one may expect for quantum features, the quadratic boost is fragile in some aspects: First, we observe for large $N$ a crossover to linear scaling (filled orange circles and blue triangles), which we can link to the fact that for fixed temperatures and increasing $N$ , the most productive cycles with $m\\approx M\\approx N/3$ are no longer populated significantly (insets display stationary populations for $N=8$ and $N=80$ ).",
"For the limit of an infinitely hot work reservoir (blue triangles), this can even be understood analytically (dashed magenta curves, see App.",
"REF ).",
"Second, a more severe restriction appears when we relax the assumption of collective couplings by allowing for random phases in the coupling operators (red symbols).",
"Then, the Liouvillian no longer decouples the subspaces of different Casimir operator eigenvalues, and already for weak deviations from the collective limit, the steady-state current no longer scales quadratically.",
"This is the case as major parts of the steady state populate less productive subspaces.",
"When the deviations from the collective limit are small, at least two time scales will emerge: A fast one describes the evolution within the subspaces of constant Casimir operator eigenvalue, and the slow one(s) will describe the leakage between the subspaces of different Casimir operator eigenvalues.",
"Thus, initializing the working fluid in a permutationally symmetric state like $\\left|0;0\\right>$ and operating the device only for a finite time, between these two time scales, may sufficiently populate states near $\\left|N/3;N/3\\right>$ and then still yield a quantum boost.",
"However, to stabilize the quantum enhancement at steady state and to resist small perturbations, fine-tuned interactions may be required, as exemplified at the end of this section.",
"To analyze the device from the thermodynamic perspective, we note that LGKS and the derived rate equations are thermodynamically fully consistent.",
"At steady state, the first law just implies that the stationary energy currents add to zero $\\sum _\\nu \\bar{I}_E^\\nu =0$ – see Eq.",
"(REF ), and the second law implies that the stationary irreversible entropy production rate is non-negative $\\bar{\\sigma }_{\\mathrm {i}} = -\\sum _\\nu \\beta _\\nu \\bar{I}_E^\\nu \\ge 0$ [53].",
"As specifically in the symmetric subspace the currents are tightly coupled (REF ), the coefficient of performance of the device is given by $\\kappa \\equiv \\frac{\\bar{I}_E^c}{\\bar{I}_E^w} \\Theta (\\bar{I}_E^c)= \\frac{\\delta }{\\Delta -\\delta } \\Theta (\\bar{I}_E^c)$ .",
"One can check that in the regions of cooling functionality ($\\bar{I}_E^c >0$ ), the coefficient of performance is bound by its Carnot value via the condition that $\\bar{\\sigma }_\\mathrm {i}\\ge 0$ .",
"Tighter bounds can be obtained by considering the thermodynamic uncertainty relation [57], [58], [59] $\\bar{\\sigma }_{\\mathrm {i}} \\frac{\\bar{S}_{I_E^\\nu }}{(\\bar{I}_E^\\nu )^2} \\ge 2\\,.$ Using that all currents are tightly coupled, we use it to lower-bound the fluctuations for e.g.",
"the cooling current as $\\bar{S}_{I_E^c} \\ge \\frac{\\delta \\bar{I}_E^c}{\\beta _h \\Delta - \\beta _c \\delta -\\beta _w (\\Delta -\\delta )}\\,,$ which proves that the fluctuations inherit the super-extensive scaling of the current.",
"As the eigenvalues of the Pauli rate equation cannot scale faster than $N^2$ , we also find that the fluctuations cannot grow faster than $N^2$ .",
"Therefore, the relative fluctuations [60] $\\sqrt{\\bar{S}_{I_E^c}}/\\bar{I}_E^c$ must vanish for large $N$ .",
"Due to the tight-coupling relations (REF ), we found the relative fluctuations to be alike for all reservoirs [61].",
"Thus, if in practice one would like a device with a large cooling power, a look at Fig.",
"REF suggests that our device should be operated at $\\beta _c \\gtrapprox \\beta _h$ .",
"Figure: Contour plot of the symmetric subspace cooling current for N=6N=6 vs. inverse dimensionless temperatures of hot δβ h \\delta \\beta _h and cold δβ c \\delta \\beta _c reservoirs.",
"Cooling is achieved in the region bounded by the outer solid lines which represent condition ().",
"Parameters: Γ c =Γ h =Γ w \\Gamma _c=\\Gamma _h=\\Gamma _w, Δ=10δ\\Delta =10\\delta , solid contours: n w =100n_w=100, dashed contours: n w →∞n_w\\rightarrow \\infty .Likewise, one would would like to have a device with small fluctuations.",
"By analyzing the uncertainty relation we find that the fluctuations are smallest at the other boundary of the cooling region, where $\\beta _c \\lessapprox \\beta _h \\frac{\\Delta }{\\delta }-\\beta _w \\frac{\\Delta -\\delta }{\\delta }$ , see Fig.",
"REF .",
"Figure: Contour plot analogous to Fig.",
"but showing the thermodynamic uncertainty relation () in the cooling region.",
"Parameters as in Fig.",
".Thus, our model recovers the usual trade-off between accuracy and performance.",
"By adding specifically taylored interactions to the system, we can energetically favor the maximally symmetric subspace and also the cycle with the maximum current.",
"One example for such an interaction could be $\\Delta \\hat{H}_S &= \\alpha _C \\left[\\frac{N(N+3)}{3}-\\hat{C}_2\\right]+ \\alpha _P \\Bigg [\\left(\\frac{N}{3}-\\hat{N}_\\Delta \\right)^2\\nonumber \\\\&\\quad +\\left(\\frac{N}{3}-\\hat{N}_\\delta \\right)^2+\\left(\\frac{N}{3}-\\hat{N}_\\Delta \\right)\\left(\\frac{N}{3}-\\hat{N}_\\delta \\right)\\Bigg ]$ with coefficients $\\alpha _C>0$ and $\\alpha _P>0$ penalizing the deviation from the maximum Casimir sector and the central triangle, respectively (all operators in square brackets are positive semidefinite).",
"For the case where $N=3k+1$ with integer $k$ (for other configurations one may adapt the penalty Hamiltonian accordingly), the three states of the maximum symmetry sector $\\left|k;k\\right>$ , $\\left|k;k+1\\right>$ , and $\\left|k+1;k\\right>$ have the same minimal energy penalty, see the blue triangle in Fig.",
"REF .",
"Figure: Visualization of H ^ S +ΔH ^ S \\hat{H}_S + \\Delta \\hat{H}_S for N=10N=10 and within the maximally symmetric subspace.",
"The bottom plane depicts the energy landscape in absence of a penalty Hamiltonian, and the energetically most favorable triangle (red) is at lowest excitation numbers.",
"The curved surface demonstrates the minimal penalization of the most productive triangle (blue), the plane defined by it is parallel to the plane without interactions.",
"Parameters: Δ=10δ\\Delta =10\\delta , α P =20δ\\alpha _P=20 \\delta (the curved surface has been shifted upwards for clarity).Therefore, it cancels out in the difference of energies, and the corresponding transition rates are the same as without any interaction.",
"Provided that the energy penalty is strong enough $\\beta _\\nu \\alpha _{C/P} \\gg 1$ , the other states with larger energy penalties can be omitted from the considerations, and we obtain the current for a single QAR as discussed in Eq.",
"(REF ) but with boosted rates $\\Gamma _\\nu \\rightarrow \\Gamma _\\nu \\left(\\frac{(N+2)}{3}\\right)^2$ , which results from the enhanced Clebsch-Gordan coefficients (REF ) in the central cycle.",
"Accordingly, for $N$ qutrits the cooling current will be boosted $\\bar{I}_{E}^{c,N} \\approx \\left(\\frac{(N+2)}{3}\\right)^2 \\bar{I}_E^{c,1}$ compared to the current from a single QAR $\\bar{I}_E^{c,1}$ .",
"As we scale all coupling constants, the fluctuations (and all higher cumulants) would increase likewise, such that both $\\bar{I}_E^c \\propto N^2$ and $\\bar{S}_{I_E^c} \\propto N^2$ ."
],
[
"Conclusions",
"We analyzed an ensemble of $N$ identical qutrits for its suitability as a QAR using different approaches.",
"The most striking effect is a quadratic scaling of the output (cooling current) with the number of qutrits for a permutationally symmetric configuration and collective system-reservoir couplings.",
"This collective phenomenon can be understood analogously to Dicke superradiant decay of two level systems – here put to use in an engine by collective couplings.",
"In contrast to quantum computers subjected to decoherence, the performance of the QAR is only reduced to linear (classical) scaling for too large $N$ (changing other parameters may increase the critical size) or for a not-perfectly-collective coupling.",
"For non-identical qutrits we would expect a similar breakdown [62].",
"Compared to a qubit implementation [43], we want to summarize some advantages of collective qutrits: First, the initialization in the proper subspace requires just a permutationally symmetric state (no entangled states like in [43]) as e.g.",
"$\\left|0\\ldots 0\\right>$ .",
"Second, to see the speedup, we do not require fine-tuned inter-qutrit interactions (although this can be used to stabilize the setup against perturbations).",
"Third, in contrast to the interacting qubit system, the (non-interacting) multi-qutrit system only provides three transition frequencies, such that a selective driving appears easier to implement.",
"On the technical side, we have confirmed the effect with a variety of methods.",
"Using a bosonization technique is helpful to compute the transition rates analytically.",
"While the behaviour of the collective limit can be well understood with analytic approximations and allowed us to treat fairly large systems, treating the non-collective limit requires significant resources.",
"We have benchmarked our results with considering also the non-LGKS Redfield master equation.",
"We found that LGKS and Redfield results did not differ much in the considered weak-coupling regime.",
"Beyond stronger system-reservoir coupling strengths, one may also expect larger deviations outside the tight-coupling regime, where the reservoirs no longer drive the individual transitions.",
"As an outlook, we think it could be interesting to consider laser-driven QARs [63], [64] or investigate the fluctuations of energy currents beyond the tight-coupling regime [65] or for detuned levels [66] in greater detail.",
"One may also be tempted to address the strong-coupling regime beyond phenomenologic models [67], e.g.",
"using reaction coordinates [68].",
"This however should be performed while maintaining a lower spectral bound on the global Hamiltonian for all coupling strengths and $N$ , in this case we expect modifiations to Eq.",
"(REF ) [46].",
"D. K. acknowledges support by the HZDR summer student program.",
"G. S. has been supported by the Helmholtz high potential program and the DFG (CRC 1242, project-ID 278162697)."
],
[
"A single three-level QAR",
"For the case of a single qutrit $N=1$ one may derive a rate equation $\\dot{\\vec{P}} = R \\vec{P}$ for the occupation probabilities $\\vec{P}=(P_0, P_1, P_2)^{\\rm T}$ of these states (Redfield, LGKS, and Pauli rate equations then yield the same dynamics for these probabilities).",
"For the normal configuration shown in Fig.",
"REF bottom left panel, where the cold reservoir drives transitions between the lowest two states $\\left|0\\right>\\leftrightarrow \\left|1\\right>$ , the hot between lowest and highest $\\left|0\\right>\\leftrightarrow \\left|2\\right>$ , and the work reservoir transitions between the two excited states $\\left|1\\right>\\leftrightarrow \\left|2\\right>$ , the rate matrix becomes $R^+ =\\left(\\begin{array}{ccc}R_{00}^+ & \\Gamma _c(1+n_c) & \\Gamma _h (1+n_h)\\\\\\Gamma _c n_c & R_{11}^+ & \\Gamma _w(1+n_w)\\\\\\Gamma _h n_h & \\Gamma _w n_w & R_{22}^+\\end{array}\\right)\\,,$ where $n_\\nu \\equiv n_\\nu (\\Omega _\\nu )$ and the negative diagonal elements are fixed by demanding vanishing of the column sums (this implements overall probability conservation).",
"In the rate matrix, we can clearly distinguish the contributions of the individual reservoirs.",
"From this, the energy current entering from, for example, the cold reservoir can be obtained by standard techniques: One may compute the current via $I_E^c(t) = \\sum _{ij} (E_i-E_j) R_{ij,c}^+ P_j(t)$ , where $R_c^+=R^+|_{\\Gamma _h=\\Gamma _w=0}$ is the cold reservoir rate matrix.",
"Alternatively, one may introduce energy counting fields $R^+\\rightarrow R^+(\\chi )$ by replacing $\\Gamma _c(1+n_c)\\rightarrow \\Gamma _c(1+n_c) e^{-\\mathrm {i}\\delta \\chi }$ and $\\Gamma _c n_c\\rightarrow \\Gamma _c n_c e^{+\\mathrm {i}\\delta \\chi }$ in the off-diagonal elements of the rate matrix and compute the current via $I_E^c(t) = -\\mathrm {i}(1,1,1)(\\partial _\\chi R^+(\\chi ))|_{\\chi =0} \\vec{P}(t)$ , see also App. .",
"With this formalism, it is possible to compute also higher cumulants of the distribution of energy transfers [69].",
"In the limit of $n_w\\rightarrow \\infty $ , the long-term (stationary) current reduces to $\\bar{I}_{E+}^{c,\\infty } = \\frac{\\Gamma _c \\Gamma _h \\delta (n_c-n_h)}{\\Gamma _c (1+3 n_c) + \\Gamma _h(1+3 n_h)}\\,.$ For the case where cold and work reservoir are exchanged $E_0=0$ , $E_1=\\Delta -\\delta $ , $E_2=\\Delta $ depicted in Fig.",
"REF top left panel, the rate matrix is given by $R^-=\\left(\\begin{array}{ccc}R_{00}^- & \\Gamma _w(1+n_w) & \\Gamma _h (1+n_h)\\\\\\Gamma _w n_w & R_{11}^- & \\Gamma _c(1+n_c)\\\\\\Gamma _h n_h & \\Gamma _c n_c & R_{22}^-\\end{array}\\right)\\,,$ and an analogous calculation leads in the case of $n_w\\rightarrow \\infty $ to a slightly different cooling current $\\bar{I}_{E-}^{c,\\infty } = \\frac{\\Gamma _c \\Gamma _h \\delta (n_c-n_h)}{\\Gamma _c (2+3 n_c) + \\Gamma _h(2+3 n_h)}\\,.$ Both currents have the same cooling condition $n_c > n_h \\qquad \\Leftrightarrow \\qquad \\beta _c < \\beta _h \\Delta /\\delta \\,,$ which – together with $\\beta _c>\\beta _h$ – defines an operational window for cooling the coldest reservoir.",
"This is a sufficient and necessary condition for $n_w \\rightarrow \\infty $ and $N=1$ .",
"For finite $n_w$ , the calculations are a bit lengthier but fully analogous.",
"One obtains for both configurations the sufficient and necessary cooling condition $\\beta _w (\\Delta -\\delta ) + \\beta _c \\delta < \\beta _h \\Delta \\,,$ which reduces to the previous condition as $\\beta _w\\rightarrow 0$ .",
"Together with the underlying assumption $\\beta _c > \\beta _h$ , this defines the bounds on the window of cooling also for larger $N$ (compare Fig.",
"REF in the main text)."
],
[
"Collective spin representation",
"For a single qutrit, we can express the system Hamiltonian as $\\hat{H}_S^1 &= (\\Delta -\\delta ) \\frac{\\hat{\\lambda }^3}{2} + (\\Delta +\\delta ) \\frac{\\hat{\\lambda }^8}{2\\sqrt{3}} + (\\Delta +\\delta ) \\frac{}{3}\\,,$ where the $\\hat{\\lambda }^{3/8}$ are the two diagonal Gell-Mann matrices.",
"With using the collective qutrit operators (REF ), we can write the system Hamiltonian as $\\hat{H}_S = (\\Delta -\\delta ) \\hat{J}_3 + \\frac{1}{\\sqrt{3}} (\\Delta +\\delta ) \\hat{J}_8 + \\frac{N}{3} (\\Delta +\\delta ) \\,.$ Furthermore, we define collective raising and lowering operators $\\hat{J}^h_\\pm &= \\sum _{i=1}^N \\hat{\\lambda }^h_{\\pm ,i} = \\hat{J}_4 \\pm \\mathrm {i}\\hat{J}_5\\,,\\nonumber \\\\\\hat{J}^c_\\pm &= \\sum _{i=1}^N \\hat{\\lambda }^c_{\\pm ,i} = \\hat{J}_6 \\pm \\mathrm {i}\\hat{J}_7\\,,\\nonumber \\\\\\hat{J}^w_\\pm &= \\sum _{i=1}^N \\hat{\\lambda }^w_{\\pm ,i} = \\hat{J}_1 \\pm \\mathrm {i}\\hat{J}_2\\,,$ for which we find an interaction picture dynamics analogous to Eq.",
"(REF ).",
"In the collective limit (where $\\hat{S}^\\nu \\rightarrow \\hat{J}^\\nu = \\hat{J}^\\nu _+ + \\hat{J}^ \\nu _-$ ), it follows that the quadratic and cubic Casimir operators of the $su(3)$ $\\hat{C}_2 &= \\sum _{\\alpha =1}^8 \\hat{J}_\\alpha ^2\\,,\\nonumber \\\\\\hat{C}_3 &=\\sum _{\\alpha \\beta \\gamma } {\\rm Tr}\\left\\lbrace \\left\\lbrace \\hat{J}_\\alpha , \\hat{J}_\\beta \\right\\rbrace \\hat{J}_\\gamma \\right\\rbrace \\hat{J}_\\alpha \\hat{J}_\\beta \\hat{J}_\\gamma $ will be automatically conserved, to all orders in the system-reservoir interaction Hamiltonian.",
"Depending on the initial condition, this strongly reduces the Hilbert space dimension that needs to be treated explicitly.",
"For example, assuming that our system is prepared e.g.",
"in the collective ground state (REF ), we can constrain ourselves to the subspace of $(N+1)(N+2)/2$ permutationally symmetric states, which is significantly less demanding than treating $3^N$ basis states [70], [71].",
"We label these states analogously to the maximum angular momentum Dicke states (known from collective two-level systems) by $\\left|M;m\\right>$ with $0 \\le M$ large and $0 \\le m$ small excitations such that $0 \\le M+m \\le N$ , specific examples are provided in App. .",
"These states are eigenstates of $\\hat{N}_\\Delta $ and $\\hat{N}_\\delta $ (or alternatively $\\hat{J}_3$ and $\\hat{J}_8$ ) and the Casimir operators, in particular we have $\\hat{N}_\\Delta \\left|M;m\\right> &= M \\left|M;m\\right>\\,,\\qquad \\hat{N}_\\delta \\left|M;m\\right> = m \\left|M;m\\right>\\,,\\nonumber \\\\\\hat{J}_3 \\left|M;m\\right> &= \\left[\\frac{M-m}{2}\\right] \\left|M;m\\right>\\,,\\nonumber \\\\\\hat{J}_8 \\left|M;m\\right> &= \\left[\\frac{\\sqrt{3}}{2} (M+m)-\\frac{N}{\\sqrt{3}}\\right] \\left|M;m\\right>\\,,\\nonumber \\\\\\hat{C}_2 \\left|M;m\\right> &= \\frac{N(N+3)}{3} \\left|M;m\\right>\\,,\\nonumber \\\\\\hat{C}_3 \\left|M;m\\right> &= \\frac{N(N+3)(2N+3)}{18} \\left|M;m\\right>\\,.$ Clearly, they are also eigenstates of the system Hamiltonian $\\hat{H}_S \\left|M;m\\right> = \\left(M \\Delta + m \\delta \\right) \\left|M;m\\right>$ ."
],
[
"Example states for finite $N$",
"In total, the Hilbert space dimension for $N$ qutrits is $D=3^N$ .",
"This can be decomposed by counting the number of states with $M$ large and $m$ small excitations $N_{Nm} = \\frac{N!}{(N-M-m)!",
"M!",
"m!",
"}\\,,$ and indeed we have $3^N = \\sum \\limits _{M=0}^N \\sum \\limits _{m=0}^{N-M} N_{Mm}$ .",
"However, the subspace of completely symmetric states under permutations has only $N_{\\rm symm} = \\frac{(N+1)(N+2)}{2}$ elements, which are closed under the action of $\\hat{J}_{\\nu \\pm }$ .",
"This subspace is characterized by the largest Casimir operator eigenvalues.",
"Therefore, many states belong to subspaces which are not symmetric under permutations, see below for $N=2$ and $N=3$ , where we take the convention of labeling the local eigenstates of the single-qutrit Gell-Mann matrices $\\hat{\\lambda }^3$ and $\\hat{\\lambda }^8$ as $\\left|0\\right>$ , $\\left|1\\right>$ , and $\\left|2\\right>$ ."
],
[
"Example states for $N=2$",
"For $N=2$ , we have 6 out of the $9=3^2$ states in total belonging to the completely symmetric subspace $\\left|0;0\\right> &= \\left|00\\right>\\,,\\qquad \\left|0;2\\right> = \\left|11\\right>\\,,\\qquad \\left|2;0\\right> = \\left|22\\right>\\,,\\nonumber \\\\\\left|0;1\\right> &= \\frac{1}{\\sqrt{2}} \\left[\\left|01\\right>+\\left|10\\right>\\right]\\,,\\qquad \\left|1;0\\right> = \\frac{1}{\\sqrt{2}} \\left[\\left|02\\right>+\\left|20\\right>\\right]\\,,\\nonumber \\\\\\left|1;1\\right> &= \\frac{1}{\\sqrt{2}} \\left[\\left|21\\right>+\\left|12\\right>\\right]\\,,$ and due to the complete permutational symmetry of the $\\hat{J}_{\\nu \\pm }$ the evolution of these six states is closed under collective couplings.",
"These states are eigenstates of $\\hat{C}_2$ and $\\hat{C}_3$ with eigenvalues $10/3$ and $35/9$ , respectively.",
"Trivially, cyclic permutations leave these states invariant.",
"The two-boson representation from App.",
"with $N_a=2$ and $N_b=0$ suffices to represent this subspace with $M$ and $m$ representing the eigenvalues of $\\hat{N}_\\Delta $ and $\\hat{N}_\\delta $ , respectively.",
"One can check the action of the ladder operators (REF ) among them.",
"Additionally, we have the three antisymmetric states $\\left|\\Psi _7\\right> &= \\frac{1}{\\sqrt{2}}\\left[\\left|01\\right>-\\left|10\\right>\\right]\\,,\\nonumber \\\\\\left|\\Psi _8\\right>\\ &= \\frac{1}{\\sqrt{2}} \\left[\\left|21\\right>-\\left|12\\right>\\right] = \\hat{J}^h_+ \\left|\\Psi _7\\right>\\,,\\nonumber \\\\\\left|\\Psi _9\\right> &= \\frac{1}{\\sqrt{2}} \\left[\\left|02\\right>-\\left|20\\right>\\right] = \\hat{J}^w_+ \\left|\\Psi _7\\right>\\,,$ whose evolution is also closed under collective couplings.",
"These states are eigenstates of the Casimir operators $\\hat{C}_2$ and $\\hat{C}_3$ with eigenvalues $4/3$ and $-10/9$ , respectively.",
"Cyclic permutations of the qutrits equip these states with a phase factor of $-1=e^{\\mathrm {i}2\\pi /2}$ .",
"In the four-boson representation $\\left|M,m,Q,q\\right>$ with $N_a=0$ and $N_b=1$ we would identify them with the states $\\left|\\Psi _7\\right>=\\left|0,0,0,0\\right>$ , $\\left|\\Psi _8\\right> = \\left|0,0,1,0\\right>$ , and $\\left|\\Psi _9\\right> = -\\left|0,0,0,1\\right>$ , compare Eq.",
"(REF )."
],
[
"Example states for $N=3$",
"For $N=3$ , we can build the completely symmetric subspace by starting from the representative state $\\left|\\Psi _1\\right> = \\left|0;0\\right>=\\left|000\\right>$ (all atoms in the ground state), and we can generate e.g.",
"$\\left|0;1\\right> &= \\frac{1}{\\sqrt{3}} \\left[\\left|001\\right>+\\left|010\\right>+\\left|100\\right>\\right]\\,,\\nonumber \\\\&= \\frac{1}{\\sqrt{3}} \\hat{J}^c_+ \\left|0;0\\right>\\,,\\nonumber \\\\\\left|1;1\\right> &=\\frac{1}{\\sqrt{6}}\\big [\\left|201\\right>+\\left|012\\right>+\\left|120\\right>\\nonumber \\\\&\\qquad +\\left|021\\right>+\\left|210\\right>+\\left|102\\right>\\big ]\\nonumber \\\\&= \\frac{1}{\\sqrt{2}} \\hat{J}^h_+ \\left|0;1\\right>\\,,\\nonumber \\\\\\left|1;2\\right> &= \\frac{1}{\\sqrt{3}}\\left[\\left|211\\right>+\\left|112\\right>+\\left|121\\right>\\right]\\nonumber \\\\&= \\frac{1}{\\sqrt{2}} \\hat{J}^c_+ \\left|1;1\\right>$ and further states.",
"That way, we obtain $(3+1)(3+2)/2=10$ completely symmetric states, and we can check the coefficients between them given in Eq.",
"(REF ).",
"These are eigenstates of the Casimir operators $\\hat{C}_2$ and $\\hat{C}_3$ with eigenvalues 6 and 9, respectively, and can be represented by two bosonic modes $\\left|M;m\\right>$ with $0 \\le M+m \\le N_a=N=3$ and $N_b=Q=q=0$ .",
"Arbitrary permutations leave these states invariant (in other words, they equip them with a phase of 1).",
"In addition to these, we have states with the same number of excitations as $\\left|0;1\\right>$ but which are orthogonal to each other and also to $\\left|0;1\\right>$ .",
"From these, we can also build further sets by acting with the ladder operators.",
"For example, starting from $\\left|\\Psi _{11}\\right>$ that is orthogonal to $\\left|0;1\\right>$ , we get in total 7 states $\\left|\\Psi _{11}\\right> &= \\frac{1}{\\sqrt{3}}\\left[\\left|001\\right>+e^{\\mathrm {i}2\\pi /3} \\left|010\\right>+e^{\\mathrm {i}4\\pi /3} \\left|100\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^c_+ \\left|\\Psi _{11}\\right> &= \\frac{e^{\\mathrm {i}\\pi }}{\\sqrt{3}}\\left[\\left|110\\right> + e^{\\mathrm {i}\\pi 2/3} \\left|101\\right>+e^{\\mathrm {i}\\pi 4/3} \\left|011\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\left|\\Psi _{11}\\right> &= \\frac{\\sqrt{2}}{\\sqrt{6}} \\Big [\\left|201\\right>+e^{\\mathrm {i}2\\pi /3} \\left|012\\right>+ e^{\\mathrm {i}4\\pi /3} \\left|120\\right>\\nonumber \\\\&\\qquad +\\left|021\\right>+e^{\\mathrm {i}2\\pi /3} \\left|210\\right>+e^{\\mathrm {i}4\\pi /3} \\left|102\\right>\\Big ]\\,,\\nonumber \\\\\\hat{J}^w_+ \\left|\\Psi _{11}\\right> &= \\frac{1}{\\sqrt{3}}\\left[\\left|002\\right>+e^{\\mathrm {i}2\\pi /3} \\left|020\\right>+e^{\\mathrm {i}4\\pi /3} \\left|200\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\hat{J}^c_+ \\left|\\Psi _{11}\\right> &= \\frac{e^{\\mathrm {i}\\pi } }{\\sqrt{3}}\\left[\\left|112\\right> + e^{\\pi \\mathrm {i}2/3} \\left|121\\right> + e^{\\mathrm {i}\\pi 4/3} \\left|211\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\hat{J}^w_+ \\left|\\Psi _{11}\\right> &= \\frac{e^{\\mathrm {i}\\pi } }{\\sqrt{3}}\\left[\\left|220\\right> + e^{\\mathrm {i}\\pi 2/3} \\left|202\\right> + e^{\\mathrm {i}\\pi 4/3} \\left|022\\right> \\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\hat{J}^h_+ \\left|\\Psi _{11}\\right> &= \\frac{2}{\\sqrt{3}}\\left[\\left|221\\right>+e^{\\mathrm {i}2\\pi /3} \\left|212\\right> + e^{\\mathrm {i}4\\pi /3} \\left|122\\right>\\right]\\,.$ Cyclic permutations equip these with a phase of $e^{\\mathrm {i}2\\pi /3}$ .",
"Further states can be generated when starting from $\\left|\\Psi _{18}\\right>$ that is orthogonal to $\\left|\\Psi _{11}\\right>$ and $\\left|0;1\\right>$ $\\left|\\Psi _{18}\\right> &= \\frac{1}{\\sqrt{3}}\\left[\\left|001\\right>+e^{\\mathrm {i}4\\pi /3} \\left|010\\right>+e^{\\mathrm {i}\\pi 2/3} \\left|100\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^c_+ \\left|\\Psi _{18}\\right> &= \\frac{e^{\\mathrm {i}\\pi }}{\\sqrt{3}} \\left(\\left|110\\right> + e^{\\mathrm {i}\\pi 4/3} \\left|101\\right> + e^{+\\mathrm {i}\\pi 2/3}\\left|011\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\left|\\Psi _{18}\\right> &= \\frac{\\sqrt{2}}{\\sqrt{6}} \\Big [\\left|201\\right>+e^{\\mathrm {i}\\pi 4/3} \\left|012\\right>+ e^{\\mathrm {i}\\pi 2/3} \\left|120\\right>\\nonumber \\\\&\\qquad +\\left|021\\right>+e^{\\mathrm {i}\\pi 4/3} \\left|210\\right>+ e^{\\mathrm {i}\\pi 2/3} \\left|102\\right>\\Big ]\\,,\\nonumber \\\\\\hat{J}^w_+ \\left|\\Psi _{18}\\right> &= \\frac{1}{\\sqrt{3}} \\left[\\left|002\\right>+e^{\\mathrm {i}\\pi 4/3} \\left|020\\right> + e^{\\mathrm {i}\\pi 2/3} \\left|200\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\hat{J}^c_+ \\left|\\Psi _{18}\\right> &= \\frac{e^{\\mathrm {i}\\pi }}{\\sqrt{3}} \\left[\\left|112\\right> + e^{\\pi \\mathrm {i}4/3} \\left|121\\right> + e^{+\\pi \\mathrm {i}2/3}\\left|211\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\hat{J}^h_+ \\left|\\Psi _{18}\\right> &= \\frac{2}{\\sqrt{3}} \\left[\\left|221\\right> + e^{\\mathrm {i}\\pi 4/3} \\left|212\\right> + e^{\\mathrm {i}\\pi 2/3} \\left|122\\right>\\right]\\,,\\nonumber \\\\\\hat{J}^h_+ \\hat{J}^w_+ \\left|\\Psi _{18}\\right> &= \\frac{e^{\\mathrm {i}\\pi }}{\\sqrt{3}}\\left[\\left|220\\right> + e^{\\pi \\mathrm {i}4/3} \\left|202\\right> + e^{\\pi \\mathrm {i}2/3} \\left|022\\right>\\right]\\,.$ Under cyclic permutations, these get a phase of $e^{\\mathrm {i}4\\pi /3}$ .",
"We have two additional states with exactly one small and one large excitation that are orthogonal to $\\left|1;1\\right>$ , $\\hat{J}^h_+ \\left|\\Psi _{11}\\right>$ , $\\hat{J}^h_+ \\left|\\Psi _{18}\\right>$ and to each other and having a similar behavior under cyclic permutations.",
"These are $\\left|\\Psi _{25}\\right> &=\\frac{1}{\\sqrt{6}} \\Big [\\left|201\\right>+e^{\\mathrm {i}\\pi 2/3}\\left|012\\right>+e^{\\mathrm {i}\\pi 4/3} \\left|120\\right>\\nonumber \\\\&\\qquad -\\left|021\\right>-e^{\\mathrm {i}\\pi 2/3}\\left|210\\right>-e^{\\mathrm {i}\\pi 4/3}\\left|102\\right>\\Big ]\\,,\\nonumber \\\\\\left|\\Psi _{26}\\right> &= \\frac{1}{\\sqrt{6}} \\Big [\\left|201\\right>+e^{\\mathrm {i}\\pi 4/3} \\left|012\\right>+ e^{\\mathrm {i}\\pi 2/3} \\left|120\\right>\\nonumber \\\\&\\qquad -\\left|021\\right>-e^{\\mathrm {i}\\pi 4/3} \\left|210\\right>- e^{\\mathrm {i}\\pi 2/3} \\left|102\\right>\\Big ]\\,.$ The first one gets a phase $e^{\\mathrm {i}2\\pi /3}$ under cyclic permutations, the second the phase $e^{\\mathrm {i}4\\pi /3}$ .",
"Additionally, one can see that the states $\\hat{J}^h_+\\left|\\Psi _{11}\\right>$ , $\\hat{J}^h_+\\left|\\Psi _{18}\\right>$ are symmetric under the state-conditional permutation $0\\leftrightarrow 2$ , whereas the states $\\left|\\Psi _{25}\\right>$ and $\\left|\\Psi _{26}\\right>$ are antisymmetric.",
"One can get to $\\left|\\Psi _{25}\\right>$ e.g.",
"via $\\hat{J}^w_+ \\hat{J}^c_+ \\left|\\Psi _{11}\\right>$ and subsequent orthonormalization.",
"Analogously, $\\left|\\Psi _{26}\\right>$ can be reached by $\\hat{J}^w_+ \\hat{J}^c_+ \\left|\\Psi _{18}\\right>$ and subsequent orthonormalization.",
"Thus, the $16=2\\times 8$ states $\\left|\\Psi _{11}\\right>, \\ldots ,\\left|\\Psi _{26}\\right>$ are closed under the action of a collective reservoir, they have eigenvalues of Casimir operators $\\hat{C}_2$ and $\\hat{C}_3$ of 3 and 0, respectively.",
"In the bosonic four-mode representation $\\left|M,m,Q,q\\right>$ , the 8 states for this Casimir subspace and given behaviour under cyclic permutations can be constructed from the subspaces with $N_a=1$ and $N_b=1$ .",
"Formally, the bosonic four-mode representation yields via $(M,m),(Q,q)\\in \\lbrace (0,0),(0,1),(1,0)\\rbrace $ in total 9 (6 non-degenerate and 3 degenerate) possible states, but one superposition of the three degenerate states $\\left|0,0,1,0\\right>$ , $\\left|0,1,0,1\\right>$ , and $\\left|1,0,0,0\\right>$ has a different Casimir operator eigenvalue and can be decoupled.",
"Finally, the last state is $\\left|\\Psi _{27}\\right> &=\\frac{1}{\\sqrt{6}}\\Big [\\left|012\\right>+\\left|120\\right>+\\left|201\\right>\\nonumber \\\\&\\qquad -\\left|021\\right>-\\left|210\\right>-\\left|102\\right>\\Big ]\\,.$ It is annihilated by $\\hat{J}^\\nu _{\\pm }$ , has the smallest eigenvalue 0 of both Casimir operators $\\hat{C}_i$ , and is also inert under cyclic permutations.",
"It is fully antisymmetric under the exchange of any two qutrits, and corresponds in the bosonic four-mode representation to the state $\\left|0,0,0,0\\right>$ with $N_a=0$ and $N_b=0$ .",
"We note that in the collective limit, states like $\\left|\\Psi _{27}\\right>$ remain fully inert, they are dark states [72], [73].",
"From the combinatorics of states, we may expect dark states with $m=M=N/3$ for any number $N$ divisible by 3.",
"These dark states will then have minimum, i.e., vanishing, Casimir operator eigenvalue $\\left< \\hat{C}_2 \\right>=0$ .",
"In particular, since already the case $N=6$ hosts more than one dark state, superpositions of these could be used to form e.g.",
"a decoherence-protected logical qubit in presence of all possible collective system-reservoir interactions."
],
[
"States for $N=4$",
"For $N=4$ , we may consider Fig.",
"REF right panel as the largest (top) layer of all states that can be grouped in a pyramid-like structure as shown in Fig.",
"REF .",
"Figure: Map of all 3 4 =813^4=81 collective basis states for N=4N=4 versus small and large numbers of excitations (horizontal directions) and versus Casimir operator C ^ 2 \\hat{C}_2 eigenvalue.The top layer corresponds to the 15 non-degenerate states from the completely permutationally symmetric subspace with largest Casimir operator eigenvalue as shown in Fig.",
"right panel.It evolves independently from the other states for fully collective transitions.The other levels are grouped according to decreasing Casimir operator eigenvalue, with numbers aside symbols indicating the degeneracy."
],
[
"Redfield master equation",
"We detail the derivation of a master equation only for a single reservoir here, as in the weak-coupling regime the dissipators act additively [50].",
"For a single coupling operator per reservoir, i.e., an interaction Hamiltonian of the form $\\hat{H}_I^\\nu = \\hat{S}^\\nu \\otimes \\hat{B}^\\nu $ , the Redfield-II master equation can be written as $\\dot{} &= -\\int _0^\\infty d\\tau \\sum _\\nu C_\\nu (+\\tau ) \\left[(t), (t-\\tau )(t)\\right]\\nonumber \\\\&\\qquad -\\int _0^\\infty d\\tau \\sum _\\nu C_\\nu (-\\tau )\\left[(t)(t-\\tau ), (t)\\right]\\,,$ where we have denoted the interaction picture by bold symbols and introduced the reservoir correlation function $C_\\nu (\\tau )={\\rm Tr_B}\\left\\lbrace e^{+\\mathrm {i}\\hat{H}_B^\\nu \\tau } \\hat{B}^\\nu e^{-\\mathrm {i}\\hat{H}_B^\\nu \\tau } \\hat{B}^\\nu \\bar{\\rho }_B^\\nu \\right\\rbrace $ .",
"In particular for the non-interacting system Hamiltonian (REF ) and coupling operators (REF ), the transformation into the interaction picture can be made very explicit $e^{+\\mathrm {i}\\hat{H}_S t} \\hat{S}^c_\\pm e^{-\\mathrm {i}\\hat{H}_S t} &= e^{\\pm \\mathrm {i}\\delta t} \\hat{S}^c_\\pm \\,,\\nonumber \\\\e^{+\\mathrm {i}\\hat{H}_S t} \\hat{S}^h_\\pm e^{-\\mathrm {i}\\hat{H}_S t} &= e^{\\pm \\mathrm {i}\\Delta t} \\hat{S}^h_\\pm \\,,\\nonumber \\\\e^{+\\mathrm {i}\\hat{H}_S t} \\hat{S}^w_\\pm e^{-\\mathrm {i}\\hat{H}_S t} &= e^{\\pm \\mathrm {i}(\\Delta -\\delta ) t} \\hat{S}^w_\\pm \\,.$ Thus, writing the Redfield equation in the Schrödinger picture, we get (compare e.g.",
"[74]) $\\dot{\\rho } &= -\\mathrm {i}\\left[\\hat{H}_S, \\rho (t)\\right]-\\sum _\\nu \\int _0^\\infty d\\tau C_\\nu (+\\tau ) \\left[\\hat{S}^\\nu , (-\\tau )\\rho (t)\\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty d\\tau C_\\nu (-\\tau )\\left[\\rho (t)(-\\tau ), \\hat{S}^\\nu \\right]\\nonumber \\\\&= -\\mathrm {i}\\left[\\hat{H}_S, \\rho (t)\\right]-\\sum _\\nu \\int _0^\\infty C_\\nu (+\\tau ) e^{-\\mathrm {i}\\Omega _\\nu \\tau } d\\tau \\left[\\hat{S}^\\nu , \\hat{S}^\\nu _+ \\rho \\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty C_\\nu (+\\tau ) e^{+\\mathrm {i}\\Omega _\\nu \\tau } d\\tau \\left[\\hat{S}^\\nu , \\hat{S}^\\nu _-\\rho \\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty C_\\nu (-\\tau ) e^{-\\mathrm {i}\\Omega _\\nu \\tau } d\\tau \\left[\\rho \\hat{S}^\\nu _+, \\hat{S}^\\nu \\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty C_\\nu (-\\tau ) e^{+\\mathrm {i}\\Omega _\\nu \\tau } d\\tau \\left[\\rho \\hat{S}^\\nu _-, \\hat{S}^\\nu \\right]\\,.$ To simplify this expression, one may express the integral prefactors by inserting the Fourier transform of the correlation function $C_\\nu (\\tau ) = \\frac{1}{2\\pi } \\int \\gamma _\\nu (\\omega ) e^{-\\mathrm {i}\\omega \\tau } d\\omega $ and afterwards invoking the Sokhotskij-Plemelj theorem $\\frac{1}{2\\pi } \\int _0^\\infty e^{+\\mathrm {i}\\omega \\tau } d\\tau = \\frac{1}{2}\\delta (\\omega ) + \\frac{\\mathrm {i}}{2\\pi } {\\cal P} \\frac{1}{\\omega }\\,,$ where ${\\cal P}$ denotes the Cauchy principal value.",
"This allows to write the integrals in terms of Hermitian (real) and anti-Hermitian (imaginary) parts, e.g.",
"$\\int _0^\\infty C_\\nu (+\\tau ) e^{+\\mathrm {i}\\Omega _\\nu \\tau } d\\tau &= \\frac{1}{2} \\gamma _\\nu (+\\Omega _\\nu ) + \\frac{1}{2} \\sigma _\\nu (+\\Omega _\\nu )\\,,\\nonumber \\\\\\int _0^\\infty C_\\nu (-\\tau ) e^{+\\mathrm {i}\\Omega _\\nu \\tau } d\\tau &= \\frac{1}{2} \\gamma _\\nu (-\\Omega _\\nu ) - \\frac{1}{2} \\sigma _\\nu (-\\Omega _\\nu )\\,,$ and analogously for the terms with $\\Omega _\\nu \\rightarrow -\\Omega _\\nu $ .",
"In the equations above, the functions on the r.h.s.",
"are then the even and odd Fourier transforms of the reservoir correlation functions $\\gamma _\\nu (\\omega ) &= \\int C_\\nu (\\tau ) e^{+\\mathrm {i}\\omega \\tau } d\\tau \\,,\\nonumber \\\\\\sigma _\\nu (\\omega ) &= \\int C_\\nu (\\tau ) {\\rm sgn}(\\tau ) e^{+\\mathrm {i}\\omega \\tau } d\\tau \\,.$ The Lamb-shift parts $\\sigma _\\nu (\\pm \\Omega _\\nu )$ are negligible in comparison to $\\hat{H}_S$ , such that it is common practice to neglect them.",
"Eventually, this transforms Eq.",
"(REF ) into Eq.",
"(REF ) in the main text.",
"The Redfield equation conserves trace and hermiticity, but not necessarily the positivity of the density matrix.",
"For general operators $\\hat{O}$ one can rewrite it in terms of expectation values $\\frac{d}{dt}\\left< \\hat{O} \\right> &= +\\mathrm {i}\\left< \\left[\\hat{H}_S, \\hat{O}\\right] \\right>\\nonumber \\\\&\\qquad +\\sum _\\nu \\frac{\\gamma _\\nu (-\\Omega _\\nu )}{2}\\Big [\\left< \\left[\\hat{S}^\\nu , \\hat{O}\\right] \\hat{S}_+^\\nu \\right>\\nonumber \\\\&\\qquad \\qquad +\\left< \\hat{S}^\\nu _-\\left[\\hat{O}, \\hat{S}^\\nu \\right] \\right>\\Big ]\\nonumber \\\\&\\qquad +\\sum _\\nu \\frac{\\gamma _\\nu (+\\Omega _\\nu )}{2}\\Big [\\left< \\left[\\hat{S}^\\nu , \\hat{O}\\right] \\hat{S}_-^\\nu \\right>\\nonumber \\\\&\\qquad \\qquad +\\left< \\hat{S}^\\nu _+\\left[\\hat{O}, \\hat{S}^\\nu \\right] \\right>\\Big ]\\,.$ Thus, it is easy to see that the above equation conserves all operators that commute with both $\\hat{H}_S$ and $\\hat{S}^\\nu $ , i.e., if $[\\hat{O}, \\hat{H}_S]=[\\hat{O}, \\hat{S}^\\nu ]=0$ , one also has ${\\rm Tr}\\left\\lbrace \\hat{O} \\dot{\\rho } \\right\\rbrace =0$ .",
"This is the case in the collective limit, where $\\hat{S}^\\nu \\rightarrow \\hat{J}^\\nu $ e.g.",
"for the Casimir operators $\\hat{O}\\rightarrow \\hat{C}_i$ .",
"All these things can be made explicit by employing a special model for the spectral coupling density.",
"For example, with using $\\Gamma _\\nu (\\omega ) &= \\Gamma _\\nu \\frac{\\omega }{\\epsilon _\\nu } \\frac{\\delta _\\nu ^4}{(\\omega ^2-\\epsilon _\\nu ^2)^2+\\delta _\\nu ^4}\\,,$ the correlation function $C_\\nu (\\tau )$ as well as the Lamb-shift contribution $\\sigma _\\nu (\\omega ) = \\frac{\\mathrm {i}}{\\pi } {\\cal P} \\int \\frac{\\gamma _\\nu (\\omega ^{\\prime })}{\\omega -\\omega ^{\\prime }} d\\omega ^{\\prime }$ may be computed analytically (but yielding extremely lengthy expressions).",
"In our calculations in the main text, we used resonant reservoirs with $\\epsilon _c=\\delta $ , $\\epsilon _h=\\Delta $ and $\\epsilon _w = \\Delta -\\delta $ .",
"For the parameters in Fig.",
"REF we found the effects of the Lamb-shift contributions to be negligible.",
"Similar to the rate equation discussion from App.",
", the cold reservoir energy current may now be computed using different options: Taking the view of the system perspective, one may obtain it by computing the system energy balance from (REF ).",
"The energy current entering from the cold reservoir then becomes $I_{E,S}^c &= \\frac{\\Gamma _c n_c}{2} {\\rm Tr}\\left\\lbrace \\hat{H}_S\\left(\\left[\\hat{S}^c_+ \\rho , \\hat{S}^c\\right]+\\left[\\hat{S}^c, \\rho \\hat{S}^c_-\\right]\\right) \\right\\rbrace \\\\&\\quad +\\frac{\\Gamma _c (1+n_c)}{2} {\\rm Tr}\\left\\lbrace \\hat{H}_S\\left(\\left[\\hat{S}^c, \\rho \\hat{S}^c_+\\right]+\\left[\\hat{S}^c_- \\rho , \\hat{S}^c\\right]\\right) \\right\\rbrace \\,,\\nonumber $ and analogously for the other reservoirs.",
"Alternatively, we may microscopically derive energy counting fields [75], [74], [50], which takes the perspective of energy leaving the reservoir and yields a tilted Liouvillian ${\\cal L}\\rightarrow {\\cal L}(\\chi )$ .",
"In Eq.",
"(REF ) in the main text, this would effectively lead to the replacements $\\left(\\left[\\hat{S}^c, \\rho \\hat{S}^c_+\\right]+\\left[\\hat{S}^c_-\\rho ,\\hat{S}^c\\right]\\right)\\rightarrow \\left(\\hat{S}^c \\rho \\hat{S}^c_+ e^{-\\mathrm {i}\\delta \\chi } + \\hat{S}^c_-\\rho \\hat{S}^c e^{-\\mathrm {i}\\delta \\chi } - \\rho \\hat{S}^c_+ \\hat{S}^c - \\hat{S}^c \\hat{S}^c_- \\rho \\right)$ and the analogous one $\\left(\\left[\\hat{S}^c_+ \\rho , \\hat{S}^c\\right] + \\left[\\hat{S}^c, \\rho \\hat{S}^c_-\\right]\\right) \\rightarrow \\left(\\hat{S}^c_+ \\rho \\hat{S}^c e^{+\\mathrm {i}\\delta \\chi } + \\hat{S}^c \\rho \\hat{S}^c_- e^{-\\mathrm {i}\\delta \\chi } - \\hat{S}^c \\hat{S}^c_+ \\rho - \\rho \\hat{S}^c_- \\hat{S}^c\\right)$ .",
"The current (and noise) can then be computed by the methods in App. .",
"This yields for the current leaving the cold reservoir $I_{E}^c &= -\\frac{\\Gamma _c\\delta (1+n_c)}{2} {\\rm Tr}\\left\\lbrace \\hat{S}^c \\rho \\hat{S}^c_+ + \\hat{S}^c_- \\rho \\hat{S}^c \\right\\rbrace \\nonumber \\\\&\\qquad + \\frac{\\Gamma _c \\delta n_c}{2} {\\rm Tr}\\left\\lbrace \\hat{S}^c_+ \\rho \\hat{S}^c + \\hat{S}^c \\rho \\hat{S}^c_- \\right\\rbrace \\,.$ In general, we have $I_{E}^\\nu \\ne I_{E,S}^\\nu $ for the Redfield equation.",
"Moreover, while by construction the stationary currents for the system add to zero at steady state $\\sum _\\nu \\bar{I}_{E,S}^\\nu = 0$ , this is only approximately true in the Redfield approach for the energy currents leaving the reservoir.",
"In the weak-coupling regime, where for our model Redfield and LGKS solutions agree, we observe no violations."
],
[
"Secular master equations",
"Under a secular approximation, Eq.",
"(REF ) simplifies into $\\dot{} &\\approx -\\sum _\\nu \\int _0^\\infty d\\tau C_\\nu (+\\tau ) e^{-\\mathrm {i}\\Omega _\\nu \\tau } \\left[\\hat{S}^\\nu _-, \\hat{S}^\\nu _+(t)\\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty d\\tau C_\\nu (+\\tau ) e^{+\\mathrm {i}\\Omega _\\nu \\tau } \\left[\\hat{S}^\\nu _+, \\hat{S}^\\nu _-(t)\\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty d\\tau C_\\nu (-\\tau ) e^{+\\mathrm {i}\\Omega _\\nu \\tau } \\left[(t)\\hat{S}^\\nu _-, \\hat{S}^\\nu _+\\right]\\nonumber \\\\&\\qquad -\\sum _\\nu \\int _0^\\infty d\\tau C_\\nu (-\\tau ) e^{-\\mathrm {i}\\Omega _\\nu \\tau } \\left[(t)\\hat{S}^\\nu _+, \\hat{S}^\\nu _-\\right]\\nonumber \\\\&= -\\mathrm {i}\\sum _\\nu \\left[\\frac{\\sigma _\\nu (-\\Omega _\\nu )}{2\\mathrm {i}} \\hat{S}^\\nu _- \\hat{S}^\\nu _++ \\frac{\\sigma _\\nu (+\\Omega _\\nu )}{2\\mathrm {i}} \\hat{S}^\\nu _+ \\hat{S}^\\nu _-, \\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\gamma _\\nu (+\\Omega _\\nu )\\left[\\hat{S}^\\nu _- \\hat{S}^\\nu _+ - \\frac{1}{2}\\left\\lbrace \\hat{S}^\\nu _+ \\hat{S}^\\nu _-, \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\gamma _\\nu (-\\Omega _\\nu )\\left[\\hat{S}^\\nu _+ \\hat{S}^\\nu _- - \\frac{1}{2}\\left\\lbrace \\hat{S}^\\nu _- \\hat{S}^\\nu _+, \\right\\rbrace \\right]\\,.$ Back in the Schrödinger picture, we thus have $\\dot{\\rho } &= -\\mathrm {i}\\left[\\hat{H}_S+ \\sum _\\nu \\frac{\\sigma _\\nu (-\\Omega _\\nu )}{2\\mathrm {i}} \\hat{S}^\\nu _- \\hat{S}^\\nu _+ + \\frac{\\sigma _\\nu (+\\Omega _\\nu )}{2\\mathrm {i}} \\hat{S}^\\nu _+ \\hat{S}^\\nu _-, \\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\gamma _\\nu (+\\Omega _\\nu )\\left[\\hat{S}^\\nu _- \\rho \\hat{S}^\\nu _+ - \\frac{1}{2}\\left\\lbrace \\hat{S}^\\nu _+ \\hat{S}^\\nu _-, \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +\\sum _\\nu \\gamma _\\nu (-\\Omega _\\nu )\\left[\\hat{S}^\\nu _+ \\rho \\hat{S}^\\nu _- - \\frac{1}{2}\\left\\lbrace \\hat{S}^\\nu _- \\hat{S}^\\nu _+, \\rho \\right\\rbrace \\right]\\,,$ which upon neglecting the Lamb shift $\\sigma _\\nu (\\pm \\Omega _\\nu )$ reduces to Eq.",
"(REF ) from the main text.",
"This equation is of LGKS form and thermalizes the system when all reservoirs are at the same equilibrium temperature.",
"Furthermore, when evaluating only the populations of $\\rho $ in the system energy eigenbasis for situations where the system Hamiltonian is non-degenerate (at least within a conserved subspace), we obtain the simple Pauli-type rate equation as exemplified in App.",
"REF .",
"In the collective limit, the associated effective non-Hermitian Hamiltonian $H_{\\rm eff} = \\hat{H}_S - \\frac{\\mathrm {i}}{2} \\sum _\\nu \\left[\\Gamma _\\nu (1+n_\\nu ) \\hat{J}^\\nu _+ \\hat{J}^\\nu _- + \\Gamma _\\nu n_\\nu \\hat{J}^\\nu _+ \\hat{J}^\\nu _-\\right]$ is diagonal in the maximum symmetry subspace $H_{\\rm eff} \\left|M;m\\right> = \\lambda _{Mm} \\left|M;m\\right>$ .",
"Its eigenvalues can be calculated analytically via (REF ), and their imaginary part – related to the waiting time distribution [76] between any two jump events [77] – indicates that for fixed thermal reservoirs $n_\\nu $ and coupling strengths $\\Gamma _\\nu $ , the waiting time is minimized for $M\\approx m\\approx N/3$ (maximum engine activity).",
"Specifically, from the imaginary part of $\\lambda _{\\frac{N}{3},\\frac{N}{3}} = \\frac{N}{3}(\\Delta +\\delta ) - \\frac{\\mathrm {i}}{2} \\frac{N}{3}\\left(\\frac{N}{3}+1\\right) \\left[\\sum _\\nu \\Gamma _\\nu (1+2n_\\nu )\\right]$ we can conclude that the lifetime of the states participating in the central cycle scales as $\\tau \\propto N^{-2}$ .",
"The currents can now be computed in analogy to the previous section, i.e., either by the energy flowing into the system from the cold reservoir $I_{E,S}^c &= \\Gamma _c(1+n_c) {\\rm Tr}\\left\\lbrace \\hat{H}_S \\left[\\hat{S}^c_- \\rho \\hat{S}^c_+ - \\frac{1}{2}\\left\\lbrace \\hat{S}^c_+ \\hat{S}^c_-, \\rho \\right\\rbrace \\right] \\right\\rbrace \\nonumber \\\\&\\qquad +\\Gamma _c n_c {\\rm Tr}\\left\\lbrace \\hat{H}_S \\left[\\hat{S}^c_+ \\rho \\hat{S}^c_- - \\frac{1}{2}\\left\\lbrace \\hat{S}^c_- \\hat{S}^c_+, \\rho \\right\\rbrace \\right] \\right\\rbrace \\,,$ or by assessing the energy flow out of the cold reservoir with counting fields $\\hat{S}^c_- \\rho \\hat{S}^c_+\\rightarrow \\hat{S}^c_- \\rho \\hat{S}^c_+ e^{-\\mathrm {i}\\delta \\chi }$ and $\\hat{S}^c_+ \\rho \\hat{S}^c_-\\rightarrow \\hat{S}^c_+ \\rho \\hat{S}^c_- e^{+\\mathrm {i}\\delta \\chi }$ and then using the methods of Sec. .",
"The current leaving the cold reservoir can then be obtained via $I_{E}^c(t) = -\\mathrm {i}{\\rm Tr}\\left\\lbrace {\\cal L}^{\\prime }(0)\\vec{\\rho }(t) \\right\\rbrace $ and becomes $I_{E}^c &= \\Gamma _c \\delta n_c {\\rm Tr}\\left\\lbrace \\hat{S}^c_+ \\rho \\hat{S}^c_- \\right\\rbrace - \\Gamma _c\\delta (1+n_c){\\rm Tr}\\left\\lbrace \\hat{S}^c_- \\rho \\hat{S}^c_+ \\right\\rbrace \\,.$ In contrast to the Redfield case discussed above, we find $I_{E}^\\nu =I_{E,S}^\\nu $ , such that we can unambiguously write the first law of thermodynamics as $\\frac{d}{dt} {\\rm Tr}\\left\\lbrace \\hat{H}_S \\rho \\right\\rbrace = \\sum _\\nu I_E^\\nu (t)\\,.$ With using Spohn's inequality [78], one can also establish the second law of thermodynamics $\\frac{d}{dt} {\\rm Tr}\\left\\lbrace -\\rho \\ln \\rho \\right\\rbrace -\\sum _\\nu \\beta _\\nu I_E^\\nu (t) \\ge 0\\,.$ These bound the coefficient of performance of the QAR by its Carnot value."
],
[
"Pauli rate equation",
"For the fully symmetric case, the derivation of a rate equation becomes particularly simple: We can formally insert a diagonal density matrix in the fully symmetric subspace $\\rho = \\sum _{M,m} P_{Mm} \\left|M;m\\right>\\left<M;m\\right|\\,,$ into (REF ), which with the help of Eq.",
"(REF ) yields $\\dot{P}_{Mm} &= \\gamma _c(-\\delta )(N-M-m+1)m P_{M,m-1}\\nonumber \\\\&\\qquad -\\gamma _c(-\\delta ) (N-M-m)(m+1) P_{M,m}\\nonumber \\\\&\\qquad +\\gamma _c(+\\delta ) (N-M-m)(m+1) P_{M,m+1}\\nonumber \\\\&\\qquad -\\gamma _c(+\\delta ) (N-M-m+1)m P_{M,m}\\nonumber \\\\&\\qquad +\\gamma _h(-\\Delta ) (N-M-m+1)M P_{M-1,m}\\nonumber \\\\&\\qquad -\\gamma _h(-\\Delta ) (N-M-m)(M+1) P_{M,m}\\nonumber \\\\&\\qquad +\\gamma _h(+\\Delta ) (N-M-m)(M+1) P_{M+1,m}\\nonumber \\\\&\\qquad -\\gamma _h(+\\Delta ) (N-M-m+1)M P_{M,m}\\nonumber \\\\&\\qquad +\\gamma _w(-\\Delta +\\delta ) M(m+1) P_{M-1,m+1}\\nonumber \\\\&\\qquad -\\gamma _w(-\\Delta +\\delta ) (M+1)m P_{M,m}\\nonumber \\\\&\\qquad +\\gamma _w(+\\Delta -\\delta ) (M+1)m P_{M+1,m-1}\\nonumber \\\\&\\qquad -\\gamma _w(+\\Delta -\\delta ) M(m+1) P_{M,m}\\,.$ This is precisely the Pauli-type rate equation (REF ) in the main text, where we note that rates outside allowed states do naturally vanish.",
"In particular, we can identify the transitions that increase the system energy as $R_{(M,m+1),(M,m)} &= \\Gamma _c n_c \\cdot (N-M-m)(m+1)\\,,\\nonumber \\\\R_{(M+1,m),(M,m)} &= \\Gamma _h n_h \\cdot (N-M-m)(M+1)\\,,\\nonumber \\\\R_{(M+1,m-1),(M,m)} &= \\Gamma _w n_w \\cdot m (M+1)\\,,$ while transitions that decrease it are always a bit larger due to local detailed balance $R_{(M,m),(M,m+1)} &=\\Gamma _c (1+n_c) \\cdot (N-M-m)(m+1)\\,,\\nonumber \\\\R_{(M,m),(M+1,m)} &= \\Gamma _h (1+n_h) \\cdot (N-M-m)(M+1)\\,,\\nonumber \\\\R_{(M,m),(M+1,m-1)} &= \\Gamma _w (1+n_w) \\cdot m (M+1)\\,.$ Currents can be obtained in analogy to the case $N=1$ discussed in App.",
": Either one computes them via the energy balance of the system which, for a system subject to the rate equation $\\dot{P}_i = \\sum _{ij} \\sum _\\nu \\left[R_{ij}^\\nu P_j - R_{ji}^\\nu P_i\\right]$ with $R_{ij}^\\nu $ denoting the transition rate between system energy eigenstate $j\\rightarrow i$ triggered by reservoir $\\nu $ , generically yields the expression $I_E^\\nu = \\sum _{ij} (E_i-E_j) R_{ij}^\\nu P_j$ .",
"Alternatively, one may also insert counting fields (or use the ones from the LGKS description), which effectively corresponds to the replacement $\\Gamma _c n_c \\rightarrow \\Gamma _c n_c e^{+\\mathrm {i}\\delta \\chi }$ and $\\Gamma _c (1+n_c) \\rightarrow \\Gamma _c (1+n_c) e^{-\\mathrm {i}\\delta \\chi }$ in the off-diagonal matrix elements of the rate matrix and then use the methods of App. .",
"As with the previous section, these two approaches are equivalent and thermodynamically consistent.",
"For multi-qutrit systems the exact computation of the steady state is cumbersome already for the collective limit and a Pauli rate equation (REF ), such that we do not provide analytic results for the currents.",
"However, in App.",
"we perform a cycle decomposition to establish tight-coupling relations between the currents."
],
[
"Coarse-grained rate equation",
"For the case where the work reservoir is infinitely hot $n_w\\rightarrow \\infty $ , the associated green transitions in Fig.",
"REF become dominant, and along them the populations become approximately equal (but remain still conditioned on the total number of excitations).",
"This allows us to coarse-grain [79], [56] the populations by summarizing all states with same total number of excitations $Q_n = \\sum _{M,m} \\delta _{M+m,n} P_{M,m}\\,.$ As the states connected by work reservoir transitions equilibrate much faster in this limit, we have for their conditional probability $\\lim _{n_w\\rightarrow \\infty } \\left.\\frac{P_{M,m}}{Q_n}\\right|_{M+m=n} = \\frac{1}{n+1}\\,.$ Summing over the respective states of Eq.",
"(REF ) allows us to obtain a coarse-grained one-dimensional Markovian rate equation $\\dot{Q}_n &= {\\sum _{M,m}\\delta _{M+m,n}}\\Big [\\gamma _c(-\\delta )(N-M-m+1)m \\frac{P_{M,m-1}}{Q_{n-1}} Q_{n-1}\\nonumber \\\\&\\qquad -\\gamma _c(-\\delta ) (N-M-m)(m+1) \\frac{P_{M,m}}{Q_n} Q_n\\nonumber \\\\&\\qquad +\\gamma _c(+\\delta ) (N-M-m)(m+1) \\frac{P_{M,m+1}}{Q_{n+1}} Q_{n+1}\\nonumber \\\\&\\qquad -\\gamma _c(+\\delta ) (N-M-m+1)m \\frac{P_{M,m}}{Q_n} Q_n\\nonumber \\\\&\\qquad +\\gamma _h(-\\Delta ) (N-M-m+1)M \\frac{P_{M-1,m}}{Q_{n-1}} Q_{n-1}\\nonumber \\\\&\\qquad -\\gamma _h(-\\Delta ) (N-M-m)(M+1) \\frac{P_{M,m}}{Q_n} Q_n\\nonumber \\\\&\\qquad +\\gamma _h(+\\Delta ) (N-M-m)(M+1) \\frac{P_{M+1,m}}{Q_{n+1}} Q_{n+1}\\nonumber \\\\&\\qquad -\\gamma _h(+\\Delta ) (N-M-m+1)M \\frac{P_{M,m}}{Q_n} Q_n\\nonumber \\\\&\\qquad +\\gamma _w(-\\Delta +\\delta ) M(m+1) \\frac{P_{M-1,m+1}}{Q_n} Q_n\\nonumber \\\\&\\qquad -\\gamma _w(-\\Delta +\\delta ) (M+1)m \\frac{P_{M,m}}{Q_n} Q_n\\nonumber \\\\&\\qquad +\\gamma _w(+\\Delta -\\delta ) (M+1)m \\frac{P_{M+1,m-1}}{Q_n} Q_n\\nonumber \\\\&\\qquad -\\gamma _w(+\\Delta -\\delta ) M(m+1) \\frac{P_{M,m}}{Q_n} Q_n\\Big ]\\nonumber \\\\&\\stackrel{n_w\\rightarrow \\infty }{\\approx }R_{n,n+1}^{\\rm cg} Q_{n+1} + R_{n,n-1}^{\\rm cg} Q_{n-1}\\nonumber \\\\&\\qquad - [R_{n-1,n}^{\\rm cg} + R_{n+1,n}^{\\rm cg}] Q_n\\,,$ where $n\\in \\lbrace 0,1,\\ldots ,N\\rbrace $ denotes the total number of excitations and we have inserted the conditional probabilities.",
"In the coarse-grained rates, the mesostate-internal transitions due to the work reservoir cancel out, whereas the other reservoirs determine the transitions between mesostates $R_{n,n+1}^{\\rm cg} &= \\sum _{M,m} \\frac{\\delta _{M+m,n}}{n+2} \\Big [\\Gamma _c (1+n_c) (N-M-m)(m+1)\\nonumber \\\\&\\qquad +\\Gamma _h (1+n_h) (N-M-m)(M+1)\\Big ]\\,,\\nonumber \\\\R_{n,n-1}^{\\rm cg} &= \\sum _{M,m} \\frac{\\delta _{M+m,n}}{n} \\Big [\\Gamma _c n_c (N-M-m+1)m\\nonumber \\\\&\\qquad +\\Gamma _h n_h (N-M-m+1)M\\Big ]\\,,$ which evaluate to Eq.",
"(REF ) in the main text.",
"Note that the case $R_{0,-1}^{\\rm cg}=0$ has to be treated separately as the state with zero excitations does not need to be coarse-grained.",
"As a sanity check, we remark that for $N=1$ , the proper coarse-grained rates (which one gets by applying an analogous procedure to the rate matrices in Sec. )",
"for a single QAR are reproduced.",
"We note that the resulting effective rates for cold and hot reservoir no longer obey local detailed balance, which allows energy to flow out of the cold reservoir.",
"The simple tri-diagonal form of this effective rate equation has the advantage that the stationary state can be computed analytically, since it obeys $\\bar{Q}_n = \\frac{R_{n,n-1}^{\\rm cg}}{R_{n-1,n}^{\\rm cg}} \\frac{R_{n-1,n-2}^{\\rm cg}}{R_{n-2,n-1}^{\\rm cg}} \\cdot \\ldots \\cdot \\frac{R_{1,0}^{\\rm cg}}{R_{0,1}^{\\rm cg}} \\bar{Q}_0\\,,\\qquad \\sum _n \\bar{Q}_n = 1\\,.$ Making the ratios explicit with (REF ), we can determine $\\bar{Q}_0$ from the normalization condition and from that write the stationary cooling current with $\\bar{n} = \\frac{\\Gamma _c n_c + \\Gamma _h n_h}{\\Gamma _c+\\Gamma _h}$ and $\\alpha _{\\bar{n}} = \\frac{\\bar{n}}{\\bar{n} +1}$ and $R_{ij}^c \\equiv R_{ij}^{\\rm cg}|_{\\Gamma _h\\rightarrow 0}$ $\\bar{I}_E^c &= \\delta \\sum _{n=1}^{N} R_{n,n-1}^c \\bar{Q}_{n-1} - \\delta \\sum _{n=0}^{N-1} R_{n,n+1}^c \\bar{Q}_{n+1}\\nonumber \\\\&= \\frac{\\Gamma _c \\delta }{2} \\sum _{n=0}^N \\Big [(n+2) (N-n) n_c\\nonumber \\\\&\\qquad \\qquad - n (N-n+1)(1+n_c)\\Big ] \\bar{Q}_n\\nonumber \\\\&= \\frac{\\Gamma _c \\Gamma _h \\delta }{2(\\Gamma _c+\\Gamma _h)} (n_c-n_h) f_N(\\bar{n})\\,,\\nonumber \\\\f_N(\\bar{n}) &\\equiv \\frac{g_N(\\bar{n})+\\alpha _{\\bar{n}}^{N+1} h_N(\\bar{n})}{\\bar{n} + 1 - \\alpha _{\\bar{n}}^{N+1} \\left[2+\\bar{n} + N\\right]}\\,,\\nonumber \\\\g_N(\\bar{n}) &\\equiv 2(N-3\\bar{n})(\\bar{n} + 1)\\,,\\nonumber \\\\h_N(\\bar{n}) &\\equiv N^2+(5+4\\bar{n})N+6(\\bar{n} + 1)^2\\,.$ In the current, only the term $(n_c-n_h)$ can turn negative, such that we recover the original cooling condition for $n_w\\rightarrow \\infty $ given in (REF ).",
"Apart from that, the scaling factor $f_N(\\bar{n})>0$ can be analyzed for various limits: First of all, we have $f_1(\\bar{n}) = \\frac{2}{1+3 \\bar{n}}$ , which yields the same current as (REF ).",
"Second, for very large $N\\gg \\bar{n}$ , we can drop the terms with powers of $\\alpha _{\\bar{n}} < 1$ , such that $f_N(\\bar{n}) \\rightarrow 2 N$ , and the current eventually will scale just linearly $\\bar{I}_E^c \\approx \\frac{\\Gamma _c \\Gamma _h \\delta }{\\Gamma _c+\\Gamma _h} (n_c-n_h)N$ .",
"Third, for $N \\ll \\bar{n}$ we have $f_N(\\bar{n}) \\rightarrow \\frac{N(N+3)}{6\\bar{n}}$ , and we maintain a quadratic scaling for the current $\\bar{I}_E^c \\approx \\frac{\\Gamma _c \\Gamma _h \\delta (n_c-n_h)}{12(\\Gamma _c n_c + \\Gamma _h n_h)} N(N+3)$ .",
"The crossover system size $N^*$ between these regimes can be found by simply equating the limits and yields $N^* = 12 \\bar{n} - 3$ .",
"These limits can be seen well in Fig.",
"REF (dashed curves).",
"The crossover between two scaling regimes is thus quite analogous to previous results for collective qubit systems [80].",
"We can see that the current becomes maximal when $n_h\\rightarrow 0$ (which can be reached by $\\Delta \\rightarrow \\infty $ and then implies $n_c \\rightarrow \\frac{\\Gamma _c+\\Gamma _h}{\\Gamma _c} \\bar{n}$ ).",
"Then, we can numerically maximize $(n_c-n_h) f_N(\\bar{n}) \\rightarrow \\frac{\\Gamma _c+\\Gamma _h}{\\Gamma _c} \\bar{n} f_N(\\bar{n})$ as a function of $\\bar{n}$ only.",
"The position of this maximum is for large $N$ roughly at $\\bar{n} \\approx N/6$ , and the current scales quadratically in $N$ at this maximum."
],
[
"Interacting Pauli rate equation",
"Pauli rate equations can also be derived for interactions present in the system Hamiltonian $\\hat{H}_S$ .",
"If these interactions can be expressed by the permutationally symmetric operators $\\hat{C}_2$ , $\\hat{N}_\\delta $ and $\\hat{N}_\\Delta $ (or equivalently by $\\sum _\\alpha \\hat{J}_\\alpha ^2$ , $\\hat{J}_3$ and $\\hat{J}_8$ ) like in Eq.",
"(REF ), it follows that the same eigenstates can be used for the representation of the problem, and only the eigenvalues change.",
"Labelling the eigenstates of $\\hat{H}_S$ with the multi-index $i$ , the Pauli rate equation will then generically have the structure [2] $\\dot{P}_i &= \\sum _\\nu \\sum _j \\left[R_{ij}^\\nu P_j - R_{ji}^\\nu P_i\\right]\\,,\\nonumber \\\\R_{ij}^\\nu &= \\gamma _\\nu (E_j-E_i) {\\left| \\left<i\\right| \\hat{S}^\\nu \\left|j\\right> \\right|}^2\\,.$ The inherent local detailed balance for the transition rates then exponentially suppresses excitations into undesired Casimir subspaces and undesired excitation numbers, whereas relaxation from the excited eigenvalues down to the desired most productive cycle is still possible.",
"Thus, for $N=3k+1$ with integer $k$ and sufficiently large penalty parameters $\\beta _\\nu \\alpha _{C/P}\\gg 1$ , we can neglect the excited states and constrain our considerations to the states from the maximum symmetry (Casimir) sector with $\\left< \\hat{C}_2 \\right> = N(N+3)/3$ : $\\left|k;k\\right>$ , $\\left|k;k+1\\right>$ , and $\\left|k+1;k\\right>$ .",
"These three states then have the energies $E_0 &= k(\\Delta +\\delta )+\\alpha _P/3\\,,\\nonumber \\\\E_1 &= k\\Delta +(k+1)\\delta +\\alpha _P/3\\,,\\nonumber \\\\E_2 &= (k+1)\\Delta + k\\delta +\\alpha _P/3\\,,$ such that the penalty $\\alpha _P$ cancels out in their differences.",
"As the eigenstates $\\left|M;m\\right>$ remain the same, the quadratic enhancement from Eq.",
"(REF ) is preserved, and we obtain the current (REF ) with $\\Gamma _\\nu \\rightarrow \\Gamma _\\nu \\left(\\frac{N+2}{3}\\right)^2$ , leading to the corresponding enhancement (REF ) in the main text."
],
[
"Bosonization",
"The 8 operators $\\lbrace \\hat{J}_3, \\hat{J}_8, \\hat{J}^h_+, \\hat{J}^h_-, \\hat{J}^c_+, \\hat{J}^c_-, \\hat{J}^w_+, \\hat{J}^w_-\\rbrace $ inherit a closed algebra from the associated single-qutrit versions $\\left[\\hat{J}_3, \\hat{J}^h_+\\right] &= \\frac{\\hat{J}^h_+}{2}\\,,\\;\\left[\\hat{J}_3, \\hat{J}^c_+\\right] = -\\frac{\\hat{J}^c_+}{2}\\,,\\;\\left[\\hat{J}_3, \\hat{J}^w_+\\right] = \\hat{J}^w_+\\,,\\nonumber \\\\\\left[\\hat{J}_8, \\hat{J}^h_+\\right] &= \\frac{\\sqrt{3} \\hat{J}^h_+}{2}\\,,\\;\\left[\\hat{J}_8, \\hat{J}^c_+\\right] = \\frac{\\sqrt{3} \\hat{J}^c_+}{2}\\,,\\nonumber \\\\\\left[\\hat{J}^h_+, \\hat{J}^h_-\\right] &= \\sqrt{3}\\hat{J}_8+\\hat{J}_3\\,,\\;\\left[\\hat{J}^c_+, \\hat{J}^c_-\\right] = \\sqrt{3}\\hat{J}_8-\\hat{J}_3\\,,\\nonumber \\\\\\left[\\hat{J}^w_+, \\hat{J}^w_-\\right] &= 2 \\hat{J}_3\\,,\\nonumber \\\\\\left[\\hat{J}^h_+, \\hat{J}^c_-\\right] &= \\hat{J}^w_+\\,,\\;\\left[\\hat{J}^h_+, \\hat{J}^w_-\\right] = -\\hat{J}^c_+\\,,\\nonumber \\\\\\left[\\hat{J}^c_+, \\hat{J}^w_+\\right] &= -\\hat{J}^h_+\\,.$ Other independent (which do not follow from Hermitian conjugation) commutators just vanish."
],
[
"Four-mode Holstein-Primakoff transform",
"The above commutation relations can be realized with four bosonic modes with annihilation operators $\\hat{a}_\\Delta $ , $\\hat{a}_\\delta $ , $\\hat{b}_\\Delta $ , and $\\hat{b}_\\sigma $ obeying the usual bosonic commutation relations by using a generalization [81] of the Holstein-Primakoff-transform $\\hat{J}_3 &= \\hat{b}_\\sigma ^\\dagger \\hat{b}_\\sigma +\\frac{1}{2} \\hat{b}_\\Delta ^\\dagger \\hat{b}_\\Delta + \\frac{1}{2}\\left(\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta - \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta \\right)-\\frac{N_b}{2}\\,,\\nonumber \\\\\\hat{J}_8 &= \\frac{\\sqrt{3}}{2} \\hat{b}_\\Delta ^\\dagger \\hat{b}_\\Delta +\\frac{\\sqrt{3}}{2} \\left(\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta \\right) - \\frac{N_a}{\\sqrt{3}}-\\frac{N_b}{2\\sqrt{3}}\\,,\\nonumber \\\\\\hat{J}^h_+ &= \\hat{a}_\\Delta ^\\dagger \\sqrt{N_a - \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta - \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta }\\nonumber \\\\&\\qquad +\\hat{b}_\\Delta ^\\dagger \\sqrt{N_b - \\hat{b}_\\Delta ^\\dagger \\hat{b}_\\Delta - \\hat{b}_\\sigma ^\\dagger \\hat{b}_\\sigma }\\,,\\nonumber \\\\\\hat{J}^c_+ &= \\hat{a}_\\delta ^\\dagger \\sqrt{N_a - \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta - \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta }+\\hat{b}_\\Delta ^\\dagger \\hat{b}_\\sigma \\,,\\nonumber \\\\\\hat{J}^w_+ &= \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\delta -\\hat{b}_\\sigma ^\\dagger \\sqrt{N_b-\\hat{b}_\\Delta ^\\dagger \\hat{b}_\\Delta - \\hat{b}_\\sigma ^\\dagger \\hat{b}_\\sigma }\\,.$ Here, the integer numbers $N_a\\ge 0$ and $N_b\\ge 0$ determine the physically admissable states, i.e., they have to be adjusted to match the behaviour of the collective qutrit operators such as e.g.",
"the Hamiltonian $\\hat{H}_S &= \\Delta \\left[\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\hat{b}_\\Delta ^\\dagger \\hat{b}_\\Delta +\\hat{b}_\\sigma ^\\dagger \\hat{b}_\\sigma + \\frac{N-N_a-2N_b}{3}\\right]\\nonumber \\\\&\\qquad +\\delta \\left[\\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta - \\hat{b}_\\sigma ^\\dagger \\hat{b}_\\sigma + \\frac{N-N_a+N_b}{3}\\right]\\,.$ In particular, the Fock states $\\left|M,m,Q,q\\right>$ with $M$ , $m$ , $Q$ , and $q$ denoting the eigenvalues of $\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta $ , $\\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta $ , $\\hat{b}_\\Delta ^\\dagger \\hat{b}_\\Delta $ , and $\\hat{b}_\\sigma ^\\dagger \\hat{b}_\\sigma $ , respectively, are physically admissable when $&[(N_a = M+m) \\vee (Q=0)] \\wedge \\nonumber \\\\&[(N_b = Q+q) \\vee (M=0)] \\wedge \\nonumber \\\\&[(q=0) \\vee (m=0)]\\,.$ For example, the special case $Q=q=N_b=0$ fulfils these conditions and admits an even simpler representation of the algebra with just two bosonic modes, which we discuss below, and that corresponds to the fully symmetric subspace discussed in the main text.",
"For the subspace with second largest Casimir operator eigenvalue one has to use the four-boson representation instead.",
"As one always has $N_a+N_b \\le N$ and $0 \\le M+m \\le N_a$ as well as $0 \\le Q+q \\le N_b$ , it follows that largest Clebsch-Gordan coefficients (and therefore the largest currents) may originate from the fully symmetric subspace."
],
[
"Two-mode Holstein-Primakoff transform",
"When $Q=q=N_b=0$ , the transformation requires only two bosonic modes [82], [72] with a non-negative integer $N_a$ $\\hat{J}_3 &= \\frac{1}{2} \\left(\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta - \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta \\right)\\,,\\nonumber \\\\\\hat{J}_8 &= \\frac{\\sqrt{3}}{2} \\left(\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta \\right) - \\frac{N_a}{\\sqrt{3}}\\,,\\nonumber \\\\\\hat{J}^h_+ &= \\hat{a}_\\Delta ^\\dagger \\sqrt{N_a - (\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta )}\\,,\\nonumber \\\\\\hat{J}^c_+ &= \\hat{a}_\\delta ^\\dagger \\sqrt{N_a - (\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta )}\\,,\\nonumber \\\\\\hat{J}^w_+ &= \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\delta \\,,$ and analogously for the lowering operators.",
"It turns out that the fully symmetric subspace can be covered by the choice $N_a=N$ (number of qutrits).",
"Then, the fully symmetric states discussed in the main text are equivalent to the Fock state representation with just two bosonic modes $\\left|M;m\\right> \\equiv \\left|M,m,0,0\\right>$ .",
"The system Hamiltonian (REF ) then simply assumes the form $\\hat{H}_S = \\Delta \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\delta \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta $ .",
"As a sanity check, representing the Casimir operator in terms of the bosons we get the maximum eigenvalue valid for the fully symmetric subspace $\\left< \\hat{C}_2 \\right> = \\frac{N(N+3)}{3}$ .",
"From the bosonic properties it is then straightforward to compute the Clebsch-Gordan coefficients (REF ) of the symmetric subspace discussed in the main text."
],
[
"Master equation for large $N$",
"The master equation (REF ) can in the symmetric subspace and for large $N$ such that $\\left< a_\\nu ^\\dagger a_\\nu \\right> \\ll N$ be simplified by expanding the roots in (REF ), which yields $\\dot{\\rho }&= -\\mathrm {i}\\left[\\Delta \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta + \\delta \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta , \\rho \\right]\\\\&\\qquad +N\\Gamma _c(1+n_c) \\left[\\hat{a}_\\delta \\rho \\hat{a}_\\delta ^\\dagger - \\frac{1}{2} \\left\\lbrace \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta , \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +N\\Gamma _c n_c \\left[\\hat{a}_\\delta ^\\dagger \\rho \\hat{a}_\\delta - \\frac{1}{2} \\left\\lbrace \\hat{a}_\\delta \\hat{a}_\\delta ^\\dagger , \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +N\\Gamma _h(1+n_h) \\left[\\hat{a}_\\Delta \\rho \\hat{a}_\\Delta ^\\dagger - \\frac{1}{2} \\left\\lbrace \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\Delta , \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +N\\Gamma _h n_h \\left[\\hat{a}_\\Delta ^\\dagger \\rho \\hat{a}_\\Delta - \\frac{1}{2} \\left\\lbrace \\hat{a}_\\Delta \\hat{a}_\\Delta ^\\dagger , \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +\\Gamma _w(1+n_w) \\left[\\hat{a}_\\delta ^\\dagger \\hat{a}_\\Delta \\rho \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\delta - \\frac{1}{2}\\left\\lbrace \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\delta \\hat{a}_\\delta ^\\dagger \\hat{a}_\\Delta , \\rho \\right\\rbrace \\right]\\nonumber \\\\&\\qquad +\\Gamma _w n_w \\left[\\hat{a}_\\Delta ^\\dagger \\hat{a}_\\delta \\rho \\hat{a}_\\delta ^\\dagger \\hat{a}_\\Delta - \\frac{1}{2}\\left\\lbrace \\hat{a}_\\delta ^\\dagger \\hat{a}_\\Delta \\hat{a}_\\Delta ^\\dagger \\hat{a}_\\delta , \\rho \\right\\rbrace \\right]\\,.\\nonumber $ In absence of the work reservoir $\\Gamma _w\\rightarrow 0$ , we find that relaxation to the steady state is therefore scaling with $N t$ $\\left< \\hat{a}_\\delta ^\\dagger \\hat{a}_\\delta \\right>_t = N_\\delta ^0 e^{-N \\Gamma _c t} + n_c (1-e^{-N \\Gamma _c t})\\,,$ and analogously for the hot reservoir.",
"The above also manifests superradiant decay as the relaxation time is inversely proportional to $N$ .",
"However, the above master equation is valid only in regimes where $\\left< a_\\nu ^\\dagger a_\\nu \\right> \\ll N$ , such that we cannot describe the regime of boosted cooling power with it."
],
[
"Full Counting Statistics",
"The starting point for the determination of currents and their fluctuations is a generalized (or tilted) Liouvillian (or rate matrix) equation of the form $\\dot{\\rho } = {\\cal L}(\\chi ) \\rho \\,,$ where $\\rho $ is the vectorized part of interest of the density matrix (this could be the complete system density matrix or just the populations of a relevant subspace) and ${\\cal L}(\\chi )$ the matrix representing the corresponding part of the Redfield or LGKS dissipator or of the rate matrix that depends on the counting field $\\chi $ .",
"In the vectorized space, the trace is computed via ${\\rm Tr}\\left\\lbrace \\rho \\right\\rbrace = \\vec{}^{\\rm T} \\cdot \\vec{\\rho }$ , and trace conservation then implies that ${\\rm Tr}\\left\\lbrace {\\cal L}(0) \\sigma \\right\\rbrace = 0$ for any operator $\\sigma $ .",
"The moments of the conjugate variable $n$ to the counting field can be obtained by taking derivatives of the moment-generating function $M(\\chi ,t) = {\\rm Tr}\\left\\lbrace e^{{\\cal L}(\\chi ) t} \\rho _0 \\right\\rbrace $ via $\\left< n^\\alpha \\right>_t = (-\\mathrm {i}\\partial _\\chi )^\\alpha M(\\chi ,t)|_{\\chi =0}$ .",
"Analogously, one may obtain cumulants from the cumulant-generating function $C(\\chi ,t) = \\ln M(\\chi ,t)$ by acting with the corresponding derivatives on it.",
"In the long-term limit and for systems with a unique stationary state, one can show that $C(\\chi ,t)\\rightarrow \\lambda (\\chi ) t$ , where $\\lambda (\\chi )$ is the dominant eigenvalue of ${\\cal L}(\\chi )$ (the one with largest real part, that fulfils $\\lambda (0)=0$ ).",
"We are interested in the long-term limit of the lowest two cumulants of the current, i.e., in the current $\\bar{I} = \\lim _{t\\rightarrow \\infty }\\frac{d}{dt} \\left< n \\right>_t = -\\mathrm {i}\\lambda ^{\\prime }(0)$ and the noise $\\bar{S}_I=\\lim _{t\\rightarrow \\infty }\\frac{d}{dt}\\left[\\left< n^2 \\right>_t-\\left< n \\right>_t^2\\right] = -\\lambda ^{\\prime \\prime }(0)$ .",
"Unfortunately, for our problems, the dominant eigenvalue is not analytically known.",
"Second order perturbation theory for non-Hermitian matrices is rather non-trivial [83] and for larger $N$ numerical differentiation [74] is not stable.",
"Therefore, to compute the stationary current and noise, we use a different approach [50], derived from the counting statistics of time-dependent conductors [84].",
"Trace conservation implies that the stationary current can be obtained via $\\bar{I} = -\\mathrm {i}{\\rm Tr}\\left\\lbrace {\\cal L}^{\\prime }(0) \\bar{\\rho } \\right\\rbrace \\,,$ where $\\bar{\\rho }$ is the solution to the equation ${\\cal L}(0)\\bar{\\rho }=0$ normalized to ${\\rm Tr}\\left\\lbrace \\bar{\\rho } \\right\\rbrace =1$ (i.e., the steady state).",
"With that, we can also compute the stationary noise $\\bar{S}_{I} = -{\\rm Tr}\\left\\lbrace {\\cal L}^{\\prime \\prime }(0)\\bar{\\rho } \\right\\rbrace -2\\mathrm {i}{\\rm Tr}\\left\\lbrace {\\cal L}^{\\prime }(0)\\bar{\\sigma } \\right\\rbrace \\,,$ where the auxiliary variable $\\bar{\\sigma }$ is the solution to the equation ${\\cal L}(0)\\bar{\\sigma }= \\mathrm {i}{\\cal L}^{\\prime }(0)\\bar{\\rho }+ \\bar{I} \\bar{\\rho }$ normalized to ${\\rm Tr}\\left\\lbrace \\bar{\\sigma } \\right\\rbrace =0$ ."
],
[
"Tight-coupling of energy currents",
"The Pauli rate equation (REF ) has the form $\\dot{P}_i = \\sum _\\nu \\sum _j \\left(R_{ij}^\\nu P_j - R_{ji}^\\nu P_i\\right)$ , where $P_i$ are occupation probabilities of energy eigenstate $i$ (which in our state corresponds to the state $\\left|M;m\\right>$ of the fully symmetric sector) and $\\nu \\in \\lbrace c,h,w\\rbrace $ labels the reservoirs.",
"We can split the stationary populations into the contributions from the individual cycles that couple to state $i$ $\\bar{P}_i = \\sum _{\\cal C} \\bar{P}_i^{\\cal C}\\,,$ where e.g.",
"in Fig.",
"REF right panel the corner states take part in just one cycle, states on the facets take part in three cycles, and states in the interior take part in six cycles.",
"Then, the steady-state condition can be rewritten as $0 = \\sum _{\\cal C} \\left[\\sum _\\nu \\sum _j \\left(R_{ij}^{\\nu ,\\cal C} \\bar{P}_j^{\\cal C} - R_{ji}^{\\nu ,\\cal C} \\bar{P}_i^{\\cal C}\\right)\\right]\\,,$ and fulfilling it for every cycle obviously fulfills the complete steady-state condition.",
"The individual conditions for a normally oriented ($+$ ) cycle read $0 &= \\Gamma _h^{\\cal C} (1+n_h) \\bar{P}_2^{\\cal C} + \\Gamma _c^{\\cal C} (1+n_c) \\bar{P}_1^{\\cal C}\\nonumber \\\\&\\qquad - \\left(\\Gamma _h^{\\cal C} n_h + \\Gamma _c^{\\cal C} n_c\\right) \\bar{P}_0^{\\cal C}\\,,\\nonumber \\\\0 &= \\Gamma _w^{\\cal C} (1+n_w) \\bar{P}_2^{\\cal C} + \\Gamma _c^{\\cal C} n_c \\bar{P}_0^{\\cal C}\\nonumber \\\\&\\qquad - \\left(\\Gamma _w^{\\cal C} n_w + \\Gamma _c^{\\cal C} (1+n_c)\\right) \\bar{P}_1^{\\cal C}\\,,\\nonumber \\\\0 &= \\Gamma _w^{\\cal C} n_w \\bar{P}_1^{\\cal C} + \\Gamma _h^{\\cal C} n_h \\bar{P}_0^{\\cal C}\\nonumber \\\\&\\qquad - \\left(\\Gamma _w^{\\cal C} (1+n_w) + \\Gamma _h^{\\cal C} (1+n_h)\\right) \\bar{P}_2^{\\cal C}\\,,$ where $\\Gamma _\\nu ^{\\cal C}$ are cycle-dependent as they are given by the bare $\\Gamma _\\nu $ rates multiplied with the squared Clebsch-Gordan coefficients from Eq.",
"(REF ).",
"Analogous equations can be written down for the negatively-oriented ($-$ ) cycles.",
"A solution to these equations exists, as these are just the steady-state conditions for a single QAR (see App. )",
"without the normalization constraint.",
"A fully analogous decomposition applies to the energy currents $\\bar{I}_E^\\nu &= \\sum _{ij} (E_i-E_j) R_{ij}^\\nu \\bar{P}_j\\nonumber \\\\&= \\sum _{\\cal C} \\sum _{ij} (E_i-E_j) R_{ij}^{\\nu ,{\\cal C}} \\bar{P}_j^{\\cal C}\\,,$ and we get for the individual currents the cycle-resolved expressions (for positively oriented $+$ cycles) $\\bar{I}_E^c &= \\delta \\sum _{\\cal C} \\left[\\Gamma _c^{\\cal C} n_c \\bar{P}_0^{\\cal C} - \\Gamma _c^{\\cal C} (1+n_c) \\bar{P}_1^{\\cal C}\\right]\\,,\\nonumber \\\\\\bar{I}_E^h &= \\Delta \\sum _{\\cal C} \\left[\\Gamma _h^{\\cal C} n_h \\bar{P}_0^{\\cal C} - \\Gamma _h^{\\cal C} (1+n_h) \\bar{P}_2^{\\cal C}\\right]\\,,\\nonumber \\\\\\bar{I}_E^w &= (\\Delta -\\delta ) \\sum _{\\cal C} \\left[\\Gamma _w^{\\cal C} n_w \\bar{P}_1^{\\cal C} - \\Gamma _w^{\\cal C} (1+n_w) \\bar{P}_2^{\\cal C}\\right]\\,.$ By eliminating one of the probabilities using (REF ) and comparing the terms in square brackets we can thus confirm the tight-coupling relations (REF ) in the main text.",
"Furthermore, the cooling condition $I_E^c > 0$ then also implies $I_E^w>0$ and $I_E^h<0$ , such that from Eq.",
"(REF ) we get the conditions $n_c \\bar{P}_0^{\\cal C} > (1+n_c) \\bar{P}_1^{\\cal C}$ , $(1+n_h) \\bar{P}_2^{\\cal C} > n_h \\bar{P}_0^{\\cal C}$ , and $n_w \\bar{P}_1^{\\cal C} > (1+n_w) \\bar{P}_2^{\\cal C}$ .",
"Multiplying these conditions then eventually eliminates the dependence on the steady-state occupations $n_c(1+n_h) n_w > (1+n_c) n_h (1+n_w)\\,,$ which is equivalent to Eq.",
"(REF ).",
"Alternatively, Eq.",
"(REF ) may also be obtained from the positivity of the entropy production rate $\\bar{\\sigma }_{\\mathrm {i}} = -\\sum _\\nu \\beta _\\nu \\bar{I}_E^\\nu \\ge 0$ and the tight-coupling relations between the currents.",
"Departing from the tight-coupling limit will also alter the cooling condition [85]."
]
] | 2210.07844 |
[
[
"Effect of influence in voter models and its application in detecting\n significant interference in political elections"
],
[
"Abstract In this article, we study the effect of vector-valued interventions in votes under a binary voter model, where each voter expresses their vote as a $0-1$ valued random variable to choose between two candidates.",
"We assume that the outcome is determined by the majority function, which is true for a democratic system.",
"The term intervention includes cases of counting errors, reporting irregularities, electoral malpractice etc.",
"Our focus is to analyze the effect of the intervention on the final outcome.",
"We construct statistical tests to detect significant irregularities in elections under two scenarios, one where exit poll data is available and more broadly under the assumption of a cost function associated with causing the interventions.",
"Relevant theoretical results on the consistency of the test procedures are also derived.",
"Through a detailed simulation study, we show that the test procedure has good power and is robust across various settings.",
"We also implement our method on three real-life data sets.",
"The applications provide results consistent with existing knowledge and establish that the method can be adopted for crucial problems related to political elections."
],
[
"Introduction",
"In the modern era, opinions of individuals hold great power over decision-making at multiple levels, going up to the functioning of the government and the society.",
"As such, the cumulative or interactive behavior within those opinions hold sway over these power structures.",
"Opinion dynamics focuses on the way different options compete in a population, giving rise to either consensus (every individual holding the same opinion or option) or coexistence of several opinions.",
"Our focus is particularly on the electoral system, and the effects of influence (we shall use the term intervention interchangeably) in them.",
"The voter model has been studied extensively as an opinion dynamics model in this regard.",
"We shall use the voter model in this paper for a two-party democratic system, which is prevalent in many countries, e.g.",
"United States of America (USA).",
"Our objective is to understand the change in the outcome of the vote when there is intentional or unintentional external influence, such as counting errors, vote rigging, reporting issues etc.",
"We shall primarily construct statistical hypothesis tests, based on exit poll data, to identify whether significant such interventions have happened.",
"We also provide a cost function based approach to detect the same, under the situation when exit poll or opinion poll data is not available."
],
[
"Background and relevant literature",
"Government forms a very important part of our society.",
"There have been various forms of government at different times in our history.",
"In ancient times, monarchy prevailed in our society.",
"After that, came the era of autocracy, in which supreme power over a state is concentrated in the hands of one person, whose decisions are subject to neither external legal restraints nor regularized mechanisms of popular control.",
"Thereafter, dictatorship showed its ugly presence in the society where the government was characterized by a single leader or group of leaders and little or no toleration for political pluralism or independent media.",
"The evils of dictatorship was badly experienced by the society then, which finally paved the way for democracy.",
"According to the Oxford dictionary, democracy is “government by the people in which the supreme power is vested in the people and exercised directly by them or by their elected agents under a free electoral system”.",
"In the phrase of Abraham Lincoln as mentioned in his biography, democracy is a government “of the people, by the people, and for the people”.",
"Our focus on this article is on an immensely crucial aspect of the democratic system.",
"In most of the democratic countries across the globe today, the government is elected based on which candidate has received the maximum number or the majority of votes.",
"Majority rule is the binary decision rule used most often in influential decision-making bodies, including all the legislatures of democratic nations.",
"Several works have been done on majority rule, see for example, [24], [14] and [10].",
"According to [20], majority rule is the only binary decision rule that has the following properties: fairness in terms of anonymity and neutrality, decisiveness and monotonicity.",
"Other forms of binary decision rules do not satisfy the above.",
"For example, another commonly discussed decision function is the dictator function, which works under the assumption that the election result is completely determined by the choice of one person, known as the `dictator'.",
"This rule does not satisfy the aforementioned properties.",
"Refer to [2], [7] and [31] for relevant reading on dictator functions.",
"An important issue that comes with the democratic election system and the majority rule is the problem of electoral irregularities, i.e.",
"intentional or unintentional interference with the process of an election, subsequently increasing the vote share of a particular candidate, depressing the vote share of rival candidates, or both.",
"In [13] and [35], it has been discussed how it is impossible to ensure that electoral systems are completely devoid of manipulations.",
"In fact, the occurrences of electoral frauds is not at all uncommon in our society.",
"For instance, in Georgia, an election server central to a legal battle over the integrity of Georgia elections was left exposed to the open internet for at least six months ([3]).",
"There are several other instances of electoral fraud in other countries in modern times, see for example, [21], [19], [29], [32], [6], and [27].",
"In light of the above, it becomes extremely important to come up with methods of identifying the occurrence of electoral anomalies.",
"One of the most popular approaches in this regard is the Benford law.",
"The works by [11], [30] and [4] are some relevant studies.",
"[17] also hypothesized that if election results are manipulated or forged, then, due to the well-known human attraction to round numbers, the frequency of reported round percentages can be increased.",
"This hypothesis was tested by analyzing raw data from seven federal elections held in the Russian Federation during the period from 2000 to 2012.",
"[34], on the other hand, used a technique based on resampled kernel density methods to detect irregularities in the pattern of vote-shares.",
"During the last decade, machine learning techniques have also been used for detecting election anomalies.",
"[5], [23], [39] and the references therein are some recommended readings in this context."
],
[
"Our contribution",
"All of the above-cited studies make various attempts to detect the presence of electoral irregularities.",
"However, not much concentration has been given to confirm whether the interference actually causes a change in the true majority.",
"To explain this further, consider a two-candidate election where the first candidate is likely to get more than 70% votes.",
"In this scenario, even if the second candidate intentionally intervenes up to 20% votes, the majority function does not change.",
"We shall call this type of influences `insignificant'.",
"Along the same line, an influence is termed `significant' if there is a high probability of a shift in majority on the commitment of the intervention.",
"Our focus in this paper is to develop statistical tests which can detect the presence of significant electoral intervention under various real-life scenarios.",
"This new method of testing whether any malpractice has occurred in the election is developed under two scenarios.",
"More broadly, we consider the scenario where we are given only the final election result.",
"Here, we make appropriate assumptions about the cost associated with intervention to develop the test.",
"In a more specific case, in addition to the final election result, we also have an exit poll data that would help us in drawing relevant inference.",
"The main focus of our paper is this scenario with the added information of prior data.",
"It is well-established that an exit poll can give an early indication as to how an election has turned out, especially because the counting process can be very time consuming in many elections.",
"Polling is the only way of collecting pertinent information as the voters are anonymous.",
"Exit polls have been historically used throughout the world to identify the degree of potential election fraud.",
"Some examples of this are the 2004 Venezuelan recall referendum, and the 2004 Ukrainian presidential election, both of which will be discussed in greater detail later in this paper.",
"We also point out that opinion polls can provide prior information as well, albeit they serve as a much weaker predictor of the election result, primarily because they are carried out before the election takes place.",
"As an application of the proposed approach, we first perform an in-depth analysis for the 2016 USA Presidential Elections.",
"We check for the presence of statistically significant intervention in each of the states of USA.",
"The exit poll data is obtained from [1] and the final-election result is obtained from [25].",
"fig1 shows the exit poll results and the final election results side-by-side.",
"It is interesting to note that Michigan, Nevada, North Carolina, Pennsylvania, Wisconsin are the only five states where the two results do not match.",
"We shall use our proposed methodology to investigate these five states in more detail.",
"Figure: Comparison of Final election result and exit poll predictions for 2016 USA Presidential election.",
"Dem and Rep refer to the Democratic Party and Republican Party respectively.To further demonstrate the usefulness of the proposed approach, we perform statistical analysis for testing the presence of significant intervention in the 2004 Ukrainian Presidential election and 2004 Venezuelan recall referendum, both being known to have experienced electoral frauds.",
"The official final election results and the exit poll data for these two studies are obtained from [29] and [32], respectively."
],
[
"Organization",
"The paper is divided into the following sections.",
"sec:model discusses the voter model in depth, as we introduce the intervention in a formal way.",
"The main results in this section are the distribution of votes after intervention.",
"Later in the section we compute the distribution of the intervened maximum.",
"Next, rigging provides the method for testing for anomalies in elections under the two scenarios mentioned in the previous section.",
"In sec:simulation, we perform a detailed simulation study under various scenarios and discuss the findings.",
"Real data applications are presented in sec:application, while some concluding remarks and scopes of future research are summarized in sec:conclusions."
],
[
"Model",
"Throughout this article, $\\mathbb {I}\\lbrace \\cdot \\rbrace $ denotes the indicator function, i.e.",
"$\\mathbb {I}\\lbrace A\\rbrace =1$ if $A$ is true and is 0 otherwise.",
"For a real number $a$ , $\\mathrm {sgn}(a)$ is the signum function which takes the value $1, 0, -1$ depending on if $a$ is positive, zero or negative.",
"The notations $\\xrightarrow{}$ and $\\xrightarrow{}$ indicate convergence in probability and convergence in law (distribution), respectively.",
"The inner product of two vectors is denoted by $\\langle \\cdot ,\\cdot \\rangle $ .",
"A $k$ -variate normal distribution with mean $\\theta $ and dispersion matrix $\\Psi $ is denoted by $\\mathcal {N}_k(\\theta ,\\Psi )$ .",
"For a univariate normal distribution, we drop the subscript $k$ for convenience.",
"Our work is motivated primarily by electoral systems involving two candidates.",
"Along the lines of [9], we assume multiple voters who cast their preferences independently.",
"We also assume that there is an overall popularity of each of the candidates, given by a proportion.",
"Since there are two candidates, the sum of the two proportions would add up to one.",
"Let us use $p$ to denote the overall popularity of the first candidate before the votes are cast, and we treat it as a random variable.",
"Clearly, if $p_0$ is a realization of $p$ , the proportion of votes in favour of the second candidate is $(1-p_0)$ .",
"These values are true parameters by which the voters independently exercise their choices.",
"Every voter can choose one, and only one candidate.",
"Henceforth, the opinion of each voter is given by a two-dimensional vector, with one entry as 1 and the other as 0.",
"If a voter chooses the first candidate then it takes the value $(1,0)$ , and otherwise it would be $(0,1)$ .",
"We assume in total there are $n$ voters.",
"The mathematical model depicting this scenario is given in the next paragraph.",
"Let $X_i$ be a two-dimensional vector which denotes the initial opinion of the $i^{th}$ voter, for $i=1,\\hdots ,n$ .",
"Throughout, we assume that $X_i$ 's are independent and identically distributed (iid).",
"We use $X_i=(1,0)$ (respectively $(0,1)$ ) if the $i^{th}$ voter originally supports the first candidate (respectively the second candidate).",
"Thus, initially the distribution of $X_i$ is given as follows: $\\mathbb {P}(X_i=t)={\\left\\lbrace \\begin{array}{ll} p_0 & \\text{ when } t=(1,0), \\\\1-p_0 & \\text{ when } t=(0,1).\\end{array}\\right.",
"}$ Let us now introduce the notion of intervention on votes.",
"This is a form of outside influence.",
"The primary motivation behind the idea of intervention is to study whether any electoral malpractice has happened during or after the casting of votes and before the vote-counting.",
"Our objective is to statistically test whether the observed results are significantly different from what would have happened in the absence of this external influence.",
"We assume a fixed form of this influence.",
"In modeling this, we follow the notion of intervention introduced in [15].",
"Mathematically, it is denoted by a vector with all entries positive and is a transformation (calculated by the inner product) applied to all the voters.",
"Depending on the relative magnitude of the entries which quantify the strength of the intervention, it may or may not alter the votes.",
"In what follows, the probability of any voter supporting the first candidate after the intervention is assumed to be $p^{\\prime }$ , as opposed to its original value of $p_0$ before intervention.",
"The main results of this section are on the distribution of individual votes after the application of an intervention.",
"Let the intervention vector $v=(\\alpha ,\\beta )$ , with $\\alpha ,\\beta >0,$ be applied to each voter with probability $\\pi _0$ .",
"Then, there are two cases: If $v$ acts on $(1,0)$ , the intervened vector is given by, $(1,0)+\\langle \\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {},\\mathopen {}\\left( {\\alpha ,\\beta }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\rangle (\\alpha ,\\beta )$ , which is same as $(\\alpha ^2+1,\\alpha \\beta )$ .",
"If $v$ acts on $(0,1)$ , the intervened vector is given by, $(0,1)+\\langle \\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {},\\mathopen {}\\left( {\\alpha ,\\beta }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\rangle (\\alpha ,\\beta )$ , which is same as $(\\alpha \\beta ,\\beta ^2+1)$ .",
"Once we have the intervened vector, it is transformed to an opinion vector by looking at the maximum value between the two coordinates.",
"To elaborate, we shall say that the updated opinion vector is $(1,0)$ if the first coordinate of the intervened vector is greater than the second one and is $(0,1)$ otherwise.",
"Note that the effect of the intervention in switching a vote depends on the magnitude of $\\alpha ,\\beta $ .",
"For instance, if $v$ acts on $\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and if $\\alpha ^2+1 \\geqslant \\alpha \\beta $ , then the opinion vector remains the same i.e.",
"$\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ .",
"However, if $\\alpha ^2+1 < \\alpha \\beta $ , the opinion vector switches to $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ .",
"Similarly, if the intervention acts on $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and $\\beta ^2+1 \\geqslant \\alpha \\beta $ , then the opinion stays the same and otherwise, it switches to $\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ .",
"Lemma 1 It is impossible to have an intervention that changes the opinion vector for all voters.",
"Suppose that there exists an intervention vector $v=\\mathopen {}\\left( {\\alpha ,\\beta }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ , which switches both $\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ to $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ to $\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ .",
"Then, we must have: $\\alpha ^2+1<\\alpha \\beta \\text{ and }\\beta ^2+1<\\alpha \\beta ,$ subsequently implying $(\\alpha -\\beta )^2+2<0$ , which is impossible for real $\\alpha ,\\beta $ .",
"Hence all opinions upon which the intervention acts cannot get switched.",
"The implication of the above lemma is pivotal for the following discussion, as we aim to develop a test for detecting irregularities in an election.",
"It establishes that there cannot be an intervention vector which would corrupt the true opinion of all the voters.",
"We discuss next to what extent an intervention can influence the choice of a particular voter.",
"Consider the following notations.",
"As mentioned earlier, there are $n$ voters under our consideration, whose initial opinion vectors are given by $X_i$ , for $i=1,\\hdots ,n$ .",
"The probability that a particular voter supports the first candidate is denoted as $p_0$ .",
"In other words, all $X_i$ 's are assumed to be iid $\\mathrm {Bernoulli}(p_0)$ type random variables, where $p_0$ denotes the probability of the event $(1,0)$ .",
"Now, suppose that $\\pi _0$ proportion of voters have been acted upon by an intervention $v=\\mathopen {}\\left( {\\alpha ,\\beta }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ , where $\\alpha ,\\beta \\geqslant 0$ , and let the updated opinion vectors be given by $X_i^{\\prime }$ , for $i=1,\\hdots ,n$ .",
"We should note that the proportion of voters who have been acted upon by the intervention $v$ is a random variable $\\pi $ whose support is on $[0,1]$ .",
"Here, $\\pi _0$ is a realization of $\\pi $ .",
"Also, let $p^{\\prime }$ be the true post-intervention probability of supporting the first candidate for a randomly selected voter.",
"In line with the above, it is easy to argue that all $X_i^{\\prime }$ are iid $\\mathrm {Bernoulli}$ type random variables with parameter $p^{\\prime }$ .",
"The relationship of $p^{\\prime }$ with $p_0$ and $\\pi _0$ is discussed in the following result.",
"Lemma 2 Depending on the values of $\\alpha ,\\beta $ , there are three possible cases: (a) if $\\alpha ^2+1 \\geqslant \\alpha \\beta $ , $\\beta ^2+1 \\geqslant \\alpha \\beta $ , $p^{\\prime }=p_0$ ; (b) if $\\alpha ^2+1 < \\alpha \\beta $ , $\\beta ^2+1 \\geqslant \\alpha \\beta $ , $p^{\\prime }=p_0-p_0\\pi _0$ ; and (c) if $\\alpha ^2+1 \\geqslant \\alpha \\beta $ , $\\beta ^2+1 < \\alpha \\beta $ , $p^{\\prime }=p_0+\\pi _0-p_0\\pi _0$ .",
"For $\\alpha ^2+1 \\geqslant \\alpha \\beta $ , $\\beta ^2+1 \\geqslant \\alpha \\beta $ , we observed above that none of the opinion vectors changes, which proves part (a).",
"Under the conditions of part (b), $v$ switches $\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ to $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ , but there is no change if $v$ acts on $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ .",
"Thus, $X_i^{\\prime }=X_i$ with probability $(1-\\pi _0)$ and is equal to $(0,1)$ with probability $\\pi _0$ .",
"It subsequently implies that $\\mathbb {P}\\left[X_1^{\\prime }= (1,0)\\right] = \\mathbb {P}\\left[X_1^{\\prime }=X_1\\right]\\mathbb {P}\\left[X_1=(1,0)\\right] = \\mathopen {}\\left( {1-\\pi _0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}p_0.$ Hence, we see that in this case, $p^{\\prime }=p_0-p_0\\pi _0$ .",
"Part (c) follows in a similar fashion by noting that $v$ can change only the opinion vector $(0,1)$ .",
"In the current work, our focus is on the democratic method of election, where the majority function determines the winning candidate.",
"That is to say, the candidate who obtains more votes wins the election.",
"Following the Condorcet Jury theorem in [9], where voting is considered as an aggregation procedure and where the effectiveness of the majority opinion is shown asymptotically, we go with the same function here.",
"For the following discussion, we make slight notational changes.",
"Let us denote the opinion vector $\\mathopen {}\\left( {1,0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ by 1 and the opinion vector $\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ by $-1$ , for convenience.",
"Let $m\\mathopen {}\\left( {\\cdot }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ denote the majority function, i.e.",
"$m\\mathopen {}\\left( {\\cdot }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=1$ if the first candidate gets the majority and $-1$ otherwise.",
"For $\\mathbf {X}=\\mathopen {}\\left( {X_1,X_2,\\hdots ,X_n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and $\\mathbf {X^{\\prime }}=\\mathopen {}\\left( {X_1^{\\prime },X_2^{\\prime },...,X_n^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ , we shall use $m(\\mathbf {X})$ to denote the initial majority (before any kind of intervention) and $m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ to denote the post-intervention majority between the two candidates.",
"The focus of this section is the computation of the distribution of the post-intervention majority, and how it is related to the original majority.",
"Based on $p_0$ and $p^{\\prime }$ defined earlier, we want to calculate the probability of $m(\\mathbf {X^{\\prime }})$ remaining same as $m(\\mathbf {X})$ .",
"Naturally, the lower this probability is, the higher is the chance that the final outcome of the election changes because of the intervention.",
"In rigging, we develop a statistical test based on this probability, the computation of which is given by the following proposition.",
"The proof is deferred to sec:proofs.",
"Proposition 1 Let $f\\mathopen {}\\left( {\\mu ,\\Sigma ,n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ be the probability that the two components of a $\\mathcal {N}_2(\\mu ,\\Sigma /n)$ distribution are of same sign.",
"If the intervention vector is $v=\\mathopen {}\\left( {\\alpha ,\\beta }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ with $\\alpha ,\\beta \\geqslant 0$ , then depending on the values of $\\alpha ,\\beta $ , the following results hold.",
"(a) If $\\alpha ^2+1 \\geqslant \\alpha \\beta $ , $\\beta ^2+1 \\geqslant \\alpha \\beta $ , $\\mathbb {P}\\mathopen {}\\left( {m\\mathopen {}\\left( {\\mathbf {X}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=1$ .",
"(b) If $\\alpha ^2+1 < \\alpha \\beta $ , $\\beta ^2+1 \\geqslant \\alpha \\beta $ , $\\mathbb {P}\\mathopen {}\\left( {m\\mathopen {}\\left( {\\mathbf {X}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=f\\mathopen {}\\left( {\\mu _2,\\Sigma _2,n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} $ .",
"(c) If $\\alpha ^2+1 \\geqslant \\alpha \\beta $ , $\\beta ^2+1 < \\alpha \\beta $ , $\\mathbb {P}\\mathopen {}\\left( {m\\mathopen {}\\left( {\\mathbf {X}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=f\\mathopen {}\\left( {\\mu _3,\\Sigma _3,n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} $ .",
"In the above, $\\mu _2=\\mu _3=\\begin{pmatrix} 2p_0-1 \\\\ 2p^{\\prime }-1 \\end{pmatrix}, \\; \\Sigma _2=\\begin{bmatrix}4p_0-4p_0^2 & 4p^{\\prime }-4p_0p^{\\prime } \\\\4p^{\\prime }-4p_0p^{\\prime } & 4p^{\\prime }-4p^{\\prime 2}\\end{bmatrix}, \\;\\Sigma _3=\\begin{bmatrix}4p_0-4p_0^2 & 4p_0-4p_0p^{\\prime } \\\\4p_0-4p_0p^{\\prime } & 4p^{\\prime }-4p^{\\prime 2}\\end{bmatrix}.$"
],
[
"Testing for irregularity in elections",
"thm:intervention is key to develop a test for detecting significant interventions in elections.",
"Recall the notations $p_0$ and $p^{\\prime }$ and suppose that $\\hat{p}^{\\prime }$ is the observed proportion of votes in favour of the first candidate.",
"Without loss of generality, we assume that the second candidate has won the election, i.e.",
"$\\hat{p}^{\\prime } < 0.5$ .",
"In other words, we develop the theory under the assumptions of part (b) in thm:intervention.",
"The other cases would follow exactly similarly.",
"Our objective is to check if there was any external influence involved in the victory of the second candidate.",
"To put it in a more formal way, we wish to perform a test where the two hypotheses are framed as follows: $\\begin{split}& H_0^n: \\text{Significant electoral intervention has not occurred}, \\\\& H_1^n: \\text{Significant electoral intervention has occurred}.\\end{split}$ We maintain the superscript $n$ to reflect the number of voters in the data.",
"An intervention is termed as `significant' if there is a high probability of the majority being changed following the intervention i.e.",
"if $f(\\mu _2,\\Sigma _2,n)<\\tau _{c}$ , for some pre-defined critical value $\\tau _c \\in (0,1)$ .",
"Generally, for simulation studies and for application to real data, we shall use $\\tau _c=0.5$ .",
"As mentioned before, we work under two different frameworks and they are described in the following subsections."
],
[
"Test procedure without prior data",
"Under the aforementioned assumptions, note that $\\mathbb {P}(m(\\mathbf {X})=m(\\mathbf {X^{\\prime }}))=f(\\mu _2,\\Sigma _2,n)$ , which is a function of $p_0,p^{\\prime }$ and $n$ .",
"Equivalently, we can also think of it as a function of $\\pi _0,p^{\\prime }$ and $n$ .",
"Thus, we can say that, $f(\\mu _2,\\Sigma _2,n)=\\eta (p_0,p^{\\prime },n)=\\xi (\\pi _0,p^{\\prime },n)$ for suitable functions $\\eta (\\cdot )$ and $\\xi (\\cdot )$ .",
"We wish to find a $100(1-\\alpha ) \\%$ confidence interval for $f(\\mu _2,\\Sigma _2,n)$ .",
"This confidence interval can then be used to construct the test for detecting irregularity in the election.",
"Since no representative data on $p_0$ or $\\pi _0$ is available, in order to develop the test procedure, we assume that the prior distribution of the random variable $\\pi $ is known (note that as already mentioned in sec:model, $\\pi _0$ is a realization of $\\pi $ ).",
"Let the probability density function (pdf) of $\\pi $ be given by $h(\\cdot )$ , a continuous function supported on $\\mathopen {}\\left[ {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right]\\mathclose {}$ , and let $H(\\cdot )$ denote the corresponding distribution function (cdf).",
"This density $h(\\cdot )$ will be referred to as cost function in our paper.",
"One can interpret it as the cost associated with the intervention, which typically would be an increasing function of the proportion of votes influenced by the second candidate.",
"In other words, the probability that $\\pi $ is high should be quite low.",
"From lem:intervention-vote, we know that, ideally, $p_0-(p^{\\prime }/(1-\\pi _0))=0$ .",
"Since $\\pi $ is distributed as $h(\\cdot )$ in this case, we take into account the following assumption.",
"Assumption 1 The cost function $h(\\cdot )$ satisfies the following: $\\int _{0}^{1-p^{\\prime }}\\mathopen {}\\left( {\\frac{\\pi _0-\\pi }{1-\\pi }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi =0.$ It can be argued (shown in the proof of thm:confidence-interval in sec:proofs) that asm-originalprop implies that, for large population size, the original proportion of voters voting for the first candidate ($p_0$ ) matches with the expected value of $p$ when the final proportion of voters voting for the first candidate is $\\hat{p}^{\\prime }$ .",
"Naturally, it is a sensible assumption which attempts to estimate $p_0$ by leveraging appropriate cost functions.",
"The cost function $h(\\cdot )$ can be a decreasing pdf like that of a truncated exponential distribution with parameter $\\lambda $ or like a beta distribution with parameter $(1,\\beta )$ .",
"Since the initial $p_0$ of the population before any intervention is unknown, we try to estimate $p_0$ from $\\hat{p}^{\\prime }$ using the cost function.",
"To that end, define $\\phi \\mathopen {}\\left( {{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=\\frac{{p}^{\\prime }}{H\\mathopen {}\\left( {1-{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}\\int _{0}^{1-{p}^{\\prime }}\\frac{1}{1-\\pi }h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi .$ We note that $\\phi ({p}^{\\prime })$ is the expected value of $p$ when the final proportion of voters voting for the first candidate (final election result) is ${p}^{\\prime }$ .",
"Hence, $\\phi (\\hat{p}^{\\prime })$ is an estimate of $p_0$ that we get from $\\hat{p}^{\\prime }$ using the cost function, $h(\\cdot )$ .",
"Let $\\gamma _n(\\hat{p}^{\\prime })=\\sqrt{\\hat{p}^{\\prime }(1-\\hat{p}^{\\prime })}z_{\\beta /2}/\\sqrt{n}$ , where $(1-\\beta )^2=1-\\alpha $ , and $z_{a}$ is the upper-$a$ quantile of a standard normal distribution.",
"We consider the following two intervals, $S_1\\left(\\hat{p}^{\\prime }\\right) &= \\left(\\hat{p}^{\\prime }-\\gamma _n\\left(\\hat{p}^{\\prime }\\right),\\hat{p}^{\\prime }+\\gamma _n\\left(\\hat{p}^{\\prime }\\right)\\right), \\\\S_2\\left(\\hat{p}^{\\prime }\\right) &= \\left(\\phi \\left(\\hat{p}^{\\prime }\\right)-\\left|\\phi ^{\\prime }\\left(\\hat{p}^{\\prime }\\right) \\right|\\gamma _n\\left(\\hat{p}^{\\prime }\\right),\\phi \\left(\\hat{p}^{\\prime }\\right)+\\left|\\phi ^{\\prime }\\left(\\hat{p}^{\\prime }\\right) \\right|\\gamma _n\\left(\\hat{p}^{\\prime }\\right)\\right),$ We shall prove in sec:proofs that $S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and $S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ are the $100(1-\\beta ) \\%$ confidence intervals for $p^{\\prime }$ and $\\phi (p^{\\prime })$ respectively.",
"The following theorem illustrates how to compute the $100(1-\\alpha ) \\%$ confidence interval for $f(\\mu _2,\\Sigma _2,n)$ when the distribution of $\\pi $ is known to us.",
"We know that if any value lies outside the confidence interval of confidence coefficient $100(1-\\alpha )\\%$ that value is rejected at the level of significance $\\alpha $ .",
"Keeping this in mind, we define our test statistics for the problem given by eq:hypothesis.",
"Theorem 1 Suppose the density of $\\pi $ is given by $h(\\cdot )$ , where $h(\\cdot )$ is a continuous density supported on $\\mathopen {}\\left[ {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right]\\mathclose {}$ , and the cdf of $\\pi $ is given by $H(\\cdot )$ .",
"Then, under asm-originalprop, $(m,M)$ is a $100(1-\\alpha )\\%$ confidence interval for $f(\\mu _2,\\Sigma _2,n)$ , where $\\begin{split}& m=\\inf _{\\left\\lbrace p^{\\prime } \\in S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}, \\; p_0 \\in S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\right\\rbrace }\\eta \\mathopen {}\\left( {p_0,p^{\\prime },n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}, \\\\& M=\\sup _{\\left\\lbrace p^{\\prime } \\in S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}, \\; p_0 \\in S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\right\\rbrace }\\eta \\mathopen {}\\left( {p_0,p^{\\prime },n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}.\\end{split}$ Further, if we use the decision rule that the hypothesis $H_0^n$ (see eq:hypothesis) is rejected for $M<\\tau _c$ , then it is a consistent test at level of significance $\\alpha $ .",
"Thus, if we know the prior density of $\\pi $ or if we can estimate it from past elections, we can perform the test for the presence of significant electoral intervention using thm:confidence-interval.",
"However, in practice, due to insufficient information, it might be quite difficult to obtain an estimate of the density of $\\pi $ in most situations.",
"In sec:simulation, we shall conduct a detailed simulation study assuming various distributions on $\\pi $ .",
"Examining the performance of our test under various scenarios, we subsequently recommend some default distributions on $\\pi $ that one can use to perform this test so that the type-1 error is controlled and the power is good.",
"We next turn our attention to see how the test discussed in thm:confidence-interval can be improved in case we have an exit poll data for the concerned election."
],
[
"Test procedure using exit poll data",
"Consider the scenario where both the final election result and some prior information of $p_0$ are available.",
"If the prior density of $p$ is known to us or can be estimated from past elections, then one can easily modify the steps from sec:final data available to test for significant electoral intervention.",
"The following remark captures this discussion and the proof follows directly from thm:confidence-interval.",
"Remark 1 Assume that for large population size, the true intervention probability $\\pi _0$ matches with the expected value of $\\pi $ when the final proportion of voters voting for the first candidate is $\\hat{p}^{\\prime }$ .",
"Then, assuming that the prior distribution of $p$ is known or can be estimated, the test for the occurrence of significant electoral intervention can be performed in the same spirit as thm:confidence-interval.",
"One only needs to replace $\\eta \\mathopen {}\\left( {\\cdot }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ with $\\xi \\mathopen {}\\left( {\\cdot }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and $\\phi (\\hat{p}^{\\prime })$ with the expected value of $\\pi $ given the final proportion of voters voting for the first candidate is $\\hat{p}^{\\prime }$ in the definitions of $S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ and $S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ .",
"From a pragmatic standpoint, obtaining an estimate of the prior density of $p$ might be tricky.",
"However, we can leverage the exit poll data to conduct the test in a slightly different way which would obviate the need for estimating the density of $p$ .",
"It is assumed that the exit poll is drawn uniformly from the entire population.",
"The total number of voters under consideration is $n$ , and we consider that the exit poll data consists of a small portion of the total population.",
"Let it be of size $k < n$ .",
"Clearly, for a fixed $n$ , the accuracy of the testing method should improve with a larger value of $k$ .",
"Suppose, $\\hat{p}_k$ is the observed proportion of voters voting for the first candidate in the exit poll data.",
"We do not require any other assumption in this case.",
"For the hypothesis testing problem in eq:hypothesis, we can follow a similar procedure as in thm:confidence-interval.",
"Define the test statistic ${T}_1$ which is calculated identically to $M$ from eq:defnm1M1, but with $S_2(\\hat{p}^{\\prime })$ therein replaced by $C_2 \\left({\\hat{p}_k}\\right) = \\left(\\hat{p}_k-\\gamma _k \\left(\\hat{p}_k\\right), \\hat{p}_k+\\gamma _k\\left(\\hat{p}_k\\right)\\right).$ Note that $C_2(\\hat{p}_k)$ provides a more accurate confidence interval for $p_0$ as it is based on a representative data of the entire population.",
"It also explains why no other assumption is needed in this case.",
"We can adopt the decision rule to reject $H_0^n$ if ${T}_1<\\tau _c$ i.e.",
"if the probability of the majority remaining unchanged is small.",
"By following identical steps as in sec:final data available, it can be shown that this results in a consistent level-$\\alpha $ test too.",
"The following corollary provides the necessary details.",
"The proof is elaborated in sec:proofs.",
"Corollary 1 If we test the hypothesis given by eq:hypothesis using the statistic ${T}_1$ as outlined above, then it is a level-$\\alpha $ test i.e.",
"$\\mathbb {P}_{H_0}({T}_1<\\tau _c) \\leqslant \\alpha $ .",
"This is a consistent test as well."
],
[
"Simulation study",
"In this section, we provide a detailed simulation study to illustrate the theoretical results discussed so far as well as to compute the type-1 error and the power of the test across various setups."
],
[
"Performance of the test procedure without prior data",
"Consider the setup of sec:final data available.",
"In practice, it is impossible to know the true distribution of the intervention probability $\\pi $ , and without prior knowledge or relevant data, one can only `guess' the nature of $h(\\cdot )$ .",
"Let us denote this as $\\hat{h}(\\cdot )$ .",
"We have already shown that under asm-originalprop, if $\\hat{h}(\\cdot )$ is the same as true $h(\\cdot )$ , then the type-1 error remains under $\\alpha $ and the power goes to 1 as the number of voters goes to infinity.",
"Here we demonstrate the behavior of the type-1 error and the power when $\\hat{h}(\\cdot )$ is possibly different from $h(\\cdot )$ .",
"Throughout, we use the population size $n=100,000$ and simulate data under asm-originalprop.",
"Both $h(\\cdot )$ and $\\hat{h}(\\cdot )$ are chosen from $\\mathrm {Exponential}(\\lambda )$ and $\\mathrm {Beta}(1,\\beta )$ , where $\\lambda $ and $\\beta $ are taken from the set $\\lbrace 10,20,30,50,100\\rbrace $ .",
"For each such scenario, 200 simulations are conducted and subsequently, empirical type-1 error and power are computed.",
"All tests are carried out at level $0.05$ using $\\tau _c=0.5$ (in accordance with the notation used in sec:final data available).",
"These results are shown in fig2 and fig4.",
"The columns in both figures represent the true distribution $h(\\cdot )$ while the rows represent the assumed distribution $\\hat{h}(\\cdot )$ .",
"The parameter values used in the graphs refer to the rate parameter $\\lambda $ (respectively the shape parameter $\\beta $ ) if the distribution is exponential (respectively beta).",
"Note that the parameter value of $h(\\cdot )$ is given along the $x$ -axis whereas the same for $\\hat{h}(\\cdot )$ is indicated by different colors.",
"We also point out that for the aforementioned choices of the distributions, it is ensured that the probability of intervention on all or most voters is not substantially big.",
"This is a realistic scenario as discussed earlier.",
"Figure: Empirical type-1 error of the test procedure with different cost functions.Figure: Empirical power of the test procedure with different cost functions.We note that both the type-1 error and power decrease with the increase in steepness (rate/shape parameter) of the distribution assumed for the cost function.",
"Hence, higher the parameter of $\\hat{h}(\\cdot )$ , lower is the chance of occurrence of any false positives.",
"We observe from fig2 that, when the parameter of $\\hat{h}(\\cdot )$ is above 20, the type-1 error always stays below $0.05$ .",
"Thus, these can be possible choices for $\\hat{h}(\\cdot )$ .",
"We further observe in fig4 that in all situations, the power is about 70% or more if the parameter of the assumed distribution is around 20 to 30.",
"It however becomes quite low when the parameter of $\\hat{h}(\\cdot )$ is 50 or more.",
"From the above, we can infer that if the original distribution of $\\pi $ can be estimated through some prior knowledge, then one can use that in the cost function.",
"This would keep the type-1 error below the chosen level $\\alpha $ and give significantly high power.",
"However, in case the prior density of $\\pi $ is not available, and if there is no statistical way to estimate that density, then the parameter of $\\hat{h}(\\cdot )$ may be chosen to be a number between 20 and 50.",
"In our real-life applications in sec:application, we shall assume that the distribution for the cost function is $\\mathrm {Exponential}(30)$ or $\\mathrm {Beta}(1,30)$ .",
"Henceforth, the test statistic based on these two cost functions are referred to as ${T}_2$ and ${T}_3$ , respectively."
],
[
"Performance of the test procedure using exit poll data",
"sec:final and exit available laid out the methodology of conducting the test for detection of electoral intervention when prior information about $p_0$ , possibly obtained from an exit poll, is available.",
"This subsection considers such cases and demonstrates how the power of the test varies with the size of the exit poll ($k$ ), the total population size ($n$ ), the true proportion of voters voting for the first candidate ($p_0$ ) and the value of the same after intervention ($p^{\\prime }$ ).",
"The computation of the power is done under nine different combinations of $n$ and $k$ .",
"Three different sizes of total population are taken viz.",
"$100,000$ , $200,000$ and $1,000,000$ .",
"For each of these, $k$ is varied between $10,000$ , $20,000$ and $50,000$ .",
"Now, for each of these nine combinations, 200 experiments are carried out and subsequently we calculate the empirical type-1 error and the empirical power of the test.",
"The type-1 error, as expected, lies within the acceptable range and we omit those results for conciseness of the paper.",
"The power values are plotted in fig3 against the initial proportion of voters who voted for the first candidate.",
"The color of the power plot represents the final proportion of voters who voted for the first candidate.",
"We show the results for three different values of $p^{\\prime }$ , viz.",
"$0.45,0.49,0.495$ , to understand the effect of the same on the power of the test.",
"All of the experiments are conducted using the critical value $\\tau _c=0.5$ and at level $0.05$ .",
"Figure: Power curves of the test for different combinations of initial and final (after intervention) proportion of voters voting for first candidate.",
"nn represents the total population size and kk represents the size of the exit poll data.We observe from fig3 that, if the size of the exit poll data is kept fixed, the power curves more or less stay the same.",
"However, they change drastically when we change the exit poll size.",
"They become steeper, and therefore better on more occasions, on increasing $k$ .",
"On the other hand, for $n=100,000$ , the power curves are slightly different from the others.",
"With total population size $200,000$ or above, we can see that the curves hardly vary.",
"We can deduce some important conclusions from these observations.",
"The power of the test primarily depends on the absolute value of the exit poll size and not so much on what proportion of the total population the exit poll data is.",
"The test becomes more accurate with the increase in size of the exit poll data.",
"Another thing to note is that when the total population size is large $(n \\geqslant 200,000)$ , the power curves of the three different colors almost coincide with each other.",
"It suggests that the test does not depend significantly on the final proportion of the voters who voted for the first candidate, but mainly depends on the initial proportion of voters who would have voted for the first candidate.",
"For an exit poll size of 20,000 or more, the test procedure has high power even when the true proportions of voters voting for the two candidates are very close (equivalent to saying $p_0$ is just above 0.5).",
"In light of this, we can also argue that an exit poll data of size at-least $20,000$ can ensure accurate detection of significant influence with a very high probability.",
"This can be explained theoretically by an application of usual central limit theorem which says that for sample size 20,000 or more, $\\hat{p}_k$ can estimate $p_0$ within $\\pm 1\\%$ error with a very high probability."
],
[
"2016 USA Presidential Election",
"The 2016 USA presidential election was the 58th quadrennial presidential election, held on Tuesday, November 8, 2016 in which, the Republican candidate Donald Trump defeated the Democratic candidate Hillary Clinton.",
"We select this data for a few reasons.",
"First, it is the most recent presidential election in the USA in which the winning candidate lost the popular vote.",
"Second, standard procedure of exit poll data collection took place in 2016, which is not the same due to the ongoing COVID-19 pandemic in 2020 USA Presidential Election.",
"And most importantly, a popular theory suggested that illegal interference did take place in this election.",
"On October 7, 2016, [16] issued a joint statement that the intelligence community is confident that the Russian government directed the recent compromises of e-mails from a few USA persons and institutions, including some political organizations.",
"A Special Counsel began in May 2017 in order to investigate the alleged collusion between Russia and the republican party led by Trump.",
"The counsel ended in March 2019.",
"According to [26], the following conclusion was reached by the investigation: the Russians interfered “in sweeping and systematic fashion” to favor Trump's candidacy but it “did not establish that members of the Trump campaign conspired or coordinated with the Russian government”.",
"Nonetheless, some discernible changes were observed in the outcomes of a few states.",
"For the first time since 1984, Wisconsin was won by the Republican party.",
"Michigan and Pennsylvania were also won by them for the first time since 1988.",
"Jill Stein, the presidential candidate of the Green party, petitioned for a recount in these three states ([37]).",
"Around the same time, as reported by [8], Rocky De La Fuente, the presidential candidate of American Delta Party/Reform Party, was granted a partial recount in Nevada.",
"Only minor changes to vote tallies were detected in the recounts in Wisconsin and Nevada, as reported in [36] and [33] respectively.",
"A partial recount of Michigan ballot revealed some unbalanced precincts in Detroit and they were corrected as well.",
"The state audit that followed came to the conclusion that the unbalanced precincts were a result of errors committed by the precinct workers and not a result of some major voter fraud ([28]).",
"Thus in spite of so many petitions, the recounts did not alter the outcome of the election.",
"In the language of this article, we can say that these recounts did not identify significant electoral irregularities in any of the states.",
"The above informations regarding the 2016 USA Presidential election (in the above two paragraphs) were obtained from [38] and then cross-verified from various news articles, reports that are cited above.",
"To statistically investigate the same problem through our proposed approach, we primarily use the exit poll data obtained from [1] and the test statistic ${T}_1$ .",
"Following the discussion in sec:simulation, we also check for robustness of our method by using the test statistics ${T}_2$ and ${T}_3$ , which correspond to the cost functions $\\mathrm {Exponential}(30)$ and $\\mathrm {Beta}(1,30)$ respectively.",
"We point out that the exit poll data had multiple polls conducted for each state in various time points.",
"Assuming that all the polls conducted for a particular state were disjoint, we combine them into a single exit poll.",
"We note that the size of the exit poll is above $20,000$ for almost all the states, thereby ensuring high power of our test.",
"Akin to the earlier sections, we use the critical value $\\tau _c=0.5$ .",
"The individuals who neither voted for Trump nor for Clinton are removed from both the election result data and the exit poll data so that we are in the set-up of two-candidates voter model.",
"We rescale the proportions accordingly.",
"On making a naive comparison of the exit poll data with the final election results, we observe that there are five states where the results from the two sources do not match.",
"This has been discussed briefly in sec:introduction (see fig1) as well.",
"These five states are Michigan, Pennsylvania, Wisconsin, North Carolina and Nevada.",
"In the first four states, Clinton was predicted to win and in the fifth state, Trump was predicted to win according to the exit poll data.",
"Naturally, it makes sense to investigate these five states in more detail.",
"We however start with a succinct account of the results for all of the other states.",
"The value of the statistics ${T}_1,{T}_2$ and ${T}_3$ are all found out to be 1 for all of these states, which provides absolutely no indication of any type of statistical evidence for electoral irregularity to make a case for recounting in these states.",
"It is critical to observe that even without the information from exit polls, as we adopt the cost function approach, the inference remains exactly the same.",
"We now take a detailed look at the aforementioned five states, where the value of the test statistics are found to be less than 1.",
"Relevant details for these states, along with the value of the test statistic and the conclusion, are presented in tab:us-election.",
"For analysis purposes, we mainly pay attention to ${T}_1$ , as it has already been shown to have much higher power and less type-1 error than the other two methods.",
"Table: Results for the electoral irregularity detection method for five states in 2016 USA Presidential Elections.We observe that the values of the test statistic ${T}_1$ for the states of Michigan, Nevada, Pennsylvania and Wisconsin are very close to zero indicating the presence of significant electoral anomalies in these four states which may have led to a change in the true majority.",
"This gives sufficient statistical evidence to make a case for recounting in these four states.",
"On the contrary, a very interesting situation occurs for the state of North Carolina.",
"Here, although the exit poll prediction does not match with the final election result, the value of our test statistic comes out to be $0.99711$ .",
"Hence, $H_0^n$ is not rejected for this state.",
"Thus we cannot say that significant electoral anomaly has been observed to make a case for recounting in this state.",
"It is worth specifying that the proportions in the two samples (exit poll data and final results data) are indeed significantly different; but our test procedure does not detect significant illegal interference to change the majority function.",
"It strengthens the importance of our test statistic over mere comparison of the exit poll data and the final election result.",
"This also substantiates the fact that unlike the other four states no claim or petition of recount was made for North Carolina.",
"In conclusion, significant evidence of influence is detected at 5% level of significance in only four states, thereby providing statistical evidence for recounting.",
"Of these four states, in Nevada, the final majority went in favor of the Democratic Party whereas for Michigan, Pennsylvania and Wisconsin, it was the other way round.",
"No significant electoral anomaly is detected in any of the other states.",
"It is imperative to declare that the the test statistics ${T}_2$ and ${T}_3$ render identical results in all states.",
"Clearly, the decisions based on the cost function approach are the same for the entire election data.",
"It establishes that even without sufficient prior information, the proposed approach works well and furnishes concrete support to the results obtained in this application."
],
[
"Fraudulent Presidential Elections from Ukraine and Venezuela",
"Our second application revolves around the 2004 Ukrainian Presidential election and the 2004 Venezuelan recall referendum, both of which are known instances of electoral irregularities.",
"Therefore, it would allow us to objectively evaluate whether our test procedure is able to detect irregularities in similar settings.",
"According to the electoral law of Ukraine, the President is elected by a two-round system in which a candidate must win a majority (50% or more) of all ballots cast.",
"The first round of voting in 2004 election was held on 31st October.",
"Since no candidate had 50% or more of the votes cast, a run-off ballot was held on 21st November between the two candidates with maximum number of votes, Viktor Yushchenko and Viktor Yanukovych.",
"The run-off election was won by the latter according to the official results announced on 23rd November.",
"The results of the second round were protested by the opposition with an allegation of illegal falsifications.",
"Massive street protests in support of the opposition as well as the blockade and picketing of the government buildings (the so-called “Orange Revolution”) with the demands to cancel the results of the elections went off in the country.",
"The Supreme Court annulled the November runoff election and ordered the third round of election (a rerun of the second round) which took place on 26th December, 2004.",
"This time, Yushchenko won the election with 52% of the votes.",
"The above information and other pertinent discussions are reported in a detailed manner in [29], [12] and [18].",
"Analogous to the previous application, here also we primarily use the exit poll data to reconfirm the presence of interference in the outcome of the second round of 2004 Ukrainian Presidential Election.",
"The exit poll was originally conducted by KIIS and the Razumkov Center with the organizational support of the Democratic Initiatives Fund, and was carried out nationwide by a secret ballot method.",
"The details of the poll can be obtained from [29].",
"The sample consisted of 750 polling stations with about 28000 respondents and a response rate of 79%.",
"Hence, the size of the exit poll data is roughly 22120, which ensures high power for the test.",
"On the other hand, the Venezuelan Recall Referendum (RR) of 15th August 2004 was a referendum to decide whether or not the then President Hugo Chávez should be removed from office.",
"In order to activate the RR, on 28th November, 2003, signatures and fingerprints were collected in a four-day event organized by the Consejo Nacional Electoral (CNE), with witnesses from all political parties.",
"The Organization of American States (OAS) sent a delegation chaired by its Secretary General to negotiate a solution.",
"The Carter Center, led by President Jimmy Carter himself, played an important role in getting the government and the opposition to agree on a course of action.",
"CNE was the official body in charge of the organization of the RR.",
"The result of the referendum was not to recall Chávez (approximately 59% voted against him).",
"However, there have been allegations of fraud shortly after ([32], [22]).",
"Since the RR was seen by all parties as a pivotal event, several organizations set up schemes to collect exit poll data to address the allegations.",
"The two exit polls considered here were conducted independently by Súmate, a nongovernmental organization, and Primero Justicia, a political party.",
"Assuming that the two polls were disjoint, we combine them to get a sample of 36174 observations.",
"These data are obtained from [32].",
"For convenience, for both of the datasets, let us denote the candidate who has won the final election as $W$ and the corresponding losing candidate as $L$ .",
"As before, we consider a two candidate setup and rescale the data as needed.",
"Critical value of the test is $\\tau _c=0.5$ , and the test is carried out at 5% level of significance.",
"The details of the exit poll data, final election results, and the outcome of the test procedure through ${T}_1$ are displayed in tab:ukrivene-election.",
"Table: Results for the electoral irregularity detection method for 2004 Ukrainian Presidential election and 2004 Venezuelan recall referendum.We observe that the value of ${T}_1$ is 0 for both Ukraine and Venezuela, indicating significant interference on both elections.",
"This reaffirms the occurrence of major electoral irregularities in these two elections as has been discussed in the papers alluded to above.",
"We also point out that the values of the other two statistics ${T}_2$ and ${T}_3$ , which rely on the cost function based approach, match with that of ${T}_1$ for 2004 Ukrainian Presidential Election.",
"In other words, it demonstrates the robustness of the proposed test procedure once again."
],
[
"Conclusion",
"In summary, this paper provides a new method for leveraging the exit poll data to detect the occurrence of significant intervention in the outcome of a two-candidate democratic electoral system.",
"We must note that not all intervention would lead to a change in the majority function of the voter model.",
"Therefore, simple comparison of the final election result and the exit poll data is likely to render misleading conclusions.",
"To that end, we work with a probabilistic voter model and develop a test procedure with a solid theoretical understanding.",
"Through a detailed simulation study, we demonstrate the performance of the proposed test statistic under various settings.",
"It establishes that when the size of the exit poll data is more than 20000, the test achieves high power.",
"However, one should remember that a key assumption of the test is that the exit poll is conducted in a scientific way where the sample is representative of the entire population.",
"Thus, the test may not give good results if the exit poll data involves biased selection of people from the population.",
"In such cases, or in the absence of any prior information, we also provide a cost function based approach that can lead to powerful tests as well.",
"As real life applications, we consider three different examples.",
"In the first one, our method detects the presence of significant electoral intervention in four states of USA in the 2016 Presidential Election.",
"The fact that recounting was done in all of these states substantiates our findings.",
"It is also observed that for one of the states, the conclusion of the proposed test differs from that of a two sample proportion comparison.",
"It clearly exhibits that our method is able to identify significant intervention that affects the overall outcome of a electoral process, thereby avoiding misleading inferences.",
"Apart from that, we also detect evidence of significant electoral fraud in 2004 Ukrainian Presidential election and 2004 Venezuelan recall referendum, both of which are in line with existing knowledge.",
"Overall, the results suggest that the procedure works well and thus can be used by news channels, political analysts and others to detect the presence of significant electoral intervention.",
"We conclude this article with a couple of interesting future directions.",
"Albeit one can detect the presence of irregularities on the overall outcome of the election (for example, on state-level), it does not provide additional idea of which sub-unit (for example, which county) is more likely to have been exposed to the irregularities.",
"It would be interesting to work on a unified approach which leverages individual county or district level data to detect if the result of the state has been significantly altered (note that the majority is decided based on the data from the entire state) and if so, which counties or districts might have caused that.",
"Another possible extension to the current work is to consider the case of multiple candidates instead of a binary voter model.",
"It would allow us to study various elections in several other countries where there are more than two major political parties competing against each other."
],
[
"Funding details",
"There are no funding bodies to thank relating to the creation of this article.",
"The authors declare no conflict of interest.",
"All of the data used in this article are publicly available.",
"Data related to the 2016 USA Presidential Election are obtained from Harvard Dataverse (link: https://dataverse.harvard.edu/file.xhtml?fileId=4788675) and from Data world (link: https://data.world/databeats/2016-us-presidential-election).",
"Data related to the other applications are obtained from [29] and [32]."
],
[
"Proofs",
"[Proof of thm:intervention] For the first case, when $\\alpha ^2+1 \\geqslant \\alpha \\beta $ and $\\beta ^2+1 \\geqslant \\alpha \\beta $ , by lem:intervention-vote, since none of the opinion vectors change on being acted upon by the intervention vector $v$ , it is easy to argue that $\\mathbb {P}(m(\\mathbf {X})=m(\\mathbf {X^{\\prime }}))=1$ .",
"Now, let us focus on the second case.",
"With $\\mathbb {I}\\lbrace \\cdot \\rbrace $ denoting the indicator function, we have the relation $\\mathbb {I}\\lbrace m(\\mathbf {X})=m(\\mathbf {X^{\\prime }})\\rbrace = (1+m(\\mathbf {X})m(\\mathbf {X^{\\prime }}))/2$ , which subsequently implies the following: $\\mathbb {P}\\mathopen {}\\left( {m\\mathopen {}\\left( {\\mathbf {X}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=\\frac{1}{2}\\mathopen {}\\left( {1+\\mathbb {E}\\mathopen {}\\left( {m\\mathopen {}\\left( {\\mathbf {X}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}.$ Note that $\\mathbb {E}(X_1)=2p_0-1$ , $\\mathbb {E}(X_1^{\\prime })=2p^{\\prime }-1$ , $\\mathrm {Var}(X_1)=4p_0-4p_0^2$ , $\\mathrm {Var}(X_1^{\\prime })=4p^{\\prime }-4p^{\\prime 2}$ , and $\\mathrm {Cov}(X_1,X_1^{\\prime })=4p^{\\prime }-4p_0p^{\\prime }$ .",
"Next, for $i=1,\\hdots ,n$ , letting $Y_i = (X_i,X_i^{\\prime })^T$ , we can write $\\mathbb {E}(Y_i)=\\begin{pmatrix} 2p_0-1 \\\\ 2p^{\\prime }-1 \\end{pmatrix}=\\mu _2, \\; \\mathrm {Var}\\mathopen {}\\left( {Y_i}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=\\begin{bmatrix}4p_0-4p_0^2 & 4p^{\\prime }-4p_0p^{\\prime } \\\\4p^{\\prime }-4p_0p^{\\prime } & 4p^{\\prime }-4p^{\\prime 2}\\end{bmatrix}=\\Sigma _2.$ Also, let $\\bar{Y}_n = \\sum _{i=1}^n Y_i/n = (\\bar{Y}_{n1},\\bar{Y}_{n2})$ .",
"An application of the multivariate central limit theorem implies that, as $n\\rightarrow \\infty $ , $\\sqrt{n}\\mathopen {}\\left( {\\bar{Y}_n-\\mu _2}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\xrightarrow{}\\mathcal {N}_2\\mathopen {}\\left( {0,\\Sigma _2}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}.$ In other words, for large $n$ , $(\\bar{Y}_{n1},\\bar{Y}_{n2})$ is approximately distributed as $\\mathcal {N}_2(\\mu _2,\\Sigma _2/n)$ .",
"On the other hand, $\\mathbb {E}(m(\\mathbf {X})m(\\mathbf {X^{\\prime }}))$ can be evaluated as $\\mathbb {E}[\\mathrm {sgn}(\\bar{Y}_{n1})\\mathrm {sgn}(\\bar{Y}_{n2})]=2\\mathbb {P}[\\mathrm {sgn}(\\bar{Y}_{n1}) = \\mathrm {sgn}(\\bar{Y}_{n2})]-1$ .",
"Combining it with eq:m=m', straightforward calculation leads to the following: $\\mathbb {P}\\mathopen {}\\left( {m\\mathopen {}\\left( {\\mathbf {X}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=m\\mathopen {}\\left( {\\mathbf {X^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} = \\mathbb {P}\\mathopen {}\\left( {\\mathrm {sgn}(\\bar{Y}_{n1}) = \\mathrm {sgn}(\\bar{Y}_{n2})}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} = f\\mathopen {}\\left( {\\mu _2,\\Sigma _2,n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {},$ where $f(\\mu _2,\\Sigma _2,n)$ is the probability that the two components of a $\\mathcal {N}_2(\\mu _2,\\Sigma _2/n)$ distribution are of same sign.",
"This completes the discussion for the second case.",
"For the third part of the theorem, defining $Y_i$ similarly as before, we can obtain $\\mathbb {E}(Y_i)=\\begin{pmatrix} 2p_0-1 \\\\ 2p^{\\prime }-1 \\end{pmatrix}=\\mu _3, \\; \\mathrm {Var}(Y_i)=\\begin{bmatrix}4p_0-4p_0^2 & 4p_0-4p_0p^{\\prime } \\\\4p_0-4p_0p^{\\prime } & 4p^{\\prime }-4p^{\\prime 2}\\end{bmatrix}=\\Sigma _3.$ Then, the rest of the proof follows in an identical fashion as in the second case.",
"[Proof of thm:confidence-interval] We have the following distributional convergence for $\\hat{p}^{\\prime }$ : $\\frac{\\sqrt{n}\\mathopen {}\\left( {\\hat{p}^{\\prime }-p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}{\\sqrt{p^{\\prime }\\mathopen {}\\left( {1-p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}} \\xrightarrow{}\\mathcal {N}\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} .$ We know that $\\hat{p}^{\\prime } \\xrightarrow{}p^{\\prime } \\text{ as } n\\rightarrow \\infty $ from the Weak Law of Large Numbers (WLLN).",
"Using Slutsky's theorem, we get the following convergence equation: $\\frac{\\sqrt{n}\\mathopen {}\\left( {\\hat{p}^{\\prime }-p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}{\\sqrt{\\hat{p}^{\\prime }\\mathopen {}\\left( {1-\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}} \\xrightarrow{}\\mathcal {N}\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {},$ which implies $\\mathbb {P}\\mathopen {}\\left( {p^{\\prime } \\in S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=1-\\beta .$ From asm-originalprop, we have the following, $\\nonumber \\begin{split}\\int _{0}^{1-p^{\\prime }}\\mathopen {}\\left( {\\frac{\\pi _0-\\pi }{1-\\pi }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi =0 & \\Rightarrow \\int _{0}^{1-p^{\\prime }}\\mathopen {}\\left( {1-\\frac{1-\\pi _0}{1-\\pi }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi =0 \\\\& \\Rightarrow p_0\\int _{0}^{1-p^{\\prime }}h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi =\\int _{0}^{1-p^{\\prime }}\\mathopen {}\\left( {\\frac{p_0(1-\\pi _0)}{1-\\pi }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi \\\\& \\Rightarrow p_0=\\frac{1}{H\\mathopen {}\\left( {1-{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}\\int _{0}^{1-{p}^{\\prime }}\\frac{{p}^{\\prime }}{1-\\pi }h\\mathopen {}\\left( {\\pi }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}d\\pi .\\end{split}$ In the above deduction, we have used the fact that $p_0(1-\\pi _0)=p^{\\prime }$ .",
"Thus, asm-originalprop implies that $p_0=\\phi (p^{\\prime })$ i.e.",
"$p_0$ matches with the expected value of $p$ when the final proportion of voters voting for the first candidate is ${p}^{\\prime }$ .",
"It can be easily shown that $\\phi (.",
")$ is a continuously differentiable function.",
"Applying Delta Theorem on (REF ), we get the following, $\\frac{\\sqrt{n}\\mathopen {}\\left( {\\phi \\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}-\\phi \\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}{\\sqrt{p^{\\prime }\\mathopen {}\\left( {1-p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\left|\\phi ^{\\prime }\\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\right|}} \\xrightarrow{}\\mathcal {N}\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} .$ We also have the following convergence in probability, $\\frac{\\sqrt{p^{\\prime }\\mathopen {}\\left( {1-p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}\\left|\\phi ^{\\prime }\\mathopen {}\\left( {{p^{\\prime }}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\right|}{\\sqrt{\\hat{p}^{\\prime }\\mathopen {}\\left( {1-\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}\\left|\\phi ^{\\prime }\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\right|} \\xrightarrow{}1 .$ Multiplying (REF ) and (REF ) by Slutsky's theorem, we get, $\\frac{\\sqrt{n}\\mathopen {}\\left( {\\phi \\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}-\\phi \\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}{\\sqrt{\\hat{p}^{\\prime }\\mathopen {}\\left( {1-\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\left|\\phi ^{\\prime }\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\right|}} \\xrightarrow{}\\mathcal {N}\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}.$ Hence, from (REF ), $\\mathbb {P}\\mathopen {}\\left( {\\phi \\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\in S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=1-\\beta .$ Next, because of the choice of $\\beta $ , we get the following equation: $\\mathbb {P}\\mathopen {}\\left( {p^{\\prime } \\in S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}, \\phi \\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\in S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\geqslant \\mathopen {}\\left( {1-\\beta }_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}^2=1-\\alpha .$ Let $m,M$ be defined as in (REF ).",
"Using asm-originalprop and the above, $\\nonumber \\begin{split}\\mathbb {P}\\mathopen {}\\left( {m \\leqslant \\eta \\mathopen {}\\left( {p_0,p^{\\prime },n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\leqslant M}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} & = \\mathbb {P}\\mathopen {}\\left( {m \\leqslant \\eta \\mathopen {}\\left( {\\phi \\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {},p^{\\prime },n}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\leqslant M}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\\\& \\geqslant \\mathbb {P}\\mathopen {}\\left( {p^{\\prime } \\in S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}, \\phi \\mathopen {}\\left( {p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\in S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\\\& \\geqslant 1-\\alpha .\\end{split}$ Thus, $(m,M)$ is a $100(1-\\alpha ) \\%$ confidence interval for $\\eta (p_0,p^{\\prime },n)$ and hence for $f(\\mu _2,\\Sigma _2,n)$ .",
"This completes the proof of the confidence interval part of the theorem.",
"To prove the consistency of the test, consider the following definitions of $\\Theta _0^n$ and $\\Theta _1^n$ : $\\Theta _0^n=\\left\\lbrace p_0,p^{\\prime }|\\eta (p_0,p^{\\prime },n) \\geqslant \\tau _c \\right\\rbrace , \\qquad \\Theta _1^n=\\left\\lbrace p_0,p^{\\prime }|\\eta (p_0,p^{\\prime },n) < \\tau _c \\right\\rbrace .$ The region $\\Theta _0^n$ corresponds to the null hypothesis $H_0^n$ and the region $\\Theta _1^n$ corresponds to the alternate hypothesis $H_1^n$ .",
"Electoral intervention is termed as “significant” if there is a high probability $\\mathopen {}\\left( {>\\tau _c}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}$ of the majority being changed on performing the intervention.",
"We know that if any value lies outside the confidence interval mentioned in thm:confidence-interval, that value is rejected at the level of significance $\\alpha $ .",
"Keeping this in mind, we define our test statistic to be $M$ , as defined in the confidence interval part of thm:confidence-interval.",
"We shall reject $H_0^n$ if $M < \\tau _c$ .",
"Let us calculate the type-1 error of this test.",
"Suppose, $\\mathopen {}\\left( {p_0,p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\in \\Theta _0^n$ i.e.",
"$\\eta (p_0,p^{\\prime },n) \\geqslant \\tau _c$ .",
"Hence, we have, $\\mathbb {P}_{p_0,p^{\\prime }}\\mathopen {}\\left( {H_0^n\\text{ is rejected}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=\\mathbb {P}_{p_0,p^{\\prime }}\\mathopen {}\\left( {M < \\tau _c}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=\\mathbb {P}_{p_0,p^{\\prime }}\\mathopen {}\\left( {\\eta (p_0,p^{\\prime },n) \\notin \\mathopen {}\\left( {m,M}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}\\leqslant \\alpha .$ Since this is true for all $(p_0,p^{\\prime }) \\in \\Theta _0^n$ , we can say the following: $\\sup _{\\mathopen {}\\left( {p_0,p^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\in \\Theta _0^n}\\mathbb {P}_{p_0,p^{\\prime }}\\mathopen {}\\left( {H_0^n\\text{ is rejected}}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\leqslant \\alpha $ Thus, we have shown that the maximum type-1 error is bounded by $\\alpha $ .",
"Note that the theoretical probability is given by $\\mathbb {P}(m(\\mathbf {X})=m(\\mathbf {X^{\\prime }}))$ = $f(\\mu _2,\\Sigma _2,n)$ , and as $n \\rightarrow \\infty $ , $f(\\mu _2,\\Sigma _2,n) \\rightarrow f(\\mu _2,\\Sigma _2,\\infty )$ .",
"Now, $f(\\mu _2,\\Sigma _2,\\infty )=\\mathbb {P}(\\mathrm {sgn}(P)=\\mathrm {sgn}(Q))$ where $(P,Q) \\sim \\mathcal {N}_2(\\mu _2,0)$ .",
"In other words, $P=\\mu _{2,1}$ and $Q=\\mu _{2,2}$ almost surely.",
"Since $\\mu _2=(2p_0-1,2p^{\\prime }-1)$ , $P=2p_0-1$ and $Q=2p^{\\prime }-1$ almost surely.",
"Thus, we conclude that, $\\eta (p_0,p^{\\prime },n) \\rightarrow 1$ as $n \\rightarrow \\infty $ if $(2p_0-1)(2p^{\\prime }-1)>0$ and $\\eta (p_0,p^{\\prime },n) \\rightarrow 0$ as $n \\rightarrow \\infty $ if $(2p_0-1)(2p^{\\prime }-1)<0$ .",
"It implies that $\\Theta _0^\\infty =\\left\\lbrace p_0,p^{\\prime }|(2p_0-1)(2p^{\\prime }-1)>0\\right\\rbrace , \\; \\Theta _1^\\infty =\\left\\lbrace p_0,p^{\\prime }|(2p_0-1)(2p^{\\prime }-1)<0\\right\\rbrace .$ We have already discussed that as $n\\rightarrow \\infty $ , $\\hat{p}^{\\prime } \\xrightarrow{}p^{\\prime }$ and $\\phi (\\hat{p}^{\\prime }) \\xrightarrow{}p$ .",
"Hence, as $n\\rightarrow \\infty $ , $S_1\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\rightarrow \\left\\lbrace p^{\\prime }\\right\\rbrace , \\;S_2\\mathopen {}\\left( {\\hat{p}^{\\prime }}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} \\rightarrow \\left\\lbrace p_0\\right\\rbrace .$ Thus, in this case, $m=M=\\eta (p_0,p^{\\prime },\\infty )$ .",
"Under $\\Theta _0^\\infty $ , $(2p_0-1)(2p^{\\prime }-1)>0$ and hence, $M=1$ .",
"Under $\\Theta _1^\\infty $ , $(2p_0-1)(2p^{\\prime }-1)<0$ and hence, $M=0 \\leqslant \\theta $ .",
"Clearly, under $\\Theta _1^\\infty $ , the test rejects $\\Theta _0^\\infty $ with probability 1, and that proves the consistency of the test.",
"[Proof of thm:exit poll theorem] For large $n$ , $k\\hat{p}_k \\sim \\mathrm {Binomial}(k,p_0)$ .",
"A straightforward application of central limit theorem suggests $\\frac{\\sqrt{n}\\mathopen {}\\left( {\\hat{p}_k-p_0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}{\\sqrt{p_0\\mathopen {}\\left( {1-p_0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}} \\xrightarrow{}\\mathcal {N}\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} .$ From the WLLN, we know that $\\hat{p}_k \\xrightarrow{}p_0$ as $k\\rightarrow \\infty $ .",
"Using Slutsky's theorem, $\\frac{\\sqrt{n}\\mathopen {}\\left( {\\hat{p}_k-p_0}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}{\\sqrt{\\hat{p}_k\\mathopen {}\\left( {1-\\hat{p}_k}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}}} \\xrightarrow{}\\mathcal {N}\\mathopen {}\\left( {0,1}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {} .$ Subsequently, we have the following: $\\mathbb {P}\\mathopen {}\\left( {p_0 \\in C_2({\\hat{p}_k})}_{{}_{}}\\,\\hspace{0.0pt}\\right)\\mathclose {}=1-\\beta .$ Thereafter, one can mimic the steps as in sec:final data available to show that $\\mathbb {P}_{H_0}({T}_1<\\tau _c) \\leqslant \\alpha $ .",
"Next, for showing that the test is consistent, note that $k \\rightarrow n,\\; n \\rightarrow \\infty $ implies that $k \\rightarrow \\infty $ and $\\hat{p}_k\\xrightarrow{}p_0$ , and in that case, from the WLLN, $C_2(\\hat{p}_k) \\rightarrow \\left\\lbrace p_0\\right\\rbrace $ .",
"Then, one can adopt identical steps as in the proof of the consistency part in thm:confidence-interval and obtain the required result."
]
] | 2210.07744 |
[
[
"PrivMVMF: Privacy-Preserving Multi-View Matrix Factorization for\n Recommender Systems"
],
[
"Abstract With an increasing focus on data privacy, there have been pilot studies on recommender systems in a federated learning (FL) framework, where multiple parties collaboratively train a model without sharing their data.",
"Most of these studies assume that the conventional FL framework can fully protect user privacy.",
"However, there are serious privacy risks in matrix factorization in federated recommender systems based on our study.",
"This paper first provides a rigorous theoretical analysis of the server reconstruction attack in four scenarios in federated recommender systems, followed by comprehensive experiments.",
"The empirical results demonstrate that the FL server could infer users' information with accuracy >80% based on the uploaded gradients from FL nodes.",
"The robustness analysis suggests that our reconstruction attack analysis outperforms the random guess by >30% under Laplace noises with b no larger than 0.5 for all scenarios.",
"Then, the paper proposes a new privacy-preserving framework based on homomorphic encryption, Privacy-Preserving Multi-View Matrix Factorization (PrivMVMF), to enhance user data privacy protection in federated recommender systems.",
"The proposed PrivMVMF is successfully implemented and tested thoroughly with the MovieLens dataset."
],
[
"Introduction",
"The recommendation system relies on collecting users' personal information, such as purchase history, explicit feedback, social relationship, and so on.",
"Recently, some laws and regulations have been enacted to protect user privacy, which places constraints on the collection and exchange of users' personal data.",
"To protect user privacy, one way is to develop a recommendation system in federated learning (FL) framework that enables the clients to jointly train a model without sharing their data.",
"In the FL setting, each client computes the updated gradient locally and sends the model update instead of the original data to a central server.",
"The server then aggregates the gradients and updates the global model [1].",
"Collaborative filtering (CF) is one of the most effective approaches in recommendation systems [2], and matrix factorization (MF) is a popular technique in CF algorithms.",
"MF decomposes a user-item interaction matrix into two low-rank matrices: user latent factors and item latent factors, which are used to generate the preference prediction [3].",
"One disadvantage of MF-based recommendation is the cold-start problem: if an item or user has no rating information, the model cannot generate a latent factor representation for it and thus suffers difficulty in performing MF recommendation.",
"A solution to the cold-start issue is to incorporate side information, i.e., user and item attributes, into matrix factorization.",
"Various approaches have been proposed for centralized recommender systems [4], [5], [6], [7].",
"However, few studies have researched the topic in the federated setting.",
"To the best of our knowledge, Flanagan et al.",
"[8] is the first to propose a federated multi-view matrix factorization (MVMF) to address this problem.",
"However, this method assumed that the conventional FL framework could fully protect user privacy.",
"However, severe privacy risks exist in the federated MVMF recommender system, which is susceptible to server reconstruction attacks, i.e., the attack to recover users' sensitive information.",
"To fill this gap, this paper first provides a theoretical analysis of the privacy threat of the federated MVMF method.",
"In theoretical analysis, we develop server reconstruction attacks in four scenarios based on different treatments on unobserved ratings and methods to update user latent factors.",
"The empirical study results indicate that the original federated MVMF method could leak users' personal information.",
"Then, we design a privacy-preserving federated MVMF framework using homomorphic encryption (HE) to enhance the user data privacy protection in federated recommender systems.",
"The main contributions of this paper are twofold: (1) To the best of our knowledge, we are the first to provide a rigorous theoretical analysis of server reconstruction attacks in the federated MVMF recommender system.",
"We also conducted comprehensive experiments, which show that the server could infer users' sensitive information with accuracy $>80\\%$ using such attacks, and the attack is effective under a small amount of noise.",
"(2) To overcome the information leakage problem, we propose PrivMVMF, a privacy-preserving federated MVMF framework enhanced with HE.",
"The proposed framework has two advantages: a) To balance the tradeoff between efficiency and privacy protection, it adopts a strategy in which some unrated items are randomly sampled and assigned a weight on their gradients.",
"b) To reduce complexity, it allocates some decrypting clients to decrypt and transmit the aggregated gradients to the server.",
"A prototype of PrivMVMF is implemented and tested on the movielens dataset."
],
[
"Literature Review",
"Federated Matrix Factorization: Federated recommender systems enable parties to collaboratively train the model without putting all data on a centralized server.",
"Several federation methods for recommender systems have been introduced in recent works.",
"Ammad-ud-din et al.",
"[9] proposed a federated matrix factorization method for implicit feedback.",
"Each client updates the user latent factor locally and sends back item latent factor gradient to the server for aggregation and update.",
"Duriakova et al.",
"[10] presented a decentralized approach to matrix factorization without a central server, where each user exchanges the gradients with their neighbors.",
"Lin et al.",
"[11] provided a matrix factorization framework based on federated meta learning by generating private item embedding and rating prediction model.",
"The above works haven't considered cold-start recommendation.",
"To address the problem, Flanagan et al.",
"[8] devised a federated multi-view matrix factorization based on the implicit feedback (e.g., clicks), where three matrices are factorized simultaneously with sharing latent factors.",
"Cryptographic Techniques in Federated Recommender System: Some studies used encryption schemes to develop privacy-preserving recommendation systems.",
"Chai et al.",
"[12] introduced FedMF, a secure federated matrix factorization framework.",
"To increase security, each client can encrypt the gradient uploaded to the server with HE.",
"Shmueli et al.",
"[13] proposed multi-party protocols for item-based collaborative filtering of vertical distribution settings.",
"In the online phase, the parties communicate only with a mediator that performs computation on encrypted data, which reduces communication costs and allows each party to make recommendations independent of other parties.",
"Although both [12] and our paper adopt HE to enhance the security, our work extends the method by introducing decrypting clients and sampling of unrated items.",
"The decrypting clients improve the efficiency to perform parameters updates, and the unrated items sampling strikes a balance between efficiency and privacy protection.",
"To the best of our knowledge, Flanagan et al.",
"[8] is the first to devise a federated multi-view matrix factorization to address the cold-start problem, where the users directly upload the plaintext gradients to the server, and no work has considered the information leakage from the gradients.",
"This paper first demonstrates the feasibility of server reconstruction attack, and then proposes a framework to enhance privacy protection.",
"The study is conducted based on the assumption of honest clients and an honest-but-curious server [14]."
],
[
"Federated MVMF",
"The federated MVMF proposed by Flanagan et al.",
"[8] is based on implicit feedback.",
"In this section, we extend the framework to explicit feedback."
],
[
"Notations",
"Table REF lists the notations and their descriptions used throughout this paper.",
"Table: Notations Used in the Paper"
],
[
"Multi-view Matrix Factorization",
"Multi-view matrix factorization is performed on the three data sources: the rating matrix $R_{n\\times m}$ , the user attribute matrix $X_{n\\times l_x}$ , and the item content matrix $ Y_{m\\times l_y}$ , for $n$ users with $l_x$ features, and m items with $l_y$ features.",
"The decomposition of the three matrices is given as: $R\\approx PQ^T,X \\approx PU^T,Y\\approx QV^T$ where ${P=P}_{n\\times K}$ , ${Q=Q}_{m\\times K}$ , ${U=U}_{l_x\\times K}$ , ${V=V}_{l_y\\times K}$ with $K$ representing the number of latent factors.",
"For $P$ and $Q$ , each row represents the latent factors for each user and item respectively.",
"For $U$ and $V$ , each row represents the latent factors for each feature of user and item respectively.",
"The predicted rating of user u on item i is given as: ${\\hat{r}}_{u,i}=p_u^Tq_i$ The latent factor representation is learned by minimizing the following cost function: ${\\begin{array}{c}J=\\sum _{i}\\sum _{j}{c_{i,j}(r_{i,j}-p_iq_j^T)^2}\\\\+\\lambda _1(\\sum _{i}\\sum _{d_u}(x_{i,d_u}-p_iu_{d_u}^T)^2+\\sum _{j}\\sum _{d_y}(y_{j,d_y}-q_jv_{d_y}^T)^2)\\\\+\\lambda _2(\\sum _{i}||p_i||^2+\\sum _{j}||q_j||^2+\\sum _{d_u}||u_{d_u}||^2+\\sum _{d_y}||v_{d_y}||^2)\\end{array}}$ where $\\lambda _1$ is used to adjust how much information the model should learn from side data, and $\\lambda _2$ is a regularization term to prevent overfitting.",
"$r_{i,j}=0$ if the rating is unobserved, and $r_{i,j}>0$ otherwise.",
"$c_{i,j}$ could be treated as a weight on the error term for each rating record.",
"This paper considers two definitions of $c_{i,j}$ : ObsOnly: $c_{i,j}=1$ if $r_{i,j}>0$ , and $c_{i,j}=0$ if $r_{i,j}=0$ .",
"Then the loss function only minimize the square error on the observed ratings.",
"InclUnc: $c_{i,j}=1$ if $r_{i,j}>0$ , and $c_{i,j}=\\alpha $ if $r_{i,j}=0$ , where $0<\\alpha <1$ is an uncertainty coefficient on the unobserved ratings.",
"This case assigns a lower weight on the loss for unobserved ratings.",
"The matrix factorization for explicit feedback typically employs the first definition to reduce the bias of unobserved interaction and improve efficiency.",
"However, employing the second definition reveals less information to the FL server.",
"Furthermore, as is shown in section and , adopting the second definition would present a challenge for the server attack.",
"Therefore, we will consider both cases when designing the server attack."
],
[
"Federated Implementation",
"The federated setting consists of three parties: clients, FL server, and item server.",
"Each client holds their ratings and attributes locally and performs local update of $P$ .",
"FL server receives the gradients from clients and item server, and updates $U$ and $Q$ .",
"Item server is introduced to facilitate the training process.",
"It stores the item features and conducts update of $V$ .",
"The following explains the details in the updates of each latent factor matrix.",
"User feature latent factor $U$ is updated on the FL server with the formula as: $u_{d_u}^t=u_{d_u}^{t-1}-\\gamma \\frac{\\partial J}{\\partial u_{d_u}}$ where: ${\\begin{array}{c}\\frac{\\partial J}{\\partial u_{d_u}}=-2\\sum _{i}f(i,d_u)+2\\lambda _2u_{d_u}\\end{array}}$ where $f(i,d_u)=\\left(x_{i,d_u}-p_iu_{d_u}^T\\right)p_i$ is computed on each user locally.",
"Item latent factor $Q$ is updated on the FL server with the formula as: $q_j^t=q_j^{t-1}-\\gamma \\frac{\\partial J}{\\partial q_j}$ where: ${\\begin{array}{c}\\frac{\\partial J}{\\partial q_j}=-2\\sum _{i}f(i,j)-2\\lambda _1\\sum _{d_y}f(j,d_y)+2\\lambda _2q_j\\end{array}}$ where $f(j,d_y)=(y_{j,d_y}-v_{d_y}q_j^T)v_{d_y}$ is computed on the item server, and $f(i,j)=c_{ij}(r_{i,j}-p_iq_j^T)p_i$ is computed on each user locally.",
"Noted that for ObsOnly, if $c_{ij}=0$ , the user only computes and sends the gradients of items with $c_{ij}>0$ , i.e., the rated items.",
"For InclUnc, the gradients for all items will be sent to the server.",
"Both user latent factor $P$ and item feature latent factor $V$ adopt two updating methods: Semi-Alternating Least Squares (SemiALS): Optimal $P$ and $V$ are computed using closed form formula under fixed $U$ and $Q$ .",
"Other parameters are updated using gradient descent method.",
"Stochastic Gradient Descent (SGD): All of the parameters are updated using gradient descent method.",
"The time complexity for SemiALS is $O(mK^2+K^3)$ per iteration, higher than that of SGD.",
"However, SGD requires more iterations to achieve the optimum.",
"[15] User latent factor $P$ is updated on each client locally.",
"For SemiALS, it's updated with the formula as: $p_i^\\ast =(r_iC^{(i)}Q+\\lambda _1x_iU)(Q^TC^{(i)}Q+\\lambda _1U^TU+\\lambda _2I)^{-1}$ where $C^{(i)}$ is a $m\\times m$ diagnal matrix with $C_{jj}^{(i)}=c_{i,j}$ .",
"For SGD, it's updated with the formula as: $p_i^t=p_i^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_i}$ where: ${\\begin{array}{c}\\frac{\\partial J}{\\partial p_i}=-2\\sum _{i}{c_{ij}(r_{i,j}-p_iq_j^T)q_j}\\\\-2\\lambda _1\\sum _{d_u}{(x_{i,d_u}-p_iu_{d_u}^T)u_{d_u}}+2\\lambda _2p_i\\end{array}}$ Item feature latent factor $V$ is updated on the item server.",
"For SemiALS, it's updated with the formula as: $v_{d_y}^\\ast =(y_{d_y}Q)(Q^TQ+\\frac{\\lambda _2}{\\lambda _1}I)^{-1}$ For SGD, it's updated with the formula as: $v_{d_y}^t=v_{d_y}^{t-1}-\\gamma \\frac{\\partial J}{\\partial v_{d_y}}$ where: ${\\begin{array}{c}\\frac{\\partial J}{\\partial v_{d_y}}=-2\\sum _{i}{(y_{i,d_v}-q_jv_{d_v}^T)q_j}+2\\lambda _2v_{d_v}\\end{array}}$ Algorithm REF outlines the federated implementation of MVMF (FedMVMF).",
"The gradient descent of U and Q are performed using Adaptive Moment Estimation (Adam) method to stabilize the convergence.",
"[htp] FedMVMF FL Server: Initialize $U$ and $Q$ .",
"t = 1 to T Receive and aggregate $f(i,j)$ and $f(i,d_u)$ from user $i$ for $i\\in [1,\\ n]$ .",
"Receive $f(j,d_v)$ from item server.",
"Update $U$ using equation (REF ).",
"Update $Q$ using equation (REF ).",
"Item Server: True Receive $Q$ from FL server.",
"Compute local $V$ using equation (REF ).",
"Compute item latent factor gradients $f(j,d_v)$ .",
"Transmit gradients to server.",
"Client: True Receive $U$ and $Q$ from server.",
"Compute local $p_i$ using equation (REF ).",
"Compute $U$ gradients $f(i,d_u)$ for $d_u\\in [1,l_x]$ .",
"Compute $Q$ gradients $f(i,j)$ for $j\\in [1,m]$ .",
"Transmit gradients to server."
],
[
"Cold-start recommendation",
"The recommendation for new users and items is discussed as followed.",
"Cold-start user recommendation: for any new user $i$ , the system first generates the user latent factor $p_i$ based on the user’s attribute $x_i$ and the user feature latent factor matrix $U$ .",
"Then the predicted rating of user $i$ on item $j$ is given by the inner product of $p_i$ and $q_i$ .",
"$p_i$ is calculated by minimizing the loss function: $J=\\lambda _1\\sum _{d_u}(x_{i,d_u}-p_iu_{d_u}^T)^2+\\lambda _2\\sum _{d_u}||u_{d_u}||^2$ The optimal solution of $p_i$ is defined as: $p_i^\\ast =x_iU(U^TU+\\frac{\\lambda _2}{\\lambda _1}I)^{-1}$ Cold-start item recommendation: given a new item $j$ , the system first generates the user latent factor $q_j$ based on the item’s feature $y_j$ and the item feature latent factor matrix $V$ .",
"The estimated $q_i$ is then used to compute the predicted rating.",
"$q_j$ is calculated by minimizing the loss function: $J=\\lambda _1\\sum _{d_y}(y_{j,d_y}-q_jv_{d_y}^T)^2+\\lambda _2\\sum _{d_y}||v_{d_y}||^2$ The optimal solution of $q_j$ is defined as: $q_j^\\ast =y_jV(V^TV+\\frac{\\lambda _2}{\\lambda _1}I)^{-1}$ In FedMVMF, the FL server could reconstruct the user ratings and attributes based on the gradients they received.",
"In this section, we consider the attacks for both SemiALS and SGD updates on user latent factor.",
"Within each case, the attacks are slightly different between ObsOnly and InclUnc.",
"The analysis is based on the assumption of honest clients and an honest-but-curious server."
],
[
"Reconstruction Attack for ",
"For SemiALS the FL server is able to recover the user information within only one epoch given that the server has access to $U$ and $Q$ .",
"Attack for ObsOnly: In this case, the clients only upload the gradients for items with observed ratings.",
"Therefore, for any user $i$ , the gradients which the FL server receives is given by: $ {\\begin{array}{c}f(i,j)=(r_{i,j}-p_iq_j^T)p_i, j \\in O_i\\\\f(i,d_u)=(x_{i,d_u}-p_iu_{d_u}^T)p_i, d_u \\in [1, l_x]\\\\\\end{array}}$ where $f(i,j)$ and $f(i,d_u)$ denote the vector of gradient with length $K$ , $O_i$ denotes the collection of items rated by user $i$ , and $l_x$ denote the number of user attributes.",
"In SemiALS, $p_i$ is updated by equation (REF ).",
"Given that $c_{i,j}=0$ when $r_{i,j}=0$ , the formula could be reduced to: $p_i=(r_iQ_i+\\lambda _1x_iU)(Q_i^TQ_i+\\lambda _1U^TU+\\lambda _2I)^{-1}$ where $r_i$ is the vector of observed ratings, and $Q_i=Q_{|O_i|\\times K}$ is the latent factors for items rated by user $i$ .",
"Let $A_i = Q_i(Q_i^TQ_i+\\lambda _1U^TU+\\lambda _2I)^{-1}$ , and $B_i=\\lambda _1U(Q_i^TQ_i+\\lambda _1U^TU+\\lambda _2I)^{-1}$ , both of which could be computed on the FL server.",
"Then $p_i$ could be written as $p_i=r_iA_i + x_i B_i$ .",
"Plugging into equation (REF ), we have: $\\left\\lbrace \\begin{array}{l}{\\begin{array}{c}f^Q_i=r_i^Tr_iA_i+ r_i^Tx_iB_i- Q_i(A_i^TR_rA_i\\\\+A_i^TR_xB_i+B_i^TX_rA_i+B_i^TX_xB_i)\\\\f^U_i=x^T_ir_iA_i+ x^T_ix_iB_i- U(A_i^TR_rA_i\\\\+A_i^TR_xB_i+B_i^TX_rA_i+B_i^TX_xB_i)\\\\\\end{array}}\\end{array}\\right.$ where $f^Q_i={f^Q_i}_{|O_i|\\times K}$ with $j^{th}$ row being $f(i,j)$ , $f^U_i={f^U_i}_{l_x\\times K}$ with $d_u^{th}$ row being $f(i,d_u)$ , and: ${\\begin{array}{c}R_r=r_i^Tr_i, R_x=r_i^Tx_i, \\\\X_r=x^T_ir_i, X_x=x^T_ix_i\\end{array}}$ Then the FL server obtain a second order non-linear system with $(l_x+|O_i|)\\times K$ equations, consisting of $(l_x+|O_i|)$ variables, $r_i$ and $x_i$ .",
"Therefore, it's plausible to find the solution of user ratings $r_i$ and user attributes $x_i$ using methods such as Newton-Raphson algorithm.",
"To reconcile the number of equations and variables, we choose a random factor $n \\in [1,K]$ , and solve the equation systems under the fixed $n$ .",
"Attack for InclUnc: In this case the client sends gradients of all items to the FL server, multiplied by a uncertainty coefficient $c_{i,j}$ .",
"For any user $i$ , the gradients the FL server is given by: $ {\\begin{array}{c}f(i,j)=c_{i,j}(r_{i,j}-p_iq_j^T)p_i, j \\in [1, m]\\\\f(i,d_u)=(x_{i,d_u}-p_iu_{d_u}^T)p_i, d_u \\in [1, l_x]\\\\\\end{array}}$ Let $A_i^{\\prime } = C^{(i)}Q(Q^TC^{(i)}Q+\\lambda _1U^TU+\\lambda _2I)^{-1}$ , and $B_i^{\\prime }=\\lambda _1U(Q^TC^{(i)}Q+\\lambda _1U^TU+\\lambda _2I)^{-1}$ .",
"Then $p_i$ can be written as: ${\\begin{array}{c}p_i = r_iA_i^{\\prime }+x_iB_i^{\\prime }\\end{array}}$ Plugging into equation (REF ), we can obtain the final equation system: $\\left\\lbrace \\begin{array}{l}{\\begin{array}{c}f^Q_i=C^{(i)}\\bigl (r_i^Tr_iA_i^{\\prime }+ r_i^Tx_iB_i- Q((A_i^{\\prime })^TR_rA_i^{\\prime }\\\\+(A_i^{\\prime })^TR_xB_i^{\\prime }+(B_i^{\\prime })^TX_rA_i^{\\prime }+(B_i^{\\prime })^TX_xB_i^{\\prime })\\bigr )\\\\f^U_i=x^T_ir_iA_i^{\\prime }+ x^T_ix_iB_i^{\\prime }- U((A_i^{\\prime })^TR_rA_i^{\\prime }\\\\+(A_i^{\\prime })^TR_xB_i^{\\prime }+(B_i^{\\prime })^TX_rD_i+(B_i^{\\prime })^TX_xB_i^{\\prime })\\\\\\end{array}}\\end{array}\\right.$ where $r_i$ is user $i$ 's ratings for all items, and $A_i^{\\prime } = C^{(i)}Q(Q^TC^{(i)}Q+\\lambda _1U^TU+\\lambda _2I)^{-1}$ and $B_i^{\\prime }=\\lambda _1U(Q^TC^{(i)}Q+\\lambda _1U^TU+\\lambda _2I)^{-1}$ are dependent on $r_i$ .",
"Since $C^{(i)}$ is a function of $r_i$ , the system consists of $l_x+m$ variables and $(l_x+m)\\times K$ equations.",
"Therefore, it's possible to recover the user information by solving the equation system.",
"Similarly, a random factor $n \\in [1,K]$ is fixed to align the number of equations and variables."
],
[
"Reconstruction Attack for ",
"For SGD, the FL server is able to recover the user information within only two epochs given that the server has access to $U$ and $Q$ .",
"Attack for ObsOnly: After two epochs, the gradients FL server receives from users $i$ is given by: $ {\\begin{array}{c}f^t(i,j)=(r_{i,j}-p^t_i{(q_j^t)}^{\\prime })p^t_i, j \\in O_i\\\\f^{t-1}(i,j)=(r_{i,j}-p^{t-1}_i{(q_j^{t-1})}^{\\prime })p^{t-1}_i, j \\in O_i\\\\f^t(i,d_u)=(x_{i,d_u}-p^t_i{(u^t_{d_u})}^{\\prime })p^t_i, d_u \\in [1, l_x]\\\\f^{t-1}(i,d_u)=(x_{i,d_u}-p^{t-1}_i{(u^{t-1}_{d_u})}^{\\prime })p^{t-1}_i, d_u \\in [1, l_x]\\\\\\end{array}}$ In pure SGD, the user latent factor is updated using equation (REF ) and (REF ).",
"Plugging into the first gradient of equation (REF ), we have: ${\\begin{array}{c}f_n^t(i,j)=\\bigl (r_{ij}-(p_i^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_i^{t-1}})(q_j^{t-1}+\\Delta q_j^t)^{\\prime }\\bigr )\\\\\\times (p_{in}^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_{in}^{t-1}})\\end{array}}$ where $\\Delta q_j^t=q_j^t-q_j^{t-1}$ , $f_n^t(i,j)$ denote the $n^{th}$ element of $f^t(i,j)$ , and $p_{in}^{t-1}$ denote the $n^{th}$ element of $p_i^{t-1}$ .",
"Equation (REF ) is a multiplication of two terms.",
"By looking at the first term, we have: ${\\begin{array}{c}r_{ij}-(p_i^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_i^{t-1}})(q_j^{t-1}+\\Delta q_j^t)^{\\prime }=r_{ij}\\\\-p_i^{t-1}{(q_j^{t-1})}^{\\prime }-p_i^{t-1}(\\Delta q_j^t)^{\\prime } + \\gamma \\frac{\\partial J}{\\partial p_{in}^{t-1}} (q_j^t)^{\\prime }\\\\=\\frac{G_n(j)}{p_{in}^{t-1}}+p_i^{t-1}g\\end{array}}$ where: ${\\begin{array}{c}G_n(j)=f_n^{t-1}(i,j)-2\\gamma (\\sum _{k\\in O_i}{f_n^{t-1}(i,k)q_k^{t-1}}\\\\+\\lambda _1\\sum _{d_u}{f_n^{t-1}(i,d_u)u_{d_u}^{t-1}}){(q_j^t)}^{\\prime }\\\\g=2\\gamma \\lambda _2{(q_j^t)}^{\\prime }-(\\Delta q_j^t)^{\\prime }\\end{array}}$ Then we look at the second term of equation (REF ), which is given by: ${\\begin{array}{c}p_{in}^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_{in}^{t-1}}=p_{in}^{t-1}(1-2\\gamma \\lambda _2)+\\frac{2\\gamma }{p_{in}^{t-1}}\\\\\\times [\\sum _{k\\in O_i}{f_n^{t-1}(i,k)q_{kn}^{t-1}}+\\lambda _1\\sum _{d_u}{f_n^{t-1}(i,d_u)u_{d_n}^{t-1}}]\\\\=p_{in}^{t-1}(1-2\\gamma \\lambda _2)+\\frac{F_n}{p_{in}^{t-1}}\\end{array}}$ where: ${\\begin{array}{c}F_n=2\\gamma [\\sum _{k\\in O_i}{f_n^{t-1}(i,k)q_{kn}^{t-1}}+\\\\\\lambda _1\\sum _{d_u}{f_n^{t-1}(i,d_u)u_{d_un}^{t-1}}]\\end{array}}$ Then equation (REF ) can be written as: ${\\begin{array}{c}f_n^t\\left(i,j\\right)=\\bigl (\\frac{G_n(j)}{p_{in}^{t-1}}+p_i^{t-1}g\\bigr )\\bigl (p_{in}^{t-1}(1-2\\gamma \\lambda _2)+\\frac{F_n}{p_{in}^{t-1}}\\bigr )\\end{array}}$ For $n \\in [1,K]$ , $j \\in O_i$ , where $p^{t-1}_i$ is the variable to solve.",
"Noted that $G_n(j)$ , $g$ , and $F_n$ could be computed on the FL server.",
"Since there are $K$ variables and $K\\times |O_i|$ equations, there should exist a solution $p^{t-1}_i$ satisfy the system (REF ).",
"To reconcile the number of equations and variables, we choose a random item $j \\in O_i$ , and solve the equation systems under the fixed $j$ .",
"After obtaining $p^{t-1}_i$ , the server could compute $r_{i,j}$ and $x_{i,d_u}$ as followed: ${\\begin{array}{c}r_{ij}=\\frac{f_n^{t-1}(i,j)}{p_{in}^{t-1}}+p_i^{t-1}{(q_j^{t-1})}^{\\prime }\\\\x_{id_u}=\\frac{f_n^{t-1}(i,d_u)}{p_{in}^{t-1}}+p_i^{t-1}{(u_{d_u}^{t-1})}^{\\prime }\\end{array}}$ Attack for InclUnc: Similarly, the FL server first obtain the equation system for $p^{t-1}_i$ given by: ${\\begin{array}{c}f_n(i,j)=\\bigl (\\frac{f_n^{t-1}(i,j)}{p_{in}^{t-1}}+ \\frac{c_{i,j}G_n^{\\prime }(j)}{p_{in}^{t-1}} + c_{i,j}p_i^{t-1}g^{\\prime } \\bigr )\\\\\\bigl (\\frac{F_n^{\\prime }}{p_{in}^{t-1}}+p_{in}^{t-1}(1-2\\gamma \\lambda _2)\\bigr ), n \\in [1,K], j \\in [1,m]\\end{array}}$ where: ${\\begin{array}{c}G_n^{\\prime }(j)=-2\\gamma (\\sum _k{f_n^{t-1}(i,k)q_k^{t-1}}\\\\+\\lambda _1\\sum _{d_u}{f_n^{t-1}(i,d_u)u_{d_u}^{t-1}}){(q_j^t)}^{\\prime }\\\\F_n^{\\prime } = 2\\gamma (\\sum _k f_n^{t-1}(i,k)q_{kn}^{t-1} +\\lambda _1\\sum _{d_u}f_n^{t-1}(i,d_u)u_{d_u,n}^{t-1}\\\\g^{\\prime }=2\\lambda _2 \\gamma p_{j}^{t-1}-\\Delta p_j^t\\end{array}}$ For detail derivation of equation (REF ) refer to appendix REF .",
"Noted that $c_{i,j}$ is a function of $r_{i,j}$ , which is dependent on $p^{t-1}_i$ based on equation (REF ).",
"Therefore, $c_{i,j}$ is linked with $p^{t-1}_i$ .",
"Given $K$ variables and $K\\times m$ equations, the server should be able to find a solution $p^{t-1}_i$ for the system.",
"Similarly, a random item $j \\in [1, m]$ is fixed when solving the equation system.",
"Then the rating and user attributes could be computed as: ${\\begin{array}{c}r_{i,j} = \\frac{f_n^{t-1}(i,j)p_i^t(q_j^t)^{\\prime }p_{in}^t-f_n^{t}(i,j)p_i^{t-1}(q_j^{t-1})^{\\prime }p_{in}^{t-1}}{f_n^{t-1}(i,j)p_{in}^t-f_n^{t}(i,j)p_{in}^{t-1}}\\\\x_{i,d_u}=\\frac{f_n^{t-1}(i,d_u)}{p_{in}^{t-1}}+p_i^{t-1}{(u_{d_u}^{t-1})}^{\\prime }\\end{array}}$ where $p_{in}^t$ and $p_i^t$ can be obtained from formula (REF ) and (REF )."
],
[
"Privacy-Preserving MVMF (PriMVMF)",
"To prevent information leakage, we develop PrivMVMF, a privacy-preserving federated MVMF framework enhanced with homomorphic encryption (HE).",
"In this framework, the client encrypts the gradients before sending them to the server, and the server can perform computation on the encoded gradients.",
"The above attacks are based on access to individual gradients, while in HE, these gradients are sent to the server in encrypted form, rendering the reconstruction attacks infeasible."
],
[
"Paillier Cryptosystem",
"This study utilized a partially HE scheme - Paillier cryptosystem [16], which consists of three parts: key generation, encryption, and decryption.",
"Key generation: Based on the $keysize$ , $(sk,pk)=Gen(keysize)$ returns the public key $pk$ shared among all participants, and secret key $sk$ distributed only among the clients.",
"Before the training process, one of the users generates a key pair.",
"Encryption: $c=Enc(m,pk)$ encrypts message $m$ to cyphertext $c$ using public key $pk$ .",
"Decryption: $m=Dec(c,sk)$ reverses cyphertext $c$ to message $m$ using secret key $sk$ .",
"Given two plaintexts $m_1$ and $m_2$ , Paillier cryptosystem $E$ has the following properties: Addition: $E(m_1)\\cdot E(m_2)=E(m_1+m_2)$ .",
"Multiplication: ${E(m_1)}^{m_2}=E(m_1\\cdot m_2)$ Number Encoding Scheme: Paillier encryption is only defined for non-negative integer, but the recommendation system contains float and negative numbers.",
"The study follows Chai et al.’s method to convert floating points and negative numbers into unsigned integer [12].",
"Sampling of Unrated Item: For the treatment of unrated item, this framework strikes a balance between efficiency and privacy protection.",
"The ObsOnly method is efficient while it reveals what items has been rated by the user.",
"The InclUnc method leaks no information but is computation intensive.",
"To reconcile the two objectives, we design a strategy to randomly sample a portion of unrated items.",
"Then the $c_{i,j}$ is given as followed: $c_{i,j}=\\left\\lbrace \\begin{array}{ll}1, & r_{i,j}>0\\\\\\alpha , & r_{i,j}=0\\ and\\ samp_{i,j}=1\\\\0, & r_{i,j}=0\\ and\\ samp_{i,j}=0\\end{array}\\right.$ where $0<\\alpha <1$ , ${samp}_{i,j}=1$ if item $j$ appears in the sampled unrated items for user $i$ , and ${samp}_{i,j}=0$ otherwise.",
"Users only send the gradients with $c_{i,j}>0$ .",
"For each user, we determine the number of sampled unrated items as a multiple of his rated items, denoted by $\\rho $ .",
"Then the upper-bound probability that the FL server could correctly infer whether a given item is rated by the user is given by $\\frac{1}{\\rho +1}$ .",
"Decrypting Clients: It's time-consuming to perform the update using the encrypted gradients.",
"To reduce complexity, the server sends the aggregated gradient to some decrypting users for decryption, and uses the plaintext aggregated gradients to update the parameters.",
"Algorithms: The detailed steps of PrivMVM are shown in Algorithm REF .",
"Noted that for the update of user latent factor $P$ and item feature latent factor $V$ , we adopt the SemiALS strategy for the following reason: although SemiALS has higher time complexity per iteration, it requires fewer iterations to achieve the optimum and thus fewer encryption and decryption operations, the bottleneck of the HE scheme.",
"Privacy Analysis: The privacy of the algorithm is analyzed in terms of information leakage, which is characterized into two forms: i) original information, the observed user data , and ii) latent information, properties of user data [17].",
"We assume an honest-but-curious server for the analysis, i.e., the server will not deviate from the defined protocol but attempt to learn information from legitimately received messages.",
"During the training of PrivMVMF, the individual gradients are sent to the server in the encrypted form, and only the plaintext aggregated gradients are available to the server.",
"The following shows that given the aggregated gradients, it leaks trivial original information about user data to the server.",
"Let $f(j)$ , $f(d_u)$ be the aggregated gradients for item $j$ and user feature $d_u$ , given by: ${\\begin{array}{c}f(j)=\\sum _i f(i,j) = \\sum _{i:j\\in O_i^{\\prime }} c_{i,j} (r_{ij}-p_iq_j^T)p_i, j\\in [1,m]\\\\f(d_u)=\\sum _i f(i,d_u) = \\sum _i (x_{i,d_u}-p_iu_{d_u}^T)p_i, d_u\\in [1,l_x]\\end{array}}$ where $O_i^{\\prime }$ denotes the set of items rated by or appeared in the sampled unrated items for user $i$ .",
"In PrivMVMF, $p_i$ is updated by: $p_i = (r_i C^{(i)}Q_i^{\\prime } + \\lambda _1 x_i U)((Q_i^{\\prime })^TC^{(i)}Q_i^{\\prime }+\\lambda _1U^TU+\\lambda _2 I)^{-1}$ where $Q_i^{\\prime }=Q_{|O_i^{\\prime }\\times K|}$ is the latent factors for items in $O_i^{\\prime }$ .",
"Let $A_i^{\\prime } = C^{(i)}Q_i^{\\prime }((Q_i^{\\prime })^TC^{(i)}Q_i^{\\prime }+\\lambda _1U^TU+\\lambda _2I)^{-1}$ , and $B_i^{\\prime }=\\lambda _1U((Q_i)^{\\prime T}C^{(i)}Q_i^{\\prime }+\\lambda _1U^TU+\\lambda _2I)^{-1}$ .",
"Then $p_i$ can be written as: ${\\begin{array}{c}p_i = r_iA_i^{\\prime }+x_iB_i^{\\prime }\\end{array}}$ Plugging into equation (REF ), we can obtain the equation system as followed: ${\\begin{array}{c}f(j) = \\sum _i c_{i,j}(r_{ij}r_iA_i^{\\prime }+r_{ij}x_iB_i^{\\prime } -q_j((A_i^{\\prime })^T R_r A_i^{\\prime } \\\\+(A_i^{\\prime })^TR_xB_i^{\\prime }+(B_i^{\\prime })^TX_rA_i^{\\prime }+(B_i^{\\prime })^TX_xB_i^{\\prime })), j\\in [1,m]\\\\f(d_u) = \\sum _i (x_{i,d_u}r_iA_i^{\\prime }+x_{i,d_u}x_iB_i^{\\prime } -u_{d_u}((A_i^{\\prime })^T R_r A_i^{\\prime } \\\\+ (A_i^{\\prime })^TR_xB_i^{\\prime }+(B_i^{\\prime })^TX_rA_i^{\\prime }+(B_i^{\\prime })^TX_xB_i^{\\prime })), d_u\\in [1,l_x]\\\\\\end{array}}$ where: ${\\begin{array}{c}R_r=r_i^Tr_i, R_x=r_i^Tx_i, X_r=x^T_ir_i, X_x=x^T_ix_i\\end{array}}$ The non-linear system consists of $(m+l_x)\\times K$ equations and $\\sum _iO_i^{\\prime }+nl_x$ variables.",
"When $\\sum _iO_i^{\\prime }+nl_x>>(m+l_x)\\times K$ , i.e., the user size is large enough, it's hard for the server to derive the original information of users.",
"[htp] PrivMVM Ramdomly select some clients as decrypters FL Server: Initialize $U$ and $Q$ .",
"t = 1 to T Receive and aggregate encrypted $f(i,j)$ and $f(i,d_u)$ from user $i$ for $i\\in [1,\\ n]$ .",
"Send encrypted $\\sum _i f(i,j)$ and $\\sum _i f(i,d_u)$ to decrypters.",
"Receive decrypted $\\sum _i f(i,j)$ and $\\sum _i f(i,d_u)$ from decrypters.",
"Receive $f(j,d_v)$ from item server.",
"Update $U$ using equation (REF ).",
"Update $Q$ using equation (REF ).",
"Item Server: True Receive $Q$ from FL server.",
"Compute local $V$ using equation (REF ).",
"Compute item latent factor gradients $f(j,d_v)$ .",
"Transmit gradients to server.",
"Client: True Receive $U$ and $Q$ from server.",
"Compute local $p_i$ using equation (REF ).",
"Compute $U$ gradients $f(i,d_u)$ for $d_u\\in [1,l_x]$ .",
"Compute $Q$ gradients $f(i,j)$ for $j\\in [1,m]$ .",
"Transmit gradients to server.",
"Decrypter: True Receive encoded $\\sum _i f(i,j)$ and $\\sum _i f(i,d_u)$ from FL server.",
"Decrypt and transmit $\\sum _i f(i,j)$ and $\\sum _i f(i,d_u)$ to FL server.",
"The experiment is performed on MovieLens-1M datasethttps://grouplens.org/datasets/movielens/1m/.",
"The dataset contains 914676 ratings from 6040 users on 3952 movies, with each user submitting at least 20 ratings.",
"The experiment is implemented on Ubuntu Linux 20.04 server with 32-core CPU and 128GB RAM, where the programming language is Python.",
"We construct the rating matrix based on the explicit ratings, where the missing values are set to zero.",
"The following user attributes are considered: Age, Gender, Occupation and Zipcode.",
"Age is discretized into seven groups with equal interval, and Zipcode is linked to the US region.",
"The movie features are described by the tag genome dataset containing $1,128$ tags for $9,734$ movies.",
"To reduce dimensionality, we take the first 20 principal components for the tags features.",
"We use Bayesian optimization [18] approach based on four-fold cross validation to optimize the hyperparameters.",
"Table REF summarizes the hyperparameters for the experiment.",
"Table: Hyperparameter for MVMF on Movielens Dataset"
],
[
"Server Attack",
"Solving Nonlinear System: To perform a server reconstruction attack, we first developed the equation systems described in section .",
"To solve the non-linear systems, we experiment with the four methods[19], [20], [21], [22]: modified Powell's hybrid method, Broyden’s bad method, Scalar Jacobian approximation, and Anderson mixing, and select the best method within each scenario using a sample of 100 users.",
"Refer to table REF for the selected method within each scenario.",
"Smoothing $c_{i,j}$ as a Function of $r_{i,j}$ for InclUnc: In InclUnc, $c_{i,j}=f(r_{i,j})$ is not a continuous function, while Jacobian matrix is needed for most of the iterative methods.",
"To smooth $f$ , we design the following function: $c_{i,j}=\\left\\lbrace \\begin{array}{ll}1, & r_{i,j}>1\\\\r_{i,j}, & 1 \\ge r_{i,j} > 0\\\\0, & 0 \\ge r_{i,j}\\end{array}\\right.$ Evaluation metrics: We employ accuracy to measure the performance of server inference, with steps as follows.",
"After obtaining the estimation of $\\hat{r}(i)$ and $\\hat{x}_i$ for each user $i$ , we clipped $\\hat{r}(i)$ within $[0, R_{max}]$ and $\\hat{x}_i$ within $[0, 1]$ ($x_i$ are all dummy variables), and then rounded the estimations to the nearest integers.",
"The accuracy for user ratings and attributes is computed as follows: ${\\begin{array}{c}Accuracy\\ for\\ Rating=\\frac{1}{|usr|} \\sum _i \\frac{|\\hat{r}_i=r_i|}{|r_i|}\\\\Accuracy\\ for\\ Attribute=\\frac{1}{|usr|} \\sum _i \\frac{|\\hat{x}_i=x_i|}{|x_i|}\\end{array}}$ where $usr$ denotes the set of all users, and $\\hat{r}_i$ and $\\hat{x}_i$ denote the transformed estimation of $r_i$ and $x_i$ .",
"Result and Analysis: Table REF reports the accuracy of server reconstruction attack in four scenarios, from which we can make the following observations: (1) In all cases, the server is able to recover the user's private information with accuracy $> 80\\%$ , which is a non-negligible privacy concern.",
"(2) For both SemiALS and SGD, including an uncertainty coefficient deteriorates the performance of server attack.",
"One explanation is that $c_{i,j}$ is not a differentiable function of $r_{i,j}$ , posing a challenge to obtaining the Jacobian matrix of the system.",
"(3) Using SGD method to update $P$ makes it harder for the server to infer user information given by the reduced accuracy.",
"Table: Accuracy of Server Reconstruction AttackRobustness Check: We consider the case when a small amount of noise is added to the gradients.",
"With perturbed gradients, the equation systems are solved using the following steps: For SemiALS (SGD with ObsOnly / SGD with InclUnc), compute the set of solutions for each $n\\in [1,K]$ ($j \\in O_i$ / $j$ from a randomly chosen set of items).",
"Clip the solutions to the correct domain.",
"Take the average of the solutions and round the estimations to integers.",
"We conduct the experiment under the four scenarios, where Laplace noises are added to the gradients with scale $b$ ranging from 0 to 2.",
"The performance is given by fig.",
"REF and fig.",
"REF .",
"Baseline refers to the theoretical accuracy under the random guessing strategy.",
"For both SemiALS and SGD, the reconstruction accuracy are stable and above the baseline by more than 50% under the InclUnc scenario.",
"Refer to fig.",
"REF in appendix for the accuracy with noise scale up to $10^4$ .",
"For ObsOnly, the rating accuracy approaches the baseline under both SemiALS and SGD when the noise scale increases to 2, and the user attribute accuracy is below the baseline under SemiALS when the noise scale exceeds 1.",
"In all scenarios, the attack accuracy is more than 30% above the random guessing baseline for noise scale less than $0.5$ .",
"The results reveal the following: (1) The reconstruction attack is effective under small amount of noises ($b\\le 0.5$ ).",
"(2) Although including an uncertainty coefficient hides the unrated ratings, it makes the attack more resilient to random noises.",
"Figure: Accuracy for ratings under four cases after adding Laplace noises from 0 to 2.Figure: Accuracy for user attributes under four cases after adding Laplace noises from 0 to 2."
],
[
"Evaluation metrics: The study adopted the following four evaluation metrics: Normalized Discounted Cumulative Gain at 10 (NDCG@10), Precision, Recall, and F1.",
"The accurate prediction is defined as an item recommended rated above a threshold by the given user[23].",
"Refer to appendix REF for the details of evaluation metrics.",
"Scenarios of Testing: Both approaches can provide recommendations for new users and items.",
"Table REF presents the performance of the three scenarios: Existing User and Item, Cold-start Item, and Cold-start User.",
"For Existing User and Item, the items of each user are randomly divided into 80% training set and 20% testing set.",
"The items and users in the testing set are supposed to have a rating history.",
"For Cold-start Item, the items are randomly divided into 90% training set and 10% testing set.",
"The testing items are treated as new items without a rating history.",
"For Cold-start User, a random subset of 10% users is held out as new users for testing.",
"Table: Test Accuracy of FedMVMF and PrivMVMFAccuracy: Table REF compares the testing accuracy between FedMVMF and PrivMVMF, with each approach running for 5 rounds.",
"Noted that the FedMVMF adopts the same sampling strategy as PrivMVMF for consistency, slightly different from that in section .",
"It can be observed that the difference in testing accuracy is trivial in all scenarios, suggesting that the proposed framework is lossless.",
"Efficiency: The model training in each stage can be divided into four phases: local update, aggregation, decryption, and server update.",
"The study evaluates the time consumption in these four phases respectively.",
"Local update: Clients compute the gradients and encryption them with the public key.",
"Aggregation: Server receives and aggregates the encrypted gradients from clients.",
"Decryption: Decrypting clients decrypts the aggregated gradients.",
"Server update: Server updates the latent factor matrix using decrypted aggregated gradients.",
"Table REF presents the computation time in each epoch.",
"It can be observed that the aggregation and decryption process take up most of the time.",
"It can improve efficiency if the decryption workload is distributed to several clients instead of only one user.",
"More work can be done to reduce the complexity of the operation of the encrypted gradient as well.",
"Table: Time Consumption in Each Phase (seconds)"
],
[
"Conclusions and future work",
"To understand the privacy risks in federated MVMF recommender systems, this paper provides a theoretical analysis of the server reconstruction attack in four scenarios.",
"It also proposes PrivMVMF, a privacy-preserving federated MVMF framework enhanced with HE, to overcome the information leakage problem.",
"Empirical studies on MovieLens-1M dataset show that: (1) In FedMVMF, the FL server could infer users' rating and attribute with accuracy $>80\\%$ using plaintext gradients.",
"(2) For OnbOnly, the reconstruction attack is effective under Laplace noise with $b\\le 0.5$ ; for InclUnc, it is effective with noise $b\\le 10$ .",
"(3) PrivMVMF can protect user privacy well compared wth FedMVMF.",
"(4) Aggregation and decryption process occupy most of the time in PrivMVMF.",
"Future work involves the following directions.",
"Firstly, communication time could be investigated in PrivMVMF framework.",
"Secondly, it's interesting to improve the efficiency of HE since it's time-consuming to perform the operation on the encrypted gradients."
],
[
"Attack for ",
"After two epochs, the gradients FL server receives from user $i$ is given by: $ {\\begin{array}{c}f^t(i,j)=c_{i,j}(r_{i,j}-p^t_i{(q_j^t)}^{\\prime })p^t_i, j \\in [1, m]\\\\f^{t-1}(i,j)=c_{i,j}(r_{i,j}-p^{t-1}_i{(q_j^{t-1})}^{\\prime })p^{t-1}_i, j \\in [1, m]\\\\f^t(i,d_u)=(x_{i,d_u}-p^t_i{(u^t_{d_u})}^{\\prime })p^t_i, d_u \\in [1, l_x]\\\\f^{t-1}(i,d_u)=(x_{i,d_u}-p^{t-1}_i{(u^{t-1}_{d_u})}^{\\prime })p^{t-1}_i, d_u \\in [1, l_x]\\\\\\end{array}}$ In SGD, the user latent factor is updated using equation (REF ) and (REF ).",
"Plugging into the first gradient of equation (REF ), we have: ${\\begin{array}{c}f_n^t(i,j)=c_{i,j}\\bigl (r_{ij}-(p_i^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_i^{t-1}})(q_j^{t-1}+\\Delta q_j^t)^{\\prime }\\bigr )\\\\\\times (p_{in}^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_{in}^{t-1}})\\end{array}}$ Equation (REF ) is a multiplication of two terms.",
"By looking at the first term, we have: ${\\begin{array}{c}c_{i,j}\\bigl (r_{ij}-(p_i^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_i^{t-1}})(q_j^{t-1}+\\Delta q_j^t)^{\\prime }\\bigr )\\\\=\\bigl (\\frac{f_n^{t-1}(i,j)}{p_{in}^{t-1}}+ \\frac{c_{i,j}G_n^{\\prime }(j)}{p_{in}^{t-1}} + c_{i,j}p_i^{t-1}g^{\\prime } \\bigr )\\end{array}}$ where: ${\\begin{array}{c}G_n^{\\prime }(j)=-2\\gamma (\\sum _k{f_n^{t-1}(i,k)q_k^{t-1}}\\\\+\\lambda _1\\sum _{d_u}{f_n^{t-1}(i,d_u)u_{d_u}^{t-1}}){(q_j^t)}^{\\prime }\\\\g^{\\prime }=2\\lambda _2 \\gamma p_{j}^{t-1}-\\Delta p_j^t\\end{array}}$ The second term of equation (REF ) is given by: ${\\begin{array}{c}p_{in}^{t-1}-\\gamma \\frac{\\partial J}{\\partial p_{in}^{t-1}}=\\frac{F_n^{\\prime }}{p_{in}^{t-1}}+p_{in}^{t-1}(1-2\\gamma \\lambda _2)\\end{array}}$ where: ${\\begin{array}{c}F_n^{\\prime } = 2\\gamma (\\sum _k f_n^{t-1}(i,k)q_{kn}^{t-1} +\\lambda _1\\sum _{d_u}f_n^{t-1}(i,d_u)u_{d_u,n}^{t-1}\\\\\\end{array}}$ Then the multiplication gives equation system (REF )."
],
[
"Evaluation Metrics for ",
"The study sets the rating threshold to be 4, and the number of items recommended to be 10 per user.",
"The metrics are defined as: $NDCG@10=\\frac{DCG@10}{iDCG@10}$ $Precision=\\frac{1}{|usr|}\\sum _{i}\\frac{t_p^i}{t_p^u+f_p^i}$ $Recall=\\frac{1}{|usr|}\\sum _{i}\\frac{t_p^i}{t_p^i+f_n^i}$ $F1=\\frac{2\\times Precision\\times Recall}{Precision + Recall}$ where $t_p^i$ denotes the true positive for user $i$ , $f_p^i$ denotes the false positive for user $i$ , $f_n^i$ denotes the false negative for user $i$ , $iDCG@10$ is the maximum possible $DCG@10$ , and $DCG@10$ is given by: $DCG@10=\\sum _{i=1}^5\\frac{2^{x_i-1}}{\\log _2(i+1)}$ where $i$ is the item with the $i^{th}$ highest predicted rating, and $x_i$ is the actual rating for the item received by a given user."
]
] | 2210.07775 |
[
[
"Greedy Transaction Fee Mechanisms for (Non-)myopic Miners"
],
[
"Abstract Decentralized cryptocurrencies are payment systems that rely on aligning the incentives of users and miners to operate correctly and offer a high quality of service to their users.",
"Recent literature studies the mechanism design problem of the auction serving as the transaction fee mechanism (TFM).",
"We show that while the protocol that requires a user to \"pay as bid\" and greedily chooses among available transactions based on their fees is not dominant strategy incentive-compatible (DSIC) for users, it has a Bayesian-Nash equilibrium (BNE) where bids are slightly shaded.",
"Relaxing this incentive compatibility requirement circumvents the impossibility result of [16] and allows for an approximately revenue and welfare optimal, myopic miners incentive-compatibility (MMIC), and off-chain-agreement (OCA)-proof mechanism.",
"We prove its guarantees using different benchmarks, and in particular, show it is the revenue optimal Bayesian incentive-compatible (BIC), MMIC and 1-OCA-proof mechanism among a large class of mechanisms.",
"We move beyond the myopic model to a model where users offer transaction fees for their transaction to be accepted, as well as report their urgency level by specifying the time to live (TTL) of the transaction, after which it expires.",
"We show guarantees provided by the greedy allocation rule, as well as a better-performing non-myopic rule.",
"The above analysis is stated in terms of a cryptocurrency TFM, but applies to other settings, such as cloud computing and decentralized \"gig\" economy, as well."
],
[
"acmcopyright 2022 2022 XXXXXXX.XXXXXXX [WWW '23]ACM Web ConferenceApril 30–May 04, 2023Austin, TX 15.00 978-1-4503-XXXX-X/18/06 colorlinks, linkcolor=red!50!black, citecolor=blue!50!black, urlcolor=blue!80!black definitionDefinitionDefinitions theoremTheoremTheorems claimClaimClaims lemmalemmalemmas corollaryCorollaryCorollarys exampleExampleExamples remarkRemarkRemarks codeCodeCode [acronym]long-short EVMEVMEthereum virtual machine DeFiDeFidecentralized finance PoWPoWProof-of-Work PoSPoSProof-of-Stake MEVMEVminer-extractable value block-DAGblock-DAGsblock directed-acyclic-graph DAADAAdifficulty-adjustment algorithm MDPMDPMarkov decision-process DQLDQLDeep-Q-learning RLRLreinforcement learning MLMLmachine learning AIAIsartificial intelligence PDFPDFprobability density function CDFCDFcumulative density function AMMAMMautomated market maker USDUSDUnited States Dollar IPIPInternet Protocol LPLPLiquidity Provider LTLTLiquidity Taker APYAPYannual percentage yield PIDPIDproportional integral derivative UTXOUTXOunspent transaction output YAMLYAMLYAML Ain't Markup Language TDTDtotal difficulty gethgethGo Ethereum WETHWETHWrapped Ethereum ASICASICApplication Specific Integrated Circuit RPCRPCremote procedure call RUMRUMriskless uncle maker PUMPUMpreemptive uncle maker URLURLuniform resource locator SSDSSDsolid state drive EIPEIPEthereum improvement proposal CPUCPUcentral processing unit RAMRAMrandom-access memory TFMTFMtransaction fee mechanism TFETFEtransaction fee estimator iidi.i.d.independent and identically distributed wrtw.r.t.with regards to wlogw.l.o.g.without loss of generality TTLTTLtime to live DSICDSICdominant strategy incentive-compatible EPICEPICex-post incentive-compatible BICBICBayesian incentive-compatible MMICMMICmyopic miners incentive-compatibility OCAOCAoff-chain-agreement SCPSCPside-contract-proofness PABGAPABGApay-as-bid greedy auction UPGAUPGAuniform-price greedy auction BNEBNEBayesian-Nash equilibrium EPIREPIRex-post individually rational EPBBEPBBex-post burn balanced QoSQoSquality of service [description= The symbol for the Bitcoin cryptocurrency.",
"]bitcoin $\\leavevmode \\vtop {B}\\hfil \\hspace{-0.29999pt}\\vrule height .3ex width .15ex\\hspace{0.80002pt}\\vrule height .3ex width .15ex\\hfil $ $$ [description= The symbol for the Ethereum cryptocurrency.",
"]ethereum $\\Xi $ [description= A main-chain block.",
"]block $b$ [description= A transaction.",
"]tx $tx$ [description= The [hyper=true]TTL of a transaction.",
"]ttl $t$ [description= The reserve price of an auction.",
"]reserve $r$ [description= The memory pool of past transactions.",
"]mempool $\\mu $ [description= Immediacy ratio for our non-myopic allocation rule.",
"]imratio $\\ell $ [description= The adversary function.",
"]adversary $\\psi $ [description= The allocation function.",
"]allocation $x$ [description= The size of transaction ${[hyper=true]{tx}}$ , in bytes.",
"]txsize $w$ [description= The transaction fee of some transaction, in tokens.",
"]fee $\\phi $ [description= The maximal [hyper=true]TTL of transactions in the blockchain, in blocks.",
"]maxttl $d$ [description= The fee-density of transaction ${[hyper=true]{tx}}$ , in tokens per byte.",
"]txfeedensity $\\phi $ [description= The maximal block-size, in bytes.",
"]blocksize $B$ [description= The total amount of fees paid in block ${[hyper=true]{block}}$ , in tokens per bytes.",
"]blockfee $\\Phi $ [description= The duration of the attack, in blocks.",
"]totalblocks $\\beta $ [description= A TFE.",
"]tfe $f$ [description= An attack strategy.",
"]strategy $s$ [description= Attacker's utility, in tokens.",
"]utility $U$ [description= Miner's relative mining power, as a fraction.",
"]minerratio $\\alpha $ [description= Miner's discount factor.",
"]discount $\\lambda $ [description= Miner's planning horizon.",
"]horizon $T$ e s o >1|m E112*434[3]4 mm122 [1][]1 Greedy Transaction Fee Mechanisms for (Non-)myopic Miners Yotam Gafni [email protected] 0000-0002-2144-655X Technion Haifa Israel Aviv Yaish [email protected] 0000-0002-7971-2494 The Hebrew University Jerusalem Israel <ccs2012> <concept> <conceptid>10010405.10003550.10003551</conceptid> <conceptdesc>Applied computing Digital cash</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> <concept> <conceptid>10003752.10010070.10010099.10010101</conceptid> <conceptdesc>Theory of computation Algorithmic mechanism design</conceptdesc> <conceptsignificance>500</conceptsignificance> </concept> </ccs2012> [500]Applied computing Digital cash [500]Theory of computation Algorithmic mechanism design Optimal Auctions, Blockchain, Mechanism Design, Transaction Fee Mechanisms Decentralized cryptocurrencies are payment systems that rely on aligning the incentives of users and miners to operate correctly and offer a high [hyper=true]QoS to their users.",
"Recent literature studies the mechanism design problem of the auction serving as the [hyper=true]TFM.",
"We show that while the protocol that requires a user to “pay as bid” and greedily chooses among available transactions based on their fees is not [hyper=true]DSIC for users, it has a [hyper=true]BNE where bids are slightly shaded.",
"Relaxing this incentive compatibility requirement circumvents the impossibility result of [16] and allows for an approximately revenue and welfare optimal, [hyper=true]MMIC, and [hyper=true]OCA-proof mechanism.",
"We prove its guarantees using different benchmarks, and in particular, show it is the revenue optimal [hyper=true]BIC, [hyper=true]MMIC and 1-[hyper=true]OCA-proof mechanism among a large class of mechanisms.",
"We move beyond the myopic model to a model where users offer transaction fees for their transaction to be accepted, as well as report their urgency level by specifying the [hyper=true]TTL of the transaction, after which it expires.",
"We show guarantees provided by the greedy allocation rule, as well as a better-performing non-myopic rule.",
"The above analysis is stated in terms of a cryptocurrency [hyper=true]TFM, but applies to other settings, such as cloud computing and decentralized “gig” economy, as well.",
"body appendix"
]
] | 2210.07793 |
[
[
"Multiple Choice Hard Thresholding Pursuit (MCHTP) for Simultaneous\n Sparse Recovery and Sparsity Order Estimation"
],
[
"Abstract We address the problem of sparse recovery using greedy compressed sensing recovery algorithms, without explicit knowledge of the sparsity.",
"Estimating the sparsity order is a crucial problem in many practical scenarios, e.g., wireless communications, where exact value of the sparsity order of the unknown channel may be unavailable a priori.",
"In this paper we have proposed a new greedy algorithm, referred to as the Multiple Choice Hard Thresholding Pursuit (MCHTP), which modifies the popular hard thresholding pursuit (HTP) suitably to iteratively recover the unknown sparse vector along with the sparsity order of the unknown vector.",
"We provide provable performance guarantees which ensures that MCHTP can estimate the sparsity order exactly, along with recovering the unknown sparse vector exactly with noiseless measurements.",
"The simulation results corroborate the theoretical findings, demonstrating that even without exact sparsity knowledge, with only the knowledge of a loose upper bound of the sparsity, MCHTP exhibits outstanding recovery performance, which is almost identical to that of the conventional HTP with exact sparsity knowledge.",
"Furthermore, simulation results demonstrate much lower computational complexity of MCHTP compared to other state-of-the-art techniques like MSP."
],
[
"Introduction",
"Sparse recovery is a signal processing technique of significant research interest in diverse practical problems, ranging from wireless communication, image processing, radar signal processing, to name a few.",
"At the core of any sparse recovery problem lies the ill-posed inverse problem of solving an under-determined system of linear equations.",
"However, the crux of addressing such ill-posed problems lies in assuming that the unknown vector is sparse, i.e., many of its entries are 0.",
"Mathematically, the problem becomes the following: $\\begin{aligned}\\min _{x\\in \\mathbb {R}^N:\\Vert x\\Vert _0\\le K}\\left\\Vert y-\\Phi x\\right\\Vert _2^2,\\end{aligned}$ where $\\left\\Vert x\\right\\Vert _{0}$ is the sparsity of $x$ , denoting the number of non-zero entries of ${x}$ , and $K$ is an upper bound of the sparsity.",
"The formulation (REF ) is at the cornerstone of the so called field of Compressed Sensing (CS) [1] and a large body of research has been devoted obtaining fast and low complexity algorithms, enjoying optimal or sub-optimal recovery guarantees for the solution.",
"The problem (REF ) is inherently difficult to solve as the constraint set, $\\lbrace x:\\left\\Vert x\\right\\Vert _{0}\\le K\\rbrace $ is non-convex.",
"Although convex relaxation approaches can be employed to obtain provably sub-optimal algorithms, they are often slow for large dimensions.",
"Greedy algorithms provide an alternate route to solve such difficult problems.",
"Although heuristic in nature, such methods provide fast alternative to convex relaxation, often with competitive and provably sub-optimal recovery guarantees.",
"Some canonical examples of such greedy methods are: orthogonal matching pursuit (OMP) [2], iterated hard thresholding (IHT) [3], subspace pursuit (SP) [4], compressive sampling matching pursuit (CoSAMP) [5], hard thresholding pursuit (HTP) [6], to name a few.",
"However, execution of many of these greedy methods require explicit knowledge of the exact sparsity of the unknown vector.",
"While orthogonal matching pursuit does not strictly require to know the sparsity for its execution, an exact knowledge of sparsity provides better recovery guarantee.",
"On the other hand, although the algorithms HTP, SP or CoSAMP can work with an upper bound of the sparsity, the computational complexity of these algorithms increase substantially if that upper bound is quite large.",
"Furthermore, in many practical problems, e.g., Delay-Doppler (DD) path estimation in orthogonal time frequency space (OTFS) modulation, the knowledge of exact number of DD paths is difficult to come by, rendering most of the above greedy methods of little use in such contexts.",
"This necessitates sparsity order estimation (SOE) as a crucial preprocessing for greedy sparse recovery algorithms.",
"The SOE problem has been addressed by several recent research works [7], [8], [9], [10], [11], [12].",
"However, most of these works have used either various heuristic methods [13], [8], [9] with no provable performance analysis, or statistical as well as methods based on asymptotic random matrix theory [10], [11], [12], which lack provable non-asymptotic performance analysis for SOE.",
"Furthermore, most of these works do not use greedy sparse recovery algorithms.",
"One of the first works which has used a greedy sparse recovery algorithm in conjunction with unknown sparsity estimation is the modified subspace pursuit (MSP) [14].",
"MSP gradually increments the sparsity and runs the SP algorithm until convergence and stops only when the convergence error is small enough.",
"However, since MSP requires SP to run repeatedly for many sparsity estimates, then it can suffer from significant computational burden.",
"Graded HTP (GHTP) [15] is another method which recovers a sparse vector without explicit knowledge of sparsity.",
"However, GHTP does not provide an explicit estimate of the sparsity and rather the sparsity has to be looked up from the iteration number of the algorithm when the algorithm stops.",
"Furthermore, the stopping criterion of GHTP is set in such a way that it can overestimate the sparsity significantly for large sparsity values.",
"In this paper, we propose the multiple choice hard thresholding pursuit, abbreviated as MCHTP, which simultaneously estimates sparsity order as well as the unknown sparse vector until convergence.",
"The salient features of our work are listed as below: To our knowledge, MCHTP is the first greedy method that can simultaneously provide both sparsity estimate as well as estimate of sparse vector in each iteration.",
"This simultaneous execution helps in saving computational burden significantly.",
"We theoretically provide a complete characterization of the evolution of the sparsity sequence estimated by MCHTP and provide theoretical bounds that ensure exact recovery for noiseless measurements.",
"Our simulation results clearly demonstrate the efficacy of the proposed MCHTP in terms of fast sparsity order estimation performance and sparse recovery performance which is at par with the sparse recovery performance of HTP with sparsity knowledge."
],
[
"Notation",
"The following notations have been used throughout the paper :`$\\top $ ' in superscript indicates transposition of matrices / vectors.",
"The entries of a vector $x\\in \\mathbb {R}^N$ are denoted by $x_1,\\cdots , x_N$ .",
"For any $S\\subseteq \\lbrace 1,2,\\cdots , N\\rbrace $ , $x_S$ denotes the vector $x$ restricted to $S$ , i.e., $x_S$ consists of those entries of $x$ that have indices belonging to $S$ .",
"The operator $H_K(\\cdot )$ returns the $K$ -best approximation of a vector, i.e., for any vector $x\\in \\mathbb {R}^N$ , $H_K(x)=\\arg \\min _{z\\in \\mathbb {R}^N:\\left\\Vert z\\right\\Vert _{0}\\le K}\\left\\Vert z - x\\right\\Vert _{2}$ .",
"Similarly, $L_K(x)$ returns the support of the $K$ -best approximation of vector $x$ .",
"The symmetric difference $\\Delta $ , between two sets $A, B$ , is defined as $A\\Delta B: = (A\\setminus B)\\cup (B\\setminus A)$ .",
"[t!]",
"Multiple Choice HTP (MCHTP) [1] A sparsity upper bound $\\bar{K}$ , an initial estimate $x^0=0$ (so that $K_0=0$ ), measurement vector $y$ , measurement matrix $\\Phi $ , step size $\\mu >0$ , number of iterations to run $T$ and a factor $\\epsilon >0$ .",
"$t=1,\\cdots ,\\ T$ Set $K_{t,0} = K_{t-1}$ and sample $K_{t,1}$ uniformly randomly from the set $\\lbrace 1,\\cdots , \\bar{K}\\rbrace \\setminus \\lbrace K_{t,0}\\rbrace $ .",
"$i=0,1$ $\\Lambda _{t,i} = L_{K_{t,i}}\\left(x^{t-1} + \\mu \\Phi ^\\top \\left(y - \\Phi x^{t-1}\\right) \\right)$ $x^t_{i} = \\arg \\min _{z:\\texttt {supp}(z)=\\Lambda _{t,i}}\\left\\Vert y-\\Phi z\\right\\Vert _2$ $E_{t,i}=\\left\\Vert y-\\Phi x^t_{i}\\right\\Vert _2^2$ $\\Delta E_t = \\left|E_{t,1}-E_{t,0}\\right|$ $\\Delta E_t> \\epsilon $ $i_t = \\operatornamewithlimits{arg\\,min}_{i\\in \\lbrace 0,1\\rbrace }E_{t,i}$ $i_t = \\operatornamewithlimits{arg\\,max}_{i\\in \\lbrace 0,1\\rbrace }E_{t,i}$ $K_t = K_{t,i_t}$ , $x^t = x^t_{i_t}$ The Multiple Choice HTP (MCHTP) algorithm is elaborated in Table .",
"The main idea of MCHTP is to evaluate the HTP computations iteratively on two different choices of sparsity and select one using a suitable criterion.",
"We explain this in the following paragraph.",
"At each iteration $t\\ge 1$ of MCHTP, at step 2, we begin with two guesses for the sparsity order, $K_{t,0}$ and $K_{t,1}$ .",
"We set $K_{t,0} = K_{t-1}$ , where $K_{t-1}$ is the sparsity order estimated at the end of the last iteration $t-1$ , whereas, $K_{t,1}$ is chosen uniformly randomly from the set $[\\bar{K}]\\setminus \\lbrace K_{t,0}\\rbrace $ , where $[\\bar{K}]:=\\lbrace 1,2,\\cdots , \\bar{K}\\rbrace $ .",
"In steps $4-5$ , one iteration of HTP is executed for each of these sparsity orders starting with the same initial estimated vector $x^{t-1}$ and two estimates $x^{t}_{i}, i=0,1$ are produced and the corresponding estimation errors $E_{t,i}, i=0,1$ are calculated in step 6.",
"At this step, it should be recalled that HTP ensures that the larger sparsity will yield smaller estimation error.",
"These errors are now used to select sparsity order in the following way: always select the smaller sparsity (with larger error) unless the absolute difference in errors, i.e, $\\Delta E_t = \\left|E_{t,1}-E_{t,0}\\right|$ is larger than the predefined threshold $\\epsilon $ .",
"In the former case, the estimation errors do not differ “much”, i.e., by less than $\\epsilon $ , and the smaller sparsity corresponding to the slightly larger estimation error is chosen.",
"In the latter case, the error corresponding to the smaller sparsity is to large to choose the corresponding sparsity and therefore, the larger sparsity with smaller error is estimated."
],
[
"Convergence Analysis of MCHTP",
"In this section, we present a theoretical convergence analysis of the proposed MCHTP algorithm.",
"For the sake of simplicity, we consider the noiseless measurement model $y=\\Phi x.$ Furthermore, we consider $\\mu = 1$ , although the analysis can be generalized in a straightforward way for general $\\mu >0$ .",
"Before embarking on the mathematical analysis, let us provide an intuitive outline of the analysis.",
"We first specify a key result which states that if the sensing matrix satisfies certain conditions and the predefined threshold $\\epsilon $ is chosen properly then an iteration $t$ of MCHTP satisfies $\\Delta E_t>\\epsilon $ as long as $\\min _{i} K_{t,i}<K$ , ensuring that $K_t = \\max _i K_{t,i}$ .",
"This initiates the first phase of MCHTP consisting of a no-decreasing sequence of sparsity estimates $\\lbrace K_t\\rbrace $ , as long as $K_t< K$ .",
"Once $K_t\\ge K$ , the phase two begins.",
"In phase two, one always has $\\max _iK_{t,i}\\ge K$ , although $\\Delta E_t$ might fluctuate around $\\epsilon $ .",
"In the third and final phase, $\\Delta E_t$ is always smaller than $\\epsilon $ , ensuring that $\\lbrace K_t\\rbrace $ is a decreasing sequence, albeit not smaller than $K$ .",
"This ensures that the sparsity tracked by the algorithm gradually converges to $K$ from above.",
"We also establish a decay inequality of the sequence of estimated vectors $\\lbrace x^{t-1}\\rbrace $ , which provides, along with the estimated sparsity sequence $\\lbrace K_t\\rbrace $ , a provable convergence guarantees for MCHTP.",
"We now state and prove below a crucial lemma about the evolution of the intermediate estimates $\\mathbf {x}_i^t$ : Proposition 3.1 At any iteration $t$ , for each $i=0,1$ , the estimate $x_i^t$ satisfies the following decay inequality: $\\left\\Vert x^t_i - x\\right\\Vert _2 & \\le \\rho _{t,i}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma _{t,i}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2,$ while $x^t$ satisfies the following: $\\left\\Vert x^t - x\\right\\Vert _2 & \\le \\rho _{t}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma _{t}\\left\\Vert x_{\\overline{\\Gamma _{t}}}\\right\\Vert _2,$ where, $\\rho _{t,i} = \\frac{\\sqrt{2}\\delta _{K_{t,i}+K_{t-1}+K}}{\\sqrt{1-\\delta _{K_{t,i}+K}^2}}, && \\rho _{t} = \\frac{\\sqrt{2}\\delta _{K_{t}+K_{t-1}+K}}{\\sqrt{1-\\delta _{K_{t}+K}^2}},\\\\\\gamma _{t,i} = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta _{K_{t,i}+K}^2}}, && \\gamma _{t} = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta _{K_{t}+K}^2}},\\\\$ Moreover, $\\Gamma _{t,i}$ (resp.",
"$\\Gamma _t$ ) is the support of the (magnitude-wise) top $K_{t,i}$ (resp.",
"$K_t$ ) entries of $x$ .",
"An upper bound of $\\left\\Vert x^t_i-x\\right\\Vert _2$ can be obtained by employing the analysis technique of HTP [6].",
"However, this is not straightforward as the sparsity estimates are different at different time instants and are in general unequal to the original sparsity.",
"Therefore, we will modify the analysis of [6] suitably to obtain the desired bounds.",
"The detailed proof can be found in Appendix .",
"We now state a central result which ensures that, for a suitable choice of $\\epsilon $ , if the sensing matrix satisfies certain condition, then the estimate sparsity sequence $\\lbrace K_t\\rbrace $ is non-decreasing in the Phase I as long as $\\min _{i}K_{t,i}<K$ .",
"Proposition 3.2 Let us denote $\\delta : = \\delta _{2\\bar{K}+K}$ which satisfies the following: $\\delta & < \\frac{1}{1+(3+5R)\\sqrt{2K}}$ where $R = \\frac{x_{\\max }}{x_{\\min }}$ , where $x_{\\min }=\\min \\lbrace \\left|x_j\\right|:j\\in \\Lambda \\rbrace ,\\ x_{\\max }=\\max \\lbrace \\left|x_j\\right|:j\\in \\Lambda \\rbrace $ .",
"At an iteration $t$ , let $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}<K$ .",
"Then, $K_{t} = \\max _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}$ if the following is satisfied along with (REF ): $0<\\epsilon & < \\frac{(1-\\delta )}{(1+\\delta )^2}\\left(a_K(\\delta )x_{\\min } - b(\\delta )x_{\\max }\\right)^2,$ where $a_K(\\delta ) = \\frac{1-\\delta }{\\sqrt{K}}-3\\sqrt{2}\\delta , & b(\\delta ) = 5\\sqrt{2}\\delta .$ The idea of the proof is to show that if at any iteration $t$ , $\\min _{i\\in \\lbrace 0,1\\rbrace }\\lbrace K_{t,i}\\rbrace <K$ , and if $\\epsilon $ is small enough, i.e., if $\\epsilon $ satisfies the bound (REF ), then $\\Delta E_t>\\epsilon $ .",
"Then, by the step 11 of MCHTP (Algorithm ), we have $K_t=\\max _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}$ .",
"The detailed proof can be found in Appendix .",
"Discussion: The explicit dependence of the conditions (REF ) and (REF ) on the signal structure through the relative signal magnitude, expressed by $R=\\frac{x_{\\max }}{x_{\\min }}$ has interesting implications for different types of signals, e.g., flat, decaying, etc.",
"We discuss below the effect of some the important signal structures on the conditions (REF ) and (REF ).",
"Flat signal structure: In this case, the nonzero entries of $x$ are all of the same magnitude and differ only by sign, implying $x_{\\min }=x_{\\max } = \\frac{\\left\\Vert x\\right\\Vert _2}{\\sqrt{K}}$ and $R=1$ .",
"Therefore, the condition for successful completion of phase I for flat signals in the noiseless setting is ensured by the following two conditions: $\\delta & < \\frac{1}{1+8\\sqrt{2K}},\\\\0<\\epsilon & < \\frac{(1-\\delta )\\left\\Vert x\\right\\Vert _2^2}{K(1+\\delta )^2}\\left(a_K(\\delta ) - b(\\delta )\\right)^2.$ Linear signal structure: In this case, the nonzero entries follow a linear profile, i.e., the $j^\\mathrm {th}$ largest entry is of the form $\\alpha j$ , for $j=1,\\cdots , K$ , where $\\alpha = \\sqrt{\\frac{6\\left\\Vert x\\right\\Vert _2^2}{K(K+1)(2K+1)}}$ .",
"Therefore, $x_{\\min }=\\alpha $ and $x_{\\max } = K\\alpha $ , so that $R=K$ .",
"Consequently, the Phase I of MCHTP successfully completes if $\\delta & < \\frac{1}{1 + (3+5K)\\sqrt{2K}},\\\\0<\\epsilon & < \\frac{6(1-\\delta )\\left\\Vert x\\right\\Vert _2^2}{(1+\\delta )^2K(K+1)(2K+1)}(a_K(\\delta ) - Kb(\\delta ))^2.$ Decaying signal structure: In this case, the nonzero entries follow a geometric profile so that, $x_{\\min }=\\alpha ^{K-1}x_{\\max }$ , for some $\\alpha \\in (0,1]$ .",
"Therefore, $x_{\\max } = \\left\\Vert x\\right\\Vert _2\\sqrt{\\frac{1-\\alpha ^2}{1-\\alpha ^{2K}}}$ , and $R=\\alpha ^{1-K}$ .",
"Consequently, the phase I of MCHTP is ensured to succeed in this case for noiseless setting if the following conditions are satisfied: $\\delta & < \\frac{1}{1+(3+5\\alpha ^{1-K})\\sqrt{2K}},\\\\0<\\epsilon & < \\frac{(1-\\delta )(1-\\alpha ^2)\\left(a_K(\\delta )\\alpha ^{K-1} - b(\\delta )\\right)^2}{K(1+\\delta )^2(1-\\alpha ^{2K})}\\left\\Vert x\\right\\Vert _2^2.$ It can be easily observed that the conditions (REF ) and () are considerably more prohibitive than the conditions (REF ) and ().",
"This indicates that successful completion of phase I of MCHTP might be much easier for flat signals than for the decaying signals.",
"Proposition REF is a significant tool to analyze the evolution of the sequence $\\lbrace K_t\\rbrace $ .",
"Using this, we proceed to obtain a characterization of the Phase I of MCHTP.",
"We first define the duration of the Phase I as below: Definition 3.1 The phase I of MCHTP is defined to consist of the time slots $\\lbrace 0,1,\\cdots , T_1-1\\rbrace $ , where $T_1$ is a random time defined as below: $T_1 & =\\min \\lbrace t\\ge 1:K_t\\ge K\\rbrace .$ We now proceed to provide a characterization of $T_1$ .",
"Lemma 3.1 Under the satisfaction of the conditions (REF ) and (REF ), the sequence $\\lbrace K_t\\rbrace $ is non-decreasing during Phase I and $K_t\\ge K,\\ \\forall t\\ge T_1$ .",
"Furthermore, $T_1$ is characterized as below: $\\mathbb {P}\\left(T_1 = t\\right) = \\left\\lbrace \\begin{array}{ll}1 - p, & t = 1\\\\pq^{t-2}(1-q), & t> 1,\\end{array}\\right.$ where $p=\\frac{K-1}{\\bar{K}}$ and $q = \\frac{K-2}{\\bar{K}-1}$ .",
"To prove the first part of the claim, note that, by definition, during the phase I of MCHTP, we have $K_t<K$ , so that $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}<K$ .",
"Therefore, by Proposition REF , if the conditions (REF ) and (REF ) are satisfied, then during phase I, $K_t = \\max _{i\\in \\lbrace 0,1\\rbrace }K_{t,i},\\ 1\\le t\\le T_1$ .",
"By the description of MCHTP in table , it follows that during Phase I of MCHTP, i.e., for $1\\le t\\le T_1$ , we have $K_t & = \\max \\lbrace K_{t-1},K_{t,1}\\rbrace = \\max \\lbrace \\max \\lbrace K_{t-2},K_{t-1,1}\\rbrace , K_{t,1}\\rbrace \\nonumber \\\\\\ & = \\max \\lbrace K_{t-2},K_{t-1,1},K_{t,1}\\rbrace \\nonumber \\\\\\ & = \\cdots = \\max \\lbrace K_0,K_{1,1},\\cdots , K_{t,1}\\rbrace = \\max _{1\\le s\\le t}K_{s,1},$ since $K_0=0$ and $K_t\\ge 1,\\ \\forall t\\ge 1$ .",
"Therefore, $\\lbrace K_t\\rbrace $ is non-decreasing during Phase I of MCHTP.",
"To see that $K_{t}\\ge K,\\ \\forall t\\ge T_1$ , we proceed via induction.",
"First note that $K_{T_1}\\ge K$ by definition.",
"Then assume that it holds for $T_1,\\cdots , t-1$ for some $t\\ge T_1+1$ .",
"We have to prove that the claim holds for $t$ .",
"If $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}\\ge K$ , then trivially, $K_t \\ge K$ .",
"So let us assume that $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}< K$ .",
"If the conditions (REF ) and (REF ) are satisfied, it follows from Proposition REF that $K_t=\\max \\lbrace K_{t-1},K_{t,1}\\rbrace $ .",
"Note that by assumption $K_s\\ge K$ , for $s=T_1,\\cdots , t-1$ , so that $K_{t-1}\\ge K$ .",
"Since we have assumed that $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i} = \\min {K_{t-1},K_{t,1}}<K$ , we must have $K_{t,1}<K$ since $K_{t-1}\\ge K$ by assumption.",
"Therefore, $K_t=\\max \\lbrace K_{t-1},K_{t,1}\\rbrace \\ge K$ .",
"This completes the induction argument.",
"To obtain a characterization of $T_1$ , note that as the sequence $\\lbrace K_t\\rbrace $ is non-decreasing in Phase I, it follows that $K_t=\\max _{s=1,\\cdots , t}K_{s,1}$ .",
"Therefore, it follows from the definition of $T_1$ that $T_1=\\min \\lbrace t\\ge 1: K_{t,1}\\ge K\\rbrace $ .",
"Therefore $\\mathbb {P}\\left(T_1 = 1\\right) & = \\mathbb {P}\\left(K_{1,1}\\ge K\\right) = \\frac{\\bar{K}-K+1}{\\bar{K}}.$ On the other hand, for $t> 1$ , $\\mathbb {P}\\left(T_1 = t\\right) & = \\mathbb {P}\\left(\\max _{1\\le s\\le t-1}K_{s,1}< K, K_{t,1}\\ge K\\right)\\nonumber \\\\\\ & = \\left(\\frac{K-1}{\\bar{K}}\\right)\\left(\\frac{K-2}{\\bar{K}-1}\\right)^{t-2}\\left(\\frac{\\bar{K}-K+1}{\\bar{K}-1}\\right).$ The last expression follows since given $K_{t-1},\\ t>1$ , $K_{t,1}$ is sampled uniformly randomly from the set $\\lbrace 1,\\cdots , \\bar{K}\\rbrace \\setminus \\lbrace K_{t-1}\\rbrace $ .",
"Using the above result, we will now state and prove our first result on the convergence of MCHTP.",
"Theorem 3.1 At any iteration $t$ , the iterate $x^t$ satisfies the following decay inequality: $\\left\\Vert x^t - x\\right\\Vert _2 & \\le \\rho \\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma \\left\\Vert x_{\\overline{\\Gamma _{t}}}\\right\\Vert _2,\\ 1\\le t<T_1,\\\\\\left\\Vert x^t - x\\right\\Vert _2 & \\le \\rho \\left\\Vert x^{t-1}-x\\right\\Vert _2,\\ t\\ge T_1,$ where $\\rho = \\frac{\\sqrt{2}\\delta }{\\sqrt{1-\\delta ^2}},\\ \\gamma = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta ^2}}$ .",
"Consequently, if $\\rho <1$ , $x^t\\rightarrow x$ if $T_1$ is finite with probability 1.",
"The proof relies on Proposition REF as well as the Lemma REF .",
"The inequality (REF ) trivially implies the inequality (REF ).",
"Now, by the Lemma REF , $K_t\\ge K$ for $t\\ge T_1$ .",
"Therefore, by the definition of $\\Gamma _t$ in Proposition REF we have that $\\Lambda \\subset \\Gamma _t,\\ \\forall t\\ge T_1$ , where $\\Lambda $ is the true support of $x$ .",
"Consequently, for all $t\\ge T_1$ , $\\left\\Vert x_{\\overline{\\Gamma }_t}\\right\\Vert _2=0$ , which proves the inequality ().",
"Finally, if $\\rho <1$ and $T_1$ is finite with probability 1 (which is true due to Lemma REF ), it is trivial to see that $\\lim _{t\\rightarrow \\infty }\\left\\Vert x^t-x\\right\\Vert _2=0\\Rightarrow x^t\\rightarrow x$ .",
"On the completion of the first $T_1$ iterations, MCHTP enters its phase II during which the estimated sparsity sequence might be increasing or decreasing, albeit remaining $\\ge K$ .",
"The following lemma is key to provide an estimate of the duration of the phase II of MCHTP.",
"Before proceeding, we first define the Phase II formally.",
"Definition 3.2 The Phase II of MCHTP is defined to be the time duration consisting of the time slots $T_1,\\cdots , T_1+T_2-1$ , where $T_2 = \\min \\left\\lbrace t\\ge 1: \\lbrace K_{s+T_1}\\rbrace _{s\\ge t}\\ \\mbox{\\emph {is a non-increasing sequence}}\\right\\rbrace .$ Proposition 3.3 If $\\epsilon $ satisfies conditions (REF ) and (REF ) then the phase II of MCHTP consists of $T_2$ iterations, where $T_2 & \\le \\left\\lfloor \\left(\\frac{\\ln \\left[\\frac{\\sqrt{(1-\\delta )\\epsilon }}{(1+\\delta )\\left\\Vert x\\right\\Vert _2}\\right]}{\\ln \\rho }\\right)^+\\right\\rfloor ,$ where $\\rho =\\frac{\\sqrt{2}\\delta }{\\sqrt{1-\\delta ^2}}$ and $\\left(x\\right)^+=\\max \\lbrace x,0\\rbrace $ for any $x\\in \\mathbb {R}$ .",
"The main observation is that to end the Phase II and begin the Phase III in the immediate next slot, one must make the estimated sparsity non-increasing.",
"By step 11 of MCHTP in Table , this can occur at slot $t$ , if $\\Delta E_t<\\epsilon $ .",
"However, under the conditions (REF ) and (REF ), $\\Delta E_t\\ge \\epsilon $ , whenever $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}<K$ .",
"Since $K_{t-1}\\ge K$ throughout after the end of Phase I, it is therefore ensured that after the Phase II ends, $\\lbrace K_t\\rbrace $ remains unchanged whenever $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}<K$ .",
"Consequently, it remains to investigate bounds which ensure that $\\Delta E_t<\\epsilon $ , whenever $\\min _{i\\in \\lbrace 0,1\\rbrace }K_{t,i}\\ge K$ .",
"The proof follows by deriving a bound ensuring this condition.",
"For a detailed proof, refer to Appendix .",
"The phase III of MCHTP commences when phase II ends.",
"With the satisfaction of the conditions (REF ) and (REF ) by virtue of Proposition REF , the absolute error difference is sufficiently small, which in turn ensures that the estimated sparsity sequence is non-increasing.",
"We say that the phase III stops, when the estimated sparsity sequence converges to $K$ .",
"Then, it is immediate that the phase III stops whenever the randomly chosen sparsity at a time step is equal to $K$ .",
"This allows us to easily estimate the duration $T_3$ of phase III as stated below: Lemma 3.2 The phase III of MCHTP has duration $T_3$ , where, $\\mathbb {P}\\left(T_3=t\\right) & = r(1-r)^{t-1}, t\\ge 1,$ where $r = \\frac{1}{\\bar{K}-1}$ .",
"Note that the phase III ends as soon as, following Phase II, one encounters $K_{t,1}=K$ .",
"Specifically, $T_3$ can be precisely defined to be the following: $\\ & T_3 = \\min \\bigg \\lbrace t\\ge 1: K_{\\tau ,1}\\ne K, K_{\\tau + 1, 1}\\ne K,\\cdots ,\\nonumber \\\\\\ & K_{t-1,1}\\ne K, K_{t,1}=K\\bigg \\rbrace .$ Since the sequence $K_{\\tau ,1},\\cdots K_{t,1}$ constitute a DTMC, it follows that, $\\mathbb {P}\\left(T_3=t\\right) & = \\mathbb {P}\\left(K_{\\tau ,1}\\ne K,\\cdots , K_{t-1,1}\\ne K, K_{t,1}=K\\right)\\nonumber \\\\\\ & = \\left(\\frac{\\bar{K}-2}{\\bar{K}-1}\\right)^{t-1}\\left(\\frac{1}{\\bar{K}-1}\\right)\\nonumber \\\\\\ & = (1-r)^{t-1}r,$ where, $r = \\frac{1}{\\bar{K}-1}$ .",
"Consequently, the expected waiting time $W$ , to estimate the correct sparsity is obtained as below: ${\\mathbb {E}\\left[W\\right] = \\mathbb {E}\\left[T_1\\right]+\\mathbb {E}\\left[T_2\\right]+\\mathbb {E}\\left[T_3\\right]} & &\\nonumber \\\\\\ & \\le \\frac{p}{1-q} + \\left\\lfloor \\left(\\frac{\\ln \\left[\\frac{\\sqrt{(1-\\delta )\\epsilon }}{(1+\\delta )\\left\\Vert x\\right\\Vert _2}\\right]}{\\ln \\rho }\\right)^+\\right\\rfloor + \\frac{1}{r}\\nonumber \\\\\\ & = \\left(1-\\frac{1}{\\bar{K}}\\right)\\frac{K-1}{\\bar{K}-K+1} + \\left\\lfloor \\left(\\frac{\\ln \\left[\\frac{\\sqrt{(1-\\delta )\\epsilon }}{(1+\\delta )\\left\\Vert x\\right\\Vert _2}\\right]}{\\ln \\rho }\\right)^+\\right\\rfloor \\nonumber \\\\\\ & + \\bar{K} - 1.$"
],
[
"Numerical Experiments",
"In this section we numerically evaluate both the sparse recovery and sparsity order estimation performance of MCHTP and compare it with other state-of-the-art results.",
"For this purpose we have chosen the parameters $N=512, M=256, \\mu = 0.3$ .",
"We have fixed $K_{\\max } = M/2$ , which is a reasonable bound since theoretically perfect recovery for a $K$ sparse vector is possible only if $K\\le M/2$ .",
"Each experiment is repeated over 50 independent instances and the results are averaged."
],
[
"Conclusion",
"In this paper we have proposed a novel greedy sparse recovery algorithm referred to as MCHTP that can simultaneously recover an unknown sparse vector along with its unknown sparsity by using a novel decision criterion and HTP calculations at each iteration.",
"Assuming noiseless measurements, we have provided theoretical analysis of the evolution of the sparsity sequence through phases and have provided theoretical bounds on the sensing matrix as well as the signal structure, which ensure perfect recovery of the vector as well as its unknown sparsity.",
"Finally, the theoretical results are corroborated with numerical simulations that demonstrate the superior convergence of MCHTP as well as its unique sparsity estimation capability compared to the popular techniques.",
"Proof of Proposition REF We revisit the analysis of Theorem 3.5 of .",
"One can verify that the first step of the proof, i.e., the analysis up to Eq.",
"(3.11) in therein is applicable here without any modification and results in the following: $\\left\\Vert x^t_i - x\\right\\Vert _2^2 & \\le \\frac{1}{1-\\delta _{K_{t,i}+K}^2}\\left\\Vert \\big (x^t_i - x\\big )_{\\overline{\\Lambda _{t,i}}}\\right\\Vert _2^2.$ On the other hand, applying the second step of the proof requires a little more care.",
"Let us define $\\Gamma _{t,i}$ as the support corresponding to the (magnitude-wise) top $K_{t,i}$ entries of $x$ .",
"We note that ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2} & &\\nonumber \\\\\\ & \\le \\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2.$ We will now separately obtain lower and upper bounds of the left hand side (LHS) and the right hand side (RHS), respec., of the inequality (REF ).",
"First note that the RHS of inequality (REF ) can be upper bounded as below: ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & = \\left\\Vert (x + (I-\\Phi ^\\top \\Phi )(x^{t-1}-x))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2 \\nonumber \\\\\\ & \\le \\left\\Vert x_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2 + \\left\\Vert ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2.$ On the other hand, the LHS of inequality (REF ) can be lower bounded as below: ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & = \\left\\Vert \\big (x^t_i - x-(I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\big (x^t_i - x\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & - \\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2.$ Combining the bounds (REF ) and (REF ), one obtains, ${\\left\\Vert \\big (x^t_i - x\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & \\le \\sqrt{2}\\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\Delta \\Lambda _{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & + \\left\\Vert x_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2.$ Therefore, ${\\left\\Vert \\big (x^t_i - x\\big )_{\\overline{\\Lambda _{t,i}}}\\right\\Vert _2} & & \\nonumber \\\\\\ & \\le \\left\\Vert \\big (x^t_i - x\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2 + \\left\\Vert x_{\\overline{\\Gamma _{t,i}\\cup \\Lambda _{t,i}}}\\right\\Vert _2\\nonumber \\\\\\ & \\le \\sqrt{2}\\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\Delta \\Lambda _{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & + \\left\\Vert x_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2 + \\left\\Vert x_{\\overline{\\Gamma _{t,i}\\cup \\Lambda _{t,i}}}\\right\\Vert _2 \\nonumber \\\\\\ & \\le \\sqrt{2}\\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\Delta \\Lambda _{t,i}}\\right\\Vert _2 + \\sqrt{2}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2\\nonumber \\\\\\ & \\le \\sqrt{2}\\delta _{K_{t,i}+K_{t-1}+K}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\sqrt{2}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2.$ Combining the bounds (REF ) and (REF ), we obtain the following: $\\left\\Vert x^t_i - x\\right\\Vert _2 & \\le \\rho _{t,i}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma _{t,i}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2,$ where $\\rho _{t,i} = \\frac{\\sqrt{2}\\delta _{K_{t,i}+K_{t-1}+K}}{\\sqrt{1-\\delta _{K_{t,i}+K}^2}}$ and $\\gamma _{t,i} = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta _{K_{t,i}+K}^2}}$ .",
"It is now clear that the following holds true: $\\left\\Vert x^t - x\\right\\Vert _2 & \\le \\rho _{t}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma _{t}\\left\\Vert x_{\\overline{\\Gamma _{t}}}\\right\\Vert _2,$ where $\\rho _{t} = \\frac{\\sqrt{2}\\delta _{K_{t}+K_{t-1}+K}}{\\sqrt{1-\\delta _{K_{t}+K}^2}}$ and $\\gamma _{t} = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta _{K_{t}+K}^2}}$ , and $\\Gamma _{t}$ is the support corresponding to the (magnitude-wise) top $K_{t}$ entries of $x$ .",
"Proof of Proposition REF Let us denote, at time $t\\ge 0$ , $i=\\operatornamewithlimits{arg\\,min}_{j\\in \\lbrace 0,1\\rbrace }K_{t,j},\\ l = \\operatornamewithlimits{arg\\,max}_{j\\in \\lbrace 0,1\\rbrace }K_{t,j}$ .",
"The corresponding estimates produced are $x^t_i$ and $x^{t}_l$ with the respective supports $\\Lambda _{t,i}$ and $\\Lambda _{t,l}$ and errors $E_{t,i}=\\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2$ and $E_{t,l}=\\left\\Vert \\mathbf {P}_{\\Lambda _{t,l}}^\\perp y\\right\\Vert _2^2$ .",
"Step 4 of MCHTP guarantees that $\\Lambda _{t,i}\\subset \\Lambda _{t,l}$ .",
"Let us denote $\\Omega _{t}=\\Lambda _{t,l}\\setminus \\Lambda _{t,i}$ .",
"Then one finds that, $\\left\\Vert \\mathbf {P}_{\\Lambda _{t,l}}^\\perp y\\right\\Vert _2^2 & = \\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2 - \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}y\\right\\Vert _2^2,$ where $\\widetilde{\\mathbf {P}_{\\Omega _{t}}} & =\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^\\dagger \\nonumber \\\\\\ & =\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^{-1}\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp $ Let us denote, $\\Delta E_t = E_{t,i}-E_{t,l}$ .",
"Then, it is clear that $\\Delta E_t = \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}y\\right\\Vert _2^2$ .",
"We will now obtain a lower bound on $\\Delta E_{t}$ .",
"First note that, using the lower bound on the eigenvalues of $\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^{-1}\\Phi _{\\Omega _{t}}^\\top $ , one obtains, $\\Delta E_t & = \\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^{-1}\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2 \\nonumber \\\\\\ & \\ge \\frac{\\left\\Vert \\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2}{(1+\\delta _{K_{t,l}})}.$ We will now derive a lower bound of $\\left\\Vert \\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2$ .",
"In order to do so, first observe that $\\mathbf {P}_{\\Lambda _{t,i}}^\\perp y = y - \\Phi x^t_i$ .",
"Using this, adding and subtracting $x^{t-1}$ and using the reverse triangle inequality, one obtains, $\\left\\Vert \\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 & \\ge \\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Omega _t}\\right\\Vert _2\\nonumber \\\\\\ & - \\left\\Vert ((I-\\Phi ^\\top \\Phi )(x^t_i-x^{t-1}))_{\\Omega _t}\\right\\Vert _2,$ where in the last step we have used the fact that $[x^t_i]_j = 0,\\ \\forall j\\in \\Omega _t$ .",
"Now observe that the step 4 of MCHTP (Algorithm ) ensures that the sets $\\Lambda _{t,i}$ and $\\Lambda _{t,l}$ contain the top (magnitude wise) $K_{t,i}$ and $K_{t,l}$ entries, respectively, of the vector $x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1})$ .",
"Now depending on whether or not $K_{t,l}-K_{t,i}> K$ , there are two cases to consider.",
"If $K_{t,l}-K_{t,i}\\le K$ , then, for all $j=1,2,\\cdots , K_{t,l}-K_{t,i}$ , the $j^\\mathrm {th}$ maximum of the vector $x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1})$ from the set $\\Omega _t$ is greater than or equal to the $j^\\mathrm {th}$ maximum of $\\left|(x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))\\right|$ from the set $\\Lambda \\setminus \\Lambda _{t,i}$ .",
"On the other hand, if $K_{t,l}-K_{t,i}> K$ , then all the entries of the corresponding vector from the set $\\Omega _t$ are larger than the entries from the set $\\Lambda \\setminus \\Lambda _{t,i}$ .",
"Therefore, in any case, we have the following: ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Omega _t}\\right\\Vert _2} & & \\nonumber \\\\\\ & \\ge \\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{{\\Gamma }_{t,l,i}}\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{{\\Gamma }_{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & -\\left\\Vert ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1}))_{{\\Gamma }_{t,l,i}}\\right\\Vert _2,$ where $\\Gamma _{t,l,i}$ is the subset of $\\Lambda \\setminus \\Lambda _{t,i}$ which contains the $K_{t,l,i}:=\\min \\lbrace K_{t,l}-K_{t,i},K\\rbrace $ (magnitudewise) largest entries of $x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1})$ .",
"We have furthermore used the fact that $[x^t_i]_j = 0,\\ \\forall j\\in \\Lambda \\setminus \\Lambda _{t,i}$ .Therefore, we obtain, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2} & &\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & -\\left\\Vert ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1}))_{\\Gamma _{t,l,i}}\\right\\Vert _2\\nonumber \\\\\\ & - \\left\\Vert ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1}))_{\\Omega _t}\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & - \\sqrt{2}\\left\\Vert \\bigg ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1})\\bigg )_{\\Omega _t\\cup \\Gamma _{t,l,i}}\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & - \\sqrt{2}\\delta _{K_{t,l}-K_{t,i}+K_{t-1}+K}\\left\\Vert x_i^t-x^{t-1}\\right\\Vert _2.$ Now note that ${\\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2} & & \\nonumber \\\\\\ & \\ge \\frac{\\sqrt{\\left|\\Gamma _{t,l,i}\\right|}\\left\\Vert ((\\Phi ^\\top (y-\\Phi x_i^t)))_{\\Lambda \\setminus \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{\\left|\\Lambda \\setminus \\Lambda _{t,i}\\right|}}\\nonumber \\\\\\ & \\stackrel{(a)}{\\ge } \\frac{\\left\\Vert ((\\Phi ^\\top (y-\\Phi x_i^t)))_{\\Lambda \\setminus \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{K}}\\nonumber \\\\\\ & = \\frac{\\left\\Vert ((\\Phi ^\\top \\Phi (x_i^t-x)))_{\\Lambda \\setminus \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{K}}\\nonumber \\\\\\ & \\stackrel{(b)}{=}\\frac{\\left\\Vert ((\\Phi ^\\top \\Phi (x_i^t-x)))_{\\Lambda \\cup \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{K}}\\nonumber \\\\\\ & \\ge \\frac{(1-\\delta _{K_{t,i}+K})\\left\\Vert x_i^t-x\\right\\Vert _2}{\\sqrt{K}},$ where the step $(a)$ follows from the fact that $\\left|\\Lambda \\setminus \\Lambda _{t,i}\\right|\\le \\left|\\Lambda \\right|=K$ and $\\left|\\Gamma _{t,l,i}\\right|=K_{t,l,i}=\\min \\lbrace K_{t,l}-K_{t,i},K\\rbrace \\ge 1$ .",
"Furthermore, step $(b)$ follows from the fact that $\\Phi ^\\top _{\\Lambda _{t,i}}(y-\\Phi x^t_i)=0_{\\Lambda _{t,i}}$ .",
"Moreover, $\\left\\Vert x_i^t-x\\right\\Vert _2 & \\ge \\min _{\\begin{array}{c}z\\in \\mathbb {R}^N:\\\\ \\left\\Vert z\\right\\Vert _{0}\\le K_{t,i}\\end{array}} \\left\\Vert z-x\\right\\Vert _2 = \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2.$ On the other hand, using Eq.",
"(REF ), we obtain, $\\left\\Vert x^t_i-x^{t-1}\\right\\Vert _2 & \\le \\left\\Vert x^t_i-x\\right\\Vert _2 + \\left\\Vert x^{t-1}-x\\right\\Vert _2.$ Therefore, we obtain, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y \\right\\Vert _2} & &\\nonumber \\\\\\ & \\ge \\left(\\frac{1-\\delta _{K_{t,i}+K}}{\\sqrt{K}} - \\sqrt{2}\\delta _{K_{t,l}-K_{t,i}+K_{t-1}+K}\\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & - \\sqrt{2}\\delta _{K_{t,l}-K_{t,i}+K_{t-1}+K}\\left\\Vert x^{t-1}-x\\right\\Vert _2.$ Note that $K_{t,i}\\ge 0,\\ K_{t,l},K_{t-1}\\le \\bar{K}$ .",
"Therefore, for simplicity, we denote $\\delta : = \\delta _{2\\bar{K}+K}$ and further lower bound the above as ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta \\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2} & &\\nonumber \\\\\\ & - \\sqrt{2}\\delta \\left\\Vert x^{t-1}-x\\right\\Vert _2.$ We now use the bound (REF ) recursively, and the fact that $x_0=0$ , to obtain the following upper bound of $\\left\\Vert x^{t-1}-x\\right\\Vert _2$ : $\\left\\Vert x^{t-1}-x\\right\\Vert _2 & \\le \\rho ^{t-1} \\left\\Vert x\\right\\Vert _2 + \\gamma \\sum _{j=1}^{t-1}\\rho ^{t-1-j}\\left\\Vert x_{\\overline{\\Gamma }_{j}}\\right\\Vert _2,$ where we have used the bound (REF ) of Proposition REF and have denoted $\\rho = \\frac{\\sqrt{2}\\delta }{\\sqrt{1-\\delta ^2}},\\ \\gamma = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta ^2}}$ Therefore, taking the bounds (REF ) and (REF ), we obtain ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta \\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\bigg (\\rho ^{t-1} \\left\\Vert x\\right\\Vert _2 + \\gamma \\sum _{j=1}^{t-1}\\rho ^{t-1-j}\\left\\Vert x_{\\overline{\\Gamma }_{j}}\\right\\Vert _2\\bigg ).$ Consider the first step of MCHTP.",
"In this step, $t=1$ and $x^0=0,\\Gamma _0=\\emptyset $ since $K_0=0$ , so that $x_{\\overline{\\Gamma }_{0}}=\\left\\Vert x\\right\\Vert _2$ .",
"Furthermore, $K_{1,i}=K_0=0$ .",
"Therefore, the bound (REF ) reduces to $\\left\\Vert \\Phi _{\\Omega _1}^\\top \\mathbf {P}_{\\Lambda _{1,i}}^\\perp y\\right\\Vert _2 & \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - 2\\sqrt{2}\\delta \\right) \\left\\Vert x\\right\\Vert _2.$ Therefore, at $t=1$ , $K_1=K_{1,l}$ if $\\Delta E_1\\ge \\epsilon $ , which always holds if $\\left(\\frac{1-\\delta }{\\sqrt{K}} - 2\\sqrt{2}\\delta \\right) \\left\\Vert x\\right\\Vert _2 & \\ge \\sqrt{\\epsilon (1+\\delta )}.$ Obviously, we require, $\\frac{1-\\delta }{\\sqrt{K}} - 2\\sqrt{2}\\delta >0$ , which is satisfied by the following: $\\delta < \\frac{1}{2\\sqrt{2K}+1}.$ Therefore, the conditions (REF ) and (REF ) ensure that $\\Delta E_1>0$ so that $K_{1}=K_{1,l}$ .",
"Now consider $t>1$ .",
"Using the fact that $\\Gamma _{t,i}\\subset \\Gamma _{t-1}$ since $ K_{t,i}\\le K_{t-1}$ , we obtain from the bound (REF ) the following: ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta (1+\\gamma )\\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\bigg (\\rho ^{t-1} \\left\\Vert x\\right\\Vert _2 + \\rho \\gamma \\sum _{j=1}^{t-2}\\rho ^{t-2-j}\\left\\Vert x_{\\overline{\\Gamma }_{j}}\\right\\Vert _2\\bigg ).$ For any $t>1$ , we can use the bounds $\\left\\Vert x_{\\overline{\\Gamma }_t}\\right\\Vert _2\\le \\sqrt{1-\\frac{K_{t}}{K}}\\left\\Vert x\\right\\Vert _2\\le \\sqrt{1-\\frac{1}{K}}\\left\\Vert x\\right\\Vert _2$ , since $K_t\\ge 1$ and $\\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2\\ge \\sqrt{K-K_{t,i}}x_{\\min }\\ge x_{\\min }$ since $K_{t,i}\\le K-1$ .",
"Furthermore, assuming $\\delta <\\frac{1}{\\sqrt{2K}+1}<\\frac{1}{\\sqrt{3}}$ , we have $\\rho <1$ (since we $K\\ge 1$ ) (we will later obtain conditions which indeed implies this bound on $\\delta $ ).",
"With the above observations, we simplify the lower bound of (REF ) to the following, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2} & & \\nonumber \\\\\\ & \\ge x_{\\min }\\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta (1+\\gamma )\\right) \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\left\\Vert x\\right\\Vert _2\\bigg (\\rho ^{t-1} + \\frac{\\rho \\gamma \\sqrt{1-\\frac{1}{K}}(1-\\rho ^{t-2})}{1-\\rho }\\nonumber \\\\\\ & + \\frac{\\rho \\left\\Vert e\\right\\Vert _2}{(1-\\rho )\\left\\Vert x\\right\\Vert _2 }\\bigg ).$ Now, note that using the bound $\\delta <\\frac{1}{\\sqrt{2K+1}}$ , we obtain, for $K\\ge 2$ , $\\frac{\\gamma \\sqrt{1-\\frac{1}{K}}}{1-\\rho } & = \\frac{\\sqrt{2}\\sqrt{1-\\frac{1}{K}}}{\\sqrt{1-\\delta ^2}-\\sqrt{2}\\delta }\\nonumber \\\\\\ & \\le \\frac{\\sqrt{2K+1}\\sqrt{K-1}}{\\sqrt{K}(\\sqrt{K}-1)}< 4,$ and $\\frac{1}{1-\\rho } = \\frac{\\sqrt{1-\\delta ^2}}{\\sqrt{1-\\delta ^2}-\\sqrt{2}\\delta } & \\le \\frac{\\sqrt{K}}{\\sqrt{K}-1}\\le 2 + \\sqrt{2}<4,$ and, $\\gamma & = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta ^2}} \\le \\frac{\\sqrt{2K+1}}{\\sqrt{K}}\\le \\sqrt{2.5}<2,\\nonumber \\\\\\rho & = \\frac{\\sqrt{2}\\delta }{\\sqrt{1-\\delta ^2}} \\le \\frac{\\sqrt{2}}{\\sqrt{2K}} = \\frac{1}{\\sqrt{K}}.$ Therefore from (REF ), we obtain, for $t>1$ , ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge x_{\\min }\\left(\\frac{1-\\delta }{\\sqrt{K-1}} - 3\\sqrt{2}\\delta \\right)} & & \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\frac{\\left\\Vert x\\right\\Vert _2}{\\sqrt{K}}\\bigg (\\frac{1}{K^{t/2-1}} + 4 + \\frac{4\\left\\Vert e\\right\\Vert _2}{\\left\\Vert x\\right\\Vert _2}\\bigg ),$ which implies that, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 > x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg )} & &\\nonumber \\\\\\ & - x_{\\max }\\sqrt{2}\\delta \\bigg (4+\\frac{1}{K^{t/2-1}}\\bigg ).$ We can further lower bound the right hand side of the above inequality by using $K_{t-1}\\le K - 1$ and $K^{t/2-1}\\ge 1$ for $t\\ge 2$ , to obtain the following lower bound whenever $K_{t,i}<K$ : $\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 > x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg ) & - x_{\\max }5\\sqrt{2}\\delta .$ Therefore, for $t>1$ , if $K_{t,i}<K$ , then $K_t=K_{t,l}$ if $\\Delta E_t\\ge \\epsilon $ , which always holds if $\\ & x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg ) - 5\\sqrt{2}\\delta x_{\\max }\\ge \\sqrt{\\epsilon (1+\\delta )}.$ To make the LHS of (REF ) positive, we obviously require $x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg ) - x_{\\max }5\\sqrt{2}\\delta >0$ , which is ensured by the following: $\\delta & < \\frac{1}{1+(3+5R)\\sqrt{2K}},$ where $R = \\frac{x_{\\max }}{x_{\\min }}$ .",
"Therefore, the conditions (REF ) and (REF ) ensure that, for any $t>1$ , if $K_{t,i}<K$ , then $\\Delta E_t>0$ and $K_t= K_{t,l}$ .",
"Furthermore, observe that the condition (REF ) implies the conditions (REF ).",
"Therefore, under the condition (REF ), we have the following lower bound for $\\Delta E_t$ $\\Delta E_t & \\ge \\frac{(1-\\delta )}{(1+\\delta )^2}\\left(a_K(\\delta )x_{\\min } - b(\\delta )x_{\\max }\\right)^2,$ where $a_K(\\delta ),b(\\delta )$ are defined in Eq.",
"(REF ).",
"Therefore, the condition (REF ) ensures that $\\Delta E_t>\\epsilon $ , which in turn guarantees that $K_t = K_{t,l}$ , i.e., the larger sparsity value is chosen for time step $t$ .",
"Proof of Proposition REF The Phase II continues until the sequence of estimated sparsity becomes a non-increasing sequence.",
"Now, by the step 11 of MCHTP, if the current sparsity estimate is $K_{t-1}$ and $K_{t,1}$ is the sparsity sampled at time $t$ , $K_t = \\min \\lbrace K_{t-1}, K_{t,1}\\rbrace $ if $\\Delta E_t<\\epsilon $ .",
"However, if $K_{t,1}<K$ , thanks to Proposition REF , $\\Delta E_t\\ge \\epsilon $ and $K_{t}=K_{t-1}$ since $K_{t-1}\\ge K$ after Phase I.",
"On the other hand, if $K_{t,1}\\ge K$ , then $K_t=\\min \\lbrace K_{t-1}, K_{t,1}\\rbrace $ if one has $\\Delta E_t< \\epsilon $ .",
"Therefore, it is enough to find conditions which guarantee that $\\Delta E_t<\\epsilon $ , whenever $K_{t,1}\\ge K$ , to ensure that the sequence $K_t$ after the end of Phase II, is a non-increasing sequence.",
"Now note that, ${\\sqrt{\\Delta E_t} = \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}y\\right\\Vert _2 = \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}\\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2\\le \\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2} & &\\nonumber \\\\\\ & =\\left\\Vert \\Phi (x\\right\\Vert _2 \\le \\sqrt{1+\\delta }\\left\\Vert x^{t}_i - x\\right\\Vert _2.$ Note that Phase II of MCHTP ends at time $T_1+T_2-1$ and Phase III begins at time $T_1+T_2$ when $\\Delta E_t\\le \\epsilon $ for all $t\\ge T_1+T_2$ , whenever $K_{t,1}\\ge K$ .",
"This is ensured if the following is satisfied at such a time instant $t$ : $\\left\\Vert x^{t}_i - x\\right\\Vert _2 & \\le \\frac{\\sqrt{\\epsilon }}{\\sqrt{1+\\delta }}.$ Since $K_{t,1}\\ge K$ , and $K_{t-1}\\ge K$ by Porposition REF , we have $K_{t,i}\\ge K$ .",
"Therefore, from Proposition REF we obtain $\\left\\Vert x^{t}_i - x\\right\\Vert _2 & \\le \\rho \\left\\Vert x^{t-1}-x\\right\\Vert _2\\nonumber \\\\\\ & \\le \\rho ^{t-T_1+1}\\left\\Vert x^{T_1-1}-x\\right\\Vert _2\\nonumber \\\\\\ & \\le \\rho ^{t-T_1+1}\\sqrt{\\frac{1+\\delta }{1-\\delta }}\\left\\Vert x\\right\\Vert _2.$ Consequently, it follows that, if $T_2$ is the duration of Phase II of MCHTP, then $T_2\\le \\tau $ , where, $\\tau & =\\min \\left\\lbrace t\\ge 0:\\rho ^{t+1}\\sqrt{\\frac{1+\\delta }{1-\\delta }}\\left\\Vert x\\right\\Vert _2\\right.\\nonumber \\\\\\ & \\le \\left.\\frac{{\\sqrt{\\epsilon }}}{\\sqrt{1+\\delta }}\\right\\rbrace = \\min \\left\\lbrace t\\ge 0:\\rho ^{t+1}\\le G\\right\\rbrace ,$ where $G & = \\frac{\\sqrt{\\epsilon (1-\\delta )}}{(1+\\delta )\\left\\Vert x\\right\\Vert _2}$"
],
[
"Proof of Proposition ",
"We revisit the analysis of Theorem 3.5 of .",
"One can verify that the first step of the proof, i.e., the analysis up to Eq.",
"(3.11) in therein is applicable here without any modification and results in the following: $\\left\\Vert x^t_i - x\\right\\Vert _2^2 & \\le \\frac{1}{1-\\delta _{K_{t,i}+K}^2}\\left\\Vert \\big (x^t_i - x\\big )_{\\overline{\\Lambda _{t,i}}}\\right\\Vert _2^2.$ On the other hand, applying the second step of the proof requires a little more care.",
"Let us define $\\Gamma _{t,i}$ as the support corresponding to the (magnitude-wise) top $K_{t,i}$ entries of $x$ .",
"We note that ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2} & &\\nonumber \\\\\\ & \\le \\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2.$ We will now separately obtain lower and upper bounds of the left hand side (LHS) and the right hand side (RHS), respec., of the inequality (REF ).",
"First note that the RHS of inequality (REF ) can be upper bounded as below: ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & = \\left\\Vert (x + (I-\\Phi ^\\top \\Phi )(x^{t-1}-x))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2 \\nonumber \\\\\\ & \\le \\left\\Vert x_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2 + \\left\\Vert ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x))_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2.$ On the other hand, the LHS of inequality (REF ) can be lower bounded as below: ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & = \\left\\Vert \\big (x^t_i - x-(I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\big (x^t_i - x\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & - \\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2.$ Combining the bounds (REF ) and (REF ), one obtains, ${\\left\\Vert \\big (x^t_i - x\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & \\le \\sqrt{2}\\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\Delta \\Lambda _{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & + \\left\\Vert x_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2.$ Therefore, ${\\left\\Vert \\big (x^t_i - x\\big )_{\\overline{\\Lambda _{t,i}}}\\right\\Vert _2} & & \\nonumber \\\\\\ & \\le \\left\\Vert \\big (x^t_i - x\\big )_{\\Gamma _{t,i}\\setminus \\Lambda _{t,i}}\\right\\Vert _2 + \\left\\Vert x_{\\overline{\\Gamma _{t,i}\\cup \\Lambda _{t,i}}}\\right\\Vert _2\\nonumber \\\\\\ & \\le \\sqrt{2}\\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\Delta \\Lambda _{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & + \\left\\Vert x_{\\Lambda _{t,i}\\setminus \\Gamma _{t,i}}\\right\\Vert _2 + \\left\\Vert x_{\\overline{\\Gamma _{t,i}\\cup \\Lambda _{t,i}}}\\right\\Vert _2 \\nonumber \\\\\\ & \\le \\sqrt{2}\\left\\Vert \\big ((I-\\Phi ^\\top \\Phi )(x^{t-1}-x)\\big )_{\\Gamma _{t,i}\\Delta \\Lambda _{t,i}}\\right\\Vert _2 + \\sqrt{2}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2\\nonumber \\\\\\ & \\le \\sqrt{2}\\delta _{K_{t,i}+K_{t-1}+K}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\sqrt{2}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2.$ Combining the bounds (REF ) and (REF ), we obtain the following: $\\left\\Vert x^t_i - x\\right\\Vert _2 & \\le \\rho _{t,i}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma _{t,i}\\left\\Vert x_{\\overline{\\Gamma _{t,i}}}\\right\\Vert _2,$ where $\\rho _{t,i} = \\frac{\\sqrt{2}\\delta _{K_{t,i}+K_{t-1}+K}}{\\sqrt{1-\\delta _{K_{t,i}+K}^2}}$ and $\\gamma _{t,i} = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta _{K_{t,i}+K}^2}}$ .",
"It is now clear that the following holds true: $\\left\\Vert x^t - x\\right\\Vert _2 & \\le \\rho _{t}\\left\\Vert x^{t-1}-x\\right\\Vert _2 + \\gamma _{t}\\left\\Vert x_{\\overline{\\Gamma _{t}}}\\right\\Vert _2,$ where $\\rho _{t} = \\frac{\\sqrt{2}\\delta _{K_{t}+K_{t-1}+K}}{\\sqrt{1-\\delta _{K_{t}+K}^2}}$ and $\\gamma _{t} = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta _{K_{t}+K}^2}}$ , and $\\Gamma _{t}$ is the support corresponding to the (magnitude-wise) top $K_{t}$ entries of $x$ ."
],
[
"Proof of Proposition ",
"Let us denote, at time $t\\ge 0$ , $i=\\operatornamewithlimits{arg\\,min}_{j\\in \\lbrace 0,1\\rbrace }K_{t,j},\\ l = \\operatornamewithlimits{arg\\,max}_{j\\in \\lbrace 0,1\\rbrace }K_{t,j}$ .",
"The corresponding estimates produced are $x^t_i$ and $x^{t}_l$ with the respective supports $\\Lambda _{t,i}$ and $\\Lambda _{t,l}$ and errors $E_{t,i}=\\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2$ and $E_{t,l}=\\left\\Vert \\mathbf {P}_{\\Lambda _{t,l}}^\\perp y\\right\\Vert _2^2$ .",
"Step 4 of MCHTP guarantees that $\\Lambda _{t,i}\\subset \\Lambda _{t,l}$ .",
"Let us denote $\\Omega _{t}=\\Lambda _{t,l}\\setminus \\Lambda _{t,i}$ .",
"Then one finds that, $\\left\\Vert \\mathbf {P}_{\\Lambda _{t,l}}^\\perp y\\right\\Vert _2^2 & = \\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2 - \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}y\\right\\Vert _2^2,$ where $\\widetilde{\\mathbf {P}_{\\Omega _{t}}} & =\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^\\dagger \\nonumber \\\\\\ & =\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^{-1}\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp $ Let us denote, $\\Delta E_t = E_{t,i}-E_{t,l}$ .",
"Then, it is clear that $\\Delta E_t = \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}y\\right\\Vert _2^2$ .",
"We will now obtain a lower bound on $\\Delta E_{t}$ .",
"First note that, using the lower bound on the eigenvalues of $\\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^{-1}\\Phi _{\\Omega _{t}}^\\top $ , one obtains, $\\Delta E_t & = \\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}}(\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp \\Phi _{\\Omega _{t}})^{-1}\\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2 \\nonumber \\\\\\ & \\ge \\frac{\\left\\Vert \\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2^2}{(1+\\delta _{K_{t,l}})}.$ We will now derive a lower bound of $\\left\\Vert \\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2$ .",
"In order to do so, first observe that $\\mathbf {P}_{\\Lambda _{t,i}}^\\perp y = y - \\Phi x^t_i$ .",
"Using this, adding and subtracting $x^{t-1}$ and using the reverse triangle inequality, one obtains, $\\left\\Vert \\Phi _{\\Omega _{t}}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 & \\ge \\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Omega _t}\\right\\Vert _2\\nonumber \\\\\\ & - \\left\\Vert ((I-\\Phi ^\\top \\Phi )(x^t_i-x^{t-1}))_{\\Omega _t}\\right\\Vert _2,$ where in the last step we have used the fact that $[x^t_i]_j = 0,\\ \\forall j\\in \\Omega _t$ .",
"Now observe that the step 4 of MCHTP (Algorithm ) ensures that the sets $\\Lambda _{t,i}$ and $\\Lambda _{t,l}$ contain the top (magnitude wise) $K_{t,i}$ and $K_{t,l}$ entries, respectively, of the vector $x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1})$ .",
"Now depending on whether or not $K_{t,l}-K_{t,i}> K$ , there are two cases to consider.",
"If $K_{t,l}-K_{t,i}\\le K$ , then, for all $j=1,2,\\cdots , K_{t,l}-K_{t,i}$ , the $j^\\mathrm {th}$ maximum of the vector $x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1})$ from the set $\\Omega _t$ is greater than or equal to the $j^\\mathrm {th}$ maximum of $\\left|(x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))\\right|$ from the set $\\Lambda \\setminus \\Lambda _{t,i}$ .",
"On the other hand, if $K_{t,l}-K_{t,i}> K$ , then all the entries of the corresponding vector from the set $\\Omega _t$ are larger than the entries from the set $\\Lambda \\setminus \\Lambda _{t,i}$ .",
"Therefore, in any case, we have the following: ${\\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{\\Omega _t}\\right\\Vert _2} & & \\nonumber \\\\\\ & \\ge \\left\\Vert (x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1}))_{{\\Gamma }_{t,l,i}}\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{{\\Gamma }_{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & -\\left\\Vert ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1}))_{{\\Gamma }_{t,l,i}}\\right\\Vert _2,$ where $\\Gamma _{t,l,i}$ is the subset of $\\Lambda \\setminus \\Lambda _{t,i}$ which contains the $K_{t,l,i}:=\\min \\lbrace K_{t,l}-K_{t,i},K\\rbrace $ (magnitudewise) largest entries of $x^{t-1}+\\Phi ^\\top (y-\\Phi x^{t-1})$ .",
"We have furthermore used the fact that $[x^t_i]_j = 0,\\ \\forall j\\in \\Lambda \\setminus \\Lambda _{t,i}$ .Therefore, we obtain, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2} & &\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & -\\left\\Vert ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1}))_{\\Gamma _{t,l,i}}\\right\\Vert _2\\nonumber \\\\\\ & - \\left\\Vert ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1}))_{\\Omega _t}\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & - \\sqrt{2}\\left\\Vert \\bigg ((I-\\Phi ^\\top \\Phi )(x_i^t-x^{t-1})\\bigg )_{\\Omega _t\\cup \\Gamma _{t,l,i}}\\right\\Vert _2\\nonumber \\\\\\ & \\ge \\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2\\nonumber \\\\\\ & - \\sqrt{2}\\delta _{K_{t,l}-K_{t,i}+K_{t-1}+K}\\left\\Vert x_i^t-x^{t-1}\\right\\Vert _2.$ Now note that ${\\left\\Vert \\Phi _{\\Gamma _{t,l,i}}^\\top (y-\\Phi x_i^t))\\right\\Vert _2} & & \\nonumber \\\\\\ & \\ge \\frac{\\sqrt{\\left|\\Gamma _{t,l,i}\\right|}\\left\\Vert ((\\Phi ^\\top (y-\\Phi x_i^t)))_{\\Lambda \\setminus \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{\\left|\\Lambda \\setminus \\Lambda _{t,i}\\right|}}\\nonumber \\\\\\ & \\stackrel{(a)}{\\ge } \\frac{\\left\\Vert ((\\Phi ^\\top (y-\\Phi x_i^t)))_{\\Lambda \\setminus \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{K}}\\nonumber \\\\\\ & = \\frac{\\left\\Vert ((\\Phi ^\\top \\Phi (x_i^t-x)))_{\\Lambda \\setminus \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{K}}\\nonumber \\\\\\ & \\stackrel{(b)}{=}\\frac{\\left\\Vert ((\\Phi ^\\top \\Phi (x_i^t-x)))_{\\Lambda \\cup \\Lambda _{t,i}}\\right\\Vert _2}{\\sqrt{K}}\\nonumber \\\\\\ & \\ge \\frac{(1-\\delta _{K_{t,i}+K})\\left\\Vert x_i^t-x\\right\\Vert _2}{\\sqrt{K}},$ where the step $(a)$ follows from the fact that $\\left|\\Lambda \\setminus \\Lambda _{t,i}\\right|\\le \\left|\\Lambda \\right|=K$ and $\\left|\\Gamma _{t,l,i}\\right|=K_{t,l,i}=\\min \\lbrace K_{t,l}-K_{t,i},K\\rbrace \\ge 1$ .",
"Furthermore, step $(b)$ follows from the fact that $\\Phi ^\\top _{\\Lambda _{t,i}}(y-\\Phi x^t_i)=0_{\\Lambda _{t,i}}$ .",
"Moreover, $\\left\\Vert x_i^t-x\\right\\Vert _2 & \\ge \\min _{\\begin{array}{c}z\\in \\mathbb {R}^N:\\\\ \\left\\Vert z\\right\\Vert _{0}\\le K_{t,i}\\end{array}} \\left\\Vert z-x\\right\\Vert _2 = \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2.$ On the other hand, using Eq.",
"(REF ), we obtain, $\\left\\Vert x^t_i-x^{t-1}\\right\\Vert _2 & \\le \\left\\Vert x^t_i-x\\right\\Vert _2 + \\left\\Vert x^{t-1}-x\\right\\Vert _2.$ Therefore, we obtain, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y \\right\\Vert _2} & &\\nonumber \\\\\\ & \\ge \\left(\\frac{1-\\delta _{K_{t,i}+K}}{\\sqrt{K}} - \\sqrt{2}\\delta _{K_{t,l}-K_{t,i}+K_{t-1}+K}\\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2\\nonumber \\\\\\ & - \\sqrt{2}\\delta _{K_{t,l}-K_{t,i}+K_{t-1}+K}\\left\\Vert x^{t-1}-x\\right\\Vert _2.$ Note that $K_{t,i}\\ge 0,\\ K_{t,l},K_{t-1}\\le \\bar{K}$ .",
"Therefore, for simplicity, we denote $\\delta : = \\delta _{2\\bar{K}+K}$ and further lower bound the above as ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta \\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2} & &\\nonumber \\\\\\ & - \\sqrt{2}\\delta \\left\\Vert x^{t-1}-x\\right\\Vert _2.$ We now use the bound (REF ) recursively, and the fact that $x_0=0$ , to obtain the following upper bound of $\\left\\Vert x^{t-1}-x\\right\\Vert _2$ : $\\left\\Vert x^{t-1}-x\\right\\Vert _2 & \\le \\rho ^{t-1} \\left\\Vert x\\right\\Vert _2 + \\gamma \\sum _{j=1}^{t-1}\\rho ^{t-1-j}\\left\\Vert x_{\\overline{\\Gamma }_{j}}\\right\\Vert _2,$ where we have used the bound (REF ) of Proposition REF and have denoted $\\rho = \\frac{\\sqrt{2}\\delta }{\\sqrt{1-\\delta ^2}},\\ \\gamma = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta ^2}}$ Therefore, taking the bounds (REF ) and (REF ), we obtain ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta \\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\bigg (\\rho ^{t-1} \\left\\Vert x\\right\\Vert _2 + \\gamma \\sum _{j=1}^{t-1}\\rho ^{t-1-j}\\left\\Vert x_{\\overline{\\Gamma }_{j}}\\right\\Vert _2\\bigg ).$ Consider the first step of MCHTP.",
"In this step, $t=1$ and $x^0=0,\\Gamma _0=\\emptyset $ since $K_0=0$ , so that $x_{\\overline{\\Gamma }_{0}}=\\left\\Vert x\\right\\Vert _2$ .",
"Furthermore, $K_{1,i}=K_0=0$ .",
"Therefore, the bound (REF ) reduces to $\\left\\Vert \\Phi _{\\Omega _1}^\\top \\mathbf {P}_{\\Lambda _{1,i}}^\\perp y\\right\\Vert _2 & \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - 2\\sqrt{2}\\delta \\right) \\left\\Vert x\\right\\Vert _2.$ Therefore, at $t=1$ , $K_1=K_{1,l}$ if $\\Delta E_1\\ge \\epsilon $ , which always holds if $\\left(\\frac{1-\\delta }{\\sqrt{K}} - 2\\sqrt{2}\\delta \\right) \\left\\Vert x\\right\\Vert _2 & \\ge \\sqrt{\\epsilon (1+\\delta )}.$ Obviously, we require, $\\frac{1-\\delta }{\\sqrt{K}} - 2\\sqrt{2}\\delta >0$ , which is satisfied by the following: $\\delta < \\frac{1}{2\\sqrt{2K}+1}.$ Therefore, the conditions (REF ) and (REF ) ensure that $\\Delta E_1>0$ so that $K_{1}=K_{1,l}$ .",
"Now consider $t>1$ .",
"Using the fact that $\\Gamma _{t,i}\\subset \\Gamma _{t-1}$ since $ K_{t,i}\\le K_{t-1}$ , we obtain from the bound (REF ) the following: ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge \\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta (1+\\gamma )\\right) \\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2} & & \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\bigg (\\rho ^{t-1} \\left\\Vert x\\right\\Vert _2 + \\rho \\gamma \\sum _{j=1}^{t-2}\\rho ^{t-2-j}\\left\\Vert x_{\\overline{\\Gamma }_{j}}\\right\\Vert _2\\bigg ).$ For any $t>1$ , we can use the bounds $\\left\\Vert x_{\\overline{\\Gamma }_t}\\right\\Vert _2\\le \\sqrt{1-\\frac{K_{t}}{K}}\\left\\Vert x\\right\\Vert _2\\le \\sqrt{1-\\frac{1}{K}}\\left\\Vert x\\right\\Vert _2$ , since $K_t\\ge 1$ and $\\left\\Vert x_{\\overline{\\Gamma }_{t,i}}\\right\\Vert _2\\ge \\sqrt{K-K_{t,i}}x_{\\min }\\ge x_{\\min }$ since $K_{t,i}\\le K-1$ .",
"Furthermore, assuming $\\delta <\\frac{1}{\\sqrt{2K}+1}<\\frac{1}{\\sqrt{3}}$ , we have $\\rho <1$ (since we $K\\ge 1$ ) (we will later obtain conditions which indeed implies this bound on $\\delta $ ).",
"With the above observations, we simplify the lower bound of (REF ) to the following, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2} & & \\nonumber \\\\\\ & \\ge x_{\\min }\\left(\\frac{1-\\delta }{\\sqrt{K}} - \\sqrt{2}\\delta (1+\\gamma )\\right) \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\left\\Vert x\\right\\Vert _2\\bigg (\\rho ^{t-1} + \\frac{\\rho \\gamma \\sqrt{1-\\frac{1}{K}}(1-\\rho ^{t-2})}{1-\\rho }\\nonumber \\\\\\ & + \\frac{\\rho \\left\\Vert e\\right\\Vert _2}{(1-\\rho )\\left\\Vert x\\right\\Vert _2 }\\bigg ).$ Now, note that using the bound $\\delta <\\frac{1}{\\sqrt{2K+1}}$ , we obtain, for $K\\ge 2$ , $\\frac{\\gamma \\sqrt{1-\\frac{1}{K}}}{1-\\rho } & = \\frac{\\sqrt{2}\\sqrt{1-\\frac{1}{K}}}{\\sqrt{1-\\delta ^2}-\\sqrt{2}\\delta }\\nonumber \\\\\\ & \\le \\frac{\\sqrt{2K+1}\\sqrt{K-1}}{\\sqrt{K}(\\sqrt{K}-1)}< 4,$ and $\\frac{1}{1-\\rho } = \\frac{\\sqrt{1-\\delta ^2}}{\\sqrt{1-\\delta ^2}-\\sqrt{2}\\delta } & \\le \\frac{\\sqrt{K}}{\\sqrt{K}-1}\\le 2 + \\sqrt{2}<4,$ and, $\\gamma & = \\frac{\\sqrt{2}}{\\sqrt{1-\\delta ^2}} \\le \\frac{\\sqrt{2K+1}}{\\sqrt{K}}\\le \\sqrt{2.5}<2,\\nonumber \\\\\\rho & = \\frac{\\sqrt{2}\\delta }{\\sqrt{1-\\delta ^2}} \\le \\frac{\\sqrt{2}}{\\sqrt{2K}} = \\frac{1}{\\sqrt{K}}.$ Therefore from (REF ), we obtain, for $t>1$ , ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 \\ge x_{\\min }\\left(\\frac{1-\\delta }{\\sqrt{K-1}} - 3\\sqrt{2}\\delta \\right)} & & \\nonumber \\\\\\ & - \\sqrt{2}\\delta \\frac{\\left\\Vert x\\right\\Vert _2}{\\sqrt{K}}\\bigg (\\frac{1}{K^{t/2-1}} + 4 + \\frac{4\\left\\Vert e\\right\\Vert _2}{\\left\\Vert x\\right\\Vert _2}\\bigg ),$ which implies that, ${\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 > x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg )} & &\\nonumber \\\\\\ & - x_{\\max }\\sqrt{2}\\delta \\bigg (4+\\frac{1}{K^{t/2-1}}\\bigg ).$ We can further lower bound the right hand side of the above inequality by using $K_{t-1}\\le K - 1$ and $K^{t/2-1}\\ge 1$ for $t\\ge 2$ , to obtain the following lower bound whenever $K_{t,i}<K$ : $\\left\\Vert \\Phi _{\\Omega _t}^\\top \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2 > x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg ) & - x_{\\max }5\\sqrt{2}\\delta .$ Therefore, for $t>1$ , if $K_{t,i}<K$ , then $K_t=K_{t,l}$ if $\\Delta E_t\\ge \\epsilon $ , which always holds if $\\ & x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg ) - 5\\sqrt{2}\\delta x_{\\max }\\ge \\sqrt{\\epsilon (1+\\delta )}.$ To make the LHS of (REF ) positive, we obviously require $x_{\\min }\\bigg (\\frac{1-\\delta }{\\sqrt{K}} - 3\\sqrt{2}\\delta \\bigg ) - x_{\\max }5\\sqrt{2}\\delta >0$ , which is ensured by the following: $\\delta & < \\frac{1}{1+(3+5R)\\sqrt{2K}},$ where $R = \\frac{x_{\\max }}{x_{\\min }}$ .",
"Therefore, the conditions (REF ) and (REF ) ensure that, for any $t>1$ , if $K_{t,i}<K$ , then $\\Delta E_t>0$ and $K_t= K_{t,l}$ .",
"Furthermore, observe that the condition (REF ) implies the conditions (REF ).",
"Therefore, under the condition (REF ), we have the following lower bound for $\\Delta E_t$ $\\Delta E_t & \\ge \\frac{(1-\\delta )}{(1+\\delta )^2}\\left(a_K(\\delta )x_{\\min } - b(\\delta )x_{\\max }\\right)^2,$ where $a_K(\\delta ),b(\\delta )$ are defined in Eq.",
"(REF ).",
"Therefore, the condition (REF ) ensures that $\\Delta E_t>\\epsilon $ , which in turn guarantees that $K_t = K_{t,l}$ , i.e., the larger sparsity value is chosen for time step $t$ ."
],
[
"Proof of Proposition ",
"The Phase II continues until the sequence of estimated sparsity becomes a non-increasing sequence.",
"Now, by the step 11 of MCHTP, if the current sparsity estimate is $K_{t-1}$ and $K_{t,1}$ is the sparsity sampled at time $t$ , $K_t = \\min \\lbrace K_{t-1}, K_{t,1}\\rbrace $ if $\\Delta E_t<\\epsilon $ .",
"However, if $K_{t,1}<K$ , thanks to Proposition REF , $\\Delta E_t\\ge \\epsilon $ and $K_{t}=K_{t-1}$ since $K_{t-1}\\ge K$ after Phase I.",
"On the other hand, if $K_{t,1}\\ge K$ , then $K_t=\\min \\lbrace K_{t-1}, K_{t,1}\\rbrace $ if one has $\\Delta E_t< \\epsilon $ .",
"Therefore, it is enough to find conditions which guarantee that $\\Delta E_t<\\epsilon $ , whenever $K_{t,1}\\ge K$ , to ensure that the sequence $K_t$ after the end of Phase II, is a non-increasing sequence.",
"Now note that, ${\\sqrt{\\Delta E_t} = \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}y\\right\\Vert _2 = \\left\\Vert \\widetilde{\\mathbf {P}_{\\Omega _{t}}}\\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2\\le \\left\\Vert \\mathbf {P}_{\\Lambda _{t,i}}^\\perp y\\right\\Vert _2} & &\\nonumber \\\\\\ & =\\left\\Vert \\Phi (x\\right\\Vert _2 \\le \\sqrt{1+\\delta }\\left\\Vert x^{t}_i - x\\right\\Vert _2.$ Note that Phase II of MCHTP ends at time $T_1+T_2-1$ and Phase III begins at time $T_1+T_2$ when $\\Delta E_t\\le \\epsilon $ for all $t\\ge T_1+T_2$ , whenever $K_{t,1}\\ge K$ .",
"This is ensured if the following is satisfied at such a time instant $t$ : $\\left\\Vert x^{t}_i - x\\right\\Vert _2 & \\le \\frac{\\sqrt{\\epsilon }}{\\sqrt{1+\\delta }}.$ Since $K_{t,1}\\ge K$ , and $K_{t-1}\\ge K$ by Porposition REF , we have $K_{t,i}\\ge K$ .",
"Therefore, from Proposition REF we obtain $\\left\\Vert x^{t}_i - x\\right\\Vert _2 & \\le \\rho \\left\\Vert x^{t-1}-x\\right\\Vert _2\\nonumber \\\\\\ & \\le \\rho ^{t-T_1+1}\\left\\Vert x^{T_1-1}-x\\right\\Vert _2\\nonumber \\\\\\ & \\le \\rho ^{t-T_1+1}\\sqrt{\\frac{1+\\delta }{1-\\delta }}\\left\\Vert x\\right\\Vert _2.$ Consequently, it follows that, if $T_2$ is the duration of Phase II of MCHTP, then $T_2\\le \\tau $ , where, $\\tau & =\\min \\left\\lbrace t\\ge 0:\\rho ^{t+1}\\sqrt{\\frac{1+\\delta }{1-\\delta }}\\left\\Vert x\\right\\Vert _2\\right.\\nonumber \\\\\\ & \\le \\left.\\frac{{\\sqrt{\\epsilon }}}{\\sqrt{1+\\delta }}\\right\\rbrace = \\min \\left\\lbrace t\\ge 0:\\rho ^{t+1}\\le G\\right\\rbrace ,$ where $G & = \\frac{\\sqrt{\\epsilon (1-\\delta )}}{(1+\\delta )\\left\\Vert x\\right\\Vert _2}$"
]
] | 2210.07800 |
[
[
"Learning to Jointly Transcribe and Subtitle for End-to-End Spontaneous\n Speech Recognition"
],
[
"Abstract TV subtitles are a rich source of transcriptions of many types of speech, ranging from read speech in news reports to conversational and spontaneous speech in talk shows and soaps.",
"However, subtitles are not verbatim (i.e.",
"exact) transcriptions of speech, so they cannot be used directly to improve an Automatic Speech Recognition (ASR) model.",
"We propose a multitask dual-decoder Transformer model that jointly performs ASR and automatic subtitling.",
"The ASR decoder (possibly pre-trained) predicts the verbatim output and the subtitle decoder generates a subtitle, while sharing the encoder.",
"The two decoders can be independent or connected.",
"The model is trained to perform both tasks jointly, and is able to effectively use subtitle data.",
"We show improvements on regular ASR and on spontaneous and conversational ASR by incorporating the additional subtitle decoder.",
"The method does not require preprocessing (aligning, filtering, pseudo-labeling, ...) of the subtitles."
],
[
"Introduction",
"Speech recognition aims to identify the words in a spoken audio fragment and transcribe what has been said into written text.",
"Traditionally, Automatic Speech Recognition (ASR) systems made use of a limited vocabulary of words and pronunciations, and consisted of several components to model speech statistically.",
"More recently, data-driven end-to-end ASR systems have been able to directly convert a sequence of audio features into a sequence of (sub)words or characters.",
"ASR requires a large dataset with careful transcriptions of spoken utterances.",
"While TV subtitles seem the obvious candidate, as they are so abundant and manually annotated, they are not verbatim (i.e.",
"exact) transcriptions of what was said.",
"Subtitles are optimised for readability on screen, and are therefore often shorter and cleaner (e.g.",
"no repetitions, no hesitations).",
"Consequently, the timings of subtitles on screen can be inaccurate compared to the spoken sentences.",
"On top of that, subtitles are “clean\", with edits to correct non-grammatical sentences and poor or dialectal word choice.",
"Nevertheless, being able to use TV subtitles would have many advantages.",
"Subtitles can cover a very broad domain, from read speech in broadcast news to spontaneous and conversational speech in talk shows, interviews, soaps, sitcoms, etc.",
"This is in contrast to the labeled speech datasets which often contain prepared speech (e.g.",
"audiobooks in LibriSpeech [1]).",
"Subtitles cover many languages, and they have been manually annotated by human annotators.",
"Additionally, accented speech and strong dialects are more common among free speech on TV, and less covered in current speech datasets.",
"Reducing the need for well-labeled speech data is a long-standing problem.",
"The performance of ASR has seen tremendous improvements from pre-training on unlabeled data in a self-supervised manner [2], [3], [4].",
"Unlabeled speech data is plentifully available, and can even be (pseudo-)labeled by pre-trained models to iteratively improve models by self-training [5].",
"Furthermore, pre-training reduces the need for large amounts of labeled data and has significantly advanced performance in low resourced languages, on account of cross- and multilingual positive transfer [6] from learned high-level acoustic feature representations to the target language.",
"In this work, we propose a model that jointly performs automatic (verbatim) speech recognition and automatic subtitling.",
"Besides solving two useful tasks at once, the model is able to make use of the subtitle data.",
"We apply a multitask learning (MTL) framework [7], [8] with a shared encoder and two decoders, inspired by end-to-end speech translation where machine translation and ASR are additional subtasks with independent [9] or interconnected decoders [10], [11] for each task.",
"We show that the introduction of the subtitle decoder and subtitle data improves the performance of the end-to-end ASR model on both a regular ASR benchmark and on a spontaneous speech dataset, without construction of a spontaneous ASR dataset.",
"Experiments are performed on Belgian Dutch (Flemish), a medium-resourced language with strong regional dialects.",
"Finally, we note that parallel work on pre-training with unlabeled data for learning representative features (e.g.",
"for Belgian Dutch [12]), is complementary to this work, and these pre-trained feature representations can be plugged into the front-end of an ASR system."
],
[
"Related work",
"Most research on the use of subtitle data can be divided into two strands.",
"On the one hand, subtitles (either detected on screen in video's [13] or gathered separately) are refined and strongly filtered to build a corpus for ASR [14], [15], e.g.",
"based on matching alignment [16], [17], [18] or thresholding on posterior probability [19].",
"On the other hand, the output of ASR systems can be adapted with models into readable subtitles, by compression [20] or with a (genre-dependent) language model [21], or directly translated into subtitles in a different language [22].",
"The Multi-Genre Broadcast (MGB) challenge inspired some of these works, with the MGB-1 challenge [23] providing English subtitles and the MGB-2 challenge [24] Arabic subtitles, and related competitions, e.g.",
"the IberSpeech-RTVE challenge on Spanish Broadcast media [25].",
"Finally, there has also been research on using weak supervision from contextual text information in social media video's to improve ASR [26], however these annotations are further away from verbatim transcriptions than subtitles.",
"The work in this paper distinguishes itself by concurrently solving both the ASR and subtitling task, and, without having to adapt or refine the data (which can be done complementary to our method in a pre-processing stage, e.g.",
"subtitle alignment [27]), improving the current ASR system in all domains.",
"No parallel data is used, i.e.",
"data for which there are both verbatim and subtitle transcriptions, as it is not generally available in most languages."
],
[
"Method",
"The proposed model adapts the end-to-end hybrid CTC-Attention ASR model, introduced in [28], to the multitask learning framework, by attaching a second decoder for subtitle generation and sharing the encoder.",
"Figure: Model architecture: comparison of (a) a general end-to-end CTC/Attention hybrid ASR model, and (b) the proposed multitask model with two decoders.",
"In (c) the proposed multitask model has additional cross-connections between both decoders."
],
[
"End-to-end ASR",
"End-to-end (E2E) models solve the ASR problem as a sequence-to-sequence problem.",
"First of all, an encoder computes feature representations from the input sequence.",
"The feature extractor in this case is a Transformer [29], relying on self-attention [30] to compute feature representations.",
"In general, end-to-end architectures either use Connectionist Temporal Classification (CTC) [31] to generate an output sequence from the encoder states, or an attention-based mechanism that can attend to different parts of the encoder output to align the input features with the output sequence.",
"Both methods have their strengths and drawbacks.",
"CTC, although efficient, relies on independent Markov assumptions and can only predict monotonic alignments.",
"Pure attention-based methods are hard to train and optimise as there is no constraint on possible alignments.",
"In this work, we adopt the hybrid CTC/attention architecture [28], that combines both methods, in the baseline ASR system.",
"Figure REF (a) shows the hybrid model.",
"CTC regularises the encoder by constraining a monotonic alignment and predicting an aligned sequence of output tokens including blanks from the encoder outputs.",
"The encoder outputs could resemble small acoustic units because of the regularisation.",
"The attention-based decoder [29] learns the alignment between the encoder states and the elements of the output sequence.",
"It can attend to the entire encoder output and during decoding the predicted tokens are generated autoregressively, i.e.",
"one-by-one.",
"The training objective is a weighted sum of the label smoothed cross-entropy classification loss of the tokens generated by the attention decoder and the CTC objective, respectively denoted by $\\mathcal {L}_{att,asr}$ and $\\mathcal {L}_{ctc}$ .",
"$\\mathcal {L}_{asr} = (1-\\lambda _{ctc}) \\mathcal {L}_{att,asr} + \\lambda _{ctc} \\mathcal {L}_{ctc}$ During decoding, the CTC and attention scores are combined in a joint beam search where the produced hypotheses are pre-scored by CTC."
],
[
"Dual decoder - MTL",
"The proposed model is depicted in Figure REF (b).",
"An additional attention-based decoder branch is added to the encoder.",
"In this multitask model, the encoder is shared but the decoders are completely independent and they both attend to the encoder's states (as the multitask model in [9]).",
"When an auxiliary task is imposed to a model, the performance on the original task might increase if the additional task is useful.",
"The inclusion of a second independent decoder could benefit the learned representation in the shared encoder.",
"The first decoder, the ASR Decoder, is the same as in the previous section and generates a verbatim transcription.",
"The second decoder, the Subtitle Decoder, generates a subtitle for the input features.",
"Due to summarisation and compression, subtitles align differently than verbatim transcriptions.",
"Hence we use an attention-based loss without CTC for the subtitles.",
"The final training objective $\\mathcal {L}_{tot}$ is now a weighted sum of both decoders' attention losses and the CTC objective, with $\\mathcal {L}_{asr}$ as defined in Equation REF .",
"$\\mathcal {L}_{tot} = \\lambda _{asr}\\mathcal {L}_{asr} + \\lambda _{subs} \\mathcal {L}_{att,subs}$ During training, batches are filled with utterances from the ASR dataset (i.e.",
"verbatim targets) and utterances from the subtitle dataset, equally mixed per batch.",
"The ASR decoder and CTC losses are backpropagated for the verbatim utterances and masked for the subtitled utterances.",
"Likewise, the subtitle decoder loss is backpropagated for the subtitled data and masked for the ASR data."
],
[
"Connected decoders",
"As ASR and subtitling are related tasks, it would seem useful to have some kind of connection between both decoders.",
"Knowing what was said is useful to generate a subtitle, and vice versa.",
"Chaining or cascading the decoders one after another, e.g.",
"an ASR decoder followed by a subtitle decoder, is difficult with autoregressive decoders.",
"Therefore, we investigate the Cross Dual-decoder Transformer proposed in [10].",
"In this model, there are additional cross-attention blocks added between the decoders, where the decoders can also attend to the previous outputs of the other decoder.",
"Both autoregressive decoders process the data simultaneously and work in a synchronous way.",
"The output of the cross-attention with the other decoder is summed together with or concatenated to the outputs of the decoder's own attention operations [10].",
"The model with cross-connected decoders, depicted in Figure REF (c), requires parallel data for both the ASR and the subtitle decoder to learn useful cross-attention weights.",
"Therefore, we first generate pseudo-labels with a pre-trained multitask model with independent decoders, i.e.",
"verbatim labels for the subtitle data and subtitles for the ASR data.",
"Second, we finetune the model with additional cross-connections between the decoders on the parallel (pseudo-)labeled data.",
"This is an extension of iterative self-training [5] with pseudo-labels.",
"During decoding, the ASR and subtitle hypotheses are expanded together as tuples of (ASR, subtitle) combinations in one joint beam.",
"Only valid combinations for the next tokens are considered.",
"The scores per tuple are computed as a weighted combination of the ASR and subtitle hypotheses' scores during the beam search.",
"Equal weights for the ASR and subtitles appear to give the best results."
],
[
"Data",
"The multitask model is trained on both verbatim ASR data, from an official database, and subtitled data gathered in-house.",
"Verbatim.",
"The Corpus Gesproken Nederlands (CGN) [32], or Spoken Dutch Corpus, is a speech database containing 270 hours of Belgian Dutch (Flemish) speech.",
"The complete dataset includes read speech, news reports, lectures, commentaries, and also narrowband telephone speech (8 kHz resampled to 16 kHz) and spontaneous conversational speech (VL-train-all in [12]).",
"For evaluation, we use the dev-other set of 15h of speech, named cgn-dev, which does not contain telephone and conversational speech, and has no overlap in speakers with the training set.",
"Subtitled.",
"The subtitled dataset contains 720 hours of speech, segmented based on the subtitle timings on screen.",
"The data consists of 190 hours of broadcast TV news, 230 hours of soap TV series, and the remaining 300 hours of speech originate from various talk shows about news, politics, entertainment, etc.",
"For evaluation, we manually annotated a set of 6 hours of speech (around 6500 utterances), named subs-annot, from a mixture of show types, and keep it held out of the training set.",
"Finally, for one experiment we have extended the subtitled dataset to 1700 hours of speech gathered from the same and similar sources."
],
[
"Model configuration",
"The models are implemented with the ESPnet2 library [33] and follow the standard encoder-decoder architecture of end-to-end ASR models.",
"The encoder is a 12-layered Transformer [29] with a Conv2D input layer to transform the input features, and an output dimension of 256.",
"The ASR decoder and the subtitle decoder are both 6-layered Transformer decoders.",
"Every Transformer block has 4 attention heads and 2048 linear units.",
"The input features are 80-dimensional Mel filterbank features, concatenated with 3-dimensional pitch features, extracted from windows of 25ms and with a frame shift of 10ms.",
"Utterance-level mean-variance normalisation and SpecAugment [34] augmentations are applied to the inputs.",
"The targets are 5000 unigram BPE's, and the BPE (Byte-Pair Encoding) model is trained on both the ASR transcriptions and the subtitles.",
"No language model is applied, not during and not after decoding."
],
[
"Training details",
"All models are trained for 100 epochs with a batch size of 32 and gradient accumulation over 8 steps.",
"We use the Adam optimizer [35] with a learning rate schedule for Transformers [29] with 25000 warm-up steps and a peak learning rate of 0.004 for models trained from scratch and 0.0004 for ASR-initialised models.",
"For evaluation, we average the 10 models with the best ASR accuracy on the validation set.",
"The CTC weight $\\lambda _{ctc}$ is set to 0.3, both during training and decoding.",
"The cross-entropy label smoothing weight is set to 0.1.",
"Decoding is implemented as beam search with beam size 20."
],
[
"Results",
"In this section we will describe the experiments conducted to evaluate the proposed model and discuss the results.",
"We examine the performance of the multitask model on a speech recognition task, and compare it to the ASR model without subtitle decoder.",
"The models are evaluated based on the Word Error Rate (WER) on a regular ASR test set cgn-dev and on the spontaneous test set subs-annot.",
"Furthermore, the capabilities of automatic subtitling of the multitask model are explored by comparing the output hypothesis of the subtitle decoder to the real subtitle on screen.",
"As metric, we utilise the smoothed BLEU-4 score [36] with uniform weights.",
"Finally, we investigate if finetuning with generated pseudo-labels can yield further improvements.",
"We compare the multitask model with independent decoders to the multitask model with cross-attention connected decoders on ASR and the subtitling task."
],
[
"Multitask model from scratch",
"First of all, we examine the effect of adding the additional subtitle decoder to the ASR model.",
"The importance of both decoders can be weighted in the loss function (Equation REF ) with the hyperparameters $\\lambda _{subs}$ and $\\lambda _{asr}$ .",
"Table REF shows the results for different combinations of both weights.",
"The first line, i.e.",
"the model with $\\lambda _{subs}=0$ , is the baseline ASR model.",
"It is important to note that training on the joint set of ASR and subtitle data, as if it were all verbatim data, always diverges, irrespective of the learning rate.",
"We postulate that the inexact transcripts could impair the CTC alignment and the classification, and the inaccurate timings are difficult to incorporate in verbatim speech recognition.",
"This is the key advantage of the proposed multitask model.",
"Table REF shows that the additional decoder can slightly improve the WER on the regular cgn-dev and strongly improve the WER on the spontaneous subs-annot, because the encoder is improved by incorporating the subtitle data for the subtitling task.",
"Using equal weights in the loss function seems most beneficial in this setup.",
"If the weight of the subtitle decoder $\\lambda _{subs}$ is too large, the model tends to diverge.",
"It is a hyperparameter that needs some tuning, but always improves results when chosen between a wide range.",
"Table: WER for multitask models trained from scratch with varying weights in the objective function.",
"The first row shows the single decoder ASR model as baseline.",
"Diverged models are annotated with (div).To get a sense of the difference between the ASR and subtitle decoder output, Table REF shows a sample utterance transcription from the spontaneous test set and the generated hypotheses from both decoders, with the model in Table REF using $\\lambda _{subs}=0.5$ .",
"Although for this utterance the reference subtitle is much shorter and extremely hard to predict, it is clear that the subtitle decoder output is cleaner, e.g.",
"no apostrophies for abbreviating words and no hesitations (`uh').",
"We found this to be a recurring theme during analysis of the subtitle output, with additional normalisation of dialectal words (e.g.",
"`gij' becomes `jij') and the decoder leaving out repetitions due to stammering.",
"Table: Comparison of the reference ASR transcription and the reference subtitle transcription to the hypotheses generated by the outputs of the ASR decoder (ASR) and the subtitle decoder (SUB).",
"The utterance is part of the subs-annot test set.",
"Translation in English is added for the reader."
],
[
"Adapting pre-trained ASR models to multitask models",
"Second, we examine if a pre-trained single-decoder ASR model can be improved with the additional decoder, or if it would harm the result on the ASR task.",
"The ASR model is a state-of-the-art E2E ASR model [28].",
"Table REF shows the results when the encoder is initialised with a pre-trained ASR encoder, and the ASR and/or the subtitle decoder branch are initialised with the parameters from the decoder of a pre-trained ASR model.",
"The baseline model of the previous section is used as pre-trained ASR model to initialise the weights.",
"The WER on both sets strongly improves by extending a pre-trained ASR model with the additional subtitle decoder and using the subtitle data.",
"Both the ASR decoder and subtitle decoder branch can be initialised with an ASR model.",
"Table: WER for multitask models with initialisation of the ASR decoder (ASR) and/or subtitle decoder (Subs) with the weights of a pre-trained ASR decoder.",
"The models in the first two rows are not initialised and trained from scratch.",
"The models in the bottom two rows both use a pre-trained ASR encoder.",
"The subtitle dataset contains 700h of data."
],
[
"Using more subtitled data",
"Finally, we examine if using more subtitled data would additionally improve the results.",
"We compare the ASR performance of a model using 700 hours of subtitled data in Table REF to a model using 1700 hours of subtitled data in Table REF .",
"The verbatim ASR dataset remains the same.",
"The results on 1700 hours (using the same settings as in the previous experiments) are comparable to the results on 700 hours of subtitled data, but not better.",
"We notice the same trend of improvements across the board for the different models.",
"The first line of the table suggests that further finetuning of $\\lambda _{asr}$ might yield further improvements.",
"Table: WER for multitask models trained on the extended 1700h subtitle dataset, with or without pre-trained initialisation."
],
[
"Subtitling",
"We evaluate the output of the subtitle decoder with a BLEU score against the reference on-screen subtitle in the test set.",
"Table REF summarises the results of all previous models.",
"For the baseline (i.e.",
"the first row), we use the output of the ASR decoder.",
"This gives a notice of how close the subtitles are to verbatim transcriptions.",
"For almost all cases, the improvements in BLEU score follow the improvements in WER as in the previous section.",
"The double-initialised dual-decoder model achieves the highest BLEU scores.",
"Additionally, we notice that for most cases there is no improvement with the model using 1700 hours of subtitle data over the model using 700 hours of subtitle data.",
"This might suggest that the additional 1000 hours of data, mostly coming from different sources, is slightly mismatched with the test set or contains more non-Dutch transcriptions.",
"Table: BLEU scores for multitask models with varying weights in the objective function and possible initialisation.",
"The first row shows the single-decoder ASR model.",
"Diverged models are annotated with (div)."
],
[
"Connected decoders",
"Section REF showed a method to connect both decoders of the multitask model.",
"Because the connected decoders require parallel data, we generate pseudo-labels with the ASR decoder for the subtitled data and with the subtitle decoder for the verbatim data.",
"For pseudo-labeling we decode with the best-performing model, i.e.",
"the model of the last row in Table REF , which is also used for initialisation of all weights except for the weights in the cross-attention part between the decoders.",
"The loss weights $\\lambda _{asr}$ and $\\lambda _{subs}$ are both kept as 0.5.",
"We compare the multitask model with cross-connected decoders to the multitask model with independent decoders (as in previous sections), starting from the same initialisation and further trained with pseudo-labels.",
"Table REF shows the ASR results and Table REF shows the subtitling results.",
"Further finetuning of the multitask model with independent decoders on the generated pseudo-labels (row 2 in Table REF ), starting from the pre-trained multitask model (row 1 in Table REF ), slightly improves the WER on both sets, at the expense of a drop in BLEU score.",
"Finetuning the pre-trained multitask model (row 1 in Table REF ) with additional cross-attention connections between the decoders is not able to improve the WER on the regular cgn-dev test set.",
"This might be because the ASR decoder cross-attention and the Subtitle-to-ASR cross-attention are merged (either summed up or concatenated and linearly projected), which could implicitly give an extra weight to the subtitles in the ASR decoder branch, which might have a slight negative impact on the ASR decoder in very clean settings due to the domain mismatch.",
"However, when the encoder is kept frozen during finetuning and only the decoders are finetuned, we do obtain the best ASR performance on the spontaneous subs-annot test set and by far the best BLEU score.",
"Table: WER for multitask models with independent decoders and with cross-connected decoders.",
"The decoder attention outputs and the inter-decoder cross-attention outputs can be concatenated or summed.",
"The models can be further trained with pseudo-labels (PL) generated by the decoders for missing targets.",
"The encoder can be kept frozen during finetuning with the pseudo-labels.Table: BLEU scores for multitask models with independent decoders and with cross-connected decoders.",
"The decoder attention outputs and the inter-decoder cross-attention outputs can be concatenated or summed.",
"The models can be further trained with pseudo-labels (PL) generated by the decoders for missing targets.",
"The encoder can be kept frozen during finetuning with the pseudo-labels."
],
[
"Conclusion",
"We have proposed a model for joint automatic speech recognition and subtitle generation, that is able to leverage subtitled data to improve ASR performance on both regular and spontaneous speech through multitask learning of both tasks.",
"A pre-trained ASR model can easily be adapted with a second decoder, and its performance can be improved even with the inaccurate subtitle transcriptions.",
"In this work, we use a dual-decoder model with independent autoregressive decoders and a dual-decoder model with synchronously cross-connected autoregressive decoders.",
"In future work, we will investigate non-autoregressive decoders to chain the decoders in a cascaded way.",
"We will also look at the MGB challenge for evaluation of the models.",
"The multitask models might also be improved through better alignment of the subtitles' endpoints with the input speech."
],
[
"Acknowledgement",
"This research received funding from the Flemish Government under the “Onderzoeksprogramma Artificiële Intelligentie (AI) Vlaanderen\" programme and was supported by Research Foundation Flanders (FWO) under grant S004923N of the SBO programme."
]
] | 2210.07771 |
[
[
"Simpson's Paradox in Recommender Fairness: Reconciling differences\n between per-user and aggregated evaluations"
],
[
"Abstract There has been a flurry of research in recent years on notions of fairness in ranking and recommender systems, particularly on how to evaluate if a recommender allocates exposure equally across groups of relevant items (also known as provider fairness).",
"While this research has laid an important foundation, it gave rise to different approaches depending on whether relevant items are compared per-user/per-query or aggregated across users.",
"Despite both being established and intuitive, we discover that these two notions can lead to opposite conclusions, a form of Simpson's Paradox.",
"We reconcile these notions and show that the tension is due to differences in distributions of users where items are relevant, and break down the important factors of the user's recommendations.",
"Based on this new understanding, practitioners might be interested in either notions, but might face challenges with the per-user metric due to partial observability of the relevance and user satisfaction, typical in real-world recommenders.",
"We describe a technique based on distribution matching to estimate it in such a scenario.",
"We demonstrate on simulated and real-world recommender data the effectiveness and usefulness of such an approach."
],
[
"Introduction",
"How can we understand and measure recommender performance and provider fairness across the broad diversity of users and requests that they serve?",
"In recent years, the research community has recognized the importance and challenge of evaluating the fairness of recommender systems, with numerous proposals being put forth [41], [7], [10], [4], [47], [8], [33], [9].",
"In this paper, we focus on the goal of provider-side fairness [41], [7], [10], [4], [47]: we assume that items or providers have different sensitive groups and want to measure whether the recommender provides equally good ranking/exposure for items of different groups.",
"Across many domains, from music recommendation [33] to advertising [12] to job search [14], [46], such a property is of deep importance to the items and their producers.",
"We study a recommender system which is responsible for ranking items for different users.",
"Here, the term “user” refers to the recommendation event and is a similar concept as “requests” or “queries” in a recommender system.",
"As users typically are exposed to the highest ranked items, the recommender plays an important role to allocate the traffic to items of different providers.",
"In this context, there has recently been a proliferation of metrics proposed to capture the provider-side fairness of recommender systems [4], [34], [41], [7], [10], [47].",
"Many papers analyze the exposure allocated to items and propose that items from different groups should have equal exposure, usually with a relation to their relevance.",
"However we note one significant difference between previous metrics, which previous papers and reviews [36] have overlooked: some prior work compares the exposure of items per-user (or \"per-query\") [41], [11], [7], [47], [14] while others aggregate across users [4], [34].",
"Our first contribution is to highlight how this decision can lead to significant practical differences.",
"Figure: Simpson's paradox (two contradictory observations): For each user, Orange seems disadvantaged (ranked lower than Blue).",
"However, aggregated across users, Orange is advantaged (Orange's mean positionis 4.3 compared to 4.7 for Blue).Explanation: Some of the user factors might affect exposure.",
"In particular, the system ranks the relevant items (both Orange and Blue) in low position for casual users, for which Blue are frequent.For simplification of this example, we use the mean position as the exposure of a given group Let's make it more concrete through a motivating example in which we consider two commonly used metrics from the fairness literature: Per-user: For each user, relevant items created by Group 0 providers should have an equal chance of exposure as relevant items from Group 1 providers [41], [11], [7], [47], [14].",
"Aggregate: Aggregated across all users, relevant items created by Group 0 providers should have an equal chance of exposure as relevant items from Group 1 providers [4], [34].",
"As both of these metrics seem intuitively similar and are supported by past literature, practitioners might choose between them arbitrarily.",
"The first key contribution of this paper is the surprising finding that these metrics can actually point in opposite directions in practice due to a Simpson’s paradox.",
"Indeed, in Figure REF , according to the per-user metric, Group 0 providers are advantaged (they are ranked higher than items from Group 1 on any user) while the aggregate metric indicates that Group 0 providers are disadvantaged (these items have lower exposure in aggregate).",
"How can we explain these seemingly-contradictory observations?",
"We point out that the aggregate metric is influenced by which users that each group is relevant.",
"Indeed, when aggregating, we are comparing two data samples (the set of relevant items from each group) which follow different user distributions.",
"Hence these differences of user distributions might influence the comparison, similar to a confounding variable in causal literature [40], [44], [6].",
"For example, if one group of items is more often relevant to frequent users than casual users (where the recommender has lower performance, i.e., putting relevant items in low positions), that group of items could seem to be advantaged in an aggregate evaluation while still facing a per-user gap, which is the scenario of Figure REF .",
"Alternatively, if one group of items is more relevant to users with broad interests for who many items are relevant (hence competing for exposure), the aggregate evaluation may suggest a bias against that group, when this is in fact due to this higher competition.",
"We derive theoretical equations to establish that, indeed, the aggregate metric is influenced by the per-user metric but also by the user distributions (the competition and performance factors).",
"Given this new understanding, one may be interested in either per-user or aggregate evaluations depending on the application.",
"While the aggregate metric can be computed under most scenarios, past literature on per-user evaluation [41], [11], [7], [47], [14] generally assumes that the relevance for every item is observed.",
"However, in real-world recommenders, we often get user feedback on one, or a few items, out of a set of recommendations, as each user at each moment in time is unique.",
"As a result, we rarely get to observe the relevance of multiple items on the same user, which does not allow to directly compute the per-user metric.",
"To enable practitioners, we describe an approach to overcome this partial observability: we show that we can estimate the per-user metric by matching the user distributions of the two samples being compared, and present a solution based on a sample matching strategy.",
"Finally, we study the conflict between these two metrics empirically and assess the quality of our technique to estimate the per-user metric under partial observability.",
"First, we use a simulated environment to demonstrate that distributional differences of the users can indeed lead to conflicting conclusions between the metrics.",
"We show that our approach can accurately estimate the per-user metric under partial observability.",
"Second, we analyze a real-world recommender, similar to past work [4], [17], [31], and demonstrate that the tension between both metrics is observable in such a system, as each group of items is relevant to different users.",
"This demonstrates the importance of taking a more holistic view to study fairness in recommender systems.",
"To summarize, our contributions are: Simpson's Paradox in Ranking Fairness: We highlight a conflict between two well-established fairness metrics in recommendations – the per-user and aggregate metrics.",
"We highlight that the aggregate metric is indeed influenced by differences in user distribution across the two groups, a form of Simpson's Paradox.",
"We point out that the influence of the user distributions on the aggregate metric can be captured by two factors of the user's recommendations, competition and performance, and formalize this by characterizing mathematically the aggregate metric as a function of the per-user gap, the competition, and the performance (Section ).",
"Method to estimate the per-user metric under partial observability: In order to overcome the partial observability of relevance, we employ approaches from the causal literature to match the distribution of users across groups and estimate the per-user metric (Section ).",
"Experimental Evidence: We demonstrate on multiple datasets that the tension between the two metrics might occur in both simulated and real-world environments, similar to the Simpson's paradox mentioned.",
"Then we show the efficiency of our technique to recover the per-user metric (Sec.",
"6 and 7).",
"To support the development of recommender systems, there has been much research to develop efficient ranking metrics.",
"Offline evaluation methods [39], [2], [15], [30], [3] use logged data to measure how well a recommender exposes items to users.",
"These approaches rely on a notion of relevant items [13] and measure how well the model is able to predict the relevance of each item (e.g.",
"Mean Squared Error [12]) or how well it ranks relevant items (e.g.",
"Normalized Discounted Cumulative Gain [23], [22], [23], [45]).",
"The \"relevance\" can be based on human ratings but, for real-world personalized systems where such ratings are unavailable, we typically utilize implicit feedback and define relevance based on user engagement and satisfaction.",
"This data collection from an already deployed recommender can lead to sampling biases [43], [24], and offline evaluations might therefore not reflect well the behavior of the actual system [21].",
"To correct for this bias, prior work has utilized random experiments where items are randomly exposed to users, a setting also known as logged bandit feedback [43], [24].",
"While these methods have laid important foundations for fairness metrics (e.g.",
"the notion of relevance, the understanding on data biases), they focus on comparing recommender models, and do not extend directly to comparing groups of items or providers, which is necessary for group fairness evaluation."
],
[
"Fairness, Recommendation, and Ranking",
"There has been a growing interest to ensure that recommenders provide fair recommendations for groups of items or providers from different demographic groups [41], [7], [10], [4], [47].",
"Most of these papers follow the notion of Equality of Exposure, where Exposure is typically defined as a function of the position of the items over the rankings [23], such as NDCG [7], impression rate [11] or pairwise order [4], [34].",
"Some papers suggest that the exposure should be allocated equally between groups [29], mapping to Statistical Parity.",
"Others follow the perspective of Equality of Opportunity [16], accounting for relevance [41], [7], [10], [4], [47].",
"For instance, [41], [7] propose that the exposure of an item or group of items should be proportional to its average relevance.",
"A variant to this proportionality constraint is to ensure that equally relevant items should have equal exposure [4], [47].",
"We will focus on this last option, however our observations transfer to the proportionality-based metrics.",
"Within this notion of Equality of Exposure, we group the past literature into two different categories: some only compare the exposure of relevant items per user [41], [11], [14], [27], while others compare the exposure when aggregated across users [4], [34].",
"As we show via the Simpson's paradox, these two different notions can lead to different conclusions: the difference is due to the fact that, when aggregating across users, the difference in user distributions between the two groups might have an influence on the gap.",
"We will articulate this influence of the user by highlighting two specific factors related to the user: the performance, i.e., whether the system is able to rank relevant higher than non-relevant items for this user, and the competition, i.e., how many relevant items the user is interested in.",
"We then prove that the aggregate metric is the combination of the per-user metric and these two factors.",
"This decomposition helps to structure prior literature.",
"The first branch of per-user comparisons [41], [11], [7], [47], [14] measures a narrower concept and does not incorporate differences due to the other factors.",
"For instance, it will not reflect a scenario where a group is relevant primarily to users where the system has poor performance.",
"On the other hand, the aggregate comparison [4], [34] encapsulates these scenarios but without any disentanglement.",
"Interestingly some prior work [47], [7] takes an intermediate path: they compare the exposure in aggregate but use an exposure normalized by the relevance of other items.",
"The most common example is the Normalized Discounted cumulative gain (NDCG) [47], [7], where the denominator accounts for how many items are also relevant for this user.",
"This effectively controls for the impact of the competition factor on the metric.",
"This framework can provide more insights to the practitioner when evaluating their recommender system.",
"However the per-user metric, as well as the ones using normalized exposure, require knowing the relevance of all items for all users, and therefore are challenging to apply to real-world recommender.",
"Indeed such systems often define relevance based on post-click signals, such as dwell-time [48] or ratings [28], and users typically engage with no more than 1 item.",
"This lack of observability is exacerbated by the use of random experiments [24], [4], where an item is sampled from the corpus and shown.",
"This process enables us to collect an unbiased sample of items with user feedback but limits as well the set of relevant items to maximum one per user (the boosted item if engaged).",
"We define partial observability, as the situation where the relevance is known for only a few instances.",
"We analyze the case of partial observability and suggest a technique, based on distribution matching, to estimate the per-user metric.",
"To the best of our knowledge, there has been no solution proposed before under this situation."
],
[
"Distribution Matching",
"We connect this challenge to a difference in underlying distributions where each group is sampled from different distributions of users.",
"This relates to Simpson's paradox [40], [44], where a confounding variable can have an effect on the measured trend.",
"A classic example comes from graduate admissions at UC Berkeley [6]: The overall rates showed that men were more likely to be admitted, but men were applying to departments with higher admission rates.",
"When controlling for the department, the data showed a slight advantage towards women.",
"A common solution to control for this effect is to equalize the distributions of both groups.",
"To that end, our paper builds on past reweighing techniques, based on Inverse Propensity Weighting or Scoring, commonly used to match two distributions in the causal literature [42], [38], [37], [20] and in the offline recommender evaluation work [43], [24]."
],
[
"Notations",
"A recommender or ranking system is responsible for presenting the relevant items to the users.",
"More formally, a recommender system ranks items $i$ from a catalog $I$ for any user $u$ , where the user encapsulates both user and contextual information [1], e.g.",
"time of the day, user features, and is a similar concept to \"query\" in search systems.",
"Note that, in some other applications, user might be referred as event, request, query, context, or home page.",
"One concrete example of such system is a comments section of a news site: When a user reads a certain news article, the recommender system would select the most relevant comments to present.",
"In this example, the items are all the comments available and the user contains properties of the article (e.g.",
"topic, language) as well as properties of the viewer (e.g.",
"past history, country).",
"The recommender considers all possible comments, ranks them and only the top ones are shown.",
"Another example is a social media platform, where the user is shown the most relevant pieces of content on their page.",
"Each item $i$ is a piece of content that the user might be interested in, the user is some combination of the user's profile and historic activity.",
"The relevance of an item $i$ for a given user $u$ is defined as $Rel(i, u)$ , and is typically defined from user feedback [13], such as clicks [18], [32], ratings [28] or dwell-time [48].",
"It is important to note that the relevance is not a function of an item only but instead is defined for every item-cross-user $(i,u) \\in I \\times U$ , representing a personalized and contextualized recommender system.",
"However throughout this paper, we might use loosely the wording \"relevant items\" to designate all tuples $(i,u)$ where an item $i$ is relevant for the user $u$ .",
"For simplicity, we binarize the notion of relevance, $Rel(i,u) \\in \\lbrace 0, 1\\rbrace $ , but it can be bucketed as well to reflect different levels of relevance, similar to [22], [4], [35].",
"Initially and until Section , we assume that $Rel$ is known for all items and users $(i,u)$ .",
"For every user, the recommender ranks all items and assigns them to a ranked-position $\\mathit {Position}(i,u)$ .",
"Then items are exposed to the user based on this ranking.",
"While our work can be generalized to various models for $Exposure$ , we decide here to model it as a function of the item's position: $\\mathit {Exposure}(i, u) = f(\\mathit {Position}(i,u))$ .",
"The function $f$ can for instance be a position-decayed function, as in NDCG or an impression prior [23], to represent the probability that the item is seen by the user."
],
[
"Two fairness metrics for recommenders",
"We now assume that each item $i$ has a sensitive attribute $S(i) \\in \\lbrace 0, 1\\rbrace $ and our goal is to measure if items from different groups are treated equally by the recommender system.",
"For example, in the news article comments setting, $S$ may be whether the comment's author belongs to a protected demographic group, and in the social media setting, $S$ may similarly correspond to the item producer's demographic group (e.g.",
"gender) or, for a non-fairness use case, its topic.",
"In this paper, we follow the perspective of Equality of Opportunity from the classification literature [16] which has been adapted for ranking [41], [7], [10], [4], [47].",
"[41], [7] propose that the exposure of a group should be proportional to its aggregated relevance, while [4], [34] uses conditioning to capture that items, when relevant, should have equal exposure.",
"We will use this latter notion for our definitions, but we expect the observations transfer to the other definitions.",
"We will now describe definitions following both frameworks for use in the rest of the paper.",
"For a user, the per-user gap is defined as the difference between groups in expected exposure over items that are relevant for this given user.",
"$PerUserGap (u) &= \\mathop {{}\\mathbb {E}}_{i\\sim I} [Exposure(i, u) | Rel(i, u) = 1, S(i) = 1] \\\\&- \\mathop {{}\\mathbb {E}}_{i\\sim I} [Exposure(i, u) | Rel(i, u) = 1, S(i) = 0]$ As this metric is defined for each user, we can capture the full system gap by averaging over users: $PerUserGap = \\mathop {{}\\mathbb {E}}_u PerUserGap(u)$ .",
"The aggregated gap is defined as the difference between groups in average exposure over over items that are relevant for any user.",
"$AggregateGap &= \\mathop {{}\\mathbb {E}}_{(i,u)\\sim I \\times U} [Exposure(i, u) | Rel(i, u) = 1, S(i) = 1] \\\\&- \\mathop {{}\\mathbb {E}}_{(i,u) \\sim I \\times U} [Exposure(i, u) | Rel(i, u) = 1, S(i) = 0]$ Equivalently, we could remove the conditioning and rewrite the metric as the difference of exposure over two uniform samples from $D^0$ and $D^1$ , with $ D^s = \\lbrace (i,u) | Rel(i,u) = 1, S(i) = s \\rbrace $ .",
"This metric targets that, aggregated over the instances where an item is relevant to a user, the exposure should not depend on the group.",
"Each definition is supported by prior work and seems intuitive when taken separately.",
"However we will see that they represent different notions and articulate those."
],
[
"Simpson's paradox as a motivating example",
"Let's first go through the example of Fig.",
"REF to articulate the potential tension between these two metrics.",
"Let's imagine that this is a recommender in a social media setting, which has to rank items for two different users: the first one is a frequent user who uses the social media regularly, while the other one is a casual user.",
"We display items in Orange/Blue (depending on their group) if they are relevant for that user, and in Grey if they are not.",
"To simplify the computations for this example, let $Exposure(i, u)$ simply be $Position(i,u)$ , such that the exposure gap measures the difference in average ranked position for the relevant items from each group.",
"The Simpson's paradox comes from making two seemingly contradictory observations.",
"(1) For any given user, Orange is ranked lower than Blue, i.e., Orange seems disadvantaged according to the per-user metric in Def.",
"REF .",
"(2) When aggregating across users, the Orange group has better exposure; the average position of Orange items is 4.3, compared to 4.7 for Blue, i.e., Orange seems advantaged according to the aggregate metric in Def.",
"REF .",
"Intuitively in this example, this paradox is due to the effect of the users on the exposure of items – in our example, the recommender does a better job at ranking the relevant items (Blue & Orange) above non-relevant ones (Grey) for active users than for casual ones; this could be due to having less historical data for the latter.",
"Combined with the fact that the Blue group has more relevant content for casual users, this offsets the fact that Blue is always ranked above Orange within the same user."
],
[
"Qualitative analysis",
"In this section, we show that the aggregate metric is broader as it is influenced by the per-user gap but also by two additional factors per-user: the performance and the competition.",
"This influence is due to the fact that this aggregate gap compares groups over two different user distributions."
],
[
"The aggregate metric compares two samples with different user distributions.",
"In the aggregate metric, we are comparing two samples of relevant items, $D^0$ and $D^1$ with $ D^s = \\lbrace (i,u) | Rel(i,u) = 1, S(i) = s \\rbrace $ , which have different marginal distributions over users.",
"Indeed, as each sample is defined as a uniform sample from $D^s$ , the marginal distribution over users will be proportional to $Count(i | Rel(i, u)=1, S(i) =s)$ .",
"Due to these different marginal distributions, the aggregated exposure of each item/provider group will be differently impacted by the users where it is relevant.",
"This user influence on the aggregate metric acts like a \"confound\" in causal literature [19], [6].",
"This is what we saw in the example before, where the Blue group is more frequent in casual users, where relevant items have on average lower exposure."
],
[
"Decomposing the user influence into two factors.",
"Let's break down the exposure of relevant items for a given user into an intuitive set of factors; The first one is the per-user gap as defined above (we will define them rigorously in the next paragraph), while the other two are additional per-user factors.",
"Per-user: As in Def.",
"REF , this is the difference in exposure between relevant items of different groups on a given user.",
"Performance: The capacity of the recommender to rank relevant items higher than non-relevant ones on a given user.",
"If the system has higher performance for one user than for another, items relevant to the first user will have higher exposure.",
"Competition: The number of items that are relevant to a given user.",
"As the exposure is limited (only a few items are shown to the user), the average exposure of relevant items is reduced when many items are relevant to a given user.",
"This breakdown allows us to identify three separate scenarios under which a Group (let's say Orange) might have lower exposure according to the aggregate metric.",
"Fig REF shows a situation where there is a per-user gap, which translates into an aggregate gap.",
"However, Fig REF and REF point to different phenomena: while both do not display any per-user gap, the aggregate metric will indicate that Orange is disadvantaged.",
"Indeed, in Fig.",
"REF , the recommender system is not as good at allocating exposure to relevant items for users 1 and 2, and it turns out that the Orange group appears primarily on these users.",
"As a result, Orange still receives lower exposure because it appears primarily on users with lower performance.",
"Competition suggests another scenario displayed in Figure REF : there is more competition on users 1 and 2 (two relevant items, versus just one) and it turns out that the Orange group is primarily relevant to these users.",
"As a result, Orange gets lower exposure because it appears primarily on users with higher competition.",
"From this, we see that lower exposure in aggregate can be due to any of these three factors.",
"Figure: Three separate scenarios with an AggregateGapAggregateGap against Orange.",
"Only the first one is associated with a Per-user gap."
],
[
"Theoretical analysis",
"We now present theoretical evidence to prove that the qualitative decomposition of the aggregate metric into these three factors is well-founded.",
"Let's start with some preliminary definitions: We define the performance for a given user as the difference of exposure between relevant and irrelevant items.",
"$\\mathit {Performance}(u) &= \\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i, u) = 1] \\\\&- \\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i, u) = 0]$ We define the competition as the number of relevant items from group $s$ for a given user $u$ .",
"$Comp_s (u) = Count(i| Rel(i,u) = 1, S(i) = s)$"
],
[
"Decomposition of the aggregate metric",
"We now prove that the competition and performance are sufficient to capture the additional influence of the user distribution on the aggregate metric.",
"We rely on the following theorem, as the key theoretical foundation, which decomposes the aggregate metric as a function of the 3 terms.",
"We can rewrite the $\\mathit {AggregateGap}$ as follows: $&\\mathit {AggregateGap} = \\sum _{u} \\Biggl ( Z*\\frac{Comp_0(u)* Comp_1(u)}{Comp_1(u) + Comp_0(u)} * PerUserGap(u) \\\\& + (\\frac{Comp_1(u)}{\\sum _{u} Comp_1(u)} - \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)}) * NR(u) * \\mathit {Performance}(u)\\Biggr ) \\\\$ where $NR(u) = (1 - \\tfrac{Comp_0 (u) + Comp_1 (u)}{|I|})$ is the proportion of items that are not relevant for user $u$ , and $Z = \\frac{1}{\\sum _{u} Comp_1(u)} + \\frac{1}{\\sum _{u} Comp_0(u)} $ is a constant.",
"The full proof is included in the appendix.",
"This theorem highlights how the $PerUserGap$ , combined with the performance and competition, form the $AggregateGap$ .",
"We can share some high-level intuition about the different terms.",
"The first term represents the contribution of the $PerUserGap$ .",
"Users with a balanced amount of relevant Orange and Blue items (i.e., $ Comp_0(u) \\cdot Comp_1(u)$ is high) will be weighted heavily, as they are the ones where relevant items from both groups compete the most for exposure.",
"The second term represents the contribution of the performance.",
"The first factor $(\\frac{Comp_1(u)}{\\sum _{u} Comp_1(u)} - \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)})$ indicates that high performance contributes more to a positive gap when items from group 1 are more frequently relevant to the user.",
"The second factor $NR(u)$ shows that performance matters more when there are more non-relevant items.",
"Interestingly, our factor decomposition gives us a new framework to grasp conceptual differences within past literature.",
"In prior work using the aggregate version [4], [34], the per-user gap, the performance and the competition will all affect the metric without any disentanglement, as displayed in Fig.",
"REF .",
"On the other hand, the past work on per-user (or per-query) metric [41], [11], [14] ignores the potential differences in the performance and competition and their effect.",
"Finally, some of the literature has incorporated knowledge from information retrieval and used an exposure normalized by the relevance of other items.",
"The most common example is the Normalized Discounted cumulative gain (NDCG) [47], [7], where the denominator accounts for how many items are also relevant for this user.",
"This effectively controls for the impact of the competition factor on the gap, but as we will discuss in the next section is hard to use in practice.",
"We hope that this decomposition can guide the practitioners to understand their evaluation choices more precisely."
],
[
"Challenge of Partial Observability",
"In the previous section, we provided some decomposition to reconcile the two metrics: we outlined that the $AggregateGap$ was combined multiple effects – the $PerUserGap$ itself, but also the performance and the competition.",
"While we leave the decision to each application and practitioner, we now want to provide them with some technical tools for the metric of their choice in any situations.",
"The challenge lies in the data collection for real-world recommenders for two main reasons.",
"First, the relevance is typically defined from user feedback including post-click feedback (e.g.",
"are users are satisfied with the item when recommended) [48], [28] and we typically observe the post-click behavior typically on only 1 item per user.",
"Secondly, to remove any data bias, we typically measure this user feedback through random experiments where an item is sampled from the corpus and shown .",
"This process, highlighted by prior work [24], [4], enables us to collect a sample of items with user feedback but limits as well the set of relevant items to maximum one per user (the boosted item if engaged).",
"Since this process leads to a uniform sample over $I \\times U$ , we can compute directly $AggregateGap$ .",
"However per-user metrics require knowing the relevance of all items (or at least multiple items) for all users and can therefore not be directly applied to real-world recommenders.",
"We define partial observability, as the situation where the relevance is known for only a few instances ($Rel(i, u)$ is sparse).",
"The rest of this section focuses on how we can estimate the per-user metric under such situations, which has not been studied yet."
],
[
"Sampling from a common user distribution.",
"We now propose a framework to isolate the per-user gap under partial observability.",
"As we have seen, the difference of marginal distributions is the reason why some factors of the users, namely the performance and competition, are affecting the gap.",
"A natural idea therefore is to match the user distribution of both groups, which is supported by the following theorem.",
"Let's assume that our samples of relevant items follow the same marginal distribution over users ( $D^s_{common}$ parameterized by $w_{common}$we will discuss later reasonable choices for $w_{common}$).",
"Then we can estimate the $PerUserGap$ by comparing the average exposure over these two data samples.",
"We are taking the difference of two expectations over the same distribution of users and can use the linearity of the expectation to re-arrange the term.",
"$&\\mathop {{}\\mathbb {E}}_{(i,u) \\sim D^1_{common}} [Exposure(i, u)| Rel(i,u) = 1, S(i) = 1]\\\\ &-\\mathop {{}\\mathbb {E}}_{(i,u) \\sim D^0_{common}} [Exposure(i, u)| Rel(i,u) = 1, S(i) = 0] \\\\&= \\mathop {{}\\mathbb {E}}_{u \\sim w_{common}} \\mathop {{}\\mathbb {E}}_i [Exposure(i, u)| Rel(i,u) = 1, S(i) = 1] \\\\&- \\mathop {{}\\mathbb {E}}_{u \\sim w_{common}} \\mathop {{}\\mathbb {E}}_i [Exposure(i, u)| Rel(i,u) = 1, S(i) = 0] \\\\&= \\mathop {{}\\mathbb {E}}_{u \\sim w_{common}} PerUserGap(u)$ This theorem is intuitive.",
"If our data samples have the same distribution over users, then the user factors will not influence the metric.",
"Matching the distribution of groups over some variables is a common technique in the causal literature to control for the effect of confounding variables [42], [38], [37], [20]."
],
[
"Reweighting technique to match distributions",
"The previous theorem gives us a recipe to estimate the per-user metric by ensuring that our two data samples have the same distribution.",
"As modifying the data collection might be challenging, we instead suggest a reweighting approach, similar to Inverse Propensity Scoring (IPS), to match the distributions offline.",
"We consider some features of a user such as user activity, time or country.",
"The choice of these variables is application-dependent and needs to be interpreted in the particular practical setting, similar to the discussion of selecting resolving variables in [26], [25]: Some might want to use a fine-grained representation of the user to control for any potential influence on the gap, while some others might select only a subset of the user characteristics.",
"We now assume that we have a list of user variables $V$ that we want to control for.",
"As mentioned earlier, the two groups have different distributions over $V$ , which therefore influences the measure of the gap.",
"The distribution over the features $V$ for one group $s \\in \\lbrace 0, 1\\rbrace $ can be written as follows: $\\forall \\ v\\ \\in \\ V, w_s(v) & = \\mathbb {P}_{i, u}(V(u) = v\\ |\\ Rel(i,u) = 1, S(i)= s) \\\\&\\propto \\mathbb {P}_{i, u}(V(u) = v,\\ Rel(i,u) =1,\\ S(i)= s)$ Our method consists in matching the two distributions, $w_0$ and $w_{1}$ , by reweighting the data based on IPS.",
"This technique has been used as to remove data biases in offline recommender evaluation [43], [24], or to control for the effect of confounding variables in the causal literature [38], [37], [20].",
"To match the two distributions, we start by selecting a target distribution over the variables V. While various target distributions could be usedFor instance, in addition to the uniform distribution presented in this paper, one might choose to use the empirical distribution of the sensitive group., we take here a simple approach and assume a uniform target distribution.",
"In other words, we want to match each group such that the distribution over V is uniform: $ \\forall \\ v\\ \\in \\ V, w_{common}(v) = 1$ .",
"We can then use IPS to match the data to this target distribution by re-weighting the collected data of each group $s$ by $w_{common}(v) / w_s(v)$ .",
"Estimation of the weight.",
"If V is simple (e.g.",
"only a few variables that can be bucketized easily), $w_s(v)$ should be computable by empirical counts such as: $&\\forall \\ v\\ \\in \\ V, w_s(v) = \\frac{|(i,u) | Rel(i,u) =1, S(i)= s, V(u) = v)|}{|(i,u) | Rel(i,u) =1, S(i)= s|}$ If v is more complex (e.g.",
"we want to correct for a large number of user characteristics, or v can not be bucketized), this estimation might not be possible due to data sparsity.",
"In this case, we can rewrite the source distribution as follows: $w_s(v) &\\propto \\mathbb {P}_{i, u}(V(u) = v, Rel(i,u) =1, S(i)= s) \\\\&\\propto \\mathbb {P}_{i,u}(V(u) = v, S= s) * \\mathbb {P}_{i,u}(Rel(i,u) =1|V(u) = v, S(i)= s)$ As the first term not depend on relevant items, it does not suffer from the partial observability issue and can be computed exactly from the data.",
"The second term is the most challenging term and we can use a model to estimate it.",
"To that purpose, we can use the random data, take as input all confounding variables V and predict the probability of an item being relevantIf V is a fine grained representation of the user, the prediction task might be challenging and the model might be imperfect (it is close to re-engineering the recommender system).",
"However the main role of this model is to correct for potential confounding variables, rather than matching perfectly the target distribution.."
],
[
"Simulation",
"In this section, we use a simulated recommender system to demonstrate that the Simpson's paradox is observable under reasonable conditions.",
"In particular, we show that $PerUserGap$ and $AggregateGap$ can point in opposite directions, and more generally that varying each of the three factors individually can cause large variations in $AggregateGap$ .",
"Furthermore, we show that our technique enables us to estimate the $PerUserGap$ under partial observability."
],
[
"Simulation setup",
"We first explain at a high level the setup of the recommender.",
"This design enables us to vary the per-user gap, the performance, and the competition as needed, and then observe the resulting aggregated gap in the system.",
"To simulate these three factors, we have parameters for the density, the noise, and the model bias.",
"High-level description: We simulate a recommender system composed of N users and M items.",
"We categorize these items into 2 different groups.",
"We generate the data from a low-rank representation, similar to past work [5]: Each user (resp.",
"item) can be represented with a normalized r-dimensional vector $U_i$ (resp.",
"$V_j$ ).",
"The relevance of an item for a user is then equal to the product of both representations: $U_i * V_j$ .",
"To map it to our problem, we binarize the relevance by setting a threshold: $Rel = 1$ if $U_i * V_j > T$ else 0.",
"We arbitrarily set a threshold of T=0.3, which empirically maps to about 20% of the items being relevant.",
"Finally, the recommender scores are a noisy and biased estimate of the true scores: $Score[i, j] = U_i * V_j + noise[i, j] + bias[j]$ .",
"The recommender system ranks the items based on $Score[i, j]$ .",
"We represent the exposure allocated to an item as the discounted cumulative gain of a position: $exposure(position)=\\tfrac{1}{log_2(1+ position)}$ .",
"Finally, we reproduce the setup of the random experiment: for each user, one item is selected at random and we observe the true relevance.",
"The aggregate metric is computed as the difference of exposure between the relevant items of each groups.",
"Choice 1: How to generate the item representations [affecting the competition factor] For users, we simply sample uniformly in $[-1, 1]^r$ .",
"However, for items, it is key to ensure that the two groups have different distributions of item representations.",
"There are many ways to represent two different distributions over the latent space and we decide here to sample items along two different directions in the latent space.",
"More precisely, for each group s in {0, 1}, we generate items from Gaussian($\\mu _s$ , $\\frac{1}{d_s}$ ).",
"We choose arbitrarily $\\mu _0$ and $\\mu _1$ to be orthogonal ([1, 0, .., 0] and [1, 1, .., 0]) but findings would adapt to other choices.",
"More importantly, $d_s$ controls the density of each group and affects the competition factor.",
"Indeed if a group is dense ($d_s$ large), all items are similar to each other and a user interested in a given item will also be interested in many other ones (high competition).",
"We set $d_0$ =1.5 (giving some overlap between the group distributions) and vary only $d_1$ .",
"Choice 2: How to generate the noise [affecting the performance factor] We also want our simulation to represent a recommender that has different levels of performance for the two groups.",
"To that end, we use a default level of noise, $\\sigma $ , but make it vary alongside the direction $\\mu _1$ , such that the typical items from group 1 (aligned with $\\mu _1$ ) will have much higher level of noise.",
"This difference of noise is controlled by a parameter $\\sigma _1$ .",
"More precisely: $Noise[i, j ] = Gaussian( 0,\\ \\sigma * [1 + f(U_i * \\mu _1)])$ .",
"We set $\\sigma =0.05$ (arbitrarily), and F is a linear function so that f(-1)=0 mapping to items that are the opposite of typical items from group 1 and f(1)=$\\sigma _1$ mapping to typical items from group 1.",
"As a result, the level of noise will vary between $\\sigma $ and $\\sigma * (1+ \\sigma _1)$ for typical items of group 1.",
"Choice 3: How to represent the model bias [affecting the per-user metric] We represent the model bias by adding a constant term to the predictions when items are from group 1."
],
[
"Variations of PerUserGap, Performance and Competition changes the AggregateGap",
"First, we conduct some analysis to verify that our simulation enables us to affect the performance, the competition and the $PerUserGap$ of the recommender system.",
"We vary the parameters of the system (the model bias, the level of noise and the density) and see their effect on the three factors (per-user gap, performance, competition).",
"While these factors are defined for every user $u \\in U$ , we extend their definition to an aggregate level via the following methodology: (1) the per-user gap factor is reported as the average PerUserGap(u) over all users where at least one item from each group is relevant; (2) the competition and performance factors are computed for a hypothetical user that would be aligned with $\\mu _1$ , to reflect the characteristics that will dominate for the items from group 1.",
"Results are displayed in Fig REF .",
"We confirm that the 3 system parameters are effective in varying the corresponding 3 factors.",
"Indeed, the first graph shows that we can affect the per-user gap by changing the model bias and holding the other 2 parameters constant.",
"The two other graphs make similar observations for the performance and competition.",
"Figure: Each parameter (model bias, noise, density) varies the targeted factors (per-user gap, performance, competition).Then we emphasize in Fig.",
"REF how the $AggregateGap$ is affected by these three factors.",
"For instance, we see (left graph) that increasing the model bias (i.e., increasing the $PerUserGap$ ) results in an increase of the aggregate metric.",
"This experiment provides empirical evidence that the performance, competition, and per-user gap affect the aggregate metric.",
"In particular, the $AggregateGap$ can vary greatly even when the $PerUserGap$ is zero or fixed.",
"Figure: Each parameter will influence AggregateGapAggregateGap."
],
[
"Estimating the per-user metric",
"We now want to confirm that the method described in Section can estimate the per-user metric well.",
"To do so, we reweigh the data based on eq.",
"REF : the first term is constant ($U_i$ is sampled uniformly) and we estimate the second term by a model that takes as input the user embedding $U_i$ and predicts the average relevance of items of each group for that user.",
"We first conduct an experiment to verify that the method is effective even when the competition or performance affect the aggregate gap significantly.",
"To that end, we arbitrarily set a constant model bias of 0.03 and then vary the level of noise and density.",
"Results are shown in Figure REF .",
"On the left graph, we observe that as the density increases, the aggregate gap increases as well.",
"We then plot the per-user gap (both true and estimated) and observe that our estimate is close to the true value.",
"Similar observations can be made in the right graph when we vary the level of density.",
"This experiment shows that our estimate of the per-user gap is reliable even when the two other factors have a large influence on the gap.",
"Figure: True vs Predicted PerUserGap depending on density/noise.We further assess the reliability of our estimate by varying performance and competition at the same time.",
"We again set a constant model bias of 0.03 and vary the other two parameters over a grid of values (model noise from 0 to 10 and density from 1.5 to 4.).",
"Over 160 simulations, we see that the $AggregateGap$ is significantly different from the true $PerUserGap$ (4.0pp difference on average), which reflects again that the two other factors have a strong impact on the gap.",
"Comparatively, our estimated $PerUserGap$ is a relatively robust estimate of the true value (0.9pp difference on average).",
"This show that our method is able to estimate $PerUserGap$ effectively."
],
[
"Real-world recommender systems",
"We now study a real-world recommender system: for every user (here the \"user\" includes contextual features such as time), the system predicts the relevance of all items in the corpus based on past interactions (click and post-clicks signals) and returns a ranked list of items.",
"This is a similar setup to [4], [17], [31].",
"Only the top items are displayed to the user.",
"We have available the group information of a subset of providers and can split them into a binary groups $S=\\lbrace 0, 1\\rbrace $ .",
"Our goal is to evaluate offline whether the system provides equal treatment based on the sensitive group $S$ .",
"We want to estimate the aggregate metric and define the terms as follows.",
"Relevance.",
"Per product definition, an item is relevant for a user if, had the item been recommended, the user would have clicked on and been satisfied with the item.",
"In order to access this in practice, we rely on random experiments where, for a subset of requests, an item is picked randomly and boosted high in the ranking.",
"We then filter to boosted items with high user satisfaction to build our data sample.",
"Exposure.",
"We estimate the exposure of items that were boosted and engaged based on the initial position before boosting.",
"We use the impression-prior aggregated over historical data to estimate the chance of impression at this initial position.",
"We report in Fig.",
"REF the aggregate exposure for items that the user engages with when recommended to them, depending on the sensitive group.",
"We observe that relevant items from $S=1$ receive lower exposure on average, mapping to an $AggregateGap$ against this group.",
"Note that due to the sensitive nature, we cannot report absolute exposure numbers.",
"However we keep the y axis constant, so that the graphs can indicate the direction and scale of the difference in exposure between groups.",
"Then we want to understand if the $AggregateGap$ is affected by differences in users for the groups, as illustrated by the Simpson's paradox of this paper.",
"As the data collection relies on random experiment, we do not observe the relevance for every items (conditions of partial observability) and therefore followed the approach described in Section .",
"Based on product experience, we identified some key characteristics of a user (e.g.",
"some features related to the user past behavior or the related query) and used the technique presented to control for the influence of these variables.",
"Figure: Exposure of 2 groups in a real-word recommender.",
"Left: AggregateGapAggregateGap.",
"Right: Estimation of PerUserGapPerUserGap by reweighting.We report in Figure REF (right part) the exposure of each group after we control for these variables.",
"The difference between both groups, which maps to the estimated per-user metric, is inverted compared to the gap from the $AggregateGap$ : when comparing over users with the same characteristics, defined by the set of variables, relevant items from $S=1$ have actually higher exposure.",
"These results represent a real-world illustration of the Simpson's paradox, which our methodology has been able to exhibit: relevant items from $S=1$ have lower $AggregateGap$ , but have have higher exposure when compared on requests with similar characteristics which is the estimated $PerUserGap$ ."
],
[
"Conclusion",
"Past literature has established two different fairness notions, comparing the exposure of relevant items either on a given user, or aggregated across users.",
"While both notions seem intuitive, we illustrate the tension between them through a Simpson's paradox, where they lead to opposite conclusions.",
"We provide a framework to articulate these differences: as the aggregate metric compares two groups with different user distributions, it will be influenced by the users.",
"We show that the aggregate metric can be expressed as a function of the per-user metric and two user factors (competition and performance).",
"This decomposition helps us to reconcile the conceptual differences within past literature.",
"We then propose a method to estimate the per-user metric in the common situation of only partially observable relevance.",
"This technique matches the distribution of users over the groups, which controls for the effect of performance and competition.",
"We use simulations to show that this method is effective to estimate the per-user gap in practice.",
"This research shows that studying fairness in recommender system is complex and multi-faceted.",
"While this work presents some foundations, it opens the door for more targeted remediation.",
"It is indeed possible that one might take different actions depending on whether a gap is due to the performance, competition, or per-user factors.",
"Acknowledgements: The authors would like to thank Fernando Diaz and Konstantina Christakopoulou for their valuable feedback on this paper."
],
[
"Appendix: Proof of Theorem ",
"Step 1: Exposure of relevant (vs irrelevant) items on a given user.",
"The performance controls how the total exposure available (noted as ExposureAvailable) is allocated to relevant/irrelevant items, as we see in the following equations (using the definition of Performance).",
"For a given user $u$ , $ExposureAvailable =& \\ (Comp_0(u) + Comp_1(u)) * \\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1] \\\\&+ (|I| - Comp_0(u) - Comp_1(u)) *\\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 0] \\\\ExposureAvailable =& \\ |I| * \\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1] \\\\&- (|I| - Comp_0(u) - Comp_1(u)) * Performance(u)$ Finally, we have: $\\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1] =& \\ \\frac{ExposureAvailable}{|I|} \\\\&+ (1 - \\frac{Comp_0(u) + Comp_1(u))}{|I|}) * Performance(u))$ Step 2: Exposure of relevant items from a given group on a given user.",
"The expected exposure of relevant content is a weighted sum over the exposure of the two groups.",
"The following equations follow (using the definition of Per-user Gap).",
"For a given user $u$ , $\\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1] =& \\ \\frac{Comp_1(u)}{Comp_1(u) + Comp_0(u)} * \\mathop {{}\\mathbb {E}}_i [Exp(i,u) | Rel(i,u) = 1, S = 1] \\\\& + \\frac{Comp_0(u)}{Comp_1(u) + Comp_0(u)} * \\mathop {{}\\mathbb {E}}_i [Exp(i,u) | Rel(i,u) = 1, S = 0] \\\\\\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1] =& \\ \\mathop {{}\\mathbb {E}}_i [Exp(i,u) | Rel(i,u) = 1, S = 1] - \\frac{Comp_0(u)}{Comp_1(u) + Comp_0(u)} * PerUserGap(u)$ Finally we have: $\\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1, S = 1] = & \\ [Exposure(i, u) | Rel(i,u) = 1] + \\frac{Comp_0(u)}{Comp_1(u) + Comp_0(u)} * PerUserGap(u)$ This equation is relatively intuitive.",
"If there is no per-context bias, the relevant content from the sensitive group will get the same exposure.",
"Otherwise, the exposure might be higher or lower depending on the bias.",
"We can then write the exposure of a given group as a function of the per-user bias, the performance and the competition.",
"$\\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1, S = 1]= & \\ \\frac{ExposureAvailable}{|I|} \\\\&+ (1 - \\frac{Comp_0(u) + Comp_1(u)}{|I|}) * Performance(u)\\\\&+ \\frac{Comp_0(u)}{Comp_1(u) + Comp_0(u)} * PerUserGap(u)$ Step 3: Gap aggregated across users.",
"$AggregateGap =& \\ \\frac{1}{\\sum _{u} Comp_1(u)} \\sum _{u} Comp_1(u) * \\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1, S = 1] \\\\&- \\frac{1}{\\sum _{u} Comp_0(u)} \\sum _{u} Comp_0(u) * \\mathop {{}\\mathbb {E}}_i [Exposure(i, u) | Rel(i,u) = 1, S = 0] \\\\AggregateGap =& \\ \\sum _{u} \\Biggl (\\\\\\frac{Comp_1(u)}{\\sum _{u} Comp_1(u)} * \\biggl (&\\frac{ExposureAvailable}{|I|}+ (1 - \\frac{Comp_0(u) + Comp_1(u)}{|I|}) * Performance(u)+ \\frac{Comp_0(u)}{Comp_1(u) + Comp_0(u)} * PerUserGap(u)\\biggr ) \\\\- \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)} * \\biggl (&\\frac{ExposureAvailable}{|I|}+ (1 - \\frac{Comp_0(u) + Comp_1(u)}{|I|}) * Performance(u)- \\frac{Comp_1(u)}{Comp_1(u) + Comp_0(u)} * PerUserGap(u)\\biggr ) \\Biggr ) \\\\AggregateGap =& \\ \\sum _{u} \\Biggl (\\\\&\\frac{ExposureAvailable}{|I|} * \\biggl (\\frac{Comp_1(u)}{\\sum _{u} Comp_1(u)} - \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)} \\biggr ) \\\\& + Performance(u) * \\biggl (\\frac{Comp_1(u)}{\\sum _{u} Comp_1(u)} - \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)} \\biggr ) * \\biggl (1 - \\frac{Comp_0 (u) + Comp_1 (u)}{|I|} \\biggr ) \\\\& + PerUserGap(u) * \\biggl (\\frac{Comp_1(u)}{\\sum _{u} Comp_1(u)} * \\frac{Comp_0(u)}{Comp_1(u) + Comp_0(u)} + \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)} * \\frac{Comp_1(u)}{Comp_1(u) + Comp_0(u)} \\biggr ) \\Biggr ) \\\\AggregateGap =& \\ \\sum _{u} \\Biggl (\\\\& + Performance(u) * \\biggl (\\frac{ Comp_1(u)}{\\sum _{u} Comp_1(u)} - \\frac{Comp_0(u)}{\\sum _{u} Comp_0(u)}\\biggr ) * \\biggl (1 - \\frac{Comp_0 (u) + Comp_1 (u)}{|I|} \\biggr ) \\\\& + PerUserGap(u) * \\biggl (\\frac{1}{\\sum _{u} Comp_1(u)} + \\frac{1}{\\sum _{u} Comp_0(u)} \\biggr ) * \\frac{Comp_0(u) * Comp_1(u)}{Comp_1(u) + Comp_0(u)} \\Biggr ) \\\\$"
]
] | 2210.07755 |
[
[
"Bethe-Salpeter Bound-State Solutions: Examining Semirelativistic\n Approaches"
],
[
"Abstract Within the formalism of relativistic quantum field theory an adequate framework for the description of two-particle bound states, such as, for instance, all conventional (i.e., non-exotic) mesons, is provided by the Poincar\\'e-covariant homogeneous Bethe-Salpeter equation.",
"In applications, however, this approach usually proves to be rather involved, whence it is not always quite easy to extract the predictions sought.",
"In view of this, a coarse idea of the bound-state spectrum to be expected might be gained by adhering to some simplifying approximations - which constitutes an entirely legitimate first step.",
"The reliability of the insights inferred from the arising simpler bound-state equation may be straightforwardly examined by taking into account a couple of rigorous constraints on the obtained discrete spectrum.",
"Application of these tools is illustrated for popular potentials."
],
[
"Bound States of Spinless Constituents: Semirelativistic\nApproach",
"A main issue in both relativistic quantum physics and quantum field theories, such as quantum electrodynamics and quantum chromodynamics, is to devise appropriate (and, if manageable, convenient) approaches to the bound states expected within specific settings.",
"In this context, a very useful tool is provided by the spinless Salpeter equation, i.e., the eigenvalue equation of a Hamiltonian operator composed of the relativistic kinetic energies of the involved bound-state constituents and some static interaction potential $V$ assumed to reflect the bound-state relevant aspects of any underlying quantum (field) theory.",
"If $V$ meets these expectations, this approach should return a rough idea of the bound-state spectrum.",
"For two bound-state constituents with relative coordinates $\\mathbf {x}$ and $\\mathbf {p}$ in configuration and momentum space, this Hamiltonian becomes $H=\\sqrt{\\mathbf {p}^2+m_1^2}+\\sqrt{\\mathbf {p}^2+m_2^2}+V(\\mathbf {x})$ (where $m_1$ and $m_2$ denote both interacting particles' masses) and, for $m_1=m_2=m$ , reduces to $\\widetilde{H}=2\\,\\sqrt{\\mathbf {p}^2+m^2}+V(\\mathbf {x})\\ .$ Conceptually, the spinless Salpeter equation can be formulated along two opposite directions: On the one hand, the spinless Salpeter equation can be understood as an improvement of the nonrelativistic Schrödinger equation that is achieved upon allowing, in the Hamiltonian, the kinetic part to retain its proper relativistic form instead of sticking to its nonrelativistic limit.",
"On the other hand, upon starting from the Poincaré-covariant Bethe–Salpeter formalism [1], representing an appropriate framework for the description of bound states within the realms of relativistic quantum field theories, the spinless Salpeter equation can be easily derived by applying several simplifying approximations to the homogeneous Bethe–Salpeter equation.",
"The detailed recipe reads: discard all timelike coordinates (which leads to the instantaneous Bethe–Salpeter formalism [2]), require free propagation of bound-state constituents (which produces the Salpeter equation [3]), drop all negative-energy contributions (which results in what is called the reduced Salpeter equation) and, finally, ignore the spin degree of freedom of all bound-state constituents in order to eventually arrive at the spinless Salpeter equation."
],
[
"Analysis of Semirelativistic Hamiltonian Operators: Some\nInsights",
"The nonlocality of the Hamiltonian (REF ), induced by the square-root operators of the relativistic kinetic energies, impedes an entirely analytic treatment of any emerging bound-state problem.",
"Nonetheless, by utilization of instruments supplied by functional analysis a qualitative or even semiquantitive picture of expectable solutions of the spinless Salpeter equation may be drawn.",
"In particular, establishing the mere existence of discrete bound states by proving boundedness from below of the spectrum of Hamiltonian operators is, beyond doubt, of utmost importance."
],
[
"Spectrum of Semirelativistic Hamiltonian Operator:\nBoundedness from Below",
"Quite generally, well-definedness of a Hamiltonian requires, among others, this operator to be bounded from below.",
"Noting the manifest positivity of the relativistic kinetic-energy operator, for semirelativistic Hamiltonians – such as those presented in Eqs.",
"(REF ) or (REF ) – the crucial issue is the possibly singular behaviour of the interaction potentials showing up in the Hamiltonian.",
"In such a context, the (in some sense) worst-case scenario originates in the Coulomb potential: $V_{\\rm C}(\\mathbf {x})=V_{\\rm C}(|\\mathbf {x}|)\\equiv -\\frac{\\alpha }{|\\mathbf {x}|}\\ ,\\qquad \\alpha >0\\ .$ For the spinless relativistic Coulomb problem, in turn, the spectral theory has been thoroughly worked out [4], [5]: A semirelativistic Hamiltonian (REF ) with a Coulomb interaction potential (REF ) is bounded from below if and only if the involved coupling parameter $\\alpha $ satisfies the constraint $\\alpha <\\frac{4}{\\pi }=1.273239\\dots \\ .$ Only for these strengths $\\alpha $ , the lower bound to the spectrum $\\sigma (\\widetilde{H})$ of the Hamiltonian (REF ) reads $\\sigma (\\widetilde{H})\\ge 2\\,m\\,\\sqrt{1-\\left(\\frac{\\pi \\,\\alpha }{4}\\right)^{\\!2}}\\ .$ For a more refined range, $\\alpha \\le 1$ , a somewhat increased lower bound on $\\sigma (\\widetilde{H})$ can be found [6]: $\\sigma (\\widetilde{H})\\ge 2\\,m\\,\\sqrt{\\frac{1+\\sqrt{1-\\alpha ^2}}{2}}\\ .$"
],
[
"Spinless-Salpeter Hamiltonian Operators: Number of\nDiscrete Eigenstates {{cite:d7b5f74edcbf6f0df6045f6ec4e9da5b0f231546}}",
"Rigorous upper limits on the total number of bound states described by some spinless Salpeter equation may be formulated [7] in terms of (Lebesgue) spaces of measurable functions, $L^p$ , on the Euclidean space ${\\mathbb {R}}^3$ , for $p=3/2,3$ .",
"For a Hamiltonian operator acting on the Hilbert space $L^2({\\mathbb {R}}^3)$ and a potential that is a negative smooth function of compact support, hence, subject to $V(\\mathbf {x})\\in C^\\infty _0({\\mathbb {R}}^3)\\ ,\\qquad V(\\mathbf {x})\\le 0\\ ,$ in equal-mass two-particle problems (REF ) the total number of bound states, $N$ , is constrained by $N\\le \\frac{C}{48\\,\\pi ^2}\\int {\\rm d}^3x\\left[|V(\\mathbf {x})|\\left(|V(\\mathbf {x})|+4\\,m\\right)\\right]^{3/2}\\ ,\\quad C=\\left\\lbrace \\begin{array}{rl}14.107590867&\\quad \\mbox{if}\\ \\ \\ m>0\\ ,\\\\[.5ex]6.074898097&\\quad \\mbox{if}\\ \\ \\ m=0\\ ,\\end{array}\\right.$ with numerical results for $C$ taken from Ref. [8].",
"Finiteness forces the potential $V(\\mathbf {x})$ to satisfy $V(\\mathbf {x})\\in L^{3/2}({\\mathbb {R}}^3)\\cap L^3({\\mathbb {R}}^3)\\ .$"
],
[
"Semirelativistic Hamiltonian Operators: Upper Limits on\nDiscrete Eigenvalues",
"Upon having succeeded to establish – in the delicate instance of singular interaction potentials by rather shameless abuse of the findings gained for the spinless relativistic Coulomb problem [4], [5] (reviewed in Subsect.",
"REF ) – the rigorous boundedness from below of the semirelativistic Hamiltonian in one's focus of interest, that is, of that operator's spectrum, most likely one will be tempted to try, as a next step, to acquire (at least, limited) information on the actual location of its lowest-lying discrete eigenvalues.",
"Such goal can be achieved by, for instance, narrowing down the conceivable range of any of these eigenvalues by finding an upper bound to its range.",
"The theoretical foundation of any attempt of this kind is the minimum–maximum theorem [9]: Consider some self-adjoint operator, $H$ , bounded from below, with its (ordered) eigenvalues $E_0\\le E_1\\le E_2\\le \\cdots \\ .$ Define the operator $\\widehat{H}$ by restricting $H$ to some $d$ -dimensional subspace of the domain of $H$ .",
"Let $\\widehat{E}_0,\\widehat{E}_1,\\dots ,\\widehat{E}_{d-1}$ be the $d$ eigenvalues of the restriction $\\widehat{H}$ , likewise ordered according to $\\widehat{E}_0\\le \\widehat{E}_1\\le \\widehat{E}_2\\le \\cdots \\le \\widehat{E}_{d-1}\\ ;$ these form upper bounds to the first $d$ eigenvalues of $H$ below its essential spectrum's onset: $E_k\\le \\widehat{E}_k\\qquad \\forall \\quad k=0,1,2,\\dots ,d-1\\ .$ Below, in Sect.",
", we will show slight predilection for interactions governed by spherically symmetric potentials.",
"If dealing with central potentials, mere convenience dictates to span the $d$ -dimensional subspace required by the minimum–maximum theorem by a basis the elements of which are functions, $\\psi _{k\\ell }(\\mathbf {x})$ , that are products of a spherical harmonic ${\\cal Y}_\\ell $ for orbital angular momentum $\\ell $ and a radial factor involving the generalized-Laguerre polynomials $L_k^{(\\gamma )}$ [10], [11]: $\\psi _{k\\ell }(\\mathbf {x})\\propto |\\mathbf {x}|^{\\ell +\\beta -1}\\exp (-\\mu \\,|\\mathbf {x}|)\\,L_k^{(2\\ell +2\\beta )}(2\\,\\mu \\,|\\mathbf {x}|)\\,{\\cal Y}_\\ell (\\Omega _{\\mathbf {x}})\\ ,\\qquad k\\in {\\mathbb {N}}_0\\ ,\\qquad \\ell \\in {\\mathbb {N}}_0\\ ,&\\\\L_k^{(\\gamma )}(x)\\equiv \\sum _{t=0}^k\\, \\binom{k+\\gamma }{k-t}\\,\\frac{(-x)^t}{t!",
"}\\ ,\\qquad \\mu >0\\ ,\\qquad \\beta >-\\frac{1}{2}\\ ;&\\nonumber $ herein, $\\Omega _{\\mathbf {x}}$ indicates the $\\mathbf {x}$ -space solid angle, and $\\mu $ and $\\beta $ represent two variational parameters."
],
[
"Spinless-Salpeter Hamiltonian Operator: Quality\nAssurance of its Eigenstates",
"Irrespective of the actual origin of approximate eigenstates $|\\chi \\rangle $ of a Hamiltonian operator, their accuracy [12], [13] can be judged (or even quantified) by their degree of fulfilment of the master or relativistic virial theorem [14], [15] relating the expectation values of the radial derivatives of all kinetic terms and of the potential.",
"This relation comprises, of course, the well-known virial theorem of nonrelativistic quantum theory as a special case.",
"For the operators (REF ), it reduces to $\\left\\langle \\chi \\left|\\frac{\\mathbf {p}^2}{\\sqrt{\\mathbf {p}^2+m_1^2}}+\\frac{\\mathbf {p}^2}{\\sqrt{\\mathbf {p}^2+m_2^2}}\\right|\\chi \\right\\rangle =\\left\\langle \\chi \\left|\\,\\mathbf {x}\\cdot \\frac{\\partial \\,V}{\\partial \\mathbf {x}}(\\mathbf {x})\\,\\right|\\chi \\right\\rangle .$"
],
[
"Trivial Upper Bounds: Nonrelativistic Limit of\nSpinless-Salpeter Hamiltonians",
"The (undeniable) concavity of the square-root operator of the proper relativistic expression for the kinetic energy regarded as a function of $\\mathbf {p}^2$ implies that its nonrelativistic limit is tangent to its relativistic precursor at the point of tangency $\\mathbf {p}^2=0$ .",
"Thus, the nonrelativistic Hamiltonian $H_{\\rm NR}=m_1+m_2+\\frac{\\mathbf {p}^2}{2\\,\\hat{m}}+V(\\mathbf {x})\\ ,\\qquad \\hat{m}\\equiv \\frac{m_1\\,m_2}{m_1+m_2}\\ ,$ forms an upper bound to its originator (REF ), with expectable impact on their sets of eigenvalues: $H\\le H_{\\rm NR}\\qquad \\Longrightarrow \\qquad \\ E_k\\le E_{{\\rm NR},k}\\ ,\\qquad k\\in {\\mathbb {N}}_0\\ .$ With respect to the total number $N$ of bound states of the spinless Salpeter equation, the above observation implies that this number can never be less than its nonrelativistic counterpart [16]."
],
[
"Informative Application to Two Spinless Semirelativistic\nProblems",
"In the past, the effortlessness of applying all the findings collected in Sect.",
"to semirelativistic bound-state problems has been demonstrated [17], [18], [19], [20], [8], [21], [22] for a number of frequently employed, rather popular interaction potentials.",
"Therefore, let's take a brief look at two of these analyses, where simplicity motivated us to confine ourselves to spherically symmetric central potentials $V(\\mathbf {x})=V(r)\\ ,\\qquad r\\equiv |\\mathbf {x}|\\ .$"
],
[
"Short-Range Singular Example: Spinless Relativistic\nHulthén Problem {{cite:102dfbf7f075e66691ede82904a9dde074345d42}}, {{cite:dd192249456c5fedffb262ce7921d6bc00b96fa6}}",
"The Hulthén potential $V_{\\rm S}(r)$ , originally introduced in nuclear physics but since then in physics widely used, is defined by two parameters, the coupling strength $\\eta $ and an exponential range $b$ : $V_{\\rm S}(r)\\equiv -\\frac{\\eta }{\\exp (b\\,r)-1}\\ ,\\qquad b>0\\ ,\\qquad \\eta \\ge 0\\ .$ As shown in Fig.",
"REF , the potential $V_{\\rm S}(r)$ approaches, for short distances, from above a Coulomb potential $V_{\\rm C}(r)$ with strength $\\eta /b$ (i.e., exhibits Coulombic behaviour) and, for large distances, from below a negative exponential potential $V_{\\rm E}(r)$ , or increases exponentially damped to zero: $V_{\\rm S}(r)\\xrightarrow[r\\rightarrow 0]{}V_{\\rm C}(r)\\equiv -\\frac{\\eta }{b\\,r}\\ ,\\qquad V_{\\rm S}(r)\\xrightarrow[r\\rightarrow \\infty ]{}V_{\\rm E}(r)\\equiv -\\eta \\exp (-b\\,r)\\ .$ Conversely, the Hulthén potential (REF ) is bounded from below by the Coulomb potential (REF ) iff $\\alpha \\ge \\frac{\\eta }{b}\\ .$ Consequently, the Hulthén potential (REF ) inevitably exhibits a Coulomb-like singularity at spatial origin $r=0$ .",
"Nevertheless, according to Subsect.",
"REF the assigned Hamiltonian (REF ) and bound-state spectrum are bounded from below if the two involved potential parameters satisfy $\\frac{\\eta }{b}<\\frac{4}{\\pi }\\ .$ Figure: Hulthén potential V S (r)V_{\\rm S}(r) (solid black line), Eq.",
"(), and asymptotes(): negative exponential potential V E (r)V_{\\rm E}(r) (dottedblue line) and Coulomb potential V C (r)V_{\\rm C}(r) (dashed magentaline), for strength η=1\\eta =1.For the nonrelativistic limit (REF ) of the Hamiltonian (REF ) with Hulthén potential (REF ) and all states of orbital angular momentum $\\ell =0$ , the eigenvalues $E_{\\rm NR}$ can be found analytically [23]: $E_{{\\rm NR},n}=m_1+m_2-\\frac{\\left(2\\,\\hat{m}\\,\\eta -n^2\\,b^2\\right)^2}{8\\,\\hat{m}\\,n^2\\,b^2}\\ ,\\qquad \\hat{m}\\equiv \\frac{m_1\\,m_2}{m_1+m_2}\\ ,\\qquad n\\in {\\mathbb {N}}\\ ,$ where the radial quantum number $n$ is constrained by the two potential characteristics $\\eta $ and $b$ , $n\\le \\frac{\\sqrt{2\\,\\hat{m}\\,\\eta }}{b}\\ .$ In the spirit of Subsect.",
"REF , the expressions (REF ) may serve to provide, at least, a vague idea of the actual location of the discrete eigenstates of any spinless semirelativistic Hulthén problem."
],
[
"Category of Spinless Semirelativistic\nGeneralized-Hellmann Problems {{cite:5096692165975706f7a55579a36ce68680363606}}, {{cite:c2c5b393e0e8f238ca62750f4489c86392514f3a}}",
"Generalized Hellmann potentials [21], [22] establish a class of central interaction potentials that encompasses the shape introduced, some time ago [24], [25], for applications in atomic-physics theory.",
"Each member of this set is defined to be a linear combination of an attractive Coulomb contribution $V_{\\rm C}(r)$ and an optionally either attractive or repulsive Yukawa contribution $V_{\\rm Y}(r)$ , $V_{\\rm H}(r)\\equiv V_{\\rm C}(r)+V_{\\rm Y}(r)=-\\frac{\\kappa }{r}-\\upsilon \\,\\frac{\\exp (-b\\,r)}{r}\\ ,$ Coulomb coupling $\\kappa $ , Yukawa coupling $\\upsilon $ , and exponential-range parameter $b$ hence satisfying $\\kappa \\ge 0\\ ,\\qquad \\upsilon \\gtreqqless 0\\ ,\\qquad b>0\\ .$ (Note that, in the notation (REF ), the interaction potential originally proposed by Hellmann [24] corresponds to assuming for the Yukawa coupling parameter $\\upsilon $ a strictly negative value $\\upsilon <0$ .)",
"At large distances $r$ , all generalized Hellmann potentials approach their Coulomb component, $V_{\\rm H}(r)\\xrightarrow[r\\rightarrow \\infty ]{}V_{\\rm C}(r)\\ .$ This guarantees that any discrete eigenvalue $E_k$ of $H$ is nonpositive, thus bounded from above: $E_k\\le 0\\qquad \\forall \\quad k\\in {\\mathbb {N}}_0\\ .$ Due to the asymptotic approach (REF ) to Coulomb-type behaviour, every generalized Hellmann potential (REF ) fails to meet all requirements imposed in Subsect.",
"REF , particularly, because [21] $V_{\\rm H}(r)\\notin L^{3/2}({\\mathbb {R}}^3)\\cap L^3({\\mathbb {R}}^3)\\ ,$ whence no finite upper bound (REF ) on the total number of its discrete bound states can be found.",
"Nonsingular generalized Hellmann potentials, in the classification of Table REF identified by $\\kappa +\\upsilon \\le 0$ , are bounded from below, as likewise (because of $\\sqrt{\\mathbf {p}^2+m^2}\\ge 0$ ) their Hamiltonians: $H\\ge V_{\\rm H}(r)&\\ge \\min _{0\\le r<\\infty }V_{\\rm H}(r)>-\\infty &\\qquad \\mbox{for}\\qquad &\\upsilon <-\\kappa \\ ,&\\\\[.3ex]H\\ge V_{\\rm H}(r)&\\ge V_{\\rm H}(0)=\\upsilon \\,b&\\qquad \\mbox{for}\\qquad &\\upsilon =-\\kappa \\ .&$ Singular generalized Hellmann potentials, in their systematics of Table REF characterized by $\\kappa +\\upsilon >0$ , develop negative singularities at $r=0$ bounded from below by a Coulomb potential: $V_{\\rm H}(r)\\ge -\\frac{\\alpha }{r}\\qquad \\mbox{with}\\qquad \\left\\lbrace \\begin{array}{ll}\\alpha =\\kappa +\\upsilon &\\qquad \\mbox{for}\\qquad \\upsilon >0\\ ,\\\\[.5ex]\\alpha =\\kappa &\\qquad \\mbox{for}\\qquad \\upsilon \\le 0\\ .\\end{array}\\right.$ As consequence of this, mutatis mutandis the relevant insights collected in Subsect.",
"REF apply.",
"Most importantly, spinless relativistic generalized-Hellmann problems are well defined [21] if $\\kappa +\\upsilon \\le \\frac{4}{\\pi }\\ .$ Table: Characterization of the seventypes of generalized Hellmann potentials V H (r)V_{\\rm H}(r)distinguished by comparing the Yukawa coupling υ\\upsilon with the(by assumption nonnegative) Coulomb coupling κ≩0\\kappa \\gneqq 0.Merely for illustrative purposes, Fig.",
"REF depicts a typical representative of four of our seven classes of generalized Hellmann potentials (REF ) identified in Table REF .",
"There is no need to show the trivial case of the pure Coulomb potential arising for vanishing Yukawa coupling strength.",
"In order to exemplify the calculation of upper bounds on energies along the lines sketched in Subsect.",
"REF , Table REF provides, for few low-lying eigenstates of three generalized-Hellmann problems and a fixed choice of the numerical values of the parameters $\\mu $ and $\\beta $ in the basis (REF ) of some $d$ -dimensional subspace, upper bounds on the associated binding energies, defined by $B_k\\equiv E_k-2\\,m\\ ,\\qquad k\\in {\\mathbb {N}}_0\\ .$ The upper bounds in Table REF can straightforwardly be optimized variationally, in two respects: definitely by increasing the dimension $d$ of the adopted subspace, and potentially by trying out different values for (i.e., varying) the parameters $\\mu $ and/or $\\beta $ defining the subspace's basis (REF ).",
"Figure: Generic representatives (solid black lines)of four of the only seven distinguishable categories in oursystematic classification (Table ) of generalizedHellmann potentials V H (r)V_{\\rm H}(r) defined in Eq.",
"() bylinear combinations of Coulombic (dashed magenta lines) and Yukawa(dotted blue lines) contributions, choosing κ=1\\kappa =1 and (a)υ=10\\upsilon =10, (b) υ=1\\upsilon =1, (c) υ=-1\\upsilon =-1, or (d)υ=-2\\upsilon =-2 for the involved coupling strengths."
]
] | 2210.07720 |
[
[
"A Sequence-Aware Recommendation Method Based on Complex Networks"
],
[
"Abstract Online stores and service providers rely heavily on recommendation softwares to guide users through the vast amount of available products.",
"Consequently, the field of recommender systems has attracted increased attention from the industry and academia alike, but despite this joint effort, the field still faces several challenges.",
"For instance, most existing work models the recommendation problem as a matrix completion problem to predict the user preference for an item.",
"This abstraction prevents the system from utilizing the rich information from the ordered sequence of user actions logged in online sessions.",
"To address this limitation, researchers have recently developed a promising new breed of algorithms called sequence-aware recommender systems to predict the user's next action by utilizing the time series composed of the sequence of actions in an ongoing user session.",
"This paper proposes a novel sequence-aware recommendation approach based on a complex network generated by the hidden metric space model, which combines node similarity and popularity to generate links.",
"We build a network model from data and then use it to predict the user's subsequent actions.",
"The network model provides an additional source of information that improves the accuracy of the recommendations.",
"The proposed method is implemented and tested experimentally on a large dataset.",
"The results prove that the proposed approach performs better than state-of-the-art recommendation methods."
],
[
"Introduction",
"A recommendation system (RS) is a software tool that uses various techniques and algorithms to filter the relevant information from the vast information found in an online platform based on multiple factors to provide users with item recommendations that most likely suit their preferences.",
"RSs have been applied in various domains, including travel and hotel industries, online shopping, books, and movie recommendations [1].",
"Conventionally, the recommendation problem is abstracted as a matrix-completion problem where users correspond to rows, items correspond to columns, and the numerical cell values indicate the user-item ratings.",
"The goal of matrix completion is to predict the ratings of unseen items for a given user based on historical data [2].",
"Although this abstraction has proved helpful in various ways, it suffers from the limitation of not utilizing the sequence of user interaction logs that are often available in real applications [3].",
"Moreover, conventional recommendation systems assume user profile availability and long-term historical data.",
"However, such long-term data does not always exist for many reasons, such as the user being new to the system, having opted not to log in, or the user is enabling tracking countermeasures [4], [5].",
"For this, sequence-aware recommendation systems (SARS) have recently been developed to harness the rich information from logged users' interactions with the system.",
"The goal is to derive predictions for subsequent user actions based on the recent series of actions in the ongoing user session, thus, bringing highly relevant and practical computational tasks to real-life applications.",
"Despite the ongoing efforts to improve SARS, the accuracy of recommendations remains an open challenge.",
"Most of the existing complex models, including deep learning and matrix factorization, are outperformed by straightforward trivial approaches such as the $k$ nearest neighbors' approach [6], [7].",
"One of the most promising directions to remedy this is using graph (or network) models to generate recommendations.",
"In network-based recommenders, where nodes represent users or items and the weighted links between the nodes represent relevancy, the analysis derived from the graph structure can produce accurate predictions.",
"Many network-based recommendation approaches have been introduced in the literature, including approaches based on real-world networks [8], [9], [10].",
"However, most approaches use networks without underlying models or established properties.",
"Using complex network models with well-understood and proven properties can improve the quality of recommendations and constitutes a promising research direction.",
"One of the most relevant recent advances in complex network modeling to the field of sequence-aware recommendation is the development of efficient navigation and routing algorithms [11], [12].",
"Since the interaction between the user and items in SARS can be seen as navigation in the space of items, this hints at the possibility of using complex network models to guide the recommendation process.",
"In this paper, we propose a novel sequence-aware recommender system approach that takes advantage of complex network models, primarily the hidden metric space model [13], to generate more accurate recommendations.",
"The hidden metric space model is a complex network model that suggests the existence of a hidden metric space underlying any observed complex network.",
"The distance between the nodes abstracts their similarities, whereas their degrees represent popularity [12].",
"Our method utilizes the rich navigation information within the large logs of sequentially ordered actions to improve recommendation accuracy.",
"The resulting algorithm is trained and tested on a large public dataset and evaluated using standard performance measures to compare its performance to state-of-art methods.",
"Figure: A high-level representation of sequence-aware recommender systems process.The remainder of this paper is organized as follows.",
"Section covers background and related work.",
"Section describes the details of the proposed method.",
"Section presents the experimental setup and performance evaluation results.",
"Finally, Section summarizes the work and presents future research directions."
],
[
"Background and Related Work",
"In this section, we briefly introduce recommender systems, particulary sequence-aware recommender systems, followed by an overview of the complex networks and the hidden metric space model."
],
[
"Sequence-aware Recommender Systems",
"The goal of sequence-aware recommender systems is to predict the evolution of the user's current session.",
"They achieve this goal by predicting the user's next action using extracted features from historical sessions' logs tracked by the application [4].",
"Fig.",
"REF shows a schematic illustration of the recommendation process under the sequence-aware setting.",
"The recommender system's input takes the form of an ordered set of users' logged actions (sessions).",
"These actions consist of various user interactions towards an item such as 'clicks', 'views', or 'purchases', or actions towards the application such as 'searches' or 'applies filters'.",
"The computational task of sequence-aware recommenders is mainly to process the input to build a model that attempts to identify recurring patterns in the sequences of actions.",
"For example, this pattern may reflect the co-occurrence of actions or their sequential ordering.",
"Finally, the system's output is the predictions for the user's next action (or set of actions) derived from the identified pattern based on the current user session.",
"Although the output from traditional RS and SARS aims at the same goal: to provide an item recommendation for the users, their settings and characteristics are distinct in several ways.",
"First, the standard input for traditional RS takes the form of user, item, and rating tuples without any information about the user's behavior or interactions with the items or the application.",
"In contrast, the primary input for SARS is the rich sequential session data.",
"Moreover, a standard RS takes the form of a matrix-completion problem, as illustrated in Fig.",
"REF , in which the task is to fill the predicted ratings given by users to items based solely on the long-term observed user behavior and preferences.",
"This formulation, however, can not accommodate the sequential input representation [2].",
"Figure: Representation of the user-item matrix in traditional recommender systems.",
"The goal of the recommender is to complete this matrix.The problem of sequence-aware recommendation has been increasingly attracting researchers.",
"Early research efforts proposed oversimplified model-free attempts to provide recommendations driven by data pattern mining and association rules techniques.",
"However, since the 2010s, machine learning and time-series techniques have shifted the algorithmic approaches towards adopting more sophisticated models to tackle the problem [14].",
"Models such as Markov chain models [15], [16], graph-based models [17], [18], [19], factorization-based models [20], recurrent neural networks [21], [22], [23] and attention mechanisms [24] contributed to improving sequence-aware recommendation systems [6].",
"Several graph-based approaches for sequence-aware recommenders exist in the literature that extends the work proposed for traditional recommender systems.",
"In these methods, the graph is often modeled as a bipartite graph having items and users on separate sides.",
"However, this poses a limitation as these approaches fail to exploit the rich information in sessions.",
"To overcome this issue, authors in [17] proposed the session-based temporal graph (STG) method that incorporates temporal dynamics and short- and long-term preferences.",
"The STG graph is represented as a two-sided bipartite graph (tripartite), with one side of the graph connecting users with items they have interacted with before (i.e., long-term interactions) and the other side connecting the current user session with session items (i.e., short-term interactions).",
"The graph is then traversed to recommend items via random walk.",
"Similarly, the Geographical-Temporal influences Aware Graph (GTAG) method [18] uses a tripartite graph representation.",
"However, GTAG incorporates geographical location to enhance the point of interest recommendations and traverse the resulting graph model in a breadth-first propagation manner.",
"Both methods, however, as in all multipartite graphs models, which do not allow edges between nodes on the same side, fail to exploit information on any independent side, for example, item-to-item data.",
"Additionally, both methods construct a simple graph without an underlying model.",
"Conversely, the proposed method in this article relies on constructing a graph with an underlying model and established properties to guide the recommendations.",
"In recent years, researchers have attempted to achieve accurate recommendations by providing explicit interpretations of the challenges and tasks in the SARS and then attempting to tackle the identified challenges by designing appropriate algorithms.",
"Quadrana et al.",
"[3] suggest that the main tasks of sequence-aware recommender systems are adopting the recommendation context, such as the current weather or the current user location, detecting the shared and individual trends, identifying the repeated patterns in user behavior, and finally identifying the order constraints of the sequence in the session.",
"Jannach et al.",
"[25] state the price discounts and current offers on items as a success factor for SARS and emphasize the impact of the user’s short-term intentions rather than only long-term ones.",
"Wang et al.",
"[14] further focus on understanding the inner interactions within the session and the interactions between the sessions as the two key challenges to reducing the complexity of SARS structure."
],
[
"Complex Networks and Hidden Metric Space Model",
"Complex networks refer to a class of graphs that exhibit nontrivial topological features that can be observed in real-life networked systems.",
"Those features are not observable in simple graphs such as regular and purely random graphs but in networks resulting from complex natural phenomena such as social, biological, technological, and physical systems [26].",
"For example, when modeling a real-life social network, where the nodes represent the people and the edges between the nodes represent the relationships between them, the resulting graph possesses a complex network structure.",
"Its topology reveals some fundamental complex network properties and characteristics [27], such as the small-world [28] and the scale-free properties.",
"Scale-free networks gained significant interest within the network science community upon the publication of the Barabási and Albert model [29].",
"The latter generates scale-free networks with short paths and highly connected nodes as hubs and is considered one of the earliest models to capture real-life natural network properties [30].",
"Efforts to design network models that capture more faithfully the properties of real-life networks have continued after the work of Barabási and Albert model [29].",
"The hidden metric space model [13] suggests the existence of a hidden metric space underlying any scale-free complex network and contains all the nodes of the observed network.",
"The nodes are positioned and linked in the hidden metric space based on their similarity and popularity.",
"The distance between nodes abstracts the nodes' similarities, and their degrees reflect their popularity.",
"Thus, the shorter the distance between nodes in the hidden metric space and the more popular they are, the higher probability of them connecting in the observed network.",
"One of the successful applications of the hidden metric space model is its capacity to guide the routing function in networks by greedily moving toward the node nearest to the target [31].",
"Unlike many complex network models, such as the small-world model or Barabási and Albert model, the hidden metric space model generates an embedding of the network, which provides a useful representation of the items for downstream applications.",
"Furthermore, the model not only aims at capturing the network's structural properties but also describes how these properties are connected with essential functionalities.",
"The successful use of the model in various domains, including developing efficient information routing algorithms [11], network modeling [12], information routing and signaling [31], link prediction [32], [33], [34] and recommender systems [35] hints at the possibility of applying it to guide the recommendation in a SARS settings.",
"Figure: Illustration of the proposed approach.",
"Items are modeled as nodes in a complex network and are connected based on similarity and popularity."
],
[
"Proposed Method",
"In this paper, we propose a sequence-aware recommendation method based on complex networks where items are modeled as a graph embedded in $D$ -dimensional space with a scale-free network structure controlled by an underlying hidden metric space.",
"Since the goal of a SARS is to predict the next click the user will perform in a session, the problem is defined as the task of ranking the most probable next item.",
"The output of the recommender is an ordered list of items $r \\in R^*$ for each session $s$ , where $R^*$ is a subset of the group of all possible items permutations, and each predicted ordered list of recommendations $r$ consists of $i$ items of length between 0 and $t$ .",
"First, each item in the system $i$ is assigned a point $x^i$ in a $D$ -dimensional Euclidean space: $x^i=\\left(x^i_1, x^i_2, \\ldots x^i_D\\right)^T , i = 1,\\ldots n,$ where $n$ denotes the number of items.",
"The position of the items reflects their description or features.",
"Hence, each dimension represents a latent feature or a combination of features relevant to the item.",
"The probability $p_{ij}$ of connection between two items nodes $i$ and $j$ is based on two factors, their similarity reflected by their distance and popularity (or degree) as derived from the hidden metric model: ${p}_{ij}={\\left(1+\\dfrac{d^2(x^i,x^j)}{{\\kappa _i\\kappa _j}}\\right)}^{-\\alpha },$ where $\\kappa _i$ and $\\kappa _j$ denotes the hidden degree (popularity) of items $i$ and $j$ , respectively, and $d^2(x^i,x^j)$ is the squared distance between them.",
"The parameter $\\alpha > 1 $ controls the weights given to distance and popularity.",
"According to Eq.",
"(REF ), the connection probability between nodes in the generated network increases when popularity increases and decreases with the squared hidden distance (dissimilarity) increase.",
"The reason behind this choice is that generated network will have three characteristics observed in real-life complex networks.",
"First, a pair of popular nodes will have a higher probability of being connected even if they are far away from each other or dissimilar.",
"A pair of nodes with moderate average popularity will be connected if the distance is moderate, and a pair of nodes with low popularity on average will get connected if the distance is short.",
"As illustrated in Fig.",
"REF , items are embedded in a $D$ -dimensional space, where the distance between them encodes their similarity, which, combined with item popularity, results in a network model.",
"The model can be used to compute the probability of a pair of nodes being connected and predict the next clicked item in a sequence.",
"We need to calculate the item's popularity and similarity to construct the model.",
"The node's degree represents its popularity and can therefore be directly calculated from the number of clicks or interactions the item has in historical sessions.",
"Next, we assign each node a position in $D$ -dimensional space to calculate the distances.",
"Since we cannot readily obtain the positions of the nodes, we infer them from the history of interaction sessions.",
"We first estimate the link probabilities directly from historical data and then use them to infer the items' positions.",
"There are arguably many ways to estimate connection probabilities.",
"However, the method we adopt in this work is to assume that the connection probability is proportional to the number of items co-occurrence in sessions calculated by the cosine similarity as follows: ${p}_{ij} \\propto sim(i,j) =\\frac{\\displaystyle \\sum _{s\\in S}i_s j_s}{\\sqrt{\\displaystyle \\sum _{s\\in S}{i_s^2}}{\\sqrt{ \\displaystyle \\sum _{s\\in S}{j_s^2}}}},$ where $i_s$ is set to 1 if item $i$ is present in session $s$ and 0 otherwise.",
"From that, we obtain the set $P$ of all connections probabilities $p_{ij}$ such that $p_{ij}>0$ .",
"It is now possible to deduce the distance $d^2_{ij}$ in Eq.",
"(REF ) as follows: $d^2_{ij} =\\kappa _i \\kappa _j \\left(p_{ij}^{-1/\\alpha }-1\\right).$ Hence, the problem of estimating the items' positions is reduced to finding coordinates $x^i$ , such that the resulting Euclidean distances are as close as possible to the distances prescribed by Eq.",
"(REF ).",
"This problem falls into the category of nonlinear optimization problems that can be solved by minimizing an objective function that measures the difference between the computed and observed distances: $\\sum _{p_{ij} \\in P} \\left( \\left\\Vert x^i - x^j\\right\\Vert _2^2- d_{ij}^2\\right)^2,$ where $\\Vert \\cdot \\Vert _2$ stands for the $L_2$ norm.",
"In order to avoid overfitting, a regularization term is added, resulting in the following objective function: $f(x^1, \\ldots , x^n)= \\sum _{p_{ij} \\in P} \\left( \\left\\Vert x^i - x^j\\right\\Vert _2^2- d_{ij}^2\\right)^2\\\\+ \\lambda \\sum _{i=1}^n \\left\\Vert x^i\\right\\Vert _2^2.$ Since this objective function is not convex, only local minima can be computed.",
"Several algorithms for local optimization exist to solve this minimization problem, and most of them require computing the gradient of the objective function: $\\frac{\\partial f}{\\partial x^i} = \\sum _{p_{ij}, p_{ji} \\in P} 4 \\left(x^i - x^j\\right) \\left( \\left\\Vert x^i - x^j\\right\\Vert _2^2- d_{ij}^2\\right) + 2 \\lambda x^i.$ The proposed method's objective is to predict a list of items ordered by the likelihood that the user will click next in the active session.",
"Once we have the item embeddings, we can use different techniques to capture the user's short-term interest in the current session to predict the next item.",
"We propose to use the $K$ nearest neighbors technique, where distance is measured in terms of connection probability.",
"Thus, the method recommends the items with the highest connection probability to the most popular item the user interacted with in the active session.",
"In other words, the method generates a list of $t$ items ordered decreasingly by the probability of connection to the most popular item in the active session as calculated by Eq.",
"(REF )."
],
[
"Experimental Evaluation",
"This section presents the experimental framework used to evaluate the proposed method's performance.",
"We present the experimental setup and describe the dataset, the dataset preparation techniques, the competing methods, the evaluation metrics, the training method, and the parameter tuning technique.",
"Finally, we compare the overall performance results against competing methods."
],
[
"Dataset Description",
"The dataset used in this paper is the Trivago dataset [36], [37] presented in the 2019 RecSys challenge obtainable online from a publicly available data source (https://recsys2019data.trivago.com).",
"The dataset contains sequences of various users actions on the Trivago hotels booking website.",
"It contains 981,655 users, 927,142 hotels, and 1,202,064 sessions containing a total of 19,715,327 user actions.",
"The dataset is pre-split into train and test sets, the splits were used as they are in this experiment.",
"We cleaned the dataset by removing sessions that do not lead to a hotel booking.",
"Each session in the test set ends with a hidden hotel booking action and a list of a maximum of 25 hotels in the impressions list, and the goal is to reorder the list of impressions in decreasing order by the likelihood of user clicks."
],
[
"Evaluation Criteria",
"The proposed approach is evaluated using the test set by applying standard metrics for the sequence-aware recommender systems evaluation.",
"There are different evaluation techniques and accuracy measures for sequence-aware recommender methods in the literature.",
"However, since the output of a sequence-aware recommender system typically takes the form of ordered lists, it is applicable to use standard information retrieval accuracy metrics [6] such as mean reciprocal rank (MRR) and mean average precision (MAP).",
"The MRR measures the place of the correct item in the predicted list.",
"Given a test set $S^{Test}$ containing previously unseen sessions, the MRR is defined as: $MRR = \\frac{1}{|S^{Test}|} \\sum _{s \\in S^{Test}} \\frac{1}{Rank_s}$ where $Rank_s$ is the place of the correct item in session $s$ .",
"From Eq.",
"(REF ), we can see that, at most, the MRR reaches 1 if the algorithm consistently predicts the item correctly by placing it first in the list and reaches $1/t$ , where $t$ is the length of the recommended list when the recommender consistently predicts the item the last in the list.",
"The MAP, on the other hand, evaluates the predicted list up to a specific cut-off point $N$ .",
"MAP$@N$ does not consider the order of the list.",
"However, it checks if the correct item is present within the first $N$ items in the list.",
"$MAP@N = \\frac{1}{|S^{Test}|} \\sum _{s \\in S^{Test}} \\frac{Top(s,N)}{N}$ where $Top(s,N)$ is a function that returns 1 if the correct item for session $s$ is in the top $N$ items in the predicted list and returns 0 otherwise.",
"In this experiment, we use the following values for $N$ : 1, 3, 5 and 10."
],
[
"Implementation",
"The experimental framework, competing methods, data preprocessing, and experimental evaluation are implemented using python with standard numerical and data manipulation libraries.",
"Several specialized software libraries can be used for solving the optimization problem, which is the most crucial step in our proposed method.",
"Interior Point OPTimizer (IPOPT) [38] is a stable and well-tested advanced nonlinear programming (NLP) solver based on an interior-point filter with a line-search algorithm for large-scale nonlinear optimization.",
"Efficient and effective linear solvers are required by IPOPT and are essential for solving the optimization problem.",
"Therefore, The solver MA57, part of the HSL package [39], was used in the experiment.",
"IPOPT NLP solver requires, at each iteration, essential information about the optimization problem in order to solve and proceed iteratively toward the solution.",
"At each step, the following information is required given the current values of the unknowns: The objective function as defined in Eq.",
"(REF ).",
"The gradient of the objective function (vector of first derivatives with respect to all unknowns) as defined in Eq.",
"(REF ).",
"The Hessian of the objective function (matrix of all second derivatives).",
"This matrix is very large.",
"However, the library offers the possibility of approximating the Hessian numerically.",
"This option has been used in order to avoid large memory consumption."
],
[
"Competing Methods",
"To assess the effectiveness of the proposed method, we compare it with a set of commonly used baselines and state-of-the-art methods: Random: Even though this is not a recommendation method, it provides insight into the lowest acceptable value that other methods must achieve on this dataset.",
"Items Popularity (I-POP): A naive baseline predictor that consistently recommends a list of the most popular items in the training set without considering user actions or similarities.",
"Regardless of its simplicity, it often provides a stable baseline in sequence-aware recommender systems.",
"Items Click-out Popularity (IC-POP): Similar to I-POP, however, only clicked-out items are considered when determining the item's popularity.",
"Items Metadata $K$ -Nearest Neighbors (IM-KNN): An implementation of a content-based filtering method that predicts the current user's next action based on the $k$ -most similar neighbors to the previously clicked item.",
"The similarity between the items is determined by the cosine similarity of the item's metadata.",
"Items Co-occurrence $K$ -Nearest Neighbors (IC-KNN): An implementation of an item-based collaborative filtering method that predicts the current user's next action based on the $k$ -most similar neighbors to the previously clicked item.",
"The similarity between the items is determined by the number of co-occurrence between them in sessions, calculated by cosine similarity.",
"Logistic Regression (LR) [36] : A method that predicts whether the item is clicked-out or not (i.e., binary classification).",
"The method requires the extraction of specific features.",
"We adopted the same problem formalization and feature selection as in [36].",
"MLP Regressor (MLP): Multilayer perceptron is a neural network model that trains using backpropagation with no activation function in the output layer.",
"We used the same features as in the LR model."
],
[
"Parameters Settings and Tuning",
"For the proposed method, several adjustable parameters require tuning.",
"The number of dimensions $D$ , the regularization coefficient $\\lambda $ , and the constant $\\alpha $ .",
"In this experiment, we used grid search to determine those parameters.",
"The number of dimensions $D$ is tried with $\\lbrace 5, 10, 20\\rbrace $ .",
"The regularization $\\lambda $ is selected from the set $\\lbrace 0.1, 0.01\\rbrace $ .",
"Finally, the constant $\\alpha $ is chosen from $\\lbrace 1, 2, 3\\rbrace $ .",
"An important issue when fitting our model to data is the initialization step.",
"Since the objective function is non-convex, only local minima can be found, and consequently, the initial values of the unknowns significantly affect the quality of the solution.",
"In particular, initialization with the same values, such as setting all coordinates to 0, causes the gradient in Eq.",
"(REF ) to vanish and the optimizer to stop immediately.",
"Hence, a random initialization is essential to avoid the trivial solution where all items are assigned the same position."
],
[
"Experimental Results",
"Table REF summarizes the results in terms of MRR and MAP.",
"The results are also displayed in Figure REF for the case baseline methods and Figure REF for state-of-the-art methods.",
"The results show that the proposed method produces better scores than other competing methods under all considered metrics.",
"As expected, popularity-based methods (I-POP and IC-POP) score weak results as they rely solely on an item's popularity and ignore the user features.",
"In other words, those methods are expected not to achieve well as they do not produce personalized recommendations.",
"Interestingly, the proposed method performed better not only against KNN-based methods (IM-KNN and IC-KNN), which rely on similarity measures without requiring feature extractions, but also against feature-based methods (MLP and LR), which require considerable effort in the feature engineering process.",
"This is particularly important given that the Trivago dataset is rich in information, which may not be true for all datasets.",
"Since the proposed approach does not involve the additional semantic information available in Trivago datasets, we expect it to perform very well on datasets with limited information.",
"Table: Performance results obtained on the Trivago dataset.Figure: Comparison of the performance of the proposed approach against baseline methods on the Trivago dataset.Figure: Comparison of the performance of the proposed approach against state-of-the-art methods on the Trivago dataset."
],
[
"Conclusion and Future Work",
"This paper demonstrates the potential of using complex networks with underlying similarity-popularity models for a sequence-aware recommendation system.",
"The experimental results show that the proposed method performs better than state-of-the-art methods in terms of MRR and MAP.",
"Furthermore, the proposed method does not rely on feature extraction, which makes it suitable for generalization to datasets with limited information (similar to KNN-based approaches) while providing better results.",
"In future work, we propose testing the method on other datasets from various domains to confirm its generalization capacity.",
"Moreover, additional similarity-popularity models other than the hidden metric space model can be explored.",
"Such models range from basic similarity-popularity dot-product to more recent and sophisticated models found in the literature.",
"In this work, once the items are embedded, we use the $k$ -nearest neighbors technique to capture the user's current session and predict the next item.",
"However, it is possible to use more advanced techniques such as deep neural networks, namely the LSTM or GRU variants of recurrent neural networks, by feeding the network the user's current sequence of actions and item embeddings as input and then training it to predict the next item."
],
[
"Acknowledgment",
"This research work is supported by the Research Center, CCIS, King Saud University, Riyadh, Saudi Arabia.",
"20"
]
] | 2210.07814 |
[
[
"Spheroidal expansion and freeze-out geometry of heavy-ion collisions in\n the few-GeV energy regime"
],
[
"Abstract A spheroidal model of the expansion of hadronic matter produced in heavy-ion collisions in the few-GeV energy regime is proposed.",
"It constitutes an extension of the spherically symmetric Siemens-Rasmussen blast-wave model used in our previous works.",
"The spheroidal form of the expansion, combined with a single-freeze-out scenario, allows for a significantly improved description of both the transverse-mass and the rapidity distributions of the produced particles.",
"With the model parameters determined by the hadronic abundances and spectra, we make further predictions of the pion HBT correlation radii that turn out to be in a qualitative agreement with the measured ones.",
"The overall successful description of the data supports the concept of spheroidal symmetry of the produced hadronic systems in this energy range."
],
[
"Introduction",
"Thermal models of hadron production, based on the idea of statistical hadronization, have been very successful in describing hadron yields and phase-space distributions in various collision processes, especially in ultrarelativistic heavy-ion collisions (UrHIC) in a wide range of beam energies and for different projectile-target systems (see, e.g., Refs.",
", , , , , , , , ).",
"The reasons for studying thermal aspects of hadron production in heavy-ion collisions are manifold.",
"The hadron abundances can be explained over several orders of magnitude of multiplicity by fixing a small number of thermodynamic parameters.",
"Moreover, the assumption of local thermalization of the expanding dense and hot matter formed in the collision (called a fireball) allows for the application of hydrodynamic concepts , , , to describe its evolution and emissivity of electromagnetic radiation .",
"Such an approach has been very successful in describing UrHIC and helped to identify landmarks in the phase diagram of Quantum Chromodynamics (QCD) in the region of vanishing net-baryon density, which is also accessible by lattice-QCD calculations , .",
"Heavy-ion collisions (HIC) at lower beam energies provide access to the strongly interacting matter at high net-baryon densities where a rich structure in the QCD phase diagram is predicted , , but lattice QCD is not applicable.",
"The question of whether the fireball formed in the few-GeV beam energy range (where in head-on collisions essentially all nucleons are stopped in the center-of-mass) is thermalized remains still a matter of debate , , , .",
"The study of hadron spectra is crucial to answering this question.",
"However, in a thermal analysis it must first be demonstrated that the experimental hadron yields can be well described with a few thermodynamic parameters such as temperature, $T$ , and the baryon chemical potential, $\\mu _{B}$ .",
"Only in the second step the transverse-mass spectra, which are typically falling off exponentially, have to be reproduced.",
"One should recall that collective radial expansion (specified by the radial velocity $v$ ) and resonance decays also affect the momentum distribution of hadrons , , .",
"The two physical aspects mentioned above are unified in a single-freeze-out model , , which identifies the chemical and kinetic freeze-outs by neglecting hadron rescattering processes (after the chemical freeze-out).",
"This model assumes a sudden freeze-out governed by local thermodynamic conditions.",
"This concept is implemented in the THERMINATOR Monte-Carlo generator , , which allows for studies of hadron production taking place on arbitrary freeze-out hypersurfaces defined in the four-dimensional space-time.",
"The most popular parametrization of such a freeze-out hypersurface , dubbed the blast-wave model, assumes the symmetry of boost invariance (along the beam axis) which was observed in UrHIC.",
"In fact, it was introduced as a modification of the original blast-wave model formulated by Siemens and Rasmussen (SR) , which instead of the boost invariance, employed a spherical symmetry of the freeze-out geometry .",
"In our previous work , using the SR model we introduced a novel approach toward a consistent simultaneous description of hadron yields and transverse-mass spectra.",
"This framework offered an alternative interpretation of experimental results in the analyzed energy domain, as it was based on the thermal equilibrium concept, as compared to commonly used non-equilibrium transport model approaches , , , , .",
"We assumed the spherical symmetry of a fireball to be a natural approximation at low energies, where the colliding nuclei are not transparent to each other (the energy dependence of this effect is shown in Ref.",
"and recently also systematical investigated in terms od stopping in: Ref. ).",
"The agreement between the transverse-mass spectra of particles predicted by the model and measured by the HADES collaboration in Au-Au collisions at $\\sqrt{s_{\\rm NN}}= 2.4$ GeV was of the order of 20% .",
"However, rapidity distributions turned out to be systematically narrower in the model than in the experiment.",
"This indicated that our assumption of the spherical symmetry was not exactly fulfilled, at least in the momentum space.",
"In this work, we extend our approach by allowing for spheroidal symmetry of the system, parameterized by the two eccentricities in the longitudinal (“beam”, $z$ ) direction, separately in the momentum and the position space.",
"For a more systematic study, we present the results obtained with the two sets of chemical freeze-out parameters, as found in Refs.",
"and .",
"They are denoted below as Case A and Case B, respectively.",
"As before, we select a reaction centrality class where thermalization is most likely to occur, i.e., we consider the 10% most central collisions only.",
"The decay of $\\Delta (1232)$ is included via density function obtained in Ref.",
"from the pion-nucleon phase shift analysis.",
"In Case B the same excited nuclear states are included in the present calculations as those used in Ref.",
"."
],
[
"Spheroidal Siemens-Rasmussen model",
"As in our previous study , the basis for this model remains the Cooper-Frye formula that describes the invariant momentum spectrum of particles emitted from an expanding source, $E_p \\frac{dN}{d^3p} = \\int d^3\\Sigma (x) \\cdot p \\,\\, f(x,p).$ Here $f$ is the phase-space distribution function of particles, $p$ is their four-momentum with the mass-shell energy, $p^0 = E_p = \\sqrt{m^2 + p^2}$ , and $d^3\\Sigma _\\mu (x)$ is the element of a three-dimensional freeze-out hypersurface from which particles are emitted.Three-vectors are shown in bold font (unless stated otherwise) and a dot is used to denote the scalar product of both four- and three-vectors.",
"The metric convention is “mostly minuses”, $g_{\\mu \\nu } = \\hbox{diag}(+1,-1,-1,-1)$ .",
"In the calculations of the total particle yields, one can exchange the order of performing the integrals over the momentum space and the freeze-out hypersurface, $N = \\int d^3\\Sigma _\\mu (x) \\int \\frac{d^3p}{E_p} p^\\mu f(x,p).",
"$ Since the equilibrium distribution function depends on the product $p \\cdot u$ and thermodynamic parameters only (which are assumed to be constant on the freeze-out hypersurface), see Eq.",
"(6) in Ref.",
", we can further write $N\\!&=&\\!n(T,\\Upsilon ) \\int d^3\\Sigma (x) \\cdot u(x) \\equiv n(T,\\Upsilon ) {\\cal V}, \\\\\\nonumber $ where the invariant volume ${\\cal V}=\\int d^3\\Sigma (x) \\cdot u(x)$ is defined by the integral in the middle part of Eq.",
"(REF ).",
"The fugacity $\\Upsilon $ is defined as $\\Upsilon =\\gamma _q^{N_q+N_{\\bar{q}}} \\gamma _s^{N_s+N_{\\bar{s}}} \\exp \\left( \\frac{\\mu }{T}\\right),$ We note that in the studies of the ratios of hadronic abundances, the invariant volume cancels out.",
"We modify the hitherto spherically-symmetric SR model by allowing the system to be stretched or squeezed in the beam direction.",
"This is taken into account by the two eccentricity parameters: $\\delta $ (for the momentum space) and $\\epsilon $ (for the position space).",
"Then, the space-time points lying on the freeze-out hypersurface have the following parametrization $x^\\mu = \\left(t, r \\sqrt{1-\\epsilon }\\sin \\theta \\,\\mathbf {\\hat{e}_\\rho }, r \\sqrt{1+\\epsilon }\\cos \\theta \\right),$ where $\\mathbf {\\hat{e}_\\rho }=(\\cos \\phi , \\sin \\phi )$ , while $\\phi $ and $\\theta $ are azimuthal and polar angles, respectively.",
"In order to completely specify the freeze-out hypersurface, a curve in the $t-r$ plane has to be defined by the mapping $\\zeta \\rightarrow \\left(t(\\zeta ),r(\\zeta )\\right)$ .",
"This curve defines the (freeze-out) times $t$ when the hadrons in the (spheroidal) shells of radius $r$ stop to interact.",
"The range of $\\zeta $ may be always restricted to the interval: $0 \\le \\zeta \\le ~1$ .",
"The resulting infinitesimal element of the spheroidal hypersurface has the form $\\!",
"d^3\\Sigma _\\mu \\!&=& (1-\\epsilon )\\!\\left(r^\\prime \\sqrt{1+\\epsilon },t^\\prime \\frac{\\sqrt{1+\\epsilon }}{\\sqrt{1-\\epsilon }}\\sin \\theta \\,\\mathbf {\\hat{e}_\\rho },t^\\prime \\cos \\theta \\right)r^2 d\\Omega \\,d\\zeta ,$ where $d\\Omega = \\sin \\theta \\,d\\theta \\,d\\phi $ is an infinitesimal element of the solid angle and the prime denotes the derivatives taken with respect to $\\zeta $ .",
"If we assume the instantaneous freeze-out, $t^\\prime =0$ , and use the parametrization $\\zeta =r$ , Eq.",
"(REF ) reduces to $\\!",
"d^3\\Sigma _\\mu \\!&=& (1-\\epsilon )\\!\\left(\\sqrt{1+\\epsilon }, 0, 0, 0\\right)d\\Omega \\,r^2 dr.$ Besides the spheroidally-symmetric hypersurface, we introduce a spheroidally-symmetric hydrodynamic flow, $u^\\mu &=& \\gamma (\\zeta ,\\theta )\\left(1,v(\\zeta ) \\sqrt{1-\\delta }\\sin \\theta \\,\\mathbf {\\hat{e}_\\rho },v(\\zeta ) \\sqrt{1+\\delta }\\cos \\theta \\right),$ Figure: Graphical representation of the flow parametrization for the three studied cases.",
"The points on the surfaces represent solutions of the equation (v x 2 +v y 2 )/(1-δ)+v z 2 /(1+δ)=v 2 (v_x^2 + v_y^2)/(1-\\delta )+v_z^2/(1+\\delta ) = v^2.where $\\gamma (\\zeta ,\\theta )$ is the Lorentz factor that, due to the normalization condition $u\\cdot u=1$ , is given by the formula $\\gamma (\\zeta ,\\theta ) = \\left[1-(1+\\delta \\cos (2 \\theta ))v^2(\\zeta )\\right]^{-1/2}$ ; for graphical representation of Eq.",
"(REF ) for the values of $\\delta $ obtained in this analysis see Fig.",
"REF .",
"Thus, we can write explicit expressions for the two dot products used in the numerical Monte-Carlo calculations, namely, the product of the four-velocity and four-momentum vectors $u \\cdot p=\\gamma (\\zeta ,\\theta ) \\left[E_p\\!-\\!p\\, v(\\zeta ) \\left(\\sqrt{1+\\delta }\\cos \\theta \\cos \\theta _p + \\sqrt{1-\\delta }\\sin \\theta \\sin \\theta _p \\cos (\\phi -\\phi _p)\\right)\\right],$ where the subscript `$p$ ' denotes quantities related to the momentum vector, and the product of the hypersurface element and the four-momentum vector $d^3\\Sigma \\cdot p = (1-\\epsilon )\\left[E_p r^\\prime \\sqrt{1+\\epsilon }- p\\, t^\\prime \\left(\\cos \\theta \\cos \\theta _p + \\frac{\\sqrt{1+\\epsilon }}{\\sqrt{1-\\epsilon }} \\sin \\theta \\sin \\theta _p \\cos (\\phi -\\phi _p)\\right)\\right]r^2 d\\Omega \\, d\\zeta .$ In the case of the instantaneous freeze-out introduced above, this expression simplifies to $d^3\\Sigma \\cdot p = (1-\\epsilon ) E_p \\sqrt{1+\\epsilon }\\, d\\Omega \\, r^2 dr.$ The invariant volume defined by Eq.",
"(REF ) takes the following form for the spheroidally-symmetric system ${\\cal V} = (1-\\epsilon )\\int \\, \\gamma (\\zeta ,\\theta ) \\left\\lbrace r^\\prime \\sqrt{1+\\epsilon }- t^\\prime v(\\zeta ) \\frac{\\sqrt{1-\\delta }}{2} \\left[\\frac{\\sqrt{1+\\delta }}{\\sqrt{1-\\delta }}+\\frac{\\sqrt{1+\\epsilon }}{\\sqrt{1-\\epsilon }} +\\left(\\frac{\\sqrt{1+\\delta }}{\\sqrt{1-\\delta }}-\\frac{\\sqrt{1+\\epsilon }}{\\sqrt{1-\\epsilon }}\\right)\\cos \\left(2\\theta \\right)\\right]\\right\\rbrace \\, r^2 d\\Omega \\, d\\zeta ,$ which in the case of the instantaneous freeze-out and the Hubble-like radial profile of the flow velocity ($v=\\tanh (Hr)$ , where $H$ is a constant parameter ) reduces to ${\\cal V} = (1-\\epsilon )\\int _0^R\\int \\, \\gamma (\\zeta ,\\theta )\\sqrt{1+\\epsilon }\\, d\\Omega \\, r^2 dr.$ Here the parameter $R$ is fixed by the abundance of one of the particle species (note that the abundances of the other species are known through the ratios set by thermodynamic parameters).",
"For $\\epsilon ~=~\\delta ~=~0$ , all the above equations reduce to the standard, spherically-symmetric version of the SR model used in Ref.",
"."
],
[
"Thermal parameters",
"Input parameters for our calculations were obtained by different fitting strategies of the particle multiplicities calculated in the grand-canonical ensemble to the ones measured by the HADES collaboration in Au-Au collisions at $\\sqrt{s_{\\rm NN}}= 2.4$ GeV, as listed in Table REF .",
"As a result of the fit, we consider two sets of thermodynamic parameters whose values are listed in the upper section of Table REF .",
"Table: Acknowledgements"
]
] | 2210.07694 |
[
[
"TechRank"
],
[
"Abstract We introduce TechRank, a recursive algorithm based on a bi-partite graph with weighted nodes.",
"We develop TechRank to link companies and technologies based on the method of reflection.",
"We allow the algorithm to incorporate exogenous variables that reflect an investor's preferences.",
"We calibrate the algorithm in the cybersecurity sector.",
"First, our results help estimate each entity's influence and explain companies' and technologies' ranking.",
"Second, they provide investors with a quantitative optimal ranking of technologies and thus, help them design their optimal portfolio.",
"We propose this method as an alternative to traditional portfolio management and, in the case of private equity investments, as a new way to price assets for which cash flows are not observable."
],
[
"Introduction",
"This work investigates the innovation structure and the dynamics underlying the life cycle of technologies.",
"We fill two research gaps.",
"The first concerns the identification of future benefits and risks of emerging technologies for the society.",
"The second regards the valuation or early-stage companies and optimal investment decisions.",
"To fill these gaps, we introduce the TechRank algorithm.",
"Our methodology assigns a score to each entities, i.e., technologies and firms, based upon their contribution to the technological ecosystem.",
"We expect this method to help stakeholders in forming optimal decisions for investment, procurement, and technology monitoring.",
"We calibrate our model in the cybersecurity sector, although TechRank could apply to any sector.",
"The cybersecurity technological landscape represents a particular challenge for this calibration, given the important share of start-ups and fast innovations it yields.[16].",
"Moreover, the important number of cyber-attacks and the increasing costs they incur has boosted cybersecurity investments.The New York Times: As Cyberattacks Surge, Security Start-Ups Reap the Rewards by Erin Woo (July 26, 2021).",
"Yahoo Finance: Microsoft Securing its Position with Cybersecurity Investments by TipRanks (July 20, 2021).",
"According to Bloomberg, the global cybersecurity market size is expected to reach USD 326.4 billion by 2027, registering a compound annual growth rate of 10.0% from 2020 to 2027.Bloomberg: Global Cybersecurity Market Could Exceed $320 Billion in Revenues by 2027 (July 29, 2020).",
"To develop the TechRank algorithm, we first model and map the ecosystem of companies and technologies from the Crunchbase dataset using a bi-partite network.",
"The bi-partite network structure accurately describes this complex and heterogeneous system.",
"We evaluate the relative influence of the network nodes in the ecosystem by adapting a recursive algorithm that estimates network-centrality.",
"This methodology should help decision-makers and investors to assess the influence of entities in the cybersecurity ecosystem, reducing investment uncertainties.",
"In fact, around 90% of startups fail and in 42% of the cases this is due to incorrect evaluation of the market demand.",
"The second reason (29%) is because they run out of funding and personal money.Findstack: The Ultimate List of Startup Statistics for 2021 by Jack Steward.",
"Christensen (1997) highlights that well-managed companies also break down, because they over-invest in new technologies[8].",
"Thus, by selecting the right technologies to invest in goes along with the optimal investment strategy.",
"Our research takes inspiration from Google's PageRank algorithm, that ranks web pages according to readers' interest[25].",
"We use a similar approach with bi-partite networks to assign a score to companies and technologies.",
"Our method is flexible and permits to incorporate investors' preferences such as location or previous funding rounds.",
"TechRank let the investor select entities' features that reflect their interests.",
"The algorithm uses their choices as input, which tweaks the entities' score to reflect them.",
"This enables investors to select a personalized portfolio strategy using a quantitative methodology.",
"The evaluation of companies and new technologies largely depends on investors' personal choices, which may lead to misread market demand.",
"This work aims to lead to more methodical decision making for investors.",
"The remainder of this article proceeds as follows.",
"Section presents the literature review and hypotheses.",
"Section details the data and the methodology.",
"Section presents the results.",
"Section concludes."
],
[
"Centrality measures",
"In network analysis, the centrality estimates the importance of nodes through ranking.",
"The most simple centrality estimate is the degree, which counts the number of neighbours of a node.",
"One of its drawbacks is that it does not show which one stands in the center of the network.",
"Two nodes may share the same degree, while being more or less peripheral.",
"Thus, the degree is a local centrality measure, which does not capture the influence across nodes within the graph.",
"Figure: Central and peripheral nodesAnother important centrality measure is the closeness, which measures how long it takes for information to spread from one node to the next.",
"Specifically, closeness is defined as the reciprocal of the farness, i.e.",
"the sum of distances of one node with respect to all other nodes.",
"The betweenness centrality of a node measures how often a node stands in the shortest path between a pair of other nodes (see, e.g., Bavelas; 1948, Saxena and Iyengar; 2020, and Freeman; 1978)[2], [28], [14]).",
"Another strand of research focuses on the top-K shortest path identification in a complex network, a topic less tackled by the literature than centrality.",
"To rank nodes, one must compute the centrality of all nodes and compare them to extract the rank, which is not always feasible due to the size of the network.",
"To overcome this problem Saxena and Iyengar (2017) attempt to estimate the global centrality of a node without analyzing the whole network[29].",
"Similarly, Bavelas (1948) develops a structural centrality measure in the context of social graphs[2].",
"Other centrality concepts include the eigenvector, Katz, or PageRank centralities[5], [25], [20].",
"Finally, Freeman (1978) creates a formal mathematical framework for centrality, which includes degree, closeness, and betweenness and advocates for the combination of different kinds of centrality measures[14]."
],
[
"Page Rank",
"Page, Brin, Motwan, and Winograd (1999) develop the PageRank algorithm[25].",
"Its primary goal is to rank web pages objectively, a challenge with the fast-growing web.",
"PageRank assigns a score to each web page based on its relations with other web pages in the graph.",
"Other fields have benefited PageRank providing modifications and improvements.",
"Xing and Ghorbani (2014) extend the algorithm and propose the weighted PageRank (WPR)[32].",
"This algorithm assigns larger rank values to more important pages, instead of dividing the rank evenly among its outlink pages.Given a web page W, an inlink of W is a link of another web page that includes a link pointing to W. An outlink of W is a link appearing in W, which points to another web page.",
"Each outlink page gets a value proportional to its popularity, taking into account the links weights.",
"On caveat of PageRank and its variants is that they do not consider n-partite structures, yet, web pages can all be linked to one another.",
"Bi-partite networks address this issue and capture this complexity, among other structures."
],
[
"Bi-partite networks",
"Networks are a fundamental tool to capture the relations between entities.",
"Graphs ($G$ ) are composed by vertices ($V$ ) and edges ($E$ ), and we denote $G=(V,E)$ .",
"We build links and mathematically analyze many properties of the whole system and of singular entities.",
"To graphically represent a real system, we synthesize its information into a simple graph framework.",
"This simplification generates an information loss in the modelling process.",
"Simple network structures might discard important information about the structure and function of the original system [23].",
"As a consequence, the failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system[6].",
"To solve the problem, extensions to the simple structure $G=(V,E)$ are added and yield graphs with more powerful features.",
"For instance, in case vertices are connected by relationships of different kinds, Battiston, Nicosia, and Latora (2014) advocate to work with multiplex networks, i.e.",
"networks where each node appears in a set of different layers, and each layer describes all the edges of a given type [1].",
"When it is possible to distinguish the nature of the edges, multiplex networks are an effective approach, which starts from embedding the edges in different layers according to their type.",
"However, even if we have two kinds of nodes, the nature of the edges is unique.",
"Therefore, a more suitable approach is bi-partite networks.",
"Bi-partite networks are for instance, a good way to describe the technological and business landscape.",
"In Figure REF , we depict two sets of nodes, companies and technologies, which are interconnected but do not present edges within the same set.",
"Figure: Bi-partite structure of companies and technologiesThere are multiple adaptations of the PageRank algorithm to bi-partite structures[3], [12], [31], [21].",
"In particular, Benzi, Estrada, and Klymko (2015), Donato, Laura, Leonardi, and Millozzi (2004), and Tu, Jiang, Song, and Zhang (2018) extend the PageRank algorithm to multiplex networks.",
"They assume that only some clusters of the graph are multiplex networks and extend the PageRank algorithm only to analyze the sub-graph centrality.",
"Bi-partite networks are used to transform directed into undirected networks with twice the number of vertices.",
"Klein, Maillart, and Chuang (2015) extend PageRank in the Wikipedia editors and articles context [21].",
"The application of this algorithm to the case of interactions between companies and technologies is straightforward.",
"A major benefit of this approach is that it starts from an unweighted graph, linking authors and articles.",
"They develop a recursive algorithm in which the two entities contribute to the quality (for articles) or the expertise (for authors) of each other.",
"They develop a bi-partied random walker by building the adjacency matrix $M_{e,a}$ that takes value 1 if editor $e$ has edited article $a$ and 0 otherwise, which tracks all the editors' contributions.",
"They obtain $M_{e,a} \\in \\mathbb {R}^{n_e, n_a}$ , where $n_e$ and $n_a$ are the number of editors and articles.",
"They sort editors by the number of articles' contributions and assign a contribution (quality) value to each editor (article) based on their degree.",
"The expertise $w_e^{0}$ (quality $w_a^{0}$ ) is given by the number of articles (editors) they have worked on (have received modifications).",
"The second part of the algorithm follows a Markov process in its iterations.",
"The step $w^{n}$ ($w^{n} = w^{n}(\\alpha , \\beta )$ ) only depends on information available at $w^{n-1}$ .",
"At each step, the algorithm incorporates information about the expertise of editors and the quality of articles, within the bi-partite network structure.",
"The process is a random walker with jumps, whose transition probability is zero in the case $M_{e,a}=0$ .",
"Next, the authors define two variables for the transition probability, $G_{e,a}(\\beta )$ and $G_{e,a}(\\alpha ))$ .",
"$G_{e,a}(\\beta )$ represents the probability of jumping from article $a$ to editor $e$ and $G_{a,e}(\\alpha )$ represents the probability to jump from an editor to an article.",
"Both parameters depend on initial conditions and the selection of optimal parameters is done through a grid search that maximizes the Spearman rank-correlation between the rank given by the model and a ground-truth metrics obtained independently.",
"Finally Klein et al.",
"(2015) observe a less-is-more situation since the presence of too many editors working on an article is detrimental to its quality.",
"Studying different categories of Wikipedia articles they find $\\alpha $ to remain constant, while $\\beta $ varies significantly across categories.",
"Estimating the global rank of a node starting from local information and centrality measures is still an open research question in many sectors [28].",
"In particular, no research to our knowledge use this approach for investment's decision and portfolio optimization.",
"Yet, this approach could help overcome the limitations of standards financial models in private equity, in which the network structure is easily obtainable, whereas the cash flow process is not."
],
[
"Private equity valuation",
"Private firms are not required to publicly disclose their financial statements, which makes it difficult to measure their past performance and estimate their expected returns, without insider information.",
"Moreover, since they are not listed on exchanges we do not observe expectations of market participants.",
"Thus, standard asset pricing methods fall short in this context.",
"Similarly, private equity analysts must rely on insider or private information to value private firms.",
"These valuations generally occur around a financing round and impose to take into account the capital dilution to compute realized returns on the firm[17].",
"One approach that attempts to overcome these limitations and estimate the expected returns and risk in venture capital is Cochrane (2005)[9].",
"He uses a maximum likelihood estimation method to obtain these values at the market and sector levels, such as healthcare, biotechnology, technology companies, and retail services.",
"He finds a mean arithmetic return of 59%, an alpha of 32%, a beta of 1.9 and a volatility of 86% (equivalent to a 4.7% daily volatility).",
"Given that the distribution of returns is heavily positively skewed in venture capital, he adopts a logarithmic model that also accounts for the inherent selection bias of this asset class.Most of venture capital data is private and available data are more often related to successful firms than under-performing ones.",
"Ewens (2009) updates this method on returns computed from financing round to the next[13].",
"He adopts a three-regime mixture model (failure, medium returns, and home-runs).",
"He also corrects for the selection bias and obtains an alpha of 27% and a beta of 2.4.",
"He finds that 60% of all venture capital investments have a negative log return.",
"Altogether the results are similar, venture capital investments exhibit positive alpha, large beta, and a high volatility.",
"Other attempts to evaluate the market parameters of the venture capital asset class includes Korteweg and Nagel (2016) and Moskowitz and Vissing-Jorgensen (2010) with results in line with the aforementioned studies[22], [24].",
"Another strand of research attempts to index and benchmark the private equity market.",
"Peng (2001) builds a venture capital index from 1987 to 1999 from about 13,000 financing rounds targeting over 5,600 firms[26].",
"He addresses the problems of missing data, censored data, and sample selection using a re-weighting procedure and method of moment regressions.",
"From the index perspective, the results are qualitatively the same, that is, high and volatile returns to venture capital (average return of 55.18% per year).",
"He finds his index to display a much higher volatility than the S&P 500 and NASDAQ indices and high exposure to these indices (betas of 2.4 and 4.7, respectively).",
"Other venture capital indices construction includes Hwang, Quigley, and Woodward (2005), Schmidt (2006), and Cuming, Has and Schweizer (2013), all obtaining results on par with the aforementioned studies.",
"One limitation of the above studies is that they only estimate these parameters at an aggregate level.",
"An investor could form her investment's decisions and portfolio choices segregating among sectors, but not obtain the actual firms' parameters.",
"One exception is Schwartz and Moon (2000), who provide an approach based upon real-options theory to price individual firms[30].",
"However, this method requires the observations of cash flows and they only provide one calibration example with Amazon.",
"Thus, there remains a caveat in methodology to help investors' forming optimal decisions using all the information available.",
"Given the recent venture capital boom, Zhong, Chuanren, Zhong, and Xiong (2018) advocates for the use of quantitative methodologies of screening and evaluation[33].",
"However, there is a clear research gap on methodologies enabling to value early stage companies and form optimal portfolios.",
"These methodologies either only enable to value a sector, instead of a specific company, or require the use of cash flows that are unobservable.",
"Non-financial features and relations between companies and technologies are instead numerous and easily observable[10].",
"We thus formulate our hypotheses as follows, $H_1$ Using a bi-partite network structure allows to create an algorithm that rank companies based on their links with technologies.",
"$H_2$ This algorithm and its ranking may be improved and tilted towards investors' preferences.",
"$H_3$ The performance of the ranking is independent from the sector considered."
],
[
"Data",
"We use Crunchbase data.Crunchbase website: https://www.crunchbase.com/ Crunchbase is a commercial database that provides access to financial and managerial data on private and public companies globally.",
"It was created in 2007 by TechCrunch, which is a source of information about start-up activities and their financing within and across countries.",
"This database has been largely adopted by both academics and industry practitioners[4].",
"It is also used by international organizations such as the OECD[11].",
"Crunchbase is made of data collected with a multifaceted approach that combines crowd-sourcing (through venture programs or direct contributions), machine learning, in-house processing, and aggregation of third-party providers data.",
"Crunchbase updates and revises data on a daily basis, which is organized into several entities such as organizations, people, events, acquisitions, or IPOs.",
"The primary focus of Crunchbase is the technology industry, although it also includes data on other sectors.",
"Data can be accessed in two ways: using an API or downloading a *.csv file directly from the Crunchbase website.",
"Data is split in several databases depending on their type.",
"We provide a non-exhaustive list in Table REF .Crunchbase daily CSV export from https://data.crunchbase.com/docs/daily-csv-export.",
"Data downloaded on April 28, 2021.",
"Table: Crunchbase files descriptionWe first analyze the Crunchbase dataset dedicated to investors.",
"With a total of 185,784 investors divided into 78,001 (41.98%) organisations and 107,783 (58.02%) persons, there are more investors than target companies.",
"Figure REF shows that the majority of investors are pure investors (87.11%).",
"Some organisations are both investee and investor (12.65%).",
"The remainder of the sample are typically universities.",
"Crunchbase ranks the top 1,000 investors through its proprietary algorithm.",
"Figure REF indicates that the majority of investors are located in the USA (29.62%).",
"In particular, there is a wide gap between the first and second country, China, where 7.04% of investors are located."
],
[
"Adaptation of the work by Klein et al. (2015).",
"In this research we use a bi-partite network that describes the relations among companies and the technologies they are involved in.",
"Figure REF describes the typical bi-partite network structure.",
"This structure benefits from advances in fields such as network theory, Markov chains, and machine learning.",
"We adapt the recursive algorithm with the method of Hidalgo, Hausmann, and Dasgupta (2009)[18].",
"We expect the resulting rank to incorporate the positive influence of well-established companies on technologies and, at the same time, the positive impact of new companies that explore new fields.",
"We build the adjacency matrix $M^{CT}_{c,t}\\in \\mathbb {R}^{n^c, n^t}$ , which takes value 1 if a company $c$ works on a technology $t$ and 0 otherwise.",
"$n^c$ and $n^t$ represent the number of companies and technologies.",
"We assume that well-established companies have more means to diversify their expertise and therefore, that an entity has a relatively high number of neighbours[7], [15].",
"Thus, we initialize the algorithm with the degree, i.e.",
"counting the neighbours, of each entity, ${\\left\\lbrace \\begin{array}{ll}w_c^{0} &= \\sum _{t=1}^{n^t} M^{CT}_{c,t} = k_c \\\\w_t^{0} &= \\sum _{c=1}^{n^c} M^{CT}_{c,t} = k_t\\end{array}\\right.",
"}$ The algorithm is a random walker that incorporates information about company expertise and technology relevance at each step.",
"The transition probabilities, $G_{c,t}$ and $G_{t,c}$ , describe the extent to which the entities weights change over the iterations.",
"If the relation between $c$ and $t$ increases (decreases) the value, the entity weight increases (decreases) in proportion with the transition probabilities.",
"We define $G_{c,t}$ and $G_{t,c}$ , ${\\left\\lbrace \\begin{array}{ll}G_{c,t}(\\beta ) &= \\frac{M^{CT}_{c,t}k_c^{-\\beta }}{\\sum _{c^{\\prime }=1}^{n^c} M^{CT}_{c^{\\prime },t}k_c^{\\prime -\\beta }} \\\\G_{t,c}(\\alpha ) &= \\frac{M^{CT}_{c,t}k_t^{-\\alpha }}{\\sum _{t^{\\prime }=1}^{n^t} M^{CT}_{c,t^{\\prime }}k_t^{\\prime -\\alpha }},\\end{array}\\right.",
"}$ where $\\alpha $ and $\\beta $ inform how coordination generates value.",
"Next, we define the recursive step, ${\\left\\lbrace \\begin{array}{ll}w_c^{n+1} &= \\sum _{t=1}^{n^t} G_{c,t}(\\beta ) w_t^{n} \\\\w_t^{n+1} &= \\sum _{c=1}^{n^c} G_{t,c}(\\alpha ) w_c^{n}\\end{array}\\right.",
"}$ As in PageRank, the recursion ends when the algorithm converges.",
"Our algorithm allows to consider the market complexity and feedback loops (investments' impact on companies and on technologies).",
"We discuss this feature and the optimization of $\\alpha $ and $\\beta $ after the addition of exogenous factors."
],
[
"Inclusion of exogenous factors",
"We include exogenous factors as ground truth in the parameters' calibration step.",
"This allows to keep the algorithm tractable, while letting it capture the technological structure.",
"We use this ground truth to compute the Spearman correlation, $\\rho _c$ for companies and $\\rho _t$ for technologies.",
"Because $\\rho _c$ and $\\rho _t$ depend on $\\alpha $ and $\\beta $ [see Eq.",
"(REF )], we find the parameters that maximize these correlations, ${\\left\\lbrace \\begin{array}{ll}(\\alpha ^*, \\beta ^*) = _{\\alpha , \\beta } \\rho _c(\\alpha , \\beta ) \\\\(\\alpha ^*, \\beta ^*) = _{\\alpha , \\beta } \\rho _t(\\alpha , \\beta ),\\end{array}\\right.",
"}$ and we solve this optimization problem with a grid search.",
"Eq.",
"(REF ) shows that parameters depend on both companies and technologies.",
"This dependence enables to create the structure of the bi-partite graph.",
"To obtain the correlation between the TechRank score, which assigns a weight $w_c$ ($w_t$ ) to each company (technology) and the ground truth evaluation, which assigns $\\hat{w_c}$ ($\\hat{w_t}$ ) to each company (technology), we normalize both TechRank results and the exogenous measure in the same range $[0,1]$ .",
"Investors use the entities' features to select companies and the investment amount they want to allocate.",
"We suppose that an investor has $n^{(C)}$ features to pick from, denoted as $f^{(C)}_1,\\ldots ,f^{(C)}_{n^{(C)}}$ , where $C$ ($T$ ) represents the association with the companies (technologies).",
"Each feature $f^{(C)}_i$ is associated with a percentage of interest $p^{(C)}_i$ and $\\sum _{i=0}^{n^{(C)}} p^{(C)}_i = 1$ .",
"For instance, if a company's features are the amount of previous investments and its geographical proximity to the investor, $n^{(C)}=2$ .",
"An investor may then decide to be interested at 80% in the first feature and at 20% in the second, by selecting $p^{(C)}_1 = 0.8$ and $p^{(C)}_2 = 0.2$ .",
"Investors may also be pushed back by a feature, in which case we multiply it by -1.",
"We define all notations in Table REF .",
"We convert quantitative and qualitative properties into a number $f^{(C)}_i \\in [0,1]$ .",
"Once we have created all the factors $f^{(C)} = f^{(C)}_1,\\ldots ,f^{(C)}_{n^{(C)}}$ , the exogenous evaluation $\\hat{w_c}$ is given by, $\\hat{w_c} = \\sum _{i=1}^{n^{(C)}} p^{(C)}_i f^{(C)}_i = p^{(C)} \\cdot f^{(C)}.$ Considering $\\sum _{i=0}^{n^{(C)}} p^{(C)}_i = 1$ and that $f^{(C)}_i \\in [0,1]$ for each company $i$ , we have $\\hat{w_c} \\in [0,1]$ .",
"The same holds for $\\hat{w_t}$ .",
"Finally, we have ${\\left\\lbrace \\begin{array}{ll}\\hat{w_c} = p^{(C)} \\cdot f^{(C)}\\\\\\hat{w_t} = p^{(T)} \\cdot f^{(T)}\\\\\\sum _{i=0}^{n^{(C)}} p^{(C)}_i = 1 \\\\\\sum _{i=0}^{n^{(T)}} p^{(T)}_i = 1,\\end{array}\\right.",
"}$ where $n^{(C)}$ ($n^{(T)}$ ) is the number of the company- (technology-)related features and $f^{(C)} = (f^{(C)}_1,\\ldots ,f^{(C)}_{n^{(C)}})$ .",
"To select the features, we use Crunchbase data about companies and investors (see in Table REF )."
],
[
"Previous Investments",
"We assume that previous investments is an essential factor to evaluate companies.",
"Investors may be willing to invest in companies which have already received capital or look for higher returns, targeting newer firms.",
"To compute this factor, we use the Crunchbase field funding_rounds, which reports the amount of all funding rounds from an investor $i$ to a company $c$ .",
"We capture this structure with another bi-partite network that describes the links among investors (I) and companies (C).",
"In this case, we weight the edges by the sum of all previous investments from investor $i$ to company $c$ , until the current period ($\\mathcal {T}$ ), and compute the adjacency matrix $M^{IC}$ .",
"We define the amount of a single investment from $i$ to $c$ at time $t$ by $\\gamma ^{i,c}_{t}$ .",
"The weight of the edge $i-c$ is given by $e_{i,c} = \\sum _{t=0}^{\\mathcal {T}} \\gamma ^{i,c}_{t}$ (see Table REF ).",
"We then sum the contribution of all investors to find the attribute $f^{C}_c \\in [0,1]$ for a company $c$ .Note that here, $f_c^{C}$ represents the factor related to a company.",
"Next, we normalize and divide all investments by the maximum investment.",
"Figure: Investors-companies bi-partite networkFigure REF depicts the investment structure as an example.",
"We consider two investors $i_1$ and $i_2$ and three companies $c_1$ , $c_2$ , and $c_3$ .",
"We compute the maximum $e_{\\text{max}}$ as $\\max \\lbrace e_{11}+e_{21}, e_{22}, e_{23}\\rbrace $ and the features related to the investments for each company as, ${\\left\\lbrace \\begin{array}{ll}f^{C}_1 = \\frac{e_{11}+e_{21}}{e_{\\text{max}}}\\\\f^{C}_2 = \\frac{e_{22}}{e_{\\text{max}}}\\\\f^{C}_3 = \\frac{e_{23}}{e_{\\text{max}}},\\end{array}\\right.",
"}$ where $n^{i}$ ($n^{c}$ ) is the total number of investors (companies).",
"Generalizing, we get, ${\\left\\lbrace \\begin{array}{ll}e^{IC}_{i,c} = \\sum _{t=0}^{\\mathcal {T}} \\gamma ^{i,c}_{t} \\hspace{39.83368pt} \\forall i, c \\\\e^C_{c} = \\sum _{i=1}^{n^{i}} e_{i,c} M^{IC}_{i,c} \\hspace{25.6073pt} \\forall c \\\\e_{\\text{max}} = \\underset{c}{\\max {}} e^C_c \\\\f^{(C)}_c = e^C_{c} / e_{\\text{max}},\\end{array}\\right.",
"}$ for each $c \\in {1,\\ldots , n^{c}}$ .",
"We present the corresponding algorithm in the Appendix REF .",
"With Eq.",
"(REF ), for each company we have a factor between 0 and 1 that summarizes the amount of previous investments.",
"Figure: Tripartite structure of investors, companies, and technologiesWe link the two bi-partite structures investment-companies and companies-technoéogies to obtain an I-C-T tri-partite structure depicted in Figure REF .",
"This structure allows to assign some features to technologies from companies (direct link) or investors (indirect link).",
"Thus, we can find the amount of previous investments on a technology through companies' funding rounds.",
"The previous investments' factor for technology is given by, ${\\left\\lbrace \\begin{array}{ll}e^{(I,C)}_{i,c} = \\sum _{t=0}^{(T)} \\gamma ^{t}_{i,c} \\hspace{17.07182pt}\\forall i, c \\\\e^C_{c} = \\sum _{i=1}^{n^{i}} e_{i,c} \\hspace{31.2982pt}\\forall c \\\\e^T_{t} = \\sum _{c=1}^{n^{c}} e_{c} M^{CT}_{c,t} \\\\e_{\\text{max}} = \\underset{t}{\\max {}} e^T_t\\\\f^{(T)}_t = e^T_{t} / e_{\\text{max}}\\end{array}\\right.",
"}.$ We provide the algorithm of this methodology in the Appendix REF ."
],
[
"Location",
"The second feature we consider is the distance between investors and companies' locations.",
"We retrieve the addresses of firms and investors from Crunchbase ($c\\_address$ ) and map them to geographic coordinates.",
"We compute the Haversine approximation to measure the distance.",
"We detail the Haversine approximation in the Appendix REF .",
"Investors may prefer short-distance investments or places with high potential.",
"If they face some investment's restrictions, we filter the companies based on the criteria before applying the algorithm.",
"Otherwise, we add a distance factor to the algorithm.",
"We use the Haversine distance $h$ to obtain a factor $f_c^{(C)} \\in [0,1]$ for each company.",
"We consider the distance $h_{i,c}$ between the company $c$ and the investor $i$ .",
"We assume that the factor is the proximity so that $f_c^{(C)}$ tends to one as the distance decreases, $f_c^{(C)} \\rightarrow 1 \\hspace{14.22636pt} \\text{when} \\hspace{14.22636pt} h_{i,c} \\rightarrow 0.$ To compute $f_c^{(C)}$ , we first find $h_{i,c}$ for each company and identify the maximum distance $h_{\\text{max}}$ among all companies.",
"We normalize by the maximum to obtain a distance that lies in the $[0,1]$ range with, $f_c^{(C)} = 1 - h_{i,c}/h_{\\text{max}}$ , so that a distance of zero corresponds to a value of $f_c^{(C)} = 1$ .",
"We report the algorithm in the Appendix REF .",
"We implement the algorithm and run the experiments using Python and the NumPy, Pandas, NetworkX, Matplotlib, and Seaborn libraries."
],
[
"Cybersecurity field",
"We select all the companies whose description contains at least two cybersecurity-related terms and obtain 2,429 companies and 477 technologies.The word list is in the Appendix REF Figure REF display the structure of the bi-partite network between technologies and companies.",
"We assume that investors are only interested in previous investments, both for technologies and companies.",
"We examine how the parameters' calibration step changes when we change the investors' preferences using a smaller sample of companies.",
"Figure REF shows the optimization in which the correlations $\\rho _c$ and $\\rho _t$ change according to $\\alpha $ and $\\beta $ .",
"In Table REF We identify the optimal $\\alpha ^*$ and $\\beta ^*$ to be of 0.04 and -1.88 for companies and 0.48 and -2.00 for technologies, respectively.",
"Next, we plug these values in the recursive algorithm.",
"Figure: Grid search of parameters α\\alpha and β\\beta Table: Optimal parameters in cybersecurityWe illustrate the evolution of the TechRank random walker in Figure REF .",
"While the entities' positions significantly change over the first steps, they gradually stabilize.",
"With the 2,429 companies and 477 technologies, the algorithm requires 723 (1,120) iterations for companies (technologies) to converge.",
"Entities starting with a high score (the initialisation is the degree of the node) do not significantly change rank and remain among the best ones.",
"Thus, the algorithm assigns good scores to entities with many neighbours.",
"Instead, entities starting with a low degree may significantly change their score, especially in the case of technologies.",
"TechRank does not only recognize the importance of the most established entities, it also enables to identify emerging technologies.",
"Figure REF shows the first classified entities in cybersecurity.",
"Figure: TechRank scores evolution in cybersecurityWe check how TechRank performs when we change the number of companies and technologies.",
"We fix the number of companies $n^c$ , which yields a resulting number of technologies $n^t$ .",
"For instance, in the cybersecurity field, by selecting 10 companies randomly, we get 26 technologies.",
"Considering that there are 2,429 cybesecurity-related companies on Crunchbase, we study the runtime running the algorithm for 10, 100, 499, 997, 1494, 1990, and 2,429 companies and 26, 134, 306, 372, 431, 456, and 477 technologies respectively.",
"Figure REF displays the results of TechRank applied on a subset of 10 cybersecurity companies.",
"We note that AppOmni's position does not change over the iteration, while two of its technologies, Software as a Service (Saas) and cloud management increase their scores.",
"In Figure REF we display this restricted network of 10 companies, that shows that SaaS and cloud management do not have other links.",
"Hence, the strength of this company depends on its ability to combine important technologies (software, cyber security, and cloud security) with more exotic fields.",
"Similarly, Integrity Market Group is the single involved in some fields (marketing, digital marketing, and advertising).",
"This company does not use more established technologies and thus does not improve its score.",
"Again, in Figure REF we observe that these technologies lie out of the main network.",
"Conversely, Lacework and Acronis follow an opposite trend.",
"Lacework (Acronis) significantly increases (decreases) its score.",
"One explanation for this behaviour is the fact that Acronis is involved in a lot of technologies, most of which are not explored by other companies.",
"On the other hand, Lacework relies on recognized technologies (security, cloud security, and software).",
"Interestingly, the compliance technology, benefits from its connections, increasing its rank by three positions.",
"Figure: TechRank scores evolution of 10 companies in cybersecurityTable REF reports the number of algorithm iterations before reaching convergence.",
"The number of iterations needed appears to be independent from the number of entities.",
"Technologies need more iterations than companies, which we explain by the fact that there are many more companies than technologies.",
"Since each company has at least one edge, the technology nodes have a higher degree than the companies on average.",
"Thus, we expect the structure and the dynamics related to technologies to be more complex.",
"The algorithm complexity does not only depend on the number of entities, but also on the network structure.",
"Table: TechRank convergence"
],
[
"Investment strategy",
"We investigate how investors can select the strategy that reflects their preferences.",
"If investors prefer to focus on technologies, they should choose companies working on the best technologies as selected by the (highest) TechRank score.",
"This decision implies many criteria such as the number of technologies they want to be invested in, the capital allocation for each company, and the diversification.",
"We sketch the procedure to solve this decision process quantitatively in the Appendix REF ."
],
[
"Comparison with the Crunchbase rank",
"Crunchbase assigns a rank to the top companies of each industry, that takes into account the entity's strength of relationships, funding events, news articles, and acquisitions.https://about.crunchbase.com/blog/influential-companies/ We compare our results in the cybersecurity sector with the Crunchbase rank and investigate the strength of the association between the two scores using Spearman's correlation.",
"To make the ranks comparable, we convert our algorithm's output into a ranking.",
"The resulting Spearman's correlation of 1.4% indicates that the two ranks are uncorrelated.",
"We explain these differences by the fact that the Crunchbase rank is fixed, while TechRank is customizable according to investors' preferences.",
"Moreover, the Crunchbase rank focuses on the company's level of activity and not on its market influence.",
"Furthermore, the Crunchbase rank results from an algorithm that involves all the companies, while we focus only on a subset.",
"We attempt to change the investors' preferences and never obtain correlation coefficients above 2%.",
"Other explanations we propose for this divergence includes the fact we assign a weight which identifies the distance between entities in the ranking.",
"In the same line, TechRank allows decision-makers to set a threshold as starting parameter before running the algorithm.",
"Finally, the Crunchbase algorithm is not open source and we do not know its mechanism, which makes the identification of the divergence's source difficult."
],
[
"Runtime",
"We detail the code related to the TechRank algorithm in the Appendix REF .",
"We run it on a machine with a 16-cores Intel Xeon CPU E5-2620 v4 @ 2.10GHz and with 126GB of memory.",
"We investigate the variations in runtime given changes in the number of companies and technologies.",
"The runtime is a positive function of the number of entities.",
"For technologies the curve is much steeper than that for companies.",
"However, considering that the number of technologies is a direct link to the number of companies, we repeat the experiment treating companies and technologies together.",
"The random walk phase lines represent the runtime to convergence.",
"There is a strong similarity between the runtime for companies and technologies, which is surprising given their different numbers.",
"This also shows how strongly they are correlated and supports the capability of TechRank to capture the complexity of the cybersecurity technological landscape.",
"Table REF reports all the runtimes and we report the corresponding runtime comparisons for technologies and companies of both cybersecurity and medical field in the Appendix REF ."
],
[
"Exogenous factors",
"We conduct a sensitivity analysis based on investors' preferences.",
"We restrict the analysis to 1,000 companies only, given the to long runtime required.",
"We assume investors to be interested in firm location only and consider the case of an investor based in New York City and in San Francisco, in turn.",
"In Table REF we report the outcome in terms of location for the five top ranked companies in both cases.",
"We uncover a location change in the company ranking, with the first one being in the state of New York (investor based in New York City) and in the state of California (investor based in San Francisco), respectively.",
"The companies with lower rank also reflects these geographical preferences, albeit with some exceptions (Singapore and Beijing).",
"This implies that, even if remote companies are disadvantaged, their other attributes overcome this flaw.",
"Table: Companies TechRank scores with location"
],
[
"Robustness tests",
"To test the robustness of our algorithm and benchmark the cybersecurity sector, we apply TechRank for companies in the medical sector.",
"We choose this sector given the important number of companies (twice the number of companies working in cybersecurity).",
"We select the companies with the same methodology, which yields a total of 4,996 companies and 437 technologies.",
"Figure REF shows the results of TechRank in the medical sector.",
"The runtime for these companies, reported in the Appendix REF , is on par with those of the cybersecurity sector.",
"To make the two fields comparable, we set as x-label the number of entities, for both companies and technologies.",
"The results reveal that the runtime of the two fields, for both the parameter calibration and the random walker steps, follow the same behaviour, for both companies and technologies.",
"Increasing the number of entities does not yield any significant change both in terms of convergence and runtime behaviour.",
"Finally, unlike Klein et al.",
"(2015), for which the $\\alpha $ remains constant and $\\beta $ changes significantly, we observe that all of our parameters significantly change across sectors[21].",
"Figure: TechRank scores evolution in the medical field"
],
[
"Limitations",
"We choose technologies related to cybersecurity according to Crunchbase description, e.g., security, privacy, or confidentiality.",
"Since these words may overlap other fields, we require the description to contain at least two of them to classify a company as cybersecurity.",
"This naive strategy could be improved with more sophisticated techniques, such as natural language processing NLP.",
"We also face limitation given the lack of information about companies' resources allocation towards technologies.",
"We only have a list of technologies for each company without more information.",
"Instead, it would be helpful to observe the amount of expenditures towards each technologies.",
"Our algorithm is static as we do not have access to time series and it would be interesting to study how the bi-partite network changes.",
"Finally, we are well aware that introducing exogenous variables may induce a bias given potential outliers.",
"Our normalization procedure divides the factors by their maximum, which may lead to unproportionate results if the maximum is an outlier.",
"However, we do not believe that removing outliers is a viable solution, since this would lead to overlook potentially profitable opportunities."
],
[
"Further research",
"This research can be expanded with time series, to investigate, for instance, the outcome of a company divesting from a technology, or investing in a new one.",
"This would also enables to assess which new technologies are successful.",
"This would give more insights about investment choices towards too recent ideas.",
"A focus on percolation theory could help assess the effects of a node disappearance in the network [27].",
"For this purpose, machine learning methods could also be employed.",
"Further research should be devoted to investigate additional exogenous factors in TechRank, enabling investors the widest range of features available possible.",
"Alternative exogenous factors include the inception date of the company, number of employees, social networks activity, and even the Crunchbase rank, which is itself based upon entity's strength of relations, funding events, news articles, or acquisitions.",
"By the same token, further investigation should confirm the pertinence of the TechRank algorithm in fields that include more entities, as an increase in nodes could lead to coordination problems.",
"Finally, it would be crucial to assess the long-term effects of the TechRank algorithm on investments returns and technologies' development through back-testing, that once again, requires time series."
],
[
"Conclusion",
"We introduce TechRank, an algorithm that assigns a score to companies and technologies in complex systems.",
"This methodology constitutes the first step towards a new data-driven investment strategy, which enables investors to follow their preferences while benefiting from a quantitative approach.",
"We include investors' preference based upon a case-by-case study.",
"Our algorithm convergence depends on the number of entities and the complexity of the relationships within the bi-partite network.",
"Using a restricted number of companies in cybersecurity, we analyze the TechRank scores and we explain the score variations of entities over iterations.",
"We also explore how results change depending on the company's location.",
"Finally, we conduct robustness tests in the medical field, for which our results are qualitatively similar.",
"We believe that our approach brings value to help investors' form their decisions.",
"Moreover, our algorithm's flexibility allows to include exogenous factors and preferences, which is impossible in alternative existing company ranks, such as that of Crunchbase.",
"Given our algorithm performance for cybersecurity, a highly complex market, as a case study, we believe that our algorithm would perform well in all markets.",
"TechRank is a complementary, if not alternative, way to look at portfolio choices."
],
[
"List of words used to identify companies' sectors",
"List of words related to cybersecurity: cybersecurity, confidentiality, integrity, availability, secure, security, safe, reliability, dependability, confidential, confidentiality, integrity, availability, defence, defensive, privacy.",
"List of words related to the medical field: cure, medicine, surgery, doctors, nurses, hospital, medication, prescription, pill, health, cancer, antibiotic, HIV, cancers, disease, resonance, rays, CAT, blood, blood transfusion, accident, injuries, emergency, poison, transplant, biotechnology, health care, healthcare, health-tech, genetics, DNA, RNA, lab, heart, lung, lungs, kidneys, brain, gynaecologist, cholesterol, diabetes, stroke, infections, infection, ECG, sonogram.",
"Figure: Runtime comparisons[h!]",
"[1] $e^C \\leftarrow [0] \\cdot len(c\\_names)$ $c \\in range(c\\_names)$ $i \\in range(i\\_names)$ $c \\in range(i\\_names)$ $e^{IC}_{i,c} \\leftarrow \\sum _{t=0}^{\\mathcal {T}} \\gamma ^{i,c}_t $ $\\gamma ^{t}_{i,c}$ is the amount of the investment from $i$ to $c$ at time $t$ $ e^C[c] \\leftarrow e^C[c] + e^{IC}_{i,c} $ $e^C_{max} \\leftarrow \\max {(e^C)}$ $f^C \\leftarrow e^C/e_{max}$ $f^C$ : list of previous investments for each technology return $f^C$ Previous investments factor for companies [h!]",
"[1] $e^C \\leftarrow [0] \\cdot len(c\\_names)$ $c \\in range(c\\_names)$ $i \\in range(i\\_names)$ $e^{IC}_{i,c} \\leftarrow \\sum _{t=0}^{(T)} \\gamma ^{IC}_t $ $\\gamma ^{t}_{i,c}$ is the amount of the investment from $i$ to $c$ at time $t$ $ e^C[c] \\leftarrow e^C[c] + e^{IC}_{i,c} $ $e^T \\leftarrow e^C \\cdot M^{CT}$ Matrix multiplication $e_{max} \\leftarrow \\max {(e^T)}$ $f^T \\leftarrow e^T/e_{max}$ $f^T$ : list of previous investments for each technology return $f^T$ Previous investments factor for technologies [h!]",
"[1] $h\\_dict \\leftarrow \\lbrace \\rbrace $ $c\\_name, c\\_address \\in c\\_locations$ $lat \\leftarrow c\\_address.latitude$ $lon \\leftarrow c\\_address.longitude$ $h \\leftarrow haver\\_dist(lat, lon, lat\\_inv, lon\\_in)$ haver_dist is a function we have created $h\\_dict[c\\_name] \\leftarrow 1/h$ $h\\_max \\leftarrow \\max {(h\\_dict)}$ $c\\_name, h \\in h\\_dict$ $h\\_dict[c\\_name] \\leftarrow 1 - h/h\\_max$ return $h\\_dict$ Geographic coordinates factor We obtain the distance between two points on earth with the Haversine approximation ($(\\theta )$ ), using latitude and longitude of the locations[19].",
"Let $(\\lambda _1, \\phi _1)$ and $(\\lambda _2, \\phi _2)$ be the longitude and latitude in radiance of two points on a sphere and $\\theta $ the central angle given by the spherical law of cosines, the Haversine distance writes, $h = (\\theta ) = (\\phi _2- \\phi _1) + \\cos {\\phi _1}\\cos {\\phi _2} (\\lambda _2-\\lambda _1).$"
]
] | 2210.07824 |
[
[
"Does personality impact requirements engineering Activities?"
],
[
"Abstract Context: Requirements engineering (RE) is an important part of Software Engineering (SE), consisting of various human-centric activities that require frequent collaboration of a variety of roles.",
"Prior research has shown that personality is one such human aspect that has a huge impact on the success of a software project.",
"However, a limited number of empirical studies exist focusing on the impact of personality on RE activities.",
"Objective: The objective of this study is to explore and identify the impact of personality on RE activities, provide a better understanding of these impacts, and to provide guidance on how to better handle these impacts in RE.",
"Method: We used a mixed-methods approach, including a personality test-based survey (50 participants) and an in-depth interview study (15 participants) with software practitioners from around the world involved in RE activities.",
"Results: Through personality profiles, we found a majority of the practitioners scored as statistically significant (high-scored) on agreeableness and conscientiousness traits and average on extraversion and neuroticism traits.",
"Through analysis of the interviews, we found a range of impacts related to the personality traits of software practitioners, their team members, and external stakeholders.",
"These impacts can be positive or negative, depending on the RE activities, the overall software development process, and the people involved in these activities.",
"Moreover, we found a set of strategies that can be applied to mitigate the negative impact of personality on RE activities.",
"Conclusion: Our identified impacts of personality on RE activities and mitigation strategies serve to provide guidance to software practitioners on handling such possible personality impacts on RE activities and for researchers to investigate these impacts in greater depth in future."
],
[
"Introduction",
"Personality is a human aspect with no one universally accepted definition.",
"Although many theories have been developed related to personality, it is commonly referred to as individual differences [1].",
"Our preferred definition is based on Mischel et.",
"al.",
"[2], who defines personality as \"a set of individual differences including personal habits, skills, memories, behaviours and social relationships that can be affected by the socio-cultural development of individuals\".",
"Personality clashes or incompatibilities can affect the efficacy of collaboration and lead people to perform less effectively[3] [4] [5].",
"Investigating the impact of personality in software engineering (SE) has been an ongoing topic over many years [6] [7] [8].",
"Various studies have examined the impact of the personality of software practitioners in the context of SE in general, and for specific SE activities or contexts, such as development [9], pair programming [10], testing [11], software team composition [12], team climate [13] to name a few.",
"A majority of these studies have focused on software practitioners involved in software development or testing in industrial settings.",
"Some have used undergraduate/postgraduate students as participants [14] [15] [16] [14].",
"Many studies were limited to particular organizations, countries, and geographic areas [13] [17].",
"Based on the key findings of our previously conducted systematic literature review (SLR) on impacts of human aspects on requirements engineering (RE)[18], we identified personality as an important human aspect that needs to be further investigated in relation to RE.",
"RE activities, such as eliciting, analysing, prioritizing, managing software requirements play a vital role in SE and good requirements are considered to be one of the most critical and challenging parts of SE.",
"Arguably, RE activities can be considered the most human-centric and socio-technically intensive activity in SE as they require extensive collaboration and understanding of the individuals involved [19].",
"However, from the literature, we identified that there are very limited studies that directly focus on the personality of the software practitioners involved in RE activities [18].",
"Hence, we want to gain a more comprehensive understanding of how the personality of individuals influences RE activities in SE.",
"Our broad research question is: How does the personality of software practitioners influence requirements engineering activities?",
"To answer this research question, we conducted a mixed methods study involving a personality test based survey of 50 software practitioners involved in RE activities, followed by 15 in-depth interviews with those willing to discuss their experiences in-depth.",
"To gather their personality profiles, we used the standard IPIP NEO-120 Personality test based on the well-known five-factor model (FFM) of personality.",
"We used the ranking scores defined in IPIP NEO-120 test to analyse personality test data [20] and socio-technical grounded theory (STGT) for data analysis to analyse the interview data [21].",
"The main contributions of this study are as follows: We identified some of the possible impacts of personality on RE activities related to software practitioners, their team members and their external stakeholders (customers/clients/end-users); We developed a set of guidelines for software professionals, software teams and stakeholders, as well as academic and industry researchers, who want to better understand the impact of personality on RE activities (both positive & negative) and how they might go about mitigating the negative impacts; and We identified a set of recommendations for future research into the impact of personality on RE activities.",
"Numerous personality models have been formulated based on various personality theories to assess the personalities of individuals by characterizing human behaviours into a set of traits [1] [22].",
"The Five-Factor Model (FFM) is one of the most widely accepted personality models by the psychologists and now used in several SE studies on personality [23] [9].",
"The FFM integrates all personality characteristics into five main traits.These traits are Openness to experience, Conscientiousness, Extroversion, Agreeableness and Neuroticism.",
"These five traits represent one's personality at the broadest level of abstraction, and each trait summarizes a large number of distinct, more specific personality characteristics [24].",
"The five main traits can be explained as follows [25]; $\\square $ Openness to Experience: relates to individuals' intellectual, cultural or creative interests.",
"Statistically significant (high-scored) individuals on openness to experience tend to be imaginative, broad-minded and curious.",
"In contrast, those at the opposite end of this spectrum usually show a lack of aesthetic sensibilities, favouring conservative values and preferring routine.",
"$\\square $ Conscientiousness: refers to individuals' focus on achievements.",
"High-scored individuals tend to be hardworking, organized, able to complete tasks thoroughly on time, and reliable.",
"Low-scored individuals on conscientiousness relate to negative traits such as being irresponsible, impulsive and disorganized.",
"$\\square $ Extraversion: relates to the degree of sociability, activeness, talkativeness, and assertiveness.",
"A person is considered an extravert if they are friendly, comfortable in social relationships, active, assertive and outgoing.",
"The opposite end of this spectrum shows a lack of social involvement, shyness, and prefer to be alone more than extraverted people.",
"But, this does not mean that they are unfriendly or antisocial; rather, they are reserved in social situations.",
"$\\square $ Agreeableness: refers to positive traits such as cooperativeness, kindness, trust and warmth; agreeable individuals value getting along with others.",
"Low-scored individuals on agreeableness tend to be sceptical, selfish and hostile.",
"$\\square $ Neuroticism: refers to the state of emotional stability of individuals.",
"Low-scored individuals on neuroticism tend to be calm, confident and secure, whereas high-scored individuals on neuroticism tend to be moody, anxious, nervous and insecure.",
"These traits can be narrowed down to what is known as \"facets\" which consists six per trait (altogether 30 facets).",
"The facets help to increase the precision and scope of FFM, thus enabling more accurate predictions [26].",
"Numerous instruments were developed to operationalize FFM.",
"Among them the International Personality Item Pool (IPIP) is a freely available personality assessment instrument developed based on FFM.",
"IPIP-NEO 120 is one of the short versions of the IPIP instrument and designed to measure the five personality traits including 30 facets efficiently.",
"This short version possesses acceptable measurement of reliability and most recent studies focus on using IPIP-NEO 120 version due to its acceptable reliability and practicality [20] [27] [28].",
"Hence, we have used IPIP-NEO 120, the personality assessment instrument in our research study that is based on FFM (figure REF ) [20].",
"Figure: Five Dimensions and Thirty Facets of FFM (IPIP-NEO 120)"
],
[
"Personality research in Software Engineering",
"A considerable amount of studies have been conducted aiming to identify the impact of personality in software engineering domain and significantly increased after 2002 [6].",
"The systematic mapping study conducted by Cruz at al.",
"[6], analysed 90 articles published between 1970 - 2010, and identified that 83% of research reported empirical findings on the role of personality in SE where pair programming and education were most recurring research topics.",
"Soomro et al.",
"[22] conducted an SLR on the effect of the personalities of software engineers and how it is associated with team climate and team performance.",
"The findings reported a relationship between software engineers' personality and team performance, and revealed that software team characteristics significantly impact software team performance.",
"Another SLR was conducted by Barroso et al.",
"[23], focused on the influence of personality in SE, and found that the most extracted primary studies focused on software designers' and programmers' personalities.",
"Salleh et al.",
"[10] investigated the effect of personality traits in pair programming on higher education with five formal experiments.",
"They identified that \"openness to experience\" significantly differentiates paired students’ academic performance.",
"An empirical study by Kanij et al.",
"[11] investigated the impact of personality of software testers by collecting personality profiles of 182 software practitioners.",
"45.1% of them were software testers and the rest were programmers.",
"The results indicated that software testers scored high on the \"conscientiousness\" trait than other software practitioners.",
"Xia et al.",
"[29], conducted a study with software professionals to identify the relationship between project managers' personalities and team personality composition and project success by investigating 28 completed software projects with 346 software professionals.",
"The results indicated that project manager personality and team personality affect the success of software projects.",
"They suggest focusing on relationships between personality and SE activities as their study only demonstrates the link between personality and overall project success.",
"Kosti et al.",
"[14] and Acua et al.",
"[30] conducted empirical studies by collecting personality profiles of SE students.",
"They identified significant relationships of personality with work preference and job satisfaction respectively.",
"Further, they suggested conducting the research for real-world software teams.",
"Vishnubhotla et al.",
"[13] investigated the relationship between personality and team climate focusing on software professionals in agile teams.",
"The findings indicated a significant positive relationship between certain personality traits and team climate factors.",
"Mendes et al.",
"[27] conducted a survey study with 63 software engineers investigating the relationship between decision-making style and personality within the context of software development.",
"They identified seven statistically significant correlations between decision-making style and personality and built a regression model considering the decision-making style as the response variable and personality factors as independent variables.",
"Requirements Engineering (RE)-related activities in SE are considered to be highly human-centred as they involve working with a diverse range of people such as stakeholders, software development team members, and other requirements engineers [31] [32] [33].",
"From our prior SLR study [18], we identified that the impact of various human aspects needs to be studied more related to RE, and personality was identified as one such human aspect that has been considered often in SE studies, but RE has been considered as just one part of it or mainly limited to requirements elicitation [34] [12] [35] [36].",
"Being motivated by our SLR, we surveyed 111 software practitioners involved in RE activities to better understand the perspective of human aspects, including personality.",
"There we found the software practitioners' personality needs to be considered when they are involved in RE activities [37].",
"Having identified the importance of personality and its potential impact on RE as a key area worth investigating in our prior studies [18] [37], acknowledging the research gap in this area motivated us to design and conduct this study to answer our above-mentioned research question.",
"Figure: Overview of the mixed methods research design"
],
[
"Research Methodology",
"This study aimed to understand participants’ perspectives on the impact of personality when involved in RE activities."
],
[
"Study Design",
"Figure REF shows the design of our study.",
"We applied a mixed-methods approach when conducting this study.",
"Mixed here refers to both qualitative and quantitative approaches being utilized.",
"Our previous studies [18] [37] identified personality as a human aspect that software practitioners believe greatly impacts RE activities.",
"To obtain an in-depth understanding of the impact of personality on RE, we designed a personality test-based survey and an interview study targeting software practitioners involved in RE activities.",
"The first step in our data collection was to obtain personality profiles of software practitioners involved in RE activities aiming to get a large scale idea of personalities of people doing RE activities and to see if any particular personality trait/facets get mentioned a lot in the interviews referring to its' impact on RE.",
"We used the standard IPIP-NEO 120 test for personality assessment (section REF ).",
"It consists of 120 items/statements, and participants are required to indicate how each statement best describes them.",
"For example, the first statement is \"I worry about things\", where participants should indicate how much it related to themselves via a Likert scale from \"very inaccurate\" to \"very accurate\".",
"Apart from the personality test, the survey consists of several demographic questions such as their age range, gender, country of residence, education qualifications, and work experience to identify their involvement in RE activities.",
"We collected their employment details such as job role/title, a summary of their main job responsibilities, how often they are involved in RE activities, and their domain and software development methods.",
"An open-ended question was used to collect their overall opinion on the impact of personality on RE activities.",
"We used the Qualtrics platform to design and distribute the survey (appendix A)."
],
[
"Interview-based study",
"The second stage of our data collection used an interview-based study to get an in-depth understanding of the impact of personality on RE activities.",
"To do this we designed a semi-structured interview schedule consisting of three sections.",
"The first section collected detailed demographic information about the participants, including their involvement in RE activities.",
"The second section collected the participant's views on the influence of personality in RE activities.",
"The views on the influence of their personality, their team members' personalities and their external stakeholders' personalities were focused on related to RE activities and how they mitigate any negative influences of personality (if any) when involved in RE activities.",
"The last section asked participant opinions on the impact of other human aspects besides personality on RE activities (if any) based on their experiences.",
"Our semi-structured interview schedule can be found in appendix B."
],
[
"Pilot study",
"After designing the interview-based study and personality test-based survey, we conducted a pilot study with two software practitioners from our professional networks to validate the clarity and understandability of the questions; the time reported to complete the studies and to get their suggestions on improving both studies.",
"Both provided feedback on the survey and interview study, where we modified the questions to improve the clarity based on their suggestions, and finalized the survey and interview studies."
],
[
"Data Collection",
"The target population of our study was software practitioners involved in RE activities.",
"After obtaining the required ethics approvalMonash Ethics Review Manager (ERM) reference number: 29072, we advertised our study on social media (LinkedIn, Twitter) and within our own professional networks.",
"Participation was voluntary, and we recruited 15 practitioners with experience in RE activities worldwide.",
"The personality test-based survey was carried out as a pre-interview questionnaire which took around 15-20 minutes to complete.",
"Then, the semi-structured interviews were carried out, with each interview lasting 50-60 minutes.",
"Due to the pandemic, all the interviews were conducted online, and were audio recorded.",
"Table REF shows the detailed demographic information of the participants.",
"We wanted to compare personality traits of those involved in RE activities.",
"As we had obtained only 15 personality profiles of the software practitioners interviewed, we advertised our personality test-based survey on the Prolific platform [38].",
"We used the built-in options in the prolific platform to filter participants based on our target audience.",
"We applied the filter options for participants' employment status as \"full-time\", employment-sector as \"information technology\", and industry as \"software\" and specifically mentioned that we are looking for software practitioners involved in RE activities to obtain our target participants.",
"We recruited 35 participants via Prolific.",
"Each of them were rewarded 8.72 AUD after completing the survey.",
"This resulted in a total of 50 personality profiles to analyse."
],
[
"Data Analysis",
"Both qualitative and quantitative data were collected from our study.",
"Quantitative data analysis was carried out with the data collected through personality test-based survey.",
"We followed the standard statistical analysis method mentioned in IPIP-NEO 120 personality test to analyze personality test data via Microsoft Excel.",
"The qualitative data analysis was carried out with interview data.",
"We used socio-technical grounded theory (STGT) for data analysis [21], which is particularly suited for analyzing open-ended data.",
"We transcribed the interview sessions using otter.ai with the consent of the participants and stored & analyzed the data using NViVO.",
"The data collection and analysis were iterative where we conducted ten interviews in the first iteration and five in the second iteration (table REF ).",
"We followed the open coding approach to generate concepts and categories with constant comparison and memoing techniques in STGT.",
"Thus, we wrote memos to record key insights while following the open coding activities.",
"Below is an example of a memo that we recorded related to the negative impact of the neuroticism dimension on RE activities.",
"We have discussed these insights generated from memoing in section .",
"Memo on \"Negative impact of neuroticism trait on RE activities\": The majority of the interview participants (n = 12 out of 15) mentioned that the neuroticism characteristics of the people negatively impacts RE.",
"They mentioned that when people involved in RE activities are \"anxious\", \"insecure\", \"overthinking\", \"moody\", and \"reserved\", it negatively impacts RE activities, specifically in requirements elicitation.",
"All these adjectives emphasize the neuroticism trait, indicating that it negatively impacts RE activities.",
"Interestingly, many of them mentioned these related to their team members or customers' characteristics and did not particularly mention their own neuroticism characteristics.",
"E.g., INT01, a lead business analyst, mentioned that their team members' \" anxious\" and\" overthinking\" behaviour creates confusion between customers and the team in requirements elicitation, resulting in getting incomplete requirements during the project.",
"However, they tend to explain the impact related to their team members' characteristics rather than their own characteristics, which also can occur due to the nature of their individual personalities.",
"Hence, the relationship between software practitioners' personalities and their team members'/customers' personalities can also be further investigated related to RE/SE.",
"Table: Participants' Demographics Information"
],
[
"Demographics of the participants",
"Our study consists of a personality test-based survey (50 software practitioners) and an interview study (15 software practitioners).",
"The majority of the participants were male (66%), with ages ranging between 31 to 40 years (40%).",
"The most common roles are software engineer (40%), followed by IT project manager (12%) and business analyst lead (10%), where the majority of the participants (60%) possess a bachelor's degree in computer science or software engineering.",
"Table REF summarises all the demographic information of the participants.",
"The majority of the participants were from the United Kingdom (20%), followed by South Africa (12%), Australia (8%) and Sri Lanka (8%).",
"An equal number of participants (34%) have 1-5 years and 5-10 years of work experience in the software industry, whereas 28% have more than ten years of work experience.",
"When specifically considering interview participants (15 software practitioners), their work experience in the software industry expands from 3 to 26+ years.",
"Most of them have been fully involved in RE activities throughout these years.",
"Table REF provides detailed demographic information of the interview participants.",
"Most participants (68%) use agile software development methods, such as scrum, kanban, and XP, and 20% use both agile and traditional (waterfall) software development methods.",
"The rest of the 12% only use traditional (waterfall) software development methods.",
"Among the majority who use agile software development, 55.8% use scrum, and the rest use other methods such as kanban, XP and lean software development.",
"The participants have experience in a variety of project domains where the majority of them (28%) were finance domain, followed by health (24%), transport & logistics (18%), IT (10%), government services (8%), manufacturing (8%), education (6%), real estate (4%), IoT & telecommunication (4%), utility services (4%), and insurance (2%).",
"Figure: Participants' involvement in RE activitiesOur target participants were software practitioners involved in RE activities.",
"We confirmed this by asking participants to rate how often they were involved in major RE activities when they completed the personality test-based survey.",
"An equal percentage of participants (38% each) indicated that their involvement in RE activities such as eliciting/analyzing/prioritizing/ managing software requirements is almost every day or a couple of times a week.",
"20% of the remaining participants mentioned that they were involved in these RE activities a couple of times a month.",
"In contrast, only 4% of the participants mentioned their involvement in RE activities is rare.",
"We also provided a set of major RE activities to identify further how much the participants were involved in RE as a part of their job.",
"As shown in figure REF , the majority of the participants are either always or very often involved in these RE activities.",
"Among them, the majority (72%) are always or very often involved in requirements management throughout the project.",
"68% of participants either always or very often collaborate with stakeholders to elicit requirements and to document them in software requirements specifications.",
"64% of the participants always or very often participate in requirements prioritization, whereas 62% always or very often lead requirements analysis and verification.",
"The least amount of participants mentioned that they were never involved in these given RE activities, which is a maximum of 4% for activity \" lead requirements analysis & verification\".",
"For all the other RE activities given, only 2% of participants mentioned that they were never involved in those activities.",
"We also asked them to mention other job responsibilities they are involved in, where 10% are involved in the implementation and 4% in testing/bug fixing, architecture designing and creating user guides after feature development as the tasks apart from the given list of activities.",
"It shows that we were able to recruit our target participant group for this research study."
],
[
"Personality Characteristics of the practitioners involved in RE activities",
"We collected 50 personality profiles from software practitioners involved in RE activities.",
"We used the standard IPIP-NEO 120 personality assessment test, designed based on FFM, as our personality test (section 2.1).",
"It includes 120 items/statements that help to analyze a person's personality, referring to five broad dimensions and 30 facets (figure REF ).",
"A sample personality profile of a participant can be seen in Appendix C. By analyzing the 50 personality profiles that we have collected, we identified that although every individual possesses a mix of personality traits, some personality traits are statistically more significant among others.",
"As shown in Figure REF , agreeableness (78%) and conscientiousness (70%) are identified as statistically significant (high-scored) personality traits among the majority of our participants.",
"An equal number of participants (50% each) have a high and average score related to the openness to experience trait.",
"The majority of the participants have an average score in their extraversion and neuroticism traits, where 72% are average in extraversion and 80% are average in neuroticism.",
"Figure: Variation of Personality traits/dimensions of the participantsThe 'high', 'average' and 'low' categories depicted in Figure REF are decided based on each individual's scores through analyzing their personality tests.",
"We identified that the majority of the participants have high agreeableness and conscientiousness traits and average extraversion and neuroticism traits.",
"None of the participants is low in agreeableness, conscientiousness, openness to experience and extraversion traits, and only one participant (2%) is identified as low in neuroticism.",
"Each personality trait consists of six facets that can be used to describe each personality trait in detail.",
"Hence, an individual's personality can be described with five personality traits and thirty facets.",
"As shown in figure REF , most of the facets related to agreeableness, conscientiousness and openness to experience are significantly high among participants.",
"In contrast, the majority of the facets related to extraversion and neuroticism have average scores.",
"Considering the agreeableness trait, the most common trait among the participants, its' straightforwardness (88%), cooperation (84%), and altruism (82%) facets are significantly high in the majority of the participants compared to sympathy, trust and modesty facets.",
"This shows that the participants' personalities related to agreeableness can be best described as straightforward, cooperative and altruistic individuals.",
"Figure: Variation of the facets of the Personality traits/dimensions of the participantsConsidering conscientiousness, participants obtained high scores in competence (86%), dutifulness (84%) and achievement striving (80%) facets, making it the second highest personality trait that can be seen among the participants.",
"However, compared to facets of the other three personality traits, most of the facets of agreeableness and conscientiousness are scored high among participants, indicating that these two personality traits are the most common traits among the participants.",
"Among the facets of openness to experience, the trait that has an equal percentage of high and average scores, ideas/intellect (74%), feelings/emotionality (68%), aesthetics (66%), and values/liberalism (62%) facets have high scores compared to the imagination and adventurousness facets.",
"Table: Demographics of the Interview ParticipantsHowever, none of the facets of openness to experience reach the level of scores of agreeableness and conscientiousness.",
"This indicates that participants tend to have more openness to experience behaviours than extraversion and neuroticism, but not as high as agreeableness or conscientiousness behaviours.",
"Referring to the facets of extraversion trait, only cheerfulness (70%) and assertiveness (68%) are considerably high among participants.",
"In contrast, friendliness, gregariousness, activity level and excitement-seeking can be seen as average-scored facets among the participants.",
"Neuroticism is the trait where most facets have average scores among the participants, indicating that neuroticism behaviour is mainly average among participants.",
"Among their average neuroticism behaviour, the majority of the participants seem to be impulsive (70%), vulnerable (68%), angry (64%) and self-conscientious (64%).",
"Compared with other personality traits, it can be identified that a considerable amount of participants are low in neuroticism facets such as depression (30%), vulnerability (12%), anger (12%), and anxiety (8%).",
"In contrast, the majority of the others facets are hardly seen in a low category.",
"By analysing these 50 personality profiles, we identified that the participants tend to be statistically significant in high agreeableness, conscientiousness and openness to experience than in extroversion and neuroticism."
],
[
"Impact of personality in RE activities",
"We conducted 15 detailed interviews with a range of software practitioners involved in RE activities.",
"Their demographics are summarised in Table REF .",
"Through this we obtained in-depth insights into the impact of personality on RE activities.",
"Throughout this section, we include several original quotes from the interviews to illustrate our findings.",
"Almost every participant mentioned their view on the impact of personality on RE activities.",
"The majority of participants agree that there is a positive or negative influence of personality in RE, the importance of identifying the impact and how the negative impacts can be mitigated.",
"Only 3 participants claimed there was little to no impact of personality, mainly referring to clients/customers/end users' personalities \"I think you can gather requirements from end users irrespective of their personality\" - INT09 [lead human-centered designer/information architecture] rather than those of team members or their own.",
"Most of our interview participants experienced the impact of personality when doing RE activities irrespective of the people involved – their own personality, team members' personalities, or clients/customers/end users personalities even in a virtual environment: \"Even in these unprecedented times, where all the things are virtual, and you do not get a chance to work with people at office, still the various personalities of the people actually, really important to conduct our work\" - INT05 [software engineer/application consultant].",
"Referring to the impact of personality on RE activities, we identified that the impact can be either positive or negative on the outcome of the RE activities, the overall outcome of the project, and/or the performance of the team and people involved.",
"For example, according to INT05, a software engineer & application consultant, personality highly impacts the most important RE activity: \"'my personal opinion in the requirement engineering, the personality mainly impacts on the requirements gathering part, which is the most important part, not the other side.. giving required output..\".",
"INT13, a lead business analyst, explained that personality impacts the individual performance as well as the performance of the team: \"personality makes a huge difference for a person as well as for the team because when they are involved in requirements engineering activities with various personalities, that affects a lot for you and the people you're working with to do your best.",
"Eventually it improves our performance\".",
"Through in-depth analysis of the interviews, we have identified that these positive and negative personality impacts can be further explained related to software practitioners' personalities, their team members' personalities and their clients/customers/end users' personalities.",
"Impacts related to software practitioners' individual personality Participants identified various personality characteristics and how these characteristics positively or negatively impact them when they are involved in RE activities.",
"Table REF summarizes these participant reported impacts, relating impact to individual personalities, and highlighting whether it is positive (+) or negative (-).",
"We have also identified that these characteristics can directly or indirectly be mapped to the five personality traits/dimensions and their facets.",
"Some important aspects are discussed as follows: Ability to 'work the way around': The majority of the participants (13) mentioned that their ability to 'work the way around' has a large impact when involved in RE activities.",
"Individuals with this characteristic know how to get things done to achieve what they need to.",
"They do this by prioritizing their work while working with their team members and customers.",
"This personality facet characteristic helps them to work with different customers during requirements elicitation in order to obtain the actual software system requirements.",
"It also helps them in their dealings with internal team members (other developers and project managers) when translating specifications into technical tasks to reduce future system complications.",
"For example: \"so now I am dealing with him in a very different way.",
"I make things simple for him.",
"I only asked him the key questions first and then gradually gathered requirements with smaller discussions\" - INT12 [Senior project manager].",
"\"But When you take it positively, when you know how to get the thing right at the first phase, you can reduce that number of errors, bugs, complications on whatever that is to come with the dev team when giving specifications as technical tasks\"- INT13 [Lead BA/Scrum master].",
"Hardworking nature: 6 participants mentioned that their hardworking nature personality facet positively impacts them when they are involved in RE activities.",
"Through this hardworking nature, they can provide the best solution to their customers, resulting in high customer satisfaction and completing their work tasks within the scheduled time, saving budget and time.",
"It also results in delivering reliable work quickly and highlights that they can identify more requirements via their hardworking nature, leading them to dig deeper in the relevant domain.",
"\"I will be doing more research on the area and then get back to them with some sort of a plan, then carry out the work with the team.",
"At the end, they are very happy about the work\" - INT02 [BA/ Project manager].",
"The above two characteristics can be directly mapped with the conscientiousness trait, where the \"ability to work the way around\" is related to the competence facet and\" hardworking nature\" is related to the achievement-striving facet.",
"Considerate of others: 14 participants said that their personality facet of concern towards others involved in RE/SE-related activities impacts them when they are involved in RE activities.",
"Consideration towards other team members leads them to focus on their allocated tasks and solve conflicts between customers and the team when involved in RE activities.",
"Moreover, it helps to gather opinions from everyone, which leads us to get the best solution among them.",
"\"I am highly concerned about my team members, I personally talk to them to get their opinions and present them in the team discussions.",
"It helps us to produce the best solution for our customers\" -INT08 [Software engineer].",
"Concern towards their customers helps them better to understand their behaviours and change the requirements elicitation techniques accordingly.",
"For example, if the customer tends to be more reserved and is not expressive of what they need, changing the requirement elicitation technique will help get the best outcome: \"As a lead, I tend to consider their behaviours, so that I can apply relevant requirement elicitation technique for them.",
"For example, we had a very reserved client, not talking much, so we changed our way of collecting requirements, and it was a success.",
"We only asked main points in the meeting and when we need more information, we emailed him\" - INT12 [Senior project manager].",
"Table: Impact of personality related to software practitioner's individual personalityCooperative nature: Participants highlighted that their personality facet of a cooperative nature is beneficial when they are involved in RE activities, both within their team and with their customers.",
"Cooperativeness greatly impacts requirements specification, prioritization, and handling of requirement changes, taking the team and the clients to one page: \"It did not help to do requirement engineering unless you have qualities such as cooperativeness, it is really helpful when we prioritize the requirements and handling requirement changes in later phase\" - INT04 [Business Analyst].",
"These characteristics can be directly mapped with the agreeableness dimension, where the\" considerate of others\" can be referred to as altruism and \"cooperative nature\" can be directly referred to as compliance/ cooperation.",
"Enjoy in-depth conversations: Individuals who have a personality facet of enjoy having in-depth conversations said that it positively impacts them when they carry out RE activities.",
"It was mentioned that without having in-depth discussions with the customers, getting actual requirements is difficult.",
"In-depth discussions connect the team with the customers from the ground level, leading to fewer conflicts in the later part of the project.",
"It is also helpful to know in and out of the project, which will assist in handling change requests from the development team: \"if you want to know, understand the system they want, then you have to have a long fruitful conversation with the clients, you can't just sit and expect them to give you the requirement.",
"I always enjoy having long conversations with them, and then I know what they actually want\" - INT10 [Business Analyst].",
"Willingness to try new things: Many participants mentioned that they are always open to trying new things, and their personality facet of having an exploratory nature has a high impact when they are involved in RE activities.",
"They said it enhances the individual learning about the projects, improves the project's design with advanced functionalities and increases customer satisfaction with new approaches, which will benefit them: \"I am always open to try new things, specially when designing each project.",
"Mostly, it has positively affected to my team and customers, they are very satisfied with new functionalities\" - INT14 [Business Analyst].",
"Both of these characteristics indicate that individuals with the openness to experience trait positively impact RE activities where the\" enjoy in-depth conversations\" can be mapped with ideas/ intellect facet and\" willingness to try new things\" can be mapped with actions/ adventurous facet.",
"Ability to interact with various roles: 9 participants mentioned that their personality facet of an ability to interact with various people/roles greatly benefits them when involved in RE activities.",
"Interactions with internal and external parties are required.",
"The ability to interact with various people greatly impacts the success of a particular task.",
"For example, individuals with high interaction skills are needed to initiate conversations with customers, which will be helpful in having a positive connection with the customers, and result in getting good support from them throughout the project.",
"\"It is needed when you start conversations with your clients, creates some kind of bond with them.",
"If you have a good bond with your clients, they will support you with everything.",
"So having high interaction skills, I think, it is like one of the strong skills that you need as in personality\" - INT04 [Business Analyst] Prefer to control/take the lead: It was mentioned that individuals' personality facet of a preference to take control or take the lead has a mixed impact when they are involved in RE activities.",
"For example, when dealing with a difficult customer who is having issues with the team, it is important to take the lead and intermediate with them.",
"This is by directly contacting the customers to identify issues and provide immediate solutions required to maintain the continuation of the project without having interruptions for the final output.",
"It also ensures that everyone in the team gets a voice: \" I will directly contact them and interact with them.",
"Try to identify the problem and make sure it will not impact our work and the final project\" - INT03 [IT development & re-engineering lead].",
"However, we identified that this personality facet sometimes negatively impacts developers when carrying out RE activities.",
"For example, if there is an over-controlling lead in the team, it will directly impact the team's performance.",
"It reduces team members' interest in the project and they start to complete the tasks just to get by which will eventually impact the project's outcome, leading to customer dissatisfaction: \"However, if you try to over control them, it will definitely impact as others will just do the tasks and it will be clearly seen in their performance\" - INT06 [Technical team lead].",
"These characteristics can be mapped onto the extraversion dimension where the \"ability to interact with various roles\" can be referred to friendliness and gregariousness facets, and \"prefer to control/ take the lead\" can be referred to as the assertiveness facet.",
"Getting anxious: Individuals with a personality facet of getting anxious tends to have a negative impact when they are involved in RE activities.",
"Individuals' confidence level, deadlines, and reluctant to change make them anxious, result in creating confusion between customers and the internal team, not understanding what customers actually need and inability to complete tasks on time which will eventually affect their performance: \"I know personally, within my experience, I know that there are times when I'm just anxious, because, probably, I'm not confident about the work, maybe I think I've not done my homework well, so it does affect my performance\" - INT10 [Business Analyst].",
"Insecure feeling: Individuals' personality facet of having insecure feelings also has a negative impact when they are involved in RE activities.",
"A feeling of insecurity comes from their self-doubts, seeing other team members' performance, or when doing changes to their allocated tasks.",
"It will also directly impact their performance and the team's performance, increasing the conflicts between team members.",
"Overall, it will impact the outcome of the project: \"A few years back, I was kind of insecure because I have self-doubt of myself, seeing all my colleagues working as pros, so you tend to feel insecure, and it really impacts my work.",
"I couldn't give what the team is asking from me\" - INT07 [Software engineer].",
"Both of these characteristics can be directly mapped onto the neuroticism dimension, where the majority of the participants mentioned that it has a negative impact when doing RE activities.",
"The \"getting anxious\" characteristic can be directly referred to anxiety facet, whereas the\" insecure feeling\" can be referred to\" vulnerability\" facet.",
"Impacts related to software practitioners' team members' personality Almost every participant provided insights on how the personality of their team members impacts them when they are involved in RE activities.",
"This included how the team members' personalities positively or negatively impact RE activities, the overall outcome of the project and their team performance.",
"We have identified several personality characteristics that positively influence RE activities that every participant wants to see within their team members and some negatively impacted characteristics that they want to mitigate from their team members.",
"Table REF summarizes the set of important characteristics, mapping with respective personality dimensions/traits and facets accordingly and discussed as follows; Table: Impact of personality related to software practitioners' team members' personalityFriendly/ outgoing nature: Team members with a personality facet of having a friendly/ outgoing nature positively influences the team when dealing with customers.",
"They are the key people who initiate and conduct discussions with customers, making them comfortable with the team to provide actual, in-depth requirements.",
"Having friendly team members are helpful in dealing with customers with constant requirement changes and providing unrealistic or impractical requirements where they can lead the team in such circumstances to reduce the conflicts between the team and the customers.",
"Moreover, they are beneficial in keeping the team's energy and carrying forward the teamwork positively: \"I have been observing this particular team member of my team who is always friendly and has some energy around him.",
"He works with customers in a very friendly manner.",
"So we didn't have many conflicts with the customer because he handles them\" - INT01 [Lead BA].",
"\"The outgoing members in the team can really drive the rest of the team, keeping the energy of the team\" - INT04 [Business Analyst].",
"However, it was also mentioned that a team member's friendly/ outgoing nature personality facet can sometimes negatively impact them when they are involved in RE activities.",
"This can reduce the customers' involvement in RE activities where they might not get enough opportunity to express their opinions.",
"This may lead to incomplete requirements, creating requirements gaps and conflicts between the customers and the team.",
"\"when the client is not getting enough chance to speak because this person is very friendly and always talking.",
"So it can also impact creating gaps between the team and the clients\" - INT10 [Business Analyst].",
"Leading & controlling nature: Leading & controlling team members has a mixed impact when involved in RE activities where sometimes their personality facet of a leading & controlling nature impacts positively.",
"For example, according to INT02, who is working as a business analyst & project manager, mentioned that it is a really important aspect to manage the internal team and lead the whole team towards one goal: \"So when people are less working as a team, I find it is important to have that one person in the team who is working as a team leader, managing the team and taking them to one page\" - INT02 [Business analyst/Project manager]) However, many participants pointed out that they have experienced negative impacts of having team member(s) with a leading & controlling personality facet, such as losing interest in the project, internal team conflicts, and not getting the best outcome due to over-power others' opinions: (E.g.",
"\"we are working on one particular project, and we were doing equal contribution.",
"But, he tried to stamp out my opinions.",
"So, you tend to lose interest\" - INT08 [Software engineer]).",
"These characteristics relate to the extraversion dimension, where 'friendly/ outgoing' nature can be mapped onto the friendliness facet, whereas 'leading & controlling' nature can be mapped into the assertiveness facet.",
"Coping with others: Team members' personality facet of an ability to cope with others has a positive impact when they are involved in RE activities.",
"It specifically impacts the teamwork and overall performance of the team as when team members are willing to cope with others to work; they come up with the best outcome.",
"Moreover, managing the team and handling the customers is easier when the team is willing to cope with others: \"The members should be able to cope with others, you know, that improve our work, and we were able to give what client exactly expected\" - INT11 [Lead BA].",
"Trustworthiness: Trust is needed within the team and with the customers and this personality facet impacts the team positively when involved in RE activities.",
"Trustworthiness of the team members leads to delivering the projects on time without delays.",
"It helps to resolve internal conflicts within the team for the leads and the management.",
"Trust between the team and the customers leads to getting complete requirements as customers tend to be more open when they trust the team: \"some kind of trust as well, we trust that they will deliver their tasks on time so that we can complete our project on time.",
"Actually, that happened in most of my projects\" - INT07 [Software engineer].",
"Both \"coping with others\" and ' trustworthiness\" characteristics can be mapped with the agreeableness dimension, where these two characteristics can be linked with cooperation and trust facets, respectively.",
"Organized nature: Organized team members are very important when they are involved in RE activities.",
"It helps to complete projects with strict budgets and timelines as the team is well prepared and organized, impacting the quality of the deliverables.",
"A personality facet of an organized nature leads the team to work with a plan and is always prepared: \"you need to be organized so that your team has a plan and can work for it.",
"In that project, we were able to complete it on time\" -INT09 [Lead human-centred designer].",
"Responsible nature: Responsible team members are also a key to successful project delivery as they tend to complete their tasks well on time.",
"Team members with a responsible nature personality facet are important when there are critical releases, as they tend to work with the team until the work is done.",
"For example, INT07, a software engineer, mentioned that he had experienced the consequence of having an irresponsible team member who was always postponing the schedule, resulting in poor project implementation and customer dissatisfaction: \"We suggested some features, and he was like, he will do later, and he is postponing the schedule, but never implementing a good project, and the customer was not happy at the end\" - INT07 [Software engineer].",
"These characteristics can be mapped onto the conscientiousness dimension, where the \"organized nature\" can be mapped to the orderliness facet, and the \"responsible nature\" can be mapped to the dutifulness facet.",
"Willingness to discuss: Team members' who have a willingness to discuss personality facet have a positive impact on RE activities.",
"Their willingness to have discussions within the team helps to maintain the project scope with time.",
"For example, when all the members do not appear to be in the same direction, it would be great to have team discussions constantly to drive the team within the project scope: \"as a team, moments where we don't appear to be all are going in the same direction, we will have conversations often and lead the team to complete what is within the scope\" - INT09 [Lead human-centred designer].",
"Willingness to change: Team members' willingness to change personality facet also positively impacts them when they are involved in RE activities.",
"This leads them to explore more, do experiments and change their approaches accordingly.",
"For example, when there is a technology change (upgrade) within the team, these team members accept the change as they are willing to explore and learn new things: \" if you are actually doing requirement engineering, you need to have the ability to change, you should know how to go forward with new technologies, techniques and you have to have that desire to learn and cope with new things\" - INT05 [Software engineer & application consultant].",
"The characteristics of \"willingness to discuss\" and \"willingness to change\" can be mapped onto the openness to experience dimension, which can be related to the ideas/ intellect facet and actions/ adventurousness facet, respectively.",
"Moody & anxious behaviour: 7 participants mentioned that moody & anxious personality facet team members negatively impact them when they are involved in RE activities.",
"They tend to overthink requirements (e.g.",
"how will we implement this?)",
"in the first phase, not focusing on gathering complete requirements.",
"They may also create confusion between the team and the customers and conflicts among internal team members: \" if a person is moody or anxious, that may negatively impact carrying out requirement engineering activities, like they will be anxious and think about implementation work all the time so that they may miss important requirements\" - INT02 [Business analyst & Project manager].",
"Easily getting annoyed: RE activities have been negatively impacted due to team members who are easily annoyed when they are involved in RE activities.",
"This personality facet leads them to loose valuable customers as they tend to interact with customers in a rude/aggressive manner.",
"Also, it creates lots of conflicts among the team members.",
"When questioned about their less cooperation with the team, they easily get annoyed, which will eventually impact the team's overall performance: \"there was a guy; he was very aggressive towards the problems and the people, he will get angry at customers.",
"So the customers didn't want to work with him\" - INT05 [Software engineer & application consultant] Both characteristics can be related to the neuroticism dimension where \"moody & anxious behaviour\" can be mapped with anxiety and \"easily getting annoyed\" can be mapped with anger facets, respectively.",
"Impact related to clients/customers/end users' personality Twelve participants mentioned that clients/ customers/ end users' personalities positively or negatively impact them when they are involved in RE activities, while the rest (3) mentioned that they do not believe that clients/ customers/end-users personalities have an impact when they are involved in RE activities.",
"However, their insights are limited and quite similar to what they have expressed about the impact of their team members' personalities on RE activities.",
"Some of the key findings related to the impacts of customers/clients/end users' personalities on RE activities are discussed as follows; Willingness to collaborate: This can be mapped onto the cooperation facet in the agreeableness dimension, where customers' interest in working collaboratively with the team has a mixed impact on conducting RE activities.",
"It helps to have a good discussion with the customers' where practitioners can understand what they actually need and gather accurate, complete requirements.",
"It reduces the number of requirements changes in the later phase of development as they are involved in the project from the beginning: \" It's very important in requirement engineering to have good discussions with the users; when they are working with us together, we can understand what they want, we can get all the requirements\" - INT03 [IT development & re-engineering lead].",
"However, it was also mentioned that the customers' collaborative nature can sometimes negatively impact RE activities.",
"Due to their collaborative nature, they can often change the requirements, resulting in disruptions to the ongoing work and frustrating the software team. \"",
"I have sometimes experienced their over-connection with the team disturbing our work.",
"You know, they want to involve in everything, and then they keep changing and demanding, sometimes it is frustrating at the end of the day\" - INT14 [Lead BA].",
"Working with a plan: Customers who always work with a plan impact RE activities positively, making the work easier for software practitioners.",
"These customers have already thought of the requirements and know what they want, which leads the software practitioners to get clear requirements: \"So it's always great to work with clients who have planned what they want, they've thought it through, and we get the requirements easily\" - INT10 [Business Analyst].",
"\"I worked with a foreign customer who is very organized, she has planned everything from the beginning, you know, she is like, we want these, these, very clear and it is easy for us.",
"We can directly get the requirements and start the work\" - INT15 [Senior software engineer].",
"This characteristic can be mapped with the self-discipline facet in the conscientiousness dimension.",
"Willingness to change: Customers who do not always stick to their beliefs and are open to change tend to positively impact work when they are involved in RE activities.",
"Customers with this personality facet are helpful in changing requirements due to technical feasibility.",
"They are open to considering suggestions from software teams, such as changing impractical requirements.",
"This characteristic can be mapped with the actions/ adventurousness facet in the openness to experience dimension: \"There are some clients who are happy with what we suggest; they understand technical difficulties and accept changes.",
"then we also try to give our best\" - INT12 [Senipr project manager].",
"Expressive nature: Customers with an expressive nature personality facet positively impact the carrying out of RE activities.",
"It helps the team to get clearly defined requirements from the customers' end, and practitioners do not want to put extra effort into digging deep into the requirements to find out more about missing requirements.",
"They are not reluctant to say what they want and always work for it: \"if your customer is an expressive person, you really don't have to push them to give you information or detailed requirements.",
"Because they would go out of their way to share it with you\" - INT10 [Business analyst].",
"However, it was also mentioned that this expressive nature personality facet can sometimes impacts negatively as it leads to project scope creep.",
"This characteristic can be related to the activity level facet in the extraversion dimension: \"customer who knows what he wants, keep expressing, keep demanding that this needs to be done and followed by B, C, D, and that could lead to scope creep, and we have to deal with those situations\" - INT10 [Business analyst].",
"Reserved nature: Having a reserved customer usually negatively impacts the team when they are involved in RE activities.",
"The software practitioners involved with customers with this personality facet need to put extra effort into gathering requirements, which will also be time-consuming.",
"The reserved customers tend not to provide their opinions, leading to incomplete requirements and conflicts with the software team members.",
"This can be mapped with the self-consciousness facet in the neuroticism dimension: \"users with high emotionally and reserved, usually, they only say about one aspect at one time, so lots of time has to be given\" -INT03 [IT development & re-engineering lead].",
"\"I think sometimes they didn't at least raise their voice and ask questions or give their opinion.",
"We didn't know whether complete what they expected and there were lots of confusions\" - INT06 [Technical team lead].",
"The interviewed practitioners who believe there was no impact of customers' personality on RE explained the reasons for their opinions.",
"This included that they need to do their job and should not get affected by the behaviours of external parties: \"they are external parties, and we cannot get affected by their behaviours\" - INT05 [Software engineer & application consult].",
"\"it's nothing major, you know, related to clients, no major complex, I think it is a minor thing, and our job is to give what they want\" - INT06 [Technical team lead].",
"However, they consider their personality and team members' personalities to be important and these do impact RE activities compared to their customers' personalities."
],
[
"Mitigation strategies to reduce the negative impacts",
"Some practitioners shared the strategies that they use to mitigate the negative impact of personality in RE activities.",
"These mitigation strategies (MS) are used from the initial phase of the requirement engineering to the later phase of the projects and are mainly used within their teams.",
"Not all of the strategies may be useful, or even applicable, in different contexts and for different teams, but sharing them here collectively will help practitioners understand what might work for them in their contexts.",
"A summary of key mitigation strategies they reported are as follows: MS 1: Changing the task allocations: The majority of the interviewed practitioners (10 out of 12) mentioned changing task allocations of the team members as their initial strategy to mitigate the negative impact of personality on RE activities.",
"This can be done by resetting team responsibilities and assigning tasks that fit their personality.",
"For example, INT04, a business analyst, mentioned that sometimes resetting the whole team's responsibilities and bringing the team back to a different state is important to deal with the negative impacts of the personalities of the team.",
"INT09, a lead human-centred designer, mentioned that assigning tasks that fit team members' personalities would be beneficial in gathering requirements without conflicts: \"I think because a lot of my work tends to involve interviewing people to gather requirements, some of the more introverted team members are sometimes shy about talking to other people, especially since they're not confident in dealing with senior stakeholders and gathering their requirements.",
"And sometimes, they're not confident in speaking to users.",
"So in that degree, a more extrovert personality is good for those types of tasks\" - INT09.",
"Creating a working environment to reduce conflicts by applying different approaches to deal with differently opinionated people internally and externally also helps to mitigate negative impacts.",
"However, it was highlighted that changes to the task allocations should be done with proper consideration: \"We ended up giving her a different role...she had done something similar in her previous role.",
"So it was probably something where she felt a little bit more comfortable\" -INT12 [Senior project manager].",
"MS 2: Conducting detailed discussions: Some of the participants (INT04, INT07, INT09, INT12 & INT15) mentioned that conducting in-depth, cordial conversations with team members help reduce the negative impacts of personality on RE activities.",
"It helps identify the disconnections within the team, understand each team member's problems and bring everyone on the same page, resulting in solving team conflicts and making them complete the tasks within the scope: \"my initial strategy is, identifying that there was a disconnection between the team and trying to just sit everyone down around the table and go……as a team, moments where we don't appear to be all going in the same direction, we will have conversations often, and lead the team to complete what is within the scope\" - INT09 [Lead human-centred designer].",
"MS 3: Right utilisation of diverse personalities: Ten out of twelve participants also mentioned that the 'right' utilization of various personalities in the team as one of the strategies used by practitioners to reduce the negative impact of personality on RE activities.",
"They highlighted the importance of having a variety of personalities in their team.",
"If they do not, it can also have a negative impact: \"if everyone within the team is an extrovert, you just end up with people talking over each other, and it is not good for the project either\" - INT09 [Lead human-centred designer] MS 4: Disseminate responsibilities: A few senior practitioners (INT01, INT12 & INT15) shared that disseminating responsibilities and making the team members accountable for allocated tasks is another strategy practitioners use to mitigate the negative impacts of personality on RE activities.",
"By doing this, they try to give the team members a sense of accountability, so they tend to work together to complete the allocated tasks successfully.",
"It improves team collaboration and, eventually, team performance as well: \"When we have meetings where we agree on what to do, I take action items, hold people to account, give them the responsibility and then they will work together to do things correctly\" - INT12 [Senior project manager].",
"MS 5: Increase collaborations: Some participants (INT01, INT07, INT09, INT12, & INT15) highlighted that increasing collaboration is another strategy that can be used to mitigate the negative consequences of personality on RE.",
"Here, they tend to increase the collaboration among team members by inculcating mutual trust and assigning tasks together to find a solution: \"try and instil a higher level of mutual trust and collaboration.",
"That's always my first strategy, and most of the time that works\" - INT09 [Lead human-centred designer].",
"MS 6: Asking for management/higher-level support: Seeking management or higher-level support when necessary is another strategy practitioners use to mitigate the negative impact of personality on RE activities (INT01, INT02, INT04, INT08 & INT13).",
"Especially in situations such as dealing with anxious/ insecure team members or dealing with tough customers, seeking management or higher-level support would reduce the negative impact on RE activities and the overall project.",
"However, it was also highlighted that developers should be able to identify and ask for support at the correct time before it is too late to be resolved: \" I made an inquiry to my project manager and also my immediate supervisor about this concern, because that particular resource was very anxious and heavily thinking, that made the customers kind of guilty and started not to reveal the requirements actually\" - INT01 [Lead BA].",
"MS 7: Removing certain personalities in rare cases: A few participants (INT01, INT09 & INT11) mentioned that treating according to the situation is one of their strategies used to mitigate the negative impacts of personality on RE activities.",
"This can vary from handling issues smoothly within the team and taking the necessary breaks to remove a person from the team.",
"However, the ability to identify the correct situation is a must, and practitioners should take action accordingly: \"it had been a commercial situation, or would you just remove that person from the project, which is essentially what I did in that situation....",
"But my strategy is always to try and solve it in the first instance, where I try to solve it with the team smoothly.",
"If the situation is not under control, then I have to act accordingly, and it's very rare that it happens\" - INT09 [Lead human-centred designer].",
"Figure: Overview of impact of personality on RE/SE activities: The direction of the arrows indicates whether the impact is increasing () or decreasing () and the colour of each arrow indicates the respective personality trait that causes the impact"
],
[
"Discussion",
"In this section, we discuss the key findings of our study, related to personality test-based survey and in-depth interviews along with an overview of the impact of personality in RE activities based on the roles involved, such as software practitioners, their team members or customers/clients/ end users."
],
[
"Key findings",
"Firstly, by analyzing the 50 personality profiles and data from 15 interviews, we have identified that personality characteristics of the people involved in RE activities impacts the successful completion of RE activities.",
"From the personality profiles, we have identified that the majority of software practitioners involved in RE activities are statistically significant (high-scored) in agreeableness and conscientiousness characteristics.",
"This indicates that these practitioners have a strong interest in others' needs and well-being, are pleasant, sympathetic, cooperative, reliable, hardworking, set clear goals, and pursue them with determination.",
"Extraversion and neuroticism characteristics are found at an average level (average-scored), indicating that some enjoy time with others while others prefer to spend time alone.",
"This also means that stressful and frustrating situations are somewhat upsetting to them.",
"However, they can generally get over them and cope with the situations.",
"An equal percentage of practitioners are found in either high or average levels related to openness to experience characteristics indicating that 50% of them enjoy novelty, variety, and change as they are curious, imaginative, and creative, and the other 50% of the practitioners' thinking is neither simple nor complex.",
"They prefer tradition but are also willing to try new things.",
"Secondly, by analyzing our in-depth interviews, we have identified various personality characteristics related to software practitioners' personalities, their team members' personalities and customers' personalities.",
"The majority of these characteristics can be mapped onto the five personality dimensions and related facets in the five-factor model (IPIP NEO-120).",
"With that, we also identified how these personality characteristics are reported to impact RE activities and the overall software development process.",
"As shown in figure REF , we have summarized the key impacts of personality related to software practitioners', their team members and customers' personality characteristics.",
"These impacts can be positive or negative and are categorized into three areas; namely impacts related to requirements engineering (RE-related), impacts related to the overall software development process (overall SE-related) and impacts related to people involved in RE/SE (human-related).",
"Although extraversion characteristics – such as friendliness, outgoing and active nature – are important in building a good relationship with customers, it is not the most important characteristic for RE activities (section REF ).",
"It was only considered as one such plus point, and other characteristics such as cooperativeness, open mindedness, and hardworking nature are considered far more important (section REF ).",
"\"extraversion is not necessary; it's a plus trait; if you have it, it's good.",
"But, if you don't, then it's perfectly fine; you still can do good in requirements gathering\" - INT04 [Business analyst].",
"Similarly, it was identified that the majority of neuroticism characteristics such as anxiousness, insecure feeling, and anger negatively impact software practitioners and external stakeholders when involved in RE activities as well as the overall software development process.",
"However, it was also mentioned that the right utilization of neuroticism characteristics of software practitioners can be beneficial as they tend to be deep-thinkers who might help the design phase.",
"Apart from personality, we also identified several other factors that affect the identified positive or negative impacts.",
"The majority of our practitioners (11) mentioned individual's culture as an important factor as they consider culture having a strong connection with individuals' personalities.",
"The work experience of the practitioners, their project domain and empathy have been mentioned as other factors that impact when involved in RE activities Here, the work experience and project domain are interrelated as it was mentioned that more experience increases the exposure to areas/domains where individuals can handle various impacts of their previous experiences.",
"Empathy was usually referred to when dealing with external people such as customers/ clients/ end-users when involved in RE activities.",
"It was highlighted that empathizing with them makes the involvement in RE activities easier: (\"even to create the empathy because that has a huge factor I have seen, even myself in this learning curve, I practice being empathized with our clients, because they are dealing with other things, especially in the health sector and I think getting requirements are much more easier then\"- INT13 [Lead BA]).",
"Our study also identified several mitigation strategies that can be used to reduce the negative impact of personality on RE/SE activities.",
"Seven mitigation strategies were identified that could be applied to software teams, and the use of these strategies can vary with the project phase (section REF ).",
"For example, MS 4: disseminating responsibilities, can be used in the initial phases of the project to make the team members accountable for each task and reduce incomplete tasks at the end."
],
[
"Recommendations",
"Based on the findings from our study, we have identified several key challenges in the area of personality impacting RE, which need more focus.",
"We have framed these as a set of recommendations, shared below, for software practitioners involved in RE activities and the wider SE research community for further research into the personality impact of RE/SE.",
"1.",
"Need for a balanced mix of personalities within software teams: According to our study findings, there is no 100% positive or negative impact of personality characteristics on RE activities.",
"The majority of participants mentioned that having balanced personality characteristics in the team will help deliver the best output.",
"Having the right mix of personalities in a team would make the whole software development process successful, including RE activities.",
"Hence, it is important to pay more attention to having a diversity of personalities within the teams by software practitioners and team leads/ project managers, while the SE/RE researchers can conduct more research on identifying the best personality combinations for software development teams while identifying approaches to mitigate the negative impacts.",
"2.",
"Focusing on impact of personality along with culture: We identified that along with personality characteristics, most software practitioners mentioned that it is important to consider the culture of the people involved in RE activities as it impacts individual personality.",
"For example, it was mentioned that the culture of a particular country has a major impact on people's personalities.",
"Some prefer to work with guidelines, and others prefer to work without rules and guidelines.",
"These characteristics depend on their culture.",
"Hence, it is important to further investigate the relationship between personality and culture through multi-national research.",
"It would be interesting to study individuals from various cultures/ mixed cultures and analyze their implications in a global context.",
"3.",
"Apply and further investigate mitigation strategies: We also identified a set of mitigation strategies (MS) used by software practitioners to reduce the negative impact of personality when involved in RE activities.",
"We have identified seven mitigation strategies (section REF ) that are used during RE related activities, and these strategies are mainly for software team members.",
"Hence, we recommend carrying out more research studies identifying and investigating mitigation strategies that can be used for other software development activities and various contexts, for the benefit of industry practitioners.",
"4.",
"Focusing on the impact of personality on individual and team performance: According to our practitioners, one of the major impacts of personality is their individual and team performance.",
"They have mentioned that different personalities impact their overall work performance, and the impact can be positive or negative related to personality characteristics.",
"Hence, we suggest further investigating the impact of personality on individual and team performance, focusing on RE activities to identify the relationship between personality and performance when involved in RE activities.",
"5.",
"Consideration of the impact of other factors when involved in RE activities: While investigating the personality impact on RE activities, we identified a few other factors that could positively or negatively impact RE activities along with personality.",
"As mentioned in recommendation 2, culture is the most mentioned aspect.",
"Apart from that, practitioners have mentioned other human aspects such as empathy, geographic distribution, gender, communication skills, and other factors such as project domain and work experience.",
"A very limited number of practitioners pointed out the impact of these aspects (one or two practitioners), and some of the practitioners believe that there is no impact of gender and age on RE activities: \"I've worked with people of different genders and ages.",
"I don't believe that has a big impact\" -INT08 [Software engineer].",
"However, we recommend investigating these aspects in further empirical studies to get a more detailed understanding of the impact of these aspects on RE activities.",
"6.",
"Performing empirical case studies in different software organizations: Since our findings are mainly based on our in-depth interview study, we suggest conducting more empirical case studies, revealing personality characteristics of individuals in various software teams in various software organizations globally to understand better how industry practitioners and organizations handle these impacts and come up with better approaches to improve the positive impacts and mitigate the negative impacts."
],
[
"Limitations and Threats to validity",
"Threats to external validity: Our data collection does not possess an equal distribution of participants worldwide and does not represent the entire community of software practitioners involved in RE activities.",
"The majority of interview participants are from Sri Lanka, Australia and India.",
"The details of all the participants and their organizations have been kept confidential as per the human ethics guidelines followed in this study.",
"Hence, the findings of our study may be biased toward participants' organizations and their country of residence and limit generalizing to the entire global SE community.",
"However, in practice, such generalization is unlikely achievable [39].",
"Regarding participants' job roles, there were 14 job roles/titles, and the interpretations of these roles can differ based on the organization.",
"In the personality test-based survey, we used a set of key RE-related job responsibilities to rate the practitioners' involvement (\"Never\" to \"Always\") to see their involvement in RE activities.",
"The majority of the participants were always or very often involved in given key-related job responsibilities.",
"Interview participants described them in detail, indicating that we collected data from our target participants.",
"Since our main focus was to gather personality profiles using the survey, we included limited demographic questions with the standard personality test with all the instructions to complete the personality test.",
"In the interview study, we mainly focused on asking in-depth questions about participants' involvement in RE activities and their perspectives and experience on the impact of personality when involved in RE activities.",
"The participants have various experiences related to various domains where their involvement in RE may vary with the context.",
"Hence, we suggest conducting more empirical case studies focusing on different software organizations to investigate its impact on RE activities along with personality (section REF - recommendation 6) Threats to internal validity: Following the STGT for qualitative data analysis, we have generated concepts and categories based on the codes.",
"All these codes, concepts and categories were collaboratively discussed and finalized by all the authors to overcome any potential biases.",
"Using payment for data collection of the personality test-based survey can also be a threat to the internal validity of the research as there maybe fake responses when using participants recruitment platforms.",
"However, we have decided to use the Prolific tool after careful consideration following [38] and to avoid collecting fake/ incomplete responses, we only approved the payments for the participants after examining their responses to check whether they belong to our target participant group and if only they responded to each question (e.g selected the ones with proper open-ended responses).",
"We have provided all the standard instructions for the personality test and explained the personality-related terminologies during interviews with the participants.",
"However, participants' understanding of personality can vary and based on that the experiences they share during the interview may differ.",
"The impacts of personality that we have identified may depend on their understanding of personality and the mitigation strategies we have proposed to deal with the negative impacts of personality may depend on how they involve in such situations.",
"These may also depend on particular contexts that they involve in or their organizations.",
"Hence, by conducting more empirical case studies with software practitioners, more data can be collected related to organizations and contexts where software practitioners are involved in RE activities.",
"This can eventually improve our findings."
],
[
"Conclusion",
"The findings of our study contribute to identifying the impact of personality of software practitioners, their team members and their external stakeholders (customers/clients/end-users) on RE activities.",
"By analyzing personality profiles, we have identified that the majority of participants are high-scored in conscientiousness and agreeable nature, and average in extraversion and neuroticism nature.",
"The scores of the facets related to each personality dimension vary, and almost all the facets related to agreeableness and conscientiousness scored high among participants.",
"In the extraversion dimension, only cheerfulness and assertiveness are considerably high-scored, and all the other facets are average among participants, whereas, in neuroticism, almost all the facets have an average score among participants.",
"This shows that the majority of the practitioners are highly determined, hard-working, cooperative, sympathetic individuals who have a strong interest in others' well-being.",
"It also indicates that the majority of them are neither highly extravert nor introvert individuals and tends to be that they enjoy time with others, but also time alone.",
"An equal percentage of high and average-scored participants were identified related to openness to experience nature.",
"Regarding the openness to experience facets, the ideas/intellect, feelings, aesthetics and values/liberalism are high-scored facets compared to imagination and adventurousness facets indicating that there is a balanced mix of high and average personality characteristics related to openness to experience dimension among the practitioners.",
"Through in-depth interviews, we have identified various personality impacts on RE activities due to software practitioners' individual personalities, their team members or customers/ clients/end-users personalities.",
"These impacts are mainly found as positive and negative impacts and categorized as the impact on the RE activities (RE-related), impact on the overall software development (overall SE-related) or impact related to people involved in RE/SE (human-related).",
"From these impacts, it shows that the majority of personality characteristics such as conscientiousness, agreeableness, openness to experience and extraversion positively impact on these three aspects and neuroticism characteristics negatively impact on these aspects.",
"We have also identified a set of mitigation strategies that can be applied to reduce the negative impact of personality when involved in RE activities and the importance of focusing on other factors such as culture, work experience, project domain and empathy along with personality.",
"The findings of this study will be beneficial for understanding the impact of personality on RE activities.",
"Whether agile or traditional, it is important to have a balanced/ diverse software team to complete RE/SE-related activities successfully.",
"However, it would be interesting to observe various software teams in the industry to see how they handle these situations in real world scenarios to make the project success."
],
[
"Acknowledgments",
"Hidellaarachchi is supported by Monash Faculty of IT PhD scholarships.",
"Grundy is supported by ARC Laureate Fellowship FL190100035 and this work is also partially supported by ARC Discovery Project DP200100020.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION"
]
] | 2210.07807 |
[
[
"The XRISM Pipeline Software System: Connecting Continents, Processes,\n Testing, and Scientists"
],
[
"Abstract XRISM (X-Ray Imaging and Spectroscopy Mission), with the Resolve high-resolution spectrometer and the Xtend wide-field imager on-board, is designed to build on the successes of the abbreviated Hitomi mission to address outstanding astrophysical questions using high resolution X-ray spectroscopy.",
"In preparation for launch, the XRISM Science Data Center (SDC) is constructing and testing an integrated and automated system for data transfer and processing based upon the Hitomi framework, introducing improvements informed by previous experience and internal collaboration.",
"The XRISM pipeline ingests FITS files transferred from Japan that contain data converted from spacecraft telemetry, processes (calibrates and screens) the data, creates data products, and transfers data and metadata used to populate data archives in the U.S. and Japan.",
"Improvement and rigorous testing of the system are conducted from the single-task level through fully-integrated levels.",
"We provide an overview of the XRISM pipeline system, with a focus on the data processing, and how new and improved documentation and testing are creating accessible and effective software tools for future XRISM data."
],
[
"INTRODUCTION",
"The X-Ray Imaging and Spectroscopy Mission (XRISM) is the next X-ray observatory to be launched no earlier than Japanese fiscal year 2022.",
"This mission is a joint operation between the Japanese Aerospace Exploration Agency (JAXA), the National Aeronautical and Space Administration (NASA), and the European Space Agency (ESA).",
"The XRISM mission is a re-fly of a previous mission, Hitomi, which met an unfortunate and early end to its tenure.",
"Before Hitomi was lost however, unprecedented high-resolution X-ray observations were obtained during commissioning, notably those of the Perseus galaxy cluster, which fueled the motion to re-fly the mission and recapture the science.",
"The XRISM satellite will be flown with two (out of the original four) instruments, the Resolve spectrograph (micro-calorimeter) and Xtend wide field imager, which is a charge coupled device (CCD), both of which are sensitive in soft X-ray wavelengths; the high energy instruments from Hitomi were scrubbed for the XRISM mission.",
"As part of the larger XRISM mission team in the US, the XRISM Science Data Center (SDC) is responsible for the main processing pipeline, reprocessing tools, and quick-look analysis tools for processing the science and calibration data.",
"We work with both Resolve (US and Japan) and Xtend (Japan) instrument teams to obtain the latest calibration data to improve our existing tools and pipeline, as well as the calibration database (CalDB).",
"Although much of the software is reused from Hitomi, updates have resulted from instrument changes or new calibration information.",
"A more rigorous testing program has uncovered a few bugs.",
"Individual programs have been refactored to accommodate these changes, along with the routine adaptations required to shift to a new mission.",
"We have used lessons learned from Hitomi, refined our software engineering practices, and taken advantage of the additional time afforded by supporting a new mission.",
"The result is an efficient and well-documented pipeline for processing XRISM science and calibration data.",
"We discuss in this manuscript an overview of the full XRISM pipeline in §, the data sets used to test the pipeline and individual tasks in §, the improved framework the XRISM SDC has set up to test the individual tasks in §, the improved and new documentation of both code and user help in §, and finally the improvements we have made since Hitomi to make XRISM even more successful in §."
],
[
"The XRISM Pipeline",
"The XRISM pipeline is a daemon-driven collection of scripts that processes the data delivered from the pre-pipeline (PPL) from Japan and produces a data set, associated with an observation (henceforth called a sequence), that is placed in data archives and available to scientists.",
"The pre-pipeline [1] (see Eguchi et al., these proceedings) obtains telemetry data from the spacecraft and converts this data into first FITS files (FFFs), which are then transmitted to the pipeline (PL) in the US.",
"The XRISM SDC and PL are responsible for ingesting and processing the FFFs, and archiving and encrypting the processed data.",
"The XRISM pipeline processing center is located on the Goddard Space Flight Center (GSFC) campus of NASA in Greenbelt, MD, USA.",
"The pre-pipeline processing center is located at the Institute of Space and Aeronautical Science (ISAS) on the Sagamihara campus of JAXA in the Kanagawa prefecture in Japan.",
"The XRISM pipeline is executed on a series of Linux virtual machines (VMs), which includes a small network of pipeline machines: five processing machines, which can run the processing script (the core pipeline processing of data) in five separate instances, and the main processing machine, which runs all the other daemons not associated with the processing script, including fetching and archiving the data.",
"Parallelization is achieved by processing a single and distinct sequence on each of the five processing machines simultaneously.",
"The XRISM pipeline is a collection of Perl modules linked together and executed by a series of daemons.",
"The pipeline is triggered when the pre-fetch daemon, which is a cron job, notices that data is ready to be transmitted from the PPL.",
"The daemons are always running in the background, unless stopped by a pipeline operator, so once the pre-fetch cron job is invoked, the pipeline runs autonomously through the following daemons, described in more detail below: fetch, stream, post_proc, pre_archive, archive, and deleteit.",
"The high-level overview of the pipeline processing steps from the receiving of FFF data from the PPL to the distribution of processed data to the archives is as follows: pre-fetch daemon (cron job): XRISM version of the data transfer system (xDTS) is executed to transfer data from the ISAS PPL to the GSFC PL pre-fetch daemon: data archive system (DAS) invoked to copy the FFFs to the deep archive fetch daemon: data is verified in the pre-processing (pre-proc) area, and prepared for processing through the pipeline script stream daemon: data is processed through the pipeline script in the processing (proc) area post_proc daemon: validate processed data and check for errors from the pipeline processing script in the post-processing (post-proc) area pre_archive daemon: build sub-directories for archive, compress and encrypt necessary data archive daemon: move processed data to local archive (deep archive) and remote archive hosted at GSFC archive daemon: xDTS is executed again to move processed and encrypted data from the PL back to the ISAS archive archive daemon: once archived, data can be provided to scientists based on PI (proprietary), or public availability deleteit daemon: cleans up post-processing area DTS and DAS are inherited from previous high-energy missions.",
"xDTS is the XRISM-adapted version of DTS, which is used in the current XRISM pipeline instead of the canonical DTS, to account for updates and changes in communication channels between ISAS and GSFC.",
"The software tasks used in the main part of the processing pipeline (by means of the stream daemon) are part of a larger suite of software tools used for general FITS file manipulation, known as FTOOLS [2], [3].",
"The FTOOLS software package is part of the broader High Energy Astrophysics Software (HEASoft [4], [3]) suite, which is hosted and maintained by the High Energy Astrophysics Science Archive Research Center (HEASARC [5]) at NASA/GSFC.",
"The HEASARC is also where archived XRISM data is hosted in the US.",
"HEASoft is a collection of FTOOLS and high-energy mission-specific software tasks including software for both legacy and current missions, such as Suzaku, NICER, Swift, and IXPE.",
"The software tasks used in the processing pipeline are both the general FTOOLS and XRISM mission-specific tasks.",
"The XRISM mission-specific tasks will be incorporated into the HEASoft suite near or soon after launch.",
"These mission-specific tasks are used for processing and re-processing (by a user) and include the following key functionalities, among others: converting pulse height amplitude (PHA) to pulse invariant (PI), creating response matrix functions (RMFs), secondary corrections, modulated X-ray source (MXS) operations, and event flagging.",
"For more specifics on some of the SDC software tasks, see the SDC SPIE 2020 paper [6].",
"These tasks use information from auxiliary and mission-dependent files, calibration files, and housekeeping files to perform their prescribed function.",
"The XRISM SDC is responsible for 62 tasks, 15 under the general HEASoft umbrella and 47 under the XRISM-specific tasks.",
"Under the XRISM-specific tasks, we maintain 16 Resolve tasks, 5 Xtend tasks, 10 general tasks, and 16 multi-instrument tasks.",
"Additionally, the XRISM SDC is responsible for maintaining 16 libraries, 4 in the general HEASoft area and 12 in the XRISM-specific area.",
"These libraries are used by multiple tasks throughout the XRISM software.",
"Figure: Data flow and processing steps applied by the SDC starting with the acquisition of FFF, and concluding with data delivery to scientific investigators.",
"Intermediate steps where the data is copied for backup storage are also indicated, but not discussed further.We discuss here the more detailed flow of pipeline processes, which can also be seen in the flow chart in Fig.",
"REF .",
"The pipeline is always “on” and waiting, as it checks periodically for data transmissions.",
"The pre-fetch daemon, which is the start, or trigger, of the pipeline, is a cron job that waits for input data.",
"The pipeline is triggered by a data delivery from the PPL, which is checked for by the pre-fetch daemon.",
"Once there is data ready to be processed, data is transferred from JAXA/ISAS, after processing through the PPL, via xDTS to GSFC.",
"DAS is then invoked and notifies GSFC that there is data ready for transmission to and processing through the pipeline, while simultaneously copying the data to the local deep archive at GSFC, where a copy of the PPL output data is stored.",
"Additionally, in the pre-fetch stage, the databases associated with the observation are also transferred from Japan via xDTS.",
"The fetch daemon then validates and copies the data to the pre-proc area.",
"The fetch daemon also validates and copies the databases to the ingest archive and installs them in the working directory.",
"Next, the stream daemon is activated.",
"The stream daemon first copies the data from the pre-proc area to the proc area.",
"The stream daemon executes the pipeline processing script in the proc area, which is the script that executes the series of FTOOLS and XRISM-specific tasks that perform the calibration on, and process the data.",
"The stream daemon processes a single sequence, associated with an observation.",
"However, the stream daemon can be run on separate machines to process different sequences simultaneously.",
"After the sequence has been processed by the pipeline script, the stream daemon checks for severe errors, copies the output to the post-proc area (if no severe errors have occurred), and cleans up the processing area.",
"The post_proc daemon then validates the processed data in the post-proc area and updates the databases accordingly.",
"The pre_archive daemon then creates the sub-directory structure for archiving the data, compresses and encrypts data as configured, and updates the catalog file.",
"The archive daemon then copies the compressed data into the local archive and invokes DTS to initiate the transfer of data to the JAXA/ISAS Data ARchives and Transmission System (DARTS [7]) and HEASARC at NASA/GSFC.",
"Finally, the deleteit daemon deletes the data from the post-proc area and cleans up any other extraneous files.",
"The pipeline processing script not only produces processed data, but also preview products for each XRISM instrument.",
"The SDC has improved and expanded the generations of preview products since Hitomi.",
"These preview products include spectra, light curves, and images, for both Resolve and Xtend instruments, plotted at preset wavelength ranges.",
"The purpose of these preview products is for scientists to be able to quickly browse an observation to determine if any features of great interest exist.",
"The pipeline products will be available as thumbnails when browsing XRISM data in the HEASARC archive.",
"Fig.",
"REF displays an example of an image, spectrum, and light curve for XRISM/Resolve, while Fig.",
"REF displays the same for XRISM/Xtend.",
"Figure: XRISM/Resolve pipeline products from the the Hitomi observation 100050020, converted to be compatible with XRISM software, see § for more details.Figure: XRISM/Xtend pipeline products from the the Hitomi observation 100050020, converted to be compatible with XRISM software, see § for more details."
],
[
"Coordination with JAXA/ISAS",
"There has been a considerable pre-launch, bi-continental, and collaborative effort to test the communication of the PL with the PPL.",
"This effort has involved producing a working PL, a working PPL, and connecting the two with email communication, xDTS, DAS, and team member communications.",
"This effort has involved the cooperation of both GSFC pipeline operators and ISAS PPL operators.",
"The process of testing the data flow from Japan to the US and back to Japan, including PPL and PL processing, is a procedure we've termed an “end-to-end test”.",
"First at ISAS, the PPL is executed and FFFs are produced for a test sequence.",
"Then there is a test of communications, which notifies GSFC of data ready to be transmitted from ISAS.",
"The data readiness automatically triggers the PL to start at GSFC.",
"The sequence is then processed through the PL and the data is transferred back to the archive at ISAS.",
"Not all functionality of the end-to-end test is fully functional at this point pre-launch, but there is enough functionality to transfer data and validate both PPL and PL output, which both teams have conducted with each end-to-end test.",
"Validation efforts are completed with each SDC internal build release of the pre-launch XRISM-specific software tasks and with each end-to-end test.",
"We have produced test data sets, discussed in §, with which we can (a) validate the output from the PPL and the interface between the PPL and PL, (b) validate the output from the PL, and (c) ensure the communication and transfer of data is working properly.",
"We have recently been able to test the archiving function of the pipeline, compressing and encrypting necessary data, and distributing data into sub-directories in the archives.",
"We are beginning to conduct the interfacing with the archives themselves, both DARTS and HEASARC.",
"The next steps are to test database creation and functionality and full archive testing."
],
[
"Mini Mission Data",
"The mini mission data are a set of sequences used for testing the PL, user reprocessing scripts, and individual XRISM-specific software tasks.",
"The main overall purpose of the mini mission data is: (1) for internal (SDC only) testing of software tasks and pipeline, (2) to set up an interface between the PPL team and the SDC for what the PL is expecting the FFFs to contain and for correct formatting, and (3) provide test data for the broader XRISM team to use for testing of software, learning how to use the XRISM software, and for comparison to ground system processing.",
"The mini mission data consists of four types of data sets: (1) archival Hitomi data converted to XRISM software-compatible data, (2) pre-pipeline output from ISAS, (3) Resolve instrument ground testing and calibration data, and (4) Xtend instrument ground testing and calibration data.",
"Each type of data set is generally used for a different testing purpose.",
"The archival Hitomi data converted to XRISM software-compatible data was the easiest and most accessible, since all Hitomi data is publicly available on HEASARC.",
"Since the Hitomi/SXS instrument is nearly identical to XRISM/Resolve and Hitomi/SXI is nearly identical to XRISM/Xtend, the conversion from Hitomi to XRISM data was straightforward.",
"Most of the conversion involved shifting times and dates five years ahead to account for the new mission epoch (2014 for Hitomi changed to 2019 for XRISM) and updating instrument-specific keywords and file names to match those of XRISM instead of Hitomi.",
"We used five sequences from the archival Hitomi data, both the pipeline-processed data that was in the public archive, and the FFFs from the PPL, which were in the SDC internal archive (obtained from the deep archive).",
"This provided test data to use both on the PL and on XRISM-specific tasks and user pipeline reprocessing scripts.",
"These data were also updated in accordance with the XRISM Science FITS ground system interface control document.",
"By creating this data set, we were able to use this as an interface to the PPL team to provide them with the formatting and contents of what the PL is expecting.",
"As previously discussed, the XRISM SDC has been conducting end-to-end tests with ISAS.",
"The ISAS and GSFC teams have been working together and increasing the complexity of data transferred over the past couple of years.",
"The first tests used Hitomi data as a simple test of the transfer of data and allowed us to perfect the interfacing of the PL and PPL systems.",
"We then progressed to an early version of the Hitomi-converted to XRISM software-compatible mini mission data, which we artificially placed on PPL outgoing servers, transferred to the PL servers, and processed the data through the PL.",
"We are currently using data output from the current pre-launch state of the PPL, transferred to the PL and processed through the PL, then sent back to ISAS.",
"The SDC has been validating the output PPL data and checking for compliance with the XRISM Science FITS and archive interface documents, and the interfacing and communication of the PPL with the PL.",
"The third set of mini mission data is the Resolve ground testing and calibration data.",
"This is data we have received from the Resolve instrument team, which has been used for ground calibration and testing.",
"The Resolve ground data is best suited for testing our user pipeline reprocessing scripts; however it is necessary to add in supplementary files and make minor modifications to the existing files in order to create a data set that can be run through the reprocessing scripts.",
"Since this data is taken on the ground, it does not have corresponding spacecraft auxiliary files such as attitude and orbit, so these either have to be artificially created, or use Hitomi-converted data, with times updated to match the ground data timing.",
"The data provided to the SDC by the Resolve instrument team is also a good check of updates, additions, and changes of keywords and such that have been discussed and agreed upon by the two teams.",
"This data aids in communication and interfacing between the Resolve instrument team and the SDC.",
"The fourth set of mini mission data is the Xtend ground calibration and testing data.",
"Similar to the Resolve ground data, this data was provided to the SDC by the Xtend instrument team following ground calibration and testing.",
"There are also similar caveats to the Xtend ground data including having to supplement the data with converted auxiliary files.",
"Similar to the Resolve ground data, we have tried to keep the Xtend data as intact as possible, updating times of converted auxiliary files to match those of the ground data.",
"The Xtend ground data is used to interface with the Xtend team and to aid in the testing and validation of Xtend-specific software tasks.",
"The conversion of the Hitomi archival data sets into XRISM software-compatible data was achieved by using a number of FTOOLS and FITS manipulation functions linked together in Python scripts.",
"The PPL and ground data conversions were conducted in a similar manner as the archival data, but fewer modifications were necessary.",
"We used tools to add columns, update header keyword values and comments, add in new keywords, delete keywords, update timing keywords and columns, and so forth.",
"Internal documentation has been produced to record the changes performed on the original data and input data, output data, and the scripts produced have been made available to the team."
],
[
"Unit Test Framework",
"We have been working to greatly improve the unit test framework that was first set up by Hitomi.",
"The automated unit test execution is performed by a HEASoft task called aht.",
"The aht task can create a unit test, and execute a unit test and report a pass or fail.",
"We have written a wrapper script that can run all or a specified set of tests and produce a succinct output of the outcome of all tests that were executed.",
"The major issue for our team with the aht framework is the multiple copies of data aht creates when creating and validating unit tests.",
"Some of the XRISM data is quite large in size and the tests use shared files from the CalDB that are repeated in several tests.",
"Additionally, some tests have large input/output files that have multiple copies stored for a single test.",
"These large files caused our git repository to become bloated with extra data.",
"We are currently investigating an addition to the aht framework that would allow for shared directories of read-only input files, instead of each unit test keeping a separate directory of input files.",
"Originally, all unit tests for all tasks under the XRISM SDC purview, with the exception of information only tasks, were stored in a single, local (not GitHub) githttps://git-scm.com/ repository.",
"This repository contained sub-directories generally dividing groups of tests based on some similarity, i.e., instrument-specific tasks, as can be seen in Fig.",
"REF .",
"Some of our unit tests contain large input/output data sets and therefore bloat the git repository.",
"We decided to separate the unit tests into multiple git repositories, to keep the size down and make the repositories easier for transfer to developer machines and for updating and running the full set of unit tests.",
"The new structure of our git repositories for all XRISM and Hitomi unit tests is displayed in Fig.",
"REF .",
"Figure: Original unit test repository setup, where all unit tests were contained in a single git repository.Figure: New unit test multiple git repository setup.",
"The tasks that have their own repositories have large data sets and therefore needed to be separated from the main XRISM repository.Additionally, we decided to set up a new framework of unit test sets.",
"We created a set of limited functional tests (LFT) and a set of comprehensive functional tests (CFT).",
"The CFT is the full set of unit tests for all 62 tasks and includes the full version of all of the git repositories.",
"The CFT is only for internal use, mostly for preparing for build releases of our software and regression testing.",
"The CFT includes very long-running and large data set tests, which are more difficult to provide to a general user.",
"The LFT will be deployed with our XRISM software to aid in user support.",
"The LFT is a light-weight subset of the CFT, which still contains tests for all 62 tasks, but only select tests for each task.",
"A general comparison of the CFT and LFT are shown in Fig.",
"REF .",
"Figure: Quick comparison of CFT vs. LFT to highlight the major differences.The original unit tests were obtained from the Hitomi mission.",
"Most of the mission-specific and instrument-specific unit tests were copied and converted to XRISM unit tests.",
"For example, unit tests from the Hitomi/SXS sxspha2pi task were copied and recreated for the corresponding XRISM/Resolve rslpha2pi task.",
"Similarly for general Hitomi ah tasks like ahmkehk, the unit tests were copied and converted to unit tests for the general XRISM xamkehk task.",
"The Hitomi unit tests have been retained for regression testing, particularly after a XRISM task has been updated to deviate from the original Hitomi task.",
"Nearly all of the XRISM unit tests have recently been reviewed, validated, updated, and in some cases replaced; the validation process is ongoing.",
"New unit tests have been added with formal development of XRISM tasks, or introduction of new tasks.",
"Each task is currently undergoing, or has undergone, a careful review to ensure that the set of unit tests for the particular task covers all functionality as laid out in the requirements of the help file (see §REF for more on help files)."
],
[
"Documentation",
"Documentation is extremely important in software development, both for the developers, and for the users.",
"We strive to provide documentation with ease of readability and understanding, coherence, and consistency for developers, XRISM team members, and general users.",
"In this section, we describe the three types of documentation the XRISM SDC has employed to try to achieve the aforementioned goal.",
"We discuss user documentation of our software tools (§REF ), documentation of the code itself (§REF ), and internal documentation for interfacing and recording necessary information for team use (§REF )."
],
[
"User Documentation: Help Files",
"Each task within the XRISM software suite has a corresponding help file.",
"Help files are a construct of all HEASoft tasks, which provide a general description of the task, general usage of the task, examples of how to run the task, and lays out all the input/output parameters of the task.",
"Help files can be accessed on the HEASoft website, as well as on the command line (after the HEASoft environment has been initialized), using the command fhelp taskname, where taskname is the name of the task.",
"The parameters are described in detail in the help files, with options for parameters that have a limited number of options, and provides functionality for when a parameter is set to a specific value.",
"The help file also displays the parameters that are required for execution of the task, and those that are optional.",
"The XRISM SDC conducted a thorough review of all the help files of the tasks under its purview, and has updated the files for understanding and accuracy.",
"The task descriptions were carefully reviewed, and in some cases rewritten, for ease of readability and understanding.",
"The examples were updated with XRISM data input/output file names, to allow for cohesiveness of the XRISM software and example data.",
"We reviewed the help files for consistency in formatting and coherence in content as well.",
"Since the help files essentially lay out the functionality of the task, we used these as a guide for the results that a particular task should achieve, given particular input parameters and/or input data.",
"This provided us with a baseline for how to test each task with unit tests.",
"As discussed previously in §, the help files were used as a guide for creating unit tests for each task, to ensure that essential functionality was properly tested, and that the task performed as expected, according to the prescribed functionality."
],
[
"Code Documentation with Doxygen",
"The previous Hitomi code included Doxygen [8] markup, but was not consistent enough or fully implemented throughout all tasks.",
"For XRISM, we have implemented consistent Doxygen markup in all code files for each task; this includes *.c, *.cxx, *.h, *.pm, *.pl, and *.dox files.",
"We performed an overhaul of the Doxygen documentation within the code modules to create consistent and comprehensive documentation that would be presented in a coherent manner.",
"The header at the beginning of each code file consists of the following documentation keys: file name, brief description, author, date of last code update, version number, definition of code groups, longer description, source files, subroutines (for Perl only), tool dependencies, library dependencies, author history, and modification history.",
"Doxygen markup is also used throughout the code to document classes, functions, structures, variables, etc.",
"The C and C$^{++}$ code is automatically assembled by Doxygen, but the Perl integration is still a work in progress, so we have created a general workaround by duplicating the Doxygen header information from any Perl modules or scripts into a corresponding .dox file.",
"The .dox files are picked up by Doxygen in a similar manner to the C/C$^{++}$ code files, however the full functionality is not equivalent for the Perl scripts.",
"We use the HTML option for output from executing Doxygen on the XRISM code files.",
"We packaged the HTML bundle and it has been released with pre-launch software builds.",
"The HTML is local to the user's machine, so it does not need to be hosted and can remain internal, at least for now, and tied to a specific software release.",
"The Doxygen HTML layout follows the directory structure of tasks within HEASoft, therefore making it easier for the user to follow.",
"The documentation is fully clickable through functions and the code itself.",
"The Doxygen output provides a way for users who want to delve a bit deeper into the code or the algorithms, to be able to easily navigate the code without having to slog through the full source code.",
"Fig.",
"REF displays an example of the main pages of the Doxygen HTML output as well as some screenshots of what a specific task would look like for a user.",
"Figure: Here we show an example of some of the components of a task in the Doxygen HTML output."
],
[
"Internal Documentation",
"We use an internal document tracking and storage system to post, review, and update major documents, which can be accessed by the entire NASA/GSFC-based XRISM team.",
"The XRISM SDC has produced a collection of documents, including reference documents from Hitomi, and new and updated documents for XRISM referring to the pipeline, its processes, products, and interfacing between JAXA/ISAS and NASA/GSFC.",
"We also produce engineering change request (ECR) documents, which can either describe a completely new tool, or modifications to an existing tool.",
"In the latter case, the ECR provides details of the issue, background, suggested code changes to solve the issue, and proposed tests for an update or modification to the code.",
"These documents are written by one or more of the SDC scientists and are implemented by one or more SDC developers, depending on the magnitude of code changes.",
"The scientists and developers work together through implementation of the ECR modifications to the code and testing.",
"A reconciliation document is then produced by a developer or developers, whereby the changes line-by-line in the code are tracked, any comments about implementation are recorded, and tests are recorded with expected outcomes and actual outcomes.",
"We also produce SDC-only internal documentation such as documenting the mini mission data creation, unit test creation, pipeline validation, pipeline processing script processing steps, and more.",
"Many of these are on shared websites for easy access to collaborative editing, with hard copies generally on our shared server.",
"We produce these detailed documents in order to maintain our processes, enable reproducible results, document communication and decisions made, and to have a “paper trail” of software modifications."
],
[
"Improvements Since Hitomi",
"As noted previously, the XRISM SDC acquired the legacy Hitomi software.",
"Over the past 4+ years, we have created a suite of software tasks that will be ready for the XRISM launch.",
"The suite is currently composed of what we call the dual build, which is to say that there exists some code that has not changed from Hitomi and is therefore common between Hitomi and XRISM, but also Hitomi and XRISM tasks that have now deviated from one another and are wholly separate.",
"For tasks with this “common code”, the Hitomi and XRISM counterparts contain the exact same code and behave exactly the same, with only essential mission-specific keywords and parameters diverging.",
"We still retain some common code, but many of the tasks have now been split.",
"Bugs found in the Hitomi code will eventually be corrected but currently, only the XRISM versions have been corrected, as a priority.",
"The XRISM SDC has made modifications to 12 Resolve tasks, 3 Xtend tasks, 11 mission-specific tasks, and 3 other tasks, a total of 29 out of 62 total tasks under the XRISM SDC purview.",
"We have also deleted some tasks, which are no longer relevant to the XRISM mission.",
"The Resolve and Xtend task updates have been informed by the respective instrument teams, whether there has been updates to the instrumentation itself, or the analysis of the calibration data.",
"We work closely with the Resolve instrument teams, both in the US and in Japan, as well as the Xtend instrument team in Japan.",
"We take input from scientists, the mission operations team, and the science operations team.",
"We have forged strong ties with these teams in order to create a communicative environment for updating software, validating data, and testing the pipeline.",
"We have also performed instrument-driven updates to for CalDB files.",
"Major tool improvements already completed include the following: Run time and functionality improvements for our raytracing program, xrtraytrace, which traces X-ray photons through a telescope A first pass at run time optimization (more planned) and introduction of new components and functionality for the Resolve RMF generator Near completion of the X-ray Spectral Line IDentifier and Explorer (XSLIDE) tool, which is a quick-look spectral analysis tool for XRISM data [9] (see Braun et al., these proceedings) Improvements to the user pipeline scripts, xapipeline, which reprocesses both Resolve and Xtend data, rslpipeline, which reprocesses only Resolve data, and xtdpipeline, which reprocesses only Xtend data Significant improvements and algorithm changes have also been performed for flagging pixels in both Resolve (rslflagpix) and Xtend (xtdflagpix), pulse invariant calculations for both Resolve (rslpha2pi) and Xtend (xtdpi) We have also performed some bug fixes, which have been found either during formal development, reported by a team member, or discovered in our intensive testing process.",
"Several minor modifications have also been performed for many of the XRISM tasks."
],
[
"Future Improvements",
"The XRISM SDC has plans for future improvements to tasks and new tools, mostly for supporting post-pipeline data analysis and creating simulated data.",
"We plan to optimize the runtime in order to create a more efficient RMF generator.",
"We will continue to make CalDB updates and improvements with better calibration and data from the Resolve and Xtend instrument teams.",
"CalDB updates will continue after launch, when there are likely to be minor tweaks due to instrument performance that is different in-flight, compared to what it was during ground testing.",
"We have started development of a raytracing driver tool, which will allow more flexibility and functionality for simulations that trace X-ray photons through an X-ray telescope and either of the detectors.",
"We intend to create a point spread function (PSF) library for Resolve, which would contain pre-computed PSFs for use by scientists, and would alleviate the burden of having to execute extremely time-consuming tasks for generating the effective area and/or PSF for arbitrary satellite pointing parameters, and detector selection regions.",
"We are also considering an RMF library, which would contain pre-computed RMFs for users, again so a user would not have to run extremely time-consuming tasks to create large RMFs.",
"We also have plans to make improvements to the efficiency and functionality of the auxiliary response function (ARF) generator [10].",
"A new tool will also be developed that will account for the effect of intervening galactic dust halos on the effective area and PSF.",
"Many of these planned improvements will be slated for after launch, to ensure the core software and the pipeline are ready for data processing."
]
] | 2210.07840 |
[
[
"Counting Statistics of Single Electron Transport in Bilayer Graphene\n Quantum Dots"
],
[
"Abstract We measure telegraph noise of current fluctuations in an electrostatically defined quantum dot in bilayer graphene by real-time detection of single electron tunneling with a capacitively coupled neighboring quantum dot.",
"Suppression of the second and third cumulant (related to shot noise) in a tunable graphene quantum dot is demonstrated experimentally.",
"With this method we demonstrate the ability to measure very low current and noise levels.",
"Furthermore, we use this method to investigate the first spin excited state, an essential prerequisite to measure spin relaxation."
],
[
"Acknowledgement",
"We are grateful for the technical support by Peter Märki, Thomas Bähler.",
"We acknowledge financial support by the European Graphene Flagship, the ERC Synergy Grant Quantropy, the European Union’s Horizon 2020 research and innovation programme under grant agreement number 862660/QUANTUM E LEAPS and NCCR QSIT (Swiss National Science Foundation, grant number 51NF40-185902) and under the Marie Sklodowska-Curie Grant Agreement Number 766025.",
"K.W.",
"and T.T.",
"acknowledge support from the Element Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI Grant Number JP20H00354 and the CREST(JPMJCR15F3), JST."
]
] | 2210.07759 |
[
[
"ISTA-Inspired Network for Image Super-Resolution"
],
[
"Abstract Deep learning for image super-resolution (SR) has been investigated by numerous researchers in recent years.",
"Most of the works concentrate on effective block designs and improve the network representation but lack interpretation.",
"There are also iterative optimization-inspired networks for image SR, which take the solution step as a whole without giving an explicit optimization step.",
"This paper proposes an unfolding iterative shrinkage thresholding algorithm (ISTA) inspired network for interpretable image SR.",
"Specifically, we analyze the problem of image SR and propose a solution based on the ISTA method.",
"Inspired by the mathematical analysis, the ISTA block is developed to conduct the optimization in an end-to-end manner.",
"To make the exploration more effective, a multi-scale exploitation block and multi-scale attention mechanism are devised to build the ISTA block.",
"Experimental results show the proposed ISTA-inspired restoration network (ISTAR) achieves competitive or better performances than other optimization-inspired works with fewer parameters and lower computation complexity."
],
[
"Introduction",
"Image super-resolution (SR), as one of the traditional image restoration tasks, has been widely investigated by researchers [1], [2].",
"Given a low-resolution (LR) image, the task of image SR is to restore a corresponding high-resolution (HR) instance with more details.",
"There are numerous applications considering the image SR, such as video deinterlacing [3] and compression [4], remote sensing [5], [6], [7], EGG analysis [8], and spatiospectral analysis [9].",
"Deep learning has demonstrated its amazing performance in image restoration.",
"There are numerous convolutional neural networks (CNNs) specially designed for image SR. SRCNN [10] is the first CNN-based method for image SR. After that, deeper and wider networks show their effectiveness with better performance, such as VDSR [11], EDSR [12], RDN [13] and RCAN [14].",
"Recent image SR networks usually develop effective blocks for improving the network representation.",
"IMDN [15] and EFDN [16] utilize information distillation mechanisms to build an efficient network for fast and accurate image SR. Cross-SRN [17] builds an edge-preserving network with cross convolution.",
"However, these works concentrate on the block designs but lack interpretation, which limits the performance.",
"Since image SR can be regarded as a classical optimization task [18], there are also works considering building the image SR network from the optimization perspective.",
"IRCNN [18] provides an iterative solution for the general image restoration task and designs a CNN-based network to solve the denoising prior.",
"ISRN [19] develops an iterative network with the help of the half-quadratic splitting (HQS) strategy.",
"DPSR [20] and USRNet [21] also achieve good performance on image SR inspired by the HQS strategy.",
"There are also works building the network by alternating direction method of multipliers (ADMM).",
"Plug-and-Play ADMM [22] regards the denoiser as a network prior for different image restoration tasks.",
"ADMMNet [23] provides an end-to-end network for the compression sensing task.",
"PSRI-Net [24] considers ADMM for SAR image SR.",
"Although these works provide an interpretable network design, the CNN architectures just task the solution step as a whole, without giving an explicit optimization step on how to solve the denoising problem.",
"Figure: An example comparison among different image super-resolution methods.In this paper, we develop an unfolding network based on the iterative shrinkage thresholding algorithm (ISTA).",
"Different from designing the CNN to directly solve the optimization step, ISTA blocks are specially designed to conduct the image restoration following the ISTA steps.",
"In the ISTA block, CNNs are utilized to adaptively learn the functions in the feature space and speed up the optimization steps.",
"To improve the network representation, multi-scale exploration (MSE) and multi-scale attention (MSA) mechanisms are utilized to build the ISTA block.",
"An ISTA-inspired restoration network (ISTAR) is developed based on the ISTA block for effective image SR.",
"Experimental results show the proposed ISTAR can achieve competitive or better performance than other works.",
"Compared with other optimization-inspired methods, ISTAR achieves better performances with much fewer parameters and lower computation complexity.",
"Figure REF shows an example comparison among different image super-resolution methods.",
"Compared with state-of-the-art methods, our proposed ISTAR can generate more satisfying textures that close to the HR image.",
"The contributions of this paper can be concluded as follows: We analyze the image super-resolution task from the optimization perspective and develop an ISTA block for image super-resolution.",
"We develop the multi-scale exploration and multi-scale attention mechanism in the ISTA block, which improves the network representation and boosts the performance.",
"Experimental results show the proposed network achieves competitive or better performance than other optimization-based works with much fewer parameters and lower computation complexity.",
"Deep learning has demonstrated its amazing performance on various computer vision tasks.",
"There are numerous convolutional neural networks (CNNs) specially designed for image super-resolution (SR).",
"SRCNN [10] is the first CNN-based image SR method composed of three convolution layers, which follows a sparse-coding manner.",
"After SRCNN, deeper and wider networks has proposed to improve the restoration performance.",
"FSRCNN [25] increases the network depth and decreases the input resolution, which makes the method faster and more effective.",
"VDSR [11] develops a very deep network with residual connection to restore the high-resolution (HR) images.",
"EDSR [12] then utilizes the residual blocks in the network and improves the network capacity.",
"ESPCN [26] provides a different upsampling strategy to restore the HR images, which is more effective than the deconvolution operation.",
"Recently, researchers concentrate more on effective block design for better restoration performance.",
"RDN [13] combines the residual connection [27] and densely connection [28], and develops a residual dense block for image SR. After that, the researchers introduce the residual-in-residual design with channel attention [29] for image SR and build an effective network termed RCAN [14].",
"RFANet [30] expands the residual connection and aggregates the residual features for better information transmission.",
"IMDN [15] and RFDN [16] build the lightweight networks with the help of an information distillation mechanism.",
"SHSR [31] and MSRN [32] utilize hierarchical exploration to further investigate the image features.",
"These works usually concentrate on the effective block designs but neglect to analyze the image SR from the optimization perspective."
],
[
"Optimization-Inspired Image Super-Resolution",
"There are also optimization-inspired networks for interpretable image SR. ADMM-Net [23] provides a good example of dealing with the image restoration problem by the optimization strategy and develops a CNN-based denoiser for plug-and-play restoration.",
"IRCNN [18] then analyzes the image restoration with the help of the half-quadratic splitting (HQS) strategy and recovers the image with a CNN-based denoiser prior.",
"After IRCNN, there are numerous HQS-based methods for effective image SR. DPSR [20] proposes a different observation model for image SR and uses kernel estimation and CNN denoiser for plug-and-play image SR. USRNet [21] develops an end-to-end network for different image SR tasks.",
"ISRN [19] devises an effective network for image SR under the guidance of HQS and maximum likelihood estimation (MLE).",
"HSRNet [33] also investigates the HQS strategy and develops a network for aliasing suppression image SR.",
"However, these works just take the solution as a whole and calculate it directly by CNNs, without giving an explicit optimization step for each iteration.",
"Figure: Network design of ISTAR"
],
[
"Methodology",
"In this section, we first analyze the image super-resolution (SR) from the optimization perspective and propose an iterative solution with the help of ISTA.",
"Then, we introduce the designed end-to-end network ISTAR.",
"After that, we discuss the design of the ISTA block.",
"Finally, the network settings are described in detail."
],
[
"ISTA for Image Super-Resolution",
"Given an low-resolution (LR) image $I^{LR}$ , the task of image SR is to find a corresponding image $I^{SR}$ , satisfying $I^{SR} = \\arg \\min _{I^{LR}}||DI^{SR}-I^{LR}||_\\ell ^2+\\lambda ||I^{LR}||_1,$ where $D$ is the down-sampling matrix, and $\\lambda $ is a weighting factor.",
"The prior term $\\lambda ||I^{LR}||_1$ is utilized to introduce the sparsity of the natural image.",
"To solve this function, we use ISTA to convert it into an iterative manner.",
"Then, the solution is $I^{SR}_{k+1}=\\mathcal {T}_{\\lambda \\alpha _k}(I^{SR}_k-\\alpha _k D^T(DI^{SR}_k-I^{LR})),$ where $\\alpha _k$ is the weighting factor for the $k$ -th iteration and $\\mathcal {T}(\\cdot )$ is the soft-thresholding operation.",
"It can be found that the right hand side of Equation REF has two independent variables $I^{SR}_k$ and $I^{LR}$ .",
"To make it clear for understanding, we re-write Equation REF as $I^{SR}_{k+1}=\\mathcal {T}_{\\lambda \\alpha _k}((E-\\alpha _k D^TD)I^{SR}_k-\\alpha _k D^TI^{LR}),$ where $E$ is the identity matrix.",
"In Equation REF , we can find that $D^TI^{LR}$ is shared for every iteration.",
"In this point of view, we can calculate this term before the ISTA optimization, and regard it as an invariant to speed up the optimization."
],
[
"Network Design",
"Figure REF shows the entire network design of our ISTAR.",
"Firstly, the input image $I^{LR}$ is converted into the feature space by one convolutional layer as $\\hat{I}^{LR} = Conv(I^{LR}).$ Then, two convolutional layers and one ReLU activation are utilized to calculate the $D^TI^{LR}$ for ISTA steps, as shown in the figure.",
"There are $K$ steps for ISTA optimization.",
"For the $k$ -th step, there is $\\hat{I}^{SR}_k=ISTABlock(D^TI^{LR}, \\hat{I}^{SR}_{k-1}),$ where $ISTABlock(\\cdot )$ is the designed ISTA block.",
"After $K$ iterations, the output $\\hat{I}^{SR}_K$ and $\\hat{I}^{LR}$ are organized in a skip connection manner as $\\hat{I}^{SR} = \\hat{I}^{LR} + Padding(\\hat{I}^{SR}_K),$ where $Padding(\\cdot )$ aims to introduce the non-linearity.",
"The padding structure is composed of two convolutional layers and one ReLU activation.",
"Finally, the restored image $I^{SR}$ is generated from the SR feature $\\hat{I}^{SR}$ as $I^{SR} = Upscaling(\\hat{I}^{SR}),$ where $Upscaling(\\cdot )$ is the upscaling module.",
"The upscaling module is composed of one convolutional layer and a sub-pixel convolution."
],
[
"Design of ISTA Block",
"Figure REF shows the ISTA block design.",
"The inputs of $k$ -th ISTA block are $\\hat{I}^{SR}_{k-1}$ and $D^TI^{LR}$ , and the output of the block is $\\hat{I}^{SR}_k$ .",
"Multi-scale exploration (MSE) block and multi-scale attention (MSA) mechanism are utilized to generate $(E-\\alpha _k D^TD)\\hat{I}^{SR}_k$ from $\\hat{I}$ .",
"Then, one $1\\times 1$ convolution combines the information from the $(E-\\alpha _k D^TD)\\hat{I}^{SR}_k$ and the $D^TI^{LR}$ , and the soft-thresholding block (ST Block) jointly explores the features following the Equation REF .",
"A padding structure is introduced to the ISTA block with skip connection for better gradient transmission.",
"The padding structure is composed of two convolutional layers and a ReLU activation.",
"Figure: Multi-scale exploration (MSE) block designFigure REF shows the block design of MSE.",
"Multi-scale design has proved to be an effective structure for image SR.",
"In this block, three different scales ($1\\times 1$ , $3\\times 3$ , and $5\\times 5$ ) are considered to explore the hierarchical information from the features.",
"To make the exploration more efficient, the $5\\times 5$ exploration is separated into two $3\\times 3$ convolutions with a ReLU activation, which hold the same receptive field.",
"After the multi-scale feature extraction, one ReLU activation and a convolutional layer concatenate the hierarchical features and generate the final output of the MSE.",
"Figure: Multi-scale attention (MSA) mechanism designFigure REF shows the block design of multi-scale attention (MSA) mechanism.",
"It can be found that the MSA has a similar multi-scale exploration design as MSE.",
"Hierarchical features from three different scales are jointly explored by two convolutional layers and two ReLU activation operations.",
"Then, the Sigmoid activation is introduced to generate the non-negative attention information.",
"Figure: Soft thresholding block (ST Block) designFigure REF shows the soft-thresholding block (ST Block) design.",
"The ST Block is composed of two $1\\times 1$ convolutional layers, one ReLU activatiion and a Sigmoid activation.",
"The ST Block calculates the hyper parameters from the input features by the network design, and performs the soft-thresholding action with the learned parameters."
],
[
"Network Details",
"In the network, all convolution kernels are set with size $3\\times 3$ except for the MSE and the ST Block.",
"The convolutions in ST Block are set with kernel size as $1\\times 1$ .",
"The filters of all convolutional layers are set with 64 except for the ST Block and the Upscaling module.",
"There are $K=16$ ISTA blocks in the ISTAR.",
"The loss function is chosen as $\\ell _1$ -norm between the SR and HR images."
],
[
"Settings",
"The network is trained with DIV2K [34] dataset, which contains 900 high resolution images.",
"We choose first 800 images for training, and last 5 image for validation.",
"Five common benchmarks are chosen for comparing the restoration effectiveness: Set5 [35], Set14 [36], B100 [37], Urban100 [38] and Manga109 [39].",
"We update our ISTAR for 1000 epochs by the Adam [40] optimizer with learning rate $10^{-4}$ .",
"The learning rate is halved for every 200 epochs.",
"The scaling factors are chosen as $\\times 2$ , $\\times 3$ and $\\times 4$ .",
"The patch size for training is chosen as $48\\times 48$ for LR images.",
"All other settings are same as RDN [13].",
"The objective indicators are chosen as peak signal-to-noise ratio [41] (PSNR) and structural similarity [42] (SSIM).",
"According to the ISTA optimization, the result becomes more accurate with the increase of iteration times.",
"To investigate the effectiveness of the iteration times, we compare the performances with different iteration time $K$ .",
"Table REF shows the PSNR/SSIM comparisons among different $K$ with scaling factor $\\times 4$ .",
"For a fair comparison, all the testing models are updated for 200 epochs under the same settings.",
"In the table, we can find that the PSNR and SSIM rises with the increase of iteration time $K$ .",
"When $K=16$ , the network achieves the best performance.",
"In this point of view, a deeper network leads to a better performance.",
"When $K$ increases from 12 to 16, the PSNR and SSIM gets small improvement.",
"To balance the performance and the computation complexity, we choose $K=16$ to build the ISTAR."
],
[
"Investigation on the Multi-Scale Exploration",
"In MSE, we utilize three different scales to explore the hierarchical information.",
"To investigate the effectiveness of multi-scale exploration, we conduct the experiments with scaling combination as 1, $(1+3)$ and $(1+3+5)$ .",
"Table REF shows the PSNR/SSIM comparisons among different scales combinations.",
"In the figure, we can find that the multi-scale exploration brings 0.08 dB PSNR improvement on Urban100 dataset and 0.05 db improvement on Manga109 dataset.",
"Compared with $S=1$ , the combination $(1+3)$ brings 0.09 dB PSNR improvement on Manga109 dataset.",
"In this point of view, the multi-scale exploration is an effective design for restoration."
],
[
"Investigation on the Multi-Scale Attention",
"In ISTA block, MSA is considered for better network representation.",
"To show the effectiveness of MSA, we compare the objective performances with and without MSA on different benchmarks.",
"Table REF shows the PSNR/SSIM comparisons after training 1000 epochs.",
"In the figure, we can find that the network with MSA has 0.1 dB improvement on Manga109 dataset and 0.06 dB improvement on Set14 and Urban100 dataset.",
"In this point of view, the MSA proves to be an effective component for image restoration and improves the network capacity.",
"We compare our ISTAR with several traditional and recent CNN-based image SR works: SRCNN [10], FSRCNN [25], VDSR [11], DRCN [43], CNF [44], LapSRN [45], DRRN [46], BTSRN [47], MemNet [48], SelNet [49], SRDenseNet [57], CARN [50], IMDN [15], RAN [51], DNCL [52], FilterNet [53], MRFN [54], SeaNet [55], DEGREE [56] and Cross-SRN [17].",
"Table REF shows the PSNR/SSIM comparisons on five testing benchmarks with scaling factor $\\times 2$ , $\\times 3$ and $\\times 4$ .",
"In the figure, we can find that our ISTAR achieves the best performance on all testing benchmarks with all scaling factors.",
"Compared with Cross-SRN, our ISTAR achieves near 0.4 dB, 0.3 dB and 0.2 dB improvement on Urban100 dataset with scaling factor $\\times 2$ , $\\times 3$ and $\\times 4$ separately.",
"When sacling factor is $\\times 4$ , ISTAR achieves 0.34 dB PSNR higher than Cross-SRN on Manga109 dataset.",
"It should be noticed that Urban100 and Manga109 are two representative datasets with plentiful edges and lines.",
"In this point of view, ISTAR can effectively restore the structural information than other works.",
"Table: PSNR/SSIM, parameters and MACs comparisons with optimization-inspired networks with scaling factor ×2\\times 2.To further investigate the effectiveness of ISTAR, we compare the network with several optimization-inspired methods.",
"Table REF shows the PSNR/SSIM comparisons with differnet optimization-inspired networks.",
"DBPN [58] is developed by the iterative back projection algorithm, while USRNet [21] and HSRNet [33] are inspired by the half-quadratic splitting (HQS) strategy.",
"The MACs is calculated by the same method as the HSRNet.",
"In the table, we can find that our ISTAR acheives better performance than USRNet and HSRNet.",
"Compared with HSRNet, our method achieves 0.1 dB PSNR improvement on Set5 and Urban100 datasets.",
"Furthrmore, ISTAR achieves better performance than USRNet with near 29.7% parameters and 13.7% computation complexity.",
"DBPN is one of the state-of-the-art image SR methods.",
"Compared with DBPN, our method achieves competitive or better PSNR/SSIM results with similar parameters and 68.9% MACs off.",
"In this point of view, ISTAR proves to be an effective optimization scheme for image SR.",
"Figure: Visualization comparisons on Urban100 dataset with scaling factor ×4\\times 4Figure REF shows the visualization comparisons with different recnet image SR works (Cross-SRN [17], HSRNet [33] and MSRN [32]) on Urban100 dataset with scaling factor $\\times 4$ .",
"In the first row of the figure, we can find that the building restored by ISTAR is closest to the ground-truth than other methods.",
"Similarly, in the second row of the figure, ISTAR can restore the lines and girds more effectively than other works with much higher PSNR/SSIM results.",
"In this point of view, ISTAR can generate more satisfying subjective results than other works."
],
[
"Conclusion",
"In this paper, we proposed an ISTA-inspired restoration network, termed ISTAR, for effective image super-resolution.",
"We analyzed the image super-resolution task from the optimization perspective and proposed an iterative solution based on the ISTA.",
"According to the formulation of the solution, an end-to-end network with optimization-inspired blocks was developed for effective image super-resolution.",
"Multi-scale exploration and multi-scale attention mechanism were specifically devised to boost the network capacity.",
"Experimental results show the proposed ISTAR achieves better subjective and objective performances than other state-of-the-art works.",
"Compared with other optimization-inspired methods, ISTAR achieves competitive or better performance with much fewer parameters and lower computation complexity."
]
] | 2210.07818 |
[
[
"An Efficient FPGA Accelerator for Point Cloud"
],
[
"Abstract Deep learning-based point cloud processing plays an important role in various vision tasks, such as autonomous driving, virtual reality (VR), and augmented reality (AR).",
"The submanifold sparse convolutional network (SSCN) has been widely used for the point cloud due to its unique advantages in terms of visual results.",
"However, existing convolutional neural network accelerators suffer from non-trivial performance degradation when employed to accelerate SSCN because of the extreme and unstructured sparsity, and the complex computational dependency between the sparsity of the central activation and the neighborhood ones.",
"In this paper, we propose a high performance FPGA-based accelerator for SSCN.",
"Firstly, we develop a zero removing strategy to remove the coarse-grained redundant regions, thus significantly improving computational efficiency.",
"Secondly, we propose a concise encoding scheme to obtain the matching information for efficient point-wise multiplications.",
"Thirdly, we develop a sparse data matching unit and a computing core based on the proposed encoding scheme, which can convert the irregular sparse operations into regular multiply-accumulate operations.",
"Finally, an efficient hardware architecture for the submanifold sparse convolutional layer is developed and implemented on the Xilinx ZCU102 field-programmable gate array board, where the 3D submanifold sparse U-Net is taken as the benchmark.",
"The experimental results demonstrate that our design drastically improves computational efficiency, and can dramatically improve the power efficiency by 51 times compared to GPU."
],
[
"Introduction",
"Three dimensions (3D) point cloud is the inherently sparse data acquired from 3D sensors and can provide rich geometric, shape, and scale information [1].",
"Compared with two dimensions (2D) RGB images, 3D point cloud preserves a better understanding of the original geometric information in 3D space for deep learning-based vision tasks.",
"While the biggest challenge of computing on the 3D point cloud comes from its extremely sparse nature.",
"What’s more, the sparsity of point cloud is fundamentally different from that in traditional convolutional neural networks (CNNs).",
"For CNNs, the sparsity is usually caused by the activation functions.",
"But for point cloud, its sparsity reflects the 3D composition of the real world.",
"How to reduce the redundant computation caused by high sparsity This work was supported in part by the National Natural Science Foundation of China under Grant 62174084, 62104097 and in part by the High-Level Personnel Project of Jiangsu Province under Grant JSSCBS20210034, the Key Research Plan of Jiangsu Province of China under Grant BE2019003-4.",
"(Corresponding author: Zhongfeng Wang; Jun Lin.)",
"Figure: An example of the point cloud application .becomes the key to the processing of point cloud.",
"Prior works have proposed deep learning-based methods.",
"For instance, [3], [4], [5] projected the 3D point cloud into 2D to compress the data dimensions and reduce computational complexity, then applied 2D CNNs on the 2D point cloud.",
"[6], [7], [8] directly leveraged the Multi-Layer Perceptrons (MLPs) operation for the original points to extract semantic features from the sparse point cloud, without voxelizing point cloud into 3D grids.",
"[9], [10], [11], [12] converted the point cloud into sparse discrete representation, then applied modified 3D CNNs for different tasks.",
"Furthermore, authors in [12] proposed submanifold sparse convolution (Sub-Conv) to reduce memory and computational costs of computing on the point cloud by restricting the computation of convolution to be related to nonzero activations.",
"The submanifold sparse convolutional network (SSCN) [12] achieves remarkable results compared to other deep learning-based methods [13].",
"Consequently, SSCN plays an important role in point cloud-based deep learning applications, motivating its deployment on resource constraint edge devices and corresponding dedicated accelerators.",
"Nowadays, to accelerate CNNs, some specifically designed hardware accelerators [14], [15], [16] are proposed.",
"Eyeriss [14] presented a general dataflow to minimize data movement.",
"GoSPA [16] proposed an intersection method to optimize the dataflow when the activations and weights had sparsity.",
"However, when these accelerators for CNNs are directly used for SSCN, they suffer from severe performance degradation because they can not perform the matching operation of explicitly determining each nonzero activation and searching its nonzero neighbors, which is the core operation of the Sub-Conv layer.",
"Therefore, a dedicated accelerator for SSCN is highly desired to promote its deployment.",
"Currently, several works presented solutions for point cloud-based networks.",
"[17] and [18] introduced ASIC-based accelerators for PonitNet++ and proposed optimization schemes to the neighbor point search.",
"[19] designed a low-power FPGA-based accelerator, which optimized the nonlinear implementations in PointNet.",
"The above works are based on the PointNet and PointNet++ networks, and thus cannot be directly applied to the acceleration of SSCN.",
"PointAcc [20] proposed an ASIC-based accelerator that unified diverse mapping operations into a multiply-accumulate operation through coordinate transformation to be compatible with different point cloud networks.",
"Other hardware solutions such as GPUs can be deployed to accelerate the point cloud networks.",
"However, GPUs are not suitable for resource constraint edge devices because of their high power consumption, and the matching operation also limits their performance.",
"Concentrating on the SSCN, we propose an FPGA-based efficient SSCN accelerator, ESCA, to support the matching operation and corresponding computations.",
"This work makes the following contributions: A tile-based zero removing strategy is proposed to improve computational efficiency.",
"The strategy reduces the processing time of the sparse information significantly, which also alleviates the computational load imbalance.",
"An encoding scheme is introduced to efficiently support the matching operation.",
"Based on the above scheme, a matching method is proposed to execute the matching operation for each nonzero activation, which solves the problems of explicit representation in the matching operation.",
"A dedicated SSCN accelerator is proposed to support the matching operation and corresponding computations.",
"The proposed design is implemented in the Xilinx ZCU102 platform and achieves significant improvement in terms of GOPS and power efficiency compared with GPU."
],
[
"Background",
"The computation rules of Sub-Conv are fundamentally different from that of traditional convolution.",
"Fig.",
"2(a) shows the results of traditional convolution for sparse features, and Fig.",
"2(b) shows the matching process of Sub-Conv.",
"In traditional convolution, the input feature map is traversed by a kernel, and multiply-accumulate operations are performed in order.",
"Even if the feature map has sparsity, as long as the convolution parameters, such as stride, kernel size, etc., are determined, the computation rules and correspondences in the convolution are explicitly determined.",
"As a result, the sparse data in the output feature map dilates [12], so it is not suitable for point cloud-based computation.",
"For Sub-Conv [12], the fields of the feature map involved in the convolution operations are strictly limited to the neighbors of the nonzero activations, and the output feature map maintains the same sparsity as the input feature map.",
"As shown in Fig.",
"2(b), five nonzero activations mean that this feature map only needs to perform five convolution operations with the corresponding kernel, and the positions are strictly limited to the fields where the central activation is nonzero.",
"Because the Sub-Conv layer can keep the same pattern of sparsity between the input feature map and the output feature map, it shows satisfying visual results when is applied to the point cloud with high sparsity.",
"However, because the restricted computation pattern of the Sub-Conv layer leads to irregular sparse matching Figure: Illustration of traditional convolution and Sub-Conv.",
"(a) Traditional convolution: the feature map is traversed by the kernel, and the sparsity in the output feature map dilates.",
"(b) Sub-Conv: The kernel only calculates with the fields where the center activation is non-zero.operations, traditional convolution accelerators suffer from performance degradation when they are directly applied to it [20].",
"Therefore, efficient accelerators for SSCN are urgently needed, and the bottleneck lies in the extreme and unstructured sparsity, and the complex computational dependency between the sparsity of the central activation and the neighborhood ones."
],
[
"Tile-based Zero Removing Strategy",
"Voxelized point cloud has huge sparsity.",
"Directly processing on the original feature map results in large memory overhead and computation cost, and dramatically reduce computational efficiency.",
"Take the ShapeNet dataset [21] as an example, it has nearly 99.9% sparsity, resulting in many regions without nonzero activations.",
"Since the computation depends on the sparsity of the central activation, removing the all-zero regions has no effect on the result.",
"To tackle this problem, we propose an effective tile-based zero removing strategy to remove the coarse-grained redundant sparse regions.",
"Figure: The process of zero removing strategy.",
"(a) The original input feature map is first divided into tiles of fixed size.",
"(b) The fully sparse tiles of the input are removed, keeping only tiles containing nonzero activations.",
"(c) Due to the nature of Sub-Conv, the removal of fully sparse tiles does not affect the output.As illustrated in Fig.",
"3(a), the original 3D feature map is divided into tiles of size ${N \\times M \\times L}$ , where ${N}$ , ${M}$ and ${L}$ are configurable parameters, and the sparsity in each tile is detected.",
"If all the activations are zero in the tile, the tile is fully sparse and will be removed from the original feature map as shown in Fig.",
"3(b).",
"Because the fully sparse tile is irrelevant to the computation of the Sub-Conv, the output feature map, as depicted in Fig.",
"3(c), still maintains the same sparsity.",
"Then the processed feature map is only composed of active tiles, which contain at least one nonzero activation, and will be sequentially matched and computed.",
"With this zero removing strategy, the time overhead when processing sparse information is significantly reduced, and the problem of computational imbalance is also alleviated."
],
[
"Matching Operation and Encoding Scheme",
"Matching operation is the procedure to locate each nonzero activation and search its nonzero neighbors, which is crucial for the computation of SSCN, and the position information recording the geometric distribution of nonzero activations is required to support the matching operation.",
"Thus, an encoding scheme is proposed, which encodes the feature map into two types of data: index mask and valid data.",
"Index Mask.",
"The index mask is used to explicitly represent the sparsity distributions of the feature map and is dynamically traversed during computation.",
"The relationship between features, masks, and nonzero activations is shown in Fig.",
"4.",
"Mask is a one-bit signal with only two states of 0 and 1, which represents that the activation is zero or not, respectively, and it is stored in the mask buffer.",
"It also has a strong correlation with the sparse distribution of the point cloud, so the computation relationship between input feature maps and matching operation can be established explicitly.",
"Valid Data.",
"Valid data are the nonzero activations and the corresponding weights, as shown in Fig.",
"4.",
"As valid data, the activations and weights are stored in the corresponding buffers, and can be read from the buffers under the guidance of the index mask.",
"Thus, the matching operation can be performed through the process of interaction between the index mask and the valid data.",
"Figure: Composition of the index mask and the vaild data.Figure: Illustration of the matching operation and match group.Figure: Description of the SDMU.",
"The Acc in the state index generator corresponds to the accumulation operation."
],
[
"Sparse Data Matching Unit",
"The matching operation and the composition of match group are elaborated in Fig.",
"5.",
"A match group contains the nonzero activations and corresponding weights for each convolution calculation based on the central nonzero activation.",
"Also a set of elements in a match group is called a match.",
"Thus, after determining all the match groups for each nonzero activation, the matching operation is completed for one feature map.",
"Meanwhile, the computation of the Sub-Conv layer is decomposed into point-wise multiply-accumulate operations for each match group.",
"To support the matching operation and search all match groups efficiently for the Sub-Conv layer, we propose the sparse data matching unit (SDMU), which is shown in Fig.",
"6.",
"The mask judger and the decoder perform the matching operation and generate the match groups from the buffers.",
"For the convolution with the kernel size of ${K \\times K \\times K}$ , the index masks of each column are read sequentially.",
"So the parallelism of the decoder in SDMU is ${K^2}$ , which corresponds to the number of columns.",
"Then the FIFO group stores the match groups in column order.",
"Finally, the multiplexer (MUX) selects matches from the FIFO group and sends them to the computing core for point-wise multiply-accumulation.",
"To coordinate the computation rules, activations and the ones that are in their neighbor field need to be explicitly acquired at the same time.",
"Therefore, ${K^compute array3}$ masks are required for determination.",
"This area is called the sparse receptive field (SRF).",
"For each nonzero activation, the matching operation and the acquisition of the match group are limited to the SRF.",
"The process of matching operation is described in Fig.",
"7(a).",
"In this case, it is presented in 2D and can be smoothly extended to 3D.",
"The kernel size is ${3^2}$ , so the parallelism is 3.",
"The following steps of the matching method, read masks, Figure: Examples of the matching steps in the SDMU.",
"(a) The process of obtaining match groups through masks.",
"(b) Pipeline representation when executing the matching operation.judge state, generate state index, and fetch activations are presented to conduct the matching operation.",
"Read masks: The index mask is read from the mask buffer for each SRF and sent to the mask judger.",
"Judge state: The mask is judged whether to perform the convolution for the SRF by the mask judger.",
"If the center mask corresponds to a nonzero activation, then this SRF is active, and the match group is fetched from buffers according to the generate state index step and fetch activations step.",
"Otherwise, it is non-active and the fetch activations step will be skipped.",
"Generate state index: In this step, the relative position of nonzero activations is generated for each SRF and is called the state index.",
"It can be regarded as an array (A, B).",
"The index A records the nonzero activations accumulated in each column and it is cumulated for each SRF.",
"The index B represents the number of activations in each column for each SRF if the state is active, otherwise, index B equals 0.",
"Thus, the index A marks the highest address of the activation in the activation buffer for each match group.",
"And the index B corresponds to the address length of the activation involved in the computation in each column.",
"Fetch activations: The address fragment for nonzero activations of each column can be represented by (A, A-B).",
"It is generated in the address generator and contains addresses for all activations in each match group.",
"Then the corresponding activations are read from the activation buffer.",
"If the mask of the central site is zero, which indicates the matching operation will not be implemented, the fetch activations step for this SRF will be skipped accordingly.",
"These matching steps are executed in a pipeline, as shown in Fig.",
"7(b).",
"Since weights and activations have a positional correspondence in each match group, the weights that need to participate in the computation can also be obtained by state index synchronously, and the corresponding activations and weights are concatenated when read from buffers.",
"In summary, the state index obtained by traversing the index mask can establish a matching relationship with valid data, through which the match group can be collected.",
"In the matching steps, parallel processing is performed according to the column dimension in every SFR to maintain the synchronization of explicit representations of each match group.",
"Therefore, after obtaining the match group from ${K^2}$ columns, which is decided by the kernel size, a FIFO group is applied to store them.",
"The FIFO group consists of ${K^2}$ identical FIFOs, and each FIFO stores the matches belonging to one column.",
"In each cycle, the controller in the decoder selects a match from a FIFO based on the calculation order, and MUX sends it to the computing core."
],
[
"Computing Core",
"Since the sparse data are already transformed into match groups in the SDMU, the computing core (CC) is designed to implement dense point-wise multiply-accumulate operations.",
"The CC contains a computing array and an accumulator.",
"In each cycle, the input to the computing array is a match belonging to a match group.",
"In order to improve throughput, the computing array is divided into $m+1$ computing units (CUs), each of which performs the computation of $n+1$ input channels (ICs), and the output of each CU is the partial sum of the corresponding output channel (OC), so the total parallelism of the computing array is $(m+1)×(n+1)$ .",
"Fig.",
"8(b) illustrates the inputs and outputs of the computing array.",
"The activations of the $n+1$ ICs are broadcast to all CUs.",
"$A_{[n]}$ represents activations belonging to IC $n$.",
"$W_{[n][m]}$ represents weights belonging to IC $n$, OC $m$.",
"For example, the result of CU $m$ is equal to the partial sum of the $n$ ICs on the $m^{th}$ OC.",
"The detailed structure of the computing unit is shown in Fig.",
"8(c).",
"The partial sum of nonzero activations for different OCs can be obtained through the computing array, then the partial sum is sent to the accumulator and the output of each SRF is obtained.",
"Figure: Illustration of loop unrolling and the composition of computing array.",
"(a) The process of loop unrolling.",
"(b) The description of the computing array.",
"(c) The structure of the computing unit in the computing array.Figure: Description of overall hardware architecture.The details of the loops are shown in Fig.",
"8(a).",
"Each active tile is traversed in turn.",
"The obtained data are fed to the CC in the order of matched nonzero activations and weights, and the IC and OC dimensions are completed sequentially according to the parallelism of the proposed computing array.",
"Finally, the partial sum of each match group is accumulated to obtain the outputs corresponding to nonzero activations.",
"The SDMU and CC are executed in pipeline to increase resource utilization and the system throughput."
],
[
"Overall Hardware Architecture",
"The overall hardware architecture is shown in Fig.",
"9, mainly containing a main controller, an SDMU, a CC, and corresponding buffers on the on-chip logic.",
"Main Controller.",
"The main controller is responsible for ensuring that the SDMU and the CC are executed in the right order.",
"SDMU.",
"In SDMU, the mask judger and the decoder perform the matching operation.",
"The obtained match groups are stored in the corresponding FIFOs, so as to read them under the control of the FIFO group and MUX, and the matched data are sent to the computing array in order.",
"CC.",
"In the computing array of CC, computation is performed in the IC and OC dimensions, and the partial sum is generated in the OC dimension.",
"Then the partial sum is accumulated in the accumulator and finally sent to the output buffer.",
"In our structure, the parallelism is set to 16 both in the OC and IC dimensions.",
"There are four buffers to store data, whose basic unit is block RAM.",
"The mask buffer stores the mask, while the activation buffer and weight buffer store activations and weights respectively.",
"The output buffer stores the outputs and sends them to the off-chip DRAM."
],
[
"Experimental Setup",
"We adopt the 3D submanifold sparse U-Net (SS U-Net) [12] to evaluate our ESCA.",
"SS U-Net can perform the semantic segmentation task of the point cloud with satisfactory visual results.",
"The pre-trained network parameters are 8bit quantized, and the activations are 16bit quantized.",
"The kernel size of the Sub-Conv in the SS U-Net is ${3\\times 3\\times 3}$ , so the parallelism of SDMU and the number of FIFOs in the FIFO group are set to ${3^2}$ .",
"The whole system is implemented with Vivado Design Suite.",
"The performance of the GPU baseline is measured by NVIDIA System Management Interface."
],
[
"Analysis of Zero Removing Strategy",
"We comprehensively evaluate the zero removing strategy on two representative point cloud datasets, ShapeNet dataset [21] and NYU Depth dataset (v2) [22].",
"The feature maps are normalized to the size of ${192 \\times 192 \\times 192}$ after voxelization.",
"We test the effect of different tiling sizes on the sparsity and the number of remaining active tiles.",
"The experimental results are shown in Table I.",
"With different tiling sizes, this strategy achieves up to 99.82% zero reduction in the ShapeNet [21], and up to 99.85% in the NYU [22].",
"A more fine-grained tile size increases the removing ratio of zeros, it also increases the computational complexity.",
"We use the tile size of ${8 \\times 8 \\times 8}$ .",
"Table: Analysis of Zero Removing Strategy"
],
[
"Results Comparison",
"The proposed ESCA architecture is implemented on the Zynq UltraScale+ ZCU102 FPGA at 270MHz.",
"The hardware resource utilization is reported in Table II.",
"Table: FPGA Frequency and Resource UtilizationESCA is compared with Tesla P100 GPU and Intel Xeon Gold 6148 CPU, which are existing hardware acceleration solutions for SSCN.",
"As shown in Fig.",
"10, our ESCA outperforms the CPU and GPU implementation by around 8.41 times and 1.89 times in terms of speedup.",
"Since the computation of SSCN depends on the sparsity of the center activation and its neighborhood ones, the GPU and CPU cannot recognize Figure: Comparison with CPU and GPU in terms of time consumption when processing a Sub-Conv layer.Table: Comparison with Other Implementations for Point Cloudthis correspondence, resulting in a large number of redundant computations.",
"While in ESCA, the matching operation is executed efficiently.",
"The detailed comparisons between GPU and our design are summarized in Table III.",
"Our design achieves 17.73 GOPS and 5.14 GOPS/W in terms of performance and power efficiency, which outperforms GPU by around 1.88 times and 51 times.",
"Note that the GOPS is effective performance containing only non-zero multiply-accumulate operations for a fair and clear comparison with other implementations.",
"To further evaluate the performance of ESCA, it is also compared with an FPGA-based accelerator[19], which targets the optimized PointNet (O-Pointnet) and leverages the MLP operations for point clouds.",
"Compared with[19], our accelerator has a significant improvement in both performance and power efficiency as shown in Table III.",
"To sum up, the higher performance of ESCA comes from two aspects.",
"On one hand, the zero removing strategy and encoding scheme optimize the data structure to facilitate the match operation.",
"On the other hand, the on-chip logic efficiently performs matching operation and multiply-add computations by the SDMU and CC."
],
[
"Conclusion",
"In this paper, we present ESCA, an efficient FPGA-based accelerator that supports SSCN.",
"A zero removing strategy is introduced to remove the coarse-grained redundant regions and an encoding scheme is proposed to simplify the matching operation.",
"Based on the encoding scheme, the sparse data matching unit (SDMU) and the computation core (CC) are developed.",
"The 3D submanifold sparse U-Net is considered for the experiment.",
"The proposed design is implemented on Xilinx ZCU102.",
"The experimental results show that our work outperforms the GPU by around 1.88 times and 51 times in terms of performance and power efficiency."
]
] | 2210.07803 |
[
[
"It takes two to know one: Computing accurate one-point PDF covariances\n from effective two-point PDF models"
],
[
"Abstract One-point probability distribution functions (PDFs) of the cosmic matter density are powerful cosmological probes that extract non-Gaussian properties of the matter distribution and complement two-point statistics.",
"Computing the covariance of one-point PDFs is key for building a robust galaxy survey analysis for upcoming surveys like Euclid and the Rubin Observatory LSST and requires good models for the two-point PDFs characterising spatial correlations.",
"In this work, we obtain accurate PDF covariances using effective shifted lognormal two-point PDF models for the mildly non-Gaussian weak lensing convergence and validate our predictions against large sets of Gaussian and non-Gaussian maps.",
"We show how the dominant effects in the covariance matrix capturing super-sample covariance arise from a large-separation expansion of the two-point PDF and discuss differences between the covariances obtained from small patches and full sky maps.",
"Finally, we describe how our formalism can be extended to characterise the PDF covariance for 3D-dimensional spectroscopic fields using the 3D matter PDF as an example.",
"We describe how covariances from simulated boxes with fixed overall density can be supplemented with the missing super-sample covariance effect by relying on theoretical predictions validated against separate-universe style simulations."
],
[
"Introduction",
"The recent rise of non-Gaussian statistics for galaxy clustering and weak lensing has highlighted the need for accurate models of the likelihood function and, in particular, the covariance matrices of such statistics.",
"Aside from the challenge to model non-Gaussian statistics and capture their response to changing cosmological parameters in the nonlinear regime, getting accurate covariance matrices can prove difficult and require thousands of numerical simulations to ensure convergence , .",
"For statistics relying on higher-order correlation functions involving $N$ points, modelling the covariance generally requires assumptions for the shape of $2N$ -point correlations.",
"Even for Gaussian fields, this task has only been achieved recently at all orders $N$ .",
"To go beyond the assumption of Gaussianity, log-normal models of the cosmic density field have become a tool of choice in higher-order analyses such as studies of the bispectrum , , moments of cosmic random fields , density split statistics , , the full probability distribution function (PDF) of cosmic random fields , , , and even map-based inference approaches ().",
"The reason why (shifted) log-normal simulations have become so popular is that they obey a similar hierarchy between variance and skewness as the physical cosmic density field.",
"In particular, they can be tuned to match both a desired power spectrum at all scales and a desired skewness at one given scale (c.f.",
"Section 5.2 and Figure 4 of ).",
"This however also highlights a limitation of those simulations: they potentially fail to realistically capture the hierarchy of moments beyond the skewness [2] as well as the full shape of higher order N-point functions.",
"With the large data vectors that future multi-probe analyses of upcoming surveys like Euclid , [1] and the Rubin Observatory LSST are aiming for, estimated covariances - whether from log-normal simulations or other mock data - can lead to a huge degradation of cosmological constraining power , , .",
"This makes it desirable to have an analytic understanding of measurement uncertainties.",
"One-point statistics such as the PDF of the weak lensing convergence have the advantage that both their dependence on cosmological parameters and their covariance, being given in terms of the two-point PDF, can be modelled theoretically , .",
"In this work, we will focus on the weak lensing convergence smoothed on mildly nonlinear scales (of order $10^{\\prime }$ at source redshifts of $z\\simeq 1-2$ ), where the bulk of the PDF and its cosmology dependence can be predicted at the percent level using large deviation statistics [10], [9], [4], and its covariance is well-approximated by a shifted lognormal distribution .",
"In the opposite regime for small smoothing scales, the high-convergence tail of the weak lensing convergence PDF and its covariance have been modelled using a halo-model formalism in , finding qualitative agreement with simulations but significant quantitative discrepancies attributed to strong sensitivity to small-scale effects.",
"For galaxy clustering, the error on and correlation among the factorial moments of galaxy counts in cells $F_k= \\langle N(N-1)\\ldots (N-k)\\rangle $ have been computed for hierarchical models in and .",
"Those studies included cosmic errors due to a finite volume and discreteness errors due to a finite number of tracers relevant for clustering but not the weak lensing convergence, which is instead affected by shape noise due to shear measurements.",
"In , the variance and covariance among galaxy counts in cells was obtained for the case of a constant separation-independent cell correlation, which has previously been considered in .",
"As we explain here, and derive in Appendix , the large-separation expansion in terms of bias functions adopted in , , , can be generalised to the physically more realistic case of a separation-dependent correlation.",
"The large-separation two-point PDF and the resulting super-sample covariance effect can also be related to the idea of the position-dependent matter PDF studied in .",
"Recent work of [8] determined covariances of density PDFs in hierarchical models, which can be used to understand the dominant terms in the covariance matrix of cosmological density fields in the strongly non-Gaussian regime.",
"In this work, we build on this foundation but focus on the weakly non-Gaussian weak lensing PDF for which hierarchical models are less suitable and better results can be achieved using shifted lognormal models."
],
[
"Structure",
"In Section , we discuss the general covariance modelling for the two-dimensional case of the weak lensing (or photometric galaxy clustering) PDF based on a model for the joint two-point PDF.",
"In Section , we use a large-separation expansion for this two-point PDF and link it to dominant terms in an eigendecomposition of the covariance matrix.",
"In Section , we validate our covariance models using several quantitative tests.",
"In Section , we present the basics for generalising our findings to the case of the three-dimensional matter (or spectroscopic galaxy clustering) PDF.",
"We conclude in Section and provide an outlook for how our results can be applied to related one-point observables, include tomography and information from two-point statistics.",
"We develop a theoretical model for the covariance of measurements of the one-point weak lensing convergence PDF in finite bins.",
"Our model is based on an integration over the two-point PDF (REF ) at different separations and we rely on a joint shifted lognormal model for that two-point PDF (REF ) tuned to match the PDF skewness and shown to capture the density-dependence of the two-point correlation (REF ) in simulations, as illustrated in Figure REF .",
"We establish expressions for the finite sampling (REF ) and super-sample covariance (REF ) contributions to the PDF covariance and their scaling with the inverse of the survey area.",
"We validate our predictions against large sets of simulated convergence maps both from N-body simulations and using the FLASK tool for generating log-normal random fields .",
"To verify the quality of our covariance model we employ a range of tests including visual comparisons, an eigendecomposition of the covariance, a band-decomposition of the precision matrix, $\\chi ^2$ tests and Fisher forecasts, with Figures REF and REF summarising two main results.",
"In Figure REF we show the measured covariance for the central region of the binned $\\kappa $ -PDF for a smoothing radius of $\\theta _s=7.5^{\\prime }$ at source redshift $z_s=2$ from $N_{\\rm sim}=108$ simulated full sky $\\kappa $ maps from (upper triangle) along with the shifted lognormal prediction (lower triangle, using equations (REF ) and (REF )).",
"For the predictions, we used the cosmological parameters of the simulation along with the shift parameter $s\\simeq 0.085$ obtained from the second and third moments measured from the PDF.",
"In Figure REF in the main text we show that the environment-dependent clustering that enters the eigendecomposition of the covariance is well captured by the shifted lognormal model.",
"Aside from the good visual agreement, we quantify their consistency using two tests.",
"First, we test the compatibility of the diagonal elements of the two covariances shown in Figure REF with a $\\chi ^2$ test, taking into account their uncertainty and correlation among each otherNote that typically only the diagonal elements of estimated covariances are well described with multi-variate Gaussian noise.",
"(covariance of the covariance).",
"We use the covariance between the numerically estimated diagonal covariance elements $\\hat{C}_{ii}$ and $\\hat{C}_{jj}$ as given in equation (REF ) following , where $N_{\\rm sim} = 108$ is the number of simulations used to estimate and the covariance elements $C_{ij}$ are evaluated with our theoretical model.",
"Using the above expression to compute a $\\chi ^2$ between the diagonal of the estimated and predicted covariance we find $\\chi ^2=36.8$ for a number of 26 bins.",
"This corresponds to a marginal $\\sim 1.50\\sigma $ deviation between the two matrix diagonals.",
"Finally, we assess the quality of the off-diagonal elements of our covariance model by performing a Fisher forecast.",
"Figure REF compares Fisher contours on the parameters $\\sigma _8$ and $\\Omega _{\\mathrm {cdm}}$ derived from a number of different covariance matrices: our covariance model, the covariance estimated from the N-body simulations of and the covariance estimated from log-normal simulations.",
"The differences in the width and alignment of the constraints obtained from these matrices is negligible compared to the overall parameter uncertainties.",
"Hence, our covariance model is sufficiently accurate for application to observational data.",
"We demonstrate the flexibility of our covariance model by adapting it to predict super sample covariance effects for the 3-dimensional matter PDF paving the way to spectroscopic galaxy clustering."
],
[
"Weak lensing PDF covariance",
"The weak lensing convergence PDF is a well-suited complementary summary statistic that can extract non-Gaussian information from cosmic shear observations, thus complementing two-point statistics.",
"In this work, we focus on mildly nonlinear scales, where weak lensing convergence PDF and its response to fundamental cosmological parameters can be predicted at percent-level accuracy from large-deviation statistics as shown in [4], .",
"In the same work it was shown how one can use shifted lognormal convergence fields generated by FLASK to match the convergence PDF and obtain precise covariance estimates in good agreement with maps from N-body simulations, of which often only a very limited number is available.",
"Following this work, we will study the covariance of the central region of the one-point PDF of the weak lensing convergence smoothed with a top-hat filter of radius $\\theta _s=7.3^{\\prime }$ at a fixed source redshift $z_s=2$ using 26 linearly spaced bins in the range $-0.024<\\kappa <0.026$ .",
"In this Section we will use both Gaussian and shifted lognormal fields generated with FLASK to test our theoretical framework for predicting the weak lensing PDF covariance based on different models for the two-point PDF of the weak lensing convergence in cells of a given separation.",
"While we focus on the two-dimensional case of the weak lensing PDF here, the formalism can be adapted to describe the three-dimensional clustering PDF as we lay out in Section ."
],
[
"Covariance prediction from joint 2-point PDF",
"Before calculating the covariance matrix of a PDF measurement, let us first specify how this measurement is conducted.",
"Let us assume that we have observed the gravitational lensing convergence field on a part of the sky with the area $A_{\\mathrm {survey}}$ .",
"Now let us smooth this convergence field with a circular aperture of some angular radius $\\theta _s$ (potentially cutting away parts of the edges of the survey, so that each aperture falls fully inside the survey area), and let us pixelise the resulting map.",
"Each pixel of that map contains the value of the average convergence within a radius $\\theta _s$ around the pixel center.",
"We would now like to estimate the probability density of a certain value $\\kappa $ for the smoothed convergence field.",
"In practice one will need to compute this PDF within a set of finite bins, say around a set of central values $\\kappa _i$ and with constant bin width $\\Delta \\kappa $ , which results in the bins $\\mathrm {bin}_i = [\\kappa _i - \\Delta \\kappa /2, \\kappa _i + \\Delta \\kappa /2]$ .",
"The PDF at each $\\kappa _i$ can then be approximated as ${\\cal P}(\\kappa _i) \\approx \\frac{P(\\kappa \\in [\\kappa _i - \\Delta \\kappa /2, \\kappa _i + \\Delta \\kappa /2])}{\\Delta \\kappa }\\ ,$ where ${\\cal P}(\\kappa )$ is the PDF (obtained in the limit $\\Delta \\kappa \\rightarrow 0$ ) and $P(A)$ is the probability of statement $A$ .",
"With our pixelised map we can estimate the right-hand side of this to obtain the measurement $\\hat{{\\cal P}}(\\kappa _i) &\\equiv \\frac{\\# \\lbrace \\mathrm {pix\\ with}\\ |\\kappa - \\kappa _i| \\le \\Delta \\kappa /2 \\rbrace }{\\# \\lbrace \\mathrm {pix}\\rbrace \\Delta \\kappa } \\ .$ Following [8], we introduce the weight function $\\chi _i(\\kappa ) = {\\left\\lbrace \\begin{array}{ll} 1 & \\mathrm {if}\\ \\kappa \\in [\\kappa _i - \\Delta \\kappa /2, \\kappa _i + \\Delta \\kappa /2] \\\\ 0 & \\mathrm {else} \\end{array}\\right.",
"}\\,,$ such that the estimator $\\hat{{\\cal P}}(\\kappa _i)$ is given by $\\hat{{\\cal P}}(\\kappa _i)=\\frac{\\sum _P \\chi _i(\\kappa _{\\rm P})}{N_{\\rm P} \\Delta \\kappa }\\approx \\frac{1}{A_{\\mathrm {survey}}} \\underset{\\mathrm {survey}}{\\int } d^2 \\mathbf {\\theta }\\ \\frac{\\chi _i(\\kappa (\\mathbf {\\theta }))}{\\Delta \\kappa }\\ ,$ where P labels the $N_{\\rm P}=A_{\\rm survey}/A_{\\rm pix}$ pixels of area $A_{\\mathrm {pix}}$ , $\\mathbf {\\theta }$ are angular coordinates on the sky, and $\\kappa (\\mathbf {\\theta })$ is the smoothed, continuous convergence field at that location.",
"The second moment of our above estimator is given by $& \\langle \\hat{{\\cal P}}(\\kappa _i) \\hat{{\\cal P}}(\\kappa _j) \\rangle \\nonumber \\\\&\\approx \\frac{1}{A_{\\mathrm {surv.",
"}}^2} \\underset{\\mathrm {surv.",
"}}{\\int }d^2 \\theta _1 \\underset{\\mathrm {surv.",
"}}{\\int } d^2 \\theta _2\\ \\frac{\\langle \\chi _i(\\kappa (\\mathbf {\\theta }_1)) \\chi _j(\\kappa (\\mathbf {\\theta }_2))\\rangle }{(\\Delta \\kappa )^2}\\nonumber \\\\&\\approx \\frac{1}{ A_{\\mathrm {surv.",
"}}^2} \\underset{\\mathrm {surv.",
"}}{\\int }d^2 \\theta _1 \\underset{\\mathrm {surv.",
"}}{\\int } d^2 \\theta _2\\ {\\cal P}(\\kappa _i, \\kappa _j ; |\\mathbf {\\theta _1} - \\mathbf {\\theta _2}|)\\ ,$ where ${\\cal P}(\\kappa _i,\\kappa _j;\\theta )$ is the joint PDF of two points of the smoothed convergence field at distance $\\theta $ .",
"This equation can be simplified to $\\langle {\\cal P}(\\kappa _i){\\cal P}(\\kappa _j)\\rangle = \\int d\\theta \\, P_d(\\theta ) {\\cal P}(\\kappa _i,\\kappa _j;\\theta )\\ ,$ with $P_d(\\theta )$ indicating the distribution of angular distances in a given survey area.",
"The covariance of the PDF estimator can then be computed from this moment via $\\rm {cov}({\\cal P}(\\kappa _i),{\\cal P}(\\kappa _j)) =\\langle \\hat{{\\cal P}}(\\kappa _i)\\hat{{\\cal P}}(\\kappa _j)\\rangle -\\bar{{\\cal P}}(\\kappa _i)\\bar{{\\cal P}}(\\kappa _j)\\ ,$ where $\\bar{{\\cal P}}\\equiv \\langle \\hat{{\\cal P}}\\rangle \\approx {\\cal P}$ .",
"We see that covariance predictions rely on two key ingredients: the distance distribution of angular separations and the two-point PDF of the lensing convergence at those separations, which will be discussed in more detail in Section REF .",
"Distance distributions.",
"Figure: Comparison of the continuous distribution of relative distances in a square Cartesian survey patch of side length Θ side =5\\Theta _{\\rm side}=5deg=300 ' =300^{\\prime } (black) and the discrete version for a coarse grid corresponding to non-overlapping circular cells of radius 7.3 ' 7.3^{\\prime } (blue).",
"We also show the distribution of relative distances on the full sky with maximum Θ side =π=180\\Theta _{\\rm side}=\\pi =180deg (red).The relevant distance distributions for weak lensing describe a two-dimensional field.",
"For the full sky, the uniform distribution of angular separations (measured in radians) reads $P_{d,\\text{full sky}}(\\theta )=\\frac{1}{2}\\sin \\theta \\,,$ with a maximum at $\\theta _{\\rm max}=\\pi /2$ .",
"This distribution when written in terms of distances relative to the maximal distance $\\Theta _{\\rm side}=\\pi $ is illustrated as red line in Figure REF .",
"For a square survey patch, the distribution of angular distances relative to the size of the survey patch, $\\hat{\\theta }=\\theta /\\Theta _{\\rm side}$ , is $&P_{d,\\text{square}}(\\hat{\\theta })= \\\\&{\\left\\lbrace \\begin{array}{ll} 2\\hat{\\theta }[\\pi +(\\hat{\\theta }-4)\\hat{\\theta }] & \\hat{\\theta }\\in [0,1[\\,, \\\\2\\hat{\\theta }[\\pi -2+4\\sqrt{\\hat{\\theta }^2-1}-\\hat{\\theta }^2-4\\rm {sec}^{-1}(\\hat{\\theta })]& \\hat{\\theta }\\in [1,\\sqrt{2}]\\,.\\end{array}\\right.",
"}$ This distribution is illustrated as black line in Figure REF , with a maximum at roughly half the patch size where $\\hat{\\theta }_{\\rm max}\\simeq 0.479$ and a linear growth for small distances, $P_{d,\\text{square}}(\\hat{\\theta })\\simeq 2\\pi \\hat{\\theta }$ .",
"Before looking at PDF covariance matrix examples in Section REF and introducing models for the two-point PDF in Section REF , let us discuss the main universal effects affecting the PDF covariance: i) the shot-noise effect related to the finite sampling of cells and ii) the super-sample covariance effect related to a finite survey area.",
"Previously we discussed the covariance of the PDF in the continuum limit.",
"In the more realistic discrete case, we estimate the PDF in a bin centred at $\\kappa _i$ with width $\\Delta _i$ (in our case linearly spaced bins $\\Delta _i=\\Delta $ ), so $\\kappa \\in [\\kappa _i-\\Delta _i/2,\\kappa _i+\\Delta _i/2]$ using a finite sample of $N_T$ cells that are placed on a regular grid in the survey area.",
"The histogram-value of the PDF in a bin is obtained from averaging the continuous PDF ${\\cal P}(\\kappa )$ over that bin ${\\cal P}_i=\\int _{-\\Delta _i/2}^{\\Delta _i/2} \\hspace{-21.33955pt} d\\delta \\kappa _i \\frac{{\\cal P}(\\kappa _i+\\delta \\kappa _i)}{\\Delta _i} \\simeq {\\cal P}(\\kappa _i)\\,,$ and for small bin sizes close to the value of the continuous PDF at the bin center $\\bar{\\kappa }_i$ .",
"To compute the covariance for the discrete case, we can discretise equation (REF ) by averaging the continuous joint PDF over the $\\kappa $ -bins and replacing the integration over the distance distribution by a sum-based average over the set of $N_d=N_T^2=\\sum _{\\theta \\in \\Theta } N_d(\\theta )$ distances $\\Theta $ of all grid cells with multiplicities $N_d(\\theta )$ that are part of the survey patch $\\langle {\\cal P}_i{\\cal P}_j\\rangle &= \\sum _{\\theta \\in \\Theta } \\frac{N_d(\\theta )}{N_d}\\!\\!\\!",
"\\int _{-\\Delta _i/2}^{\\Delta _i/2} \\hspace{-21.33955pt} d\\delta \\kappa _i \\int _{-\\Delta _j/2}^{\\Delta _j/2} \\hspace{-21.33955pt} d\\delta \\kappa _j \\frac{{\\cal P}(\\kappa _i+\\delta \\kappa _i,\\kappa _j+\\delta \\kappa _j;\\theta )}{\\Delta _i\\Delta _j}\\\\&= \\sum _{\\theta \\in \\Theta } \\frac{N_d(\\theta )}{N_d} {\\cal P}_{ij}(\\theta )\\,,$ where for small enough bins ${\\cal P}_{ij}(\\theta )\\simeq {\\cal P}(\\kappa _i,\\kappa _j;\\theta )$ .",
"In Figure REF we show a comparison between the continuous distribution of distances in a square patch and the discrete case for a coarser grid of non-overlapping cells.",
"We now split this sum over cell separations showing that the contribution from zero separation leads to a finite sampling or shot-noise term, and the remainder resembles the super-sample covariance contribution that dominates for the case of well-separated cells.",
"Since the one-point PDF can be obtained from the two-point PDF by marginalisation, ${\\cal P}(\\kappa _i)=\\int d\\kappa _j {\\cal P}(\\kappa _i,\\kappa _j;\\theta )$ , for zero separation we require that ${\\cal P}(\\kappa _i,\\kappa _j;\\theta =0)={\\cal P}(\\kappa _i)\\delta _D(\\kappa _i-\\kappa _j)$ .",
"In the discrete case the joint two-point PDF at zero separation is given by ${\\cal P}_{ij}(\\theta =0)=\\bar{{\\cal P}}_i \\delta _{ij}/\\Delta _i$ , required by the relation $\\bar{{\\cal P}}_i = \\sum _j \\Delta _j {\\cal P}_{ij}(\\theta =0)$ .",
"The zero separation multiplicity equals the cell number $N_d(\\theta =0)=N_T$ , such that the zero-separation term in equation (REF ) produces a finite sampling (or shot-noise) covariance contributionThis shot-noise term was already described in .",
"Even if there was no spatial correlation in the sample, in the Poisson limit the covariance of finding $N_i$ cells in convergence bin $\\text{cov}_{\\rm SN}(N_i,N_j) = \\bar{N}_i \\delta _{ij}$ , where $\\bar{N}_i$ is the mean number of cells expected to be in the convergence bin centred on $\\bar{\\kappa }_i$ .",
"The number of cells, $N_i$ in a given convergence bin of width $\\Delta _i$ centered on $\\kappa _i$ is related to the histogram-value of the binned PDF from equation (REF ) such that $N_i=N_T\\Delta _i {\\cal P}_i$ and similarly $\\bar{N}_i=N_T\\Delta _i\\bar{{\\cal P}}_i$ .",
"$\\text{cov}_{\\rm FS}({\\cal P}_i,{\\cal P}_j) =\\frac{N_T}{N_d} {\\cal P}_{ij}(\\theta =0)= \\frac{\\bar{{\\cal P}}_i}{\\Delta _i N_T} \\delta _{ij}\\,.$ Unsurprisingly, this finite sampling term is inversely proportional to the number of cells.",
"If the cells are placed at a fixed separation, an increase of the area corresponds to a suppression of the shot-noise covariance term as $\\sigma ^2_{\\rm FS}({\\cal P})\\propto N_T^{-1}\\propto \\Theta _{\\rm side}^{-2}\\propto A_{\\rm survey}^{-1}$ , which is inversely proportional to the area.",
"While in principle the finite sampling term can be rendered arbitrarily small by using an increasing number of cells, this comes at the price of heavy cell overlaps at small separations.",
"In this regime, a small-scale expansion with small $\\Delta _\\xi ^{1/2}(\\theta )=\\sqrt{\\sigma ^2-\\xi (\\theta )}\\ll \\xi (\\theta )\\ll 1$ can be used to capture effects of heavy cell overlaps as discussed in [8].",
"In the closed-form two-point PDF models discussed in Section REF those overlap effects will be automatically included."
],
[
"Super-sample covariance effects",
"Beyond assuming specific (log-)normal models for the bivariate PDF, one can obtain accurate estimates of the two-point PDF in the large separation regime illustrated in Figure REF .",
"A large-scale expansion with small correlation $\\xi (\\theta )\\ll \\Delta _\\xi (\\theta )\\ll 1$ is suitable for well-separated cells where the joint PDF at leading order can be approximated as ${\\cal P}(\\kappa _i,\\kappa _j;\\theta \\gtrsim 2\\theta _{\\rm s}) &\\simeq {\\cal P}(\\kappa _i){\\cal P}(\\kappa _j)[1+\\xi (\\theta )b_1(\\kappa _i)b_1(\\kappa _j) ]\\,,$ with some bias function $b_1$ , which describes how the two-point correlation at a given separation is modulated by the local weak-lensing convergences.",
"This functional has been found to be very robust and the previous result extends what is expected in cosmological density fields [7].",
"When cells are at sufficiently large separations and the initial conditions are Gaussian, this modulation is independent of separation.",
"We will introduce a discretised estimator in equation (REF ) and derive the functional form of those bias functions for specific forms of the two-point PDF discussed in Section REF by an expansion in the correlation $\\xi (\\theta )$ .",
"For the related case of densities in spheres and cylinders, this bias function can be accurately predicted by spherical or cylindrical collapse, respectively.",
"The large-deviation statistics techniques used to predict the one-point weak lensing PDF [4], based on cylindrical collapse for thin redshift slices could be extended to derive the functional form of this bias, although this is beyond the scope of this work.",
"This large-separation expansion predicts a leading order super-sample covariance term and further sub-leading contributions that we will describe in Section .",
"When all cells are at sufficiently wide separations $\\theta \\gtrsim 2\\theta _s$ , the covariance is obtained from integrating the large-separation limit of the two-point PDF (REF ) to obtain a super-sample covariance (SSC) covariance term $\\text{cov}_{\\rm SSC}({\\cal P}(\\kappa _i),{\\cal P}(\\kappa _j)) = \\bar{\\xi }\\, (b_1{\\cal P})(\\kappa _i) (b_1{\\cal P})(\\kappa _j) \\,,$ where we defined the mean correlation $\\bar{\\xi }=\\int ^{\\theta _{\\rm max}} \\hspace{-14.22636pt} d\\theta \\, P_d(\\theta ) \\xi (\\theta )\\,,$ with $P_d(\\theta )$ being the probability distribution of angular distances and $\\theta _{\\rm \\max }$ being the maximum distance set by the size of the survey patch, respectively.",
"This mean correlation agrees with the variance of the mean of $\\kappa $ measured across different patches, so $\\bar{\\xi }=\\sigma _{\\bar{\\kappa }}^2$ .",
"For a square patch of side length $\\Theta _{\\rm side}$ , the maximal distance in the patch is $\\theta _{\\rm max}\\simeq \\sqrt{2} \\Theta _{\\rm side}$ where the normalised distance distribution $P_d(\\hat{\\theta })$ is given in Equation (REF ), by conservation of probability we have $P_d(\\theta )=P_d(\\hat{\\theta }) \\frac{d\\hat{\\theta }}{d\\theta }= P_d\\left(\\hat{\\theta }=\\frac{\\theta }{\\Theta _{\\rm side}}\\right) \\frac{1}{\\Theta _{\\rm side}} \\stackrel{\\hat{\\theta }\\ll 1}{\\propto } \\Theta _{\\rm side}^{-2} \\,.$ The main contribution in the mean correlation integral comes from small angles $\\theta $ , where the correlation function is largest and $P_d(\\hat{\\theta })\\propto \\hat{\\theta }$ .",
"Hence, the SSC covariance term from equation (REF ) scales as $\\text{cov}_{\\rm SSC}({\\cal P}_i,{\\cal P}_j)\\propto \\Theta _{\\rm side}^{-2} \\propto A_{\\rm survey}^{-1}$ , which is inversely proportional to the survey area.",
"In the discrete case, the covariance can be expressed as a sum of the previously discussed finite sampling term (REF ) and a super-sample covariance like term $\\rm {cov}({\\cal P}_i,{\\cal P}_j) &=\\langle {\\cal P}_j{\\cal P}_j\\rangle -\\bar{{\\cal P}}_i\\bar{{\\cal P}}_j \\\\&= \\text{cov}_{\\rm FS}({\\cal P}_i,{\\cal P}_j) \\\\&+\\frac{1}{N_d} \\sum _{\\theta \\in \\Theta \\setminus \\lbrace 0\\rbrace } \\Big [{\\cal P}_{ij}(\\theta )-\\bar{{\\cal P}}_i\\bar{{\\cal P}}_j \\underbrace{\\frac{N_d}{N_T(N_T-1)}}_{\\simeq 1}\\Big ]\\ \\,.$ If cells at separation $\\theta $ were completely independent, such that ${\\cal P}_{ij}(\\theta )=\\bar{{\\cal P}}_i\\bar{{\\cal P}}_j$ , then only the shot-noise term from the finite sampling of cells would remain.",
"The second term in equation (REF ) includes significant contributions from cells at wide separation $\\theta \\gtrsim 2\\theta _s$ , where we can use the large-separation PDF expansion (REF ).",
"This leads to a super-sample covariance term analogous to equation (REF ) ${\\rm cov}_{\\rm SSC}({\\cal P}_i,{\\cal P}_j)=\\bar{\\xi }(b_1{\\cal P})_i(b_1{\\cal P})_j\\,,$ where $(b_1{\\cal P})_i$ are obtained from a bin-average of the involved functions resembling equation (REF ) and the mean correlation can be obtained as $\\bar{\\xi }=\\sum _{\\theta \\in \\Theta \\setminus \\lbrace 0\\rbrace } \\frac{N_d(\\theta )}{N_d} \\xi (\\theta )\\,.$ Using the relationship $\\left\\langle \\kappa ^{\\prime }_j|\\kappa _i\\right\\rangle _{\\kappa ^{\\prime }_j}=b_1(\\kappa _i)/\\bar{\\xi }$ , the value of the bias function in the bin $\\kappa _i$ can be estimated by comparing the mean of the convergence at separation $\\theta =2\\theta _s$ , $\\kappa ^{\\prime }_j$ , from this given convergence $\\kappa _i$ to the average correlation $\\bar{\\xi }$ using a sum over the 6 nearest neighbour (NN) pixels at that separation $\\hat{b}_1(\\kappa _i) = \\frac{1}{\\bar{\\xi }}\\frac{\\sum _{\\rm P} \\sum _{\\rm P^{\\prime }\\in NN(2\\theta _s)}\\chi _i(\\kappa _{\\rm P})\\kappa _{\\rm P^{\\prime }}}{6\\sum _{\\rm P}\\chi _i(\\kappa _{\\rm P})}\\,.$ This definition is the analogue of the estimator for densities in spheres and cylinders , , .",
"Since the correlation function decreases with increasing angular separation and the probability distribution of cell distances peaks at roughly half the patch size, this super-sample covariance term is the most relevant for small patches while it is heavily suppressed for full sky maps.",
"In some cases, like for simulation-based analysis or estimation of covariances, it is desirable to construct small patches which closely resemble the full sky behaviour.",
"The main contribution from the SSC covariance term can be reduced by subtracting the mean $\\kappa $ in every small patch, thus reducing the mean correlation $\\bar{\\xi }$ and modifying the shape of the bias function $b$ as we will discuss in Section REF .",
"Figure: Two-point correlation of circular cells of radius θ s ≃7.3 ' \\theta _s\\simeq 7.3^{\\prime } (solid black) in comparison to the matter correlation (dashed black), the variance in cells (dashed blue) and the difference Δ ξ (θ)\\Delta _\\xi (\\theta ) (red line) that vanishes in the limit of small separations.",
"The blue dotted lines indicate the point at which the cell separation equals twice the smoothing scale, while the green dotted lines indicate the point where the correlation is half of the variance, corresponding to slightly smaller scales.To validate our covariance models, we generate a large set of Gaussian as well as shifted lognormal (FLASK) $\\kappa $ maps (with the shift parameter tuned to fit the $\\kappa $ -PDF skewness).",
"We start from full-sky convergence maps (healpix $N_{\\rm side}=4096$ corresponding to $0.85^{\\prime }$ resolution) that we divide into 1200 non-overlapping square patches with side-length of 5deg.",
"We build a data vector $\\vec{S}$ from the values of the PDF histogram in the central region $-0.024<\\kappa <0.026$ measured from the top-hat smoothed $\\kappa $ map by considering the regular grid of all cells (overlapping) or only cells separated by at least twice the smoothing radius (non-overlapping).",
"The covariance matrix of the binned $\\kappa $ -PDF across all those patches is obtained using the formula $C_{ij} = \\langle (S_i-\\bar{S}_i)(S_j - \\bar{S}_j) \\rangle \\,,\\quad \\bar{S}_i = \\langle S_i \\rangle ~\\,,$ where $\\langle \\cdot \\rangle $ indicates an ensemble average over different map realisations for the full sky or patches, respectively.",
"We rescale all covariances to mimic a total Euclid-like survey area $A_{\\rm survey}=15,000$ deg$^2$ using the rescaling factor $A_{\\rm patch}/A_{\\rm survey}$ for the patches and $A_{\\rm fullsky}/A_{\\rm survey}$ for the full-sky maps.",
"We verified the validity of this rescaling using covariance measurements from full sky maps in addition to the theoretical arguments in Section REF .",
"For visualisation purposes, it is often useful to normalise the covariance matrix, with components $C_{ij}$ , by its diagonal components, which defines the correlation matrix $R$ such that $R_{ij} = \\frac{C_{ij}}{\\sqrt{C_{ii}C_{jj}}}\\,.$ In Figures REF and REF we illustrate two effects that impact the overall size and structure of the PDF covariance and the derived correlation matrix, finite sampling (shot noise) and super-sample covariance.",
"In Figure REF we show the diagonal of the covariance matrix for FLASK patches (solid lines) and Gaussian patches (dashed lines).",
"In Figure REF we show the corresponding correlation matrices ordered by decreasing covariance, with the upper triangle showing the FLASK patches and the lower triangle showing the Gaussian patches.",
"Figure: Diagonal of the covariance matrix for the non-Gaussian FLASK PDFs (solid lines) and the Gaussian version (dashed lines) measured from small patches with non-overlapping cells (red), overlapping cells (blue) and overlapping cells with mean-subtraction in every patch (purple).",
"The Gaussian case shows a symmetric covariance contribution as expected.",
"The corresponding correlation matrices are shown in Figure .",
"Additionally, we show our theoretical predictions for the Gaussian case (dotted lines, partially obscured by the dashed lines).",
"The dotted vertical lines indicate the range of κ\\kappa values considered for the covariance matrices and Fisher forecasts.Figure: Correlation matrix between the bins of the weak lensing convergence κ\\kappa PDF measured at a smoothing scale ≃7.3 ' \\simeq 7.3^{\\prime } from (5 deg) 2 ^2 patches of a full sky FLASK map of κ\\kappa including the nonzero mean in the patches (left and middle panel) with non-overlapping (NO) cells (left) and overlapping cells (middle) and κ-κ ¯\\kappa -\\bar{\\kappa } after subtracting the mean with overlapping cells (right panel).",
"The upper triangles labelled by FLASK show results from shifted-lognormal realisations, while the lower triangles those of Gaussian realisations.",
"The diagonal of the covariance matrix is shown in Figure .We consider the three cases for the covariance in small patches of side length 5deg: raw $\\kappa $ measured from non-overlapping cells (red lines in Figure REF , left panel in Figure REF ), raw $\\kappa $ measured from overlapping cells (blue lines in Figure REF , middle panel in Figure REF ), and mean-subtracted $\\tilde{\\kappa }=\\kappa -\\bar{\\kappa }$ from overlapping cells (purple lines in Figure REF , right panel in Figure REF ) .",
"We see that while the lognormal maps have a visible asymmetry in the covariance diagonal compared to the Gaussian maps, both lead to very similar correlation matrices.",
"Comparing the red and blue lines shows that given a fixed total area, increasing the number of cells for the PDF measurement decreases the finite sampling (or shot noise term) that determines the covariance matrix diagonal, as we explained in Section REF .",
"The suppression of the diagonal for the case of an increased number of cells comes at the price of making cells overlap which creates a strong positive correlation band around the diagonal, as can be seen from comparing the left and middle panel in Figure REF .",
"Comparing the blue and purple lines in Figure REF illustrates that subtracting the mean in small patches significantly reduces the impact of the super-sample covariance term discussed in Section REF and causes the 4-tiled pattern in the middle panel of Figure REF to transform to a 9-tiled pattern in the right panel.",
"Additionally, the removal of this dominant term causes the overlap contribution in the band along the diagonal to become more apparent.",
"As we will see later, the mean-subtracted convergence in small patches provides a proxy for the convergence on the full sky, as evident from Figure REF .",
"While most of our discussion is focused on the underlying weak lensing convergence field, we discuss the inclusion of shape noise arising from cosmic shear measurements in Section REF ."
],
[
"Specific models for the joint two-point PDF",
"As evident from equation (REF ), the PDF covariance can be computed directly if the two-point PDF ${\\cal P}(\\kappa _i,\\kappa _j;\\theta )$ is known a priori for all separations.",
"In the following we will discuss three possible cases relying on closed-form expressions for that joint PDF, a bivariate Gaussian (normal) PDF, a bivariate shifted lognormal PDF and a bivariate PDF built from a marginal one-point PDF and a correlation structure, for example specified by an expansion for large separations or heavy overlaps."
],
[
"Bivariate Gaussian",
"A Gaussian joint PDF is parametrised by the one-point PDF variance $\\sigma ^2$ and the two-point correlation between cells $\\xi =\\xi _{12}(\\theta )$ such that ${\\cal P}_{\\rm G}(\\kappa _1,\\kappa _2;\\theta )&={\\cal P}_G(\\kappa _1,\\kappa _2;\\sigma ^2,\\xi _{12}(\\theta ))\\\\&= \\frac{\\exp \\left[-\\frac{\\sigma ^2}{2d}[{\\kappa _1}^2+{\\kappa _2}^2]+\\frac{\\xi _{12}(\\theta )}{d}\\kappa _1\\kappa _2\\right]}{2\\pi \\sqrt{d}}\\,,$ with the determinant $d=\\sigma ^4-{\\xi _{12}}^2=\\sigma ^4[1-(\\xi _{12}/\\sigma ^2)^2]$ .",
"One can perform a large-separation expansion for $\\xi _{12}/\\sigma ^2\\ll 1$ valid in the regime of no overlaps as shown in Figure REF .",
"At leading order, one can approximate $d\\simeq \\sigma ^4$ and expand the remaining $\\xi _{12}(\\theta )$ term in the exponential to obtain $\\frac{{\\cal P}_{\\rm G}(\\kappa _1,\\kappa _2;\\theta )}{{\\cal P}_{\\rm G}(\\kappa _1){\\cal P}_{\\rm G}(\\kappa _2)}&= 1+\\xi _{12}(\\theta )\\frac{\\kappa _1}{\\sigma ^2} \\frac{\\kappa _2}{\\sigma ^2} +\\mathcal {O}(\\xi ^2)\\,,$ which is a meaningful expansion in the regime $\\kappa \\lesssim \\mathcal {O}(\\sigma )$ and reproduces the linear `Kaiser' bias function , , $b_{1,\\rm G}(\\kappa )=\\kappa /\\sigma ^2\\,.$ We will discuss an expansion up to higher orders in $\\xi (\\theta )/\\sigma ^2$ in the context of the large-separation expansion in Section REF .",
"The impact of mean-subtraction.",
"For small patches, the mean convergence $\\kappa _s$ will significantly fluctuate from patch to patch with a sizeable average correlation $\\bar{\\xi }=\\sigma _{\\kappa _s}^2$ causing the super-sample covariance.",
"To better emulate measurements of modern surveys on a significant fraction of the full sky, subtracting this mean convergence in each patch in the spirit of the peak-background split [3], , can be desirable.",
"The two-point PDF of the mean-subtracted lensing convergence $\\tilde{\\kappa }_i=\\kappa _i-\\bar{\\kappa }$ is obtained from a trivariate Gaussian PDF for $\\kappa _1,\\kappa _2$ and the background value in the patch $\\kappa _s=\\bar{\\kappa }$ , which is then integrated out ${\\cal P}_{\\rm G}(\\tilde{\\kappa }_1,\\tilde{\\kappa }_2)=\\!\\int \\!\\!",
"d\\kappa _s {\\cal P}_{\\rm G}(\\kappa _1=\\tilde{\\kappa }_1+\\kappa _s,\\kappa _2=\\tilde{\\kappa }_2+\\kappa _s,\\kappa _s)\\,.$ The covariance matrix is of the form $\\Sigma _{\\kappa _1,\\kappa _2,\\kappa _s}=\\begin{pmatrix} \\sigma ^2 & \\xi _{12} & \\xi _{1s} \\\\\\xi _{12} & \\sigma ^2 & \\xi _{2s}\\\\\\xi _{1s} & \\xi _{2s} & \\sigma ^2_s\\end{pmatrix}\\,,$ where we assume $\\bar{\\xi }=\\sigma ^2_s=\\xi _{1s}=\\xi _{2s}$ , which holds for large enough patches compared to the cell size.",
"The integration yields a bivariate Gaussian PDF for the mean-subtracted convergence ${\\cal P}(\\tilde{\\kappa }_1,\\tilde{\\kappa }_2)$ with reduced variance $\\sigma _{\\tilde{\\kappa }}^2=\\sigma ^2-\\bar{\\xi }$ and reduced correlation $\\xi _{\\tilde{\\kappa }}(\\theta )=\\xi (\\theta )-\\bar{\\xi }$ , where the reduction is set by the variance at the patch scale, $\\bar{\\xi }=\\sigma _s^2$ .",
"The mean-subtracted correlation $\\xi -\\bar{\\xi }$ now leads to a vanishing leading order super-sample covariance term from equation (REF ) and creates a new effective leading order term that we will discuss in Section REF .",
"Obviously this above described formalism has a long history which goes back to early papers on constrained Gaussian random fields e.g and Covariance.",
"When obtaining the covariance according to equation (REF ) from the large-separation expansion of the Gaussian (REF ), the leading 1 disappears and the correlation $\\xi _{12}(\\theta )$ is integrated over the distance distribution to yield $\\bar{\\xi }$ which sets the amplitude of a super-sample covariance like term we will discuss further in Section REF $\\text{cov}_{\\rm G}({\\cal P}(\\kappa _i),{\\cal P}(\\kappa _j))\\!=\\!",
"\\bar{\\xi } (b_{1}{\\cal P})_{\\rm G}(\\kappa _i)(b_{1}{\\cal P})_{\\rm G}(\\kappa _j)+\\mathcal {O}(\\overline{\\xi ^2})\\,.$ In Figure REF we compare the correlation matrices as measured (upper triangle) and predicted (lower triangle) for Gaussian fields of the three cases discussed before (generated with FLASK, with the same initial power spectrum and smoothing as before), from non-overlapping cells with raw $\\kappa $ (left) along with overlapping cells of raw $\\kappa $ (middle) and mean-subtracted $\\tilde{\\kappa }=\\kappa -\\bar{\\kappa }$ (right panel).",
"In the non-overlapping case on the left-hand side, we can see how the finite sampling term (REF ) causes the diagonal to be most pronounced with a sub-dominant 4-tile pattern showing the leading order super-sample covariance like term from equation (REF ).",
"When making cells overlap as shown in the middle panel, the finite sampling term becomes irrelevant and the 4-tile pattern [due to the factorisation of that covariance into a product of two independent bias functions with one zero-crossing each] as seen in equation (REF )] becomes more pronounced, while the resulting cell overlap causes a significant increase of the correlation in nearby bins as can be seen in a band along the diagonal.",
"Considering the mean-subtracted convergence shown in the right panel will remove the leading order, super-sample covariance term from equation (REF ) and instead lead to a 9-tile pattern [due to the factorisation of that covariance contribution into a products of two bias functions with two zero-crossings each, see equation (REF )].",
"Figure: Correlation matrix between the bins of the Gaussian weak lensing convergence κ\\kappa PDF at a smoothing scale ≃7.3 ' \\simeq 7.3^{\\prime } measured from 1200 (5 deg) 2 ^2 patches for each of 10 full sky Gaussian maps (upper triangle) and the theory prediction (lower triangle) for κ\\kappa including the nonzero mean in the patches (left and middle panel) with non-overlapping (NO) cells (left) and overlapping cells (middle) and κ-κ ¯\\kappa -\\bar{\\kappa } after subtracting the mean with overlapping cells (right panel).",
"The predictions for the diagonal of the covariance matrices are shown in Figure ."
],
[
"Bivariate shifted lognormal",
"A shifted lognormal joint PDF for the lensing convergence $\\kappa $ is parametrised by the variance $\\sigma ^2_\\kappa $ and two-point correlation $\\xi _\\kappa =\\xi _{12,\\kappa }(\\theta )$ along with a shift parameter $s$ related to the skewness, which we will keep fixed throughout here.",
"We can define a Gaussian-distributed variable $g(\\kappa )$ with zero-mean $g(\\kappa )=\\ln \\left(\\frac{\\kappa }{s}+1\\right)-\\mu _{\\rm G} =\\ln \\left(\\frac{\\kappa }{s}+1\\right)+\\frac{\\sigma _{\\rm G}^2}{2}\\,,$ where $\\mu _{\\rm G}=-\\sigma _{\\rm G}^2/2$ is fixed to ensure $\\langle \\kappa \\rangle =0$ and the Gaussian variance is $\\sigma _{\\rm G}^2=\\ln \\left(1+\\sigma ^2_\\kappa /s^2\\right)$ .",
"This can be used to construct the shifted lognormal univariate PDF as ${\\cal P}_{\\rm LN}(\\kappa ;\\sigma ^2_G,s) = \\frac{\\Theta (\\kappa +s)}{\\sqrt{2\\pi }\\sigma _{\\rm G}(\\kappa +s)}\\exp \\left[\\frac{-g(\\kappa )^2}{2\\sigma _{\\rm G}^2}\\right]\\,,$ where $\\Theta $ indicates the Heaviside step function, such that the PDF is zero if $\\kappa \\le -s$ .",
"The shift parameter, $s$ , can be set to replicate the desired skewness $s &= \\frac{\\sigma _\\kappa }{\\tilde{\\mu }_3}\\left(1+y(\\tilde{\\mu }_3)^{-1}+y(\\tilde{\\mu }_3)\\right)-\\langle \\kappa \\rangle \\approx 3\\frac{\\sigma _\\kappa ^4}{\\langle \\kappa ^3\\rangle }, \\\\y(\\tilde{\\mu }_3) &= \\@root 3 \\of {\\frac{2+\\tilde{\\mu }_3^2+\\tilde{\\mu }_3\\sqrt{4+\\tilde{\\mu }_3^2}}{2}}\\approx 1\\,, $ where $\\langle \\kappa \\rangle $ is the desired mean, $\\sigma $ the target variance and $\\tilde{\\mu }_3=\\langle \\kappa ^3\\rangle /\\sigma _\\kappa ^3$ the skewness.",
"The shift parameter is typically positive and for our fiducial weak lensing convergence case we use $s=0.1145$ throughout the main text as done in .",
"In the executive summary, we use a different shift parameter because the simulations have a different cosmology.",
"The Gaussian limit can be reproduced by sending $s/\\sigma _\\kappa \\rightarrow \\infty $ .",
"The bivariate shifted lognormal distribution also depends on the correlation $\\xi _{\\rm G}=\\ln \\left(1+\\xi _\\kappa /s^2\\right)$ and the determinant $d=\\sigma ^4_{\\rm G}-\\xi _{\\rm G}^2$ ${\\cal P}_{\\rm LN}&(\\kappa _1,\\kappa _2;\\sigma ^2_G,\\xi _G,s)=\\frac{\\Theta (\\kappa _1+s)\\Theta (\\kappa _2+s)}{2\\pi \\sqrt{d}(\\kappa _1+s)(\\kappa _2+s)}\\\\&\\times \\exp \\left[-\\frac{\\sigma _{\\rm G}^2}{2d}[g(\\kappa _1)^2+g(\\kappa _2)^2]+\\frac{\\xi _G}{d} g(\\kappa _1)g(\\kappa _2)\\right]\\,.$ The joint shifted lognormal PDF can be expanded at large separations where $\\xi _G(\\xi _\\kappa )/\\sigma _G^2\\ll 1$ to obtain up to leading order in $\\xi _\\kappa $ $\\frac{{\\cal P}_{\\rm LN}(\\kappa _1,\\kappa _2;\\theta )}{{\\cal P}_{\\rm LN}(\\kappa _1){\\cal P}_{\\rm LN}(\\kappa _2)}&=1+\\frac{\\xi _\\kappa (\\theta )}{s^2}\\frac{g(\\kappa _1)}{\\sigma _G^2}\\frac{g(\\kappa _2)}{\\sigma _G^2}+\\mathcal {O}(\\xi _\\kappa ^2)\\,.$ In particular, it is useful to define a leading order bias function that resembles the Gaussian expression $b_{1,\\rm G}=\\kappa /\\sigma ^2$ $b_{1,\\rm LN}(\\kappa )&=\\frac{g(\\kappa )}{s\\sigma _G^2}\\simeq \\frac{sg(\\kappa )}{\\sigma _{\\kappa }^2}\\,,$ where for $\\kappa \\ll s$ we have $sg(\\kappa )\\simeq \\kappa +\\sigma _\\kappa ^2/(2s)$ and hence reproduce the Gaussian result modulo a small shift.",
"For larger $\\kappa $ the shifted lognormal result deviates from the Gaussian one to closely resemble measurements from simulated $\\kappa $ maps using the eestimator (REF ) as illustrated in Figure REF .",
"The impact of mean-subtraction.",
"When describing small patches which will feature different background `sample' convergence values $\\kappa _s$ , one needs to integrate over a trivariate PDF of the convergences $\\kappa _{1/2}$ and the sample value $\\kappa _s$ , similarly as for the mean-subtraction in the Gaussian case (REF ) ${\\cal P}(\\tilde{\\kappa }_1,\\tilde{\\kappa }_2)&=\\!\\!\\int \\!\\!",
"d\\kappa _s {\\cal P}_{\\rm LN}(\\kappa _1\\!=\\!\\tilde{\\kappa }_1+\\kappa _s,\\kappa _2\\!=\\!\\tilde{\\kappa }_2+\\kappa _s,\\kappa _s)\\,.$ Figure: κ\\kappa -dependent clustering as measured by the cell bias from one full-sky map in the Takahashi simulationin comparison to the theoretical prediction for a shifted lognormal field from equation ().To evaluate this, let us assume that $\\tilde{\\kappa }$ is a shifted log-normal random variable , i.e.",
"given in terms of a zero-mean Gaussian random variable $\\tilde{g}=\\ln (\\tilde{\\kappa }/\\tilde{s}+1)+\\tilde{\\sigma }_G^2/2$ with variance $\\tilde{\\sigma }_{\\rm G}^2=\\ln \\left(1+\\sigma _{\\tilde{\\kappa }}^2/\\tilde{s}^2\\right)$ and a new lognormal shift $\\tilde{s}$ .",
"The correlation of the Gaussian variables is given by $\\tilde{\\xi }_{\\rm G}=\\ln (1+\\xi _{\\tilde{\\kappa }}/\\tilde{s}^2)$ .",
"To find an effective shifted lognormal approximation for $\\tilde{\\kappa }$ , we must determine variance, correlation and the shift parameter $\\tilde{s}$ .",
"Recall that we know the variance of $\\kappa \\equiv \\kappa _s + \\tilde{\\kappa }$ from any procedure for calculating the matter power spectrum.",
"In the same way, we can also know the covariance $\\langle \\kappa _i \\kappa _s\\rangle =\\xi _{is}$ and the variance $\\langle \\kappa _s^2 \\rangle =\\sigma ^2_{s}$ , which we will approximate as identical $\\xi _{is}=\\sigma ^2_s=\\bar{\\xi }$ .",
"So we get the following variance and correlation for the mean-subtracted $\\tilde{\\kappa }$ $\\sigma _{\\tilde{\\kappa }}^2 &= \\langle \\tilde{\\kappa }^2 \\rangle = \\langle \\kappa ^2 \\rangle - 2\\langle \\kappa \\kappa _s\\rangle +\\langle \\kappa _s^2 \\rangle \\approx \\sigma _\\kappa ^2-\\bar{\\xi }\\,,\\\\\\xi _{\\tilde{\\kappa }} &= \\langle \\tilde{\\kappa }_1\\tilde{\\kappa }_2 \\rangle =\\langle \\kappa _1 \\kappa _2\\rangle - \\langle \\kappa _1\\kappa _s \\rangle - \\langle \\kappa _2\\kappa _s \\rangle + \\langle \\kappa _s^2 \\rangle \\nonumber \\\\&\\approx \\xi _\\kappa -\\bar{\\xi }\\,,$ and hence the Gaussian variance $\\tilde{\\sigma }_{\\rm G}^2= \\ln [ 1 + (\\sigma _\\kappa ^2-\\bar{\\xi })/\\tilde{s}^2]$ and correlation $\\tilde{\\xi }_{\\rm G} = \\ln [ 1 + (\\xi _\\kappa -\\bar{\\xi })/{\\tilde{s}^2}]$ .",
"The shift parameter can be related to the skewness as described in equation B.11 of $\\langle \\tilde{\\kappa }^3 \\rangle = \\frac{3}{\\tilde{s}} \\langle \\tilde{\\kappa }^2 \\rangle ^2 + \\frac{1}{\\tilde{s}^3} \\langle \\tilde{\\kappa }^2 \\rangle ^3 \\approx \\frac{3}{\\tilde{s}} \\langle \\tilde{\\kappa }^2 \\rangle ^2 \\Rightarrow \\tilde{s}\\approx \\frac{3\\langle \\tilde{\\kappa }^2 \\rangle ^2}{\\langle \\tilde{\\kappa }^3 \\rangle }\\,.$ Following we use the shift parameter $\\tilde{s}$ to match the skewness of $\\tilde{\\kappa }$ , which is obtained from the third moment of the difference $\\langle \\tilde{\\kappa }^3 \\rangle =&\\ \\langle (\\kappa - \\kappa _s)^3 \\rangle = \\langle \\kappa ^3 \\rangle - 3\\langle \\kappa ^2\\kappa _s \\rangle + 3\\langle \\kappa \\kappa _s^2 \\rangle \\,.$ The third moment of $\\kappa $ can be expressed in terms of the variance $\\sigma _\\kappa $ and original shift parameter $s$ using $\\langle \\kappa ^3\\rangle \\approx 3\\sigma _\\kappa ^4/s$ .",
"Additionally, we can compute the relevant mixed moments of a zero-mean normal random variable $Y$ and a lognormal variable $Z=s_z(e^X-1)$ given in terms of a normal variable $X$ .",
"The relevant relations are $\\langle Y(e^X-1)\\rangle =\\xi _{XY}$ and $\\langle Y^2(e^X-1)\\rangle =\\xi _{XY}^2 \\left(\\sigma _X^2 + 1 - \\sigma _X^{-2}\\right)$ which yields $\\langle Y(e^X-1)^2\\rangle =0$ .",
"This can be used to obtain $\\langle \\kappa ^2\\kappa _s \\rangle =0$ and $\\langle \\kappa \\kappa _s^2 \\rangle \\approx \\bar{\\xi }^2 \\left(\\sigma _{\\rm G}^2 + 1 - \\sigma _{\\rm G}^{-2}\\right)/s$ .",
"Inserting this into equations (REF ) and (REF ) leads to a relation between the shift parameters $s$ and $\\tilde{s}$ , which we found to be practically identical in our case.",
"Covariance.",
"Again, we can integrate the large-separation expansion of the shifted lognormal (REF ) over the distance distribution to obtain the covariance from equation (REF ) and eventually get at leading order $\\text{cov}_{\\rm LN}({\\cal P}(\\kappa _i),{\\cal P}(\\kappa _j))= \\bar{\\xi } (b_{1}{\\cal P})_{\\rm LN}(\\kappa _i)(b_{1}{\\cal P})_{\\rm LN}(\\kappa _j)+\\mathcal {O}(\\overline{\\xi ^2})\\,,$ which is in form analogous to the Gaussian result (REF ) despite the modified ingredients.",
"This explains why the correlation matrix from small patches (with sizable $\\bar{\\xi }$ ) in the FLASK case closely resembles the Gaussian case, as seen in Figure REF while the diagonal is modified due to the non-Gaussian shape of the final convergence PDF as seen in Figure REF .",
"In Figure REF we compare the measurements from FLASK maps to the theoretical predictions for shifted lognormal fields.",
"Figure: (Top) Correlation matrix between the bins of the shifted lognormal weak lensing convergence κ\\kappa PDF at a smoothing scale ≃7.3 ' \\simeq 7.3^{\\prime } comparing FLASK measurements (upper triangles) to the theory predictions (lower triangles).",
"(left) Measurements from 1200 square patches of 5deg size cut from 10 full sky realisations in comparison to the lognormal theory prediction in patches.",
"While the first order bias function b 1, LN b_{1,\\rm LN} () creates the 4-tile pattern, cell overlaps create the positive correlation band along the diagonal.",
"(right) Measurements from 500 full sky realisations in comparison to the lognormal theory prediction on the full sky.",
"The 9-tile pattern is set by the second order bias function b 2, LN b_{2,\\rm LN} (), while cell overlaps create the strong correlation along the diagonal.",
"(Bottom) Diagonal of the covariance matrix for the non-Gaussian FLASK PDFs measured from the full sky (green solid) and as predicted by the lognormal theory (green dotted).",
"We also show the result for the FLASK patches without/with mean subtraction (purple/blue solid) and the lognormal theory computed for the patches without/with mean-subtraction (purple/blue dotted).",
"The shaded bands are indicating estimated errors)In the left upper panel we show the correlation matrix obtained from 1200 FLASK patches of 5deg each cut from 20 full sky realisations (upper triangle) along with the theoretical prediction (lower triangle) finding excellent agreement.",
"We can see that the correlation matrix has a 4-tile structure.",
"This is because the mean correlation $\\bar{\\xi }$ in the patches is sizeable such that the first order covariance term (REF ) dominates the visual appearance.",
"In the upper right panel we show the correlation matrix obtained from 500 full sky FLASK maps (upper triangle) along with the theoretical prediction (lower triangle) finding excellent agreement.",
"We can see that the correlation matrix closely resembles the case of the mean-subtracted convergence measured in small patches, shown in the lower panel of Figure REF .",
"This is because the mean correlation $\\bar{\\xi }$ on a full sky is so small that the first order covariance term (REF ) becomes subdominant compared to the second order term that we will discuss in Section REF .",
"The latter term also appears as leading order contribution in the mean-subtracted case addressed in Section REF .",
"In the bottom panel we focus on the diagonal of the covariance matrix, comparing FLASK maps and theoretical predictions for different cases.",
"This plot shows that while the PDF covariance of the raw $\\kappa $ in small patches looks significantly different, the mean-subtracted $\\tilde{\\kappa }$ in small patches resembles the full sky covariance.",
"The theoretical prediction captures all three behaviours quite well, Considering relative 1-$\\sigma $ errors on the diagonal of the covariance matrix (shaded bands) following from the covariance between the numerically estimated diagonal covariance elements $\\hat{C}_{ii}$ and $\\hat{C}_{jj}$ as given in $\\mathrm {Cov}\\left(\\hat{C}_{ii},\\hat{C}_{jj}\\right) = \\frac{2 C_{ij}^2}{N_{\\rm sim} - 1}\\ ,$ the theoretical predictions capture all three behaviours quite well.",
"We notice that while the lognormal prediction for the full sky and the raw $\\kappa $ in patches agree extremely well with the predictions, there are some minor discrepancies for the mean-subtracted $\\tilde{\\kappa }$ in small patches.",
"As this discrepancy was absent for Gaussian fields shown in Figure REF , we suspect that it is caused by a more complicated patch-to-patch correlation induced by cutting multiple patches from one full sky, which is unaccounted for in the theory describing the mean-subtracted convergence as another shifted lognormal random field."
],
[
"Bivariate Gaussian copula with general marginals",
"In case neither the bivariate normal nor the shifted lognormal distributions can provide a good description of the target PDF as its univariate marginal, one can design a bivariate PDF by specifying the marginal and correlation structure independently.",
"A joint PDF can be built from a given marginal one-point PDF ${\\cal P}(\\kappa )$ and a correlation structure encoded in a copula density $c_{12}$ which quantifies the correlation between ranked variables obtained from the cumulative distribution function $\\mathcal {C}(\\kappa )=\\int ^{\\kappa }\\!",
"d\\tilde{\\kappa }\\, \\mathcal {P}(\\tilde{\\kappa })$ , such that ${\\cal P}(\\kappa _1,\\kappa _2;\\theta )={\\cal P}(\\kappa _1){\\cal P}(\\kappa _2) c_{12}(\\mathcal {C}(\\kappa _1),\\mathcal {C}(\\kappa _2);\\theta )\\,,$ where $\\theta $ indicates angular separation as before.",
"If the $\\kappa _i$ and $\\kappa _2$ are independent, then the copula density is unity $c_{12,\\rm indep}=1$ .",
"A Gaussian copula only depends on the distance through the pairwise correlation $\\xi _{12}(\\theta )$ , $c_{12,\\rm G}(\\mathcal {C}(\\kappa _1),\\mathcal {C}(\\kappa _2);\\xi _{12}(\\theta ))$ and is a known function which can be computed numerically although it has no analytical expression.",
"Beyond those two cases, systematically constructing the dependence structure is possible in the large-separation regime, where the copula can be expanded in powers of the pairwise correlation $\\xi _{12}(\\theta )$ as we will further discuss in Section ."
],
[
"Inclusion of shape noise",
"Observationally, the weak lensing convergence map is obtained from cosmic shear measurements, which are subject to shape noise often constituting the dominant source of noise.",
"Since galaxies are intrinsically elliptical, the observed shear contains a contribution from this intrinsic signal, quantified by the variance of the intrinsic ellipticity $\\sigma _\\epsilon $ .",
"In simulated maps, shape noise can be included by adding a white noise – modeled by a Gaussian random map $\\kappa _{\\rm SN}$ – to the `raw' simulated convergence map $\\kappa $ before smoothing.",
"The standard deviation of this Gaussian is typically assumed to be $\\sigma _{SN}= \\frac{\\sigma _\\epsilon }{\\sqrt{n_g \\cdot \\Omega _{\\theta _s}}} \\propto \\theta _s^{-1}\\,,$ with the shape-noise parameter $\\sigma _\\epsilon $ (here taken to be $\\sigma _\\epsilon =0.30$ ), the galaxy density $n_g$ (here taken to be 30 arcmin$^{-2}$ ) and the solid angle $\\Omega _{\\theta _s}$ (here taken to correspond to circular apertures of radius $\\theta _s=7.3^{\\prime }$ ).",
"Smoothing induced shape noise correlations and covariance.",
"At the raw $\\kappa $ map level, the shape noise values in different pixels are uncorrelated and $\\xi \\equiv 0$ .",
"Top-hat smoothing the field at angular size $\\theta _s$ will correlate shape noise values in overlapping cells (being at separation $\\theta <2\\theta _s$ ).",
"Their correlation is related to the area of the cell overlap (which is a symmetric lens) $A(\\theta <2\\theta _s)=2\\theta _s^2\\arccos \\left(\\frac{\\theta }{2\\theta _s}\\right)-\\frac{\\theta }{2}\\sqrt{4\\theta _s^2-\\theta ^2}\\,.$ The resulting correlation of the top-hat smoothed shot noise contribution to the weak lensing convergence is $\\xi _{\\rm SN}(\\theta < 2\\theta _s)=\\sigma _{\\rm SN}^2A(\\theta )/(\\pi \\theta _s^2)\\,,\\ \\xi _{\\rm SN}(\\theta \\ge 2\\theta _s)=0\\,.$ To obtain the covariance of the shot noise contribution alone, we integrate the bivariate Gaussian with variance $\\sigma _{\\rm SN}^2$ and covariance (REF ) over distances following equation (REF ) to get $\\langle {\\cal P}_{\\rm SN}(\\kappa _i){\\cal P}_{\\rm SN}(\\kappa _j)\\rangle &= \\int _0^{2\\theta _s}\\!\\!\\!\\!\\!\\!",
"d\\theta \\, P_d(\\theta ) {\\cal P}_{\\rm G}(\\kappa _{i},\\kappa _{j};\\sigma _{\\rm SN}^2,\\xi _{\\rm SN}(\\theta )) \\\\&+ {\\cal P}_{\\rm G} (\\kappa _{i};\\sigma _{\\rm SN}^2){\\cal P}_{\\rm G} (\\kappa _j;\\sigma _{\\rm SN}^2) \\int _{2\\theta _s}^\\infty \\!\\!\\!\\!\\!\\!",
"d\\theta \\, P_d(\\theta ) \\,,$ where $P_d(\\theta )$ indicates the distribution of angular distances in the survey area, which is approximately linear for small distances $P_d(\\theta )\\approx 2\\pi \\theta $ .",
"This gives the PDF covariance for the case of pure shape noise as $\\rm {cov}_{\\rm SN}({\\cal P}(\\kappa _i),{\\cal P}(\\kappa _j))&= \\int _0^{2\\theta _s}\\!\\!\\!\\!\\!\\!",
"d\\theta \\, P_d(\\theta ) {\\cal P}_{\\rm G}(\\kappa _{i},\\kappa _{j};\\sigma _{\\rm SN}^2,\\xi _{\\rm SN}(\\theta )) \\\\&-{\\cal P}_{\\rm G} (\\kappa _{i};\\sigma _{\\rm SN}^2){\\cal P}_{\\rm G} (\\kappa _j;\\sigma _{\\rm SN}^2) \\int _0^{2\\theta _s}\\!\\!\\!\\!\\!\\!",
"d\\theta \\, P_d(\\theta )\\,.$ In Figure REF we compare this theoretical result to a measurement across 1000 shape noise realisations for square maps of length 5deg with a top-hat smoothing of $7.3^{\\prime }$ finding excellent agreement.",
"Figure: PDF covariance diagonal (upper panel) and correlation matrix (lower panel) for the pure shape-noise lensing convergence in circular apertures of radius θ s \\theta _s comparing the prediction from equation () to the measurement from shape noise realisations.",
"The slight apparent asymmetry between positive and negative κ\\kappa is caused by the noise due to the finite number of realisations.Shape noise induced convergence PDF convolution.",
"Shape noise impacts the one-point weak lensing convergence PDF as if it was convolved with a zero-centred Gaussian $\\mathcal {P}_{\\kappa +\\rm SN}(\\hat{\\kappa }) &= ({\\cal P}_{\\rm G}(\\sigma _{\\rm SN})*{\\cal P})(\\hat{\\kappa }) \\\\&=\\frac{1}{\\sqrt{2\\pi }\\sigma _{SN}}\\int d\\kappa \\exp \\left(-\\frac{(\\hat{\\kappa }-\\kappa )^2}{2\\sigma _{SN}^2}\\right) \\mathcal {P}(\\kappa )\\,,$ where the shape noise standard deviation $\\sigma _{SN}$ is given in equation (REF ).",
"For the two-point PDF of the smoothed weak lensing we similarly have $\\hat{\\kappa }_i=\\kappa _i+\\kappa _{i,SN}$ and the bivariate PDF is given in terms of a 2D-convolution ${\\cal P}_{\\kappa +\\rm SN}(\\hat{\\kappa }_1,\\hat{\\kappa }_2;\\theta )=\\mathcal {(}{\\cal P}_{\\rm G}(\\sigma _{\\rm SN},\\xi _{\\rm SN}(\\theta ))*{\\cal P})(\\hat{\\kappa }_1,\\hat{\\kappa }_2;\\theta ) \\,,$ where the shape noise contribution is described by a bivariate Gaussian PDF from equation (REF ) with standard $\\sigma _{\\rm SN}$ (REF ) and correlation $\\xi _{\\rm SN}(\\theta )$ (REF ).",
"If the joint PDF of raw $\\kappa _i$ was Gaussian, then the noisy PDF of $\\hat{\\kappa }_i$ arising from the convolution would be another zero-mean Gaussian distribution with covariance matrix $\\Sigma _{\\kappa +\\rm SN}=\\Sigma _{\\kappa }+\\Sigma _{\\rm SN}$ .",
"For the case of a shifted lognormal $\\kappa $ PDF, we recompute the covariance from the convolved two-point PDF (REF ) to find that the main impact of adding shape noise is to change the diagonal of the covariance matrix, shown in Figure REF , while leaving the correlation matrix largely similar.",
"This result was obtained from $N_{\\rm sim}=500$ FLASK maps of the full sky with a smoothing of $7.3^{\\prime }$ applied to the raw $\\kappa $ and the shape-noise added $\\hat{\\kappa }$ , respectively.",
"Considering relative 1-$\\sigma $ errors on the diagonal of the covariance matrix (REF ) evaluated for $N_{\\rm sim}$ the predictions are in good agreement with the measurements.",
"Figure: PDF covariance diagonal for the noise-free lensing convergence as measured from 500 FLASK full sky maps (green dashed, shaded band indicating estimated errors) and predicted for the shifted lognormal (green dotted) in comparison to the noisy lensing convergence as measured from FLASK (green solid, shaded band indicating estimated errors) and predicted from equation () with with the noisy two-point PDF () (green dot-dashed)."
],
[
"Large separation covariance expansions",
"In this Section we will perform an expansion of the joint two-point PDF in powers of the two-point cell correlation function $\\xi _{12}(\\theta )$ , partially with the help of the cumulant generating function (CGF).",
"We will pay particular attention to the leading and next-to-leading order contributions to the covariance matrix, which reveals a super-sample covariance term that is controlled by the average correlation in the survey patch.",
"We will compute the expansion to all orders in $\\xi _{12}$ for a Gaussian field and the two leading order terms for the general non-Gaussian case, illustrated by results for normal and shifted lognormal fields.",
"This will naturally lead to an expansion of the covariance matrix that resembles an eigendecomposition, which we will discuss in Section REF ."
],
[
"Leading order determined by $\\kappa $ -dependent clustering",
"Let us first recall that at leading order in the large-separation regime, the joint two-point PDF can be approximated in terms of the one-point PDF and the `bias' functions indicating the density-dependent two-point clustering ${\\cal P}(\\kappa _1,\\kappa _2,\\theta )\\simeq {\\cal P}(\\kappa _1){\\cal P}(\\kappa _2)[1+\\xi (\\theta )b(\\kappa _1)b(\\kappa _2)]\\,.$ Hence, we have for the covariance of the binned probability $\\text{cov}({\\cal P}(\\kappa _i),{\\cal P}(\\kappa _j)) = \\bar{\\xi }(\\bar{{\\cal P}}b)(\\kappa _i) \\bar{(}{\\cal P}b)(\\kappa _j) + \\delta _{ij} \\frac{\\bar{{\\cal P}}(\\kappa _i)}{\\Delta _i N_T}\\,,$ where $N_T$ is the total number of cells and the mean correlation $\\bar{\\xi }$ is given by equation (REF ).",
"Note that the shot-noise term only acts on the diagonal.",
"For the diagonal, the leading-order prediction for covariance diagonal is then $\\text{var}({\\cal P}(\\kappa )) = \\bar{\\xi }(b\\bar{{\\cal P}})^2(\\kappa )+\\frac{\\bar{{\\cal P}}(\\kappa )}{\\Delta N_T}\\,,$ where $\\Delta $ is the PDF bin width.",
"For a Gaussian field, the bias function is given by linear Kaiser bias $b(\\kappa )={\\kappa }/{\\sigma _\\kappa ^2}$ , which seems to be a decent approximation for the central region of $\\kappa $ -values, see Figure REF .",
"Note that the cell bias $b(\\kappa )$ has at least one zero crossing, for the case of raw $\\kappa $ , around $\\kappa \\simeq 0$ , while for the mean-subtracted case this is modified as described in Section REF .",
"Let us also recall that for non-overlapping cells, typically the `finite sampling' effect dominates (well-described by Poisson).",
"For overlapping cells, the number of cells is extremely large thus effectively removing the finite sampling term, but requiring the treatment of overlaps.",
"The impact of overlapping cells can be captured by either closed-form expressions for the two-point PDF valid at small distances (like for the Gaussian and shifted-lognormal case discussed here) or a heavy overlap expansion complementary to the large separation one as described in [8]."
],
[
"Beyond leading order: Gaussian case",
"For a Gaussian field of zero mean, we have the following simple structure of the joint two-point cumulant generating function in terms of the two-point correlation $\\xi =\\xi _{12}(\\theta )$ as follows $\\varphi _{\\rm G}(\\lambda _1,\\lambda _2)&=\\varphi _{0,\\rm G}(\\lambda _1)+\\varphi _{0,\\rm G}(\\lambda _2) + \\xi \\varphi _{1,\\rm G}(\\lambda _1)\\varphi _{1,\\rm G}(\\lambda _2)\\,,$ where $\\varphi _{0,\\rm G}=\\lambda ^2\\sigma ^2/2$ , $\\varphi _{1,\\rm G}(\\lambda )=\\lambda $ and $\\varphi _{n\\ge 2, \\rm G}(\\lambda )=0$ .",
"By expanding the last term $\\exp [\\xi \\lambda _1\\lambda _2]$ in a power series and converting $\\lambda _i$ to derivatives w.r.t.",
"$\\kappa _i$ , we obtain $\\frac{{\\cal P}_{\\rm G}(\\kappa _1,\\kappa _2,\\theta )}{{\\cal P}_{\\rm G}(\\kappa _1){\\cal P}_{\\rm G}(\\kappa _2)}&= 1+\\sum _{n=1}^\\infty \\frac{\\xi ^n(\\theta )}{n!",
"}b_{n,\\rm G}(\\kappa _1)b_{n,\\rm G}(\\kappa _2)\\\\b_{n,\\rm G}(\\kappa )&=\\frac{(-1)^n}{{\\cal P}_{\\rm G}(\\kappa )}\\frac{\\partial ^n{\\cal P}_{\\rm G}(\\kappa )}{\\partial \\kappa ^n} = \\frac{1}{\\sigma ^n} He_n\\left(\\frac{\\kappa }{\\sigma }\\right)\\,,$ which can be written in terms of the probabilist's Hermite polynomials $He_n$ $He_n(x)=(-1)^n\\exp \\left[\\frac{x^2}{2}\\right]\\frac{d}{dx^n}\\exp \\left[-\\frac{x^2}{2}\\right]\\,,$ when using $x=\\kappa /\\sigma $ .",
"The first two terms read $b_{1,\\rm G}(\\kappa )&=\\frac{\\kappa }{\\sigma ^2}\\,,\\quad b_{2,\\rm G}(\\kappa )=\\frac{\\kappa ^2-\\sigma ^2}{\\sigma ^4}=b_{1,\\rm G}^2-\\frac{1}{\\sigma ^2}\\,,$ which can be also obtained from a direct expansion of the joint PDF (REF ).",
"This leads to the following expansion for the covariance following equation (REF ) ${\\rm cov}&({\\cal P}_{\\rm G}(\\kappa _1),{\\cal P}_{\\rm G}(\\kappa _2))=\\sum _{n=1}^\\infty \\frac{\\overline{\\xi ^n}}{n!}",
"\\frac{\\partial ^n{\\cal P}_{\\rm G}(\\kappa _1)}{\\partial {\\kappa _1}^n} \\frac{\\partial ^n{\\cal P}_{\\rm G}(\\kappa _2)}{\\partial {\\kappa _2}^n}\\\\&=\\sum _{n=1}^\\infty \\frac{\\overline{\\xi ^n}}{n!\\sigma ^{2n}} He_n\\left(\\frac{\\kappa _1}{\\sigma }\\right){\\cal P}_{\\rm G}(\\kappa _1) He_n\\left(\\frac{\\kappa _2}{\\sigma }\\right){\\cal P}_{\\rm G}(\\kappa _2)\\,,\\\\\\overline{\\xi ^n} &:= \\int d\\theta P_d(\\theta )\\xi ^n(\\theta )\\simeq \\sum _{\\theta \\in \\Theta \\setminus \\lbrace 0\\rbrace } \\frac{N_d(\\theta )}{N_d} \\xi ^n(\\theta )\\,,$ where in the second line we have defined the average of powers of the two-point correlation function over the survey patch in analogy to the mean from equation (REF ).",
"The averages of powers of the correlation function $\\overline{\\xi ^n}$ compared to powers of the variance $\\sigma ^{2n}$ control the large-separation expansion of the covariance and their hierarchy is illustrated in Figure REF .",
"This plot also shows that the covariance contributions scale inverse proportional to the survey area $\\overline{\\xi ^n}\\propto A^{-1}$ .",
"Figure: Scaling of the average of powers of the two-point cell correlation function compared to the variance for the raw κ\\kappa maps in 5deg patches (red plusses) and the full sky result rescaled by a factor of A patch /A fullsky A_{\\rm patch}/A_{\\rm fullsky} (blue plusses).",
"Aside from the removal of the first order term, those hierarchies closely resemble each other as well as the mean-subtracted κ ˜\\tilde{\\kappa } maps in patches (red crosses).Due to the hierarchical ordering of averages of the correlation function we expect the two leading order terms in the regime $\\kappa \\lesssim \\mathcal {O}(\\sigma )$ to be $\\frac{{\\rm cov}({\\cal P}_{\\rm G}(\\kappa _1),{\\cal P}_{\\rm G}(\\kappa _2))}{{\\cal P}_{\\rm G}(\\kappa _1){\\cal P}_{\\rm G}(\\kappa _2)} &=\\frac{\\bar{\\xi }}{\\sigma ^2} \\frac{\\kappa _1}{\\sigma }\\frac{\\kappa _2}{\\sigma } \\\\&+\\frac{\\overline{\\xi ^2}}{2\\sigma ^4} \\frac{\\kappa _1^2-\\sigma ^2}{\\sigma ^2} \\frac{\\kappa _2^2-\\sigma ^2}{\\sigma ^2}\\\\&+\\mathcal {O}\\left(\\frac{\\overline{\\xi ^3}}{\\sigma ^6}\\right)\\,,$ which predicts a super-sample covariance leading order term similar to the separate universe picture (REF ), being proportional to the variance of the mean $\\kappa $ in different patches, $\\bar{\\xi }=\\sigma ^2(\\kappa _s)$ , and the derivative of the PDF.",
"This large-separation expansion of the covariance will be connected to an eigendecomposition in Section REF .",
"Here, the hierarchical ordering of $\\overline{\\xi ^n}/\\sigma ^{2n}$ will reflect in the eigenvalues and the bias functions multiplied by the Gaussian PDF evaluated at a given $\\kappa $ -value will yield the eigenvectors (after orthogonalisation and appropriate normalisation).",
"In this context, the predicted zero-crossings of the first bias function $b_{1,\\rm G}(\\kappa )$ at $\\kappa =0$ and of the second $b_{2,\\rm G}(\\kappa )$ at $\\kappa =\\sigma ^2$ will translate into the approximate zero crossings of the first two eigenvectors as we will show later in Figure REF ."
],
[
"Beyond leading order: general non-Gaussian case",
"Higher-order terms are obtained from expanding the joint two-point cumulant generating function in powers of the two-point correlation $\\xi =\\xi _{12}(\\theta )$ as follows $\\varphi (\\lambda _1,\\lambda _2)&=\\varphi _0(\\lambda _1)+\\varphi _0(\\lambda _2) + \\xi \\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)\\\\&+\\frac{\\xi ^2}{2} [\\varphi _1^2(\\lambda _1)\\varphi _2(\\lambda _2)+\\varphi _2(\\lambda _1)\\varphi _1^2(\\lambda _2)] + \\mathcal {O}(\\xi ^3)$ We are then in position to compute the (connected part) of the joint PDF, $&{\\cal P}(\\kappa _1,\\kappa _2)-{\\cal P}(\\kappa _1){\\cal P}(\\kappa _2)=\\int \\frac{\\lambda _1}{2\\pi \\hbox{i}}\\int \\frac{\\lambda _2}{2\\pi \\hbox{i}}\\nonumber \\\\& \\exp \\left[-\\lambda _1\\kappa _1-\\lambda _2\\kappa _2+\\varphi (\\lambda _1,-\\lambda _1)+\\varphi (\\lambda _2,-\\lambda _2)\\right]\\times \\nonumber \\\\& \\Big (\\xi \\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)+\\frac{\\xi ^2}{2}\\varphi _1^2(\\lambda _1)\\varphi _1^2(\\lambda _2)\\\\&+ \\frac{\\xi ^2}{2} [\\varphi _1^2(\\lambda _1)\\varphi _2(\\lambda _2)+\\varphi _2(\\lambda _1)\\varphi _1^2(\\lambda _2)] +\\mathcal {O}(\\xi ^3)\\Big )$ We obtain the covariance according to equation (REF ) after integrating over $\\lambda _i$ and the distance distribution and symmetrising $&\\frac{{\\rm cov}({\\cal P}(\\kappa _1),{\\cal P}(\\kappa _2))}{{\\cal P}(\\kappa _1){\\cal P}(\\kappa _2)} =\\bar{\\xi }b_1(\\kappa _1)b_{1}(\\kappa _2) \\\\&+\\frac{\\overline{\\xi ^2}}{2}\\left[(b_{2}+q_1)(\\kappa _1)(b_{2}+q_1)(\\kappa _2)-q_1(\\kappa _1)q_1(\\kappa _2)\\right]\\\\&+\\mathcal {O}(\\overline{\\xi ^3})\\,,$ where we defined $(b_1\\mathcal {P})(\\kappa )&=\\int \\frac{d\\lambda }{2\\pi \\hbox{i}}\\varphi _1(\\lambda )\\exp [-\\lambda \\kappa +\\varphi (\\lambda )] \\\\(b_2\\mathcal {P})(\\kappa )&=\\int \\frac{d\\lambda }{2\\pi \\hbox{i}}\\varphi _1(\\lambda )^2\\exp [-\\lambda \\kappa +\\varphi (\\lambda )] \\\\(q_1\\mathcal {P})(\\kappa )&=\\int \\frac{d\\lambda }{2\\pi \\hbox{i}}\\varphi _2(\\lambda )\\exp [-\\lambda \\kappa +\\varphi (\\lambda )]\\,.$ For a saddle-point approximation of the integral, we would typically expect $b_2\\simeq b_1^2$ , although that does not reproduce the Gaussian or shifted lognormal result.",
"The expression (REF ) consists of dyadic products that resemble the structure of an eigendecomposition, which we will look at in Section REF , although eigenvectors are additionally pairwise orthogonal."
],
[
"Beyond leading order: shifted lognormal case",
"The recipe described in the previous paragraph for general non-Gaussian distributions is not valid for a lognormal PDF, which is not well-described in terms of a CGF.",
"Here we detail an approach that can be used to obtain lognormal bias functions by applying a large-separation expansion to the shifted lognormal two-point PDF (REF ).",
"This bias expansion is similar in spirit to the Gaussian case, but requires an additional expansion of the Gaussian field correlation $\\xi _G$ in terms of the underlying $\\kappa $ correlation $\\xi _\\kappa $ .",
"The joint shifted lognormal PDF can be expanded at large separations where $\\xi _G(\\xi _\\kappa )/\\sigma _G^2\\ll 1$ to obtain up to next-to-leading order in $\\xi _\\kappa $ $\\frac{{\\cal P}_{\\rm LN}(\\kappa _1,\\kappa _2;\\xi _\\kappa )}{{\\cal P}_{\\rm LN}(\\kappa _1){\\cal P}_{\\rm LN}(\\kappa _2)}&=1+\\xi _G(\\xi _\\kappa )\\frac{g(\\kappa _1)}{\\sigma _G^2}\\frac{g(\\kappa _2)}{\\sigma _{\\rm G}^2}\\\\&+\\frac{\\xi _{\\rm G}^2(\\xi _\\kappa )}{2}\\frac{\\left(g(\\kappa _1)^2 -\\sigma _{\\rm G}^2\\right)}{\\sigma _{\\rm G}^4}\\frac{\\left(g(\\kappa _2)^2-\\sigma _{\\rm G}^2\\right)}{\\sigma _{\\rm G}^4}\\\\&+\\mathcal {O}(\\xi _G^3)\\\\&=1+\\left(\\frac{\\xi _\\kappa }{s^2}-\\frac{\\xi _\\kappa ^2}{2s^4}\\right)\\frac{g(\\kappa _1)}{\\sigma _G^2}\\frac{g(\\kappa _2)}{\\sigma _G^2}\\\\&+\\frac{\\xi _\\kappa ^2}{2s^4}\\frac{\\left(g(\\kappa _1)^2 -\\sigma _{\\rm G}^2\\right)}{\\sigma _{\\rm G}^4}\\frac{\\left(g(\\kappa _2)^2-\\sigma _{\\rm G}^2\\right)}{\\sigma _{\\rm G}^4}\\\\&+\\mathcal {O}(\\xi _\\kappa ^3)\\,,$ In particular, it is useful to define two additional bias functions $q_{1,\\rm LN}$ and $b_{2,\\rm LN}$ following the previous expansion (REF ), with their combination resembling the Gaussian expression from equation (REF ) $(b_2+q_1)_{\\rm LN}(\\kappa )&=\\frac{g(\\kappa )^2-\\sigma _G^2}{s^2\\sigma _G^4}\\simeq \\frac{[sg(\\kappa )]^2-\\sigma ^2_\\kappa }{\\sigma _{\\kappa }^4} \\\\q_{1,\\rm LN}(\\kappa )&=\\frac{1}{s} b_{1,\\rm LN}(\\kappa )=\\frac{g(\\kappa )}{s^2\\sigma _G^2}\\simeq \\frac{g(\\kappa )}{\\sigma _{\\kappa }^2}\\,,$ where we used the result for $b_{1,\\rm LN}$ from equation (REF ).",
"We see that in the limit of $s\\rightarrow \\infty $ we recover $sg(\\kappa )\\rightarrow \\kappa $ and the Gaussian result with $q_{1,\\rm G}=0$ , while for mildly non-Gaussian fields, we typically have $q_1\\ll b_2$ .",
"In a small variance expansion where $\\sigma /s\\ll 1$ , we obtain the limiting behaviour $b_{2,\\rm LN}(\\kappa )\\stackrel{\\frac{\\sigma }{s}\\ll 1}{\\longrightarrow } \\frac{s^2\\ln \\left(1+\\frac{\\kappa }{s}\\right)^2-\\sigma ^2}{\\sigma ^4}-\\frac{1}{4s^2}$ Using the all-order results for $b_{n,\\rm G}$ () together with the expansion of $\\xi _G(\\xi _\\kappa )=\\sum _{n=1}\\frac{(-1)^{n-1}}{n} \\left(\\frac{\\xi _\\kappa }{s^2}\\right)^n$ allows to compute all order expressions for the lognormal bias functions.",
"Note that due to the difference between $\\xi _G$ and $\\xi _\\kappa $ , the higher-order terms receive contributions from lower-order bias functions, as predicted by the CGF-based expansion from before.",
"In Figure REF we show a comparison of the lognormal terms $b_2$ (solid) and $b_2+q_1$ (dashed) to the Gaussian result for the shift parameter adopted in this work and different variances.",
"We can see that for our case (closely corresponding to the green line), the $q_1$ contribution is negligible.",
"The crossing between the solid $b_2$ line and the dotted line of constant $\\sigma ^2$ indicates the zero-crossing of $b_2(\\kappa )$ , which is reflected in the second eigenvector for the $\\kappa $ covariance shown in the lower panel of Figure REF .",
"For the case of a survey area being a significant portion of the sky, the mean correlation $\\bar{\\xi }$ will be extremely small and this second order bias function $b_2(\\kappa )$ will determine the covariance structure.",
"The two zero-crossings of $b_2$ cause the 9-tiled correlation matrix structure displayed earlier, notably in the upper right panel of Figure REF .",
"Figure: A comparison of a rescaled version of the second order bias functions from the shifted lognormal model used here b 2, LN b_{2,\\rm LN} (solid) along with (b 2 +q 1 ) LN (b_2+q_1)_{\\rm LN} (dot-dashed) from equations () (colours for different variances) and the limiting behaviour () (black solid) in comparison to the Gaussian result (black dashed).",
"We also show lines of constant σ 2 \\sigma ^2 (dotted) to indicate the zero-crossings of b 2, LN (κ)b_{2,\\rm LN}(\\kappa ) which appear in the second eigenvectors for the κ\\kappa PDF covariance in the lower panel of Figure ."
],
[
"Covariance expansion for mean-subtracted $\\tilde{\\kappa }=\\kappa -\\kappa _s$",
"To determine the PDF covariance for the case of a mean-subtracted $\\kappa $ field, one needs to consider the 3 variables $\\kappa _1$ , $\\kappa _2$ and $\\kappa _s$ , where $\\kappa _{1/2}$ are as before and $\\kappa _s$ is the mean convergence in the survey patch.",
"Like the convergence field, those variables all have zero expectation value and we are then interested in the joint PDF of the mean-subtracted $\\tilde{\\kappa }_i$ ${\\tilde{\\kappa }}_1=\\kappa _1-\\kappa _s\\,, \\quad {\\tilde{\\kappa }}_2=\\kappa _2-\\kappa _s\\,.$ Let us note that this tilde variable is formally equivalent to the slope variable that was introduced in , .",
"The CGF for the three variables defined above is written here up to linear order in the two-point function at the survey scale, $\\xi _s$ , and up to quadratic order in the two-point correlation at the given cell separation $\\xi _{12}$ $\\varphi (\\lambda _1,\\lambda _2,\\lambda _s)&=\\frac{1}{2}\\sigma ^2_s\\lambda _s^2+\\varphi _0(\\lambda _1)+\\varphi _0(\\lambda _2)+\\xi _{12}\\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)\\nonumber \\\\&+\\lambda _s\\left[\\xi _{1s}\\varphi _1(\\lambda _1)+\\xi _{2s}\\varphi _1(\\lambda _2)\\right]\\\\&+ \\frac{\\xi _{12}^2}{2} [\\varphi _1^2(\\lambda _1)\\varphi _2(\\lambda _2)+\\varphi _2(\\lambda _1)\\varphi _1^2(\\lambda _2)] + \\mathcal {O}(\\xi _{12}^3)\\\\\\varphi (\\lambda _1,\\lambda _s)&=\\frac{1}{2}\\xi _s\\lambda _s^2+\\varphi _0(\\lambda _1)+\\lambda _s\\xi _{1s}\\varphi _1(\\lambda _1)\\,,$ where we assumed that the patches are big enough such that the $\\kappa _s$ PDF is Gaussian.",
"The one- and two-point PDFs are obtained from this CGF by performing the change of variable (REF ) and then marginalising over the background $\\kappa _s$ , leading to ${\\cal P}({\\tilde{\\kappa }}_1)&=\\int \\frac{\\lambda _1}{2\\pi \\hbox{i}}e^{-\\lambda _1{\\tilde{\\kappa }}_1+\\varphi (\\lambda _1,-\\lambda _1)}\\\\{\\cal P}({\\tilde{\\kappa }}_1,{\\tilde{\\kappa }}_2)&=\\!\\int \\!\\!\\!\\int \\!\\frac{\\lambda _1}{2\\pi \\hbox{i}}\\frac{\\lambda _2}{2\\pi \\hbox{i}}e^{-\\lambda _1{\\tilde{\\kappa }}_1-\\lambda _2{\\tilde{\\kappa }}_2+\\varphi (\\lambda _1,\\lambda _2,-\\lambda _1-\\lambda _2)}\\,.$ When all computations are made at linear order in $\\xi $ , we can set $\\sigma ^2_s=\\xi _{1s}=\\xi _{2s}=\\xi _{12}=\\xi (\\theta )$ .",
"To obtain the second order, one needs to start from $\\bar{\\xi }=\\sigma ^2_s=\\xi _{1s}=\\xi _{2s} \\ne \\xi _{12}=\\xi (\\theta )$ , which we will use to illustrate the similarities and differences between the Gaussian and non-Gaussian cases."
],
[
"Gaussian case",
"For a Gaussian field, the two terms $\\lambda _s^2$ and $\\lambda _s[\\varphi _1(\\lambda _1)+\\varphi _1(\\lambda _2)]$ can be summarised and act to reduce the variance and correlation function by $\\bar{\\xi }$ , such that joint PDF ${\\cal P}(\\tilde{\\kappa }_1,\\tilde{\\kappa }_2)$ is a Gaussian with variance $\\sigma _{\\tilde{\\kappa }}^2=\\sigma ^2-\\bar{\\xi }$ and correlation $\\xi _{\\tilde{\\kappa }}(\\theta )=\\xi (\\theta )-\\bar{\\xi }$ .",
"Hence, the covariance can be obtained in full analogy to the previous computation that led to equation (REF ) which is modified to read ${\\rm cov}({\\cal P}_{\\rm G}(\\tilde{\\kappa }_1),{\\cal P}_{\\rm G}(\\tilde{\\kappa }_2)) &=\\sum _{n=2}^\\infty \\frac{\\overline{(\\xi -\\bar{\\xi })^n}}{n!}",
"\\frac{\\partial ^n{\\cal P}_{\\rm G}(\\tilde{\\kappa }_1)}{\\partial \\tilde{\\kappa }_1^n} \\frac{\\partial ^n{\\cal P}_{\\rm G}(\\tilde{\\kappa }_2)}{\\partial {\\tilde{\\kappa }_2}^n}\\,,$ where due to the definition of the average of powers of the correlation function () the $n=1$ term disappears.",
"Note that ${\\cal P}_{\\rm G}({\\tilde{\\kappa }}_1)$ does not denote the same PDF as ${\\cal P}_{\\rm G}(\\kappa _1)$ , for the Gaussian case they differ by the variance being $\\sigma _{\\tilde{\\kappa }}^2=\\sigma _{\\kappa }^2-\\bar{\\xi }$ rather than $\\sigma _{\\kappa }^2$ .",
"Aside from this (small) difference and the removal of the $n=1$ term, we now have a different prefactor consisting of averages of powers of $\\xi -\\bar{\\xi }$ rather than $\\xi $ .",
"Those averages of powers of the correlation function $\\overline{\\xi ^n}$ and $\\overline{(\\xi -\\bar{\\xi })^n}$ compared to powers of the variances $\\sigma ^{2n}$ and $(\\sigma ^2-\\bar{\\xi })^n$ control the large-separation expansion of the covariance (in the Gaussian case given in equation (REF ) for raw $\\kappa $ and in equation (REF ) for the mean-subtracted $\\tilde{\\kappa }=\\kappa -\\bar{\\kappa }$ ) follow similar hierarchies as illustrated in Figure REF .",
"In particular, the leading order term will be $\\frac{{\\rm cov}({\\cal P}_{\\rm G}(\\tilde{\\kappa }_1),{\\cal P}_{\\rm G}(\\tilde{\\kappa }_2))}{{\\cal P}_{\\rm G}(\\tilde{\\kappa }_1){\\cal P}_{\\rm G}(\\tilde{\\kappa }_2)} &= \\frac{\\overline{(\\xi -\\bar{\\xi })^2}}{2\\sigma _{\\tilde{\\kappa }}^4} \\underbrace{\\left(\\frac{{\\tilde{\\kappa }}_1^2}{\\sigma _{\\tilde{\\kappa }}^2}-1\\right)\\left(\\frac{{\\tilde{\\kappa }}_2^2}{\\sigma _{\\tilde{\\kappa }}^2}-1\\right)}_{\\sigma _{\\tilde{\\kappa }}^4b_{2,\\rm G}(\\tilde{\\kappa }_1)b_{2,\\rm G}(\\tilde{\\kappa }_2)}\\\\&+ \\mathcal {O}\\left(\\frac{\\overline{(\\xi -\\bar{\\xi })^3}}{\\sigma _{\\tilde{\\kappa }}^6}\\right)\\,,$ which predicts a first eigenvalue/-vector similar to the second eigenvalue/-vector of the Gaussian case without mean-subtraction () because $\\overline{(\\xi -\\bar{\\xi })^2}/\\sigma _{\\tilde{\\kappa }}^4 \\simeq \\overline{\\xi ^2}/{\\sigma ^4}$ as shown in Figure REF .",
"The zero-crossings of the first eigenvector will be at $\\tilde{\\kappa }\\approx \\sigma ^2-\\bar{\\xi }\\approx \\sigma ^2$ , closely following the result for full sky maps where $\\bar{\\xi }$ is negligibly small and eigenvectors are shown as green/olive lines in the middle panel of Figure REF , as discussed in Section REF .",
"Note that as before the joint two-point PDF of the mean-subtracted $\\tilde{\\kappa }$ values could be written in terms of the marginal Gaussian PDFs and a series of Hermite polynomials as in equation () with prefactors set by averages of powers of $(\\xi -\\bar{\\xi })/\\sigma _{\\tilde{\\kappa }}^2$ ."
],
[
"Leading order: general non-Gaussian case",
"Similarly as for the Gaussian case, we will rewrite the joint CGF of the three variables $\\lambda _1,\\lambda _2,\\lambda _s$ in terms of the joint CGF for just two variables $\\lambda _i$ and $\\lambda _s$ , where $\\lambda _s=-(\\lambda _1+\\lambda _2)$ .",
"We will first discuss the leading order result which reads $\\varphi &(\\lambda _1,\\lambda _2,-\\lambda _1-\\lambda _2)-\\varphi (\\lambda _1,-\\lambda _1)-\\varphi (\\lambda _2,-\\lambda _2)\\\\=&\\xi \\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)+\\bar{\\xi }[\\lambda _1\\lambda _2-\\lambda _1\\varphi _1(\\lambda _2)-\\lambda _2\\varphi _1(\\lambda _1) ]+ \\mathcal {O}(\\xi ^2)\\\\=&(\\xi -\\bar{\\xi })\\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)+\\bar{\\xi }[\\varphi _1(\\lambda _1)-\\lambda _1][\\varphi _1(\\lambda _2)-\\lambda _2]+ \\mathcal {O}(\\xi ^2)\\,,$ where we used $\\xi =\\xi (\\theta )$ for brevity and reorganised the formula such that the for a Gaussian only the first term remains, because $\\varphi _1(\\lambda )=\\lambda $ .",
"We are then in position to compute the connected part of the joint PDF, given by $&{\\cal P}({\\tilde{\\kappa }}_1,{\\tilde{\\kappa }}_2)-{\\cal P}({\\tilde{\\kappa }}_1){\\cal P}({\\tilde{\\kappa }}_2) \\nonumber \\\\&=\\!\\int \\!\\!\\!\\int \\!\\frac{\\lambda _1}{2\\pi \\hbox{i}}\\frac{\\lambda _2}{2\\pi \\hbox{i}}e^{-\\lambda _1{\\tilde{\\kappa }}_1-\\lambda _2{\\tilde{\\kappa }}_2+\\varphi (\\lambda _1,-\\lambda _1)+\\varphi (\\lambda _2,-\\lambda _2)} \\nonumber \\\\& \\times \\Big [(\\xi -\\bar{\\xi })\\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)+\\bar{\\xi }[\\varphi _1(\\lambda _1)-\\lambda _1][\\varphi _1(\\lambda _2)-\\lambda _2]\\\\&\\quad \\nonumber +\\mathcal {O}(\\lbrace \\overline{\\xi ^2},\\bar{\\xi }^2,\\overline{(\\xi -\\bar{\\xi })^2}\\rbrace )\\Big ]\\,.$ When integrated over the distance distribution, the term linear in $\\xi -\\bar{\\xi }$ will vanish and the term involving products of $\\varphi _1(\\lambda _i)-\\lambda _i$ will create a new bias function $\\tilde{b}_{1,\\rm NG}(\\tilde{\\kappa })$ for the mean-subtracted convergence $\\tilde{\\kappa }$ only present for non-Gaussian fields.",
"In terms of this, the PDF covariance can be written as ${\\rm cov}({\\cal P}(\\tilde{\\kappa }_1),{\\cal P}(\\tilde{\\kappa }_2)) =&\\bar{\\xi }(\\tilde{b}_{1,\\rm NG}{\\cal P})(\\tilde{\\kappa }_1)(\\tilde{b}_{1,\\rm NG}{\\cal P})(\\tilde{\\kappa }_2) \\\\&+ \\mathcal {O}(\\lbrace \\overline{\\xi ^2},\\bar{\\xi }^2,\\overline{(\\xi -\\bar{\\xi })^2}\\rbrace )\\,,$ where we introduced the leading order composite bias $\\tilde{b}_{1,\\rm NG}({\\tilde{\\kappa }})=b_1({\\tilde{\\kappa }})+\\frac{\\partial }{\\partial {\\tilde{\\kappa }}}\\log {\\cal P}({\\tilde{\\kappa }})\\,.$ Figure: Additional first-order bias b ˜ 1, NG (κ ˜)\\tilde{b}_{1,\\rm NG}(\\tilde{\\kappa }) (green) of the mean-subtracted κ ˜=κ-κ ¯\\tilde{\\kappa }=\\kappa -\\bar{\\kappa } obtained from the sum of the `raw' bias b 1 b_1 (green) and the response of the PDF ∂log𝒫 ¯/∂κ\\partial \\log \\bar{{\\cal P}}/\\partial \\kappa (red) as given in equation ().",
"We also show predictions from the shifted lognormal case (dashed) relying on the PDF from equation () and the bias b 1, LN b_{1,\\rm LN} from equation ()In Figure REF we illustrate the first composite non-Gaussian bias function (green) along with its ingredients, the first order bias $b_1$ (blue) and the logarithmic derivative of the PDF (red) as measured in one FLASK realisation (data points) and predicted for a shifted lognormal (dashed lines).",
"We can see that for a mildly non-Gaussian field, this composite non-Gaussian bias function is more than an order of magnitude smaller than the raw bias $b_1$ , such that there is no guarantee that our result at leading order in $\\bar{\\xi }$ constitutes the most relevant correction.",
"This can be established analytically for a shifted lognormal distribution using equations (REF ) and (REF ), which gives $\\tilde{b}_{1,\\rm LN}({\\tilde{\\kappa }})=-\\frac{1}{{\\tilde{\\kappa }}+s}+\\frac{x({\\tilde{\\kappa }})}{s\\sigma ^2_{\\rm G}(\\sigma _{{\\tilde{\\kappa }}}^2)}\\left(1-\\frac{1}{1+{\\tilde{\\kappa }}/s}\\right)\\,.$ The minimum value is $\\tilde{b}_{1,\\rm LN}(\\tilde{\\kappa }\\simeq 0)\\simeq -1/s$ and in the limit of $s\\rightarrow \\infty $ we recover the Gaussian result with vanishing $\\tilde{b}_{1}$ ."
],
[
"Next-to-leading order: general non-Gaussian case",
"We now generalise the previous result up to next-to-leading order, where we find $\\varphi &(\\lambda _1,\\lambda _2,-\\lambda _1-\\lambda _2)-\\varphi (\\lambda _1,-\\lambda _1)-\\varphi (\\lambda _2,-\\lambda _2)\\\\=&(\\xi -\\bar{\\xi })\\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)+\\bar{\\xi }[\\varphi _1(\\lambda _1)-\\lambda _1][\\varphi _1(\\lambda _2)-\\lambda _2]\\\\&+ \\frac{\\xi ^2}{2} [\\varphi _1^2(\\lambda _1)\\varphi _2(\\lambda _2)+\\varphi _2(\\lambda _1)\\varphi _1^2(\\lambda _2)]+ \\mathcal {O}(\\xi ^3)\\,.$ Again, we compute the connected part of the joint PDF, given by $&{\\cal P}({\\tilde{\\kappa }}_1,{\\tilde{\\kappa }}_2)-{\\cal P}({\\tilde{\\kappa }}_1){\\cal P}({\\tilde{\\kappa }}_2) \\nonumber \\\\&=\\!\\int \\!\\!\\!\\int \\!\\frac{\\lambda _1}{2\\pi \\hbox{i}}\\frac{\\lambda _2}{2\\pi \\hbox{i}}e^{-\\lambda _1{\\tilde{\\kappa }}_1-\\lambda _2{\\tilde{\\kappa }}_2+\\varphi (\\lambda _1,-\\lambda _1)+\\varphi (\\lambda _2,-\\lambda _2)}\\nonumber \\\\\\nonumber &\\times \\Big [(\\xi -\\bar{\\xi })\\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2)+\\bar{\\xi }[\\varphi _1(\\lambda _1)-\\lambda _1][\\varphi _1(\\lambda _2)-\\lambda _2]\\\\&\\quad +\\frac{\\bar{\\xi }^2}{2} [\\varphi _1(\\lambda _1)-\\lambda _1]^2[\\varphi _1(\\lambda _2)-\\lambda _2]^2\\\\&\\quad + \\bar{\\xi }(\\xi -\\bar{\\xi }) \\varphi _1(\\lambda _1)\\varphi _1(\\lambda _2) [\\varphi _1(\\lambda _1)-\\lambda _1][\\varphi _1(\\lambda _2)-\\lambda _2]\\nonumber \\\\&\\quad + \\frac{\\xi ^2}{2} [\\varphi _1^2(\\lambda _1)\\varphi _2(\\lambda _2)+\\varphi _2(\\lambda _1)\\varphi _1^2(\\lambda _2)]\\\\&\\quad + \\frac{(\\xi -\\bar{\\xi })^2}{2}\\varphi _1^2(\\lambda _1)\\varphi _1^2(\\lambda _2) +\\mathcal {O}(\\xi ^3)\\Big ] \\,.\\nonumber $ When integrated over the distance distribution, all terms linear in $\\xi -\\bar{\\xi }$ will vanish.",
"The remaining contributions are terms in the first two lines, involving powers of $\\varphi _1(\\lambda )-\\lambda $ creating new bias functions, $\\tilde{b}_{n,\\rm NG}$ , only present for non-Gaussian fields.At higher orders, this will generalise to a class of terms with products of positive powers of $\\varphi _1(\\lambda -\\lambda )$ and powers of $\\varphi _1(\\lambda )^2$ and/or $\\varphi _{n\\ge 2}(\\lambda )$ .",
"terms in the last two lines containing products of powers of $\\varphi _n$ that yield the bias functions encountered for the covariance of the raw $\\kappa $ PDF, at second order $b_2$ from equation () and $q_1$ from equation ().",
"This yields $\\nonumber &{\\rm cov}({\\cal P}(\\tilde{\\kappa }_1),{\\cal P}(\\tilde{\\kappa }_2)) =\\bar{\\xi }(\\tilde{b}_{1,\\rm NG}{\\cal P})(\\tilde{\\kappa }_1)(\\tilde{b}_{1,\\rm NG}{\\cal P})(\\tilde{\\kappa }_2)\\\\&+\\frac{\\bar{\\xi }^2}{2}(\\tilde{b}_{2,\\rm NG}{\\cal P})(\\tilde{\\kappa }_1)(\\tilde{b}_{2,\\rm NG}{\\cal P})(\\tilde{\\kappa }_2)\\\\\\nonumber &+\\frac{\\overline{\\xi ^2}}{2} [(b_2{\\cal P})(\\tilde{\\kappa }_1)(q_1{\\cal P})(\\tilde{\\kappa }_2)+(q_1{\\cal P})(\\tilde{\\kappa }_1)(b_2{\\cal P})(\\tilde{\\kappa }_2)]\\\\&+\\frac{\\overline{(\\xi -\\bar{\\xi })^2}}{2}(b_{2}{\\cal P})(\\tilde{\\kappa }_1)(b_{2}{\\cal P})(\\tilde{\\kappa }_2)\\\\&+ \\mathcal {O}(\\lbrace \\overline{\\xi ^3},\\bar{\\xi }\\,\\overline{\\xi ^2},\\bar{\\xi }^3,\\bar{\\xi }\\, \\overline{(\\xi -\\bar{\\xi })^2},\\overline{(\\xi -\\bar{\\xi })^3}\\rbrace )\\,,$ where we defined the following NLO composite bias function only present for non-Gaussian fields $\\tilde{b}_{2,\\rm NG}({\\tilde{\\kappa }})&=b_2({\\tilde{\\kappa }})+2\\frac{1}{{\\cal P}({\\tilde{\\kappa }})}\\frac{\\partial (b_1{\\cal P})}{\\partial {\\tilde{\\kappa }}}+\\frac{1}{{\\cal P}({\\tilde{\\kappa }})}\\frac{\\partial ^2{\\cal P}}{\\partial {\\tilde{\\kappa }}^2}\\,.$ For a Gaussian field, the first three lines in equation (REF ) do not contribute because $\\varphi _1(\\lambda )=\\lambda $ and $\\varphi _{n\\ge 2}=0$ , so we recover the leading order of equation (REF ).",
"Note that while we ordered terms according to powers of $\\xi $ here, there is no guarantee that the leading order term proportional to $\\bar{\\xi }$ will be larger than the next-to-leading order terms proportional to $\\overline{\\xi ^2}$ and $\\overline{(\\xi -\\bar{\\xi })^2}$ , as highlighted by the Gaussian case where the leading order term vanishes and the shifted lognormal case where we found the term to be subdominant.",
"Typically, power of averages of the mean correlation are much smaller than averages of powers, so $\\bar{\\xi }^2\\ll \\overline{\\xi ^2}\\approx \\overline{(\\xi -\\bar{\\xi })^2}$ as evident from Figure REF and for a weakly non-Gaussian field the $\\tilde{b}_{2,\\rm NG}$ term will only give a small contribution.",
"For the mildly non-Gaussian $\\kappa $ PDF we are considering here, we observe that the first order term is effectively removed by the mean-subtraction, as evident from the visual inspection of the covariance matrix shown in Figure REF and the dominant eigenvectors, which resemble the full sky case shown in green/olive in the lower panel of Figure REF .",
"Given the similarity between the Gaussian and FLASK correlation matrices, we can deduce that the $b_2$ term given by equation (REF ) for the shifted lognormal case is in fact the dominant contribution.",
"Our results from this section can largely be carried over to the covariance modelling for the 3-dimensional matter PDF on mildly nonlinear scales (roughly 10Mpc$/h$ at low redshifts), as we will discuss in Section below.",
"Typically, the matter PDF features stronger non-Gaussianities, such that the lognormal shift parameter will be close to unity $s_{\\rm 3D}\\lesssim 1$ .",
"In Appendix we additionally discuss tree models as a useful class of non-Gaussian models for which all order covariance expansion results can be obtained."
],
[
"Quantitative covariance model tests",
"Beyond performing a visual comparison between the covariances obtained from simulated maps and the theoretical models, we quantitatively compare their accuracy using an eigendecomposition of the covariance validating the set of eigenvalues and leading order eigenvectors, a comparison of the dominant bands in the almost band-diagonal precision matrix as inverse of the covariance, a $\\chi ^2$ -test to establish the Gaussianity of the PDF data vector and accuracy of the model covariance, and a Fisher forecast for two cosmological parameters $\\lbrace \\Omega _m,\\sigma _8\\rbrace $ .",
"When comparing our measurements to the predictions in tests involving the inverse covariance or precision matrix, we account for that fact that matrix inversion is a non-linear operation, so the noise in the estimate of the covariance elements will lead to a bias in the elements of the precision matrix.",
"This bias from covariance estimation noise turns out to be just a factor multiplying the entire inverse covariance matrix, the so called Kaufman-Hartlap factor , , $h=(N_{\\rm sim} - N_d- 2 )/(N_{\\rm sim} - 1) \\ ,$ where $N_{\\rm sim}$ is the number of samples used and $N_d$ is the length of the data vector.",
"This effect closely resembles the impact of the non-Gaussian distribution of a data vector that is a consequence of a finite number of realisations and of size $(N_{\\rm sim}-N_d + N_p-1)/(N_{\\rm sim}-1)$ , where $N_p$ is the number of parameters.",
"Another effect is due to the nonlinear relationship between the precision matrix and the resulting parameter covariance matrix , which can partially counteract the previous effect, but was found to be negligible in our setting.",
"Beyond direct effects for the width of the likelihood contours, noise also leads to an uncertainty for their location in parameter space , which is only negligible if $N_{\\rm sim}-N_d \\gg N_d-N_p$ as is the case for our simplified setting with a short data vector."
],
[
"Eigendecomposition of the covariance matrix",
"The eigendecomposition of the covariance matrix of dimension $n\\times n$ (set by the length $n$ of the data vector) can be written in terms of a complete set of eigenvalues $\\lambda _i$ (for convenience ordered in descending order, so $\\lambda _1>\\ldots >\\lambda _n$ ) and associated normalised eigenvectors $\\mathbf {e_i}$ arranged in an orthogonal matrix $O$ ${\\rm cov}({\\cal P},{\\cal P})&=O\\cdot {\\rm diag}(\\lambda _1,\\ldots ,\\lambda _n)\\cdot O^T\\,,\\ O=\\begin{pmatrix} \\mathbf {e_1}, \\ldots ,\\mathbf {e_n}\\end{pmatrix}\\\\&=\\sum _{k=1}^n \\lambda _k \\mathbf {e_k}\\mathbf {e_k}^T\\,.$ In terms of components, we can write this as ${\\rm cov}({\\cal P}_i,{\\cal P}_j)&=\\sum _{k=1}^n \\lambda _k \\left(\\mathbf {e_k}\\mathbf {e_k}^T\\right)_{ij}=\\sum _{k=1}^n \\lambda _k \\left(\\mathbf {e_k}\\right)_i\\left(\\mathbf {e_k}\\right)_j\\,,$ where $\\left(\\mathbf {e_k}\\right)_i$ denotes the $i$ -th component of the eigenvector $\\mathbf {e_k}$ .",
"The eigendecomposition can be understood in the context of the analytical results from the large-separation expansion in Section .",
"For the Gaussian case, we obtained an expansion consisting of dyadic products with building blocks of the form $He_n(x)\\exp (-x^2/2)$ with $x=\\kappa /\\sigma $ corresponding to $(b_n{\\cal P})(\\kappa )$ .",
"While the Hermite polynomials are orthogonal with respect to the Gaussian weight function $\\exp (-x^2/2)$ corresponding to the PDF ${\\cal P}_G(\\kappa )$ , the dyadic product building blocks are not generally orthogonal as they carry a combined weight of $\\exp (-x^2)$ .",
"However, the orthogonality of neighbouring bias contributions $b_n{\\cal P}$ and $b_{n+1}{\\cal P}$ is ensured by the alternating symmetry of the Hermite functions.",
"Considering the binned PDF means that there is just a finite set of eigenvectors, which receive contributions from a whole series of $b_n{\\cal P}$ terms.",
"We illustrate that in Figure REF by comparing the first three eigenvectors (solid/dashed/dotted) of the PDF covariance for a Gaussian $\\kappa $ field obtained from the full theory expression (cyan) with the normalised bias functions (light grey).",
"The leading order bias contribution $b_1{\\cal P}$ agrees with the first eigenvector almost perfectly, while $b_2{\\cal P}$ resembles the overall shape of the second eigenvector with slightly modified zero crossings.",
"This agreement reflects that there is a hierarchy in the contributions from different powers of the correlation function compared to the variance $\\overline{\\xi ^n}/\\sigma ^{2n}$ as shown in Figure REF .",
"The slight difference between $b_2{\\cal P}$ and the second eigenvector is due to contributions from higher-order terms as we illustrate by showing the eigenvectors of the Hermite expansions (REF ) up to a finite $n_{\\rm max}$ (shaded from dark grey to cyan with increasing $n_{\\rm max}$ ).",
"The large difference between $b_3{\\cal P}$ and the third eigenvector comes from its non-orthogonality with $b_1{\\cal P}$ as can be seen from the difference between the dotted lines in light and darker grey.",
"Figure: First (solid), second (dashed) and third (dotted) eigenvectors of the Gaussian κ\\kappa PDF covariance predicted for small patches (cyan) and the normalised bias functions (b n 𝒫) G (κ)(b_n\\mathcal {P})_{\\rm G}(\\kappa ) (light grey).",
"We also show the eigenvectors obtained from a Hermite expansion () up to order n max n_{\\rm max} (shaded from dark grey to cyan).",
"The vertical black dotted lines indicate where κ=±σ\\kappa =\\pm \\sigma .For the case of the full sky (or the mean-subtracted $\\kappa $ in patches), the smallness (or vanishing) of the mean correlation $\\bar{\\xi }$ will practically remove the contribution of $b_1{\\cal P}$ such that the first and second eigenvectors will be resembling $b_2{\\cal P}$ and $b_3{\\cal P}$ , respectively.",
"This qualitative statement remains true for mildly non-Gaussian fields such as our shifted lognormal (FLASK) case.",
"We show the eigenvalues and dominant eigenvectors for the Gaussian and FLASK maps in Figure REF .",
"Figure: (Upper panel) Eigenvalues of the Gaussian κ\\kappa PDF covariance for κ\\kappa from small patches (measurement in blue, prediction in cyan) and the full sky (measurement in green, prediction in olive).",
"(Middle and lower panel) The dominant eigenvectors of the Gaussian (middle) and FLASK (lower) κ\\kappa PDF covariance, including the 1st eigenvector (solid lines), 2nd eigenvector (dashed lines) and 3rd eigenvector for raw κ\\kappa from small patches (dotted lines).",
"The move from small patches to the full sky (solid and dashed lines) essentially removes the first term in the eigendecomposition and replaces it with the second term as evident from the agreement of the first green cross with the second blue cross, and the agreement of the solid green and the dashed blue lines.The FLASK results are qualitatively similar to the Gaussian case except for subtle shifts in the location of the zero-crossings and the asymmetry between positive and negative κ\\kappa , both related to the non-Gaussianity of the underlying field.The eigenvalues in the upper panel of Figure REF show a good agreement between measurements and theory and suggest a clear ordering between contributions from different bias functions arising from a large-separation expansion, qualitatively resembling the behaviour seen in Figure REF for the averages of powers of the cell correlation in comparison with the powers of the variance.",
"The middle and lower panels of Figure REF show the leading three eigenvectors for Gaussian (middle) and lognormal (bottom) $\\kappa $ fields in small patches (blue/cyan) and the full sky (green/olive).",
"The first eigenvector (solid) is determined by $b_1{\\cal P}$ and hence shows one zero-crossing close to $\\kappa \\simeq 0$ as predicted by $b_{1,\\rm G}$ (REF ) and $b_{1,\\rm LN}$ (REF ).",
"The second eigenvector (dashed) is driven by $b_2{\\cal P}$ with two zero-crossings around $\\kappa \\simeq \\sigma _\\kappa $ as predicted by $b_{2,\\rm G}$ (REF ) and $b_{2,\\rm G}$ (REF ).",
"The third eigenvector (dotted) involves $b_3{\\cal P}$ , which is predicted to have three zero-crossings for the Gaussian case from equation (), but is also sensitive to higher-order terms as we illustrated in Figure REF .",
"For the covariance of the $\\kappa $ PDF on the full sky we show the first two eigenvectors (measurements in green, predictions in olive).",
"We observe that the first eigenvector closely resembles the second eigenvector of the $\\kappa $ PDF from small patches (measurements in blue, predictions in cyan), as predicted for Gaussian fields in equation (REF ) and for mildly non-Gaussian fields in equation (REF ).",
"Overall, the theoretical predictions are in good agreement with the measurements from Gaussian and FLASK maps.",
"The Gaussian and the lognormal cases are qualitatively similar with a few important differences appearing mostly as shifts towards negative $\\kappa $ (reflecting the PDF peak location) and an asymmetry between positive and negative $\\kappa $ reflecting the skewness of the PDF.",
"Due to the nature of the eigendecomposition, the contribution of the finite sampling term on the diagonal of the covariance matrix (REF ) to individual eigenvectors is typically small, such that the leading order results for the non-overlapping case (not shown) resemble the overlapping case aside from additional noise caused by the small number of sampling cells.",
"In principle, one can reconstruct the covariance matrix from the terms in the eigendecomposition, although in the presence of an incomplete set of eigenvectors (a number less than the length of the data vector), this matrix will not be invertible and hence unsuitable for statistical analysis.",
"For the case of non-overlapping cells, there is a large contribution to the diagonal from shot noise, so one can attempt a covariance reconstruction using only the first few eigenvectors along with a matched diagonal mostly arising from shot noise.",
"In practice, we opt for the approach to use an invertible covariance baseline model, such as the ones obtained from the Gaussian or shifted lognormal model.",
"If this was found to be insufficiently accurate, it could be successively improved using the precision matrix expansion or iteratively corrected by adjusting the dominant eigenvectors to their desired values as predicted by a large-scale expansion discussed in Section ."
],
[
"Bands of the precision matrix",
"The accuracy of cosmological parameter estimation relies on an accurate modelling of the precision (inverse covariance) matrix which enter the $\\chi ^2$ tests and Fisher forecasts .",
"We show this precision matrix in the case of shifted lognormal fields from the full sky (rescaled to the survey area) in the upper panel of Figure REF as measured in FLASK maps (upper triangle) and predicted (lower triangle).",
"We notice that the inverse covariance is close to band-diagonal with alternating signs.",
"In the lower panel we focus on the three dominant bands (black, blue, red) comparing the prediction (solid) to the measurements (dotted).",
"We find similar results for the precision matrix shape obtained from FLASK patches, and for both the raw $\\kappa $ and the mean-subtracted $\\tilde{\\kappa }=\\kappa -\\bar{\\kappa }$ from overlapping cells.",
"We can qualitatively understand the agreement between $\\kappa $ and $\\tilde{\\kappa }$ in terms of the previously discussed eigendecomposition.",
"In principle, the precision matrix can be obtained from the eigenvectors and eigenvalues of the covariance matrix as ${\\rm cov}^{-1}({\\cal P},{\\cal P})&=O\\cdot {\\rm diag}(\\lambda _1^{-1},\\ldots ,\\lambda _n^{-1})\\cdot O^T\\,,$ which means eigenvectors are retained but their order is reversed.",
"From the upper panel of Figure REF we expect a flat hierarchy of the highest eigenvalues that determine the inverse, with the lowest order eigenvectors (associated with the largest eigenvalues) playing a minor role for the shape of the precision matrix.",
"Figure: (Upper panel) Impression of the approximately band-diagonal precision matrix obtained as inverse of the κ\\kappa covariance matrix for overlapping cells for shifted lognormal fields as predicted (lower triangle) and measured in 500 FLASK full sky realisations corrected by the Hartlap factor () (upper triangle).",
"This matrix is close to penta-diagonal, featuring 5 central bands, one diagonal band (black) and 2 neighbouring bands above/below (blue, red).",
"We highlight those bands with with solid/dotted lines for the measured and predicted case.",
"On the diagonal we show the mean of the FLASK measurements and the prediction.",
"The measurement from a finite number of realisations in the upper panel exhibits additional noise off the diagonal.",
"(Lower panel) Comparison of the bands in the precision matrix between the the shifted lognormal theory prediction (dotted lines) and the measurement from FLASK realisations (solid lines).The band-diagonal structure of the precision matrix is mainly driven by the strong correlations of neighbouring bins induced by the overlap of cells arising at small separations.",
"This can be inferred from a comparison with the non-overlapping case and could be formalised by evaluating the small separation contributions described in [8] for the analogous case of densities.",
"We can illustrate this effect in a simpler setting by computing the covariance between 1D self-centered cells for a Poisson (constant) power spectrum with no correlation between non-overlapping cells.",
"Thus for two cells of half-length $x_i$ and $x_j$ , their covariance is given by the integral over the (constant) power spectrum multiplied by the 1D top-hat window function of each cell.",
"We thus have ${\\rm cov}_{\\rm 1D}(x_i,x_j) \\propto \\!\\!\\!\\int \\!",
"{\\rm d}k \\frac{\\sin (kx_i)}{kx_i}\\frac{\\sin (kx_j)}{kx_j}\\propto \\left[\\max (x_i,x_j)\\right]^{-1}\\,.$ This special covariance shape is called a hook matrix $\\rm {cov}_{\\rm 1D} \\propto \\begin{pmatrix}1/x_1 & 1/x_2 & 1/x_3 & \\dots & 1/x_n \\\\1/x_2 & 1/x_2 & 1/x_3 & \\dots & 1/x_n \\\\1/x_3 & 1/x_3 & 1/x_3 & \\dots & 1/x_n \\\\\\vdots & & & & \\vdots \\\\1/x_n & & \\dots & & 1/x_n\\end{pmatrix}.$ This matrix can be inverted by obtaining the cofactor matrix $C$ which determines the inverse as $\\rm {cov}_{\\rm 1D}^{-1}=C^T/\\det \\rm {cov}_{\\rm 1D}$ .",
"The elements of the cofactor matrix are given by Cramer's rule in terms of determinants of submatrices obtained from $\\rm {cov}_{\\rm 1D}$ by deleting the $i$ th row and $j$ th column $C_{i,j}=(-1)^{i+j}\\det (\\rm {cov}_{\\rm 1D}|_{i,j})\\,.$ It can then be shown that indeed the cofactor elements vanish everywhere except on the diagonal and the directly adjacent bands to the main diagonal.",
"This is because the removal of a row $i$ and column $j$ in the covariance (REF ) leads to linearly dependent rows/columns and hence $C_{i,j}=0$ for $|i-j|<2$ .",
"As the covariance is symmetric, so is the cofactor matrix and hence the inverse of the covariance will have a band-diagonal structure.",
"When the values of the radii $x_i$ (resembling $\\kappa _i$ ) are an increasing sequence, the diagonal will have a positive sign and the band adjacent to the diagonal will have a negative sign."
],
[
"$\\chi ^2$ tests",
"We will use a $\\chi ^2$ -test to establish the Gaussianity of the data vector and assess the accuracy of our predicted covariances.",
"Let us consider the $\\chi ^2$ associated with the original $N$ samples of the data vector.",
"For every sample, labelled by $i$ , of the data vector, $D_{\\rm org}[i]$ , of length $n$ (encoding the number of degrees of freedom), one computes the $\\chi ^2$ $\\chi ^2_{\\rm org}[i]=(D_{\\rm org}[i]-\\mu )^T \\cdot C^{-1} \\cdot (D_{\\rm org}[i]-\\mu )\\,,$ using the mean data vector, $\\mu $ (also of length $n$ ) and data covariance $C$ (of dimension $n\\times n$ ).",
"The distribution of $\\chi ^2$ follows a $\\chi ^2$ distribution with the appropriate number of degrees of freedom $n$ (here the number of PDF bins).",
"To test the Gaussianity of the data vector distribution, one can create another sample of data vectors, $D_{\\rm G}$ , by drawing $N$ times from a multivariate Gaussian using the mean data vector, $\\mu $ and covariance $C$ .",
"Then one recomputes the $\\chi ^2$ for every sample using the covariance to check whether it follows the same distribution $\\chi ^2_{\\rm G}[i]=(D_{\\rm G}[i]-\\mu )^T \\cdot C^{-1} \\cdot (D_{\\rm G}[i]-\\mu )\\,.$ If so, one can assert that the data vector is `close enough' to Gaussian distributed around the mean with the measured covariance.",
"Both of those quantities will have the mean $\\overline{\\chi ^2}=\\overline{\\chi ^2_{\\rm org}}=n$ , since the covariance matrix is identical and is reproduced in the expectation value of the fluctuations of the data vector around the mean.",
"Figure: χ 2 \\chi ^2-tests for the validation of our predicted covariances and the Gaussianity of our data vector showing the χ 2 \\chi ^2 distribution (black line) and the result for a data vector drawn from a Gaussian with the measured covariance (black histogram).",
"(Upper panel) Covariance matrix test for the PDF of the convergence κ\\kappa (blue) or mean-subtracted convergence κ-κ ¯\\kappa -\\bar{\\kappa } (purple) measured from overlapping cells computed in small patches comparing the FLASK measurements (solid) to the predictions from a shifted lognormal (dotted).",
"(Lower panel) Covariance matrix test for overlapping cells computed for the full sky comparing the measurements from 500 FLASK realisations (solid) to the lognormal prediction (dotted).One can also use a $\\chi ^2$ test to compare the closeness of two covariances, where $C$ denotes the original `true' (or target) covariance and $C_{\\rm mod}$ the model (or approximated) covariance.",
"By recomputing the $\\chi ^2$ using the model covariance $\\chi ^2_{\\rm mod}[i]=(D_{\\rm org}[i]-\\mu )^T \\cdot (C_{\\rm mod})^{-1} \\cdot (D_{\\rm org}[i]-\\mu )\\,$ and checking its distribution compared to the original one $\\chi ^2_{\\rm org}$ mentioned above, one can assess how well the model and true covariance agree.",
"We found that the covariance of non-overlapping cells from the FLASK patches is well captured in both the Gaussian and shifted lognormal theory, as it is dominated by the finite sampling term and thus not shown here.",
"The covariance of heavily overlapping cells from FLASK patches was visually different from the Gaussian case, mainly driven by the asymmetry in the diagonal of the covariance matrix shown in Figure REF .",
"In Figure REF we show the $\\chi ^2$ test results for the overlapping case with FLASK covariances for patches (upper panel) and the full sky (lower panel).",
"The agreement of the $\\chi ^2$ obtained from the measured covariances (coloured solid) and a draw from a Gaussian with the same variance (black) shows that the data vectors are close to Gaussian distributed.",
"We find good agreement between the FLASK covariance measurements (coloured solid) and the shifted lognormal predictions (coloured dotted).",
"The results for patches (upper panel) show a slight (6-9%) bias of the theory towards higher $\\chi ^2$ , which could be caused by the patch-to-patch correlation arising from cutting multiple patches from each FLASK full sky that is unaccounted for in the theory.",
"Interestingly, we find no evidence for a corresponding discrepancy in the Fisher forecasts for those two cases, demonstrating the complementarity of the two tests adopted in ."
],
[
"Fisher forecast validation",
"To forecast the errors on a set of cosmological parameters, $\\vec{p}$ , we use the Fisher matrix formalism.",
"The Fisher matrix given a (set of) summary statistics in the data vector $\\vec{D}$ is defined as $F_{ij}= \\sum _{\\alpha ,\\beta }\\frac{\\partial D_\\alpha }{\\partial p_i}(C^{-1})_{\\alpha \\beta }\\frac{\\partial D_\\beta }{\\partial p_j}~, \\quad $ where $D_\\alpha $ is the $\\alpha $ -th element of the data vector $\\vec{D}$ , $C^{-1}$ denotes the precision matrix introduced in Section REF , which is the matrix-inverse of the data covariance $C$ given in equation (REF ) and is was assumed that the covariance matrix is independent of cosmology.",
"The Fisher formalism rests on the assumption that the summary data vector is Gaussian distributed which we have explicitly checked in the last subsection for the central region of the weak lensing PDF considered here.",
"The parameter covariance matrix $\\mathbf {C}(\\vec{p})$ is then obtained as inverse of the Fisher matrix, $C^{\\rm P}_{ij}=(F^{-1})_{ij}$ .",
"In the Fisher formalism, marginalisation over a subset of parameters is achieved by simply selecting the appropriate sub-elements of the parameter covariance.",
"To further validate our theoretical covariances, we perform simple Fisher forecasts for two cosmological parameters $\\lbrace \\Omega _{\\rm cdm},\\sigma _8\\rbrace $ with the measured and predicted covariance matrices, while keeping the data vector and its dependence on cosmological parameters in (REF ) as predicted from large-deviation statistics .The predicted derivatives rely on large-deviations theory for the convergence PDF [4].",
"Using CLASS , linear and non-linear matter density variances are calculated to rescale the matter density CGF in each redshift slice.",
"The convergence CGF is obtained by integration over the redshift slices, and translated to the final PDF by an inverse Laplace transform.",
"Figure: Fisher forecasts comparing different FLASK-generated covariances using non-overlapping cells (cyan), overlapping cells (blue), overlapping cells with mean-subtraction (purple) and overlapping cells on full sky maps (green).",
"All covariances were rescaled to mimic the same survey area.The Fisher forecasts in Figure REF demonstrate that measuring non-overlapping cells only (cyan) leads to much wider parameter errors than including cell overlaps (blue), such that capturing cell overlaps is imperative to avoid artificially inflating the error budgets .",
"Additionally we show that when measuring covariances from small patches obtained from FLASK and subtracting the mean in each small patch (purple) this reproduces the result obtained from full sky maps rescaled to the survey area (green).",
"Figure: Fisher forecasts validating the predicted covariances (dotted) against FLASK measurements (solid) for shifted lognormal fields on small patches (purple), the full sky (green) and including shape noise (orange).In Figure REF we show that our shifted lognormal predicted covariances (dotted) for small patches with mean-subtraction (purple) and the full sky (green) agree well with the FLASK measurements (solid).",
"We also illustrate the impact of shape noise (orange) and validate our convolution approach from equation (REF ) against FLASK measurements with added shape noise."
],
[
"Extension to 3D clustering PDF covariance",
"In this Section, we describe the essential steps needed to adapt our covariance model to 3-dimensional fields using the 3D matter PDF as an example.",
"While the line-of-sight projection involved in computing the weak lensing convergence and photometric galaxy density on mildly nonlinear scales tends to Gaussianise the underlying field, 3D matter and spectroscopic galaxy densities typically show a greater degree of non-Gaussianity on mildly nonlinear scales.",
"This might complicate the two-point PDF modelling, but we will show that the previously discussed shifted lognormal model from Section REF has flexibility to match the skewness of the one-point distribution and capture the clustering properties.",
"Finally, we will focus on the theoretical modelling of the super-sample covariance, which can be used to complement simulated covariances from periodic boxes missing this effect."
],
[
"Applicability of the shifted lognormal model",
"The three-dimensional matter PDF is typically extracted in terms of the normalised cell density $\\hat{\\rho }=\\rho /\\rho _s$ as ratio of the physical cell density $\\rho $ and the mean sample density $\\rho _s$ , or the associated zero-mean density contrast $\\delta =\\hat{\\rho }-1$ .",
"For illustration purposes, we use previously obtained measurements for the matter PDF at redshift $z=0$ in spheres of radius $R=10$ Mpc$/h$ in the Quijote simulation suite .",
"The one-point PDF of the normalised densityensuring the compatibility of measurements in simulations with different particle numbers (changing the mean density of particles in a sphere).",
"$\\hat{\\rho }$ was extracted in logarithmically spaced bins designed to equally capture under- and overdensities in the strongly non-Gaussian distribution and means and covariances obtained from 15,000 realisations at the fiducial cosmology.",
"Figure: 3D matter density PDF at redshift z=0z=0 and smoothing radius R=10R=10 Mpc/h/h as measured from the Quijote simulations (data points, with error bars indicating the sample variance), predicted by large-deviation theory (black line) and approximated with a lognormal distribution (black dotted) and a shifted lognormal distribution with tuned shift parameter (black dashed).In Figure REF we give a visual impression of the mean PDF for the normalised matter density $\\hat{\\rho }$ as measured from the simulations (data points) and large-deviation theory prediction , .",
"To describe clustering observables with the shifted lognormal model, we can simply replace the weak lensing convergence $\\kappa $ with the density contrast $\\delta $ in equations (REF ) and (REF ).",
"Choosing a unit shift parameter $s=1$ corresponds to the usual lognormal approximation .",
"This already provides a decent phenomenological fit to first principle predictions for the matter PDF from large deviation statistics , , , and can be improved by including a shift factor below unity (dashed black line).The lognormal model is however insufficient to accurately predict the response to cosmological parameters .",
"The good match for the fiducial PDF in Figure REF and the leading order bias in Figure REF suggests that the shifted lognormal model can be used to obtain accurate analytical PDF covariance estimates.",
"This can be done by computing the joint PDF (REF ) following a similar lognormal matching procedure as sketched for the mean-subtracted weak lensing convergence $\\tilde{\\kappa }$ .",
"For 3D observables for which the shifted lognormal model would not yield a good match of the PDF and leading order bias, the class of tree models discussed in Appendix and [8] offer an alternative with the capability of skewness matching."
],
[
"Relevance and prediction of super-sample covariance",
"Measurements.",
"The covariance measured from simulated boxes with fixed mean density (due to a fixed particle number) $\\bar{\\rho }$ cannot capture the super-sample effect created by having different sample densities $\\rho _s$ .",
"To estimate this super-sample covariance effect for the Quijote simulation suite, separate-universe style `DC' runs emulating a background density contrast $\\delta _b=\\pm 0.035$ have been performed using changed cosmological parameters and simulation snapshot times from the separate universe approach .",
"The super-sample covariance between two data vector entries $D_i$ and $D_j$ can be estimated by $\\text{cov}^{\\rm SSC}_{\\rm SU}(D_i,D_j)= \\sigma _b^2 \\frac{\\partial D_i}{\\partial \\delta _b}\\frac{\\partial D_j}{\\partial \\delta _b},$ where $\\sigma _b^2$ is the variance of $\\delta _b$ (here just $\\delta _b^2$ ), and the other two terms encode the linear response of the data vector, which can be determined from the simulations using finite differences.",
"In the lower triangle of the lower panel of Figure REF we show the covariance matrix of the matter density PDF showing the correlation between different density bins in the central region of the PDF.",
"We see that, as expected, neighbouring bins are positively correlated, while intermediate underdense and overdense bins are anticorrelated with each other.",
"The strong positive correlation in the bands along the diagonal is induced by cell overlaps, similarly as discussed previously for the analogous weak lensing case.",
"We illustrate the separate universe super-sample effect on the PDF covariance in the upper triangle of Figure REF , showing that the super-sample covariance term completely dominates the overall covariance for a volume of $1($ Gpc$/h)^3$ and will still lead to a significant effect for a volume of $\\mathcal {O}(10)$ (Gpc$/h)^3$ .",
"Figure: Comparison of the PDF covariance matrix contributions for z=0z=0, R=10R=10 Mpc/h/h as measured from the 1(1(Gpc/h) 3 /h)^3 Quijote simulations and the super-sample covariance contribution constructed from the DC mode following equation ().The upper triangle shows the super-sample covariance contribution while the lower triangle shows the measured covariance from the simulations enhanced by a factor of 10, while the diagonal from the upper panel is masked.Predictions.",
"Predicting the covariance of the PDF of the normalised matter density $\\hat{\\rho }$ in a three-dimensional volume is conceptually analogous to the previous two-dimensional case relevant for weak lensing, so we can write the covariance in terms of the two-point PDF as in equation (REF ) by replacing the weak lensing convergence with the density $\\kappa \\rightarrow \\rho $ and angular by 3D-distances $\\theta \\rightarrow r$ , where $P_d(r)$ indicates the distribution of distances $r$ in a given survey volume.",
"The distance distribution in a cubic box is easily obtained from the one for a 3D unit cube that corresponds to the problem of cube line picking for which a closed-form expression is available , .",
"Similarly to the weak lensing case, where the mean-subtraction leads to a modification of the relevant bias functions determining the covariance, the relative density definition will result in modified bias functions.",
"This can be predicted by considering the matter PDF in different sample densities $\\rho _s$ , where the joint PDF of normalised densities $\\hat{\\rho }_i$ is obtained from the trivariate PDF of physical densities $\\rho _i=\\rho _s\\hat{\\rho }_i$ and the background density $\\rho _s$ by integrating out the latter ${\\cal P}(\\hat{\\rho }_1,\\hat{\\rho }_2)=\\int d\\rho _s\\, \\rho _s^2 {\\cal P}(\\rho _s,\\rho _1=\\rho _s \\hat{\\rho }_1,\\rho _2=\\rho _s\\hat{\\rho }_2)\\,.$ The super-sample covariance effect (REF ) for the PDF of the normalised matter density $\\hat{\\rho }$ is obtained from the leading order effective bias function in the resulting covariance, given by equation (28) in [8] $\\hat{b}_1(\\hat{\\rho })=1+\\frac{\\partial \\log {\\cal P}(\\hat{\\rho })}{\\partial \\log \\hat{\\rho }} + b_1(\\hat{\\rho }) \\,.$ Note that while the expression looks similar to the one for the mean-subtracted weak lensing case $\\tilde{b}_{1,\\rm NG}(\\tilde{\\kappa })$ from equation (REF ), this contribution does not vanish even for a Gaussian field.",
"Predictions for the ingredients, the one-point PDF ${\\cal P}(\\rho )$ and the sphere bias $b_1(\\rho )$ , can be obtained from large-deviation statistics , , shifted lognormal models (Section REF ) and tree models [8].",
"Comparison.",
"In Figure REF we demonstrate that the derivative of the PDF with respect to the background density (green data points) determining the super-sample covariance (REF ) is well predicted by the effective bias $\\hat{b}_1$ from equation (REF ) using theoretical ingredients (green solid line) or simulated ones (green dotted line).",
"We also show the $\\hat{b}_1$ ingredients consisting of the first order sphere bias function $b_1$ as measured from the correlation between neighbouring spheres in one realisation (blue data points) in comparison to the large-deviation theory prediction and the lognormal approximation (blue dashed line), and the logarithmic derivative of the one-point PDF as measured from finite differences (red data points) and predicted from large-deviation theory (red solid line).",
"We conclude that the super-sample covariance effect can be robustly predicted by the theory presented here and that the shifted lognormal model holds promise to compute analytical covariances for the PDF of 3D clustering observables.",
"Applications.",
"The total covariance matrix is a sum of the simulated covariance, $C_{\\rm sim}$ and the super-sample covariance term consisting of a dyadic product, $C_{\\rm SSC}=v_{\\rm SSC}v_{\\rm SSC}^\\textsf {T}$ with $v_{\\rm SSC}=\\sqrt{\\bar{\\xi }}(\\hat{b}_1{\\cal P})(\\hat{\\rho })$ .",
"This sum can be inverted using the Sherman-Morrison formula to obtain the precision matrix $\\left(C_{\\rm sim} + C_{\\rm SSC}\\right)^{-1} = C_{\\rm sim}^{-1} - \\frac{C^{-1}_{\\rm sim} C_{\\rm SSC}C_{\\rm sim}^{-1}}{1 + v_{\\rm SSC}^\\textsf {T}C_{\\rm sim}^{-1}v_{\\rm SSC}}\\,.$ The result can be used to predict the super-sample covariance impact on statistical tests like $\\chi ^2$ and Fisher contours, which generally leads to a degradation of parameter constraints as detailed in for the case of Euclid-like photometric galaxy clustering observables.",
"Figure: Comparison of the theoretically predicted sphere bias (blue solid), the lognormal approximation (blue dashed) and the measurement in one realisation of the fiducial cosmology of Quijote (blue data points) at radius R=10R=10 Mpc/h/h, distance d=20d=20 Mpc/h/h and redshift z=0z=0.Additionally shown is the logarithmic derivative of the PDF (red) as predicted (solid) and measured (data points) and the resulting prediction from the effective bias () using the theoretical ingredients (green solid) or the simulated ones (green dotted).",
"The predictions agrees very well with the measurements from separate universe simulations (green data points).The vertical dotted line indicates the peak location of the PDF.",
"The ρ\\rho -range shown equals the one used in the covariance plots in Figure ."
],
[
"Conclusion & Outlook",
"Conclusion.",
"In this work, we have shown how to compute covariances for the one-point PDF of the weak lensing convergence on mildly nonlinear scales.",
"Our covariance predictions (REF ) rely on the two-point PDF, which is integrated over all separations according to the distribution of distances in the survey area.",
"Our formalism includes predictions for the key effects including finite sampling (REF ) and super-sample covariance (REF ) which were shown to scale with the inverse survey area.",
"We presented effective models for this joint PDF for a Gaussian field (REF ), a shifted lognormal field (REF ) matching the PDF skewness and density-dependence of the two-point clustering (Figure REF ), and a field affected by shape noise (REF ) from galaxy shape measurements.",
"Going beyond (log-)normal models, we laid out a large-separation expansion for the two-point PDF that captures the key effects of the PDF covariance for general non-Gaussian fields.",
"The validity of our predictions has been established through a set of tests including visual inspection of the covariance, leading order eigenvalues and eigenvectors (Figure REF ), the precision matrix (Figure REF ), $\\chi ^2$ tests (Figure REF ) and a two-parameter Fisher forecast (Figure REF ).",
"With this suite of tests we have demonstrated that our PDF covariance model not only accurately describes large sets of simulated FLASK maps, but also measurements of realistic full sky convergence maps obtained from full $N$ -body simulations .",
"Finally, we have sketched how the formalism described here for mildly non-Gaussian weak lensing observables is also applicable to more strongly non-Gaussian 3D clustering quantities.",
"Most importantly, we showed how this can yield predictions for the super-sample covariance effect (Figures REF and REF ) to complement covariance measurements from finite-box simulations.",
"Outlook.",
"We expect our formalism can be straightforwardly adapted to describe the one-point PDF of related weak-lensing observables such as the CMB weak lensing convergence [5] and aperture mass [6], as well as photometric galaxy clustering.",
"Our model can be extended to describe the cross-correlations between weak lensing PDFs at different smoothing scales and/or tomographic redshift bins.",
"Based on our PDF estimator, one can further attempt to model cross-covariances between the one-point PDF and 2-point statistics, which in the mildly non-linear regime are dominated by the correlation between the PDF and the variance .",
"Note that our covariance model and its possible extensions can serve as a starting point for improving covariance estimates from simulations, even in regimes where the model itself becomes inaccurate.",
"The Gaussian and shifted lognormal covariance models can be used as a starting point for a precision matrix expansion in which the leading order eigenvectors could iteratively refine this baseline covariance model.",
"Additionally, analytical models can be used to construct priors for a Bayesian covariance estimation with the help of cheap surrogates ."
],
[
"Acknowledgements",
"This work has made use of the Infinity Cluster hosted by Institut d'Astrophysique de Paris.",
"We thank Stéphane Rouberol for smoothly running this cluster for us.",
"We thank the authors of for their publicly available lensing maps, for their publicly available simulation suite products, and for their publicly available FLASK code.",
"The figures in this work were created with matplotlib making use of the numpy , scipy , sci-kit image and ChainConsumer Python packages.",
"CU is supported by the STFC Astronomy Theory Consolidated Grant ST/W001020/1 from UK Research & Innovation.",
"CU's work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611 and partially supported by a grant from the Simons Foundation.",
"AG is supported by an EPSRC studentship under Project 2441314 from UK Research & Innovation.",
"SC thanks Fondation Merac and the French Agence Nationale de la Recherche (grant ANR-18-CE31-0009) for funding.",
"The authors thank the Euclid work packages for additional galaxy clustering probes and higher-order weak lensing statistics, as well as the Rubin LSST:DESC higher-order statistics group for useful discussions."
]
] | 2210.07819 |
[
[
"CHANG-ES XXVI: Insights into cosmic-ray transport from radio halos in\n edge-on galaxies"
],
[
"Abstract Galactic winds play a key role in regulating the evolution of galaxies over cosmic time.",
"In recent years, the role of cosmic rays (CR) in the formation of the galactic wind has increasingly gained attention.",
"Therefore, we use radio continuum data to analyse the cosmic ray transport in edge-on galaxies.",
"Data from the LOFAR Two-metre Sky Survey (LoTSS) data release 2 at 144 MHz (HBA) and reprocessed VLA data at 1.6 GHz (L-band) from the Continuum Halos in Nearby Galaxies - an EVLA Survey (CHANG-ES) enable us to increase the extent of the analysed radio continuum profile significantly (up to a factor of 2) compared to previous studies.",
"We compute thermal emission maps using a mixture approach of H-alpha and near infrared data, which is then subtracted to yield radio synchrotron emission maps.",
"Then we compile non-thermal spectral index maps and compute intensity profiles using a box integration approach.",
"Lastly, we perform 1D cosmic ray transport modelling.",
"The non-thermal spectral index maps show evidence that the LoTSS maps are affected by thermal absorption, in star forming regions.",
"The scale height analysis reveals that most of the galaxies are equally well fitted with an one-component instead of a twocomponent exponential profile.",
"We find a bi-modality within our sample.",
"While NGC 3432 and NGC 4013 have similar scale heights in the L-band and HBA, the low-frequency scale heights of NGC 891, NGC 4157, and NGC 4631 exceed their high-frequency counterpart significantly.",
"The 1D CR transport modelling shows agreement of the predicted magnetic field strength and the magnetic field strength estimates of equipartition measurements.",
"Additionally we find an increasing difference of wind velocities (with increasing height over the galactic disk) between central and outer regions of the analysed galaxies."
],
[
"Introduction",
"The evolution of galaxies is a highly complex and not yet fully understood process.",
"The circumgalactic medium (CGM) builds the venue where most of the driving factors interplay.",
"There are multiple processes that directly influence the CGM.",
"Gas is introduced by cosmic accretion of the intergalactic medium (IGM), satellite galaxies and their winds add gas from outside of the galaxy halo [29], thermal instabilities convert hot to cold gas [16], and galactic winds of the host galaxy itself also transport gas from the disk into its halo.",
"The constituents of this interplay need to be analysed separately and discussed together in order to derive a holistic view.",
"Analyses of the CGM show that the CGM fuels the galaxy's star formation and moderates feedback processes [92].",
"Therefore, analysing details of the CGM (e.g.",
"multiphase structure, gas dynamics, metallicity) is key to a better understanding of the evolution of galaxies.",
"Besides all these dynamical and thermal processes, simulations show that there are also non-thermal processes like magnetic fields and cosmic rays (CR) that directly influence the CGM [45], [94].",
"In this article, we focus on galactic winds of the host galaxy that drive the processed interstellar medium (ISM) into the galactic halo.",
"In recent years, cosmic rays gained increasing attention from observers as well as theorists, as their influence on the ISM and galactic evolution in general might have been overlooked before.",
"The interactions of CRs with the ISM are manifold and complex.",
"CRs propagate from their origin, supernova (SN) remnants, through the galactic disk and into the galactic halo.",
"Several propagation mechanisms can operate in this environment, namely advection, diffusion, and streaming [86], [17], [74], [73].",
"CRs interact mainly with the galactic magnetic field and can therefore transfer some of their momentum to the thermal gas through scattering in magnetic field irregularities with the force directed down the CR pressure gradient.",
"This then leads to a fluid description of the interaction between CRs and thermal gas and is sometimes referred to as `cosmic ray hydrodynamics' [109].",
"In this context, the CR electrons (CREs) are used as proxies for the whole CR population (CREs and CR protons (CRPs)), which peaks at a few GeV.",
"This is motivated by the similar shape of the CRE and CRP spectrumThe CRPs have a much higher energy density than CREs but due to the higher rest mass of the protons, CRPs with energies of a few GeV are not observable because they emit most of their synchrotron radiation at very low and therefore non-observable frequencies [108]..",
"This hydrodynamic description has been implemented in galactic wind models which show that the mass-loss rate, gaseous distribution and even the existence of winds are dependent on CR transport [14], [26], [93], [76], [75], [28].",
"It is the properties (i.e.",
"wind speed, implied magnetic field strength in the galactic halo) of these winds, driven by stellar feedback, that we would like to explore further in this work.",
"Radio continuum observations offer us the possibility to study the transport of CRs using the electrons as a tracer [34].",
"With the non-thermal component of the radio continuum emission, one can follow CREs from their origin, SNe in star forming regions, into the galactic halo.",
"On their way through the galaxy, these charged particles gyrate around the magnetic field lines and therefore emit synchrotron radiation.",
"Among the questions we would like to answer are whether these winds exist in the first place, and if so, whether they are able to overcome the galactic gravitational potential and therefore enrich the CGM.",
"Edge-on spiral galaxies are good laboratories to study galactic winds because only here, the path of the galactic wind into the halo can be examined.",
"[104] show that such radio halos can be detected using deep radio continuum data, they even detect radio halos with vertical extents larger than the diameter of the disk.",
"[92] note that the CGM reaches several hundreds of kilo parsecs (kpc) (based on absorption studies).",
"Such a scale cannot be probed with today's radio continuum observations.",
"Here, the emission reaches about ten kpc [32].",
"However, as pointed out before, the analysis of these galactic winds is very important for a better understanding of the CGM properties, since the material of the galactic disk can only reach the CGM if and only if a bona fide wind develops.",
"Since synchrotron radiation preferably depletes the highest energy electrons, the non-thermal spectral index (SPIX) of the emission is expected to steepen on its way from the disk into the galaxy halo.",
"This steepening of the radio spectrum is usually referred to as spectral ageing.",
"The spectral index $\\alpha $ $(\\text{I}_\\nu \\propto \\nu ^\\alpha )$ in this work is defined as: $\\alpha \\ = \\frac{\\log \\left(S_{\\nu _1}/S_{\\nu _2} \\right)}{\\log \\left( \\nu _1/\\nu _2 \\right)}.$ Such a steepening is observed in radio galaxies [19], [35] as well as in star-forming galaxies [77], [61], [59], [83], [32].",
"For star forming galaxies, the steepening of the SPIX can be used to further analyse the transport of CRs in galactic winds.",
"As CRs are accelerated by SNe in the galactic disk, there is a gradient in the CR pressure pointing from the galactic disk into the halo.",
"As pointed out by [75] and [32], this gradient influences the background plasma by adding an additional pressure term, PCR, to the overall pressure, which can exceed the gravitational pull and therefore launch a CR-driven wind.",
"The use of low-frequency radio data to analyse non-thermal galactic winds via CR ageing offers a twofold advantage.",
"First, for such an analysis, a reliable spectral index measurement is essential.",
"From Eq.",
"(REF ), one can deduce that the accuracy of the SPIX measurement grows with the relative distance of the two analysed spectral bands.",
"Therefore combining low- and high-frequency data sets increases the quality of the SPIX measurement [85], [59], [83].",
"Second, with low-frequency data one can trace the older CR population that has already travelled far into the galactic halo, and therefore increase the extent to which the halo is observable.",
"On basis of such data sets (high- and low-frequency radio continuum maps), one can reliably distinguish whether advection or diffusion is the dominant transport mechanism.",
"Advection-dominated transport seems to produce large radio halos, while diffusion-dominated transport results in much smaller halos [36], [82].",
"To conclude this section, we note that the importance of CRs for the galactic winds and for the evolution of galaxies as a whole is now widely recognised.",
"However, the interaction of CRs with the ISM and the galactic magnetic field are not yet understood in detail.",
"In particular, the potential of CRs to drive galactic winds is the subject of ongoing discussion.",
"A crucial observational experiment would be to measure both the outflow velocity of the ionised gas and the CR advection speed.",
"With CR-driven winds, we would expect the CRs to be transported faster than the gas, which is the basic mechanism that can offset the adiabatic cooling of the gas in the hydrodynamic model of CR-driven winds [109].",
"Such a differential velocity could be explained for instance by CR streaming with the Alfvén velocity or by CR diffusion down the density gradient [106].",
"Observationally, this is a difficult task as it is hard to measure both velocities due to geometry.",
"While edge-on galaxies allow us to study the radio halo and the cosmic-ray advection speed, face-on galaxies are better suited to study outflow speeds of the ionised gas with optical or ultra-violet spectroscopy [33].",
"Therefore, as an alternative, we aim to build up a coherent picture of both CR and gaseous transport in galactic halos in a sample of galaxies with similar properties over a range of fundamental parameters to better understand the influence of CR transport mechanisms on stellar feedback-driven winds [34].",
"To expand this picture with new deep radio observations is the scope of this work.",
"This paper is structured as follows: In Sect.",
", we describe the galaxies that are analysed in this work and put them into context by comparing them to other nearby galaxies.",
"Section describes the data that are used and Sect.",
"describes the processing of the data.",
"We then present the first results in Sect.",
"before we further analyse the radio halo profiles in Sect.",
"and derive physical properties of the non-thermal galactic wind by applying a 1D cosmic ray transport model in Sect.",
".",
"Lastly, we conclude the paper in Sects.",
"and ."
],
[
"Sample Description",
"In order to analyse the interaction of the galactic disk with its halo, the orientation of the galaxy is an important factor.",
"Edge-on galaxies allow us to trace the CREs from their origin in the disk over several kpc into the galaxy halo.",
"Therefore, the data acquired for the Continuum Halos in Nearby Galaxies – an EVLA Survey (CHANG-ES), is the starting point for our analysis.",
"The CHANG-ES survey consists of 35 nearby (d $\\le $ 42 Mpc) edge-on (76$^\\circ \\le \\text{i}\\le \\text{90}^\\circ $ ) galaxies [41], [104].",
"As previously outlined, adding low-frequency data to the information drawn from the VLA maps highly improves the quality of SPIX maps, which is a crucial for the planned analysis.",
"Therefore, we only choose the 17 galaxies that lie in the footprint of the LoTSS-DR2.",
"Radio halos of edge-on galaxies can be identified by comparing the minor-to-major axis ratios in the radio continuum and optical emission.",
"If a radio halo is present, we expect the ratio in the radio continuum to be significantly larger than the optical one.",
"In Fig.",
"REF , we show such a comparison for the LoTSS-DR2 sample.",
"As the CHANG-ES galaxies have high inclination angles, they lie on the left side of the plot, clearly separated from the other galaxies with moderate inclination angles.",
"We find that of the 17 CHANG-ES galaxies in LoTSS-DR2, nine galaxies have axis ratios with $(a/b)_{\\rm radio}\\gtrsim 2(a/b)_{\\rm opt}$ .",
"These galaxies are NGC 2820, NGC 4157, NGC 5297, NGC 3079, NGC 4631, NGC 891, NGC 3556, NGC 2683, and NGC 4217 (listed with increasing $(a/b)_{\\rm opt}$ ).",
"Hence, these galaxies are most promising to study galactic winds.",
"Of these galaxies, two were studied previously with LOFAR, namely NGC 891 [62] and NGC 3556 [59], where CR transport was specifically modelled only in NGC 3556.",
"Of the remaining galaxies, NGC 4631, NGC 891, NGC 2820, and NGC 4157 have the highest radio-to-optical ratios.",
"We chose to omit NGC 2820 as previous analyses (e.g.",
":[95] (VLA 1.5 GHz), [104] (CHANG-ES D-configuration), [42] (CHANG-ES B-configuration), [47] (GMRT), and [39] (LOFAR LoTSS)) show a strong radio bridge, reaching from the galaxy to its closest interaction partner NGC 2814, which will most probably disturb the radio halo and therefore prevent the planned analysis.",
"As NGC 891 and NGC 4157 are fairly fast rotating, compared to NGC 4631, we decided to also include NGC 3432, which has a less prominent radio halo, to have another slow rotating galaxy in our sample.",
"We further include NGC 4013, which was already studied by [83] who found a diffusion-dominated radio halo.",
"We chose to re-analyse this galaxy with consistent methods, as we expect a more accurate measurement of the radio halo intensity profile due to the newly processed data sets.",
"Hence, we chose five galaxies from LoTSS-DR2 for further analysis, namely NGC 891, NGC 3423, NGC 4013, NGC 4157, and NGC 4631.",
"Table REF lists some fundamental parameters of each analysed galaxy.",
"Figure: Total Intensity maps + contours from CHANG-ES L-band (left column) and LOFAR HBA (right column) after subtracting/masking point sources in the vicinity of the galaxy for NGC 891 (top row), NGC 3432 (middle row), and NGC 4013 (bottom row).",
"Contours start at 3σ\\sigma above the background noise with an increment of 2.",
"The beam is displayed in the bottom right corner of each map as a dark red circle.Figure: Continuation of Figure .",
"Total Intensity maps from CHANG-ES L-band (left column) and LOFAR HBA (right column) after subtracting/masking point sources in the vicinity of the galaxy for NGC 4157 (top row), NGC 4631(bottom row)."
],
[
"Radio-Continuum Data",
"In order to avoid losses of diffuse emission in these extended galaxies and their halos, we combine C-configuration and D-configuration L-band data.",
"All VLA data sets that are used in this project where observed for the CHANG-ES project (Project Code 10C-119).",
"The observation and calibration strategy is fully described for the D-configuration data in [105] The data are publicly available: https://www.queensu.ca/changes.",
"In Table REF , we list the relevant information for the VLA C-configuration L-band observations.",
"Phase and flux density calibration have been performed using the Common Astronomy Software Applications package (CASA) [58].",
"Self calibration was incorporated in the calibration process if it increased the data quality.",
"The frequency range for all targets spans from 1.247 GHz to 1.503 GHz and from 1.647 GHz to 1.903 GHz.",
"Table: Basic information about the galaxies EVLA C-Array observations.Cleaning is performed using WSClean [66].",
"We incorporate multiscale cleaning, automatic masking and a Briggs robust weighting of 0.4.",
"Additionally, a Gaussian taper of 6 is applied to the uv-weights and the restoring beam size is set to 20 .",
"The LOFAR data are based on observations from the LOFAR Two-metre Sky Survey [79], [80], using data from LoTSS-DR2 [78].",
"These data are taken with the high-band antenna (HBA) system with an effective frequency of 144 MHz.",
"The data were presented in [39], the data release paper of nearby galaxies in LoTSS-DR2.",
"Besides following standard observing and data reduction procedures for LoTSS data as described by [79], [80], we use the re-calibration pipeline described by [97] which results in new $(u,v)$ data sets that are specifically tailored to our target galaxies.",
"The maps are cleaned using WSClean with a Briggs weighting of $-0.25$ and a Gaussian taper of 10 with a fixed restoring beam size of 20 [39].",
"The imaging process of NGC 891 was challenging due to its proximity to a $\\sim 10$ Jy radio source [48], the radio galaxy 3C 66, and some fine tuning of parameters was necessary.",
"Therefore, we use a slightly different strategy when creating a new 144 MHz map for NGC 891 with WSClean.",
"Here, we incorporate a Briggs weighting of $-0.15$ together with multi-scale CLEAN.",
"In order to accurately account for the influence of 3C 66, which lies to the north of NGC 891, the map spans over $\\sim 1.9^{\\circ } \\times 1.9^{\\circ }$ .",
"A Gaussian taper of 7 is applied in the ($uv$ )-plane and the final beam size is set to 20 .",
"This map has also been published by [39].",
"To estimate the noise levels of all radio maps, we run PyBDSF [60], using standard parameters (source detection threshold 5$\\sigma $ ; island boundary threshold 3$\\sigma $ ).",
"The resulting noise estimates are listed in Table REF ."
],
[
"Additional Data Sets",
"As in [99], we use resolution enhanced (124) WISE 22 µm images to correct for thermal emission in our CHANG-ES galaxies [44], combined with H$$ data.",
"Since the resulting WISE point spread function (PSF) may deviate significantly from a purely Gaussian PSF, we use an empirical PSF model when convolving the maps to the resolution of the radio maps [43].",
"This model was compiled by stacking stars in a large field to extract the shape of the PSF [43].",
"H$$ images of all CHANG-ES galaxies have been analysed in [100].",
"For this analysis, 25 of the 35 CHANG-ES galaxies (including NGC 3432, NGC 4013, and NGC 4157) were observed with the 3.5 m telescope of the Apache Point Observatory [100].",
"For NGC 4631, [100] use an H$$ map that was observed with the 2.1 m telescope at the Kitt Peak National Observatory as ancillary data for the Spitzer Infrared Nearby Galaxies Survey [22], [23].",
"The H$$ map of NGC 891 was obtained from the ancillary data set for the WSRT Hydrogen Accretion in LOcal GAlaxieS (HALOGAS) survey [31], [30]."
],
[
"Point Source Removal",
"In order to extract the radio profile, background and foreground point-like sources need to be removed.",
"We detect compact sources by using PyBDSFwith an adaptive_rms_box and a maximum detection island size.",
"The detected sources are then fitted as Gaussian components and subtracted from the original image.",
"Only large, extended sources remain in the residual image.",
"We further inspect the images manually to mask out previously undetected or nested sources by using the corresponding LoTSS 6map, published in [39], for an exact positioning of the masks.",
"With this approach, the profile of the galaxy can be reliably followed, without being affected by back- or foreground sources."
],
[
"Thermal Correction",
"In star forming galaxies, radio continuum emission is a superposition of thermal and non-thermal processes: $S_{\\mathrm {tot}}(\\nu ) = S_{\\mathrm {th}}(\\nu _0)\\left(\\frac{\\nu }{\\nu _0}\\right)^{-0.1} + S_{\\mathrm {nth}}(\\nu _0)\\left(\\frac{\\nu }{\\nu _0}\\right)^{\\alpha _{\\mathrm {nth}}},$ where $S$ denotes the flux density at a given frequency $\\nu _0$ .",
"The standard picture is that the thermal emission follows a power-law with a spectral index of $-0.1$ .",
"Overall, the measured combined spectral index is about $\\alpha _{\\mathrm {tot}} \\approx -0.7$ .",
"Therefore, in order to analyse non-thermal processes such as winds driven by stellar feedback, the thermal emission needs to be estimated and subtracted.",
"However, from Eq.",
"(REF ), one can deduce that the fraction of the thermal emission drops when reaching lower frequencies and hence that such a correction might not be needed for low-frequency data.",
"It is also important to note that there are significant deviations from a constant non-thermal spectral index when reaching higher frequencies [50].",
"In the literature, there are several approaches to separate the synchrotron emission from the thermal emission, e.g.",
"assuming a constant non-thermal SPIX [49], estimating the thermal emission from H ii regions [7], or using de-reddened H$$ emission [88], [89].",
"For this project, we re-implementedThe Python3 package can be downloaded via https://github.com/msteinastro/thermal_maps_chang-es_xxvi the mixture-method technique developed by [99], which is specifically designed to estimate the thermal emission in edge-on galaxies.",
"This technique is based on the combination of H$$ and mid-infrared (WISE 22 µm) images to compile a star formation map and then convert this map to a thermal emission map.",
"This technique also has been applied in the second CHANG-ES data release paper [100].",
"For in depth details of the technique, we refer to [99], [100].",
"Here, we shortly describe the overall workflow and point out where we deviate from the current implementation.",
"In order to combine H$$ with mid-infrared data and later apply them to a radio map, we first need to match resolutions.",
"As the mid-infrared images have an resolution of $\\sim $ 124 but deviate significantly from a Gaussian PSF, it is not advisable to convolve the original images with a purely Gaussian kernel as the result will not be a Gaussian PSF.",
"This is especially important when the target resolution of the radio maps is only slightly above the FWHM of the mid-infrared images.",
"Therefore, it is imperative to use individually tuned homogenisation kernels.",
"The former implementation used convolution kernels provided by [3], where one could select from a variety of kernels to produce a Gaussian PSF with a predefined FWHM.",
"As this approach limits analyses to certain resolutions, we now use the WISE PSF model described in Sect.",
"and the Python package PyPHERhttps://github.com/aboucaud/pypher [13], [12] to convolve the mid-infrared image to the target resolution.",
"The benefit of such an approach is that one can select any type of target kernel, rather than having to choose from a couple of pre-selected kernels.",
"As the H$$ observations are mainly seeing limited, their resolution is of the order of $\\sim $ 15.",
"Since the target resolution of any study that uses the mid-infrared data needs to be above 124, the resulting PSF for the H$$ image will be dominated by the convolution kernel, so that we can convolve the H$$ images with a purely Gaussian kernel.",
"The resolution-matched and calibrated (erg s-1 cm-2) images can then be further processed.",
"As the following method relies on 24 µm data, the WISE band 4 data, which are centred on 22 µm, need to be corrected.",
"As shown by [104], there is a tight linear relationship between 22 µm and 24 µm emission and therefore, the data can be scaled using a factor of 1.03.",
"Following [18], we can use the mid-infrared data to correct the H$$ maps for dust extinction: $L_{\\mathrm {H\\alpha \\_corr}}=L_{\\mathrm {H\\alpha \\_obs}} + 0.042\\cdot L_{24\\mathrm {\\mu m}}.$ The factor 0.042 was derived in [99] and is only valid for edge-on or very dusty galaxies and corrects the original factor by [18] with an additional factor of $1.36$ .",
"The corrected H$$ emission can then be related to a star formation rate and hence to the thermal radio emission at a frequency, $\\nu $ , [65], [99]: $\\mathrm {SFR_{mix}\\,[M_\\odot /yr]} = 5.37\\cdot 10^{-42}\\cdot L_{\\mathrm {H\\alpha \\_corr\\,[erg\\,s^{-1}]}},$ $L_\\nu ^{\\mathrm {T}}\\,\\mathrm {[erg\\,s^{-1}\\,Hz^{-1}]} = 2.2\\cdot 10^{27}\\left( \\frac{T_{e}\\,[\\mathrm {K}]}{10^{4}} \\right)^{0.45} \\left(\\nu \\,[\\mathrm {GHz}]^{-0.1} \\right)\\mathrm {SFR_{mix}}.$ Here, we assume an average electron temperature of $T_e = 10^{4}$ K [69], [18].",
"With Eq.",
"(REF ) we compute thermal emission maps for 144 MHz as well as for 1.5 GHz.",
"These thermal maps can then be subtracted from the radio maps to obtain the pure synchrotron emission.",
"As a concluding remark, we note that several assumptions and systematic uncertainties influence the estimate of the thermal emission (e.g.",
"a varying H$$ /N [ii] ratio in the diffuse ionised gas [70], which makes accounting for N [ii] contamination in the H$$ images more complicated, or varying electron temperatures) that dominate the statistical uncertainties coming from the H$$ and mid-infrared background noises.",
"[99] present a detailed analysis of all relevant factors and conclude that the thermal emission estimates have an overall uncertainty of approximately 14%.",
"Considering the fact that the contribution of thermal emission to the overall radio emission in L-band in the observed galaxies is of the order of approximately 30% in star forming regions, the introduced uncertainties onto the radio data are expected to be approximately 4% (even less than the HBA data) and are therefore smaller than standard calibration errors.",
"Figure: Non-thermal spectral index (L-band corrected for thermal emission & uncorrected HBA) maps.",
"We use a custom colormap, where a flat SI is coloured blue to plum and a steep SI orange to yellow.",
"The beam is displayed in the bottom right corner of each map as black circle."
],
[
"Total Intensity",
"Table REF lists the integrated flux density measurements as well as the noise estimates for all analysed galaxies in LOFAR HBA and VLA (CHANG-ES) L-band.",
"The uncertainties of the flux density measurements $\\Delta _{\\text{S}_{\\nu }}$ with a flux density of $\\text{S}_{\\nu }$ , are computed via: $\\Delta _{\\text{S}_{\\nu }} = \\sqrt{\\left( \\sigma \\sqrt{N_{\\text{beams}}} \\right)^2 + \\left( \\epsilon S_{\\nu } \\right)^2}.$ Here, $\\sigma $ denotes the background noise of the individual map and $N$beams the number of telescope beams that fit into the defined aperture.",
"As an additional error term, one needs to account for calibration errors.",
"This is implemented as a relative error term by setting $\\epsilon =0.1$ for the HBA measurements [79] and setting $\\epsilon =0.05$ for the L-band measurements.",
"Calibration errors are typically much more severe in the low-frequency regime, that is why we choose different weightings for the relative error terms.",
"Since all the galaxies that are analysed in this work are relatively bright, the error terms are dominated by the calibration error rather than the background noise.",
"In Fig.",
"REF and REF , we present the radio maps after the subtraction and masking of background or foreground sources.",
"All galaxies show a distinct extent orthogonal to the galactic disk.",
"Comparing the extent of the outermost (3$\\sigma $ ) contour, we detect larger extents in the HBA maps compared to the L-band contours (for NGC 3432, this comparison is more challenging as the overall structure of the contours is more complex compared to the other galaxies).",
"Of course, comparing the outermost contours is not a perfect way to determine the extent of a radio halo, as this measurement is limited only by the sensitivity of the particular data set.",
"However, one can already assume that this larger extent in the lower frequency data set reflects the ageing process of the CREs.",
"Another interesting point to discuss is that the size of the individual radio halos varies strongly between the galaxies when compared to the galaxy's major axis.",
"This is best observed in the low-frequency maps.",
"The physical extents of the galactic disk and the halo based on the $3\\sigma $ -contours of the HBA maps are listed in Table REF .",
"While the radio halo of NGC 4013 shows only a small extent in $z$ direction, the radio halos of NGC 891, NGC 3432, and NGC 4157 reach already far up into the galactic halo.",
"The most extreme case in this sample is certainly NGC 4631.",
"Here, the extent in $z$ -direction almost exceeds the galaxy's major axis.",
"Such a comparison between the galaxies can be made because all LOFAR maps have a similar background noise (cf.",
"Table REF ).",
"At this point, we can conclude that the dominating CRE transport mechanisms of the galaxies studied here are different.",
"In a previous study, [83] have shown that the intensity and SPIX profiles of NGC 4013 might best be modelled with a diffusion dominated CRE transport, which might result in a smaller vertical extent of the galactic halo."
],
[
"Thermal & Non-Thermal Emission",
"Here we present the result of the thermal emission correction, using the mixture method approach described in Sect.",
".",
"Table REF lists the integrated thermal flux densities integrated over defined aperture as well as the fraction of non-thermal emission (NTF), which is the remaining synchrotron radiation.",
"As expected, the contribution of thermal emission to the total radio emission decreases when lower frequencies are reached.",
"Table: Integrated thermal flux densities and non-thermal fractions in the analysed sample.As an integrated value, the thermal emission in the L-band is on the order of magnitude of the uncertainty due to the calibration error.",
"However, the importance of the thermal emission changes when looking at the resolved fraction of thermal emission.",
"To keep this work concise, we refer the reader to App.",
"for the thermal emission and thermal fraction maps.",
"In the thermal fraction maps, one can see that both the intensity of the thermal emission and the fraction of thermal emission in the overall radio emission decrease strongly when leaving the galactic disk.",
"In L-band the thermal emission constitutes $\\sim 30\\%$ of the overall observed radio emission in the galactic disk (for NGC 4631, the thermal contribution reaches even up to $\\sim 40\\%$ in the galactic disk).",
"However, the thermal fraction in the HBA data is much smaller.",
"Here, the thermal emission normally does not exceed 10% of the total emission (only NGC 4631 shows some regions where the thermal fraction reaches 15%).",
"Considering the overall calibration error of 10% in the LoTSS data, the effect of thermal emission does not seem to be significant at the observed frequency.",
"Therefore, we decided to correct only the L-band data for thermal emission in the further analysis."
],
[
"Spectral Index",
"In Fig.",
"REF , we present the non-thermal SPIX mapsCustom colour maps can be downloaded at: https://github.com/mlarichardson/CosmosCanvas/ for all analysed galaxies.",
"As mentioned before, only L-band has been corrected for thermal emission, since the thermal contamination in the HBA is not significant when considering calibration errors.",
"Overall, the galaxies show a common trend.",
"In the galactic plane, we detect a non-thermal SPIX of approximately -0.5.",
"Similar values have been reported in studies of NGC 3556 [59], NGC 5775 [32], and previous studies of NGC 891 [77], NGC 4013 [83] and NGC 4631[61], [102].",
"NGC 4631 shows in the outer part of the galactic disk similar SPIX values compared to the other galaxies of our sample, but deviates from the other galaxies in the centre.",
"Here, we report a SPIX of approximately -0.4.",
"This area coincides with the region of highest star formation and thermal radio emission (cf.",
"bottom left map in Fig.",
"REF ).",
"In Sec.",
"REF , these results will be discussed in more detail.",
"NGC 891, NGC 4157, and NGC 4631 show a strong and uniform gradient (declining SPIX) from the galactic disk into the halo.",
"Such a gradient can be attributed to the ageing of CREs.",
"In NGC 4013, we also detect a gradient in the non-thremal SPIX.",
"However, we find an extended region in the centre of the galaxy with very flat SPIX values ($\\alpha _{\\mathrm {nth}} < -0.7$ ) and the gradient in the outer regions seems to be less prominent.",
"The case of NGC 3432 is much less ordered.",
"A gradient in the SPIX is not clearly visible and and only further analysis will show if CRE ageing is observable in this galaxy.",
"However, one could have expected such a confused SPIX map, considering the fact that NGC 3432 is closely interacting with the dwarf galaxy UGC 5983UGC 5983 is located at 10h52m16.749s +36d35m40.24s (coordinates taken from NASA NED) but does not show radio continuum emission..",
"Therefore the dynamics of the galaxy might have disturbed the propagation of the CREs.",
"Another process that could explain the non-detection of a gradient in the spectral index is CR diffusion with a very strong energy dependence ($\\mu =1$ )In this paper, the energy dependence of the CR diffusion is described as $D(E) = D_0\\cdot (E/\\mathrm {GeV})^\\mu $.",
"The theoretical implications for a $\\mu =1$ energy dependence of the CR diffusion is discussed in [15].",
"This transport type would lead to a homogeneous non-thermal SPIX map, as the effects of CR transport and CR ageing would counterbalance each other.",
"However, a $\\mu > 0.7$ would require extreme conditions of the ISM turbulence or CR energies of about $10^4$ GeV [52].",
"By comparing their models to radio data, [15] obtain a limit of $\\mu < 0.4$ for the energy dependence of the CR diffusion.",
"[9] report an energy dependence of $D(E) \\propto E^{0.7}$ for the energy range $10\\,\\mathrm {GeV} < E <200\\,\\mathrm {GeV}$The energy dependence of the CR diffusion strongly depends on the observed energy range.. For energies slightly below 10 GeV, recent studies point to an even weaker ($\\mu \\sim 0.3$ ) energy dependence or no energy dependence of the CR diffusion at all [52], [24]."
],
[
"Box Integration & Scale Height Fitting",
"To trace the synchrotron emission further into the galactic halo, we average horizontally over large areas of the galaxy.",
"In the literature, there is some variety in the number of strips into which the integration areas are divided.",
"While [32] split up the galaxy into quadrants by choosing two vertical strips and splitting those again into northern and southern strips, there are also more detailed approaches (e.g.",
": [83]: five strips, [77]: five and seven strips).",
"The number of strips used is obviously dependent on the spatial extend of the target compared to the spatial resolution of the observation, and less strips makes it possible to follow the galaxy's profile a bit further.",
"We place three strips on each galaxy (left, middle, right) and analyse the upper and lower sections separately.",
"Here, the idea is to average as much data as possible while accounting for the different conditions in the galaxy regions (e.g.",
"stronger magnetic field in the centre of the galaxy; localised star formation regions, etc.).",
"Separating the central part of the galaxy from the outer regions is very valuable for our study as we know from our Galaxy that AGN activity can heavily influence the galactic halo (e.g.",
"Fermi-Bubbles [87]).",
"Box integration is performed using the BoxModels method of NOD3https://gitlab.mpifr-bonn.mpg.de/peter/NOD3 [63], [64].",
"We use the 3$\\sigma $ contour of the HBA map as outline for defining the width of the boxes.",
"The box setup for all galaxies is described in Table REF .",
"Concerning the box height, we follow the approach of [51] and [83] and set it to half of the beam width.",
"As an example, we show the box setup for NGC 891 on the HBA map in Fig.",
"REF .",
"Figure: Strip setup on the HBA map of NGC 891.",
"The map has been point source subtracted and masked.",
"The six strips (UL, UM, UR, LW, LM, LR) are indicated with thick black lines.",
"A similar setup is used on all galaxies during box integration with NOD3.",
"Central boxes are part of the lower as well as the upper strips.",
"Left strips have a positive offset, right strips have a negative one.",
"Individual boxes (thin white lines) have a height of 10and a width of 177(as listed in Table ).Table: Box widths (W) used for box integration, radio beam FWHM converted to physical scales, assumed flux tube radius (R0) used for cosmic ray transport modelling for each galaxy; SFR, SFR surface density, and average equipartition magnetic field strength in the galactic disk for individual stripes; gravitational pull for central and outer stripes (as we assume the gravitational potential to be symmetric, there is no distinction for left and right stripes).In order to derive reliable scale heights for each strip, we fit each strip with four models of decreasing complexity and evaluate the Akaike information criterion (AIC) [2] to select the best-fitting model while accounting for the different model complexities.",
"As the number of data points are quite limited per strip, we apply the small sample correction to the AIC (AICc).",
"For models that are generated by least square fitting routines, the AICc is computed in the following way: $\\mathrm {AICc} = 2k + n \\cdot \\mathrm {ln}(\\mathrm {RSS})+\\frac{2k^2+2k}{n-k-1}.$ Here $k$ denotes the number of model parameters, $n$ the number of data points and RSS the residual sum of squares.",
"Overall, we fit exponential profiles, however we need to account for the radio beam.",
"Therefore, the models that are actually fitted are convolutions of exponential profiles with Gaussian kernels [25].",
"Intrinsic exponential profiles with an intensity $w_0$ and a scale height $z_0$ : $\\mathrm {w}(z)=w_0 \\mathrm {exp}\\left(\\frac{-z}{z_0}\\right)$ need to be convoluted with a Gaussian kernel with a standard deviation $\\sigma $ : $\\mathrm {g}(z) = \\frac{1}{\\sqrt{2\\pi \\sigma ^2}}\\mathrm {exp}\\left(\\frac{-z^2}{2\\sigma ^2}\\right)$ to account for the limited resolution of the images.",
"We tried to fit different profiles, including Gaussian profiles, but the analysed intensity profiles are overall better fitted with exponential functions.",
"We use those fits to get the scale heights used in the following analysis.",
"The most complex model we fit to the vertical profiles is a two-component exponential profile with intensities $w_0$ , $w_1$ and scale heights $z_0$ , $z_1$ : $\\mathrm {w_{dual}}(z) = \\frac{w_0}{2}\\mathrm {exp}\\left(\\frac{{-z}^2}{2\\sigma ^2}\\right) &\\left[\\mathrm {exp}\\left(\\left( \\frac{\\sigma ^2 - z z_0}{\\sqrt{2}\\sigma z_0}\\right)^2\\right) \\mathrm {erfc} \\left( \\frac{\\sigma ^2 - z z_0}{\\sqrt{2}\\sigma z_0}\\right)\\right.\\\\+&\\left.",
"\\mathrm {exp}\\left( \\left(\\frac{\\sigma ^2 + z z_0}{\\sqrt{2}\\sigma z_0}\\right)^2\\right) \\mathrm {erfc} \\left( \\frac{\\sigma ^2 + z z_0}{\\sqrt{2}\\sigma z_0}\\right) \\right]\\\\+ \\frac{w_1}{2}\\mathrm {exp}\\left(\\frac{{-z}^2}{2\\sigma ^2}\\right)& \\left[\\mathrm {exp}\\left(\\left( \\frac{\\sigma ^2 - z z_1}{\\sqrt{2}\\sigma z_1}\\right)^2\\right) \\mathrm {erfc} \\left( \\frac{\\sigma ^2 - z z_1}{\\sqrt{2}\\sigma z_1}\\right) \\right.\\\\+ & \\left.",
"\\mathrm {exp}\\left(\\left( \\frac{\\sigma ^2 + z z_1}{\\sqrt{2}\\sigma z_1}\\right)^2\\right) \\mathrm {erfc} \\left( \\frac{\\sigma ^2 + z z_1}{\\sqrt{2}\\sigma z_1}\\right) \\right].$ Here, erfc is the complementary error function.",
"With such a two-component model, one can try to decompose galactic disks and halos.",
"However, such a distinction is not always possible.",
"Furthermore, the intensities $w_0$ and $w_1$ as well as the scale heights $z_0$ and $z_1$ in the two-component model are degenerate.",
"This can result in very large uncertainties in the model parameters, especially when fitting the model to only a limited number of noisy data points.",
"When the estimated uncertainty of a fitted model parameter exceeds the parameter value, we reduce the complexity of the model to obtain well-constrained fitting parameters by using a one-component exponential profile.",
"$\\mathrm {w}_{\\mathrm {single}}(z) = \\frac{w_0}{2}\\mathrm {exp}\\left(\\frac{{-z}^2}{2\\sigma ^2}\\right) &\\left[\\mathrm {exp}\\left( \\left( \\frac{\\sigma ^2 - z z_0}{\\sqrt{2}\\sigma z_0}\\right)^2\\right) \\mathrm {erfc} \\left( \\frac{\\sigma ^2 - z z_0}{\\sqrt{2}\\sigma z_0}\\right)\\right.\\\\+&\\left.",
"\\mathrm {exp}\\left(\\left( \\frac{\\sigma ^2 + z z_0}{\\sqrt{2}\\sigma z_0}\\right)^2\\right) \\mathrm {erfc} \\left( \\frac{\\sigma ^2 + z z_0}{\\sqrt{2}\\sigma z_0}\\right) \\right].$ With these models, we can fit radio halo scale heights in each strip.",
"We additionally allow for a shift in z-direction using the following coordinate transformation: $z^\\prime = z - Z_0.$ To summarise our scale height fitting process: Each strip is fitted with four models: A two-component exponential with shift in $z$ -direction (two), a two-component exponential without shift in $z$ -direction (two_fo), a one-component exponential with shift in $z$ -direction (one), and a one-component exponential without shift in $z$ -direction (one_fo).",
"The model selection is based on the AICc and the model complexity is reduced if the uncertainty of the best fit parameter exceeds the parameter itself."
],
[
"Scale Heights",
"To keep this paper concise, we refer the reader to App.",
", where we list the fitted model parameters and present the fitted models for each strip.",
"The scale heights averaged per galaxy are listed in Table REF .",
"As an example, we present the fitted profiles of NGC 891 (Fig.",
"REF ) and the corresponding fitting parameters (Table REF ).",
"Figure: Vertical radio continuum intensity profiles in the three strips of NGC 891.",
"Left panels show 144-MHz LOFAR data right panels show 1.51.5-GHz JVLA data.",
"Best-fitting double exponential profiles are shown as dashed lines.Table: Parameters from box integration model fitting of NGC 891: frequency of fitted data, model identifier, strip identifier, peak intensity and scale height of the first exponential component, peak intensity and scale height of the second exponential component (if applicable), offset in z-direction (if applicable), reduced χ-square.",
"The corresponding profiles are shown in Fig.",
".",
"A complete list of all galaxies can be found in Table .Table: Fitted radio scale heights per galaxy.",
"One and two component model are listed separately.",
"For the two component models, we distinguish disk and halo scale heights.",
"The Number of stripes (N) that have been fitted with each model is also indicated.",
"If only one stripe has been fitted with a certain model, the uncertainty of the scale heights is computed within χ 2 \\chi ^2-minimisation.",
"If multiple stripes have been been fitted with a model, the uncertainty is computed via the standard deviation of the sample.The intensity profile of NGC 891 has already been analysed in the previous works by [62] (MU18) and [77].",
"While [77] analysed a combination of VLA C-band and L-band data, MU18 also analysed the galaxy with help of data from the LOFAR telescope.",
"Both studies extract intensity profiles up to a height of $\\sim $ 3.5 kpc.",
"With the data used in this work, we can follow the intensity profiles much farther into the halo.",
"The newly cleaned CHANG-ES L-band allows us to extract accurate intensity profiles up to $\\sim $ 8 kpc for the upper half and to 6 kpc for the lower half of NGC 891.",
"Therefore, we can nearly double the extent of the analysed halo in the L-band data.",
"The improvement in data quality is even more pronounced for the low-frequency data set.",
"Here, we can follow the intensity profile to a height of $\\sim $ 11 kpc and therefore more than triple the extent of the analysed halo.",
"When averaging over strips where the data allowed to fit two exponential components (four out of six strips for HBA), we derive mean scale heights $z_{\\mathrm {disk}}=0.76\\pm 0.25$ kpc and $z_{\\mathrm {halo}}=2.9\\pm 0.5$ kpc for the galactic disk and the halo of NGC 891 in the HBA data.",
"Therefore, the derived scale height of the disk in this work is slightly larger than that derived by MU18 ($z_{\\mathrm {disk}}=0.32\\pm 0.08$ kpc; $z_{\\mathrm {halo}}=2.3\\pm 0.7$ kpc), while the halo scale heights are similar.",
"Both studies correct for the effect of smoothing by the radio beam during the fitting routine.",
"Nevertheless, especially the difference in the measurement of the disk scale height might result from different resolutions (12 in MU18, 20 in this work).",
"Due to the larger extent of the strips analysed in this work, the derived halo scale heights might be more reliable compared to previous studies.",
"Judging from the AICc model assessment, none of L-band data strips shows sufficient complexity to be fit with a two-component model.",
"Averaging over all L-band strips that were fitted with one component, we derive a mean scale height of NGC 891 of $z_{\\mathrm {1comp}}=1.09\\pm 0.16$ kpc.",
"Scale heights of NGC 4631 have been analysed before, by [61] (MP19) in the 6 GHz and 1.5 GHz bands.",
"With the strips that were fitted with two components in our analysis, we find larger radio halo scale heights than MP19.",
"However, both measurements have fairly large uncertainties ($z_{\\mathrm {halo}}=2.1 \\pm 0.3$ kpc (this work), $z_{\\mathrm {halo}}=1.75 \\pm 0.27$ kpc (MP19)).",
"In the HBA data, we report a mean scale height of the galactic halo of $z_{\\mathrm {halo}}=4.9 \\pm 0.4$ kpc.",
"This is more than double the scale height compared to the higher frequency data.",
"While MP19 extracted profiles of $\\sim 6.6$ kpc reaching from south to north of the galaxy, we follow the intensity profile $\\sim 8-10$ kpc in either direction of the galactic disk, which most probably increases the reliability of our analysis.",
"In the HBA data, the extent of the galactic halo can be analysed on unprecedented scales.",
"Here, we can follow the intensity profile to a height of 14 kpc[32] reached similar extents in their analysis of NGC 5775 but had a lower physical resolution of 2.2 kpc..",
"Comparing our results of NGC 4013 to [83] (ST19) is more difficult than in the case of NGC 891 or NGC 4631, since their profile fitting is based on Gaussian instead of exponential profiles.",
"ST19 report disk (halo) scale heights of $0.36\\pm 0.05$ kpc ($2.0\\pm 0.1$ kpc) in VLA L-band and $0.47\\pm 0.10$ kpc ($3.1\\pm 0.3$ kpc) in LOFAR HBA.",
"Our model comparison based on the AICc shows again that the strips are equally well fitted with just a single exponential fit rather than a two-component approach.",
"The radio halo of NGC 4013 will be discussed in more detail in Sec.",
"REF .",
"Scale heights of NGC 3432 and NGC 4157 have been fitted with two-component exponential functions in C-band and L-band by [51] (L-band: NGC 3432: $z_{\\mathrm {halo}}=0.92\\pm 0.25$ kpc, NGC 4157: $z_{\\mathrm {halo}}=1.08\\pm 0.07$ kpc)The authors node that their derived disk scale heights are limited in their reliability, because of the resolution of the observations.",
"This is in agreement with the values we find, considering the fact that we only fit one-component models for NGC 3432.",
"Now, we analyse the scale heights of all galaxies as a sample.",
"The model comparison with the AICc shows that most strips can be fitted, with the chosen box integration setup (spatial resolution and box size), with one-component models.",
"For the following analysis we use results from the one-component fits only, since mixing scale heights of one- and two-component models is not advisable at all (also strips that showed enough complexity to be fitted with two components are re-fitted with one-component models for this part of the analysis).",
"In Fig.",
"REF we compare the radio scale heights of each strip in the L-band and HBA data.",
"In their analysis of 13 galaxies in L- and C-band, [51] report overall similar scale heights in both spectral bands.",
"This picture changes drastically when comparing L-band to HBA profiles.",
"Here, one can see a bi-modality in the data.",
"Galaxies that showed a strong and uniform gradient in the non-thermal SPIX maps (NGC 891, 4157, and 4631) have much larger scale heights in HBA than in L-band and galaxies without such a gradient (NGC 3432) or with a less prominent SPIX gradient (NGC 4013) lie much closer to the one-to-one line.",
"The relative scattering of the individual strips of a galaxy is largest in NGC 4157.",
"Comparing the HBA and L-band contour lines of NGC 4157 in Fig.",
"REF , one can notice that not only the minor axis, but also the major axis is larger in HBA than in L-band.",
"Therefore, the comparison of the box profiles might show a larger variety.Assuming a rather symmetric galactic magnetic field structure, one interpretation explaining the scatter in the measured scale-height ratios of a given galaxy could also be a space-dependent CR transport mechanism [27].",
"In the literature, there are also studies that have observed larger scale heights in the low-frequency data.",
"[62] report a scale height ratio of $1.7\\pm 0.3$ for NGC 891 and [32] find a ratio of $1.2\\pm 0.3$ in NGC 5775 when comparing L-band and HBA scale heights.",
"We report the mean scale height ratio (average of the ratios of individual strips) for each galaxy in Table REF .",
"These ratios reflect the results from the non-thermal SPIX maps.",
"Those galaxies that show a clear gradient in the SPIX maps also have significantly larger scale heights in HBA than in L-band, as expected.",
"Table: One component radio scale height ratios of HBA and L-band profiles.",
"Uncertainties represent the spread of the different strips of a galaxy.Following the argumentation of [51], the observed scale height ratio can predict the dominating CR transport mechanism in the galaxy.",
"First of all, loss-dominated and escape dominated halos need to be distinguished.",
"In a loss-dominated halo, the synchrotron emission falls below the detection limit because the CREs have lost so much energy through radiation that they can no longer be detected.",
"In escape-dominated halos, the synchrotron radiation drops below the detection limit because of the decreasing number of CREs and lower magnetic field strengths.",
"Of course, the observed expansion of the galaxy halo is actually limited by a combination of both effects, which makes it difficult to argue on the basis of scale heights alone.",
"Nevertheless, for a loss-dominated halo, [51] report the following frequency dependencies on the advection length lcon and diffusion length ldiff for diffusion with a non-energy dependent diffusion coefficient D ($\\mu $ =0), energy-dependent diffusion with D$\\propto $ E0.5 ($\\mu $ =0.5) and advection, which can than be translated into expected scale height ratios r for the frequencies analysed in this paper (144 MHz & 1.5 GHz).",
"We list the derived proportionality and expected ration in Table REF .",
"Table: Derived proportionalities of the diffusion- / advection-lengths and the observed radio frequency (as derived by and expected scale height ratios for HBA and L-band.We show these relations in Fig.",
"REF to compare them to the measured scale heights.",
"Most of the scale heights measured in NGC 3432 and NGC 4013 fall even below the energy-dependent diffusion line, while NGC 891 and NGC 4631 seem to be well represented by the $\\mu $ =0-diffusion transport.",
"The aforementioned CRE escape process drives the measured scale ratios closer to the one-to-one relation.",
"Because of that, [59] find in the 1D CR transport modelling that advection models are the best-fitting transport mechanism in NGC 3556, although they find an HBA/L-band ratio of 1.8 in the halo, which would point to energy-independent diffusion if only the scale height ratio is considered.",
"Therefore, it seems likely that the halo of NGC 3556 is not loss-dominated.",
"Figure: Comparison of fitted non-thermal radio scale heights in VLA L-band and LOFAR HBA.",
"For each strip of each galaxy the radio scale height in the HBA data is plotted against its L-band counterpart.",
"Individual galaxies are colour coded as indicated in the legend.",
"The solid black line represents the identity relation which one would expect in escape-dominated halos.",
"Dashed lines indicate expected scale height ratios for loss-dominated halos: dark-blue: energy independent diffusion; gold: energy dependent diffusion; red: advection.",
"[38] performed a correlation analysis to find possible relations between the fitted radio scale heights and other measured properties such as SFR, SFR surface density, and the rotation velocity.",
"They did not find a strong correlation between the radio scale height and any of these properties.",
"In Fig.",
"REF , we present the fitted HBA radio scale heights for one- and two-component models separately in comparison to the diameter of the galactic disk, measured on resolution enhanced WISE 22 µm images, and the SFR within the individual strips.",
"In the two-component plots, only the halo scale heights are shown, which neither show a clear trend with the galaxy diameter nor the SFR.",
"Figure: Individual radio halo scale heights plotted against galaxy diameter (left), and star formation within the individual strip (right).",
"Two-component and one-component fits are displayed separately.",
"In the plots of the two-component fits, only the halo scale heights are shown.",
"Dots represent the central strips and x-symbols the outer strips of each galaxy.",
"Colour coding: NGC 891 blue; NGC 3432 orange; NGC 4013 green; NGC 4157 grey; NGC 4631 magenta (same as Fig.",
").For the one-component models, one might expect a positive trend for the galaxy diameter as well as the SFR.",
"Therefore, we perform a more detailed correlation analysis where all strips have been fitted with one-component models in Fig.",
"REF .",
"Obviously there is a strong link between the HBA radio scale height, L-band scale height and the HBA/L-band ratio (r).",
"However, here we are more interested in a link of the radio measurements of the halo with other quantities such as SFR or the magnetic field.",
"The diameter of the galactic disk seems to be a quite good predictor for the measured HBA scale height (Spearman-r: 0.67, p-value: 5e-5)It is important to note that the p-value for such small samples is not completely reliable.",
"However, the measured correlation coefficients are large enough to be reliable and the trends are also visible in the scatter plots.",
"The SFR is equally strongly linked to $z_0$ , however, the scatter plot does show outlier values with much larger SFRs than the average trend.",
"We also checked if the specific star formation rate (sSFR) (based on the mass estimates in Table REF ) is a better predictor but the sSFR does not perform better than the SFR.",
"Therefore we just present the SFR values in Fig.",
"REF .",
"We do not find significant correlations between z0 and the SFR surface density nor the magnetic field strength.",
"Overall, the L-band scale height (and therefore also the ratio r) is less strongly linked to the diameter and SFR compared to the HBA scale height.",
"The strongest correlations we find in this study are a) the link between the equipartition magnetic field strength and the SFR in the individual strips, and b) between the SFR surface density and the gravitational potential.",
"A correlation between the SFR and the magnetic field strength has already been reported in studies of dwarf galaxies [20], the KINGFISHER nearby galaxy sample [90], and nearby edge-on galaxies [38].",
"However, the relation found in this study is most probably affected by the study setup.",
"Meaning that we simply measure higher SFRs and magnetic field strengths in the centre of a galaxy, because of the radial dependence and not only because of a link between the SFR and the galactic magnetic field.",
"The same is also most likely true for the link between the SFR surface density and the gravitational pull.",
"Concluding this section, we find that even the predictors that were found to be best in this study - measured SFR within a strip and the galaxy diameter - are not suited to reliably constrain the radio halo size.",
"The formation of a galactic wind seems to be too complex to be predicted by a single parameter in such a small sample.",
"We continue the analysis of galactic winds driven by stellar feedback by fitting the extracted intensity profiles with a 1D CR-transport model.",
"Figure: Correlation analysis of several physical properties in comparison to the fitted HBA and L-band radio scale heights (z0, HBA, z0, L-band) and the ratio of HBA and L-band (r).",
"The main diagonal displays the distribution of z0 and r as well as the other analysed quantities (SFR, SFR surface density, and equipartition magnetic field strength) as histograms and kernel density estimates.",
"Bottom: bivariate scatter plots for pairs of the analysed quantities with interpolation line.",
"Top: Colour coded (blue: negative, red: positive) Spearman rank correlation coefficients."
],
[
"Stellar Feedback Driven Winds",
"In this section, we extend the analysis of the extracted profiles in order to derive physical parameters of the galactic wind.",
"The section is structured as follows: First, we introduce the models that are used to analyse the CR transport mechanisms in the galaxies and the input and output parameters for each galaxy.",
"Further, we motivate the model choice for each galaxy and explain the exclusion of NGC 3432 from the further analysis.",
"In Sec.",
"REF the results of the diffusion model fitting of NGC 4013 are displayed.",
"In the last subsection (Sec.",
"REF ), we present the results of the completed advection model fitting for NGC 891, NGC 4157, and NGC 4631."
],
[
"Cosmic Ray Transport Models",
"The 1D cosmic ray transport modelling is performed with the SPectral INdex Numeral Analysis of K(c)osmic-ray Electron Radio-emission (SPINNAKER) code [36], [38].",
"SPINNAKER fits vertical intensity profiles of multiple frequencies and the resulting SPIX profiles to several wind models (diffusion or advection).",
"The latest SPINNAKER extension for advection-dominated galactic winds, the so called flux tube model, has been published by [32].",
"In the literature, several studies already made use of synchrotron radiation to analyse the cosmic ray transport of nearby edge-on galaxies [77], [59], [83].",
"Some of those studies analysed galaxies that are also part of the present work.",
"However, as pointed out by [32], the new flux tube model has several benefits compared to the previous approaches (e.g.",
"magnetic and velocity scale heights are fitted simultaneously in the flux tube model, whereas they had to be fitted separately before).",
"Therefore, we apply the advection flux tube model (accounting for advection losses) introduced by [32].",
"The model assumes an hyperboloidal tuneable flux tube, an isothermal wind, and exponentially declining gravitational acceleration for increasing heights above the galactic disk.",
"Therefore, the cross sectional area $A$ of the flux tube above the disk is described as: $A(z) = A_0 \\left[1 + \\left(\\frac{z}{z_0}\\right)^\\beta \\right].$ The model results in a hyperboloid if the free parameter $\\beta $ is set to $\\beta = 0$ .",
"Further, [32] assume a magnetic field that declines with the midplane flow radius $r$ and the advection speed $v$ : $B(r,v) = B_0 \\left(\\frac{r}{r_0}\\right)^{-1} \\left( \\frac{v}{v_0}\\right)^{-1}.$ Here, $B_0$ [$\\mu $ G] describes the magnetic field strength in the galactic plane.",
"For further details about the model, we refer to [32].",
"To describe the hyperbolodial flux tube, we introduce the base radius of the flux tube $R_0$ , its scale height $z_0$ and the power law index that describes its opening angle $\\beta $ .",
"With this model, we fit the vertical height $z_c$ of the critical point, the wind speed at the critical point $_c$ , and the power law index of the CRE injection spectrum $\\gamma $ .",
"The critical point denotes the transition of a subsonic to a supersonic galactic wind.",
"As input parameters, we provide the $z=0$ radius of the flux tube $R_0$ (in this work defined as the strip width of each galaxy), rotational velocities, and the magnetic field strength $B_0$ within the galactic disk.",
"$B_0$ is determined as the average magnetic field strength for each strip measured on equipartition magnetic field maps, published in [37] in 20 wide boxes (see Table REF for physical extents).",
"[37] use the revised equipartition formula of [8] to compile their magnetic field strength maps.",
"In order to derive magnetic field strength values from radio continuum data, some assumptions have to be made.",
"First of all, one assumes equipartition between the CRs and the total magnetic energy which is similar to a minimisation of the total energy density.",
"Another important value that needs to be estimated to predict the magnetic field strength via the equipartition approach, is the proton-electron ratio $K_0$ .",
"[37] use $K_0=100$ .",
"The applied value of the proton-electron ratio bases on measurements in our Galaxy [21].",
"We fit for optimal solutions by incorporating a grid-search algorithm implemented in the interactive extension of SPINNAKER, SPINTERACTIVE [59].",
"As shown by ST19, the CR transport of NGC 4013 is most probably dominated by diffusion.",
"Therefore we recheck for all strips of NGC 4013 if the diffusion model (as described in [40] and [83]) still better fits the data, compared to the newly developed flux tube advection model.",
"Here, we only consider energy independent ($\\mu =0$ ) diffusionWe note that energy independent diffusion refers to the CR energy range that is probed with radio synchrotron observations (hundreds of MeV to a few GeV).",
"Energy independent diffusion has found to be a good fitting model for CR transport probed with data that is similar to ours in M 51 [24].",
"For higher energies, the CR diffusion coefficient is expected to be energy dependent.. All other galaxies (NGC 891, NGC 3432, NGC 4157, and NGC 4631) are only fitted with the flux tube advection model from [32].",
"Input parameters for the galaxies can be found in Table REF ($$rot) and Table REF ($B_0$ , $R_0$ , beam resolution).",
"The resulting output parameters of the individual strips for all SPINNAKER advection fits are listed in Table REF and the results from the diffusion fits are summarised in Table REF .",
"As an example, in Fig.",
"REF we show best fitting models for one strip each of NGC 891, NGC 4013, and NGC 4631.",
"All other SPINNAKER fits are displayed in Appendix .",
"To summarise the SPINNAKER-fitting results, we report that all strips of NGC 891 and NGC 4631 have been successfully fitted with the flux tube advection model.",
"For NGC 4157, we fitted four strips with the advection model and excluded the upper and lower right strips from the further analysis as the resulting SPIX profiles did not allow a proper fitting of the model.",
"As expected from the SPIX map of NGC 3432, we do not find a clear gradient in the SPIX profiles of this galaxy (which is a needed requirement for every CR-propagation process).",
"Nonetheless, to be consistent we present to best fitting advection model (for the two middle strips), but the wind velocity is basically unconstrained.",
"We therefore exclude NGC 3432 from further analysis, as its interaction with UGC 5983 has most likely disrupted its radio halo.",
"Table: Output parameters from SPINNAKER advection-model fitting: the maximal height of the fitting region z max z_{\\mathrm {max}}; power law index of the cosmic ray injection spectrum γ\\gamma ; wind speed at the critical point c _c, scale height of the flux tube z 0 z_0; power law index of the flux tube β\\beta ; height of the critical point z c z_c; reduced chi square of the best fitting model χ ν 2 \\chi ^2_\\nu computed as the quadratic mean of the two intensity profiles and the SPIX profile.Table: Output parameters from SPINNAKER energy-independent diffusion-model fitting: the maximal height of the fitting region z max z_{\\mathrm {max}}; power law index of the cosmic ray injection spectrum γ\\gamma ; energy-independent diffusion coefficient D 0 D_0; exponential scale height of the magnetic field h z h_z; reduced chi square of best the fitting model χ ν 2 \\chi ^2_\\nu computed as the quadratic mean of the two intensity profiles and the SPIX profile.Figure: Best fitting profiles for NGC 891 UM (top panel), NGC 4013 UL (middle panel), and NGC 4631 UM (bottom panel).",
"For each galaxy, SPINNAKER fits the LOFAR HBA (top graph) and L-band (middle graph) intensity profiles as well as the spectral index profile (bottom graph) simultaneously.",
"The model is shown as a red line.",
"The advection model is applied to NGC 891 and NGC 4631, while the presented strip of NGC 4013 is fitted with a diffusion model."
],
[
"Cosmic Ray Diffusion",
"NGC 4013 shows mixed results.",
"We find that the upper half of the halo fits better with the energy independent diffusion model while the lower halo seems to be dominated by advection.",
"Due to the limited extend of the halo, each profile of NGC 4013 is sampled with only five data points (see middle panel of Fig.",
"REF ).",
"Therefore, we reduce the complexity of the advection model by excluding $\\beta $ from the fitting process.",
"Only two of the three strips (UM and UL) of NGC 4013 that have been fitted with a diffusion model, result in a reasonably constrained diffusion coefficient $D_0$ .",
"Figure: Velocity (left) and magnetic field (right) profiles for all fitted strips of a galaxy.",
"From top to bottom panel: NGC 891, NGC 4157, NGC 4631.",
"The extent of the galactic disk (estimated via the scale height of the HBA disk components (Table )) is indicated with a black bar in the velocity profile plots.",
"Dashed lines represent strips pointing downward, solid lines represent upward strips.",
"Lines are best fitting models and shaded areas indicate one-σ\\sigma uncertainties.",
"Escape velocities are computed for the central strip of each galaxy via Eq.",
", assuming a truncated dark matter halo size of 30 kpc."
],
[
"Cosmic Ray Advection",
"For the galaxies that have been overall successfully fitted with the advection flux tube model (NGC 891, NGC 4157, and NGC 4631), we present the velocity and magnetic field profiles in Fig.",
"REF .",
"In the left column of Fig.",
"REF , we show the fitted wind velocities.",
"First of all, we note that although the SPINNAKER wind profiles start at an altitude of $z=0$ kpc, the motions within the galactic disk are predominantly turbulent.",
"From there on, the wind starts to evolve and will eventually start to dominate the overall motion.",
"Therefore, we indicate the extent of the galactic disk, approximated from the disk component scale height of the fitted two-component models in each galaxy.",
"At some point within this range, we expect the wind to form and dominate over turbulent motions.",
"Since the wind solutions studied in this paper are accelerated winds, the wind material in this context will eventually leave the gravitational potential.",
"However, in order to better understand the variety of CR-driven winds, we compare the wind velocities to the escape velocity of a spherical, truncated iso-thermal dark matter halo with a assumed size of $R_{\\mathrm {max}}=30$ kpcWe note that dark matter halo shapes and profiles have shown to be more complex and that the expected halo size strongly depends on the galaxy environment.",
"However, we choose this simplified assumption to be comparable to previous studies.",
"For more details about halo sizes we refer to [72], [56], [81].",
"Following [101] and [59], the escape velocity vesc at radius r can be computed for known rotation velocities vrot: $_{\\mathrm {esc}}(r) = \\sqrt{2} \\cdot _{\\mathrm {rot}}\\sqrt{1 + \\ln \\left( \\frac{R_{\\mathrm {max}}}{r}\\right)}.$ The escape velocities for the middle strip of each galaxy is shown in black in Fig.",
"REF , for comparison.",
"Overall, we find that the wind velocities of the central strips are higher than in outer strips.",
"For NGC 4157, the split between inner and outer strips is very distinct.",
"Here, the wind in the upper middle strip reaches the escape velocity at 6 kpc.",
"The spread of the velocity profiles of NGC 891 is much smaller compared to NGC 4157 and the wind does not reach the escape velocity within the first 8 kpc of its path.",
"As the rotation velocity of NGC 4631 is much slower compared to NGC 891 and NGC 4157, the computed escape velocity is also lower.",
"Therefore, the galactic wind of NGC 4631 reaches escape velocity in the UM strip at 8 kpc.",
"The right column of Fig.",
"REF displays the magnetic field strength models for the fitted strips.",
"In addition, we compare these profiles to the magnetic field strength measured on the equipartition maps published by [37].",
"Here, we can only present equipartition profiles for the central strips as the extent of the maps is not large enough to also properly sample the outer strips.",
"We find good agreement between the equipartition and the modelled magnetic field strength profiles.",
"In the case of NGC 4157 and NGC 4631, one might suspect a flattening of the equipartition profile around a value of approximately 4 µG but deeper data sets would be needed to confirm such a conjecture."
],
[
"Discussion",
"In the following, we discuss our results of the non-thermal SPIX analysis regarding the influence of thermal absorption, the substructure that was found in the intensity and SPIX profiles, the CR transport of NGC4013 in particular, as well as the results of the advection flux tube modelling, and we compare our findings to relevant previous works."
],
[
"Non-Thermal Spectral Index",
"For the observed frequencies, the power-law index of the CRE injection spectrum is expected to be $\\gamma _\\mathrm {inj}\\approx -2.2$ , which translates to an observed radio SPIX of $\\alpha _\\mathrm {inj} \\approx -0.6$ [34].",
"In the sample we studied, the observed spectral indices within the galactic disks are flatter than the expected value, with NGC 4631 being the most extreme example.",
"Here, a region within the galactic disk (slightly to the left of the galaxy centre) has a non-thermal SPIX of $\\alpha _\\mathrm {nth}=-0.4$ .",
"In the following, we call this area the region of interest (ROI).",
"We consider two possible rationales for this finding.",
"First of all, the thermal emission within the star forming regions in the galactic disk might have been underestimated.",
"This would lead to a higher flux density in the L-band measurement and therefore to a flatter SPIX.",
"However, to change the measured SPIX from $-0.4$ to $-0.6$ , the thermal contamination of the synchrotron emission in the L-band would still have to be of the order of 40% after thermal correction, which seems very unlikely considering that this is the maximum total fraction of thermal emission found in the sample.",
"The second mechanism to consider is thermal absorption in the low-frequency measurement.",
"Due to thermal absorption, the low-frequency pure synchrotron flux density measurement would be underestimated and result in a flatter SPIX.",
"Based on measurements within our Galaxy, [68] predict the the peak of thy synchrotron spectrum for the inner Galaxy to be at approximately 100 MHz, considering that ROI is the region with high SFR, the synchrotron peak will at higher frequencies, which makes a influence of thermal absorption on the low frequency probable.",
"Whether our measurements are affected by thermal absorption can be checked by comparing our spectral index map to the findings of MP19 and [102].",
"MP19 show a non-thermal SPIX map based on VLA 1.6 GHz (L-band) and 6 GHz (C-band) data and [102] present a SPIX map based on 315 MHz and 745 MHz data from the upgraded Giant Metrewave Radio Telescope (uGMRT)As they use 315 MHz and 745 MHz data, the influence of thermal emission can be ruled out.. For clarity, we summarise the non-thermal SPIX values for the ROI (read from the published SPIX maps) of MP19, our study and [102] here: MP19 (1.5 GHz and 6 GHz): $\\alpha _\\mathrm {nth}=-0.6$ (expected value for freshly injected CREs).",
"Our study (144 MHz and 1.5 GHz): $\\alpha _\\mathrm {nth}=-0.4$ .",
"[102] (315 MHz and 745 MHz): $-0.2 < \\alpha _\\mathrm {nth} < -0.4$ .",
"The MP19 finding allows us to conclude that the L-band data are not affected by thermal absorption.",
"As the SPIX for the ROI read from the uGMRT-SPIX map is even flatter than the SPIX value from our analysis, one can suspect that the peak of the synchrotron spectrum for the high star formation regions lies between 315 MHz and 745 Mhz, which is the reason for the flat SPIX measurement.",
"We further underline our findings in Fig.",
"REF .",
"Here, we fit the emission measure (EM) of the region, as described in [1].",
"In the fit we include an opacity $\\tau $ that adds an absorption term to the assumed power law of the pure snychrotron emission: $S = S_0 \\left( \\frac{\\nu }{\\nu _0}\\right)^\\alpha \\mathrm {e}^{-\\tau }.$ Following [107] and [1], for free-free absorption, the absorption coefficient is given by: $\\tau = \\frac{8.2\\cdot 10^{-2}\\nu ^{-2.1} \\mathrm {EM}}{T_e^{1.35}}.$ Under these assumptions, the peak of the synchrotron spectrum is at $\\sim 200$ MHz.",
"For this region, the best fitting EM is $\\sim 3.9\\cdot 10^5$ pc cm-6, which is comparable to the EM in the core region of M 82 of $\\sim 3.2\\cdot 10^5$ pc cm-6 [1].",
"Another possible way to explain the hardening of the observed CR spectrum are re-acceleration processes as described by [91], [53], [54].",
"[53] conclude that re-acceleration processes can change the SPIX between low and high frequency radio data by $\\Delta \\alpha \\le 0.5$ .",
"Figure: Double logarithmic diagram displaying the flux measurements of the region with a very flat spectral index in NGC 4631 at the following frequencies: LOFAR HBA (144 MHz), uGMRT (315 MHz and 745 MHz) , and VLA L-Band (1.5 GHz).",
"The orange line displays the synchrotron spectrum, while accounting for thermal absorption with the best fitting emission measure.Overall, we find thermal absorption to be the more likely process to explain the flat SPIX values in this region.",
"Although [6] conclude that thermal absorption has no significant influence on integrated low-frequency radio measurements of star-forming galaxies, they nevertheless suggest that it may play a role in compact regions close to the galaxy centre.",
"Furthermore, VLBI measurements of the nuclear starburst of M 82 also show that, for compact regions, thermal absorption influences the low frequency radio data [107], [98].",
"We therefore deduce that the LOFAR measurements, within the galactic disk, might be affected by thermal absorption.",
"In order to check if our scale height or CR transport analysis is affected by this phenomenon, we reran the scale height- and SPINNAKER-fitting by removing the $z=0$ flux measurement, but the results did not change significantly.",
"We therefore do not expect thermal absorption to have a strong influence on our analysis of the radio halo."
],
[
"Substructure in Intensity and Non-Thermal Spectral Index Profiles",
"The analysis of the box integration profiles shows that the intensity profiles are preferably fitted with one- or two-component exponential profiles rather than with Gaussian profiles.",
"Still, for some strips of NGC 891 and NGC 4631, the box integration reveals a significant substructure that is imprinted onto the intensity profiles.",
"[71] find in their analysis of 30 high-resolution cosmological zoom simulations of Milky Way-like galaxies that the vertical magnetic field strength profiles in the inner and outer parts of the galaxy each follow different exponential profiles.",
"They report that the break of the profiles typically occur at a height of $5-15$ kpc.",
"Such a break in the magnetic field strength will certainly affect the observed synchrotron profile, which could be at least part of an explanation for the observed substructure.",
"However, the substructure seems to be more complex than a single break in the magnetic field strength profile could explain.",
"For the UR strip of NGC 891 (see top left panel of Fig REF ), this deviation is most probably caused by the residual of an incompletely subtracted background- or foreground source.",
"The substructures imprinted on the profiles of NGC 4631 do not seem to result from point source residuals.",
"A careful manual inspection of the total power image did not reveal any background sources.",
"The observed substructure of the intensity profiles propagates directly into the observed SPIX profiles.",
"We find plateaus in the SPIX profiles (e. g. Fig.",
"REF top panel: NGC 891 UM) as well as an actual increase of the SPIX values (e. g. Fig.",
"REF bottom panel: NGC 4631 UM).",
"While NGC 891 and NGC 4631 show a lot of substructure in their intensity and SPIX profiles (see Appendix ), the fitted profiles of NGC 3432, NGC 4013, and NGC 4157 follow the predicted SPINNAKER models without major deviations.",
"A flattening or an actual increase of the spectral index profile points to a rejuvenation process of the CREs, meaning that there needs to be a process that allows the CREs to loose less energy, or even gain energy.",
"Currently, there is no model implemented in SPINNAKER that accounts for such a rejuvenation of CREs.",
"In the ideal magneto-hydrodynamic simulations of a rotating Milky Way-type disk galaxy of [103], a plateau in the vertical synchrotron radiation profiles is found, which becomes more pronounced when the simulation runs with higher SFRs.",
"In a low-frequency uGMRT follow-up study, [102] find evidence for such a plateau in NGC 4631 at a $2-3$ kpc height and suggest that the plateau might be attributed to CRE re-acceleration in shocks of the galactic outflow.",
"However, we do not find clear evidence for the reported plateau in our data.",
"On the contrary, we find that most of the substructure found in our low-frequency profiles start after 4 kpc.",
"Another interesting observation is that we detect a much larger radio halo within our LOFAR 144 MHz data than shown by [102], even though the spectral bands are comparable (this study: 144 MHz; [102]: 315 MHz).",
"Additionally, the simulations by [103] run for about 10 Myr, which might not be enough to result in an equilibrium state.",
"Additionally, some of the assumptions made in the simulations might not apply.",
"To conclude this section, we note that the intensity and SPIX profiles analysed in this work show significant substructures at distances larger than $\\sim $ 2 kpc for NGC 891 and $\\sim $ 4 kpc for NGC 4631 that are yet not well explained by current model predictions.",
"One possible line of explanation could be that variations in the SFR history change the parameters (gas density, CRE number density, and velocity) of the galactic wind and are therefore imprinted in the radio continuum profile.",
"However, further studies and new models are needed to fully understand the structures found in the galactic halos."
],
[
"Cosmic Ray Diffusion in NGC 4013?",
"Comparing our diffusion fitting results of NGC 4013 to the ST19 analysis, we find a higher diffusion coefficient (this study: $1.4 \\cdot 10^{28}$ cm s-1; ST19: $0.55-0.65\\cdot 10^{28}$ cm s-1), a slightly lower injection index for the CREs (this study: 2.2; ST19: 2.6) and higher disk scale heights of the assumed exponential profile of the magnetic field (this study: 1.2 kpc; ST19: 0.1 kpc).",
"The diffusion coefficients in the analysis of ST19 are based on VLA L- and C-band (6 GHz) data, which makes a direct comparison to our results more difficult as the CREs probed in this work are much older than the CREs traced by the higher frequency measurements.",
"Additionally, ST19 find advection to dominate when using 144 MHz and 1.6 GHz data for the CR-transport analysis.",
"However, we find much higher wind velocities (this study: $\\sim 150$ km s-1; ST19: $18-22$ km s-1).",
"Following [75] and [34], one can estimate the location where the transition from diffusion-dominated to advection-dominated CR-transport happens: $z_\\star \\approx 1.2\\frac{D/10^{28}\\,\\mathrm {cm^2 s^{-1}}}{/100\\,\\mathrm {km s^{-1}}}\\,\\mathrm {kpc},$ where $D$ is the diffusion coefficient and $$ the advection wind speed.",
"Using the diffusion coefficient from the UM and UL strips and the wind velocity of the LR and LM strips (taken from Table REF and REF ), we find that the transition will happen at $z_\\star =1.5$ kpc.",
"Of course, this is only a zero-order estimate as we get the diffusion coefficients and wind velocities from different locations of the galaxy.",
"The accuracy of the estimate is also affected by the relatively poor sampling of the radio halo, and because diffusion and advection profiles only can be reliably distinguished by analysing extended halo profiles (i.e.",
"distinguishing between the two is difficult for small halos).",
"ST19 report the position of this transition to be at $z_\\star =1-2$ kpcST19 use a slightly modified version of Eq.",
"REF , with a factor ranging from 0.3 to 0.6, instead of 1.2, which comes from different assumptions of the diffusion process.",
"One-dimensional diffusion would lead to 0.3, while isotropic 3D-diffusion leads to 1.2., which is consistent with our estimate.",
"Another galaxy that has been found to be diffusion-dominated is NGC 4565 [40], [77].",
"While there is some scatter in the diffusion coefficients of this galaxy, our results fall in the same range.",
"From the diffusion coefficient, one can derive the size of the diffusion halo $L$ , if the CRE lifetime is known [34]: $L = \\sqrt{\\frac{D \\cdot \\tau }{0.75\\cdot 10^{29}}}\\,\\mathrm {kpc}$ Here, $D$ is the derived diffusion coefficient in units of $10^{28}\\,\\mathrm {cm^2 s^{-1}}$ and $\\tau $ is the CRE lifetime which is typically about 100 Myr for the energy range that is traced with the LOFAR HBA data.",
"With this assumption we derive a diffusion halo size for NGC 4013 of $L=4.3$ kpc (UR stripe excluded).",
"A similar halo size is measured within our own Galaxy [68], [67].",
"To summarise our results about NGC 4013, we note that our analysis remains inconclusive.",
"Overall, the small extent of the galaxy halo makes it very difficult to analyse its CRE transport in great detail.",
"New data reaching farther into the halo would be necessary in order to reliably distinguish the CRE transport mechanisms."
],
[
"Advection Dominated Galactic Winds",
"Of the analysed sample, NGC 891 has been previously modelled with SPINNAKER advection models.",
"[77] (SC19) analyse the radio halo of NGC 891 with 1.5 GHz and 6 GHz VLA CHANG-ES (+Effelsberg) data.",
"They apply purely advective or diffusive transport models, but restrict the modelling region to a maximal height of 3 kpc.",
"With the new LoTSS data and the newly cleaned CHANG-ES L-band data, we can now nearly double the extent of the modelling regions.",
"Furthermore, we analyse six strips (three northern, three southern) instead of two strips (one northern, one southern).",
"SC19 compare two models, one advection dominated and one diffusion dominated and find that an advection model is a much better fit.",
"With their advection model, they find a mid-plane wind velocity of 150 km s-1 for an accelerated galactic wind model.",
"A direct comparison between the model used by SC19 and the flux tube model introduced by [32] is difficult because the newer model includes additional processes (e.g.",
"advection losses) and also because the modelling technique has changed.",
"Nevertheless, we find similar velocities in proximity of the galactic disk, while we detect a split between central and outer strips farther out into the halo, where central strips have higher velocities than outer strips.",
"[77] predict that the galactic wind in NGC 891 reaches the halo escape velocity at a height of $9-17$ kpc.",
"Our analysis does not allow a confirmation of their results.",
"New, more sensitive data are needed to probe such scales.",
"One might suspect that the UM strip does exceed the escape velocity in this range, as our model uncertainty allow for higher wind velocities compared to the escape velocity at heights larger than 7 kpc, but this prediction remains uncertain.",
"As reported by MP19 and [102], we expect advection to dominate the CR transport of NGC 4631.",
"Nevertheless, we expect the wind to be much slower than estimated by MP19.",
"MP19 estimate the wind to reach a velocity of 300 km s-1 at a height of 3 kpc, which is much higher than the models in our analysis predict, although the uncertainties are high.",
"Judging from our velocity profile, the galactic wind in NGC 4631 will reach the escape velocity much later (after 8 kpc in the UM strip, which has the fastest wind), compared to the prediction of MP19.",
"Comparing our results from the advection flux tube modelling to the analysis of NGC 5775 by [32], our analysis of NGC 891, NGC 4157, and NGC 4631 points towards lower wind velocities.",
"[32] report wind velocities of more than 600 km/s.",
"However, NGC 5775 also has a much higher integrated SFR (7.5 $\\mathrm {M}_\\odot $ yr-1) and also a higher SFR surface density ($9.4 \\times 10^{-3}$ $\\mathrm {M}_\\odot $ yr-1 kpc-2) than the galaxies that we have analysed, which most likely influences the CR-driven wind since these quantities can be interpreted as the driving source of the wind.",
"Concerning the magnetic field strength profiles, we can confirm the results of [32].",
"As in the case of NGC 5775, the analysed galaxies generally show a good agreement between the modelled magnetic field strength profiles from SPINNAKER and the equipartition measurements.",
"However, for heights above 6 kpc, the model profiles of NGC 5775 deviate to lower field strength values compared to the equipartition measurements.",
"We find a similar trend for the LM strip of NGC 891 at heights larger than 2 kpc, which should be investigated in future studies.",
"[26] analyse the galactic wind of the Milky Way by applying a hybrid wind model, where the wind is driven thermally as well as by CR pressure.",
"They find an initial wind velocity of $_{0}=173$ km s-1 and the position of the critical point at a height of $z_c=2.4$ kpc.",
"At the critical point, they predict a sound speed $c_{\\star }=251$ km s-1.",
"From their model, [26] expect the wind velocity to be rather constant until it reaches a height of approximately 1 kpc and then to accelerate quickly to surpass the sound speed.",
"Therefore the overall shape of the wind profile differs from the wind profiles shown in Fig.",
"REF , where our velocity increases more gradually, and the predicted wind velocities for the Milky Way seem to lie between our predictions for NGC 4631 and NGC 891.",
"One of the key benefits of our study, compared to the analysis of NGC 5775 by [32], is the chosen three-strip setup during box integration.",
"This allows us to compare the properties of the galactic wind in the inner and outer part of each galaxy.",
"In Fig.",
"REF , we present the difference of predicted magnetic field strength and wind velocity between the middle and the outer strips for NGC 891, NGC 4157, and NGC 4631.",
"While we have not found a significant difference between middle and outer strips in the radio scale height analysis, we do see a split in the modelled wind velocity and magnetic field strength.",
"Since the magnetic field strength at $z=0$ is set as the average field strength from the equipartition measurement in each strip, there is an offset between the middle and outer strips.",
"Interestingly, the magnetic field strength in the central regions declines faster than in the outer regions of the galaxy.",
"This results in an approximate equality of the predicted magnetic field strength for high $z$ values in central and outer strips.",
"A similar behaviour is also visible in the cosmological magnetohydrodynamical simulations [57] and simulations of isolated disk galaxies [84].",
"With regard to wind velocity, the opposite behaviour can be observed.",
"Here the difference in wind velocity between central and outer strips predominately increases (wind velocity increases faster in central strips than in outer ones) in the radio halo.",
"For some strips of NGC 4631, central and outer wind velocities converge, but here the predicted uncertainties are very high.",
"Figure: Difference of magnetic field strength (top row) and wind velocity (bottom row) between central and outer, left and right, strips for NGC 891 (left column), NGC 4157 (middle column, and NGC 4631 (right column).",
"Black lines indicate equal magnetic filed strengths/wind velocities in the central and outer strips."
],
[
"Summary & Outlook",
"In this paper, we have reprocessed VLA L-band data in D-configuration and C-configuration from the CHANG-ES project and combined them with newly published low-frequency data from the LOFAR LoTSS DR2 for an analysis of an edge-on galaxy sample.",
"We investigated resolved non-thermal SPIX maps, intensity profiles perpendicular to the galactic disks, and modelled the CR transport in the galactic halos.",
"We demonstrated that new reduction techniques highly increase data quality, which allowed us to double the extent of the visible radio halo for most of the analysed galaxies.",
"Additionally, we have re-implemented the 'mixture-method' developed by [99] to estimate the resolved thermal emission of each galaxy and made it publicly available, using common python libraries.",
"We then analysed the resolved non-thermal spectral index and radio scale heights.",
"Lastly we modelled the CR transport using the 1D advection and diffusion models of SPINNAKER.",
"Our main results can be summarised as follows: We find that the LOFAR LoTSS DR2 maps are affected by thermal absorption within the galactic disk, especially in star forming regions, as the non-thermal spectral index maps exceed the expected limit for freshly induced CRE of $\\alpha _{\\mathrm {nth}}=-0.6$ in these areas.",
"The scale height analysis based on the model comparison with the AICc shows that most of the profiles are equally well fitted with one- as with two-component exponential models.",
"Additionally, we find substructures in the intensity and SPIX profiles that are not well explained by contemporary CRE transport models.",
"We do not find a strong correlation of the radio halo scale height and physical properties such as galactic diameter, gravitational potential, SFR, SFR surface density, specific SFR, and magnetic field strength measured within the galactic disk.",
"The modelling of the galactic wind indicates a split of magnetic field strength and wind velocity in the immediate vicinity of the galactic disks for inner and outer strips of the galaxy.",
"While the difference in magnetic field strength evens out in the radio halo, the difference in wind speed increases.",
"In this paper, we successfully analysed the CRE transport of NGC 891, NGC 4157, and NGC 4631, while the analysis of NGC 3432 and NGC 4013 remained inconclusive.",
"Our results suggest that there are significant differences in CR transport depending on the location within the galactic disk, which should be accounted for in future CRE transport models.",
"To better understand the CRE transport within the radio halo of NGC 4013, a possible approach would be to also include higher frequency data, as done by [83], to trace the younger CREs in the halo and then simultaneously fit three frequencies within SPINNAKER.",
"Adding a third frequency to the analysis would lead to higher precision of the model predictions, as it increases the number of profiles that are fitted by SPINNAKER from three (two intensities, one SPIX) to five (three intensities, two SPIXs).",
"However, the relatively small extent of the radio halo of NGC 4013 will most likely always be a limiting factor of the CR transport analysis.",
"Adding information about the magnetic field structure from polarisation measurements and implementing it in new CRE transport models would further help to better understand the role of CRs within galactic feedback processes.",
"More detailed modelling approaches are certainly needed, and will be possible, as the Square Kilometre Array Observatory (SKAO) precursors and pathfinders and eventually the SKAO itself enable us to analyse CR transport within galaxies on an unprecedented scale, both in terms of sensitivity and resolution.",
"We thank the anonymous referee for a constructive report that helped to improve our paper.",
"M. S., R.-J.",
"D., B.",
"A., and D. J.",
"B. acknowledge funding from the German Science Foundation DFG, within the Collaborative Research Center SFB1491 \"Cosmic Interacting Matters - From Source to Signal.",
"M.B.",
"acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 ‘Quantum Universe’ – 390833306.",
"TW acknowledges financial support from The coordination of the participation in SKA-SPAIN, funded by the Ministry of Science and Innovation (MCIN), from the State Agency for Research of the Spanish Ministry of Science, Innovation and Universities through the \"Center of Excellence Severo Ochoa\" awarded to the Instituto de Astrofísica de Andalucía (SEV-2017-0709) We thank Tom Jarrett for kindly providing us the resolution enhanced 22 µm WISE maps.",
"This research was supported in part by the National Science Foundation under Grant No.",
"NSF PHY-1748958.",
"LOFAR data products were provided by the LOFAR Surveys Key Science project (LSKSP; https://lofar-surveys.org/) and were derived from observations with the International LOFAR Telescope (ILT).",
"LOFAR [96] is the Low Frequency Array designed and constructed by ASTRON.",
"It has observing, data processing, and data storage facilities in several countries, which are owned by various parties (each with their own funding sources), and which are collectively operated by the ILT foundation under a joint scientific policy.",
"The efforts of the LSKSP have benefited from funding from the European Research Council, NOVA, NWO, CNRS-INSU, the SURF Co-operative, the UK Science and Technology Funding Council, the Ministry of Science and Higher Education, Poland, and the Jülich Supercomputing Centre.",
"The following software packages have been used in this work: Astropy [5], [4].",
"This research has made use of \"Aladin sky atlas\" developed at CDS, Strasbourg Observatory, France [11], [10]; SAOImage DS9 [46]"
],
[
"Thermal Fraction Maps",
"The thermal emission maps in for the VLA L-band and LOFAR HBA data as well as the corresponding thermal fraction maps are displayed in this section.",
"In the thermal fraction maps, we plot contours at factors of 0.25, 0.5, and 0.75 of the peak thermal fraction that was measured for each galaxy.",
"Figure: Thermal emission at 1.5 GHz (CHANG-ES L-band) (left column) and thermal fraction maps of the L-band data (right column) of NGC 891 (top row), NGC 3432 (middle row), and NGC 4013 (bottom row).",
"Thermal emission maps use a power law scaling with a power law index of 0.5.",
"Thermal fraction maps are clipped below 3σ\\sigma above the background noise.",
"The beam is displayed in the bottom right corner of each map as a white (thermal emission maps) or red (thermal fraction maps) circle.Figure: Fig.",
"continued: Thermal emission at 1.5 GHz (CHANG-ES L-band) (left column) and thermal fraction maps of the L-band data (right column) of NGC 4157 (top row) and NGC 4631 (bottom row).Figure: Thermal emission at 144 MHz (LOFAR HBA) (left column) and thermal fraction maps of the HBA data (right column) of NGC 891 (top row), NGC 3432 (middle row), and NGC 4013 (bottom row).",
"Thermal emission maps use a power law scaling with a power law index of 0.5.",
"Thermal fraction maps are clipped below 3σ\\sigma above the background noise.",
"The beam is displayed in the bottom right corner of each map as a white (thermal emission maps) or red (thermal fraction maps) circle.Figure: Fig.",
"continued: Thermal emission at 144 MHz (LOFAR HBA) (left column) and thermal fraction maps of the HBA data (right column) of NGC 4157 (top row), and NGC 4631 (bottom row)."
],
[
"Intensity Profiles",
"In this section, we present the best fitting model parameters from $\\chi ^2$ -minimisation for one- and two-component exponential profiles.",
"The model comparison was performed with the AICc in Table REF .",
"Four models were tested (listed from fewest to most free model parameters): one-component exponential without offset in $z-$ direction (one_fo), one-component exponential with offset in $z-$ direction (one), two-component exponential without offset in $z$ -direction (two_fo), and two-component exponential with offset in $z$ -direction (two).",
"We also display the intensity profiles from the box-integration technique for LOFAR HBA and VLA L-band as well as the best fitting (based on the AICc model comparison) model as a black dashed line.",
"Table: Parameters from box integration model fitting: Galaxy name, frequency of fitted data, model identifier, strip identifier, intensity and scale height of the first exponential component, intensity and scale height of the second exponential component (if applicable), offset in zz-direction (if applicable), reduced χ\\chi -square.Figure: Intensity Profiles of NGC 3432Figure: Intensity Profiles of NGC 4013Figure: Intensity Profiles of NGC 4157Figure: Intensity Profiles of NGC 4631"
],
[
"SPINNAKER Profiles",
"Best fitting SPINNAKER-models (advection or diffusion) for all galaxies (NGC 891, NGC 3432, NGC 4013, NGC 4157, and NGC 4631) are displayed in this section.",
"The structure of the SPINNAKER output is explained in Fig.",
"REF .",
"Figure: SPINNAKER profiles of NGC 891 (all advection).",
"Strips are presented as follows: top left: LR, top right: UR, middle left: LM, middle right: UM, bottom left: LL, and bottom right: UL.Figure: SPINNAKER profiles of NGC 3432 (all advection).",
"Strips are presented as follows: left: LM and right: UM.Figure: SPINNAKER profiles of NGC 4013.",
"Strips are presented as follows: top left: LR (advection), top right: UR (diffusion), middle left: LM (advection), middle right: UM (diffusion), bottom left: LL (advection), and bottom right: UL (diffusion).Figure: SPINNAKER profiles of NGC 4157 (all advection).",
"Stripes are present as follows: top left: LM, top right: UM, bottom left: LL, and bottom right: UL.Figure: SPINNAKER profiles of NGC 4631 (all advection).",
"Strips are presented as follows: top left: LR, top right: UR, middle left: LM, middle right: UM, bottom left: LL, and bottom right: UL."
]
] | 2210.07709 |
[
[
"First moments of a polyhedron clipped by a paraboloid"
],
[
"Abstract We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid.",
"These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron's face with the paraboloid.",
"We provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence.",
"Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations.",
"The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms."
],
[
"Introduction",
"Many computational methods and applications, ranging from finite-element , , discontinuous Galerkin , and immersed isometric analysis methods , , to the initialization , , , , and transport of interfaces for simulating gas-liquid flows, require estimating integrals over polyhedra that are clipped by curved surfaces.",
"These applications have engendered multiple dedicated quadrature rules and integration strategies, most of which focusing on estimating the first few moments of these clipped polyhedra, thus considering polynomial integrands only.",
"The numerical approaches employed to estimate these moments vary greatly in terms of accuracy, computational cost, and robustness.",
"Monte-Carlo methods , are extremely robust and straightforward to implement, however, they suffer from a poor convergence rate, hence their cost/accuracy ratio is significant.",
"Approaches based on octree subdivision , , or surface triangulation/tesselation exhibit better convergence rates, yet their computational cost remains prohibitive for numerical applications requiring on-the-fly moment estimations.",
"A number of recent approaches rely on successive applications of the divergence theorem, converting the first moments of a clipped polyhedron into two- and/or one-dimensional integrals.",
"These integrals can then be numerically integrated at low computational cost or, for specific surface types, even be derived into closed-form expressions.",
"Bnà et al.",
"estimate the volume (zeroth moment) of a cube clipped by an implicit surface represented with a level-set function through integrating the local height of the surface using a two-dimensional Gauss-Legendre quadrature rule.",
"This work has been extended to the first moments of a clipped cuboid by Chierici et al. .",
"For the similar purpose of estimating the zeroth moment of a polyhedron clipped by an implicit surface, Jones et al.",
"decompose the clipped polyhedron into a set of simplices, themselves split into a reference polyhedron whose volume is computed analytically, and a set of fundamental curved domains whose volumes are estimated using a two-dimensional Gauss-Legendre quadrature rule.",
"Chin and Sukumar use a Duvanant quadrature rule for integrating over the faces of a polyhedron bounded by rational or non-rational Bézier and B-spline patches.",
"For non-rational surface parametrizations, this yields exact integral estimations of polynomial integrands, provided that the order of the Duvanant rule is high enough.",
"Kromer and Bothe , estimate the zeroth moment of a polyhedron clipped by an implicit surface by locally approximating the implicit surface as a paraboloid and applying the divergence theorem twice, converting the clipped volume into a sum of one-dimensional integrals, which are then estimated with a Gauss-Legendre quadrature rule.",
"Finally, using Berstein basis functions instead of monomial ones, Antolin and Hirschler recently showed that, following successive applications of the divergence theorem, polynomial integrands can be integrated in a straightforward and analytical manner over curved polyhedra bounded by non-rational Bézier or B-spline surfaces.",
"This manuscript is concerned with estimating the first moments of a specific type of curved polyhedra, that are planar non-convex polyhedra clipped by a paraboloid surface (as in , ).",
"Moreover, we require this estimation to (a) reach machine accuracy, while (b) maintaining a computational expense that is low enough to enable its on-the-fly execution in typical numerical applications (e.g., the simulation of two-phase flows with finite-volumes), and (c) being robust to singular configurations (e.g., paraboloid surfaces being parabolic cylinder or planes, and/or ambiguous discrete intersection topologies).",
"These choices and requirements, although mainly motivated by the use of these moments for simulating two-phase flows, may also find applications in the numerical fields listed above.",
"A main difficulty in clipping a polyhedron with a paraboloid lies in the fact that the faces of the clipped polyhedron cannot systematically be represented with non-rational or rational Bézier patches , .",
"This prevents the use of recently proposed integration strategies designed for curved polyhedra bounded by Bézier or B-spline surfaces , .",
"By successive applications of the divergence theorem, we show that the first moments of the clipped polyhedron can be expressed as a sum of one-dimensional integrals over straight line segments and conic section arcs.",
"With a parametrization of the latter into rational Bézier arcs, we derive closed-form expressions for the first moments, rendering their numerical estimation exact within machine accuracy.",
"Implemented within the half-edge data structure of the open-source Interface Reconstruction Library The Interface Reconstruction Library source code is available under Mozilla Public License 2.0 at https://github.com/robert-chiodi/interface-reconstruction-library., the computational cost of these moment estimations is kept within an order of magnitude of that of clipping a polyhedron with a half-space.",
"Finally, our choice of arc parametrization, in conjuction with the detection and treatment of ambiguous discrete topologies, allows for robust moment estimates even in degenerate configurations.",
"The remainder of this manuscript is organized as follows: sec:problem introduces the problem that we address and the notations employed throughout the manuscript.",
"The closed-form expressions of the clipped polyhedron's first moments are derived in sec:moments.",
"sec:surface touches upon the integration of quantities (e.g., the moments) of the clipped polyhedron's curved face(s).",
"sec:robustness details the procedure employed for preventing ambiguous clipped polyhedron topologies.",
"Finally, the accuracy, efficiency, and robustness of our proposed integration stategy are assessed in sec:tests, and we draw conclusions in sec:conclusions."
],
[
"Problem statement",
"Consider the two following subsets of $\\smash{\\mathbb {R}^3}$ : A polyhedron $\\smash{\\mathcal {P}}$ delimited by $\\smash{n_{\\!\\mathcal {F}}}$ planar polygonal faces $\\smash{\\mathcal {F}_i, i \\in \\lbrace 1,\\ldots ,n_{\\!\\mathcal {F}}\\rbrace }$ , each equipped with an outward-pointing normal vector $\\smash{\\mathbf {n}_i}$ (e.g., see fig:poly,fig:polywithfaces).",
"Neither $\\smash{\\mathcal {P}}$ nor its faces $\\smash{\\mathcal {F}_i}$ are required to be convex.",
"The region $\\smash{\\mathcal {Q}}$ , located on one side of a paraboloid $\\smash{\\mathcal {S}}$ (e.g., see fig:polyandpara).",
"Without loss of generality, we assume to be working in a Cartesian coordinate system equipped with the orthonormal basis $\\smash{(\\mathbf {e}_x, \\mathbf {e}_y,\\mathbf {e}_z)}$ , within which the position vector reads $\\smash{\\mathbf {x} = \\begin{bmatrix}x & y & z\\end{bmatrix}^\\intercal }$ , and where $\\smash{\\mathcal {Q}}$ and $\\smash{\\mathcal {S}}$ are implicitly defined as $\\mathcal {Q} & = \\lbrace \\mathbf {x} \\in \\mathbb {R}^3 : \\phi (\\mathbf {x}) \\le 0 \\rbrace \\, , \\\\\\mathcal {S} & = \\lbrace \\mathbf {x} \\in \\mathbb {R}^3 : \\phi (\\mathbf {x}) = 0 \\rbrace \\, , $ with Figure: A polyhedron 𝒫⊂ℝ 3 \\smash{\\mathcal {P}\\subset \\mathbb {R}^3} intersected by the clipping region 𝒬⊂ℝ 3 \\smash{\\mathcal {Q}\\subset \\mathbb {R}^3} located below a paraboloid surface 𝒮\\smash{\\mathcal {S}}.$\\phi : \\left\\lbrace \\begin{array}{lcl}\\mathbb {R}^3 & \\!\\rightarrow & \\mathbb {R} \\\\\\mathbf {x} & \\!\\mapsto & \\alpha x^2 + \\beta y^2 + z \\,,\\end{array}\\right.",
"\\quad (\\alpha ,\\beta ) \\in \\mathbb {R}^2 \\, .",
"$ These assumptions do not restrict $\\smash{\\mathcal {Q}}$ and $\\smash{\\mathcal {S}}$ since, for any paraboloid-bounded clipping region in $\\smash{\\mathbb {R}^3}$ , there exists a combination of rotations and translations of the canonical coordinate system resulting in such implicit definitions of $\\smash{\\mathcal {Q}}$ and $\\smash{\\mathcal {S}}$ .",
"For the sake of clarity and conciseness, we introduce the following notations: The subscript $\\smash{\\square _i}$ refers to a topological element or quantity related to the $i$ th face of the polyhedron $\\smash{\\mathcal {P}}$ .",
"The superscript $\\smash{\\hat{\\square }}$ implies an intersection with the clipping region $\\smash{\\mathcal {Q}}$ , e.g., $\\smash{\\hat{\\mathcal {P}} \\equiv \\mathcal {P}\\cap \\mathcal {Q}}$ or $\\smash{\\hat{\\mathcal {F}}_i \\equiv \\mathcal {F}_i\\cap \\mathcal {Q}}$ .",
"The superscript $\\smash{\\tilde{\\square }}$ implies an intersection with the polyhedron $\\smash{\\mathcal {P}}$ , e.g., $\\smash{\\tilde{\\mathcal {S}} \\equiv \\mathcal {S}\\cap \\mathcal {P}}$ .",
"This means that $\\smash{\\hat{\\mathcal {P}} \\equiv \\tilde{\\mathcal {Q}}}$ .",
"As mentioned in sec:introduction, we are interested in calculating the zeroth and first moments of $\\smash{\\hat{\\mathcal {P}}=\\mathcal {P}\\cap \\mathcal {Q}}$ (e.g., see fig:clippedpoly), i.e., its volume and volume-weighted centroid, given as $\\mathrm {M}_0^{\\hat{\\mathcal {P}}} & = \\int _{\\hat{\\mathcal {P}}} 1 \\ \\mathrm {d}\\mathbf {x} \\, , \\quad \\quad \\text{and} \\quad \\quad \\mathbf {M}^{\\hat{\\mathcal {P}}}_1 = \\int _{\\hat{\\mathcal {P}}} \\mathbf {x} \\ \\mathrm {d}\\mathbf {x} \\, .$ In the remainder of this work, we shall refer to these quantities as “the first moments” or “the moments” of $\\smash{\\hat{\\mathcal {P}}}$ , which we group into the vector $\\mathcal {M}^{\\hat{\\mathcal {P}}} & = \\begin{bmatrix} \\mathrm {M}_0^{\\hat{\\mathcal {P}}} \\\\ \\mathbf {M}^{\\hat{\\mathcal {P}}}_1 \\end{bmatrix} = \\int _{\\hat{\\mathcal {P}}} \\Upsilon (\\mathbf {x}) \\ \\mathrm {d}\\mathbf {x} \\, , $ where $\\Upsilon : \\left\\lbrace \\begin{array}{lcl}\\mathbb {R}^3 & \\!\\rightarrow & \\mathbb {R}^4 \\\\\\mathbf {x} & \\!\\mapsto & \\begin{bmatrix} 1 & x & y & z \\end{bmatrix}^\\intercal \\end{array}\\right.",
"\\, .$"
],
[
"Moments derivation",
"Using the divergence theorem, Eq.",
"eq:moments can be rewritten as $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\int _{\\hat{\\mathcal {P}}} \\nabla \\cdot \\left( \\Phi (\\mathbf {x}) \\otimes \\mathbf {e}_z \\right) \\ \\mathrm {d} \\mathbf {x}= \\int _{\\partial \\hat{\\mathcal {P}}} \\Phi (\\mathbf {x}) (\\mathbf {n}_{\\partial \\hat{\\mathcal {P}}} \\cdot \\mathbf {e}_z) \\ \\mathrm {d} \\mathbf {a} \\, , $ where $\\smash{\\mathrm {d}\\mathbf {a}}$ is an infinitesimal surface element on $\\smash{\\partial \\hat{\\mathcal {P}} = \\cup _{i} \\hat{\\mathcal {F}}_i \\cup \\tilde{\\mathcal {S}}}$ , the boundary of $\\smash{\\hat{\\mathcal {P}}}$ , $\\smash{\\mathbf {n}_{\\partial \\hat{\\mathcal {P}}}}$ is the normal to $\\smash{\\partial \\hat{\\mathcal {P}}}$ pointing towards the outside of $\\smash{\\hat{\\mathcal {P}}}$ , and $\\smash{\\Phi }$ is defined as $\\Phi : \\left\\lbrace \\begin{array}{lcl}\\mathbb {R}^3 & \\!\\rightarrow & \\mathbb {R}^4 \\\\\\mathbf {x} & \\!\\mapsto & {\\displaystyle \\int _0^z} \\Upsilon (\\mathbf {x}) \\ \\mathrm {d}z \\ \\ \\ \\left(= \\begin{bmatrix} z & xz & yz & \\frac{1}{2}z^2 \\end{bmatrix}^\\intercal \\right) \\end{array}\\right.",
"\\, .$ Eq.",
"eq:volafterdiv1 can be split into the following sum of integrals, $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\int _{\\tilde{\\mathcal {S}}}\\Phi (\\mathbf {x}) (\\mathbf {n}_{\\tilde{\\mathcal {S}}} \\cdot \\mathbf {e}_z) \\ \\mathrm {d}\\mathbf {a} + \\sum \\limits _{i=1}^{n_\\mathcal {F}} \\int _{\\hat{\\mathcal {F}}_i} \\Phi (\\mathbf {x}) (\\mathbf {n}_{i} \\cdot \\mathbf {e}_z) \\ \\mathrm {d}\\mathbf {a} \\, , $ where $\\smash{\\tilde{\\mathcal {S}} = \\mathcal {S} \\cap \\mathcal {P}}$ is the portion of the paraboloid $\\smash{\\mathcal {S}}$ inside the polyhedron $\\smash{\\mathcal {P}}$ (e.g., see fig:polyintersurf), $\\smash{\\mathbf {n}_{\\tilde{\\mathcal {S}}}}$ is the normal to $\\smash{\\tilde{\\mathcal {S}}}$ pointing outwards of $\\smash{\\mathcal {Q}}$ (i.e., $\\smash{\\mathbf {n}_{\\tilde{\\mathcal {S}}} \\cdot \\mathbf {e}_z \\ge 0}$ ), and $\\smash{\\hat{\\mathcal {F}}_i = \\mathcal {F}_i \\cap \\mathcal {Q}}$ is the portion of the face $\\smash{\\mathcal {F}_i}$ inside the clipping region $\\smash{\\mathcal {Q}}$ (e.g., see fig:clippedpolywithfaces).",
"Owing to the definitions of $\\smash{\\mathcal {Q}}$ and $\\smash{\\mathcal {S}}$ , as given in Eqs.",
"def:paraboloid,def:halfspace, the normal to $\\smash{\\mathcal {S}}$ pointing outwards of $\\smash{\\mathcal {Q}}$ reads as $\\mathbf {n}_{{\\mathcal {S}}} = \\frac{\\nabla \\phi }{\\left\\Vert \\nabla \\phi \\right\\Vert } \\, ,$ yielding $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\int _{\\tilde{\\mathcal {S}}} \\Phi (\\mathbf {x}) \\left(\\frac{\\nabla \\phi (\\mathbf {x})\\cdot \\mathbf {e}_z}{\\left\\Vert \\nabla \\phi (\\mathbf {x})\\right\\Vert }\\right) \\ \\mathrm {d}\\mathbf {a} + \\sum \\limits _{i=1}^{n_\\mathcal {F}} \\int _{\\hat{\\mathcal {F}}_i} \\Phi (\\mathbf {x}) (\\mathbf {n}_{i} \\cdot \\mathbf {e}_z) \\ \\mathrm {d}\\mathbf {a} \\, .",
"$ The surface $\\smash{\\tilde{\\mathcal {S}}}$ can be expressed in the parametric form $\\tilde{\\mathcal {S}} = \\left\\lbrace \\begin{bmatrix} x \\\\ y \\\\ - \\alpha x^2 - \\beta y^2\\end{bmatrix} , (x,y) \\in \\tilde{\\mathcal {S}}^{\\perp } \\right\\rbrace \\, , $ with $\\alpha $ and $\\beta $ the coefficients introduced in Eq.",
"def:psi, and with $\\smash{\\tilde{\\mathcal {S}}^{\\perp }}$ the projection of $\\smash{\\tilde{\\mathcal {S}}}$ onto the $xy$ -plane.",
"Under the assumption that $\\smash{\\mathbf {n}_{i} \\cdot \\mathbf {e}_z \\ne 0}$ , each face $\\smash{\\hat{\\mathcal {F}}_i}$ , $\\smash{i \\in \\lbrace 1,\\ldots ,n_{\\!\\mathcal {F}}\\rbrace }$ , can also be expressed in the parametric form $\\hat{\\mathcal {F}}_i = \\left\\lbrace \\begin{bmatrix} x \\\\ y \\\\ \\delta _i - \\lambda _i x - \\tau _i y \\end{bmatrix}, (x,y) \\in \\hat{\\mathcal {F}}_{i}^{\\perp } \\right\\rbrace \\, , $ with $\\delta _i & = \\frac{\\mathbf {n}_i\\cdot \\mathbf {x}_{\\mathcal {F}_i}}{\\mathbf {n}_i \\cdot \\mathbf {e}_z} , \\quad \\text{for any } \\mathbf {x}_{\\mathcal {F}_i} \\in \\mathcal {F}_i \\, , \\\\\\lambda _i & = \\frac{\\mathbf {n}_i \\cdot \\mathbf {e}_x}{\\mathbf {n}_i \\cdot \\mathbf {e}_z} \\, , \\\\\\tau _i & = \\frac{\\mathbf {n}_i \\cdot \\mathbf {e}_y}{\\mathbf {n}_i \\cdot \\mathbf {e}_z} \\, ,$ and with $\\smash{\\hat{\\mathcal {F}}_{i}^{\\perp }}$ the projection of $\\smash{\\hat{\\mathcal {F}}_{i}}$ onto the $xy$ -plane.",
"These explicit parametrizations yield $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\int _{\\tilde{\\mathcal {S}}^{\\perp }} \\Phi _{\\!\\mathcal {S}}(x,y) \\ \\mathrm {d}\\mathbf {a}^\\perp + \\sum \\limits _{i=1}^{n_\\mathcal {F}} \\text{sign}(\\mathbf {n}_{i} \\cdot \\mathbf {e}_z)\\int _{\\hat{\\mathcal {F}}_{i}^{\\perp }} \\Phi _{\\!\\mathcal {F}_i}(x,y) \\ \\mathrm {d}\\mathbf {a}^\\perp \\, , $ where $\\smash{\\Phi _{\\!\\mathcal {S}}}$ and $\\smash{\\Phi _{\\!\\mathcal {F}_i}}$ are the function vectors $\\Phi _{\\!\\mathcal {S}}(x,y) = \\begin{bmatrix} - \\alpha x^2 - \\beta y^2 \\\\ - \\alpha x^3 - \\beta x y^2 \\\\ - \\alpha y x^2 - \\beta y^3 \\\\ \\frac{1}{2}\\left(\\alpha x^2 + \\beta y^2\\right)^2 \\end{bmatrix} \\, , \\quad \\quad \\Phi _{\\!\\mathcal {F}_i}(x,y) = \\begin{bmatrix} \\delta _i - \\lambda _i x - \\tau _i y \\\\ \\delta _i x - \\lambda _i x^2 - \\tau _i x y \\\\ \\delta _i y - \\lambda _i x y - \\tau _i y^2 \\\\ \\frac{1}{2} \\left(\\delta _i - \\lambda _i x - \\tau _i y \\right)^2 \\end{bmatrix} \\, , $ and $\\smash{\\mathrm {d}\\mathbf {a}^\\perp = \\mathrm {d}x\\mathrm {d}y}$ is an infinitesimal surface element on the $xy$ -plane.",
"Note that in order to simplify Eq.",
"eq:volafterdiv3 into eq:volafterdiv4, we have used the fact that $\\smash{\\left\\Vert \\nabla \\phi (\\mathbf {x})\\right\\Vert }$ is the determinant of the parametrization def:paramS of $\\smash{\\tilde{\\mathcal {S}}}$ , and that $\\smash{\\left|\\mathbf {n}_{i} \\cdot \\mathbf {e}_z\\right|^{-1}}$ is the determinant of the parametrization def:paramFk of $\\smash{\\hat{\\mathcal {F}}_i}$ .",
"The projected integration domains $\\smash{\\tilde{\\mathcal {S}}^\\perp }$ and $\\smash{\\hat{\\mathcal {F}}_i^\\perp }$ corresponding to the configuration of fig:intropoly are illustrated in fig:projection.",
"Figure: Illustration of the one-dimensional integration domains used for integrating the first, second, and third contributions to the moments.Eq.",
"eq:volafterdiv4 can be rewritten as $\\mathcal {M}^{\\hat{\\mathcal {P}}} & = \\int _{\\tilde{\\mathcal {S}}^{\\perp }} \\nabla \\cdot \\left( \\Psi _{\\mathcal {S}}(x,y) \\otimes \\mathbf {e}_x\\right) \\ \\mathrm {d}\\mathbf {a}^\\perp \\\\& \\quad \\quad \\quad \\quad \\quad \\quad + \\sum \\limits _{i=1}^{n_\\mathcal {F}} \\text{sign}(\\mathbf {n}_{i} \\cdot \\mathbf {e}_z)\\int _{\\hat{\\mathcal {F}}_{i}^{\\perp }} \\nabla \\cdot \\left(\\Psi _{\\mathcal {F}_i}(x,y) \\otimes \\mathbf {e}_x\\right)\\ \\mathrm {d}\\mathbf {a}^\\perp \\, , \\nonumber $ where $\\smash{\\Psi _{\\mathcal {S}}}$ and $\\smash{\\Psi _{\\mathcal {F}_i}}$ are defined as $\\Psi _{\\mathcal {S}} : \\left\\lbrace \\begin{array}{lcl}\\mathbb {R}^2 & \\!\\rightarrow & \\mathbb {R}^4 \\\\(x,y) & \\!\\mapsto & {\\displaystyle \\int _0^x} \\Phi _{\\!\\mathcal {S}}(x,y) \\ \\mathrm {d}x \\end{array}\\right.",
"\\, .$ and $\\smash{\\forall i \\in \\lbrace 1,\\ldots ,n_{\\!\\mathcal {F}}\\rbrace }$ , $\\Psi _{\\mathcal {F}_i} : \\left\\lbrace \\begin{array}{lcl}\\mathbb {R}^2 & \\!\\rightarrow & \\mathbb {R}^4 \\\\(x,y) & \\!\\mapsto & {\\displaystyle \\int _0^x} \\Phi _{\\!\\mathcal {F}_i}(x,y) \\ \\mathrm {d}x \\end{array}\\right.",
"\\, .$ Note that the choice made here of integrating $\\smash{\\Phi _{\\!\\mathcal {S}}}$ and $\\smash{\\Phi _{\\!\\mathcal {F}_i}}$ with respect to $x$ is arbitrary, and that we could have equivalently integrated them with respect to $y$ , requiring to replace $\\smash{\\mathbf {e}_x}$ by $\\smash{\\mathbf {e}_y}$ in Eq. eq:divsecondtime.",
"Using the divergence theorem once again, this gives $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\int _{\\partial \\tilde{\\mathcal {S}}^{\\perp }} \\Psi _{\\mathcal {S}}(x,y) \\left(\\mathbf {n}_{\\partial \\tilde{\\mathcal {S}}^{\\perp }} \\cdot \\mathbf {e}_x\\right) \\ \\mathrm {d}l + \\sum \\limits _{i=1}^{n_\\mathcal {F}} \\int _{\\partial \\hat{\\mathcal {F}}_{i}^{\\perp }} \\Psi _{\\mathcal {F}_i}(x,y) (\\mathbf {n}_{\\partial \\hat{\\mathcal {F}}_{i}^{\\perp }} \\cdot \\mathbf {e}_x) \\ \\mathrm {d}l \\, , $ where $\\smash{\\mathrm {d}l}$ is an infinitesimal line element on the integration domains $\\smash{\\partial \\tilde{\\mathcal {S}}^{\\perp }}$ and $\\smash{\\partial \\hat{\\mathcal {F}}_{i}^{\\perp }}$ , which are the boundaries of the projections of the faces of $\\smash{\\hat{\\mathcal {P}}}$ onto the $xy$ -plane.",
"As such, they consist of closed curves in the $xy$ -plane, that are successions of conic section arcs and/or line segments (e.g., see fig:projviewxy,fig:projf11).",
"Note that the term “$\\smash{\\text{sign}(\\mathbf {n}_{i} \\cdot \\mathbf {e}_z)}$ ” is now implicitly accounted for, as the closed curves $\\smash{\\partial \\hat{\\mathcal {F}}_i}$ (and therefore their projection onto the $xy$ -plane, $\\smash{\\partial \\hat{\\mathcal {F}}_i^\\perp }$ ) are oriented so as to produce a normal vector pointing towards the outside of $\\smash{\\hat{\\mathcal {P}}}$ .",
"It should also be noted that the integration domains $\\smash{\\partial \\tilde{\\mathcal {S}}^{\\perp }}$ and $\\smash{\\partial \\hat{\\mathcal {F}}_{i}^{\\perp }}$ do not necessarily consist of one closed curve each – they may each be the union of several non-intersecting oriented closed curves.",
"Let us assume that a parametrization $\\mathbf {x}_{i,j}(t) = x_{i,j}(t) \\mathbf {e}_x + y_{i,j}(t) \\mathbf {e}_y + z_{i,j}(t) \\mathbf {e}_z, \\quad t \\in [0,1] , \\quad j \\in \\lbrace 1,\\ldots ,n_{\\partial \\hat{\\mathcal {F}_i}}\\rbrace \\, , $ is known for each of the $\\smash{n_{\\partial \\hat{\\mathcal {F}_i}}}$ arcs of $\\smash{\\partial \\hat{\\mathcal {F}}_{i}}$ , where the functions $\\smash{x_{i,j}}$ , $\\smash{y_{i,j}}$ , and $\\smash{z_{i,j}}$ belong to $\\mathcal {C}^1([0,1])$ .",
"Moreover, let us note that each parametrized conic section arc belonging to $\\smash{\\partial \\tilde{\\mathcal {S}}}$ is necessarily present in one and only one of the clipped face boundaries $\\smash{\\partial \\hat{\\mathcal {F}}_i}$ , where it is traversed in the opposite direction for integrating $\\smash{\\Psi _{\\!\\mathcal {F}_i}}$ .",
"Eq.",
"eq:volafterdiv6 can then be written as $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} \\int _{0}^{1} \\left(\\Psi _{\\mathcal {F}_i}(x_{i,j}(t),y_{i,j}(t)) - 1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\, \\Psi _{\\mathcal {S}}(x_{i,j}(t),y_{i,j}(t)) \\right) y^\\prime _{i,j}(t) \\ \\mathrm {d}t \\, , $ where $1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} & = \\left\\lbrace \\begin{array}{ll} 1 & \\text{if the $j$th arc of $\\partial \\hat{\\mathcal {F}}_i$ also belongs to $\\partial \\tilde{\\mathcal {S}}$} \\\\ 0 & \\text{otherwise} \\end{array}\\right.",
"\\, .$ Note that in Eq.",
"eq:volafterdiv7 and in the remainder of this manuscript, the superscript $\\smash{\\square ^\\prime }$ indicates that a function has been differentiated with respect to its unique variable.",
"A closed-form expression can be derived for the integral in Eq.",
"eq:volafterdiv7, however, its use for numerically calculating the moments is undesirable for two main reasons: the expression contains many terms, rendering its numerical calculation expensive; the expression depends on $\\smash{\\delta _i}$ , $\\smash{\\lambda _i}$ , and $\\smash{\\tau _i}$ , which all tend towards infinity as $\\mathbf {n}_i \\cdot \\mathbf {e}_z$ tends towards zero, leading to large round-off errors in the context of floating-point arithmetics.",
"Instead, the introduction of a twin parametrization of the arcs of $\\smash{\\partial \\partial \\hat{\\mathcal {P}}}$ and the judicious splitting of the integral in Eq.",
"eq:volafterdiv7 can both reduce the complexity of its closed-form expression and remove its direct dependency on the potentially singular coefficients $\\smash{\\delta _i}$ , $\\smash{\\lambda _i}$ , and $\\smash{\\tau _i}$ .",
"Let us then introduce the parametrization $\\bar{\\mathbf {x}}_{i,j}(t) = \\bar{x}_{i,j}(t) \\mathbf {e}_x + \\bar{y}_{i,j}(t) \\mathbf {e}_y + \\bar{z}_{i,j}(t) \\mathbf {e}_z, \\quad t \\in [0,1] , \\quad j \\in \\lbrace 1,\\ldots ,n_{\\partial \\hat{\\mathcal {F}_i}}\\rbrace \\, , $ which links $\\smash{{\\mathbf {x}}_{i,j}(0)}$ to $\\smash{{\\mathbf {x}}_{i,j}(1)}$ by a straight line.",
"For the sake of conciseness, we shall refer to these two points as $\\smash{{\\mathbf {x}}_{i,j}(0) = {\\mathbf {x}}_{i,j,0}}$ and $\\smash{{\\mathbf {x}}_{i,j}(1) = {\\mathbf {x}}_{i,j,1}}$ in the remainder of this work.",
"The twin parametrization of each arc is simply given as $\\bar{\\mathbf {x}}_{i,j}(t) = (1-t) {\\mathbf {x}}_{i,j,0} + t {\\mathbf {x}}_{i,j,1}, \\quad t \\in [0,1] , \\quad j \\in \\lbrace 1,\\ldots ,n_{\\partial \\hat{\\mathcal {F}_i}}\\rbrace \\, .",
"$ If the $j$ th arc of $\\smash{\\partial \\hat{\\mathcal {F}}_i}$ does not belong to $\\smash{\\tilde{\\mathcal {S}}}$ , then $\\smash{x_{i,j} \\equiv \\bar{x}_{i,j}}$ , $\\smash{y_{i,j} \\equiv \\bar{y}_{i,j}}$ and $\\smash{z_{i,j} \\equiv \\bar{z}_{i,j}}$ .",
"We can then re-organize Eq.",
"eq:volafterdiv7 as $\\mathcal {M}^{\\hat{\\mathcal {P}}} & = \\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} \\left[ \\; \\int _{0}^{1} \\Psi _{\\mathcal {F}_i}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t))\\bar{y}^\\prime _{i,j}(t) \\ \\mathrm {d}t \\right.",
"\\\\& \\quad \\quad \\quad \\quad \\quad \\quad - 1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\int _{0}^{1} \\Psi _{\\mathcal {S}}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t))\\bar{y}^\\prime _{i,j}(t) \\ \\mathrm {d}t \\nonumber \\\\& \\quad \\quad \\quad \\quad \\quad \\quad - 1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\int _{0}^{1} \\left( \\phantom{\\frac{1}{1}\\!\\!\\!\\!",
"}\\Psi _{\\mathcal {F}_i}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t)) - \\Psi _{\\mathcal {S}}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t)) \\right) \\bar{y}^\\prime _{i,j}(t) \\nonumber \\\\& \\quad \\quad \\quad \\quad \\quad \\quad \\left.",
"\\phantom{\\int _{0}^{1}} + \\left(\\phantom{\\frac{1}{1}\\!\\!\\!\\!",
"}\\Psi _{\\mathcal {S}}(x_{i,j}(t),y_{i,j}(t)) - \\Psi _{\\mathcal {F}_i}(x_{i,j}(t),y_{i,j}(t)) \\right)y^\\prime _{i,j}(t) \\ \\mathrm {d}t \\; \\right]\\, , \\nonumber $ The moments can thus be described as the sum of three distinct contributions, i.e., $\\mathcal {M}^{\\hat{\\mathcal {P}}} = \\mathcal {M}^{\\hat{\\mathcal {P}}_1} + \\mathcal {M}^{\\hat{\\mathcal {P}}_2} + \\mathcal {M}^{\\hat{\\mathcal {P}}_3} = \\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\mathcal {M}^{\\hat{\\mathcal {P}}_1}_{i} + \\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i} + \\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\mathcal {M}^{\\hat{\\mathcal {P}}_3}_{i} \\, ,$ where $\\mathcal {M}^{\\hat{\\mathcal {P}}_1}_{i} &= \\int _{0}^{1} \\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} \\Psi _{\\mathcal {F}_i}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t))\\bar{y}^\\prime _{i,j}(t) \\ \\mathrm {d}t \\, , \\\\\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i} & = \\int _{0}^{1} \\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} -1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\Psi _{\\mathcal {S}}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t))\\bar{y}^\\prime _{i,j}(t) \\ \\mathrm {d}t \\, , \\\\\\mathcal {M}^{\\hat{\\mathcal {P}}_3}_{i} &= \\int _{0}^{1} \\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} 1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\left(\\phantom{\\frac{1}{1}\\!\\!\\!\\!\\!\\!",
"}\\left( \\Psi _{\\mathcal {S}}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t)) - \\Psi _{\\mathcal {F}_i}(\\bar{x}_{i,j}(t),\\bar{y}_{i,j}(t)) \\right)\\bar{y}^\\prime _{i,j}(t) \\right.",
"\\\\& \\quad \\quad \\quad \\quad \\quad \\quad \\left.",
"+ \\left(\\Psi _{\\mathcal {F}_i}(x_{i,j}(t),y_{i,j}(t)) - \\Psi _{\\mathcal {S}}(x_{i,j}(t),y_{i,j}(t)) \\right)y^\\prime _{i,j}(t) \\phantom{\\frac{1}{1}\\!\\!\\!\\!\\!\\!",
"}\\right) \\ \\mathrm {d}t \\, .",
"\\nonumber $ The contributions $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_1}}$ and $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_2}}$ require the integration of the paraboloid and plane primitives over straight lines only (e.g., see fig:projf11m1,fig:projf11m2), hence are straightforward to derive.",
"The contribution $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_3}}$ , on the other hand, requires the parametrization of the conic section arcs in $\\smash{\\partial \\tilde{\\mathcal {S}}}$ (e.g., see fig:projf11m3).",
"It should also be noted that each arc of the clipped faces $\\smash{\\partial \\hat{\\mathcal {F}}_i}$ contributes to $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_1}}$ , whereas only the conic section arcs in those faces (originating from the intersection of $\\smash{\\partial \\mathcal {P}}$ with $\\smash{\\mathcal {S}}$ ) contribute to $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_2}}$ and $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_3}}$ , owing to the presence of the coefficient $\\smash{1^{\\partial \\tilde{\\mathcal {S}}}_{i,j}}$ ."
],
[
"First term: $\\mathcal {M}^{\\hat{\\mathcal {P}}_1}$",
"Let us be reminded that the boundary of each clipped face, $\\smash{\\partial \\hat{\\mathcal {F}}_i}$ , is a succession of $\\smash{n_{\\partial \\hat{\\mathcal {F}}_i}}$ conic section arcs and/or straight line segments that each link a start point $\\smash{{\\mathbf {x}}_{i,j,0}}$ to an end point $\\smash{{\\mathbf {x}}_{i,j,1}}$ .",
"Now recall that we aim to derive expressions that are free of the coefficents $\\smash{\\delta _i}$ , $\\smash{\\lambda _i}$ , and $\\smash{\\tau _i}$ , so as to avoid round-off errors in the numerical calculation of the moments.",
"To do so, we assign to each face $\\smash{\\hat{\\mathcal {F}}_i}$ a reference point $\\smash{\\mathbf {x}_{i,\\text{ref}}}$ whose only requirement is to belong to the plane containing $\\smash{\\hat{\\mathcal {F}}_i}$ , e.g., $\\smash{\\mathbf {x}_{i,\\text{ref}} = {\\mathbf {x}}_{i,1,0}}$ .",
"For each arc of each clipped face, rather than integrating on the straight line linking $\\smash{{\\mathbf {x}}_{i,j,0}}$ to $\\smash{{\\mathbf {x}}_{i,j,1}}$ , we integrate instead on the oriented triangle $\\smash{T^{(1)}_{i,j} = \\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},{\\mathbf {x}}_{i,\\text{ref}}\\right)}$ .",
"Since $\\smash{\\partial \\hat{\\mathcal {F}}_i}$ is the union of closed curves, the start point of each of its constituting arcs is necessarily the end point of another arc, and the sum of all these triangle integrals is equal to the sum of the straight arc integrals.",
"This yields $\\mathcal {M}^{\\hat{\\mathcal {P}}_1}_i & = \\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} \\mathcal {A}\\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},{\\mathbf {x}}_{i,\\text{ref}}\\right) \\mathcal {B}^{(1)}\\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},{\\mathbf {x}}_{i,\\text{ref}}\\right) \\, ,$ where $\\mathcal {A}$ is the operator for calculating the signed projected area of a triangle from the knowledge of its three corners, i.e., $\\mathcal {A} : \\left\\lbrace \\begin{array}{ccl} \\mathbb {R}^3\\times \\mathbb {R}^3\\times \\mathbb {R}^3 & \\rightarrow & \\mathbb {R} \\\\ \\left(\\mathbf {x}_{a},\\mathbf {x}_{b},\\mathbf {x}_{c}\\right) & \\mapsto & \\frac{1}{2} \\left(x_a(y_b-y_c) + x_b(y_c-y_a) + x_c(y_a-y_b) \\right) \\end{array}\\right.",
"\\, , $ and $\\smash{\\mathcal {B}^{(1)} : \\mathbb {R}^3\\times \\mathbb {R}^3\\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^4}$ reads as $\\mathcal {B}^{(1)}\\left(\\mathbf {x}_{a},\\mathbf {x}_{b},\\mathbf {x}_{c}\\right) = \\frac{1}{12}\\begin{bmatrix} 4 \\left( z_{a} + z_{b} + z_{c} \\right) \\\\\\left( z_{a} + z_{b} + z_{c} \\right) \\left( x_{a} + x_{b} + x_{c} \\right) + x_{a} z_{a} + x_{b} z_{b} + x_{c} z_{c}\\\\\\left( z_{a} + z_{b} + z_{c} \\right) \\left( y_{a} + y_{b} + y_{c} \\right) + y_{a} z_{a} + y_{b} z_{b} + y_{c} z_{c}\\\\ z_{a}^2 + z_{b}^2 +z_{c}^2 + z_{b} z_{c} +z_{a} z_{b} + z_{a} z_{c}\\end{bmatrix} \\, .$"
],
[
"Second term: $\\mathcal {M}^{\\hat{\\mathcal {P}}_2}$",
"For computing $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_2}}$ , similarly as in sec:firstcontri, we choose a reference point $\\smash{\\mathbf {x}_{\\mathcal {S}}}$ belonging to $\\smash{\\mathcal {S}}$ .",
"An obvious choice for this reference point is the origin of our coordinate system, i.e., $\\smash{\\mathbf {x}_{\\mathcal {S}}=0}$ .",
"For each conic section arc of each clipped face, rather than integrating on the straight line linking $\\smash{{\\mathbf {x}}_{i,j,0}}$ to $\\smash{{\\mathbf {x}}_{i,j,1}}$ , we integrate instead on the oriented triangle $\\smash{T^{(2)}_{i,j} = \\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},\\mathbf {x}_{\\mathcal {S}}\\right)}$ .",
"For each $i$ th face of $\\smash{\\mathcal {P}}$ , this yields the moment contribution $\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i,\\bigtriangleup } & = \\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} 1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\mathcal {A}\\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},0\\right) \\mathcal {B}^{(2)}\\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1}\\right) \\, ,$ where $\\smash{\\mathcal {A}}$ is the signed projected triangle area operator defined in Eq.",
"eq:trianglearea and the operator $\\smash{\\mathcal {B}^{(2)} : \\mathbb {R}^3\\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^4}$ reads as $\\mathcal {B}^{(2)}\\left(\\mathbf {x}_{a},\\mathbf {x}_{b}\\right) = {\\small \\frac{1}{90}\\begin{bmatrix}[1.1] 15 \\left( \\alpha x_{a} x_{b} + \\beta y_{a} y_{b} - z_{a} - z_{b} \\right) \\\\6 \\beta (y_{a} - y_{b})(x_{b} y_{a} - x_{a} y_{b}) +9 (x_{a} + x_{b}) (z_{a} + z_{b})\\\\6 \\alpha (x_{a} - x_{b})(y_{b} x_{a} - y_{a} x_{b}) +9 (y_{a} + y_{b}) (z_{a} + z_{b})\\\\ 2 \\alpha \\beta (x_{b} y_{a} - x_{a} y_{b})^2 +3 (z_{a} + z_{b}) (\\alpha x_{a} x_{b} + \\beta y_{a} y_{b} - z_{a} - z_{b})+ 3 z_{a} z_{b}\\end{bmatrix}} \\, .$ Note that $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i,\\bigtriangleup } \\ne \\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i}}$ , owing to our choice of integrating over triangles rather than the individual arcs, however their sum over all faces is equal, yielding $\\mathcal {M}^{\\hat{\\mathcal {P}}_2} = \\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i,\\bigtriangleup } \\, .$"
],
[
"Third term: $\\mathcal {M}^{\\hat{\\mathcal {P}}_3}$",
"To derive a closed-form expression for $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_3}}$ , a parametrization of the conic section arcs in $\\smash{\\partial \\tilde{\\mathcal {S}}}$ must be provided.",
"For the elliptic and hyperbolic cases, traditional parametrizations using trigonometric functions are obvious choices, however they can yield significant round-off errors due to very large values of their constitutive parameters (e.g., the semi-major and semi-minor axes).",
"This is illustrated in fig:singularparam where the semi-major axis of the conic section generated by the intersection of a paraboloid with a plane is plotted as this plane is rotated about the $\\mathbf {e}_y$ basis vector.",
"Figure: Semi-major axis of the conic section generated by the intersection of the paraboloid 𝒮\\mathcal {S}, as defined in Eq.",
"def:paraboloid, when α=|β|=1\\smash{\\alpha =|\\beta |=1}, with the plane implicitely defined by 𝐧·𝐱=-𝐧·𝐞 z \\smash{\\mathbf {n}\\cdot \\mathbf {x} = -\\mathbf {n}\\cdot \\mathbf {e}_z}, with 𝐧=cos(θ)0sin(θ) ⊺ \\smash{\\mathbf {n} = \\begin{bmatrix} \\cos (\\theta ) & 0 & \\sin (\\theta )\\end{bmatrix}^\\intercal }, as a function of the angle θ\\theta .",
"The semi-major axis of the conic section tends to infinity when θ\\theta tends to a multiple of π\\pi , meaning that a parametrization of any arc of this conic section with trigonometric functions becomes singular if 𝐧·𝐞 z →0\\smash{\\mathbf {n}\\cdot \\mathbf {e}_z \\rightarrow 0}.To avoid singular arc parametrizations, we express each conic section arc as a rational Bézier curve .",
"This provides a general parametrization that is valid over all conic section cases (i.e., elliptic, hyperbolic, and parabolic) and allows a seamless and smooth transition between cases.",
"A conic section arc linking a start point $\\smash{{\\mathbf {x}}_{i,j,0}}$ to an end point $\\smash{{\\mathbf {x}}_{i,j,1}}$ can be exactly represented by the rational quadratic Bézier curve parametrically defined as $\\mathbf {x}_{i,j}(t) = \\dfrac{B_0(t) \\, {\\mathbf {x}}_{i,j,0} + w_{i,j} \\, B_1(t) \\, {\\mathbf {x}}_{i,j}^{\\star } + B_2(t) \\, {\\mathbf {x}}_{i,j,1}}{B_0(t) + w_{i,j} \\, B_1(t) + B_2(t)} \\, , \\quad t \\in [0,1] \\, ,$ where $B_0(t) & = (1-t)^2 \\, ,\\\\B_1(t) & = 2(1-t)t \\, ,\\\\B_2(t) & = t^2 \\, ,$ are the Bernstein polynomials of degree 2, and $\\smash{w_{i,j}}$ is a weight associated with the control point $\\smash{{\\mathbf {x}}_{i,j}^{\\star }}$ .",
"This control point is located at the intersection of the tangents to the conic section at the start and end points.",
"In the case of the intersection of a planar face $\\smash{\\mathcal {F}_i}$ with a paraboloid surface $\\smash{\\mathcal {S}}$ , these tangents are obtained by the intersection of the planes tangent to $\\smash{\\mathcal {S}}$ at the start and end points, with the plane containing $\\smash{\\mathcal {F}_i}$ .",
"The point $\\smash{\\mathbf {x}_{i,j}\\left(\\tfrac{1}{2}\\right)}$ is, by definition, located at the intersection of $\\smash{\\mathcal {S}}$ with the segment linking $\\smash{\\bar{\\mathbf {x}}_{i,j}\\left(\\tfrac{1}{2}\\right)}$ to $\\smash{{\\mathbf {x}}_{i,j}^{\\star }}$ .",
"Substituting $t$ by $\\smash{\\tfrac{1}{2}}$ in Eq.",
"eq:ratbezier, it follows that $w_{i,j} \\left(\\mathbf {x}_{i,j}\\left(\\tfrac{1}{2}\\right) - {\\mathbf {x}}_{i,j}^{\\star }\\right) = \\bar{\\mathbf {x}}_{i,j}\\left(\\tfrac{1}{2}\\right) - \\mathbf {x}_{i,j}\\left(\\tfrac{1}{2}\\right) \\, ,$ from which $\\smash{w_{i,j}}$ can be deducted.",
"If $\\smash{\\left|w_{i,j}\\right|<1}$ , the rational Bézier curve is an arc of an ellipse, if $\\smash{\\left|w_{i,j}\\right|=1}$ , the rational Bézier curve is an arc of a parabola, and if $\\smash{\\left|w_{i,j}\\right|>1}$ , the rational Bézier curve is an arc of a hyperbola (e.g., see fig:bezierarc).",
"Figure: Three rational Bézier arcs linking a start point 𝐱 i,j,0 \\smash{\\mathbf {x}_{i,j,0}} to an end point 𝐱 i,j,1 \\smash{\\mathbf {x}_{i,j,1}}.",
"Also shown are the control point 𝐱 i,j ☆ \\smash{\\mathbf {x}^{\\star }_{i,j}} and weight w i,j \\smash{w_{i,j}}, along with the points 𝐱 ¯ i,j 1 2\\smash{\\bar{\\mathbf {x}}_{i,j}\\left(\\tfrac{1}{2}\\right)} and 𝐱 i,j 1 2\\smash{{\\mathbf {x}}_{i,j}\\left(\\tfrac{1}{2}\\right)} used for determining w i,j \\smash{w_{i,j}}.Note that, in order to prevent round-off errors in the numerical calculation of $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_3}}$ , we limit our implementation to cases where $\\smash{w_{i,j} \\ge 0}$ .",
"As a consequence, conic section arcs that would result in negative rational Bézier weights are recursively split until positive weights are found.",
"With such a parametrization of the conic section arcs, it can be shown that $\\mathcal {M}^{\\hat{\\mathcal {P}}_3}_{i} & = \\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} 1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\mathcal {A}\\left({\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},\\mathbf {x}_{i,j}^{\\star }\\right) \\mathcal {B}^{(3)}\\left(w_{i,j},{\\mathbf {x}}_{i,j,0},{\\mathbf {x}}_{i,j,1},\\mathbf {x}_{i,j}^{\\star }\\right) \\, , $ where $\\smash{\\mathcal {A}}$ is the signed projected triangle area operator defined in Eq.",
"eq:trianglearea and the operator $\\smash{\\mathcal {B}^{(3)}: \\mathbb {R}^{+} \\times \\mathbb {R}^3 \\times \\mathbb {R}^3 \\times \\mathbb {R}^3 \\rightarrow \\mathbb {R}^4}$ is given in apdx:M3."
],
[
"A special elliptic case",
"When $\\smash{\\alpha \\beta > 0}$ (i.e., $\\smash{\\mathcal {S}}$ is an elliptic paraboloid) and the normal to $\\smash{\\mathcal {F}_i}$ is such that $\\smash{\\mathbf {n}_i \\cdot \\mathbf {e}_z \\ne 0}$ , the intersection of the plane containing the face $\\smash{\\mathcal {F}_i}$ with the surface $\\smash{\\mathcal {S}}$ is an ellipse.",
"The intersection of $\\smash{\\mathcal {F}_i}$ with $\\smash{\\mathcal {S}}$ can then be: empty, a collection of arcs of this ellipse, or the entire ellipse.",
"In the latter case, the sum of the contributions $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i}}$ and $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}_3}_{i}}$ can be directly calculated by integrating $\\smash{\\Psi _\\mathcal {S}}$ and $\\smash{\\Psi _{\\mathcal {F}_i}}$ over the full ellipse, which yields the more concise expression $\\mathcal {M}^{\\hat{\\mathcal {P}}_2}_{i}+\\mathcal {M}^{\\hat{\\mathcal {P}}_3}_{i} = -\\mathrm {sign}\\left(\\mathbf {n}_i \\cdot \\mathbf {e}_z\\right)\\pi \\frac{(\\tau _i^2 \\alpha + \\lambda _i^2 \\beta - 4 \\alpha \\beta \\delta _i)^2}{32\\left(\\alpha \\beta \\right)^{5/2}}\\begin{bmatrix}[1.2]1 \\\\\\frac{\\lambda _i }{2\\alpha } \\\\\\frac{\\tau _i }{2\\beta } \\\\\\frac{5\\tau _i^2}{12\\beta } + \\frac{5\\lambda _i^2}{12\\alpha } - \\frac{2\\delta _i}{3}\\end{bmatrix} \\, ,$ where $\\smash{\\delta _i}$ , $\\smash{\\lambda _i}$ , and $\\smash{\\tau _i}$ have been defined in Eqs. eq:distance,eq:lambda,eq:tau.",
"Note that this case only occurs for $\\smash{\\mathbf {n}_i \\cdot \\mathbf {e}_z \\ne 0}$ , hence these coefficients are here non-singular.",
"Although this manuscript is concerned with estimating the first moments of $\\smash{\\hat{\\mathcal {P}}}$ , the integration domains and parametrizations introduced in sec:moments can also be used for integrating quantities associated with the clipped surface $\\smash{\\tilde{\\mathcal {S}}}$ , e.g., its moments.",
"The area of $\\tilde{\\mathcal {S}}$ , for instance, given as $\\mathrm {M}_0^{\\tilde{\\mathcal {S}}} & = \\int _{\\tilde{\\mathcal {S}}} 1 \\ \\mathrm {d}\\mathbf {x} \\, ,$ also reads after application of the divergence theorem as $\\mathrm {M}_0^{\\tilde{\\mathcal {S}}} & =\\sum \\limits _{i=1}^{n_{\\!\\mathcal {F_{\\phantom{k}\\!\\!}}}}",
"\\sum \\limits _{j = 1}^{n_{\\partial \\hat{\\mathcal {F}}_i}} -1^{\\partial \\tilde{\\mathcal {S}}}_{i,j} \\int _0^1 \\gamma (x_{i,j}(t),y_{i,j}(t)) y^\\prime _{i,j}(t) \\ \\mathrm {d}t \\, , $ where $\\gamma : \\left\\lbrace \\begin{array}{lcl}\\mathbb {R}^2 & \\!\\rightarrow & \\mathbb {R} \\\\(x,y) & \\!\\mapsto & {\\displaystyle \\int _0^x} \\sqrt{1+4\\alpha ^2x^2 + 4\\beta ^2y^2} \\ \\mathrm {d}x \\end{array} \\right.",
"\\, .$ The first moments of $\\smash{\\tilde{\\mathcal {S}}}$ , as well as the average normal vector, average Gaussian curvature and average mean curvature of $\\smash{\\tilde{\\mathcal {S}}}$ , for example, can be expressed similarly as sums of one-dimensional integrals over the parametric arcs of $\\smash{\\partial \\tilde{\\mathcal {S}}^\\perp }$ .",
"Contrary to the first moments of $\\smash{\\hat{\\mathcal {P}}}$ , however, they need to be estimated using a numerical quadrature rule, as closed-form expressions cannot be derived."
],
[
"On floating-point arithmetics and robustness",
"In the context of floating-point arithmetics, there exist cases for which the computed topology of the clipped faces $\\smash{\\hat{\\mathcal {F}}_i}$ may be ill-posed, preventing the accurate calculation of the moments of $\\smash{\\hat{\\mathcal {P}}}$ .",
"This occurs when, in the discrete sense: The surface $\\smash{\\mathcal {S}}$ is tangent to one or more edges of the polyhedron $\\smash{\\mathcal {P}}$ ; At least one corner or vertex of the polyhedron $\\smash{\\mathcal {P}}$ belongs to the surface $\\smash{\\mathcal {S}}$ .",
"For any given face of $\\smash{\\mathcal {P}}$ , the former case is numerically detected by computing the absolute value of the dot product between the normalized tangent to $\\smash{\\mathcal {S}}$ and the normalized edge from which the tangent originates, and checking whether it lies within $\\epsilon _\\text{tangent}$ of unity.",
"The latter case is detected by checking whether the intersection of an edge with $\\smash{\\mathcal {S}}$ lies within $\\epsilon _\\text{corner}$ of a corner or vertex of the polyhedron $\\mathcal {P}$ .",
"When any of these configurations is detected, the polyhedron $\\smash{\\mathcal {P}}$ is translated in the direction of the normal to $\\smash{\\mathcal {S}}$ at the location of the intersection, by a distance $\\smash{\\epsilon _\\text{nudge}}$ , and the clipped face discrete topologies are re-computed.",
"In the current work, where we use “double precision”, we find the values $\\smash{\\epsilon _\\text{nudge} = \\epsilon _\\text{corner} = 15 \\epsilon _{\\mathrm {64}}}$ and $\\smash{\\epsilon _\\text{tangent} = 100 \\epsilon _{\\mathrm {64}}}$ , with $\\smash{\\epsilon _{\\mathrm {64}} = 2^{-52}}$ the upper bound of the relative approximation error in 64-bit floating-point arithmetics, to prevent the computation of any ill-posed topologies in all tests presented in sec:tests (for which more than $\\smash{5\\times 10^7}$ occurences of the current “nudging” procedure are forced to occur).",
"Choosing lower values for these tolerances may result in the generation of non-valid discrete topologies and/or erroneous moments.",
"It should be noted that this procedure introduces a relative error in the calculation of the moments of $\\smash{\\hat{\\mathcal {P}}}$ which, for a polyhedron $\\smash{\\mathcal {P}}$ with volume $\\smash{\\mathrm {M}^\\mathcal {P}_0 = \\mathcal {O}(1)}$ , is of the order of $\\smash{\\epsilon _\\text{nudge}}$ .",
"However, this bares little significance, since the rate of occurence of the cases triggering this correction procedure is also extremely small (that is, for non-engineered intersection configurations)."
],
[
"Verification",
"In this section, the closed-form expressions derived in sec:moments, the approach of sec:surface for integrating on the clipped surface, and the corrective procedure of sec:robustness are tested on a wide variety of engineered and random intersection configurations.",
"When analytical expressions of the exact moments are not available, we recursively split the faces of the polyhedron $\\mathcal {P}$ so as to approximate $\\smash{\\partial {\\mathcal {P}}\\cap \\mathcal {Q}}$ and $\\smash{\\partial \\mathcal {P}\\cap (\\mathbb {R}^3\\setminus \\mathcal {Q})}$ by collections of oriented triangles.",
"We refer to this procedure as the adaptive mesh refinement (AMR) of the faces of the polyhedron $\\mathcal {P}$ .",
"We then exactly integrate $\\smash{\\Phi _{\\mathcal {S}}}$ on the triangulated approximation of $\\smash{\\partial \\mathcal {P}\\cap (\\mathbb {R}^3\\setminus \\mathcal {Q})}$ and $\\smash{\\Phi _{\\mathcal {F}_i}}$ on the triangulated approximation of each clipped face $\\smash{\\hat{\\mathcal {F}}_i}$ , effectively approximating Eq. eq:volafterdiv4.",
"For each case, we ensure that enough levels of recursive refinement are employed in order to reach machine-zero.",
"Accumulated errors due to the summation of the contributions of all triangles are avoided by the use of compensated summation, also known as Kahan summation ."
],
[
"Unit cube translating along $\\mathbf {e}_z$",
"In a first test, we consider the elliptic paraboloid defined by Eq.",
"def:paraboloid with $\\alpha = \\beta = 1$ , intersecting with the unit cube centered at $\\smash{\\mathbf {x}_c = \\begin{bmatrix} 1/2 &1/2 & 1/2 - k \\end{bmatrix}^\\intercal }$ (as illustrated in fig:testcube).",
"For this case, the zeroth and first moments of $\\smash{\\hat{\\mathcal {P}}}$ , as well as the zeroth moment of $\\smash{\\tilde{\\mathcal {S}}}$ , can be derived as analytical functions of $k$ .",
"We compare these exact moments against those computed using the closed-form expressions derived in sec:moments for estimating the moments of $\\smash{\\hat{\\mathcal {P}}}$ , and by integrating Eq.",
"eq:surfacearea numerically with an adaptive Gauss-Legendre quadrature rule for estimating the zeroth moment of $\\smash{\\tilde{\\mathcal {S}}}$ .",
"The parameter $k$ is regularly sampled on $[0,3]$ with a uniform spacing $\\smash{\\Delta k = 10^{-3}}$ .",
"Figure: Unit cube centred at 1/21/21/2-k ⊺ \\begin{bmatrix} 1/2 & 1/2 & 1/2 - k \\end{bmatrix}^\\intercal clipped by the elliptic paraboloid parametrically defined as z=-x 2 -y 2 z = -x^2 - y^2.Figure: Moments of the unit cube centred at 1/21/21/2-k ⊺ \\begin{bmatrix} 1/2 & 1/2 & 1/2 - k \\end{bmatrix}^\\intercal clipped by the elliptic paraboloid parametrically defined as z=-x 2 -y 2 z = -x^2 - y^2, and the error in their estimation with 64-bit floating point arithmetics.",
"The volume moments and their associated estimation errors are scaled with respect to the moments at k=3k=3.",
"The surface area and its associated estimation error is scaled with respect to the surface area at k=1k=1.",
"The volumetric moments are computed from the analytical expressions derived in sec:moments, whereas the surface area is computed from Eq.",
"eq:surfacearea using an adaptive Gauss-Legendre quadrature rule.",
"The 64-bit machine epsilon ϵ 64 =2 -52 \\epsilon _{64} = 2^{-52} is shown as the dashed red line.The left-hand graph of fig:resultscube shows the exact moments of $\\smash{\\hat{\\mathcal {P}}}$ , scaled by their maximum values, as well as the exact zeroth moment of $\\smash{\\tilde{\\mathcal {S}}}$ , scaled by its value at $k=1$ .",
"The right-hand side of fig:resultscube shows the errors associated with their estimation, scaled similarly.",
"These errors are all contained within an order of magnitude of $\\smash{\\epsilon _{64} = 2^{-52}}$ , the upper bound of the relative approximation error in 64-bit floating-point arithmetics, except for the cases $k=1$ and $k=2$ which produce slightly higher errors.",
"This is due to the occurence of ambiguous topologies at $k = 1$ and $k=2$ , for which $\\smash{\\mathcal {S}}$ is tangent to edges of the cube and/or contains corners of the cube, hence triggering the correction procedure described in sec:robustness."
],
[
"Parameter sweep for several geometries",
"In a second test, we consider a selection of convex polyhedra (a regular tetrahedron, a cube, and a regular dodecahedron) and non-convex polyhedra (a hollow cube and the triangulated Stanford bunny ).",
"These polyhedra, whose properties are summarized in tab:geometries, are scaled so as to have a unit volume (i.e., $\\smash{\\mathrm {M}_0^\\mathcal {P} = 1}$ ), and a centroid initially located at the origin (i.e., $\\smash{\\mathbf {M}_1^\\mathcal {P} = \\mathbf {0}}$ ).",
"They are then translated along $\\mathbf {t} = \\begin{bmatrix} t_x & t_y & t_z \\end{bmatrix}^\\intercal $ , and rotated about the three basis vectors $\\smash{(\\mathbf {e}_x, \\mathbf {e}_y,\\mathbf {e}_z)}$ with the angles $\\theta _x$ , $\\theta _y$ , and $\\theta _z$ , respectively.",
"Throughout this section, $(t_x,t_y,t_z)$ are varied in $[-\\tfrac{1}{2},\\tfrac{1}{2}]^3$ , $(\\theta _x,\\theta _y,\\theta _z)$ are varied in $[-\\pi ,\\pi ]^3$ , and the paraboloid coefficients $(\\alpha ,\\beta )$ are varied in $[-5,5]^2$ .",
"Table: Five polyhedra are considered: a regular tetrahedron, a cube, a regular dodecahedron, a hollow cube, and the triangulated Stanford bunny .",
"The first three polyhedra are convex, whereas the last two are not.",
"The hollow cube contains non-convex faces.A random parameter sweep is first conducted by uniform random sampling of the eight parameters ($t_x, t_y, t_z, \\theta _x, \\theta _y, \\theta _z, \\alpha , \\beta $ ) in the parameter space $[-\\tfrac{1}{2},\\tfrac{1}{2}]^3 \\times [-\\pi ,\\pi ]^3 \\times [-\\tfrac{1}{2},\\tfrac{1}{2}]^2$ , totalling more than $\\smash{2\\times 10^8}$ realizations.",
"A graded parameter sweep is then conducted, in which the eight parameters are chosen in the discrete parameter space $\\lbrace -\\tfrac{1}{2},-\\tfrac{1}{4},0,\\tfrac{1}{4},\\tfrac{1}{2}\\rbrace ^3 \\times \\lbrace -\\pi ,-\\tfrac{\\pi }{2},0,\\tfrac{\\pi }{2},\\pi \\rbrace ^3 \\times \\lbrace -5,-4,\\ldots ,4,5\\rbrace ^2$ , resulting in $5^3\\times 5^3\\times 11^2 = 1,890,625$ distinct realizations for each geometryNote that we do not present the results of a graded parameter sweep on the Stanford bunny, since the “organic” nature of this polyhedron renders a graded parameter sweep equivalent to a random one..",
"The graded parameter sweep differs from the random one in that it raises many singular intersection configurations (e.g., degenerate conic sections that consist of parallel or intersecting line segments, or conic sections that are parabolas) and/or ambiguous discrete topologies that arise from polyhedron vertices lying on the paraboloid or edges of the polyhedron being tangent to the paraboloid, therefore testing the robustness of our implementation as well as of the procedure described in sec:robustness.",
"For each case of the random and graded parameter sweeps, the reference value of $\\smash{\\mathcal {M}^{\\hat{\\mathcal {P}}}}$ for calculating the moment errors is obtained by adaptive mesh refinement (AMR) of the faces of $\\smash{\\mathcal {P}}$ into triangles, followed by the (exact) integration of $\\smash{\\Phi _\\mathcal {S}}$ and $\\smash{\\Phi _{\\mathcal {F}_i}}$ , given in Eq.",
"eq:primitives, over the triangles above and below the paraboloid, respectively.",
"Examples of random intersection cases and their associated AMR are shown in fig:testcases, whereas examples of singular cases raised during the graded parameter sweep are shown in fig:degeneratecases.",
"Figure: Examples of random intersection cases between the five considered polyhedra and a paraboloid.",
"The left column shows the full polyhedra; the central column shows the polyhedra clipped by a paraboloid; the right column shows the edges of the AMR of the polyhedra used for calculating the reference moments.Figure: Examples of cases with singular intersection configurations and/or ambiguous topologies covered by the graded parameter sweep: (a) the elliptic paraboloid is tangent to four edges of the polyhedron; (b) the parabolic cylinder is tangent to one face of the polyhedron, resulting in a degenerate conic section intersection that is made of two overlapping parallel lines; (c) the intersection of the hyperbolic paraboloid with one face of the polyhedron results in a degenerate hyperbola that is the intersection of two straight lines.The maximum and average moment errors obtained during the random parameter sweep are given in tab:randomsweep.",
"For the tetrahedron, cube, dodecahedron, and hollow cube, the average moment error is of the order $\\smash{\\epsilon _{64} = 2^{-52} \\sim \\mathcal {O}(10^{-16})}$ , whereas the maximum error is about one order of magnitude larger.",
"The average and maximum moment errors for the Stanford bunny, which contains $\\mathcal {O}(10^5)$ faces, are each about one order of magnitude larger than for the other polyhedra but still close to machine-zero.",
"Table: Third contribution to the moments"
]
] | 2210.07772 |
[
[
"Existence and uniqueness of a weak solution to fractional\n single-phase-lag heat equation"
],
[
"Abstract In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation.",
"This model contains the terms $\\cal{D}_t^\\alpha(u_t)$ and $\\cal{D}_t^\\alpha u $ (with $\\alpha \\in(0,1)$), where $\\cal{D}_t^\\alpha$ denotes the Caputo fractional derivative in time of constant order $\\alpha\\in(0,1)$.",
"We consider homogeneous Dirichlet boundary data for the temperature.",
"We rigorously show the existence of a unique weak solution under low regularity assumptions on the data.",
"Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe's method.",
"We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution.",
"The variational approach is employed to show the uniqueness of this weak solution to the problem.",
"We also consider the one-dimensional problem and derive a representation formula for the solution.",
"We establish bounds on this explicit solution and its time derivative by extending properties of the multinomial Mittag-Leffler function."
],
[
"Problem formulation",
"Consider a material contained in a bounded domain $\\Omega \\subset \\mathbb {R}^d, d\\in \\mathbb {N}$ with a Lipschitz continuous boundary $\\partial \\Omega .$ Set $Q_T :=\\Omega \\times (0,T]$ and $\\Sigma _T:= \\Gamma \\times (0,T],$ where $T >0$ denotes the final time.",
"The function $u(\\mathbf {x},t)$ represents the temperature at a material point $\\mathbf {x}\\in \\Omega $ at time $t.$ The heat flux is denoted by $\\mathbf {q}(\\mathbf {x},t).$ Let $\\rho $ and $c$ be positive constants denoting the material's density and specific heat, respectively.",
"Moreover, let $\\mbox{${k}$}$ be the thermal conductivity, which may be space-dependent.",
"In [36], a modification of the classical Fourier law for heat conduction processes, $\\mathbf {q}(\\mathbf {x},t+\\tau _q) = -\\mbox{${k}$}(\\mathbf {x}) \\nabla u (\\mathbf {x}, t+\\tau _T),$ was proposed to overcome the presence of infinite speed of heat propagation.",
"This model links the heat flux $\\mathbf {q}$ to the temperature gradient $\\nabla u$ at the spatial point $\\mathbf {x}$ at different times, allowing for a delay in the build-up of the heat flux or temperature gradient.",
"Here $\\tau _q$ and $\\tau _T$ are the phase-lag parameters.",
"These parameters are material properties and represent the relaxation and delay time, respectively.",
"In case $\\tau _q>0$ and $\\tau _T= 0,$ the law is of single-phase-lag (SPL) type.",
"If $\\tau _q>0$ and $\\tau _T>0$ , the law is called to be of dual-phase-lag (DPL) type.",
"The classical Fourier law is recovered if both lagging parameters equal zero.",
"In this contribution, we consider the model obtained by first-order Taylor approximation of (REF ) if $\\tau _T=0$ , where we neglect the higher order terms and replace the time derivative by its fractional variant of Caputo type with order $\\alpha \\in (0,1)$ .",
"Hence, the adapted form of (REF ) reads as $\\left(1+\\tau _q^\\alpha \\mathcal {D}_t^\\alpha \\right) \\mathbf {q}(\\mathbf {x},t) = -\\mbox{${k}$}(\\mathbf {x})\\nabla u (\\mathbf {x},t),$ where the parameter $\\tau _q^\\alpha $ is introduced to maintain the dimensions.",
"The Caputo derivative $\\mathcal {D}_t^\\gamma $ of order $\\gamma \\in (0,1)$ is defined by [3], [1] $\\mathcal {D}_t^\\gamma f(t)= \\partial _t \\int _0^t \\frac{(t-s)^{-\\gamma }}{\\Gamma (1-\\gamma )} \\left( f(s) - f(0)\\right) \\operatorname{d\\hspace{-1.5pt}}s, \\quad t \\in [0,T],$ where $\\Gamma $ denotes the Gamma function.",
"The energy conservation law is given as in the classical case $\\rho c \\partial _tu(\\mathbf {x},t) + \\nabla \\cdot \\mathbf {q}(\\mathbf {x},t) = G(\\mathbf {x},t),$ where $G$ involves the heat sources and sinks.",
"Following [21], [31], [11], we can write $\\left(1+\\tau _q^\\alpha \\mathcal {D}_t^\\alpha \\right) G(\\mathbf {x},t) = F(\\mathbf {x},t) - a\\left(1+\\tau _q^\\alpha \\mathcal {D}_t^\\alpha \\right) u(\\mathbf {x},t),$ where $a\\ge 0$ is a constant and $F$ is a function representing the sources.",
"Applying the operator $\\mathbb {I}+\\tau _q^\\alpha \\mathcal {D}_t^\\alpha $ on (REF ), and employing (REF ) and (REF ), we obtain the following equation $\\rho c \\tau _q^\\alpha \\mathcal {D}_t^\\alpha \\partial _tu (\\mathbf {x},t) + a \\tau _q^\\alpha \\mathcal {D}_t^\\alpha u(\\mathbf {x},t) + \\rho c \\partial _tu(\\mathbf {x},t) + \\mathfrak {L}u(\\mathbf {x},t) = F(\\mathbf {x},t),$ for $(\\mathbf {x},t) \\in Q_T,$ where the operator $\\mathfrak {L}$ is the following second-order linear differential operator $\\mathfrak {L}u(\\mathbf {x},t) := au(\\mathbf {x},t) + \\nabla \\cdot \\left( - \\mbox{${k}$}(\\mathbf {x}) \\nabla u (\\mathbf {x},t)\\right).$ The main target of this paper is to study the existence and uniqueness questions for the single-phase lag heat equation (REF ) subjected to given initial data and homogeneous Dirichlet boundary conditions.",
"This SPL problem is given by $\\left\\lbrace \\begin{array}{rll}\\rho c \\tau _q^\\alpha \\mathcal {D}_t^\\alpha \\partial _tu(\\mathbf {x},t) + a\\tau _q^\\alpha \\mathcal {D}_t^\\alpha u(\\mathbf {x},t)&&\\\\ + \\rho c \\partial _tu(\\mathbf {x},t) + \\mathfrak {L} u(\\mathbf {x},t) &= F(\\mathbf {x},t), & (\\mathbf {x},t) \\in Q_T \\\\u(\\mathbf {x},0) &= U_0(\\mathbf {x}), & \\mathbf {x}\\in \\Omega \\\\\\partial _tu (\\mathbf {x},0) &= V_0(\\mathbf {x}), & \\mathbf {x}\\in \\Omega \\\\u(\\mathbf {x},t) & = 0, & (\\mathbf {x},t) \\in \\Sigma _T.\\end{array}\\right.$"
],
[
"Literature overview, new aspects and outline",
"First, we provide an overview of the main results when considering classical derivatives.",
"In [27], the stability of the corresponding problem was investigated.",
"Models obtained by first-order approximation for $u$ and second-order approximation for $\\mathbf {q}$ in (REF ) yield a hyperbolic equation which is (exponentially) stable if $\\tau _q< 2\\tau _T,$ whereas second-order approximation for both $u$ and $\\mathbf {q}$ yield stability if $\\tau _T>(2-\\sqrt{3})\\tau _q$ or $\\tau _T\\le (2-\\sqrt{3})\\tau _q$ , and the (restrictive) condition [27] hold.",
"For the first-order approximation considered in this paper, their result shows no conditions on the phase-lag parameters for stability.",
"The well-posedness of these problems on smoothly bounded domains and convex domains was studied in [17], [28] with the aid of the semigroup theory.",
"In the case of first-order approximation for both $u$ and $\\mathbf {q}$ , the well-posedness on a bounded domain was investigated in [45] for different boundary conditions and extended to higher dimensions in [44].",
"Fractional calculus has attracted many researchers because of the nonlocal property of the fractional derivatives and their use in the modelling part of complex systems, see, e.g.",
"[35] and references therein.",
"Fractional wave and diffusion equations (with constant order fractional derivatives) were studied in e.g.",
"[18], [29], [20], [8], [10], [24] by means of eigenfunction expansions and in e.g.",
"[38], [39], [40], [41] by means of the Rothe method.",
"For well-posedness results about equations of the form $\\mathcal {D}_t^\\alpha u + \\partial _tu(\\mathbf {x},t) + Lu =f$ , we refer to e.g.",
"[25], [32].",
"Similar results related to multi-term fractional equations can be found in [19], [34], [16], [46], [49], the fractional telegraph equation [4], [48], [6], [12], [51] and the recent works about variable-order operators [43], [50], [42].",
"To the best of our knowledge, there is no general well-posedness result known for the SPL problem (REF ) under low regularity conditions.",
"The goal of this paper is to fill in this gap.",
"Note that the inverse source problems considered in the work [22] assume the well-posedness of problem (REF ).",
"This paper is organised as follows.",
"In Section , we fix the notations and assumptions, and give some valuable properties of the convolution kernel together with some technical results.",
"The Fourier method will be used to obtain a representation formula for the solution in 1D in Section .",
"Section covers the existence of the solution.",
"It includes the setup of the weak formulation and the time discretisation, as well as the a priori estimates and the Rothe functions.",
"The uniqueness part is dealt with in Subsection REF ."
],
[
"Notations and assumptions",
"We start by introducing the function spaces used throughout this work.",
"The classical $\\operatorname{L}^2$ -inner product is denoted by $\\left(\\cdot ,\\cdot \\right)$ and the corresponding norm by $\\left\\Vert \\cdot \\right\\Vert .$ Let $(X,\\left\\Vert \\cdot \\right\\Vert _X)$ be a Banach space.",
"Its dual space is denoted by $X^\\ast .$ The duality pairing between ${\\operatorname{H}^{1}_0(\\Omega )}^\\ast $ and $\\operatorname{H}^{1}_0(\\Omega )$ is denoted by $\\left\\langle \\cdot ,\\cdot \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )}$ and is seen as a continuous extension of $\\left(\\cdot ,\\cdot \\right).$ For $p\\ge 1$ the space $\\operatorname{L}^{p}\\left((0,T),X\\right)$ is the space of all measurable functions $u \\colon (0,T)\\rightarrow X$ such that $\\left\\Vert u \\right\\Vert _{\\operatorname{L}^{p}\\left((0,T),X\\right)}^p := \\int _0^T \\left\\Vert u(t) \\right\\Vert _X^p \\operatorname{d\\hspace{-1.5pt}}t< \\infty .$ The space $\\operatorname{L}^{\\infty }\\left((0,T),X\\right)$ is the space of all measurable functions $u\\colon (0,T)\\rightarrow X$ that are essentially bounded, i.e.",
"$\\left\\Vert u \\right\\Vert _{\\operatorname{L}^{\\infty }\\left((0,T),X\\right)} := \\operatornamewithlimits{ess\\,sup}\\limits _{t \\in [0,T]} \\left\\Vert u(t) \\right\\Vert _X < \\infty .$ The space $\\operatorname{C}\\left(I,X\\right)$ consists of all continuous functions $u\\colon [0,T]\\rightarrow X$ such that $\\left\\Vert u \\right\\Vert _{\\operatorname{C}\\left(I,X\\right)} := \\max _{t \\in [0,T]} \\left\\Vert u(t) \\right\\Vert _X < \\infty .$ The space $\\operatorname{H}^k\\left((0,T),X\\right)$ consists of all functions $u:[0,T]\\rightarrow X$ such that the weak derivative with respect to $t$ up to order $k$ exists and $\\left\\Vert u \\right\\Vert _{\\operatorname{H}^k\\left((0,T),X\\right)}^2 := \\int _0^T \\left(\\sum _{i=0}^k \\left\\Vert u^{(i)}(t) \\right\\Vert ^2_X \\right) \\operatorname{d\\hspace{-1.5pt}}t< \\infty .$ Similar notations will be used for the case of vector-valued functions.",
"Additionally, the values of $C, \\varepsilon $ and $C_\\varepsilon $ are considered generic and positive constants.",
"Their value can differ from place to place, but their meaning should be clear from the context.",
"These constants are independent of the time discretisation parameter, where $\\varepsilon $ is arbitrarily small and $C_\\varepsilon $ arbitrarily large, i.e.",
"$C_\\varepsilon = C\\left(1+\\varepsilon + \\frac{1}{\\varepsilon }\\right).$ Now, we review some properties related to the operator $\\mathfrak {L}$ .",
"The bilinear form $\\mathcal {L}$ associated to the operator $\\mathfrak {L}$ is given by $\\mathcal {L}\\left(u(t),\\varphi \\right) := \\left(\\mbox{${k}$}\\nabla u(t),\\nabla \\varphi \\right) + a\\left(u(t),\\varphi \\right), \\quad u(t),\\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$ for $u, \\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$ Here $a\\ge 0$ and the matrix $\\mbox{${k}$}$ is a symmetric matrix-valued function on $\\Omega $ consisting of essentially bounded functions, i.e.",
"$\\mbox{${k}$} = \\left(k_{i,j}(\\mathbf {x})\\right) \\in {\\bf L}^{\\infty } (\\Omega ):=\\left(\\operatorname{L}^{\\infty }({\\Omega })\\right)^{d \\times d}, \\quad \\mbox{${k}$} = \\mbox{${k}$}^\\top .$ We assume $\\mbox{${k}$}$ to be uniformly elliptic, that is there exists a constant $\\widetilde{k}>0$ such that for all $\\mathbf {x}\\in \\Omega $ and all $\\mathbf {\\xi } = \\left(\\xi _1,\\dots ,\\xi _d\\right)^\\top \\in {\\mathbb {R}}^d$ it holds that $\\mathbf {\\xi }^\\top \\cdot \\mbox{${k}$}(\\mathbf {x}) \\mathbf {\\xi } = \\sum _{i,j=1}^{d} k_{i,j}(\\mathbf {x}) \\xi _i \\xi _j \\ge \\widetilde{k} \\left\\Vert \\mathbf {\\xi } \\right\\Vert ^2.$ These assumptions yield the following type of inequalities for $\\mathcal {L}$ : $\\mathcal {L}\\left(u,\\varphi \\right) &\\le C\\left\\Vert u \\right\\Vert _{\\operatorname{H}^{1}(\\Omega )}\\left\\Vert \\varphi \\right\\Vert _{\\operatorname{H}^{1}(\\Omega )}\\nonumber \\\\\\mathcal {L}\\left(\\varphi ,\\varphi \\right) & \\ge \\widetilde{k} \\left\\Vert \\nabla \\varphi \\right\\Vert ^2,$ for $u,\\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$ It follows that $\\mathcal {L}$ is $\\operatorname{H}^{1}_0(\\Omega )$ -elliptic by applying the Friedrichs' inequality on (REF ).",
"Finally, we remember that the fractional order appearing in (REF ) satisfies $0<\\alpha <1$ and that the fractional phase-lag parameters $ \\tau _q^\\alpha $ is considered to be positive."
],
[
"Properties of the kernel",
"We denote by ${g_{\\gamma }}$ the Riemann-Liouville kernel ${g_{\\gamma }}(t) = \\frac{t^{-\\gamma }}{\\Gamma (1-\\gamma )}, \\quad t>0, \\gamma \\in (0,1).$ The Caputo derivative operator $\\mathcal {D}_t^\\gamma $ can be rewritten as a convolution $\\mathcal {D}_t^\\gamma f(t) = \\partial _t\\left({g_{\\gamma }}\\ast (f-f(0))\\right)(t),$ where $\\ast $ denotes the Laplace convolution of two functions $(k\\ast z)(t) = \\int _0^t k(t-s)z(s)\\operatorname{d\\hspace{-1.5pt}}s.$ The ensuing lemma contains several properties of the kernel ${g_{\\gamma }},$ see e.g.",
"[40], [40] and [10].",
"Lemma 2.1 The function ${g_{\\gamma }}(t), t>0, 0<\\gamma <1$ satisfies ${g_{\\gamma }}\\in \\operatorname{L}^1(0,T)$ ; ${g_{\\gamma }}$ is decreasing in time $t$ and ${g_{\\gamma }}(t) \\ge \\frac{\\min \\lbrace 1,T^{-\\gamma }\\rbrace }{\\Gamma (1-\\gamma )} >0, \\quad t\\in (0,T];$ $\\partial _t{g_{\\gamma }}\\in \\operatorname{L}^1_{\\text{loc}}\\left(0,T\\right)$ ; ${g_{\\gamma }}(t) \\ge 0, \\partial _t{g_{\\gamma }}(t) \\le 0, \\partial _{tt}{g_{\\gamma }}(t) \\ge 0$ for all $t > 0$ and $\\partial _t{g_{\\gamma }}(t) \\lnot \\equiv 0,$ thus ${g_{\\gamma }}$ is strongly positive definite, i.e.",
"for all $v \\in \\operatorname{L}^2_{\\textit {loc}}\\left( (0,\\infty ), \\operatorname{L}^{2} (\\Omega )\\right)$ it holds that $\\int _0^t \\left(\\left({g_{\\gamma }}\\ast v\\right)(s),v(s)\\right)\\operatorname{d\\hspace{-1.5pt}}s\\ge 0, \\quad t \\ge 0;$ For any $v \\colon [0,T] \\rightarrow \\operatorname{L}^{2} (\\Omega )$ satisfying $v \\in \\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ with ${g_{\\gamma }}\\ast v \\in \\operatorname{H}^1\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right),$ it holds for all $t \\in [0,T]$ that $\\int _0^t \\left(\\partial _t\\left({g_{\\gamma }}\\ast v \\right)(s),v(s)\\right)\\operatorname{d\\hspace{-1.5pt}}s\\ge \\frac{g_\\gamma (T)}{2} \\int _0^t \\left\\Vert v(s) \\right\\Vert ^2_{\\operatorname{L}^{2} (\\Omega )}\\operatorname{d\\hspace{-1.5pt}}s.$ Fractional integration $I^\\alpha f(t) = \\frac{1}{\\Gamma (\\alpha )}\\int _0^t (t-s)^{\\alpha -1} f(s)\\operatorname{d\\hspace{-1.5pt}}s= \\left(g_{1-\\alpha } \\ast f\\right)(t), \\quad \\text{for } \\alpha > 0, f\\in \\operatorname{L}^1(0,T),$ has the property [30] $\\left(I^{\\alpha _1} (I^{\\alpha _2} f)\\right)(t) = \\left(I^{\\alpha _1+\\alpha _2}f\\right)(t), \\quad \\forall t\\in I,$ if $\\alpha _1> 0, \\alpha _2> 0$ and $f \\in \\operatorname{C}([0,T]).$ This property holds true for a.e.",
"$t\\in I$ if $f\\in \\operatorname{L}^1(0,T)$ and it leads to the following consequence.",
"Lemma 2.2 Let $u$ be absolutely continuous on $I$ and assume that $\\gamma \\in (0,1)$ .",
"Then for a.a. $t\\in I$ it holds that $\\partial _t({g_{\\gamma }}\\ast u) (t) = {g_{\\gamma }}(t) u(0) + ({g_{\\gamma }}\\ast \\partial _tu)(t).$ We follow the lines of [9].",
"Since $u$ is absolutely continuous (then $\\partial _tu \\in \\operatorname{L}^1 (0,T)$ exists and $u(t)=u(a)+\\int _a^t \\partial _tu(s)\\operatorname{d\\hspace{-1.5pt}}s$ ), we may write that $\\left({g_{\\gamma }}\\ast u\\right)(t) &= \\left({g_{\\gamma }}\\ast u(0)\\right)(t) + \\left(g_{\\beta }\\ast \\int _0^{\\bullet } \\partial _tu (s)\\operatorname{d\\hspace{-1.5pt}}s\\right)(t)\\\\&= \\left(\\int _0^t {g_{\\gamma }}(s)\\operatorname{d\\hspace{-1.5pt}}s\\right)u(0) + \\int _0^t {g_{\\gamma }}(s)\\int _0^{t-s}\\partial _tu(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\operatorname{d\\hspace{-1.5pt}}s,$ where the notation $\\bullet $ is a placeholder for the independent variable.",
"As $\\partial _s \\int _0^{t-s}\\partial _tu(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau = - \\partial _{t-s} \\int _0^{t-s}\\partial _tu(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau =- \\partial _tu(t-s)$ , we get via partial integration that $\\left({g_{\\gamma }}\\ast u\\right)(t) = \\left(\\int _0^t {g_{\\gamma }}(s)\\operatorname{d\\hspace{-1.5pt}}s\\right)u(0) + \\int _0^t \\left(\\int _0^s {g_{\\gamma }}(\\tau ) \\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\partial _tu(t-s) \\operatorname{d\\hspace{-1.5pt}}s,$ since the boundary term vanishes.",
"Note that $\\int _0^s {g_{\\gamma }}(\\tau ) \\operatorname{d\\hspace{-1.5pt}}\\tau = \\int _0^s \\frac{\\tau ^{-\\gamma }}{\\Gamma (1-\\gamma )} \\operatorname{d\\hspace{-1.5pt}}\\tau = \\frac{s^{1-\\gamma }}{\\Gamma (2-\\gamma )} = {g_{\\gamma -1}}(s).$ Therefore, $\\left({g_{\\gamma }}\\ast u\\right)(t) =\\left(\\int _0^t {g_{\\gamma }}(s)\\operatorname{d\\hspace{-1.5pt}}s\\right)u(0) + ({g_{\\gamma -1}}\\ast \\partial _tu)(t).$ From (REF ), it follows that $({g_{\\gamma -1}}\\ast \\partial _tu)(t) = \\left(I^{2-\\gamma } \\partial _tu\\right)(t) = \\left(I (I^{1-\\gamma } \\partial _tu)\\right)(t) = \\left( I ({g_{\\gamma }}\\ast \\partial _tu)\\right)(t).$ Hence, $\\partial _t({g_{\\gamma -1}}\\ast \\partial _tu)(t) &= \\partial _tI ({g_{\\gamma }}\\ast \\partial _tu)(t) = ({g_{\\gamma }}\\ast \\partial _tu)(t).$ Therefore, the result follows from differentiating (REF ) with respect to $t$ ."
],
[
"Solution via Fourier method in 1D",
"In this part, we consider (REF ) with $k(x)=\\bar{k}>0$ on the domain $\\Omega =(0,L), L>0,$ and we suppose that the solution $u$ can be written as $u(x,t) =X(x)T(t).$ Employing the separation of variables technique leads to the eigenvalue problem $\\left\\lbrace \\begin{array}{rll}\\bar{k} X^{\\prime \\prime }(x) + (\\sigma - a) X(x) &= 0, & \\quad x \\in (0,L)\\\\X(0) = X(L) &= 0,&\\end{array}\\right.$ where $\\sigma $ is the separation constant and $a>0$ .",
"The normalised solutions to this problem are given by $ X_n(x) = \\sqrt{\\frac{2}{L}} \\sin \\left(\\sqrt{\\frac{\\sigma _n -a}{\\bar{k}} } x\\right) = \\sqrt{\\frac{2}{L}} \\sin \\left(\\frac{n\\pi }{L}x\\right),$ where $\\sigma _n = a + \\bar{k}\\left(\\frac{n\\pi }{L}\\right)^2$ for $n \\in \\mathbb {N}.$ The functions $T_n(t)$ satisfy a fractional differential equation of the form $\\rho c \\tau _q^\\alpha \\mathcal {D}_t^\\alpha \\partial _tT + \\rho c \\partial _tT + a\\tau _q^\\alpha \\mathcal {D}_t^\\alpha T + \\sigma _n T = 0.$ By means of the Laplace transform, we find that $\\mathcal {L}\\left[ T_n(t)\\right](s) = T_n(0) \\left(\\rho c \\tau _q^\\alpha s^\\alpha + a \\tau _q^\\alpha s^{\\alpha -1} + \\rho c\\right) g(s) + T^{\\prime }_n(0)\\rho c \\tau _q^\\alpha s^{\\alpha - 1} g(s),$ where by [26] the function $g(s)$ is given by $g(s) = \\frac{1}{\\rho c \\tau _q^\\alpha } \\sum _{m=0}^{\\infty } (-1)^m \\left(\\frac{\\sigma _n}{\\rho c \\tau ^\\alpha _q}\\right)^m \\sum _{k=0}^m \\binom{m}{k} \\left(\\frac{a \\tau _q^\\alpha }{\\sigma _n}\\right)^k \\frac{s^{\\alpha k - m - 1}}{\\left(s^\\alpha + \\frac{1}{\\tau _q^\\alpha }\\right)^{m+1}},$ and forms a Laplace pair with $G(t) = \\frac{1}{\\rho c \\tau _q^\\alpha } \\sum _{m=0}^\\infty \\frac{1}{m!}",
"\\left(\\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha }\\right)^m \\sum _{k = 0}^m \\binom{m}{k} \\left(\\frac{a \\tau _q^\\alpha }{\\sigma _n}\\right)^k \\\\ \\quad \\times t^{(\\alpha +1)(m+1) - \\alpha k - 1} E^{(m)}_{\\alpha , \\alpha + 1 + m - \\alpha k} \\left(-\\frac{1}{\\tau _q^\\alpha } t^\\alpha \\right).$ Here $E_{\\gamma , \\delta }(z) = \\sum _{k=0}^\\infty \\frac{z^k}{\\Gamma (\\beta + \\alpha k)}$ is the two parameter Mittag-Leffler function, see [26].",
"Using properties of the Laplace transform related to fractional derivatives [26], we find that $\\mathcal {L}\\left[\\mathcal {D}_t^\\alpha G(t) \\right](s) = s^\\alpha g(s) \\quad \\text{ and } \\quad \\mathcal {L}\\left[ \\int _0^t \\mathcal {D}_t^\\alpha G(z)\\mathrm {d}z \\right](s) = s^{\\alpha - 1} g(s),$ noting that $G(0) = 0.$ The multinomial Mittag-Leffler function, see [5], is given by $ E_{(\\alpha _1, \\dots , \\alpha _m),\\beta }(z_1, \\dots , z_m) = \\sum _{k=0}^\\infty \\sum _{\\begin{array}{c} k_1+\\dots k_m = k\\\\ k_j\\ge 0\\end{array}} \\binom{k}{k_1,\\dots , k_m} \\frac{\\prod _{j=1}^m z_j^{k_j}}{ \\Gamma (\\beta + \\sum _{j=1}^m \\alpha _j k_j)},$ where $\\binom{k}{k_1,\\dots ,k_m}$ is the multinomial coefficient.",
"Let us remark that (REF ) is invariant under (the same) permutation on the parameters $\\alpha _j$ and the variables $z_j$ with $j=1,\\dots ,m.$ Following the approach from [14], we can rewrite (REF ) by means of multinomial Mittag-Leffler functions.",
"After relabelling, we obtain $G(t) = \\frac{t^\\alpha }{\\rho c \\tau _q^\\alpha } E_{(\\alpha +1, \\alpha ,1), \\alpha +1}\\left(\\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1},\\frac{-1}{\\tau _q^\\alpha }t^\\alpha , \\frac{-a}{\\rho c} t\\right).$ We now list some useful lemmas, the first one is a slight generalisation of [15], which we recover by setting $\\gamma = \\beta .$ Lemma 3.1 Let $\\alpha _1,\\dots , \\alpha _m$ be positive constants and $q_1, \\dots , q_m, \\beta $ real.",
"Then, for real $\\gamma $ , we have that $&\\frac{\\mathrm {d}}{\\mathrm {d}t} \\left[ t^\\gamma E_{(\\alpha _1, \\dots , \\alpha _m), \\beta +1}\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right)\\right]\\\\ &\\quad = t^{\\gamma -1} E_{(\\alpha _1, \\dots , \\alpha _m),\\beta }\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right) \\\\ & \\qquad +(\\gamma -\\beta ) t^{\\gamma -1} E_{(\\alpha _1,\\dots ,\\alpha _m), \\beta +1}\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right).$ First, we calculate $& \\frac{\\mathrm {d}}{\\mathrm {d}t} \\left[t^\\beta E_{(\\alpha _1, \\dots , \\alpha _m),\\beta +1}\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right)\\right] \\\\&\\quad = \\frac{\\mathrm {d}}{\\mathrm {d}t} \\left[ \\sum _{k=0}^\\infty \\sum _{k_1+\\dots + k_m=k} \\binom{k}{k_1,\\dots , k_m} \\prod _{j=1}^m q_j^{k_j} \\frac{t^{\\beta + \\sum _{j=1}^m \\alpha _j k_j}}{\\Gamma \\left(\\beta + 1+ \\sum _{j=1}^m \\alpha _j k_j\\right)} \\right] \\\\&\\quad = \\sum _{k=0}^\\infty \\sum _{k_1+ \\dots + k_m = k} \\binom{k}{k_1,\\dots , k_m} \\prod _{j=1}^m q_j^{k_j} \\frac{t^{\\beta - 1 + \\sum _{j=1}^m \\alpha _j k_j}}{\\Gamma \\left(\\beta + \\sum _{j=1}^m \\alpha _j k_j\\right)} \\\\&\\quad = t^{\\beta -1} E_{(\\alpha _1,\\dots , \\alpha _m), \\beta }\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right).$ Then, an application of the chain rule shows that $& \\frac{\\mathrm {d}}{\\mathrm {d}t} \\left[t^\\gamma E_{(\\alpha _1,\\dots , \\alpha _m), \\beta +1}\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right) \\right] \\\\&\\quad = \\frac{\\mathrm {d}}{\\mathrm {d}t} \\left[ t^{\\gamma - \\beta } t^{\\beta } E_{(\\alpha _1,\\dots , \\alpha _m),\\beta +1}\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right) \\right] \\\\&\\quad = t^{\\gamma - \\beta } t^{\\beta -1} E_{(\\alpha _1, \\dots , \\alpha _m), \\beta }\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right)\\\\&\\qquad + (\\gamma - \\beta ) t^{\\gamma -\\beta -1} t^{\\beta } E_{(\\alpha _1, \\dots , \\alpha _m), \\beta +1}\\left(q_1 t^{\\alpha _1}, \\dots , q_m t^{\\alpha _m}\\right),$ which proves the result.",
"The next result is a generalization of the property $E_{\\alpha , 0}(z) = z E_{\\alpha , \\alpha }(z)$ for the two-parameter Mittag-Leffler function.",
"Lemma 3.2 Let $\\alpha _1, \\dots , \\alpha _m$ be positive constants, $z_1,\\dots ,z_m \\in \\mathbb {C}$ and $0<\\beta <2$ be fixed.",
"Then $\\sum _{j=1}^m z_j E_{(\\alpha _1, \\dots , \\alpha _m), \\beta + \\alpha _j}\\left(z_1, \\dots , z_m\\right) + \\frac{1}{\\Gamma (\\beta )} = E_{(\\alpha _1, \\dots , \\alpha _m),\\beta }\\left(z_1, \\dots , z_m\\right).$ The proof can be found in [15].",
"Let us remark that this result remains valid in the limit case $\\beta \\rightarrow 0,$ as $1/\\Gamma (\\beta ) \\rightarrow 0.$ Next, we extend [15] such that this lemmata can be used in our problem setting (multinomial Mittag-Leffler function (REF ) with $\\alpha _1 \\in [1,2)$ ) as well.",
"Lemma 3.3 Let $\\beta >0$ and $\\alpha _1>\\alpha _2 > \\dots > \\alpha _m>0$ be given with $\\alpha _1 \\in (0,2)$ .",
"Assume that $\\alpha _1 \\pi /2 < \\mu < \\min \\lbrace \\alpha _1\\pi , \\pi \\rbrace , \\mu \\le \\left|\\operatorname{arg}(z_1) \\right| \\le \\pi $ and there exists $K>0$ such that $-K \\le z_j<0$ for $j=2,\\dots ,m.$ Then there exists a constant $C>0$ depending only on $\\mu , K, \\alpha _j, j=1,\\dots ,m$ and $\\beta $ such that $\\left|E_{(\\alpha _1, \\dots ,\\alpha _m), \\beta }\\left(z_1,\\dots ,z_m\\right) \\right| \\le \\frac{C}{1+ \\left|z_1 \\right|}.$ The proof goes along the same lines as that of [15], therefore we point out some key steps and comment on the differences.",
"The starting point is the contour representation of the reciprocal of the gamma function, [26] $ \\frac{1}{\\Gamma (z)} = \\frac{1}{2\\pi \\alpha _1 i} \\int _{\\gamma (R,\\theta )} \\exp \\left(\\zeta ^{1/\\alpha _1}\\right) \\zeta ^{(1-z-\\alpha _1)/\\alpha _1} \\mathrm {d}\\zeta ,$ which is applicable since $\\alpha _1 < 2$ and $\\pi \\alpha _1 /2 < \\mu < \\min \\lbrace \\alpha _1\\pi , \\pi \\rbrace .$ Here $\\gamma (R,\\theta )$ is the contour consisting of an circular arc $\\lbrace \\zeta \\in {\\mathbb {C}}: \\left|\\zeta \\right| =R, \\left|\\operatorname{arg}(\\zeta ) \\right| \\le \\theta \\rbrace $ and two half lines $\\lbrace \\zeta \\in {\\mathbb {C}}: \\left|\\zeta \\right|>R, \\left|\\operatorname{arg}(\\zeta ) \\right| = \\pm \\theta \\rbrace .",
"$ The radius $R$ is to be fixed later and $\\theta $ is chosen such that $ \\alpha _1 \\pi / 2 < \\theta < \\mu .$ By means of (REF ), we can rewrite $&E_{(\\alpha _1,\\dots , \\alpha _m),\\beta }(z_1,\\dots ,z_m) \\\\&\\quad = \\frac{1}{2\\pi \\alpha _1 i } \\int _{\\gamma (R,\\theta )} \\exp \\left(\\zeta ^{1/\\alpha _1}\\right) \\zeta ^{\\frac{1-\\beta }{\\alpha _1} - 1} \\sum _{k = 0}^\\infty \\sum _{k_1 + \\dots + k_m = k} \\binom{k}{k_1, \\dots , k_m} \\\\ &\\qquad \\qquad \\qquad \\qquad \\times \\prod _{j=1}^m z_j^{k_j} \\zeta ^{- \\sum _{j=1}^m \\frac{\\alpha _j}{\\alpha _1}k_j} \\mathrm {d} \\zeta \\\\&\\quad = \\frac{1}{2\\pi \\alpha _1 i } \\int _{\\gamma (R,\\theta )} \\exp \\left(\\zeta ^{1/\\alpha _1}\\right) \\zeta ^{ \\frac{1-\\beta }{\\alpha _1} - 1} \\sum _{k=0}^{\\infty } \\left( \\frac{z_1}{\\zeta } + \\sum _{j=2}^m z_j \\zeta ^{-\\alpha _j/\\alpha _1}\\right)^k \\mathrm {d}\\zeta ,$ by isolating the index $j=1$ and applying the multinomial theorem.",
"Note that $1- \\frac{\\alpha _j}{\\alpha _1} \\in (0,1)$ for $j =2,\\dots ,m.$ Therefore, taking $R > \\left|z_1 \\right| + K \\sum _{j=2}^m R^{1-\\frac{\\alpha _j}{\\alpha _1}}$ ensures convergence of the series.",
"In case all $\\left|z_j \\right|\\le K, j = 1,\\dots ,m$ we can fix $R$ as a constant depending only on $K$ and $\\alpha _1,\\dots , \\alpha _m.$ In that case we deduce that $E_{(\\alpha _1,\\dots , \\alpha _m), \\beta } (z_1,\\dots ,z_m) = \\frac{1}{2\\pi \\alpha _1 i} \\int _{\\gamma (R,\\theta )} \\frac{\\exp \\left(\\zeta ^{1/\\alpha _1}\\right) \\zeta ^{\\frac{1-\\beta }{\\alpha _1}}}{\\zeta - z_1 - \\sum _{j=2}^m z_j \\zeta ^{1-\\frac{\\alpha _j}{\\alpha _1}}}\\mathrm {d}\\zeta .$ Repeating the arguments of [15] we obtain in case $\\left|z_1 \\right|>R$ and $\\mu \\le \\left|\\operatorname{arg}(z_1) \\right| \\le \\pi $ that $\\left|E_{(\\alpha _1, \\dots , \\alpha _m), \\beta }(z_1,\\dots ,z_m) \\right| \\le \\frac{C}{\\left|z_1 \\right|},$ and in case $\\left|z_1 \\right| \\le R, \\mu \\le \\left|\\operatorname{arg}(z_1) \\right| \\le \\pi $ that $\\left|E_{(\\alpha _1,\\dots ,\\alpha _m),\\beta }(z_1,\\dots ,z_m) \\right| \\le C E_{\\alpha _m ,\\beta } \\left(R + (m-1)K\\right),$ where $C \\ge \\max \\lbrace 1, \\frac{\\beta +\\alpha _1}{\\alpha _m}\\rbrace .$ To proof this last inequality, we used the bound $ \\frac{1}{\\Gamma (\\beta + \\sum _{j=1}^m \\alpha _j k_j)} \\le \\frac{\\max \\lbrace 1, \\frac{\\beta + \\alpha _1}{\\alpha _m}\\rbrace }{\\Gamma (\\beta + \\alpha _mk)}.$ This last result (REF ) can be shown in the following way.",
"For $k =0$ there is nothing to show, so assume $k\\ge 1.$ For $\\beta \\ge 1,$ we have $\\Gamma \\left(\\beta + \\sum _{j=1}^m \\alpha _j k_j\\right) > \\Gamma \\left(\\beta + \\alpha _m k\\right), $ since $\\Gamma (z) $ is monotonically increasing for $z\\ge 1$ and $\\alpha _1>\\alpha _2>\\dots >\\alpha _m>0.$ For $0< \\beta < 1$ , we use that $\\Gamma (z+1) = z\\Gamma (z)$ to write $\\frac{\\Gamma \\left(\\beta + \\alpha _m k \\right)}{\\Gamma \\left(\\beta + \\sum _{j = 1}^m \\alpha _j k_j\\right)} &= \\frac{\\Gamma \\left(\\beta + 1 + \\alpha _m k \\right) \\left(\\beta + \\sum _{j=1}^m \\alpha _jk_j\\right)}{\\Gamma \\left(\\beta + 1 + \\sum _{j=1}^m \\alpha _j k_j\\right)\\left(\\beta + \\alpha _m k\\right)} \\\\&\\le \\frac{\\beta + \\sum _{j=1}^m \\alpha _j k_j}{\\beta + \\alpha _m k } \\le \\frac{\\beta }{\\alpha _mk} + \\frac{\\alpha _1}{\\alpha _m} \\\\&\\le \\frac{\\beta + \\alpha _1}{\\alpha _m},$ which is a constant that can be brought out of the summations.",
"Using (REF ), the solution $T_n(t)$ takes the form $T_n(t) = c_n T_n^1(t) + d_n T_n^2(t), \\quad c_n,d_n\\in {\\mathbb {R}},$ with $T_n^1(t) &= \\rho c\\tau _q^\\alpha \\mathcal {D}_t^\\alpha G(t) + a \\tau _q^\\alpha \\int _0^t \\mathcal {D}_t^\\alpha G(s)\\mathrm {d}s + \\rho c G(t) \\\\T_n^2(t) &= \\rho c \\tau _q^\\alpha \\int _0^t \\mathcal {D}_t^\\alpha G(s)\\mathrm {d}s,$ where $G(t)$ is given by (REF ).",
"The results in [2] show that $\\mathcal {D}_t^\\alpha G(t) &= \\frac{1}{\\rho c \\tau _q^\\alpha } E_{(\\alpha +1, \\alpha ,1),1} \\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^\\alpha , \\frac{-a}{\\rho c}t\\right) \\\\\\int _0^t\\mathcal {D}_t^\\alpha G(s)\\mathrm {d}s &= \\frac{t}{\\rho c \\tau _q^\\alpha } E_{(\\alpha +1,\\alpha ,1),2} \\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right).$ Hence, we find that $T_n^1(t) = E_{(\\alpha +1,\\alpha ,1),1}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right) \\\\+ \\frac{a}{\\rho c} tE_{(\\alpha +1,\\alpha ,1),2}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right) \\\\+ \\frac{t^\\alpha }{\\tau _q^\\alpha } E_{(\\alpha +1,\\alpha ,1),\\alpha +1}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right) \\nonumber $ and $T_n^2(t) = t E_{(\\alpha +1,\\alpha ,1),2}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right).$ From Lemma REF , we have that $E_{(\\alpha +1,\\alpha ,1),1}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right)\\\\= 1 - \\frac{\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1} E_{(\\alpha +1,\\alpha ,1),\\alpha +2}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right) \\\\-\\frac{1}{\\tau _q^\\alpha }t^{\\alpha } E_{(\\alpha +1,\\alpha ,1),\\alpha +1}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right) \\\\-\\frac{a}{\\rho c}t E_{(\\alpha +1,\\alpha ,1),2}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right).$ Hence, $T_n^1$ simplifies to $T_n^1(t) = 1 - \\frac{\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1} E_{(\\alpha +1,\\alpha ,1),\\alpha +2}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right).$ Moreover, from Lemma REF , it follows that $\\max _{t\\in I}\\left|T_n^i(t) \\right| \\le T_i, \\quad \\forall n\\in {\\mathbb {N}}, \\quad i=1,2.$ The unknown function $u(x,t)$ can be represented as the series $u(x,t) = \\sum _{n=1}^{\\infty } X_n(x)\\left[(U_0,X_n)T_n^1(t) + (V_0,X_n) T_n^2(t)\\right].$ Hence, we have for all $t\\in I$ that $\\left\\Vert u(t) \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 &= \\sum _{n=1}^{\\infty } \\left|(U_0,X_n)T_n^1(t) + (V_0,X_n) T_n^2(t) \\right|^2 \\\\& \\le 2\\max \\lbrace T_1^2,T_2^2\\rbrace \\sum _{n=1}^{\\infty } \\left[ \\left|(U_0,X_n) \\right|^2 + \\left|(V_0,X_n) \\right|^2 \\right] \\\\& \\le 2\\max \\lbrace T_1^2,T_2^2\\rbrace \\left(\\left\\Vert U_0 \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 + \\left\\Vert V_0 \\right\\Vert _{\\operatorname{L}^2(0,L)}^2\\right).$ Using Lemma REF , we obtain that $\\left(T^1_n\\right)^\\prime (t) = \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^\\alpha E_{(\\alpha +1, \\alpha , 1), \\alpha +1}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right).$ and $\\left(T^2_n\\right)^\\prime (t) = E_{(\\alpha +1,\\alpha ,1),1}\\left( \\frac{-\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}, \\frac{-1}{\\tau _q^\\alpha }t^{\\alpha }, \\frac{-a}{\\rho c}t\\right).$ Therefore, from Lemma REF , it follows for all $n\\in {\\mathbb {N}}$ that $\\max _{t\\in I}\\left|\\left(T^1_n\\right)^\\prime (t) \\right| \\le \\left(\\frac{\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha -1} \\right)^{\\frac{1}{2}} \\frac{\\left(\\frac{\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1} \\right)^\\frac{1}{2}C}{1+\\frac{\\sigma _n}{\\rho c \\tau _q^\\alpha } t^{\\alpha +1}} \\le \\sigma _n^{\\frac{1}{2}} t^{\\frac{\\alpha -1}{2}} \\dot{T}_1$ and $\\max _{t\\in I}\\left|\\left(T^2_n\\right)^\\prime (t) \\right| \\le \\dot{T}_2.$ Using these bounds, we have for all $t\\in I$ that $\\left\\Vert \\partial _tu(t) \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 &= \\sum _{n=1}^{\\infty } \\left|(U_0,X_n)\\left(T^1_n\\right)^\\prime (t) + (V_0,X_n) \\left(T^2_n\\right)^\\prime (t) \\right|^2 \\\\& \\le 2 \\dot{T}_1^2 t^{\\alpha -1} \\sum _{n=1}^{\\infty } \\sigma _n \\left|(U_0, X_n) \\right|^2 + 2 \\dot{T}_2^2 \\sum _{n=1}^\\infty \\left|(V_0,X_n) \\right|^2 \\\\&\\le 2\\max \\lbrace \\dot{T}_1^2,\\dot{T}_2^2\\rbrace \\left( t^{\\alpha -1} \\left\\Vert U_0 \\right\\Vert _{\\operatorname{H}^1_0(0,L)}^2+ \\left\\Vert V_0 \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 \\right).$ Hence, we can conclude that $\\max _{t\\in I} \\left\\Vert u(t) \\right\\Vert _{\\operatorname{L}^2(0,L)} \\le C \\quad \\text{ and } \\quad \\left\\Vert \\partial _tu(t) \\right\\Vert _{\\operatorname{L}^2(0,L)} \\le C \\left(1+t^{\\frac{\\alpha -1}{2}}\\right)$ if $U_0\\in \\operatorname{H}_0^1(0,L)$ and $V_0\\in \\operatorname{L}^2(0,L)$ .",
"Note that from Lemma REF it also follows that $\\max _{t\\in I}\\left|\\left(T^1_n\\right)^\\prime (t) \\right| \\le \\sigma _n \\dot{T}_1, \\quad \\forall n\\in {\\mathbb {N}}.$ Consequently, we can obtain for all $t\\in I$ that $\\left\\Vert \\partial _tu(t) \\right\\Vert _{\\operatorname{L}^2(0,L)}^2& \\le 2\\max \\lbrace \\dot{T}_1^2,\\dot{T}_2^2\\rbrace \\sum _{n=1}^{\\infty } \\left[ \\sigma _n^2\\left|(U_0,X_n) \\right|^2 + \\left|(V_0,X_n) \\right|^2 \\right] \\\\&\\le 2\\max \\lbrace \\dot{T}_1^2,\\dot{T}_2^2\\rbrace \\left( \\left\\Vert U_0 \\right\\Vert _{\\operatorname{H}^2(0,L) \\cap \\operatorname{H}^1_0(0,L)}^2+ \\left\\Vert V_0 \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 \\right),$ i.e.",
"$\\max _{t\\in I} \\left\\Vert u(t) \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 + \\max _{t\\in I} \\left\\Vert \\partial _tu(t) \\right\\Vert _{\\operatorname{L}^2(0,L)}^2 \\le C$ if $U_0\\in \\operatorname{H}^2(0,L) \\cap \\operatorname{H}_0^1(0,L)$ and $V_0\\in \\operatorname{L}^2(0,L)$ ."
],
[
"The single-phase-lag problem",
"This section is devoted to constructing a weak solution to (REF ) through Rothe's method [7].",
"This section is structured as follows.",
"First, the variational formulation and its discrete analogue are posed.",
"Next, the existence and uniqueness of a solution $u_i$ to the discrete variational problem at each time slice $i$ is deduced.",
"Given those solutions, we prove some a priori estimates in appropriate norms.",
"Finally, we construct the Rothe functions and show that they possess a converging subsequence whose limit constitutes a weak solution of (REF ).",
"In the following section, the uniqueness of a weak solution will be established, implying that the whole Rothe sequence converges to this weak solution."
],
[
"Weak formulation",
"Note that the convolution map $f \\mapsto {g_{\\alpha }}\\ast f$ is a bounded operator $\\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)\\rightarrow \\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ with norm bounded by $\\left\\Vert {g_{\\alpha }} \\right\\Vert _{\\operatorname{L}^1(0,T)},$ as can be seen from Young's inequality for convolutions.",
"Therefore, if $\\partial _tu \\in \\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ , by Lemma REF , we have that ${g_{\\alpha }}\\ast \\partial _tu = \\partial _t\\left( {g_{\\alpha }}\\ast (u-U_0)\\right)$ is an element of $\\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right).$ Now, the weak formulation can be stated as follows.",
"Definition 4.1 (Weak formulation (SPL)) Find $u\\in \\operatorname{L}^{\\infty }\\left((0,T),\\operatorname{H}^{1}_0(\\Omega )\\right)\\cap \\operatorname{C}\\left(I,\\operatorname{L}^{2} (\\Omega )\\right)$ with $\\partial _tu \\in \\operatorname{L}^{2}\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right) $ and $\\partial _t\\left({g_{\\alpha }}\\ast \\left(\\partial _tu - V_0\\right)\\right) \\in \\operatorname{L}^{2}\\left((0,T),{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\right)$ such that for a.a. $t \\in (0,T)$ it holds that $\\rho c \\tau _q^\\alpha \\left\\langle \\partial _t\\left({g_{\\alpha }}\\ast \\left(\\partial _tu - V_0\\right)\\right)(t),\\varphi \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )}+ a \\tau _q^\\alpha \\left(\\left({g_{\\alpha }}\\ast \\partial _tu \\right)(t),\\varphi \\right) \\\\ + \\rho c \\left(\\partial _tu(t),\\varphi \\right) + \\mathcal {L}\\left(u(t),\\varphi \\right) = \\left(F(t),\\varphi \\right),$ for all $\\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$"
],
[
"Time discretisation",
"Let $n\\in {\\mathbb {N}}$ be given.",
"We discretise the time interval $(0,T)$ in $n$ subintervals according to the nodes $t_i = i\\tau $ for $i=0,1,\\dots ,n$ where $\\tau = T/n$ is the time step.",
"We define $\\lceil t\\rceil _{\\tau } = t_i$ for $t\\in (t_{i-1},t_i]$ .",
"For a function $z(t)$ we will write $z_i = z(t_i)$ for its evaluation at the time steps.",
"The backward Euler method is used for approximating the first and second order derivatives at each $t_i$ , i.e.",
"$\\delta z_i = \\frac{z_i - z_{i-1}}{\\tau } \\quad \\text{ and } \\quad \\delta ^2 z_i = \\frac{\\delta z_i - \\delta z_{i-1}}{\\tau } = \\frac{z_i - z_{i-1} }{\\tau ^2} - \\frac{\\delta z_{i-1}}{\\tau }.$ We set $u_0 = U_0$ and $\\delta u_0 = V_0$ according to the initial conditions.",
"Given a kernel $\\kappa \\colon (0,T] \\rightarrow \\mathbb {R}$ and a function $z \\colon [0,T]\\rightarrow \\mathbb {R}$ , we define the following discrete convolution approximations of $(\\kappa \\ast z)(t_i)$ : $\\left(\\kappa \\ast z \\right)^{c}_i &:= \\sum _{\\ell =1}^i \\kappa _{i+1-\\ell }z_\\ell \\tau .",
"$ If $z_0=0$ , then we can properly define $\\left(\\kappa \\ast z \\right)^{c}_0:=0.$ We record the following two lemmas which will be needed and crucial in the estimates below.",
"It is a discrete version of the inequality (REF ).",
"For the proof, we refer to [33].",
"Lemma 4.1 Let $\\tau = T/n$ be the time step, where $n\\in {\\mathbb {N}}$ is the number of time discretisation intervals.",
"Let $\\lbrace z_i\\rbrace _{i\\in {\\mathbb {N}}\\cup \\lbrace 0\\rbrace }$ and $\\lbrace \\kappa _i\\rbrace _{i\\in {\\mathbb {N}}}$ be two sequences of real numbers.",
"Assume that $\\kappa _{i+1} \\le \\kappa _{i}$ for all $i\\in {\\mathbb {N}}$ and $(\\kappa \\ast z)^{c}_0 = 0$ .",
"Then $2 \\delta (\\kappa \\ast z)^{c}_i z_i \\ge \\delta \\left(\\kappa \\ast z^2\\right)^{c}_i + \\kappa _i z_i^2$ As a consequence, we get the following lemma.",
"Lemma 4.2 Let the assumptions of Lemma REF be fulfilled.",
"Then, it holds that $2\\sum _{i=1}^j \\delta (\\kappa \\ast z)^{c}_i z_i \\ge \\left(\\kappa \\ast z^2\\right)^{c}_j + \\sum _{i=1}^j \\kappa _{i} z_i^2.$ assuming $\\left( \\kappa \\ast \\left\\Vert z \\right\\Vert ^2\\right)^{c}_0 =0$ .",
"For convenience, we state the summation by parts formula for bilinear mappings.",
"Lemma 4.3 Let $b\\colon V \\times V \\rightarrow {\\mathbb {R}}$ be a bilinear form on a vector space $V$ and let $\\lbrace z_i\\rbrace _{i\\in {\\mathbb {N}}}, \\lbrace w_i\\rbrace _{i\\in {\\mathbb {N}}}$ be two sequences in $V.$ Then $\\sum _{i=1}^j b\\left(z_i,w_i-w_{i-1}\\right) = b\\left(z_j,w_j\\right) - b\\left(z_0,w_0\\right) - \\sum _{i=1}^j b\\left( \\delta z_i, w_{i-1}\\right) \\tau .$"
],
[
"Existence at each time step",
"The discrete variational formulation is obtained by approximating (REF ) at time $t =t_i$ by means of (REF ) and (REF ).",
"Definition 4.2 (Discrete weak formulation (SPL)) Find $u_i \\in \\operatorname{H}^{1}_0(\\Omega ), i =1,\\dots ,n,$ such that $\\rho c\\tau _q^\\alpha \\left(\\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_i,\\varphi \\right) + a\\tau _q^\\alpha \\left(\\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_i,\\varphi \\right) \\\\ + \\rho c \\left(\\delta u_i,\\varphi \\right) + \\mathcal {L}\\left(u_i,\\varphi \\right) = \\left(F_i,\\varphi \\right),$ for all $\\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$ The existence of a solution on a single time step is established in the following lemma.",
"Lemma 4.4 Suppose that $F \\in \\operatorname{L}^2\\left([0,T],\\operatorname{L}^{2} (\\Omega )\\right)$ , $U_0 \\in \\operatorname{L}^{2} (\\Omega )$ and $V_0 \\in \\operatorname{L}^{2} (\\Omega ).$ Then for any $i=1,\\dots ,n$ there exists a unique solution $u_i \\in \\operatorname{H}^{1}_0(\\Omega )$ to (REF ).",
"We have for $i\\ge 1$ that $\\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_i = \\left({g_{\\alpha }}\\ast \\delta (\\delta u-V_0)\\right)^{c}_i = \\left({g_{\\alpha }}\\ast \\delta ^2 u\\right)^{c}_i.$ Hence, using (REF ) and (REF ), the discrete problem (REF ) can be rewritten into the form $\\mathcal {A}\\left(u_i,\\varphi \\right) = \\mathcal {F}_i(\\varphi ) ,\\quad \\text{ for all } \\varphi \\in \\operatorname{H}^{1}_0(\\Omega ),$ where $\\mathcal {A}$ is the $\\operatorname{H}^{1}_0(\\Omega )$ -elliptic and continuous bilinear form given by $\\mathcal {A} \\left(u_i,\\varphi \\right):=\\left(\\frac{\\rho c \\tau _q^\\alpha }{\\tau } {g_{\\alpha }}(\\tau ) + a \\tau _q^\\alpha {g_{\\alpha }}(\\tau ) + \\frac{\\rho c}{\\tau }\\right) \\left(u_i,\\varphi \\right) + \\mathcal {L}\\left(u_i,\\varphi \\right),$ and $\\mathcal {F}_i(\\varphi ):= \\left(F_i,\\varphi \\right) + \\rho c \\tau _q^\\alpha {g_{\\alpha }}(\\tau ) \\left(\\frac{u_{i-1}}{\\tau } +\\delta u_{i-1},\\varphi \\right)+\\frac{\\rho c}{\\tau } \\left(u_{i-1},\\varphi \\right) \\\\- \\rho c \\tau _q^\\alpha \\sum _{k=1}^{i-1}{g_{\\alpha }}(t_{i+1-k}) \\left(\\delta ^2 u_k,\\varphi \\right)\\tau + a\\tau _q^\\alpha {g_{\\alpha }}(\\tau ) \\left(u_{i-1},\\varphi \\right)\\\\ - a\\tau _q^\\alpha \\sum _{k=1}^{i-1} {g_{\\alpha }}(t_{i+1-k}) \\left(\\delta u_k,\\varphi \\right)\\tau ,$ where in case $i = 1$ the summations yield no contribution.",
"The boundedness of $\\mathcal {F}_i$ follows inductively under the assumptions $F \\in \\operatorname{L}^{2}\\left((0,T),{\\operatorname{L}^{2} (\\Omega )}\\right)$ and $U_0 \\in \\operatorname{H}^{1}_0(\\Omega ), V_0 \\in \\operatorname{L}^{2} (\\Omega ).$ An application of the Lax-Milgram lemma [47] consecutively gives the existence and uniqueness of $u_i \\in \\operatorname{H}^{1}_0(\\Omega ), i =1,\\dots ,n$ to the discrete problem (REF )."
],
[
"A priori estimates",
"Our next goal is to derive estimates on the discrete solutions $u_i$ and various related objects.",
"These estimates will be crucial in the convergence study in the following subsection.",
"Lemma 4.5 Suppose that $F \\in \\operatorname{L}^2\\left([0,T],\\operatorname{L}^{2} (\\Omega )\\right)$ , $U_0 \\in \\operatorname{H}^{1}_0(\\Omega )$ and $V_0 \\in \\operatorname{L}^{2} (\\Omega ).$ Then, there exist a positive constant $C$ such that for every $j = 1,\\dots ,n$ the following estimate holds $\\left( {g_{\\alpha }}\\ast \\left\\Vert \\delta u-V_0 \\right\\Vert ^2\\right)^{c}_j + \\sum _{i =1}^j \\left\\Vert \\delta u_i \\right\\Vert ^2 \\tau + \\left\\Vert u_j \\right\\Vert _{\\operatorname{H}^{1}(\\Omega )}^2 + \\sum _{i=1}^{j} \\left\\Vert u_i - u_{i-1} \\right\\Vert _{\\operatorname{H}^{1}(\\Omega )}^2 \\le C.$ Take $\\varphi = \\delta u_i \\tau $ in (REF ) and sum the resulting equations for $i = 1,\\dots ,j$ with $1 \\le j \\le n,$ we obtain that $\\rho c \\tau _q^\\alpha \\sum _{i =1}^{j} \\left(\\delta \\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_i,\\delta u_i\\right)\\tau + a \\tau _q^\\alpha \\sum _{i =1}^{j} \\left(\\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_i,\\delta u_i\\right)\\tau \\\\ + \\rho c \\sum _{i =1}^{j} \\left\\Vert \\delta u_i \\right\\Vert ^2 \\tau + \\sum _{i=1}^{j} \\mathcal {L}\\left(u_i,\\delta u_i\\right)\\tau = \\sum _{i=1}^{j} \\left(F_i,\\delta u_i\\right)\\tau .$ The proof now proceeds by handling each term of the above expression.",
"The first convolution term on the left-hand side of (REF ) can be rewritten as $\\sum _{i =1}^{j} \\left(\\delta \\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_i,\\delta u_i\\right)\\tau = \\sum _{i =1}^{j} \\left(\\delta \\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_i,\\delta u_i-V_0\\right)\\tau \\\\+ \\left(\\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_j,V_0\\right).$ The first term on the right-hand side of (REF ) is bounded from below by $\\sum _{i =1}^{j} \\left(\\delta \\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_i,\\delta u_i-V_0\\right)\\tau \\ge \\frac{1}{2} \\left({g_{\\alpha }}\\ast \\left\\Vert \\delta u -V_0 \\right\\Vert ^2\\right)^{c}_j,$ as an application of Lemma REF shows.",
"The last term of (REF ) is moved to the right-hand side of (REF ), where it is bounded by the Cauchy Schwarz and $\\varepsilon $ -Young inequality as follows $\\left|\\left(\\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_j,V_0\\right) \\right|\\le C_{\\varepsilon _1} \\left\\Vert V_0 \\right\\Vert ^2 + \\varepsilon _1 \\left\\Vert \\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_j \\right\\Vert ^2.$ Since $\\sum _{l=1}^j (g_\\alpha )_{j-l+1}\\tau = \\sum _{l=1}^j g_\\alpha (t_j-t_{l-1})\\tau \\le \\int _0^{t_j}g_\\alpha (t_j-s) \\operatorname{d\\hspace{-1.5pt}}s\\le \\left\\Vert {g_{\\alpha }} \\right\\Vert _{\\operatorname{L}^1(0,T)},$ we obtain that $\\left|\\left(\\left({g_{\\alpha }}\\ast \\left( \\delta u-V_0\\right)\\right)^{c}_j,V_0\\right) \\right|\\le C_{\\varepsilon _1} + \\varepsilon _1 \\left({g_{\\alpha }}\\ast \\left\\Vert \\delta u -V_0 \\right\\Vert ^2\\right)^{c}_j,$ where the constant $C_{\\varepsilon _1}$ only depends on $\\left\\Vert V_0 \\right\\Vert .$ By [32], we have that $\\sum _{i=1}^{j} \\left(\\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_i,\\delta u_i\\right)\\tau \\ge 0.$ For the term involving the bilinear form $\\mathcal {L}$ , we use that $\\mbox{${k}$}$ is uniformly elliptic and symmetric.",
"Lemma REF shows that $\\sum _{i=1}^{j} \\left(\\mbox{${k}$} \\nabla u_i,\\delta \\nabla u_i\\right)\\tau &= \\frac{1}{2} \\left(\\mbox{${k}$} \\nabla u_j,\\nabla u_j\\right) - \\frac{1}{2} \\left(\\mbox{${k}$} \\nabla U_0,\\nabla U_0\\right)\\nonumber \\\\ & \\quad + \\frac{1}{2} \\sum _{i=1}^{j} \\left(\\mbox{${k}$}\\left(\\nabla u_i - \\nabla u_{i-1}\\right),\\nabla u_i-\\nabla u_{i-1}\\right) \\nonumber \\\\& \\ge \\frac{\\widetilde{k}}{2} \\left\\Vert \\nabla u_j \\right\\Vert ^2 -C + \\frac{\\widetilde{k}}{2} \\sum _{i\\le j} \\left\\Vert \\nabla u_i - \\nabla u_{i-1} \\right\\Vert ^2$ and (as $a\\ge 0$ ) $a \\sum _{i=1}^{j}\\left(u_i,\\delta u_i\\right)\\tau = \\frac{a}{2}\\left(\\left\\Vert u_j \\right\\Vert ^2 - \\left\\Vert U_0 \\right\\Vert ^2 + \\sum _{i=1}^{j} \\left\\Vert u_i- u_{i-1} \\right\\Vert ^2 \\right) \\ge -\\frac{a}{2} \\left\\Vert U_0 \\right\\Vert ^2,$ where $C$ depends on $\\left\\Vert \\mbox{${k}$} \\right\\Vert _{{\\bf L}^{\\infty } (\\Omega )}$ and $\\left\\Vert U_0 \\right\\Vert _{\\operatorname{H}^{1}(\\Omega )}.$ The right-hand side of (REF ) is estimated by use of the $\\varepsilon $ -Young inequality as $\\left|\\sum _{i=1}^{j} \\left(F_i,\\delta u_i\\right) \\tau \\right| \\le C_{\\varepsilon _2} + \\varepsilon _2 \\sum _{i=1}^{j} \\left\\Vert \\delta u_i \\right\\Vert ^2 \\tau ,$ where $C_{\\varepsilon _2}$ only depends on $\\left\\Vert F \\right\\Vert _{\\operatorname{L}^2\\left([0,T],\\operatorname{L}^{2} (\\Omega )\\right)}.$ Summarising the above results, we obtain the estimate $\\rho c \\tau _q^\\alpha \\left( \\frac{1}{2} -\\varepsilon _1\\right) \\left( {g_{\\alpha }}\\ast \\left\\Vert \\delta u-V_0 \\right\\Vert ^2\\right)^{c}_j + \\left(\\rho c -\\varepsilon _2\\right) \\sum _{i=1}^{j} \\left\\Vert \\delta u_i \\right\\Vert ^2 \\tau \\\\+ \\frac{\\widetilde{k}}{2} \\left\\Vert \\nabla u_j \\right\\Vert ^2 + \\frac{\\widetilde{k}}{2} \\sum _{i=1}^{j} \\left\\Vert \\nabla u_i - \\nabla u_{i-1} \\right\\Vert ^2\\le C + C_{\\varepsilon _1} + C_{\\varepsilon _2}.$ Fixing $\\varepsilon _1$ and $\\varepsilon _2$ sufficiently small and applying the Friedrichs' inequality yield the stated result.",
"Lemma 4.6 Let the assumptions of Lemma REF be fulfilled.",
"Then, there exists a positive constant $C$ such that $\\sum _{j=1}^n \\left\\Vert \\left({g_{\\alpha }}\\ast \\delta u \\right)^{c}_j \\right\\Vert ^2 \\tau \\le C.$ Note that by the discrete Young inequality, we have that $\\sum _{j=1}^n \\left\\Vert \\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_j \\right\\Vert ^2 \\tau & = \\int _\\Omega \\sum _{j=1}^n \\left|\\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_j(\\mathbf {x}) \\right|^2 \\tau \\operatorname{d\\hspace{-1.5pt}}\\mathbf {x}\\\\& \\le \\left(\\sum _{j=0}^{n-1} \\left({g_{\\alpha }}\\right)_{j+1} \\tau \\right)^2 \\sum _{j=1}^n \\left\\Vert \\delta u_j \\right\\Vert ^2\\tau \\\\&\\le C \\left\\Vert {g_{\\alpha }} \\right\\Vert ^2_{\\operatorname{L}^1(0,T)}$ by means of Lemma REF .",
"Lemma 4.7 Let the assumptions of Lemma REF be satisfied.",
"Then, there exists a positive constant $C$ such that $\\sum _{j=1}^n \\left\\Vert \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_j \\right\\Vert _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast }^2 \\tau \\le C.$ Note that $\\left\\Vert \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_j \\right\\Vert _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast } &= \\sup \\limits _{\\left\\Vert \\varphi \\right\\Vert _{\\operatorname{H}^{1}_0(\\Omega )}= 1} \\left|\\left\\langle \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_j,\\varphi \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )} \\right|\\nonumber \\\\&=\\frac{1}{\\rho c \\tau _q^\\alpha } \\sup \\limits _{\\left\\Vert \\varphi \\right\\Vert _{\\operatorname{H}^{1}_0(\\Omega )}=1} \\left|\\left(F_j,\\varphi \\right) - \\mathcal {L}\\left(u_j,\\varphi \\right) \\right.\\nonumber \\\\& \\qquad \\qquad \\qquad - \\rho c \\left(\\delta u_j,\\varphi \\right) -a\\tau _q^\\alpha \\left.\\left(\\left({g_{\\alpha }}\\ast \\delta u\\right)_j,\\varphi \\right)\\right|\\nonumber \\\\& \\le C \\left(\\left\\Vert F_j \\right\\Vert + \\left\\Vert u_j \\right\\Vert _{\\operatorname{H}^{1}(\\Omega )} + \\left\\Vert \\delta u_j \\right\\Vert + \\left\\Vert \\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_j \\right\\Vert \\right).$ Applying the rule $(a+b)^2 \\le 2a^2 + 2b^2$ yields $\\sum _{j = 1}^n \\left\\Vert \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_j \\right\\Vert ^2 \\tau \\\\ \\le C \\left(\\sum _{j=1}^n \\left\\Vert F_j \\right\\Vert ^2\\tau + \\sum _{j=1}^n \\left\\Vert u_j \\right\\Vert ^2_{\\operatorname{H}^{1}(\\Omega )} \\tau + \\sum _{j= 1}^n \\left\\Vert \\delta u_j \\right\\Vert ^2 \\tau + \\sum _{j=1}^n \\left\\Vert \\left({g_{\\alpha }}\\ast \\delta u\\right)^{c}_j \\right\\Vert ^2\\tau \\right),$ which is uniformly bounded by Lemma REF and REF ."
],
[
"Existence",
"The existence of a weak solution in shown in this subsection.",
"To achieve this goal, we introduce the Rothe functions $[0,T]\\rightarrow \\operatorname{L}^{2} (\\Omega )$ , which are build from the solutions $u_i, i = 0,\\dots ,n$ at each time slice as follows, $v_n \\colon t &\\mapsto {\\left\\lbrace \\begin{array}{ll}U_0, & t =0 \\\\ u_{i-1} + (t-t_{i-1}) \\delta u_i, & t \\in (t_{i-1},t_i], \\quad 1 \\le i \\le n,\\end{array}\\right.",
"}\\\\\\overline{v}_n \\colon t & \\mapsto {\\left\\lbrace \\begin{array}{ll}U_0, & t = 0 \\\\ u_i, & t \\in (t_{i-1},t_i], \\quad 1 \\le i \\le n,\\end{array}\\right.}",
"\\\\\\overline{w}_n \\colon t& \\mapsto {\\left\\lbrace \\begin{array}{ll}V_0, & t = 0 \\\\ \\delta u_i, & t\\in (t_{i-1},t_i], \\quad 1 \\le i \\le n.\\end{array}\\right.",
"}$ Note that $\\partial _tv_n = \\overline{w}_n.$ Similar notations are used for $\\overline{F}_n$ and $ \\overline{{g_{\\gamma }}}.$ Additionally, we define for two functions $\\kappa $ and $z$ [38]: $C_{\\kappa ,z}^n \\colon t &\\mapsto {\\left\\lbrace \\begin{array}{ll}0, & t = 0 \\\\\\left(\\kappa \\ast z \\right)^{c}_{i-1} + (t-t_{i-1})\\delta \\left(\\kappa \\ast z \\right)^{c}_i, & t \\in (t_{i-1}, t_i],\\end{array}\\right.",
"}\\\\\\overline{C}_{\\kappa ,z}^n \\colon t & \\mapsto {\\left\\lbrace \\begin{array}{ll}0, & t = 0 \\\\\\left(\\kappa \\ast z \\right)^{c}_{i}, & t \\in (t_{i-1},t_i].\\end{array}\\right.",
"}$ Note that for $t \\in (t_{i-1},t_i]$ , we have that $\\partial _t\\left(C^n_{\\overline{{g_{\\alpha }}}_n, \\overline{w}_n-V_0} \\right)(t) = \\delta \\left(\\overline{{g_{\\alpha }}}_n \\ast (\\overline{w}_n-V_0)\\right)^{c}_i = \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_i$ as $\\overline{w}_n(t_k) = \\delta u_k.$ Moreover, we have that $\\left(\\overline{{g_{\\alpha }}}_{n} \\ast \\overline{w}_{n}\\right)(\\lceil t\\rceil _\\tau ) = \\int _0^{t_i} \\overline{{g_{\\alpha }}}_{n}(t_i-s) \\overline{w}_{n}(s)\\operatorname{d\\hspace{-1.5pt}}s& = \\sum _{j = 1}^i \\int _{t_{j-1}}^{t_j} \\overline{{g_{\\alpha }}}_{n}(t_i - s)\\delta u_j \\operatorname{d\\hspace{-1.5pt}}s\\\\ & = \\sum _{j=1}^{i } {g_{\\alpha }}(t_{i-j+1}) \\delta u_j \\tau $ as for $s\\in (t_{j-1},t_j]$ , we have $t_i - s \\in [t_{i-j}, t_{i-j+1})$ and so $\\overline{{g_{\\alpha }}}_{n}(t_i-s) = {g_{\\alpha }}(t_{i-j+1}).$ Now, we extend (REF ) to the whole time frame.",
"In terms of the Rothe functions, the discrete variational formulation now reads as $\\rho c \\tau _q^\\alpha \\left(\\partial _t\\left(C^n_{\\overline{{g_{\\alpha }}}_n, \\overline{w}_n-V_0}\\right)(t) ,\\varphi \\right) + a\\tau _q^\\alpha \\left((\\overline{{g_{\\alpha }}}_n \\ast \\overline{w}_n)(\\lceil t\\rceil _{\\tau }) ,\\varphi \\right) \\\\ + \\rho c \\left(\\partial _tv_n(t),\\varphi \\right) + \\mathcal {L}\\left(\\overline{v}_n(t),\\varphi \\right) = \\left(\\overline{F}_n(t),\\varphi \\right), \\quad \\forall \\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$ In the remainder of this subsection, we show that a subsequence of the Rothe functions converges to a weak solution.",
"The following subsection is then devoted to proving the uniqueness of a solution.",
"Theorem 4.1 (Existence (SPL)) Suppose that $F \\in \\operatorname{L}^2\\left([0,T],\\operatorname{L}^{2} (\\Omega )\\right)$ , $U_0 \\in \\operatorname{H}^{1}_0(\\Omega )$ and $V_0 \\in \\operatorname{L}^{2} (\\Omega ).$ Then, there exists a weak solution $u$ to (REF ) with $u \\in \\operatorname{L}^{\\infty }\\left((0,T),\\operatorname{H}^{1}_0(\\Omega )\\right)\\cap \\operatorname{C}\\left(I,\\operatorname{L}^{2} (\\Omega )\\right)$ for which $\\partial _tu \\in \\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ and $\\partial _t\\left({g_{\\alpha }}\\ast \\left(\\partial _tu - V_0 \\right)\\right) \\in \\operatorname{L}^{2}\\left((0,T),{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\right).$ The Rellich-Kondrachov theorem [23] gives us the compact embedding of $\\operatorname{H}^{1}_0(\\Omega )$ in $\\operatorname{L}^{2} (\\Omega ).$ The estimates from Lemma REF show in particular that $\\left\\Vert \\overline{v}_n(t) \\right\\Vert _{\\operatorname{H}^{1}_0(\\Omega )} \\le C, \\quad \\forall t \\in [0,T],$ and $\\int _0^T \\left\\Vert \\partial _tv_n(t) \\right\\Vert _{\\operatorname{L}^{2} (\\Omega )}^2 \\operatorname{d\\hspace{-1.5pt}}t= \\sum _{i=1}^n \\left\\Vert \\delta u_i \\right\\Vert _{\\operatorname{L}^{2} (\\Omega )}^2 \\tau \\le C.$ Hence, by the Aubin-Lions lemma [7], we find a function $v \\in \\operatorname{C}\\left(I,\\operatorname{L}^{2} (\\Omega )\\right) \\cap \\operatorname{L}^{\\infty }\\left((0,T),\\operatorname{H}^{1}_0(\\Omega )\\right)$ with $\\partial _tv \\in \\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ and a corresponding subsequence $\\lbrace v_{n_k}\\rbrace _{k \\in {\\mathbb {N}}}$ of $\\lbrace v_n\\rbrace _{n\\in {\\mathbb {N}}}$ having the following properties, as $k \\rightarrow \\infty ,$ $\\begin{array}{rl}v_{n_k} \\rightarrow v & \\text{ in } \\operatorname{C}\\left(I,\\operatorname{L}^{2} (\\Omega )\\right)\\\\v_{n_k}(t) \\rightharpoonup v(t) & \\text{ in } \\operatorname{H}^{1}_0(\\Omega ), \\text{ for all } t \\in [0,T]\\\\\\overline{v}_{n_k}(t) \\rightharpoonup v(t) & \\text{ in } \\operatorname{H}^{1}_0(\\Omega ), \\text{ for all } t \\in [0,T]\\\\\\partial _tv_{n_k} = \\overline{w}_{n_k} \\rightharpoonup \\partial _tv & \\text{ in } \\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right).\\end{array}$ Note that $\\int _0^T \\left\\Vert \\nabla v_{n_k}(t) - \\nabla \\overline{v}_{n_k}(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t& = \\sum _{i=1}^{n_k} \\int _{t_{i-1}}^{t_i} \\left\\Vert (t-t_i)\\delta \\nabla u_i \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t\\\\&= \\frac{\\tau _{n_k}}{3} \\sum _{i=1}^{n_k} \\left\\Vert \\nabla \\delta u_i \\right\\Vert ^2 \\tau _{n_k}^2 \\\\& = \\frac{\\tau _{n_k}}{3} \\sum _{i=1}^{n_k} \\left\\Vert \\nabla u_i - \\nabla u_{i-1} \\right\\Vert ^2 \\le C \\tau _{n_k},$ where we have set $\\tau _{n_k} = T/n_k.$ As a result, $\\lbrace v_{n_k}\\rbrace _{k\\in {\\mathbb {N}}}$ and $\\lbrace \\overline{v}_{n_k}\\rbrace _{k\\in {\\mathbb {N}}}$ have the same limit $v$ in $\\operatorname{L}^{2}\\left((0,T),\\operatorname{H}^{1}_0(\\Omega )\\right).$ Next, we integrate (REF ) over the interval $(0,\\eta ) \\subset (0,T)$ to arrive at $\\rho c \\tau _q^\\alpha \\left( C^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k},\\overline{w}_{n_k}-V_0}(\\eta ),\\varphi \\right) + a\\tau _q^\\alpha \\int _0^\\eta \\left((\\overline{{g_{\\alpha }}}_{n_k} \\ast \\overline{w}_{n_k})(\\lceil t\\rceil _{\\tau }) ,\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\\\ + \\rho c \\int _0^\\eta \\left(\\partial _tv_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t+ \\int _0^\\eta \\mathcal {L}\\left(\\overline{v}_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t= \\int _0^\\eta \\left(\\overline{F}_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t.$ Now, we will first discuss some convergence results.",
"Using Lemma REF , we can estimate $&\\left| \\int _0^T \\left(C^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k}, \\overline{w}_{n_k}-V_0}(t) - \\overline{C}^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k},\\overline{w}_{n_k}-V_0}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t \\right| \\\\&= \\sum _{i = 1}^{n_k} \\int _{t_{i-1}}^{t_i} \\left|\\left((t-t_i)\\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_i,\\varphi \\right) \\right| \\operatorname{d\\hspace{-1.5pt}}t\\\\&\\le \\sum _{i=1}^{n_k} \\tau _{n_k}^2\\left|\\left( \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_i,\\varphi \\right) \\right| \\operatorname{d\\hspace{-1.5pt}}t\\\\& \\le C\\tau _{n_k}\\sqrt{\\sum _{i=1}^{n_k} \\left\\Vert \\delta \\left({g_{\\alpha }}\\ast (\\delta u-V_0)\\right)^{c}_i \\right\\Vert _{{\\operatorname{H}^{1}(\\Omega )}^{*}}^2 \\tau _{n_k}} \\\\& \\le C \\tau _{n_k} \\rightarrow 0 \\quad \\text{ as } k \\rightarrow \\infty ,$ We note that $\\overline{C}^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k}, \\overline{w}_{n_k}-V_0}(t) = \\sum _{j=1}^i {g_{\\alpha }}(t_{i+1-j}) (\\delta u_j-V_0) \\tau _{n_k} = \\left(\\overline{{g_{\\alpha }}}_{n_k} \\ast (\\overline{w}_{n_k}-V_0)\\right) \\left(\\lceil t\\rceil _\\tau \\right).$ The same arguments as in [40] yield the limit transition $\\left| \\int _0^\\xi \\left(\\overline{C}^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k}, \\overline{w}_{n_k}-V_0}(\\eta ) - \\left({g_{\\alpha }}\\ast (\\overline{w}_{n_k}-V_0)\\right)(\\eta ),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}\\eta \\right| \\rightarrow 0, \\quad \\text{ as } k \\rightarrow \\infty ,$ for any $\\xi \\in (0,T].$ It is here used that $\\overline{{g_{\\alpha }}}_{n_k} \\rightarrow {g_{\\alpha }}$ pointwise in $(0,T)$ and that the estimate $\\left( {g_{\\alpha }}\\ast \\left\\Vert \\delta u-V_0 \\right\\Vert ^2\\right)^{c}_j + \\sum _{i=1}^j \\left\\Vert \\delta u_i \\right\\Vert ^2\\tau _{n_k} \\le C$ is available (see Lemma REF ).",
"The operator $w \\mapsto \\int _0^\\eta \\left(\\left({g_{\\gamma }}\\ast w\\right)(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t,$ with $\\gamma \\in (0,1)$ and $\\eta \\in (0,T)$ fixed, is a bounded linear functional on $\\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ since by Young's inequality for convolutions it holds that $\\left|\\int _0^\\eta \\left(\\left(g_{\\gamma } \\ast w\\right)(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t \\right| \\le \\sqrt{T} \\left\\Vert g_{\\gamma } \\right\\Vert _{\\operatorname{L}^1(0,T)} \\left\\Vert \\varphi \\right\\Vert \\left\\Vert w \\right\\Vert _{\\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)}.$ Therefore, using $\\overline{w}_{n_k} \\rightharpoonup \\partial _tv$ in $\\operatorname{L}^2\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ it finally holds that $\\left| \\int _0^\\xi \\left({C}^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k}, \\overline{w}_{n_k}-V_0}(\\eta ) - \\left({g_{\\alpha }}\\ast (\\partial _tv-V_0)\\right)(\\eta ),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}\\eta \\right| \\rightarrow 0, \\quad \\text{ as } k \\rightarrow \\infty ,$ for any $\\xi \\in (0,T).$ Next, we note that $\\overline{C}^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k}, \\overline{w}_{n_k}}(t) = \\left(\\overline{{g_{\\alpha }}}_{n_k} \\ast \\overline{w}_{n_k}\\right)(\\lceil t\\rceil _{\\tau _{n_k}})$ Hence, we immediately see that $\\left|\\int _0^\\xi \\left(\\overline{C}^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k}, \\overline{w}_{n_k}}(t) - \\left( {g_{\\alpha }}\\ast \\partial _tv \\right)(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t \\right| \\rightarrow 0 \\quad \\text{ as } k\\rightarrow \\infty .$ Furthermore, we have that $\\left|\\int _0^\\eta \\mathcal {L}\\left(\\overline{v}_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t- \\int _0^\\eta \\mathcal {L}\\left(v(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t \\right| &\\rightarrow 0 \\quad \\text{ as } k \\rightarrow \\infty , \\\\\\left|\\int _0^\\eta \\left(\\overline{F}_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t- \\int _0^\\eta \\left(F(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t \\right|& \\rightarrow 0 \\quad \\text{ as } k \\rightarrow \\infty .$ Before making the limit transition, we need to integrate (REF ) in time over $\\eta \\in (0,\\xi ) \\subset (0,T)$ .",
"We obtain that $\\rho c \\tau _q^\\alpha \\int _0^\\xi \\left( C^{n_k}_{\\overline{{g_{\\alpha }}}_{n_k},\\overline{w}_{n_k}-V_0}(\\eta ),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}\\eta + a\\tau _q^\\alpha \\int _0^\\xi \\int _0^\\eta \\left((\\overline{{g_{\\alpha }}}_{n_k} \\ast \\overline{w}_{n_k})(\\lceil t\\rceil _{\\tau }) ,\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta \\\\ + \\rho c \\int _0^\\xi \\int _0^\\eta \\left(\\partial _tv_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta + \\int _0^\\xi \\int _0^\\eta \\mathcal {L}\\left(\\overline{v}_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta \\\\= \\int _0^\\xi \\int _0^\\eta \\left(\\overline{F}_{n_k}(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta .$ Using the previously obtained results, we can make the limit transition in (REF ) to receive that $\\rho c \\tau _q^\\alpha \\int _0^\\xi \\left(\\left({g_{\\alpha }}\\ast (\\partial _tv-V_0)\\right)(\\eta ),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}\\eta + a\\tau _q^\\alpha \\int _0^\\xi \\int _0^\\eta \\left(\\left({g_{\\alpha }}\\ast \\partial _tv\\right)(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta \\\\+ \\rho c \\int _0^\\xi \\int _0^\\eta \\left(\\partial _tv(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta + \\int _0^\\xi \\int _0^\\eta \\mathcal {L}\\left(v(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta \\\\ = \\int _0^\\xi \\int _0^\\eta \\left(F(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\operatorname{d\\hspace{-1.5pt}}\\eta .$ Differentiation of (REF ) with respect to $\\xi $ yields $\\rho c \\tau _q^\\alpha \\left(\\left({g_{\\alpha }}\\ast (\\partial _tv-V_0)\\right)(\\xi ),\\varphi \\right)+a \\tau _q^\\alpha \\int _0^\\xi \\left(\\left({g_{\\alpha }}\\ast \\partial _tv)\\right)(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t\\\\ + \\rho c \\int _0^\\xi \\left(\\partial _tv(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t+ \\int _0^\\xi \\mathcal {L}\\left(v(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t= \\int _0^\\xi \\left(F(t),\\varphi \\right) \\operatorname{d\\hspace{-1.5pt}}t.$ From (REF ), it follows that $\\lim _{\\xi \\searrow 0} \\left(\\left({g_{\\alpha }}\\ast (\\partial _tv - V_0)\\right)(\\xi ),\\varphi \\right) = 0.",
"$ Hence, differentiating (REF ) again w.r.t.",
"$\\xi $ we obtain that $\\rho c \\tau _q^\\alpha \\left\\langle \\partial _t\\left({g_{\\alpha }}\\ast \\left(\\partial _tv- V_0\\right)\\right)(\\xi ),\\varphi \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )} + a\\tau _q^\\alpha \\left(\\left({g_{\\alpha }}\\ast \\partial _tv\\right)(\\xi ),\\varphi \\right) \\\\ + \\rho c \\left(\\partial _tv(\\xi ),\\varphi \\right) + \\mathcal {L}\\left(v(\\xi ),\\varphi \\right) = \\left(F(\\xi ),\\varphi \\right).$ Therefore, $v$ satisfies equation (REF )."
],
[
"Uniqueness",
"In this subsection, we show the uniqueness of the solution to problem (REF ) under the additional assumption that $\\partial _tu \\in \\operatorname{L}^{2}\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ satisfies $\\left\\Vert \\partial _tu(t) \\right\\Vert \\le C \\left(1+t^{\\frac{\\alpha -1}{2}}\\right)$ This assumption is reasonable considering the regularity of the solution to the SPL problem in one dimension, see Subsection .",
"Theorem 4.2 (Uniqueness SPL) Suppose that $F \\in \\operatorname{L}^2\\left([0,T],\\operatorname{L}^{2} (\\Omega )\\right)$ , $U_0 \\in \\operatorname{H}^{1}_0(\\Omega )$ and $V_0 \\in \\operatorname{L}^{2} (\\Omega ).$ Then, the weak solution to problem (REF ) satisfying $u \\in \\operatorname{L}^{\\infty }\\left((0,T),\\operatorname{H}^{1}_0(\\Omega )\\right)\\cap \\operatorname{C}\\left(I,\\operatorname{L}^{2} (\\Omega )\\right)$ , $\\partial _tu \\in \\operatorname{L}^{2}\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ with $\\left\\Vert \\partial _tu(t) \\right\\Vert \\le C \\left(1+t^{\\frac{\\alpha -1}{2}}\\right),$ and $\\partial _t\\left({g_{\\alpha }}\\ast \\left(\\partial _tu - V_0 \\right)\\right) \\in \\operatorname{L}^{2}\\left((0,T),{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\right)$ is unique.",
"Let $u_1$ and $u_2$ be solutions to the problem (REF ) and consider their difference $u = u_1 - u_2.$ Then $u$ is a solution to the homogeneous problem (i.e.",
"$F = 0$ and $U_0 = 0 = V_0$ ) with vanishing Dirichlet boundary conditions.",
"By Lemma REF , the function $u$ satisfies $\\rho c\\tau _q^\\alpha \\left\\langle \\partial _t\\left({g_{\\alpha }}\\ast \\partial _tu\\right)(t),\\varphi \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )} + a \\tau _q^\\alpha \\left(\\partial _t\\left({g_{\\alpha }}\\ast u \\right)(t),\\varphi \\right) \\\\ + \\rho c \\left(\\partial _tu(t),\\varphi \\right) + \\mathcal {L}\\left(u(t),\\varphi \\right) = 0,$ for all $\\varphi \\in \\operatorname{H}^{1}_0(\\Omega ).$ Note that we can not choose $\\varphi = \\partial _tu(t)$ as $\\partial _tu(t) \\notin \\operatorname{H}^{1}_0(\\Omega ).$ We use a method due to Ladyzhenskaya (see [13] and [37]).",
"Fix $s\\in (0,T)$ and let $\\varphi = v(t)$ where $v(t)$ is given by $v(t) = {\\left\\lbrace \\begin{array}{ll}\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau & \\text{ if } 0 \\le t \\le s \\\\0 &\\text{ otherwise}.\\end{array}\\right.",
"}$ The crucial property of $v$ is that $v^\\prime (t) = -u(t)$ if $t\\le s.$ We substitute $\\varphi = v(t)$ in (REF ) and integrate $t$ over $(0,s)$ to obtain that $\\rho c \\tau _q^\\alpha \\int _0^s \\left\\langle \\partial _t\\left(g_{\\alpha }\\ast \\partial _tu\\right)(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )} \\operatorname{d\\hspace{-1.5pt}}t\\\\ + a\\tau _q^\\alpha \\int _0^s \\left(\\partial _t\\left(g_{\\alpha } \\ast u \\right)(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t+ \\rho c \\int _0^s\\left(\\partial _tu(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t\\\\ + \\int _0^s \\mathcal {L}\\left(u(t), \\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) = 0.$ The additional assumption on $\\partial _tu$ implies that $\\lim \\limits _{t\\underset{>}{\\rightarrow }0} \\left\\Vert \\left({g_{\\alpha }}\\ast \\partial _tu\\right)(t) \\right\\Vert \\le \\lim \\limits _{t\\underset{>}{\\rightarrow }0} \\left({g_{\\alpha }}\\ast \\left\\Vert \\partial _tu \\right\\Vert \\right)(t) & \\le \\lim \\limits _{t\\underset{>}{\\rightarrow }0} C\\int _0^t (t-s)^{-\\alpha } \\left(1+s^{\\frac{\\alpha -1}{2}}\\right) \\operatorname{d\\hspace{-1.5pt}}s\\\\& \\le \\lim \\limits _{t\\underset{>}{\\rightarrow }0} C \\left(t^{1-\\alpha } + t^{\\frac{1-\\alpha }{2}} \\right) = 0.$ Hence, partial integration in the first fractional term yields $& \\int _0^s \\left\\langle \\partial _t\\left({g_{\\alpha }}\\ast \\partial _tu\\right)(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right\\rangle _{{\\operatorname{H}^{1}_0(\\Omega )}^\\ast \\times \\operatorname{H}^{1}_0(\\Omega )} \\operatorname{d\\hspace{-1.5pt}}t\\\\& \\quad = \\left.\\left(\\left({g_{\\alpha }}\\ast \\partial _tu\\right)(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right)\\right|_{t=0}^{t=s} + \\int _0^s \\left(\\left({g_{\\alpha }}\\ast \\partial _tu\\right)(t),u(t)\\right) \\operatorname{d\\hspace{-1.5pt}}t\\\\& \\quad = \\int _0^s \\left(\\partial _t\\left({g_{\\alpha }}\\ast u\\right)(t),u(t)\\right) \\operatorname{d\\hspace{-1.5pt}}t\\\\& \\quad \\ge \\frac{T^{-\\alpha }}{2\\Gamma (1-\\alpha )}\\int _0^s \\left\\Vert u(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t,$ where the last inequality follows from Lemma REF (v).",
"Analogously, for the second fractional term, partial integration shows that $\\int _0^s \\left(\\partial _t\\left({g_{\\alpha }}\\ast u\\right)(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t= \\int _0^s \\left(\\left({g_{\\alpha }}\\ast u\\right)(t),u(t)\\right) \\operatorname{d\\hspace{-1.5pt}}t\\ge 0,$ which is positive by Lemma REF (iv).",
"Since $\\frac{\\mathrm {d}}{ \\operatorname{d\\hspace{-1.5pt}}t} \\left(u(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) &= \\left(\\partial _tu(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) + \\left(u(t),\\partial _t\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right)\\\\ &= \\left(\\partial _tu(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) - \\left\\Vert u(t) \\right\\Vert ^2,$ we find that $\\int _0^s \\left(\\partial _tu(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t&= \\int _0^s \\frac{\\mathrm {d}}{ \\operatorname{d\\hspace{-1.5pt}}t} \\left(u(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t+ \\int _0^s \\left\\Vert u(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t\\\\ &=- \\left(u(0),\\int _0^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) + \\int _0^s \\left\\Vert u(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t,$ and therefore $\\rho c \\int _0^s \\left(\\partial _tu(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t= \\rho c \\int _0^s \\left\\Vert u(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t.$ Next, we have that $a \\int _0^s \\left(u(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t&= -a \\int _0^s \\left(-u(t),\\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\operatorname{d\\hspace{-1.5pt}}t\\\\ & = -\\frac{a}{2}\\int _0^s \\partial _t\\left\\Vert \\int _t^s u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t\\\\ & = \\frac{a}{2} \\left\\Vert \\int _0^s u(t) \\operatorname{d\\hspace{-1.5pt}}t \\right\\Vert ^2 \\ge 0.$ Similarly, by the symmetry of $\\mbox{${k}$}$ , we obtain that $\\int _0^s \\left(\\mbox{${k}$} \\nabla u(t),\\int _t^s \\nabla u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) &= -\\int _0^s \\left(-\\mbox{${k}$} \\nabla u(t),\\int _t^s \\nabla u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right)\\\\ &= -\\frac{1}{2} \\int _0^s \\partial _t\\left(\\mbox{${k}$} \\int _t^s \\nabla u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau ,\\int _t^s \\nabla u(\\tau )\\operatorname{d\\hspace{-1.5pt}}\\tau \\right) \\\\ & \\ge \\frac{\\widetilde{k}}{2} \\left\\Vert \\int _0^s\\nabla u(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t.$ Summarising, we obtain the inequality $\\rho c \\left(1 + \\tau _q^\\alpha \\frac{T^{-\\alpha }}{2\\Gamma (1-\\alpha )}\\right) \\int _0^s \\left\\Vert u(t) \\right\\Vert ^2 \\operatorname{d\\hspace{-1.5pt}}t+ \\frac{\\widetilde{k}}{2} \\left\\Vert \\int _0^s \\nabla u(t) \\operatorname{d\\hspace{-1.5pt}}t \\right\\Vert ^2 \\le 0.$ Therefore, $u= 0$ a.e.",
"in $Q_T$ , which shows that $u_1 = u_2$ a.e.",
"in $Q_T$ .",
"Remark 4.3 From theoretical viewpoint, Theorem REF stays valid when the condition on $\\left\\Vert \\partial _tu(t) \\right\\Vert $ is replaced by $\\left\\Vert \\partial _tu(t) \\right\\Vert \\le C \\left(1+t^{-\\beta }\\right),\\quad 0 < \\beta < \\min \\left\\lbrace \\frac{1}{2},1-\\alpha \\right\\rbrace .$"
],
[
"Conclusion",
"In this article, we have investigated the existence and uniqueness of a weak solution to the fractional single-phase-lag heat equation.",
"Rothe's method has been employed to show the existence of a solution satisfying $u \\in \\operatorname{L}^{\\infty }\\left((0,T),\\operatorname{H}^{1}_0(\\Omega )\\right)\\cap \\operatorname{C}\\left(I,\\operatorname{L}^{2} (\\Omega )\\right)$ and $\\partial _tu \\in \\operatorname{L}^{2}\\left((0,T),\\operatorname{L}^{2} (\\Omega )\\right)$ .",
"Under the additional assumption $ \\left\\Vert \\partial _tu(t) \\right\\Vert \\le C \\left(1+t^{\\frac{\\alpha -1}{2}}\\right)$ , the uniqueness of a solution has been shown by contradiction.",
"Moreover, we have derived an explicit solution to the one-dimensional problem.",
"Bounds on the solution and its time derivative have been obtained by generalising the uniform boundedness property of the multinomial Mittag-Leffler function.",
"Future work will concern the existence of a unique solution to the DPL model corresponding with (REF )."
]
] | 2210.07794 |
[
[
"i13DR: A Real-Time Demand Response Infrastructure for Integrating\n Renewable Energy Resources"
],
[
"Abstract With the ongoing integration of Renewable Energy Sources (RES), the complexity of power grids is increasing.",
"Due to the fluctuating nature of RES, ensuring the reliability of power grids can be challenging.",
"One possible approach for addressing these challenges is Demand Response (DR) which is described as matching the demand for electrical energy according to the changes and the availability of supply.",
"However, implementing a DR system to monitor and control a broad set of electrical appliances in real-time introduces several new complications, including ensuring the reliability and financial feasibility of the system.",
"In this work, we address these issues by designing and implementing a distributed real-time DR infrastructure for laptops, which estimates and controls the power consumption of a network of connected laptops in response to the fast, irregular changes of RES.",
"Furthermore, since our approach is entirely software-based, we dramatically reduce the initial costs of the demand side participants.",
"The result of our field experiments confirms that our system successfully schedules and executes rapid and effective DR events.",
"However, the accuracy of the estimated power consumption of all participating laptops is relatively low, directly caused by our software-based approach."
],
[
"Introduction",
"Reduction of greenhouse gasses is one of the significant concerns of the global community, where the continuous development of Renewable Energy Sources (RES) plays a crucial role [25].",
"Despite several apparent advantages, integrating RES into existing transmission and distribution grids is challenging.",
"One main issue is the fluctuating nature of RES, which increases the complexity and vibrancy of the power grids [29].",
"Several works address these issues by proposing a broad range of solutions, ranging from developing more energy-efficient appliances and materials to developing Smart Grids (SG) that actively match the demand side's load with the available energy supply.",
"This concept is known as Demand Response (DR) [24], [26], [5].",
"A minimal DR infrastructure consists of a few essential components, including a power measurement and control component for estimating and managing the load on the demand side, an overlay network for connecting the demand-side participants with the electrical utilities who aggregate the power consumptions and schedule DR events to match the demand with the available supply.",
"We define the DR event as the specific schedule during which the appliance's power consumption must be managed to match the available supply.",
"Participation of many electric appliances on the demand side plays a crucial role in implementing a robust DR infrastructure.",
"However, managing many distributed devices requires a sophisticated and resilient system.",
"This complexity increases with the integration of immediate fluctuating RES.",
"Another challenge is providing the demand-side participants with attractive incentives to join the DR system.",
"The participants are required to initially pay for enabling their appliances with power measurement and control devices and communication systems.",
"This initial cost discourages several potential participants, mainly residential consumers, who might not receive a significant financial gain from participating in the DR events [1].",
"In this work, we propose a design and implementation of a distributed DS infrastructure for balancing the power consumption of a network of laptops with the intermittent supplies of RES in real time.",
"The primary reason for choosing laptops as our target appliances is the extensive prevalence of laptops in everyday use.",
"Furthermore, we use laptops' computational resources and communicational capabilities to offer an entirely software-oriented approach for performing DR tasks and events, including monitoring, estimating, and controlling the power consumption of laptops and interacting with the utilities.",
"Furthermore, we design and conduct field experiments with some participating laptops and a simulated RES to evaluate the performance of our design for performing DR events.",
"Several DR infrastructures have already been realized and implemented [12], [23], [10], but they require a significant time ahead for scheduling and performing DR events.",
"Our approach's novelty is developing a responsive DR system where DR schedules are created and executed immediately upon a request, e.g., when the production of RES suddenly drops.",
"Furthermore, since our approach is entirely software based on the demand side, we dramatically reduce the initial cost of demand-side participants to zero cents.",
"We recognize the following contributions to this work: We offer a design and implementation of a responsive DR system for matching the power consumption of several connected laptops with the fluctuating supply of RES.",
"Our systems schedule and creates DR events immediately when changes in RES occur and notify the participating devices in real time.",
"To increase the motivation of the demand-side participants to join the DR system, we offer an entirely software-based approach, which significantly reduces the initial cost to zero.",
"Our demand-side approach is capable of measuring laptops' current power consumption using regression models.",
"Our system also controls the power consumption of laptops by using the built-in functionalities of operating systems (OS).",
"The majority of the academic research on the different aspects of DR is based on mathematical simulations without any empirical measurements of real-world experiments [28], [31].",
"We offer a DR system that fills this gap and enables the researchers with a testbed for experimenting with different DR optimization and scheduling approaches.",
"We organize the rest of the paper as follows: First, in Section , we review the existing literature on DR infrastructures and power modeling.",
"Then, we analyze the realized requirements of our system and explain our implementation in Section .",
"Afterward, in Section we explain the conducted experiments to evaluate the performance of our system.",
"We discuss our findings and the limitations of our approach in Section .",
"Finally, in Section , we summarize our work and discuss future works."
],
[
"Related Work",
"RES accounts for 28% of electricity production in Germany, and it has been steadily increasing over the past two decades .",
"Nevertheless, the fluctuating nature of RES, commonly the wind and the sun, causes many concerns with the irregularity, uncertainty, and unreliability of supplies [29], [11], [18].",
"One solution for increasing the predictability of the RES is a weather forecast which is not without errors [19].",
"The typical solution is using on-site or emergency generators, which are expensive, polluting, and face many governmental restrictive regulations in many countries [24].",
"Also, since RES is often connected to the power grid at the distribution level, e.g., in Germany, 90% of installed RES is directly connected t the distribution grid, the unpredictability of the power consumption on the demand-side increases the complexity of the system.",
"Several studies confirm DR as a useful solution for addressing these issues [3], [4], [20], [30].",
"In [3], authors offer an approach for optimizing the integration of RES to the grid by using energy management systems and offering real-time pricing, where the simulation study yields good results for matching the demand with available RES.",
"In [20], [30], the authors offer scheduling and optimization approaches for maximizing the benefits of DR in the presence of volatile RES and reducing the electricity cost.",
"Their simulation study also confirms DR as a good candidate for integrating RES into the grid.",
"We extend the previous works by implementing a real-world DR infrastructure where different optimization and scheduling can be realized through field experiments.",
"Moreover, we provide a real-world evaluation of the approach, which was missing in the literature.",
"Residential and small commercial consumers account for about 40 percent of total electricity consumption in Germany , and their share will increase with the adaptation of electric vehicles [13].",
"They can contribute to a DR system by curtailing or shifting their demand loads to create a more reliable distribution and transmission grid [18], [21].",
"Reducing the initial cost of joining the DR system is a significant incentive for residential and commercial consumers.",
"Therefore, we use the built-in functionalities of consumers' laptops to reduce the initial cost.",
"We achieve this by measuring the power consumption of laptops by using mathematical power models.",
"Several studies have explored the power consumption models for server environments and data centers [7], [22], [6], [14].",
"However, we can not directly apply these works to laptops because server environments and laptops consume energy differently.",
"Studies on constructing power models for laptops and battery-operated electrical appliances are relatively limited [8], mainly because of the laptops' limited consumption in comparison to server environments.",
"According to [22], an appropriate mathematical model for estimating power consumption must satisfy several requirements.",
"These constraints include the accuracy of the power estimation, the speed of prediction, the generality of the model when applied to various systems with different hardware and software specifications, affordable, non-intrusive measuring devices, and the simplicity of the design.",
"Furthermore, [22] classifies real-time power modeling approaches into two groups: detailed analytical power modeling and high-level black-box modeling.",
"The analytical power modeling exploits the CPU performance counters for accurately estimating the power.",
"However, this approach only applies to particular processors that are not portable from one system to another.",
"The high-level black-box modeling technique uses system metrics, such as CPU, disk, and memory utilization, for constructing linear or multiple regression power estimation models.",
"This type of modeling is less accurate than analytical modeling; however, black-box modeling is more general and portable due to its independence from system specifications.",
"Furthermore, [22] constructs and evaluates five different power consumption models by using Mantis [9], a non-intrusive power modeling system.",
"Mantis uses a one-time model fitting, during which system utilization metrics are fitted to power readings of an external AC power meter.",
"Evaluations of the generated power models indicate that the utilization-based regression models perform better than other models.",
"We should mention that the discussed studies fit and generate power models using external assistance like AC power meters or a second computer.",
"In contrast, [8] introduces an approach for self-constructing power modeling systems for laptops and mobile devices.",
"This approach exploits the smart battery interface of laptops for self-power measurement.",
"Despite the accurately estimated power consumption, the constructed power models only measure the power consumption of laptops without considering the AC battery charger.",
"In contrast, measuring the power consumed by laptops' AC chargers is essential for our work.",
"In this study, we follow the same approach for collecting system utilization metrics and actual power consumption of laptops by using external power meters for creating linear regression power consumption models for laptops.",
"However, we deploy the generated models on other laptops without refitting the models to examine the model's portability.",
"We differentiate ourselves from previous works in integrating the RES with a real-time responsive DR system and with non-intrusive power consumption modeling, i.e., we only estimate the power consumption by measuring the laptop utilization metrics.",
"Finally, in contrast to other works, we conduct field experiments to evaluate the performance of implemented DR system and power models."
],
[
"Design and Implementation of 13DR",
"In the following section, we discuss the realized use cases and explain the implementation of our DR infrastructure in detail.",
"For the rest of the paper, we call our proposed DR system i13DR, where i13 refers to the location where this study has been conducted, and DR stands for Demand Response.",
"We open-sourced the code https://github.com/i13DR"
],
[
"Overview",
"The proposed i13DR design consists of two major parts, a demand-side manager application and a bundle of server-side applications that manage the DR system, as Figure REF displays.",
"For the rest of the paper, we refer to the demand-side application as i13DM, which stands for i13 Demand Manager and we call the server-side DR provider i13DRP, that stands for i13 Demand Response Provider.",
"We design and develop the i13DM to encapsulate the fundamental functionalities for performing an efficient DR event, including the features for measuring and limiting the power consumption of laptops and communicating with the i13DRP.",
"On the server side, the i13DRP consists of two primary subsystems, the DR provider and the real-time database.",
"The DR provider is responsible for managing the laptops and communicating with RES to inquire about the availability of supplies.",
"Furthermore, i13DRP performs the scheduling and execution of the DR events with the cooperation of a real-time database.",
"The real-time database persists the DR-related data and can distribute the data in real time.",
"Figure: Overview of i13DR infrastructure design."
],
[
"Requirements Analysis of i13DR",
"We realize several use cases for i13DR, and to better understand the scope of i13DR, we utilize a use case diagram to summarize the use cases as Figure REF displays.",
"A use case describes the system's functionality that the user comprehends [2].",
"Typically an actor is described as an entity interacting with the system, including a user, another system, or the system's physical environment.",
"However, to enhance the understandability of our diagram, since i13DR internally consists of two parts of i13DM and i13DRP interacting with one another, we recognize both i13DM and i13DRP as the studied systems as well as actors.",
"Figure REF shows that we have three external actors.",
"On the left, we have the DR participant who directly interacts with i13DM.",
"On the right side of the diagram is the administrator of i13DRP and the RES interacting with i13DRP, and finally, in the middle, i13DRP and i13DM communicate with each other.",
"Figure: Use case diagram of i13DR.For performing the DR-related activities, i13DR relies on two primary datasets of Location Profiles and Power Profiles of laptops which the i13DMs provide and continuously update.",
"Location profile provides the i13DRP with an approximate location of the laptop during the week, i.e., i13DRP, with some probability, knows about the laptop's location at the specific time of the week.",
"The power profile of each i13DM provides an estimate of laptops' power consumption in standard mode and power save mode during a week.",
"We define the power save mode as the time when the i13DM activates the load control mechanism on the laptop to reduce the power consumption, which we achieve by taking advantage of the power saving functionalities of OS.",
"We require managing the profiles because we assume people follow a weekly routine, e.g., the laptops are located five days a week at the users' works and two days at the users' homes.",
"Thereby, we can create context-aware DR events depending on the requirements of the grid in different locations.",
"In the following, we describe the five most important use cases in detail:"
],
[
"Perform DR Event",
"This use case explains the steps for executing a successful DR event in the i13DR system.",
"The i13DRP continuously monitors the current production of the RES.",
"When i13DRP detects a reduction of supply, i13DRP selects the candidate laptops according to existing power profiles and location profiles of laptops connected to the same power grid as RES.",
"Then i13DRP uses the scheduling algorithm (any optimization and scheduling algorithm can be plugged and play, we do not implement a specific scheduling approach for this study) to estimate the power reduction that candidate laptops can offer in a way that the estimated reduction matches the current supply.",
"Once i13DRP decides on the participating laptops, i13DRP creates a DR schedule for the event containing the start time and duration of the DR event and submits the schedule to the real-time database.",
"Finally, i13DM on the candidate laptops is immediately notified about the new schedule through the real-time database.",
"The i13DMs fetch the schedule from i13DRP, update their local schedule accordingly, and activate the power control at the appropriate time."
],
[
"Initial Profiling and Analysis of Laptop",
"This use case only executes once when i13DM is launching for the first time on the user's laptop to collect the necessary data about the laptop's specifications and create the initial location and power profiles.",
"Upon the initial startup, i13DM executes several platform-specific commands to extract the hardware and software specifications.",
"Once the analysis is over, i13DM submits the extracted data to i13DRP.",
"This data is later used for estimating power consumption.",
"Afterward, i13DM fetches the previously constructed generic power model from i13DRP for calculating the power consumption of the laptop in normal mode and power save mode.",
"Then, i13DM creates a power profile dataset containing an object for every minute of the day and every day of the week.",
"After, for every object in the dataset, i13DM estimates the normal mode and power save mode power consumptions calculated based on the power models.",
"Also, i13DM fixes the probability of the laptop running, the i13DM application running, and the laptop being connected to the AC adapter to 50%.",
"After, 13DM stores the power profile in the local database and submits the profile to the i13DRP.",
"After, i13DM uses geolocation APIs to locate the current position of the laptop.",
"Then, i13DM creates a location profile dataset containing an object for every quarter of an hour and every day of the week.",
"Afterward, for every object in the dataset, i13DM sets the longitude, latitude, measurement accuracy, and the location's zip code.",
"Furthermore, i13DM fixes the probability of the laptop being presented in this area to 100%.",
"Finally, once the location profiling is over, i13DM stores the location profile in the local database and submits the location profile to i13DRP.",
"After finishing the initial system analysis and profiling, the i13DM keeps running continuously in the background."
],
[
"Update Location Profile",
"i13DM regularly updates the location profile of the laptop.",
"Every 90 seconds, i13DM locates the current geographical position of the laptop by using geolocation API and inserts the longitude, latitude, accuracy of measurements, and zip code into its local database.",
"After, i13DM queries the local database to retrieve the locations with the time of the day and the weekday equal to the last location record inserted.",
"Then, i13DM groups the retrieved records by zip code and selects the longitude and latitude of the group with the highest number of records and sets the probability of the laptop being present in this longitude and latitude by dividing the number of records in the group by the number of all fetched records.",
"Finally, i13DM updates the corresponding location profile in the local database and submits the new location profile to i13DPR."
],
[
"Update Power Profile",
"i13DM continuously updates the power consumption profile of the laptops.",
"Every few seconds (one-second interval on Ubuntu and three seconds interval on Windows machines.",
"We use different intervals due to some platform-related limitations), i13DM uses the power model to estimate the current power consumption of the laptop in normal mode and power save mode and checks if the laptop is connected to the AC charger.",
"Then, i13DM inserts the record into the local database.",
"After, i13DM queries the local database to retrieve the stored power readings, which have the time of the day and the weekday equal to the last observed power reading, and groups the records based on the status of being connected to the AC charger.",
"After, i13DM calculates the arithmetical mean of power consumption in normal mode and power save mode for all retrieved records in each group and sets the probability of the laptop being connected to the AC charger by dividing the number of records in each group by the total number of fetched rows.",
"Finally, i13DM updates the corresponding power profile record in the local database with new measurements and submits the updated power profile to i13DRP."
],
[
"Update Running Profile",
"Every 90 seconds, i13DM inserts an activity record to the database, indicating that i13DM and laptop are running.",
"Afterward, i13DM retrieves the two most recent activity records and calculates the time difference between the two retrieved records.",
"If the time difference is more significant than 90 seconds, then i13DM inserts several new activity records into the database containing a status flag indicating that i13DM and the laptop were not running within the calculated time difference.",
"After, i13DM queries and groups the stored activity record for every minute of the day and every day of the week.",
"Then, for every group, i13DM calculates the probability of i13DM and laptop running by dividing the number of records with the status flag by the total number of rows in that group.",
"Finally, i13DM updates the corresponding power consumption profiles at the same time of the day and the day of the week in the local database and submits the updated power profile to i13DRP.",
"While designing the i13DM, we ensure that i13DM performs DR-related tasks independently without any required interaction with the user.",
"Although i13DM does not interrupt the user's activity, we provide the user with the option of preventing any DR activity when desired.",
"We also have to emphasize that we only collect the laptops and users' data that we specifically require for performing DR activities.",
"We sufficiently anonymize the collected data and transfer the data to i13DRP through secure and encrypted connections.",
"Although we designed the i13DRP so that DR-related activities can be performed through direct interaction with utilities and RES without the supervision of any administrators, we provide a control panel for the administrator to manage the registered laptops, their data, and the DR-related tasks.",
"We describe the structure of i13DR by using a class diagram based on the discussed use cases and requirements, as Figure REF displays.",
"Classes are abstractions of the behaviors and attributes of the system, and objects are entities that encapsulate behavior and state of components [2].",
"On the left side of Figure REF , we present the i13DM classes, and on the right side, we include the classes for i13DRP.",
"The i13DM classes characterize the system required for power and location profiling and performing DR events.",
"MonitorResources in association with Locator and PowerModeler initiates the activities for profiling and monitoring resources.",
"Furthermore, Watcher, subscribes to Publisher/Subscriber for receiving power models and DR schedules, and PowerController is responsible for activating and deactivating the power controls according to the schedules.",
"On the i13DRP side, i13DMProfiling subscribes to Publisher/Subscriber for obtaining the power consumption and location profile from i13DM and Scheduler is responsible for scheduling a DR events.",
"Figure: Class diagram of i13DR."
],
[
"Implementation of i13DR",
"Before implementing the i13DR, to decrease the complexity of our design, we break down the i13DM into smaller resizeable subsystems according to the recognized use cases.",
"We decompose the i13DR into subsystems with low coupling and high coherence to decrease the dependencies between two or multiple subsystems and increase the dependencies among classes within a subsystem, as Figure REF illustrates.",
"DRManager, DeviceManager and DeviceAnalyzer are the fundamental parts of i13DM carrying out the necessary tasks of a DR event.",
"DeviceAnalyzer contains the entities responsible for estimating power consumption, location profiling, and initial analysis.",
"DeviceManager consists of classes responsible for the general behavior of the application.",
"DRManager holds all the classes for managing a DR event.",
"As previously stated, i13DRP consists of several server-side applications, but we do not present the boundaries to keep the subsystem decomposition diagram clear.",
"The primary task of i13DRP is managing the participating laptops and performing DR events.",
"DRManager carries out the essential scheduling operations in association with i13DMProfiler, which is responsible for aggregating the power consumption and location profiles of laptops.",
"i13DMManager provides the administrator of i13DRP with the necessary operations for managing the registered laptops.",
"Furthermore, several other subsystems facilitate i13DRP operations including CrashManager handling the reported i13DM crashes, i13DMUpdater and DevOpsManager maintain building and releasing new versions of i13DM and i13DRP and finally, AuthN/AuthZ maintains the authentication/authorization and security policies of i13DM and i13DRP.",
"Furthermore, in the appendix, we include a few snapshots of the implemented i13DM and i13DRP.",
"Figure: Subsystem decomposition of i13DR."
],
[
"Implementation of i13DM",
"We develop and package the discussed functionalities of i13DM into an cross-platform desktop application for Microsoft Windows 7/8/10 and Ubuntu 16.04 LTS based on Electron Frameworkhttps://electron.atom.io/ (a JavaScript framework for creating desktop applications) and several Node.jshttps://nodejs.org/en/ packges.",
"To reduce laptops' power consumption, we use the built-in power management features of OS.",
"The ideal would be disconnecting the AC battery charger from the plug and only draining power from the battery.",
"However, the OS does not provide any functionalities for this purpose.",
"Hence, we use the OS API to put the laptop into power save mode.",
"On Windows, we rely on powercfghttps://msdn.microsoft.com/en-us//library/hh824902.aspx to import, activate and deactivate Windows-specific energy save scheme, which we previously generated and exported from a Windows 10 machine and we ship it with i13DM.",
"Ubuntu is equipped with very few built-in functionalities for controlling power consumption.",
"For this reason, to save power, we only dim the screen and turn the screen off when the laptop is idle.",
"To perform the initial analysis of laptops, we use built-in functionalities of OS to extract the hardware and software specifications.",
"On Windows, we heavily make use of Windows Management Instrumentation (WMI)https://msdn.microsoft.com/en-us/library/aa384642(v=vs.85).aspx commands to query the system specification.",
"However, on Ubuntu we rely on several commands, including but not limited to lshw, dmidecode, lshw, lspci and lscpu to collect the necessary system specifications."
],
[
"Implementation of i13DRP",
"i13DRP comprises several server applications which emphasize real-time communication and scalability.",
"We implement the primary functionalities of i13DRP based on Meteorhttps://www.meteor.com/ and Angular 2+https://angular.io/, which are JavaScript frameworks for developing real-time back-end and front-end server applications.",
"We also use NGIXhttps://nginx.org/, a high-performance HTTP and reverse proxy server running on a Ubuntu 16.04 LTS to server the applications.",
"The main storage provider of i13DRP is Firebasehttps://firebase.google.com/ providing us with a real-time cloud-hosted NoSQL database for storing and retrieving the i13DM and i13DRP data in JSON format.",
"Firebase also offers cloud storage, which we use to distribute the constructed power models to i13DM in real time.",
"The main reason for choosing this platform is the high scalability of the offered services, which are empowered by Google infrastructure to support several thousand simultaneous connections, reasonable prices, high security, and availability."
],
[
"Power Model Construction",
"We propose an approach for generating regression models, which i13DM uses to estimate the power consumption of laptops in real time based on the system metrics reading.",
"The laptop's power consumption is positively correlated with the circuit's load.",
"In order words, when the amount of power consumed is either load on the processors, networking traffic, or the screen's brightness increases.",
"Also, while the laptop's battery is charging, the AC charger causes an extra load on the regular power consumption.",
"Therefore, when estimating the power consumption, we extract the system metrics which highly correlate with the power consumption.",
"Afterward, we feed the derived parameters into a previously built regression model to calculate the total power consumption.",
"Equation REF shows the mathematical relationship between estimated power consumption $E_{estimated}$ and the power model $f()$, where $m_1(t)$ to $m_n(t)$ are the extracted system metrics at the time $t$.",
"$ E_{estimated} = f(m_1(t), ....m_n(t))$ Since i13DM is executable on Windows and Ubuntu and both OS have different specifications, we require creating four different power models: A model for Windows in normal mode, a model for Windows in power save mode, a model for Ubuntu in normal mode, and a model for Ubuntu in power save mode.",
"Our modeling procedure is based on six distinct steps, as Figure REF displays.",
"Figure: Power model generation procedure.First, we require collecting power consumption data for fitting the models.",
"We use a Lenovo ThinkPad L540 with specifications described in Table REF as SUT.",
"We connect the SUT to MEDAL [15], a custom-built low-cost measurement system for high-frequency energy data built upon a voltage-sensing circuit, current sensors, and a single-board PC as a data aggregator.",
"Medal collects the active power consumption of SUT in watts with a sampling rate of up to 50 kHz.",
"Meanwhile, we log the system metrics of SUT with a one-second interval on Ubuntu and a three-second interval on Windows.",
"We use different intervals on different OS because of some platform-specific limitation that prevents us from faster reading.",
"The logged system metrics include CPU Utilization in percentage, display brightness in percentage, power drain rate of battery in watts, the charging/discharging status, the remaining capacity of the battery in percentage, memory usage in percentage, disk read/write KB per seconds and request and network download/upload rate in KB per seconds.",
"We collect the data over four days for each OS and power consumption mode while the user works with laptops performing the common daily activities.",
"Table: Lenovo ThinkPad L540 SpecificationsAfter we preprocess, normalize and clean the data and remove the outliers from the raw collected data.",
"We consider the power readings below 8 watts and above 65 watts (the maximum output of the used AC charger of the SUT) to be outliers and remove them from the dataset.",
"Before training the linear regression model, to improve the accuracy and reduce the complexity of the constructed model, we examine the quality of different combinations of features to select a subset of features with the best accuracy.",
"For this reason, we perform an exhaustive search to determine the best subset of power-related features.",
"We include the detailed results of the exhaustive search of all models in the appendix.",
"We perform the calculations by making use of R Programming Languagehttps://www.r-project.org/.",
"According to the results, for the Windows in normal and power save mode, we select the following features: Battery charging/discharging rate Battery charging/discharging rate squared Interaction of battery charging/discharging rate with display brightness Interaction of battery charging/discharging rate with the remaining capacity of the battery Charging status CPU usage Memory usage Remaining capacity of the battery Download/upload rate in KB Disk read/write request per second We also select the following features for Ubuntu in normal and power save modes: Battery charging/discharging rate Battery charging/discharging rate squared Interaction of battery charging/discharging rate with the remaining capacity of the battery Charging status CPU usage Memory usage Remaining capacity of the battery Download/upload rate in KB Disk read/write request per second Before training the regression model with four datasets, first, we split each dataset into two smaller subgroups.",
"One subgroup holds 80% of the dataset that we use for fitting the model, and the other contains 20% of the dataset meant for testing the model's accuracy in predicting out-of-sample values, which we discuss in the next section.",
"Then, we use the selected features to fit a simple linear regression model by using the lm method in R. We should mention that we use the same power model generated on the SUT on different participating laptops."
],
[
"Evaluation",
"In the following section, we evaluate the performance and accuracy of the i13DR and the constructed power models.",
"First, we explain the methodologies we use for evaluating power models, followed by an experiment that we designed and conducted for quantifying the abilities of i13DR infrastructure.",
"Then, we describe the objectives and hypotheses we expect to realize, followed by presenting the results gathered from the experiments.",
"Afterward, we interpret and discuss the findings.",
"Finally, we mention the limitation of our system."
],
[
"Power Models Evaluation",
"We evaluated the four generated power models by using the test dataset of the corresponding model based on different metrics including Adjusted R-Squared, MAPE, and Min/Max Accuracy of constructed models and also the Correlation Accuracy between the actual and predicted values, as Table REF summarizes.",
"The Adjusted R-Squaredhttp://r-statistics.co/Linear-Regression.html represents the proportion of the variation in the measured real power explained by the model when taking the number of features in the model into consideration.",
"A higher Adjusted R-Squared implies a better model.",
"Mean Absolute Percentage Error (MAPE), defined as Equation REF , measures the prediction error, and a lower MAPE value indicates the better models' quality.",
"Furthermore, Min/Max Accuracy is defined as Equation REF , and it measures how far the model prediction is from the actual real power readings.",
"The better the quality of the regression model is, the higher the Min/Max Accuracy will be.",
"Finally, Correlation Accuracy is a simple correlation between predicted and actual real powers, and as the correlation among the values increases, it indicates that both values have similar directional moves.",
"On average, the results report that power models on Ubuntu measure the power consumption with an accuracy of up to 95% and in Windows with an accuracy of up to 85%.",
"We achieved better accuracy for Ubuntu because we had a shorter sampling rate, hence more accurate input.",
"We also include a more detailed summary of all models in the appendix.",
"$ \\begin{aligned}MinMaxAccuracy = \\\\mean\\Bigg (\\frac{min(actualRealPowers, predictedRealPowers)}{max(actualRealPowers, predictedRealPowers)}\\Bigg )\\end{aligned}$ $ \\begin{aligned}MAPE = \\\\ mean\\Bigg (\\frac{abs(predictedRealPowers - actualRealPowers)}{actualRealPowers}\\Bigg )\\end{aligned}$ Table: Power Model Evaluation ResultsFinally, we perform 5-folds cross-validations on all four models.",
"Figure REF illustrates all the resulting cross-validations side by side where the minor symbols are predicted real powers and bigger ones are actual real powers.",
"Additionally, the mean squared error of the models varies from 8% to 54%, with power models of Ubuntu having a better performance than Windows models, as Table REF represents the average squared errors of five folds.",
"According to the plots, we verify that model's prediction accuracies are approximately uniform among different samples, and the slopes and level of fitted lines have relatively low variations.",
"Figure: Power models cross-validations.Table: Power Models Cross-validations Mean Squared Error"
],
[
"i13DR Evaluation",
"We designed a comprehensive experiment consisting of two parts to evaluate the performance of the i13DR infrastructure.",
"The first part of the experiment is designed to estimate the i13DM capability to reduce the laptop's power consumption while connected to the electricity grid.",
"For the second part of the experiment, we perform a demo scenario of a DR event with a few participating laptops to investigate the effect of the DR schedule on the demand load."
],
[
"Power Control Functionality Evaluation",
"In our approach, we explained that our power consumption mechanism relies on the built-in functionalities of OS.",
"To determine the impacts of i13DM's power control techniques on the laptop load curtailment, we executed various workloads on different OS with different power consumption modes.",
"We used two workload generator applications to stress various components of the SUT.",
"The SUT was fully charged to eliminate the impact of battery charging on the demand load.",
"We executed a five-minute long workload on both OS; three times while the laptop was running in normal power consumption mode and three times while running on power save mode activated by i13DM.",
"During the execution of workloads, we monitored and recorded the real power consumption in watts and consumed energy in kWh through a connected power meter.",
"We used stresshttp://manpages.ubuntu.com/manpages/zesty/man1/stress.1.html on Ubuntu to create workloads and on Windows, we made use of HeavyLoad v3.4http://www.jam-software.com/heavyload.",
"We calculated the average power consumption drop in watts and energy consumption in kWh, as Table REF summarizes.",
"The results indicate that i13DM can decrease the power consumption on Windows by up to 27% and up to 7% on Ubuntu.",
"The main reason for such a significantly better performance is the availability of more extensive built-in power management features on Windows compared to Ubuntu.",
"However, the drop in real power consumption on Windows is muchly noticeable than on Ubuntu.",
"The reduction in total consumed energy is relatively in the same range for Windows and Ubuntu, with 27% and 22%, respectively.",
"We believe the measurement device's low accuracy and low consumption of laptops over the short period of sampling are causing the similarity of observed energy consumption.",
"For this reason, we argue that the drop in power consumption is a better indicator of i13DM's ability to control the electrical power consumption of laptops.",
"Table: Average Drop in Power and Energy Consumption"
],
[
"DR Event Scheduling Evaluation",
"To investigate the effectiveness of i13DR in scheduling and performing DR events, we designed and conducted a field experiment for executing a DR event with a few participating laptops.",
"We developed a scheduling mock-up component for i13DRP where the administrator can schedule and manage multiple DR events and monitor the participating laptops.",
"Because i13DR was not connected to any RES supplies while performing the experiments, the mock-up component simulated the behavior of a wind turbine integrated with the local power grid.",
"We also developed a scenario for scheduling a DR event initiated by an administrator.",
"For planning a DR event, the administrator is first required to determine the position of the wind turbine.",
"Afterward, the administrator specified the possible reduction of the wind turbine's electrical output in watts.",
"The administrator was also required to provide the length of the DR event in minutes.",
"The administrator also had the option to either start the DR event immediately or provide the start time of the DR event.",
"Once all required inputs are provided, I13DRP starts the scheduling procedure.",
"First, i13DRP queried all the currently online laptops to find the laptops within a 1000 meters radius of the provided location of the wind turbine.",
"Next, for each retrieved laptop, i13DRP fetched the power consumption profiles from 20 minutes before the start time of the DR event up to the start time.",
"Then, i13DRP accumulates the reported difference between real power consumption on average and power save modes from estimating the amount of power one specific laptop can contribute to power reduction.",
"Finally, i13DRP created a schedule with the provided start time, or if the start time was not specified, i13DRP scheduled an event that immediately began and lasted as defined by the administrator.",
"When all the schedules were created, i13DRP submitted the schedules to Firebase to be downloaded by i13DM on the filtered laptops.",
"Afterward, i13DMs fetched the new schedule and activated and deactivated the power control according to the new timetable.",
"Moreover, i13DMs sent a status code to Firebase announcing that they either joined or left the DR event, which the administrator could observe."
],
[
"Experimental Setup of the DR Event",
"We conducted the DR scheduling scenario on three laptops with the hardware specification in Table REF .",
"All the participating laptops were fully charged and connected to the MEDAL power meters, which recorded the power consumption during the experiments.",
"We conducted the experiments five times which two DR events lasted for ten minutes and three events for five minutes.",
"Table: Hardware Specification of Participating Laptops"
],
[
"Results of DR Events",
"To evaluate the responsiveness of i13DR to the immediate changes in RES supply, we measured the latency of publishing a DR schedule to the real-time database after a request for a DR event was initiated.",
"Furthermore, we measured the latency of all filtered laptops joining the DR event (fetching the DR schedule and activating the power control) after submitting the DR schedule to the real-time database by i13DRP.",
"To evaluate the total impact of i13DR on curtailing the laptop demand load, we obtained the estimated demand reduction according to the laptops' power profiles and compared the estimated power reduction with the real power reduction provided from the readings of the MEDAL power meter.",
"We also include the demand load plots for the participating laptops during the execution of each DR event in the appendix.",
"On average, we observed a power reduction of 14.23 watts, although we expected an 8.92 watts reduction during the events, as Table REF illustrates.",
"We also observed an average latency of 658 milliseconds for scheduling an event and an average latency of 164 milliseconds for all the selected laptops to join the DR event, as Table Table REF summarizes.",
"Table: Summary of Experiments Average Power ReductionTable: Latency of DR EventsBased on the results, we verify that i13DR can rapidly reduce the real power consumption of a group of laptops in a matter of milliseconds.",
"In general, we observed a min/max accuracy of 66% and a mean absolute percentage error of 72% of the estimated demand reduction against the arithmetic mean of measured power consumption.",
"However, the accuracy of our predictions is relatively low, which we discuss the reasons in the next section."
],
[
"Discussion and Limitations",
"Our findings show that the designed i13DR can successfully schedule and perform fast DR events.",
"However, they also highlight some remarks and potential pitfalls.",
"Although on average, the measured demand load reduction exceeds the estimated reduction, the accuracy of our estimation is relatively low at 66% against the arithmetic mean of the measured power.",
"The main reason for this relatively low accuracy is the low accuracy of deployed power models used for creating the power consumption profiles.",
"All participating laptops used identical power models to estimate the power consumption, which is fitted based on the data collected from the SUT.",
"Even though the power model yields relatively high accuracy for predicting real power consumption on SUT, the power model fails to measure the power on other laptops.",
"Therefore, we argue that it is significantly vital to consider different hardware specifications when fitting the power consumption models.",
"We ignore this factor in our design because we require additional measurement devices installed on the demand side, which are expensive for demand-side participants.",
"Because one of our primary objectives is reducing the initial costs for participants, we sacrifice accuracy to reach our financial goals.",
"Since we used two Windows laptops and one Ubuntu laptop during the experiments, we expect to observe a 19.96% reduction on average, according to previous evaluations.",
"However, we observe a 17.80% reduction according to the arithmetic mean of measured power consumption reductions, indicating a roughly 89% accurate estimate of the effectiveness of power control approaches.",
"During the experiments, we observed that i13DR could quickly respond to the immediate changes in RES.",
"However, the results also imply that i13DR significantly has higher latency for selecting the participating laptops and scheduling the DR events than the latency of all the laptops participating in the DR event.",
"The primary reason for the more significant latency is the quality of the mock-up component developed for the experiments.",
"The mock-up is created with AngularJS and is executed on the administrator's web browser.",
"Therefore, the performance is limited to the administrator's laptop's performance (the SUT).",
"Finally, we suggest developing the scheduling components as a server application that can offer to schedule a DR event as a service to the administrator.",
"Finally, we notice that the amount of power reduction that laptops can contribute to the power grids using operating systems' built-in power management features is relatively low.",
"Therefore, compensating for a substantial loss of RES supplies solely based on a limited number of participating laptops may not be feasible."
],
[
"Conclusions",
"In this study, we offer a design and an implementation of DR infrastructure for integrating the fluctuating into the grid.",
"Furthermore, our approach offers no initial cost for demand-side participants and laptops, and our system can be used as a testbed for researching different scheduling and optimization approaches.",
"The evaluation of our field experiments verifies that our system successfully schedules and executes DR events.",
"Furthermore, we construct power models for estimating the power consumption of laptops with accuracy up to 95% on Ubuntu and 85% on Windows on the SUT.",
"However, the accuracy of our estimation of demand load reduction is about 67% indicating relatively low reliability.",
"The main reason for the low accuracy is our entire software-based approach, which sacrifices high accuracy to eliminate initial DR costs.",
"In future work, we suggest improving power modeling techniques, power consumption control methods, and DR event scheduling and optimization approaches.",
"Our approach toward power modeling is solely based on linear regression techniques.",
"However, there are several other machine learning approaches worth further studies, such as k-nearest neighbors algorithm.",
"Furthermore, we need to implement a production-ready scheduler and optimizer for minimizing the electrical load consumption on i13DRP, preferably based on the optimization algorithms such as Mixed Integer Linear Programming, Convex Optimization Problem and Particle Swarm Optimization.",
"Finally, as previously described, our power control approaches heavily depend on the built-in power management features of the OS.",
"Although Windows offers extensive features in that regard, Ubuntu suffers from insufficient power management options.",
"Therefore, we suggest investigating other techniques for managing power consumption on Ubuntu.",
"We sincerely thank Jose Rivera and Martin Jergler for their valuable input to this work and for advising on the master's thesis on which this paper is built [16]."
],
[
"Snapshots of i13DR",
"We include a few snapshots of i13DR where Figure REF displays the i13DRP administrative page.",
"Figure REF and Figure REF illustrate the i13DM's panel, where the user can see some reports on the laptop's power consumption and modify the behavior of i13DM.",
"Figure: A snapshot of i13DRP administrator panel with power profile of one laptop.Figure: A snapshot of i13DM's settings window on Ubuntu.Figure: A snapshot of i13DM's status window on Windows."
],
[
"All Subsets Regressions of Power Related Data",
"We use R package leapshttps://cran.r-project.org/web/packages/leaps/leaps.pdf to perform an exhaustive search to determine the best subset of power-related features.",
"Figure REF , Figure REF , Figure REF , and Figure REF illustrates the result of an exhaustive search on all subsets of the regression model for the dataset collected from different OS in normal power consumption mode and power save mode.",
"leaps sorts the results by Adjusted R-Squared and Bayesian Information Criterion (BIC)http://r-statistics.co/Linear-Regression.html and examines them for the five best models reported for each subset size (one feature, two features and so on).",
"The Adjusted R-Squared represents the proportion of variation in the measured power consumption that the model has explained by considering the number of features in the model.",
"BIC measures the goodness of a model based on the maximized value of a likelihood function.",
"The higher the calculated value for Adjusted R-squared, the better the constructed model, whereas as the value of BIC decreases, the model improves."
],
[
"Power Models Summary",
"Figure REF , Figure REF and Figure REF display the summary of each power model for different OS and power consumption mode, created by the summery() method of R. Figure: Summary of power model for Windows on normal power mode.Figure: Summary of power model for Windows on power save mode.Figure: Summary of power model for Ubuntu on normal power mode.Figure: Summary of power model for Ubuntu on power save mode."
],
[
"Demand Load Plots of Experimental DR Events",
"Figure REF , Figure REF , Figure REF , Figure REF , and Figure REF illustrate the demand load plots for the participating laptops during the execution of each experiment, where the dotted line represent the moment power control is activated.",
"Every plot includes the demand load of every laptop separately, named as Laptop 1, Laptop 2 and Laptop 3 as well as the accumulated demand loads of all laptops, called as All.",
"Each plot demonstrates the demand load for ten minutes before and after the activation of power control independent of the length of the DR event.",
"Figure: First DR event.Figure: Second DR event.Figure: Third DR event.Figure: Fourth DR event.Figure: Fifth DR event."
]
] | 2210.07789 |
[
[
"Language Generation Models Can Cause Harm: So What Can We Do About It?\n An Actionable Survey"
],
[
"Abstract Recent advances in the capacity of large language models to generate human-like text have resulted in their increased adoption in user-facing settings.",
"In parallel, these improvements have prompted a heated discourse around the risks of societal harms they introduce, whether inadvertent or malicious.",
"Several studies have identified potential causes of these harms and called for their mitigation via development of safer and fairer models.",
"Going beyond enumerating the risks of harms, this work provides a survey of practical methods for addressing potential threats and societal harms from language generation models.",
"We draw on several prior works' taxonomies of language model risks to present a structured overview of strategies for detecting and ameliorating different kinds of risks/harms of language generators.",
"Bridging diverse strands of research, this survey aims to serve as a practical guide for both LM researchers and practitioners with explanations of motivations behind different mitigation strategies, their limitations, and open problems for future research."
],
[
"Introduction",
"The new wave of large language models [18], [227] has been beating benchmarks hitherto considered impossible [32], and generating text so fluent and rich that it can be easily confused with human-written language [166], [222]While the majority of these models are trained on English language datasets, recent studies have also obtained similar gains on other languages [121], [183].",
"Built on top of such LMs, user-facing language generation technologies are ubiquitous—autocomplete, machine translation, summarization, question answering, dialogue generation, story generation, etc.",
"Since language is inherently a tool of power—the primary means by which people and societies perpetuate stereotypes [9], [205], [134], [34] and manipulate opinions [31], [176], [15]—LMs that generate language and are deployed to millions of users also hold similar power.",
"However, while language generation models have rapidly gained power to influence users, our understanding of potential risks and harms associated with their deployment has lagged behind [10].",
"Figure: Overview of Intervention Strategies.",
"A typical ML/NLP model development process involves data collection/curation, model training and design, inference, and finally application deployment.",
"For each phase of this development cycle, different techniques can be adopted to mitigate harms.",
"Our survey presents a taxonomy of intervention strategies organized around the different phases where they can be applied.Indeed, widespread adoption of language generation technologies has revealed that models introduce vulnerabilities and threats – both inadvertent and malicious – to individual users, social groups, and content integrity.",
"Without social context and content control, deployed language generators have quickly derailed to racist, homophobic, hateful comments [85], [90], [210], [192], compromised user privacy [24], spread disinformation [181], and even encouraged users to commit suicide [41], just to name a few.",
"Prior works have outlined these risks [138], [182], [203], proposed taxonomies [204], discussed their points of origin, and advocated for future research on ethical development of LMs [10], [185].",
"However, there is little work that summarizes actionable approaches and technological solutions to preventing or mitigating these potential harms.",
"In this survey, we provide a comprehensive taxonomy of relevant mitigation strategies proposed in prior literature, specifically focusing on language generation models.",
"Following the structure of recent surveys listing the risks of LMs (§), this survey focuses on automated identification/explanation of potential harms and technical interventions (§) aimed to mitigate these harms at different stages of LM development: interventions in data collection, modeling, decoding, and post-facto editing.",
"We highlight the motivations behind these approaches and their limitations.",
"Within each of these categories, our taxonomy brings together previously disjoint application areas targeting different types of language generation harms (toxic/biased language and misinformation) under a unifying framework.",
"We do this because intervention techniques are not disjoint, and future work can benefit from leveraging techniques across application areas.",
"While this survey focuses on technical and algorithmic solutions, there are other risk mitigation approaches such as policy decisions, which we briefly discuss in the end (§).",
"Our survey's contribution is a unified taxonomy of prior research centered around technical mitigation strategies.",
"It draws new connections between prior approaches and identifies their gaps, highlighting directions for future research.",
"These include incorporating sociocultural context to produce socially-sensitive interventions, detecting the intent of generations (inadvertent vs. malicious) and developing appropriate interventions for each, and going beyond an English, Western/US-centric view of the research to account for the challenges of ethics in multilingual language generation."
],
[
"Background",
"Since we focus on language generation, we use the term language models (LMs) to refer to their classic definition as generative models (or decoders), which predict the next token given the preceding generated context.For the purposes of this survey, This paradigm also subsumes conditional (or sequence-to-sequence) language models conditioned on inputs from different modalities such as text, image, or speech via an encoder.While many different strategies to (pre-)train encoder LMs have been introduced in the literature [45], [161], they are generally not conducive to generating text and are out of scope in this survey.",
"Unless otherwise specified, we assume that (1) the LM decoder is parameterized by a transformer architecture [191], and (2) the LM is first pretrained on a large amounts of text (usually containing from 100-billions to trillions of tokens), which, together with their large number of parameters, have earned such models the name large language models.",
"After pretraining, LMs are either used in a zero- or few-shot manner [18], or modified for specific tasks via finetuning all or some of their parameters [126].",
"The generation tasks this survey focuses on can be broadly categorized as either (1) transformation tasks, where a given input is transformed into a textual output (such as machine translation, abstractive summarization, data-to-text generation, and stylistic re-writing, among others) [168], [226], [3], [163], (2) or open-ended tasks (such as dialogue generation, prompt-based autocompletion, story generation, and more) [2], [18], [68]."
],
[
"Risks in Language Generation",
"Before diving into mitigation techniques (§), the main focus of this paper, we briefly outline potential harms that LMs can cause, following [203]'s taxonomy.",
"Discrimination, Toxicity, and Exclusion The scope of linguistic diversity in human communication is enormous and is often linked to various personal, social, and cultural factors [80], [53], [35], [25].",
"As such, language produced in the real world reflects sociocultural stereotypes and presuppositions [9].",
"LMs trained on such language can overfit to and amplify stereotypes [228], [189].",
"This can lead to several risks of harms from models: [(1)] Stereotyping and discrimination occurs when generated text reinforces discriminatory stereotypes and perpetuates biases against disadvantaged groups, based on factors like gender, race, age, religion, sexuality, profession [10], and intersectional identities [36].",
"Evidence for this behavior has been substantially corroborated in NLP literature [147], [151], [84], [74], [14], [118], [57].",
"Toxicity is generated language that is rude, disrespectful, offensive, threatening, violent, or otherwise harmful [61], [167], [1].",
"Toxicity has many shades and can range from overtly toxic content, such as violent hate speech, to more subtle, veiled toxicity, such as microaggressions.",
"Exclusion refers to the disparate performance of text generation systems across language variations.",
"Models may fail to comprehend or generate “non-standard” dialects and sociolects, essentially excluding their speakers from their user base [95], [103], [208].",
"Factual Errors, Misinformation, and Disinformation LMs learn complex statistical correlations between different tokens, and as a result, they can generate fluent outputs that users may easily mistake for human-written language [223].",
"However, at the same time, they often assign high probabilities to utterances that are factually incorrect or misleading [138].",
"This can lead to problematic usages of LMs, whether malicious (disinformation) or unintentional (misinformation), with models producing misleading content, conspiracy theories, and falsehoods [216], [181], [140], [119].",
"Examples of the tangible harm that can be caused by factually incorrect generations are found in many domains.",
"In high-stakes applications like medicine, LMs may misdiagnose, suggest wrong medication dosages [12], or even suggest suicide [41].",
"Additionally, LMs can be used with malicious intent to create, for example, Internet bots that spread propaganda worldwide [16], [11]: with improved language modeling tools, malicious users can create realistic propaganda and disinformation articles at scale [19].",
"Privacy Violations and Other Underexplored Issues LMs have been shown to learn sensitive details about people via leaks from user databases containing personal identifiable information, for example, SSNs, phone numbers, and bank account information.",
"When prompted, LMs can generate these details, leading to privacy violations [100], [23].",
"Although this can pose serious security risks, privacy-related issues of LMs are underexplored in NLP.",
"In addition to the above risks, [203] discusses a variety of other malicious applications of LMs, as well as economical and environmental impacts.",
"While extremely important, mitigating these risks requires not only technological innovation, but also the development of regulatory practices and guidelines in an interdisciplinary effort.",
"In this survey, we focus on algorithmic solutions, leaving this discussion for future work.",
"Taxonomy of Intervention Strategies In this section, we delve into actionable methods that can be used to mitigate language model harms.",
"The development pipeline of a typical Machine Learning (ML) model involves several critical decisions where risk of different harms can inadvertently arise.",
"Different stakeholders have access to different components in this pipeline and therefore can employ different intervention strategies to mitigate harms.",
"For example, while a researcher might be involved in data curation and can intervene before (pre)training, a practitioner building an application might only have access to a pretrained model and may only be able to employ interventions during inference.",
"We present a taxonomy of such intervention strategies organized by the different stages of training and deploying an NLP model (Fig.",
"REF ), aiming to show a variety of intervention measures that different stakeholders can employ at different stages of the cycle.",
"We step backwards through the LM development pipeline, beginning with interventions on inference-time model outputs and peeling back the layers to data-level interventions.",
"Application Level Interventions Harm Detection In order to mitigate harms, we first need to be able to detect problematic, incorrect, and unreliable model outputs [169].",
"User-facing language applications can employ detectors in order to intervene before harmful text reaches a user.",
"Such detectors are typically coarse, binary classifiers that are trained to identify problems in generated text.",
"They are often trained for a single task, such as predicting toxicity [150], [40], [215], or factuality [108], [65], [194] of the outputs.",
"Early approaches for detecting problematic text focused on building linear classification models using hand-designed textual features like lexicons (such as hatebase) [214], [37], [20], [129], $n$ -grams, capitalization/punctuation details [29], [201], [150], among others [219], [21].",
"In misinformation detection, features like the presence of novel words in generated document summaries can indicate hallucination [229], [101].",
"While such classifiers support quick development and are interpretable, they tend to overfit to lexical features, are prone to false positives, and are easy for malicious users to bypass [113].",
"Neural text classifiers, on the other hand, can incorporate contextual information and have shown to be more robust [60], [162].",
"When built by finetuning pretrained LMs instead of training from scratch, they naturally lead to even better performance [51], [215].",
"Training supervised text classifiers naturally relies on labeled datasets.",
"For tasks like toxicity detection, annotated datasets in several domains, especially to detect overtly toxic text, have been collected for English [39], [201], [207], [160], [146], [145].",
"Such resources in other languages are still lacking [154].",
"Human annotation efforts for more subtle types of toxicity like microaggressions, however, have not been as successful due to annotators' own biases [17].",
"Hence, unsupervised or distantly supervised approaches have been adopted to detect it [105], [58], [152].",
"Information-related harms can arise either inadvertently (due to model errors) or deliberately (due to malicious uses of LMs, such as neural fake news or misinformation bots).",
"Malicious manipulations can be detected by employing approaches for automated fact checking, propaganda, or fake news detection using existing annotated datasets [190], [153], [136], [230], [69].",
"However, human annotations for non-factual text have been shown to be unreliable, for both machine generated and human written text [87].",
"Hence, prior works instead have relied on synthetic datasets of non-factual or machine generated text [223].",
"Another commonly used approach is to detect whether text has been machine-generated [52], [62], empowering users themselves to understand the source of the text they are reading [91].",
"In the case of inadvertent misinformation harms, representative annotated datasets are hard to obtain.",
"[108] construct synthetic datasets by introducing heuristic errors into factually correct text to create an incorrect version.",
"Such synthetic datasets can be used to train classifiers [108], [65], but their generalizability is limited, and they have low correlations with human judgements [157].",
"Challenges: While binary classifiers enable us to detect problematic model outputs, they are only effective if they can achieve high accuracy on tasks that are often inherently hard, highly contextual, and subjective in nature.",
"For toxicity detection, factors like region, political views, and social and cultural background of the users affect whether they will perceive some text as toxic [212].",
"Existing datasets are often biased due to their curation process [49], [206], [64], [179], [108] and can have unreliable annotations [175], [59], [157].",
"Further, as with many black-box models, coarse classifiers overfit to spurious artifacts [71], [139], [112] and amplify biases in their training data [228], [189].",
"For instance, toxicity detectors have been shown to rely on racial markers, disproportionately flagging text with African-American English as toxic [178].",
"Therefore, it is challenging to both construct sound, unbiased datasets and develop accurate models for harm detection.",
"Redacting or Flagging Harmful Text After detecting when harmful text is generated, the most straightforward way a user-facing application can prevent such text from causing harm to users is to not display the text at all (redacting) or to display it with a warning sign (flagging) [218].",
"Product designers can thus use the detection methods described previously to redact or flag problematic text generated by their applications.",
"Even when the detectors are imperfect, explicitly flagging problematic outputs is still useful because it signals users to take all model outputs with a grain of salt.",
"However, this strategy is not always applicable: for example, in speech-based dialogue agents, “displaying” a warning sign is a nontrivial UX decision, and in writing or auto-complete assistants (such as in Gmail Smart Compose), redacting is not an option and simply warning may not dissuade users from accepting the generated text.",
"Output Level Interventions Increasingly, many practitioners are building applications using LMs as APIs without explicit knowledge of how the model was trained or what training data was used.see https://gpt3demo.com/ for examples Such APIs may vary in how much information developers can see: some APIs allow access to all LM parameters, while black box APIs such as GPT3 [41] limit access to model outputs only.",
"Hence, multiple solutions have been proposed in prior literature for intervening on problematic model outputs, ranging from modifying decoding algorithms to editing the outputs with auxiliary models.",
"Decoding Methods Prior works have proposed various search and sampling algorithms to improve the quality of text generated by LMs [67], [56], [81], [141].",
"Recently, [224] introduced methods for controlling these decoding algorithms by promoting or demoting specific properties in the output text.",
"The motivating principle is that while an LM's training data may have contained harmful/non-factual text, it also contained clean data, so by constraining the decoding process to suppress problematic outputs, good outputs emerge.",
"Controls for decoding are implemented using risk detection approaches we discuss in §REF , such as toxicity/bias classifiers [38], [106], [123], factuality metrics [108], [65], or functions that quantify similarity between source and target [172], [173].",
"A simple way in which such detectors can be used to control outputs is rejection sampling or reranking: for a given input, multiple outputs can be generated by beam search or sampling, which can then be reranked using detectors to discard dubious outputs [107], [101].",
"However, this is not effective for multifaceted controls like factuality or for combinations of many controls, since all the generated candidates might be rejected.",
"To tackle these issues, a class of algorithms that we call guided-autoregressive decoding aims to incorporate control by modifying output distributions at every decoding step.",
"One branch of work adopts logical controls, in which developers directly specify sets of words that should or should not appear in the output [132], [159].",
"[211] apply this method to zero out the probabilities of offensive terms, and [102], [131] improve factuality by up-weighting generation probabilities of entities present in the source, using heuristics for transformation tasks.",
"A second branch of work uses the probabilities of harmful text detectors directly, composing them with LM output probabilities to up-weight or down-weight the probabilities of each token based on generated left context [220], [123], [38], [106], [180].",
"More recent work has also explored ways to induce sentence-level control via non-autoregressive controlled decoding.",
"Such decoding algorithms incorporate control using Monte Carlo Markov Chain (MCMC) techniques [79], [165], [143], in which a full sequence is initialized and then iteratively refined.",
"These methods have been applied for reducing toxicity [111], as well as improving fidelity in translation systems [110].",
"While extremely promising, these techniques suffer from slower decoding speed and need further exploration in order to be adopted in products.",
"Post-factum Editing Model Outputs Rather than modifying the decoding algorithms, recent studies have also explored ways to edit or revise model-generated text to remove harmful content.",
"Text editing is a decades-old subfield of NLP that has traditionally focused on fixing errors in machine translation [30], [184], [28] or grammar in human-written text [200].",
"While many of these text editing approaches are applicable to post-editing LM outputs, for the purposes of this survey, we highlight recent work related to rewriting harmful text.",
"The first set of works treat the task of rewriting as a sequence labeling task, where each token in the output sequence is either substituted, deleted, or kept the same [164], [76].",
"This, however, can be limiting in cases where the entire output needs rewriting.",
"For transformation tasks, especially those with supervision, parallel data for editing can be very easily obtained by converting source-target pairs to source-output-target triplets, along with an additional signal indicating errors (which can be obtained using automatic evaluators or human judgment).",
"For more open-ended tasks where this is not feasible, prior works have explored unsupervised solutions to remove bias [133] and semi-supervised solutions [22], [114], [7] to correct factuality errors.",
"These methods create pseudo-parallel data using various heuristics to induce errors in clean text, then train a model to correct them.",
"Challenges Decoding interventions rely on accurate detectors to prevent harmful outputs, so as a result, the challenges in designing robust detectors (§REF ) impact decoding algorithms as well.",
"For example, [217] show that toxicity avoidance algorithms refrain from generating African American English , thereby causing another harm (exclusion) while trying to address the first (toxicity).",
"Additionally, detecting misinformation and factuality can be extremely hard using simple detectors that do not provide a useful signal to guide the decoding process, so prior works have primarily employed heuristics.",
"Finally, controlled decoding algorithms are double-edged in that controls can be reversed by malicious users to inflict harm – to generate hateful messages, or to do targeted manipulation by copying users' personas.",
"However, this risk should not discourage research in decoding algorithms; rather, research on detecting such malicious uses should be conducted in parallel.",
"Model Level Interventions Several recent studies have provided evidence that different model design choices and optimization procedures can result in harmful generations downstream.",
"In this section, we describe approaches for modifying LM parameters to prevent risky generations.",
"We organize these approaches into two categories based on their requirements for computational resources: architecture/training interventions, which require training LMs from scratch and can thus incur heavy costs, and finetuning/model editing interventions, which use limited resources.",
"Architecture and Training Algorithms An LM trained only to predict the next token given left context has no checks on whether the predicted token is grounded in truth, which can lead to misinformation.",
"Thus, several studies explore modifying LM training objectives to incorporate factual information using either knowledge bases (KBs) or graphs [221]: each token prediction is scored not only on its likelihood given the previously generated context, but also on whether the generation is grounded in facts in the KBs [199].Knowledge-augmented LMs is a rich field where most existing work focuses on masked LMs [232] for solving understanding tasks.",
"Here we highlight papers in generation.",
"However, manually curating structured KBs is an arduous and expensive process, so most existing KBs are limited in size.",
"As an alternative, [127] propose using automatically generated KBs to train LMs.",
"In contrast, [116], [43], [88], [82] use unstructured text as knowledge.",
"Known as retrieval-augmented LMs, they are trained with a two-stage approach of first retrieving a document from the training set, then using it as additional context while generating, essentially providing evidence for what the LM generates.",
"Such techniques have already resulted in significant improvements in many NLP tasks: they have improved factual consistency in summarization models [83], fidelity in translation [8], and reliability in dialogue agents [48], [55], [225].",
"Another benefit of retrieval-augmented LMs is that, since they generate using the KBs at test time, these LMs can be dynamically updated by modifying the KBs with new information [99], [75].",
"[198], [92] also use KBs to embed commonsense knowledge in LMs.",
"These existing solutions tackle content-related harms, but future work in reducing discrimination and toxicity in LMs may also benefit from KBs that encode social [27], cultural, [78], and moral norms [77], [93].",
"Relying on KBs at inference time, however, can be computationally prohibitive.",
"Thus, rather than using explicit knowledge, [46] propose to train LMs with time stamps for each document, and show that this helps them discern current facts from outdated ones, as well as makes them amenable to further training with more data in the future.Along similar lines, [98], [70], [26] adopt a setup of training class-conditional LMs where each LM accepts \"control codes\" as an additional input.",
"When trained with data annotated for toxicity or bias, the LMs can be prompted to not generate toxic or biased outputs.",
"More recently, another paradigm gaining popularity in LM training is instruction-based learning, where in addition to the objective to predict next token, models are also trained to solve several NLP tasks with instructions written in natural language [202], [177].",
"These models have been shown to be capable of performing well with new instructions in zero-shot settings.",
"Providing explicit instructions to not generate certain kinds of text has shown some promise [156], [202] and is an interesting avenue for future work.",
"Text transformation tasks, where the goal is to produce text that is factually consistent with the input without hallucinating information, has seen the most innovation in architectural modifications.",
"Studies on these tasks have explored (1) modifying attention mechanisms to specifically capture relationships between entities [148], [231], (2) improving coreference to mitigate gender bias in translation [115], and (3) using text entailment to develop loss functions to improve fidelity [54], [117].",
"Some other notable directions in this space involve fact-aware pretraining [54], [193] and structured learning frameworks [209].",
"Finetuning and Model Editing The methods discussed so far have focused on mitigating harmful generations by designing and training models from scratch, which can incur heavy environmental and resource costs.",
"In contrast, an alternative branch of work has developed methods for modifying the model parameters of already-trained LMs, which requires much fewer resources.",
"An elementary way of doing this is finetuning (a subset of) an LM's parameters on small, curated datasets which are balanced with respect to demographics, and filtered for nontoxicity [70], [26], [126].",
"Recent work has also shown success via prompt-based learning [197].",
"Prompt-based learning makes use of the generative power of large LMs, while at the same time nudging the distribution of generated text to be more balanced.",
"These approaches have successfully been used to reduce toxicity [61] and exclusion [33], [109].",
"However, balanced data curation may be nontrivial for many settings, especially transformation tasks [220].",
"Finetuning LMs with reinforcement learning (RL) has been suggested as a better alternative [4], [128], [156], [187], [130], [170] for training modern LMs, as RL models do not require carefully curated, balanced datasets and can instead learn from discrete rewards such as human feedback [188].",
"Another less-explored alternative to finetuning is model surgery or editing, which uses harm detection methods we discussed in §REF to identify a specific set of neurons that contribute to harmful generations.",
"Culling such parameters has been shown to reduce toxicity [63].",
"In a similar vein, [42], [144] systematically edit model parameters by adding KBs to LMs trained without them, in order to revise facts memorized by the model.",
"These under-studied approaches hold great promise for more computationally practical approaches to risk mitigation.",
"Challenges An undeniable argument against LMs from scratch or augmenting them with knowledge is its cost, making these interventions infeasible for most researchers and practitioners.",
"However, even for organizations with access to large computing resources, research on training safer LMs lags behind research on training ever-larger LMs on raw data.",
"We attribute this to the difficulty of curating KBs, as well as the decreased training and inference speed that comes with such modifications.",
"Finetuning, on the other hand, is less costly, but may reduce the general purpose utility of LMs and has not been shown to be useful in reducing information-related harms.",
"Future work may benefit from drawing on continual learning [46] and reinforcement learning [156].",
"Data Level Interventions Training any machine learning model requires data, so a natural approach to creating fairer, more reliable LMs is carefully creating balanced training sets that are broadly representative of different worldviews.",
"This requires dedicated efforts in data curation [86], [94], [96] and novel data pipelines [44].",
"However, as modern LMs practically consume the entire public Internet to train, this is infeasible.",
"Existing works tackling this issue instead devise semi-automated solutions, which we categorize as follows.",
"Data Filtration The main idea here is simple: identify problematic documents in the training set, remove them, and train a safer model on the clean training set.",
"As the training dataset can be extremely large, sophisticated neural filters can be prohibitively slow to apply.",
"Hence, most work has utilized simple filters, such as the presence of \"unwanted\" words [168] or the predictions of linear classifiers [18].",
"Due to their simplistic setup, these approaches admit many false negatives (failing to detect documents with subtler toxicity) and false positives (erroneously flagging documents that discuss sensitive topics and use hateful speech as examples; additionally, removing data from different dialects like African-American Vernacular English, ironically exacerbating risks of marginalization and exclusion [50]).",
"[149] adopt a more sophisticated explanation-based approach: they train an LM on raw data, then feed the LM manually-curated toxic prompts and filter out documents to which the LM assigns high probability, and then retrain the LM on the filtered corpus.",
"Data Augmentation While data filtration aims to remove problematic training samples, data augmentation aims to offset the effect of problematic data by adding safer/healthier examples to existing datasets.",
"[137] explore adding counterspeech to datasets in order to balance out the hate speech already present in web data.",
"Augmentation with synthetically generated data has also been explored for gender bias mitigation in dialogue [47], [124] and translation models [186].",
"While prior work has shown that data augmentation methods are effective, these methods are extremely difficult to scale on large datasets with evolving distributions and definitions of biases.",
"Challenges In addition to the difficulties with filtering and augmentation interventions we describe above, data interventions are primarily designed to address population-centric risks such as discrimination, toxicity, and, to an extent, exclusion and privacy—but not factuality.",
"Even though one could also try to filter non-factual content [66], it is challenging to define [5] and detect facts [6] in the wild, making data interventions insufficient for addressing misinformation and factuality-related harms.",
"Since language, identity, and society are tightly intertwined, aggressive data filtering methods risk further imbalancing already imbalanced data.",
"Besides, even if we were able to perfectly filter out all problematic examples in the training data, and we could train an LM that has never seen any toxic or biased inputs, the model may still degrade when toxic inputs are provided to it.",
"Discussion and Open Challenges We have discussed several interventions for prevent harms that have been proposed in the literature and achieved some success, despite their challenges.",
"However, there is a lot of room for improvement in harm mitigation strategies.",
"Owing to the blackbox nature of neural models, many risks and their repercussions are still not well understood.",
"Besides, the field of LM research is still nascent, and LMs are bound to scale to larger datasets, more languages, and new tasks in the near future.",
"Hence, along with summarizing current research in risk mitigation, we also discuss open problems and avenues for future work.",
"Trade-offs: Where should one intervene?",
"Our primary goal in organizing intervention strategies into a taxonomy is to understand the impact, cost, and limitations of interventions at different stages, as well as to identify which methods can mitigate which kinds of risks.",
"However, different strategies have vastly different computational requirements and may not be feasible for every stakeholder.",
"A significant chunk of the responsibility to develop safer LMs falls on NLP researchers and organizations with access to substantial resources; these stakeholders may be able to implement data or modeling interventions.",
"On the other hand, developers creating applications on top of LMs may have access to neither the training data nor computing resources to design and train an LM from scratch on cleaned data.",
"In such cases, flagging and decoding approaches are more accessible.",
"In fact, there is no clear evidence that one step of intervention is sufficient—rather, much evidence exists that mitigating one type of risk actually exacerbates another (such as in data filtration).",
"In practice, a combination of interventions may be required.",
"Risks exist in LMs in all languages Most research on large LMs, their uses, and their risks is West-centric and primarily conducted on English.",
"There have been various efforts to accommodate linguistic diversity with LMs that can understand or generate text in several languages [120].",
"However, research on associated risks and their mitigation in these multilingual models has not even remotely caught up [158].",
"Further, in principle, many of these techniques are applicable to LMs trained on any language, but the definitions of risks themselves change with context and across cultures (for example, boundaries between what is polite/offensive vary across cultures, and some topics or words common to one culture are considered taboo in others).",
"Hence, there is a dire need to develop cross-cultural, cross-lingual analyses as well as mitigation tools.",
"Harm detection beyond simple classifiers In most of the intervention strategies we discuss, accurately detecting harms in machine-generated text not only informs the solutions, but is also essential in evaluating their effectiveness.",
"Thus, many of the shortcomings of interventions throughout the model development process are at their root due to poorly or under-defined risk detection methods.",
"Current detection methods are primarily based on classifying whether generated text is harmful or not on various axes like toxicity and factuality, but we recommend researchers and practitioners to move beyond simplistic coarse classifiers and towards more fine-grained, interpretable, and explainable analyses [122], [89].",
"Examples of such methods include: [(1)] span-level classifiers that provide local explanations of undesired behavior.",
"These have been applied to detect toxic [215] and non-factual phrases [65] in generated text.",
"Similarly, [135] developed an annotated challenge set for propaganda detection in human-written text, which can be used to train models that can evaluate misinformation in machine generated text.",
"Model probing techniques providing general trends of errors made by models.",
"For example, [157] use human annotators to devise a taxonomy of factual errors made by text summarization models, providing insights for targeted ways to fix them.",
"Similarly, probing techniques previously used for studying the linguistic capabilities of LMs can be adapted to evaluate toxic content in pretrained LMs [155].",
"Instance attribution methods that generate global explanations by finding examples in the training set that influenced certain model decisions [104].",
"Recent work has shown that these techniques can uncover training examples that contain subtly offensive terms, which models learn to amplify [72], [73].",
"And finally, analysis and visualization tools that allow developers to debug how LMs make predictions [62].",
"Systematic evaluation frameworks for mitigation strategies The rapid advancement of LM performance in recent years can, in part, be attributed to the development of systematic evaluation frameworks [196], [195], [13].",
"Such benchmarks generally present a simplistic one-dimensional view of performance, communicated only through accuracy numbers [125].",
"However, practices for evaluating harms in LM generated text, as well the effectiveness of mitigation strategies, are not systematized, with most research papers proposing their own evaluation criteria.",
"While there is an emerging body of work dedicated to benchmarking LM harms (and mitigation approaches) [171], the space of potential harms is huge and intersectional, and existing work only covers a fraction of it—primarily focusing on discrimination and toxicity in a limited societal and application context.",
"Developing a suite of evaluations or augmenting existing generation benchmarks [142] with axes of risk evaluations [174] will encourage development of holistic solutions, bridging discrimination/toxicity and information-related harms—two related directions in which researchers have often developed similar solutions.",
"Since harm detection and evaluation metrics are also an active area of research, this setup can further be improved by adopting a bidimensional setting [97] where new strategies are evaluated on harm prevention tasks, and the evaluations themselves are evaluated against human judgements.",
"Conclusions We present a survey of practical methods and techniques for addressing the societal harms and safety risks of language generation models.",
"Our structured taxonomy covers a wide variety of interventions that can be applied at different stages of the model development pipeline to mitigate harms.",
"This work bridges multiple strands of research and presents an actionable overview on methods for preventing harms from language generation models.",
"Limitations The goal of this survey was to present current research on analysing and mitigating harms of language generation.",
"There are multiple documented and anticipated harms that these models perpetuate, and it is not feasible to address intervention strategies for each of them.",
"We aimed to generalize multiple proposed solutions and present them in a structured form, considering a few popularly studied harms as case studies.",
"Inevitably, certain harms and their mitigation strategies might not have been considered for this survey.",
"Current research in this field is nascent but fast moving.",
"While this survey enlists techniques and approaches that are popular now, there is a potential for them to be replaced with newer research.",
"We anticipate that this survey may need to be updated or even redone to incorporate new research.",
"Ethics Statement In this survey, we present and discuss various risk analysis and intervention strategies to prevent societal harms from LMs.",
"We also comment on common themes across approaches for detecting and resolving population-centric harms (such as toxicity and discrimination) and misinformation related harms, and we recommend future work combining them.",
"First, many datasets and resources we discuss may contain biases, and using them in downstream applications can lead to risks as we have outlined.",
"Second, many techniques we discuss have limitations or are known to exacerbate other kinds of harms [213], and thus, applying them to newer problems may lead to unseen issues.",
"Finally, the interventions we identify to raise general awareness have potential for misuse: a malicious user can further imbalance the data to train even more harmful models, use the models and decoding algorithms to generate fake news, and target marginalized populations.",
"This, however, should not discourage development of mitigation strategies; rather, more work should be done to detect and ban malicious users.",
"This requires not only technological solutions in NLP, but also in social science, social network analysis, and public policy.",
"Organized References See Tab.",
"REF Table: References for strategies adopted for mitigating different kinds of harms."
],
[
"Taxonomy of Intervention Strategies",
"In this section, we delve into actionable methods that can be used to mitigate language model harms.",
"The development pipeline of a typical Machine Learning (ML) model involves several critical decisions where risk of different harms can inadvertently arise.",
"Different stakeholders have access to different components in this pipeline and therefore can employ different intervention strategies to mitigate harms.",
"For example, while a researcher might be involved in data curation and can intervene before (pre)training, a practitioner building an application might only have access to a pretrained model and may only be able to employ interventions during inference.",
"We present a taxonomy of such intervention strategies organized by the different stages of training and deploying an NLP model (Fig.",
"REF ), aiming to show a variety of intervention measures that different stakeholders can employ at different stages of the cycle.",
"We step backwards through the LM development pipeline, beginning with interventions on inference-time model outputs and peeling back the layers to data-level interventions."
],
[
"Harm Detection",
"In order to mitigate harms, we first need to be able to detect problematic, incorrect, and unreliable model outputs [169].",
"User-facing language applications can employ detectors in order to intervene before harmful text reaches a user.",
"Such detectors are typically coarse, binary classifiers that are trained to identify problems in generated text.",
"They are often trained for a single task, such as predicting toxicity [150], [40], [215], or factuality [108], [65], [194] of the outputs.",
"Early approaches for detecting problematic text focused on building linear classification models using hand-designed textual features like lexicons (such as hatebase) [214], [37], [20], [129], $n$ -grams, capitalization/punctuation details [29], [201], [150], among others [219], [21].",
"In misinformation detection, features like the presence of novel words in generated document summaries can indicate hallucination [229], [101].",
"While such classifiers support quick development and are interpretable, they tend to overfit to lexical features, are prone to false positives, and are easy for malicious users to bypass [113].",
"Neural text classifiers, on the other hand, can incorporate contextual information and have shown to be more robust [60], [162].",
"When built by finetuning pretrained LMs instead of training from scratch, they naturally lead to even better performance [51], [215].",
"Training supervised text classifiers naturally relies on labeled datasets.",
"For tasks like toxicity detection, annotated datasets in several domains, especially to detect overtly toxic text, have been collected for English [39], [201], [207], [160], [146], [145].",
"Such resources in other languages are still lacking [154].",
"Human annotation efforts for more subtle types of toxicity like microaggressions, however, have not been as successful due to annotators' own biases [17].",
"Hence, unsupervised or distantly supervised approaches have been adopted to detect it [105], [58], [152].",
"Information-related harms can arise either inadvertently (due to model errors) or deliberately (due to malicious uses of LMs, such as neural fake news or misinformation bots).",
"Malicious manipulations can be detected by employing approaches for automated fact checking, propaganda, or fake news detection using existing annotated datasets [190], [153], [136], [230], [69].",
"However, human annotations for non-factual text have been shown to be unreliable, for both machine generated and human written text [87].",
"Hence, prior works instead have relied on synthetic datasets of non-factual or machine generated text [223].",
"Another commonly used approach is to detect whether text has been machine-generated [52], [62], empowering users themselves to understand the source of the text they are reading [91].",
"In the case of inadvertent misinformation harms, representative annotated datasets are hard to obtain.",
"[108] construct synthetic datasets by introducing heuristic errors into factually correct text to create an incorrect version.",
"Such synthetic datasets can be used to train classifiers [108], [65], but their generalizability is limited, and they have low correlations with human judgements [157]."
],
[
"Challenges:",
"While binary classifiers enable us to detect problematic model outputs, they are only effective if they can achieve high accuracy on tasks that are often inherently hard, highly contextual, and subjective in nature.",
"For toxicity detection, factors like region, political views, and social and cultural background of the users affect whether they will perceive some text as toxic [212].",
"Existing datasets are often biased due to their curation process [49], [206], [64], [179], [108] and can have unreliable annotations [175], [59], [157].",
"Further, as with many black-box models, coarse classifiers overfit to spurious artifacts [71], [139], [112] and amplify biases in their training data [228], [189].",
"For instance, toxicity detectors have been shown to rely on racial markers, disproportionately flagging text with African-American English as toxic [178].",
"Therefore, it is challenging to both construct sound, unbiased datasets and develop accurate models for harm detection."
],
[
"Redacting or Flagging Harmful Text",
"After detecting when harmful text is generated, the most straightforward way a user-facing application can prevent such text from causing harm to users is to not display the text at all (redacting) or to display it with a warning sign (flagging) [218].",
"Product designers can thus use the detection methods described previously to redact or flag problematic text generated by their applications.",
"Even when the detectors are imperfect, explicitly flagging problematic outputs is still useful because it signals users to take all model outputs with a grain of salt.",
"However, this strategy is not always applicable: for example, in speech-based dialogue agents, “displaying” a warning sign is a nontrivial UX decision, and in writing or auto-complete assistants (such as in Gmail Smart Compose), redacting is not an option and simply warning may not dissuade users from accepting the generated text."
],
[
"Output Level Interventions",
"Increasingly, many practitioners are building applications using LMs as APIs without explicit knowledge of how the model was trained or what training data was used.see https://gpt3demo.com/ for examples Such APIs may vary in how much information developers can see: some APIs allow access to all LM parameters, while black box APIs such as GPT3 [41] limit access to model outputs only.",
"Hence, multiple solutions have been proposed in prior literature for intervening on problematic model outputs, ranging from modifying decoding algorithms to editing the outputs with auxiliary models."
],
[
"Decoding Methods",
"Prior works have proposed various search and sampling algorithms to improve the quality of text generated by LMs [67], [56], [81], [141].",
"Recently, [224] introduced methods for controlling these decoding algorithms by promoting or demoting specific properties in the output text.",
"The motivating principle is that while an LM's training data may have contained harmful/non-factual text, it also contained clean data, so by constraining the decoding process to suppress problematic outputs, good outputs emerge.",
"Controls for decoding are implemented using risk detection approaches we discuss in §REF , such as toxicity/bias classifiers [38], [106], [123], factuality metrics [108], [65], or functions that quantify similarity between source and target [172], [173].",
"A simple way in which such detectors can be used to control outputs is rejection sampling or reranking: for a given input, multiple outputs can be generated by beam search or sampling, which can then be reranked using detectors to discard dubious outputs [107], [101].",
"However, this is not effective for multifaceted controls like factuality or for combinations of many controls, since all the generated candidates might be rejected.",
"To tackle these issues, a class of algorithms that we call guided-autoregressive decoding aims to incorporate control by modifying output distributions at every decoding step.",
"One branch of work adopts logical controls, in which developers directly specify sets of words that should or should not appear in the output [132], [159].",
"[211] apply this method to zero out the probabilities of offensive terms, and [102], [131] improve factuality by up-weighting generation probabilities of entities present in the source, using heuristics for transformation tasks.",
"A second branch of work uses the probabilities of harmful text detectors directly, composing them with LM output probabilities to up-weight or down-weight the probabilities of each token based on generated left context [220], [123], [38], [106], [180].",
"More recent work has also explored ways to induce sentence-level control via non-autoregressive controlled decoding.",
"Such decoding algorithms incorporate control using Monte Carlo Markov Chain (MCMC) techniques [79], [165], [143], in which a full sequence is initialized and then iteratively refined.",
"These methods have been applied for reducing toxicity [111], as well as improving fidelity in translation systems [110].",
"While extremely promising, these techniques suffer from slower decoding speed and need further exploration in order to be adopted in products."
],
[
"Post-factum Editing Model Outputs",
"Rather than modifying the decoding algorithms, recent studies have also explored ways to edit or revise model-generated text to remove harmful content.",
"Text editing is a decades-old subfield of NLP that has traditionally focused on fixing errors in machine translation [30], [184], [28] or grammar in human-written text [200].",
"While many of these text editing approaches are applicable to post-editing LM outputs, for the purposes of this survey, we highlight recent work related to rewriting harmful text.",
"The first set of works treat the task of rewriting as a sequence labeling task, where each token in the output sequence is either substituted, deleted, or kept the same [164], [76].",
"This, however, can be limiting in cases where the entire output needs rewriting.",
"For transformation tasks, especially those with supervision, parallel data for editing can be very easily obtained by converting source-target pairs to source-output-target triplets, along with an additional signal indicating errors (which can be obtained using automatic evaluators or human judgment).",
"For more open-ended tasks where this is not feasible, prior works have explored unsupervised solutions to remove bias [133] and semi-supervised solutions [22], [114], [7] to correct factuality errors.",
"These methods create pseudo-parallel data using various heuristics to induce errors in clean text, then train a model to correct them."
],
[
"Challenges",
"Decoding interventions rely on accurate detectors to prevent harmful outputs, so as a result, the challenges in designing robust detectors (§REF ) impact decoding algorithms as well.",
"For example, [217] show that toxicity avoidance algorithms refrain from generating African American English , thereby causing another harm (exclusion) while trying to address the first (toxicity).",
"Additionally, detecting misinformation and factuality can be extremely hard using simple detectors that do not provide a useful signal to guide the decoding process, so prior works have primarily employed heuristics.",
"Finally, controlled decoding algorithms are double-edged in that controls can be reversed by malicious users to inflict harm – to generate hateful messages, or to do targeted manipulation by copying users' personas.",
"However, this risk should not discourage research in decoding algorithms; rather, research on detecting such malicious uses should be conducted in parallel."
],
[
"Model Level Interventions",
"Several recent studies have provided evidence that different model design choices and optimization procedures can result in harmful generations downstream.",
"In this section, we describe approaches for modifying LM parameters to prevent risky generations.",
"We organize these approaches into two categories based on their requirements for computational resources: architecture/training interventions, which require training LMs from scratch and can thus incur heavy costs, and finetuning/model editing interventions, which use limited resources."
],
[
"Architecture and Training Algorithms",
"An LM trained only to predict the next token given left context has no checks on whether the predicted token is grounded in truth, which can lead to misinformation.",
"Thus, several studies explore modifying LM training objectives to incorporate factual information using either knowledge bases (KBs) or graphs [221]: each token prediction is scored not only on its likelihood given the previously generated context, but also on whether the generation is grounded in facts in the KBs [199].Knowledge-augmented LMs is a rich field where most existing work focuses on masked LMs [232] for solving understanding tasks.",
"Here we highlight papers in generation.",
"However, manually curating structured KBs is an arduous and expensive process, so most existing KBs are limited in size.",
"As an alternative, [127] propose using automatically generated KBs to train LMs.",
"In contrast, [116], [43], [88], [82] use unstructured text as knowledge.",
"Known as retrieval-augmented LMs, they are trained with a two-stage approach of first retrieving a document from the training set, then using it as additional context while generating, essentially providing evidence for what the LM generates.",
"Such techniques have already resulted in significant improvements in many NLP tasks: they have improved factual consistency in summarization models [83], fidelity in translation [8], and reliability in dialogue agents [48], [55], [225].",
"Another benefit of retrieval-augmented LMs is that, since they generate using the KBs at test time, these LMs can be dynamically updated by modifying the KBs with new information [99], [75].",
"[198], [92] also use KBs to embed commonsense knowledge in LMs.",
"These existing solutions tackle content-related harms, but future work in reducing discrimination and toxicity in LMs may also benefit from KBs that encode social [27], cultural, [78], and moral norms [77], [93].",
"Relying on KBs at inference time, however, can be computationally prohibitive.",
"Thus, rather than using explicit knowledge, [46] propose to train LMs with time stamps for each document, and show that this helps them discern current facts from outdated ones, as well as makes them amenable to further training with more data in the future.Along similar lines, [98], [70], [26] adopt a setup of training class-conditional LMs where each LM accepts \"control codes\" as an additional input.",
"When trained with data annotated for toxicity or bias, the LMs can be prompted to not generate toxic or biased outputs.",
"More recently, another paradigm gaining popularity in LM training is instruction-based learning, where in addition to the objective to predict next token, models are also trained to solve several NLP tasks with instructions written in natural language [202], [177].",
"These models have been shown to be capable of performing well with new instructions in zero-shot settings.",
"Providing explicit instructions to not generate certain kinds of text has shown some promise [156], [202] and is an interesting avenue for future work.",
"Text transformation tasks, where the goal is to produce text that is factually consistent with the input without hallucinating information, has seen the most innovation in architectural modifications.",
"Studies on these tasks have explored (1) modifying attention mechanisms to specifically capture relationships between entities [148], [231], (2) improving coreference to mitigate gender bias in translation [115], and (3) using text entailment to develop loss functions to improve fidelity [54], [117].",
"Some other notable directions in this space involve fact-aware pretraining [54], [193] and structured learning frameworks [209]."
],
[
"Finetuning and Model Editing",
"The methods discussed so far have focused on mitigating harmful generations by designing and training models from scratch, which can incur heavy environmental and resource costs.",
"In contrast, an alternative branch of work has developed methods for modifying the model parameters of already-trained LMs, which requires much fewer resources.",
"An elementary way of doing this is finetuning (a subset of) an LM's parameters on small, curated datasets which are balanced with respect to demographics, and filtered for nontoxicity [70], [26], [126].",
"Recent work has also shown success via prompt-based learning [197].",
"Prompt-based learning makes use of the generative power of large LMs, while at the same time nudging the distribution of generated text to be more balanced.",
"These approaches have successfully been used to reduce toxicity [61] and exclusion [33], [109].",
"However, balanced data curation may be nontrivial for many settings, especially transformation tasks [220].",
"Finetuning LMs with reinforcement learning (RL) has been suggested as a better alternative [4], [128], [156], [187], [130], [170] for training modern LMs, as RL models do not require carefully curated, balanced datasets and can instead learn from discrete rewards such as human feedback [188].",
"Another less-explored alternative to finetuning is model surgery or editing, which uses harm detection methods we discussed in §REF to identify a specific set of neurons that contribute to harmful generations.",
"Culling such parameters has been shown to reduce toxicity [63].",
"In a similar vein, [42], [144] systematically edit model parameters by adding KBs to LMs trained without them, in order to revise facts memorized by the model.",
"These under-studied approaches hold great promise for more computationally practical approaches to risk mitigation."
],
[
"Challenges",
"An undeniable argument against LMs from scratch or augmenting them with knowledge is its cost, making these interventions infeasible for most researchers and practitioners.",
"However, even for organizations with access to large computing resources, research on training safer LMs lags behind research on training ever-larger LMs on raw data.",
"We attribute this to the difficulty of curating KBs, as well as the decreased training and inference speed that comes with such modifications.",
"Finetuning, on the other hand, is less costly, but may reduce the general purpose utility of LMs and has not been shown to be useful in reducing information-related harms.",
"Future work may benefit from drawing on continual learning [46] and reinforcement learning [156]."
],
[
"Data Level Interventions",
"Training any machine learning model requires data, so a natural approach to creating fairer, more reliable LMs is carefully creating balanced training sets that are broadly representative of different worldviews.",
"This requires dedicated efforts in data curation [86], [94], [96] and novel data pipelines [44].",
"However, as modern LMs practically consume the entire public Internet to train, this is infeasible.",
"Existing works tackling this issue instead devise semi-automated solutions, which we categorize as follows."
],
[
"Data Filtration",
"The main idea here is simple: identify problematic documents in the training set, remove them, and train a safer model on the clean training set.",
"As the training dataset can be extremely large, sophisticated neural filters can be prohibitively slow to apply.",
"Hence, most work has utilized simple filters, such as the presence of \"unwanted\" words [168] or the predictions of linear classifiers [18].",
"Due to their simplistic setup, these approaches admit many false negatives (failing to detect documents with subtler toxicity) and false positives (erroneously flagging documents that discuss sensitive topics and use hateful speech as examples; additionally, removing data from different dialects like African-American Vernacular English, ironically exacerbating risks of marginalization and exclusion [50]).",
"[149] adopt a more sophisticated explanation-based approach: they train an LM on raw data, then feed the LM manually-curated toxic prompts and filter out documents to which the LM assigns high probability, and then retrain the LM on the filtered corpus."
],
[
"Data Augmentation",
"While data filtration aims to remove problematic training samples, data augmentation aims to offset the effect of problematic data by adding safer/healthier examples to existing datasets.",
"[137] explore adding counterspeech to datasets in order to balance out the hate speech already present in web data.",
"Augmentation with synthetically generated data has also been explored for gender bias mitigation in dialogue [47], [124] and translation models [186].",
"While prior work has shown that data augmentation methods are effective, these methods are extremely difficult to scale on large datasets with evolving distributions and definitions of biases."
],
[
"Challenges",
"In addition to the difficulties with filtering and augmentation interventions we describe above, data interventions are primarily designed to address population-centric risks such as discrimination, toxicity, and, to an extent, exclusion and privacy—but not factuality.",
"Even though one could also try to filter non-factual content [66], it is challenging to define [5] and detect facts [6] in the wild, making data interventions insufficient for addressing misinformation and factuality-related harms.",
"Since language, identity, and society are tightly intertwined, aggressive data filtering methods risk further imbalancing already imbalanced data.",
"Besides, even if we were able to perfectly filter out all problematic examples in the training data, and we could train an LM that has never seen any toxic or biased inputs, the model may still degrade when toxic inputs are provided to it."
],
[
"Discussion and Open Challenges",
"We have discussed several interventions for prevent harms that have been proposed in the literature and achieved some success, despite their challenges.",
"However, there is a lot of room for improvement in harm mitigation strategies.",
"Owing to the blackbox nature of neural models, many risks and their repercussions are still not well understood.",
"Besides, the field of LM research is still nascent, and LMs are bound to scale to larger datasets, more languages, and new tasks in the near future.",
"Hence, along with summarizing current research in risk mitigation, we also discuss open problems and avenues for future work."
],
[
"Trade-offs: Where should one intervene?",
"Our primary goal in organizing intervention strategies into a taxonomy is to understand the impact, cost, and limitations of interventions at different stages, as well as to identify which methods can mitigate which kinds of risks.",
"However, different strategies have vastly different computational requirements and may not be feasible for every stakeholder.",
"A significant chunk of the responsibility to develop safer LMs falls on NLP researchers and organizations with access to substantial resources; these stakeholders may be able to implement data or modeling interventions.",
"On the other hand, developers creating applications on top of LMs may have access to neither the training data nor computing resources to design and train an LM from scratch on cleaned data.",
"In such cases, flagging and decoding approaches are more accessible.",
"In fact, there is no clear evidence that one step of intervention is sufficient—rather, much evidence exists that mitigating one type of risk actually exacerbates another (such as in data filtration).",
"In practice, a combination of interventions may be required."
],
[
"Risks exist in LMs in all languages",
"Most research on large LMs, their uses, and their risks is West-centric and primarily conducted on English.",
"There have been various efforts to accommodate linguistic diversity with LMs that can understand or generate text in several languages [120].",
"However, research on associated risks and their mitigation in these multilingual models has not even remotely caught up [158].",
"Further, in principle, many of these techniques are applicable to LMs trained on any language, but the definitions of risks themselves change with context and across cultures (for example, boundaries between what is polite/offensive vary across cultures, and some topics or words common to one culture are considered taboo in others).",
"Hence, there is a dire need to develop cross-cultural, cross-lingual analyses as well as mitigation tools."
],
[
"Harm detection beyond simple classifiers",
"In most of the intervention strategies we discuss, accurately detecting harms in machine-generated text not only informs the solutions, but is also essential in evaluating their effectiveness.",
"Thus, many of the shortcomings of interventions throughout the model development process are at their root due to poorly or under-defined risk detection methods.",
"Current detection methods are primarily based on classifying whether generated text is harmful or not on various axes like toxicity and factuality, but we recommend researchers and practitioners to move beyond simplistic coarse classifiers and towards more fine-grained, interpretable, and explainable analyses [122], [89].",
"Examples of such methods include: [(1)] span-level classifiers that provide local explanations of undesired behavior.",
"These have been applied to detect toxic [215] and non-factual phrases [65] in generated text.",
"Similarly, [135] developed an annotated challenge set for propaganda detection in human-written text, which can be used to train models that can evaluate misinformation in machine generated text.",
"Model probing techniques providing general trends of errors made by models.",
"For example, [157] use human annotators to devise a taxonomy of factual errors made by text summarization models, providing insights for targeted ways to fix them.",
"Similarly, probing techniques previously used for studying the linguistic capabilities of LMs can be adapted to evaluate toxic content in pretrained LMs [155].",
"Instance attribution methods that generate global explanations by finding examples in the training set that influenced certain model decisions [104].",
"Recent work has shown that these techniques can uncover training examples that contain subtly offensive terms, which models learn to amplify [72], [73].",
"And finally, analysis and visualization tools that allow developers to debug how LMs make predictions [62].",
"Systematic evaluation frameworks for mitigation strategies The rapid advancement of LM performance in recent years can, in part, be attributed to the development of systematic evaluation frameworks [196], [195], [13].",
"Such benchmarks generally present a simplistic one-dimensional view of performance, communicated only through accuracy numbers [125].",
"However, practices for evaluating harms in LM generated text, as well the effectiveness of mitigation strategies, are not systematized, with most research papers proposing their own evaluation criteria.",
"While there is an emerging body of work dedicated to benchmarking LM harms (and mitigation approaches) [171], the space of potential harms is huge and intersectional, and existing work only covers a fraction of it—primarily focusing on discrimination and toxicity in a limited societal and application context.",
"Developing a suite of evaluations or augmenting existing generation benchmarks [142] with axes of risk evaluations [174] will encourage development of holistic solutions, bridging discrimination/toxicity and information-related harms—two related directions in which researchers have often developed similar solutions.",
"Since harm detection and evaluation metrics are also an active area of research, this setup can further be improved by adopting a bidimensional setting [97] where new strategies are evaluated on harm prevention tasks, and the evaluations themselves are evaluated against human judgements.",
"Conclusions We present a survey of practical methods and techniques for addressing the societal harms and safety risks of language generation models.",
"Our structured taxonomy covers a wide variety of interventions that can be applied at different stages of the model development pipeline to mitigate harms.",
"This work bridges multiple strands of research and presents an actionable overview on methods for preventing harms from language generation models.",
"Limitations The goal of this survey was to present current research on analysing and mitigating harms of language generation.",
"There are multiple documented and anticipated harms that these models perpetuate, and it is not feasible to address intervention strategies for each of them.",
"We aimed to generalize multiple proposed solutions and present them in a structured form, considering a few popularly studied harms as case studies.",
"Inevitably, certain harms and their mitigation strategies might not have been considered for this survey.",
"Current research in this field is nascent but fast moving.",
"While this survey enlists techniques and approaches that are popular now, there is a potential for them to be replaced with newer research.",
"We anticipate that this survey may need to be updated or even redone to incorporate new research.",
"Ethics Statement In this survey, we present and discuss various risk analysis and intervention strategies to prevent societal harms from LMs.",
"We also comment on common themes across approaches for detecting and resolving population-centric harms (such as toxicity and discrimination) and misinformation related harms, and we recommend future work combining them.",
"First, many datasets and resources we discuss may contain biases, and using them in downstream applications can lead to risks as we have outlined.",
"Second, many techniques we discuss have limitations or are known to exacerbate other kinds of harms [213], and thus, applying them to newer problems may lead to unseen issues.",
"Finally, the interventions we identify to raise general awareness have potential for misuse: a malicious user can further imbalance the data to train even more harmful models, use the models and decoding algorithms to generate fake news, and target marginalized populations.",
"This, however, should not discourage development of mitigation strategies; rather, more work should be done to detect and ban malicious users.",
"This requires not only technological solutions in NLP, but also in social science, social network analysis, and public policy.",
"Organized References See Tab.",
"REF Table: References for strategies adopted for mitigating different kinds of harms."
],
[
"Conclusions",
"We present a survey of practical methods and techniques for addressing the societal harms and safety risks of language generation models.",
"Our structured taxonomy covers a wide variety of interventions that can be applied at different stages of the model development pipeline to mitigate harms.",
"This work bridges multiple strands of research and presents an actionable overview on methods for preventing harms from language generation models."
],
[
"Limitations",
"The goal of this survey was to present current research on analysing and mitigating harms of language generation.",
"There are multiple documented and anticipated harms that these models perpetuate, and it is not feasible to address intervention strategies for each of them.",
"We aimed to generalize multiple proposed solutions and present them in a structured form, considering a few popularly studied harms as case studies.",
"Inevitably, certain harms and their mitigation strategies might not have been considered for this survey.",
"Current research in this field is nascent but fast moving.",
"While this survey enlists techniques and approaches that are popular now, there is a potential for them to be replaced with newer research.",
"We anticipate that this survey may need to be updated or even redone to incorporate new research.",
"In this survey, we present and discuss various risk analysis and intervention strategies to prevent societal harms from LMs.",
"We also comment on common themes across approaches for detecting and resolving population-centric harms (such as toxicity and discrimination) and misinformation related harms, and we recommend future work combining them.",
"First, many datasets and resources we discuss may contain biases, and using them in downstream applications can lead to risks as we have outlined.",
"Second, many techniques we discuss have limitations or are known to exacerbate other kinds of harms [213], and thus, applying them to newer problems may lead to unseen issues.",
"Finally, the interventions we identify to raise general awareness have potential for misuse: a malicious user can further imbalance the data to train even more harmful models, use the models and decoding algorithms to generate fake news, and target marginalized populations.",
"This, however, should not discourage development of mitigation strategies; rather, more work should be done to detect and ban malicious users.",
"This requires not only technological solutions in NLP, but also in social science, social network analysis, and public policy."
],
[
"Organized References",
"See Tab.",
"REF Table: References for strategies adopted for mitigating different kinds of harms."
]
] | 2210.07700 |
[
[
"A radio-detected thermonuclear supernova from a single-degenerate\n progenitor with a helium star donor"
],
[
"Abstract Type Ia supernovae are thought to be carbon-oxygen white dwarf stars that explode after accreting material from a companion star, but despite extensive studies the nature of the companion star is still poorly understood, as is the explosion mechanism.",
"In the single degenerate scenario, the companion is a non-degenerate star that loses material through winds and/or binary interaction, and a few Type Ia supernovae have shown evidence for hydrogen-rich circumstellar material.",
"We present here the study of SN 2020eyj, a unique Type Ia supernova showing delayed interaction with helium-rich, but hydrogen-poor, circumstellar material.",
"This material surrounding SN 2020eyj is revealed by its unusual light curve and infrared emission, narrow helium emission lines and, for the first time ever in a Type Ia supernova, also a radio counterpart.",
"The circumstellar material likely originates from the companion star, providing the first direct evidence for a, so far hypothesized, single degenerate progenitor system composed of a white dwarf and a helium donor star."
],
[
"Optical spectroscopy",
"The first optical spectrum of SN 2020eyj was obtained with the Spectral Energy Distribution Machine (SEDM) [24] mounted on the Palomar 60-inch telescope (P60) [23], +25 days after discovery.",
"All SEDM spectra are automatically reduced and calibrated with pysedm [74], and the quality of the SEDM spectrum of SN 2020eyj was further improved using hypergal (Lezmy et al., in prep).",
"Follow-up spectroscopy was obtained from +131 days onward with the Low-Resolution Imaging Spectrometer (LRIS) [75] on the Keck I telescope, and the Alhambra Faint Object Spectrograph and Camera (ALFOSC) on the Nordic Optical Telescope (NOT) [76].",
"A host spectrum was obtained at +678 days, after the SN had fully faded from view.",
"The spectra were reduced in a standard manner, using lpipe [77] and PypeIt [78], [79] for Keck/LRIS and NOT/ALFOSC, respectively.",
"A log of the obtained spectra is provided in Table REF , and the epochs of spectroscopy are indicated by the diamond markers on top of the light curves in Fig.",
"REF .",
"The spectra were absolute flux-calibrated against the $r$ -band magnitudes using the Gaussian Process interpolated magnitudes and then corrected for Milky Way (MW) extinction.",
"All spectral data and corresponding information will be made available via WISeREP$\\endcsname $https://wiserep.weizmann.ac.il [80].",
"We present the peak SEDM spectrum in Fig.",
"REF and the later sequence of spectra in Fig.",
"REF .",
"The initial spectrum obtained with SEDM is characterized by broad absorption features (Sect. ).",
"The later spectra are shaped by broad Fe lines, in particular the quasi-continuum blueward of 5700 Å, and a prominent Ca ii near-IR triplet.",
"Superimposed on the continuum are narrow He emission lines, as well as H$\\alpha $ .",
"We measure FWHM velocities of the He emission lines and H$\\alpha $ in the spectra obtained with Keck at +131 and +251 days, by fitting a Lorentzian profile to the complete lines, as well as to just the blue wings.",
"The receding red wings in the He and H$\\alpha $ lines are significantly attenuated (Fig.",
"REF and Sect.",
"), so the intrinsic FWHM velocities are better represented by (double) the blue wing FWHM.",
"We report these FWHM velocities in Table REF .",
"The asymmetric line profile we associate with the SN also applies to the H$\\alpha $ emission line, suggesting the presence of H in the CSM.",
"In the spectrum obtained at +131 days, H$\\alpha $ has an equivalent width (EW) of 14 Å, not corrected for contribution by the host.",
"By comparison, the He i emission lines at 5876, 6678 and 7065 Å in the same spectrum have EWs of 47 Å, 43 Å and 61 Å, respectively.",
"As H is easier to ionize than He, the more prominent He lines means that the CSM must predominantly consist of He.",
"By epoch +329 days, the H$\\alpha $ luminosity has dropped to the luminosity of the H$\\alpha $ narrow emission line in the host spectrum obtained at +678 days (Sect.",
")."
],
[
"Optical photometry",
"Follow-up photometry was obtained as part of public and partnership ZTF survey observations [81] with the ZTF camera [82] on the P48 telescope in the $g$ and $r$ bands, and later phases were also covered in the $i$ band.",
"The P48 data were reduced and host subtracted using the ZTF reduction and image subtraction pipeline [83], which makes use of the ZOGY algorithm [84] for reference image subtraction.",
"Following the rationale illustrated in [85], we apply the difference image zero point magnitude to convert fluxes from units in detector data number (DN) to $\\mu $ Jy, and translate fluxes to AB magnitudes.",
"We apply a detection threshold of S/N $\\gtrsim $ 3, and for non-detections we compute 5 sigma upper limits.",
"Table REF lists the ZTF magnitudes and upper limits.",
"Additional photometric epochs were obtained with the Liverpool Telescope (LT) [86], the SEDM on the P60, the LCO telescopes (program id.",
"NOAO2020B-012), and ALFOSC on the NOT, with data reduced and host subtracted using the pipelines described in [87], [88] or standard methods.",
"In this work we also make use of the forced photometry service from the ATLAS survey [89], [90], which contained valuable photometry in the $o$ and $c$ bands.",
"One $i$ -band epoch was obtained from the Pan-STARRS1 data archive [91].",
"ZTF and ATLAS also obtained observations of the location of SN 2020eyj in the nights immediately preceding the first detection, with limiting magnitudes in the ZTF $g$ band on 2020 March 5 and 6 UT of 20.8 and 19.7, respectively, and the (binned) observations in $o$ band from ATLAS on March 5 UT correspond to a limiting magnitude of 20.2.",
"Phases in this study are relative to the first ZTF detection (MJD = 58915.212, 2020 March 7 UT) in rest-frame days, unless stated otherwise.",
"Given the excellent constraints on the nights before, this epoch is also close to the time of first light.",
"All magnitudes are reported in the AB system.",
"The reddening corrections are applied using the [92] extinction law with $R_V=3.1$ .",
"The extinction in the MW was obtained from [93] as E(B$-$ V) = 0.024 mag.",
"The photometric magnitudes of SN 2020eyj are listed in Table REF .",
"The ATLAS and P48 light curves are shown in Fig.",
"REF , binned into 1-night bins to enhance the signal to noise ratio (S/N)."
],
[
"Infrared photometry",
"Following a report [94] of a mid-IR detection of SN 2020eyj in the 2021 data release of the NEOWISE Reactivation (NEOWISE-R) [95] survey, we queried the IPAC Infrared Science Archive$\\endcsname $https://irsa.ipac.caltech.edu/ for any NEOWISE-R detections at the position of SN 2020eyj.",
"After filtering out poor quality data and binning individual exposures following the method described in [96], the SN was recovered in both $W1$ and $W2$ filters (3.4 and 4.6 $\\mu $ m, respectively) in all four 2020 and 2021 epochs, with the earliest detection at +59 days after explosion (Fig.",
"REF and Table REF ).",
"The host is not detected in (stacked) WISE data prior to the SN explosion (Fig.",
"REF , top panels), so we assume the contribution from the host is negligible and all observed flux is due to the SN.",
"Figure: SN 2020eyj was accompanied by a bright mid-IR counterpart.",
"Top left panel: a coadded image of the last NEOWISE-R epoch before the SN explosion, without any sign of the SN host.",
"Top right panel: the coadded image in the W1W1 filter of the 2020 November NEOWISE-R epoch, +261d after explosion, with SN 2020eyj clearly visible.",
"Bottom panel: a mid-IR light curve comparison of SN 2020eyj in the W2W2 filter (4.6 μ\\mu m) to a sample of SNe observed with Spitzer at 4.5μ\\mu m, adapted from , including Type IIn SNe (in black), Type Ia-CSM SNe (in red), Type Ibc SNe (in blue), and Type Ibn SN 2006jc (in lilac).",
"Additionally, the light curves of a sample of SNe Ia from is plotted in green.",
"SN 2020eyj (in pink large circles) is among the brightest SNe observed in the mid-IR, at the level of the Type IIn SN 2010jl and SNe Ia-CSM 2005gj and PTF11kx.",
"The error bars represent 1σ\\sigma uncertainties."
],
[
"Radio",
"We observed SN 2020eyj with the electronic Multi-Element Radio Linked Interferometre Network (e-MERLIN) in two epochs.",
"The first epoch, with a duration on target and phase calibrator of $\\sim $ 16 hours, was conducted on 2021 November 19 (centred on MJD 59538.29), +605 days after discovery and included six e-MERLIN telescopes (Mk2, Kn, De, Cm, Da and Pi).",
"The second epoch was conducted during 6 consecutive days between 2022 April 6 and 12 (mean MJD 59678.59, +741 days after discovery).",
"Between 5 and 6 telescopes (including the Lovell) participated, with some antennae missing part of the runs due to technical problems.",
"Due to the significantly smaller field of view of the Lovell telescope, the pointing centre of the second epoch was shifted by 1 arcmin to include an inbeam calibrator in the primary beam of this telescope.",
"3C 286 and OQ 208 were used as amplitude and bandpass calibrators, respectively.",
"The phase calibrator, J1106+2812, was correlated at position $\\alpha _{\\rm J2000.0}=11^{\\rm h} 06^{\\rm m} 07.2617^{\\rm s}$ and $\\delta _{\\rm J2000.0}=$ 28°12' 47.065”, at a separation of 1.7° from the target and was detected with a flux density of 150 mJy.",
"We centered our observations at a frequency of 5.1 GHz, using a bandwidth of 512 MHz.",
"The data were correlated with the e-MERLIN correlator at Jodrell Bank Observatory (JBO), using 4 spectral windows, each of 512 channels, with 1 sec integrations and 4 polarizations.",
"We calibrated and processed the data using the e-MERLIN CASA pipeline [99] version v1.1.19 running on CASA version 5.6.2.",
"We used the 10 mJy inbeam source to self-calibrate the residual phases and amplitudes of the target source.",
"Cleaning was done with the software package wsclean [100].",
"Final images of the target were produced with a synthesized beam of 80 mas $\\times $ 35 mas at a P.A.",
"of 28°, and 94 mas $\\times $ 71 mas at a P.A.",
"of $-71$ °, in the first and second epoch, respectively.",
"The 1-$\\sigma $ rms of the images is 17 and 8 $\\mu $ Jy beam$^{-1}$ , respectively.",
"The target is detected in both epochs as an unresolved source as characterized with task IMFIT.",
"We estimate the uncertainty of the peak flux density to be a quadratic sum of the image rms and a conservative 10% amplitude scale calibration error.",
"The final flux density of the source is $80\\pm 20$ and $60\\pm 10$ $\\mu $ Jy beam$^{-1}$ in the first and second epoch, respectively.",
"The radio source is located at an average position of $\\alpha _{\\rm J2000.0}=11^{\\rm h} 11^{\\rm m} 47.1763^{\\rm s}$ and $\\delta _{\\rm J2000.0} = $ 29°23'06.45”, with an estimated uncertainty of 10 mas.",
"The average position of the e-MERLIN detections relative to the optical positions of SN 2020eyj is shown in Fig.",
"REF .",
"The radio detection is consistent with the position of the SN in the ALFOSC epoch at +382 days ($r$ band), and the position reported in GaiaAlertshttp://gsaweb.ast.cam.ac.uk/alerts of the detection of SN 2020eyj in $G$ band at +42 days.",
"Figure: The position of the radio detection is consistent with the position of SN 2020eyj in the optical.",
"Left panel: A 3 ' ×3 ' 3^{\\prime }\\times 3^{\\prime } color composite image, obtained with NOT/ALFOSC, of the compact star-forming host galaxy of SN 2020eyj and its environment.",
"Right panel: The average position of the e-MERLIN detections (black circle, 0.01 '' ^{\\prime \\prime } uncertainty), the position reported in GaiaAlerts (GG band, green circle, 0.06 '' ^{\\prime \\prime } uncertainty), and the position of SN 2020eyj in the ALFOSC epoch at +382 days (rr band, red circle, 0.1 '' ^{\\prime \\prime } uncertainty), overlaid on a 4 '' ×4 '' 4^{\\prime \\prime }\\times 4^{\\prime \\prime } Pan-STARRS1 ii-band data of the host."
],
[
"X-ray",
"On 27 April 2022, we observed SN 2020eyj for 3.8 ks with the X-ray telescope XRT between 0.3 and 10 keV aboard the Neil Gehrels Swift Observatory [101], [102].",
"We analyzed the data with the online-tools of the UK Swift teamhttps://www.swift.ac.uk/user_objects/ that use the methods described in [103], [104] and the software package HEASoft version 6.26.1.",
"SN 2020eyj evaded detection down to a count-rate of 0.003 count s$^{-1}$ ($3\\sigma $ limit).",
"To convert the count-rate limit into a flux limit, we assumed a power-law spectrum with a photon index $\\Gamma $$\\endcsname $The photon index $\\Gamma $ is defined as the power-law index of the photon flux density ($N(E)\\propto E^{-\\Gamma }$ ).",
"of 2 and a Galactic neutral hydrogen column density of $1.9\\times 10^{20}$ cm$^{-2}$ [105].",
"Between 0.3–10 keV the count-rate limit corresponds to an unabsorbed flux of $1.1\\times 10^{-13}~{\\rm erg\\,cm}^{-2}\\,{\\rm s}^{-1}$ and a luminosity of $<2.4\\times 10^{41}~{\\rm erg}\\,{\\rm s}^{-1}$ ."
],
[
"SN Ia classification",
"During the peak phase of SN 2020eyj, an optical spectrum was obtained with the low-resolution (R$\\sim $ 100) SEDM on the P60, +25 days (rest-frame) after discovery.",
"This high S/N spectrum was characterized by broad absorption features (Fig.",
"REF ), based on which SN 2020eyj was classified as a Type Ia SN at redshift $z=0.03$ [22].",
"Using SNIascore, a deep-learning-based classifier of SNe Ia based on low-resolution spectra, [106] noted that the SN could be a Type Ibc SN erroneously classified as SN Ia due to the degeneracy between peak spectra of SNe Ibc with those of SNe Ia at post-peak phases, but their classifier anyway favored a SN Ia classification.",
"In general, based on the comparison study by [31], Type Ibc SNe erroneously classified as Type Ia(-CSM/91T) are a lot less common than the inverse.",
"Here we scrutinize the SEDM spectrum using comparisons with SNe from the literature, based on spectral matching with the SuperNova IDentification (SNID) [107] and Superfit [108] classification tools, where the SNID template library has been supplemented with the Type Ibc templates from [109].",
"We adopt a $g$ -band peak epoch of MJD = 58929 $\\pm $ 2, based on the light curve fitting described in Sect.",
"REF , with the uncertainty driven by the poor sampling of our photometry around peak.",
"The top 10 SNID (rlap $>$ 10) and Superfit matches are all of Type Ia, and include normal SNe Ia such as SN 2004eo [69] and 91T-like SNe such as SN 2001V [68].",
"The best matching SN of Type Ibc (rlap = 8) is the Type Ic SN 1994I [110], [70], [111], [112].",
"The phases corresponding to the matched SNe Ia are all post-peak, ranging from +12 days to +$\\sim $ 50 days, whereas the matching SN Ibc spectra are all within a few days from peak.",
"The phase of the SEDM spectrum of SN 2020eyj is +12 days post peak, which corroborates the SN Ia classification.",
"In terms of spectral features, the SEDM spectrum shows broad absorption lines that based on the spectral comparisons can be unambiguously identified as Si ii, Fe ii and Ca ii (Fig.",
"REF ).",
"Compared to normal SNe Ia as exemplified by SN 2004eo, the Si ii features in SN 2020eyj are quite shallow.",
"Diluted Si ii absorption is common for 91T-like SNe Ia, as in the spectrum of SN 2001V.",
"Type Ia-CSM are known to show 91T-like spectra around peak [31].",
"As a SN strongly interacting with a CSM, the presence of diluted Si ii in the SEDM spectrum of SN 2020eyj is consistent with a Type Ia(-CSM) classification.",
"In terms of expansion velocity, the velocity of the Si ii $\\lambda $ 6355 absorption feature in the SEDM spectrum is $8900 \\pm 600$ km s$^{-1}$ .",
"This velocity is on the slow side for the SN Ia sample described in [113], but consistent with the [114] SN Ia sample and comparable to, for example, SN 2004eo (Fig.",
"REF ).",
"Another notable feature in the SEDM spectrum is the complete lack of O i 7774 Å absorption (Fig.",
"REF ).",
"Even though O i absorption in SNe Ia is quite common, in particular 91T-like SNe Ia can have shallow or non-existent O i [115].",
"This is clearly visible in the matched spectrum of SN 2001V.",
"In contrast, SNe Ibc that lack O i absorption are extremely uncommon, especially at $\\sim $ 12 days post peak [116], [117].",
"An absence of oxygen lines is typical for Type Ia-CSM spectra, both as an absorption feature around peak and as emission in later epochs [28], [118], as seen in the early and late spectra of PTF11kx in Figs.",
"REF and REF , respectively.",
"Similarly, the late spectra of SN 2020eyj lack any sign of O i $\\lambda $ 7774 emission (Fig.",
"REF ).",
"Other features in the late-time spectra of SN 2020eyj that are typical for Type Ia-CSM include prominent broad Ca ii emission and a high H$\\alpha $ /H$\\beta $ Balmer ratio, which indicates that the emission lines are likely produced through collisional excitation rather than recombination [28].",
"The high S/N spectrum at +251 days shows both H$\\alpha $ and H$\\beta $ emission, but after correcting for contribution by the host, only H$\\alpha $ shows some residual flux related to the transient.",
"In conclusion, based on its spectral features we classify SN 2020eyj as a Type Ia(-CSM) SN.",
"Furthermore, as we discuss in Sect.",
"REF , the light curves of SN 2020eyj show strong similarities to those of PTF11kx, the SN that cemented SNe Ia-CSM as a subclass."
],
[
"Light curve fits",
"The light curve of SN 2020eyj (Fig.",
"REF ) can be divided into two phases, similar to its spectral evolution.",
"In the first phase, lasting $\\sim $ 50 days, the light curve follows a fairly typical bell-like shape, peaking at m$\\sim $ 17.2 in both the $r$ band and the ATLAS bands, which at a luminosity distance of 131.4 Mpc (Sect. )",
"corresponds to M$\\sim -$ 18.4, not accounting for host extinction.",
"During the first phase the light curve has a red $g-r$ color, consistent with the classification spectrum.",
"The second phase, the tail phase from +50 days onward, is characterized by a slowly evolving light curve with spectra that are dominated by CSM interaction.",
"While the $r$ -band light curve continues to fade, albeit at a slower rate of $\\sim $ 0.6 mag per 100 days between day +50 and +251, the $g$ -band light curve plateaus.",
"This results in a $g-r$ color change to blue (see bottom panel of Fig.",
"REF ), which based on the spectra is driven by the pseudo-continuum blueward of 5700 Å.",
"This Fe ii feature, typical for CSM-powered spectra, is well traced by the ZTF $g$ band (4100 $-$ 5500 Å).",
"From +251 days onward, the light curve fades in all bands at a rate of $\\sim $ 1 mag per 100 days.",
"The transition between the two phases is well captured by the photometry at +50 days, when the decline in $g$ band is abruptly halted and changes to a plateau lasting $\\sim $ 200 days.",
"This divergence of the $g$ -band light curve from a smooth decline is likely the epoch where CSM interaction starts contributing (significantly) to the light curve, and where the spectra start to look like those of SNe Ibn.",
"But even though the late spectra may be similar to SNe Ibn, the light curve is unlike those of documented SNe Ibn.",
"SNe Ibn are characterized by uniform rapidly evolving blue light curves [27], peaking at $M_r\\sim -19.5$ .",
"There are a handful of long-lived, slowly evolving SNe Ibn reported in the literature, but they are either much brighter than SN 2020eyj [119], [120] or have a much longer risetime [121].",
"None of the literature SNe Ibn show a long-duration ($>$ 300 days) slowly evolving light-curve tail like the one observed in SN 2020eyj.",
"It is worth noting there have been suggestions in the literature that some SNe Ibn may come from thermonuclear explosions, hidden by a dense CSM [122].",
"The discovery of SN 2020eyj seemingly supports that notion.",
"The post-peak decline rates and peak magnitudes of SNe Ia are strongly correlated (the Phillips relation [123]), with brighter (fainter) SNe Ia evolving slower (faster).",
"We fit the first phase of the multi-band light curves with SNooPyhttps://users.obs.carnegiescience.edu/cburns/SNooPyDocs/html/, to determine if the width (stretch) of SN 2020eyj is consistent with the expected peak luminosity.",
"The light curve of SN 2020eyj up to +50 days is well described by a SN Ia light curve with an adopted stretch of $s_{BV} = 1.2 \\pm 0.1$ and an extinction of $E(B-V) = 0.5 \\pm 0.1$ mag (adopting a total-to-selective extinction ratio $R_V = 2.0$ ), resulting in a peak magnitude $\\sim $ 0.06 mag fainter than expected from the Phillips relation.",
"The required line-of-sight extinction is considerable, but is consistent with the host extinction of E(B$-$ V) = $0.54^{+0.14}_{-0.12}$ mag derived from host galaxy Balmer lines (Sect. ).",
"We apply the same fitting method to the light curve of PTF11kx, consisting of published and previously unpublished photometry.",
"For PTF11kx we adopt the same stretch factor of 1.2, and obtain an extinction of E(B$-$ V) = 0.27 $\\pm $ 0.02 mag, consistent with the extinction A$_V \\sim 0.5$ mag derived by [34].",
"After correcting for the fitted host extinction, the resulting absolute magnitude light curves of SN 2020eyj and PTF11kx are practically identical in $g$ and $r$ band for the first $\\sim $ 45 days, even though the fits are independent (Fig.",
"REF ).",
"The $r$ -band light curves peak at M$_r \\sim -19.3$ for both SNe, consistent with both SNe Ia and SNe Ia-CSM, although both SNe are on the fainter end of the sample of SNe Ia-CSM described by [28].",
"From the light-curve fits we obtain for SN 2020eyj rise times in $g$ and $r$ band of 14 $\\pm $ 2 and 16 $\\pm $ 2 days since discovery, respectively.",
"This is fast for a SN Ia [124], but similar to PTF11kx (Fig.",
"REF ).",
"Figure: The light curves of SN 2020eyj are consistent with a SN Ia and its H-rich analog SN Ia-CSM PTF11kx.",
"We simultaneously fit the gg, rr and ii light curves of the initial peak phases of both SN 2020eyj and PTF11kx with the SN Ia light curve fitter SNooPy.",
"SN 2020eyj is well fit with stretch factor 1.2 and E(B--V) = 0.5 ±\\pm 0.1 mag.",
"Similarly, PTF11kx is well fit with stretch factor 1.2 and E(B--V) = 0.25 ±\\pm 0.02 mag.",
"Panels show the absolute magnitude light curves of SN 2020eyj and PTF11kx, after correcting for the host extinction derived from the fit.",
"From left to right: gg band; rr band for SN 2020eyj and r/Rr/R band for PTF11kx; ii band.",
"Open circles indicate synthetic photometry derived from the spectra.",
"The error bars represent 1σ\\sigma uncertainties.An important caveat about the light curve fit is that the intrinsic decline rate of SN 2020eyj could appear slower because of the contribution by CSM interaction.",
"Based on the color evolution of the light curve, we know from day +50 onward that the CSM contribution is significant, but it is reasonable to assume that some CSM interaction already contributes to the light curve at earlier epochs.",
"This means that the stretch parameter we measure should be regarded as an upper limit, and as a result so is the peak luminosity of the fit.",
"The main purpose of the light curve fit was to demonstrate that the light curve evolution of SN 2020eyj can indeed be reconciled with that of a SN Ia."
],
[
"Bolometric light curve",
"The light curve of SN 2020eyj around peak has limited photometric coverage, both in wavelength and cadence, which hinders the construction of a precise, full bolometric light curve.",
"Instead, we compute the bolometric light curve based on the SN Ia light curve template fit obtained in Sect.",
"REF , for epochs when the photometry (notably $g$ band) still matches well with the fitted light curve (up to +38 days after discovery, Fig.",
"REF ).",
"From the fitted optical light curves, we flux calibrate, correct for host extinction and integrate the spectral time-series from [125] from the UV to the near-IR (1000-25000 Å).",
"For the tail phase, we integrate the Keck spectra at +131 and +251 days from 3000 to 10000 Å, and apply a linear extrapolation in the UV to zero flux at 2000 Å.",
"There is little spectroscopic (Fig.",
"REF ) and color (Fig.",
"REF ) evolution between the Keck spectrum at +251 days and the NOT spectrum at +328 days, so we extend the pseudo-bolometric light curve to the final photometric epoch at +383 days obtained with ALFOSC on the NOT assuming a constant bolometric correction applied to the $g$ -band magnitude.",
"Fig.",
"REF shows the bolometric luminosity inferred from the template fit, the Keck spectra, and the final photometric epoch.",
"The template fit to the initial peak included considerable line of sight extinction of E(B$-$ V) = 0.5 mag (Sect.",
"REF ), but for the CSM-interaction driven tail phase the extinction is poorly constrained.",
"If the line of sight extinction to the SN is primarily due to the host, as the host extinction derived from the Balmer decrement seems to indicate (Sect.",
"), the full SN light curve would be similarly affected.",
"However, if the dust extinction is local to the SN, geometrical effects and/or dust destruction could result in variable extinction between the SN Ia peak and the tail phase.",
"As such, for the tail phase we show in Fig.",
"REF two extreme cases of E(B$-$ V) = 0.5, as applied to the peak phase, and E(B$-$ V) = 0 mag.",
"Depending on the tail phase dust extinction, the total integrated energy radiated across the bolometric light curve amounts to $0.6 - 1.2\\times 10^{50}$ erg.",
"Figure: The bolometric light curve of SN 2020eyj can be described with a radioactive decay model for the peak phase, and an optically thick wind for the tail phase.",
"For the initial SN Ia peak of SN 2020eyj we adopt the bolometric light curve (solid blue line) accompanying the SN Ia template fit to the grigri photometry (Sect. ).",
"Overplotted are the associated bolometrically corrected luminosities up until +40 days.",
"From epoch +46 days onward the SN Ia template fit does not accurately describe the observed (gg band) photometry any longer (Fig. ).",
"The blue dotted line shows the continuation of the bolometric light curve of the underlying SN Ia.",
"The three measurements in the tail phase are based on the integration of the two Keck spectra, extrapolated to the UV, and a bolometrically corrected photometric ALFOSC epoch.",
"We show both the measurements assuming no line of sight extinction (in green) and an extinction of E(B-V) = 0.5 mag (in orange).",
"The dotted lines represent the fits to the tail phase measurements using the analytical model from , following the same color scheme for the level of extinction.",
"In the transition region from the diffusion peak to the CSM interaction-powered tail, between +50 and +100 days, the sum of the models would overestimate the luminosity, suggesting the CSM configuration is more complicated than a simple wind-like density profile."
],
[
"Host galaxy",
"The host of SN 2020eyj is a faint and compact galaxy with designation SDSS J111147.15+292305.9 (Fig.",
"REF ).",
"We retrieved science-ready co-added images from the Galaxy Evolution Explorer (GALEX) general release 6/7 [126], the Sloan Digital Sky Survey data release 9 (SDSS DR 9; [127]), and the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS, PS1) DR1 [91] and measured the brightness of the host using LAMBDAR$\\endcsname $https://github.com/AngusWright/LAMBDAR (Lambda Adaptive Multi-Band Deblending Algorithm in R; [128]) and the methods described in [129].",
"We augment this data set with an optical $r$ -band image obtained with ALFOSC on the NOT on 2022 May 2 and UV observations from Swift/UVOT from 2022 April 27.",
"The photometry on the UVOT images was done with uvotsource in HEASoft and an aperture encircling the entire galaxy (aperture radius $8^{\\prime \\prime }$ ).",
"Table REF lists all measurements.",
"We fit the host galaxy SED with the software package Prospector$\\endcsname $https://github.com/bd-j/prospector version 0.3 [130] to determine the host galaxy properties.",
"We assumed a Chabrier initial mass function [131] and approximated the star formation history (SFH) by a linearly increasing SFH at early times followed by an exponential decline at late times (functional form $t \\times \\exp \\left(-t/\\tau \\right)$ , where $t$ is the age of the SFH episode and $\\tau $ is the $e$ -folding timescale).",
"The model was attenuated with the [132] model.",
"The priors were set identical to [129].",
"The fit resulted in a low host-galaxy mass of $\\log (M/M_\\odot ) = 7.79^{+0.15}_{-0.34}$ .",
"We obtained a host galaxy spectrum with LRIS/Keck after SN 2020eyj had faded from view, at +678 days past discovery.",
"We identify unresolved ($\\lesssim 150$ km s$^{-1}$ ) host galaxy lines in the spectrum, such as numerous Balmer lines in emission and absorption, [N ii] $\\lambda \\lambda $ 6548,6583, [O ii] $\\lambda \\lambda $ 3726,3729, [O iii] $\\lambda \\lambda $ 4959,5007 and [S ii] $\\lambda \\lambda $ 6716,6731, based on which we measure a redshift of $z = 0.0297 \\pm 0.0001$ .",
"Adopting a flat cosmology with $H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ and $\\Omega _M$ = 0.3, this redshift corresponds to a luminosity distance to SN 2020eyj of 131.4 Mpc, which we use throughout this paper.",
"Correcting for MW extinction the adopted distance results in a host galaxy absolute magnitude of M$_r = -15.8$ .",
"Based on the Balmer decrement measured in the host spectrum, we estimate a host extinction with E(B$-$ V) = $0.54^{+0.14}_{-0.12}$ mag, in agreement with the extinction obtained by fitting the light curves of SN 2020eyj with a SN Ia template (Sect.",
"REF ).",
"The line ratios of log$_{10}$ ([O iii] $\\lambda 5007$ / H$\\beta $ ) = 0.39 and log$_{10}$ ([N ii] $\\lambda 6583$ / H$\\alpha $ ) = $-1.26$ put the host galaxy well into the regime of star forming galaxies on the BPT diagram [133].",
"Adopting the parameterisation of the empirical oxygen calibration O3N2 by [134], we obtain an oxygen abundance of 12 + log(O/H) = 8.14 $\\pm $ 0.03.",
"Such a low oxygen abundance is expected for a low mass galaxy [135].",
"The host properties of 16 SNe Ia-CSM were reported in [42], [28].",
"These authors concluded that all objects in their samples exploded in star-forming late-type galaxies (spiral and dwarf galaxies) with absolute magnitudes between M$_r=-20.6$ and $-18.1$ mag.",
"The hosts of 3 SNe in this sample evaded detection in archival SDSS images, implying an absolute magnitude of M$_r>-18$ mag.",
"SN 2020eyj exploded in a markedly low-luminosity star-forming dwarf galaxy with an absolute $r$ -band magnitude of only M$_r = -15.8$ mag (not corrected for host attenuation).",
"However, the modeling of the host galaxy SED and the Balmer decrement reveal non-negligible attenuation of $0<E(B-V)<0.55$ mag ($3 \\sigma $ confidence interval from host SED modeling) or $0.2<E(B-V)<1$ mag ($3 \\sigma $ confidence interval from the Balmer decrement), which would alleviate the apparent extremeness of the host galaxy."
],
[
"Dust properties",
"IR emission is commonly observed in interacting SNe, and can be attributed to the condensation of dust in the SN ejecta or in the shocked CSM, or to pre-existing dust in the CSM that is heated radiatively by the SN emission or by the ejecta/CSM shock interaction (e.g., [136], [137], [138], [139]).",
"The mid-IR luminosity of SN 2020eyj is at a similar level as for the most IR-luminous interacting SNe, such as Type IIn and SNe Ia-CSM, and is 2$-$ 4 magnitudes brighter than the mid-IR luminosities of Type Ia, Ib/c and Ibn SNe (Fig.",
"REF , bottom panel).",
"Assuming optically thin dust, the flux $F_{\\nu }$ can be written as [140]: $F_{\\nu } = \\frac{M_{\\textrm {d}}~B_{\\nu }(T_{\\textrm {d}})~\\kappa _{\\nu }(a)}{d^2},$ where $M_{\\textrm {d}}$ is the mass of the dust, $B_{\\nu }$ the Planck blackbody function, $T_{\\textrm {d}}$ the temperature of the dust, $\\kappa _{\\nu }(a)$ the dust absorption coefficient as function of dust particle radius $a$ , and $d$ the distance to the observer.",
"For simplicity, we assume a simple dust population of a single size composed entirely of amorphous carbon with grain size of 0.1 $\\mu $ m with the corresponding absorption coefficient $\\kappa $ as in [141], [142], and fit the WISE data to obtain an estimate of the dust temperature and mass.",
"Over the first three epochs, up to +412 days, we derive a constant dust temperature of $\\sim $ 800 K (Table REF ), consistent with a lack of color evolution in the WISE photometry (Fig.",
"REF ).",
"Only at the fourth WISE epoch (+614 days) do we see a significant drop in the dust temperature, to $608\\pm 23$ K. These dust temperatures are well below the expected evaporation temperature of dust (1500 K for silicates and 1900 K for graphite grains, e.g.",
"[139]).",
"In addition to the dust temperatures, we obtain dust mass estimates of $(1.8\\pm 0.3) \\times 10^{-3}$ $M_{\\odot }$ to $(9.9\\pm 2.1) \\times 10^{-3}$ $M_{\\odot }$ for the first and the final WISE epochs, respectively (Table REF ).",
"The dust mass estimated for the final epoch corresponds to a CSM mass of $1\\times (0.01/r_{dg})$ $M_{\\odot }$ , where $r_{dg}$ is the dust-to-gas ratio.",
"In the case of optically thin dust that we consider here, the blackbody radius can be interpreted as a lower limit to the radius at which the dust resides.",
"In the case of SN 2020eyj, the blackbody radius is $(2.5\\pm 0.2) \\times 10^{16}$ cm in the first epoch, and increases thereafter to $(6.4\\pm 0.6) \\times 10^{16}$ cm at +614 days (Table REF ).",
"Assuming a SN ejecta velocity of $10^{4}$ km s$^{-1}$ , by +59 days the ejecta would only have traveled $\\sim 20$ % of the distance inferred from the blackbody fit at that epoch.",
"Combined with the constant dust temperature, this suggests that the IR emission of SN 2020eyj is dominated by pre-existing dust getting radiatively heated by CSM interaction emission, as was the case in Type Ia-CSM SN 2005gj [142].",
"Furthermore, because the dust reached a temperature of 800 K as early as +59 days and showed no significant evolution afterwards, it is unlikely that any surrounding dust was evaporated due to the SN, because such hot dust would have dominated the IR flux.",
"For a peak SN luminosity of $\\sim 10^{43}$ erg s$^{-1}$ (Fig.",
"REF ), the dust evaporation radius is $R_{\\textrm {evap}}$ = (0.34–2.6) $\\times 10^{17}$ cm, depending on dust grain size and composition [139].",
"The lack of dust at the sublimation temperature implies that the immediate region surrounding the SN was devoid of dust, much like the CSM-free cavity inferred from the SN light curve.",
"The He i and H$\\alpha $ emission line profiles show the red wing being attenuated with time (Fig.",
"REF ).",
"Such an evolution in the line profiles has been interpreted to result from condensation of dust in the ejecta or the shocked CSM, obscuring the receding red wing of the emission line [143], [138], [144].",
"Similar line profiles have been observed in many SNe Ia-CSM [28] and in the prototypical Type Ibn SN 2006jc, where the evolution of the line profiles was attributed to dust condensing in a cool dense shell produced by the interaction of the ejecta with CSM also producing a substantial IR excess [137].",
"Interestingly, such line profile evolution has also been observed in the He nova V445 Pup, where it was attributed to dust obscuration within the shell [61].",
"In particular, for Type Ia-CSM SN 2005gj dust formation was inferred from line profiles [28], while the bulk of the IR emission was also attributed to pre-existing dust [142].",
"Finally, the prototypical Type Ia-CSM SN 2002ic was also accompanied by a bright IR counterpart, which was attributed to an IR echo arising from the pre-existing dusty CSM [136].",
"Figure: The He and Hα\\alpha emission line profiles in the late spectra of SN 2020eyj show evidence for dust formation.",
"The He i emission lines at 5876 Å, 6678 Å and 7065 Å all show strong attenuation in the receding red wings, and an apparent blue shift over time between the +131 and +251 days epochs.",
"Such line asymmetry is commonly observed in SNe Ia-CSM , and is interpreted as due to the condensation of dust in the ejecta or shocked CSM, obscuring the red wing (Sect. ).",
"The Hα\\alpha emission line at +131 days also shows asymmetry and there is a (minor) decline in flux between the two epochs shown here.",
"By +329 days, the Hα\\alpha luminosity has dropped to the level of the line emission in the host spectrum (Sect.",
")."
],
[
"Optically thick wind",
"Using progenitor models for the He star donor SN Ia channel from [14], [13] investigated the CSM properties resulting from this channel, where accretion from a non-degenerate He star allows the accompanying WD to reach the Chandresekhar limit.",
"The study by [13] focused on the low circumstellar density regime, where the CSM properties in the WD + He star systems still adhere to the stringent CSM constraints imposed by radio non-detections of SNe Ia [9], [10], [11], [12].",
"Here we explore if the models with sufficiently dense CSM, with a wind-like density profile ($\\rho \\propto r^{-2}$ ), can explain the interaction powered light-curve tail of SN 2020eyj and the detections at radio wavelengths.",
"In order to quantify the properties of the CSM, we fit the CSM interaction-powered tail of the bolometric light curve using the analytical model from [13], and use the resulting mass-transfer rates to fit the radio detections.",
"Fig.",
"REF shows the bolometric light curve of SN 2020eyj, with the initial peak described by the SN Ia template fit (solid line), and for the tail phase the luminosities inferred from the Keck spectra at +131 and +251 days and the ALFOSC epoch at +383 days (Sect.",
"REF ).",
"From the light curve described by the SN Ia component alone (solid and dotted line), it is also clear that the late-time light curve of SN 2020eyj cannot be powered by $^{56}$ Ni decay, since the flux integrated across the Keck spectrum at +131 days is already at least ten times larger than what the radioactive decay delivers.",
"Also plotted are the CSM-interaction model fits to the light curve tail, both for an E(B-V) = 0 mag and E(B-V) = 0.5 mag as discussed in Sect.",
"REF .",
"Assuming a pre-SN wind velocity of 1000 km s$^{-1}$ , the CSM-powered tail of SN 2020eyj is consistent with mass-transfer rates between $10^{-3}$ $M_{\\odot }$ yr$^{-1}$ (E(B-V) = 0 mag) and $3\\times 10^{-2}$ $M_{\\odot }$ yr$^{-1}$ (E(B-V) = 0.5 mag), which is 1-2 orders of magnitude larger than considered in the original study [13].",
"At these very high mass-transfer rates, the critical mass accretion rate by the WD is exceeded, and the excess is ejected as an optically thick wind resulting in an extended He envelope [13].",
"In the model the forward shock reaches to $\\sim 10^{17}$ cm in 800 days.",
"If we assume a wind velocity of 1000 km s$^{-1}$ , the CSM mass within $10^{17}$ cm in the models range from 0.3 $M_{\\odot }$ to 1 $M_{\\odot }$ , for E(B-V) = 0 and 0.5 mag, respectively.",
"Fig.",
"REF shows the wind model radio light curves fitted to the radio detections at 5.1 GHz, adopting $\\epsilon _{\\textrm {e}}$ = 0.1 and mass-transfer rates of $10^{-3}$ $M_{\\odot }$ yr$^{-1}$ and $3\\times 10^{-2}$ $M_{\\odot }$ yr$^{-1}$ , for E(B-V) = 0 and 0.5 mag, respectively.",
"We consider both synchrotron emission with synchrotron self-absorption (SSA) and free-free absorption (FFA), but note that at the late phase of the radio detection, FFA only has a minor impact.",
"The radio light curve with an adopted mass-transfer rate of $10^{-3}$ $M_{\\odot }$ yr$^{-1}$ is consistent with the radio detections of SN 2020eyj at 5.1 GHz, with microphysics parameter $\\epsilon _{\\textrm {B}} = 1.7\\times 10^{-3}$ .",
"For the high extinction scenario, with a mass-transfer rate of $3\\times 10^{-2}$ $M_{\\odot }$ yr$^{-1}$ , the model fits when $\\epsilon _{\\textrm {B}} = 1.5\\times 10^{-5}$ .",
"In either case, the late time evolution follows the observed power law decline rate of the observed radio luminosity of $\\beta = -1.6$ , which is comparable to that for hydrogen-free SNe Ibc [145].",
"It is worth noting that the bolometric light curve only extends to +400 days, whereas the first detection of SN 2020eyj at 5 GHz took place at +605 days.",
"Furthermore, it has been argued that the mass-transfer rates associated with the optically thick wind phase ($>10^{-7}$ $M_{\\odot }$ yr$^{-1}$ ) do not lead to SNe Ia, but rather to accretion induced collapse of the WD [146]."
],
[
"CSM shells",
"The CSM surrounding the H-rich analog of SN 2020eyj, PTF11kx, was argued to be concentrated in shells [34].",
"Models for the radio emission of SNe Ia colliding with a constant-density shell of CSM have been previously presented in the literature, along with approximate functional forms to describe the evolution of the optically thick synchrotron light curve [72].",
"Since those models assume hydrogen-rich material, for our calculations we modify $n_e = \\rho /m_p$ to $n_e = \\rho /(2m_p)$ ; otherwise we use the default parameters, notably $\\epsilon _B=0.1$ .",
"We explore shell models with a range of CSM masses $M_{\\mathrm {csm}}=(0.01-1)~M_\\odot $ and interaction end times from $t_\\mathrm {end}=328$ days (the spectrum that does not show prominent He I lines) to $t_\\mathrm {end}=763$ days (the second radio detection) – in this model, interaction must have ended before the second radio detection for the radio emission to have declined between the two observations.",
"We assume a shell inner radius of $R_\\mathrm {in} = (30,000~\\mathrm {km~s^{-1}})(50~\\mathrm {days}) = 1.3\\times 10^{16}~\\mathrm {cm}$ to close the system of equations in the model; then, the ranges of $M_\\mathrm {csm}$ and $t_\\mathrm {end}$ correspond to a range of shell widths $\\Delta R/R_\\mathrm {in}=3.4-7.5$ .",
"For each model we calculate the representative model error as $\\sigma _\\mathrm {mod} = \\max (|L_{\\nu ,\\mathrm {obs}}(t_i)-L_{\\nu ,\\mathrm {mod}}(t_i)| / \\Delta L_{\\nu ,\\mathrm {obs}}(t_i))$ , where subscripts “obs” and “mod” refer to observed and modeled values, $L_\\nu $ is spectral luminosity, and $\\Delta L_\\nu $ is the error on the luminosity (flux error only; error in distance is not included).",
"The best-fit model by this metric has $M_\\mathrm {csm} = 0.36~M_\\odot $ and $t_\\mathrm {end} = 665~\\mathrm {days}$ , which is a very similar mass to what is found for PTF11kx based on analysis of its optical spectra [147].",
"We find models with $\\sigma _\\mathrm {mod}\\le 3$ have $t_\\mathrm {end}\\sim (500-763)~\\mathrm {days}$ and $M_\\mathrm {csm} \\sim (0.2-0.5)~M_\\odot $ , while those with $\\sigma _\\mathrm {mod}\\le 1$ (i.e., a better fit) have $t_\\mathrm {end}\\gtrsim 580~\\mathrm {days}$ and $M_\\mathrm {csm} \\sim (0.3-0.4)~M_\\odot $ .",
"The best fit shell model is shown in Fig.",
"REF ."
],
[
"V445 Puppis",
"The nova outburst of V445 Pup in the year 2000 lacked any Balmer emission in the spectra of its ejecta, but instead was characterized by He and carbon emission lines [58], [59], making it the first and so far only known He nova system.",
"Based on light curve modeling, a mass ($\\ge 1.35$ $M_{\\odot }$ ) close to the Chandresekhar limit was inferred for the WD in V445 Pup [62], consistent with the observed high ejecta velocities up to 8450 km s$^{-1}$ [61].",
"Combined with a high mass-transfer rate $ > 10^{-7}$ $M_{\\odot }$ yr$^{-1}$ , where half of the accreted matter remains on the WD [62], V445 Pup is considered to be a prime candidate progenitor for the single degenerate He + WD SN Ia progenitor channel.",
"Based on infrared spectra showing prominent carbon lines [58], [60], and a rapid decline in the light curve of V445 Pup, it was shown that a carbon-rich thick dust shell must have formed in the nova ejecta [58], [59].",
"High resolution near-IR images resolved the nova event into an expanding narrow bipolar shell with bulk velocities of $\\sim $ 6700 km s$^{-1}$ , and a perpendicular central dust disc that strongly attenuates the optical He i emission lines arising from the receding shell [61].",
"Seven years after the outburst, the bipolar shell of V445 Pup, as imaged in the near-IR, extended to $\\sim 10^{17}$ cm, and the central dust torus had an outer radius (perpendicular to the lobes) of $\\gtrsim 10^{16}$ cm [61].",
"A recent study of the long-lived radio evolution of V445 Pup showed the system was continuously synchrotron luminous for years after the outburst [65].",
"The synchrotron emission originated from the inner edge of the equatorial disc, and was interpreted as interaction between a wind coming off the WD from nuclear burning, and the surviving disc.",
"The persistence of the disc through the nova outburst suggests the disc is at least comparable in mass with the mass of the nova ejecta, which was estimated to be $\\sim 10^{-4}$ $M_{\\odot }$[62].",
"In turn, the mass of the WD in V445 Pup, close to the Chandrasekhar limit, limits the ejecta mass in the system to not more than $\\sim 10^{-3}$ $M_{\\odot }$ ([62], their Fig.",
"7)."
],
[
"ISM",
"Radio emission can potentially arise from a Type Ia SN in the double-degenerate scenario as a result of interaction with the ISM.",
"We have modeled the radio light curve from such a merger scenario in the same way as in [56], [12], i.e., we assume that the supernova is the result of two merging white dwarfs with masses 0.9 and 1.1 $M_{\\odot }$ as described by [148].",
"The outermost ejecta has a density slope $\\propto \\rho ^{-n}$ with $n$ = 13 (see [56] for a discussion on $n$ ).",
"The microphysics parameters are the standard values $\\epsilon _{\\textrm {e}}$ = 0.1 and $\\epsilon _{\\textrm {B}}$ = 0.01.",
"The modeled radio emission increases with time (Fig.",
"REF ), and to agree with the observed 5.1 GHz fluxes at +605 and +741 days, the ISM electron density has to be 660 cm$^{-3}$ and 450 cm$^{-3}$ , respectively, assuming fully ionized hydrogen and helium with He/H = 0.1.",
"For $n$ = 13 and fixed $\\epsilon _{\\textrm {e}}$ , the electron density scales roughly as $\\epsilon _{\\textrm {B}}^{-0.74}$ , so other ISM densities are possible accordingly.",
"For a likely upper limit on $\\epsilon _{\\textrm {B}}$ of 0.1, the ISM density would be $n_{\\textrm {e}}$ = 85 cm$^{-3}$ to fit the flux at the second epoch, and for $\\epsilon _{\\textrm {B}}$ of 0.001, $n_{\\textrm {e}}$ = 2570 cm$^{-3}$ .",
"The increase in radio flux with time is opposite to what is observed, and is a property for all our ISM models with $n > 7.1$ .",
"Lower $n$ -values are not expected [56], and the densities required in our ISM models are much higher than normal ISM densities.",
"Moreover, for the $n$ = 13, $\\epsilon _{\\textrm {B}}$ = 0.01 model, where $n_{\\textrm {e}}$ = 450 cm$^{-3}$ , the modeled flux for the first epoch undershoots by 2 sigma (Fig.",
"REF ).",
"In summary, our radio observations and their modeling argue strongly against an ISM scenario, which arises from a double degenerate progenitor system.",
"Furthermore, the observed strong helium lines are also at odds with an ISM scenario [149].",
"We therefore conclude that SN 2020eyj did not result from the thermonuclear runaway of a WD in a DD progenitor system, leaving the SD scenario as the only viable alternative."
],
[
"Precursor search",
"The CSM surrounding SN 2020eyj could have originated from one or more novae such as observed in V445 Pup.",
"We investigate if a similar outburst at the location of SN 2020eyj can be found in ZTF data going back $>2$ years.",
"The position of SN 2020eyj was observed 772 times (after quality cuts) in the $g$ , $r$ , and $i$ bands across 202 different nights in the final 2.29 years before the SN explosion.",
"There are no significant pre-explosion detections in unbinned or binned light curves (1-day to 90-day long bins) following the search method described by [150].",
"When combining observations in week-long bins we reach a median limiting absolute magnitude of $-$ 14.28 in the $r$ band ($-$ 14.26 in the $g$ band).",
"We can hence rule out precursors that are brighter than $-$ 14 magnitude 21% of the time in the $r$ band (16% of the time in $g$ band).",
"Precursors brighter than magnitude $-$ 15 can be ruled out 49% of the time in $r$ band (39% for $g$ band) in the final 2.29 years before the SN.",
"The nova outburst of V445 Pup peaked at $m_V = 8.6$ [151], which at a distance of 8.2 kpc [61] equates to an absolute magnitude of $M_V = -1$ , far below the detection threshold of ZTF."
]
] | 2210.07725 |
[
[
"EfficientVLM: Fast and Accurate Vision-Language Models via Knowledge\n Distillation and Modal-adaptive Pruning"
],
[
"Abstract Pre-trained vision-language models (VLMs) have achieved impressive results in a range of vision-language tasks.",
"However, popular VLMs usually consist of hundreds of millions of parameters which brings challenges for fine-tuning and deployment in real-world applications due to space, memory, and latency constraints.",
"In this work, we introduce a distilling then pruning framework to compress large vision-language models into smaller, faster, and more accurate ones.",
"We first shrink the size of a pre-trained large VLM and apply knowledge distillation in the vision-language pre-training stage to obtain a task-agnostic compact VLM.",
"Then we propose a modal-adaptive pruning algorithm to automatically infer the importance of vision and language modalities for different downstream tasks and adaptively remove redundant structures and neurons in different encoders with controllable target sparsity.",
"We apply our framework to train EfficientVLM, a fast and accurate vision-language model consisting of 6 vision layers, 3 text layers, and 3 cross-modal fusion layers, accounting for only 93 million parameters in total, which is 44.3% of the teacher model.",
"EfficientVLM retains 98.4% performance of the teacher model and accelerates its inference speed by 2.2x.",
"EfficientVLM achieves a large absolute improvement over previous SoTA efficient VLMs of similar sizes by a large margin on various vision-language tasks, including VQAv2 (+4.9%), NLVR2 (+5.6%), ITR (R@1 on TR +17.2%, on IR + 15.6% ) and COCO caption generation (CIDEr +6.5), demonstrating a large potential on training lightweight VLMs."
],
[
"Introduction",
"Inspired by the success of large pre-trained language models [8], [46] in the field of natural language processing (NLP), recent studies [54], [28], [44], [24], [31] in vision-language pretraining (VLP) have advanced the state-of-the-art on various vision-language tasks such as image captioning, visual question answering, and image-text retrieval.",
"However, in both NLP and vision-language domains, large Transformer-based pre-trained models often consist of hundreds of millions, if not billions, of parameters, bringing various practical challenges for deployment.",
"As summarized in [50] and [71], large pre-trained models require large amounts of space (in terms of GPU memory and disk storage) and heavy computing for fine-tuning and inference, which is both costly and may lead to negative environmental impact.",
"Furthermore, large models inevitably lead to low latency, which poses a challenge for the production environment.",
"Recent literature revealed that BERT [8], a popular Transformer-based pre-trained language model, can be effectively compressed and accelerated via knowledge distillation [48], [22], [68], [63].",
"However, only a few prior works investigated building efficient VLMs.",
"For instance, [61] introduced MiniVLM which combines a lighter object detector with a compressed BERT [63].",
"[10] further proposed DistilVLM, which uses knowledge distillation to pre-training a compact VLM with the guidance from a large pre-trained VLM.",
"However, their approach is limited to object-feature-based VLMs.",
"As such, the vision feature extractor cannot be distilled together with the Transformer model in an end-to-end manner, which limits the potential of knowledge distillation.",
"As a result, existing compact VLMs are generally falling short compared to regular-size VLMs.",
"In this work, we investigate strategies for VLM compression and introduce a distilling then pruning framework for compressing fully Transformer-based VLMs.",
"Specifically, in the first stage, we use knowledge distillation for task-agnostic compression of a pre-trained VLM by aligning the logits, attention distribution, and hidden representations between the student model and the teacher model.",
"This results in a task-agnostic compact VLM that achieves competitive results on many downstream vision-language tasks by simply fine-tuning.",
"The general distillation stage reduces the size of all modules (i.e., vision encoder, text encoder, cross-modal encoder) equally so that the compressed model can be versatile to different downstream tasks.",
"However, our preliminary study, which is described in detail in section REF , shows that not all modules are created equal in a VLM and their importance drastically varies on different downstream vision-language tasks requiring different level of understanding on either vision and text modalities.",
"This indicates that compressing a VLM requires modal- and task-specific designs.",
"Therefore, in the second stage, we propose to prune the compact VLM when fine-tuning on different downstream tasks to flexibly adjust the model size/latency according to modal importance.",
"Concretely, we propose a modal-adaptive pruning strategy that regularizes the model with a differentiable approximation to the $L_0$ -norm regularization [38] to automatically infer the importance of vision and language modalities with controllable target sparsity.",
"In this way, our method can adaptively prune different modules in the VLM in the fine-tuning stage according to the relative importance of vision-language modalities on different downstream tasks.",
"We apply our framework to compress X-VLM [73], a recent Transformer-based VLM and train EfficientVLM, a fast and accurate vision-language model.",
"EfficientVLM consists of 6 vision layers, 3 text layers, and 3 cross-modal fusion layers, accounting for only 93 million parameters in total, which is 44.3% of the X-VLM model.",
"EfficientVLM recovers 98.4% performance of X-VLM and accelerates its inference speed by $2.2\\times $ .",
"Experimental results show that despite being trained with fewer image-text pairs, EfficientVLM achieves a large absolute improvement over DistilVLM, the previous best-performing efficient VLM with similar size and inference speed, on various vision-language tasks, including VQAv2 [15] (+6.7%), NLVR2 [55] (+7.8%), ITR-COCO [34] (R@1 on TR +19.9%, R@1 on IR + 15.6% ) and COCO caption generation [5] (CIDEr +6.5), demonstrating a large potential on training lightweight VLMs.",
"To the best of our knowledge, our work is the first attempt to (1) compress a fully Transformer-based vision-language model, and (2) combine knowledge distillation with (modal-adaptive) pruning for vision-language model compression."
],
[
"Vision-Language Pre-training",
"The existing work on vision language pre-training typically falls into two categories.",
"Most methods rely on object detection [57], [39], [30], [54], [28], [6], [32], [13], [31], [70], [36], [33], where an image is represented by dozens of object-centric features.",
"However, the object detection process requires high-resolution images as model input and is very time-consuming.",
"Moreover, most works under this category utilize pre-trained object detectors [47], [2], and do not optimize the model in an end-to-end manner, yielding sub-optimal performance.",
"Therefore, recent works turn to encoding images by convolutional network [21], [20], [19], [62] or vision transformer [24], [29], largely improving the inference speed.",
"Nevertheless, some recent work [74], [73] shows that understanding fine-grained vision language alignments (e.g.",
"object-level) is critical for some downstream tasks such as visual reasoning and visual grounding.",
"Figure: The distilling then pruning framework for training EfficientVLM.",
"In the pre-training stage, we apply knowledge distillation with a pre-trained X-VLM model as the teacher.",
"During fine-tuning, we use a modal-adaptive pruning method to adaptively prune encoders of different modalities."
],
[
"Pre-trained Model Compression",
"Prior work has shown that BERT [8], a popular encoder-only pre-trained Transformer [59], can be effectively compressed and accelerated.",
"As summarized in [71], popular BERT compression techniques include knowledge distillation [18], [48], [56], [22], [63], [76], [69] which trains a compact student network to mimic the behavior of the original teacher model, pruning [27], [41], [14], [49], [26], [65], [66] which prunes redundant neurons or structures in the original model, module replacing [68] which train compact successor sub-modules to replace that in the original model, and quantization [53], [72] that compresses a neural network by reducing the number of bits used to represent its parameters.",
"On the other hand, a number of work also investigated efficient inference with BERT-like models with early exit [58], [67], [35], [51], [75] or adaptive computation time [16], [9].",
"In contrast, only a few prior work investigated methods to compress a pre-trained vision-language model.",
"[10] explored distilling a pre-trained vision-language model into a more compact student model and proposed a teacher adaptation method that aligns object feature proposal.",
"However, their approach is limited to the use of an object detection based vision-language model, which makes end-to-end distillation infeasible and results in unsatisfactory performance compared to recent state-of-the-art.",
"[64] explored distilling a vision-language model with a cross-modal fusion module to a dual-encoder model for efficient retrieval.",
"Moreover, [12] explored the lottery ticket hypothesis [11] in vision-language models and find that sparse winning tickets exist in pre-trained VLMs.",
"However, the process of finding and re-training winning tickets are less efficient compared to other compression methods."
],
[
"EfficientVLM",
"In this section, we present EfficientVLM, a fast and accurate vision-language model trained with our distilling then pruning framework.",
"We choose X-VLM [73], one of the state-of-the-art vision-language model, as the teacher model.",
"In practice, our proposed method suits for any VLMs that equipped with modal-specific module such as VLMo [3] or ALBEF [29]."
],
[
"Model Overview",
"EfficientVLM is a compressed version of X-VLM, a fully Transformer-based VLM.",
"X-VLM has the same architecture as ALBEF [29], which consists of an image encoder, a text encoder, and a cross-modal encoder.",
"The image encoder contains 12 transformer layers, while the text encoder and the cross-modal encoder consist of 6 transformer layers respectively.",
"The cross-modal encoder fuses the vision features with the text features through cross-attention at each layer.",
"EfficientVLM shrinks the size of X-VLM by half, thus consisting of 6 vision layers, 3 text layers, and 3 cross-modal layers, accounting for only 92 million parameters in total, which is 43.6% of the X-VLM model.",
"The teacher model is optimized by: 1) aligning the texts and visual concepts, where the alignments are in multi-granularity using a contrastive loss $\\mathcal {L}_\\text{ITC}$ , a matching loss $\\mathcal {L}_\\text{ITM}$ , and a masked language modeling loss $\\mathcal {L}_\\text{MLM}$ ; 2) in the meantime locating visual concepts in the image given the corresponding texts by bounding box prediction loss $\\mathcal {L}_\\text{BBOX}$ .",
"Overall, the vision language pre-training loss is: $\\mathcal {L}_{\\text{VLP}} = \\mathcal {L}_\\text{ITC} + \\mathcal {L}_\\text{ITM} + \\mathcal {L}_\\text{MLM} + \\mathcal {L}_\\text{BBOX}$"
],
[
"Pre-training with Knowledge Distillation",
"We initialize EfficientVLM with a pre-trained X-VLM and shrink its size by half by only retaining the even-numbered layers.",
"Then we pre-train EfficientVLM on image-text pairs with both the original vision-language pre-training objectives of X-VLM and knowledge distillation objective with the pre-trained X-VLM as the teacher model.",
"The knowledge distillation objective consists of attention distillation, hidden states distillation, and logits distillation."
],
[
"Attention Distillation",
"Prior work [22] on BERT distillation have shown the effectiveness of transferring the latent knowledge in self-attention matrices: $\\mathbf {A}=\\operatorname{softmax(}{\\mathbf {Q}\\cdot \\mathbf {K}}/{\\sqrt{d_k}}\\text{)}.$ where $\\mathbf {Q}$ and $\\mathbf {K}$ denotes the query and key matrix in the attention layer of a transformer block.",
"$\\mathbf {d_k}$ is the dimension of the key matrix as a scaling factor.",
"We formulate attention distillation loss by minimizing the mean square error between the self-attention matrices of the teacher and the student: $\\mathcal {L}_{\\text{attn}} = \\frac{1}{h}\\sum \\nolimits ^{L}_{j=1}\\sum \\nolimits ^{h}_{i=1} \\operatorname{MSE}(\\mathbf {A}_{i,j}^{S}, \\mathbf {A}_{i,2j}^{T})$ where $L$ denotes the number of layer in each encoder of the student, $h$ is the number of attention heads, $\\mathbf {A}_i$ refers to the normalized attention matrix corresponding to the $i$ -th head in $j$ -th layer of the student and in $2j$ -th layer of the teacher.",
"The attention matrix is in shape of $\\mathbf {A} \\in \\mathbb {R}^{l\\times p}$ .",
"$l$ and $p$ are the length of query and key, respectivelyIn the cross-attention module of cross-modal encoder, $p$ represents the length of patch sequence of vision encoder otherwise $l$ and $p$ are equal."
],
[
"Hidden States Distillation",
"Following Transformer distillation in TinyBERT [22], we also adopt the hidden states distillation to better utilize the information from the teacher model.",
"The loss function is defined as follows: $\\mathcal {L}_{\\text{hid}} = \\sum \\nolimits ^{L}_{i=1}\\text{MSE}(\\mathbf {H}_{i}^{S}, \\mathbf {H}_{2i}^{T}),$ $\\mathbf {H}^{S} \\in \\mathbb {R}^{l\\times d^{\\prime }}$ and $\\mathbf {H}^{T} \\in \\mathbb {R}^{l \\times d}$ refer to the hidden states of student and teacher networks in the corresponding layer."
],
[
"Logits Distillation",
"In addition to imitating the behaviors of intermediate layers, we also use the knowledge distillation to fit the predictions of teacher model as in [18].",
"We adopt KL divergence as the optimization objective:"
],
[
"Pre-training",
"We formulate the final loss by combing the original vision-language pre-training loss with general distillation loss.",
"$&\\mathcal {L}_\\text{KD} = \\alpha \\mathcal {L}_\\text{attn} + \\beta \\mathcal {L}_\\text{hid} + \\gamma \\mathcal {L}_\\text{logits} \\\\&\\mathcal {L}_\\text{pretrain} = \\lambda \\mathcal {L}_\\text{VLP} + (1-\\lambda )\\mathcal {L}_\\text{KD}$ where $\\alpha $ , $\\beta $ , $\\gamma $ and $\\lambda $ are the weights of the loss terms.",
"We only adjust the weights to scale the losses to similar values so that the optimization process can perform more robust."
],
[
"Fine-tuning with Pruning",
"To flexibly adjust the efficiency-performance trade-off of EfficientVLM on different downstream tasks according to varying resource constraints, we propose a modal-adaptive pruning method to further compress EfficientVLM to a desired size in the fine-tuning stage.",
"Figure: Empirical study of modal-encoders importance on NLVR2 and ITR-COCO tasks."
],
[
"Are All Modalities Created Equal in VLMs?",
"Unlike prior work [26] on BERT pruning where there is only one Transformer encoder, pruning VLMs are more challenging because the importance of vision and language clues may not be equally important [4].",
"This is also verified by our preliminary experiments where we prune 40% attention heads in each encoder and find that the performance drops drastically, which is contrary to prior findings on pruning BERT [41].",
"To this end, we conduct an empirical study to investigate whether encoders for vision / language modalities have similar importance across different vision-language tasks.",
"We prune each encoder in a fine-tuned teacher model at one time while leaving other encoders untouched.",
"From Figure REF , we observe that: (1) the encoders of different modalities have different sensitivity with respect to head pruning, and (2) the difference in sensitivity varies on different downstream tasks.",
"Specifically, on ITR-COCO task, pruning 40% heads in text encoder and cross-modal encoder does not significantly impact performance while pruning vision encoder causes a large performance drop.",
"However, the results on NLVR2 show that text encoder is as important as image encoder in this task while cross-modal encoder are not very sensitive to head pruning.",
"These results suggest that encoders of different modalities are not created equal in a vision-language model, motivating us to explore modal-specific pruning methods for VLMs."
],
[
"Modal-adaptive pruning",
"A naive way to achieve modal-specific pruning is to manually adjust the pruning percentage of different encoders based on the prior observation.",
"Specifically, we consider a baseline that prune 30% parameters out of each encoder as the baseline.",
"Then for ITR-COCO, we prune 10% parameters in the vision encoder while pruning 40% parameters in the text and the cross-modal encoder.",
"For NLVR2, we set this percentage to 10%, 10%, and 60% for image, text, and cross-modal encoders, respectively.",
"These percentages are heuristically adjusted according to the previous findings and the empirical performance.",
"Moreover, the relative sparsity is set to ensure the overall sparsity of the model is similar.",
"Table: Modal-specific pruning results on NLVR2 and ITR-COCO.",
"All models are trained with pruning and knowledge distillation.The results are shown in Table REF .",
"We find that manually specifying sparsity levels for different encoders according to their “importance” leads to substantial improvements, demonstrating the effectiveness of modal-specific pruning.",
"However, manually determining the sparsity for different encoders could be laborious and sub-optimal.",
"Therefore, we propose modal-adaptive pruning, an end-to-end pruning algorithm using a differentiable approximation of $L_0$ regularization [38] to automatically infer the importance of vision and language modalities and adaptively remove redundant structures and neurons in different encoders with controllable target sparsity.",
"Consider a given neural network model $f(\\cdot ; \\mathbf {\\theta })$ parameterized by $\\mathbf {\\theta }= \\lbrace \\theta _j \\rbrace _{j=1}^{n}$ , where each $\\theta _j$ represents an individual parameter weight or a block of weights (e.g.",
"a column of a weight matrix) and $n$ denotes the number of blocks.",
"A pruning strategy of the model can be parameterized by introducing additional binary variables $\\mathbf {z}= \\lbrace z_j \\rbrace _{j=1}^{n}$ such that $z_j \\in \\lbrace 0, 1\\rbrace $ and $\\tilde{\\mathbf {\\theta }} = \\mathbf {\\theta }\\odot \\mathbf {z}\\quad \\qquad \\forall j \\ \\ \\tilde{\\theta }_j = \\theta _j \\, z_j .$ Here $\\tilde{\\mathbf {\\theta }} = \\lbrace \\tilde{\\theta }_j \\rbrace $ denotes the set of model parameters after pruning and its $L_0$ norm, $\\Vert \\tilde{\\mathbf {\\theta }}\\Vert _0 = \\sum _{j=1}^n z_j$ , measures the effective size of the pruned model.",
"The optimization during training can be formulated as minimizing the objective below $\\mathbb {E}_{\\textbf {z}} \\left[ \\, \\frac{1}{D} \\sum _{i=1}^D \\mathcal {L}\\left(\\mathbf {x}_i, \\mathbf {y}_i; \\tilde{\\mathbf {\\theta }} \\right) + \\lambda \\Vert \\tilde{\\mathbf {\\theta }}\\Vert _0 \\, \\right]$ where $\\lbrace \\mathbf {x}_i, \\mathbf {y}_i\\rbrace _{i=1}^D$ are training examples, $\\mathcal {L}$ is the training loss function and $\\lambda > 0$ is a constant hyper-parameter.",
"During training, the masking variables $\\mathbf {z}$ are learned as real numbers in range [0, 1] while during inference all the variable that below a threshold are set to 0 so that our pruned model can achieve the expected sparsity.",
"See Appendix for more details.",
"We also adopt knowledge distillation at fine-tuning with pruning stage to help the student model better preserving capacity on downstream tasks.",
"The final training objective is as follows: $\\mathcal {L}_\\text{ft} = \\lambda \\mathcal {L}_\\text{VL} + (1-\\lambda )\\mathcal {L}_\\text{KD} + \\mathcal {L}_\\text{Lgr}$ where $\\mathcal {L}_\\text{VL}$ represents the task-specific fine-tuning loss brought by the re-parameterized student model, the $\\mathcal {L}_\\text{KD}$ is the task-specific knowledge distillation loss and $\\mathcal {L}_\\text{Lgr}$ infers to the lagrangian loss."
],
[
"Baselines",
"We mainly compare EfficientVLM with two baselines: MiniVLM [61], a compact VLM consists of a lightweight object detection model and a compact Transformers-based vision-language encoder, which is initialized by MiniLM [63], a compressed pre-trained language model; and DistillVLM[10], which adopts the same model architecture with MiniVLM and apply knowledge distillation for further boosting model's performance.",
"For reference, we also include the performance of DistilDualEnc [64], ViLT [24] and X-VLM$_\\text{small}$ in our comparison.",
"DistillDualEnc is a dual-encoder VLM distilled from a fusion-based VLM, ViLT is a single-stream VLM that feeds vision features without using region features nor deep convolutional visual embedders and X-VLM$_\\text{small}$ use the same initialization as EfficientVLM but trained without knowledge distillation or pruning.",
"To make our comparison clearer, we present the size and inference speed of compared models in Table REF .",
"We test model inference time on both GPU and CPU devices which are Nvidia Tesla V100 GPU and Intel(R) Xeon(R) Platinum 8260 CPU @2.40GHz, respectively.",
"Since the number of FLOPs is affected by the input sequence length, we show the input image token length and average text length of each model in their settings in the table.",
"We can see that despite the fully Transformer-based visual feature extractor being heavier on model size, it consumes much less time during inference comparing to MiniVLM.",
"As for the Transformer-based text/fusion module, EfficientVLM is slightly larger than MiniVLM and DistilVLM while much faster thanks to the parallel nature of image and text encoders in its architecture.",
"Despite the extremely efficient vision module of ViLT, it consume more time because of its heavy text and fusion encoder.",
"Specifically, when comparing with their corresponding teacher model, DistilVLM only reduces the inference time of the Transformer encoder by around 15% on GPU, while EfficientVLM achieves a speed-up ratio of 1.9$\\times $ on GPU and 2.2$\\times $ on CPU.",
"Table: Main resultson various downstream vision-language tasks.",
"The top group are teacher models and the 98% performance of them.",
"The bottom group contains previous efficient VLMs and the X-VLM small _\\text{small} baseline."
],
[
"Pre-training datasets",
"We construct our pre-training dataset following [73] 4M-setting using two in-domain datasets, COCO [34] and Visual Genome (VG) [25], and two out-of-domain datasets, SBU Captions [42] and Conceptual Captions (CC) [52].",
"Note that we have cleaned the pre-training datasets to avoid data leaks since downstream V+L tasks have overlaps in images with COCO and Visual Genome.",
"The statistics of our pre-training dataset are presented in the Appendix In practice, the text encoder can be run in parallel with image encoder while being much faster.",
"Therefore, the inference time of text encoders does not actually contribute to the overall actual inference time of the model.",
".",
"Image-Text Retrieval There are two subtasks: text retrieval (TR) and image retrieval (IR).",
"We evaluate X-VLM on MSCOCO datasets.",
"We adopt the widely used Karpathy split [23] datasets.",
"Following ALBEF and X-VLM, we optimize $\\mathcal {L}_\\mathrm {ITC}$ and $\\mathcal {L}_\\mathrm {ITM}$ and fine-tune the model for 10 epochs.",
"During inference, we first compute $s(I,T)$ for all images and texts, and then take the top-$k$ candidates and calculate ${p}^\\textrm {match}(I,T)$ for ranking.",
"$k$ is set to 256 for MSCOCO following [73].",
"Visual Question Answering (VQA 2.0) [15] It requires the model to predict an answer given an image and a question.",
"Following ALBEF and X-VLM, we use a three-layer Transformer decoder initialized by the cross-modal encoder of EfficientVLM to generate answers based on the outputs of the cross-modal encoder.",
"We fine-tune the model for 10 epochs.",
"During inference, we constrain the decoder to only generate from the 3,129 candidate answers following [73], [29].",
"Natural Language for Visual Reasoning (NLVR2 [55]) The task prescribe the model to predict whether a text describes the relations between two images.",
"Following ALBEF and X-VLM, we extend the cross-modal encoder to enable reasoning over two images and performs a domain pre-training step for two epochs.",
"We then fine-tune the model for 10 epochs.",
"Image Captioning The task requires a model to generate textual descriptions of input images.",
"We evaluate X-VLM on the COCO Captioning dataset [5].",
"We report BLEU-4 [43], METEOR [7], SPICE [1] and CIDEr [60] scores on the Karparthy test split.",
"Following [73], we simply adapt EfficientVLM to a multi-modal decoder for caption generation.",
"We train EfficientVLM with language modeling loss for two epoch on 4M data.",
"Then, we fine-tune it on the COCO Captioning dataset for 10 epochs."
],
[
"Experiment Setup",
"Teacher Models We initialized the teacher X-VLM model by a pre-trained CLIP ViT[45] and a pre-trained BERT.",
"We pre-train the X-VLM on 4 million image-text pairs for 20w steps.",
"Then we fine-tune the teacher model on downstream tasks following [73].",
"Pre-training We pre-train EfficientVLM on the aforementioned 4 million image-text pairs for 40w steps with 16$\\times $ V100 32G GPU.",
"We adopt AdamW[37] optimizer and set the learning rate and weight decay as 1e-4 and 0.01 respectively.",
"The batch size is set to 1024.",
"Fine-tuning We combine the modal-adaptive pruning algorithm with knowledge distillation from the fine-tuned teacher models.",
"We set pruning sparsity at 25%.",
"Other fine-tuning hyper-parameters are presented in the Appendix .",
"Table: Ablation study results.",
"The top group shows the effects of gradually adding different distilled knowledge at pre-training stage.",
"We take checkpoints at 10w training steps for evaluation.",
"The bottom group presents ablation experiments of pruning and knowledge distillation at fine-tuning stage."
],
[
"Main Results",
"We present the main results in Table REF .",
"The top group of models denotes the base-size VLMs used as the teacher model for different compact VLMs.",
"We also list the 98% performance of these models for better comparison.",
"Specifically, X-VLM$_\\text{clip}$ We adopted the first version of XVLM model as teacher instead of the latest one that using Swin-Transformer as its vision encoder because the model architecture of Swin-Transformer maskes the general distillation more difficult.",
"is the teacher of EfficientVLM while OSCAR$_\\text{B}$ is the teacher of DistillVLM.",
"In the bottom group, we compare EfficientVLM with other efficient vision-language models as well as the X-VLM$_\\text{small}$ baseline.",
"We can see that EfficientVLM substantially outperforms all compared models by a large margin despite DistilVLM and MiniVLM are trained with 7 million image-text pairs while EfficientVLM is only trained with 4 million image-text pairs.",
"Specifically, EfficientVLM achieves a R@1 of 78.7% and 60.6% on Image Retrieval and Text Retrieval respectively, accounting for a large absolute improvement of of 17.2% and 15.6% compared to the previous compact SoTA VLMs.",
"We also achieve 81.83% and 81.72% accuracy on validation set and test-P set of NLVR2, respectively, surpassing prior efficient VLMs by a large margin.",
"Similar observation can also be found on VQA 2.0 and COCO Captioning, where EfficientVLM achieves 76.2% accuracy and 76.28 on test-dev set and test-std set, and 127.3 CIDEr score, respectively.",
"EfficientVLM also consistently outperforms X-VLM$_\\text{small}$ by a large margin on all datasets despite being more compact and efficient, demonstrating the effectiveness of the proposed distilling then pruning framework.",
"Moreover, we find that EfficientVLM surpasses 98% performance of the teacher model on most datasets.",
"In contrast, DistillVLM underperforms the 98% OSCAR$_\\text{B}$ baseline by a large margin.",
"Actually, EfficientVLM recovers 98.4% performance of X-VLM$_\\text{clip}$ on average, while DistilVLM only retains 89.3% performance of OSCAR$_\\text{B}$ on average.",
"This further confirms the effectiveness of our method."
],
[
"Ablation Study",
"We also conduct a series of ablation study to better understand the effectiveness of EfficientVLM.",
"Impact of Knowledge Distillation We first investigate the impact of different distillation objectives by starting with a small-size X-VLM model pre-trained with its original objectives only.",
"We then gradually add logits distillation, hidden states distillation and attention distillation.",
"The results are shown in the top group of Table REF .",
"We find that adding each component improves the overall performance, demonstrating the effectiveness of combing these components for pre-train distillation.",
"Impact of Fine-tuning Objectives We then study the effect of modal-adaptive pruning and knowledge distillation in the fine-tuning stage.",
"The results are shown in Table REF .",
"First, by comparing the results of EfficientVLM and that in Table REF , we can see that modal-adaptive pruning with learned sparsity for encoders of each modality substantially outperforms manually tuned sparsity.",
"We also find that EfficientVLM performs similarly to the KD-only variant.",
"These results confirm the effectiveness of modal-adaptive pruning.",
"We also find that pruning without distillation results in worse results, demonstrating the necessity of knowledge distillation during fine-tuning.",
"Finally, we can see that simply fine-tuning the compact task-agnostic pre-trained EfficientVLM performs not as well.",
"However, it still outperforms existing baselines by a very large margin.",
"This shows that EfficientVLM can also be used as a good compact task-agnostic VLM.",
"Figure: Ablation study results with different sparsity ranging from 10% to 80% on NLVR2 and COCO Captioning datasets.Impact of Pruning Sparsity We also investigate the performance of our modal-adaptive pruning methods with different target sparsity ranging from 10% to 80%.",
"The results are shown in Figure REF .",
"We can see that EfficientVLM retains over 95% performance of the teacher model with a sparsity of 50% and 40% on NLVR2 and COCO Captioning, respectively.",
"EfficientVLM also outperforms previous best results of compact VLMs with a sparsity up to 70% and 60% on these tasks.",
"This shows EfficientVLM also performs well with larger sparsity."
],
[
"Conclusion",
"We introduce EfficientVLM, a fast and accurate vision-language model trained with a distilling then pruning framework.",
"Empirical results show that EfficientVLM retains 98.4% performance of the base-size teacher model while only preserving 44.3% parameters and achieving a speed-up ratio of $2.2\\times $ .",
"EfficientVLM also achieves a large absolute improvement over previous efficient VLMs such as DistilVLM and MiniVLM, demonstrating a large potential towards lightweight VLMs."
],
[
"Limitations",
"EfficientVLM is applied on X-VLM.",
"However, there are also many recent fully Transformer VLMs achieving comparable or better performance.",
"Therefore, applying our distilling then pruning framework on other state-of-the-art VLMs can be interesting.",
"Also, we do not apply quantization or matrix decomposition, which are also prevalent model compression techniques.",
"Our method is used to compress VLMs.",
"Therefore, ethic considerations of VLMs generally apply to our method.",
"We encourage users to assess potential biases before deploying EfficientVLM."
],
[
"Differentiable $L_0$ -Norm Regularization",
"The formulation of Equation REF is still hard for gradient-based optimization by the discrete nature of masks, but the expectation provides some guidance for empirically effective relaxations.",
"Following prior work[38], [65], [17], we apply Hard-Concrete distribution [40] to relax $\\mathbf {z}$ into continuous space $[0, 1]^d$ .",
"Specifically, $\\mathbf {z}$ is now defined to be a deterministic and (sub)differentiable function of a sample $\\mathbf {u}$ from a uniform distribution, $& \\mathbf {u} \\sim U(0,1) \\\\& \\ \\mathbf {s} = \\text{sigmoid}(\\log \\mathbf {u} - \\log (1-\\mathbf {u}) + \\mathbf {\\alpha }) \\\\& \\ \\bar{\\mathbf {s}} = \\mathbf {s} \\times (r - l) + l \\\\& \\ \\mathbf {z}= \\min (1, \\max (0, \\bar{\\mathbf {s}}))$ Here $l < 0$ and $r > 1$ are two constants used to stretch $\\mathbf {s}$ into the interval $(l, r)^d$ before it is clamped to $[0, 1]^d$ with the $\\min (\\mathbf {1}, \\max (\\mathbf {0}, \\cdot ))$ operation.",
"In this case we have a differentiable closed-form expression for the expected $L_0$ -norm, $\\mathbb {E}\\left[ \\Vert \\tilde{\\mathbf {\\theta }}\\Vert _0 \\right] &= \\sum _{j=1}^n \\mathbb {E}\\left[ z_j > 0\\right] \\nonumber \\\\&= \\sum _{j=1}^n \\text{sigmoid}\\left(\\alpha _j - \\log \\frac{-l}{r}\\right)$ To better control the expected sparsity of the student model, we follow [65] to replace the vanilla $l_0$ objective with a Lagrangian multiplier.",
"Let $t$ be the target model size and $s(\\mathbf {\\alpha })$ be the constrained model size determined by the Hard Concrete parameter $\\mathbf {\\alpha }$ .",
"The Lagrangian method imposes an equality constraint $s(\\mathbf {\\alpha })=t$ by introducing a violation penalty, $\\mathcal {L}_\\text{Lgr} = \\lambda _1 \\cdot (s(\\mathbf {\\alpha })-t) + \\lambda _2 \\cdot (s(\\mathbf {\\alpha })-t)^2$ where $\\lambda _1, \\lambda _2 \\in \\mathbb {R}$ are two Lagrangian multipliers that will be jointly updated during training."
],
[
"Hyperparameters",
"The hyperparameters to reproduce fine-tuning results are in Table REF .Tasks with $^{*}$ need two-stage fine-tuning.",
"Table: Hyper-parameters for fine-tuning on downstream tasks."
]
] | 2210.07795 |
[
[
"Computational Design of Active Kinesthetic Garments"
],
[
"Abstract Garments with the ability to provide kinesthetic force-feedback on-demand can augment human capabilities in a non-obtrusive way, enabling numerous applications in VR haptics, motion assistance, and robotic control.",
"However, designing such garments is a complex, and often manual task, particularly when the goal is to resist multiple motions with a single design.",
"In this work, we propose a computational pipeline for designing connecting structures between active components - one of the central challenges in this context.",
"We focus on electrostatic (ES) clutches that are compliant in their passive state while strongly resisting elongation when activated.",
"Our method automatically computes optimized connecting structures that efficiently resist a range of pre-defined body motions on demand.",
"We propose a novel dual-objective optimization approach to simultaneously maximize the resistance to motion when clutches are active, while minimizing resistance when inactive.",
"We demonstrate our method on a set of problems involving different body sites and a range of motions.",
"We further fabricate and evaluate a subset of our automatically created designs against manually created baselines using mechanical testing and in a VR pointing study."
],
[
"Introduction",
"Kinesthetic garments are an efficient and non-obtrusive way of providing force feedback for human body motion.",
"By augmenting stretchable fabric with strategically designed reinforcements, they offer targeted resistance to motions along specific directions [40].",
"They are part of an emerging trend of soft robotic garments [35] that have the potential to assist human wearers in various ways such as during locomotion [19], [20], rehabilitation [1], and increasing immersion in mixed reality [14], [34], [2].",
"However, relying only on passive mechanical structure for feedback prevents their use in such applications because they require active feedback.",
"In this work, we propose a computational approach for designing active kinesthetic garments that can resist user-defined motions on demand.",
"To implement such adaptive resistance, we rely on electrostatic clutches [15], i.e., pre-fabricated components that provide extremely high stiffness contrast between their active and inactive states.",
"Designing active kinesthetic garments then amounts to determining clutch placements, typically placed over high-strain areas, and finding a passive structure that connects and anchors the active components.",
"Crucially, this layout should result in the garment providing maximal resistance when clutches are active, but minimally interfere with motion otherwise.",
"Designing effective connecting structures requires the understanding of the interaction between stretchable garments in multiple states sliding over a deforming body in multiple poses, a very difficult and unintuitive task.",
"To address this challenge, we formalize the design of active kinesthetic garments as an on-body topology optimization problem whose objective function explicitly balances the opposing goals for active and inactive states.",
"By maximizing the difference in elastic energy between active and inactive states, our formulation encourages layouts in which clutches link disconnected parts of the passive structure.",
"In this way, clutches leverage the passive structure to establish strong, load-carrying paths when active while maintaining freedom of movement otherwise.",
"We implement our formulation within a standard evolutionary optimization algorithm, and produce a set of active kinesthetic garment designs that each target multiple motions spanning different body sites.",
"Our results indicate that designs produced with our approach are highly effective and outperform manually-designed alternatives by significant margins.",
"To further substantiate this analysis, we manufacture a subset of our designs for experimental evaluation.",
"Both mechanical testing and a VR pointing task indicate clear advantages for the designs created with our method.",
"To summarize, we make the following contributions: A computational design pipeline for the automatic creation of active kinesthetic garments that includes a novel objective function that considers active components and multiple motions.",
"A set of fabricated active kinesthetic garments built on compliant material integrating ES clutches as kinesthetic feedback components.",
"A comprehensive evaluation showing the effectiveness of our method in simulation, in a physical validation, and in a VR user study against manually-designed and visual baselines."
],
[
"Related Work",
"We summarize works in the areas of computational methods in garment modeling and augmentation, intersecting with hardware and devices capable of providing body-scale kinesthetic feedback."
],
[
"Body-scale Kinesthetic Haptic Feedback Systems",
"Early work to provide kinesthetic feedback to the body used motors and hydraulic pistons to actuate heavy bulky haptic platforms.",
"More recently, several wearable body kinesthetic feedback systems have been developed, mostly based on electromagnetic motors [8], [38] with rods [36], [4] or cables [3], [12] transmission, and based on pneumatic actuators [14], [10] which are soft and more comfortable at the detriment of a bulkier equipment (pumps, compressors, valves).",
"An alternative way to provide body kinesthetic feedback are passive blocking mechanisms like vacuum jamming [9] (still requiring pumps) and ES clutches [11], [16], [33], [32].",
"In particular, ES clutches offer the advantages of being ultra-thin, light, and soft enabling the design of compliant kinesthetic garment designs.",
"Such kinesthetic systems are typically manually designed to specifically fit a limb/joint and block a certain motion.",
"In contrast, we leverage an automatic design method that models and simulates clutches, allowing us to accommodate any set of motions and body areas.",
"Topology optimization is a powerful method used in engineering disciplines to most efficiently distribute a finite amount of material, typically to minimize compliance [5], [41].",
"The graphics community has also combined compliance minimization with user guided input [25], [37].",
"It has also been demonstrated on elastic materials [39] as well as structures undergoing large displacements [7].",
"Closer to our work, topology optimization has moved into the on-body domain where it has been used for personalized cast design [43] and casts designed for thermal comfort [42].",
"Most recently, Vechev et al.",
"demonstrated the design of kinesthetic garments, which are passively reinforced garments designed to resist a single motion [40].",
"However, this work only formulates a single compliance minimization objective, and thus cannot be used in a setting that leverages active components.",
"We extend this approach in two important ways, first by the addition of a dual objective that considers the active and inactive states of our components.",
"A second important contribution is a formulation that enables optimization for multiple motions.",
"The intelligent design of garments is an emerging discipline with important applications for the general population.",
"Computational design approaches to garment design have recently started to consider motion as a fundamental design quantity in so-called 4D garments [22] that minimize friction and pressure via integrated knitting maps.",
"In addition to minimizing friction during motion, Montes et al.",
"also optimize for pressure distributions and body fit by employing a physically based model of skin-tight garments on the body [26].",
"Vechev et al.",
"augment existing skin-tight clothing with passive reinforced materials to resist a single given motion, employing a more flexible model of the garment that allows cloth to slide and lift-off from the body [40].",
"Optimization of component placement has also been used in soft-robotic garments, in combining elastic cords, clutches, and dampers to reduce the force and power required by a person to generate lower body motion [28].",
"Evolutionary optimization techniques were employed by Gholami et al.",
"for designing garments with optimally placed fabric sensors [13].",
"Muthukumarana et al.",
"integrated combinations of active shape-memory based components into garments allowing for actuation on clothing [27].",
"In our work, we augment garments with active components that generate kinesthetic feedback and design supporting optimization objectives to create efficient structures connecting them.",
"Figure: Pipeline overview, from left to right (a) Input: Designers specify N motions with a single rest pose, and a garment (shirt).",
"The simulated behaviour of the garment is shown, with light blue indicating high strain.",
"Clutches are then placed on the garment over high strain areas.",
"(b) Automatic Design: We simulate the garment under each motion with all clutches ON (top), or OFF (bottom), noting that the energy difference between these states increases until the target coverage is reached.",
"Designs can be sampled at any point in the evolutionary progress.",
"(c) Fabricate: We assemble the ES clutches made of flexible strips sliding in a stretchable textile guide and fabricate the connecting structures attaching everything together to make the final active kinesthetic garment (right)."
],
[
"Computational Design Pipeline",
"Our method supports designers in the task of creating active kinesthetic garments that can resist any motion from a predefined set of movements.",
"The design goals of our pipeline are to enable kinesthetic garments that maximize the feedback felt by users when ES Clutches are active, while minimizing interference with their motion when inactive.",
"Our pipeline consists of three main phases: 1) input — where designers specify motions, garment designs, and clutch placements; 2) automatic design — where our method automatically links clutches with stiff material to satisfy the above design goals; and 3) a fabrication phase.",
"The full computational pipeline is illustrated in Fig REF ."
],
[
"Input",
"Our pipeline requires three components as input: a set of body motion, a base garment, and a predefined number of ES clutches."
],
[
"Motions",
"are specified using the STAR/SMPL parametric human body model [29], [23] which produces a surface mesh $v$ with N = 6890 vertices $\\in \\mathbb {R}^{3n}$ based on a 72 pose $\\mathbf {\\theta }$ parameters.",
"To create target poses, we sample from the AMASS dataset [24] and make individual adjustments to the $\\mathbf {\\theta }$ as needed.",
"We define a motion as a single rest pose $\\bar{\\mathbf {v}}$ and an accompanying deformed pose $\\mathbf {v}$ .",
"A set of motions is defined with $\\mathbf {v}=(\\mathbf {v}_1,\\ldots ,\\mathbf {v}_i)$ deformed poses, and a common rest pose $\\bar{\\mathbf {v}}$ .",
"are modeled as 2D mesh surfaces embedded on the body, initialized with the same rest and deformed nodal positions as the underlying body mesh.",
"A garment in its rest state is defined through nodal positions $\\bar{\\mathbf {x}}=(\\bar{\\mathbf {x}}_1,\\ldots ,\\bar{\\mathbf {x}}_n)\\in \\mathbb {R}^{3}$ and $\\mathbf {x}=(\\mathbf {x}_1,\\ldots ,\\mathbf {x}_n)\\in \\mathbb {R}^{3n}$ when deformed.",
"The connecting structure of the garment is modeled using a bi-material distribution where each triangle element $e$ of the garment mesh is assigned a specific material property.",
"This property is set through the design variable $d^e \\in [{0, 1}]$ for each element $e$ , where 0 and 1 correspond to cloth and reinforced cloth respectively.",
"In our formulation, ES clutches are modeled as rectangular surface meshes that are attached to the garment at a predefined set of vertices.",
"A key requirement for optimal ES clutch operation is that they are initialized in a taut state, that is, all slack must be removed from the system before forces are felt at the endpoints.",
"We create a low-dimensional parametrization of ES clutches that is defined by the following variables: a starting point, a surface direction, and a length and width.",
"From this, we procedurally generate a spline, and extrude a mesh (see Appendix ) with rest vertices $\\mathbf {q}=(\\mathbf {q}_1,\\ldots ,\\mathbf {q}_m)\\in \\mathbb {R}^{3n}$ and $\\bar{\\mathbf {q}}=(\\bar{\\mathbf {q}}_1,\\ldots ,\\bar{\\mathbf {q}}_m)\\in \\mathbb {R}^{3n}$ when deformed.",
"The endpoints $q^c$ of the clutch mesh (three at each end) are connected to the garment mesh using simple quadratic penalty functions, which allows for firm attachment."
],
[
"Automatic Design",
"Finding a passive mechanical structure that optimally connects electrostatic clutches placed by the user is a key challenge in the design of active kinesthetic garments.",
"Recent work by Vechev et al.",
"[40] demonstrated a method for on-body topology optimization using a single compliance-minimization objective (summarized in Appendix ).",
"Such an objective cannot be applied in our setting, as it has no notion of component states, and the single objective does not sufficiently capture the high-level goal of minimizing motion interference when components are inactive.",
"Therefore, we propose to extend this formulation from passive reinforcements to our setting of active kinesthetic garments by (1) distinguishing between active and inactive clutch states by extending the simulation model with stateful components, (2) reconciling the different design goals for active and inactive states through a new state-dependant dual-objective, and (3) accounting for multiple motions."
],
[
"Active Component Model and Simulation",
"ES Clutch stiffness varies according to their state, thus, we model their behaviour using a bi-modal material.",
"We implement this as a neo-Hookean material that resists compression and changes modes depending on the activation vector $\\gamma = [\\gamma _0, \\gamma _1,..., \\gamma _n], \\quad \\gamma _n \\in [{0, 1}]$ .",
"Each $\\gamma _i$ determines the state of clutch $i$ , with 0 and 1 corresponding to inactive and active states, respectively.",
"The Young's modulus of the clutch material is then set to $Y_\\mathrm {clutch}^i=\\gamma _iY_\\mathrm {stiff}+(1-\\gamma _i)Y_\\mathrm {cloth}$ .",
"The elastic energy stored in the clutches during deformation is defined as $E_\\mathrm {clutches}(q, \\gamma )$ .",
"We define penalty terms $E_\\mathrm {body}(v, \\mathbf {q})$ preventing clutches from entering the body, and an additional term $E_\\mathrm {attach} = \\frac{1}{2}k(q^c-x^x_c)^T(q^c-x^x_c)$ that attaches the six endpoint vertices to their respective locations $x_c$ on the garment.",
"Throughout all examples, we set constant values for Young's Modulus to $Y_\\mathrm {cloth} = 0.5\\textrm {MPa}$ , $Y_\\mathrm {reinforced\\_cloth} = 0.5\\textrm {GPa}$ , and $Y_\\mathrm {stiff} = 3.0\\textrm {GPa}$ .",
"We use a Poisson’s ratio of 0.33 for all materials.",
"We combine our active component model, with the garment-on-body model described in [40].",
"The terms $E_\\mathrm {garment}(\\mathbf {x}, d)$ , $E_\\mathrm {body}(v, \\mathbf {x})$ , and $E_\\mathrm {attach}$ are summarized in Appendix .",
"With the model and energies defined above, we perform a quasi-static simulation by solving an unconstrained optimization problem, $\\begin{split}\\mathbf {x}^*, \\mathbf {q}^* = \\arg \\min _{\\mathbf {x}, \\mathbf {q}} \\quad E_\\mathrm {garment}(\\mathbf {x}, d) + E_\\mathrm {body}(v, \\mathbf {x}) + E_\\mathrm {attach}(\\mathbf {x}) \\ + \\\\E_\\mathrm {clutches}(\\mathbf {q}, \\gamma ) + E_\\mathrm {body}(v, \\mathbf {q}) + E_\\mathrm {attach}(\\mathbf {q}) \\ ,\\end{split}$ using the GPU-based L-BFGS [21] optimizer provided by PyTorch [31].",
"We take advantage of GPU parallelism by simulating all states (poses) simultaneously.",
"We consider simulations converged once the gradient norm of (REF ) reaches 1e-7.",
"A central goal for the structural optimization step is to find a material layout such that the garment resists the specified motions as strongly as possible when clutches are active, while showing minimal resistance otherwise.",
"Assuming all-elastic materials, we translate this goal into the requirement that the stored energy of the garment should be maximized when clutches are active, and minimized when they are inactive.",
"Our key insight is to introduce an energy differential objective that combines these opposing goals as $\\begin{split}\\mathbf {d}^* = \\arg \\max _{\\mathbf {d}} \\quad E_\\mathrm {garment}(x_{ON}^*(\\mathbf {d, q, \\gamma }), \\mathbf {d}) \\\\- E_\\mathrm {garment}(x_{OFF}^*(\\mathbf {d, q, \\gamma }), \\mathbf {d}) \\\\ \\textrm {s.t.}",
"\\quad \\sum _e{A_e} d_e=A^*\\ ,\\quad \\mathbf {f}(x_{ON}^*)= \\mathbf {0}, \\quad \\mathbf {f}(x_{OFF}^*)= \\mathbf {0} \\end{split} \\ ,$ where $\\mathbf {q}$ holds the variables of all clutches, and $x_{ON}^*$ , $x_{OFF}^*$ are distinct equilibrium states corresponding to all clutches being active ($\\gamma _i=1 \\forall i$ ) and inactive ($\\gamma _i=0 \\forall i$ ), respectively.",
"To solve this optimization problem with the BESO algorithm, we must compute the per-element sensitivities, i.e., the partial derivatives of the objective function with respect to per-element material assignment variables $d^e$ .",
"Following (REF ), we simply have to sum the sensitivity values for the active and inactive states to obtain a single value that is used in the BESO ranking procedure.",
"Everything else follows the procedure described in [40] and is summarized in Appendix .",
"Whereas the method described in [40] computes static reinforcements for a single target motion, we ultimately want to move towards programmable garments that can resist many motions by use of their active components.",
"To this end, we extend (REF ) to the multi-motion setting by summing contributions for all poses as $\\begin{split}\\mathbf {d}^* = \\arg \\max _{\\mathbf {d}} \\quad \\sum _k \\hat{E}_\\mathrm {garment}^k(x_{k, ON}^*(\\mathbf {d, q, \\gamma }), \\mathbf {d}) \\\\- \\sum _k \\hat{E}_\\mathrm {garment}^k(x_{k, OFF}^*(\\mathbf {d, q, \\gamma }), \\mathbf {d}) \\\\\\textrm {s.t.}",
"\\quad \\sum _e{A^e} d^e=A^*\\ ,\\quad \\mathbf {f}(x_{k, ON}^*)= \\mathbf {0}, \\quad \\mathbf {f}(x_{k, OFF}^*)= \\mathbf {0} \\ \\forall k \\ ,\\end{split}$ where $k$ runs over all input poses.",
"A problem with this simple approach is that the optimization may receive larger rewards for increasing an already good performance for a given pose instead of improving the worst-performing case.",
"We address this problem by normalizing the strain energy density for each pose in a pre-processing step $\\hat{E}_{\\mathrm {garment}}^k = \\sum _e t^e A^e\\hat{W}_\\mathrm {garment}^{k,e}(x^*, d^e), \\ \\hat{W}^{k,e} = \\frac{W^{k,e}}{\\max _e(W^{k,e})} \\ .$ In this way, each pose is given the same importance, irrespective of its initial strain energy, thus encouraging material layouts that more evenly distribute the garment's performance across all input motions."
],
[
"Hardware Details and Fabrication",
"In the last step of the pipeline, designs are fabricated."
],
[
"ES Clutches",
"provide resistance to elongation when active [15], while remaining stretchable with low resistance when inactive.",
"They are thin, light and flexible which make them highly compliant and consume very low power when engaged (e.g.",
"one 14cm by 1cm clutch consumes 12 mW at 350V).",
"The ES clutches from [15] were modified for better integration by making them stiffer to reduce bending, packaging them in elastic guides to keep them fully self-retractable and safer for on-body use.",
"Each ES clutch is composed of 3 parts: an electrode strip, an insulating strip, and a stretchable textile guide.",
"Strips are made of 125 $\\mu $ m metalized polyester films from McMaster-Carr.",
"Films are laser cut into long 1cm wide strips of various lengths.",
"Additionally, insulating strips are covered with a 25$\\mu $ m thick layer of poly (vinylidene fluoride-trifluoroethylene-chlorotrifluoroethylene) from Piezotech-Arkema [15].",
"All designs are exported as meshes and manually processed in Blender.",
"We simplify geometry, and unroll the designs onto flat surfaces using the Paper model plugin (without changing area).",
"As our connecting material, we attached a layer of polyurethane (Siser EasyWeed Stretch) onto 100% cotton fabric.",
"This material combination enables much higher forces than in [40].",
"As base garments we used stretchable GripGrab UV sleeves and Nike Dri-Fit Pro Compression shirts.",
"The different parts of the connecting structure were cut with a Trotec 300 laser cutter and glued onto base garments following marks taken on an experimenter wearing the garment.",
"Next, pressure buttons are riveted at locations where ES clutches connect.",
"Finally, ES clutches are fixed onto the garments using pressure buttons and connected with thin wires to a custom voltage power supply powered by a USB power bank and controlled by Bluetooth (see Fig.",
"REF b).",
"The overall modular system can accommodate different sizes of clutches and slight variations in body sizes."
],
[
"Evaluation",
"We conduct a multi-faceted evaluation of our method showing results for different types of motions and garments in simulation, a mechanical force study, and a VR pointing task."
],
[
"Automatic Designs",
"We show a range of designs produced by our method for a variety of motions and garment types.",
"For all experiments, we use a common rest pose with the body in an A-pose, and sample from a set of motions that include Arms Forward, Arms Raise, Arm Flexion, Arm Extension, Bend Forwards (see Fig.REF ).",
"Three garments are designed using our tool to cover a variety of body sites: a short-sleeve shirt, an arm-sleeve, and a long-sleeve shirt.",
"All clutches are placed manually on the garments, typically over high strain energy areas of the garments (see Fig.",
"REF a).",
"We set the following standard BESO parameters for all experiments: evolutionary rate $ER = 1.5\\%$ , maximum material added per iteration $AR = 1.5\\%$ , material interpolation $p=1.6$ .",
"Similarly, we set the material budget to $A^*= 15\\%$ for all examples except for arm flexion and extension, where we use $A^*$ = 20%.",
"As our primary metric, we use the relative energy density $\\rho (\\gamma ,\\mathbf {d}) = \\frac{E_\\mathrm {garment}(x^*(\\gamma ,\\mathbf {d}),\\mathbf {d})\\cdot A_\\mathrm {dense}}{E_\\mathrm {garment}(x^*(\\gamma ,\\mathbf {1}),\\mathbf {1})\\cdot A_\\mathrm {opt}} \\ ,$ i.e., the ratio between energy density for the optimized and fully dense designs.",
"As the optimization progresses, we expect to see a widening gap in this metric between active and inactive clutch states (see Fig.",
"REF for a visualization)."
],
[
"Single-Motion Designs",
"We begin by showing results for the single motion cases of our method.",
"We target Arm Flexion with a single (8cm) clutch on the elbow, and Arm Extension also with a single (8cm) clutch on the inside of the forearm.",
"We show two separate results in Fig.",
"REF a, and b.",
"Relative to the fully dense design, we see that energy density increases to 1.14 for flexion, and 1.74 for extension when clutches are active.",
"When clutches are inactive, relative energy density decreases to 0.34 and 0.54, respectively.",
"Next, we target single motions on the upper body using three clutches of 15cm length.",
"Fig.",
"REF shows results for Arms Forward, Arms Raise, and Bend Forwards, with increases in relative energy density of 2.13, 1.51, and 2.47 respectively.",
"For deactivated clutches, we observe that relative energy density decreases to 0.48, 0.66, and 0.73 for each design.",
"Figure: Single-Motion designs for (a) Arm Flexion, (b) Arm Extension, (c) Arms Forwards, (d) Arms Raise, and (e) Bend Forwards.",
"Color coding indicates energy density.The ability to resist multiple motions with a single design is an important step towards programmable active kinesthetic garments.",
"We used our method to create three such designs, starting with an arm sleeve design (Fig.",
"REF ) that combines Flexion and Extension.",
"It uses the same 20% material budget as in the single motion designs, but now this material must be distributed to balance performance for two distinct motions.",
"The optimized design achieves relative energy densities of 0.88 and 1.27 for Flexion and Extension, respectively, which is 77% and 73% of the corresponding single-motion designs.",
"For perspective, when evaluating the single-motion designs for Flexion/Extension on the Extension/Flexion motion, the relative efficiency is only 2%/5%.",
"These results are not unexpected as Flexion and Extension are orthogonal motions such that designs optimized for only one of them are ineffective for the other one.",
"Figure: Multi-motion design for simultaneously optimized for (a) Arm Flexion and (b) Arm Extension.",
"This design effectively integrates the single motion designs of Fig.",
"into an intertwined structure (c).Our second design is a shirt that combines three upper body motions as shown in Fig.",
"REF .",
"Many of the features observed in the single-motion versions can be seen here, with clutches linking disconnected reinforcements.",
"It is worth noting that each of these motions leads to a distinct load path (light green/yellow) running through at least one of the clutches.",
"We also compare the performance of the multi-motion design to the single-motion versions.",
"As can be seen in Table.",
"REF , the multi-motion design is within 83%, 72%, and 65% as efficient as the single-motion designs, and yet using the same material budget.",
"The performance of the single-motion designs on motions for which they were not optimized is, again, significantly lower.",
"Additionally, for each motion we show the progress plots of the evolutionary optimization in Fig.",
"REF .",
"As our automatic design method removes material, we see a clear separation in relative energy density for active and inactive states for all three motions.",
"In the inactive mode, the relative energy densities of the garment for each motion are decreased by 0.62, 0.72, and 0.5, showing that our method is able to consistently achieve its minimization objective.",
"Figure: Active kinesthetic shirt designed for the three motions: (a) Arms Forwards, (b) Arms Raise, and (c) Bend Forwards.",
"Strain energy density is shown in color-coding with increasing intensity from dark blue to red.Figure: Progress of the evolutionary optimization algorithm for the Shirt Design for (a) Arms Forwards, (b) Arms Raise, and (c) Bend Forwards.Table: Comparison of garments optimized for a single motion against a garment optimized for all three motions.",
"A higher number corresponds to an increase in relative energy density when clutches are active.Our final example investigates the scalability of our method to more complex scenarios involving five clutches and five motions.",
"The performance of this design exhibits relative energy density increases of 1.41, 0.85, 1.62, 0.64, and 1.16 for the motions Arms Forwards, Arms Raise, Bend Forwards, Arm Flexion, and Arm Extension, respectively.",
"These numbers are comparable to the results obtained for our other multi-motion garments, especially as the allotted per-motion coverage has decreased overall.",
"In general, the more motions a given design supports with the same material coverage target (i.e.",
"15%), the material available per motion will decrease and thus be less energy-dense in the ON state.",
"In this case, the material coverage target can be increased, or the designer can sample from an earlier progression step with higher coverage.",
"Figure: Active kinesthetic long-sleeve shirt with five clutches designed for five motions as indicated."
],
[
"Comparison to Manual Designs",
"We conducted a pilot study to provide a manual baseline for our automatically generated designs.",
"A central question in this context is whether users converge towards particular designs and if those designs exhibit features found in automatically generated ones.",
"We recruited six participants (5M, 1F), two of whom were experts in structural optimization techniques (P2, P3).",
"Using our interactive tool, we asked users to 'draw' stiff material on garment meshes, connecting a set of already placed clutches.",
"Participants were asked to distribute material in such a way as to maximally resist the set of specified motions when clutches are activated.",
"Each participant created two designs, a 2-clutch, 2-motion arm sleeve, covering no more than 20% of the available area, and a 3-clutch 3-motion shirt with a coverage budget of 15%.",
"Each of these designs corresponds to an automatically generated designs shown in the previous section.",
"The secondary goal of minimizing energy when the clutch is inactive was not assigned.",
"Figure: Manually-designed garments for 2-motion arm sleeve (top), and 3-motion shirt (bottom) for participants P1 (left) to P6 (right) with clutches shown in green.",
"Note the large variance among the designs, particularly in the shirt case.",
"The sleeve from P1 and the shirt from P2 were chosen for fabrication.",
"* Denotes expert designed garments.The manually-created results shown in Fig.",
"REF exhibit large variety in their designs.",
"While most examples can be expected to perform reasonably, none of them resembled the automatically generated designs.",
"Compared to the fully dense version, manually-created designs were only 0.48x and 0.27x as energy-dense for the arm sleeve and shirt, respectively.",
"Automatic designs, on the other hand, showed a 1.1x and 1.48x higher energy density.",
"We can see that in the case of designing for a larger number of motions, the effectiveness of user designs drops drastically, while automatically generated designs can maintain a relatively high energy density.",
"Looking at only designs from expert users, we see relative average energy densities of 0.54 for the sleeve, and 0.34 for the shirt, still much lower than our automatic designs.",
"Non-expert designs on the other hand had average relative energy densities of 0.44 for the sleeve and 0.23 for the shirt, showing a much larger drop in performance for the more complex shirt design.",
"Thus, automatic design methods can be especially useful for such users.",
"Table REF summarizes these findings.",
"Table: Performance summary of manually-created designs.",
"We report the energy density of the garment relative to the fully dense design.",
"Note that the automatic design has the highest energy densities across all motions.",
"* Denotes designs by expert users."
],
[
"Physical Validation",
"We seek to quantify the resistive force of our automatically designed garments under the motions for which they were optimized, and compare them against Manual-Design counterparts.",
"We selected two designs for fabrication - the multi-motion arm sleeve and multi-motion short-sleeve shirt.",
"We fabricated both, the designs produced by our automatic method and the corresponding manually-designed garments.",
"For the shirt, we selected the clearly highest performing garment, which was from P2, while for the sleeve, we selected the design from P1.",
"This sleeve design represents a common (line) design seen in literature [33], [32], [11], while having similar performance as other designs.",
"In order to best isolate the impact of the connecting structure, we replace clutches with flexible plastic strips that connect to a force sensor as shown in Fig.",
"REF .",
"For the Arms Forward motion, we mount the force sensor in the upper back, while for the Arms Raise and Bend Forward motions, we mount it on the bottom left clutch location.",
"The target motion is then slowly performed by the experimenter wearing the garment (three trials per motion), while the force is measured using a 10kg DYLY-108 force sensor with an HX711 load cell amplifier (see the Video Figure for visual demonstration).",
"The results shown in Fig.",
"REF indicate that, relative to the manual design, the designs generated by our method were on average two times and up to four times more efficient in terms of force output.",
"These measurements confirm our observations made on simulation results in which, as for the experimental case, the largest difference in relative energy density was observed for the Arms Raise motion.",
"Figure: Physical force measurements.",
"Left: experimental setup with force with clutch replaced by a stand-in equipped with a force sensor.",
"Right: maximum force (N) readings when blocking different movements as labeled for the manually-designed (blue) and automatically generated (green) shirt and sleeve garments."
],
[
"User Evaluation",
"To quantitatively evaluate the ability of our active kinesthetic garments to efficiently block motion, we conduct a user study based on a VR pointing task in which participants were asked to reach targets from a predefined set of locations within their reach.",
"The hypothesis that we seek to test is that, when wearing our optimized designs, users generally need more time to reach targets when clutches are active compared to when they are inactive.",
"A secondary hypothesis is that our automatically generated designs lead to higher blocking efficiency than a user-generated baseline."
],
[
"Procedure and Setup",
"Six healthy adult subjects ($M$ =28.1; $SD$ =4.14;) were recruited.",
"Since we only fabricated one size of our designs, participants were all male and similar in size to the template STAR mesh.",
"All participants wore noise cancelling headphones.",
"The procedure and tasks were described and an introduction to the garments and the active components was given.",
"After donning the garments (shirt and sleeve), clutches were attached and adjusted according to participant size to achieve sufficient pre-tension.",
"The left hand of participants was rested on a tripod such for stability.",
"Participants were then introduced to the VR setting and asked to practice touching the spherical targets with and without clutch activation until they felt comfortable proceeding.",
"The study was implemented in Unity 2021 using a Meta Quest 2 relying on the built-in hand-tracking functionality.",
"Figure: User study setup.",
"Participant wearing Auto-Designed garment and reaching for target (left), and their corresponding motion in the virtual environment (right).We use a within-subject design with two independent variables: Feedback Type {via Auto-Designed, via Manual-Designed, Visual Only} and Target Placement: {Forward, Raise, Bend, Flex, Ext}.",
"Each target is placed to elicit a specific motion from the user, and is color-coded to 4 to participants which target they should touch (see Fig.REF ).",
"As a dependent variable, we measure Time, which starts automatically when the participant's hand leaves the starting position (white sphere), and ends as soon as they touch it again.",
"The main task was to touch a given target in one continuous ballistic back-and-forth motion at a natural speed.",
"For each target placement, three trials were collected for a total of 30 trials, one half with clutches active, the other half with clutches deactivated (Visual).",
"The order of clutch activation was randomized and participants were not told if the clutch was on or off.",
"The order of the feedback type was also randomized.",
"At the end of the study, participants were free to comment on their experience using each garment design.",
"The mean time to reach targets were 1.68s ($\\sigma =0.77)$ for the Auto-Designed condition, 1.32s ($\\sigma =0.43)$ for Manual-Designed, and 1.33s ($\\sigma =0.43)$ for Visual.",
"A longer reach time indicates more impact on the participant's ability to reach the target.",
"The full results are visualized in Figure REF .",
"A two-way repeated-measures ANOVA resulted in a significant effect on feedback type ($F(2, 5) = 29.82,\\ p<.001$ ), target placement ($F(4, 5) = 34.45, \\ p<.001$ ) and interaction ($F(8, 5) = 5.72, \\ p=.004$ ).",
"We conducted a Holm-corrected post-hoc test and found significant differences for feedback type.",
"Our Automatic Design method significantly impacted participant movement time compared to both Manual Design feedback ($p<.001$ ) and Visual feedback ($p<.001$ ).",
"We found no significant difference between Manual Design feedback and Visual feedback.",
"When looking at times across target placements, our Automatic Design method significantly impacted participant movement time for the Bend and Raise motions when comparing to both Visual and Manual Design baselines (both $p<.001$ ).",
"From these results, we see a trend that the automatically designed garments performed better in terms of limiting user motion, particularly when the motions involved larger movements in the upper body.",
"We observe that our Auto-Designed garments performed substantially better in larger motions than the Manual-Design counterparts, results which are in-line with both simulated and force-characterization data.",
"The exception is the Forward motion, where we observed a less substantial impact, possibly due to the fact that participants could twist their body to reach that target.",
"The low performance of the Flex and Ext methods could be due to the fact that we use smaller ES clutches for these motions, and the force was too small compared to the force produced by larger motions (see Fig.",
"REF ), and thus, below a critical threshold that would have an impact on user motion.",
"Thus, our first hypothesis was confirmed for two of the three larger upper body motions.",
"What is surprising is that the performance of the Manual-Design baseline was nearly indistinguishable from the Visual baseline, even for larger motions.",
"In relation to this, two participants commented that they had trouble perceiving any resisting effects of the Manual-Design.This shows that, even with the same active components, our optimization-based approach for designing connecting structures can indeed make the difference between a system having clear or negligible impact on user motion.",
"Figure: User study results showing the average trial time for each feedback type and target location."
],
[
"Example Applications",
"We show four applications enabled by the ability of active kinesthetic garments to selectively and dynamically engage clutches with a single design.",
"Figure: Applications in (a) Workplace Training, (b) Posture Correction, (c) Resistance Training, and (d) VR Gaming."
],
[
"Workplace Training",
"When picking up a virtual box, we activate all ES clutches on the shirt to provide stability to the upper-body, preventing the arms from going through the box, and preventing the user from overly bending their back.",
"More complex motion control could also provide further training and guidance in combination with a complex control loop (i.e.",
"using body pose as an input).",
"Bad posture is a very common problem when sitting at a desk, and many posture correcting shirts already exist to help this issue.",
"However, only active kinesthetic garments can periodically allow the user to go into a slouching posture on demand, in addition to keeping other limbs completely unrestricted (i.e.",
"elbow).",
"Multiple clutches can be selectively activated to resist a target motion, the upper clutch in the case of arm extension, and the lower clutch in the case of arm flexion.",
"The opposing clutch is meanwhile disabled, to prevent full arm-locking.",
"This shows how a single garment can be re-configured at run-time for resisting multiple motions, potentially encompassing a user's entire workout.",
"VR immersion can be increased significantly by providing physical forces when users make contact with the world.",
"In this game, a user practices hitting tennis balls out of the air, and only the upper back clutch is activated on contact, noting that the elbow clutch remains off and does not prevent natural elbow bending during such sports movements."
],
[
"Discussion and Future Work",
"Our user study results indicate that automatically-designed active kinesthetic garments were able to have a significant impact on user motion, whereas the manually designed counterparts could not meet this threshold, indicating the need for automated methods to assist designers in such tasks."
],
[
"Emergent Structural Properties",
"We found in our evaluation three emergent structural properties: 1) no connecting material is isolated from the main structure (no disparate island) 2) all active components are at junctures of connecting material, and 3) overlapping, yet distinct load paths are created for each specific motion.",
"When comparing designs using our dual-objective directly to the single compliance minimization objective in [40] (b), we find that these same properties do not emerge (See Fig.",
"REF ).",
"Each property plays an important role — for example, if clutches are not at junctures, then their activation will have no effect on the user.",
"Similarly, unbalanced load paths and islands of disconnected material may degrade performance for particular motions and comfort respectively.",
"Users on the other hand performed well in terms of connecting clutches, but struggled to balance load paths, leading to very poor performance in particular motions.",
"Figure: Emergent structural properties for the multi-motion shirt designed with dual-objective minimization (a) vs single compliance minimization from (b)."
],
[
"Limitations",
"The main work of the designer in our tool is in the manual placement of ES clutches.",
"As the number of active components and motions grows, the requirement for manual clutch placement may become more and more challenging.",
"Our method can be extended to optimize for ES clutch placement, thereby freeing designers from this task, potentially increasing the relative efficiency of the design.",
"Our method is also limited to a simple mode of activation, where clutches are either all active or inactive.",
"However, clutch activations can be controlled individually and continuously through voltage input that affect the degree of resistance.",
"Accounting for these degrees of freedom during design could further improve efficiency and allow for more targeted resistance to selected motions.",
"While our active kinesthetic garments are fully wearable and mobile, they do not have any sensing capabilities.",
"Integrating sensing could be done via capacitive sensors, which could be optimized based on the same strain-maximization principle used for connecting clutches.",
"Our study was limited to only six male participants and the type of feedback collected was mainly quantitative.",
"Our method could be used to create garments for female users and even personalized garments by simply changing the $\\beta $ and gender parameters in the SMPL/STAR models.",
"Richer VR interaction opportunities can be explored in the future by moving beyond simple button presses and object intersections, for example, by integrating body-pose sensing into the control loop."
],
[
"Conclusion",
"We presented a computational approach for automatic design of active kinesthetic garments that block user-specified body motion on demand.",
"As our core technical contribution, we cast the design of reinforcing structures that connect and anchor individual clutches as an on-body topology optimization problem and introduced a novel objective term that encourages maximum resistance of the garment when clutches are active while minimizing interference with body motion when they are inactive.",
"Our experiments indicate that our designs are highly effective and consistently and significantly outperform user-created designs.",
"The structure optimization techniques developed here have the potential to be useful in the routing and placement of other types of active components such as actuators and sensors.",
"By laying out a theoretical and algorithmic basis for this central problem, we hope that our work will serve as a step toward computational design of highly integrated multi-modal wearable interfaces in the future.",
"This work was supported in part by grants from the Hasler Foundation (Switzerland) and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme grant agreement No 717054."
],
[
"Procedural Clutch Generation",
"ES clutches are defined by the following variables: a starting point, a surface direction, and a length and width.",
"The starting point is defined using barycentric coordinates $(u_e, v_e)$ on a particular element $e$ of the garment mesh.",
"The surface direction is a vector in barycentric space $(\\overrightarrow{u_e}, \\overrightarrow{v_e})$ from the starting point to another barycentric coordinate on the same element $e$ .",
"We start by tracing out a piece-wise linear path of the desired length in the direction of $(\\overrightarrow{u_e}, \\overrightarrow{v_e})$ until an edge is encountered, whereby the vector is converted to Euclidean space $\\in \\mathbb {R}^{3n}$ and rotated to lie on the surface of the next triangle $e_i$ .",
"This is repeated until the length of the vector is exhausted.",
"Two endpoints are produced, one at the starting point, and one at the last barycentric coordinate of where the path finishes.",
"From this path, a mesh is triangulated by creating center vertices at edge intersections and projecting side vertices to the left and right of the path based on $\\overrightarrow{e_n} \\times \\overrightarrow{p_{xyz}}$ , the cross product of the element normal and the path direction in world space respectively.",
"This is scaled by the $width$ parameter.",
"The resulting mesh has rest vertices $\\mathbf {q}=(\\mathbf {q}_1,\\ldots ,\\mathbf {q}_m)\\in \\mathbb {R}^{3n}$ and $\\bar{\\mathbf {q}}=(\\bar{\\mathbf {q}}_1,\\ldots ,\\bar{\\mathbf {q}}_m)\\in \\mathbb {R}^{3n}$ when deformed.",
"We give special treatment to the side vertices of the two endpoints by walking them in an orthogonal direction to the main path using the same walking algorithm outlined above.",
"The endpoints $q^c$ of the clutch mesh (3 at each end) are connected to the garment mesh using simple quadratic penalty functions, which allows for firm attachment.",
"The full path walking and meshing algorithm is fast enough to work in real-time, allowing for rapid user placement and re-positioning of ES clutches."
],
[
"Garment-on-Body Model",
"As our garment model, we use a compressible neo-Hookean material model [6] adapted with a relaxed energy under wrinkling as in [40].",
"This allows the garment to wrinkle under compression without producing geometric artifacts.",
"This results in the garment energy $E_\\mathrm {garment}(\\mathbf {x}, d)$ , which is a function of the garment design $d$ , and the deformed nodal positions $\\mathbf {x}$ .",
"We similarly convert the discrete body mesh to a continuous implicit signed distance field [30], resulting in the energy $E_\\mathrm {body}(v, \\mathbf {x})$ , which pushes back on the garment vertices $\\mathbf {x}$ away from the body.",
"This allows the garment to smoothly slide on top of the body and to lift-off from its surface.",
"To attach the garment to the body in specific areas, we introduce a simple coupling potential, $E_\\mathrm {attach} = \\frac{1}{2}k(x^c-x^v_c)^T(x^c-x^v_c)$ , attracting elements of the garment mesh $x^c$ to corresponding elements $v_c$ on the body mesh.",
"As the garment mesh is initialized from the SMPL mesh, for more accurate simulation, we subdivide the garment mesh until it has 16x the resolution of the base SMPL template mesh."
],
[
"On-Body Topology Optimization",
"To design passive reinforcement structures, Vechev et al.",
"use a bi-directional evolutionary structural optimization (BESO) algorithm [17], [18] to solve the constrained optimization problem with a single objective, $\\begin{split}\\mathbf {d}^* = \\arg \\max _{\\mathbf {d}} \\quad E_\\mathrm {garment}(x^*, \\mathbf {d}) \\\\\\textrm {s.t.}",
"\\quad \\sum _e{A^e} d^e=A^*\\ ,\\quad \\mathbf {f}(x^*)= \\mathbf {0} .\\end{split}$ The goal of this formulation is to find an optimal per-element material assignment $\\mathbf {d}^*$ that maximizes the energy of the garment in its equilibrium state $\\mathbf {x}^*$ while satisfying constraints on force equilibrium, $\\mathbf {f}(x^*)= \\mathbf {0}$ , and material budget, $\\sum _e{A_e} d_e=A^*$ .",
"The strain energy of the garment is defined per element as $E_{\\mathrm {garment}} = \\sum _e t^e A^eW_\\mathrm {garment}^e(x^*, d^e)\\ ,$ where $W_\\mathrm {garment}^e$ is the elemental strain energy density, and $t^e, A^e$ are the thickness and area of the element respectively."
]
] | 2210.07689 |
[
[
"A centra-limit theorem for conservative fragmentation chain"
],
[
"Abstract We are interested in a fragmentation process.",
"We observe fragments frozen when their sizes are less than {\\epsilon} ({\\epsilon} > 0).",
"It is known ([BM05]) that the empirical measure of these fragments converges in law, under some renormalization.",
"In [HK11], the authors show a bound for the rate of convergence.",
"Here, we show a central-limit theorem, under some assumptions.",
"This gives us an exact rate of convergence."
],
[
"Scientific and economic context",
"One of the main goals in the mining industry is to extract blocks of metallic ore and then separate the metal from the valueless material.",
"To do so, rock is fragmented into smaller and smaller rocks.",
"This is carried out in a series of steps, the first one being blasting, after which the material goes through a sequence of crushers.",
"At each step, the particles are screened, and if they are smaller than the diameter of the mesh of a classifying grid, they go to the next crusher.",
"The process stops when the material has a sufficiently small size (more precisely, small enough to enable physicochemical processing).",
"This fragmentation process is energetically costly (each crusher consumes a certain quantity of energy to crush the material it is fed).",
"One of the problems that faces the mining industry is that of minimizing the energy used.",
"The optimization parameters are the number of crushers and the technical specifications of these crushers.",
"In [4], the authors propose a mathematical model of what happens in a crusher.",
"In this model, the rock pieces/fragments are fragmented independently of each other, in a random and auto-similar manner.",
"This is consistent with what is observed in the industry, and this is supported by the following publications : [19], [9], [24], [22].",
"Each fragment has a size $s$ (in $\\mathbb {R}^{+}$ ) and is then fragmented into smaller fragments of sizes $s_{1}$ , $s_{2}$ , ... such that the sequence $(s_{1}/s,s_{2}/s,\\dots )$ has a law $\\nu $ which does not depend on $s$ (which is why the fragmentation is said to be auto-similar).",
"This law $\\nu $ is called the dislocation measure (each crusher has its own dislocation measure).",
"The dynamic of the fragmentation process is thus modeled in a stochastic way.",
"In each crusher, the rock pieces are fragmented repetitively until they are small enough to slide through a mesh whose holes have a fixed diameter.",
"So the fragmentation process stops for each fragment when its size is smaller than the diameter of the mesh, which we denote by $ \\varepsilon $ ($ \\varepsilon >0$ ).",
"We are interested in the statistical distribution of the fragments coming out of a crusher.",
"If we renormalize the sizes of these fragments by dividing them by $ \\varepsilon $ , we obtain a measure $\\gamma _{-\\log ( \\varepsilon )}$ , which we call the empirical measure (the reason for the index $-\\log ( \\varepsilon )$ instead of $ \\varepsilon $ will be made clear later).",
"In [4], the authors show that the energy consumed by the crusher to reduce the rock pieces to fragments whose diameters are smaller than $ \\varepsilon $ can be computed as an integral of a bounded function against the measure $\\gamma _{-\\log ( \\varepsilon )}$ (they cite [5], [6], [25] on this particular subject).",
"For each crusher, the empirical measure $\\gamma _{-\\log ( \\varepsilon )}$ is one of the two only observable variables (the other one being the size of the pieces pushed into the grinder).",
"The specifications of a crusher are summarized in $ \\varepsilon $ and $\\nu $ ."
],
[
"State of the art",
"In [4], the authors show that the energy consumed by a crusher to reduce rock pieces of a fixed size into fragments whose diameter are smaller than $ \\varepsilon $ behaves asymptotically like a power of $ \\varepsilon $ when $ \\varepsilon $ goes to zero.",
"More precisely, this energy multiplied by a power of $ \\varepsilon $ converges towards a constant of the form $\\kappa =\\nu (\\varphi )$ (the integral of $\\nu $ , the dislocation measure, against a bounded function $\\varphi $ ).",
"In [4], the authors also show a law of large numbers for the empirical measure $\\gamma _{-\\log ( \\varepsilon )}$ .",
"More precisely, if $f$ is bounded continuous, $\\gamma _{-\\log ( \\varepsilon )}(f)$ converges in law, when $ \\varepsilon $ goes to zero, towards an integral of $f$ against a measure related to $\\nu $ (this result also appears in [15], p. 399).",
"We set $\\gamma _{\\infty }(f)$ to be this limit (check Equations (REF ), (REF ), (REF ) to get an exact formula).",
"The empirical measure $\\gamma _{-\\log ( \\varepsilon )}$ thus contains information relative to $\\nu $ and one could extract from it an estimation of $\\kappa $ or of an integral of any function against $\\nu $ .",
"It is worth noting that by studying what happens in various crushers, we could study a family $(\\nu _{i}(f_{j}))_{i\\in I,j\\in J}$ (with an index $i$ for the number of the crusher and the index $j$ for the $j$ -th test function in a well-chosen basis).",
"Using statistical learning methods, one could from there make a prediction for $\\nu (f_{j}$ ) for a new crusher for which we know only the mechanical specifications (shape, power, frequencies of the rotating parts ...).",
"It would evidently be interesting to know $\\nu $ before even building the crusher.",
"In the same spirit, [14] studies the energy efficiency of two crushers used after one another.",
"When the final size of the fragments tends to zero, this paper tells us wether it is more efficient enrgywise to use one crusher or two crushers in a row (another asymptotic is also considered in the paper).",
"In [16], the authors prove a convergence result for the empirical measure similar to the one in [4], the convergence in law being replaced by an almost sure convergence.",
"In [15], the authors give a bound on the rate of this convergence, in a $L^{2}$ sense, under the assumption that the fragmentation is conservative.",
"This assumption means there is no loss of mass due to the formation of dust during the fragmentation process.french Figure: americanState of the art.So we have convergence results ([4], [16]) of an empirical quantity towards constants of interest (a different constant for each test function $f$ ).",
"Using some transformations, these constants could be used to estimate the constant $\\kappa $ .",
"Thus it is natural to ask what is the exact rate of convergence in this estimation, if only to be able to build confidence intervals.",
"In [15], we only have a bound on the rate.",
"When a sequence of empirical measures converges to some measure, it is natural to study the fluctuations, which often turn out to be Gaussian.",
"For such results in the case of empirical measures related to the mollified Boltzmann equation, one can cite [18], [23], [13].",
"When interested in the limit of a $n$ -tuple as in Equation (REF ) below, we say we are looking at the convergence of a $U$ -statistics.",
"Textbooks deal with the case where the points defining the empirical measure are independent or with a known correlation (see [7], [8], [17]).",
"The problem is more complex when the points defining the empirical measure are in interaction with each other like it is the case here."
],
[
"Goal of the paper",
"As explained above, we want to obtain the rate of convergence in the convergence of $\\gamma _{-\\log ( \\varepsilon )}$ when $ \\varepsilon $ goes to zero.",
"We want to produce a central-limit theorem of the kind : for a bounded continuous $f$ , $ \\varepsilon ^{\\beta }(\\gamma _{-\\log ( \\varepsilon )}(f)-\\gamma _{\\infty }(f))$ converges towards a non-trivial measure when $ \\varepsilon $ goes to zero (the limiting measure will in fact be Gaussian), for some exponent $\\beta $ .",
"The technics used will allow us to prove the convergence towards a multivariate Gaussian of a vector of the kind $ \\varepsilon ^{\\beta }(\\gamma _{-\\log ( \\varepsilon )}(f_{1})-\\gamma _{\\infty }(f_{1}),\\dots ,\\gamma _{-\\log ( \\varepsilon )}(f_{n})-\\gamma _{\\infty }(f_{n}))$ for functions $f_{1}$ , ..., $f_{n}$ .",
"More precisely, if by $Z_{1}$ , $Z_{2}$ , ..., $Z_{N}$ we denote the fragments sizes that go out from a crusher (with mesh diameter equal to $ \\varepsilon $ ).",
"We would like to show that for a bounded continuous $f$ , $\\gamma _{-\\log ( \\varepsilon )}(f):=\\sum _{i=1}^{N}Z_{i}f(Z_{i}/ \\varepsilon )\\longrightarrow \\gamma _{\\infty }(f)\\text{, almost surely, when } \\varepsilon \\rightarrow 0\\,,$ and that for all $n$ , and $f_{1}$ , ...,$f_{n}$ bounded continuous function such that $\\gamma _{\\infty }(f_{i})=0$ , $ \\varepsilon ^{\\beta }(\\gamma _{-\\log ( \\varepsilon )}(f_{1}),\\dots ,\\gamma _{-\\log ( \\varepsilon )}(f_{n}))$ converges in law towards a multivariate Gaussian when $ \\varepsilon $ goes to zero.",
"The exact results are stated in Proposition REF and Theorem REF ."
],
[
"Outline of the paper",
"We will state our assumptions along the way (Assumptions REF , REF , REF , REF ).",
"Assumption REF can be found at the beginning of Section .",
"We define our model in Section .",
"The main idea is that we want to follow tags during the fragmentation process.",
"Let us imagine the fragmentation is the process of breaking a stick (modeled by $[0,1]$ ) into smaller sticks.",
"We suppose that the original stick has painted dots and that during the fragmentation process, we take note of the sizes of the sticks supporting the painted dots.",
"When the sizes of these sticks get smaller than $ \\varepsilon $ ($ \\varepsilon >0$ ), the fragmentation is stopped for them and we call them the painted sticks.",
"In Section , we make use of classical results on renewal processes and of [21] to show that the size of one painted stick has an asymptotic behavior when $ \\varepsilon $ goes to zero and that we have a bound on the rate with which it reaches this behavior.",
"Section is the most technical.",
"There we study the asymptotics of symmetric functionals of the sizes of the painted sticks (always when $ \\varepsilon $ goes to zero).",
"In Section , we precisely define the measure we are interested in ($\\gamma _{T}$ with $T=-\\log ( \\varepsilon )$ ).",
"Using the results of Section , it is then easy to show a law of large numbers for $\\gamma _{T}$ (Proposition REF ) and a central-limit Theorem (Theorem REF ).",
"Proposition REF and Theorem REF are our two main results.",
"The proof of Theorem REF is based on a simple computation involving characteristic functions (the same technique was already used in [10], [11], [12], [20])."
],
[
"Notations",
"For $x$ in $\\mathbb {R}$ , we set $\\lceil x\\rceil =\\inf \\lbrace n\\in \\mathbb {Z}\\,:\\,n\\ge x\\rbrace $ , $\\lfloor x\\rfloor =\\sup \\lbrace n\\in \\mathbb {Z}\\,:\\,n\\le x\\rbrace $ .",
"The symbol $\\sqcup $ means “disjoint union”.",
"For $n$ in $\\mathbb {N}^{*}$ , we set $[n]=\\lbrace 1,2,\\dots ,n\\rbrace $ .",
"For $f$ an application from a set $E$ to a set $F$ , we write $f:E\\hookrightarrow F$ if $f$ is injective and, for $k$ in $\\mathbb {N}^{*}$ , if $F=E$ , we set $f^{\\circ k}=\\underset{k\\mbox{ times }}{\\underbrace{f\\circ f\\circ \\dots \\circ f}}\\,.$ For any set $E$ , we set $\\mathcal {P}(E)$ to be the set of subsets of $E$ ."
],
[
"Fragmentation chains",
"Let $ \\varepsilon >0$ .",
"Like in [15], we start with the space $\\mathcal {S}^{\\downarrow }=\\left\\lbrace \\mathbf {s}=(s_{1},s_{2},\\dots ),\\,s_{1}\\ge s_{2}\\ge \\dots \\ge 0,\\,\\sum _{i=1}^{+\\infty }s_{i}\\le 1\\right\\rbrace \\,.$ A fragmentation chain is a process in $\\mathcal {S}^{\\downarrow }$ characterized by a dislocation measure $\\nu $ which is a finite measure on $\\mathcal {S}^{\\downarrow }$ , a description of the law of the times between fragmentations.",
"A fragmentation chain with dislocation measure $\\nu $ is a Markov process $X=(X(t),t\\ge 0)$ with values in $\\mathcal {S}^{\\downarrow }$ .",
"Its evolution can be described as follows : a fragment with size $x$ lives for some time (which may or may not be random) then splits and gives rise to a family of smaller fragments distributed as $x\\xi $ , where $\\xi $ is distributed according to $\\nu (.",
")/\\nu (\\mathcal {S}^{\\downarrow })$ .",
"We suppose the life-time of a fragment of size $x$ is an exponential time of parameter $x^{\\alpha }\\nu (\\mathcal {S}^{\\downarrow })$ , for some $\\alpha $ .",
"We could here make different assumptions on the life-time of fragments, but this would not change our results.",
"Indeed, as we are interested in the sizes of the fragments frozen as soon as they are smaller than $ \\varepsilon $ , the time they need to become this small is not important.",
"We denote by $\\mathbb {P}_{m}$ the law of $X$ started from the initial configuration $(m,0,0,\\dots )$ with $m$ in $(0,1]$ .",
"The law of $X$ is entirely determined by $\\alpha $ and $\\nu (.",
")$ (Theorem 3 of [2]).",
"We make the same assumption as in [15] and we will call it Assumption REF .",
"Assumption A We have $\\nu (\\mathcal {S}^{\\downarrow })=1$ and $\\nu (s_{1}\\in ]0;1[)=1$ .",
"Let $\\mathcal {U}:=\\lbrace 0\\rbrace \\cup \\bigcup _{n=1}^{+\\infty }(\\mathbb {N}^{*})^{n}$ denote the infinite genealogical tree.",
"For $u$ in $\\mathcal {U}$ , we use the conventional notation $u=()$ if $u=\\lbrace 0\\rbrace $ and $u=(u_{1},\\dots ,u_{n})$ if $u\\in (\\mathbb {N}^{*})^{n}$ with $n\\in \\mathbb {N}^{*}.$ This way, any $u$ in $\\mathcal {U}$ can be denoted by $u=(u_{1},\\dots ,u_{n})$ , for some $u_{1},\\dots ,u_{n}$ and with $n$ in $\\mathbb {N}$ .",
"Now, for $u=(u_{1},\\dots ,u_{n})\\in \\mathcal {U}$ and $i\\in \\mathbb {N}^{*}$ , we say that $u$ is in the $n$ -th generation and we write $|u|=n$ , and we write $ui=(u_{1},\\dots ,u_{n},i)$ , $u(k)=(u_{1},\\dots ,u_{k})$ for all $k\\in [n]$ .",
"For any $u=(u_{1},\\dots ,u_{n})$ and $v=ui$ ($i\\in \\mathbb {N}^{*}$ ), we say that $u$ is the mother of $v$ .",
"For any $u$ in $\\mathcal {U}\\backslash \\lbrace 0\\rbrace $ ($\\mathcal {U}$ deprived of its root), $u$ has exactly one mother and we denote it by $m(u)$ .",
"The set $\\mathcal {U}$ is ordered alphanumerically : If $u$ and $v$ are in $\\mathcal {U}$ and $|u|<|v|$ then $u<v$ .",
"If $u$ and $v$ are in $\\mathcal {U}$ and $|u|=|v|=n$ and $u=(u_{1},\\dots ,u_{n})$ , $v=(v_{1},\\dots ,v_{n})$ with $u_{1}=v_{1}$ , ... , $u_{k}=v_{k}$ , $u_{k+1}<v_{k+1}$ then $u<v$ .",
"Suppose we have a process $X$ which has the law $\\mathbb {P}_{m}$ .",
"For all $\\omega $ , we can index the fragments that are formed by the process $X$ with elements of $\\mathcal {U}$ in a recursive way.",
"We start with a fragment of size $m$ indexed by $u=()$ .",
"If a fragment $x$ , with a birth-time $t_{1}$ and a split-time $t_{2}$ , is indexed by $u$ in $\\mathcal {U}$ .",
"At time $t_{2}$ , this fragment splits into smaller fragments of sizes $(xs_{1},xs_{2},\\dots )$ with $(s_{1},s_{2},\\dots )$ of law $\\nu (.",
")/\\nu (\\mathcal {S}^{\\downarrow })$ .",
"We index the fragment of size $xs_{1}$ by $u1$ , we index the fragment of size $xs_{2}$ by $u2$ , and so on.",
"A mark is an application from $\\mathcal {U}$ to some other set.",
"We associate a mark $\\xi _{\\dots }$ on the tree $\\mathcal {U}$ to each path of the process $X$ .",
"The mark at node $u$ is $\\xi _{u}$ , where $\\xi _{u}$ is the size of the fragment indexed by $u$ .",
"The distribution of this random mark can be described recursively as follows.",
"Proposition 2.1 (Consequence of Proposition 1.3, p. 25, [3]) There exists a family of i.i.d.",
"variables indexed by the nodes of the genealogical tree, $((\\widetilde{\\xi }_{ui})_{i\\in \\mathbb {N}^{*}},u\\in \\mathcal {U})$ , where each $(\\widetilde{\\xi }_{ui})_{i\\in \\mathbb {N}^{*}}$ is distributed according to the law $\\nu (.",
")/\\nu (\\mathcal {S}^{\\downarrow })$ , and such that the following holds : Given the marks $(\\xi _{v},|v|\\le n)$ of the first $n$ generations, the marks at generation $n+1$ are given by $\\xi _{ui}=\\widetilde{\\xi }_{ui}\\xi _{u}\\,,$ where $u=(u_{1,}\\dots ,u_{n})$ and $ui=(u_{1},\\dots ,u_{n},i)$ is the $i-th$ child of $u$ ."
],
[
"Tagged fragments",
"From now on, we suppose that we start with a block of size $m=1$ .",
"We assume that the total mass of the fragments remains constant through time, as follows.",
"Assumption B (Conservative property).",
"We have $\\nu (\\sum _{i=1}^{+\\infty }s_{i}=1)=1$ .",
"This assumption was already present in [15].",
"We observe that the Malthusian exponent of [3] (p. 27) is equal to 1 under our assumptions.",
"Without this assumption, the link between the empirical measure $\\gamma _{-\\log ( \\varepsilon )}$ and the tagged fragments (Equation (REF )) vanishes and our proofs of Proposition REF and Theorem REF fail.",
"We can now define tagged fragments.",
"We use the representation of fragmentation chains as random infinite marked tree to define a fragmentation chain with $q$ tags.",
"Suppose we have a fragmentation process $X$ of law $\\mathbb {P}_{1}$ .",
"We take $(Y_{1},Y_{2},\\dots ,Y_{q})$ to be $q$ i.i.d.",
"variables of law $\\mathcal {U}([0,1])$ .",
"We set, for all $u$ in $\\mathcal {U}$ , $(\\xi _{u},A_{u},I_{u})$ with $\\xi _{u}$ defined as above.",
"The random variables $A_{u}$ take values in the subsets of $[q]$ .",
"The random variables $I_{u}$ are intervals.",
"These variables are defined as follows.",
"We set $A_{\\lbrace 0\\rbrace }=[q]$ , $I_{\\lbrace 0\\rbrace }=(0,1]$ ($I_{\\lbrace 0\\rbrace }$ is of length $\\xi _{\\lbrace 0\\rbrace }=1$ ) For all $n\\in \\mathbb {N}$ .",
"Suppose we are given the marks of the first $n$ generations.",
"Suppose that, for $u$ in the $n$ -th generation, $I_{u}=(a_{u},a_{u}+\\xi _{u}]$ for some $a_{u}\\in \\mathbb {R}$ (it is of length $\\xi _{u}$ ).",
"Then the marks at generation $n+1$ are given by Proposition REF (concerning $\\xi _{.",
"}$ ) and, for all $u$ such that $|u|=n$ and for all $i$ in $\\mathbb {N}^{*}$ $I_{ui}=(a_{u}+\\xi _{u}(\\widetilde{\\xi }_{u1}+\\dots +\\widetilde{\\xi }_{u(i-1)}),a_{u}+\\xi _{u}(\\widetilde{\\xi }_{u1}+\\dots +\\widetilde{\\xi }_{ui})]\\,,$ $k\\in A_{ui}\\text{ if and only if }Y_{k}\\in I_{ui}\\,,$ ($I_{ui}$ is then of length $\\xi _{ui}$ ).",
"We observe that for all $j\\in [q]$ , $u\\in \\mathcal {U}$ , $i\\in \\mathbb {N}^{*}$ , $\\mathbb {P}(j\\in A_{ui}|j\\in A_{u},\\widetilde{\\xi }_{ui})=\\widetilde{\\xi }_{ui}\\,.$ In this definition, we imagine having $q$ dots on the interval $[0,1]$ and we impose that the dot $j$ has the position $Y_{j}$ (for all $j$ in $[q]$ ).",
"During the fragmentation process, if we know that the dot $j$ is in the interval $I_{u}$ of length $\\xi _{u}$ , then the probability that this dot is on $I_{ui}$ (for some $i$ in $\\mathbb {N}^{*}$ , $I_{ui}$ of length $\\xi _{ui}$ ) is equal to $\\xi _{ui}/\\xi _{u}=\\widetilde{\\xi }_{ui}$ .",
"In the case $q=1$ , the branch $\\lbrace u\\in \\mathcal {U}\\,:\\,A_{u}\\ne \\emptyset \\rbrace $ has the same law as the randomly tagged branch of Section 1.2.3 of [3].",
"The presentation is simpler in our case because the Malthusian exponent is 1 under Assumption REF ."
],
[
"Observation scheme",
"We freeze the process when the fragments become smaller than a given threshold $ \\varepsilon >0$ .",
"That is, we have the following data $(\\xi _{u})_{u\\in \\mathcal {U}_{ \\varepsilon }}\\,,$ where $\\mathcal {U}_{ \\varepsilon }=\\lbrace u\\in \\mathcal {U},\\,\\xi _{m(u)}\\ge \\varepsilon ,\\,\\xi _{u}< \\varepsilon \\rbrace \\,.$ We now look at $q$ tagged fragments ($q\\in \\mathbb {N}^{*}$ ).",
"For each $i$ in $[q]$ , we call $L_{0}^{(i)}=1$ , $L_{1}^{(i)}$ , $L_{2}^{(i)}$ ... the successive sizes of the fragment having the tag $i$ .",
"More precisely, for each $n\\in \\mathbb {N}^{*}$ , there is almost surely exactly one $u\\in \\mathcal {U}$ such that $|u|=n$ and $i\\in A_{u}$ ; and so, $L_{n}^{(i)}=\\xi _{u}$ .",
"For each $i$ , the process $S_{0}^{(i)}=-\\log (L_{0}^{(i)})=0\\le S_{1}^{(i)}=-\\log (L_{1}^{(i)})\\le \\dots $ is a renewal process without delay, with waiting-time following a law $\\pi $ (see [1], Chapter V for an introduction to renewal processes).",
"The waiting times are (for $i$ in $[q]$ ) : $S_{0}^{(i)}$ , $S_{1}^{(i)}-S_{0}^{(i)}$ , $S_{2}^{(i)}-S_{1}^{(i)}$ , ...",
"The renewal times are (for $i$ in $[q]$ ) : $S_{0}^{(i)}$ , $S_{1}^{(i)}$ , $S_{2}^{(i)}$ , ...",
"The law $\\pi $ is defined by the following.",
"$\\text{For all bounded measurable }f:[0,1]\\rightarrow [0,+\\infty )\\,,\\,\\int _{\\mathcal {S}^{\\downarrow }}\\sum _{i=1}^{+\\infty }s_{i}f(s_{i})\\nu (ds)=\\int _{0}^{+\\infty }f(e^{-x})\\pi (dx)\\,,$ (see Proposition 1.6, p. 34 of [3], or Equations (3), (4), p. 398 of [15]).",
"Under Assumption REF and Assumption REF , Proposition 1.6 of [3] is true, even without the Malthusian Hypothesis of [3].",
"We make the following assumption on $\\pi $ .",
"Assumption Ca There exist $a$ and $b$ ($0<a<b<+\\infty $ ) such that the support of $\\pi $ is $[a,b]$ .",
"We set $\\delta =e^{-b}$ .",
"We added a comment about the above Assumption in Remark REF .",
"We believe that we could replace the above Assumption by the following.",
"Assumption Cb The support of $\\pi $ is $(0,+\\infty )$ .",
"However, this would lead to difficult computations.",
"We set $T=-\\log ( \\varepsilon )\\,.$ We set, for all $i\\in [q]$ , $t\\ge 0$ , $N_{t}^{(i)}=\\inf \\lbrace j\\,:\\,S_{j}^{(i)}>t\\rbrace \\,,$ $B_{t}^{(i)}=S_{N_{t}^{(i)}}^{(i)}-t\\,,$ $C_{t}^{(i)}=t-S_{N_{t}^{(i)}-1}^{(i)}\\,,$ Figure: Process B (1) B^{(1)} and C (1) C^{(1)}.",
"(see Figure REF for an illustration).",
"The processes $B^{(i)}$ , $C^{(i)}$ , $N^{(i)}$ are time-homogeneous Markov processes (Proposition 1.5 p. 141 of [1]).",
"All of them are cà dlà g (i.e.",
"right-continuous with a left-hand side limit).",
"We call $B^{(i)}$ the residual lifetime of the fragment tagged by $i$ .",
"We call $C^{(i)}$ the age of the fragment tagged by $i$ .",
"We call $N^{(i)}$ the number of renewal before $t$ .",
"In the following, we will treat $t$ as a time parameter.",
"This has nothing to do with the time in which the fragmentation process $X$ evolves.",
"We observe that, for all $t$ , $(B_{t}^{(1)},\\dots ,B_{t}^{(q)})$ is exchangeable (meaning that for all $\\sigma $ in the symmetric group of order $q$ , $(B_{t}^{(\\sigma (1))},\\dots ,B_{t}^{(\\sigma (q))})$ has the same law as $(B_{t}^{(1)},\\dots ,B_{t}^{(q)})$ ).",
"When we look at the fragments of sizes $(\\xi _{u},\\,u\\in \\mathcal {U}_{ \\varepsilon }\\,:\\,A_{u}\\ne \\emptyset )$ , we have almost the same information as when we look at $(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)})$ .",
"We say almost because knowing $(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)})$ does not give exactly the number of $u$ in $\\mathcal {U}_{ \\varepsilon }$ such that $A_{u}$ is not empty.",
"In the remaining of Section , we define processes that will be useful when we will describe the asymptotics of our model (in Section )."
],
[
"Stationary renewal processes ($\\overline{B}^{(1)}$ , {{formula:69a64b89-96f5-4b67-b462-4bcc6f665e39}} ).\n",
"We define $\\widetilde{X}$ to be an independent copy of $X$ .",
"We suppose it has $q$ tagged fragments.",
"Therefore it has a mark $(\\widetilde{\\xi },\\widetilde{A})$ and a renewal processes $(\\widetilde{S}_{k}^{(i)})_{k\\ge 0}$ (for all $i$ in $[q]$ ) defined in the same way as for $X$ .",
"We let $(\\widetilde{B}^{(1)},\\widetilde{B}^{(2)})$ be the residual lifetimes of the fragments tagged by 1 and 2.",
"Let $\\mu =\\int _{0}^{+\\infty }x\\pi (dx)$ and let $\\pi _{1}$ be the distribution with density $x\\mapsto x/\\mu $ with respect to $\\pi $ .",
"We set $\\overline{C}$ to be a random variable of law $\\pi _{1}$ .",
"We set $U$ to be independent of $\\overline{C}$ and uniform on $(0,1)$ .",
"We set $\\widetilde{S}_{-1}=\\overline{C}(1-U)$ .",
"The process $\\overline{S}_{0}=\\widetilde{S}_{-1}$ , $\\overline{S}_{1}=\\widetilde{S}_{-1}+\\widetilde{S}_{0}^{(1)}$ , $\\overline{S}_{2}=\\widetilde{S}_{-1}+\\widetilde{S}_{1}^{(1)}$ , $\\overline{S}_{2}=\\widetilde{S}_{-1}+\\widetilde{S}_{2}^{(1)}$ , ... is a renewal process with delay $\\pi _{1}$ (with waiting times $\\overline{S}_{0}$ , $\\overline{S}_{1}-\\overline{S}_{0}$ , ... all smaller than $b$ by Assumption REF ).",
"The renewal times are $\\overline{S}_{0}$ , $\\overline{S}_{1}$ , $\\overline{S}_{2}$ , ... We set $(\\overline{B}_{t}^{(1)})_{t\\ge 0}$ to be the residual lifetime process of this renewal process : $\\overline{B}_{t}^{(1)}={\\left\\lbrace \\begin{array}{ll}\\overline{C}(1-U)-t & \\mbox{ if }t<\\overline{S}_{0}\\,,\\\\\\inf _{n\\ge 0}\\lbrace \\overline{S}_{n}\\,:\\,\\overline{S}_{n}>t\\rbrace -t & \\mbox{ if }t\\ge \\overline{S}_{0}\\,,\\end{array}\\right.",
"}$ and we define $(\\overline{C}_{t}^{(1)})_{t\\ge 0}$ : $\\overline{C}_{t}^{(1)}={\\left\\lbrace \\begin{array}{ll}\\overline{C}U+t & \\mbox{ if }t<\\overline{S}_{0}\\,,\\\\t-\\sup _{n\\ge 0}\\lbrace \\overline{S}_{n}\\,:\\,\\overline{S}_{n}\\le t\\rbrace & \\mbox{ if }t\\ge \\overline{S}_{0}\\,,\\end{array}\\right.",
"}$ (we call it the age process of our renewal process) and we set $\\overline{N}_{t}^{(1)}=\\inf \\lbrace j\\,:\\,\\overline{S}_{j}>t\\rbrace \\,.$ Fact 2.2 Theorem 3.3 p.151 of [1] tells us that $(\\overline{B}_{t}^{(1)},\\overline{C}_{t}^{(1)})_{t\\ge 0}$ has the same transition as $(B_{t}^{(1)},C_{t}^{(1)})_{t\\ge 0}$ defined above and that $(\\overline{B}_{t}^{(1)},\\overline{C}_{t}^{(1)})_{t\\ge 0}$ is stationary.",
"In particular, this means that the law of $\\overline{B}_{t}^{(1)}$ does not depend on $t$ .",
"We see in Figure REF a graphic representation of $\\overline{B}_{.",
"}^{(1)}$ .",
"Figure: Renewal process with delayThis might be counter-intuitive to start with $\\overline{B}_{0}^{(1)}$ having a law which is not $\\pi $ in order to get a stationary process, but Corollary 3.6 p. 153 of [1] is clear on this point : a delayed renewal process (with waiting-time of law $\\pi $ ) is stationary if and only if the distribution of the initial delay is $\\eta $ (defined below).",
"We define a measure $\\eta $ on $\\mathbb {R}^{+}$ by its action on bounded measurable functions : $\\text{For all bounded measurable }f\\,:\\,\\mathbb {R}^{+}\\rightarrow \\mathbb {R}\\,,\\,\\eta (f)=\\frac{1}{\\mu }\\int _{\\mathbb {R}^{+}}\\mathbb {E}(f(Y-s)\\mathbb {1}_{\\lbrace Y-s>0\\rbrace })ds\\,,\\,(Y\\sim \\pi )\\,.$ Lemma 2.3 The measure $\\eta $ is the law of $\\overline{B}_{t}^{(1)}$ (for any $t\\ge 0$ ).",
"It is also the law of $(\\overline{C}_{t}^{(1)})$ (for any $t$ ).",
"We write the proof for $\\overline{B}_{t}^{(1)}$ only.",
"Let $\\xi \\ge 0$ .",
"We set $f(y)=\\mathbb {1}_{y\\ge \\xi }$ , for all $y$ in $\\mathbb {R}$ .",
"We have (with $Y$ of law $\\pi $ ) $\\frac{1}{\\mu }\\int _{\\mathbb {R}^{+}}\\mathbb {E}(f(Y-s)\\mathbb {1}_{Y-s>0})ds & = & \\frac{1}{\\mu }\\int _{\\mathbb {R}^{+}}\\left(\\int _{0}^{y}\\mathbb {1}_{y-s\\ge \\xi }ds\\right)\\pi (dy)\\\\& = & \\frac{1}{\\mu }\\int _{\\mathbb {R}^{+}}(y-\\xi )_{+}\\pi (dy)\\\\& = & \\int _{\\xi }^{+\\infty }\\left(1-\\frac{\\xi }{y}\\right)\\frac{y}{\\mu }\\pi (dy)\\\\& = & \\mathbb {P}(\\overline{C}(1-U)\\ge \\xi )\\,.$ We set $\\eta _{2}$ to be the law of $(\\overline{C}_{0}^{(1)},\\overline{B}_{0}^{(1)})=(\\overline{C}U,\\overline{C}(1-U)).$ The support of $\\eta _{2}$ is $\\mathcal {C}:=\\lbrace (u,v)\\in [0,b]^{2}\\,:\\,a\\le u+v\\le b\\rbrace $ .",
"For $v$ in $\\mathbb {R}$ , we now want to define a process $(\\overline{C}_{t}^{(1),v},\\overline{B}_{t}^{(1),v})_{t\\ge v-2b}\\text{ having the same transition as }(C_{t}^{(1)},B_{t}^{(1)})\\text{ and being stationary.",
"}$ We set $(\\overline{C}_{v-2b}^{(1),v},\\overline{B}_{v-2b}^{(1),v})$ such that it has the law $\\eta _{2}$ .",
"As we have given its transition, the process $(\\overline{C}_{t}^{(1),v},\\overline{B}_{t}^{(1),v})_{t\\ge v-2b}$ is well defined in law.",
"In addition, we suppose that it is independent of all the other processes.",
"By Fact REF , the process $(\\overline{C}_{t}^{(1),v},\\overline{B}_{t}^{(1),v})_{t\\ge v-2b}$ is stationary.",
"We define the renewal times of $\\overline{B}^{(1),v}$ by : $\\overline{S}_{1}^{(1),v}=\\inf \\lbrace t\\ge v-2b\\,:\\,\\overline{B}_{t+}^{(1),v}\\ne \\overline{B}_{t-}^{(1),v}\\rbrace \\,,$ $\\text{and by recurrence,}\\,\\overline{S}_{k}^{(1),v}=\\inf \\lbrace t>\\overline{S}_{k-1}^{(1),v}\\,:\\,\\overline{B}_{t+}^{(1),v}\\ne \\overline{B}_{t-}^{(1),v}\\rbrace \\,.$ We also define, for all $t$ , $\\overline{N}_{t}^{(1),v}=\\inf \\lbrace j\\,:\\,\\overline{S}_{j}^{(1),v}>t\\rbrace \\,.\\text{}$ As will be seen later, the processes $\\overline{B}^{(1),v}$ and $\\overline{B}^{(2),v}$ are used to define asymptotic quantities (see, for example, Proposition REF ) and we need them to be defined on an interval $[v,+\\infty )$ with $v$ possibly in $\\mathbb {R}^{-}$ .",
"The process $\\overline{B}^{(2),v}$ is defined below (Section REF )."
],
[
"Tagged fragments conditioned to split\nup ($(\\widehat{B}^{(1),v},\\widehat{B}^{(2),v})$ ).",
"For $v$ in $[0,+\\infty )$ , we define a process $(\\widehat{B}_{t}^{(1),v},\\widehat{B}_{t}^{(2),v})_{t\\ge 0}$ such that $\\widehat{B}^{(1),v}=B^{(1)}\\text{ and, with }B^{(1)}\\text{ fixed, }\\widehat{B}^{(2),v}\\text{ has the law of }B^{(2)}\\text{ conditioned on }\\\\\\forall u\\in \\mathcal {U}\\,,\\,1\\in A_{u}\\Rightarrow [2\\in A_{u}\\Leftrightarrow -\\log (\\xi _{u})\\le v]\\,,$ which reads as follows : the tag 2 remains on the fragment bearing the tag 1 until the size of the fragment is smaller than $e^{-v}$ .",
"We observe that, conditionally on $\\widehat{B}_{v}^{(1),v}$ , $\\widehat{B}_{v}^{(2),v}$ : $(\\widehat{B}_{v+\\widehat{B}_{v}^{(1),v}+t}^{(1),v})_{t\\ge 0}$ and $(\\widehat{B}_{v+\\widehat{B}_{v}^{(2),v}+t}^{(2),v})_{t\\ge 0}$ are independent.",
"We also define $\\widehat{C}^{(1),v}=C^{(1)}$ .",
"There is an algorithmic way to define $\\widehat{B}^{(1),v}$ and $\\widehat{B}^{(2),v}$ , which is illustrated in Figure REF .",
"Remember $\\widehat{B}^{(1),v}=B^{(1)}$ and the definition of the mark $(\\xi _{u},A_{u},I_{u})_{u\\in \\mathcal {U}}$ in Section REF .",
"Figure: Processes B ^ (1),v \\widehat{B}^{(1),v}, B ^ (2),v \\widehat{B}^{(2),v}.We call $(\\widehat{S}_{j}^{(i)})_{i=1,2;j\\ge 1}$ the renewal times of these processes (as before, they can be defined as the times when the right-hand side and left-hand side limits are not the same).",
"If $\\widehat{S}_{j}^{(1)}\\le v$ then $\\widehat{S}_{j}^{(2)}=\\widehat{S}_{j}^{(1)}$ .",
"If $k$ is such that $\\widehat{S}_{k-1}^{(1)}\\le v$ and $\\widehat{S}_{k}^{(1)}>v$ , we remember that $\\exp (\\widehat{S}_{k}^{(1)}-\\widehat{S}_{k-1}^{(1)})=\\widetilde{\\xi }_{ui}$ for some $u$ in $\\mathcal {U}$ with $|u|=k-1$ and some $i$ in $\\mathbb {N}^{*}$ (because $\\widehat{B}^{(1),v}=B^{(1)}$ ).",
"We have points $Y_{1},$$Y_{2}\\in [0,1]$ such that $Y_{1},Y_{2}$ are in $I_{u}$ of length $\\xi _{u}$ .",
"Conditionally on $\\lbrace Y_{1},Y_{2}\\in I_{u}\\rbrace $ , $Y_{1}$ and $Y_{2}$ are independent and uniformly distributed on $I_{u}$ .",
"The interval $I_{ui}$ , of length $\\xi _{u}\\widetilde{\\xi }_{ui}$ , is a sub-interval of $I_{u}$ such that $Y_{1}\\in I_{ui}$ (because of Equation (REF ) above).",
"Then, for $r\\in \\mathbb {N}^{*}\\backslash \\lbrace i\\rbrace $ , we want $Y_{2}$ to be in $I_{ur}$ with probability $\\widetilde{\\xi }_{ur}/(1-\\widetilde{\\xi }_{ui})$ (because we want : $2\\notin A_{ui}$ ).",
"So we take $\\widehat{S}_{k}^{(2)}=\\widehat{S}_{k-1}^{(1)}-\\log \\widetilde{\\xi }_{ur}$ with probability $\\widetilde{\\xi }_{ur}/(1-\\widetilde{\\xi }_{ui})$ ($r\\in \\mathbb {N}^{*}\\backslash \\lbrace i\\rbrace $ ).",
"Fact 2.4 The knowledge of the couple $(\\widehat{S}_{N_{v}^{(1)}-1}^{(1)},\\widehat{B}_{v}^{(1),v})$ is equivalent to the knowledge of the couple $(\\widehat{C}_{v}^{(1),v},\\widehat{B}_{v}^{(1),v})$ .",
"The law of $B_{v}^{(1)}$ knowing $C_{v}^{(1)}$ is $\\pi -C_{v}^{(1)}$ with $\\pi $ conditioned to be bigger than $C_{v}^{(1)}$ , we call it $\\eta _{1}(\\dots |C_{v}^{(1)})$ .",
"As $\\widehat{B}^{(1),v}=B^{(1)}$ and $\\widehat{C}^{(1),v}=C^{(1)}$ , we also have that the law of $\\widehat{B}_{v}^{(1),v}$ knowing $\\widehat{C}_{v}^{(1),v}$ is $\\eta _{1}(\\dots |\\widehat{C}_{v}^{(1),v})$ .",
"The law of $\\widehat{B}_{v}^{(2),v}$ knowing $(\\widehat{C}_{v}^{(1),v},\\widehat{B}_{v}^{(1),v})$ does not depend on $v$ and we denote it by $\\eta ^{\\prime }(\\dots |\\widehat{C}_{v}^{(1),v},\\widehat{B}_{v}^{(1),v})$ .",
"The subsequent waiting-time $\\widehat{S}_{k+1}^{(1)}-\\widehat{S}_{k}^{(1)}$ , $\\widehat{S}_{k+1}^{(2)}-\\widehat{S}_{k}^{(2)}$ , ... are chosen independently of each other, each of them having the law $\\pi $ .",
"For $j$ equal to 1 or 2 and $t$ in $[0,+\\infty )$ , we define $\\widehat{N}_{t}^{(j)}=\\inf \\lbrace i\\,:\\,\\widehat{S}_{i}^{(j)}>t\\rbrace \\,.$ We observe that for $t\\ge 2b$ , $\\widehat{N}_{t}^{(1)}$ is bigger than 2 (because of Assumption REF )."
],
[
"Two stationary processes after a split-up ($\\overline{B}^{(1),v},\\overline{B}^{(2),v}$ ).",
"Let $k$ be an integer bigger than 2 ($k\\ge 2$ ) and such that $k\\times (b-a)\\ge b\\,.$ Now we state a small Lemma that will be useful below.",
"Remember that $(\\overline{C}_{t}^{(1),v},\\overline{B}_{t}^{(1),v})_{t\\ge v-2b}$ is defined in Equation (REF ).",
"The process $(\\widehat{C}_{t}^{(1),v},\\widehat{B}_{t}^{(1),v})_{t\\ge 0}$ is defined in the previous Section.",
"Lemma 2.5 Let $v$ be in $\\mathbb {R}.$ The variables $(\\overline{C}_{v}^{(1),v},\\overline{B}_{v}^{(1),v})$ and $(\\widehat{C}_{kb}^{(1),kb},\\widehat{B}_{kb}^{(1),kb})$ have the same support (and it is $\\mathcal {C}$ , defined below Lemma REF ).",
"The law $\\eta _{2}$ is the law of $(\\overline{C}_{0}^{(1)},\\overline{B}_{0}^{(1)})$ ($\\eta _{2}$ is defined below Lemma REF ).",
"As said before, the support of $\\eta _{2}$ is $\\mathcal {C}$ ; and so (by stationarity) the support of $(\\overline{C}_{v}^{(1),v},\\overline{B}_{v}^{(1),v})$ is $\\mathcal {C}$ .",
"Keep in mind that $\\widehat{B}^{(1),v}=B^{(1)}$ , $\\widehat{C}^{(1),v}=C^{(1)}$ .",
"By Assumption REF , the support of $S_{k}^{(1)}$ is $[ka,kb]$ and the support of $S_{k+1}^{(1)}-S_{k}^{(1)}$ is $[a,b]$ .",
"If $S_{k+1}^{(1)}>kb$ then $B_{kb}^{(1)}=S_{k+1}^{(1)}-S_{k}^{(1)}-(kb-S_{k}^{(1)})$ and $C_{kb}^{(1)}=kb-S_{k}^{(1)}$ (see Figure REF ).",
"Figure: B kb (1) B_{kb}^{(1)} and C kb (1) C_{kb}^{(1)}The support of $S_{k}^{(1)}$ is $[ka,kb]$ and $kb-ka\\ge b$ (Equation (REF )), so, as $S_{k}^{(1)}$ and $S_{k+1}^{(1)}-S_{k}^{(1)}$ are independent, we get that the support of $(C_{kb}^{(1)},S_{k+1}^{(1)}-S_{k}^{(1)})$ includes $\\lbrace (u,w)\\in [0;b]^{2}\\,:\\,w\\ge \\sup (a,u)\\rbrace $ .",
"And so, the support of $(C_{kb}^{(1)},B_{kb}^{(1)})=(C_{kb}^{(1)},S_{k+1}^{(1)}-S_{k}^{(1)}-C_{kb}^{(1)})$ includes $\\mathcal {C}$ .",
"As this support is included in $\\mathcal {C}$ , we have proved the desired result.",
"For $v$ in $\\mathbb {R}$ , we define a process $(\\overline{B}_{t}^{(2),v})_{t\\ge v}$ .",
"We start by : $\\overline{B}_{v}^{(2),v}\\text{ has the law }\\eta ^{\\prime }(\\dots |\\overline{C}_{v}^{(1),v},\\overline{B}_{v}^{(1),v})$ (remember $\\eta ^{\\prime }$ is defined in Fact REF ).",
"This conditioning is correct because the law of $(\\overline{C}_{v}^{(1),v},\\overline{B}_{v}^{(1),v})$ is $\\eta _{2}$ and its support is included in the support of the law of $(\\widehat{C}_{kb}^{(1),kb},\\widehat{B}_{kb}^{(1),kb})$ , which is $\\eta _{2}$ (see the Lemma above and below Equation (REF )).",
"We then let the process $(\\overline{B}_{t}^{(1),v},\\overline{B}_{t}^{(2),v})_{t\\ge v}$ run its course as a Markov process having the same transition as $(\\widehat{B}_{t-v+kb}^{(1),kb},\\widehat{B}_{t-v+kb}^{(2),kb})_{t\\ge v}$ .",
"This means that, after time $v$ , $\\overline{B}_{t}^{(1),v}$ and $\\overline{B}_{t}^{(2),v}$ decrease linearly (with a slope $-1$ ).",
"Until they reach 0.",
"When they reach 0, each of these two processes makes a jump of law $\\pi $ , independently of the other one.",
"After what, they decrease linearly ... and so on.",
"Fact 2.6 The process $(\\overline{B}_{t}^{(1),v},\\overline{B}_{t}^{(2),v})_{t\\ge v}$ is supposed independent from all the other processes (until now, we have defined its law and said that that $\\overline{B}^{(1),v}$ is independent from all the other processes)."
],
[
"Rate of convergence in the Key Renewal\nTheorem",
"We need the following regularity assumption.",
"Assumption D The probability $\\pi (dx)$ is absolutely continuous with respect to the Lebesgue measure (we will write $\\pi (dx)=\\pi (x)dx$ ).",
"The density function $x\\mapsto \\pi (x)$ is continuous on $(0;+\\infty )$ .",
"Fact 3.1 Let $\\theta >1$ ($\\theta $ is fixed in the rest of the paper).",
"The density $\\pi $ satisfies $\\limsup _{x\\rightarrow +\\infty }\\exp (\\theta x)\\pi (x)<+\\infty \\,.$ For $\\varphi $ a nonnegative Borel-measurable function on $\\mathbb {R}$ , we set $S(\\varphi )$ to be the set of complex-valued measures $\\rho $ (on the Borelian sets) such that $\\int _{\\mathbb {R}}\\varphi (x)|\\rho |(dx)<\\infty $ , where $|\\rho |$ stands for the total variation norm.",
"If $\\rho $ is a finite complex-valued measure on the Borelian sets of $\\mathbb {R}$ , we define $\\mathcal {T}\\rho $ to be the $\\sigma $ -finite measure with the density $v(x)={\\left\\lbrace \\begin{array}{ll}\\rho ((x,+\\infty )) & \\text{ if }x\\ge 0\\,,\\\\-\\rho ((-\\infty ,x]) & \\text{ if }x<0\\,.\\end{array}\\right.",
"}$ Let $F$ be the cumulative distribution function of $\\pi $ .",
"We set $B_{t}=B_{t}^{(1)}$ (see Equation (REF ) for the definition of $B^{(1)}$ , $B^{(2)}$ , ...).",
"By Theorem 3.3 p. 151 and Theorem 4.3 p. 156 of [1], we know that $B_{t}$ converges in law to a random variable $B_{\\infty }$ (of law $\\eta $ ) and that $C_{t}$ converges in law to a random variable $C_{\\infty }$ (of law $\\eta $ ).",
"The following Theorem is a consequence of [21], Theorem 5.1, p. 2429.",
"It shows there is actually a rate of convergence for these convergences in law.",
"Theorem 3.2 Let $ \\varepsilon ^{\\prime }\\in (0,\\theta )$ .",
"Let $M\\in (0,+\\infty )$ .",
"Let $\\varphi (x)={\\left\\lbrace \\begin{array}{ll}e^{(\\theta - \\varepsilon ^{\\prime })x} & \\text{ if }x\\ge 0\\,,\\\\1 & \\text{ if }x<0\\,.\\end{array}\\right.",
"}$ If $Y$ is a random variable of law $\\pi $ then $\\sup _{\\alpha \\,:\\,\\Vert \\alpha \\Vert _{\\infty }\\le M}\\left|\\mathbb {E}(\\alpha (B_{t}))-\\frac{1}{\\mu }\\int _{\\mathbb {R}^{+}}\\mathbb {E}(\\alpha (Y-s)\\mathbb {1}_{\\lbrace Y-s>0\\rbrace })ds\\right|=o\\left(\\frac{1}{\\varphi (t)}\\right)$ as $t$ approaches $+\\infty $ outside a set of Lebesgue measure zero (the supremum is taken on $\\alpha $ in the set of Borel-measurable functions on $\\mathbb {R}$ ), and $\\sup _{\\alpha \\,:\\,\\Vert \\alpha \\Vert _{\\infty }\\le M}\\left|\\mathbb {E}(\\alpha (C_{t}))-\\frac{1}{\\mu }\\int _{\\mathbb {R}^{+}}\\mathbb {E}(\\alpha (Y-s)\\mathbb {1}_{\\lbrace Y-s>0\\rbrace })ds\\right|=o\\left(\\frac{1}{\\varphi (t)}\\right)$ as $t$ approaches $+\\infty $ outside a set of Lebesgue measure zero (the supremum is taken on $\\alpha $ in the set of Borel-measurable functions on $\\mathbb {R}$ ) .",
"We write the proof of Equation (REF ) only.",
"The proof of Equation (REF ) is very similar.",
"Let $*$ stands for the convolution product.",
"We define the renewal measure $U(dx)=\\sum _{n=0}^{+\\infty }\\pi ^{*n}(dx)$ (notations : $\\pi ^{*0}(dx)=\\delta _{0}$ , the Dirac mass at 0, $\\pi ^{*n}=\\pi *\\pi *\\dots *\\pi $ ($n$ times)).",
"We take i.i.d.",
"variables $X,X_{1},X_{2}\\dots $ of law $\\pi $ .",
"Let $f:\\,\\mathbb {R}\\rightarrow \\mathbb {R}$ be a measurable function such that $\\Vert f\\Vert _{\\infty }\\le M$ .",
"We have, for all $t\\ge 0$ , $\\mathbb {E}(f(B_{t})) & = & \\mathbb {E}\\left(\\sum _{n=0}^{+\\infty }f(X_{1}+X_{2}+\\dots +X_{n+1}-t)\\mathbb {1}_{\\lbrace X_{1}+\\dots +X_{n}\\le t<X_{1}+\\dots +X_{n+1}\\rbrace }\\right)\\\\& = & \\int _{0}^{t}\\mathbb {E}(f(s+X-t)\\mathbb {1}_{\\lbrace s+X-t>0\\rbrace })U(ds)\\,.$ We set $g(t)={\\left\\lbrace \\begin{array}{ll}\\mathbb {E}(f(X-t)\\mathbb {1}_{\\lbrace X-t>0\\rbrace }) & \\text{ if }t\\ge 0\\,,\\\\0 & \\text{ if }t<0\\,.\\end{array}\\right.",
"}$ We observe that $\\Vert g\\Vert _{\\infty }\\le M$ .",
"We have, for all $t\\ge 0$ , $\\left|\\mathbb {E}(f(X-t)\\mathbb {1}_{\\lbrace X-t>0\\rbrace })\\right| & \\le & \\Vert f\\Vert _{\\infty }\\mathbb {P}(X>t)\\\\& \\le & \\Vert f\\Vert _{\\infty }e^{-(\\theta -\\frac{ \\varepsilon ^{\\prime }}{2})t}\\mathbb {E}(e^{(\\theta -\\frac{ \\varepsilon ^{\\prime }}{2})X})\\,.$ We have (by Fact REF ) : $\\mathbb {E}(e^{(\\theta -\\frac{ \\varepsilon ^{\\prime }}{2})X})<\\infty $ .",
"The function $\\varphi $ is submultiplicative and it is such that $\\lim _{x\\rightarrow -\\infty }\\frac{\\log (\\varphi (x))}{x}=0\\le \\lim _{x\\rightarrow +\\infty }\\frac{\\log (\\varphi (x))}{x}=\\theta - \\varepsilon ^{\\prime }\\,.$ The function $g$ is in $L^{1}(\\mathbb {R})$ .",
"The function $g.\\varphi $ is in $L^{\\infty }(\\mathbb {R})$ .",
"We have $g(x)\\varphi (x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .",
"We have $\\varphi (t)\\int _{t}^{+\\infty }|g(x)|dx\\underset{t\\rightarrow +\\infty }{\\longrightarrow }0\\,,\\,\\varphi (t)\\int _{-\\infty }^{t}|g(x)|dx\\underset{t\\rightarrow -\\infty }{\\longrightarrow }0\\,.$ We have $\\mathcal {T}^{\\circ 2}(\\pi )\\in S(\\varphi )$ .",
"Let us now take a function $\\alpha $ such that $\\Vert \\alpha \\Vert _{\\infty }\\le M$ .",
"We set $\\widehat{\\alpha }(t)={\\left\\lbrace \\begin{array}{ll}\\mathbb {E}(\\alpha (X-t)\\mathbb {1}_{\\lbrace X-t\\ge 0\\rbrace }) & \\text{ if }t\\ge 0\\,,\\\\0 & \\text{ if }t<0\\,.\\end{array}\\right.",
"}$ Then we have $\\Vert \\widehat{\\alpha }\\Vert _{\\infty }\\le M$ and (computing as above for $f$ ) $\\mathbb {E}(\\alpha (B_{t})) & = & \\widehat{\\alpha }*U(t)$ In the case where $f$ is constant equal to $M$ , we have $\\Vert g\\Vert _{\\infty }=M$ .",
"So, by [21], Theorem 5.1 (applied to the case $f\\equiv M$ ), we have proved the desired result.",
"Corollary 3.3 There exists a constant $\\Gamma _{1}$ bigger than 1 such that : for any bounded measurable function $F$ on $\\mathbb {R}$ such that $\\eta (F)=0$ , $|\\mathbb {E}(F(B_{t}))|\\le \\Vert F\\Vert _{\\infty }\\times \\frac{\\Gamma _{1}}{\\varphi (t)}$ for $t$ outside a set of Lebesgue measure zero, and $|\\mathbb {E}(F(C_{t}))|\\le \\Vert F\\Vert _{\\infty }\\times \\frac{\\Gamma _{1}}{\\varphi (t)}$ for $t$ outside a set of Lebesgue measure zero.",
"We write the proof of Equation (REF ) only.",
"The proof of Equation (REF ) is very similar.",
"We take $M=1$ in the above Theorem.",
"Keep in mind that $\\eta $ is defined in Equation (REF ).",
"By the above Theorem, there exists a constant $\\Gamma _{1}$ such that : for all measurable function $\\alpha $ such that $\\Vert \\alpha \\Vert _{\\infty }\\le 1$ , $\\left|\\mathbb {E}(\\alpha (B_{t}))-\\eta (\\alpha )\\right|\\le \\frac{\\Gamma _{1}}{\\varphi (t)}\\,\\text{(for }t\\text{ outside a set of Lebesque measure zero).",
"}$ Let us now take a bounded measurable $F$ such that $\\eta (F)=0$ .",
"By Equation (REF ), we have (for $t$ outside a set of Lebesgue measure zero) $\\left|\\mathbb {E}\\left(\\frac{F(B_{t})}{\\Vert F\\Vert _{\\infty }}\\right)-\\eta \\left(\\frac{F}{\\Vert F\\Vert _{\\infty }}\\right)\\right| & \\le & \\frac{\\Gamma _{1}}{\\varphi (t)}\\\\|\\mathbb {E}(F(B_{t}))| & \\le & \\Vert F\\Vert _{\\infty }\\times \\frac{\\Gamma _{1}}{\\varphi (t)}\\,.$"
],
[
"Notations",
"We fix $q\\in \\mathbb {N}^{*}$ .",
"We set $\\mathcal {S}_{q}$ to be the symmetric group of order $q$ .",
"A function $F\\,:\\,\\mathbb {R}^{q}\\rightarrow \\mathbb {R}$ is symmetric if $\\forall \\sigma \\in \\mathcal {S}_{q}\\,,\\,\\forall (x_{1},\\dots ,x_{q})\\in \\mathbb {R}^{q}\\,,\\,F(x_{\\sigma (1)},x_{\\sigma (2)},\\dots ,x_{\\sigma (q)})=F(x_{1},x_{2},\\dots ,x_{q})\\,.$ For $F:\\mathbb {R}^{q}\\rightarrow \\mathbb {R}$ , we define a symmetric version of $F$ by $F_{\\text{sym}}(x_{1},\\dots ,x_{q})=\\frac{1}{q!",
"}\\sum _{\\sigma \\in \\mathcal {S}_{q}}F(x_{\\sigma (1)},\\dots ,x_{\\sigma (q)})\\,,\\,\\text{for all }(x_{1},\\dots ,x_{q})\\in \\mathbb {R}^{q}\\,.$ We set $\\mathcal {B}_{\\text{sym }}(q)$ to be the set of bounded, measurable, symmetric functions $F$ on $\\mathbb {R}^{q}$ , and we set $\\mathcal {B}_{\\text{sym }}^{0}(q)$ to be the $F$ of $\\mathcal {B}_{\\text{sym }}(q)$ such that $\\int _{x_{1}}F(x_{1},x_{2},\\dots ,x_{q})\\eta (dx_{1})=0\\,,\\,\\forall (x_{2},\\dots ,x_{q})\\in \\mathbb {R}^{q-1}\\,.$ Suppose that $k$ is in $[q]$ and $l\\ge 1$ .",
"For $t$ in $[0,T]$ , we consider the following collections of nodes of $\\mathcal {U}$ (remember $T=-\\log \\varepsilon $ and $\\mathcal {U}$ , $\\mathbf {m}(.",
")$ defined in Section REF ) : $\\mathcal {T}_{1}=\\lbrace u\\in \\mathcal {U}\\backslash \\lbrace 0\\rbrace \\,:\\,A_{u}\\ne \\emptyset \\,,\\,\\xi _{m(u)}\\ge \\varepsilon \\rbrace \\cup \\lbrace 0\\rbrace \\,,$ $S(t)=\\lbrace u\\in \\mathcal {T}_{1}\\,:\\,-\\log (\\xi _{m(u)})\\le t\\,,\\,-\\log (\\xi _{u})>t\\rbrace =\\mathcal {U}_{e^{-t}}\\,,$ $L_{t}=\\sum _{u\\in S(t)\\,:\\,A_{u}\\ne \\emptyset }(\\#A_{u}-1)\\,.$ We set $\\mathcal {L}_{1}$ to be the set of leaves in the tree $\\mathcal {T}_{1}$ .",
"For $t$ in $[0,T]$ and $i$ in $[q]$ , there exists one and only one $u$ in $S(t)$ such that $i\\in A_{u}$ .",
"We call it $u\\lbrace t,i\\rbrace $ .",
"Under Assumption REF , there exists a constant bounding the numbers vertices of $\\mathcal {T}_{1}$ almost surely.",
"Let us look at an example in Figure REF .",
"Figure: Tree and marksHere, we have a graphic representation of a realization of $\\mathcal {T}_{1}$ .",
"Each node $u$ of $\\mathcal {T}_{1}$ is written above a rectangular box in which we read $A_{u}$ ; the right side of the box has the coordinate $-\\log (\\xi _{u})$ on the $X$ -axis.",
"For simplicity, the node $(1,1)$ is designated by 11, the node $(1,2)$ is designated by 12, and so on.",
"In this example : $\\mathcal {T}_{1}=\\lbrace (0),(1),(2),(1,1),(2,1),(1,2),(1,1,1),(2,1,1),(1,1,1,1),(1,2,1)\\rbrace $ , $\\mathcal {L}_{1}=\\lbrace (2,1,1),(1,1,1,1),(1,2,1)\\rbrace $ , $A_{(1)}=\\lbrace 1,2,3\\rbrace $ , $A_{(1,2)}=\\lbrace 1,2\\rbrace $ , ... $S(t)=\\lbrace (1,2),(1,1),(2,1)\\rbrace $ , $u\\lbrace t,1\\rbrace =(1,2)$ , $u\\lbrace t,2\\rbrace =(1,2)$ , $u\\lbrace t,3\\rbrace =(1,1)$ , $u\\lbrace t,4\\rbrace =(2,1)$ .",
"For $k$ , $l$ in $\\mathbb {N}$ and $t$ in $[0,T]$ , we define the event $C_{k,l}(t)=\\lbrace \\sum _{u\\in S(t)}\\mathbb {1}_{\\#A_{u}=1}=k\\,,\\,\\sum _{u\\in S(t)}(\\#A_{u}-1)=l\\rbrace \\,.$ For example, in Figure REF , we are in the event $C_{2,1}(t)$ .",
"We define $\\mathcal {T}_{2}=\\lbrace u\\in \\mathcal {T}_{1}\\backslash \\lbrace 0\\rbrace \\,:\\,\\#A_{m(u)}\\ge 2\\rbrace \\cup \\lbrace 0\\rbrace \\,,$ $m_{2}\\,:\\,u\\in \\mathcal {T}_{2}\\mapsto (\\xi _{u},\\inf \\lbrace i,i\\in A_{u}\\rbrace )\\,.$ For example, in Figure REF , $\\mathcal {T}_{2}=\\lbrace (0),(1),(2),(1,1),(1,2),(1,2,1)\\rbrace $ .",
"Let $\\alpha $ be in $(0,1)$ .",
"Fact 4.1 We can always suppose that $(1-\\alpha )T>b$ because we are interested in $T$ going to infinity.",
"So, in the following, we suppose $(1-\\alpha )T>b$ .",
"For any $t$ , we can compute $\\sum _{u\\in S(t)}(\\#A_{u}-1)$ if we know $\\sum _{u\\in S(t)}\\mathbb {1}_{\\#A_{u}=1}$ and $\\#S(t)$ .",
"As $T-\\alpha T>b$ , any $u$ in $S(\\alpha T)$ satisfies $\\#A_{u}\\ge 2$ if and only if $u$ is the mother of some $v$ in $\\mathcal {T}_{2}$ .",
"So we deduce that $C_{k,l}(\\alpha T)$ is measurable with respect to $(\\mathcal {T}_{2},m_{2})$ .",
"We set, for all $u$ in $\\mathcal {T}_{2}$ , $T_{u}=-\\log (\\xi _{u})\\,.$ For any $i$ in $[q]$ , $t\\mapsto u\\lbrace t,i\\rbrace $ is piece-wise constant and the the ordered sequence of its jump times is $S_{_{1}}^{(i)}<S_{2}^{(i)}<\\dots $ (the $S_{\\dots }^{(i)}$ are defined in Section REF ).",
"We simply have that 1, $e^{-S_{1}^{(i)}}$ , $e^{-S_{2}^{i)}}$ , ... are the successive sizes of the fragment supporting the tag $i$ .",
"For example, in Figure REF , we have $S_{1}^{(1)}=-\\log (\\xi _{1})\\,,\\,S_{2}^{(1)}=-\\log (\\xi _{(1,2)})\\,,\\,S_{3}^{(1)}=-\\log (\\xi _{(1,2,1)})\\,,\\dots $ Let $\\mathcal {L}_{2}$ be the set of leaves $u$ in the tree $\\mathcal {T}_{2}$ such that the set $A_{u}$ has a single element $n_{u}$ .",
"For example, in Figure REF , $\\mathcal {L}_{2}=\\lbrace (2),(1,1)\\rbrace $ .",
"We observe that $\\#\\mathcal {L}_{1}=q\\Leftrightarrow \\#\\mathcal {L}_{2}=q$ , and thus $\\lbrace \\mathcal {L}_{1}=q\\rbrace \\in \\sigma (\\mathcal {L}_{2})\\,.$ We summarize the definition of $n_{u}$ in the following equation $\\#A_{u}=1\\Rightarrow A_{u}=\\lbrace n_{u}\\rbrace \\,.$ For $q$ even ($q=2p$ ) and for all $t$ in $[0,T]$ , we define the events $G_{t}=\\lbrace \\forall i\\in [p]\\,,\\,\\exists u_{i}\\in \\mathcal {U}\\,:\\,\\xi _{u_{i}}<e^{-t}\\,,\\,\\xi _{m(u_{i})}\\ge e^{-t}\\,,\\,A_{u_{i}}=\\lbrace 2i-1,2i\\rbrace \\rbrace \\,,$ $\\forall i\\in [p]\\,,\\,G_{i,i+1}(t)=\\lbrace \\exists u\\in S(t)\\,:\\,\\lbrace 2i-1,2i\\rbrace \\subset A_{u}\\rbrace \\,.$ We set, for all $t$ in $[0,T]$ , $\\mathcal {F}_{S(t)}=\\sigma (S(t),(\\xi _{u},A_{u})_{u\\in S(t)})\\,.$"
],
[
"Intermediate results ",
"The reader has to keep in mind that $T=-\\log ( \\varepsilon )$ (Equation (REF )) and that $\\delta $ is defined in Assumption REF .",
"The set $\\mathcal {B}_{\\text{sym}}^0(q)$ is defined in Section REF .",
"Lemma 4.2 We suppose that $F$ is in $\\mathcal {B}_{\\text{sym}}^0(q)$ and that $F$ is of the form $F=(f_{1}\\otimes f_{2}\\otimes \\dots \\otimes f_{q})_{\\text{sym}}$ , with $f_{1}$ , $f_{2}$ , ... , $f_{q}$ in $\\mathcal {B}_{\\text{sym}}^0(1)$ .",
"Let $A$ be in $\\sigma (\\mathcal {L}_{2})$ .",
"For any $\\alpha $ in $]0,1[$ , $k$ in $[q]$ and $l$ in $\\lbrace 0,1,\\dots ,(q-k-1)_{+}\\rbrace $ , we have $|\\mathbb {E}(\\mathbb {1}_{C_{k,l}(\\alpha T)}\\mathbb {1}_{A}F(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)}))|\\le \\Vert F\\Vert _{\\infty }\\Gamma _{1}^{q}C_{tree}(q)\\left(\\frac{1}{\\delta }\\right)^{q} \\varepsilon ^{q/2}\\,,$ (for a constant $C_{tree}(q)$ defined below in the proof and $\\Gamma _{1}$ defined in Corollary REF ) and $$ $ \\varepsilon ^{-q/2}\\mathbb {E}(\\mathbb {1}_{C_{k,l}(\\alpha T)}\\mathbb {1}_{A}F(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)}))\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }0\\,.$ Let $A$ be in $\\sigma (\\mathcal {L}_{2})$ .",
"We have $\\mathbb {E}(\\mathbb {1}_{C_{k,l}(\\alpha T)}\\mathbb {1}_{A}F(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)}))\\\\=\\mathbb {E}(\\mathbb {1}_{A}\\sum _{f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])\\text{ s.t.",
"\\dots }}\\mathbb {E}(F(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2}))\\,\\\\\\text{(}\\mathcal {P}\\text{ defined in Section \\ref {subsec:Notations})}$ where we sum on the $f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])$ such that ${\\left\\lbrace \\begin{array}{ll}f(u)=\\sqcup _{v:m(v)=u}f(v), & \\text{ for all u in $\\mathcal {T}_{2}$,}\\\\\\sum _{u\\in S(\\alpha T)}\\mathbb {1}_{\\#f(u)=1}=k\\text{ and }\\sum _{u\\in S(\\alpha T)}(\\#f(u)-1)=l\\end{array}\\right.",
"}$ We remind the reader that $\\sqcup $ is defined in Section REF (disjoint union), $m(\\dots )$ is defined in Section REF (mother), $S(\\dots )$ is defined in Equation (REF ).",
"Here, we mean that we sum over the $f$ compatible with a description of tagged fragments.",
"If $u$ in $\\mathcal {L}_{2}$ and if $T_{u}<T$ , then, conditionally on $\\mathcal {T}_{2}$ , $m_{2}$ , $B_{T}^{(n_{u})}$ is independent of all the other variables and has the same law as $B_{T-T_{u}}^{(1)}$ ($T_{u}$ is defined in Equation (REF ), $n_{u}$ is defined in Equation (REF )).",
"Thus, using Theorem REF and Corollary REF , we get, for any $ \\varepsilon ^{\\prime }\\in (0,\\theta -1)$ , $u\\in \\mathcal {L}_{2}$ , $|\\mathbb {E}(f_{n_{u}}(B_{T}^{(n_{u})})|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})|\\le \\Gamma _{1}\\Vert f_{n_{u}}\\Vert _{\\infty }e^{-(\\theta - \\varepsilon ^{\\prime })(T-T_{u})_{+}}\\,,\\,\\\\\\text{for }T-T_{u}\\notin Z_{0}\\text{ where $Z_{0}$ is of Lebesgue measure zero.",
"}$ Thus we get $\\frac{|\\mathbb {E}(\\mathbb {1}_{C_{k,l}(\\alpha T)}\\mathbb {1}_{A}F(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)}))|}{\\Vert F\\Vert _{\\infty }\\Gamma _{1}^{q}}\\\\\\text{(since }F\\text{ is of the form }F=(f_{1}\\otimes \\dots \\otimes f_{q})_{\\text{sym}}\\text{,}\\\\\\text{since, conditionally on }u\\in \\mathcal {L}_{2}\\text{, }\\text{the distribution of }T_{u}\\text{ is absolutely continuous }\\\\\\text{with respect to the Lebesgue measure)}\\\\\\text{ }\\\\\\le \\mathbb {E}\\left(\\sum _{f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])\\text{ s.t.",
"\\dots }}\\left[\\prod _{u\\in \\mathcal {L}_{2}}e^{-(\\theta - \\varepsilon ^{\\prime })(T-T_{u})_{+}}\\times \\mathbb {1}_{A}\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\right]\\right)\\\\\\text{(because of Assumption \\ref {hyp:delta-step} and because }\\theta - \\varepsilon ^{\\prime }>1\\text{)}\\\\\\le \\mathbb {E}\\left(\\sum _{f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])\\text{ s.t.",
"\\dots }}\\left[\\prod _{u\\in \\mathcal {L}_{2}}e^{-(T-T_{m(u)})-\\log (\\delta )}\\times \\mathbb {1}_{A}\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\right]\\right)\\\\\\text{(because of Equation (\\ref {eq:proba-rester-dans-fragment}), see full proof in Section \\ref {subsec:Detailed-proof-of})}\\\\\\le \\mathbb {E}\\left(\\sum _{f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])\\text{ s.t.",
"\\dots }}\\mathbb {1}_{A}\\left[\\prod _{u\\in \\mathcal {L}_{2}}e^{-(T-T_{m(u)})-\\log (\\delta )}\\times \\prod _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace }e^{-(\\#f(u)-1)(T_{u}-T_{m(u)})}\\right]\\right)\\,.$ For a fixed $\\omega $ and a fixed $f$ , we have $\\prod _{u\\in \\mathcal {L}_{2}}e^{-(T-T_{m(u)})-\\log (\\delta )}\\times \\prod _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace }e^{-(\\#f(u)-1)(T_{u}-T_{m(u)})}=\\left(\\frac{1}{\\delta }\\right)^{\\#\\mathcal {L}_{2}}\\exp \\left(-\\int _{0}^{T}a(s)ds\\right)\\,,$ where, for all $s$ , $a(s) & = & \\sum _{u\\in \\mathcal {L}_{2}\\backslash \\lbrace 0\\rbrace \\,:\\,T_{m(u)}\\le s<T}\\mathbb {1}_{\\#f(u)=1}+\\sum _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace \\,:\\,T_{m(u)}\\le s\\le T_{u}}(\\#f(u)-1)\\\\& & \\text{(if }u\\in \\mathcal {T}_{2}\\backslash \\mathcal {L}_{2}\\text{, }\\mathbb {1}_{\\#f(u)=1}=0\\text{) }\\\\& = & \\sum _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace \\,:\\,T_{m(u)}\\le s<T}\\mathbb {1}_{\\#f(u)=1}+\\sum _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace \\,:\\,T_{m(u)}\\le s\\le T_{u}}(\\#f(u)-1)\\\\& & \\text{(}S(.",
")\\text{ defined in Equation (\\ref {eq:def_S(t)}))}\\\\& \\ge & \\sum _{u\\in S(s)}\\mathbb {1}_{\\#f(u)=1}+\\sum _{u\\in S(s)}(\\#f(u)-1)\\,.$ We observe that, under Equation (REF ) : $a(t)\\ge \\left\\lceil \\frac{q}{2}\\right\\rceil \\,,\\,\\forall t\\,,$ $a(\\alpha T)\\ge k+l,$ and if $t$ is such that $\\sum _{u\\in S(t)}\\mathbb {1}_{\\#f(u)=1}=k^{\\prime }\\,,\\,\\sum _{u\\in S(t)}(\\#f(u)-1)=l^{\\prime }\\,,$ for some integers $k^{\\prime }$ , $l^{\\prime }$ , then for all $s\\ge t$ , $a(s)\\ge k^{\\prime }+\\left\\lceil \\frac{q-k^{\\prime }}{2}\\right\\rceil \\,.$ We observe that, under Assumption REF , there exists a constant which bounds $\\#\\mathcal {T}_{2}$ almost surely (because for all $u$ in $\\mathcal {U}\\backslash \\lbrace 0\\rbrace $ , $-\\log (\\xi _{u})+\\log (\\xi _{\\mathbf {m}(u)})\\ge a$ ) and so there exists a constant $C_{tree}(q)$ which bounds $\\#\\lbrace f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])\\rbrace $ almost surely.",
"So, we have $|\\mathbb {E}(\\mathbb {1}_{A}F(B_{T}^{(1)},B_{T}^{(2)},\\dots ,B_{T}^{(q)}))|\\\\\\le \\Vert F\\Vert _{\\infty }\\Gamma _{1}^{q}\\mathbb {E}\\left(\\sum _{f:\\mathcal {T}_{2}\\rightarrow \\mathcal {P}([q])\\text{ s.t.",
"\\dots }}\\mathbb {1}_{A}\\left(\\frac{1}{\\delta }\\right)^{\\#\\mathcal {L}_{2}}e^{-\\lceil q/2\\rceil \\alpha T}e^{-(k+\\left\\lceil \\frac{q-k}{2}\\right\\rceil )(T-\\alpha T)}\\right)\\\\\\le \\Vert F\\Vert _{\\infty }\\Gamma _{1}^{q}C_{tree}(q)\\left(\\frac{1}{\\delta }\\right)^{q}e^{-\\lceil q/2\\rceil \\alpha T}e^{-(k+\\left\\lceil \\frac{q-k}{2}\\right\\rceil )(1-\\alpha )T}\\,.$ As $k\\ge 1$ , then $k+\\left\\lceil \\frac{q-k}{2}\\right\\rceil >\\frac{q}{2}$ , and so we have proved the desired result (remember that $T=-\\log \\varepsilon $ ).",
"Remark 4.3 If we replaced Assumption REF by Assumption REF , we would have difficulties adapting the above proof.",
"In the second line of Equation (REF ) above, the $1/\\delta $ becomes $e^{T_{u}-T_{m(u)}}$ .",
"In addition, the tree $\\mathcal {T}_{2}$ is not a.s. finite anymore.",
"So the expectation on the second line of (REF ) could certainly be bounded, but for a high price (a lot more computations, maybe assumptions on the tails of $\\pi $ , and so on).",
"This is why we stick to Assumption REF .",
"Remember that $L_{t}$ ($t\\ge 0$ ) is defined in Equation (REF ).",
"Lemma 4.4 Let $k$ be an integer $\\ge q/2$ .",
"Let $\\alpha \\in [q/(2k),1]$ .",
"We have $\\mathbb {P}(L_{\\alpha T}\\ge k)\\le K_{1}(q) \\varepsilon ^{q/2}\\,,$ where $K_{1}(q)=\\sum _{i\\in [q]}\\frac{q!}{(q-i)!",
"}\\times i^{q-i}$ .",
"Let $k$ be an integer $>q/2$ .",
"Let $\\alpha \\in (q/(2k),1)$ .",
"We have $ \\varepsilon ^{-q/2}\\mathbb {P}(L_{\\alpha T}\\ge k)\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }0\\,.$ (We remind the reader that $T=-\\log ( \\varepsilon )$ .)",
"Let $k$ be an integer $\\ge q/2$ and let $\\alpha \\in [q/(2k),1]$ .",
"Remember that $S(\\dots )$ is defined in Equation (REF ).",
"Observe that : $\\#S(\\alpha T)=i$ if, and only if $L_{\\alpha T}=q-i$ (see Equation (REF )).",
"We decompose $\\lbrace L_{\\alpha T}\\ge k\\rbrace & = & \\lbrace L_{\\alpha T}\\in \\lbrace k,k+1,\\dots ,q-1\\rbrace \\rbrace \\\\& = & \\cup _{i\\in [q-k]}\\lbrace \\#S(\\alpha T)=i\\rbrace \\\\& = & \\cup _{i\\in [q-k]}\\cup _{m:[i]\\hookrightarrow [q]}(F(i,m)\\cap \\lbrace \\#S(\\alpha T)=i\\rbrace )\\,,$ (remember that “$\\hookrightarrow $ ” means we are summing on injections, see Section REF ) where $F(i,m)=\\lbrace i_{1},i_{2}\\in [i]\\text{ with }i_{1}\\ne i_{2}\\Rightarrow \\exists u_{1},u_{2}\\in S(\\alpha T),u_{1}\\ne u_{2},m(i_{1})\\in A_{u_{1}}\\,,\\,m(i_{2})\\in A_{u_{2}}\\rbrace $ (to make the above equations easier to understand, observe that if $\\#S(\\alpha T)=i$ , we have for each $j\\in [i]$ , an index $m(j)$ in $A_{u}$ for some $u\\in S(\\alpha T)$ , and we can choose $m$ such that we are in the event $F(i,m)$ ).",
"Suppose we are in the event $F(i,m)$ .",
"For $u\\in S(\\alpha T)$ and for all $j$ in $[i]$ such that $m(j)\\in A_{u}$ , we define (remember $|u|$ and $\\mathbf {m}$ are defined in Section REF ) $T_{|u|}^{(j)}=-\\log (\\xi _{u})\\,,\\,T_{|u|-1}^{(j)}=-\\log (\\xi _{m(u)})\\,,\\,\\dots \\,,\\,T_{1}^{(j)}=-\\log (\\xi _{m^{\\circ (|u|-1)}(u)})\\,,T_{0}^{(j)}=0\\,,$ $l(j)=|u|\\,,\\,v(j)=u\\,.$ We have $\\mathbb {P}(L_{\\alpha T}\\ge k)\\le \\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\mathbb {P}(F(i,m)\\cap \\lbrace \\#S(\\alpha T)=i\\rbrace )\\\\=\\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\mathbb {E}(\\mathbb {1}_{F(i,m)}\\mathbb {E}(\\mathbb {1}_{\\#S(\\alpha T)=i}|F(i,m),(T_{p}^{(j)})_{j\\in [i],p\\in [l(j)]},(v(j))_{j\\in [i]}))\\\\\\text{(below, we sum over the partitions $\\mathcal {B}$ of }[q]\\backslash m([i])\\text{ into }i\\text{ subsets $\\mathcal {B}_{1},\\mathcal {B}_{2},\\dots ,\\mathcal {B}_{i}$)}\\\\=\\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\sum _{\\mathcal {B}}\\mathbb {E}(\\mathbb {1}_{F(i,m)}\\mathbb {E}(\\prod _{j\\in [i]}\\prod _{r\\in \\mathcal {B}_{j}}\\mathbb {1}_{r\\in A_{v(j)}}|F(i,m),(T_{p}^{(j)})_{j\\in [i],p\\in [l(j)]},(v(j))_{j\\in [i]}))\\\\\\text{(as }Y_{1},\\dots ,Y_{q}\\text{ defined in Section \\ref {subsec:Tagged-fragments} are independant)}\\\\=\\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\sum _{\\mathcal {B}}\\mathbb {E}(\\mathbb {1}_{F(i,m)}\\prod _{j\\in [i]}\\prod _{r\\in \\mathcal {B}_{j}}\\mathbb {E}(\\mathbb {1}_{r\\in A_{v(j)}}|F(i,m),(T_{p}^{(j)})_{j\\in [i],p\\in [l(j)]},(v(j))_{j\\in [i]}))\\\\\\mbox{(because of Equation (\\ref {eq:proba-rester-dans-fragment}) and Equation (\\ref {eq:def-T_u}))}\\\\=\\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\sum _{\\mathcal {B}}\\mathbb {E}\\left(\\mathbb {1}_{F(i,m)}\\prod _{j\\in [i]}\\prod _{r\\in \\mathcal {B}_{j}}\\prod _{s=1}^{l(j)}\\exp ((-T_{s}^{(j)}+T_{s-1}^{(j)}))\\right)\\\\\\text{(as }v(j)\\in S(\\alpha T)\\text{)}\\\\\\le \\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\sum _{\\mathcal {B}}\\prod _{j\\in [i]}\\prod _{r\\in \\mathcal {B}_{j}}e^{-\\alpha T}\\\\=\\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\sum _{\\mathcal {B}}e^{-\\alpha (q-i)T}\\\\\\le \\sum _{i\\in [q-k]}\\sum _{m:[i]\\hookrightarrow [q]}\\sum _{\\mathcal {B}}e^{-k\\alpha T}\\le e^{-k\\alpha T}\\times \\sum _{i\\in [q]}\\frac{q!}{(q-i)!",
"}i^{q-i}\\,.$ If we suppose that $k>q/2$ and $\\alpha \\in (q/(2k),1)$ , then $\\exp \\left(\\frac{qT}{2}\\right)\\exp (-k\\alpha T)\\underset{T\\rightarrow +\\infty }{\\longrightarrow }0\\,.$ Immediate consequences of the two above lemmas are the following Corollaries.",
"Corollary 4.5 If $q$ is odd and if $F\\in \\mathcal {B}_{\\text{sym}}^{0}(q)$ is of the form $F=(f_{1}\\otimes \\dots \\otimes f_{q})_{\\text{sym}}$ , then $$ $ \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q})\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }0\\,.$ ($\\mathcal {B}_{\\text{sym}}^{0}$ and $\\mathcal {L}_{1}$ are defined in Section REF .)",
"We take $\\alpha \\in \\left(\\frac{q}{2}\\left\\lceil \\frac{q}{2}\\right\\rceil ^{-1},1\\right)$ .",
"We observe that for $k$ in $[q]$ , $t$ in $(0,T)$ , $\\sum _{u\\in S(t)}\\mathbb {1}_{\\#A_{u}=1}=k\\Rightarrow \\sum _{u\\in S(t)}(\\#A_{u}-1)\\in \\lbrace 0,1,\\dots ,(q-k-1)_{+}\\rbrace $ and ($L_{t}$ defined in Equation (REF )) $\\sum _{u\\in S(t)}\\mathbb {1}_{\\#A_{u}=1}=0\\Rightarrow L_{t}\\ge \\lceil \\frac{q}{2}\\rceil \\,.$ So we can decompose $ \\varepsilon ^{-q/2}\\left|\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q})\\right|\\\\=| \\varepsilon ^{-q/2}\\sum _{k\\in [q]}\\sum _{l\\in \\lbrace 0,1,\\dots ,(q-k-1)_{+}\\rbrace }\\mathbb {E}(\\mathbb {1}_{C_{k,l}(\\alpha T)}\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}F(B_{T}^{(1)},\\dots ,B_{T}^{(q)}))\\\\+ \\varepsilon ^{-q/2}\\mathbb {E}(\\mathbb {1}_{L_{\\alpha T}\\ge \\left\\lceil q/2\\right\\rceil }\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}F(B_{T}^{(1)},\\dots ,B_{T}^{(q)}))|\\\\\\text{(by (\\ref {eq:L1(q)_in_sigma...}) and Lemmas \\ref {lem:conv-nouille-libre}, \\ref {lem:conv-nouilles-liees}) }\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }0\\,$ ($\\mathcal {L}_{1}$ and $\\mathcal {L}_{2}$ defined in Section REF ).",
"Corollary 4.6 Suppose $F\\in \\mathcal {B}_{sym}^{0}(q)$ is of the form $F=(f_{1}\\otimes \\dots \\otimes f_{q})_{\\text{sym}}$ .",
"Let $A$ in $\\sigma (\\mathcal {L}_{2})$ .",
"Then $|\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{A})|\\le \\Vert F\\Vert _{\\infty } \\varepsilon ^{q/2}\\left\\lbrace K_{1}(q)+\\Gamma _{1}^{q}C_{\\text{tree}}(q)\\left(\\frac{1}{\\delta }\\right)^{q}(q+1)^{2}\\right\\rbrace \\,$ We get (as in Equation (REF ) above) $|\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{A})|=|\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{A}(\\mathbb {1}_{L_{\\alpha T}\\ge q/2}+\\sum _{k^{\\prime }\\in [q]}\\sum _{0\\le l\\le (q-k^{\\prime }-1)_{+}}\\mathbb {1}_{C_{k^{\\prime },l}(\\alpha T)}))|\\\\\\text{(from Lemmas \\ref {lem:conv-nouille-libre}, \\ref {lem:conv-nouilles-liees})}\\\\\\le \\Vert F\\Vert _{\\infty } \\varepsilon ^{q/2}\\left\\lbrace K_{1}(q)+\\Gamma _{1}^{q}C_{\\text{tree}}(q)\\left(\\frac{1}{\\delta }\\right)^{q}\\sum _{k^{\\prime }\\in [q]}1+(q-k^{\\prime }-1)_{+}\\right\\rbrace $ and $\\sum _{k^{\\prime }\\in [q]}1+(q-k^{\\prime }-1)_{+}\\le (q+1)^{2}$ (see Section REF in the Appendix for a detailed proof).",
"We now want to find the limit of $ \\varepsilon ^{-q/2}\\mathbb {E}(\\mathbb {1}_{L_{T}\\le q/2}\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}F(B_{T}^{(1)},\\dots ,B_{T}^{(q)}))$ when $ \\varepsilon $ goes to 0, for $q$ even.",
"First we need a technical lemma.",
"For any $i$ , the process $(B_{t}^{(i)})$ has a stationary law (see Theorem 3.3 p. 151 of [1]).",
"Let $B_{\\infty }$ be a random variable having this stationary law $\\eta $ (it has already appeared in Section ).",
"We can always suppose that it is independent of all the other variables.",
"Fact 4.7 From now on, when we have an $\\alpha $ in $(0,1)$ , we suppose that $\\alpha T-\\log (\\delta )<(T+\\alpha T)/2$ , $(T+\\alpha T)/2-\\log (\\delta )<T$ (this is true if $T$ is large enough).",
"(The constant $\\delta $ is defined in Assumption REF .)",
"Lemma 4.8 Let $f_{1}$ , $f_{2}$ be in $\\mathcal {B}_{sym}^{0}(1)$ .",
"Let $\\alpha $ belong to $(0,1)$ .",
"Let $ \\varepsilon ^{\\prime }$ belong to $(0,\\theta -1)$ ($\\theta $ is defined in Fact REF ).",
"We have $\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}|\\mathbb {E}(f_{1}(\\overline{B}_{0}^{(1),v})f_{2}(\\overline{B}_{0}^{(2),v}))|dv<\\infty $ and (a.s.), for $T$ large enough, $\\left|e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\mathbb {E}(f_{1}\\otimes f_{2}(B_{T,}^{(1)}B_{T}^{(2)})\\mathbb {1}_{G_{1,2}(T)^{\\complement }}|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\right.\\\\\\left.-\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f_{1}(\\overline{B}_{0}^{(1),v})f_{2}(\\overline{B}_{0}^{(2),v}))dv\\right|\\\\\\le \\Gamma _{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }\\exp \\left(-(T-\\alpha T)\\left(\\frac{\\theta - \\varepsilon ^{\\prime }-1}{2}\\right)\\right)\\,,$ where $\\Gamma _{2}=\\frac{\\Gamma _{1}^{2}}{\\delta ^{2+2(\\theta - \\varepsilon ^{\\prime })}(2(\\theta - \\varepsilon ^{\\prime })-1)}+\\frac{\\Gamma _{1}}{\\delta ^{\\theta - \\varepsilon ^{\\prime }}}+\\frac{\\Gamma _{1}^{2}}{\\delta ^{2(\\theta - \\varepsilon ^{\\prime })}(2(\\theta - \\varepsilon ^{\\prime })-1)}\\,.$ (The processes $B^{(1)}$ , $B^{(2)}$ , $\\overline{B}^{(1),v}$ , $\\overline{B}^{(2),v}$ are defined in Sections REF , REF , REF .)",
"We have, for all $s$ in $[\\alpha T+B_{\\alpha T}^{(1)},T]$ , (because of Equation (REF ) and Equation (REF )) $\\mathbb {P}(u\\lbrace s,2\\rbrace =u\\lbrace s,1\\rbrace |\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})=\\exp (-(s+B_{s}^{(1)}-(\\alpha T+B_{\\alpha T}^{(1)}))$ (we remind the reader that $u\\lbrace s,1\\rbrace $ , $G_{1,2}$ , ... are defined in Section REF , below Equation (REF )).",
"Let us introduce the breaking time $\\tau _{1,2}$ between 1 and 2 as a random variable having the following property : conditionally on $\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1}$ , $\\tau _{1,2}$ has the density $s\\in \\mathbb {R}\\mapsto \\mathbb {1}_{[\\alpha T+B_{\\alpha T}^{(1)},+\\infty )}(s)e^{-(s-(\\alpha T+B_{T}^{(1)}))}$ (this is a translation of an exponential law).",
"We have the equalities : $\\alpha T+B_{\\alpha T}^{(1)}=S_{j_{0}}^{(1)}$ for some $j_{0}$ , $T+B_{T}^{(1)}=S_{i_{0}}^{(1)}$ for some $i_{0}$ .",
"Here, we have to make a comment on the definitions of Section REF .",
"In Figure REF , we have : $-\\log (\\xi _{(1,2)})=S_{2}^{(1)}$ (as in Equation (REF )), $S(S_{2}^{(1)})=\\lbrace (1,2,1),(1,1,1),(2,1)\\rbrace $ , $u\\lbrace -\\log (\\xi _{(1,2)}),1\\rbrace =\\lbrace (1,2,1)\\rbrace $ .",
"It is important to understand this example before reading what follows.",
"The breaking time $\\tau _{1,2}$ has the following interesting property (for all $k\\ge j_{0}$ ) $\\mathbb {P}(u\\lbrace S_{k}^{(1)},2\\rbrace \\ne u\\lbrace S_{k}^{(1)},1\\rbrace |\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})\\\\=\\mathbb {P}(\\tau _{1,2}\\in [\\alpha T+B_{\\alpha T}^{(1)},S_{k}^{(1)}]|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})\\,.$ Just because we can, we impose, for all $k\\ge j_{0}$ , conditionally on $\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1}$ , $\\lbrace u\\lbrace S_{k}^{(1)},2\\rbrace \\ne u\\lbrace S_{k}^{(2)},1\\rbrace \\rbrace =\\lbrace \\tau _{1,2}\\in [\\alpha T+B_{\\alpha T}^{(1)},S_{k}^{(1)}]\\rbrace \\,.$ Now, let $v$ be in $[\\alpha T+B_{\\alpha T}^{(1)},T+B_{T}^{(1)}]$ .",
"We observe that, for all $v$ in $[\\alpha T+B_{\\alpha T}^{(1)},T+B_{T}^{(1)}]$ , $\\mathbb {E}(f_{1}\\otimes f_{2}(B_{T,}^{(1)}B_{T}^{(2)})\\mathbb {1}_{G_{1,2}(T)^{\\complement }}|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1},\\tau _{1,2}=v)\\\\\\text{(because of Equation (\\ref {eq:def-B-chapeau}))}\\\\=\\mathbb {E}(f_{1}\\otimes f_{2}(\\widehat{B}_{T,}^{(1),v}\\widehat{B}_{T}^{(2),v})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})\\,.$ And so, $\\mathbb {E}(f_{1}\\otimes f_{2}(B_{T,}^{(1)}B_{T}^{(2)})\\mathbb {1}_{G_{1,2}(T)^{\\complement }}|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\\\=\\mathbb {E}\\left(\\mathbb {E}(f_{1}\\otimes f_{2}(B_{T,}^{(1)}B_{T}^{(2)})\\mathbb {1}_{G_{1,2}(T)^{\\complement }}|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}\\right)\\\\=\\mathbb {E}(\\mathbb {E}(\\mathbb {E}(f_{1}\\otimes f_{2}(B_{T,}^{(1)}B_{T}^{(2)})\\mathbb {1}_{G_{1,2}(T)^{\\complement }}|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1},\\tau _{1,2})\\\\|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\\\\\text{(keep in mind that }\\widehat{B}^{(1),v}=B^{(1)}\\text{ for all }v\\text{)}\\\\=\\mathbb {E}(\\mathbb {E}(\\int _{\\alpha T+B_{\\alpha T}^{(1)}}^{T+B_{T}^{(1)}}e^{-(v-\\alpha T-\\widehat{B}_{\\alpha T}^{(1),v})}\\mathbb {E}(f_{1}\\otimes f_{2}(B_{T,}^{(1)}B_{T}^{(2)})\\mathbb {1}_{G_{1,2}(T)^{\\complement }}|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1},\\tau _{1,2}=v)dv\\\\|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\\\=\\mathbb {E}(\\mathbb {E}(\\int _{\\alpha T+B_{\\alpha T}^{(1)}}^{T+B_{T}^{(1)}}e^{-(v-\\alpha T-\\widehat{B}_{\\alpha T}^{(1),v})}\\mathbb {E}(f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})dv\\\\|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T},(S_{j}^{(1)})_{j\\ge 1})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\\\=\\mathbb {E}(\\int _{\\alpha T+B_{\\alpha T}^{(1)}}^{T+B_{T}^{(1)}}e^{-(v-\\alpha T-\\widehat{B}_{\\alpha T}^{(1),v})}f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})dv|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})$ Let us split the above integral into two parts and multiply them by $e^{T-\\alpha T-B_{\\alpha T}^{(1)}}$ .",
"We have (this is the first part) $\\left|e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\mathbb {E}\\left(\\int _{\\alpha T+B_{\\alpha T}^{(1)}}^{(T+\\alpha T)/2}e^{-(v-\\alpha T-\\widehat{B}_{\\alpha T}^{(1),v})}f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})dv|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}\\right)\\right|\\\\=e^{T-\\alpha T-B_{\\alpha T}^{(1)}}|\\mathbb {E}(\\int _{\\alpha T+B_{\\alpha T}^{(1)}}^{(T+\\alpha T)/2}e^{-(v-\\alpha T-\\widehat{B}_{\\alpha T}^{(1),v})}\\\\\\times \\mathbb {E}(f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})|\\widehat{B}_{v}^{(1),v},\\widehat{B}_{v}^{(2),v},\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})dv\\\\|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})|\\\\\\text{(using the fact that }\\widehat{B}_{T}^{(1),v}\\text{ and }\\widehat{B}_{T}^{(2),v}\\text{ are independant }\\\\\\text{conditionally on }\\lbrace \\widehat{B}_{v}^{(1),v},\\widehat{B}_{v}^{(2),v},\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}\\rbrace \\text{ if }T\\ge v-\\log (\\delta )\\text{, }\\\\\\text{we get, by Theorem \\ref {lem:sgibnev} and Corollary \\ref {cor:sgibnev} and Fact \\ref {fact:From-now-on,})}\\\\\\le e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\\\\\times \\mathbb {E}(\\int _{\\alpha T+B_{\\alpha T}^{(1)}}^{(T+\\alpha T)/2}e^{-(v-\\alpha T-\\widehat{B}_{\\alpha T}^{(1)})}(\\Gamma _{1}\\Vert f_{1}\\Vert _{\\infty }e^{-(\\theta - \\varepsilon ^{\\prime })(T-v-\\widehat{B}_{v}^{(1),v})_{+}}\\times \\Gamma _{1}\\Vert f_{2}\\Vert _{\\infty }e^{-(\\theta - \\varepsilon ^{\\prime })(T-v-\\widehat{B}_{v}^{(2),v})_{+}})dv\\\\|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\\\\\text{(using Assumption \\ref {hyp:delta-step})}\\\\\\le \\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }e^{T-\\alpha T-\\log (\\delta )}\\int _{\\alpha T}^{(T+\\alpha T)/2}e^{-(v-\\alpha T+\\log (\\delta ))}e^{-2(\\theta - \\varepsilon ^{\\prime })(T-v+\\log (\\delta ))}dv\\\\=\\frac{\\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }}{\\delta ^{2+2(\\theta - \\varepsilon ^{\\prime })}}e^{T-2(\\theta - \\varepsilon ^{\\prime })T}\\left[\\frac{e^{(2(\\theta - \\varepsilon ^{\\prime })-1)v}}{2(\\theta - \\varepsilon ^{\\prime })-1}\\right]_{\\alpha T}^{(T+\\alpha T)/2}\\\\\\le \\frac{\\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }}{\\delta ^{2+2(\\theta - \\varepsilon ^{\\prime })}}\\frac{\\exp \\left(-(2(\\theta - \\varepsilon ^{\\prime })-1)T+(2(\\theta - \\varepsilon ^{\\prime })-1)\\frac{(T+\\alpha T)}{2}\\right)}{2(\\theta - \\varepsilon ^{\\prime })-1}\\\\=\\frac{\\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }}{\\delta ^{2+2(\\theta - \\varepsilon ^{\\prime })}}\\frac{\\exp \\left(-(2(\\theta - \\varepsilon ^{\\prime })-1)\\left(\\frac{T-\\alpha T}{2}\\right)\\right)}{2(\\theta - \\varepsilon ^{\\prime })-1}\\,.$ We have (this is the second part, minus some other term) $\\left|\\begin{array}{c}\\underbrace{e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\mathbb {E}\\left(\\int _{(T+\\alpha T)/2}^{T+B_{T}^{(1)}}e^{-(v-\\alpha T-B_{\\alpha T}^{(1)})}f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})dv|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}\\right)}\\\\\\text{second part}\\end{array}\\right.\\\\\\left.-\\begin{array}{c}\\underbrace{\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{-(v-T)}\\mathbb {E}(\\mathbb {1}_{v\\le T+\\overline{B}_{T}^{(1),v}}f_{1}(\\overline{B}_{T}^{(1),v})f_{2}(\\overline{B}_{T}^{(2),v}))dv}\\\\(\\heartsuit )\\end{array}\\right|\\\\=\\left|e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\mathbb {E}(\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{-(v-\\alpha T-B_{\\alpha T}^{(1)})}\\mathbb {1}_{v\\le T+B_{T}^{(1)}}f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})dv|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})\\right.\\\\\\left.-e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\mathbb {E}(\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{-(v-\\alpha T-B_{\\alpha T}^{(1)})}\\mathbb {1}_{v\\le T+\\overline{B}_{T}^{(1),v}}f_{1}(\\overline{B}_{T}^{(1),v})f_{2}(\\overline{B}_{T}^{(2),v})dv)\\right|\\\\=e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\left|\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{-(v-\\alpha T-B_{\\alpha T}^{(1)})}\\mathbb {E}(\\mathbb {E}(\\mathbb {1}_{v\\le T+B_{T}^{(1)}}f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})\\right.\\\\|\\widehat{C}_{v}^{(1),v},\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})dv\\\\-\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{-(v-\\alpha T-B_{\\alpha T}^{(1)})}\\mathbb {E}(\\mathbb {E}(\\mathbb {1}_{v\\le T+\\overline{B}_{T}^{(1),v}}f_{1}(\\overline{B}_{T}^{(1),v})f_{2}(\\overline{B}_{T}^{(2),v})\\left.|\\overline{C}_{v}^{(1),v}))dv\\right|$ We observe that, for all $v$ in $[(T+\\alpha T)/2,T-\\log (\\delta )]$ , once $\\widehat{C}_{v}^{(1),v}$ is fixed, we can make a simulation of $\\widehat{B}_{T}^{(1),v}=B_{T}^{(1)}$ , $\\widehat{B}_{T}^{(2),v}$ (these processes are independent of $\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}$ conditionally on $\\widehat{C}_{v}^{(1),v}$ ).",
"Indeed, we draw $\\widehat{B}_{v}^{(1),v}$ conditionally on $\\widehat{C}_{v}^{(1),v}$ (with law $\\eta _{1}(\\dots |\\widehat{C}_{v}^{(1),v})$ defined in Fact REF ), then we draw $\\widehat{B}_{v}^{(2),v}$ conditionally on $\\widehat{B}_{v}^{(1),v}$ and $\\widehat{C}_{v}^{(1),v}$ (with law $\\eta ^{\\prime }(\\dots |\\widehat{B}_{v}^{(1),v},\\widehat{C}_{v}^{(1),v})$ , see Fact REF ).",
"Then, $(\\widehat{B}_{t}^{(1),v})_{t\\ge v}$ , $(\\widehat{B}_{t}^{(2),v})_{t\\ge v}$ run their courses as independent Markov processes, until we get $\\widehat{B}_{T}^{(1),v}$ , $\\widehat{B}_{T}^{(2),v}$ .",
"In the same way (for all $v$ in $[(T+\\alpha T)/2,T-\\log (\\delta )]$ ), we observe that the process $(\\overline{C}^{(1),v},\\overline{B}^{(1),v})$ starts at time $v-2b$ and has the same transition as $(C^{(1)},B^{(1)})$ (see Equation (REF )).",
"By Assumption REF , there exists the following time : $S=\\sup \\lbrace t\\,:\\,v-b\\le t\\le v\\,,\\,\\overline{C}_{t}^{(1),v}=0\\rbrace $ .",
"We then have $v-S=\\overline{C}_{v}^{(1),v}$ .",
"When $\\overline{C}_{v}^{(1),v}$ is fixed, this entails that $\\overline{B}_{v}^{(1),v}$ has the law $\\eta _{1}(\\dots |\\overline{C}_{v}^{(1),v})$ .",
"We have $\\overline{B}_{v}^{(2),v}$ of law $\\eta ^{\\prime }(\\dots |\\overline{C}_{v}^{(1),v},\\overline{B}_{v}^{(1),v})$ (by Equation (REF )).",
"As said before, we then let the process $(\\overline{B}_{t}^{(1),v},\\overline{B}_{t}^{(2),v})_{t\\ge v}$ run its course as a Markov process having the same transition as $(\\widehat{B}_{t-v+kb}^{(1),kb},\\widehat{B}_{t-v+kb}^{(2),kb})_{t\\ge v}$ until we get $\\overline{B}_{T}^{(1),v}$ , $\\overline{B}_{T}^{(2),v}$ .",
"So we get that (for all $v$ in $[(T+\\alpha T)/2,T-\\log (\\delta )]$ ) $\\mathbb {E}(\\mathbb {1}_{v\\le T+B_{T}^{(1)}}f_{1}(\\widehat{B}_{T}^{(1),v})f_{2}(\\widehat{B}_{T}^{(2),v})|\\widehat{C}_{v}^{(1),v},\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T})=\\Psi (\\widehat{C}_{v}^{(1),v})\\,,$ $\\mathbb {E}(\\mathbb {1}_{v\\le T+\\overline{B}_{T}^{(1),v}}f_{1}(\\overline{B}_{T}^{(1),v})f_{2}(\\overline{B}_{T}^{(2),v})|\\overline{C}_{v}^{(1),v})=\\Psi (\\overline{C}_{v}^{(1),v})\\overset{\\text{law}}{=}\\Psi (C_{\\infty })\\,,$ for some function $\\Psi $ , the same on both lines, such that $\\Vert \\Psi \\Vert _{\\infty }\\le \\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }$ ($C_{\\infty }$ defined in Section ).",
"So, by Theorem REF and Corollary REF applied on the time interval $[\\alpha T+B_{\\alpha T}^{(1)},v]$ , the quantity in Equation (REF ) can be bounded by (remember that $\\widehat{C}^{(1),v}=C^{(1)}$ , Section REF ) $e^{T-\\alpha T-B_{\\alpha T}^{(1)}}\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{-(v-\\alpha T-B_{\\alpha T}^{(1)})}\\Gamma _{1}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }e^{-(\\theta - \\varepsilon ^{\\prime })(v-\\alpha T-B_{\\alpha T}^{(1)})}dv$ (coming from Corollary REF there is an integral over a set of Lebesgue measure zero in the above bound, but this term vanishes).",
"The above bound can in turn be bounded by : $\\Gamma _{1}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }\\delta ^{-(\\theta - \\varepsilon ^{\\prime })}e^{T}\\int _{(T+\\alpha T)/2}^{T-\\log (\\delta )}e^{(\\theta - \\varepsilon ^{\\prime })\\alpha T}e^{-(\\theta - \\varepsilon ^{\\prime }+1)v}dv\\\\\\text{(as }\\theta - \\varepsilon ^{\\prime }+1>1\\text{) }\\\\\\le \\Gamma _{1}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }\\delta ^{-(\\theta - \\varepsilon ^{\\prime })}e^{T+\\alpha T(\\theta - \\varepsilon ^{\\prime })}\\exp (-(\\theta - \\varepsilon ^{\\prime }+1)(\\frac{T+\\alpha T}{2}))\\\\=\\Gamma _{1}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }\\delta ^{-(\\theta - \\varepsilon ^{\\prime })}\\exp \\left(-(\\theta - \\varepsilon ^{\\prime }-1)\\left(\\frac{T-\\alpha T}{2}\\right)\\right)\\,.$ We have $\\int _{\\frac{T+\\alpha T}{2}}^{T-\\log (\\delta )}e^{-(v-T)}\\mathbb {E}(\\mathbb {1}_{v\\le T+\\overline{B}_{T}^{(1),v}}f_{1}(\\overline{B}_{T}^{(1),v})f_{2}(\\overline{B}_{T}^{(2),v}))dv\\\\\\text{(as }(\\overline{B}_{T}^{(1),v},\\overline{B}_{T}^{(2),v})\\text{ and }(\\overline{B}_{0}^{(1),v-T},\\overline{B}_{0}^{(2),v-T})\\text{ have same law)}\\\\=\\int _{\\frac{T+\\alpha T}{2}}^{T-\\log (\\delta )}e^{-(v-T)}\\mathbb {E}(\\mathbb {1}_{v-T\\le \\overline{B}_{0}^{(1),v-T}}f_{1}(\\overline{B}_{0}^{(1),v-T})f_{2}(\\overline{B}_{0}^{(2),v-T}))dv\\\\\\text{(change of variable }v^{\\prime }=v-T\\text{)}\\\\=\\mathbb {E}\\left(\\int _{-\\left(\\frac{T-\\alpha T}{2}\\right)}^{-\\log (\\delta )}e^{-v^{\\prime }}\\mathbb {1}_{v^{\\prime }\\le \\overline{B}_{0}^{(1),v^{\\prime }}}f_{1}(\\overline{B}_{0}^{(1),v^{\\prime }})f_{2}(\\overline{B}_{0}^{(1),v^{\\prime }})dv^{\\prime }\\right)$ and $\\int _{-\\infty }^{-\\frac{(T-\\alpha T)}{2}}e^{-v}|\\mathbb {E}(f_{1}(\\overline{B}_{0}^{(1),v})f_{2}(\\overline{B}_{0}^{(2),v}))|dv\\\\\\text{(since }\\overline{B}_{0}^{(1),v}\\text{ and }\\overline{B}_{0}^{(2),v}\\text{ are independant conditionnaly on }\\overline{B}_{v}^{(1),v}\\text{, }\\overline{B}_{v}^{(2),v}\\\\\\text{if }v-\\log (\\delta )\\le 0\\text{)}\\\\\\text{(using Theorem \\ref {lem:sgibnev} and Corollary \\ref {cor:sgibnev}) }\\\\\\le \\int _{-\\infty }^{-\\frac{(T-\\alpha T)}{2}}e^{-v}\\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }\\mathbb {E}(e^{-(\\theta - \\varepsilon ^{\\prime })(-v-\\overline{B}_{v}^{(1),v})_{+}}e^{-(\\theta - \\varepsilon ^{\\prime })(-v-\\overline{B}_{v}^{(2),v})_{+}})dv\\\\\\text{(again, coming from Corollary \\ref {cor:sgibnev} there is an integral }\\\\\\text{over a set of Lebesgue measure zero in the above bound, but this term vanishes)}\\\\\\le \\int _{-\\infty }^{-\\frac{(T-\\alpha T)}{2}}e^{-v}\\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }e^{-2(\\theta - \\varepsilon ^{\\prime })(-v+\\log (\\delta ))}dv\\\\=\\frac{\\Gamma _{1}^{2}\\Vert f_{1}\\Vert _{\\infty }\\Vert f_{2}\\Vert _{\\infty }}{\\delta ^{2(\\theta - \\varepsilon ^{\\prime })}}\\frac{\\exp \\left(-(2(\\theta - \\varepsilon ^{\\prime })-1)\\frac{(T-\\alpha T)}{2}\\right)}{2(\\theta - \\varepsilon ^{\\prime })-1}\\,.$ Equations (REF ) and (REF ) give us Equation (REF ).",
"Equations (REF ), (REF ), (REF ) and (REF ) give us the desired result (see Figure REF to understand the puzzle).",
"Figure: PuzzleLemma 4.9 Let $k$ in $\\lbrace 0,1,2,\\dots ,p\\rbrace $ .",
"We suppose $q$ is even and $q=2p$ .",
"Let $\\alpha \\in (q/(q+2),1)$ .",
"We suppose $F=f_{1}\\otimes f_{2}\\otimes \\dots \\otimes f_{q}$ , with $f_{1}$ , ... , $f_{q}$ in $\\mathcal {B}_{sym}^{0}(1)$ .",
"We then have : $ \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T,}^{(1)}\\dots ,B_{T}^{(q)})\\mathbb {1}_{G_{\\alpha T}}\\mathbb {1}_{\\#\\mathcal {L}_{1}=q})\\\\\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }\\prod _{i=1}^{p}\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f_{2i-1}(\\overline{B}_{0}^{(1),v})f_{2i}(\\overline{B}_{0}^{(2),v}))dv\\,.$ (Remember that $T=-\\log \\varepsilon $ .)",
"By Fact REF , we have $T>\\alpha T-\\log (\\delta )$ .",
"We have (remember the definitions just before Section REF ) $G_{\\alpha T}\\cap \\lbrace \\#\\mathcal {L}_{1}=q\\rbrace =G_{\\alpha T}\\cap \\underset{1\\le i\\le p}{\\bigcap }G_{2i-1,2i}(T)^{\\complement }\\,.$ We have (remember $T=-\\log ( \\varepsilon )$ ) $ \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T,}^{(1)}\\dots ,B_{T}^{(q)})\\mathbb {1}_{G_{\\alpha T}}\\mathbb {1}_{\\#\\mathcal {L}_{1}=q})\\\\=e^{pT}\\mathbb {E}\\left(\\mathbb {1}_{G_{\\alpha T}}\\mathbb {E}\\left(\\left.\\prod _{i=1}^{p}f_{2i-1}\\otimes f_{2i}(B_{T}^{(2i-1)},B_{T}^{(2i)})\\mathbb {1}_{G_{2i-1,2i}(T)^{\\complement }}\\right|\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}\\right)\\right)\\\\\\text{(as }(B_{T}^{(1)},B_{T}^{(2)},\\mathbb {1}_{G_{1,2}(T)})\\text{, }(B_{T}^{(3)},B_{T}^{(4)},\\mathbb {1}_{G_{3,4}(T)})\\text{, \\dots \\ are independant conditionally on }\\mathcal {F}_{S(\\alpha T)},G_{\\alpha T}\\\\\\text{due to Fact \\ref {fact:From-now-on,})}\\\\=\\mathbb {E}\\left(\\mathbb {1}_{G_{\\alpha T}}\\prod _{i=1}^{p}e^{T}\\mathbb {E}\\left(\\left.f_{2i-1}\\otimes f_{2i}(B_{T}^{(2i-1)},B_{T}^{(2i)})\\mathbb {1}_{G_{2i-1,2i}(T)^{\\complement }}\\right|\\mathcal {F}_{S(\\alpha T)},G_{2i-1,2i}(\\alpha T)\\right)\\right)\\\\\\text{(by Lemma \\ref {lem:terme-b} and as $(B^{(1)},\\dots ,B^{(q)})$ is exchangeable)}\\\\=\\mathbb {E}\\left(\\mathbb {1}_{G_{\\alpha T}}\\prod _{i=1}^{p}e^{\\alpha T+B_{\\alpha T}^{(2i-1)}}\\left(\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f_{2i-1}(\\overline{B}_{0}^{(1),v})f_{2i}(\\overline{B}_{0}^{(2),v}))dv+R_{2i-1,2i}\\right)\\right)\\,,$ with (a.s.) $|R_{2i-1,2i}|\\le \\Gamma _{2}\\Vert f_{2i-1}\\Vert _{\\infty }\\Vert f_{2i}\\Vert _{\\infty }e^{-(T-\\alpha T)\\frac{(\\theta - \\varepsilon ^{\\prime }-1)}{2}}\\,.$ We introduce the events (for $t\\in [0,T]$ )($u\\lbrace \\dots \\rbrace $ defined below Equation (REF )) $O_{t}=\\left\\lbrace \\#\\lbrace u\\lbrace t,2i-1\\rbrace ,1\\le i\\le p\\rbrace =p\\right\\rbrace \\,,$ and the tribes (for $i$ in $[q]$ , $t\\in [0,T]$ ) $\\mathcal {F}_{t,i}=\\sigma (u\\lbrace t,i\\rbrace ,\\xi _{u\\lbrace t,i\\rbrace })\\,.$ As $G_{\\alpha T}=O_{\\alpha T}\\cap \\bigcap _{1\\le i\\le p}\\lbrace u\\lbrace \\alpha T,2i-1\\rbrace =u\\lbrace \\alpha T,2i\\rbrace \\rbrace $ , we have : $\\mathbb {E}(\\mathbb {1}_{G_{\\alpha T}}\\prod _{i=1}^{p}e^{B_{\\alpha T}^{(2i-1)}+\\alpha T})=\\mathbb {E}(\\mathbb {1}_{O_{\\alpha T}}\\prod _{i=1}^{p}e^{B_{\\alpha T}^{(2i-1)}+\\alpha T}\\mathbb {E}(\\prod _{i=1}^{p}\\mathbb {1}_{u\\lbrace \\alpha T,2i-1\\rbrace =u\\lbrace \\alpha T,2i\\rbrace }|\\vee _{1\\le i\\le p}\\mathcal {F}_{\\alpha T,2i-1}))\\\\\\text{(by Proposition \\ref {prop:(reformulation-of-Proposition} and Equation (\\ref {eq:proba-rester-dans-fragment}))}=\\mathbb {E}(\\mathbb {1}_{O_{\\alpha T}})\\,.$ We then observe that $O_{\\alpha T}^{\\complement }=\\cup _{i\\in [p]}\\cup _{j\\in [p],j\\ne i}\\lbrace u\\lbrace \\alpha T,2i-1\\rbrace =u\\lbrace \\alpha T,2j-1\\rbrace \\rbrace \\,,$ and, for $i\\ne j$ , $\\mathbb {P}(u\\lbrace \\alpha T,2i-1\\rbrace =u\\lbrace \\alpha T,2j-1\\rbrace ) & = & \\mathbb {E}(\\mathbb {E}(\\mathbb {1}_{u\\lbrace \\alpha T,2i-1\\rbrace =u\\lbrace \\alpha T,2j-1\\rbrace }|\\mathcal {F}_{\\alpha T,2i-1}))\\\\\\text{(by Proposition \\ref {prop:(reformulation-of-Proposition} and Equation (\\ref {eq:proba-rester-dans-fragment}))} & = & \\mathbb {E}(e^{-\\alpha T-B_{\\alpha T}^{(2i-1)}})\\\\\\mbox{(because of Assumption (\\ref {hyp:delta-step}))} & \\le & e^{-\\alpha T-\\log (\\delta )}\\,.$ So $\\mathbb {P}(O_{\\alpha T})\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }1\\,.$ This gives us enough material to finish the proof of Equation (REF ).",
"Indeed, starting from Equation (REF ), we have $ \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T,}^{(1)}\\dots ,B_{T}^{(q)})\\mathbb {1}_{G_{\\alpha T}}\\mathbb {1}_{\\#\\mathcal {L}_{1}=q})\\\\=\\mathbb {E}\\left(\\mathbb {1}_{G_{\\alpha T}}\\prod _{i=1}^{p}e^{\\alpha T+B_{\\alpha T}^{(2i-1)}}\\right)\\times \\prod _{i=1}^{p}\\left(\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f_{2i-1}(\\overline{B}_{0}^{(1),v})f_{2i}(\\overline{B}_{0}^{(2),v}))dv\\right)\\\\+\\mathbb {E}\\left(\\mathbb {1}_{G_{\\alpha T}}\\prod _{i=1}^{p}e^{\\alpha T+B_{\\alpha T}^{(2i-1)}}R_{2i-1,2i}\\right)=:(1)+(2)\\,.$ By Equation (REF ), $(1)=\\mathbb {P}(O_{\\alpha t})\\prod _{i=1}^{p}\\left(\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f_{2i-1}(\\overline{B}_{0}^{(1),v})f_{2i}(\\overline{B}_{0}^{(2),v}))dv\\right)\\\\\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }\\prod _{i=1}^{p}\\left(\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f_{2i-1}(\\overline{B}_{0}^{(1),v})f_{2i}(\\overline{B}_{0}^{(2),v}))dv\\right)\\,.$ And, by Equation (REF ), $(2)\\le \\mathbb {P}(O_{\\alpha t})\\prod _{i=1}^{p}\\left(\\Gamma _{2}\\Vert f_{2i-1}\\Vert _{\\infty }\\Vert f_{2i}\\Vert _{\\infty }e^{-(T-\\alpha T)\\frac{(\\theta - \\varepsilon ^{\\prime }-1)}{2}}\\right)\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }0\\,.$"
],
[
"Convergence result",
"For $f$ and $g$ bounded measurable functions, we set $V(f,g)=\\int _{-\\infty }^{-\\log (\\delta )}e^{-v}\\mathbb {E}(\\mathbb {1}_{v\\le \\overline{B}_{0}^{(1),v}}f(\\overline{B}_{0}^{(1),v})g(\\overline{B}_{0}^{(2),v}))dv\\,.$ For $q$ even, we set $\\mathcal {I}_{q}$ to be the set of partitions of $[q]$ into subsets of cardinality 2.",
"We have $\\#\\mathcal {I}_{q}=\\frac{q!",
"}{\\left(q/2\\right)!2^{q/2}}\\,.$ For $I$ in $\\mathcal {I}_{q}$ and $t$ in $[0,T]$ , we introduce $G_{t,I}=\\lbrace \\forall \\lbrace i,j\\rbrace \\in I\\,,\\,\\exists u\\in \\mathcal {U}\\text{ such that }\\xi _{u}<e^{-t}\\,,\\,\\xi _{m(u)}\\ge e^{-t}\\,,\\,A_{u}=\\lbrace i,j\\rbrace \\rbrace \\,.$ For $t$ in $[0,T]$ , we define $\\mathcal {P}_{t}=\\cup _{I\\in \\mathcal {I}_{q}}G_{t,I}\\,.$ The above event can be understood as “at time $t$ , the dots are paired on different fragments”.",
"As before, the reader has to keep in mind that $T=-\\log ( \\varepsilon )$ (Equation (REF )).",
"Proposition 4.10 Let $q$ be in $\\mathbb {N}^{*}$ .",
"Let $F=(f_{1}\\otimes \\dots \\otimes f_{q})_{\\text{sym }}$ with $f_{1}$ , ..., $f_{q}$ in $\\mathcal {B}_{\\text{sym}}^0(1)$ ($(\\dots )_{\\text{sym}}$ defined in Equation (REF )).",
"If $q$ is even ($q=2p$ ) then $ \\varepsilon ^{q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q})\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }\\sum _{I\\in \\mathcal {I}_{q}}\\prod _{\\lbrace a,b\\rbrace \\in I}V(f_{a},f_{b})\\,.$ Let $\\alpha $ be in $(q/(q+2),1)$ .",
"We have $ \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}) & = & \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}(\\mathbb {1}_{\\mathcal {P}_{\\alpha T}}+\\mathbb {1}_{\\mathcal {P}_{\\alpha T}^{\\complement }}))\\,.$ Remember that the events of the form $C_{k,l}(t)$ , $L_{t}$ are defined in Section REF .",
"The set $\\mathcal {P}_{\\alpha T}^{\\complement }$ is a disjoint union of sets of the form $C_{k,l}(\\alpha T)$ (with $k\\ge 1$ ) and $\\lbrace L_{\\alpha T}>q/2\\rbrace $ (this can be understood heuristically by : “if the dots are not paired on fragments than some of them are alone on their fragment, or none of them is alone on a fragment and some are a group of at least three on a fragment”).",
"As said before, the event $\\lbrace \\#\\mathcal {L}_{1}=q\\rbrace $ is measurable with respect to $\\mathcal {L}_{2}$ (see Equation (REF )).",
"So, by Lemma REF and Lemma REF , we have that $\\lim _{ \\varepsilon \\rightarrow 0} \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}\\mathbb {1}_{\\mathcal {P}_{\\alpha T}^{\\complement }})=0\\,.$ We compute : $ \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}\\mathbb {1}_{\\mathcal {P}_{\\alpha T}})= \\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}\\sum _{I_{q}\\in \\mathcal {I}_{q}}\\mathbb {1}_{G_{\\alpha T,I_{q}}})\\\\\\text{(as }F\\text{ is symmetric and }(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\text{ is exchangeable)}\\\\=\\frac{q!}{2^{q/2}\\left(\\frac{q}{2}\\right)!}",
"\\varepsilon ^{-q/2}\\mathbb {E}(F(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}\\mathbb {1}_{G_{\\alpha T}})\\\\=\\frac{q!",
"\\varepsilon ^{-q/2}}{2^{q/2}\\left(\\frac{q}{2}\\right)!}\\frac{1}{q!",
"}\\sum _{\\sigma \\in \\mathcal {S}_{q}}\\mathbb {E}((f_{\\sigma (1)}\\otimes \\dots \\otimes f_{\\sigma (q)})(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}=q}\\mathbb {1}_{G_{\\alpha T}})\\\\\\text{(by Lemma \\ref {lem:calcul-exact-limite}) }\\\\\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }\\frac{1}{2^{q/2}\\left(\\frac{q}{2}\\right)!",
"}\\sum _{\\sigma \\in \\mathcal {S}_{q}}\\prod _{i=1}^{p}V(f_{\\sigma (2i-1)},f_{\\sigma (2i)})=\\sum _{I\\in \\mathcal {I}_{q}}\\prod _{\\lbrace a,b\\rbrace \\in I}V(f_{a},f_{b})\\,.$ $$"
],
[
"Results",
"We are interested in the probability measure $\\gamma _{T}$ defined by its action on bounded measurable functions $F\\,:\\,[0,1]\\rightarrow \\mathbb {R}$ by $\\gamma _{T}(F)=\\sum _{u\\in \\mathcal {U}_{ \\varepsilon }}\\xi {}_{u}F\\left(\\frac{\\xi {}_{u}}{ \\varepsilon }\\right)\\,.$ We define, for all $q$ in $\\mathbb {N}^{*}$ , $F$ from $[0,1]^{q}$ to $\\mathbb {R}$ , $\\gamma _{T}^{\\otimes q}(F)=\\sum _{a\\,:\\,[q]\\rightarrow \\mathcal {U_{ \\varepsilon }}}\\xi {}_{a(1)}\\dots \\xi {}_{a(q)}F\\left(\\frac{\\xi _{a(1)}}{ \\varepsilon },\\dots ,\\frac{\\xi _{a(q)}}{ \\varepsilon }\\right)\\,,$ $\\gamma _{T}^{\\odot q}(F)=\\sum _{a\\,:\\,[q]\\hookrightarrow \\mathcal {U_{ \\varepsilon }}}\\xi _{a(1)}\\dots \\xi _{a(q)}F\\left(\\frac{\\xi _{a(1)}}{ \\varepsilon },\\dots ,\\frac{\\xi _{a(q)}}{ \\varepsilon }\\right)\\,,$ where the last sum is taken over all the injective applications $a$ from $[q]$ to $\\mathcal {U}_ \\varepsilon $ .",
"We set $\\Phi (F)\\,:\\,(y_{1},\\dots ,y_{q})\\in \\mathbb {R}^{+}\\mapsto F(e^{-y_{1}},\\dots ,e^{-y_{q}})\\,,$ The law $\\gamma ^{\\otimes q}$ is the law of $q$ fragments picked in $\\mathcal {U}_{ \\varepsilon }$ with replacement.",
"For each fragment, the probability to be picked is its size.",
"The measure $\\gamma ^{\\odot q}$ is not a law : $\\gamma ^{\\odot q}(F)$ is an expectation over $q$ fragments picked in $\\mathcal {U}_{ \\varepsilon }$ with replacement (for each fragment, the probability to be picked is its size), in this expectation, we multiply the integrand by zero if two fragments are the same (and by one otherwise).",
"The definition of Section REF says that we can define the tagged fragment by painting colored dots on the stick $[0,1]$ ($q$ dots of different colors, these are the $Y_{1}$ , ..., $Y_{q}$ ) and then by looking on which fragments of $\\mathcal {U}_{ \\varepsilon }$ we have these dots.",
"So, we get (remember $T=-\\log \\varepsilon $ ) $\\mathbb {E}(\\gamma _{T}^{\\otimes q}(F))=\\mathbb {E}(\\Phi (F)(B_{T}^{(1)},\\dots ,B_{T}^{(q)}))\\,,$ $\\mathbb {E}(\\gamma _{T}^{\\odot q}(F))=\\mathbb {E}(\\Phi (F)(B_{T}^{(1)},\\dots ,B_{T}^{(q)})\\mathbb {1}_{\\#\\mathcal {L}_{1}}=q)\\,.$ We define, for all bounded continuous $f\\,:\\,\\mathbb {R}^{+}\\rightarrow \\mathbb {R}$ , $\\gamma _{\\infty }(f)=\\eta (\\Phi (f))\\,.$ Proposition 5.1 (Law of large numbers) We remind the reader that we have Fact REF and that we are under Hypothesis REF , REF , REF , REF .",
"Let f be a continuous function from $[0,1]$ to $\\mathbb {R}$ .",
"We have : $\\gamma _{T}(f)\\underset{T\\rightarrow +\\infty }{\\overset{\\text{a.s.}}{\\longrightarrow }}\\gamma _{\\infty }(f)\\,.$ (Remember $T=-\\log \\varepsilon $ .)",
"We take a bounded measurable function $f\\,:\\,[0,1]\\rightarrow \\mathbb {R}$ .",
"We define $\\overline{f}=f-\\eta (\\Phi (f))$ .",
"We take an integer $q\\ge 2$ .",
"We introduce the notation : $\\forall g\\,:\\,\\mathbb {R}^{+}\\rightarrow \\mathbb {R}\\,,\\,\\forall (x_{1},\\dots ,x_{q})\\in \\mathbb {R}^{q}\\,,\\,g^{\\otimes q}(x_{1},\\dots ,x_{q})=g(x_{1})g(x_{2})\\dots g(x_{q})\\,.$ We have $\\mathbb {E}((\\gamma _{T}(f)-\\eta (\\Phi (f)))^{q}) & = & \\mathbb {E}((\\gamma _{T}(\\overline{f}))^{q})\\\\& = & \\mathbb {E}(\\gamma _{t}^{\\otimes q}(\\overline{f}^{\\otimes q}))\\\\& & \\text{(as }(B^{(1)},\\dots ,B^{(q)})\\text{ is exchangeable)}\\\\& = & \\mathbb {E}(\\gamma _{t}^{\\otimes q}((\\overline{f}^{\\otimes q})_{\\text{sym}}))\\\\\\text{(by Corollary \\ref {cor:maj-fonction-centree})} & \\le & \\Vert \\overline{f}\\Vert _{\\infty }^{q} \\varepsilon ^{q/2}\\left\\lbrace K_{1}(q)+\\Gamma _{1}^{q}C_{\\text{tree}}(q)\\left(\\frac{1}{\\delta }\\right)^{q}(q+1)^{2}\\right\\rbrace \\,.$ We now take sequences $(T_{n}=\\log (n))_{n\\ge 1}$ , $( \\varepsilon _{n}=1/n)_{n\\ge 1}$ .",
"We then have, for all $n$ and for all $\\iota >0$ , $\\mathbb {P}([\\gamma _{T_{n}}(f)-\\eta (\\Phi (f))]^{4}\\ge \\iota )\\le \\frac{\\Vert \\overline{f}\\Vert _{\\infty }^{4}}{\\iota n^{2}}\\left\\lbrace K_{1}(4)+\\Gamma _{1}^{4}C_{\\text{tree}}(4)\\left(\\frac{1}{\\delta }\\right)^{4}\\times 25\\right\\rbrace \\,.$ So, by Borell-Cantelli's Lemma, $\\gamma _{T_{n}}(f)\\underset{n\\rightarrow +\\infty }{\\underset{\\longrightarrow }{\\text{a.s.}}}\\eta (\\Phi (f))\\,.$ We now have to work a little bit more to get to the result.",
"Let $n$ be in $\\mathbb {N}^{*}$ .",
"We can decompose ($\\mathcal {U}_{ \\varepsilon }$ defined in Section REF , $\\sqcup $ stands for “disjoint union” and is defined in Section REF ) $\\mathcal {U}_{ \\varepsilon _{n}}=\\mathcal {U}_{ \\varepsilon _{n}}^{(1)}\\sqcup \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}\\text{ where }\\mathcal {U}_{ \\varepsilon _{n}}^{(1)}=\\mathcal {U}_{ \\varepsilon _{n}}\\cap \\mathcal {U}_{ \\varepsilon _{n+1}}=\\mathcal {U}_{ \\varepsilon _{n+1}}\\,,\\,\\mathcal {U}_{ \\varepsilon _{n}}^{(2)}=\\mathcal {U}_{ \\varepsilon _{n}}\\backslash \\mathcal {U}_{ \\varepsilon _{n+1}}\\,.$ For $u$ in $\\mathcal {U}_{ \\varepsilon _{n}}\\backslash \\mathcal {U}_{ \\varepsilon _{n+1}}$ , we set $d(u)=\\lbrace v\\,:\\,u=\\mathbf {\\mathbf {m}}(v)\\rbrace $ ($\\mathbf {m}$ is defined in Section REF ) and we observe that, for all $u$ ($T_{u}$ defined in Equation (REF )) $\\sum _{v\\in d(u)}\\xi _{v}=\\xi _{u}\\,.$ We can then write $\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}}\\xi _{u}f\\left(\\frac{\\xi _{u}}{ \\varepsilon _{n}}\\right)=\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}f(n\\xi _{u})+\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\xi _{u}f(n\\xi _{u})\\,.$ There exists $n_{1}$ such that, for $n$ bigger than $n_{1}$ , $e^{-a}< \\varepsilon _{n+1}/ \\varepsilon _{n}$ (remember Assumption REF ).",
"We suppose $n\\ge n_{1}$ , we then have, for all $u$ in $\\mathcal {U}_{ \\varepsilon _{n}}^{(2)}$ , $ \\varepsilon _{n}>\\xi _{u}\\ge \\varepsilon _{n+1}$ and, for any $v$ in $\\mathbf {d}(u)$ , $\\xi _{v}\\le \\varepsilon _{n}e^{-a}$ , $\\xi _{v}< \\varepsilon _{n+1}$ .",
"So we get $\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n+1}}}\\xi _{u}f\\left(\\frac{\\xi _{u}}{ \\varepsilon _{n+1}}\\right)=\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}f((n+1)\\xi _{u})+\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}\\xi _{v}f((n+1)\\xi _{v})\\,.$ So we have, for $n\\ge n_{1}$ , $\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}\\xi _{v}f((n+1)\\xi _{v})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\xi _{u}f(n\\xi _{u})\\right|\\\\\\le |\\gamma _{T_{n+1}}(f)-\\gamma _{T_{n}}(f)|+\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}f((n+1)\\xi _{u})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}f(n\\xi _{u})\\right|\\,.$ If we take $f=\\operatornamewithlimits{Id}$ , the terms in the equation above can be bounded : $\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}\\xi _{v}f((n+1)\\xi _{v})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\xi _{u}f(n\\xi _{u})\\right|\\\\\\ge \\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\left(\\xi _{u}f(n\\xi _{u})-\\sum _{v\\in d(u)}\\xi _{v}f(n\\xi _{v})\\right)\\right|-\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}(\\xi _{v}f(n\\xi _{v})-\\xi _{v}f((n+1)\\xi _{v}))\\right|\\\\\\text{(by Assumption \\ref {hyp:delta-step})}\\\\\\ge \\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\left(\\xi _{u}f(n\\xi _{u})-\\sum _{v\\in d(u)}\\xi _{v}f(n\\xi _{u})e^{-a}\\right)-\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}(\\xi _{v}f(n\\xi _{v})-\\xi _{v}f((n+1)\\xi _{v}))\\right|\\\\\\text{(by Equation (\\ref {eq:d-subset-a})) }\\ge \\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\xi _{u}(1-e^{-a})\\frac{n}{n+1}-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}\\xi _{v}\\frac{1}{n+1}\\,,$ $|\\gamma _{T_{n+1}}(f)-\\gamma _{T_{n}}(f)|+\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}f((n+1)\\xi _{u})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}f(n\\xi _{u})\\right|\\\\\\le |\\gamma _{T_{n+1}}(f)-\\gamma _{T_{n}}(f)|+\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}\\frac{1}{n}\\,.$ Let $\\iota >0$.",
"We fix $\\omega $ in $\\Omega $ .",
"By Equation (REF ), almost surely, there exists $n_{2}$ such that, for $n\\ge n_{2}$ , $|\\gamma _{T_{n+1}}(f)-\\gamma _{T_{n}}(f)|<\\iota $ .",
"For $n\\ge n_{1}\\vee n_{2}$ , we can then write : $\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\xi _{u}\\,\\,\\,\\text{(by Equations (\\ref {eq:terme-a}), (\\ref {eq:terme-b}), (\\ref {eq:terme-c}))}\\\\\\le \\frac{n+1}{n(1-e^{-a})}\\left(\\iota +\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}}\\sum _{v\\in d(u)}\\xi _{v}\\frac{1}{n+1}+\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}}\\xi _{u}\\frac{1}{n}\\right)\\\\\\text{(by Equation (\\ref {eq:d-subset-a}))}\\le \\frac{n+1}{n(1-e^{-a})}\\left(\\iota +\\frac{1}{n}\\right)\\,.$ Let $n\\ge n_{1}\\vee n_{2}$ and $t$ in $(T_{n},T_{n+1})$ .",
"We can decompose $\\mathcal {U}_{ \\varepsilon _{n}}=\\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)\\sqcup \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)\\text{ where }\\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)=\\mathcal {U}_{ \\varepsilon _{n}}\\cap \\mathcal {U}_{e^{-t}}=\\mathcal {U}_{e^{-t}}\\,,\\,\\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)=\\mathcal {U}_{ \\varepsilon _{n}}\\backslash \\mathcal {U}_{e^{-t}}\\,.$ For $u$ in $\\mathcal {U}_{ \\varepsilon _{n}}\\backslash \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)$ , we set $d(u,t)=\\lbrace v\\in \\mathcal {U}_{e^{-t}}\\,:\\,u=\\mathbf {m}(v)\\rbrace $ .",
"As $n\\ge n_{1}$ , $\\mathbf {d}(u,t)=\\mathbf {d}(u)$ and we have $\\sum _{v\\in d(u,t)}\\xi _{v}=\\xi _{u}\\,.$ Similar to Equation (REF ), we have $\\sum _{u\\in \\mathcal {U}_{e^{-t}}}\\xi _{u}f(e^{t}\\xi _{u})=\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}f(e^{t}\\xi _{u})+\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)}\\sum _{v\\in d(u,t)}\\xi _{v}f(e^{t}\\xi _{v})\\,.$ We fix $f$ continuous from $[0,1]$ to $\\mathcal {\\mathbb {R}}$ , there exists $n_{3}\\in \\mathbb {N}^{*}$ such that, for all $x,y\\in [0,1]$ , $|x-y|\\le 1/n_{3}\\Rightarrow |f(x)-f(y)|<\\iota $ .",
"Suppose that $n\\ge n_{1}\\vee n_{2}\\vee n_{3}$ .",
"Then, using Equation (REF ) and Equation (REF ), we have (for all $t\\in [T_{n},T_{n+1}]$ ), $|\\gamma _{t}(f)-\\gamma _{T_{n}}(f)|=\\left|\\sum _{u\\in \\mathcal {U}_{e^{-t}}}\\xi _{u}f(e^{t}\\xi _{u})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}f(n\\xi _{u})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)}\\xi _{u}f(n\\xi _{u})\\right|\\\\=\\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}f(e^{t}\\xi _{u})+\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)}\\sum _{v\\in d(u,t)}\\xi _{v}f(e^{t}\\xi _{v})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}f(n\\xi _{u})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)}\\xi _{u}f(n\\xi _{u})\\right|\\\\\\le \\left|\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}f(e^{t}\\xi _{u})-\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}f(n\\xi _{u})\\right|+2\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)}\\xi _{u}\\Vert f\\Vert _{\\infty }\\\\\\le \\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(1)}(t)}\\xi _{u}\\iota +2\\sum _{u\\in \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)}\\xi _{u}\\Vert f\\Vert _{\\infty }\\\\\\text{(using Equation (\\ref {eq:maj_U-2}) and as $\\mathcal {U}_{ \\varepsilon _{n}}^{(2)}(t)\\subset \\mathcal {U}_{ \\varepsilon _{n}}^{(2)}$)}\\le \\iota +2\\Vert f\\Vert _{\\infty }\\frac{n+1}{n(1-e^{-a})}\\left(\\iota +\\frac{1}{n}\\right)\\,.$ Equations (REF ) and (REF ) prove the desired result.",
"The set $\\mathcal {B}_{\\text{sym}}^0(1)$ is defined in Section REF .",
"Theorem 5.2 (Central-limit Theorem) We remember we have Fact REF and we are under Hypothesis REF , REF , REF , REF .",
"Let $q$ be in $\\mathbb {N}^{*}$ .",
"For functions $f_{1}$ , ..., $f_{q}$ which are continuous and in $\\mathcal {B}_{\\text{sym}}^0(1)$ , we have $ \\varepsilon ^{-q/2}(\\gamma _{T}(f_{1}),\\dots ,\\gamma _{T}(f_{q}))\\underset{T\\rightarrow +\\infty }{\\overset{\\text{law}}{\\longrightarrow }}\\mathcal {N}(0,(K(f_{i},f_{j}))_{1\\le i,j\\le q})\\,(\\underline{ \\varepsilon =e^{-T}})$ ($K$ is given in Equation (REF )).",
"Let $f_{1}$ , ..., $f_{q}$ $\\mathcal {B}_{\\text{sym}}^0(1)$ and $v_{1},\\dots ,v_{q}\\in \\mathbb {R}$ .",
"First, we develop the product below (remember that for $u$ in $\\mathcal {U}_{ \\varepsilon }$ , $\\xi _{u}/ \\varepsilon <1$ a.s.) $\\prod _{u\\in \\mathcal {U}_{ \\varepsilon }}\\left(1+\\sqrt{ \\varepsilon }\\frac{\\xi _{u}}{ \\varepsilon }(iv_{1}f_{1}+\\dots +iv_{q}f_{q})\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)\\right)=\\\\\\exp \\left(\\sum _{u\\in \\mathcal {U}_{ \\varepsilon }}\\log \\left[1+\\sqrt{ \\varepsilon }\\operatornamewithlimits{Id}\\times (iv_{1}f_{1}+\\dots +iv_{q}f_{q})\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)\\right]\\right)=\\\\\\text{(for } \\varepsilon \\text{ small enough)}\\\\\\exp \\left(\\sum _{u\\in \\mathcal {U_{ \\varepsilon }}}\\sum _{k\\ge 1}\\frac{(-1)^{k+1}}{k} \\varepsilon ^{k/2}(\\operatornamewithlimits{Id}\\times (iv_{1}f_{1}+\\dots +iv_{q}f_{q}))^{k}\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)\\right)=\\\\\\exp \\left(\\frac{1}{\\sqrt{ \\varepsilon }}\\gamma _{T}(iv_{1}f_{1}+\\dots +iv_{q}f_{q})+\\frac{1}{2}\\gamma _{T}(\\operatornamewithlimits{Id}\\times (v_{1}f_{1}+\\dots +v_{q}f_{q})^{2})+R_{ \\varepsilon }\\right)\\,,$ where $R_{ \\varepsilon } & = & \\sum _{k\\ge 3}\\sum _{u\\in \\mathcal {U}_{ \\varepsilon }}\\frac{(-1)^{k+1}}{k} \\varepsilon ^{k/2-1}\\xi _{u}\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)^{k-1}(iv_{1}f_{1}+\\dots +iv_{q}f_{q})^{k}\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)\\\\& = & \\sum _{k\\ge 3}\\frac{(-1)^{k+1}}{k} \\varepsilon ^{k/2-1}\\gamma _{T}((\\operatornamewithlimits{Id})^{k-1}(iv_{1}f_{1}+\\dots +iv_{q}f_{q})^{k})\\,,$ $|R_{ \\varepsilon }|\\le \\sum _{k\\ge 3}\\frac{ \\varepsilon ^{k/2-1}}{k}(|v_{1}|\\Vert f_{1}\\Vert _{\\infty }+\\dots +|v_{q}|\\Vert f_{q}\\Vert _{\\infty })^{k}=O(\\sqrt{ \\varepsilon })\\,.$ We have, for some constant $C$ , (we use that : $x\\in \\mathbb {R}\\Rightarrow |e^{ix}|=1$ ) $\\mathbb {E}\\left(\\left|\\exp \\left(\\frac{1}{\\sqrt{ \\varepsilon }}\\gamma _{T}(iv_{1}f_{1}+\\dots +iv_{q}f_{q})+\\frac{1}{2}\\gamma _{T}(\\operatornamewithlimits{Id}\\times (v_{1}f_{1}+\\dots +v_{q}f_{q})^{2})+R_{ \\varepsilon }\\right)\\right.\\right.\\\\\\left.\\left.-\\exp \\left(\\frac{1}{\\sqrt{ \\varepsilon }}\\gamma _{T}(iv_{1}f_{1}+\\dots +iv_{q}f_{q})+\\frac{1}{2}\\eta (\\Phi (\\operatornamewithlimits{Id}\\times (v_{1}f_{1}+\\dots +v_{q}f_{q})^{2})\\right)\\right|\\right)\\\\\\le \\mathbb {E}\\left(C\\left|\\frac{1}{2}\\gamma _{T}(\\operatornamewithlimits{Id}\\times (v_{1}f_{1}+\\dots +v_{q}f_{q})^{2})-\\frac{1}{2}\\eta (\\Phi (\\operatornamewithlimits{Id}\\times (v_{1}f_{1}+\\dots +v_{q}f_{q})^{2})+R_{ \\varepsilon }\\right|\\right)\\\\\\text{(by Proposition \\ref {prop:convergence-ps})}\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }0\\,.$ Second, we develop the same product in a different manner.",
"We have (the order on $\\mathcal {U}$ is defined in Section REF ) $\\prod _{u\\in \\mathcal {U}_{ \\varepsilon }}\\left(1+\\sqrt{ \\varepsilon }\\frac{\\xi _{u}}{ \\varepsilon }(iv_{1}f_{1}+\\dots +iv_{q}f_{q})\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)\\right)=\\\\\\sum _{k\\ge 0} \\varepsilon ^{-k/2}i^{k}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{\\begin{array}{c}u_{1},\\dots ,u_{k}\\in \\mathcal {U_{ \\varepsilon }}\\\\u_{1}<\\dots <u_{k}\\end{array}}\\xi _{u_{1}\\dots }\\xi _{u_{k}}f_{j_{1}}\\left(\\frac{\\xi _{u_{1}}}{ \\varepsilon }\\right)\\dots f_{j_{k}}\\left(\\frac{\\xi _{u_{k}}}{ \\varepsilon }\\right)=\\\\\\text{(a detailed proof can be found in Section \\ref {subsec:Detailed-proof-of-2})}\\\\\\sum _{k\\ge 0} \\varepsilon ^{-k/2}i^{k}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\frac{1}{k!",
"}\\gamma _{T}^{\\odot k}(f_{j_{1}}\\otimes \\dots \\otimes f_{j_{k}})\\,.$ We have, for all $k$ , $\\left| \\varepsilon ^{-k/2}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\frac{1}{k!",
"}\\mathbb {E}(\\gamma _{T}^{\\odot k}(f_{j_{1}}\\otimes \\dots \\otimes f_{j_{k}}))\\right|\\\\\\le \\varepsilon ^{-k/2}\\times \\frac{q^{k}\\sup (|v_{1}|,\\dots ,|v_{q}|)^{k}\\sup (\\Vert f_{1}\\Vert _{\\infty },\\dots ,\\Vert f_{q}\\Vert _{\\infty })^{k}}{k!",
"}.$ So, by Corollary REF , Proposition REF and Equation (REF ), we get that $\\mathbb {E}\\left(\\prod _{u\\in \\mathcal {U}_{ \\varepsilon }}\\left(1+\\sqrt{ \\varepsilon }\\frac{\\xi _{u}}{ \\varepsilon }(iv_{1}f_{1}+\\dots +iv_{q}f_{q})\\left(\\frac{\\xi _{u}}{ \\varepsilon }\\right)\\right)\\right)\\\\\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }\\sum _{\\begin{array}{c}k\\ge 0\\\\k\\text{ even}\\end{array}}(-1)^{k/2}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}\\frac{1}{k!",
"}\\sum _{I\\in I_{k}}\\prod _{\\lbrace a,b\\rbrace \\in I}V(v_{j_{a}}f_{j_{a}},v_{j_{b}}f_{j_{b}})\\\\\\text{(a detailed proof can be found in Section \\ref {subsec:Detailed-proof-of-1})}\\\\\\text{(using Equation (\\ref {eq:card-I-q}))}=\\sum _{\\begin{array}{c}k\\ge 0\\\\k\\text{ even}\\end{array}}\\frac{(-1)^{k/2}}{2^{k/2}(k/2)!",
"}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}V(v_{j_{1}}f_{j_{1}},v_{j_{2}}f_{j_{2}})\\dots V(v_{j_{k-1}}f_{j_{k-1}},v_{j_{k}}f_{j_{k}})\\\\=\\sum _{\\begin{array}{c}k\\ge 0\\\\k\\text{ even}\\end{array}}\\frac{(-1)^{k/2}}{2^{k/2}(k/2)!",
"}\\left(\\sum _{1\\le j_{1},j_{2}\\le q}v_{j_{1}}v_{j_{2}}V(f_{j_{1}},f_{j_{2}})\\right)^{k/2}\\\\=\\exp \\left(-\\frac{1}{2}\\sum _{1\\le j_{1},j_{2}\\le q}v_{j_{1}}v_{j_{2}}V(f_{j_{1}},f_{j_{2}})\\right)\\,.$ In conclusion, we have $\\mathbb {E}\\left(\\exp \\left(\\frac{1}{\\sqrt{ \\varepsilon }}\\gamma _{T}(iv_{1}f_{1}+\\dots +iv_{q}f_{q})\\right)\\right)\\\\\\underset{ \\varepsilon \\rightarrow 0}{\\longrightarrow }\\exp \\left(-\\frac{1}{2}\\eta (\\Phi (\\operatornamewithlimits{Id}\\times (v_{1}f_{1}+\\dots +v_{q}f_{q})^{2}))-\\frac{1}{2}\\sum _{1\\le j_{1},j_{2}\\le q}v_{j_{1}}v_{j_{2}}V(f_{j_{1}},f_{j_{2}})\\right)\\,.$ So we get the desired result with, for all $f$ , $g$ , $K(f,g)=\\eta (\\Phi (\\operatornamewithlimits{Id}\\times fg)+V(f,g))$ ($V$ is defined in Equation (REF ))."
],
[
"Detailed proof of a bound appearing\nin the proof of Lemma ",
"Lemma 6.1 We have (for any $f$ appearing in the proof of Lemma REF ) $\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\le \\prod _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace }e^{-(\\#f(u)-1)(T_{u}-T_{m(u)})}\\,.$ We want to show this by recurrence on the cardinality of $\\mathcal {T}_{2}$ .",
"If $\\#\\mathcal {T}_{2}=1$ , then $\\mathcal {T}_{2}=\\lbrace 0\\rbrace $ and the claim is true.",
"Suppose now that $\\#\\mathcal {T}_{2}=k$ and the claim is true up to the cardinality $k-1$ .",
"There exists $v$ in $\\mathcal {T}_{2}$ such that $(v,i)$ is not in $\\mathcal {T}_{2}$ , for any $i$ in $\\mathbb {N}^{*}$ .",
"We set $\\mathcal {T}_{2}^{\\prime }=\\mathcal {T}_{2}\\backslash \\lbrace v\\rbrace $ , $\\mathcal {L}_{2}^{\\prime }=\\mathcal {L}_{2}\\backslash \\lbrace v\\rbrace $ , $m_{2}^{\\prime }\\,:\\,u\\in \\mathcal {T}_{2}^{\\prime }\\rightarrow (\\xi _{u},\\inf \\lbrace i,i\\in A_{u}\\rbrace )$ .",
"We set $f(v)=\\lbrace i_{1},\\dots ,i_{p}\\rbrace $ (with $i_{1}<\\dots <i_{p}$ ), $f(m(v))=\\lbrace i_{1},\\dots ,i_{p},i_{p+1},\\dots ,i_{q}\\rbrace $ (with $i_{p+1}<\\dots <i_{q}$ ).",
"We suppose $m_{2}(v)=(\\xi _{v},i_{1})$ because if $m_{2}(v)=(\\xi _{v},j)$ with $j\\ne i_{1}$ then $A_{v}\\ne f(v)$ for all $\\omega $ , and then the left-hand side of the inequality above is zero.",
"We have $\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\=\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}\\mathbb {E}(\\mathbb {1}_{A_{v=f(v)}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2},(A_{u},u\\in \\mathcal {T}_{2}^{\\prime }))|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\=\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}\\mathbb {E}(\\mathbb {1}_{i_{1},\\dots ,i_{p}\\in A_{v}}\\mathbb {1}_{i_{p+1},\\dots ,i_{q}\\notin A_{v}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2},(A_{u},u\\in \\mathcal {T}_{2}^{\\prime }))|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\\\text{(remember we condition on $m_{2}$, so the $\\mathbb {1}_{i_{1},\\dots ,i_{p}}$can be replaced by $\\mathbb {1}_{i_{2},\\dots ,i_{p}}$)}\\\\=\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}\\mathbb {E}(\\mathbb {1}_{i_{2},\\dots ,i_{p}\\in A_{v}}\\mathbb {1}_{i_{p+1},\\dots ,i_{q}\\notin A_{v}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2},(A_{u},u\\in \\mathcal {T}_{2}^{\\prime }))|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\\\le \\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}\\mathbb {E}(\\mathbb {1}_{i_{2},\\dots ,i_{p}\\in A_{v}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2},(A_{u},u\\in \\mathcal {T}_{2}^{\\prime }))|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\\\text{(because the $(Y_{j})$ introduced in Section \\ref {subsec:Tagged-fragments} are independant) }\\\\=\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}\\prod _{r=2}^{p}\\mathbb {E}(\\mathbb {1}_{i_{r}\\in A_{v}}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2},(A_{u},u\\in \\mathcal {T}_{2}^{\\prime }))|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\\\text{(because of Equation (\\ref {eq:proba-rester-dans-fragment}))}\\\\\\text{(if $v\\in \\mathcal {L}_{2}$ then $\\prod _{r=2}^{p}\\dots $ is empty and thus $=1$)}\\\\=\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}\\prod _{r=2}^{p}\\widetilde{\\xi }_{v}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\\\text{(by Equation (\\ref {eq:def-T_u}) and Proposition \\ref {prop:(reformulation-of-Proposition})}\\\\=e^{-(\\#f(v)-1)(T_{v}-T_{m(v)})}\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\=e^{-(\\#f(v)-1)(T_{v}-T_{m(v)})}\\mathbb {E}(\\mathbb {E}(\\mathbb {1}_{A_{u}=f(u),\\forall u\\in \\mathcal {T}_{2}^{\\prime }}|\\mathcal {L}_{2}^{\\prime },\\mathcal {T}_{2}^{\\prime },m_{2}^{\\prime })|\\mathcal {L}_{2},\\mathcal {T}_{2},m_{2})\\\\\\text{(by recurrence)}\\\\\\le \\prod _{u\\in \\mathcal {T}_{2}\\backslash \\lbrace 0\\rbrace }e^{-(\\#f(u)-1)(T_{u}-T_{m(u)})}\\,.$"
],
[
"Detailed proof of a bound appearing\nin the proof of Corollary ",
"Lemma 6.2 Let $q$ be in $\\mathbb {N}$ , we have $\\sum _{k^{\\prime }\\in [q]}1+(q-k^{\\prime }-1)_{+}\\le (q+1)^{2}\\,.$ We have $\\sum _{k^{\\prime }\\in [q]}1+(q-k^{\\prime }-1)_{+} & = & q+\\sum _{k^{\\prime }\\in [q-2]}(q-k^{\\prime }-1)\\\\& = & q+\\sum _{i=1}^{q-2}i\\\\& \\le & \\frac{q(q+1)}{2}\\\\& \\le & (q+1)^{2}\\,.$"
],
[
"Detailed proof of an equality\nappearing in the proof of Theorem ",
"Lemma 6.3 Let $q\\in \\mathbb {N}^{*}$ .",
"Suppose we have $q$ functions $g_{1}$ , ..., $g_{q}$ in $\\mathcal {B}_{\\text{sym}}^0(1)$ .",
"Then, for all $k$ even ($k$ in $\\mathbb {N}$ ) $\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}\\sum _{I\\in \\mathcal {I}_{k}}\\prod _{\\lbrace a,b\\rbrace \\in I}V(g_{j_{a}},g_{j_{b}})=\\frac{k!}{2^{k/2}(k/2)!",
"}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}V(g_{j_{1}},g_{j_{2}})\\dots V(g_{j_{k-1}},g_{j_{k}})\\,.$ We set $\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}\\sum _{I\\in \\mathcal {I}_{k}}\\prod _{\\lbrace a,b\\rbrace \\in I}V(g_{j_{a}},g_{j_{b}})=(1)\\,,$ $\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}V(g_{j_{1}},g_{j_{2}})\\dots V(g_{j_{k-1}},g_{j_{k}})=(2)\\,.$ Suppose, for some $k$ , we have $i_{1}$ , ..., $i_{k}$ in $[q]$ , distinct.",
"There exists $N_{1}$ , $N_{2}$ such that : the term (1) has $N_{1}$ terms $V(g_{i_{1}},g_{i_{2}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ (up to permutations, that is we consider that $V(g_{i_{3}},g_{i_{4}})V(g_{i_{2}},g_{i_{1}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ and $V(g_{i_{1}},g_{i_{2}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ are the same term), the term (2) has $N_{2}$ terms $V(g_{i_{1}},g_{i_{2}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ (again, up to permutations).",
"These numbers $N_{1}$ , $N_{2}$ do not depend on $i_{1}$ , ..., $i_{k}$ .",
"In the case where the indexes $i_{1}$ , ..., $i_{k}$ are not distinct, we can find easily the number of terms equal to $V(g_{i_{1}},g_{i_{2}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ in terms (1), (2).",
"For example, if $i_{2}=i_{1}$ and $i_{1},\\,i_{3},\\dots ,i_{k}$ are distinct, then the term (1) has $2N_{1}$ terms $V(g_{i_{1}},g_{i_{2}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ , the term (2) has $2N_{2}$ terms $V(g_{i_{1}},g_{i_{2}})\\dots V(g_{i_{k-1}},g_{i_{k}})$ (we multiply simply by the number of $\\sigma $ in $\\mathcal {S}_{k}$ such that $(i_{1},i_{2},\\dots ,i_{k})=(i_{\\sigma (1)},i_{\\sigma (2)},\\dots ,i_{\\sigma (k)})$ ).",
"We do not need to know $N_{1}$ and $N_{2}$ but we need to know $N_{1}/N_{2}$ .",
"By taking $V(g,f)$ to be 1 for all $g$ , $f$ , we see that $N_{1}/N_{2}=\\#\\mathcal {I}_{k}=k!/(2^{k/2}(k/2)!",
")$ ."
],
[
"Detailed proof of an equality\nappearing in the proof of Theorem ",
"Lemma 6.4 Let $f_{1}$ , ..., $f_{q}$ $\\mathcal {B}_{\\text{sym}}^0(1)$ , $k$ in $\\mathbb {N}$ and $v_{1},\\dots ,v_{q}\\in \\mathbb {R}$ .",
"We have $\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\gamma _{T}^{\\odot k}(f_{j_{1}}\\otimes \\dots \\otimes f_{j_{k}})\\\\=k!\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{\\begin{array}{c}u_{1},\\dots ,u_{k}\\in \\mathcal {U_{ \\varepsilon }}\\\\u_{1}<\\dots <u_{k}\\end{array}}\\xi _{u_{1}\\dots }\\xi _{u_{k}}f_{j_{1}}\\left(\\frac{\\xi _{u_{1}}}{ \\varepsilon }\\right)\\dots f_{j_{k}}\\left(\\frac{\\xi _{u_{k}}}{ \\varepsilon }\\right)\\,.$ We have $\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\gamma _{T}^{\\odot k}(f_{j_{1}}\\otimes \\dots \\otimes f_{j_{k}})\\\\=\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{a:[k]\\hookrightarrow \\mathcal {U}_{ \\varepsilon }}\\xi _{a(1)}\\dots \\xi _{a(k)}f_{j_{1}}\\left(\\frac{\\xi _{a(1)}}{ \\varepsilon }\\right)\\dots f_{j_{k}}\\left(\\frac{\\xi _{a(k)}}{ \\varepsilon }\\right)\\\\\\text{(for all injection }a\\text{, there is exactly one }\\sigma _{a}\\in \\mathcal {S}_{k}\\text{ such that }a(\\sigma _{a}(1))<\\dots <a(\\sigma _{a}(k))\\text{)}\\\\=\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{a:[k]\\hookrightarrow \\mathcal {U}_{ \\varepsilon }}\\xi _{a(\\sigma _{a}(1))}\\dots \\xi _{a(\\sigma _{a}(k))}f_{j_{\\sigma _{a}(1)}}\\left(\\frac{\\xi _{a(\\sigma _{a}(1))}}{ \\varepsilon }\\right)\\dots f_{j_{\\sigma _{a}(k)}}\\left(\\frac{\\xi _{a(\\sigma _{a}(k))}}{ \\varepsilon }\\right)\\\\\\text{(for }\\tau \\in \\mathcal {S}_{k}\\text{, we set }\\mathcal {E}(\\tau )=\\lbrace a:[k]\\hookrightarrow \\mathcal {U}_{ \\varepsilon }\\,:\\,\\sigma _{a}=\\tau \\rbrace \\text{)}\\\\=\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{\\tau \\in \\mathcal {S}_{k}}\\sum _{a\\in \\mathcal {E}(\\tau )}\\xi _{a(\\tau (1))}\\dots \\xi _{a(\\tau (k))}f_{j_{\\tau (1)}}\\left(\\frac{\\xi _{a(\\tau (1))}}{ \\varepsilon }\\right)\\dots f_{j_{\\tau (k)}}\\left(\\frac{\\xi _{a(\\tau (k))}}{ \\varepsilon }\\right)\\\\=\\sum _{\\tau \\in \\mathcal {S}_{k}}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{a\\in \\mathcal {E}(\\tau )}\\xi _{a(\\tau (1))}\\dots \\xi _{a(\\tau (k))}f_{j_{\\tau (1)}}\\left(\\frac{\\xi _{a(\\tau (1))}}{ \\varepsilon }\\right)\\dots f_{j_{\\tau (k)}}\\left(\\frac{\\xi _{a(\\tau (k))}}{ \\varepsilon }\\right)\\\\=\\sum _{\\tau \\in \\mathcal {S}_{k}}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{\\tau (1)}}\\dots v_{j_{\\tau (k)}}\\sum _{a\\in \\mathcal {E}(\\tau )}\\xi _{a(\\tau (1))}\\dots \\xi _{a(\\tau (k))}f_{j_{\\tau (1)}}\\left(\\frac{\\xi _{a(\\tau (1))}}{ \\varepsilon }\\right)\\dots f_{j_{\\tau (k)}}\\left(\\frac{\\xi _{a(\\tau (k))}}{ \\varepsilon }\\right)\\\\=\\sum _{\\tau \\in \\mathcal {S}_{k}}\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{\\tau (1)}}\\dots v_{j_{\\tau (k)}}\\sum _{\\overset{u_{1},\\dots ,u_{k}\\in \\mathcal {U}_{ \\varepsilon }}{u_{1}<\\dots <u_{k}}}\\xi _{u_{1}}\\dots \\xi _{u_{k}}f_{j_{\\tau (1)}}\\left(\\frac{\\xi _{u_{1}}}{ \\varepsilon }\\right)\\dots f_{j_{\\tau (k)}}\\left(\\frac{\\xi _{u_{k}}}{ \\varepsilon }\\right)$ The application (“$\\hookrightarrow $ ” means that an application is injective) $(a\\,:\\,[k]\\rightarrow [q]\\,,\\,\\tau \\,:\\,[k]\\hookrightarrow [k])\\overset{\\Theta }{\\longrightarrow }a\\circ \\tau \\,$ is such that $\\forall b\\,:\\,[k]\\rightarrow [q]\\,,\\,\\#\\Theta ^{-1}(\\lbrace b\\rbrace )=k!\\,.$ So the above quantity is equal to $k!\\sum _{1\\le j_{1},\\dots ,j_{k}\\le q}v_{j_{1}}\\dots v_{j_{k}}\\sum _{\\overset{u_{1},\\dots ,u_{k}\\in \\mathcal {U}_{ \\varepsilon }}{u_{1}<\\dots <u_{k}}}\\xi _{u_{1}}\\dots \\xi _{u_{k}}f_{j_{1}}\\left(\\frac{\\xi _{u_{1}}}{ \\varepsilon }\\right)\\dots f_{j_{k}}\\left(\\frac{\\xi _{u_{k}}}{ \\varepsilon }\\right)\\,.$"
]
] | 2210.07791 |
[
[
"Physics of Soil and Other Natural Porous Media"
],
[
"Abstract The purpose of this monograph is to review the early theoretical basis of what is known today as soil physics and to serve as a textbook for intermediate porous media physics or transport in porous media graduate courses."
],
[
"colorlinks=true, linkcolor=blue, filecolor=blue, citecolor = black, urlcolor=red, tocchapterPreface The purpose of this monograph is to review the early theoretical basis of what is known today as soil physics and to serve as a textbook for intermediate level porous media physics or transport in porous media graduate courses.",
"In writing this material we tried to avoid as much as possible the presentation and discussion of empirical methods lacking a rigorous physical basis.",
"Because much of agricultural soil physics is still based on empirical methods, it was not possible to avoid discussing water retention and unsaturated hydraulic conductivity equations, which still lack a rigorous theoretical basis.",
"The fully empirical soil resistance equation is also discussed in a few sections.",
"We also avoided relying on soil physics books opting to consult original sources in fluid mechanics, mathematics, engineering and porous media physics, from which much of soil physics is largely derived.",
"Because of that we understand that “Physics of Soil and Other Natural Porous Media” is a more appropriately title than “Soil Physics”, which imply restriction in methods and applications.",
"We are deeply indebted in writing this manuscript to the early predecessors that laid the foundations for porous media physics and fluid dynamics, namely the mathematicians L. Euler, G.G.",
"Stokes, H. Lamb, G.I.",
"Taylor and G.I.",
"Batchelor, the physicists I. Newton, L.F. Richardson, E. Buckingham, M. Muskat, L.A. Richards, L.D.",
"Landau, E.M. Lifshitz, A.E.",
"Scheidegger, the engineers H. Darcy, J.",
"Bear and his collaborators, G. de Josselin de Jong, J. Philip.",
"We also became aware of a profound contributions to transport phenomena in porous media of a class unknown to me and largely unsung in the soil science community, the chemical engineers, listing here R. Aris, H. Brenner, F.A.L.",
"Dullien, J.C. Slattery, R.B.",
"Bird, W.E.",
"Stewart and E.N.",
"Lightfoot.",
"Not forgotten is the dedication of many mathematicians, computer scientists and physicist who worked on numerical and computational methods.",
"Of course there are many people with significant contributions to theoretical soil physics and transport phenomena in porous media, scattered in many publications for at least two centuries, too many to mention.",
"We plan in gradually introducing newer literature and expanding the mathematical foundations in later editions, always aiming at maintaining the physical basis of the processes and phenomena and avoiding purely empirical methods.",
"Because of the nature of this material and the amount of mathematics required, many mistakes and misspellings are expected in earlier drafts.",
"Potential users are encouraged to reports mistakes and to provide suggestions and corrections to improve the material.",
"Drafts will be indicated as fractional numbers on the cover, while main releases are indicated as integers.",
"A quantity is said to be a scalar if it is not associated with both magnitude and direction.",
"One example of a scalar is the temperature at a given point in space.",
"If you are interested in the temperature at a given point at a given time, the measurement is associated with a single value.",
"In this case, if we use the symbol T to represent temperature and the measured value is in Kelvin units, it can be represented as $T = 376.7 K$ Now if you record several instances of the temperature in space and/or time you have a set of all the temperatures recorded and temperature is now a variable.",
"$T = \\left\\lbrace 376.7 , 375.6 , 370.7 , 369.9 , ..., N \\right\\rbrace $ where $N$ is the $n^{th}$ temperature measurement.",
"In general you would want to associate the variable of interest with a system of spacial coordinates and/or time measurements.",
"Imagine that your thermometer is placed in a fixed position on a weather station and records temperature at regular intervals throughout the year.",
"Each temperature measurement is now associated with a time measurement, usually the date and the time in hours, minutes and seconds.",
"A table of such measurements would look like Table REF .",
"Table: Temperature measurements recorded at different time intervals.Now we can assign the temperature as a function of time at that point in space.",
"In reality, temperature is not a function of time, nothing really is a function of time, temperature varies because the position of the sun in the sky and its influence on incident radiation, season of the year, cloud cover, convective air flow and many other factors, but sometimes it is easier for us to model it simply as a function of time.",
"Temperature as a function of time can be represented as $T = f(t)$ or simply as $T(t)$ .",
"One class of empirical functions that can be used to model temperature is $T(t) = T_{0} + \\alpha \\cos {(\\omega t + \\phi )}$ In this equation temperature and time are variables since each represents a set of values, while $T_{0}$ , $\\alpha $ , $\\omega $ and $\\phi $ are parameters which can be obtained by fitting the equation to data.",
"Fitting can be achieved using statistical techniques such as least-squares regression or could also be done visually for such a simplistic equation, as long as one is aware of the potential limitations of doing so.",
"The parameters values obtained by fitting the equation to observed data are usually refer to fitting parameters or empirical parameters.",
"In this case each has a specific mathematical meaning.",
"$\\alpha $ is the amplitude of oscillation of the sinusoidal wave produced, $T_{0}$ is the value in which the predictions are centered, $\\omega $ is the angular frequency and $ \\phi $ is the phase lag.",
"Equation 1.3 is said to be univariate as temperature $T$ depends on time alone.",
"Notice that the parameters determine the shape of the function but $T$ is not a function of the parameters since they do not vary.",
"$T$ is a functions of time, $t$ , alone in this case (Figure REF ).",
"Figure: Example of a sinusoidal function of time with parameters T 0 T_{0} = 15.5, α\\alpha = 10, ω\\omega = 2 and φ\\phi = 10.Given any parameterization of Equation 1.3, one might be interested in the rate of variation of temperature as a function of time.",
"The rate of variation of a given variable as a function of the same independent variable can be calculated using the first derivative of the function.",
"There are many symbolic representations for the derivative of a function.",
"For example, given a generic function $y = f(x) $ its first derivative can be represented using Lagrange's notation as $y^{\\prime }(x)$ and by Leibniz's notation as $dy/dx$ .",
"Newton's dot notation, common in classical mechanics is usually reserved for position as a function of time, $x(t)$ for example, such that its velocity is $v = \\dot{x}(t) $ and its acceleration is $ a = \\dot{v}(t)= \\ddot{x}(t) $ .",
"In classical mechanics $x$ , $v$ and $a$ are usually vector quantities which were not denoted as such here because we haven't talked about vectors yet.",
"We shall use Leibniz's notation in this book and as such the first derivative of Equation 1.3 is $\\frac{ dT(t)}{dt} =-A \\omega \\sin {(\\omega t + \\phi )}$ If one is interested on the rate of variarion of the rate of variation of the temperature as a function of time, what is needed is the second derivative of the function, represented using the same notation as $\\frac{ dT^{2}(t)}{dt^{2}} =-A \\omega ^{2} \\cos {(\\omega t + \\phi )}$ and so forth.",
"For now you don't have to worry about the rules of differentiation, we will get to that in a bit, what you should understand first is the geometrical meaning of the derivative.",
"Observe the graph of the first derivative of Figure REF .",
"The dashed line in Figure REF is the rate of change of temperature as a function of time.",
"Notice that at the maximum and minimum values of the original function (Point A), the value of the derivative is zero (Point B) while at the inflection points, where the curvature of the first graphs changes from upwards to downwards and vice-versa, the rate of change, or the derivative function, is at its absolute maximum values.",
"Let's now analyze a simple example from classical mechanics.",
"Suppose an object is relased from rest from a height of 10 meters.",
"Following Newton's laws the governing equation is $Y(t) = Y_{0} + v_{0} t - \\frac{1}{2}g t^{2}$ The only variables in this equation are $Y$ and $t$ , everything else is either a constant or can assumed to be constant if the height of fall is small compared to the radius of Earth, as is the case with gravitational acceleration.",
"As before, we are treating gravitation acceleration as a scalar, since we are assuming it as a constant and are only interested in its effect in the $Y$ direction.",
"$Y_{0}$ is the initial height from which the object is released, $v_{0}$ is the initial velocity, Y is the predicted height and t is time.",
"If the initial height is $Y_{0} = 10 \\ m $ and if the object is released from rest, such that $v_{0} = 0~m~s^{-1} $ , we have $Y(t) = 10 - \\frac{1}{2}g t^{2}$ Notice that the height $Y$ decreases as a function of time, as it is expected, since the object is falling on Earth's gravitational field.",
"At one point $Y$ will reach zero, which is when the object reaches the surface.",
"It is also important to notice that the equation predicts negative heights.",
"This means only that the object has passed a surface that we assumed as a reference point, such that all gravitational potential energy considerations are made with reference to that point, if we were interested in that.",
"It turns out we aren't right now but that will be fundamental when we study the energy of water in porous media.",
"If we plot $Y(t)$ as a function of time you will see that position $Y$ decreases quadratically with time, since time is raised to the power of two in this equation.",
"$Y$ is a polynomial function of time in this case.",
"Extracting the roots of such polynomials can provide important information in physics problems, as we shall discuss later.",
"For now suppose you want the velocity as a function of time.",
"We know that the velocity is the rate of variation of distance over time and this can be expressed in derivative form as $\\frac{dY(t)}{dt} = v(t) = - g t$ The velocity is a linear function of time, meaning that it increases at a constant rate as the object falls.",
"As you might expect the rate with which the velocity increases is the gravitational acceleration of Earth, $g$ .",
"To further make that clear let us calculate the rate of change of velocity with time, or the acceleration $\\frac{dY^{2}(t)}{dt^{2}} = \\frac{dv(t)}{dt} = a = - g$ With that, one can be convinced that an object falling under Earth's gravitational field falls with an acceleration equal to Earth's gravitational acceleration.",
"Suppose now that all you known is that an object is falling under a gravitational field equal to $g$ and you have no equation of motion.",
"From Equation 1.9, we know that the rate of variation of velocity of the object is $\\frac{dv(t)}{dt} = - g$ As before we also know that the object was released from a height of 10 m with initial velocity equal to zero.",
"We can rearrange Equation 1.10 by multiplying both sides of the equality by $dt$ $dv(t) = - g dt$ This is one of the simplest possible cases of what is called a differential equation and to remove the differential from both sides all we need to do is integrate it $\\int dv(t) = \\int - g dt = - g \\int dt$ Since g is a constant.",
"As this is an indefinite integral, the solution includes a constant of integration $C_{1}$ .",
"$v(t) = - g t + C_{1}$ Constants will routinely appear in the solutions of differential equations in physical problems.",
"To find the value of the constant one needs specific conditions of the problem under evaluation.",
"These can be called initial conditions or, in many applications, boundary conditions.",
"We know here that when the velocity is zero, the time is also zero, in other words when the experiment was set, the very moment when the object was released from rest it was at a height of 10 m. Therefore, our initial conditions are $v = 0,\\ t = 0$ Substitution in Equation 1.13 results $0 = - g 0 + C_{1} \\Rightarrow C_{1} = 0$ Now from Equations 1.8, 1.13 and 1.15 $dY(t) = v(t)dt = - g t dt$ By repeating the integration procedure we arrive at $Y(t) = - \\frac{1}{2}g t^{2} + C_{2}$ Since we know that the height is 10 m when time is zero we arrive at $Y(t) = - \\frac{1}{2}g t^{2} + 10 = 10 - \\frac{1}{2}g t^{2}$ Which is the exact same equation we started with (Equation 1.7).",
"The objective of the steps from Equations 1.6 to 1.18 were two-fold, first to show the relationship between derivation and integration and how such operations can appear on the solutions of physical problems, and second, mainly on the steps covered from Equations 1.10 to 1.18, to show the logic behind what most students outside the fields of physics and mathematics understand by mathematical modeling.",
"In a broad sense, mathematical modeling is viewed as the solution of a differential equation applied to a set of initial or boundary conditions.",
"As physicists and mathematicians know well simple solutions to these problems are rare if they exist at all.",
"Nevertheless, the strategy used here can be applied to simplifications of mechanics and dynamics problems, such as uniform flux through an isotropic medium with constant boundary conditions.",
"Derivation and integration are at the core of physics and applied mathematics.",
"These techniques were developed along with the disciplines of physics and mathematics during the “Scientific Revolution\" on around the years 1600-1700s to solve real problems in mechanics by the likes of Newton and Lebniz, followed by improvements and innovations by mathematicians such as Laplace, Lagrange and Gauss.",
"The discipline of fluid dynamics was developed shortly after, based on the theoretical insights provided by Isaac Newton.",
"Leonard Euler, contemporary of Laplace and Lagrange laid the foundation of much of what is now known as fluid dynamics, followed by fundamental developments by George G. Stokes and Claude-Louis Navier which fathered the equation at the heart of fluid mechanics, the Navier-Stokes equations.",
"Now that we have an idea how differentiation and integration can be used to solve physical problems, lets look at some of the basic rules and symbols associated with these techniques.",
"For a function $f(x)$ the following basic rules of differentiation apply.",
"The derivative of a constant is zero $\\frac{d}{dx}c = 0$ In other words, there can be no rate of variation of a constant function $f(x) = c$ , the value remains constant with time and is equal to c. The most basic rule of differentiation can be written as $\\frac{d}{dx} x^{n} = nx^{n-1}$ For any $ n \\in \\mathbb {R} $ .",
"A constant multiplied by any function can be written out of the differentiation $\\frac{d}{dx} cf(x) = c \\frac{d}{dx} f(x)$ For $ c $ constant.",
"If we have two functions of $x$ , say $u(x)$ and $v(x)$ the following sum, product and quotient rules apply.",
"$\\frac{d}{dx} (u + v) = \\frac{du}{dx} + \\frac{dv}{dx}$ $\\frac{d}{dx} (uv) = u\\frac{dv}{dx} + v\\frac{du}{dx}$ $\\frac{d}{dx} (\\frac{u}{v}) = \\frac{v\\frac{du}{dx} - u\\frac{dv}{dx}}{v^{2}}$ A few examples are $\\frac{d}{dx}2 = 0$ $\\frac{d}{dx} x^{3} = 3x^{3-1} = 3x^{2}$ $\\frac{d}{dx} 2f = 2 \\frac{d}{dx}x^{3} = 6x^{2}$ $\\frac{d}{dx} (2x^{3} + 3x^{2}) = 6x^{2} + 6x$ $\\frac{d}{dx} (x^{4}x^{3}) = x^{4}3x^{2} + 4x^{3}x^{3} = 3x^{6} + 4x^{6} = x^{4}x^{3}$ It should be obvious that $x^{4}x^{3}$ could have been written as $x^{7}$ and derived as such, but we wish to limit our examples to the derivatives of univariate functions.",
"We will use a similar approach below for simplicity.",
"$\\frac{d}{dx} (\\frac{x^{4}}{x^{3}}) = \\frac{x^{3}4x^{3}-x^{4}3x^{2}}{(x^{3})^{2}} = \\frac{x^{6}}{x^{6}} = 1$ Other simple derivatives that might not directly follow these rules are $\\frac{d}{dx} \\sin {x} = \\cos {x}$ $\\frac{d}{dx} \\cos {x} = -\\sin {x}$ $\\frac{d}{dx} \\tan {x} = sec^{2} x$ $\\frac{d}{dx} e^{x} = e^{x}$ $\\frac{d}{dx} \\log {x} = \\frac{1}{x}$ $\\frac{d}{dx} \\frac{1}{x} = -\\frac{1}{x^{2}}$ Our objective here is not to teach elementary calculus, as you should have been exposed to that in intro math courses, but to review the symbols that should be necessary in later chapters.",
"The relationship between integration and differentiation is summarized by the fundamental theorem of calculus $F(x) = \\int _{a}^{x} f(t)dt$ $\\frac{d}{dx}F(x) = \\frac{d}{dx} \\int _{a}^{x} f(t)dt = f(x)$ and $\\int _{a}^{b} f(x)dx = F(a) - F(b)$ For $f(x)$ continuous on the interval $[a,b]$ and $F(x)$ continuous on the interval $[a,b]$ and differentiable in $(a,b)$ .",
"$F(x)$ in this case is called the primitive of $f(x)$ .",
"In other words $F(x)$ is any function that when differentiated results in $f(x)$ .",
"For our purposes, what you have to take from the theorem is that, for a function $f(x)$ which follows the conditions stated above, its integral is $F(x)$ , which can be reverted back to $f(x)$ by differentiation.",
"We have demonstrated an example of this process in Equations 1.6 to 1.18, but bear in mind that this discussion is as far as possible from a mathematical proof, which we will not provide here.",
"The interested reader can find more about proofs in mathematical books on the field of analysis.",
"Now, intuitively, integration can be understood as the “inverse path\" of differentiation, so if the derivative of any function $x^{n}$ is given by $\\frac{d}{dx} x^{n} = nx^{n-1}$ for $n \\ne -1$ , we can retrieve the original function by adding 1 on the exponent and dividing the derivative by $n$ such that $\\frac{nx^{n-1+1}}{n-1+1} = x^{n}$ and thus, at least operationally, the integral of a function $x^{n}$ can be computed as $\\int x^{n} dx = \\frac{x^{n+1}}{n+1} + C$ Where $C$ is an integration constant and $dx$ indicates that the integration is performed on the variable $x$ .",
"The integral in equation (1.42) is said to be unbounded because there are no limits of integration.",
"By rule (1.19) any constant when differentiated will be equal to zero and in this case if we move from right to left in equation (1.42) by differentiation is not difficult to see that the original function will result if differentiation is carried out.",
"Rarely in physics and applied mathematics we will encounter such simple functions.",
"In many cases the functions to be integrated are much more complicated and often, especially when dealing with differential equations, might not have an analytical or closed form solution.",
"In water and other transport phenomena such equations are the rule, and they are solved using a different approach that uses approximations and computational methods.",
"These numerical solutions are at the core of not only fluid dynamics in the form of computational fluid dynamics (CFD), but are also an essential component of modern soil physics, hydrology and hydrogeology.",
"For simple univariate integrals the rules and techniques below apply $\\int x^{n}dx = \\frac{x^{n+1}}{n+1} + C$ $\\int x^{2}dx = \\frac{x^{3}}{3} + C$ $\\int dx = x + C$ $\\int x^{3}dx = \\frac{x^{4}}{4} + C$ $\\int \\sin {x} \\ dx = -\\cos {x} + C$ $\\int \\cos {x} \\ dx = sin{x} + C$ $\\int \\tan {x} \\ dx = -\\log {\\cos {x}} + C$ $\\int e^{x} \\ dx = e^{x} + C$ $\\int \\log {x} \\ dx = x \\log {x} - x + C$ $\\int \\frac{1}{x} \\ dx = \\log {x} + C$ In introductory calculus books you learn about integration as a Riemann sum and as the area under a curve $\\lim _{n\\rightarrow \\infty } \\sum _{k=1}^{n} f(c_{k}) \\Delta {x} = J = \\int _{a}^{b} f(x) dx$ In this equation, when the length of the intervals $\\Delta {x}$ goes to zero and the number of $k$ intervals goes to infinite, the sum can be interpreted as an integral over the entire interval from $a$ to $b$ and the result is the area under a curve, $J$ .",
"We will not discuss this right now but it is important that you get familiarized with sums and their role in calculus, because they make frequent appearance in properties of granular materials and on the solutions of differential equations in advanced soil physics.",
"Consider a very simple example of a line described by (REF ) $f(x) = y = \\frac{x}{2} + 2$ Figure: Graph of the function f(x)=y=x 2+2f(x) = y = \\frac{x}{2} + 2 .If you wish to calculate the area of the curve on the interval from $x = 2$ to $x = 8$ you could simply calculate the area of the trapeze bounded on its upper end by the line and in its lower end by the x axis and at its left and right limits by the lines $x = 2$ and $x = 8$ resulting in 120.",
"Notice that you would get the same result if you divide the interval from 2 to 8 in 6 equal intervals of length 1 in $x$ , calculate the area of each and sum the areas.",
"This is exactly what it is done by the left hand side of equation (1.53).",
"Now by integration, the area under the line is given by (REF ) $J = \\int _{2}^{8} 2x + 10 dx = [x^{2} + 10x]\\Biggr |_{2}^{8} = [8^{2} + 10\\times 8] - [2^{2} + 10\\times 2] = 120$ It is obvious from the previous discussion that you don't need integration to calculate the area under a straight line, but it is also clear that you cannot use the formulas created for regular geometric figures for calculating the area under curves and this is the appeal of integration.",
"You should be familiarized with integration techniques from your intro math courses such as substitution, integration by parts and partial fractions.",
"If you haven't had any college level math you are urged to take at least a semester to study it before any serious endeavor in soil physics.",
"True soil physics requires vector calculus and mathematical methods of physics, and whoever tells you - “you don't need these subjects to understand soil physics\" - does not know what soil physics is.",
"When integration is performed in a bounded interval such as in equation (1.55), the integral is said to be a definite integral and there is no need for a constant of integration as is the case with indefinite integrals.",
"Some properties of the indefinite integral are $\\int _{b}^{a} f(x) dx = -\\int _{a}^{b} f(x) dx$ $\\int _{a}^{a} f(x) dx = 0$ $\\int _{a}^{b} kf(x) dx = k\\int _{a}^{b} f(x) dx$ $\\int _{a}^{b} (f(x) \\pm g(x))dx = \\int _{a}^{b} f(x) dx \\pm \\int _{a}^{b} g(x) dx$ $\\int _{a}^{b} f(x) dx + \\int _{b}^{c} f(x) dx = \\int _{a}^{c} f(x) dx$ Where $f(x)$ and $g(x)$ are both univariate functions of $x$ , and $k$ is a constant.",
"You must have learned in high school physics that a vector differs from a scalar in a sense that it has both direction and magnitude.",
"Now if we think of soil physical properties we might think of a range of properties that are scalars such as soil bulk density, soil temperature and soil water content.",
"Now when we think of flow of water and gases inside porous media, most often than not it is as important to know the magnitude of the water or gas being transported, or in other words the bulk mass being transported by unit of time, as it is important to know the direction in which they are being transported.",
"Suppose you have an irrigation project and you calculate exactly the volume of water that need to be provided for each plant for adequate growth and productivity, now suppose that you provide this water below the root zone so that it is mostly lost by downwards flow, you would be then providing the right magnitude of water but the direction of the flow would be inadequate.",
"Suppose you have a rainfall and you are concerned with aquifer recharge and sediment loss by erosion.",
"If most of the water flows vertically it is likely that groundwater recharge would be maximized and erosion minimized, and thus you wanna take measures for this to happen such as conserving natural vegetation or using sustainable land management practices.",
"Let's say $x$ is a scalar, if we say that $x = 10$ that is all we need to describe it, meaning that its magnitude is 10.",
"Now before talking about direction is important to define a system of coordinates.",
"The familiar system of coordinates that you might be familiar with is the two dimensional Cartesian space or the plane.",
"Suppose you and your friend are pushing a large box on an empty room.",
"Imagine that you are pushing the box by applying a force parallel to two of the walls and your friend is applying a force perpendicular to the force you are applying.",
"Let's call the direction which you are applying the force x so that the wall parallel to that directions has the coordinates of x, while the direction in which your friend is applying the force is y and the wall parallel to that direction defines the y axis.",
"If none of the forces applied is zero, the object will move at an angle to each of the applied forces because the vector sum of the forces has a resulting force $\\mathbf {F}$ defined as $\\mathbf {F} = F_{x} \\mathbf {i} + F_{y} \\mathbf {j}$ In this equation $F_{x}$ and $F_{y}$ are the components of the force F in the directions x and y or the forces applied by you and your friend.",
"The symbols $\\mathbf {i}$ and $\\mathbf {j}$ are used to indicate unit vectors, and their role is to indicate in which directions the components of the vector are acting.",
"The unit vectors have magnitude 1 and can be represented by different notations.",
"If we have three forces being applied in a three dimensional space, with coordinates x, y and z equation (1.61) can be written as $\\mathbf {F} = F_{x} \\mathbf {i} + F_{y} \\mathbf {j} + F_{z} \\mathbf {k}$ In fluid dynamics is not unusual that the unit vectors are represented using a different notation.",
"A velocity vector in a three dimensional space is usually written as $\\mathbf {v} = v_{x} \\mathbf {x} + v_{y} \\mathbf {y} + v_{z} \\mathbf {z}$ Where $\\mathbf {i} = \\mathbf {x}$ , $\\mathbf {j} = \\mathbf {y}$ and $\\mathbf {k} = \\mathbf {z}$ .",
"Now in your handwritten notebook or in a board it is not convenient to write letters as boldface as notation for representing vectors.",
"In handwritten form but also in a few books, vectors are represented with arrows draw above the symbol while unit vectors are represented by the circumflex accent over the symbol, known in mathematics and colloquial language as hat “”.",
"Using this notation, equations (1.62) and (1.63) can be written as $\\vec{F} = F_{x} \\hat{i} + F_{y} \\hat{j} + F_{z} \\hat{k}$ Another notation commonly seen in fluid dynamics is $\\vec{v} = v_{x} \\hat{x} + v_{y} \\hat{y} + v_{z} \\hat{z}$ In this material I will use the boldface representation because I feel it looks more elegant in print and electronic format.",
"In more advanced physics and mathematics books these vectors could also be represented as $\\mathbf {F} = \\langle F_{x}, F_{y}, F_{z} \\rangle $ $\\mathbf {v} = \\langle v_{x}, v_{y}, v_{z} \\rangle $ and we might make use of this notation for higher dimension vectors (i.e.",
"higher than three dimensions).",
"Suppose now that in Cartesian space there is a vector a defined as $\\mathbf {a} = a_{x} \\mathbf {i} + a_{y} \\mathbf {j}$ as represented in Figure REF .",
"Figure: Vector 𝐚\\mathbf {a} with components 𝐚 𝐱 \\mathbf {a_{x}} and 𝐚 𝐲 \\mathbf {a _{y}}.You will notice that the components of the vector are the projections of $\\mathbf {a}$ over the axis in the directions $\\mathbf {i}$ and $\\mathbf {j}$ .",
"The length of the vector or its magnitude is given by $|\\mathbf {a}| = \\sqrt{a_{x}^2 + a_{y}^2}$ The proof is immediate from elementary mathematics and the user is encouraged to consult calculus and linear algebra books for further information.",
"A few properties of vectors are listed below $\\mathbf {a} + \\mathbf {b} = \\mathbf {b} + \\mathbf {a}$ $\\mathbf {a} + \\mathbf {0} = \\mathbf {a}$ $k(\\mathbf {a} + \\mathbf {b}) = k\\mathbf {a} + k\\mathbf {b}$ $cd\\mathbf {a} = c(d\\mathbf {a})$ $\\mathbf {a} + (\\mathbf {b} + \\mathbf {c}) = (\\mathbf {a} + \\mathbf {b}) + \\mathbf {c}$ $\\mathbf {a} + (-\\mathbf {a}) = \\mathbf {0}$ $(c + d)\\mathbf {a} = c\\mathbf {a} + d\\mathbf {a}$ $1\\mathbf {a} = \\mathbf {a}$ Where $\\mathbf {a}$ , $\\mathbf {b}$ and $\\mathbf {c}$ are vectors and and $c$ and $d$ are scalars.",
"In all vectors shown until now the components are treated as scalars as the direction is specified by the unit vectors.",
"In many cases the components themselves are functions in x, y and z such that $\\mathbf {V}(x,y,z) = u(x,y,z) \\ \\mathbf {i} + v(x,y,z) \\ \\mathbf {j} + w(x,y,z) \\ \\mathbf {k}$ or $\\mathbf {V}(x,y,z) = \\langle u(x,y,z), v(x,y,z), w(x,y,z) \\rangle $ In rectangular coordinates each of these components can represent an equation which is itself a function of $x$ , $y$ and $z$ .",
"Equation (1.78, 1.79) represent vector fields where in each point in three dimensional space a vector can be drawn specifying the field.",
"A simple example would be a vector field where $u(x,y,z) = 2xyz$ $v(x,y,z) = x^{2} + yz$ $w(x,y,z) = x - y + 2z$ which can be written as $\\mathbf {V}(x,y,z) = 2xyz \\ \\mathbf {i} + (x^{2} + yz) \\ \\mathbf {j} + (x - y + 2z) \\ \\mathbf {k}$ If we define two vectors $\\mathbf {a}$ and $\\mathbf {b}$ $\\mathbf {a} = a_{x} \\mathbf {i} + a_{y} \\ \\mathbf {j} + a_{z} \\ \\mathbf {k}$ $\\mathbf {b} = b_{x} \\mathbf {i} + b_{y} \\ \\mathbf {j} + b_{z} \\ \\mathbf {k}$ regular mathematical operations of sum and subtractions of these two vectors and multiplication by scalars are defined by properties (1.70) to (1.77), see for example property (1.76) $(c + d)\\mathbf {a} = (c + d)(a_{x} \\mathbf {i} + a_{y} \\ \\mathbf {j} + a_{z} \\ \\mathbf {k})\\\\= (c + d)a_{x} \\mathbf {i} + (c + d) a_{y} \\ \\mathbf {j} + (c + d)a_{z} \\ \\mathbf {k}\\\\= ca_{x} \\mathbf {i} + c a_{y} \\ \\mathbf {j} + ca_{z} \\ \\mathbf {k} + da_{x} \\mathbf {i} + d a_{y} \\ \\mathbf {j} + da_{z} \\ \\mathbf {k}\\\\= c \\mathbf {a} + d \\mathbf {a}$ You might have noticed that nothing is said about multiplication of vectors by vectors in those properties.",
"This is because regular multiplication and division operations used for scalars are not valid for vectors, they require their own rules.",
"The first of such rules is the scalar or dot product, represented by the symbol “$\\cdot {}$ ” $\\mathbf {a} \\cdot \\mathbf {b} = a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z}$ The demonstration requires some further explanation as you might not be familiar or might not recall the properties of unit vectors $\\mathbf {a} \\cdot \\mathbf {b} =(a_{x}\\mathbf {i} + a_{y}\\mathbf {j} + a_{z}\\mathbf {k}) \\cdot (b_{x}\\mathbf {i} + b_{y}\\mathbf {j} + b_{z}\\mathbf {k}) \\\\= a_{x}\\mathbf {i} \\cdot b_{x}\\mathbf {i} +a_{x}\\mathbf {i} \\cdot b_{y}\\mathbf {j} +a_{x}\\mathbf {i} \\cdot b_{z}\\mathbf {k} +a_{y}\\mathbf {i} \\cdot b_{x}\\mathbf {i} +a_{y}\\mathbf {i} \\cdot b_{y}\\mathbf {j} + \\\\+ a_{y}\\mathbf {i} \\cdot b_{z}\\mathbf {k} +a_{z}\\mathbf {i} \\cdot b_{x}\\mathbf {i} +a_{z}\\mathbf {i} \\cdot b_{y}\\mathbf {j} +a_{z}\\mathbf {i} \\cdot b_{z}\\mathbf {k} \\\\= a_{x}b_{x}\\mathbf {i} \\cdot \\mathbf {i} +a_{x}b_{y}\\mathbf {i} \\cdot \\mathbf {j} +a_{x}b_{z}\\mathbf {i} \\cdot \\mathbf {k} +a_{y}b_{x}\\mathbf {i} \\cdot \\mathbf {i} +a_{y}b_{y}\\mathbf {i} \\cdot \\mathbf {j} + \\\\+ a_{y}b_{z}\\mathbf {i} \\cdot \\mathbf {k} +a_{z}b_{x}\\mathbf {i} \\cdot \\mathbf {i} +a_{z}b_{y}\\mathbf {i} \\cdot \\mathbf {j} +a_{z}b_{z}\\mathbf {i} \\cdot \\mathbf {k} \\\\= a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z}$ This is because the dot product operates following a distributive property, and for the unit vectors defined as $i = \\langle 1,0,0 \\rangle $ , $j = \\langle 0,1,0 \\rangle $ and $k = \\langle 0,0,1 \\rangle $ $\\mathbf {i} \\cdot \\mathbf {i} = \\mathbf {i} \\cdot \\mathbf {i} = \\mathbf {i} \\cdot \\mathbf {i} = 1$ $\\mathbf {i} \\cdot \\mathbf {j} = \\mathbf {i} \\cdot \\mathbf {k} = \\mathbf {j} \\cdot \\mathbf {k} = 0$ A special case of the dot product in two dimensions is the Pythagoras theorem, which as we already seen it provides the length of a vector $\\mathbf {a} \\cdot \\mathbf {a} = |\\mathbf {a}|^{2} = a_{x}^2 + a_{y}^2$ Back to three dimensions, the dot product can be written as $\\mathbf {a} \\cdot \\mathbf {b} = |\\mathbf {a}| |\\mathbf {b}| \\cos {\\theta }$ Where $\\cos {\\theta }$ is the angle between the two vectors.",
"There is much more information about the geometric interpretation of the dot product in calculus and linear algebra books.",
"This information is important for the interpretation of vector quantities in soil physics and groundwater hydrology but it would be beyond the scope of this book to present it.",
"What is important now is to understand the definitions of these vector operators because they are required for understanding vector calculus operators and the fundamental transport equations.",
"The second operator is the cross or vector product, represented by the symbol “$\\times $ ” $\\mathbf {a} \\times \\mathbf {b} =(a_{y} b_{z} - a_{z} b_{y}) \\mathbf {i} +(a_{z} b_{x} - a_{x} b_{z}) \\mathbf {j} +(a_{x} b_{y} - a_{y} b_{x}) \\mathbf {k}$ or $\\mathbf {a} \\times \\mathbf {b} = \\ \\langle a_{y}b_{z} - a_{z}b_{y}, a_{z}b_{x} - a_{x}b_{z}, a_{x}b_{y} - a_{y}b_{x} \\rangle $ The demonstration can be done in a similar way as we did for the dot product $\\mathbf {a} \\times \\mathbf {b} =(a_{x}\\mathbf {i} + a_{y}\\mathbf {j} + a_{z}\\mathbf {k}) \\times (b_{x}\\mathbf {i} + b_{y}\\mathbf {j} + b_{z}\\mathbf {k}) \\\\= a_{x}\\mathbf {i} \\times b_{x}\\mathbf {i} +a_{x}\\mathbf {i} \\times b_{y}\\mathbf {j} +a_{x}\\mathbf {i} \\times b_{z}\\mathbf {k} +a_{y}\\mathbf {i} \\times b_{x}\\mathbf {i} +a_{y}\\mathbf {i} \\times b_{y}\\mathbf {j} + \\\\+ a_{y}\\mathbf {i} \\times b_{z}\\mathbf {k} +a_{z}\\mathbf {i} \\times b_{x}\\mathbf {i} +a_{z}\\mathbf {i} \\times b_{y}\\mathbf {j} +a_{z}\\mathbf {i} \\times b_{z}\\mathbf {k} \\\\= a_{x}b_{x}\\mathbf {i} \\times \\mathbf {i} +a_{x}b_{y}\\mathbf {i} \\times \\mathbf {j} +a_{x}b_{z}\\mathbf {i} \\times \\mathbf {k} +a_{y}b_{x}\\mathbf {j} \\times \\mathbf {i} +a_{y}b_{y}\\mathbf {j} \\times \\mathbf {j} + \\\\+ a_{y}b_{z}\\mathbf {j} \\times \\mathbf {k} +a_{z}b_{x}\\mathbf {k} \\times \\mathbf {i} +a_{z}b_{y}\\mathbf {k} \\times \\mathbf {j} +a_{z}b_{z}\\mathbf {k} \\times \\mathbf {k} \\\\$ and because the geometrical properties of the cross product are different than those of the dot product such that $\\mathbf {i} \\times \\mathbf {i} = \\mathbf {j} \\times \\mathbf {j} = \\mathbf {k} \\times \\mathbf {k} = 0$ $\\mathbf {i} \\times \\mathbf {j} = \\mathbf {k}$ $\\mathbf {j} \\times \\mathbf {k} = \\mathbf {i}$ $\\mathbf {k} \\times \\mathbf {i} = \\mathbf {j}$ $\\mathbf {j} \\times \\mathbf {i} = -\\mathbf {k}$ $\\mathbf {k} \\times \\mathbf {j} = -\\mathbf {i}$ $\\mathbf {i} \\times \\mathbf {k} = -\\mathbf {j}$ we have $\\mathbf {a} \\times \\mathbf {b} =a_{x}b_{x} 0 +a_{x}b_{y}\\mathbf {k} +a_{x}b_{z}(-\\mathbf {j}) +a_{y}b_{x}(-\\mathbf {k}) +a_{y}b_{y}0 + \\\\+ a_{y}b_{z}\\mathbf {i} +a_{z}b_{x}\\mathbf {j} +a_{z}b_{y}(-\\mathbf {i}) +a_{z}b_{z}0 \\\\= (a_{y} b_{z} - a_{z} b_{y}) \\mathbf {i} +(a_{z} b_{x} - a_{x} b_{z}) \\mathbf {j} +(a_{x} b_{y} - a_{y} b_{x}) \\mathbf {k}$ Notice that the result of scalar product is, as the name implies, a scalar while the vector product is, as you might have guessed, a vector.",
"In practice the vector product might be calculated from $\\mathbf {a} \\times \\mathbf {b} = \\begin{vmatrix}\\mathbf {i} & \\mathbf {j} & \\mathbf {k} \\\\a_{x} & a_{y} & a_{z} \\\\b_{x} & b_{y} & b_{z} \\\\\\end{vmatrix}$ such that $\\mathbf {a} \\times \\mathbf {b} =a_{y}b_{z} \\mathbf {i} +a_{z}b_{x} \\mathbf {j} +a_{x}b_{y} \\mathbf {k} -a_{z}b_{y} \\mathbf {i} -a_{x}b_{z} \\mathbf {j} -a_{y}b_{x} \\mathbf {k} \\\\= (a_{y} b_{z} - a_{z} b_{y}) \\mathbf {i} +(a_{z} b_{x} - a_{x} b_{z}) \\mathbf {j} +(a_{x} b_{y} - a_{y} b_{x}) \\mathbf {k}$ Note that there are different approaches to calculating the determinant of a matrix so we are omitting a few steps.",
"You can consult any calculus or linear algebra book you if you are unfamiliar with the procedure.",
"Notice that in many countries such procedures are taught in secondary school so you might want to consult such books as well for a more didactic introduction.",
"The vector product can also be written as $\\mathbf {a} \\times \\mathbf {b} = |\\mathbf {a}| |\\mathbf {b}| \\sin {\\theta }$ Where $\\sin {\\theta }$ is the sin of the angle between the two vectors.",
"As with the dot product the reader is encouraged to consult its geometrical properties and interpretation.",
"The properties of the scalar and vector products are summarized below $\\mathbf {a} \\cdot \\mathbf {a} = |\\mathbf {a}|^{2}$ $\\mathbf {a} \\cdot (\\mathbf {b} + \\mathbf {c}) = \\mathbf {a} \\cdot \\mathbf {b} + \\mathbf {a} \\cdot \\mathbf {c}$ $\\mathbf {a} \\cdot \\mathbf {b} = \\mathbf {b} \\cdot \\mathbf {a}$ $(c\\mathbf {a}) \\cdot \\mathbf {b} = c(\\mathbf {a} \\cdot \\mathbf {b}) = \\mathbf {a} \\cdot (c\\mathbf {b})$ $\\mathbf {0} \\cdot \\mathbf {a} = 0$ $\\mathbf {a} \\times \\mathbf {b} = -\\mathbf {b} \\times \\mathbf {a}$ $(c\\mathbf {a}) \\times \\mathbf {b} = c(\\mathbf {a} \\times \\mathbf {b}) = \\mathbf {a} \\times (c\\mathbf {b})$ $\\mathbf {a} \\times (\\mathbf {b} + \\mathbf {c}) =\\mathbf {a} \\times \\mathbf {b} + \\mathbf {a} \\times \\mathbf {c}$ $(\\mathbf {a} \\times \\mathbf {b}) + \\mathbf {c} =\\mathbf {a} \\times \\mathbf {c} + \\mathbf {b} \\times \\mathbf {c}$ $\\mathbf {a} \\cdot (\\mathbf {b} + \\mathbf {c}) =(\\mathbf {a} \\times \\mathbf {b}) \\cdot \\mathbf {c}$ $\\mathbf {a} \\times (\\mathbf {b} + \\mathbf {c}) =(\\mathbf {a} \\cdot \\mathbf {c})\\mathbf {b} - (\\mathbf {a} \\cdot \\mathbf {b})\\mathbf {c}$ Before delving into vector calculus operators we need to take a step back and look what how derivation and integration work when you have a multivariate function.",
"As we've seen, an univariate function has a dependent variable that it is a function of a single independent variable as is the case with the equation $y = x$ .",
"A multivariate function is one which is a function of two or more independent variables, for example, each component of the vector field in (1.83) is a multivariate function of the variables $x$ , $y$ and $z$ as shown in (1.80), (1.81) and (1.82).",
"You already know a few basic rules to differentiate univariate functions, but how to proceed with multivariate functions?",
"The tools for solving such problems are in the realm of multivariate calculus and are the partial derivatives and multiple integrals.",
"Most engineering, mathematics and physics undergraduate programs devote an entire course for these subjects, usually calculus III or multivariate calculus.",
"The idea behind partial differentiation is apparently simple if we ignore rigorous mathematical proofs and theorems behind it as we will do in this material.",
"If you have a multivariate function you can take the derivative with respect to a single variable and treat the other variables as constants as in (1.21).",
"When taking partial derivatives the symbol “$\\partial $ ” is used instead of “$d$ ”.",
"For example, if we wish to differentiate (1.80) with respect to $x$ we have $\\frac{\\partial }{\\partial x} u(x,y,z) = \\frac{\\partial }{\\partial x}2xyz = yz \\frac{\\partial x}{\\partial x} 2x = 2yz$ Now let's say you want to derivate (1.81) with respect to $y$ $\\frac{\\partial }{\\partial y} v(x,y,z) = \\frac{\\partial }{\\partial y} (x^{2} + yz) = \\frac{\\partial }{\\partial y} x^{2} + \\frac{\\partial }{\\partial y} yz = 0 + z = z$ Because here $x$ is a constant and any constant squared is still a constant, and $z$ is another constant.",
"Other representations of a partial derivative are for example $(\\frac{\\partial }{\\partial x} u)_{y,z}$ and $\\frac{\\partial }{\\partial x} u \\Bigm | _{y,z}$ Where in both cases the differentiation is performed with respect to $x$ with $y$ and $z$ held constant.",
"Such notation is most commonly used Thermodynamics, a field which makes extensive use of multivariate calculus and interlaces with both fluid dynamics and soil physics.",
"Earlier we saw that the first derivative is the rate of variation of a function, intuitively, for a straight line $y = ax + b$ The first derivative is the slope of the line $\\frac{dy}{dx} = a$ For a second degree polynomial, the rate of variation is not a constant, but it is a straight line so that the slope of the quadratic curve varies with $x$ $y = ax^{2} + bx + c$ The first derivative is the slope of the line $\\frac{dy}{dx} = 2ax + b$ Notice that the derivatives can be used to identify critical points of functions as you might learn in any intro calculus books.",
"A function in two or more dimensions will have slopes in each particular direction as represented by the partial derivatives.",
"Lets consider a function $w(x,y,z)$ $w = x^{2} + 2xy + z^{3}$ with partial derivatives in $x$ , $y$ and $z$ $\\frac{\\partial w}{\\partial x} = 2x + 2y$ $\\frac{\\partial w}{\\partial y} = 2x$ $\\frac{\\partial w}{\\partial z} = 3z^{2}$ Which are the slopes of the function in the directions $x$ and $y$ .",
"From our previous discussions it should be apparent that these partial derivatives represent the slopes of the function on the directions $ \\mathbf {i}$ , $ \\mathbf {j}$ and $ \\mathbf {k}$ and as such we can represent these partial derivatives as a vector function at any point in space (if it exists) as $\\nabla {w(x,y,z)} =\\frac{\\partial w}{\\partial x} = (2x + 2y) \\mathbf {i} +\\frac{\\partial w}{\\partial y} = (2x) \\mathbf {j} +\\frac{\\partial w}{\\partial z} = (3z^{2}) \\mathbf {k}$ Where the symbol “$\\nabla $ ” is called “nabla” or “del” and is a representation for a mathematical operator called gradient.",
"The gradient of a scalar or vector function is a vector function and can be generalized as $\\boxed{\\nabla {} =\\frac{\\partial w}{\\partial x} \\mathbf {i} +\\frac{\\partial w}{\\partial y} \\mathbf {j} +\\frac{\\partial w}{\\partial z} \\mathbf {k}}$ applied to any scalar function $f$ results $\\boxed{\\nabla {f} =\\frac{\\partial w}{\\partial x}f \\ \\mathbf {i} +\\frac{\\partial w}{\\partial y}f \\ \\mathbf {j} +\\frac{\\partial w}{\\partial z}f \\ \\mathbf {k}}$ Now for a vector $\\mathbf {u} = u \\ \\mathbf {i} + v \\ \\mathbf {j} + w \\ \\mathbf {k}$ The gradient in rectangular coordinates is a second rank tensor $\\nabla {\\mathbf {u}} =\\begin{pmatrix}\\frac{\\partial u}{\\partial x} & \\frac{\\partial v}{\\partial x} & \\frac{\\partial w}{\\partial x}\\\\\\frac{\\partial u}{\\partial y} & \\frac{\\partial v}{\\partial y} & \\frac{\\partial w}{\\partial y} \\\\\\frac{\\partial u}{\\partial z} & \\frac{\\partial v}{\\partial z} & \\frac{\\partial w}{\\partial z}\\end{pmatrix}$ Although this vector gradient makes an appearance in Navier-Stokes equations, it is often not necessary to expand it.",
"The reader of this material should focus on the gradient of scalar functions whose importance will be apparent when we deal of water flux in porous media.",
"Now the dot product of the gradient operator over a vector, say $u$ , defines the divergence of the vector $\\boxed{\\nabla {\\cdot \\mathbf {u}} =\\frac{\\partial u}{\\partial x} +\\frac{\\partial v}{\\partial y} +\\frac{\\partial w}{\\partial z} }$ Which is a scalar field.",
"The demonstration is straightforward from the definition of $\\nabla $ and $\\mathbf {u}$ and from the dot product properties $\\nabla {\\cdot \\mathbf {u}} =(\\frac{\\partial }{\\partial x} \\mathbf {i} +\\frac{\\partial }{\\partial y} \\mathbf {j} +\\frac{\\partial }{\\partial z} \\mathbf {k}) \\cdot (u \\mathbf {i} + v \\mathbf {j} + w \\mathbf {k}) \\\\= \\frac{\\partial u}{\\partial x} \\mathbf {i} \\cdot \\mathbf {i} +\\frac{\\partial v}{\\partial y} \\mathbf {j} \\cdot \\mathbf {j}+\\frac{\\partial w}{\\partial z} \\mathbf {k} \\cdot \\mathbf {k} =\\frac{\\partial u}{\\partial x} +\\frac{\\partial v}{\\partial y} +\\frac{\\partial w}{\\partial z}$ Following the same logic we can apply the vector product of the gradient to any vector function yielding the curl of a vector, $\\mathbf {u}$ for example $\\boxed{\\nabla {\\times \\mathbf {u}} =(\\frac{\\partial w}{\\partial y}-\\frac{\\partial v}{\\partial z}) \\mathbf {i} +(\\frac{\\partial u}{\\partial z}-\\frac{\\partial w}{\\partial x}) \\mathbf {j} +(\\frac{\\partial v}{\\partial x}-\\frac{\\partial u}{\\partial y}) \\mathbf {k}}$ Which is a also vector field.",
"As we learned with the dot product the determinant form is the easiest way to remember how to calculate the curl of a vector function $\\nabla {\\times \\mathbf {u}} =\\begin{vmatrix}\\mathbf {i} & \\mathbf {j} & \\mathbf {k}\\\\\\frac{\\partial }{\\partial x} & \\frac{\\partial }{\\partial y} & \\frac{\\partial }{\\partial z} \\\\u & v & w\\end{vmatrix}$ As with the scalar and vector products there are several properties that define operations for the gradient, divergence and curl, including operations among them.",
"One such operation which appears very frequently in applied mathematics and physics is the divergence of a gradient.",
"$\\boxed{\\nabla {\\cdot (\\nabla {f)} =\\frac{\\partial ^{2} f}{\\partial x^{2}} +\\frac{\\partial ^{2} f}{\\partial y^{2}} +\\frac{\\partial ^{2} f}{\\partial z^{2}} }}$ The demonstration is also somewhat trivial $\\nabla {\\cdot (\\nabla {f})} =(\\frac{\\partial }{\\partial x} \\mathbf {i}+\\frac{\\partial }{\\partial y} \\mathbf {j}+\\frac{\\partial }{\\partial z} \\mathbf {k}) \\cdot (\\frac{\\partial f}{\\partial x} \\mathbf {i}+\\frac{\\partial f}{\\partial y} \\mathbf {j}+\\frac{\\partial f}{\\partial z} \\mathbf {k}) \\\\= \\frac{\\partial ^{2} f}{\\partial x^{2}} \\mathbf {i} \\cdot \\mathbf {i} +\\frac{\\partial f ^{2}}{\\partial y^{2}} \\mathbf {j} \\cdot \\mathbf {j}+\\frac{\\partial w^{2}}{\\partial z^{2}} \\mathbf {k} \\cdot \\mathbf {k} \\\\= \\frac{\\partial ^{2} f}{\\partial x^{2}} +\\frac{\\partial ^{2} f}{\\partial y^{2}} +\\frac{\\partial ^{2} f}{\\partial z^{2}}$ The operator $\\nabla {}^{2} = \\nabla {} \\cdot \\nabla {}$ is called the Laplace operator because of the equation of the form $\\nabla {}^{2}f = \\frac{\\partial ^{2} f}{\\partial x^{2}} +\\frac{\\partial ^{2} f}{\\partial y^{2}} +\\frac{\\partial ^{2} f}{\\partial z^{2}} = 0$ which is of the utmost importance not only for soil physics and fluid mechanics but to most areas of physics.",
"The Laplace operator also operates in vectors such that $\\nabla {}^{2} \\mathbf {u}= \\nabla {}^{2} u \\ \\mathbf {i} + \\nabla {}^{2} v \\ \\mathbf {i} + \\nabla {}^{2} w \\ \\mathbf {i}$ Understanding the gradient, divergence, curl and Laplacian is fundamental for the understanding of fluid mechanics and therefore soil physics and underground hydrology.",
"Understanding these operators along with their properties and interrelations and the fundamental theorems of vectors calculus is almost all that is needed before presenting the fundamental equations of fluid mechanics.",
"If you decide to consult the literature be aware that in many older books the operators are presented as del($f$ ), grad($f$ ) or grad $f$ for the gradient of a scalar function $f$ , div($\\mathbf {u}$ ) or div $\\mathbf {u}$ for the divergence of a vector function $\\mathbf {u}$ , curl($\\mathbf {u}$ ), curl $\\mathbf {u}$ , rot($\\mathbf {u}$ ) or rot $\\mathbf {u}$ for the curl of a vector $\\mathbf {u}$ and div(grad $f$ ) for the Laplacian of a scalar function $f$ .",
"The “rot” in curl is because in many countries the curl is called rotational or one its translated similar words.",
"The word rotational as you might imagine comes from rotation and the divergence is obviously related to something that diverges.",
"The meaning is exactly as implied, these operators were developed in the realm of the physics of fluids.",
"Most of the mathematics used in fluid dynamics was used to describe the pervasive ether (or æther) before Einstein and his contemporaries proved that it was not a necessary entity for the description of the the cosmos.",
"A few vector operators identities are offered without proof $\\nabla {(fg)} = f (\\nabla {g}) + g (\\nabla {f})$ $\\nabla {(\\mathbf {A} \\cdot \\mathbf {B} )} = \\mathbf {A} \\times (\\nabla \\times \\mathbf {B}) + \\mathbf {B} \\times (\\nabla \\times \\mathbf {A}) + (\\mathbf {A} \\cdot \\nabla ) \\mathbf {B} + (\\mathbf {B} \\cdot \\nabla )\\mathbf {A}$ $\\nabla \\cdot (f \\mathbf {A}) = f(\\nabla \\cdot \\mathbf {A}) + \\mathbf {A} \\cdot (\\nabla {f})$ $\\nabla \\cdot (\\mathbf {A} \\times \\mathbf {B}) =\\mathbf {B} \\cdot (\\nabla \\times \\mathbf {A}) - \\mathbf {A} \\cdot (\\nabla \\times \\mathbf {B})$ $\\nabla \\times (f\\mathbf {A}) =f(\\nabla \\times \\mathbf {A}) - \\mathbf {A} \\times (\\nabla f)$ $\\nabla \\times (\\mathbf {A} \\times \\mathbf {B}) = (\\mathbf {B} \\cdot \\nabla )\\mathbf {A} - (\\mathbf {A} \\cdot \\nabla ) \\mathbf {B} + \\mathbf {A}(\\nabla \\cdot \\mathbf {B}) - \\mathbf {B}(\\nabla \\cdot \\mathbf {A})$ $\\nabla \\cdot (\\nabla \\times \\mathbf {A}) = 0$ $\\nabla \\times (\\nabla f) = 0$ $\\nabla \\times (\\nabla \\times \\mathbf {A}) = \\nabla (\\nabla \\times \\mathbf {A}) - \\nabla ^{2}\\mathbf {A}$ For now it is also important to say that integration can also be performed in multidimensional space, we have seen that integration in one dimension can be represented as $\\int x dx = \\frac{x^{2}}{2} + C$ now integration in two dimensions can be understood in terms of a surface integral, for example $\\int \\int xy dx dy$ Similar for what was discussed for partial derivatives, the integration is performed on each variable per step, considering the others as a constant $\\int \\int xy dx dy = \\frac{x^{2}}{2}\\int y dy + C_{1} \\\\= \\frac{x^{2}}{2}\\frac{y^{2}}{2} + C_{1}y + C_{2} \\\\= \\frac{x^{2}y^{2}}{4} + C_{1}y + C_{2}$ The same principles being applied to simple integrals in three or more dimensions.",
"An integral in three dimensions is an integral in volume, for example $\\int \\int \\int xyz dx dy dz = C_{1} x y + C_{2} y + C_{3} + 1/8 x^{2} y^{2} z^{2}$ Of course integrals of such simple functions are rarely seen in physics and applied mathematics.",
"More often than not more advanced integration techniques and system of coordinates transformation are needed for solving the equations.",
"The most common systems of coordinates used being polar, cylindrical and spherical coordinates.",
"The user is encouraged to consult the literature for transformations in system of coordinates and for the gradient, divergence, curl and Laplacian in those coordinate systems.",
"Another integral that is common in physics and applied math is the integral over a curve or line integral, in which the integration is performed over a curve $\\int _{a}^{b} \\mathbf {v} \\cdot d \\mathbf {l}$ which for a closed loop is usually represented as $\\oint \\mathbf {v} \\cdot d \\mathbf {l}$ Integrals over lines, regions (surface) and volumes (domains) can also be represented by the symbols $\\oint _C, \\ \\ \\int _R, \\ \\ \\int _S, \\ \\ \\int _V, \\ \\ \\int _D,$ Such representations appear in three fundamental theorems of vector calculus which are used very often in fluid mechanics, electromagnetism and other areas of applied mathematics and physics [49], the divergence (or Gauss) theorem $\\boxed{\\iint _S \\mathbf {F} \\cdot \\mathbf {n} \\ d\\sigma = \\iiint _D \\nabla \\cdot \\mathbf {F} \\ dV}$ Green's theorem in normal form $\\boxed{\\oint _C \\mathbf {F} \\cdot \\mathbf {n} \\ ds = \\iint _R \\nabla \\cdot \\mathbf {F} \\ dA}$ and in tangential form $\\boxed{\\oint _C \\mathbf {F} \\cdot d \\mathbf {r} = \\iint _R \\nabla \\times \\mathbf {F} \\ \\cdot \\mathbf {k} \\ dA}$ and Stokes' theorem $\\boxed{\\oint _C \\mathbf {F} \\cdot d \\mathbf {r} = \\iint _S \\nabla \\times \\mathbf {F} \\ \\cdot \\mathbf {n} \\ d \\sigma }$ When working with the algebra of general relativity and other theoretical physics topics, Albert Einstein introduced a notation aimed at simplifying the representation of summation terms.",
"As you might know, a sum is represented by the Greek letter $\\Sigma $ , so a sum of three $x$ is terms can be represented as $y = \\sum _{i=1}^n x_i = x_1 + x_2 + x_3$ Now imagine working through hundreds or thousands of equation and having to repeat this symbology over and over.",
"To simplify the notation Einstein omitted the $\\Sigma $ symbol from the sums.",
"Thus, in Einstein's notation, the sum above can be simplified to $y = x_i = x_1 + x_2 + x_3 + ...$ as the number of terms in the sum can be omitted for most practical purposes.",
"If there is a single index in the equation, say $i$ , the sum is over $i$ $y =a_i x_i = a_1 x_1 + a_2x_2 +a_2 x_3 + ...$ If there are two or more indexes, say $i$ and $j$ i, the summation is on the repeated indexes, for example [14] $y_i = a_{ij} x_j = a_{i1} x_1 + a_{i2} x_2 + a_{i3} x_3 + ...$ $\\frac{\\partial u}{\\partial x_i}\\frac{\\partial x_i}{\\partial y} = \\frac{\\partial u}{\\partial x_1}\\frac{\\partial x_1}{\\partial y}+\\frac{\\partial u}{\\partial x_2}\\frac{\\partial x_2}{\\partial y}+ \\frac{\\partial u}{\\partial x_3}\\frac{\\partial x_3}{\\partial y}+ ...$ Although Einstein's notation might take some time to get used to, it is very convenient to deal with a large number of equations, especially when dealing with tensors, and it will be used extensively in this material.",
"As a final warning to the reader, if most or all of the content in this chapter is new to you, this material is far from enough for you to learn the subject as it condenses material that is usually seen in three one semester undergraduate calculus courses.",
"You should study and review the subject on your own if you wish to be proficient in porous media physics.",
"If you are learning soil physics for the first time as a soil scientist, agronomist or other life sciences field you should use this chapter to become familiar with the notation and language used in this book and you can consult it whenever necessity arises throughout this book.",
"ll $ T $ Temperature $ f $ Generalized function $ T(t) $ Time dependent temperature $ T_0 $ Temperature where the function is centered $ \\alpha $ Amplitude $ \\omega $ Angular frequency $ \\phi $ Phase lag $ t $ Time $ Y $ Height $ Y_0 $ Initial height $ v_0 $ Initial velocity $ g $ Earth's gravitational acceleration $ k, c, C, C_1, C_2 $ Constants $ n $ Integer number $ x, y, v $ Generalized position and velocity scalar variables $ \\dot{x} $ First derivative of position, or velocity $ \\ddot{x} $ Second derivative of position, or acceleration $a$ $ F $ Generalized function $ F(x) $ Generalized function of x $ f(t) $ Generalized function of t $ f(x), g(x) $ Generalized functions of x $ J $ Area under a curve $ \\vec{F} $ Generalized vector function $ F_x, F_y, F_z $ Components of $\\vec{F}$ in Cartesian coordinates $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k}, \\hat{i}, \\hat{j}, \\hat{k}, \\hat{x}, \\hat{y}, \\hat{z}, $ Different representations of unit vectors in Cartesian coordinates $ \\vec{v} $ Velocity vector $ v_x, v_y, v_z $ Components of $\\vec{v}$ in Cartesian coordinates $ u, v, w $ Alternative representation of components of $\\vec{v}$ in Cartesian coordinates $ \\mathbf {a}, \\mathbf {b}, \\mathbf {A}, \\mathbf {B}, \\mathbf {V} $ Generalized vectors $ c, d $ Generalized scalars $ a_x, a_y $ Cartesian components of $\\mathbf {a}$ $ |\\mathbf {a}| $ Norm of $\\mathbf {a}$ $ |\\mathbf {b}| $ Norm of $\\mathbf {b}$ $ d $ Univariate derivative operator $ \\partial $ Partial derivative operator $ \\int $ Integral operator $ \\sum $ Summation operator $ \\nabla $ Gradient $ \\nabla \\cdot $ Divergence $ \\nabla \\times $ Curl $ \\nabla ^2 $ Laplacian $ \\mathbf {n} $ Unit normal vector $ \\mathbf {r} $ Generalized position vector $ A $ Area element $ V $ Volume element $ \\sigma $ Surface area element The first requirement for a material to be considered a porous media is obviously that it is composed of a solid but not necessarily rigid matrix that is not continuous in space such that the spaces between the solid fractions are pores.",
"Right now we are considering those pores as empty spaces, disregarding the possibility that they can be filled by a material of a different nature than that of the solid matrix.",
"There is a large amount of natural and artificial porous materials on Earth and by a large extent on the universe.",
"A common bath or kitchen sponge is a perfect example of a porous material whose matrix is not rigid and not granular.",
"These set of distinctions regarding the mechanical nature of the matrix, i.e.",
"rigid, elastic, plastic, porous, etc., and the nature of the pores, i.e.",
"connected or non-connected, is fundamental for the understanding of the physical properties of the media and we will talk some of them throughout this book.",
"Natural porous media include tissues from animals, vegetables, fungi and a range of other organisms, while artificial porous media of significance for large scale transport phenomena in nature include concrete, asphalt and other building materials.",
"A lot of the theory discussed in this book might apply to these artificial materials, especially regarding water and contaminant transport and storage.",
"Asphalt and concrete used for pavement can under certain circumstances be considered low porosity and low permeability porous media and thus control the infiltration of water in urban environments and roads.",
"The understanding of water storage and dynamics in foods, animal feed, grains, paper, and a variety of material is fundamental for their conservation and theories regarding water storage and determination in soils and rocks can be readily applied to food products.",
"These materials are however beyond the scope of this book.",
"We are interested here in three types of natural porous materials that cover immense extensions in both area and volume on Earth and are the basis of water and chemical elements cycles, water storage, agriculture, and environmental impacting activities, the materials we are concerned are soils, sediments and rocks.",
"As with the last chapter a true introduction to these topics would require several one semester courses covering each of the subjects, including intro soil science and geology courses, soil genesis, mineralogy (both soil and rock), igneous and metamorphic petrology, and sedimentology and stratigraphy courses.",
"Thus, we can only give a brief introduction and provide the reader with a terminology to be used in the rest of the book.",
"While the last chapter might be superficial to the student of mathematics and physics and might be too advanced to the life sciences student, in this chapter the opposite is true, the challenge here is to present a palatable introduction on the subjects of soil science and geology for those who might not have had any introduction to the subject, as with the last chapter this is not without the risk of displeasing both groups.",
"There are different types of porous media, as was mentioned before a common kitchen sponge is an example of a nonrigid and nongranular porous media.",
"This means that the geometry and volume of the pores might not be constant as the material is wetted and dried, even excluding here the possibility that the sponge is compressed by the hand.",
"Because water is a polar, almost non-compressible media, it can apply forces to the sponge surfaces that might cause it to deform.",
"In the saturated case, water can transmit forces and stresses on the material that holds it, and as it dries the water molecules exert capillary forces which might cause surfaces to come close together.",
"Soils and sediments are a class of porous materials which are both granular and nonrigid, meaning that the grains can be reorganized in response to applied stresses, altering pore volume and pore sizes distribution.",
"Whether you subscribe to a philosophical view of an objective reality or not, this entire book deals with an idealized representation of what a porous medium is.",
"Throughout it we will make a series of simplifications and assumptions in order to be create a conceptual and physical representation of porous materials that is suitable for mathematical treatment.",
"A representation is a simplification of a material which is often infinitely complex and impossible to be represented mathematically if every shape, every composition, every topological aspect is uniquely considered.",
"One of the fundamental assumptions that lie at the core of solid and fluid mechanics and that is inherited into soil physics is the continuum hypothesis.",
"Water, air, soils and rocks are composed of discrete particles.",
"In the case of water, these particles are water molecules that could be further discretized into hydrogen and oxygen atoms which could be further discretized and so onElementary particles physics and quantum effects and uncertainty will not be dealt in this book unless in brief explanatory notes such as this one.",
"For the most part, and maybe for the time being, soil physics can be entirely formulated within a classical mechanics framework.. Soils, rocks and sedimentary deposits are for the most part composed of minerals, minerals on the other hand are composed of atoms occupying positions in a regular lattice.",
"On a nanoscopic scales, if we measured a cross sectional area of a still frame of water or gas flowing or of a mineral and calculate the areal occupation of space we could fall into a molecule or atom, empty space or a boundary between the two.",
"For most materials the areal density at this scale would be highly irregular depending on where the area is evaluated.",
"As we increase the measured area, moving into macroscopic measured areas, the areal density will tend to a representation of the average areal occupation of the material (Figure REF ).",
"The principle holds if we consider a given property being measured over a volume of the material.",
"The minimum volume in which the measured property is a representation of the statistical properties of the macroscopic area is the representative elementary volume (REV) and corresponds to, by analogy, the approximate point on the plot in Figure REF where the line oscillations are negligible.",
"Figure: Illustration of the Representative Elementary Volume (REV) concept.",
"AA is the measured area and the circles represent solid particles.",
"As the measured area increases, the areal density will tend to the average of the system.Compared to a pure liquid or gas, a porous medium is a much more complicated system as we often have multiple phases interacting.",
"Each phase can be treated as a continuous medium on its own which interacts with the surrounding continuous phases.",
"For some properties the entire system can be treated as a single continuous medium, including all phases.",
"Often, when dealing with flow problems, fluids and gases are treated as dynamical phases while the solid matrix is treated as static, and the overall flow properties are defined by the interactions between the phases, while for measuring properties such as bulk mass, density, or temperature at a given time, for example, the individual composition of each phase might be neglected and the system might be treated as a whole.",
"A system can be closed, isolated or open, closed systems do not interact with their surroundings, and there is no exchange of matter or energy, in closed systems there might be exchange of heat or other forms of energy through the walls of the system and its surroundings and in open systems there might be exchange of matter and energy of the system and its surroundings.",
"Depending on the scenario, any representation can apply to natural porous media.",
"When modeling transport phenomena, the system is usually considered open, such open system are sometimes understood to be analogous to the concept of control volume in engineering.",
"We will use the traditional definitions of homogeneity and isotropy to classify porous media as a system.",
"First a homogeneous system can be defined as one with only one component and one phase, and where its physical properties do not vary with space.",
"Pure water in liquid state at 25 $^{o}$ C is a homogeneous substance.",
"Of course, achieving a complete state of purity in water might not be feasible, and defining a completely homogeneous density and temperature system for water might also not be realistic, for one because of the constraints of the continuum hypothesis discussed before.",
"But this would lead us back to the philosophical argument which, if explored to the full extent, might making writing this book impossible.",
"So we will established that a substance that can be thought as composed of only one constituent and a single phase is homogeneous.",
"Substances composed of two or more phases and/or two or more separate constituents are heterogeneous.",
"A mixture of water and ice is heterogeneous while a mixture of water and salt is homogeneous if temperature, density etc.",
"do not vary within the system.",
"Soils, rocks and sediments are invariable heterogeneous systems, although it is often useful to assume that soils where the physical in which the properties are at near equilibrium everywhere are homogeneous systems.",
"Even dry sand composed only of quartz grains might be thought as heterogeneous because it is a mixture of quartz and air.",
"In nature, most soils are heterogeneous mixtures composed of various minerals, amorphous inorganic compounds, organic matter, air and a solution composed of water and various dissolved substances.",
"In later chapters we will discuss homogeneity in respect to dynamical physical properties, but that will be in a different context.",
"A porous media can therefore be thought as a system which can be subdivided into subsystems.",
"For many purposes these systems can be modeled using thermodynamics.",
"Classical thermodynamics is concerned with the macroscopic properties of matter.",
"In this sense it usually deals with properties that represent an average of microscopic systems.",
"The bridge between classical thermodynamics and the microscopic world is given by the discipline of statistical mechanics.",
"Many derivations in fluid mechanics and soil physics are grounded in classical thermodynamics.",
"The entire theory of water retention in soils is derived from classical thermodynamics, although we are led to believe that it could be derived from first principles from statistical mechanics.",
"Classical thermodynamics does not take into account quantum effects which can be studied in statistical mechanics.",
"On a side note, general relativity can also be accounted in fluid mechanics, especially when dealing with gas ejects in astrophysics, in the discipline of relativistic fluid mechanics.",
"However, water and other fluids in porous media in Earth's gravitational fields, move slowly enough and are studied in scales large enough that classical mechanics and thermodynamics usually suffice.",
"You could however use the theories presented in this book to study fluid flow in Mars or other planets, as long as the conditions of scale and velocity hold.",
"In thermodynamicsClassical thermodynamics if not specified otherwise.",
"simple systems can be macroscopically characterized by extensive properties such as internal energy, volume and number of moles.",
"What is meant by extensive variable is that if the system is subdivided into subsystems, the magnitude of the extensive variable into each subsystem is less than the total.",
"Extensive variables depend on the extension of the system.",
"Intensive variables can be defined as ratios of extensive variables and do not vary with extension or location where the measurements are taken in an homogeneous system in equilibrium.",
"Thus, temperature can be defined as the partial derivative of internal energy, $U$ , with respect to entropy, $S$ , while volume, $V$ , and number of moles, N$_r$ , of dissolved substances are held constant, represented as $T = \\left( \\frac{\\partial U}{\\partial S} \\right)_{V, N_1, ..., N_r}$ The others being pressure, $P$ , defined as the partial derivative of internal energy with respect to volume, V, with S and N$_r$ constant, and the electrochemical potential of a given component, defined as the internal energy divided by the number of moles of the component, N$_j$ , while keeping the number of moles of other components, N$_r$ , constant, along with $S$ and $V$ .",
"$P = -\\left( \\frac{\\partial U}{\\partial V} \\right)_{S, N_1, ..., N_r}$ $\\mu _j = -\\left( \\frac{\\partial U}{\\partial N_j} \\right)_{S, N_1, ..., N_r}$ We can generalize the concept of intensive and extensive variable to commonly measured properties of materials, such as mass and volume, generating a number of well known intensive properties, the most common of which probably being density, represented in this book by the Greek letter $\\rho $ (Appendix A), defined by the mass of a substance, $M$ , divided by its volume, $V$ $\\rho = \\frac{M}{V}$ It is often convenient to define an area density as some extensive property divided by the area (as we did when explaining the REV concept), and a linear density as some extensive property divided by length.",
"For example, a thin wire can have its linear density calculated by dividing its weight by its length.",
"Measurements and calculations of physical systems are often linked to some system of units.",
"The most common systems are the International System of Units (SI), the CGS (centimeter-gram-seconds) and to some extent the British Imperial Units.",
"In this book, whenever possible we will adhere to the international system of units.",
"The SI uses seven base units, the second (s) for time, the meter (m) for length, the kilogram (kg) for mass, the ampere (A) for electric current, the kelvin (K) for temperature, the mole (mol) for amount of substance and the candela (cd) for luminous intensity.",
"A number of derived units can be defined from the SI, for example the newton (N = kg m s$^{-2}$ ) for force, the pascal (Pa = kg m$^{-1}$ s$^{-2}$ ) for pressure and stress and the joule (J = kg m$^{2}$ s$^{-2}$ ) for energy, work and heat.",
"It is convenient, however, for many derivations and for checking consistency or units, to express the dimensions of physical quantities independent of a system of units.",
"For elegance, in this book, whenever dimensions are considered when expressing a physical quantity, square brackets [] will be used.",
"The elementary dimensions and symbols most often used in soil physics and groundwater hydrology are length [L], mass [M], time [T] and absolute temperature [$\\Theta $ ].",
"Therefore, using the base length dimension L, length is represented as [L], area [L$^2$ ] and volume [L$^3$ ].",
"The consistence of dimensions and units can be calculated for a physical quantity.",
"For pressure for example we have $P = \\frac{\\text{force}}{\\text{area}} = \\frac{N}{m^2} = \\frac{kg~m~s^{-2}}{m^2} = \\frac{kg~m}{m^2~s^{2}} = \\frac{kg}{m~s^{2}} = \\text{Pa}$ Alternatively, in terms of dimensions $P = \\frac{[M~L~T^{-2}]}{[L^2]} = \\frac{[M~L]}{[L^2 T^{2}]} = \\frac{[M]}{[L~T^{2}]} = [M L^{-1} T^{-2}]$ The last two terms on the right being representations of the same thing, any of which can be chosen for convenience.",
"A range of intensive properties are defined in soil physics, soil mechanics and geology which are useful to characterize porous materials.",
"We will, for the most part, use the formulas and definitions encountered in soil physics but several are analogous or are simple adaptations from soil mechanics or chemistry.",
"Dry bulk densitybecause wet bulk density is rarely considered in soil physics, the term “bulk density\" often refers to “dry bulk density\".",
"is the mass of dried material over volume.",
"Dry bulk density can be measured by collecting an undisturbed sample of the material, by using a coring device and measuring the overall volume of the core.",
"The sample is usually oven dried at 105 $^{o}$ C for 24h to 48h to determine its dried mass be weighing.",
"The measured volume includes pores and therefore, the more porous the material is the lower its bulk density is.",
"$\\rho _b = \\frac{m_s}{V_t}$ In which $\\rho _b$ is the dry bulk density in [M L$^{-3}$ ], $m_s$ is the dry soil mass [M] and V$_t$ is the total volume [L$^3$ ].",
"Bulk density can be modified by artificial compaction or consolidation due to applied loads or by natural processes such as pore filling with finer particles or diagenetic precipitation or consolidation in sedimentary deposits.",
"Due to low natural bulk densities soils can be easily compacted by agricultural machinery, grazing, a process which is directly linked to soil degradation.",
"Bulk density is determined by porosity, grain size and mineralogy of the solid particles.",
"In most soils, bulk densities ranges from less than 1000 $kg~m^3$ in oxidic and organic soils under natural conditions to more than 1300 $kg~m^3$ in sandy or severely compacted clay soils.",
"In sedimentary deposits the density will depend on factors such as mineralogical composition, age of deposition and depth below overburden layers.",
"Particle density is analogous to the “true\" density of the material, considering only the mass and volume occupied by the solid particles.",
"For accurate measurement of particle density, accurate measurement of solid particles volume is needed.",
"This is usually achieved by liquid or gas displacement methods, where the volume occupied by solid particles is equivalent to the displaced volume of liquid or gas at a given temperature.",
"$\\rho _s = \\frac{m_s}{V_s}$ In which $\\rho _s$ is the (solid) particle density in [M L$^{-3}$ ], $m_s$ is the dry soil mass [M] and V$_p$ is the volume occupied by solid particles [L$^3$ ].",
"Particle density represents a physical averaging of the individual densities of the components of the solid phase.",
"Soil and sediment particle density is often assumed as 2650 kg m$^{-3}$ , the density of the mineral quartz, which is abundant in many soils and rocks.",
"However the density can be lower in materials rich in organic particles and higher in materials rich in certain heavy iron and titanium oxides.",
"The volumetric porosity is the volume of pores per unit total volume of soil including solids and pores.",
"It is the fraction of the material that is occupied by voids.",
"In this calculation we are not concerned if the voids are occupied by gas or water.",
"$\\phi = \\frac{V_v}{V_t}$ In which $\\phi $ is the total porosity [L$^{3}$ L$^{-3}$ ] (meaning volume of pores per total volume and thus the units are often not canceled), $V_v$ volume of voids, or pores [L$^3$ ] and V$_t$ is the total volume of the porous media [L$^3$ ].",
"Because it might be difficult to accurately measure the volume of voids in soil and as particle and bulk densities are routinely measured, it is convenient to calculate the total porosity as $\\phi = 1 - \\frac{\\rho _b}{\\rho _b}$ Equation 2.10 can be obtained by isolating the volumes in Equations 2.7 and 2.8, replacing into Equation 2.9 and assuming that the dry soil masses are the same.",
"Bulk density and total porosity are sensitive to compaction, pore clogging and alterations in pore structure, either anthropogenic or caused by natural processes.",
"Traffic by machinery, humans and animals and loads by buildings and roads can directly affect bulk density and porosity, which will in turn affect fluxes of matter and energy in porous materials.",
"Particle distribution, shape and sorting can also affect bulk density and porosity.",
"Density and porosity are important parts of the characterization of the solid matrix and pore space.",
"Another important property used for the characterization of porous materials is the specific surface area (SSA).",
"$SSA = \\frac{A_s}{m_s}$ In which $A_s$ is the surface area of the particles and $m_s$ is the mass of particles.",
"The specific surface area is in [L$^{2}$ M$^{-1}$ ] and represents the surface area of solid particles per unit mass of particles.",
"It is usually expressed in $m^{2}~g^{-1}$ or $m^{2}~kg^{-1}$ , although the former is usually preferred because it results in smaller numbers.",
"Surface area varies with grain size, roundness and surface rugosity, and with granulometric distribution of a material, and materials composed of finer particles will usually have larger specific surfaces areas.",
"Surface area influences the reactivity of the material to water and contaminant retention via adsorption and other mechanisms and in the natural accumulation of organic matter.",
"Like many other properties discussed in this book, SSA is an operational concept that depends on the method used for its determination.",
"It can be determined by adsorption of nitrogen, water, ethylene glycol monoethyl ether (EGME) and other substances in vapor phase, and values in the range of 1 to 200 $m^{2}~g^{-1}$ can be found in soils depending on grain size distribution [61].",
"The most common method found in the literature is nitrogen ($N_2$ ) adsorption using BETFrom the authors of the study that presented the adsorption isotherms used Brunauer, Emmett and Teller method.",
"However, BET could underestimate SSA in certain materials because the $N_2$ molecule might be too large to enter interlayer spaces and some very small pores [61].",
"Equally important in soil physics in the quantification of the the amount and volume of the fraction occupied by fluid.",
"In theory the fraction occupied by different fluids can be calculated in a similar fashion.",
"Fluids of interest in porous media physics are, in addition to water, air, natural gas, petroleum, and a wide range of contaminants such as DNAPLDense non-aqueous phase liquids, LNAPLLight non-aqueous phase liquids, industrial oils and a range of compounds dissolved in water at various concentrations.",
"We will use the symbol $\\theta $ for representing water mass and volume content in porous media because it has been traditionally used in soil physics, but the calculation can be extended to other dense fluids, and to some extent to gases in the case of volumetric saturation.",
"The gravimetric water content is the mass of water per mass of dry soil or sediment material.",
"It is usually measured by weighing (gravimetrically) a portion of material before and after drying in an oven at 105 $^{o}$ C for 24h to 48h.",
"$\\theta _g = \\frac{m_w}{m_s}$ In which $\\theta _g$ is the gravimetric water content [M M$^{-1}$ ], meaning mass of water per mass of dry soil or sediment and thus the units are often not canceled, m$_a$ is the mass of water [M] and m$_s$ is mass of dry soil or sediment [M].",
"Gravimetric water content in excess of 1 kg kg$^{-1}$ is possible for highly porous materials such as volcanic ash and materials rich in organic matter.",
"Because gravimetric water content does not take into consideration the volume available for water storage and total pore space, it is much more useful to define a volumetric water content, or the volume of water per unit volume of porous media $\\theta = \\frac{V_w}{V_t}$ In which $\\theta $ is the volumetric water content [L$^{3}$ L$^{-3}$ ], meaning volume of water divided by total volume of soil or sediment and thus the units are often not canceled, V$_a$ volume of water [L$^{3}$ ] and V$_t$ is the total volume of dry soil or sediment [L$^{3}$ ], including pores.",
"Defining the density of water as $\\rho _w = \\frac{m_w}{V_w}$ and isolating the volumes in Equations 2.8 and 2.14 and replacing into Equation 2.13 we have $\\theta = \\frac{m_w/\\rho _w}{m_s/\\rho _b} = \\frac{\\rho _b}{\\rho _w} \\frac{m_w}{m_s}$ which with Equation 2.9 can be written as $\\theta = \\frac{\\rho _b}{\\rho _w} \\theta _g$ For simplicity, water content is usually calculated from this equation because bulk density and gravimetric water content are routinely measured in laboratory.",
"The density of water can be calculated from temperature, as we will see in later chaptersAlthough it is often assumed as 1 g cm$^{-3}$ or 1000 kg m$^{-3}$ , depending on the unit system adopted.. Another quantity seen in soil physics, soil mechanics and petroleum engineering is the degree of saturation.",
"The degree of saturation is the volume of water divided by the volume of voids $S_w = \\frac{V_w}{V_v} = \\frac{\\theta }{\\phi }$ The degree of saturation can theoretically vary from 0 in a dry media to 1 in a completely water saturated media and it is usually represented as a dimensionless number.",
"Another quantity used in soil mechanics is the void ratio $e = \\frac{V_v}{V_s}$ In which V$_s$ is the volume of solids [L$^{3}$ ].",
"It is left to the reader to prove the following relationships $e = \\frac{\\phi }{1 - \\phi }$ $\\phi = \\frac{e}{1 + e}$ The void ratio is used in geotechnical engineering to evaluate volume changing in foundations under loading.",
"ll $ T $ Temperature $ U $ Internal energy $ S $ Entropy $ N_r $ Number of moles of the rth substance $ P $ Pressure $ V$ Volume $ \\mu _j$ Chemical potential of the jth substance $ \\rho $ Generalized density $ M$ Generalized mass $ [L]$ Generalized length unit $ [M]$ Generalized mass unit $ [T]$ Generalized time unit $ [\\Theta ]$ Generalized temperature unit $ \\rho _b$ Dry bulk density $ m_s$ Dry soil (or sediment) mass $ V_t$ Total volume $ \\rho _s$ Solid particles density $ V_s$ Solid particles volume $ \\phi $ Volumetric porosity $ V_v$ Volume of voids (pores) $ SSA$ Specific surface area $ A_s$ Surface area $ \\theta _g$ Gravimetric water content $ m_w$ Mass of water $ \\theta $ Volumetric water content $ \\rho _w$ Water density $ V_w$ Volume of water $ S_w$ Degree of saturation $ e$ Void ratio Conceptually, a soil is a material that has been subjected to the process, or set of processes, known as pedogenesis.",
"Large fractions of Earth's crust is composed of rocks, these rocks are usually formed under intense pressure and/or temperature conditions.",
"Rocks are divided into three main types igneous, metamorphic and sedimentary.",
"Igneous rocks are formed when magma is cooled, either within Earth's crust or when it flows to the surface of Earth in the form of lava.",
"Metamorphic rocks are formed when any type of rock is subjected to intense pressure and/or temperature conditions within the crust such that minerals are subjected to mineralogical and petrological transformations without complete fusion of all minerals of the rock at high temperatures.",
"Sedimentary rocks are those formed by sediments that are usually transported to an accumulation area and then subjected to pressure and temperature within ranges of values usually much lower than those required for the formation of metamorphic rocks, finally transforming into a sedimentary rock by a set of processes called diagenesis.",
"Because those rocks are formed at high temperatures and pressures when compared to those at Earth's surface they tend to be thermodynamically unstable when exposed to lower pressures, temperatures and chemical environment close to the surface.",
"In addition rocks near surface are often exposed to rainwater, yearly and daily temperature variation and a variety of compounds and forces produced by organisms in the interface with the atmosphere.",
"The transformations related to the conditions at Earth's surface caused by all these processes is called weathering.",
"At the right conditions, weathering can evolve into pedogenesis and the minerals present in the rocks can be altered or reprecipitated into forms stable at atmospheric conditions.",
"While rocks can be composed of a range of minerals, usually dependent on rock genesis, soils usually concentrate phylossilicate clays, iron and aluminum oxides and tectossilicates such as quartz and feldspars, depending on weathering conditions.",
"These forms tend to be much more stable in soil environment than other minerals found in rocks.",
"Of course soils are a complex material with a large variety of minerals of different species, but most soils will have a predominance of oxides, phylossilicate clays and quartz in different proportions in addition to the organic phase.",
"Because soils and sedimentary deposits and to some extent partially weathered rocks of all types tend to be more porous, compressible and with an interconnected pore network than other rock types, the material presented in this book tends to apply more directly to these materials.",
"However, fractures and pores of different geometries can be found in other types of rocks and with the necessary theoretical constraints, the models can be applied to these materials as well.",
"Soils and sedimentary deposits, in particular, are unconsolidated media and therefore, the individuals particles and aggregates can be easily disturbed by a series of processes such as compaction, penetration of roots and organisms, traffic, and many others, while rocks are consolidated materials being able to withstand mechanical stresses of larger magnitude.",
"The degree of consolidation of rocks varies with genesis and mineralogical composition, but it is safe to say that most rocks under normal conditions will act as a rigid matrix when considering flow and transport phenomena.",
"We mentioned that soils are formed from weathering and pedogenesis of rocks due to the minerals present in rocks being thermodynamically unstable in Earth's superficial environment.",
"Technically the process is a lot more complex, soils can not only form from rocks but also from sediments and from previously existing soils, and under influence of anthropogenic activities such as deposition of matter and superficial disturbance.",
"A given soil is in equilibrium with its environment, including climate, natural vegetation and geomorphological setting.",
"If any of these factors is significantly altered a soil can evolve, devolve, be complete buried or completely destroyed by erosional processes over time.",
"Likewise, sediment and rock formation is very complex and dependent on several geological and geomorphological processes and on factors such as Earth's internal dynamic and climate.",
"Because soil, rock and sediment forming processes are very complex and covered in different fields of soil science, geology and physical geography, discussing each one of them is beyond the scope of this book.",
"An important issue in natural porous media physics is the composition of solid phase where mechanical and transport phenomena are being considered.",
"Rocks, sedimentary deposits and soils of interest are largely composed of minerals.",
"A mineral is a compound that has a more or less constant chemical composition between samples of the same mineral, has an internal structure that is organized and is formed naturally in Earth or other astronomical objects.",
"A rock is composed of one or more minerals and is consolidated, i.e.",
"cannot be easily disassembled by hand or using manual tools such as spades or a geological hammer.",
"From the soil physics point of view, soils are much more complicated materials than rocks because they are very heterogeneous in composition and are not consolidated, being easily deformed and disaggregated with much less effort than that needed for most rocks.",
"Soils possess varying grain size distributions, proportions of inorganic mineral and amorphous materials, organic phases, and gas and solution.",
"Large variability in the composition of the mineral phase can be found depending on climate, parent material and other soil forming processes.",
"The other factors that makes soils so complex is that the colloidal solid phase is most often electrically charged, mainly due to inorganic and organic superficial groups and isomorphic substitution in crystalline lattices.",
"Because of that, the small particles have the ability to interact with charged ions in solution.",
"Modeling contaminant retention and transport in soils can be much more challenging in soils than in sedimentary rocks composed of a single coarse grained mineral phase, such as in sandstones and sand deposits, where the density of charges is negligible and the specific surface area is very small compared to that of most soils.",
"Because soils are so complex, they cannot be fully characterized and studied in detail in laboratory or in lattice, physicochemical or thermodynamic simulations as geologists do with individual minerals and simple rocks and physicists do with synthetic crystals.",
"Although soil physics is traditionally constrained to the domains of fluid phenomena and soil mechanics, much can be done by using the same approach and techniques used by condensed matter physicists.",
"The fact that the study and characterization of soils has been almost entirely relegated to soil science and engineering during most of its history reflects the fact that the complexity and heterogeneity of the material prevents, for the most part, the physicist's approach which is often concerned with materials that are simple enough that they can be investigated and modeled using available laboratory and mathematical methods.",
"With appropriate simplifications and with modern physical methods, soils can be studied in the framework of condensed matter physics.",
"Although we mentioned that soils are more complex from the physical point of view than most rocks and sedimentary deposits, there are geological settings where the line between soils, rocks and sediments is blurred, either because of a physical mixing of different materials due to pedogenetic and geological conditions and/or because a system were mechanical stresses and transport phenomena is considered has interfaces between different materials.",
"One example of a material intermediary between soils and rocks is the saprolite which is a partially weathered rock theoretically in its way to become a soil.",
"Another example is sedimentary rocks and deposits composed of phylossilicate clays.",
"These materials can result in chemical behavior when interacting with contaminants more closely related to soils than to inert materials.",
"It is common in natural environments that soils occupy the most superficial layers in Earth's crust followed by saprolite, slightly weathered or fragmented or fractured rock followed by consolidated rock.",
"A dissolved contaminant, for example, will have to interact with soil and saprolite before it reaches deeper saturated layers in porous rock, which can compose aquifers where water is being extracted.",
"Ultimately the mineralogical composition of soils, rocks and saprolite will influence, if not control, the interaction with contaminants, water transport and storage properties as well as mechanical behavior.",
"Other than inorganic materials, organic deposits, which constitute some soils and geological deposits, present an unique challenge from the physics and engineering point of view, as they tend to be very loose, compressible, can be easily lost due to decomposition, and can be very reactive when it comes to interacting with chemicals, depending on the nature of the organic fractions involved.",
"Water retention and transport is particularly challenging to be modeled in these materials as the tend to have very low densities, can be very porous and can absorb water on a content greater than the solid phase on a mass base.",
"We will focus on materials that are mostly inorganic throughout this book, as these are materials comprise the Earth's crust upper layer in most of the planet, while organic deposits are restricted to certain geological and pedogenetic settings.",
"One of the disciplines in which condensed matter physics and mineralogy intersect is crystallography.",
"Most of us have an intuitive knowledge of what a crystal is, it is usually associated with “rock crystals” such as quartz.",
"Early physicists or natural philosophers observed that many minerals had an external shape that was more or less constant between different samples of the same mineral [44].",
"They speculated that the external shape is the result of the internal organization or the packing of minuscule particles which compose matter.",
"Later, these minuscule particles would be recognized as atoms and the internal organization of these atoms would be experimentally confirmed by x-ray diffraction [18].",
"You might have noticed that most varieties of quartz that people collect are formed by elongated crystals.",
"If you count the number of faces in these crystals and measure the internal angle between faces you will find six and approximately 120$^{o}$ .",
"This is reflect of the internal atomic structure of quartz, which in ideal conditions is composed entirely of silicon and oxygen atoms.",
"Crystals are then materials that have a highly organized internal structure which repeats in space, and crystallography is the branch of science dedicated to the study of crystals.",
"Materials which have a random or non-organized arrangement of atoms are called amorphous.",
"Crystals can be synthesized in laboratory or can be formed naturally such as in soil and rock forming processes where atoms reorganized in response to a change in thermodynamic conditions.",
"These inorganic crystals that are formed under natural conditions and which can be identified as having a more or less constant chemical composition between different species and whose internal arrangement and physical properties make it possible to recognized as a distinct species from other crystals in the environment are called minerals.",
"Minerals in rocks originate from rock forming processes or they can be inherited from a different rock when they pass through the rock cycle.",
"Rocks are primarily classified based on origin as igneous, sedimentary and metamorphic.",
"Igneous rocks are formed from cooling of magma within Earth or lava in Earth's surface.",
"As the materials cools, bonds are formed between atoms and the structure organizes itself based on the affinity between atoms, remaining concentration and atomic radius relationships.",
"The issue of radius ratio is usually modeled using the sphere models presented Chapter REF , but for atoms instead of mineral particles or aggregates as in that case.",
"Because igneous rocks form from magma or lava which are molten materials, the minerals are newly formed when the rock finishes cooling.",
"This is not always the case in other types of rocks.",
"Sedimentary rocks are formed from erosion, transport, deposition and consolidation of sediments which came from another source, usually either a rock that was weathered or a soil or sedimentary deposit which was eroded.",
"In reality, sedimentary rocks can be also formed from precipitation of ions in solution with or without the influence of organisms.",
"Sedimentary rocks can be composed of minerals that were formed in other rocks or other conditions and were weathered and transported forming a new rock via consolidation without altering the chemical composition and crystalline structure or by minerals that were formed during diagenesis.",
"Metamorphic rocks are formed by solid state transformations in which part or all of the minerals in a rock of any type are transformed into new minerals by the action of pressure and/or temperature at greater intensities than those that occur during the genesis of sedimentary rocks.",
"During formation of metamorphic rocks there can be re-composition and/or reorganization of the internal structure of minerals, thus forming new species.",
"Partial fusion of some minerals is allowed, but if the rock is completely molten and later recrystallized it would be classified as an igneous rock.",
"Rocks are composed of minerals, there are rocks composed of only one mineral species, such as some marbles and sandstones and rocks composed of more than one type of mineral such as basalts and granites.",
"There are minerals that are more commonly found in each type of rock and this is used in the classification and identification of rocks.",
"The chemical composition of the most common minerals reflects the composition of Earth's continental crust in which oxygen, silicon, aluminum, iron, calcium, sodium, potassium and magnesium account for approximately 98% of Earth's crust on a mass basis.",
"Oxygen and silicon alone account for more than 70% of the mass.",
"Minerals are classified based on the chemical composition usually represented as a chemical formula.",
"Although some minerals are composed by a single element bonded via ionic or covalent bond, diamond (C), graphite (C), sulfur (S), gold (Au) being a few examples, most minerals of interest in rocks, sediments and soils are composed of one or more cations bonded to an anion or anionic group.",
"These cations and anions will bond to form stable configurations based on valence, affinity, abundance and radius ratio and according to thermodynamic laws.",
"The two most common elements on Earth's crust are an anion and a cation, oxygen ($O^{2-}$ ) and silicon ($Si^{4+}$ ).",
"Based on this configuration one would expect that a mineral with formula $SiO_2$ with one silicon bonding with two oxygens would be a stable configuration.",
"Although $SiO_2$ is the formula of a common silicate mineral, the radius ratio precludes the coordination of only two oxygen atoms around the silicon.",
"In reality, each silicon atom coordinates four oxygen atoms and shares one electron with each oxygen via a predominately covalent bond.",
"This structure with one silicon coordinating four oxygen atoms is represented as a polyhedron called the silicon tetrahedron (Figure REF (a)) and forms the basis of the silicate minerals which are extremely common in soils, sediments and rocks.",
"Because each oxygen has only one electron shared with the silicon atom, one more electron needs to be shared in each corner of the tetrahedron for physicochemical stability to occur.",
"In silicates the stability can be achieved by other cations forming “bridges” between tetrahedrons defining the class of nesossilicates, by sharing the apical oxygen between two tetrahedra forming a double tetrahedra in the sorossilicates, by sharing two oxygens between adjacent tetrahedra forming ring of three or six tetrahedra in the cyclossilicates, by sharing two or three oxygens forming chains in the inossilicates, by sharing the three basal oxygens forming sheets of tetrahedrons in the phyillossilicates (Figure REF (b)) or by sharing all four oxygens forming complex tridimensional networks in the tectossilicates.",
"The chemical formula in each of these classes is defined by the occurrence of $Si_nO_m$ in the formula in which $n$ and $m$ are whole numbers or fractions.",
"The class of phylossilicates is particularly important in soils and a few sedimentary rocks as these clay minerals can define or heavily influence the physical and chemical behavior or the material in which they occur, they can also have surface charges and can interact with ions in solution.",
"A few examples of silicates common in rocks are quartz ($SiO_2$ ), orthoclase feldspar ($KAlSi_3O_8$ ), plagioclase feldspars ranging from calcium the plagioclase anorthite ($NaAlSi_3O_8$ ) to the sodium plagioclase albite ($CaAl_2Si_2O_8$ ), olivine ($(Mg, Fe)_2SiO_4$ ) and various pyroxenes.",
"These minerals are commonly found in igneous rocks and can appear in sedimentary and metamorphic rocks as an inheritance from the original material which either suffered diagenesis or metamorphism.",
"Other mineral classes are found in many types of rocks usually in smaller quantities, a few examples being oxides such as zircon ($ZrSiO_4$ ), ilmenite ($FeTiO_3$ ) and rutile ($TiO_2$ ) and sulfides such as spharelite ($ZnS$ ) and galene ($PbS$ ).",
"In sedimentary rocks, besides quartz which is common in sandstones and also in sedimentary deposits, it is also relatively common to find minerals that formed in solution via precipitaion such as calcite and aragonite (both $CaCO_3$ , but with different internal structures), dolomite ($CaMg(CO_3)_2$ ), gypsum ($CaSO_4\\cdot 2H_2O$ ), anhydrite ($CaSO_4$ ) and salts such as halite ($NaCl$ ) and sylvite ($NaCl$ ).",
"These minerals are usually composed primarily by ionic bonds and tend to be less resistant to weathering.",
"Figure: Single silicon tetrahedron (a) and tetrahedral sheet (b) and sigle octahedra (c) and octahedral sheet (d) (Adapted from ).Soils are formed by weathering and pedogenesis over rocks, sedimentary deposits or either previously formed soils.",
"They can inherit minerals from the soil forming material that have gone through no discernible mineralogical transformation except size reduction and fragmentation.",
"These minerals are called primary minerals and the amount in soil will depend on parent material and climate.",
"Common primary minerals found in soils are quartz, feldspars, primary micas (a group of phylossilicates).",
"Heavy minerals such as rutile, anatase (also $TiO_2$ ) and magnetite ($Fe_3O_4$ ), tend to occur in very minor quantities in some soils.",
"More often than not new minerals are created during pedogenesis and these minerals control how the soil functions, its porosity, pore geometry, mechanical properties and how it interacts with water, gas and with solutes.",
"The minerals that are formed during pedogenesis are often referred to as secondary minerals while the minerals inherited from parent material are the primary minerals.",
"On the tropics where average annual temperatures and precipitation are elevated, at least in part of the year, soils tend to accumulate iron and aluminum oxides, oxi-hydroxides and hydroxides as secondary minerals, commonly hematite ($Fe_2O_3$ ), goethite ($FeOOH$ ) and gibbsite ($Al(OH)_3$ ), and usually referred to as a group as iron and aluminum oxides.",
"Iron oxides usually give strong red, orange and yellow hues to soils and are usually not restrictive to agricultural use of soils from a physical standpoint but can be severely restrictive from a soil fertility standpoint under natural conditions.",
"Soils rich in oxides are challenging for modeling water and contaminant transport and storage because they usually are associated with unique pore geometry and distribution and because surface hydroxi groups ($OH$ ) these minerals are subjected to protonation and deprotonation according to solution pH following the reactions [89] $M - OH + H^+ \\rightleftharpoons M - OH_2^+$ $M - OH \\rightleftharpoons M - O^- + H^+$ Thus, under acidic conditions the OH groups protonate, generating a positive charges surfaces with ability to attract anions while at higher pHs the OH groups deprotonate generating negatively charged surfaces with the ability to attract cations.",
"At an equilibrium point where there is an equilibrium between positive and negative sites the points of zero charge (PCZ) of the compound can be found.",
"In soils this behavior occurs in iron and aluminum oxides such as hematite, goethite and gibbsite where M represents iron (Fe) or aluminum (Al) metals usually on octahedral sites on the mineral structure.",
"However these variable charged sites can be found in organic compounds and in silicates to some degree.",
"Roughly speaking, the PZC of a soil represents a weighed average of PZCs of individual components.",
"Cation exchange capacity (CEC) and anion exchange capacity (AEC) in soils are defined by the predominance of negative or positive surface sites.",
"AEC is mainly observed in highly weathered oxidic soils in the tropics and can revert to CEC as soil pH increases.",
"The other major group of secondary minerals in soils are the phylossilicates or clay minerals.",
"Although these minerals can occur as primary minerals in sedimentary deposits and rocks, they occur in almost all soils in various quantities depending on parent material and pedogenetic conditions.",
"Clay minerals along with iron and aluminum oxides can be found on the clay fraction of soils while primary minerals usually occur in the silt and sand fractions.",
"Phylossilicate minerals in soils are layered or sheet minerals where there is an alternation of sheets formed by silica tetrahedrons and octahedral layers where metallic cations such as aluminum or magnesium coordinate six hydroxides forming an octahedral polyhedron of coordination (Figure REF (c)) in which some of its hydroxides are shared among adjacent octahedrons (Figure REF (d)).",
"Phylossilicate clays are classified according to the proportion of octahedral (Figure REF (a)) to tetrahedral (Figure REF (b)) layers.",
"In 2:2 layer silicates, often simplified to 1:1, two octahedral layers bound to two tetrahedral layers (Figure REF (b)), while in 4:2 layer silicates, often simplified to 2:1, two tetrahedral layers are each bound to one octahedral layer forming two pairs of 2:1 structures (Figure REF (c)).",
"The space between the pairs of sheets in the 1:1 configuration and pair of groups of sheets in the 2:1 configuration is called in the interlayer space and the distance between the groups of 1:1 or 2:1 sheets can vary according to environmental conditions in some types of minerals.",
"The most common 1:1 layer mineral in tropical soils is kaolinite ($Al_2Si_2O_5(OH)_4$ ) and because the space between layers does not vary and because it has little isomorphic substitution it has relatively low specific surface area, ion retention and exchange capacity.",
"The structure of kaolinite is presented in Figure REF illustrating the tetrahedral and octahedral sheets .",
"The distance between the basal planes of the two 1:1 layers does not vary with environmental conditions and is around 0.7 nm.",
"On the octahedral sheets some of the hydroxides are replaced with oxygen and these are shared with silicon tetrahedrons forming the mineral structure (Figure REF ).",
"Isomorphic substitution is a process that occurs during the mineral formation in which the central cation is replaced by a cation of similar atomic radios but with different valence.",
"Thus if aluminum ($Al^{3+}$ ) is replaced by iron in its reduced state ($Fe^{2+}$ ) in specific point within the octahedral sheet, a charge deficit will appear due to the resulting unbalanced negative charges in hydroxides.",
"These charges are the basis of the chemical reactivity and will control the dynamics of nutrients and contaminants in soils and other materials rich in phylossilicate clays.",
"The most common 2:1 phylossilicate minerals in soils are from the smectite group, particularly montmorillonite $((Na,Ca)_{0.33}(Al,Mg)_2(Si_4O_{10})(OH)_2 \\cdot nH_2O)$ , and vermiculite $((Mg,Fe^{2+},Fe^{3+})_3[(Al,Si)_4O_{10}](OH)_2 \\cdot 4H_2O$ ).",
"Montmorillonite is highly expansible, meaning that the interlayer space can be increase when water enters the space, has high specific surface area, is very reactive towards adsorption of water and solutes and can have shrinking swelling behavior according to soil water content, being extremely hard when dry and extremely plastic and sticky when wet.",
"Smectite's structure is illustrated in Figure REF showing the range of variation of the distance between the two 2:1 structure.",
"Isomorphic substitution on cation on the octahedral layer is common in smectites (Figure REF ) and other 2:1 minerals and because of that the surface charge and CEC of these minerals is much larger than oxides and 1:1 minerals.",
"The interlayer can be filled with water and cations under wet conditions, expanding the mineral on a microscopic scale and consequently the material on a macroscopic scale.",
"Another less common 2:1 mineral is vermiculite which is less expansive than montmorillonite.",
"Illite $ (K,H_3O)(Al,Mg,Fe)_2(Si,Al)_4O_{10}[(OH)_2,(H_2O)]$ ) can also occur in some soils in minor amounts depending on genesis and parent material, but the interlayer space in this mineral usually does not vary and it tends to be less reactive than either montmorillonite or vermiculite.",
"There are also hydroxi-interlayered minerals in which the interlayer space is occupied by octahedral sheets or natural polymers, but these are of restricted occurrence in terms of amount and representative and are of interest only in specific cases.",
"Anyone interested in soil physics or in unsaturated zone hydrogeology must be proficient in soil chemistry and mineralogy.",
"It is impossible to study these soils and sediments without understanding how the solid phase interacts with fluids and gases, these disciplines cannot be dissociated.",
"The reader is strongly recommended to consult books on the subject of soil chemistry and mineralogy [91], [36].",
"Figure: Representation of the tetrahedral (a) and octahedral (b) sheets and 1:1 (c) and 2:1 (d) layer minerals.Figure: Kaolinite structure (Adapted from ).Figure: Smectite structure (Adapted from ).Because soil secondary minerals, colloidal organic matter and some amorphous compounds are charged, an electric field around those compounds results which can attract ions in solution.",
"The liquid phase solution in this case is in the pores around solid particles and the volume occupied by it can range from complete saturation of the pores to thin films over solid surfaces.",
"Ions in solution will form a distribution based on the equilibrium of the electrostatic attractive and repulsive forces in particle surfaces and the concentration gradient in solution.",
"The interaction of the charged surfaces and ions results in a structure called Electric Double Layer (EDL).",
"The presence of an EDL and its relationship with the different types of surface sites in soil minerals and organic compounds will influence the behavior of clay particles in aqueous solution as in the case of grain size analysis and in the stability of the soils in the field.",
"The mechanisms of dispersion and coagulation play a fundamental role in soil dispersion both in laboratory and under field conditions.",
"The surface chemistry will also determine parameters for solute transport and retention in soils.",
"The solutes of interest include inorganic and organic compounds in fertilizers, polar and non-polar pesticides, viruses, bacteria and other pathogenic microorganisms, heavy metals, radionuclides and many other.",
"Solutes can also be transported attached to dispersed clays and other colloids in what is called colloid-facilitated transport.",
"The EDL can be modeled using the diffuse double layer theory, which assumes the mineral surfaces as a plane with an even distribution of charges.",
"Under these conditions, the distribution of charges at a distance $x$ from the surface plane under equilibrium conditions follows the Boltzmann distribution [6], [39] $n_i = n_{0i} \\exp {(\\frac{-z e \\psi (x)}{ kT})}$ In which $n_i$ is the ion concentration of the ith species at a distance $x$ from the surface, $\\psi (x)$ is the electrical potential at the same distance, $n_{0i}$ is the ion concentration of the ith species in bulk solution, $z$ is the ion valence, $e$ is the elementary charge, $k$ is the Boltzmann constant and $T$ is the absolute temperature.",
"The charge density at any point in the diffuse layer is given by $\\rho = \\sum z_i e n_i$ In which $i$ indicates a summation term over every ion species.",
"The variation of the electrical field strength with distance is given by a differential equation know as the Poisson equation $\\frac{d^2 \\psi }{dx^2} = - \\frac{4 \\pi }{\\epsilon } \\rho $ in which $\\epsilon $ is the absolute permittivity of the medium.",
"For symmetrical electrolytes, the Poisson equation can be expressed in terms of ion concentration and charge density by combining it with the concentration distribution and charge density equations, resulting in the Poisson-Boltzmann equation $\\frac{d^2 \\psi }{dx^2} = - \\frac{4 \\pi }{\\epsilon } \\sum z_i e n_{0i} \\exp {(-\\frac{z_i e \\psi }{kT})}$ For suspensions consisting of a single symmetrical electrolyte the summation expands to This can be achieved by considering the valence of the positively charged ions as $z_{+} = +|z|$ and the valence of the negatively charged ions as $z_{-} = -|z|$ .",
"Note that a symmetrical electrolyte has the same proportion of cations and anions, the most simple example being NaCl which dissociated to Na$^{+}$ and Cl$^{-}$ , therefore, their concentrations in solution are also equal, i.e.",
"$n_{0+} = n_{0-}$ $\\frac{d^2 \\psi }{dx^2} = - \\frac{4 \\pi |z_i| n_{0i}}{\\epsilon } [ \\exp {(-\\frac{|z_i| e \\psi }{kT})} - \\exp {(\\frac{|z_i| e \\psi }{kT})} ]$ To solve this differential equation several simplifications and manipulations are necessary.",
"The first is to define $y = \\frac{|z_i| e \\psi }{kT}$ resulting in $\\frac{d^2 y}{dy^2} = - \\frac{4 \\pi z_i^2 e^2 n_{0i}}{\\epsilon k T} [ \\exp {(-y)} - \\exp {(y)} ]$ It is also useful to expand the exponential term using Taylor series Note that $\\exp {(-y)} - \\exp {(y)}$ can also be written as $-2 \\sinh (y) $ and $\\sinh (y)$ can be written as a Taylor series expansion, as it will be used again later $\\exp {(-y)} - \\exp {(y)} = -2y - \\frac{y^3}{3} - \\frac{y^5}{60} + ...$ If the solution is dilute the Debye-Hückel $-\\frac{|z_i| e \\psi }{kT} \\ll 1$ approximation is used and higher order terms on the series can be neglected.",
"Under such conditions the expansion can be simplified to $\\exp {(-y)} - \\exp {(y)} \\approx -2y$ Replacing the expanded exponential into the original equation $\\frac{d^2 y}{dy^2} = - \\frac{4 \\pi z_i^2 e^2 n_{0i}}{\\epsilon k T} (-2y) = \\frac{8 \\pi z_i^2 e^2 n_{0i}}{\\epsilon k T} y$ We can now define $\\kappa ^2 = \\frac{8 \\pi z_i^2 e^2 n_{0i}^2}{\\epsilon k T}$ The parameter $\\kappa $ is an important parameter in surface chemistry and its inverse $\\kappa ^{-1}$ is the Debye length or distance over which the electric potential decays to 1/e e here is Euler’s number e = 2.71828....",
"The differential equation to be solved then simplifies to $\\frac{d^2 y}{dy^2} = \\kappa ^2 y$ This is a second order linear ordinary differential equation with general solution The solution to this differential equation would require a lengthy detour into differential equations theory and will be provided in future editions $y = c_1 \\exp (\\kappa x) + c_2 \\exp (-\\kappa x)$ The constants can be determined by applying the boundary conditions to the problem.",
"The first boundary condition states that the electrical potential $\\psi = 0 $ at $x = \\infty $ .",
"This implies that $y = 0$ at $x = \\infty $ $0 = c_1 \\exp (\\kappa \\infty ) + c_2 \\exp (-\\kappa \\infty ) = c_1 \\exp (\\kappa \\infty ) + 0$ $c_1 = 0$ The second is that the $\\psi = \\psi _0$ and thus $y = y_0$ at x = 0 $y_0 = 0 \\exp (\\kappa 0) + c_2 \\exp (-\\kappa 0) = c_2$ Replacing the constant on the general solution $y = y_0 \\exp (-\\kappa x)$ Therefore $\\psi = \\psi _0 \\exp (-\\kappa x)$ This is the electrical potential at the electric double layer following the Gouy-Chapman model with the Debye-Hückel approximation.",
"Notice that this solution is valid for the potential in relation to an infinite plane.",
"For different geometries and where the solute concentration is not low enough for the Debye-Hückel the solution for the Poisson-Boltzmann equation is not trivial (see for example [91] Chapter 5).",
"The potential is $\\psi _0$ at the surface and deceases exponentially as the distance $x$ from the surface increases.",
"It is a simplistic model but it is usually to explain double layer compression under high ionic strengths and the theory of surface forces balance that controls dispersion and coagulation of clays.",
"This theory plays a fundamental role in the physicochemistry of salt affected soils, erosion and sedimentation and grain size analysis.",
"Let us now consider an alternative treatments to Equation REF .",
"From hyperbolic geometry identities, the hyperbolic sin function can be written in terms of exponential functions as $\\ \\sinh y = - \\frac{\\exp {(-y)} - \\exp {(y)}}{2}$ Thus, replacing it into Equation REF results in $\\frac{d^2 y}{dx^2} = \\frac{8 \\pi z_i^2 e^2 n_{0i}}{\\epsilon k T} \\sinh y$ And from the definition of $\\kappa $ $\\frac{d^2 y}{dx^2} = \\kappa ^2 \\sinh y$ This is a nonlinear second-order differential equation and fortunately we only need to find how it reduces to a first-order equation on integration.",
"The reason will become apparent in further steps.",
"To reduce the equation to first order we multiply both sides by $dy/dx$ and integrate [14] $\\int \\frac{d^2 y}{dx^2}\\frac{d y}{dx} = \\int \\kappa ^2 \\sinh y \\frac{d y}{dx}$ Substituting $u = dy/dx$ we have $du = d^2y/dx^2$ on the left side and making $v = y$ we have $dv = d^2y/dx^2$ on the left side $\\int u du = \\int \\kappa ^2 \\sinh v dv$ which results in $\\frac{u^2}{2} = \\kappa ^2 \\cosh v + C_1$ Where $C_1$ is an integration constant.",
"Replacing u and v we have $(\\frac{d y}{dx})^2 = 2 \\kappa ^2 \\cosh y + C_2$ in which $C_2 = 2C_1$ and it does not alter the solution to the problem.",
"Applying the square root to both sides $\\frac{d y}{dx} = \\sqrt{ 2 \\kappa ^2 \\cosh y + C_2}$ Applying the boundary condition that the electrical potential and thus $y$ is zero when $x = \\infty $ i.e.",
"bulk solution and that $dy/dx = 0$ at that point, we find that $C_2 = -2\\kappa ^2$ and $\\frac{d y}{dx} = \\sqrt{ 2 \\kappa ^2 \\cosh y - 2 \\kappa ^2}$ Finally we consider another geometric identity of the hyperbolic functions, the half-angle formula $\\sinh \\frac{y}{2} = sgn(y) \\sqrt{ \\frac{\\cosh y - 1}{2}}$ In which $sgn$ is the sign function with the following properties $sgn (x)={\\left\\lbrace \\begin{array}{ll}-1 & \\text{if} ~ x < 0\\\\0 & \\text{if} ~ x = 0\\\\1 & \\text{if} ~ x > 0\\end{array}\\right.",
"}$ Considering that we are concerned with electrical potential and thus $y$ negative and rearranging and replacing the half-angle formula we have $\\frac{d y}{dx} = - 2 \\kappa \\sinh \\frac{y}{2}$ This formula to will be useful to derive an equation for net charge on the diffuse layer, $\\sigma $ .",
"The net charge is the integral of the charge density on the $x$ interval of zero to infinity.",
"$\\sigma = \\int _0^\\infty \\rho dx$ Replacing the Poisson-Boltzmann equation into the surface charge equation we have $\\sigma = \\int _0^\\infty (- \\frac{\\epsilon }{4 \\pi } \\frac{d^2 \\psi }{dx^2}) dx = - \\frac{\\epsilon }{4 \\pi } \\int _0^\\infty \\frac{d^2 \\psi }{dx^2} dx$ The integration is over the second derivative, resulting $\\sigma = -\\frac{\\epsilon }{4 \\pi } \\bigg (\\frac{d \\psi }{dx}\\bigg )\\bigg |_{x=0}^{x=\\infty }$ But the electrical potential is constant at $x=\\infty $ such that the term $d\\psi /dx$ vanishes when evaluated at infinity, resulting $\\sigma = \\frac{\\epsilon }{4 \\pi } \\bigg (\\frac{d \\psi }{dx}\\bigg )_{x=0}$ Expressing sigma in terms of $y$ $\\sigma = \\frac{\\epsilon k T }{4 \\pi |z_i| e } \\bigg (\\frac{d y}{dx}\\bigg )_{x=0}$ Now we can replace the equation for $dy/dx$ derived previously and the condition that $y = y_0$ at $x = 0$ resulting in $\\sigma = -\\frac{\\epsilon k T \\kappa }{2 \\pi |z_i| e } \\sinh \\frac{y_0}{2}$ Which in terms of electrical potential and replacing $\\kappa $ is $\\sigma = -\\frac{\\epsilon k T }{2 \\pi |z_i| e } \\sqrt{\\frac{8 \\pi z_i^2 e^2 n_{0i}^2}{\\epsilon k T}} \\sinh \\frac{\\frac{|z_i| e \\psi _0}{kT}}{2} = - \\sqrt{\\frac{2 \\epsilon k T n_{0i}^2}{\\pi }} \\sinh \\frac{|z_i| e \\psi _0}{2kT}$ The surface charge, $\\sigma _0$ has the same magnitude of the sum over the diffuse layer with opposite sign $\\sigma _0 = \\sqrt{\\frac{2 \\epsilon k T n_{0i}^2}{\\pi }} \\sinh \\frac{|z_i| e \\psi _0}{2kT}$ This is the Gouy-Chapman relationship for the surface charge density [39].",
"The charge density if a function of the surface potential and ion concentration among other factors.",
"In variable charged oxides the surface potential is not constant and varies with pH.",
"Considering small surface potential, the last equation can be simplifiedNote that the hyperbolic sin expands to $ \\sinh y = y + y^3/3!",
"+ y^5/5!",
"+ ...$ thus, for small values of $y$ higher order terms may be neglected to [39] $\\sigma _0 \\approx \\frac{\\epsilon \\kappa }{4 \\pi } \\psi _0$ and the surface potential can be determined by the activity of the potential determining ion $a_{pdi}$ in relation to its activity at the point of zero charge (pzc) $\\psi _0 \\pm \\frac{RT}{zF} \\ln {\\frac{a_{pdi}}{(a_{pdi})_{pzc}}}$ If $H^+$ is the potential determining ion as is the case in oxide surfaces, the relationship between pH and hydrogen activity is $ pH = - log[H^+]$ and the equation can be written as $\\psi _0 = \\frac{-2.303 RT}{F} (pH - pH_{pzc})$ Combining the electrical potential and surface charge equations, the following equation is valid $\\sigma _0 \\approx \\frac{-2.303 RT \\epsilon \\kappa }{4\\pi F} (pH - pH_{pzc})$ In which $F$ is the Faraday constant.",
"On a side note, the rigorous derivation above was provided by Kenneth L. Babcock one of the fathers of soil chemistry [6].",
"Much of soil chemistry interfaces with soil physics on the subjects of thermodynamics and transport and retention of solutes.",
"One of the most important soil chemists in recent history Garrison Sposito was a student of Kenneth Babcock.",
"Sposito gave fundamental contributions to soil physics no only on thermodynamics and solute chemistry but also regarding water transport and scaling of soil properties and processes.",
"We have illustrated that soils and rock differ in terms of their mineralogical composition and by the fact that rocks are consolidated materials.",
"Rocks differ from sedimentary deposits in terms of genesis and mineralogy and by the fact that soils have organic matter in their composition.",
"The amount of organic matter varies with soil type, but according to soil science criteria, mineral soils have around 5% organic matter on mass basis for average soils.",
"The concept of organic soils is a somewhat loose definition based on a more or less arbitrary limit of organic matter or carbon content [87].",
"The classification of organic soils varies according to engineering classification systems [70] and can vary depending on several specific criteria within a soil classification system used in soil science [87].",
"Because soils occupy only an thin layer of the Earth surface, usually a meter or two to a few meters in depth at the most in most cases, they are under direct influence of the weather and organisms.",
"Soils are usually rich in organisms, both macro and micro, and plants and are continually affected by their actions.",
"Soils are dynamic and porosity and structure are continually being altered by the actions of roots, ants, termites and other burrowing animals of larger size.",
"Rocks and sedimentary deposits can reach kilometers in depth on the Earth's crust and are very little affected by organisms and surface conditions in most cases.",
"The transition zone between soils and subsurface materials is called the saprolite and can reach several meters in depth under extreme weathering condition on the tropics over certain rocks.",
"Soils constitute for the most part the unsaturated zone, where the pores are saturated by both gas and water, although it is possible that the water table can be located within the soil profile.",
"Below the water table is the saturated zone where the pores are saturated with water.",
"Most aquifers are located in sedimentary deposits and to some extent porous or fractured rocks.",
"Transport and retention of water, contaminants, oil and gas in rocks and sedimentary deposits is usually the object of study of hydrogeology and petroleum engineering while soil physics is often concerned with soils and other unsaturated media.",
"However, the mathematical analytical and numerical framework between these disciplines is similar since the equations that govern transport processes in these materials have the same physical basis if they are not the same.",
"Much of the governing equations of processes of transport originated from the realm of mechanics and later chemical engineering.",
"In the next chapter we will try to provide a mathematical framework for the equations that govern transport processes in soils and other natural porous media.",
"ll $ \\psi $ Electric potential $ \\psi _0 $ Electric potential at the surface of the charged plane $ x $ Distance from the charged plane $ z $ Ion valence $ e $ Elementary charge $ k $ Boltzmann constant $ T $ Absolute temperature $ \\rho $ Charge density $ \\epsilon $ Absolute permittivity of the medium $ \\kappa ^{-1} $ Debye length $ \\text{e} $ Euler's number $ ln $ Logarithm base e $ log $ Logarithm base 10 $ c_1, c_2, C_1, C_2 $ General constants for differential equations solutions $ y $ General variable for mathematical manipulations $ y_0 $ Value of $y$ at $y = 0$ $ u, v $ Variables for integration by substitution $ sgn $ Sign function $ \\sigma $ Net charge at the diffuse layer $ \\sigma _0 $ Surface charge $ R $ Molar gas constant $ F $ Faraday constant $ a_{pdi} $ Activity of a potential determining ion (pdi) $ pH_{pzc} $ pH at the point of zero charge (pzc) Up until now we have been discussing properties of the solid matrix of porous media.",
"Some of these properties are directly affected by water content and can be altered by engineering, hydrological, agricultural as well as a wide range of natural processes, both on short and long term.",
"We have not discussed so far any type of flux of liquid of gaseous phases through porous materials.",
"Detailed discussions of mass flow through porous media are reserved to the chapters dealing with saturated and unsaturated hydrodynamics of porous media, and other transport phenomena.",
"Technically speaking, we will deal with another property of the solid matrix on the next chapter, the grain size distribution, so there should not be an immediate need for equations governing fluid flow.",
"However, the majority of methods for determination of grain size distribution of porous media rely on settling in liquid and, occasionally, gaseous materials.",
"To understand the dynamics of settling we need to understand the fundamental equations of fluid dynamics which control how settling occurs in a viscous media.",
"The knowledge discussed in this chapter will also be the basis for modeling flow problems in open conducts and to a large extent flow in porous media.",
"This chapter will introduce you to the beauty and brutality that is the mathematics involved in fluid mechanics and in theoretical soil physics.",
"If you are already familiar with tensor analysis in areas such as continuum mechanics or general relativity you should be able to follow this chapter without major issues.",
"Consulting the bibliography might help with issues specific to fluid mechanics that might arise.",
"If you are seeing the application of tensor analysis for the first time it might be worth consulting an introductory text on tensors.",
"When dealing with fluid flow, two formulations are usually discussed in the literature, the Lagrangian and the Eulerian specifications, after the great mathematicians Joseph-Louis Lagrange and Leonard Euler Although Euler was responsible for both formulations, see for example Landau and Lifshitz (1987, p. 5) [57]..",
"The Lagrangian formulation follows from the idea of tracking individual particles or the center of mass of a group of particles from classical mechanics.",
"In the context of fluid flow the movement of a single water molecule could be tracked in relation to an initial time $t_0$ and position $\\mathbf {s}$ , where, as seen in Chapter REF .",
"$\\mathbf {s} = s_x \\mathbf {i} + s_y \\mathbf {j} + s_z \\mathbf {k}$ Thus, in the Lagrangian formulation, the velocity of a tracked particle or center of mass of a group of particles is $\\mathbf {v}(\\mathbf {s}, t)$ or, considering the vector components $\\mathbf {v}(s_x, s_y, s_z, t)$ .",
"It is a consensus that the Lagrangian formulation is not mathematically convenient for the majority of fluid flow problems and thus, it will not be discussed further.",
"The formulation normally adopted is more akin to that used in electromagnetism in the sense that the flow quantities are defined as a vector field with independent variables position in space and time, $\\mathbf {x}$ , and t. The vector velocity field in the Eulerian formulation is traditionally represented as $\\mathbf {u}(\\mathbf {x}, t) $ , where $\\mathbf {u} = u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}$ often also represented as $\\mathbf {u} = u \\mathbf {x} + v \\mathbf {y} + w \\mathbf {z}$ such that $\\mathbf {u}(\\mathbf {x}, t) $ can be written as $\\mathbf {u}(u_x, u_y, u_z,t) $ in the first case and $\\mathbf {u}(u, v, w,t) $ in the second.",
"The difference between $\\mathbf {v}$ and $\\mathbf {u}$ is that in the latter we are now no longer interested in an individual particle or the center of mass of a group of particles, what we have is a velocity field in space and time.",
"Now at any position in space and at a given time there is a velocity value assigned.",
"Imagine a 2D river or stream seen from above, in the Lagrangian river we would track the position of individual particles as a function of time, while on the Eulerian river we are not worried about individual particles but we have a velocity field that covers the entire space of the river, such that if we chose an arbitrary point $\\mathbf {x_1}$ and $t_1$ on the surface of the river, the velocity at that point is given by $\\mathbf {u}(\\mathbf {x_1}, t_1)$ .",
"One advantage of the Eulerian formulation is now obvious as there is no need to track an exorbitant number of particles to model the flow of a river or stream.",
"Another major advantage of the Eulerian formulation is that it can be used to model many other important fluid properties such as density and pressure over space and time, however it is important to point out that these are scalar fields given by $\\rho (\\mathbf {x}, t)$ and $P(\\mathbf {x}, t)$ ,C.L.",
"for density and pressure, respectively, noting that $\\mathbf {u}$ at any point in space and time is itself a vector, while $\\rho $ and $P$ are scalars.",
"From the Eulerian formulation we now find the acceleration of a fluid element.",
"The acceleration of the fluid element is not given directly by $ \\partial \\mathbf {u}/ \\partial t $ as each fluid element travels through the vector field as times varies, being subjected to velocity variation in space along the vector field in addition to time.",
"The correct formulation of the acceleration of a material element is given by $\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u}$ Using the notation introduced in Chapter REF and because $\\mathbf {u}$ is a vector the equation above can be written as $\\frac{\\partial }{\\partial t} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) \\cdot [\\frac{\\partial }{\\partial x}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) \\mathbf {i} + \\\\ + \\frac{\\partial }{\\partial y}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) \\mathbf {j} + \\frac{\\partial }{\\partial z} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k})\\mathbf {k}]$ which, considering the dot product $\\mathbf {u} \\cdot \\nabla \\mathbf {u}$ , can be written as $\\frac{\\partial }{\\partial t} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + [u_x \\frac{\\partial }{\\partial x}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + u_y \\frac{\\partial }{\\partial y}(u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) + \\\\ + u_z \\frac{\\partial }{\\partial z} (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k})]$ With some effortAs opposed to \"it can easily be shown\" it can be shown that the vector equation above can be written as a set of equations.",
"You will see that most of the fluid dynamics equations in two and three-dimensional spaces are actually sets of equations and it is important that you familiarize yourself with the steps in converting from vector form to components and vice-versa when doableIn several fluid equations, multiple tensor components are involved, and it is not practical to do these operations without using vector and tensor identities.",
"Thus in Cartesian coordinates, the components in $x$ , $y$ and $z$ are $\\frac{\\partial u_x}{\\partial t} + [u_x \\frac{\\partial u_x}{\\partial x} + u_y \\frac{\\partial u_x}{\\partial y} + u_z \\frac{\\partial u_x}{\\partial z} ] \\\\\\frac{\\partial u_y}{\\partial t} + [u_x \\frac{\\partial u_y}{\\partial x} + u_y \\frac{\\partial u_y}{\\partial y} + u_z \\frac{\\partial u_y}{\\partial z} ]\\\\\\frac{\\partial u_z}{\\partial t} + [u_x \\frac{\\partial u_z}{\\partial x} + u_y \\frac{\\partial u_z}{\\partial y} + u_z \\frac{\\partial u_z}{\\partial z} ]$ The quantity $\\mathbf {u} \\cdot \\nabla \\mathbf {u} $ is the dot product of a vector (first order tensor) and a second order tensor and is a vector, such that $ \\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u} $ is the sum of two vectors and is itself a vector.",
"The derivative form found in Equation 4.4 has such an importance in fluid dynamics that it constitutes a mathematical operator $\\boxed{\\frac{D}{D t} = \\frac{\\partial }{\\partial t} + \\mathbf {u} \\cdot \\nabla }$ such that for any scalar or vector quantity $f$ we have $\\frac{D f}{D t} = \\frac{\\partial f}{\\partial t} + \\mathbf {u} \\cdot \\nabla f$ In which $ \\frac{\\partial f}{\\partial t} $ is the temporal rate of change at a specific location and $\\mathbf {u} \\cdot \\nabla f $ is the convective rate of change due to transport of the fluid element from one point to another in space.",
"If $f$ is constant throughout the space under consideration $\\frac{D f}{D t} = \\frac{\\partial f}{\\partial t} + \\mathbf {u} \\cdot \\nabla f = 0$ and there is no acceleration of the material elements of fluid.",
"The conservation equation, or mass conservation equation, is often referred to the continuity equation on the literature.",
"In agreement with G. K. Batchelor, we do not think that continuity is an appropriate term and will refrain from using it in this text.",
"The idea of conservation of mass, be it a gas, a fluid or a particulate solid, is a very intuitive one.",
"Imagine a garden hose, if you consider a given length of the hose, say one meter, once you turn on the faucet and the flux out of the faucet becomes constant, and if there are no leaks along the hose, the amount of water that enters on one end is equal to the amount that exits on the other end, per unit time.",
"This condition in which the amounts in and out of the system are constant and equal is called steady state and is very important in groundwater hydrology.",
"This simple example is a basic application of the conservation equation, if there are no losses or gains within the system, matter is conserved.",
"Suppose now you have a steady state condition and there are leaks in your garden hose within the length considered.",
"The mass conservation equation still applies, the amount in is equal to the sum of the amounts on the other end of the hose plus any leaks, per unit time.",
"In real scenarios we often have to deal with sources or sinks within the volume considered, this is especially true in the case of reactive and/or volatile transport in which there can be loss of gain within the consider volume due to adsorption, condensation, degradation and many other physico-chemical processes.",
"A simple example of a sink (loss) would be evaporation of water in a reservoir.",
"Let us now treat this problem from a mathematical point of view.",
"There are at least two approaches to treat this problem, one is deceitfully simple and direct and is based on vector calculus.",
"The other is based on algebraAs in algebraic manipulations in a broader sense and appears to be very complex although being very simple.",
"The first one can be found in advanced fluid dynamics and mechanics, physics and applied mathematics books, while the other is preferred in introductory engineering and in soil physics books.",
"The conceptual model is the same in both cases, an element of volume in space, in which flow is considered through an element of surface area.",
"It is easy for us to consider water, but you will see later that the conservation equation applies to any substance in transport processes in porous media.",
"In the vector treatment we are not initially concerned with the geometry of the volume, other than it should be a closed space.",
"Initially, however, it should be much easier to imagine the volume as a cube or as a sphere or other similar closed surfacesIn mathematics, the topology of the space is of interest, and can have implications in terms of the mathematical description.",
"This is, however, beyond the scope of this text.",
"Starting with a volume $V$ enclosed by a surface $A$ completely filled with a liquid of density $\\rho $ , the mass of water within the volume is given by $\\int \\rho dV$ This should come of no surprise, as by integrating the volume elements $dV$ over the entire volume we are left with $V$ , and obviously the water density in [M L$^{-3}$ ] multiplied by the total volume in [L$^{3}$ ] is the mass M of water within the enclosed volume.",
"The amount of water flowing through an element of surface area $dA$ $\\int \\rho \\mathbf {u} \\cdot \\mathbf {n} dA$ Where $\\mathbf {n}$ is the unit normal in dAC.L..",
"Note that the velocity, $\\mathbf {u}$ , in [L T$^{-1}$ ] times water density, $\\rho $ , in [M L$^{-3}$ ], over the entire surface, $A$ , in [L$^{2}$ ] is the rate of mass transfer across the surface in [M T$^{-1}$ ].",
"Now the rate of variation of the mass within the volume with time can be written in differential form as $\\frac{\\partial }{\\partial t} \\int \\rho dV$ in units of [M T$^{-1}$ ].",
"If this doesn't make much sense to you don't worry, you can find more information about this construction in vector calculus or calculus 3 books.",
"Note that as we defined Equation 4.8 as the total mass within the volume, Equation 4.10 is nothing more than the variation of mass with time within the volume.",
"Now, from our discussion of conservation laws we can agree that the variation of mass within a volume is the net rate of flow across its surface, and thus the conservation of mass implies that $\\frac{\\partial }{\\partial t} \\int \\rho dV = - \\int \\rho \\mathbf {u} \\cdot \\mathbf {n} dA$ So that $\\frac{\\partial }{\\partial t} \\int \\rho dV + \\int \\rho \\mathbf {u} \\cdot \\mathbf {n} dA = 0$ Now all the vector calculus tools discussed in Chapter REF will come in handy.",
"We can use the divergence theorem to transform a surface integral into a volume integral.",
"Applying it to the second term in Equation 4.12 results in $\\frac{\\partial }{\\partial t} \\int \\rho dV + \\int \\nabla \\cdot (\\rho \\mathbf {u}) dV = 0$ Which can be written as $\\int [ \\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mathbf {u})] dV = 0$ Because the integration must hold for any volume, the integrand must vanish and we are left with one of the fundamental equations of fluid dynamics, the mass conservation (or continuity) equation $\\boxed{ \\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mathbf {u}) = 0 }$ Now for the derivation that does not need vector calculus.",
"Imagine a cube with sides $\\Delta x$ , $\\Delta y$ and $\\Delta z$ .",
"Consider a velocity component $u_x$ in the direction $x$ .",
"The area which is perpendicular to the velocity component $u_x$ is $\\Delta y\\Delta z$ .",
"In other words, the velocity component $u_x$ enters into the cube in the side $\\Delta y\\Delta z$ .",
"Now, the rate of mass accumulation in the direction $x$ is the amount of mass that enters on one end on the direction $x$ and leaves on the other, in that same direction.",
"The amount of mass that enters in one end is equal to the velocity component $u_x$ times the density of the fluid $\\rho $ times the area $ \\Delta y \\Delta z$ , note that $\\frac{m_{in-x}}{t} = \\rho u_x \\Delta y \\Delta z = \\frac{\\mathrm {M}}{L^3} \\frac{\\mathrm {L}}{T} L^2 = \\frac{\\mathrm {M}}{T}$ The amount that exits on the other end will depend on the variation of the velocity of the fluid through the direction $x$ within the volume, if the fluid slows down, the velocity decreases and vice-versa.",
"Therefore, in the outflow we have to consider the rate of variation of velocity within the volume such that, the outflow mass in the $x$ direction is $\\frac{m_{out-x}}{t} = ( \\rho u_x + \\frac{\\partial \\rho u_x}{\\partial x}\\Delta x) \\Delta y \\Delta z$ Therefore, the amount of mass that accumulates or is lost per unit time due to flux in the $x$ direction is given by $\\frac{\\partial }{\\partial t} M_{x} = \\rho u_x \\Delta y \\Delta z - (\\rho u_x + \\frac{\\partial \\rho u_x}{\\partial x}\\Delta x) \\Delta y \\Delta z$ Which results in $\\frac{\\partial }{\\partial t} M_{x} = - \\frac{\\partial \\rho u_x}{\\partial x}\\Delta x \\Delta y \\Delta z$ By symmetry it is not difficult to show that the mass lost or accumulated due to flux on the directions $y$ and $z$ is $\\frac{\\partial }{\\partial t} M_{y} = - \\rho \\frac{\\partial \\rho u_y}{\\partial y}\\Delta x \\Delta y \\Delta z$ $\\frac{\\partial }{\\partial t} M_{z} = - \\rho \\frac{\\partial \\rho u_z}{\\partial z}\\Delta x \\Delta y \\Delta z$ and that the total mass accumulated or loss within the cube per unit time due to flow in the three directions is $\\frac{\\partial }{\\partial t} M_{xyz} = - (\\frac{\\partial \\rho u_x}{\\partial x}+ \\frac{\\partial \\rho u_y}{\\partial y}+ \\frac{\\partial \\rho u_z}{\\partial z})\\Delta x \\Delta y \\Delta z$ Since the density of the cube is $\\rho $ and its volume is $\\Delta x \\Delta y \\Delta z$ and rearranging $ \\rho = M/V$ to $ M = \\rho V$ we have $\\frac{\\partial }{\\partial t} \\rho \\Delta x \\Delta y \\Delta z = - (\\frac{\\partial \\rho u_x}{\\partial x}+\\frac{\\partial \\rho u_y}{\\partial y}+\\frac{\\partial \\rho u_z}{\\partial z})\\Delta x \\Delta y \\Delta z$ or $\\frac{\\partial \\rho }{\\partial t} = - (\\frac{\\partial \\rho u_x}{\\partial x}+\\frac{\\partial \\rho u_y}{\\partial y}+\\frac{\\partial \\rho u_z}{\\partial z})$ Using the vector operators, this equation can be written as $\\frac{\\partial \\rho }{\\partial t} + \\nabla \\cdot (\\rho \\mathbf {u}) = 0$ which is the same as Equation 4.15.",
"The derivation using geometric considerations is favored in soil physics and hydrology books while the vector calculus derivation is found in fluid mechanics and dynamics books for physicists and mathematicians.",
"If the density of the fluid is constant, $\\rho $ does not vary with time and $\\partial \\rho /\\partial t = 0 $ and we arrive at an important case of the conservation equation $\\boxed{\\nabla \\cdot \\mathbf {u} = 0}$ Stating in words, the divergence of the velocity field is zero for an incompressible fluid.",
"For most practical purposes, water can be assumed as incompressible and Equation 4.23 will be valid for many flow problems.",
"Much like for mass, the conservation laws are valid for other extensive properties in fluid mechanics including energy and momentum and this will allow us to derive the equations at the core of fluid mechanics and transport phenomena.",
"Recall from introductory physics that Newton's second law can be written either as $\\mathbf {F} = m\\mathbf {a}$ or as $\\mathbf {F} = \\frac{d \\mathbf {p}}{dt}$ where the vector quantity $\\mathbf {p}$ is the linear momentum and is equal to $ m \\mathbf {v} $ , for an object of mass $m$ moving at velocity $\\mathbf {v}$ .",
"Considering an element of fluid of volume $V$ , the momentum corresponding to the entire volume is given by $\\int \\mathbf {u} \\rho dV$ Notice that velocity is in [L T$^{-1}$ ], density is in [M L$^{-3}$ ] and volume is in [L$^{3}$ ] such that the units are consistent with those of momentum [M L T$^{-1}$ ].",
"The rate of change of momentum for the elementary volume can be represented by the derivative of the relationship above with respect to time, the material derivative in this case.",
"If the density can be assumed as constant $\\int \\frac{D\\mathbf {u}}{Dt} \\rho dV$ Now, as specified by Newton's second law of motion, the rate of change of momentum in an object is equal to the sum of forces acting on the object.",
"There are two important points here to be considered because we are dealing with fluid mechanics and not with a point particle.",
"The first is that the total momentum is the sum of the individual momentum of each infinitesimal element of fluid.",
"If the system was modeled as a system of individual particles the integral on the previous equations could be replaced by a summation of the momenta of the $n$ particles that compose the system.",
"The second point is that the forces acting in a volume of fluid can be divided into body (or volume) forces such as those generated by gravity and electromagnetic fields and surface (of contact) forces, which act directly over the surface enclosing the volume element.",
"Surface forces are ubiquitous in continuum mechanics and are an important concept in soil physics both in treating the solid and liquid phases and their interactions.",
"To understand surface forces it is convenient to explore the concept of the stress tensor.",
"Imagine an element of volume of fluid immersed in a fluid or other continuous matter, the matter around the element of fluid exerts a contact force across the entire surface of the fluid element.",
"For the sake of simplicity, consider a volume element of cubic shape in Cartesian coordinates system.",
"Each face can be subjected to forces that are perpendicular (i.e.",
"normal) to the surface, in one direction, or parallel (tangential) to the surface in two directions.",
"For example, the face that lies on the $x-y$ plane can be subjected to a perpendicular force on the $z$ direction and to two parallel forces on the $x$ and $y$ directions.",
"Thus, the cube can be subjected to three forces in each face for each of its six faces.",
"If each surface of the cube is characterized by a surface area, we can define each component of the stress acting on the surface as force over area.",
"Thus, on the $z$ direction lies the $x-y$ plane with a normal stress component $\\sigma _{zz}$ and two perpendicular stress components, one in the $x$ direction, represented by $\\sigma _{zx}$ and another in the $y$ direction represented by $\\sigma _{zy}$ .",
"This leaves us with three unique stress components in each direction, $x$ , $y$ and $z$ , with a total of nine stress components completely defining the stresses acting on the element.",
"The nine elements define the stress tensor $ \\sigma _{ij} =\\begin{bmatrix}\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\\end{bmatrix}$ The components where $i = j$ (i.e.",
"$\\sigma _{xx}$ , $\\sigma _{yy}$ , $\\sigma _{zz}$ ) are the normal stress components, while the components where $i \\ne j$ are the tangential stress components.",
"The tangential stress components correspond to shearing stresses seen in solids which can cause isovolumetric deformations.",
"It is convenient now to introduce the Kronecker delta operator $\\delta _{ij} ={\\left\\lbrace \\begin{array}{ll}0 & \\text{if} ~ i \\ne j, \\\\1 & \\text{if} ~ i = j \\\\\\end{array}\\right.",
"}$ Such that the normal stresses on the fluid element can be represented as $\\sigma _{ij} = \\delta _{ij} \\sigma _{ij}$ It is important to point out that if the fluid element is at rest, the static fluid pressure is defined as $\\sigma _{ij} = -p \\delta _{ij}$ Applying this equation to the stress tensor results in $ \\sigma _{ij} =\\begin{bmatrix}-p & 0 & 0 \\\\0 & -p & 0 \\\\0 & 0 & -p \\\\\\end{bmatrix}$ Which is nothing more than the well known principle from hydrostatics which states that for a small volume immersed in a fluid (such as in water for example), the pressure is constant across its surface.",
"Keep in mind that for a moving fluid element the story is quite different.",
"Now the total surface force exerted on the volume element is the sum of the stresses acting in each surface element with area $dS$ and normal $n_j$ and can be written as $\\int \\sigma _{ij} n_{j} dS$ which can be transformed to a volume integral using the divergence theorem $\\int \\frac{\\partial \\sigma _{ij}}{\\partial x_j} dV$ From Newton's law expressed as conservation of momentum, for the element of volume of the fluid, we can write the total momentum as the sum of forces acting on the fluid, including the summation of all body forces and the surface forces $\\int \\frac{D u_i}{Dt} \\rho dV = \\int F_i \\rho dV + \\int \\frac{\\partial \\sigma _{ij}}{\\partial x_j} dV$ As usual, the integral vanishes identically and we are left with $\\frac{D u_i}{Dt} \\rho = F_i \\rho + \\frac{\\partial \\sigma _{ij}}{\\partial x_j}$ In which $F_i$ is the sum of all body forces acting on the fluid element, i.e.",
"gravitational, electromagnetic, etc.",
"Expanding this equation in three dimensional Cartesian coordinates using Einstein's summation convention we have $\\frac{D u_x}{Dt} \\rho = F_x \\rho + \\frac{\\partial \\sigma _{xx}}{\\partial x} + \\frac{\\partial \\sigma _{xy}}{\\partial y} + \\frac{\\partial \\sigma _{xz}}{\\partial z}\\\\\\frac{D u_y}{Dt} \\rho = F_y \\rho + \\frac{\\partial \\sigma _{yx}}{\\partial x} + \\frac{\\partial \\sigma _{yy}}{\\partial y} + \\frac{\\partial \\sigma _{yz}}{\\partial z}\\\\\\frac{D u_z}{Dt} \\rho = F_z \\rho + \\frac{\\partial \\sigma _{zx}}{\\partial x} + \\frac{\\partial \\sigma _{zy}}{\\partial y} + \\frac{\\partial \\sigma _{zz}}{\\partial z}$ These equations show that the acceleration in each direction is caused by the forces acting on that direction and the components of the stress tensor acting on the face perpendicular to the direction of the acceleration.",
"With some inspection it can also become clear that the components of the stress tensor can cause deformation during motion depending of the nature of the shearing components (i.e.",
"$i \\ne j$ ).",
"With that in mind, the stress tensor can be divided into two components, one equivalent to the static pressure experienced by a stationary volume of fluid and a non-isotropic component which originates due to motion of the fluid element $\\sigma _{ij} = -p \\delta _{ij} + d_{ij}$ Which can be represented as $\\begin{bmatrix}\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\\end{bmatrix}=\\begin{bmatrix}-p & 0 & 0 \\\\0 & -p & 0 \\\\0 & 0 & -p \\\\\\end{bmatrix}+\\begin{bmatrix}d_{xx} & d_{xy} & d_{xz} \\\\d_{yx} & d_{yy} & d_{yz} \\\\d_{zx} & d_{zy} & d_{zz} \\\\\\end{bmatrix}$ Now $d_{ij}$ is called the deviatoric stress tensor and is the non-isotropic, tangential, component of stresses.",
"Considering the components of the stress tensor and expanding the material derivative operator, the unabridged equation of motion for an element of fluid in three-dimensional Cartesian coordinates is $\\rho (\\frac{\\partial u_x}{\\partial t}+u_x \\frac{\\partial u_x}{\\partial x}+u_y\\frac{\\partial u_x}{\\partial y}+u_z\\frac{\\partial u_x}{\\partial z}) = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\frac{\\partial d_{xx}}{\\partial x} + \\frac{\\partial d_{xy}}{\\partial y} + \\frac{\\partial d_{xz}}{\\partial z}\\\\\\rho (\\frac{\\partial u_y}{\\partial t}+u_x \\frac{\\partial u_y}{\\partial x}+u_y\\frac{\\partial u_y}{\\partial y}+u_z\\frac{\\partial u_y}{\\partial z}) = F_y \\rho -\\frac{\\partial p}{\\partial y}+ \\frac{\\partial d_{yx}}{\\partial x} + \\frac{\\partial d_{yy}}{\\partial y} + \\frac{\\partial d_{yz}}{\\partial z}\\\\\\rho (\\frac{\\partial u_z}{\\partial t}+u_x \\frac{\\partial u_z}{\\partial x}+u_y\\frac{\\partial u_z}{\\partial y}+u_z\\frac{\\partial u_z}{\\partial z}) = F_z \\rho -\\frac{\\partial p}{\\partial z} + \\frac{\\partial d_{zx}}{\\partial x} + \\frac{\\partial d_{zy}}{\\partial y} + \\frac{\\partial d_{zz}}{\\partial z}$ The vector form, being simpler and more elegant, is referred to as the Cauchy momentum equation after Augustin-Louis Cauchy $\\boxed{\\rho \\frac{D \\mathbf {u}}{D t} = \\mathbf {F} \\rho - \\nabla p + \\nabla \\cdot \\mathbf {d}}$ The full momentum equation is a highly nonlinear partial differential equation and knowledge of the stress tensor and body forces to which the fluid volume are subjected would be necessary to find a solution for this equation, if at all possible.",
"Two of the most fundamental equations of fluid mechanics work with simplifications and assumptions over the momentum equation.",
"The first developed from Newton's laws was derived by Euler and a more fundamental one which is thought to account for most fluid flow phenomena was developed by Claude-Louis Navier and Georges Gabriel Stokes less than a hundred years later.",
"The Euler equations were derived by Leonhard Euler himself in 1755 for the motion of an inviscid fluid, or a fluid without viscosity.",
"The complete set of equations of motion, including the viscosity term, were derived in the XIX century, culminating from the works of Claude-Louis Navier and Sir George Gabriel Stokes, after which they are named.",
"The Euler equations can be understood as a special case of the Navier-Stokes equations and both sets of equations are applications of the Newton laws of motion to fluid flow, and both can be understood as special cases of the Cauchy equation of motion within the Newtonian framework.",
"Up until now we have not discussed viscosity on any broader sense, most of us know the colloquial definition of viscosity used in daily language.",
"A so called viscous fluid has more resistance to flow as we might have experienced with grease and honey as compared to water, for example.",
"A higher viscosity reflects, in essence, higher internal friction on the fluid, or a larger magnitude of the forces between its elemental particles.",
"Mathematically, viscosity can be represented as the coefficient of proportionality between a tangential stress and the rate of variation of velocity of a fluid in a direction perpendicular to that of the applied stress.",
"The most simple explanation is that found on most of the literature and we can make use of the stress tensor already introduced.",
"Imagine you have a two-dimensional element of fluid within a system of horizontal and vertical axes $x$ and $y$ .",
"Now imagine that in the upper surface of the fluid element a stress $\\sigma _{xy}$ parallel to the surface and perpendicular to the $y$ axis is applied.",
"If the fluid has any internal friction, any imaginary fluid layer parallel to the surface layer where the stress is being applied will want to move with it.",
"For a rigid material with infinite viscosity over a frictionless surface, every imaginary layer will move at the same velocity and the material will move as a block.",
"Now if the material is not rigid the gradient of velocity in the $y$ direction will be proportional to the stress applied and as mentioned above, the viscosity is the coefficient of proportionality $\\sigma _{xy} = \\mu \\frac{d }{dy} u_x$ The more viscous the material, the more a tangential stress applied will propagate to the lower layersC.L.. One of the most important assumptions is that this relationship is linear, or alternatively, that the viscosity is constant in this equation.",
"This is the case for the so called Newtonian fluids, if the relationship is nonlinear the fluid is called non-Newtonian and a different framework needs to be adopted for finding a general equation of motion.",
"Now suppose an ideal fluid has no viscosity, in this case there is no sense in speaking of tangential components because there is no internal friction, the deviatoric stress term in the Cauchy equation is zero and we are left with $\\boxed{\\rho \\frac{D \\mathbf {u}}{D t} = \\mathbf {F} \\rho - \\nabla p }$ This is Euler's equation of motion and is valid for ideal fluids.",
"We can further simplify Euler's equation disregarding body forces, when possible, so that acceleration is caused only by surface forces $\\rho \\frac{D \\mathbf {u}}{D t} = - \\nabla p$ Expanding the material derivative term $\\rho (\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u}) = - \\nabla p$ Euler's equation can be also used to derive the equation for hydrostatic pressure in a fluid in equilibrium.",
"In this cases there is no motion and the acceleration term represented by the material derivative is zero.",
"For the hydrostatic pressure to exist it is necessary the action of a body force, represented here by gravitation force $ \\mathbf {F} = \\mathbf {g} $ $\\nabla p = \\rho \\mathbf {g}$ Expanding $\\frac{\\partial p}{\\partial x} \\mathbf {i} + \\frac{\\partial p}{\\partial y} \\mathbf {j} + \\frac{\\partial p}{\\partial z} \\mathbf {k} = \\rho (g_x \\mathbf {i} + g_y \\mathbf {j} + g_z \\mathbf {k})$ And considering that Earth's gravitational force acts in $z$ only, and considering $z$ positive upwards $\\frac{\\partial p}{\\partial z} = -\\rho g$ Which can be readily integrated as for $\\rho $ a $g$ constant as $\\int \\partial p = - \\rho g \\int \\partial z$ Resulting in $p = - \\rho g z + \\text{constant}$ Applying the boundary condition that $p = p_0$ when $z = 0$ (fluid surface) results $p = p_0 - \\rho g z$ Which will increase considering that the absolute value of $z$ increases with depth and that $z$ is negative in the coordinate system adopted.",
"The problem with Euler's equation, except for special applications, is that most liquids will have some degree of viscosity and the non-isotropic components of the stress tensor need to be considered on the analysis.",
"The Navier-Stokes equations are the most general form of the governing equation of fluid motion and can be interpreted as a special case of Cauchy momentum equation.",
"The transition from the Cauchy momentum equation to the Navier-Stokes equations requires that the non-isotropic stress tensor is expressed in a form amenable to the solution of the differential equation using analytical or numerical methods.",
"Following the language of Batchelor [8], it is necessary to find a relationship between the deviatoric stress tensor and the local properties of the fluid.",
"As with a lot of transport relations in physics, one possible simplification is to assume a linear relationship between the deviatoric stress and the velocity gradient.",
"Keep in mind that this assumption carries profound implications in all areas of physics that deal with transport phenomena, either of matter, energy, momentum, electrical current, heat, ... and should not be taken lightly, I urge you to consult the references for details.",
"The hypothesis of a linear relation between the deviatoric stress tensor and the velocity gradients can be expressed analytically as $d_{ij} = A_{ijkl} \\frac{\\partial u_k}{\\partial x_l}$ Now $A_{ijkl}$ is a fourth order tensor with 81 individual components, from $A_{xxxx}$ to $A_{zzzz}$ in Cartesian $x$ , $y$ and $z$ coordinates.",
"The full expansion of this equation is not practical in this text due to space and formatting considerations, but a reduced form is provided, considering Einstein's summation convention $& d_{xx} = A_{xxxx} \\frac{\\partial u_x}{\\partial x} + A_{xxxy} \\frac{\\partial u_x}{\\partial y} + A_{xxxz} \\frac{\\partial u_x}{\\partial z} + \\cdots + A_{xxzz} \\frac{\\partial u_z}{\\partial z}\\\\& d_{xy} = A_{xyxx} \\frac{\\partial u_x}{\\partial x} + A_{xyxy} \\frac{\\partial u_x}{\\partial y} + A_{xyxz} \\frac{\\partial u_x}{\\partial z} + \\cdots + A_{xyzz} \\frac{\\partial u_z}{\\partial z}\\\\& \\vdots \\\\& d_{zz} = A_{zzxx} \\frac{\\partial u_x}{\\partial x} + A_{zzxy} \\frac{\\partial u_x}{\\partial y} + A_{zzxz} \\frac{\\partial u_x}{\\partial z} + \\cdots + A_{zzzz} \\frac{\\partial u_z}{\\partial z}$ There are nine coefficients in each line and nine lines, one for each component of the deviatoric stress tensor, totaling the 81 coefficients of the $A_{ijkl}$ fourth-order tensor.",
"As we have done before, it is convenient to write the deviatoric stress tensor as the sum of its symmetric and antisymmetric counterparts $d_{ij} = A_{ijkl} d_{ij}^{s} + A_{ijkl} d_{ij}^{a}$ or $d_{ij} = A_{ijkl} e_{kl} + A_{ijkl} \\xi _{ij}$ where $e_{kl} = \\frac{1}{2}(\\frac{\\partial u_k}{\\partial u_l} + \\frac{\\partial u_l}{\\partial u_k})$ is a symmetric second-order tensor, which in $x$ , $y$ and $z$ Cartesian coordinates can be represented as $\\frac{1}{2}\\begin{bmatrix}(\\frac{\\partial u_x}{\\partial x} + \\frac{\\partial u_x}{\\partial x}) & (\\frac{\\partial u_x}{\\partial y} + \\frac{\\partial u_y}{\\partial x}) & (\\frac{\\partial u_x}{\\partial z} + \\frac{\\partial u_z}{\\partial x}) \\\\(\\frac{\\partial u_y}{\\partial x} + \\frac{\\partial u_x}{\\partial y}) & (\\frac{\\partial u_y}{\\partial y} + \\frac{\\partial u_y}{\\partial y}) & (\\frac{\\partial u_y}{\\partial z} + \\frac{\\partial u_z}{\\partial y}) \\\\(\\frac{\\partial u_z}{\\partial x} + \\frac{\\partial u_x}{\\partial z}) & (\\frac{\\partial u_z}{\\partial y} + \\frac{\\partial u_y}{\\partial z}) & (\\frac{\\partial u_z}{\\partial z} + \\frac{\\partial u_z}{\\partial z}) \\\\\\end{bmatrix}$ Verify that the quantities that are opposed to each other across the diagonal are the same, and this is what is mean be symmetric in this case.",
"The antisymmetric component is $\\xi _{kl} = \\frac{1}{2}(\\frac{\\partial u_k}{\\partial u_l} - \\frac{\\partial u_l}{\\partial u_k})$ and can be represented as $\\frac{1}{2}\\begin{bmatrix}(\\frac{\\partial u_x}{\\partial x} - \\frac{\\partial u_x}{\\partial x}) & (\\frac{\\partial u_x}{\\partial y} - \\frac{\\partial u_y}{\\partial x}) & (\\frac{\\partial u_x}{\\partial z} - \\frac{\\partial u_z}{\\partial x}) \\\\(\\frac{\\partial u_y}{\\partial x} - \\frac{\\partial u_x}{\\partial y}) & (\\frac{\\partial u_y}{\\partial y} - \\frac{\\partial u_y}{\\partial y}) & (\\frac{\\partial u_y}{\\partial z} - \\frac{\\partial u_z}{\\partial y}) \\\\(\\frac{\\partial u_z}{\\partial x} - \\frac{\\partial u_x}{\\partial z}) & (\\frac{\\partial u_z}{\\partial y} - \\frac{\\partial u_y}{\\partial z}) & (\\frac{\\partial u_z}{\\partial z} - \\frac{\\partial u_z}{\\partial z}) \\\\\\end{bmatrix}$ Verify that the quantities across the diagonals are different (reverse sign) and this is what is meant by antisymmetric in this context.",
"Verify also that the sum of the symmetric and antisymmetric counterparts reduces to the second-order tensor $\\frac{\\partial u_k}{\\partial x_l}$ .",
"The antisymmetric component can further be written as $\\xi _{kl} = -\\frac{1}{2}A_{ijkl} \\epsilon _{klm} \\omega _{m}$ The operator $\\epsilon _{klm}$ is a third-order tensor, with 27 components, where only six are non-zero, called the Levi-Civita symbol with the following property: $\\epsilon _{klm} ={\\left\\lbrace \\begin{array}{ll}+1 & \\text{if} ~(k, l, m)~\\text{is an even permutation} \\\\-1 & \\text{if} ~(k, l, m)~\\text{is an odd permutation} \\\\0 & \\text{if any index is repeated}\\end{array}\\right.",
"}$ Notice that the choice of $k, l, m$ is incidental to the problem, most often on the literature $i, j, k$ or numbers are used.",
"Even permutations in this case are $(k,l,m)$ , $(l,m,k)$ and $(m, k, l)$ and odd permutations are $(m, l, k)$ , $(l, k, m)$ and $(k, m, l)$ .",
"The vector $\\omega _{m}$ has a fundamental role in fluid mechanics, and it is called the vorticity of the fluid.",
"The components of the vorticity vector in $x$ , $y$ and $z$ Cartesian coordinates are $\\omega _x = \\frac{\\partial u_z}{\\partial y} - \\frac{\\partial u_y}{\\partial z} \\\\\\omega _y = \\frac{\\partial u_x}{\\partial z} - \\frac{\\partial u_z}{\\partial x} \\\\\\omega _z = \\frac{\\partial u_y}{\\partial x} - \\frac{\\partial u_x}{\\partial y}$ Which can be obtained from the curl of the $\\textbf {u}$ vector field (recall Chapter REF ) $\\omega = \\nabla \\times \\mathbf {u}$ A fluid with zero curl is called irrotational.",
"The mathematical concept of rotation is directly linked to the physical idea of rotation in fluid flow.",
"The idea of rotational and irrotational flows is rarely discussed in soil physics, largely because of the assumptions adopted related to the nature of the flow as we will discuss later.",
"Now the deviatoric stress tensor can be rewritten as $d_{ij} = A_{ijkl} e_{kl} - \\frac{1}{2} A_{ijkl} \\epsilon _{klm} \\omega _m$ In simple liquids, $A_{ijkl}$ , is an isotropic tensor, that is, the molecular structure of the fluid is isotropic.",
"In this case, $A_{ijkl}$ , can be written as a summation of products of delta tensorsObviously, there is much more to this, the reader is encouraged to consult a book on tensor analysis and Aris [4] $A_{ijkl} = \\mu \\delta _{ik}\\delta _{jl} + \\mu ^{\\prime } \\delta _{il}\\delta _{jk} + \\mu ^{\\prime \\prime } \\delta _{ij}\\delta _{kl}$ Because $A_{ijkl}$ is symmtrical in $i$ and $j$ , it is required that $\\mu = \\mu ^{\\prime } $ such that $A_{ijkl} = 2\\mu \\delta _{ik}\\delta _{jl} + \\mu ^{\\prime \\prime } \\delta _{ij}\\delta _{kl}$ The requirement that $A_{ijkl}$ is also symmetrical in $k$ and $l$ requires that the term $\\omega $ is dropped in the deviatoric stress tensor, so that by replacing into it we have $d_{ij} = 2\\mu \\delta _{ik}\\delta _{jl} e_{kl} + \\mu ^{\\prime \\prime } \\delta _{ij}\\delta _{kl} e_{kl}$ Because the rules of manipulation of Kronecker delta tensors, the relationship can be simplified to $d_{ij} = 2\\mu e_{ij} + \\mu ^{\\prime \\prime } \\nabla \\cdot e_{kk}~\\delta _{ij}$ Where $e_{kk} = \\nabla \\cdot \\mathbf {u} $ is the rate of expansion.",
"Since the deviatoric stress tensor does not contribute to the mean normal stresses $d_{ii} = (2\\mu + 3\\mu ^{\\prime \\prime }) e_{ii} = 0$ Such that $(2\\mu + 3\\mu ^{\\prime \\prime }) = 0$ Solving for $\\mu ^{\\prime \\prime }$ and replacing into the deviatoric stress tensor C.L.",
"results $d_{ij} = 2\\mu (e_{ij} - \\frac{1}{3} e_{ii}~\\delta _{ij})$ We now have an expression for the deviatoric stress tensor which can be used to define the stress tensor.",
"Replacing into the stress tensorC.L.",
"results $\\sigma _{ij} = -p \\delta _{ij} + 2\\mu (e_{ij} - \\frac{1}{3} e_{ii}~\\delta _{ij})$ Which can then be replaced into the equation of motion resulting in the set of equations $\\rho \\frac{D u_x}{D t} = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\frac{\\partial }{\\partial x}[ 2 \\mu (e_{xx} - \\frac{1}{3} e_{xx} \\delta _{xx})]+ \\\\ \\frac{\\partial }{\\partial y}[ 2 \\mu (e_{xy} - \\frac{1}{3} e_{xx} \\delta _{xy})]+ \\frac{\\partial }{\\partial z}[ 2 \\mu (e_{xz} - \\frac{1}{3} e_{xx} \\delta _{xz})]$ $\\rho \\frac{D u_y}{D t} = F_y \\rho -\\frac{\\partial p}{\\partial y}+ \\frac{\\partial }{\\partial x}[ 2 \\mu (e_{yx} - \\frac{1}{3} e_{yy} \\delta _{yx})]+ \\\\ \\frac{\\partial }{\\partial y}[ 2 \\mu (e_{yy} - \\frac{1}{3} e_{yy} \\delta _{yy})]+ \\frac{\\partial }{\\partial z}[ 2 \\mu (e_{yz} - \\frac{1}{3} e_{yy} \\delta _{yz})]$ $\\rho \\frac{D u_z}{D t} = F_z \\rho -\\frac{\\partial p}{\\partial z}+ \\frac{\\partial }{\\partial x}[ 2 \\mu (e_{zx} - \\frac{1}{3} e_{zz} \\delta _{zx})]+ \\\\ \\frac{\\partial }{\\partial y}[ 2 \\mu (e_{zy} - \\frac{1}{3} e_{zz} \\delta _{zy})]+ \\frac{\\partial }{\\partial z}[ 2 \\mu (e_{zz} - \\frac{1}{3} e_{zz} \\delta _{zz})]$ These are the Navier-Stokes equations and they account for the description of viscous fluid flow.",
"The viscosity term in the full form presented above is not independent of position and the equation is valid for compressible fluids as well as incompressible fluids.",
"If we consider that the viscosity is constant throughout the flow field, the term $\\mu $ can be placed outside the derivative terms.",
"Another simplification that can be used for modeling flow in porous media is incompressibility.",
"For an incompressible fluid $\\nabla \\cdot \\mathbf {u} = e_{ii} = 0 $ and the term $ e_{ii} \\delta _{ij} $ is dropped, resulting in $\\rho \\frac{D u_x}{D t} = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\mu \\frac{\\partial }{\\partial x} (e_{xx})+\\mu \\frac{\\partial }{\\partial y}(e_{xy})+ \\mu \\frac{\\partial }{\\partial z} (e_{xz})\\\\\\rho \\frac{D u_y}{D t} = F_y \\rho -\\frac{\\partial p}{\\partial y}+ \\mu \\frac{\\partial }{\\partial x}(e_{yx})+\\mu \\frac{\\partial }{\\partial y}(e_{yy})+ \\mu \\frac{\\partial }{\\partial z}(e_{yz})\\\\\\rho \\frac{D u_z}{D t} = F_z \\rho -\\frac{\\partial p}{\\partial z}+ \\mu \\frac{\\partial }{\\partial x}(e_{zx})+\\mu \\frac{\\partial }{\\partial y}(e_{zy})+ \\mu \\frac{\\partial }{\\partial z}(e_{zz})$ Note that, in this case, each partial derivative of $e_{ij}$ can be written as $\\frac{\\partial }{\\partial x_j} e_{ij} = \\frac{\\partial }{\\partial x_j} \\frac{1}{2}(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}) = \\frac{1}{2}\\frac{\\partial ^2 u_i}{\\partial x_j\\partial x_j} + \\frac{1}{2}\\frac{\\partial }{\\partial x_i}\\frac{\\partial u_j}{\\partial x_j} = \\frac{1}{2} \\nabla ^2 u_i + \\frac{1}{2} \\frac{\\partial }{\\partial x_i}(\\nabla \\cdot \\mathbf {u})$ the incompressibility condition reduces the Navier-Stokes equations to $\\rho \\frac{D u_x}{D t} = F_x \\rho -\\frac{\\partial p}{\\partial x} + \\mu ( \\frac{\\partial ^2 u_x}{\\partial x^2} + \\frac{\\partial ^2 u_x}{\\partial y^2} + \\frac{\\partial ^2 u_x}{\\partial z^2})\\\\\\rho \\frac{D u_y}{D t} = F_y \\rho -\\frac{\\partial p}{\\partial y} + \\mu ( \\frac{\\partial ^2 u_y}{\\partial x^2} + \\frac{\\partial ^2 u_y}{\\partial y^2} + \\frac{\\partial ^2 u_y}{\\partial z^2})\\\\\\rho \\frac{D u_z}{D t} = F_z \\rho -\\frac{\\partial p}{\\partial z} + \\mu ( \\frac{\\partial ^2 u_z}{\\partial x^2} + \\frac{\\partial ^2 u_z}{\\partial y^2} + \\frac{\\partial ^2 u_z}{\\partial z^2})$ which can be written in vector form and by expanding the material derivative as $\\boxed{\\rho (\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u} ) = \\mathbf {F} \\rho - \\nabla p + \\mu \\nabla ^2 \\mathbf {u}}$ This is the fundamental equation that governs the flow of incompressible fluids and many important results can be derived by solving it for a specified set of boundary conditions.",
"We will used it over subsequent chapters to investigate settling and water flow phenomena.",
"What we will call Stokes equation is a special case of the Navier-Stokes equation for a set of specified conditions.",
"Because we will use Stokes equation as a starting point for particle settling and flow in porous media derivations, it will be specified here.",
"Under steady-state flow or flows in which the velocity variation is small enough that it can be considered constant and in conditions where the inertia is small as compared to the pressure and viscous terms the acceleration term on the left side of the Navier-Stokes equation can be neglected.",
"If the body forces can also be neglected due to being small or the direction of flow in relation to a gravitational field for example, the governing equations become $\\boxed{\\nabla p = \\mu \\nabla ^2 \\mathbf {u}}$ $\\nabla \\cdot \\mathbf {u} = 0$ The second of those being the conservation law described previously.",
"Bernoulli equation can be derived from Euler equations by using the simplifying assumptions of steady and isentropic flow.",
"Steady flow means that the velocity is constant in any particular point in the flow field, i.e.",
"does not vary with time, such that $ \\frac{\\partial \\mathbf {u}}{\\partial t} = 0 $ .",
"Isentropic means that the entropy $s$ is constant throughout the flow field.",
"To derive Bernoulli equation we need to first express Euler equation in terms of enthalpy.",
"In thermodynamics, specific enthalpy can be expressed as $dh = Tds + vdP$ But, because entropy is constant, $ds = 0 $ and $dh = vdp$ In this equation $h$ is the specific enthalpy or enthalpy per unit mass $h = H/M$ , $s$ is the specific entropy, $s = S/M$ and $v$ is the specific volume $v = V/M$ .",
"Since we know that density of a fluid is mass per unit volume, the specific volume can be written as $v = 1/\\rho $ and $dh = \\frac{1}{\\rho }dp$ which can be generalized to $\\nabla h = \\frac{1}{\\rho } \\nabla p$ and replaced into Euler equation resulting $\\frac{\\partial \\mathbf {u}}{\\partial t} + \\mathbf {u} \\cdot \\nabla \\mathbf {u} = - \\nabla h + \\mathbf {F} \\rho $ We now use a vector calculus identity referred to as the vector dot del operator $\\mathbf {u} \\cdot \\nabla \\mathbf {u} = \\frac{1}{2} \\nabla u^2 - \\mathbf {u} \\times (\\nabla \\times \\mathbf {u} )$ by replacing into the “enthalpy” Euler equation, and with the condition that the velocity does not very in time specified previously, we have $\\mathbf {u} \\times (\\nabla \\times \\mathbf {u} ) = - \\nabla (h + \\frac{1}{2} u^2) + \\mathbf {F}$ Flow conditions require that the curl terms on the left side is zero.",
"We can also rewrite the force term as the negative of the gradient of a potential $\\Psi $ $0 = - \\nabla (h + \\frac{1}{2} u^2) - \\nabla \\Psi $ and lump the terms into a single gradient operator $\\nabla (h + \\frac{1}{2} u^2 + \\Psi ) = 0$ Replacing the expression for the enthalpy previously defined and considering $\\Psi $ as the force potential to Earth's gravitational field $\\Psi = \\rho g$ , we are left with $\\boxed{\\frac{p}{\\rho } + \\frac{1}{2} u^2 + \\rho g = \\text{constant}}$ This equation states that the left hand sum is constant along any streamline.",
"A streamline is a line whose tangent at any point gives the direction of the velocity vector at that point.",
"Streamlines are defined by the system of equations $\\frac{dx}{u_x} = \\frac{dy}{u_y} = \\frac{dz}{u_z}$ Reynolds number is known in soil physics and applied hydraulics and hydrology as a means of establishing if a given flow regime is laminar or turbulent.",
"It is much more than that, it derives from writing the Navier-Stokes equation in a dimensionless form and conveys important information about the flow regime and scaling of flow processes and can be used in laboratory to design experiments to measure flow parameters that could not be measured otherwise, on account of the law of similarity.",
"We start with the Navier-Stokes equations in an incompressible and homogeneous fluid with regard to density.",
"The body force is ignored, but the same principles applied to the other terms can be used if one wishes to proceed including the body force $\\rho (\\frac{\\partial u_i}{\\partial t} + u_j \\frac{\\partial u_i}{\\partial x_j} ) = -\\frac{\\partial p}{\\partial x_i} + \\mu \\frac{\\partial ^2 u_i}{\\partial x_j \\partial x_i}$ Each quantity on the Navier-Stokes equation is now written in a dimensionless form by defining some representative length L and velocity U, with the new dimensionless variables $\\mathbf {u}^{\\prime }$ , $t^{\\prime }$ and $\\mathbf {x}^{\\prime }$ $\\mathbf {u}^{\\prime } = \\frac{\\mathbf {u}}{U}$ $\\mathbf {x}^{\\prime } = \\frac{\\mathbf {x}}{L}$ $t^{\\prime } = \\frac{tU}{L}$ Note that U/L has units of 1/time since [L T$^{-1}$ ] / [L] = [T$^{-1}$ ].",
"Pressure can also be normalized using $p^{\\prime } = \\frac{p}{\\rho U^2}$ Since $\\rho U^2$ has units of [M L$^{-3}$ ] [L T$^{-1}$ ]$^2$ = [M L$^{-1}$ T$^{-2}$ ] which is equivalent to pressure.",
"Substituting the dimensionless quantities on the Navier-Stokes $\\rho (\\frac{\\partial u^{\\prime }_i U}{\\partial t^{\\prime }L/U} + u^{\\prime }_j U \\frac{\\partial u^{\\prime }_i U}{\\partial x^{\\prime }_j L} ) = -\\frac{\\partial p^{\\prime }\\rho U^2}{\\partial x^{\\prime }_i L} + \\mu \\frac{\\partial ^2 u^{\\prime }_i U}{\\partial x^{\\prime }_j L \\partial x^{\\prime }_i L}$ Remembering that constant terms can be brought out of the derivatives and reorganizing $\\frac{U^2}{L}\\rho (\\frac{\\partial u^{\\prime }_i}{\\partial t^{\\prime }} + u^{\\prime }_j \\frac{\\partial u^{\\prime }_i}{\\partial x^{\\prime }_j} ) = -\\frac{U^2}{L}\\frac{\\partial p^{\\prime }\\rho }{\\partial x^{\\prime }_i} + \\mu \\frac{U}{L^2}\\frac{\\partial ^2 u^{\\prime }_i}{\\partial x^{\\prime }_j \\partial x^{\\prime }_i} \\\\\\frac{\\partial u^{\\prime }_i}{\\partial t^{\\prime }} + u^{\\prime }_j \\frac{\\partial u^{\\prime }_i}{\\partial x^{\\prime }_j} = -\\frac{\\partial p^{\\prime }}{\\partial x^{\\prime }_i} + \\frac{\\mu }{\\rho U L}\\frac{\\partial ^2 u^{\\prime }_i}{\\partial x^{\\prime }_j \\partial x^{\\prime }_i}$ The dimensionless quantity $\\boxed{Re = \\frac{\\rho U L}{\\mu }}$ is called the Reynolds number, being introduced by Stokes and later named after Osborne Reynolds who popularized its use.",
"The dimensionless form of the Navier-Stokes equation can be written in vector notation as $\\frac{\\partial \\mathbf {u}^{\\prime }}{\\partial t^{\\prime }} + \\mathbf {u}^{\\prime }\\cdot \\nabla \\mathbf {u}^{\\prime } = -\\nabla p^{\\prime } + \\frac{1}{Re} \\nabla ^2 \\mathbf {u}^{\\prime }$ The Reynolds number is a ratio of inertial forces, that tend to keep the fluid in motion, and viscous forces, which resist movement.",
"In fluids in which the Reynolds number is low, there is large contribution of inertial forces.",
"If inertial forces are much larger than viscous forces the flow can become unstable and vortices and other flow instabilities can develop.",
"Under such conditions the assumption that the flow is smooth, constant and “continuous” is no longer valid and the flow can no longer be considered laminar.",
"If inertial forces are very high in respect to viscous forces, fully turbulent flow might develop.",
"ll $ \\mathbf {s} $ Vector position function $ \\mathbf {u}, \\mathbf {v} $ Vector velocity functions $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k}, $ Unit vectors in Cartesian coordinates $ \\mathbf {x}, \\mathbf {y}, \\mathbf {z}, $ Unit vectors in Cartesian coordinates $ s_x, s_y, s_z $ Components of the position vector in Cartesian coordinates $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ u, v, w $ Components of the velocity vector in Cartesian coordinates $ t $ Time $ \\frac{D}{D t} $ Substantial (or material) derivative $ f $ Generalized function $ \\rho $ Density $ V $ Volume $ A $ Area $ \\mathbf {n} $ Normal unit vector $ m, M $ Mass $ x, y, z, $ Generalized Cartesian coordinates $ \\mathbf {F} $ Force vector $ \\mathbf {a} $ Acceleration vector $ \\mathbf {p} $ Momentum vector $ \\sigma $ Stress $ \\delta _{ij} $ Kroenecker delta $ p $ Pressure $ S $ surface $ d_{ij} $ Deviatoric stress tensor $ \\mathbf {d} $ Deviatoric stress tensor, vector form $ \\mathbf {g} $ Gravitational body force vector $ g_x, g_y, g_z $ Components of the gravitational body force vector in Cartesian coordinates $ A_{ijkl} $ Fourth-order tensor $ e_{kl} $ Symmetric component of the deviatoric stress tensor $ \\xi _{kl} $ Antisymmetric component of the deviatoric stress tensor $ \\epsilon _{klm} $ Levi-Civita symbol $ \\omega $ Vorticity vector $ \\omega _x, \\omega _y, \\omega _w $ Components of the vorticity vector in Cartesian coordinates $ e_{kk} $ Rate of expansion $ \\mu $ Viscosity $ h $ Specific enthalpy $ s $ Specific entropy $ v $ Specific volume $ T $ Temperature $ P $ Pressure $ S $ Entropy $ H $ Enthalpy $ \\Psi $ Potential $ \\mathbf {u^{\\prime }} $ Dimensionless velocity vector $ \\mathbf {x^{\\prime }} $ Dimensionless position vector $ t^{\\prime } $ Dimensionless time $ p^{\\prime } $ Dimensionless pressure $ Re $ Reynolds number $ L $ Length $ U $ Velocity There are two basic methods for the determination of particle size distribution of soils and sediments.",
"For coarser particles such as gravel and cobbles, and sands to some extent, grain size distributions or grain size fractions can be determined using sieves.",
"In essence what is done is to disaggregate the particles, if they are aggregated at all, using physical and or chemical methods, pass the material through a set of sieves of decreasing apertures and to weigh the dried fraction retained in each sieve and to express this as a percent of the total dried mass of the material used.",
"The sieve aperture used for each class interval is a convention, and the textural classes vary between disciplines in soil science, Earth science and engineering.",
"For particles finer than sand, sieving does not work adequately, first because it might not be feasible to create sieves with apertures small enough for the particles, which might have diameters on the micrometer range, second because it might be virtually impossible to disaggregate these particles in the dried state, and third because even if these particles could be disaggregated and precision micrometer aperture sized sieves created, they might cause the clogging of the apertures of the sieve which at that size range would act analogous as pores.",
"For smaller particles, most methods of particle size determination rely on settling of the dispersed particles in a fluid, usually water.",
"Particles are assumed as perfectly spherical, non-interacting, slowly settling in an infinitely dispersed solution.",
"Under these conditions the Navier-Stokes equation solution for a sphere settling in a viscous fluid is the solution to the problem.",
"The fact that the settling is slow under low Reynolds number guarantees that the flow around the sphere is laminar (Figure REF a).",
"Under turbulent flows, analytical solutions to the Navier-Stokes equations, such as the one presented below, might not be feasible due to the complexity of the chaotic fluctuations of the flow regime.",
"In any case even before turbulence developed, the settling velocity might be large enough such that the inertial component of the Navier-Stokes equation is not negligible, and analytic solutions might not be achievable.",
"Figure: Illustration of sphere settling under laminar regime.The solution of the sphere settling problem was provided by G.G.",
"Stokes [92], many authors present interpretations of Stokes solution which are more or less similar to the original and to each-other [66], [57], [8], [1].",
"The solution given by Acheson [1] is presented here for reasons of clarity and conciseness.",
"For slow, steady-state settling conditions the mathematical problems is to find an analytical solution to the Stokes and conservation equations presented in Chapter REF .",
"$\\nabla p = \\mu \\nabla ^2 \\mathbf {u}$ $\\nabla \\cdot \\mathbf {u} = 0$ Not surprisingly, the problem of flow past a sphere is solved by using spherical coordinates.",
"Thus the vector $\\mathbf {u} = \\langle u_x, u_y, u_z \\rangle = u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}$ is written in spherical coordinates as $\\mathbf {u} = \\langle u_r(r, \\theta ), u_{\\theta }(r, \\theta ), 0 \\rangle $ The conservation equation $ \\nabla \\cdot \\mathbf {u}$ is satisfied by a Stokes stream function $\\Psi (r , \\theta )$ which satisfies the conditions $u_r = \\frac{1}{r^2 \\sin {\\theta }} \\frac{\\partial \\Psi }{\\partial \\theta }, ~ u_{\\theta } = \\frac{1}{r \\sin {\\theta }} \\frac{\\partial \\Psi }{\\partial r}$ Now, the curl of $\\mathbf {u}$ in spherical coordinates (consult Chapter REF ) is $\\nabla \\times \\mathbf {u} = \\langle 0, 0, \\frac{1}{r \\sin {\\theta }} E^2 \\Psi \\rangle $ With $E^2$ representing the differential operator $E^2 = \\frac{\\partial ^2}{\\partial r^2} + \\frac{\\sin {\\theta }}{r^2} \\frac{\\partial }{\\partial \\theta } (\\frac{1}{\\sin {\\theta }} \\frac{\\partial }{\\partial \\theta } )$ By virtue of vector identities, Stokes equation can be written as $\\nabla p = -\\mu \\nabla \\times (\\nabla \\times \\mathbf {u})$ in which the gradient operators in $p$ and after $\\mu $ can be written in spherical coordinates as $\\nabla = \\langle \\frac{\\partial }{\\partial r}, \\frac{1}{r} \\frac{\\partial }{\\partial \\theta }, \\frac{1}{r \\sin {\\theta }} \\frac{\\partial }{\\partial \\phi } \\rangle $ from Equations 5.3, 5.5. and 5.6 the components in $\\mathbf {r}$ and $\\theta $ can be obtained $\\frac{\\partial p}{\\partial r} = \\frac{\\mu }{r^2 \\sin {\\theta }} \\frac{\\partial }{\\partial \\theta } E^2 \\Psi ,~\\frac{1}{r} \\frac{\\partial p}{\\partial \\theta } = \\frac{-\\mu }{r \\sin {\\theta }} \\frac{\\partial }{\\partial r} E^2 \\Psi $ The detailed procedure for cross differentiation considering a stream function $\\Psi $ is given by Lamb [66].",
"Acheson [1] explains it in terms of the pressure being eliminated by cross-differentiation which implies that $E(E^2\\Psi ) = 0$ such thatObviously there is much more to this, the user is encouraged to consult Stokes (1901)[92] and Lamb (1932)[66], for an in depth discussion.",
"Stokes (1901) is a reprint of the original and Lamb's discussion draws directly from the source and the level of detail provided is unmatched in modern literature (Horace Lamb was a student of Sir George Stokes at Trinity College, University of Cambridge).",
"There is a very elegant derivation in Landau and Lifshitz [57] based largely on vector calculus operators.",
"Although precise, Landau and Lifshitz and Batchelor [8] derivations rely heavily in implicit arguments.",
"We tried to find a compromise between conciseness and clarity and most likely have not been entirely successful to say the least.",
"$[\\frac{\\partial ^2}{\\partial r^2} + \\frac{\\sin {\\theta }}{r^2} \\frac{\\partial }{\\partial \\theta } (\\frac{1}{\\sin {\\theta }} \\frac{\\partial }{\\partial \\theta } ) ]^2 \\Psi = 0$ The boundary conditions on the settling sphere are $u_r = u_{\\theta } = 0 ~\\text{at} ~r = a$ $u_r = U\\cos {\\theta },~u_{\\theta } = -U\\sin {\\theta } ~\\text{as} ~r \\rightarrow a$ Because the decomposition of the velocity components in $\\mathbf {r}$ and $\\theta $ .",
"The latter conditions along with Equation 5.2, results, by integration in the condition $\\Psi = \\frac{U}{2} r^2 \\sin ^2{\\theta } ~\\text{as} ~r \\rightarrow a$ Now we search for a solution to the partial differential equation 5.8.",
"The theory on solution to partial differential equations plays a fundamental role in analytical methods in fluid mechanics and soil physics, reader unfamiliar with it are encouraged to consult Boas [14] for a concise introduction and Asmar [5] for a more detailed discussion.",
"Equation 5.8 is satisfied by a solution of the form $\\Psi = \\sin ^2{\\theta } f(r)$ provided that $[\\frac{d^2}{dr^2} - \\frac{2}{r^2}] f(r) = 0$ with solution of the form $f(r) = \\frac{A}{r} + Br + Cr^2 + Dr^4$ with A, B, C and D constants.",
"Substituting Equation 5.14 into 5.12 results $\\Psi = \\sin ^2{\\theta } [ \\frac{A}{r} + Br + Cr^2 + Dr^4]$ The values of the constant can be obtained by applying the boundary conditions previously defined to Equation 5.15, resulting $\\Psi = \\frac{U}{2}[ r^2 + \\frac{a^3}{2r} - \\frac{3ar}{2}] \\sin ^2{\\theta }$ By combining $\\mathbf {r}$ component of Equation 5.7 , Equation 5.4 and Equation 5.16 and integrating from $r = 0$ from $r = \\infty $ a pressure function is obtained $p = p_{\\infty } - \\frac{3}{2} \\frac{\\mu U a}{r^2} \\cos {\\theta }$ The stress components on a sphere settling in a viscous fluid are $\\tau _r = -p + 2\\mu \\frac{\\partial u_r}{\\partial r}, ~ \\tau _{\\theta } = \\mu r \\frac{\\partial }{\\partial r}(\\frac{u_{\\theta }}{r}) + \\frac{\\mu }{r} \\frac{\\partial u_r}{\\partial \\theta }, ~\\tau _{\\phi } = 0$ From Equations 5.2, 5.16 and 5.18 we have $\\tau _r = -p_{\\infty } + \\frac{3}{2} \\frac{\\mu U}{a}\\cos {\\theta }$ $\\tau _{\\theta } = - \\frac{3}{2} \\frac{\\mu U}{a}\\sin {\\theta }$ The resulting stress vector on the sphere is $\\tau _{net} = \\tau _r \\cos {\\theta } - \\tau _{\\theta }\\sin {\\theta } = -p_{\\infty }\\cos {\\theta } + \\frac{3}{2} \\frac{\\mu U}{a}$ and the total drag force on the sphere can be calculated by integrating the net force in spherical coordinates $F_{D} = \\int _0^{2\\pi } \\int _0^{\\pi } a^2~\\tau _{net}~\\sin {\\theta }~d \\theta d \\phi $ $F_{D} = \\int _0^{2\\pi } \\int _0^{\\pi } a^2~(-p_{\\infty }\\cos {\\theta } + \\frac{3}{2} \\frac{\\mu U}{a})~\\sin {\\theta }~d \\theta d \\phi \\\\F_{D} = \\int _0^{2\\pi } \\int _0^{\\pi } a^2~(-p_{\\infty } ~\\sin {\\theta }~\\cos {\\theta })~d \\theta d \\phi + \\int _0^{2\\pi } \\int _0^{\\pi } (a~\\frac{3}{2} \\mu U~\\sin {\\theta })~d \\theta d \\phi \\\\F_{D} = \\int _0^{2\\pi } a^2~(-p_{\\infty } ~\\frac{\\sin ^2{\\theta }}{2})\\bigg |_0^{\\pi }~d \\phi + \\int _0^{2\\pi } -(a~\\frac{3}{2} \\mu U~\\cos {\\theta })\\bigg |_0^{\\pi } ~ d \\phi \\\\$ Remembering that $\\sin {0}$ = 0 and $\\sin {\\pi } = 0$ and that $\\cos {0}$ = 1 and $\\cos {\\pi } = -1$ , the first term on the right side is equal to zero $F_{D} = \\int _0^{2\\pi } (2 a~\\frac{3}{2} \\mu U) ~ d \\phi \\\\F_{D} = \\int _0^{2\\pi } (a 3 \\mu U) ~ d \\phi \\\\F_{D} = (a 3 \\mu U) \\phi \\bigg |_0^{2\\pi } \\\\$ $\\boxed{F_{D} = 6 \\pi \\mu U a}$ This is Stokes formula for the drag on a sphere settling at low velocities in a viscous fluid.",
"It is valid for creep flow, often called Stokes flow, in which the inertial forces are much smaller than the viscous forces.",
"Flow is laminar, meaning that fluid particles follow smooth trajectories, without turbulent mixing.",
"It is often conceptualized as continuous flow lines around the object which is moving through the fluid or around which the fluid is moving.",
"The terminal settling velocity can be found by the force balance in the $z$ direction since the sphere is settling in direction of the gravitational force, against viscosity and buoyancy in a stationary fluid.",
"$\\sum F_z = F_D + F_b + F_g = ma$ Buoyancy force is proportional to the mass of fluid displaced by the settling object $F_b = m_{fluid} g$ While the gravitational force is $F_g = m_{sphere} g$ But the volume of displaced fluid and the volume of the sphere are the same and equal to $V_{sphere} = \\frac{4}{3} \\pi R^3$ If the density of the fluid is $\\rho _{fluid} = \\frac{m_{fluid}}{V_{fluid}}$ and that of the sphere is $\\rho _{sphere} = \\frac{m_{sphere}}{V_{sphere}}$ the buoyancy force can be written as $F_b = \\rho _{fluid} \\frac{4}{3} \\pi R^3 g$ while the gravitational force is $F_g = \\rho _{sphere} \\frac{4}{3} \\pi R^3 g$ Now in the drag force equation $a$ is R, the radius of the sphere and the terminal translational velocity can be written as $v_s$ such that $\\boxed{F_{D} = 6 \\pi \\mu v_s R}$ Replacing all the forces in Equation 5.24 and remembering that the terminal velocity is constant, i.e., there is no acceleration, $ma = 0$ , and $\\sum F_z = 6 \\pi \\mu v_s R + \\rho _{fluid} \\frac{4}{3} \\pi R^3 g - \\rho _{sphere} \\frac{4}{3} \\pi R^3 g = 0$ Where the coordinates systems establishes that downward forces have negative sign.",
"Solving for $v_s$ $6 \\pi \\mu v_s R = \\rho _{sphere} \\frac{4}{3} \\pi R^3 g - \\rho _{fluid} \\frac{4}{3} \\pi R^3 g \\\\6 \\pi \\mu v_s R = \\frac{4}{3}(\\rho _{sphere} - \\rho _{fluid}) \\pi R^3 g \\\\v_s = \\frac{2}{9} \\frac{ (\\rho _{sphere} - \\rho _{fluid}) g}{\\mu } R^2$ Since the sphere in this case is the solid particles settling, or solid minerals, we can rewrite using notation from previous chapters as $\\boxed{v_s = \\frac{2}{9} \\frac{ (\\rho _{p} - \\rho _{f}) g}{\\mu } R^2}$ This equation is know as Stokes law of settling and it is used to model the settling behavior of infinitely dispersed spherical solid particles settling in laminar regime.",
"Terminal velocity is a function of the squared radius of the particle, therefore, everything else constant in a viscous fluid, larger particles settle faster.",
"Settling velocity is also controlled by the densities of the fluid and the solid, viscosity and gravity.",
"Fluid density and viscosity are controlled by temperature and empirical laws should need to be used to estimate these quantities.",
"Stokes law of settling is valid for laminar flow, where $Re \\ll 1$ The limits are determined empirically so the critical value of the Reynolds number might depend on the nature of the problem being studied.",
"More specifically, there might be transitional flow regimes between laminar and fully turbulent.",
"The Reynolds number a settling spherical particle is $Re = \\frac{2\\rho _f R v_s}{\\mu }$ Considering a maximum Reynolds number for laminar flow in settling of $Re \\approx 1$ (this might be a conservative value), the maximum radius of a smooth spherical particle falling under laminar flow regime is $\\frac{\\mu }{2\\rho _{f} R} = \\frac{2}{9} \\frac{ (\\rho _{p} - \\rho _{f}) g}{\\mu } R^2 \\\\$ $R = \\@root 3 \\of { \\frac{9 \\mu ^2}{4 \\rho _f(\\rho _s-\\rho _f)g }}$ For water density of $\\rho \\approx 1000 $ kg m$^{-3}$ , and viscosity at 25 $^{o}$ C of $\\mu \\approx 0.00089 $ Pa s$^{-1}$ , the maximum radius of a smooth spherical quartz particle with $\\rho _s \\approx 2650 $ kg m$^{-3}$ and $g = 9.80665 m~s^{-2}$ falling under laminar flow regime is approximately $R = 4.793 \\times 10^{-5}~m $ or $R = 0.048~mm$ , which translates into a diameter of approximately $0.1~mm$ .",
"On the other hand of the spectrum, for colloidal particles, Stokes law is no longer valid because under real circumstances, the solution is not infinitely dispersed and the particles are subjected to thermal motion (also known as Brownian motion).",
"As smaller particles settle, a concentration gradient is established along the length of the settling column and a concentration gradient will keep particles in suspension as long as there is no coagulation or aggregation in suspension [60], [76].",
"In other words, the thermal motion is able to overcome gravitational forcesAlbert Einstein's Ph.D. dissertation was on Brownian motion, before he was made famous by his studies in relativity and the photoelectric effect.",
"The Péclet number is another dimensionless number in hydrodynamics which expresses the ratio between advective and diffusive transport rates [76].",
"If $Pe \\ll 1$ , diffusive transport predominates and settling is perturbed by thermal motion.",
"Considering $Pe = 1$ the radius of a sphere settling is given by $R = [\\frac{3 k T}{4 \\pi (\\rho _s - \\rho _f)g}]^{\\frac{1}{4}}$ In which $k = 1.380649 \\times 10^{-23}~J~K^{-1}$ is the Boltzmann constant and all other variables were previously defined.",
"Using the values for a spherical silicate mineral settling in water presented for the previous calculations, the radius above which diffusive transports can be consider to start affecting settling is $ r = 4.96 \\times 10^{-7} ~m$ or $r = 0.4965~\\mu m$ , or approximately $1~\\mu m$ .",
"The conclusion is that Stokes law for an average spherical mineral particle settling in water at room temperature is valid for diameters above that of colloidal particles and below that of fine sand particles.",
"Below the colloidal limit settling is hindered by thermal motion and above the fine sand limit settling is under turbulent regime.",
"Stokes law only applies to creeping (laminar) flow.",
"In reality, the dynamics of particles on the clay size range is much more complicated because of surface forces among the interacting particles.",
"These forces can cause aggregation of particles in suspension to which Stokes law applies to some degree and settling via coagulation and other mechanisms.",
"Stokes law in particle size characterization of soils and fine sediments assumes that aggregation and coagulation do not apply and the analysis should be disregarded if these processes are observed.",
"We saw in Chapter REF that when charged particles are put in contact with an electrolyte solution an Electric Double Layer (EDL) develops with charged ions adsorbed onto the surface by electrostatic forces.",
"The concentration of ions is larger at the surface and at certain concentrations the EDL extends far enough that repulsive forces predominated between charged particles and the colloidal suspension remains dispersed.",
"However, this is not the entire story.",
"Between particles, atoms and molecules, dispersion forcesThe term dispersion forces here refers to intermolecular and interatomic forces and not to particles in dispersion originating from quantum fluctuations arise.",
"These dispersion forces are grouped as what is called van der Waals forces (vdW) and for the systems we are concerned are attractive in nature.",
"If the electrolyte concentration is increased, the EDL is compressed and the vdW forces which are stronger than the repulsive forces in the short range predominate and the colloidal particles can aggregate and coagulate.",
"This can happen with dispersed soil solutions depending on the valence and concentration of the electrolyte as clays are charged particles of small diameter.",
"The theory of balance of these attractive and repulsive forces in colloidal suspensions is called DLVO after Boris Derjaguin, Lev Landau, Evert Verwey and Theodoor Overbeek who developed it, the first two working together and the last two independently.",
"Considering the interaction of two charged spheres of radius R in a 1:1 electrolyte solution, the energy of interaction due to electrostatic double layer forces is given by [52] $W_{EDL} = \\frac{64 \\pi k T R \\rho _{\\infty }}{\\kappa } \\tanh ^2 (\\frac{z e \\psi _0 }{4kT}) \\exp {(-\\kappa D^{\\prime })}$ in which the eletrochemical parameters are the same as in Chapter 3, i.e.",
"$\\rho _{\\infty }$ is the electrolyte concentration in bulk solution, $\\psi _0$ is the surface potential, $z$ is the valence of the ions, $k$ is the Boltzmann constant, $T$ is the absolute temperature, $\\kappa $ is a parameter related to the thickness of the double layer, and $e$ is the electronic charge, and $R$ is the spheres radius.",
"While the attractive vdW force is given for two spheres of the same radius as [52] $W_{vdW} = \\frac{A_HR}{12 D^{\\prime }}$ in which $D^{\\prime }$ is the separation distance usually in $nm$ or angstrons (Å) and $A_H$ is the Hamaker constant $A_H = \\pi ^2 C \\rho _1 \\rho _2$ In which $\\rho _1$ and $\\rho _1$ are the number densities of the interacting spheres and $C$ is the London constantFor the time being we will refrain from exploring the derivations of these relationships as they might require exploring quantum mechanical perturbation theory.",
"In the case of equal particles $\\rho _1 = \\rho _2$ and $\\rho _1\\rho _2 = \\rho ^2$ .",
"The Hamaker constant has an important role in the adsorption of water films in surfaces and is fundamental for modeling dry water adsorption in soils.",
"The pair interaction energy considering the two forces is $W(D^{\\prime }) = W_{EDL} - W_{vdW}$ Negative values of the pair interaction energy (usually given in J) indicate potential for van der Waals attraction and coagulation.",
"However the relationship is complex and depending on several of the factors seen in the preceding equations including temperature, which affects the kinetic energy of the colloidal particles, surface potential, which might change with factors such as pH as we discussed in Chapter REF and others.",
"The proper analysis must be done in terms of the energy barrier that needs to be overcome for coagulation to occur.",
"The electrolyte concentration at which coagulation occurs is the critical coagulation concentration (ccc) which plays a critical role in grain size analysis of soils, especially in the grain size analysis of oxidic soils which are variable charged and in which the analysis is almost always performed under dispersion using sodium hydroxide or other sodium salts [60].",
"If the constraints of settling velocity, lower limit of settling and dispersing agent concentration are observed, then grain size analysis by settling methods can be performed normally.",
"For the determination of particle size distribution and the proportion of the particle fractions we need some more or less arbitrary scale for classification of grain sizes (Table REF ).",
"For soils, the soil must be first dispersed chemically and or physically using salt solutions such as sodium hydroxide and/or sodium hexametaphoshate at concentrations high enough to create repulsive EDL forces but not above the ccc.",
"Mechanical energy is applied to the samples by stirring, ultrasonic waves, gas bubbles or shaking.",
"Several combinations of chemical and physical methods are possible, some more efficient than others [40].",
"Table: Classification of soil particles based on diameter .After dispersion, the sample is again vigorously stirred and the amount of particles in suspension is measured at prescribed times directly, by collecting a volumetric sample of the dispersion, or indirectly, by using a hydrometer or by transmission/diffraction to electromagnetic radiation methods as in laser and x-ray methods.",
"The most common direct method is the pipette method in which a sample is taken and oven dried and the mass in suspension measured using a precision scale.",
"The mass in suspension is then extrapolated for the entire volume.",
"As for the indirect methods the most commonly used is the hydrometer method.",
"Hydrometers measure the amount or particles in suspension by buoyancy, the greater the mass in suspension the greater the buoyancy force, as particles settle, the hydrometer will tend gradually sink in response to settling of particles until settling stops, when the hydrometer will be reading the amount of particles in suspension under equilibrium, or the density of the dispersing solution if the material is composed only of coarser particles.",
"Most hydrometers are nothing more than a weighed glass tube with a bulbous shape and a scale.",
"The scale is calibrated at the solution level and will read zero for pure water at a given temperature and as the mass of particles in suspension increases, the reading will increase in the downward direction indicating the mass of particles in suspension, usually in grams, per unit volume, usually liters.",
"For any settling based method, the settling velocity is given by Stokes law and the mass of particles greater than a given radius (or diameter) in suspension will depend on the sampling depth and on the calculated settling time.",
"From basic physics the velocity of a particle is given by $v_s = \\frac{\\Delta S}{\\Delta t}$ Because we are concerned with a spherical particle settling under Stokes regime, and therefore at a constant terminal velocity, $v_s$ is the settling velocity, $S$ is the settling depth and $t$ is the time for the particle to reach $S$ .",
"Replacing the velocity in Stokes law $\\frac{\\Delta S}{\\Delta t} = \\frac{2}{9} \\frac{ (\\rho _{p} - \\rho _{f}) g}{\\mu } R^2$ $\\Delta t = \\frac{9 \\mu \\Delta S}{2 (\\rho _{p} - \\rho _{f}) g R^2}$ and because it is convenient to set $t_0$ and $S_0$ to zero we have $\\boxed{t = \\frac{9 \\mu S}{2 (\\rho _{p} - \\rho _{f}) g R^2}}$ This equation implies that at time $t$ , all particles with radius greater than $R$ will have settled to a depth greater than $S$ .",
"In addition to the settling depth and the particle radius, the time for a reading for determining particle size distribution using a given method is dependent on the fluid viscosity, the settling mineral particle density, the density of the fluid and gravitation acceleration.",
"Particle density, discussed in Chapter REF , can be measured with relatively simple methods, while fluid viscosity and density will require more specialized high precision techniques.",
"It is much more convenient to perform the measurements at a standard temperature to which water viscosity and density have tabulated values or to use published empirical relationships for water viscosity and density, for example $\\mu = 1 \\times 10^{-6} [280.68 (\\frac{T}{300})^{-1.9} + 511.45 (\\frac{T}{300})^{-7.7} + 61.131 (\\frac{T}{300})^{-19.6} + 0.45903 (\\frac{T}{300})^{-40.0} ]$ in which $\\mu $ is water dynamic viscosity in Pa s and T is the temperature in Kelvin in the range of 253.15 K to 383.15 K [51], and $\\rho _f = 999.974950 [1 - \\frac{(Tc - 3.983035)^2(Tc + 301.797)}{522528.9(Tc + 69.34881)}]$ in which $\\rho _f $ is the density of water in kg m$^{-3}$ and Tc is the temperature in $^{o}$ C in the range of 0 $^{o}$ C to 40 $^{o}$ C [94].",
"For example, using the equations above at 25°C = 298.15 K, $\\mu = 0.000889997~ Pa~ s$ , $\\rho _f = 997.047~ kg~ m^{-3}$ , and using $\\rho _s $ and $g$ as in previous calculations, for the silt $D = 0.05~mm$ and clay $D = 0.002~mm$ limits, the time for all the particles with diameter greater than $D$ to settle 0.1 m (10 cm) will be $t_{clay} = 6.86~h$ and $ t_{silt} = 39.53~s $ .",
"Thus, a reading at $t_{silt}$ will indicate the amount of particles finer than 0.05 mm in suspension and the reading at $t_{clay}$ will indicate the amount of particles finer than 0.002 mm in diameter in suspension.",
"The particle size distribution curve is the measured cumulative mass fraction of particles in suspension (usually in %) at a given time plotted as a function of the logarithm of particle diameter corresponding to each time, given by $D = \\sqrt{\\frac{18 \\mu S}{ (\\rho _{p} - \\rho _{f}) g t}}$ Examples of hypothetical granulometric curves for clay and sandy soils are illustrated in Figure REF Figure: Grain size distribution curves for hypothetical clay and sandy soils.In soil science, and soil physics to some degree, once the mass based fraction of particles in the general classes clay ($D< 0.002~mm$ ), silt ($0.002~mm \\le D < 0.05~mm$ ) and sand ($0.05~ mm \\le D < 2.00~ mm$ ) is determined, the data can be entered into a ternary plot generally referred to as textural triangle and the nominal soil texture class determined (Figure REF ).",
"The nominal texture classes are a relic from soil pedology and are an attempt to classify soils using the “feeling” method in the field.",
"The feeling method can be useful to discern the general texture classes after extensive training of the field technician.",
"In broader terms, the soil sample is disagregatted, moistened to provide some degree of dispersion by manipulating it using the fingers and rubbed between the index and thumb so that the soil texture is literally felt by the person.",
"It can be useful as an expedite field method for estimating soil texture and for soil classification where laboratory methods might not be available.",
"Figure: Textural triangle for determination of soil texture class.",
"The direction of data entry for each class is indicated by the arrows.Grain size distribution and soil texture are important properties of soils because they will directly or indirectly affect almost all soil physical and chemical phenomena.",
"Aggregation and pore size distribution will affect water transport and storage as well as aeration and susceptibility to compaction.",
"The amount of clay will affect surface area the reactivity of soils as we discussed previously.",
"For a given mineralogy, increasing clay content, will tend to increase plasticity and susceptibility to compaction and compression under certain conditions.",
"ll $ p $ Pressure $ \\mu $ Viscosity $ \\mathbf {u} $ Vector velocity function $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k}$ Unit vectors in Cartesian coordinates $ \\mathbf {r}, \\theta , \\phi $ Unit vectors in spherical coordinates $ u_r, u_{\\theta }, u_{\\phi } $ Components of the velocity vector in spherical coordinates $ r, \\theta , phi $ Length and angle components in spherical coordinates $ \\Psi $ Stokes stream function $ E^2 $ Differential operator $ a $ Radius $ A, B, C, D $ Constants $ \\tau $ Stress components $ F_D $ Drag force $ U $ Velocity $ F_b $ Buoyancy force $ F_g $ Gravitational force $ m $ Mass $ \\rho $ Density $ R $ Radius $ v_s $ Settling velocity $ Re $ Reynolds number $ Pe $ Péclet number $ T $ Temperature $ k $ Boltzmann constant $ W_{EDL} $ Energy of interaction due to double layer forces $ \\rho _{\\infty } $ Electrolyte concentration in bulk solution $ \\kappa $ Parameter related to the thickness of the double layer $ z $ Valence of the ion $ e $ Electronic charge $ W_{vdW} $ Attractive van der Waals force $ A_H $ Hamaker constant $ D^{\\prime } $ Separation distance $ \\rho _1, \\rho _2 $ Number densities of interacting spheres $ C $ London constant $ W(D^{\\prime }) $ Pair interaction energy $ S $ Settling distance $ T_c $ Temperature in Celsius $ D $ Particle diameter Because soils and sedimentary deposits are composed of solid, unconsolidated particles, mechanical stresses caused by foundations, plant growth, traffic of machines, animals and people, overburden of confining layers, water in pores, clay expansion and contraction and many other natural and man-made processes can cause reorganization of particles and depending on the circumstances, compaction, consolidation or compression or increase in pore volume by processes such as loosening.",
"Before we consider aggregation of soil primary particles and organic matter, it is convenient to treat porous media as an idealized stacking of spheres.",
"This is directly applied to round, homogeneous, coarse grained particles in a few soils and sedimentary deposits and can be an important tool in modeling water retention and transport phenomena, but it is less representative to heterogeneous particle size distributions and as the particle shapes deviate from an ideal sphere.",
"Treating porous media as a set of equally sized spherical particles, the porosity or pore fraction of the material will depend on how the spheres are organized in space.",
"There are more and less efficient ways of packing spheres, a pile of oranges in a supermarket tray is an example of a packing system.",
"The packing models are organized in lattices which can be defined as a repetitive arrangement in space.",
"The lattices seem in this chapter are directly applied to the organization of atoms in space seen in Chapter REF and the structure of minerals is organized in lattices, often following the systems below, but usually with more than one type of chemical element, each element represented by different spheres [44].",
"The systems described below can be applied to soil aggregates to some extent.",
"Soil aggregates are usually less resistant to soil stresses than most primary particles and will be discussed in this chapter.",
"The most common sphere packing arrangements are described below.",
"The packing density is derived from geometric arguments.",
"I. Cubic Lattice In the case of cubic lattice, or simple cubic lattice, the spheres touch and the distance between the spheres centers is two times the radius ($R$ ).",
"The volume of the unit cell considered is the cube that unites the points at the centers of the eight spheres (Figure REF ).",
"Considering the side of the cube as $A = 2R$ , the volume of the unit cell, the volume occupied by spheres and the packing density, given by the volume occupied by the spheres over the unit cell volume are given by Figure: Cubic lattice.$V_{cell} = A^3 = (2R)^3 = 8R^3 \\\\V_{spheres} = 8 \\cdot \\frac{1}{8} \\cdot \\frac{4}{3} \\pi R^3 = \\frac{4}{3} \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{\\frac{4}{3} \\pi R^3}{8R^3}$ $\\eta = \\frac{\\pi }{6} = 0.52359 ...$ In which $V_{cell}$ is the volume of the unit cell, $V_{sphere}$ is the corresponding volume of spheres located within the unit cell and $\\eta $ is dimensionless packing density.",
"Notice that within the unit cell there are eight $1/8$ of spheres such that the total volume of spheres within the unit cell is $8 \\cdot 1/8 = 1$ sphere.",
"Similar logic will be used in the following packing schemes.",
"II.",
"Face-Centered Cubic In this arrangement, an additional sphere is located on the center of each face (Figure REF ).",
"Now the side $A$ is no longer $2R$ as additional space is needed between the spheres in the corners to accommodate the spheres on the faces.",
"Notice that the simplified ball-and-stick representation is used, but each corner and face is occupied by a large sphere of radius $R$ .",
"The color red in indicates the corner spheres and the light green indicates the face spheres (Figure REF ).",
"Figure: Face-centered cubic packing.Because of the additional space, the side of the unit cell is given by $A = 2 \\sqrt{2} R$ and the volumes and packing density can be calculated as $V_{cell} = A \\cdot A \\cdot A = (2 \\sqrt{2} R)^3 = 16 \\sqrt{2} R^3\\\\V_{spheres} = (8 \\cdot \\frac{1}{8} + 6 \\cdot \\frac{1}{2}) \\cdot \\frac{4}{3} \\pi R^3 = \\frac{16}{3} \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{\\frac{16}{3} \\pi R^3}{16 \\sqrt{2} R^3}$ $\\eta = \\frac{\\pi }{3 \\sqrt{2}} = 0.74048 ...$ Notice that the packing density is considerably higher and the porosity, given by $ 1 - \\eta $ is considerably lower, around 25%, when compared to the simple cubic lattice.",
"III.",
"Body-Centered Cubic In this case there is one sphere in each corner and one sphere (represented in blue) in the center of the structure (Figure REF ).",
"Notice that the sides $A$ are still equal on the cube, but the diagonals are formed by the radius of the spheres in the transverse corners plus the diameter of the sphere in the center ($2R$ ) totaling $4R$ .",
"From the triangle formed by the height $A$ , the diagonal $4R$ , the diagonals in each face can be calculated from simple geometric arguments resulting in $d = \\sqrt{16 R^2 - A^2}$ .",
"Again the relationship between $A$ and $R$ can be calculated from the diagonal of each face and the sides $A$ using simple geometric arguments resulting in $ A = 4R/\\sqrt{3}$ .",
"The volumes and packing density can then be calculated as before Figure: Body-centered cubic packing.$V_{cell} = (\\frac{4R}{\\sqrt{3}})^3 = \\frac{64}{3\\sqrt{3}} R^3 \\\\V_{spheres} = (8 \\cdot \\frac{1}{8} + 1) \\cdot \\frac{4}{3} \\pi R^3 = \\frac{8}{3} \\pi R^3\\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{\\frac{8}{3} \\pi R^3}{\\frac{64}{3\\sqrt{3}} R^3}$ $\\eta = \\frac{\\pi \\sqrt{3}}{8} = 0.68017 ...$ IV.",
"Hexagonal Lattice Now on the two hexagonal packing schemes presented here, the geometric polyhedron considered is no longer a cube but an hexagonal prism.",
"Looking at one layer of packed spheres from above it is clear that the center of the spheres can be linked to form an imaginary hexagon (Figure REF ).",
"The height of the hexagonal prism will depend of how the additional layers of spheres are stacked.",
"If the additional layer of spheres is staked directly on top of the first layer, an hexagonal packing lattice is formed (Figure REF ).",
"Notice that because the spheres on the base touch, the base edge, $s$ , is $2R$ and because the next layer is stacked directly above the first layer, the height of the hexagonal prism linking the center of the group of spheres is also $2R$ .",
"Recalling that the volume of an hexagonal prism is given by $ (3 \\sqrt{3}/2) s^2 h $ , the volume of the unit cell is the volume of the hexagonal prism and the packing density can be calculated as $V_{cell} = \\frac{3}{2} \\cdot \\sqrt{3} \\cdot s^2 \\cdot h = \\frac{3}{2} \\cdot \\sqrt{3} \\cdot (2R)^2 \\cdot (2R) = 12 \\sqrt{3} R^3\\\\V_{spheres} = 3 \\cdot \\frac{4}{3} \\pi R^3 = 4 \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{4 \\pi R^3}{12 \\sqrt{3} R^3}$ $\\eta = \\frac{\\pi }{3 \\sqrt{3}} = 0.64599 ...$ Notice that there are 12 spheres in the corners of the hexagonal prism and each one contributes to 1/6th of the volume within the unit cell, plus two half spheres in the upper and lower faces, thus we have $ 12 \\cdot 1/6 + 2 \\cdot 1/2 = 3$ spheres.",
"Figure: Spheres plane in an hexagonal arrangement.Figure: Hexagonal lattice.V.",
"Hexagonal Close Packing The hexagonal close packing is similar to hexagonal packing but there are three spheres located in between the two staked planes (Figure REF a).",
"It is important to note that these three spheres will also form a plane of spheres when multiple layers are considered in three dimensional space, in addition to the two planes of spheres represented in red (Figure REF a).",
"The calculation of the volume of the unit cell is more complicated than in the hexagonal packing lattice because the distance between the centers of the spheres in the two planes is no longer $2R$ .",
"Consider a tetrahedron formed by the central sphere in the base plus two spheres on the edges forming the base of the tetrahedron and one of the spheres in the middle plane as the upper vertex (Figure REF b).",
"Considering the sides of the base of the tetrahedron as $a$ , the distance to the point of projection of the vertex sphere $b$ can be calculated from the internal angle and the position at the middle point of the side of the base $a/2$ (Figure REF c) Figure: Hexagonal close packing.$\\cos {30^{o}} &= \\frac{a/2}{b}\\\\b &= \\frac{a}{\\sqrt{3}}$ The height $d$ to the plane of spheres in the middle of the unit cell is then given by (Figure REF d) $a^2 &= d^2 + b^2 = d^2 + (\\frac{a}{\\sqrt{3}})^2 = d^2 + (\\frac{a^2}{3})\\\\d &= a \\frac{\\sqrt{2}}{\\sqrt{3}}$ Because the sides of edge of the base of the hexagonal prism is $a = 2R$ and the total distance between the two planes in red in Figure REF a is $2d = 2 \\cdot 2R \\frac{\\sqrt{2}}{\\sqrt{3}}$ , the total height of the hexagonal prism are $h^{\\prime }_T = 4R \\frac{\\sqrt{2}}{\\sqrt{3}}$ and the volumes and packing density of the hexagonal close packing lattice are given by $V_{cell} = \\frac{3}{2} \\cdot \\sqrt{3} \\cdot (2R)^2 \\cdot (4R\\frac{\\sqrt{2}}{\\sqrt{3}}) = 24R^3\\sqrt{2} \\\\V_{spheres} = 6 \\cdot \\frac{4}{3} \\pi R^3 = 8 \\pi R^3 \\\\\\eta = \\frac{V_{sphere}}{V_{cell}} = \\frac{8 \\pi R^3}{24R^3\\sqrt{2}}$ $\\eta = \\frac{\\pi }{3 \\sqrt{2}} = 0.74048 ...$ Notice that the packing density is the same as the Face Centered Cubic.",
"In essence the difference between these two systems is geometrical, as in the way the layers of spheres are conceptualized and organized.",
"One system can be transformed into the other by rotation of coordinates.",
"In soil science, aggregation refers to the union of primaryThe meaning here is not as in primary minerals but in the basic building blocks of soil aggregates mineral particles clay, silt and sand with organic compounds and various amorphous materials forming clusters with shapes and sizes that are more or less constant in certain naturally occurring soil types and depths, in response to soils forming processes.",
"Soil structure generally refers to the shape, size and structural stability of these aggregates and larger domains formed by the set of aggregates in a given soil.",
"In natural soils, structure is a direct product of soil forming processes, namely climate, organisms, relief, parent material and time.",
"In fine textured soils, close to surface, the structure tends to be composed of larger aggregates and with higher porosity, especially in humid tropical climates where there is higher biological activity of plants, animals and other organisms.",
"Aggregates in natural soils also tend to be more stable than in areas disturbed by human activity.",
"Human activity, especially agriculture, tends to disturb soil structure, destroying large size aggregates due to tillage and chemical modification on soil upper layers.",
"The destruction of aggregates and soil structure is usually a physical and a chemical process.",
"Traffic of animals, humans and machinery tends to compress and break soil aggregates, causing increases in bulk density and decrease in porosity, creating an environment that is less adequate to biological activity which further causes reduction in creation of new aggregates and pores, and loss of organic matter.",
"Agricultural practices also employ a variety of tillage implements which in some cases are directly designed to destroy soil structure for sowing operations, while others tend to destroy soil structure indirectly.",
"Chemically, the input of ionic chemicals such as fertilizers and other contaminants can cause dispersion by the surface chemistry mechanisms discussed in Chapter REF which decreases soil structural stability on one end while on the other some compounds can also kill soil organisms, contributing to loss of organic matter and biological activity, further increasing soil structural degradation.",
"We will not discuss tillage systems and soil quality in this book.",
"There are a lot of sociological, ecological, philosophical and political issues related to soil quality, tillage and agriculture that rely on personal opinion of different groups none of which we have any interest in supporting.",
"Water retention due to capillary phenomena tends to be controlled by the pore size distribution and pore shape in soils and these are to a larger extent controlled by the size and shape of soil aggregates.",
"In most soils aggregation creates at least two pore domains, the intra-aggregate pores and inter-aggregate pores, intra-aggregate pores tends to be larger and usually hold water by capillary phenomena at lower energies of retention when compared to inter-aggregates smaller pores.",
"In soils with high amounts of clay, these inter-aggregate pores tend to be very small and as such, water retention in them can have contribution of both capillary and adsorption phenomena.",
"Water retention can be modeled in terms of aggregation and pore size distribution in self-similar and fractal models for example [77].",
"Formation of aggregates is a slow and complex process dependent on several factors.",
"There are several theories of soil aggregate formation (e.g.",
"[95]), but overall it is thought that aggregation in most soils is dependent on biological activity and organic matter which creates physical and chemical cementing agents contributing to uniting primary particles in larger structural domains.",
"Biological activity can directly contribute to formation of aggregates as some organisms such as termites and earthworms produce aggregate like structures by ingesting or manipulating soil organic matter and mineral particles.",
"From the chemical point of view, charged surfaces in soil particles, colloidal organic matter, and ions can create aggregating and dispersing forces during aggregate formation.",
"Higher valence cations tend to favor aggregation while monovalent high hydrated radius such as sodium tend to favor dispersionAs we seen, this depends also on the concentration in solution.",
"Physically, stresses caused by roots and organisms and mechanical stresses caused by natural processes such as soil drying, expansion and shrinking of soil minerals due to loss of interlayer water as well as capillary forces related to drying can also contribute to aggregation of primary particles.",
"Soil structure, shapes, sizes and stability of aggregates, total porosity and pore size distribution are all affected by stresses applied to soils by human activities.",
"The modifications caused by these activities will in turn affect soil functioning.",
"In many engineering applications such as in the construction of earth dams and foundations, the increase in bulk density and reduction of porosity to a minimum are usually desired while in agriculture, where the soil is viewed as a medium to plant growth, porosity, pore size and distribution must be at an ideal state for root growth, water storage, water and air circulation and biological activity.",
"Much of the theory behind soil porosity reduction and increase in bulk density in response to applied stresses comes form geotechnical engineering, specific from the discipline of Soil Mechanics.",
"In soil mechanics, compaction is the increase in density of a soil by reduction of the pore volume occupied by air, consolidation is a reduction in volume of a fully saturated, low permeability soil due to drainage of soil water on the pores, and compression is a reduction of volume of a soil under compressive stress [29], [33].",
"Although we have not discussed soil permeability to water, it can be understood as a coefficient which measured how easy it is for water to flow out of the soil under a pressure gradient.",
"On the geotechnical engineering literature, consolidation applies to low permeability soils, such as soils with high phylossilicate clays content, such that there is a time lag between the application of the load and the extrusion of water from pores and soil settlement [33].",
"We have studied the concept of stress tensor applied to fluids in porous media in Chapter REF and now we need to apply the same concept to stresses in a solid (Figure REF ).",
"Imagining a volume of soil shaped like a cube, forces can be applied to each surface can be parallel or perpendicular to each face.",
"The force component divided by the area over which it is acting acting defines the stress $\\sigma $ (Figure REF ).",
"As with the stress tensor in a fluid, the stress components in each direction, $x$ , $y$ and $z$ can be further decomposed into three components, one for each original direction.",
"Therefore, the stress components acting on the face perpendicular to the direction $x$ are $\\sigma _{xx}$ , $\\sigma _{xy}$ and $\\sigma _{xz}$ and so forth for the other three directions.",
"The full stress tensor is, as before $\\sigma _{ij} =\\begin{bmatrix}\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\\end{bmatrix}$ The perpendicular (or normal) stresses $ \\sigma _{ii} = \\sigma _{xx} = \\sigma _{yy} = \\sigma _{zz} $ are equivalent to pressures, or force over area ($F/A$ ) acting over each surfaces while the stresses parallel to each face $ \\sigma _{ij}, ~ i \\ne j$ are called the shearing stresses, also $F/A$ , in units analogous to pressure units.",
"Stresses applied by foundations, traffic of animals and machinery, tillage and other agricultural operations are usually multi-directional and have normal and shearing components, especially in non-rigid, granular porous materials such as soils and sedimentary deposits.",
"In some cases the stresses can be modeled as unidimensional pressures as in modeling compaction by agricultural machinery and animal traffic.",
"Figure: Components of stress in a tridimensional solid.If we imagine a foundation, a tire or an animal foot applying a vertical force to a soil, the response of the soil will depend on the magnitude of the applied force and on the mechanical strength of the soil.",
"The mechanical strength of the soil will depend directly and indirectly on a series of factors such as soil structure, grain and aggregate strength, grain size distribution, porosity, water content, mineralogy, story of previous applied loads and many other internal mechanisms.",
"If the applied stress is greater than the mechanical strength of the soil, deformation will occur.",
"Considering only unidimensional compressive stresses on a soil, deformation will correspond to a decrease in porosity.",
"There are different mechanisms and regimes of deformation in soils, a soil can suffer elastic deformation when the soil returns to its original state after the force is removed or plastic deformation when the soils does not returns to its original state after the force is removed.",
"On a macro scale, in soils subjected to unequal stresses in multiple directions, rupture or compressive failure can also occur, although these concepts are more applied to rigid solids.",
"However, internally, rupture of aggregates due to compressive stresses is an important mechanism of soil compaction, even under uniaxial loads.",
"As larger aggregates break the overall strength of the soil decreases and the volume of pores also decreases, as the structural units are destroyed and the pores filled with smaller aggregates or loose particles.",
"Dry soils are more subjected to elastic deformation and rupture while wet soils tend to become plastic above a given water content depending on specific characteristics of the soil, like grain size distribution and mineralogy.",
"Compaction in dry soils can be caused by reorganization of aggregates and individual particles, as seen in the packing models at the beginning of this chapter, rupture of aggregates and in some cases of individual mineral particles under load, and elastic deformation of aggregates and individual particles (usually sand and coarser fractions) under elevated loads.",
"Usually aggregates in dry soil tend to be more brittle, i.e.",
"they can fracture with little deformation, while as the water content increases, water molecules act as lubricants within particles and aggregates, contributing to plastic deformation.",
"The set of critical water contents at which a fine soil transition to different consistency states is called Atterberg limits.",
"The Atterberg limits are used in engineering and in agricultural sciences to predict soil behavior in the field for construction, traffic and tillage.",
"In simple terms the principal soil conceptual consistency limits considered in soil physics and soil mechanics are [33] Shrinkage limit: Most fine texture soils, especially soils rich in smectites, shrink as the soil dries.",
"This shrinking can be thought to be related to two main mechanisms, first in phylossilicate clays, water is lost from the interlayer planes causing shrinkage at the mineral structure level which reflects on the macrostructure, second, drying tends to cause internal stresses between particles and aggregates due to forces related to meniscus in pore water water.",
"These forces can pull the particles together as the soil dries and the radius of the meniscus becomes smaller increasing the capillary forces.",
"The shrinkage limit is the water content below which decrease in water content will not cause further volume reduction of the soil (i.e.",
"shrinkage).",
"A soil in a water content below the shrinkage limit is said to be in a solid state and it is said to be in a semisolid state at water contents above the shrinkage limit and below the plastic limit.",
"Plastic limit: The water content below which the soil ceases to display plastic behavior.",
"In practical terms the soil will start to crumble or fracture if manipulated when below the plastic limit.",
"Above the plastic limit the soil is subjected to plastic deformation and is in a plastic state.",
"Liquid limit: It is the water content above which the soil changes from a plastic state to a liquid sate.",
"In practical terms, above the liquid limit the water content is such that the soil will begin to flow similar to a liquid under stress.",
"Shrinkage limit: Most fine texture soils, especially soils rich in smectites, shrink as the soil dries.",
"This shrinking can be thought to be related to two main mechanisms, first in phylossilicate clays, water is lost from the interlayer planes causing shrinkage at the mineral structure level which reflects on the macrostructure, second, drying tends to cause internal stresses between particles and aggregates due to forces related to meniscus in pore water water.",
"These forces can pull the particles together as the soil dries and the radius of the meniscus becomes smaller increasing the capillary forces.",
"The shrinkage limit is the water content below which decrease in water content will not cause further volume reduction of the soil (i.e.",
"shrinkage).",
"A soil in a water content below the shrinkage limit is said to be in a solid state and it is said to be in a semisolid state at water contents above the shrinkage limit and below the plastic limit.",
"Plastic limit: The water content below which the soil ceases to display plastic behavior.",
"In practical terms the soil will start to crumble or fracture if manipulated when below the plastic limit.",
"Above the plastic limit the soil is subjected to plastic deformation and is in a plastic state.",
"Liquid limit: It is the water content above which the soil changes from a plastic state to a liquid sate.",
"In practical terms, above the liquid limit the water content is such that the soil will begin to flow similar to a liquid under stress.",
"Atterberg limits are strongly dependent on mineralogy and soil texture.",
"The main goal in agricultural soils is that traffic and other agricultural operations are to be performed below the plastic limit.",
"Plastic deformations are related to structural degradation and compaction that is usually expensive and labor intensive to recover.",
"Above the liquid limit machinery can sink and become stuck in some conditions.",
"In practice however, as the soil dries the mechanical resistance can become such that it is impossible or not economically viable to perform tillage or sowing operations.",
"Thus, the limits for trafficability and workability in soils can be narrow, either due to excess water content, turning the soil plastic and subjected to compaction or consolidation, or due to water content that is too low for efficient tillage operations.",
"Soils that are too plastic, such as those rich in smectites, are often too sticky, requiring high amounts of energy for operations or even being impossible to till under certain conditions In many practical applications, either agricultural, military or geotechnical engineering, a simple and easy approach to estimate soil mechanical resistance is the use of cone penetrometers.",
"The principle is very simple, a rod with a cone at its end is inserted into the soil and the mechanical resistance in terms of stress or force required is measured.",
"Devices based on this principle are called penetrometers.",
"There are many different designs of penetrometers, either analogical or with digital sensors, for measuring force and data collecting.",
"Soil mechanical resistance measured by a penetrometer can give an idea of the mechanical resistance of the soil to withstand loads and can be used to estimate trafficability.",
"In agricultural sciences, the resistance to penetration or cone index is used as a proxy to the mechanical resistance encountered by plant roots.",
"The mechanisms of root growth are complex and multiple and it is wrong to attribute critical limits of soil resistance to root growth based on penetrometer readings.",
"In practical terms, a penetrometer can be used as one parameters, among many, for assessing soil compaction and decision making in terms of compaction remediation measures.",
"Penetrometer resistance measurements should never be used without checking soil water content and bulk density as it is an exponential function of these two soil variables (Figure REF ).",
"An empirical relationship between soil penetration resistance, bulk density and water content usually encountered on the literature is [62] $SR = a_0 \\rho ^{a_1} \\theta ^{a_2}$ In which $SR$ is the soil mechanical resistance, usually in stress (analogous to pressure) units, $\\rho $ is soil bulk density, $\\theta $ is the volumetric water content, and $a0$ , $a1$ and $a2$ are empirical parameters commonly fitted using regression techniques such as least squares regression.",
"This equation has useful practical applications for adjusting $SR$ to a given water content and bulk density so that comparisons between different areas and conditions can be made, and also has pedagogical applications, illustrating the exponential relationship between $SR$ and $\\theta $ and $\\rho $ .",
"However, you must be aware that it is an empirical relationship and has no physical basis and was not derived based on first principles and thus does not explain the physical basis of the relationship between resistance, density and water content.",
"As such, it cannot be extrapolated to different soils, or to conditions not observed on the fitting data.",
"First principles derivations of the $SR = f(\\rho , \\theta )$ relationship should be possible accounting for internal stresses of the soil provided by cohesive forces between particles and within aggregates, and adhesive forces generated by water in contact with solid surfaces.",
"These forces will also depend on soil texture, mineralogy, organic matter content among other soil properties.",
"Figure: Soil penetration resistance as a function of bulk density and volumetric water content based on empirical data.",
"The parameters of Equation 6.8 used are within the ranges for real soils, a 0 =0.01a_{0} = 0.01, a 1 =5.0a_{1} = 5.0 and a 2 =-1.0a_{2} = -1.0.ll $ V_{cell} $ Volume of the unit cell $ V_{spheres} $ Volume of spheres inside the unit cell $ A $ Side length in cubic unit cells $ R $ Radius of spheres $ \\eta $ Packing density $ s $ Side length in an hexagonal prism $ h $ Height of the hexagonal prism $ a, b, c $ Parameters for calculation in an hexagonal close packing system $ h^{\\prime }_T $ Height of the unit cell in an hexagonal close packing system $ \\sigma _{ij} $ Stress tensor $ SR $ Soil penetration resistance $ \\rho $ Soil bulk density $ \\theta $ Volumetric water content $ a_0, a_1, a_2 $ Empirical parameters on the soil penetration resistance function On the bulk of a pure, static liquid in thermal equilibrium, all molecules are subjected to the same forces, they interact with their neighbors through intermolecular surface forces which, when averaged over time, are constant in all directions and thus, the resultant of these forces is zero.",
"This is not true when there is a discontinuity between the fluid and a different fluid, liquid, solid or gas.",
"On an interface with a gas, the balance of forces is altered such that a resulting force on the direction of the interior of the fluid appears.",
"Because the molecules at the interface are surrounded by other liquid molecules at the liquid end, but not at the gas end, resulting cohesion forces pulling the molecules in direction of the liquid appear.",
"Intermolecular forces also generate the adhesion and adsorption forces between the liquid and solid surfaces.",
"The surface tension at the interface can be conceptualized using the thermodynamics or the mechanics frameworks.",
"The mechanics framework states that, because of these interfacial forces, the external work at constant volume and temperature, $\\delta W$ , required to increase the surface area by an infinitesimal amount $dA$ is directly proportional to the increase in area with the proportionality coefficient being called surface tensionThe symbol $\\delta $ with respect to W instead of d is because of thermodynamic considerations, $d\\hspace*{-0.80002pt}\\bar{}\\hspace*{1.00006pt}$ is also often used.",
"Heat and work in thermodynamics are imperfect differentials.",
"See Callen (2005).",
"$\\delta W = \\gamma dA$ In which if the energy $W$ is in $J$ and the area in $m^2$ , and the surface tension coefficient needs to have units of $J~m^{-2}$ , which is consistent with the definition of energy per unit surface area.",
"The surface tension is very often conceptualized using the wire and soap bar analogy and experiment.",
"Imagine that you have a square wire with one of the sides movable, now imagine a soap bubble is created within the square, if the square has a fixed dimension $l$ and a movable dimension $x$ , the area can be increased by increasing or decreasing the length of $x$ , such that $dA = ldx$ this implies that $\\delta W = \\gamma l dx$ In this case, the surface tension coefficient is usually conceptualized as a force per unit length in $N~m^{-1}$ .",
"Note that $1~J = 1~N~m$ such that $1~N~m^{-1}$ = $1~J~m^{-2}$ Now lets consider the liquid, the gas and the interface as a thermodynamical system.",
"Because the variables of interest in this system are temperature and volume, the thermodynamic potential of interest here is the Helmholtz free energy, $\\mathcal {F}$ [24], [75] $d\\mathcal {F} = -SdT - pdV + \\mu _i dN_i$ Since the temperature and number of moles are kept constant, $dT$ and $dN_i$ are zero, and the equation reduces to $d\\mathcal {F} = - pdV$ Since we know from thermodynamics that $\\delta W = - pdV$ we have, for an interface $d\\mathcal {F} = d\\mathcal {F}_1 + d\\mathcal {F}_2 + d\\mathcal {F}_3 = - \\delta W_1 - \\delta W_2 - \\delta W_3$ in which 1 and 2 refer to the solid and liquid phases, respectively, and 3 to the interface phase.",
"Considering that Equation 7.2 is known experimentally, and that the work in that case is negative because it is carried by the interface instead of by an external agent we have, after some reorganizing $- \\delta W_1 - \\delta W_2 - \\delta W_3 = - p_1dV_1 - p_2dV_2 + \\gamma dA \\\\- p_1dV_1 - p_2dV_2 + d F_3 = - p_1dV_1 - p_2dV_2 + \\gamma dA \\\\d \\mathcal {F}_3 = \\gamma dA \\\\\\gamma = \\frac{d \\mathcal {F}_3}{dA}$ Rewriting in a more elegant symbology $\\boxed{\\gamma = \\bigg (\\frac{\\partial \\mathcal {F}_{int}}{\\partial A}\\bigg )_T}$ The surface tension coefficient is the change in Helmholtz free energy of the interface ($\\mathcal {F}_{int}$ ) per unit area at constant temperature [75].",
"Notice that we have not yet discussed the concept of thermodynamic potential because it would take us to a long detour which is not strictly necessary at this point.",
"This long detour will be necessary when we discuss the concept of soil water potentials later in this chapter.",
"Capillary theory is based on developments by Thomas YoungYoung, T. 1805.",
"An essay on the cohesion of fluids.",
"Transactions of the Royal Society of London.",
"95:65-87. who laid the descriptive foundations of the process, Pierre LaplaceLaplace, M. 1805.",
"Traité de mécanique céleste.",
"Tome Quatrième.",
"Courcier, Paris.",
"who provided a rigorous initial mathematization and Carl Fredrich GaussGavss, C.F.",
"1830.",
"Principia generalia theoriae figvrae flvidorvm in statv aeqvilibrii.",
"Gottingae.",
"who unified both descriptions in terms of the virtual work principle, based on the principle of least action.",
"The works are based on a rigorous mathematical description and occupy several pages of works in French for Laplace and Latin for Gauss.",
"Like most of what is now known as the core of soil physics, simplifications and ad-hoc derivations came later in order to provide more palatable descriptions.",
"The Young-Laplace equation is a differential equation of the form $\\Delta p = \\gamma \\nabla \\cdot \\mathbf {n}$ in which $\\nabla \\cdot \\mathbf {n}$ represents the mean curvature and can be calculated using differential geometry for different shapes of the interfacial surface.",
"For reasons of simplicity and elegance we will present here the derivations of Landau and Lifshitz [57] followed by that of Defay and Prigogine [34].",
"Suppose you have a curved surface with two principal radius of curvature $R_1$ and $R_2$ on an interface between two media.",
"A limiting case would be a sphere in which $R_1 \\equiv R_2$ .",
"An element of surface area $dA$ is defined in the interface between the two mediums (Figure REF a).",
"The area element can be calculated as the product of two elements of circumference of a circle, defined in terms of the two radii $R_1$ and $R_2$ as $dl_1$ and $dl_2$ .",
"Suppose now that the surface area is increased by displacing the surface by an amount $\\delta \\zeta $ in the direction of the normal of the surface $dA$ (Figure REF ).",
"The two elements of circumference are now $dl_1^{\\prime }$ and $dl_2^{\\prime }$ and the surface area is increased from $dA = dl_1dl_2$ to $dA^{\\prime } = dl_1^{\\prime }dl_2^{\\prime }$ .",
"An element of volume between the surfaces is $dV = \\delta \\zeta dA$ .",
"If the pressures on the two media are $p_1$ and $p_2$ , the work necessary for the volume increase is related to the pressure difference between the two surfaces by $\\delta W_v = \\int p dV = \\int (-p_1 + p_2) dV = \\int (-p_1 + p_2) \\delta \\zeta dA$ and the total work is the work necessary for the volume increase added to the work necessary for surface expansion, as described on the previous section, thus $\\delta W = \\delta W_v + \\delta W_s = \\int p dV + \\gamma \\delta A = -\\int (p_1 - p_2) \\delta \\zeta dA + \\gamma \\delta A$ in which $\\delta A$ is a change in the area of surface.",
"As we have two areas $dA$ and $dA^{\\prime }$ , the change in surface area after displacement, $\\delta A$ , is needed.",
"Suppose that the length element $dl_1$ is increased to $dl_1^{\\prime }$ after the $\\delta \\zeta $ displacement in the direction of the normal.",
"Considering an element of angle $\\theta $ , and using the circumference of a circle formula (Figure REF b) we have $dl_1 = \\theta R_1$ and $dl_1^{\\prime } = \\theta (R_1 + \\delta \\zeta )$ Isolating $\\theta $ in the first equation and replacing into the second we have $dl_1^{\\prime } = \\frac{dl_1}{R_1} (R_1 + \\delta \\zeta ) = (1 + \\frac{\\delta \\zeta }{R_1}) dl_1$ Using the same argument in $dl_2$ $dl_2^{\\prime } = \\frac{dl_2}{R_2} (R_2 + \\delta \\zeta ) = (1 + \\frac{\\delta \\zeta }{R_2}) dl_2$ Now $dA^{\\prime }$ can be calculated in terms of $dl_1$ and $dl_2$ $dA^{\\prime } & = dl_1^{\\prime }dl_2^{\\prime } \\\\& = (1 + \\frac{\\delta \\zeta }{R_1}) dl_1 (1 + \\frac{\\delta \\zeta }{R_2}) dl_2 \\\\& = dl_1dl_2 + dl_1dl_2 \\frac{\\delta \\zeta }{R_2} + dl_1dl_2 \\frac{\\delta \\zeta }{R_1} +dl_1dl_2 \\frac{\\delta \\zeta }{R_1} \\frac{\\delta \\zeta }{R_2}$ Neglecting the much smaller second order term $\\delta \\zeta \\delta \\zeta = \\delta \\zeta ^2$ $dA^{\\prime } = (1 + \\frac{\\delta \\zeta }{R_1} + \\frac{\\delta \\zeta }{R_2}) dl_1dl_2 \\\\$ Now the element change in area as the surface is displaced is $da & = dA^{\\prime } - dA \\\\& = (1 + \\frac{\\delta \\zeta }{R_1} + \\frac{\\delta \\zeta }{R_2}) dl_1dl_2 - dl_1dl_2 \\\\& = (\\frac{\\delta \\zeta }{R_1} + \\frac{\\delta \\zeta }{R_2}) dl_1dl_2$ The total change in area of the surface of separation is $\\delta A = \\int da = \\int (\\frac{1}{R_1} + \\frac{1}{R_2}) \\delta \\zeta dA$ and replacing into Equation 7.10 $\\delta W = -\\int (p_1 - p_2) \\delta \\zeta dA + \\gamma \\int (\\frac{1}{R_1} + \\frac{1}{R_2}) \\delta \\zeta dA$ Because the displacement is required to occur in thermodynamic equilibrium, $\\delta W = 0$ and we can write $\\int [ (p_1 - p_2) - \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2}) ] \\delta \\zeta dA = 0$ In which, as in other cases seen before, the integral vanishes identically, thus $\\boxed{\\Delta p = p_1 - p_2 = \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2})}$ This is the Young-Laplace equation for a curved interface with constant radii $R_1$ and $R_2$ between two media.",
"For a sphere in which $R \\equiv R_1 \\equiv R_2$ it reduces to $\\boxed{\\Delta p = p_1 - p_2 = \\frac{2 \\gamma }{R}}$ Figure: Schematics for the energy derivation of the Young-Laplace equation.The derivation from Defay and Prigogine [34] is simpler and relies on mechanical and geometrical arguments.",
"Consider a non-spherical cap with principal radii of curvature $R_1$ and $R_2$ (Figure REF ).",
"Because of surface tension, a line element $\\delta l$ at A is subjected to a force $F_A = \\gamma \\delta l$ .",
"The projection of $F_A$ on the normal P-N is then $\\sin {\\phi } = \\frac{proj_{F_A}}{\\gamma \\delta l} \\\\proj_{F_A} = \\gamma \\delta l \\sin {\\phi }$ Using the small angle approximation on the triangle A-P-N it is possible to define $\\sin {\\phi } \\approx \\tan {\\phi } \\approx \\phi \\approx \\frac{r}{R_1}$ and $proj_{F_A} = \\gamma \\frac{r}{R_1} \\delta l$ The same argument for points B, C and D results in $proj_{F_B} = \\gamma \\frac{r}{R_1} \\delta l$ $proj_{F_C} = \\gamma \\frac{r}{R_2} \\delta l$ $proj_{F_D} = \\gamma \\frac{r}{R_2} \\delta l$ The resulting force on the N direction on the cap is the sum of the force components of each point in the N direction $F = proj_{F_A} + proj_{F_B} + proj_{F_C} + proj_{F_D} = \\gamma \\frac{r}{R_1} \\delta l + \\gamma \\frac{r}{R_1} \\delta l + \\gamma \\frac{r}{R_2} \\delta l + \\gamma \\frac{r}{R_2} \\delta l$ $F = 2 r \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2}) \\delta l$ Figure: Schematics for the mechanics derivation of the Young-Laplace equation (Adapted from ).The total force around the border of the cap can be calculated by integrating in $\\delta l$ .",
"Since there are four quadrants and the perimeter is treated as that of a circle, the integration is made in the 0 to $\\pi /2$ interval.",
"The element of length needs to be transformed into an element of angle using the arc length formula $\\delta l = r \\delta \\theta $ .",
"$F_{Do} = \\int _0^{\\frac{\\pi }{2}} 2r\\gamma (\\frac{1}{R_1} + \\frac{1}{R_2}) r \\delta \\theta = \\gamma \\pi r^2 (\\frac{1}{R_1} + \\frac{1}{R_2})$ For the cap to be in equilibrium, the force downward caused by surface tension has to be in equilibrium with the upward force which in this case is caused by the pressure gradient between the two sides of the interface $F_U = (p_1 - p_2) \\pi r^2$ Note that the pressure difference is force over area and $\\pi r^2$ is an area such that we have the resulting force.",
"On equilibrium $F_U = F_{Do}$ $F_U = F_{Do} = (p_1 - p_2) \\pi r^2 = \\gamma \\pi r^2 (\\frac{1}{R_1} + \\frac{1}{R_2})$ And we again arrive at the Young-Laplace equation $p_1 - p_2 = \\gamma (\\frac{1}{R_1} + \\frac{1}{R_2})$ The Young-Dupré equation defines the equilibrium condition, more specifically the contact angle, when three homogeneous phases are placed in contact and are in thermodynamic equilibrium.",
"The condition applies to an interface solid-liquid-gas where the shape of a liquid drop is defined by the balance of the surface tensions on the three interfaces, liquid-solid, liquid-gas and gas solid.",
"In general liquids in contact with solids can behave as wetting or nonwetting phases.",
"If the fluid behaves as a wetting phase it will tend to maximize the surface area of contact with the solid (Figure REF a) while nonwetting phases will tend to minimize the contact area with the solid (Figure REF b).",
"Water over a hydrophilic surface is an example of a wetting phase while it is a nonwetting phase when in contact with hydrophobic surfaces such as surfaces covered with wax.",
"The simplest derivation of the force balance is based on the force equilibrium on the $x$ plane (Figure REF ).",
"For a wetting phase, for example, the surface tension on the direction of the solid-gas interface $\\gamma _{13}$ is balanced by summation of the forces on the opposite direction, the surface tension on the solid-liquid interface $\\gamma _{12}$ and the $x$ component of the surface tension on the liquid-gas interface, $\\gamma _{23} \\cos {\\theta }$ $\\boxed{\\gamma _{13} = \\gamma _{12} + \\gamma _{23} \\cos {\\theta }}$ $\\theta $ in this case is the contact angle between the liquid-gas and solid-liquid interfaces.",
"Figure: Angle of contact on an interface solid (1) - liquid (2) - gas (3).",
"(a) Is a wetting liquid θ<90 o \\theta < 90 ^{o} and (b) is a nonwetting liquid θ>90 o \\theta > 90 ^{o}.The physical interpretation of the capillary rise phenomenon is a controversial issue.",
"The force balance interpretation given in many textbooks is criticized in terms of accuracy, alternative mechanical and thermodynamic interpretations being preferred.",
"The force balance interpretations due to pressure difference will be presented here due to its simplicity.",
"Consider water rising to the level h in a fine glass tube (Figure REF ), the pressures at points A and C are equal to atmospheric pressure being exerted at the surface of the liquid, $P_A = P_C = P_{atm}$ , and following Pascal's principle, $P_C = P_D = P_{atm}$ .",
"Capillary rise and the Young-Laplace equation indicate that there is a pressure difference between the two sides of the curved surface represented by the meniscus between A and B.",
"Because there is capillary rise, the pressure is lower below the surface of the meniscus, in B, than above the meniscus, in A, such that $P_B < P_A$ .",
"The capillary rise also indicates that the pressure in D is greater than in B, such that a pressure gradient exists producing an upwards force.",
"The condition of equilibrium in terms of force is $(P_D - P_B - \\rho g h) \\pi r^2 = 0$ In other words, the upwards force $P_D \\pi r^2 $ = force/area $\\cdot $ area = force, is balanced by the weight of the water column $\\rho g h \\pi r^2$ = P $\\cdot $ area = force/area $\\cdot $ area = force, and the force exerted at B.",
"Because of the equilibrium condition, $P_D - P_B = P_A - P_B $ and $P_A - P_B = \\rho g h$ substituting the Young-Laplace equation for the pressure gradient on the interface $\\frac{2 \\gamma }{R} = \\rho g h$ This equation can be written in terms of the radius of the capillary tube instead of the radius of the spherical meniscus by considering the geometrical relation in Figure REF $R = \\frac{r}{\\cos {\\theta }}$ The capillary rise equation results from replacing $R$ into the Young-Laplace equation $\\boxed{h = \\frac{2 \\gamma \\cos {\\theta }}{\\rho g r}}$ The rise considered is up to the lowest point of the meniscus, the volume of water above that point being neglected from the weight of the column because it is usually much less than the weight of the rest of the column.",
"Figure: Capillary rise.We will leave capillary rise and curved interfaces aside for a little while in order to discuss the theory of soil water potentials.",
"Potential theory of soil water is nothing more than an application of one of the most fundamental principles in physics - and in nature for that matter -, the principle of conservation of energy, to soil systems.",
"We already discussed several phenomena in terms of energy throughout the book, now we need to formalize how energy plays a role in transport and storage of water in porous media.",
"Energy is a fundamental concept on the ontology of physics and it is often easier to be understood intuitively than to be given an actual definition.",
"The concept of energy in classical mechanics is a good starting point for an intuitive discussion.",
"We all know that a body will fall under the force of gravity if it is released close to Earth's surface.",
"The nature of gravitational force which accelerates the body in direction of the center of Earth is complex and the most accepted explanation is given in terms of Einstein's general relativity and curvature of space-time.",
"The theory is beyond the scope of this book and for the sake of simplicity we can accept that such force exists as has been experimentally verified.",
"One of the most important ideas in the concept of energy is that it can only be defined in relation some reference state or condition.",
"When we lift a body 10 m into the air close to Earth's surface we are defining Earth's surface itself as the reference level.",
"We could lift a body 10 m above the surface in Mars, on the Moon or above some point in an inertial space station trillions of kilometers away from the nearest star or planet and use any of these points as a reference level and calculate stored energy.",
"The gravitational force near Earth is defined experimentally as $\\mathbf {F} = -m \\mathbf {g}$ The negative sign indicates that the force is downwards in a coordinate system where upwards is defined as positive.",
"If we defined Earth's surface at a particular point and we wish to increase the elevation of an object with mass, work needs to be done against this force.",
"If the object is then released, work is done by the gravitational field as the object falls to a new equilibrium condition.",
"The work done against gravitational force by elevating an object with mass m from the surface reference point where h is set to zero to a height h is $W = \\int _{0}^{h} \\mathbf {F} dh = \\int _{0}^{h} -m \\mathbf {g} dh = -m \\mathbf {g} \\bigg \\vert _{0}^{h} = -m \\mathbf {g} h - (-m \\mathbf {g}0) = -m \\mathbf {g} h$ If the force is conservative, or path independent, the work done is directly related to an energy gradient as $W = -\\Delta U$ Thus the gravitational potential energy of a body lifted to a height h above a reference level where height is arbitrarily set as zero is $U = mgh$ A body falls from height because it has mechanical energy stored.",
"Water in rivers, streams, pipes, and pores in soils and rocks also moves because of an energy gradient.",
"In pipes, rivers and streams, the flow can be fast enough that there is a contribution from kinetic energy to the overall energy conservation calculations, as evidenced by the use of the Bernoulli equation, but this is usually not the case in small pores where the flow is slow enough that the kinetic energy contribution can be ignored.",
"In water in soils and rocks, many different forms of energy can be involved.",
"In Chapter REF , where we discussed the fundamental equations of fluid mechanics, we saw that pressure gradients are involved in the flow of fluids, and we also saw that gravitational forces can be involved along with pressure gradients.",
"In porous media, especially in unsaturated porous media, things can be a little more complicated, in both statics and dynamics, a variety of other forces are involved, including complex intermolecular and surface forces.",
"Because of the irregularity and often random distribution of pores and solids, other governing laws in addition to the Navier-Stokes equations might be necessary for modeling such systems.",
"Although in this section we discuss the role of potentials in statics, a truly static condition is virtually impossible in natural porous media.",
"Even a block of wet soil in laboratory conditions might be constantly subjected to potential gradients with the surrounding air and thermal fluctuations within the soil mass, which will cause energy gradients and water movement in response to these gradients.",
"However, the assumption of a transitory static condition is very important for the use of methods of determination of potential and water content in porous materials.",
"To discuss the state of energy of water in contact with solids in rocks and soils we need to address a different form of energy, other than the mechanical forms gravitational potential energy and kinetic energy, the internal energy.",
"The internal energy, contrary to forms of energy which are related to external fields and motion of the system as whole, is the energy within the system as defined by a particular state in comparison to a reference state in which the energy is arbitrarily taken as zero.",
"In these equilibrium states, the system can be completely characterized macroscopically by the internal energy, U, the volume, V, and the number of moles of its chemical components, $N_{1}$ , $N_{2}$ , ..., $N_{i}$ .",
"Another extensive function that can be posited for equilibrium states of a system is the entropy, SInterpretations of the concept of entropy in popular media and introductory courses are most often inadequate, if not incorrect.",
"Entropy must be defined on a physical and mathematical basis, and different definitions can be given based on different subfields of physics, we will address the issue in later editions.",
"For now refer to [42], [24].",
"The entropy is a function of the other extensive properties $S = S(U, V, N_i)$ and because the entropy function is continuous, differentiable and is a monotonically increasing function of energy, it can be solved for energy resulting in $U = U(S, V, N_i)$ Now the internal energy and entropy can be stated for a multiphased porous media as long as it is in a state of equilibriumNon-equilibrium thermodynamics plays an important role in transport phenomena, but for the sake of simplicity it is necessary to assume that states of equilibrium are attained for a reasonable simple treatment of the problem, and in such condition they contain all information necessary to characterize the system.",
"By computing the differential of the energy form we have [24] $dU = \\left(\\frac{\\partial U}{\\partial S}\\right)_{V, N_i} dS + \\left(\\frac{\\partial U}{\\partial V}\\right)_{S, N_i} dV + \\sum _i \\left(\\frac{\\partial U}{\\partial N_i}\\right)_{V, N_j} dN_i$ From this equation the well known intensive properties temperature, pressure and chemical potential of the $i^{th}$ chemical species can be defined by the following partial derivatives [24] $T = \\left(\\frac{\\partial U}{\\partial S}\\right)_{V, N_i}$ $-P = \\left(\\frac{\\partial U}{\\partial V}\\right)_{S, N_i}$ $\\mu _i = \\left(\\frac{\\partial U}{\\partial N_i}\\right)_{S, V, N_j}$ The internal energy relationship can then be written as $dU = T dS -P dV + \\mu _i dN_i$ For a partially saturated porous media composed of a solid, liquid and a gas, the chemical potential can be defined for each phase, assuming that there are no different species within each phase, such that $dU = T dS -P dV + \\mu _{solid} dN_{solid} + \\mu _{liquid} dN_{liquid} + \\mu _{gas} dN_{gas}$ T, P and $\\mu _i$ are partial derivatives of functions of S, V and $N_i$ and because of that they are themselves functions of these quantities.",
"These functional relationships are called equations of state.",
"For a natural porous media system we are interested in the equation of state which describes the chemical potential of pore water $\\mu _w = \\mu _w(S, V, N_1, N_2, ...)$ However, $S$ , $V$ and $N_i$ are inconvenient variables to control in experimental studies in partially saturated soils and rocks.",
"Since the chemical potential of water in this case was derived from the energy function, these variables also appear on the equation state.",
"One alternative is to change the independent variables on the water chemical potential equation of state by performing a Legendre transformation in order to obtain new functions called thermodynamic potentials which can be a function of one or more more easily measured and controlled intensive parameters under experimental conditions.",
"Examples of thermodynamic potentials are the internal energy itself, Helmholtz free energy, enthalpy and Gibbs free energy, although others can be derived.",
"A convenient potential for porous media studies is the Gibbs (free) energy $G$ .",
"In the case of the Gibbs energy we want to replace both S and V as independent variables and in this case G is defined by differential of $U$ minus the two conjugate pairs one wishes to replace, $TS$ and $-PV$ [42] $dG = d(U - TS - (-PV)) = d(U -TS + PV) = dU - TdS - SdT + PdV + VdP$ Recalling dU = TdS - pdV $dG = TdS - PdV + \\sum _i \\mu _i dN_i - TdS - SdT + PdV + VdP$ arriving at $dG = - SdT + VdP + \\sum _i \\mu _i dN_i$ In which the Gibbs free energy is a function of $T$ , $P$ and $N_i$ $G = G(T, P, N_i)$ As with the internal energy function, we can also define intensive parameters and equations of state for the Gibbs free energy $-S = \\left(\\frac{\\partial G}{\\partial T}\\right)_{P, N_i}$ $V = \\left(\\frac{\\partial G}{\\partial P}\\right)_{T, N_i}$ $\\mu _i = \\left(\\frac{\\partial G}{\\partial N_i}\\right)_{T, P, N_j}$ This last term is the partial molar Gibbs energy of component $i$ and is defined at constant T, P and number of moles of any other constituents, $N_j$ [15].",
"Thus, for the component $i$ , the equation of state is a function of the form $\\mu _i = \\mu _i(T, P, N_j)$ with differential $d\\mu _i = \\left(\\frac{\\partial \\mu _i}{\\partial T}\\right)_{P, N_j} dT + \\left(\\frac{\\partial \\mu _i}{\\partial P}\\right)_{T, N_j} dP + \\sum _j \\left(\\frac{\\partial \\mu _i}{\\partial N_j}\\right)_{T, P} dN_j$ There are many ways of using these equations in thermodynamic systems, the user having freedom within the constraints of the theory to choose the independent variables.",
"In soils and other porous systems we are usually concerned with chemical potential of the water in the pores and surfaces $\\mu _w$ .",
"It is convenient to modify the chemical potential to include the mas based or gravimetric water content $\\theta _g$ , which is essentially the mass based water fraction.",
"Under these conditions, the chemical potential of water can be defined as a function of $T$ , $P$ , $\\theta _g$ and the number of moles of each dissolved solute fraction $N_j$ [7], [15], [95]An alternative derivation can be found in [90] $\\mu _w = \\mu _w(T, P, \\theta _g, N_j)$ thus, $d\\mu _w = \\left(\\frac{\\partial \\mu _w}{\\partial T}\\right)_{P, \\theta _g, N_j} dT + \\left(\\frac{\\partial \\mu _w}{\\partial P}\\right)_{T, \\theta _g, N_j} dP + \\left(\\frac{\\partial \\mu _w}{\\partial \\theta _g}\\right)_{T, P, N_j} d\\theta _g + \\sum _j \\left(\\frac{\\partial \\mu _w}{\\partial N_j}\\right)_{T, P, \\theta _g} dN_j$ In the soil physics literature what is defined as water potential ($\\psi _w$ ) is the chemical potential of water in soil in reference to a Standard State, usually pure water at 293.15 K and 101325 Pa ($\\mu _w^0$ ) $\\boxed{\\psi _w \\equiv d\\mu _w \\equiv \\mu _w - \\mu _w^0}$ Make sure to understand this concept, it is one of the most fundamental principles in applied soils physics and, along with other forms energy and potential gradients, defines how water is stored and moves in soils, and the principles of operation of various field and laboratory devices such as tensiometers, pressure plate apparatus, psychrometers and what the water retention curve is.",
"Most soil physicists and vadose zone hydrologists take this basic knowledge for granted and it is impossible to have a firm grasp on anything related to soil water phenomena without understanding these concepts.",
"Since $\\mu _w$ is an intensive property in units of energy/amount (usually $J~kg^{-1}$ or $J~mol^{-1}$ ) the water potential has the same units.",
"Having defined what the water potential $\\psi _w$ is we now have to define its components as usually encountered on the soil physics literatureWe mention soil physics because these concepts evolved within and along with early soil physics.",
"It is interesting to follow the evolution of the concepts and discussions on journal articles published by N.E.",
"Edlefsen and A.B.C.",
"Anderson, the Gardner brothers, G.H.",
"Bolt, K.L.",
"Babcock, R.D.",
"Miller, S. Takagi and many of their collaborators and students during the 1940s, 50s and 60s.",
"Because$j$ here has the same function as $i$ from when this function was first presented $\\mu _w = \\left(\\frac{\\partial G}{\\partial N_j}\\right)$ the coefficients in $dP$ and in $dT$ in Equation 7.54 can be transformed by maintaining the appropriate quantities constant and changing the order of differentiation as [15], [24] $\\frac{\\partial \\mu _w}{\\partial T} = \\frac{\\partial }{\\partial T}(\\frac{\\partial G}{\\partial N_j}) = \\frac{\\partial }{\\partial N_j} (\\frac{\\partial G}{\\partial T}) = \\frac{\\partial }{\\partial N_j}(-S) = -\\frac{\\partial S}{\\partial N_j} = -S_w$ $\\frac{\\partial \\mu _w}{\\partial P} = \\frac{\\partial }{\\partial P} (\\frac{\\partial G}{\\partial N_j}) = \\frac{\\partial }{\\partial N_j} (\\frac{\\partial G}{\\partial P}) = \\frac{\\partial }{\\partial N_j}(V) = \\frac{\\partial V}{\\partial N_j} = V_w$ in which $V_w$ is the partial molar volume of water and $S_w$ is the partial molar entropy of pore water.",
"Replacing into the original formula we arrive at $\\boxed{d\\mu _w = -S_w dT + V_w dP + \\left(\\frac{\\partial \\mu _w}{\\partial \\theta _g}\\right)_{T, P, N_j} d\\theta _g + \\sum _j \\left(\\frac{\\partial \\mu _w}{\\partial N_j}\\right)_{T, P, \\theta _g} dN_j}$ This is the thermodynamic equation that defines the potentials of soil water, it is possible to arrive at this equation using different thermodynamic potentials although Gibbs free energy and the Groenevelt-Parlange [41] potentials are most often employed in the soil physics literature [90].",
"Using the soil physics $\\psi $ notation, this equation is usually written as [95] $\\boxed{\\psi _w = -S_w dT + \\psi _p + \\psi _m + \\psi _o}$ Considering isothermal conditions, $dT = 0$ , and the equation reduces to $\\boxed{\\psi _w =\\psi _p + \\psi _m + \\psi _o}$ In reality, many field and laboratory applications of these equations must consider isothermal conditions, not doing so would require some entropy function for the temperature potential, $\\psi _T$ , to be considered, which would cause great mathematical complications.",
"The $\\psi $ terms in this equation are the most commonly defined components of soil water potential $\\psi _w$ , the pressure potential $\\psi _p = V_w dP$ the matric potential $\\psi _m = \\left(\\frac{\\partial \\mu _w}{\\partial \\theta _g}\\right)_{T, P, N_j} d\\theta _g$ and the osmotic potential $\\psi _o = \\sum _j \\left(\\frac{\\partial \\mu _w}{\\partial N_j}\\right)_{T, P, \\theta _g} dN_j$ The pressure potential is usually associated with hydraulic pressure in saturated soils below the water table, however it can be positive or negativeA body of water can be under tension, and not only under compressive stresses.",
"Under the water table the pressure potential corresponds to the hydraulic pressure plus the atmospheric pressure, at the water table the hydraulic pressure is zero and the second component of pressure is the atmospheric pressure.",
"Because we define the standard state at atmospheric pressure, the pressure potential is zero at the water table.",
"In unsaturated soils the liquid pressure is zero and the soil water is at atmospheric pressure.",
"The soil matric potential is negative and is related to the capillary and adsorptive forces exerted by the soil solid phase on soil water.",
"Everything else constant, the water potential in an unsaturated soil is lower than that of free water at the same height and elevation, so that the state of energy of soil water is lower than that of free water.",
"In simpler terms, water molecules in contact with soil solid particles are “less free to move” because of the forces exerted by these surfaces.",
"At higher water contents, the adsortion forces of water molecules with surfaces play a small role when compared to capillary forces.",
"As the soil dries and the water is restricted to thin films around the particles and very small pores, the adsorption component cannot be neglected and the matric potential will be some function of the capillary potential and the adsorption potential.",
"Detailed discussion of the components of capillary and adsorptive forces can be found in [72], [96].",
"The osmotic (or solute) potential is the reduction of water potential in contact with dissolved solids, or more specifically, solutes.",
"Water molecules in contact with dissolved solids have lower energy, thus, as with cells and in other semi-permeable membranes, a larger concentration within the porous membrane or the cell will tend to draw water from the exterior until a pressure equilibrium is achieved in response to the osmotic forces.",
"Different authors report different soil water potentials.",
"This is a complex issue and as we mentioned before, the potentials will depend on thermodynamic theoretical considerations such as the thermodynamic potential used to derive the water potentials, see for example the discussions on pneumatic, submergence and overburden potentials, among others in [95], [90].",
"Because we initially expressed the water potential as a partial molar Gibbs energy $\\partial G/\\partial N_j$ , the units of the water potential are in $J~mol^{-1}$ , and this water potential can be defined as the molar water potential.",
"However, this unit is not often used in soil physics and groundwater hydrology, the most used units are derived from this basic unit.",
"Considering the molar mass of water $M_w$ in $kg~mol^{-1}$ , we can divide the soil molar water potential by it, defining the water potential in units of $J~kg^{-1}$ $\\frac{\\psi _w}{M_w} \\rightarrow \\frac{J~mol^{-1}}{kg~mol^{-1}} = \\frac{J}{kg}$ Perhaps the most common form encountered in soil physics is in terms of pressure units, or the soil water pressure.",
"The water potential in joules per mass is converted to pressure units by multiplying by water density $\\psi _w \\rho _w \\rightarrow \\frac{J }{kg} \\frac{kg}{m^{-3}} = \\frac{J}{m^{3}} = \\frac{N~m}{m^{3}} = \\frac{N}{m^{2}} = Pa$ Pascal ($Pa$ ) and its equivalents $kPa$ , $hPa$ and $MPa$ are frequently used in soil physics, especially when dealing with matric potentials measured using tensiometers and pressure plates.",
"The third most commonly used conversion of water potential is in terms of head units or length, or the soil water head.",
"To convert to length units, the water potential in joules per mass is divided by the gravitational acceleration, $g$ $\\frac{\\psi _w}{g} \\rightarrow \\frac{J~kg^{-1} }{m~s^{-2}} = \\frac{N~m~kg^{-1} }{m~s^{-2}} = \\frac{N~kg^{-1} }{ s^{-2}} = \\frac{kg~m~s^{-2}~kg^{-1} }{ s^{-2}} = m$ Recall for the above derivations that $1~J = 1~N~m$ and that $1~N = 1~kg~m~s^{-2}$ .",
"The head units are common in hydrogeology and to some extent in soil physics, especially for the pressure potential.",
"Sposito [90] warned that the different units should not obscure the physical interpretation of the problem.",
"No matter in what unit it is expressed, the soil water potential represent an gradient of energy per unit amount of water which drives movement of soil water.",
"Because water close to Earth's surface is subjected to a gravitational field and gravitational forces, the description of the potentials is incomplete without considering the effect of gravity.",
"For a system under the effect of gravity the gravitational potential $ \\phi = gz$ must be added to the chemical potential of water to completely define the state of said water.",
"The complete treatment of the inclusion of the gravitational potential into the thermodynamic treatment of soil water is complex and a more detailed analysis of the methods can be found in Babcock [6].",
"The general theoretical account of including the gravitational potential with the chemical potential can be found in Guggenheim [42].",
"For nowA complete treatment is planned to be included in later editions and for simplicity we will follow the approach of [95] and simply add the soil water gravitational potential to the other potentials asNotice that this relationship can be easily demonstrated to hold by considering the gravitational potential energy $ U = mgh$ and solving the density of water $\\rho _w = m_w/V$ for $m_w$ and replacing into the gravitational potential energy.",
"The soil water gravitational potential then is simply $U/V \\rightarrow J~m^{-3} = N~m^{-2} = Pa $ $\\psi _g = \\rho _w g z$ In which $z$ is the vertical elevation.",
"Considering now the potentials in pressure units, the total potential of soil water is the sum of the components of water potential and the gravitational potential, plus any other potentials that might be relevant under specific conditions $\\boxed{\\psi _t = \\psi _p + \\psi _m + \\psi _o + \\psi _g + ....}$ Remember that a potential gradient can be understood as a potential to perform work, movement of water will occur as a response to a gradient of total potential.",
"In a water body under static conditions the total potential must be the same everywhere.",
"Notice that the individual components might not be equal, but the sum of the components must be the same everywhere, otherwise there is a potential gradient and the water will flow in response to such gradient.",
"Consider the points $A$ , $B$ and $C$ at heights $a$ , $b$ and $c$ on the vessel in Figure REF .",
"If the water level is at $h$ and we consider the gravitational reference at the dotted line at the bottom, the gravitational potential, in head units, at each point is $\\psi _{gA} = a \\\\\\psi _{gB} = b \\\\\\psi _{gC} = c$ and the pressure potential is $\\psi _{pA} = h - a \\\\\\psi _{pB} = h - b \\\\\\psi _{pC} = h - c$ Since only these two potentials are being considered, the total potential $\\psi _t = \\psi _p + \\psi _g$ is $\\psi _{tA} = (h - a) + a = h \\\\\\psi _{tB} = (h - b) + b = h\\\\\\psi _{tC} = (h - c) + c = h$ Thus the total potential is constant and equal to $h$ everywhere.",
"The same principle is valid for saturated soils, porous rocks and sedimentary deposits, if the total potential is constant everywhere the fluid is static.",
"Figure: Total potential in a vessel.Consider now a porous medium which is half saturated and half unsaturated (Figure REF ).",
"Above the water table the water column is under tension following the mechanisms described for a capillary tube.",
"We showed that the height of capillary rise was given by $h = \\frac{2 \\gamma \\cos {\\theta }}{\\rho g r}$ Figure: Total potential in a partially saturated porous medium.Because water at each point above the water table is under tension as demonstrated for the capillary rise equation derivation, the capillary potential in head units at any point above the water table can be shown to be equivalent to the height above the water tableThis offers no mathematical or physical proof of any kind and is presented as a statement.",
"A mathematical proof is being pursued $\\psi _c = -h$ Considering that the matric potential can be approximated by the capillary potential $\\psi _m \\approx \\psi _c$ , the matric potential, in head units, will correspond to the negative of the height of water column above the water table.",
"Thus, for a porous medium that is half saturated and half unsaturated (Figure REF ) we can calculate the gravitational, pressure, matric and total potentials at the points $A$ , $B$ and $C$ .",
"The gravitational potential is $\\psi _{gA} = a \\\\\\psi _{gB} = b \\\\\\psi _{gC} = c$ the pressure potential is $\\psi _{pA} = b - a \\\\\\psi _{pB} = 0 \\\\\\psi _{pC} = 0$ the matric potential is $\\psi _{mA} = 0 \\\\\\psi _{mB} = 0 \\\\\\psi _{mC} = -(c - b)$ such that the total potential, $\\psi _t = \\psi _g + \\psi _g + \\psi _m $ , is $\\psi _{tA} = a + (b - a) + 0 = b\\\\\\psi _{tB} = b + 0 + 0 = b\\\\\\psi _{tC} = c + 0 -(c - b) = b$ Thus the total potential is constant and equal to $b$ on the whole system, as expected.",
"Most devices for measuring soil water potential fall into four categories: 1) suction devices, such as Haines apparatus or tension tables, 2) tensiometers, 3) pressure plates or pressure chamber apparatus and 4) psychrometers [31].",
"Methods 1, 2 and 3 rely on similar physical principles, the soil sample is put into contact with a porous membrane which is contact with a water reservoir.",
"In 1 the pressure in the water reservoir is then decreased by decreasing the elevation of the opening of a water column in contact with the porous membrane which in turn is in contact with the soil sample.",
"As the pressure decreases, water in an initially saturated soil sample will flow to the reservoir, and from there to the outflow tube, until equilibrium is reached.",
"In head units, the capillary potential and thus the matric potential are effectively the negative of the decrease in elevation of the opening of the water column with respect to the porous membrane.",
"$\\psi _m \\approx \\psi _c = -h$ If the soil sample is relatively small in height, less than one to two cm [31], at equilibrium the matric potential of soil water is the same as the porous membrane.",
"Different porous membranes can be used, depending on the application of the method and the desired matric potential range.",
"Fine sand, kaolin clay, paper, porous stone and others can be found on the literature.",
"Often in the literature the term tension is used instead or matric potential, tension ($\\tau $ ) being simply the negative value of the matric potential $\\tau = -\\psi _m$ Tensiometers (category 2) function in a similar way, except that the pressure on the water reservoir is not controlled by the user as in the case of hanging water column devices.",
"The device is usually composed of a tube full of water with a porous cup on one end.",
"The porous cup is put into contact with soil and if the soil is unsaturated, water will flow from the device and into the soil until equilibrium is reached.",
"When equilibrium is reached, the water within the tube is in a state of tension and the tension or pressure can be measured using a variety of methods.",
"Historically, the tension was measured by the decrease in height of the water level in a tube connected to the porous medium, however, because this would require a very long column, a liquid a much greater density, mercury (Hg) was used in contact with a water reservoir, resulting in the need of much smaller columns and in devices that could be conveniently deployed to the field.",
"Later, due to the extreme environmental toxicity or Hg, and for convenience, analog pressure gauges were used.",
"Although analog gauges are still used, with the decrease in cost of electronics in the last decades, digital pressure and tension sensors connected to data logging devices are now common in research and in agricultural and environmental monitoring applications.",
"At equilibrim, and considering the water reservoir and soil at the same elevation, the sum of the pressure and matric potentials are going to be the same on the soil and on the water reservoir [95] $\\psi _{ps} + \\psi _{ms} = \\psi _{pr} + \\psi _{mr}$ Where the subscripts $s$ and $r$ stand for soil and water reservoir, respectively.",
"However, the pressure potential is zero if the soil is unsaturated and the matric potential is zero in the water reservoir as there is no soil matrix such that $\\psi _{ms} = \\psi _{pr}$ Therefore, the (negative) pressure reading corresponding to the water on the reservoir is the soil matric potential.",
"Notice that atmospheric pressure still acts on the unsaturated soil, but since the pressure reading on the device is with respect to the atmospheric pressure (by definition the gauge pressure), the two terms cancel out and there is no need to consider atmospheric pressure.",
"Because the boiling point of water at pressures below atmospheric pressures decreases as the pressure difference increases at constant temperature, the maximum practical tension possible at the sea level and at average surface temperature is around $8~m$ , although in laboratory conditions, using de-areated water, higher tensions are possible [102].",
"In order to increase the range of pressures (or tensions), the soil sample and porous membrane can be put inside a pressurized chamber (category 3).",
"As the pressure inside the chamber increases, the capillary pressure of the water in the porous membrane decreases as the pores will gradually drain according to their radius.",
"At higher pressures, water exists in smaller pores of small radius, such that the radius of curvature of the water meniscus is also very small, corresponding to very small capillary pressures (higher numbers in absolute value, or in terms of tension).",
"The soil samples in hydraulic contact with the porous membrane will lose water until equilibrium is reached True equilibrium is never reached in reality, but fluxes are microscopic only or gradients small enough so that they can be disregarded.",
"If the height of the soil sample is small, the gravitational potential can be neglected and the tension on the soil sample is the gauge pressure within the chamber.",
"The porous membrane, usually a porous ceramic plate, is in contact with a water reservoir which is open to the atmosphere such that, as before $\\psi _{ps} + \\psi _{ms} = \\psi _{pr} + \\psi _{mr}$ and the matric and pressure potential are now zero on the water reservoir such that $\\psi _{ms} = -\\psi _{ps}$ In which $\\psi _{ps}$ is effectively the gauge pressure within the chamber.",
"Pressure chamber devices can reach tensions well beyond those of tension devices.",
"The limit usually considered is more or less arbitrarily set at $15~m$In part because of the idea that there is some universal permanent wilting point.",
"At pressures above this value, adsorption forces start to play a more prominent role and the pressure gradient alone is not enough to further remove water from the soil.",
"The water that cannot be removed by pressure plate apparatus is often referred to residual water content and can often be removed using other methods.",
"Psychromters, hygrometes and potentiameters (category 4) are, generally speaking, devices that measure the water potential based on the Kelvin equation $\\psi _{w} = \\frac{R_g T}{V_w} \\ln {\\frac{p}{p_0}}$ in which the water potential measured is the sum of the matric and osmotic potential, $R_g$ is the universal gas constant, T is the absolute temperature, $V_w$ is the molar volume of water, $p$ is the air actual vapor pressure and $p_0$ is the air saturation vapor pressure, and $p/p_0$ is the relative humidity of the air in contact with the sample.",
"The water activity is also used in some devices and is, by definition, $a_w \\equiv p/p_0$ .",
"The water potential measured by these devices will not necessarily correspond to the water potential measured by pressure and suction devices, since the nature of the components measured are different.",
"One of the main advantages of psychrometric and other similar devices is that they can measure potentials at extremely low water contents, well beyond what is possible with the pressure chamber, well below -1.5 MPa.",
"Details on these devices can be found in [3], [82] The gravimetric method, based on weighing a soil or porous material sample, oven drying at 105 °C for at least 24 h and reweighing the oven dried sample and calculating the mass of water divided the mass of oven dried material is the standard method for measuring water content and it is used to calibrate indirect water content estimation methods.",
"The gravimetric method is a direct method because the mass of water is directly measured by weighing.",
"Because it is a direct method, if conducted properly using scales with high enough accuracy and precision, results in errors, both systematic and random, much smaller than indirect methods.",
"In indirect methods, the water content is estimated indirectly and they are usually calibrated for a particular soil or condition using the gravimetric method.",
"Errors related to calibration, or lack of thereof, and instrumental errors, both on the sensors and electronics usually result in lower precision and/or accuracy when compared to direct methods.",
"There are several different types of indirect methods for measuring water content, the two most common types in use today being methods using radioactive sources, such as neutrons and gamma radiation, and general electromagnetic methods including time domain reflectometry (TDR), capacitance and other devices [31].",
"The neutron thermalization method historically has been an important method for water content measurement and monitoring in the field.",
"Once calibrated it provides fast and accurate measurements of soil water content.",
"Physically, the method relies on a radioctive source that emits fast neutrons, with kinetic energy on the MeVeV = electron-volt; 1 $eV = 1.60217646 \\cdot 10^{-19}~J$ range and speed on the $10^4~km~s^{-1}$ order of magnitude, and a detector of slow neutrons, with energies less than about 1 eV [45].",
"The fast neutrons emitted by the source collide with hydrogen atoms (H) in water molecules.",
"Because H has a high scattering cross section, it can absorb or scatter fast neutrons, causing attenuation of the energies and correspondingly the velocity of the neutrons.",
"The higher the water content, more hydrogen nuclei are available for collisions, attenuating fast neutrons and producing slow neutrons which are then counted by the detector.",
"Of course other atoms can attenuate neutrons, but because H present in water is particularly effective in doing so in comparison to other atoms commonly found in soils, it can be used to water content estimation in most soils.",
"The calibration is performed by direct measurement of volumetric water content around the sensing volume and corresponding slow neutron count ratios [45] $C_R = \\frac{x}{x_s}$ in which $x$ is the count in the measured material and $x_s$ is a standard count.",
"The count ratio can then be related to the volumetric water content by a calibration equation of the form $\\theta _v = b_0 + b_1 C_R$ in which $b_0$ and $b_1$ are empirical parameters obtained from statistical or mathematical fitting procedures and $\\theta _v$ is the volumetric water content [45].",
"Other forms of calibration equations can be found on the literature for different materials and water content ranges.",
"The principle of operation of most electromagnetic methods including TDR, frequency domain reflectometry (FDR), ground penetrating radar (GPR), capacitance, impedance and others are based on the principle that the relative permittivityReferred to as the dielectric constant in older literature of water is much greater than that of other soil components, including air, minerals and organic matter.",
"The relative permittivity of a material is a ratio of the absolute permittivity of the material to that of vacuum $\\epsilon _r = \\frac{\\epsilon }{\\epsilon _0}$ Because the absolute permittivity of the material, $\\epsilon $ , and that of vacuum have both the same units, usually $F~m^{-1}$ , and the permittivity of vacuum is a physical constant $\\epsilon _0 = 8.8541878128(13)\\cdot 10^{-12} ~ F~m^{-1}$ , $\\epsilon _r$ is a dimensionless normalized quantity, with values around 80 for water, 1 for air and around 5 for quartz rich soils [80].",
"The relative permittivity of the porous media as a whole will be some form of average of the individual permittivities of each phase, the term apparent permittivity ($\\epsilon _{r(app)}$ ) being used on the literature [63].",
"Because the solid phase is usually constant in fractional volume for most porous media applications and because the relativity permittivity of one of the dynamic phases, water, is much greater than that of the other, air, under unsaturated conditions the apparent permittivity will be a function of the volumetric water fraction, i.e.",
"the volumetric water content.",
"Thus, for most practical applications, water content measured using direct methods, usually gravimetric, can be related to apparent relativity permittivity measurements obtained using electromagnetic measuring devices, generating calibration equations.",
"The most common empirical calibration equations found in the literature are third degree polynomials, square root and logarithmic [64] $\\theta _v = a_0 + a_1 \\epsilon _{r(app)} + a_2 \\epsilon _{r(app)}^2 + a_3 \\epsilon _{r(app)}^3$ $\\theta _v = a_0 + a_1 \\sqrt{\\epsilon _{r(app)}}$ $\\theta _v = a_0 + a_1 \\log {(\\epsilon _{r(app)})}$ in which $a_0$ to $a_3$ are empirical fitting parameters, fitted for each individual equation and $\\log $ is the natural logarithm.",
"Most commercial devices come with “universal” calibration equations from the manufacturers, but it is almost always recommended that the device should be calibrated for the individual soils or porous materials in which is going to be used.",
"Water content can also be estimated using more physically based models based on effective medium approximations theory [58], [63].",
"There are several different designs and electronics for electromagnetic water content measuring sensors.",
"The relative permittivity is in reality an imaginary number and is highly dependent on the frequency of operation of the device in terms of the frequency of the electromagnetic radiation emitted.",
"For most soils, but not all, the relative apparent permittivity can be approximated by the real component of the complex permittivity.",
"A detailed discussion of measuring water content with electromagnetic devices is beyond the scope of this book.",
"The user is highly encouraged to study electromagnetic theory before using any electromagnetic device for soil and porous media measurements, there are many misconceptions related to these devices.",
"For more information the reader is referred to [99], [31], [63].",
"There are many other methods used for estimating water content, including methods based on remote sensors, thermal methods and other ionizing radiation methods.",
"These methods will not be discussed in this book, as they are not as broadly employed as the methods discussed above.",
"The user interested in these specific methods should consult [31] and the references listed therein.",
"For a set of tubes of different radius $r_i$ , there is an exact relationship between the percent saturation and the capillary pressure.",
"At the capillary pressure $Pc_i$ , all tubes of radius greater than $r_i$ will have drained, following the Young-Laplace equation $Pc_{i} =\\frac{ 2\\gamma }{r_i}$ thus the volume of water retained will correspond to the sum of the volumes of water retained in all tubes of radius smaller or equal than $r_i$ .",
"But the volume will also depend on the capillary rise for each individual tube which is related to the radius by the capillary rise equation For the angle $\\theta = 0$ and $\\cos theta = 1$.",
"$h_{i} =\\frac{ 2\\gamma }{\\rho g r_i}$ Considering the volume of each tube as that of a cylinder $V_{i} =\\pi r_i^2 h_i$ and replacing the capillary rise equation $V_{i} = \\pi r_i^2 \\frac{ 2\\gamma }{\\rho g r_i} = \\frac{ 2\\gamma \\pi r_i}{\\rho g }$ Now the volume of water is the sum of the volumes of water in all $i$ pores with radius equal or smaller than $r_i$ $V_{water} =\\sum _i \\pi r_i^2 \\frac{ 2\\gamma }{\\rho g r_i} = \\sum _i \\frac{ 2\\gamma \\pi r_i}{\\rho g }$ and the fractional water content is simply the volume of water divided by the total volume of all tubes in the system $\\theta _{f} =\\frac{ V_{water}}{V_{t} }$ Thus, for each capillary pressure $Pc_{j}$ corresponding to a radius $r_j$ , the fractional water content can be directly calculated from the sum of the volumes of water in all pores with radius less or equal than $r_j$ An explicit mathematical relationship can be derived by manipulating summation terms, this will be explored in following editions.",
"This is what is called the capillary bundle model for water retention.",
"It is also an important model for hydraulic conductivity, as we will discuss in later chapters.",
"The capillary bundle model is a useful tool to model and understand transport and retention of water in porous media.",
"In this approach, the pore space is simplified as a set of continuous tubes of constant radius in each pore.",
"Because pores are irregular, do not have constant radius, are usually not cylindrical, and because forces related to other than simple capillary phenomena act in complex natural porous systems, there is no direct way of calculating water saturation from matric potential and empirical and semi-empirical models are necessary.",
"Empirical models rely on fitting to observed data of water content and matric potential, in other words, there is no analytical expression to calculate the volumetric water content by knowing only the matric potential, water content data is necessary.",
"The water retention curve is nothing more than the empirical relationship between water content and matric potential obtained from laboratory and field data.",
"The relationship has been observed to follow a characteristic sigmoid function (S-shaped) when the matric potential is considered in log scale.",
"To avoid mathematical and graphing difficulties it is convenient to consider the absolute values of matric potential ($|\\psi _m|$ ), which as we have seen is a negative quantity, when dealing with water retention equations.",
"There has been several efforts to derive analytical expressions for the water retention curve.",
"These have important practical applications in irrigation and environmental modeling and because, in most cases, an unsaturated hydraulic conductivity function was derived from the water retention curve (e.g.",
"[19], [25], [97], [56]).",
"Some of the models most often found on the literature are presented below.",
"We will leave the related unsaturated hydraulic conductivity functions for later chapters.",
"One of the earliest models that saw wide application is that of Brooks and Corey (BC) [19] For simplicity and to be consistent we will refer to soil water potential as $\\psi $ on the water retention equations $\\theta _v ={\\left\\lbrace \\begin{array}{ll}\\theta _r + (\\theta _s - \\theta _r) (\\frac{\\psi _b}{|\\psi |})^\\lambda , & \\text{for } |\\psi | > \\psi _b \\\\\\theta _s, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.",
"}$ The BC equation, like many water retention models, is a piecewise equation, meaning that it is segmented into two parts.",
"Its parameters are the saturation water content, $\\theta _s$ , the residual water content, $\\theta _r$ the air entry value, often called bubbling pressure, $\\psi _b$ , and the pore size distribution parameter, $\\lambda $ .",
"Most water retention models can be defined in terms of saturation and residual water contents.",
"The saturation water content correspond to water content at a matric potential of zero, whenever it is defined, where in theory the pore space is completely water saturated.",
"The residual water content is the remaining water content at very high values of matric tensionOr very low values if you are thinking in potential, remember that tension is defined as the negative of the potential, where further increase in tension does not cause decrease in water content.",
"In most models, residual water content is an asymptotic parameter reached by taking tension to infinity.",
"The bubbling pressure is the matric tension in which air starts to enter the pore system, usually at relatively low matric tensions for most soils.",
"Below the bubbling pressure, increase in tension does not cause a corresponding decrease in water content.",
"Most water retention models will have parameters related to pore size distribution since, as we saw, water retention is controlled by the distribution of pores.",
"In the BC model the piecewise condition is defined by the $\\psi _b$ parameter.",
"At tensions below or equal to $\\psi _b$ the water content is constant and equal to saturation water content, while at tension above $\\psi _b$ water content decreases following a power law of $\\lambda $ .",
"Most water retention equations can be expressed in terms of the effective saturation, given by $S_e = \\frac{\\theta _v - \\theta _r}{\\theta _s - \\theta _r}$ The BC equation, for example, can be expressed as $S_e ={\\left\\lbrace \\begin{array}{ll}(\\frac{\\psi _b}{|\\psi |})^\\lambda , & \\text{for } |\\psi | > \\psi _b \\\\1, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.",
"}$ Using the effective saturation normalizes the water content between zero and one, making it possible to compare the shapes of the curves for different materials.",
"Using the effective saturation can, in some cases, make fitting easier using computer methods by reducing the number of fitting parameters.",
"The Campbell equation is in essence the BC equation, but defining the effective saturation as $\\theta /\\theta _s$ [25] $\\theta ={\\left\\lbrace \\begin{array}{ll}\\theta _s (\\frac{\\psi _b}{|\\psi |})^b, & \\text{for } |\\psi | > \\psi _b \\\\\\theta _s, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.",
"}$ In which the physical meaning is the same as the BC equation, and $b$ is essentially $\\lambda $ .",
"Brutsaert [21] proposed an equation of the form $S_e = \\frac{a}{a + \\psi ^b}$ where a and b are empirical parameters.",
"Brutsaert's equation was later improved upon by van Genuchten, resulting in one of the most frequently used water retention equations in soil physics and hydrology [97] $\\theta _v = \\theta _r + \\frac{(\\theta _s - \\theta _r)}{[1 + (\\alpha \\psi )^n]^m}$ In which $m$ and $n$ are shape parameters related to the pore size distribution.",
"It is often convenient to define the Mualem [67] restriction as $m = 1 - \\frac{1}{n}$ Reducing the number of empirical parameters to four.",
"Kosugui [56] derived an equation based on a lognormal pore size distribution $\\theta _v ={\\left\\lbrace \\begin{array}{ll}\\theta _r + \\frac{(\\theta _s - \\theta _r)}{2} \\text{erfc}[ \\frac{\\log (\\psi _b-\\psi )/(\\psi _b - \\psi _0) - \\sigma ^2}{\\sqrt{2} \\sigma }] , & \\text{for } |\\psi | > \\psi _b \\\\\\theta _s, & \\text{for } |\\psi | \\le \\psi _b\\end{array}\\right.",
"}$ with two new parameters, $\\sigma $ , the standard deviation of the transformed pore capillary pressure distribution, and $\\psi _0$ , the capillary pressure at the inflection point.",
"Kosugui's model uses the complementary error function (erfc), defined as [14] $\\text{erf}(x) = \\frac{2}{\\sqrt{\\pi }}\\int _0^x e^{-t^2} dt$ $\\text{erfc}(x) = 1 - \\text{erf}(x) = \\frac{2}{\\sqrt{\\pi }}\\int _x^\\infty e^{-t^2} dt$ In which erf is the error function.",
"However, the erfc is calculated numerically in almost all practical applications and is available as a function in most programming languages.",
"Thus, the user does not need to use the integral form when fitting water retention data.",
"The BC, Brutsaert-van Genuchten and Kosugui equation are fitted to simulated oxisol data and shown in Figure REF Figure: Commonly used water retention models fitted to simulated oxisol data.ll $ W $ Thermodynamical work $ \\gamma $ Surface tension $ A, A^{\\prime }, a $ Area $ l $ length $ dx $ Infinitesimal increase in $x$ $ \\mathcal {F} $ Helmholtz (free) energy $ S $ Entropy $ p, P $ Pressure, except when specified otherwise $ \\mu $ Chemical potential of a substance $ N_{i} $ Number of moles of a substance $ T $ Temperature $ \\mathbf {n} $ Unit normal vector $ R, r $ Radius $ l_1, l_2, l^{\\prime }_1, l^{\\prime }_2 $ Elements of arc $ \\zeta $ Unspecified displacement $ \\theta , \\phi $ Angles $ proj_F $ Component of a force $ F $ Force $ \\rho $ Generalized density $ \\rho , \\rho _w $ Water density $ g $ Earth's gravitational acceleration $ U $ Potential of a force $ m $ Mass $ h $ Height $ G $ Gibbs (free) energy $ \\theta _g $ Gravimetric water content $ \\theta _v $ Volumetric water content $ V_w $ Partial molar volume of water $ S_w $ Partial molar entropy of water $ \\psi $ Soil potential $ \\psi _w $ Soil water potential $ \\psi _p $ Soil pressure potential $ \\psi _m $ Soil matric potential $ \\psi _o $ Soil osmotic potential $ \\psi _c, P_c $ Soil capillary potential $ \\psi _g $ Soil water gravitational potential $ \\psi _t $ Total soil potential $ z $ Elevation $ \\tau $ Soil water tension $ p, p0 $ Actual air vapor pressure and air saturation vapor pressure on Kelvin's equation $ R_g $ Universal gas constant on Kelvin's equation $ a w $ Water activity on Kelvin's equation $ C_R $ Count rate $ a, b, a_0, a_1, _2, A_3, b_0, b_1, m, n $ Empirical parameters $ \\epsilon _r $ Relative permittivity $ \\epsilon $ Absolute permittivity of a material $ \\epsilon _0 $ Permittivity of free space $ \\theta _f $ Fractional water content $ V_{water} $ Volume of water $ V_t $ Total volume $ \\theta _r $ Residual volumetric water content $ \\theta _s $ Saturation volumetric water content $ \\psi _b $ Air entry value $ \\lambda $ pore size distribution parameter $ S_e $ Effective saturation $ \\alpha $ Inverse air entry value $ \\psi _0 $ Capillary pressure at the inflection point of Kosugi's equation $ \\sigma $ Standard deviation of transformed capillary pressure distribution in Kosugi's equation $ \\text{erf} $ Error function $ t $ Generalized variable in the definition of erf $ \\text{erfc} $ Complementary error function As you will see in the following sections, although there are analytical solutions for the Navier-Stokes equations to simple cases of laminar flow in open conduits, for flow in porous media, the statement given by Morris Muskat ([69] Muskat, 1937, p.55) summarizes the problem “Unfortunately, however, in spite of the justifiable simplification of neglecting the inertial forces - due to the small velocities generally characteristic of flow through porous media - the mathematical difficulties of applying these equations to porous media are for practical purposes entirely unsurmountable.\"",
"Although theoretical derivations of Darcy's law from the Navier-Stokes equations have been proposed over the last decades, a solution for random porous media will invariable depend on some stochastic mathematical method, such as averaging or homogenization theory, a direct analytical derivation for a random and irregular porous medium remains a problem of difficulty on par with that of modelling turbulence.",
"In simpler words, the random nature of the behaviour of fluid flows cannot be directly solved analytically in all but the simplest cases.",
"We will be mentioning Muskat's work on several occasion throughout this chapter.",
"He was a trained theoretical physicist who earned his Ph.D. at California Institute of Technology in 1929 with a thesis on “The Continuous Spectra of Hydrogen Like Atoms” [68].",
"After graduating, he joined the petroleum industry were he applied his training in problems of multi-phase flow in porous media, making important contributions to the mathematics of flow in porous media in the process, especially by modifying and generalizing the original form of Darcy's equation.",
"His original contributions remain an important topic not only in engineering and applied mathematics, but also in pure mathematics (e.g.",
"[2]).",
"His point is that there are no analytical solutions of the Navier-Stokes equation for true random porous media.",
"On top of that, there is no guarantee that a solution exists for many fluid mechanics problems.",
"This remains an open problem in mathematics in the form of the Navier-Stokes existence and smoothness problem one of the so called millennium problems for which you can win a large sum of money you manage to solve it.",
"Getting back to Darcy's law, it is an empirical law, initially based on experimental data.",
"There has been many a posteriori attempts to derive Darcy's law from first principles, including averaging the Navier-Stokes equations, but the fact remain that most of these make several assumptions along the way and are for the most part largely heuristic, there are no first principles derivation from a direct analytical solution of the Navier-Stokes equations and the hydraulic conductivity remains an empirical constant.",
"Before we get too much ahead of ourselves, as you haven't been formally introduced to Darcy's law, let's look at a few cases of flow through free conduits where there is nothing but the fluid going through the conduit.",
"In porous media, as we have discussed, there is a solid porous matrix inside a volume through which the fluid is moving.",
"Under these conditions, the fluid does not occupy the entire inner volume and is forced to flow through an often very complex geometry of pores.",
"This is one of the main factors which makes flow thorough porous media so mathematically complex.",
"In this section, two analytical solutions of the Navier-Stokes will be discussed.",
"The first is that for flow through two infinite and flat surfaces parallel to each-other and the second is for flow through a cylindrical conduit.",
"The second is often seen in soil physics and transport of fluid through porous media books as an introduction to Darcy's law, given that the pore network of a porous material can be generalized to a network of tubes, from which an approximation to Darcy's law can be obtained Which as an approximation, does not adequatelly describes real porous media.",
"Case I.",
"Flow between two infinite plates Let us start be recalling the Navier-Stokes equations in vector form $\\frac{\\partial \\mathbf {u}}{ \\partial t} + (\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} = \\mathbf {F} -\\frac{1}{\\rho } \\nabla p + \\frac{\\mu }{\\rho } \\nabla ^{2} \\mathbf {u}$ For steady state flow, the velocity field does not vary with time, so $\\mathbf {u}$ is constant with time everywhere such that $\\frac{\\partial \\mathbf {u}}{ \\partial t} = 0$ and $(\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} = \\mathbf {F} -\\frac{1}{\\rho } \\nabla p + \\frac{\\mu }{\\rho } \\nabla ^{2} \\mathbf {u}$ There are many conditions in which body forces can be neglected, such as in purely horizontal flow of a fluid in which the horizontal dimension is much larger than the vertical direction in which a gravitational field acts or if the contact forces are much larger than the body forces.",
"Ignoring the body forces results in $(\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} = -\\frac{1}{\\rho } \\nabla p + \\frac{\\mu }{\\rho } \\nabla ^{2} \\mathbf {u}$ Now we have to take a look at each term, the term $ (\\mathbf {u} \\cdot \\nabla ) \\mathbf {u} $ was discussed earlier in the context of the material derivative, based on that discussion, it is possible to show that it represents the set of equations $u_x \\frac{\\partial u_x}{\\partial x} + u_y \\frac{\\partial u_x}{\\partial y} + u_z \\frac{\\partial u_x}{\\partial z}$ $u_x \\frac{\\partial u_y}{\\partial x} + u_y \\frac{\\partial u_y}{\\partial y} + u_z \\frac{\\partial u_y}{\\partial z}$ $u_x \\frac{\\partial u_z}{\\partial x} + u_y \\frac{\\partial u_z}{\\partial y} + u_z \\frac{\\partial u_z}{\\partial z}$ and it is called the inertial or convective term, due to change of velocity as the fluid element is transported to different positions in the velocity field.",
"The first term on the right side is the pressure gradient and represent the contact forces which cause movement of the fluid if the gradient is not equal to zero.",
"Recalling the gradient operator, $ \\nabla p$ can be represented as $\\nabla p = \\frac{\\partial p}{\\partial x} \\mathbf {i} + \\frac{\\partial p}{\\partial y} \\mathbf {j} + \\frac{\\partial p}{\\partial x} \\mathbf {k}$ Do not worry about the density $\\rho $ , it is assumed as constant and thus can be manipulated to either side of the equation.",
"To have an intuitive understanding of the pressure gradient, think about a garden hose connected to a reservoir a few meters above ground, it is the pressure gradient between both ends of the hose that causes the water to flow out of the end where the overall pressure is lower (assuming the hose is perpendicular in relation to the gravitational field), but in this case the gradient is unidimensional, say $\\partial p/\\partial x$ .",
"The second term on the right is the viscosity or diffusion term, recalling the Laplacian operator, it can be written as $\\mu \\nabla ^2 \\mathbf {u} = \\mu [\\nabla \\cdot \\nabla (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k})]$ which in Cartesian coordinates resolves in three components on the directions $\\mathbf {i}$ , $\\mathbf {j}$ and $\\mathbf {k}$ $\\mu (\\frac{\\partial ^2 u_x}{\\partial x^2} + \\frac{\\partial ^2 u_x}{\\partial y^2} + \\frac{\\partial ^2 u_x}{\\partial z^2} )$ $\\mu (\\frac{\\partial ^2 u_y}{\\partial x^2} + \\frac{\\partial ^2 u_y}{\\partial y^2} + \\frac{\\partial ^2 u_y}{\\partial z^2} )$ $\\mu (\\frac{\\partial ^2 u_z}{\\partial x^2} + \\frac{\\partial ^2 u_z}{\\partial y^2} + \\frac{\\partial ^2 u_z}{\\partial z^2} )$ or, simply $\\mu \\nabla ^2 \\mathbf {u} = \\mu (\\nabla ^2 u_x \\mathbf {i} + \\nabla ^2 u_y \\mathbf {j} + \\nabla ^2 u_x \\mathbf {k})$ For a fluid moving slowly, the inertial forces are much smaller than the viscous forces, such that $ \\mu \\nabla ^2 \\mathbf {u} \\gg \\mathbf {u} \\cdot \\nabla \\mathbf {u} $ and the latter can be ignored resulting in a much simpler form of the Navier-Stokes equations $\\nabla p = \\mu \\nabla ^{2} \\mathbf {u}$ Or, in Cartesian coordinates $\\frac{\\partial p}{\\partial x} \\mathbf {i} + \\frac{\\partial p}{\\partial y} \\mathbf {j} + \\frac{\\partial p}{\\partial z} \\mathbf {k} = \\mu [ \\nabla \\cdot \\nabla (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) ]$ Now many analytical solutions of the Navier-Stokes equations can be derived from this simplification, for a broad range of experimental conditions.",
"The important point here is that at low flow velocities, when the viscous forces are much greater than the inertial forces the flow is often laminar (for many fluids found in daily life, the most important being water, this is true, it might not be if viscosity is a lot larger or smaller than expected values).",
"If the flow velocity is too high and/or viscosity too low a phenomenon known as turbulence might occur and it might become impossible to solve the differential equations analytically.",
"We can finally state our problem as the “laminar flow of water between two smooth parallel plates in which the pressure gradient is in the $x$ direction, the plates are infinitely long on the $y$ direction and the distance $d$ between the plates, through which the water flows, is oriented in the $z$ direction”.",
"If you are not familiar with solutions of physics problems in textbooks and theoretical models, you might wonder why would anyone assume anything as infinite and how on earth would this translate into a practical application.",
"The infinite plates simplification is a common occurrence in electromagnetism and other areas of physics and engineering as it becomes much easier to solve the differential equations that arise from the geometry of the problem.",
"The simple fact of assuming flow through a square tube results in a partial differential equation with series solution Check any partial differential equations or mathematical methods for physicists book and see how convoluted the whole thing is.",
"First if the pressure gradient is in the $x$ direction, there are no pressure gradients on the $y$ or $z$ directions and the equation further simplifies to $\\frac{\\partial p}{\\partial x} \\mathbf {i} = \\mu [ \\nabla ^2 (u_x \\mathbf {i} + u_y \\mathbf {j} + u_z \\mathbf {k}) ]$ Second, if the flow is in the $x$ direction, and is steady-state, the velocity does not vary on that direction, and if the plates are infinite in $y$ , as stated it also does not vary in $y$ such that $\\frac{\\partial p}{\\partial x} \\mathbf {i} = \\mu \\frac{\\partial ^2 u_x}{\\partial z^2} \\mathbf {i}$ Showing that $ \\mu \\nabla ^2 \\mathbf {u} = \\mu \\frac{\\partial ^2 u_z}{\\partial z^2} \\mathbf {i} $ when the flow velocity does not vary in $x$ or $y$ is left as an exercise to the reader.",
"A simple solution can be found if the pressure gradient in the $x$ direction is constant.",
"Omitting the unit vectors the equation to be solved is $\\frac{\\partial p}{\\partial x} = \\mu \\frac{\\partial ^2 u_x}{\\partial z^2}$ Whose solution can be found by separating the variables and integrating $\\int \\partial ^2 u_z = \\int \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\partial z^2$ $\\int \\partial u_z = \\int (\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} z + C_1) \\partial z$ $u_z = \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{z^2}{2} + C_1 z + C_2$ As always, we need the boundary conditions to solve the equation, in this case, because of the viscosity condition, the velocity is zero at the surface of the plates $u_z = 0 ~\\text{at} ~z = 0, ~u_z = 0 ~\\text{at} ~z = h{^{\\prime }}$ The first condition gives $C_2 = 0$ , while applying the second $0 = \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}^2}{2} + C_1 h{^{\\prime }} + 0 \\\\C_1 h = -\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}^2}{2}$ $C_1 = -\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}}{2}$ Replacing into Equation 8.5 results $u_z = \\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{z^2}{2} -\\frac{1}{\\mu } \\frac{\\partial p}{\\partial x} \\frac{h{^{\\prime }}}{2} z$ $\\boxed{u_z = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} z(z - h{^{\\prime }})}$ The discharge (or volumetric flux) passing through a section can be calculated by integrating the flow velocity over the area (or length in this case) $Q = \\int _0^{h{^{\\prime }}} u dz \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} \\int _0^{h{^{\\prime }}} z(z - h{^{\\prime }}) dz \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} (\\int _0^{h{^{\\prime }}} z^2 dz - \\int _0^{h{^{\\prime }}} zh{^{\\prime }} dz) \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} ( \\frac{z^3}{3}\\bigg |_0^{h{^{\\prime }}} - h^{\\prime } \\frac{z^2}{2}\\bigg |_0^{h{^{\\prime }}}) \\\\Q = \\frac{1}{2\\mu } \\frac{\\partial p}{\\partial x} ( \\frac{h{^{\\prime }}^3}{3} - \\frac{h{^{\\prime }}^3}{2}) \\\\$ $\\boxed{Q = - \\frac{1}{12\\mu } \\frac{\\partial p}{\\partial x} h{^{\\prime }}^3 \\\\}$ Dimensional analysis shows that in this case the formula results in an “areal discharge” in [L$^2$ T$^{-1}$ ].",
"Note that this has to be the case because of the infinite plates assumption.",
"Case II.",
"Flow in a cylindrical conduit - Poiseuille flow We can start with Equation 8.5, the process before hand is the same as discussed in the previous section.",
"The pressure gradient driving the flow is still in the $x$ direction but now the circular geometry in the cylider cross section requires both $y$ and $z$ components $\\frac{\\partial p}{\\partial x} \\mathbf {i} = \\mu (\\frac{\\partial ^2 u_y}{\\partial y^2}\\mathbf {j} + \\frac{\\partial ^2 u_z}{\\partial z^2} \\mathbf {k})$ This equation can be much more easily solved in cylindrical coordinates in which the Laplacian operator can be written as $\\nabla ^2 = \\frac{1}{r} \\frac{\\partial }{\\partial r}( r \\frac{\\partial }{\\partial r}) + \\frac{1}{r^2} \\frac{\\partial ^2 }{\\partial \\theta ^2} + \\frac{\\partial ^2 }{\\partial z^2}$ However, the flow velocity components do not vary in $z$ nor with the polar angle $\\theta $ along the cylindrical conduit, varying only with the radius of the tube $r$ , such that $\\nabla ^2 = \\frac{1}{r} \\frac{\\partial }{\\partial r}( r \\frac{\\partial }{\\partial r})$ In this case, Equation 8.10 can be written in cylindrical coordinates as $\\frac{\\partial p}{\\partial x} = \\mu \\frac{1}{r} \\frac{\\partial }{\\partial r}( r \\frac{\\partial u_r}{\\partial r})$ Now integration can be performed in a single variable $r$ instead of on $y$ and $z$ which simplifies the process.",
"$\\int \\partial ( r \\frac{\\partial u_r}{\\partial r}) = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\int r \\partial r\\\\r \\frac{\\partial u_r}{\\partial r} = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{r^2}{2} + C_1\\\\\\int \\partial u_r = \\int (\\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{r}{2} + \\frac{C_1}{r}) \\partial r \\\\$ $u_r = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{r^2}{4} + C_1 \\log {r} + C_2 \\\\$ The boundary conditions in this case are that the velocity is zero at the walls of the tube, as before, because of the viscosity condition, the velocity is maximum at center of the tube, at $r = 0$ .",
"$u_r = 0 ~\\text{at} ~r = a, ~u_r = u_{max} ~\\text{at} ~ r = 0$ Another important condition is that the velocity is finite at the center of the tube.",
"This implies that $C_1$ has to be zero, otherwise $C_1 \\log {0}$ results in a singularity.",
"Applying the first boundary condition and with $C_1 = 0 $ $0 = \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{a^2}{4} + 0 + C_2 \\\\$ $C_2 = - \\frac{1}{\\mu }\\frac{\\partial p}{\\partial x} \\frac{a^2}{4}$ Replacing into Equation 8.14 $u_r = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} (r^2 - a^2)$ The discharge (volumetric flux) can now be calculated by integrating the velocity function in $y$ and $z$ $Q = \\int _0^y \\int _0^z u dydz$ But again, because of the circular geometry of the tube's cross section, this is much more easily done by converting to polar coordinates, taking into account that $dydz$ is an area element $dA$ , which in polar coordinates is $ r dr d\\phi $ .",
"There should be no mystery here, if you take a circle of radius r and rotate this radius around 360° you will draw an area.",
"Rewriting the volumetric flux equation and solving $Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } \\int _0^a (r^2 - a^2) r dr d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } \\int _0^a (r^3 - r a^2) dr d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } (\\frac{r^4}{4} - \\frac{r^2}{2} a^2)\\bigg |_0^a d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } (-\\frac{a^4}{4}) d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} \\int _0^{2\\pi } (-\\frac{a^4}{4}) d\\phi \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} (-\\frac{a^4}{4}) \\phi \\bigg |_0^{2\\pi } \\\\Q = \\frac{1}{4\\mu }\\frac{\\partial p}{\\partial x} (-\\frac{a^4}{4}) 2\\pi \\\\$ $\\boxed{Q = -\\frac{\\pi }{8\\mu }\\frac{\\partial p}{\\partial x} a^4 \\\\}$ in which $Q$ is the discharge in [L$^3$ T$^{-1}$ ] as expected.",
"Henry Darcy was a french engineer who, while working on water distribution system for the city of Dijon in the 1800s, wrote a monumental report concerning various aspects of water supply and distribution [32].",
"In one of the appendixes of the document he presents an equation that describes the flow of water through a column filled with sand under a hydraulic gradient (Figure REF ) $q = k \\frac{s}{e} [h + e \\mp h_0]$ In which $q$ was the water flux, $e$ is the height of the sand column, $s$ is the cross sectional area of the column, $h$ is the water height in the upper part of the column (on the inlet), $h_0$ is the water height on the inferior part of the column (on the outlet).",
"Figure: Schematics of sand column used in the original Darcy experiment (adapted from ).",
"(a) is the water inlet, (b) and (d) are manometers, (c) is the sand column, (e) is the water outlet, (f) is the support, and (g) is a reservoir for water collecting and measuring outflow volume.If the water height is zero at the outlet, $h_0 = 0$ $q = k \\frac{s}{e} [h + e]$ He realized that the water flux through the sand column was proportional to the pressure difference between the inlet and the outlet and he defined the constant of proportionality as a coefficient k, dependent on the permeability of the sand (“dependant de la permeabilité de la couche”).",
"So $k$ in this context was defined as a permeability, dependent on the physical characteristics of the solid phase.",
"In contemporary terms, the height of water at the inlet and outlet is effectively the pressure potential in terms of height of water.",
"Considering the potentials at the inlet and outlet, and if we establish the gravitational reference at the bottom of the column, we have at the inlet (top of the column): $\\psi _{gi} = e \\\\\\psi _{pi} = h + P_0\\\\\\psi _{ti} = e + h + P_0$ At the outlet (bottom of the column): $\\psi _{go} = 0 \\\\\\psi _{po} = h_0 + P_0\\\\\\psi _{to} = h_0 + P_0$ in which $P_0$ is the atmospheric pressure.",
"Therefore, Darcy's law can be expressed as $q = k A\\frac{\\psi _{ti} - \\psi _{to}}{L} = k s \\frac{(e + h + P_0) - (h_0 + P_0)}{e} \\\\= k s\\frac{(e + h + P_0) - (h_0 + P_0)}{e} = k s\\frac{(e + h)}{e}$ for $L = e$ , the height of the column, $A = s$ the cross sectional area of the column and $h_0 = 0$ .",
"The unidimensional form of Darcy's law, as applied to a column filled with saturated porous media is represented in modern notation in introductory soil physics and groundwater hydrology books as $\\boxed{Q = K_s A\\frac{\\psi _{ti} - \\psi _{to}}{L} = Q = K_s A\\frac{\\Delta \\psi _{t}}{L}}$ for total potential expressed in terms of height of water.",
"A more precise general representation is in terms of differentials, thus for a one dimensional column oriented in some arbitrary $x$ direction, Darcy's law is $\\boxed{Q = -K_s A\\frac{d \\psi _{t}}{dx}}$ in which $Q$ is the water discharge in L$^3$ T$^{-1}$ , $K_s$ is now defined as the saturated hydraulic conductivity in L T$^{-1}$ and the term $\\frac{d \\psi _{t}}{dx}$ is termed the hydraulic gradient or the potential difference over the length between the two points in which potential is being measured, in L L$^{-1}$ .",
"The negative sign, source of much confusion in soil physics, means that the flow is opposite to the direction of the gradient.",
"In other words, the flow is always from the high total potential to the low total potential.",
"As with the original Darcy formulation, these equations can be expressed in terms of height of water on the inlet and outlet of a saturated column filled with some porous material.",
"If the column is oriented on the $z$ direction, here defined as in the direction of the gravitational field, we have $Q = -K_s A \\frac{d \\psi _{t}}{dz} = -K_s A \\frac{ (\\psi _{pi} + \\psi _{gi}) - (\\psi _{po} + \\psi _{go}) }{dz}$ If the length of the column is $z$ and establishing the gravitational potential at the bottom of the column at $z = 0$ , we have $Q = -K_s A \\frac{ (\\psi _{pi} + z) - (\\psi _{po} + 0) }{z}$ Representing the pressure potential in terms of height $h_i$ $Q = -K_s A \\frac{ (h_i-h_o) + z}{z} = -K_s A (\\frac{ h_i-h_o}{z} + 1)$ If the pressure potential is zero on the outlet $Q = -K_s A (\\frac{ h_i}{z} + 1)$ It is easy to show that if the column is oriented perpendicular to the gravitational field in some arbitrary $x$ (or $y$ ) direction, Darcy's law in terms of the height of water on the inlet and outlet will be $Q = -K_s A (\\frac{ h_i - h_o}{x})$ For $h_o$ = 0 this reduces to $Q = -K_s A \\frac{ h_i}{x}$ where $x$ (or $y$ ) represent the horizontal length of the column and $z$ the vertical length.",
"It is clear that Darcy's law is a linear relationship between discharge and gradient with slope equal to the saturated hydraulic conductivity.",
"Darcy's law, along with Fourier's law of heat conduction, Ohm's law in electromagnetism, Fick's law of molecular diffusion and Poiseuille's law is one of the linear gradient transport laws discovered in the 1800s [84].",
"Although Darcy only mentions Poiseuille's law in his report, it is very likely that he was aware and was influenced by the physics of the time in creating his empirical law [84].",
"Darcy's law can be written in terms of a flux density $q$ , similar to the other laws of physics mentioned, by dividing the discharge $Q$ by the area $A$ $q = \\frac{Q}{A} = -K_s A \\frac{d \\psi _t}{dx}$ Darcy obviously knew about and investigated Poiseuille's, as he was an engineer concerned with water distribution, which would obviously require studying water flow in pipes.",
"Analyzing Poiseuille's solution for flow in a cylindrical conduit, $Q = -\\frac{\\pi }{8\\mu }\\frac{\\partial p}{\\partial x} a^4$ we can rewrite this equation by reorganizing it $Q = -\\frac{\\pi a^2 a^2}{8\\mu }\\frac{\\partial p}{\\partial x}$ and introducing $A = \\pi a^2$ as the area of the tube $Q = - A\\frac{\\pi a^2}{8\\mu }\\frac{\\partial p}{\\partial x}$ Rewriting the term $\\frac{\\pi a^2}{8\\mu }$ as some proportionality constant $k^*$ $Q = - A k^*\\frac{\\partial p}{\\partial x}$ Poiseulle's law can be interpreted as some limiting case of Darcy's law for when there is no porous media within the pipe.",
"The presence of a porous medium causes additional stresses in the fluid that decrease the discharge under the same potential gradient.",
"Because natural porous media usually have complex grain size distribution, grain shape and rugosity, resulting in complex pore size distribution and geometry, the empirical constant in Darcy's law, $K_s$ , has to be determined experimentally, there is no analytical solution of Darcy's law from the Navier-Stokes equation.",
"There are a few studies that use stochastic considerations to derive Darcy's law from the Navier-Stokes equations, but as discussed previously, they must rely in some averaging mechanism, unless the porous media model is extremelly simplistic.",
"In a stratified media, such as a soil profile, sedimentary rock or deposit, or laboratory column, where the direction of flux is perpendicular to the layering, the flow will be determined by some form of averaging of the individual hydraulic conductivity of the layers Here as a direct average, in a much more simplistic way than what was discussed previously for averaging of the Navier-Stokes equation at the pore scale.",
"Consider a composite column, composed of two independently homogeneous but different materials $A$ and $B$ with heights $x_A$ and $x_B$ (REF ).",
"The flux is in the $z$ direction and the total potential is measured on the points $a$ , $b$ and $c$ as $\\psi _{ta}$ , $\\psi _{tb}$ and $\\psi _{tc}$ .",
"Darcy's law is defined for material $A$ in terms of the flux between points $a$ and $b$ $q_{ab} = -K_{sA} \\frac{\\psi _{tA} - \\psi _{tB}}{x_A}$ and for material $B$ in terms of the flux between points $b$ and $c$ $q_{bc} = -K_{sB} \\frac{\\psi _{tB} - \\psi _{tC}}{x_B}$ Figure: Darcy's law in a composite column with individually homogeneous but different materials AA and BB.As point $b$ is located in the interface between the two materials, the total potential is equal and we can solve both equations for $\\psi _{tB}$ .",
"$\\psi _{tB} = \\psi _{tA} + \\frac{q_{ab}x_A}{ K_{sA}}$ $\\psi _{tB} = \\psi _{tC} - \\frac{q_{bc}x_B}{ K_{sB}}$ Replacing $\\psi _{tB}$ in any of the equations and solving for the total potential difference between the column's inlet in $a$ and the outlet in $c$ $\\psi _{tA} - \\psi _{tC} = - (\\frac{q_{bc}x_B}{ K_{sB}} + \\frac{q_{ab}x_A}{ K_{sA}})$ For constant (steady-state) saturated flow, for which Darcy's law is being considered, the flux in any point within the length of the composite column is constant, and thus $q_{ab} = q_{bc} = q_{ac} = q$ , and $\\psi _{tA} - \\psi _{tC} = - q(\\frac{x_B}{ K_{sB}} + \\frac{x_A}{ K_{sA}})$ Finally solving for $q$ we arrive at an expression for a composite material $q = -\\frac{ (\\psi _{tA} - \\psi _{tC})}{(\\frac{x_B}{ K_{sB}} + \\frac{x_A}{ K_{sA}})}$ The term $\\frac{1}{(\\frac{x_B}{ K_{sB}} + \\frac{x_A}{ K_{sA}})} = \\frac{K_{sA}K_{sB}}{x_BK_{sA} + x_AK_{sB}}$ is a form of weighed mean of the hydraulic conductivity of the layers with the height of each layer acting as a weighing term.",
"In practical terms, especially in laboratory, saturated hydraulic conductivity can be determined using constant head and falling head permeametersAs we will see, hydraulic conductivity and permeability are different things, however, these devices historically have been called permeameters.",
"As they can also be used to measure the permeability of the medium, we will maintain the use of the terminology.",
"In constant head permeameters, the total potential or hydraulic head is kept constant in both ends of the column.",
"The hydraulic head must be equal or greater than zero on both ends.",
"If the total potential is negative the water will be under tension, the water content might be below saturation and hydraulic conductivity is no longer constant.",
"Saturated hydraulic conductivity is calculated by measuring discharge or flux density as $K_s = -\\frac{QL}{A(h_i - h_o)}$ for a horizontal column, and $K_s = -\\frac{Q}{A[(h_i - h_o)/L +1]}$ for a vertical column with the gravitational reference chosen at the outlet.",
"Note that the flow can be in any direction including upwards, the direction being indicated by the sign of the hydraulic conductivity.",
"The second common type of laboratory method for measuring saturated hydraulic conductivity are falling head permeameters.",
"In the falling head method the hydraulic head is not constant but falls from a height $h_0$ to a height $h$ in time.",
"Falling head permeameters are convenient when high hydraulic heads are required [10].",
"To derive the expression for measuring saturated hydraulic conductivity, consider a pemeameter connected to a tube filled with water on its base (Figure REF ).",
"The flow is upwards and the discharge is measured at the outlet on the water level on the upper part of the column.",
"The tube has diameter $d_t$ , radius $r_t$ and area $A_t$ and the column filled with porous media has diameter $d_c$ , radius $r_c$ and area $A_c$ .",
"The inflow on the smaller tube is given by [37] $Q_{in} = -A_t \\frac{dh}{dt}$ while on the column the flow obeys Darcy's law $Q_{out} = \\frac{K A_c h}{L}$ Under saturated flow conditions the discharge into the column has to be equal to the discharge out of the column $Q_{in} = Q_{out}$ $-A_t \\frac{dh}{dt} = \\frac{K A_c h}{L}$ Reorganizing the terms and integrating $\\frac{dh}{h} = - \\frac{A_c K}{A_t L} dt$ $\\int \\frac{dh}{h} = - \\int \\frac{A_c K}{A_t L} dt$ $\\log {|h|} \\bigg |_{h}^{h_{0}} = - \\frac{A_c K}{A_t L} t \\bigg |_{t_{0}}^t$ By applying the boundary conditions $t = 0$ at $h = h_0$ , and considering $h$ as always positive we arrive at $\\log {h} - \\log {h_{0}} = - \\frac{A_c K}{A_t L} t$ Solving for $K$ and considering the area of the small tube as $A_t = \\pi r_t^2 $ and the area of the soil column as $A_t = \\pi r_t^2$ Remember that $\\log x - \\log y = \\log {x/y}$ and notice that we multiplied both sides by $-1$ to eliminate the negative sign in $K$ .",
"$K = \\frac{r_t^2 L}{r_c^2 t} \\log \\frac{h_0}{h}$ or, in terms of diameter $K = \\frac{d_t^2 L}{d_c^2 t} \\log \\frac{h_0}{h}$ For alternative methods and equipment for measuring saturated hydraulic conductivity on the field and on the laboratory, the user is referred to [31].",
"Figure: Diagram illustrating a generic the falling head permeameter.Up until now we have avoided dealing with vectors by working with flow in one dimension.",
"A column filled with a porous medium saturated with water, where the diameter of the column is small and where the walls are impermeable can be treated as an unidimensional system and the flux density vector is oriented in the general direction of water flow.",
"In soils and sedimentary deposits, aquifers, porous rocks and many other natural materials, flow is often not constrained to one dimension.",
"This is especially true on the field, where water and contaminant transport occur in large scales and in response to total potential gradients, but also in laboratory, for large enough samples and when heterogeneity cannot be disregarded.",
"Under such conditions, not only the flux density is a vector but the gradient is also a vector, and often hydraulic conductivity is a second order tensor.",
"Recalling Chapter REF , Darcy's law can be generalized as $\\mathbf {q} = \\mathbf {K} \\cdot \\nabla \\psi _t$ In three-dimensional Cartesian coordinates, the flux density is decomposed as $\\mathbf {q} = q_x \\mathbf {i} + q_y \\mathbf {j} + q_x \\mathbf {k}$ while the three-dimensional gradient is $\\nabla = \\frac{\\partial \\psi _t}{\\partial x} \\mathbf {i} + \\frac{\\partial \\psi _t}{\\partial y} \\mathbf {j} + \\frac{\\partial \\psi _t}{\\partial z} \\mathbf {k}$ Under conditions of anisotropy, where hydraulic conductivity varies with direction in the material, saturated hydraulic conductivity is a second rank tensor with $3^2 = 9$ elements $\\mathbf {K} = K_{ij} =\\begin{pmatrix}K_{xx} & K_{xy} & K_{xz}\\\\K_{yx} & K_{yy} & K_{yz}\\\\K_{zx} & K_{zy} & K_{zz}\\end{pmatrix}$ With the conductivity tensor, Darcy's law can be written as $\\begin{pmatrix}q_{x}\\\\q_{y} \\\\q_{z}\\end{pmatrix}=\\begin{pmatrix}K_{xx} & K_{xy} & K_{xz}\\\\K_{yx} & K_{yy} & K_{yz}\\\\K_{zx} & K_{zy} & K_{zz}\\end{pmatrix}\\cdot \\begin{pmatrix}\\partial \\psi _t/\\partial x\\\\\\partial \\psi _t/\\partial y \\\\\\partial \\psi _t/\\partial z\\end{pmatrix}$ Therefore, the components of flux density in $x$ , $y$ and $z$ are $q_x = K_{xx} \\frac{\\partial \\psi _t}{\\partial x} + K_{xy} \\frac{\\partial \\psi _t}{\\partial y} + K_{xz}\\frac{\\partial \\psi _t}{\\partial z}$ $q_y = K_{yx} \\frac{\\partial \\psi _t}{\\partial x} + K_{yy} \\frac{\\partial \\psi _t}{\\partial y} + K_{yz}\\frac{\\partial \\psi _t}{\\partial z}$ $q_z = K_{zx} \\frac{\\partial \\psi _t}{\\partial x} + K_{zy} \\frac{\\partial \\psi _t}{\\partial y} + K_{zz}\\frac{\\partial \\psi _t}{\\partial z}$ The hydraulic conductivity tensor is a symmetrical tensor, meaning that $K_{ij} = K_{ji}$ such that the tensor reduces to six different components.",
"There are profound implications related to these equations.",
"The fact that the conductivity tensor is symmetrical is by no means a trivial issue and is a property of most of the transport equations and related to the theory know as the Onsager reciprocal relations, in the realm of non-equilibrium thermodynamics.",
"Another important property of the conductivity tensor is that it is assumed as orthotropic, meaning that there are three principal directions for the conductivity vectors which are orthogonal to eachother.",
"If this condition is valid, it is possible to orientate the axis in such a way that the conductivity tensor is reduced to three principal directions, such that $K_ij = 0$ for $i \\ne j$ and $K_{ii}$ for $i = j$ $\\mathbf {K} = K_{ij} =\\begin{pmatrix}K_{xx} & 0 & 0\\\\0 & K_{yy} & 0\\\\0 & 0 & K_{zz}\\end{pmatrix}$ In practice, this means that it is possible to orientate a sample of an anisotropic media in laboratory conditions such that only the three principal permeability directions need to be measured.",
"If the medium is isotropic, there is no difference between $K_ii$ measured in any direction and $K$ is reduced to the constant hydraulic conductivity scalar $\\mathbf {K} = K_{ij} = K\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{pmatrix}$ Suppose now that the saturated hydraulic conductivity does not vary with direction at each point in space, but the value, even being equal in all directions, is different in different points in space.",
"In this case, the saturated hydraulic conductivity is a scalar function in space $K(x,y,z)$ and Darcy's law is $\\mathbf {q} = -K(x,y,z) \\nabla \\psi _t$ A medium where the hydraulic conductivity does not vary with direction at a given point in space is said to be isotropic and if the scalar hydraulic conductivity does vary in space is called heterogeneous.",
"If the saturated hydraulic conductivity does not vary with direction and is the same in all points in space the medium is said to be homogenous and isotropic and Darcy's law simplifies to $\\mathbf {q} = -K \\nabla \\psi _t$ A consistent representation of Darcy's law requires that the potentials are expressed in terms of hydraulic height [L].",
"This guarantees that the gradient $\\frac{\\partial \\psi _t}{\\psi x}$ or $y$ or $z$ is nondimentional [L L$^{-1}]$ and that the flux density has the same units of the saturated hydraulic conductivity [L T$^{-1}$ ].",
"It is convenient now to fully introduce the notation used before for the total potential in height units as $h_t = h_p + h_g$ in which $h_t$ is the total potential, $h_p$ is the pressure potential and $h_g$ is the gravitational potential, all in [L] units.",
"Using this notation and considering the $z$ direction as the direction of the gravitational force, the components of Darcy's law in $x$ , $y$ and $z$ are $q_x = -K \\frac{\\partial h_t}{\\partial x} = -K \\frac{h_{ti} - h_{to}}{x - x_0} = -K \\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{x - x_0}$ $q_y = -K \\frac{\\partial h_t}{\\partial y} = -K \\frac{h_{ti} - h_{to}}{y - y_0} = -K \\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{y - y_0}$ $q_z = -K \\frac{\\partial h_t}{\\partial y} = -K \\frac{h_{ti} - h_{to}}{z - z_0} = -K \\frac{(h_{pi} + h_{gi}) - (h_{po} + h_{go})}{z - z_0}$ where the subscripts $i$ and $o$ stand for inlet and outlet.",
"Setting the gravitational reference at $x_0$ , $y_0$ and $z_0$ , on the outlet, the gravitational potential will be $z_0$ in both the inlet and outlet in $x$ and $y$ and in the outlet only in $z$ .",
"If the column height is $z$ , and the inlet is located in $z$ this will be the gravitational potential in the inlet for the vertical column.",
"Using these conditions, the components of Darcy's law in three dimensions are $q_x = -K \\frac{(h_{pi} + z_0) - (h_{po} + z_0)}{x - x_0} = \\frac{\\partial h_{p}}{\\partial x}$ $q_y = -K \\frac{(h_{pi} + z_0) - (h_{po} + z_0)}{y - y_0} = \\frac{\\partial h_{p}}{\\partial y}$ $q_z = -K \\frac{(h_{pi} + z) - (h_{po} + z_0)}{z - z_0} = -K \\frac{(h_{pi} - h_{po}) + (z - z_0)}{z - z_0} = -K (\\frac{\\partial h_{p} + \\partial z}{\\partial z})$ $q_z = -K (\\frac{\\partial h_{p}}{\\partial z} + 1)$ Suppose now the total potential is expressed in pressure units.",
"Using the $z$ coordinate for convenience $q_z = -K \\frac{\\partial \\psi _t}{\\partial z}$ it becomes clear that the the gradient is no longer dimensionless and the equation is not consistent in terms of units.",
"Expressing the total potential in terms of the sum of the potentials $q_z = -K \\frac{\\partial (\\psi _p + \\psi _g)}{\\partial z} = -K \\frac{\\partial (\\rho _w g h + \\rho _w g z)}{\\partial z}$ and using the definitions from Chapter REF , it becomes clear that for consistence of units to be achieved, the saturated hydraulic conductivity would have to be divided by $\\rho _w g$ $q_z = -\\frac{K}{\\rho _w g} \\frac{\\partial (\\rho _w g h + \\rho _w g z)}{\\partial z}$ Therefore, for a liquid of constant density, Darcy's law in its original form is retrieved by placing the constant terms outside of the derivative $q_z = -\\frac{K \\rho _w g}{\\rho _w g} \\frac{\\partial (h + z)}{\\partial z} = -K \\frac{\\partial (h + z)}{\\partial z}$ Although stated otherwise in his report, the saturated hydraulic conductivity as defined by Darcy in his original experiment is a composite property of the solid and liquid phases [69].",
"Although this is convenient in soil physics where the phases concerned are usually soil and water, it might not be convenient in groundwater hydrology and petroleum engineering, where different and often multiple phases are concerned and where the characteristics of the porous material might be of concern.",
"Suppose you define a saturated conductivity term for water using a column experiment.",
"You could repeat the same experiment with other fluids, for example oil, salt solution, gasoline, and for each fluid you would obtain one saturated conductivity.",
"To circumvent this issue, and using dimensional analysis, the saturated conductivity term was rewritten in terms of the properties of the fluid being used, namely viscosity and density and a new coefficient, called intrinsic permeability or simply permeability $\\kappa $ , which depends on the solid phase alone, thus [73], [101], [69] $K = \\frac{\\kappa \\rho _f g}{\\mu }$ In which $\\rho _f$ and $\\mu $ are the density and viscosity of a fluid moving through the porous materialThese laws should be valid for viscous, incompressible newtonian fluids in general.",
"Thus, Darcy's law can be expressed in terms of pressure using the permeability as $q_z = -\\frac{\\kappa \\rho _f g}{\\mu } \\frac{1}{\\rho _f g} \\frac{\\partial (\\rho _f g h + \\rho _f g z)}{\\partial z} = -\\frac{\\kappa }{\\mu } \\frac{\\partial (\\rho _f g h + \\rho _f g z)}{\\partial z}$ and in terms of length for the potential as $q_z = -\\frac{\\kappa \\rho _f g}{\\mu } \\frac{\\partial ( h + z)}{\\partial z}$ For an arbitrary $x$ direction perpendicular to Earth's gravitational field these reduce to $q_x = - \\frac{\\kappa }{\\mu } \\frac{\\partial (\\rho _f g h)}{\\partial z} = - \\frac{\\kappa }{\\mu } \\frac{\\partial P}{\\partial z}$ and $q_x = -\\frac{\\kappa \\rho _f g}{\\mu } \\frac{\\partial h}{\\partial x}$ Note that the similarity between Darcy's law and Poiseuille's equation is much more evident when the pressure form is considered.",
"In general, the permeability is a tensor with the same properties of the saturated hydraulic conductivity, i.e.",
"for an anisotropic medium $\\mathbf {k} = k_{ij} =\\begin{pmatrix}k_{xx} & k_{xy} & k_{xz}\\\\k_{yx} & k_{yy} & k_{yz}\\\\k_{zx} & k_{zy} & k_{zz}\\end{pmatrix}$ which can be orientated in the direction of the principal axes $\\mathbf {k} = k_{ij} =\\begin{pmatrix}k_{xx} & 0 & 0\\\\0 & k_{yy} & 0\\\\0 & 0 & k_{zz}\\end{pmatrix}$ and for an isotropic medium $\\mathbf {k} = k_{ij} = k\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{pmatrix}$ We have seen that the flux density vector in Darcy's law is the discharge over area.",
"Because the pore geometry is complex, a variety of shapes and sizes of pores are available for flow in porous materials.",
"The flux density in Darcy's law is not the velocity of flow of fluid in the pores.",
"Because the flow only takes place within the pores, the average flow velocity is the flux density divided by the porosity.",
"$v = \\frac{Q}{An} = \\frac{q}{n}$ in which $n$ is the porosity.",
"This relationship is valid because for an homogeneous porous material, area porosity is equivalent to the volumetric porosity [10].",
"In other words, considering a cross section of the column, only part of the area is available for flow and for a principle equivalent to Bernoulli's law, the flux velocity at each point will be higher for smaller pores, within the limitations of viscosity and wall effects.",
"The difference in velocities in a porous material will be an important mechanism in contaminant transport.",
"Like many other equations and laws in fluid mechanics, Darcy's law is valid for laminar flow.",
"Laminar flow here meaning the same as we discussed when analyzing the problem of a sphere settling under gravity.",
"The flow is slow enough that there is no turbulence or vortexes.",
"Poiseuille's law also has a restriction of laminar flow.",
"As we seen for the sphere problem, one way of identifying an approximate limit between laminar and turbulent flow is by using the Reynolds number.",
"For flow in a cylindrical conduit, the Reynolds number is $Re = \\frac{\\rho _f u d_h}{\\mu }$ In which $d_h$ is the hydraulic diameter of the pipe, simple the inner diameter if the tube is circular.",
"For Darcy's law, this can be approximated by $Re = \\frac{\\rho _f u d^{\\prime }}{\\mu }$ In which $d^{\\prime }$ is some representative length dimension of the porous matrix representing the elementary channels of the porous medium, usually a parameter related to average grain diameter or other representative grain diameters [10] such as the diameter with 10$\\%$ , 30$\\%$ or or 50$\\%$ of the material passing through a sieve of mesh equal to the diameter of interest.",
"Darcy's law implies a linear relationship between flux density and hydraulic gradient.",
"The experimentally determined $R_e$ values for the transition zone from laminar to turbulent flow for Darcy's law ranges from 1 to 10, with laminar flow for values less than 1, turbulent flow for values above 10 [10].",
"For values above 10 the relationship between flux density and gradient is no longer linear and Darcy's law cannot be used.",
"As the flow velocity increases, inertia terms are no longer negligible.",
"At the transition zone, the Darcy-Forchheimer equation can be used, presented here in one dimension as [86].",
"$-\\frac{\\partial p}{\\partial x} = \\frac{\\mu }{\\kappa }q + \\beta \\rho _f q^2$ In which $\\beta $ is the Forchheimer coefficient or the inertial permeability.",
"Note the quadratic term on the flux density.",
"If the flow is too slow, for very small gradients and for some very fine soils and sediments, surface forces (Chapter REF ) can become important as well as viscous effects.",
"It has been speculated that under such conditions water can display non-newtonian behavior and non-darcyan flow is observed [53], [10].",
"Saturated hydraulic conductivity is a function of total porosity and pore size distribution.",
"As such it is a function of bulk density and grain size distribution.",
"Compaction and consolidations will decrease the total porosity and will tend to reduce saturated hydraulic conductivity.",
"In soil science saturated hydraulic conductivity can be used as an indicator of soil degradation.",
"Grain size distribution controls not only pore size distribution but also influences bulk density.",
"Coarse grained materials can have lower total porosity but the average pore diameter will be greater, resulting in larger values of saturated hydraulic conductivity.",
"For soils rich in 2:1 layer silicates, high clay content will result in lower saturated hydraulic conductivity, because of the smaller average pore radius and in some cases because of expansion of the silicate clays.",
"For soils rich in iron and aluminum oxides, the saturated hydraulic conductivity is often very high, due to the microaggregate structure with two pore domains, micropores and mesopores, and macropores.",
"Under saturated conditions the bulk of the water transport will take part in macropores.",
"In sediments and rocks, grain size distribution and degree of consolidation will control saturated hydraulic conductivity.",
"In highly consolidated igneous and metamorphic rocks, water transport can occur in fractures and hydraulic conductivity can be highly directional.",
"Saturated hydraulic conductivity in the field is not a normally distributed variable and this should be accounted for when analyzing data.",
"Bear [10] reports saturated hydraulic conductivity values ranging from $10^{2}~cm~s^{-1}$ for clean gravel to $10^{-11}~cm~s^{-1}$ for impervious rocks.",
"The same source reports values ranging from $10^{-3}~cm~s^{-1}$ for sandy soils to $10^{-11}~cm~s^{-1}$ for dispersive clay soils.",
"Because of the relationship of saturated hydraulic conductivity with porous media parameters, various attempts have been made to derive equations to calculate the saturated hydraulic conductivity from physical properties of the material.",
"These models vary from physically based but approximated models based on Poiuseuille's equation to purely empirical models based on statistical regression.",
"One model commonly found in engineering and hydrogeology is the Kozeny-Carman equationDerivations based on the original sources should be provided in later editions $k = \\frac{n^3}{(1-n)^2} \\frac{c_0}{M_s^2}$ in which $\\kappa $ is the permeability, $M_s$ the the specific surface area with respect to unit volume of porous medium, $n$ is the porosity and $c_0$ is the Kozeny's constant, dependent on the geometrical form of the channels on the model [10].",
"The term $f(n) = \\frac{n^3}{(1-n)^2}$ is the porosity factor.",
"It has been found that $c_0 = 1/5$ resulted good agreement with measured data, resulting in $k = \\frac{n^3}{(1-n)^2} \\frac{1}{5M_s^2}$ Defining the mean particle size as $d_m = M_s/6$ , the Kozeny-Carman equation can be written in terms mean of particle diameter as $k = \\frac{n^3}{(1-n)^2} \\frac{d_m^2}{180}$ In Chapter REF we derived a mass conservation equation (continuity equation) of the form $\\frac{\\partial \\rho }{\\partial t} = - \\nabla \\cdot (\\rho \\mathbf {u})$ Suppose that water is being accumulated in the pore space of the representative volume of interest.",
"If the media is completely saturated and the density of water is considered constant, $\\frac{\\partial \\rho }{\\partial t} = 0$ and $\\nabla \\cdot (\\rho \\mathbf {u}) = \\rho \\nabla \\cdot \\mathbf {u}$ and, as before, we arrive at $\\nabla \\cdot \\mathbf {u} = 0$ For flow in porous media, the velocity term is replaced by the flux density $\\nabla \\cdot \\mathbf {q} = 0$ We already know that the flux density is represented by Darcy's law, thus, expanding the components of flux and replacing Darcy's law $(\\frac{\\partial }{\\partial x} \\mathbf {i} + \\frac{\\partial }{\\partial y} \\mathbf {j} + \\frac{\\partial }{\\partial y} \\mathbf {k}) \\cdot (q_x \\mathbf {i} + q_y \\mathbf {j} + q_z \\mathbf {k}) = 0\\\\(\\frac{\\partial q_x}{\\partial x} + \\frac{\\partial q_y}{\\partial y} + \\frac{\\partial q_z}{\\partial z}) = 0 \\\\(\\frac{\\partial }{\\partial x}(-K_x \\frac{\\partial h}{\\partial x}) + \\frac{\\partial }{\\partial y}(-K_y \\frac{\\partial h}{\\partial y}) + \\frac{\\partial }{\\partial z} (-K_z \\frac{\\partial h}{\\partial z}) ) = 0\\\\$ dropping the parentheses and multiplying both sides by $-1$ results in $\\frac{\\partial }{\\partial x}(K_x \\frac{\\partial h}{\\partial x}) + \\frac{\\partial }{\\partial y}(K_y \\frac{\\partial h}{\\partial y}) + \\frac{\\partial }{\\partial z} (K_z \\frac{\\partial h}{\\partial z}) ) = 0$ This equation is valid for a porous media where the unsaturated hydraulic conductivity cannot be assumed as constant.",
"Because for a truly anisotropic media, the saturated hydraulic conductivity is a second rank tensor, a more general representation would be $\\nabla \\cdot (\\mathbf {K} \\cdot \\nabla {h}) = 0$ Back to more general cases, if the saturated hydraulic conductivities in each direction does not vary with the position in space, but the conductivities in each direction are different, we have $ K_x \\ne K_y \\ne K_z $ all constant, and $K_x \\frac{\\partial h^2}{\\partial ^2 x} + K_y \\frac{\\partial ^2 h}{\\partial y^2} + K_z \\frac{\\partial ^2 h}{\\partial z^2} = 0$ Finally, if the saturated hydraulic conductivity is constant everywhere, independent of direction, such that $ K_x \\ne K_y \\ne K_z = K $ we have $K \\frac{\\partial h^2}{\\partial ^2 x} + K \\frac{\\partial ^2 h}{\\partial y^2} + K \\frac{\\partial ^2 h}{\\partial z^2} = 0$ and dividing both sides by $K$ we arrive at $\\boxed{\\frac{\\partial h^2}{\\partial ^2 x} + \\frac{\\partial ^2 h}{\\partial y^2} + \\frac{\\partial ^2 h}{\\partial z^2} = 0}$ This is a partial differential equation know as Laplace equation.",
"It is ubiquitous in physics and mathematics and can be represented using the aptly named Laplace operator ($\\nabla ^2$ ), of which we have made extensive usage in this text $\\boxed{\\nabla ^2 h = 0}$ Laplace's equation is a second-order partial differential equation that appears in electrostatics, gravitation, head conduction, transport phenomena and many other problems in applied physics, mathematics and engineering.",
"As discussed previously, there is vast amount of literature covering methods of solution of partial differential equations, the user being encourage to consult the references.",
"Analytical solutions in three and two dimensions can be found in the literature.",
"As an example we will show how Laplace's equation can be solved in one-dimension to find the distribution of $h$ along a length of saturated porous media.",
"Let us consider flow in the $x$ directionTo avoid any complication with implicit gravitational field, such that h is constant in $y$ and $z$ and the partial derivatives are zero $\\frac{\\partial h^2}{\\partial ^2 x} = 0 \\\\\\int \\partial h^2 = \\int 0 \\partial ^2 x \\\\\\int \\partial h = \\int C_1 \\partial x \\\\$ $h = C_1 x + C_2 \\\\$ To find the values of the constants one just needs to repeat the same procedure of applying the boundary conditions.",
"Supposing, for example that $h = h_1$ at $x = 0$ and $h = h_2$ at $x = h$ , we have $h = \\frac{(h_2 - h_1)}{h} x + h_1 \\\\$ This shows that $h$ varies linearly along the length of the $x$ gradient.",
"Laplace's equation solutions in 2D and 3D are highly dependent on the geometry of the problem, and can be challenging.",
"For example, A general solution of Laplace's equation in a square geometry with boundary conditions with $h = 0$ in all sides except the right vertical wall will have the form $h(x, y) = \\sum _{n = 1}^\\infty c_n \\sinh (\\frac{n \\pi x}{b}) sin(\\frac{n \\pi y}{b})$ with $c_n = \\frac{2}{b \\sinh (\\frac{n \\pi a}{b})} \\int _0^b g_2(y) sin(\\frac{n \\pi y}{b}) dy$ in which $g_2(y)$ is a function that defines the nonzero boundary conditionhttp://www.math.ubc.ca/~peirce/M257_316_2012_Lecture_24.pdf.",
"ll $ p $ Pressure $ \\mu $ Viscosity $ \\mathbf {u} $ Vector velocity function $ u_x, u_y, u_z $ Components of the velocity vector in Cartesian coordinates $ \\mathbf {i}, \\mathbf {j}, \\mathbf {k} $ Unit vectors in Cartesian coordinates $ x, y, z$ Generalized Cartesian coordinates $ C_1, C_2 $ Constants $ h $ Distance between plates $ Q $ Discharge $ r $ Radial dimension $ \\phi $ Angle in polar coordinates $ q $ Water flux $ h^{\\prime } $ Height of water above a given point (hydraulic head) $ A, s $ Cross section area $ L, e $ Length $ \\psi $ Water potential $ P $ Pressure $ k , K_s $ Saturated hydraulic conductivity $ d_t $ Tube diameter $ r_t $ Tube radius $ A_t $ Tube cross sectional area $ d_c $ Column diameter $ r_c $ Column radius $ A_c $ Column cross sectional area $ t $ Time $ \\mathbf {q} $ Flux density vector $ \\mathbf {K} $ Saturated hydraulic conductivity tensor, vector form $ q_x, q_y, q_z $ Components of the flux density vector in Cartesian coordinates $ K_{ij} $ Saturated hydraulic conductivity tensor $ \\rho _w $ Water density $ \\rho _f $ Fluid density $ g $ Earth's gravitational acceleration $ k $ Intrinsic permeability $ \\mathbf {k} $ Permeability tensor, vector form $ k_{ij} $ Permeability tensor $ n $ Porosity $ Re $ Reynolds number $ \\beta $ Forchheimer coefficient $ M_s $ Specific surface area $ c_0 $ Kozeny's constant $ d_m $ Mean particle size $ \\rho $ Generalized density Edgar Buckingham, one of the forefathers of soil physics, wrote a 61 page report to the U.S. Department of agriculture which set the foundations of what today is the theory of unsaturated flow in soils [22].",
"Using an analogy to other physical transport laws, he postulated that the capillary current density (equivalent to $q$ in modern notation) is proportional to the capillary potential gradient.",
"Adapting to the notation used in this book and in one dimension His $\\lambda $ was replaced with $\\beta $ as we will need the former later $q = \\beta \\frac{\\partial \\psi _c}{\\partial x}$ Based on his observations and theoretical analysis he concluded that the capillary conductivity, $\\beta $ , was not constant and was a function of water content $\\beta = f(\\theta )$ .",
"His conductivity law can be understood as a generalization of Darcy's law to unsaturated porous media, thus, writing the total potential in terms of hydraulic head, $h$ , and $\\beta $ as an unsaturated hydraulic conductivity, which is no longer constant but dependent on water content, Darcy's law can be written, for the one dimensional case as $\\boxed{q = K(\\theta ) \\frac{\\partial h}{\\partial x}}$ This is the one dimensional Darcy-Buckingham equation.",
"Contrary to the saturated case, the unsaturated hydraulic conductivity is an empirical function of water content, and not a constant which can be determined experimentally.",
"There has been a lot of effort in developing these empirical functions, with a lot of the fundamental equations being derived in the XX century.",
"Many of the water retention functions presented in Chapter REF were essentially derived to arrive at an unsaturated hydraulic conductivity function.",
"One of the fundamental differences between Darcy's law and the Darcy-Buckingham equation is that in all, or at least in some part of the porous media, the water content is below saturation.",
"This implies that the pressure potential is zero and the matric potential is the component of interest along with the gravitational potential in the total potential.",
"Instead of applying pressure, represented by a height of water column on the inlet and outlet of the column filled with soil or other porous media of interest, the outlet and inlet can be under tension or suction, by placing the water level below the height at the point of interest.",
"Generally speaking, the pressure potential can be positive on the inlet and negative on the outlet which causes the column to be partially saturated in part of its extension and saturated on the rest.",
"The most common example, given in many soil physics textbooks is a column which is under tension on both ends (Figure REF ).",
"Usually, a porous plate is used on both ends of the column, similar to the system on a tensiometer.",
"Figure: Unsaturated horizontal column.Flow is from point A from point B (right to left).",
"By fixing the gravitational reference ($z = 0$ ) at the center of the column, where the inlet and outlet tubes are located, the matric potential is equal to the distance between the center of the column and the water level on the recipients connected to the inlet and outlet.",
"The recipient connected to the outlet is fed from a water source to guarantee that the level of water is constant.",
"Another way of achieving this, which is commonly used in laboratory, is to use a Mariotte's bottle.",
"The scheme illustrated in Figure REF is valid as long as the column's height is small enough not to create significant gravitational potentials on the $z$ direction, this principle also being valid for saturated columns.",
"The problem is critical in unsaturated soils because gravitational potential gradients can induce total potential gradients on the $z$ direction which induce hydraulic conductivity and velocity gradients.",
"The gradient on the $x$ direction follows the decrease in water content along the column.",
"For simple illustrative examples, if the gradient small and the column can be assumed as small enough on the $x$ length, the unsaturated hydraulic conductivity can be approximated as the average between the inlet and outlet.",
"In real world applications an unsaturated hydraulic conductivity function is necessary.",
"Because unsaturated hydraulic conductivity was theorized to be a function of water content and because water content has been experimentally shown to be a function of matric potential, hydraulic conductivity can be expressed as a function of matric potential $\\boxed{q = -K(h_m) \\frac{\\partial h}{\\partial x}}$ in which $h_m = \\psi _m$ is, as before, the matric potential expressed in height units.",
"Expanding the total potential, we arrive at similar equations as for the saturated hydraulic conductivity in a general horizontal $x$ or $y$ direction $\\boxed{q = -K(h_m) \\frac{\\partial h_m}{\\partial x}}$ $\\boxed{q = -K(h_m) \\frac{\\partial h_m}{\\partial y}}$ and on the $z$ direction, parallel to the gravitational field $\\boxed{q = -K(h_m) (\\frac{\\partial h_m}{\\partial z} + 1)}$ Applying the Darcy-Buckingham equation to Figure REF , and expressing the total potential as $h_t$ for clarity $q = -K(h_m) (\\frac{h_{t,A} - h_{t,B}}{x_A - x_B}) = -K(h_m) (\\frac{(h_{m,A}-h_{g,A}) - (h_{m,B}-h_{g,B})}{x_A - x_B})$ Because $h_{g,A} = h_{g,B} = 0$ for a horizontal column and making $x_A - x_B = L$ the length of the column $q = -K(h_m) (\\frac{(h_{A}-0) - (h_{B}-0)}{L}) = -K(h_m) (\\frac{h_{A} - h_{B}}{L})$ where $h_A$ and $h_B$ are the distances between the center of the column, where the inlet and outlet are located, and the respective water levels at the reservoirs.",
"We now know the transport equation for unsaturated porous media.",
"The relationship indicated by the Darcy-Buckingham equation is physically and mathematically more complex than Darcy's law because introduces a hitherto unknown relationship between unsaturated hydraulic conductivity and water content or matric potential.",
"Early experimental investigations indicated a nonlinear relationship between unsaturated hydraulic conductivity and water content or matric potential.",
"As we have already indicated, unsaturated hydraulic conductivity functions have been derived from the water retention curve function.",
"The unsaturated hydraulic conductivity functions can be derived from the water retention function using the frameworks proposed by Burdine [23], [19], [97] Remember from the previous chapter that $\\Theta $ is the relative saturation equal to $(\\theta - \\theta _r)/(\\theta _s - \\theta _r)$ $K_r(\\Theta ) = \\Theta ^2 \\int _0^{\\Theta } \\frac{d \\Theta }{\\psi _m^2(\\Theta )}\\bigg / \\int _0^1 \\frac{d \\Theta }{\\psi _m^2(\\Theta )}$ and Mualem [67], [97] $K_r(\\Theta ) = \\Theta ^{1/2} [\\int _0^{\\Theta } \\frac{d \\Theta }{\\psi _m(\\Theta )}\\bigg / \\int _0^1 \\frac{d \\Theta }{\\psi _m(\\Theta )}]^2$ Details about the derivation of these formulas can be found in [23], [19], [67].",
"They are essentially based on pore size distribution considerations.",
"The symbology used in these equations is the same as the one used in the previous chapter.",
"Note that we are using $\\psi _m$ here so not to avoid being constrained by units, since these equations do not necessarily require matric potential in height units.",
"The new term $K_r$ is the relative unsaturated hydraulic conductivity $K_r(\\Theta ) = \\frac{K(\\Theta )}{K_s}$ varying from 0 to 1, and in which $K_s $ is the saturated hydraulic conductivity.",
"To obtain the unsaturated hydraulic conductivity function it is necessary to solve the water retention function for $\\psi _m$ , insert it on the Burdine or Mualem equations and solve the integrals.",
"For the Brooks and Corey equation [19], the process is straightforward.",
"Expressing the Brooks and Corey in terms of the relative saturation $\\Theta $ $\\Theta = (\\frac{\\psi _b}{\\psi _m})^{\\lambda }$ we first solve for the matric potential $\\psi _m = (\\frac{\\psi _b}{\\Theta ^{1/\\lambda }})$ then plug into the Burdine equation $K_r(\\Theta ) = \\Theta ^2 \\int _0^{\\Theta } \\frac{d \\Theta }{(\\frac{\\psi _b}{\\Theta ^{1/\\lambda }})^2}\\bigg / \\int _0^1 \\frac{d \\Theta }{(\\frac{\\psi _b}{\\Theta ^{1/\\lambda }})^2}\\\\= \\Theta ^2 \\int _0^{\\Theta } \\frac{\\Theta ^{2/\\lambda } d \\Theta }{\\psi _b^2} \\bigg / \\int _0^1 \\frac{\\Theta ^{2/\\lambda } d \\Theta }{\\psi _b^2}$ because $\\psi _b$ is constant $K_r(\\Theta ) = \\Theta ^2 \\int _0^{\\Theta } \\Theta ^{2/\\lambda } d \\Theta \\bigg / \\int _0^1 \\Theta ^{2/\\lambda } d \\Theta $ The solution of the integral form in the numerator and denominator is trivial, i.e.",
"$\\int _a^b x^{2/c} dx = \\frac{x^{2/c+1}}{2/c+1} \\bigg |_a^b$ thus $K_r(\\Theta ) = \\Theta ^2 \\frac{[\\Theta ^{2/\\lambda +1}]_0^{\\Theta }}{[\\Theta ^{2/\\lambda +1}]_0^1}$ Applying the integration limits $K_r(\\Theta ) = \\Theta ^2 \\frac{\\Theta ^{2/\\lambda +1}}{1} = \\Theta ^2 \\Theta ^{2/\\lambda +1}$ or $K_r(\\Theta ) = \\Theta ^{2/\\lambda +3}$ Recalling that $\\Theta $ is the effective saturation and $K_r$ is the relative unsaturated conductivity, the equation can be written in terms of volumetric water content and unsaturated hydraulic conductivity $K(\\theta ) = K_s (\\frac{\\theta - \\theta _r}{\\theta _s - \\theta _r})^{2/\\lambda +3}$ To retrieve the matric potential form of unsaturated hydraulic conductivity just replace the original Brooks and Corey function for $\\Theta $ $K(\\psi _m) = K_s (\\frac{\\psi _b}{\\psi _m})^{2 +3 \\lambda }$ These last two equations are the Brooks and Corey unsaturated hydraulic conductivity equations derived using Burdine's approach.",
"The formulas can also be derived using Mualem's formulas by the same process.",
"For Brutsaert-van Genuchten type equations the process is not as straightforward.",
"Following [97], we can solve the effective saturation form of the water retention function resulting in $\\psi _m = [\\frac{1}{\\alpha } (\\frac{1}{\\Theta ^{1/m}} - 1)^{1/n}]$ Replacing into the Mualem function $K_r(\\Theta ) = \\Theta ^{1/2} [\\int _0^{\\Theta } \\frac{d \\Theta }{[\\frac{1}{\\alpha } (\\frac{1}{\\Theta ^{1/m}} - 1)^{1/n}]}\\bigg / \\int _0^1 \\frac{d \\Theta }{[\\frac{1}{\\alpha } (\\frac{1}{\\Theta ^{1/m}} - 1)^{1/n}]}]^2$ Doing some simple algebraic manipulations and placing the constant $1/\\alpha $ term outside of the integral $K_r(\\Theta ) = \\Theta ^{1/2} [\\int _0^{\\Theta } (\\frac{\\Theta ^{1/m} }{1 - \\Theta ^{1/m}})^{1/n} d \\Theta \\bigg / \\int _0^1 (\\frac{\\Theta ^{1/m} }{1 - \\Theta ^{1/m}})^{1/n} d \\Theta ]^2$ The issue here is the solution of the integral $\\int (\\frac{\\Theta ^{1/m} }{1 - \\Theta ^{1/m}})^{1/n} d \\Theta $ The integral on the denominator is the same equation with different bounds.",
"This equations does not have a closed form analytical solution.",
"This means that the solution is in form of special functions, there is no simple expression in which you can plug the constants and obtain results without using numerical methods or approximations to the special functions.",
"In the last chapter we encountered the complementary error function when dealing with water retention functions.",
"Other important special functions are the gamma and beta functions and their incomplete counterparts [14].",
"To find an analytical solution to this equation van Genuchten [97] used manipulations of the incomplete beta function [14].",
"The first step is apply the transformation $\\Theta = y^m \\\\d\\Theta = m y^{m-1} dy$ which after replaced into the Equation REF results $\\int _0^{y^m} (\\frac{(y^m)^{1/m} }{1 - (y^m)^{1/m}})^{1/n} m y^{m-1} dy$ simplifying and rearranging $m\\int _0^{y^m} y^{1/n + m -1} (1-y)^{-1/n} dy$ This equation is a form of the incomplete beta function [14].",
"These functions have a very important role in physics and mathematics and have been studied for hundreds of years, dating at least to the time of Euler and Legendre.",
"There is no closed form analytical solution for the general case, but it is possible to introduce simplifications such that a solution can be found.",
"The next simplification was to propose that for all integers $k = m-1+1/n$ the integration could be performed directly.",
"For the particular case $k=0$ $m = 1-1/n$ We already seen this term in the last chapter when studying the Brutsaert-van Genuchten equation.",
"It was then called the Mualem restriction, the reason why being apparent now.",
"Applying this restriction to the incomplete beta function representation allows a straightforward solution of the integral.",
"For simplicity we will replace $n$ rewriting the equation in terms of $m$ , thus $m\\int _0^{y^m} y^{m -m} (1-y)^{m-1} dy = m\\int _0^{y^m} (1-y)^{m-1} dy$ The solution to this equation is very simple and can be found using the $u$ substitution technique studied in introductory calculus books.",
"Let $u = 1-y \\\\du = -dy$ we have $-m\\int _0^{y^m} u^{m-1} du$ Integrating in $u$ $-m \\frac{u^{m-1+1}}{m-1+1} |_0^{y^m} = u^{m} |_0^{y^m}$ Substituting $y$ back $-(1-y)^{m} |_0^{y^m}$ At the beginning of the derivation we established that $\\Theta = y^m$ , replacing the original variable $-(1-\\Theta ^{1/m})^{m} |_0^\\Theta = 1-(1-\\Theta ^{1/m})^{m}$ The exact same steps apply to the denominator, however the limits of integration result in $-(1-\\Theta ^{1/m})^{m} |_0^1 = 1$ Replacing the solutions of the integrals into the Mualem expression (Equation REF ) $K_r(\\Theta ) = \\Theta ^{1/2} [\\frac{1-(1-\\Theta ^{1/m})^{m} }{1}]^2$ and finally $K_r(\\Theta ) = \\Theta ^{1/2} [1-(1-\\Theta ^{1/m})^{m} ]^2$ with $m = 1-1/n \\\\0 < m < 1$ The van Genuchten-Mualem hydraulic conductivity function can be written in terms of $\\theta $ and $K(\\theta )$ as $K(\\theta ) = K_s (\\frac{\\theta -\\theta _r}{\\theta _s-\\theta _r})^{1/2} [1-(1-(\\frac{\\theta -\\theta _r}{\\theta _s-\\theta _r})^{1/m})^{m} ]^2$ The matric potential form can be retrieved by replacing the Brutsaert-van Genuchten in terms of $\\Theta $ Be careful not to mix the the subscript “m” on the matric potential symbol $\\psi _m$ with the empirical parameter “m” $K_r(\\psi _m) = \\frac{[1-(\\alpha \\psi _m)^{nm}((1+(\\alpha \\psi _m)^n)^{-m}]^2}{[1+(\\alpha \\psi _m)^n]^{m/2}}$ the derivation being left to the reader as an exercise.",
"The van Genuchten unsaturated hydraulic conductivity functions can also be derived from the Burdine expression and the Brooks and Corey equations can be derived from the Mualem expression.",
"In fact many new equations have been derived using these expressions.",
"There are other approaches to derive unsaturated hydraulic conductivity functions.",
"A comparison of the Brooks and Corey [19] and van Genuchten [97] relative unsaturated conductivity functions is shown in Figure REF for $\\lambda = 0.47$ and $m = 0.77$ .",
"Figure: Relative unsaturated hydraulic conductivity as a function of effective saturation for the van Genuchten equation with m=0.77m = 0.77 and the Brooks and Corey equation with λ=0.47\\lambda = 0.47.In the previous chapter we derived the Laplace equation for a saturated porous media from the mass conservation equation (continuity equation).",
"To derive the conservation equation for unsaturated porous media we again start from the generalized mass conservation equation presented in Chapter REF .",
"$\\frac{\\partial \\rho }{\\partial t} = - \\nabla \\cdot (\\rho \\mathbf {u})$ The original derivation of this equation considered a control volume full of water or other Newtonian viscous fluid.",
"We retrieved Laplace's equation on the last chapter by replacing the density of the fluid as the density of water $\\rho _w$ and by considering water as an incompressible fluid such that its density is constant resulting in it vanishing from the conservation equation.",
"Note that for a saturated porous medium, the volume available for water storage is less than the total volume of interest and is equal to the porosity of the medium.",
"For saturated porous media, assuming the porosity and water density as constant has the same implications as if the media was simply a volume filled with water, the rate of variation of water content with time is always zero.",
"For unsaturated porous media these assumptions might not be valid as the volume (or mass) of water within the pore space might vary with time.",
"In this case we are not immediately concerned with the variation of the density of fluid within the volume of interest, but we are concerned with the variation of the concentration of the fluid within the porous media, thus we can rewrite the conservation equation as $\\frac{\\partial c_w}{\\partial t} = - \\nabla \\cdot (\\rho _w \\mathbf {u})$ in which $c_w = m_w/V_t$ is now the water concentration in mass $m_w$ per unit of total volume of the porous medium $V_t$ $\\frac{\\partial \\frac{m_w}{V_t}}{\\partial t} = - \\nabla \\cdot (\\rho _w \\mathbf {u})$ We know that the density of water is $\\rho _w = m_w/V_w$ , thus solving for $m_w$ and replacing into the conservation equation $\\frac{\\partial \\frac{\\rho _w V_w}{V_t}}{\\partial t} = - \\nabla \\cdot (\\rho _w \\mathbf {u})$ Assuming that the density of water is constant, it can be placed outside the derivative and divergence terms $\\rho _w \\frac{\\partial \\frac{V_w}{V_t}}{\\partial t} = - \\rho _w \\nabla \\cdot \\mathbf {u}$ Recalling that $V_w/V_t$ is the volumetric water content $\\theta $ , and assuming that for a porous medium the velocity term $\\mathbf {u}$ can be represented by the flux density $\\mathbf {q}$ $\\frac{\\partial \\theta }{\\partial t} = - \\nabla \\cdot \\mathbf {q}$ With volumetric water content varying with time, this is a partial differential equation that is difficult to solve analytically without certain assumptions and simplifications.",
"If the water content does not vary with time $\\partial \\theta / \\partial t = 0$ and $\\nabla \\cdot \\mathbf {q} = 0$ Note, however, that many of the simplifying assumptions used for saturated flow are not valid because $K$ and $\\theta $ are no longer constant in space, even under steady state conditions for a partially saturated porous medium.",
"Before deriving the general equation that governs unsaturated flow in porous media we need to find a general form of the Darcy-Buckingham's equation.",
"Theoretically, the most general form of Darcy-Buckingham's equation for an anisotropic, heterogeneous porous media would include a vector valued function in space and matric potential for $K$ The tensor form for anisotropic and heterogeneous porous media is speculative, the mathematical implications and the calculations have not being investigated in depth by the author $\\mathbf {q} = -\\mathbf {K}(\\mathbf {r}, \\psi _m) \\cdot \\nabla \\psi _t$ in which $\\mathbf {r} = x~ \\mathbf {i} + y~ \\mathbf {j}+ z~\\mathbf {k}$ For an homogeneous and anisotropic medium, unsaturated hydraulic conductivity does not vary in space, only with direction at a given point.",
"In this case the components of the conductivity tensor are scalar functions of matric potential alone $\\mathbf {K}(\\psi _m) = -K(\\psi _m)_{ij} =\\begin{pmatrix}K(\\psi )_{xx} & K(\\psi _m)_{xy} & K(\\psi _m)_{xz}\\\\K(\\psi )_{yx} & K(\\psi _m)_{yy} & K(\\psi _m)_{yz}\\\\K(\\psi )_{zx} & K(\\psi _m)_{zy} & K(\\psi _m)_{zz}\\end{pmatrix}$ For an heterogeneous and isotropic system, the unsaturated hydraulic conductivity is no longer a tensor but a scalar function of space and matric potential, such that $K(x,y,z,\\psi _m)$ and $\\mathbf {q} = -K(x, y, z, \\psi _m) \\nabla \\psi _t$ The simplest case is when hydraulic conductivity is a function of matric potential only everywhere $\\mathbf {q} = -K(\\psi _m) \\nabla \\psi _t$ Richards [78] derived an equation governing the fluids in porous media by replacing the Darcy-Buckingham equation into the mass conservation equation for water $\\boxed{\\frac{\\partial \\theta }{\\partial t} = \\nabla \\cdot ( K(\\psi _m) \\nabla \\psi _t )}$ This is known as the Richards or Richardson-Richards equation.",
"Richardson derived an analogous equations and published it in a small section of a book dealing with numerical methods for weather prediction [79].",
"The equation derived by Richardson was not derived from Darcy's law but included a “conductance of the channels connecting opposite faces” in a volume of soil.",
"Whether Richards was aware of Richardson's work is unknown, although some authors believe he wasn't [55] Richards did not cite Richardson in his original work, but it is entirely possible that his ideas could have reached Richards, ideas seem to float through several means within a scientific epoch.",
"Expansion of the divergence and gradient terms results in $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(\\psi _m) \\frac{\\partial \\psi _t}{\\partial x}] + \\frac{\\partial }{\\partial y} [K(\\psi _m) \\frac{\\partial \\psi _t}{\\partial y}] + \\frac{\\partial }{\\partial z} [K(\\psi _m) \\frac{\\partial \\psi _t}{\\partial z}]$ The unsaturated hydraulic conductivity is not constant and cannot be placed outside the derivative.",
"Writing in terms of the total potential in height of water $h = h_m + h_g$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) \\frac{\\partial h}{\\partial x}] + \\frac{\\partial }{\\partial y} [K(h_m) \\frac{\\partial h}{\\partial y}] + \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial h}{\\partial z}]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) \\frac{\\partial (h_m + h_g)}{\\partial x}] + \\frac{\\partial }{\\partial y} [K(h_m) \\frac{\\partial (h_m + h_g)}{\\partial y}] + \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial (h_m + h_g)}{\\partial z}]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) (\\frac{\\partial h_m}{\\partial x} + \\frac{\\partial h_g}{\\partial x})] + \\frac{\\partial }{\\partial y} [K(h_m) (\\frac{\\partial h_m}{\\partial y} + \\frac{\\partial h_g}{\\partial y})] + \\frac{\\partial }{\\partial z} [K(h_m) (\\frac{\\partial h_m}{\\partial z} + \\frac{\\partial h_g}{\\partial z})]$ Considering that the gradient of the gravitational potential is zero in $x$ and $y$ , i.e.",
"the gravitational potential is constant such that $\\partial h_g/\\partial x = 0 $ and $\\partial h_g/\\partial y = 0$ and that in length units the gravitational potential difference in $z$ is equal to $\\partial z$ , Richardson-Richards equations in $x$ , $y$ and $z$ are $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [K(h_m) \\frac{\\partial h_m}{\\partial x} ]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial y} [K(h_m) \\frac{\\partial h_m}{\\partial y}]$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [K(h_m) (\\frac{\\partial h_m}{\\partial z} + \\frac{\\partial z}{\\partial z})] = \\frac{\\partial }{\\partial z} [K(h_m) (\\frac{\\partial h_m}{\\partial z} + 1)]$ Richardson-Richards equation is a nonlinear differential equation which has analytical solutions only in very specific cases which might be physically realistic.",
"One idea to put the equation in a form which could be solved more easily is to include a diffusivity term.",
"Diffusion type equations have been studied for centuries and before high processing power computers were available, any simplification would help immensely in terms of simplifying an equation to find analytical solutions or to solve it numerically.",
"Klute [54] noted an artifice used by Buckingham [22] and later by [27], [28] in order to write the Darcy-Buckingham and Richardson-Richards equations in terms of a hydraulic diffusivity term.",
"First let us rewrite Richardson-Richards equation in the $z$ direction by simple using the distributive property of multiplication and recalling that the unsaturated hydraulic conductivity can be written as a function of volumetric water content $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [K(\\theta ) \\frac{\\partial h_m}{\\partial z}] + \\frac{\\partial K(\\theta ) }{\\partial z}$ Note that in the gradient term $\\frac{\\partial h_m}{\\partial z}$ if we can assume that the matric potential is a unique function of water content [71], then, by the chain rule for an implicit function [14] $\\frac{\\partial h_m}{\\partial z} =\\frac{\\partial h_m}{\\partial \\theta } \\frac{\\partial \\theta }{\\partial z}$ Replacing the expanded differential into the Richardson-Richards equation $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [K(\\theta ) \\frac{\\partial h_m}{\\partial \\theta } \\frac{\\partial \\theta }{\\partial z}]+ \\frac{\\partial K(\\theta ) }{\\partial z}$ The term $D(\\theta ) = K(\\theta ) \\frac{\\partial h_m}{\\partial \\theta }$ is now called the hydraulic diffusivity or soil water diffusivity [28], [54] and the Richardson-Richards equation can be written in water content form as $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} [D(\\theta ) \\frac{\\partial \\theta }{\\partial x}] + \\frac{\\partial K(\\theta ) }{\\partial z}$ Conversely, Richardson-Richards equation can also be written in the matric potential form by defining the implicit differential for $\\partial \\theta /\\partial t$ and replacing into the left side of the equation in terms of matric potential [50] $\\frac{\\partial \\theta }{\\partial h_m}\\frac{\\partial h_m}{\\partial t} = \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial h_m}{\\partial z}] + \\frac{\\partial K(h_m) }{\\partial z}$ By defining a new function $C(\\theta ) = \\frac{\\partial \\theta }{\\partial h_m}$ called the hydraulic capacity, we can write the Richardson-Richards equation in terms of matric potential only as $C(h_m)\\frac{\\partial h_m}{\\partial t} = \\frac{\\partial }{\\partial z} [K(h_m) \\frac{\\partial h_m}{\\partial z}] + \\frac{\\partial K(h_m) }{\\partial z}$ The hydraulic capacity is the slope of the water retention function.",
"The reasoning for reducing the number of variables in these equations is that it should simplify the process of solution either analytically, when an analytical solution is possible, or numerically.",
"Because of the functional relationship between water content and matric potential, and unsaturated hydraulic conductivity, the functions discussed in Chapter REF and in this chapter can be used when solving these equations.",
"Because these functions exist, there is also a functional relationship between hydraulic diffusivity and water content.",
"Notice that based on the discussion on the previous paragraphs, the hydraulic diffusivity can be written as [50] $D(\\theta ) = \\frac{K(\\theta )}{C(\\theta )}$ It is also worth noticing that the Darcy-Buckingham equation can be written in terms of diffusivity.",
"In a generalized horizontal $x$ direction $q = D(\\theta ) \\frac{\\partial \\theta }{\\partial x}$ Beware, however, of the interpretation of the hydraulic diffusivity [53], [71].",
"The introduction of the diffusivity form made it easier to solve the equation in certain cases, but the hydraulic diffusivity does not have the same meaning as a diffusion term in heat and solute dispersion processes.",
"Keep in mind that many of these equations require a unique relationship between the variables unsaturated hydraulic conductivity, diffusivity and water content and/or matric potential, and between water content and matric potential.",
"This assumption is not physically realistic because of hysteresis, the relationship between water content and matric potential is not unique in real natural porous media and dependents on the history of the processes, either if a drying or wetting processe is being considering.",
"In addition, for wetting-drying cycles within a given water content range, there might by multiple possible paths (hysteresis loops).",
"In other words, there is a non-unique relationship between water content and matric potential for real porous media, for a given matric potential there might be at least two corresponding water content values depending on the wetting history.",
"Bruce and Klute [20] described a procedure do find the diffusivity function following a procedure that Ludwig Boltzmann [16] used to solve diffusion type equations, now known as the Boltzmann transformation.",
"Recall the diffusivity form of the Richardson-Richards equation in a generalized horizontal $x$ direction $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial x} [D(\\theta ) \\frac{\\partial \\theta }{\\partial x} ]$ Boltzmann [16] showed that diffusion type equations can be solved by introducing a new variable $\\Lambda = x t^{-1/2}$ For the diffusivity equation of concern in soil and porous media transport, the volumetric water content is now a function of the new variable [53] $\\theta = f(\\Lambda )$ The idea now is to replace $\\Lambda $ into the Richardson-Richards equation in order to reduce it from a partial differential equation in $\\theta $ , $t$ and $x$ to a ordinary differential equation in $\\Lambda $ and $\\theta $ .",
"The first step is to calculate the partial derivatives of $\\Lambda $ in $x$ and $t$ $\\frac{\\partial \\Lambda }{\\partial x} = \\frac{1}{\\sqrt{t}}$ $\\frac{\\partial \\Lambda }{\\partial t} = -\\frac{x}{2t^{3/2}} = -\\frac{\\Lambda }{2t}$ Next, the water content partial derivatives can be expanded by the chain rule.",
"Recall that, by the definition of $\\Lambda $ , the unsaturated hydraulic conductivity is an implicit function of it such that $\\frac{\\partial \\theta }{\\partial x} = \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\partial \\Lambda }{\\partial x}$ $\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\partial \\Lambda }{\\partial t}$ Since we have already calculated the partial derivatives of $\\Lambda $ it is just a matter of replacing them into the two equations above $\\frac{\\partial \\theta }{\\partial x} = \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{1}{\\sqrt{t}}$ $\\frac{\\partial \\theta }{\\partial t} = -\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t}$ Replacing these into the Richardson-Richards equation results in $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{\\partial }{\\partial x} [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{1}{\\sqrt{t}}]$ Because the partial derivatives on the right side no longer depend on $t$ they can be placed outside of the differentials $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{1}{\\sqrt{t}} \\frac{\\partial }{\\partial x} [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ We can now expand the partial derivative in $x$ by the chain rule $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{1}{\\sqrt{t}} \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ] \\frac{\\Lambda }{\\partial x}$ and, since we already know $\\partial \\Lambda / \\partial x$ $-\\frac{\\partial \\theta }{\\partial \\Lambda } \\frac{\\Lambda }{2t} = \\frac{1}{\\sqrt{t}} \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]\\frac{1}{\\sqrt{t}}$ and finally [20] $-\\frac{\\Lambda }{2} \\frac{\\partial \\theta }{\\partial \\Lambda } = \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ An experimentally based solution to this ordinary differential equation was proposed by [20] for the boundary conditions $\\theta = \\theta _i~ \\text{for}~ x > 0, t = 0 \\\\\\theta = \\theta _s~ \\text{for}~ x = 0, t \\ge 0$ in which $\\theta _i$ is the initial water content of the system and $\\theta _s$ is the saturated water content.",
"These conditions are valid for an infinitely long column, such that the wetting front will not reach the other end.",
"In practical terms, a long column filled with porous material at constant initial water content is placed in contact with a water source, such that the water content will increase with time.",
"After an elapsed time $t$ , the $\\Lambda $ function can be evaluated along the length of the column.",
"The $\\Lambda $ function would later be called the sorptivity of the soil or porous media.",
"Under these conditions, the $\\Lambda $ form of Richardson-Richards equation can be integrated in the limits of initial water content to water content at a position $x$ and from $\\Lambda $ at the initial condition to $\\Lambda $ evaluated at $x$ $-\\int _{\\theta _i}^{\\theta _x}\\frac{\\Lambda }{2} \\frac{\\partial \\theta }{\\partial \\Lambda } = \\int _{\\Lambda _i}^{\\Lambda _x} \\frac{\\partial }{\\partial \\Lambda } [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ which immediately simplifies to $-\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = \\int _{\\Lambda _i}^{\\Lambda _x} \\partial [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ Noticing that $\\Lambda _i = \\Lambda _\\infty $ corresponds to $\\theta _i$ and that $\\Lambda _x$ corresponds to $\\theta _x$ , a change of variables can be performed on the right integral $-\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = \\int _{\\theta _i}^{\\theta _x} \\partial [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]$ or $-\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = [D(\\theta ) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _i}^{\\theta _x} = [D(\\theta _x) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _x} - [D(\\theta _i) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _i}$ Because $\\theta _i$ is constant, the partial derivative $\\partial \\theta / \\partial \\Lambda $ evaluated at $\\theta _i$ is zero $\\frac{1}{2}\\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta = - [D(\\theta _x) \\frac{\\partial \\theta }{\\partial \\Lambda } ]_{\\theta _x}$ Thus, the diffusivity can be calculated numerically and experimentally by [20] $D(\\theta _x) = -\\frac{1}{2} (\\frac{\\partial \\Lambda }{\\partial \\theta } )_{\\theta _x} \\int _{\\theta _i}^{\\theta _x} \\Lambda \\partial \\theta $ The procedure is described in [53] and on the original publication, among others [31].",
"You have seen that the mathematical complexity of the models for describing unsaturated flow is substantially greater than that of saturated flow equations.",
"The partial differential equations used includes the variables $K$ , $D$ , $\\theta $ and $h_m$ in addition to spatial coordinates.",
"These variables are usually related to each other by nonlinear functions, which must be considered when searching for the solution.",
"In many real world applications, in transient flow conditions and in dimensions greater that one, these equations might not have analytical solutions or can only be solved analytically using unrealistic boundary conditions.",
"One example of simplification was described on the previous section.",
"When analytical solutions are not possible, numerical solutions are used.",
"In numerical methods the equations are solved numerically through approximation methods.",
"In fact, much of fluid mechanics and many areas of applied physics and groundwater hydrology now rely heavily on numerical methods.",
"A whole subarea of fluid mechanics called computational fluid mechanics is now on the frontier of knowledge in the field.",
"Simple numerical solutions can be computed by hand but today and for real world scenarios, most applications will use a computer.",
"Knowledge of informatics and programming languages is essential to applied numerical methods in soil physics and groundwater hydrology.",
"ll $\\beta $ Capillary conductivity defined by Buckingham [22] $\\psi _c$ Capillary potential $x$ Generalized horizontal direction or coordinate $q$ Flux density $K(\\theta )$ Unsaturated hydraulic conductivity as a function of volumetric water content $h$ Total potential in length units $h_m$ Matric potential in length units $K(h_m)$ Unsaturated hydraulic conductivity as a function of matric potential $z$ Generalized vertical direction or coordinate $h_t$ Total potential in length units $h_g$ Gravitational potential in length units $K_r(\\Theta ) $ Unsaturated hydraulic conductivity as a function of effective saturation $\\Theta $ Effective saturation $= (\\theta - \\theta _r)/(\\theta _s - \\theta _r)$ $\\psi _m$ Matric potential in general units $\\lambda $ Pore distribution parameter of Brooks and Corey [19] $a, b, c$ Arbitrary constants $\\theta $ Volumetric water content $\\theta _r$ Residual water content $K_s$ Saturated hydraulic conductivity $y$ Parameter for transformation on the van Genuchten [97] solution or arbitrary horizontal direction or coordinate elsewhere $k$ Generic parameter for solving the Mualem equation in the van Genuchten framework $m$ Shape parameter for the van Genuchten equation $n$ Shape parameter for the van Genuchten equation $\\alpha $ Inverse of air entry pressure for the van Genuchten equation $\\rho $ Arbitrary density of a material for the generalized conservation equation $\\mathbf {u}$ Velocity vector for fluid flow in general $c_w$ Water concentration within a porous medium $\\rho _w$ Water density $t$ Time $m_w$ Water mass $V_w$ Water volume within a porous medium $\\mathbf {q}$ Flux density in a porous medium $\\mathbf {K}(\\mathbf {r}, \\psi _m) $ Unsaturated hydraulic conductivity tensor $\\mathbf {r}$ Position vector $K(x,y,z, \\psi _m) $ Unsaturated hydraulic conductivity scalar function $D(\\theta )$ Diffusivity function $C(\\theta )$ Water (hydraulic) capacity function $ = \\partial \\theta /\\partial \\psi _m$ $\\Lambda $ Variable introduced for the Boltzmann transformation In the last two chapters we studied the equation of conservation of water for porous media under saturated and unsaturated conditions.",
"To study conservation of other types of mass in porous media, similar principles can be applied.",
"Soluble solids are of immense interest in groundwater hydrology and soil physics because a large part of the groundwater and soil contaminants are soluble solids.",
"Examples include pesticides, organic molecules, viruses, bacteria, radionuclides, ammonium, nitrates, phosphates and many others.",
"In agriculture, fertilizers and other plant nutrients can be also transported through soil with water fluxes and by diffusion.",
"Many compounds can act as either plant nutrients or contaminants, depending on where they are and on the amount and on intrinsic factors of soils and aquifers.",
"The process can be also complicated as many solids are volatile and/or can be adsorbed onto the solid phase, suffer transformations, decay and many other processes.",
"Suppose you have a volume of a porous media either soil, an aquifer or other geologic material.",
"Suppose now that we are concerned with the dynamics of a single soluble solid.",
"The concentration of the solid within the porous media is $C$ .",
"One of the mechanisms by which the contaminant can be transported in and out of the volume of interest is by advection.",
"Advective transport is the mass transport with the flow of the fluid in which it is dissolved.",
"When the conservation equation was derived, we saw that the flux density was given by $\\mathbf {q}$ .",
"The mass flux of the dissolved solid being transported in an unsaturated medium is given by $\\mathbf {J}_{adv} = \\theta \\mathbf {v} C$ in which $\\theta $ is the volumetric water content [L$^3$ L$^{-3}$ ], $\\mathbf {v}$ is the water velocity within the pores [L T$^{-1}$ ], as discussed in the previous chapter and $C$ is the concentration of the dissolved solid in the liquid phase [M L$^{-3}$ ].",
"Dimensional analysis indicates that the vector $\\mathbf {J}_{adv}$ is the mass of the dissolved solid entering (or exiting) per unit area per unit time [M L$^{-2}$ T$^{-1}$ ].",
"If there was no diffusion or hydrodynamic dispersion, an unrealistic scenario for dissolved solids in porous media, the conservation equation would be of the form $\\frac{\\partial (\\theta C)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C) $ However, dissolved solids are subject to diffusion due to concentration gradients.",
"As we discussed, regions of high concentration of dissolved solids will result in lower water potential, in this scenario diffusion tends to create equilibrium of concentration throughout the fluid, subjected that there are no other constraints regarding other potentials e.g.",
"thermal and electrostatic.",
"Considering an element of static, incompressible fluid, not within a porous medium, the diffusion of a soluble solid in the fluid can be modeled using Fick's law $\\mathbf {J}_{diff} = - D_d \\nabla C$ $\\mathbf {J}_{diff}$ is the diffusive flux [M L$^{-2}$ T$^{-1}$ ], $D_d$ is the diffusion coefficient in [L$^{2}$ T$^{-1}$ ] and $\\nabla C$ is the concentration gradient.",
"Within a porous media, not all the space is available for diffusion.",
"In a saturated porous media, the space occupied by solid particles is not available for diffusion, only the water saturated pore space.",
"Even within a saturated porous medium, it might be that not all of the pore space is available to transport due to the fact that not all pores might be connected, and dead end pores [38].",
"In unsaturated porous media, only the fraction of space saturated with water is available for transport [12].",
"To account for the geometry of the pore space available for fluid flow and its modification with changing water content, a second rank tortuosity tensor is included into Fick's law of diffusion [11], [12] $\\mathbf {J}_{diff} = - D_d \\mathbf {T}(\\theta ) \\cdot \\nabla C$ The product of the diffusion coefficient by the tortuosity tensor can be written as a tensor, defined as the coefficient of diffusion for a a given phase in a partially saturated porous medium [12] $\\mathbf {J}_{diff} = - \\mathbf {D}^*(\\theta ) \\cdot \\nabla C$ where $ \\mathbf {D}^*(\\theta ) = D_d\\mathbf {T}(\\theta )$ An analysis of the diffusion and tortuosity tensors will be left for later editions, the interested reader should consult [11], [12].",
"In essence this equation shows that the concentration of soluble solids will still tend to equilibrum due to concentration gradients in partially saturated porous media, but the volume available for diffusive movement will be limited by the volume occupied by water, and the path for movement will be constrained by the geometry of the porous space.",
"Because of the geometry of the solid and pore space, the velocity and paths of water flux in porous media tends to be highly heterogeneous.",
"In general, the more heterogeneous the porous media is in terms of grain size distribution and solid particles shapes and surface rugosity, the more heterogeneous the flux behavior will be, specially if the shapes, and rugosity distributions are more or less random.",
"Grain and pore size distributions are still random but tend to follow certain statistical distributions and this has been taken advantage for modeling flow and transport using stochastic models.",
"Because of the irregular distribution of velocities within the pores, the irregularity of the pore paths, a third transport mechanism called mechanical dispersion is also important.",
"Even within a single pore, because of viscosity and surface forces, the velocity will be maximum at the center of the pore and zero at the surface if the pore is modelled as two infinite, parallel plates or as a cylinder, as we have seen when deriving Hagen-Poiseuille's equation in Chapter REF .",
"All these sub-mechanisms cause mixing of the dissolved solid when the water in which it is dissolved is flowing through a porous medium.",
"Dispersion has been shown experimentally to follow a Fick's diffusion type law [12] $\\mathbf {J}_{disp} = - \\mathbf {D} \\cdot \\nabla C$ in which $\\mathbf {D}$ is the dispersion coefficient, also a second rank tensor.",
"Considering the three forms of transport, the total flux of a dissolved solid in a partially saturated porous medium is $\\mathbf {J}_{tot} =\\mathbf {J}_{adv} + \\mathbf {J}_{diff} + \\mathbf {J}_{disp} = \\theta \\mathbf {v} C - (\\mathbf {D}^*(\\theta ) \\cdot \\nabla C + \\mathbf {D} \\cdot \\nabla C)$ The diffusion and mechanical dispersion components can be lumped into a single tensor called the coefficient of hydrodynamic dispersion $\\mathbf {D}_h$ $\\mathbf {D}_h = \\mathbf {D} + \\mathbf {D}^*(\\theta )$ such that the total flux can be written as $\\mathbf {J}_{tot} =\\mathbf {J}_{adv} + \\mathbf {J}_{diff} + \\mathbf {J}_{disp} = \\theta \\mathbf {v} C - \\mathbf {D}_h \\cdot \\nabla C$ In theory our work is almost be done, considering the conservation of mass and the total flux of the dissolved solid, the conservation equation should be something like $\\frac{\\partial (\\theta C)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C - \\mathbf {D}^*(\\theta ) \\cdot \\nabla C - \\mathbf {D} \\cdot \\nabla C)$ In reality, the behavior of chemicals in porous media is much more complicated than that.",
"Many chemicals can transition from one phase to another, for example, several compounds exist as a solid or as a liquid or as both depending on chemical equilibrium conditions, other chemicals can be adsorbed onto solid surfaces by several different mechanisms, including those discussed in Chapter REF , while some are subjected to both adsorption and volatilization.",
"In addition to that, some chemicals can be created or destroyed while being transported, for example, radionuclides can decay generating different isotopes or elements, organic matter can be transformed or destroyed by organisms and other bio-degradation processes, and both mechanisms can be effectively creating new chemicals in the environment.",
"Because of that, a general equation of transport of chemicals in porous media must include terms for exchanges among phases, and sources and sinks of compounds, a source meaning a term for creation of a substance and a sink a term for the elimination of a given substance Obviously conservation laws still apply, a substance is eliminated by being transformed into other forms of matter or energy and created from other forms of matter or energy.",
"Following the language of [12] the mass balance equation for a single chemical species in a porous media including sources and sinks is $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C - \\mathbf {D}^*(\\theta ) \\cdot \\nabla C - \\mathbf {D} \\cdot \\nabla C) + \\theta \\rho _{f} \\Gamma _{f} + \\rho \\Gamma _s$ in which the total mass balance term in time $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t}$ now accounts for the rate of variation in time of the amount adsorbed into the solid phase in the form of the component $\\frac{\\partial \\rho F}{\\partial t}$ where $F$ is the mass adsorbed into the solid phase per unit mass of solid phase and $\\rho $ is the bulk density of the porous medium.",
"The source (or sink) term of the chemical in the liquid phase is given by $\\theta \\rho _f \\Gamma _f$ in which $\\theta $ is, as before, the volumetric water content, $\\rho _f$ is the density of water, or of the liquid phase in general, and $\\Gamma _f$ is the rate of creation (or destruction if the signal is negative) of the chemical per unit mass of water [M T$^{-1}$ M$^{-1}$ ].",
"The source (or sink) term for the chemical in the solid phase is given by $\\rho \\Gamma _s$ in which $\\Gamma _s$ is the rate of creation (or destruction if the signal is negative) per unit mass of solid phase [M T$^{-1}$ M$^{-1}$ ].",
"In theory, the transport equation can be further expanded to include transport in the gas phase, by including a source term in the gas phase and the vapor diffusive flux [47].",
"The form without the gas transport is commonly used in soil physics and hydrology and has been called the advection-dispersion or advective-dispersive equation when the lumped hydrodynamic dispersion term is considered $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t} = -\\nabla \\cdot (\\theta \\mathbf {v} C - \\mathbf {D}_h \\cdot \\nabla C) + \\theta \\rho _{f} \\Gamma _f + \\rho \\Gamma _s$ The full equation with the tensor components in three dimensions is obviously too complex for any analytical treatment.",
"Solutions for several cases are available on the literature, in general [12], [38] and for particular cases [98] Another fundamental reference for the mathematics of diffusion and numerical methods is the book by John Crank [30], Crank along with Phyllis Nicolson also created an important finite differences scheme for solving differential equations.",
"One obvious simplification, especially useful for laboratory column experiments and field studies of soil profiles or geological strata is to consider a single dimension.",
"Thus, in a generalized horizontal $x$ direction $\\frac{\\partial (\\theta C + \\rho F)}{\\partial t} = - \\frac{\\partial }{\\partial x} (\\theta \\mathbf {v} C) + \\frac{\\partial }{\\partial x} (\\mathbf {D}_h \\frac{\\partial C}{\\partial x}) + \\theta \\rho _{f} \\Gamma _f + \\rho \\Gamma _s$ For one dimensional flow of a non-adsorbing, chemical which suffers no degradation and is not created within the porous media the sources the terms can be discarded.",
"An example would be dissolved sodium chloride (NaCl) being transported in water through a column composed of silica sand.",
"A further simplification can be applied if the hydrodynamic dispersion coefficient is constant for the fluid and can be placed outside the partial derivative.",
"With all these simplifications, the conservation equation is $\\frac{\\partial (\\theta C)}{\\partial t} = - \\frac{\\partial }{\\partial x} (\\theta v_x C) + D_h \\frac{\\partial ^2 C}{\\partial x^2}$ If the water content and velocity are constant they can be placed outside the derivative terms and $\\frac{\\partial C}{\\partial t} = - v_x \\frac{\\partial C}{\\partial x} + D \\frac{\\partial ^2 C}{\\partial x^2}$ where $D = D_h/\\theta $ .",
"This equation can be solved for different boundary conditions.",
"For an infinite uni-dimensional domain saturated with water with initial chemical concentration $C = 0$ in which a solution containing a chemical (or a tracer) is introduced in the inlet with concentration $C = C_0$ subjected to the following the initial condition $C(x,0) = 0 ~ \\text{at}~ x \\ge 0$ and boundary conditions $C(0,t) = C_0~ \\text{for}~ t \\ge 0 \\\\C(\\infty ,t) = 0~ \\text{for}~ t \\ge 0$ the solution is provided by Ogata and Banks [74] The solution spans at least three pages in the original and might be included in further editions $\\frac{C(x,t)}{C_0} = \\frac{1}{2} [\\text{erfc}(\\frac{x-v_xt}{2\\sqrt{Dt}}) + \\exp {(\\frac{v_x x}{D})}\\text{erfc}(\\frac{x+v_xt}{2\\sqrt{Dt}})]$ where erfc is the complementary error function presented in Chapter REF .",
"For a cylinder filled with a static fluid in which a pulse of solution with concentration $C_0$ is injected at $x=0$ , the concentration will spread symmetrically from the point of application with time following [30] $C(t) = \\frac{M}{2\\sqrt{\\pi D t}} \\exp {(-\\frac{x^2}{4Dt})}$ in which $M$ is the mass of substance deposited at time $t= 0$ at $x = 0$ .",
"This solution has been adapted to a cylinder filled with porous media in which the fluid is moving with velocity $v_x$ by considering a relative displacement term $x - v_x t$ [10], [81] $\\frac{C(x,t)}{C_0} = \\frac{1}{2\\sqrt{\\pi D t}} \\exp {[-\\frac{(x-v_x t)^2}{4Dt}]}$ Jury and Sposito [48] working within a different framework arrived at a solution of similar form with the exception of a cubic time term and an additional $x$ length term $\\frac{C(x,t)}{C_0} = \\frac{x}{2\\sqrt{\\pi D t^3}} \\exp {[-\\frac{(x-v_x t)^2}{4Dt}]}$ Equations REF , REF and REF can be applied to finite column of length $L$ to predict the relative concentration $C/C_0$ at $x = L$ with time $t$ .",
"Simulated values of $C/C_0$ as a function of time for a $100~cm$ saturated horizontal column with solution moving with velocity $v_x = 2.0~cm~h^{-1}$ for the hydrodynamic dispersion coefficients $D = D_h/\\theta $ of $0.5~cm^2~h^{-1}$ , $5.0~cm^2~h^{-1}$ , $10.0~cm^2~h^{-1}$ and $50.0~cm^2~h^{-1}$ are presented for each of the solutions.",
"Equation REF predicts an increase in the concentration outlet from zero to one with the time for solute to first appear on the outlet depending on the diffusivity (Figure REF ).",
"These types curves are called breakthrough curves because the model predicts the time and behavior for the solute to break through the column.",
"With increasing hydrodynamic dispersion coefficient, the contribution of dispersion and diffusion to solute transport increases, ultimately resulting in some of the solute breaking through earlier than the average velocity of the fluid alone would predict.",
"Dispersion and diffusion also cause a dilution of the front with respect to the concentration of the applied pulse.",
"Small values of $D$ indicate little contribution of diffusion and dispersion as can be observed from the increase in the slope of the curve at lower values of $D$ .",
"At a theoretical $D=0$ the solute pulse would move exactly at the velocity of the fluid flow and would appear at once at the end of the column without change in concentration.",
"This type of theoretical flow is called piston flow, because it would mimic the behavior of a fluid being pushed by a piston, as in a syringe for example.",
"Figure: Breakthrough curves for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~ h^{-1} and different hydrodynamic dispersion parameter values.For the same simulation parameters used for Equation REF Equations REF and REF predict the behavior of a pulse of a solution of concentration $C_0$ with time (Figures REF , REF ).",
"The two equations predict similar results as $D$ decreases.",
"Equation REF tends to predict higher concentrations as $D$ is increased when compared to Equation REF .",
"For a theoretical $D = 0$ both equations would predict a single instantaneous pulse, sometimes modeled as a dirac delta function.",
"As $D$ increases, an increase in the spreading of the pulse is observed, the outlet concentration reaching smaller maximum output concentrations.",
"Figure: Behaviour of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm ~h^{-1} and different hydrodynamic dispersion parameter values .Figure: Behavior of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~h^{-1} and different hydrodynamic dispersion parameter values .Suppose now you have a chemical that does not have any sources of sinks but can be adsorbed into the solid phase, as it is the case with ions in solution and charged compounds in soils.",
"In one dimension, the mass balance equation for a single species can be written as [47], [12] $\\frac{\\partial (\\theta C)}{\\partial t} + \\frac{\\partial \\rho F}{\\partial t} = \\frac{\\partial }{\\partial x} D_h \\frac{\\partial C}{\\partial x} - \\frac{\\partial }{\\partial x} (\\theta v_x C)$ For a saturated medium, the volumetric water content $\\theta $ is constant and equal to the total porosity $\\phi $ and can be placed outside of the partial derivatives.",
"Assuming also the hydrodynamic dispersion coefficient and flow velocity as constant, the equation simplifies to $\\phi \\frac{\\partial C}{\\partial t} + \\frac{\\partial \\rho F}{\\partial t} = D_h \\frac{\\partial ^2 C}{\\partial x^2} - \\phi v_x \\frac{\\partial C}{\\partial x}$ If the mass adsorbed onto the solid phase is a linear function of the concentration in the liquid phase, we can define the following relationship $F = K_d C$ This equation describes a linear adsorption isotherm with $K_d$ being the partition coefficient between the solid and liquid phases, and it is the slope of the relationship between $F$ and $C$ .",
"Replacing it into the mass balance equation $\\phi \\frac{\\partial C}{\\partial t} + \\frac{\\partial \\rho K_d C }{\\partial t} = D_h \\frac{\\partial ^2 C}{\\partial x^2} - \\phi v_x \\frac{\\partial C}{\\partial x}$ Because $K_d$ does not depend on the concentration and the bulk density of the porous medium it is assumed as constant $\\phi \\frac{\\partial C}{\\partial t} + \\rho K_d\\frac{\\partial C }{\\partial t} = D_h \\frac{\\partial ^2 C}{\\partial x^2} - \\phi v_x \\frac{\\partial C}{\\partial x}$ Reorganizing and rearranging $(1 + \\frac{\\rho K_d}{\\phi })\\frac{\\partial C }{\\partial t} = D \\frac{\\partial ^2 C}{\\partial x^2} - v_x \\frac{\\partial C}{\\partial x}$ In which $D_h = D/\\phi $ .",
"Defining a coefficient of retardation $R_d$ [12] $R_d \\equiv 1 + \\frac{\\rho K_d}{\\phi }$ the mass balance equation is $R_d\\frac{\\partial C }{\\partial t} = D \\frac{\\partial ^2 C}{\\partial x^2} - v_x \\frac{\\partial C}{\\partial x}$ The retardation coefficient is a number equal to or greater than zero.",
"There are different adsorption isotherms, including nonlinear relationships between $F$ and $C$ and in the mass balance equation needs to be derived for each specific case.",
"For linear adsorption isotherms, the retardation coefficient appears in the breakthrough and pulse solutions as a linear term [85] $\\frac{C(x,t)}{C_0} = \\frac{1}{2} [\\text{erfc}(\\frac{R_d x-v_xt}{2\\sqrt{D R_d t}}) + \\exp {(\\frac{v_x x}{D})}\\text{erfc}(\\frac{R_d x+v_xt}{2\\sqrt{DR_d t}})]$ $\\frac{C(x,t)}{C_0} = \\frac{x}{2\\sqrt{\\pi D t^3 / R_d }} \\exp {(-\\frac{(R_d x-v_x t)^2}{4DR_d t})}$ The retardation coefficient, as the name indicates, controls the rate of retardation or a delay in the transport of the chemical throughout the column and will tend to increase the width of the pulse.",
"The effect of the retardation coefficient in Equation REF is illustrated for $R_d = 0.25$ , $0.50$ and $1.0$ (Figure REF ).",
"Figure: Behavior of a pulse of a chemical for a 100.0cm100.0~cm long saturated horizontal column with flow velocity v=2.0cmh -1 v = 2.0~cm~h^{-1} and different retardation factors.Note that for unsaturated porous media the retardation factor is a function of volumetric water content.",
"There are numerous other cases and solutions that could be covered including unsaturated cases, presence of mobile and immobile fluid phases, partition among gas, solid and liquid phases, radioactive decay and others.",
"For the time being the interested reader is encouraged to consult [98], [12], [38], [85].",
"The framework for discussing gas transport in porous media is the same as that of general chemical transport discussed in the previous section.",
"Technically, gas transport could have been included with chemical transport, especially for gases that partition between liquid and gas phases and can be adsorbed into solids.",
"In this section we will provide a brief discussion of the specifics of gas transport.",
"As with other types of mass transport, gases can be transported by advection and diffusion.",
"Gas diffusion is controlled by Fick's law, one of the general transport laws in physics discussed in previous sections.",
"For a binary mixture of gases, the mole flux of the component of interest is given by [100] $\\mathbf {J}_g^{mole} = -C_g \\mathbf {D}_{g(ij)} \\nabla \\chi _g$ in which $C_g$ is the concentration of the gas, $\\mathbf {D}_g$ is the diffusion coefficient tensor and $\\chi _g$ is the mole fraction of the component gas of interest.",
"This relationship can be written in terms of a mass flux as $\\mathbf {J}_g^{mass} = -\\rho _g \\mathbf {D}_{g(ij)} \\nabla \\omega _g$ in which $\\omega _g$ is the mass fraction of the component of interest.",
"Now $\\mathbf {J}_g$ and $\\mathbf {J}_g^{mole}$ indicate the flux relative to the average mole or mass velocity of the mixture and not to stationary coordinates [13], [100].",
"Supposing that the diffusion coefficient is constant through the medium, it does not need to be treated as a tensor and $\\mathbf {J}_g^{mass} = -\\rho _g D_g \\nabla \\omega _g$ As we have discussed for water and chemicals, the general law for a volume devoid of solid phase is not valid when the diffusion or advection is taking place within a porous medium.",
"If the porous medium was completely devoid of liquid, i.e.",
"completely dry, the solid particles would still occupy part of the volume and generate a complex and tortuous pore space through which the gas would need to navigate.",
"For a completely saturated porous medium, the gas phase would need to diffuse through the liquid, the rates being often orders of magnitude lower than that in air.",
"If the pore space is partially saturated with water, only a fraction of the pore space inversely proportional to water content is available for diffusion and advection.",
"Thus, Fick's law needs to be adapted to porous media.",
"Under such conditions the diffusion coefficient is replaced by an effective diffusion coefficient $D_g^* = \\beta D_g$ in which $\\beta $ is a porous media factor $\\beta = \\phi S_g \\tau $ where $\\phi $ is the porosity, $S_g$ is the effective gas saturation, from zero to one, and $\\tau $ is the tortuosity factor.",
"The tortuosity factor can be written as $\\tau = \\phi ^{1/3} S_g^{7/3}$ which for a media completely saturated with gas simplifies to $\\tau = \\phi ^{1/3}$ In soil physics it is more common that the gradient is expressed in terms of concentration, which is effectively the density multiplied by the mass fraction, thus $\\mathbf {J}_g = - D_g^* \\nabla C_g$ For gas diffusion in soil, the soil gas diffusion coefficient is $ D_g^*=\\tau D_g$ and the tortuosity if often expressed as $ \\tau = 0.66 \\theta _a $ in which $\\theta _a$ is the soil volumetric air content [53], [47].",
"For advective gas transport, Darcy's law is valid [47], [100] $\\mathbf {q}_g = - \\frac{\\kappa _g}{\\mu _g} (\\nabla P_g - \\rho _g \\mathbf {g})$ here written in terms of the permeability of the media to the gas ($\\kappa _g$ ), the viscosity of the gas ($\\mu _g$ ), the density of the gas ($\\rho _g$ ) and the pressure gradient ($\\nabla P_g$ ) and in which, as defined before, $q_g$ is the flux density, but for the gas in this case.",
"The formula can be also written in terms of a gas conductivity $K_g = k_g/\\mu _g$ parameter $\\mathbf {q}_g = - K_g (\\nabla P_g - \\rho _g \\mathbf {g})$ In general, the gravity term can be neglected because the density of most gases is low, thus $\\mathbf {q}_g = - K_g \\nabla P_g$ for pressure in length units.",
"Because advective flow of gases is usually not an important transport mechanism in soil physics and groundwater hydrology, except in very specific conditions, transport of gases in porous media is modeled as a diffusion processBeware that this might not be the case when modeling natural gas reservoirs.",
"We can now rewrite the general transport equation for a non-adsorbing gas for a generalized direction $z$ , neglecting advective transport, as $\\frac{\\partial (\\theta _g C_g)}{\\partial t} = \\frac{\\partial }{\\partial z} \\mathbf {J}_g + \\theta _g \\rho _{g} \\Gamma _g$ $\\frac{\\partial (\\theta _g C)}{\\partial t} = \\frac{\\partial }{\\partial z} (\\mathbf {D}^*_g \\frac{\\partial C^g}{\\partial x}) + \\theta _g \\rho _{g} \\Gamma _g$ and if the volumetric air content and the soil gas diffusion coefficient are constant $\\theta _g \\frac{\\partial C_g}{\\partial t} = D^*_g \\frac{\\partial ^2 C_g}{\\partial z^2} + \\theta _g \\rho _{g} \\Gamma $ In many cases this is a very simple differential equation whose techniques of solution were generally shown in Chapter REF .",
"Thus, for transport of a inert gas (no sources of sinks) through a column of length $L$ saturated with inert gas with concentration $C_0$ at the inlet and zero at the outlet, the solution is [47] $C_g(z) = C_0 (1 - \\frac{z}{L})$ while for steady-state transport, in which the gas concentration does not vary with time (i.e.",
"$ \\partial C_g/\\partial t = 0 $ ) with a consumption term (e.g.",
"oxygen in soil environment) the solution is [47] $C_g(z) = C_0 + \\frac{\\theta _g \\rho _g \\Gamma }{2 D^*_g} (z^2 + 2Lz)$ Because the calculations are simple the previous derivations are left as an exercise to reader.",
"The same conservation principles for mass applied in the previous sections generally apply for heat transport.",
"However, heat transport is a form of energy transport and some particularities need to be consider.",
"Like before, heat can be transported by diffusion and advection, but in the case of advection, heat is transported with transport of a solid, gas or liquid.",
"The amount of heat stored in a material is given by [53] $Q = \\rho c V T$ in which $Q$ is the amount of thermal energy stored in the material, in joules or calories, $\\rho $ is the density of the material in [M L$^{-3}$ ], $c$ is the mass based specific heat capacity or the heat capacity $C_h$ divided by the amount of mass of the material, $V$ is the volume of material and $T$ is the temperature.",
"Because the heat capacity is given in energy per temperature, the mass based specific heat capacity is in energy times temperature$^{-1}$ mass$^{-1}$ , representing thus the amount of energy necessary to increase the temperature of one unit mass of the material in one unit of temperature, usually in (calories) cal $g^{-1}$ $^{o}C^{-1}$ or in $J$ $kg^{-1}$ $K^{-1}$ .",
"As with the other transport laws discussed throughout this material, thermal conduction law is given by $\\mathbf {J}_T = -\\frac{\\kappa _T}{\\rho c} \\nabla T$ in which $\\kappa _T$ is the thermal conductivity.",
"Defining now a thermal diffusivity term as $D_T = -\\frac{\\kappa _T}{\\rho c}$ $\\mathbf {J}_T = - D_T \\nabla T$ This is Fourier's law of heat conduction and it is analogous to Fick's law for gases, Darcy's law for fluids and Ohm's law for electricity.",
"The driver gradient is now not a gradient in terms of concentration or potential but in terms of temperature.",
"As before, we can define a general conservation equation for heat [47] $\\frac{\\partial T}{\\partial t} = - \\frac{\\partial J_T}{\\partial z} + \\gamma _T $ in which $\\gamma _T$ is a general heat source or sink term.",
"Following the procedures for the other conservation laws, and considering the thermal diffusivity term as constant, the heat flux $J_T$ can be replaced into the conservation equation, and ignoring heat sources and sinks $\\frac{\\partial T}{\\partial t} = D_T \\frac{\\partial ^2 T}{\\partial z^2} $ Because the temperature at surface ($z = 0$ ) varies sinusoidally due to daily and annual variations of temperature and solar heat influx, a solution for temperature variation in depth is of the form [53] $T(z, t) = T_a + A \\exp {(-\\sqrt{\\frac{\\omega }{2D_T}}z)} \\sin {[\\omega t - \\sqrt{\\frac{\\omega }{2D_T}}z + \\xi ]} $ in which $z$ is the depth within the material or porous media, $t$ is time, $T_a$ is the average temperature, $A$ is the amplitude of the sinusoidal wave components, $\\omega = 2\\pi /P$ , where $P$ is the period, and $\\xi $ is the phase constant.",
"The temperature variation with depth up to $100.0~cm$ , for a soil with thermal diffusivity $D_T = 0.015~cm^2~s^{-1}$ , phase constant $\\xi = -7\\pi /12$ , $T_a = 25.0~^{o} C$ and $A = 10.0~^{o} C$ is illustrated in Figure REF , where $\\omega $ was calculated considering a period of $24~h$ or $86400~ s$ .",
"Note that the amplitude of temperature is damped with depth, being close to zero at $100~ cm$ and maximum at surface.",
"Figure: Variation of temperature with time over a period of two days up to a depth of 100cm100~ cm for a soil with D T =0.015cm 2 s -1 D_T = 0.015~cm^2~s^{-1} .ll $\\mathbf {J}_{adv} $ Advective flux density for a dissolved solid being transported $\\theta $ Volumetric water content $\\mathbf {v} $ Fluid velocity vector $C $ Dissolved chemical concentration $D_d $ Generalized diffusion coefficient $\\mathbf {J}_{diff} $ Diffusive flux density for a dissolved solid being transported $\\mathbf {T}(\\theta ) $ Tortuosity tensor $\\mathbf {D}^*(\\theta ) $ Coefficient of diffusion for a dissolved solid being transported in porous media $\\mathbf {D} $ Dispersion tensor in porous media $\\mathbf {D}_{h} $ Hydrodynamic dispersion tensor $\\mathbf {J}_{tot} $ Total flux density for a dissolved solid being transported in porous media $\\mathbf {J}_{disp} $ Dispersive flux density for a dissolved solid being transported in porous media $F $ Mass of chemical absorbed per unit mass of solid phase $\\rho _f $ Fluid density $\\rho $ Bulk density of the porous media $\\Gamma $ Rate of creation or destruction of the chemical in the liquid phase $\\Gamma _s $ Rate of creation of destruction of the chemical adsorbed into the solid phase $v_x $ Velocity in a generalized $x$ direction $D_h $ Hydrodynamic dispersion coefficient $D $ Dispersion coefficient corrected to volumetric water content $= D_h/\\theta $ $\\phi $ Total porosity $K_d $ Coefficient of partition $R_d $ Retardation term $t $ Time $\\mathbf {J}_{g}^{mole} $ Mole flux for a gaseous component $\\mathbf {D}_{g(ij)} $ Gas diffusion coefficient tensor $ \\chi _g$ Mole fraction of the gaseous component of interest $ \\omega $ Mass fraction of the component of interest $\\rho _g $ Density of the gas $ D_g$ Gas diffusion coefficient $ D^*_g $ Effective gas diffusion coefficient in porous media $ \\beta $ Porous media factor $ S_g $ Effective gas saturation $ \\tau $ Tortuosity factor $ \\kappa _g $ Permeability of the media to a gas $ \\mu _g $ Viscosity of the gas $ P_g $ Gas pressure $ \\mathbf {g} $ Acceleration of gravity $ \\mathbf {q}_g $ Advective flux density of a gas in porous media $ K_g $ Gas conductivity $\\Gamma _g $ Rate of creation of destruction of the gas $C_g $ Concentration of the gas $ L $ Length of a column filled with porous media $\\theta _g$ Gas volumetric content $ Q $ Energy stored in the form of heat $ c $ Specific heat capacity $ V$ Volume of a material $ T$ Temperature $\\mathbf {J}_{T} $ Heat flux $ \\kappa _T $ Thermal conductivity $ D_T $ Thermal diffusivity $ \\omega $ $ = 2 \\pi / P $ , where $P$ is the period $ T_a$ Average temperature $ A $ Sinusoidal wave amplitude $ \\xi $ Phase constant As we have seen throughout this book, many of the models that describe physical phenomena are in the form of partial differential equations which might not have have simple analytical solutions for modeling real world phenomena, and very often might not have any analytical solutions at all.",
"When analytical solutions are not available, numerical approximations are an extremely useful tool in modeling real world phenomena.",
"With today's computers availability and processing power, numerical methods can be easily implemented using computational tools.",
"Large computational codes, both open source and proprietary, are used for modeling subsurface flow and transport in soils, geological deposits and watersheds.",
"Even larger models for matter and energy balance are available for Earth's scale modeling, but these are somewhat beyond the scope of this book.",
"One example of numerical model that all of us encounter in daily life is weather prediction.",
"Weather prediction is based on a complicated set of differential equations, based on a large set of parameters which cannot be solved analytically for larger domains such as a city, state, country or the whole world.",
"For weather prediction the variables available are the past observations and current observations.",
"These values will be then the initial conditions of the model, as we have seen for the solution of many differential equations.",
"The domains might also be subjected to a set of prescribed boundary conditions.",
"Given past predictions, and boundary and initial conditions, a numerical model can predict the weather in the future.",
"This type of prediction based on initial and boundary conditions using numerical solutions to differential equations is called forward modeling.",
"The stability, accuracy and range of the predictions is dependent on a series of factors including, accuracy of past and current predictions, nature of the differential equations being solved, numerical method being used and many others.",
"Another type of modeling frequently used in soil physics and groundwater hydrology is called inverse modeling.",
"In inverse modeling we have a set of observations and we wish to fit a particular empirical model to data to make predictions in the same system from which the data was obtained or in similar systems.",
"Inverse modeling can be used to make predictions in time and/or space but it is fundamentally different from numerical forward modeling because the empirical data is necessary a priori.",
"One example is a dataset of pairs water content and matric potential values which is fit to a water retention curve equation.",
"Fitting is usually achieved using statistical techniques such as least squares, maximum likelihood or Bayesian regression, among others.",
"The parameterized model can be then used to monitor water content from matric potential data in soils, vadose zone or other media for agricultural applications and environmental modeling.",
"Both types of models can be improved, calibrated and validated using independent data.",
"In the weather prediction model, once data has been collected, it can be compared to the numerical predictions and error indicators can be computed.",
"The model used and computation technique can be adapted accordingly.",
"In the case of the inverse model, once the water retention equation is fit to data, it can be compared to data collected in nearby sites using the same or similar laboratory techniques and the results be compared.",
"Once a large enough data set is compiled, stochastic estimation of the empirical parameters that are representative of the area based on more or less arbitrary statistical and mathematical indicators can be determined.",
"Over the next sections we will provide an extremely condensed account of curve fitting techniques and numerical methods used in soil physics and groundwater hydrology.",
"For the time being only finite differences schemes will discussed for numerical methods for solving differential equations, because they are somewhat simpler from the mathematical point of view and can be easily implemented in most programming languages.",
"The equations presented in this material are used to model real world data.",
"Consider for example the water retention curve and soil mechanical resistance equations.",
"In the first case data from the water retention curve can be obtained using the gravimetric method for water content, and pressure plates or tensiometers for soil matric potential.",
"For the soil mechanical resistance to penetration equation the soil resistance can be obtained by cone penetrometers, bulk density by the core method and water content by the gravimetric method.",
"After data collection, the equations can be fit to data using statistical or mathematical techniques.",
"Most of the equations encountered in soil physics and groundwater hydrology are nonlinear.",
"The meaning of nonlinear is not directly related to the shape of the curve when data is plotted, nonlinear is related to the parameters of the equation.",
"Consider a general linear equation, essentially a first degree polynomial $y = a x + b$ where $y$ is the dependent variable, $x$ is the independent variable and $a$ and $b$ are fitting parameters.",
"For pairs of $y$ and $x$ the procedure to fit a linear regression model is exact.",
"In other words, there are exact analytical formulas that can be used to obtain the values of $a$ and $b$ A theoretical analysis is beyond the scope of this material and can be found in introductory statistics and generalized linear models textbooks $a = \\frac{\\sum (x_i - \\bar{x})(y_i - \\bar{y})}{\\sum (x_i - \\bar{x})^2}$ $b = \\frac{\\sum y_i - a \\sum x_i }{n}$ in which $x_i$ and $y_i$ are the $i^{th}$ observation and $\\bar{x}$ and $\\bar{y}$ the averages for the variables $x$ and $y$ , and $n$ is the number of observations.",
"For polynomials of degree greater than one, although the shape of the curve of a plot is not linear, the fitting procedure is still exact and the equation is not considered a nonlinear equation.",
"The definition of a nonlinear regression model is related to nonlinearity in relation to the parameters.",
"In other words, a true nonlinear function is nonlinear with respect to its parameters.",
"Consider a second degree polynomial.",
"$y = a x^2 + b x + c$ The partial derivatives with respect to the parameters are $\\frac{\\partial y}{\\partial a} = x^2$ $\\frac{\\partial y}{\\partial b} = x$ None of the derivatives are functions of the parameters.",
"Consider now a water retention function of the form $\\Theta = \\frac{A}{A + |\\psi _m|^B}$ in which $\\Theta = S_e$ .",
"The partial derivatives with respect to the parameters are $\\frac{\\partial \\Theta }{\\partial A} = \\frac{|\\psi _m|^B}{(A + |\\psi _m|^B)^2}$ $\\frac{\\partial \\Theta }{\\partial B} = -\\frac{A|\\psi _m|^B \\log {|\\psi _m|}}{(A + |\\psi _m|^B)^2}$ It is easy to see that the partial derivatives with respect to the each parameter are functions of the parameters.",
"These types of functions cannot be fit to data by straightforward analytical and computational methods as in the case of linear and polynomial regression.",
"Nonlinear regression methods are required to fit nonlinear models.",
"Nonlinear regression methods rely on numerical methods based on successive approximations using iterative methods such as the Newton-Raphson, Gauss-Newton and Levenberg-Marquardt methods.",
"In each case an initial guess of the values of the parameters is provided by the user and the procedure tries to converge to an optimal parameter vector where the sum of the squares of the differences between the observed and predicted $y$ values is minimized.",
"This is a general description of one of the most common methods for fitting nonlinear regression models, nonlinear least squares [9].",
"Although there are other methods for fitting nonlinear models such as maximum likelihood, Bayesian estimation and other robust methods [9], nonlinear least squares is still the most common methods in commercial and noncommercial software packages.",
"A generalized nonlinear model can be written as a function of not only the independent vector $\\mathbf {x} $ but also of a parameter vector $\\mathbf {\\Omega } $ [9] $y_i = f(\\mathbf {x}; \\mathbf {\\Omega }) + \\epsilon _i$ in which $\\epsilon _i$ is random error term with mathematical expectation zero In practical terms the average is zero.",
"The procedure used must find a least squares estimate of $\\mathbf {\\Omega } $ , denoted by $\\hat{\\mathbf {\\Omega }} $ which minimizes the sum of squares [9], [83] $SSQ(\\mathbf {\\Omega }) = || \\mathbf {y} - \\mathbf {x} \\mathbf {\\Omega } ||^2 = \\sum _{i=1}^ {n} [y_i - f(x_i; \\mathbf {\\Omega })]^2$ In simple terms, $y_i$ are the measured values and $f(x_i; \\mathbf {\\Omega })$ are the values predicted by the function.",
"Thus, $y_i - f(x_i; \\mathbf {\\Omega })$ are the differences between measured and predicted values, often called residuals or errors which are sought to be minimized.",
"Because of their numerical nature, nonlinear regression models can be unstable and convergence to an optimal solution might not always happen.",
"As the complexity of the model and the number of parameters increases, the likelihood that the search procedure will diverge and not find the minimum on the parameters surface increases.",
"Because of that, a topic of research in statistics, numerical methods and computation is to find algorithms with adequate properties for nonlinear regression.",
"The search procedures discussed here try to find an optimal parameter vector by an iterative procedure of the type [9], [83] $\\mathbf {\\Omega } ^{(\\phi +1)} = \\mathbf {\\Omega } ^{(\\phi )} + \\delta ^{(\\phi )}$ In which $\\phi $ can be understood as the iteration number, thus the estimate of the $\\mathbf {\\Omega }$ vector at $\\phi +1$ is conditionally based on the estimate at $\\phi $ plus a step term $\\delta $ at the iteration $\\phi $ .",
"For the Gauss-Newton method [9] $\\delta ^{(\\phi )} = - (\\mathbf {J}^{(\\phi )T} \\mathbf {J}^{(\\phi )})^{-1} \\mathbf {J}^{(\\phi )T} \\mathbf {r}^{(\\phi )T}$ $\\mathbf {J}$ is the Jacobian matrix estimates, the Jacobian is the matrix of the derivatives with respect to the parameters, $T$ indicates a transposed vector or matrix, $-1$ indicates the inverse of the matrix and $\\mathbf {r}$ is a vector of residuals, i.e.",
"observed minus predicted values at a given iteration level.",
"Obviously there is much more to this, although the actual application to a simple function might not be very complicated from the programming standpoint.",
"A simple implementation of the Gauss-Newton method will be provided next.",
"Consider the full implementation of the iterative procedure for the parameter vector using the Gauss-Newton step $\\mathbf {\\Omega } ^{(\\phi +1)} = \\mathbf {\\Omega } ^{(\\phi )} - (\\mathbf {J}^{(\\phi )T} \\mathbf {J}^{(\\phi )})^{-1} \\mathbf {J}^{(\\phi )T} \\mathbf {r}^{(\\phi )T}$ Suppose now we want to fit a simple water retention function of the form $\\theta = \\frac{a}{|\\psi _m|^b}$ in which $\\theta $ is the volumetric water content and $\\psi _m$ is the matric potential, in generalized units not relevant for this example, and $a$ and $b$ are fitting parameters.",
"We wish to fit the model to the fictitious data set $\\theta = \\lbrace 0.4, 0.38, 0.35, 0.3, 0.28, 0.23, 0.22, 0.21 \\rbrace $ $\\psi _m = \\lbrace -1, -5, -10, -50, -100, -1000, -5000, -10000 \\rbrace $ $\\theta $ and $\\psi _m$ are vectors containing the observed water content and matric potential estimates.",
"We start the Gauss-Newton method with an initial guess for the parameter vector, say $a = 0.5$ and $b = 0.2$ .",
"Ideally the initial parameter vector should be defined taking into consideration the physical meaning of the parameters and/or the user should plot predicted and estimated values in the same graph and try to estimate the approximate best fit visually, so the parameter vector is close to the optimum values.",
"The parameter vector at the zeroth iteration contains the initial guess, thus $\\Omega ^{(0)} =\\begin{bmatrix}0.50\\\\0.20\\end{bmatrix}$ The Jacobian matrix is the matrix containg the partial derivatives of the residuals vector $\\mathbf {r} = \\theta - \\hat{\\theta }$ with respect to the parameters, where $\\hat{\\theta }$ is the vector containing the model estimates based on the parameter vector.",
"In simpler term, for each observation $i$ , the residuals function is $r_i = \\theta _i- \\frac{a}{|\\psi _{m,i}|^b}$ Thus the partial derivatives with respect to the parameters are $\\frac{\\partial r_i}{\\partial a} = -|\\psi _{m,i}|^{-b}$ $\\frac{\\partial r_i}{\\partial b} = a|\\psi _{m,i}|^{-b}\\log {|\\psi _{m,i}|}$ The Jacobian matrix is then populated as $J_{i,2} = \\lbrace \\frac{\\partial r_i}{\\partial a}, \\frac{\\partial r_i}{\\partial b} \\rbrace $ ThusWe are approximating everything to four decimals whenever possible to save space, however, it is almost never a good idea to work with low precision approximations in numerical methods $\\mathbf {J}^{(0)} =\\begin{bmatrix}-1.00000 & 0.00000 \\\\-0.72478 & 0.58324 \\\\-0.63096 & 0.72642 \\\\-0.45731 & 0.89449 \\\\-0.39811, & 0.91668 \\\\-0.25119, & 0.86757 \\\\-0.18206 & 0.77530 \\\\-0.15849, & 0.72987\\end{bmatrix}$ with transpose $\\begin{split}& \\mathbf {J}^{(0)T} = \\\\&\\begin{bmatrix}-1.0000& -0.7248& -0.6310& -0.4573& -0.3981& -0.2512& -0.1821& -0.1585 \\\\0.0000& 0.5832& 0.7264& 0.8945& 0.9167& 0.8676& 0.7753& 0.7299\\end{bmatrix}\\end{split}$ The product $\\mathbf {J}^{(0)T}\\mathbf {J}^{(0)}$ is then a $ 2 \\times 2$ matrix At this point it might convenient for the reader to review an introductory linear algebra book $\\mathbf {J}^{(0)T}\\mathbf {J}^{(0)} =\\begin{bmatrix}2.4124 & -2.1298 \\\\-2.1298 & 4.3948\\end{bmatrix}$ With inverse likewise calculated using linear algebra procedures $(\\mathbf {J}^{(0)T}\\mathbf {J}^{(0)})^{-1} =\\begin{bmatrix}0.7245 & 0.3511 \\\\0.3511 & 0.3977\\end{bmatrix}$ The residuals vector is $\\mathbf {r}^{(0)} =\\begin{bmatrix}-0.1000 \\\\0.0176 \\\\0.0345 \\\\0.0713 \\\\0.0809 \\\\0.1044 \\\\0.1290 \\\\0.1308 \\\\\\end{bmatrix}$ resulting in a Gauss-Newton step vector $\\delta ^{0} = (\\mathbf {J}^{(0)T} \\mathbf {J}^{(0)})^{-1} \\mathbf {J}^{(0)T} \\mathbf {r}^{(0)T}$ of $\\delta ^{(0)} =\\begin{bmatrix}0.1107 \\\\0.1582\\end{bmatrix}$ Noting that $\\mathbf {\\Omega } ^{(1)} = \\Omega ^{(0)} - \\delta ^{(0)}$ results in $\\mathbf {\\Omega } ^{(1)} =\\begin{bmatrix}0.3893 \\\\0.0418\\end{bmatrix}$ with $SSQ = 0.067$ .",
"Repeating the procedure for five iterations results in $a = 0.4099$ and $b = 0.0761$ and $SSQ = 0.00075715$ .",
"For comparison, a numerical procedure from a popular free software spreadsheet resulted in the same values after 40 iterations.",
"The Levenberg-Marquardt method improves upon the Gauss-Newton method allowing for search in two direction and for singular or ill conditioned matrices [9], [83] $\\delta ^{(\\phi )} = - (\\mathbf {J}^{(\\phi )T} \\mathbf {J}^{(\\phi )} + \\eta ^{(\\phi )} \\mathbf {D}^{(\\phi )} )^{-1} \\mathbf {J}^{(\\phi )T} \\mathbf {r}^{(\\phi )T}$ in which $\\mathbf {D}^{(\\phi )}$ is a diagonal matrix with positive elements and $\\eta $ is a damping factor.",
"The Levenberg-Marquardt procedure is usually efficient for Brutsaert-van Genuchten type equations, although the Gauss-Newton procedure is usually adequate if the number of parameters is reduced and if the initial guess is reasonably close to optimal values.",
"A program for fitting the Brutsaert-van Genuchten to soil water retention data is presented in Appendix REF (Figure REF ).",
"The Levenberg-Marquardt is invoked from a program module and thus was not written from scratch.",
"Figure: Brutsaert-van Genuchten equation (BvG) fit to simulated water retention data from the program in Appendix .The procedures described above can be also used in soil mechanical resistance equations of the form $SR = a \\rho ^b \\theta ^c$ Note that the partial derivatives indicate that this function is nonlinear $\\frac{\\partial SR}{\\partial a} = \\rho ^b \\theta ^c$ $\\frac{\\partial SR}{\\partial b} = ba\\theta ^c\\rho ^{(b-1)}$ $\\frac{\\partial SR}{\\partial c} = ca\\rho ^b \\theta ^{(c-1)}$ However, this equation is intrinsically linear since it can be transformed into a linear function.",
"In this case the transformation is achieved by applying logarithms at both sides $\\log {SR} = \\log {a} + b \\log {\\rho } + c \\log { \\theta }$ which can be fitted by multiple linear regression with fitting parameters $a$ , $b$ and $c$ .",
"Note, however, that linearization of intrinsically linear functions can result in undesirable statistical properties [9], [59] In order to approximate differential equations using finite difference methods, it is necessary to use a mathematical tool called Taylor series.",
"Taylor series allow us to expand a infinitely differentiable function around a point using its derivatives.",
"It is a fundamental tool in physics and mathematics, having applications in several different areas, especially in the approximation of functions which are too difficult or impossible to solve analytically.",
"Consider a real of imaginary valued function $f(x)$ which we want to expand around the point $a$ , in which $a$ is a real or complex number.",
"The Taylor series expansion is $f(x) = f(a) + \\frac{(x-a)}{1!}",
"\\frac{d }{dx} f(a) + \\frac{(x-a)^2}{2!}",
"\\frac{d^2 }{dx^2} f(a) + \\frac{(x-a)^3}{3!",
"}\\frac{d^3 }{dx ^3} f(a)+ ...$ which can be written in summation form as $f(x) = \\sum _{0}^{\\infty } \\frac{(x-a)^n}{n!}",
"\\frac{d^n }{dx ^n} f(a)$ An important case of the Taylor series is when the expansion is done around zero, being called Maclaurin series $f(x) = f(0) + \\frac{x}{1!}",
"\\frac{d }{dx} f(0)+ \\frac{x^2}{2!}",
"\\frac{d^2 }{dx^2}f(0) + \\frac{x^3}{3!",
"}\\frac{d^3 }{dx ^3} f(0)+ ...$ For most applications in physics, the higher order terms (higher powers) are small and can be neglected, thus resulting in a simple and often precise approximation of mathematical functions.",
"This was particularly important before the advent of electronic calculators and digital computers, when the estimations of functions which today are viewed as trivial such as the value of $\\pi $ where difficult tasks.",
"Examples of Taylor series approximations for a few simple functions are given below [93] $e^x = 1 + x + \\frac{x^2}{2} + \\frac{x^3}{6} + \\frac{x^4}{24} + ...$ $\\log (1 + x) = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + ...$ $(1 + x)^n = 1 + nx + \\frac{n(n-1)}{2} x^2 + \\frac{n(n-1)(n-2)}{6} x^3 + ...$ We rewrite Laplace equation for total water potential for a two dimensional domain $\\frac{\\partial ^2 h}{\\partial x^2} + \\frac{\\partial ^2 h}{\\partial y^2} = 0$ As we have discussed, this equation does have an analytical solution in terms of series.",
"However, the numerical solution of this equation provides a convenient introduction to finite difference methods.",
"Our goal here is to approximate the second order partial derivatives within a domain.",
"Let's define the domain as a region in space, here in the horizontal plane, with dimensions $L_x$ and $L_y$ .",
"The domain can be discretized in the form of a regular grid with nodes spaced $\\Delta x$ in the $x$ direction and $\\Delta y$ in the $y$ direction (Figure REF ).",
"Figure: Discretization for the finite differences solution of the 2D Laplace equation.The second order partial derivatives in the $x$ direction can be approximated between two adjacent points, where the next point is located in the positive $x$ direction of $h^j_i$ , by Taylor series expansion [46] $\\begin{split}h^j_{i+1} = & h^j_{i} + (x^j_{i+1} - x^j_{i}) \\frac{\\partial h}{\\partial x} + \\frac{(x^j_{i+1} - x^j_{i})^2}{2!}",
"\\frac{\\partial ^2 h}{\\partial x^2} + \\\\ & \\frac{(x^j_{i+1} - x^j_{i})^3}{3!}",
"\\frac{\\partial ^3 h}{\\partial x^3} + ...\\end{split}$ and for points is located in the negative $x$ direction of $h(i,j)$ by $\\begin{split}h^j_{i-1} = & h^j_{i} - (x^j_{i-1} - x^j_{i}) \\frac{\\partial h}{\\partial x} + \\frac{(x^j_{i-1} - x^j_{i})^2}{2!}",
"\\frac{\\partial ^2 h}{\\partial x^2} - \\\\ & \\frac{(x^j_{i-1} - x^j_{i})^3}{3!}",
"\\frac{\\partial ^3 h}{\\partial x^3} + ...\\end{split}$ Adding the two equations and noting that for our regular grid $(x^j_{i+1} - x^j_{i}) = \\Delta x$ $(x^j_{i-1} - x^j_{i}) = -\\Delta x$ results in $h^j_{i+1} + h^j_{i-1} = 2h^j_{i} + (\\Delta x)^2 \\frac{\\partial ^2 h}{\\partial x^2} + \\mathcal {O}((\\Delta x)^3)$ In which $\\mathcal {O}((\\Delta x)^n)$ are higher order terms represented using the big “o” notation $\\mathcal {O}$ .",
"If we ignore error terms with order greater than two, which tends to be increasingly smaller as the order of the term increases, the approximation to the second order derivative in $x$ is $\\frac{\\partial ^2 h}{\\partial x^2} = \\frac{h^j_{i+1} + h^j_{i-1} - 2h^j_{i}}{(\\Delta x)^2}$ The same procedure results in the approximation in $y$ $\\frac{\\partial ^2 h}{\\partial y^2} = \\frac{h^{j+1}_{i} + h^{j-1}_{i} - 2h^{j}_{i}}{(\\Delta y)^2}$ Replacing the approximations into the Laplace equation and solving for $h^j_i $ allows us to approximate the values of the total potential inside the domain, given the boundary conditions of fixed total potentials at the upper, lower, right and left boundaries using the formula $h^j_{i} = \\frac{1}{4} [h^j_{i+1} + h^j_{i-1} + h^{j+1}_{i} + h^{j-1}_{i}]$ The solution is obtained by numerical methods.",
"A program for solving the 2D Laplace equation using the Jacobi iterative method is presented in Appendix REF .",
"A surface plot The figure was produced with Gnuplot: http://www.gnuplot.info/ of the solution for $h_{left} = 20~cm$ , $h_{right} = 1~cm$ , $h_{top} = 5~cm$ and $h_{bottom} = 0~cm$ is presented in Figure REF .",
"Figure: Numerical solution of the 2D Laplace equation from the program in Appendix .The discretization of an 1D partial differential equation in time and space follows a similar scheme as 2D discretization in space.",
"In diffusion type equations (e.g.",
"advective-dispersive, Richards-Richardson, heat equation) besides the second order partial derivative in space, there is a first order derivative in time.",
"Recall the 1D advective-dispersive equation for constant flux velocity and diffusion coefficient $\\frac{\\partial C}{\\partial t} = D \\frac{\\partial C}{\\partial x} - v \\frac{\\partial ^2 C}{\\partial x^2}$ We already know from the previous section that the second order partial derivatives in space can be approximated by $\\frac{\\partial ^2 C}{\\partial x^2} = \\frac{C^j_{i+1} + C^j_{i-1} - 2C^j_{i}}{(\\Delta x)^2}$ Obviously for concentration $C$ in this case.",
"To find the approximation of first order partial derivatives, the Taylor series expansion equations as in Equations REF REF are subtracted.",
"Ignoring higher order terms and solving for the partial derivatives of interest, the discretization of the first order partial derivatives in space and time for the advective-dispersive equation are $\\frac{\\partial C}{\\partial x} = \\frac{C^j_{i+1} + C^j_{i-1}}{(2\\Delta x)}$ $\\frac{\\partial C}{\\partial t} = \\frac{C^j_{i+1} + C^j_{i-1}}{(2\\Delta t)}$ The discretization is now in space and time, where $i$ is the increment in space and $j$ the increment in time.",
"Care must be taking in writing programs regarding the order of $i$ and $j$ in loops.",
"For the discretization in time, the condition at $t=0$ is now the initial contition and the conditions in the inlet and outlet of the 1D domain are the boundary conditions (Figure REF ).",
"Figure: Discretization for the finite differences solution of the 1D advective-dispersive equation.By replacing the partial derivatives in time and space, the finite differences equation to be solved for the 1D advective-dispersive equation is $C^j_{i} = \\Delta t C^{j-1}_{i} + D\\frac{\\Delta t}{\\Delta x^2} (C^{j-1}_{i-1} - 2C^{j-1}_{i} + C^{j-1}_{i+1} ) - v\\frac{\\Delta t }{2 \\Delta x} (C^{j-1}_{i+1} - C^{j-1}_{i-1} )$ One of the issues of the numerical solution of the advective-dispersive equation is that it is valid for infinitely long columns.",
"To circumvent this issue a program was written to solve the equation for a long column, but the outlet concentration measurements were made a fixed point along the length $x$ .",
"The comparison of the numerical and analytical solutions of the advective-dispersive equation for concentration measured at $50~cm$ from the inlet is presented in Figure REF .",
"The column length for the numerical simulations was $L = 100~cm$ , the flow velocity was $v = 2.0~cm~h^{-1} $ , the dispersion coefficient $D = D_h/\\theta = 5.0~cm^2~h^{-1} $ and the simulation time was $100~h$ .",
"The program is listed in the Appendix REF .",
"Figure: Comparison of numerical and analytical solutions of the 1D advective-dispersive equation from the program in Appendix .By the same principles illustrated for the Laplace and advective-dispersive equations, the Richardson-Richards can be discretized to be solved by explicit methods.",
"Recalling the 1D matric potential (here represented as $\\psi $ for simplicity) form of the Richardson-Richards equation in the general coordinate $z$ [50] $C^*(\\psi )\\frac{\\partial \\psi }{\\partial t} = \\frac{\\partial }{\\partial z} [K(\\psi ) (\\frac{\\partial \\psi }{\\partial z} + 1 )]$ The main issue here is that the unsaturated hydraulic conductivity is not constant and cannot be placed outside of the derivative terms.",
"The hydraulic capacity function also needs to be defined for this equation to be solved numerically.",
"Following the discussion in Chapter REF we will define the following water retention and unsaturated hydraulic conductivity functions [50] $\\theta (\\psi ) = \\theta _r + \\frac{\\alpha (\\theta _s - \\theta _r)}{\\alpha + |\\psi |^{\\beta }}$ $K(\\psi ) = \\frac{A}{A + |\\psi |^B}$ Recalling that the hydraulic capacity function is $C^*(\\psi ) = [\\alpha h_m (\\theta _s - \\theta _r) \\psi ^{\\beta -1}] \\frac{\\beta (\\alpha + \\psi ^ {\\beta })}{\\alpha + \\psi ^{\\beta }}$ The discretization for the explicit solution is [50] $\\begin{split}{\\psi }^{j+1}_{i} = & {\\psi }^{j}_{i} + \\frac{\\Delta t}{{C^*}^{j}_{i} \\Delta z} [ K^{j}_{i+1/2} (\\frac{{\\psi }^{j}_{i+1} - {\\psi }^{j}_{i}}{\\Delta z} -1 ) \\\\& - K^{j}_{i-1/2} (\\frac{{\\psi }^{j}_{i} - {\\psi }^{j}_{i-1}}{\\Delta z} -1 ) ]\\end{split}$ where the unsaturated hydraulic conductivity at intermediary point in nodes is $K^{j}_{i+1/2} = \\frac{K^{j}_{i+1}+ K^{j}_{i}}{2}$ $K^{j}_{i-1/2} = \\frac{K^{j}_{i}+ K^{j}_{i-1}}{2}$ Thus, the numerical solution can be achieved by applying the discretization schemes above while simultaneously evaluating the functions $K(\\psi )$ and $^*C(\\psi )$ at the $\\psi ^j_i$ points.",
"The grid is analogous to the one described for the advective-dispersive equation.",
"A program for solving the 1D Richardson-Richards equation using an explicit scheme is presented in Appendix REF .",
"The results for the parameters $\\alpha = 1.611 \\times 10^{6}~cm^{-1}$ , $\\theta _s = 0.287~cm^3~cm^{-3} $ , $\\theta _r = 0.075~cm^3~cm^{-3}= $ and $\\beta = 3.96$ are compared to the Philip's quasi-analytical solution presented by [50] for an $80~cm$ 1D soil column initially at $\\psi = -61.5~cm$ ($t=0$ ) and with $\\psi = -20.73~cm$ at soil surface and $\\psi = -61.5~cm$ at the bottom of the column (or profile) for $t > 0$ (Figure REF ).",
"The unsaturated hydraulic conductivity parameters were $ A = 0.00944$ and $B = 4.74$ .",
"Figure: Comparison of numerical and quasi-analytical solutions of the 1D Richardson-Richards equation from the program in Appendix .",
"The quasi-analytical solution follows Philip's method and is presented in .Although explicit methods are mathematically simpler and easier to implement from the computational point of view, they are notoriously unstable for most real world problems.",
"The discretization scheme is then limited by the stability of the method.",
"For most application implicit methods are preferred, these methods result in a set of equations which needs to be solved using matrix manipulation algorithms.",
"In the implicit discretization of the 1D equation, there is no simpler equation which can be inserted into a code to solve the problem, a simultaneous solution for the different nodes is sought.",
"Two popular choices of discretization are the Backward Time Centered Space (BTCS) $(-\\frac{\\gamma }{\\Delta x^2}) u^{j+1}_{i-1} + (\\frac{1}{\\Delta t} + \\frac{2\\gamma }{\\Delta x^2}) u^{j+1}_{i} + (-\\frac{\\gamma }{\\Delta x^2}) u^{j+1}_{i+1} = \\frac{1}{\\Delta t} u^{j}_{i}$ and the Crank-Nicolson scheme [30] A very good introduction to these methods can be found in https://web.cecs.pdx.edu/~gerry/class/ME448/ from which these discretization schemes were transcribed.",
"$\\begin{split}(-\\frac{\\gamma }{2\\Delta x^2}) u^{j+1}_{i-1} + (\\frac{1}{\\Delta t} + & \\frac{\\gamma }{\\Delta x^2}) u^{j+1}_{i} + (-\\frac{\\gamma }{2\\Delta x^2}) u^{j+1}_{i+1} = \\\\& \\frac{\\gamma }{2\\Delta x^2}) u^{j}_{i-1} + (\\frac{1}{\\Delta t} - \\frac{\\gamma }{\\Delta x^2}) u^{j}_{i} + \\frac{\\gamma }{2\\Delta x^2} u^{j}_{i+1}\\end{split}$ In which $u^j_i$ is the temperature at each grid point, $\\gamma = \\kappa _T/ (\\rho c_p)$ is the thermal diffusivity and the other parameters are the same as previously defined.",
"These numerical schemes can be applied to the Richardson-Richards and other partial differential equations.",
"One common strategy of solution is the diagonal matrix solver, often called the Thomas algorithm [26].",
"Technically, the explicit scheme can be thought as an special case of the discretization of the heat equation called the Backward Time Centered Space (BTCS).",
"The Crank-Nicolson and other more complex schemes are recommended over BTCS and (Forward Time Ceneterd Space) FTCS for real world applications.",
"The main advantages of simpler methods, specially the FTCS is that it can be used as a teaching tool for a first introduction to finite differences methods, as it is done in most numerical methods courses and books.",
"In many problems other even more complicated schemes, both mathematically and computationally are employed, namely Finite Elements, Finite Volume and Lattice Boltzmann methods A simple introduction to these should be included in further editions.",
"Most modern computational fluid mechanics, soil physics and groundwater hydrology numerical codes are based on one or more of these methods.",
"ll $ y $ Generic dependent variable $ x $ Generic independent variable $ a, b, c, d, e $ Generic fitting parameters $ y_0 $ Generic $y$ coordinate $ x_0 $ Generic $x$ coordinate $ \\Theta $ Effective saturation $ = (\\theta - \\theta _s)/(\\theta _s - \\theta _r)$ $ \\theta $ Volumetric water content $ \\theta _s $ Saturation volumetric water content $ \\theta _r $ Residual volumetric water content $ A, B $ Unsaturated hydraulic conductivity function empirical parameters $ \\psi _m $ Matric potential in unspecified units $ SSQ $ Sum of squares of the residuals $ \\Omega $ Parameter vector $ \\mathbf { y} $ Observed dependent variable values vector $ \\mathbf { x} $ Observed independent variable values vector $ \\delta $ Step in nonlinear regression procedures $ \\mathbf { J} $ Jacobian matrix $ \\mathbf { r} $ Predicted values vector $ \\mathbf { D} $ Diagonal matrix $ \\phi $ Iteration level $ \\eta $ Damping factor $ SR $ Soil mechanical resistance to penetration $ \\rho $ Bulk density $ h $ Total potential in length units $ i, j$ Integer variables for step in space and/or time $ \\Delta x, \\Delta y, \\Delta t $ Discretization interval in time and space $ C $ Concentration $ v $ Flow velocity $ D $ Dispersion coefficient corrected to volumetric water content $= D_h/\\theta $ $ t $ Time $ h_m $ Matric potential in length units $ \\psi $ Matric potential in length units $ C^*(h_m) $ Hydraulic capacity function $ = \\partial \\theta /\\partial h_m$ $ \\theta (h_m) $ Water retention function $ K(h_m) $ Unsaturated hydraulic conductivity function $ \\alpha , \\beta $ Water retention function fitting parameters $ u$ Temperature at specified grid points $ y^j_i$ Specifies the value of the property at $i$ and $j$ for any function $y$ $ \\gamma $ Thermal diffusivity $ \\kappa _T $ Thermal conductivity $ c_p$ Specific heat capacity tocchapterBibliography $\\alpha $ , $A $ : alpha $\\beta $ , $B $ : beta $\\gamma $ , $\\Gamma $ : gamma $\\delta $ , $\\Delta $ : delta $\\epsilon $ ,$\\varepsilon $ , $E $ : epsilon $\\zeta $ , $ Z $ : zeta $\\eta $ , $H $ : eta $\\theta $ , $\\Theta $ : theta $\\iota $ , $I $ : iota $\\kappa $ , $ K $ : kappa $\\lambda $ , $\\Lambda $ : lambda $\\mu $ , $ M $ : mu $\\nu $ , $ N $ : nu $\\xi $ , $ \\Xi $ : xi $o$ , $ O $ : omicron $\\pi $ , $ \\Pi $ : pi $\\rho $ , $ R $ : rho $\\sigma $ , $ \\varsigma $ , $ \\Sigma $ : sigma $\\tau $ , $ T $ : tau $\\upsilon $ , $ Y $ : upsilon $\\phi $ , $\\varphi $ , $ \\Phi $ : phi $\\chi $ , $X$ : chi $\\psi $ , $ \\Psi $ : psi $\\omega $ , $\\Omega $ , $ \\Phi $ : omega T.P.",
"Leão in Oct 5 2021, light revision in Aug 29 2022 Fits the Brutsaert-van Genuchten equation to data using the Leverberg-Marquardt algorithm Language: Python 3.8.10 This code does not check for regression diagnostics Import the necessary packages import numpy as np from scipy.optimize import curvefit, leastsquares import matplotlib.pyplot as plt Data psi = np.array([20, 40, 60, 80, 100, 200, 500, 800, 1000, 3000, 5000, 15000]) theta = np.array([0.41, 0.409, 0.406, 0.41268, 0.41045, 0.39708, 0.3899, 0.3525, 0.3203, 0.18478, 0.14497, 0.13]) Define initial parameters (Initial guess for the nonlinear regression procedure) thetas = 0.4 thetar = 0.1 alpha = 0.001 n = 2.0 m = 0.5 Defines the nonlinear regression model to be fit def vg(x, qs, qr, alpha, n, m): return qr + (qs - qr)/(1+ (alpha*psi)**n)**m Puts the initial parametes into the \"InitialParams\" vector InitPar = [thetas, thetar, alpha, n, m] Curve fitting procedure - \"lm\" is the Leverberg-Marquardt algorithm popt, pcov = curvefit(vg, psi, theta, p0=InitPar, method='lm') Creates values of psi for plotting the predicted data xd2 = np.linspace(10, 30000, 10000) Uses the parameters from the fit to create the data for the predicted plot thetap2 = popt[1] + (popt[0] - popt[1])/(1+ (popt[2]*xd2)**popt[3])**popt[4] computes the standard deviation of the parameters perr = np.sqrt(np.diag(pcov)) print(\"Best fit paramaters, thetas, thetar, alpha, n and m \", popt) print(\"Covariance of the parameters \", popt) print(\"Standard deviation of the parameters \", perr) Plot of the data and best fit line plt.plot(psi, theta, \"o\", label = \"Simulated data\") plt.plot(xd2, thetap2, \":\", label = \"van Genuchten (1980)\") plt.ylim(0, 0.45) plt.ylabel('Volumetric water content [vol/vol]') plt.xlabel('Matric potential [|cm|]') legend = plt.legend(loc='lower left', shadow=True, fontsize='large') plt.xscale(\"log\") plt.show() // Solution to the laplace equation 2D using an explicit finite differences scheme // T.P.",
"Leao on 5 out 2021 light revision on Aug 29 2022 // Language: C++ include<iostream> include<cmath> include<iomanip> include<stdlib.h> int main() // define the simulation parameters double dx = 5.0; double dy = 5.0; double X = 100.00; // x length double Y = 100.00; // y length int nx = X/dx+1; int ny = Y/dy+1; int i, j; // define the grid double h[nx][ny]; // initialize the array with zeros for(i=0; i < nx; i++) for(j=0; j < ny; j++) h[i][j] = 0.0; std::cout << \"————————————-\" << std::endl; // define the boundary conditions for(i=0; i < nx; i++) h[i][0] = 5.0; h[i][ny-1] = 1.0; for(j=1; j < ny; j++) h[0][j] = 20.0; h[nx-1][j] = 0; // Solves the system double d[1000], diff; double tol = 1e-5; std::cout << \"Iterations and diff \" << std::endl; for (int iter = 0; iter <100; iter++) // Maximum number of iterations for(i=1; i < nx-1; i++) for(j=1; j < ny-1; j++) h[i][j] = 0.25 * (h[i+1][j] + h[i-1][j] + h[i][j+1] + h[i][j-1]); d[iter] = h[1][1]; diff = d[iter] - d[iter-1]; if (diff < tol) break; std::cout << iter << \"\" << diff << std::endl; // Print iteration and difference between sucessive iterations at h[1][1] std::cout << \"Solution in gnuplot format \" << std::endl; for(i=0; i < nx; i++) for(j=0; j < ny; j++) std::cout << std::setprecision(6) << h[i][j] << \"\"; // Prints solution in terminal std::cout << std::endl; return 0; // Solution to the advective dispersive 1D equation using an explicit method // T.P.",
"Leao 01 out 2021 light revision 29 aug 2022 // Language: C++ include<iostream> include<cmath> include<iomanip> include<stdlib.h> int main() // define the simulation parameters double dx = 1.0; double dtau = 0.1; double TMAX = 500.0; double L = 100.0; double D = 5; double V = 2.0; int nx = L/dx+1; int ntau = TMAX/dtau+1; int i, j; // define the grid double C[ntau][nx]; // initialize the array with zeros for(i=0; i < nx; i++) for(j=0; j < ntau; j++) C[j][i] = 0.0; std::cout << \"————————————-\" << std::endl; // define the initial conditions for(i=0; i < nx; i++) C[0][i] = 0.0; // define the boundary conditions // for(j=1; j < ntau; j++) C[j][0] = 1.0; C[j][nx-1] = 0.0; // this is the explicit solution for(j=0; j < ntau-1; j++) for(i=1; i < nx-1; i++) C[j+1][i] = C[j][i] + dtau/pow(dx,2)*D*(C[j][i-1] -2*C[j][i] +C[j][i+1]) - V *dtau/(2*dx)*(C[j][i+1] - C[j][i-1]); std::cout << \" Explicit solution to the 1D advective-dispersion equation in time \"; std::cout << \"\"; std::cout << \" \"; std::cout << \"\"; for(j=0; j < ntau; j++) std::cout << \"| \"; for(i=0; i < nx; i++) std::cout << C[j][i] << \" \" ; return 0; // Solves the 1D richards equatio using and explicit finite differences scheme // This code has been validaded using Haverkamp 1977 analytical solutions using Philip's model // The discretization also follows Haverkamp 77 // In 2 out 2021 - TPL, light revision 29 aug 2022 // Language: C++ include<iostream> include<cmath> include <fstream> include <cstdlib> // static parameters for Warwick hydraulic functions, see Zarba 1988 MIT Masters define thetas 0.287 define thetar 0.075 define alpha 1.611e06 define beta1 3.96 define Ks 9.44e-3 define A 1.175e06 define B 4.74 using namespace std; // define the functions for water retention and hydraulic conductivity double theta(double h) double theta; theta = alpha * (thetas-thetar)/(alpha+ pow(abs(h), beta1)) + thetar; return theta; double K(double h) double K; K = Ks * A/(A + (pow(abs(h), B))); return K; double C(double h) double C; C = alpha*(thetas-thetar) * pow(abs(h), beta1-1) * beta1/((alpha+ pow(abs(h), beta1))* (alpha+ pow(abs(h), beta1))); return C; int main() double dz = 1.0; double dt = 0.4; int j, i; double L = 80.0; double tmax = 360; int nx = L/dz + 1; int ntau = tmax/dt + 1; double h[ntau][nx];//, K[ntau][nx], C[ntau][nx], q[ntau][nx]; for (j = 1; j < ntau; j++) h[j][0] = -20.73; h[j][nx-1] = -61.5; for (i = 0; i < nx; i++) h[0][i] = -61.5; for (j=0; j < ntau-1; j++) for (i=1; i < nx-1; i++) h[j+1][i] = h[j][i] + dt*1.0/(C(h[j][i])*dz) * ( (K(h[j][i+1])+K(h[j][i]) )/2.0 * ((h[j][i+1]-h[j][i])/dz -1.0 ) - (K(h[j][i])+K(h[j][i-1]) )/2.0 * ((h[j][i]-h[j][i-1])/dz - 1.0 )); for (j = 0; j < ntau; j++) for (i = 0; i < nx; i++) cout << i << \"\" << h[j][i]<<\"\"; cout << endl; // saves output to *.dat file std::ofstream outdata; std::ofstream output; outdata.open(\"rich1d.dat\"); for (int j=0; j < ntau; j++)outdata << std::endl; //std::cout << j << std::endl; for (int i=0; i < nx; i++) outdata << h[j][i] <<\"\"; outdata.close(); cout << endl; cout << \"—————————\" << endl; cout << \"Results saved to rich1d.dat\" << endl; return 0;"
]
] | 2210.07716 |
[
[
"HGARN: Hierarchical Graph Attention Recurrent Network for Human Mobility\n Prediction"
],
[
"Abstract Human mobility prediction is a fundamental task essential for various applications, including urban planning, transportation services, and location recommendation.",
"Existing approaches often ignore activity information crucial for reasoning human preferences and routines, or adopt a simplified representation of the dependencies between time, activities and locations.",
"To address these issues, we present Hierarchical Graph Attention Recurrent Network (HGARN) for human mobility prediction.",
"Specifically, we construct a hierarchical graph based on all users' history mobility records and employ a Hierarchical Graph Attention Module to capture complex time-activity-location dependencies.",
"This way, HGARN can learn representations with rich contextual semantics to model user preferences at the global level.",
"We also propose a model-agnostic history-enhanced confidence (MaHec) label to focus our model on each user's individual-level preferences.",
"Finally, we introduce a Recurrent Encoder-Decoder Module, which employs recurrent structures to jointly predict users' next activities (as an auxiliary task) and locations.",
"For model evaluation, we test the performances of our Hgarn against existing SOTAs in recurring and explorative settings.",
"The recurring setting focuses more on assessing models' capabilities to capture users' individual-level preferences.",
"In contrast, the results in the explorative setting tend to reflect the power of different models to learn users' global-level preferences.",
"Overall, our model outperforms other baselines significantly in the main, recurring, and explorative settings based on two real-world human mobility data benchmarks.",
"Source codes of HGARN are available at https://github.com/YihongT/HGARN."
],
[
"Introduction",
"Human mobility is critical for various downstream applications such as urban planning, transportation services, location recommendation, and epidemic management.",
"The ability to model and accurately predict future human mobility can inform us about important public policy decisions for society's betterment, such as promoting social integration, encouraging innovation, and maximizing productivity [29].",
"Central to human mobility modeling is the problem of next location prediction, i.e., the prediction of where an individual is going next, which has received great attention.",
"On the one hand, the increasing prevalence of mobile devices and the popularity of location-based social networks (LBSNs) provide unprecedented data sources for mining individual-level mobility traces and preferences [38].",
"On the other hand, the advancement of AI and machine learning offers a plethora of analytical tools for modeling human mobility.",
"These innovations have supercharged the development of human mobility models in the past decade, especially for the next location prediction.",
"Figure: An illustration of two human mobility trajectories.",
"Activities are essential in affecting human travel decisions and in modeling multiple dependencies.Traditional approaches studied human mobility over sequential transitions, where several approaches explored using Markov Chains (Mc) [10], [28], [25] to model transition patterns over location sequences.",
"Later, recurrent neural network-based [13] models demonstrated better predictive power over Mc-based methods, where pioneering works employed recurrent structures to model temporal periodicity [9] and spatial regularity [36].",
"Some other Rnn-based methods incorporate spatial and temporal contexts [21], [30] into the Rnn update process to boost models' performances.",
"In addition, due to the great success of the Transformer architecture [31], the attention mechanism has also been adopted in several works [22], [12] to better model sequences and obtain good prediction results.",
"In recent years, graph-based approaches leveraged graph representation learning [37], [3] and graph neural networks (Gnns) [15] to model user preferences [20] and the spatial-temporal relationships [8], [22] between locations, obtaining rich representations [33], [27] to improve the performance of the next location prediction.",
"While these works rely on mining individual location sequences for human mobility prediction, they rarely consider the underlying travel behavior structure and activity information.",
"In the travel behavior literature, an individual's decision to visit a location is usually considered to be derived from their daily activity patterns [2].",
"As a result, important inter-dependencies exist between activity participation and location visitation behaviors, which are often overlooked.",
"Furthermore, the number of possible activity categories is much smaller than the number of locations, making it easier to obtain a robust estimate for the former.",
"With data regarding human activity patterns becoming increasingly accessible, it is both behaviorally meaningful and computationally efficient to incorporate such activity information and its spatial-temporal dependencies in human mobility modeling.",
"Figure REF shows a couple of human mobility trajectories reflecting time-activity-location dependencies.",
"For example, when the time is approaching noon, one user may decide to dine at a nearby restaurant, and another may go to the movie theater for a specific starting time.",
"These examples illustrate that activities are usually scheduled according to the time of day, affecting the choice of which location to visit next, as also evidenced by Figure REF , where people usually go to restaurants at noon and mostly return home or go to the gym at night.",
"However, only a few works have considered adding activity information (e.g., location categories) to improve next location prediction.",
"Yu et al.",
"[39] used activities along with spatial distances to reduce search space (i.e., candidate locations).",
"Huang et al.",
"[14] proposed a Cslsl which comprises an Rnn-based [7] structure, where the time, activity, and location are predicted sequentially to model human travel decision logic.",
"However, the design of Cslsl oversimplifies the time-activity-location dependencies.",
"Given data sparsity and behavioral uncertainties [44], the time prediction tends to be more challenging [43], which can further compromise the prediction of activities and locations.",
"To the best of our knowledge, no efforts have been made to model the sophisticated time-activity-location dependencies and leverage them to improve the next location prediction.",
"In this paper, we present Hierarchical Graph Attention Recurrent Network (Hgarn) for the next location prediction.",
"Specifically, we first construct a hierarchical graph based on time-activity-location history mobility records of all users and employ a Hierarchical Graph Attention Module to capture complex temporal-view activity-activity, affiliated-view activity-location, spatial-view location-location dependencies.",
"In this way, Hgarn can learn representations with rich contextual semantics to model user preferences at the global level.",
"Correspondingly, a model-agnostic history-enhanced confidence (MaHec) label is proposed to guide our model to learn each user’s individual-level preferences.",
"We finally introduce a Recurrent Encoder-Decoder Module.",
"The module takes the sequence of user embeddings combined with the learned hierarchical graph representations as input to jointly predict a user's next activity (as the auxiliary task) and location (as the main task).",
"Specifically, the former can be residually incorporated into the latter.",
"In such a design, our model can leverage the learned time-activity-location dependencies to benefit both global- and individual-level human mobility modeling, and use predicted next activity distribution to facilitate model performance for personalized next location prediction.",
"In summary, our paper makes the following contributions: [leftmargin=*] We propose a Hierarchical Graph to represent the temporal-view activity-activity, affiliated-view activity-location, spatial-view location-location dependencies.",
"To the best of our knowledge, among the few approaches incorporating activity information into the next location prediction, this is the first work to model the dependencies of time, activities and locations by leveraging a Hierarchical Graph.",
"We design a Hierarchical Graph Attention Recurrent Network (i.e., Hgarn), which contains a hierarchical graph attention module to model dependencies between time, activities, and locations to capture users' global-level preferences, and a recurrent encoder-decoder module to incorporate the hierarchical graph representations into sequence modeling and utilize next activity prediction to boost the next location prediction.",
"We introduce a simple yet effective model-agnostic history-enhanced confidence (MaHec) label to guide our model to learning each user's individual-level preferences, which enables the model to focus more on relevant locations in their history trajectories when predicting their next locations.",
"Extensive experiments are conducted using two real-world location-based social networks (LBSN) check-in datasets.",
"Specifically, we evaluate the prediction performance of our Hgarn against existing SOTAs in the main, recurring, and explorative settings.",
"Our work is the first to separately evaluate the next location prediction models' recurring and explorative prediction performances.",
"The results show that the Hgarn can significantly outperform existing methods in all experimental settings.",
"Figure: A workflow of the proposed Hgarn."
],
[
"Related work",
"The next location prediction is essentially about sequence modeling since successive location visits are usually correlated [5], [40].",
"Traditional Mc-based methods often incorporate other techniques, such as matrix factorization [28] and activity-based modeling [25] to capture mobility patterns and make predictions.",
"However, Mc-based methods are limited in capturing long-term dependencies or predicting exploratvie human mobility.",
"Deep learning-based approaches consider the next location prediction a sequence-to-sequence task and obtain better prediction results than traditional methods.",
"The majority of existing deep models are based on Recurrent neural networks [13], [6] (Rnn).",
"Strnn [21] is a pioneering work that incorporates spatial-temporal features between consecutive human visits into Rnn models to predict human mobility.",
"Stgn [42] adds spatial and temporal gates to Lstm to learn users interests.",
"Flashback [36] leverages spatial and temporal intervals to compute an aggregated past Rnn hidden states for predictions.",
"Lstpm [30] introduces a non-local network and a geo-dilated Lstm to model users’ long- and short-term preferences.",
"Some approaches use the attention mechanism to improve their performances.",
"DeepMove [9] leveraged attention mechanisms combined with an Rnn module to capture users’ long- and short-term preferences.",
"Arnn [12] uses a knowledge graph to find related neighbor locations and model the sequential regularity of check-ins through attentional Rnn.",
"Stan [22] extracts relative spatial-temporal information between consecutive and non-consecutive locations through a spatio-temporal attention network for locations predictions.",
"In addition, some efforts incorporate contextual information [17] such as geographical information [18], dynamic-static [24], text content about locations [4] into sequence modeling.",
"Recently, graph-based models have been designed to enrich contextual features, which can reflect global preferences.",
"LBSN2Vec [37] performs random walks on a hypergraph to learn embeddings for next location and friendship predictions.",
"STP-UDGAT [20] uses Gat to learn location relationships from both local and global views based on constructed spatial, temporal, and preference graphs.",
"HMT-GRN [19] learned several user-region matrices of different granularity levels to alleviate the data sparsity issue and efficiently predict next locations.",
"Gcdan [8] uses dual attention to model the high-order sequential dependencies and graph convolutions to mitigate the data sparsity issue.",
"Graph-Flashback [27] uses knowledge graph embedding to construct transition graph, and applies Gcn to refine graph representations and combines with Flashback to make predictions.",
"Some other works focused on different aspects, activity-aware methods like [1] model the activities with a weighted category hierarchy (WCH), CatDM [39] uses activities along with spatial distances to reduce search space, Cslsl [14] proposed an Rnn-based causal structure to capture human travel decision logic.",
"However, most existing methods ignore activity information and cannot effectively model the time-activity-location dependencies, which are essential for predicting and understanding human mobility."
],
[
"Preliminaries",
"This section introduces definitions relevant to this study and then formulates the next location prediction problem.",
"We use notations $U=\\left\\lbrace u^{i} \\right\\rbrace _{i=1}^{|U|}$ , $L=\\lbrace l^{i}\\rbrace _{i=1}^{|L|}$ , $C=\\lbrace c^{i}\\rbrace _{i=1}^{|C|}$ , and $T=\\lbrace t^{i}\\rbrace _{i=1}^{|T|}$ to denote the sets of users, locations, activities and time series, respectively.",
"In particular, we denote a user $u\\in U$ 's sets of locations, activities and time series in a temporal order as $L_{u}=\\lbrace l^{i}_{u} \\rbrace _{i=1}^{|L_{u}|}$ , $C_{u}=\\lbrace c^{i}_{u} \\rbrace _{i=1}^{|C_{u}|}$ , and $T_{u}=\\lbrace t^{i}_{u} \\rbrace _{i=1}^{|T_{u}|}$ , where $\\delta _{u}^{i}$ may not equal to $\\delta ^{i}$ , $\\delta \\in \\lbrace U, L, C, T\\rbrace $ .",
"Definition 1 (Mobility Record) We use the notation $r$ to denote a single human mobility record.",
"Specifically, the $i$ th record of a given user $u \\in U$ is represented by a tuple $r_{u}^{i}=(u, c_{u}^{i}, l_{u}^{i}, t_{u}^{i})$ .",
"Each record tuple comprises a user $u \\in U$ , an activity $c_{u}^{i} \\in C_{u}$ , a location $l_{u}^{i} \\in L_{u}$ and the visit time $t_{u}^{i} \\in T_{u}$ .",
"Definition 2 (Trajectory) A trajectory is a sequence of mobility records that belongs to a user $u \\in U$ , denoted by $R_{u}=\\lbrace r_{u}^{i}\\rbrace _{i=1}^{|R_{u}|}$ .",
"Each trajectory $R_{u}$ can be divided into a history trajectory $R_{u}^{his}=\\lbrace r_{u}^{i}\\rbrace _{i=1}^{|R_{u}|-1}$ and the final record $r_{u}^{|R_{u}|}$ .",
"In this way, we can obtain the user $u$ 's activity trajectories $R_{u}^{C}={\\lbrace c_{u}^{i}\\rbrace }_{i=1}^{|R_{u}|}$ , location trajectories $R_{u}^{L}={\\lbrace l_{u}^{i}\\rbrace }_{i=1}^{|R^{u}|}$ , and time trajectories $R_{u}^{T}={\\lbrace t_{u}^{i}\\rbrace }_{i=1}^{|R_{u}|}$ with their history trajectories $R_{u}^{C, his}$ , $R_{u}^{L, his}$ and $R_{u}^{T, his}$ .",
"Given the definitions above, the next location prediction problem can be formulated as follows: Problem 1 (Next Location Prediction) Given a user $u \\in U$ 's history trajectory $R_{u}^{his}$ as input, we consider $u$ 's next record as its future state.",
"The human mobility prediction task $\\mathcal {T}$ maps $u$ 's history trajectory $R_{u}^{his}$ to $u$ 's next location $l_{u}^{|R_{u}|}$ in the future.",
"The above-described process can be summarized as follows: $R_{u}^{his}\\stackrel{\\mathcal {T}(\\cdot ;\\theta )}{\\longrightarrow }l_{u}^{|R_{u}|},$ where $\\theta $ is the parameters of mapping $\\mathcal {T}$ ."
],
[
"Methodology",
"The Hgarn's workflow is demonstrated in Figure REF .",
"The raw data is first encoded in the embedding layer and then input to the hierarchical graph attention module to model multi-dependencies.",
"Finally, the user's personalized embeddings are fused with the learned hierarchical graph representations and input to the recurrent encoder-decoder module to make predictions.",
"In the following sections, we will elaborate on the details of our Hgarn."
],
[
"Embedding Layer",
"We first introduce our embedding layer, which aims to assign trainable embeddings to each unique mobility record element, i.e., the user, activity, location and time.",
"It is worth noting that all these elements are discrete values except for time, and it is necessary to split the continuous time into discrete intervals to facilitate the encoding of time.",
"In this work we discretize the 24 hours of the day into $|T^{h}|$ time slots $T^{h}=\\lbrace h\\rbrace _{h=1}^{|T^{h}|}$ and another variable $T^{w}=\\lbrace w\\rbrace _{w=1}^{|T^{w}|}$ indicating the day of the week.",
"Note that all $t \\in T$ can be written in the form of $t=(h, w)$ .",
"We obtain the embeddings of each user, activity, location and time $e_{u} \\in \\mathbb {R}^{d^u}$ , $e_{c}, e_{l} \\in \\mathbb {R}^{d}$ , $e_{t} \\in \\mathbb {R}^{d^t}$ by multiplying their one-hot vectors with the corresponding trainable transformation matrices, where $d^u$ , $d$ and $d^t$ represent embedding dimensions.",
"$e_{U} \\in \\mathbb {R}^{|U| \\times {d^u}}$ , $e_{C} \\in \\mathbb {R}^{|C| \\times {d}}$ , $e_{L} \\in \\mathbb {R}^{|L| \\times {d}}$ and $e_{T} \\in \\mathbb {R}^{|T| \\times {d^t}}$ are used to denote the embedding matrices for the user, activity, location and time, respectively.",
"$e_{r_{u}^{i}}=\\left(e_{u}, e_{c_{u}^{i}}, e_{l_{u}^{i}}, e_{t_{u}^{i}}\\right)$ is used to denote a user $u$ 's embeddings of $i$ th record."
],
[
"Hierarchical Graph Attention Module",
"The hierarchical graph attention module consists of two parts: hierarchical graph construction and hierarchical graph attention (HGA) networks on the graph for multi-dependencies modeling.",
"Figure: The design of hierarchical graph attention module."
],
[
"Hierarchical Graph Construction",
"Given that a spatial network is naturally a graph and recent advances in graph neural networks [23], [32], we choose to model the location-location, location-activity and activity-activity dependencies with a hierarchical graph.",
"The constructed hierarchical graph is shown in Figure REF , which includes three layers: location layer, localized-activity layer, and activity layer, where the localized-activity layer is adopted to suppress noise aggregated from the location layer.",
"We formally describe the hierarchical graph with notation $G=(V,E)$ , where $V$ $=$ $V^{L} \\cup V^{C} \\cup V^{C^\\prime }$ and $E=\\lbrace A^{L}$ ,$A^{C}$ ,$A^{LC^\\prime }$ ,$A^{CC^\\prime }\\rbrace $ .",
"Specifically, $V^{L}$ and $V^{C}$ represents the location node-set and the activity node-set.",
"$V^{C^\\prime }$ indicates the localized-activity node-set.",
"$E$ comprises four adjacency matrices that denote connectivity between (1) two location nodes, (2) two activity nodes, (3) a location node and a localized-activity node, (4) an activity node and a localized-activity node.",
"For the location adjacency matrix $A^{L}$ , we utilize the geographical distance to determine whether two nodes are linked.",
"We employ the haversine formula to compute distances between locations.",
"Given $l^{i}$ and $l^{j}$ and their GPS data, $A^{L} \\in \\mathbb {R}^{|L| \\times |L|}$ is as: $A^{L}_{l^{i}, l^{j}}={\\left\\lbrace \\begin{array}{ll}1, & Haversine \\left(\\text{GPS}^{i}, \\text{GPS}^{j} \\right) < D^{h}\\\\0, & \\text{otherwise}\\end{array}\\right.",
"},$ where $D^{h}$ is a hyperparameter to control the distance threshold to influence the connectivity between locations.",
"The construction of $A^{C}$ is based on history trajectories of all users.",
"Intuitively, the dependency between two activities can be measured by the frequency of co-occurrence in the same time interval $h \\in T^{h}$ .",
"However, if we directly consider the activity co-occurrence frequency based on all trajectories (regardless of the user), it may lead to unrelated activities being interrelated (e.g., check-in at subway stations and gyms both often occur in the evening), due to the difference in user preferences.",
"Instead, a more reasonable way is to learn the inter-activity dependencies based on activity co-occurrence within individual-level trajectory sets.",
"Therefore, we can traverse the history trajectories of each user $u \\in U$ and calculate the co-occurrence matrix $M^{C} \\in \\mathbb {R}^{|C| \\times |C|}$ : $M^{C}_{c_{u}^{i}, c_{u}^{j}}=\\sum _{u \\in U}\\sum _{i=1}^{|R_{u}^{his}|}\\sum ^{|R_{u}^{his}|}_{j=i+1} \\mathbf {1}_{h}\\left(h_{u}^{i}, h_{u}^{j}\\right),$ $\\mathbf {1}_{h}\\left(h_{u}^{i}, h_{u}^{j}\\right)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } {h}_{u}^{i} = {h}_{u}^{j} \\\\0, & \\text{otherwise}\\end{array}\\right.",
"},$ where $\\mathbf {1}_{h}(\\cdot )$ is an indicator function.",
"Larger elements in the $M^{C}$ indicate stronger dependencies between the corresponding two activities.",
"For the sake of simplicity and suppression of correlation between some of the less relevant activities, we calculate the adjacency between each activity pair $(c^{i}, c^{j})$ to attain $A^{C} \\in \\mathbb {R}^{|C| \\times |C|}$ : $A^{C}_{c^{i}, c^{j}}={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } M^{C}_{c^{i}, c^{j}} > \\text{mean}\\left(M^{C}\\right) \\\\0, & \\text{otherwise}\\end{array}\\right.",
"},$ The matrix $A^{LC^{\\prime }}$ defines the adjacency between nodes of the location layer and nodes of the localized-activity layer.",
"Each node of $V^{L}$ is linked to only one node of $V^{C^\\prime }$ , representing the corresponding activity category at that location.",
"In contrast, each node of $V^{C^\\prime }$ may be linked to multiple nodes of $V^{L}$ , as several locations can share the same activity.",
"Formally, we define $A^{LC^{\\prime }}_{L} \\in \\mathbb {R}^{|L| \\times |C|}$ based on the affiliations of locations and activities, where each row corresponds to a location and each column an activity.",
"Additionally, we construct the adjacency matrix $A^{LC^{\\prime }} \\in \\mathbb {R}^{(|L|+|C|) \\times (|L|+|C|)}$ based on $A^{LC^{\\prime }}_{L}$ in the following block matrix form: $A^{LC^{\\prime }}=\\begin{bmatrix}O_{L} & A^{LC^{\\prime }}_{L} \\\\{(A^{LC^{\\prime }}_{L})}^{\\top } & O_{C}\\end{bmatrix},$ where ${(A^{LC^{\\prime }})}^{\\top }$ is the transpose of $A^{LC^{\\prime }}$ , $O_{L} \\in \\mathbb {R}^{|L| \\times |L|}$ and $O_{C} \\in \\mathbb {R}^{|C| \\times |C|}$ are two zero matrices.",
"The function of the localized-activity layer is to suppress noise from the location layer aggregated to the activity layer, so we simply define that each node in the localized-activity layer is connected to the node in the activity layer representing the same activity.",
"Mathematically, similar to Equation REF , we have $A^{CC^\\prime } \\in \\mathbb {R}^{2|C| \\times 2|C|}$ : $A^{CC^\\prime }=\\begin{bmatrix}O_{C} & I_{C} \\\\I_{C} & O_{C}\\end{bmatrix},$ where $I_{C} \\in \\mathbb {R}^{|C| \\times |C|}$ is an identity matrix."
],
[
"Hierarchical Graph Attention Networks",
"Inspired by recent work conducted deep learning on hierarchical graphs [11], [34], [41], we design a hierarchical graph attention network (HGA) to model the hierarchical graph we just defined.",
"Since the importance of locations within a certain distance is different for a specific location, we employ graph attention mechanisms to model spatial-view location-location dependencies: $H^{L}=\\textsc {\\textsc {Gat}}_{L}\\left(e_{L}, A^{L}\\right),$ where $\\textsc {\\textsc {Gat}}(\\cdot )$ is a standard implementation of [32], $H^{L} \\in \\mathbb {R}^{|L| \\times d^{g}}$ is the learned representations as the output of the $\\textsc {Gat}_{L}$ .",
"To integrate location information into representation learning of activities and suppress the noise aggregated to the nodes of the activity layer, we introduce the localized-activity layer to pre-aggregate location embeddings.",
"We first concatenate $e_L$ and $e_C$ to obtain the fused embedding matrix $e_{LC} \\in \\mathbb {R}^{(|L|+|C|) \\times d} = {\\begin{bmatrix}e_L & e_C\\end{bmatrix}}^{\\top }$ .",
"Then the localized-activity process is implemented as: $H^{LC^{\\prime }}=\\textsc {Gat}_{LC}\\left(e_{LC}, A^{LC^{\\prime }}\\right),$ where $H^{LC^{\\prime }} \\in \\mathbb {R}^{(|L|+|C|) \\times d^{g}}$ is the output of $\\textsc {Gat}_{LC}$ , we further slice out the first $|L|$ rows of matrix $H^{LC^{\\prime }}$ to obtain the pre-aggregated representation matrix ${H^{C^{\\prime }}} \\in \\mathbb {R}^{|C| \\times d^{g}}$ .",
"The learned representation $H^{C^{\\prime }}$ is again concatenated with activity embeddings $e_C$ as the $\\textsc {Gat}_C$ 's input $e_{CC}={\\begin{bmatrix}e_C & H^{C^{\\prime }}\\end{bmatrix}}^{\\top }$ .",
"It is worth noting that for all nodes in the activity layer, we could simultaneously aggregate information from neighbors in the localized-activity layer and neighbors in the same layer by simply modifying the matrix $A^{CC^\\prime }$ to: $A^{CC^\\prime }_{new}=\\begin{bmatrix}A^{C} & I_{C} \\\\I_{C} & O_{C}\\end{bmatrix}.$ We still employ a similar strategy to update the representation of the nodes in the activity layer: ${H^{C}}^\\prime =\\textsc {Gat}_{C}\\left(e_{CC}, A^{CC^\\prime }_{new}\\right),$ where ${H^{C}}^\\prime \\in \\mathbb {R}^{2|C| \\times d^{g}}$ is the learned representation from $\\textsc {Gat}_{C}$ , we slice out the last $|C|$ rows of matrix ${H^{C}}^\\prime $ to obtain the updated activity node representation $H^{C} \\in \\mathbb {R}^{|C| \\times d^{g}}$ ."
],
[
"Recurrent Encoder-Decoder Module",
"The recurrent encoder-decoder module 's structure is shown in Figure REF .",
"Given a user $u \\in U$ 's history trajectory, the input of the activity and the location encoder at $i$ th iteration are implemented as: $X^{C,i}_{u}= {e_{u}} \\Vert e_{t_{u}^{i}} \\Vert e_{c_{u}^{i}} \\Vert H^{C}_{c_{u}^{i}},$ $X^{L,i}_{u}= {e_{u}} \\Vert e_{t_{u}^{i}} \\Vert e_{l_{u}^{i}} \\Vert H^{C}_{c_{u}^{i}} \\Vert H^{L}_{l_{u}^{i}},$ where $\\Vert $ is the concatenation operation, $H^{C}_{c_{u}^{i}}$ and $H^{L}_{l_{u}^{i}}$ are the learned hierarchical graph representation of the activity node $c_{u}^{i}$ and the location node $l_{u}^{i}$ , the concatenated $X^{C,i}_{u}$ and $X^{L,i}_{u}$ are the inputs of the two encoders at the $i$ th iteration.",
"Figure: Recurrent encoder-decoder module's architecture.Both activity and location encoders are implemented with Lstm, where hidden states updating process between $h^{\\cdot ,i}_{u}$ and $h^{\\cdot ,i-1}_{u}$ can be illustrated as: $\\begin{aligned}p_{u}^{i}, f_{u}^{i}, g_{u}^{i}, o_{u}^{i} =\\text{split} ( ( & W_{x}X^{\\cdot ,i}_{u}+b_{x} ) + ( W_{h}h^{\\cdot ,i-1}_{u}+b_{h} ) ),\\end{aligned}$ $c^{\\cdot , i}_{u} = \\sigma \\left(f_{u}^{i}\\right) \\odot c^{\\cdot , i-1}_{u} + \\sigma \\left(p_{u}^{i}\\right) \\odot \\tanh \\left(g_{u}^{i}\\right),$ $h^{\\cdot , i}_{u} = \\sigma \\left(o_{u}^{i}\\right) \\odot \\tanh \\left(c^{\\cdot , i}_{u}\\right),$ where “$\\cdot $ ” can be either $C$ or $L$ .",
"$p_{u}^{i}$ , $f_{u}^{i}$ , $g_{u}^{i}$ , $o_{u}^{i}$ are the input, forget, cell and output gates.",
"$c^{\\cdot , i}_{u}$ is the cell state at iteration $i$ .",
"$\\sigma (\\cdot )$ and $\\tanh (\\cdot )$ are the sigmoid and tanh activation functions.",
"$W_{x}$ , $W_{h}$ , $b_{x}$ and $b_{h}$ are the trainable weights shared by inputs and $\\odot $ is the Hadamard product.",
"Initial hidden states for $h^{\\cdot , 1}_{u}$ and $c^{\\cdot , 1}_{u}$ are set to zeros.",
"After obtaining final hidden states of activity and location encoder as $h^{C}_{u}$ and $h^{L}_{u}$ , we implement our activity decoder as a multi-layer perceptron (Mlp) to get the next activity logits $\\widetilde{h^{C}_{u}} \\in \\mathbb {R}^{|C|}$ : $\\widetilde{h^{C}_{u}}=\\textsc {Mlp}_{C}\\left(h^{C}_{u}\\right).$ We finally residually combine the obtained activity logits with the encoded $h^{L}_{u}$ to learn the location logits $\\widetilde{h^{L}_{u}} \\in \\mathbb {R}^{|L|}$ : $\\begin{aligned}\\widetilde{h^{L}_{u}}=& \\lambda _{r}\\cdot \\textsc {Mlp}_{L}^{r}\\left(h^{L}_{u}\\right)+ (1-\\lambda _{r})\\cdot \\textsc {Mlp}_{L}\\left(\\textsc {Mlp}_{L}^{h}\\left(h^{L}_{u} \\Vert h^{C}_{u}\\right) \\Vert \\widetilde{h^{C}_{u}}\\right),\\end{aligned}$ where $\\lambda _{r}$ is a factor that controls the residual connections to trades off different features."
],
[
"In this section we introduce the model-agnostic history-enhanced confidence label and explain the optimization process of Hgarn."
],
[
"Model-Agnostic History-Enhanced Confidence Label Construction",
"As human mobility trajectories exhibit repetitive nature (i.e., recurring mobility), existing models [9], [8] often try to capture periodicity by applying attention mechanisms to all user trajectories, have limited performances and interpretability.",
"To overcome this, we modify the original label to get the model-agnostic history-enhanced confidence label (MaHec label, in short), which can guide our model to focus on relevant user trajectories.",
"Specifically, for each location $l^{i}\\in L$ , we differentiate its confidence for a user $u$ 's next location in two types: ${\\textsc {MaHec}}_{l^{i}}^{u}={\\left\\lbrace \\begin{array}{ll}w^{c}, & \\text{if } l^{i}=l_{u}^{|R_{u}|} \\\\\\left(1-w^{c}\\right) \\cdot f_{l^{i}}^{u}, & \\text{otherwise}\\end{array}\\right.",
"},$ $f_{l^{i}}^{u}=\\frac{1}{|R_{u}^{his}|}\\sum _{{l^{i}}\\in L} \\mathbf {1}_{R^{L,his}_{u}}\\left(l^{i}\\right),\\text{ }\\text{ }\\mathbf {1}_{R^{L,his}_{u}}\\left(l^{i}\\right)={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } l^{i} \\in R^{L,his}_{u} \\\\0, & \\text{if } l^{i} \\notin R^{L,his}_{u}\\end{array}\\right.",
"},$ where $\\mathbf {1}_{R^{L,his}_{u}}(\\cdot )$ is an indicator function and $w^{c}\\in [0,1]$ is a hyperparameter that indicates the confidence of the $u$ 's ground truth label and $f_{l^{i}}^{u}$ denotes the user $u$ 's history visit frequency to $l^{i}$ .",
"Then the MaHec label for the user $u$ 's next location is as: ${\\textsc {MaHec}}_{L}^{u}=\\left({\\textsc {MaHec}}_{l^{i}}^{u},\\right)_{i=1}^{|L|}\\in \\mathbb {R}^{|L|},$ where each element in ${\\textsc {MaHec}}_{L}^{u}$ represents the confidence that the user $u$ decides to pick as its next location.",
"Similarly, we conduct the same operations for user activity trajectories to obtain ${\\textsc {MaHec}}_{C}^{u}$ ."
],
[
"Model Optimization",
"Since next location prediction is a classification problem, we transform $\\widetilde{h^{L}_{u}}$ to the probability distribution $\\widehat{h^{L}_{u}} \\in \\mathbb {R}^{|L|}$ of all locations by $\\widehat{h^{L}_{u}}=\\text{Softmax}\\left(\\widetilde{h^{L}_{u}}\\right)$ .",
"Given ${\\textsc {MaHec}}_{L}^{u}$ and $\\widehat{h^{L}_{u}}$ , we compute $\\mathcal {L}_L$ based on cross-entropy loss: $\\mathcal {L}_L=- \\frac{1}{|U|}\\sum _{u \\in U} \\sum _{i=1}^{|L|}{\\textsc {MaHec}}^{u}_{l^{i}}\\cdot \\log \\left(\\widehat{h^{L, i}_{u}}\\right),$ where $\\widehat{h^{L, i}_{u}}$ is the $i$ th element of $\\widehat{h^{L}_{u}}$ and we compute the next activity loss $\\mathcal {L}_{C}$ based on same above described operations.",
"Finally, we could train our Hgarn end-to-end with a total loss function: $\\mathcal {L}=\\lambda _{L}\\cdot \\mathcal {L}_{L}+\\lambda _{C}\\cdot \\mathcal {L}_{C}$ $\\lambda _{L}$ and $\\lambda _{C}$ are hyperparameters that trade off different loss terms."
],
[
"Experiments",
"In this section, we compare our Hgarn with existing SOTAs on two real-world LBSN check-in datasets."
],
[
"Datasets",
"We adopt two LBSN datasets [38] containing Foursquare check-in records in New York City (NYC) and Tokyo (TKY) from April 12, 2012 to February 16, 2013, including 227,428 check-ins for NYC and 573,703 check-ins for TKY.",
"The location distributions of NYC and TKY datasets are shown in Figure REF in the appendix.",
"Users and locations with less than 10 records are removed following previous work, and after cleaning, NYC and TKY have 308 and 286 activities, respectively.",
"We divide the data into a training and testing sets in a ratio of 8:2, following the setting of [42].",
"Key data summary statistics are listed in Table REF .",
"Table: Main performance comparisons of different methods in two real-world datasets."
],
[
"Baselines & Experimental Details",
"The following baseline methods are considered in experiments: [leftmargin=*] Mc [28]: Mc is a widely used sequential prediction approach which models transition patterns based on visited locations.",
"Strnn [21]: Strnn is an Rnn-based model that incorporates the spatial-temporal contexts by leveraging transition matrices.",
"DeepMove [9]: DeepMove uses attention mechanisms and an Rnn module to capture users' long- and shot-term preferences.",
"Lstpm [30]: Lstpm introduces a non-local network and a geo-dilated Lstm to model users' long- and short-term preferences.",
"Flashback [36]: Flashback is an Rnn-based model that leverages spatial and temporal intervals to compute an aggregated hidden state from past hidden states for prediction.",
"Plspl [35]: Plspl incorporates activity information to learn user preferences and utilizes two Lstms to capture long-term and short-term preferences.",
"PG2Net [16]: PG2Net learns users' group and personalized preferences with spatial-temporal dependencies and attention-based Bi-Lstm.",
"Gcdan [8]: Gcdan leverages graph convolution to learn spatial-temporal representations and use dual-attention to model the sequential dependencies.",
"Cslsl [14]: Cslsl employs multi-task learning to model decision logic and two Lstms to capture long- and short-term preferences.",
"Graph-Flashback [27]: Graph-Flashback adds Gcn to Flashback to enrich learned transition graph representations constructed based on defined similarity functions over embeddings from the existing Knowledge Graph Embedding method.",
"For all baselines and our method, we adopt two commonly-employed metrics in prior works: Rec@K (Recall) and NDCG@K (Normalized Discounted Cumulative Gain).",
"Specific formulas of these two metrics are defined as: $Recall@K=\\frac{1}{|U|}\\sum _{u \\in U}\\frac{|l_{u}^{|R_{u}|} \\cap \\widehat{l_{u, K}^{|R_{u}|}}|}{|l_{u}^{|R_{u}|}|}$ and $NDCG@K=\\frac{1}{|U|}\\sum _{u \\in U}\\sum _{i=1}^{K}\\frac{|l_{u}^{|R_{u}|} \\cap \\widehat{l_{u, i}^{|R_{u}|}}|}{\\log (i+1)}$ , where $\\widehat{l^{\\cdot }_{u, k}}$ indicates the top $k$ predicted locations.",
"We port all the baselines to our run time environment for fair comparisons based on their open-source codes.",
"We carefully tuned their hyperparameters to get the best results.",
"Additionally, unlike previous works that only evaluate overall model performances (under the main setting), we also conduct experiments under the recurring and explorative settings for more comprehensive performance evaluation against the existing SOTAs.",
"For the main and recurring settings, we choose $K=\\lbrace 1, 5, 10\\rbrace $ for evaluation.",
"As the performance is generally poorer under the explorative setting, we set $K=\\lbrace 10, 20\\rbrace $ .",
"For the choice of hyperparameters, we set both $\\lambda _{L}$ and $\\lambda _{C}$ to 1, $\\lambda _{r}$ to 0.6 for both datasets.For embedding dimensions, we set $d=200$ , $d^{u}=10$ , $d^{t}=30$ , $d^{g}=50$ and the dimension of encoders' hidden states are set to 600.",
"Detailed reproducibility information can be found in supplementary materials."
],
[
"Main Results",
"Table REF shows the main performance comparisons of different methods for the next location prediction.",
"Our Hgarn achieves state-of-the-art performances on all metrics for both datasets.",
"Specifically, Hgarn outperforms the best baseline approach by 12-19% on Recall@$K$ and NDCG@$K$ for NYC, and 11-20% for TKY.",
"Its advantages become more significant as $K$ increases, validating the effectiveness of the hierarchical graph modeling and MaHec label for the next location prediction task.",
"Table: Results under the recurring setting.In addition, we also evaluate different models separately in the recurring and explorative settings.",
"In the recurring setting, based on results in Table REF , our model shows an improvement of (10.8%, 23.4%, 31.6%, 10.8%, 19.1%, 20.1%) and (9.4%, 19.2%, 19.7%, 9.4%, 16.5%, 16.9%) on all metrics (R@1, R@5, R@10, N@1, N@5, N@10) compared to the best baseline on the two datasets.",
"Hgarn outperforms all baselines significantly in the recurring setting.",
"Table: Results under the explorative setting.As shown in Table REFThe Mc's results are all zeros and thus deleted from the table., the overall results in the explorative setting are much lower than those in the main and recurring settings, which is intuitive because of the inherent difficulty of predicting unseen locations.",
"A possible approach to improving the prediction in the explorative setting is to model the dependencies between locations.",
"In addition, due to the larger number of locations in the TKY dataset, the hierarchical graph modeling may introduce noise, making our model less effective in ranking the predicted locations.",
"The above hypotheses may also be why our model performs better in Recall but are not consistently better in NDCG."
],
[
"Ablation Study",
"To study each component of Hgarn, we conduct an ablation study considering the following four variants of Hgarn: (1) Hgarn w/o HGat contains only the recurrent encoder-decoder module for the next location prediction.",
"(2) Hgarn w/o AGat: This variant's hierarchical graph attention module contains only the location layer and corresponding graph attention networks.",
"(3) Hgarn w/o Res is the variant that removes the residual connection of the recurrent encoder-decoder module.",
"(4) Hgarn w/o MaHec is the variant that leverages original labels to optimize our model.",
"The ablation study results are shown in Figure REF .",
"It is found that all metrics get improved as more components are included.",
"The gradual increase in prediction results from Hgarn w/o HGat to Hgarn w/o AGat to Hgarn is a fine-grained demonstration of the effectiveness of Gat in each layer of the hierarchical graph.",
"In addition, the model's performance improvements by adding the MaHec label is significant when $K$ are large, verifying its effectiveness under simplicity."
],
[
"Hyperparameter Sensitivity Analysis",
"We further study the sensitivity of a few key parameters by varying each parameter while keeping others constant.",
"(1) $D^{h}$ affects the location dependencies, too high or too low will lead to decreased performances.",
"The best results were obtained at $D^{h}=1$ km on NYC and optimal at $D^{h}=0.1$ km on TKY, this is probably because the geographic space and the distance between locations are larger in NYC than in TKY, as in Figure REF .",
"(2) $w^{c}$ A affects the model's attention to history locations.",
"The results show an upward and then downward trend as $w^{c}$ rises on both datasets, indicating that both too large and too small dependence on locations in the history trajectory is detrimental to the model's performance.",
"(3) $\\lambda _{r}$ affects how much activity information is fused in predicting the next location.",
"Intuitively, a $\\lambda _{r}$ too high would introduce noise, and too low may result in ineffective utilization of activity information.",
"The results align with our conjecture that the model achieves optimal performance with $\\lambda _{r}$ at 0.6 for both NYC and TKY datasets."
],
[
"How does ",
"To understand the mechanism of MaHec label, we randomly selected two human trajectories from two datasets and visualized the predicted probability difference of “Hgarn” and “Hgarn w/o MaHec”.",
"Figure REF shows the probability change, where the purple line represents the locations in the current user's history trajectory.",
"With MaHec labels, the probability distribution of the next location predicted by the model increases in most of the visited locations, demonstrating that our MaHec labels can effectively guide the model to pay attention to the user's history trajectory when predicting the next location.",
"The mechanism of MaHec can also interpret, to some extent, why the prediction performances of our Hgarn can far exceed that of the baseline methods, especially under the recurring setting."
],
[
"What the Hierarchical Graph learned?",
"Unlike other methods that have difficulty interpreting learned higher-order spatial-temporal dependencies, our Hgarn can somewhat understand the dependencies between activities through the learned Hierarchical Graph.",
"We visualize one attention head of $\\textsc {Gat}_{C}$ 's sliced attention matrix to analyze the learned activity-activity dependencies.",
"In Figure REF , We select four activity pairs to show the related activities and their corresponding attention scores.",
"These activity pairs are intuitive and consistent with common sense human travel, such as the high dependencies between Gyms and Stadiums, Bus Stops and Travel Lounges.",
"These results have important implications for understanding human activity patterns and predicting the next location."
],
[
"Conclusion",
"Both travel behavior theories and empirical evidence suggest that human mobility patterns largely depend on the need to participate in activities at different times of the day.",
"Therefore, it is crucial to consider the latter when modeling the former.",
"In this paper, we propose a Hierarchical Graph Attention Recurrent Network (Hgarn) for activity-aware human mobility prediction.",
"Specifically, Hgarn introduces hierarchical graph attention mechanisms on a constructed hierarchical graph to model time-activity-location dependencies, and employ next activity prediction as an auxiliary task to further improve the main task of the next location prediction.",
"In addition, we propose a simple yet powerful MaHec label that can guide our model to flexibly tradeoff the importance of history locations when predicting future locations.",
"Finally, we perform comprehensive experiments to demonstrate the superiority of Hgarn, this is the first work to evaluate existing models and our Hgarn under recurring and explorative settings to our best knowledge.",
"We find that introducing activity information can effectively improve the model's prediction accuracy.",
"In addition, our results show that the existing models have unsatisfactory prediction performances under the explorative setting.",
"We hope our work could spark the attention of future work on explorative mobility prediction.",
"For future work, we consider adding a ranking-based component to organize high-probability candidate locations and incorporate human travel decision logic modeling to make more interpretable predictions."
],
[
"Appendix",
"In this section, we provide more details, visualizations and experimental results."
],
[
"Experimental Supplementary",
"Figure REF demonstrated the location distribution of NYC and TKY datasets.",
"Figure: Location distributions of NYC and TKY.Figure REF displayed two more probability distribution change cases with the effect of MaHec labels.",
"Figure: More visualizations of predicted locations probaility distributions on two datasets."
],
[
"Introduction's supplementary",
"Given the recent emphasis on modeling contextual information (e.g., spatio-temporal) in mobility prediction approaches, we find little work analyzing changes in model prediction performances other than adding or removing relevant components (i.e., ablation studies) and parameter sensitivities.",
"However, it is well-known that human mobility exhibits two different tendencies, i.e., returning to \"old\" places and exploring \"new\" ones [26].",
"Inspired by this dichotomy, we can define recurring and explorative mobility based on whether a user' next location is seen before, as illustrated in Figure REF .",
"Human mobility trajectories contain both recurring and explorative tendencies, although different users may have different mixtures of the two.",
"When data is sparse, the latter may seem more pronounced, exacerbating the challenge of the next location prediction.",
"Therefore, when evaluating those mobility prediction models, it is necessary to assess their predictive power under both recurring and explorative settings in addition to the main setting.",
"Figure: Activity distribution statistics in New York and Tokyo at two different time periods.Figure REF demonstrated a strong temporal dependency between activities, and we remove TKY's top-1 activity Train Station for better visualization."
],
[
"Reproducibility",
"In this section, to address the reproducibility of our experiments, we provide the specific information about computing devices and detailed hyperparameter settings used in our experiments.",
"All models (our Hgarn and other baselines) with learnable parameters are trained on a desktop with Intel Core i9-10990K CPU @3.7GHz $\\times $ 10, 2666MHz $\\times $ 2 $\\times $ 16GB RAM, GeForce RTX 3080 $\\times $ 2, 500GB SSD.",
"We implement Hgarn based on Pytorch.",
"Parameters of Hgarn are randomly initialized and optimized by the Adam optimizer with a learning rate of 2e-4, decaying by 0.8 with each epoch and we train Hgarn by 80 epochs.",
"For hyperparameter settings, we set $\\lambda _{L}$ =$\\lambda _{C}$ =1, and dimensions $d^{g}=50$ , $d=200$ , $d^{u}=20$ , $d^{t}=30$ .",
"Location and activity encoders have hidden states with dimensions of 600.",
"The above settings remain the same in all experiment settings.",
"For the main and recurring settings of all datasets, we employ 2 attention heads and set the dropout to 0.1, while for the explorative setting of all datasets, we let the number of attention heads be 1 and set the dropout to 0.6 to prevent overfitting.",
"For the NYC main and recurring settings, we set $D^{h}$ to 1, $w^{c}$ to 0.8, and $\\lambda _{r}$ to 0.6, while for the explorative setting, $D^{h}$ , $w^{c}$ , and $\\lambda _{r}$ is set to 0.1, 0.9, and 1, respectively.",
"for the TKY main and recurring settings, $D^{h}$ , $w^{c}$ , and $\\lambda _{r}$ are set to 0.1, 0.6 and 0.6 for TKY's main and recurring, respectively, and 0.1, 1, 0.5 for the explorative setting.",
"(All notations mentioned above, their corresponding explanations could be found in the main text)"
],
[
"Full Numerical Results",
"In this section, we show the complete numerical results for the figures presented in the main part.",
"Table REF contains complete numerical results for $D^{h}$ 's sensitivity experiments.",
"Table REF contains complete numerical results for $\\lambda _{r}$ 's sensitivity experiments.",
"Table REF contains complete numerical results for $w^{c}$ 's sensitivity experiments.",
"Table REF contains complete numerical results for our ablation study.",
"Detailed analysis of the relevant experimental results can be found in the experimental section of the main part.",
"Table: Full results of w c w^{c}'s sensitivity experiments.Table: Full results of D h D^{h}'s sensitivity experiments.Table: Full results of λ r \\lambda _{r}'s sensitivity experiments.Table: Full results of the ablation study."
]
] | 2210.07765 |
[
[
"The Triangle Condition for the Marked Random Connection Model"
],
[
"Abstract We investigate a spatial random graph model whose vertices are given as a marked Poisson process on $\\mathbb{R}^d$.",
"Edges are inserted between any pair of points independently with probability depending on the spatial displacement of the two endpoints and on their marks.",
"Upon variation of the Poisson density, a percolation phase transition occurs under mild conditions: for low density there are finite connected components only, whilst for large density there is an infinite component almost surely.",
"Our focus is on the transition between the low- and high-density phase, where the system is critical.",
"We prove that if the dimension is high enough and the edge probability function satisfies certain conditions, then an infrared bound for the critical connection function is valid.",
"This implies the triangle condition, thus indicating mean-field behaviour.",
"We achieve this result through combining the recently established lace expansion for Poisson processes with spectral estimates."
],
[
"Phase transition in percolation",
"For percolation, each edge of the hypercubic lattice is retained independently with fixed probability $p\\in [0,1]$ .",
"It is well-known that the model undergoes a phase transition: in any dimension $d\\ge 2$ there is a $p_c\\in (0,1)$ such that for $p<p_c$ there is no infinite cluster and if $p>p_c$ there is a unique infinite cluster.",
"Behaviour of the model at (and near) the critical threshold $p_c$ is a major theme in probability theory.",
"Indeed, we have good control on the critical behaviour in two regimes: if $d=2$ , then planarity and the link with complex analysis allows a refined description of the critical behaviour (to some extent only on the triangular lattice), see [43], [44], [5] and references therein; if $d$ is large (at least $d>6$ ), then Hara and Slade [25] established the famous triangle condition, which implies that various critical exponents exist and take on their mean-field values [1], [3], [37], [30], [31], [20], [28], see also [23] for a corresponding result for site percolation.",
"While it is generally believed that $d>6$ is sufficient, current proof techniques only allow the proof for $d\\ge 11$ [11].",
"Already [25] show it for $d>6$ under an additional spread-out assumption.",
"The focus of the present paper is on the second case, which we coin the “mean-field regime”.",
"The triangle condition, and hence the mean-field value of various critical exponents, has also been established for long-range percolation in lower dimension provided that the edge occupation probability is decaying slowly enough [22], [6], [7].",
"The triangle condition for the random connection model, a continuum percolation model where vertices are given as a Poisson point process, was proven recently [21].",
"Similarly, it has been established for non-amenable graphs under varying assumptions [39], [40], [27].",
"All of the above papers crucially rely on the independence of edge occupation.",
"In the present work, we establish the triangle condition for dependent percolation models.",
"Mean-field bounds on some critical exponents were proven recently in the context of Boolean percolation [9], but a proof of the triangle condition in such situations is new."
],
[
"Complex networks",
"Random geometric graphs are versatile stochastic models for real networks.",
"As many real-world networks exhibit certain stylised facts such as scale-free degrees, short distances and geometric clustering, it is clear that mathematical models of complex networks must share such features.",
"While numerous different mathematical models for complex networks exist, it has been pointed out by [13] that the weight-dependent random connection model provides a unified approach that generalises many models for (infinite) geometric random graphs.",
"This is a continuum random graph model whose vertices are given as a Poisson point process on $\\mathbb {R}^d\\times (0,1)$ .",
"If $x,y\\in \\mathbb {R}^d$ and $s,t\\in (0,1)$ , where edges $\\big \\lbrace (x,s),(y,t)\\rbrace $ are inserted with probability given by a connection function $\\varphi ((x,s),(y,t))$ .",
"A standard assumption is that $\\varphi $ is decreasing in the spatial distance $|x-y|$ as well as in both marks $s$ and $t$ .",
"Relevance of this model comes from the fact that different choices for the connection function $\\varphi $ generalise various well-known random graph models such as Boolean percolation, scale-free Gilbert graph, scale-free percolation, soft random geometric graphs, long-range percolation, and spatial preferential attachment graphs.",
"Results proven in the generalised setup are thus valid in great generality.",
"For some choices of $\\varphi $ , these models are known to be robust, i.e., infinite components exist for all edge densities; see [15], [16].",
"Our interest is in regimes where these graphs are non-robust and thus a proper percolation phase transition exist: if $\\varphi $ is small (scaled by a certain parameter) then there is no infinite cluster, but if $\\varphi $ is large, then an infinite cluster does appear.",
"Investigation of such transitions is classical in percolation theory, and of great importance in modern probability.",
"The present paper is the first one to prove an infrared bound for an inhomogeneous spatial percolation model.",
"While such results are well-known for the homogeneous case (i.e., without marks), there are hardly any results for the percolation phase transition for inhomogeneous models.",
"A notable exception is Boolean percolation, where sharp phase transition is known [8] and the mean-field bounds on three critical exponents is identified [9].",
"The key challenge is therefore, to overcome the dependencies imposed by the vertex weights.",
"We prove the infrared bound under specific conditions, and verify these conditions for various concrete examples."
],
[
"The Marked Random Connection Model",
"In this paper we consider a generalisation of the weight-dependent random connection model, which we first describe informally and refer to Section REF for a formal construction.",
"It is for our purposes more convenient to generalise the setup by considering a general measurable mark space $\\mathcal {E}$ and thus define the model on the space $\\mathbb {X} = \\mathbb {R}^d\\times \\mathcal {E}$ .",
"Let $\\nu = {\\rm Leb}\\times \\mathcal {P}$ be a $\\sigma $ -finite measure on $\\mathbb {X}$ where ${\\rm Leb}$ denotes the Lebesgue measure on $\\mathbb {R}^d$ and $\\mathcal {P}$ is a probability measure on $\\mathcal {E}$ .",
"Let $\\lambda >0$ , and then $\\lambda \\nu $ is an intensity measure on $\\mathbb {X}$ .",
"The connection function is given by the measurable and symmetric $\\varphi \\colon \\mathbb {X}^2\\rightarrow \\left[0,1\\right]$ .",
"We furthermore require that $\\varphi $ is translation and reflection invariant in the position components.",
"That is, if we denote $x=\\left(\\overline{x},a\\right)$ and $y=\\left(\\overline{y},b\\right)$ for $\\overline{x},\\overline{y}\\in \\mathbb {R}^d$ and $a,b\\in \\mathcal {E}$ , we require that $\\varphi \\left(x,y\\right) = \\varphi (\\overline{x}-\\overline{y};a,b) = \\varphi (-\\overline{x}+\\overline{y};a,b)$ .",
"We denote by $\\eta $ the Poisson point process with intensity measure $\\lambda \\nu $ (which represents the vertex set of the random graph) and by $\\xi $ the full random graph including additional randomness for the formation of edges.",
"Once we have the concept of two vertices being adjacent, we can consider the concept of two vertices being connected.",
"We say $x,y\\in \\mathbb {X}$ are connected, and write $x\\longleftrightarrow y$ , if they are equal or there exists a path in $\\xi $ connecting the two vertices.",
"By this we mean that there exists a sequence of distinct vertices $x = u_0, u_1, \\ldots , u_k, u_{k+1} = y \\in \\eta $ (with $k \\in \\mathbb {N}_0$ ) such that $u_i \\sim u_{i+1}$ for all $0 \\le i \\le k$ .",
"This definition can also be extended to connectedness on the RCM augmented by the vertex $x\\in \\mathbb {X}$ (resp.",
"$x,y\\in \\mathbb {X}$ ), where $\\left(\\eta ,\\xi \\right)$ is replaced by $\\left(\\eta ^x,\\xi ^x\\right)$ (resp.",
"$\\left(\\eta ^{x,y},\\xi ^{x,y}\\right)$ ).",
"Therefore given a pair of points $x,y\\in \\mathbb {X}$ we can consider the event $\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ , and in turn the probability of this event.",
"We define the pair-connectedness (or two-point) function $\\tau _\\lambda \\colon \\mathbb {X}^2 \\rightarrow \\left[0,1\\right]$ to be $\\tau _\\lambda \\left(x,y\\right) := \\mathbb {P}_\\lambda \\left(x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right).$ Using the connection and pair-connectedness functions $\\varphi ,\\tau _\\lambda \\colon \\mathbb {X}^2 \\rightarrow \\left[0,1\\right]$ as kernel functions, we construct associated linear integral operators.",
"Where $L^2\\left(\\mathbb {X}\\right)$ is the Hilbert space of square integrable functions on $\\mathbb {X}$ , we define the connection operator, $\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ , and the pair-connectedness operator, $\\mathcal {T}_\\lambda \\colon L^2\\left(\\mathbb {X}\\right)\\rightarrow L^2\\left(\\mathbb {X}\\right)$ .",
"For $f\\in L^2\\left(\\mathbb {X}\\right)$ , we have $f \\left(x\\right) = \\int \\varphi (x,y)f(y)\\nu \\left(\\mathrm {d}y\\right), \\qquad \\mathcal {T}_\\lambda f \\left(x\\right) = \\int \\tau _\\lambda (x,y)f(y)\\nu \\left(\\mathrm {d}y\\right),$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .",
"Note that the composition of two of these operators can be seen as a “convolution over $\\mathbb {X}$ \" of the kernel functions.",
"Where we rarely use it, we reserve the $\\star $ notation for straight-forward convolution over $\\mathbb {R}^d$ only.",
"For example, given $\\overline{x}\\in \\mathbb {R}^d$ and $a_1,a_2,a_3,a_4\\in \\mathcal {E}$ , $\\left(\\varphi \\left(\\cdot ;a_1,a_2\\right)\\star \\varphi \\left(\\cdot ;a_3,a_4\\right)\\right)\\left(\\overline{x}\\right) = \\int _{\\mathbb {R}^d}\\varphi \\left(\\overline{x}-\\overline{y};a_1,a_2\\right)\\varphi \\left(\\overline{y};a_3,a_4\\right) \\mathrm {d}\\overline{y}.$ We will also consider Fourier transformed versions of these operators.",
"For a Lebesgue-integrable function $g\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ , we define the Fourier transform of $g$ to be $\\widehat{g}(k) = \\int \\text{e}^{ik\\cdot \\overline{x}}g\\left(\\overline{x}\\right) \\mathrm {d}\\overline{x}$ for $k\\in \\mathbb {R}^d$ where $k\\cdot \\overline{x}= \\sum ^d_{j=1}k_j\\overline{x}_j$ denotes the standard inner product on $\\mathbb {R}^d$ .",
"For given marks $a,b\\in \\mathcal {E}$ , we define $\\widehat{\\varphi }(k;a,b)$ and $\\widehat{\\tau }_\\lambda (k;a,b)$ as the Fourier transforms of $\\varphi (\\overline{x};a,b)$ and $\\tau _\\lambda (\\overline{x};a,b)$ respectively.",
"We use these $\\mathcal {E}^2\\rightarrow \\mathbb {R}$ functions to define linear operators on $L^2(\\mathcal {E})$ , the space of square integrable functions on $\\mathcal {E}$ .",
"For each $k\\in \\mathbb {R}^d$ we define the operators $\\widehat{}(k),\\widehat{\\mathcal {T}}_\\lambda (k)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ by their action on $f\\in L^2(\\mathcal {E})$ : $\\left[\\widehat{}(k) f\\right] \\left(a\\right) = \\int \\widehat{\\varphi }(k;a,b)f(b)\\mathcal {P}\\left(\\mathrm {d}b\\right), \\qquad \\left[\\widehat{\\mathcal {T}}_\\lambda (k) f\\right] \\left(a\\right) = \\int \\tau _\\lambda (k;a,b)f(b)\\mathcal {P}\\left(\\mathrm {d}b\\right),$ for $\\mathcal {P}$ -almost every $a$ .",
"Our main aims of this paper are to derive an infrared bound for the two-point function (via its integral operators), and to prove that the so-called triangle condition is satisfied by a large family of models in sufficiently high dimension.",
"For $\\lambda >0$ we define $\\triangle _\\lambda := \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int \\tau _\\lambda (x,u)\\tau _\\lambda (u,v)\\tau _\\lambda (v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right).$ It is relatively simple to show that $\\triangle _\\lambda $ is finite for sub-critical $\\lambda $ and infinite for super-critical $\\lambda $ , but a model is said to satisfy the triangle condition if $\\triangle _\\lambda $ is finite at criticality.",
"An aside: the essential supremum taken in (REF ) is with respect to the measure $\\nu $ on $\\mathbb {X}$ .",
"We will be taking the essential supremum many times in this paper over various spaces, and each space will naturally will have a different measure implicitly associated with it.",
"It should be clear from the context what that measure is."
],
[
"Overview",
"In Section we provide the main results of the paper.",
"We begin by making precise the idea of a critical intensity in Section REF .",
"These propositions are largely independent of the rest of the paper and their proofs can be found in Appendix .",
"The main concern of the paper is in proving the results of Section REF .",
"Given the set of assumption REF , REF , and REF , we prove that the triangle condition is satisfied and we provide an infrared bound on the two-point operator in terms of the adjacency operator.",
"Some examples of models that satisfy these assumptions are provided in Section REF , with the proofs that they satisfy them given in Appendix .",
"Before we begin with the proof proper, we give some preliminaries in Section .",
"First Section REF gives a formal construction of the RCM.",
"Then we have Sections REF and REF , which give standard but important results in probability and linear algebra respectively that we will frequently use in our arguments.",
"The probabilistic lemmas will be familiar to students of point process theory and percolation theory, with the caveat that we require them to apply with the augmented space $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ taking the place of the more usual $\\mathbb {R}^d$ .",
"This adds no complication.",
"The linear algebra results may well be foreign to a probabilistic audience, but will nevertheless be fundamental to our proof.",
"Here we provide a sufficient technical description of the linear operators we construct and the Hilbert spaces upon which they act.",
"Perhaps the most important result we call upon from this section is the Spectral Theorem (Theorem REF ), which allows us to relate bounded self-adjoint linear operators to a multiplication operator by some unitary map.",
"Multiplication operators are far simpler to work with, and we use the theorem many times towards the end of our proof.",
"For operators on finite dimensional vector spaces, the spectral theorem is equivalent to saying that a Hermitian matrix can be diagonalized over $\\mathbb {R}$ .",
"A slightly more unusual lemma in this section is Lemma REF , in which we relate the spectra of the raw operators $$ and $\\mathcal {T}_\\lambda $ to the spectra of the Fourier transformed operators $\\widehat{}(k)$ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ respectively.",
"We conclude Section with Section REF , in which we show that the two-point operator and its derived operators are differentiable with respect to $\\lambda $ in the sub-critical regime and we provide bounds on their derivatives.",
"This will be used to show equicontinuity in a step of the proof of Proposition REF , in which we use a forbidden-region argument to bound a bootstrap function.",
"At a broad description level of the proof, we begin by deriving a finite lace expansion for the sub-critical two-point operator and show that this converges to an Ornstein-Zernike equation (OZE).",
"Via a bootstrap argument this will also prove that various expressions - including $\\triangle _\\lambda $ - are bounded uniformly for subcritical $\\lambda $ , and therefore we will be able to extend these bounds and the OZE to critical $\\lambda $ by using monotone and dominated convergence-like arguments.",
"We will then be able to take full advantage of the operator description to solve the linear OZE to get the critical two-point operator in terms of the adjacency operator and a lace-expansion-coefficient operator.",
"Section deals with finding the finite lace expansion for the sub-critical $\\lambda $ .",
"Formulating this expansion in terms of the relevant functions is essentially the same as for the lace expansion argument appearing in [21] for the unmarked random connection model, except we replace $\\mathbb {R}^d$ with $\\mathbb {X}$ .",
"For this reason we give only a very brief description of the argument - presenting the important lemmas and ideas - and refer the reader to that previous work for the details.",
"The step of formulating the finite lace expansion in terms of operators (done in Section REF ) is then simple.",
"To show that the finite lace expansion converges, we require bounds on various operators arising from it.",
"We begin in Section by bounding the operator norms of these operators by fairly long and complicated integrals of $\\tau _\\lambda $ and $\\varphi $ (see Proposition REF ).",
"We are then able to divide these complicated integrals into smaller and more manageable elementary `diagrams' - it is here that $\\triangle _\\lambda $ first makes an appearance.",
"It is also here that we have our first significant detour from the analogous arguments for the proof for the unmarked models.",
"We not only reformulate the proof strategy to be more straight-forward, direct, and transparent than it as appeared previously, but also reduce the reliance on the assumption analogous to the bounds on the sets $B(x)$ defined in (REF ).",
"In [21] this was used multiple times to bound diagrams arising from both the displaced and non-displaced expansion coefficients, but for us this part of the assumption is only ever used to control one simple diagram in Proposition REF .",
"Unfortunately, it does indeed seem that this particular diagram would need to be controlled via a specific assumption, and the bounds on the sets $B(x)$ appear to be relatively easy to check for many models.",
"We also end up with more elementary diagrams than would appear in the unmarked case.",
"The term we describe here as $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ appearing in Propositions REF and REF would simplify into another diagram in the unmarked case, but the inhomogeneity that the marks introduce means that we cannot necessarily commute terms and it does not simplify in our case.",
"Providing bounds for these more elementary diagrams that are uniform in $\\lambda $ in the subcritical regime is now a much more realistic task, one we achieve in Section by a bootstrap argument.",
"Many of the elementary diagrams contain instances of the two-point function, which is not explicitly known.",
"In Section REF we therefore replace each of them with an explicitly known Green's function $G_{\\mu _\\lambda }$ at the cost of one factor of a bootstrap function $f(\\lambda )$ , and in Section REF we use a bootstrap argument with a forbidden-region argument to prove that $f$ is uniformly bounded for subcritical $\\lambda $ .",
"This then proves that the OZE is satisfied subcritically and that the elementary diagrams - including $\\triangle _\\lambda $ - are uniformly bounded for subcritical $\\lambda $ .",
"Now we have the OZE and uniform bounds for subcritical $\\lambda $ , we want to extend these to critical $\\lambda $ .",
"We begin Section by using monotone convergence to extend our uniform bound on $\\triangle _\\lambda $ to include critical $\\lambda $ .",
"Using similar monotone and dominated convergence ideas we are able to also show that OZEs for the Fourier transformed operators hold at criticality.",
"We are then finally able to find an infrared bound for the two-point operator at criticality by solving these OZEs using our linear operator formalism."
],
[
"Results",
"As we will be working with operators, we will require some concept of size for these operators.",
"This will take a number of forms, as some will have advantages in various scenarios.",
"Let $\\left(\\mathcal {X},\\mu \\right)\\in \\left\\lbrace \\left(\\mathbb {X},\\nu \\right),\\left(\\mathcal {E},\\mathcal {P}\\right)\\right\\rbrace $ , and let $L^2\\left(\\mathcal {X}\\right)\\in \\left\\lbrace L^2\\left(\\mathbb {X}\\right),L^2\\left(\\mathcal {E}\\right)\\right\\rbrace $ denote the corresponding space of square integrable functions.",
"In practice it will be clear which is being used.",
"We will require five various ideas of size.",
"First suppose $H\\colon L^2\\left(\\mathcal {X}\\right) \\rightarrow L^2\\left(\\mathcal {X}\\right)$ is a bounded linear operator.",
"Then the operator norm is defined by $\\left\\Vert H\\right\\Vert _{\\rm op} := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{*{Hf}_2}{*{f}_2},$ where $*{\\cdot }_2$ is the standard $L^2$ -norm on $L^2\\left(\\mathcal {X}\\right)$ .",
"In particular, we have $\\left\\Vert H\\right\\Vert _{\\rm op} = \\sup \\left\\lbrace *{z}:z\\in \\sigma (H)\\right\\rbrace $ , where $\\sigma \\left(H\\right)\\subset \\mathbb {C}$ denotes the spectrum of $H$ .",
"Let ${\\cdot }{\\cdot }$ be the inner product on $L^2\\left(\\mathcal {X}\\right)$ defined by (REF ).",
"If $H$ is also self-adjoint, then ${f}{Hf}\\in \\mathbb {R}$ and we define the spectral supremum $\\mathbb {S}\\left(H\\right) := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{{f}{Hf}}{{f}{f}}.$ In particular, the self-adjointness of $H$ implies that $\\sigma (H)\\subset \\mathbb {R}$ and $\\mathbb {S}\\left(H\\right) = \\sup \\left\\lbrace z:z\\in \\sigma (H)\\right\\rbrace $ .",
"Note that $\\mathbb {S}\\left(\\cdot \\right)$ is not a norm: it is not necessarily non-negative.",
"Now suppose $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ is a bounded linear integral operator with measurable kernel function $h\\colon \\mathcal {X}^2 \\rightarrow \\mathbb {C}$ .",
"Then we have the explicit norms: $\\left\\Vert H\\right\\Vert _{1,\\infty } &:= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}*{h\\left(\\cdot ,y\\right)}_1 = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}\\int *{h\\left(x,y\\right)}\\mu \\left(\\mathrm {d}x\\right), \\\\\\left\\Vert H\\right\\Vert _{2,\\infty } &:= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}*{h\\left(\\cdot ,y\\right)}_2 = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}\\left(\\int *{h\\left(x,y\\right)}^2\\mu \\left(\\mathrm {d}x\\right)\\right)^\\frac{1}{2},\\\\\\left\\Vert H\\right\\Vert _{\\infty ,\\infty } &:= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathcal {X}}*{h\\left(\\cdot ,y\\right)}_\\infty = \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathcal {X}}*{h\\left(x,y\\right)} .$"
],
[
"Critical Intensities",
"Before we move on to our results describing behaviour at and around criticality, we will first explore how we are defining criticality.",
"In our main results in Section REF we use the operator critical intensity $\\lambda _O$ , defined by $\\lambda _O := \\inf \\left\\lbrace \\lambda >0:\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}=\\infty \\right\\rbrace .$ Whilst the divergence of a two-point operator has been used previously to define a critical threshold in random graphs before (for example in [26]), at first glance it may seem to be quite far removed from the standard interpretations of a percolation transition.",
"A variety of transition thresholds for the continuum Boolean disc model are discussed in [38], [17], [8].",
"However, we are able to show that under certain conditions the operator critical intensity coincides with more familiar interpretations.",
"For $\\lambda \\ge 0$ , we define the two functions $\\theta _\\lambda \\colon \\mathcal {E}\\mapsto \\left[0,1\\right]$ and $\\chi _\\lambda \\colon \\mathcal {E}\\mapsto \\left[0,+\\infty \\right]$ as $\\theta _\\lambda (a) &:= \\mathbb {P}_\\lambda \\left(*{\\overline{0},a)}=\\infty \\right),\\\\\\chi _\\lambda (a) &:= \\mathbb {E}_\\lambda \\left[*{\\overline{0},a)}\\right].$ For each $p\\in \\left[1,+\\infty \\right]$ , we define corresponding functions $\\theta ^{(p)}\\colon \\lambda \\mapsto {\\theta _\\lambda }_p$ and $\\chi ^{(p)}\\colon \\lambda \\mapsto {\\chi _\\lambda }_p$ .",
"These suggest the critical intensities $\\lambda _c^{(p)} &:= \\inf \\left\\lbrace \\lambda \\ge 0: {\\theta _\\lambda }_p >0\\right\\rbrace ,\\\\\\lambda _T^{(p)} &:= \\inf \\left\\lbrace \\lambda \\ge 0: {\\chi _\\lambda }_p =\\infty \\right\\rbrace .$ Some relations between these intensities can be swiftly deduced without requiring the RCM to satisfy any special conditions.",
"Their definitions clearly show that $\\theta _\\lambda (a)>0$ implies $\\chi _\\lambda (a)=\\infty $ .",
"Therefore $\\lambda ^{(p)}_c \\ge \\lambda ^{(p)}_T$ for all $p\\in \\left[1,\\infty \\right]$ .",
"If $*{\\theta _\\lambda }_p>0$ for some $p\\in \\left[1,\\infty \\right]$ , then it is non-zero on some $\\mathcal {P}$ -positive set, and therefore $*{\\theta _\\lambda }_q>0$ for all $q\\in \\left[1,\\infty \\right]$ .",
"Therefore $\\lambda _c^{(p)} = \\lambda _c^{(1)}$ for all $p\\in \\left[1,+\\infty \\right]$ , and we denote this value simply by $\\lambda _c$ .",
"By Jensen's inequality, $1\\le p_1 \\le p_2 \\le \\infty $ implies the ordering $*{\\chi _\\lambda }_1 \\le *{\\chi _\\lambda }_{p_1} \\le *{\\chi _\\lambda }_{p_2} \\le *{\\chi _\\lambda }_\\infty $ .",
"Therefore we have $\\lambda _T^{(1)} \\ge \\lambda ^{(p_1)}_T \\ge \\lambda ^{(p_2)}_T \\ge \\lambda ^{(\\infty )}_T$ .",
"With some of the techniques developed in this paper it is possible to relate $\\lambda _O$ to the intensities $\\lambda ^{(1)}_T$ , $\\lambda ^{(\\infty )}_T$ , and $\\lambda _c$ described here.",
"Proposition 2.1 The critical intensities satisfy $\\lambda ^{(\\infty )}_T \\le \\lambda _O \\le \\lambda ^{(1)}_T \\le \\lambda _c$ .",
"If $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi (\\overline{x};a,b)\\mathrm {d}\\overline{x}<\\infty ,$ then we also have $\\lambda ^{(p)}_T = \\lambda _O$ for all $p\\in \\left[1,\\infty \\right]$ .",
"This proof of Proposition REF uses Mecke's equation, Lemma REF , Lemma REF , and some simple probabilistic ideas.",
"The details can be found in Appendix .",
"The coincidence of critical intensities for percolation in marked random connection models has also been considered by [35], [17], [8] in the specific case of a family of Boolean defect models, including Boolean disc models with random bounded radii.",
"In our terminology they introduced a new mark, $\\dagger $ , which forms connections as if it had radius 0, and proved that $\\theta _\\lambda \\left(\\dagger \\right) =0$ if and only if $\\chi _\\lambda \\left(\\dagger \\right) < \\infty $ .",
"Therefore if we define $\\lambda _c\\left(\\dagger \\right) := \\inf \\left\\lbrace \\lambda \\ge 0: \\theta _\\lambda \\left(\\dagger \\right) >0\\right\\rbrace , \\qquad \\lambda _T\\left(\\dagger \\right) := \\inf \\left\\lbrace \\lambda \\ge 0: \\chi _\\lambda \\left(\\dagger \\right) = \\infty \\right\\rbrace ,$ this gives $\\lambda _c\\left(\\dagger \\right) = \\lambda _T\\left(\\dagger \\right)$ .",
"The following Proposition uses this to prove that for the Boolean disc model with bounded radii, all the critical intensities we have considered are equal.",
"Proposition 2.2 In the case of the Boolean disc model with bounded radii, $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right) = \\lambda _c = \\lambda _O = \\lambda _T^{(p)},$ for all $p\\in \\left[1,\\infty \\right]$ .",
"To prove Proposition REF we show $\\lambda _T\\left(\\dagger \\right) \\le \\lambda ^{(p)}_T$ by considering points in $\\eta $ adjacent to the root point with mark $\\dagger $ .",
"The details can be found in Appendix ."
],
[
"Two-Point Operator",
"The majority of this paper is concerned with describing the two-point function and the associated two-point operator.",
"Our results will require Assumption REF described below to hold.",
"This can be considered to be a generalisation of the “finite variance\" models described by [21] - extending to cases with multiple marks.",
"HAssumption H We require $\\varphi $ to satisfy the following properties: For all dimensions $d$ we have $\\mathbb {S}\\left(\\widehat{}(0)\\right) < \\infty $ , and there exists a $d$ -independent constant $C>0$ such that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } &\\le C \\mathbb {S}\\left(\\widehat{}(0)\\right),\\\\\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } &\\le C \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ There exist $d$ -independent constants $C_1\\in \\left(0,1\\right)$ and $C_2>0$ such that $\\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(0\\right)\\right)} \\le C_1 \\vee \\left(1 - C_2{k}^2\\right).$ There exists a function $g\\colon \\mathbb {N} \\rightarrow \\mathbb {R}_{\\ge 0}$ with the following three properties.",
"Firstly, that $g(d)\\rightarrow 0$ as $d\\rightarrow \\infty $ .",
"Secondly, that $\\mathbb {S}\\left(\\widehat{}(0)\\right)^{-3}\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6)\\right)\\left(\\overline{x}\\right) \\le g(d).$ Thirdly, that the family of sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ given by $B\\left(x\\right) := \\left\\lbrace y\\in \\mathbb {X}: \\mathbb {S}\\left(\\widehat{}(0)\\right)^{-2}\\int \\varphi \\left(x,u\\right)\\varphi \\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right) > g\\left(d\\right)\\right\\rbrace $ satisfy $\\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\nu \\left(B\\left(x\\right)\\right) \\le g\\left(d\\right)$ .",
"Remark 2.3 We have some remarks regarding these assumptions.",
"By Jensen's inequality, the condition (REF ) in REF also implies that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty } \\le C \\mathbb {S}\\left(\\widehat{}(0)\\right).$ We will often use these bounds (the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ bound in particular) more directly than (REF ) itself.",
"From (REF ) and (REF ) we can use a supremum bound to deduce that analogous bounds hold for the convolution of $m$ adjacency functions (for $m\\ge 3$ ) with corresponding bound $C^{m-3}g(d)$ .",
"The sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ appearing in REF are all measurable - this follows from the measurability of $\\varphi $ .",
"Furthermore, the symmetry of $\\varphi $ implies that $y\\in B\\left(x\\right)\\iff x\\in B\\left(y\\right)$ .",
"In stating our theorems, we will use a parameter $\\beta =\\beta (\\varphi , d)$ that depends upon $g(d)$ .",
"The definition depends upon the asymptotic properties of $g(d)$ in that $\\beta (d) := {\\left\\lbrace \\begin{array}{ll}g(d)^{\\frac{1}{4} -\\frac{3}{2d}}d^{-\\frac{3}{2}} &: \\lim _{d\\rightarrow \\infty }g(d)\\rho ^{-d}\\Gamma \\left(\\frac{d}{2}+1\\right)^2 = 0 \\qquad \\forall \\rho >0,\\\\g(d)^\\frac{1}{4}&: \\text{otherwise}.\\end{array}\\right.",
"}$ That is, $\\beta (d) = g(d)^\\frac{1}{4}$ unless $g(d)$ approaches zero particularly quickly.",
"The parameter $\\beta $ will reappear many times in this paper - often in the case where $\\beta $ is `small.'",
"This corresponds to `large' dimension $d$ , and in particular, where the Landau notation $\\mathcal {O}\\left(\\beta \\right)$ is used the asymptotics are as $d\\rightarrow \\infty $ .",
"Theorem 2.4 Suppose $\\varphi $ and $\\nu $ satisfy the assumptions REF , REF , and REF .",
"Then there exists a $\\lambda $ -independent $d^*>6$ and $C>0$ such that for $\\lambda \\in \\left[0,\\lambda _O\\right]$ there exists an operator $_\\lambda $ with $\\left\\Vert _\\lambda \\right\\Vert _{\\rm op}\\le C\\beta $ and a family of operators $\\left\\lbrace \\widehat{}_{\\lambda }(k)\\right\\rbrace _{k\\in \\mathbb {R}^d}$ with $\\left\\Vert \\widehat{}_{\\lambda }(k)\\right\\Vert _{\\rm op}\\le C\\beta $ , such that for all $d\\ge d^*$ the following statements are satisfied.",
"For $\\lambda \\in \\left[0,\\lambda _O\\right)$ the bounded operator $\\mathcal {T}_\\lambda $ satisfies the Ornstein–Zernike equation: $\\mathcal {T}_\\lambda = + _\\lambda + \\lambda \\mathcal {T}_\\lambda \\left( + _\\lambda \\right).$ For $\\lambda \\in \\left[0,\\lambda _O\\right)$ and $k\\in \\mathbb {R}^d$ , or $\\lambda = \\lambda _O$ and $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ , the bounded operator $\\widehat{\\mathcal {T}}_\\lambda (k)$ satisfies the Ornstein–Zernike equation in Fourier form: $\\widehat{\\mathcal {T}}_\\lambda (k) = \\widehat{}(k) + \\widehat{}_{\\lambda }(k) + \\lambda \\widehat{\\mathcal {T}}_\\lambda (k) \\left( \\widehat{}(k) + \\widehat{}_{\\lambda }(k) \\right).$ Theorem 2.5 If $\\varphi $ and $\\nu $ satisfy the assumptions REF , REF , and REF , and $\\lambda \\in \\left[0,\\lambda _O\\right]$ , then there exist $\\lambda $ -independent $d^*>6$ and $C>0$ such that for all $d\\ge d^*$ $\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)+ C \\beta }{\\mathbb {S}\\left(\\widehat{}\\left(0\\right)\\right) - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\vee 1 \\qquad \\forall k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace .$ Furthermore, we have $\\triangle _\\lambda \\le C\\beta ^2$ and $*{\\theta _{\\lambda }}_p=0$ for all $p\\in \\left[1,\\infty \\right]$ and $\\lambda \\in \\left[0,\\lambda _O\\right]$ .",
"This infrared bound can also be related directly to the two-point function by Lemma REF via $\\int \\widehat{\\tau }_\\lambda \\left(k;a,b\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a, \\mathrm {d}b\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}.$ Remark 2.6 Note that in Theorem REF we say there exists some $d^*>6$ such that the results hold for all $d\\ge d^*$ .",
"If we were willing to say that there exists some $d^*>10$ instead (which would not change the strict logical content of the result), then the proof would be far simpler and Assumptions REF and REF could be weakened.",
"The $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ in (REF ) and () could be replaced with $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ and $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ respectively, and (REF ) could be replaced with $\\left\\Vert ^3\\right\\Vert _{\\infty ,\\infty }\\le g(d)\\mathbb {S}\\left(\\widehat{}(0)\\right)^3$ .",
"We do not pursue this approach because it is expected that a very similar argument to that presented here (with context appropriate changes to the assumptions) would prove the corresponding result for the spread-out model in $d>6$ .",
"For the simpler argument and weaker conditions, we would only get the spread-out result in $d>10$ , and this would be a logically weaker statement."
],
[
"Examples of Models",
"It is natural to ask if there are any models for which $\\varphi $ and $\\nu $ do indeed satisfy REF , REF , and REF .",
"All the models we describe here also satisfy the condition (REF ) for every dimension $d$ .",
"We prove in Appendix that all the models below do satisfy our assumptions."
],
[
"Single Mark Finite Variance Model.",
"Suppose $\\mathcal {E}$ is a singleton, so essentially $\\mathbb {X}=\\mathbb {R}^d$ and $\\nu = {\\rm Leb}$ .",
"Then the finite variance models described in [21] all satisfy our conditions.",
"These include the Poisson blob model (or the spherical Boolean model) in which $\\varphi (\\overline{x}) = {1}_{\\lbrace *{\\overline{x}} \\le r_d\\rbrace }$ where $r_d$ is the radius of the unit volume $d$ -ball, and the Gaussian connection model with $\\smash{\\varphi }(\\overline{x}) = \\left(2\\pi \\right)^{-d/2}\\exp \\big (-\\tfrac{1}{2}*{\\overline{x}}^2\\big )$ ."
],
[
"Double Mark Finite Variance Model.",
"Suppose $\\mathcal {E}= \\lbrace a_1,a_2\\rbrace $ has only two distinct elements with $\\mathcal {P}= p\\delta _{a_1} + q\\delta _{a_2}$ for some $p = 1-q \\in \\left(0,1\\right)$ , and that all the three independent functions $\\overline{x}\\mapsto \\varphi (\\overline{x};a,b)$ (for $a,b\\in \\mathcal {E}$ ) each satisfy the conditions required of a single mark finite variance model.",
"Further suppose that (without loss of generality) $p^2\\varphi (\\overline{x};a_1,a_1) \\ge q^2\\varphi (\\overline{x};a_2,a_2)$ for almost every $\\overline{x}\\in \\mathbb {R}^d$ .",
"Then our conditions are satisfied."
],
[
"Space-Mark Factorisation Model",
"Suppose now that the connection function admits the factorisation $\\varphi (\\overline{x};a,b) = \\overline{\\varphi }(\\overline{x})K(a,b)$ .",
"We require that $\\overline{\\varphi }$ satisfies the conditions of a single mark finite variance model, and that the ($d$ -independent) mark-kernel $K\\colon \\mathcal {E}^2 \\rightarrow \\left[0,1\\right]$ is symmetric and measurable.",
"Note that this means that the integral operator $\\mathcal {K}\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ with kernel function $K$ is a bounded self-adjoint operator with operator norm $\\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}\\le 1$ .",
"We further suppose that $\\mathbb {S}\\left(\\mathcal {K}\\right) = \\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}$ .",
"Then the connection operator is a bounded operator on the mark-space multiplied by a scalar function of spatial position.",
"This formulation places no constraint on the number of marks, permitting even models with uncountably many marks."
],
[
"Marked Multivariate Gaussian Model.",
"Let $\\Sigma \\colon \\mathcal {E}^2\\rightarrow \\mathbb {R}^{d\\times d}$ be a measurable map where for every $a,b\\in \\mathcal {E}$ , $\\Sigma \\left(a,b\\right)$ is itself a symmetric positive definite covariance matrix.",
"We further require that there exists $\\Sigma _{\\max }<\\infty $ and $\\Sigma _{\\min } = \\Sigma _{\\min }(d) > 0$ such that the spectrum $\\sigma (\\Sigma (a,b)) \\subset \\left[\\Sigma _{\\min },\\Sigma _{\\max }\\right]$ for all $a,b\\in \\mathcal {E}$ and dimension $d$ .",
"Then for $\\mathcal {A} = \\mathcal {A}(d)>0$ such that $\\mathcal {A}^2 \\le \\left(2\\pi \\right)^d\\det \\Sigma (a,b)$ for all $a,b\\in \\mathcal {E}$ and $\\limsup _{d\\rightarrow \\infty } \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2} = 0$ , let the connection function be given by $\\varphi (\\overline{x};a,b) = \\mathcal {A}\\left(2\\pi \\right)^{-d/2}\\left(\\det \\Sigma (a,b)\\right)^{-1/2}\\exp \\left(-\\frac{1}{2}\\overline{x}^{\\intercal }\\Sigma (a,b)^{-1}\\overline{x}\\right).$ This class of model includes examples of models with (uncountably) infinitely many marks, and for which the mark and spatial behaviours are truly coupled.",
"For a concrete example, consider $\\mathcal {E}=\\left[\\tfrac{1}{4\\pi },1\\right]$ with $\\Sigma (a,b) = (a+b)1$ and $\\mathcal {A} = 1$ ."
],
[
"Bounded-Volume Boolean Disc Model.",
"Let $\\left(\\left[0,1\\right], \\Sigma _{\\left[0,1\\right]}, \\mathcal {P}\\right)$ be a separable probability space and $\\left\\lbrace R_d\\right\\rbrace _{d\\ge 1}$ be a sequence of measurable functions $R_d\\colon \\left[0,1\\right]\\rightarrow \\mathbb {R}_+$ such that there exist $ 0<c_1<1/\\sqrt{8\\pi \\text{e}}$ and $V^{({\\rm max})}>0$ such that $R^{({\\rm min})}_d:= c_1 d^{\\frac{1}{2}} \\le R_d\\left(a\\right) \\le \\frac{1}{2\\sqrt{\\pi }}\\left(V^{({\\rm max})}\\right)^{\\frac{1}{d}} \\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d} =: R^{({\\rm max})}_d,$ for all $a\\in [0,1]$ .",
"Note that as $d\\rightarrow \\infty $ , Stirling's formula gives that the upper bound $R^{({\\rm max})}_d\\sim \\sqrt{d/8\\pi \\text{e}}$ .",
"It is also required that there exist $\\epsilon >0$ and $c_2>0$ such that $\\mathcal {P}\\left(R_d > \\left(1-\\frac{c_2}{d}\\right)R^{({\\rm max})}_d\\right)>\\epsilon $ for all $d$ .",
"Then let the connection function be defined by $\\varphi _d(\\overline{x};a,b) = {1}\\left\\lbrace *{\\overline{x}} \\le R_d(a) + R_d(b)\\right\\rbrace .$ It is also perhaps worth mentioning a model that does not satisfy condition REF .",
"Such an example would be a Boolean disk model with random radii for which $\\mathcal {P}$ gives positive measure to arbitrarily large radii.",
"No matter the decay rate of $\\mathcal {P}$ for large radii, the presence of the suprema in (REF ) and () ensure that REF cannot be satisfied.",
"Similarly the condition (REF ) is also not satisfied for such a model."
],
[
"Outlook",
"Theorem REF establishes the triangle condition, which is a strong indication of mean-field behaviour.",
"The identification of mean-field critical exponents in the inhomogeneous setting is postponed to future work.",
"We believe that the infrared bound is valid beyond the restricted setting of Hypothesis REF , in particular for the Boolean model with unbounded radii as well as models for complex networks admitting power-law degree graphs (under suitable moment assumptions).",
"However, our current proof does not admit such generalisation, as explained above."
],
[
"Formal Construction of the RCM",
"We choose to construct our RCM in largely the same way as [21], with appropriate replacements of $\\mathbb {R}^d$ with the marked space $\\mathbb {X}$ .",
"As they did, we follow [34] in defining the RCM, $\\xi $ , as an independent edge-marking of a Poisson point process $\\eta $ .",
"Recall $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ .",
"We denote by $\\mathbf {N}\\left(\\mathbb {X}\\right)$ the set of all at most countably infinite subsets of $\\mathbb {X}$ and accompany it with the $\\sigma $ -algebra $\\mathcal {N}(\\mathbb {X})$ generated by the sets $\\lbrace A\\in \\mathbf {N}\\left(\\mathbb {X}\\right):|A \\cap B| = k, B \\in \\Sigma _\\mathbb {X}, k \\in \\mathbb {N}_0\\rbrace $ , where $*{A\\cap B}$ denotes the cardinality of $A\\cap B$ .",
"Then a point process on $\\mathbb {X}$ is a measurable mapping $\\zeta \\colon \\Omega \\rightarrow \\mathbf {N}\\left(\\mathbb {X}\\right)$ for some underlying probability space $\\left(\\Omega ,\\Sigma _\\Omega ,\\mathbb {P}\\right)$ .",
"The intensity measure of such a point process is then the measure on $\\mathbb {X}$ given by $B\\mapsto \\mathbb {E}\\left[*{\\zeta \\cap B}\\right]$ for $B\\in \\Sigma _\\mathbb {X}$ .",
"Let $\\zeta $ be a point process on $\\mathbb {X}$ with a $\\sigma $ -finite intensity measure.",
"Then there exist measurable $\\pi _i\\colon \\mathbf {N} \\left(\\mathbb {X}\\right) \\rightarrow \\mathbb {X}$ , $i\\in \\mathbb {N}$ such that almost surely $\\zeta = \\left\\lbrace \\pi _i\\left(\\zeta \\right): i\\le *{\\zeta \\cap \\mathbb {X}}\\right\\rbrace .$ For a $\\sigma $ -finite non-atomic measure on $\\mathbb {X}$ , say $\\nu $ , a Poisson point process (PPP) on $\\mathbb {X}$ with intensity measure $\\nu $ is a point process $\\zeta $ such that the number of points $*{\\zeta \\cap B}$ is $\\text{Poi}(\\nu (B))$ -distributed for each $B \\in \\Sigma _\\mathbb {X}$ , and the random variables $*{\\zeta \\cap B_1}, \\ldots , *{\\zeta \\cap B_m}$ are independent whenever $B_1, \\ldots , B_m\\in \\Sigma _\\mathbb {X}$ are pairwise disjoint.",
"In this paper we will be considering a PPP $\\eta $ in $\\mathbb {X}$ with intensity measure $\\lambda \\nu $ , where $\\nu ={\\rm Leb}\\otimes \\mathcal {P}$ and $\\lambda >0$ .",
"We write $\\eta =\\left\\lbrace X_i:i\\in \\mathbb {N}\\right\\rbrace $ , where $X_i=\\pi _i\\left(\\eta \\right)$ for $i\\in \\mathbb {N}$ .",
"We now construct the RCM as a (non-Poissionian) point process on the space $\\mathbb {X}^{[2]}\\times \\left[0,1\\right]$ .",
"We denote by $\\mathbb {X}^{[2]}$ the set of all subsets of $\\mathbb {X}$ containing precisely two elements, and each $e\\in \\mathbb {X}^{[2]}$ is then a potential edge of the RCM.",
"Also let $\\left\\lbrace U_{i,j}\\right\\rbrace _{i,j\\in \\mathbb {N}}$ be a double sequence of independent identically distributed random variables uniformly distributed on $\\left[0,1\\right]$ , such that the sequence is independent of the PPP $\\eta $ .",
"We define $\\xi = \\left\\lbrace \\left(\\left\\lbrace X_i,X_j\\right\\rbrace ,U_{i,j}\\right):1\\le i\\le j\\right\\rbrace ,$ where we recall $X_i=\\pi _i(\\eta )$ .",
"Thus $\\xi $ is a random point process on $\\mathbb {X}^{[2]}\\times \\left[0,1\\right]$ , which we interpret as a random marked graph and say that $\\xi $ is an independent edge-marking of $\\eta $ .",
"The independent edge-marking $\\xi $ of $\\eta $ can then be related by a deterministic functional, $\\Gamma _\\varphi $ , to a random graph on $\\mathbb {X}$ by defining its vertex and edge sets: $V(\\Gamma _\\varphi (\\xi )) = \\eta = \\lbrace X_i:i\\in \\mathbb {N}\\rbrace ,\\qquad E(\\Gamma _\\varphi (\\xi )) = \\lbrace \\lbrace X_i,X_j\\rbrace :U_{i,j} \\le \\varphi (X_i,X_j), 1\\le i\\le j\\rbrace .$ Since the sequence $U_{i,j}$ is independent of the Poisson process $\\eta $ , the ordering in (REF ) does not change the distribution of the resulting random graph.",
"This interpretation of the RCM also allows for the introduction of additional points.",
"This will be required many times in this paper, not least in our definition of the two-point function itself.",
"For additional points $x_1,x_2\\in \\mathbb {X}$ , the point process $\\eta $ is augmented to make $\\eta ^{x_1}:=\\eta \\cup \\lbrace x_1\\rbrace ,\\quad \\eta ^{x_1,x_2}:=\\eta \\cup \\lbrace x_1,x_2\\rbrace .$ To extend these point processes on $\\mathbb {X}$ to a RCM, we extend the sequence $\\left\\lbrace U_{m,n}\\right\\rbrace _{m,n \\ge 1}$ to a sequence $\\left\\lbrace U_{m,n}\\right\\rbrace _{m,n \\ge -1}$ of independent random variables uniformly distributed on $[0,1]$ , independent of the Poisson process $\\eta $ .",
"We then define $\\xi ^{x_1,x_2}$ on $\\mathbb {X}^{[2]}\\times [0,1]$ as $\\xi ^{x_1,x_2}:=\\lbrace (\\lbrace X_i,X_j\\rbrace ,U_{i,j}):-1\\le i\\le j\\rbrace ,$ where $X_0 = x_1$ and $X_{-1} = x_2$ .",
"It is straightforward to define $\\xi ^{x_1, \\ldots , x_m}$ for arbitrary $m \\ge 3$ by the same idea, and to define $\\xi ^{x_1}$ by removing all (marked) edges incident to $x_2$ from $\\xi ^{x_1,x_2}$ .",
"We assume implicitly that the augmented points are properly coupled such that, e.g., $\\xi ^{x_1}$ obtained in this way from $\\xi ^{x_1,x_2}$ (by removing all edges adjacent to $x_2$ ) is the same as if obtained from $\\xi ^{x_1,x_3}$ (by removing all edges adjacent to $x_3$ ); see [21] for further details."
],
[
"Probabilistic Lemmas",
"We introduce here a number of important lemmas that are standard in point process theory or percolation theory."
],
[
"The Mecke Equation",
"An important part of our analysis will be a version of the Mecke equation - for a discussion see [33].",
"This result allows us to make sums over the random sum of points in $\\eta $ more tractable.",
"Given $m\\in \\mathbb {N}$ and a measurable function $f\\colon \\mathbf {N}\\left((\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1]\\right) \\times \\mathbb {X}^m \\rightarrow \\mathbb {R}_{\\ge 0}$ , the Mecke equation for $\\xi $ states that $\\mathbb {E}_\\lambda \\left[ \\sum _{\\vec{x} \\in \\eta ^{(m)}} f(\\xi , \\vec{x})\\right] = \\lambda ^m \\int \\mathbb {E}_\\lambda \\left[ f\\left(\\xi ^{x_1, \\ldots , x_m}, \\vec{x}\\right)\\right] \\nu ^{\\otimes m}\\left(\\mathrm {d}\\vec{x}\\right), $ where $\\vec{x}=(x_1,\\ldots ,x_m)$ , $\\eta ^{(m)}=\\lbrace (x_1,\\ldots ,x_m): x_i \\in \\eta , x_i \\ne x_j \\text{ for } i \\ne j\\rbrace $ , and $\\nu ^{\\otimes m}$ is the $m$ -product measure of $\\nu $ on $\\mathbb {X}^m$ .",
"We will only need (REF ) for $m\\le 3$ , and largely only use it for $m=1$ ."
],
[
"The Margulis-Russo Formula",
"An often used and very important tool in (discrete) percolation theory, the Margulis-Russo formula will be widely used in our analysis as well.",
"The version we use follows from a slight adjustment to the more general result given in [34].",
"Writing $\\mathbf {N}:=\\mathbf {N}\\left((\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1]\\right)$ , let $\\Lambda \\subset \\mathbb {X}$ be $\\nu $ -finite, $\\zeta \\in \\mathbf {N}$ , and define $\\zeta _\\Lambda := \\lbrace (\\lbrace (x,v),(y,w)\\rbrace ,u)\\in \\zeta :\\lbrace x,y\\rbrace \\subseteq \\Lambda \\rbrace .",
"$ We call $\\zeta _\\Lambda $ the restriction of $\\zeta $ to $\\Lambda $ .",
"We say that $f\\colon \\mathbf {N}\\rightarrow \\mathbb {R}$ lives on $\\Lambda $ if $f(\\zeta ) = f(\\zeta _\\Lambda )$ for every $\\zeta \\in \\mathbf {N}$ .",
"Assume that there exists a $\\nu $ -finite set $\\Lambda \\subset \\mathbb {X}$ such that $f$ lives on $\\Lambda $ .",
"Moreover, assume that there exists $\\lambda _0>0$ such that $\\mathbb {E}_{\\lambda _0} \\left[*{f(\\xi )}\\right] < \\infty $ .",
"Then the Margulis-Russo formula states that, for all $\\lambda \\le \\lambda _0$ , $\\frac{\\partial }{\\partial \\lambda } \\mathbb {E}_\\lambda [f(\\xi )] = \\int _\\Lambda \\mathbb {E}_\\lambda \\left[f(\\xi ^x) - f(\\xi )\\right] \\nu \\left(\\mathrm {d}x\\right).",
"$ The result in [34] requires that $\\mathbb {X}=\\mathbb {R}^d$ and that $\\Lambda \\subset \\mathbb {R}^d$ is compact, but [32] allows us to make the necessary adjustments to permit $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ ."
],
[
"The BK Inequality",
"The idea of the BK inequality is that we acquire a simple upper bound for the probability that two increasing events happen, and that they do so on disjoint subsets of the space.",
"For example, we may want to bound the probability that there exist two disjoint paths (up to the start and end points) in $\\xi $ between two given points.",
"The BK inequality we present here is slightly more general than the usual BK inequality in that the point process $\\xi $ is augmented with independent random variables.",
"It differs only slightly from the version appearing in [21].",
"For notational clarity, we write $\\mathbf {N}:=\\mathbf {N}((\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1])$ and denote the $\\sigma $ -algebra on $\\mathbf {N}$ by $\\mathcal {N}$ .",
"We call a set $E\\subset \\mathbf {N}$ increasing if $\\mu \\in E$ implies $\\nu \\in E$ for each $\\nu \\in \\mathbf {N}$ with $\\mu \\subseteq \\nu $ .",
"Let $(\\mathbb {Y}_1,\\mathcal {Y}_1)$ , $(\\mathbb {Y}_2,\\mathcal {Y}_2)$ be two measurable spaces.",
"We say that a set $E_i\\subset \\mathbf {N}\\times \\mathbb {Y}_i$ is increasing if $\\mathbb {E}^z_i:=\\lbrace \\mu \\in \\mathbf {N}:(\\mu ,z)\\in E_i\\rbrace $ is increasing for each $z\\in \\mathbb {X}_i$ .",
"For Borel $\\Lambda \\in \\mathcal {B}\\left(\\mathbb {R}^d\\right)$ and $\\mu \\in \\mathbf {N}$ , we define $\\mu _\\Lambda $ , the restriction of $\\mu $ to all edges completely contained in $\\left(\\Lambda \\times \\mathcal {E}\\right)\\times \\mathbb {M}$ , analogously to (REF ).",
"Also let $\\mathcal {R}$ denote the ring of all finite unions of half-open $d$ -dimensional rectangles in $\\mathbb {R}^d$ with rational coordinates.",
"Then for increasing $E_i\\in \\mathcal {N}\\otimes \\mathcal {Y}_i$ , we define $E_1 \\circ E_2 := \\left\\lbrace (\\mu ,z_1,z_2)\\in \\mathbf {N}\\times \\mathbb {Y}_1\\times \\mathbb {Y}_2: \\exists K_1,K_2 \\in \\mathcal {R} \\text{ s.t.~} \\right.\\\\\\left.K_1\\cap K_2 = \\varnothing , (\\mu _{K_1},z_1)\\in E_1,(\\mu _{K_2},z_2)\\in E_2 \\right\\rbrace .",
"$ A set $E_i\\in \\mathcal {N}\\otimes \\mathcal {Y}_i$ lives on $\\Lambda $ if $1_{E_i}(\\mu ,z)=1_{E_i}(\\mu _\\Lambda ,z)$ for each $(\\mu ,z)\\in \\mathbf {N}\\times \\mathbb {Y}_i$ .",
"We consider random elements $W_1,W_2$ of $\\mathbb {Y}_1$ and $\\mathbb {Y}_2$ , respectively, and assume that $\\xi $ , $W_1$ , and $W_2$ are independent.",
"Theorem 3.1 (BK inequality) Let $E_1\\in \\mathcal {N}\\otimes \\mathcal {Y}_1$ and $E_2\\in \\mathcal {N}\\otimes \\mathcal {Y}_2$ be increasing events that live on $\\Lambda \\times \\mathcal {E}$ for some bounded set $\\Lambda \\in \\mathcal {B}\\left(\\mathbb {R}^d\\right)$ .",
"Then $\\mathbb {P}_\\lambda ((\\xi ,W_1,W_2)\\in E_1 \\circ E_2) \\le \\mathbb {P}_\\lambda ((\\xi ,W_1)\\in E_1)\\mathbb {P}_\\lambda ((\\xi ,W_2)\\in E_2).$ A proof for this theorem can be found in [21].",
"The presence of the marked space $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ rather than just $\\mathbb {R}^d$ makes no difference to the validity of their proof.",
"It is worth noting that the operation $\\circ $ is commutative and associative, which allows repeated application of the inequality."
],
[
"The FKG Inequality",
"In contrast to the BK inequality, the FKG inequality gives us a simple lower bound on the probability of two increasing events occurring.",
"Given two increasing and integrable functions $f,g$ on $(\\mathbb {X}\\times \\mathbb {M})^{[2]}\\times [0,1]$ , we have $\\mathbb {E}_\\lambda \\left[f(\\xi ) g(\\xi )\\right] = \\mathbb {E}_\\lambda \\left[ \\mathbb {E}_\\lambda [f(\\xi )g(\\xi ) \\mid \\eta ] \\right] \\ge \\mathbb {E}_\\lambda \\left[ \\mathbb {E}_\\lambda \\left[f(\\xi )\\mid \\eta \\right] \\; \\mathbb {E}_\\lambda \\left[g(\\xi )\\mid \\eta \\right] \\right]\\ge \\mathbb {E}_\\lambda \\left[f(\\xi )\\right] \\; \\mathbb {E}_\\lambda \\left[g(\\xi )\\right].",
"$ The first inequality was obtained by applying FKG to the random graph conditioned to have $\\eta $ as its vertex set, the second inequality by applying FKG for point processes (see [33] for details)."
],
[
"Linear Operator Lemmas",
"We shall formulate our problem as one of linear operators acting on Hilbert spaces.",
"We will want to define analogous objects on the full marked space $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ and on the mark space $\\mathcal {E}$ itself.",
"Let $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)\\in \\left\\lbrace \\left(\\mathbb {X},\\Sigma _\\mathbb {X},\\nu \\right),\\left(\\mathcal {E},\\Sigma _\\mathcal {E},\\mathcal {P}\\right)\\right\\rbrace $ , and $L^2\\left(\\mathcal {X}\\right)\\in \\left\\lbrace L^2\\left(\\mathbb {X}\\right),L^2\\left(\\mathcal {E}\\right)\\right\\rbrace $ correspondingly.",
"Given the measure space $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ , we consider the Banach space of square-integrable functions $L^2(\\mathcal {X}) = L^2\\left(\\mathcal {X},\\Sigma _{\\mathcal {X}},\\mu \\right)$ .",
"We augment this with the inner product defined by ${f}{g} := \\int \\overline{f\\left(x\\right)}g\\left(x\\right) \\mu \\left(\\mathrm {d}x \\right),$ for all $f,g\\in L^2(\\mathcal {X})$ , where $\\overline{f(x)}$ is the complex conjugate of $f(x)$ .",
"With this inner product, $L^2(\\mathcal {X})$ is a Hilbert space.",
"The measure space $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ is separable if there exists a countable family $\\left\\lbrace E_k\\right\\rbrace _{k=1}^\\infty \\subset \\Sigma _\\mathcal {X}$ such that given $E\\in \\Sigma _\\mathcal {X}$ with $\\mu (E)<\\infty $ there exists an $E$ -dependent subsequence $\\lbrace n_k\\rbrace _{k=1}^\\infty $ such that $\\mu \\left(E\\Delta E_{n_k}\\right) \\rightarrow 0, \\qquad \\text{ as }k\\rightarrow \\infty ,$ where $\\Delta $ here is the symmetric difference.",
"A well-known example of a separable measure space is $\\left(\\mathbb {R}^d,\\mathfrak {B},{\\rm Leb}\\right)$ - where $\\mathfrak {B}$ is the Borel $\\sigma $ -algebra - and therefore if the probability measure space $\\left(\\mathcal {E},\\Sigma _{\\mathcal {E}},\\mathcal {P}\\right)$ is separable then so is the product measure space $\\left(\\mathbb {X},\\Sigma _\\mathbb {X},\\nu \\right)$ .",
"A topological space is called separable if there exists a countable dense subset.",
"If the measure space $\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ is separable, then the function spaces $L^p\\left(\\mathcal {X}\\right) = L^p\\left(\\mathcal {X},\\Sigma _\\mathcal {X},\\mu \\right)$ with $p\\in \\left[1,\\infty \\right)$ are all separable (see, for example, [42]).",
"Since we have assumed that $\\left(\\mathcal {E},\\Sigma _\\mathcal {E},\\mathcal {P}\\right)$ is separable, we know that both $L^2(\\mathbb {X})$ and $L^2(\\mathcal {E})$ are separable Hilbert spaces.",
"The inner product for $L^2(\\mathcal {X})$ allows us to define the $L^2$ -norm for $f\\in L^2(\\mathcal {X})$ : $*{f}_2 = {f}{f}^{\\frac{1}{2}}.$ This allows us to define a norm on the space of bounded linear operators on $L^2(\\mathcal {X})$ .",
"The space of bounded linear operators on a Banach space is itself a Banach space when augmented with the operator norm defined by $\\left\\Vert H\\right\\Vert _{\\rm op} := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{*{Hf}_2}{*{f}_2},$ for a linear operator $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ .",
"Since $H$ is linear, the supremum can equivalently be taken over $f\\in L^2(\\mathcal {X})$ such that $*{f}_2=1$ .",
"Note that $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ is sub-multiplicative, in the sense that for linear operators $G,H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ we have $\\left\\Vert GH\\right\\Vert _{\\rm op} \\le \\left\\Vert G\\right\\Vert _{\\rm op}\\left\\Vert H\\right\\Vert _{\\rm op}.$ Given a linear operator $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ on Hilbert spaces, the adjoint of $H$ is the operator $H^\\dagger \\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ such that ${f}{Hg} = {H^\\dagger f}{g}$ for all $f,g\\in L^2(\\mathcal {X})$ .",
"An operator is self-adjoint if $H=H^\\dagger $ .",
"It is worth noting that a bounded integral operator $H$ with kernel function $h$ is self-adjoint if and only if $h(x,y) = \\overline{h(y,x)}$ for $\\mu $ -a.e.",
"$x,y\\in \\mathcal {X}$ .",
"If $h$ is real valued, this is equivalent to $h$ being symmetric on such a set of $x,y$ .",
"A simple calculation shows that if $H$ is bounded and self-adjoint, then ${f}{Hf}\\in \\mathbb {R}$ for all $f\\in L^2(\\mathcal {X})$ .",
"This allows us to define the following object.",
"Given a bounded and self-adjoint linear $H\\colon L^2(\\mathcal {X})\\rightarrow L^2(\\mathcal {X})$ , define $\\mathbb {S}\\left(H\\right) := \\sup _{f\\in L^2(\\mathcal {X}):f\\ne 0}\\frac{{f}{Hf}}{{f}{f}}.$ Note that, in contrast to $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ , the spectral supremum $\\mathbb {S}\\left(\\cdot \\right)$ is not a norm - it is possible for it to be negative.",
"The objects $\\left\\Vert H\\right\\Vert _{\\rm op}$ and $\\mathbb {S}\\left(H\\right)$ can also be understood via the spectrum of $H$ .",
"The spectrum, denoted $\\sigma (H)$ , is the set of $z\\in \\mathbb {C}$ such that $z1- H$ does not have a bounded linear inverse.",
"For linear operators on finite dimensional vector spaces this coincides with the set of eigenvalues of $H$ .",
"If $H$ is self-adjoint, then $\\sigma (H)\\subset \\mathbb {R}$ .",
"We can then express $\\left\\Vert H\\right\\Vert _{\\rm op}$ and $\\mathbb {S}\\left(H\\right)$ in terms of the spectrum: $\\left\\Vert H\\right\\Vert _{\\rm op} = \\sup \\left\\lbrace *{z}:z\\in \\sigma (H)\\right\\rbrace ,\\qquad \\mathbb {S}\\left(H\\right) = \\sup \\left\\lbrace z:z\\in \\sigma (H)\\right\\rbrace .$ Note that for self-adjoint operators the operator norm is equal to the spectral radius of the operator.",
"While $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ and $\\mathbb {S}\\left(\\cdot \\right)$ are very important properties of the operators we will be considering, they are not necessarily simple to compute for an integral operator given its kernel function.",
"Recall that in (REF )-() we defined the three integral operator norms $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ .",
"These have the advantage that they are explicitly defined in terms of the kernel function.",
"We now provide some results regarding the four norms (plus $\\mathbb {S}\\left(\\cdot \\right)$ ), including relations between some of them.",
"Lemma 3.2 (Bounds on the Operator Norm) Let $H\\colon L^2\\left(\\mathcal {X}\\right)\\rightarrow L^2\\left(\\mathcal {X}\\right)$ be a bounded self-adjoint integral operator.",
"Then we can bound the operator norm: $*{\\mathbb {S}\\left(H\\right)} \\le \\left\\Vert H\\right\\Vert _{\\rm op} \\le \\left\\Vert H\\right\\Vert _{1,\\infty }.$ Furthermore, if $\\mathcal {X}=\\mathcal {E}$ and $H$ has kernel function $h$ , then $\\int h\\left(a,b\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\mathbb {S}\\left(H\\right)$ The first inequality in (REF ) holds from the expressions of $\\mathbb {S}\\left(\\cdot \\right)$ and $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ in terms of the spectrum (see (REF )).",
"The boundedness and self-adjointness of $H$ implies that the second inequality follows from an application of the Schur Test and the conjugate symmetry of its kernel function (see, for example [24]).",
"For the lower bound on $\\mathbb {S}\\left(\\cdot \\right)$ , we use the definition (REF ) and consider the test function $f(a) = 1$ .",
"Lemma 3.3 Suppose $h,g\\colon \\mathcal {X}^2\\rightarrow \\mathbb {C}$ are kernel functions for the self-adjoint linear operators $H,G\\colon L^2\\left(\\mathcal {X}\\right) \\rightarrow L^2\\left(\\mathcal {X}\\right)$ , such that $h\\left(x,y\\right)\\in \\mathbb {R}_{\\ge 0}$ and $h\\left(x,y\\right) \\ge *{g\\left(x,y\\right)}$ for $\\mu $ -a.e.",
"$x,y\\in \\mathcal {X}$ .",
"Then $\\mathbb {S}\\left(H\\right)\\ge \\mathbb {S}\\left(G\\right),\\quad \\left\\Vert H\\right\\Vert _{\\rm op}\\ge \\left\\Vert G\\right\\Vert _{\\rm op}, \\quad \\left\\Vert H\\right\\Vert _{1,\\infty }\\ge \\left\\Vert G\\right\\Vert _{1,\\infty }, \\quad \\left\\Vert H\\right\\Vert _{2,\\infty }\\ge \\left\\Vert G\\right\\Vert _{2,\\infty }, \\quad \\left\\Vert H\\right\\Vert _{\\infty ,\\infty }\\ge \\left\\Vert G\\right\\Vert _{\\infty ,\\infty }.$ For the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ norms the inequality is clear from the definitions.",
"For the $\\left\\Vert \\cdot \\right\\Vert _{\\rm op}$ version, let $f\\in L^2\\left(\\mathcal {X}\\right)$ , and use $\\rm Abs$ to denote the operator $f\\mapsto *{f}$ .",
"While this operator is not linear, it is clear that $*{{\\rm Abs}f}_2 = *{f}_2$ .",
"With this setup, $*{Gf}^2_2 = \\int *{\\int g\\left(x,y\\right) f\\left(y\\right) \\nu \\left(\\mathrm {d}y\\right)}^2\\nu \\left(\\mathrm {d}x\\right) \\le \\int \\left(\\int *{g\\left(x,y\\right)} *{f\\left(y\\right)} \\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\\\\\le \\int \\left(\\int h\\left(x,y\\right) *{f\\left(y\\right)} \\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) = *{H{\\rm Abs}f}^2_2.$ Therefore $\\left\\Vert G\\right\\Vert _{\\rm op} = \\sup _f\\frac{*{Gf}_2}{*{f}_2} \\le \\sup _f\\frac{*{H{\\rm Abs}f}_2}{*{f}_2} = \\sup _f\\frac{*{H{\\rm Abs}f}_2}{*{{\\rm Abs}f}_2}\\le \\sup _f\\frac{*{Hf}_2}{*{f}_2} =\\left\\Vert H\\right\\Vert _{\\rm op}.$ Here we used that the image of $\\rm Abs$ is a subset of $L^2\\left(\\mathcal {X}\\right)$ .",
"A similar argument works for the $\\mathbb {S}\\left(\\cdot \\right)$ inequality.",
"Given $f\\in L^2(\\mathcal {X})$ , $*{\\left<f,Gf\\right>} \\le \\int *{f(x)}*{g(x,y)}*{f(y)}\\nu ^{\\otimes 2}\\left(\\mathrm {d}x,\\mathrm {d}y\\right) \\le \\left<{\\rm Abs}f,H{\\rm Abs}f\\right>.$ Therefore $\\mathbb {S}\\left(G\\right) = \\sup _f\\frac{\\left<f,Gf\\right>}{*{f}^2_2} \\le \\sup _f\\frac{\\left<{\\rm Abs}f,H{\\rm Abs}f\\right>}{*{f}^2_2} = \\sup _f\\frac{\\left<{\\rm Abs}f,H{\\rm Abs}f\\right>}{*{{\\rm Abs}f}^2_2}\\le \\sup _f\\frac{\\left<f,Hf\\right>}{*{f}^2_2} =\\mathbb {S}\\left(H\\right).$ Lemma 3.4 Let $G,H$ be bounded self-adjoint linear operators on a separable Hilbert space.",
"Then $*{\\mathbb {S}\\left(G+H\\right) - \\mathbb {S}\\left(G\\right)} \\le \\left\\Vert H\\right\\Vert _{\\rm op}.$ It is a well-known result (for example see [29]) that $\\sigma (G+H)$ is contained in the closed $\\left\\Vert H\\right\\Vert _{\\rm op}$ -neighbourhood of $\\sigma (G)$ .",
"Perturbing $G$ by $+H$ , and perturbing $G+H$ by $-H$ then implies the result.",
"The main advantage of the Hilbert space formulation for the operators is that it allows us to use the following theorem to better understand the operators.",
"Although the “Multiplication Operator\" version presented here may not provide as much information as the “Projection-Valued Measure\" or “Direct Integral\" versions (see [19]), it has the advantage of being simpler to state and is sufficient for our purposes.",
"Theorem 3.5 (Spectral Theorem - Multiplication Operator Version) Let $\\left\\lbrace H_i\\right\\rbrace _{i\\in I}$ be a commutative family of bounded, self-adjoint operators on a separable Hilbert space $\\mathbb {H}$ .",
"Then there exists a $\\sigma $ -finite measure space $\\left(\\Omega ,\\Sigma _\\Omega , \\mu \\right)$ , a family of bounded, measurable, real-valued functions $\\left\\lbrace h_i\\right\\rbrace _{i\\in I}$ on $\\Omega $ , and a unitary map $U \\colon \\mathbb {H}\\rightarrow L^2\\left(\\Omega ,\\Sigma _\\Omega , \\mu \\right)$ such that for $\\mu $ -a.e.",
"$\\omega $ , $\\left(UH_iU^{-1}f\\right)\\left(\\omega \\right) = h_i\\left(\\omega \\right)f\\left(\\omega \\right)$ for all $f\\in L^2\\left(\\Omega ,\\Sigma _\\Omega .",
"\\mu \\right)$ and $i\\in I$ .",
"For a proof in more generality, see [12].",
"In the context of a finite dimensional Hilbert space $\\mathbb {H}$ , the Spectral Theorem is effectively just the statement that Hermitian matrices can be diagonalized over $\\mathbb {R}$ .",
"Here $U$ is represented by a unitary matrix composed from the orthonormal eigenbasis of $\\left\\lbrace H_i\\right\\rbrace $ , and the entries of the resulting diagonal matrix corresponding to $H_i$ are the values of the function $h_i(\\omega )$ .",
"One famous application of the Spectral Theorem for infinite dimensional Hilbert spaces is in the context of convolution operators on $\\mathbb {R}^d$ with Lebesgue measure.",
"Suppose the operator $G\\colon L^2\\left(\\mathbb {R}^d\\right) \\rightarrow L^2\\left(\\mathbb {R}^d\\right)$ acts on the function $f$ by $Gf(\\overline{x}) = \\int g(\\overline{x}-\\overline{y})f(\\overline{y}) \\mathrm {d}\\overline{y}$ for some function $g\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ .",
"If $g$ is integrable then Schur's test proves that $G$ is bounded, and if $g$ is reflection invariant (that is, $g(\\overline{x})=g(-\\overline{x})$ ) then expanding the inner product proves that $G$ is self-adjoint.",
"Supposing that these conditions hold, then the Fourier transform of $g$ is given by the bounded real function $\\widehat{g}\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\widehat{g}(k) = \\int \\text{e}^{i\\overline{x}\\cdot k}g(\\overline{x})\\mathrm {d}\\overline{x}.$ Furthermore, the Plancherel Theorem states that for a function $f\\in L^2\\left(\\mathbb {R}^d\\right)\\cap L^1\\left(\\mathbb {R}^d\\right)$ then the $L^2$ -norm of $f$ is equal to the $L^2$ -norm of $\\widehat{f}$ .",
"This - along with the clear linearity of the Fourier transform map - shows that the Fourier transform map can be considered a bounded linear operator $L^2\\left(\\mathbb {R}^d\\right) \\rightarrow L^2\\left(\\mathbb {R}^d\\right)$ .",
"The Fourier Inversion Theorem also relates the Fourier transform map to its inverse and its adjoint, and this can be used to show that the Fourier transform map is in fact a unitary linear operator.",
"The famous Convolution Theorem says that the Fourier transform of the convolution of two functions is the pointwise product of their individual Fourier transforms.",
"This means that $\\widehat{Gf}(k) = \\widehat{g}(k)\\widehat{f}(k).$ This can then be understood as an instance of the Spectral Theorem.",
"In this case, both the Hilbert spaces are $L^2\\left(\\mathbb {R}^d\\right)$ , the unitary map is (the inverse of) the Fourier transform map, and the function $h$ takes the values of the Fourier function, $\\widehat{g}$ , on $\\mathbb {R}^d$ .",
"Part of the power of the Fourier transform lies in that fact that this single unitary map simultaneously “diagonalizes\" a large class of operators.",
"The translation and reflection invariance of the Lebesgue measure on $\\mathbb {R}^d$ means that for integrable functions $g,h\\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ , $\\int g(\\overline{x}-\\overline{y})h(\\overline{y})\\mathrm {d}\\overline{y}= \\int h(\\overline{x}-\\overline{u})g(\\overline{u})\\mathrm {d}\\overline{u}$ (using the substitution $\\overline{u}=\\overline{x}-\\overline{y}$ ).",
"Therefore the family of convolution operators described here form a commutative family of operators.",
"We also present here a lemma that relates the operator norm and spectral supremum for operators on $L^2(\\mathbb {X})$ to operators on $L^2(\\mathcal {E})$ .",
"Given a family $\\left\\lbrace A_k\\right\\rbrace _{k\\in \\mathbb {R}^d}$ of subsets of $\\mathbb {C}$ , we define the essential union of these sets to be ${\\rm ess.",
"}\\bigcup _{k\\in \\mathbb {R}^d}A_k := \\left\\lbrace z\\in \\mathbb {C}:\\forall \\epsilon >0, {\\rm Leb}\\left(k\\in \\mathbb {R}^d:\\exists x\\in A_k \\text{ s.t.",
"}*{x-z}<\\epsilon \\right)>0\\right\\rbrace .$ This can be understood as the union of $A_k$ over $k$ neglecting null sets of $k$ .",
"Lemma 3.6 We have $\\sigma \\left(\\right) = {\\rm ess.",
"}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{}(k)\\right) , \\qquad \\sigma \\left(\\mathcal {T}_\\lambda \\right) = {\\rm ess.",
"}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right).$ It then follows that $\\left\\Vert \\right\\Vert _{\\rm op} = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\left\\Vert \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op}, \\qquad \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op},$ $\\mathbb {S}\\left(\\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\mathbb {S}\\left(\\widehat{}(k)\\right), \\qquad \\mathbb {S}\\left(\\mathcal {T}_\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right).$ We present the proof of (REF ) for the connection operator case.",
"The two-point operator case follows similarly.",
"First suppose $\\zeta \\in {\\rm ess.",
"}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{}(k)\\right)$ .",
"Therefore for all $\\epsilon >0$ there exists $P_\\epsilon \\subset \\mathbb {R}^d$ such that ${\\rm Leb}\\left(P_\\epsilon \\right)>0$ and for all $k\\in P_\\epsilon $ there exists $z\\in \\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ such that $*{\\zeta -z}<\\epsilon $ .",
"Furthermore, given $z\\in \\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ there exists $g\\in L^2\\left(\\mathcal {E}\\right)$ with $*{g}_2=1$ such that $*{\\left(z1- \\widehat{}\\left(k\\right)\\right)g}_2<\\epsilon $ .",
"For all $k\\in P_\\epsilon $ we let $z^{(k)}$ and $g^{(k)}$ denote such choices.",
"Now let $\\widehat{f}$ be any element of $L^2\\left(\\mathbb {R}^d\\right)$ supported on $P_\\epsilon $ with $*{\\widehat{f}}_2=1$ .",
"Then define $h\\left(\\overline{x},a\\right) := \\int _{P_\\epsilon } \\text{e}^{-i k\\cdot \\overline{x}}\\widehat{f}(k)g^{(k)}(a)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}.$ By the Fourier Inversion Theorem, the Fourier transform of $h$ is given by $\\widehat{h}\\left(k,a\\right) = \\widehat{f}(k)g^{(k)}(a)$ for $k\\in P_\\epsilon $ and vanishes elsewhere.",
"By Plancherel's Theorem (that is, the Fourier transform is unitary), $h$ and $\\widehat{h}$ have the same $L^2$ -norm: $*{h}^2_2= *{\\widehat{h}}^2_2 = \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\left(\\int *{g^{(k)}(a)}^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\\\= \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2*{g^{(k)}}^2_2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} = \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} = *{\\widehat{f}}^2_2 = 1.$ By applying Plancherel's Theorem once again, we can write $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\int *{\\zeta g^{(k)}(a) - \\int \\widehat{\\varphi }(k;a,b)g^{(k)}(b)\\mathcal {P}\\left(\\mathrm {d}b\\right)}^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\\\= \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2*{\\left(\\zeta 1- \\widehat{}(k)\\right)g^{(k)}}^2_2 \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} < \\int _{P_\\epsilon }*{\\widehat{f}(k)}^2\\left(\\epsilon + \\underbrace{*{\\left(z(k)1- \\widehat{}(k)\\right)g^{(k)}}_2}_{<\\epsilon }\\right)^2 \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} < 4\\epsilon ^2.$ Here we use the triangle inequality and $*{g^{(k)}}_2=1$ to replace $\\zeta $ with $z(k)$ at the cost of an $\\epsilon $ .",
"Since for each $\\epsilon >0$ we have constructed a unit $h\\in L^2\\left(\\mathbb {X}\\right)$ such that $*{\\left(\\zeta 1- \\right)h}_2<2\\epsilon $ , we have proven that $\\left(\\zeta 1- \\right)$ has no bounded linear inverse and therefore $\\zeta \\in \\sigma \\left(\\right)$ .",
"We now show the converse.",
"Suppose $\\zeta \\in \\sigma \\left(\\right)$ .",
"Then for all $\\epsilon >0$ there exists $h \\in L^2\\left(\\mathbb {X}\\right)$ such that $*{h}_2=1$ and $*{\\left(\\zeta 1- \\right)h}_2 < \\epsilon $ .",
"By using Plancherel's Theorem to replace the original functions with their Fourier transforms, we get $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int *{\\zeta h\\left(\\overline{x},a\\right) - \\int \\varphi \\left(\\overline{x}-\\overline{y};a,b\\right)h\\left(\\overline{y},b\\right)\\mathrm {d}\\overline{y}\\mathcal {P}\\left(\\mathrm {d}b\\right)}^2\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\= \\int *{\\zeta \\widehat{h}\\left(k,a\\right) - \\int \\widehat{\\varphi }\\left(k;a,b\\right)\\widehat{h}\\left(k,b\\right)\\mathcal {P}\\left(\\mathrm {d}b\\right)}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\mathcal {P}\\left(\\mathrm {d}a\\right).$ Now let us define the family of functions $g^{(k)}\\colon \\mathcal {E}\\rightarrow \\mathbb {C}$ by $g^{(k)}\\left(a\\right) := \\widehat{h}\\left(k,a\\right)$ for all $k\\in \\mathbb {R}^d$ .",
"Then we can again re-write the equality above as $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int *{\\left(\\zeta 1- \\widehat{}\\left(k\\right)\\right)g^{(k)}}^2_2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ For each $k\\in \\mathbb {R}^d$ , $\\widehat{}(k)$ is self-adjoint and therefore by the Spectral Theorem (Theorem REF ) there exists a Hilbert space $L^2\\left(\\mathfrak {E}_k,\\mu _k\\right)$ , a unitary operator $U_k\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathfrak {E}_k\\right)$ , and a measurable function $e\\mapsto \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)$ such that $\\widehat{}(k)$ is unitarily equivalent to pointwise multiplication by $\\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)$ on $L^2\\left(\\mathfrak {E}_k\\right)$ .",
"If we define $f^{\\left(k\\right)\\left(e\\right)} := \\left(U_kg^{\\left(k\\right)}\\right)\\left(e\\right)\\in \\mathbb {C}$ , then (REF ) becomes $*{\\left(\\zeta 1- \\right)h}^2_2 = \\int \\left(\\int *{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}^2*{f^{(k)(e)}}^2\\mu _k\\left(\\mathrm {d}e\\right)\\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ Note that since the Fourier transform and the maps $\\left\\lbrace U_k\\right\\rbrace _k$ are all unitary, the normalisation of $h$ passes on to $g^{(k)}$ and on to $f^{(k)(e)}$ so that $\\int \\left(\\int *{f^{(k)(e)}}^2\\mu _k\\left(\\mathrm {d}e\\right)\\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}=1.$ Now suppose for contradiction that $*{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}\\ge \\epsilon $ for $\\mu _k$ -a.e.",
"$e\\in \\mathfrak {E}_k$ , for ${\\rm Leb}$ -a.e.",
"$k\\in \\mathbb {R}^d$ .",
"Then the normalisation of $f^{(k)(e)}$ implies that $\\int \\left(\\int *{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}^2*{f^{(k)(e)}}^2\\mu _k\\left(\\mathrm {d}e\\right)\\right)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\ge \\epsilon ^2.$ However this contradicts the condition that $*{\\left(\\zeta 1- \\right)h}_2 < \\epsilon $ .",
"Instead, for each $\\epsilon >0$ there exists $P_\\epsilon \\subset \\mathbb {R}^d$ with ${\\rm Leb}\\left(P_\\epsilon \\right)>0$ such that for all $k\\in P_\\epsilon $ there exists $Q^{(k)}_\\epsilon \\subset \\mathfrak {E}_k$ with $\\mu _k\\left(Q^{(k)}_\\epsilon \\right)>0$ such that for all $e\\in Q^{(k)}_\\epsilon $ we have $*{\\zeta - \\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)}< \\epsilon $ .",
"For each $k\\in \\mathbb {R}^d$ , the operator $U_k\\widehat{}(k)U_k^{-1}$ is a multiplication operator on $L^2\\left(\\mathfrak {E}_k\\right)$ with function $e\\mapsto \\widetilde{\\varphi }(k)(e)$ , and therefore it is easy to calculate its spectrum.",
"Since the operator is unchanged by changing $\\widetilde{\\varphi }(k)(e)$ on a $\\mu _k$ -null set, the spectrum is given by the essential image of $\\widetilde{\\varphi }(k)$ : ${\\rm ess.Im}\\left(\\widetilde{\\varphi }\\left(k\\right)\\right) = \\left\\lbrace z\\in \\mathbb {C}:\\forall \\epsilon >0,\\mu _k\\left(e\\in \\mathfrak {E}_k:*{\\widetilde{\\varphi }\\left(k\\right)\\left(e\\right)-z}>\\epsilon \\right)>0\\right\\rbrace .$ Since the unitary equivalence preserves the spectrum, this gives us an expression for $\\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ .",
"In particular, we can write our previous conclusion as the following.",
"For all $\\epsilon >0$ there exists $P_\\epsilon \\subset \\mathbb {R}^d$ with ${\\rm Leb}\\left(P_\\epsilon \\right)>0$ such that for all $k\\in P_\\epsilon $ there exists $z\\in \\sigma \\left(\\widehat{}\\left(k\\right)\\right)$ such that $*{\\zeta -z}<\\epsilon $ .",
"Equivalently, this means that $\\zeta \\in {\\rm ess.",
"}\\bigcup _{k\\in \\mathbb {R}^d}\\sigma \\left(\\widehat{}(k)\\right)$ .",
"Since $\\left\\Vert A\\right\\Vert _{\\rm op} = \\sup \\left\\lbrace *{z}:z\\in \\sigma \\left(A\\right)\\right\\rbrace $ and $\\mathbb {S}\\left(A\\right) = \\sup \\left\\lbrace z:z\\in \\sigma \\left(A\\right)\\subset \\mathbb {R}\\right\\rbrace $ for self-adjoint operator $A$ , the remaining equalities follow from (REF )."
],
[
"Differentiating the Two-Point Operator",
"The ideas of this section can be found in [21], but we will need extra care to account for the fact that we are dealing with operators and marks and not just functions on $\\mathbb {R}^d$ .",
"Recall $\\tau _\\lambda \\left(x,y\\right) := \\mathbb {P}_\\lambda (x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y})$ .",
"Using the notation $x = \\left(\\overline{x},a\\right)\\in \\mathbb {R}^d\\times \\mathcal {E}$ , define $\\Lambda _n\\left(x\\right) = \\left(\\overline{x} + \\left[-n,n\\right)^d\\right)\\times \\mathcal {E}$ .",
"Then we can define the truncated two-point function $\\tau _\\lambda ^n\\left(x,y\\right) := \\mathbb {P}_\\lambda \\left(x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}_{\\Lambda _n\\left(y\\right)}\\right).$ We will want to give meaning to the event $\\left\\lbrace x \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^x_{\\Lambda _n\\left(y\\right)}\\right\\rbrace $ .",
"To do so we add a “ghost vertex” $g$ in the same way we added deterministic vertices, and add an edge between $v \\in \\xi _{\\Lambda _n\\left(y\\right)}$ and $g$ with probability $1-\\exp \\left(-\\int _{\\Lambda _n^c\\left(y\\right)} \\varphi (u,v) \\nu \\left(\\mathrm {d}u\\right)\\right)$ .",
"We now identify $\\Lambda _n^c\\left(y\\right)$ with $g$ .",
"Lemma 3.7 (Differentiability of $\\tau _\\lambda $ ) Let $x,y\\in \\mathbb {X}$ and $\\varepsilon >0$ be arbitrary.",
"The function $\\lambda \\mapsto \\tau _\\lambda ^n(x,y)$ is differentiable on $[0, \\lambda _O-\\varepsilon ]$ for any $n\\in \\mathbb {N}$ .",
"Furthermore, $ \\tau _\\lambda ^n(x,y)$ converges to $\\tau _\\lambda (x,y)$ uniformly in $\\lambda $ and $\\tfrac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y)$ converges to a limit uniformly in $\\lambda $ .",
"Consequently, $\\tau _\\lambda (x,y)$ is differentiable w.r.t.",
"$\\lambda $ on $[0,\\lambda _O)$ and $\\lim _{n\\rightarrow \\infty } \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) = \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\tau _\\lambda (x,y) =\\int \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y,u}, y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y}) \\nu \\left(\\mathrm {d}u\\right).$ For $S \\subseteq \\mathbb {X}$ , let $\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y} \\textrm { through } S \\rbrace $ be the event that $x$ is connected to $b$ in $\\xi ^{x,y}$ , and all such paths use a vertex from $S$ .",
"Also, let $\\lbrace x \\longleftrightarrow S\\textrm { in } \\xi ^x\\rbrace $ be the event that there exists $y \\in \\eta ^x\\cap S$ that is connected to $x$ .",
"The convergence $\\tau _\\lambda ^n(x,y) \\rightarrow \\tau _\\lambda (x,y)$ is uniform for $\\lambda \\in [0,\\lambda _O-\\varepsilon ]$ , as $|\\tau _\\lambda (x,y) - \\tau _\\lambda ^n(x,y)| &= \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\textrm { through } \\Lambda _n^c\\left(y\\right)) \\\\& \\le \\mathbb {P}_{\\lambda _O-\\varepsilon }(y \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{y})\\xrightarrow{} 0.$ We now go about showing $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) \\xrightarrow{} f_\\lambda (x,y) := \\int \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x,}, y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}) \\nu \\left(\\mathrm {d}u\\right) $ uniformly in $\\lambda $ .",
"Note that $\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}_{\\Lambda _n\\left(y\\right)}\\rbrace $ lives on the bounded set $\\Lambda _n\\left(y\\right)$ , and so the the Margulis-Russo formula gives us differentiability and the following explicit expression for the derivative $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) = \\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda \\left( y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x}_{\\Lambda _n\\left(y\\right)}, y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}_{\\Lambda _n\\left(y\\right)} \\right) \\nu \\left(\\mathrm {d}u\\right).",
"$ Using this we can write $\\left|\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) - f_\\lambda (x,y)\\right|&= \\left|\\int _{\\Lambda _n} \\Big ( \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x}) -\\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}) \\right.\\nonumber \\\\& \\qquad \\qquad -\\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x}_{\\Lambda _n\\left(y\\right)}) +\\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}_{\\Lambda _n\\left(y\\right)}) \\Big ) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& \\qquad \\qquad \\qquad \\left.+ \\int _{\\Lambda _n^c\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x,}, y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}) \\nu \\left(\\mathrm {d}u\\right) \\right|\\\\& \\le \\left|\\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x} \\textrm { through } \\Lambda _n^c\\left(y\\right))\\right.\\nonumber \\\\&\\qquad \\qquad - \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\textrm { through } \\Lambda _n^c\\left(y\\right)) \\nu \\left(\\mathrm {d}u\\right) \\bigg \\vert \\nonumber \\\\& \\qquad \\qquad \\qquad + \\int _{\\Lambda _n^c\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow u\\textrm { in } \\xi ^{y,u} ) \\nu \\left(\\mathrm {d}u\\right) \\\\& \\le \\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda (y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x} \\textrm { through } \\Lambda _n^c\\left(y\\right) \\text{ and through } u) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& \\qquad \\qquad + \\int _{\\Lambda _n^c\\left(y\\right)} \\tau _{\\lambda _O-\\varepsilon }(u,y) \\nu \\left(\\mathrm {d}u\\right) .$ We now have two main observations.",
"First, that the event $\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi ^{y,u,x} \\textrm { through } \\Lambda _n^c\\left(y\\right) \\text{ and through } u\\rbrace $ is contained in $& \\left( \\lbrace y \\longleftrightarrow u\\textrm { in } \\xi ^{y,u}\\rbrace \\circ \\lbrace u \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{u}\\rbrace \\circ \\lbrace \\Lambda _n^c\\left(y\\right) \\longleftrightarrow x\\textrm { in } \\xi ^{x}\\rbrace \\right) \\nonumber \\\\&\\qquad \\cup \\left( \\lbrace y \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{y}\\rbrace \\circ \\lbrace \\Lambda _n^c\\left(y\\right) \\longleftrightarrow u\\textrm { in } \\xi ^{u}\\rbrace \\circ \\lbrace u \\longleftrightarrow x\\textrm { in } \\xi ^{u,x}\\rbrace \\right).$ Secondly, from $\\lambda <\\lambda _O$ we have $\\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} <\\infty .$ This means that for $\\nu $ -almost every $y\\in \\mathbb {X}$ we have $\\int _{\\Lambda _n^c\\left(y\\right)} \\tau _{\\lambda _O-\\varepsilon }(u,y) \\nu \\left(\\mathrm {d}u\\right)\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"Applying the BK inequality then gives $\\left|\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) - f_\\lambda (x,y)\\right|& \\le \\int _{\\Lambda _n\\left(y\\right)} \\mathbb {P}_\\lambda (u \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^u) \\Big [\\tau _\\lambda (u,y)\\mathbb {P}_\\lambda (x \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^x) \\nonumber \\\\&\\qquad \\qquad + \\tau _\\lambda (x,u)\\mathbb {P}_\\lambda (y \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^{y}) \\Big ] \\nu \\left(\\mathrm {d}u\\right) +o(1) \\\\& \\le 2 \\max _{z\\in \\lbrace y,x\\rbrace } \\mathbb {P}_{\\lambda _O-\\varepsilon }\\big (z \\longleftrightarrow \\Lambda _n^c\\left(y\\right)\\textrm { in } \\xi ^z\\big ) \\int \\tau _{\\lambda _O-\\varepsilon }(u,y) \\nu \\left(\\mathrm {d}u\\right) + o(1).$ The remaining integral is again finite, and the remaining probability tends to zero uniformly in $\\lambda $ .",
"The uniform convergence justifies the exchange of limit and derivative in (REF ).",
"A family of operators $\\left\\lbrace H_x\\right\\rbrace _{x\\in \\mathbb {R}}$ is differentiable at $x$ if there exists an operator $F_x$ such that $\\lim _{\\varepsilon \\rightarrow 0}\\left\\Vert \\frac{1}{\\varepsilon }\\left(H_{x+\\varepsilon }-H_x\\right) - F_x\\right\\Vert _{\\rm op} = 0.$ Using the notation $x = \\left(\\overline{x},a\\right)\\in \\mathbb {R}^d\\times \\mathcal {E}$ , recall that we require that the model has $\\mathbb {R}^d$ -translation invariance and so we can write $\\tau _\\lambda \\left(x,y\\right) = \\tau _\\lambda \\left(\\overline{x}-\\overline{y};a,b\\right)$ .",
"Also recall that we defined the Fourier transform $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel $\\widehat{\\tau }_\\lambda \\left(k;a,b\\right) = \\int \\cos \\left(k\\cdot \\overline{x}\\right)\\tau _\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}$ .",
"We now define the displaced Fourier transform $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel $\\widehat{\\tau }_{\\lambda ,k}\\left(l;a,b\\right) = \\int \\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\cos \\left(l\\cdot \\overline{x}\\right)\\tau _\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}$ .",
"Corollary 3.8 (Differentiability of $\\mathcal {T}_\\lambda $ ) The operators $\\mathcal {T}_\\lambda $ , $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)$ , and $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)$ are differentiable w.r.t.",
"$\\lambda $ on $\\left(0,\\lambda _O\\right)$ and their derivatives are the bounded linear operators $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ , $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ , and $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ with kernel functions $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\tau _\\lambda \\left(x,y\\right)$ , $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\tau }_\\lambda \\left(k;a,b\\right)$ , and $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\tau }_{\\lambda ,k}\\left(l;a,b\\right)$ respectively.",
"Specifically, we have the following bounds on the derivatives: $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} &\\le \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}^2,\\\\\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} &\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2,\\\\\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} &\\le 4\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}.",
"$ We use Lemma REF in conjunction with the Leibniz integral rule to differentiate the two-point operator.",
"Fix $f\\in L^2\\left(\\mathbb {X}\\right)$ and let $\\varepsilon \\in \\left(0,\\lambda _O\\right)$ .",
"From the reasoning of (REF ) we know that the subcritical integral $\\int \\tau _{\\lambda _O-\\varepsilon }(\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{(\\otimes 2)}\\left(\\mathrm {d}a,\\mathrm {d}b\\right)<\\infty $ , and therefore $\\int \\tau _{\\lambda _O-\\varepsilon }\\left(x,u\\right)^2\\nu \\left(\\mathrm {d}u\\right) \\le \\int \\tau _{\\lambda _O-\\varepsilon }\\left(x,u\\right)\\nu \\left(\\mathrm {d}u\\right) < \\infty ,$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .",
"Here we have also used that $\\tau _\\lambda (x,u)$ is $\\left[0,1\\right]$ -valued.",
"This implies that $\\tau _{\\lambda _O-\\varepsilon }\\left(x,\\cdot \\right)\\in L^2\\left(\\mathbb {X}\\right)$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .",
"In conjunction with $f\\in L^2\\left(\\mathbb {X}\\right)$ , Hölder's inequality implies that the function $\\Theta _{\\varepsilon ,x}\\colon y\\mapsto \\tau _{\\lambda _O-\\varepsilon }\\left(x,y\\right)*{f\\left(y\\right)}$ is in $L^1\\left(\\mathbb {X}\\right)$ for $\\nu $ -almost every $x\\in \\mathbb {X}$ .",
"Furthermore, $\\tau _\\lambda \\left(x,y\\right)$ is non-decreasing in $\\lambda $ and so for all $\\lambda <\\lambda _O$ there exists $\\varepsilon >0$ such that $\\Theta _{\\varepsilon ,x}$ dominates $y\\mapsto \\tau _\\lambda \\left(x,y\\right)f\\left(y\\right)$ .",
"This domination by an $L^1\\left(\\mathbb {X}\\right)$ function allows us to use the Leibniz integral rule to exchange the integral and derivative to get (for $\\nu $ -almost every $x\\in \\mathbb {X}$ ) $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\left(\\left[\\mathcal {T}_\\lambda f\\right]\\left(x\\right)\\right) = \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\int \\tau _\\lambda \\left(x,u\\right)f\\left(u\\right)\\nu \\left(\\mathrm {d}u\\right) = \\int \\left(\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda \\left(x,u\\right)\\right)f\\left(u\\right)\\nu \\left(\\mathrm {d}u\\right) = \\left[\\left(\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right)f\\right]\\left(x\\right),$ where Lemma REF gives the differentiability of the function $\\tau _\\lambda \\left(x,y\\right)$ .",
"We now prove that the operator $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda $ is bounded.",
"From Lemma REF and the BK inequality, we get the bound $0\\le \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(x,y) \\le \\int \\mathbb {P}_\\lambda \\left( \\lbrace y \\longleftrightarrow u\\textrm { in } \\xi ^{y,u}_{\\Lambda _n\\left(y\\right)} \\rbrace \\circ \\lbrace u \\longleftrightarrow x\\textrm { in } \\xi ^{u,x}_{\\Lambda _n\\left(y\\right)} \\rbrace \\right) \\nu \\left(\\mathrm {d}u\\right) \\\\\\le \\int \\tau _\\lambda ^n(u,y)\\mathbb {P}_\\lambda \\left(u \\longleftrightarrow x\\textrm { in } \\xi ^{u,x}_{\\Lambda _n\\left(y\\right)}\\right) \\nu \\left(\\mathrm {d}u\\right) \\le \\int \\tau _\\lambda \\left(u,y\\right)\\tau _\\lambda \\left(x,u\\right)\\nu \\left(\\mathrm {d}u\\right),$ where the last inequality holds because we are removing the $\\Lambda _n\\left(y\\right)$ restriction.",
"Note that the right hand side of this equation is the kernel function of the operator $\\mathcal {T}_\\lambda ^2$ .",
"Choose $f\\in L^2\\left(\\mathbb {X}\\right)$ , and use $\\rm Abs$ to denote the map $f\\mapsto *{f}$ .",
"Now the above inequality implies that $*{\\left[\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right] f}^2_2 & \\le \\int \\left(\\int *{\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\tau _\\lambda \\left(x,y\\right)}*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2 \\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\& \\le \\int \\left(\\int \\int \\tau _\\lambda \\left(x,u\\right)\\tau _\\lambda \\left(u,y\\right)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}u\\right)\\nu \\left(\\mathrm {d}y\\right)\\right)^2 \\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\int \\left(\\left[\\mathcal {T}_\\lambda ^2 {\\rm Abs}{f}\\right]\\left(x\\right)\\right)^2 \\nu \\left(\\mathrm {d}x\\right) = *{\\mathcal {T}_\\lambda ^2 {\\rm Abs}f}^2_2.$ Since $*{{\\rm Abs}f}_2 = *{f}_2$ , this then gives us $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} \\le \\sup _f\\frac{*{\\mathcal {T}_\\lambda ^2 {\\rm Abs}f}_2}{*{f}_2} = \\sup _f\\frac{*{\\mathcal {T}_\\lambda ^2 {\\rm Abs}f}_2}{*{{\\rm Abs}f}_2}\\le \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\rm op} = \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}^2 < \\infty .$ Now we address the Fourier transform.",
"Pick $\\varepsilon \\in (0,\\lambda _O-\\lambda )$ .",
"Recall $*{\\text{e}^{ik\\cdot \\overline{x}}\\tau _\\lambda \\left(\\overline{x};a,b\\right)}$ is bounded by $\\tau _{\\lambda _O-\\varepsilon }\\left(\\overline{x};a,b\\right)$ for all $\\overline{x}\\in \\mathbb {R}^d$ and $a,b\\in \\mathcal {E}$ .",
"From the reasoning of (REF ), we know that $\\tau _{\\lambda _O-\\varepsilon }\\left(\\cdot ;a,b\\right)$ is Lebesgue integrable for $\\mathcal {P}$ -a.e.",
"$a,b\\in \\mathcal {E}$ .",
"We are then justified in using the Leibniz integral rule to say that for every $b\\in \\mathcal {E}$ and $\\mathcal {P}$ -a.e.",
"$a\\in \\mathcal {E}$ , $\\tfrac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\int \\text{e}^{ik\\cdot \\overline{x}} \\tau _\\lambda (\\overline{x};a,b) \\mathrm {d}\\overline{x}= \\int \\text{e}^{ik\\cdot \\overline{x}}\\tfrac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda (\\overline{x};a,b) \\mathrm {d}\\overline{x}$ .",
"Applying Lemma REF as well as (REF ), we derive that for such $a,b$ we have $*{\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\widehat{\\tau }_\\lambda (k;a,b)} \\le \\int \\lim _{n\\rightarrow \\infty } \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda } \\tau _\\lambda ^n(\\overline{x};a,b) \\mathrm {d}\\overline{x}\\le \\int \\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) \\,\\mathrm {d}\\overline{x}\\,\\mathrm {d}\\overline{y}\\,\\mathcal {P}\\left(\\mathrm {d}c \\right).$ Note that the right hand side of this equation is the kernel function of the operator $\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)^2$ .",
"By repeating the argument for $\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\mathcal {T}_\\lambda $ - with $\\mathbb {X}$ replaced by $\\mathcal {E}$ - we similarly arrive at $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2.$ For the displaced Fourier transform, note that $0\\le 1-\\cos \\left(k\\cdot \\overline{x}\\right)\\le 2$ , and so the same argument as for the Fourier transform allows us to use the Leibniz integral rule to exchange derivative and integral.",
"We can also improve on the immediate bound of $2\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2$ for the operator norm of the derivative.",
"Let $f\\in L^2\\left(\\mathcal {E}\\right)$ , $*{\\left[\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right]f}^2_2 &\\le \\int \\left(\\int *{\\text{e}^{il\\cdot \\overline{x}}}\\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) *{f\\left(b\\right)}\\mathrm {d}\\overline{x}\\,\\mathrm {d}\\overline{y}\\,\\mathcal {P}\\left(\\mathrm {d}c \\right)\\mathcal {P}\\left(\\mathrm {d}b\\right)\\right)^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\& \\le \\int \\left(\\int 2\\Big [\\left(1 - \\cos \\left(k\\cdot \\left(\\overline{x}-\\overline{y}\\right)\\right)\\right)\\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) \\right.",
"\\nonumber \\\\&\\qquad + \\left(1 - \\cos \\left(k\\cdot \\overline{y}\\right)\\right)\\tau _\\lambda (\\overline{x}-\\overline{y};a,c) \\tau _\\lambda (\\overline{y};c,b) \\Big ]*{f\\left(b\\right)}\\,\\mathrm {d}\\overline{x}\\,\\mathrm {d}\\overline{y}\\,\\mathcal {P}\\left(\\mathrm {d}c \\right)\\mathcal {P}\\left(\\mathrm {d}b\\right)\\Big )^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\& = 4 *{\\left[\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) + \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)\\right]{\\rm Abs} f}^2_2,$ where we have used the cosine-splitting lemma (Lemma REF ) to divide the displacement factor $\\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)$ into one spanning $\\overline{x}-\\overline{y}$ and one spanning $\\overline{y}$ .",
"Using the above approach of (REF ) and the triangle inequality then produces the required bound.",
"This is the cosine-splitting lemma referenced in the previous proof.",
"Lemma 3.9 (Split of cosines, [10]) Let $t_i \\in \\mathbb {R}$ for $i=1, \\ldots , m$ and $t = \\sum _{i=1}^{m} t_i$ .",
"Then $1-\\cos (t) \\le m \\sum _{i=1}^{m} [1- \\cos (t_i)].",
"$ We use the following lemma in the proof of Proposition REF to bound the displacement on the two-point operator using the displacement of the connection operator when working in the sub-critical regime away from the critical threshold.",
"Lemma 3.10 For $\\lambda <\\lambda _O$ , $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\text{e}^{4\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}}\\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op}.$ For the inequality removing $l$ , let $f\\in L^2\\left(\\mathcal {E}\\right)$ .",
"Then $*{\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right)f}^2_2 &\\le \\int \\left(\\int *{\\text{e}^{il\\cdot \\overline{x}}}\\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\tau _\\lambda (\\overline{x};a,b) *{f\\left(b\\right)}\\,\\mathrm {d}\\overline{x}\\, \\mathcal {P}\\left(\\mathrm {d}b\\right)\\right)^2\\mathcal {P}\\left(\\mathrm {d}a\\right) \\nonumber \\\\& = *{\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right){\\rm Abs} f}^2_2.$ Since $*{{\\rm Abs}f}_2=*{f}_2$ , we can use the approach of (REF ) to get the first inequality.",
"For the second inequality, we first note that $\\widehat{\\mathcal {T}}_0\\left(k\\right) = \\widehat{}\\left(k\\right)$ .",
"We then use the differential inequality () to extend to $\\lambda <\\lambda _O$ .",
"Using the reverse triangle inequality, we have $\\limsup _{\\varepsilon \\downarrow 0}\\frac{1}{\\varepsilon }*{\\left\\Vert \\widehat{\\mathcal {T}}_{s+\\varepsilon }\\left(0\\right) - \\widehat{\\mathcal {T}}_{s+\\varepsilon }\\left(k\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right) - \\widehat{\\mathcal {T}}_{s}\\left(k\\right)\\right\\Vert _{\\rm op}} &\\le \\left\\Vert \\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\left(\\widehat{\\mathcal {T}}_{\\lambda }\\left(0\\right) - \\widehat{\\mathcal {T}}_{\\lambda }\\left(k\\right)\\right)\\right|_{\\lambda =s}\\right\\Vert _{\\rm op} \\nonumber \\\\&\\le 4 \\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right) - \\widehat{\\mathcal {T}}_{s}\\left(k\\right)\\right\\Vert _{\\rm op}.$ Since $\\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right)\\right\\Vert _{\\rm op}$ is monotone increasing for $s \\in \\left[0,\\lambda \\right]$ , the right hand side of the inequality is bounded by $4 \\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda }\\left(0\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_{s}\\left(0\\right) - \\widehat{\\mathcal {T}}_{s}\\left(k\\right)\\right\\Vert _{\\rm op}$ , giving the exponential factor in the result."
],
[
"The Expansion",
"There are two parts to this section.",
"First we perform an expansion of the two-point function.",
"This argument is essentially identical to the corresponding step in [21], so we refer to this reference for most of the details.",
"The novelty in this section is rather in the formulation of this expansion in terms of operators, contained in the second part."
],
[
"Function-level expansion",
"The expansion of the two-point function $\\tau _\\lambda $ proceeds in essentially the same way as in [21].",
"The similarity holds because the argument uses quite general properties of Poisson point processes and connection models - the change from $\\mathbb {R}^d$ to $\\mathbb {X}=\\mathbb {R}^d\\times \\mathcal {E}$ adds no further complication.",
"We give here a very brief overview of the derivation - the details can be found in that reference.",
"We first introduce some notation.",
"It will be important to consider thinning events and pivotal points.",
"Definition 4.1 Let $x,y \\in \\mathbb {X}$ , and let $A \\subset \\mathbb {X}$ be locally finite and of cardinality ${A}$ .",
"(1) Set $\\bar{\\varphi }(A,x) :=\\prod _{y\\in A}(1-\\varphi (x,y)) $ and define $\\eta _{\\langle A \\rangle }$ as a $\\bar{\\varphi }(A, \\cdot )$ -thinning of $\\eta $ (or simply $A$ -thinning of $\\eta $) as follows.",
"We keep a point $w \\in \\eta $ as a point of $\\eta _{\\langle A \\rangle }$ with probability $\\bar{\\varphi }(A,w)$ independently of all other points of $\\eta $ .",
"We similarly define $\\eta _{\\langle A \\rangle }^x$ as a $\\bar{\\varphi }(A,\\cdot )$ -thinning of $\\eta ^x$ using the marks in $\\xi ^x$ .",
"(2) We write $x {\\,\\,A\\,\\,} y\\textrm { in } \\xi $ if both $\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi \\rbrace $ and $\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi [\\eta _{\\langle A \\rangle } \\cup \\lbrace x\\rbrace ] \\rbrace $ take place.",
"In words, $\\lbrace x {\\,\\,A\\,\\,} y\\textrm { in } \\xi \\rbrace $ is the event that $x,y\\in \\eta $ and $x$ is connected to $y$ in $\\xi $ , but that this connection is broken by an $A$ -thinning of $\\eta \\setminus \\lbrace x\\rbrace $ .",
"In particular, the connection does not survive if $y$ is thinned out.",
"(3) We define $\\tau _\\lambda ^{A}(x,y) = \\mathbb {P}_\\lambda \\left(y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y}[\\eta ^y_{\\langle A \\rangle }\\cup \\lbrace x\\rbrace ] \\right).",
"$ In words, $\\tau _\\lambda ^A(x,y)$ is the probability of the event that there exists an open path between $x$ and $y$ in an RCM driven by an $A$ -thinning of $\\eta ^y$ , where the point $x$ is fixed to be present (but $y$ is not).",
"(4) Given $x,y\\in \\mathbb {X}$ and edge-marking $\\xi $ , we say $u\\in \\mathbb {X}$ is pivotal and $u\\in \\textsf {Piv}(y,x,\\xi )$ if $\\left\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ and yet $\\left\\lbrace y \\longleftrightarrow x\\textrm { in } \\xi \\left[\\eta \\setminus \\left\\lbrace u\\right\\rbrace \\right]\\right\\rbrace $ .",
"That is, every path on $\\xi ^{x,y}$ connecting $x$ and $y$ uses the vertex $u$ .",
"Note that the end points $x$ and $y$ are never pivotal.",
"(5) We will also use the set $E(x,y;A,\\xi ) := \\lbrace x {\\,\\,A\\,\\,} y\\textrm { in } \\xi \\rbrace \\cap \\lbrace \\nexists w \\in \\textsf {Piv}(x,y;\\xi ): x {\\,\\,A\\,\\,} w\\textrm { in } \\xi \\rbrace , $ for a locally finite set $A \\subset \\mathbb {X}$ , and $x,y\\in \\mathbb {R}^d$ .",
"If we consider the pivotal points of the $x$ to $y$ connection in $\\xi $ in sequence, then this is the event that an $A$ -thinning breaks the connection after the last pivotal point, but not before.",
"Now we can state a continuum version of a standard lemma often used in discrete models - see, for example, [25].",
"In bond percolation, it has the name “Cutting-bond lemma”.",
"The proof of this continuum version can be found in [21].",
"Lemma 4.2 (Cutting-point lemma) Let $\\lambda \\ge 0$ and let $v,u,x \\in \\mathbb {X}$ with $u \\ne x$ and let $A\\subset \\mathbb {X}$ be locally finite.",
"Then $\\mathbb {E}_\\lambda \\big [1_{E(v,u;A,\\xi ^{v,u,x})} 1_{\\lbrace u \\in \\textsf {\\textup {Piv}}(v,x;\\xi ^{v,u,x})\\rbrace } \\big ]= \\mathbb {E}_\\lambda \\left[1_{E(v,u;A,\\xi ^{v,u})} \\cdot \\tau _\\lambda ^{v,\\xi ^v) }(x,u) \\right].$ Moreover, $\\mathbb {P}_\\lambda \\left(y \\Longleftrightarrow u\\textrm { in } \\xi ^{y,u,x}, u \\in \\textsf {\\textup {Piv}}(y,x; \\xi ^{y,u,x})\\right) = \\mathbb {E}_\\lambda \\left[1_{\\lbrace y \\Longleftrightarrow u\\textrm { in } \\xi ^{y,u}\\rbrace } \\cdot \\tau _\\lambda ^{y,\\xi ^{y}) }(x,u) \\right].$ We now define the lace expansion coefficient function.",
"Definition 4.3 (Lace-expansion function coefficients) For $n\\in \\mathbb {N}$ and $x,y\\in \\mathbb {X}$ , we define $\\pi _\\lambda ^{(0)}(x,y) &:= \\mathbb {P}_\\lambda \\left(y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) - \\varphi (x,y), \\\\\\pi _\\lambda ^{(n)}(x,y) &:= \\lambda ^n \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y \\Longleftrightarrow u_0\\textrm { in } \\xi ^{y, u_0}_{0}\\right\\rbrace \\cap \\bigcap _{i=1}^{n} E\\left(u_{i-1},u_i; {i-1}, \\xi ^{u_{i-1}, u_i}_{i}\\right) \\right) \\nu ^{\\otimes n}\\left(\\mathrm {d}\\vec{u}_{[0,n-1]}\\right) , $ where $u_n=x$ and ${i} = u_{i-1}, \\xi ^{u_{i-1}}_{i})$ is the cluster of $u_{i-1}$ in $\\xi ^{u_{i-1}}_i$ .",
"Further define the remainder functions $r_{\\lambda , 0} (x,y) &:= - \\lambda \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y \\Longleftrightarrow u_0\\textrm { in } \\xi ^{y, u_0}_0\\right\\rbrace \\cap \\left\\lbrace u_0 {\\,\\,0\\,\\,} x\\textrm { in } \\xi ^{u_0,x}_1\\right\\rbrace \\right) \\nu \\left(\\mathrm {d}u_0\\right), \\\\r_{\\lambda , n}(x,y) &:= (-\\lambda )^{n+1} \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y \\Longleftrightarrow u_0\\textrm { in } \\xi ^{y, u_0}_0\\right\\rbrace \\cap \\bigcap _{i=1}^{n} E\\left(u_{i-1},u_i; {i-1}, \\xi ^{u_{i-1}, u_i}_{i}\\right)\\right.",
"\\nonumber \\\\& \\hspace{170.71652pt}\\left.\\cap \\left\\lbrace u_n {\\,\\,n\\,\\,} x\\textrm { in } \\xi ^{u_n,x}_{n+1} \\right\\rbrace \\right) \\nu ^{\\otimes \\left(n+1\\right)}\\left(\\mathrm {d}\\vec{u}_{[0,n]}\\right).$ Additionally, define $\\pi _{\\lambda , n}$ as the alternating partial sum $\\pi _{\\lambda , n}(x,y) := \\sum _{m=0}^{n} (-1)^m \\pi _\\lambda ^{(m)}(x,y).",
"$ We will also be requiring Fourier transforms of these functions.",
"Since the whole model is spatially translation invariant, so are the functions defined above.",
"Therefore for each pair of marks $(a,b)$ we perform the Fourier transform on the spatial displacement to get the functions $\\widehat{\\pi }^{(n)}_\\lambda (k;a,b)$ , $\\widehat{\\pi }_{\\lambda ,n}(k;a,b)$ , and $\\widehat{r}_{\\lambda ,n}(k;a,b)$ for $n\\ge 0$ .",
"Proposition 4.4 (Lace expansion: function-level) Let $\\lambda \\in [0,\\lambda _O)$ and $n\\ge 0$ .",
"Then for $x,y\\in \\mathbb {X}$ , and for $k\\in \\mathbb {R}^d$ and $a,b\\in \\mathcal {E}$ , $\\tau _\\lambda \\left(x,y\\right) &= \\varphi \\left(x,y\\right) + \\pi _{\\lambda , n}\\left(x,y\\right) + \\lambda \\int \\tau _\\lambda \\left(x,u\\right)\\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right) + r_{\\lambda , n}\\left(x,y\\right), \\\\\\widehat{\\tau }_\\lambda \\left(k;a,b\\right) &= \\widehat{\\varphi }\\left(k;a,b\\right) + \\widehat{\\pi }_{\\lambda , n}\\left(k;a,b\\right) + \\lambda \\int \\widehat{\\tau }_\\lambda \\left(k;a,c\\right)\\left(\\widehat{\\varphi }+ \\widehat{\\pi }_{\\lambda , n}\\right)\\left(k;c,b\\right)\\mathcal {P}\\left(\\mathrm {d}c\\right) + \\widehat{r}_{\\lambda , n}\\left(k;a,b\\right).",
"$ We outline the proof of (REF ) here.",
"The equation () then follows by applying the Fourier transform.",
"The details of the proof of (REF ) - with appropriate contextual changes - can be found in [21].",
"Firstly, the definitions give $\\tau _\\lambda (x,y) = \\varphi (x,y) + \\pi _\\lambda ^{(0)}(x,y) + \\mathbb {P}_\\lambda \\left(y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}, y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right).",
"$ With an appropriate event partition, Mecke's equation and the Cutting-point lemma allow us to re-write the probability term as $\\mathbb {P}_\\lambda \\left(y \\longleftrightarrow x\\textrm { in } \\xi ^{y,x}, y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) = \\lambda \\int \\mathbb {E}_\\lambda \\left[ 1_{\\lbrace y \\Longleftrightarrow u\\textrm { in } \\xi ^{y,u}\\rbrace } \\tau _\\lambda ^{0}(x,u) \\right] \\mathrm {d}u$ Now we use the identity $\\tau _\\lambda ^{A}(x,u) = \\tau _\\lambda (x,u) - \\mathbb {P}_\\lambda \\left( u {\\,\\,A\\,\\,} x\\textrm { in } \\xi ^{u,x} \\right), $ to extract the deterministic function $\\tau _\\lambda (x,u)$ from the expectation at the cost of an extra term.",
"The $\\tau _\\lambda $ term then gives us the second $\\pi ^{(0)}_\\lambda $ term and the extra term becomes the remainder $r_{\\lambda ,0}$ .",
"Repeated use of (REF ) with Mecke's equation and the Cutting-point lemma gives an alternating series that produces the various $\\pi _\\lambda ^{(n)}$ and $r_{\\lambda ,n}$ terms.",
"Operator-level expansion Definition 4.5 (Lace-expansion operator coefficients) For $n\\in \\mathbb {N}$ , we define $_\\lambda ^{(n)}\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator with kernel function $\\pi ^{(n)}_\\lambda $ , and $R_{\\lambda ,n}\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator with kernel function $r_{\\lambda ,n}$ .",
"We also define $_{\\lambda ,n}\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator with kernel function $\\pi _{\\lambda , n}$ .",
"It follows that $_{\\lambda , n} = \\sum _{m=0}^{n} (-1)^m _\\lambda ^{(m)}.$ We also use the kernel functions $\\widehat{\\pi }^{(n)}_\\lambda (k;a,b)$ , $\\widehat{\\pi }_{\\lambda ,n}(k;a,b)$ , and $\\widehat{r}_{\\lambda ,n}(k;a,b)$ to define the operators $\\widehat{}^{(n)}_{\\lambda }(k),\\widehat{}_{\\lambda ,n}(k),\\widehat{R}_{\\lambda ,n}(k)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ respectively for each $k\\in \\mathbb {R}^d$ .",
"Proposition 4.6 (Lace expansion: operator-level) Let $\\lambda \\in [0,\\lambda _O)$ .",
"Then for $n \\ge 0$ and $k\\in \\mathbb {R}^d$ , $\\mathcal {T}_\\lambda &= + _{\\lambda , n} + \\lambda \\mathcal {T}_\\lambda \\left(+ _{\\lambda , n}\\right) + R_{\\lambda , n},\\\\\\widehat{\\mathcal {T}}_\\lambda (k) &= \\left(\\widehat{}+ \\widehat{}_{\\lambda , n}\\right)(k) + \\lambda \\widehat{\\mathcal {T}}_\\lambda (k)\\left(\\widehat{}+ \\widehat{}_{\\lambda , n}\\right)(k) + \\widehat{R}_{\\lambda , n}(k).$ Let $f\\in L^2\\left(\\mathbb {X}\\right)$ be a test function.",
"Then Proposition REF gives for $x\\in \\mathbb {X}$ , $\\mathcal {T}_\\lambda f\\left(x\\right) &= \\int \\tau _\\lambda \\left(x,y\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right)\\nonumber \\\\& = \\int \\left(\\varphi \\left(x,y\\right) + \\pi _{\\lambda , n}\\left(x,y\\right) + \\lambda \\int \\tau _\\lambda \\left(x,u\\right)\\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right) + r_{\\lambda , n}\\left(x,y\\right)\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right)\\nonumber \\\\& = \\left(+ _{\\lambda , n} + \\lambda \\mathcal {T}_\\lambda \\left(+ _{\\lambda , n}\\right) + R_{\\lambda , n}\\right)f\\left(x\\right).$ Note that in this last equality we used $\\int \\int \\tau _\\lambda \\left(x,u\\right)\\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)\\nu \\left(\\mathrm {d}u\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right) &= \\int \\tau _\\lambda \\left(x,u\\right)\\int \\left(\\varphi + \\pi _{\\lambda , n}\\right)\\left(u,y\\right)f\\left(y\\right)\\nu \\left(\\mathrm {d}y\\right)\\nu \\left(\\mathrm {d}u\\right)\\nonumber \\\\& = \\mathcal {T}_\\lambda \\left(+ _{\\lambda , n}\\right)f\\left(x\\right).$ This exchange of integrals is valid by Fubini's theorem because $\\lambda <\\lambda _O$ ensures the integrals are finite.",
"The same argument applies for the Fourier transformed operators to complete the proof.",
"Diagrammatic bounds For our argument, we will require bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}$ for all $n\\in \\mathbb {N}$ and $k\\in \\mathbb {R}^d$ .",
"However, it turns out that we can bound the last two remainder terms using others.",
"Lemma 5.1 For all $n\\in \\mathbb {N}$ , $k\\in \\mathbb {R}^d$ , and $\\lambda >0$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op},\\\\\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(0)\\right\\Vert _{\\rm op}.$ As $\\mathbb {P}_\\lambda (u_n {\\,\\,A\\,\\,} x\\textrm { in } \\xi ^{u_n,x}_{n+1} ) \\le \\tau _\\lambda (x,u_n)$ for an arbitrary locally finite set $A$ , the definition () gives $*{r_{\\lambda ,n}(x)} \\le \\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right).$ Let $f\\in L^2\\left(\\mathbb {X}\\right)$ .",
"Then this gives $*{R_{\\lambda ,n}f}^2_2 &\\le \\int \\left(\\int *{r_{\\lambda ,n}(x,y)}*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&\\le \\lambda ^2\\int \\left(\\int \\left(\\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right)\\right)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\lambda ^2\\int \\left(\\int \\tau _\\lambda (x,u)\\left(\\int \\pi _\\lambda ^{(n)}(u,y)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right) \\nu \\left(\\mathrm {d}u\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\& = \\lambda ^2 *{\\mathcal {T}_\\lambda _\\lambda ^{(n)}{\\rm Abs}f}^2_2.$ Note that we were able to use Tonelli's theorem to swap the $u$ and $y$ integrals because $\\tau _\\lambda $ and $\\pi ^{(n)}_\\lambda $ are both non-negative and measurable.",
"Finally, the approach of (REF ) implies the result.",
"For each $n\\ge 0$ , either $r_{\\lambda ,n}(x)\\ge 0$ for all $x\\in \\mathbb {X}$ or $r_{\\lambda ,n}(x)\\le 0$ for all $x\\in \\mathbb {X}$ .",
"Therefore $*{\\widehat{r}_{\\lambda ,n}(k;a,b)}\\le *{\\widehat{r}_{\\lambda ,n}(0;a,b)}$ for all $a,b\\in \\mathcal {E}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{R}_{\\lambda ,n}(0)\\right\\Vert _{\\rm op}$ (by Lemma REF ) for all $k\\in \\mathbb {R}^d$ .",
"From (REF ) we have $*{\\widehat{r}_{\\lambda ,n}(0;a,b)} \\le \\int \\widehat{\\tau }_\\lambda (0;a,c)\\widehat{\\pi }_\\lambda ^{(n)}(0;c,b) \\mathcal {P}\\left(\\mathrm {d}c\\right).$ Therefore the same argument as above gives the second bound in the result.",
"Because the $n=0$ cases for our bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ are so simple, we give the proof here before moving onto $n\\ge 1$ .",
"Recall $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right) = \\widehat{\\mathcal {T}}_\\lambda \\left(k+l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right)$ .",
"We will find it convenient to also define $\\widehat{}_k\\left(l\\right) = \\widehat{}\\left(k+l\\right) - \\widehat{}\\left(l\\right)$ .",
"Proposition 5.2 (Bounds for $n=0$ ) Let $\\lambda \\in [0,\\lambda _O)$ .",
"Then $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }.$ For $k\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op} &\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le \\lambda ^2\\left(\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } + \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\right).$ Our first comment relates to these bounds and every bound we perform hereafter in Section .",
"From the positivity of $\\pi ^{(0)}_{\\lambda }(x,y)$ and Lemma REF we have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op}$ , and by Lemma REF we have $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty }$ .",
"However from the definitions of the norm we also have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty } = \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ , and so we will be interested in bounding this last norm.",
"Now note that for the event $\\lbrace y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\rbrace $ to hold, either there is a direct edge between $y$ and $x$ , or there exist vertices $w,z$ in $\\eta $ that are direct neighbours of the origin and have respective disjoint paths to $x$ that both do not contain $y$ .",
"Hence, by the Mecke equation (REF ), $\\mathbb {P}_\\lambda ( y \\Longleftrightarrow x\\textrm { in } \\xi ^{\\mathbf {0},x}) &\\le \\varphi (x,y) + \\tfrac{1}{2} \\mathbb {E}_\\lambda \\Big [ \\sum _{(w,z) \\in \\eta ^{(2)}} 1_{(\\lbrace y\\sim w \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^x\\rbrace ) \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^x \\rbrace )} \\Big ] \\nonumber \\\\& = \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\int \\mathbb {P}_\\lambda \\big ( (\\lbrace y\\sim y \\text{ in } \\xi ^{y, w}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{x,w}\\rbrace )\\nonumber \\\\& \\hspace{99.58464pt} \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y, z}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^{x,z} \\rbrace ) \\big ) \\nu ^{\\otimes 2}\\left(\\mathrm {d}w,\\mathrm {d}z\\right).$ After applying the BK inequality to the above probability, $\\mathbb {P}_\\lambda \\left( y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) \\le \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left( \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y\\sim w \\text{ in } \\xi ^{y,w}\\right\\rbrace \\cap \\left\\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{w,x}\\right\\rbrace \\right) \\nu \\left(\\mathrm {d}w\\right) \\right)^2 \\\\= \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2.",
"$ Thus, recalling that $\\pi _\\lambda ^{(0)}(x,y) = \\mathbb {P}_\\lambda (y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x})-\\varphi (x,y) \\ge 0$ , and using the symmetry of $\\varphi $ and $\\tau _\\lambda $ , $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty } &= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\pi ^{(0)}_\\lambda \\left(x,y\\right)\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi \\left(y,u\\right)\\tau _\\lambda \\left(u,x\\right)\\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}w,\\mathrm {d}u\\right)\\nonumber \\\\&\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty }.$ We then use a supremum bound on this integral to split one of the $$ off from the others, and bound the other from above with $\\mathcal {T}_\\lambda $ to get the first two bounds of the result.",
"For the last bound of Proposition REF , we apply (REF ) and obtain $&\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{1,\\infty }\\nonumber \\\\&\\qquad = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(0)}_\\lambda \\left(\\overline{x};a,b\\right) \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\left(\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\mathrm {d}\\overline{u}\\mathcal {P}\\left(\\mathrm {d}c\\right)\\right)^2 \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b,b^\\prime \\in \\mathcal {E}}\\int \\Big [\\left(1 - \\cos \\left(k\\cdot \\left(\\overline{x}-\\overline{u}\\right)\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b^\\prime ,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{85.35826pt}+ \\left(1 - \\cos \\left(k\\cdot \\overline{u}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\Big ] \\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}+ \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty }.$ Here we have again used the symmetry of $\\varphi $ and $\\tau _\\lambda $ , and also used Lemma REF to split the cosine factor over $\\overline{x}-\\overline{u}$ and $\\overline{u}$ .",
"We can consider each term individually by using the triangle inequality.",
"By using a supremum bound, the symmetry of $\\varphi $ , and by bounding $\\varphi $ with $\\tau _\\lambda $ , we get the bound $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty }.$ Bounds on expansion coefficients In this section we aim to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ .",
"Lemmas REF and REF along with the positivity of $\\pi ^{(n)}_\\lambda (x,y)$ prove that it is sufficient to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty }$ to get bounds on both of these.",
"Much of the hard work for this has been done previously - for example in [21].",
"The novelty here is to see how the argument can be naturally written in terms of operators.",
"We inherit the following notation from [21].",
"Note that we make use of the Dirac delta function - specifically the one that holds with respect to the measure $\\nu $ .",
"They are used here primarily for convenience and to make the argument more readable.",
"In particular, they appear when applying the Mecke equation (REF ) to obtain $\\mathbb {E}\\left[ \\sum _{y\\in \\eta ^u} f(y, \\xi ^u) \\right] = \\int \\left(\\lambda +\\delta _{y,u}\\right) \\mathbb {E}_\\lambda \\left[ f\\left(y,\\xi ^{u,y}\\right)\\right] \\nu \\left(\\mathrm {d}y\\right),$ and so the factor $\\lambda +\\delta _{y,u}$ encodes a case distinction of whether point $y$ coincides with $u$ or not.",
"Definition 5.3 (The $\\psi $ functions) Let $r,s,u,w,x,y\\in \\mathbb {X}$ .",
"We first set $\\tau _\\lambda ^\\circ (x,y) := \\lambda ^{-1}\\delta _{x,y} + \\tau _\\lambda (x,y)$ .",
"Also define $\\psi _0^{(1)}(w,u,y) &:= \\lambda ^2\\tau _\\lambda (y,u)\\tau _\\lambda (u,w)\\tau _\\lambda (w,y),\\\\\\psi _0^{(2)}(w,u,y) &:=\\lambda ^2 \\delta _{w,y} \\tau _\\lambda (y,u)\\int \\tau _\\lambda (u,t)\\tau _\\lambda (t,y) \\nu \\left(\\mathrm {d}t\\right),\\\\\\psi _0^{(3)}(w,u,y) &:= \\lambda ^2\\varphi (u,y) \\left(\\lambda ^{-1}\\delta _{w,y}\\right),\\\\\\psi ^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda ^\\circ (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\psi ^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\left(\\lambda ^{-1}\\delta _{w,s}\\right)\\tau _\\lambda (u,r),\\\\\\psi _n^{(1)} (x,r,s) &:= \\lambda ^3\\int \\tau _\\lambda ^\\circ (t,s)\\tau _\\lambda (z,r)\\tau _\\lambda (t,z)\\tau _\\lambda (z,x)\\tau _\\lambda (x,t)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right),\\\\\\psi _n^{(2)}(x,r,s) &:=\\lambda \\tau _\\lambda (x,s)\\tau _\\lambda (x,r),$ and set $\\psi _0 := \\psi _0^{(1)}+\\psi _0^{(2)}+\\psi _0^{(3)}$ , $\\psi _n := \\psi _n^{(1)} + \\psi _n^{(2)}$ , and $\\psi := \\psi ^{(1)}+\\psi ^{(2)}+\\psi ^{(3)} + \\psi ^{(4)}$ .",
"Note that by using $\\lambda ^{-1}\\delta _{\\left(\\cdot ,\\cdot \\right)}$ in $\\tau _\\lambda ^\\circ $ and in $\\psi ^{(3)}_0$ , we ensure that each integral that will be truly performed (that is, not an integration over a Dirac delta function accounting for a coincidence of points) has a $\\lambda $ factor associated with it when the bound $\\pi ^{(n)}_\\lambda $ is calculated.",
"Diagrammatic representations of these functions can be found in Figure REF .",
"In these diagrams, the declared variables are represented by (0,0) circle (2pt) vertices whereas the variables that are integrated over are represented by (0,0) circle (2pt) vertices.",
"If $\\tau _\\lambda $ connects two variables then a standard edge (0,0) – (1,0); connects their vertices, if $\\tau _\\lambda ^\\circ $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; connects their vertices, and if $\\varphi $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\sim $; connects their vertices.",
"Figure: Diagrams of the ψ 0 \\psi _0, ψ\\psi , and ψ n \\psi _n functions.Definition 5.4 (The $\\Psi $ operators) Here we turn the functions of Definition REF into linear operators, with the $\\psi $ functions acting as their kernels.",
"For $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , we define $\\Psi ^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\Psi ^{(j)}_0 f\\left(w,u\\right) = \\int \\psi ^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right).$ For $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , we define $\\Psi ^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)} f\\left(w,u\\right) = \\int \\psi ^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ , we define $\\Psi ^{(j)}_n\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)}_n f\\left(x\\right) = \\int \\psi ^{(j)}_n(x,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ We can then naturally define $\\Psi _0\\colon L^2(\\mathbb {X})\\rightarrow L^2(\\mathbb {X}^2)$ , $\\Psi \\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X}^2)$ , and $\\Psi _n\\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X})$ as the sums of $\\Psi ^{(j)}_0$ , $\\Psi ^{(j)}$ , and $\\Psi ^{(j)}_n$ respectively, or equivalently as those operators having kernel functions $\\psi _0$ , $\\psi $ , and $\\psi _n$ respectively.",
"Proposition 5.5 (Bound in terms of $\\psi $ and $\\Psi $ ) Let $n \\ge 1$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in [0,\\lambda _O)$ .",
"Then $\\lambda \\pi _\\lambda ^{(n)}(x,y) \\le \\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes \\left(2n\\right)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right),$ where $\\vec{v}_i = (w_i,u_i,w_{i-1},u_{i-1})$ .",
"In operator terms, this means $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n \\Psi ^{n-1} \\Psi _0\\right\\Vert _{1,\\infty }.$ The argument for (REF ) is essentially identical to [21].",
"It uses only general properties of Poisson point process and connection models - for example thinning edges and Mecke's equation.",
"The inequality (REF ) then follows from the clear sequential structure of (REF ).",
"The inequality (REF ) suggests that we will bound the lace coefficient operators with some product of $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ terms.",
"This is nearly correct.",
"Unfortunately the norms $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ will not have the decay in $d$ that we will require.",
"We will have to consider the norms of pairs of operators.",
"Definition 5.6 Define $\\triangle _\\lambda &:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^\\circ _\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^{\\circ \\circ }_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda ^\\circ \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1,\\\\\\overline{\\triangle ^{\\circ \\circ }_\\lambda }&:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda ^\\circ \\left(\\overline{u}-\\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}.$ We can think of $\\triangle _\\lambda $ as an $\\mathbb {X}$ -convolution of three $\\tau _\\lambda $ functions with suprema taken over the end vertices.",
"Then $\\triangle ^\\circ _\\lambda $ and $\\triangle ^{\\circ \\circ }_\\lambda $ are produced by adding on the $\\mathbb {X}$ -convolution of two $\\tau _\\lambda $ functions and one $\\tau _\\lambda $ function (which will trivially take value 1 once the suprema are taken).",
"The object $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the marks at adjacent $\\tau _\\lambda $ functions need not be equal, and then we take the supremum over all the marks - not just the end ones.",
"We then use these elementary diagrams to produce composite objects: $U_\\lambda := \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }+ \\overline{\\triangle ^{\\circ \\circ }_\\lambda }^2 + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }, \\qquad V_\\lambda := \\left(\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }U_\\lambda \\right)^\\frac{1}{2}.$ We note a few relations.",
"Firstly, since all the terms in $U_\\lambda $ are non-negative we have $\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le U_\\lambda $ and thus $V_\\lambda \\le U_\\lambda $ .",
"Also note that $1\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ and thus $U_\\lambda \\ge \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .",
"In particular this means that $U_\\lambda \\ge 1$ and $U_\\lambda \\ge \\triangle ^\\circ _\\lambda $ .",
"The following lemma tells us that we can use $U_\\lambda $ to bound single operators, and that we can use $V_\\lambda ^2$ to bound pairs of operators.",
"To aid the reader's understanding, bear in mind that later (in Section ) we will prove that $U_\\lambda = \\mathcal {O}\\left(1\\right)$ , whilst $V_\\lambda $ is much smaller and is $\\mathcal {O}\\left(\\beta \\right)$ .",
"Lemma 5.7 For all $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j,j^{\\prime }\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $U_\\lambda &\\ge \\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\right\\Vert _{1,\\infty },\\\\V_\\lambda ^2 &\\ge \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j^{\\prime })}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }.$ We begin with the single operators.",
"For $\\Psi ^{(1)}$ we find $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } &= \\lambda ^4\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (w,u)\\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 4}\\left(\\mathrm {d}z, \\mathrm {d}t, \\mathrm {d}w, \\mathrm {d}u\\right), \\nonumber \\\\& \\le \\lambda ^4\\left(\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (r,z) \\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (t,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t \\right)\\right) \\nonumber \\\\&\\hspace{142.26378pt} \\times \\left(\\operatornamewithlimits{ess\\,sup}_{t^{\\prime },z^{\\prime }\\in \\mathbb {X}}\\int \\tau _\\lambda (t^{\\prime },w)\\tau _\\lambda (w,u)\\tau _\\lambda (u,z^{\\prime })\\nu ^{\\otimes 2}\\left(\\mathrm {d}w, \\mathrm {d}u\\right)\\right),\\nonumber \\\\& = \\triangle ^\\circ _\\lambda \\triangle _\\lambda ,$ where we have used a supremum bound on the $t$ and $z$ integrals to split them into $L^1$ and $L^\\infty $ bounds.",
"It is also clear that $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty }= \\triangle _\\lambda $ and $\\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\le \\triangle ^\\circ _\\lambda $ .",
"The $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ bound is more involved and we return to it in a moment.",
"We are able to represent these $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ norms and their bounds pictorially.",
"The vertices of the diagrams represent the variables $w,u,z,t,a,b$ , etc.",
"appearing in the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ expression.",
"The variables over which a $\\nu $ -integral is taken are represented by the (0,0) circle (2pt) vertices, and the variables over which a supremum is taken are represented by the (0,0) circle (2pt) vertices.",
"The presence of a $\\tau _\\lambda $ connecting two variables is then represented by a standard edge (0,0) – (1,0); between their vertices, and the presence of a $\\tau _\\lambda ^\\circ $ connecting two variables is correspondingly represented by an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; between their vertices.",
"When using a supremum bound to split the diagrams, an integrated vertex gets split into two new vertices, precisely one integrated vertex and one supremum vertex.",
"Each edge connected to such a split vertex can independently choose which of the new vertices to associate to.",
"The calculations producing the bounds on $\\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty }$ for $j=1,3,4$ can then be represented by: $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } \\quad &=\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\\\\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } + \\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad .$ Our strategy for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ is to use the spatial translation invariance to shift the origin to the bottom right vertex, and then use a supremum bound to split the diagram in two.",
"Unfortunately, this spatial shift decouples the spatial and mark components of the vertices so - for example - at some vertices we end up taking suprema over marks whilst integrating over space.",
"Let us introduce some more notation.",
"The vertex mark size=2pt] at (0,0) square* indicates that the spatial component is integrated over whilst the mark component is fixed and its supremum is taken after all the integrals.",
"We also introduce the edge-vertex combination [dashed] (0,0) – (1,0); [fill=white] (0,0) circle (2pt) which indicates that there is no term connecting the associated vertices, but there is a supremum taken over the spatial displacement.",
"Note that in principle we could use a vertex that had a spatial supremum and a mark integral, but we will always bound that probability integral by the supremum.",
"The bound for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ can therefore be expressed in the following: $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1);(0,-0.6) -- (1,-0.6);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (1.5,-0.1) circle (2pt);(1.5,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);[fill=white] (0,0) circle (2pt);\\end{tikzpicture}}.$ To clarify the calculation for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ , we write out the integral here: $&\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}-\\overline{s};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}- \\overline{w}; a_4,a_3)\\tau _\\lambda (\\overline{u}-\\overline{t};a_6,a_4)\\tau _\\lambda (\\overline{z}-\\overline{u};a_5,a_6) \\nonumber \\\\&\\hspace{142.26378pt}\\times \\tau _\\lambda (\\overline{t}- \\overline{z}; a_4,a_5)\\tau _\\lambda (\\overline{z}; a_5,a_2) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5, \\mathrm {d}a_6\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\left(\\operatornamewithlimits{ess\\,sup}_{\\overline{s},\\overline{t}^{\\prime },\\overline{z}^{\\prime }\\in \\mathbb {R}^d,a_1,a_2,a_4,a_5\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathcal {P}\\left(\\mathrm {d}a_3\\right)\\right)\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_6\\in \\mathcal {E}}\\int \\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6)\\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\right).$ Then the two components of the first factor that are joined by the $\\raisebox {2pt}{{[dashed] (0,0) -- (1,0); [fill=white] (0,0) circle (2pt)}}$ edge can be spatially translated together to form three consecutive edges.",
"Note however, that the marks on either side of this join are not necessarily equal.",
"We can therefore bound this first term by $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .",
"In summary, these manipulations then imply that $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\triangle _\\lambda $ .",
"For $\\Psi ^{(1)}_0$ and $\\Psi ^{(2)}_0$ the calculations are identical to each other, and we can once again represent the calculations pictorially: $\\left\\Vert \\Psi ^{(1)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_0\\right\\Vert _{1,\\infty } =\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- cycle;[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad = \\triangle _\\lambda ,$ where splitting the single supremum into two produces an upper bound.",
"For the third case we have $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) -- (1,0);(0.5,0) circle (0pt) node[above]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad = \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }.$ The diagrams representing the calculations for $\\Psi _n$ are as follows: $\\left\\Vert \\Psi _n^{(1)}\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(1,-0.6) -- (0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{113.81102pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\qquad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda , \\nonumber \\\\\\left\\Vert \\Psi ^{(2)}_n\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda .$ In the first case we expanded a $\\tau _\\lambda ^\\circ $ edge into a $\\tau _\\lambda $ edge and a contraction.",
"Then in the contracted case we bound $\\tau _\\lambda \\le 1$ on the diagonal edge.",
"We have now proven the bounds for the single operator norms.",
"For most of the operator pairs, we can then use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ to immediately get sufficient bounds.",
"The only pairs for which this strategy fails are: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ .",
"We now deal with these cases by hand: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- (0,0.6);(0,0) circle (0pt) node[above, rotate=90]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda , \\\\\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda .$ For the first bound we used $\\varphi \\le \\tau _\\lambda $ , and then split the single supremum into two.",
"For the second bound we used $\\tau _\\lambda \\le 1$ on the diagonal edge.",
"Proposition 5.8 For $n\\ge 1$ , $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le 6\\times 4^{n-1}U_\\lambda V_\\lambda ^n.$ Recall from Lemma REF that $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n\\Psi ^{n-1}\\Psi _0\\right\\Vert _{1,\\infty }$ .",
"We therefore aim to bound $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ for $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ .",
"We have two cases depending upon the parity of $n$ .",
"If $n$ is odd then we have an even number of operators.",
"We pair them off, use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , and Lemma REF to get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le \\left(V_\\lambda ^2\\right)^{\\frac{n+1}{2}} = V_\\lambda ^{n+1}.$ If $n$ is even, then we pair off the first $n$ operators and treat the last $\\Psi ^{(j_n)}_n$ operator by itself.",
"We then get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le U_\\lambda \\left(V_\\lambda ^2\\right)^{\\frac{n}{2}} = U_\\lambda V_\\lambda ^{n}.$ Since $U_\\lambda \\ge V_\\lambda $ , we have the bound $U_\\lambda V_\\lambda ^{n}$ for all $n\\ge 1$ .",
"Since there are 3 choices for $j_0$ , 4 choices for each of $j_1,\\ldots ,j_{n-1}$ , and 2 choices for $j_n$ , we get the pre-factor of $6\\times 4^{n-1}$ in our result by applying the triangle inequality.",
"Bounds on displaced expansion coefficients In this section we will be aiming to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ for $n\\ge 1$ .",
"Our strategy draws inspiration from the corresponding step in [21], whilst being adapted to account for inhomogeneous marks in a similar way to Section REF .",
"The central idea is to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(n)}_\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right),$ and bound $\\pi ^{(n)}_\\lambda $ using the function expression from Proposition REF .",
"This produces an integral (or diagram) composed from simpler segments with the displacement factor, $\\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)$ , spanning the whole length of the diagram.",
"We can then use the Cosine-Splitting result (Lemma REF ) to get a sum of diagrams where the displacement factor only spans a single segment of each diagram.",
"To demonstrate this, consider the diagram corresponding to $\\Psi ^{(2)}_n\\Psi ^{(1)}\\Psi ^{(1)}_0$ with the displacement factor: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8) -- (3,0.8) -- (4,0.2);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\le 3\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\right.",
"\\\\ \\left.\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (1,0.8) -- (3,0.8);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right).$ In these diagrams and hereafter we use [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; to denote that a displacement factor is connecting the two indicated vertices.",
"As above, the path the [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; takes will also suggest the route along which we will intend to use cosine-splitting.",
"For the sake of simplicity, we will be taking the displacement path across the `top' of the diagram from the perspective of the usual orientation of the $\\Psi _0$ segment.",
"However - as can be seen in the above example - the way the operators compose with each other means that the orientation of each segment inverts when compared to its neighbours.",
"Therefore the displacement may be across the `top' or the `bottom' of the $\\Psi $ and $\\Psi _n$ segments (in their usual orientation) depending upon the parity of its place in the sequence of segments.",
"Our general strategy will be to isolate the displaced segment (and perhaps a neighbouring segment) by splitting off `earlier' and `later' segments.",
"When we split off earlier segments (and sometimes later segments), we will use the $\\Psi $ structure we have been using so far.",
"However, we will sometimes want to group vertices and edges slightly differently when we split off `later' segments.",
"We use an observation from [21] to conveniently group these.",
"Let us define the kernel functions $\\overline{\\psi }^{(1)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\tau _\\lambda (y,r)\\tau _\\lambda (r,s)\\tau _\\lambda (s,x),\\\\\\overline{\\psi }^{(2)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\left(\\lambda ^{-1}\\delta _{y,r}\\right)\\left(\\lambda ^{-1}\\delta _{y,s}\\right),\\\\\\overline{\\psi }^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\overline{\\psi }^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\left(\\lambda ^{-1}\\delta _{u,r}\\right),$ and set $\\overline{\\psi }_0 := \\overline{\\psi }^{(1)}_0+\\overline{\\psi }^{(2)}_0$ and $\\overline{\\psi }:= \\overline{\\psi }^{(1)}+\\overline{\\psi }^{(2)}+\\overline{\\psi }^{(3)} + \\overline{\\psi }^{(4)}$ .",
"Diagrammatic representations of these functions can be found in Figure REF .",
"Note the similarity to the $\\psi $ functions, with the $\\tau _\\lambda ^\\circ $ edge appearing elsewhere in the integral.",
"Figure: Diagrams of the ψ ¯ 0 \\overline{\\psi }_0, and ψ ¯\\overline{\\psi } functions.For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\overline{\\Psi }^{(j)}_0 f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right),$ and for $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\overline{\\Psi }^{(j)} f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ In our terminology, the important observation of [21] was that for some $m\\ge 0$ the `later' segments of each diagram can be bounded using $\\big (\\overline{\\Psi }^m\\overline{\\Psi }_0\\big )^\\dagger $ , the adjoint of $\\overline{\\Psi }^m\\overline{\\Psi }_0$ .",
"In terms of the kernel functions, taking the adjoint in this case amounts to reflecting the `input' and `output' arguments, and for the diagrams this amounts to reflecting in the vertical plane.",
"Lemma 5.9 For $n\\ge 1$ , $m\\in \\left\\lbrace 0,\\ldots ,n\\right\\rbrace $ , $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^m,\\\\\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^{n-m},\\\\\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le {\\left\\lbrace \\begin{array}{ll}U_\\lambda &: m=0\\\\U_\\lambda ^2V_\\lambda ^{m-1} &: m\\ge 1.\\end{array}\\right.",
"}$ The argument for $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty }$ is the same as that in Lemma REF and Proposition REF .",
"We consider bounds for single operators and pairs of operators, and then use sub-multiplicity to get a bound for each diagram.",
"For $\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty }$ the argument is a little more complicated.",
"For $m=0,1$ we can easily check each diagram satisfies the bound.",
"For $m\\ge 2$ we have the issue that there exists one pair of operators which we can't bound by $V_\\lambda ^2$ : $\\left\\Vert \\overline{\\Psi }^{(4)}\\overline{\\Psi }^{(2)}_0\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) to [out=10,in=170] (1,0) to [out=190,in=350] (0,0);[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda \\le U_\\lambda ^2.$ If $m\\ge 2$ is even (an odd number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }_0$ term which we bound with $U_\\lambda $ .",
"We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda V_\\lambda ^m$ .",
"If $m\\ge 2$ is odd (an even number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }\\overline{\\Psi }_0$ pair which we bound with $U_\\lambda ^2$ .",
"We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda ^2V_\\lambda ^{m-1}$ .",
"Definition 5.10 In addition to the terms defined in Definition REF , we will now use $W_k&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int \\tau _{\\lambda ,k}(x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right) = \\lambda \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty },\\\\\\overline{W_k}&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _{\\lambda ,k}(\\overline{x}-\\overline{u};a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4)\\mathrm {d}u,\\\\\\overline{H_k}&:= \\lambda ^5 \\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,a_1,\\ldots ,a_{16}\\in \\mathcal {E}}\\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8)\\nonumber \\\\& \\hspace{42.67912pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\tau _{\\lambda ,k}(\\overline{t}-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u}.$ Note that we only use $\\overline{W_k}$ in Propositions REF and REF because $\\overline{W_k}\\ge W_k$ , but in the proof we will sometimes use $W_k$ to make the derivation easier to follow.",
"The diagrams for $\\overline{W_k}$ and $\\overline{H_k}$ are in Figure REF .",
"Also recall the sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ assumed to exist in Assumption REF .",
"Then define $\\mathbb {B} &:= \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\nu \\left(B\\left(x\\right)\\right),\\\\\\triangle ^{(B)}_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}: r\\notin {B\\left(s\\right)}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right).$ Note that these are only required to deal with one specific diagram in the following $n=1$ case.",
"Figure: Diagrams representing W k ¯\\overline{W_k} and H k ¯\\overline{H_k}.",
"In this figure, the marks on edges incident to filled square vertices need not be equal.Proposition 5.11 For $k\\in \\mathbb {R}^d$ and $n\\ge 2$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}\\left(n+1\\right)^24^{n-1}\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^4 V_\\lambda ^{n-4} &: n\\ge 4\\\\16\\times 16\\left(50\\overline{W_k}V_\\lambda ^2 + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=3\\\\9\\times 4\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=2.\\end{array}\\right.",
"}$ Proposition 5.12 For $k\\in \\mathbb {R}^d$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(1)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(1)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le 13\\overline{W_k}V_\\lambda U_\\lambda + \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\left(V_\\lambda + \\triangle ^{(B)}_\\lambda \\right) + \\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + 4\\lambda \\mathbb {B}\\overline{W_k}.$ [Proof of Proposition REF ] In bounding the diagrams with one displacement segment, we have three broad cases.",
"Let $i\\in \\left\\lbrace 0,1,\\ldots ,n\\right\\rbrace $ denote which segment the displacement lies upon.",
"Our three cases are then: the displacement lies on the $\\psi _n$ segment (i.e.",
"$i=n$ ), the displacement lies on the $\\psi _0$ segment (i.e.",
"$i=0$ ), the displacement lies on a $\\psi $ segment (i.e.",
"$i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ ).",
"We will aim to get a single bound for each diagram having a displacement crossing a single segment.",
"This will be uniform in the sequence $\\left(j_0,j_1,\\ldots ,j_n\\right)$ .",
"We then account for the number of possible choices of $\\left(j_0,j_1,\\ldots ,j_n\\right)$ , and finally include the factor of $(n+1)^2$ arising from using cosine-splitting (the explicit factor and the number of diagrams).",
"Case REF is the simplest.",
"We use a supremum bound to pull off the displaced $\\Psi _n$ term.",
"We then bound this by hand and use Lemma REF to bound the remainder.",
"In representing this scheme we use a shorthand $\\Psi ^{n-1}\\Psi _0$ to denote a particular sequence $\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0$ rather than actually the operators $\\Psi $ and $\\Psi _0$ defined above.",
"We also use labelled grey shapes to represent these groupings of diagrams, highlighting only end vertices or vertices that connect to neighbouring segments.",
"This scheme look like: $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8) -- (4,0.2);(2.8,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (2,0.8) -- (3,0.2);(2.5,0.5) circle (0pt) node[rotate = -30]{\\times };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{28.45274pt}\\le U_\\lambda V_\\lambda ^{n-1} \\left(2\\left( \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 2U_\\lambda V_\\lambda ^{n-1}\\left(\\overline{W_k}\\triangle ^{\\circ \\circ }_\\lambda + W_k\\triangle ^\\circ _\\lambda \\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 4\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that this bound also holds if the displacement runs along the bottom of the $\\Psi _n$ segment.",
"Since there are $3\\times 4^{n-1}$ possible sequences of indices for the $\\Psi ^{n-1}\\Psi _0$ diagram, this case contributes that as a pre-factor.",
"For Case REF , we need a few more tricks.",
"Since we are taking the displacement across the top of the diagram, the only term that will make a contribution is $\\Psi ^{(1)}_0$ .",
"To tackle this case we first use the observation of [21] that the diagram we want to bound can in turn be bounded by a diagram using $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ .",
"As above, we use $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ as a shorthand for a particular sequence $\\overline{\\Psi }^{(j_{n-1})}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0$ .",
"We are also imprecise in this notation over whether it is this operator or its adjoint.",
"Nevertheless the diagrams we use are clear on which vertices are integrated over and which have suprema, and this will avoid ambiguity.",
"We then spatially translate the supremum to the far end and pull off the $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram using a supremum bound.",
"This looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}\\\\\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (3,-0.6) to [out=100,in=260] (3,0.6) to [out=190,in=350] (2,0.6);\\node at (2.5,0) {\\overline{\\Psi }};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0};(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}.$ For $\\overline{\\Psi }^{(j)}$ with $j=1,2,3$ , a simple supremum bound shows that this starting diagram can be bounded by $\\overline{W_k}\\triangle ^\\circ _\\lambda \\left\\Vert \\overline{\\Psi }^{(j)}\\right\\Vert _{1,\\infty }\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2$ .",
"For $j=4$ a little more care is needed: $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ We therefore find that $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le {\\left\\lbrace \\begin{array}{ll}\\overline{W_k}V_\\lambda ^{n-1}U_\\lambda ^2 &: n\\ge 3\\\\\\overline{W_k}V_\\lambda ^2U_\\lambda &: n=2.\\end{array}\\right.",
"}$ Since there are 4 choices for the appended $\\overline{\\Psi }^{(j)}$ term, and $4^{n-2}\\times 2$ for the remaining $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram, this case contributes $2\\times 4^{n-1}$ as a pre-factor.",
"For Case REF , we shall find we have 3 sub-cases, depending not only on the $j$ -index of the displaced segment, but on which term we are looking at once it has been expanded.",
"Sub-cases $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ deal with $j=1,2,3$ .",
"In Appendix we expand out the various displaced segments using cosine-splitting and by expanding $\\tau _\\lambda ^\\circ $ edges into $\\tau _\\lambda $ edges and points, and label each of the resulting diagrams as sub-case $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ .",
"These sub-cases are distinguished by the strategy we use to bound them.",
"Sub-case $\\mathrm {\\left(III\\right)}$ then addresses the case with $j=4$ .",
"For sub-case $\\mathrm {\\left(I\\right)}$ , we just use the $\\Psi $ structure of the diagrams.",
"We first split off segments from the left of the displaced term like we did in Case REF .",
"If $i=n-1$ we can then bound the remaining two pair of segments with $\\overline{W_k}V_\\lambda ^2$ (Appendix explains how this bound is found).",
"This can be described diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda V_\\lambda ^{n}.$ If $i\\le n-2$ , then we also split off the segments to the right of this pair, and then bound the pair by $\\overline{W_k}V_\\lambda ^2$ again.",
"Diagrammatically this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (6,-0.6) to [out=40,in=200] (7,0) to [out=160,in=320] (6,0.6) to [out=190,in=350] (4,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };\\node at (5,0) {\\Psi _n\\Psi ^{n-i-2}};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(7,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda ^2V_\\lambda ^{n-3} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that the since $U_\\lambda \\ge V_\\lambda $ , the bound in (REF ) is greater than or equal to the bound in (REF ).",
"It will be convenient to find an $i$ -independent bound.",
"To this end, since the $i\\le n-2$ case only exists for $n\\ge 3$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(I\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}&: n\\ge 3\\\\\\overline{W_k}U_\\lambda V_\\lambda ^{2}&: n=2,\\end{array}\\right.",
"}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .",
"For sub-case $\\mathrm {\\left(II\\right)}$ , we use a different strategy.",
"First we split off the earlier (left) segments as we did for the first sub-case and in Case REF .",
"However we arrange the later terms differently - more like we did in Case REF .",
"We associate our displaced $\\Psi $ term with $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges from the subsequent term.",
"If the subsequent term would be $\\Psi ^{(2)}_n$ , these are actually two $\\tau _\\lambda $ edges, but in our schematic diagrams we will draw a $\\tau _\\lambda ^\\circ $ edge and bear this special case in mind when it is relevant.",
"While the displacement portrayed on the `top' of the segment could equally well be on the `bottom', the $\\tau _\\lambda ^\\circ $ edge will always be attached to the bottom in the usual orientation.",
"Everything to the right of these $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges can then be described as a sequence of $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ terms.",
"If $i=n-1$ , then we pair our augmented displaced segment with the subsequent $\\overline{\\Psi }_0$ segment.",
"We then perform a spatial translation to make the right-most vertex a supremum vertex like we did in (REF ).",
"The remaining diagram of a $\\Psi ^{(j)}$ segment and $\\overline{\\Psi }_0$ segment connected by a $\\tau _\\lambda ^\\circ $ and $\\tau _\\lambda $ edge can then be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ).",
"Recall that if $\\overline{\\Psi }_0$ is in fact $\\overline{\\Psi }^{(2)}_0$ then we have two $\\tau _\\lambda $ edges connecting the terms, not a $\\tau _\\lambda $ edge and a $\\tau _\\lambda ^\\circ $ edge.",
"We need to note this to get the $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ bound.",
"The scheme is then represented diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(6,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda V_\\lambda ^{n-2}.$ If $i\\le n-2$ , we also perform a spatial translation to move a supremum to the right-most vertex, but then also split off all $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ segments after the $\\overline{\\Psi }$ segment immediately following the displaced term.",
"Once again the remaining pair can be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ) and the split-off diagrams bounded using Lemma REF .",
"The scheme can be represented diagrammatically as $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);[fill=gray!50] (5,0.6) to [out=280,in=80] (5,-0.6) to [out=10,in=170] (7,-0.6) to [out=40,in=200] (8,0) to [out=160,in=320] (7,0.6) to [out=190,in=350] (5,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};\\node at (6,0) {\\overline{\\Psi }^{n-i-2}\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(5,0.6) circle (2pt);(5,-0.6) circle (2pt);(8,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\hspace{170.71652pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times {\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.}",
"\\nonumber \\\\&\\hspace{170.71652pt}\\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right){\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.",
"}$ Again for simplicity, we would like to get and $i$ -independent bound for such diagrams.",
"Since the $i = n-2$ case only exists for $n\\ge 3$ , and the $i\\le n-3$ case only exists for $n\\ge 4$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(II\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^3V_\\lambda ^{n-4}&: n\\ge 4\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^2&: n=3\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda &: n=2,\\end{array}\\right.",
"}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .",
"We now address sub-case $\\mathrm {\\left(III\\right)}$ , that is when the displacement crosses a $\\Psi ^{(4)}$ segment.",
"This is the simplest sub-case because when we take a displacement across $\\Psi ^{(4)}$ in isolation, we get exactly $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad =W_k.$ We will be able to use the $\\Psi $ -arrangement of segments here.",
"For $n\\ge 4$ we split off the earlier and later segments without pairing the displaced term with any of them.",
"With Lemma REF this produces the bound $\\left(U_\\lambda V_\\lambda ^{i-1}\\right)W_k\\left(U_\\lambda V_\\lambda ^{n-i-1}\\right) = \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-2} \\qquad :n\\ge 4.$ For $n=3$ we split off two segments from one side and one from the other.",
"This gives the finer bound $W_kU_\\lambda V_\\lambda ^2 \\quad :n=3.$ For $n=2$ , we use $j_0$ to indicate which $\\Psi _0$ term we have and $j_2$ to denote which $\\Psi _n$ term we have.",
"We note that if $j_0=1,2$ then $\\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ , and if $j_2 = 1$ then $\\left\\Vert \\Psi ^{(j_2)}_n\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ .",
"Therefore if $\\left(j_0,j_2\\right) \\ne \\left(3,2\\right)$ we have the bound $W_kU_\\lambda V_\\lambda $ .",
"If $\\left(j_0,j_2\\right) = \\left(3,2\\right)$ we proceed by hand.",
"We bound $\\varphi \\le \\tau _\\lambda $ and spatially shift one vertex to show $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };(1,0) circle (0pt) node[rotate=90, above]{\\sim };(0,0) circle (0pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (1,0.6) {{square*}};(1,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda .$ Carefully comparing the above bounds gives the required bounds for $n\\le 2$ .",
"Including the multiplicity of each diagram (including that arising from cosine-splitting), we find that we can bound the displaced and subsequent non-displaced pair by $\\left(24\\overline{W_k}V_\\lambda ^2 + 2\\overline{H_k}\\right)$ .",
"Note that this includes summing over the $j$ -index of the displaced term but not the index of the subsequent term.",
"Since there are $3\\times 4^{n-2}\\times 2$ options for the $j$ -indices of the other segments, this case contributes this to the counting pre-factors in our result.",
"[Proof of Proposition REF ] For $n=1$ , we can consider each of the six cases by hand.",
"Most of the diagrams can be dealt with routinely using cosine-splitting, supremum bounds, and spatial shifts.",
"We enumerate the diagrams by their $\\left(j_0,j_1\\right)$ values corresponding to the constituent segments.",
"$(1,1):& \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (0,-0.2) -- (1,-0.8) -- (2,-0.8) -- (3,-0.2);(1.5,-0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 3\\left(\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda + \\triangle _\\lambda \\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(1,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (0,0.2) -- (1,0.8) -- (2,0.2);(0.8,0.68) circle (0pt) node[rotate=30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda W_k\\right)\\\\(2,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (1,0.8) -- (2,0.8) -- (3,0.2);(1.8,0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\triangle _\\lambda \\overline{W_k}\\triangle ^\\circ _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(2,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (1,0.8) -- (2,0.2);(1.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle _\\lambda W_k\\\\(3,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6) -- cycle;(1,0.6) -- (1,-0.6);[<->] (0,0.8) -- (1,0.8) -- (2,0.2);(0.8,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(2,0) circle (2pt);(0,-0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\overline{W_k}\\triangle _\\lambda + \\overline{W_k}\\triangle _\\lambda + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\triangle _\\lambda W_k+ \\triangle _\\lambda W_k\\right)$ These terms sum to no more than $13\\overline{W_k}V_\\lambda U_\\lambda .$ For the $\\left(3,2\\right)$ diagram, we use a different `trick.'",
"In Lemma REF we will prove the bound $\\tau _\\lambda \\left(x,y\\right) \\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right)$ .",
"When applied to this diagram, this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (1.2,0.6) -- (1.2,-0.6);(1.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right),$ Here we have also used the cosine-splitting result on the second part.",
"For the first of these 4-vertex diagrams, we split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\lambda \\triangle _\\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }$ .",
"For the second of the 4-vertex diagrams, we also split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ .",
"For the 3-vertex diagram we condition on whether two of the vertices are in each other's $B$ -set (defined in Assumption REF ).",
"In the following diagrams [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\in B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\in B\\left(u\\right)\\right\\rbrace }$ relates two variables, and [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\notin B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\notin B\\left(u\\right)\\right\\rbrace }$ relates them: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.7) -- (0,-0.8) -- (1,-0.2);(-0.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-20pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,-0.8) -- (1,-0.2);(0.5,-0.5) circle (0pt) node[rotate=30]{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right) \\nonumber \\\\&\\le \\left(\\quad \\raisebox {-7pt}{\\begin{tikzpicture}(0,0) -- (1,0);[<->] (0,0.2) -- (1,0.2);(0.5,0.2) circle (0pt) node{\\times };(0.5,0) circle (0pt) node[below]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node[above, rotate=90]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) +4\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);[<->] (0,-0.2) -- (1,-0.2);(0.5,-0.2) circle (0pt) node[below]{\\scriptsize \\in B};\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right)\\nonumber \\\\& \\le \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\triangle ^{(B)}_\\lambda + 4\\lambda \\mathbb {B}W_k.$ Bootstrap analysis In this section we will be assuming that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = 1.$ By Lemma REF this also implies $\\mathbb {S}\\left(\\right)\\le 1$ .",
"A re-scaling argument similar to [36] explains how this can be done while maintaining generality.",
"First note that since $\\widehat{\\varphi }(0;a,b)>0$ for a positive $\\mathcal {P}$ -measure set of $a,b\\in \\mathcal {E}$ , we have $\\mathbb {S}\\left(\\widehat{}(0)\\right)>0$ .",
"Now suppose we scale $\\mathbb {R}^d$ by a factor of $q^{-1/d}$ (that is, the new unit radius ball is the previous ball of radius $q^{1/d}$ ), and leave the mark space $\\mathcal {E}$ unchanged.",
"Then we find that the system we are left with has the distribution of an RCM model with the new parameters $\\lambda ^* = \\mathbb {E}_\\lambda \\left[*{\\eta \\cap \\left[0, q^{\\frac{1}{d}}\\right]^d }\\right] = \\lambda q, \\qquad \\varphi ^*\\left(\\overline{x};a,b\\right) = \\varphi \\left(q^{\\frac{1}{d}}\\overline{x};a,b \\right).$ Therefore for all $a,b\\in \\mathcal {E}$ we have $\\widehat{\\varphi }^*(\\overline{x};a,b) = \\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\widehat{\\varphi }(\\overline{x};a,b),$ and $\\mathbb {S}\\left(\\widehat{}^*(0)\\right) = q^{-1} \\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"We can therefore achieve our assumption by choosing $q=\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"Furthermore, under this scaling we find that the old event $\\left\\lbrace \\left(\\overline{x},a\\right) \\longleftrightarrow \\left(\\overline{y},b\\right)\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ becomes the new event $\\left\\lbrace \\left(q^{-1/d}\\overline{x},a\\right) \\longleftrightarrow \\left(q^{-1/d}\\overline{y},b\\right)\\textrm { in } \\xi ^{\\left(q^{-1/d}\\overline{x},a\\right),\\left(q^{-1/d}\\overline{y},b\\right)} \\right\\rbrace $ , and we get $\\tau ^*_{\\lambda ^*}(\\overline{x};a,b) = \\tau _\\lambda \\left(q^\\frac{1}{d}\\overline{x};a,b\\right)$ where $\\tau ^*_{\\lambda ^*}$ is the two-point function in the RCM governed by the connection function $\\varphi ^*$ and intensity $\\lambda ^*$ .",
"It is also worth noting that the scaling choice to have $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ means that REF reduces to the condition that there exists a $d$ -independent constant $C>0$ such that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } &\\le C,\\\\\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } &\\le C \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Note that the inequalities relating $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ follow from Jensen's inequality and a supremum bound on a probability space.",
"The Bootstrap Function To show that the bounds arising from the diagrams in Section are sufficient for our purposes, we make use of a bootstrap argument.",
"Define $\\mu _\\lambda & = 1- \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)},\\\\\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) &= \\frac{1}{1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)},\\\\\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) &=\\left(1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\left(\\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right)\\right).$ From these we can then define the bootstrap function $f= f_1\\vee f_2\\vee f_3$ , where $f_1\\left(\\lambda \\right) = \\lambda , \\qquad f_2\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}, \\qquad f_3\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)}.$ Let us now outline some initial properties of $\\mu _\\lambda $ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .",
"Lemma 6.1 The map $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous and bounded below by $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ on $\\left[0,\\lambda _O\\right)$ .",
"Therefore the map $\\lambda \\mapsto \\mu _\\lambda $ is a continuous map from $\\left[0,\\lambda _O\\right)\\rightarrow \\left[0,1\\right)$ .",
"Furthermore, $\\widehat{G}_{\\mu _\\lambda }(k)$ is bounded below by $\\left(1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)^{-1}$ for all $k\\in \\mathbb {R}^d$ .",
"We first recall that $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (0)$ is differentiable with respect to the operator norm from Corollary REF , and is therefore continuous.",
"Lemma REF then implies that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous.",
"For the bounded below property, we use Lemma REF to get $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge \\mathbb {S}\\left(\\widehat{}(0)\\right) = 1$ .",
"The continuity of $\\mu _\\lambda $ follows directly from the continuity and positivity of $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ .",
"The bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ bounds $\\mu _\\lambda \\ge 0$ , and $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ shows that $\\mu _\\lambda <1$ for all $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"The case $\\mu _0 = 0$ follows immediately from $\\mathcal {T}_0 = $ .",
"Because $\\mu _\\lambda \\in \\left[0,1\\right)$ , to minimise $\\widehat{G}_{\\mu _\\lambda }(k)$ we want to find a lower bound for $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ .",
"The $k$ -uniform bound on $\\widehat{G}_{\\mu _\\lambda }(k)$ then follows from $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\ge -\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\rm op} \\ge -\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}$ .",
"Using the Bootstrap function to bound terms Bounding the Lace Operator We have managed to bound the lace operator and its displacement in terms of a variety of objects.",
"For the lace operator itself (and its associated Fourier transformed operators), these are: $\\lambda , \\triangle _\\lambda , \\overline{\\triangle ^{\\circ \\circ }_\\lambda }, \\left\\Vert \\right\\Vert _{1,\\infty }.$ As a guide, we expect $\\lambda $ , $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ , and $\\left\\Vert \\right\\Vert _{1,\\infty }$ to be bounded uniformly in $d$ and $\\lambda <\\lambda _O$ , whereas we expect $\\triangle _\\lambda $ to decay uniformly in $\\lambda <\\lambda _O$ as $d\\rightarrow \\infty $ .",
"For some, these properties are easy to prove.",
"The boundedness of $\\left\\Vert \\right\\Vert _{1,\\infty }$ follows from $\\left\\Vert \\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we have used REF via (REF ).",
"Regarding $\\lambda $ , since we are only interested in $\\lambda <\\lambda _O$ the question is perhaps more clearly written as “Is $\\lambda _O$ uniformly bounded in $d$ ?",
"\".",
"While there may be more elementary ways of proving this directly, we will get it for free from the bootstrap argument.",
"Since we have the bound $\\lambda \\le f(\\lambda )$ , proving the uniform boundedness of $f(\\lambda )$ with respect to $d$ and $\\lambda $ for $\\lambda <\\lambda _O$ will prove the same for $\\lambda _O$ .",
"This will leave us with only $\\triangle _\\lambda $ and $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .",
"Recall that we can write $\\triangle _\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }, \\quad \\triangle ^{\\circ \\circ }_\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1.$ Then $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the supremum is taken over all the marks, and adjacent $\\tau _\\lambda $ can have different marks.",
"To get the desired decay and boundedness behaviour, we will therefore only need to prove that $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ decays and to bound $&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v},\\\\&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}.$ We begin by proving the decay of $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ .",
"Lemma 6.2 Let $x,y \\in \\mathbb {X}$ .",
"Then $\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\varphi (x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right),\\\\\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right).$ Then for $n\\ge 2$ , $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le \\sum _{m=0}^n\\genfrac(){0.0pt}0{n}{m}\\lambda ^{m} \\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }.$ By combining Mecke's formula and the BK inequality, we obtain $\\tau _\\lambda (x,y) & \\le \\varphi (x,y) + \\mathbb {E}_\\lambda \\left[\\sum _{u \\in \\eta } 1_{\\lbrace x \\sim u \\text{ in } \\xi ^{x} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y}\\rbrace } \\right] \\nonumber \\\\& = \\varphi (x,y) + \\lambda \\int \\mathbb {P}_\\lambda \\left(\\lbrace x \\sim u \\text{ in } \\xi ^{x,u} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y,u}\\rbrace \\right) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& = \\varphi (x,y) +\\lambda \\int \\varphi (x,u) \\tau _\\lambda (u,y) \\nu \\left(\\mathrm {d}u\\right).$ In the last line, we have used that the two intersected events are independent.",
"This is due to the fact that $\\nu $ is non-atomic and $x\\notin \\eta ^{y,u}$ a.s.",
"The second inequality is then proven nearly identically, but with the adjacency to the intermediate point $u$ holding for $y$ instead of $x$ .",
"To get (REF ), we first use () to expand out the right-most $\\mathcal {T}_\\lambda $ .",
"This produces one term with $$ replacing the $\\mathcal {T}_\\lambda $ , and one term with $\\lambda \\mathcal {T}_\\lambda $ replacing it.",
"We then repeatedly use (REF ) to expand the left-most $\\mathcal {T}_\\lambda $ to get a $$ factor and a $\\lambda \\mathcal {T}_\\lambda $ factor until we have a total of $n$ factors of $$ in the expression.",
"Counting the number of ways to get to each possible end term then gives the binomial $\\genfrac(){0.0pt}1{n}{m}$ factor as required.",
"The following lemma will allow us to describe $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ using Hilbert space language.",
"Lemma 6.3 Let $H\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ be an integral linear operator with kernel function $h\\colon \\mathcal {E}^2\\rightarrow \\mathbb {R}$ .",
"Then for all $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ and $\\varepsilon >0$ , there exist $\\mathcal {P}$ -positive sets $E_1,E_2\\subset \\mathcal {E}$ such that $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ satisfy $*{h\\left(a,b\\right) - {g_1}{Hg_2}} \\le \\varepsilon .$ Since $h$ is measurable, the set $A=\\left\\lbrace c,c^{\\prime }\\in \\mathcal {E}:*{h\\left(c,c^{\\prime }\\right)-h\\left(a,b\\right)}\\le \\varepsilon \\right\\rbrace $ is measurable.",
"Since $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ , it has positive measure: $\\mathcal {P}\\left(A\\right)>0$ .",
"Furthermore, by considering the $\\pi $ -system of product sets $\\left\\lbrace E_1\\times E_2: E_1,E_2\\subset \\mathcal {E}\\text{ measurable}\\right\\rbrace $ that generates the $\\sigma $ -algebra on $\\mathcal {E}^2$ , we can find $B = E_1\\times E_2$ such that $E_1,E_2\\subset \\mathcal {E}$ are both measurable, $B\\subset A$ , and $\\mathcal {P}^{\\otimes 2}\\left(B\\right) = \\mathcal {P}\\left(E_1\\right)\\mathcal {P}\\left(E_2\\right)>0$ .",
"Then use $E_1,E_2$ to define $g_1,g_2$ as in the statement on the lemma.",
"Since $E_1,E_2$ are both measurable and have positive measure, both $g_1,g_2\\in L^2\\left(\\mathcal {E}\\right)$ .",
"We also have ${g_1}{Hg_2} = \\frac{1}{\\mathcal {P}^{\\otimes 2}\\left(B\\right)}\\int _B h\\left(c,c^{\\prime }\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right) \\in \\left[h(a,b) - \\varepsilon ,h(a,b) +\\varepsilon \\right],$ proving the result.",
"Let us define the function $\\overline{\\epsilon }\\colon \\mathbb {N}\\rightarrow \\mathbb {R}_+$ , where $\\overline{\\epsilon }(d) := {\\left\\lbrace \\begin{array}{ll}\\frac{1}{d} + \\frac{\\log d}{\\log g(d)} &: \\lim _{d\\rightarrow \\infty }g(d)\\rho ^{-d}\\Gamma \\left(\\frac{d}{2}+1\\right)^2 = 0 \\qquad \\forall \\rho >0,\\\\0&: \\text{otherwise}.\\end{array}\\right.",
"}$ Recall $g(d)$ is the function defined in REF .",
"In particular, $\\overline{\\epsilon }(d) = 0$ unless $g(d)$ approaches zero particularly quickly, and $\\beta (d) = g(d)^{\\frac{1}{4}-\\frac{3}{2}\\overline{\\epsilon }(d)}$ .",
"Lemma 6.4 Let $\\lambda <\\lambda _O$ , $n\\ge 2$ , and $d>2m$ .",
"Then there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=2\\\\c_fg(d)^{\\frac{1}{2}-m\\overline{\\epsilon }(d)} &: n\\ge 3.\\end{array}\\right.",
"}$ The condition $\\lambda < \\lambda _O$ here is required to allow us to swap integrals over the mark space with integrals over the Fourier space.",
"We are able to use Fubini's Theorem because sub-criticality implies that the relevant integrals are finite.",
"For clarity, let us introduce the notation $\\varphi ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\varphi (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),\\quad \\tau _\\lambda ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\tau _\\lambda (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),$ where $u_0=x$ and $u_n=y$ .",
"Furthermore, for each $\\overline{x}\\in \\mathbb {R}^d$ let us define the integral linear operator $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) \\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $(a,b)\\mapsto \\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) := \\int \\varphi ^{(n-1)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda ^{(m)}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime }\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ Given $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $a,b\\in \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right)\\in {\\rm ess.Im}\\left(\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};\\cdot ,\\cdot \\right)\\right)$ , we can use Lemma REF to get measurable and $\\mathcal {P}$ -positive sets $E^{(\\overline{x},a,b,\\delta )}_1,E^{(\\overline{x},a,b,\\delta )}_2\\subset \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) \\le *{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2} + \\delta ,$ where $g^{(\\overline{x},a,b,\\delta )}_i(c) = \\frac{1}{\\mathcal {P}\\left(E^{(\\overline{x},a,b,\\delta )}_i\\right)}1_{E^{(\\overline{x},a,b,\\delta )}_i}\\left(c\\right)$ for $i=1,2$ .",
"Now let $E_1,E_2\\subset \\mathcal {E}$ be arbitrary measurable $\\mathcal {P}$ -positive sets and $g_1,g_2$ be their associated functions.",
"We aim to produce bounds that are independent of the choice of $E_1,E_2$ .",
"We first use the Fourier inversion theorem to reformulate the position behaviour in terms of a $k$ -integral of the Fourier transforms, and use REF and $\\lambda <\\lambda _O$ (so the relevant integrals are finite) to allow us to swap the $k$ -integral and the mark integrals.",
"This produces $*{*{g_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g_2}} = *{\\int \\text{e}^{-i\\overline{x}\\cdot k}*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}}\\\\ \\le \\int *{*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ Recall that $\\widehat{}(k)$ is self-adjoint, so we can move $\\widehat{}(k)^{n-1}$ to the other side of the inner product, and then we can use Cauchy-Schwarz and the definition of the operator norm to get $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} = *{\\widehat{}(k)^{n-1}g_1}{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\\\\\le *{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}_2 \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}^m*{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{}(k)g_2}_2.$ We can then use the definition of the bootstrap function $f(\\lambda )$ to replace the $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ factors with factors of $f(\\lambda )$ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .",
"Also writing the norm $*{\\cdot }_2$ as the square root of an inner product and using the self-adjoint property of $\\widehat{}(k)$ gives $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\le f(\\lambda )^m\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}.$ We therefore want to derive bounds for the integral $\\int \\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ We first derive pointwise and integral bounds for the $\\widehat{}$ factors.",
"Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\widehat{\\varphi }^{(2N)}(k;c,c^{\\prime })\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right)^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2}.$ By using Cauchy-Schwarz and supremum bounds we can split this into norms of $\\widehat{}(k)$ only, then use Lemma REF to replace $k$ with 0, and finally use REF to bound these terms: $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2}\\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty }^{N-1} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }^{N-1} \\le C^N.$ For the integral bound, we apply Cauchy-Schwarz to the $k$ -integral and swap the $k$ -integrals with the mark integrals to un-do the Fourier transforms.",
"We get $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\left(\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2}\\left(\\int *{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2} \\\\= *{g_1}{^{(2n-2)}\\left(\\overline{0}\\right)g_1}^\\frac{1}{2}*{g_2}{^{(2)}\\left(\\overline{0}\\right)g_2}^\\frac{1}{2},$ where $^{(N)}\\left(\\overline{0}\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ is the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\varphi ^{(N)}\\left(\\overline{0};a,b\\right)$ .",
"Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{^{(2N)}\\left(\\overline{0}\\right)g} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\varphi ^{(2N)}\\left(\\overline{0};c,c^{\\prime }\\right)\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right) \\le \\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }.$ Here for $N\\ge 2$ we bound with $\\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert ^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le g(d)^\\frac{1}{2}$ from REF .",
"For $N=1$ we bound using $\\varphi \\left(\\overline{x};a,b\\right)\\in \\left[0,1\\right]$ and REF : $\\left\\Vert ^{(2)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,c\\right)\\varphi \\left(-\\overline{x};c,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}c\\right) \\le \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\\\= \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }\\le C.$ Therefore $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le {\\left\\lbrace \\begin{array}{ll}C &: n=2,\\\\C^\\frac{1}{2}g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.",
"}$ We now return to bounding (REF ).",
"To perform the $k$ -integral, we split the domain.",
"We first consider $B_\\varepsilon (0) = \\lbrace k\\in \\mathbb {R}^d: *{k} < \\varepsilon \\rbrace $ , where we choose $\\varepsilon >0$ such that $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1,C_2$ coming from the assumption REF .",
"From REF , we have $\\widehat{G}_{\\mu _\\lambda }(k) = \\left(1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)^{-1}\\le 1/C_2*{k}^2$ on $B_\\varepsilon (0)$ and $\\widehat{G}_{\\mu _\\lambda }(k)\\le 1/\\left(C_2\\varepsilon ^2\\right)$ on $B_\\varepsilon (0)^c$ .",
"Our pointwise bound on the $\\widehat{}$ factors then gives $\\int _{B_\\varepsilon (0)}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\int ^\\varepsilon _0\\frac{1}{r^{2m}}r^{d-1}\\mathrm {d}r = \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{d-2m}\\frac{\\varepsilon ^{d-2m}}{\\left(2\\pi \\right)^d},$ where $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.",
"Note that for all fixed $\\varepsilon >0$ this term vanishes in the $d\\rightarrow \\infty $ limit.",
"To deal with the integral over $B_\\varepsilon (0)^c$ , we find the upper bound $\\int _{B_\\varepsilon (0)^c}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{1}{C^m_2\\varepsilon ^{2m}}\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\frac{C^\\frac{1}{2}}{C^m_2\\varepsilon ^{2m}}{\\left\\lbrace \\begin{array}{ll}C^\\frac{1}{2} &: n=2\\\\g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.",
"}$ Note that the bounds we found were independent of the sets $E_1,E_2$ , and therefore these bounds also apply to $*{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2}$ uniformly over every $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $\\mathcal {P}$ -almost every $a,b\\in \\mathcal {E}$ .",
"Since we can take $\\delta \\rightarrow 0$ , our bound also applies to $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }$ .",
"For $n=2$ , the $B_\\varepsilon (0)^c$ integral dominates the $B_\\varepsilon (0)$ integral, and so the result is proven.",
"For $n\\ge 3$ , both parts of the integral approach 0 as $d\\rightarrow \\infty $ .",
"If both are of the same order or if the $B_\\varepsilon (0)^c$ integral dominates, then we can fix $\\varepsilon >0$ and get the result.",
"On the other hand, if $g(d)^\\frac{1}{2} \\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^{d}$ for all fixed $\\varepsilon >0$ and the $B_\\varepsilon (0)$ integral always dominates, we can improve the overall bound by letting $\\varepsilon =\\varepsilon (d)$ and having $\\varepsilon (d)\\rightarrow 0$ as $d\\rightarrow \\infty $ .",
"Having a smaller value of $\\varepsilon $ produces a smaller bound for the $B_\\varepsilon (0)$ integral, but a larger bound for the $B_\\varepsilon (0)^c$ integral.",
"To get an optimal $\\varepsilon $ we can set both terms to be of the same order - that is by having $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{2m}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .",
"For this choice, we have both integrals of the order $\\varepsilon ^{-2m}g(d)^\\frac{1}{2} = 2^{-2m}\\pi ^{-m}A^{-\\frac{2m}{d}} g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\left(1-\\frac{2m}{d}\\right)^{-\\frac{2m}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{2m}{d}} \\\\= \\left(\\frac{\\text{e}}{2\\pi }\\right)^m g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\frac{1}{d^m}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .",
"Proposition 6.5 (Bound for the $n$ -gon diagrams) Let $\\lambda < \\lambda _O$ .",
"Then for each $n$ such that $d>2n$ , there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=1,2,\\\\c_fg(d)^{\\frac{1}{2}-n\\overline{\\epsilon }(d)}&: n\\ge 3.\\end{array}\\right.",
"}$ The $n=1$ case is trivial because $\\tau _\\lambda \\left(\\overline{x},a,b\\right)\\in \\left[0,1\\right]$ .",
"For $n=2$ , we first bound $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty }$ .",
"We get $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int *{\\varphi (x,u)\\varphi (u,y)}\\nu (\\mathrm {d}u) \\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}^2\\nu (\\mathrm {d}u) \\\\\\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}\\nu (\\mathrm {d}u) = \\left\\Vert \\right\\Vert _{1,\\infty },$ by using Cauchy-Schwarz and symmetry of $\\varphi $ in the first inequality and $\\varphi (x,u)\\in \\left[0,1\\right]$ in the second.",
"We then use REF to bound this by the constant $C$ .",
"We use Lemma REF to get the required bound for the remaining terms arising from (REF ) in Lemma REF .",
"For $n\\ge 3$ , we once again bound the terms in the relevant expansion in (REF ).",
"For $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }$ we use REF to get $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }= O\\left(g(d)^\\frac{1}{2}\\right)$ .",
"We then use Lemma REF to bound the remaining terms.",
"For our proofs of Lemma REF and Lemma REF , we adopt the following notation.",
"Given $g(\\overline{x};a,b)$ and $h(\\overline{x};a,b)$ , we define $(gh)(\\overline{x};a,b) := \\int g(\\overline{x}-\\overline{y};a,c)h(\\overline{y};c,b)\\mathrm {d}\\overline{y}\\mathcal {P}(\\mathrm {d}c).$ This notational convention is associative and therefore can generalise to three or more terms unambiguously.",
"Lemma 6.6 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}\\le c_f,\\\\\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le c_f,$ and therefore $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le 1 + 2c_f.$ By applying (REF ) and (), we can get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ We will in fact prove the more general result for the convolution of $m\\ge 2$ $\\tau _\\lambda $ -functions in dimensions $d>2m$ .",
"By bounding $\\lambda \\le f(\\lambda )$ , and using Lemma REF and the Fourier inversion theorem we can realise that we only need to get further bounds for the following objects.",
"For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^m$ , we want to bound $\\int \\prod ^m_{i=1} P^{j_i}_{2i-1,2i}(k)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{p,q}(k) := {\\left\\lbrace \\begin{array}{ll}*{*{g_p}{\\widehat{}(k) g_q}} &:j=1\\\\*{*{g_p}{\\widehat{}(k)^2g_q}} &:j=2\\\\*{*{g_p}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_q}} &: j=3,\\end{array}\\right.",
"}$ where $\\left\\lbrace g_i\\right\\rbrace _{i=1}^{2m}$ are the functions of the form $g_i(a) = \\frac{1}{\\mathcal {P}(E_i)}1_{E_i}(a)$ arising from the application of Lemma REF .",
"We now bound the following terms in the same way as (REF ): $*{*{g_1}{\\widehat{}(k)^2g_2}} &\\le *{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}\\\\*{*{g_1}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_2}} &\\le f\\left(\\lambda \\right)\\widehat{G}_{\\mu _\\lambda }(k)*{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}.$ In terms of pointwise uniform bounds for the $\\widehat{}$ terms, we have $*{*{g_1}{\\widehat{}(k)g_2}}\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C$ and $*{g}{\\widehat{}(k)^2g}\\le C^2$ (the latter also having been used in the proof of Lemma REF ).",
"As in Lemma REF , we will require various integral bounds for these $\\widehat{}$ terms.",
"Suppose we have $r \\in \\left\\lbrace 1,\\ldots ,m-1\\right\\rbrace $ instances of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and $m-r$ instances of $P^{1}_{p,q}(k)$ .",
"Then by applying Cauchy-Schwarz and the definitions of the $g_i$ we get $&\\int \\prod ^{r}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\prod ^{m}_{i^{\\prime }=r+1}*{*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int \\prod ^{r}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\prod ^{m}_{i^{\\prime }=r+1}*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2r-1},a_{2r})\\right.\\right.\\nonumber \\\\&\\hspace{199.16928pt}\\left.\\left.\\star \\varphi (\\cdot ;a_{2r+1},a_{2r+2})\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m+r}.$ In this last inequality we have used $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\le C$ and $\\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }^2 \\le C^2$ to extract off $\\varphi $ and $\\varphi ^{(2)}$ via supremum bounds on the spatial position.",
"We will also require integral bounds where there are $m$ factors of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and where there are $m$ factors of $P^{1}_{p,q}(k)$ : $&\\int \\prod ^{m}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\left(\\int \\prod ^{m}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{2m}.\\\\&\\int \\prod ^{m}_{i=1}*{*{g_{2i-1}}{\\widehat{}(k)g_{2i}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)g_2}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\left(\\prod ^{m}_{i=2}*{g_{2i-1}}{\\widehat{}(k)g_{2i}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m}.$ Note that for $m\\ge 3$ , better bounds are available via REF , but will not be required here.",
"We can now proceed to bound each of the expressions of the form (REF ) in much the same way as we did in the proof of Lemma REF .",
"We set $\\varepsilon >0$ and partition the integral over $\\mathbb {R}^d$ into one over $B_\\varepsilon (0)$ and one over $B_\\varepsilon (0)^c$ .",
"For the $B_\\varepsilon (0)$ integral we use the uniform bounds on the $\\widehat{}$ terms and perform the integral of the $\\widehat{G}_{\\mu _\\lambda }$ terms (if there are any) as before.",
"Note that we require $d>6$ for all of these to be finite.",
"For the $B_\\varepsilon (0)^c$ integrals we uniformly bound $\\widehat{G}_{\\mu _\\lambda }(k) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and use the integral bounds we calculated above.",
"Since we are only asking for a constant bound, we don't need to worry about having $\\varepsilon \\rightarrow 0$ as $d\\rightarrow \\infty $ .",
"Bounding the Lace Operator Displacement We now consider the objects required to bound the displacement of the lace operator.",
"In addition to the objects used to bound the lace operator itself, we require bounds on $\\left\\Vert _k\\right\\Vert _{1,\\infty }, \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }, \\overline{W_k}, \\overline{H_k}, \\triangle ^{(B)}_\\lambda , \\mathbb {B}.$ The decay $\\mathbb {B}\\le g(d)$ follows from Assumption REF , and the following Observation REF demonstrates the decay of $\\triangle ^{(B)}_\\lambda $ .",
"Observation 6.7 Note that the inequality (REF ) allows us to bound $\\triangle ^{(B)}_\\lambda $ in terms of other terms.",
"By using the inequality twice and appropriately bounding $\\varphi $ with $\\tau _\\lambda $ , we get $\\int \\tau _\\lambda (r,u)\\tau _\\lambda (u,s)\\nu (\\mathrm {d}u) \\le \\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) + 2\\lambda \\int \\tau _\\lambda (r,v)\\tau _\\lambda (v,u)\\tau _\\lambda (u,s)\\nu ^{\\otimes 2}(\\mathrm {d}u, \\mathrm {d}v)$ for any $r,s\\in \\mathbb {X}$ .",
"Therefore $\\triangle ^{(B)}_\\lambda \\le 3\\triangle _\\lambda + \\lambda \\operatornamewithlimits{ess\\,sup}_{r\\notin B(s)}\\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) \\le 3\\triangle _\\lambda + \\lambda g(d),$ where we have used REF .",
"Therefore Proposition REF implies that for $d>6$ there exists $c_f$ such that $\\triangle ^{(B)}_\\lambda \\le c_f\\beta ^2.$ We are then left with $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ , $\\overline{W_k}$ , and $\\overline{H_k}$ , and we will deal them in this order.",
"First note that $\\left\\Vert _k\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi _k(x,y)\\nu \\left(\\mathrm {d}x\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int (1-\\cos \\left(k\\cdot \\overline{x}\\right))\\varphi (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}(\\mathrm {d}a) \\\\ = \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty }.$ The following lemma therefore also allows us to bound $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ .",
"Its extra generality will be required later.",
"Lemma 6.8 For all $k,l\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\infty ,\\infty } \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where $C>0$ is the $d$ -independent constant given in REF .",
"The first two inequalities in (REF ) holds from the general inequalities that holds for these norms (on a probability space for the second inequality).",
"Then we can remove the $l$ -dependence by performing the following calculation: $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+ k\\right)\\right\\Vert _{\\infty ,\\infty } \\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\text{e}^{il\\cdot \\overline{x}}}*{\\left(1-\\cos \\left( k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}\\\\= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}= \\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left( k\\right)\\right\\Vert _{\\infty ,\\infty }.$ The final step relating the $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ -norm to the difference of the spectral suprema is precisely that given by () in REF .",
"We now address $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ .",
"Lemma 6.9 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ First note that for $k=0$ the kernel function $\\tau _{\\lambda ,k}\\left(x,y\\right)=0$ for all $x,y\\in \\mathbb {X}$ , and so the required inequality holds trivially.",
"In this proof we hereafter may assume $k\\ne 0$ .",
"We use (REF ) to write $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }.$ We first get the bound on $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ , using a similar approach to Lemma REF .",
"For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda \\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}^{\\prime }-\\overline{u}^{\\prime \\prime };c^{\\prime },c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime \\prime };c^{\\prime \\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathrm {d}\\overline{u}^{\\prime \\prime }\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime },\\mathrm {d}c^{\\prime \\prime }\\right).$ Recall the notation $\\varphi ^{(2)}$ used in (REF ).",
"As in the proof of Lemma REF , we let $E_1,E_2\\subset \\mathcal {E}$ be measurable and $\\mathcal {P}$ -positive sets, and aim to bound ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ (where $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ ) independently of the choice of $E_1,E_2$ .",
"If we do this we will have proven the result.",
"As before, we use the Fourier inversion theorem to write ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ as the integral of an inner product of Fourier transformed operators over the Fourier argument.",
"Having $\\lambda <\\lambda _O$ allows us to swap the integrals in this step.",
"Then we can use Cauchy-Schwarz and the definitions of the operator norm and the bootstrap functions to extract factors of $\\widehat{G}_{\\mu _\\lambda }(l)$ and $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ from the inner product.",
"The net result is $*{g_1}{\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} &\\le \\int *{*{g_1}{\\widehat{}(l)^{2}\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l)g_2}}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\le f(\\lambda )^2\\int \\widehat{G}_{\\mu _\\lambda }(l) \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ The $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ We partition our integral using open $\\varepsilon $ -balls around the poles where $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1$ and $C_2$ coming from REF .",
"Let $B_{\\varepsilon }(p)$ denote the open $\\varepsilon $ -ball around $p\\in \\mathbb {R}^d$ .",
"For $n\\in \\left\\lbrace 0,1,2,3\\right\\rbrace $ , let $A_n := \\left\\lbrace l\\in \\mathbb {R}^d: l\\in B_\\varepsilon (p) \\text{ for precisely $n$ elements }p\\in \\left\\lbrace k,0,-k\\right\\rbrace \\right\\rbrace .$ Firstly it is possible for $A_3,A_2 = \\emptyset $ if $*{k}$ is sufficiently large compared to $\\varepsilon $ .",
"It is also easy to see that $A_3\\subset B_{\\varepsilon }(0)$ and $A_2\\subset B_{2\\varepsilon }(0)$ .",
"We will partition $\\mathbb {R}^d$ into $A_3$ , $A_2$ , $A_1$ , and $A_0$ and bound the $l$ -integral on each part.",
"Recall that REF implies that $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2*{l}^{-2}$ for $*{l}\\le \\varepsilon $ and $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2\\varepsilon ^2$ for $*{l}\\ge \\varepsilon $ .",
"Also recall the pointwise (REF ) and integral (REF ) bounds we derived for the $\\widehat{}$ factors.",
"We treat $A_3$ first: $&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)\\cap B_\\varepsilon (k)}\\frac{1}{*{l}^4*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\qquad \\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\\\&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{A_3}\\frac{1}{*{l+k}^2*{l}^2*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (-k)}\\frac{1}{*{l+k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ Recall $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.",
"In these calculations we used Hölder's inequality and increased the domain of integration to get upper bounds.",
"These integrals are finite for $d>6$ , and approach 0 if we take bounded $\\varepsilon $ .",
"For $A_2$ the counting of cases is the only extra complication.",
"A precise counting is entirely possible, but unnecessary for our purposes.",
"It is simple to see that the arguments outlined for $A_3$ can be applied to get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^6} + \\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a pair of two distinct points from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with three, two, and one factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on these two points.",
"This bound is finite for $d>6$ , and approaches 0 if we take bounded $\\varepsilon $ .",
"Repeating this for $A_1$ , we get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} + \\frac{1}{\\varepsilon ^6} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a single point from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with two, one, and zero factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on this single point.",
"This bound is finite for $d>4$ , and approaches 0 if we take bounded $\\varepsilon $ .",
"The $A_0$ integral is qualitatively different.",
"By taking the integral bound for the $\\widehat{}$ factors and a uniform (on $A_0$ ) bound for the factors of $\\widehat{G}_{\\mu _\\lambda }$ , we get the bound $\\frac{C^3}{C^3_2}\\varepsilon ^{-6}g(d)^\\frac{1}{2}.$ As in Lemma REF , if the $g(d)^\\frac{1}{2}$ term dominates, then we have a sufficient bound for our result.",
"The largest of the other terms (for $d$ sufficiently large and $\\varepsilon \\le 1$ ) is the $\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ term.",
"Like we did in Lemma REF , we can make these other terms smaller by decreasing $\\varepsilon $ at the cost of making the $g(d)^\\frac{1}{2}$ bound greater.",
"The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{6}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .",
"For this choice, we have both integrals of the order $\\varepsilon ^{-6}g(d)^\\frac{1}{2} = 2^{-6}\\pi ^{-3}A^{-\\frac{6}{d}}g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\left(1-\\frac{6}{d}\\right)^{-\\frac{6}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{6}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^3 g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\frac{1}{d^3}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .",
"Bounding the $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ term is similar.",
"For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ We once again take $\\mathcal {P}$ -positive sets $E_1,E_2$ and corresponding $g_1,g_2$ .",
"Then we use the Fourier inversion theorem and Cauchy-Schwarz to get $*{g_1}{\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} \\le f(\\lambda )\\int \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Then once again the $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Counting the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ centred on each of the poles $\\left\\lbrace k,0,-k\\right\\rbrace $ then tells us that there is a uniform bound that is some integer multiple of $\\frac{C^2}{C^2_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^{4}} + \\frac{1}{\\varepsilon ^2*{l}^{2}} + \\frac{1}{\\varepsilon ^4}\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} + \\varepsilon ^{-4}g(d)^\\frac{1}{2}\\right) \\\\= \\frac{C^2}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-4} + \\frac{C^2}{C^2_2}\\varepsilon ^{-4}g(d)^\\frac{1}{2}.$ Once again, if the $g(d)^\\frac{1}{2}$ term dominates the result is proven.",
"Otherwise, we let $\\varepsilon $ vary with $d$ .",
"The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{4}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .",
"For this choice, we have both integrals of the order $\\varepsilon ^{-4}\\beta ^2 = 2^{-4}\\pi ^{-2}A^{-\\frac{4}{d}} g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(1-\\frac{4}{d}\\right)^{-\\frac{4}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{4}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^2 g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\frac{1}{d^2}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .",
"For large $d$ , this is dominated by $g(d)^{\\frac{1}{2}-\\frac{3}{d}}d^{-3}$ and therefore $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ is dominated by $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ and we prove the result.",
"The following lemma allows us to deal with occurrences of $\\overline{W_k}$ .",
"Lemma 6.10 Let $x,y \\in \\mathbb {X}$ .",
"Then $\\tau _{\\lambda ,k}(x,y) \\le \\varphi _k(x,y) + 2\\lambda \\int \\left(\\tau _\\lambda (x,u)\\varphi _k(u,y) + \\tau _{\\lambda ,k}(x,u)\\varphi (u,y)\\right)\\nu \\left(\\mathrm {d}u\\right).",
"$ For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{W_k}\\le c_f\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The first inequality follows by applying the cosine-splitting result (Lemma REF ) to ().",
"For the $\\tau _\\lambda $ term in $\\overline{W_k}$ we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of (REF ), we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we can bound $\\overline{W_k}$ by bounding the following objects.",
"For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace \\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {W}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l)Q^{j_2}_{3,4}(l;k)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{1,2}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_1}{\\widehat{}(l) g_2}} &:j=1\\\\*{*{g_1}{\\widehat{}(l)^2g_2}} &:j=2\\\\*{*{g_1}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_2}} &: j=3,\\end{array}\\right.",
"}\\\\Q^{j}_{3,4}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} &:j=1\\\\*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} &:j=2\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{}_k(l) g_{4}}} &:j=3\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &:j=4\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} &:j=5,\\end{array}\\right.",
"}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{4}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .",
"For the $P^j_{1,2}(l)$ factor we can get bounds as we did in the proof of Lemma REF .",
"For $Q^{j}_{3,4}(l;k)$ we need to do a bit more processing.",
"For $j=1$ , $*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} \\le \\frac{1}{\\mathcal {P}(E_{3})\\mathcal {P}(E_{4})}\\int _{E_{3}\\times E_{4}}*{\\widehat{\\varphi }_k(l;a,b)}\\mathcal {P}^{\\otimes 2}(\\mathrm {d}a, \\mathrm {d}b) \\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right),$ where in the last inequality we used Lemma REF .",
"For $j=2$ , we use Cauchy-Schwarz to get $*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} \\le *{g_3}{\\widehat{}_k(l)^2 g_3}^{\\frac{1}{2}} *{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}} \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}.$ An identical argument gives the same bound for $j=3$ .",
"For $j=4$ , we can use the self-adjointness of $\\widehat{}(l)$ and $\\widehat{\\mathcal {T}}_\\lambda (l)$ and the Cauchy-Schwarz inequality to get $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &\\le *{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)^2\\widehat{}(l)g_{3}}^{\\frac{1}{2}} *{g_{4}}{\\widehat{}_k(l)^2g_{4}}^{\\frac{1}{2}} \\nonumber \\\\&\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}}\\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }\\nonumber \\\\&\\le Cf(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}},$ and similarly $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} \\le f(\\lambda )\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2} \\\\ \\le Cf(\\lambda )\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2},$ where we write for convenience $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k) = \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3-k) + \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k) + \\widehat{G}_{\\mu _\\lambda }(l_3-k)\\widehat{G}_{\\mu _\\lambda }(l_3+k).$ Note that all the bounds for the $Q^{j}_{3,4}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.",
"So we only need to prove that the remaining $l$ -integral of $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors is bounded.",
"As in our previous proofs, we partition $\\mathbb {R}^d$ into the sets $A_n$ defined earlier in (REF ).",
"For the $A_1$ , $A_2$ , and $A_3$ parts we then use the bounds $C$ and $C^2$ for the various $\\widehat{}$ factors and $1/\\left(C_2*{l}^2\\right)$ to bound the $\\widehat{G}_{\\mu _\\lambda }$ factors.",
"By using the techniques used previously in this section, we can show that these contributions is bounded by some $\\varepsilon $ and $d$ (only) dependent constant.",
"Since we don't need to show decay, we don't need to worry about taking $\\varepsilon $ to 0.",
"For the $A_0$ contribution, we bound $\\widehat{G}_{\\mu _\\lambda }$ factors by $1/\\left(C_2\\varepsilon ^2\\right)$ and show that the integral of the $\\widehat{}$ factors over all of $\\mathbb {R}^d$ is bounded.",
"For $j_2\\ne 1$ this proceeds similarly to the argument in the proof of Lemma REF .",
"For $j_2=1$ we need to take a step back.",
"These versions of (REF ) arise from trying to bound one particular type of term, which we now treat separately.",
"We bound $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\operatornamewithlimits{ess\\,sup}_{a_3,a_4\\in \\mathcal {E}}\\int \\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{\\infty ,\\infty }.$ Then Lemma REF gives the result.",
"We are now only left with bounding $\\overline{H_k}$ .",
"lemmaMartiniBound For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{H_k}\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The proof of the lemma uses the same techniques and tools as our previous lemmas, but is much more complicated in terms of keeping track of cases and components of integrals.",
"To avoid overburdening a reader on an early reading, the proof has been placed in Appendix .",
"The Bootstrap Argument In this section we bring together the diagrammatic bounds of Section with the estimates of Section REF to bound the relevant operators in such a way that the finite lace expansion of Proposition REF converges to an Ornstein-Zernike equation (OZE) for both the raw operators on $L^2\\left(\\mathbb {X}\\right)$ and the Fourier-transformed operators on $L^2\\left(\\mathcal {E}\\right)$ .",
"Proposition REF does this for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , with the caveat that there is a constant $c_f$ that is increasing in the bootstrap function $f$ (but independent of $d$ ) appearing in the bounds.",
"Observation REF then highlights that if $f$ is uniformly bounded on $\\left[0,\\lambda _O\\right)$ then so is $c_f$ and the associated bounds from Proposition REF .",
"The crucial step to the bootstrap argument is then Proposition REF .",
"Here a forbidden-region argument shows that $f$ is continuous, that there is an interval of finite values that it cannot take (the interval being independent of $\\lambda $ ), and that it starts below this interval.",
"This proves that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , and so are our bounds in Proposition REF .",
"Proposition 6.11 (Convergence of the operator lace expansion and OZE) Let $\\lambda \\in [0, \\lambda _O)$ and $d>6$ be sufficiently large.",
"Then there exists $c_f=c(f(\\lambda ))$ (which is increasing in $f$ and independent of $d$ ) such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f\\beta , &\\qquad \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}\\le c_f\\beta , \\\\ \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le c_f\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right)-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\beta , \\\\\\left\\Vert R_{\\lambda , n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} c_f\\beta ^{n}.",
"$ Furthermore, the limit $_\\lambda := \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}_\\lambda ^{(n)}$ exists and is a bounded operator with Fourier transform $\\widehat{}_{\\lambda }\\left(k\\right) = \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}\\widehat{}_{\\lambda }^{(n)}\\left(k\\right)$ for $k\\in \\mathbb {R}^d$ .",
"The operators $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ satisfy operator Ornstein-Zernike equations, taking the form $\\mathcal {T}_\\lambda = + _\\lambda + \\lambda \\mathcal {T}_\\lambda (+_\\lambda ), $ and $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right) + \\lambda \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\left(\\widehat{}\\left(k\\right)+\\widehat{}_{\\lambda }\\left(k\\right)\\right), $ for all $\\lambda < \\lambda _O$ and $k\\in \\mathbb {R}^d$ .",
"Before we prove this proposition, we make the following observation, which will guide the rest of the section.",
"Observation 6.12 (Uniform convergence of the lace expansion) Suppose that there exists finite $C>0$ such that $f \\le C$ on $[0,\\lambda _O)$ .",
"Then for $d>6$ sufficiently large there exits $c>0$ (independent of $\\lambda $ and $d$ ) such that the bounds (REF ), (), () hold with $c_f$ replaced by $c$ .",
"We now return to prove Proposition REF .",
"[Proof of Proposition REF ] So far in this section we have provided bounds for the building blocks of the bounds appearing in Propositions REF , REF , REF , and REF .",
"We still have to convert these into bounds for the composite terms $U_\\lambda $ and $V_\\lambda $ .",
"From their definitions and the bounds on the building blocks, it is simple to see that there exists $c_f^{\\prime }$ such that for $d>6$ $U_\\lambda \\le c_f^{\\prime }, \\qquad V_\\lambda \\le c_f^{\\prime }\\beta .$ This then implies that there exists $c_f^{\\prime \\prime }$ such that for $d>6$ $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime }\\beta ^2 &: n=0,\\\\c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n &: n\\ge 1.\\end{array}\\right.",
"}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime }\\left(\\beta ^2 + \\frac{c_f^{\\prime \\prime }\\beta }{1-c_f^{\\prime \\prime }\\beta }\\right) \\le c_f\\beta .$ When we want to consider the displaced terms, we also make use of Observation REF , and Lemmas REF , REF , REF , and REF to give us the existence of $c_f^{\\prime \\prime \\prime }$ such that for $d>6$ $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime \\prime }\\beta ^2\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=0,3,\\\\c_f^{\\prime \\prime \\prime }\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=1,2,\\\\\\left(n+1\\right)^2c_f^{\\prime \\prime \\prime }\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^{n-3}\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n\\ge 4.\\end{array}\\right.",
"}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime \\prime }\\left(2\\beta ^2 + 2\\beta + \\frac{16\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^3-39\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^2 + 25c_f^{\\prime \\prime \\prime }\\beta }{\\left(1-c_f^{\\prime \\prime \\prime }\\beta \\right)^3}\\right)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\\\\\le c_f\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Lemma REF then allows us to use (REF ) to bound $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ .",
"This immediately gives the required result for $n\\ge 1$ .",
"For $n=0$ , we simply additionally require $d$ to be large enough that $\\beta <1$ and therefore $\\beta ^2<\\beta ^0$ .",
"Note that the dual space of a Banach space (endowed with the operator norm) is also a Banach space.",
"In particular, it is complete.",
"Since $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n$ for $n\\ge 1$ , the sequence $_{\\lambda ,n}$ is a Cauchy sequence in the dual space of $L^2(\\mathbb {X})$ for sufficiently large $d$ .",
"Therefore the limit $_\\lambda $ exists and is a bounded linear operator on $L^2(\\mathbb {X})$ .",
"Since the Fourier transform is a unitary linear operator it is bounded and continuous, and therefore the same argument says that the limit $\\widehat{}_{\\lambda }(k)$ exists and is indeed the Fourier transform of $_\\lambda $ .",
"Since $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}$ is finite for $\\lambda <\\lambda _O$ , we have $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}\\rightarrow 0$ as $n\\rightarrow \\infty $ if $d$ is sufficiently large.",
"As a a consequence of this and equation (REF ) from Proposition REF , the $\\mathcal {T}_\\lambda $ operator satisfies the operator Ornstein-Zernike equation.",
"The following lemma has two main uses in what follows.",
"Firstly, it takes the OZEs from Proposition REF and uses it to write the two-point operators in terms of the sum of the adjacency operators and the lace expansion coefficient operators.",
"Secondly, it uses this and an intermediate value theorem argument to bound the spectral supremum of the sum of the adjacency operators and the lace expansion coefficient operators.",
"Lemma 6.13 Let $\\lambda \\in \\left[0,\\lambda _O\\right)$ and $d>6$ be sufficiently large.",
"Then the operators $\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)$ and $\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)$ all have bounded linear inverses and $\\mathcal {T}_\\lambda &= \\left(+ _\\lambda \\right)\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)^{-1}, \\\\\\widehat{\\mathcal {T}}_\\lambda (k) &= \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1}, \\qquad \\forall k\\in \\mathbb {R}^d.",
"$ Furthermore, $\\lambda \\mapsto _\\lambda $ and $\\lambda \\mapsto \\widehat{}_{\\lambda }(k)$ are all continuous in the operator norm topology and $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) &< 1,\\\\\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &< 1, \\qquad \\forall k\\in \\mathbb {R}^d.$ We first prove that $1- \\lambda \\left(+ _\\lambda \\right)$ has a bounded linear inverse.",
"This is clear for $\\lambda = 0$ , so we only need to consider $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"Suppose for contradiction that $1- \\lambda \\left(+ _\\lambda \\right)$ does not have a bounded linear inverse.",
"Then there exists a sequence $f_n\\in L^2(\\mathbb {X})$ such that $*{f_n}_2 = 1$ and $*{\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)f_n}_2\\rightarrow 0$ .",
"This also means that $\\lambda *{\\left(+ _\\lambda \\right)f_n}\\rightarrow 1$ .",
"Then from Lemma REF and sub-criticality we have $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ , and therefore the OZE equation (REF ) implies that $*{\\left(+ _\\lambda \\right)f_n}\\rightarrow 0$ , a contradiction.",
"A similar argument proves the corresponding statement for the Fourier transformed operators.",
"Once we know that the inverses exist, it is only a matter of rearranging the OZE equations to get the expressions for $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ in terms of the other operators.",
"Since $+ _\\lambda $ is a bounded operator, a similar argument proves that $1+ \\lambda \\mathcal {T}_\\lambda $ has a bounded inverse, and therefore $+ _\\lambda = \\left(1+ \\lambda \\mathcal {T}_\\lambda \\right)^{-1}\\mathcal {T}_\\lambda .$ From Corollary REF we have the continuity of $\\lambda \\mapsto \\mathcal {T}_\\lambda $ , and therefore the continuity $+ _\\lambda $ (and therefore $_\\lambda $ ).",
"We also have the continuity of $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (k)$ from Corollary REF , and we similarly get the continuity of $\\widehat{}_{\\lambda }(k)$ .",
"The continuity of $+ _\\lambda $ implies the continuity of $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right)$ (via Lemma REF ).",
"At $\\lambda = 0$ it is clear that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) = 0$ , and our above argument implies that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) \\ne 1$ .",
"Therefore an intermediate value theorem argument proves that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) < 1$ .",
"This argument also works for $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .",
"It now remains to show that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ .",
"We do this by performing a forbidden-region argument.",
"In Proposition REF we prove that $f(0)$ is bounded and $f$ is continuous on $\\left[0,\\lambda _O\\right)$ .",
"However we also prove that $f$ is never in the region $\\left(\\kappa ,\\kappa +1\\right]$ , where $\\kappa := 10 C \\left(1+C\\right)^2,$ $C$ being the constant appearing in REF .",
"Note that (REF ) in REF requires that $C\\ge 1$ .",
"The intermediate value theorem then implies that $\\kappa $ acts as an upper bound on the whole domain $\\left[0,\\lambda _O\\right)$ .",
"One should not read too much into the value of $\\kappa $ here.",
"It is sufficient for our purposes, but not remotely optimal.",
"Proposition 6.14 (The forbidden-region argument) The following three statements are all true: $f$ satisfies $f(0) \\le \\kappa $ .",
"$f(\\lambda ) \\notin (\\kappa ,\\kappa +1]$ for all $\\lambda \\in [0,\\lambda _O)$ provided that $d$ is sufficiently large.",
"$f$ is continuous on $[0, \\lambda _O)$ .",
"Therefore $f(\\lambda ) \\le \\kappa $ holds uniformly in $\\lambda <\\lambda _O$ for all $d$ sufficiently large.",
"We will deal with each of the three statements in Proposition REF in turn.",
"[Proof of Proposition REF -REF ] Firstly, it is clear $f_1\\left(0\\right) = 0$ .",
"Then note that $\\mathcal {T}_0 = $ , and hence $\\left\\Vert \\widehat{\\mathcal {T}_0}\\left(k\\right)\\right\\Vert _{\\rm op} = \\left\\Vert \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we use REF (via (REF )) in the last inequality.",
"Furthermore $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_0(0)\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) =1$ , and therefore $\\widehat{G}_{\\mu _{0}}\\left(k\\right) = \\widehat{G}_0\\left(k\\right) = 1$ and $f_2\\left(0\\right) \\le C$ .",
"Similarly we find that $f_3\\left(0\\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op}}{3\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]}.$ We use then Lemma REF to get the bound $\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Hence $f_3\\left(0\\right) \\le \\tfrac{2}{3}C$ .",
"[Proof of Proposition REF -REF ] We will assume that $f\\le \\kappa +1$ and show that this implies that $f\\le \\kappa $ .",
"Most crucially, the assumption $f\\le \\kappa +1$ allows us to apply Observation REF .",
"To highlight the $\\kappa $ -dependence, we write $c_{\\kappa +1}$ as the constant arising from this observation.",
"It will be convenient to introduce some temporary notation for this section.",
"Define $\\widehat{a}\\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as $\\widehat{a}\\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)$ for all $k\\in \\mathbb {R}^d$ .",
"We will also make use of the functions $\\delta _\\lambda \\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\delta _\\lambda \\left(k\\right) := \\mathbb {S}\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)\\right) - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ It is clear from the triangle inequality and Lemma REF that ${\\delta _\\lambda \\left(k\\right)}\\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k\\right)\\right\\Vert _{\\rm op}\\le c_{\\kappa +1}\\beta ,$ but we will also need the following inequality.",
"We will prove it at the end of this section.",
"Lemma 6.15 Let $k_1,k_2\\in \\mathbb {R}^d$ .",
"Then $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2 \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right) - \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ Figure: If 𝒯 ^ λ (k)\\widehat{\\mathcal {T}}_\\lambda (k) and a ^(k)\\widehat{a} (k) commute, then they are simultaneously diagonalizable in the sense of Theorem .",
"The arguments of their diagonal functions (τ ˜ λ (k)\\widetilde{\\tau }_\\lambda (k) and a ˜(k)\\widetilde{a}(k) respectively) are related by the monotone increasing function x↦x 1-λxx\\mapsto \\tfrac{x}{1-\\lambda x} depicted here.It will be important to note how $\\widehat{a}(k)$ relates to $\\widehat{\\mathcal {T}}_\\lambda (k)$ .",
"From Observation REF , the OZE gives us the following expression involving the commutator $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right] = \\widehat{\\mathcal {T}}_\\lambda (k) \\widehat{a}(k) - \\widehat{a}(k)\\widehat{\\mathcal {T}}_\\lambda (k)$ : $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right]\\left(1- \\lambda \\widehat{a}(k)\\right) = 0.$ Lemma REF implies that $\\left(1- \\lambda \\widehat{a}(k)\\right)$ has a bounded linear inverse and with (REF ) implies that $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ commute.",
"Therefore there exists a single unitary map that `diagonalizes' them both in the sense of Theorem REF .",
"By considering the diagonal functions, the OZE then implies that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\mathbb {S}\\left(\\widehat{a}(k)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)\\right)$ .",
"In particular, this relation allows us to write an alternative expression for $\\mu _\\lambda $ : $\\mu _\\lambda := 1 - \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} = \\lambda + \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)-1}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)} = \\lambda + \\frac{\\delta _\\lambda \\left(0\\right)}{1+\\delta _\\lambda \\left(0\\right)}.$ We first address $f_1$ .",
"From the above discussion relating $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ , we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\ge \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) = \\mathbb {S}\\left(\\widehat{a}(0)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(0)\\right)\\right)$ .",
"From Lemma REF and our choice of scaling we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right) \\ge 1 - \\left\\Vert \\widehat{}_{\\lambda }(0)\\right\\Vert _{\\rm op} \\ge 1 - c_{\\kappa +1}\\beta $ , and therefore we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right)>0$ for sufficiently large $d$ and $\\lambda \\le \\frac{1}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}\\left(1 - \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\right) \\le \\frac{1}{1-c_{\\kappa +1}\\beta } \\le 1 + 2c_{\\kappa +1}\\beta .$ Here we have used an idea that we will use very often in this part of the proof.",
"For $d$ large enough that $c_{\\kappa +1}\\beta <1$ we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1} = \\sum ^\\infty _{n=0}\\left(c_{\\kappa +1}\\beta \\right)^n$ , and for $d$ sufficiently large enough ($c_{\\kappa +1}\\beta \\le 1/2$ ) we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1}\\le 1 + 2c_{\\kappa +1}\\beta $ .",
"We also use a similar idea of neglecting higher order terms by increasing an earlier term's coefficient in other cases.",
"For $f_2$ we first replace $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ .",
"Since $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a} (k)$ commute, they are simultaneously diagonalizable in the sense of Theorem REF , and therefore their spectra are related by the function $x\\mapsto \\tfrac{x}{1-\\lambda x}$ (monotone increasing on $\\left(-\\infty ,1/\\lambda \\right)$ ).",
"Therefore we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op} = \\sup _{x\\in \\sigma \\left(\\widehat{a} (k)\\right)}*{\\frac{x}{1-\\lambda x}}$ .",
"Now (perhaps with the aid of Figure REF ) it is easy to see that $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\vee \\mathbb {S}\\left(-\\widehat{a}(k)\\right) = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) \\vee \\mathbb {S}\\left(-\\widehat{a} (k)\\right).$ Now since $\\mathbb {S}\\left(-\\widehat{a}(k)\\right) \\le \\left\\Vert \\widehat{a}(k)\\right\\Vert _{\\rm op}\\le 1 + c_{\\kappa +1}\\beta $ , and $\\widehat{G}_{\\mu _\\lambda }(k)^{-1} \\le 1+C$ (recall Lemma REF and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C$ from REF ), we only need to bound $\\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We now partition $k\\in \\mathbb {R}^d$ into two.",
"Define the subset of the Fourier space $A_{\\frac{1}{2}} := \\left\\lbrace k\\in \\mathbb {R}^d:\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\le \\frac{1}{2}\\right\\rbrace .$ Then we only need to bound the two functions $f_4,f_5$ , where $f_4\\left(\\lambda \\right) := \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)},\\qquad \\qquad f_5\\left(\\lambda \\right):= \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}^c} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We first deal with the simpler $f_4$ .",
"We consider $k\\in A_{\\frac{1}{2}}$ and choose $\\beta $ small enough that $1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right) \\ge 1 - \\frac{\\frac{1}{2}+c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta } \\ge \\frac{1}{2} - 2c_{\\kappa +1}\\beta > 0.$ Then $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\frac{1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}}{\\frac{1}{2} - 2c_{\\kappa +1}\\beta } \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right).",
"$ Since we have $\\mathbb {S}\\left(\\widehat{a}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{a}(0)\\right) \\le 1 + c_{\\kappa +1}\\beta $ , we then get $f_4(\\lambda ) \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right)\\left(1 + c_{\\kappa +1}\\beta \\right) \\le 2\\left(1+C\\right)\\left(1+7c_{\\kappa +1}\\beta \\right).$ To address $f_5$ we will have $k\\in A^c_\\rho $ .",
"We define: $\\widehat{N}(k) = \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{F}(k) = \\frac{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{Q}(k) = \\frac{1+\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}.$ Note that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\widehat{N}(k) / \\widehat{F}(k)$ .",
"Rearranging gets us to $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &= \\widehat{N}(k) \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)}\\nonumber \\\\&= \\widehat{Q}(k) + \\frac{\\widehat{N}(k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)} \\left(1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - \\frac{\\widehat{Q}(k)}{\\widehat{N}(k)}\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)\\right) .$ Since $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ (by our scaling choice), the extracted term $\\widehat{Q}(k)$ satisfies $*{\\widehat{Q}(k)} \\le 1 + 3c_{\\kappa +1}\\beta $ .",
"We further observe that $\\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{Q}(k)}{\\widehat{N}(k)} = \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\left(1+ \\delta _\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) +\\delta _\\lambda (k)} = 1 - \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right) \\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} =: 1- \\widehat{b}(k).$ Recalling identity (REF ) for $\\mu _\\lambda $ , we can rewrite the quantity $\\left[1- \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)- (1-\\widehat{b}(k)) \\widehat{F}(k)\\right]$ , appearing in (REF ), as $\\frac{1+\\delta _\\lambda (0) - \\left[\\lambda + \\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right] \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - 1 + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)+\\delta _\\lambda (k)\\right) + \\widehat{b}(k)\\left(1-\\lambda \\widehat{a}(k)\\right)}{1+\\delta _\\lambda (0)} \\\\= \\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}+ \\frac{\\widehat{b}(k)(1-\\lambda \\widehat{a}(k))}{1+\\delta _\\lambda (0)}.$ Noting that ${\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)} \\le 2c_{\\kappa +1}\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\beta $ by Lemma REF and Observation REF , the first term can be bounded $&*{\\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}} \\nonumber \\\\&\\qquad \\le \\frac{ c_{\\kappa +1}\\beta + \\left(1 + 2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta + 2\\left(1+2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta }{1 - c_{\\kappa +1}\\beta }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 5c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 6c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Using (REF ), the last term is $*{\\frac{\\widehat{b}(k)(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right))}{1+\\delta _\\lambda (0)}} &= *{ \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\delta _\\lambda (k)}{\\left(1+\\delta _\\lambda (0)\\right)\\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)\\right)}- \\frac{\\lambda \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\delta _\\lambda (k)}{1+\\delta _\\lambda (0)}} \\nonumber \\\\& = \\frac{*{\\delta _{\\lambda }(k)}}{*{1+\\delta _\\lambda (0)}}*{ \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} -\\lambda }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le \\frac{c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta }\\left( \\frac{1}{\\frac{1}{2}-c_{\\kappa +1}\\beta } + \\frac{1}{1-c_{\\kappa +1}\\beta }\\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\le c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left(3+8c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le 4c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Putting these bounds back into (REF ), we can find constants $M$ and $\\tilde{M}$ such that $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &\\le \\widehat{Q}(k) + 10c_{\\kappa +1}\\beta \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\frac{\\widehat{N}(k)}{\\widehat{F}(k)} \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\ \\\\& \\le 1+ 3c_{\\kappa +1}\\beta + 20c_{\\kappa +1}\\beta \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)} \\\\& \\le 1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta .",
"$ Note that we have used the bound $\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\le \\left[1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]$ (since $\\mu _\\lambda \\le 1$ and $\\mathbb {S}\\left(\\widehat{}(k)\\right)>\\frac{1}{2} >0$ ) to get from $\\widehat{G}_1(k)^{-1}$ to $\\widehat{G}_{\\mu _\\lambda }(k)^{-1}$ .",
"As $\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\in \\left(\\frac{1}{2}, 1\\right]$ , this concludes the improvement of $f_5$ and hence of $f_2$ .",
"Before we treat $f_3$ , we introduce $f_6$ given by $f_6(\\lambda ) := \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}\\left(k\\right)\\right)}.$ We show that $f(\\lambda ) \\le \\kappa +1$ implies $f_6(\\lambda ) \\le M$ uniformly for some $\\kappa $ -independent constant $M$ .",
"For $k\\in A_\\frac{1}{2}$ we showed in (REF ) that we can bound uniformly with $3\\left(1+C\\right)$ for sufficiently large small $\\beta $ .",
"For $k \\in A_{\\frac{1}{2}}^c$ , we have $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{G}_{\\mu _\\lambda }(k)} \\cdot \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) *{ 1- \\frac{\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}} \\\\ \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) \\left(1+\\frac{c_{\\kappa +1}\\beta }{\\frac{1}{2}- c_{\\kappa +1}\\beta } \\right) \\le 1+ \\left(20\\kappa + 27\\right)c_{\\kappa +1}\\beta .",
"$ Note that for the first bound in (REF ), we used the estimate established in (REF ), which is stronger than a bound on $f_5$ .",
"We therefore have the uniform bound $f_6(\\lambda ) \\le 3 + 3C$ for $\\beta $ sufficiently small.",
"Let us now improve the bound on $f_3$ .",
"It will be notationally convenient to denote $A\\left(k\\right) := \\left[1- \\lambda \\widehat{a}\\left(k\\right)\\right]^{-1}$ .",
"Since $\\widehat{a}\\left(k\\right)$ is self-adjoint, it can be “diagonalized\" in the sense of Theorem REF .",
"Therefore $A\\left(k\\right)$ can be “diagonalized\" over the same space and with the same unitary operator - it will have the diagonal function $\\widetilde{A}(k)(e) = 1/\\left(1-\\widetilde{a}\\left(k\\right)(e)\\right)$ .",
"It is then clear that $A\\left(k\\right)$ and $\\widehat{a}\\left(k\\right)$ commute (for each $k\\in \\mathbb {R}^d$ ).",
"Using Observation REF , we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} &= \\left\\Vert \\widehat{a}\\left(l\\right)A\\left(l\\right) - \\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op}\\nonumber \\\\&= \\left\\Vert A\\left(l\\right)\\widehat{a}\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l+k\\right)\\right]A\\left(l+k\\right) - A\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l\\right)\\right]\\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op} \\nonumber \\\\&= \\left\\Vert A(l)\\left(\\widehat{a}(l)-\\widehat{a}(l+k)\\right)A(l+k)\\right\\Vert _{\\rm op} \\nonumber \\\\&\\le \\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}\\left\\Vert A(l)\\right\\Vert _{\\rm op}\\left\\Vert A(l+k)\\right\\Vert _{\\rm op}.",
"$ For the $\\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}$ factor, we use the triangle inequality to separate this into a term of $\\widehat{}$ and $\\widehat{}_{\\lambda }$ .",
"Lemma REF immediately gives $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ for the $C>0$ arising from REF .",
"For the $\\widehat{}_{\\lambda }$ term, a similar argument to (REF ) gives $\\left\\Vert \\widehat{}_{\\lambda }\\left(l\\right) - \\widehat{}_{\\lambda }\\left(1+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(0\\right) - \\widehat{}_{\\lambda }\\left( k\\right)\\right\\Vert _{1,\\infty } \\le c_{\\kappa +1}\\beta \\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where the last inequality follows from Observation REF .",
"We then deal with the remaining factors of (REF ) by noting that for all $k\\in \\mathbb {R}^d$ , $\\left\\Vert A(k)\\right\\Vert _{\\rm op} = \\frac{1}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\widehat{G}_{\\mu _\\lambda }(k) \\le 3\\left(1+C\\right) \\widehat{G}_{\\mu _\\lambda }(k), $ employing our improved bound on $f_6$ .",
"The result of this is that we have the bound $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C + c_{\\kappa +1}\\beta \\right)\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right).$ Then since $\\widehat{G}_{\\mu _\\lambda }(k)\\ge 0$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C+c_{\\kappa +1}\\beta \\right)\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right).$ [Proof of Proposition REF -REF ] The continuity of $f_1$ is obvious.",
"For the other two functions we prove equicontinuity of a family of functions and use [41] to show the continuity of the desired functions.",
"The procedure is outlined here.",
"Suppose we wish to show that $H\\left(\\lambda \\right) := \\sup _{\\alpha \\in B}\\left\\Vert h_\\alpha \\left(\\lambda \\right)\\right\\Vert _{\\rm op}$ is continuous on $\\left[0,\\lambda _O\\right)$ .",
"For our purposes the parameter $\\alpha $ will be either $k$ or $\\left(k,l\\right)$ and thus $B=\\mathbb {R}^d$ or $B=\\mathbb {R}^{2d}$ .",
"The continuity on the half-open interval is implied by having continuity on the closed interval $\\left[0,\\lambda _O-\\rho \\right]$ for any $\\rho >0$ .",
"For closed intervals, [41] gives continuity if the family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous and $H\\left(\\lambda \\right)<+\\infty $ for all $\\lambda $ in the closed interval.",
"The family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous if for all $\\varepsilon >0$ there exists $\\delta >0$ such that $|s-t|<\\delta $ implies $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\varepsilon $ uniformly in $\\alpha \\in B$ .",
"In fact, the reverse triangle inequality implies that $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\left\\Vert h_\\alpha (s)-h_\\alpha (t)\\right\\Vert _{\\rm op},$ and therefore we only need to prove equicontinuity for the un-normed family $\\left\\lbrace h_\\alpha \\right\\rbrace _{\\alpha \\in B}$ .",
"We prove equicontinuity by bounding the `near-derivative' $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert h_\\alpha (\\lambda +\\varepsilon )-h_\\alpha (\\lambda )\\right\\Vert _{\\rm op}$ uniformly in $\\alpha \\in B$ for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ and arbitrary $\\rho >0$ .",
"For $f_2$ the operator-valued functions $h_\\alpha $ are $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)/\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)$ .",
"By using a variation on the chain rule, we get $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)$ is differentiable with the bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2$ .",
"We also have $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ from Lemma REF and $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ immediately from the definition.",
"It remains to deal with the $\\mu _\\lambda $ term.",
"Recall that $\\mu _\\lambda := 1 - \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^{-1}$ .",
"Therefore, using the reverse triangle inequality, $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda } = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)\\right)-\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} \\\\\\le \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)-\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2,$ where we have used $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ from Lemma REF in the last inequality.",
"Therefore $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + *{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2\\\\\\le \\left(1 + C\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + C\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^3.$ We therefore have a finite $k$ -independent bound.",
"This has proven the equicontinuity.",
"For the uniform boundedness, we note that for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}<\\infty $ .",
"In conjunction with $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ we have the required uniform boundedness and therefore the continuity of $f_2$ .",
"We repeat this approach for $f_3$ .",
"The corresponding step to (REF ) now reads $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda +\\varepsilon }}\\left(k,l\\right)} - \\frac{\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)$ is differentiable with the operator norm bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le 4\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}$ .",
"The operator itself has the similar bound $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}$ .",
"We would like to have the bound (REF ), but this was proven under the assumption $f\\le \\kappa +1$ , which we no longer assume.",
"Fortunately, we only require our bound for $\\lambda < \\lambda _O -\\rho $ .",
"Therefore we can use Lemma REF and Lemma REF to get $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\text{e}^{4\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\\\\\le C\\text{e}^{4\\left(\\lambda _O-\\rho \\right)\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}}\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ From this and $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge \\left(1+C\\right)^{-1}$ , there exists a constant $\\widetilde{M}>0$ such that both $\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M} \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}, \\qquad \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M},$ for sufficiently large $d$ .",
"Now recall $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ and $\\left(1+C\\right)^{-1}\\le \\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\le \\left(1-\\mu _\\lambda \\right)^{-1} = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}$ .",
"Therefore an application of the chain rule gives the bound $*{\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }} \\le 6\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^4\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op},$ uniformly in $k$ and $l$ .",
"These properties, along with the bound on $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }$ from above, are sufficient to prove equicontinuity and the uniform boundedness for the required $\\lambda $ .",
"Therefore we have proved that $f_3\\left(\\lambda \\right)$ is continuous.",
"It just remains to prove that we can indeed bound the difference of $\\delta _\\lambda (k)$ with the difference of $\\widehat{}_{\\lambda }(k)$ as we claimed.",
"[Proof of Lemma REF ] First note that there is nothing to prove if $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}=\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}=0$ , because the triangle inequality then forces $\\delta _\\lambda (k_1) = \\delta _\\lambda (k_2)=0$ .",
"Without loss of generality, assume $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ with $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}>0$ .",
"Then $\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2) = \\frac{\\delta _\\lambda (k_1)}{\\left\\Vert \\widehat{}_{\\lambda }(k_1)\\right\\Vert _{\\rm op}}\\left(\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}\\right) + \\left[\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)\\right]$ The second factor is the more troublesome: $*{\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)} = \\frac{1}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\delta _\\lambda (k_1)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op} - \\delta _\\lambda (k_2)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} \\\\\\le \\frac{*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}.$ Since $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ , we have $*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}$ .",
"Therefore $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}} \\le 2\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)-\\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op},$ where the last inequality holds because of the reverse triangle inequality.",
"Proof of Main Theorems Proposition 7.1 Let $d>6$ be sufficiently large.",
"Then for $\\lambda \\le \\lambda _O$ there exists $c>0$ such that $\\triangle _\\lambda < c\\beta ^2$ .",
"Proposition REF gives us that there exists $c_f$ (increasing in $f$ ) such that $\\triangle _\\lambda \\le c_f\\beta ^2$ for all $\\lambda <\\lambda _O$ .",
"Proposition REF then implies that there exists $c$ such that $c_f\\le c$ uniformly for all $\\lambda < \\lambda _O$ .",
"It remains to prove the assertion for $\\lambda =\\lambda _O$ .",
"Recall from Section REF the definition of $\\tau _\\lambda ^n(x,y)$ and $\\Lambda _n(x)$ .",
"Furthermore, recall that if $x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}$ then such a connection is achieved in finitely many steps and the bound in $\\ref {Assump:2ndMoment}$ then implies that $\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right\\rbrace = \\bigcup ^\\infty _{n=1}\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}_{\\Lambda _n(y)}\\right\\rbrace .$ Therefore by monotone convergence we have the pointwise limit $\\tau _\\lambda ^n(x,y) \\rightarrow \\tau _\\lambda (x,y)$ as $n\\rightarrow \\infty $ for all $\\lambda >0$ and $x,y\\in \\mathbb {X}$ .",
"Since $\\tau _\\lambda ^n(x,y)$ only depends upon the finite region $\\Lambda _n(y)$ , the functions $\\lambda \\mapsto \\tau _\\lambda ^n(x,y)$ are continuous for all $x,y\\in \\mathbb {X}$ .",
"This, with the monotonicity $\\tau _\\lambda ^n(x,y) \\le \\tau _\\lambda ^{n+1}(x,y)$ , implies that the function $\\lambda \\mapsto \\tau _\\lambda (x,y)$ is lower-semicontinuous.",
"Since $\\tau _\\lambda (x,y)$ is non-decreasing in $\\lambda $ , this lower semi-continuity implies that the pointwise limit $\\tau _\\lambda (x,y)\\uparrow \\tau _{\\lambda _O}(x,y)$ as $\\lambda \\uparrow \\lambda _O$ holds, and the limit holds monotonically.",
"This monotonic pointwise convergence implies that the integral $\\int \\tau _\\lambda (x,u)\\tau _\\lambda (u,v)\\tau _\\lambda (v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) \\rightarrow \\int \\tau _{\\lambda _O}(x,u)\\tau _{\\lambda _O}(u,v)\\tau _{\\lambda _O}(v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right)$ for all $x,y\\in \\mathbb {X}$ as $\\lambda \\uparrow \\lambda _O$ , and that this convergence is monotone increasing.",
"If we take the supremum over $x,y\\in \\mathbb {X}$ , we get that $\\lim _{\\lambda \\uparrow \\lambda _O}\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } = \\left\\Vert \\mathcal {T}_{\\lambda _O}^3\\right\\Vert _{\\infty ,\\infty }$ .",
"The uniform bound for $\\lambda <\\lambda _O$ then implies that this bound also holds at $\\lambda =\\lambda _O$ .",
"The following proposition uses the bound $\\triangle _{\\lambda _O} < c\\beta ^2$ to show that percolation does not occur at criticality.",
"Another element of the proof is that there is almost surely at most one infinite cluster.",
"There are general considerations that show that this is the case.",
"Indeed [14] establishes that on the discrete space $\\mathbb {Z}^d$ there is at most one infinite cluster if the edge occupation measure is stationary and obeys the ‘finite energy property’, and an analogous result for Poisson processes in the continuum applies in our case (see [4], [36]).",
"Proposition 7.2 The critical percolation function $\\theta _{\\lambda _O}(a)=0$ for $\\mathcal {P}$ -almost every $a\\in \\mathcal {E}$ .",
"Assume that $(\\overline{0},a),(\\overline{x},b)\\in \\mathbb {X}$ are in the (a.s. unique) infinite cluster, and are therefore connected.",
"Then the FKG inequality implies that $\\theta _{\\lambda _O}(a)\\theta _{\\lambda _O}(b)\\le \\mathbb {P}_{\\lambda _O}\\big (|\\overline{0},a)|=\\infty ,|\\overline{x},b)|=\\infty \\big )\\le \\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x},b)).$ Since $\\triangle _{\\lambda _O} \\le c\\beta ^2$ , for $\\mathcal {P}$ -a.e.",
"$a,b\\in \\mathcal {E}$ there exists a sequence $\\overline{x}^{(a,b)}_n$ such that $\\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x}^{(a,b)}_n,b))\\rightarrow 0$ .",
"Therefore for $\\mathcal {P}$ -a.e.",
"$a\\in \\mathcal {E}$ we have $\\theta _{\\lambda _O}(a)=0$ .",
"We can define the linear operator $\\mathcal {T}_{\\lambda _O}$ by its action on $f\\in L^2(\\mathbb {X})$ : $\\mathcal {T}_{\\lambda _O}f(x) = \\int \\tau _{\\lambda _O}(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right).$ Contrary to $\\mathcal {T}_\\lambda $ for $\\lambda <\\lambda _O$ , the operator $\\mathcal {T}_{\\lambda _O}$ may be an unbounded linear operator.",
"Proposition 7.3 (The operator OZE at the critical point) Let $d>6$ be sufficiently large.",
"The Ornstein-Zernike equation then extends to $\\lambda _O$ in the sense that the sum of (unbounded) linear operators vanishes: $\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) = 0.$ Furthermore, for $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ the linear operator $\\widehat{\\mathcal {T}}_{\\lambda _O}(k)$ is a bounded linear operator and the following equality holds: $\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right) - \\widehat{}\\left(k\\right) - \\widehat{}_{\\lambda _O}\\left(k\\right) - \\lambda _O\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right)\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda _O}\\left(k\\right)\\right) = 0.$ Finally, there exists $c>0$ such that for all $k\\in \\mathbb {R}^d$ $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le c\\beta , \\qquad \\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}\\le c\\beta .$ Let $f\\in L^2(\\mathbb {X})$ .",
"Then by the triangle inequality, for $\\lambda <\\lambda _O$ ${\\left(\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right)\\right)f}_2 \\le {\\left(\\mathcal {T}_\\lambda - - _\\lambda - \\lambda \\mathcal {T}_\\lambda \\left(+ _\\lambda \\right)\\right)f}_2 \\\\+ {\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda - _{\\lambda _O}+ _\\lambda - \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) + \\lambda \\mathcal {T}_\\lambda \\left(+_\\lambda \\right)\\right)f}_2.$ The first norm on the right hand side vanishes, since the OZE holds for $\\lambda <\\lambda _O$ .",
"To prove that the left hand side vanishes, we aim to show that the second norm on the right hand side vanishes as $\\lambda \\uparrow \\lambda _O$ .",
"To do this, the triangle inequality implies that we only need to show that the following limits hold as $\\lambda \\uparrow \\lambda _O$ : ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2 &\\rightarrow 0, \\\\{\\left(_{\\lambda _O}- _\\lambda \\right)f}_2 &\\rightarrow 0,\\\\{\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2 &\\rightarrow 0.$ We begin with (REF ).",
"Recall from the proof of Proposition REF that $\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y) := \\tau _{\\lambda _O}(x,y) - \\tau _\\lambda (x,y)$ converges monotonically and pointwise to zero.",
"From the definition of the ${\\cdot }_2$ norm, ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\\\&\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes (3)}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right).$ The monotone convergence of $\\left(\\tau _{\\lambda _O}-\\tau _\\lambda \\right)(x,y)\\downarrow 0$ then implies that the limit (REF ) holds.",
"For (), we use the functions $h^{(n)}_\\lambda \\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ defined by $h^{(n)}_\\lambda (x,y) := {\\left\\lbrace \\begin{array}{ll}\\lambda ^{-1}\\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes (2n)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right) &:n\\ge 1\\\\\\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2 &: n=0.\\end{array}\\right.",
"}$ From Proposition REF and the proof of Proposition REF , we know that $0\\le \\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_\\lambda (x,y)$ for all $n\\ge 0$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in \\left(0,\\lambda _O\\right)$ - the non-negativity follows from the definition of $\\pi ^{(n)}_\\lambda $ .",
"The functions $ h^{(n)}_\\lambda $ can also be defined for $\\lambda =\\lambda _O$ , and since they are monotone increasing in $\\lambda $ , we have the $\\lambda $ -independent bound $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda <\\lambda _O$ and $x,y\\in \\mathbb {X}$ .",
"For the single-mark version of the model it is proven in [21] that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.",
"This same argument works for our multi-mark version, and therefore $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda \\le \\lambda _O$ .",
"The triangle inequality then implies that $*{\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)} = *{\\sum ^\\infty _{n=0}\\left(-1\\right)^n\\left( \\pi ^{(n)}_{\\lambda _O}(x,y) - \\pi ^{(n)}_{\\lambda }(x,y)\\right)} \\le 2\\sum ^\\infty _{n=0}h^{(n)}_{\\lambda _O}(x,y).$ It will be convenient to define $h_{\\lambda }\\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ as $h_\\lambda (x,y) := \\sum ^\\infty _{n=0}h^{(n)}_{\\lambda }(x,y)$ .",
"Then ${\\left(_{\\lambda _O}- _\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,z)f(y)f(z)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right) \\\\& \\le 4\\int h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right)\\nonumber \\\\& = 4\\int \\left(\\int h_{\\lambda _O}(x,y)*{f(y)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right).",
"$ By Schur's test, this integral is finite for all $f\\in L^2(\\mathbb {X})$ if both the values $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right) \\qquad \\text{ and } \\qquad \\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ are finite.",
"To prove this for the former, we repeat the arguments of Proposition REF and Section REF to bound it in terms of $\\lambda _O$ , the triangle diagrams, and other $\\lambda $ -independent terms.",
"The argument of Proposition REF with the uniform bounds of Observation REF then proves that $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ is indeed finite.",
"A similar argument also holds for $\\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ .",
"We now have an integrable function $h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}$ that dominates the integrand of (REF ).",
"Recall that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.",
"Thus the dominated convergence theorem implies that the limit () holds.",
"For (), we can use $h_{\\lambda _O}$ to get the bound ${\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2^2 \\\\\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,u)h_{\\lambda _O}(u,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,v)h_{\\lambda _O}(v,z)*{f(y)}*{f(z)} \\nu ^{\\otimes 5}\\left(\\mathrm {d}u,\\mathrm {d}v, \\mathrm {d}x, \\mathrm {d}y, \\mathrm {d}z\\right).$ The integrand converges monotonically to zero, and therefore the integral vanishes in the limit.",
"We now consider the Fourier version of the OZE.",
"We use the notation that, given a bounded linear operator $A\\colon L^2(\\mathbb {X}) \\rightarrow L^2(\\mathbb {X})$ , the Fourier transform is given by $\\mathcal {F}\\left( A \\right)(k)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ .",
"By hypothesis, $$ is a bounded operator, and the finiteness of $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ implies that $_{\\lambda _O}$ is also bounded.",
"Then (REF ) implies that $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) = \\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)$ for all $k\\in \\mathbb {R}^d$ .",
"We now make the claim that $\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)$ has a bounded linear inverse for all $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ .",
"This is proven in Lemma REF below.",
"This claim then implies that for $k\\ne 0$ we have $&\\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) \\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)^{-1} \\right)(k) = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k).$ That is, $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k)$ is a bounded linear operator for $k\\ne 0$ .",
"The limit () also allows us to extend our sub-critical bound on $_\\lambda $ to apply at criticality.",
"The limit () may be phrased as saying that the function $\\lambda \\mapsto *{_\\lambda f}_2$ is continuous at $\\lambda =\\lambda _O$ (from the left) for all $f\\in L^2\\left(\\mathbb {X}\\right)$ .",
"Since $\\lambda \\mapsto \\left\\Vert _\\lambda \\right\\Vert _{\\rm op}$ can be written as a supremum of such functions, it is a lower semi-continuous function.",
"Our bound $\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta $ for $\\lambda < \\lambda _O$ (from Proposition REF ) then implies $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le \\liminf _{\\lambda \\uparrow \\lambda _O}\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta .$ The limit $*{\\left(\\widehat{}_{\\lambda _O}^{(n)}(0) - \\widehat{}_{\\lambda }^{(n)}(0)\\right)f}_2 \\rightarrow 0$ for each $f\\in L^2\\left(\\mathcal {E}\\right)$ can be derived in essentially the same way as () above.",
"The above lower semi-continuity argument can then be used to show that the bound from Proposition REF extends to also hold at $\\lambda =\\lambda _O$ .",
"From Lemma REF and the positivity of $\\pi ^{(n)}_{\\lambda _O}(x,y)$ this bound also holds for all $k\\in \\mathbb {R}^d$ .",
"By summing over $n$ , we then get the desired bound on $\\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}$ .",
"Lemma 7.4 For $\\lambda = \\lambda _O$ and $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ , the bounded linear operator $\\left(1- \\lambda _O\\left(\\widehat{}(k) + \\widehat{}_{\\lambda _O}(k)\\right)\\right)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ has a bounded linear inverse.",
"We first introduce some notation.",
"Let $A\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ be a self-adjoint linear operator.",
"Then we define the minimum modulus to be $\\left\\Vert A\\right\\Vert _{\\rm min} := \\inf _{0\\ne f\\in L^2(\\mathcal {E})}\\frac{*{Af}_2}{{f}_2}.$ Note that it follows from the Spectral Theorem (Theorem REF ) that $A$ has a bounded linear inverse if and only if $\\left\\Vert A\\right\\Vert _{\\rm min}>0$ .",
"Now suppose that $A=A_\\lambda $ , and that for all $f\\in L^2(\\mathcal {E})$ the map $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .",
"Then the map $\\lambda \\mapsto \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}$ is upper semicontinuous on $\\left[0,\\lambda _O\\right]$ .",
"That is, for any sequence $\\lambda _n$ in $\\left[0,\\lambda _O\\right]$ such that $\\lambda _n\\rightarrow \\lambda $ , $\\limsup _{n\\rightarrow \\infty }\\left\\Vert A_{\\lambda _n}\\right\\Vert _{\\rm min} \\le \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}.$ We now consider $A_\\lambda = 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .",
"A similar argument to the way we proved () shows that $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .",
"Our result is then proven if we can show that for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ we have an $\\varepsilon = \\varepsilon (k)>0$ such that $\\left\\Vert 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right\\Vert _{\\rm min} \\ge \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"We will in fact show the stronger result that $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"Recalling the definition of $\\delta _\\lambda (k)$ from (REF ), we have $1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &= 1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\lambda \\delta _{\\lambda }(k) \\nonumber \\\\&= \\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{>0 \\text{ from Lemma~\\ref {lem:Optlam-a}}} + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) + \\lambda \\left(\\delta _{\\lambda }(0) - \\delta _{\\lambda }(k)\\right) \\nonumber \\\\& > \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1+\\mathcal {O}\\left(\\beta \\right)\\right).$ Here we have used Lemma REF to bound the difference of $\\delta $ with the difference of $\\widehat{}_{\\lambda _O}$ .",
"For sufficiently large $d$ , REF then proves $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"Note that in conjunction, Proposition REF , Proposition REF and Proposition REF prove the OZE equations of Theorem REF .",
"In particular, the former two prove that the OZE holds in the subcritical regime, and the latter extends this to criticality where possible.",
"Furthermore, Proposition REF gives the bounds on $_\\lambda $ and $\\widehat{}_{\\lambda }(k)$ subcritically whilst Proposition REF extends this bound to criticality.",
"Regarding Theorem REF , the bound on $\\triangle _\\lambda $ and the lack of percolation at criticality were proven at the beginning of this section.",
"The infrared bound component of the theorem is proven below.",
"[Proof of Theorem REF ] First note that there is nothing to prove for $k=0$ .",
"From Proposition REF and Lemma REF , we can write $\\widehat{\\mathcal {T}}_\\lambda (k) = \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1},$ for all $k\\in \\mathbb {R}^d$ if $\\lambda <\\lambda _O$ , and for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ if $\\lambda = \\lambda _O$ .",
"If we use the Spectral Theorem (Theorem REF ) to `diagonalize' the operator $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , we get a multiplication operator on some Hilbert space $L^2(\\mathfrak {E}_k,\\mu _k)$ taking values $\\widetilde{a}(k)(e)$ for $e\\in \\mathfrak {E}_k$ .",
"Since, $\\widehat{\\mathcal {T}}_\\lambda (k)$ is formed from $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , it can also be diagonalized over $L^2(\\mathfrak {E}_k,\\mu _k)$ , and takes values $\\widetilde{\\tau }_\\lambda (k)(e) = \\frac{\\widetilde{a}(k)(e)}{1 - \\lambda \\widetilde{a}(k)(e)}.$ Since $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) < 1$ (from the proof of Lemma REF ), we know $\\lambda \\widetilde{a}(k)(e) >1$ and $\\lambda \\widetilde{\\tau }_\\lambda (k)(e) \\ge -1$ for $\\mu _k$ -almost every $e\\in \\mathfrak {E}_k$ .",
"It therefore only remains to bound $\\widetilde{\\tau }_\\lambda (k)(e)$ from above.",
"Equivalently, we only need to bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)$ from above.",
"We also note that the proof of () can be adapted to show that $\\lambda \\mapsto \\left<f,\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) f\\right>$ is continuous on $\\left[0,\\lambda _O\\right]$ for all $f\\in L^2(\\mathcal {E})$ .",
"Then because $\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) = \\sup _{f}\\tfrac{\\left<f,\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) f\\right>}{\\left<f,f\\right>}$ , this proves that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)$ is lower-semicontinuous on $\\left[0,\\lambda _O\\right]$ .",
"The bound $\\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) <1$ for $\\lambda < \\lambda _O$ (from Lemma REF ) then implies that $\\lambda _O\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda _O}(0)\\right)\\le 1$ .",
"From (REF ) the calculation proceeds: $\\lambda \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) &= \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& = \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{\\ge 0} + \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) + \\delta _{\\lambda }(0) - \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\delta _{\\lambda }(k)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1 + \\mathcal {O}\\left(\\beta \\right)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)},$ where we have bounded $\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)$ using Lemma REF and Observation REF .",
"Critical Intensities [Proof of Proposition REF ] By the Mecke equation (REF ), we find $*{\\chi _\\lambda }_1 & = 1 + \\lambda \\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) = 1 + \\lambda \\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right),\\\\*{\\chi _\\lambda }_\\infty & = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\widehat{\\tau }_\\lambda (0;a,b) \\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }.$ From Lemma REF we have $\\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }$ .",
"Therefore $\\lambda ^{(1)}_T \\ge \\lambda _O \\ge \\lambda _T^{(\\infty )}$ .",
"Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(1)}_T$ .",
"Therefore we only now need to prove that $\\lambda ^{(1)}_T = \\lambda ^{(\\infty )}_T$ .",
"We do this by proving that if $*{\\chi _\\lambda }_1<\\infty $ then $*{\\chi _\\lambda }_\\infty <\\infty $ .",
"Fix $b\\in \\mathcal {E}$ .",
"Then by considering the vertices in the cluster $\\overline{0},b)$ adjacent to $\\left(\\overline{0},b\\right)$ , we find $\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\mathbb {E}_\\lambda \\left[\\sum _{\\left(\\overline{x},a\\right)\\in \\eta :\\left(\\overline{x},a\\right)\\sim \\left(\\overline{0},b\\right)}*{\\overline{x},a)}\\right] = 1 + \\lambda \\int \\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right]\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right).$ In this equality we have used Mecke's equation (REF ).",
"From the spatial translation invariance of the model, $\\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right] = \\mathbb {E}_\\lambda \\left[*{\\overline{0},a)}\\right]$ for all $\\overline{x}\\in \\mathbb {R}^d$ .",
"Therefore an application of a supremum bound to the $a$ -integral gives $*{\\chi _\\lambda }_\\infty = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)\\int \\mathbb {E}_\\lambda \\left[*{\\overline{0},c)}\\right]\\mathcal {P}\\left(\\mathrm {d}c\\right)\\\\= 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)*{\\chi _\\lambda }_1.$ From the finiteness of the parenthesised factor, $*{\\chi _\\lambda }_1<\\infty $ implies $*{\\chi _\\lambda }_\\infty <\\infty $ and the result is proven.",
"[Proof of Proposition REF ] The equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ holds from [35].",
"The equality $\\lambda _O = \\lambda _T^{(p)}$ follows from Proposition REF because the condition (REF ) clearly holds for the bounded radii model.",
"Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(p)}_T$ for all $p\\in \\left[1,\\infty \\right]$ .",
"Now note that for all $a,b\\in \\mathcal {E}$ we have that if $\\left(\\overline{0},\\dagger \\right)\\sim \\left(\\overline{x},b\\right)$ then $\\left(\\overline{0},a\\right)\\sim \\left(\\overline{x},b\\right)$ .",
"Therefore $\\lambda _c\\left(\\dagger \\right) \\ge \\lambda _c$ , and the equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ then means we only need to prove $\\lambda _T\\left(\\dagger \\right) \\le \\lambda _T^{(p)}$ for some $p\\in \\left[1,\\infty \\right]$ .",
"We will do so for $p=1$ .",
"Suppose that $\\lambda >\\lambda _T^{(1)}$ , so that $\\int \\mathbb {E}_\\lambda \\left[(\\overline{0};a\\right)\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\infty $ .",
"That is, the expected cluster size of a vertex with a random ($\\mathcal {P}$ -distributed) mark is infinite.",
"Note that we have an ordering of the marks in that if two radii $a,b$ satisfy $a\\ge b$ , then $\\varphi \\left(\\overline{x};a,c\\right) \\ge \\varphi \\left(\\overline{x};b,c\\right)$ for all $\\overline{x}\\in \\mathbb {R}^d$ and $c\\in \\mathcal {E}$ .",
"This also implies that $a\\mapsto \\mathbb {E}_\\lambda \\left[*{(\\overline{0},a}\\right]$ is a non-decreasing function.",
"This in turn implies that if $\\lambda >\\lambda _T^{(1)}$ and $a$ satisfies $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , then $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right) = \\infty $ .",
"Given some radius $a\\in \\mathcal {E}$ such that $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we define a new mark, $\\star _a$ , that forms connections according to $\\varphi \\left(\\overline{x};\\star _a,b\\right) = 1\\left\\lbrace *{\\overline{x}}\\le a\\right\\rbrace ,$ for all $b\\in \\mathcal {E}$ .",
"Note that if $b\\ge a$ then $\\varphi \\left(\\overline{x};\\star _a,c\\right) \\le \\varphi \\left(\\overline{x};b,c\\right)$ for all $c\\in \\mathcal {E}$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\le \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]$ .",
"We now consider the expected cluster size of vertices adjacent to $\\left(\\overline{0},\\star _a\\right)$ with radius $\\ge a$ .",
"We also restrict to cases for which there is an unique neighbour, and that neighbour has radius $\\ge a$ .",
"Since the probability of a connection is independent of the radius of the proposed neighbour, the distribution of the unique neighbour is equal to $\\mathcal {P}$ - conditioned upon having radius $\\ge a$ .",
"Therefore we have the lower bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\ge 1 + \\mathbb {P}_\\lambda \\left(\\exists !",
"\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right).$ As noted above, $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)=\\infty $ .",
"Furthermore, as $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , the random variable $\\#\\left\\lbrace \\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right\\rbrace $ obeys a Poisson distribution with strictly positive mean.",
"Therefore $\\mathbb {P}_\\lambda \\left(\\exists !",
"\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)>0$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right]=\\infty $ .",
"This in turn implies that $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right] =\\infty $ for all $a>0$ .",
"To relate this to $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]$ , we perform a similar lower bound.",
"Given a radius $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we restrict to cases where there is an unique neighbour of $\\left(\\overline{0},\\dagger \\right)$ with radius $\\ge a$ and bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right] \\ge 1+ \\mathbb {P}_\\lambda \\left(\\exists !",
"\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\dagger \\right),b\\ge a\\right) \\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right].$ As before, the probability factor is strictly positive and the expected cluster size term is infinite.",
"Hence $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]=\\infty $ and $\\lambda \\ge \\lambda \\left(\\dagger \\right)$ .",
"Therefore $\\lambda _T\\left(\\dagger \\right)\\le \\lambda ^{(1)}_T$ as required.",
"Model Properties Here we will prove that the models outlined in Section REF do indeed satisfy Assumption REF .",
"First we consider the single mark “finite variance models” considered by [21].",
"As proven in that reference, this includes the Poisson blob and Gaussian connection models.",
"Lemma B.1 Single mark finite variance models satisfy the conditions of Assumption REF .",
"First note that since $\\mathcal {E}$ is a singleton, the operators $\\widehat{}(k)$ are simply scalars.",
"In particular, this means that the conditions (REF ), and () of REF hold trivially.",
"The finiteness of $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is also equivalent to the finiteness of $\\int \\varphi (x)\\mathrm {d}x$ required by [21].",
"Since $\\widehat{}(k)$ is simply a scalar, assumption REF follows directly from the third finite variance condition of [21].",
"The condition REF is similarly a generalisation of the second finite variance condition, and therefore follows.",
"Specifically, the sets $B(x)$ are the $\\varepsilon $ -balls around $x\\in \\mathbb {R}^d$ where $0<\\varepsilon <r_d$ where $r_d$ is the radius of the ball of volume 1.",
"Lemma B.2 Double mark finite variance models satisfy the conditions of Assumption REF .",
"Since $\\mathcal {E}= \\lbrace a_1,a_2\\rbrace $ , let us denote the functions $\\varphi _{i,j}(\\overline{x}) = \\varphi (\\overline{x};a_i,a_j)$ and their Fourier transforms $\\widehat{\\varphi }_{i,j}(k) = \\widehat{\\varphi }(k;a_i,a_j)$ for $i,j\\in \\lbrace 1,2\\rbrace $ .",
"Since $\\widehat{}(k)$ is on operator on a finite dimensional vector space, it is clear that REF will hold (note that each of the entries will be finite).",
"We now represent $\\widehat{}(k)$ with a matrix.",
"Given $f\\in L^2(\\mathcal {E})$ - taking values $f_1,f_2$ - we have $\\left<f,\\widehat{}(k)f\\right> = p^2\\widehat{\\varphi }_{1,1}(k)f_1^2 + 2pq\\widehat{\\varphi }_{1,2}(k)f_1f_2 + q^2\\widehat{\\varphi }_{2,2}(k)f_2^2,$ and therefore the spectrum of $\\widehat{}(k)$ is equal to the set of eigenvalues of the matrix $\\begin{pmatrix}p^2\\widehat{\\varphi }_{1,1}(k) & pq\\widehat{\\varphi }_{1,2}(k)\\\\pq\\widehat{\\varphi }_{1,2}(k) & q^2\\widehat{\\varphi }_{2,2}(k)\\end{pmatrix}.$ This $2\\times 2$ matrix has characteristic polynomial given by $z \\mapsto z^2 - \\left(p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)\\right) z + p^2q^2\\left( \\widehat{\\varphi }_{1,1}(k)\\widehat{\\varphi }_{2,2}(k) - \\widehat{\\varphi }_{1,2}(k)^2\\right).$ Solving for roots of this quadratic then gives $\\mathbb {S}\\left(\\widehat{}(k)\\right) = \\frac{p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)}{2} + \\sqrt{\\left(\\frac{p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)}{2}\\right)^2 + p^2q^2\\widehat{\\varphi }_{1,2}(k)^2},$ where the square root takes the positive value.",
"Note that by the definition of the double mark finite variance model $p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)$ is the Fourier transform of a non-negative (or non-positive) function, and therefore $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ is a sum of terms maximised at $k=0$ .",
"Furthermore, each approaches that maximum quadratically and at least one of $\\widehat{\\varphi }_{1,1}(k)$ , $\\widehat{\\varphi }_{2,2}(k)$ , and $\\widehat{\\varphi }_{1,2}(k)$ uniquely achieves its maximum at $k=0$ (otherwise we have $\\varphi = 0$ almost everywhere).",
"Therefore REF is satisfied.",
"Since we have only finitely many marks, there are only finitely many sequences of marks of a given length.",
"Therefore since each of the $\\varphi _{i,j}$ individually satisfy the 1-mark analog of REF , the 2-mark version is satisfied by $\\varphi $ too.",
"It is not inconceivable that the above argument could be generalised to 3 or 4 mark systems.",
"The degree 3 and 4 characteristic polynomials arising from the matrices have roots that can be solved for in radicals.",
"It may be then that a particular clever rearrangement of the expression allows the required properties to be deduced like above.",
"Such an investigation is currently beyond the authors' patience.",
"Of course, the famous Abel-Ruffini Theorem states that no such solution in radicals exists for arbitrary polynomials of degree 5 and higher, and so this approach seems doomed for systems with such families of marks.",
"Lemma B.3 Space-mark factorisation models satisfy the conditions of Assumption REF .",
"Note that $\\widehat{}(k) = \\widehat{\\overline{\\varphi }}(k)\\mathcal {K}$ , where $\\mathcal {K}\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ is the self-adjoint linear integral operator with kernel function $K$ .",
"Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b}\\int K(a,b)\\mathcal {P}(\\mathrm {d}a)\\le 1$ , $\\mathcal {K}$ is a bounded operator and $\\mathbb {S}\\left(\\widehat{}(k)\\right) = {\\left\\lbrace \\begin{array}{ll}\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) \\ge 0\\\\-\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(-\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) < 0.\\end{array}\\right.",
"}$ Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}\\le 1$ , the finiteness of $\\widehat{\\overline{\\varphi }}(0)$ implies that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite, and the $d$ -independence of $\\mathcal {K}$ implies that REF holds.",
"Since $\\overline{\\varphi }(\\overline{x})\\ge 0$ , we have $*{\\widehat{\\overline{\\varphi }}(k)} < \\widehat{\\overline{\\varphi }}(0)$ for all $k\\ne 0$ .",
"Furthermore, $\\mathbb {S}\\left(\\mathcal {K}\\right) = \\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}$ implies that $\\mathbb {S}\\left(\\mathcal {K}\\right)\\ge \\mathbb {S}\\left(-\\mathcal {K}\\right)$ .",
"Then $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ inherits the properties required by REF from $\\widehat{\\overline{\\varphi }}(k)$ .",
"For REF , we note that the $d$ -independence of $\\mathcal {K}$ means that required decay properties are again inherited from $\\widehat{\\overline{\\varphi }}(k)$ .",
"Lemma B.4 The marked multivariate Gaussian model satisfies the conditions of Assumption REF .",
"Firstly note that for each $\\left(a,b\\right)\\in \\mathcal {E}^2$ the Gaussian structure of $\\varphi $ means that it factorises over the $d$ eigenvector directions of $\\Sigma (a,b)$ .",
"This factorisation ensures that that the $d$ -dimensional Fourier transform is the product of the 1-dimensional Fourier transforms.",
"We have $\\widehat{\\varphi }(k;a,b) = \\mathcal {A} \\exp \\left(-\\frac{\\pi ^2}{2}k^{\\intercal }\\Sigma (a,b)^{-1}k\\right).$ In particular, when $k=0$ the Fourier transform is $a,b$ -independent.",
"This means $\\widehat{}(0) = \\mathcal {A}\\mathbf {1}$ where $\\mathbf {1}$ is the integral operator with constant kernel function 1.",
"For $f\\in L^2(\\mathcal {E})$ we have ${f}{\\mathbf {1}f} &= \\int f(a) \\left(\\int f(b) \\mathcal {P}\\left(\\mathrm {d}b\\right)\\right) \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f\\right]^2,\\\\{f}{f} &= \\int f(a)^2 \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f^2\\right].$ By Jensen's inequality we therefore have $\\mathbb {S}\\left(\\mathbf {1}\\right) = 1$ (equality follows by considering the test function $f(a)\\equiv 1$ ).",
"It therefore follows that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = \\mathcal {A}<\\infty $ .",
"From (REF ) we have $\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } = \\mathcal {A}\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\exp \\left(-\\pi ^2k^{\\intercal }\\Sigma (a,b)^{-1}k\\right) \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This bound then implies the condition (REF ) of REF holds with $C=1$ .",
"Before we address () of REF , we turn to REF .",
"From our above bounds we already have $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) \\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This exponential term ensures that REF is satisfied.",
"We now return to () of REF .",
"For all $x\\in \\mathbb {R}$ we have $1-\\cos (x) \\le \\frac{1}{2}x^2$ , and therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\frac{1}{2}*{k}^2 \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x},$ for some unit vector $e\\in \\mathbb {R}^d$ .",
"Since the vector $e$ picks out the second moment of $\\varphi $ in only one direction, a standard calculation gives $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\Sigma _{\\max }^2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }.$ Therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty }$ is bounded by a $d$ -independent quadratic function.",
"In conjunction with the quadratic bound of REF , this then proves () in the $k\\rightarrow 0$ regime.",
"The remaining regime then holds because the triangle inequality implies $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le 2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } = 2\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"To address REF , recall that the spatial convolution of two multivariate Gaussian functions with means $\\overline{0}$ and covariance matrices $\\Sigma _1$ and $\\Sigma _2$ is a multivariate Gaussian function with mean $\\overline{0}$ and covariance matrix $\\Sigma _1+\\Sigma _2$ .",
"Therefore for all $\\overline{x}\\in \\mathbb {R}^d$ we have $&\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\int \\varphi (\\overline{x}-\\overline{y};a_1,a_2)\\varphi (\\overline{y};a_3,a_4)\\mathrm {d}\\overline{y}\\nonumber \\\\&\\hspace{14.22636pt}= \\mathcal {A}^2\\left(2\\pi \\right)^{-d/2}\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\left(\\det \\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)\\right)^{-1/2}\\exp \\left(-\\frac{1}{2}\\overline{x}^{\\intercal }\\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)^{-1}\\overline{x}\\right) \\nonumber \\\\& \\hspace{14.22636pt} \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2}\\exp \\left(-\\frac{*{\\overline{x}}^2}{4\\Sigma _{\\min }}\\right) \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2} \\rightarrow 0.$ In particular, this bounds the integral appearing in the definition of the sets $B(x)$ (see (REF )) and therefore proves they are empty.",
"For the bound of the convolution of three $\\varphi $ functions, we can get this immediately from the above calculation and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"Specifically REF holds with $g(d) = \\left(4\\pi \\Sigma _{\\min }(d)\\right)^{-d/2}$ and the sets $B(x) = \\emptyset $ .",
"Lemma B.5 The bounded-volume Boolean disc model satisfies the conditions of Assumption REF .",
"Crucial to the conditions of REF holding is the upper bound on $R_d(a)$ in (REF ).",
"We use the shorthand $R_{a,b} := R_d(a) + R_d(b),\\qquad V_{a,b} := \\frac{\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R^d_{a,b},$ so $V_{a,b}$ equals the Lebesgue volume of the $d$ -dimensional Euclidean ball with radius $R_{a,b}$ .",
"Note that the upper bound $R^{({\\rm max})}_d$ on $R_d(a)$ in (REF ) ensures $V_{a,b} \\le V^{({\\rm max})}$ uniformly in $d$ .",
"We first find $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int V_{a,b}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\le V^{({\\rm max})}.$ This proves the statement that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite in REF .",
"From the condition (REF ) we also get $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\ge \\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a, \\mathrm {d}b\\right) \\ge \\epsilon ^2\\frac{d\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)} \\int ^{2\\left(1-\\frac{c_2}{d}\\right)R^{({\\rm max})}_d}_0 r^{d-1}\\mathrm {d}r \\\\= \\epsilon ^2V^{({\\rm max})}\\left(1-\\frac{c_2}{d}\\right)^d \\rightarrow \\epsilon ^2\\text{e}^{-c_2}V^{({\\rm max})},$ where we have used the test function $f(a) \\equiv 1$ to lower bound $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"We then prove (REF ) by bounding $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } =\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}V_{a,b} \\le V^{({\\rm max})}.$ We will return for () after addressing REF .",
"For REF , it will be useful to have an expression for $\\widehat{\\varphi }(k;a,b)$ for each $a,b\\in \\left[0,1\\right]$ .",
"Suppose we are able to find $C_1\\in \\left(0,1\\right)$ and $C_2 >0$ such that for all $a,b\\in \\left[0,1\\right]$ $*{\\widehat{\\varphi }(k;a,b)} \\le \\widehat{\\varphi }(0;a,b) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right).$ Then since the $k$ -dependent factor is $a,b$ -independent we can use Lemma REF to show that $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{}(0)\\right) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right),$ and therefore prove that REF is satisfied.",
"In finding an expression for $\\widehat{\\varphi }(k;a,b)$ we are assisted by the spherical symmetry of the connection function and follow [18].",
"It transpires that $\\widehat{\\varphi }(k;a,b) = \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}}\\left(R_{a,b}*{k}\\right),$ where $J_{\\frac{d}{2}}$ is the Bessel function of the first kind of order $\\frac{d}{2}$ .",
"This function has the expansion $J_{\\frac{d}{2}}\\left(r\\right) = \\sum ^\\infty _{m=0} \\frac{\\left(-1\\right)^{m}}{m!\\Gamma \\left(\\frac{d}{2}+m+1\\right)}\\left(\\frac{r}{2}\\right)^{\\frac{d}{2}+2m},$ converging for all $r\\ge 0$ .",
"We will consider three different regimes for $*{k}$ .",
"For $r \\ll \\sqrt{d}$ the expansion (REF ) is asymptotic, and since $R^{({\\rm max})}_d$ and $R^{({\\rm min})}_d$ are both asymptotically proportional to $\\sqrt{d}$ we have $\\widehat{\\varphi }(k;a,b) \\sim \\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R_{a,b}^d - \\frac{\\pi ^\\frac{d}{2}}{4\\Gamma \\left(\\frac{d}{2}+2\\right)}R_{a,b}^{d+2}*{k}^2 = V_{a,b}\\left(1 - \\frac{R^2_{a,b}}{2\\left(d+2\\right)}*{k}^2\\right)$ for $*{k}\\ll 1$ .",
"This proves the quadratic part of the desired bound.",
"The Bessel function $J_{\\frac{d}{2}}$ is bounded and achieves its global maximum (in absolute value) at its first non-zero stationary point, denoted $j^{\\prime }_{\\frac{d}{2},1}$ .",
"From [2], we have $j^{\\prime }_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _1 \\approx 0.81$ , and $*{J_{\\frac{d}{2}}\\left(j^{\\prime }_{\\frac{d}{2},1}\\right)} \\le M d^{-\\frac{1}{3}}$ for a $d$ -independent $M>0$ .",
"Since $R_{a,b} \\in \\left[2c_1,1/\\sqrt{2\\pi \\text{e}}+o(1)\\right]d^\\frac{1}{2}$ (for $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right]$ from (REF )) and $j^{\\prime }_{\\frac{d}{2},1} / R_{a,b} \\in \\left[\\sqrt{\\frac{\\pi \\text{e}}{2}}+o(1),\\frac{1}{4c_1}+o(1)\\right]d^{\\frac{1}{2}}$ we have $*{\\widehat{\\varphi }(k;a,b)} \\le \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\le \\left(2e^{-1} + o(1)\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\rightarrow 0$ for $*{k} \\ge j^{\\prime }_{\\frac{d}{2},1} / R_{a,b}$ .",
"We are now left with the intermediate range for $*{k}$ .",
"The first positive zero of $J_\\frac{d}{2}$ occurs at $j_{\\frac{d}{2},1}$ , where $j_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _2 \\approx 1.86$ (see [2]).",
"In particular, we will always have $j^{\\prime }_{\\frac{d}{2},1} < j_{\\frac{d}{2},1}$ .",
"From differential identities relating Bessel functions (see [18]), we have $\\frac{\\mathrm {d}}{\\mathrm {d}*{k}} \\widehat{\\varphi }(k;a,b) = -R_{a,b}\\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}+1}\\left(R_{a,b}*{k}\\right).$ The Bessel function $J_{\\frac{d}{2}+1}(r)$ is positive for $r>0$ until its first positive zero at $j_{\\frac{d}{2}+1,1}$ , and $j_{\\frac{d}{2}+1,1} > j_{\\frac{d}{2},1}$ .",
"Therefore $\\widehat{\\varphi }(k;a,b)$ is positive and decreasing on the whole region $*{k}\\in \\left(0,j_{\\frac{d}{2},1}/R_{a,b}\\right]$ .",
"Since this overlaps with the high $*{k}$ range, and we have the uniform quadratic behaviour near $k=0$ , the function $k\\mapsto *{\\widehat{\\varphi }(k;a,b)}$ can never increase and approach $\\widehat{\\varphi }(0;a,b)$ again after leaving a neighbourhood of $k=0$ .",
"We therefore have the bound (REF ) and thus have proven that REF is satisfied.",
"Figure: Sketch of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) against *k*{k}.",
"It approaches its maximum quadratically as *k→0*{k}\\rightarrow 0.",
"The first local maximum of J d 2 J_{\\frac{d}{2}} occurs at j d 2,1 ' ∼d 2+γ 1 d 2 1 3 j^{\\prime }_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3}.",
"The first zero of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) occurs at *kR a,b =j d 2,1 ∼d 2+γ 2 d 2 1 3 *{k}R_{a,b} = j_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} where γ 2 >γ 1 \\gamma _2>\\gamma _1.",
"Furthermore, ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) is strictly decreasing until *kR a,b =j d 2+1,1 ∼d 2+γ 2 d 2 1 3 +1 2*{k}R_{a,b} = j_{\\frac{d}{2}+1,1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\frac{1}{2}.We return to () in REF .",
"We first calculate the second moment: $\\int *{\\overline{x}}^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{d}{d+2}V_{a,b}R^2_{a,b}.$ From $*{\\overline{x}}^2 = \\overline{x}_1^2 + \\ldots + \\overline{x}^2_d$ (for an orthonormal basis) and the rotational symmetry, we have $\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{1}{d+2}V_{a,b}R^2_{a,b}$ for all unit vectors $e\\in \\mathbb {R}^d$ .",
"Therefore $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\frac{4}{d+2}V^{({\\rm max})}\\left(R^{({\\rm max})}_d\\right)^2.$ While $V^{({\\rm max})}$ is a constant, $R^{({\\rm max})}_d\\sim \\sqrt{d/8\\pi \\text{e}}$ and therefore we have a bound uniform in $d$ .",
"The argument now proceeds similarly to the corresponding part of the proof for the multivariate Gaussian model.",
"From (REF ) and REF we have both sides of () satisfying quadratic bounds around $k=0$ , and the triangle inequality with our other bounds prove that the bound also holds on the remainder.",
"For REF we first want to bound the convolution of three $\\varphi $ functions.",
"Given marks $a_1,\\ldots ,a_6\\in \\left[0,1\\right]$ and $\\overline{x}\\in \\mathbb {R}^d$ , let $\\mathbb {B}_{a,b}(\\overline{x})$ denote the ball of radius $R_{a,b}$ centred on $\\overline{x}$ , and let $1_{a,b}\\left(\\overline{x}\\right) = 1\\left\\lbrace *{\\overline{x}}\\le R_{a,b}\\right\\rbrace $ .",
"Then $\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6)\\right)\\left(\\overline{x}\\right)&= \\int 1_{a_1,a_2}(\\overline{y})1_{a_3,a_4}(\\overline{z}-\\overline{y})1_{a_5,a_6}(-\\overline{z})\\mathrm {d}\\overline{y}\\mathrm {d}\\overline{z}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)}\\left(\\int _{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)} 1_{a_3,a_4}(\\overline{z}-\\overline{y})\\mathrm {d}\\overline{z}\\right) \\mathrm {d}\\overline{y}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}.$ Then for $\\delta \\in \\left(0,1\\right)$ we have $&\\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}\\nonumber \\\\&\\qquad \\le \\int _{\\delta \\mathbb {B}_{a,b}\\left(\\overline{0}\\right)}*{\\mathbb {B}_{a_3,a_4}\\left(\\overline{0}\\right)}\\mathrm {d}\\overline{y}+ *{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} \\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} \\nonumber \\\\& \\qquad \\le \\left(V^{({\\rm max})}\\right)^2\\delta ^d + V^{({\\rm max})}\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}.$ Let $\\overline{e}_1\\in \\mathbb {R}^d$ be a unit vector.",
"Then by spherical symmetry we have $\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} = *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)}.$ We bound the volume of this intersection with the volume of the $d$ -ball with radius equal to the radius of the $d-1$ -sphere formed by the intersection of their boundaries.",
"This radius is equal to the length $h$ in Figure REF , which is maximised when $R_{a_5,a_6}$ and $R_{a_3,a_4}$ are maximised and $\\delta R_{a_1,a_2}$ is minimised.",
"Recall we defined the constant $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right)$ when we defined the model in Section REF .",
"Since we have $R_{a_5,a_6},R_{a_3,a_4} \\le 2R^{({\\rm max})}_d$ and $\\delta R_{a_1,a_2} \\ge \\delta c_1 d^\\frac{1}{2}$ , we have $h\\le \\sqrt{\\left(2 R^{({\\rm max})}_d\\right)^2 - \\delta ^2 c^2_1 d}$ - we have used the fact that the triangle is isosceles if both the $R_{a_5,a_6}$ and $R_{a_3,a_4}$ edges attain their maximum.",
"The ball with this radius then has the volume $\\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}h^d \\le V^{({\\rm max})}\\left(1 - \\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}}\\right)^\\frac{d}{2}.$ Since $\\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}} = 2\\pi \\text{e}\\delta ^2c_1^2\\left(1+o(1)\\right)$ and $c_1>0$ , this shows that $*{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)} \\le V^{({\\rm max})}\\left(1 - \\pi \\text{e}\\delta ^2c_1^2\\right)^\\frac{d}{2}$ for $d$ sufficiently large.",
"Sine we now have a mark independent bound with the required decay, we have proved the bound for the convolution of three $\\varphi $ functions.",
"Figure: For both the arguments, we bound the intersection of two dd-balls with a new dd-ball that has the radius hh in these diagrams.For finding the sets $B(x)$ in (REF ), we write $\\int \\varphi \\left(\\overline{x}- \\overline{y};a,c\\right)\\varphi \\left(\\overline{y};c,b\\right)\\mathrm {d}\\overline{y}\\mathcal {P}\\left(\\mathrm {d}c\\right) = \\int *{\\mathbb {B}_{a,c}\\left(\\overline{x}\\right)\\cap \\mathbb {B}_{c,b}\\left(\\overline{0}\\right)}\\mathcal {P}\\left(\\mathrm {d}c \\right).$ As before, we can bound the volume of this intersection independently of the mark $c$ .",
"We can bound it with the volume of the ball with radius $h$ where $h\\le \\sqrt{\\left(2R^{({\\rm max})}_d\\right)^2 - \\frac{1}{4}*{\\overline{x}}^2}$ (as can be seen in Figure REF ).",
"As before, if $*{\\overline{x}} \\ge \\kappa d^\\frac{1}{2}$ for some $\\kappa >0$ , then the volume of the ball with radius $h$ vanishes in the $d\\rightarrow \\infty $ limit.",
"Furthermore, if $\\kappa \\le \\frac{1}{\\sqrt{2\\pi \\text{e}}}$ then the volume of the ball with radius $\\kappa d^\\frac{1}{2}$ vanishes in the $d\\rightarrow \\infty $ limit.",
"Therefore assumption REF is satisfied with the sets $B(x)\\equiv B\\left(\\overline{x},a\\right)=\\left\\lbrace \\left(\\overline{y},b\\right)\\in \\mathbb {X}:*{\\overline{x}-\\overline{y}} \\le \\kappa d^\\frac{1}{2}\\right\\rbrace $ for any $\\kappa \\in \\left(0,\\frac{1}{\\sqrt{2\\pi \\text{e}}}\\right)$ .",
"Displacement Diagram Case Distinctions This section provides the details omitted from the proof of Proposition REF .",
"Here we show which diagrams arise from displacements crossing a $\\Psi $ segment, and whether they can be dealt with by the first sub-case, $\\mathrm {\\left(I\\right)}$ , or by the second sub-case, $\\mathrm {\\left(II\\right)}$ .",
"Recall that $\\mathrm {\\left(I\\right)}$ are those that use a $\\Psi $ decomposition of the larger diagram only (and the displacement is not crossing a $\\Psi ^{(4)}$ segment), while $\\mathrm {\\left(II\\right)}$ are those that will require a combination of a $\\Psi $ decomposition and a $\\overline{\\Psi }$ decomposition.",
"We first expand out the displaced $\\Psi $ terms and label which are in sub-case $\\mathrm {\\left(I\\right)}$ and which are in sub-case $\\mathrm {\\left(II\\right)}$ .",
"Note that the displacement may be across the `top' or the `bottom' of the segments in the usual orientation.",
"If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(1)}$ term, we use cosine-splitting and sometimes expand $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (2,0.8);(1,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[below]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad \\right)\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}\\quad \\right).$ Using the same ideas, if we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(2)}$ term we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };(0.75,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\right.\\nonumber \\\\&\\hspace{28.45274pt}\\left.\\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\quad \\right).$ If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(3)}$ term, then by symmetry in both cases we get $\\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}.$ We first show the bounds for the single segments in sub-case $\\mathrm {\\left(I\\right)}$ .",
"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2.$ We first consider the displaced $\\Psi ^{(j)}$ term by itself.",
"By using supremum bounds and spatial translations we arrive at $\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , \\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda .$ For most of these, it is sufficient to separate the pair using a supremum bound and use the individual bound above for the displaced $\\Psi ^{(j)}$ term and the bound on the $\\Psi $ or $\\Psi _n$ term from Lemma REF .",
"The only pairs that need a more careful approach are those that have the bound $W_k\\triangle ^\\circ _\\lambda $ above for the displaced $\\Psi ^{(j)}$ , followed by a $\\Psi ^{(4)}$ or $\\Psi ^{(2)}_n$ term.",
"In these cases we consider the pair together and get the diagram bounds $\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2,\\qquad \\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2.$ In both of these cases we bounded a $\\tau _\\lambda $ edge by 1 (essentially omitting the edge) before applying a supremum bound.",
"In the first we omit the diagonal edge and split the upper supremum vertex in two.",
"In the second we omit the left-most vertical edge and apply a supremum bound at the bottom right vertex.",
"We now turn our attention to sub-case $\\mathrm {\\left(II\\right)}$ .",
"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda .$ Also recall that if the $\\overline{\\Psi }_0$ term is actually a $\\overline{\\Psi }^{(2)}_0$ term, then the $\\tau _\\lambda ^\\circ $ edge connecting them is actually just a $\\tau _\\lambda $ edge.",
"We first investigate bounds for the displaced $\\Psi ^{(j)}$ term with the connecting $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges.",
"By using supremum bounds, spatial translations, splitting supremum vertices in two, and sometimes expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction, we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(3,0.6) -- (2,0.6) -- (2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);\\node [mark size=2pt] at (0,-0.6) {{square*}};[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(2,-0.6) -- (3,-0.6);(1.5,0.6) -- (3,0.6);[dashed] (3,0.6) -- (3,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (3,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(1.5,0.6) -- (3,0.6);[dashed] (0,0.6) -- (0,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\le \\overline{H_k}+ W_k\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda .$ For most of these, it is clear from a crude supremum bound that we have the bound $\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda $ for the pair of these with the subsequent $\\overline{\\Psi }$ or $\\overline{\\Psi }_0$ term (like we did for sub-case $\\mathrm {\\left(I\\right)}$ ).",
"The only pairs that need a more careful approach are those that have the bound $\\overline{W_k}\\triangle ^\\circ _\\lambda $ or $\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda $ above, followed by a $\\overline{\\Psi }^{(4)}$ or $\\overline{\\Psi }^{(2)}_0$ term.",
"We first consider the cases where they are followed by a $\\overline{\\Psi }^{(4)}$ term.",
"Here we use supremum bounds, spatial translations, and expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2.$ We now consider the cases where they are followed by a $\\overline{\\Psi }^{(2)}_0$ term.",
"Recall that for this case, the connecting $\\tau _\\lambda ^\\circ $ edge is in fact only a $\\tau _\\lambda $ edge.",
"We then get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ The remaining sub-case $\\mathrm {\\left(III\\right)}$ was fully dealt with in the main text.",
"Proof of Lemma REF Recall that we wish to prove the following result: * We begin by defining a sightly different object to $\\overline{H_k}$ .",
"Given $\\vec{a}=\\left(a_1,\\ldots ,a_{16}\\right)\\in \\mathcal {E}^{16}$ , define $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right) :=& \\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8) \\nonumber \\\\&\\hspace{28.45274pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _{\\lambda ,k}(\\overline{t}+ \\overline{x}_3-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u},$ so we have $\\overline{H_k}\\le f(\\lambda )^5\\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,\\vec{a}\\in \\mathcal {E}^{16}}H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{0};\\vec{a};k\\right).$ We then proceed to bound $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right)$ to get our result.",
"This form is preferred because it more easily indicates what the Fourier arguments should be when we take a Fourier transform.",
"A schematic Fourier diagram for this can be found in [21].",
"We then expand out the $\\tau _\\lambda $ and $\\tau _{\\lambda ,k}$ terms.",
"For each of the $\\tau _\\lambda $ terms we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of the cosine-splitting lemma, we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we only need to get appropriate bounds for the following objects.",
"For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^7\\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {H}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l_1)P^{j_2}_{3,4}(l_1)P^{j_3}_{5,6}(l_2)P^{j_4}_{7,8}(l_2)P^{j_5}_{9,10}(l_1-l_2)P^{j_6}_{11,12}(l_1-l_3)P^{j_7}_{13,14}(l_2-l_3) \\\\ \\times Q^{j_8}_{15,16}(l_3;k)\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}},$ where $P^{j}_{m,n}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_m}{\\widehat{}(l) g_n}} &:j=1\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &:j=2\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &: j=3,\\end{array}\\right.}",
"\\qquad \\text{for }m,n\\in \\left\\lbrace 1,2,\\ldots ,14\\right\\rbrace ,\\\\Q^{j}_{15,16}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &:j=1\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &:j=2\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{}_k(l) g_{16}}} &:j=3\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &:j=4\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &:j=5,\\end{array}\\right.",
"}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{16}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .",
"If we can bound these $\\mathcal {H}(k;\\vec{j})$ appropriately, then we can combine them to get the required bound.",
"We now recall pointwise bounds for the $P^{j}_{m,n}(l)$ and $Q^{j}_{15,16}(l)$ terms.",
"In the proof of Lemma REF we derived $*{*{g_m}{\\widehat{}(l)g_n}} &\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &\\le *{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}}\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}},$ and $*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{16}}{\\widehat{}(l)^2g_{16}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{15}}{\\widehat{}(l)^2g_{15}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(l)*{g_{15}}{\\widehat{}(l)^2g_{15}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k)*{g_{15}}{\\widehat{}(l)^2g_{15}}^\\frac{1}{2}*{g_{16}}{\\widehat{}(l)^2g_{16}}^\\frac{1}{2}$ where $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k)\\widehat{G}_{\\mu _\\lambda }(l+k).$ Also recall that we can bound $*{*{g_m}{\\widehat{}(l)g_n}}\\le C$ and $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ if required.",
"Note that all the bounds for the $Q^{j}_{15,16}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.",
"We therefore only need to show that the remaining integrals of $\\widehat{}$ and $\\widehat{G}_{\\mu _\\lambda }$ terms can all be bounded by some constant multiple of $\\beta ^2$ .",
"Also note that the bounds for $Q^{1}_{15,16}(l)$ , $Q^{2}_{15,16}(l)$ , and $Q^{3}_{15,16}(l)$ only differ by a uniform constant (after we bound $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ ), and so once we have dealt with one of these we have dealt with the other two.",
"Let us first consider $\\vec{j}= \\left(1,1,1,1,1,1,1,1\\right)$ .",
"After applying the uniform pointwise bound for $Q^{1}_{15,16}(l)$ , we use the Cauchy-Schwarz inequality to bound $&\\int *{*{g_1}{\\widehat{}(l_1)g_2}}*{*{g_3}{\\widehat{}(l_1)g_4}}*{*{g_5}{\\widehat{}(l_2)g_6}}*{*{g_7}{\\widehat{}(l_2)g_8}}*{*{g_9}{\\widehat{}(l_1-l_2)g_{10}}} \\nonumber \\\\&\\hspace{199.16928pt}\\times *{*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}}*{*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\&\\qquad \\le \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int *{g_7}{\\widehat{}(l_2)g_8}^2*{g_9}{\\widehat{}(l_1-l_2)g_{10}}^2*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.",
"$ To deal with these parentheses, we use a volume-preserving change of variables and factorise each into three integrals.",
"For example, $\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\\\= \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2\\frac{\\mathrm {d}l_1}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_5}{\\widehat{}(l_2)g_6}^2\\frac{\\mathrm {d}l_2}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_{13}}{\\widehat{}(l_3^{\\prime })g_{14}}^2\\frac{\\mathrm {d}l_3^{\\prime }}{\\left(2\\pi \\right)^{d}}\\right).$ There are now two types of integral we need to bound: $&\\int *{g_m}{\\widehat{}(l)g_n}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int \\left(\\int _{E_m\\times E_n \\times E_m \\times E_n} \\widehat{\\varphi }(l;a,b)\\widehat{\\varphi }(l;c,d) \\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\right) \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int _{E_m\\times E_n \\times E_m \\times E_n} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right)\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b)\\varphi (-\\overline{x};c,d)\\mathrm {d}\\overline{x}\\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b) \\mathrm {d}\\overline{x}\\le C, \\\\&\\int *{g_1}{\\widehat{}(l)g_2}^2 *{g_3}{\\widehat{}(l)g_4}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\le g(d).",
"$ In (REF ) we used $\\varphi \\left(-\\overline{x};c,d\\right)\\in \\left[0,1\\right]$ and REF .",
"In () we bounded the convolution of four adjacency functions using REF .",
"The result of this is that we can bound (REF ) by $C^{\\frac{5}{2}}g(d)^{\\frac{1}{2}}$ .",
"Since $g(d)\\le \\beta ^4$ , this satisfies the bound we require.",
"As noted above, this argument also proves the required bound for $\\vec{j}= \\left(1,1,1,1,1,1,1,2\\right)$ and $\\vec{j}= \\left(1,1,1,1,1,1,1,3\\right)$ .",
"Now suppose that we change some subset of $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace $ from having value 1 to having value 2.",
"The argument proceeds similarly, except that after we have factorised our integrals we now want to also bound integrals of the following types: $&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int \\left(\\int _{E_1\\times E_1\\times E_2\\times E_2}\\widehat{\\varphi }^{(2)}(l;a,b)\\widehat{\\varphi }^{(2)}(l;c,d)\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d)\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int _{E_1\\times E_1\\times E_2\\times E_2}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)(\\overline{0})\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d) \\nonumber \\\\& \\hspace{56.9055pt}\\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\le g(d), \\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)g_4}^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^2 g(d)\\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)^2g_3} *{g_4}{\\widehat{}(l)^2g_4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi ^{(2)}(\\cdot ;a_5,a_6) \\star \\varphi ^{(2)}(\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^4 g(d).$ Up to a constant, these are all less than or equal to the bounds we used for the $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace = \\left\\lbrace 1,\\ldots ,1\\right\\rbrace $ case, and therefore that case dominates.",
"These other cases would behave as constant multiple of one of $g(d)$ , $g(d)^{\\frac{3}{2}}$ , $g(d)^2$ , $g(d)^{\\frac{5}{2}}$ , and $g(d)^3$ .",
"We have now dealt with all the cases which have no factors of $\\widehat{G}_{\\mu _\\lambda }$ .",
"To demonstrate how we will deal with these, we first consider $\\vec{j}=\\left\\lbrace 3,3,3,3,3,3,3,5\\right\\rbrace $ .",
"For notational compactness, we define $\\left<\\widehat{}(l)^2\\right>_{I} := \\prod _{m\\in I}*{g_m}{\\widehat{}(l)^2g_m},$ where $I\\subset \\mathbb {N}$ and $l\\in \\mathbb {R}^d$ .",
"Since we have already extracted the factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ via the pointwise bound on $Q^{5}_{15,16}(l;k)$ , we are left trying to bound the following integral: $\\int \\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\\\ \\qquad \\times \\left<\\widehat{}(l_1)^2\\right>^{\\frac{1}{2}}_{1,2,3,4} \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_2)^2\\right>^{\\frac{1}{2}}_{9,10} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}.$ To bound this integral we need to partition our space.",
"Fix $\\varepsilon >0$ and define $B_1 &:= \\left\\lbrace *{l_1}< \\varepsilon \\right\\rbrace \\\\B_2 &:= \\left\\lbrace *{l_2}< \\varepsilon \\right\\rbrace \\\\B_3 &:= \\left\\lbrace *{l_1-l_3}< \\varepsilon \\right\\rbrace \\\\B_4 &:= \\left\\lbrace *{l_1-l_2}< \\varepsilon \\right\\rbrace \\\\B_5 &:= \\left\\lbrace *{l_2-l_3}< \\varepsilon \\right\\rbrace \\\\B_6 &:= \\left\\lbrace *{l_3}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3-k}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3+k}< \\varepsilon \\right\\rbrace ,$ and for $n=0,1,\\ldots ,6$ define $A_n := \\left\\lbrace (l_1,l_2,l_3)\\in \\left(\\mathbb {R}^{d}\\right)^3 \\colon \\#\\left\\lbrace m\\colon (l_1,l_2,l_3)\\in B_m \\right\\rbrace =n\\right\\rbrace .$ In words, $A_n$ is the set of points within $\\varepsilon $ (in Euclidean distance) of precisely $n$ of the hyperplanes on which the $\\widehat{G}_{\\mu _\\lambda }$ factors diverge.",
"If we are working with a factor of $\\widehat{G}_{\\mu _\\lambda }$ on the set $B_m$ associated with its relevant hyperplane, we use the bound $\\widehat{G}_{\\mu _\\lambda }(l)\\le 1/\\left(C_2*{l}^2\\right)$ , and if we are working off this set we use the bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ .",
"When we integrate over $A_0$ , we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ for all such factors and recycle the bound on the “only $\\widehat{}$ \" integrals from above.",
"The contribution from $A_0$ is therefore bounded by some constant multiple of $\\frac{1}{\\varepsilon ^{18}}g(d)^3.$ For $A_1$ , first suppose we are considering $A_1\\cap B_m$ for $m\\ne 6$ .",
"First we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\le 3/\\left(C_2\\varepsilon ^2\\right)$ for the “irrelevant\" directions, and use the pointwise bound for $\\left<\\widehat{}(l)^2\\right>_{m,n}$ in the “relevant\" direction.",
"Then we can use the Cauchy-Schwarz and factorise argument we used above and recycle the integral bounds for the $\\widehat{}$ factors as we did above.",
"However we are careful about the way we split apart terms in the Cauchy-Schwarz step.",
"We split the $\\widehat{G}_{\\mu _\\lambda }$ terms across both factors and arrange the remaining $\\widehat{}$ terms so that each factor has “directions\" spanning $\\left(\\mathbb {R}^d\\right)^3$ .",
"To demonstrate we consider $A_1\\cap B_1$ : $&\\int _{A_1\\cap B_1} \\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12} \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\& \\hspace{56.9055pt} \\le C^2\\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\left<\\widehat{}(l_2)^2\\right>_{5,6}\\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\& \\hspace{142.26378pt}\\times \\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>_{7,8} \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.$ After a change of variables, factorising and using our integral bounds of $\\widehat{}$ terms from above, we are only left needing to bound $\\int _{*{l}<\\varepsilon } \\widehat{G}_{\\mu _\\lambda }(l)^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\le \\frac{1}{C^2_2} \\int _{*{l}<\\varepsilon } \\frac{1}{*{l}^4} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-4}}{d-4}.$ Therefore the contribution from $A_1\\cap B_m$ for $m\\ne 6$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ .",
"For $A_1\\cap B_6$ we need to be slightly more careful because for $*{k}<2\\varepsilon $ the constituent parts of $B_6$ overlap - this is the same considerations we had to account for in the proof of Lemma REF .",
"We can overcome it in the same way by using Hölder's inequality (actually just Cauchy-Schwarz in this case) and then re-centring the integrals.",
"The net result is that the bound of a constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ also applies to $A_1\\cap B_6$ .",
"In summary, the contribution from $A_1$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2.$ The $A_2$ case proceeds similarly.",
"After partitioning the set according to which are the overlapping hyperplanes, we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions and one $\\widehat{}$ factor from a linearly independent direction, and pointwise bound all the other $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors.",
"The contribution from $A_2$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-4\\right)^2}g(d),$ arising from the $A_1\\cap B_1\\cap B_2$ , $A_1\\cap B_1\\cap B_6$ , and $A_1\\cap B_2\\cap B_6$ integrals.",
"The $A_3$ case is more complicated because there are two qualitatively different sub-cases to consider.",
"First suppose that the three normal vectors to the overlapping hyperplanes are linearly independent.",
"Then we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions, pointwise bound the other $\\widehat{G}_{\\mu _\\lambda }$ factors and all the $\\widehat{}$ factors.",
"The contribution from this sub-case of $A_3$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3}$ , arising from the $A_1\\cap B_1\\cap B_2\\cap B_6$ integral.",
"On the other hand, if the three normal vectors to the overlapping hyperplanes are linearly dependent, then we need to retain a $\\widehat{}$ from the remaining linearly independent direction.",
"To demonstrate, let us consider the $A_3\\cap B_1 \\cap B_2 \\cap B_4$ integral.",
"After removing unnecessary terms via uniform bounds, $&\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\& \\qquad \\le \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}.$ Applying a change of variables, factorising the integral, and applying Hölder's inequality allows us to bound these parentheses: $& \\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)\\left(\\int \\left<\\widehat{}(l)^2\\right>_{13,14}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^4}\\frac{1}{*{l_2}^4}\\frac{1}{*{l_1-l_2}^2} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right) g(d)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^6}\\frac{1}{*{l_2}^3} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{2}{3}}\\left(\\int _{\\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_2}^6}\\frac{1}{*{l_1-l_2}^6} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{1}{3}} g(d)\\nonumber \\\\& \\qquad = \\frac{1}{C^5_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{4}{3}}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-3}\\frac{\\varepsilon ^{d-3}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}g(d).$ For all of these linearly dependent cases, we can bound the integral by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d)$ .",
"In summary, the contribution from $A_3$ is bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3} + \\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d).$ For $A_4$ , $A_5$ , and $A_6$ we don't have the two linearly dependent/independent cases because any four of the directions spans $\\left(\\mathbb {R}^d\\right)^3$ .",
"We therefore don't need factors of $\\widehat{}$ to control the unbounded directions.",
"Instead, we need to be careful that when we split our integral at the Cauchy-Schwarz step the resulting parentheses have sufficiently few factors of $\\widehat{G}_{\\mu _\\lambda }$ that their bounds will be finite for $d>6$ .",
"The way we perform this split will be different for the different parts of $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k)$ .",
"Let us define $\\widehat{D}(l;k) := \\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k).$ We first consider $A_6$ .",
"By writing $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)$ , we have two integrals we want to bound.",
"First $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.",
"$ We then perform the $l_i$ integrals in specific orders.",
"For the first factor we integrate over $l_3$ (which factorises out already), then we integrate over $l_2$ for fixed $l_1$ , and finally integrate over $l_1$ .",
"For the second factor we integrate over $l_1$ for fixed $l_2$ and $l_3$ , then we integrate over $l_3$ for fixed $l_2$ , and finally we integrate over $l_2$ .",
"There are therefore three forms of integral we need to bound: $\\int _{*{l}<\\varepsilon }\\widehat{G}_{\\mu _\\lambda }(l)^3\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l}^2}\\frac{1}{*{l-l^{\\prime }}^4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\&\\le \\frac{1}{C^3_2}\\left(\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{1}{3}\\left(\\int _{*{l-l^{\\prime }}<\\varepsilon }\\frac{1}{*{l-l^{\\prime }}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{2}{3}\\nonumber \\\\& = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ In this last inequality, $C^{\\prime }>0$ is some uniform constant.",
"The last inequality is derived by applying Hölder's inequality in much the same way as for the second inequality, taking care to see when the three sets that are integrated over are overlapping.",
"The calculation uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .",
"The second integral we want to bound for $A_6$ is $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3-l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.",
"$ We then perform the $l_i$ integrals in specific orders.",
"For the first factor we integrate over $l_1$ (which factorises out already), then we integrate over $l_2$ for fixed $l_3$ , and finally integrate over $l_3$ .",
"For the second factor we integrate over $l_3$ for fixed $l_1$ and $l_2$ , then we integrate over $l_1$ for fixed $l_2$ , and finally we integrate over $l_2$ .",
"The new integrals we need to bound are: $\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}$ $\\int _{\\left(\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace \\right)\\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4}\\right)\\varepsilon ^{d-6}.$ The calculation of these two bounds uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .",
"The result of these bounds is that the integral in (REF ) is bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ For $A_4$ and $A_5$ we can repeat the argument for $A_6$ , but note that in some places the bound $1/\\left(C_2*{l}^2\\right)$ will be replaced by $1/\\left(C_2\\varepsilon ^2\\right)$ .",
"The net result of this is that the factor $\\left(d-6\\right)^3$ can be replaced by $\\left(d-6\\right)^{\\alpha _0}\\left(d-4\\right)^{\\alpha _1}\\left(d-2\\right)^{\\alpha _2} d^{\\alpha _3}$ for some $\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _3\\le 0$ such that $\\alpha _0+\\alpha _1+\\alpha _2+\\alpha _3 =3$ .",
"The net result is that the contribution from both $A_4$ and $A_5$ can also be bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ Above we have established bounds for the various parts of the integral when $\\vec{j}=\\left(3,3,3,3,3,3,3,5\\right)$ .",
"Having $j_8=4$ can also be dealt with by very similar arguments to that outlined above - the $l_3$ integrals are in fact simpler.",
"The above bounds still hold with the $\\varepsilon ^{-18}$ becoming a $\\varepsilon ^{-16}$ .",
"Since $\\varepsilon <1$ , this produces a larger bound.",
"Recall that up to a constant factor, having $j_m=1$ produces a larger bound than $j_m=2$ for $m=1,\\ldots ,7$ .",
"If we replace $j_m=3$ with $j_m=1$ , then after ignoring constant values we lose a factor of $\\varepsilon ^{-2}$ , but we may also lose a factor of $g(d)^\\frac{1}{2}$ unless that was for $m=1,2$ (corresponding to $l_1$ direction) or $m=3,4$ (corresponding to $l_2$ direction).",
"Note that doing for both directions will lose a $g(d)^\\frac{1}{2}$ factor.",
"If $\\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d= O\\left(g(d)^{\\frac{1}{2}}\\right)$ for all fixed $\\varepsilon >0$ , then it is clear that the dominating bound is $g(d)^\\frac{1}{2}$ .",
"This arises, for example from the bound for $\\vec{j}=\\left(1,1,1,1,1,1,1,1\\right)$ .",
"The situation is more complicated if $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , because we can choose to have $\\varepsilon =\\varepsilon (d)$ and take it to 0 as $d\\rightarrow \\infty $ .",
"It is easy to see that all the $A_1$ components (and $A_2,\\ldots ,A_6$ components) for each $\\vec{j}$ can be bounded by some constant multiple of $\\frac{1}{d}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\varepsilon ^d$ (provided $d>6$ ).",
"We therefore only need to compare this to the contributions from $A_0$ .",
"The bound we get from the above arguments for each $A_0$ contribution are of the form $\\varepsilon ^{-2m}g(d)^\\frac{n}{2}$ , where $m$ is the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ and $n$ is related to the multiplicity of $$ terms in each Fourier direction.",
"To find the terms with the largest bound, we want to have as many factors of $\\widehat{G}_{\\mu _\\lambda }$ as possible without producing extra factors of $g(d)^\\frac{1}{2}$ .",
"Since $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , increasing $n$ here instantly produces a smaller bound.",
"By changing $j_8$ from 1 to 4 or 5, or any other element from 1 to 3, we do gain a factor of $\\widehat{G}_{\\mu _\\lambda }$ but we also increase the number of $$ factors and therefore in most places we acquire extra factors of $g(d)^{\\frac{1}{2}}$ .",
"The exception to this is for $j_1$ and $j_2$ , which correspond to the $l_1$ directions, since they already produce a factor of $g(d)^{\\frac{1}{2}}$ .",
"The $A_0$ component of the $\\vec{j}=\\left(3,3,1,1,1,1,1,1\\right)$ term can be bounded by constant multiple of $\\varepsilon ^{-4}g(d)^{\\frac{1}{2}}$ .",
"We then optimise our choice of $\\varepsilon $ to have our two dominant bounds be of the same order.",
"This produces $\\varepsilon (d)^2 = g(d)^{\\frac{1}{d}}\\frac{2\\pi d}{\\text{e}}\\left(1+o(1)\\right)$ , and therefore our overall bound is given by $\\varepsilon ^{-4}g(d)^\\frac{1}{2} = g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(\\frac{\\text{e}}{2\\pi }\\right)^2\\frac{1}{d^2}\\left(1+o(1)\\right).$ Since $g(d)^{\\frac{1}{2}-\\frac{2}{d}} \\le g(d)^{\\frac{1}{2}-\\frac{3}{d}}$ for sufficiently large $d$ , this bound is of order $\\beta ^2$ and our bound is proven.",
"Acknowledgements.",
"This work is supported by Deutsche Forschungsgemeinschaft (project number 443880457) through priority program “Random Geometric Systems” (SPP 2265).",
"The authors thank the Centre de recherches mathématiques Montreal for hospitality during a research visit in spring 2022 through the Simons-CRM scholar-in-residence program."
],
[
"Diagrammatic bounds",
"For our argument, we will require bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}$ for all $n\\in \\mathbb {N}$ and $k\\in \\mathbb {R}^d$ .",
"However, it turns out that we can bound the last two remainder terms using others.",
"Lemma 5.1 For all $n\\in \\mathbb {N}$ , $k\\in \\mathbb {R}^d$ , and $\\lambda >0$ , $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op},\\\\\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(0)\\right\\Vert _{\\rm op}.$ As $\\mathbb {P}_\\lambda (u_n {\\,\\,A\\,\\,} x\\textrm { in } \\xi ^{u_n,x}_{n+1} ) \\le \\tau _\\lambda (x,u_n)$ for an arbitrary locally finite set $A$ , the definition () gives $*{r_{\\lambda ,n}(x)} \\le \\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right).$ Let $f\\in L^2\\left(\\mathbb {X}\\right)$ .",
"Then this gives $*{R_{\\lambda ,n}f}^2_2 &\\le \\int \\left(\\int *{r_{\\lambda ,n}(x,y)}*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&\\le \\lambda ^2\\int \\left(\\int \\left(\\int \\tau _\\lambda (x,u)\\pi _\\lambda ^{(n)}(u,y) \\nu \\left(\\mathrm {d}u\\right)\\right)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\lambda ^2\\int \\left(\\int \\tau _\\lambda (x,u)\\left(\\int \\pi _\\lambda ^{(n)}(u,y)*{f\\left(y\\right)}\\nu \\left(\\mathrm {d}y\\right)\\right) \\nu \\left(\\mathrm {d}u\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\& = \\lambda ^2 *{\\mathcal {T}_\\lambda _\\lambda ^{(n)}{\\rm Abs}f}^2_2.$ Note that we were able to use Tonelli's theorem to swap the $u$ and $y$ integrals because $\\tau _\\lambda $ and $\\pi ^{(n)}_\\lambda $ are both non-negative and measurable.",
"Finally, the approach of (REF ) implies the result.",
"For each $n\\ge 0$ , either $r_{\\lambda ,n}(x)\\ge 0$ for all $x\\in \\mathbb {X}$ or $r_{\\lambda ,n}(x)\\le 0$ for all $x\\in \\mathbb {X}$ .",
"Therefore $*{\\widehat{r}_{\\lambda ,n}(k;a,b)}\\le *{\\widehat{r}_{\\lambda ,n}(0;a,b)}$ for all $a,b\\in \\mathcal {E}$ and $\\left\\Vert \\widehat{R}_{\\lambda ,n}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{R}_{\\lambda ,n}(0)\\right\\Vert _{\\rm op}$ (by Lemma REF ) for all $k\\in \\mathbb {R}^d$ .",
"From (REF ) we have $*{\\widehat{r}_{\\lambda ,n}(0;a,b)} \\le \\int \\widehat{\\tau }_\\lambda (0;a,c)\\widehat{\\pi }_\\lambda ^{(n)}(0;c,b) \\mathcal {P}\\left(\\mathrm {d}c\\right).$ Therefore the same argument as above gives the second bound in the result.",
"Because the $n=0$ cases for our bounds on $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ , and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ are so simple, we give the proof here before moving onto $n\\ge 1$ .",
"Recall $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right) = \\widehat{\\mathcal {T}}_\\lambda \\left(k+l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right)$ .",
"We will find it convenient to also define $\\widehat{}_k\\left(l\\right) = \\widehat{}\\left(k+l\\right) - \\widehat{}\\left(l\\right)$ .",
"Proposition 5.2 (Bounds for $n=0$ ) Let $\\lambda \\in [0,\\lambda _O)$ .",
"Then $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } \\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }.$ For $k\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op} &\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\right\\Vert _{1,\\infty }\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le \\lambda ^2\\left(\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } + \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\right).$ Our first comment relates to these bounds and every bound we perform hereafter in Section .",
"From the positivity of $\\pi ^{(0)}_{\\lambda }(x,y)$ and Lemma REF we have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(k)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op}$ , and by Lemma REF we have $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{\\rm op} \\le \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty }$ .",
"However from the definitions of the norm we also have $\\left\\Vert \\widehat{}_{\\lambda }^{(0)}(0)\\right\\Vert _{1,\\infty } = \\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty }$ , and so we will be interested in bounding this last norm.",
"Now note that for the event $\\lbrace y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x} \\rbrace $ to hold, either there is a direct edge between $y$ and $x$ , or there exist vertices $w,z$ in $\\eta $ that are direct neighbours of the origin and have respective disjoint paths to $x$ that both do not contain $y$ .",
"Hence, by the Mecke equation (REF ), $\\mathbb {P}_\\lambda ( y \\Longleftrightarrow x\\textrm { in } \\xi ^{\\mathbf {0},x}) &\\le \\varphi (x,y) + \\tfrac{1}{2} \\mathbb {E}_\\lambda \\Big [ \\sum _{(w,z) \\in \\eta ^{(2)}} 1_{(\\lbrace y\\sim w \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^x\\rbrace ) \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^x \\rbrace )} \\Big ] \\nonumber \\\\& = \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\int \\mathbb {P}_\\lambda \\big ( (\\lbrace y\\sim y \\text{ in } \\xi ^{y, w}\\rbrace \\cap \\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{x,w}\\rbrace )\\nonumber \\\\& \\hspace{99.58464pt} \\circ (\\lbrace y \\sim z \\text{ in } \\xi ^{y, z}\\rbrace \\cap \\lbrace z \\longleftrightarrow x\\textrm { in } \\xi ^{x,z} \\rbrace ) \\big ) \\nu ^{\\otimes 2}\\left(\\mathrm {d}w,\\mathrm {d}z\\right).$ After applying the BK inequality to the above probability, $\\mathbb {P}_\\lambda \\left( y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x}\\right) \\le \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left( \\int \\mathbb {P}_\\lambda \\left( \\left\\lbrace y\\sim w \\text{ in } \\xi ^{y,w}\\right\\rbrace \\cap \\left\\lbrace w \\longleftrightarrow x\\textrm { in } \\xi ^{w,x}\\right\\rbrace \\right) \\nu \\left(\\mathrm {d}w\\right) \\right)^2 \\\\= \\varphi (x,y) + \\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2.",
"$ Thus, recalling that $\\pi _\\lambda ^{(0)}(x,y) = \\mathbb {P}_\\lambda (y \\Longleftrightarrow x\\textrm { in } \\xi ^{y,x})-\\varphi (x,y) \\ge 0$ , and using the symmetry of $\\varphi $ and $\\tau _\\lambda $ , $\\left\\Vert _\\lambda ^{(0)}\\right\\Vert _{1,\\infty } &= \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\pi ^{(0)}_\\lambda \\left(x,y\\right)\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\nonumber \\\\& = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi \\left(y,u\\right)\\tau _\\lambda \\left(u,x\\right)\\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}w,\\mathrm {d}u\\right)\\nonumber \\\\&\\le \\tfrac{1}{2}\\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty }.$ We then use a supremum bound on this integral to split one of the $$ off from the others, and bound the other from above with $\\mathcal {T}_\\lambda $ to get the first two bounds of the result.",
"For the last bound of Proposition REF , we apply (REF ) and obtain $&\\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(0)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(0)}\\left(k\\right)\\right\\Vert _{1,\\infty }\\nonumber \\\\&\\qquad = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(0)}_\\lambda \\left(\\overline{x};a,b\\right) \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\left(\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\mathrm {d}\\overline{u}\\mathcal {P}\\left(\\mathrm {d}c\\right)\\right)^2 \\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\tfrac{1}{2}\\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1 - \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad \\le \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{b,b^\\prime \\in \\mathcal {E}}\\int \\Big [\\left(1 - \\cos \\left(k\\cdot \\left(\\overline{x}-\\overline{u}\\right)\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b^\\prime ,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\nonumber \\\\&\\qquad \\hspace{85.35826pt}+ \\left(1 - \\cos \\left(k\\cdot \\overline{u}\\right)\\right)\\varphi \\left(-\\overline{u}^\\prime ;b,c^\\prime \\right)\\tau _\\lambda \\left(\\overline{u}^\\prime -\\overline{x};c^\\prime ,a\\right)\\tau _\\lambda \\left(\\overline{x}-\\overline{u};a,c\\right)\\varphi \\left(\\overline{u};c,b\\right)\\Big ] \\nonumber \\\\&\\qquad \\hspace{284.52756pt}\\times \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^\\prime \\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^\\prime ,\\mathrm {d}a\\right)\\nonumber \\\\& \\qquad = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}+ \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty }.$ Here we have again used the symmetry of $\\varphi $ and $\\tau _\\lambda $ , and also used Lemma REF to split the cosine factor over $\\overline{x}-\\overline{u}$ and $\\overline{u}$ .",
"We can consider each term individually by using the triangle inequality.",
"By using a supremum bound, the symmetry of $\\varphi $ , and by bounding $\\varphi $ with $\\tau _\\lambda $ , we get the bound $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }\\left\\Vert _k\\right\\Vert _{1,\\infty }.$"
],
[
"Bounds on expansion coefficients",
"In this section we aim to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}$ and $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}$ .",
"Lemmas REF and REF along with the positivity of $\\pi ^{(n)}_\\lambda (x,y)$ prove that it is sufficient to bound $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty }$ to get bounds on both of these.",
"Much of the hard work for this has been done previously - for example in [21].",
"The novelty here is to see how the argument can be naturally written in terms of operators.",
"We inherit the following notation from [21].",
"Note that we make use of the Dirac delta function - specifically the one that holds with respect to the measure $\\nu $ .",
"They are used here primarily for convenience and to make the argument more readable.",
"In particular, they appear when applying the Mecke equation (REF ) to obtain $\\mathbb {E}\\left[ \\sum _{y\\in \\eta ^u} f(y, \\xi ^u) \\right] = \\int \\left(\\lambda +\\delta _{y,u}\\right) \\mathbb {E}_\\lambda \\left[ f\\left(y,\\xi ^{u,y}\\right)\\right] \\nu \\left(\\mathrm {d}y\\right),$ and so the factor $\\lambda +\\delta _{y,u}$ encodes a case distinction of whether point $y$ coincides with $u$ or not.",
"Definition 5.3 (The $\\psi $ functions) Let $r,s,u,w,x,y\\in \\mathbb {X}$ .",
"We first set $\\tau _\\lambda ^\\circ (x,y) := \\lambda ^{-1}\\delta _{x,y} + \\tau _\\lambda (x,y)$ .",
"Also define $\\psi _0^{(1)}(w,u,y) &:= \\lambda ^2\\tau _\\lambda (y,u)\\tau _\\lambda (u,w)\\tau _\\lambda (w,y),\\\\\\psi _0^{(2)}(w,u,y) &:=\\lambda ^2 \\delta _{w,y} \\tau _\\lambda (y,u)\\int \\tau _\\lambda (u,t)\\tau _\\lambda (t,y) \\nu \\left(\\mathrm {d}t\\right),\\\\\\psi _0^{(3)}(w,u,y) &:= \\lambda ^2\\varphi (u,y) \\left(\\lambda ^{-1}\\delta _{w,y}\\right),\\\\\\psi ^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda ^\\circ (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\psi ^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\psi ^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\left(\\lambda ^{-1}\\delta _{w,s}\\right)\\tau _\\lambda (u,r),\\\\\\psi _n^{(1)} (x,r,s) &:= \\lambda ^3\\int \\tau _\\lambda ^\\circ (t,s)\\tau _\\lambda (z,r)\\tau _\\lambda (t,z)\\tau _\\lambda (z,x)\\tau _\\lambda (x,t)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right),\\\\\\psi _n^{(2)}(x,r,s) &:=\\lambda \\tau _\\lambda (x,s)\\tau _\\lambda (x,r),$ and set $\\psi _0 := \\psi _0^{(1)}+\\psi _0^{(2)}+\\psi _0^{(3)}$ , $\\psi _n := \\psi _n^{(1)} + \\psi _n^{(2)}$ , and $\\psi := \\psi ^{(1)}+\\psi ^{(2)}+\\psi ^{(3)} + \\psi ^{(4)}$ .",
"Note that by using $\\lambda ^{-1}\\delta _{\\left(\\cdot ,\\cdot \\right)}$ in $\\tau _\\lambda ^\\circ $ and in $\\psi ^{(3)}_0$ , we ensure that each integral that will be truly performed (that is, not an integration over a Dirac delta function accounting for a coincidence of points) has a $\\lambda $ factor associated with it when the bound $\\pi ^{(n)}_\\lambda $ is calculated.",
"Diagrammatic representations of these functions can be found in Figure REF .",
"In these diagrams, the declared variables are represented by (0,0) circle (2pt) vertices whereas the variables that are integrated over are represented by (0,0) circle (2pt) vertices.",
"If $\\tau _\\lambda $ connects two variables then a standard edge (0,0) – (1,0); connects their vertices, if $\\tau _\\lambda ^\\circ $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; connects their vertices, and if $\\varphi $ connects two variables then an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\sim $; connects their vertices.",
"Figure: Diagrams of the ψ 0 \\psi _0, ψ\\psi , and ψ n \\psi _n functions.Definition 5.4 (The $\\Psi $ operators) Here we turn the functions of Definition REF into linear operators, with the $\\psi $ functions acting as their kernels.",
"For $j\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , we define $\\Psi ^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\Psi ^{(j)}_0 f\\left(w,u\\right) = \\int \\psi ^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right).$ For $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , we define $\\Psi ^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)} f\\left(w,u\\right) = \\int \\psi ^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ , we define $\\Psi ^{(j)}_n\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}\\right)$ as the linear operator acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\Psi ^{(j)}_n f\\left(x\\right) = \\int \\psi ^{(j)}_n(x,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ We can then naturally define $\\Psi _0\\colon L^2(\\mathbb {X})\\rightarrow L^2(\\mathbb {X}^2)$ , $\\Psi \\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X}^2)$ , and $\\Psi _n\\colon L^2(\\mathbb {X}^2)\\rightarrow L^2(\\mathbb {X})$ as the sums of $\\Psi ^{(j)}_0$ , $\\Psi ^{(j)}$ , and $\\Psi ^{(j)}_n$ respectively, or equivalently as those operators having kernel functions $\\psi _0$ , $\\psi $ , and $\\psi _n$ respectively.",
"Proposition 5.5 (Bound in terms of $\\psi $ and $\\Psi $ ) Let $n \\ge 1$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in [0,\\lambda _O)$ .",
"Then $\\lambda \\pi _\\lambda ^{(n)}(x,y) \\le \\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes \\left(2n\\right)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right),$ where $\\vec{v}_i = (w_i,u_i,w_{i-1},u_{i-1})$ .",
"In operator terms, this means $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n \\Psi ^{n-1} \\Psi _0\\right\\Vert _{1,\\infty }.$ The argument for (REF ) is essentially identical to [21].",
"It uses only general properties of Poisson point process and connection models - for example thinning edges and Mecke's equation.",
"The inequality (REF ) then follows from the clear sequential structure of (REF ).",
"The inequality (REF ) suggests that we will bound the lace coefficient operators with some product of $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ terms.",
"This is nearly correct.",
"Unfortunately the norms $\\left\\Vert \\Psi _n\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi \\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi _0\\right\\Vert _{1,\\infty }$ will not have the decay in $d$ that we will require.",
"We will have to consider the norms of pairs of operators.",
"Definition 5.6 Define $\\triangle _\\lambda &:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^\\circ _\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty },\\\\\\triangle ^{\\circ \\circ }_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda ^\\circ \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1,\\\\\\overline{\\triangle ^{\\circ \\circ }_\\lambda }&:= \\lambda ^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda ^\\circ \\left(\\overline{u}-\\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}.$ We can think of $\\triangle _\\lambda $ as an $\\mathbb {X}$ -convolution of three $\\tau _\\lambda $ functions with suprema taken over the end vertices.",
"Then $\\triangle ^\\circ _\\lambda $ and $\\triangle ^{\\circ \\circ }_\\lambda $ are produced by adding on the $\\mathbb {X}$ -convolution of two $\\tau _\\lambda $ functions and one $\\tau _\\lambda $ function (which will trivially take value 1 once the suprema are taken).",
"The object $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the marks at adjacent $\\tau _\\lambda $ functions need not be equal, and then we take the supremum over all the marks - not just the end ones.",
"We then use these elementary diagrams to produce composite objects: $U_\\lambda := \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }+ \\overline{\\triangle ^{\\circ \\circ }_\\lambda }^2 + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }, \\qquad V_\\lambda := \\left(\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }U_\\lambda \\right)^\\frac{1}{2}.$ We note a few relations.",
"Firstly, since all the terms in $U_\\lambda $ are non-negative we have $\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le U_\\lambda $ and thus $V_\\lambda \\le U_\\lambda $ .",
"Also note that $1\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ and thus $U_\\lambda \\ge \\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .",
"In particular this means that $U_\\lambda \\ge 1$ and $U_\\lambda \\ge \\triangle ^\\circ _\\lambda $ .",
"The following lemma tells us that we can use $U_\\lambda $ to bound single operators, and that we can use $V_\\lambda ^2$ to bound pairs of operators.",
"To aid the reader's understanding, bear in mind that later (in Section ) we will prove that $U_\\lambda = \\mathcal {O}\\left(1\\right)$ , whilst $V_\\lambda $ is much smaller and is $\\mathcal {O}\\left(\\beta \\right)$ .",
"Lemma 5.7 For all $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j,j^{\\prime }\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $U_\\lambda &\\ge \\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\right\\Vert _{1,\\infty },\\\\V_\\lambda ^2 &\\ge \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j)}\\Psi ^{(j^{\\prime })}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j)}\\right\\Vert _{1,\\infty } \\vee \\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }.$ We begin with the single operators.",
"For $\\Psi ^{(1)}$ we find $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } &= \\lambda ^4\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (w,u)\\tau _\\lambda ^\\circ (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda (z,r)\\nu ^{\\otimes 4}\\left(\\mathrm {d}z, \\mathrm {d}t, \\mathrm {d}w, \\mathrm {d}u\\right), \\nonumber \\\\& \\le \\lambda ^4\\left(\\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}}\\int \\tau _\\lambda (r,z) \\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (t,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t \\right)\\right) \\nonumber \\\\&\\hspace{142.26378pt} \\times \\left(\\operatornamewithlimits{ess\\,sup}_{t^{\\prime },z^{\\prime }\\in \\mathbb {X}}\\int \\tau _\\lambda (t^{\\prime },w)\\tau _\\lambda (w,u)\\tau _\\lambda (u,z^{\\prime })\\nu ^{\\otimes 2}\\left(\\mathrm {d}w, \\mathrm {d}u\\right)\\right),\\nonumber \\\\& = \\triangle ^\\circ _\\lambda \\triangle _\\lambda ,$ where we have used a supremum bound on the $t$ and $z$ integrals to split them into $L^1$ and $L^\\infty $ bounds.",
"It is also clear that $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty }= \\triangle _\\lambda $ and $\\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\le \\triangle ^\\circ _\\lambda $ .",
"The $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ bound is more involved and we return to it in a moment.",
"We are able to represent these $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ norms and their bounds pictorially.",
"The vertices of the diagrams represent the variables $w,u,z,t,a,b$ , etc.",
"appearing in the $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ expression.",
"The variables over which a $\\nu $ -integral is taken are represented by the (0,0) circle (2pt) vertices, and the variables over which a supremum is taken are represented by the (0,0) circle (2pt) vertices.",
"The presence of a $\\tau _\\lambda $ connecting two variables is then represented by a standard edge (0,0) – (1,0); between their vertices, and the presence of a $\\tau _\\lambda ^\\circ $ connecting two variables is correspondingly represented by an edge (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\circ $; between their vertices.",
"When using a supremum bound to split the diagrams, an integrated vertex gets split into two new vertices, precisely one integrated vertex and one supremum vertex.",
"Each edge connected to such a split vertex can independently choose which of the new vertices to associate to.",
"The calculations producing the bounds on $\\left\\Vert \\Psi ^{(j)}\\right\\Vert _{1,\\infty }$ for $j=1,3,4$ can then be represented by: $\\left\\Vert \\Psi ^{(1)}\\right\\Vert _{1,\\infty } \\quad &=\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\\\\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } + \\left\\Vert \\Psi ^{(4)}\\right\\Vert _{1,\\infty } \\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad .$ Our strategy for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ is to use the spatial translation invariance to shift the origin to the bottom right vertex, and then use a supremum bound to split the diagram in two.",
"Unfortunately, this spatial shift decouples the spatial and mark components of the vertices so - for example - at some vertices we end up taking suprema over marks whilst integrating over space.",
"Let us introduce some more notation.",
"The vertex mark size=2pt] at (0,0) square* indicates that the spatial component is integrated over whilst the mark component is fixed and its supremum is taken after all the integrals.",
"We also introduce the edge-vertex combination [dashed] (0,0) – (1,0); [fill=white] (0,0) circle (2pt) which indicates that there is no term connecting the associated vertices, but there is a supremum taken over the spatial displacement.",
"Note that in principle we could use a vertex that had a spatial supremum and a mark integral, but we will always bound that probability integral by the supremum.",
"The bound for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ can therefore be expressed in the following: $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}\\node [mark size=2pt] at (0,-0.6) {{square*}};(0,0.6) -- (1.5,0.6) -- (1.5,-0.1);(0,-0.6) -- (1,-0.6);[dashed] (0,0.6) -- (0,-0.6);(0.75,0.6) circle (0pt) node[above]{\\circ };(1.5,0.25) circle (0pt) node[left]{\\circ };[fill=white] (1.5,-0.1) circle (2pt);(1.5,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);[fill=white] (0,0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);[fill=white] (0,0) circle (2pt);\\end{tikzpicture}}.$ To clarify the calculation for $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }$ , we write out the integral here: $&\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}-\\overline{s};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}- \\overline{w}; a_4,a_3)\\tau _\\lambda (\\overline{u}-\\overline{t};a_6,a_4)\\tau _\\lambda (\\overline{z}-\\overline{u};a_5,a_6) \\nonumber \\\\&\\hspace{142.26378pt}\\times \\tau _\\lambda (\\overline{t}- \\overline{z}; a_4,a_5)\\tau _\\lambda (\\overline{z}; a_5,a_2) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5, \\mathrm {d}a_6\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\operatornamewithlimits{ess\\,sup}_{\\overline{s}\\in \\mathbb {R}^d,a_1,a_2,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}a_3, \\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\nonumber \\\\&\\hspace{14.22636pt}\\le \\left(\\operatornamewithlimits{ess\\,sup}_{\\overline{s},\\overline{t}^{\\prime },\\overline{z}^{\\prime }\\in \\mathbb {R}^d,a_1,a_2,a_4,a_5\\in \\mathcal {E}}\\int \\tau _\\lambda ^\\circ (\\overline{w}^{\\prime }-\\overline{s}+\\overline{u};a_3,a_1)\\tau _\\lambda ^\\circ (\\overline{t}^{\\prime } - \\overline{w}^{\\prime }; a_4,a_3)\\tau _\\lambda (\\overline{z}^{\\prime } + \\overline{u}; a_5,a_2) \\mathrm {d}\\overline{u}\\mathrm {d}\\overline{w}^{\\prime } \\mathcal {P}\\left(\\mathrm {d}a_3\\right)\\right)\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_6\\in \\mathcal {E}}\\int \\tau _\\lambda (-\\overline{t}^{\\prime };a_6,a_4)\\tau _\\lambda (\\overline{z}^{\\prime };a_5,a_6)\\tau _\\lambda (\\overline{t}^{\\prime } - \\overline{z}^{\\prime }; a_4,a_5) \\mathrm {d}\\overline{t}^{\\prime } \\mathrm {d}\\overline{z}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a_4, \\mathrm {d}a_5\\right)\\right).$ Then the two components of the first factor that are joined by the $\\raisebox {2pt}{{[dashed] (0,0) -- (1,0); [fill=white] (0,0) circle (2pt)}}$ edge can be spatially translated together to form three consecutive edges.",
"Note however, that the marks on either side of this join are not necessarily equal.",
"We can therefore bound this first term by $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .",
"In summary, these manipulations then imply that $\\left\\Vert \\Psi ^{(2)}\\right\\Vert _{1,\\infty }\\le \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\triangle _\\lambda $ .",
"For $\\Psi ^{(1)}_0$ and $\\Psi ^{(2)}_0$ the calculations are identical to each other, and we can once again represent the calculations pictorially: $\\left\\Vert \\Psi ^{(1)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_0\\right\\Vert _{1,\\infty } =\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- cycle;[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad = \\triangle _\\lambda ,$ where splitting the single supremum into two produces an upper bound.",
"For the third case we have $\\left\\Vert \\Psi ^{(3)}\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) -- (1,0);(0.5,0) circle (0pt) node[above]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad = \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }.$ The diagrams representing the calculations for $\\Psi _n$ are as follows: $\\left\\Vert \\Psi _n^{(1)}\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(0.5,0.6) circle (0pt) node[above]{\\circ };(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);(1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(1,-0.6) -- (0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{113.81102pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\qquad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda , \\nonumber \\\\\\left\\Vert \\Psi ^{(2)}_n\\right\\Vert _{1,\\infty } & = \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda .$ In the first case we expanded a $\\tau _\\lambda ^\\circ $ edge into a $\\tau _\\lambda $ edge and a contraction.",
"Then in the contracted case we bound $\\tau _\\lambda \\le 1$ on the diagonal edge.",
"We have now proven the bounds for the single operator norms.",
"For most of the operator pairs, we can then use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ to immediately get sufficient bounds.",
"The only pairs for which this strategy fails are: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ , and $\\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty }$ .",
"We now deal with these cases by hand: $\\left\\Vert \\Psi ^{(4)}\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(3)}_0\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- (0,0.6);(0,0) circle (0pt) node[above, rotate=90]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda , \\\\\\left\\Vert \\Psi ^{(4)}\\Psi ^{(4)}\\right\\Vert _{1,\\infty } = \\left\\Vert \\Psi ^{(2)}_n\\Psi ^{(4)}\\right\\Vert _{1,\\infty } &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle _\\lambda .$ For the first bound we used $\\varphi \\le \\tau _\\lambda $ , and then split the single supremum into two.",
"For the second bound we used $\\tau _\\lambda \\le 1$ on the diagonal edge.",
"Proposition 5.8 For $n\\ge 1$ , $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le 6\\times 4^{n-1}U_\\lambda V_\\lambda ^n.$ Recall from Lemma REF that $\\lambda \\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\Psi _n\\Psi ^{n-1}\\Psi _0\\right\\Vert _{1,\\infty }$ .",
"We therefore aim to bound $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ for $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ .",
"We have two cases depending upon the parity of $n$ .",
"If $n$ is odd then we have an even number of operators.",
"We pair them off, use the sub-multiplicity of $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , and Lemma REF to get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le \\left(V_\\lambda ^2\\right)^{\\frac{n+1}{2}} = V_\\lambda ^{n+1}.$ If $n$ is even, then we pair off the first $n$ operators and treat the last $\\Psi ^{(j_n)}_n$ operator by itself.",
"We then get $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } \\le U_\\lambda \\left(V_\\lambda ^2\\right)^{\\frac{n}{2}} = U_\\lambda V_\\lambda ^{n}.$ Since $U_\\lambda \\ge V_\\lambda $ , we have the bound $U_\\lambda V_\\lambda ^{n}$ for all $n\\ge 1$ .",
"Since there are 3 choices for $j_0$ , 4 choices for each of $j_1,\\ldots ,j_{n-1}$ , and 2 choices for $j_n$ , we get the pre-factor of $6\\times 4^{n-1}$ in our result by applying the triangle inequality."
],
[
"Bounds on displaced expansion coefficients",
"In this section we will be aiming to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op}$ for $n\\ge 1$ .",
"Our strategy draws inspiration from the corresponding step in [21], whilst being adapted to account for inhomogeneous marks in a similar way to Section REF .",
"The central idea is to bound $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\pi ^{(n)}_\\lambda \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right),$ and bound $\\pi ^{(n)}_\\lambda $ using the function expression from Proposition REF .",
"This produces an integral (or diagram) composed from simpler segments with the displacement factor, $\\left(1- \\cos \\left(k\\cdot \\overline{x}\\right)\\right)$ , spanning the whole length of the diagram.",
"We can then use the Cosine-Splitting result (Lemma REF ) to get a sum of diagrams where the displacement factor only spans a single segment of each diagram.",
"To demonstrate this, consider the diagram corresponding to $\\Psi ^{(2)}_n\\Psi ^{(1)}\\Psi ^{(1)}_0$ with the displacement factor: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8) -- (3,0.8) -- (4,0.2);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\le 3\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\right.",
"\\\\ \\left.\\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (1,0.8) -- (3,0.8);(2,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right).$ In these diagrams and hereafter we use [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; to denote that a displacement factor is connecting the two indicated vertices.",
"As above, the path the [<->] (0,0) – (1,0); (0.5,0) node[cross=3pt]; takes will also suggest the route along which we will intend to use cosine-splitting.",
"For the sake of simplicity, we will be taking the displacement path across the `top' of the diagram from the perspective of the usual orientation of the $\\Psi _0$ segment.",
"However - as can be seen in the above example - the way the operators compose with each other means that the orientation of each segment inverts when compared to its neighbours.",
"Therefore the displacement may be across the `top' or the `bottom' of the $\\Psi $ and $\\Psi _n$ segments (in their usual orientation) depending upon the parity of its place in the sequence of segments.",
"Our general strategy will be to isolate the displaced segment (and perhaps a neighbouring segment) by splitting off `earlier' and `later' segments.",
"When we split off earlier segments (and sometimes later segments), we will use the $\\Psi $ structure we have been using so far.",
"However, we will sometimes want to group vertices and edges slightly differently when we split off `later' segments.",
"We use an observation from [21] to conveniently group these.",
"Let us define the kernel functions $\\overline{\\psi }^{(1)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\tau _\\lambda (y,r)\\tau _\\lambda (r,s)\\tau _\\lambda (s,x),\\\\\\overline{\\psi }^{(2)}_0\\left(w,u,x\\right) &:= \\lambda ^2\\left(\\lambda ^{-1}\\delta _{y,r}\\right)\\left(\\lambda ^{-1}\\delta _{y,s}\\right),\\\\\\overline{\\psi }^{(1)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,u)\\int \\tau _\\lambda (t,s) \\tau _\\lambda (t,w)\\tau _\\lambda (u,z)\\tau _\\lambda (z,t)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(2)}(w,u,r,s) &:= \\lambda ^4\\tau _\\lambda (w,s)\\int \\tau _\\lambda (t,z)\\tau _\\lambda (z,u)\\tau _\\lambda (u,t)\\tau _\\lambda ^\\circ (t,w)\\tau _\\lambda ^\\circ (z,r)\\nu ^{\\otimes 2}\\left(\\mathrm {d}z, \\mathrm {d}t\\right), \\\\\\overline{\\psi }^{(3)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\tau _\\lambda (u,r),\\\\\\overline{\\psi }^{(4)}(w,u,r,s) &:= \\lambda ^2\\tau _\\lambda (u,w)\\tau _\\lambda (w,s)\\left(\\lambda ^{-1}\\delta _{u,r}\\right),$ and set $\\overline{\\psi }_0 := \\overline{\\psi }^{(1)}_0+\\overline{\\psi }^{(2)}_0$ and $\\overline{\\psi }:= \\overline{\\psi }^{(1)}+\\overline{\\psi }^{(2)}+\\overline{\\psi }^{(3)} + \\overline{\\psi }^{(4)}$ .",
"Diagrammatic representations of these functions can be found in Figure REF .",
"Note the similarity to the $\\psi $ functions, with the $\\tau _\\lambda ^\\circ $ edge appearing elsewhere in the integral.",
"Figure: Diagrams of the ψ ¯ 0 \\overline{\\psi }_0, and ψ ¯\\overline{\\psi } functions.For $j\\in \\left\\lbrace 1,2\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}_0\\colon L^2\\left(\\mathbb {X}\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}\\right)$ as $\\overline{\\Psi }^{(j)}_0 f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}_0(w,u,y) f(y)\\nu \\left(\\mathrm {d}y\\right),$ and for $j\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ we define $\\overline{\\Psi }^{(j)}\\colon L^2\\left(\\mathbb {X}^2\\right) \\rightarrow L^2\\left(\\mathbb {X}^2\\right)$ to be the linear operators acting on $f\\in L^2\\left(\\mathbb {X}^2\\right)$ as $\\overline{\\Psi }^{(j)} f\\left(w,u\\right) = \\int \\overline{\\psi }^{(j)}(w,u,r,s) f(r,s)\\nu ^{\\otimes 2}\\left(\\mathrm {d}r, \\mathrm {d}s\\right).$ In our terminology, the important observation of [21] was that for some $m\\ge 0$ the `later' segments of each diagram can be bounded using $\\big (\\overline{\\Psi }^m\\overline{\\Psi }_0\\big )^\\dagger $ , the adjoint of $\\overline{\\Psi }^m\\overline{\\Psi }_0$ .",
"In terms of the kernel functions, taking the adjoint in this case amounts to reflecting the `input' and `output' arguments, and for the diagrams this amounts to reflecting in the vertical plane.",
"Lemma 5.9 For $n\\ge 1$ , $m\\in \\left\\lbrace 0,\\ldots ,n\\right\\rbrace $ , $j_0\\in \\left\\lbrace 1,2,3\\right\\rbrace $ , $j_1,\\ldots ,j_{n-1}\\in \\left\\lbrace 1,2,3,4\\right\\rbrace $ , and $j_n\\in \\left\\lbrace 1,2\\right\\rbrace $ , $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^m,\\\\\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty } &\\le U_\\lambda V_\\lambda ^{n-m},\\\\\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty } &\\le {\\left\\lbrace \\begin{array}{ll}U_\\lambda &: m=0\\\\U_\\lambda ^2V_\\lambda ^{m-1} &: m\\ge 1.\\end{array}\\right.",
"}$ The argument for $\\left\\Vert \\Psi ^{(j_m)}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }$ and $\\left\\Vert \\Psi ^{(j_n)}_n\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_{m})}\\right\\Vert _{1,\\infty }$ is the same as that in Lemma REF and Proposition REF .",
"We consider bounds for single operators and pairs of operators, and then use sub-multiplicity to get a bound for each diagram.",
"For $\\left\\Vert \\overline{\\Psi }^{(j_m)}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0\\right\\Vert _{1,\\infty }$ the argument is a little more complicated.",
"For $m=0,1$ we can easily check each diagram satisfies the bound.",
"For $m\\ge 2$ we have the issue that there exists one pair of operators which we can't bound by $V_\\lambda ^2$ : $\\left\\Vert \\overline{\\Psi }^{(4)}\\overline{\\Psi }^{(2)}_0\\right\\Vert _{1,\\infty } = \\quad \\raisebox {0pt}{\\begin{tikzpicture}(0,0) to [out=10,in=170] (1,0) to [out=190,in=350] (0,0);[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\triangle ^\\circ _\\lambda \\le U_\\lambda ^2.$ If $m\\ge 2$ is even (an odd number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }_0$ term which we bound with $U_\\lambda $ .",
"We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda V_\\lambda ^m$ .",
"If $m\\ge 2$ is odd (an even number of operators) then we use sub-multiplicity to pull off the $\\overline{\\Psi }\\overline{\\Psi }_0$ pair which we bound with $U_\\lambda ^2$ .",
"We then pull off pairs which we can bound with $V_\\lambda ^2$ to get the bound $U_\\lambda ^2V_\\lambda ^{m-1}$ .",
"Definition 5.10 In addition to the terms defined in Definition REF , we will now use $W_k&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int \\tau _{\\lambda ,k}(x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right) = \\lambda \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty },\\\\\\overline{W_k}&:= \\lambda \\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _{\\lambda ,k}(\\overline{x}-\\overline{u};a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4)\\mathrm {d}u,\\\\\\overline{H_k}&:= \\lambda ^5 \\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,a_1,\\ldots ,a_{16}\\in \\mathcal {E}}\\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8)\\nonumber \\\\& \\hspace{42.67912pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\tau _{\\lambda ,k}(\\overline{t}-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u}.$ Note that we only use $\\overline{W_k}$ in Propositions REF and REF because $\\overline{W_k}\\ge W_k$ , but in the proof we will sometimes use $W_k$ to make the derivation easier to follow.",
"The diagrams for $\\overline{W_k}$ and $\\overline{H_k}$ are in Figure REF .",
"Also recall the sets $\\left\\lbrace B\\left(x\\right)\\right\\rbrace _{x\\in \\mathbb {X}}$ assumed to exist in Assumption REF .",
"Then define $\\mathbb {B} &:= \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\nu \\left(B\\left(x\\right)\\right),\\\\\\triangle ^{(B)}_\\lambda &:= \\lambda ^2 \\operatornamewithlimits{ess\\,sup}_{r,s\\in \\mathbb {X}: r\\notin {B\\left(s\\right)}}\\int \\tau _\\lambda ^\\circ \\left(r,u\\right)\\tau _\\lambda \\left(u,v\\right)\\tau _\\lambda \\left(v,s\\right)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right).$ Note that these are only required to deal with one specific diagram in the following $n=1$ case.",
"Figure: Diagrams representing W k ¯\\overline{W_k} and H k ¯\\overline{H_k}.",
"In this figure, the marks on edges incident to filled square vertices need not be equal.Proposition 5.11 For $k\\in \\mathbb {R}^d$ and $n\\ge 2$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}\\left(n+1\\right)^24^{n-1}\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^4 V_\\lambda ^{n-4} &: n\\ge 4\\\\16\\times 16\\left(50\\overline{W_k}V_\\lambda ^2 + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=3\\\\9\\times 4\\left(50\\overline{W_k}V_\\lambda + 6\\overline{H_k}\\right) U_\\lambda ^2 &: n=2.\\end{array}\\right.",
"}$ Proposition 5.12 For $k\\in \\mathbb {R}^d$ , $\\lambda \\left\\Vert \\widehat{}_{\\lambda }^{(1)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(1)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le 13\\overline{W_k}V_\\lambda U_\\lambda + \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\left(V_\\lambda + \\triangle ^{(B)}_\\lambda \\right) + \\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + 4\\lambda \\mathbb {B}\\overline{W_k}.$ [Proof of Proposition REF ] In bounding the diagrams with one displacement segment, we have three broad cases.",
"Let $i\\in \\left\\lbrace 0,1,\\ldots ,n\\right\\rbrace $ denote which segment the displacement lies upon.",
"Our three cases are then: the displacement lies on the $\\psi _n$ segment (i.e.",
"$i=n$ ), the displacement lies on the $\\psi _0$ segment (i.e.",
"$i=0$ ), the displacement lies on a $\\psi $ segment (i.e.",
"$i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ ).",
"We will aim to get a single bound for each diagram having a displacement crossing a single segment.",
"This will be uniform in the sequence $\\left(j_0,j_1,\\ldots ,j_n\\right)$ .",
"We then account for the number of possible choices of $\\left(j_0,j_1,\\ldots ,j_n\\right)$ , and finally include the factor of $(n+1)^2$ arising from using cosine-splitting (the explicit factor and the number of diagrams).",
"Case REF is the simplest.",
"We use a supremum bound to pull off the displaced $\\Psi _n$ term.",
"We then bound this by hand and use Lemma REF to bound the remainder.",
"In representing this scheme we use a shorthand $\\Psi ^{n-1}\\Psi _0$ to denote a particular sequence $\\Psi ^{(j_{n-1})}\\ldots \\Psi ^{(j_1)}\\Psi ^{(j_0)}_0$ rather than actually the operators $\\Psi $ and $\\Psi _0$ defined above.",
"We also use labelled grey shapes to represent these groupings of diagrams, highlighting only end vertices or vertices that connect to neighbouring segments.",
"This scheme look like: $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8) -- (4,0.2);(2.8,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (2,0.8) -- (3,0.2);(2.5,0.5) circle (0pt) node[rotate = -30]{\\times };\\node at (1,0) {\\Psi ^{n-1}\\Psi _0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\hspace{28.45274pt}\\le U_\\lambda V_\\lambda ^{n-1} \\left(2\\left( \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- (1,0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(2,0.6) -- (3,0.6) -- (3,-0.6) -- (2,-0.6);(3,0.6) -- (4,0) -- (3,-0.6);[<->] (3,0.8) -- (4,0.2);(3.5,0.5) circle (0pt) node[rotate = -30]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(4,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 2U_\\lambda V_\\lambda ^{n-1}\\left(\\overline{W_k}\\triangle ^{\\circ \\circ }_\\lambda + W_k\\triangle ^\\circ _\\lambda \\right)\\nonumber \\\\&\\hspace{28.45274pt}\\le 4\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that this bound also holds if the displacement runs along the bottom of the $\\Psi _n$ segment.",
"Since there are $3\\times 4^{n-1}$ possible sequences of indices for the $\\Psi ^{n-1}\\Psi _0$ diagram, this case contributes that as a pre-factor.",
"For Case REF , we need a few more tricks.",
"Since we are taking the displacement across the top of the diagram, the only term that will make a contribution is $\\Psi ^{(1)}_0$ .",
"To tackle this case we first use the observation of [21] that the diagram we want to bound can in turn be bounded by a diagram using $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ .",
"As above, we use $\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0$ as a shorthand for a particular sequence $\\overline{\\Psi }^{(j_{n-1})}\\ldots \\overline{\\Psi }^{(j_1)}\\overline{\\Psi }^{(j_0)}_0$ .",
"We are also imprecise in this notation over whether it is this operator or its adjoint.",
"Nevertheless the diagrams we use are clear on which vertices are integrated over and which have suprema, and this will avoid ambiguity.",
"We then spatially translate the supremum to the far end and pull off the $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram using a supremum bound.",
"This looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}\\\\\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (3,-0.6) to [out=100,in=260] (3,0.6) to [out=190,in=350] (2,0.6);\\node at (2.5,0) {\\overline{\\Psi }};\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0};(2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (5,0) circle (2pt);\\end{tikzpicture}}.$ For $\\overline{\\Psi }^{(j)}$ with $j=1,2,3$ , a simple supremum bound shows that this starting diagram can be bounded by $\\overline{W_k}\\triangle ^\\circ _\\lambda \\left\\Vert \\overline{\\Psi }^{(j)}\\right\\Vert _{1,\\infty }\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2$ .",
"For $j=4$ a little more care is needed: $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (2,-0.6) -- (3,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);(2,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };\\node [mark size=2pt] at (0,0) {{square*}};(1,0.6) circle (2pt);(1,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ We therefore find that $\\raisebox {-15pt}{\\begin{tikzpicture}(1,-0.6) -- (0,0) -- (1,0.6);(2,0.6) -- (1,0.6) -- (1,-0.6) -- (2,-0.6);[<->] (0,0.2) -- (1,0.8);(0.5,0.5) circle (0pt) node[rotate = 30]{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=gray!50] (2,0.6) to [out=280,in=80] (2,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (2,0.6);\\node at (3,0) {\\overline{\\Psi }^{n-1}\\overline{\\Psi }_0};[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le {\\left\\lbrace \\begin{array}{ll}\\overline{W_k}V_\\lambda ^{n-1}U_\\lambda ^2 &: n\\ge 3\\\\\\overline{W_k}V_\\lambda ^2U_\\lambda &: n=2.\\end{array}\\right.",
"}$ Since there are 4 choices for the appended $\\overline{\\Psi }^{(j)}$ term, and $4^{n-2}\\times 2$ for the remaining $\\overline{\\Psi }^{n-2}\\overline{\\Psi }_0$ diagram, this case contributes $2\\times 4^{n-1}$ as a pre-factor.",
"For Case REF , we shall find we have 3 sub-cases, depending not only on the $j$ -index of the displaced segment, but on which term we are looking at once it has been expanded.",
"Sub-cases $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ deal with $j=1,2,3$ .",
"In Appendix we expand out the various displaced segments using cosine-splitting and by expanding $\\tau _\\lambda ^\\circ $ edges into $\\tau _\\lambda $ edges and points, and label each of the resulting diagrams as sub-case $\\mathrm {\\left(I\\right)}$ and $\\mathrm {\\left(II\\right)}$ .",
"These sub-cases are distinguished by the strategy we use to bound them.",
"Sub-case $\\mathrm {\\left(III\\right)}$ then addresses the case with $j=4$ .",
"For sub-case $\\mathrm {\\left(I\\right)}$ , we just use the $\\Psi $ structure of the diagrams.",
"We first split off segments from the left of the displaced term like we did in Case REF .",
"If $i=n-1$ we can then bound the remaining two pair of segments with $\\overline{W_k}V_\\lambda ^2$ (Appendix explains how this bound is found).",
"This can be described diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda V_\\lambda ^{n}.$ If $i\\le n-2$ , then we also split off the segments to the right of this pair, and then bound the pair by $\\overline{W_k}V_\\lambda ^2$ again.",
"Diagrammatically this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (6,-0.6) to [out=40,in=200] (7,0) to [out=160,in=320] (6,0.6) to [out=190,in=350] (4,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };\\node at (5,0) {\\Psi _n\\Psi ^{n-i-2}};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(7,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda ^2V_\\lambda ^{n-3} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}.$ Note that the since $U_\\lambda \\ge V_\\lambda $ , the bound in (REF ) is greater than or equal to the bound in (REF ).",
"It will be convenient to find an $i$ -independent bound.",
"To this end, since the $i\\le n-2$ case only exists for $n\\ge 3$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(I\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-1}&: n\\ge 3\\\\\\overline{W_k}U_\\lambda V_\\lambda ^{2}&: n=2,\\end{array}\\right.",
"}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .",
"For sub-case $\\mathrm {\\left(II\\right)}$ , we use a different strategy.",
"First we split off the earlier (left) segments as we did for the first sub-case and in Case REF .",
"However we arrange the later terms differently - more like we did in Case REF .",
"We associate our displaced $\\Psi $ term with $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges from the subsequent term.",
"If the subsequent term would be $\\Psi ^{(2)}_n$ , these are actually two $\\tau _\\lambda $ edges, but in our schematic diagrams we will draw a $\\tau _\\lambda ^\\circ $ edge and bear this special case in mind when it is relevant.",
"While the displacement portrayed on the `top' of the segment could equally well be on the `bottom', the $\\tau _\\lambda ^\\circ $ edge will always be attached to the bottom in the usual orientation.",
"Everything to the right of these $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges can then be described as a sequence of $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ terms.",
"If $i=n-1$ , then we pair our augmented displaced segment with the subsequent $\\overline{\\Psi }_0$ segment.",
"We then perform a spatial translation to make the right-most vertex a supremum vertex like we did in (REF ).",
"The remaining diagram of a $\\Psi ^{(j)}$ segment and $\\overline{\\Psi }_0$ segment connected by a $\\tau _\\lambda ^\\circ $ and $\\tau _\\lambda $ edge can then be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ).",
"Recall that if $\\overline{\\Psi }_0$ is in fact $\\overline{\\Psi }^{(2)}_0$ then we have two $\\tau _\\lambda $ edges connecting the terms, not a $\\tau _\\lambda $ edge and a $\\tau _\\lambda ^\\circ $ edge.",
"We need to note this to get the $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ bound.",
"The scheme is then represented diagrammatically as $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{n-2}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(6,0) circle (2pt);\\end{tikzpicture}}\\quad \\le U_\\lambda V_\\lambda ^{n-2} \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\\\ \\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda V_\\lambda ^{n-2}.$ If $i\\le n-2$ , we also perform a spatial translation to move a supremum to the right-most vertex, but then also split off all $\\overline{\\Psi }$ and $\\overline{\\Psi }_0$ segments after the $\\overline{\\Psi }$ segment immediately following the displaced term.",
"Once again the remaining pair can be bounded by $\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)$ (see Appendix ) and the split-off diagrams bounded using Lemma REF .",
"The scheme can be represented diagrammatically as $&\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=260,in=100] (2,-0.6) to [out=170,in=10] (0,-0.6) to [out=140,in=340] (-1,0) to [out=20,in=220] (0,0.6) to [out=350,in=190] (2,0.6);[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);[fill=gray!50] (5,0.6) to [out=280,in=80] (5,-0.6) to [out=10,in=170] (7,-0.6) to [out=40,in=200] (8,0) to [out=160,in=320] (7,0.6) to [out=190,in=350] (5,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (1,0) {\\Psi ^{i-1}\\Psi _0};\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};\\node at (6,0) {\\overline{\\Psi }^{n-i-2}\\overline{\\Psi }_0};[fill=white] (-1,0) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);(5,0.6) circle (2pt);(5,-0.6) circle (2pt);(8,0) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\hspace{170.71652pt}\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\times {\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.}",
"\\nonumber \\\\&\\hspace{170.71652pt}\\le \\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right){\\left\\lbrace \\begin{array}{ll}U_\\lambda ^2 V_\\lambda ^{n-4}&: i\\le n-3 \\\\U_\\lambda V_\\lambda ^{n-3}&: i=n-2\\end{array}\\right.",
"}$ Again for simplicity, we would like to get and $i$ -independent bound for such diagrams.",
"Since the $i = n-2$ case only exists for $n\\ge 3$ , and the $i\\le n-3$ case only exists for $n\\ge 4$ , we can bound an instance of a displaced diagram in Case REF , sub-case $\\mathrm {\\left(II\\right)}$ with ${\\left\\lbrace \\begin{array}{ll}\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^3V_\\lambda ^{n-4}&: n\\ge 4\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda ^2&: n=3\\\\\\left(\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda \\right)U_\\lambda &: n=2,\\end{array}\\right.",
"}$ uniformly in $i\\in \\left\\lbrace 1,\\ldots ,n-1\\right\\rbrace $ .",
"We now address sub-case $\\mathrm {\\left(III\\right)}$ , that is when the displacement crosses a $\\Psi ^{(4)}$ segment.",
"This is the simplest sub-case because when we take a displacement across $\\Psi ^{(4)}$ in isolation, we get exactly $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) circle (2pt);[fill=white] (1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad =W_k.$ We will be able to use the $\\Psi $ -arrangement of segments here.",
"For $n\\ge 4$ we split off the earlier and later segments without pairing the displaced term with any of them.",
"With Lemma REF this produces the bound $\\left(U_\\lambda V_\\lambda ^{i-1}\\right)W_k\\left(U_\\lambda V_\\lambda ^{n-i-1}\\right) = \\overline{W_k}U_\\lambda ^2V_\\lambda ^{n-2} \\qquad :n\\ge 4.$ For $n=3$ we split off two segments from one side and one from the other.",
"This gives the finer bound $W_kU_\\lambda V_\\lambda ^2 \\quad :n=3.$ For $n=2$ , we use $j_0$ to indicate which $\\Psi _0$ term we have and $j_2$ to denote which $\\Psi _n$ term we have.",
"We note that if $j_0=1,2$ then $\\left\\Vert \\Psi ^{(j_0)}_0\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ , and if $j_2 = 1$ then $\\left\\Vert \\Psi ^{(j_2)}_n\\right\\Vert _{1,\\infty }\\le \\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le V_\\lambda $ .",
"Therefore if $\\left(j_0,j_2\\right) \\ne \\left(3,2\\right)$ we have the bound $W_kU_\\lambda V_\\lambda $ .",
"If $\\left(j_0,j_2\\right) = \\left(3,2\\right)$ we proceed by hand.",
"We bound $\\varphi \\le \\tau _\\lambda $ and spatially shift one vertex to show $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };(1,0) circle (0pt) node[rotate=90, above]{\\sim };(0,0) circle (0pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(1,0.6) -- (2,0.6) --(1,-0.6) -- (2,-0.6) -- (2,0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (1,0.6) {{square*}};(1,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda .$ Carefully comparing the above bounds gives the required bounds for $n\\le 2$ .",
"Including the multiplicity of each diagram (including that arising from cosine-splitting), we find that we can bound the displaced and subsequent non-displaced pair by $\\left(24\\overline{W_k}V_\\lambda ^2 + 2\\overline{H_k}\\right)$ .",
"Note that this includes summing over the $j$ -index of the displaced term but not the index of the subsequent term.",
"Since there are $3\\times 4^{n-2}\\times 2$ options for the $j$ -indices of the other segments, this case contributes this to the counting pre-factors in our result.",
"[Proof of Proposition REF ] For $n=1$ , we can consider each of the six cases by hand.",
"Most of the diagrams can be dealt with routinely using cosine-splitting, supremum bounds, and spatial shifts.",
"We enumerate the diagrams by their $\\left(j_0,j_1\\right)$ values corresponding to the constituent segments.",
"$(1,1):& \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (0,-0.2) -- (1,-0.8) -- (2,-0.8) -- (3,-0.2);(1.5,-0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 3\\left(\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle _\\lambda + \\triangle _\\lambda \\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(1,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (0,0.2) -- (1,0.8) -- (2,0.2);(0.8,0.68) circle (0pt) node[rotate=30]{\\times };[fill=white] (0,0) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\overline{W_k}\\triangle _\\lambda + \\triangle _\\lambda W_k\\right)\\\\(2,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0.6) --(2,-0.6) -- (1,-0.6);(2,0.6) -- (3,0) -- (2,-0.6);[<->] (1,0.8) -- (2,0.8) -- (3,0.2);(1.8,0.8) circle (0pt) node{\\times };(1.5,-0.6) circle (0pt) node[above]{\\circ };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\triangle _\\lambda \\overline{W_k}\\triangle ^\\circ _\\lambda + \\triangle _\\lambda \\triangle ^\\circ _\\lambda W_k\\right)\\\\(2,2):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0) -- (1,0.6) -- (1,-0.6) -- (0,0);(1,0.6) -- (2,0) -- (1,-0.6);[<->] (1,0.8) -- (2,0.2);(1.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (2pt);[fill=white] (1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle _\\lambda W_k\\\\(3,1):& \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (2,0) -- (1,-0.6) -- (0,-0.6) -- cycle;(1,0.6) -- (1,-0.6);[<->] (0,0.8) -- (1,0.8) -- (2,0.2);(0.8,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(2,0) circle (2pt);(0,-0.6) circle (2pt);(1,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\overline{W_k}\\triangle _\\lambda + \\overline{W_k}\\triangle _\\lambda + \\lambda \\left\\Vert \\right\\Vert _{1,\\infty }\\triangle _\\lambda W_k+ \\triangle _\\lambda W_k\\right)$ These terms sum to no more than $13\\overline{W_k}V_\\lambda U_\\lambda .$ For the $\\left(3,2\\right)$ diagram, we use a different `trick.'",
"In Lemma REF we will prove the bound $\\tau _\\lambda \\left(x,y\\right) \\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right)$ .",
"When applied to this diagram, this looks like $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6) -- cycle;[<->] (1.2,0.6) -- (1.2,-0.6);(1.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.6) circle (0pt) node[above]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right),$ Here we have also used the cosine-splitting result on the second part.",
"For the first of these 4-vertex diagrams, we split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_\\lambda _k\\right\\Vert _{\\infty ,\\infty } \\le \\lambda \\triangle _\\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }$ .",
"For the second of the 4-vertex diagrams, we also split the supremum in two and get the bound $\\lambda ^3\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ .",
"For the 3-vertex diagram we condition on whether two of the vertices are in each other's $B$ -set (defined in Assumption REF ).",
"In the following diagrams [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\in B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\in B\\left(u\\right)\\right\\rbrace }$ relates two variables, and [<->] (0,0) – (1,0); (0.5,0) circle (0pt) node[above]$\\notin B$; indicates that an indicator of the form ${1}_{\\left\\lbrace w \\notin B\\left(u\\right)\\right\\rbrace }$ relates them: $\\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,above]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &= \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.7) -- (0,-0.8) -- (1,-0.2);(-0.2,0) circle (0pt) node{\\times };(0,0) circle (0pt) node[rotate=90,below]{\\sim };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\ &\\le \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,1) -- (1,0.4);(0.5,0.5) circle (0pt) node[rotate=-30]{\\times };(0.5,0.3) circle (0pt) node[rotate=-30,below]{\\sim };(0.5,0.7) circle (0pt) node[above, rotate=-30]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + 2\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad + \\quad \\raisebox {-20pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6) -- cycle;[<->] (0,0.8) -- (1,0.2);[<->] (0,-0.8) -- (1,-0.2);(0.5,-0.5) circle (0pt) node[rotate=30]{\\times };(0.5,0.5) circle (0pt) node[above, rotate=-30]{\\scriptsize \\in B};[fill=white] (0,0.6) circle (2pt);(1,0) circle (2pt);(0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\right) \\nonumber \\\\&\\le \\left(\\quad \\raisebox {-7pt}{\\begin{tikzpicture}(0,0) -- (1,0);[<->] (0,0.2) -- (1,0.2);(0.5,0.2) circle (0pt) node{\\times };(0.5,0) circle (0pt) node[below]{\\sim };[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (-0.2,0.6) -- (-0.2,-0.6);(-0.2,0) circle (0pt) node[above, rotate=90]{\\scriptsize \\notin B};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right) +4\\left(\\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=white] (0,0) circle (2pt);(1,0) circle (2pt);[<->] (0,-0.2) -- (1,-0.2);(0.5,-0.2) circle (0pt) node[below]{\\scriptsize \\in B};\\end{tikzpicture}}\\quad \\times \\quad \\raisebox {-15pt}{\\begin{tikzpicture}(0,0.6) -- (1,0) -- (0,-0.6);[<->] (0,0.8) -- (1,0.2);(0.5,0.5) circle (0pt) node[rotate=-30]{\\mathbf {\\times }};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);(1,0) circle (2pt);\\end{tikzpicture}}\\quad \\right)\\nonumber \\\\& \\le \\lambda \\left\\Vert _k\\right\\Vert _{1,\\infty }\\triangle ^{(B)}_\\lambda + 4\\lambda \\mathbb {B}W_k.$"
],
[
"Bootstrap analysis",
"In this section we will be assuming that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = 1.$ By Lemma REF this also implies $\\mathbb {S}\\left(\\right)\\le 1$ .",
"A re-scaling argument similar to [36] explains how this can be done while maintaining generality.",
"First note that since $\\widehat{\\varphi }(0;a,b)>0$ for a positive $\\mathcal {P}$ -measure set of $a,b\\in \\mathcal {E}$ , we have $\\mathbb {S}\\left(\\widehat{}(0)\\right)>0$ .",
"Now suppose we scale $\\mathbb {R}^d$ by a factor of $q^{-1/d}$ (that is, the new unit radius ball is the previous ball of radius $q^{1/d}$ ), and leave the mark space $\\mathcal {E}$ unchanged.",
"Then we find that the system we are left with has the distribution of an RCM model with the new parameters $\\lambda ^* = \\mathbb {E}_\\lambda \\left[*{\\eta \\cap \\left[0, q^{\\frac{1}{d}}\\right]^d }\\right] = \\lambda q, \\qquad \\varphi ^*\\left(\\overline{x};a,b\\right) = \\varphi \\left(q^{\\frac{1}{d}}\\overline{x};a,b \\right).$ Therefore for all $a,b\\in \\mathcal {E}$ we have $\\widehat{\\varphi }^*(\\overline{x};a,b) = \\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\int \\varphi ^*(\\overline{x};a,b) \\mathrm {d}\\overline{x}= q^{-1}\\widehat{\\varphi }(\\overline{x};a,b),$ and $\\mathbb {S}\\left(\\widehat{}^*(0)\\right) = q^{-1} \\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"We can therefore achieve our assumption by choosing $q=\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"Furthermore, under this scaling we find that the old event $\\left\\lbrace \\left(\\overline{x},a\\right) \\longleftrightarrow \\left(\\overline{y},b\\right)\\textrm { in } \\xi ^{x,y}\\right\\rbrace $ becomes the new event $\\left\\lbrace \\left(q^{-1/d}\\overline{x},a\\right) \\longleftrightarrow \\left(q^{-1/d}\\overline{y},b\\right)\\textrm { in } \\xi ^{\\left(q^{-1/d}\\overline{x},a\\right),\\left(q^{-1/d}\\overline{y},b\\right)} \\right\\rbrace $ , and we get $\\tau ^*_{\\lambda ^*}(\\overline{x};a,b) = \\tau _\\lambda \\left(q^\\frac{1}{d}\\overline{x};a,b\\right)$ where $\\tau ^*_{\\lambda ^*}$ is the two-point function in the RCM governed by the connection function $\\varphi ^*$ and intensity $\\lambda ^*$ .",
"It is also worth noting that the scaling choice to have $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ means that REF reduces to the condition that there exists a $d$ -independent constant $C>0$ such that $\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } &\\le C,\\\\\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } &\\le C \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Note that the inequalities relating $\\left\\Vert \\cdot \\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\cdot \\right\\Vert _{2,\\infty }$ , and $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ follow from Jensen's inequality and a supremum bound on a probability space."
],
[
"The Bootstrap Function",
"To show that the bounds arising from the diagrams in Section are sufficient for our purposes, we make use of a bootstrap argument.",
"Define $\\mu _\\lambda & = 1- \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)},\\\\\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) &= \\frac{1}{1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)},\\\\\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) &=\\left(1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\left(\\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right) + \\widehat{G}_{\\mu _\\lambda }\\left(l-k\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right)\\right).$ From these we can then define the bootstrap function $f= f_1\\vee f_2\\vee f_3$ , where $f_1\\left(\\lambda \\right) = \\lambda , \\qquad f_2\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}, \\qquad f_3\\left(\\lambda \\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)}.$ Let us now outline some initial properties of $\\mu _\\lambda $ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .",
"Lemma 6.1 The map $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous and bounded below by $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ on $\\left[0,\\lambda _O\\right)$ .",
"Therefore the map $\\lambda \\mapsto \\mu _\\lambda $ is a continuous map from $\\left[0,\\lambda _O\\right)\\rightarrow \\left[0,1\\right)$ .",
"Furthermore, $\\widehat{G}_{\\mu _\\lambda }(k)$ is bounded below by $\\left(1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)^{-1}$ for all $k\\in \\mathbb {R}^d$ .",
"We first recall that $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (0)$ is differentiable with respect to the operator norm from Corollary REF , and is therefore continuous.",
"Lemma REF then implies that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ is continuous.",
"For the bounded below property, we use Lemma REF to get $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge \\mathbb {S}\\left(\\widehat{}(0)\\right) = 1$ .",
"The continuity of $\\mu _\\lambda $ follows directly from the continuity and positivity of $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)$ .",
"The bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ bounds $\\mu _\\lambda \\ge 0$ , and $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ shows that $\\mu _\\lambda <1$ for all $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"The case $\\mu _0 = 0$ follows immediately from $\\mathcal {T}_0 = $ .",
"Because $\\mu _\\lambda \\in \\left[0,1\\right)$ , to minimise $\\widehat{G}_{\\mu _\\lambda }(k)$ we want to find a lower bound for $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ .",
"The $k$ -uniform bound on $\\widehat{G}_{\\mu _\\lambda }(k)$ then follows from $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\ge -\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\rm op} \\ge -\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}$ ."
],
[
"Bounding the Lace Operator",
"We have managed to bound the lace operator and its displacement in terms of a variety of objects.",
"For the lace operator itself (and its associated Fourier transformed operators), these are: $\\lambda , \\triangle _\\lambda , \\overline{\\triangle ^{\\circ \\circ }_\\lambda }, \\left\\Vert \\right\\Vert _{1,\\infty }.$ As a guide, we expect $\\lambda $ , $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ , and $\\left\\Vert \\right\\Vert _{1,\\infty }$ to be bounded uniformly in $d$ and $\\lambda <\\lambda _O$ , whereas we expect $\\triangle _\\lambda $ to decay uniformly in $\\lambda <\\lambda _O$ as $d\\rightarrow \\infty $ .",
"For some, these properties are easy to prove.",
"The boundedness of $\\left\\Vert \\right\\Vert _{1,\\infty }$ follows from $\\left\\Vert \\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we have used REF via (REF ).",
"Regarding $\\lambda $ , since we are only interested in $\\lambda <\\lambda _O$ the question is perhaps more clearly written as “Is $\\lambda _O$ uniformly bounded in $d$ ?",
"\".",
"While there may be more elementary ways of proving this directly, we will get it for free from the bootstrap argument.",
"Since we have the bound $\\lambda \\le f(\\lambda )$ , proving the uniform boundedness of $f(\\lambda )$ with respect to $d$ and $\\lambda $ for $\\lambda <\\lambda _O$ will prove the same for $\\lambda _O$ .",
"This will leave us with only $\\triangle _\\lambda $ and $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ .",
"Recall that we can write $\\triangle _\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }, \\quad \\triangle ^{\\circ \\circ }_\\lambda = \\lambda ^2\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert \\mathcal {T}_\\lambda ^2\\right\\Vert _{\\infty ,\\infty } + 1.$ Then $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }$ differs from $\\triangle ^{\\circ \\circ }_\\lambda $ in that now the supremum is taken over all the marks, and adjacent $\\tau _\\lambda $ can have different marks.",
"To get the desired decay and boundedness behaviour, we will therefore only need to prove that $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ decays and to bound $&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v},\\\\&\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}.$ We begin by proving the decay of $\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty }$ .",
"Lemma 6.2 Let $x,y \\in \\mathbb {X}$ .",
"Then $\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\varphi (x,u)\\tau _\\lambda (u,y)\\nu \\left(\\mathrm {d}u\\right),\\\\\\tau _\\lambda (x,y) &\\le \\varphi (x,y) + \\lambda \\int \\tau _\\lambda (x,u)\\varphi (u,y)\\nu \\left(\\mathrm {d}u\\right).$ Then for $n\\ge 2$ , $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le \\sum _{m=0}^n\\genfrac(){0.0pt}0{n}{m}\\lambda ^{m} \\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }.$ By combining Mecke's formula and the BK inequality, we obtain $\\tau _\\lambda (x,y) & \\le \\varphi (x,y) + \\mathbb {E}_\\lambda \\left[\\sum _{u \\in \\eta } 1_{\\lbrace x \\sim u \\text{ in } \\xi ^{x} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y}\\rbrace } \\right] \\nonumber \\\\& = \\varphi (x,y) + \\lambda \\int \\mathbb {P}_\\lambda \\left(\\lbrace x \\sim u \\text{ in } \\xi ^{x,u} \\rbrace \\cap \\lbrace u \\longleftrightarrow y\\textrm { in } \\xi ^{y,u}\\rbrace \\right) \\nu \\left(\\mathrm {d}u\\right) \\nonumber \\\\& = \\varphi (x,y) +\\lambda \\int \\varphi (x,u) \\tau _\\lambda (u,y) \\nu \\left(\\mathrm {d}u\\right).$ In the last line, we have used that the two intersected events are independent.",
"This is due to the fact that $\\nu $ is non-atomic and $x\\notin \\eta ^{y,u}$ a.s.",
"The second inequality is then proven nearly identically, but with the adjacency to the intermediate point $u$ holding for $y$ instead of $x$ .",
"To get (REF ), we first use () to expand out the right-most $\\mathcal {T}_\\lambda $ .",
"This produces one term with $$ replacing the $\\mathcal {T}_\\lambda $ , and one term with $\\lambda \\mathcal {T}_\\lambda $ replacing it.",
"We then repeatedly use (REF ) to expand the left-most $\\mathcal {T}_\\lambda $ to get a $$ factor and a $\\lambda \\mathcal {T}_\\lambda $ factor until we have a total of $n$ factors of $$ in the expression.",
"Counting the number of ways to get to each possible end term then gives the binomial $\\genfrac(){0.0pt}1{n}{m}$ factor as required.",
"The following lemma will allow us to describe $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ using Hilbert space language.",
"Lemma 6.3 Let $H\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ be an integral linear operator with kernel function $h\\colon \\mathcal {E}^2\\rightarrow \\mathbb {R}$ .",
"Then for all $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ and $\\varepsilon >0$ , there exist $\\mathcal {P}$ -positive sets $E_1,E_2\\subset \\mathcal {E}$ such that $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ satisfy $*{h\\left(a,b\\right) - {g_1}{Hg_2}} \\le \\varepsilon .$ Since $h$ is measurable, the set $A=\\left\\lbrace c,c^{\\prime }\\in \\mathcal {E}:*{h\\left(c,c^{\\prime }\\right)-h\\left(a,b\\right)}\\le \\varepsilon \\right\\rbrace $ is measurable.",
"Since $h\\left(a,b\\right)\\in {\\rm ess.Im}\\left(h\\right)$ , it has positive measure: $\\mathcal {P}\\left(A\\right)>0$ .",
"Furthermore, by considering the $\\pi $ -system of product sets $\\left\\lbrace E_1\\times E_2: E_1,E_2\\subset \\mathcal {E}\\text{ measurable}\\right\\rbrace $ that generates the $\\sigma $ -algebra on $\\mathcal {E}^2$ , we can find $B = E_1\\times E_2$ such that $E_1,E_2\\subset \\mathcal {E}$ are both measurable, $B\\subset A$ , and $\\mathcal {P}^{\\otimes 2}\\left(B\\right) = \\mathcal {P}\\left(E_1\\right)\\mathcal {P}\\left(E_2\\right)>0$ .",
"Then use $E_1,E_2$ to define $g_1,g_2$ as in the statement on the lemma.",
"Since $E_1,E_2$ are both measurable and have positive measure, both $g_1,g_2\\in L^2\\left(\\mathcal {E}\\right)$ .",
"We also have ${g_1}{Hg_2} = \\frac{1}{\\mathcal {P}^{\\otimes 2}\\left(B\\right)}\\int _B h\\left(c,c^{\\prime }\\right)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right) \\in \\left[h(a,b) - \\varepsilon ,h(a,b) +\\varepsilon \\right],$ proving the result.",
"Let us define the function $\\overline{\\epsilon }\\colon \\mathbb {N}\\rightarrow \\mathbb {R}_+$ , where $\\overline{\\epsilon }(d) := {\\left\\lbrace \\begin{array}{ll}\\frac{1}{d} + \\frac{\\log d}{\\log g(d)} &: \\lim _{d\\rightarrow \\infty }g(d)\\rho ^{-d}\\Gamma \\left(\\frac{d}{2}+1\\right)^2 = 0 \\qquad \\forall \\rho >0,\\\\0&: \\text{otherwise}.\\end{array}\\right.",
"}$ Recall $g(d)$ is the function defined in REF .",
"In particular, $\\overline{\\epsilon }(d) = 0$ unless $g(d)$ approaches zero particularly quickly, and $\\beta (d) = g(d)^{\\frac{1}{4}-\\frac{3}{2}\\overline{\\epsilon }(d)}$ .",
"Lemma 6.4 Let $\\lambda <\\lambda _O$ , $n\\ge 2$ , and $d>2m$ .",
"Then there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=2\\\\c_fg(d)^{\\frac{1}{2}-m\\overline{\\epsilon }(d)} &: n\\ge 3.\\end{array}\\right.",
"}$ The condition $\\lambda < \\lambda _O$ here is required to allow us to swap integrals over the mark space with integrals over the Fourier space.",
"We are able to use Fubini's Theorem because sub-criticality implies that the relevant integrals are finite.",
"For clarity, let us introduce the notation $\\varphi ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\varphi (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),\\quad \\tau _\\lambda ^{(n)}(x,y) = \\int \\prod ^{n}_{j=1}\\tau _\\lambda (u_{j-1},u_{j})\\nu ^{\\otimes (n-1)}\\left(\\mathrm {d}u_{[1,n-1]}\\right),$ where $u_0=x$ and $u_n=y$ .",
"Furthermore, for each $\\overline{x}\\in \\mathbb {R}^d$ let us define the integral linear operator $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) \\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $(a,b)\\mapsto \\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) := \\int \\varphi ^{(n-1)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda ^{(m)}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime }\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ Given $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $a,b\\in \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right)\\in {\\rm ess.Im}\\left(\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};\\cdot ,\\cdot \\right)\\right)$ , we can use Lemma REF to get measurable and $\\mathcal {P}$ -positive sets $E^{(\\overline{x},a,b,\\delta )}_1,E^{(\\overline{x},a,b,\\delta )}_2\\subset \\mathcal {E}$ such that $\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x};a,b\\right) \\le *{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2} + \\delta ,$ where $g^{(\\overline{x},a,b,\\delta )}_i(c) = \\frac{1}{\\mathcal {P}\\left(E^{(\\overline{x},a,b,\\delta )}_i\\right)}1_{E^{(\\overline{x},a,b,\\delta )}_i}\\left(c\\right)$ for $i=1,2$ .",
"Now let $E_1,E_2\\subset \\mathcal {E}$ be arbitrary measurable $\\mathcal {P}$ -positive sets and $g_1,g_2$ be their associated functions.",
"We aim to produce bounds that are independent of the choice of $E_1,E_2$ .",
"We first use the Fourier inversion theorem to reformulate the position behaviour in terms of a $k$ -integral of the Fourier transforms, and use REF and $\\lambda <\\lambda _O$ (so the relevant integrals are finite) to allow us to swap the $k$ -integral and the mark integrals.",
"This produces $*{*{g_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g_2}} = *{\\int \\text{e}^{-i\\overline{x}\\cdot k}*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}}\\\\ \\le \\int *{*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ Recall that $\\widehat{}(k)$ is self-adjoint, so we can move $\\widehat{}(k)^{n-1}$ to the other side of the inner product, and then we can use Cauchy-Schwarz and the definition of the operator norm to get $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} = *{\\widehat{}(k)^{n-1}g_1}{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\\\\\le *{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2}_2 \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}^m*{\\widehat{}(k)^{n-1}g_1}_2*{\\widehat{}(k)g_2}_2.$ We can then use the definition of the bootstrap function $f(\\lambda )$ to replace the $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ factors with factors of $f(\\lambda )$ and $\\widehat{G}_{\\mu _\\lambda }(k)$ .",
"Also writing the norm $*{\\cdot }_2$ as the square root of an inner product and using the self-adjoint property of $\\widehat{}(k)$ gives $*{g_1}{\\widehat{}(k)^{n-1}\\widehat{\\mathcal {T}}_\\lambda (k)^m\\widehat{}(k)g_2} \\le f(\\lambda )^m\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}.$ We therefore want to derive bounds for the integral $\\int \\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}.$ We first derive pointwise and integral bounds for the $\\widehat{}$ factors.",
"Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\widehat{\\varphi }^{(2N)}(k;c,c^{\\prime })\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right)^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2}.$ By using Cauchy-Schwarz and supremum bounds we can split this into norms of $\\widehat{}(k)$ only, then use Lemma REF to replace $k$ with 0, and finally use REF to bound these terms: $*{g}{\\widehat{}(k)^{2N}g}^\\frac{1}{2}\\le \\left\\Vert \\widehat{}(k)^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty }^{N-1} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }\\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }^{N-1} \\le C^N.$ For the integral bound, we apply Cauchy-Schwarz to the $k$ -integral and swap the $k$ -integrals with the mark integrals to un-do the Fourier transforms.",
"We get $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\left(\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2}\\left(\\int *{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d}\\right)^\\frac{1}{2} \\\\= *{g_1}{^{(2n-2)}\\left(\\overline{0}\\right)g_1}^\\frac{1}{2}*{g_2}{^{(2)}\\left(\\overline{0}\\right)g_2}^\\frac{1}{2},$ where $^{(N)}\\left(\\overline{0}\\right)\\colon L^2\\left(\\mathcal {E}\\right) \\rightarrow L^2\\left(\\mathcal {E}\\right)$ is the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\varphi ^{(N)}\\left(\\overline{0};a,b\\right)$ .",
"Given $N\\ge 1$ , and $g(c)=\\frac{1}{\\mathcal {P}\\left(E\\right)}1_E(c)$ for some measurable and $\\mathcal {P}$ -positive set $E\\subset \\mathcal {E}$ , $*{g}{^{(2N)}\\left(\\overline{0}\\right)g} = \\left(\\frac{1}{\\mathcal {P}(E)^2}\\int _E\\int _E\\varphi ^{(2N)}\\left(\\overline{0};c,c^{\\prime }\\right)\\mathcal {P}(\\mathrm {d}c)\\mathcal {P}(\\mathrm {d}c^{\\prime })\\right) \\le \\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }.$ Here for $N\\ge 2$ we bound with $\\left\\Vert ^{(2N)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le \\left\\Vert ^{2N}\\right\\Vert _{\\infty ,\\infty }^\\frac{1}{2} \\le g(d)^\\frac{1}{2}$ from REF .",
"For $N=1$ we bound using $\\varphi \\left(\\overline{x};a,b\\right)\\in \\left[0,1\\right]$ and REF : $\\left\\Vert ^{(2)}\\left(\\overline{0}\\right)\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,c\\right)\\varphi \\left(-\\overline{x};c,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}c\\right) \\le \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\\\= \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty }\\le C.$ Therefore $\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le {\\left\\lbrace \\begin{array}{ll}C &: n=2,\\\\C^\\frac{1}{2}g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.",
"}$ We now return to bounding (REF ).",
"To perform the $k$ -integral, we split the domain.",
"We first consider $B_\\varepsilon (0) = \\lbrace k\\in \\mathbb {R}^d: *{k} < \\varepsilon \\rbrace $ , where we choose $\\varepsilon >0$ such that $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1,C_2$ coming from the assumption REF .",
"From REF , we have $\\widehat{G}_{\\mu _\\lambda }(k) = \\left(1 - \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)^{-1}\\le 1/C_2*{k}^2$ on $B_\\varepsilon (0)$ and $\\widehat{G}_{\\mu _\\lambda }(k)\\le 1/\\left(C_2\\varepsilon ^2\\right)$ on $B_\\varepsilon (0)^c$ .",
"Our pointwise bound on the $\\widehat{}$ factors then gives $\\int _{B_\\varepsilon (0)}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\int ^\\varepsilon _0\\frac{1}{r^{2m}}r^{d-1}\\mathrm {d}r = \\frac{C^n}{C^m_2}\\frac{\\mathfrak {S}_{d-1}}{d-2m}\\frac{\\varepsilon ^{d-2m}}{\\left(2\\pi \\right)^d},$ where $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.",
"Note that for all fixed $\\varepsilon >0$ this term vanishes in the $d\\rightarrow \\infty $ limit.",
"To deal with the integral over $B_\\varepsilon (0)^c$ , we find the upper bound $\\int _{B_\\varepsilon (0)^c}\\widehat{G}_{\\mu _\\lambda }(k)^m*{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\\\\\le \\frac{1}{C^m_2\\varepsilon ^{2m}}\\int *{g_1}{\\widehat{}(k)^{2n-2}g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(k)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^d} \\le \\frac{C^\\frac{1}{2}}{C^m_2\\varepsilon ^{2m}}{\\left\\lbrace \\begin{array}{ll}C^\\frac{1}{2} &: n=2\\\\g(d)^\\frac{1}{2} &: n\\ge 3.\\end{array}\\right.",
"}$ Note that the bounds we found were independent of the sets $E_1,E_2$ , and therefore these bounds also apply to $*{g^{(\\overline{x},a,b,\\delta )}_1}{\\left(^{n-1}\\mathcal {T}_\\lambda ^m\\right)\\left(\\overline{x}\\right) g^{(\\overline{x},a,b,\\delta )}_2}$ uniformly over every $\\delta >0$ , $\\overline{x}\\in \\mathbb {R}^d$ , and $\\mathcal {P}$ -almost every $a,b\\in \\mathcal {E}$ .",
"Since we can take $\\delta \\rightarrow 0$ , our bound also applies to $\\left\\Vert ^{n-1}\\mathcal {T}_\\lambda ^m\\right\\Vert _{\\infty ,\\infty }$ .",
"For $n=2$ , the $B_\\varepsilon (0)^c$ integral dominates the $B_\\varepsilon (0)$ integral, and so the result is proven.",
"For $n\\ge 3$ , both parts of the integral approach 0 as $d\\rightarrow \\infty $ .",
"If both are of the same order or if the $B_\\varepsilon (0)^c$ integral dominates, then we can fix $\\varepsilon >0$ and get the result.",
"On the other hand, if $g(d)^\\frac{1}{2} \\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^{d}$ for all fixed $\\varepsilon >0$ and the $B_\\varepsilon (0)$ integral always dominates, we can improve the overall bound by letting $\\varepsilon =\\varepsilon (d)$ and having $\\varepsilon (d)\\rightarrow 0$ as $d\\rightarrow \\infty $ .",
"Having a smaller value of $\\varepsilon $ produces a smaller bound for the $B_\\varepsilon (0)$ integral, but a larger bound for the $B_\\varepsilon (0)^c$ integral.",
"To get an optimal $\\varepsilon $ we can set both terms to be of the same order - that is by having $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{2m}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .",
"For this choice, we have both integrals of the order $\\varepsilon ^{-2m}g(d)^\\frac{1}{2} = 2^{-2m}\\pi ^{-m}A^{-\\frac{2m}{d}} g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\left(1-\\frac{2m}{d}\\right)^{-\\frac{2m}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{2m}{d}} \\\\= \\left(\\frac{\\text{e}}{2\\pi }\\right)^m g(d)^{\\frac{1}{2}-\\frac{m}{d}}\\frac{1}{d^m}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .",
"Proposition 6.5 (Bound for the $n$ -gon diagrams) Let $\\lambda < \\lambda _O$ .",
"Then for each $n$ such that $d>2n$ , there exists finite $c_f= c_f(m,n,f(\\lambda ))$ (increasing in $f(\\lambda )$ and independent of $d$ ) such that $\\left\\Vert \\mathcal {T}_\\lambda ^n\\right\\Vert _{\\infty ,\\infty } \\le {\\left\\lbrace \\begin{array}{ll}c_f&: n=1,2,\\\\c_fg(d)^{\\frac{1}{2}-n\\overline{\\epsilon }(d)}&: n\\ge 3.\\end{array}\\right.",
"}$ The $n=1$ case is trivial because $\\tau _\\lambda \\left(\\overline{x},a,b\\right)\\in \\left[0,1\\right]$ .",
"For $n=2$ , we first bound $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty }$ .",
"We get $\\left\\Vert ^2\\right\\Vert _{\\infty ,\\infty } = \\operatornamewithlimits{ess\\,sup}_{x,y\\in \\mathbb {X}}\\int *{\\varphi (x,u)\\varphi (u,y)}\\nu (\\mathrm {d}u) \\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}^2\\nu (\\mathrm {d}u) \\\\\\le \\operatornamewithlimits{ess\\,sup}_{x\\in \\mathbb {X}}\\int *{\\varphi (x,u)}\\nu (\\mathrm {d}u) = \\left\\Vert \\right\\Vert _{1,\\infty },$ by using Cauchy-Schwarz and symmetry of $\\varphi $ in the first inequality and $\\varphi (x,u)\\in \\left[0,1\\right]$ in the second.",
"We then use REF to bound this by the constant $C$ .",
"We use Lemma REF to get the required bound for the remaining terms arising from (REF ) in Lemma REF .",
"For $n\\ge 3$ , we once again bound the terms in the relevant expansion in (REF ).",
"For $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }$ we use REF to get $\\left\\Vert ^n\\right\\Vert _{\\infty ,\\infty }= O\\left(g(d)^\\frac{1}{2}\\right)$ .",
"We then use Lemma REF to bound the remaining terms.",
"For our proofs of Lemma REF and Lemma REF , we adopt the following notation.",
"Given $g(\\overline{x};a,b)$ and $h(\\overline{x};a,b)$ , we define $(gh)(\\overline{x};a,b) := \\int g(\\overline{x}-\\overline{y};a,c)h(\\overline{y};c,b)\\mathrm {d}\\overline{y}\\mathcal {P}(\\mathrm {d}c).$ This notational convention is associative and therefore can generalise to three or more terms unambiguously.",
"Lemma 6.6 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_6\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u}- \\overline{v};a_3,a_4\\right)\\tau _\\lambda \\left(\\overline{v};a_5,a_6\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{v}\\le c_f,\\\\\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}&\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\tau _\\lambda \\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le c_f,$ and therefore $\\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le 1 + 2c_f.$ By applying (REF ) and (), we can get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ We will in fact prove the more general result for the convolution of $m\\ge 2$ $\\tau _\\lambda $ -functions in dimensions $d>2m$ .",
"By bounding $\\lambda \\le f(\\lambda )$ , and using Lemma REF and the Fourier inversion theorem we can realise that we only need to get further bounds for the following objects.",
"For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^m$ , we want to bound $\\int \\prod ^m_{i=1} P^{j_i}_{2i-1,2i}(k)\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{p,q}(k) := {\\left\\lbrace \\begin{array}{ll}*{*{g_p}{\\widehat{}(k) g_q}} &:j=1\\\\*{*{g_p}{\\widehat{}(k)^2g_q}} &:j=2\\\\*{*{g_p}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_q}} &: j=3,\\end{array}\\right.",
"}$ where $\\left\\lbrace g_i\\right\\rbrace _{i=1}^{2m}$ are the functions of the form $g_i(a) = \\frac{1}{\\mathcal {P}(E_i)}1_{E_i}(a)$ arising from the application of Lemma REF .",
"We now bound the following terms in the same way as (REF ): $*{*{g_1}{\\widehat{}(k)^2g_2}} &\\le *{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}\\\\*{*{g_1}{\\widehat{}(k)\\widehat{\\mathcal {T}}_\\lambda (k)\\widehat{}(k)g_2}} &\\le f\\left(\\lambda \\right)\\widehat{G}_{\\mu _\\lambda }(k)*{g_1}{\\widehat{}(k)^2g_1}^{\\frac{1}{2}}*{g_2}{\\widehat{}(k)^2g_2}^{\\frac{1}{2}}.$ In terms of pointwise uniform bounds for the $\\widehat{}$ terms, we have $*{*{g_1}{\\widehat{}(k)g_2}}\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C$ and $*{g}{\\widehat{}(k)^2g}\\le C^2$ (the latter also having been used in the proof of Lemma REF ).",
"As in Lemma REF , we will require various integral bounds for these $\\widehat{}$ terms.",
"Suppose we have $r \\in \\left\\lbrace 1,\\ldots ,m-1\\right\\rbrace $ instances of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and $m-r$ instances of $P^{1}_{p,q}(k)$ .",
"Then by applying Cauchy-Schwarz and the definitions of the $g_i$ we get $&\\int \\prod ^{r}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\prod ^{m}_{i^{\\prime }=r+1}*{*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int \\prod ^{r}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\prod ^{m}_{i^{\\prime }=r+1}*{g_{2i^{\\prime }-1}}{\\widehat{}(k)g_{2i^{\\prime }}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2r-1},a_{2r})\\right.\\right.\\nonumber \\\\&\\hspace{199.16928pt}\\left.\\left.\\star \\varphi (\\cdot ;a_{2r+1},a_{2r+2})\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m+r}.$ In this last inequality we have used $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\le C$ and $\\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{2,\\infty }^2 \\le C^2$ to extract off $\\varphi $ and $\\varphi ^{(2)}$ via supremum bounds on the spatial position.",
"We will also require integral bounds where there are $m$ factors of $P^{2}_{p,q}(k)$ and $P^{3}_{p,q}(k)$ , and where there are $m$ factors of $P^{1}_{p,q}(k)$ : $&\\int \\prod ^{m}_{i=1} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}^{\\frac{1}{2}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}^{\\frac{1}{2}} \\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)^2g_1}*{g_2}{\\widehat{}(k)^2g_2}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}}\\left(\\int \\prod ^{m}_{i=2} *{g_{2i-1}}{\\widehat{}(k)^2g_{2i-1}}*{g_{2i}}{\\widehat{}(k)^2g_{2i}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi ^{(2)}(\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{2m}.\\\\&\\int \\prod ^{m}_{i=1}*{*{g_{2i-1}}{\\widehat{}(k)g_{2i}}}\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\&\\hspace{28.45274pt}\\le \\left(\\int *{g_1}{\\widehat{}(k)g_2}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\left(\\prod ^{m}_{i=2}*{g_{2i-1}}{\\widehat{}(k)g_{2i}}^2\\frac{\\mathrm {d}k}{\\left(2\\pi \\right)^{d}}\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le \\left(\\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4)\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{170.71652pt}\\times \\left(\\operatornamewithlimits{ess\\,sup}_{a_5,\\ldots ,a_{2m}\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_5,a_6)\\star \\ldots \\star \\varphi (\\cdot ;a_{2m-1},a_{2m})\\right)\\left(\\overline{0}\\right)\\right)^{\\frac{1}{2}} \\nonumber \\\\&\\hspace{28.45274pt} \\le C^{m}.$ Note that for $m\\ge 3$ , better bounds are available via REF , but will not be required here.",
"We can now proceed to bound each of the expressions of the form (REF ) in much the same way as we did in the proof of Lemma REF .",
"We set $\\varepsilon >0$ and partition the integral over $\\mathbb {R}^d$ into one over $B_\\varepsilon (0)$ and one over $B_\\varepsilon (0)^c$ .",
"For the $B_\\varepsilon (0)$ integral we use the uniform bounds on the $\\widehat{}$ terms and perform the integral of the $\\widehat{G}_{\\mu _\\lambda }$ terms (if there are any) as before.",
"Note that we require $d>6$ for all of these to be finite.",
"For the $B_\\varepsilon (0)^c$ integrals we uniformly bound $\\widehat{G}_{\\mu _\\lambda }(k) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and use the integral bounds we calculated above.",
"Since we are only asking for a constant bound, we don't need to worry about having $\\varepsilon \\rightarrow 0$ as $d\\rightarrow \\infty $ ."
],
[
"Bounding the Lace Operator Displacement",
"We now consider the objects required to bound the displacement of the lace operator.",
"In addition to the objects used to bound the lace operator itself, we require bounds on $\\left\\Vert _k\\right\\Vert _{1,\\infty }, \\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }, \\overline{W_k}, \\overline{H_k}, \\triangle ^{(B)}_\\lambda , \\mathbb {B}.$ The decay $\\mathbb {B}\\le g(d)$ follows from Assumption REF , and the following Observation REF demonstrates the decay of $\\triangle ^{(B)}_\\lambda $ .",
"Observation 6.7 Note that the inequality (REF ) allows us to bound $\\triangle ^{(B)}_\\lambda $ in terms of other terms.",
"By using the inequality twice and appropriately bounding $\\varphi $ with $\\tau _\\lambda $ , we get $\\int \\tau _\\lambda (r,u)\\tau _\\lambda (u,s)\\nu (\\mathrm {d}u) \\le \\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) + 2\\lambda \\int \\tau _\\lambda (r,v)\\tau _\\lambda (v,u)\\tau _\\lambda (u,s)\\nu ^{\\otimes 2}(\\mathrm {d}u, \\mathrm {d}v)$ for any $r,s\\in \\mathbb {X}$ .",
"Therefore $\\triangle ^{(B)}_\\lambda \\le 3\\triangle _\\lambda + \\lambda \\operatornamewithlimits{ess\\,sup}_{r\\notin B(s)}\\int \\varphi (r,u)\\varphi (u,s)\\nu (\\mathrm {d}u) \\le 3\\triangle _\\lambda + \\lambda g(d),$ where we have used REF .",
"Therefore Proposition REF implies that for $d>6$ there exists $c_f$ such that $\\triangle ^{(B)}_\\lambda \\le c_f\\beta ^2.$ We are then left with $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ , $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ , $\\overline{W_k}$ , and $\\overline{H_k}$ , and we will deal them in this order.",
"First note that $\\left\\Vert _k\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{y\\in \\mathbb {X}}\\int \\varphi _k(x,y)\\nu \\left(\\mathrm {d}x\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\int (1-\\cos \\left(k\\cdot \\overline{x}\\right))\\varphi (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}(\\mathrm {d}a) \\\\ = \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{1,\\infty } = \\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{1,\\infty }.$ The following lemma therefore also allows us to bound $\\left\\Vert _k\\right\\Vert _{1,\\infty }$ .",
"Its extra generality will be required later.",
"Lemma 6.8 For all $k,l\\in \\mathbb {R}^d$ , $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\infty ,\\infty } \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where $C>0$ is the $d$ -independent constant given in REF .",
"The first two inequalities in (REF ) holds from the general inequalities that holds for these norms (on a probability space for the second inequality).",
"Then we can remove the $l$ -dependence by performing the following calculation: $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+ k\\right)\\right\\Vert _{\\infty ,\\infty } \\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\text{e}^{il\\cdot \\overline{x}}}*{\\left(1-\\cos \\left( k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}\\\\= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int *{\\left(1-\\cos \\left(k\\cdot \\overline{x}\\right)\\right)\\varphi \\left(\\overline{x};a,b\\right)}\\mathrm {d}\\overline{x}= \\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left( k\\right)\\right\\Vert _{\\infty ,\\infty }.$ The final step relating the $\\left\\Vert \\cdot \\right\\Vert _{\\infty ,\\infty }$ -norm to the difference of the spectral suprema is precisely that given by () in REF .",
"We now address $\\left\\Vert \\mathcal {T}_{\\lambda ,k}\\mathcal {T}_\\lambda \\right\\Vert _{\\infty ,\\infty }$ .",
"Lemma 6.9 For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ First note that for $k=0$ the kernel function $\\tau _{\\lambda ,k}\\left(x,y\\right)=0$ for all $x,y\\in \\mathbb {X}$ , and so the required inequality holds trivially.",
"In this proof we hereafter may assume $k\\ne 0$ .",
"We use (REF ) to write $\\left\\Vert \\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } \\le \\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty } + \\lambda \\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }.$ We first get the bound on $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ , using a similar approach to Lemma REF .",
"For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _\\lambda \\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime }\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}^{\\prime }-\\overline{u}^{\\prime \\prime };c^{\\prime },c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime \\prime };c^{\\prime \\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathrm {d}\\overline{u}^{\\prime \\prime }\\mathcal {P}^{\\otimes 3}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime },\\mathrm {d}c^{\\prime \\prime }\\right).$ Recall the notation $\\varphi ^{(2)}$ used in (REF ).",
"As in the proof of Lemma REF , we let $E_1,E_2\\subset \\mathcal {E}$ be measurable and $\\mathcal {P}$ -positive sets, and aim to bound ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ (where $g_i(c) := \\frac{1}{\\mathcal {P}\\left(E_i\\right)}1_{E_i}\\left(c\\right)$ for $i=1,2$ ) independently of the choice of $E_1,E_2$ .",
"If we do this we will have proven the result.",
"As before, we use the Fourier inversion theorem to write ${g_1}{\\left(\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2}$ as the integral of an inner product of Fourier transformed operators over the Fourier argument.",
"Having $\\lambda <\\lambda _O$ allows us to swap the integrals in this step.",
"Then we can use Cauchy-Schwarz and the definitions of the operator norm and the bootstrap functions to extract factors of $\\widehat{G}_{\\mu _\\lambda }(l)$ and $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ from the inner product.",
"The net result is $*{g_1}{\\left(^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} &\\le \\int *{*{g_1}{\\widehat{}(l)^{2}\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l)g_2}}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\le f(\\lambda )^2\\int \\widehat{G}_{\\mu _\\lambda }(l) \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ The $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ We partition our integral using open $\\varepsilon $ -balls around the poles where $\\varepsilon ^2 \\le \\tfrac{1-C_1}{C_2}$ - the constants $C_1$ and $C_2$ coming from REF .",
"Let $B_{\\varepsilon }(p)$ denote the open $\\varepsilon $ -ball around $p\\in \\mathbb {R}^d$ .",
"For $n\\in \\left\\lbrace 0,1,2,3\\right\\rbrace $ , let $A_n := \\left\\lbrace l\\in \\mathbb {R}^d: l\\in B_\\varepsilon (p) \\text{ for precisely $n$ elements }p\\in \\left\\lbrace k,0,-k\\right\\rbrace \\right\\rbrace .$ Firstly it is possible for $A_3,A_2 = \\emptyset $ if $*{k}$ is sufficiently large compared to $\\varepsilon $ .",
"It is also easy to see that $A_3\\subset B_{\\varepsilon }(0)$ and $A_2\\subset B_{2\\varepsilon }(0)$ .",
"We will partition $\\mathbb {R}^d$ into $A_3$ , $A_2$ , $A_1$ , and $A_0$ and bound the $l$ -integral on each part.",
"Recall that REF implies that $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2*{l}^{-2}$ for $*{l}\\le \\varepsilon $ and $\\widehat{G}_{\\mu _\\lambda }(l) \\le C^{-1}_2\\varepsilon ^2$ for $*{l}\\ge \\varepsilon $ .",
"Also recall the pointwise (REF ) and integral (REF ) bounds we derived for the $\\widehat{}$ factors.",
"We treat $A_3$ first: $&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l)^2\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)\\cap B_\\varepsilon (k)}\\frac{1}{*{l}^4*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\qquad \\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\\\&\\int _{A_3}\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{A_3}\\frac{1}{*{l+k}^2*{l}^2*{l-k}^2} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (-k)}\\frac{1}{*{l+k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}}\\left(\\int _{B_\\varepsilon (k)}\\frac{1}{*{l-k}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right)^{\\frac{1}{3}} \\nonumber \\\\&\\hspace{56.9055pt}\\le \\frac{C^3}{C^3_2}\\int _{B_\\varepsilon (0)}\\frac{1}{*{l}^6} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ Recall $\\mathfrak {S}_{d-1}= d\\pi ^\\frac{d}{2}/\\Gamma \\left(\\tfrac{d}{2}+1\\right)$ is the surface area of an unit $d$ -sphere.",
"In these calculations we used Hölder's inequality and increased the domain of integration to get upper bounds.",
"These integrals are finite for $d>6$ , and approach 0 if we take bounded $\\varepsilon $ .",
"For $A_2$ the counting of cases is the only extra complication.",
"A precise counting is entirely possible, but unnecessary for our purposes.",
"It is simple to see that the arguments outlined for $A_3$ can be applied to get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^6} + \\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a pair of two distinct points from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with three, two, and one factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on these two points.",
"This bound is finite for $d>6$ , and approaches 0 if we take bounded $\\varepsilon $ .",
"Repeating this for $A_1$ , we get the bound as an integer multiple of $\\frac{C^3}{C^3_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{\\varepsilon ^2*{l}^4} + \\frac{1}{\\varepsilon ^4*{l}^2} + \\frac{1}{\\varepsilon ^6} \\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\right) = \\frac{C^3}{C^3_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ Essentially, these terms arise because given a single point from $\\left\\lbrace k,0,-k\\right\\rbrace $ , there are terms with two, one, and zero factor(s) of $\\widehat{G}_{\\mu _\\lambda }$ centred on this single point.",
"This bound is finite for $d>4$ , and approaches 0 if we take bounded $\\varepsilon $ .",
"The $A_0$ integral is qualitatively different.",
"By taking the integral bound for the $\\widehat{}$ factors and a uniform (on $A_0$ ) bound for the factors of $\\widehat{G}_{\\mu _\\lambda }$ , we get the bound $\\frac{C^3}{C^3_2}\\varepsilon ^{-6}g(d)^\\frac{1}{2}.$ As in Lemma REF , if the $g(d)^\\frac{1}{2}$ term dominates, then we have a sufficient bound for our result.",
"The largest of the other terms (for $d$ sufficiently large and $\\varepsilon \\le 1$ ) is the $\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}$ term.",
"Like we did in Lemma REF , we can make these other terms smaller by decreasing $\\varepsilon $ at the cost of making the $g(d)^\\frac{1}{2}$ bound greater.",
"The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{6}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .",
"For this choice, we have both integrals of the order $\\varepsilon ^{-6}g(d)^\\frac{1}{2} = 2^{-6}\\pi ^{-3}A^{-\\frac{6}{d}}g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\left(1-\\frac{6}{d}\\right)^{-\\frac{6}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{6}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^3 g(d)^{\\frac{1}{2}-\\frac{3}{d}}\\frac{1}{d^3}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .",
"Bounding the $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ term is similar.",
"For each $\\overline{x}\\in \\mathbb {R}^d$ we define the integral linear operator $\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as the integral operator with kernel function $\\left(a,b\\right)\\mapsto \\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x};a,b\\right) \\\\:= \\int \\varphi ^{(2)}\\left(\\overline{x}-\\overline{u};a,c\\right)\\tau _{\\lambda ,k}\\left(\\overline{u}-\\overline{u}^{\\prime };c,c^{\\prime \\prime }\\right)\\varphi \\left(\\overline{u}^{\\prime };c^{\\prime },b\\right)\\mathrm {d}\\overline{u}\\mathrm {d}\\overline{u}^{\\prime } \\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}c,\\mathrm {d}c^{\\prime }\\right).$ We once again take $\\mathcal {P}$ -positive sets $E_1,E_2$ and corresponding $g_1,g_2$ .",
"Then we use the Fourier inversion theorem and Cauchy-Schwarz to get $*{g_1}{\\left(^2\\mathcal {T}_{\\lambda ,k}\\right)\\left(\\overline{x}\\right)g_2} \\le f(\\lambda )\\int \\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right) *{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Then once again the $\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)$ term produces the required factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ , so (after using symmetry in $k$ ) we only need to bound the expression $\\int \\left(2\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)\\right)*{g_1}{\\widehat{}(l)^4g_1}^\\frac{1}{2}*{g_2}{\\widehat{}(l)^2g_2}^\\frac{1}{2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}.$ Counting the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ centred on each of the poles $\\left\\lbrace k,0,-k\\right\\rbrace $ then tells us that there is a uniform bound that is some integer multiple of $\\frac{C^2}{C^2_2}\\left(\\int _{B_\\varepsilon (0)}\\left(\\frac{1}{*{l}^{4}} + \\frac{1}{\\varepsilon ^2*{l}^{2}} + \\frac{1}{\\varepsilon ^4}\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} + \\varepsilon ^{-4}g(d)^\\frac{1}{2}\\right) \\\\= \\frac{C^2}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\left(\\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-4} + \\frac{C^2}{C^2_2}\\varepsilon ^{-4}g(d)^\\frac{1}{2}.$ Once again, if the $g(d)^\\frac{1}{2}$ term dominates the result is proven.",
"Otherwise, we let $\\varepsilon $ vary with $d$ .",
"The largest term can be minimised (up to a constant) by taking $\\varepsilon (d) = 2\\pi ^{\\frac{1}{2}}A^\\frac{1}{d}g(d)^\\frac{1}{2d}\\left(1-\\frac{4}{d}\\right)^\\frac{1}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{1}{d}$ for some fixed $A>0$ .",
"For this choice, we have both integrals of the order $\\varepsilon ^{-4}\\beta ^2 = 2^{-4}\\pi ^{-2}A^{-\\frac{4}{d}} g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(1-\\frac{4}{d}\\right)^{-\\frac{4}{d}}\\Gamma \\left(\\frac{d}{2}+1\\right)^{-\\frac{4}{d}} = \\left(\\frac{\\text{e}}{2\\pi }\\right)^2 g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\frac{1}{d^2}\\left(1+o\\left(1\\right)\\right),$ as $d\\rightarrow \\infty $ .",
"For large $d$ , this is dominated by $g(d)^{\\frac{1}{2}-\\frac{3}{d}}d^{-3}$ and therefore $\\left\\Vert ^2\\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ is dominated by $\\left\\Vert ^2\\mathcal {T}_\\lambda \\mathcal {T}_{\\lambda ,k}\\right\\Vert _{\\infty ,\\infty }$ and we prove the result.",
"The following lemma allows us to deal with occurrences of $\\overline{W_k}$ .",
"Lemma 6.10 Let $x,y \\in \\mathbb {X}$ .",
"Then $\\tau _{\\lambda ,k}(x,y) \\le \\varphi _k(x,y) + 2\\lambda \\int \\left(\\tau _\\lambda (x,u)\\varphi _k(u,y) + \\tau _{\\lambda ,k}(x,u)\\varphi (u,y)\\right)\\nu \\left(\\mathrm {d}u\\right).",
"$ For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{W_k}\\le c_f\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The first inequality follows by applying the cosine-splitting result (Lemma REF ) to ().",
"For the $\\tau _\\lambda $ term in $\\overline{W_k}$ we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of (REF ), we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we can bound $\\overline{W_k}$ by bounding the following objects.",
"For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace \\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {W}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l)Q^{j_2}_{3,4}(l;k)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}},$ where $P^{j}_{1,2}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_1}{\\widehat{}(l) g_2}} &:j=1\\\\*{*{g_1}{\\widehat{}(l)^2g_2}} &:j=2\\\\*{*{g_1}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_2}} &: j=3,\\end{array}\\right.",
"}\\\\Q^{j}_{3,4}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} &:j=1\\\\*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} &:j=2\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{}_k(l) g_{4}}} &:j=3\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &:j=4\\\\*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} &:j=5,\\end{array}\\right.",
"}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{4}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .",
"For the $P^j_{1,2}(l)$ factor we can get bounds as we did in the proof of Lemma REF .",
"For $Q^{j}_{3,4}(l;k)$ we need to do a bit more processing.",
"For $j=1$ , $*{*{g_{3}}{\\widehat{}_k(l) g_{4}}} \\le \\frac{1}{\\mathcal {P}(E_{3})\\mathcal {P}(E_{4})}\\int _{E_{3}\\times E_{4}}*{\\widehat{\\varphi }_k(l;a,b)}\\mathcal {P}^{\\otimes 2}(\\mathrm {d}a, \\mathrm {d}b) \\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right),$ where in the last inequality we used Lemma REF .",
"For $j=2$ , we use Cauchy-Schwarz to get $*{*{g_{3}}{\\widehat{}_k(l)\\widehat{}(l) g_{4}}} \\le *{g_3}{\\widehat{}_k(l)^2 g_3}^{\\frac{1}{2}} *{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}} \\\\\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_4}{\\widehat{}(l)^2 g_4}^{\\frac{1}{2}}.$ An identical argument gives the same bound for $j=3$ .",
"For $j=4$ , we can use the self-adjointness of $\\widehat{}(l)$ and $\\widehat{\\mathcal {T}}_\\lambda (l)$ and the Cauchy-Schwarz inequality to get $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{4}}} &\\le *{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)^2\\widehat{}(l)g_{3}}^{\\frac{1}{2}} *{g_{4}}{\\widehat{}_k(l)^2g_{4}}^{\\frac{1}{2}} \\nonumber \\\\&\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}}\\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty }\\nonumber \\\\&\\le Cf(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_{3}}{\\widehat{}(l)^2g_{3}}^{\\frac{1}{2}},$ and similarly $*{*{g_{3}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{4}}} \\le f(\\lambda )\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2} \\\\ \\le Cf(\\lambda )\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)*{g_3}{\\widehat{}(l)^2g_3}^\\frac{1}{2}*{g_4}{\\widehat{}(l)^2g_4}^\\frac{1}{2},$ where we write for convenience $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k) = \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3-k) + \\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k) + \\widehat{G}_{\\mu _\\lambda }(l_3-k)\\widehat{G}_{\\mu _\\lambda }(l_3+k).$ Note that all the bounds for the $Q^{j}_{3,4}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.",
"So we only need to prove that the remaining $l$ -integral of $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors is bounded.",
"As in our previous proofs, we partition $\\mathbb {R}^d$ into the sets $A_n$ defined earlier in (REF ).",
"For the $A_1$ , $A_2$ , and $A_3$ parts we then use the bounds $C$ and $C^2$ for the various $\\widehat{}$ factors and $1/\\left(C_2*{l}^2\\right)$ to bound the $\\widehat{G}_{\\mu _\\lambda }$ factors.",
"By using the techniques used previously in this section, we can show that these contributions is bounded by some $\\varepsilon $ and $d$ (only) dependent constant.",
"Since we don't need to show decay, we don't need to worry about taking $\\varepsilon $ to 0.",
"For the $A_0$ contribution, we bound $\\widehat{G}_{\\mu _\\lambda }$ factors by $1/\\left(C_2\\varepsilon ^2\\right)$ and show that the integral of the $\\widehat{}$ factors over all of $\\mathbb {R}^d$ is bounded.",
"For $j_2\\ne 1$ this proceeds similarly to the argument in the proof of Lemma REF .",
"For $j_2=1$ we need to take a step back.",
"These versions of (REF ) arise from trying to bound one particular type of term, which we now treat separately.",
"We bound $\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\int \\tau _\\lambda \\left(\\overline{x}-\\overline{u};a_1,a_2\\right)\\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\operatornamewithlimits{ess\\,sup}_{a_3,a_4\\in \\mathcal {E}}\\int \\varphi _k\\left(\\overline{u};a_3,a_4\\right)\\mathrm {d}\\overline{u}\\le \\left\\Vert \\widehat{}_k(0)\\right\\Vert _{\\infty ,\\infty }.$ Then Lemma REF gives the result.",
"We are now only left with bounding $\\overline{H_k}$ .",
"lemmaMartiniBound For $\\lambda <\\lambda _O$ , there exists $c_f$ (increasing in $f$ and independent of $d$ ) such that for $d>6$ , $\\overline{H_k}\\le c_f\\beta ^2\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ The proof of the lemma uses the same techniques and tools as our previous lemmas, but is much more complicated in terms of keeping track of cases and components of integrals.",
"To avoid overburdening a reader on an early reading, the proof has been placed in Appendix ."
],
[
"The Bootstrap Argument",
"In this section we bring together the diagrammatic bounds of Section with the estimates of Section REF to bound the relevant operators in such a way that the finite lace expansion of Proposition REF converges to an Ornstein-Zernike equation (OZE) for both the raw operators on $L^2\\left(\\mathbb {X}\\right)$ and the Fourier-transformed operators on $L^2\\left(\\mathcal {E}\\right)$ .",
"Proposition REF does this for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , with the caveat that there is a constant $c_f$ that is increasing in the bootstrap function $f$ (but independent of $d$ ) appearing in the bounds.",
"Observation REF then highlights that if $f$ is uniformly bounded on $\\left[0,\\lambda _O\\right)$ then so is $c_f$ and the associated bounds from Proposition REF .",
"The crucial step to the bootstrap argument is then Proposition REF .",
"Here a forbidden-region argument shows that $f$ is continuous, that there is an interval of finite values that it cannot take (the interval being independent of $\\lambda $ ), and that it starts below this interval.",
"This proves that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ , and so are our bounds in Proposition REF .",
"Proposition 6.11 (Convergence of the operator lace expansion and OZE) Let $\\lambda \\in [0, \\lambda _O)$ and $d>6$ be sufficiently large.",
"Then there exists $c_f=c(f(\\lambda ))$ (which is increasing in $f$ and independent of $d$ ) such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f\\beta , &\\qquad \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}(k)\\right\\Vert _{\\rm op}\\le c_f\\beta , \\\\ \\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} &\\le c_f\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right)-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\beta , \\\\\\left\\Vert R_{\\lambda , n}\\right\\Vert _{\\rm op} &\\le \\lambda \\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} c_f\\beta ^{n}.",
"$ Furthermore, the limit $_\\lambda := \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}_\\lambda ^{(n)}$ exists and is a bounded operator with Fourier transform $\\widehat{}_{\\lambda }\\left(k\\right) = \\lim _{M\\rightarrow \\infty } \\sum ^M_{n=0}\\widehat{}_{\\lambda }^{(n)}\\left(k\\right)$ for $k\\in \\mathbb {R}^d$ .",
"The operators $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ satisfy operator Ornstein-Zernike equations, taking the form $\\mathcal {T}_\\lambda = + _\\lambda + \\lambda \\mathcal {T}_\\lambda (+_\\lambda ), $ and $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right) + \\lambda \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\left(\\widehat{}\\left(k\\right)+\\widehat{}_{\\lambda }\\left(k\\right)\\right), $ for all $\\lambda < \\lambda _O$ and $k\\in \\mathbb {R}^d$ .",
"Before we prove this proposition, we make the following observation, which will guide the rest of the section.",
"Observation 6.12 (Uniform convergence of the lace expansion) Suppose that there exists finite $C>0$ such that $f \\le C$ on $[0,\\lambda _O)$ .",
"Then for $d>6$ sufficiently large there exits $c>0$ (independent of $\\lambda $ and $d$ ) such that the bounds (REF ), (), () hold with $c_f$ replaced by $c$ .",
"We now return to prove Proposition REF .",
"[Proof of Proposition REF ] So far in this section we have provided bounds for the building blocks of the bounds appearing in Propositions REF , REF , REF , and REF .",
"We still have to convert these into bounds for the composite terms $U_\\lambda $ and $V_\\lambda $ .",
"From their definitions and the bounds on the building blocks, it is simple to see that there exists $c_f^{\\prime }$ such that for $d>6$ $U_\\lambda \\le c_f^{\\prime }, \\qquad V_\\lambda \\le c_f^{\\prime }\\beta .$ This then implies that there exists $c_f^{\\prime \\prime }$ such that for $d>6$ $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime }\\beta ^2 &: n=0,\\\\c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n &: n\\ge 1.\\end{array}\\right.",
"}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime }\\left(\\beta ^2 + \\frac{c_f^{\\prime \\prime }\\beta }{1-c_f^{\\prime \\prime }\\beta }\\right) \\le c_f\\beta .$ When we want to consider the displaced terms, we also make use of Observation REF , and Lemmas REF , REF , REF , and REF to give us the existence of $c_f^{\\prime \\prime \\prime }$ such that for $d>6$ $\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le {\\left\\lbrace \\begin{array}{ll}c_f^{\\prime \\prime \\prime }\\beta ^2\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=0,3,\\\\c_f^{\\prime \\prime \\prime }\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n=1,2,\\\\\\left(n+1\\right)^2c_f^{\\prime \\prime \\prime }\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^{n-3}\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) &: n\\ge 4.\\end{array}\\right.",
"}$ If $d$ is sufficiently large we have $c_f^{\\prime \\prime \\prime }\\beta <1/2$ (for example), and therefore there exists $c_f$ such that $\\sum _{n\\ge 0}\\left\\Vert \\widehat{}_{\\lambda }^{(n)}\\left(0\\right) - \\widehat{}_{\\lambda }^{(n)}\\left(k\\right)\\right\\Vert _{\\rm op} \\le c_f^{\\prime \\prime \\prime }\\left(2\\beta ^2 + 2\\beta + \\frac{16\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^3-39\\left(c_f^{\\prime \\prime \\prime }\\beta \\right)^2 + 25c_f^{\\prime \\prime \\prime }\\beta }{\\left(1-c_f^{\\prime \\prime \\prime }\\beta \\right)^3}\\right)\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) \\\\\\le c_f\\beta \\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right).$ Lemma REF then allows us to use (REF ) to bound $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}$ .",
"This immediately gives the required result for $n\\ge 1$ .",
"For $n=0$ , we simply additionally require $d$ to be large enough that $\\beta <1$ and therefore $\\beta ^2<\\beta ^0$ .",
"Note that the dual space of a Banach space (endowed with the operator norm) is also a Banach space.",
"In particular, it is complete.",
"Since $\\left\\Vert _\\lambda ^{(n)}\\right\\Vert _{\\rm op}\\le c_f^{\\prime \\prime }\\left(c_f^{\\prime \\prime }\\beta \\right)^n$ for $n\\ge 1$ , the sequence $_{\\lambda ,n}$ is a Cauchy sequence in the dual space of $L^2(\\mathbb {X})$ for sufficiently large $d$ .",
"Therefore the limit $_\\lambda $ exists and is a bounded linear operator on $L^2(\\mathbb {X})$ .",
"Since the Fourier transform is a unitary linear operator it is bounded and continuous, and therefore the same argument says that the limit $\\widehat{}_{\\lambda }(k)$ exists and is indeed the Fourier transform of $_\\lambda $ .",
"Since $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op}$ is finite for $\\lambda <\\lambda _O$ , we have $\\left\\Vert R_{\\lambda ,n}\\right\\Vert _{\\rm op}\\rightarrow 0$ as $n\\rightarrow \\infty $ if $d$ is sufficiently large.",
"As a a consequence of this and equation (REF ) from Proposition REF , the $\\mathcal {T}_\\lambda $ operator satisfies the operator Ornstein-Zernike equation.",
"The following lemma has two main uses in what follows.",
"Firstly, it takes the OZEs from Proposition REF and uses it to write the two-point operators in terms of the sum of the adjacency operators and the lace expansion coefficient operators.",
"Secondly, it uses this and an intermediate value theorem argument to bound the spectral supremum of the sum of the adjacency operators and the lace expansion coefficient operators.",
"Lemma 6.13 Let $\\lambda \\in \\left[0,\\lambda _O\\right)$ and $d>6$ be sufficiently large.",
"Then the operators $\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)$ and $\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)$ all have bounded linear inverses and $\\mathcal {T}_\\lambda &= \\left(+ _\\lambda \\right)\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)^{-1}, \\\\\\widehat{\\mathcal {T}}_\\lambda (k) &= \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1}, \\qquad \\forall k\\in \\mathbb {R}^d.",
"$ Furthermore, $\\lambda \\mapsto _\\lambda $ and $\\lambda \\mapsto \\widehat{}_{\\lambda }(k)$ are all continuous in the operator norm topology and $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) &< 1,\\\\\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &< 1, \\qquad \\forall k\\in \\mathbb {R}^d.$ We first prove that $1- \\lambda \\left(+ _\\lambda \\right)$ has a bounded linear inverse.",
"This is clear for $\\lambda = 0$ , so we only need to consider $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"Suppose for contradiction that $1- \\lambda \\left(+ _\\lambda \\right)$ does not have a bounded linear inverse.",
"Then there exists a sequence $f_n\\in L^2(\\mathbb {X})$ such that $*{f_n}_2 = 1$ and $*{\\left(1- \\lambda \\left(+ _\\lambda \\right)\\right)f_n}_2\\rightarrow 0$ .",
"This also means that $\\lambda *{\\left(+ _\\lambda \\right)f_n}\\rightarrow 1$ .",
"Then from Lemma REF and sub-criticality we have $\\left\\Vert \\mathcal {T}_\\lambda \\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} < \\infty $ , and therefore the OZE equation (REF ) implies that $*{\\left(+ _\\lambda \\right)f_n}\\rightarrow 0$ , a contradiction.",
"A similar argument proves the corresponding statement for the Fourier transformed operators.",
"Once we know that the inverses exist, it is only a matter of rearranging the OZE equations to get the expressions for $\\mathcal {T}_\\lambda $ and $\\widehat{\\mathcal {T}}_\\lambda (k)$ in terms of the other operators.",
"Since $+ _\\lambda $ is a bounded operator, a similar argument proves that $1+ \\lambda \\mathcal {T}_\\lambda $ has a bounded inverse, and therefore $+ _\\lambda = \\left(1+ \\lambda \\mathcal {T}_\\lambda \\right)^{-1}\\mathcal {T}_\\lambda .$ From Corollary REF we have the continuity of $\\lambda \\mapsto \\mathcal {T}_\\lambda $ , and therefore the continuity $+ _\\lambda $ (and therefore $_\\lambda $ ).",
"We also have the continuity of $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda (k)$ from Corollary REF , and we similarly get the continuity of $\\widehat{}_{\\lambda }(k)$ .",
"The continuity of $+ _\\lambda $ implies the continuity of $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right)$ (via Lemma REF ).",
"At $\\lambda = 0$ it is clear that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) = 0$ , and our above argument implies that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) \\ne 1$ .",
"Therefore an intermediate value theorem argument proves that $\\lambda \\mathbb {S}\\left(+ _\\lambda \\right) < 1$ .",
"This argument also works for $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .",
"It now remains to show that $f$ is indeed uniformly bounded for $\\lambda \\in \\left[0,\\lambda _O\\right)$ .",
"We do this by performing a forbidden-region argument.",
"In Proposition REF we prove that $f(0)$ is bounded and $f$ is continuous on $\\left[0,\\lambda _O\\right)$ .",
"However we also prove that $f$ is never in the region $\\left(\\kappa ,\\kappa +1\\right]$ , where $\\kappa := 10 C \\left(1+C\\right)^2,$ $C$ being the constant appearing in REF .",
"Note that (REF ) in REF requires that $C\\ge 1$ .",
"The intermediate value theorem then implies that $\\kappa $ acts as an upper bound on the whole domain $\\left[0,\\lambda _O\\right)$ .",
"One should not read too much into the value of $\\kappa $ here.",
"It is sufficient for our purposes, but not remotely optimal.",
"Proposition 6.14 (The forbidden-region argument) The following three statements are all true: $f$ satisfies $f(0) \\le \\kappa $ .",
"$f(\\lambda ) \\notin (\\kappa ,\\kappa +1]$ for all $\\lambda \\in [0,\\lambda _O)$ provided that $d$ is sufficiently large.",
"$f$ is continuous on $[0, \\lambda _O)$ .",
"Therefore $f(\\lambda ) \\le \\kappa $ holds uniformly in $\\lambda <\\lambda _O$ for all $d$ sufficiently large.",
"We will deal with each of the three statements in Proposition REF in turn.",
"[Proof of Proposition REF -REF ] Firstly, it is clear $f_1\\left(0\\right) = 0$ .",
"Then note that $\\mathcal {T}_0 = $ , and hence $\\left\\Vert \\widehat{\\mathcal {T}_0}\\left(k\\right)\\right\\Vert _{\\rm op} = \\left\\Vert \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C,$ where we use REF (via (REF )) in the last inequality.",
"Furthermore $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_0(0)\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) =1$ , and therefore $\\widehat{G}_{\\mu _{0}}\\left(k\\right) = \\widehat{G}_0\\left(k\\right) = 1$ and $f_2\\left(0\\right) \\le C$ .",
"Similarly we find that $f_3\\left(0\\right) = \\operatornamewithlimits{ess\\,sup}_{k,l\\in \\mathbb {R}^d}\\frac{\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op}}{3\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]}.$ We use then Lemma REF to get the bound $\\left\\Vert \\widehat{}_k\\left(l\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Hence $f_3\\left(0\\right) \\le \\tfrac{2}{3}C$ .",
"[Proof of Proposition REF -REF ] We will assume that $f\\le \\kappa +1$ and show that this implies that $f\\le \\kappa $ .",
"Most crucially, the assumption $f\\le \\kappa +1$ allows us to apply Observation REF .",
"To highlight the $\\kappa $ -dependence, we write $c_{\\kappa +1}$ as the constant arising from this observation.",
"It will be convenient to introduce some temporary notation for this section.",
"Define $\\widehat{a}\\left(k\\right)\\colon L^2\\left(\\mathcal {E}\\right)\\rightarrow L^2\\left(\\mathcal {E}\\right)$ as $\\widehat{a}\\left(k\\right) = \\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)$ for all $k\\in \\mathbb {R}^d$ .",
"We will also make use of the functions $\\delta _\\lambda \\colon \\mathbb {R}^d\\rightarrow \\mathbb {R}$ defined by $\\delta _\\lambda \\left(k\\right) := \\mathbb {S}\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda }\\left(k\\right)\\right) - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ It is clear from the triangle inequality and Lemma REF that ${\\delta _\\lambda \\left(k\\right)}\\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k\\right)\\right\\Vert _{\\rm op}\\le c_{\\kappa +1}\\beta ,$ but we will also need the following inequality.",
"We will prove it at the end of this section.",
"Lemma 6.15 Let $k_1,k_2\\in \\mathbb {R}^d$ .",
"Then $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2 \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right) - \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ Figure: If 𝒯 ^ λ (k)\\widehat{\\mathcal {T}}_\\lambda (k) and a ^(k)\\widehat{a} (k) commute, then they are simultaneously diagonalizable in the sense of Theorem .",
"The arguments of their diagonal functions (τ ˜ λ (k)\\widetilde{\\tau }_\\lambda (k) and a ˜(k)\\widetilde{a}(k) respectively) are related by the monotone increasing function x↦x 1-λxx\\mapsto \\tfrac{x}{1-\\lambda x} depicted here.It will be important to note how $\\widehat{a}(k)$ relates to $\\widehat{\\mathcal {T}}_\\lambda (k)$ .",
"From Observation REF , the OZE gives us the following expression involving the commutator $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right] = \\widehat{\\mathcal {T}}_\\lambda (k) \\widehat{a}(k) - \\widehat{a}(k)\\widehat{\\mathcal {T}}_\\lambda (k)$ : $\\left[\\widehat{\\mathcal {T}}_\\lambda (k),\\widehat{a}(k)\\right]\\left(1- \\lambda \\widehat{a}(k)\\right) = 0.$ Lemma REF implies that $\\left(1- \\lambda \\widehat{a}(k)\\right)$ has a bounded linear inverse and with (REF ) implies that $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ commute.",
"Therefore there exists a single unitary map that `diagonalizes' them both in the sense of Theorem REF .",
"By considering the diagonal functions, the OZE then implies that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\mathbb {S}\\left(\\widehat{a}(k)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)\\right)$ .",
"In particular, this relation allows us to write an alternative expression for $\\mu _\\lambda $ : $\\mu _\\lambda := 1 - \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} = \\lambda + \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)-1}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)} = \\lambda + \\frac{\\delta _\\lambda \\left(0\\right)}{1+\\delta _\\lambda \\left(0\\right)}.$ We first address $f_1$ .",
"From the above discussion relating $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a}(k)$ , we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\ge \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) = \\mathbb {S}\\left(\\widehat{a}(0)\\right)/\\left(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(0)\\right)\\right)$ .",
"From Lemma REF and our choice of scaling we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right) \\ge 1 - \\left\\Vert \\widehat{}_{\\lambda }(0)\\right\\Vert _{\\rm op} \\ge 1 - c_{\\kappa +1}\\beta $ , and therefore we have $\\mathbb {S}\\left(\\widehat{a}(0)\\right)>0$ for sufficiently large $d$ and $\\lambda \\le \\frac{1}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}\\left(1 - \\frac{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\right) \\le \\frac{1}{1-c_{\\kappa +1}\\beta } \\le 1 + 2c_{\\kappa +1}\\beta .$ Here we have used an idea that we will use very often in this part of the proof.",
"For $d$ large enough that $c_{\\kappa +1}\\beta <1$ we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1} = \\sum ^\\infty _{n=0}\\left(c_{\\kappa +1}\\beta \\right)^n$ , and for $d$ sufficiently large enough ($c_{\\kappa +1}\\beta \\le 1/2$ ) we have $\\left(1-c_{\\kappa +1}\\beta \\right)^{-1}\\le 1 + 2c_{\\kappa +1}\\beta $ .",
"We also use a similar idea of neglecting higher order terms by increasing an earlier term's coefficient in other cases.",
"For $f_2$ we first replace $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op}$ .",
"Since $\\widehat{\\mathcal {T}}_\\lambda (k)$ and $\\widehat{a} (k)$ commute, they are simultaneously diagonalizable in the sense of Theorem REF , and therefore their spectra are related by the function $x\\mapsto \\tfrac{x}{1-\\lambda x}$ (monotone increasing on $\\left(-\\infty ,1/\\lambda \\right)$ ).",
"Therefore we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (k)\\right\\Vert _{\\rm op} = \\sup _{x\\in \\sigma \\left(\\widehat{a} (k)\\right)}*{\\frac{x}{1-\\lambda x}}$ .",
"Now (perhaps with the aid of Figure REF ) it is easy to see that $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\vee \\mathbb {S}\\left(-\\widehat{a}(k)\\right) = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) \\vee \\mathbb {S}\\left(-\\widehat{a} (k)\\right).$ Now since $\\mathbb {S}\\left(-\\widehat{a}(k)\\right) \\le \\left\\Vert \\widehat{a}(k)\\right\\Vert _{\\rm op}\\le 1 + c_{\\kappa +1}\\beta $ , and $\\widehat{G}_{\\mu _\\lambda }(k)^{-1} \\le 1+C$ (recall Lemma REF and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } \\le C$ from REF ), we only need to bound $\\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d}\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We now partition $k\\in \\mathbb {R}^d$ into two.",
"Define the subset of the Fourier space $A_{\\frac{1}{2}} := \\left\\lbrace k\\in \\mathbb {R}^d:\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\le \\frac{1}{2}\\right\\rbrace .$ Then we only need to bound the two functions $f_4,f_5$ , where $f_4\\left(\\lambda \\right) := \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)},\\qquad \\qquad f_5\\left(\\lambda \\right):= \\operatornamewithlimits{ess\\,sup}_{k\\in A_{\\frac{1}{2}}^c} \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)}.$ We first deal with the simpler $f_4$ .",
"We consider $k\\in A_{\\frac{1}{2}}$ and choose $\\beta $ small enough that $1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right) \\ge 1 - \\frac{\\frac{1}{2}+c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta } \\ge \\frac{1}{2} - 2c_{\\kappa +1}\\beta > 0.$ Then $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\frac{1+\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}}{\\frac{1}{2} - 2c_{\\kappa +1}\\beta } \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right).",
"$ Since we have $\\mathbb {S}\\left(\\widehat{a}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{a}(0)\\right) \\le 1 + c_{\\kappa +1}\\beta $ , we then get $f_4(\\lambda ) \\le 2\\left(1+C\\right)\\left(1+5c_{\\kappa +1}\\beta \\right)\\left(1 + c_{\\kappa +1}\\beta \\right) \\le 2\\left(1+C\\right)\\left(1+7c_{\\kappa +1}\\beta \\right).$ To address $f_5$ we will have $k\\in A^c_\\rho $ .",
"We define: $\\widehat{N}(k) = \\frac{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{F}(k) = \\frac{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}, \\qquad \\widehat{Q}(k) = \\frac{1+\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(0)\\right)}.$ Note that $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) = \\widehat{N}(k) / \\widehat{F}(k)$ .",
"Rearranging gets us to $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &= \\widehat{N}(k) \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)}\\nonumber \\\\&= \\widehat{Q}(k) + \\frac{\\widehat{N}(k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)} \\left(1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - \\frac{\\widehat{Q}(k)}{\\widehat{N}(k)}\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{F}(k)\\right) .$ Since $\\mathbb {S}\\left(\\widehat{}(0)\\right)=1$ (by our scaling choice), the extracted term $\\widehat{Q}(k)$ satisfies $*{\\widehat{Q}(k)} \\le 1 + 3c_{\\kappa +1}\\beta $ .",
"We further observe that $\\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{Q}(k)}{\\widehat{N}(k)} = \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\left(1+ \\delta _\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) +\\delta _\\lambda (k)} = 1 - \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right) \\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} =: 1- \\widehat{b}(k).$ Recalling identity (REF ) for $\\mu _\\lambda $ , we can rewrite the quantity $\\left[1- \\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)- (1-\\widehat{b}(k)) \\widehat{F}(k)\\right]$ , appearing in (REF ), as $\\frac{1+\\delta _\\lambda (0) - \\left[\\lambda + \\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right] \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) - 1 + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)+\\delta _\\lambda (k)\\right) + \\widehat{b}(k)\\left(1-\\lambda \\widehat{a}(k)\\right)}{1+\\delta _\\lambda (0)} \\\\= \\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}+ \\frac{\\widehat{b}(k)(1-\\lambda \\widehat{a}(k))}{1+\\delta _\\lambda (0)}.$ Noting that ${\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)} \\le 2c_{\\kappa +1}\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\beta $ by Lemma REF and Observation REF , the first term can be bounded $&*{\\frac{ \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\left(\\delta _\\lambda (0) + \\lambda \\delta _\\lambda (0)\\right) + \\lambda \\left[\\delta _\\lambda (k)-\\delta _\\lambda (0)\\right]}{1+\\delta _\\lambda (0)}} \\nonumber \\\\&\\qquad \\le \\frac{ c_{\\kappa +1}\\beta + \\left(1 + 2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta + 2\\left(1+2c_{\\kappa +1}\\beta \\right)c_{\\kappa +1}\\beta }{1 - c_{\\kappa +1}\\beta }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 5c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\qquad \\le 6c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Using (REF ), the last term is $*{\\frac{\\widehat{b}(k)(1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right))}{1+\\delta _\\lambda (0)}} &= *{ \\frac{\\left(1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right)\\delta _\\lambda (k)}{\\left(1+\\delta _\\lambda (0)\\right)\\left(\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)\\right)}- \\frac{\\lambda \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\delta _\\lambda (k)}{1+\\delta _\\lambda (0)}} \\nonumber \\\\& = \\frac{*{\\delta _{\\lambda }(k)}}{*{1+\\delta _\\lambda (0)}}*{ \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) + \\delta _\\lambda (k)} -\\lambda }\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le \\frac{c_{\\kappa +1}\\beta }{1-c_{\\kappa +1}\\beta }\\left( \\frac{1}{\\frac{1}{2}-c_{\\kappa +1}\\beta } + \\frac{1}{1-c_{\\kappa +1}\\beta }\\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\&\\le c_{\\kappa +1}\\beta \\left(1+2c_{\\kappa +1}\\beta \\right)\\left(3+8c_{\\kappa +1}\\beta \\right)\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\nonumber \\\\& \\le 4c_{\\kappa +1}\\beta \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ Putting these bounds back into (REF ), we can find constants $M$ and $\\tilde{M}$ such that $\\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(k)} &\\le \\widehat{Q}(k) + 10c_{\\kappa +1}\\beta \\frac{1}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\frac{\\widehat{N}(k)}{\\widehat{F}(k)} \\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\ \\\\& \\le 1+ 3c_{\\kappa +1}\\beta + 20c_{\\kappa +1}\\beta \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\widehat{G}_{\\mu _\\lambda }(k)} \\\\& \\le 1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta .",
"$ Note that we have used the bound $\\left[1-\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right] \\le \\left[1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]$ (since $\\mu _\\lambda \\le 1$ and $\\mathbb {S}\\left(\\widehat{}(k)\\right)>\\frac{1}{2} >0$ ) to get from $\\widehat{G}_1(k)^{-1}$ to $\\widehat{G}_{\\mu _\\lambda }(k)^{-1}$ .",
"As $\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\in \\left(\\frac{1}{2}, 1\\right]$ , this concludes the improvement of $f_5$ and hence of $f_2$ .",
"Before we treat $f_3$ , we introduce $f_6$ given by $f_6(\\lambda ) := \\operatornamewithlimits{ess\\,sup}_{k\\in \\mathbb {R}^d} \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1- \\lambda \\mathbb {S}\\left(\\widehat{a}\\left(k\\right)\\right)}.$ We show that $f(\\lambda ) \\le \\kappa +1$ implies $f_6(\\lambda ) \\le M$ uniformly for some $\\kappa $ -independent constant $M$ .",
"For $k\\in A_\\frac{1}{2}$ we showed in (REF ) that we can bound uniformly with $3\\left(1+C\\right)$ for sufficiently large small $\\beta $ .",
"For $k \\in A_{\\frac{1}{2}}^c$ , we have $\\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right) \\widehat{G}_{\\mu _\\lambda }(k)} \\cdot \\frac{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) *{ 1- \\frac{\\delta _\\lambda (k)}{\\mathbb {S}\\left(\\widehat{a}(k)\\right)}} \\\\ \\le \\left(1+ \\left(20\\kappa + 23\\right)c_{\\kappa +1}\\beta \\right) \\left(1+\\frac{c_{\\kappa +1}\\beta }{\\frac{1}{2}- c_{\\kappa +1}\\beta } \\right) \\le 1+ \\left(20\\kappa + 27\\right)c_{\\kappa +1}\\beta .",
"$ Note that for the first bound in (REF ), we used the estimate established in (REF ), which is stronger than a bound on $f_5$ .",
"We therefore have the uniform bound $f_6(\\lambda ) \\le 3 + 3C$ for $\\beta $ sufficiently small.",
"Let us now improve the bound on $f_3$ .",
"It will be notationally convenient to denote $A\\left(k\\right) := \\left[1- \\lambda \\widehat{a}\\left(k\\right)\\right]^{-1}$ .",
"Since $\\widehat{a}\\left(k\\right)$ is self-adjoint, it can be “diagonalized\" in the sense of Theorem REF .",
"Therefore $A\\left(k\\right)$ can be “diagonalized\" over the same space and with the same unitary operator - it will have the diagonal function $\\widetilde{A}(k)(e) = 1/\\left(1-\\widetilde{a}\\left(k\\right)(e)\\right)$ .",
"It is then clear that $A\\left(k\\right)$ and $\\widehat{a}\\left(k\\right)$ commute (for each $k\\in \\mathbb {R}^d$ ).",
"Using Observation REF , we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} &= \\left\\Vert \\widehat{a}\\left(l\\right)A\\left(l\\right) - \\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op}\\nonumber \\\\&= \\left\\Vert A\\left(l\\right)\\widehat{a}\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l+k\\right)\\right]A\\left(l+k\\right) - A\\left(l\\right)\\left[1- \\lambda \\widehat{a}\\left(l\\right)\\right]\\widehat{a}\\left(l+k\\right)A\\left(l+k\\right)\\right\\Vert _{\\rm op} \\nonumber \\\\&= \\left\\Vert A(l)\\left(\\widehat{a}(l)-\\widehat{a}(l+k)\\right)A(l+k)\\right\\Vert _{\\rm op} \\nonumber \\\\&\\le \\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}\\left\\Vert A(l)\\right\\Vert _{\\rm op}\\left\\Vert A(l+k)\\right\\Vert _{\\rm op}.",
"$ For the $\\left\\Vert \\widehat{a}(l)-\\widehat{a}(l+k)\\right\\Vert _{\\rm op}$ factor, we use the triangle inequality to separate this into a term of $\\widehat{}$ and $\\widehat{}_{\\lambda }$ .",
"Lemma REF immediately gives $\\left\\Vert \\widehat{}\\left(l\\right) - \\widehat{}\\left(l+k\\right)\\right\\Vert _{\\rm op} \\le C\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ for the $C>0$ arising from REF .",
"For the $\\widehat{}_{\\lambda }$ term, a similar argument to (REF ) gives $\\left\\Vert \\widehat{}_{\\lambda }\\left(l\\right) - \\widehat{}_{\\lambda }\\left(1+k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(0\\right) - \\widehat{}_{\\lambda }\\left( k\\right)\\right\\Vert _{1,\\infty } \\le c_{\\kappa +1}\\beta \\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right],$ where the last inequality follows from Observation REF .",
"We then deal with the remaining factors of (REF ) by noting that for all $k\\in \\mathbb {R}^d$ , $\\left\\Vert A(k)\\right\\Vert _{\\rm op} = \\frac{1}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} = \\frac{1-\\mu _\\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right)}{1-\\lambda \\mathbb {S}\\left(\\widehat{a}(k)\\right)} \\widehat{G}_{\\mu _\\lambda }(k) \\le 3\\left(1+C\\right) \\widehat{G}_{\\mu _\\lambda }(k), $ employing our improved bound on $f_6$ .",
"The result of this is that we have the bound $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C + c_{\\kappa +1}\\beta \\right)\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right]\\widehat{G}_{\\mu _\\lambda }\\left(l\\right)\\widehat{G}_{\\mu _\\lambda }\\left(l+k\\right).$ Then since $\\widehat{G}_{\\mu _\\lambda }(k)\\ge 0$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le 9\\left(1+C\\right)^2\\left(C+c_{\\kappa +1}\\beta \\right)\\widehat{J}_{\\mu _\\lambda }\\left(k,l\\right).$ [Proof of Proposition REF -REF ] The continuity of $f_1$ is obvious.",
"For the other two functions we prove equicontinuity of a family of functions and use [41] to show the continuity of the desired functions.",
"The procedure is outlined here.",
"Suppose we wish to show that $H\\left(\\lambda \\right) := \\sup _{\\alpha \\in B}\\left\\Vert h_\\alpha \\left(\\lambda \\right)\\right\\Vert _{\\rm op}$ is continuous on $\\left[0,\\lambda _O\\right)$ .",
"For our purposes the parameter $\\alpha $ will be either $k$ or $\\left(k,l\\right)$ and thus $B=\\mathbb {R}^d$ or $B=\\mathbb {R}^{2d}$ .",
"The continuity on the half-open interval is implied by having continuity on the closed interval $\\left[0,\\lambda _O-\\rho \\right]$ for any $\\rho >0$ .",
"For closed intervals, [41] gives continuity if the family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous and $H\\left(\\lambda \\right)<+\\infty $ for all $\\lambda $ in the closed interval.",
"The family $\\left\\lbrace \\left\\Vert h_\\alpha \\right\\Vert _{\\rm op}\\right\\rbrace _{\\alpha \\in B}$ is equicontinuous if for all $\\varepsilon >0$ there exists $\\delta >0$ such that $|s-t|<\\delta $ implies $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\varepsilon $ uniformly in $\\alpha \\in B$ .",
"In fact, the reverse triangle inequality implies that $*{\\left\\Vert h_\\alpha (s)\\right\\Vert _{\\rm op} - \\left\\Vert h_\\alpha (t)\\right\\Vert _{\\rm op}} \\le \\left\\Vert h_\\alpha (s)-h_\\alpha (t)\\right\\Vert _{\\rm op},$ and therefore we only need to prove equicontinuity for the un-normed family $\\left\\lbrace h_\\alpha \\right\\rbrace _{\\alpha \\in B}$ .",
"We prove equicontinuity by bounding the `near-derivative' $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert h_\\alpha (\\lambda +\\varepsilon )-h_\\alpha (\\lambda )\\right\\Vert _{\\rm op}$ uniformly in $\\alpha \\in B$ for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ and arbitrary $\\rho >0$ .",
"For $f_2$ the operator-valued functions $h_\\alpha $ are $\\lambda \\mapsto \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)/\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)$ .",
"By using a variation on the chain rule, we get $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)$ is differentiable with the bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2$ .",
"We also have $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ from Lemma REF and $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ immediately from the definition.",
"It remains to deal with the $\\mu _\\lambda $ term.",
"Recall that $\\mu _\\lambda := 1 - \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^{-1}$ .",
"Therefore, using the reverse triangle inequality, $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda } = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)\\right)-\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)} \\\\\\le \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }(0)-\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} = \\frac{1}{\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right)^2}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2,$ where we have used $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\ge 1$ from Lemma REF in the last inequality.",
"Therefore $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon }\\left(k\\right)}{\\widehat{G}_{\\mu _{\\lambda +\\varepsilon }}\\left(k\\right)} - \\frac{\\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)}{\\widehat{G}_{\\mu _\\lambda }\\left(k\\right)}\\right\\Vert _{\\rm op} \\le \\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + *{\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^2\\\\\\le \\left(1 + C\\right)\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}^2 + C\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^3.$ We therefore have a finite $k$ -independent bound.",
"This has proven the equicontinuity.",
"For the uniform boundedness, we note that for $\\lambda \\in \\left[0,\\lambda _O-\\rho \\right]$ we have $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}<\\infty $ .",
"In conjunction with $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge 1/\\left(1 + \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op}\\right)$ we have the required uniform boundedness and therefore the continuity of $f_2$ .",
"We repeat this approach for $f_3$ .",
"The corresponding step to (REF ) now reads $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\left\\Vert \\frac{\\widehat{\\mathcal {T}}_{\\lambda +\\varepsilon ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda +\\varepsilon }}\\left(k,l\\right)} - \\frac{\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\right\\Vert _{\\rm op} \\le \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\\\+ \\frac{\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)^2}*{\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }}\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }.$ Recall from Corollary REF that $\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)$ is differentiable with the operator norm bound $\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le 4\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}$ .",
"The operator itself has the similar bound $\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op}\\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(k\\right)\\right\\Vert _{\\rm op}$ .",
"We would like to have the bound (REF ), but this was proven under the assumption $f\\le \\kappa +1$ , which we no longer assume.",
"Fortunately, we only require our bound for $\\lambda < \\lambda _O -\\rho $ .",
"Therefore we can use Lemma REF and Lemma REF to get $\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(l\\right) - \\widehat{\\mathcal {T}}_\\lambda \\left(l+k\\right)\\right\\Vert _{\\rm op} \\le \\text{e}^{4\\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}}\\left\\Vert \\widehat{}\\left(0\\right) - \\widehat{}\\left(k\\right)\\right\\Vert _{\\rm op} \\\\\\le C\\text{e}^{4\\left(\\lambda _O-\\rho \\right)\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda _O-\\rho }(0)\\right\\Vert _{\\rm op}}\\left[1 - \\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)\\right].$ From this and $\\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\ge \\left(1+C\\right)^{-1}$ , there exists a constant $\\widetilde{M}>0$ such that both $\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\frac{\\mathrm {d}}{\\mathrm {d}\\lambda }\\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M} \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda \\left(0\\right)\\right\\Vert _{\\rm op}, \\qquad \\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left\\Vert \\widehat{\\mathcal {T}}_{\\lambda ,k}\\left(l\\right)\\right\\Vert _{\\rm op} \\le \\widetilde{M},$ for sufficiently large $d$ .",
"Now recall $\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{G}_\\mu \\left(k\\right)\\right|_{\\mu =\\mu _\\lambda } = \\widehat{G}_{\\mu _\\lambda }\\left(k\\right)^2\\mathbb {S}\\left(\\widehat{}\\left(k\\right)\\right)$ and $\\left(1+C\\right)^{-1}\\le \\widehat{G}_{\\mu _\\lambda }\\left(k\\right) \\le \\left(1-\\mu _\\lambda \\right)^{-1} = \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (0)\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}$ .",
"Therefore an application of the chain rule gives the bound $*{\\frac{1}{\\widehat{J}_{\\mu _{\\lambda }}\\left(k,l\\right)}\\left.\\frac{\\mathrm {d}}{\\mathrm {d}\\mu }\\widehat{J}_{\\mu }\\left(k,l\\right)\\right|_{\\mu =\\mu _\\lambda }} \\le 6\\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op}^4\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\rm op},$ uniformly in $k$ and $l$ .",
"These properties, along with the bound on $\\limsup _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }*{\\mu _{\\lambda +\\varepsilon } - \\mu _\\lambda }$ from above, are sufficient to prove equicontinuity and the uniform boundedness for the required $\\lambda $ .",
"Therefore we have proved that $f_3\\left(\\lambda \\right)$ is continuous.",
"It just remains to prove that we can indeed bound the difference of $\\delta _\\lambda (k)$ with the difference of $\\widehat{}_{\\lambda }(k)$ as we claimed.",
"[Proof of Lemma REF ] First note that there is nothing to prove if $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}=\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}=0$ , because the triangle inequality then forces $\\delta _\\lambda (k_1) = \\delta _\\lambda (k_2)=0$ .",
"Without loss of generality, assume $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ with $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}>0$ .",
"Then $\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2) = \\frac{\\delta _\\lambda (k_1)}{\\left\\Vert \\widehat{}_{\\lambda }(k_1)\\right\\Vert _{\\rm op}}\\left(\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}\\right) + \\left[\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)\\right]$ The second factor is the more troublesome: $*{\\delta _\\lambda (k_1)\\frac{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} - \\delta _\\lambda (k_2)} = \\frac{1}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\delta _\\lambda (k_1)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op} - \\delta _\\lambda (k_2)\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}} \\\\\\le \\frac{*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)}}{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}}*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}}.$ Since $\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} \\ge \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}$ , we have $*{\\delta _\\lambda (k_1)} \\vee *{\\delta _\\lambda (k_2)} \\le \\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op}$ .",
"Therefore $*{\\delta _\\lambda (k_1)-\\delta _\\lambda (k_2)} \\le 2*{\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op} - \\left\\Vert \\widehat{}_{\\lambda }\\left(k_2\\right)\\right\\Vert _{\\rm op}} \\le 2\\left\\Vert \\widehat{}_{\\lambda }\\left(k_1\\right)-\\widehat{}_{\\lambda }\\left(k_1\\right)\\right\\Vert _{\\rm op},$ where the last inequality holds because of the reverse triangle inequality."
],
[
"Proof of Main Theorems",
"Proposition 7.1 Let $d>6$ be sufficiently large.",
"Then for $\\lambda \\le \\lambda _O$ there exists $c>0$ such that $\\triangle _\\lambda < c\\beta ^2$ .",
"Proposition REF gives us that there exists $c_f$ (increasing in $f$ ) such that $\\triangle _\\lambda \\le c_f\\beta ^2$ for all $\\lambda <\\lambda _O$ .",
"Proposition REF then implies that there exists $c$ such that $c_f\\le c$ uniformly for all $\\lambda < \\lambda _O$ .",
"It remains to prove the assertion for $\\lambda =\\lambda _O$ .",
"Recall from Section REF the definition of $\\tau _\\lambda ^n(x,y)$ and $\\Lambda _n(x)$ .",
"Furthermore, recall that if $x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}$ then such a connection is achieved in finitely many steps and the bound in $\\ref {Assump:2ndMoment}$ then implies that $\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}\\right\\rbrace = \\bigcup ^\\infty _{n=1}\\left\\lbrace x \\longleftrightarrow y\\textrm { in } \\xi ^{x,y}_{\\Lambda _n(y)}\\right\\rbrace .$ Therefore by monotone convergence we have the pointwise limit $\\tau _\\lambda ^n(x,y) \\rightarrow \\tau _\\lambda (x,y)$ as $n\\rightarrow \\infty $ for all $\\lambda >0$ and $x,y\\in \\mathbb {X}$ .",
"Since $\\tau _\\lambda ^n(x,y)$ only depends upon the finite region $\\Lambda _n(y)$ , the functions $\\lambda \\mapsto \\tau _\\lambda ^n(x,y)$ are continuous for all $x,y\\in \\mathbb {X}$ .",
"This, with the monotonicity $\\tau _\\lambda ^n(x,y) \\le \\tau _\\lambda ^{n+1}(x,y)$ , implies that the function $\\lambda \\mapsto \\tau _\\lambda (x,y)$ is lower-semicontinuous.",
"Since $\\tau _\\lambda (x,y)$ is non-decreasing in $\\lambda $ , this lower semi-continuity implies that the pointwise limit $\\tau _\\lambda (x,y)\\uparrow \\tau _{\\lambda _O}(x,y)$ as $\\lambda \\uparrow \\lambda _O$ holds, and the limit holds monotonically.",
"This monotonic pointwise convergence implies that the integral $\\int \\tau _\\lambda (x,u)\\tau _\\lambda (u,v)\\tau _\\lambda (v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right) \\rightarrow \\int \\tau _{\\lambda _O}(x,u)\\tau _{\\lambda _O}(u,v)\\tau _{\\lambda _O}(v,y)\\nu ^{\\otimes 2}\\left(\\mathrm {d}u, \\mathrm {d}v\\right)$ for all $x,y\\in \\mathbb {X}$ as $\\lambda \\uparrow \\lambda _O$ , and that this convergence is monotone increasing.",
"If we take the supremum over $x,y\\in \\mathbb {X}$ , we get that $\\lim _{\\lambda \\uparrow \\lambda _O}\\left\\Vert \\mathcal {T}_\\lambda ^3\\right\\Vert _{\\infty ,\\infty } = \\left\\Vert \\mathcal {T}_{\\lambda _O}^3\\right\\Vert _{\\infty ,\\infty }$ .",
"The uniform bound for $\\lambda <\\lambda _O$ then implies that this bound also holds at $\\lambda =\\lambda _O$ .",
"The following proposition uses the bound $\\triangle _{\\lambda _O} < c\\beta ^2$ to show that percolation does not occur at criticality.",
"Another element of the proof is that there is almost surely at most one infinite cluster.",
"There are general considerations that show that this is the case.",
"Indeed [14] establishes that on the discrete space $\\mathbb {Z}^d$ there is at most one infinite cluster if the edge occupation measure is stationary and obeys the ‘finite energy property’, and an analogous result for Poisson processes in the continuum applies in our case (see [4], [36]).",
"Proposition 7.2 The critical percolation function $\\theta _{\\lambda _O}(a)=0$ for $\\mathcal {P}$ -almost every $a\\in \\mathcal {E}$ .",
"Assume that $(\\overline{0},a),(\\overline{x},b)\\in \\mathbb {X}$ are in the (a.s. unique) infinite cluster, and are therefore connected.",
"Then the FKG inequality implies that $\\theta _{\\lambda _O}(a)\\theta _{\\lambda _O}(b)\\le \\mathbb {P}_{\\lambda _O}\\big (|\\overline{0},a)|=\\infty ,|\\overline{x},b)|=\\infty \\big )\\le \\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x},b)).$ Since $\\triangle _{\\lambda _O} \\le c\\beta ^2$ , for $\\mathcal {P}$ -a.e.",
"$a,b\\in \\mathcal {E}$ there exists a sequence $\\overline{x}^{(a,b)}_n$ such that $\\tau _{\\lambda _O}((\\overline{0},a),(\\overline{x}^{(a,b)}_n,b))\\rightarrow 0$ .",
"Therefore for $\\mathcal {P}$ -a.e.",
"$a\\in \\mathcal {E}$ we have $\\theta _{\\lambda _O}(a)=0$ .",
"We can define the linear operator $\\mathcal {T}_{\\lambda _O}$ by its action on $f\\in L^2(\\mathbb {X})$ : $\\mathcal {T}_{\\lambda _O}f(x) = \\int \\tau _{\\lambda _O}(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right).$ Contrary to $\\mathcal {T}_\\lambda $ for $\\lambda <\\lambda _O$ , the operator $\\mathcal {T}_{\\lambda _O}$ may be an unbounded linear operator.",
"Proposition 7.3 (The operator OZE at the critical point) Let $d>6$ be sufficiently large.",
"The Ornstein-Zernike equation then extends to $\\lambda _O$ in the sense that the sum of (unbounded) linear operators vanishes: $\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) = 0.$ Furthermore, for $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ the linear operator $\\widehat{\\mathcal {T}}_{\\lambda _O}(k)$ is a bounded linear operator and the following equality holds: $\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right) - \\widehat{}\\left(k\\right) - \\widehat{}_{\\lambda _O}\\left(k\\right) - \\lambda _O\\widehat{\\mathcal {T}}_{\\lambda _O}\\left(k\\right)\\left(\\widehat{}\\left(k\\right) + \\widehat{}_{\\lambda _O}\\left(k\\right)\\right) = 0.$ Finally, there exists $c>0$ such that for all $k\\in \\mathbb {R}^d$ $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le c\\beta , \\qquad \\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}\\le c\\beta .$ Let $f\\in L^2(\\mathbb {X})$ .",
"Then by the triangle inequality, for $\\lambda <\\lambda _O$ ${\\left(\\mathcal {T}_{\\lambda _O}- - _{\\lambda _O}- \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right)\\right)f}_2 \\le {\\left(\\mathcal {T}_\\lambda - - _\\lambda - \\lambda \\mathcal {T}_\\lambda \\left(+ _\\lambda \\right)\\right)f}_2 \\\\+ {\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda - _{\\lambda _O}+ _\\lambda - \\lambda _O\\mathcal {T}_{\\lambda _O}\\left(+ _{\\lambda _O}\\right) + \\lambda \\mathcal {T}_\\lambda \\left(+_\\lambda \\right)\\right)f}_2.$ The first norm on the right hand side vanishes, since the OZE holds for $\\lambda <\\lambda _O$ .",
"To prove that the left hand side vanishes, we aim to show that the second norm on the right hand side vanishes as $\\lambda \\uparrow \\lambda _O$ .",
"To do this, the triangle inequality implies that we only need to show that the following limits hold as $\\lambda \\uparrow \\lambda _O$ : ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2 &\\rightarrow 0, \\\\{\\left(_{\\lambda _O}- _\\lambda \\right)f}_2 &\\rightarrow 0,\\\\{\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2 &\\rightarrow 0.$ We begin with (REF ).",
"Recall from the proof of Proposition REF that $\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y) := \\tau _{\\lambda _O}(x,y) - \\tau _\\lambda (x,y)$ converges monotonically and pointwise to zero.",
"From the definition of the ${\\cdot }_2$ norm, ${\\left(\\mathcal {T}_{\\lambda _O}- \\mathcal {T}_\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right)\\\\&\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes (3)}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right).$ The monotone convergence of $\\left(\\tau _{\\lambda _O}-\\tau _\\lambda \\right)(x,y)\\downarrow 0$ then implies that the limit (REF ) holds.",
"For (), we use the functions $h^{(n)}_\\lambda \\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ defined by $h^{(n)}_\\lambda (x,y) := {\\left\\lbrace \\begin{array}{ll}\\lambda ^{-1}\\int \\psi _n(x,w_{n-1},u_{n-1}) \\left( \\prod _{i=1}^{n-1} \\psi (\\vec{v}_i) \\right) \\psi _0(w_0,u_0,y) \\nu ^{\\otimes (2n)}\\left(\\mathrm {d}\\left( \\left(\\vec{w}, \\vec{u}\\right)_{[0,n-1]} \\right)\\right) &:n\\ge 1\\\\\\tfrac{1}{2} \\lambda ^2 \\left(\\int \\tau _\\lambda \\left(x,w\\right)\\varphi \\left(w,y\\right)\\nu \\left(\\mathrm {d}w\\right)\\right)^2 &: n=0.\\end{array}\\right.",
"}$ From Proposition REF and the proof of Proposition REF , we know that $0\\le \\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_\\lambda (x,y)$ for all $n\\ge 0$ , $x,y\\in \\mathbb {X}$ , and $\\lambda \\in \\left(0,\\lambda _O\\right)$ - the non-negativity follows from the definition of $\\pi ^{(n)}_\\lambda $ .",
"The functions $ h^{(n)}_\\lambda $ can also be defined for $\\lambda =\\lambda _O$ , and since they are monotone increasing in $\\lambda $ , we have the $\\lambda $ -independent bound $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda <\\lambda _O$ and $x,y\\in \\mathbb {X}$ .",
"For the single-mark version of the model it is proven in [21] that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.",
"This same argument works for our multi-mark version, and therefore $\\pi ^{(n)}_\\lambda (x,y) \\le h^{(n)}_{\\lambda _O}(x,y)$ for all $\\lambda \\le \\lambda _O$ .",
"The triangle inequality then implies that $*{\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)} = *{\\sum ^\\infty _{n=0}\\left(-1\\right)^n\\left( \\pi ^{(n)}_{\\lambda _O}(x,y) - \\pi ^{(n)}_{\\lambda }(x,y)\\right)} \\le 2\\sum ^\\infty _{n=0}h^{(n)}_{\\lambda _O}(x,y).$ It will be convenient to define $h_{\\lambda }\\colon \\mathbb {X}^2\\rightarrow \\mathbb {R}_+$ as $h_\\lambda (x,y) := \\sum ^\\infty _{n=0}h^{(n)}_{\\lambda }(x,y)$ .",
"Then ${\\left(_{\\lambda _O}- _\\lambda \\right)f}_2^2 &= \\int \\left(\\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)f(y)\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right) \\nonumber \\\\&= \\int \\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,y)\\left(\\pi _{\\lambda _O} - \\pi _{\\lambda }\\right)(x,z)f(y)f(z)\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right) \\\\& \\le 4\\int h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}\\nu ^{\\otimes 3}\\left(\\mathrm {d}x,\\mathrm {d}y, \\mathrm {d}z\\right)\\nonumber \\\\& = 4\\int \\left(\\int h_{\\lambda _O}(x,y)*{f(y)}\\nu \\left(\\mathrm {d}y\\right)\\right)^2\\nu \\left(\\mathrm {d}x\\right).",
"$ By Schur's test, this integral is finite for all $f\\in L^2(\\mathbb {X})$ if both the values $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right) \\qquad \\text{ and } \\qquad \\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ are finite.",
"To prove this for the former, we repeat the arguments of Proposition REF and Section REF to bound it in terms of $\\lambda _O$ , the triangle diagrams, and other $\\lambda $ -independent terms.",
"The argument of Proposition REF with the uniform bounds of Observation REF then proves that $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ is indeed finite.",
"A similar argument also holds for $\\operatornamewithlimits{ess\\,sup}_{x}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}y\\right)$ .",
"We now have an integrable function $h_{\\lambda _O}(x,y)h_{\\lambda _O}(x,z)*{f(y)}*{f(z)}$ that dominates the integrand of (REF ).",
"Recall that $\\pi _{\\lambda _O}-\\pi _{\\lambda }$ converges pointwise to zero.",
"Thus the dominated convergence theorem implies that the limit () holds.",
"For (), we can use $h_{\\lambda _O}$ to get the bound ${\\left(\\mathcal {T}_{\\lambda _O}_{\\lambda _O}- \\mathcal {T}_\\lambda _\\lambda \\right)f}_2^2 \\\\\\le \\int \\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,u)h_{\\lambda _O}(u,y)\\left(\\tau _{\\lambda _O} - \\tau _\\lambda \\right)(x,v)h_{\\lambda _O}(v,z)*{f(y)}*{f(z)} \\nu ^{\\otimes 5}\\left(\\mathrm {d}u,\\mathrm {d}v, \\mathrm {d}x, \\mathrm {d}y, \\mathrm {d}z\\right).$ The integrand converges monotonically to zero, and therefore the integral vanishes in the limit.",
"We now consider the Fourier version of the OZE.",
"We use the notation that, given a bounded linear operator $A\\colon L^2(\\mathbb {X}) \\rightarrow L^2(\\mathbb {X})$ , the Fourier transform is given by $\\mathcal {F}\\left( A \\right)(k)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ .",
"By hypothesis, $$ is a bounded operator, and the finiteness of $\\operatornamewithlimits{ess\\,sup}_{y}\\int h_{\\lambda _O}(x,y) \\nu \\left(\\mathrm {d}x\\right)$ implies that $_{\\lambda _O}$ is also bounded.",
"Then (REF ) implies that $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) = \\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)$ for all $k\\in \\mathbb {R}^d$ .",
"We now make the claim that $\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)$ has a bounded linear inverse for all $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ .",
"This is proven in Lemma REF below.",
"This claim then implies that for $k\\ne 0$ we have $&\\mathcal {F}\\left( + _{\\lambda _O} \\right)(k)\\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right) \\right)(k) \\mathcal {F}\\left( 1- \\lambda _O\\left(+ _{\\lambda _O}\\right) \\right)(k)^{-1} \\nonumber \\\\& \\hspace{85.35826pt} = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O}\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)\\left(1- \\lambda _O\\left(+ _{\\lambda _O}\\right)\\right)^{-1} \\right)(k) = \\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k).$ That is, $\\mathcal {F}\\left( \\mathcal {T}_{\\lambda _O} \\right)(k)$ is a bounded linear operator for $k\\ne 0$ .",
"The limit () also allows us to extend our sub-critical bound on $_\\lambda $ to apply at criticality.",
"The limit () may be phrased as saying that the function $\\lambda \\mapsto *{_\\lambda f}_2$ is continuous at $\\lambda =\\lambda _O$ (from the left) for all $f\\in L^2\\left(\\mathbb {X}\\right)$ .",
"Since $\\lambda \\mapsto \\left\\Vert _\\lambda \\right\\Vert _{\\rm op}$ can be written as a supremum of such functions, it is a lower semi-continuous function.",
"Our bound $\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta $ for $\\lambda < \\lambda _O$ (from Proposition REF ) then implies $\\left\\Vert _{\\lambda _O}\\right\\Vert _{\\rm op} \\le \\liminf _{\\lambda \\uparrow \\lambda _O}\\left\\Vert _\\lambda \\right\\Vert _{\\rm op} \\le c\\beta .$ The limit $*{\\left(\\widehat{}_{\\lambda _O}^{(n)}(0) - \\widehat{}_{\\lambda }^{(n)}(0)\\right)f}_2 \\rightarrow 0$ for each $f\\in L^2\\left(\\mathcal {E}\\right)$ can be derived in essentially the same way as () above.",
"The above lower semi-continuity argument can then be used to show that the bound from Proposition REF extends to also hold at $\\lambda =\\lambda _O$ .",
"From Lemma REF and the positivity of $\\pi ^{(n)}_{\\lambda _O}(x,y)$ this bound also holds for all $k\\in \\mathbb {R}^d$ .",
"By summing over $n$ , we then get the desired bound on $\\left\\Vert \\widehat{}_{\\lambda _O}(k)\\right\\Vert _{\\rm op}$ .",
"Lemma 7.4 For $\\lambda = \\lambda _O$ and $k\\in \\mathbb {R}^d\\setminus \\left\\lbrace 0\\right\\rbrace $ , the bounded linear operator $\\left(1- \\lambda _O\\left(\\widehat{}(k) + \\widehat{}_{\\lambda _O}(k)\\right)\\right)\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ has a bounded linear inverse.",
"We first introduce some notation.",
"Let $A\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ be a self-adjoint linear operator.",
"Then we define the minimum modulus to be $\\left\\Vert A\\right\\Vert _{\\rm min} := \\inf _{0\\ne f\\in L^2(\\mathcal {E})}\\frac{*{Af}_2}{{f}_2}.$ Note that it follows from the Spectral Theorem (Theorem REF ) that $A$ has a bounded linear inverse if and only if $\\left\\Vert A\\right\\Vert _{\\rm min}>0$ .",
"Now suppose that $A=A_\\lambda $ , and that for all $f\\in L^2(\\mathcal {E})$ the map $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .",
"Then the map $\\lambda \\mapsto \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}$ is upper semicontinuous on $\\left[0,\\lambda _O\\right]$ .",
"That is, for any sequence $\\lambda _n$ in $\\left[0,\\lambda _O\\right]$ such that $\\lambda _n\\rightarrow \\lambda $ , $\\limsup _{n\\rightarrow \\infty }\\left\\Vert A_{\\lambda _n}\\right\\Vert _{\\rm min} \\le \\left\\Vert A_\\lambda \\right\\Vert _{\\rm min}.$ We now consider $A_\\lambda = 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)$ .",
"A similar argument to the way we proved () shows that $\\lambda \\mapsto *{A_\\lambda f}$ is continuous on $\\left[0,\\lambda _O\\right]$ .",
"Our result is then proven if we can show that for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ we have an $\\varepsilon = \\varepsilon (k)>0$ such that $\\left\\Vert 1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right\\Vert _{\\rm min} \\ge \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"We will in fact show the stronger result that $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"Recalling the definition of $\\delta _\\lambda (k)$ from (REF ), we have $1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) &= 1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\lambda \\delta _{\\lambda }(k) \\nonumber \\\\&= \\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{>0 \\text{ from Lemma~\\ref {lem:Optlam-a}}} + \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) + \\lambda \\left(\\delta _{\\lambda }(0) - \\delta _{\\lambda }(k)\\right) \\nonumber \\\\& > \\lambda \\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1+\\mathcal {O}\\left(\\beta \\right)\\right).$ Here we have used Lemma REF to bound the difference of $\\delta $ with the difference of $\\widehat{}_{\\lambda _O}$ .",
"For sufficiently large $d$ , REF then proves $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) \\le 1 - \\varepsilon $ uniformly in $\\lambda \\in \\left(0,\\lambda _O\\right)$ .",
"Note that in conjunction, Proposition REF , Proposition REF and Proposition REF prove the OZE equations of Theorem REF .",
"In particular, the former two prove that the OZE holds in the subcritical regime, and the latter extends this to criticality where possible.",
"Furthermore, Proposition REF gives the bounds on $_\\lambda $ and $\\widehat{}_{\\lambda }(k)$ subcritically whilst Proposition REF extends this bound to criticality.",
"Regarding Theorem REF , the bound on $\\triangle _\\lambda $ and the lack of percolation at criticality were proven at the beginning of this section.",
"The infrared bound component of the theorem is proven below.",
"[Proof of Theorem REF ] First note that there is nothing to prove for $k=0$ .",
"From Proposition REF and Lemma REF , we can write $\\widehat{\\mathcal {T}}_\\lambda (k) = \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\left(1- \\lambda \\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)\\right)^{-1},$ for all $k\\in \\mathbb {R}^d$ if $\\lambda <\\lambda _O$ , and for all $k\\in \\mathbb {R}^d\\setminus \\lbrace 0\\rbrace $ if $\\lambda = \\lambda _O$ .",
"If we use the Spectral Theorem (Theorem REF ) to `diagonalize' the operator $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , we get a multiplication operator on some Hilbert space $L^2(\\mathfrak {E}_k,\\mu _k)$ taking values $\\widetilde{a}(k)(e)$ for $e\\in \\mathfrak {E}_k$ .",
"Since, $\\widehat{\\mathcal {T}}_\\lambda (k)$ is formed from $\\widehat{a}(k) =\\widehat{}(k) + \\widehat{}_{\\lambda }(k)$ , it can also be diagonalized over $L^2(\\mathfrak {E}_k,\\mu _k)$ , and takes values $\\widetilde{\\tau }_\\lambda (k)(e) = \\frac{\\widetilde{a}(k)(e)}{1 - \\lambda \\widetilde{a}(k)(e)}.$ Since $\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right) < 1$ (from the proof of Lemma REF ), we know $\\lambda \\widetilde{a}(k)(e) >1$ and $\\lambda \\widetilde{\\tau }_\\lambda (k)(e) \\ge -1$ for $\\mu _k$ -almost every $e\\in \\mathfrak {E}_k$ .",
"It therefore only remains to bound $\\widetilde{\\tau }_\\lambda (k)(e)$ from above.",
"Equivalently, we only need to bound $\\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right)$ from above.",
"We also note that the proof of () can be adapted to show that $\\lambda \\mapsto \\left<f,\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) f\\right>$ is continuous on $\\left[0,\\lambda _O\\right]$ for all $f\\in L^2(\\mathcal {E})$ .",
"Then because $\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) = \\sup _{f}\\tfrac{\\left<f,\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) f\\right>}{\\left<f,f\\right>}$ , this proves that $\\lambda \\mapsto \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)$ is lower-semicontinuous on $\\left[0,\\lambda _O\\right]$ .",
"The bound $\\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) <1$ for $\\lambda < \\lambda _O$ (from Lemma REF ) then implies that $\\lambda _O\\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda _O}(0)\\right)\\le 1$ .",
"From (REF ) the calculation proceeds: $\\lambda \\mathbb {S}\\left(\\widehat{\\mathcal {T}}_\\lambda (k)\\right) &= \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& = \\frac{\\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\underbrace{1 - \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right)}_{\\ge 0} + \\lambda \\mathbb {S}\\left(\\widehat{}(0) + \\widehat{}_{\\lambda }(0)\\right) - \\lambda \\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k) + \\widehat{}_{\\lambda }(k)\\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) + \\delta _{\\lambda }(0) - \\mathbb {S}\\left(\\widehat{}(k)\\right) - \\delta _{\\lambda }(k)} \\nonumber \\\\& \\le \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\left(\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\left(1 + \\mathcal {O}\\left(\\beta \\right)\\right)} = \\frac{\\mathbb {S}\\left(\\widehat{}(k)\\right) + \\mathcal {O}\\left(\\beta \\right)}{\\mathbb {S}\\left(\\widehat{}(0)\\right) - \\mathbb {S}\\left(\\widehat{}(k)\\right)},$ where we have bounded $\\delta _{\\lambda }(0)-\\delta _{\\lambda }(k)$ using Lemma REF and Observation REF ."
],
[
"Critical Intensities",
"[Proof of Proposition REF ] By the Mecke equation (REF ), we find $*{\\chi _\\lambda }_1 & = 1 + \\lambda \\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) = 1 + \\lambda \\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right),\\\\*{\\chi _\\lambda }_\\infty & = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\tau _\\lambda (\\overline{x};a,b)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\operatornamewithlimits{ess\\,sup}_{b}\\int \\widehat{\\tau }_\\lambda (0;a,b) \\mathcal {P}\\left(\\mathrm {d}a\\right) = 1 + \\lambda \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }.$ From Lemma REF we have $\\int \\widehat{\\tau }_\\lambda (0;a,b)\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a,\\mathrm {d}b\\right) \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{\\rm op} \\le \\left\\Vert \\widehat{\\mathcal {T}}_\\lambda (0)\\right\\Vert _{1,\\infty }$ .",
"Therefore $\\lambda ^{(1)}_T \\ge \\lambda _O \\ge \\lambda _T^{(\\infty )}$ .",
"Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(1)}_T$ .",
"Therefore we only now need to prove that $\\lambda ^{(1)}_T = \\lambda ^{(\\infty )}_T$ .",
"We do this by proving that if $*{\\chi _\\lambda }_1<\\infty $ then $*{\\chi _\\lambda }_\\infty <\\infty $ .",
"Fix $b\\in \\mathcal {E}$ .",
"Then by considering the vertices in the cluster $\\overline{0},b)$ adjacent to $\\left(\\overline{0},b\\right)$ , we find $\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\mathbb {E}_\\lambda \\left[\\sum _{\\left(\\overline{x},a\\right)\\in \\eta :\\left(\\overline{x},a\\right)\\sim \\left(\\overline{0},b\\right)}*{\\overline{x},a)}\\right] = 1 + \\lambda \\int \\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right]\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right).$ In this equality we have used Mecke's equation (REF ).",
"From the spatial translation invariance of the model, $\\mathbb {E}_\\lambda \\left[*{\\overline{x},a)}\\right] = \\mathbb {E}_\\lambda \\left[*{\\overline{0},a)}\\right]$ for all $\\overline{x}\\in \\mathbb {R}^d$ .",
"Therefore an application of a supremum bound to the $a$ -integral gives $*{\\chi _\\lambda }_\\infty = \\operatornamewithlimits{ess\\,sup}_{b\\in \\mathcal {E}}\\mathbb {E}_\\lambda \\left[*{\\overline{0},b)}\\right] \\le 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)\\int \\mathbb {E}_\\lambda \\left[*{\\overline{0},c)}\\right]\\mathcal {P}\\left(\\mathrm {d}c\\right)\\\\= 1 + \\lambda \\left(\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\right)*{\\chi _\\lambda }_1.$ From the finiteness of the parenthesised factor, $*{\\chi _\\lambda }_1<\\infty $ implies $*{\\chi _\\lambda }_\\infty <\\infty $ and the result is proven.",
"[Proof of Proposition REF ] The equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ holds from [35].",
"The equality $\\lambda _O = \\lambda _T^{(p)}$ follows from Proposition REF because the condition (REF ) clearly holds for the bounded radii model.",
"Also recall from the discussion in Section REF that $\\lambda _c \\ge \\lambda ^{(p)}_T$ for all $p\\in \\left[1,\\infty \\right]$ .",
"Now note that for all $a,b\\in \\mathcal {E}$ we have that if $\\left(\\overline{0},\\dagger \\right)\\sim \\left(\\overline{x},b\\right)$ then $\\left(\\overline{0},a\\right)\\sim \\left(\\overline{x},b\\right)$ .",
"Therefore $\\lambda _c\\left(\\dagger \\right) \\ge \\lambda _c$ , and the equality $\\lambda _c\\left(\\dagger \\right)= \\lambda _T\\left(\\dagger \\right)$ then means we only need to prove $\\lambda _T\\left(\\dagger \\right) \\le \\lambda _T^{(p)}$ for some $p\\in \\left[1,\\infty \\right]$ .",
"We will do so for $p=1$ .",
"Suppose that $\\lambda >\\lambda _T^{(1)}$ , so that $\\int \\mathbb {E}_\\lambda \\left[(\\overline{0};a\\right)\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\infty $ .",
"That is, the expected cluster size of a vertex with a random ($\\mathcal {P}$ -distributed) mark is infinite.",
"Note that we have an ordering of the marks in that if two radii $a,b$ satisfy $a\\ge b$ , then $\\varphi \\left(\\overline{x};a,c\\right) \\ge \\varphi \\left(\\overline{x};b,c\\right)$ for all $\\overline{x}\\in \\mathbb {R}^d$ and $c\\in \\mathcal {E}$ .",
"This also implies that $a\\mapsto \\mathbb {E}_\\lambda \\left[*{(\\overline{0},a}\\right]$ is a non-decreasing function.",
"This in turn implies that if $\\lambda >\\lambda _T^{(1)}$ and $a$ satisfies $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , then $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right) = \\infty $ .",
"Given some radius $a\\in \\mathcal {E}$ such that $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we define a new mark, $\\star _a$ , that forms connections according to $\\varphi \\left(\\overline{x};\\star _a,b\\right) = 1\\left\\lbrace *{\\overline{x}}\\le a\\right\\rbrace ,$ for all $b\\in \\mathcal {E}$ .",
"Note that if $b\\ge a$ then $\\varphi \\left(\\overline{x};\\star _a,c\\right) \\le \\varphi \\left(\\overline{x};b,c\\right)$ for all $c\\in \\mathcal {E}$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\le \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]$ .",
"We now consider the expected cluster size of vertices adjacent to $\\left(\\overline{0},\\star _a\\right)$ with radius $\\ge a$ .",
"We also restrict to cases for which there is an unique neighbour, and that neighbour has radius $\\ge a$ .",
"Since the probability of a connection is independent of the radius of the proposed neighbour, the distribution of the unique neighbour is equal to $\\mathcal {P}$ - conditioned upon having radius $\\ge a$ .",
"Therefore we have the lower bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right] \\ge 1 + \\mathbb {P}_\\lambda \\left(\\exists !",
"\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right).$ As noted above, $\\int \\mathbb {E}_\\lambda \\left[*{(\\overline{0};b}\\right]1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)=\\infty $ .",
"Furthermore, as $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , the random variable $\\#\\left\\lbrace \\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right\\rbrace $ obeys a Poisson distribution with strictly positive mean.",
"Therefore $\\mathbb {P}_\\lambda \\left(\\exists !",
"\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\star _a\\right),b\\ge a\\right)>0$ and $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};\\star _a}\\right]=\\infty $ .",
"This in turn implies that $\\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right] =\\infty $ for all $a>0$ .",
"To relate this to $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]$ , we perform a similar lower bound.",
"Given a radius $a>0$ and $\\int 1\\left\\lbrace b\\ge a\\right\\rbrace \\mathcal {P}\\left(\\mathrm {d}b\\right)>0$ , we restrict to cases where there is an unique neighbour of $\\left(\\overline{0},\\dagger \\right)$ with radius $\\ge a$ and bound $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right] \\ge 1+ \\mathbb {P}_\\lambda \\left(\\exists !",
"\\left(\\overline{x},b\\right)\\in \\eta :\\left(\\overline{x},b\\right)\\sim \\left(\\overline{0},\\dagger \\right),b\\ge a\\right) \\mathbb {E}_\\lambda \\left[*{(\\overline{0};a}\\right].$ As before, the probability factor is strictly positive and the expected cluster size term is infinite.",
"Hence $\\mathbb {E}_\\lambda \\left[*{(\\overline{0},\\dagger }\\right]=\\infty $ and $\\lambda \\ge \\lambda \\left(\\dagger \\right)$ .",
"Therefore $\\lambda _T\\left(\\dagger \\right)\\le \\lambda ^{(1)}_T$ as required."
],
[
"Model Properties",
"Here we will prove that the models outlined in Section REF do indeed satisfy Assumption REF .",
"First we consider the single mark “finite variance models” considered by [21].",
"As proven in that reference, this includes the Poisson blob and Gaussian connection models.",
"Lemma B.1 Single mark finite variance models satisfy the conditions of Assumption REF .",
"First note that since $\\mathcal {E}$ is a singleton, the operators $\\widehat{}(k)$ are simply scalars.",
"In particular, this means that the conditions (REF ), and () of REF hold trivially.",
"The finiteness of $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is also equivalent to the finiteness of $\\int \\varphi (x)\\mathrm {d}x$ required by [21].",
"Since $\\widehat{}(k)$ is simply a scalar, assumption REF follows directly from the third finite variance condition of [21].",
"The condition REF is similarly a generalisation of the second finite variance condition, and therefore follows.",
"Specifically, the sets $B(x)$ are the $\\varepsilon $ -balls around $x\\in \\mathbb {R}^d$ where $0<\\varepsilon <r_d$ where $r_d$ is the radius of the ball of volume 1.",
"Lemma B.2 Double mark finite variance models satisfy the conditions of Assumption REF .",
"Since $\\mathcal {E}= \\lbrace a_1,a_2\\rbrace $ , let us denote the functions $\\varphi _{i,j}(\\overline{x}) = \\varphi (\\overline{x};a_i,a_j)$ and their Fourier transforms $\\widehat{\\varphi }_{i,j}(k) = \\widehat{\\varphi }(k;a_i,a_j)$ for $i,j\\in \\lbrace 1,2\\rbrace $ .",
"Since $\\widehat{}(k)$ is on operator on a finite dimensional vector space, it is clear that REF will hold (note that each of the entries will be finite).",
"We now represent $\\widehat{}(k)$ with a matrix.",
"Given $f\\in L^2(\\mathcal {E})$ - taking values $f_1,f_2$ - we have $\\left<f,\\widehat{}(k)f\\right> = p^2\\widehat{\\varphi }_{1,1}(k)f_1^2 + 2pq\\widehat{\\varphi }_{1,2}(k)f_1f_2 + q^2\\widehat{\\varphi }_{2,2}(k)f_2^2,$ and therefore the spectrum of $\\widehat{}(k)$ is equal to the set of eigenvalues of the matrix $\\begin{pmatrix}p^2\\widehat{\\varphi }_{1,1}(k) & pq\\widehat{\\varphi }_{1,2}(k)\\\\pq\\widehat{\\varphi }_{1,2}(k) & q^2\\widehat{\\varphi }_{2,2}(k)\\end{pmatrix}.$ This $2\\times 2$ matrix has characteristic polynomial given by $z \\mapsto z^2 - \\left(p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)\\right) z + p^2q^2\\left( \\widehat{\\varphi }_{1,1}(k)\\widehat{\\varphi }_{2,2}(k) - \\widehat{\\varphi }_{1,2}(k)^2\\right).$ Solving for roots of this quadratic then gives $\\mathbb {S}\\left(\\widehat{}(k)\\right) = \\frac{p^2\\widehat{\\varphi }_{1,1}(k) + q^2\\widehat{\\varphi }_{2,2}(k)}{2} + \\sqrt{\\left(\\frac{p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)}{2}\\right)^2 + p^2q^2\\widehat{\\varphi }_{1,2}(k)^2},$ where the square root takes the positive value.",
"Note that by the definition of the double mark finite variance model $p^2\\widehat{\\varphi }_{1,1}(k) - q^2\\widehat{\\varphi }_{2,2}(k)$ is the Fourier transform of a non-negative (or non-positive) function, and therefore $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ is a sum of terms maximised at $k=0$ .",
"Furthermore, each approaches that maximum quadratically and at least one of $\\widehat{\\varphi }_{1,1}(k)$ , $\\widehat{\\varphi }_{2,2}(k)$ , and $\\widehat{\\varphi }_{1,2}(k)$ uniquely achieves its maximum at $k=0$ (otherwise we have $\\varphi = 0$ almost everywhere).",
"Therefore REF is satisfied.",
"Since we have only finitely many marks, there are only finitely many sequences of marks of a given length.",
"Therefore since each of the $\\varphi _{i,j}$ individually satisfy the 1-mark analog of REF , the 2-mark version is satisfied by $\\varphi $ too.",
"It is not inconceivable that the above argument could be generalised to 3 or 4 mark systems.",
"The degree 3 and 4 characteristic polynomials arising from the matrices have roots that can be solved for in radicals.",
"It may be then that a particular clever rearrangement of the expression allows the required properties to be deduced like above.",
"Such an investigation is currently beyond the authors' patience.",
"Of course, the famous Abel-Ruffini Theorem states that no such solution in radicals exists for arbitrary polynomials of degree 5 and higher, and so this approach seems doomed for systems with such families of marks.",
"Lemma B.3 Space-mark factorisation models satisfy the conditions of Assumption REF .",
"Note that $\\widehat{}(k) = \\widehat{\\overline{\\varphi }}(k)\\mathcal {K}$ , where $\\mathcal {K}\\colon L^2(\\mathcal {E}) \\rightarrow L^2(\\mathcal {E})$ is the self-adjoint linear integral operator with kernel function $K$ .",
"Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b}\\int K(a,b)\\mathcal {P}(\\mathrm {d}a)\\le 1$ , $\\mathcal {K}$ is a bounded operator and $\\mathbb {S}\\left(\\widehat{}(k)\\right) = {\\left\\lbrace \\begin{array}{ll}\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) \\ge 0\\\\-\\widehat{\\overline{\\varphi }}(k)\\mathbb {S}\\left(-\\mathcal {K}\\right) &: \\widehat{\\overline{\\varphi }}(k) < 0.\\end{array}\\right.",
"}$ Since $\\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}\\le 1$ , the finiteness of $\\widehat{\\overline{\\varphi }}(0)$ implies that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite, and the $d$ -independence of $\\mathcal {K}$ implies that REF holds.",
"Since $\\overline{\\varphi }(\\overline{x})\\ge 0$ , we have $*{\\widehat{\\overline{\\varphi }}(k)} < \\widehat{\\overline{\\varphi }}(0)$ for all $k\\ne 0$ .",
"Furthermore, $\\mathbb {S}\\left(\\mathcal {K}\\right) = \\left\\Vert \\mathcal {K}\\right\\Vert _{\\rm op}$ implies that $\\mathbb {S}\\left(\\mathcal {K}\\right)\\ge \\mathbb {S}\\left(-\\mathcal {K}\\right)$ .",
"Then $\\mathbb {S}\\left(\\widehat{}(k)\\right)$ inherits the properties required by REF from $\\widehat{\\overline{\\varphi }}(k)$ .",
"For REF , we note that the $d$ -independence of $\\mathcal {K}$ means that required decay properties are again inherited from $\\widehat{\\overline{\\varphi }}(k)$ .",
"Lemma B.4 The marked multivariate Gaussian model satisfies the conditions of Assumption REF .",
"Firstly note that for each $\\left(a,b\\right)\\in \\mathcal {E}^2$ the Gaussian structure of $\\varphi $ means that it factorises over the $d$ eigenvector directions of $\\Sigma (a,b)$ .",
"This factorisation ensures that that the $d$ -dimensional Fourier transform is the product of the 1-dimensional Fourier transforms.",
"We have $\\widehat{\\varphi }(k;a,b) = \\mathcal {A} \\exp \\left(-\\frac{\\pi ^2}{2}k^{\\intercal }\\Sigma (a,b)^{-1}k\\right).$ In particular, when $k=0$ the Fourier transform is $a,b$ -independent.",
"This means $\\widehat{}(0) = \\mathcal {A}\\mathbf {1}$ where $\\mathbf {1}$ is the integral operator with constant kernel function 1.",
"For $f\\in L^2(\\mathcal {E})$ we have ${f}{\\mathbf {1}f} &= \\int f(a) \\left(\\int f(b) \\mathcal {P}\\left(\\mathrm {d}b\\right)\\right) \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f\\right]^2,\\\\{f}{f} &= \\int f(a)^2 \\mathcal {P}\\left(\\mathrm {d}a\\right) = \\mathbb {E}_\\mathcal {P}\\left[f^2\\right].$ By Jensen's inequality we therefore have $\\mathbb {S}\\left(\\mathbf {1}\\right) = 1$ (equality follows by considering the test function $f(a)\\equiv 1$ ).",
"It therefore follows that $\\mathbb {S}\\left(\\widehat{}(0)\\right) = \\mathcal {A}<\\infty $ .",
"From (REF ) we have $\\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } = \\mathcal {A}\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\exp \\left(-\\pi ^2k^{\\intercal }\\Sigma (a,b)^{-1}k\\right) \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This bound then implies the condition (REF ) of REF holds with $C=1$ .",
"Before we address () of REF , we turn to REF .",
"From our above bounds we already have $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{1,\\infty } \\le \\left\\Vert \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\mathcal {A}\\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right) = \\mathbb {S}\\left(\\widehat{}(0)\\right) \\exp \\left(-\\frac{\\pi ^2}{2\\Sigma _{\\max }}*{k}^2\\right).$ This exponential term ensures that REF is satisfied.",
"We now return to () of REF .",
"For all $x\\in \\mathbb {R}$ we have $1-\\cos (x) \\le \\frac{1}{2}x^2$ , and therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le \\frac{1}{2}*{k}^2 \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x},$ for some unit vector $e\\in \\mathbb {R}^d$ .",
"Since the vector $e$ picks out the second moment of $\\varphi $ in only one direction, a standard calculation gives $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\Sigma _{\\max }^2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }.$ Therefore $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty }$ is bounded by a $d$ -independent quadratic function.",
"In conjunction with the quadratic bound of REF , this then proves () in the $k\\rightarrow 0$ regime.",
"The remaining regime then holds because the triangle inequality implies $\\left\\Vert \\widehat{}(0) - \\widehat{}(k)\\right\\Vert _{\\infty ,\\infty } \\le 2\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } = 2\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"To address REF , recall that the spatial convolution of two multivariate Gaussian functions with means $\\overline{0}$ and covariance matrices $\\Sigma _1$ and $\\Sigma _2$ is a multivariate Gaussian function with mean $\\overline{0}$ and covariance matrix $\\Sigma _1+\\Sigma _2$ .",
"Therefore for all $\\overline{x}\\in \\mathbb {R}^d$ we have $&\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\int \\varphi (\\overline{x}-\\overline{y};a_1,a_2)\\varphi (\\overline{y};a_3,a_4)\\mathrm {d}\\overline{y}\\nonumber \\\\&\\hspace{14.22636pt}= \\mathcal {A}^2\\left(2\\pi \\right)^{-d/2}\\operatornamewithlimits{ess\\,sup}_{a_1,a_2,a_3,a_4\\in \\mathcal {E}}\\left(\\det \\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)\\right)^{-1/2}\\exp \\left(-\\frac{1}{2}\\overline{x}^{\\intercal }\\left(\\Sigma (a_1,a_2) + \\Sigma (a_3,a_4)\\right)^{-1}\\overline{x}\\right) \\nonumber \\\\& \\hspace{14.22636pt} \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2}\\exp \\left(-\\frac{*{\\overline{x}}^2}{4\\Sigma _{\\min }}\\right) \\le \\mathcal {A}^2\\left(4\\pi \\Sigma _{\\min }\\right)^{-d/2} \\rightarrow 0.$ In particular, this bounds the integral appearing in the definition of the sets $B(x)$ (see (REF )) and therefore proves they are empty.",
"For the bound of the convolution of three $\\varphi $ functions, we can get this immediately from the above calculation and $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } \\le C\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"Specifically REF holds with $g(d) = \\left(4\\pi \\Sigma _{\\min }(d)\\right)^{-d/2}$ and the sets $B(x) = \\emptyset $ .",
"Lemma B.5 The bounded-volume Boolean disc model satisfies the conditions of Assumption REF .",
"Crucial to the conditions of REF holding is the upper bound on $R_d(a)$ in (REF ).",
"We use the shorthand $R_{a,b} := R_d(a) + R_d(b),\\qquad V_{a,b} := \\frac{\\pi ^{\\frac{d}{2}}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R^d_{a,b},$ so $V_{a,b}$ equals the Lebesgue volume of the $d$ -dimensional Euclidean ball with radius $R_{a,b}$ .",
"Note that the upper bound $R^{({\\rm max})}_d$ on $R_d(a)$ in (REF ) ensures $V_{a,b} \\le V^{({\\rm max})}$ uniformly in $d$ .",
"We first find $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{1,\\infty } = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}\\left(\\mathrm {d}a\\right) = \\operatornamewithlimits{ess\\,sup}_{b\\in \\left[0,1\\right]}\\int V_{a,b}\\mathcal {P}\\left(\\mathrm {d}a\\right) \\le V^{({\\rm max})}.$ This proves the statement that $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ is finite in REF .",
"From the condition (REF ) we also get $\\mathbb {S}\\left(\\widehat{}(0)\\right) \\ge \\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\mathcal {P}^{\\otimes 2}\\left(\\mathrm {d}a, \\mathrm {d}b\\right) \\ge \\epsilon ^2\\frac{d\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)} \\int ^{2\\left(1-\\frac{c_2}{d}\\right)R^{({\\rm max})}_d}_0 r^{d-1}\\mathrm {d}r \\\\= \\epsilon ^2V^{({\\rm max})}\\left(1-\\frac{c_2}{d}\\right)^d \\rightarrow \\epsilon ^2\\text{e}^{-c_2}V^{({\\rm max})},$ where we have used the test function $f(a) \\equiv 1$ to lower bound $\\mathbb {S}\\left(\\widehat{}(0)\\right)$ .",
"We then prove (REF ) by bounding $\\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty } =\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}\\int \\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\left[0,1\\right]}V_{a,b} \\le V^{({\\rm max})}.$ We will return for () after addressing REF .",
"For REF , it will be useful to have an expression for $\\widehat{\\varphi }(k;a,b)$ for each $a,b\\in \\left[0,1\\right]$ .",
"Suppose we are able to find $C_1\\in \\left(0,1\\right)$ and $C_2 >0$ such that for all $a,b\\in \\left[0,1\\right]$ $*{\\widehat{\\varphi }(k;a,b)} \\le \\widehat{\\varphi }(0;a,b) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right).$ Then since the $k$ -dependent factor is $a,b$ -independent we can use Lemma REF to show that $\\mathbb {S}\\left(\\widehat{}(k)\\right) \\le \\mathbb {S}\\left(\\widehat{}(0)\\right) \\left(C_1 \\vee \\left(1 - C_2*{k}^2\\right)\\right),$ and therefore prove that REF is satisfied.",
"In finding an expression for $\\widehat{\\varphi }(k;a,b)$ we are assisted by the spherical symmetry of the connection function and follow [18].",
"It transpires that $\\widehat{\\varphi }(k;a,b) = \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}}\\left(R_{a,b}*{k}\\right),$ where $J_{\\frac{d}{2}}$ is the Bessel function of the first kind of order $\\frac{d}{2}$ .",
"This function has the expansion $J_{\\frac{d}{2}}\\left(r\\right) = \\sum ^\\infty _{m=0} \\frac{\\left(-1\\right)^{m}}{m!\\Gamma \\left(\\frac{d}{2}+m+1\\right)}\\left(\\frac{r}{2}\\right)^{\\frac{d}{2}+2m},$ converging for all $r\\ge 0$ .",
"We will consider three different regimes for $*{k}$ .",
"For $r \\ll \\sqrt{d}$ the expansion (REF ) is asymptotic, and since $R^{({\\rm max})}_d$ and $R^{({\\rm min})}_d$ are both asymptotically proportional to $\\sqrt{d}$ we have $\\widehat{\\varphi }(k;a,b) \\sim \\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}R_{a,b}^d - \\frac{\\pi ^\\frac{d}{2}}{4\\Gamma \\left(\\frac{d}{2}+2\\right)}R_{a,b}^{d+2}*{k}^2 = V_{a,b}\\left(1 - \\frac{R^2_{a,b}}{2\\left(d+2\\right)}*{k}^2\\right)$ for $*{k}\\ll 1$ .",
"This proves the quadratic part of the desired bound.",
"The Bessel function $J_{\\frac{d}{2}}$ is bounded and achieves its global maximum (in absolute value) at its first non-zero stationary point, denoted $j^{\\prime }_{\\frac{d}{2},1}$ .",
"From [2], we have $j^{\\prime }_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _1 \\approx 0.81$ , and $*{J_{\\frac{d}{2}}\\left(j^{\\prime }_{\\frac{d}{2},1}\\right)} \\le M d^{-\\frac{1}{3}}$ for a $d$ -independent $M>0$ .",
"Since $R_{a,b} \\in \\left[2c_1,1/\\sqrt{2\\pi \\text{e}}+o(1)\\right]d^\\frac{1}{2}$ (for $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right]$ from (REF )) and $j^{\\prime }_{\\frac{d}{2},1} / R_{a,b} \\in \\left[\\sqrt{\\frac{\\pi \\text{e}}{2}}+o(1),\\frac{1}{4c_1}+o(1)\\right]d^{\\frac{1}{2}}$ we have $*{\\widehat{\\varphi }(k;a,b)} \\le \\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\le \\left(2e^{-1} + o(1)\\right)^\\frac{d}{2} M d^{-\\frac{1}{3}} \\rightarrow 0$ for $*{k} \\ge j^{\\prime }_{\\frac{d}{2},1} / R_{a,b}$ .",
"We are now left with the intermediate range for $*{k}$ .",
"The first positive zero of $J_\\frac{d}{2}$ occurs at $j_{\\frac{d}{2},1}$ , where $j_{\\frac{d}{2},1} = \\frac{d}{2} + \\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\mathcal {O}\\left(d^{-\\frac{1}{3}}\\right)$ for a given $\\gamma _2 \\approx 1.86$ (see [2]).",
"In particular, we will always have $j^{\\prime }_{\\frac{d}{2},1} < j_{\\frac{d}{2},1}$ .",
"From differential identities relating Bessel functions (see [18]), we have $\\frac{\\mathrm {d}}{\\mathrm {d}*{k}} \\widehat{\\varphi }(k;a,b) = -R_{a,b}\\left(\\frac{2\\pi R_{a,b}}{*{k}}\\right)^\\frac{d}{2}J_{\\frac{d}{2}+1}\\left(R_{a,b}*{k}\\right).$ The Bessel function $J_{\\frac{d}{2}+1}(r)$ is positive for $r>0$ until its first positive zero at $j_{\\frac{d}{2}+1,1}$ , and $j_{\\frac{d}{2}+1,1} > j_{\\frac{d}{2},1}$ .",
"Therefore $\\widehat{\\varphi }(k;a,b)$ is positive and decreasing on the whole region $*{k}\\in \\left(0,j_{\\frac{d}{2},1}/R_{a,b}\\right]$ .",
"Since this overlaps with the high $*{k}$ range, and we have the uniform quadratic behaviour near $k=0$ , the function $k\\mapsto *{\\widehat{\\varphi }(k;a,b)}$ can never increase and approach $\\widehat{\\varphi }(0;a,b)$ again after leaving a neighbourhood of $k=0$ .",
"We therefore have the bound (REF ) and thus have proven that REF is satisfied.",
"Figure: Sketch of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) against *k*{k}.",
"It approaches its maximum quadratically as *k→0*{k}\\rightarrow 0.",
"The first local maximum of J d 2 J_{\\frac{d}{2}} occurs at j d 2,1 ' ∼d 2+γ 1 d 2 1 3 j^{\\prime }_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _1\\left(\\frac{d}{2}\\right)^\\frac{1}{3}.",
"The first zero of ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) occurs at *kR a,b =j d 2,1 ∼d 2+γ 2 d 2 1 3 *{k}R_{a,b} = j_{\\frac{d}{2},1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} where γ 2 >γ 1 \\gamma _2>\\gamma _1.",
"Furthermore, ϕ ^k;a,b\\widehat{\\varphi }\\left(k;a,b\\right) is strictly decreasing until *kR a,b =j d 2+1,1 ∼d 2+γ 2 d 2 1 3 +1 2*{k}R_{a,b} = j_{\\frac{d}{2}+1,1}\\sim \\frac{d}{2}+\\gamma _2\\left(\\frac{d}{2}\\right)^\\frac{1}{3} + \\frac{1}{2}.We return to () in REF .",
"We first calculate the second moment: $\\int *{\\overline{x}}^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{d}{d+2}V_{a,b}R^2_{a,b}.$ From $*{\\overline{x}}^2 = \\overline{x}_1^2 + \\ldots + \\overline{x}^2_d$ (for an orthonormal basis) and the rotational symmetry, we have $\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}= \\frac{1}{d+2}V_{a,b}R^2_{a,b}$ for all unit vectors $e\\in \\mathbb {R}^d$ .",
"Therefore $\\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}}\\int \\left(e\\cdot \\overline{x}\\right)^2\\varphi \\left(\\overline{x};a,b\\right)\\mathrm {d}\\overline{x}\\le \\frac{4}{d+2}V^{({\\rm max})}\\left(R^{({\\rm max})}_d\\right)^2.$ While $V^{({\\rm max})}$ is a constant, $R^{({\\rm max})}_d\\sim \\sqrt{d/8\\pi \\text{e}}$ and therefore we have a bound uniform in $d$ .",
"The argument now proceeds similarly to the corresponding part of the proof for the multivariate Gaussian model.",
"From (REF ) and REF we have both sides of () satisfying quadratic bounds around $k=0$ , and the triangle inequality with our other bounds prove that the bound also holds on the remainder.",
"For REF we first want to bound the convolution of three $\\varphi $ functions.",
"Given marks $a_1,\\ldots ,a_6\\in \\left[0,1\\right]$ and $\\overline{x}\\in \\mathbb {R}^d$ , let $\\mathbb {B}_{a,b}(\\overline{x})$ denote the ball of radius $R_{a,b}$ centred on $\\overline{x}$ , and let $1_{a,b}\\left(\\overline{x}\\right) = 1\\left\\lbrace *{\\overline{x}}\\le R_{a,b}\\right\\rbrace $ .",
"Then $\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6)\\right)\\left(\\overline{x}\\right)&= \\int 1_{a_1,a_2}(\\overline{y})1_{a_3,a_4}(\\overline{z}-\\overline{y})1_{a_5,a_6}(-\\overline{z})\\mathrm {d}\\overline{y}\\mathrm {d}\\overline{z}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)}\\left(\\int _{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)} 1_{a_3,a_4}(\\overline{z}-\\overline{y})\\mathrm {d}\\overline{z}\\right) \\mathrm {d}\\overline{y}\\nonumber \\\\&= \\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}.$ Then for $\\delta \\in \\left(0,1\\right)$ we have $&\\int _{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}\\mathrm {d}\\overline{y}\\nonumber \\\\&\\qquad \\le \\int _{\\delta \\mathbb {B}_{a,b}\\left(\\overline{0}\\right)}*{\\mathbb {B}_{a_3,a_4}\\left(\\overline{0}\\right)}\\mathrm {d}\\overline{y}+ *{\\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} \\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} \\nonumber \\\\& \\qquad \\le \\left(V^{({\\rm max})}\\right)^2\\delta ^d + V^{({\\rm max})}\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)}.$ Let $\\overline{e}_1\\in \\mathbb {R}^d$ be a unit vector.",
"Then by spherical symmetry we have $\\operatornamewithlimits{ess\\,sup}_{\\overline{y}\\in \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)\\setminus \\delta \\mathbb {B}_{a_1,a_2}\\left(\\overline{0}\\right)} *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\overline{y}\\right)} = *{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)}.$ We bound the volume of this intersection with the volume of the $d$ -ball with radius equal to the radius of the $d-1$ -sphere formed by the intersection of their boundaries.",
"This radius is equal to the length $h$ in Figure REF , which is maximised when $R_{a_5,a_6}$ and $R_{a_3,a_4}$ are maximised and $\\delta R_{a_1,a_2}$ is minimised.",
"Recall we defined the constant $c_1\\in \\left(0,1/\\sqrt{8\\pi \\text{e}}\\right)$ when we defined the model in Section REF .",
"Since we have $R_{a_5,a_6},R_{a_3,a_4} \\le 2R^{({\\rm max})}_d$ and $\\delta R_{a_1,a_2} \\ge \\delta c_1 d^\\frac{1}{2}$ , we have $h\\le \\sqrt{\\left(2 R^{({\\rm max})}_d\\right)^2 - \\delta ^2 c^2_1 d}$ - we have used the fact that the triangle is isosceles if both the $R_{a_5,a_6}$ and $R_{a_3,a_4}$ edges attain their maximum.",
"The ball with this radius then has the volume $\\frac{\\pi ^\\frac{d}{2}}{\\Gamma \\left(\\frac{d}{2}+1\\right)}h^d \\le V^{({\\rm max})}\\left(1 - \\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}}\\right)^\\frac{d}{2}.$ Since $\\frac{\\pi \\delta ^2 c^2_1 d}{\\left(V^{({\\rm max})}\\right)^\\frac{2}{d}\\Gamma \\left(\\frac{d}{2}+1\\right)^\\frac{2}{d}} = 2\\pi \\text{e}\\delta ^2c_1^2\\left(1+o(1)\\right)$ and $c_1>0$ , this shows that $*{\\mathbb {B}_{a_5,a_6}\\left(\\overline{0}\\right)\\cap \\mathbb {B}_{a_3,a_4}\\left(\\delta R_{a_1,a_2}\\overline{e}_1\\right)} \\le V^{({\\rm max})}\\left(1 - \\pi \\text{e}\\delta ^2c_1^2\\right)^\\frac{d}{2}$ for $d$ sufficiently large.",
"Sine we now have a mark independent bound with the required decay, we have proved the bound for the convolution of three $\\varphi $ functions.",
"Figure: For both the arguments, we bound the intersection of two dd-balls with a new dd-ball that has the radius hh in these diagrams.For finding the sets $B(x)$ in (REF ), we write $\\int \\varphi \\left(\\overline{x}- \\overline{y};a,c\\right)\\varphi \\left(\\overline{y};c,b\\right)\\mathrm {d}\\overline{y}\\mathcal {P}\\left(\\mathrm {d}c\\right) = \\int *{\\mathbb {B}_{a,c}\\left(\\overline{x}\\right)\\cap \\mathbb {B}_{c,b}\\left(\\overline{0}\\right)}\\mathcal {P}\\left(\\mathrm {d}c \\right).$ As before, we can bound the volume of this intersection independently of the mark $c$ .",
"We can bound it with the volume of the ball with radius $h$ where $h\\le \\sqrt{\\left(2R^{({\\rm max})}_d\\right)^2 - \\frac{1}{4}*{\\overline{x}}^2}$ (as can be seen in Figure REF ).",
"As before, if $*{\\overline{x}} \\ge \\kappa d^\\frac{1}{2}$ for some $\\kappa >0$ , then the volume of the ball with radius $h$ vanishes in the $d\\rightarrow \\infty $ limit.",
"Furthermore, if $\\kappa \\le \\frac{1}{\\sqrt{2\\pi \\text{e}}}$ then the volume of the ball with radius $\\kappa d^\\frac{1}{2}$ vanishes in the $d\\rightarrow \\infty $ limit.",
"Therefore assumption REF is satisfied with the sets $B(x)\\equiv B\\left(\\overline{x},a\\right)=\\left\\lbrace \\left(\\overline{y},b\\right)\\in \\mathbb {X}:*{\\overline{x}-\\overline{y}} \\le \\kappa d^\\frac{1}{2}\\right\\rbrace $ for any $\\kappa \\in \\left(0,\\frac{1}{\\sqrt{2\\pi \\text{e}}}\\right)$ ."
],
[
"Displacement Diagram Case Distinctions",
"This section provides the details omitted from the proof of Proposition REF .",
"Here we show which diagrams arise from displacements crossing a $\\Psi $ segment, and whether they can be dealt with by the first sub-case, $\\mathrm {\\left(I\\right)}$ , or by the second sub-case, $\\mathrm {\\left(II\\right)}$ .",
"Recall that $\\mathrm {\\left(I\\right)}$ are those that use a $\\Psi $ decomposition of the larger diagram only (and the displacement is not crossing a $\\Psi ^{(4)}$ segment), while $\\mathrm {\\left(II\\right)}$ are those that will require a combination of a $\\Psi $ decomposition and a $\\overline{\\Psi }$ decomposition.",
"We first expand out the displaced $\\Psi $ terms and label which are in sub-case $\\mathrm {\\left(I\\right)}$ and which are in sub-case $\\mathrm {\\left(II\\right)}$ .",
"Note that the displacement may be across the `top' or the `bottom' of the segments in the usual orientation.",
"If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(1)}$ term, we use cosine-splitting and sometimes expand $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (2,0.8);(1,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[below]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad \\right)\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}\\quad \\right).$ Using the same ideas, if we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(2)}$ term we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad + \\quad \\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (2,-0.8);(1,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.25) circle (0pt) node[right]{\\circ };(0.75,0.6) circle (0pt) node[above]{\\circ };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le 2\\left(\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\right.\\nonumber \\\\&\\hspace{28.45274pt}\\left.\\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}} \\quad +\\quad 2\\times \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}} \\quad +\\quad \\underbrace{\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(I\\right)}}\\quad \\right).$ If we consider a displacement across the `top' or the `bottom' of a $\\Psi ^{(3)}$ term, then by symmetry in both cases we get $\\underbrace{\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}}_{\\mathrm {\\left(II\\right)}}.$ We first show the bounds for the single segments in sub-case $\\mathrm {\\left(I\\right)}$ .",
"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=280,in=80] (3,-0.6) to [out=10,in=170] (4,-0.6) to [out=40,in=200] (5,0) to [out=160,in=320] (4,0.6) to [out=190,in=350] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi _n};[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(5,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=80,in=280] (2,0.6);[fill=gray!50] (3,0.6) to [out=350,in=190] (4,0.6) to [out=260,in=100] (4,-0.6) to [out=170,in=10] (3,-0.6) to [out=80,in=280] (3,0.6);[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (3.5,0) {\\Psi };[fill=white] (2,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2.$ We first consider the displaced $\\Psi ^{(j)}$ term by itself.",
"By using supremum bounds and spatial translations we arrive at $\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}\\triangle _\\lambda , \\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda , & \\raisebox {-25pt}{\\begin{tikzpicture}(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(1.5,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };[fill=white] (1.5,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle ^\\circ _\\lambda .$ For most of these, it is sufficient to separate the pair using a supremum bound and use the individual bound above for the displaced $\\Psi ^{(j)}$ term and the bound on the $\\Psi $ or $\\Psi _n$ term from Lemma REF .",
"The only pairs that need a more careful approach are those that have the bound $W_k\\triangle ^\\circ _\\lambda $ above for the displaced $\\Psi ^{(j)}$ , followed by a $\\Psi ^{(4)}$ or $\\Psi ^{(2)}_n$ term.",
"In these cases we consider the pair together and get the diagram bounds $\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2,\\qquad \\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1,0.6);(1,0.6) -- (2,0.6) -- (2,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,-0.6) circle (2pt);(2,-0.6) circle (2pt);(2,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (1,0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le W_k\\triangle _\\lambda \\le W_kV_\\lambda ^2.$ In both of these cases we bounded a $\\tau _\\lambda $ edge by 1 (essentially omitting the edge) before applying a supremum bound.",
"In the first we omit the diagonal edge and split the upper supremum vertex in two.",
"In the second we omit the left-most vertical edge and apply a supremum bound at the bottom right vertex.",
"We now turn our attention to sub-case $\\mathrm {\\left(II\\right)}$ .",
"Recall from the proof of Proposition REF that in this sub-case the aim is prove that the schematic diagrams can be bounded: $\\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=280,in=80] (4,-0.6) to [out=10,in=170] (5,-0.6) to [out=40,in=200] (6,0) to [out=160,in=320] (5,0.6) to [out=190,in=350] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }_0};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(2,-0.6) circle (2pt);(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (6,0) circle (2pt);\\end{tikzpicture}}\\quad \\vee \\quad \\raisebox {-15pt}{\\begin{tikzpicture}[fill=gray!50] (2,0.6) to [out=350,in=190] (3,0.6) to [out=260,in=100] (3,-0.6) to [out=170,in=10] (2,-0.6) to [out=70,in=290] (2,0.6);[fill=gray!50] (4,0.6) to [out=350,in=190] (5,0.6) to [out=260,in=100] (5,-0.6) to [out=170,in=10] (4,-0.6) to [out=80,in=280] (4,0.6);(3,0.6) -- (4,0.6);(3,-0.6) -- (4,-0.6);(3.5,-0.6) circle (0pt) node[above]{\\circ };[<->] (2,0.8) -- (3,0.8);(2.5,0.8) circle (0pt) node[rotate = 0]{\\times };\\node at (2.5,0) {\\Psi ^{(j)}};\\node at (4.5,0) {\\overline{\\Psi }};[dashed] (2,0.6) -- (2,-0.6);[fill=white] (2,0.6) circle (2pt);\\node [mark size=2pt] at (2,-0.6) {{square*}};(3,0.6) circle (2pt);(3,-0.6) circle (2pt);(4,0.6) circle (2pt);(4,-0.6) circle (2pt);[fill=white] (5,0.6) circle (2pt);[fill=white] (5,-0.6) circle (2pt);\\end{tikzpicture}}\\quad \\le \\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda .$ Also recall that if the $\\overline{\\Psi }_0$ term is actually a $\\overline{\\Psi }^{(2)}_0$ term, then the $\\tau _\\lambda ^\\circ $ edge connecting them is actually just a $\\tau _\\lambda $ edge.",
"We first investigate bounds for the displaced $\\Psi ^{(j)}$ term with the connecting $\\tau _\\lambda $ and $\\tau _\\lambda ^\\circ $ edges.",
"By using supremum bounds, spatial translations, splitting supremum vertices in two, and sometimes expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction, we get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);(3,0.6) -- (2,0.6) -- (2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);[fill=white] (0,0.6) circle (2pt);\\node [mark size=2pt] at (0,-0.6) {{square*}};[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(2.5,-0.6) circle (0pt) node[above]{\\circ };[fill=white] (3,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,0.8) -- (1.5,0.8);(0.75,0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^{\\circ \\circ }_\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1.5,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(2,-0.6) -- (3,-0.6);(1.5,0.6) -- (3,0.6);[dashed] (3,0.6) -- (3,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (3,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (0,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);(2,-0.6) circle (2pt);\\end{tikzpicture}}\\quad +\\quad \\raisebox {-25pt}{\\begin{tikzpicture}(0,0.6) -- (1.5,0.6) -- (1.5,-0.1) -- (1,-0.6);(0,-0.6) -- (1,-0.6) -- (2,-0.6) -- (1.5,-0.1);(1.5,0.6) -- (3,0.6);[dashed] (0,0.6) -- (0,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node{\\times };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);(1.5,0.6) circle (2pt);(1,-0.6) circle (2pt);(1.5,-0.1) circle (2pt);[fill=white] (2,-0.6) circle (2pt);\\end{tikzpicture}}\\nonumber \\\\&\\le \\overline{H_k}+ W_k\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda },\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda ,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle ^\\circ _\\lambda .$ For most of these, it is clear from a crude supremum bound that we have the bound $\\overline{W_k}V_\\lambda ^2 + \\overline{H_k}U_\\lambda $ for the pair of these with the subsequent $\\overline{\\Psi }$ or $\\overline{\\Psi }_0$ term (like we did for sub-case $\\mathrm {\\left(I\\right)}$ ).",
"The only pairs that need a more careful approach are those that have the bound $\\overline{W_k}\\triangle ^\\circ _\\lambda $ or $\\overline{W_k}\\triangle ^\\circ _\\lambda \\triangle ^\\circ _\\lambda $ above, followed by a $\\overline{\\Psi }^{(4)}$ or $\\overline{\\Psi }^{(2)}_0$ term.",
"We first consider the cases where they are followed by a $\\overline{\\Psi }^{(4)}$ term.",
"Here we use supremum bounds, spatial translations, and expanding $\\tau _\\lambda ^\\circ $ edges into a $\\tau _\\lambda $ edge and a contraction to get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\triangle ^\\circ _\\lambda \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0.6);(2,-0.6) -- (3,-0.6);[dashed] (0,0.6) -- (0,-0.6);(2.5,-0.6) circle (0pt) node[above]{\\circ };(3,0.6) -- (3,-0.6) -- (4,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(3,-0.6) circle (2pt);[fill=white] (3,0.6) circle (2pt);[fill=white] (4,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\triangle ^{\\circ \\circ }_\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda + \\overline{W_k}\\triangle _\\lambda = \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0.6);(1,-0.6) -- (2,-0.6);[dashed] (0,0.6) -- (0,-0.6);(1.5,-0.6) circle (0pt) node[above]{\\circ };(2,0.6)-- (2,-0.6) -- (3,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);(2,-0.6) circle (2pt);[fill=white] (3,-0.6) circle (2pt);[fill=white] (2,0.6) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\overline{\\triangle ^{\\circ \\circ }_\\lambda }\\le \\overline{W_k}V_\\lambda ^2.$ We now consider the cases where they are followed by a $\\overline{\\Psi }^{(2)}_0$ term.",
"Recall that for this case, the connecting $\\tau _\\lambda ^\\circ $ edge is in fact only a $\\tau _\\lambda $ edge.",
"We then get $\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,0.8) -- (2,0.8);(1.5,0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(2,0.6) circle (2pt);(2,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);(1,0.6) -- (2,0.6) -- (2,-0.6) -- (1,-0.6);[<->] (1,-0.8) -- (2,-0.8);(1.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(0.5,0.6) circle (0pt) node[above]{\\circ };(2,0.6) -- (3,0);(2,-0.6) -- (3,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (3,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\triangle ^\\circ _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-15pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,0.8) -- (1,0.8);(0.5,0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2,\\\\\\raisebox {-25pt}{\\begin{tikzpicture}(1,0.6) circle (2pt);(1,-0.6) circle (2pt);(0,0.6) -- (1,0.6) -- (1,-0.6) -- (0,-0.6);[<->] (0,-0.8) -- (1,-0.8);(0.5,-0.8) circle (0pt) node[rotate = 0]{\\times };(1,0.6) -- (2,0);(1,-0.6) -- (2,0);[dashed] (0,0.6) -- (0,-0.6);\\node [mark size=2pt] at (0,-0.6) {{square*}};[fill=white] (0,0.6) circle (2pt);[fill=white] (2,0) circle (2pt);\\end{tikzpicture}}\\quad &\\le \\overline{W_k}\\triangle _\\lambda \\le \\overline{W_k}V_\\lambda ^2.$ The remaining sub-case $\\mathrm {\\left(III\\right)}$ was fully dealt with in the main text."
],
[
"Proof of Lemma ",
"Recall that we wish to prove the following result: * We begin by defining a sightly different object to $\\overline{H_k}$ .",
"Given $\\vec{a}=\\left(a_1,\\ldots ,a_{16}\\right)\\in \\mathcal {E}^{16}$ , define $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right) :=& \\int \\tau _\\lambda (\\overline{s}-\\overline{x}_1;a_1,a_2)\\tau _\\lambda (\\overline{u};a_3,a_4) \\tau _\\lambda (\\overline{v}-\\overline{s};a_5,a_6) \\tau _\\lambda (\\overline{v}+\\overline{x}_2-\\overline{t};a_7,a_8) \\nonumber \\\\&\\hspace{28.45274pt}\\times \\tau _\\lambda (\\overline{s}-\\overline{w};a_9,a_{10})\\tau _\\lambda (\\overline{w}-\\overline{u};a_{11},a_{12}) \\tau _\\lambda (\\overline{t}-\\overline{w};a_{13},a_{14}) \\nonumber \\\\&\\hspace{56.9055pt}\\times \\tau _{\\lambda ,k}(\\overline{t}+ \\overline{x}_3-\\overline{u};a_{15},a_{16}) \\mathrm {d}\\overline{t}\\mathrm {d}\\overline{w}\\mathrm {d}\\overline{z}\\mathrm {d}\\overline{u},$ so we have $\\overline{H_k}\\le f(\\lambda )^5\\operatornamewithlimits{ess\\,sup}_{\\overline{x}_1,\\overline{x}_2\\in \\mathbb {R}^d,\\vec{a}\\in \\mathcal {E}^{16}}H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{0};\\vec{a};k\\right).$ We then proceed to bound $H^{\\prime }\\left(\\overline{x}_1,\\overline{x}_2,\\overline{x}_3;\\vec{a};k\\right)$ to get our result.",
"This form is preferred because it more easily indicates what the Fourier arguments should be when we take a Fourier transform.",
"A schematic Fourier diagram for this can be found in [21].",
"We then expand out the $\\tau _\\lambda $ and $\\tau _{\\lambda ,k}$ terms.",
"For each of the $\\tau _\\lambda $ terms we apply (REF ) and () to get $\\tau _\\lambda (\\overline{x};a,b) \\le \\varphi (\\overline{x};a,b) + \\lambda (\\varphi \\varphi )(\\overline{x};a,b)+ \\lambda ^2(\\varphi \\tau _\\lambda \\varphi )(\\overline{x};a,b).$ Similarly, with the additional use of the cosine-splitting lemma, we get $\\tau _{\\lambda ,k}(\\overline{x};a,b) \\le \\varphi _k(\\overline{x};a,b) + 2\\lambda \\left( (\\varphi _k\\varphi )(\\overline{x};a,b) +(\\varphi \\varphi _k)(\\overline{x};a,b)\\right)+ 4\\lambda ^2(\\varphi \\tau _\\lambda \\varphi _k)(\\overline{x};a,b) \\\\ + 4\\lambda ^2(\\varphi \\tau _{\\lambda ,k}\\varphi )(\\overline{x};a,b).$ By using Lemma REF , the Fourier inversion theorem, and $\\lambda \\le f(\\lambda )$ , we can realise that we only need to get appropriate bounds for the following objects.",
"For $\\vec{j}\\in \\left\\lbrace 1,2,3\\right\\rbrace ^7\\times \\left\\lbrace 1,2,3,4,5\\right\\rbrace $ , define $\\mathcal {H}(k;\\vec{j}) := \\int P^{j_1}_{1,2}(l_1)P^{j_2}_{3,4}(l_1)P^{j_3}_{5,6}(l_2)P^{j_4}_{7,8}(l_2)P^{j_5}_{9,10}(l_1-l_2)P^{j_6}_{11,12}(l_1-l_3)P^{j_7}_{13,14}(l_2-l_3) \\\\ \\times Q^{j_8}_{15,16}(l_3;k)\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}},$ where $P^{j}_{m,n}(l) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_m}{\\widehat{}(l) g_n}} &:j=1\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &:j=2\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &: j=3,\\end{array}\\right.}",
"\\qquad \\text{for }m,n\\in \\left\\lbrace 1,2,\\ldots ,14\\right\\rbrace ,\\\\Q^{j}_{15,16}(l;k) &:= {\\left\\lbrace \\begin{array}{ll}*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &:j=1\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &:j=2\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{}_k(l) g_{16}}} &:j=3\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &:j=4\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &:j=5,\\end{array}\\right.",
"}$ where $\\left\\lbrace g_m\\right\\rbrace _{m=1}^{16}$ are the functions of the form $g_m(a) = \\frac{1}{\\mathcal {P}(E_m)}1_{E_m}(a)$ arising from the application of Lemma REF .",
"If we can bound these $\\mathcal {H}(k;\\vec{j})$ appropriately, then we can combine them to get the required bound.",
"We now recall pointwise bounds for the $P^{j}_{m,n}(l)$ and $Q^{j}_{15,16}(l)$ terms.",
"In the proof of Lemma REF we derived $*{*{g_m}{\\widehat{}(l)g_n}} &\\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }\\\\*{*{g_m}{\\widehat{}(l)^2g_n}} &\\le *{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}}\\\\*{*{g_m}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}(l)g_n}} &\\le f(\\lambda )\\widehat{G}_{\\mu _\\lambda }(l)*{g_m}{\\widehat{}(l)^2g_m}^{\\frac{1}{2}}*{g_n}{\\widehat{}(l)^2g_n}^{\\frac{1}{2}},$ and $*{*{g_{15}}{\\widehat{}_k(l) g_{16}}} &\\le \\left\\Vert \\widehat{}_k(l)\\right\\Vert _{\\infty ,\\infty } \\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{16}}{\\widehat{}(l)^2g_{16}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}_k(l)\\widehat{}(l) g_{16}}} &\\le C\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right) *{g_{15}}{\\widehat{}(l)^2g_{15}}^{\\frac{1}{2}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_\\lambda (l)\\widehat{}_k(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{G}_{\\mu _\\lambda }(l)*{g_{15}}{\\widehat{}(l)^2g_{15}}\\\\*{*{g_{15}}{\\widehat{}(l)\\widehat{\\mathcal {T}}_{\\lambda ,k}(l)\\widehat{}(l) g_{16}}} &\\le Cf(\\lambda )\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k)*{g_{15}}{\\widehat{}(l)^2g_{15}}^\\frac{1}{2}*{g_{16}}{\\widehat{}(l)^2g_{16}}^\\frac{1}{2}$ where $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-k) + \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k)\\widehat{G}_{\\mu _\\lambda }(l+k).$ Also recall that we can bound $*{*{g_m}{\\widehat{}(l)g_n}}\\le C$ and $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ if required.",
"Note that all the bounds for the $Q^{j}_{15,16}(l)$ terms have a $\\left(1 - \\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ factor.",
"We therefore only need to show that the remaining integrals of $\\widehat{}$ and $\\widehat{G}_{\\mu _\\lambda }$ terms can all be bounded by some constant multiple of $\\beta ^2$ .",
"Also note that the bounds for $Q^{1}_{15,16}(l)$ , $Q^{2}_{15,16}(l)$ , and $Q^{3}_{15,16}(l)$ only differ by a uniform constant (after we bound $*{g_m}{\\widehat{}(l)^2g_m}\\le C^2$ ), and so once we have dealt with one of these we have dealt with the other two.",
"Let us first consider $\\vec{j}= \\left(1,1,1,1,1,1,1,1\\right)$ .",
"After applying the uniform pointwise bound for $Q^{1}_{15,16}(l)$ , we use the Cauchy-Schwarz inequality to bound $&\\int *{*{g_1}{\\widehat{}(l_1)g_2}}*{*{g_3}{\\widehat{}(l_1)g_4}}*{*{g_5}{\\widehat{}(l_2)g_6}}*{*{g_7}{\\widehat{}(l_2)g_8}}*{*{g_9}{\\widehat{}(l_1-l_2)g_{10}}} \\nonumber \\\\&\\hspace{199.16928pt}\\times *{*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}}*{*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\&\\qquad \\le \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\hspace{56.9055pt}\\times \\left(\\int *{g_7}{\\widehat{}(l_2)g_8}^2*{g_9}{\\widehat{}(l_1-l_2)g_{10}}^2*{g_{11}}{\\widehat{}(l_1-l_3)g_{12}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.",
"$ To deal with these parentheses, we use a volume-preserving change of variables and factorise each into three integrals.",
"For example, $\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2*{g_5}{\\widehat{}(l_2)g_6}^2*{g_{13}}{\\widehat{}(l_2-l_3)g_{14}}^2\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\\\= \\left(\\int *{g_1}{\\widehat{}(l_1)g_2}^2*{g_3}{\\widehat{}(l_1)g_4}^2\\frac{\\mathrm {d}l_1}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_5}{\\widehat{}(l_2)g_6}^2\\frac{\\mathrm {d}l_2}{\\left(2\\pi \\right)^{d}}\\right)\\left(\\int *{g_{13}}{\\widehat{}(l_3^{\\prime })g_{14}}^2\\frac{\\mathrm {d}l_3^{\\prime }}{\\left(2\\pi \\right)^{d}}\\right).$ There are now two types of integral we need to bound: $&\\int *{g_m}{\\widehat{}(l)g_n}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d}\\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int \\left(\\int _{E_m\\times E_n \\times E_m \\times E_n} \\widehat{\\varphi }(l;a,b)\\widehat{\\varphi }(l;c,d) \\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\right) \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} = \\frac{1}{\\mathcal {P}\\left(E_m\\right)^2\\mathcal {P}\\left(E_n\\right)^2}\\int _{E_m\\times E_n \\times E_m \\times E_n} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right)\\mathcal {P}^{\\otimes 4}\\left(\\mathrm {d}a,\\mathrm {d}b,\\mathrm {d}c,\\mathrm {d}d\\right)\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\left(\\varphi (\\cdot ;a,b)\\star \\varphi (\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b)\\varphi (-\\overline{x};c,d)\\mathrm {d}\\overline{x}\\le \\operatornamewithlimits{ess\\,sup}_{a,b\\in \\mathcal {E}} \\int \\varphi (\\overline{x};a,b) \\mathrm {d}\\overline{x}\\le C, \\\\&\\int *{g_1}{\\widehat{}(l)g_2}^2 *{g_3}{\\widehat{}(l)g_4}^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^d} \\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi (\\cdot ;a_1,a_2)\\star \\varphi (\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\le g(d).",
"$ In (REF ) we used $\\varphi \\left(-\\overline{x};c,d\\right)\\in \\left[0,1\\right]$ and REF .",
"In () we bounded the convolution of four adjacency functions using REF .",
"The result of this is that we can bound (REF ) by $C^{\\frac{5}{2}}g(d)^{\\frac{1}{2}}$ .",
"Since $g(d)\\le \\beta ^4$ , this satisfies the bound we require.",
"As noted above, this argument also proves the required bound for $\\vec{j}= \\left(1,1,1,1,1,1,1,2\\right)$ and $\\vec{j}= \\left(1,1,1,1,1,1,1,3\\right)$ .",
"Now suppose that we change some subset of $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace $ from having value 1 to having value 2.",
"The argument proceeds similarly, except that after we have factorised our integrals we now want to also bound integrals of the following types: $&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int \\left(\\int _{E_1\\times E_1\\times E_2\\times E_2}\\widehat{\\varphi }^{(2)}(l;a,b)\\widehat{\\varphi }^{(2)}(l;c,d)\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d)\\right)\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\nonumber \\\\& \\hspace{56.9055pt}= \\frac{1}{\\mathcal {P}\\left(E_1\\right)^2\\mathcal {P}\\left(E_2\\right)^2}\\int _{E_1\\times E_1\\times E_2\\times E_2}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)(\\overline{0})\\mathcal {P}^{\\otimes 4}(\\mathrm {d}a,\\mathrm {d}b, \\mathrm {d}c, \\mathrm {d}d) \\nonumber \\\\& \\hspace{56.9055pt}\\le \\operatornamewithlimits{ess\\,sup}_{a,b,c,d\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a,b)\\star \\varphi ^{(2)}(\\cdot ;c,d)\\right)\\left(\\overline{0}\\right) \\le g(d), \\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)g_4}^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi (\\cdot ;a_5,a_6) \\star \\varphi (\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^2 g(d)\\\\&\\int *{g_1}{\\widehat{}(l)^2g_1} *{g_2}{\\widehat{}(l)^2g_2}*{g_3}{\\widehat{}(l)^2g_3} *{g_4}{\\widehat{}(l)^2g_4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\& \\hspace{56.9055pt} \\le \\operatornamewithlimits{ess\\,sup}_{a_1,\\ldots ,a_8\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4) \\star \\varphi ^{(2)}(\\cdot ;a_5,a_6) \\star \\varphi ^{(2)}(\\cdot ;a_7,a_8)\\right)\\left(\\overline{0}\\right) \\nonumber \\\\& \\hspace{56.9055pt} \\le \\left\\Vert \\widehat{}(0)^2\\right\\Vert _{\\infty ,\\infty }^2\\operatornamewithlimits{ess\\,sup}_{\\overline{x}\\in \\mathbb {R}^d,a_1,\\ldots ,a_4\\in \\mathcal {E}}\\left(\\varphi ^{(2)}(\\cdot ;a_1,a_2)\\star \\varphi ^{(2)}(\\cdot ;a_3,a_4)\\right)\\left(\\overline{x}\\right)\\le C^4 g(d).$ Up to a constant, these are all less than or equal to the bounds we used for the $\\left\\lbrace j_1,\\ldots ,j_7\\right\\rbrace = \\left\\lbrace 1,\\ldots ,1\\right\\rbrace $ case, and therefore that case dominates.",
"These other cases would behave as constant multiple of one of $g(d)$ , $g(d)^{\\frac{3}{2}}$ , $g(d)^2$ , $g(d)^{\\frac{5}{2}}$ , and $g(d)^3$ .",
"We have now dealt with all the cases which have no factors of $\\widehat{G}_{\\mu _\\lambda }$ .",
"To demonstrate how we will deal with these, we first consider $\\vec{j}=\\left\\lbrace 3,3,3,3,3,3,3,5\\right\\rbrace $ .",
"For notational compactness, we define $\\left<\\widehat{}(l)^2\\right>_{I} := \\prod _{m\\in I}*{g_m}{\\widehat{}(l)^2g_m},$ where $I\\subset \\mathbb {N}$ and $l\\in \\mathbb {R}^d$ .",
"Since we have already extracted the factor of $\\left(1-\\mathbb {S}\\left(\\widehat{}(k)\\right)\\right)$ via the pointwise bound on $Q^{5}_{15,16}(l;k)$ , we are left trying to bound the following integral: $\\int \\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\\\ \\qquad \\times \\left<\\widehat{}(l_1)^2\\right>^{\\frac{1}{2}}_{1,2,3,4} \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_2)^2\\right>^{\\frac{1}{2}}_{9,10} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}.$ To bound this integral we need to partition our space.",
"Fix $\\varepsilon >0$ and define $B_1 &:= \\left\\lbrace *{l_1}< \\varepsilon \\right\\rbrace \\\\B_2 &:= \\left\\lbrace *{l_2}< \\varepsilon \\right\\rbrace \\\\B_3 &:= \\left\\lbrace *{l_1-l_3}< \\varepsilon \\right\\rbrace \\\\B_4 &:= \\left\\lbrace *{l_1-l_2}< \\varepsilon \\right\\rbrace \\\\B_5 &:= \\left\\lbrace *{l_2-l_3}< \\varepsilon \\right\\rbrace \\\\B_6 &:= \\left\\lbrace *{l_3}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3-k}< \\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l_3+k}< \\varepsilon \\right\\rbrace ,$ and for $n=0,1,\\ldots ,6$ define $A_n := \\left\\lbrace (l_1,l_2,l_3)\\in \\left(\\mathbb {R}^{d}\\right)^3 \\colon \\#\\left\\lbrace m\\colon (l_1,l_2,l_3)\\in B_m \\right\\rbrace =n\\right\\rbrace .$ In words, $A_n$ is the set of points within $\\varepsilon $ (in Euclidean distance) of precisely $n$ of the hyperplanes on which the $\\widehat{G}_{\\mu _\\lambda }$ factors diverge.",
"If we are working with a factor of $\\widehat{G}_{\\mu _\\lambda }$ on the set $B_m$ associated with its relevant hyperplane, we use the bound $\\widehat{G}_{\\mu _\\lambda }(l)\\le 1/\\left(C_2*{l}^2\\right)$ , and if we are working off this set we use the bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ .",
"When we integrate over $A_0$ , we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ for all such factors and recycle the bound on the “only $\\widehat{}$ \" integrals from above.",
"The contribution from $A_0$ is therefore bounded by some constant multiple of $\\frac{1}{\\varepsilon ^{18}}g(d)^3.$ For $A_1$ , first suppose we are considering $A_1\\cap B_m$ for $m\\ne 6$ .",
"First we bound $\\widehat{G}_{\\mu _\\lambda }(l) \\le 1/\\left(C_2\\varepsilon ^2\\right)$ and $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) \\le 3/\\left(C_2\\varepsilon ^2\\right)$ for the “irrelevant\" directions, and use the pointwise bound for $\\left<\\widehat{}(l)^2\\right>_{m,n}$ in the “relevant\" direction.",
"Then we can use the Cauchy-Schwarz and factorise argument we used above and recycle the integral bounds for the $\\widehat{}$ factors as we did above.",
"However we are careful about the way we split apart terms in the Cauchy-Schwarz step.",
"We split the $\\widehat{G}_{\\mu _\\lambda }$ terms across both factors and arrange the remaining $\\widehat{}$ terms so that each factor has “directions\" spanning $\\left(\\mathbb {R}^d\\right)^3$ .",
"To demonstrate we consider $A_1\\cap B_1$ : $&\\int _{A_1\\cap B_1} \\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>^{\\frac{1}{2}}_{5,6,7,8} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12} \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}} \\nonumber \\\\& \\hspace{56.9055pt} \\le C^2\\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\left<\\widehat{}(l_2)^2\\right>_{5,6}\\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\& \\hspace{142.26378pt}\\times \\left(\\int _{A_1\\cap B_1}\\widehat{G}_{\\mu _\\lambda }(l_1)^2 \\left<\\widehat{}(l_2)^2\\right>_{7,8} \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.$ After a change of variables, factorising and using our integral bounds of $\\widehat{}$ terms from above, we are only left needing to bound $\\int _{*{l}<\\varepsilon } \\widehat{G}_{\\mu _\\lambda }(l)^2 \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\le \\frac{1}{C^2_2} \\int _{*{l}<\\varepsilon } \\frac{1}{*{l}^4} \\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^2_2}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-4}}{d-4}.$ Therefore the contribution from $A_1\\cap B_m$ for $m\\ne 6$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ .",
"For $A_1\\cap B_6$ we need to be slightly more careful because for $*{k}<2\\varepsilon $ the constituent parts of $B_6$ overlap - this is the same considerations we had to account for in the proof of Lemma REF .",
"We can overcome it in the same way by using Hölder's inequality (actually just Cauchy-Schwarz in this case) and then re-centring the integrals.",
"The net result is that the bound of a constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2$ also applies to $A_1\\cap B_6$ .",
"In summary, the contribution from $A_1$ is bounded by some constant multiple of $\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\frac{\\varepsilon ^{d-18}}{d-4}g(d)^2.$ The $A_2$ case proceeds similarly.",
"After partitioning the set according to which are the overlapping hyperplanes, we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions and one $\\widehat{}$ factor from a linearly independent direction, and pointwise bound all the other $\\widehat{G}_{\\mu _\\lambda }$ and $\\widehat{}$ factors.",
"The contribution from $A_2$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-4\\right)^2}g(d),$ arising from the $A_1\\cap B_1\\cap B_2$ , $A_1\\cap B_1\\cap B_6$ , and $A_1\\cap B_2\\cap B_6$ integrals.",
"The $A_3$ case is more complicated because there are two qualitatively different sub-cases to consider.",
"First suppose that the three normal vectors to the overlapping hyperplanes are linearly independent.",
"Then we take the relevant $\\widehat{G}_{\\mu _\\lambda }$ factors from these directions, pointwise bound the other $\\widehat{G}_{\\mu _\\lambda }$ factors and all the $\\widehat{}$ factors.",
"The contribution from this sub-case of $A_3$ is then bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3}$ , arising from the $A_1\\cap B_1\\cap B_2\\cap B_6$ integral.",
"On the other hand, if the three normal vectors to the overlapping hyperplanes are linearly dependent, then we need to retain a $\\widehat{}$ from the remaining linearly independent direction.",
"To demonstrate, let us consider the $A_3\\cap B_1 \\cap B_2 \\cap B_4$ integral.",
"After removing unnecessary terms via uniform bounds, $&\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>^{\\frac{1}{2}}_{13,14} \\left<\\widehat{}(l_1-l_3)^2\\right>^{\\frac{1}{2}}_{11,12}\\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\& \\qquad \\le \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_1-l_3)^2\\right>_{11,12} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^\\frac{1}{2}.$ Applying a change of variables, factorising the integral, and applying Hölder's inequality allows us to bound these parentheses: $& \\int _{A_3\\cap B_1 \\cap B_2 \\cap B_4}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\left<\\widehat{}(l_2-l_3)^2\\right>_{13,14} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)\\left(\\int \\left<\\widehat{}(l)^2\\right>_{13,14}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^4}\\frac{1}{*{l_2}^4}\\frac{1}{*{l_1-l_2}^2} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right) g(d)\\nonumber \\\\& \\qquad \\le \\frac{1}{C^5_2}\\left(\\int _{\\left\\lbrace *{l_1}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_1}^6}\\frac{1}{*{l_2}^3} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{2}{3}}\\left(\\int _{\\left\\lbrace *{l_2}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l_1-l_2}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l_2}^6}\\frac{1}{*{l_1-l_2}^6} \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2}{\\left(2\\pi \\right)^{2d}}\\right)^{\\frac{1}{3}} g(d)\\nonumber \\\\& \\qquad = \\frac{1}{C^5_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-6}\\frac{\\varepsilon ^{d-6}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{4}{3}}\\left(\\frac{\\mathfrak {S}_{d-1}}{d-3}\\frac{\\varepsilon ^{d-3}}{\\left(2\\pi \\right)^d}\\right)^{\\frac{2}{3}}g(d).$ For all of these linearly dependent cases, we can bound the integral by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d)$ .",
"In summary, the contribution from $A_3$ is bounded by some constant multiple of $\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\frac{\\varepsilon ^{3d-18}}{\\left(d-4\\right)^3} + \\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^2\\frac{\\varepsilon ^{2d-18}}{\\left(d-6\\right)^2}g(d).$ For $A_4$ , $A_5$ , and $A_6$ we don't have the two linearly dependent/independent cases because any four of the directions spans $\\left(\\mathbb {R}^d\\right)^3$ .",
"We therefore don't need factors of $\\widehat{}$ to control the unbounded directions.",
"Instead, we need to be careful that when we split our integral at the Cauchy-Schwarz step the resulting parentheses have sufficiently few factors of $\\widehat{G}_{\\mu _\\lambda }$ that their bounds will be finite for $d>6$ .",
"The way we perform this split will be different for the different parts of $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l_3;k)$ .",
"Let us define $\\widehat{D}(l;k) := \\widehat{G}_{\\mu _\\lambda }(l+k) + \\widehat{G}_{\\mu _\\lambda }(l-k).$ We first consider $A_6$ .",
"By writing $\\widehat{\\mathcal {G}}_{\\mu _\\lambda }(l;k) = \\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k) + \\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)$ , we have two integrals we want to bound.",
"First $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{D}(l_3;k)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)^2\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.",
"$ We then perform the $l_i$ integrals in specific orders.",
"For the first factor we integrate over $l_3$ (which factorises out already), then we integrate over $l_2$ for fixed $l_1$ , and finally integrate over $l_1$ .",
"For the second factor we integrate over $l_1$ for fixed $l_2$ and $l_3$ , then we integrate over $l_3$ for fixed $l_2$ , and finally we integrate over $l_2$ .",
"There are therefore three forms of integral we need to bound: $\\int _{*{l}<\\varepsilon }\\widehat{G}_{\\mu _\\lambda }(l)^3\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{1}{C^3_2}\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\frac{1}{*{l}^2}\\frac{1}{*{l-l^{\\prime }}^4}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\nonumber \\\\&\\le \\frac{1}{C^3_2}\\left(\\int _{*{l}<\\varepsilon }\\frac{1}{*{l}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{1}{3}\\left(\\int _{*{l-l^{\\prime }}<\\varepsilon }\\frac{1}{*{l-l^{\\prime }}^6}\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}}\\right)^\\frac{2}{3}\\nonumber \\\\& = \\frac{1}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\frac{\\varepsilon ^{d-6}}{d-6}\\\\\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l)\\widehat{D}(l;k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} &\\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}.$ In this last inequality, $C^{\\prime }>0$ is some uniform constant.",
"The last inequality is derived by applying Hölder's inequality in much the same way as for the second inequality, taking care to see when the three sets that are integrated over are overlapping.",
"The calculation uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .",
"The second integral we want to bound for $A_6$ is $&\\int _{A_6}\\widehat{G}_{\\mu _\\lambda }(l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_2)^2\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)\\widehat{G}_{\\mu _\\lambda }(l_1-l_3)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k) \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\nonumber \\\\&\\qquad \\le \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\widehat{G}_{\\mu _\\lambda }(l_3-k)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)\\widehat{G}_{\\mu _\\lambda }(l_2-l_3)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)^3\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}\\nonumber \\\\&\\qquad \\qquad \\times \\left(\\int _{A_6}\\left[\\widehat{G}_{\\mu _\\lambda }(l_2)^3\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_1)\\widehat{G}_{\\mu _\\lambda }(l_1-l_2)^2\\right]\\left[\\widehat{G}_{\\mu _\\lambda }(l_3-l_1)^2\\widehat{G}_{\\mu _\\lambda }(l_3+k)\\right] \\frac{\\mathrm {d}l_1 \\mathrm {d}l_2 \\mathrm {d}l_3}{\\left(2\\pi \\right)^{3d}}\\right)^{\\frac{1}{2}}.",
"$ We then perform the $l_i$ integrals in specific orders.",
"For the first factor we integrate over $l_1$ (which factorises out already), then we integrate over $l_2$ for fixed $l_3$ , and finally integrate over $l_3$ .",
"For the second factor we integrate over $l_3$ for fixed $l_1$ and $l_2$ , then we integrate over $l_1$ for fixed $l_2$ , and finally we integrate over $l_2$ .",
"The new integrals we need to bound are: $\\int _{\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-k)^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4} + \\frac{1}{d-2} + \\frac{1}{d}\\right)\\varepsilon ^{d-6}$ $\\int _{\\left(\\left\\lbrace *{l}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l-k}<\\varepsilon \\right\\rbrace \\cup \\left\\lbrace *{l+k}<\\varepsilon \\right\\rbrace \\right)\\cap \\left\\lbrace *{l-l^{\\prime }}<\\varepsilon \\right\\rbrace }\\widehat{G}_{\\mu _\\lambda }(l+k)\\widehat{G}_{\\mu _\\lambda }(l-l^{\\prime })^2\\frac{\\mathrm {d}l}{\\left(2\\pi \\right)^{d}} \\\\ \\le \\frac{C^{\\prime }}{C^3_2}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)\\left(\\frac{1}{d-6} + \\frac{1}{d-4}\\right)\\varepsilon ^{d-6}.$ The calculation of these two bounds uses Hölder's inequality and a partition of the space, and is similar to the calculation performed in the proof of Lemma REF .",
"The result of these bounds is that the integral in (REF ) is bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ For $A_4$ and $A_5$ we can repeat the argument for $A_6$ , but note that in some places the bound $1/\\left(C_2*{l}^2\\right)$ will be replaced by $1/\\left(C_2\\varepsilon ^2\\right)$ .",
"The net result of this is that the factor $\\left(d-6\\right)^3$ can be replaced by $\\left(d-6\\right)^{\\alpha _0}\\left(d-4\\right)^{\\alpha _1}\\left(d-2\\right)^{\\alpha _2} d^{\\alpha _3}$ for some $\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _3\\le 0$ such that $\\alpha _0+\\alpha _1+\\alpha _2+\\alpha _3 =3$ .",
"The net result is that the contribution from both $A_4$ and $A_5$ can also be bounded by some constant multiple of $\\frac{1}{\\left(d-6\\right)^3}\\left(\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\right)^3\\varepsilon ^{3d-18}.$ Above we have established bounds for the various parts of the integral when $\\vec{j}=\\left(3,3,3,3,3,3,3,5\\right)$ .",
"Having $j_8=4$ can also be dealt with by very similar arguments to that outlined above - the $l_3$ integrals are in fact simpler.",
"The above bounds still hold with the $\\varepsilon ^{-18}$ becoming a $\\varepsilon ^{-16}$ .",
"Since $\\varepsilon <1$ , this produces a larger bound.",
"Recall that up to a constant factor, having $j_m=1$ produces a larger bound than $j_m=2$ for $m=1,\\ldots ,7$ .",
"If we replace $j_m=3$ with $j_m=1$ , then after ignoring constant values we lose a factor of $\\varepsilon ^{-2}$ , but we may also lose a factor of $g(d)^\\frac{1}{2}$ unless that was for $m=1,2$ (corresponding to $l_1$ direction) or $m=3,4$ (corresponding to $l_2$ direction).",
"Note that doing for both directions will lose a $g(d)^\\frac{1}{2}$ factor.",
"If $\\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d= O\\left(g(d)^{\\frac{1}{2}}\\right)$ for all fixed $\\varepsilon >0$ , then it is clear that the dominating bound is $g(d)^\\frac{1}{2}$ .",
"This arises, for example from the bound for $\\vec{j}=\\left(1,1,1,1,1,1,1,1\\right)$ .",
"The situation is more complicated if $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , because we can choose to have $\\varepsilon =\\varepsilon (d)$ and take it to 0 as $d\\rightarrow \\infty $ .",
"It is easy to see that all the $A_1$ components (and $A_2,\\ldots ,A_6$ components) for each $\\vec{j}$ can be bounded by some constant multiple of $\\frac{1}{d}\\frac{\\mathfrak {S}_{d-1}}{\\left(2\\pi \\right)^d}\\varepsilon ^d$ (provided $d>6$ ).",
"We therefore only need to compare this to the contributions from $A_0$ .",
"The bound we get from the above arguments for each $A_0$ contribution are of the form $\\varepsilon ^{-2m}g(d)^\\frac{n}{2}$ , where $m$ is the number of factors of $\\widehat{G}_{\\mu _\\lambda }$ and $n$ is related to the multiplicity of $$ terms in each Fourier direction.",
"To find the terms with the largest bound, we want to have as many factors of $\\widehat{G}_{\\mu _\\lambda }$ as possible without producing extra factors of $g(d)^\\frac{1}{2}$ .",
"Since $g(d)^{\\frac{1}{2}}\\ll \\frac{1}{d}\\mathfrak {S}_{d-1}\\varepsilon ^d$ for fixed $\\varepsilon $ , increasing $n$ here instantly produces a smaller bound.",
"By changing $j_8$ from 1 to 4 or 5, or any other element from 1 to 3, we do gain a factor of $\\widehat{G}_{\\mu _\\lambda }$ but we also increase the number of $$ factors and therefore in most places we acquire extra factors of $g(d)^{\\frac{1}{2}}$ .",
"The exception to this is for $j_1$ and $j_2$ , which correspond to the $l_1$ directions, since they already produce a factor of $g(d)^{\\frac{1}{2}}$ .",
"The $A_0$ component of the $\\vec{j}=\\left(3,3,1,1,1,1,1,1\\right)$ term can be bounded by constant multiple of $\\varepsilon ^{-4}g(d)^{\\frac{1}{2}}$ .",
"We then optimise our choice of $\\varepsilon $ to have our two dominant bounds be of the same order.",
"This produces $\\varepsilon (d)^2 = g(d)^{\\frac{1}{d}}\\frac{2\\pi d}{\\text{e}}\\left(1+o(1)\\right)$ , and therefore our overall bound is given by $\\varepsilon ^{-4}g(d)^\\frac{1}{2} = g(d)^{\\frac{1}{2}-\\frac{2}{d}}\\left(\\frac{\\text{e}}{2\\pi }\\right)^2\\frac{1}{d^2}\\left(1+o(1)\\right).$ Since $g(d)^{\\frac{1}{2}-\\frac{2}{d}} \\le g(d)^{\\frac{1}{2}-\\frac{3}{d}}$ for sufficiently large $d$ , this bound is of order $\\beta ^2$ and our bound is proven."
],
[
"Acknowledgements.",
"This work is supported by Deutsche Forschungsgemeinschaft (project number 443880457) through priority program “Random Geometric Systems” (SPP 2265).",
"The authors thank the Centre de recherches mathématiques Montreal for hospitality during a research visit in spring 2022 through the Simons-CRM scholar-in-residence program."
]
] | 2210.07727 |
[
[
"Object-Category Aware Reinforcement Learning"
],
[
"Abstract Object-oriented reinforcement learning (OORL) is a promising way to improve the sample efficiency and generalization ability over standard RL.",
"Recent works that try to solve OORL tasks without additional feature engineering mainly focus on learning the object representations and then solving tasks via reasoning based on these object representations.",
"However, none of these works tries to explicitly model the inherent similarity between different object instances of the same category.",
"Objects of the same category should share similar functionalities; therefore, the category is the most critical property of an object.",
"Following this insight, we propose a novel framework named Object-Category Aware Reinforcement Learning (OCARL), which utilizes the category information of objects to facilitate both perception and reasoning.",
"OCARL consists of three parts: (1) Category-Aware Unsupervised Object Discovery (UOD), which discovers the objects as well as their corresponding categories; (2) Object-Category Aware Perception, which encodes the category information and is also robust to the incompleteness of (1) at the same time; (3) Object-Centric Modular Reasoning, which adopts multiple independent and object-category-specific networks when reasoning based on objects.",
"Our experiments show that OCARL can improve both the sample efficiency and generalization in the OORL domain."
],
[
"Introduction",
"Reinforcement Learning (RL) has achieved impressive progress in recent years, such as results in Atari [atari] and Go [go] in which RL agents even perform better than human beings.",
"Despite its successes, conventional RL is also known to be of low sample efficiency [rlintro] and poor generalization ability [procgen].",
"Object-oriented RL (OORL) [rmdp, oomdp, d-oomdp,schema,rrl] is a promising way to deal with these limitations.",
"Inspired by studies in cognitive science [piaget] that objects are the basic units of recognizing the world, OORL focuses on learning the invariance of objects' functionalities in different scenarios to achieve better generalization ability.",
"In OORL, the agent's observation is a set of object representations, and the task can be solved via reasoning based on these objects.",
"Most previous works [rmdp, schema, oomdp] in OORL employ hand-crafted object features given in advance, which require human expertise and result in low generality, limiting the applications of these methods.",
"Recent works [rrl, smorl, ics] try to broaden the scope of OORL's applications by avoiding using additional feature engineering.",
"Their works can be divided into either top-down approaches [rrl] which purely rely on reward signals to ground representations to objects, or bottom-up approaches [smorl, ics] which utilize unsupervised object discovery (UOD) technologies to provide structural object representations.",
"However, none of these works [rrl, smorl, ics] tries to explicitly model the inherent similarity between different object instances of the same category when reasoning based on objects.",
"In most cases, objects of the same category should share the same functionalities, such as obeying similar behaviour patterns or having the same effects when interacting with other objects.",
"Therefore, an intelligent agent should recognize an object's category to be aware of its functionalities.",
"Since objects from different categories differ in their functionalities, the objects' functionalities can also be separately modelled (according to their categories), leading to modularity which is beneficial for generalization [modularity, modularity2,naturetaskcluster] in general.",
"Following the above insight, we propose a framework named Object-Category Aware Reinforcement Learning (OCARL), which utilizes the category information of objects to facilitate both perception and reasoning.",
"OCARL consists of three parts: (1) category-aware UOD, which can automatically discover the objects as well as their corresponding categories via UOD and unsupervised clustering.",
"(2) Object-Category Aware Perception (OCAP), a perception module that takes the object category information obtained in (1) as an additional supervision signal.",
"OCAP can encode the category information into representations and is also robust to the incompleteness of UOD at the same time.",
"(3) Object-Centric Modular Reasoning (OCMR), a reasoning module that adopts multiple independent and object-category-specific networks, each of which processes the object features of the same corresponding category.",
"Such a modular mechanism can enhance further the generalization ability of OCARL.",
"Our experiments show that OCARL can improve the sample efficiency on several tasks and also enable the agent to generalize to unseen combinations of objects where other baselines fail."
],
[
"Unsupervised Object Discovery",
"Unsupervised object discovery (UOD) methods try to automatically discover objects without additional supervision.",
"Roughly speaking, there are two main categories in this area: spatial attention models and spatial mixture models.",
"Spatial attention models such as SCALOR [scalor] and TBA [tba] explicitly factorize the scene into a set of object properties such as position, scale and presence.",
"Then these properties are utilized by a spatial transformer network to select small patches in the original image, which constitute a set of object proposals.",
"These methods can deal with a flexible number of objects and thus are prominent in the UOD tasks.",
"Spacial mixture models such as Slot Attention [slotatten], IODINE [idonet] and MONet [monet] decompose scenes via clustering pixels that belong to the same object (often in an iterative manner).",
"However, they often assume a fixed maximum number of objects and thus cannot deal with a large number of objects.",
"SPACE [space], a UOD method used in this paper, combines both the spatial attention model and spatial mixture model.",
"The spatial attention model extracts objects from the scene, whereas the spacial mixture model is responsible for decomposing the remaining background.",
"Such a combination enables SPACE to distinguish salient objects from relative complex backgrounds."
],
[
"Object-Oriented Reinforcement Learning",
"Object-oriented reinforcement learning (OORL) has been widely studied in RL community.",
"There are many different structural assumptions on the underlying MDP in OORL, such as relational MDP [rmdp], propositional object-oriented MDPs [oomdp], and deictic object-oriented MDP [d-oomdp].",
"Although these assumptions differ in detail, they all share a common spirit that the state-space of MDPs can be represented in terms of objects.",
"Following these assumptions, [objq] proposes object-focused Q-learning to improve sample efficiency.",
"[schema] learns an object-based casual model for regression planning.",
"[newobj] improves the generalization ability to novel objects by leveraging relation types between objects that are given in advance.",
"Although these works have been demonstrated to be useful in their corresponding domain, they require explicit encodings of object representations, which limits their applications.",
"There are other lines of work in OORL such as COBRA [cobra], OODP [oodp], and OP3 [op3] that try to solve OORL tasks in an end-to-end fashion.",
"Most of these works fall into a model-based paradigm in which an object-centric model is trained and then utilized to plan.",
"Although these methods are effective in their corresponding domains, in this paper, we restrict ourselves to the model-free setting, which can generally achieve better asymptotic performance than model-based methods [rlsurvey, imagine].",
"The most related works should be RRL [rrl] and SMORL [smorl].",
"RRL proposes an attention-based neural network to introduce a relational inductive bias into RL agents.",
"However, since RRL relies purely on this bias, it may fail to capture the objects from the observations.",
"On the other hand, SMORL utilizes an advanced UOD method to extract object representations from raw images, and then these representations are directly used as the whole observations.",
"However, SMORL adopts a relatively simple reasoning module, which limits its applications in multi-object scenarios."
],
[
"Method",
"OCARL consists of three parts: (1) category-aware unsupervised object discovery (category-aware UOD) module, (2) object-category aware perception (OCAP), and (3) object-centric modular reasoning (OCMR) module.",
"The overall architecture is shown in Figure REF .",
"We will introduce (1)(2)(3) in Section REF , Section REF , and Section REF respectively."
],
[
"Unsupervised Object Discovery",
"In this work, SPACE [space] is utilized to discover objects from raw images.",
"In SPACE, an image $\\mathbf {x}$ is decomposed into two latent representations: foreground $\\mathbf {z}^{fg}$ and background $\\mathbf {z}^{bg}$ .",
"For simplicity, we only introduce the inference model of the foreground latent $\\mathbf {z}^{fg}$ , which is a set of object representations.",
"We encourage the readers to refer to [space] for more details.",
"To obtain $\\mathbf {z}^{fg}$ , $\\mathbf {x}$ is treated as if it were divided into $H\\times W$ cells and each cell is tasked with modelling at most one (nearby) object.",
"Therefore, $\\mathbf {z}^{fg}$ consists of a set of $H\\times W$ object representations (i.e.",
"$\\mathbf {z}^{fg} = \\lbrace \\mathbf {z}_{ij}^{fg}\\rbrace _{i=1}^H{}_{j=1}^W$ ), each of which is a 3-tuplewe omit $\\mathbf {z}_{ij}^{depth}$ for simplicity, because we only consider 2-D environment without object occlusions in this paper.",
"$\\mathbf {z}_{ij}^{fg} = (z_{ij}^{pres}, \\mathbf {z}_{ij}^{where}, \\mathbf {z}_{ij}^{what})$ .",
"$z_{ij}^{pres}\\in \\lbrace 0,1\\rbrace $ is a 1-D variable that indicates the presence of any object in cell $(i,j)$ , $\\mathbf {z}_{ij}^{where}$ encodes $\\mathbf {z}_{ij}^{fg}$ 's size and location, and $\\mathbf {z}_{ij}^{what}$ is a latent vector that identifies $\\mathbf {z}_{ij}^{fg}$ itself.",
"The whole inference model of $\\mathbf {z}^{fg}$ can be written as: $\\begin{aligned}q(\\mathbf {z}^{fg}|\\mathbf {x}) = \\prod _{i=1}^{H}\\prod _{j=1}^W q(z_{ij}^{pres}|\\mathbf {x}) (q(\\mathbf {z}_{ij}^{where}|\\mathbf {x}) q(\\mathbf {x}_{ij} | \\mathbf {z}_{ij}^{where}, \\mathbf {x}) q(\\mathbf {z}_{ij}^{what}|\\mathbf {x}_{ij}))^{z_{ij}^{pres}},\\end{aligned}$ where $\\mathbf {x}_{ij}$ is a small patch from the $\\mathbf {x}$ that is obtained by the proposal bounding box identified by $\\mathbf {z}_{ij}^{where}$ , which should contain exactly one object if $z_{ij}^{pres}=1$ ."
],
[
"Unsupervised Clustering",
"After applying SPACE on the randomly collected dataset, we can obtain a set of object representations: $D=\\lbrace z_{nij}^{pres}, \\mathbf {z}_{nij}^{where}, \\mathbf {z}_{nij}^{what}, \\mathbf {x}_{nij}\\rbrace _{n=1}^N{}_{i=1}^H{}_{j=1}^W$ , where $N$ is the size of the dataset, $z_{nij}^{pres}, \\mathbf {z}_{nij}^{where}, \\mathbf {z}_{nij}^{what}, \\mathbf {x}_{nij}$ are defined in Eq.",
"(REF ).",
"We first select object patches from $D$ that is of high object-presence probability by a threshold $\\tau $ : $D_\\tau = \\lbrace \\mathbf {x}_{nij}\\in D: p(z_{nij}^{pres}=1) > \\tau \\rbrace $ .",
"Roughly speaking, each patch $\\mathbf {x}_{nij}\\in D_\\tau $ should contain exactly one object.",
"Therefore, we first project $\\mathbf {x}_{nij}\\in D_\\tau $ into a low-dimensional latent space $\\mathbf {R}^d$ by running IncrementalPCA [ipca] on $D_\\tau $ , and then adopt KMeans clustering with a given cluster number $C$ upon the projected latent representations.",
"The object-category predictor $q(\\mathbf {z}_{ij}^{cat}|\\mathbf {x}_{ij}) = \\Psi _\\texttt {KMeans}\\circ \\Psi _\\texttt {IncrementalPCA}$ is the composition of IncrementalPCA and KMeans: $\\begin{aligned}\\mathbf {z}_{ij}^{cat} = \\Psi _{\\texttt {KMeans}}\\circ \\Psi _{\\texttt {IcreamentalPCA}} (\\mathbf {x}_{ij})\\end{aligned}$ where $\\mathbf {z}_{ij}^{cat}\\in \\mathbf {R}^C$ is an one-hot latent which indicates the category of $\\mathbf {z}_{ij}$ .",
"For simplicity, we regard the background (i.e.",
"$z_{ij}^{pres}=0$ ) as special 'object', thus $\\mathbf {z}_{ij}^{cat}$ can be written as $\\mathbf {\\hat{z}}_{ij}^{cat}=\\Psi _{\\texttt {concatenate}}([1-z^{pres}_{ij}, z^{pres}_{ij}\\cdot \\mathbf {z}_{ij}^{cat}])$ .",
"Note that $\\hat{\\mathbf {z}}_{ij}^{cat}$ is an one-hot latent $\\in \\mathbf {R}^{C+1}$ .",
"Although the object-category predictor Eq.",
"(REF ) is very simple, we find it works well in our experiments.",
"In practice, the cluster number $C$ of KMeans can be set by leveraging prior knowledge about ground-truth category number, or by analysing some clustering quality metrics such as silhouette coefficients.",
"In more complex environments, we can also rely on other advanced unsupervised clustering methods such as NVISA [cluster1], CC [cluster2], etc.",
"Figure: OCARL consists of 3 parts: (1) category-aware UOD, (2) OCAP, and (3) OCMR.",
"(1) In category-aware UOD, the objects are firstly discovered by a UOD method (SPACE) and then assigned an (extended) category label by unsupervised clustering (IcreamentalPCA + KMeans).",
"(2) In OCAP, the observation is first encoded into a set of features 𝐙\\mathbf {Z} by convolution networks.",
"Each vector in 𝐳 i =𝐙 :,i,j ∈𝐙\\mathbf {z}_i=\\mathbf {Z}_{:,i,j}\\in \\mathbf {Z} is given a task by the OCAP module to predict the corresponding (extended) object category, which is given by (1).",
"𝐙\\mathbf {Z} is then fed to the OCMR module.",
"(3) In OCMR, x-y coordinate information is added into 𝐙\\mathbf {Z}, which gives us 𝐙 ^\\mathbf {\\widehat{Z}}.",
"Then we apply self-attention on 𝐙 ^\\mathbf {\\widehat{Z}} to query related information 𝐐 ^\\mathbf {\\widehat{Q}} for each object in 𝐙 ^\\mathbf {\\widehat{Z}}.",
"𝐐 ^\\mathbf {\\widehat{Q}} and 𝐙 ^\\mathbf {\\widehat{Z}} are concatenated to form 𝐙 ˜\\mathbf {\\widetilde{Z}}.",
"Each entity in 𝐙 ˜\\mathbf {\\widetilde{Z}} will be passed through different independent neural networks according to its corresponding object category, which will give us 𝐙 out \\mathbf {Z}^{out}.",
"Finally, we perform a max operation along the H,WH,W dimension of 𝐙 out +𝐙 ^\\mathbf {Z}^{out} + \\mathbf {\\widehat{Z}}, and use the resulting feature vector to predict the value function and action probabilities."
],
[
"OCAP: Object-Category Aware Perception",
"The OCAP module is designed to robustly incorporate the object knowledge of the (category-aware) UOD model defined in Section REF .",
"In a word, OCAP is a plain convolution encoder that accepts the category information from the UOD model as additional supervision signals.",
"Such a design makes OCARL robust to the incompleteness of the UOD model: in extreme cases in which the UOD model fails to discover any objects, OCAP degenerates into a plain convolution encoder which is still able to extract useful information from the raw images with the help of reward signals.",
"Suppose the convolution encoder in OCAP maps the raw image observation $\\mathbf {X} \\in \\mathbf {R}^{3\\times H_{img}\\times W_{img}}$ to a latent representation $\\mathbf {Z}\\in \\mathbf {R}^{F\\times H\\times W}$ , where $H, W$ are the same with those in the UOD model (see Eq.",
"(REF )).",
"On the other hand, $\\mathbf {X}$ is also fed into the UOD model to get the (extended) object category information $\\hat{\\mathbf {z}}^{cat} = \\lbrace \\hat{\\mathbf {z}}_{ij}^{cat}\\rbrace _{i=1}^H{}_{j=1}^W$ via Eq.",
"(REF ).",
"The OCAP module forces the latent representation $\\mathbf {Z}$ to encode the object category information, which is implemented by training a additional category predictor $f_{cat}: \\mathbf {R}^F\\rightarrow \\texttt {Categorical}(C+1)$ .",
"$f_{cat}$ predicts $\\hat{\\mathbf {z}}_{ij}^{cat}$ given $\\mathbf {Z}_{:,i,j}$ which is a single channel in $\\mathbf {Z}$ .",
"Therefore, the additional supervision signal that incorporates the object knowledge of the UOD model is given as: $\\begin{aligned}L_{cat} = \\sum _{i=1}^H\\sum _{j=1}^W \\texttt {CrossEntropyLoss}(f_{cat}(\\mathbf {Z}_{:,i,j}); \\hat{\\mathbf {z}}_{ij}^{cat}).\\end{aligned}$ $L_{cat}$ is used for training the convolution encoder in OCAP.",
"In practice, $L_{cat}$ can be used as an auxiliary loss to any RL algorithm: $\\begin{aligned}L_{total} = L_{RL} + \\lambda _{cat} L_{cat},\\end{aligned}$ where $\\lambda _{cat}\\in \\mathbf {R}^+$ is a coefficient."
],
[
"OCMR: Object-Centric Modular Reasoning",
"OCMR is a module that takes $\\mathbf {Z}\\in \\mathbf {R}^{F\\times H\\times W}$ from the OCAP module as input and outputs a feature vector that summarizes $\\mathbf {Z}$ .",
"The key design philosophy of OCMR is to adopt multiple independent and object-category-specific networks, each of which focuses on processing the object features of the same corresponding category.",
"Compared with using a universal category-agnostic network, the processing logic of each independent network in OCMR is much simpler and therefore easier to master, which in turn allows for improved generalization as we will show in Section REF .",
"This design philosophy also agrees with the recent discovery from [naturesepnet,modularity] which says that neural modules of specialization can lead to better generalization ability.",
"Given $\\mathbf {Z}\\in \\mathbf {R}^{F\\times H\\times W}$ from OCAP, we first encode the x-y coordinates information (corresponding to the $H\\times W$ grid) to each channel, and then map the resulting tensor into $\\widehat{\\mathbf {Z}}\\in \\mathbf {R}^{HW\\times F}$ .",
"$\\widehat{\\mathbf {Z}}$ is treated as $HW$ objects with x-y coordinate information encoded.",
"To model the relations between objects, we apply a self-attention [attention] module on $\\widehat{\\mathbf {Z}}$ to query information for each object from its related objects: $\\begin{aligned}\\widehat{\\mathbf {Q}} = \\texttt {softmax}(\\widehat{\\mathbf {Z}}\\mathbf {W}_q(\\widehat{\\mathbf {Z}}\\mathbf {W}_k)^T)\\widehat{\\mathbf {Z}}\\mathbf {W}_v \\in \\mathbf {R}^{HW\\times F},\\end{aligned}$ where $\\mathbf {W}_q,\\mathbf {W}_k,\\mathbf {W}_v\\in \\mathbf {R}^{F\\times F}$ are trainable parameters.",
"Modeling the relations is important, because one object's high-level semantic feature can be derived from other objects.",
"For example, whether a door can be opened is determined by the existence of the key; therefore, the door should query other objects to check whether it is openable.",
"After the attention module, $\\widehat{\\mathbf {Q}}$ and $\\widehat{\\mathbf {Z}}$ are concatenated together to get $\\widetilde{\\mathbf {Z}} = \\Psi _\\texttt {concatenate}([\\widehat{\\mathbf {Q}},\\widehat{\\mathbf {Z}}])\\in \\mathbf {R}^{HW\\times 2F}$ .",
"Each $\\widetilde{\\mathbf {Z}}_{i,j,:}$ in $\\widetilde{\\mathbf {Z}}$ is then fed into different independent neural networks according to its corresponding object category $\\hat{\\mathbf {z}}_{ij}^{cat}$ .",
"In practice, this can be implemented as: $\\begin{aligned}\\mathbf {Z}^{out}_{i,j,:} = \\sum _{c=1}^{C+1} f_c(\\widetilde{\\mathbf {Z}}_{i,j,:}) \\cdot \\hat{z}_{ij;c}^{cat},\\end{aligned}$ where $f_c:\\mathbf {R}^{2F} \\rightarrow \\mathbf {R}^F$ , $[\\hat{z}_{ij;1}^{cat},...,\\hat{z}_{ij;C+1}^{cat}] = \\hat{\\mathbf {z}}_{ij}^{cat}$ .",
"Eq.",
"(REF ) can be computed in parallel, resulting a tensor $\\mathbf {Z}^{out}\\in \\mathbf {R}^{H W\\times F}$ .",
"Finally, the $\\mathbf {Z}^{out}$ is added to $\\widehat{\\mathbf {Z}}$ (i.e.",
"a residual connection).",
"We perform a max operation along the $H,W$ dimension, and using the resulting vector $\\in \\mathbf {R}^F$ to predict the value function and action probabilities: $\\begin{aligned}\\texttt {action\\_probs, value} = f_{ac}(\\max _{H,W}(\\mathbf {Z}^{out} + \\widehat{\\mathbf {Z}})).\\end{aligned}$"
],
[
"Task Description",
"In this work, we consider two task domains: Crafter and Hunter.",
"The observations on both tasks are raw images of shape $64\\times 64$ ."
],
[
"Crafter",
"[crafter] is a complex 2-D MineCraft-style RL task, where complex behaviors are necessary for the agent' survival.",
"The environment is procedurally generated by arranging various resources, terrain types, and objects (18 in total).",
"The agent is rewarded if it can craft new items and accomplish achievements (22 possible achievements in total)."
],
[
"Hunter",
"is much simpler than Crafter.",
"It is also procedurally generated but only contains 4 types of objects: Hunter, Cow, Zombie and Wall The images to render these objects come from Crafter [crafter], which is under MIT license..",
"The agent can control Hunter and can get positive reward (=1) if Hunter catches a Cow or kill a Zombie.",
"The agent will be given a high reward (=5) if it can accomplish this task by catching/killing all Cows and Zombies.",
"However, once Hunter is caught by a Zombie, the agent will receive a negative reward (=-1), and the episode ends.",
"In a word, the agent in Hunter should master two different behavior patterns: chase & catch the Cow, and avoid & shoot at Zombie.",
"Although Hunter is simple, it is a typical OORL task.",
"We can derive different environment instances from Hunter by setting the number of Zombies and Cows.",
"We use Hunter-ZmCn to denote an environment that spawns (m Zombies + n Cows) at the beginning of each episode, and Hunter-ZmCn/ZnCm an environment that spawns (m Zombies + n Cows) or (n Zombies + m Cows)."
],
[
"Evaluation",
"In this section, we evaluate OCARL's effect both on sample efficiency and generalization ability.",
"We consider the following baselines: PPO [ppo], RRL [rrl], and SMORL [smorl].",
"PPO is a general-purpose on-policy RL algorithm with a plain neural network (i.e.",
"Convolution + MLP).",
"RRL proposes an attention-based neural network to introduce relational inductive biases and iterated relational reasoning into RL agent, which aims to solve object-oriented RL tasks without any other external supervision signals.",
"SMORL utilizes the UOD model to decompose the observation into a set of object representations which are directly used as the agent's new observation.",
"All algorithms (RRL, SMORL, and OCARL) are (re-)implemented upon PPO to ensure a fair comparison.",
"Note that OCARL is orthogonal to the backbone RL algorithm, therefore other advanced RL methods such as [rainbow,hvf] are also applicable.",
"For more implementation details, please refer to the Appendix."
],
[
"Sample Efficiency",
"The results on Hunter domain are shown in Figure REF .",
"We can conclude three facts from this figure: (1) OCARL performs better and is more stable than other baselines on all tasks.",
"(2) RRL is competitive with OCARL on Hunter-Z4C4 but significantly worse on other simper environments.",
"This is because RRL relies solely on the reward signals to ground the visual images into objects; therefore, it requires enough (rewarded) interactions between objects, which simpler environments such as Hunter-Z1C1 fail to supply.",
"(3) SMORL performs better than other baselines (PPO, RRL) on simple environments such as Hunter-Z1C0/Z0C1, Hunter-Z1C1 and Hunter-Z4C0/Z0C4, but unable to make progress in Hunter-Z4C4 which consists of more objects.",
"This is because SMORL adopts a simple reasoning module that may not be well-suited to multi-object reasoning.",
"The results on Crafter domain are shown in Table REF and Figure REF .",
"Crafter is a complex environment in that it features sparse (rewarded) interactions between objects and large object category numbers.",
"Besides, some objects are omitted by the UOD model (see Appendix) because they almost do not appear in the training dataset due to the insufficient exploration of the random policy.",
"Due to these features, both RRL and SMORL fail to provide noteworthy results (with scores of 4.22 and 3.94, respectively).",
"On the other hand, OCARL is able to present more meaningful behaviours (with a score of 12.31), such as collecting coal, defeating zombies, making wood pickaxes and so on, as shown in Figure REF .",
"The learning curves and detailed achievement success rates are provided in the Appendix.",
"Figure: Success rates (averaged across 12 seeds) on 22 achievements in Crafter.",
"These success rates are calculated using the final 1M environment steps during training.",
"OCARL presents more meaningful behaviours than other baselines, such as collecting coal, defeating zombies, making wood pickaxes and so on."
],
[
"Generalization",
"To evaluate the generalization ability, we train agents on Hunter-Z1C0/Z0C1, Hunter-Z1C1 and Hunter-Z4C0/Z0C4 separately, and then observe their test performance on Hunter-Z4C4.",
"In Hunter-Z1C1, the agent needs to generalize from a few objects to more objects.",
"In Hunter-Z4C0/Z0C4, the agent never observes the coincidence of Cow and Zombie.",
"The generalization from Hunter-Z1C0/Z0C1 to Hunter-Z4C4 is the most difficult.",
"Such a paradigm actually follows the out-of-distribution (OOD) setting, in that the object combination (4 Zombies + 4 Cows) is never seen in the training environments.",
"Therefore, although such generalization is possible for humans, it is much harder for RL algorithms.",
"As shown in Figure REF , OCARL is the only algorithm that make significant progress on the test environment.",
"Although other baselines can improve their training performance (Figure REF ), they are unable to generalize to the test environment.",
"The generalization performance of OCARL is most impressive on Hunter-Z1C1, because agent trained on Hunter-Z1C1 even achieves slightly better than that on Hunter-Z4C4 (i.e.",
"directly trained on the test environment, see the last figure in Figure REF .",
")."
],
[
"Robustness to the Incompleteness of Object Discovery Methods",
"OCARL relies on UOD methods to discover different objects from given observations, which are not guaranteed to discover all objects, especially in complex environments [benchmarkoom].",
"Therefore, one may naturally ask to what extent OCARL's performance is affected by such incompleteness.",
"To answer this question, we randomly mask out each object (by setting $z_{ij}^{pres}$ to 0) discovered by the UOD model with a given probability $p\\in [0,1]$ , and evaluate OCARL's performance after training 5M environment steps on Hunter.",
"Such random confusion of the UOD model will affect both the OCAP and OCMR modules of OCARL.",
"Note that OCARL($p=0$ ) is OCARL itself, and OCARL($p=1$ ) is exactly RRL This is because Eq.",
"(REF ) takes no effect and Eq.",
"(REF ) also degenerates into a single universal network because all object features are passed through the same network $f_{bg}$ which is corresponding to the 'background'..",
"The results are shown in Figure REF .",
"Although increasing $p$ does hinder OCARL's performance, OCARL can still benefit from the imperfect UOD model in that its performance is still better than RRL in general.",
"Such robustness should be attributed to the fact that OCARL does not directly use the output of the UOD model as observation but instead treats it as an additional supervision signal that help the encoder to capture object-category information, which gives OCARL a second chance to find other useful information from the raw image."
],
[
"OCMR Improves Generalization",
"One key advantage of OCARL is that it can generalize to unseen combinations of objects by leveraging the invariant relations between objects learnt during training.",
"We argue that such an advantage is established via OCMR, which adopts object-category-specific networks to deal with different objects.",
"To confirm our argument, we consider a variant of OCARL (i.e.",
"OCARL w/o OCMR) in which the OCMR module is removed and replaced by an RRL-like reasoning module which adopts a universal network (instead of multiple object-category-specific networks as OCARL does).",
"The comparison between OCARL w/ and w/o OCMR is shown in Figure REF .",
"OCARL w/o OCMR (almost completely) fails to generalize from Hunter-Z1C0/Z0C1, Hunter-Z4C0/Z0C4 to Hunter-Z4C4, while OCARL w/ OCMR does.",
"In Figure REF , we also consider a variant of OCARL (i.e.",
"OCARL w/o OCAP) in which the OCAP module is disabled by setting $\\lambda _{cat}=0$ in Eq.",
"(REF ).",
"In this paradigm, only OCMR is in effect.",
"As shown in Figure REF , OCMR alone can still bring benefits without the help of OCAP (although not as consistently as OCMR+OCAP ).",
"According to these results, we can conclude that OCMR is crucial in OCARL's success.",
"Table: The Modularity of OCMR.",
"We first train agent on Hunter-Z4C4 and then disable the module f Z f_Z (corresponding to Zombie) or f C f_C (corresponding to Cow) in OCMR (see Eq.()).",
"Each agent is evaluated on both Hunter-Z4C0 and Hunter-Z0C4, and the mean and standard deviation of 12 seeds are reported .",
"The results show that the behaviour patterns (chase & catch the Cow) and (avoid & shoot at Zombie) are encoded in f C f_C and f Z f_Z respectively."
],
[
"The Modularity of OCMR",
"OCMR adopts different modules to deal with object features from different categories.",
"In this section, we will show that each module in OCMR actually encodes an object-category-specific behaviour pattern.",
"As stated in Section REF , the agent should master two useful behaviour patterns in Hunter: (1) chase & catch the Cow, and (2) avoid & shoot at Zombie.",
"The existence of these patterns can be observed by evaluating the agent in Hunter-Z0C4 and Hunter-Z4C0 respectively, because agents that master pattern (1) (pattern (2)) should achieve better performance on Hunter-Z0C4 (Hunter-Z4C0).",
"In Table REF , we first train agent on Hunter-Z4C4 and then disable the module $f_Z$ (corresponding to Zombie) or $f_C$ (corresponding to Cow) by replacing $f_C$ or $f_E$ with $f_{bg}$ (corresponding to the 'background') in Eq.",
"(REF ).",
"The results shows that disabling $f_Z$ does not affect the performance on Hunter-Z0C4 ($8.29\\rightarrow 8.13$ ), which means that $f_Z$ is not relative to behaviour pattern (1).",
"Instead, $f_Z$ is corresponding to behaviour pattern (2) because we can observe a significant performance decline ($8.27\\rightarrow 1.17$ ) on Hunter-Z4C0 when we disable $f_Z$ .",
"Similar phenomenon can also be observed on $f_C$ ."
],
[
"Conclusion",
"In this paper, we propose OCARL, which utilize the category information of objects to facilitate both perception.",
"For the perception, we propose the OCAP, which enables the encoder to distinguish the categories of objects.",
"For the reasoning, we propose the OCMR, which adopts object-category-specific networks to deal with different objects.",
"Our experiments are carried out on Crafter and Hunter.",
"Experiments show that OCARL outperforms other baselines both on sample efficiency and generalization ability.",
"We also perform several ablation studies to show that (1) OCARL is robust to the incompleteness of UOD methods, (2) OCMR is critical to improving generalization, and (3) each module in OCMR actually encodes an object-category-specific behaviour pattern in Hunter."
],
[
"Limitation",
"The main limitation is that we only test OCARL's generalization ability to unseen object combinations, but not novel object instances.",
"In order to generalize to novel object instances that looks very different from old ones, the agent should interact with the novel objects first and then infer the their underlying categories according to these interactions.",
"Such a problem is likely to be in the meta-RL domain and needs more efforts to solve, which we would like to leave for future work."
],
[
"Acknowledgements",
"This work is partially supported by the National Key Research and Development Program of China(under Grant 2018AAA0103300), the NSF of China(under Grants 61925208, 62102399, 62002338, U19B2019, 61906179, 61732020), Strategic Priority Research Program of Chinese Academy of Science (XDB32050200), Beijing Academy of Artificial Intelligence (BAAI) and Beijing Nova Program of Science and Technology (Z191100001119093) , CAS Project for Young Scientists in Basic Research(YSBR-029), Youth Innovation Promotion Association CAS and Xplore Prize.",
"numbers Checklist For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?",
"Did you describe the limitations of your work?",
"Did you discuss any potential negative societal impacts of your work?",
"Have you read the ethics review guidelines and ensured that your paper conforms to them?",
"If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?",
"Did you include complete proofs of all theoretical results?",
"If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?",
"Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?",
"see Appendix.",
"Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?",
"Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?",
"If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...",
"If your work uses existing assets, did you cite the creators?",
"Did you mention the license of the assets?",
"Partially.",
"Did you include any new assets either in the supplemental material or as a URL?",
"Did you discuss whether and how consent was obtained from people whose data you're using/curating?",
"Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?",
"If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?",
"Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?",
"Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?",
"Implementation Details The implementation of OCARL is available at https://anonymous.4open.science/r/OCARL-51BF.",
"Hyper-parameters for SPACE In this paper, we use SPACE for object discovery.",
"Most hyper-parameters are the same as the default parameters for Atari in SPACE, except that (1) we use a smaller network to deal with $64\\times 64$ images; (2) For Hunter, we disable the background module in SPACE because the observations in Hunter is quite simple.",
"(3) We set the dimension of object representations $\\mathbf {z}^{what}_{ij}$ to be 16, and (in Crafter) $\\textbf {z}^{fg}$ to be 8.",
"The training data for SPACE is obtained via running a random policy on Hunter-Z4C4 and Crafter for 100000 environment steps.",
"Hyper-parameters for PPO Our PPO implementation is based on Tianshou [tianshou] which is purely based on PyTorch.",
"We adopt the default hyper-parameters in Tianshou, which are shown in Table REF .",
"Hyper-parameters for OCARL OCARL only introduces 2 hyper-parameter $\\lambda _{cat}$ (the coefficient for $L_{cat}$ defined in Eq.",
"(REF )) and $C$ (the number of object categories).",
"We set $\\lambda _{cat}=0.01$ for all experiments, $C=4,8$ for Hunter and Crafter respectively.",
"Network Architecture In OCAP, we use the convolution encoder from IMPALA [impla], which consists of 3 residual blocks with 16, 32, and 32 channels, respectively.",
"By the encoder, the $3\\times 64\\times 64$ image observation is encoded into $32\\times 8\\times 8$ .",
"The category predictor $f_{cat}$ is a simple 2-layer MLP with hidden size 32.",
"The object-category-specific networks $f_1,...,f_{C+1}$ in the OCMR module consists of $C+1$ MLPs, each of which is of 2-layer with hidden size 64.",
"Implementation for RRL The main implementation differences between RRL and OCARL are that (1) RRL does not use the supervision signals from the UOD model (i.e.",
"$\\lambda _{cat}=0$ in Eq.",
"(REF )); (2) RRL adopts a different (although very similar) network architecture.",
"In RRL, we use the same convolution encoder as OCARL.",
"The reasoning module in RRL is similar to OCARL, except that it utilizes a non-linear universe transformation instead of multiple object-category-specific networks.",
"We also use only one relational block in RRL as OCARL does, because we found more blocks may hinder the performance in our early experiments.",
"Implementation for SMORL We use SPACE [space] instead of SCALOR [scalor] (as suggested in the SMORL paper) to obtain object representations for SMORL.",
"Since SMORL is originally designed for goal-based RL, it uses an attention model to gather information from the set of object representations, in which the goal serves as a query vector.",
"In our settings, we do not consider goal-based RL; therefore, we use $L$ learnable vectors as queries, which are also used in the original SMORL.",
"We search over $L=[1,2,4,8]$ and find that $L=4$ works best in our experiments.",
"We also find that it is better to apply an MLP to the object representations before they are fed into the attention module.",
"To be more specific, we first run SPACE to get the object representations, sharing the same procedure as OCARL.",
"The information of these objects is first processed by a 2-layer MLP with hidden size 32 and then gathered through an attention module in which the query consists of $L=4$ learnable vectors.",
"The resulting tensor (of shape $L\\times 32$ ) is flattened and then fed into a 2-layer MLP with hidden size 64, which finally output the value function and action probabilities.",
"Table: PPO hyper-parameters.",
"Details of SPACE OCARL utilizes SPACE to discovery objects from raw objects.",
"In Section REF , we have introduced the inference model of $\\mathbf {z}_{}^{fg}$ .",
"For completeness, we would like to introduce the remaining parts of SPACE in this section: (1) inference model of $\\mathbf {z}_{}^{bg}$ ; (2) generative model of the image $\\mathbf {x}$ ; and (3) the ELBO to train the model.",
"Inference model of $\\mathbf {z}_{}^{bg}$ $\\mathbf {z}_{}^{bg}$ consists of several components $\\mathbf {z}_{k}^{bg}$ , and these components are inferred from the image $\\mathbf {x}$ in an iterative manner: $q(\\mathbf {z}_{k}^{bg}|\\mathbf {x})=\\prod _{k=1}^{K}q(\\mathbf {z}_{k}^{bg}|\\mathbf {z}_{<k}^{bg},\\mathbf {x})$ .",
"Each component $\\mathbf {z}_{k}^{bg}$ is futher divided into two parts: $\\mathbf {z}_{k}^{bg}=(\\mathbf {z}_{k}^{m}, \\mathbf {z}_{k}^{c})$ , where $\\mathbf {z}_{k}^{m}$ models the mixing weight assigned to the background component (see $\\pi _k$ in the generative model of $\\mathbf {x}$ ), and $\\mathbf {z}_{k}^{c}$ models the RGB distribution ($p(\\mathbf {x}|\\mathbf {z}_{k}^{bg})$ ) of the background component.",
"Generative model of $\\mathbf {x}$ The generative model consists of two parts: $p(\\mathbf {x}|\\mathbf {z}_{}^{fg})$ and $p(\\mathbf {x}|\\mathbf {z}_{}^{bg})$ .",
"For $p(\\mathbf {x}|\\mathbf {z}_{}^{fg})$ , each $\\mathbf {z}_{ij}^{fg}$ is passed through a decoder to reconstruct the image patch determined by $\\mathbf {z}_{ij}^{where}$ .",
"For $p(\\mathbf {x}|\\mathbf {z}_{}^{bg})$ , each $\\mathbf {z}_{k}^{bg}$ is decoded into a background component and all components are mixed together to get the background.",
"Foreground and background are combined with a pixel-wise mixture model to reconstruct the original image $\\mathbf {x}$ .",
"The whole generative model of $\\mathbf {x}$ is: $\\begin{aligned}p(\\mathbf {x}) &= \\int \\int p(\\mathbf {x}|\\mathbf {z}_{}^{fg}, \\mathbf {z}_{}^{bg}) p(\\mathbf {z}_{}^{fg})p(\\mathbf {z}_{}^{bg})d\\mathbf {z}_{}^{fg}d\\mathbf {z}_{}^{bg}\\\\p(\\mathbf {x}|\\mathbf {z}_{}^{fg}, \\mathbf {z}_{}^{bg}) &= \\alpha p(\\mathbf {x}|\\mathbf {z}_{}^{fg}) + (1-\\alpha ) \\sum _{k=1}^K \\pi _k p(\\mathbf {x}|\\mathbf {z}_{k}^{bg}), \\alpha =f_\\alpha (\\mathbf {z}_{}^{fg}), \\pi _k=f_{\\pi _k}(\\mathbf {z}_{1:k}^{bg})\\\\p(\\mathbf {z}_{}^{fg}) &= \\prod _{i=1}^{H}\\prod _{j=1}^{W} p(z_{ij}^{pres}) (p(\\mathbf {z}_{ij}^{where})p(\\mathbf {z}_{ij}^{what}))^{z_{ij}^{pres}}\\\\p(\\mathbf {z}_{}^{bg}) &= \\prod _{k=1}^K p(\\mathbf {z}_{k}^{c}|\\mathbf {z}_{k}^{m}) p(\\mathbf {z}_{k}^{m}|\\mathbf {z}_{<k}^{m})\\end{aligned}$ In above equations, $z_{ij}^{pres}, \\mathbf {z}_{ij}^{where}, \\mathbf {z}_{ij}^{what}$ have been discussed in Section REF , $\\alpha $ is foreground mixing probability, and $\\pi _k$ is the mixing weight assigned to the background component $\\mathbf {z}_{k}^{bg}$ .",
"The ELBO to train the model SPACE is trained using the following ELBO via reparameterization tricks: $\\begin{aligned}\\mathcal {L}(\\mathbf {x}) = \\mathbb {E}_{q\\left(\\mathbf {z}^{fg}, \\mathbf {z}^{bg} \\mid \\mathbf {x}\\right)}\\left[p\\left(\\mathbf {x} \\mid \\mathbf {z}^{fg}, \\mathbf {z}^{bg}\\right)\\right]-D_{\\mathrm {KL}}\\left(q\\left(\\mathbf {z}^{fg} \\mid \\mathbf {x}\\right) \\Vert p\\left(\\mathbf {z}^{fg}\\right)\\right)-D_{\\mathrm {KL}}\\left(q\\left(\\mathbf {z}^{bg} \\mid \\mathbf {x}\\right) \\Vert p\\left(\\mathbf {z}^{bg}\\right)\\right)\\end{aligned}$ Figure: The learning curves on Crafter.",
"The mean returns over 12 seeds are plotted with a 95% confidence interval.",
"Both RRL and SMORL fail to make progress after the initial 5M environment steps, while OCARL does.",
"More Results on Crafter In Figure REF , we plot the learning curves on the Crafter domain.",
"Only OCARL is able to make progress continuously.",
"Both RRL and SMORL get stuck in local minimal after training for about 1M environment steps.",
"In Table REF , we give the detailed success rates for 22 achievements on Crafter, which is also reported in Figure REF .",
"As shown in Table REF , OCARL presents more meaningful behaviours such as collecting coal, defeating zombies, making wood pickaxe and so on.",
"Table: Success rates on 22 achievements in Crafter.",
"Analysis of Unsupervised Object Discovery In Figure REF , we plot the discovered object categories in Crafter and Hunter.",
"Since Hunter is quite simple, our algorithm can assign correct categories for all objects.",
"In Crafter, there exist 18 kinds of objects in total, making it much harder than Hunter.",
"Since we use only 8 categories, there are some cases that multiple objects with different ground-truth categories are predicted into the same category (such as Category 5 and 7 in Figure REF ).",
"Although there exist assignment mistakes in Crafter, OCARL is still much better than other baselines.",
"Figure REF also shows that some objects in Crafter is omitted by the UOD model, such as Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONREF ) or are treated as background by SPACE.",
"Table: The average number of objects in a single image.",
"These numbers are obtained on the training dataset of SPACE via analysing the ground truth object category label provided by Crafter.In Figure REF , we report the object discovered by SPACE.",
"The combination of SPACE and unsupervised clustering on the discovered objects (i.e.",
"Eq.",
"(REF )) can tell us not only where one object locates, but also what category it is.",
"Figure: The discovered object categories on on CrafterFigure: The objects discovered on Crafter For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?",
"Did you describe the limitations of your work?",
"Did you discuss any potential negative societal impacts of your work?",
"Have you read the ethics review guidelines and ensured that your paper conforms to them?",
"If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?",
"Did you include complete proofs of all theoretical results?",
"If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?",
"Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?",
"see Appendix.",
"Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?",
"Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?",
"If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...",
"If your work uses existing assets, did you cite the creators?",
"Did you mention the license of the assets?",
"Partially.",
"Did you include any new assets either in the supplemental material or as a URL?",
"Did you discuss whether and how consent was obtained from people whose data you're using/curating?",
"Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?",
"If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?",
"Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?",
"Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?"
],
[
"Implementation Details",
"The implementation of OCARL is available at https://anonymous.4open.science/r/OCARL-51BF."
],
[
"Hyper-parameters for SPACE",
"In this paper, we use SPACE for object discovery.",
"Most hyper-parameters are the same as the default parameters for Atari in SPACE, except that (1) we use a smaller network to deal with $64\\times 64$ images; (2) For Hunter, we disable the background module in SPACE because the observations in Hunter is quite simple.",
"(3) We set the dimension of object representations $\\mathbf {z}^{what}_{ij}$ to be 16, and (in Crafter) $\\textbf {z}^{fg}$ to be 8.",
"The training data for SPACE is obtained via running a random policy on Hunter-Z4C4 and Crafter for 100000 environment steps."
],
[
"Hyper-parameters for PPO",
"Our PPO implementation is based on Tianshou [tianshou] which is purely based on PyTorch.",
"We adopt the default hyper-parameters in Tianshou, which are shown in Table REF ."
],
[
"Hyper-parameters for OCARL",
"OCARL only introduces 2 hyper-parameter $\\lambda _{cat}$ (the coefficient for $L_{cat}$ defined in Eq.",
"(REF )) and $C$ (the number of object categories).",
"We set $\\lambda _{cat}=0.01$ for all experiments, $C=4,8$ for Hunter and Crafter respectively."
],
[
"Network Architecture",
"In OCAP, we use the convolution encoder from IMPALA [impla], which consists of 3 residual blocks with 16, 32, and 32 channels, respectively.",
"By the encoder, the $3\\times 64\\times 64$ image observation is encoded into $32\\times 8\\times 8$ .",
"The category predictor $f_{cat}$ is a simple 2-layer MLP with hidden size 32.",
"The object-category-specific networks $f_1,...,f_{C+1}$ in the OCMR module consists of $C+1$ MLPs, each of which is of 2-layer with hidden size 64."
],
[
"Implementation for RRL",
"The main implementation differences between RRL and OCARL are that (1) RRL does not use the supervision signals from the UOD model (i.e.",
"$\\lambda _{cat}=0$ in Eq.",
"(REF )); (2) RRL adopts a different (although very similar) network architecture.",
"In RRL, we use the same convolution encoder as OCARL.",
"The reasoning module in RRL is similar to OCARL, except that it utilizes a non-linear universe transformation instead of multiple object-category-specific networks.",
"We also use only one relational block in RRL as OCARL does, because we found more blocks may hinder the performance in our early experiments."
],
[
"Implementation for SMORL",
"We use SPACE [space] instead of SCALOR [scalor] (as suggested in the SMORL paper) to obtain object representations for SMORL.",
"Since SMORL is originally designed for goal-based RL, it uses an attention model to gather information from the set of object representations, in which the goal serves as a query vector.",
"In our settings, we do not consider goal-based RL; therefore, we use $L$ learnable vectors as queries, which are also used in the original SMORL.",
"We search over $L=[1,2,4,8]$ and find that $L=4$ works best in our experiments.",
"We also find that it is better to apply an MLP to the object representations before they are fed into the attention module.",
"To be more specific, we first run SPACE to get the object representations, sharing the same procedure as OCARL.",
"The information of these objects is first processed by a 2-layer MLP with hidden size 32 and then gathered through an attention module in which the query consists of $L=4$ learnable vectors.",
"The resulting tensor (of shape $L\\times 32$ ) is flattened and then fed into a 2-layer MLP with hidden size 64, which finally output the value function and action probabilities.",
"Table: PPO hyper-parameters."
],
[
"Details of SPACE",
"OCARL utilizes SPACE to discovery objects from raw objects.",
"In Section REF , we have introduced the inference model of $\\mathbf {z}_{}^{fg}$ .",
"For completeness, we would like to introduce the remaining parts of SPACE in this section: (1) inference model of $\\mathbf {z}_{}^{bg}$ ; (2) generative model of the image $\\mathbf {x}$ ; and (3) the ELBO to train the model."
],
[
"Inference model of $\\mathbf {z}_{}^{bg}$",
"$\\mathbf {z}_{}^{bg}$ consists of several components $\\mathbf {z}_{k}^{bg}$ , and these components are inferred from the image $\\mathbf {x}$ in an iterative manner: $q(\\mathbf {z}_{k}^{bg}|\\mathbf {x})=\\prod _{k=1}^{K}q(\\mathbf {z}_{k}^{bg}|\\mathbf {z}_{<k}^{bg},\\mathbf {x})$ .",
"Each component $\\mathbf {z}_{k}^{bg}$ is futher divided into two parts: $\\mathbf {z}_{k}^{bg}=(\\mathbf {z}_{k}^{m}, \\mathbf {z}_{k}^{c})$ , where $\\mathbf {z}_{k}^{m}$ models the mixing weight assigned to the background component (see $\\pi _k$ in the generative model of $\\mathbf {x}$ ), and $\\mathbf {z}_{k}^{c}$ models the RGB distribution ($p(\\mathbf {x}|\\mathbf {z}_{k}^{bg})$ ) of the background component."
],
[
"Generative model of $\\mathbf {x}$",
"The generative model consists of two parts: $p(\\mathbf {x}|\\mathbf {z}_{}^{fg})$ and $p(\\mathbf {x}|\\mathbf {z}_{}^{bg})$ .",
"For $p(\\mathbf {x}|\\mathbf {z}_{}^{fg})$ , each $\\mathbf {z}_{ij}^{fg}$ is passed through a decoder to reconstruct the image patch determined by $\\mathbf {z}_{ij}^{where}$ .",
"For $p(\\mathbf {x}|\\mathbf {z}_{}^{bg})$ , each $\\mathbf {z}_{k}^{bg}$ is decoded into a background component and all components are mixed together to get the background.",
"Foreground and background are combined with a pixel-wise mixture model to reconstruct the original image $\\mathbf {x}$ .",
"The whole generative model of $\\mathbf {x}$ is: $\\begin{aligned}p(\\mathbf {x}) &= \\int \\int p(\\mathbf {x}|\\mathbf {z}_{}^{fg}, \\mathbf {z}_{}^{bg}) p(\\mathbf {z}_{}^{fg})p(\\mathbf {z}_{}^{bg})d\\mathbf {z}_{}^{fg}d\\mathbf {z}_{}^{bg}\\\\p(\\mathbf {x}|\\mathbf {z}_{}^{fg}, \\mathbf {z}_{}^{bg}) &= \\alpha p(\\mathbf {x}|\\mathbf {z}_{}^{fg}) + (1-\\alpha ) \\sum _{k=1}^K \\pi _k p(\\mathbf {x}|\\mathbf {z}_{k}^{bg}), \\alpha =f_\\alpha (\\mathbf {z}_{}^{fg}), \\pi _k=f_{\\pi _k}(\\mathbf {z}_{1:k}^{bg})\\\\p(\\mathbf {z}_{}^{fg}) &= \\prod _{i=1}^{H}\\prod _{j=1}^{W} p(z_{ij}^{pres}) (p(\\mathbf {z}_{ij}^{where})p(\\mathbf {z}_{ij}^{what}))^{z_{ij}^{pres}}\\\\p(\\mathbf {z}_{}^{bg}) &= \\prod _{k=1}^K p(\\mathbf {z}_{k}^{c}|\\mathbf {z}_{k}^{m}) p(\\mathbf {z}_{k}^{m}|\\mathbf {z}_{<k}^{m})\\end{aligned}$ In above equations, $z_{ij}^{pres}, \\mathbf {z}_{ij}^{where}, \\mathbf {z}_{ij}^{what}$ have been discussed in Section REF , $\\alpha $ is foreground mixing probability, and $\\pi _k$ is the mixing weight assigned to the background component $\\mathbf {z}_{k}^{bg}$ ."
],
[
"The ELBO to train the model",
"SPACE is trained using the following ELBO via reparameterization tricks: $\\begin{aligned}\\mathcal {L}(\\mathbf {x}) = \\mathbb {E}_{q\\left(\\mathbf {z}^{fg}, \\mathbf {z}^{bg} \\mid \\mathbf {x}\\right)}\\left[p\\left(\\mathbf {x} \\mid \\mathbf {z}^{fg}, \\mathbf {z}^{bg}\\right)\\right]-D_{\\mathrm {KL}}\\left(q\\left(\\mathbf {z}^{fg} \\mid \\mathbf {x}\\right) \\Vert p\\left(\\mathbf {z}^{fg}\\right)\\right)-D_{\\mathrm {KL}}\\left(q\\left(\\mathbf {z}^{bg} \\mid \\mathbf {x}\\right) \\Vert p\\left(\\mathbf {z}^{bg}\\right)\\right)\\end{aligned}$"
],
[
"More Results on ",
"In Figure REF , we plot the learning curves on the Crafter domain.",
"Only OCARL is able to make progress continuously.",
"Both RRL and SMORL get stuck in local minimal after training for about 1M environment steps.",
"In Table REF , we give the detailed success rates for 22 achievements on Crafter, which is also reported in Figure REF .",
"As shown in Table REF , OCARL presents more meaningful behaviours such as collecting coal, defeating zombies, making wood pickaxe and so on.",
"Table: Success rates on 22 achievements in Crafter."
],
[
"Analysis of Unsupervised Object Discovery",
"In Figure REF , we plot the discovered object categories in Crafter and Hunter.",
"Since Hunter is quite simple, our algorithm can assign correct categories for all objects.",
"In Crafter, there exist 18 kinds of objects in total, making it much harder than Hunter.",
"Since we use only 8 categories, there are some cases that multiple objects with different ground-truth categories are predicted into the same category (such as Category 5 and 7 in Figure REF ).",
"Although there exist assignment mistakes in Crafter, OCARL is still much better than other baselines.",
"Figure REF also shows that some objects in Crafter is omitted by the UOD model, such as Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTIONREF ) or are treated as background by SPACE.",
"Table: The average number of objects in a single image.",
"These numbers are obtained on the training dataset of SPACE via analysing the ground truth object category label provided by Crafter.In Figure REF , we report the object discovered by SPACE.",
"The combination of SPACE and unsupervised clustering on the discovered objects (i.e.",
"Eq.",
"(REF )) can tell us not only where one object locates, but also what category it is.",
"Figure: The discovered object categories on on CrafterFigure: The objects discovered on Crafter"
]
] | 2210.07802 |
[
[
"Quenched invariance principle for biased random walks in random\n conductances in the sub-ballistic regime"
],
[
"Abstract We consider a biased random walk in positive random conductances on $\\mathbb{Z}^d$ for $d\\geq 5$.",
"In the sub-ballistic regime, we prove the quenched convergence of the properly rescaled random walk towards a Fractional Kinetics."
],
[
"Introduction",
"Random walks in random environment (RWRE) have been extensively studied since the mid seventies and especially so in the last two decades.",
"We refer the reader to , , , and for different surveys of the field.",
"A lot of effort has been directed towards proving invariance principles in various models.",
"The majority of those results have been obtained in isotropic settings.",
"We could mention as an example the numerous works done for the simple random walk on supercritical percolation clusters in $\\mathbb {Z}^d$ .",
"This started with results in the annealed setting in the 80s, see and (based on works of and ) which was later generalized into a quenched result, initially for $d\\ge 4$ in and later in dimensions 2 and 3 in and .",
"In models where trapping occurs, the classical invariance principle is not expected to hold.",
"However, using a different time scaling, it is still possible to obtain a convergence result where the limiting Brownian motion is replaced by a Fractional Kinetics (a Brownian motion time-changed by a stable subordinator).",
"For example, Barlow and Černý prove in subdiffusive scaling limit in both the Bouchaud trap model (BTM) and the random conductance model.",
"Mourrat gives a different proof for the result of for the Bouchaud trap model in .",
"All the results mentioned above are for isotropic models.",
"Other cases, such as models with directional transience, have attracted a lot of interest as well.",
"Annealed invariance principles have been obtained for such models on $\\mathbb {Z}$ (see ), trees (see , ) and on $\\mathbb {Z}^d$ with $d \\ge 2$ (see , ).",
"Moreover, there are a few results in the quenched setting as well on $\\mathbb {Z}$ (see ), trees (see ) and $\\mathbb {Z}^d$ (see ).",
"The models with directional transience of interest are typically studied because several of them exhibit slowdown effects due to a trapping phenomenon, such as , , , , , , , , .",
"The most precise results for an anisotropic model on $\\mathbb {Z}^d$ with $d \\ge 2$ were obtained by Fribergh and Kious in , where it is shown that under heavy tail condition for conductances we have an aging phenomenon for the walk and an annealed scaling limit related to Fractional Kinetics.",
"In light of Mourrat's result in it is natural to expect that the annealed scaling limit obtained in should hold in the quenched setting as well.",
"This is the main result of this article.",
"It provides a first example for a directionally transient RWRE which converges to a Fractional Kinetics under the quenched setting."
],
[
"Definition of the model and main result",
"Fix $d \\ge 5$ .",
"Let us consider the following random walk in random environment.",
"Let $\\mathbf {P}$ be a measure on the set of nearest neighbour edges of the square lattice $E(\\mathbb {Z}^d)$ , i.e.",
"$e \\in E(\\mathbb {Z}^d)$ if $e = [x, y]$ for $x \\sim y$ (adjacent vertices in $\\mathbb {Z}^d$ ).",
"In particular, let $\\mathbf {P}(\\cdot ) = \\mu ^{\\otimes E(\\mathbb {Z}^d)}$ , where $\\mu $ is a probability measure on $(0,+\\infty )$ .",
"For $e \\in E(\\mathbb {Z}^d)$ , we denote by $c_{*}(e)$ the random variable of the marginal of the conductance of the edge $e$ under $\\mathbf {P}$ .",
"Let us call the space of environments $\\Omega $ and a single realisation of the environment $\\omega \\in \\Omega $ .",
"To introduce the bias consider a unit vector $\\vec{\\ell } \\in \\mathbb {S}^{d-1}$ and $\\lambda > 0$ , denote $\\ell \\lambda \\vec{\\ell }$ and set for $e = [x,y] \\in E(\\mathbb {Z}^{d})$ , $c(e) = c_{*}(e)e^{(x+y)\\cdot \\ell }.$ We focus on the regime where $\\mathbf {P}\\left(c_{*}(e) \\ge x \\right) = \\mu ([x,+\\infty [) = L(x) x^{-\\gamma } \\quad \\textnormal {for any }x \\ge 0,$ with $\\gamma \\in (0,1)$ and $L(\\cdot )$ is a slowly-varying function.",
"For a fixed environment $\\omega $ , we consider the random walk $(X_n)_{n \\ge 0}$ with conductances $(c^\\omega (e))_{e \\in E(\\mathbb {Z}^d)}$ .",
"We will call $P_{x}^{\\omega }$ the quenched law of the random walk on the environment starting from $x$ and with transition probabilities $p^\\omega (x, y)$ for $x, y \\in \\mathbb {Z}^d$ $p^\\omega (x, y) = \\frac{c^\\omega ([x, y])}{\\sum \\limits _{y \\sim x} c^\\omega ([x, y])}.$ Let $\\mathbb {P}_{x} = \\mathbf {E}[P_{x}^{\\omega }(.",
")]$ be the annealed law of this random walk ($\\mathbf {E}[\\cdot ]$ being the expectation with respect to $\\mathbf {P}$ and $\\mathbf {Var}$ the corresponding variance).",
"The main theorem of this work is the following.",
"Theorem 1.1 Let $d \\ge 5$ and fix some $T > 0$ , the following statements hold for $\\mathbf {P}$ -almost every environment $\\omega \\in \\Omega $ .",
"There exists a deterministic $v_0 \\in \\mathbb {S}^{d-1}$ with $v_0 \\cdot \\vec{\\ell } > 0$ such that, under the quenched law $P^\\omega _0(\\cdot )$ , we have the following weak convergences holding $\\left( \\frac{X_{{n t}}}{n^\\gamma /L(n)} \\right)_{0 \\le t \\le T} \\overset{(\\mathrm {d})}{\\rightarrow } \\left( v_0 C_\\infty ^{-\\gamma } \\mathcal {S}_\\gamma ^{-1}(t) \\right)_{0 \\le t \\le T},$ $\\left( \\frac{X_{{n t}} - (X_{{n t}} \\cdot v_0) v_0}{\\sqrt{n^\\gamma /L(n)}} \\right)_{0 \\le t \\le T} \\overset{(\\mathrm {d})}{\\rightarrow }\\left( M_d B_{\\mathcal {S}_\\gamma ^{-1}(t)} \\right)_{0 \\le t \\le T}.$ For a deterministic constant $C_\\infty >0$ , a deterministic $d \\times d$ matrix $M_d$ of rank $d-1$ , a standard Brownian motion $B$ and a stable subordinator of index $\\gamma $ , independent of $B$ , $\\mathcal {S}_\\gamma $ .",
"The first convergence holds in the uniform topology while the second holds in the Skorohod's $J_1$ topology."
],
[
"Quenched convergence via series convergence",
"Let us explain the main strategy that we will follow.",
"In one can find the definition of a regeneration structure with suitable independence properties, in particular let $(\\tau _k)_{k \\ge 0}$ be the regeneration times.",
"See Section REF where we recall the rigorous definition of $(\\tau _k)_{k \\ge 0}$ .",
"Call $v \\mathbb {E}[X_{\\tau _2} - X_{\\tau _1} ]$ , then we introduce, for all $t \\ge 0$ $Y_n(t) = \\frac{X_{\\tau _{{tn}}}}{n}, \\quad Z_n(t) = \\frac{X_{\\tau _{{tn}}} - v n t}{n^{1/2}},\\quad \\textnormal {and} \\quad S_n(t) = \\frac{\\tau _{{tn}}}{\\mathrm {Inv}(n)},$ where $\\mathrm {Inv}(t) \\inf \\left\\lbrace s: \\mathbf {P}[c_{*} > s]\\le 1/t \\right\\rbrace ,$ is the generalised right-continuous inverse of the tail of $c_{*}$ , let us point out that $\\mathrm {Inv}(n) \\approx n^{1/\\gamma }$ .",
"Our initial aim is to prove that for $\\mathbf {P}$ -almost all environments $\\omega \\in \\Omega $ these three quantities converge weakly.",
"Then we will use this to prove Theorem REF .",
"We get the quenched convergence of $Y_n(t)$ for free from .",
"For technical reasons let us define the process $S^*_n(t) = \\frac{\\tau _{{tn}+1} - \\tau _{1}}{\\mathrm {Inv}(n)}.$ To prove quenched convergence of $S^*_n(t)$ and $Z_n(t)$ we will use alternative formulations considered in and to rephrase the problem.",
"These conditions are stated in the following theorem.",
"Let us define $F(h) = \\exp \\left(-\\lambda _1 h(t_1) - \\lambda _2(h(t_2) - h(t_1)) -... - \\lambda _m(h(t_m) - h(t_{m-1}))\\right), $ with $\\lambda _1,\\cdots ,\\lambda _m \\ge 0$ and $0 < t_1 < \\cdots < t_m \\le T$ .",
"Theorem 1.2 Let $d \\ge 5$ , then the following holds for any $F_1$ as in (REF ) $ \\sum _{n = 1 }^{+\\infty }\\mathbf {Var}\\left(E_{0}^{\\omega }\\left[F_1(S^*_{b^n})\\right]\\right)< +\\infty ,$ with $b \\in (1,2)$ .",
"Moreover, by denoting with $Z_n$ the polygonal interpolant of the quantity denoted with the same letter in (REF ), we have $ \\sum _{n = 1 }^{+\\infty }\\mathbf {Var}\\left(E_{0}^{\\omega }[F_2(Z_{b^n})]\\right)< +\\infty ,$ where $F_2$ is any bounded $K$ -Lipschitz function from $\\mathcal {C}([0, T],\\mathbb {R}^{d}) \\rightarrow \\mathbb {R}$ ."
],
[
"Sketch of proof",
"To prove Theorem REF , we can consider under each environment two independent random walks $X^1$ and $X^2$ which have the same law as $X$ under $P_0^{\\omega }$ .",
"Then the term in the second sum above is equal to $\\mathbb {E}[F_2(Z_{b^n}^1)F_2(Z_{b^n}^2)] - \\mathbb {E}[F_2(Z_{b^n}^1)]\\mathbb {E}[F_2(Z_{b^n}^2)].$ We will show that the two walks do not meet each other after reaching a distance not too far from the origin, then deduce that the quantity above goes quickly to 0.",
"Indeed, when the two walks do not meet, they behave almost as if they evolved in independent environments.",
"The important ideas can be grouped in two main parts.",
"The first one deals with joint regeneration levels while the second consists in proving the asymptotic separation using joint regeneration levels.",
"Joint regeneration levels: Section will be devoted to extend the notion of regeneration times of to the case where we have two walks.",
"We will construct joint regeneration levels: a sequence of levels such that the hitting time of these are regeneration times for both walks.",
"Asymptotic separation: Once we have constructed joint regeneration levels the idea is the following: at each regeneration level, both walks have a decent chance of staying far apart for a long time, after which it is unlikely that they will ever meet again.",
"As joint regeneration levels are common, the walks will decouple relatively quickly.",
"In this part we use a method which is inspired from .",
"However, we cannot directly use their results because our model lacks both the uniform ellipticity property of the jump probabilities and it is not ballistic."
],
[
"Notations",
"We fix an orthonormal basis $(e_1,...,e_d)$ of $\\mathbb {R}^d$ with elements of $\\mathbb {Z}^d$ such that $e_1\\cdot \\vec{\\ell } \\ge e_2\\cdot \\vec{\\ell } \\ge ... \\ge e_d\\cdot \\vec{\\ell }$ .",
"In particular we have $e_1\\cdot \\vec{\\ell } \\ge 1/\\sqrt{d}$ .",
"We also fix an orthonormal basis of $\\mathbb {R}^d$ , $(f_1,...,f_d)$ such that $f_1 = \\vec{\\ell }$ .",
"For any $L \\in \\mathbb {R}$ we set $\\mathcal {H}^{+}(L) = \\lbrace z \\in \\mathbb {Z}^d ; \\, z\\cdot \\vec{\\ell } > L\\rbrace \\,\\, \\mathrm {and} \\,\\, \\mathcal {H}^{-}(L) = \\lbrace z \\in \\mathbb {Z}^d ; \\, z\\cdot \\vec{\\ell } \\le L\\rbrace .$ Similarly, for $z\\in \\mathbb {Z}^d$ $\\mathcal {H}^{+}_{z} = \\mathcal {H}^{+}(z\\cdot \\vec{\\ell }) \\,\\, \\mathrm {and} \\,\\, \\mathcal {H}^{-}_{z} = \\mathcal {H}^{-}(z\\cdot \\vec{\\ell }).$ For any $x \\in \\mathbb {Z}^d$ we define $ \\mathcal {V}_x = \\lbrace x,x+e_1,x-e_1,\\cdots ,x+e_d,x-e_d\\rbrace .$ For any $x,y \\in \\mathbb {Z}^d$ we say that $x$ and $y$ are adjacent and write $x \\sim y$ if $x = y \\pm e_i$ where $i \\in \\lbrace 1,\\cdots ,d\\rbrace $ .",
"For $x$ in $\\mathbb {Z}^d$ and $e = [e^{+}, e^{-}]$ and edge, we write $x \\sim e$ if $x \\sim e^{-}$ or $x \\sim e^{+}$ and $x \\notin \\lbrace e^{+},e^{-}\\rbrace $ .",
"Two distinct edges $e$ and $e^{\\prime }$ are said to be adjacent and we write $e\\sim e^{\\prime }$ if they share one endpoint.",
"For $x \\in \\mathbb {Z}^d$ we define $\\mathcal {E}_x \\lbrace [x,y] \\, | \\, y\\sim x\\rbrace , \\,\\, \\text{ and for }A\\subset \\mathbb {Z}^d \\text{, }\\mathcal {E}_A = \\bigcup _{x\\in A}\\mathcal {E}_x.$ $\\mathcal {L}^x \\lbrace [y,z] \\,\\, | \\,\\, y \\cdot \\vec{\\ell } \\le x \\cdot \\vec{\\ell } \\text{ and } z \\cdot \\vec{\\ell } \\le x \\cdot \\vec{\\ell }\\rbrace \\cup \\mathcal {E}_x.$ $\\mathcal {R}^x \\lbrace [y,z] \\,\\, | \\,\\, y\\cdot \\vec{\\ell } > x\\cdot \\vec{\\ell } \\text{ or } z\\cdot \\vec{\\ell } > x\\cdot \\vec{\\ell }\\rbrace \\cup \\mathcal {E}_x.$ For any subset $V$ of $\\mathbb {Z}^d$ we define its edge-set $E(V) \\lbrace [x,y] \\text{ ; } x \\in V \\text{ and } y \\in V\\rbrace ,$ its vertex-boundary $\\partial V \\lbrace x \\notin V \\text{ ; }\\exists y \\in V, x \\sim y \\rbrace ,$ and its edge-boundary $\\partial _{E} V \\lbrace [x,y]\\in E(\\mathbb {Z}^d)\\text{ ; } y \\in V, x \\notin V \\rbrace .$ For any subset $A \\subset \\mathbb {Z}^d$ we define its width $W(A)= \\max _{1\\le i \\le d}\\left(\\max _{y \\in A}y \\cdot e_i - \\min _{y \\in A}y \\cdot e_i\\right).$ For any $L,L^{\\prime } > 0$ and $y \\in \\mathbb {Z}^d$ we define the box $\\mathcal {B}_y(L,L^{\\prime }) = \\lbrace x \\in \\mathbb {Z}^d \\text{ ; }|(x-y)\\cdot \\vec{\\ell }| \\le L \\text{ and } |(x-y)\\cdot f_i| \\le L^{\\prime } \\text{ for all } i \\in \\lbrace 2,...,d\\rbrace \\rbrace ,$ and its positive boundary $\\partial ^{+}\\mathcal {B}_y(L,L^{\\prime }) = \\lbrace x \\in \\partial \\mathcal {B}_y(L,L^{\\prime }) \\text{ s.t.",
"}|(x-y)\\cdot \\vec{\\ell }| > L\\rbrace .$ When $y = 0$ we just use the notation $\\mathcal {B}(L,L^{\\prime })$ .",
"For a general random walk on a graph $(V,E)$ we define for $B \\subset V$ the hitting times of $B$ , $T_B = \\inf \\lbrace n \\ge 0 \\text{ : } X_n \\in B\\rbrace \\text{ and } T^{+}_B = \\inf \\lbrace n \\ge 1 \\text{ : } X_n \\in B\\rbrace ,$ and the exit time of $B$ $T^{\\mathrm {ex}}_B = \\inf \\lbrace n \\ge 0 \\text{ ; } X_n \\notin B\\rbrace .$ When $B$ is a singleton $\\lbrace x\\rbrace $ we use the notation $T_x$ .",
"For $R \\in \\mathbb {R}$ we define $T_{R} = T_{\\mathcal {H}^{+}(R)}.$ We denote by $\\theta _n$ the translation of $n$ units of time.",
"For $\\mathbb {P}$ a probability measure, $A_1,\\cdots ,A_n$ events and $X$ a random variable the notation $\\mathbb {E}[A_1,\\cdots ,A_n,X]$ denotes $\\mathbb {E}[{1}_{A_1}\\cdots {1}_{A_n}X]$ .",
"In most of the paper we will add an index $i \\in \\lbrace 1,2\\rbrace $ as superscript when the event depends on $X^i$ , where $X^i$ are two realisations of the random walk (independent with respect to the quenched distribution).",
"For example, $T^{i}_R$ will be the hitting time of $\\mathcal {H}^{+}(R)$ by the $i$ -th random walk.",
"In the following we will use the notations $c$ , $C$ , $a$ to denote constants which only depend on the dimension $d$ and the strength of the bias $\\lambda $ , its direction $\\vec{\\ell }$ and a parameter $K > 0$ ."
],
[
"Probability measures.",
"Let us recall that $P^\\omega $ denotes the quenched law while $\\mathbb {P}$ denotes the annealed law of the walk.",
"As we already mentioned we will often consider two walks $X^1_{\\cdot }$ and $X^2_{\\cdot }$ , for these walks it is useful to introduce two probability measures.",
"The first is for two walks that evolve independently (according to the law defined in (REF )) in the same environment.",
"It is characterised by the formula: $\\mathbb {P}_{U_1, U_2}[X^1_{\\cdot } \\in A_1, X^2_{\\cdot } \\in A_2] \\mathbf {E}\\left[ P_{U_1}^\\omega \\left( X^1_{\\cdot } \\in A_1 \\right) \\times P_{U_2}^\\omega \\left( X^2_{\\cdot } \\in A_2 \\right) \\right].$ It is also possible to define a measure according to which the two walks evolve with the same quenched law in two independent environments $\\omega _1, \\omega _2$ sampled according to $\\mathbf {P}$ .",
"In formulas, $Q_{U_1, U_2}[X^1_{\\cdot } \\in A_1, X^2_{\\cdot } \\in A_2] &\\mathbf {E}\\left[ P_{U_1}^{\\omega _1}\\left( X^1_{\\cdot } \\in A_1 \\right) \\times P_{U_2}^{\\omega _2}\\left( X^2_{\\cdot } \\in A_2 \\right) \\right] \\\\ &= \\mathbb {P}_{U_1}\\left( X^1_{\\cdot } \\in A_1 \\right) \\times \\mathbb {P}_{U_1}\\left( X^2_{\\cdot } \\in A_2 \\right).\\nonumber $ We denote by $\\mathbb {E}_{U_1,U_2}[\\cdot ]$ the expectation with respect to the measure (REF ).",
"This is the same notation as the annealed law of one walk (with the exception of the two starting points), this because by considering the marginal of $X^1_{\\cdot }$ that is the same as the law of the single walk.",
"$\\mathbb {E}^Q[\\cdot ]$ is the expectation associated to (REF ).",
"For $a \\in \\mathbb {R}^{2d}$ and $x \\in \\mathbb {Z}^d$ we write $\\mathbb {P}_x^{a}$ (resp.",
"$\\mathbb {P}_{x,y}^{a}$ ) for the annealed law where we set the conductances in $\\mathcal {E}_x$ (resp.",
"in $\\mathcal {E}_x$ and $\\mathcal {E}_y$ ) to the value $a$ .",
"A special case is when $a = (K,\\ldots ,K)$ with $K > 0$ , in that case we write $\\mathbb {P}_x^{K}$ (resp.",
"$\\mathbb {P}_{x,y}^{K}$ ).",
"The notation $Q_{x,y}^{a}$ means the law $\\mathbb {P}_x^{a}\\otimes \\mathbb {P}_y^{a}$ .",
"In accordance with the notation of (that we will recall in Section REF ) we write $\\mathbb {P}_x^{K}(\\cdot |D=+\\infty )$ for the law of the future of the walk after a regeneration time at $x$ .",
"We also write $Q_{x,y}^K(\\cdot |D^{\\otimes }=+\\infty ) = \\mathbb {P}_x^{K}(\\cdot |D=+\\infty )\\otimes \\mathbb {P}_y^{K}(\\cdot |D=+\\infty )$ ."
],
[
"Joint regeneration levels",
"For this model a regeneration structure was introduced in .",
"It was further improved in to obtain the independence of trajectories between regeneration times.",
"For our purpose it will be more convenient to have joint regeneration levels.",
"Let us now introduce a joint regeneration structure for two walks in the same environment.",
"Figure: Two trajectories and the first joint regeneration level ℒ 1 \\mathcal {L}_1.In principle a regeneration time is simply a new maximum which will be a minimum for the future of the trajectory.",
"However, for technical reasons we want to work with a subset of those regeneration times: those happening on points with bounded conductances.",
"This is why we will adapt the construction of which uses $K$ -open ladder points.",
"The first step is to introduce the notion of joint $K$ -open ladder point and we do this in the next subsection."
],
[
"Notations and preliminary results",
"A point $z \\in \\mathbb {Z}^d$ is said to be $K$ -open if for all $e \\in \\mathcal {E}_x$ we have $c_{*}(e) \\in [1/K,K]$ (we call such edges $K$ -normal).",
"Joint $K$ -open ladder points are new maxima in the bias direction for both walks such that the points are $K$ -open and the maxima are close in the sense that they are both at roughly the same distance, say $\\approx R$ , from the origin in the bias direction.",
"Note that the distance cannot be exactly the same due to the lattice structure.",
"The main goal of this subsection is to prove that the first joint $K$ -open ladder point is close enough to the origin in the bias direction.",
"The main result, that will be used in the rest of the paper is Proposition REF which follows from Lemma REF .",
"Lemma REF which is related to Proposition REF will also be instrumental for the rest of this work.",
"The proofs and the other results are purely technical and will not be used in the rest of the paper.",
"Let us first introduce some useful quantities and results from .",
"We introduce the inner positive boundary of $\\mathcal {B}_{x}(L, L^{\\prime })$ $\\partial _{\\mathrm {in}}^+ \\mathcal {B}_{x}(L, L^{\\prime }) \\lbrace z \\in \\mathcal {B}_{x}(L, L^{\\prime }), \\text{ where } z \\sim y \\text{ with } y \\in \\partial ^+ \\mathcal {B}_{x}(L, L^{\\prime }) \\rbrace .$ We consider two walks $(X^i_n)_{n\\ge 0}$ for $i\\in \\lbrace 1,2\\rbrace $ and define the events $ A^{i}(L) = \\lbrace T^i_{\\partial \\mathcal {B}_{X^{i}_{0}}(L, L^\\alpha )} \\ge T^i_{\\partial ^+_{\\mathrm {in}} \\mathcal {B}_{X^{i}_{0}}(L, L^\\alpha )}\\rbrace , \\quad \\mathrm {and} \\quad A(L) A^{1}(L) \\cap A^{2}(L).$ In all the following we write $\\mathcal {B}_{i} = \\mathcal {B}_{X_{0}^{i}}$ and $\\mathcal {B} = \\mathcal {B}_{0}$ .",
"Lemma 3.1 (Analog of ) For $\\alpha > d + 3$ and any $L > 0$ we have $\\sup _{U_1, U_2 \\in \\mathbb {Z}^d} \\mathbb {P}_{U_1, U_2}\\left( A(L)^c \\right) \\le C e^{-c L}.$ where the constants $C,c$ only depends on $\\lambda ,\\vec{\\ell },d$ .",
"Fix any $U_1, U_2 \\in \\mathbb {Z}^d$ .",
"We aim to bound $\\mathbb {P}_{U_1, U_2}\\left( A(L)^c \\right) &= \\mathbb {P}_{U_1, U_2}\\left( A^1(L)^c \\cup A^2(L)^c \\right) \\le \\mathbb {P}_{U_1}\\left( A^1(L)^c \\right) + \\mathbb {P}_{U_2}\\left( A^2(L)^c \\right),$ where the inequality is due to a union bound.",
"By we have that $\\mathbb {P}_{0}\\left[ A^{i}(L)^c \\right] \\le C \\exp (-c L)$ , thanks to translation invariance of the environment we get $\\mathbb {P}_{U_1, U_2}\\left( A(L)^c \\right) \\le C e^{-c L},$ as desired.",
"We say that a vertex $x \\in \\mathcal {B}(L, L^\\alpha )$ is $K$ -$L$ -closed if there exists a neighbour $y \\notin \\mathcal {H}^+(L)$ of $x$ such that $c^*([x, y]) \\notin [1/K, K]$ .",
"Let $\\bar{K}_x(L)$ be the $K$ -$L$ -closed connected component of $x$ (set it to $\\lbrace x\\rbrace $ in case it is empty) and let us define the event $ B(L) = \\left\\lbrace \\textnormal {for all } x \\in \\partial _{\\mathrm {in}}^+ \\mathcal {B}_1(L, L^\\alpha ) \\cup \\partial _{\\mathrm {in}}^+ \\mathcal {B}_2(L, L^\\alpha ), \\textnormal { } |\\bar{K}_x(L)| \\le \\log (L) \\right\\rbrace .$ Lemma 3.2 (Analog of ) For any $M<\\infty $ , we can find $K_0$ such that for all $K \\ge K_0$ $\\mathbf {P}\\left( B(L)^c \\right) \\le C L^{-M}.$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"Thanks to we have the bound for the event $\\widehat{B}(L) = \\lbrace \\textnormal {for all } x \\in \\partial _{\\mathrm {in}}^+ \\mathcal {B}(L, L^\\alpha ), \\textnormal { we have } |\\bar{K}_x(L)| \\le \\log (L) \\rbrace ,$ $\\mathbf {P}\\left( \\widehat{B}(L)^c \\right) \\le C L^{-M}.$ Just as in the proof of the previous Lemma, we get the result by union bound and translation invariance of the the measure $\\mathbf {P}$ .",
"We also recall the following useful lemma.",
"Lemma 3.3 Take $G \\ne \\varnothing $ to be a finite connected subset of $\\mathbb {Z}^d$ .",
"Assume each edge $e$ of $\\mathbb {Z}^d$ is assigned a positive conductance $c(e)$ and that there exists a constant $c_1>0$ , $x \\in \\partial G$ and $y \\in G$ such that $x \\sim y$ and $c([x, y]) \\ge c_1 c(e)$ for any $e \\in \\partial _{E} G$ .",
"We have $P_y\\left( T_x \\le T_{\\partial G} \\right) \\ge \\frac{c_1}{4d}|G|^{-1},$ where $P_y$ is the law of the random walk on the family of conductances $\\lbrace c(e)\\rbrace _{e \\in E(\\mathbb {Z}^d)}$ , starting from $y$ ."
],
[
"Properties of joint $K$ -open ladder points",
"For $i \\in \\lbrace 1,2\\rbrace $ we recall the notation $T^i_R$ defined in (REF ).",
"We introduce $\\mathcal {M}^{\\bullet } = \\mathcal {M}^{\\bullet (K)} = \\inf \\Big \\lbrace R > &\\max \\lbrace |X^{1}_0\\cdot \\vec{\\ell }|, |X^{2}_0\\cdot \\vec{\\ell }|\\rbrace \\text{ ; for } i\\in \\lbrace 1,2\\rbrace ,\\text{ } X^i_{T^i_{R}} \\text{ is } K\\text{-open, } \\nonumber \\\\ &X^i_{j} \\cdot \\vec{\\ell } < X^i_{T^i_{R}-2} \\cdot \\vec{\\ell } \\text{ for } j < T^i_{R}-2,\\\\& \\nonumber X^i_{T^i_{R}} = X^i_{T^i_{R}-1} + e_1,X^i_{T^i_{R}} = X^i_{T^i_{R}-2} + 2e_1 \\Big \\rbrace .$ In general, we will put a bullet when referring to quantities related to two walks evolving in the same environment to avoid confusion with similar quantities for a single walk or two walks in independent environments.",
"Since the definition of $\\mathcal {M}^{\\bullet }$ corresponds to a level, the joint $K$ -open ladder point is then $(X^1_{T^1_{\\mathcal {M}^{\\bullet }}},X^2_{T^2_{\\mathcal {M}^{\\bullet }}})$ .",
"However, as we will show in the rest of the article, it is much more convenient to work with levels rather than time or points when we consider two walks.",
"The main proposition of this subsection is the following.",
"Proposition 3.4 For any $M > 0$ there exists $K_0 > 0$ such that for $K\\ge K_0$ we have $\\sup _{U_1, U_2 \\in \\mathcal {U}}\\mathbb {P}_{U_1,U_2}(\\mathcal {M}^{\\bullet (K)} > n) \\le Cn^{-M},$ where we have $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"To prove this result we require a technical lemma, for $1 \\le k \\le n$ , let us define the events $R^{(K)}(n k) \\left\\lbrace \\mathcal {M}^{\\bullet (K)} > kn + 3 \\right\\rbrace .$ Lemma 3.5 For any $\\varepsilon ^{\\prime } > 0$ and $M > 0$ , we can find $K_0 = K_0(\\varepsilon ^{\\prime }, M)$ large enough such that for any $U_1,U_2 \\in \\mathbb {Z}^d$ with $|U_1\\cdot \\vec{\\ell }| \\le 1$ and $|U_2\\cdot \\vec{\\ell }| \\le 1$ , all $K > K_0$ , $1< k \\le n$ we have $\\displaystyle \\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n k)\\right)\\le \\left(1-cn^{-\\varepsilon ^{\\prime }}\\right)\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n (k-1))\\right) + Cn^{-M},$ where the constants $C,c$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We postpone the proof of Lemma REF after the proof of the main result.",
"[Proof of Proposition REF ] This Proposition is the analogous of .",
"In particular, one can iterate Lemma REF to get a tail estimate on the quantity $\\mathcal {M}^{\\bullet (K)}$ .",
"Indeed, $\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n k)\\right)\\le \\left(1-cn^{-1/2}\\right)\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n (k-1))\\right) + Cn^{-M},$ implies, by induction, that $\\mathbb {P}_{U_1,U_2}\\left(R^{(K)}(n^2)\\right)\\le \\left(1-cn^{-1/2}\\right)^n + Cn^{-M + 1} \\le 2Cn^{-M + 1}.$ In the end we get $\\mathbb {P}_{U_1,U_2}\\left(\\mathcal {M}^{\\bullet (K)} > n^2 + 3\\right) \\le 2Cn^{-M + 1},$ or equivalently $\\mathbb {P}_{U_1,U_2}\\left(\\mathcal {M}^{\\bullet (K)} > n\\right) \\le Cn^{-(M + 1)/2},$ which is enough to conclude since $M$ is arbitrary.",
"Figure: Black points represent the KK-open points.",
"We look at the successive attempts to reach a joint KK-open ladder point when the walks exit boxes.",
"[Proof of Lemma REF ] The proof essentially goes the same way as .",
"Fix $U_1,U_2 \\in \\mathcal {U}$ as in the hypothesis of the Lemma.",
"For $i = 1, 2$ let us denote as $\\widetilde{\\mathcal {B}}^i_n$ the largest box of the form $\\mathcal {B}_i(m, m^\\alpha )$ with $m \\in \\mathbb {R}_+$ contained in $\\mathcal {H}^{-}(n)$ .",
"Note that $ \\mathcal {B}_i((n-1), (n-1)^\\alpha ) \\subset \\widetilde{\\mathcal {B}}^i_n \\subset \\mathcal {B}_i(n+1, (n+1)^\\alpha )$ .",
"Recall that $\\bar{K}_x(n)$ denotes the $K$ -$n$ -closed connected component of $x$ defined just before (REF ).",
"We introduce $\\mathcal {K}^i(n) \\bar{K}_{X_{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}}} (n) \\subseteq \\widetilde{\\mathcal {B}}^i_n, \\quad i = 1, 2.$ We then set $\\mathcal {K}(n) = \\mathcal {K}^1(n) \\cup \\mathcal {K}^2(n)$ .",
"In case $T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n} = + \\infty $ we set $\\mathcal {K}^i(n) = \\varnothing $ and $\\partial \\mathcal {K}^i(n) = \\varnothing $ .",
"We recall the definition of $A(L)$ , $B(L)$ in (REF ),(REF ).",
"We slightly modify them and define, with a little abuse of the notation, $A^{i}(n) = \\left\\lbrace T^i_{\\partial \\widetilde{\\mathcal {B}}_{X^{i}_{0}}(n, n^\\alpha )} \\ge T^i_{\\partial ^+_{\\mathrm {in}} \\widetilde{\\mathcal {B}}_{X^{i}_{0}}(n, n^\\alpha )}\\right\\rbrace , \\quad \\mathrm {and} \\quad A(n) A^{1}(n) \\cap A^{2}(n);$ $B(n) = \\left\\lbrace \\textnormal {for all } x \\in \\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}_{X^{1}_{0}}(n, n^\\alpha )\\cup \\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}_{X^{2}_{0}}(n, n^\\alpha ), \\textnormal { we have } |\\bar{K}_x(n)| \\le \\log (n) \\right\\rbrace .$ Notice that the estimates derived in Lemmas REF REF would still hold thanks to the fact that $ \\mathcal {B}_i((n-1), (n-1)^\\alpha ) \\subset \\widetilde{\\mathcal {B}}^i_n \\subset \\mathcal {B}_i(n+1, (n+1)^\\alpha )$ .",
"Moreover, we introduce the events: $&\\bullet \\hspace{3.00003pt}C(n) \\left\\lbrace \\text{for all } x \\in \\partial \\mathcal {K}(n)\\cap \\mathcal {H}^+(n), \\text{ the vertex } x \\text{ is } K\\textnormal {-open} \\right\\rbrace .",
"\\\\\\vspace{2.125pt}&\\bullet \\hspace{3.00003pt} D^{i}(n) \\left\\lbrace T^i_{\\partial \\mathcal {K}^{i}(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}} = T^i_{\\partial \\mathcal {K}^{i}(n) \\cap \\mathcal {H}^+(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}} \\right\\rbrace .",
"\\\\\\vspace{2.125pt}&\\bullet \\hspace{3.00003pt} E^{i}(n) \\left\\lbrace X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 2} = X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 1} + e_1 = X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n}} + 2 e_1 \\textnormal { and } X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 2}, X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} + 1} \\textnormal { are \\right.K-open}.$ Furthermore, define $D(n) = D^{1}(n) \\cap D^{2}(n)$ and $E(n) = E^{1}(n) \\cap E^{2}(n)$ .",
"Let us consider the event $A(n) \\cap C(n) \\cap D(n) \\cap E(n)$ then it is possible to notice the following: On $A(n)$ , defined in (REF ), for $i = 1, 2$ , we have $T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n} \\ge T^i_{\\partial ^+_{\\mathrm {in}} \\widetilde{\\mathcal {B}}^i_n}$ so that $T^i_{\\partial ^+ \\widetilde{\\mathcal {B}}^i_n} \\ge T^i_{\\partial \\mathcal {K}(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}} \\widetilde{\\mathcal {B}}^i_n}}.$ On $D(n)$ , for $i = 1, 2$ , we have $T_{\\partial \\mathcal {K}^{i}(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}} = T_{\\partial \\mathcal {K}^{i}(n) \\cap \\mathcal {H}^+(n)} \\circ \\theta _{T^i_{\\partial _{\\mathrm {in}}^+ \\widetilde{\\mathcal {B}}^i_n}}.$ On $C(n)$ , for $i = 1, 2$ , we have that all points in $\\partial \\mathcal {K}^i(n) \\cap \\mathcal {H}^+(n)$ are $K$ -open.",
"Putting these observations together one can see that, for $i \\in \\lbrace 1,2\\rbrace $ , $X^i_{T^i_{\\partial \\widetilde{\\mathcal {B}}^i_n}}$ is a new maximum in the direction $\\vec{\\ell }$ which is also $K$ -open and the next two steps are in the direction $e_1$ on $K$ -open points, so that $\\lbrace A(n) \\cap C(n) \\cap D(n) \\cap E(n)\\rbrace \\subset \\lbrace \\mathcal {M} \\le n + 3 \\rbrace .$ By following the steps of from Equation $(6.6)$ to $(6.10)$ we get the following $\\mathbb {P}\\left( R(kn) \\right) &\\le \\mathbb {P}\\left( A(kn)^c \\right) + \\mathbb {P}\\left( B(kn)^c \\right) + \\cdots \\\\&+ \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn)^c \\right) + \\cdots \\\\&+ \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) + \\cdots \\\\&+ \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) .$ The first two terms in (REF ) are controlled, respectively by Lemma REF and Lemma REF .",
"In order to finish the proof we need to control the terms in (), () and ().",
"Step 1: Control of the term ().",
"Recall the definition of $\\mathcal {K}(kn)$ at the start of this proof.",
"For $k \\le n$ , on the event $A(kn) \\cap B(kn)$ it holds that $|\\mathcal {K}(kn)| \\le 4 \\log (n),$ since for both $i = 1, 2$ it holds that $|\\mathcal {K}^i(kn)| \\le 2 \\log (n)$ .",
"We deduce that the term $\\mathbb {P}( R((k-1)n), A(kn), B(kn), C(kn)^c )$ can be rewritten as $&\\sum _{i =1, 2} \\sum _{\\begin{array}{c}F^i \\subset \\mathbb {Z}^d, \\\\ |F^i|\\le 2 \\log (n)\\end{array}} \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn)^c, \\mathcal {K}(kn) = F^1 \\cup F^2 \\right) \\\\& = \\sum _{\\begin{array}{c}i =1, 2, \\\\F^i \\subset \\mathbb {Z}^d, \\\\ |F^i|\\le 2 \\log (n)\\end{array}} \\mathbf {E} \\left[ P^\\omega \\left( R((k-1)n), A(kn), B(kn), \\mathcal {K}(kn) = F^1 \\cup F^2 \\right) {1}_{\\begin{array}{c}\\lbrace \\text{some }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\\\ \\text{ is closed} \\rbrace \\end{array}} \\right].$ We may notice that: $P^\\omega \\left( R((k-1)n), A(kn), B(kn), \\mathcal {K}(kn) = F^1 \\cup F^2 \\right)$ is a $\\mathbf {P}$ -random variable measurable with respect to the sigma algebra $\\sigma \\left(c([x, y]), \\,\\, x, y \\notin \\mathcal {H}^+(kn) \\right)$ .",
"The event $\\left\\lbrace \\text{some }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\text{ is closed} \\right\\rbrace $ is measurable with respect to the sigma algebra $\\sigma \\left(c([x, y]), \\,\\, x, y \\in \\mathcal {H}^+(kn)\\right)$ .",
"Hence we have $\\mathbf {P}$ -independence between these two random objects.",
"Hence, after some standard manipulation last sum can be rewritten as $\\sum _{i =1, 2} \\sum _{\\begin{array}{c}F^i \\subset \\mathbb {Z}^d,\\\\ |F^i|\\le 2 \\log (n)\\end{array}} &\\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), \\mathcal {K}(kn) = F^1 \\cup F^2 \\right) \\\\& \\times \\left( 1 - \\mathbf {P}\\left( \\text{all }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\text{ are open} \\right) \\right)$ Since $|F^{i}| \\le 2\\log (n)$ and that $|F^1 \\cup F^2| \\le |F^1| + |F^2|$ we have $\\mathbf {P}\\left( \\text{all }x \\in \\partial (F^1 \\cup F^2) \\cap \\mathcal {H}^+(kn) \\text{ are open} \\right) & \\ge \\left(1 - \\varepsilon (K)\\right)^{2d |F^1 \\cup F^2|}\\\\& \\ge \\left(1 - \\varepsilon (K)\\right)^{8d\\log (n)} \\\\& = n^{8d \\log (1 - \\varepsilon (K))},$ where $\\varepsilon (K) \\mathbf {P}\\left( c_* \\notin [1/K, K] \\right) $ .",
"For any $\\varepsilon ^{\\prime } > 0$ it is possible to fix $K$ large enough such that $8d \\log (1 - \\varepsilon (K)) \\ge - \\varepsilon ^{\\prime }$ .",
"Then we have $\\displaystyle \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn)^c \\right) \\le ( 1 - n^{-\\varepsilon ^{\\prime }}) \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn)\\right),$ which implies $n^{-\\varepsilon ^{\\prime }} \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn)\\right) \\le \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn) \\right).",
"$ Step 2: Control of the term ().",
"Notice that, on the event $A(n)$ we have that, for $i = 1, 2$ , $X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}}} \\in \\partial ^+_{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}$ and by definition of $\\mathcal {K}^{i}(kn)$ , $X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}}} \\in \\mathcal {K}^{i}(kn)$ .",
"Moreover on $B(kn)$ we also have that $|\\mathcal {K}^{i}(kn)| \\le 2 \\log (n)$ .",
"Hence $&\\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) \\\\&\\le \\sum _{i =1, 2}\\sum _{z^i} \\sum _{F^i} \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), \\bigcap _{i = 1, 2} \\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\rbrace , C(kn), D(kn)^c \\right),$ where $\\displaystyle \\sum _{z^i} \\sum _{F^i}$ is a shortcut for $\\displaystyle \\sum _{z^i \\in \\partial ^+_{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}} \\sum _{\\begin{array}{c}F^i \\subset \\mathbb {Z}^d \\\\ |F^i| \\le 2 \\log (n)\\\\ z^i \\in F^i\\end{array}}$ .",
"For $\\omega $ fixed, the events $\\displaystyle R((k-1) n ), A(kn), B(kn), \\bigcap _{i = 1, 2} \\left\\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\right\\rbrace $ are $P^\\omega $ -measurable with respect to $\\sigma \\Big (\\lbrace X^1_j, j \\le T^1_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^1_{kn}}\\rbrace , \\lbrace X^2_k, k \\le T^2_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^2_{kn}} \\rbrace \\Big )$ .",
"Thus, thanks to Lemma REF and the quenched independence of the two walks, we can write $\\mathbb {P}&\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) \\nonumber \\\\&\\le \\sum _{i =1, 2} \\sum _{z^i} \\sum _{F^i} \\mathbf {E}\\bigg [ P^\\omega \\Big ( R((k-1)n), A(kn), B(kn), \\bigcap _{i = 1, 2} \\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}^i_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\rbrace \\Big ) \\nonumber \\\\& \\times P_{z^1, z^2}^\\omega \\left(T^1_{\\partial F^1} < T^1_{\\partial F^1 \\cap \\mathcal {H}^+(kn)} \\textnormal { or } T^2_{\\partial F^2} < T^2_{\\partial F^2 \\cap \\mathcal {H}^+(kn)}\\right) {1}_{\\lbrace x \\in \\partial F \\cap \\mathcal {H}^+(kn) \\text{ is open} \\rbrace } \\bigg ].",
"$ For both $F^1$ and $F^2$ we can notice thanks to the definition of $\\mathcal {K}(kn)$ and the fact that we are considering the event $\\lbrace x \\in \\partial F \\cap \\mathcal {H}^+(kn) \\text{ is } K\\textnormal {-open} \\rbrace $ we get that all the edges $e \\in \\partial _E F$ are $K$ -normal.",
"Just as a remark also notice that $F^1$ and $F^2$ are either the same set or disjoint sets, so overall we get that $\\partial _E F^1$ and $\\partial _E F^2$ are two (possibly one) sets of normal edges.",
"We obtain a bound for $P_{z^1}^\\omega (T^1_{\\partial F^1} \\ge T^1_{\\partial F^1 \\cap \\mathcal {H}^+(kn)})$ , the case $P_{z^2}^\\omega (T^2_{\\partial F^2} \\ge T^2_{\\partial F^2 \\cap \\mathcal {H}^+(kn)})$ can be treated by following the same reasoning.",
"For any $z^1 \\in F^1 \\cap \\partial ^+_{\\text{in}} \\widetilde{\\mathcal {B}}^1_{kn}$ , there exists a neighbour $y^1 \\in \\mathcal {H}^+(kn)$ .",
"Thanks to the fact that $F^1 \\subset \\widetilde{\\mathcal {B}}_{kn}$ we have that, for any $x \\in \\partial F^1 \\cup F^1$ , $(y^1 - x) \\cdot \\vec{\\ell } \\ge - 1$ and $(z^1 - x) \\cdot \\vec{\\ell } \\ge - 2$ .",
"Then, thanks to this observation and the definition of $K$ -normal edges, we see that $c^\\omega (e) \\le K^2 e^{3 \\lambda } c^{\\omega }([z^1, y^1]), \\quad \\textnormal {for } e \\in \\partial _E F^1.$ Thus we can apply Lemma REF with $G= F^1$ (resp.",
"$F^2$ ) and obtain, thanks to quenched independence $P_{z^1, z^2}^\\omega &\\left(T^1_{\\partial F^1} < T^1_{\\partial F^1 \\cap \\mathcal {H}^+(kn)} \\textnormal { or } T^2_{\\partial F^2} < T^2_{\\partial F^2 \\cap \\mathcal {H}^+(kn)}\\right) \\\\ &\\le P_{z^1, z^2}^\\omega \\left(T^1_{\\partial F^1} < T^1_{y^1} \\textnormal { or } T^2_{\\partial F^2} < T^2_{y^2}\\right) \\\\& = \\left( 1 - P_{z^1}^\\omega \\left( T^1_{y^1} \\le T^1_{\\partial F^1}\\right) P_{z^2}^\\omega \\left( T^2_{y^2} \\le T^2_{\\partial F^2}\\right) \\right) \\\\&\\le (1 - c\\log (n)^{-2}),$ using $|F^1| \\le 2 \\log (n)$ .",
"By plugging it in Equation (REF ) we get $\\mathbb {P}&\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn)^c \\right) \\\\&\\le \\sum _{i =1, 2} \\sum _{z^i} \\sum _{F^i} \\mathbf {E}\\bigg [ P^\\omega \\bigg ( R((k-1)n), A(kn), B(kn), \\bigcap _{i = 1, 2} \\Big \\lbrace X^{i}_{T^i_{\\partial _{\\text{in}} \\widetilde{\\mathcal {B}}_{kn}}} = z^i, \\mathcal {K}^{i}(kn) = F^i \\Big \\rbrace \\bigg ) \\\\& \\times {1}_{\\lbrace x \\in \\partial F \\cap \\mathcal {H}^+(kn) \\text{ is open} \\rbrace } \\bigg ] (1 - c\\log (n)^{-2})\\\\&\\le (1 - c\\log (n)^{-2}) \\mathbb {P}\\left( R((k - 1)n ), A(n), B(n), C(n) \\right).$ Which readily implies that for some positive constant $c>0$ we get $c\\log (n)^{-2} & \\mathbb {P}\\left( R((k - 1)n ), A(n), B(n), C(n) \\right) \\nonumber \\\\&\\le \\mathbb {P}\\left( R((k-1)n), A(kn), B(kn), C(kn), D(kn) \\right).",
"$ Step 3: Control of the term ().",
"On the event $A(n)\\cap B(n) \\cap C(n) \\cap D(n)$ we know that both $X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} \\in \\partial ^+ \\widetilde{\\mathcal {B}}^1_{kn}$ and $X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} \\in \\partial ^+ \\widetilde{\\mathcal {B}}^2_{kn}$ are $K$ -open new maxima in the direction $\\vec{\\ell }$ .",
"Moreover, the event $E(kn)$ does not depend on what happens in the “double box” $\\widetilde{\\mathcal {B}}^1_{kn} \\cup \\widetilde{\\mathcal {B}}^2_{kn}$ .",
"Let us also introduce the concept of $x, y$ -open vertex, we say that $z$ is $x, y$ -open if all the edges in the set $\\mathcal {E}_{z} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{y})$ are $K$ -normal.",
"Let us also define $R^{\\prime }((k - 1)n) = R((k - 1)n) \\cap A(n)\\cap B(n) \\cap C(n) \\cap D(n).$ Recall (see (REF ) and considerations above) that on this event we have that both walks, when they hit level $kn$ , are at new $K$ -open maxima.",
"We can write $\\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) \\\\&\\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbb {P}\\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2, \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}, E(kn)^c \\right)$ $&\\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbf {E}\\bigg [ P^\\omega \\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\right) {1}_{\\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace } \\\\&\\big ( {1}_{\\lbrace \\textnormal {A vertex }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed}\\rbrace } + {1}_{\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace } \\\\&\\times P^\\omega _{x^1, x^2}\\left( X^1_1 \\ne x^1 + e_1 \\text{ or } X^1_2 \\ne x^1 + 2e_1 \\text{ or } X^2_1 \\ne x^2 + e_1 \\text{ or } X^2_2 \\ne x^2 + 2e_1 \\right) \\big ) \\bigg ].$ On the event $\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace \\cap \\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace $ one has that $P^\\omega _{x^i}\\left( X^i_1 = x^i + e_1, X^i_2 = x^i + 2e_1 \\right) \\ge c > 0$ for both $i = 1, 2$ .",
"Then $\\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) \\\\&\\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbf {E}\\bigg [ P^\\omega \\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\right) {1}_{\\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace } \\\\& \\big ( {1}_{\\lbrace \\textnormal {A vertex in }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed}\\rbrace } \\\\ &+ (1-c^2){1}_{\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace } \\big ) \\bigg ].$ The $\\mathbf {P}$ -random variables $P^\\omega \\Big ( R^{\\prime }((k - 1)n), X^1_{T_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\Big )$ and ${1}_{\\lbrace \\lbrace x^1, x^2\\rbrace \\textnormal { are }K\\textnormal {-open}\\rbrace }$ are measurable with respect to $\\sigma ( e \\in E(\\widetilde{\\mathcal {B}}^1_{kn} \\cup \\widetilde{\\mathcal {B}}^2_{kn})\\cup \\mathcal {E}_{x^1} \\cup \\mathcal {E}_{x^2})$ , we recall that $\\mathcal {E}_x, x \\in \\mathbb {Z}^d$ is defined in (REF ).",
"On the other hand the events $\\lbrace \\textnormal {A vertex in }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed}\\rbrace $ and $\\lbrace \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open}\\rbrace $ are measurable with respect to $\\sigma ( e \\notin E(\\widetilde{\\mathcal {B}}^1_{kn} \\cup \\widetilde{\\mathcal {B}}^2_{kn})\\cup \\mathcal {E}_{x^1} \\cup \\mathcal {E}_{x^2} )$ , thanks to the fact that we are summing over $x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}$ .",
"This implies $\\mathbf {P}$ -independence between these two groups of random variables.",
"This yields $\\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)^c \\right) \\\\& \\le \\sum _{i = 1, 2} \\sum _{x^i \\in \\partial ^+ \\widetilde{\\mathcal {B}}^i_{kn}} \\mathbf {E}\\left[ P^\\omega \\left( R^{\\prime }((k - 1)n), X^1_{T^1_{\\partial \\widetilde{\\mathcal {B}}^1_{kn}}} = x^1, X^2_{T^2_{\\partial \\widetilde{\\mathcal {B}}^2_{kn}}} = x^2 \\right) \\right] \\times \\\\& \\times \\bigg ( \\mathbf {P}\\left( \\textnormal {A vertex in }\\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { is }x^1, x^2\\textnormal {-closed} \\right) + \\\\&+ (1 - c^{\\prime }) \\mathbf {P}\\left( \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open} \\right) \\bigg ) \\\\& \\le (1 - c^{\\prime }) \\mathbb {P}\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn) \\right),$ since $\\mathbf {P}\\left( \\lbrace x^1 + e_1, x^1 + 2e_1, x^2 + e_1, x^2 + 2e_1\\rbrace \\textnormal { are }x^1, x^2\\textnormal {-open} \\right) > 0$ and its value does not depend on $x^1,x^2$ .",
"This implies that there exists $c>0$ such that $c \\mathbb {P}&\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn) \\right) \\nonumber \\\\ &\\le \\mathbb {P}\\left(R((k - 1)n), A(kn), B(kn), C(kn), D(kn), E(kn)\\right).",
"$ Step 4: Conclusion.",
"For any $\\varepsilon ^{\\prime } > 0$ , $M > 0$ , one can find a $K$ chosen large enough such that, putting together (REF ) with Lemma REF and Lemma REF and the estimates attained in (REF ), (REF ) and (REF ), implies that, for all $k \\in \\lbrace 2, \\dots , n \\rbrace $ $\\mathbb {P}\\left( R(kn) \\right) \\le \\mathbb {P}\\left( R((k-1)n) \\right)\\left( 1 - c\\log (n)^{-2} n^{-\\varepsilon ^{\\prime }} \\right) + n^{-M},$ which implies the result.",
"For the construction of joint regeneration times we will need for the joint $K$ -open ladder points to be close.",
"We introduce for $R > 0$ the variable $\\mathcal {M}^{\\bullet (K)}_R = \\mathcal {M^\\bullet }_R \\mathcal {M}^{\\bullet (K)} \\circ \\theta _{T_R} - R$ where we recall that $\\mathcal {M}^{\\bullet (K)}$ is defined in (REF ) and $\\theta _{T_R}$ means that the first walk is shifted by $T^{1}_R$ and the second one by $T^{2}_R$ .",
"We introduce $ \\Upsilon (n) = \\bigcap _{R = 0}^{n}\\left\\lbrace \\mathcal {M^\\bullet }_R\\le n^{\\frac{1}{2}}\\right\\rbrace ,$ under this event before reaching $\\mathcal {H}^{+}(n)$ the successive joint $K$ -open ladder points are always at distance less than $n^{\\frac{1}{2}}$ in the direction $\\vec{\\ell }$ .",
"Proposition 3.6 For any $M > 0$ there exists $K > 0$ $\\sup _{U_1, U_2 \\in \\mathcal {U}}\\mathbb {P}_{U_1, U_2}(\\Upsilon (n)^{c}) \\le Cn^{-M}.$ where we recall $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and $C$ only depends on $d,\\lambda $ .",
"The proof is similar to the proof of .",
"Let us consider the events, for $j \\in \\lbrace 0, \\dots , n\\rbrace $ $\\Upsilon _j(n) = \\bigcap _{R = 0}^{j - 1}\\left\\lbrace \\mathcal {M^\\bullet }_R\\le n^{\\frac{1}{2}}\\right\\rbrace \\quad \\textnormal {and} \\quad \\Theta _j(n) = \\left\\lbrace \\mathcal {M^\\bullet }_j > n^{\\frac{1}{2}}\\right\\rbrace .$ We can split the event over the minimal $j \\in \\lbrace 0, \\dots , n\\rbrace $ such that $\\Theta _j(n)$ is true and get $ \\mathbb {P}_{U_1, U_2}(\\Upsilon (n)^{c}) \\le \\sum _{j = 0}^n \\mathbb {P}_{U_1, U_2} \\left( \\Upsilon _j(n), \\Theta _j(n), T^1_{\\partial \\widetilde{\\mathcal {B}}^1_n} = T^1_{\\partial ^+ \\widetilde{\\mathcal {B}}^1_n} , T^2_{\\partial \\widetilde{\\mathcal {B}}^2_n} = T^2_{\\partial ^+ \\widetilde{\\mathcal {B}}^2_n} \\right) + C e^{-c n}.$ On the event $\\Upsilon _j(n) \\cap \\Theta _j(n) \\cap \\lbrace T^1_{\\partial \\widetilde{\\mathcal {B}}^1_n} = T^1_{\\partial ^+ \\widetilde{\\mathcal {B}}^1_n} \\rbrace \\cap \\lbrace T^2_{\\partial \\widetilde{\\mathcal {B}}^2_n} = T^2_{\\partial ^+ \\widetilde{\\mathcal {B}}^2_n} \\rbrace $ one is sure that $X^i_{T_{j}} \\in \\widetilde{\\mathcal {B}}^i_n$ for $i = 1, 2$ then by union bound argument $ \\nonumber \\mathbb {P}_{U_1, U_2} &\\left( \\Upsilon _j(n), \\Theta _j(n), T^1_{\\partial \\widetilde{\\mathcal {B}}^1_n} = T^1_{\\partial ^+ \\widetilde{\\mathcal {B}}^1_n}, T^2_{\\partial \\widetilde{\\mathcal {B}}^2_n} = T^2_{\\partial ^+ \\widetilde{\\mathcal {B}}^2_n} \\right) \\\\&\\nonumber \\le \\sum _{x_1, x_2 \\in \\widetilde{\\mathcal {B}}^1_n, \\widetilde{\\mathcal {B}}^2_n} \\mathbb {P}_{U_1, U_2} \\left( X^1_{T_j} = x_1, X^2_{T_j} = x_2, \\mathcal {M}_{j}^{\\bullet (K)} > n^{1/2} \\right) \\\\&\\le |\\mathcal {B}_{n+1}|^2 \\sup _{x_1, x_2}\\mathbb {P}_{x_1,x_2}\\left(\\mathcal {M}^{\\bullet (K)} - j \\ge n^{1/2}\\right),$ where we used Lemma REF in the last inequality and the $\\sup $ is over all $x_1,x_2 \\in \\widetilde{\\mathcal {B}}^1_n, \\widetilde{\\mathcal {B}}^2_n$ such that $x_1,x_2$ are compatible with $\\lbrace X^1_{T_j} = x_1, X^2_{T_j} = x_2\\rbrace $ .",
"Now we conclude using the fact that $\\mathbb {P}_{x_1,x_2}\\left(\\mathcal {M}^{\\bullet (K)} - j \\ge n^{1/2}\\right) = \\mathbb {P}_{0,x_2-x_1}\\left(\\mathcal {M}^{\\bullet (K)} - j+ x_1\\cdot \\vec{\\ell } \\ge n^{1/2}\\right).$ Where we used translation invariance of the environment, now using the fact that $|x_1\\cdot \\vec{\\ell } - j| \\le 1$ and $|(x_2-x_1)\\cdot \\vec{\\ell }| \\le 1$ , the assumptions of Proposition REF are verified.",
"We conclude that $\\sup _{x_1, x_2}\\mathbb {P}_{x_1,x_2}\\left(\\mathcal {M}^{\\bullet (K)} - j \\ge n^{1/2}\\right) \\le Cn^{-M},$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"We then conclude the domination of (REF ) using the fact that $|\\mathcal {B}_{n+1}| \\le Cn^{2d\\alpha }$ ."
],
[
"Enhanced walks",
"The traditional definition of regeneration times has the drawback that it doesn't guarantee independence since the knowledge of the conductances at $x$ influences the walk at any time it reaches a neighbour of $x$ (even when that neighbour $y$ is such that $y\\cdot \\vec{\\ell }> x \\cdot \\vec{\\ell }$ ).",
"In order to obtain stronger independence properties, which we need for our purpose, we need to introduce a variant in the construction of regeneration times.",
"Such a task was already done in using a so-called “enhanced-walk”.",
"We recall the definition of the enhanced walk which is the same as in .",
"For any environment $\\omega $ we define two transition probabilities for $x \\sim y$ $p^{\\omega }(x,y) = \\frac{c_{*}(x,y)e^{(x+y)\\cdot \\ell }}{\\displaystyle \\sum _{z \\sim x}^{} c_{*}(x,z)e^{(x+z)\\cdot \\ell }},$ and $p_K^{\\omega }(x,y) = \\frac{(c_{*}(x,y)\\wedge K^{-1})e^{(x+y)\\cdot \\ell }}{\\displaystyle \\sum _{ z \\sim x}^{} (c_{*}(x,z)\\vee K)e^{(x+z)\\cdot \\ell }},$ and 0 in both cases if $x$ and $y$ are not adjacent.",
"Note that, for any $x,y \\in \\mathbb {Z}^d$ , we have that $p^{\\omega }_{K}(x,y) \\le p^{\\omega }(x,y)$ .",
"For any $(x,z) \\in \\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ we define the Markov chain $(\\tilde{X}_{n})_{n \\ge 0} = (X_{n}, Z_{n})_{n \\ge 0}$ with law $P^\\omega _{(x,z)}$ with transitions probabilities $p^{\\omega }((x_1,z_1),(x_2,z_2))$ defined by $&1.",
"\\,\\, \\tilde{X}_{0} = (x,z), P^{\\omega }_{(x,z)}\\text{-a.s},\\\\&2.\\,\\, p^{\\omega }((x,z),(y,1)) = p_K^{\\omega }(x,y),\\\\&3.\\,\\, p^{\\omega }((x,z),(y,0)) = p^{\\omega }(x,y) - p_K^{\\omega }(x,y).$ This definition ensures that the first coordinate has the same law that our classic walk in $\\omega $ .",
"When we are talking of the annealed law for the enhanced walk we will also write $\\mathbb {P}_{(x,z)}$ .",
"We observe that the evolution of the first coordinate does not depend on the second coordinate, thus when we are talking about only the first coordinate, we will simply write $X$ instead of $\\tilde{X}$ and $P_{x}^{\\omega }$ instead of $P^{\\omega }_{(x,z)}$ ."
],
[
"Regeneration times of a single walk",
"In this subsection we recall the rigorous definition of regeneration times introduced .",
"This definition will be used at the end of the paper and we highlight the fact that the definition of joint regeneration levels in Section REF is very similar to the definition of regeneration times.",
"First we introduce $\\mathcal {M}^{(K)}= \\inf \\Big \\lbrace &i \\ge 2 \\text{ ; }X_i \\text{ is }K\\text{-open, }X_j\\cdot \\vec{\\ell } < X_{i-2} \\cdot \\vec{\\ell } \\text{ for any } j < i-2 \\\\& \\text{ and } X_i = X_{i-1} + e_1 = X_{i-2} + 2e_1 \\Big \\rbrace .$ We also introduce $& \\mathbf {BACK} = \\inf \\left\\lbrace n \\ge 1 \\text{ ; } X_n \\cdot \\vec{\\ell } \\le X_0 \\cdot \\vec{\\ell }\\right\\rbrace \\\\& \\mathbf {ORI} = \\inf \\left\\lbrace n \\ge 1 \\text{ ; }X_{n-1} \\in \\mathcal {V}_{X_0} \\text{ and } Z_n = 0\\right\\rbrace ,$ and finally $D = \\mathbf {BACK} \\wedge \\mathbf {ORI}.$ The event $\\lbrace \\mathbf {BACK} > n\\rbrace $ means that the walk $X$ does not backtrack below $X_0$ before time $n$ .",
"The event $\\lbrace \\mathbf {ORI} > n\\rbrace $ is essential to get independence between the regeneration times.",
"In particular, a coupling argument shows that, as long as $\\lbrace \\mathbf {ORI} > n\\rbrace $ occurs, the conductances around the starting point do not affect the future of the walk if the point is $K$ -open.",
"That is why we will be interested in the event $\\lbrace D = +\\infty \\rbrace $ to regenerate.",
"Now we introduce the sequence of dependent stopping times $(S_n)_{n \\ge 0}$ and the levels $(M_n)_{n \\ge 0}$ $&\\hspace{40.0pt}S_0 = 0\\text{, }M_0 = X_{0} \\cdot \\vec{\\ell } \\text{ and }\\\\& \\text{ for } n \\ge 0\\text{, } S_{n+1} = \\mathcal {M}^{(K)} \\circ \\theta _{T_{\\mathcal {H}^{+}(M_n)}} + T_{\\mathcal {H}^{+}(M_n)},$ where $M_n = \\sup \\lbrace X_k \\cdot \\vec{\\ell } \\text{ with } 0 \\le k \\le R_n \\rbrace ,$ with $R_n = S_n + D \\circ \\theta _{S_n}.$ Then, we define the first regeneration time as $\\tau _1^{(K)} = S_N \\text{ with } N = \\inf \\lbrace n \\ge 1 \\text{ ; }S_n < +\\infty \\text{ and } M_n = +\\infty \\rbrace .$ Remark 3.7 Most of the time we will forego the superscript $K$ in $\\tau _1^{(K)}$ since it will be fixed large enough.",
"Using we obtain that $\\tau _1$ is $\\mathbb {P}_0$ -almost surely finite and we can define inductively finite regeneration times by $\\tau _{k+1} = \\tau _1 + \\tau _{k}((X_{\\tau _1+\\cdot } - X_{\\tau _1}),\\omega ( \\cdot +X_{\\tau _1})).$ The key result states that the $\\tau _k$ are $\\mathbb {P}_0$ -almost surely finite and that $(X_{\\tau _{k}+\\cdot } - X_{\\tau _k})$ is independent of $(X_i)_{0\\le i \\le \\tau _k}$ and distributed as a random walk under $\\mathbb {P}_0^K( \\, \\cdot \\text{ | }D=+\\infty )$ .",
"In the next section we will generalize these notions to two walks.",
"As mentioned before, regeneration times will turn into regeneration levels and the independence property will become a Markov-type property.",
"Remark 3.8 The reader should be careful in the following since we are going to define joint regeneration levels and we will use the notation $\\mathcal {D}^{^{\\bullet }}$ which is not the same as $D$ and $D^{\\otimes }$ , which are the natural counterparts for a single walk or for two independent walks under the annealed measure."
],
[
"Enhanced walks and joint regeneration",
"We consider two enhanced random walks $\\tilde{X}^1$ and $\\tilde{X}^2$ independent under the quenched law.",
"We recall the notation set in Section REF .",
"For $i \\in \\lbrace 1,2\\rbrace $ we introduce the random variables: $&\\mathbf {BACK}^i = \\inf \\left\\lbrace n > 0 \\text{ ; } X^i_n \\cdot \\vec{\\ell } \\le X^i_0 \\cdot \\vec{\\ell } \\right\\rbrace , \\\\& \\mathbf {ORI}^i = \\inf \\left\\lbrace n > 0 \\text{ ; } X^i_{n-1} \\in \\mathcal {V}_{X^1_0} \\cup \\mathcal {V}_{X^2_0} \\text{ and } Z^i_n = 0 \\right\\rbrace .",
"$ Then we introduce $\\mathcal {D}^{\\bullet i} = \\mathbf {BACK}^i \\wedge \\mathbf {ORI}^i.$ Finally we set $\\mathcal {D}^{\\bullet } = \\mathcal {D}^{\\bullet 1} \\wedge \\mathcal {D}^{\\bullet 2}.$ Remark 3.9 The notation $\\mathbf {BACK}^i$ refers to the first time the walk $X^i$ backtracks.",
"The notation $\\mathbf {ORI}^i$ encapsulates information around the starting points in such a way that Proposition REF holds.",
"In the following we will be interested in the case $\\mathcal {D}^{\\bullet } = +\\infty $ and $X_0^1,X_0^2$ are $K$ -open.",
"In that case the walks never backtrack and they behave as if for all $e \\in \\mathcal {E}_{X_0^{1}} \\cup \\mathcal {E}_{X_0^{2}}$ , $c_{*}(e) = K$ .",
"We notice that $\\mathcal {D}^{\\bullet 1}$ is not measurable with respect to $\\sigma (X_0^{1},(\\tilde{X}^{1}_{n})_{n \\ge 1})$ but with respect to $\\sigma (X_0^{1},X_0^{2},(\\tilde{X}^{1}_{n})_{n \\ge 1})$ .",
"Remark 3.10 We can notice that under $P^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}$ , $\\mathcal {D}^{\\bullet 1}$ and $\\mathcal {D}^{\\bullet 2}$ are independent.",
"Indeed, the variables $X^1_0$ and $X^2_0$ are almost surely constant under $P^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}$ and then we use the quenched independence of the two walks.",
"The following proposition shows that under $\\mathcal {D}^{\\bullet } = +\\infty $ the walks behave as if the environment around the starting points is deterministic.",
"For $\\omega $ an environment and $(x_1,z_1),(x_2,z_2) \\in \\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ we define $P^{\\omega _K }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}$ the quenched probability such that the conductances $c_{*}(e)$ with $e \\in \\mathcal {E}_{x_1}\\cup \\mathcal {E}_{x_2}$ are set to $K$ .",
"Proposition 3.11 Fix $(x_1,z_1),(x_2,z_2) \\in \\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ .",
"If $x_1$ and $x_2$ are both $K$ open points then for any bounded $ \\sigma \\big (\\tilde{X}^{1}_{n},\\tilde{X}^{2}_{n}\\big )$ -measurable function $f(\\cdot )$ , we have the equality: $E^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right] = E^{\\omega _K }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right].$ To lighten the notation in the proof we will write $P^{\\omega }_{\\begin{array}{c}(x_1,z_1)\\\\(x_2,z_2)\\end{array}} = P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}.$ The proof goes the same way as the proof of that uses a coupling argument.",
"The idea of the proof is to construct a couple $(U^1,U^2)$ with the same law as $(\\tilde{X}^1,\\tilde{X}^2)$ under $P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}$ and a couple $(V^1,V^2)$ with the same law as $(\\tilde{X}^1,\\tilde{X}^2)$ under $P^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ .",
"In addition we will construct the random variables in such a way that $\\displaystyle (U_i^1,U_i^2)_{0\\le i \\le n} = (V_i^1,V_i^2)_{0 \\le i \\le n}$ as soon as $\\mathcal {D}^{\\bullet }(U) > n$ where $\\mathcal {D}^{\\bullet }(U)$ denotes the quantity $\\mathcal {D}$ defined in (REF ) associated to $(U^1,U^2)$ .",
"We give the construction of $(U^1_{\\cdot },V^{1}_{\\cdot })$ under a probability measure $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ .",
"We can then build $(U^2_{\\cdot },V^{2}_{\\cdot })$ independently in the same way under $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ .",
"To simplify the notations and highlight the fact that $U^1$ has the same law as $\\tilde{X}^1 = (X^1,Z^1)$ we will write $U^1 = U = (X_n,Z_n)_{n \\ge 0}$ and $V^1 = V = (Y_n,W_n)_{n\\ge 0}$ .",
"We also call $\\mathcal {D}^{\\bullet }_U = \\mathcal {D}^{\\bullet 1}(U)$ and $\\mathcal {D}^{\\bullet }_V = \\mathcal {D}^{\\bullet 1}(V)$ .",
"We set $((X_0,Z_0),(Y_0,W_0)) = ((x_1,z_1),(x_1,z_1))$ $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.",
"Given the trajectory up to time $n \\ge 0$ the conditional law of $(U_{n+1},V_{n+1})$ is given by the following rules: $1.$ If $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace $ and if $U_n = V_n = x \\in \\mathbb {Z}^d$ .",
"Then $U$ makes a step according to $P^{\\omega }$ and (a) If $x \\notin \\lbrace x_1,x_2\\rbrace $ and $x \\notin \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ then $V_{n+1}= U_{n+1}$ .",
"(b) If $x \\in \\lbrace x_1,x_2\\rbrace $ or $x \\in \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ and if $Z_{n+1} = 1$ then $V_{n+1}= U_{n+1}$ .",
"(c) If $x \\in \\lbrace x_1,x_2\\rbrace $ or $x \\in \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ and if $Z_{n+1} = 0$ then for any $j \\in \\lbrace 1,...,2d\\rbrace $ , $(Y_{n+1},W_{n+1}) = (x + e_j,0)$ with probability $\\frac{p^{\\omega _K}(x,x+e_j)-p_K^{\\omega _K}(x,x+e_j)}{1- \\displaystyle \\sum _{1 \\le j \\le 2d}^{} p_{K}^{\\omega _K}(x,x+e_j)}.$ $2.$ if $\\lbrace \\mathcal {D}^{\\bullet }_U \\le n\\rbrace \\cup \\lbrace \\mathcal {D}^{\\bullet }_V \\le n\\rbrace $ then $U$ and $V$ move independentely according to $P^{\\omega }$ and $P^{\\omega _K}$ respectively.",
"We have to show that $(U,V)$ is well defined following this construction.",
"By construction, if $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace \\cap \\lbrace U_n = V_n\\rbrace $ then $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely either $\\lbrace \\mathcal {D}^{\\bullet }_U > n+1\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n+1\\rbrace \\cap \\lbrace U_{n+1} = V_{n+1}\\rbrace $ or $\\lbrace \\mathcal {D}^{\\bullet }_U = \\mathcal {D}^{\\bullet }_V = n+1\\rbrace $ .",
"By induction it follows that: The event $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace \\cap \\lbrace U_n \\ne V_n\\rbrace $ never occurs $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.",
"Thus $U$ and $V$ are well defined.",
"Under $\\lbrace \\mathcal {D}^{\\bullet }_U > n\\rbrace \\cap \\lbrace \\mathcal {D}^{\\bullet }_V > n\\rbrace $ we have $X_k = Y_k$ for all $k \\in \\lbrace 0,...,n\\rbrace $ $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.",
"We have $\\mathcal {D}^{\\bullet }_U = \\mathcal {D}^{\\bullet }_V$ $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely.",
"Thus we have $&P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left(\\lbrace \\mathcal {D}^{\\bullet }_U = +\\infty \\rbrace \\cap \\left\\lbrace \\lbrace \\exists n \\ge 0; U_n \\ne V_n\\rbrace \\cup \\lbrace \\mathcal {D}^{\\bullet }_V < +\\infty \\rbrace \\right\\rbrace \\right) \\\\&\\le P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left(\\mathcal {D}^{\\bullet }_U \\ne \\mathcal {D}^{\\bullet }_V\\right) + \\sum _{n = 0}^{+\\infty } P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left(\\mathcal {D}^{\\bullet }_U > n+1, \\mathcal {D}^{\\bullet }_V >n+1, X_{n+1} \\ne Y_{n+1}\\right) = 0.",
"\\nonumber $ We notice that $p_{K}^{\\omega }(x,\\cdot ) = p_{K}^{\\omega _K}(x,\\cdot )$ for any $x \\in \\mathbb {Z}^d$ and $p^{\\omega }(x,\\cdot ) = p^{\\omega _K}(x,\\cdot )$ as soon as $x \\notin \\lbrace x_1,x_2\\rbrace \\cup \\lbrace x_i + e_j \\text{ | } j \\in \\lbrace 1,...,2d\\rbrace \\text{ and }i \\in \\lbrace 1,2\\rbrace \\rbrace $ , it is easy to check that the law of $U$ (resp.",
"$V$ ) under $P^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ is the law of $\\tilde{X}^1$ under $P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}$ (resp.",
"the law of $\\tilde{X}^1$ under $P^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}$ ).",
"Then take $A_0,\\cdots ,A_n$ subsets of $\\mathbb {Z}^d\\times \\lbrace 0,1\\rbrace $ where $A_i$ are in the $\\sigma $ -algebra generated by the subsets of $\\mathbb {Z}^d \\times \\lbrace 0,1\\rbrace $ and call $A = A_0 \\times \\cdots \\times A_n$ .",
"Thus we have: $&E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (\\tilde{X}^1_{0},\\cdots ,\\tilde{X}^1_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet 1} = +\\infty } \\right] \\\\&= E^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (U_{0},...,U_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet }_U = +\\infty } \\right]\\\\&= E^{\\omega ,\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (V_{0},...,V_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet }_V = +\\infty } \\right] \\\\& = E^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}\\left[{1}_{}\\lbrace (\\tilde{X}^1_{0},...,\\tilde{X}^1_{n}) \\in A\\rbrace {1}_{\\mathcal {D}^{\\bullet 1} = +\\infty } \\right],$ where the second equality follows from (REF ).",
"By a Monotone Class Theorem, for any bounded $\\sigma \\big (\\tilde{X}^{1}_{n}\\big )$ -measurable function we have $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right] = E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right].$ Then, for bounded measurable functions $f$ and $g$ we have: $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right]&= E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }{1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right] \\\\& = E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right] E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}\\left[g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right]\\\\&= E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }\\right] E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right] \\\\&=E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet 1} = +\\infty }{1}_{\\mathcal {D}^{\\bullet 2} = +\\infty }\\right] \\\\&=E^{\\omega _K }_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^1_{\\cdot })g(\\tilde{X}^2_{\\cdot }){1}_{\\mathcal {D}^{\\bullet } = +\\infty }\\right].$ To extend the previous result to bounded measurable functions of the form $f(\\cdot ,\\cdot )$ we use once again the Monotone Class Lemma."
],
[
"$K$ -good points",
"We recall the definition of a $K$ -good point taken from .",
"A point $z \\in \\mathbb {Z}^d$ is said to be $K$ -good if there exists an infinite path $(x_0 = z, x_1,x_2,\\cdots )$ such that all $x_i$ are $K$ -open and $x_{2i+1}-x_{2i} = e_1$ and $x_{2i+2}-x_{2i+1} = \\lbrace e_1,e_2,\\cdots ,e_d\\rbrace $ .",
"Proposition 3.12 There exists a constant $K_0>0$ and $\\eta > 0$ such that for $K > K_0$ we have for any $U_1,U_2 \\in \\mathbb {Z}^d$ we have $\\mathbf {P}(U_1 \\text{ and }U_2\\text{ are }K\\text{-good points}) > \\eta > 0$ .",
"For $e_1,\\cdots ,e_n \\in E(\\mathbb {Z}^d)$ distinct edges, the events $(c^{*}_{e_i} \\in [1/K,K])_{1\\le i \\le n}$ are independent and have the same probability.",
"Thus $\\displaystyle ({1}_{c^{*}_{e} \\in [1/K,K]})_{e\\in E(\\mathbb {Z}^d)}$ is a percolation on $\\mathbb {Z}^d$ .",
"Since for $x \\in \\mathbb {Z}^d$ the event $\\lbrace x \\text{ is }K\\text{-good}\\rbrace $ is an increasing event we can apply FKG inequality and obtain $\\mathbf {P}(U_1 \\text{ and }U_2\\text{ are }K\\text{-good points}) \\ge \\mathbf {P}(U_1 \\text{ is a }K\\text{-good point})\\mathbf {P}(U_2 \\text{ is a }K\\text{-good point}),$ then we conclude using ."
],
[
"Construction of joint regeneration levels and main result",
"We recall the notation $\\mathcal {M}^{\\bullet }$ defined in (REF ).",
"We set $L_0 = 0, \\quad M_{0} = X^1_0 \\cdot \\vec{\\ell } \\vee X^{2}_{0} \\cdot \\vec{\\ell },$ and, for $k\\ge 0$ , $ L_{k+1} = \\mathcal {M}^{\\bullet }\\circ \\theta _{T_{M_k}}.$ Where $ M_k = \\sup \\bigg \\lbrace \\lbrace X^1_m\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le m \\le R_k\\rbrace \\cup \\lbrace X^2_m\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le m \\le R_k\\rbrace \\bigg \\rbrace ,$ with $R_k = \\mathcal {D}^{\\bullet } \\circ \\theta _{T_{L_k}}.$ Here we set $N = \\inf \\lbrace k \\ge 0 \\text{ ; } L_k < +\\infty \\text{ and } R_k =+\\infty \\rbrace $ we define the first joint regeneration level as $\\mathcal {L}_1 = L_{N}$ .",
"With this definition the associated joint regeneration times (respectively points) for $X^1$ and $X^2$ are $T^{1}_{\\mathcal {L}_1}$ and $T^{2}_{\\mathcal {L}_1}$ (respectively $X^{1}_{T^{1}_{\\mathcal {L}_1}}$ and $X^{2}_{T^{2}_{\\mathcal {L}_1}}$ ).",
"We observe that the definition is very similar to the definition in Section REF but instead of defining a sequence of stopping times $(S_k)$ we define a sequence of levels $(L_k)$ .",
"The first regeneration level is the first level $L_k$ such that the walks do not backtrack and properly encodes the information in variables $Z^1, Z^2$ around the starting points.",
"The first thing to show is that the quantity $\\mathcal {L}_1$ , as defined above, is $\\mathbb {P}_0$ -almost surely finite, Proposition REF below shows that the first joint regeneration level happens pretty quickly.",
"Then we will focus in proving the Markov property in Theorem REF .",
"Proposition 3.13 For any $M \\in (0,+\\infty )$ there exists $K_0 >0$ such that for $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ we have $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}(\\mathcal {L}_1 \\ge n) \\le Cn^{-M},$ where the constant $C$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"The proof will be showed later, as we need several lemmas in order to be able to prove this result.",
"We recall the notation $\\mathbb {P}_{U_1,U_2}^{K}$ which is the usual annealed measure with the restriction that the conductances around $U_1,U_2$ are set to have value $K$ ."
],
[
"Proof of Proposition ",
"Lemma 3.14 There exists $K_0 > 0$ and $ \\eta > 0$ such that for $K \\ge K_0$ , for $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ we have $& \\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty ) > \\eta > 0, \\\\& \\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}^{K}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty ) > \\eta > 0,$ where $\\eta $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"We follow the lines of .",
"We only prove the result for $\\mathbb {P}_{U_1,U_2}$ since the proof for $\\mathbb {P}_{U_1,U_2}^{K}$ is similar.",
"We have $\\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty ) > c \\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }=+\\infty \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}),$ where $c = \\mathbb {P}(U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}) > \\eta > 0$ using Proposition REF and where $\\eta $ only depends on $d$ .",
"On event $\\lbrace U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}\\rbrace $ we fix $\\mathcal {P}^i$ a path given by the fact that $U_i$ is a $K$ -good point.",
"We necessarily have $\\mathcal {P}^{i}(1) = e_1$ and $\\mathcal {P}^{i}(2) = e_1 + e_{j_2}$ with $e_{j_2} \\in \\lbrace e_1,\\cdots ,e_d\\rbrace $ .",
"We introduce $L^i_{\\partial ^{+}\\mathcal {B}(n,n^2)+U_i} = \\inf \\lbrace k \\in \\mathbb {N} \\text{ ; }\\mathcal {P}^{i}(k) \\in \\partial ^{+}\\mathcal {B}(n,n^2)+U_i \\rbrace $ and $x^i = \\mathcal {P}^{i}(L^{i}_{\\partial ^{+}\\mathcal {B}(n,n^2)+ U_i})$ .",
"We introduce $&A_1^{i} = \\lbrace (X^{i}_1,Z^{i}_1) = (e_1,1)\\rbrace ,\\\\&A_2^{i} = \\lbrace (X^{i}_2,Z^{i}_2) = (e_1+e_{j_2},1)\\rbrace ,\\\\&A_3^{i} = \\lbrace X^{i}_{k} = \\mathcal {P}^{i}(k) \\text{ for } 3\\le k \\le L_{\\partial ^{+}\\mathcal {B}(n,n^2)+U_i}\\rbrace ,\\\\&A_4^{i} = \\lbrace T^{i}_{\\mathcal {H}^{-}(2)} \\circ \\theta _{T_{x^i}} = +\\infty \\rbrace .$ and define $A_k = A^1_k \\cap A^2_k$ and $\\displaystyle A = \\bigcap _{k=1}^{4}A_{k}$ .",
"We see that $A \\subset \\lbrace \\mathcal {D}^{\\bullet } =+\\infty \\rbrace $ .",
"As $U_1$ and $U_2$ are $K$ -good we have $L^{i}_{\\partial ^{+}\\mathcal {B}(n,n^2)+U_i} \\le Cn$ and since all points in $\\mathcal {P}^{i}$ are $K$ -open, we have $P_{U_1,U_2}^{\\omega }(A_1,A_2,A_3) \\ge c^{n},$ with $c > 0$ which only depend on $K$ , $d$ and $\\ell $ .",
"Thus, by Markov property we have $&\\mathbf {E}[P_{U_1,U_2}^{\\omega }(A_1,A_2,A_3,A_4) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}]\\\\&= \\mathbf {E}[P_{U_1,U_2}^{\\omega }(A_1,A_2,A_3)P^{\\omega }_{x^1,x^2}(A^1_4,A^2_4) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}]\\\\&\\ge c^n\\mathbf {E}[P^{\\omega }_{x^1,x^2}(A_4) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}]\\\\& = c^n \\mathbf {E}\\left[P^{\\omega }_{x^1,x^2}(T^{1}_{\\mathcal {H}^{-}(2)} = +\\infty ,T^{2}_{\\mathcal {H}^{-}(2)} = +\\infty ) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}\\right].$ We also have $&\\mathbf {E}\\left[P^{\\omega }_{x^1,x^2}( \\lbrace T^1_{\\mathcal {H}^{-}(2)} < +\\infty \\rbrace \\cup \\lbrace T^2_{\\mathcal {H}^{-}(2)} < +\\infty \\rbrace ) \\text{ | }U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}\\right]\\\\&\\le C\\mathbb {P}_0\\left(T_{\\mathcal {H}^{-}(-n+2)} < +\\infty \\right) \\\\&\\le C\\exp (-cn).$ Where we used a union bound, translation invariance of the law of the environment and the fact $\\mathbf {P}(U_1 \\text{ and } U_2 \\text{ are } K\\text{-good}) > \\eta > 0$ and .",
"Hence we see that there exists $n_0$ big enough which only depend on $\\lambda ,\\vec{\\ell },d,K$ such that this quantity is smaller than $1/2$ for all $n \\ge n_0$ and it concludes the proof.",
"The following lemmas correspond to the ones in .",
"We recall the definitions of $L_k$ and $M_k$ in (REF ), (REF ), then we introduce the events: $&M^{K}(n) = \\left\\lbrace \\text{for }k \\text{ such that }M_k < n \\text{ we have }L_{k+1} - M_k < n^{1/2} \\right\\rbrace , \\\\&S(n) = \\left\\lbrace \\text{for } k \\text{ such that }L_k < n \\text{ and } M_k < +\\infty \\text{ we have } M_k - L_k < n^{1/2}\\right\\rbrace .$ In the following we may drop the superscript $K$ to lighten the notations.",
"Lemma 3.15 For $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and for any $M > 0$ there exists $K > 0$ and a constant $C > 0$ such that $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}\\left(M^{K}(n)^{c}\\right) \\le Cn^{-M},$ where the constant $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"It follows from Proposition REF .",
"Indeed, $M^{K}(n)^{c}$ implies that there exists $0 \\le R \\le n$ such that $\\mathcal {M}^{\\bullet }_R \\ge n^{\\frac{1}{2}}$ .",
"We introduce $\\bar{L} = \\sup \\left\\lbrace \\lbrace X^1_n\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le n \\le \\mathcal {D}^{\\bullet }\\rbrace \\cup \\lbrace X^2_n\\cdot \\vec{\\ell } \\text{ ; with } 0 \\le n \\le \\mathcal {D}^{\\bullet }\\rbrace \\right\\rbrace .$ The following proposition shows that under $\\mathcal {D} < +\\infty $ , $\\bar{L}$ cannot be too big.",
"Lemma 3.16 Let $\\mathcal {U} = \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ , then there exists constants $C,c$ such that $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}\\left(\\bar{L} \\ge n \\text{ }|\\text{ }\\mathcal {D}^{\\bullet }<+\\infty \\right) \\le C\\exp (-cn),$ where $C,c$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"First thing to say is that $\\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet } < +\\infty ) > \\eta > 0$ with $\\eta $ which does not depends on $U_1,U_2$ .",
"Indeed $0 < \\mathbb {P}_{U_1,U_2}(X_1^1 = U_1+e_1,X^1_2 = U_1) = \\mathbb {P}_{0}(X_1 = e_1 , X_2 = 0) < \\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet } < +\\infty ),$ where we used translation invariance of the environment for the equality.",
"Thus, we only have to dominate $\\mathbb {P}_{U_1,U_2}(\\mathcal {D}^{\\bullet }<+\\infty ,\\bar{L} \\ge n)$ .",
"By union bound and symmetry, we only have to dominate $\\mathbb {P}_{U_1,U_2}\\left(\\mathcal {D}^{\\bullet }<+\\infty ,\\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } \\ge n\\right).$ Without loss of generality we can suppose $U_1= U_2 =0$ , indeed the proof is exactly the same in the general case but instead of considering the boxes $\\mathcal {B}(2^k,2^{\\alpha k})$ we should consider the boxes $\\mathcal {B}(2^k,2^{\\alpha k}) + U_i$ .",
"First we have $\\mathbb {P}_{0}\\left(2^k \\le \\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } < 2^{k+1} \\right) \\le &\\mathbb {P}_{0}\\left(T^1_{\\partial \\mathcal {B}(2^k,2^{\\alpha k})} \\ne T^1_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}\\right) +\\\\&\\mathbb {P}_{0}\\Big (T^1_{\\partial \\mathcal {B}(2^k,2^{\\alpha k})} = T^1_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})},\\\\&T^{+}_{\\mathcal {H}^{-}(0)}\\circ \\theta _{T_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}}< T^{+}_{\\mathcal {H}^{+}(2^{k+1})}\\circ \\theta _{T_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}} \\Big ).$ The first term is smaller than $C\\exp (-c2^k)$ using .",
"Then using a union bound and translation invariance of the environment the second term is dominated by $(2^{k})^{\\alpha d}\\mathbb {P}_{0}\\left(T^1_{\\partial \\mathcal {B}(2^k,2^{\\alpha k})} \\ne T^1_{\\partial ^{+} \\mathcal {B}(2^k,2^{\\alpha k})}\\right) \\le C2^{\\alpha kd}\\exp (-c2^k),$ where we have used once again Theorem $5.1$ of .",
"Then we conclude using the following domination and the previous inequality $\\mathbb {P}_{0}\\left(\\mathcal {D}^{\\bullet }<+\\infty ,\\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } \\ge n\\right) \\le C\\sum _{k,2^{k}\\ge n}^{} \\mathbb {P}_{0}\\left(2^k \\le \\sup _{j \\le \\mathcal {D}^{\\bullet }}X^1_{j}\\cdot \\vec{\\ell } < 2^{k+1}\\right).$ We recall the definition of $S(n)$ in ().",
"Lemma 3.17 Let $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ , then there exists constants $C,c > 0$ such that $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}\\left(S(n)^{c}\\right) \\le C\\exp (-cn^{\\frac{1}{2}}).$ It is an adaptation of .",
"First we notice that $\\mathbb {P}_{0}$ -almost surely, $\\# \\lbrace k \\ge 0 \\text{ such that } L_k < n\\rbrace $ is smaller than $d n$ .",
"Hence we can write $\\displaystyle \\mathbb {P}_{U_1,U_2}\\left(S(n)^{c}\\right) \\le &\\mathbb {P}_{U_1}\\left(T^1_{\\partial \\mathcal {B}_1(n,n^{\\alpha })} \\ne T^1_{ \\partial ^{+} \\mathcal {B}_1(n,n^{\\alpha })} \\right) + \\mathbb {P}_{U_2}\\left(T^2_{\\partial \\mathcal {B}_2(n,n^{\\alpha })} \\ne T^2_{ \\partial ^{+} \\mathcal {B}_2(n,n^{\\alpha })} \\right) \\\\&+ \\sum _{k \\le d n}^{} \\sum _{\\begin{array}{c}x \\in \\mathcal {B}_1(n,n^{\\alpha })\\\\y \\in \\mathcal {B}_2(n,n^{\\alpha })\\end{array}}^{} \\mathbb {P}_{U_1,U_2}\\left(M_k < +\\infty , M_k - L_k \\ge n^{\\frac{1}{2}},X^{1}_{T_{L_k}} = x, X^{2}_{T_{L_k}} = y\\right).$ The first two terms are controlled by Theorem 5.1 of and using translation invariance of the environment.",
"We can dominate the term in the sum by $& \\mathbb {P}_{U_1,U_2}\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet } \\circ \\theta _{T^1_{x}} + T^1_{x}}(X^{1}_j - x)\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet }\\circ \\theta _{T_{x,y}} < +\\infty ,X^{1}_{T_{L_k}} = x,X^{2}_{T_{L_k}} = y\\right) +\\\\&\\mathbb {P}_{U_1,U_2}\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet } \\circ \\theta _{T^2_{y}} + T^2_{y}}(X^{2}_j - y)\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet }\\circ \\theta _{T_{x,y}} < +\\infty ,,X^{1}_{T_{L_k}} = x,X^{2}_{T_{L_k}} = y\\right).$ Let us dominate the first term, the second is dominated the same way by symmetry.",
"Using Lemma REF and the invariance of the environment it is bounded by $\\mathbf {E}\\left[P_{x,y}^{\\omega }\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet }}(X^{1}_j -x)\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet } < +\\infty \\right) \\right]&\\le \\mathbf {E}\\left[P_{0,y-x}^{\\omega }\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet }} X^{1}_j\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}},\\mathcal {D}^{\\bullet } < +\\infty \\right) \\right] \\\\&\\le \\mathbb {P}_{0,y-x}\\left(\\sup _{j \\le \\mathcal {D}^{\\bullet }} X^{1}_j\\cdot \\vec{\\ell } \\ge n^{\\frac{1}{2}}\\text{ }|\\text{ }\\mathcal {D}^{\\bullet }<+\\infty \\right)\\\\& \\le \\mathbb {P}_{0,y-x}\\left(\\bar{L} > n^{\\frac{1}{2}}\\text{ }|\\text{ }\\mathcal {D}^{\\bullet }<+\\infty \\right).$ The last term is dominated using Lemma REF since we have $ |(y-x) \\cdot \\vec{\\ell }| \\le 1$ by definition of $L_k$ .",
"Taking the sum over $k,x,y$ we conclude that we have $\\displaystyle \\mathbb {P}_{U_1,U_2}(S(n)^{c}) \\le Cn^{\\beta }\\exp (-cn^{\\frac{1}{2}}).$ where $\\beta $ only depends on $d,\\alpha $ .",
"It concludes the proof by taking $C$ large enough and $c$ small enough.",
"The following lemma is the equivalent of .",
"Recall that we defined $N = \\inf \\lbrace k \\ge 0 \\text{ ; } L_k < +\\infty \\text{ and } R_k =+\\infty \\rbrace $ , and the definition of $\\mathcal {E}_x, \\mathcal {L}^x, \\mathcal {R}^x$ as in Equations (REF )-(REF ).",
"Furthermore, we define $\\mathcal {E}^{2}_x \\mathcal {E}_x \\cup \\mathcal {E}_{x-e_1}\\cup \\mathcal {E}_{x-2e_1}\\,\\, \\textnormal {the set of all edges which touch} \\,\\, \\lbrace x,x-e_1,x-2e_1\\rbrace .$ Lemma 3.18 There exists $K_0 > 0$ such that for $K \\ge K_0$ $\\sup _{U_1, U_2 \\in \\mathcal {U}}\\mathbb {P}_{U_1,U_2}(N \\ge n) \\le C\\exp (-cn).$ Where $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and $C,c > 0$ only depends on $\\lambda ,\\vec{\\ell },d$ .",
"We follow the lines of the proof of .",
"The idea of the proof is simple, each time we reach a level $L_k$ there is a positive probability to regenerate and using the way we have defined $\\mathcal {D}$ we should have good independence properties, hence $N$ should decay at least like a geometric random variable.",
"We introduce event $C(n) = \\lbrace \\text{for all }k \\le n \\text{ such that }L_k <+\\infty \\text{ we have } \\mathcal {D}^{\\bullet } \\circ \\theta _{T_{L_k}} < +\\infty \\rbrace .$ We have $\\lbrace N \\ge n\\rbrace \\subset C(n)$ , hence we only have to dominate $C(n)$ .",
"We can write $\\mathbb {P}_{U_1,U_2}(C(n+1)) \\le \\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n),\\mathcal {D}^{\\bullet }\\circ \\theta _{T_{L_k}} < +\\infty \\right)\\right].$ The event $\\lbrace X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\rbrace $ is measurable with respect to $\\sigma \\lbrace (X^{1}_n,Z_{n}^1),(X^{2}_k,Z_{k}^2)\\text{ for }0 \\le n \\le T^1_x \\text{ and }0\\le k \\le T^2_y\\rbrace $ .",
"Thus we can use Lemma REF and get $&P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n),\\mathcal {D}^{\\bullet } \\circ \\theta _{T_{L_k}} < +\\infty \\right) \\\\=& P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)P_{x,y}^{\\omega }(\\mathcal {D}^{\\bullet } < +\\infty ).$ Then, the sum is also equal to $&\\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)P_{x,y}^{\\omega }(\\mathcal {D}^{\\bullet } < +\\infty )\\right]\\\\&= \\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)P_{x,y}^{\\omega _{K}}(\\mathcal {D}^{\\bullet } < +\\infty )\\right],$ where we used the fact that both $x$ and $y$ are $K$ -open and Proposition REF .",
"Since $P_{U_1,U_2}^{\\omega }\\big (X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\big )$ is measurable with respect to $\\sigma \\big (c_{*}(e) \\text{ with } e \\in \\mathcal {L}^{x-2e_1} \\cup \\mathcal {L}^{y-2e_1} \\cup \\mathcal {E}^2_{x} \\cup \\mathcal {E}^2_{y}\\big )$ and $P_{x,y}^{\\omega _{K}}(\\mathcal {D}^{\\bullet } < +\\infty )$ is measurable with respect to $\\sigma \\big (c_{*}(e) \\text{ with } e \\in \\mathcal {R}^{x} \\cup \\mathcal {R}^{y} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{y})\\big )$ and using the fact that those sets of edges are disjoint (see Figure REF ) we can write by $\\mathbf {P}$ independence that the sum is equal to $\\sum _{x,y}^{} \\mathbf {E}\\left[P_{U_1,U_2}^{\\omega }\\left(X^1_{T_{L_{n+1}}}=x,X^2_{T_{L_{n+1}}}=y,C(n)\\right)\\right]\\mathbf {E}\\left[P_{x,y}^{\\omega _{K}}(\\mathcal {D}^{\\bullet } < +\\infty )\\right].$ Using Lemma REF with $K_0$ large enough we have $\\mathbb {P}_{x,y}^{K}(\\mathcal {D}^{\\bullet }<+\\infty ) \\le \\eta < 1$ with $\\eta $ a constant which only depends on $\\lambda ,\\vec{\\ell },d$ .",
"It follows that $\\mathbb {P}_{U_1,U_2}(C(n+1)) \\le \\eta \\mathbb {P}_{U_1,U_2}(C(n)).$ The result follows by induction.",
"We now have all the tools to prove Proposition REF .",
"Indeed combining Lemmas REF , REF and REF , it is unlikely to have the $n$ first $L_k$ finite and $L_{k+1} - L_{k} \\ge n^{\\frac{1}{2}}$ for $k \\le n$ .",
"[Proof of Proposition REF ] On $\\lbrace \\mathcal {L}_1 \\ge n\\rbrace \\cap S(n)\\cap M(n)$ we have $N \\ge n^{1/3}$ for $n$ large enough.",
"Indeed, on $\\lbrace \\mathcal {L}_1 \\ge n\\rbrace \\cap S(n)\\cap M(n)$ for $k < n$ such that $L_k < n $ and $M_k < n $ , we have $L_{k+1} - L_k < 2n^{1/2}$ , and since $\\mathcal {L}_1 \\ge n$ , as long as $L_{k+1} < n$ we have $M_{k+1} < +\\infty $ .",
"Hence it follows that we must have $N \\ge n^{\\frac{1}{3}}$ since in the opposite case we would have $\\displaystyle \\mathcal {L}_1 = L_{N} = \\sum _{k=0}^{N-1} L_{k+1}-L_k < N2n^{1/2} < 2n^{5/6}<n$ .",
"The result follows thanks to our estimates in Lemmas REF , REF and REF , putting them together we have $\\mathbb {P}_{U_1,U_2}(\\mathcal {L}_1 \\ge n) &\\le \\mathbb {P}_{U_1,U_2}(M(n)^c) + \\mathbb {P}_{U_1,U_2}(S(n)^{c}) + \\mathbb {P}_{U_1,U_2}(N \\ge n^{\\frac{1}{3}})\\\\&\\le 2C\\exp (-cn^{\\frac{1}{3}}) + C n^{-M} \\\\&\\le Cn^{-M}.$"
],
[
"Further properties of joint regeneration levels",
"We give a slight variation of Proposition REF which will be useful in the following: we extend the result under the measure $\\mathbb {P}_{U_1,U_2}^{K}$ .",
"This is the analogous of .",
"Proposition 3.19 For any $M > 0$ there exists $K_0 > 0$ such that for $K > K_0$ $\\sup _{U_1, U_2 \\in \\mathcal {U}} \\mathbb {P}_{U_1,U_2}^{K}(\\mathcal {L}_1 > n) \\le Cn^{-M}.$ Where $\\mathcal {U} \\lbrace U \\in \\mathbb {Z}^d : |U\\cdot \\vec{\\ell }|\\le 1 \\rbrace $ and the constant $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"For the same reasons as in previous proofs we restrict to the case where $U_1 = U_2 = 0$ without loss of generality, indeed we simply have to replaces boxes $\\mathcal {B}(n,n^{\\alpha })$ with $\\widetilde{\\mathcal {B}}_i(n,n^{\\alpha })$ .",
"We write: $&\\mathbb {P}_{0}^{K}(\\mathcal {L}_1 > n) \\le \\mathbb {P}^{K}_{0}\\left(T^1_{\\partial B(n/2,(n/2)^{\\alpha })} \\ne T^1_{\\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })} \\right) + \\mathbb {P}^{K}_{0}\\left(T^2_{\\partial \\mathcal {B}(n/2,(n/2)^{\\alpha })} \\ne T^2_{\\partial ^{+} \\mathcal {B}(n/2,(n/2)^{\\alpha })} \\right)\\\\&+\\sum _{x,y \\in \\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })}^{} \\mathbb {P}^{K}_{0}(X^1_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = y,T^1_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T_x} < +\\infty )\\\\&+ \\sum _{x,y \\in \\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })}^{} \\mathbb {P}^{K}_{0}(X^1_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = y,T^2_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T^2_y} < +\\infty )\\\\&+\\sum _{x,y \\in \\partial ^{+} \\mathcal {B}((n/2),(n/2)^{\\alpha })}^{} \\mathbb {P}^{K}_{0}(X^1_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial \\mathcal {B}((n/2),(n/2)^{\\alpha })}} = y,\\\\&\\hspace{128.0374pt} T^1_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T^1_x} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} \\circ \\theta _{T^2_y} = +\\infty ,\\mathcal {L}_1 \\circ \\theta _{T_{x,y}} \\ge n/2),$ where in the last sum $\\theta _{T_{x,y}}$ means that the first walk is shifted by $T_x$ and the second by $T_y$ .",
"The first two terms are dominated thanks to .",
"Let us dominate the third term.",
"The fourth can be dominated in the same way by symmetry.",
"The term in the sum can be written using Markov property $\\mathbf {E}&\\left[P^{\\omega _K}_{0}\\left(X^1_{T_{\\partial B((n/2),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial B((n/2),(n/2)^{\\alpha })}} = y\\right)P_{x}^{\\omega ^{0}_K}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty \\right)\\right] \\\\ &\\le \\mathbf {E}\\left[P_{x}^{\\omega ^{0}_K}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty \\right)\\right],$ where the notation $\\omega ^{0}_{K}$ means that we have set the conductances of $\\mathcal {E}_0$ to $K$ .",
"Then we notice that $P_{x}^{\\omega ^{0}_K}(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty )$ does not depend on the conductances of $\\mathcal {E}_0$ , so it is also equal to $P_{x}^{\\omega }(T^1_{\\mathcal {H}^{-}_{e_1}} < +\\infty )$ .",
"Using translation invariance of the environment and the sum is bounded by $Cn^{2d\\alpha }\\exp (-cn)$ .",
"Let us dominate the fifth term.",
"Using Lemma REF the term in the sum can be written $&\\mathbf {E}\\left[P_{0}^{\\omega _{K}}\\left(X^1_{T_{\\partial B((n/2)),(n/2)^{\\alpha })}} = x,X^2_{T_{\\partial B((n/2),(n/2)^{\\alpha })}} = y\\right)P_{x,y}^{\\omega ^{0}_{K}}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,\\mathcal {L}_1\\ge n\\right)\\right]\\\\&\\le \\mathbf {E}\\left[P_{x,y}^{\\omega ^{0}_{K}}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,\\mathcal {L}_1\\ge n\\right)\\right].$ Under $\\lbrace T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty \\rbrace $ both walks never meet conductances in $\\mathcal {E}_{0}$ and so we can replace $\\omega ^{0}_{K}$ by $\\omega $ .",
"Then we write $\\mathbf {E}\\left[P_{x,y}^{\\omega ^{0}_{K}}\\left(T^1_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,T^2_{\\mathcal {H}^{-}_{e_1}} = +\\infty ,\\mathcal {L}_1\\ge n\\right)\\right]&\\le \\mathbb {P}_{x,y}(\\mathcal {L}_1\\ge n) \\\\&\\le \\mathbb {P}_{0,y-x}(\\mathcal {L}_1\\ge n/4),$ where we used Proposition REF since we have $|(y-x) \\cdot \\vec{\\ell }| \\le 1$ because $x,y \\in \\partial ^{+} B((n/2),(n/2)^{\\alpha })$ , translation invariance of the environment.",
"Then we see that the sum is bounded by $ C n^{2d\\alpha } n^{-(M+2d\\alpha )} $ by taking $K$ large enough.",
"This concludes the proof.",
"We can define successive joint regeneration levels as $\\mathcal {L}_{k+1} = \\mathcal {L}_{k} \\circ \\theta _{T_{\\mathcal {L}_1}}.$ Let us define $\\mathcal {G}_k = &\\sigma \\big (\\mathcal {L}_0,\\dots ,\\mathcal {L}_k,(\\tilde{X}^1_{T^1_{\\mathcal {L}_k}\\wedge m})_{m \\ge 0},(\\tilde{X}^2_{T^2_{\\mathcal {L}_k}\\wedge m})_{m \\ge 0}, \\nonumber \\\\ & c_{*}(e) \\text{ with } e \\in E(\\mathcal {H}^{-}(\\mathcal {L}_k-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{X^1_{T^1_{\\mathcal {L}_k}}} \\cup \\mathcal {E}^{2}_{X^2_{T^2_{\\mathcal {L}_k}}}\\big ).$ The definition of $\\mathcal {G}_k$ may seem a bit complicated at first sight.",
"It represents the information that the walkers see before reaching level $\\mathcal {L}_k$ and also the whole environment on the left of $\\mathcal {L}_k-2e_1\\cdot \\vec{\\ell }$ .",
"Figure: The sigma algebra 𝒢 k \\mathcal {G}_k: In red and blue the trajectories before reaching ℒ k \\mathcal {L}_k and in orange the edges considered.Now we want to show that when we consider our walks shifted of $T_{\\mathcal {L}_k}$ they behave like new walks starting from respectively $X^1_{T^1_{\\mathcal {L}_k}}$ and $X^2_{T^2_{\\mathcal {L}_k}}$ conditionned to never backtrack and such that $c_{*}(e) = K$ for $e \\in \\mathcal {E}_{X^1_{T^1_{\\mathcal {L}_k}}} \\cup \\mathcal {E}_{X^2_{T^2_{\\mathcal {L}_k}}}$ .",
"This is the purpose of the next theorem.",
"Theorem 3.20 Fix $K$ large enough.",
"For any $k \\ge 1$ , we have $\\mathcal {L}_k < +\\infty $ $\\mathbb {P}_0$ -almost surely.",
"Now let $f, g, h_k$ be bounded functions which are measurable with respect to $\\sigma \\left( (X^1_0,X^2_0), (\\tilde{X}^1_n,\\tilde{X}^2_n)_{n \\ge 1}\\right)$ , $\\sigma \\left(c_{*}(e) \\text{ ; } e \\in \\mathcal {R}^{X^1_0} \\cup \\mathcal {R}^{X^2_0} \\backslash (\\mathcal {E}_{X^1_0} \\cup \\mathcal {E}_{X^2_0})\\right)$ and $\\mathcal {G}_k$ , respectively.",
"We have the following equality $\\mathbb {E}_{\\begin{array}{c}(0,z_1)\\\\(0,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{\\mathcal {L}_k}+\\cdot },\\tilde{X}^{2}_{T^{2}_{\\mathcal {L}_k}+\\cdot })g\\circ \\theta _{T_{\\mathcal {L}_k}}h_k\\right] = \\mathbb {E}_{\\begin{array}{c}(0,z_1)\\\\(0,z_2)\\end{array}}\\left[h_k\\mathbb {E}^K_{X^1_{T^{1}_{\\mathcal {L}_k}},X^2_{T^{2}_{\\mathcal {L}_k}}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot })g \\text{ } | \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ This proof can be done by induction.",
"We can restrict to the case $k = 1$ , but we have to prove the result with initial points $|U_1\\cdot \\vec{\\ell }| \\le 1$ and $|U_2 \\cdot \\vec{\\ell }| \\le 1$ and also when the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ are fixed to a certain value $a \\in [1/K,K]^{\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}}$ $\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{\\mathcal {L}_1}+\\cdot },\\tilde{X}^{2}_{T^{2}_{\\mathcal {L}_1}+\\cdot })g\\circ \\theta _{T_{\\mathcal {L}_1}}h_1\\right] = \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1\\mathbb {E}^K_{X^1_{T^{1}_{\\mathcal {L}_1}},X^2_{T^{2}_{\\mathcal {L}_1}}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot })g \\text{ } | \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ Here the notation $\\mathbb {E}^{a}$ means that we fix the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ to $a$ .",
"The proof with $\\mathbb {E}$ is the same.",
"By definition of $\\mathcal {L}_1$ , we have $\\displaystyle &\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{\\mathcal {L}_1}+\\cdot },\\tilde{X}^{2}_{T^{2}_{\\mathcal {L}_1}+\\cdot })g\\circ \\theta _{T_{\\mathcal {L}_1}}h_1\\right] \\\\&= \\sum _{k \\ge 0}^{} \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{L_k}+\\cdot },\\tilde{X}^{2}_{T^{2}_{L_k}+\\cdot })g\\circ \\theta _{T_{L_k}}h_1,L_k <+\\infty , R_k = +\\infty \\right]\\\\&= \\sum _{\\begin{array}{c}k \\ge 0 \\\\ p \\ge 0\\end{array}, \\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}^{} \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{p}+\\cdot },\\tilde{X}^{2}_{T^{2}_{p}+\\cdot })g\\circ \\theta _{T_p}h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_{p}} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right].$ For $p,(x,z),(x^{\\prime },z^{\\prime })$ in the sum and following the arguments of there exists $h_1^{\\prime }$ measurable with respect to $\\sigma \\left((\\tilde{X}^1_{T^1_{p}\\wedge m})_{m \\ge 0},(\\tilde{X}^2_{T^2_{p}\\wedge m})_{m \\ge 0}, c_{*}(e) \\text{ with } e \\in \\mathcal {L}^{x- 2e_1} \\cup \\mathcal {L}^{x^{\\prime }- 2e_1} \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }}\\right),$ such that on the event $\\lbrace \\tilde{X}^1_{T^{1}_p} = (x,z)\\rbrace \\cap \\lbrace \\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\rbrace \\cap \\lbrace \\mathcal {L}_1 = p\\rbrace $ the functions $h_1$ and $h^{^{\\prime }}_1$ coincide.",
"Fix $k,p \\ge 0$ and $(x,z),(x^{\\prime },z^{\\prime })$ , in the sum.",
"We can replace $h_1$ by $h_1^{\\prime }$ but to simplify notations we will still write $h_1$ and suppose it is measurable with respect to the above $\\sigma $ -algebra.",
"We can do the same with $g \\circ \\theta _{T_p}$ , indeed we can consider $g$ as a function of $(X^1_0,X^2_0,\\omega )$ and then under $\\lbrace \\mathcal {L}_1 = p,X^1_{T^1_{\\mathcal {L}_1}} = x,X^2_{T^2_{\\mathcal {L}_1}} = x^{\\prime } \\rbrace $ we can replace $g$ by a function which is measurable with respect to $\\sigma \\left(c_{*}(e) \\text{ ; } e \\in \\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }})\\right).$ The term in the sum can then be written $&\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{p}+\\cdot },\\tilde{X}^{2}_{T^{2}_{p}+\\cdot })g^{(x,x^{\\prime })}h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\\\&=\\mathbf {E}\\left[E^{\\omega _a}_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\times g^{(x,x^{\\prime })}E^{\\omega _a}_{\\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}\\left[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ In the following we will use the shortcut $E^{\\omega _a}_{\\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}\\left[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty \\right] = E^{\\omega _a}_{(\\mathbf {x}, \\mathbf {z})}\\left[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty \\right].$ Where we have used Lemma REF with $\\tilde{X}^1$ and $\\tilde{X}^2$ and with the stopping times $T^1_{p}$ and $T^2_{p}$ .",
"We recall the notation $\\omega _a$ which means we fix the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ to $a$ .",
"We want to stress that in the expression $E^{\\omega _a}_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ , with a little abuse of the notation, we fix to the value $a$ the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ and not the ones in $\\mathcal {E}_{x}\\cup \\mathcal {E}_{x^{\\prime }}$ as one might expect.",
"Moreover, thanks to the fact that $P^{\\omega _a}_{(\\mathbf {x}, \\mathbf {z})}$ -almost surely the walks never backtrack below $x$ and $x^{\\prime }$ on $\\lbrace \\mathcal {D}^{\\bullet } = +\\infty \\rbrace $ , the value of the conductances in $\\mathcal {E}_{U_1}\\cup \\mathcal {E}_{U_2}$ does not matter in the expectation so we can replace $\\omega _a$ with $\\omega $ and write $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ .",
"Then, since under $\\lbrace L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\rbrace $ the points $x$ and $x^{\\prime }$ are $K$ -open and using Proposition REF , we can replace $\\omega $ by $\\omega _K$ in $E^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ .",
"Notice that $f$ and $\\mathcal {D}$ are measurable with respect to $\\sigma \\big ( X^1_0,X^2_0,(\\tilde{X}^1_{n})_{n \\ge 1},(\\tilde{X}^2_{n})_{n\\ge 1} \\big )$ and thus can be written as $\\phi \\left(X^1_0,X^2_0,(\\tilde{X}^1_{n})_{n \\ge 1},(\\tilde{X}^2_{n})_{n\\ge 1}\\right)$ with $\\phi $ measurable.",
"Since the law of $\\left( X^1_0,X^2_0,(\\tilde{X}^1_{n})_{n \\ge 1},(\\tilde{X}^2_{n})_{n\\ge 1}\\right)$ is the same under $P^{\\omega }_{(\\mathbf {x}, \\mathbf {z})}$ and under $P^{\\omega }_{(\\mathbf {x}, 0)}$ , we replace $z,z^{\\prime }$ by 0 in $E^{\\omega _K}_{(\\mathbf {x}, \\mathbf {z})}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ .",
"We observe that $E^{\\omega _K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}[f(\\tilde{X}^{1}_{\\cdot },\\tilde{X}^{2}_{\\cdot }), \\mathcal {D}^{\\bullet } = +\\infty ]$ is measurable with respect to $\\sigma \\left(c_{*}(e) \\text{ ; } e \\in \\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }})\\right).$ Moreover, $E^{\\omega _a}_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })]$ is measurable with respect to $\\sigma \\left(c_{*}(e) \\text{ with } e \\in E(\\mathcal {H}^{-}(p-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }}\\right).$ In addition, $\\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }})$ and $E(\\mathcal {H}^{-}(p-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }}$ can be proved to be disjoint in an elementary way as illustrated in the figure below.",
"Figure: In orange E(ℋ - (p-2e 1 ·ℓ →))∪ℰ x 2 ∪ℰ x ' 2 E(\\mathcal {H}^{-}(p-2e_1\\cdot \\vec{\\ell })) \\cup \\mathcal {E}^{2}_{x} \\cup \\mathcal {E}^{2}_{x^{\\prime }} and in green ℛ x ∪ℛ x ' ∖(ℰ x ∪ℰ x ' )\\mathcal {R}^{x} \\cup \\mathcal {R}^{x^{\\prime }} \\backslash (\\mathcal {E}_{x} \\cup \\mathcal {E}_{x^{\\prime }}).By independence of the two $\\sigma $ -algebra we deduce that $&\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[f(\\tilde{X}^{1}_{T^{1}_{p}+\\cdot },\\tilde{X}^{2}_{T^{2}_{p}+\\cdot })g^{(x,x^{\\prime })}h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\\\&=\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.",
"}),\\mathcal {D}^{\\bullet } =+\\infty \\right].$ Last equality used in the particular case where $f = 1$ and $g = 1$ gives $&\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\\\&=\\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {P}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}(\\mathcal {D}^{\\bullet } =+\\infty ).$ Hence we can write $& \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.",
"}),\\mathcal {D}^{\\bullet } =+\\infty \\right] \\\\&= \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.})",
"\\text{ }| \\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right].$ Then taking the sum we obtain $&\\sum _{\\begin{array}{c}k \\ge 0 \\\\ p \\ge 0\\end{array}, \\begin{array}{c}(x,z)\\\\(x^{\\prime },z^{\\prime })\\end{array}}^{} \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1,L_k =p, R_k = +\\infty ,\\tilde{X}^1_{T^{1}_p} = (x,z),\\tilde{X}^2_{T^{2}_p} = (x^{\\prime },z^{\\prime })\\right]\\mathbb {E}^{K}_{\\begin{array}{c}x,x^{\\prime }\\end{array}}\\left[gf(\\tilde{X}^{1}_{.},\\tilde{X}^{2}_{.})",
"\\text{ }| \\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right]\\\\&= \\mathbb {E}^a_{\\begin{array}{c}(U_1,z_1)\\\\(U_2,z_2)\\end{array}}\\left[h_1\\mathbb {E}^K_{X^1_{T^{1}_{\\mathcal {L}_1}},X^2_{T^{2}_{\\mathcal {L}_1}}}\\left[f(\\tilde{X}^1_{\\cdot },\\tilde{X}^{2}_{\\cdot })g \\text{ } | \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right]\\right].$ Then for the induction, if the formula has been proved for $k$ , from Proposition REF we get that $\\mathcal {L}_{k+1}$ is almost surely finite.",
"The proof works in a similar way to go from $k$ to $k+1$ and the details of the adaptation can be found in .",
"A consequence of this theorem is the analogous of Proposition REF but in a more general setting.",
"Proposition 3.21 For any $M > 0$ there exists a constant $c$ such that for any $k > 0$ $\\mathbb {P}_{0}(\\mathcal {L}_{k+1} - \\mathcal {L}_k \\ge n) \\le cn^{-M}.$ We write $\\mathbb {P}_{0}(\\mathcal {L}_{k+1} - \\mathcal {L}_k \\ge n) &= \\sum _{p,x,x^{\\prime }}^{} \\mathbb {E}_{0}\\left[\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\prime }, \\mathcal {L}_{k+1} - \\mathcal {L}_k \\ge n\\right]\\\\& = \\sum _{p,x,x^{\\prime }}^{} \\mathbb {E}_{0}\\left[\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\prime }, \\mathcal {L}_1 \\circ \\theta _{T_{x,x^{^{\\prime }}}} - p \\ge n\\right]\\\\& = \\sum _{p,x,x^{\\prime }}^{} \\mathbb {E}_{0}\\left[\\mathcal {L}_k = p , X^{1}_{T^1_p} = x, X^{2}_{T^2_p} = x^{\\prime },\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n \\text{ } | \\text{ }\\mathcal {D}^{\\bullet } =+\\infty )\\right].$ where the third equality follows from Theorem REF .",
"We only have to dominate $\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n \\text{ } | \\text{ }\\mathcal {D}^{\\bullet } =+\\infty )$ to conclude.",
"To do so, using the fact that $\\mathbb {P}_{x,x^{\\prime }}^{K}(\\mathcal {D}^{\\bullet } =+\\infty ) > \\alpha > 0$ with $\\alpha $ which does not depend on $x$ and $x^{\\prime }$ , we can see it suffices to bound $\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n )$ .",
"Set $y = x^{\\prime } - x$ , by the invariance in law of the environment by translation we have $\\mathbb {P}^{K}_{x,x^{\\prime }}(\\mathcal {L}_1 - p \\ge n ) = \\mathbb {P}^{K}_{0,y}(\\mathcal {L}_1 -p + x\\cdot \\vec{\\ell } \\ge n ) $ .",
"Notice that we have $|x\\cdot \\vec{\\ell } - p| \\le 1$ , which follows from the fact that $X^1_{T^{1}_p} = x$ .",
"Thus the result follows from Proposition REF ."
],
[
"Asymptotic separation",
"The aim of this section is to prove the following result: the trajectories of the two walks do not cross anymore in $\\mathcal {H}^{+}_N$ with high probability.",
"We recall the notation $\\mathcal {H}^{+}_{N} = \\lbrace x \\in \\mathbb {Z}^d \\text{ }|\\text{ } x\\cdot \\vec{\\ell } > N\\rbrace $ .",
"We introduce the event that the traces of the walks in $\\mathcal {H}^{+}_N$ get close $ \\mathbf {M}_N \\bigg \\lbrace \\inf _{\\begin{array}{c}x \\in \\mathcal {H}^{+}_{N}\\cap \\lbrace X_n^1\\rbrace \\\\y \\in \\mathcal {H}^{+}_{N}\\cap \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\le 1 \\bigg \\rbrace $ Proposition 4.1 For any $N \\ge 0$ , the following inequality holds $\\mathbb {P}_{0}(\\mathbf {M}_N ) \\le CN^{-c},$ where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d$ .",
"In this part we will adapt methods from .",
"Firstly we will deal with the case when the two walks $X^1$ and $X^2$ live in independent environments.",
"Then, when the two walks live in the same environment, we will try use the fact that when they do not cross, they behave like if they were in independent environments."
],
[
"independent environments",
"First we recall some notations: The law $Q_{U_1,U_2}$ is defined as $Q_{U_1,U_2} = \\mathbb {P}_{U_1} \\otimes \\mathbb {P}_{U_2}$ .",
"Hence $X^1$ and $X^2$ are independent under $Q_{U_1,U_2}$ .",
"We also have $Q^{K}_{U_1,U_2} =\\mathbb {P}^{K}_{U_1} \\otimes \\mathbb {P}^{K}_{U_2}$ .",
"We write $Q_{U_1,U_2}(\\cdot \\, | \\, D^{\\otimes }=+\\infty ) = \\mathbb {P}_{U_1}(\\cdot \\, | \\, D=+\\infty ) \\otimes \\mathbb {P}_{U_2}(\\cdot \\, | \\, D=+\\infty )$ where $D$ is defined in in the definition of regeneration times.",
"Proposition 4.2 There exists $K_0 > 0$ such that for any $U_1, U_2 \\in \\mathbb {Z}^d$ and $K > K_0$ we have $Q^{K}_{U_1, U_2}\\bigg (\\inf _{\\begin{array}{c}x \\in \\lbrace X_n^1\\rbrace \\\\y \\in \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\le 1 \\, \\bigg | \\, \\textnormal {}D^{\\otimes }=+\\infty \\bigg ) \\le C\\Vert U_1 - U_2\\Vert ^{-c},$ where the constants $C,c > 0 $ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Remark 4.3 We observe that if $U_1 = U_2$ the above inequality remains true with the convention $\\Vert U_1-U_2\\Vert ^{-c} = +\\infty $ .",
"For $z \\in \\mathbb {Z}^d$ we define $F_i(z) = \\mathbb {P}^{K}_{U_i}(\\exists k, X_k^{i} = z \\, | \\, \\text{}D=+\\infty ),$ and $F_i^{(R)}(z) = F_i(z){1}_{\\lbrace \\Vert z-U_i\\Vert > R\\rbrace }.$"
],
[
"Step 1: bounding our quantity using $\\Vert F_i^{(R)}\\Vert _2$ and {{formula:7505ed72-501c-44cd-9caa-ec40deaa1136}} .",
"Fix $R = \\Vert U_1-U_2\\Vert /3$ .",
"If $\\Vert X^{1}_p-X^2_q\\Vert \\le 1$ then necessarily either $\\Vert U_1-X^1_p\\Vert \\ge R$ or $\\Vert U_2-X^2_q\\Vert \\ge R$ , otherwise we would have $\\Vert U_1-U_2\\Vert \\le 2/3\\Vert U_1-U_2\\Vert + 2 < \\Vert U_1-U_2\\Vert $ (as soon as $\\Vert U_1-U_2\\Vert \\ge 7$ ).",
"It follows $\\nonumber Q^{K}_{U_1,U_2} &\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\&\\le Q^{K}_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\text{ and } \\Vert U_1-X^1_p\\Vert \\ge R \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\& + Q^{K}_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\text{ and } \\Vert U_2-X^2_q\\Vert \\ge R \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right).$ We write: $Q^{K}_{U_1,U_2} &\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\text{ and } \\Vert U_1-X^1_p\\Vert \\ge R \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\&\\le \\sum _{\\Vert z-U_1\\Vert \\ge R}^{} Q^{K}_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N} \\text{ ; } X^1_p = z, X^2_q \\in \\mathcal {V}_z\\, | \\, \\text{}D^{\\otimes }=+\\infty \\right)\\\\& \\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{\\Vert z-U_1\\Vert \\ge R}^{} F_1^{(R)}(z)F_2^{}(z+e) \\\\&\\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{z \\in \\mathbb {Z}^d}^{} F_1^{(R)}(z)F_2^{}(z+e) \\\\& \\le C\\Vert F^{(R)}_1\\Vert _2\\Vert F_2\\Vert _2,$ where the third inequality comes from Cauchy-Schwarz inequality and where the constant $C > 0$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"Symmetrically, we can do the same computation for (REF ).",
"It follows that if we show that $\\Vert F_i\\Vert _2 < +\\infty $ and $\\Vert F_i^{(R)}\\Vert _2 \\le CR^{-c}$ with $C,c > 0$ constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ , the proof is finished."
],
[
"Step 2: bound on $\\Vert F_i\\Vert _2$ and {{formula:23b48c57-5a69-4a5d-8b96-9d7b1d444caf}} .",
"For $n \\in \\mathbb {N}$ we introduce $H_i(z,n) = \\mathbb {P}^{K}_{U_i}(\\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\text{, } X_k^{i} = z \\, | \\, \\text{}D=+\\infty ),$ and $H_i^{(R)}(z,n) = H_i(z,n){1}_{\\lbrace \\Vert z-U_i \\Vert > R \\rbrace }.$ Using the triangle inequality we write $\\Vert F_i\\Vert _{2} \\le \\sum _{n=1}^{+\\infty } \\Vert H_i(\\cdot ,n)\\Vert _{2},$ and $\\Vert F_i^{(R)}\\Vert _{2} \\le \\sum _{n = 1}^{+\\infty } \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2}.$ We want to bound $\\Vert H_i(\\cdot ,n)\\Vert _2$ and $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _2$ .",
"The quantity $H_i(\\cdot ,n)$ can be rewritten as a convolution because $X_{\\tau _n^{i}}^{i}$ and $\\lbrace X_{\\tau _n^{i} + \\, \\cdot }^{i} - X_{\\tau _n^{i}}^{i} \\rbrace $ are independent by .",
"Thus $H_i(\\cdot ,n) = T_i^{n} \\ast J,$ where $T_i^{n}(z) = \\mathbb {P}_{U_i}^{K}(X^{i}_{\\tau ^{i}_n} = z),$ and $J(z) = \\mathbb {P}_{0}^{K}\\left(\\exists 0 \\le k \\le \\tau _{1}^{i} \\text{, } X_k^{i} = z \\, | \\, \\text{}D=+\\infty \\right).$ Young's inequality gives $\\Vert H_i(\\cdot ,n)\\Vert _{2} \\le \\Vert T_i^{n}\\Vert _{2} \\Vert J\\Vert _{1}.$ We have $\\Gamma := \\Vert J\\Vert _{1} < +\\infty $ .",
"Indeed $\\sum _{z \\in \\mathbb {Z}^d}^{} \\mathbb {P}_{0}^{K}\\left(\\exists 0 \\le k \\le \\tau _{1}^{i} \\text{, } X_k^{i} = z \\, | \\, \\text{}D=+\\infty \\right) \\le C\\mathbb {E}_{0}^{K}\\bigg [\\sum _{z \\in \\mathbb {Z}^d}^{} {1}_{\\exists 0 \\le k \\le \\tau ^i_{1} \\text{, } X_k^{i} = z} \\bigg ],$ where $C$ is a constant which only depend on $\\lambda ,\\vec{\\ell },d,K$ using .",
"The term in the last expectation is equal to $\\#\\lbrace X_k^{i} \\text{ ; } 0\\le k \\le \\tau _1^{i}\\rbrace $ , now recall the notation $\\chi ^i = \\inf \\lbrace m \\ge 0 \\text{ ; }\\lbrace X_k^{i} \\text{ ; } 0\\le k \\le \\tau _1^{i}\\rbrace \\subset X^i_0 + \\mathcal {B}(m,m^{\\alpha })\\rbrace $ , then we clearly have $\\#\\lbrace X_k^{i} \\text{ ; } 0\\le k \\le \\tau _1^{i}\\rbrace \\le (\\chi ^i)^{\\beta }$ for $\\beta $ large enough which only depends on $d,\\alpha $ .",
"By , for $K$ large enough, $\\chi ^i$ has finite moment of order $\\beta $ .",
"It follows that $||J||_1 < +\\infty $ .",
"For the term $\\Vert T_i^{n}\\Vert _2$ , using Proposition REF we have for $K$ high enough $\\displaystyle \\sup _{z \\in \\mathbb {Z}^d}\\mathbb {P}_{U_i}^{K}(X_{\\tau _n^{i}}^{i} = z) \\le Cn^{-\\frac{d}{2}+\\varepsilon },$ where $C$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"It gives the bound $\\Vert H_i^{}(\\cdot ,n)\\Vert _{2} \\le Cn^{-\\frac{d}{4}+\\frac{\\varepsilon }{2}}.$ In dimension $d \\ge 5$ we have $\\Vert F_i\\Vert _{2} < +\\infty .$ We still have to bound $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2}$ .",
"A first bound is given by (REF ), however for $n$ small this bound is not precise enough, thus we compute another bound which is useful for small $n$ .",
"Let us bound $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1}$ and then use the inequality $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2} \\le \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1}^{1/2}$ .",
"We can rewrite $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1} &= \\sum _{||z-U_{i}|| \\ge R}\\mathbb {E}^{K}_{U_i}\\left[ {1}_{ \\left\\lbrace \\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\, ; \\, X_k^{i} = z \\right\\rbrace } \\, \\big | \\, \\text{}D=+\\infty \\right] \\nonumber \\\\ &= \\mathbb {E}^{K}_{U_i}\\left[ \\sum _{||z - U_i|| \\ge R}{1}_{ \\left\\lbrace \\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\, ; \\, X_k^{i} = z \\right\\rbrace } \\, \\big | \\, \\text{}D=+\\infty \\right].",
"$ Define $A(n,R) \\lbrace \\exists \\, \\tau _n \\le k \\le \\tau _{n+1} \\textnormal { such that } |X^i_k-X^i_0| \\ge R\\rbrace $ .",
"We introduce the notation $\\chi ^i_n = \\chi ^i(X^{i}_{\\tau _n + \\cdot })$ .",
"The term $\\displaystyle \\sum _{||z - U_i|| \\ge R}{1}_{ \\left\\lbrace \\exists \\tau _n^{i} \\le k \\le \\tau _{n+1}^{i} \\, ; \\, X_k^{i} = z \\right\\rbrace }$ in the expectation (REF ) is bounded by ${1}_{A(n,R)}\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })$ , indeed it is clearly bounded by $\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })$ and under $A(n,R)^{c}$ it is equal to 0.",
"Using the fact that $\\mathrm {diam}(\\mathcal {B}(L,L^{\\alpha })) \\le CL^{\\alpha }$ where $C$ is a constant which only depends on $\\vec{\\ell },d$ , we have the following inclusion: $A(n,R) \\subset \\bigg \\lbrace \\sum _{k=0}^{n} (C\\chi ^{i}_k)^{\\alpha } \\ge R \\bigg \\rbrace .$ Since $((\\chi ^{i}_{k})^{\\alpha })_{k \\ge 0}$ is a family of independent identically distributed random variables and for $K$ large enough $(\\chi ^{i}_0)^{\\alpha }$ has a finite first moment under $\\mathbb {P}^K_{U_i}(\\cdot \\text{ | }D=+\\infty )$ by , we have $\\mathbb {P}^{K}_{U_i}\\left(A(n,R) \\text{ | }D=+\\infty \\right) &\\le \\mathbb {P}^{K}_{U_i}\\left(\\sum _{k=0}^{n} C(\\chi ^{i}_k)^{\\alpha } \\ge R\\text{ | }D=+\\infty \\right)\\\\&\\le C\\frac{(n+1)\\, \\mathbb {E}_{U_i}^{K}\\left[(\\chi ^{i}_{1})^{\\alpha }\\text{ | }D=+\\infty \\right]}{R} = \\frac{Cn}{R},$ where $C > 0$ is a constant which only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"$\\bullet $ Now using , for any $M > 0$ there exists $K_0 > 0$ such that for $K \\ge K_0$ the variable $\\chi ^i_n$ has a moment of order $M$ .",
"By chosing $K_0$ big enough, $\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })$ has a moment of order 2.",
"Indeed $\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha }) \\le C(\\chi ^i_n)^{d\\alpha }$ with $C$ a constant which only depends on $d$ .",
"Using Cauchy-Schwarz inequality we have $\\nonumber \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{1} &\\le \\mathbb {P}_{U_i}^{K}\\left( A(n,R)\\text{ | }D=+\\infty \\right)^{\\frac{1}{2}}\\mathbb {E}^{K}_{U_i}\\left[\\#\\mathcal {B}(\\chi ^{i}_n,(\\chi ^{i}_n)^{\\alpha })^2 \\text{ | }D=+\\infty \\right]^{\\frac{1}{2}}\\\\& \\le Cn^{\\frac{1}{2}}R^{-\\frac{1}{2}},$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"It follows that we have $\\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2} \\le \\Vert H_i^{(R)}(\\cdot ,n)\\Vert ^{\\frac{1}{2}}_{1} \\le Cn^{\\frac{1}{4}}R^{-\\frac{1}{4}}.$ Pick $a > 0$ arbitrarily small.",
"We bound $\\Vert F_i^{(R)}\\Vert _{2}$ by $\\Vert F_i^{(R)}\\Vert _{2} &\\le \\sum _{n=1}^{\\lfloor R^{a} \\rfloor } \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2} +\\sum _{n=\\lfloor R^{a} \\rfloor +1}^{+\\infty } \\Vert H_i^{(R)}(\\cdot ,n)\\Vert _{2}\\\\&\\underset{(\\ref {first_bound}),(\\ref {second_bound})}{\\le } C \\left(\\displaystyle \\sum _{n = 1}^{\\lfloor R^{a} \\rfloor }n^{\\frac{1}{4}}R^{-\\frac{1}{4}} + \\sum _{n = \\lfloor R^{a} \\rfloor + 1}^{+\\infty }n^{-\\frac{d}{4}+\\frac{\\varepsilon }{2}} \\right)\\\\&\\le CR^{-\\frac{1}{4}}R^{\\frac{5}{4}a} + C(R^{a})^{-\\frac{d}{4}+1+\\frac{\\varepsilon }{2}}.$ By taking $a$ small enough such that $\\frac{5}{4}a-\\frac{1}{4} < 0$ and since $-\\frac{d}{4}+1+\\frac{\\varepsilon }{2} < 0$ we have $\\Vert F_i^{(R)}\\Vert _{2} &\\le CR^{-c},$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We introduce for $R \\ge 0$ the event $\\nonumber F(R) = &\\hspace{10.0pt}\\left\\lbrace \\exists p,q \\ge 2 \\text{, }\\Vert X^1_p- X^2_q\\Vert \\le 1\\right\\rbrace \\\\ \\nonumber & \\cap \\left\\lbrace \\forall p,q \\ge 2 \\text{, }\\Vert X^1_p-X^2_q\\Vert \\le 1 \\Rightarrow \\Vert X^1_p-X^1_0\\Vert \\wedge \\Vert X^2_q - X^1_0\\Vert \\ge R \\right\\rbrace \\\\&\\cap \\left\\lbrace \\forall p,q \\ge 2 \\text{, }\\Vert X^1_p-X^2_q\\Vert \\le 1 \\Rightarrow \\Vert X^1_p-X^2_0\\Vert \\wedge \\Vert X^2_q - X^2_0\\Vert \\ge R \\right\\rbrace .$ Roughly speaking, on the event $F(R)$ the two walks almost intersect (actually get at distance less than 1) but do not intersect at a point at distance less than $R$ from $U_1$ or $U_2$ .",
"The last proof immediately gives the following corollary.",
"Corollary 4.4 There exists $K_0 > 0$ such that, for any $U_1,U_2 \\in \\mathbb {Z}^d$ and for any $K \\ge K_0$ , we have that $Q^{K}_{U_1,U_2}\\left(F(R)\\, | \\,\\text{}D^{\\otimes }=+\\infty \\right) \\le CR^{-c},$ where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Taking the same notations as in last proof we have: $\\displaystyle Q^{K}_{U_1,U_2}\\left(F(R)\\, | \\,\\text{}D^{\\otimes }=+\\infty \\right) &\\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{\\begin{array}{c}z \\in \\mathbb {Z}^d\\\\ \\Vert z-U_1\\Vert \\ge R \\\\ \\Vert z-U_2\\Vert \\ge R\\end{array}}^{} F_1^{(R)}(z)F_2^{(R)}(z + e)\\\\&\\le \\sum _{e \\in \\mathcal {V}_{0}}^{} \\sum _{\\begin{array}{c}z \\in \\mathbb {Z}^d\\end{array}}^{} F_1^{(R)}(z)F_2^{(R - 1)}(z + e).$ We conclude as in last proof by using Cauchy-Schwarz inequality.",
"The last corollary shows that there is little probability that the two walks only intersect on points which are far from the starting points.",
"In other words, with high probability, if the two walks do intersect, then it has to be close to one of the starting points.",
"We need a last proposition before moving on to the case where both walks are in the same environment.",
"We recall the notation $T^i_{R}$ denotes the first hitting time of the set $\\mathcal {H}^{+}(R)$ for the walk $X^i$ .",
"We introduce the notation: $R_0 \\max (X^{1}_0\\cdot \\vec{\\ell },X^{2}_0 \\cdot \\vec{\\ell }).$ For $R > 0$ we define the event $ A(R) {\\left\\lbrace \\begin{array}{ll}X_p^{1}\\cdot \\vec{\\ell } > X^1_0\\cdot \\vec{\\ell } \\quad & \\forall 0 < p \\le T^1_{R+R_0} + 2,\\\\X_q^{2}\\cdot \\vec{\\ell } > X^2_0\\cdot \\vec{\\ell } \\quad &\\forall 0 < q \\le T^2_{R+R_0} + 2,\\\\\\Vert X_p^1 - X_q^2\\Vert \\ge 2 \\quad &\\forall 1 < p \\le T^1_{R+R_0} + 2 \\,\\, \\textnormal {and} \\,\\, \\forall 1 < q \\le T^2_{R+R_0} + 2.\\end{array}\\right.",
"}$ Proposition 4.5 Let $R_0 = \\max (X_0^1 \\cdot \\vec{\\ell },X_0^2 \\cdot \\vec{\\ell }) $ and $R > 0$ .",
"There exists $\\Lambda > 0$ and $K_0 > 0$ such that, for $K > K_0$ , $U_1,U_2 \\in \\mathbb {Z}^d$ such that $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $\\Vert U_1-U_2\\Vert \\ge \\Lambda $ , $Q^{K}_{U_1,U_2}(A(R)) \\ge \\rho > 0,$ where the constant $\\rho $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"We have $Q^{K}_{U_1,U_2}(A(R)) \\ge cQ^{K}_{U_1,U_2}(A(R) \\, | \\, \\text{}D=+\\infty )$ with $c = \\mathbb {P}^{K}_{0}(D=+\\infty )^2$ .",
"Thus, under the law $Q^{K}_{U_1,U_2}( \\cdot \\, | \\, \\text{}D^{\\otimes }=+\\infty )$ the event $A(R)^{c}$ implies $ \\left\\lbrace \\exists \\, p,q \\in \\mathbb {N}, \\,\\, \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\right\\rbrace $ .",
"It follows from Proposition REF that one can bound $Q^{K}_{U_1,U_2}(A(R)^{c} \\, | \\, \\text{}D^{\\otimes }=+\\infty ) &\\le Q^K_{U_1,U_2}\\left(\\exists p,q \\in \\mathbb {N}, \\,\\, \\Vert X_p^1 - X_q^2\\Vert \\le 1 \\, | \\, \\text{}D^{\\otimes }=+\\infty \\right) \\\\ &\\le C\\Vert U_1 - U_2\\Vert ^{-c}.$ The result comes by choosing $\\Lambda $ large enough."
],
[
"Identical environments",
"We will try to exploit the results attained in the last part in order to deduce estimates for two walks in the same environment.",
"When the two walks are far enough they live in independent environments and so, by using the previous subsection, we will be able to show that they have a positive probability to not hit each other anymore.",
"If they intersect, then we wait for the walks to get far enough and we iterate this way.",
"First we state the analogous of Proposition REF .",
"We recall the definition of $A(R)$ in (REF ) and the notations $\\mathcal {E}_x, \\mathcal {E}_A$ in (REF ).",
"Proposition 4.6 There exists $\\Lambda > 0$ and $K_0 > 0$ such that, for any $K > K_0$ , $R > 0$ and $U_1,U_2 \\in \\mathbb {Z}^d$ with $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $||U_1-U_2|| \\ge \\Lambda $ , $\\mathbb {P}_{U_1,U_2}^{K}(A(R) \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty )\\ge \\rho ,$ the constant $\\rho $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"We only have to bound $\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c} \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty )$ .",
"Hence, using Lemma REF we only have to bound $\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c})$ .",
"Since under $A(R)$ the paths of the two walks stay at distance more than 2 we have $\\mathbb {P}_{U_1,U_2}^{K}(A(R)^{c}) = Q_{U_1,U_2}^{K}(A(R)^{c})$ .",
"Indeed we can write $\\mathbb {P}_{U_1,U_2}^{K}(A(R)) & = \\sum _{(x_i),(y_j)}^{} \\mathbf {E}\\bigg [P^{\\omega _K}_{U_1}\\bigg ((X^1_i) = (x_i)\\bigg )P^{\\omega _K}_{U_2}\\bigg ((X^2_j) = (y_j)\\bigg )\\bigg ],$ where the paths $(x_i),(y_j)$ are chosen under $A(R)$ .",
"We highlight the fact that in $P^{\\omega _K}_{U_1}$ the conductances in $\\mathcal {E}_{U_2}$ are not set to $K$ , while in $\\mathbb {P}^{K}_{U_1,U_2}$ they are.",
"Indeed $P^{\\omega _{K}}_{U_1}\\left((X^1_i) = (x_i)\\right)$ depends on the values $c^{*}(e)$ where $e \\in \\mathcal {E}_{(x_i)}$ , using the fact that $(x_i)$ and $(y_j)$ stay at distance more than 2 we have $\\mathcal {E}_{(x_i)} \\cap \\mathcal {E}_{U_2} = \\emptyset $ .",
"Thus modifying the value of $c^{*}(e)$ where $e \\in \\mathcal {E}_{U_2}$ does not modify the value of $P^{\\omega _{K}}_{U_1}\\left((X^1_i) = (x_i)\\right)$ .",
"The term $P^{\\omega _K}_{U_1}\\left((X^1_i) = (x_i)\\right)$ is measurable with respect to $ \\sigma (c^{*}(e) \\text{ ; } e\\in \\mathcal {E}_{(x_i)})$ and $P^{\\omega _K}_{U_2}((X^2_j) = (y_j))$ is measurable with respect to $ \\sigma (c^{*}(e) \\text{ ; } e\\in \\mathcal {E}_{(y_j)})$ .",
"The two sigma algebras are independent since for all $i,j$ we have $\\Vert x_i-y_j\\Vert \\ge 2$ .",
"We conclude that we have $\\mathbb {P}_{U_1,U_2}^{K}(A(R)) & = \\sum _{(x_i),(y_j)}^{} \\mathbf {E}\\left[P^{\\omega _K}_{U_1}\\left((X^1_i) = (x_i) \\right) \\right] \\mathbf {E} \\left[P^{\\omega _K}_{U_2}\\left((X^2_j) = (y_j)\\right)\\right]\\\\& = Q_{U_1,U_2}^{K}(A(R)).$ We conclude using Proposition REF .",
"We want to extend the last result to the case when $U_1$ and $U_2$ may be close.",
"Proposition 4.7 There exists $K_0 > 0$ such that for any $K > K_0$ and any $U_1,U_2 \\in \\mathbb {Z}^d$ with $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $R > 0$ : $\\mathbb {P}_{U_1,U_2}^{K}(A(R) \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty )\\ge \\rho ,$ where the constant $\\rho > 0$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"First with the same arguments as in last proposition we see that we only have to find a lower bound for $\\mathbb {P}_{U_1,U_2}^{K}(A(R))$ .",
"Fix $\\Lambda $ as in Proposition REF .",
"We may suppose $\\Vert U_1-U_2\\Vert \\le \\Lambda $ otherwise it is just applying Proposition REF .",
"We recall the notation $R_0 = \\min (U_1\\cdot \\vec{\\ell },U_2\\cdot \\vec{\\ell })$ .",
"We build paths that start form $U_1$ and $U_2$ in the following way: Fix $V_1$ and $V_2$ such that $\\Lambda \\le ||V_1-V_2|| < 2\\Lambda $ and $ R_0+5< \\lfloor V_1\\cdot \\vec{\\ell } \\rfloor ,\\lfloor V_2\\cdot \\vec{\\ell } \\rfloor < R_0+6$ .",
"Build two finite paths $(x_i)$ and $(y_j)$ starting from $U_1$ , $U_2$ respectively and ending at $V_1$ , $V_2$ , which are: Compatible with having joint regeneration points on $(V_1,V_2)$ .",
"Of length lower than $C\\Lambda $ where $C$ is a large enough constant.",
"Compatible with $A(R)$ , i.e.",
"on those paths the walks get at distance more than 2 after the first 2 steps and stay on the right side (in terms of inner product with $\\vec{\\ell }$ ) of their starting point.",
"It is always possible to find such paths, see the figure below for a visual explanation of why these paths exist.",
"Figure: On the left, the general construction of the paths described above, the distance between U 1 U_1 (resp U 2 U_2) and V 1 V_1 (resp.",
"V 2 V_2), in the direction ℓ →\\vec{\\ell }, is between 5 and 6 because of the lattice geometry.",
"On the right, a “worst case scenario” when the two starting points coincide and the bias is in the direction of the vector e 1 e_1.",
"This is why we allow, in the definition of A(R)A(R) (), that the walks might be close after one step (in orange).We have the following inequality: $\\mathbb {P}_{U_1,U_2}^{K}\\left(A(R)\\right) \\ge \\mathbb {P}_{U_1,U_2}^{K}\\left(\\begin{array}{c}(X^1_i) = (x_i),(X^2_j)=(y_j),(x_i) \\text{ and }(y_j)\\text{ are all }K\\text{-open points}\\\\A(R) \\circ \\theta _{T_{V_1},T_{V_2}}, \\mathcal {D}^{\\bullet } \\circ \\theta _{T_{V_1},T_{V_2}}\\end{array}\\right).$ Using Theorem REF last term can be rewritten as $&\\mathbb {P}_{U_1,U_2}^{K}\\left((X^1_i) = (x_i),(X^2_j)=(y_j),(x_i) \\text{ and }(y_j)\\text{ are all }K\\text{-open points}\\right)\\\\&\\times \\mathbb {P}_{V_1,V_2}^{K}\\left(A(R) ,\\mathcal {D}^{\\bullet } = +\\infty \\right).$ The left term of the product is bounded below by a constant which only depends on $\\lambda ,\\vec{\\ell },d,K$ and the right term is bounded by a $\\rho > 0$ using Proposition REF and Lemma REF .",
"It concludes the proof.",
"We show that under $\\mathbb {P}_0$ , on $A(R)$ and when the two trajectories reach a joint regeneration level before reaching level $R$ , then it is unlikely that they meet in the future.",
"We recall the definition of $\\mathbf {M}_N$ in (REF ).",
"Proposition 4.8 We recall the notation $R_0 = \\max (U_1\\cdot \\vec{\\ell },U_2\\cdot \\vec{\\ell })$ .",
"There exists $K_0 > 0$ such that for $K > K_0$ , for $U_1,U_2 \\in \\mathbb {Z}^d$ such that $|(U_1-U_2)\\cdot \\vec{\\ell }| \\le 1$ and $R > 0$ we have $\\mathbb {P}_{U_1,U_2}^{K}\\left( A(R) \\cap \\mathrm {\\bf {M}}_{R+R_0} \\text{ }| \\text{ } \\mathcal {D}^{\\bullet } = +\\infty \\right) \\le CR^{-c},$ where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We want to use Corollary REF .",
"First notice that it suffices to bound $\\mathbb {P}_{U_1,U_2}^{K}\\big ( A(R) \\cap \\mathrm {\\bf {M}}_{R+R_0}\\cap \\mathcal {D}^{\\bullet } = +\\infty \\big )$ , indeed it follows from Lemma REF that $\\mathbb {P}_{U_1,U_2}^{K}\\big (\\mathcal {D}^{\\bullet } = +\\infty \\big )$ is bounded below by a constant $\\eta > 0$ which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Let us consider the set $\\mathcal {P}^{\\bullet }$ of all finite paths $((\\tilde{x}_i),(\\tilde{y}_j))$ for the enhanced walks $(\\tilde{X}^1_{\\cdot },\\tilde{X}^2_{\\cdot })$ (defined in Section REF ) such that $(x_i)$ and $(y_j)$ are at distance 1 at their ending points and always more than 2 (except at the starting point) otherwise and which are compatible with $\\lbrace \\mathcal {D}^{\\bullet } = +\\infty \\rbrace \\cap A(R)$ .",
"We then only have to bound $\\mathbb {P}^{K}_{U_1,U_2}\\left(\\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace \\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j) \\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\right).$ Under all paths considered, the environments each walk encounters are independent.",
"Then, the quantity above can be bounded in the following way $\\nonumber \\mathbb {P}^{K}_{U_1,U_2}\\Bigg (\\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace &\\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\Bigg ) \\\\ \\nonumber &= \\sum _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\mathbb {P}^{K}_{U_1,U_2}\\left( \\lbrace \\tilde{X}^1_i\\rbrace = \\lbrace \\tilde{x}_i\\rbrace ,\\lbrace \\tilde{X}^2_j\\rbrace = \\lbrace \\tilde{y}_j\\rbrace \\right)\\\\&\\nonumber \\le C \\sum _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} Q^{K}_{U_1,U_2}\\left( \\lbrace \\tilde{X}^1_i\\rbrace = \\lbrace \\tilde{x}_i\\rbrace ,\\lbrace \\tilde{X}^2_j\\rbrace = \\lbrace \\tilde{y}_j\\rbrace \\right)\\\\& = C Q^{K}_{U_1,U_2}\\left(\\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace \\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\right).$ Where the inequality is using the fact that the term $P^{\\omega _K}_{U_1,U_2}\\big (\\lbrace \\tilde{X}^1_i\\rbrace = \\lbrace \\tilde{x}_i\\rbrace \\big )$ is measurable with respect to $\\sigma \\big (c^{*}(e) \\text{ ; }e \\in \\mathcal {E}_{\\lbrace \\tilde{x}_i\\rbrace _{i\\ge 0}\\backslash \\lbrace \\tilde{x}_{\\infty }\\rbrace }\\big )$ and the term $P^{\\omega _K}_{U_1,U_2}\\big (\\lbrace \\tilde{X}^2_j\\rbrace = \\lbrace \\tilde{y}_j\\rbrace \\big )$ is measurable with respect to $\\sigma \\big (c^{*}(e) \\text{ ; }e \\in \\mathcal {E}_{\\lbrace \\tilde{y}_j\\rbrace _{j\\ge 0}\\backslash \\lbrace \\tilde{y}_{\\infty }\\rbrace }\\big )$ where $\\tilde{x}_{\\infty },\\tilde{y}_{\\infty }$ denotes the ending point of the paths.",
"In most cases, we simply use the fact that the two set of edges considered are disjoint and the independence of the conductances would give an equality.",
"We only have an inequality because when $\\Vert U_1 - U_2 \\Vert _{1} \\le 1$ and $\\vec{\\ell } = e_1$ , we might be forced to take the first step in the direction $e_1$ by the definition of $A(R)$ .",
"In this last case we just lose a constant because there is a positive chance that the second site visited is $K$ -open under both $\\mathbb {P}^{K}$ and $Q^{K}$ .",
"Note that we could also have an inequality in the opposite direction with a different constant if needed.",
"See the right of Figure REF for a visualisation of the “worst case scenario”.",
"Let us introduce the notation: $\\mathbf {TRAJ} = \\Big \\lbrace \\lbrace \\tilde{X}^1_{i}\\rbrace \\times \\lbrace \\tilde{X}^2_{j}\\rbrace \\in \\bigcup _{(\\tilde{x}_i),(\\tilde{y}_j)\\in \\mathcal {P}^{\\bullet }}^{} \\lbrace \\tilde{x}_i\\rbrace \\times \\lbrace \\tilde{y}_j\\rbrace \\Big \\rbrace .$ We can rewrite (REF ) as $\\displaystyle & Q^{K}_{U_1,U_2}\\left(\\mathbf {TRAJ},D^{\\otimes } < +\\infty \\right) \\\\& + Q^{K}_{U_1,U_2}\\left(\\mathbf {TRAJ},D^{\\otimes }=+\\infty \\right).$ Using the fact that $\\mathbf {TRAJ}$ is compatible with $\\mathcal {D}^{\\bullet } = +\\infty $ and that the trajectories go beyond level $R+R_0$ , if $D^1 < +\\infty $ (resp.",
"$D^2 < +\\infty )$ then we have $X^1_{\\tau _1}\\cdot \\vec{\\ell } > R + R_0$ (resp.",
"$X^2_{\\tau _1}\\cdot \\vec{\\ell } > R + R_0$ ).",
"Using , we dominate the first term by $CR^{-c}$ .",
"It remains to bound the term (REF ).",
"To do so, we define $C_i, i=1, 2$ : the walk $X^i$ regenerates between the levels $R_0 + R/2$ and $R_0 + R$ .",
"Using once again of we have $ Q^{K}_{U_1,U_2}\\left(C_1^{c} \\cup C_2^{c}\\text{ | }D^{\\otimes }=+\\infty \\right) \\le CR^{-c}.$ On the event $\\displaystyle C_1\\cap C_2 \\cap \\mathbf {TRAJ}$ , all points $(x,y)$ in the two trajectories (except with the starting points) which are at distance less than $R/2$ from $\\lbrace U_1,U_2\\rbrace $ we have $\\Vert x-y\\Vert _1 \\ge 2$ .",
"Thus $\\displaystyle C_1\\cap C_2 \\cap \\mathbf {TRAJ}$ implies $F(R/2)$ where $F(R/2)$ is defined in (REF ).",
"Using Corollary REF we conclude that $ Q^{K}_{U_1,U_2}\\left(C_1 \\cap C_2 \\cap \\mathbf {TRAJ} \\text{ | }D^{\\otimes }=+\\infty \\right) \\le Q^{K}_{U_1,U_2}\\left(F(R/2)\\right) \\le CR^{-c}.$ Finally we dominate (REF ) by $& Q^{K}_{U_1,U_2}\\left(C_1^{c} \\cup C_2^{c}\\text{ | }D^{\\otimes }=+\\infty \\right)+ Q^{K}_{U_1,U_2}\\left(C_1 \\cap C_2 \\cap \\mathbf {TRAJ} \\text{ | }D^{\\otimes }=+\\infty \\right),$ and we conclude with the estimates (REF ) and (REF )."
],
[
"Asymptotic separation proved",
"We have everything needed to prove the asymptotic separation stated in Proposition REF .",
"We fix $\\psi \\in (0, 1)$ arbitrarily small.",
"We also introduce $\\mathbf {d} = N^{1-\\psi }$ .",
"We set $\\gamma _1 = \\mathcal {L}_1$ and for $ k \\ge 2 $ we call $\\gamma _k$ the first joint regeneration level in $\\mathcal {H}^{+}(\\gamma _{k-1}+2\\mathbf {d})$ .",
"Recall the notation $A(R)$ for $R > 0$ defined in (REF ).",
"We introduce the event $S_k$ which is just event $A(\\mathbf {d})$ where the walks start respectively from $X_{T^1_{\\gamma _k}}$ and $X_{T^2_{\\gamma _k}}$ .",
"We set: $ \\forall k \\ge 0, \\text{ }S_k = A(\\mathbf {d}) \\circ \\theta _{T_{\\gamma _k}} \\text{ and }S(N) = \\bigcup _{k=1}^{\\lfloor N^{\\psi }/4 \\rfloor } S_k.$ The idea of the proof is the following: each time our walks reach a level $\\gamma _k$ we check whether $S_k$ occurs or not.",
"If it does, using Proposition REF , with high probability the walks will never meet after that that.",
"If they do meet, we wait for the walks to reach $\\gamma _{k+1}$ and repeat the procedure.",
"Using the independence property of joint regeneration levels (see Theorem REF ) we expect the first success of this process to happen very quickly (See Figure REF ).",
"Figure: Successive attempts for events S k S_k.Proposition 4.9 There exists $K_0 > 0$ such that for any $K > K_0$ , there exists $\\rho > 0$ such that for any $k \\ge 1$ we have $\\displaystyle \\mathbb {P}_{0}(S_k \\, | \\, S_0^{c},\\cdots ,S_{k-1}^{c} ) \\ge \\rho ,$ where $\\rho $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"This is a direct application of Theorem REF and Proposition REF .",
"Indeed, using Theorem REF we have $\\mathbb {P}_0(S_k,S_{k-1}^{c},\\cdots ,S_0^{c}) &=\\sum _{U_1,U_2}^{} \\mathbb {P}_{0}(S_{k-1}^{c},\\cdots ,S_0^{c},X^{1}_{T_{\\gamma _k}} = U_1,X^{2}_{T_{\\gamma _k}} = U_2)\\mathbb {P}_{U_1,U_2}^{K}(A(w) \\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty ).$ Applying Proposition REF gives the result.",
"We introduce $ \\textbf {Reg}(N) = \\bigcap _{k=1}^{N^2}\\lbrace \\mathcal {L}_{k} - \\mathcal {L}_{k-1} \\le \\mathbf {d}\\rbrace .$ which is the event under which the $N^2$ first joint regeneration levels are close enough.",
"We also recall the definition of $\\mathbf {M}_N$ in (REF ).",
"We have the following proposition.",
"Proposition 4.10 There exists $K_0 > 0$ such that for any $K > K_0$ and any $0 \\le k \\le \\lfloor N^{\\psi }/4 \\rfloor $ , we have the following inequality: $\\mathbb {P}_0(\\textnormal {\\textbf {Reg}}(N)\\cap \\mathrm {\\bf {M}}_N \\cap S_k) \\le CN^{-c(1-\\psi )}$ , where the constants $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We sum over all possible values for $(X^1_{T_{\\gamma _k}},X^2_{T_{\\gamma _k}})$ on the event $\\mathbf {Reg}(N)$ $\\mathbb {P}_0(\\textbf {Reg}(N)\\cap \\mathrm {\\bf {M}}_N \\cap S_k) &= \\sum _{U_1,U_2}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2,\\textbf {Reg}(N)\\cap \\mathrm {\\bf {M}}_N \\cap S_k\\right) \\\\&\\le \\sum _{\\begin{array}{c}U_1,U_2\\\\\\text{compatible with }\\mathbf {Reg}(N)\\end{array}}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2, \\mathrm {\\bf {M}}_N \\cap S_k\\right).$ Since $U_1,U_2$ where chosen on event $\\textbf {Reg}(N)$ we have $U_1\\cdot \\vec{\\ell } < 3N/4$ and $U_2\\cdot \\vec{\\ell } < 3N/4$ .",
"The worst case clearly happens for $k = \\lfloor N^{\\psi }/4 \\rfloor $ .",
"We claim that for every $0 \\le j \\le \\lfloor N^{\\psi }/4 \\rfloor $ on the event $\\mathbf {Reg}(N)$ , we have $\\gamma _{j+1} \\le \\gamma _j + 3\\mathbf {d}$ .",
"Indeed, between $\\gamma _j$ and $\\gamma _{j+1}$ there are at most $C \\mathbf {d}$ joint regeneration levels, for some constant depending on the dimension $d$ and $\\vec{\\ell }$ .",
"The claim follows from the fact that, for $0 \\le j \\le \\lfloor N^{\\psi }/4 \\rfloor $ , we are choosing all the $\\gamma _{j}$ between the first $N^2$ regeneration levels.",
"Then, from the claim we deduce that for $k = \\lfloor N^{\\psi }/4 \\rfloor $ we have $\\gamma _k \\le 3\\mathbf {d}N^{\\psi }/4 \\le 3N/4$ .",
"We recall $S_k = A(\\mathbf {d}) \\circ \\theta _{T_{\\gamma _k}}$ and since $\\gamma _k \\le 3N/4$ on $\\mathbf {Reg}(N)$ , $\\mathrm {\\bf {M}}_N$ implies $\\mathrm {\\bf {M}}_N \\circ \\theta _{T_{\\gamma _k}}$ .",
"We deduce that the last sum is bounded by $\\sum _{\\begin{array}{c}U_1,U_2\\\\\\text{compatible with }\\mathbf {Reg}(N)\\end{array}}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2, \\mathrm {\\bf {M}}_N \\circ \\theta _{T_{\\gamma _k}}, A(\\mathbf {d}) \\circ \\theta _{T_{\\gamma _k}}\\right).$ Using Theorem REF this sum can be rewritten as $ \\sum _{U_1,U_2}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2\\right)\\mathbb {P}_{U_1,U_2}^{K}\\left(\\mathrm {\\bf {M}}_N\\cap A(\\mathbf {d})\\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right).$ We recall $R_0 = \\max (U_1 \\cdot \\vec{\\ell },U_2 \\cdot \\vec{\\ell })$ .",
"Now, $\\mathbb {P}_{U_1,U_2}^{K}$ -almost surely $\\mathrm {\\bf {M}}_{N}$ implies $\\mathbf {M}_{R_0+\\mathbf {d}}$ .",
"Indeed, $\\displaystyle U_1\\cdot \\vec{\\ell } < 3N/4$ and $\\displaystyle U_2\\cdot \\vec{\\ell } < 3N/4$ , hence $\\mathbb {P}^{K}_{U_1,U_2}$ -almost surely $\\displaystyle R_0 + \\mathbf {d} \\le 3N/4 + N^{1-\\psi } \\le N$ which gives $\\mathbf {M}_N$ implies $\\mathbf {M}_{R_0+\\mathbf {d}}$ .",
"It follows that (REF ) is bounded by $\\sum _{U_1,U_2}^{} \\mathbb {P}_0\\left(X^1_{T_{\\gamma _k}} = U_1,X^2_{T_{\\gamma _k}} = U_2\\right)\\mathbb {P}_{U_1,U_2}^{K}\\left(\\mathrm {\\bf {M}}_{R_0+\\mathbf {d}}\\cap A(\\mathbf {d})\\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right).$ We conclude using Proposition REF .",
"Finally, we prove the asymptotic separation.",
"[Proof of Proposition REF ] We recall the definition of $S(N)$ in (REF ) and of $\\mathbf {Reg}(N)$ in (REF ).",
"Using Proposition REF we have $\\mathbb {P}_{0}\\left(S(N)^c\\right) \\le (1-\\rho )^{N^{\\psi }} $ and by Proposition REF for any $M > 0$ we can take $K$ large enough such that we have $\\mathbb {P}_{0}(\\mathbf {Reg}(N)^{c}) \\le CN^{-M}$ for any $M > 0$ where the constants $\\rho ,C>0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Hence, $\\mathbb {P}_0(\\mathrm {\\bf {M}}_N) &= \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right) + \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N \\cap (\\mathbf {Reg}(N)^c \\cup S(N)^c)\\right) \\\\ &\\le \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right) + CN^{-M} + (1-\\rho )^{N^{\\psi }}.$ We only need to bound $\\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right)$ , by the definition of $S(N)$ and union bound we get $\\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S(N)\\right) &= \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N \\cap \\mathbf {Reg}(N) \\cap \\bigg ( \\bigcup _{k=1}^{\\lfloor N^{\\psi }/4 \\rfloor } S_k\\bigg )\\right)\\\\& \\le \\sum _{k=1}^{\\lfloor N^{\\psi }/4 \\rfloor } \\mathbb {P}_0\\left(\\mathrm {\\bf {M}}_N\\cap \\mathbf {Reg}(N) \\cap S_k\\right) \\\\&\\le CN^{\\psi }N^{-c(1-\\psi )},$ where the last step is a consequence of Proposition REF .",
"We conclude by choosing $\\psi $ small enough, we notice that this choice depends on $c$ which only depend on $\\lambda ,\\vec{\\ell },d,K$ ."
],
[
"For the trajectory",
"In this part we aim to prove that $\\mathbb {E}[F(Z_n^1)F(Z_n^2)] - \\mathbb {E}[F(Z_n^1)]\\mathbb {E}[F(Z_n^2)]$ converges polynomially fast to 0 as $n \\rightarrow +\\infty $ .",
"Where $F$ is a bounded $L$ -Lipschitz function from $\\mathcal {C}([0,1],\\mathbb {R}^{d}) \\rightarrow \\mathbb {R}$ and $Z^{i}_n(t) = \\displaystyle \\frac{X^{i}_{\\tau _{\\lfloor nt \\rfloor }} - nvt}{n^{1/2}}$ for $t \\in [0,1]$ .",
"Actually, with a slight abuse of the notation, in all this section we will refer with $Z^{i}_n(t)$ to the polygonal interpolant of $\\displaystyle \\frac{X^{i}_{\\tau _{\\lfloor nt \\rfloor }} - nvt}{n^{1/2}}$ , which is needed for technical reasons.",
"Throughout this proof we assume that the function $F$ is bounded by 1 and has Lipschitz constant $L=1$ as an extension to other constants follows with the same reasoning.",
"We take $t \\in [0,1]$ as it is the same for $t \\in [0,T]$ .",
"We recall the notation: $ \\chi ^{i}_k = \\inf \\left\\lbrace m \\ge 0 \\text{ ; } \\lbrace X^i_{j}-X^i_{\\tau ^{i}_k}\\rbrace _{\\tau ^{i}_k \\le j \\le \\tau ^{i}_{k+1}} \\subset \\mathcal {B}(m,m^{\\alpha })\\right\\rbrace .$ where we recall that $\\alpha $ is a constant chosen greater than $d+3$ .",
"Let us state the main theorem of this part.",
"Theorem 5.1 There exists $K > 0$ such that the following inequality holds: $\\Big |\\mathbb {E}[F(Z_n^1)F(Z_n^2)] - \\mathbb {E}[F(Z_n^1)]\\mathbb {E}[F(Z_n^2)]\\Big | \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We postpone the proof to the end of the section but let us give some intuition.",
"Roughly speaking, on the event $\\mathbf {M}_{n^\\theta }$ (see (REF )) and after few regeneration times, the two walks do not intersect anymore.",
"It follows that they should behave like two regenerating random walks in two independent environment.",
"This is the point of Propositions REF -REF which are the main steps to prove Theorem REF .",
"The rest of the proof basically consists in proving that the beginning of the trajectories does not matter in the re-scaled trajectory $Z_n$ .",
"Fix $0 < \\theta < 1/2$ .",
"Let us start with an easy estimate which will be useful later on.",
"Proposition 5.2 The following inequality holds: $\\left|\\mathbb {E}_0\\left[F(Z_n)\\right] - \\mathbb {E}_0\\left[F(Z_n)\\text{ }|\\text{ }D=+\\infty \\right]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We introduce the scaled position of the walk starting after the first regeneration time and re-centered $\\bar{Z}_n(t) = Z_n(t+1/n) - Z_n(1/n).$ We define $B= \\big \\lbrace \\forall k \\in \\lbrace 0,\\cdots ,n+1\\rbrace \\text{ ; }\\chi _k \\le n^{\\varepsilon }\\big \\rbrace $ where $\\varepsilon > 0$ is a constant that we choose arbitrarily small.",
"Then, under $B$ , we have $\\Vert \\bar{Z}_n-Z_n\\Vert _{\\infty } \\le Cn^{-\\frac{1}{2}}n^{\\alpha \\varepsilon } \\le Cn^{-c},$ where $c = \\alpha \\varepsilon - \\frac{1}{2} < 0$ when choosing $\\varepsilon $ small enough.",
"It follows that under $B$ and using the fact that $F$ is 1-Lipschitz we get $|F(Z_n)-F(\\bar{Z}_n)| \\le Cn^{-c},$ By , for $K$ large enough we have $\\mathbb {P}_0(B_1^{c} \\cup B_2^{c}) \\le Cn^{-c}$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Using this inequality and the fact that $F$ is bounded by 1 we get that $\\left|\\mathbb {E}_0[F(Z_n)]-\\mathbb {E}_0[F(\\bar{Z}_n)]\\right| \\le Cn^{-c}.$ We conclude the proof observing that using we have that: $\\mathbb {E}_0[F(\\bar{Z}_n)] = \\mathbb {E}_0[F(Z_n)\\text{ | }D=+\\infty ].$ Let us introduce some events and estimates on them: We recall $\\mathrm {\\bf {M}}_{n^\\theta } = \\bigg \\lbrace \\displaystyle \\inf _{\\begin{array}{c}x \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^1\\rbrace \\\\y \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\ge 2 \\bigg \\rbrace .$ Using Proposition REF we have $\\mathbb {P}_{0}(\\mathrm {\\bf {M}}_{n^\\theta }^{c}) \\le Cn^{-c}$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We define $\\mathcal {R}_0$ the first joint regeneration level in $ \\mathcal {H}^{+}_{n^{\\theta }}$ .",
"For $i \\in \\lbrace 1,2\\rbrace $ we introduce $\\rho _0^{i} = T^{i}_{\\mathcal {R}_0}$ .",
"In particular, $T^i_{\\mathcal {R}_0}$ is a regeneration time for the walk $X^i$ .",
"We call $B \\lbrace \\mathcal {R}_0 \\le 2n^{\\theta }\\rbrace $ .",
"Using , there exists $K > 0$ such that $\\mathbb {P}_{0}(B^{c}) \\le Cn^{-c}$ , where $C,c$ depends on $\\theta , \\lambda ,d,\\vec{\\ell },K$ .",
"Indeed, under the event $\\lbrace \\forall i \\in \\lbrace 1,\\cdots ,dn^{\\theta }\\rbrace , \\text{ }\\mathcal {L}_{k+1}-\\mathcal {L}_{k} \\le n^{\\theta }\\rbrace $ with the proviso $\\mathcal {L}_{0} = 0$ (and using the fact that we always have $\\mathcal {L}_{k+1}-\\mathcal {L}_k \\ge \\frac{2}{\\sqrt{d}}$ ) we have $\\mathcal {R}_0 \\le 2n^{\\theta }$ .",
"The following inequality follows: $ \\mathbb {P}_{0}(B^{c}) & \\le \\mathbb {P}_{0}\\bigg (\\exists k \\in \\lbrace 1,\\cdots ,dn^{\\theta }\\rbrace , \\text{ }\\mathcal {L}_{k+1}-\\mathcal {L}_{k} \\ge n^{\\theta }\\bigg ).$ Using Proposition REF for $K$ large enough the last term is bounded above, for $M$ arbitrarily big, by $Cn^{\\theta }n^{-M\\theta }.$ We call $\\rho ^i_{n}$ the $n^{\\text{th}}$ (not joint) regeneration time after $\\rho _{0}^{i}$ for $X^{i}$ .",
"We recall the notation (REF ).",
"Then define $F_i = \\big \\lbrace \\forall k \\in \\lbrace 1,\\cdots ,n^2\\rbrace \\text{, }\\chi ^{i}_k \\le n^{\\theta }\\big \\rbrace $ .",
"Using , there exists $K > 0$ such that $\\mathbb {P}(F^{c}_{i}) \\le Cn^{-M}$ , where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"Define $E:= \\mathrm {\\bf {M}}_{n^\\theta } \\cap B \\cap F_1 \\cap F_2$ .",
"The estimates above can be summarized in the following proposition.",
"Proposition 5.3 There exists $K > 0$ such that we have: $\\mathbb {P}_{0}(E^{c}) \\le Cn^{-c},$ where the constants $C,c >0$ only depend on $\\theta , \\lambda ,\\vec{\\ell },d,K$ .",
"Let us introduce $\\tilde{X}^i_{\\cdot } = X^{i}_{\\rho _{0}^{i}+\\cdot } - X^{i}_{\\rho ^i_0}$ which is just the walk $X^i$ shifted by $\\rho _0^{i}$ and recentered, call $\\tilde{Z}^{i}_{n}(t) = (\\tilde{X}^{i}_{\\tau _{\\lfloor nt \\rfloor }} - nvt)/n^{1/2}$ .",
"Let us show that under $E$ , the trajectories $\\tilde{Z}^{i}_n$ and $Z_n^{i}$ stay close.",
"Proposition 5.4 The following inequality holds almost surely under the event $E$ $\\forall i \\in \\lbrace 1,2\\rbrace , \\text{ } \\sup _{t \\in [0, 1]}||\\tilde{Z}^{i}_n(t) - Z^{i}_n(t)||_{\\infty } \\le Cn^{-c},$ where $C,c$ depend on $\\lambda ,\\vec{\\ell }, d, \\theta $ .",
"Fix $i \\in \\lbrace 1,2\\rbrace $ and $k \\in \\lbrace 0,\\cdots ,n\\rbrace $ , let us show that $X^{i}_{\\tau _k}$ is close to $X^{i}_{\\rho _k}$ after the proper rescaling.",
"We have, $\\mathcal {R}_0 \\le 2n^{\\theta }$ and since $\\rho _{0}^{i}$ is a regeneration time, it can be written as $\\tau ^{i}_{k_0}$ with $k_0 \\in \\lbrace 0,\\cdots ,d\\lfloor n^{2\\theta } \\rfloor \\rbrace $ .",
"Indeed, at each regeneration time the walk has increased at least $2/\\sqrt{d}$ in the direction $\\vec{\\ell }$ .",
"Using the fact that under $E$ , $\\chi ^{i}_k \\le n^{\\theta }$ for all $k \\in \\lbrace 0,\\cdots ,n^2\\rbrace $ we have $\\Vert X^{i}_{\\rho _{k}} - X^i_{\\rho _{0}}- X^{i}_{\\tau _k}\\Vert _\\infty =\\Vert X^{i}_{\\tau _{k+k_0}} - X^{i}_{\\tau _{k_0}} - X^{i}_{\\tau _k}\\Vert _\\infty & \\le Ck_0n^{\\alpha \\theta }\\\\& \\le Cn^{(\\alpha +2)\\theta },$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d$ .",
"The first inequality comes from the fact that the diameter of $\\mathcal {B}(m,m^{\\alpha })$ is bounded by $Cm^{\\alpha }$ .",
"Then, we only have to chose $\\theta $ such that $(\\alpha + 2)\\theta \\le 1/4$ to conclude, indeed for all $k \\in \\lbrace 1, \\cdots , n\\rbrace $ : $\\Vert Z^{i}_n(k/n) - \\tilde{Z}^{i}_n(k/n)\\Vert _\\infty &= \\bigg \\Vert \\frac{X^{i}_{\\tau _k} +X^{i}_{\\rho _0} - {X}_{\\rho _k}^{i}}{n^{1/2}}\\bigg \\Vert _\\infty \\le Cn^{-1/4},$ we conclude by taking $c = 1/4$ .",
"We give the following corollary of the last proposition.",
"Corollary 5.5 The following inequality holds: $\\left|\\mathbb {E}_0\\left[F(Z^1_n)F(Z^2_n)\\right] - \\mathbb {E}_0\\left[F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"It follows from last Proposition REF , Proposition REF and the fact that $F$ is a bounded Lipschitz function.",
"We introduce $I = \\bigg \\lbrace \\displaystyle \\inf _{\\begin{array}{c}p,q \\ge 0\\end{array}}\\Vert X^1_{ p}-X^2_{q}\\Vert _{1}\\ge 2 \\bigg \\rbrace $ the event that the two walks never “see” the same edge.",
"The following proposition is crucial and shows that when shifted and non-intersecting anymore, we can approximate the behavior of our walks as if they lived in independent environments.",
"We also, highlight the fact that the following proposition gives an inequality.",
"One can prove it is an equality with an error term of order $Cn^{-c}$ where $C,c > 0$ only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Proposition 5.6 For any bounded positive function $f$ measurable with respect to $\\sigma \\left(\\big \\lbrace X^1_0,X^2_0,(X^1_{p},Z^1_{p}),(X^2_{q},Z^2_{q}) \\text{ }|\\text{ }p,q \\ge 1 \\big \\rbrace \\cup \\big \\lbrace c^{*}(e) \\text{ } |\\text{ }e \\in \\mathcal {E}_{(X^1_n)_{n\\ge 0}} \\cup \\mathcal {E}_{(X^2_n)_{n\\ge 0}} \\backslash (\\mathcal {E}_{X^1_0}\\cup \\mathcal {E}_{X^2_0})\\big \\rbrace \\right),$ we have the following inequality: $\\mathbb {E}_0[f \\circ \\theta _{\\rho _0} {1}_{I}\\circ \\theta _{\\rho _0}] \\le \\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }I, D^{\\otimes } = +\\infty \\bigg ]\\bigg ].$ Remark 5.7 Looking at this inequality one may be surprised.",
"Indeed, on the left term $\\rho _0$ is using the notion of joint regeneration levels while on the right term, $\\lbrace D^{\\otimes } = \\infty \\rbrace = \\lbrace D^1=+\\infty ,D^2=+\\infty \\rbrace $ refers to the classical notion of regeneration time for a single walk.",
"However, thinking about it carefully, we observe that under event $I$ , the event $\\lbrace \\mathcal {D}^{\\bullet } = +\\infty \\rbrace $ coincide with $\\lbrace D^{\\otimes } = + \\infty \\rbrace $ .",
"First using Theorem REF we can write $\\mathbb {E}_0\\left[f \\circ \\theta _{\\rho _0} {1}_{I} \\circ \\theta _{\\rho _0}\\right] = \\mathbb {E}_{0}\\left[\\mathbb {E}^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left[f{1}_{I} \\text{ }|\\text{ }\\mathcal {D}^{\\bullet } =+\\infty \\right]\\right].$ Indeed, we can first restrict to functions $f$ of type ${1}_{X^1 = (a_0,\\cdots ,a_k)}{1}_{X^2 = (b_0,\\cdots ,b_k)}g$ where $g$ is measurable with respect to $\\sigma (c^{*}(e) \\text{ }|\\text{ } e \\in \\mathcal {R}_{X^1_0} \\cup \\mathcal {R}_{X^2_0} \\backslash (\\mathcal {E}_{X^1_0 \\cup X^2_0}))$ .",
"In addition, ${1}_{X^1 = (a_0,\\cdots ,a_k)}{1}_{X^2 = (b_0,\\cdots ,b_k)}{1}_{I}$ is measurable with respect to $\\sigma ((X_0^1,X_0^2),(\\tilde{X}^1_n,\\tilde{X}^2_n)_{n \\ge 1})$ .",
"Then for the general case we use the fact that $f$ can be written as a limit of an increasing sequence of functions which are a linear combination of the above type and a monotone convergence theorem.",
"Now fix $a,b \\in \\mathbb {Z}^d$ such that $\\mathbb {P}_{0}\\left(X^1_{\\rho ^1_0}=a,X^2_{\\rho ^2_0} = b\\right) > 0$ .",
"First notice that $\\mathbb {P}^{K}_{a,b}$ -almost surely, under $I$ the event $\\mathcal {D}^{\\bullet } =+\\infty $ coincide with $\\lbrace D^{\\otimes }=+\\infty \\rbrace $ .",
"Moreover, for any bounded function $g(\\cdot )$ measurable with respect to $\\sigma \\left(\\left\\lbrace X^1_0,X^2_0,(X^1_{p},Z^1_{p}),(X^2_{q},Z^2_{q}) \\text{ }|\\text{ }p,q \\ge 1 \\right\\rbrace \\cup \\left\\lbrace c^{*}(e) \\text{ } |\\text{ }e \\in \\mathcal {E}_{(X^1_n)_{n\\ge 0}} \\cup \\mathcal {E}_{(X^2_n)_{n\\ge 0}} \\backslash (\\mathcal {E}_{X^1_0}\\cup \\mathcal {E}_{X^2_0})\\right\\rbrace \\right),$ we have that $\\mathbb {E}^K_{a,b}\\left[g{1}_{I}\\right] = \\mathbb {E}^{Q^{K}_{a,b}}\\left[g{1}_{I}\\right].$ Indeed, it follows by restricting to simple functions with disjoint set of edges for both walks and using the fact $g{1}_{I}$ can be written as an increasing limit of linear combination of functions of this type.",
"In particular, we have $\\mathbb {E}^K_{a,b}\\left[f{1}_{I}{1}_{D^1=+\\infty }{1}_{D^2=+\\infty }\\right] = \\mathbb {E}^{Q^{K}_{a,b}}\\left[f{1}_{I}{1}_{D^1=+\\infty }{1}_{D^2=+\\infty }\\right].$ We conclude the proof using the fact that $\\mathbb {P}_{a,b}^{K}\\left(I,D^{\\otimes }=+\\infty \\right)\\le \\mathbb {P}_{a,b}^{K}\\left(\\mathcal {D}^{\\bullet } =+\\infty \\right).$ The next lemma will enable us to erase the conditioning with the event $I$ .",
"Lemma 5.8 We have the following inequality: $\\mathbb {E}_{0}\\left[Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c} \\text{ }|\\text{ }D^{\\otimes }=+\\infty \\right)\\right] \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"By we have $Q^K_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(D^{\\otimes }=+\\infty \\right) = \\mathbb {P}_0(D=+\\infty )^2 > \\eta > 0$ where $\\eta $ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"Hence we only have to dominate $\\mathbb {E}_{0}\\left[Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c},D^{\\otimes }=+\\infty \\right)\\right].$ By the same argument as in last proof, it is equal to $\\mathbb {E}_{0}\\left[\\mathbb {P}^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c},\\mathcal {D}^{\\bullet } =+\\infty \\right)\\right].$ Using Lemma REF , last term is bounded above by $C\\mathbb {E}_{0}\\left[\\mathbb {P}^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\left(I^{c}\\text{ | }\\mathcal {D}^{\\bullet } =+\\infty \\right)\\right],$ where $C > 0$ is a constant which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"By Proposition REF , this expectation is equal to $\\mathbb {P}_{0}\\left(I^{c} \\circ \\theta _{\\rho _0}\\right) \\le \\mathbb {P}_0\\bigg (\\bigg \\lbrace \\displaystyle \\inf _{\\begin{array}{c}x \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^1\\rbrace \\\\y \\in \\mathcal {H}^{+}_{n^{\\theta }}\\cap \\lbrace X_n^2\\rbrace \\end{array}}\\Vert x-y\\Vert _{1}\\le 1 \\bigg \\rbrace \\bigg ) \\le Cn^{-c},$ where last inequality comes by using Proposition REF .",
"Proposition 5.9 For any bounded positive function $f$ we have the following inequality: $&\\left|\\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ } I,D^{\\otimes }=+\\infty \\bigg ]\\bigg ] - \\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }D^{\\otimes }=+\\infty \\bigg ]\\bigg ]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"We can rewrite the left term $\\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }D^{\\otimes }=+\\infty \\bigg ]\\bigg ]$ as $& \\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }I,D^{\\otimes }=+\\infty \\bigg ]Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\bigg (I\\text{ | }D^{\\otimes }=+\\infty \\bigg )\\bigg ] +\\\\&\\mathbb {E}_{0}\\bigg [\\mathbb {E}^{Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}}\\bigg [f \\text{ }|\\text{ }I^{c},D^{\\otimes }=+\\infty \\bigg ]Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\bigg (I^{c}\\text{ | }D^{\\otimes }=+\\infty \\bigg )\\bigg ].$ Using the fact that $f$ is bounded, it follows that the term in the statement of the proposition is bounded above by $C\\mathbb {E}_{0}\\bigg [Q^{K}_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}\\bigg (I^{c}\\text{ | }D^{\\otimes }=+\\infty \\bigg )\\bigg ].$ We conclude using Proposition REF .",
"Now we can conclude with the proof of Theorem REF .",
"[Proof of Theorem REF ] We recall that we want to dominate $\\mathbb {E}_0\\left[F(Z^1_n)F(Z^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right].$ First, using Corollary REF we dominate this by $Cn^{-c}+\\mathbb {E}_0\\left[F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right].$ Thanks to the fact that $\\mathbb {P}_{0}(I^{c} \\circ \\theta _{\\rho _0}) \\le Cn^{-c}$ by Proposition REF and the fact that $F$ is bounded we dominate it by $Cn^{-c}+\\mathbb {E}_0\\left[{1}_{I} \\circ \\theta _{\\rho _0} F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right].$ Let us bound above the term $\\mathbb {E}_0\\left[{1}_{I} \\circ \\theta _{\\rho _0} F(\\tilde{Z}^1_n)F(\\tilde{Z}^2_n)\\right].$ Let us introduce the function $f = F\\left(Z^1_n-Z^1_n(0)\\right)F\\left(Z^2_n-Z^2_n(0)\\right)$ , then the last term can be rewritten as $\\mathbb {E}_0\\left[{1}_{I} \\circ \\theta _{\\rho _0} f\\circ \\theta _{\\rho _0} \\right].$ Furthermore, we observe that $f$ satisfies the assumptions of Proposition REF and Proposition REF , hence last term is bounded above by $Cn^{-c} + \\mathbb {E}_0\\bigg [\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\left[f\\text{ | }D^{\\otimes }=+\\infty \\right]\\bigg ].$ Using and the form of the function $f$ we can rewrite $\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\left[F\\left(Z^1_{n}-Z^1_n(0)\\right)F\\left(Z^2_{n}-Z^2_n(0)\\right)\\text{ | }D^{\\otimes }=+\\infty \\right],$ as $\\mathbb {E}^K_0\\left[F(Z_n) \\text{ | }D=+\\infty \\right]^2.$ To summarise we have dominated the term $\\mathbb {E}_0\\left[F(Z^1_n)F(Z^2_n)\\right] - \\mathbb {E}_0\\left[F(Z^1_n)\\right]\\mathbb {E}_0\\left[F(Z^2_n)\\right],$ by $Cn^{-c} + \\mathbb {E}^K_0\\left[F(Z_n) \\text{ | }D=+\\infty \\right]^2 - \\mathbb {E}_0\\left[F(Z_n)\\right]^2.$ We conclude by using the fact that $F$ is bounded and Proposition REF ."
],
[
"For the clock process",
"We recall the notation $S^*_n(t) =\\displaystyle \\frac{\\tau _{{tn} + 1} - \\tau _1}{\\mathrm {Inv}(n)}$ where $\\mathrm {Inv}(n) = \\inf \\lbrace t \\ge 0 \\text{ ; }\\mathbf {P}[c^{*} > t] \\le 1/n\\rbrace $ .",
"In this part we aim to prove the following main result.",
"Proposition 5.10 There exists $K > 0$ such that the following inequality holds: $|\\mathbb {E}[F(S^{*1}_n)F(S^{*2}_n)] - \\mathbb {E}[F(S^{*1}_n)]\\mathbb {E}[F(S^{*2}_n)]| \\le Cn^{-c},$ where $F$ is as in (REF ), $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Before proving this result we will need to state and prove Lemma REF .",
"Let us also admit the following lemma which is crucial in the proof of Lemma REF .",
"Lemma 5.11 For all $\\eta \\in (0,1)$ and $\\rho < \\eta / \\gamma $ , the probability $\\mathbb {P}_{0} \\left( S^*_n(n^{-\\eta }) > n^{-\\rho } \\right) = \\mathbb {P}_{0} \\left( S_n(n^{-\\eta }) > n^{-\\rho } | D = + \\infty \\right)$ decays polynomially fast to 0 as $n \\rightarrow \\infty $ .",
"We let $\\rho ^i_0$ be $T^i_{\\mathcal {R}_0}$ , where $\\mathcal {R}_0$ is the first joint regeneration level in $\\mathcal {H}^+_{n^\\theta }$ .",
"Let us also introduce the random integer $k_n^{i} \\ge 1$ , which is the one such that $\\tau ^{i}_{k_n^{i}} = \\rho ^{i}_0$ .",
"Then let us introduce, for $i \\in \\lbrace 1,2\\rbrace $ $ \\widetilde{S}^{*i}_n(t) = \\mathrm {Inv}(n)^{-1}\\sum _{j = k^i_n+1}^{k^i_n + {nt}} (\\tau ^{i}_{j+1} - \\tau ^{i}_j) = S^{*i}_n\\left(\\frac{k^{i}_n}{n}+t\\right) - S^{*i}_n\\left(\\frac{k^{i}_n}{n}\\right),$ which is the value taken by $S^{*i}_n$ for the walk shifted by $\\tau _{k^{i}_n}$ .",
"Lemma 5.12 The following inequality holds: $\\left|\\mathbb {E}_0\\left[F(S^{*1}_n)F(S^{*2}_n)\\right] - \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right]\\right| \\le Cn^{-c},$ where $C,c > 0$ are constants which only depend on $\\lambda ,\\vec{\\ell },d,K$ .",
"Let us compute the difference $|F(S^{*i}_n)-F(\\widetilde{S}^{*i}_n(t))|$ .",
"Let us write the two terms properly: $& F(S^{*i}_n) = \\exp \\left(-\\lambda _1 S^{*i}_n(t_1) - \\cdots - \\lambda _m (S^{*i}_n(t_{m}) - S^{*i}_n(t_{m-1}))\\right)\\\\& F(\\tilde{S}^{*i}_n) = \\exp \\left(-\\lambda _1 \\tilde{S}^{*i}_n(t_1) - \\cdots - \\lambda _m (\\tilde{S}^{*i}_n(t_{m}) - \\tilde{S}^{*i}_n(t_{m-1}))\\right).$ Then using (REF ), and factorising by $F(S^{*i}_n)$ , the difference is dominated by $ F(S^{*i}_n) \\left[1-\\exp \\bigg (CS^{*i}_n( k^{i}_n / n)\\bigg )\\prod _{j=1}^{m}\\exp \\bigg (C\\bigg (S^{*i}_n( t_j + k^{i}_n / n) - S^{*i}_n( t_j )\\bigg )\\bigg )\\right],$ where $C = \\max \\left\\lbrace \\lambda _1,\\cdots ,\\lambda _m\\right\\rbrace $ .",
"Let us introduce the following event $ R \\mathcal {R}_0 \\le 2n^{\\theta },$ for $\\mathcal {R}_0$ being the first joint regeneration level inside $\\mathcal {H}^+_{n^\\theta }$ .",
"We already proved that $\\mathbb {P}_0(R^c)\\le Cn^{-c}$ in (REF ).",
"Let us put ourselves on the event $R$ and fix $1 > 1 - \\eta > \\theta > 0$ .",
"Then for $i \\in \\lbrace 1,2\\rbrace $ , $X^i_{\\tau ^i_{{n^{1-\\eta }}}}$ is $\\mathbb {P}_{0}$ -almost surely in $\\mathcal {H}^+_{2 n^\\theta }$ , hence $\\tau ^i_{k^i_n} < \\tau ^i_{n^{1-\\eta }}$ since $\\mathcal {R}_0 \\le 2n^{\\theta }$ .",
"It follows that event $R$ implies $C_n^{i} \\lbrace k_n^{i} \\le n^{1 - \\eta }\\rbrace $ for $i \\in \\lbrace 1,2\\rbrace $ .",
"In addition, under event $C^{i}_n$ the term $S^{*i}_n\\left(t_{j} + (k_n^i)/n)\\right) - S^{*i}_n\\left(t_{j}\\right)$ is dominated by $S^{*i}_n(t_{j} + n^{-\\eta }) - S^{*i}_n(t_{j})$ which is distributed as $S^{*i}_n(n^{-\\eta })$ under $\\mathbb {P}_0$ .",
"We define (fixing $t_0 = 0$ ): $A^i_n = \\left\\lbrace \\forall j \\in \\lbrace 0,\\cdots ,m-1\\rbrace \\text{, } S^{*i}_n\\left(t_{j} + (k_n^i)/n)\\right) - S^{*i}_n\\left(t_{j}\\right) \\le n^{\\rho }\\right\\rbrace .$ Then, using our estimate (REF ), Lemma REF , union bound and the i.i.d structure of regeneration times one gets $ &\\mathbb {P}_0\\left( (A^{i}_n)^{c} \\right) \\le m\\mathbb {P}\\left(S^{*i}_n(n^{-\\eta }) \\ge n^{\\rho } \\text{ | }D=+\\infty \\right) + \\mathbb {P}_0\\left(R^{c}\\right) \\le Cn^{-c},$ where $\\rho < \\eta / \\gamma < 1/\\gamma $ .",
"Therefore, on the event $A^{i}_n$ we have that $|F(S^{*i}_n)-F(\\widetilde{S}^{*i}_n(t))| \\le Cn^{-c},$ where $C,c$ only depend on $\\lambda ,\\vec{\\ell },d,K,\\lambda _1,\\cdots ,\\lambda _m$ .",
"Putting everything together we obtain $&\\left|\\mathbb {E}_0\\left[F(S^{*1}_n)F(S^{*2}_n)\\right] - \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right]\\right|&&\\\\ & \\hspace{100.0pt} \\le \\left|\\mathbb {E}_0\\left[F(S^{*1}_n)F(S^{*2}_n) {1}_{\\lbrace A^i_n\\rbrace }\\right] - \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n){1}_{\\lbrace A^i_n\\rbrace }\\right] \\right| + Cn^{-c} && \\\\& \\hspace{100.0pt} \\le Cn^{-c}.&&$ In the first inequality we used (REF ) and in the second one the fact that $F$ is bounded, (REF ) and the expression (REF ).",
"We are now able to prove the key result.",
"[Proof of Proposition REF ] Let us fix again $1 > 1 - \\eta > \\theta > 0$ and recall that $I = \\Big \\lbrace \\displaystyle \\inf _{\\begin{array}{c}p,q \\ge 0\\end{array}}(\\Vert X^1_{ p}-X^2_{q}\\Vert _{1})\\ge 2 \\Big \\rbrace $ .",
"Thanks to Lemma REF we get that $\\mathbb {E}[F(S^{*1}_n)F(S^{*2}_n)] - \\mathbb {E}[F(S^{*1}_n)]\\mathbb {E}[F(S^{*2}_n)] \\le Cn^{-c} + \\mathbb {E}[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)] - \\mathbb {E}[F(S^{*1}_n)]\\mathbb {E}[F(S^{*2}_n)]$ Moreover, we also get that $\\left| \\mathbb {E}_0\\left[F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right] - \\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace }F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right] \\right| &\\le \\mathbb {P}_{0}\\left(I^{c} \\circ \\theta _{\\rho _0} \\right)\\nonumber \\\\&\\le \\mathbb {P}_0\\left(\\mathbf {M}_{n^{\\theta }}\\right)\\nonumber \\\\ &\\le C n^{-c}.$ where we recall the definition of $\\mathbf {M}_N$ in (REF ) and we used Proposition REF .",
"We introduce $f = F(S^{*1}_n)F(S^{*2}_n)$ and note that $\\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace }F(\\widetilde{S}^{*1}_n)F(\\widetilde{S}^{*2}_n)\\right] = \\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace } f \\circ \\theta _{\\rho _0}\\right].$ Furthermore, $f$ satisfies the assumptions of Proposition REF and REF , hence we get $\\mathbb {E}_0\\left[{1}_{\\lbrace I \\circ \\theta _{\\rho _0}\\rbrace } f \\circ \\theta _{\\rho _0} \\right] \\le Cn^{-c} + \\mathbb {E}_0\\bigg [\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\Big [f\\text{ | }D^{\\otimes }=+\\infty \\Big ]\\bigg ].$ We can rewrite the last term in the following way $\\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\Big [f\\text{ | }D^{\\otimes }=+\\infty \\Big ] &= \\mathbb {E}^{Q_{X^1_{\\rho ^1_0},X^2_{\\rho ^2_0}}^{K}}\\Big [F(S^{*1}_n)F(S^{*2}_n)\\text{ | }D^{\\otimes }=+\\infty \\Big ]\\nonumber \\\\&= \\mathbb {E}\\left[F(S^{*}_n)\\text{ | }D=+\\infty \\right]^2 = \\mathbb {E}\\left[F(S^{*}_n)\\right]^2.$ Putting together the estimates (REF ) and (REF ) we have shown that $\\big | \\mathbb {E} \\left[ F(S^{*1}_n)F(S^{*2}_n) \\right] - \\mathbb {E} \\left[ F(S^{*1}_n)\\right] \\mathbb {E} \\left[ F(S^{*2}_n)\\right] \\big | \\le Cn^{-c},$ which finishes the proof."
],
[
"Small times do not matter",
"Lemma 5.13 (Fuk-Nagaev inequality, ) Let $(X_i)_{i \\ge 1}$ be a family of independent identically distributed random variables with regularly varying tails and the tail parameter $\\displaystyle \\gamma \\in (0, 1)$ .",
"Let $\\displaystyle {S}(n) \\sum _{i = 1}^n X_i$ and $\\displaystyle {M}(n) \\max _{i \\in \\lbrace 1,\\cdots ,n\\rbrace } X_i$ , then there exists a constant $c>0$ such that, for all $y \\le x$ $P \\left( {S}(n) > x, \\, {M}(n) < y \\right) \\stackrel{\\textnormal {F-N}}{\\le } \\left( c n \\frac{y}{x} L(y) y^{-\\gamma }\\right)^{x/y}.$ Where $L(y)$ is the slowly varying function associated to the tail of $X_1$ .",
"Let us consider $\\lbrace \\tau _1^{(i)}\\rbrace _{i \\ge 1}$ , a sequence of independent identically distributed regeneration times under the measure $\\mathbb {P}^K_{0}(\\cdot \\text{ | }D=+\\infty )$ .",
"In it is proved that the sum ${S}(n) \\sum _{i = 1}^{n} \\tau _1^{(i)}$ is in the domain of attraction of a completely asymmetric stable law of index $\\gamma $ .",
"This implies, thanks to , that $\\mathbb {P}_0^K(\\tau _1 > t \\text{ | }D=+\\infty ) = \\psi (t)t^{-\\gamma }$ for some slowly varying function $\\psi (\\cdot )$ .",
"Lemma 5.14 For all $\\eta \\in (0,1)$ set $\\displaystyle {M}(n^{1-\\eta }) = \\max _{1 \\le i \\le {n^{1-\\eta }}} \\tau _1^{(i)}$ then for all $\\rho < \\eta /\\gamma $ and all $\\varepsilon < (\\eta - \\rho \\gamma )/3\\gamma $ there exists $\\nu >0$ such that $\\mathbb {P}\\left( {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) \\le Cn^{-\\nu }.$ $\\mathbb {P}\\left( {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) &\\le n^{1 - \\eta } \\mathbb {P}_{0}^{K}\\left( \\tau _1 > n^{1/\\gamma - \\rho - 2\\varepsilon } | D=\\infty \\right) \\\\&\\le Cn^{1-\\eta } n^{-1+\\gamma \\rho + 2\\varepsilon \\gamma } \\psi (n^{1/\\gamma - \\rho - 2\\varepsilon }) \\\\&\\le Cn^{-\\eta + \\gamma \\rho + 3\\varepsilon \\gamma }\\le n^{-\\nu },$ where we used the fact that a slowly varying function can always be bounded above by any polynomial so that $\\psi (n^{1/\\gamma }) \\le C n^{\\varepsilon /\\gamma }$ (Potter's bound ).",
"Finally, it holds that $\\eta - \\gamma \\rho - 3\\varepsilon \\gamma > 0$ thanks to the choice of $\\varepsilon $ , one has that the last inequality holds for some $\\nu $ small enough.",
"Lemma 5.15 For all $\\eta \\in (0, 1)$ recall $\\displaystyle {M}(n^{1-\\eta }) = \\max _{1 \\le i \\le {n^{1-\\eta }}}^{} \\tau _1^{(i)}$ and let $ \\displaystyle {S}(n^{1-\\eta }) \\displaystyle \\sum _{i = 1}^{{n^{1-\\eta }}} \\tau _1^{(i)}$ then for all $\\rho < \\eta /\\gamma $ and all $\\varepsilon < (\\eta - \\rho \\gamma )/3\\gamma $ there exists $\\nu >0$ such that $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho } \\right) \\le n^{-\\nu }.$ First notice that, by total probability $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho } \\right) &= \\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho }, {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) \\\\ & +\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho }, {M}(n^{1-\\eta }) \\le n^{1/\\gamma - \\rho - 2\\varepsilon } \\right),$ Thanks to Lemma REF $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma -\\epsilon - \\rho }, {M}(n^{1-\\eta }) > n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) \\le Cn^{-\\nu },$ for some $\\nu >0$ .",
"So we just need to bound the second term of the sum.",
"We can appeal to Lemma REF with the couple $\\quad \\quad x = n^{1/\\gamma - \\varepsilon - \\rho }, \\quad \\quad y=n^{1/\\gamma - \\rho - 2\\varepsilon }.$ Plugging these quantities into the formula gives $\\mathbb {P}\\left( {S}(n^{1-\\eta }) > n^{1/\\gamma - \\varepsilon - \\rho }, {M} \\le n^{1/\\gamma - \\rho - 2\\varepsilon } \\right) &\\le \\left( C n^{1-\\eta } n^{-\\varepsilon } n^{-1 + \\gamma \\rho + 2\\gamma \\varepsilon } \\psi (n^{1/\\gamma - \\rho - 2\\varepsilon }) \\right)^{n^\\varepsilon }.$ Since the exponent is large we just need to check that what is inside the brackets is small.",
"This is immediate as $C n^{1-\\eta } n^{-\\varepsilon } n^{-1 + \\gamma \\rho + 2\\gamma \\varepsilon } \\psi (n^{1/\\gamma - \\rho - 2\\varepsilon })\\le C n^{2\\varepsilon \\gamma - \\eta + \\gamma \\rho } \\le e^{-1},$ as $\\psi (n^{1/\\gamma }) \\le Cn^{\\varepsilon }$ thanks to Potter's bound and $2\\varepsilon \\gamma - \\eta + \\gamma \\rho < 0$ by our choice of $\\varepsilon $ , this gives a super-polynomial bound for the desired quantity.",
"[Proof of Lemma REF ] Let us start with the first statement.",
"First let us note that, under $\\mathbb {P}_0$ , $\\lbrace \\tau _{i + 1} - \\tau _i\\rbrace _{i \\ge 1}$ are independent identically distributed random variables distributed as $\\tau _1$ under $\\mathbb {P}^K_{0}(.\\text{ | }D=+\\infty )$ .",
"Thanks to this fact we can re-write $S^*_n(n^{-\\eta }) \\stackrel{(\\mathrm {d})}{=} \\mathrm {Inv}(n)^{-1}\\sum _{i = 1}^{n^{1 - \\eta }} \\tau _{1}^{(i)},$ where $\\lbrace \\tau _{1}^{(i)}\\rbrace _{i \\ge 1}$ are i.i.d.",
"copies of $\\tau _1$ under the measure $\\mathbb {P}^{K}_{0}[ \\cdot | D = \\infty ]$ .",
"Then Lemma REF is enough to conclude together with the fact that for all $\\varepsilon >0$ $\\frac{n^{1/\\gamma - \\varepsilon }}{\\mathrm {Inv}(n)} \\rightarrow 0.$"
],
[
"Quenched convergence",
"We start this section by proving Theorem REF .",
"[Proof of Theorem REF ] The result is a straightforward consequence of Theorem REF and Proposition REF .",
"In particular one just needs to notice that $\\mathbf {Var}\\left(E_{0}^{\\omega }\\left[F_2(Z_{b^n})\\right]\\right) = \\mathbb {E}[F_2(Z_{b^n}^1)F_2(Z_{b^n}^2)] - \\mathbb {E}[F_2(Z_{b^n}^1)]\\mathbb {E}[F_2(Z_{b^n}^2)].$ The same decomposition holds for the process $S^*$ and the function $F_1$ .",
"Recall $D(\\mathbb {R})$ (resp.",
"$D(\\mathbb {R}^d)$ ) is the space of $\\mathbb {R}$ -valued (resp.",
"$\\mathbb {R}^d$ -valued) càdlàg functions.",
"In the following we consider $U$ the uniform topology and $J_1$ , $M_1$ two classical Skorohod topologies.",
"Let us recall that the vector $v$ and the matrix $\\Sigma $ have been introduced in .",
"In particular $v \\mathbb {E}_{0}^K[X_{\\tau _1}| D = \\infty ]$ while $\\Sigma $ is the covariance matrix of $X_{\\tau _1}$ under $\\mathbb {P}_{0}^{K}[ \\cdot | D = \\infty ]$ .",
"Moreover, let us introduce $v_0 = v / \\Vert v\\Vert $ , $I_d$ as the $d$ -dimensional identity matrix and $P_{v_0}$ as the projection matrix on $v_0$ .",
"Finally let $M_d C_\\infty ^{-\\gamma /2}(I_d - P_{v_0}) \\sqrt{\\Sigma }.$ Proposition 6.1 Fix any $T > 0$ , the followings holds for almost every environment $\\omega \\in \\Omega $ .",
"The law of $(Y_n(t))_{0 \\le t \\le T}$ converges under $P^\\omega _0(\\cdot )$ for the $U$ topology on $D^d$ towards the distribution of $(v t)_{0 \\le t \\le T}$ .",
"Moreover, the law of $(Z_n(t), S_n(t))_{0 \\le t \\le T}$ under $P^\\omega _0(\\cdot )$ converges in $D(\\mathbb {R}^d) \\times D(\\mathbb {R})$ for the $J_1 \\times M_1$ topology as $n \\rightarrow \\infty $ towards the law of $(\\sqrt{\\Sigma } B_t, C_\\infty \\mathcal {S}_\\gamma (t))_{0 \\le t \\le T}$ where $B$ is a standard Brownian motion and $\\mathcal {S}_\\gamma $ is a stable Subordinator of index $\\gamma $ independent of $B$ .",
"The convergence of $(Y_n(t))_{0 \\le t \\le T}$ is a consequence of .",
"We start by proving that Theorem REF together with imply the fact that $S_n$ converges in the process $M_1$ sense under the quenched law towards $\\mathcal {S}_\\gamma $ , we follow the proof of .",
"First we observe that $S_n(t) = \\frac{1}{\\mathrm {Inv}(n)} \\left( \\tau _1 + \\sum _{i = 1}^{{nt} - 1} (\\tau _{i + 1} - \\tau _i) \\right).$ We have that $\\tau _1 < + \\infty $ $\\mathbb {P}$ -a.s. thanks to , this means that for $\\mathbf {P}$ -almost all realisation of the environment $\\tau _1< + \\infty $ $P^{\\omega }$ -a.s..",
"But then, applying Slutsky's theorem, we just need to prove the quenched convergence of $S^*_n$ towards $\\mathcal {S}_\\gamma $ .",
"In order to prove convergence of the finite dimensional distributions we can restrict to prove convergence of the Laplace functionals $F$ of the form $F(h) = \\exp \\Big (-\\lambda _1h(t_1)-\\lambda _2(h(t_2)-h(t_1)) -\\cdots -\\lambda _n(h(t_n)-h(t_{n-1}))\\Big )$ Thanks to we have annealed convergence, so that Laplace functionals converge, in formulas $\\mathbb {E}\\left[F(S_n^*)\\right] = \\mathbf {E}\\left[ E^\\omega \\left[ F(S_n^*) \\right]\\right] \\underset{n \\rightarrow +\\infty }{\\rightarrow } \\mathbb {E}\\left[ F(\\mathcal {S}_\\gamma ) \\right] = \\exp \\left\\lbrace - t_1 \\psi (\\lambda _1) - \\dots - (t_m - t_{m -1})\\psi (\\lambda _m) \\right\\rbrace ,$ where $\\psi $ is the Laplace exponent of the stable Subordinator.",
"Moreover thanks to Theorem REF we know that for $b \\in (1, 2)$ $\\sum _{k = 1 }^{+\\infty }\\mathbf {Var}\\left[E_{0}^{\\omega }(F(S^*_{b^k}))\\right]< +\\infty .$ Therefore, by applying Borel-Cantelli and Chebyshev inequality to the sequence of events $|E_{0}^{\\omega }(F(S^*_{b^k})) - \\mathbb {E}_{0}\\left[ F(S^*_{b^k}) \\right]| > \\varepsilon $ and using the annealed convergence we get that $E_{0}^{\\omega }\\left[ F(S^*_{b^k}) \\right] \\rightarrow \\mathbb {E}\\left[ F(\\mathcal {S}_\\gamma ) \\right],$ almost surely along $b^k$ .",
"Moreover, the convergence holds jointly on a set of full measure for all functions $F$ of the form above for $b$ rational.",
"On the set of full measure where the joint convergence holds, we want to show that the promised convergence holds.",
"Let $k_n$ be the largest integer such that $b^{k_n} \\le n$ .",
"Thanks to the definition of $F$ , for any two increasing processes $h$ and $h^{\\prime }$ starting from 0 one gets $|F(h) - F(h^{\\prime })| \\le C \\max _{1 \\le i \\le m} |h(t_i) - h^{\\prime }(t_i)| \\wedge 1.$ Observe that, $S^*_n(t_i) = \\left( \\frac{b^{k_n}}{n} \\right)^{1/\\gamma } S^*_{b^{k_n}} \\left( \\frac{n}{b^{k_n}} t_i\\right).$ By the definition of $k_n$ and the monotonicity of $S^*_{b^{k_n}}(t)$ the latter is smaller than $S^*_{b^{k_n}}(b t_i)$ , as a consequence $(b^{-\\frac{1}{\\gamma }}-1)S^*_{b^{k_n}}(t_i) \\le S^*_n(t_i) - S^*_{b^{k_n}}(t_i) \\le S^*_{b^{k_n}}(b t_i) - S^*_{b^{k_n}}(t_i).$ The quantity $\\limsup _{n \\rightarrow \\infty } \\left| E^\\omega \\left[ F(S^*_n(t_i)) \\right] - E^\\omega \\left[ F(S^*_{b^{k_n}}(t_i)) \\right] \\right|,$ is thus bounded from above, up to a multiplicative constant, by $\\limsup _{n \\rightarrow \\infty } E^\\omega \\left[ \\max _{1 \\le i \\le m} \\bigg (|S^*_n(t_i) - S^*_{b^{k_n}}(t_i)|,|(b^{-\\frac{1}{\\gamma }}-1)S^*_{b^{k_n}}(t_i)|\\bigg ) \\wedge 1 \\right].$ Thanks (REF ) we get that this last quantity is bounded by $&\\limsup _{n \\rightarrow \\infty } E^\\omega \\left[ \\max _{1 \\le i \\le m} \\bigg (|S^*_{b^{k_n}}(b t_i) - S^*_{b^{k_n}}(t_i)|,|(b^{-\\frac{1}{\\gamma }}-1)S^*_{b^{k_n}}(t_i)|\\bigg ) \\wedge 1 \\right] \\\\&= \\mathbb {E} \\left[ \\max _{1 \\le i \\le m} \\bigg (|\\mathcal {S}_\\gamma (b t_i) - \\mathcal {S}_\\gamma (t_i)|,|(b^{-\\frac{1}{\\gamma }}-1)\\mathcal {S}_\\gamma (t_i)| \\bigg ) \\wedge 1 \\right].$ By the fact that $\\mathcal {S}_\\gamma $ is almost surely continuous at deterministic times we get that the quantity on the right converges to 0 as $b \\rightarrow 1$ along rationals by dominated convergence theorem since the quantity is bounded by 1.",
"The tightness of the process in $M_1$ comes from this f.d.d.",
"distributions convergence we just proved, thanks to the fact that all processes involved are increasing.",
"The convergence of the position $Z_n(t)$ (both seen as the polygonal interpolation and in the canonical way) is proved with the same hypotheses and thesis in .",
"We now need to prove joint convergence of $S_n$ and $Z_n$ .",
"First notice that, thanks to the previous two points and , tightness of the marginals is enough to have tightness in the product space.",
"Therefore, we have that the law $(S_n, Z_n)$ is tight on $D(\\mathbb {R}^d) \\times D(\\mathbb {R})$ endowed with the $J_1 \\times M_1$ -topology on a set of the environment $\\Omega _1 \\subset \\Omega $ of full $\\mathbf {P}$ -measure.",
"We need to prove that on a set of full measure $\\Omega _2$ , the law of the increments of the process $W_n = (S_n, Z_n)$ converge to the one of $W = (\\mathcal {S}_\\gamma , \\sqrt{\\Sigma } B)$ .",
"The proof follows the lines of the proof of the convergence of $S_n$ and the argument of .",
"Let $\\lambda _1, \\dots , \\lambda _m \\in \\mathbb {R}^{d+1}$ and $0 < t_0 < \\dots < t_m$ as it is sufficient to check on intervals that do not contain 0.",
"Let us define the function $G(w) = \\exp \\Big ( i \\lambda _1 \\cdot \\left( w(t_1) - w(t_0) \\right) + \\dots + i \\lambda _m \\cdot \\left( w(t_m) - w(t_{m - 1}) \\right) \\Big ).$ From we have that $\\mathbb {E}\\left[ G(W_n) \\right] \\underset{n \\rightarrow +\\infty }{\\longrightarrow } \\mathbb {E}\\left[ G(W) \\right].$ By proceeding in the same way as we did for Theorem REF (using all the instrumental lemmas and propositions that we proved in Section ) it is possible to prove that for all $b \\in (1, 2)$ $\\sum _{k = 1 }^{+\\infty }\\mathbf {Var}\\left[E_{0}^{\\omega }(G(W_{b^k}))\\right]< +\\infty .$ As a consequence we get that $E_{0}^{\\omega }\\left[ G(W_{b^k}) \\right] \\rightarrow \\mathbb {E}\\left[ G(W) \\right].$ Then, by following the argument that we did before, it is possible to show that for all $G$ of the form of (REF ) with $\\lambda _1, \\dots , \\lambda _m$ and $0 < t_0 < \\dots < t_m$ rationals on a set $\\Omega _2$ of full measure one gets $E_{0}^{\\omega }\\left[ G(W_{n}) \\right] \\rightarrow \\mathbb {E}\\left[ G(W) \\right].$ Finally, by right continuity of $W_n$ and $W$ and continuity of the Fourier functional it is possible to show that the limit holds for all $0 \\le t_0 \\le \\dots \\le t_m$ and for all $\\lambda _1, \\dots , \\lambda _m \\in \\mathbb {R}^{d+1}$ .",
"This finishes the proof as we showed that on a set of full measure $\\Omega _1 \\cap \\Omega _2$ the sequence $W_n$ is tight and has just one possible limit $W$ .",
"We are now able to prove the main result, the method is the same as the proof of .",
"[Proof of Theorem REF ] We fix a realisation of the environment $\\omega \\in \\Omega $ , the following convergences in distribution are meant to be understood under the quenched law $P^\\omega $ , we assume the result of Proposition REF .",
"Recall that $D(\\mathbb {R})$ and $D(\\mathbb {R}^d)$ denote the sets of càdlàg functions in dimension one (resp.",
"$d$ ).",
"Let $D_{\\uparrow }$ be the set of non-deacreasing càdlàg functions, $D_{\\uparrow \\uparrow }$ the set of strictly increasing càdlàg functions and $D_u$ the set of unbounded càdlàg functions.",
"Let $\\mathcal {C}$ be the set of continuous functions and $U$ the uniform topology.",
"It is a known fact that $\\mathcal {S}_\\gamma ^{-1}$ is almost surely continuous and increasing.",
"Firstly we get that $\\left(Y_{n^\\gamma /L(n)}\\left( S_{n^\\gamma /L(n)}^{-1}\\left(\\frac{nt}{\\mathrm {Inv}\\left(n^\\gamma /L(n)\\right)}\\right) \\right) \\right)_{t \\in [0, T]} \\rightarrow \\left( v C_\\infty ^{-\\gamma } \\mathcal {S}_\\gamma ^{-1}(t) \\right)_{t \\in [0, T]},$ in the uniform topology.",
"This thanks to the fact the right continuous inverse is continuous from $(D_{u, \\uparrow \\uparrow }, M_1)$ to $(\\mathcal {C}, U)$ thanks to , and if $(x_n, y_n) \\rightarrow (x, y)$ in $D^d \\times D_{\\uparrow }$ with $(x, y) \\in \\mathcal {C}^d \\times \\mathcal {C}_{\\uparrow }$ , then $x_n \\circ y_n \\rightarrow x \\circ y$ in the $U$ topology (see ).",
"Lastly, note that $(nt/\\mathrm {Inv}(n^\\gamma /L(n)))_{t \\in [0, T]}$ converges uniformly to $(t)_{t \\in [0, T]}$ .",
"This proves the first result.",
"Moreover, thanks to the same results it holds that $\\left(Z_{n^\\gamma /L(n)}\\left( S_{n^\\gamma /L(n)}^{-1}\\left(\\frac{nt}{\\mathrm {Inv}\\left(n^\\gamma /L(n)\\right)}\\right) \\right) \\right)_{t \\in [0, T]} \\stackrel{(\\mathrm {d})}{\\rightarrow } \\left( C_\\infty ^{-\\gamma /2} B_{\\mathcal {S}_\\gamma ^{-1}(t)} \\sqrt{\\Sigma } \\right)_{t \\in [0, T]},$ in the $J_1$ topology.",
"We now need to ensure that $\\mathbf {P}$ -a.s., with high $P^\\omega $ -probability $X_{{nt}}$ is close to $X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }}$ .",
"It is immediate to prove that $\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }$ is the smallest $\\tau _i$ such that $\\tau _i > tn$ .",
"Then we can bound the distance $X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }} - X_{{nt}}$ by a constant times the size of a regeneration block.",
"$M_n &\\max _{t \\in [0, T]} \\left\\Vert X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }} - X_{{nt}} \\right\\Vert _\\infty \\\\& \\le C \\max _{k = 1, \\dots , {nT} + 1} \\left\\Vert X_{\\tau _k} - X_{\\tau _{k - 1}} \\right\\Vert _\\infty .$ By the fact that $\\Vert X_{\\tau _k} - X_{\\tau _{k - 1}}\\Vert _\\infty \\le \\chi _k^\\alpha $ defined in (REF ), applying and using a union bound one gets that, for all $M>0$ and $\\delta >0$ $\\mathbb {P}_0\\left( M_n > \\delta n^{\\gamma /4} \\right) = \\mathbf {E}\\left[ P_0^\\omega (M_n > \\delta n^{\\gamma /4}) \\right] \\le n^{-M}.$ Markov inequality gives that $\\mathbf {P}\\left( P_0^\\omega (M_n > \\delta n^{\\gamma /4}) > n^{-1} \\right) \\le n^{-2}.$ Then Borel-Cantelli implies that $\\mathbf {P}$ -almost surely $P_0^\\omega [M_n > \\delta n^{\\gamma /4}] \\le 1/n$ for all $n$ large enough, which leads us to conclude that $X_{\\tau _{\\left\\lfloor \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}(nt/\\mathrm {Inv}(n^\\gamma /L(n)))\\right\\rfloor }} - X_{{nt}} \\le \\delta n^{\\gamma /4}$ in the uniform sense with high $P^\\omega $ -probability.",
"This implies, together with (REF ) and (REF ) and Slutsky's theorem, that $\\mathbf {P}$ -a.s., under the quenched law $P^\\omega $ the statement (REF ) holds and that $\\left(\\frac{X_{{nt}} - v \\frac{n^\\gamma }{L(n)} S_{n^\\gamma /L(n)}^{-1}\\left(\\frac{nt}{\\mathrm {Inv}\\left(n^\\gamma /L(n)\\right)}\\right)}{\\sqrt{n^\\gamma /L(n)}} \\right)_{t \\in [0, T]} \\stackrel{(\\mathrm {d})}{\\rightarrow } \\left( C_\\infty ^{-\\gamma /2} B_{\\mathcal {S}_\\gamma ^{-1}(t)} \\sqrt{\\Sigma } \\right)_{t \\in [0, T]}.$ Finally, notice that (REF ) follows by applying the linear transformation $(I_d - P_{v_0})$ to (REF ) as $(I_d - P_{v_0}) v = 0$ .",
"All linear combinations are continuous in the $J_1$ -topology at continuous functions."
],
[
"Extension of Markov property",
"The appendix gives some results which are useful in the paper but considered as tools.",
"Since in the whole paper we will consider two random walks we extend the Markov property when we have two Markov chains with the general following lemma.",
"Lemma A.1 Consider two Markov chains $(X_n)_{n \\ge 0}$ and $(Y_n)_{n \\ge 0}$ which take values in a countable set $E$ .",
"Consider the product measure $P_{(x,y)} = P_x \\otimes P_y$ .",
"We consider the canonical markov chain on $E^2$ .",
"We define for $k,k^{\\prime } \\in \\mathbb {N}$ the translation operator defined by $\\begin{array}{l|rcl}\\end{array}\\theta _{k,k^{\\prime }}: & (E^2)^{\\mathbb {N}} & \\longrightarrow & (E^2)^{\\mathbb {N}} \\\\& ((x_n),(y_n)) & \\longmapsto & ((x_{n+k}),(y_{n+k^{\\prime }})) $ .$$ Consider also $P_{x,y} = P_x \\otimes P_y$ almost surely finite stopping times $\\tau $ and $\\rho $ for $X$ and $Y$ .",
"Consider two bounded and measurable functions $f$ and $g$ which a respectively measurable with respect to $\\sigma \\Big (X_0,...,X_{\\tau },Y_0,...,Y_{\\rho }\\Big ) \\text{ and }\\sigma \\Big ((X_n)_{n \\ge 0},(Y_n)_{n \\ge 0}\\Big ).$ Then we have the following equality: $E_{x,y}\\left[f g \\circ \\theta _{\\tau ,\\rho }\\right] = E_{x,y}\\left[fE_{X_{\\tau },Y_{\\rho }}[g]\\right].$ The proof is straightforward using strong Markov property and a Monotone Class Theorem.",
"Indeed, if $f$ is of the form $f((x_n),(y_n)) = f_1((x_n))f_2((y_n))$ with $f_1$ measurable with respect to $\\sigma \\big (X_0,...,X_{\\tau }\\big )$ and $f_2$ measurable with respect to $\\sigma \\big (Y_0,...,Y_{\\rho }\\big )$ and $g((x_n),(y_n)) = g_1((x_n))g_2((y_n))$ with $g_1$ measurable with respect to $\\sigma \\big ((X_n)_{n \\ge 0}\\big )$ and $g_2$ measurable with respect to $\\sigma \\big ((Y_n)_{n \\ge 0}\\big )$ .",
"Then we have that $&E_{x,y}\\left[f g \\circ \\theta _{\\tau ,\\rho }\\right]\\\\&= E_{x,y}\\left[f_1 g_1 \\theta _{\\tau } \\cdot f_2g_2 \\circ \\theta _{\\rho }\\right]\\\\& = E_{x}\\left[f_1 g_1 \\circ \\theta _{\\tau }\\right]\\cdot E_{y}\\left[f_2 g_2 \\circ \\theta _{\\rho }\\right],$ where the last equality comes from the independence of the two walks.",
"Then using the Strong Markov property we get $&E_{x}\\left[f_1 E_{X_{\\tau }}[g_1]\\right]\\cdot E_{y}\\left[f_2 E_{Y_{\\rho }}[g_2]\\right] \\\\& = E_{x,y}\\left[f_1f_2 E_{X_{\\tau }}[g_1]E_{Y_{\\rho }}[g_2]\\right]\\\\& = E_{x,y}\\left[f E_{X_{\\tau },Y_{\\rho }}[g]\\right].$ We conclude using a Monotone Class Theorem argument."
],
[
"Polynomial decay",
"Proposition B.1 For any $\\varepsilon > 0$ , there exists $K_0 > 0$ such that for any $K > K_0$ , any $n \\in \\mathbb {N}^{*}$ and any $z \\in \\mathbb {Z}^d$ we have $\\mathbb {P}_0(X_{\\tau _n} = z) \\le Cn^{\\varepsilon -\\frac{d}{2}},$ where $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"Remark B.2 We recall that the notion of regeneration time introduced in (see the definition in Section REF ) depends on the value of $K$ , but we always take $K$ large enough.",
"First we have the following equality: $\\mathbb {P}_0(X_{\\tau _n} = z)& =\\mathbb {P}_0\\left(X_{\\tau _1} +\\sum _{i=1}^{n-1} (X_{\\tau _{i+1}}-X_{\\tau _i}) = z\\right)\\\\&\\nonumber =\\mathbb {P}_0\\left(\\sum _{i=1}^{n-1} (X_{\\tau _{i+1}}-X_{\\tau _i} - v ) = z - X_{\\tau _1} - (n-1)v\\right),$ where $v = \\mathbb {E}[X_{\\tau _{2}}-X_{\\tau _1}]$ , we introduce for $i\\ge 1$ , $\\xi _i = X_{\\tau _{i+1}}-X_{\\tau _i} - v $ .",
"The family $(\\xi _i)_{i \\ge 1}$ is an independent family of random variables identically distributed and independent of $X_{\\tau _1}$ .",
"By , for any $M > 0$ there exists $K_0 > 0$ such that for any $K > K_0$ we have $\\mathbb {E}_{0}\\left[\\Vert \\xi _1\\Vert ^{M}\\right] < +\\infty $ .",
"Thus we can write (REF ) as $\\sum _{y \\in \\mathbb {Z}^d}^{} \\mathbb {P}_0(X_{\\tau _1} = y)\\mathbb {P}_0\\bigg (\\sum _{i=1}^{n-1} \\xi _i = e - y - (n-1)v\\bigg ).$ Then we only need a bound for the value of $\\mathbb {P}_0\\bigg (\\displaystyle \\sum _{i=1}^{n-1} \\xi _i = x\\bigg )$ with $x = e - y - (n-1)v$ uniformly for $y \\in \\mathbb {Z}^d$ .",
"We define $\\begin{array}{l|rcl}\\end{array}H: & \\mathbb {R} & \\longrightarrow & \\mathbb {R} \\\\& x & \\longmapsto & x^{M} $$, then we have $ E0(H(||i||)) < +$.",
"$ H$ increases faster than $ x x2+$ and slower than $ x ex$.",
"Using \\cite [Theorem 3]{Zaitsev} we can construct a probability space and two family of random variables $ (Yi)$ and $ (Ni)$ such that:\\begin{itemize}\\item [\\bullet ] (Y_i)_{i \\ge 1} has the same law as (\\xi _i)_{i \\ge 1} under \\mathbb {P}_0.\\item [\\bullet ] (N_i)_{i \\ge 1} is a family of independent identically distributed centered gaussian vectors with covariance matrix \\Sigma where \\Sigma is the covariance matrix of \\xi _1.\\end{itemize}Moreover, they satisfy\\begin{equation*}\\mathbb {P}\\left(\\max _{\\begin{array}{c}1 \\le j \\le n\\end{array}}(|\\sum _{i=1}^{j} (Y_i - N_i)|) \\ge cn^{\\varepsilon }\\right) \\le C\\frac{n}{n^{\\varepsilon M}},\\end{equation*}where the constants $ C,c > 0$ only depend on $ ,,d,K$.",
"We write{\\begin{@align}{1}{-1}\\nonumber \\mathbb {P}_0\\bigg (\\sum _{i=1}^{n-1} \\xi _i = x\\bigg ) &= \\mathbb {P}\\bigg (\\sum _{i=1}^{n-1} Y_i = x\\bigg ) \\\\&\\nonumber \\le Cn^{1-\\varepsilon M} + \\mathbb {P}\\bigg (\\sum _{i=1}^{n-1} N_i \\in B_{\\Vert \\cdot \\Vert _{1}}(x,cn^{\\varepsilon })\\bigg ) \\\\& \\le Cn^{1-\\varepsilon M} + \\mathbb {P}\\bigg ( \\frac{\\sum _{i=1}^{n-1} N_i}{\\sqrt{n-1}} \\in B_{\\Vert \\cdot \\Vert _{1}}\\bigg (\\frac{x}{\\sqrt{n-1}},\\frac{cn^{\\varepsilon }}{\\sqrt{n-1}}\\bigg ) \\bigg ).\\end{@align}}Since $ i=1n-1 Nin-1$ is a centered Gaussian vector with covariance matrix $$, the right term in (\\ref {Gaussian}) is bounded above by{\\begin{@align*}{1}{-1}C\\mu \\left(B_{\\Vert \\cdot \\Vert _{1}}\\bigg (\\frac{x}{\\sqrt{n-1}},\\frac{cn^{\\varepsilon }}{\\sqrt{n-1}}\\bigg )\\right),\\end{@align*}}where $$ denotes the Lebesgue measure on $ Rd$ and $ C$ is a constant which depends on $$, hence on $ ,,d,K$.",
"We deduce that last term is dominated by{\\begin{@align*}{1}{-1}Cn^{d\\varepsilon -\\frac{d}{2}},\\end{@align*}}where $ C$ is a constant which only depends on $ ,,d,K$.$ Choosing $M$ large enough such that $1-\\varepsilon M < - \\frac{d}{2}$ we obtain that (REF ) is bounded above by $Cn^{1-\\varepsilon M} + Cn^{d\\varepsilon -\\frac{d}{2}} \\le C n^{\\varepsilon d-\\frac{d}{2}},$ where the constant $C$ only depends on $\\lambda ,\\vec{\\ell },d,K$ .",
"We conclude since the result is true for all $\\varepsilon > 0$ ."
],
[
"Acknowledgments",
"We would like to thank Daniel Kious for encouraging this collaboration and useful comments at various stages of the project.",
"We would like also to thank Christophe Sabot and Noam Berger for insightful discussions.",
"AF acknowledges the support of an NSERC Discovery Grant.",
"CS is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/S022945/1."
]
] | 2210.07825 |
[
[
"(1,1)-Cluster Editing is Polynomial-time Solvable"
],
[
"Abstract A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques.",
"Abu-Khzam (2017) introduced the $(a,d)$-{Cluster Editing} problem, where for fixed natural numbers $a,d$, given a graph $G$ and vertex-weights $a^*:\\ V(G)\\rightarrow \\{0,1,\\dots, a\\}$ and $d^*{}:\\ V(G)\\rightarrow \\{0,1,\\dots, d\\}$, we are to decide whether $G$ can be turned into a cluster graph by deleting at most $d^*(v)$ edges incident to every $v\\in V(G)$ and adding at most $a^*(v)$ edges incident to every $v\\in V(G)$.",
"Results by Komusiewicz and Uhlmann (2012) and Abu-Khzam (2017) provided a dichotomy of complexity (in P or NP-complete) of $(a,d)$-{Cluster Editing} for all pairs $a,d$ apart from $a=d=1.$ Abu-Khzam (2017) conjectured that $(1,1)$-{Cluster Editing} is in P. We resolve Abu-Khzam's conjecture in affirmative by (i) providing a serious of five polynomial-time reductions to $C_3$-free and $C_4$-free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$-{Cluster Editing} on $C_3$-free and $C_4$-free graphs of maximum degree at most 3."
],
[
"Introduction",
"A graph $H$ is a clique graph if $H$ is a vertex-disjoin union of cliques.",
"Addition or deletion of an edge to a graph is called an edge edition.",
"Given a graph $G$ and an integer $k \\ge 0,$ the Cluster Editing problem asks whether $k$ or less edge editions can transform $G$ into a clique graph.",
"Cluster Editing is NP-Complete [11], [12], but it is fixed-parameter tractable if parameterized by $k$ [5].",
"The parameterization has received considerable attention [2], [3], [4], [6], [7], [8], [9].",
"In particular, an $O^*(1.618^k)$ -time algorithm was designed in [2] and a kernel with at most $2k$ vertices in [7].",
"Unfortunately, $k$ is often not small for real-world instances.",
"For example, in a protein similarity data set that has been frequently used for evaluating Cluster Editing algorithms, the instances with $n \\ge 30,$ where $n$ is the number of vertices, have an average number of edge editions between $2n$ and $4n$ [3].",
"This has led to introduction of different parameterizations in [1], [10].",
"We will consider the following variation of Cluster Editing problem introduced by Abu-Khzam [1].",
"In $(a,d)$ -Cluster Editing, for fixed natural numbers $a$ and $d$ , given a graph $G$ and vertex-weights $a^*_{}:\\ V(G)\\rightarrow \\lbrace 0,1,\\dots , a\\rbrace $ and $d^*_{}:\\ V(G)\\rightarrow \\lbrace 0,1,\\dots , d\\rbrace $ , decide whether $G$ can be turned into a cluster graph by deleting at most $d^*_{}(v)$ edges incident to every $v\\in V(G)$ and adding at most $a^*_{}(v)$ edges incident to every $v\\in V(G)$ .",
"Note that there is no upper bound on the total number of additions and deletions, see Paragraph 7 of Section 2 and Paragraph 1 of Section 5.1 in [1].",
"$(a,d)$ -Cluster Editing is similar to a version of Cluster Editing studied by Komusiewicz and Uhlmann [10].",
"In particular, they proved that $(0,d)$ -Cluster Editing is NP-hard for every fixed integer $d\\ge 2.$ Abu-Khzam [1] proved additional results on $(a,d)$ -Cluster Editing and established computational complexity of $(a,d)$ -Cluster Editing for every value of $a$ and $d$ apart from $a=d=1$ : $(a,d)$ -Cluster Editing is in P when $(a,d)\\in \\lbrace (a,0), (0,1):\\ a\\in \\lbrace 0,1,2\\rbrace \\rbrace $ and NP-complete, otherwise, apart from possibly the case of $a=d=1$ .",
"In fact, Abu-Khzam [1] conjectured that $(1,1)$ -Cluster Editing is polynomial-time solvable.",
"In this paper we resolve Abu-Khzam's conjecture in affirmative by (i) providing, in Section , a serious of five polynomial-time reductions to $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3, and (ii) designing, in Section , a polynomial-time algorithm for solving $(1,1)$ -Cluster Editing on $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3.",
"An edge $uv$ of $G$ is deletable (in $G$ ) if $d^*_{}(u)=d^*_{}(v)=1$ and non-deletable, otherwise.",
"An edge $uv$ of the complement $\\overline{G}$ of $G$ is addable (to $G$ ) if $a^*_{}(u)=a^*_{}(v)=1$ and non-addable, otherwise.",
"Now we reformulate $(1,1)$ -Cluster Editing as follows: Given a graph $G$ , decide whether $G$ has a matching $D$ of edges deletable in $G$ and $\\overline{G}$ has a matching $A$ of edges addable to $G$ such that $G-D+A$ is a clique graph.",
"We call it the matching formulation of $(1,1)$ -Cluster Editing and the desirable clique graph $G-D+A$ a solution.",
"Note that we will often write a solution in the form $G-D+A$ , which defines the matchings $D$ and $A$ rather than the resulting cliques.",
"We will denote by $P_n$ ($C_n$ , respectively) a path (cycle, respectively) on $n$ vertices, by $K_n$ a complete graph on $n$ vertices, and by $K_{n,m}$ a complete bipartite graph with partite sets of sizes $n$ and $m.$ For a positive integer $p$ , $[p]=\\lbrace 1,2,\\dots , p\\rbrace .$"
],
[
"Five polynomial-time reductions",
"For two sets ${\\cal G}$ and ${\\cal H}$ of graphs, we say that $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}$ to ${\\cal H}$ if for all $G\\in {\\cal G}$ and its vertex-weights $a^*_{},d^*_{}$ , we can either decide in polynomial time in $|V(G)|$ whether $(G,a^*_{},d^*_{})$ is a YES-instance or NO-instance, or construct $H \\in {\\cal H}$ with vertex-weights $a^+_{},d^+_{}$ such that $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if $(H,a^+_{},d^+_{})$ is a YES-instance of $(1,1)$ -Cluster Editing.",
"Note that as the reduction is a polynomial-time algorithm, the size of $H$ is also bounded above by a polynomial in the size of $G$ .",
"When the vertex-weights are clear from the context, we will often say that $G$ is an instance of the problem (i.e., of $(1,1)$ -Cluster Editing).",
"We say that a graph $G$ is triangle-free if it does not contain a cycle $C_3$ as a subgraph and is quadrilateral-free if it does not contain a cycle $C_4$ as a subgraph.",
"We will now prove a number of lemmas which will culminate in a final polynomial-time algorithm for $(1,1)$ -Cluster Editing.",
"Let ${\\cal G}_1$ denote all graphs and let ${\\cal G}_2$ denote all triangle-free graphs of maximum degree at most 3.",
"Lemma 1 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_1$ to ${\\cal G}_2$ .",
"Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing.",
"Suppose that $G$ has a triangle $T\\cong C_3$ .",
"We can delete at most one edge from $T$ and add at most one edge to $T$ (after an edge was deleted).",
"Thus, if $(G,a^*_{},d^*_{})$ has a solution, it must contain $T$ as (part of) a clique.",
"In the algorithm below to shorten its description, once a deletable edge $uv$ is deleted from $G$ , we immediately set $d^*_{}(u)=d^*_{}(v)=0$ and if an addable edge is added to $G$ , we immediately set $a^*_{}(u)=a^*_{}(v)=0.$ Let $Q:=T$ and execute the following loop.",
"While there is a vertex outside $Q$ adjacent to a vertex in $Q$ , for every such vertex $v$ do the following cases in turn.",
"If there is only one edge $e$ between $v$ and $Q$ then if $e$ is an deletable edge, then $e$ is deleted and the loop is continued.",
"If there are exactly $|V(Q)|-1$ edges between $v$ and $V(Q)$ in $G$ and the edge in $\\overline{G}$ between $v$ and $V(Q)$ is addable, then add the addable edge to $G$ and continue the loop.",
"If there are $|V(Q)|$ edges between $v$ and $V(Q)$ in $G$ , then add vertex $v$ and the edges between $v$ and $Q$ to $Q$ .",
"Otherwise, $v$ has between two and $|V(Q)|-2$ edges to $Q$ and we conclude that $(G,a^*_{},d^*_{})$ is a NO-instance and stop the loop.",
"If the loop stops without concluding that $(G,a^*_{},d^*_{})$ is a NO-instance, then delete $Q$ from $G.$ We can continue as above and either eliminate all triangles from $G$ or conclude that $(G,a^*_{},d^*_{})$ is a NO-instance.",
"In the former case, we obtain a triangle-free instance $(G,a^*_{},d^*_{})$ which is a YES-instance if and only if the initial instance is.",
"Note that it takes a polynomial time to compute the triangle-free instance or conclude that the initial instance is a NO-instance.",
"Now suppose that $(G,a^*_{},d^*_{})$ is a triangle-free instance and $G$ has a vertex $v$ of degree at least four.",
"Since we can delete at most one edge incident to $v$ and $G$ is triangle-free, we cannot make a clique including $v$ without adding at least two edges incident to a neighbour of $v$ .",
"Thus, $(G,a^*_{},d^*_{})$ is a NO-instance.",
"It is not hard to verify that the algorithm runs in polynomial time.",
"Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing and let $Q \\subseteq V(G)$ be arbitrary.",
"Consider the graph $G-Q$ , where we delete $Q$ (and all edges incident with $Q$ ).",
"We define $d^*_{Q}$ such that $d^*_{Q}(v) = d^*_{}(v) - |N(v) \\cap Q|$ for all $v \\in V(G-Q)$ .",
"We define $a^*_{Q}$ to be the function $a^*_{}$ restricted to $V(G-Q)$ .",
"Note that by definition of vertex-weights, if $d^*_{Q}(v) <0$ for any $v \\in V(G-Q)$ then $(G-Q,a^*_{Q},d^*_{Q})$ is a NO-instance.",
"Similarly, if $F\\subseteq E(G)$ , then $d^*_{F}(v) = d^*_{}(v) - |E(v) \\cap F|$ , where $E(v)$ is the set of edges incident to $v$ .",
"Also, $a^*_{F}=a^*_{}.$ Again, if $d^*_{F}(v) <0$ for any $v \\in V(G)$ then $(G-F,a^*_{F},d^*_{F})$ is a NO-instance.",
"We say that a vertex set $X$ of $G$ be R-deletable if $X$ has constant size and every solution to $(G,a^*,d^*)$ (if any) puts all edges between $X$ and $V(G)-X$ in $D.$ Note that vacuously this implies that if $G$ is a NO-instance then any set $X$ of constant size is an R-deletable set.",
"Lemma 2 Let $(G,a^*_{},d^*_{})$ be an instance of $(1,1)$ -Cluster Editing.",
"If a vertex set $X$ of $G$ is R-deletable, then we can either solve $(G,a^*_{},d^*_{})$ is constant time or reduce the instance to $G-X$ .",
"Note that if $(G,a^*_{},d^*_{})$ is a NO-instance, then any arbitrary vertex set $X$ of constant size is R-deletable and therefore satisfies the above.",
"If $X=V(G)$ then we can solve the problem in constant time, so assume that $X \\ne V(G).$ Let $F$ be all edges between $X$ and $V(D)-X$ and let $G^{\\prime } =G-F.$ If any $d^*_{F}$ value drops below 0 we have a NO-instance by definition of vertex-weights.",
"Otherwise, $G$ is a YES-instance if and only if $G^{\\prime }$ is a YES-instance.",
"And as $X$ has constant size we can determine if there is a solution to $G^{\\prime }[X]$ in constant time.",
"If there is no solution to $G^{\\prime }[X]$ then $G$ is a NO-instance and if there is a solution to $G^{\\prime }[X]$ , then we can reduce $G$ by deleting $X$ , as $G$ is a YES-instance if and only if $G^{\\prime }-X=G-X$ is a YES-instance.",
"Note that the above also holds when $(G,a^*_{},d^*_{})$ is a NO-instance as then either $G[X]$ is a NO-instance (after adjustment of $d^*$ ), which can be decided in constant time, or $G-X$ is a NO-instance (after adjustment of $d^*$ ).",
"Lemma 3 Let $G$ be a triangle-free graph with maximum degree at most 3.",
"Then $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if there is a matching $D$ of deletable edges from $G$ such that every connected component $C$ of $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ or $C_4$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ and if $C\\cong C_4$ then the two chords of $C$ , which lie in $\\overline{G}$ , are addable to $G$ .",
"Let $(G,a^*_{},d^*_{})$ be a YES-instance of $(1,1)$ -Cluster Editing and $G-D+A$ is a matching solution.",
"Since $G$ is a triangle-free graph with maximum degree at most 3, each connected component of $G-D+A$ is isomorphic to $K_1$ or $K_2$ or $K_3$ or $K_4$ .",
"Now the lemma follows from the fact that $G$ is a triangle-free graph.",
"Lemma 4 Let $G$ be a triangle-free graph with maximum degree at most 3 containing a $K_{2,3}$ as a subgraph.",
"Then $(G,a^*_{},d^*_{})$ is a NO-instance of $(1,1)$ -Cluster Editing for every $(a^*,d^*)$ .",
"Let $G$ be a triangle-free graph with maximum degree at most 3 containing a $K_{2,3}$ subgraph, $R$ , where $V(R)=\\lbrace x_1,x_2,y_1,y_2,y_3\\rbrace $ and $E(R)=\\lbrace x_i y_j \\; | \\; i=1,2 \\mbox{ and } j=1,2,3\\rbrace $ .",
"Assume that $(G,a^*_{},d^*_{})$ is a YES-instance with solution $G-D+A$ .",
"By Lemma REF , no vertex in $G-D$ has degree 3.",
"Thus, we must have $x_1 y_i \\in D$ and $x_2 y_j \\in D$ for some $i \\ne j$ (as $D$ is a matching) and these are the only edges in $E(R)$ that belong to $D$ .",
"However this implies that $R-D$ is a $P_5$ , which is a contradiction by Lemma REF .",
"So, $(G,a^*_{},d^*_{})$ is a NO-instance.",
"Let ${\\cal G}_3$ denote all triangle-free graphs of maximum degree at most 3 which contain no 4-cycle, all of whose vertices have degree 3.",
"Lemma 5 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_2$ to ${\\cal G}_3$ in polynomial time.",
"Let $G \\in {\\cal G}_2$ be arbitrary with vertex-weights $(a^*,d^*)$ .",
"If there is no 4-cycle in $G$ where all vertices have degree three then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be a 4-cycle in $G$ with $d(v_i)=3$ for all $i=1,2,3,4$ .",
"Let $w_i$ be the neighbour of $v_i$ in $G$ which does not lie on $C$ .",
"Let $V_8=\\lbrace v_1,v_2,v_3,v_4,w_1,w_2,w_3,w_4\\rbrace $ .",
"We will show that $|V_8|=8$ (or we have a NO-instance), so assume for the sake of contradiction that this is not the case.",
"As $G$ is triangle-free, we must have $w_1=w_3$ or $w_2=w_4$ .",
"In either case $K_{2,3}$ is a subgraph of $G$ and we have a NO-instance by Lemma REF .",
"Therefore we may assume that $|V_8|=8$ .",
"We now prove the following claims.",
"Claim A: If $(G,a^*_{},d^*_{})$ is a YES-instance and $G-D+A$ is a solution, then at least one of the following options holds.",
"(a): $v_i w_i \\in D$ for all $i=1,2,3,4$ and no edge on $C$ belongs to $D$ .",
"(b): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_1 v_2, v_3 v_4 \\in D$ and $v_2 v_3, v_4 v_1 \\notin D$ .",
"(c): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_2 v_3, v_4 v_1 \\in D$ and $v_1 v_2, v_3 v_4 \\notin D$ .",
"Furthermore, for every such 4-cycle in $G$ either zero or two edges of the cycle belong to $D$ .",
"Proof of Claim A: Let $C^{\\prime }$ be any 4-cycle in $G$ .",
"If one edge $e \\in E(C^{\\prime })$ belongs to $D$ , then at least two edges of $C^{\\prime }$ must belong to $D$ , as otherwise $C^{\\prime } - e$ is a $P_4$ in $G-D$ , which is not part of a 4-cycle in $G-D$ , a contradiction by Lemma REF .",
"As each vertex is adjacent to at most one edge from $D$ , we note that we cannot have more than two edges from $C^{\\prime }$ in $D$ .",
"Option (a) now corresponds to the case when no edge from $C$ belongs to $D$ and Options (b) and (c) correspond to the two possible ways that we can add two (non-adjacent) edges of $C$ to $D$ .",
"This completes the proof of Claim A.",
"($\\diamond $ ) Claim B: We may assume that $d^*_{}(v_i)=a^*_{}(v_i)=1$ for all $i=1,2,3,4,$ as otherwise we can reduce our instance or solve it in polynomial time.",
"Proof of Claim B: If $d^*_{}(v_i)=0$ for some $i\\in [4]$ , then none of the options in Claim A is possible, so we have a NO-instance.",
"So we may assume that $d^*_{}(v_i)=1$ for all $i\\in [4]$ .",
"For the sake of contradiction, assume that $a^*_{}(v_1)=0$ and $(G,a^*_{},d^*_{})$ is a YES-instance with a solution $G-D+A$ .",
"In this case Option (a) of Claim A cannot hold as we cannot add the edge $v_1 v_3$ to $C$ .",
"Option (b) of Claim A cannot hold as we cannot add the edge $w_1 v_4$ to $G-D$ .",
"Also, Option (c) of Claim A cannot hold as we cannot add the edge $w_1 v_2$ to $G-D$ .",
"So in this case we have a NO-instance.",
"Analogously, we may assume that $a^*_{}(v_i)=1$ for all $i\\in [4]$ , which completes the proof of Claim B.",
"($\\diamond $ ) Let $e_1 = w_1 w_2$ , $e_2 = w_2 w_3$ , $e_3 = w_3 w_4$ and $e_4 = w_4 w_1$ be edges that may or may not exist in $G$ .",
"We now prove the following claims.",
"Claim C: We may assume that $e_1 \\notin E(G)$ and that Option (c) in Claim A cannot occur in any solution of $(G,a^*,d^*)$ , as otherwise we can reduce our instance or solve it in polynomial time.",
"Furthermore, if we know that there is no solution of $(G,a^*,d^*)$ for which Option (b) of Claim A occurs, then we can reduce or solve our instance in polynomial time.",
"Also, if we know that there is no solution of $(G,a^*,d^*)$ for which Option (a) of Claim A occurs, then we can reduce or solve our instance in polynomial time.",
"Proof of Claim C: We may assume that $G$ is connected; otherwise consider the connected component containing $V_8.$ By renaming vertices we may assume that $e_1 \\notin E(G)$ , unless $e_i \\in E(G)$ for all $i \\in [4]$ .",
"So assume that $e_i \\in E(G)$ for all $i \\in [4]$ , which, as $G$ is connected and of maximum degree three, implies that $V(G)=V_8$ .",
"In this case we can determine if we have a YES-instance (which is the case if and only if $d^*_{}(v)=a^*_{}(v)=1$ for all $v \\in V(G)$ ) or NO-instance in constant time, and so we would be done.",
"So we may indeed assume that $e_1 \\notin E(G)$ .",
"If Option (c) in Claim A occurs, then $w_1 v_1 v_2 w_2$ is a $P_4$ in $G-D$ where $w_1w_2 \\notin E(G)$ , which is not possible by Lemma REF , so Option (c) in Claim A cannot occur.",
"By Lemma REF , we may assume that, in the rest of the proof, $(G,a^*,d^*)$ is a YES-instance.",
"As we may assume that Option (c) in Claim A cannot occur, either Option (a) or Option (b) of Claim A must occur for every solution of $(G,a^*_{},d^*_{})$ .",
"First assume that we can show that Option (b) does not occur, and therefore Option (a) must occur in all solutions.",
"In this case, $w_iv_i \\in D$ for all $i \\in [4]$ and therefore $V(C)$ is a R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .",
"Now assume that Option (a) cannot occur which implies that Option (b) must occur in all solutions.",
"This implies that all edges between $V_8$ and $V(G) \\setminus V_8$ must belong to $D$ .",
"Thus, either $V(G)=V_8$ in which case we are done or $V_8$ is an R-deletable set, so we can reduce $G$ to $G-V_8$ by Lemma REF .",
"This completes the proof of Claim C.($\\diamond $ ) Figure: Graph (a) is an illustration of Claim D in the proof of Lemma , while Graph (b) is an illustration of Claim F.Claim D: We may assume that $e_2,e_4 \\in E(G)$ and $e_1,e_3 \\notin E(G),$ as otherwise we can reduce our instance or solve it in polynomial time.",
"(See Figure REF (a) for an illustration.)",
"We may also assume that $d^*_{}(w_i)=a^*_{}(w_i)=1$ for all $i \\in \\lbrace 1,2,3,4\\rbrace $ .",
"Proof of Claim D: By Claim C we may assume that $e_1 \\notin E(G)$ .",
"For the sake of contradiction assume that $e_2 \\notin E(G)$ .",
"By Claim C we note that part (c) in Claim A cannot occur.",
"Analogously, if Option (b) in Claim A occurs, then $w_2 v_2 v_3 w_3$ is a $P_4$ in $G-D$ where $w_2w_3 \\notin E(G)$ , which is not possible.",
"So, Option (b) in Claim A can also not occur and we are done by Claim C. Hence, we may assume that $e_2 \\in E(G)$ .",
"The fact that $e_4 \\in E(G)$ can be proved analogously, so we may assume that $e_2,e_4 \\in E(G)$ .",
"For the sake of contradiction assume that $e_3 \\in E(G)$ .",
"In this case Option (a) in Claim A cannot occur, as $w_2 w_3 w_4 w_1$ would then be a $P_4$ in $G-D$ where $w_1 w_2 \\notin E(G-D)$ , a contradiction.",
"So, again we are done by Claim C. So we may assume that $e_3 \\notin E(G)$ , as otherwise we either reduce our instance or solve it in constant time.",
"This completes the proof of the first part of Claim D. In order to prove the second part we consider the following two cases.",
"First assume that $d^*_{}(w_i)=0$ for some $i \\in [4]$ .",
"Then Option (a) in Claim A cannot occur, so by Claim C, Option (b) must occur.",
"As above we now see that we either can solve our instance in constant time or $V_8$ is a R-deletable set, so we can reduce $G$ to $G-V_8$ by Lemma REF .",
"Hence, we may assume that $d^*_{}(w_i)=1$ for all $i \\in [4]$ .",
"Now assume that $a^*_{}(w_i)=0$ for some $i \\in [4]$ .",
"In this case Option (b) in Claim A cannot occur, so by Claim C, Option (a) must occur.",
"In this case $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .",
"So we may assume that $a^*_{}(w_i)=1$ for all $i \\in [4]$ .",
"This completes the proof of Claim D.($\\diamond $ ) Definition: If $d(w_1)=3$ then let $z_1$ be defined such that $N(w_1)=\\lbrace z_1,v_1,w_4\\rbrace $ .",
"If $d(w_2)=3$ then let $z_2$ be defined such that $N(w_2)=\\lbrace z_2,v_2,w_3\\rbrace $ .",
"If $d(w_3)=3$ then let $z_3$ be defined such that $N(w_3)=\\lbrace z_3,v_3,w_2\\rbrace $ .",
"If $d(w_4)=3$ then let $z_4$ be defined such that $N(w_4)=\\lbrace z_4,v_4,w_1\\rbrace $ .",
"Note that the $z_i$ 's may not be distinct and we may have $z_i=w_j$ for some $i$ and $j$ .",
"But, by Claim D we have $z_1 \\ne w_2$ , $z_2 \\ne w_1$ , $z_3 \\ne w_4$ and $z_4 \\ne w_3$ .",
"Claim E: We may assume that if $z_i$ exists then $z_i \\notin V_8$ for all $i=1,2,3,4$ and if $z_i$ and $z_j$ both exist with $i \\ne j$ , then $z_i \\ne z_j$ .",
"Otherwise, we can reduce our instance or solve it in polynomial time.",
"Proof of Claim E: For the sake of contradiction assume that $z_1$ exists and $z_1 \\in V_8$ .",
"By definition there exists a $j \\in [4]$ such that $z_i = w_j$ .",
"By Claim D, we must have $j=3$ (and therefore $z_1=w_3$ and $z_3=w_1$ ).",
"First consider the case when $z_2$ exists and $z_2=w_4$ .",
"In this case, as $G$ is connected, we have $V(G)=V_8$ and we can determine if $(G, a^*,d^*)$ is a YES-instance or a NO-instance in constant time.",
"So we may assume that $w_2w_4 \\notin E(G)$ .",
"Option (a) in Claim A cannot occur, as $w_4 w_1 w_3 w_2$ would then be a $P_4$ in $G-D$ where $w_2 w_4 \\notin E(G-D)$ , a contradiction.",
"So, by Claim C, we are done in this case.",
"Analogously we can show that $z_i \\notin V_8$ for all $i=1,2,3,4$ (or we can reduce our instance or solve it in polynomial time).",
"Now for the sake of contradiction assume that $z_i$ and $z_j$ both exist and $i \\ne j$ but $z_i=z_j$ .",
"Option (b) in Claim A cannot occur, as in this case $w_i z_i \\in D$ and $w_j z_j \\in D$ , but $z_i$ can be incident with at most one edge from $D$ .",
"So, we are done by Claim C. This completes the proof of Claim E.($\\diamond $ ) Figure: The reduction in Claim G of Lemma .Claim F: If $z_1$ and $z_4$ both exist then $z_1 z_4 \\in E(G)$ .",
"Otherwise, we can reduce our instance or solve it in polynomial time.",
"Analogously, we may assume that if $z_2$ and $z_3$ both exist then $z_2 z_3 \\in E(G)$ .",
"Proof of Claim F: For the sake of contradiction assume that $z_1$ and $z_4$ both exist, but $z_1 z_4 \\notin E(G)$ (see Figure REF (b)).",
"By Claim C, either Option (a) or Option (b) in Claim A must occur.",
"Assume that Option (a) in Claim A occurs.",
"This implies that $v_i w_i \\in D$ for all $i \\in [4]$ .",
"Hence, $z_1 w_1 w_4 z_4$ is a $P_4$ in $G-D$ where $z_1 z_4 \\notin E(G)$ , a contradiction.",
"So Option (b) in Claim A must occur.",
"We are now done by Claim C. This completes the proof of Claim F.($\\diamond $ ) Claim G: If $z_1$ , $z_3$ and $z_4$ all exist, then we may assume that $z_3 z_4 z_1$ is a path in $G$ .",
"Otherwise, we can reduce our instance or solve it in polynomial time.",
"Analogously if $z_1$ , $z_2$ and $z_4$ all exist, then we may assume that $z_4 z_1 z_2$ is a path in $G$ .",
"And if $z_2$ , $z_3$ and $z_4$ all exist, then that $z_2 z_3 z_4$ is a path in $G$ .",
"And finally if $z_1$ , $z_2$ and $z_3$ all exist, then that $z_1 z_2 z_3$ is a path in $G$ .",
"Proof of Claim G: Assume that $z_1$ , $z_3$ and $z_4$ exist.",
"By Claim F we may assume that $z_1 z_4 \\in E(G)$ .",
"For the sake of contradiction assume that $z_3 z_4 \\notin E(G)$ .",
"Let $G^+$ be obtained from $G$ by deleting the vertices $v_1,v_2,v_3,v_4$ and adding the edges $w_1 w_2$ and $w_3 w_4$ (the vertex weights $d^*_{}$ and $a^*_{}$ remain unchanged).",
"See Figure REF for an illustration.",
"We will now show that $G^+$ is a YES-instance if and only if $G$ is a YES-instance.",
"First assume that $(G,a^*,d^*)$ is a YES-instance.",
"By Claim C we note that Option (a) or Option (b) in Claim A must occur.",
"If Option (a) occurs then $v_i w_i \\in D$ for all $i \\in [4]$ in $G$ .",
"In $G^+$ we use exactly the same $D$ , except we remove $v_1w_1, v_2w_2, v_3w_3, v_4w_4$ and add $w_1w_2$ and $w_3 w_4$ .",
"This way we get exactly the same components in $G$ and $G^+$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G$ .",
"Now assume that Option (b) occurs in Claim A.",
"In this case we use the same $D$ in $G^+$ and in $G$ , except we remove the edges $v_1 v_2$ and $v_3 v_4$ .",
"Now we get exactly the same components in $G$ and $G^+$ , except the 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ have been replaced by $w_1 w_2 w_3 w_4 w_1$ in $G^+$ .",
"So in all cases $G^+$ is a YES-instance.",
"Conversely assume that $G^+$ is a YES-instance.",
"We will now show that $G$ is a YES-instance.",
"Let $D^+$ be the matching in $G^+$ such that all components in $G^+-D^+$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).",
"Let $C^+$ be the 4-cycle $w_1 w_2 w_3 w_4 w_1$ in $G^+$ .",
"As in the proof of Claim A we note that $D^+$ contains no edge from $C^+$ or it contains exactly two edges from $C^+$ , which are non-adjacent.",
"Note that $w_2w_3, w_4w_1 \\in D^+$ is not possible as the path $z_4 w_4 w_3 z_3$ would be a $P_4$ in $G^+ - D^+$ and $z_3 z_4 \\notin E(G)$ .",
"So either $w_1w_2, w_3w_4 \\in D^+$ or no edge from $C^+$ is in $D^+$ .",
"First assume that $w_1w_2, w_3w_4 \\in D^+$ .",
"Then $z_1 w_1 w_4 z_4 z_1$ is a $C_4$ in $G^+ - D^+$ and $w_2 w_3 z_3$ is a $P_2$ in $G^+ - D^+$ (which may be part of a $C_4$ if $z_2$ exists).",
"In this case let $D = D^+ \\cup \\lbrace v_1 w_1, v_2w_2, v_3w_3, v_4w_4\\rbrace \\setminus \\lbrace w_1w_2, w_3w_4\\rbrace $ and note that we obtain exactly the same components in $G$ and $G^+$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G$ .",
"So we now consider the case when no edge from $C^+$ is in $D^+$ .",
"In this case $w_1 z_1, w_3 z_3, w_4 z_4 \\in D^+$ .",
"Let $D = D^+ \\cup \\lbrace v_1 v_2, v_3 v_4\\rbrace $ and note that we obtain exactly the same components in $G$ and $G^+$ , except the 4-cycle $w_1 w_2 w_3 w_4 w_1$ in $G^+$ has been replaced by the 4-cycles $w_1 v_1 v_4 w_4 w_1$ and $w_2 v_2 v_3 w_3 w_2$ in $G$ .",
"We have now proven that $G^+$ is a YES-instance if and only if $G$ is a YES-instance.",
"This completes the proof of Claim G.($\\diamond $ ) Claim H: We may assume that at most one of $z_1$ and $z_4$ exist.",
"Otherwise, we can reduce our instance or solve it in polynomial time.",
"Similarly, at most one of $z_2$ and $z_3$ exist.",
"Proof of Claim H: Assume that both $z_1$ and $z_4$ exist.",
"We first consider the case when both $z_2$ and $z_3$ also exist.",
"By Claim G, we have that $z_1 z_2 z_3 z_4 z_1$ is a 4-cycle in $G$ and therefore $G$ has 12 vertices, and we can determine if it is a YES-instance or a NO-instance in constant time.",
"So we may assume that either none of $z_2$ and $z_3$ exist or exactly one of them exist.",
"First assume that $z_3$ exists, but $z_2$ does not exist.",
"By Claim G, $z_1 z_4, z_3 z_4 \\in E(G)$ .",
"Let $V_{11}=V_8 \\cup \\lbrace z_1,z_3,z_4\\rbrace $ .",
"If $V(G)=V_{11}$ then we can solve our instance in constant time so assume that this is not the case.",
"We consider the cases when there exists a $w \\in V(G) \\setminus V_{11}$ with $\\lbrace z_1,z_3\\rbrace \\subseteq N(w)$ and when no such $w$ exists.",
"If such a $w$ exists then Option (a) in Claim A cannot occur, as if it did both the edges $wz_1$ and $wz_3$ would need to be in $D$ (as $z_1 w_1 w_4 z_4 z_1$ would be a $C_4$ in $G-D$ and $w_2 w_3 z_3$ would be a $P_3$ in $G-D$ ), a contradiction.",
"So, we are done by Claim C. We may therefore now assume that no such $w$ exists.",
"As $G$ is connected and $V(G) \\ne V_{11}$ we must have a vertex $r \\in V(G) \\setminus V_{11}$ which is adjacent to either $z_1$ or to $z_3$ .",
"Now Option (b) of Claim A cannot occur, as if it did then the edges $w_1z_1$ , $w_3z_3$ and $w_4z_4$ would need to be in $D$ , which implies that either $r z_1 z_4 z_3$ or $z_1 z_4 z_3 r$ is a $P_4$ in $G-D$ where the end-points of the path are not adjacent in $G$ (as no $w$ exists in $V(G) \\setminus V_{11}$ that is adjacent to both $z_1$ and $z_3$ ).",
"So, again we are done by Claim C. This completes all cases and implies that we can reduce when $z_3$ exists, but $z_2$ does not exist.",
"Figure: The graph when z 2 z_2 and z 3 z_3 do not exist.Analogously we can also reduce if $z_2$ exists, but $z_3$ does not exist.",
"So the last case to consider is when neither $z_2$ nor $z_3$ exist.",
"In this case the graph looks like in Figure REF .",
"If we had considered the cycle $w_1 v_1 v_4 w_4 w_1$ instead of $v_1 v_2 v_3 v_4 v_1$ , then we would be in the case where $G$ has 10 vertices (when $d(z_1)=d(z_4)=2$ ) or we would have been in a case where at least three $z$ -vertices would exist (namely $\\lbrace w_2,w_3\\rbrace \\cup {N(z_1)\\cup N(z_4)}\\setminus \\lbrace z_1,w_1,z_4,w_4\\rbrace $ ), which means we can reduce as seen above.",
"This completes the proof of Claim H.($\\diamond $ ) We now return to the proof of Lemma REF .",
"By Claim H we may assume that at most one of $z_1$ and $z_4$ exist and at most one of $z_2$ and $z_3$ exist.",
"If no $z_i$ exist then $G$ has 8 vertices, so we can solve our problem in constant time.",
"Therefore we may, without loss of generality assume that $z_1$ exists.",
"If $z_2$ exists then let $z=z_2$ and if $z_3$ exists then let $z=z_3$ and if neither $z_2$ or $z_3$ exist then let $z$ be undefined.",
"We now create $G^*$ from $G$ by removing the vertices $w_1, w_2, w_3, w_4$ and adding the edge $v_1 z_1$ and if $z$ exists also the edge $v_3 z$ .",
"We will show that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.",
"First assume that $G$ is a YES-instance.",
"By Claim C we note that Option (a) or Option (b) in Claim A must occur.",
"If Option (a) occurs then $v_i w_i \\in D$ for all $i \\in [4]$ in $G$ .",
"Let $D^* = D \\cup \\lbrace v_1 v_2, v_3 v_4\\rbrace \\setminus \\lbrace v_1w_1, v_2w_2, v_3w_3, v_4w_4 \\rbrace $ .",
"This way we get exactly the same components in $G$ and $G^*$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ , the $P_3=z_1w_1w_4$ and the $P_2$ or $P_3$ on the vertex set $\\lbrace w_2, w_3\\rbrace $ or $\\lbrace w_2, w_3, z\\rbrace $ in $G$ get replaced by the $P_3=z_1 v_1 v_4$ and either the $P_2=v_2 v_3$ (if $z$ does not exist) or the $P_3=v_2 v_3 z$ in $G^*$ .",
"Now assume that Option (b) occurs in Claim A.",
"In this case $z_1 w_1 \\in D$ and if $z$ exists then $w_2 z$ or $w_3z$ is in $D$ .",
"Also, we let $D^* = D - \\lbrace v_1 v_2, v_3 v_4\\rbrace $ , where we also replace $z_1w_1$ by $z_1v_1$ and if $z$ exists then the edge $w_2 z$ or $w_3z$ gets replaced by $v_3z$ .",
"Now we get exactly the same components in $G-D$ and $G^* - D^*$ , except the 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ have been replaced by $v_1 v_2 v_3 v_4 v_1$ in $G^*$ .",
"So in all cases $G^*$ is a YES-instance.",
"Conversely assume that $G^*$ is a YES-instance.",
"We will now show that $G$ is a YES-instance.",
"Let $D^*$ be the matching in $G^*$ such that all components in $G^* - D^*$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).",
"As in the proof of Claim A we note that $D^*$ contains no edge from $C$ or it contains exactly two edges from $C$ , which are non-adjacent.",
"First assume that $D^*$ contains no edge from $C$ .",
"Then $z_1 v_1 \\in D^*$ and if $z$ exists then $z v_3 \\in D$ .",
"In this case, let $D$ be obtained from $D^* + \\lbrace v_1v_2, v_3v_4,z_1 w_1\\rbrace - \\lbrace z_1 v_1\\rbrace $ by replacing $v_3 z$ by the edge from $\\lbrace w_2,w_3\\rbrace $ to $z$ if $z$ exists.",
"Again, we obtain exactly the same components in $G-D$ and $G^* - D^*$ , except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G^*$ has been replaced by the two 4-cycles $v_1 v_4 w_4 w_1 v_1$ and $v_2 v_3 w_3 w_2 v_2$ in $G$ .",
"So we now consider the case when $D^*$ contains exactly two edges from $C$ .",
"In this case $G^* - D^*$ contains a $P_3$ containing the edge $z_1 v_1$ and either the edge $v_1 v_2$ or $v_1 v_4$ .",
"Similarly, if $z$ exists, then $G^* - D^*$ contains a $P_3$ containing the edge $z v_3$ and either the edge $v_3 v_4$ or $v_3 v_2$ .",
"In this case let $D$ be obtained from $D^*$ by deleting these two edges on $C$ that belong to $D^*$ and adding the edges $\\lbrace v_1w_1, v_2w_2, v_3w_3, v_4w_4 \\rbrace $ instead.",
"This way we obtain exactly the same components in $G-D$ and $G^* - D^*$ , except for the following: The $P_3$ in $G^* - D^*$ containing $v_1$ now becomes the $P_2$ $z_1 w_1 w_4$ and the $P_3$ containing $z$ (if it exists) now becomes a $P_3$ containing the edge $w_2 w_3$ as well as the vertex $z$ .",
"Furthermore $G-D$ will contain the 4-cycle $C$ , which is not in $G^* - D^*$ .",
"This proves that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.",
"Let ${\\cal G}_4$ denote all triangle-free graphs of maximum degree at most 3 which contain no 4-cycle, with at least three vertices of degree three.",
"Lemma 6 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_3$ to ${\\cal G}_4$ in polynomial time.",
"Let $G \\in {\\cal G}_3$ be arbitrary with vertex-weights $(a^*,d^*)$ .",
"If there is no 4-cycle in $G$ where three of the vertices have degree three then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be a 4-cycle in $G$ with $d(v_i)=3$ for all $i=1,2,3$ and $d(v_4)=2$ .",
"Let $w_i$ be the neighbour of $v_i$ in $G$ which does not lie on $C$ for $i=1,2,3$ .",
"Let $V_7=\\lbrace v_1,v_2,v_3,v_4,w_1,w_2,w_3\\rbrace $ .",
"We will show that $|V_7|=7$ (or we have a NO-instance), so assume for the sake of contradiction that this is not the case.",
"As $G$ is triangle-free, we must have $w_1=w_3$ , which implies that $K_{2,3}$ is a subgraph of $G$ and we have a NO-instance by Lemma REF .",
"Therefore we may assume that $|V_7|=7$ .",
"The following claim can be proved analogously to Claim A in Lemma REF and thus its proof is omitted.",
"Claim A: If $(G,a^*_{},d^*_{})$ is a YES-instance and $G-D+A$ is a solution.",
"Then one of the following options hold.",
"(a): $v_i w_i \\in D$ for all $i=1,2,3$ and no edge on $C$ belongs to $D$ .",
"(b): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_1 v_2, v_3 v_4 \\in D$ and $v_2 v_3, v_4 v_1 \\notin D$ .",
"(c): $v_i w_i \\notin D$ for all $i=1,2,3,4$ and $v_2 v_3, v_4 v_1 \\in D$ and $v_1 v_2, v_3 v_4 \\notin D$ .",
"Furthermore, for every such 4-cycle in $G$ either zero or two edges of the cycle belong to $D$ .",
"Claim B: We may assume that $d^*_{}(v_i)=1$ for all $i=1,2,3,4$ .",
"Otherwise, we have a NO-instance.",
"Proof of Claim B: If $d^*_{}(v_i)=0$ for some $i\\in [4]$ , then none of the options in Claim A is possible, so we have a NO-instance.",
"This proves Claim B.",
"($\\diamond $ ) Claim C: If Option (b) in Claim A occurs, then we must have $w_2 w_3 \\in E(G)$ .",
"If Option (c) occurs in Claim A, then we must have $w_1 w_2 \\in E(G)$ .",
"Proof of Claim C: For the sake of contradiction, assume that Option (b) occurs in Claim A, but $w_2 w_3 \\notin E(G)$ .",
"In this case $w_2 v_2 v_3 w_3$ is a $P_4$ in $G-D$ where $w_2 w_3 \\notin E(G)$ , a contradiction.",
"The fact that if Option (c) occurs in Claim A, then we must have $w_1 w_2 \\in E(G)$ is proved analogously.",
"($\\diamond $ ) Claim D: We may assume that exactly one of the edges $w_1w_2$ and $w_2w_3$ belongs to $G$ .",
"Proof of Claim D: If $w_1w_2 \\notin E(G)$ and $w_2w_3 \\notin E(G)$ then, by Claim C, neither Option (b) nor Option (c) occurs in Claim A.",
"Therefore Option (a) must occur in Claim A.",
"This implies that $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .",
"We may therefore assume that at least one of $w_1w_2$ and $w_2w_3$ belong to $G$ .",
"Assume that $w_1w_2 \\in E(G)$ and $w_2w_3 \\in E(G)$ .",
"As $G \\in {\\cal G}_3$ we note that the two 4-cycles $w_1 v_1 v_2 w_2 w_1$ and $w_2 v_2 v_3 w_3 w_2$ both must contain a degree-2 vertex.",
"This implies that $d(w_1)=d(w_3)=2$ and $V(G)=V_7$ .",
"We can therefore solve the problem in constant time.",
"We may therefore assume that one of the edges $w_1w_2$ and $w_2w_3$ do not belong to $G$ .",
"This proves Claim D.($\\diamond $ ) By Claim D, we may without loss of generality assume that $w_1w_2 \\in E(G)$ and $w_2w_3 \\notin E(G)$ .",
"Claim E: Option (b) in Claim A cannot occur.",
"Furthermore, we may assume that $d(w_1) \\le 2$ or $d(w_2) \\le 2$ .",
"We may also assume that $d(w_3) \\le 2$ as otherwise we can either solve our problem in polynomial time or reduce the instance.",
"Proof of Claim E: Since $w_2w_3 \\notin E(G)$ by Claim C, Option (b) in Claim A cannot occur.",
"Note that $d(w_1)=d(w_2)=3$ is not possible as $G \\in {\\cal G}_3$ and $d(v_1)=d(v_2)=3$ .",
"For the sake of contradiction assume that $d(w_3)=3$ and that $N(w_3)=\\lbrace v_3,a,b\\rbrace $ .",
"Recall that Option (b) in Claim A cannot occur.",
"Assume that Option (c) in Claim A occurs.",
"This implies that either $v_4 v_3 w_3 a$ or $v_4 v_3 w_3 b$ is a $P_4$ in $G-D$ where the end-points are not adjacent in $G$ , a contradiction.",
"So Option (c) in Claim A cannot occur, which implies that Option (a) in Claim A must occur.",
"This implies that $V(C)$ is an R-deletable set, so we can reduce $G$ to $G-V(C)$ by Lemma REF .",
"This proves Claim E.($\\diamond $ ) We now return to the proof of Lemma REF .",
"If $d(w_1)=3$ then let $s=1$ and otherwise let $s=2$ .",
"By Claim E we note that $d(w_{3-s})=2$ .",
"Let $G^*$ be obtained from $G-\\lbrace v_1,v_2,v_3,v_4,w_{3-s}\\rbrace $ by adding a vertex $x$ and the edge $w_s w_3$ and $w_s x$ and setting $d^*_{}(x)=0$ and $a^*_{}(x)=1$ .",
"See Figure REF for an illustration.",
"We will now show that $G^*$ is a YES-instance if and only if $G$ is a YES-instance.",
"Figure: An illustration of G * G^* in the proof of Lemma when s=2s=2.First assume that $G$ is a YES-instance.",
"By Claim E we note that Option (a) or Option (c) in Claim A must occur.",
"If Option (a) occurs then $v_i w_i \\in D$ for all $i \\in [3]$ in $G$ .",
"Let $D^* = D \\cup \\lbrace w_s w_3 \\rbrace - \\lbrace v_1 w_1, v_2 w_2, v_3 w_3\\rbrace $ .",
"Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 v_2 v_3 v_4 v_1$ in $G-D$ and the component containing $w_1 w_2$ have been replaced by a component containing $x w_s$ in $G^* - D^*$ .",
"However the component containing $w_1 w_2$ in $G-D$ is isomorphic with the component containing $x w_s$ in $G^*-D^*$ .",
"So we get a solution for $G^*$ .",
"See Figure REF for an illustration.",
"Figure: An illustration of G-DG-D and G * -D * G^* - D^* when part (a) of Claim A occurs.If Option (c) occurs then $v_2 v_3, v_4 v_1 \\in D$ .",
"Let $D^* = D - \\lbrace v_2 v_3, v_4 v_1\\rbrace $ .",
"Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 w_1 w_2 v_2 v_1$ and the path $v_4 v_3 w_3$ in $G-D$ have been replaced by the path $x w_s w_3$ in $G^* - D^*$ .",
"So we get a solution for $G^*$ .",
"See Figure REF for an illustration.",
"Figure: An illustration of G-DG-D and G * -D * G^* - D^* when part (b) of Claim A occurs.Conversely assume that $G^*$ is a YES-instance.",
"We will now show that $G$ is a YES-instance.",
"Let $D^*$ be the matching in $G^*$ such that all components in $G^*-D^*$ are $P_1$ , $P_2$ , $P_3$ or $C_4$ (and the $P_3$ 's and $C_4$ 's can be made into cliques by adding a matching of addable edges).",
"As $d^*_{}(x)=0$ in $G^*$ we note that $x w_s \\notin D^*$ .",
"We now consider the case when $w_s w_3$ belongs to $D^*$ and then the case when it does not.",
"So first assume that $w_s w_3 \\in D^*$ .",
"In this case let $D = D^* + \\lbrace v_1w_1, v_2 w_2, v_3 w_3 \\rbrace - \\lbrace w_s w_3\\rbrace $ .",
"Now $G-D$ and $G^* - D^*$ have exactly the same components except the 4-cycle $v_1 v_4 w_4 w_1 v_1$ is in $G-D$ and the component containing $x w_s$ have been replaced by an isomorphic component containing $w_1 w_2$ in $G^* - D^*$ .",
"So we get a solution for $G$ (this corresponds to the reverse process to that illustrated in Figure REF ).",
"Now assume that $w_s w_3 \\notin D^*$ .",
"This implies that any edges incident with $w_s$ and $w_3$ , different from $w_s w_3$ and $x w_s$ must belong to $D^*$ (due to the path $x w_s w_3$ in $G^* - D^*$ ).",
"Let $D = D^* + \\lbrace v_2v_3, v_4 v_1 \\rbrace $ .",
"Now $G-D$ and $G^* - D^*$ have exactly the same components except the path $x w_s w_3$ in $G^* - D^*$ has been replaced by the 4-cycle $v_1 w_1 w_2 v_2 v_1$ and the path $v_4 v_3 w_3$ in $G-D$ .",
"So we get a solution for $G$ (this corresponds to the reverse process to that illustrated in Figure REF ).",
"The above reduction completes the proof of the lemma.",
"Let ${\\cal G}_5$ denote all triangle-free and quadrilateral-free graphs of maximum degree at most 3.",
"Lemma 7 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_4$ to ${\\cal G}_5$ in polynomial time.",
"Let $G \\in {\\cal G}_4$ be arbitrary with vertex-weights $(a^*,d^*)$ .",
"We may assume that $G$ is connected.",
"If there is no 4-cycle in $G$ then we are done, so let $C = v_1 v_2 v_3 v_4 v_1$ be any 4-cycle in $G$ .",
"As $G \\in {\\cal G}_4$ , we know that at most two of the vertices in $C$ have degree three.",
"If no vertex of $G$ has degree three then $V(G)=V(C)$ and we can solve our problem in constant time.",
"So we may assume that at least one vertex on $C$ has degree three.",
"If $d^*_{}(v_i)=0$ for any $i \\in [4]$ then any solution $G-D+A$ to our instance must have all edges between $C$ and $V(G)-V(C)$ in $D$ , which implies that $V(C)$ is an R-deletable set.",
"So by Lemma REF , we can reduce the instance in this case.",
"Thus, we may assume that $d^*_{}(v_i)=1$ for all $i \\in [4]$ .",
"We first consider the case when exactly one vertex on $C$ has degree three.",
"Without loss of generality assume that $d(v_1)=3$ and $d(v_2)=d(v_3)=d(v_4)=2$ .",
"Let $e$ be the edge incident with $v_1$ that does not belong to $C$ .",
"Let $G^*$ be obtained from $G^*=G-\\lbrace v_3,v_4\\rbrace $ by letting $d^*_{}(v_2)=0$ .",
"Now $G^*$ is a YES-instance if and only if $G$ is a YES-instance, due to the following.",
"Deleting $e$ from $G$ corresponds to deleting $e$ from $G^*$ and vice versa.",
"Not deleting $e$ from $G$ corresponds to deleting two edges from $C$ in $G$ which corresponds to not deleting $e$ in $G^*$ .",
"So $G$ is a YES-instance if and only if $G^*$ is a YES-instance.",
"We may therefore assume that exactly two vertices on $C$ have degree 3.",
"Let $x$ and $y$ be the two vertices on $C$ of degree 3.",
"Let $x^{\\prime }$ be the neighbour of $x$ not on $C$ and let $y^{\\prime }$ be the neighbour of $y$ not on $C$ .",
"Analogously to Claim A in both Lemma REF and Lemma REF we note that if our instance has a solution and $D$ is the set of edges that are deleted from $G$ in the solution, then one of the following holds.",
"(a): $x x^{\\prime }, y y^{\\prime } \\in D$ and no edge on $C$ belongs to $D$ .",
"(b): $x x^{\\prime }, y y^{\\prime } \\notin D$ and $v_1 v_2, v_3 v_4 \\in D$ and $v_2 v_3, v_4 v_1 \\notin D$ .",
"(c): $x x^{\\prime }, y y^{\\prime } \\notin D$ and $v_2 v_3, v_4 v_1 \\in D$ and $v_1 v_2, v_3 v_4 \\notin D$ .",
"First consider the case when $x$ and $y$ are adjacent vertices on $C$ .",
"Then without loss of generality assume that $x=v_1$ and $y=v_4$ .",
"As $G$ is triangle-free we have $x^{\\prime } \\ne y^{\\prime }$ .",
"First assume that $x^{\\prime } y^{\\prime } \\in E(G)$ .",
"Then $V(G)=\\lbrace v_1, v_2, v_3, v_4, x^{\\prime }, y^{\\prime }\\rbrace $ as $G \\in {\\cal G}_4$ , which implies that the cycle $x^{\\prime } x y y^{\\prime } x^{\\prime }$ ($=x^{\\prime } v_1 v_4 y^{\\prime } x^{\\prime }$ ) has at most two vertices of degree three, which are $x$ and $y$ .",
"So in this case we can solve our problem in constant time.",
"Therefore, we may assume that $x^{\\prime } y^{\\prime } \\notin E(G)$ .",
"This also implies that part (b) above is not possible, as if part (b) occurs then $x^{\\prime } v_1 v_4 y^{\\prime }$ is a $P_4$ in $G-D$ where $x^{\\prime } y^{\\prime } \\notin E(G)$ , a contradiction.",
"So either part (a) or part (c) occurs.",
"This is illustrated in Figure REF .",
"Figure: Possibble options for G-DG-D when xx and yy are adjacent.If $x$ and $y$ are not adjacent, then without loss of generality assume that $x=v_1$ and $y=v_3$ .",
"If $x^{\\prime }=y^{\\prime }$ then $G$ contains a $K_{2,3}$ -subgraph, which implies that our instance is a NO-instance by Lemma REF .",
"Thus, we may assume that $x^{\\prime } \\ne y^{\\prime }$ .",
"Now part (a), part (b) and part (c) are illustrated in Figure REF .",
"Figure: Possibble options for G-DG-D when xx and yy are not adjacent.So whether $x$ and $y$ is adjacent or not, we will in $G-D$ either have both edges $xx^{\\prime }$ and $yy^{\\prime }$ in $D$ or neither in $D$ .",
"And if neither is in $D$ then each of $x^{\\prime }$ and $y^{\\prime }$ is the endpoint of a $P_3$ containing two vertices from $C$ .",
"So in both cases we will remove $V(C)$ from $G$ and add the gadget $G(x,y)$ , illustrated in Figure REF , to $G$ instead.",
"Note that $V(G(x,y))=\\lbrace q_1, x, q_3, q_4, y, q_6 \\rbrace $ and $E(G(x,y))=\\lbrace q_1 x, x q_3, q_3 q_4, q_4 y, y q_5 \\rbrace $ and we add the edges $x x^{\\prime }$ and $y y^{\\prime }$ .",
"Furthermore $d^*_{}(q_1)=d^*_{}(q_6)=0$ and all other $d^*_{}$ -values and $a^*_{}$ -values are one.",
"Figure: The gadget G(x,y)G(x,y).It is not difficult to check that if the modified graph, $G^{\\prime }$ , is a YES-instance and $D^{\\prime }$ the matching that gets removed from $G^{\\prime }$ , then we must have one of the two cases illustrated in Figure REF .",
"Figure: The two options, (I) and (II), for G ' -D ' G^{\\prime }-D^{\\prime }.So $G^{\\prime \\prime }$ has either both edges $xx^{\\prime }$ and $yy^{\\prime }$ in $D^{\\prime }$ or neither in $D^{\\prime }$ .",
"And if neither are in $D^{\\prime }$ then both $x^{\\prime }$ and $y^{\\prime }$ are endpoints of a $P_3$ containing two vertices from the gadget $G(x,y)$ .",
"This was exactly the same property we had in $G$ , so $G$ is a YES-instance if and only if $G^{\\prime }$ is.",
"Furthermore, this reduction deletes a 4-cycle.",
"So even though the reduction adds an edge (that is, $|E(G^{\\prime })|=|E(G)|+1$ ) the reduction can be carried out at most as many times as there are 4-cycles in $G$ which is at most a polynomial number.",
"Therefore repeatadly performing the above reduction creates a graph in ${\\cal G}_5$ in polynomial time, as desired."
],
[
"Polynomial-time algorithm for triangle-free and quadrilateral-free graphs of maximum degree at most 3",
"Lemma REF implies the following: Lemma 8 Let $G$ be a triangle-free and quadrilateral-free graph with maximum degree at most 3.",
"Then $(G,a^*_{},d^*_{})$ is a YES-instance of $(1,1)$ -Cluster Editing if and only if there is a matching $D$ of deletable edges from $G$ such that every connected component $C$ of $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .",
"Let ${\\cal G}_6$ denote all triangle-free and quadrilateral-free graphs of maximum degree at most 3, such that the following holds.",
"All vertices $v$ of degree at least 2 have $d^*_{}(v)=1$ .",
"No vertex $v$ of degree 3 is adjacent to a vertex $w$ with $a^*_{}(w)=0$ .",
"The vertices of degree 3 form an independent set.",
"Lemma 9 $(1,1)$ -Cluster Editing can be reduced from ${\\cal G}_5$ to ${\\cal G}_6$ in polynomial time.",
"Let $G \\in {\\cal G}_5$ be arbitrary with vertex-weights $(a^*,d^*)$ .",
"Note that if some vertex $v$ has $d(v)=3$ and $d^*_{}(v)=0$ then $(G,a^*_{},d^*_{})$ is a NO-instance, so we may assume that $d^*_{}(v)=1$ for all vertices, $v$ , of degree 3.",
"If $d(v)=2$ and $d^*_{}(v)=0$ , then $(G,a^*_{},d^*_{})$ is a YES-instance if and only if both edges incident to $v$ have to remain in $G-D$ and therefore form a $P_3.$ Let $N(x)=\\lbrace u,v\\rbrace ,$ and note that $\\lbrace x,u,v\\rbrace $ is an R-deletable set.",
"So by Lemma REF , we can reduce the instance.",
"Assume that there exists a $uw \\in E(G)$ with $d(u)=3$ and $a^*_{}(w)=0$ .",
"Assume that $G$ is a YES-instance and $G-D+A$ is a cluster graph.",
"If $uw \\notin D$ , then $w$ is the endpoint of a $P_3$ in $G-D$ which is a contradiction as $a^*_{}(w)=0$ .",
"So, we must have $uw \\in D$ .",
"Since $uw \\in D$ , $d^*_{}(u)=d^*_{}({w})=1$ or $(G,a^*,d^*)$ is a NO-instance.",
"Form a new instance with $G^{\\prime }=G- uw$ and $d^*_{}(u)=d^*_{}({w})=0$ .",
"Then $G$ is a YES-instance if and only if $G^{\\prime }$ is a YES-instance.",
"This reduction allows us to get rid of all cases where a degree three vertex is adjacent to a vertex with $a^*_{}$ -value zero.",
"Now assume that there exists a $uw \\in E(G)$ with $d(u)=d(w)=3$ .",
"Assume that $G$ is a YES-instance and $G-D+A$ is a cluster graph.",
"If $uw \\notin D$ , then we obtain a $P_4$ in $G-D$ containing the edge $uw$ , a contradiction.",
"So, we must have $uw \\in D$ .",
"As $d(u)=d(w)=3$ , we have $d^*(u)=d^*(w)=1$ .",
"Now we form a new instance with $G^{\\prime }=G- uw$ and $d^*_{}(u)=d^*_{}(v)=0$ as above and note that $G$ is a YES-instance if and only if $G^{\\prime }$ is a YES-instance.",
"Thus, we may assume that $uw \\notin E(G)$ if $d(u)=d(w)=3$ .",
"This completes the proof.",
"In the proof of the next lemma, we will use the well-known result that given a graph $G$ in polynomial time either a perfect matching can be found in $G$ or we can decide that $G$ has no perfect matching.",
"Lemma 10 Let $G$ be a graph and let $Y$ and $Z$ be disjoint sets of vertices in $G$ .",
"In polynomial time, we can decide whether $G$ contains a matching covering $Y$ , but not covering $Z$ .",
"Let $G^{\\prime }=G-Z$ .",
"Observe that $G$ contains a matching covering $Y$ and not covering $Z$ if and only if $G^{\\prime }$ contains a matching covering $Y$ .",
"Let $U= V(G^{\\prime })-Y$ and if $G^{\\prime } $ has odd number of vertices then add an extra (isolated) vertex to $G^{\\prime }$ which is not in $Y$ (and therefore will be added to $U$ ).",
"Finally add edges between all pairs of non-adjacent vertices in $U$ .",
"Let the resulting graph be $G^{\\prime \\prime }.$ Now we will show that $G^{\\prime \\prime }$ has a perfect matching if and only if there is a matching in $G ^{\\prime }$ covering $Y.$ If $M^{\\prime \\prime }$ is a perfect matching in $G^{\\prime \\prime }$ then by deleting all edge with both end-points in $U$ we get the desired matching in $G^{\\prime }.$ And if $M^{\\prime }$ is the desired matching in $G^{\\prime }$ then adding as many edges as possible with both endpoints in $U$ gives us a perfect matching in $G^{\\prime \\prime }.$ Lemma 11 $(1,1)$ -Cluster Editing can be solved in polynomial time for all graphs in ${\\cal G}_6$ in polynomial time.",
"Let $G \\in {\\cal G}_6$ be arbitrary with vertex-weights $(a^*,d^*)$ .",
"Let $X_i = \\lbrace v \\; | \\; d_G(v)=i \\rbrace $ for all $i=0,1,2,3$ .",
"We may assume that $G$ is connected as otherwise we can just consider each connected component separately (it will never be advantages to add edges between different components).",
"We may clearly also assume that $G$ has at least two vertices, which implies that $X_0 = \\emptyset $ .",
"Let $X_2^{\\prime }$ denote all vertices in $X_2$ with an edge to a vertex in $X_3$ and let $X_2^{\\prime \\prime } = X_2 \\setminus X_2^{\\prime }$ .",
"Let $Z$ contain all vertices, $z$ , in $X_2^{\\prime \\prime }$ , such that $z$ is adjacent to a vertex, $w$ , with $a^*_{}(w)=0$ .",
"In other words, $Z$ contains all vertices from $X_2^{\\prime \\prime }$ which are neighbours of a vertex which we are not allowed to add edges to.",
"Define $Y$ as follows: $Y = Z \\cup X_2^{\\prime } \\cup X_3.$ We will show that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, if and only if there exists a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .",
"This will give us the desired polynomial algorithm by Lemma REF .",
"First assume that $D$ is a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .",
"We may assume that $D$ is maximal, as otherwise we just keep adding edges to $D$ (where both end-points have $d^*$ -value 1) until it becomes maximal (which is not necessarily maximum).",
"Let $G^{\\prime } = G-D$ .",
"By Lemma REF it suffices to show that every component, $C$ , in $G^{\\prime }$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .",
"As every vertex in $X_3$ belongs to $Y$ we note that $\\Delta (G^{\\prime }) \\le 2$ .",
"For the sake of contradiction assume that $uv \\in E(G^{\\prime })$ and $d_{G^{\\prime }}(u)=d_{G^{\\prime }}(v)=2$ .",
"As all vertices in $X_2^{\\prime }$ will have degree 1 in $G^{\\prime }$ we note that neither $u$ nor $v$ belong to $X_2^{\\prime }$ .",
"As all vertices in $X_3$ are only adjacent to vertices in $X_2^{\\prime }$ , by the definition of ${\\cal G}_6$ and $X_2^{\\prime }$ , we note that neither $u$ nor $v$ belong to $X_3$ .",
"So $u$ and $v$ must both belong to $X_2^{\\prime \\prime }$ .",
"By the definition of ${\\cal G}_6$ we note that $d^*_{}(u)=d^*_{}(v)=1$ .",
"However, this contradicts the fact that $D$ is maximal, as we could have added the edge $uv$ to $D$ .",
"So, no $uv \\in E(G^{\\prime })$ has $d_{G^{\\prime }}(u)=d_{G^{\\prime }}(v)=2$ .",
"This implies that all components in $G^{\\prime }$ are isomorphic to $P_1$ or $P_2$ or $P_3$ .",
"Assume that $C$ is a component in $G^{\\prime }$ isomorphic to $P_3$ and let $v_1 v_2 v_3$ be the 3-path in $C$ .",
"By the construction of $G^{\\prime }$ we note that $v_2 \\in X_3$ or $v_2 \\in X_2^{\\prime \\prime }$ .",
"If $v_2 \\in X_3$ then, by the definition of ${\\cal G}_6$ , we note that $a^*_{}(v_1)=a^*_{}(v_3)=1$ .",
"And if $v_2 \\in X_2^{\\prime \\prime }$ , then $v_2 \\notin Y$ which by the definition of $Z$ implies that $a^*_{}(v_1)=a^*_{}(v_3)=1$ .",
"So in both cases $a^*_{}(v_1)=a^*_{}(v_3)=1$ , which by Lemma REF , implies that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, as desired.",
"Now conversely assume that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing and that $D$ denotes the edges deleted from $G$ in the solution.",
"By definition $D$ is a matching in $G$ and by Lemma REF every component, $C$ , in $G-D$ is isomorphic to $P_1$ or $P_2$ or $P_3$ and if $C\\cong P_3$ then the edge of $\\overline{G}$ between the end-vertices of $C$ is addable to $G$ .",
"Let $G^* = G-D$ .",
"As $\\Delta (G^*) \\le 2$ we note that $D$ covers all vertices in $X_3$ .",
"Let $u \\in X_2^{\\prime }$ be arbitrary and let $v$ be a neighbour of $u$ in $G$ such that $v \\in X_3$ (which exists by the definition of $X_2^{\\prime }$ ).",
"Let $N(u)=\\lbrace x,v\\rbrace $ and let $N(v)=\\lbrace u,s,t\\rbrace $ , where as $G$ is triangle-free and quadrilateral-free implies that $x,v,u,s,t$ are all distinct.",
"If no edge incident with $u$ belongs to $D$ , then either $xuvs$ or $xuvt$ will be a $P_4$ in $G^*$ , a contradiction, so $u$ is covered by $D$ .",
"So all vertices in $X_2^{\\prime }$ are covered by $D$ and previously we showed that all vertices in $X_3$ are also covered by $D$ .",
"Let $z \\in Z$ be arbitrary.",
"That is $z \\in X_2^{\\prime \\prime }$ and $z$ is adjacent to a vertex, $w$ , with $a^*_{}(w)=0$ .",
"Let $N(z)=\\lbrace w,q\\rbrace $ and note that if $z$ is not covered by $D$ then $wzq$ is a $P_2$ in $G^*$ , but as $a^*_{}(w)=0$ , the edge $wq \\in \\overline{G}$ is not addable to $G$ .",
"So $z$ must be covered by $D$ .",
"This implies that $D$ is a matching covering $Y$ in $G$ .",
"Furthermore, by definition, $D$ does not cover any vertex with $d$ -value zero.",
"We have therefore shown that $(G,a^*_{},d^*_{})$ is a YES-instance to $(1,1)$ -Cluster Editing, if and only if there exists a matching in $G$ that covers all vertices in $Y$ but no vertex, $w$ , with $d^*_{}(w)=0$ .",
"Corollary 12 $(1,1)$ -Cluster Editing can be solved in polynomial time.",
"This follows immediately from Lemmas REF , REF , REF , REF , REF and REF ."
],
[
"Discussion",
"The main result of our paper concludes a complete dichotomy of complexity of $(a,d)$ -Cluster Editing.",
"We proved that $(1,1)$ -Cluster Editing can be solved in polynomial time.",
"Our proof consists of two stages: (i) providing a serious of five polynomial-time reductions to $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3, and (ii) designing a polynomial-time algorithm for solving $(1,1)$ -Cluster Editing on $C_3$ -free and $C_4$ -free graphs of maximum degree at most 3.",
"While Stage 2 is relatively short, Stage 1 is not.",
"Moreover, while our reduction from all graphs to $C_3$ -free graphs of maximum degree at most 3 is not hard, getting rid of 4-cycles required a lengthy and non-trivial series of four polynomial-time reductions.",
"It would be interesting to see whether the four reductions can be replaced by a shorter and simpler series of reductions."
]
] | 2210.07722 |
[
[
"Theory and Approximate Solvers for Branched Optimal Transport with\n Multiple Sources"
],
[
"Abstract Branched Optimal Transport (BOT) is a generalization of optimal transport in which transportation costs along an edge are subadditive.",
"This subadditivity models an increase in transport efficiency when shipping mass along the same route, favoring branched transportation networks.",
"We here study the NP-hard optimization of BOT networks connecting a finite number of sources and sinks in $\\mathbb{R}^2$.",
"First, we show how to efficiently find the best geometry of a BOT network for many sources and sinks, given a topology.",
"Second, we argue that a topology with more than three edges meeting at a branching point is never optimal.",
"Third, we show that the results obtained for the Euclidean plane generalize directly to optimal transportation networks on two-dimensional Riemannian manifolds.",
"Finally, we present a simple but effective approximate BOT solver combining geometric optimization with a combinatorial optimization of the network topology."
],
[
"Introduction",
"Optimal transport (OT) [27], [6], [23] stipulates transportation costs that increase linearly with the transported mass.",
"However, in many systems of practical and theoretical interest, a diminishing cost property is more realistic: it is more economic to jointly transport two loads with nearby destinations along the same route.",
"The optimal transportation networks under diminishing costs exhibit branching; and indeed, nature and societies are using branched networks, e.g.",
"in blood circulation, gas supply or mail delivery.",
"In this paper, we study the theory and practice of finding good or even optimal solutions in branched optimal transport (BOT).",
"More formally, we consider a finite set of sources $S$ with supplies $\\mu _S > 0$ and sinks $T$ with demands $\\mu _T > 0$ , located at fixed positions $x_S$ and $x_T$ in $\\mathbb {R}^2$ .",
"A possible transportation network is represented as a directed, edge-weighted graph $G(V,E)$ with nodes $V = S \\cup T \\cup B$ .",
"The edges $E \\subset V \\times V$ interconnect the terminals $S$ and $T$ with the help of a set of additional nodes $B$ , so-called branching points (BPs), with coordinates $x_B$ .",
"The edge direction indicates the direction of mass flow.",
"The edge weights, denoted by $m_e$ , specify the absolute flows.",
"Gilbert first proposed the BOT problem [9] in which the objective is to solve for $ \\underset{B, E, x_B, m_E}{\\arg \\min } \\ \\sum _{(i,j) \\, \\in \\, E} m_{ij}^\\alpha &\\left\\Vert x_i - x_j \\right\\Vert _2 , \\text{ subject to }\\\\ \\nonumber \\text{supply } \\ \\mu _s = \\sum \\nolimits _{k} m_{sk} &- \\sum \\nolimits _{k} m_{ks} \\ \\text{ at each source } s,\\\\ \\nonumber \\text{demand } \\ \\mu _t = \\sum \\nolimits _{k} m_{kt} &- \\sum \\nolimits _{k} m_{tk} \\ \\text{ at each sink } t,\\\\ \\nonumber \\text{conservation } \\ \\sum \\nolimits _{k} m_{kb} &= \\sum \\nolimits _{k} m_{bk} \\ \\text{ at each BP } b,$ given a single parameter $\\alpha \\in [0,1]$ .",
"The problem of BOT is interesting in that it combines combinatorial optimization (over $B$ , $E$ ) with continuous optimization (over $x_B$ , $m_E$ ).",
"Figure: Branched optimal transport (Eq. )",
"interpolates between optimal transport and the Euclidean Steiner tree problem.",
"On a toy example, shown are good BOT solutions (found by our approximate solver for α≠1\\alpha \\ne 1, see Sect. )",
"for the same set of sources (red) and sinks (blue).",
"The disk sizes indicate the demands and supplies, the edge widths the mass transported along each edge.For $\\alpha = 1$ , the BOT problem is the discrete version of the famous optimal transport problem for which optimal solutions can be found efficiently [6], [23].",
"However, due to the linearity of the cost function, OT solutions do not exhibit any branching but consist of straight lines between sources and sinks, see Fig.",
"REF .",
"In contrast, for $\\alpha \\in [0,1)$ , the subadditivity of $m \\mapsto m^\\alpha $ reflects the increased efficiency of transporting loads together, i.e.",
"$(m_1 + m_2)^\\alpha < m_1^\\alpha + m_2^\\alpha $ .",
"Thus, for $\\alpha \\in [0,1)$ , BOT solutions show a branched structure, see Fig.",
"REF -al0.",
"Unlike OT, the optimization problem of BOT is NP-hard [11].",
"In the special case of $\\alpha = 0$, BOT turns into the well-studied Euclidean Steiner tree problem (ESTP) [28], [13].",
"In the ESTP, the objective is to find the overall shortest network that interconnects all terminals (with the help of BPs), independently of the edge flows, since $m^0 = 1$ .",
"For different values of $\\alpha $ , BOT interpolates between these two optimization problems, see Fig.",
"REF ."
],
[
"Connection of BOT to machine learning.",
"Optimal transport has emerged as an important tool in machine learning [1], [5], [23].",
"BOT is a strict generalization, describing a more versatile concept and more challenging optimization problem.",
"BOT offers a mathematical formalism that is deceivingly simple (cf. Eq.",
"(REF )) and yet engenders non-trivial structure.",
"Many machine learning problems such as tracking of divisible targets (computer vision), skeletonization (image analysis), trajectory inference (bioinformatics) come with input that is essentially continuous (images, distributions) and require structured output that is discrete, e.g. graphs.",
"Arguably, this transition from continuous to discrete is one of the most interesting aspects (and an unsolved problem) in current machine learning research.",
"It is also a problem that cannot be solved by a mere upscaling of standard deep learning architectures.",
"In addition, routing problems have become a popular problem to challenge machine learning and amortized optimization algorithms with difficult optimization problems [17], [3], [15].",
"Combining combinatorial and continuous optimization, BOT is a highly instructive target for new machine learning approaches.",
"In Sect.",
"we address the generalization of BOT to higher-dimensional Euclidean space, particularly relevant for applications in data science.",
"In this paper, we make the following contributions: We generalize an existing method for constructing BOT solutions with optimal geometry to the case of multiple sources.",
"Based on this generalization, we present an analytical and numerical scheme to rule out $n$ -degree branchings with $n > 3$ .",
"Further, we demonstrate how to extend geometric and topological properties of optimal BOT solutions to two-dimensional Riemannian manifolds.",
"Lastly, we propose a more practical numerical algorithm for the geometry optimization together with a simple but compelling heuristic, addressing the optimization of the BOT topology.",
"To the best of our knowledge, no readily accessible code for finding BOT solutions is publicly available.",
"By making our code available at https://github.com/hci-unihd/BranchedOT we hope to aid the evolution of the field."
],
[
"Topology and geometry of BOT solutions",
"A BOT problem can be divided into the combinatorial optimization of the network topology, specified by the set of BPs $B$ and edges $E$ (see Sect.",
"), and the geometric optimization of the BP positions $x_B$ .",
"[2] showed that optimal BOT solutions can be assumed to be acyclic, which restricts the search for the optimal topology to trees.",
"Given $n$ terminals, WLOG, the topology can be represented as a so-called full tree topology, which has $n-2$ BPs, each of degree three.",
"Higher-degree branchings may effectively form during the geometry optimization if multiple BPs settle at the same position.",
"A set of such BPs is referred to as coupled BP, cf. Fig.",
"REF .",
"The union set of all neighbors of the individual BPs (not including the BPs themselves) is referred to as set of effective neighbors.",
"Conversely, a BP configuration in which all BPs are uncoupled and located away from the terminals is called non-degenerate, see Fig.",
"REF .",
"The number of distinct full tree topologies interconnecting $n$ terminals is given by $(2n - 5)!!",
"= (2n - 5) \\cdot (2n - 7) \\cdot ... \\cdot 3 \\cdot 1$ and hence increases super-exponentially with the number of terminals [25].",
"Given 100 terminals, one would have to consider more than $10^{18}$ possible full tree topologies, making an exhaustive search computationally intractable already for problems of modest size.",
"Fortunately, given a tree topology, the geometric optimization of the BP positions reduces to a convex optimization problem, as all edge flows $m_{ij}$ are already uniquely determined by the flow constraints in Eq.",
"(REF ).",
"The corresponding linear system can be solved in linear time by dynamic programming, called “elimination on leaves of a tree” in [26].",
"Since the Euclidean norm, like any norm, is convex, given a fixed tree topology, the cost function in Eq.",
"(REF ) becomes a convex function of the BP positions.",
"Together with the independence of the individual BPs, this implies the following lemma on the optimal substructure of BOT solutions (see App. ).",
"Definition 2.1 For a chosen topology $T$ , a BOT solution is called a relatively optimal solution (ROS of $T$ ) if its BP configuration has minimal cost.",
"The overall best BOT solution, given by the optimal topology together with its ROS, is called the globally optimal solution (GOS).",
"Lemma 2.1 (a) For a given tree topology, a BOT solution is relatively optimal if and only if every (coupled) BP connects its (effective) neighbors at minimal cost.",
"(b) In a globally optimal solution, every subsolution restricted to a connected subset of nodes solves its respective subproblem (cf. App. )",
"globally optimally.",
"Figure: Two BP configurations for the same full tree topology: A higher-degree branching may effectively form by coupling the two BPs at the same position."
],
[
"Geometric optimization of BOT solutions",
"Although the BP optimization for a given tree topology is a convex problem, as argued above, it is non-trivial, since the objective function is not everywhere differentiable.",
"Here, we present a principled geometric approach, which was first suggested by Gilbert in [9] and previously developed in the context of the ESTP [20].",
"More recently, this approach was discussed in the comprehensive work by [2], where it was applied exclusively to BOT problems with a single source.",
"A generalization to the case of multiple sources was posed as an open problem by the authors (see Problem 15.11), for which we give the solution in this section."
],
[
"Geometric solution for one source and two sinks",
"Motivated by Lem.",
"REF , we start by considering a single BP $b$ in isolation (cf. Fig.",
"REF ), following [2].",
"Given a source at positionWe will often use the node label, e.g., $a_0$ , to denote also the position of the node, instead of writing $x_{a_0}$ .",
"$a_0$ and two sinks at positions $a_1$ and $a_2$ , we aim to find the optimal position $b^*$ for the BP connecting the three terminals, i.e., the minimizer of $ \\mathcal {C}(b) = m_1^\\alpha \\vert a_1 - b \\vert + m_2^\\alpha \\vert a_2 - b \\vert + (m_1 + m_2)^\\alpha \\vert a_0 - b \\vert ,$ where $m_1$ and $m_2$ are the respective demands of the two sinks.",
"Due to the convexity of $\\mathcal {C}(b)$ , the minimum must lie either at a stationary point at which $\\nabla _{\\!",
"\\!",
"b} \\, \\mathcal {C} = 0$ or at a non-differentiable point, where $b$ coincides with one of the $a_i$ .",
"[2] showed that the gradient is equal to zero if and only if the branching angles $\\theta _i$ , see Fig.",
"REF , are given by $&\\theta ^*_1 = \\arccos \\bigg ( \\frac{k^{2\\alpha } + 1 - (1-k)^{2 \\alpha }}{2 k^\\alpha } \\bigg ) =: f(\\alpha , k), \\nonumber \\\\&\\theta ^*_2 = \\arccos \\bigg ( \\frac{(1 \\!",
"- \\!",
"k)^{2\\alpha } + 1 - k^{2 \\alpha }}{2 (1 \\!",
"- \\!",
"k)^\\alpha } \\bigg ) = f(\\alpha , 1 \\!",
"-k), \\\\&\\theta ^*_1 + \\theta ^*_2 = \\arccos \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) =: h(\\alpha , k) \\nonumber ,$ where we have defined the flow fraction $k:= m_1/(m_1 + m_2)$ and the two functions $f$ and $h$ , related via $h(\\alpha , k) = f(\\alpha , k) + f(\\alpha , 1-k)$ .",
"If a BP exists that realizes the branching angles $\\theta _i^*$ , it can be constructed geometrically based on the central angle property (see App. ).",
"It states that, given a circle through $a_1$ and $a_2$ , the angle $\\angle a_1 o a_2$ at the center $o$ is twice the angle enclosed with a point anywhere on the opposite circle arc, cf. Fig.",
"REF .",
"In particular, let us construct the so-called pivot circle with central angle $\\angle a_1 o a_2 = 2\\theta ^*_1 + 2\\theta ^*_2$ and pivot point $p$ as in Fig.",
"REF .",
"Applying the central angle property twice (once for $\\theta _1^*$ and once for $\\theta _2^*$ ), a BP located at the intersection of the lower circle arc and the connection line $\\overline{a_0p}$ realizes both angles $\\theta ^*_i$ and is therefore optimal.",
"Figure: (a) Branching point bb connecting one source and two sinks with branching angles θ 1 \\theta _1 and θ 2 \\theta _2.",
"(b) Construction of the optimal BP b * b^* applying twice the central angle property illustrated in (c).",
"(d) shows the relation of the optimal branching angles θ i * \\theta ^*_i and ϑ i * \\vartheta ^*_i, relevant for the case of asymmetric branchings (see Sect.",
").However, given the pivot point and pivot circle, $\\overline{a_0 p}$ may not intersect the lower circle arc, depending on the position of $a_0$ .",
"Accordingly, the lower half plane can be partitioned into a region for which the described construction yields an optimal Y-shaped branching and three other regions, see Fig.",
"REF .",
"For $a_0$ located in one of these regions, the optimal BP position coincides with one of the terminals, resulting in a V-shaped branching ($b^* = a_0$ ) or an L-shaped branching ($b^* \\in \\lbrace a_1,a_2\\rbrace $ ), cf. Fig.",
"REF [2]."
],
[
"Geometric construction of BOT solutions for a given topology",
"Applying the geometric construction from above in a recursive manner, one can construct the ROS (see Def.",
"REF ) for larger BOT problems, as illustrated in Fig.",
"REF .",
"Given a full tree topology $T$ , first, we determine all edge flows (see Sect. )",
"and consequently the optimal branching angles.",
"Then, a root node is chosen, arbitrarily (here $a_0$ ), and all other nodes are sorted based on the number of edges to $a_0$ (ignoring edge directions and resolving ties arbitrarily).",
"Starting from the furthest nodes and working towards the root, two nodes are recursively summarized by a pivot point, constructed from the optimal branching angles, see Fig.",
"REF -recur2.",
"Afterwards, in reversed order, the optimal BPs are placed iteratively, each as in the 1-to-2 case, see Fig.",
"REF -recur4.",
"In this manner, the optimal branching angles are realized at every BP and the resulting solution is a ROS of $T$ by Lem.",
"REF .",
"The choice of the root node induces a node ordering as described above.",
"Given this ordering, consider any BP $b$ and denote its children by $a_1$ and $a_2$ and its parent node by $a_0$ .",
"The construction of the pivot point now requires the positions of $a_1$ and $a_2$ and the optimal branching angles enclosed by the children edges $(b,a_1)$ and $(b,a_2)$ .",
"However, the branching angles do not only depend on the absolute flows $m_1$ and $m_2$ of the respective edges but also on the flow directions.",
"Given that both flows point towards $b$ or given that both flows point away from $b$ , as in Fig.",
"REF , the branching is referred to as symmetric and the optimal branching angles of interest are given by $\\theta _i^*$ , cf. Eq.",
"(REF ).",
"Note that BOT problems and their solutions are fully symmetric under complete exchange of sinks and sources (up to reversal of all flow directions).",
"On the contrary, in case of one flow pointing towards $b$ and one pointing away from $b$ , referred to as asymmetric branching, the optimal branching angles $\\vartheta _i^*$ enclosed by the children edges are calculated differently, see Fig.",
"REF .",
"However, the branching angles $\\vartheta _i^*$ can be related geometrically to the known $\\theta _i^*$ .",
"Using the functions $f$ and $h$ from Eq.",
"(REF ), we find that $\\begin{aligned}\\vartheta ^*_1 &= \\pi - \\theta ^*_1 - \\theta ^*_2 = \\pi - h\\Big (\\alpha , k=\\frac{m_2 - m_1}{m_2}\\Big ) \\\\\\vartheta ^*_2 &= \\theta ^*_1 = f\\Big (\\alpha , k = \\frac{m_2 - m_1}{m_2}\\Big ).\\end{aligned}$ After determining the two angles $\\vartheta _1^*$ and $\\vartheta _2^*$ from the flows $m_1$ and $m_2$ , the BP construction based on the central angle property works analogously to the symmetric case.",
"Crucially, this distinction of symmetric and asymmetric branching makes the recursive construction applicable also to problems with multiple sources, where asymmetric branchings may be unavoidable, consider e.g. Fig.",
"REF with $a_0$ and $a_3$ as sources and $a_1$ and $a_2$ as sinks (see App.",
"REF ).",
"Further, note that the known conditions for optimal V- and L-branching can be transferred to the asymmetric case simply by relabelling $a_0 \\rightarrow a_2$ , $a_1 \\rightarrow a_0$ and $a_2 \\rightarrow a_1$ , cf. Fig.",
"REF and Fig.",
"REF .",
"In terms of angular inequalities (derived in App.",
"), these conditions, for both branching types, are summarized in Table REF .",
"Figure: Recursive geometric construction of a relatively optimal solution, using one pivot point and pivot circle per branching point to collectively realize the optimal branching angles.Figure: Regions of optimal Y-, V- and L-branching.",
"Table: NO_CAPTION tableRelations between the L- and V-branching conditions.",
"In principle, given a full tree topology, the described method efficiently constructs the ROS in linear time.",
"However, as already pointed out by [9], the approach has some practical limitations, even after our generalization.",
"Figure REF shows how the pivot point is constructed only from the positions of two children $a_1$ and $a_2$ and the corresponding optimal branching angles.",
"However, a priori there are two possible pivot point locations, one in the upper and one in the lower half plane with respect to $\\overline{a_1a_2}$ .",
"Hence, the construction relies on knowing in which half plane the third node $a_0$ lies.",
"For larger trees, the topological parent $a_0$ may itself be a BP whose position is not yet determined.",
"In the worst case, one would thus have to try all $2^{n-2}$ possible pivot point combinations to find the ROS.",
"This pivot point degeneracy gets substantially worse in higher dimensions, making the recursive construction applicable only in $\\mathbb {R}^2$ .",
"Secondly, the geometric construction only produces solutions which are non-degenerate, i.e., solutions without edge contractions.",
"For now, the geometric construction is therefore primarily of theoretical interest; and indeed, it forms the basis of our following arguments.",
"Note that both of the aforementioned problems could be solved elegantly in the special case of $\\alpha = 0$ [12], [14]."
],
[
"Properties of optimal BOT topologies",
"Let us now consider topological modifications in order to improve the transportation cost of a BOT solution.",
"In particular, we intend to show that a topology $T$ can be improved if its ROS contains coupled BPs.",
"Let us start by considering a general BOT solution which contains a coupled 4-BP, i.e., a coupled BP with four effective neighbors, as in Fig.",
"REF .",
"Lemma REF states that a solution is not globally optimal if any subsolution is not globally optimal.",
"It will therefore suffice to study the coupled BP as an isolated subproblem."
],
[
"Non-optimality of coupled branching points",
"Given two sources and two sinks, there are two possible configurations in which the terminals can be arranged, cf. Fig.",
"REF ,fork2.",
"First we address the case in which the two sources are at opposite corners of the terminal quadrilateral, as in Fig.",
"REF .",
"Based on Lem.",
"REF , a necessary condition for the existence of a globally optimal 4-BP is that all four V-branchings between neighboring terminals are optimal.",
"This puts a lower bound on each of the angles $\\gamma _i$ , see Tab.",
"REF .",
"The general idea, also regarding the other 4-branching scenarios, is to show that the angular sum of these lower bounds already exceeds $2 \\pi $ .",
"This will immediately imply that not all V-branchings can be optimal simultaneously and thus a coupled 4-BP cannot be globally optimal.",
"Given a 4-BP as in Fig.",
"REF , all V-branchings are asymmetric (i.e.",
"neighboring flows point in opposite directions).",
"Hence, all four lower bounds (in Tab.",
"REF ) are of the form $\\gamma _i \\ge \\pi - \\theta ^*_2 = \\pi - f(\\alpha , 1 - k)$ and indeed $\\pi - f(\\alpha , 1 - k) > \\pi / 2$ , see Lem.",
"REF , so that their sum exceeds $2 \\pi $ .",
"Next, let us consider the scenario in Fig.",
"REF with two sources at neighboring corners.",
"WLOG, we use the normalization $m_1 + m_2 = 1 = m_3 + m_4$ and assume that $m_1 > m_3$ and $m_2 < m_4$ .",
"In this case, the four conditions for optimal V-branching in Tab.",
"REF read: $\\gamma _1 \\ge \\pi - f\\Big (\\alpha , 1-\\frac{m_1 - m_3}{m_1} \\Big ) = \\pi - f\\Big (\\alpha , \\frac{m_3}{m_1} \\Big )$ , $\\gamma _2 \\ge h\\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) = h(\\alpha , m_1)$ , $\\gamma _3 \\ge \\!",
"\\pi - f\\Big (\\alpha , 1 \\!",
"- \\frac{m_4 - m_2}{m_4}\\Big ) \\!",
"= \\pi - f\\Big (\\alpha , \\frac{1- m_1}{1 - m_3} \\Big )$ , $\\gamma _4 \\ge h\\Big (\\alpha , \\frac{m_3}{m_3 + m_4} \\Big ) = h (\\alpha , m_3)$ , Figure: Different scenarios of coupled 4-BPs with symmetric branching angles in blue, asymmetric ones in orange.",
"where the expressions (REF ) were plugged into the V-branching conditions in Tab.",
"REF for symmetric and asymmetric branching respectively, as indicated by the colors in Fig.",
"REF .",
"Let us show that in fact for all combinations of $\\alpha $ , $m_1$ and $m_3$ the sum of the lower bounds already exceeds $2 \\pi $ .",
"Indeed, summing the lower bounds and subtracting $2\\pi $ yields $&\\underbrace{h(m_1)}_{= \\ h(1-m_1)} + \\ h(m_3) - f\\Big (\\underbrace{\\frac{m_3}{m_1}}_{> \\, m_3} \\Big ) - f\\Big (\\underbrace{ \\frac{1 - m_1}{1 - m_3} }_{> \\, 1 - m_1} \\Big ) >h(1-m_1) + h(m_3) - f(m_3) - f(1-m_1) \\\\&= f(m_1) + f(1-m_3) > 0,$ using $h(\\alpha ,k) = f(\\alpha , k) + f(\\alpha , 1-k)$ and the fact that $f(\\alpha , k)$ is strictly decreasing with respect to $k$, see Lem.",
"REF .",
"To summarize, we have arrived at the following lemma: Lemma 4.1 A coupled 4-BP not coincident with a terminal connecting two sources and two sinks is never globally optimal.",
"Exactly the same logic applies for a coupled 4-BP connecting one source and three sinks (or equivalently 3 sources and 1 sink), as in Fig.",
"REF .",
"WLOG, in the following, we normalize the flows so that $m_1 + m_2 + m_3 =1$ .",
"We then determine the necessary conditions under which all V-branchings are optimal.",
"We again intend to show that such a 4-BP can never be globally optimal by showing that for any combination of $\\alpha $ and ${m_i}$ the sum of the lower bounds exceeds $2 \\pi $ .",
"This is equivalent to proving the following inequality (see App.",
"REF ): $h \\Big (\\frac{m_1}{m_1 + m_2} \\Big ) &- f( m_1) + h \\Big (\\frac{m_3}{m_3 + m_2} \\Big ) - f(m_3) > 0.$ Assuming a globally optimal 4-BP existed, one could continuously displace a terminal in a way such that for the resulting BOT problem a coupled 4-BP is still globally optimal.",
"Choosing different such displacements four additional inequalities can be derived (see App.",
"REF ): Proposition 4.2 Given a BOT problem with one source and three sinks, with demands $m_1, m_2, m_3$ as in Fig.",
"REF , a coupled 4-BP away from the terminals cannot be globally optimal if at least one of the following inequalities holds true: $\\Gamma &= h \\Big (\\frac{m_1}{m_1 + m_2} \\Big ) - f(m_1) + h \\Big (\\frac{m_3}{m_3 + m_2} \\Big ) - f( m_3) > 0, \\\\\\Gamma _{1,*} &= f(1 -m_*) + f\\Big (1 - \\frac{m_2}{1-m_*} \\Big ) - f(1-m_*-m_2) > 0, \\\\\\Gamma _{2,*} &= h \\Big (\\frac{m_*}{m_* + m_2} \\Big ) + f\\Big (\\frac{m_2}{1 - m_*} \\Big ) - h(m_*) > 0$ where $*=1,3$ .",
"Note that $\\Gamma = \\Gamma _{1,1} + \\Gamma _{2,1} = \\Gamma _{1,3} + \\Gamma _{2,3}$ .",
"In App.",
"REF , we prove the inequalities analytically for a large subset of the parameter space.",
"For the remainder we present a numerical argument (see App.",
"REF ).",
"In addition, we show by induction how, given that coupled 4-BPs are never globally optimal, one can further rule out coupled $n$ -BPs (with $n$ effective neighbors) for all $n > 4$ .",
"Theorem 4.3 Given a BOT problem in the Euclidean plane and assuming that coupled 4-BPs are never globally optimal, in a globally optimal BOT solution each branching point not coincident with a terminal must have degree three."
],
[
"Generalization of BOT to Riemannian manifolds",
"In this section, we extend the BOT problem together with many of the previous results to two-dimensional Riemannian manifolds $\\mathcal {M}$ embedded into $\\mathbb {R}^3$ [18].",
"This includes the sphere as important special case, particularly relevant for global transportation networks.",
"In the generalized BOT cost function (REF ) we replace the Euclidean metric by the geodesic distance $d: \\mathcal {M} \\times \\mathcal {M} \\rightarrow \\mathbb {R}^+$ , i.e.",
"$ \\mathcal {C}_M = \\sum _{(i,j) \\, \\in \\, E} m_{ij}^\\alpha \\, d(x_i,x_j).$ As we assume the manifold to be embedded, the length of a geodesics can be measured in $\\mathbb {R}^3$ .",
"First, we generalize the non-optimality of cyclic solutions.",
"The corresponding proof in [2] readily applies also to two- and higher-dimensional manifolds.",
"As before, solving a BOT problem on a curved surface can thus be separated into the combinatorial topology optimization and the continuous optimization of the BP configuration."
],
[
"Linear approximation of BOT solutions on manifolds",
"Intuitively speaking, a two-dimensional Riemannian manifold locally looks like the Euclidean plane.",
"If we zoom in on a sufficiently small region, geodesics again resemble straight lines and the geodesic distance approaches the Euclidean one.",
"This can be used to show that the branching angles which were optimal for Y-branchings in the Euclidean plane are also optimal on Riemannian manifolds.",
"Below, we summarize the main steps of the proof.",
"All details can be found in App. .",
"Given a Y-branching on a manifold, we measure the angles between the three geodesics in the tangent space $T_b \\mathcal {M}$ at the BP $b$ .",
"We now zoom in on a small neighborhood $U$ around $b$ and consider only the subsolution in $\\mathcal {M} \\cap U$ .",
"The terminals of the corresponding subproblem are projected orthogonally onto the tangent space, more specifically onto a small disk of radius $r$ , denoted by $D(r)$ , see Fig.",
"REF .",
"Let us denote the cost of the subsolution on the manifold by $\\mathcal {C}_M(b)$ and the cost of the corresponding subproblem in the flat disk by $\\mathcal {C}(b)$ .",
"Now, assuming that the angles between the geodesics deviate from the optimal branching angles, the same holds true for the projected subsolution.",
"Consequently, there exists an alternative BP $b^*$ in the disk with cheaper cost $\\mathcal {C}(b^*)$ .",
"Note that the radius of this disk becomes smaller the smaller we choose the region $\\mathcal {M} \\cap U$ of the subproblem.",
"Figure: On the flat disk D(r)⊂T b ℳD(r) \\subset T_b \\mathcal {M}, the BP bb does not solve the problem with terminals a ^ i \\hat{a}_i optimally if the angles between the dotted geodesics are not optimal.Crucially, the cost difference between a subsolution on the manifold and its projection onto the plane tends to zero quadratically in the limit of $r \\rightarrow 0$ .",
"The intuitive reason for this is that the tangent space $T_b \\mathcal {M}$ locally approximates the manifold to linear order.",
"On the contrary, the costs $\\mathcal {C}(b)$ and $\\mathcal {C}(b^*)$ in the disk scale linearly in $r$ and so does the cost improvement $\\mathcal {C}(b) - \\mathcal {C}(b^*) = M \\, r$, for some fixed $M > 0$ .",
"To conclude the proof, one projects $b^*$ onto the manifold and evaluates the cost difference of the two subsolutions there.",
"The difference is of the form $M \\, r + O(r^2)$ , with second order differences due to the projection from $D(r)$ to the manifold.",
"Consequently, a finite radius $r > 0$ must exist for which the cost difference is truly positive.",
"A BOT solution on the manifold for which the Y-branching angles deviate from the optimal branching angles can thus be improved and is not relatively optimal.",
"The logic of the proof outlined here can easily be extended to the V- and L-branching conditions as well as our results regarding the non-optimality of coupled BPs.",
"Again, improving the BOT solution locally in the tangent plane (w.r.t.",
"its geometry or topology) and projecting back to the manifold results in an improved solution on the manifold (see App.",
"REF ).",
"Theorem 5.1 Consider the solution to a generalized BOT problem on a two-dimensional Riemannian manifold embedded into $\\mathbb {R}^3$ .",
"For the solution to be relatively optimal, it is a necessary condition that each BP satisfies the optimal angle conditions for Y-, V- and L-branching, which apply for BOT in the Euclidean plane.",
"For it to be globally optimal, assuming that coupled 4-BPs are not optimal in the plane, it is a necessary condition that BPs not coincident with a terminal have degree three.",
"Though there is no readily available algorithm to solve BOT on embedded surfaces, we discuss some possible approaches in App.",
"REF ."
],
[
"Heuristics and numerical optimization",
"In this section, we present a simple but effective algorithm for the geometry optimization, followed by a compelling heuristic for the topology optimization.",
"As pointed out earlier, the difficulty of solving a BOT problem stems from the super-exponentially growing number of possible full tree topologies.",
"Obtaining an exact solution by brute-force is almost always computationally infeasible and hence fast heuristic solvers are needed.",
"For BOT problems with a single source, a branch-and-bound method is applicable [31], enabling exact solutions for up to 16 nodes.",
"However, this method does not generalize directly to the case of multiple sources.",
"While some literature exists on heuristics for BOT problems with a single source [29], we are not aware of heuristics for multiple sources, except [24].",
"The authors of [24] present a simulated annealing based optimization strategy for BOT, based on hand-crafted geometrical and topological modifications, which may require user supervision.",
"Furthermore, continuous approaches to solve BOT exist which do not rely on a subdivision into geometry and topology optimization.",
"The authors of [22] phrase BOT as a limit of functional minimization problems.",
"Since their algorithm discretizes the plane and the BOT cost function, their output is however not sparse but a discretized function."
],
[
"Numerical branching point optimization for a given topology",
"Brute-force and heuristic BOT solvers alike typically rely on the geometry optimization of many different topologies.",
"A fast and reliable BP optimization routine is therefore essential, as it determines the computational bottleneck of these algorithms.",
"For a given tree topology $T$ , all edge flows $m_{ij}$ are known (see Sect. ).",
"The objective is thus to minimize the following convex cost function: $ \\mathcal {C}(\\lbrace x_i\\rbrace ) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\left\\Vert x_i - x_j \\right\\Vert _2,$ where, for $1 \\le i \\le n$ , the $x_i$ hold the fixed coordinates of the terminals and, for $n+1 \\le i \\le n+m$ , the variable BP positions.",
"Since the cost function is not everywhere differentiable, we suggest the following generalization of Smith's algorithm developed for geometry optimization in the ESTP [26].",
"It is an effective algorithm specifically for minimizing the sum of Euclidean norms in two- and higher-dimensional Euclidean space.",
"Unlike the geometric construction in Section REF , it is applicable to all (not necessarily full) tree topologies.",
"Starting from a non-optimal, non-degenerate BP configuration, e.g.",
"from a random initialization, the gradient with respect to each BP position $x_i$ is set to zero for $n+1 \\le i \\le n+m$ , resulting in the following non-linear system of $m$ equations: $x_i = \\sum _{j \\, : \\, (i,j) \\in T} m_{ij}^\\alpha \\frac{x_j}{\\vert x_i - x_j \\vert }\\ \\Bigg / \\sum _{j \\, : \\, (i,j) \\in T} \\frac{m_{ij}^\\alpha }{\\vert x_i - x_j \\vert }.$ This system can be solved approximately, by iteratively solving the following linearized system $ &x_i^{(k + 1)} = \\!",
"\\!",
"\\sum _{j \\, : \\, (i,j) \\in T} m_{ij}^\\alpha \\frac{x_j^{(k+1)}}{\\vert x_i^{(k)} - x_j^{(k)} \\vert }\\ \\Bigg / \\sum _{j \\, : \\, (i,j) \\in T} \\frac{m_{ij}^\\alpha }{\\vert x_i^{(k)} - x_j^{(k)} \\vert }, \\hspace{11.38092pt}\\text{for } n+1 \\le i \\le n+m.$ Note that $x_i^{(k)} = x_i$ is fixed for $1 \\le i \\le n$ .",
"For each iteration, the solution can be found in linear time, again by “elimination on leaves of a tree”, similar to determining all edge flows from the flow constraints.",
"The algorithm is easily parallelized over $d$ spatial dimensions of a BOT problem so that a single iteration is of order $O(nd)$ .",
"In essence, this is an iteratively reweighted least squares (IRLS) approach [4].",
"The connection is made explicit in App.",
"REF .",
"Details on the proof of convergence, the empirical runtime of the algorithm and suitable convergence criteria can be found in App.",
"REF and in [26].",
"The arguments in [26] readily apply to our generalization.",
"Besides our method, other techniques may be used for the geometry optimization, for instance the interior point method presented in [30]."
],
[
"A greedy randomized algorithm for the topology optimization",
"Our heuristic for the optimization of the BOT topology is inspired by the idea of simulated annealing [16], which has been applied in different variants to combinatorial problems such as the Traveling Salesman Problem [19] or the ESTP [10].",
"In our heuristic, the BOT topology is iteratively modified by randomly deleting an edge and replacing it with a new one.",
"At each step, the new solution is accepted according to a criterion, which typically depends on the cost difference between the solutions and a user-chosen hyperparameter, the temperature, used to mimic a physical cooling process.",
"However, because in practice it works already sufficiently well (see Fig.",
"REF ), we refrained from designing an elaborate cooling scheme.",
"Instead, we apply the heuristic most greedily, i.e., in the zero-temperature limit, where a new state is accepted only if it decreases the cost.",
"Starting from an initial tree topology $T$ , e.g., the minimum spanning tree (mST) or the OT solutionIn particular in the regime $\\alpha \\approx 1$ , our BOT solver benefits from existing efficient OT solvers by using their solution as initial guess., we uniformly sample an edge $\\hat{e} \\in E$ and remove it from $T$ .",
"Let the incident node of $\\hat{e}$ which ended up in the smaller connected component be $\\ell $ .",
"Then, one calculates the distance $d(e, \\ell )$ between $\\ell $ and every edge $e=(i,j)$ in the larger component and samples one of these edges with probability $p(e) \\propto \\exp (-d(e, \\ell )^2 / d_{min}^2)$ , where $d_{min}$ is the distance to the closest considered edge.",
"The node $\\ell $ is then connected to the sampled edge via a new BP to produce a new tree topology.",
"For this topology, we optimize the geometry (as described in Sect.",
"REF ) and compare costs with the previous solution.",
"If the new state is rejected, start the next iteration by sampling $\\hat{e}$ without replacement until either a move is accepted and all above steps are repeated; or until no accepted move is found, upon which the search terminates.",
"Experiments for small BOT problems suggest that even in the greedy zero-temperature limit the algorithm often finds the globally optimal solution, after comparatively few iterations.",
"For this, the greedy heuristic (using the mST as initialization) was compared against exact solutions with up to nine terminals, obtained by brute-force.",
"For each $n$ , 100 BOT problems were sampled uniformly with respect to $\\alpha $ , the terminal positions and demands and supplies, cf. Alg.",
"REF .",
"The ratios of the heuristic's cost divided by the cost of the exact solution are plotted in Fig.",
"REF .",
"On average the heuristic solution is less than 0.5% worse than the brute-force solution.",
"This is impressive, considering the fact that for $n=9$ the brute-force solver requires over $10^5$ BP optimizations, whereas the simulated annealing heuristic on average required $29 \\pm 10$ iterations to converge.",
"Additional experiments (also for larger BOT problems) suggest that the number of BP optimizations until convergence scales better than $O(n^2)$ , see App.",
"REF .",
"Further, the cost ratios in Fig.",
"REF stay roughly constant as $n$ increases.",
"Additional experiments for BOT in higher dimensions (see Fig.",
"REF ) indicate that the average quality of the heuristic solution decreases only very slightly with $n$ .",
"Unfortunately, one can only speculate how this trend extends to larger BOT problems, where brute-force solutions are no longer feasible.",
"Figure REF shows heuristic solutions of a larger example problem for different values of $\\alpha $ .",
"In particular, we find that the greedy heuristic is very effective at removing higher-degree branchings and undesirable edge crossings."
],
[
"Generalization to higher-dimensional BOT",
"Optimal BOT solutions are acyclic also in $\\mathbb {R}^d$ [2].",
"Thus, for a given topology, the edge flows are known, the optimal substructure property of Lemma REF generalizes and the convex geometry optimization can be separated from the combinatorial topology optimization.",
"Though, the optimal angle conditions for Y-, V- and L-branching (see Sect. )",
"hold also in $\\mathbb {R}^d$ , the results on the degree limitation do not generalize, as the arguments rely on the fact that the angles between edges meeting at a higher-degree branching point sum up to $2\\pi $ (cf.",
"Sect.",
"REF ).",
"The numerical geometry optimization as well as the greedy algorithm for the topology optimization presented in Sect.",
"are readily applicable to BOT problems in $\\mathbb {R}^d$ (see also App.",
"REF and App.",
"REF )."
],
[
"Conclusions",
"We have studied branched optimal transport in $\\mathbb {R}^2$ from a theoretical and practical perspective.",
"First, we have tackled the geometric optimization of BOT solutions, given a tree topology.",
"We generalized the existing exact method presented in [2], [9] to the case of multiple sources.",
"Based on theory developed in the process of this generalization, we formulated a catalog of necessary and sufficient conditions for optimal BOT solutions and argued that $n$ -degree branching points for $n>3$ are never optimal.",
"Moreover, we showed that these conditions also apply for BOT on two-dimensional manifolds.",
"Lastly, we presented a greedy randomized algorithm, which optimizes the tree topology, combined with an efficient numerical branching point optimization method.",
"We compared our algorithm to the optimal solution for small examples, obtaining compelling results.",
"BOT provides a unifying framework for optimal transport and the Euclidean Steiner tree problem and is itself of great theoretical and practical interest.",
"The emergent branching in BOT can be used to simulate and study the myriad of efficient transportation systems which exhibit subadditive costs.",
"Moreover, BOT combines both combinatorial and convex optimization and could be an inspiring problem to be solved by machine learning techniques.",
"The number of optimality criteria derived in this paper can guide further research in this area and the presented approximate solvers may serve as competitive baseline for new ML-based approaches.",
"We would like to thank Jarosław Piersa for sharing his code with us for a comparison to his work.",
"Further, we thank Edouard Oudet for helpful hints on the comparison to his related work and Fabian Egersdoerfer for his improved C++ implementation of the geometry optimization.",
"This work is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), by Informatics for Life and by SIMPLAIX funded by the Klaus Tschira Foundation."
],
[
"Appendix overview",
"The appendix is subdivided into the following seven topics: app-central: Quick proof of the central angle property, used in Sect.",
"REF in the recursive construction of relatively optimal solutions based on the optimal branching angles.",
"secLV: Derivation of the conditions listed in Tab.",
"REF under which V- or L-branching are optimal.",
"sec:subopt: Formal proof of Lemma REF on the optimal substructure of BOT solutions.",
"appfh: Collection of small lemmas on the monotonicity and other properties of the analytical expression for the branching angles (cf. Eq.",
"(REF )).",
"appsec:nonopt: Technical proofs and numerical scheme to show the non-optimality of higher-degree branchings discussed in Sect.",
"REF .",
"app:mf: Formal proof of Theorem REF , which generalizes the optimal branching conditions and other properties from the Euclidean plane to embedded surfaces.",
"A sketch of the proof can be found in Sect.",
"REF .",
"appsec:algo: Additional details and experiments for the different algorithms presented in the main paper.",
"Section REF focuses on the numerical geometry optimization and Sect.",
"REF on the greedy algorithm for the topology optimization.",
"Section REF holds a few examples of the recursive geometric construction of relatively optimal solutions for BOT problems with multiple sources."
],
[
"Central angle property",
"In this section, we present a geometric proof of the central angle property used in the geometric construction of relatively optimal solutions for a given full tree topology (see Sect.",
"REF ).",
"It states that for a circle, as in Fig.",
"REF , the central angle $\\angle a_1 o a_2$ is twice the angle $\\angle a_1 q a_2 = \\theta $ for all $q$ on the lower circle arc.",
"Figure: (a) Illustration of the central angle theorem.",
"(b) Isosceles triangles used in the geometric proof.Let us start by constructing the three isosceles triangles $\\triangle o q a_1$ , $\\triangle o q a_2$ and $\\triangle o a_1 a_2$ with angles as denoted in Fig.",
"REF .",
"Now consider the angular sums in the following triangles: $\\triangle q a_1 a_2 : \\ 2 \\delta _1 + 2 \\delta _2 + 2 \\tau = 180^\\circ \\ \\text{ and } \\ \\triangle o a_1 a_2 : \\gamma + 2 \\tau = 180^\\circ \\ .$ Subtracting the two equations immediately reveals that $\\gamma = 2 \\delta _1 + 2 \\delta _2 = 2 \\theta $ and the proof is complete."
],
[
"Optimal L- and V-shaped branching",
"Below, we formally derive the conditions listed in Tab.",
"REF under which V- or L-branching provide the optimal solution to a BOT problem with one source and two sinks.",
"The proof is inspired by the approach in [21], where subdifferentials are applied to the related Fermat-Torricelli problem.",
"Definition B.1 (Subgradient and subdifferential) A vector $v \\in \\mathbb {R}^n$ is called a subgradient of a convex scalar function $g: \\mathbb {R}^n \\rightarrow \\mathbb {R}$ at a specific point $y$ if for all $x \\in \\mathbb {R}^n$ it satisfies $g(x) \\ge g(y) + \\langle v, x - y \\rangle $ The set of all subgradients of the function $g$ at a given point $y$ is called the subdifferential of $g$ at $y$ and is denoted by $\\partial g(y)$ .",
"From a geometric point of view, the subdifferential of $g$ at $y$ is the set of gradients of all straight lines which cross $g(y)$ and lie below the image of $g$ .",
"The subdifferential rule of Fermat follows immediately from the definition and states that $g$ achieves an absolute minimum at $y$ if and only if $0 \\in \\partial g(y)$ .",
"Now, for $g(b) = c \\cdot \\left\\Vert b - a \\right\\Vert _2$ with $c \\in \\mathbb {R}$ and $a \\in \\mathbb {R}^n$ the subdifferential is given by $\\partial g(b) = {\\left\\lbrace \\begin{array}{ll}\\mathbb {B}_{c} \\, , \\quad \\quad \\quad \\ \\text{ for } b = a \\\\\\bigg \\lbrace c \\frac{b - a}{\\left\\Vert b - a \\right\\Vert _2} \\bigg \\rbrace \\, , \\text{ elsewhere, }\\end{array}\\right.",
"}$ with $B_r = \\lbrace v \\in \\mathbb {R}^n : \\left\\Vert v \\right\\Vert _2 \\le r \\rbrace $ , the ball of radius $r$ .",
"Furthermore, it can be easily shown that for $g(x) = g_1(x) + g_2(x)$, one has $\\partial g(y) = \\partial g_1(y) + \\nabla g_2(y)$ , given that both $g_1$ and $g_2$ are convex functions and $g_2$ is differentiable.The sum of a vector and a set of vectors, as in $\\partial g_1(y) + \\nabla g_2(y)$ , is known as the Minkowski sum.",
"Using this, we calculate the subdifferentials of the cost function of the 1-to-2 branching in Eq.",
"(REF ).",
"The subdifferentials at $b = a_i$ are of the form $\\partial \\mathcal {C}(a_i) = \\mathbb {B}_{m_i^\\alpha } + \\sum _{j \\ne i}m_j^\\alpha \\frac{ a_i - a_j}{ \\left\\Vert a_i - a_j \\right\\Vert } \\ .$ Based on the rule of Fermat, the cost function achieves an absolute minimum at $b = a_i$ if and only if $ \\left\\Vert \\ \\sum _{j \\ne i} m_j^\\alpha \\frac{ a_i - a_j}{\\left\\Vert a_i - a_j \\right\\Vert } \\ \\right\\Vert \\le m_i^\\alpha \\ .",
"\\\\[-0.4cm]\\nonumber $"
],
[
"V-branching.",
"We square condition (REF ) and evaluate it for $i=0$ in order to determine under which condition a V-shaped branching with $b^* = a_0$ is optimal: $\\left\\Vert \\ m_1^\\alpha \\frac{a_0 - a_1}{\\left\\Vert a_0 - a_1 \\right\\Vert }+ m_2^\\alpha \\frac{a_0 - a_2}{\\left\\Vert a_0 - a_2 \\right\\Vert }\\ \\right\\Vert ^2 =m_1^{2 \\alpha } + m_2^{2 \\alpha } + 2 m_1^\\alpha m_2^\\alpha \\cos (\\psi )\\le m_0^{2\\alpha } = (m_1 + m_2)^{2 \\alpha } \\,$ where $\\psi $ denotes the angle of the terminal triangle at $a_0$ , i.e., $\\psi = \\angle a_1a_0a_2$.",
"The condition in terms of $\\psi $ reads $ \\psi \\ge \\arccos \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) = h(\\alpha , k) = \\theta ^*_1 + \\theta ^*_2 \\ ,$ where we have used the flow fraction $k = m_1/(m_1 + m_2)$ .",
"We immediately recognize the expression for the optimal branching angle $\\theta ^*_1 + \\theta ^*_2$ , cf. Eq.",
"(REF ).",
"We already know that $\\psi = \\theta ^*_1 + \\theta ^*_2$ on the lower circle arc of the pivot circle by construction.",
"And indeed one can easily check that $\\psi > \\theta ^*_1 + \\theta ^*_2$ if an only if the source $a_0$ lies inside the lower half of the pivot circle, as in Fig.",
"REF .",
"For that, we construct a line through $a_1$ and $a_0$ and the intersection of $\\overline{a_1a_0}$ with the lower pivot circle we denote by $q$ .",
"By construction of the pivot circle, $\\angle a_1 q a_2 = \\theta ^*_1 + \\theta ^*_2$ .",
"Using the angular sum in the triangle $\\triangle a_0 q a_2$ , one immediately obtains: $180^\\circ = \\delta + ( 180^\\circ - \\psi ) + (\\theta ^*_1 + \\theta ^*_2) \\ \\rightarrow \\ \\psi - (\\theta ^*_1 + \\theta ^*_2) = \\delta > 0 \\ .$ Figure: Sketch to show that ψ>θ 1 * +θ 2 * \\psi > \\theta ^*_1 + \\theta ^*_2.Hence, for $a_0$ located inside the lower half of the pivot circle indeed $\\psi > \\theta ^*_1 + \\theta ^*_2$ .",
"A similar argument can show that for any point outside the pivot circle the optimal V-branching condition is not fulfilled."
],
[
"L-branching.",
"Analogous to the above steps, one obtains conditions for optimal L1- and L2-branching, where $b^*= a_1$ and $b^*= a_2$ respectively.",
"Again squaring the general condition (REF ), now for $i=1,2$ , one eventually finds that $\\varphi &\\ge \\pi - f(\\alpha , 1-k) = \\pi - \\theta ^*_2 \\ , \\\\\\varrho &\\ge \\pi - f(\\alpha , k) = \\pi - \\theta ^*_1 \\ ,$ where the angle $\\varphi $ and $\\varrho $ denote the angles of the terminal triangle located at $a_1$ and $a_2$ , i.e.",
"$\\varphi = \\angle a_2a_1a_0$ and $\\varrho = \\angle a_0a_2a_1$ (see also Fig.",
"REF ,vll4).",
"Let us now demonstrate that these conditions are indeed fulfilled if and only if the source is located in the L1- and L2-region, as marked also in Fig.",
"REF .",
"The pivot point $p$ is constructed such that $\\angle a_1 o p = 2 \\theta ^*_1$ and $\\angle p o a_2 = 2 \\theta ^*_2$ , cf. Fig.",
"REF .",
"Besides that, by construction, we have that $\\angle p a_1 a_2 = \\theta ^*_2$ and $\\angle a_1 a_2 p = \\theta ^*_1$ (as shown in Fig.",
"REF ,vll4).",
"Then, looking at Fig.",
"REF , it is evident that indeed for any source $a_0$ inside the L1-branching sector the condition $\\varphi = \\angle a_2a_1a_0 \\ge \\pi - \\theta ^*_2$ holds.",
"The respective condition (REF ) for L2-branching holds true exactly inside the highlighted L2-region.",
"Definition B.2 (Transient and strict V- and L-branchings) A V- or L-branching for which the inequality conditions (REF )-(REF ) hold as equality is referred to as transient V- or L-branching.",
"The reason for this is that, in such a case, one of the terminal positions may be perturbed infinitesimally, so that the condition is no longer fulfilled and the optimal solution transitions to a Y-shaped branching.",
"On the contrary, if an L- or V-branching condition is fulfilled as strict inequality, we call the L- or V-branching strict.",
"Figure: After constructing the pivot point the position of the source a 0 a_0 determines whether a Y-,V- or L-shaped branching is optimal."
],
[
"Relation of V- and L-conditions between symmetric and asymmetric branchings",
"Let us briefly show that the corner cases of Y-, V- and L-branching work analogously for both flow scenarios (described in Fig.",
"REF ,case2a).",
"The conditions under which L- and V-branching are optimal in the asymmetric branching case could be again determined straightforwardly by plugging into the subdifferential condition (REF ), as before.",
"However, looking at Fig.",
"REF ,case2a, we notice that the well-known symmetric branching case can be transformed into the asymmetric case by relabeling $a_0 \\rightarrow a_2$ , $a_1 \\rightarrow a_0$ and $a_2 \\rightarrow a_1$ .",
"This relation provides a direct correspondence of the L- and V-branching conditions.",
"The conditions for the corner case are transferred according to Table REF .",
"Note that the L- and V-branching conditions for both cases are of the exact same form, only that for the asymmetric case the stationary branching angles are $\\vartheta ^*_i$ instead of $\\theta ^*_i$ .",
"Moreover, for fixed children positions $a_1$ and $a_2$ , the position of the source again distinguishes between optimal Y-, V- and L-branching.",
"The partitioning of the lower half plane into the respective regions is completely analogous to Fig.",
"REF .",
"Table: Relations between the L- and V-branching conditions (see Fig. )",
"for symmetric and asymmetric branchings."
],
[
"Optimal substructure of BOT solutions (Proof of Lem. ",
"In this section, we provide the formal proof to Lemma REF , repeated below for completeness: Lemma C.1 (a) For a given tree topology, a BOT solutions is a relatively optimal if and only if every (coupled) BP connects its (effective) neighbors at minimal cost.",
"(b) In a globally optimal solution, every subsolution restricted to a connected subset of nodes solves its respective subproblem globally optimally.",
"Let us start with the following definition, which divides a BOT problem and its possible solutions into subproblems and corresponding subsolutions.",
"Definition C.1 (Subproblems and subsolutions) A given BOT solution may be split into two subsolutions, by choosing a number of edges $\\lbrace e_i\\rbrace $ and cutting them at points $\\lbrace x_i\\rbrace $ , so that the topology is split into two connected components.",
"This procedure induces two subproblems and two subtopologies.",
"Each subproblem consist of the terminals contained in the respective component plus additional terminals at the positions $x_i$ .",
"The demands or supplies of the additional terminals at $x_i$ are equal to the amount of flow through the corresponding edge $e_i$ that was cut.",
"The terminal becomes a sink in one subproblem and a source in the other according to the direction of flow through $e_i$ .",
"The two subtopologies are given by the induced subgraph on all terminals contained in one component.",
"The subsolutions to the created subproblems are given by the subtopologies and the BP configurations of the respective subsets of branching points contained in each subproblem.",
"Note that using a number of such splits a given solution may be divided into several subsolutions, each solving their respective subproblem.",
"An illustrative example can be found in Fig REF .",
"Figure: A BOT solution (left) is split into two subsolutions (right) by cutting an edge as indicated by the dashed line."
],
[
"Proof of the optimal substructure as necessary condition for optimality.",
"The optimal substructure as necessary optimality condition follows immediately from the independence of the subproblems.",
"Given that a solution is minimal in cost (in the relative or global sense), each subproblems itself must be minimal in that sense.",
"Otherwise the cost of this subproblem could be decreased by (a) improving the BP configuration or (b) the topology.",
"In that case, also the full solution could be improved in cost and could thus not be optimal."
],
[
"Proof of the optimal substructure as sufficient condition for optimality.",
"The optimal substructure as sufficient optimality condition stems from the fact that, given a tree topology, the BOT cost function is convex with respect to the BP positions.",
"As a first step, let us prove the following lemmata: Lemma C.2 Let $b$ be a branching point with neighbors $\\lbrace a_i \\rbrace $ .",
"Let the optimal position of $b$ be at one of its neighbors with the condition (REF ) fulfilled as equality, i.e.",
"the branching is transient.",
"Then, for every $\\delta >0 $ there exists a BP location $b_\\delta \\in \\mathbb {R}^2$ away from all neighbors with $\\left\\Vert b - b_\\delta \\right\\Vert _2 < \\delta $ and a continuous function $f:\\mathbb {R^+}\\rightarrow \\mathbb {R}$ with $\\lim _{x\\rightarrow 0}f(x)=0$ , such that the gradient $\\left\\Vert \\nabla _{\\!",
"\\!",
"b} \\, \\mathcal {C}(b_\\delta ) \\right\\Vert _2 < \\epsilon $ for $0<\\epsilon <f(\\delta )$ .",
"Proof.",
"Let $\\mathcal {C}(b)$ be the cost of the subproblem with terminals $a_i$ .",
"WLOG, the condition (REF ) is fulfilled as equality for terminal $a_0$ , meaning that $\\left\\Vert v \\right\\Vert = m_0^{\\alpha } \\ \\text{ with } \\ v:= \\sum _{i \\ne 0} m_i^\\alpha \\frac{ a_0 - a_i}{\\left\\Vert a_0 - a_i \\right\\Vert } \\ .$ The optimal branching point is hence located at $a_0$ .",
"Consider the alternative branching point position $b_\\delta = a_0 - \\delta v$ with sufficiently small $\\delta >0$ , such that $b_\\delta $ is located away from all terminals.",
"The gradient with respect to the branching point position is well defined at $b_\\delta $ and reads: $\\nabla _{\\!",
"\\!",
"b} \\, \\mathcal {C}(b_\\delta ) = - m_0^\\alpha \\frac{v}{\\left\\Vert v \\right\\Vert } +\\sum _{i \\ne 0} m_i^\\alpha \\frac{ a_0 - a_i - \\delta v}{\\left\\Vert a_0 - a_i -\\delta v \\right\\Vert } \\ .$ Clearly, $\\left\\Vert \\nabla _{\\!",
"\\!",
"b} \\, \\mathcal {C}(b_\\delta ) \\right\\Vert $ can be brought arbitrarily close to zero as $\\delta \\rightarrow 0$ , since $\\left\\Vert \\nabla _{\\!",
"\\!",
"b} \\, \\mathcal {C}(b_\\delta ) \\right\\Vert ^2 = \\underbrace{m_0^{2 \\alpha } + m_0^{2 \\alpha } - 2 m_0^\\alpha \\Big \\langle v, \\frac{v}{\\left\\Vert v \\right\\Vert } \\Big \\rangle }_{= \\, 0} + O(\\delta ) \\ ,$ comprising the terms which go to zero as $\\delta \\rightarrow 0$ in $O(\\delta )$ .",
"The function absorbed in $O(\\delta )$ therefore provides the function $f$ and $b_\\delta $ is the desired branching point with arbitrarily small gradient.",
"The argument works for both coupled and uncoupled branching points.",
"$$ Lemma C.3 For a given full tree topology $T$ , let $B=\\lbrace b_i\\rbrace $ be the ROS of a BOT problem with terminals $A = \\lbrace a_i\\rbrace $ .",
"Let $B$ contain Y-branchings and transient V- or L-branchings.",
"Then, for every $\\delta >0 $ there exists a non-degenerate BP configuration $B_\\delta $ and a continuous function $f:\\mathbb {R^+}\\rightarrow \\mathbb {R}$ with $\\lim _{x\\rightarrow 0}f(x)=0$ , such that $\\left\\Vert B - B_\\delta \\right\\Vert _2 < \\delta $ and the gradient norm fulfills $\\left\\Vert \\nabla _{\\!",
"\\!",
"B} \\, \\mathcal {C}(B_\\delta ) \\right\\Vert _2 < \\epsilon $ for $0<\\epsilon <f(\\delta )$ .",
"Proof.",
"Let us split the solutions into subsolutions at branching points that exhibit optimal Y-branchings.",
"They are kept fixed $\\delta _i = 0$ and have zero gradients.",
"The resulting subproblems can then be considered independently.",
"It suffices to show that, for every individual subproblem, individual displacements of the BPs exist that yield a non-degenerate BP configuration with arbitrarily small gradients.",
"If a subproblem consist of a single branching point, Lem.",
"REF is directly applicable and the infinitesimal displacement is chosen as described there.",
"If multiple branching points are coupled in a transient branching we proceed as follows: For the given topology, define a root node and apply the recursive geometric construction with pivot points and pivot circles, as illustrated in Fig.",
"REF .",
"In the construction, all pivot circles will meet at the position of the coupled branching point, which is the geometric equivalent of condition (REF ) being fulfilled as equality, cf. Fig.",
"REF .",
"Then, starting from the branching point furthest from the root node, take an infinitesimal step of size $\\sim O(\\delta )$ towards its corresponding pivot point, exactly as in Lem.",
"REF .",
"Thereby, the optimal branching angles will almost be realized and consequently the gradient of the resulting uncoupled BP will be arbitrarily small (as shown explicitly in Lem.",
"REF ).",
"For the next BP in topological order (w.r.t.",
"the chosen root node) repeat the procedure with a step size even smaller of size $\\sim O(\\delta ^2)$ , again the optimal branching angles are almost realized and the resulting gradient vanishes as $\\delta \\rightarrow 0$ .",
"Repeating, this procedure for every of the finitely many branching points with smaller and smaller step sizes produces a non-degenerate BP configuration with arbitrarily small gradient.",
"$$ Based on Lem.",
"REF , we now prove that the optimality of each individual branching point is a sufficient condition for relatively optimal solutions.",
"Similar to the proof above we do not explicitly distinguish between coupled and uncoupled branching points.",
"Given a fixed tree topology, the BOT cost function $\\mathcal {C}(\\lbrace b_i \\rbrace )$ is a convex function of the branching point positions $\\lbrace b_i \\rbrace _{1 \\le i \\le m}$ .",
"Let us summarize all branching point coordinates in the vector $B \\in \\mathbb {R}^{2m}$ and denote the configuration in which every BP connects its neighbors at minimal cost by $B^* = \\lbrace b_i^* \\rbrace $ .",
"Due to the convexity it suffices to show that the BP configuration $\\lbrace b_i^* \\rbrace $ is a local minimum of the cost function.",
"We may distinguish the following three cases: If the BP configuration $\\lbrace b_i^* \\rbrace $ is non-degenerate, the cost function is differentiable and for each BP one has $\\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}(b_i^*) = 0$ and thus also $\\nabla _{\\!",
"\\!",
"B} \\, \\mathcal {C}(B^*) = 0$ .",
"Let the BP configuration additionally contain BPs which are strictly anchored at one of their neighbors (referred to as anchor), meaning that the inequality condition (REF ) holds strictly (see also Def.",
"REF ).",
"Denote the subset of these BPs by $\\lbrace \\tilde{b}_j \\rbrace $ .",
"We intend to show that a neighborhood around $B^*$ exists such that $B^*$ is the minimal cost configuration in it.",
"If the neighboring branching points of $\\lbrace \\tilde{b}_j \\rbrace $ are moved away from $B^*$ only by a sufficiently small $\\delta $ , the optimal position for $\\lbrace \\tilde{b}_j \\rbrace $ stays at their anchors, since the condition (REF ) will still hold.",
"Thus, we can find a sufficiently small neighborhood $U$ around $B^*$ , so that WLOG, the coordinates of $\\lbrace \\tilde{b}_j \\rbrace $ are fixed to be equal to the coordinates of their respective anchors, as other configurations would be suboptimal.",
"If a branching point is anchored at an external node, its position is fixed to the terminal position and the cost function restricted to $U$ no longer depends on them.",
"So, by choosing $U$ sufficiently small, the cost function depends only on BPs away from terminals.",
"Summarize the remaining free branching points in the vector $B_{f}$ for which the gradient at $B_{f}^* \\subset B^*$ is zero by assumption, i.e.",
"$\\nabla _{\\!",
"\\!",
"B_f} \\, \\mathcal {C}(B_f^*) = 0$ .",
"Thus, $B^*$ is a local minimum in $U$ and due to the convexity an absolute minimum.",
"Let the BP configuration additionally contain transient branchings, i.e.",
"branching points for which the condition (REF ) holds as equality.",
"Denote them by $\\lbrace \\hat{b}_k \\rbrace $ .",
"Again, the sufficiently small neighborhood around $B^*$ is constructed, as before, to remove the dependency of $\\mathcal {C}$ on $\\lbrace \\tilde{b}_j \\rbrace $ .",
"However, the cost function still depends on the position of the $\\lbrace \\hat{b}_k \\rbrace $ , and $C(B_f)$ is not differentiable at $\\lbrace \\hat{b}_k^* \\rbrace \\subset B_f^*$ .",
"Let us assume a BP configuration $B_{f,0}$ existed with cheaper cost than $B^*_{f}$ , i.e.",
"$\\mathcal {C}(B^*_{f}) = \\mathcal {C}(B_{f,0}) + \\Delta $ for some $\\Delta > 0$ .",
"Lemma REF states that we can find an arbitrarily small $\\delta $ so that $B^*_{f} + \\delta $ is non-degenerate and the gradient $\\nabla _{\\!",
"\\!",
"B_f} \\, \\mathcal {C}(B^*_{f} + \\delta )$ is arbitrarily small.",
"Due to the convexity of $\\mathcal {C}$ , the following inequality holds: $\\mathcal {C}(B_{f,0}) &\\ge \\mathcal {C}(B^*_{f} + \\delta ) + \\langle \\nabla _{\\!",
"\\!",
"B_f} \\, \\mathcal {C}(B^*_{f} + \\delta ), B_{f,0} - (B^*_{f} + \\delta ) \\rangle \\\\&= \\mathcal {C}(B^*_{f}) + \\underbrace{O( \\left\\Vert \\delta \\right\\Vert _2) + \\langle \\nabla _{\\!",
"\\!",
"B_f} \\, \\mathcal {C}(B^*_{f} + \\delta ), B_{f,0} - (B^*_{f} + \\delta ) \\rangle }_{=: \\ K(\\delta )}$ where we have used that $\\mathcal {C}(B^*_{f} + \\delta ) = \\mathcal {C}(B^*_{f}) + O( \\left\\Vert \\delta \\right\\Vert _2)$ , using Big-O notation.",
"All terms which tend to zero as $\\delta \\rightarrow 0$ have been summarized in $K(\\delta )$ .",
"$K(\\delta )$ can be brought arbitrarily close to zero by choosing a sufficiently small $\\delta $ .",
"In particular, there exists a $\\delta > 0$ so that $\\vert K(\\delta ) \\vert < \\Delta / 2$ .",
"Using the assumption $\\mathcal {C}(B^*_{f}) = \\mathcal {C}(B_{f,0}) + \\Delta $ , this leads to: $\\mathcal {C}(B_{f,0}) &\\ge \\mathcal {C}(B^*_{f}) + K(\\delta ) = \\mathcal {C}(B_{f,0}) + \\Delta + K(\\delta ) \\ge \\mathcal {C}(B_{f,0}) + \\frac{\\Delta }{2} \\ ,$ which is a contradiction.",
"Hence $B^*$ , is again a local minimum.",
"$$"
],
[
"Properties of the functions $f$ and {{formula:ae7b64eb-edf7-4367-a415-b0559ec00a0b}} describing the optimal branching angles",
"The optimal branching angles are expressed in terms of the following functions $f(\\alpha , k)$ and $h(\\alpha , k)$ , defined for $\\alpha \\in [0,1]$ and $k \\in (0,1)$ , cf. Eq.",
"(REF ): $f(\\alpha , k) &= \\arccos \\bigg ( \\frac{k^{2\\alpha } + 1 - (1-k)^{2 \\alpha }}{2 k^\\alpha } \\bigg ) \\ , \\\\h(\\alpha , k) &= \\arccos \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) \\ .$ Figure REF shows $f$ and $h$ as functions of $k$ for a number of different values of $\\alpha $ .",
"The two functions are related by $h(\\alpha , k) = f(\\alpha , k ) + f(\\alpha , 1-k)$ , so that $h$ is symmetric around $k=1/2$ .",
"Both functions are defined for inputs $\\alpha \\in [0,1]$ and $k \\in (0,1)$ .",
"For $\\alpha = 1$ , we have $f(\\alpha =1, k) = 0 = h(\\alpha =1, k)$ for all $k$ , reflecting the fact that Y-shaped branchings are never optimal in the optimal transport case.",
"On the other hand, we have $f(\\alpha =0, k) = \\pi /3$ and $h(\\alpha =0, k)= 2 \\pi / 3$ for all $k$ .",
"This limit corresponds to the Euclidean Steiner tree problem where in an optimal Y-branching all edges meet at $120^\\circ $ .",
"Moreover, considering the limits of $k \\rightarrow 0$ and $k \\rightarrow 1$ , one finds that $f(\\alpha , k \\rightarrow 0) \\rightarrow \\pi /2$ and $f(\\alpha , k \\rightarrow 1) \\rightarrow 0$ .",
"Consequently, $h(\\alpha , k) \\rightarrow \\pi /2$ for both $k \\rightarrow 0$ and for $k \\rightarrow 1$ .",
"In the following three lemmas, we investigate the monotonicities of the two functions.",
"Figure: The functions ff and hh vs. kk for different values of α\\alpha .Lemma D.1 $f(\\alpha , k)$ is strictly decreasing in $k$ for all $\\alpha \\in (0,1)$ and $f(\\alpha ,k) < \\pi /2$ for all $\\alpha \\in [0,1]$ and $k \\in (0,1)$ .",
"Proof.",
"Since the inverse cosine function decreases monotonically, it is sufficient to show that the argument is monotonically increasing.",
"Let us consider its derivative $\\frac{\\partial }{\\partial k} \\bigg ( \\frac{k^{2\\alpha } + 1 - (1-k)^{2 \\alpha }}{2 k^\\alpha } \\bigg )\\sim k^{2 \\alpha } + (1-k)^{2 \\alpha } + 2 k (1-k)^{2 \\alpha - 1} - 1 =: s(\\alpha , k) \\ .$ The $\\sim $ indicates that we have dropped overall factors which were clearly positive, as we are only interested in the sign of the derivative.",
"First, note that $s(\\alpha = 0, k) = \\frac{1+k}{1-k} > 0$ and $s(\\alpha = 1, k) = 0$ for all $k$ .",
"Secondly, $s(\\alpha , k)$ is strictly decreasing with respect to $\\alpha $ : $\\frac{\\partial s}{\\partial \\alpha } =2 k^{2\\alpha } \\log (k) + 2(1-k)^{2\\alpha } \\log (1-k) + 4k (1-k)^{2\\alpha -1} \\log (1-k) < 0 \\ .$ Each of the above terms are negative due to the fact that $k, (1-k) \\in (0,1)$ .",
"Together it follows that $s(\\alpha , k)$ is positive for any combination of $\\alpha $ and $k$ and the proof of the monotonicity is complete.",
"The fact that $f(\\alpha , k \\rightarrow 0) \\rightarrow \\pi /2$ for all $\\alpha $ then immediately implies that $f(\\alpha ,k) < \\pi /2$ .",
"$$ Lemma D.2 For $\\alpha \\le 0.5$ , $h(\\alpha , k)$ increases monotonically in $k$ for $k \\in (0,1/2]$ and decreases for larger values of $k$ .",
"Conversely, for $\\alpha \\ge 0.5$ , $h(\\alpha , k)$ increases first until in reaches a maximum at $k=1/2$ and decreases afterwards.",
"Proof.",
"The argument here follows the proof of Lemma 12.14 in [2].",
"Due to the monotonicity of the inverse cosine function, it is again sufficient to investigate the expression in the argument.",
"We rewrite this expression trivially, so that it becomes a function of the fraction $r := \\frac{k}{1-k}$ and consider the following derivative: $\\frac{\\partial }{\\partial r} \\bigg ( \\frac{(r+1 )^{2\\alpha } - r^{2 \\alpha } - 1}{2 r^\\alpha } \\bigg )\\sim -(r+1)^{2\\alpha } - r^{2\\alpha } + 1 + 2 r (r+1)^{2\\alpha - 1} =: t(\\alpha , r)$ We first note that $t(\\alpha = 0, r) = \\frac{r-1}{r+1} \\le 0$ and that $t(\\alpha = 0.5, r) = 0 $ as well as $t(\\alpha = 1, r) = 0$ for all $r$ .",
"Next, we show that $t(\\alpha , r)$ is a concave function with respect to $r$ for $r \\le 1$ , which corresponds to $k \\in (0,1/2]$ : $\\frac{\\partial ^2 t}{\\partial \\alpha ^2} = 4 (r -1) (r+1)^{2\\alpha -1} \\log ^2(r+1) - 4 r^{2\\alpha } \\log ^2(r) \\le 0 \\ ,$ since $r \\le 1$ .",
"Taken together, we conclude that $t(\\alpha ,r) \\le 0$ for all $\\alpha \\in [0,0.5]$ and that $t(\\alpha ,r) \\ge 0$ for all $\\alpha \\in [0.5,1]$ .",
"Moreover, since $r$ increases monotonically w.r.t.",
"$k$ , the monotonicity of $h(\\alpha , k)$ holds for $k \\in (0,1/2]$ .",
"Due to the symmetry of $h$ around $k = 1/2$ , the proof is complete.",
"$$ Lemma D.3 $f(\\alpha , k)$ decreases monotonically in $\\alpha $ for $\\alpha \\in [0.5,1]$ and $h(\\alpha , k)$ decreases monotonically in $\\alpha $ for all $\\alpha \\in [0,1]$ .",
"Proof.",
"As the inverse cosine function decreases monotonically, it is again sufficient to investigate the derivative of the function in the $\\arccos $ -argument.",
"For $f(\\alpha , k)$ , we consider $\\frac{\\partial }{\\partial \\alpha } \\bigg ( \\frac{k^{2\\alpha } + 1 - (1 \\!",
"- \\!",
"k)^{2\\alpha }}{2 k^a} \\bigg ) \\!= \\frac{1}{2} k^{-\\alpha } \\Big ( \\underbrace{ \\log (k)}_{< \\ 0} [ \\,\\underbrace{ k^{2 \\alpha } + (1 \\!",
"- \\!",
"k)^{2 \\alpha } - 1 }_{\\le \\ 0} \\, ] - 2 (1 \\!",
"- \\!",
"k)^{2 \\alpha } \\underbrace{\\log (1 \\!- \\!k)}_{< \\ 0} \\Big ) > 0 .$ To see that in fact the expression in the square bracket is smaller or equal to zero, we exploit that for $\\alpha \\ge 0.5$ the function $k \\mapsto k^{2\\alpha }$ is superadditive, so that $1 = 1^{2 \\alpha } = ( k + (1-k))^{2 \\alpha } \\ge k^{2 \\alpha } + (1-k)^{2 \\alpha }$ .",
"For $h(\\alpha , k)$ , we consider $\\frac{\\partial }{\\partial \\alpha } \\bigg ( \\frac{1 - k^{2 \\alpha } - (1-k)^{2 \\alpha }}{2 k^\\alpha (1-k)^\\alpha } \\bigg ) = - \\frac{1}{2} (1-k)^{-\\alpha } k^{-\\alpha } \\Big (&\\underbrace{\\log (k)}_{< \\ 0} \\, [ k^{2 \\alpha } + 1 - (1-k)^{2 \\alpha } ] + \\\\&\\underbrace{ \\log (1- k)}_{< \\ 0} \\, [ (1-k)^{2 \\alpha } + 1 - k^{2 \\alpha } ]\\Big ) > 0 \\ .$ Since $k, (1-k) \\in (0,1)$ clearly the expressions in square brackets are positive so that the overall expression is positive too.",
"In the respective regions, $f(\\alpha , k)$ and $h(\\alpha , k)$ are thus monotonically decreasing in $\\alpha $ .",
"$$"
],
[
"Non-optimality of higher-degree branchings",
"This section supplements Sect.",
"REF of the paper.",
"First, we address the third and most involved 4-branching scenario in which a coupled 4-BP connects one source and three sinks (or equivalently 3 sources and 1 sink), see Fig.",
"REF .",
"We derive the inequalities listed in Proposition REF and prove them analytically for a large subset of the parameter space.",
"For the remainder we present a numerical argument (App.",
"REF ).",
"Lastly, we show by induction how, given that coupled 4-BPs are never globally optimal, one can further rule out coupled $n$ -BPs (with $n$ effective neighbors) for all $n > 4$ (App.",
"REF )."
],
[
"Non-optimality of coupled 4-BPs between one source and three sinks",
"Let us start by providing the derivation of Proposition REF , which we repeat here for completeness: Proposition E.1 Given a BOT problem with one source and three sinks, with demands $m_1, m_2, m_3$ as in Fig.",
"REF , a coupled 4-BP away from the terminals cannot be globally optimal if at least one of the following inequalities holds true: $\\Gamma &= h \\Big (\\frac{m_1}{m_1 + m_2} \\Big ) - f(m_1) + h \\Big (\\frac{m_3}{m_3 + m_2} \\Big ) - f( m_3) > 0, \\\\\\Gamma _{1,*} &= f(1 -m_*) + f\\Big (1 - \\frac{m_2}{1-m_*} \\Big ) - f(1-m_*-m_2) > 0, \\\\\\Gamma _{2,*} &= h \\Big (\\frac{m_*}{m_* + m_2} \\Big ) + f\\Big (\\frac{m_2}{1 - m_*} \\Big ) - h(m_*) > 0$ where $*=1,3$ .",
"Note that $\\Gamma = \\Gamma _{1,1} + \\Gamma _{1,3} = \\Gamma _{2,1} + \\Gamma _{2,3}$ .",
"Note that it is an important specification that we consider coupled 4-BPs away from the terminals.",
"For instance, in the OT case where $\\alpha = 1$ , all BPs are located at the terminals and coupled BPs with arbitrary number of neighbors may be globally optimal.",
"For all following considerations, we assume that $\\alpha \\in [0,1)$ .",
"Figure: Coupled 4-BP between one source and three sinks."
],
[
"Derivation of the $\\Gamma $ -inequality",
"WLOG, let us normalize the masses so that $m_1 + m_2 + m_3 =1$ and determine the necessary conditions under which all V-branchings are optimal: $\\gamma _1 &\\ge \\pi - f\\Big (\\alpha , 1- \\frac{1-m_1}{1}\\Big ) = \\pi - f(\\alpha , m_1) \\ , \\nonumber \\\\\\gamma _2 &\\ge \\pi - f \\Big ( \\alpha , 1 - \\frac{1-m_3}{1}\\Big ) = \\pi - f(\\alpha , m_3) \\ , \\nonumber \\\\\\gamma _3 &\\ge h \\Big (\\alpha , \\frac{m_3}{m_3 + m_2} \\Big ) \\ , \\nonumber \\\\\\gamma _4 &\\ge h \\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) \\ .$ We intend to show that such a 4-BP can never be globally optimal by showing that for any combination of $\\alpha $ and the masses $m_i$ the sum of the lower bounds is already larger than $2 \\pi $ .",
"This is equivalent to proving that the following inequality holds true for all parameter combinations: $\\Gamma := h \\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) - f(\\alpha , m_1) + h \\Big (\\alpha , \\frac{m_3}{m_3 + m_2} \\Big ) - f(\\alpha , m_3) > 0 \\ .$ The inequality reflects that the problem setup is inherently symmetric under exchange of $m_1$ and $m_3$ .",
"WLOG, we assume that $m_1 \\le m_3$ ."
],
[
"Derivation of the $\\Gamma _{1,*}$ - and {{formula:2319981d-25b2-4794-a10a-81724537d3b3}} -inequalities",
"For a BOT solution to be globally optimal it means that it is the cheapest relatively optimal solution of all possible full tree topologies.",
"In our case of four terminals, there are three distinct topologies, see Fig.",
"REF .",
"Let us assume that a globally optimal 4-BP away from the terminals exists and denote the terminal positions by $a_i$ .",
"Then, for all three topologies $T_1$ , $T_2$ and $T_3$ , this branching point configuration is the ROS, since clearly for all $T_i$ a coupled 4-BP configuration can be realized by coupling the two branching points.",
"Figure: The three distinct full tree topologies connecting four terminals.Let us investigate graphically under which conditions a coupled 4-BP provides the relatively optimal solution for the different topologies.",
"Figure REF shows the pivot circle and pivot point construction for topology $T_1$ , where $a_0$ has been chosen as root node.",
"Let us refer to the line through $p_2$ and the intersection of the two pivot circles as transition line, for the following reason: If $a_0$ was positioned to the left of the transition line, the ROS of $T_1$ would be non-degenerate, as shown in Fig.",
"REF for instance.",
"For $a_0$ to the right of the transition line, the ROS of $T_1$ is given by a coupled 4-BP.",
"Hence, the transition line marks the transition between a non-generate ROS of $T_1$ and a coupled 4-BP solution.",
"Consequently, $a_0$ must lie to the right of the transition line of $T_1$ .",
"But, $a_0$ must simultaneously also lie on the appropriate side of the transition lines of the two other topologies $T_2$ and $T_3$ .",
"One can now argue that the root node $a_0$ can be moved along a continuous path onto the transition line of topology $T_1$ , without crossing any of the other transition lines.",
"In doing so, the coupled 4-branching stays relatively optimal for all three topologies.",
"Most importantly, it thereby stays globally optimal.",
"During this procedure the terminals $a_1$ , $a_2$ and $a_3$ stay fixed so that the pivot points and pivot circles as well as the transition lines stay exactly the same.",
"For topology $T_1$ we have now arrived at a special case of coupled 4-branching, in which the V- and L-branchings are transient (see Def.",
"REF ).",
"Let us refer to such a BP configuration as transient 4-branching.",
"As a consequence, the angle $\\gamma _4$ may be expressed in terms of the following branching angles (see Fig.",
"REF ) $ \\gamma _4 = \\beta _1 + \\beta _2 - \\theta _1 = h(\\alpha , m_1) - f\\Big (\\alpha , \\frac{m_2}{1-m_1} \\Big ) \\ .$ Figure: Transient coupled 4-BP with a 0 a_0 placed on the transition line of topology T 1 T_1.All in all, the above argument shows that the existence of a globally optimal coupled 4-BP necessarily implies the existence of a globally optimal transient 4-branching, which can be constructed by changing only the coordinates of one of the terminals.",
"Clearly, for the globally optimal and transient 4-BP of topology $T_1$ all four necessary conditions for optimal V-branching must still apply.",
"In order to generally rule out globally optimal coupled 4-BPs, it is therefore sufficient to show that at least one of the following two conditions is fulfilled for all parameter combinations $\\alpha $ and $m_i$ : the sum of the lower bounds on $\\gamma _i$ always exceeds $2 \\pi $ , condition (REF ) is always incompatible with Eq.",
"(REF ).",
"The first condition is equivalent to the following inequality, obtained by substituting the lower bound for $\\gamma _4$ in (REF ) by (REF ): $ \\Gamma _{1,1}(\\alpha , m_1,m_2) := f(\\alpha , 1 -m_1) + f\\Big ( \\alpha , 1 - \\frac{m_2}{1-m_1} \\Big ) - f(\\alpha , 1-m_1-m_2) > 0 \\ .$ The second condition can be expressed as inequality simply by combining (REF ) and (REF ): $\\Gamma _{2,1}(\\alpha , m_1, m_2) := h \\Big (\\alpha , \\frac{m_1}{m_1 + m_2} \\Big ) + f\\Big ( \\alpha , \\frac{m_2}{1 - m_1} \\Big ) - h(\\alpha , m_1) > 0 \\ .$ Proving one of these two inequalities already suffices to rule out globally optimal 4-branching between one source and three sinks.",
"Note that the three inequalities presented so far are not completely independent but are related via $\\Gamma = \\Gamma _{1,1} + \\Gamma _{2,1}$ .",
"The above procedure of moving the root node onto the transition line of topology $T_1$ can be repeated exactly analogously for $T_2$ .",
"This results in the inequalities $\\Gamma _{1,3}$ and $\\Gamma _{2,3}$ , which are of the exact same form except that $m_3$ appears in all places instead of $m_1$ .",
"Note that, in both cases, we have used that $a_0$ can be moved onto the transition lines of $T_1$ and $T_2$ without crossing the transition line of topology $T_3$ .",
"A justification for this is given in form of the following lemma.",
"Lemma E.2 Starting from a globally optimal coupled 4-BP connecting the terminals $\\lbrace a_i\\rbrace $ and located away from all terminals, one terminal node may be moved along a continuous path onto the transition lines of topology $T_1$ and $T_2$ without crossing the transition line of topology $T_3$ first.",
"Proof.",
"Let us give a proof by contradiction.",
"Assume that that $a_0$ could actually be moved along a continuous path onto the transition line of topology $T_3$ without touching any of the other two transition lines.",
"Then, one may also move $a_0$ infinitesimally further across the transition line of topology $T_3$ , such that the ROS of $T_3$ becomes non-degenerate.",
"At the same time, for topology $T_1$ and $T_2$ the coupled 4-branching configuration stays relatively optimal.",
"Now, since the ROS of $T_3$ deviates from the coupled 4-branching configuration, the 4-branching can no longer be globally optimal.",
"Note that a coupled 4-BP can only be globally optimal if all three topologies agree.",
"Consequently, the ROS of $T_3$ must be globally optimal.",
"However, the non-degenerate solution of $T_3$ will necessarily contain a cycle.",
"To see this, let $b_1$ be the branching point connected to $a_0$ and $b_2$ the other branching point in $T_3$ .",
"After crossing the transition line of $T_3$ , the edge $(b_1, b_2)$ at first has finite but infinitesimal length, so that either one of the edges $(b_2,a_1)$ or $(b_2,a_3)$ must intersect with $(b_1,a_2)$ , thereby creating a cycle in the ROS of $T_3$ (cf.",
"Fig.",
"REF ).",
"However, it was proven in [2] that for $\\alpha \\in [0,1)$ cyclic BOT solution cannot be globally optimal, so that we have arrived at a contradiction.",
"$$ To summarize, we have shown the following lemma: Lemma E.3 The existence of any globally optimal 4-BP connecting one source and three sinks and located away from the terminals implies the existence of two globally optimal transient 4-branchings, one for topology $T_1$ and one for $T_2$ ."
],
[
"$\\Gamma > 0 \\ $ for {{formula:b6d5d064-a7aa-4620-b08e-8d52134ebf39}} :",
"For $\\alpha \\le 0.5$ we have $h \\ge \\pi /2$ and $f < \\pi /2$ for all combinations of $m_i$ (see App. )",
"and thus clearly $\\Gamma > 0$ if $\\alpha \\le 0.5$ .",
"For all following arguments, we therefore assume that $\\alpha >0.5$ .",
"The following analytical arguments all rely on the properties of $h$ and $f$ , which are listed and proven in App.",
"."
],
[
"$\\Gamma > 0 \\ $ for {{formula:4340b88c-d10b-4164-adc8-fcac7451b1ad}} :",
"Using that $f(\\alpha , k)$ is monotonically decreasing in $k$ , the following loose lower bound suffices to demonstrate that $\\Gamma > 0$ if $m_1 \\ge 1/4$ : $\\Gamma &\\ge 2 \\ \\min _k \\, h(\\alpha , k) - 2 f(\\alpha , m_1) \\\\&= 2 \\arccos (2^{2 \\alpha - 1} - 1) - 2 \\arccos \\bigg ( \\frac{m_1^{2\\alpha } + 1 - (1-m_1)^{2 \\alpha }}{2 m_1^\\alpha } \\bigg ) \\ ,$ where we have used that $f(m_1) \\ge f(m_3)$ due to our assumption $m_1 \\le m_3$ and that $h(\\alpha , k)$ forms a minimum at $k=1/2$ .",
"For $m_1 = 1/4$ , one finds that this expression is truly positive if and only if $9^\\alpha < 2 \\cdot 4^\\alpha + 1 \\ ,$ which is fulfilled for all $\\alpha \\in [0,1)$ due to the subadditivity of the function $m \\mapsto m^\\alpha $ , namely $9^\\alpha = (4 + 4 + 1)^\\alpha < 4^\\alpha + 4^\\alpha +1$.",
"It follows that $\\Gamma > 0$ also for all $m_1 > 1/4$ due to the monotonicity of $f$ .",
"Figure: m 1 m_1-m 2 m_2 parameter space largely ruled out analytically.",
"In the white region the inequalities are checked numerically."
],
[
"$\\Gamma _{2,3} > 0 \\ $ for {{formula:1550a853-7ad7-41ff-84d6-e6ee54b09d65}} :",
"Inequalities $\\Gamma _{2,3}$ is fulfilled if $m_3 \\ge 1/2$ , since $\\Gamma _{2,3}(\\alpha , m_3, m_2) > h \\Big (\\alpha , \\underbrace{\\frac{m_3}{m_3 + m_2}}_{> \\ m_3} \\Big ) - h(\\alpha , m_3) \\ge 0 \\ \\text{ for } \\ m_3 \\ge 1/2 \\ ,$ where we have used that for $\\alpha > 0.5$ , $h(\\alpha , k)$ is monotonically increasing in $k$ for $k \\in [0.5,1)$ , as shown in Lem.",
"REF .",
"The remaining parameter region, for which none of the inequalities have been shown yet, can be characterized by the following conditions: $ \\alpha > 0.5 \\, , \\ \\ m_1 < 0.25 \\ \\text{ and } \\ m_2 \\in [0.5 - m_1, 1 - 2 m_1] \\ .$ The constraints for $m_3$ are implicitly represented, using that the normalization was chosen such that $m_1 +m_2 +m_3 = 1$ .",
"A visualization of the remaining region can be found in Fig.",
"REF .",
"Finally, we propose a simple and watertight numerical scheme, which can be used to rule out almost the entire remaining volume, already with little numerical effort."
],
[
"Numerical treatment of the remaining parameter space",
"For the remaining parameter region characterized by the conditions in (REF ), we propose a numerical scheme which checks the inequality $\\Gamma _{2,1}$ for all practically relevant parameter combinations of $m_1$ , $m_2$ and $\\alpha $ .",
"For this, we split the remaining volume into cuboids $ I = [\\check{\\alpha }, \\hat{\\alpha }] \\times [\\check{m}_1, \\hat{m}_1] \\times [\\check{m}_2, \\hat{m}_2]$ which are divided recursively into smaller cuboids based on the following octree scheme.",
"For each cuboid, we determine a lower bound of $\\Gamma _{2,1}$ .",
"This lower bound becomes tighter the smaller the cuboid $I$ .",
"If, for a cuboid, this lower bound is not yet positive, it is divided further into eight new cuboids by splitting each of the three intervals in half.",
"This procedure is iterated until for all cuboids the lower bound is truly positive.",
"For all $m_1, m_2, \\alpha $ in a cuboid $I=[\\check{\\alpha }, \\hat{\\alpha }] \\times [\\check{m}_1, \\hat{m}_1] \\times [\\check{m}_2, \\hat{m}_2]$ , the lower bound of $\\Gamma _{2,1}$ is obtained by minimizing each summand individually: $\\Gamma _{2,1}(\\alpha , m_1, m_2) &\\ge \\underset{I}{\\min } \\ h \\big (\\alpha , \\frac{m_1}{m_1 + m_2} \\big ) + \\underset{I}{\\min } \\ f\\big ( \\alpha , \\frac{m_2}{1 - m_1} \\big ) - \\underset{I}{\\max } \\ h(\\alpha , m_1) \\nonumber \\\\&= h \\Big (\\hat{\\alpha }, \\frac{\\hat{m}_1}{\\hat{m}_1 + \\check{m}_2} \\Big )+ f \\Big ( \\hat{\\alpha }, \\frac{ \\hat{m}_2}{1 - \\hat{m}_1} \\Big )- h(\\check{\\alpha }, \\check{m}_1)$ where we have used the crucial fact that for the remaining volume, characterized by (REF ), the functions $h(\\alpha , k)$ and $f(\\alpha , k)$ are monotonically decreasing in both arguments.",
"The proof of these monotonicities can be found in App. .",
"Note that the described procedure allows to rigorously confirm the inequality across a continuous region with finitely many evaluations.",
"For $\\alpha \\rightarrow 1$ or $m_1 \\rightarrow 0$ , the value of $\\Gamma _{2,1}$ approaches zero, so that the above scheme cannot be used to proof the inequality for values arbitrarily close to these limits.",
"However, if we restrict us to $m_1 > \\delta $ and $\\alpha < 1 - \\epsilon $ for finite $\\epsilon , \\delta > 0$ , the inequality $\\Gamma _{2,1}$ can be shown for practically all parameter combinations with little numerical effort.",
"For $\\epsilon = \\delta = 10^{-3}$ , using the proposed scheme, it was checked in only a few minutes that the lower bound in Eq.",
"(REF ) is larger than $10^{-4}$ everywhere in the remaining volume.",
"To be numerically on the safe side, we have stopped splitting a cuboid not if the lower bound exceeded zero but set a suitable finite threshold, in this case $10^{-4}$ .",
"In other words, we have stopped splitting a cuboid if its respective lower bound was $>10^{-4}$ .",
"The smallest terms which occur during the arithmetic operations inside the functions $h$ and $f$ are of the order $\\delta ^2$ , the largest terms are of order one.",
"It is therefore safe to say that numerical errors at the order of the machine accuracy are negligibly small against the margin of $10^{-4}$ and we may say that all together globally optimal 4-BPs are ruled out, for all practical parameter combinations.",
"In principle, the presented scheme can be used to check the inequality $\\Gamma _{2,1}$ up to even smaller $\\epsilon $ and $\\delta $ .",
"The Python code of the numerical scheme is made available at https://github.com/hci-unihd/BranchedOT."
],
[
"Non-optimality of five- and higher-degree branchings",
"In this section, we formally prove that globally optimal coupled $n$ -BPs not coincident with a terminal can be ruled out in general, given that 4-BPs are not globally optimal.",
"Lemma REF states that a solution is not globally optimal if any subsolution is not globally optimal.",
"It will therefore suffice to study the coupled BP as an isolated subproblem.",
"Let us start by proving the following corollary about the preservation of relative optimality under edge extensions for transient V- and L-branchings (see Def.",
"REF ): Corollary E.4 Consider a BOT problem with terminals $A=\\lbrace a_i\\rbrace $ and let the BP configuration $B = \\lbrace b_i \\rbrace $ be relatively optimal for a given topology $T$ .",
"Let $B$ contain a transient V-branching between two terminals, say $a_0$ and $a_1$ and denote the branching point connected to $a_0$ and $a_1$ by $b_1$ .",
"Otherwise, let $B$ not contain any strict L-or V-branchings.",
"Let us further denote the branching point to which $b_1$ is coupled in a transient V-branching by $b_2$ , as illustrated on the left side of Fig.",
"REF .",
"Then, there exists a direction in which the zero length edge $(b_1, b_2)$ can be extended to finite length $l>0$ (cf.",
"right side of Fig.",
"REF ) such that the new BOT problem (with shifted terminal positions) is solved relatively optimally by the new BP configuration (with shifted $b_1$ ).",
"Proof.",
"By assumption, the BP configuration of interest $B = \\lbrace b_i \\rbrace $ contains only Y-branchings and transient V- and L-branchings.",
"Then, according to Lemma REF , there exists a set of arbitrarily small displacements $\\delta _i$ , one for each branching point $b_i$ , so that $B + \\delta = \\lbrace b_i + \\delta _i \\rbrace $ is a non-degenerate BP configuration with arbitrarily small gradients $\\left\\Vert \\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}(B+\\delta , A) \\right\\Vert $ .",
"Note that the notation $C(B,A)$ emphasizes that the cost function also depends on the terminal positions.",
"Since the BP configuration $B + \\delta $ is non-degenerate, any edge, in our case $(b_1,b_2)$ , can be easily expanded in length without changing any of the branching angles, assuming that the extension preserves the direction of the edge and that the BPs and terminals are moved along correspondingly.",
"As the gradient $\\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}$ depends only on the directions of the edges meeting at $b_i$ (i.e.",
"the branching angles), the gradient is not changed by this procedure.",
"Let us summarize the shifted BPs by $B_{shift}$ and the shifted terminals by $A_{shift}$ .",
"Then, $\\left\\Vert \\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}(B + \\delta , A) \\right\\Vert = \\left\\Vert \\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}(B_{shift} + \\delta , A_{shift}) \\right\\Vert $ can also be made arbitrarily small as $\\delta \\rightarrow 0$ .",
"Let us prove by contradiction that $B_{shift}$ is the ROS of the BOT problem with terminals $A_{shift}$ : Assuming that a different BP configuration $B_0 \\ne B_{shift}$ was the ROS, there would exist a constant $\\Delta >0$ , so that $\\mathcal {C}(B_{shift} , A_{shift}) = \\mathcal {C}(B_0, A_{shift}) + \\Delta $ .",
"However, since the cost is a convex function w.r.t.",
"the BPs, we have $\\mathcal {C}(B_0, A_{shift}) \\ge \\mathcal {C}(B_{shift} + \\delta , A_{shift})+ \\langle \\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}(B_{shift} + \\delta , A_{shift}), B_0 - (B_{shift} + \\delta ) \\rangle \\\\= \\mathcal {C}(B_{shift} , A_{shift}) + \\underbrace{O(\\delta ) + \\langle \\nabla _{\\!",
"\\!",
"b_i} \\, \\mathcal {C}(B_{shift} + \\delta , A_{shift}), B_0 - (B_{shift} + \\delta ) \\rangle }_{=:K} \\, .$ As $\\delta \\rightarrow 0$ , the latter two terms summarized by $K$ can clearly be made arbitrarily small.",
"In particular, there exists an $\\delta > 0$ , such that $\\vert K \\vert < \\Delta /2$ , which together with $\\mathcal {C}(B_{shift} , A_{shift}) = \\mathcal {C}(B_0, A_{shift}) + \\Delta $ implies that $\\mathcal {C}(B_0, A_{shift}) \\ge \\mathcal {C}(B_{shift}, A_{shift}) + K > \\mathcal {C}(B_0, A_{shift}) + \\Delta /2$ and we have thereby arrived at a contradiction, similarly to the reasoning in part (c) of the proof in App. .",
"$$ Figure: Extension of the edge (b 1 ,b 2 )(b_1,b_2) in the direction of the pivot point p 1 p_1, preserving relative optimality.Coming back to the non-optimality of coupled $n$ -BPs, let us, for concreteness, consider a globally optimal coupled 5-BP not coincident with a terminal.",
"By induction, repeating the presented argument one can then rule out all globally optimal $n$ -BPs.",
"The proof is by contradiction, so let us assume a globally optimal 5-BP existed, between terminals $a_0$ , $a_1$ , $a_2$ , $a_3$ and $a_4$ .",
"In terms of enclosed angles $\\gamma _i$ (see Fig.",
"REF ), it is a necessary condition that all $\\gamma _i$ must exceed their respective lower bound, specified by the optimal V-branching conditions in Table REF (Section REF ).",
"Again, for a 5-BP configuration to be globally optimal all possible full tree topologies must agree on the 5-BP configuration as their relative optimal solution.",
"Starting from this configuration, one may continuously move one of the terminals, e.g.",
"$a_0$ , such that the globally optimal coupled 5-BP starts to decouple, meaning that $a_0$ is moved until for (at least) one of the possible topologies, say $\\tilde{T}$ , the 5-BP configuration no longer provides the ROS.",
"This can always be achieved, for instance by bringing $a_0$ sufficiently close to $a_1$ so that a Y-branching between the two terminals becomes optimal.",
"Similar to the 4-branching case, the ROS of such a topology $\\tilde{T}$ in this moment becomes the globally optimal solution, as the other topologies are still in the 5-branching configuration which can only be globally optimal if all topologies agree on it.",
"The globally optimal solution of topology $\\tilde{T}$ must exhibit one of the following two properties it contains one Y-branching and a coupled 4-BP or it is non-degenerate and contains only Y-branchings.",
"In the first case, it would mean that a coupled 4-BP exists which is globally optimal as subgraph of a globally optimal solution (using the necessarily optimal substructure of Lem.",
"REF ).",
"This contradicts our assumption that 4-BPs are not globally optimal.",
"Figure: Edge extension to finite length l>0l > 0 in a globally optimal and transient 5-branching with transient V-branching between a 0 a_0 and a 1 a_1.",
"The resulting ROS is split into two subsolutions, as indicted by the red dashed line.Regarding the second option, we proceed as follows.",
"Let us move $a_0$ back to the point in which the coupled 5-BP configuration was still globally optimal but the ROS of $\\tilde{T}$ is on the verge of decoupling into a non-degenerate branching configuration.",
"Denote this special location of the terminals by $\\lbrace a_i^* \\rbrace $ .",
"In this configuration an infinitesimal movement of $a_0$ away from $a_0^*$ can cause the ROS of $\\tilde{T}$ to transition from coupled 5-branching to a non-degenerate BP configuration, very much analogous to the case of the transient 4-branching illustrated in Fig.",
"REF .",
"This means that in this configuration all V- and L-branchings are transient in the ROS of $T$ and WLOG we choose the labeling of the terminals such that the V-branching between $a_0^*$ and $a_1^*$ is transient, cf. Fig.",
"REF .",
"At this point, let us split the set of all possible full tree topologies $\\mathcal {T}$ into the following subsets.",
"The subset in which the terminals $a_0$ and $a_1$ are connected to a common branching point, say $b_1$ , is denoted by $\\mathcal {T}^{(0,1)}$ .",
"Let us label the BP to which $b_1$ is connected in these topologies by $b_2$ .",
"The ROS of all topologies $T \\in \\mathcal {T}^{(0,1)}$ for the current BOT problem contains a transient V-branching at branching point $b_1$ .",
"Moreover, let us single out a specific subset in $\\mathcal {T}^{(0,1)}$ , defined by the following condition: $\\mathcal {T}^{(0,1)}_{trans} = \\lbrace T \\in \\mathcal {T}^{(0,1)} : \\text{ROS of $T$ is transient if the terminals are located at $a_i^*$} \\rbrace \\ .$ Visibly, the branching at $b_1$ appears as V-branching but it may also be seen as a Y-branching with a zero length stub.",
"Let us now extend this zero length edge between $b_1$ and $b_2$ to finite length $l>0$ into the direction of the pivot point between $a_0^*$ and $a_1^*$ , as explained in the proof of Corollary REF .",
"The two terminals $a_1$ and $a_2$ are shifted from $a_0^*$ and $a_1^*$ to $a_0(l)$ and $a_1(l)$ and Corollary REF guarantees that the resulting BP configuration for all $T \\in \\mathcal {T}^{(0,1)}_{trans}$ solves the new BOT problem relatively optimally.",
"For an illustration of the edge extension see Fig.",
"REF .",
"Note that for all $T \\in \\mathcal {T}^{(0,1)}_{trans}$ this ROS is the same.",
"Now split this ROS into two subsolutions as indicted in Fig.",
"REF .",
"This induces two subproblems and subtopologies, as described in Def.",
"REF .",
"We focus on the upper right subproblem, consisting of four terminals.",
"Note that any topology $T^{(4)}$ on this four terminal subproblem, may be induced as subtopology by a topology $T \\in \\mathcal {T}^{(0,1)}$ .",
"Let us distinguish the following two cases: a) On the four terminal subproblem the topology of the GOS, denoted by $T^{*(4)}$ , is induced by a topology $T \\in \\mathcal {T}^{(0,1)}_{trans}$ .",
"Or b) $T^{*(4)}$ is induced by a topology $T \\in \\mathcal {T}^{(0,1)} \\setminus \\mathcal {T}^{(0,1)}_{trans}$ .",
"In case a) the ROS of $T^{*(4)}$ is given by the right subsolution in Fig.",
"REF , as the edge extension preserved the relative optimality for all $T \\in \\mathcal {T}^{(0,1)}_{trans}$ .",
"Hence, the globally optimal solution on the four terminal subproblem is given by a coupled 4-BP and we have arrived at a contradiction.",
"Otherwise, in case b), we do not know the ROS of $T^{*(4)}$ a priori, but it cannot be a coupled 4-branching configuration and must hence be non-degenerate.",
"Crucially, it must be non-degenerate for any finite length extension $l > 0$ , even if we consider the limit of $l \\rightarrow 0$ .",
"But this means that the ROS of $T^{*(4)}$ already transitions from a coupled 4-BP into a non-degenerate ROS, if the terminals $a_0$ and $a_1$ are perturbed infinitesimally ($l > 0$ but infinitesimal).",
"Consequently, for $l=0$ the ROS of $T^{*(4)}$ is transient.",
"But this means that $T^{*(4)}$ can be induced as a subtopology of a transient topology $T \\in \\mathcal {T}^{(0,1)}_{trans}$ and we have arrived yet at another contradiction.",
"$$"
],
[
"BOT on two-dimensional Riemannian manifolds",
"In the following section, we prove that the optimal branching angles for Y-shaped branchings on two-dimensional manifolds are the same as the optimal branching angles derived for BOT in the Euclidean plane.",
"An outline of the proof was given in Section REF .",
"The same strategy, presented below for the optimal branching angles, can be used to generalize other necessary conditions for optimal BOT solutions to manifolds, as explained in Sect.",
"REF ."
],
[
"Optimal Y-branching on two-dimensional Riemannian manifolds",
"For a Y-shaped branching at BP $b$ connecting the terminals $a_0$ , $a_1$ and $a_2$ in Euclidean plane the cost function of BOT is given by $\\mathcal {C}(b) = \\sum _i m_i^\\alpha \\left\\Vert a_i - b \\right\\Vert _2 \\ ,$ where the $m_i$ are the known edge flows.",
"Let us consider a two-dimensional Riemannian manifold $\\mathcal {M}$ embedded into $\\mathbb {R}^3$ , for which the metric is induced by the standard Euclidean inner product in $\\mathbb {R}^3$ .",
"Let the geodesic distance be denoted by $d(x,y)$ .",
"The generalized cost function for 1-to-2 branching then reads $\\mathcal {C}_M(b) = \\sum _i m_i^\\alpha \\, d( a_i, b) \\ .$ All points $b$ and $a_i$ now lie on the manifold and are assumed to have differing positions.",
"In a solution which minimizes $\\mathcal {C}_M$ the terminals $a_i$ are connected to $b$ via geodesics.",
"For $b$ to be a valid solution to the BOT problem on the manifold, these geodesics must exist.",
"We denote the geodesic which connects $b$ and $a_i$ by $v_i(\\lambda ) \\in \\mathcal {M}$ , parametrized by the length $\\lambda $ .",
"The tangent space at $b$ is denoted by $T_b \\mathcal {M}$ .",
"Furthermore, let $\\hat{n}_i$ be the unit tangent vectors of the geodesics $v_i(\\lambda )$ at the branching point $b$ , i.e.",
"$\\hat{n}_i = \\partial _\\lambda v_i(\\lambda ) \\big \\vert _{\\lambda = 0}$ .",
"WLOG, for all following considerations let us rotate and translate the manifold so that $b \\in \\mathbb {R}^3$ lies at the origin, i.e.",
"$b=0$ , and that the tangent space $T_b \\mathcal {M}$ is equal to the $x_1$ -$x_2$ -plane of $\\mathcal {R}^3$ , i.e.",
"$T_b \\mathcal {M} = \\mathbb {R}^2 \\times \\lbrace 0\\rbrace $ ."
],
[
"Restriction to a local subsolution on the manifold.",
"Since $\\mathcal {M}$ is embedded into $\\mathbb {R}^3$ , there exists an $r > 0$ and an environment $U(r) \\subset \\mathbb {R}^3$ around $b=0$ such that the manifold $\\mathcal {M} \\cap U(r)$ can be represented as the graph of a function: $\\mathcal {M} \\cap U(r) = \\lbrace (x, u(x)): x \\in D(r) \\subset \\mathbb {R}^2 \\rbrace \\ ,$ where $u$ is a smooth, scalar function $u: D(r) \\rightarrow \\mathbb {R}$ , defined on the disk of radius $r$ , denoted by $D(r) := \\lbrace (x_1, x_2) \\in \\mathbb {R}^2 : \\left\\Vert (x_1,x_2)^T \\right\\Vert _2 \\le r \\rbrace $ .",
"Note that due to the mentioned rotation and translation of $\\mathcal {M}$ , we have $u(0) = 0$ and $\\nabla u(0) = 0$ , where $\\nabla u$ denotes the gradient of $u$ .",
"The existence of such a function $u$ is guaranteed by the implicit function theorem.",
"A formal proof can be found in John M. Lee's book [18] in Proposition 8.24.",
"Further, let us define the orthogonal projection $\\sigma $ from the manifold onto the $x_1$ -$x_2$ -plane as $\\sigma : \\mathcal {M} \\cap U(r) \\rightarrow D(r) \\times \\lbrace 0\\rbrace , \\ \\begin{pmatrix} x_1 \\\\ x_2 \\\\ u(x_1,x_2) \\end{pmatrix} \\mapsto \\begin{pmatrix} x_1 \\\\ x_2 \\\\ 0 \\end{pmatrix} \\ .$ WLOG, $r$ is chosen sufficiently small so that this projection is bijective.",
"Now, Taylor's theorem states that $u$ can be approximated by the following expansion around $x = 0$ : $ u(x) = \\underbrace{u(0)}_{= \\ 0} + \\langle \\underbrace{\\nabla u(0)}_{= \\ 0}, x \\rangle + O(\\left\\Vert x \\right\\Vert _2^2) \\in O(r^2) \\ ,$ where $\\langle \\cdot , \\cdot \\rangle $ is the standard Euclidean inner product and we have introduced the Big-O notation.",
"A term is of order $O(r^2)$ if it goes to zero for $r \\rightarrow 0$ at least as fast as $r^2$ , or more formally: $p(r) \\in O(q(r)) \\ \\Leftrightarrow \\ \\lim _{r \\rightarrow 0} \\frac{p(r)}{q(r)} = c$ for some finite constant $c$ .",
"Consequently, a point $a = (x,u(x))$ in $\\mathcal {M} \\cap U(r)$ and its projection onto the plane $\\sigma (a)$ agree to first order, i.e.",
"$ \\left\\Vert \\sigma (a) - a \\right\\Vert _2 = u(x) \\in O(r^2) \\ .$ One of the key ingredients when transferring BOT problems from two-dimensional surfaces to the tangent plane $T_b \\mathcal {M}$ is the following Lemma about the difference between the Euclidean distance and the geodesic distance: Lemma F.1 (Relation between geodesic and Euclidean distance) Let $r$ be a small radius, which characterizes the environment $U(r)$ around the origin $b = 0$ on a two-dimensional Riemannian manifold as described above.",
"Let $a \\in \\mathcal {M} \\cap U$ be a point in this environment, located at $a = (x,u(x)) \\in \\mathbb {R}^3$ for some $x \\in D(r)$ and $u$ as above.",
"Then, the geodesic distance can be expressed through the Euclidean distance as $d(a,b) = \\left\\Vert a - b \\right\\Vert _2 + O(r^3) \\ .$ Proof.",
"Let us first note that $d(a,b) \\ge \\left\\Vert a - b \\right\\Vert _2$ .",
"It is hence sufficient to show that $d(a,b) \\le \\left\\Vert a - b \\right\\Vert _2 + O(r^3)$ .",
"Let us consider the following curve $\\gamma (t) = (tx, u(tx))$ on the manifold $\\mathcal {M}$ , which for $t \\in [0,1]$ connects $a = (x,u(x))$ and $b = 0$ .",
"In general, $\\gamma (t)$ is not a geodesic between $a$ and $b$ .",
"Thus, the length of $\\gamma (t)$ provides an upper bound to $d(a,b)$ .",
"For the calculation of the length we use that $\\gamma \\hspace{0.85355pt} ^{\\prime }(t) = (x, \\langle \\nabla u(tx), x \\rangle )$ and simply integrate $\\left\\Vert \\gamma \\hspace{0.85355pt} ^{\\prime }(t) \\right\\Vert _2$ along the curve in $\\mathbb {R}^3$ , since the metric of the embedding is induced by the standard Euclidean inner product in $\\mathbb {R}^3$ : $\\frac{d(a,b)}{\\left\\Vert a - b \\right\\Vert _2} &\\le \\frac{1}{\\sqrt{\\left\\Vert x \\right\\Vert ^2_2 + u^2(x)}} \\int _0^1 \\left\\Vert \\gamma \\hspace{0.85355pt} ^{\\prime }(t) \\right\\Vert _2 \\mathrm {d} t \\\\&\\le \\frac{1}{\\left\\Vert x \\right\\Vert _2} \\int _0^1 \\sqrt{\\left\\Vert x \\right\\Vert ^2_2 + \\vert \\langle \\nabla u(tx), x \\rangle \\vert ^2} \\ \\mathrm {d} t \\\\&= \\int _0^1 \\sqrt{1 + \\left| \\Big \\langle \\nabla u(tx), \\frac{x}{\\left\\Vert x \\right\\Vert _2} \\Big \\rangle \\right|^2} \\ \\mathrm {d}t \\\\&\\le 1 + \\frac{1}{2} \\int _0^1 \\left| \\Big \\langle \\nabla u(tx), \\frac{x}{\\left\\Vert x \\right\\Vert _2} \\Big \\rangle \\right|^2 \\mathrm {d}t \\\\&\\le 1 + \\frac{1}{2} \\int _0^1 \\left\\Vert \\nabla u(tx) \\right\\Vert _2^2 \\ \\mathrm {d}t \\ .$ For the last step we have used the Cauchy-Schwarz inequality.",
"Furthermore, we have used that $\\sqrt{1 + x^2} \\le 1 + x^2 / 2$ .",
"The only thing left to show is that the remaining integral is $O(r^2)$ , as $\\left\\Vert a - b \\right\\Vert = ( \\left\\Vert x \\right\\Vert ^2_2 + u^2(x) )^{1/2} \\in O(r)$ , since $\\left\\Vert x \\right\\Vert _2 \\le r$ and $u(x) \\in O(r^2)$, cf. Eq.",
"(REF ).",
"For that, let us consider the Taylor expansion of the $i$ -th component of $\\nabla u(tx)$ around 0.",
"For $i=1,2$ one has $\\partial _i \\, u(tx) = \\underbrace{\\partial _i \\, u(0)}_{= \\ 0 } + \\partial _j \\partial _i \\, u(0) \\ t x_j + O(r^2) \\ ,$ where the sum in $j$ is implicit and runs over both indices $j=1,2$ .",
"And thus: $\\frac{1}{2} \\int _0^1 \\left\\Vert \\nabla u(tx) \\right\\Vert _2^2 \\ \\mathrm {d}t = \\frac{1}{6} \\, \\partial _j \\partial _i \\, u(0) \\ x_j \\ \\partial _k \\partial _i \\, u(0) \\ x_k + O(r^2) \\ ,$ where again the sum over all index pairs is implied.",
"Since for all components $x_i$ we have $x_i \\le r$ , this completes the proof.",
"$$"
],
[
"Projection of a subproblem onto the tangent plane.",
"Coming back to the 1-to-2 BOT problem on the manifold, let us project the geodesics $v_i$ onto the $x_1$ -$x_2$ -plane using the orthogonal projection $\\sigma $ from above, in order to transfer the BOT problem from the manifold onto the tangent plane.",
"WLOG, we choose the orientation of the $x_1$ and $x_2$ axis such that $\\hat{n}_0$ points along the $x_1$ -axis.",
"Then, one may easily check that the implicit function theorem guarantees that for sufficiently small $r$ the projected geodesic $\\sigma (v_0)$ can be represented by a graph $(x_1, w(x_1))$ with smooth $w: (-\\epsilon ,r+\\epsilon ) \\rightarrow \\mathbb {R}$ .",
"Note, that the projected geodesic $\\sigma (v_0)$ may be considered on the open interval with $\\epsilon >0$ such that all derivatives with respect to $x_1 \\in [0,r]$ are well-defined.",
"Due to our special orientation of the $x_1$ -$x_2$ -plane, we have $w(0)=0$ and $w^{\\prime }(0) = 0$ .",
"Let us now define the following two points inside the tangent plane: Define $\\hat{a}_0$ as the point which lies in the direction of $\\hat{n}_0$ at a distance $r$ away from the origin, i.e.",
"$\\hat{a}_0 = r \\cdot \\hat{n}_0 = (r,0)$ .",
"Similarly, define $\\hat{a}_i = r \\cdot \\hat{n}_i$ .",
"These will be the three terminals of the BOT subproblem of interest in the Euclidean plane.",
"For an illustration see Fig.",
"REF .",
"Define $a_{\\perp ,0} = (r, w(r))$ as special point on the projected geodesic $\\sigma (v_0)$ .",
"From an analogous construction using the projected geodesics $\\sigma (v_1)$ and $\\sigma (v_2)$ , we obtain the $a_{\\perp ,i}$ also for $i=1,2$ .",
"No explicit representation of these will be necessary.",
"What is however important is that the two defined points agree up to linear order, in the sense that $\\left\\Vert \\hat{a}_i - a_{\\perp ,i} \\right\\Vert _2 \\in O(r^2)$ for all $i=0,1,2$ .",
"This is shown for $i=0$ , by Taylor expansion of $w(x_1)$ around $x_1 = 0$ , but holds of course equally true for $i=1,2$ : $ \\left\\Vert \\hat{a}_0 - a_{\\perp ,0} \\right\\Vert _2 = w(r)= \\underbrace{w(0)}_{= \\ 0} + \\underbrace{w^{\\prime }(0)}_{= \\ 0} \\, r + O(r^2) \\in O(r^2) .$ Note that since the points $a_{\\perp ,i}$ lie on the projected geodesics, after an inverse projection, the $\\sigma ^{-1}(a_{\\perp ,i})$ mark special points on the unprojected geodesics $v_i \\in \\mathcal {M}$ .",
"These will be the terminals of the subproblem of interest on the manifold.",
"Combining Eq.",
"(REF ) and (REF ), we can relate $\\sigma ^{-1}(a_{\\perp , i}) \\in \\mathcal {M}$ to the terminals $\\hat{a}_i$ of the BOT problem of interest in the plane by $ \\left\\Vert \\sigma ^{-1}(a_{\\perp , i}) - \\hat{a}_i \\right\\Vert _2 \\in O(r^2) \\ ,$ Note that all steps above can be repeated for radii smaller than the chosen $r$ .",
"In the following we will decrease the scale $r$ of the two problems and therefore indicate the $r$ -dependence in $\\sigma ^{-1}(a_{\\perp , i}(r))$ and $\\hat{a}_i(r)$ explicitly.",
"Figure: The tangent vectors of the geodesics (dotted) in bb define the position of the terminals a ^ i \\hat{a}_i of the subproblem on the flat disk D(r)⊂T b ℳD(r) \\subset T_b \\mathcal {M}."
],
[
"Improved subsolutions on the manifold by projection from the tangent space.",
"Now, follows the main line of arguments to show that $b$ does not connect $a_i \\in \\mathcal {M}$ at minimal cost if the geodesic angles in the tangent space $T_b \\mathcal {M}$ are different from the optimal angles in the Euclidean plane.",
"We start by considering the subproblem on the manifold with terminals $\\sigma ^{-1}(a_{\\perp , i}) \\in \\mathcal {M}$ .",
"The generalized cost function of this subproblem reads: $\\mathcal {C}_M(b) &= \\sum _i m_i^\\alpha \\, d(\\sigma ^{-1}(a_{\\perp ,i}(r)), b) \\ .$ We can now use Eq.",
"(REF ) to express the terminals on the manifold through the terminals $\\hat{a}_i$ in the tangent plane and express the geodesic distance in terms of the Euclidean distance, based on Lem.",
"REF .",
"All additional terms are at least of order $O(r^2)$ and we have $ \\mathcal {C}_M(b) = \\underbrace{\\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b \\right\\Vert _2}_{=: \\ \\mathcal {C}_b(r)} + \\ O(r^2) \\ .$ Crucially, with $\\mathcal {C}_b(r)$ we have arrived at the cost of the BOT problem which purely lives on the tangent space.",
"It is the cost of the solution in which $b$ is connected via straight lines to the terminals $\\hat{a}_i(r) = r \\cdot \\hat{n}_i$ at a distance $r$ away from $b$ , as shown in Fig.",
"REF .",
"By assumption, this solution is not relatively optimally since the angles (or equivalently the directions $\\hat{n}_i$ ) are assumed to deviate from the optimal angles in the Euclidean plane.",
"This means that a branching point $b^* \\in D(r)\\times \\lbrace 0 \\rbrace $ exists which provides a better solution with cost denoted by $\\mathcal {C}_{b^*}(r) < \\mathcal {C}_b(r)$ .",
"From the definition of $\\hat{a}_i = r \\cdot \\hat{n}_i$ , we see that two BOT problems within disks of different radii $D(r)$ and $D(\\nu \\cdot r)$ are related simply by rescaling the coordinates.",
"Under rescaling of coordinates the Euclidean distance between points and consequently also the cost functions changes proportionally, i.e.",
"$\\mathcal {C} \\rightarrow \\nu \\mathcal {C}$ .",
"Thus, there exist non-negative constants $M_1$ and $M_2$ , so that $\\mathcal {C}_b(r) = M_1 \\cdot r$ and $\\mathcal {C}_{b^*}(r) = M_2 \\cdot r$ .",
"Let us distinguish the following two cases: a) $\\ \\ \\ \\mathcal {C}_{b^*}(r) = \\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b^* \\right\\Vert _2 = 0$ .",
"This special case looks unusual, but is in principle possible, e.g.",
"if all terminals $a_i$ and $b$ lie on a common geodesic.",
"In the case, where $C_{b^*}(r) = 0$ , we project $b^*$ onto the manifold using $\\sigma ^{-1}$ and go backwards in the above steps.",
"We find that $0 = \\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b^* \\right\\Vert _2 =\\underbrace{\\sum _i m_i^\\alpha \\, d( \\sigma ^{-1}(a_{\\perp ,i}(r)) , \\sigma ^{-1}(b^*))}_{= \\ \\mathcal {C}_M(\\sigma ^{-1}(b^*)))} + O(r^2) \\, .$ Clearly, a sufficiently small $r > 0$ exists so that the terms contained in $O(r^2)$ are much smaller than $M_1 \\cdot r$ , so that $\\mathcal {C}_M(\\sigma ^{-1}(b^*))) = 0 + O(r^2) < M_1 \\cdot r + O(r^2) = \\mathcal {C}_M(b) \\ .$ This proves that the projection of $b^*$ onto the manifold provides a cheaper cost solution on the manifold than $b$ .",
"b) $\\ \\ \\ \\mathcal {C}_{b^*}(r) > 0$ , and thus $M_2 >0$ .",
"In this case, let us write the ratio of the two costs as $ \\mathcal {C}_b(r)/ \\mathcal {C}_{b^*}(r) = M_1 / M_2 =: 1 + \\kappa $ for some $\\kappa > 0$ .",
"Again, $b^*$ and its projection onto the manifold agree to first order, that is $\\left\\Vert \\sigma ^{-1}(b^*) - b^* \\right\\Vert \\in O(r^2)$ .",
"We now show that $\\sigma ^{-1}(b^*)$ provides a better solution to the BOT problem on the manifold than $b$ .",
"We proceed from Eq.",
"(REF ), using that $\\mathcal {C}_{b^*}(r)$ is linear in $r$ : $\\mathcal {C}_M(b) &= C_b(r) + O(r^2) = (1 + \\kappa ) \\, \\mathcal {C}_{b^*}(r) + O(r^2) \\nonumber \\\\&= \\sum _i m_i^\\alpha \\left\\Vert \\hat{a}_i(r) - b^* \\right\\Vert _2 + \\kappa M_2 \\cdot r + O(r^2) \\nonumber \\\\&= \\underbrace{\\sum _i m_i^\\alpha \\, d( \\sigma ^{-1}(a_{\\perp , i}(r)) , \\sigma ^{-1}(b^*))}_{= \\ \\mathcal {C}_M(\\sigma ^{-1}(b^*))} + \\kappa M_2 \\cdot r + O(r^2) \\ .$ In the last step, we have projected inversely onto the manifold and replaced the Euclidean distance by the geodesic distance (all with differences at least of $O(r^2)$ ).",
"We have therefore arrived at the cost of the new solution on the manifold.",
"Since the term $\\kappa M_2 \\cdot r$ is positive, we conclude that $r$ may always be chosen sufficiently small, so that the linear term dominates over higher-order terms and we have: $\\mathcal {C}_M(\\sigma ^{-1}(b^*)) < \\mathcal {C}_M(b) \\ .$ Together with Lem.",
"REF on the necessarily optimal substructure of all subsolutions, this concludes the proof that any optimally placed branching point on a two-dimensional embedded Riemannian manifold must exhibit the same optimal branching angles as in the Euclidean plane.$$"
],
[
"Other local properties of optimal BOT solutions on manifolds",
"The logic of the proof outlined above can be easily extended to the conditions for optimal V- and L-branching.",
"For BOT problems in the Euclidean plane, Table REF lists these conditions under which the optimal BP position in a 1-to-2 branching coincides with one of the terminals.",
"Let us now consider a 1-to-2 BOT solution on a two-dimensional manifold in which $b$ is located at the terminal $a_0$ and is connected to the terminals $a_1$ and $a_2$ via the geodesics $v_1$ and $v_2$ respectively.",
"Imagine that in the tangent plane $T_b \\mathcal {M}$ the angle enclosed by these geodesics does not fulfill the optimal V-branching criterion.",
"In this case, one may again consider a sufficiently small disk of radius $r$ around $b = a_0$ in the tangent space $T_b \\mathcal {M}$ and project the corresponding subproblem from the manifold onto this disk.",
"In the Euclidean plane, an improved BP location $b^*$ must exists, since $b = a_0$ is the optimal solution if and only if the V-branching condition is fulfilled.",
"All arguments about the scaling of the cost improvement apply as described above and the projection $b^*$ back to the manifold will provide an improved solution to the subproblem on the manifold if $r$ is chosen sufficiently small.",
"Even more so, the same reasoning can also be applied to generalize our results regarding the non-optimality of higher-degree branchings.",
"Given a coupled $n$ -BP at position $b$ not coincident with a terminal, one may again consider a sufficiently small region around $b$ on the manifold and project the corresponding subsolution onto $T_b \\mathcal {M}$ .",
"Exactly analogous to the previous arguments, improving the topology locally in the plane and projecting back to the manifold eventually results in an improved solution on the manifold."
],
[
"The practical side of BOT on manifolds",
"Although much of the theory generalizes nicely to Riemannian manifolds, the generalization of the practical algorithms is highly non-trivial.",
"Unlike in Euclidean space, realizing the optimal branching angles is a necessary but no longer sufficient condition for relatively optimal solutions.",
"For instance, on the sphere the meridians of three terminals located in the southern hemisphere at longitudes $0^\\circ $ , $120^\\circ $ , $240^\\circ $ will intersect at both poles at angles of $120^\\circ $ , which is the optimal angle for $\\alpha =0$ .",
"Nonetheless, only the south pole is the optimal branching point.",
"In essence, the geometry optimization aims to assign simultaneously to each branching point the coordinates of the weighted geometric median of its neighbors, a problem that is considered in [8].",
"The topology optimization presented in Sect.",
"REF could be easily generalized to manifolds if the geodesic distance can be computed.",
"Due to these obstacles, previous works of the Steiner Tree problem ($\\alpha = 1$ ) have focused mostly on the sphere as important special case [7]."
],
[
"Hardware and code availability.",
"Python code for all experiments can be found at https://github.com/hci-unihd/BranchedOT.",
"For the different experiments, a single machine with 56 CPUs (Intel(R) Xeon(R) CPU E5-2660 v4 @ 2.00GHz) and 256GB RAM was used.",
"Execution times for all described experiments lie (at most) in the order of hours.",
"A more detailed estimate can be obtained from the performance statistics reported in Fig.",
"REF and Fig.",
"REF ."
],
[
"Geometric construction of relatively optimal solutions for BOT with multiple sources",
"In Sect.",
"REF we have presented the the exact geometric construction of relatively optimal solutions based on [2], [9] and its generalization to the case of BOT with multiple sources.",
"We have thereby solved the open problem 15.11 in [2], for which an example is illustrated in Fig.",
"REF .",
"Figure REF shows a simple BOT problem with a chosen topology for which no root node can be chosen such that all branchings are symmetric (see Sect.",
"REF ).",
"Choosing for instance the left source as a root node, the branching at $b_1$ is symmetric whereas the one at $b_2$ is asymmetric.",
"However, after having solved the asymmetric branching case analogously to the symmetric one, the relatively optimal solution can now be constructed geometrically as shown in Fig.",
"REF .",
"More examples of the geometric construction for a given topology applied BOT problems with multiple sources are shown in Fig.",
"REF .",
"Figure: Simple setup with symmetric branching and asymmetric branching, see Sect.",
".Figure: Examples for geometric construction of relatively optimal solutions with fixed topology for BOT problems with multiple sources.",
"The construction is based on the optimal branching angles and uses one pivot circle and pivot point (light blue) per branching point.",
"Sources are shown in red, sinks in blue."
],
[
"Continuity of the optimal BP configuration.",
"The limitations of the geometric construction above are discussed in Sect.",
"REF .",
"Still, it forms the basis for our theoretical arguments in the paper.",
"Moreover, from the construction of a single branching point based on the branching angles (see Fig.",
"REF ), it can be seen that its optimal position changes continuously w.r.t.",
"the neighbor positions, the edge flows or $\\alpha $ .",
"Namely, the Y-, V- and L- branching change continuously as the source $a_0$ is moved around.",
"In addition, the optimal branching angles are given by continuous functions of $k$ and $\\alpha $ (cf. Eq.",
"(REF )), making the construction of the pivot point and pivot circle continuous.",
"By transitivity and based on the optimal substructure in Lem.",
"REF , the continuity generalizes also to the construction of larger ROS for a given topology."
],
[
"Numerical algorithm for geometry optimization",
"In this section, we provide theoretical details and practical experiments for the numerical geometry optimization presented in Sect.",
"REF .",
"We have generalized this approached from the context of the ESTP [26] to BOT.",
"For a given tree topology $T$ , the BP configuration is optimized by minimizing the following cost function: $\\mathcal {C}(X) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\left\\Vert x_i - x_j \\right\\Vert _2 ,$ where the $x_i$ for $1 \\le i \\le n$ are fixed terminals and the $x_i$ for $n+1 \\le i \\le n+m$ denote the variable branching point positions.",
"All coordinates are collectively summarized by $X$ .",
"Starting from a non-optimal, non-degenerate BP configuration denoted $X^{(0)}$ , e.g.",
"from a random initialization of the branching points, the algorithm iteratively solves the following linear system of equations $ x_i^{(k + 1)} = \\!",
"\\!",
"\\sum _{j \\, : \\, (i,j) \\in T} m_{ij}^\\alpha \\frac{x_j^{(k+1)}}{\\vert x_i^{(k)} - x_j^{(k)} \\vert }\\ \\Bigg / \\sum _{j \\, : \\, (i,j) \\in T} \\frac{m_{ij}^\\alpha }{\\vert x_i^{(k)} - x_j^{(k)} \\vert }, \\hspace{11.38092pt}\\text{for } n+1 \\le i \\le n+m.$ In essence, this is an iteratively reweighted least squares (IRLS) approach [4].",
"To see this, let us rewrite the cost function in Eq.",
"(REF ) into pseudo-quadratic from: $\\mathcal {C}(X) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\vert x_i - x_j \\vert ^{(\\beta -2) + 2} = \\sum _{(i,j) \\, \\in \\, T} \\underbrace{m_{ij}^\\alpha \\vert x_i - x_j \\vert ^{\\beta -2}}_{=: w_{ij}(X)} \\vert x_i - x_j \\vert ^2 \\, ,$ with $\\beta = 1$ for BOT.",
"Clearly, the weights $w_{ij}$ depend on the branching point positions.",
"However, the idea of IRLS is to insert the coordinates $X^{(k)} = \\lbrace x_i^{(k)} \\rbrace $ at $k$ -th iteration into $w_{ij}$ to obtain a truly quadratic form: $Q^{(k)}(X) &= w_{ij}(X^{(k)}) \\vert x_i - x_j \\vert ^2 =\\sum _{ (i,j) \\, \\in \\, T} m_{ij}^\\alpha \\frac{ \\vert x_i - x_j \\vert ^2}{ \\vert x_i^{(k)} - x_j^{(k)} \\vert } \\, ,$ where we have plugged in $w_{ij}(X^{(k)})$ and $\\beta = 1$ .",
"Indeed, minimizing this quadratic form yields $X^{(k+1)}$ as in Eq.",
"(REF ).",
"The updated coordinates $X^{(k+1)}$ are then plugged into $w_{ij}(X)$ during the next iteration.",
"This iterative updating of the weights $w_{ij}$ gives IRLS its name.",
"[26] proved in detail that this iterative solver converges to the minimum cost BP configuration.",
"The reasoning in [26] is based on the following key argument: As $X^{(k+1)}$ is the minimizer of $Q^{(k)}(X)$ , surely $Q^{(k)}(X^{(k)}) \\ge Q^{(k)}(X^{(k+1)})$ .",
"Together with the fact that $\\mathcal {C} (X^{(k)}) = Q^{(k)}(X^{(k)})$ , we have $\\mathcal {C}(X^{(k)}) &= Q^{(k)}(X^{(k)}) \\ge Q^{(k)}(X^{(k+1)}) \\\\&= \\sum _{ (i,j) \\, \\in \\, T: } m_{ij}^\\alpha \\frac{\\big ( \\vert x_i^{(k)} - x_j^{(k)} \\vert +\\vert x_i^{(k+1)} - x_j^{(k+1)} \\vert - \\vert x_i^{(k)} - x_j^{(k)} \\vert \\big )^2}{ \\vert x_i^{(k)} - x_j^{(k)} \\vert } \\\\&= \\mathcal {C}(X^{(k)}) + 2 [\\mathcal {C}(X^{(k+1)}) - \\mathcal {C}(X^{(k)}) ] \\\\&\\hspace{54.06006pt} + \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\frac{ \\big ( \\vert x_i^{(k+1)} - x_j^{(k+1)} \\vert - \\vert x_i^{(k)} - x_j^{(k)} \\vert \\big )^2}{ \\vert x_i^{(k)} - x_j^{(k)} \\vert } \\ .$ Since the sum in the last expression is clearly non-negative, the inequality above implies that $\\mathcal {C}(X^{(k)}) \\ge \\mathcal {C}(X^{(k+1)})$ .",
"This means that the cost of the BP configuration decreases with each iteration which, as shown in [26], implies that the iterations defined by Eq.",
"(REF ) converge to an absolute minimum of $\\mathcal {C}(X)$ .",
"As can be seen from the above derivation, the edge flows $m_{ij}^\\alpha $ are constant coefficients which do not complicate the considerations in [26].",
"Consequently, all arguments presented there can be directly transferred from ESTP to BOT.",
"For a detailed discussion on the complexity of the algorithm and suitable convergence criteria, we refer the reader to the work of Smith and only briefly state the results here.",
"Based on an analytically tractable example, Smith claims that the algorithm requires at most $O(N/\\epsilon )$ iterations to converge to a solution of the ESTP whose cost is within $\\epsilon $ of the optimal cost.",
"As a convergence criterion, Smith suggests to stop the iteration when the angles at all branching points are sufficiently close to the optimal angle conditions.",
"In order to be able to apply the algorithm also to trees with higher-degree branching points where the optimal angle conditions are a necessary but not sufficient condition for the cost minimum, the experiments in this paper use a different criterion.",
"The algorithm is considered to have converged if from one iteration to the next the relative cost improvement $\\big (\\mathcal {C}(X^{(k)}) - \\mathcal {C}(X^{(k+1)}) \\big ) / \\mathcal {C}(X^{(k+1)})$ has dropped below a certain threshold.",
"The BP optimization routine for a given tree topology is summarized in Alg.",
"REF .",
"Note that, in practice, the denominators $\\vert x_i^{(k)} - x_j^{(k)} \\vert $ in Eq.",
"(REF ) are clipped to $10^{-7}$ to avoid numerical instabilities.",
"BP optimization routine Input: threshold $\\eta $ , tree topology $T$ , BOT problem (terminal positions $x_{1:n}$ , supplies/demands $\\mu $ , $\\alpha $ ) Output: numerical minimal cost BP configuration [1] $F \\leftarrow $ get_all_edge_flows($T,\\mu $ ) $\\triangleright \\ $ uniquely determined from flow constraints $x_{n+1:n+m} \\leftarrow $ randomly initialized.",
"$C_{old} \\leftarrow \\infty $ $C \\leftarrow $ BOT_cost$(T,x_{1:n+m},F, \\alpha )$ $\\frac{\\mathcal {C}_{old} - \\mathcal {C}}{\\mathcal {C}} > \\eta $ $C_{old} \\leftarrow C$ $x_{1:n+m} \\leftarrow $ update_BPs($x_{1:n+m},F, \\alpha $ ) $\\triangleright \\ $ solves linear system in Eq.",
"(REF ) $C \\leftarrow $ BOT_cost$(T,x_{1:n+m},F, \\alpha )$ return $x_{1:n+m}$"
],
[
"BOT with costs scaling non-linearly with the edge length.",
"Let us briefly consider BOT with a modified cost function of $\\mathcal {C}(X) = \\sum _{(i,j) \\, \\in \\, T} m_{ij}^\\alpha \\left\\Vert x_i - x_j \\right\\Vert ^\\beta _2 \\,$ including one additional parameter $\\beta $ , regulating the scaling of the cost function w.r.t.",
"edge lengths.",
"For $\\beta \\ge 1$ the cost function as a function of the branching points is still convex, as $x \\mapsto x^\\beta $ is convex and increasing and the Euclidean norm is convex.",
"Thus, it has a unique minimum.",
"As can be seen from Eq.",
"(REF ), the IRLS scheme completely absorbs the $\\beta $ into the weights $w_{ij}$ .",
"Consequently, the geometry optimization as presented here is readily applicable also to the modified cost in Eq.",
"(REF ).",
"However, for $\\beta >1$ we are not aware of any theoretical convergence guarantee to the minimum, therefore, further investigation is required.",
"Though, in the special case of $\\beta =2$ , it is clear that the geometry optimization can be solved to global optimality in just a single iteration."
],
[
"Performance test of numerical BP optimization.",
"In order to evaluate how long the BP optimization for a given topology takes, we have randomly generated 1000 BOT problems for different number of terminals using Alg.",
"REF .",
"For each of these problems, we have applied the BP optimization routine, given a uniformly sampled full tree topology.",
"Regarding the generation of problem setups, note that BOT solutions and problems are invariant under rescaling of the total demand and supply as well as under global rescaling of the coordinates, see Eq.",
"(REF ).",
"WLOG, these scales are chosen to be 1.",
"Figure REF shows the average runtime in seconds plotted against the number of terminals.",
"For BOT problems with 10 to 1000 terminals our efficient C++ implementation of the geometry optimization takes just a fraction of a second.",
"The relatively large error bars, indicating the standard deviation, are not due to an insufficient sample size but due to the natural run time variability for different problems that exists independent of the $n$ -dependence.",
"For instance, plotting the average runtime of all problems with $\\alpha \\le 0.5$ and $\\alpha > 0.5$ separately reveals that the optimization on average requires more time for problems with higher $\\alpha $ .",
"The reason for this was not investigated further, but we suspect that for larger $\\alpha $ on average more V- and L-branchings occur which may require more iterations than Y-branchings for convergence.",
"For this and all following experiments we chose the convergence threshold $\\eta $ to be $10^{-6}$ .",
"Figure: Average runtime of BP optimization routine applied to each 1000 random BOT problems of different size.",
"Average of all problems with α≤0.5\\alpha \\le 0.5 in orange (dotted) and α>0.5\\alpha > 0.5 in green (dashed)."
],
[
"Scaling of the geometry optimization",
"A single iteration of the geometry optimization, i.e.",
"solving the linear system defined by Eq.",
"(REF ) once, takes $O(nd)$ operations for a problem with $n$ terminals in $d$ spatial dimensions.",
"The elimination scheme used in our efficient C++ implementation is based on the “elimination on leaves of a tree” found in [26].",
"From Eq.",
"(REF ), one can easily see that the geometry optimization parallelizes over the spatial dimensions, as the linear system of the form $Ax = b$ shares the same matrix $A$ across the different dimensions and only the $b$ is different.",
"Although a single iteration is of order $O(nd)$ , it is a priori not clear how many iterations are required until convergence is reached.",
"Paralleling the setup of Fig.",
"REF , we have conducted an experiment where we report the number of iterations, for a batch of BOT problems.",
"We performed the same experiment in Euclidean space of different dimensions ($d \\in \\lbrace 3,4,5,10,30,100 \\rbrace $ ).",
"Figure REF shows the number of iterations until convergence plotted against the number of terminals.",
"The plot suggests that, for the investigated regime of $n$ and $d$ , the number of iterations on average scales like $\\log (n)$ .",
"However, this is merely an empirical observation and theoretical investigations are an interesting subject for future research.",
"Figure: Average number of iterations required by the BP optimization routine until convergence for each 1000 random BOT problems of different sizes nn and different dimensionality dd.",
"The x-axis is in log-scale.",
"The plot suggests that, for the investigated regime of nn and dd, the number of iterations on average scales like log(n)\\log (n), though one cannot generalize this statement.Random BOT problem generation Input: number of terminals $n$ Output: BOT problem with $n$ terminals located in $[0,1]\\times [0,1]$ , total supply and demand equal to 1 [1] $\\alpha \\sim $ Uniform($[0,1]$ ) $\\triangleright \\ $ sample uniformly from $[0,1]$ $n^+ \\sim $ Uniform($\\lbrace 1,2,...,n-1\\rbrace $ ) $\\triangleright \\ $ number of sources $n^- \\leftarrow n - n^+$ $\\mu ^+_{1:n^+} \\sim $ Uniform($[0,1]$ , size=$n^+$ ) $\\triangleright \\ $ array of supplies $\\mu ^-_{1:n^-} \\sim $ Uniform($[0,1]$ , size=$n^-$ ) $\\triangleright \\ $ array of demands $a_{1:n} \\sim $ Uniform($[0,1]$ , size=($n$ ,2)) $\\triangleright \\ $ terminal coordinates return $a_{1:n}, \\ \\mu ^+/\\mathrm {sum}(\\mu ^+), \\ \\mu ^-/\\mathrm {sum}(\\mu ^-), \\ \\alpha $"
],
[
"Greedy randomized heuristic for topology optimization",
"In this section, we give the details of the greedy randomized heuristic presented in Section REF and show the results of some additional experiments conducted with it.",
"Starting from a tree topology $T$ , a uniformly sampled edge $\\hat{e}$ is removed.",
"The node in the smaller connected component $\\ell $ is connected via a new branching point to an edge in the other component.",
"This edge is sampled according to $p(e) \\propto \\exp (-d(e, \\ell )^2 / d_{min}^2)$ , where $d(e, \\ell )$ is the distance between an edge $e=(i,j)$ and node $\\ell $ , defined by $ d(e, \\ell ) = \\underset{\\lambda \\in [0,1]}{\\min } \\ \\left\\Vert [x_i + \\lambda (x_j - x_i) ] - \\ell \\right\\Vert _2 \\ $ and $d_{min}$ is the distance to the closest considered edge.",
"The resulting new tree topology $T_{new}$ is accepted if it decreases the cost compared to the previous topology $T$ and the above procedure is iterated.",
"This greedy optimization strategy is summarized in Alg.",
"REF below.",
"Greedy randomized heuristic (zero temperature limit) Input: BOT problem $P$ , tree $T$ interconnecting terminals $a_{1:n}$ (with help of BPs) Output: heuristic BOT solution [1] $B \\leftarrow $ optimize_BP_configuration($T, P$ ) $\\triangleright \\ $ returns coordinates of BPs and terminals, see Alg.",
"REF $\\mathcal {C} \\leftarrow $ BOT_cost$(T,B,P)$ $E \\leftarrow $ list(T.edges()) $E$ not empty $S \\leftarrow $ Store current state $(T,B)$ $\\mathcal {C}_{old} \\leftarrow \\mathcal {C}$ $\\hat{e} \\sim $ Uniform($E$ ) $\\triangleright \\ $ sample uniformly from $E$ $E$ .remove($e$ ) $T$ .remove_edge($\\hat{e}$ ) $C_b \\leftarrow $ subgraph component of $T$ with more nodes $E_b \\leftarrow $ list($C_b$ .edges) $b \\leftarrow $ node of $\\hat{e}$ in $C_b$ $\\ell \\leftarrow $ node of $\\hat{e}$ not in $C_b$ degree($b$ ) == 2 $n_1,n_2 \\leftarrow $ neighbors($T, b$ ) $T$ .remove_node($b$ ) $\\triangleright \\ $ remove unnecessary BPs with degree 2 $T$ .add_edge($n_1,n_2$ ) $E_b$ .remove(($n_1,b$ ), ($n_2,b$ )) $d \\leftarrow $ Initalize array of size length($E_b$ ) edge $e \\in E_b$ $d[e] \\leftarrow $ get_distance($e,\\ell $ ) $\\triangleright \\ $ calculate distance defined by Eq.",
"(REF ) $d_{min} \\leftarrow $ min($d$ ) $e_c \\sim \\exp (-d^2/d_{min}^2)$ $\\triangleright \\ $ sample edge according to distance kernel $T$ .add_node($b_{new}$ ) $\\triangleright \\ $ Initialize a new BP $T$ .add_edges($(\\ell ,b_{new}),(b_{new}, e_c[0]),(b_{new}, e_c[1])$ ) $B \\leftarrow $ optimize_BP_configuration($T, P$ ) $\\mathcal {C} \\leftarrow $ BOT_cost$(T,X,P)$ $ \\mathcal {C} < \\mathcal {C}_{old} $ $E \\leftarrow $ list(T.edges()) $\\triangleright \\ $ accept new state $T,B \\leftarrow S$ $\\triangleright \\ $ Restore old state $(T,B)$ $\\mathcal {C} \\leftarrow \\mathcal {C}_{old}$ return $T, B$ Figure REF shows an example BOT problem with 100 terminals to which our greedy heuristic has been applied.",
"As starting point, the topology with the least possible structure was used, a star-like tree centred around a single BP of degree 100, cf. Fig.",
"REF .",
"It can be seen how the topology evolves over less than 2000 iterations to the final solution.",
"Figure REF shows the decreasing transportation cost $\\mathcal {C}$ plotted against the number of iterations."
],
[
"Performance test of heuristic topology optimization.",
"Finally, in a large experiment, we have investigated how many iterations the greedy heuristic requires on average to converge.",
"For that, the greedy heuristic was applied to a number of BOT problems with $n$ terminals and the number of iterations until convergence were counted.",
"The mean and standard deviation of the required iterations are plotted in Fig.",
"REF .",
"The reason for the relatively large standard deviations are due to intrinsic variation of the sampled problems.",
"To illustrate, for instance, the influence of the $\\alpha $ -value, the average number of iterations until convergence were plotted separately for all BOT problems with $\\alpha \\le 0.5$ and $\\alpha > 0.5$ .",
"We find that the greedy heuristic systematically needs more iterations to converge for $\\alpha > 0.5$ .",
"Figure: The greedy topology optimization applied to each 150 problems with nn terminals: Mean and standard deviation of required iterations plotted vs. nn in blue.",
"Average for problems with α≤0.5\\alpha \\le 0.5 in green (dashed) and for problems with α>0.5\\alpha > 0.5 in orange (dotted)."
],
[
"Influence of the edge sampling kernel.",
"Our heuristic topology optimization involves a number of design choices which may affect its performance.",
"A systematic investigation is beyond the scope of this work and left for future research.",
"However, one hyperparameter of particular interest is the kernel (chosen to be Gaussian) and its width (chosen to be $d_{min}$ ), which together define the replacement probability of the edges, see l. REF in Alg.",
"REF .",
"To study its influence on the performance we varied the width of the Gaussian kernel $\\exp (-d^2/(\\omega d_{min})^2)$ by tuning the parameter $\\omega $ .",
"Based on 150 random problems of various sizes $n$ , we calculated the average cost ratio of the heuristic with different $\\omega $ to the default of $\\omega = 1$ (cf.",
"Fig.",
"REF ).",
"Indeed, Fig.",
"REF shows that the default choice of $\\omega = 1$ is quite strong and relatively robust given that $\\omega = 0.5$ or $\\omega = 0.1$ work similarly well.",
"Clearly, wider kernels lead to larger (i.e.",
"less local) changes of the topology.",
"At later stages of the algorithm most of these topology changes will be unfavorable, explaining why the algorithm for wider kernels terminates with comparatively less optimal solutions.",
"Furthermore, we have investigated the influence of the kernel width on the number of iterations required (cf.",
"Fig.",
"REF ).",
"Qualitatively, Fig.",
"REF confirms that for wider kernels, which encourage exploration, more iterations are required.",
"Fitting a power law of the form $x \\mapsto ax^b$ to the curves in Fig.",
"REF , one finds that, depending on the kernel, the greedy heuristic on average scales between $O(n^{1.5})$ and $O(n^{1.7})$ .",
"Again, this is a purely empirical statement.",
"A careful theoretical analysis to obtain guarantees, also for problem sizes not covered in our experiments, is beyond the scope of this work.",
"Figure: Influence of kernel width in topology optimization based on 150 random problems of various sizes nn: (a) Ratio of the average cost of the heuristic solution with kernel width factor ω\\omega to the default of ω=1\\omega =1.",
"(b) Average number of iterations until convergence for different kernel width factors ω\\omega .",
"Depending on the kernel width the number of iterations on average scales between O(n 1.5 )O(n^{1.5}) and O(n 1.7 )O(n^{1.7}).Figure: Average cost ratios of our greedy heuristic and brute-force solutions (the closer to 1 the better) for different number of terminals nn and dimensions dd.",
"For each nn, we uniformly sampled 100 different BOT problems.",
"Though the average cost ratio increases slightly with nn, our approximate BOT solver compares very well against the ground truth solutions, independently of the dimensionality dd."
],
[
"Greedy topology optimization vs. brute-force search for higher-dimensional BOT.",
"Both the numerical geometry optimization and the greedy algorithm for the topology optimization presented in Sect.",
"are readily applicable to BOT problems in $\\mathbb {R}^d$ .",
"Paralleling the experiment presented in Fig.",
"REF , we have compared our heuristic topology optimization against brute-forced solutions for each 100 problems of size $n=5$ to $n=9$ and spatial dimension $d$ .",
"Although Fig.",
"REF suggests that the average cost ratio increases slightly with $n$ , our approximate BOT solver again compares very well against the ground truth solutions, independently of the dimensionality $d$ ."
]
] | 2210.07702 |
[
[
"An atom's worth of anonymity"
],
[
"Abstract Anonymity has gained notoriety in modern times as data about our actions and choices accumulates in the internet partly unbeknownst to us and partly by our own choice.",
"Usually people wish some data about themselves were private while some other date may be public or is even wanted to be public for publicity reasons.",
"There are different criteria which characterize the degree of anonymity of data.",
"Given data can also be anonymized by different techniques in order to increase its degree of anonymity.",
"In this paper we take a very simple \"atomic\" degree of anonymity as our starting place.",
"We axiomatize these atoms and propose the investigation of first order logic based on these atoms.",
"Considering the vast literature and the huge importance of anonymity our investigation may seem quite modest.",
"However, it is about the logic of anonymity, not about how to secure, create or break anonymity."
],
[
"I contribute this paper on anonymity to honor the birthday of John Crossley.",
"I am not only John's friend but also his grandson in the academic sense, as my doctoral supervisor was Peter Aczel, a student of John's.",
"Of course, it is my hope that this attempt to find the logic of anonymity amuses John.",
"Suppose we have data about some individuals and in the data the values of some attributes are publicly known while the values of some other attributes are sensitive personal data that should be protected from becoming public.",
"Anonymity in this context means the impossibility to derive the values of the sensitive attributes from the values of the publicly known attributes, and this impossibility should be true of each individual.",
"Anonymity has gained notoriety in modern times as data about our actions and choices accumulates in the internet partly unbeknownst to us and partly by our own choice.",
"Usually people wish some data about themselves were private while some other date may be public or is even wanted to be public for publicity reasons.",
"There are different criteria which characterize the degree of anonymity of data.",
"Given data can also be anonymized by different techniques in order to increase its degree of anonymity.",
"In this paper we take a very simple “atomic\" degree of anonymity as our starting place.",
"We axiomatize these atoms and propose the investigation of first order logic based on these atoms.",
"Considering the vast literature and the huge importance of anonymity our investigation may seem quite modest.",
"However, it is about the logic of anonymity, not about how to secure, create or break anonymity.",
"In this paper, as in database theory, variables denote attributes.",
"Thus our $x$ , $y$ , $z$ , etc stand for such attributes as “date\", “time\", “surname\", “first name\", “hometown\", “year of birth\", “gender\", “salary\", “blood type\", “systolic blood pressure\", “diastolic blood pressure\", etc.",
"It is natural to think of an assignment of values to such variables as a database record.",
"We call any set of such assignments a team.",
"Calling them teams carries no special meaning but is just a practice I adopted in my book [6], with teams of robots performing some task in mind.",
"Table: A team.In the team of Table REF the attributes hometown and salary do not reveal 100% the attribute surname because for all combinations of hometown and salary there are two different values of surname.",
"So, if we know hometown and salary we are still a little uncertain who is in question.",
"Of course, this is an extremely weak form of anonymity, and that is why we call it “atomic\" anonymity.",
"On the other hand, the attribute surname completely determines hometown and salary, so no anonymity or uncertainty there.",
"We formalize the concept of keeping part of data anonymous while publishing some other part.",
"We use $x,y,z,u,v,$ etc to denote finite sequences of attributes.",
"The concatenation of $x$ and $y$ is denoted $xy$ .",
"If $s$ is an assignment and $x=\\langle x_1,\\ldots , x_n\\rangle $ , then $\\langle s(x_1),\\ldots ,s(x_n)\\rangle $ is denoted $s(x)$ .",
"Definition 1 (Anonymity atom) A team $X$ satisfies “$x$ leaves $y$ anonymous” in the model $M$ , in symbols $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ , if $\\forall s\\in X\\exists s^{\\prime }\\in X(s(x)=s^{\\prime }(x)\\wedge s(y)\\ne s^{\\prime }(y)).$ The intuition of $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ is the following: We can publish $X\\upharpoonright x$ and still keep the anonymity of $X\\upharpoonright y$ as there is no row $s$ of $X$ (i.e.",
"no individual $s$ ) for which we can infer $s(y)$ from $s(x)$ .",
"A better anonymity is provided by the $k$ -anonymity atom which demands in Definition REF not just one but $k$ different values $s^{\\prime }(y)\\ne s(y)$ for each $s(y)$ .",
"Note that the full team (in a domain with at least two elements) and the empty team always satisfy $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .",
"Unlike ${=\\hspace{-0.66666pt}}(x,y)$ (as in [6]), the atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ is not closed downwards (or upwards).",
"We can regard $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ as a strong (maximal) denial of ${=\\hspace{-0.66666pt}}(x,y)$ .",
"But note that the independence atom $x\\ \\bot \\ y$ (as in [4]) does not imply $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ , because $x\\ \\bot \\ y$ may be true because $y$ is constant.",
"The anonymity atom is clearly closed under unions and it is first order.",
"Hence it is definable in inclusion logic [3], in fact, if $u$ is a sequence of new variables, as many as in $y$ , then $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\iff \\exists u(\\lnot u=y\\wedge xu\\subseteq xy).$ Thus `anonymity logic' is a sublogic on inclusion logic.",
"In fact, the two are equivalent [2].",
"See also [5].",
"We now ask the question, what are the general principles that propositions of the form $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ obey?",
"If we know a bunch of such propositions, what can we infer from them?",
"For example, is it true that if $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and $y\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ , then $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ ?",
"That this is not so can be seen by investigating the team of Table REF .",
"This team satisfies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and $y\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ but fails to satisfy $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ because it actually satisfies $x=z$ .",
"Table: Failure of transitivityDefinition 2 The axioms of anonymity are A1 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ implies $yx\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ and $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}uz$ .",
"(Permutation) A2 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ .",
"(Monotonicity) A3 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zy$ implies $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ .",
"(Cancellation) A4 $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}$ implies $\\bot $ .",
"$xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ implies $yx\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ and $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}uz$ .",
"(Permutation) $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zu$ .",
"(Monotonicity) $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}zy$ implies $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}z$ .",
"(Cancellation) $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}$ implies $\\bot $ .",
"From the axioms (or rules) of Definition REF we get a notion $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ of derivability of an atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ from a set $\\Sigma $ of such atoms.",
"Theorem 3 The following are equivalent for any set $\\Sigma \\cup \\lbrace x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\rbrace $ of anonymity atoms: $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .",
"$M\\models _X\\Sigma $ implies $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ for every model $M$ and team $X$ .",
"$\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .",
"$M\\models _X\\Sigma $ implies $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ for every model $M$ and team $X$ .",
"Suppose first $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and $M\\models _X\\Sigma $ .",
"All the rules (A1)-(A4) preserve truth.",
"Hence $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .",
"Suppose then $\\Sigma \\nvdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .",
"By the validity of (A3) we may assume $x\\cap y=\\emptyset $ .",
"By the validity of (A4) we may assume $y\\ne \\emptyset $ .",
"Let $x=\\langle x_1,\\ldots ,x_n\\rangle $ and $y=\\langle y_1,\\ldots ,y_m\\rangle $ .",
"Let $W$ be the set of all variables occurring in $\\Sigma \\cup \\lbrace x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\rbrace $ .",
"Let $M$ be a model with at least three elements $0,1,2$ and $\\begin{array}{lcl}X&=&\\lbrace s\\in \\lbrace 0,1,2\\rbrace ^W : \\bigvee _{1\\le i\\le n}(s(x_i)\\ne 0)\\vee \\bigwedge _{1\\le i\\le m}(s(y_i)= 0)\\rbrace \\end{array}$ First of all, $M\\lnot \\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ for if $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ and we let $s(w)=0$ for all $w\\in W$ , then there would be $s^{\\prime }\\in X$ such that $s^{\\prime }(x_i)=s(x_i)=0$ for all $i\\in \\lbrace 1,\\ldots ,n\\rbrace $ and $s^{\\prime }(y_i)\\ne s(y_i)=0$ for some $i\\in \\lbrace 1,\\ldots ,m\\rbrace $ , a contradiction.",
"We are left with proving $M\\models _X \\Sigma $ .",
"To see this, let ${u\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}v}\\in \\Sigma $ .",
"We may assume, w.l.o.g., $u\\cap v=\\emptyset $ .",
"Let $s\\in X$ .",
"We shall construct $s^{\\prime }\\in X$ such that $s^{\\prime }(u)=s(u)$ and $s^{\\prime }(v)\\ne s(v)$ .",
"Let us assume first: ${x\\lnot \\subseteq u.",
"}$ Since $x\\cap (W\\setminus u)\\ne \\emptyset $ , there is some $x_i\\notin u$ .",
"Either $x_i\\in v$ or $x_i\\notin v$ .",
"Case 1.1: $x_i\\in v$ , i.e.",
"$x_i=v_j$ for some $j\\in \\lbrace 1,\\ldots ,k\\rbrace $ .",
"Since there are three possible values for $s^{\\prime }(v_j)$ , we can make sure both $s^{\\prime }(v_j)\\ne s(v_j)$ and $s^{\\prime }(v_j)\\ne 0$ .",
"Then $s^{\\prime }(x)\\ne 0$ , since $x_i=v_j$ .",
"Thus $s^{\\prime }\\in X$ , whichever way we otherwise define $s^{\\prime }$ .",
"By construction $s^{\\prime }(v)\\ne s(v)$ .",
"Since $u\\cap v=\\emptyset $ , we are free to let $s^{\\prime }(u)=s(u)$ , and we are done.",
"Case 1.2: $x_i\\notin v$ .",
"Let $s^{\\prime }(u)=s(u)$ and let $s^{\\prime }(v)$ be such that $s^{\\prime }(v)\\ne s(v)$ .",
"Since now $x_i\\notin u\\cup v$ , we can let $s^{\\prime }(x_i)\\ne 0$ , and this guarantees $s^{\\prime }\\in X$ .",
"We are done again.",
"Let us then assume ${x\\subseteq u.",
"}$ Since $s\\in X$ , we know that $s(x)$ is not all zero or $s(y)$ is all zero.",
"Case 2.1: $s(x)$ is not all zero.",
"We let $s^{\\prime }(u)=s(u)$ .",
"Since now $s^{\\prime }(x)$ is not all zero, we have $s^{\\prime }\\in X$ whatever $s^{\\prime }$ is otherwise.",
"So we choose $s^{\\prime }(v)\\ne s(v)$ in an arbitrary way, and we are done.",
"So let us assume Case 2.2: $s(y)$ is all zero.",
"If $v\\subseteq y$ , then ${u\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}v}\\in \\Sigma $ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y\\in \\Sigma $ , by Axiom (A2), contrary to our assumption.",
"So we may pick $v_i\\in v\\setminus y$ .",
"Let $s^{\\prime }(v_i)\\ne s(v_i)$ .",
"Now we know $s^{\\prime }(v)\\ne s(v)$ .",
"Since $u\\cap v=\\emptyset $ , we can let $s^{\\prime }(u)=s(u)$ .",
"Since $v_i\\notin y$ we can let $s^{\\prime }(y)=s(y)=0$ .",
"Now $s^{\\prime }\\in X$ , whichever way $s^{\\prime }$ is otherwise defined.",
"We may combine $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ with logical operations as is done in dependence logic [6].",
"Here is a practical example: Example 4 Suppose we collect some data about individuals and we need to publish this data together with the addresses of the individuals.",
"Some individuals may, for privacy reasons, not want their address to be associated with other data about them.",
"For this purpose we allow any individual to declare their address private by setting ${\\tt publicity}={\\tt private}$ .",
"A team satisfies, according to the semantics of implication given in [6]: ${\\tt publicity}={\\tt private}\\rightarrow {\\tt data}\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address},$ iff the anonymity constraint ${\\tt data}\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address}$ holds (at least) in the part of the team that satisfies ${\\tt publicity}={\\tt private}$ .",
"Example 5 We continue the previous example.",
"A team satisfies $\\exists x (\\phi (x)\\wedge ({\\tt data}\\ x)\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address}),$ iff (possibly multiple) values can be assigned to $x$ so that the new values of $x$ satisfy $\\phi (x)$ and, in addition, the anonymity constraint $({\\tt data}\\ x)\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}{\\tt address}$ (still) holds.",
"In other words, data about the individuals can be updated with data about the attributes $x$ without violating the modest anonymity provided by the $\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}$ -atom.",
"Our anonymity atom provides the most basic form of anonymity, not sufficient for practical applications.",
"Therefore we strengthen the atom to $k$ -anonymity atom.",
"The bigger the $k$ , the more anonymity it provides.",
"The idea is that an attribute $y$ is $k$ -anonymous with respect to some other attributes $x$ if any data about $x$ in the team is associated with at least $k$ values for $y$ .",
"Intuitively, my name does not reveal me if there are $k$ different persons with the same name as me, and $k$ is relatively large.",
"In September 9, 2022, there were 4703 people alive with the last name Väänänen in Finland.",
"Thus I can remain relatively anonymous in Finland even if I reveal that my last name is Väänänen.",
"At the same moment of time there were only 7 people in Finland's population information system with the last name Crossley, providing much less anonymity if the last name was revealed.",
"Definition 6 ($k$ -anonymity atom) A team $X$ satisfies “publishing $x$ keeps $y$ $k$ -anonymous” in a model $M$ , in symbols $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ , if $\\forall s\\in X(|\\lbrace s^{\\prime }(y) : s^{\\prime }\\in X \\wedge s(x)=s^{\\prime }(x)\\rbrace |\\ge k).$ The atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ is called simple, if $|y|=1$ .",
"Note that the criterion is different from $\\forall s\\in X(|\\lbrace s\\in X : s(x)=s^{\\prime }(x)\\wedge s(y)\\ne s^{\\prime }(y)|\\ge k)$ but equivalent to $\\forall s\\in X\\exists s^{\\prime }_1\\ldots \\exists s^{\\prime }_k\\in X\\bigwedge _{1\\le i<j\\le k}(s(x)=s^{\\prime }_i(x)\\wedge s^{\\prime }_i(y)\\ne s^{\\prime }_j(y)).$ $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{1}\\hspace{1.111pt}y$ is always true and $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{2}\\hspace{1.111pt}y$ is equivalent to $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ .",
"Definition 7 The axioms of $k$ -anonymity (for varying $k$ ) are A1 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}zu$ implies $yx\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}zu$ and $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}uz$ .",
"(Permutation) A2 $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}z$ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{l}\\hspace{1.111pt}zu$ for $l\\le k$ .",
"(Monotonicity) A3 If $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}zy$ , then $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}z$ .",
"(Cancellation) A4 If $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}$ , then $\\bot $ .",
"A5 If $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_1}\\hspace{1.111pt}y_1$ , $xy_1\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_2}\\hspace{1.111pt}y_2, \\ldots , xy_1...y_{n-1}\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_n}\\hspace{1.111pt}y_n$ , then $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_1\\cdot \\ldots \\cdot k_n}\\hspace{1.111pt}y_1y_2\\ldots y_n.$ $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}zu$ implies $yx\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}zu$ and $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}uz$ .",
"(Permutation) $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}z$ implies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{l}\\hspace{1.111pt}zu$ for $l\\le k$ .",
"(Monotonicity) If $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}zy$ , then $xy\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}z$ .",
"(Cancellation) If $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}$ , then $\\bot $ .",
"If $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_1}\\hspace{1.111pt}y_1$ , $xy_1\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_2}\\hspace{1.111pt}y_2, \\ldots , xy_1...y_{n-1}\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_n}\\hspace{1.111pt}y_n$ , then $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k_1\\cdot \\ldots \\cdot k_n}\\hspace{1.111pt}y_1y_2\\ldots y_n.$ From the axioms (or rules) of Definition REF we get a notion $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ of derivability of an atom $x\\hspace{1.111pt}{\\Upsilon }\\hspace{1.111pt}y$ from a set $\\Sigma $ of such (for various $k$ ).",
"We conjecture that the above axioms are complete in the sense that $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ holds iff every team satisfying $\\Sigma $ also satisfies $x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ .",
"We now prove the conjecture for the special case of simple $k$ -anonymity atoms.",
"Then, of course (A5) is unnecessary.",
"Theorem 8 The following are equivalent for any set $\\Sigma \\cup \\lbrace x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y\\rbrace $ of simple $k$ -anonymity atoms (for varying $k$ ): $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ .",
"$M\\models _X\\Sigma $ implies $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ for every model $M$ and team $X$ .",
"$\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ .",
"$M\\models _X\\Sigma $ implies $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ for every model $M$ and team $X$ .",
"Suppose first $\\Sigma \\vdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ and $M\\models _X\\Sigma $ .",
"All the rules (A1)-(A4) preserve truth.",
"Hence $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ .",
"Suppose then $\\Sigma \\nvdash x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ , where $k\\ge 2$ .",
"By the validity of (A3) we may assume $y\\notin x$ .",
"Let $x=\\langle x_1,\\ldots ,x_n\\rangle $ .",
"Let $W$ be the set of all variables in $\\Sigma \\cup \\lbrace x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y\\rbrace $ .",
"Let $M$ be a model the domain of which contains the natural numbers, and let $X=\\lbrace s\\in {N}^W : \\bigvee _{1\\le i\\le n}(s(x_i)\\ne 0)\\vee (s(y)\\in \\lbrace 0,\\ldots , k-2\\rbrace )\\rbrace .$ This team, as well as $M$ , are infinite, but the set $\\Sigma $ may contain $u\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}z$ for all $k\\in {N}$ , so the infinity of $X$ is unavoidable.",
"As in the proof of Theorem REF we can show that $M\\lnot \\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ , namely, let $s(w)=0$ for all $w\\in W$ .",
"If $M\\models _X x\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{k}\\hspace{1.111pt}y$ , then there are $k$ assignments $s^{\\prime }\\in X$ such that $s^{\\prime }(x_i)=s(x_i)=0$ for $1\\le i\\le n$ and $s^{\\prime }(y)$ is different for each $s^{\\prime }$ .",
"But this contradicts the definition of $X$ .",
"Likewise, as in the proof of Theorem REF , we can show that $M\\models _X \\Sigma $ , namely, let ${u\\hspace{1.111pt}{\\Upsilon }\\hspace{-1.66656pt}_{l}\\hspace{1.111pt}v}\\in \\Sigma $ .",
"We may assume, w.l.o.g., $v\\notin u$ .",
"Let $s\\in X$ .",
"By considering different cases according to whether ${x\\subseteq u}$ or ${x\\lnot \\subseteq u}$ , we can in both cases construct $l$ different $s^{\\prime }\\in X$ such that $s^{\\prime }(u)=s(u)$ and $s^{\\prime }(v)$ is different for each $s^{\\prime }$ .",
"We have discussed only inferences among atomic anonymity statements.",
"What about more complex formulas?",
"Unfortunately consequence in full first order logic with anonymity atoms is badly non-axiomatizable, a consequence of results in [2].",
"We can still investigate fragments of first order logic with anonymity atoms and look for fragments of low complexity.",
"There are some early results in this direction in [1]."
]
] | 2210.07834 |
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